TPTP Problem File: ITP283^4.p
View Solutions
- Solve Problem
%------------------------------------------------------------------------------
% File : ITP283^4 : TPTP v8.2.0. Released v8.0.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer problem VEBT_BuildupMemImp 00582_027779
% 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_00582_027779 [Des22]
% Status : Theorem
% Rating : 1.00 v8.1.0
% Syntax : Number of formulae : 9994 (3009 unt; 566 typ; 0 def)
% Number of atoms : 29948 (10050 equ; 0 cnn)
% Maximal formula atoms : 71 ( 3 avg)
% Number of connectives : 199049 (2351 ~; 324 |;2355 &;180332 @)
% ( 0 <=>;13687 =>; 0 <=; 0 <~>)
% Maximal formula depth : 40 ( 7 avg)
% Number of types : 17 ( 16 usr)
% Number of type conns : 2997 (2997 >; 0 *; 0 +; 0 <<)
% Number of symbols : 553 ( 550 usr; 6 con; 0-8 aty)
% Number of variables : 28578 (2572 ^;24759 !; 817 ?;28578 :)
% ( 430 !>; 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 18:42:46.360
%------------------------------------------------------------------------------
% Could-be-implicit typings (27)
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_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_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_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_Word_Oword,type,
word: $tType > $tType ).
thf(ty_t_Real_Oreal,type,
real: $tType ).
thf(ty_t_List_Olist,type,
list: $tType > $tType ).
thf(ty_t_Set_Oset,type,
set: $tType > $tType ).
thf(ty_t_Rat_Orat,type,
rat: $tType ).
thf(ty_t_Num_Onum,type,
num: $tType ).
thf(ty_t_Nat_Onat,type,
nat: $tType ).
thf(ty_t_Int_Oint,type,
int: $tType ).
thf(ty_t_itself,type,
itself: $tType > $tType ).
% Explicit typings (539)
thf(sy_cl_Complete__Lattices_Ocomplete__linorder,type,
comple5582772986160207858norder:
!>[A: $tType] : $o ).
thf(sy_cl_Typerep_Otyperep,type,
typerep:
!>[A: $tType] : $o ).
thf(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : $o ).
thf(sy_cl_Nat_Osize,type,
size:
!>[A: $tType] : $o ).
thf(sy_cl_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_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_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_Groups_Omonoid__mult,type,
monoid_mult:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Olinorder,type,
linorder:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Opreorder,type,
preorder:
!>[A: $tType] : $o ).
thf(sy_cl_Parity_Oring__parity,type,
ring_parity:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oidom__abs__sgn,type,
idom_abs_sgn:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oordered__ring,type,
ordered_ring:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Ozero__neq__one,type,
zero_neq_one:
!>[A: $tType] : $o ).
thf(sy_cl_Fields_Ofield__char__0,type,
field_char_0:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oab__group__add,type,
ab_group_add:
!>[A: $tType] : $o ).
thf(sy_cl_Nat_Osemiring__char__0,type,
semiring_char_0:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oorder__bot,type,
order_bot:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oorder__top,type,
order_top:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Owellorder,type,
wellorder:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Ocomm__semiring,type,
comm_semiring:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Ozero__less__one,type,
zero_less_one:
!>[A: $tType] : $o ).
thf(sy_cl_Fields_Odivision__ring,type,
division_ring:
!>[A: $tType] : $o ).
thf(sy_cl_Fields_Ofield__abs__sgn,type,
field_abs_sgn:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Osemigroup__add,type,
semigroup_add:
!>[A: $tType] : $o ).
thf(sy_cl_Num_Osemiring__numeral,type,
semiring_numeral:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemidom__divide,type,
semidom_divide:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemidom__modulo,type,
semidom_modulo:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Osemigroup__mult,type,
semigroup_mult:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Odense__order,type,
dense_order:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Ocomm__semiring__0,type,
comm_semiring_0:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Ocomm__semiring__1,type,
comm_semiring_1:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__idom,type,
linordered_idom:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__ring,type,
linordered_ring:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemiring__modulo,type,
semiring_modulo:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocomm__monoid__add,type,
comm_monoid_add:
!>[A: $tType] : $o ).
thf(sy_cl_Parity_Osemiring__parity,type,
semiring_parity:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oordered__ring__abs,type,
ordered_ring_abs:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oordered__semiring,type,
ordered_semiring:
!>[A: $tType] : $o ).
thf(sy_cl_Code__Evaluation_Oterm__of,type,
code_term_of:
!>[A: $tType] : $o ).
thf(sy_cl_Fields_Olinordered__field,type,
linordered_field:
!>[A: $tType] : $o ).
thf(sy_cl_Generic__set__bit_Oset__bit,type,
generic_set_set_bit:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oab__semigroup__add,type,
ab_semigroup_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocomm__monoid__diff,type,
comm_monoid_diff:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocomm__monoid__mult,type,
comm_monoid_mult:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oalgebraic__semidom,type,
algebraic_semidom:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemiring__1__cancel,type,
semiring_1_cancel:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oab__semigroup__mult,type,
ab_semigroup_mult:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Odense__linorder,type,
dense_linorder:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__semidom,type,
linordered_semidom:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oordered__semiring__0,type,
ordered_semiring_0:
!>[A: $tType] : $o ).
thf(sy_cl_Least__significant__bit_Olsb,type,
least_6119777620449941438nt_lsb:
!>[A: $tType] : $o ).
thf(sy_cl_Real__Vector__Spaces_Obanach,type,
real_Vector_banach:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__semiring,type,
linordered_semiring:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocancel__semigroup__add,type,
cancel_semigroup_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__ab__group__add,type,
ordered_ab_group_add:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__semiring__1,type,
linord6961819062388156250ring_1:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oordered__comm__semiring,type,
ordere2520102378445227354miring:
!>[A: $tType] : $o ).
thf(sy_cl_Topological__Spaces_Ot1__space,type,
topological_t1_space:
!>[A: $tType] : $o ).
thf(sy_cl_Topological__Spaces_Ot2__space,type,
topological_t2_space:
!>[A: $tType] : $o ).
thf(sy_cl_Bit__Operations_Osemiring__bits,type,
bit_semiring_bits:
!>[A: $tType] : $o ).
thf(sy_cl_Limits_Otopological__group__add,type,
topolo1633459387980952147up_add:
!>[A: $tType] : $o ).
thf(sy_cl_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_Quickcheck__Narrowing_Opartial__term__of,type,
quickc6926020345158392990erm_of:
!>[A: $tType] : $o ).
thf(sy_cl_Real__Vector__Spaces_Oreal__normed__field,type,
real_V3459762299906320749_field:
!>[A: $tType] : $o ).
thf(sy_cl_Topological__Spaces_Odiscrete__topology,type,
topolo8865339358273720382pology:
!>[A: $tType] : $o ).
thf(sy_cl_Topological__Spaces_Olinorder__topology,type,
topolo1944317154257567458pology:
!>[A: $tType] : $o ).
thf(sy_cl_Topological__Spaces_Otopological__space,type,
topolo4958980785337419405_space:
!>[A: $tType] : $o ).
thf(sy_cl_Euclidean__Division_Oeuclidean__semiring,type,
euclid3725896446679973847miring:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
canoni5634975068530333245id_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__cancel__comm__monoid__add,type,
ordere8940638589300402666id_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ostrict__ordered__comm__monoid__add,type,
strict7427464778891057005id_add:
!>[A: $tType] : $o ).
thf(sy_cl_Real__Vector__Spaces_Oreal__normed__vector,type,
real_V822414075346904944vector:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__comm__semiring__strict,type,
linord2810124833399127020strict:
!>[A: $tType] : $o ).
thf(sy_cl_Bit__Operations_Osemiring__bit__operations,type,
bit_se359711467146920520ations:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__ab__semigroup__add__imp__le,type,
ordere2412721322843649153imp_le:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__cancel__ab__semigroup__add,type,
ordere580206878836729694up_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__cancel__comm__monoid__diff,type,
ordere1170586879665033532d_diff:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ostrict__ordered__ab__semigroup__add,type,
strict9044650504122735259up_add:
!>[A: $tType] : $o ).
thf(sy_cl_Real__Vector__Spaces_Oordered__real__vector,type,
real_V5355595471888546746vector:
!>[A: $tType] : $o ).
thf(sy_cl_Real__Vector__Spaces_Oreal__normed__algebra,type,
real_V4412858255891104859lgebra:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemiring__no__zero__divisors__cancel,type,
semiri6575147826004484403cancel:
!>[A: $tType] : $o ).
thf(sy_cl_Euclidean__Division_Oeuclidean__ring__cancel,type,
euclid8851590272496341667cancel:
!>[A: $tType] : $o ).
thf(sy_cl_Real__Vector__Spaces_Oreal__normed__algebra__1,type,
real_V2822296259951069270ebra_1:
!>[A: $tType] : $o ).
thf(sy_cl_Divides_Ounique__euclidean__semiring__numeral,type,
unique1627219031080169319umeral:
!>[A: $tType] : $o ).
thf(sy_cl_Real__Vector__Spaces_Oreal__normed__div__algebra,type,
real_V8999393235501362500lgebra:
!>[A: $tType] : $o ).
thf(sy_cl_Topological__Spaces_Ofirst__countable__topology,type,
topolo3112930676232923870pology:
!>[A: $tType] : $o ).
thf(sy_cl_Euclidean__Division_Oeuclidean__semiring__cancel,type,
euclid4440199948858584721cancel:
!>[A: $tType] : $o ).
thf(sy_cl_Euclidean__Division_Ounique__euclidean__semiring,type,
euclid3128863361964157862miring:
!>[A: $tType] : $o ).
thf(sy_cl_Topological__Spaces_Olinear__continuum__topology,type,
topolo8458572112393995274pology:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__ab__semigroup__monoid__add__imp__le,type,
ordere1937475149494474687imp_le:
!>[A: $tType] : $o ).
thf(sy_cl_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_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__Nat_Oint__of__nat,type,
code_T6385005292777649522of_nat: nat > int ).
thf(sy_c_Code__Target__Word__Base_Oquickcheck__narrowing__samples_Onarrowing__samples,type,
code_T4080844693773952564amples:
!>[A: $tType] : ( ( code_integer > ( product_prod @ A @ A ) ) > A > code_integer > ( list @ A ) ) ).
thf(sy_c_Code__Target__Word__Base_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_Ocomplex_OComplex,type,
complex2: real > real > complex ).
thf(sy_c_Complex_Ocsqrt,type,
csqrt: complex > complex ).
thf(sy_c_Complex_Oimaginary__unit,type,
imaginary_unit: complex ).
thf(sy_c_Deriv_Odifferentiable,type,
differentiable:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( filter @ A ) > $o ) ).
thf(sy_c_Deriv_Ohas__derivative,type,
has_derivative:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( A > B ) > ( filter @ A ) > $o ) ).
thf(sy_c_Deriv_Ohas__field__derivative,type,
has_field_derivative:
!>[A: $tType] : ( ( A > A ) > A > ( filter @ A ) > $o ) ).
thf(sy_c_Divides_Oadjust__div,type,
adjust_div: ( product_prod @ int @ int ) > int ).
thf(sy_c_Divides_Odivmod__nat,type,
divmod_nat: nat > nat > ( product_prod @ nat @ nat ) ).
thf(sy_c_Divides_Oeucl__rel__int,type,
eucl_rel_int: int > int > ( product_prod @ int @ int ) > $o ).
thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivides__aux,type,
unique5940410009612947441es_aux:
!>[A: $tType] : ( ( product_prod @ A @ A ) > $o ) ).
thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod,type,
unique8689654367752047608divmod:
!>[A: $tType] : ( num > num > ( product_prod @ A @ A ) ) ).
thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod__step,type,
unique1321980374590559556d_step:
!>[A: $tType] : ( num > ( product_prod @ A @ A ) > ( product_prod @ A @ A ) ) ).
thf(sy_c_Factorial_Ocomm__semiring__1__class_Opochhammer,type,
comm_s3205402744901411588hammer:
!>[A: $tType] : ( A > nat > A ) ).
thf(sy_c_Factorial_Osemiring__char__0__class_Ofact,type,
semiring_char_0_fact:
!>[A: $tType] : ( nat > A ) ).
thf(sy_c_Fields_Oinverse__class_Oinverse,type,
inverse_inverse:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Filter_Oat__bot,type,
at_bot:
!>[A: $tType] : ( filter @ A ) ).
thf(sy_c_Filter_Oat__top,type,
at_top:
!>[A: $tType] : ( filter @ A ) ).
thf(sy_c_Filter_Oeventually,type,
eventually:
!>[A: $tType] : ( ( A > $o ) > ( filter @ A ) > $o ) ).
thf(sy_c_Filter_Ofilterlim,type,
filterlim:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( filter @ B ) > ( filter @ A ) > $o ) ).
thf(sy_c_Filter_Omap__filter__on,type,
map_filter_on:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B ) > ( filter @ A ) > ( filter @ B ) ) ).
thf(sy_c_Finite__Set_Ofinite,type,
finite_finite2:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
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__Time__Monad_Oreturn,type,
heap_Time_return:
!>[A: $tType] : ( A > ( heap_Time_Heap @ A ) ) ).
thf(sy_c_If,type,
if:
!>[A: $tType] : ( $o > A > A > A ) ).
thf(sy_c_Int_Oint__ge__less__than,type,
int_ge_less_than: int > ( set @ ( product_prod @ int @ int ) ) ).
thf(sy_c_Int_Oint__ge__less__than2,type,
int_ge_less_than2: int > ( set @ ( product_prod @ int @ int ) ) ).
thf(sy_c_Int_Onat,type,
nat2: int > nat ).
thf(sy_c_Int_Oring__1__class_OInts,type,
ring_1_Ints:
!>[A: $tType] : ( set @ A ) ).
thf(sy_c_Int_Oring__1__class_Oof__int,type,
ring_1_of_int:
!>[A: $tType] : ( int > A ) ).
thf(sy_c_Lattices__Big_Olinorder__class_OMax,type,
lattic643756798349783984er_Max:
!>[A: $tType] : ( ( set @ A ) > A ) ).
thf(sy_c_Least__significant__bit_Olsb__class_Olsb,type,
least_8051144512741203767sb_lsb:
!>[A: $tType] : ( A > $o ) ).
thf(sy_c_Limits_OBfun,type,
bfun:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( filter @ A ) > $o ) ).
thf(sy_c_Limits_Oat__infinity,type,
at_infinity:
!>[A: $tType] : ( filter @ A ) ).
thf(sy_c_List_Oappend,type,
append:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Oconcat,type,
concat:
!>[A: $tType] : ( ( list @ ( list @ A ) ) > ( list @ A ) ) ).
thf(sy_c_List_Ocount__list,type,
count_list:
!>[A: $tType] : ( ( list @ A ) > A > nat ) ).
thf(sy_c_List_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_Olex,type,
lex:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) ) ).
thf(sy_c_List_Olexn,type,
lexn:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > nat > ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) ) ).
thf(sy_c_List_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_Omap,type,
map:
!>[A: $tType,Aa: $tType] : ( ( A > Aa ) > ( list @ A ) > ( list @ Aa ) ) ).
thf(sy_c_List_Olist_Oset,type,
set2:
!>[A: $tType] : ( ( list @ A ) > ( set @ A ) ) ).
thf(sy_c_List_Olist_Osize__list,type,
size_list:
!>[A: $tType] : ( ( A > nat ) > ( list @ A ) > nat ) ).
thf(sy_c_List_Olist__update,type,
list_update:
!>[A: $tType] : ( ( list @ A ) > nat > A > ( list @ A ) ) ).
thf(sy_c_List_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_Ofun__of__rel,type,
fun_of_rel:
!>[B: $tType,A: $tType] : ( ( set @ ( product_prod @ B @ A ) ) > B > A ) ).
thf(sy_c_Misc_Oinv__on,type,
inv_on:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > B > A ) ).
thf(sy_c_Misc_Opairself,type,
pairself:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( product_prod @ A @ A ) > ( product_prod @ B @ B ) ) ).
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_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_Ocase__num,type,
case_num:
!>[A: $tType] : ( A > ( num > A ) > ( num > A ) > num > A ) ).
thf(sy_c_Num_Onum_Osize__num,type,
size_num: num > nat ).
thf(sy_c_Num_Onum__of__nat,type,
num_of_nat: nat > num ).
thf(sy_c_Num_Onumeral__class_Onumeral,type,
numeral_numeral:
!>[A: $tType] : ( num > A ) ).
thf(sy_c_Num_Opow,type,
pow: num > num > num ).
thf(sy_c_Num_Opred__numeral,type,
pred_numeral: num > nat ).
thf(sy_c_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_OGreatest,type,
order_Greatest:
!>[A: $tType] : ( ( A > $o ) > A ) ).
thf(sy_c_Orderings_Oorder__class_Oantimono,type,
order_antimono:
!>[A: $tType,B: $tType] : ( ( A > B ) > $o ) ).
thf(sy_c_Orderings_Otop__class_Otop,type,
top_top:
!>[A: $tType] : A ).
thf(sy_c_Power_Opower__class_Opower,type,
power_power:
!>[A: $tType] : ( A > nat > A ) ).
thf(sy_c_Product__Type_OPair,type,
product_Pair:
!>[A: $tType,B: $tType] : ( A > B > ( product_prod @ A @ B ) ) ).
thf(sy_c_Product__Type_Oprod_Ocase__prod,type,
product_case_prod:
!>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
thf(sy_c_Product__Type_Oproduct,type,
product_product:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( set @ B ) > ( set @ ( product_prod @ A @ B ) ) ) ).
thf(sy_c_Pure_Otype,type,
type2:
!>[A: $tType] : ( itself @ A ) ).
thf(sy_c_Rat_OFrct,type,
frct: ( product_prod @ int @ int ) > rat ).
thf(sy_c_Rat_Onormalize,type,
normalize: ( product_prod @ int @ int ) > ( product_prod @ int @ int ) ).
thf(sy_c_Rat_Oof__int,type,
of_int: int > rat ).
thf(sy_c_Rat_Oquotient__of,type,
quotient_of: rat > ( product_prod @ int @ int ) ).
thf(sy_c_Real__Vector__Spaces_Obounded__linear,type,
real_V3181309239436604168linear:
!>[A: $tType,B: $tType] : ( ( A > B ) > $o ) ).
thf(sy_c_Real__Vector__Spaces_Odist__class_Odist,type,
real_V557655796197034286t_dist:
!>[A: $tType] : ( A > A > real ) ).
thf(sy_c_Real__Vector__Spaces_Onorm__class_Onorm,type,
real_V7770717601297561774m_norm:
!>[A: $tType] : ( A > real ) ).
thf(sy_c_Real__Vector__Spaces_Oof__real,type,
real_Vector_of_real:
!>[A: $tType] : ( real > A ) ).
thf(sy_c_Real__Vector__Spaces_OscaleR__class_OscaleR,type,
real_V8093663219630862766scaleR:
!>[A: $tType] : ( real > A > A ) ).
thf(sy_c_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_Othe__elem,type,
the_elem:
!>[A: $tType] : ( ( set @ A ) > A ) ).
thf(sy_c_Set__Interval_Ofold__atLeastAtMost__nat,type,
set_fo6178422350223883121st_nat:
!>[A: $tType] : ( ( nat > A > A ) > nat > nat > A > A ) ).
thf(sy_c_Set__Interval_Oord__class_OatLeast,type,
set_ord_atLeast:
!>[A: $tType] : ( A > ( set @ A ) ) ).
thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost,type,
set_or1337092689740270186AtMost:
!>[A: $tType] : ( A > A > ( set @ A ) ) ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan,type,
set_or7035219750837199246ssThan:
!>[A: $tType] : ( A > A > ( set @ A ) ) ).
thf(sy_c_Set__Interval_Oord__class_OatMost,type,
set_ord_atMost:
!>[A: $tType] : ( A > ( set @ A ) ) ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThan,type,
set_ord_greaterThan:
!>[A: $tType] : ( A > ( set @ A ) ) ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost,type,
set_or3652927894154168847AtMost:
!>[A: $tType] : ( A > A > ( set @ A ) ) ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan,type,
set_or5935395276787703475ssThan:
!>[A: $tType] : ( A > A > ( set @ A ) ) ).
thf(sy_c_Set__Interval_Oord__class_OlessThan,type,
set_ord_lessThan:
!>[A: $tType] : ( A > ( set @ A ) ) ).
thf(sy_c_Signed__Division_Osigned__division__class_Osigned__divide,type,
signed7115095781618012415divide:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Signed__Division_Osigned__division__class_Osigned__modulo,type,
signed6721504322012087516modulo:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_String_Oascii__of,type,
ascii_of: char > char ).
thf(sy_c_String_Ochar_OChar,type,
char2: $o > $o > $o > $o > $o > $o > $o > $o > char ).
thf(sy_c_String_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_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_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_H,type,
vEBT_V9176841429113362141ildupi: nat > int ).
thf(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OT__vebt__buildupi_H__rel,type,
vEBT_V3352910403632780892pi_rel: nat > nat > $o ).
thf(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OT__vebt__buildupi__rel,type,
vEBT_V2957053500504383685pi_rel: nat > nat > $o ).
thf(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OTb,type,
vEBT_VEBT_Tb: nat > int ).
thf(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OTb_H,type,
vEBT_VEBT_Tb2: nat > nat ).
thf(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OTb_H__rel,type,
vEBT_VEBT_Tb_rel: nat > nat > $o ).
thf(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OTb__rel,type,
vEBT_VEBT_Tb_rel2: nat > nat > $o ).
thf(sy_c_VEBT__BuildupMemImp_OVEBT__internal_Ohighi,type,
vEBT_VEBT_highi: nat > nat > ( heap_Time_Heap @ nat ) ).
thf(sy_c_VEBT__BuildupMemImp_OVEBT__internal_Olowi,type,
vEBT_VEBT_lowi: nat > nat > ( heap_Time_Heap @ nat ) ).
thf(sy_c_VEBT__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__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_Oof__nat,type,
of_nat:
!>[A: $tType] : ( nat > ( word @ A ) ) ).
thf(sy_c_Word_Orevcast,type,
revcast:
!>[A: $tType,B: $tType] : ( ( word @ A ) > ( word @ B ) ) ).
thf(sy_c_Word_Oring__1__class_Osigned,type,
ring_1_signed:
!>[B: $tType,A: $tType] : ( ( word @ B ) > A ) ).
thf(sy_c_Word_Osemiring__1__class_Ounsigned,type,
semiring_1_unsigned:
!>[B: $tType,A: $tType] : ( ( word @ B ) > A ) ).
thf(sy_c_Word_Osigned__cast,type,
signed_cast:
!>[A: $tType,B: $tType] : ( ( word @ A ) > ( word @ B ) ) ).
thf(sy_c_Word_Osigned__drop__bit,type,
signed_drop_bit:
!>[A: $tType] : ( nat > ( word @ A ) > ( word @ A ) ) ).
thf(sy_c_Word_Oslice,type,
slice2:
!>[A: $tType,B: $tType] : ( nat > ( word @ A ) > ( word @ B ) ) ).
thf(sy_c_Word_Oslice1,type,
slice1:
!>[A: $tType,B: $tType] : ( nat > ( word @ A ) > ( word @ B ) ) ).
thf(sy_c_Word_Othe__signed__int,type,
the_signed_int:
!>[A: $tType] : ( ( word @ A ) > int ) ).
thf(sy_c_Word_Oword__cat,type,
word_cat:
!>[A: $tType,B: $tType,C: $tType] : ( ( word @ A ) > ( word @ B ) > ( word @ C ) ) ).
thf(sy_c_Word_Oword__int__case,type,
word_int_case:
!>[B: $tType,A: $tType] : ( ( int > B ) > ( word @ A ) > B ) ).
thf(sy_c_Word_Oword__pred,type,
word_pred:
!>[A: $tType] : ( ( word @ A ) > ( word @ A ) ) ).
thf(sy_c_Word_Oword__roti,type,
word_roti:
!>[A: $tType] : ( int > ( word @ A ) > ( word @ A ) ) ).
thf(sy_c_Word_Oword__rotl,type,
word_rotl:
!>[A: $tType] : ( nat > ( word @ A ) > ( word @ A ) ) ).
thf(sy_c_Word_Oword__rotr,type,
word_rotr:
!>[A: $tType] : ( nat > ( word @ A ) > ( word @ A ) ) ).
thf(sy_c_Word_Oword__sle,type,
word_sle:
!>[A: $tType] : ( ( word @ A ) > ( word @ A ) > $o ) ).
thf(sy_c_Word_Oword__sless,type,
word_sless:
!>[A: $tType] : ( ( word @ A ) > ( word @ A ) > $o ) ).
thf(sy_c_Word_Oword__split,type,
word_split:
!>[A: $tType,B: $tType,C: $tType] : ( ( word @ A ) > ( product_prod @ ( word @ B ) @ ( word @ C ) ) ) ).
thf(sy_c_Word_Oword__succ,type,
word_succ:
!>[A: $tType] : ( ( word @ A ) > ( word @ A ) ) ).
thf(sy_c_fChoice,type,
fChoice:
!>[A: $tType] : ( ( A > $o ) > A ) ).
thf(sy_c_member,type,
member:
!>[A: $tType] : ( A > ( set @ A ) > $o ) ).
thf(sy_v_na____,type,
na: nat ).
% Relevant facts (8180)
thf(fact_0_False,axiom,
~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ na ) ).
% False
thf(fact_1_max__in__set__def,axiom,
( vEBT_VEBT_max_in_set
= ( ^ [Xs: set @ nat,X: nat] :
( ( member @ nat @ X @ Xs )
& ! [Y: nat] :
( ( member @ nat @ Y @ Xs )
=> ( ord_less_eq @ nat @ Y @ X ) ) ) ) ) ).
% max_in_set_def
thf(fact_2_min__in__set__def,axiom,
( vEBT_VEBT_min_in_set
= ( ^ [Xs: set @ nat,X: nat] :
( ( member @ nat @ X @ Xs )
& ! [Y: nat] :
( ( member @ nat @ Y @ Xs )
=> ( ord_less_eq @ nat @ X @ Y ) ) ) ) ) ).
% min_in_set_def
thf(fact_3__C3_OIH_C_I4_J,axiom,
! [X2: nat] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ na ) ) )
=> ( ( X2
= ( divide_divide @ nat @ ( suc @ ( suc @ na ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
=> ( ord_less_eq @ nat @ ( vEBT_VEBT_Tb2 @ ( suc @ X2 ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ one2 ) ) ) @ ( vEBT_VEBT_cnt2 @ ( vEBT_vebt_buildup @ ( suc @ X2 ) ) ) ) ) ) ) ).
% "3.IH"(4)
thf(fact_4__C3_OIH_C_I3_J,axiom,
! [X2: nat] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ na ) ) )
=> ( ( X2
= ( divide_divide @ nat @ ( suc @ ( suc @ na ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
=> ( ord_less_eq @ nat @ ( vEBT_VEBT_Tb2 @ X2 ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ one2 ) ) ) @ ( vEBT_VEBT_cnt2 @ ( vEBT_vebt_buildup @ X2 ) ) ) ) ) ) ).
% "3.IH"(3)
thf(fact_5__C3_OIH_C_I1_J,axiom,
! [X2: nat] :
( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ na ) ) )
=> ( ( X2
= ( divide_divide @ nat @ ( suc @ ( suc @ na ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
=> ( ord_less_eq @ nat @ ( vEBT_VEBT_Tb2 @ X2 ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ one2 ) ) ) @ ( vEBT_VEBT_cnt2 @ ( vEBT_vebt_buildup @ X2 ) ) ) ) ) ) ).
% "3.IH"(1)
thf(fact_6_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_7_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_8_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_9__C0_C,axiom,
ord_less_eq @ nat @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ one2 ) ) ) @ ( vEBT_VEBT_Tb2 @ ( suc @ ( suc @ ( divide_divide @ nat @ na @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) @ ( times_times @ nat @ ( vEBT_VEBT_Tb2 @ ( suc @ ( divide_divide @ nat @ na @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ ( divide_divide @ nat @ na @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ one2 ) ) ) @ ( vEBT_VEBT_cnt2 @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ na ) ) @ ( replicate @ vEBT_VEBT @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( divide_divide @ nat @ ( suc @ ( suc @ na ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide @ nat @ ( suc @ ( suc @ na ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( suc @ ( divide_divide @ nat @ ( suc @ ( suc @ na ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% "0"
thf(fact_10_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_11_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_12_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_13_vebt__buildup_Osimps_I3_J,axiom,
! [Va: nat] :
( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ Va ) ) )
=> ( ( vEBT_vebt_buildup @ ( suc @ ( suc @ Va ) ) )
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ Va ) ) @ ( replicate @ vEBT_VEBT @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) )
& ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ Va ) ) )
=> ( ( vEBT_vebt_buildup @ ( suc @ ( suc @ Va ) ) )
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ Va ) ) @ ( replicate @ vEBT_VEBT @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( suc @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ).
% vebt_buildup.simps(3)
thf(fact_14_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_15_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_16_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_17_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_18_even__odd__cases,axiom,
! [X2: nat] :
( ! [N2: nat] :
( X2
!= ( plus_plus @ nat @ N2 @ N2 ) )
=> ~ ! [N2: nat] :
( X2
!= ( plus_plus @ nat @ N2 @ ( suc @ N2 ) ) ) ) ).
% even_odd_cases
thf(fact_19_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_20_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_21_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_22_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_23_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_24_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_25_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_26_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_27_num__double,axiom,
! [N: num] :
( ( times_times @ num @ ( bit0 @ one2 ) @ N )
= ( bit0 @ N ) ) ).
% num_double
thf(fact_28_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_29_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_30_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_31_bit__concat__def,axiom,
( vEBT_VEBT_bit_concat
= ( ^ [H: nat,L: nat,D: nat] : ( plus_plus @ nat @ ( times_times @ nat @ H @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ D ) ) @ L ) ) ) ).
% bit_concat_def
thf(fact_32_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_33_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_34_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_35_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_36_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_37_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_38_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_39_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_40_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_41_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_42_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_43_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_44_mem__Collect__eq,axiom,
! [A: $tType,A2: A,P: A > $o] :
( ( member @ A @ A2 @ ( collect @ A @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_45_Collect__mem__eq,axiom,
! [A: $tType,A3: set @ A] :
( ( collect @ A
@ ^ [X: A] : ( member @ A @ X @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_46_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_47_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_48_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_49_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_50_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_51_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_52_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_53_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_54_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_55_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_56_le__num__One__iff,axiom,
! [X2: num] :
( ( ord_less_eq @ num @ X2 @ one2 )
= ( X2 = one2 ) ) ).
% le_num_One_iff
thf(fact_57_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_58_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_59_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_60_subset__divisors__dvd,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [C3: A] : ( dvd_dvd @ A @ C3 @ A2 ) )
@ ( collect @ A
@ ^ [C3: A] : ( dvd_dvd @ A @ C3 @ B2 ) ) )
= ( dvd_dvd @ A @ A2 @ B2 ) ) ) ).
% subset_divisors_dvd
thf(fact_61_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_62_div__mult2__numeral__eq,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [A2: A,K: num,L2: num] :
( ( divide_divide @ A @ ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ K ) ) @ ( numeral_numeral @ A @ L2 ) )
= ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( times_times @ num @ K @ L2 ) ) ) ) ) ).
% div_mult2_numeral_eq
thf(fact_63_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_64_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_65_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_66_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_67_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_68_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_69_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_70_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_71_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_72_dvd__refl,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A2: A] : ( dvd_dvd @ A @ A2 @ A2 ) ) ).
% dvd_refl
thf(fact_73_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_74_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_75_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_76_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_77_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_78_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_79_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_80_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_81_dvd__def,axiom,
! [A: $tType] :
( ( dvd @ A )
=> ( ( dvd_dvd @ A )
= ( ^ [B3: A,A4: A] :
? [K2: A] :
( A4
= ( times_times @ A @ B3 @ K2 ) ) ) ) ) ).
% dvd_def
thf(fact_82_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_83_dvdE,axiom,
! [A: $tType] :
( ( dvd @ A )
=> ! [B2: A,A2: A] :
( ( dvd_dvd @ A @ B2 @ A2 )
=> ~ ! [K3: A] :
( A2
!= ( times_times @ A @ B2 @ K3 ) ) ) ) ).
% dvdE
thf(fact_84_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_85_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_86_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_87_div__le__dividend,axiom,
! [M: nat,N: nat] : ( ord_less_eq @ nat @ ( divide_divide @ nat @ M @ N ) @ M ) ).
% div_le_dividend
thf(fact_88_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_89_div__mult2__eq,axiom,
! [M: nat,N: nat,Q2: nat] :
( ( divide_divide @ nat @ M @ ( times_times @ nat @ N @ Q2 ) )
= ( divide_divide @ nat @ ( divide_divide @ nat @ M @ N ) @ Q2 ) ) ).
% div_mult2_eq
thf(fact_90_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_91_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_92_divide__numeral__1,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A2: A] :
( ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ one2 ) )
= A2 ) ) ).
% divide_numeral_1
thf(fact_93_num_Oexhaust,axiom,
! [Y2: num] :
( ( Y2 != one2 )
=> ( ! [X22: num] :
( Y2
!= ( bit0 @ X22 ) )
=> ~ ! [X32: num] :
( Y2
!= ( bit1 @ X32 ) ) ) ) ).
% num.exhaust
thf(fact_94_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_95_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_96_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_97_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_98_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_99_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_100_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_101_VEBT__internal_OTb_H_Osimps_I3_J,axiom,
! [N: nat] :
( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( vEBT_VEBT_Tb2 @ ( suc @ ( suc @ N ) ) )
= ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ one2 ) ) ) @ ( vEBT_VEBT_Tb2 @ ( suc @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( times_times @ nat @ ( vEBT_VEBT_Tb2 @ ( suc @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) )
& ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( vEBT_VEBT_Tb2 @ ( suc @ ( suc @ N ) ) )
= ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ one2 ) ) ) @ ( vEBT_VEBT_Tb2 @ ( suc @ ( suc @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) @ ( times_times @ nat @ ( vEBT_VEBT_Tb2 @ ( suc @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.Tb'.simps(3)
thf(fact_102_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_103_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_104_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_105_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_106_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_107_evenE,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A2: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
=> ~ ! [B4: A] :
( A2
!= ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B4 ) ) ) ) ).
% evenE
thf(fact_108_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_109_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_110_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_111_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_112_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_113_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_114_sum__squares__bound,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X2: A,Y2: A] : ( ord_less_eq @ A @ ( times_times @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X2 ) @ Y2 ) @ ( plus_plus @ A @ ( power_power @ A @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% sum_squares_bound
thf(fact_115_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_116_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_117_power2__sum,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [X2: A,Y2: A] :
( ( power_power @ A @ ( plus_plus @ A @ X2 @ Y2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( plus_plus @ A @ ( plus_plus @ A @ ( power_power @ A @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( times_times @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X2 ) @ Y2 ) ) ) ) ).
% power2_sum
thf(fact_118_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_119_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_120_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_121_div2__even__ext__nat,axiom,
! [X2: nat,Y2: nat] :
( ( ( divide_divide @ nat @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( divide_divide @ nat @ Y2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
=> ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ X2 )
= ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Y2 ) )
=> ( X2 = Y2 ) ) ) ).
% div2_even_ext_nat
thf(fact_122_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_123_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_124_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_125_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_126_nat_Oinject,axiom,
! [X23: nat,Y22: nat] :
( ( ( suc @ X23 )
= ( suc @ Y22 ) )
= ( X23 = Y22 ) ) ).
% nat.inject
thf(fact_127_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_128_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_129_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_130_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_131_Suc__numeral,axiom,
! [N: num] :
( ( suc @ ( numeral_numeral @ nat @ N ) )
= ( numeral_numeral @ nat @ ( plus_plus @ num @ N @ one2 ) ) ) ).
% Suc_numeral
thf(fact_132_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_133_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_134_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_135_add__One__commute,axiom,
! [N: num] :
( ( plus_plus @ num @ one2 @ N )
= ( plus_plus @ num @ N @ one2 ) ) ).
% add_One_commute
thf(fact_136_four__x__squared,axiom,
! [X2: real] :
( ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( power_power @ real @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ X2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% four_x_squared
thf(fact_137_Suc__inject,axiom,
! [X2: nat,Y2: nat] :
( ( ( suc @ X2 )
= ( suc @ Y2 ) )
=> ( X2 = Y2 ) ) ).
% Suc_inject
thf(fact_138_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_139_le__refl,axiom,
! [N: nat] : ( ord_less_eq @ nat @ N @ N ) ).
% le_refl
thf(fact_140_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_141_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq @ nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_142_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_143_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_144_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_145_dvd__antisym,axiom,
! [M: nat,N: nat] :
( ( dvd_dvd @ nat @ M @ N )
=> ( ( dvd_dvd @ nat @ N @ M )
=> ( M = N ) ) ) ).
% dvd_antisym
thf(fact_146_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_147_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_148_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_149_power__commuting__commutes,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [X2: A,Y2: A,N: nat] :
( ( ( times_times @ A @ X2 @ Y2 )
= ( times_times @ A @ Y2 @ X2 ) )
=> ( ( times_times @ A @ ( power_power @ A @ X2 @ N ) @ Y2 )
= ( times_times @ A @ Y2 @ ( power_power @ A @ X2 @ N ) ) ) ) ) ).
% power_commuting_commutes
thf(fact_150_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_151_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_152_add__Suc,axiom,
! [M: nat,N: nat] :
( ( plus_plus @ nat @ ( suc @ M ) @ N )
= ( suc @ ( plus_plus @ nat @ M @ N ) ) ) ).
% add_Suc
thf(fact_153_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_154_Suc__leD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ ( suc @ M ) @ N )
=> ( ord_less_eq @ nat @ M @ N ) ) ).
% Suc_leD
thf(fact_155_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_156_le__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ord_less_eq @ nat @ M @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_157_Suc__le__D,axiom,
! [N: nat,M2: nat] :
( ( ord_less_eq @ nat @ ( suc @ N ) @ M2 )
=> ? [M3: nat] :
( M2
= ( suc @ M3 ) ) ) ).
% Suc_le_D
thf(fact_158_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_159_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq @ nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_160_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_161_full__nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M4: nat] :
( ( ord_less_eq @ nat @ ( suc @ M4 ) @ N2 )
=> ( P @ M4 ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% full_nat_induct
thf(fact_162_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_163_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,Z2: nat] :
( ( R @ X3 @ Y3 )
=> ( ( R @ Y3 @ Z2 )
=> ( R @ X3 @ Z2 ) ) )
=> ( ! [N2: nat] : ( R @ N2 @ ( suc @ N2 ) )
=> ( R @ M @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_164_dvd__power__same,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [X2: A,Y2: A,N: nat] :
( ( dvd_dvd @ A @ X2 @ Y2 )
=> ( dvd_dvd @ A @ ( power_power @ A @ X2 @ N ) @ ( power_power @ A @ Y2 @ N ) ) ) ) ).
% dvd_power_same
thf(fact_165_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_166_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_167_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq @ nat @ N @ ( plus_plus @ nat @ N @ M ) ) ).
% le_add1
thf(fact_168_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq @ nat @ N @ ( plus_plus @ nat @ M @ N ) ) ).
% le_add2
thf(fact_169_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_170_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_171_le__Suc__ex,axiom,
! [K: nat,L2: nat] :
( ( ord_less_eq @ nat @ K @ L2 )
=> ? [N2: nat] :
( L2
= ( plus_plus @ nat @ K @ N2 ) ) ) ).
% le_Suc_ex
thf(fact_172_add__le__mono,axiom,
! [I: nat,J: nat,K: nat,L2: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( ord_less_eq @ nat @ K @ L2 )
=> ( ord_less_eq @ nat @ ( plus_plus @ nat @ I @ K ) @ ( plus_plus @ nat @ J @ L2 ) ) ) ) ).
% add_le_mono
thf(fact_173_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_174_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_175_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_176_nat__le__iff__add,axiom,
( ( ord_less_eq @ nat )
= ( ^ [M5: nat,N3: nat] :
? [K2: nat] :
( N3
= ( plus_plus @ nat @ M5 @ K2 ) ) ) ) ).
% nat_le_iff_add
thf(fact_177_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_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_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_180_le__cube,axiom,
! [M: nat] : ( ord_less_eq @ nat @ M @ ( times_times @ nat @ M @ ( times_times @ nat @ M @ M ) ) ) ).
% le_cube
thf(fact_181_le__square,axiom,
! [M: nat] : ( ord_less_eq @ nat @ M @ ( times_times @ nat @ M @ M ) ) ).
% le_square
thf(fact_182_mult__le__mono,axiom,
! [I: nat,J: nat,K: nat,L2: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( ord_less_eq @ nat @ K @ L2 )
=> ( ord_less_eq @ nat @ ( times_times @ nat @ I @ K ) @ ( times_times @ nat @ J @ L2 ) ) ) ) ).
% mult_le_mono
thf(fact_183_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_184_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_185_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_186_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_187_lift__Suc__mono__le,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [F2: nat > A,N: nat,N4: nat] :
( ! [N2: nat] : ( ord_less_eq @ A @ ( F2 @ N2 ) @ ( F2 @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq @ nat @ N @ N4 )
=> ( ord_less_eq @ A @ ( F2 @ N ) @ ( F2 @ N4 ) ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_188_lift__Suc__antimono__le,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [F2: nat > A,N: nat,N4: nat] :
( ! [N2: nat] : ( ord_less_eq @ A @ ( F2 @ ( suc @ N2 ) ) @ ( F2 @ N2 ) )
=> ( ( ord_less_eq @ nat @ N @ N4 )
=> ( ord_less_eq @ A @ ( F2 @ N4 ) @ ( F2 @ N ) ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_189_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_190_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_191_dvd__power__le,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [X2: A,Y2: A,N: nat,M: nat] :
( ( dvd_dvd @ A @ X2 @ Y2 )
=> ( ( ord_less_eq @ nat @ N @ M )
=> ( dvd_dvd @ A @ ( power_power @ A @ X2 @ N ) @ ( power_power @ A @ Y2 @ M ) ) ) ) ) ).
% dvd_power_le
thf(fact_192_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_193_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_194_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_195_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_196_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_197_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_198_power4__eq__xxxx,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [X2: A] :
( ( power_power @ A @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) )
= ( times_times @ A @ ( times_times @ A @ ( times_times @ A @ X2 @ X2 ) @ X2 ) @ X2 ) ) ) ).
% power4_eq_xxxx
thf(fact_199_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_200_bit__eq__rec,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ( ( ^ [Y5: A,Z3: A] : Y5 = Z3 )
= ( ^ [A4: A,B3: A] :
( ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A4 )
= ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B3 ) )
& ( ( divide_divide @ A @ A4 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( divide_divide @ A @ B3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% bit_eq_rec
thf(fact_201_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_202_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_203_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_204_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_205_semiring__norm_I8_J,axiom,
! [M: num] :
( ( plus_plus @ num @ ( bit1 @ M ) @ one2 )
= ( bit0 @ ( plus_plus @ num @ M @ one2 ) ) ) ).
% semiring_norm(8)
thf(fact_206_semiring__norm_I5_J,axiom,
! [M: num] :
( ( plus_plus @ num @ ( bit0 @ M ) @ one2 )
= ( bit1 @ M ) ) ).
% semiring_norm(5)
thf(fact_207_semiring__norm_I4_J,axiom,
! [N: num] :
( ( plus_plus @ num @ one2 @ ( bit1 @ N ) )
= ( bit0 @ ( plus_plus @ num @ N @ one2 ) ) ) ).
% semiring_norm(4)
thf(fact_208_semiring__norm_I3_J,axiom,
! [N: num] :
( ( plus_plus @ num @ one2 @ ( bit0 @ N ) )
= ( bit1 @ N ) ) ).
% semiring_norm(3)
thf(fact_209_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_210_semiring__norm_I70_J,axiom,
! [M: num] :
~ ( ord_less_eq @ num @ ( bit1 @ M ) @ one2 ) ).
% semiring_norm(70)
thf(fact_211_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_212_semiring__norm_I69_J,axiom,
! [M: num] :
~ ( ord_less_eq @ num @ ( bit0 @ M ) @ one2 ) ).
% semiring_norm(69)
thf(fact_213_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_214_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_215_semiring__norm_I87_J,axiom,
! [M: num,N: num] :
( ( ( bit0 @ M )
= ( bit0 @ N ) )
= ( M = N ) ) ).
% semiring_norm(87)
thf(fact_216_semiring__norm_I90_J,axiom,
! [M: num,N: num] :
( ( ( bit1 @ M )
= ( bit1 @ N ) )
= ( M = N ) ) ).
% semiring_norm(90)
thf(fact_217_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_218_semiring__norm_I83_J,axiom,
! [N: num] :
( one2
!= ( bit0 @ N ) ) ).
% semiring_norm(83)
thf(fact_219_semiring__norm_I85_J,axiom,
! [M: num] :
( ( bit0 @ M )
!= one2 ) ).
% semiring_norm(85)
thf(fact_220_semiring__norm_I88_J,axiom,
! [M: num,N: num] :
( ( bit0 @ M )
!= ( bit1 @ N ) ) ).
% semiring_norm(88)
thf(fact_221_semiring__norm_I89_J,axiom,
! [M: num,N: num] :
( ( bit1 @ M )
!= ( bit0 @ N ) ) ).
% semiring_norm(89)
thf(fact_222_semiring__norm_I84_J,axiom,
! [N: num] :
( one2
!= ( bit1 @ N ) ) ).
% semiring_norm(84)
thf(fact_223_semiring__norm_I86_J,axiom,
! [M: num] :
( ( bit1 @ M )
!= one2 ) ).
% semiring_norm(86)
thf(fact_224_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_225_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_226_semiring__norm_I11_J,axiom,
! [M: num] :
( ( times_times @ num @ M @ one2 )
= M ) ).
% semiring_norm(11)
thf(fact_227_semiring__norm_I12_J,axiom,
! [N: num] :
( ( times_times @ num @ one2 @ N )
= N ) ).
% semiring_norm(12)
thf(fact_228_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_229_semiring__norm_I68_J,axiom,
! [N: num] : ( ord_less_eq @ num @ one2 @ N ) ).
% semiring_norm(68)
thf(fact_230_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_231_semiring__norm_I2_J,axiom,
( ( plus_plus @ num @ one2 @ one2 )
= ( bit0 @ one2 ) ) ).
% semiring_norm(2)
thf(fact_232_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_233_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_234_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_235_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_236_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_237_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_238_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_239_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_240_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_241_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_242_field__sum__of__halves,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X2: A] :
( ( plus_plus @ A @ ( divide_divide @ A @ X2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ A @ X2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
= X2 ) ) ).
% field_sum_of_halves
thf(fact_243_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_244_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_245_buildup__nothing__in__leaf,axiom,
! [N: nat,X2: nat] :
~ ( vEBT_V5719532721284313246member @ ( vEBT_vebt_buildup @ N ) @ X2 ) ).
% buildup_nothing_in_leaf
thf(fact_246_cnt__cnt__eq,axiom,
( vEBT_VEBT_cnt
= ( ^ [T: vEBT_VEBT] : ( semiring_1_of_nat @ real @ ( vEBT_VEBT_cnt2 @ T ) ) ) ) ).
% cnt_cnt_eq
thf(fact_247_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_248_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_249_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_250_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_251_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_252_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_253_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_254_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_255_of__nat__eq__of__nat__power__cancel__iff,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [B2: nat,W: nat,X2: nat] :
( ( ( power_power @ A @ ( semiring_1_of_nat @ A @ B2 ) @ W )
= ( semiring_1_of_nat @ A @ X2 ) )
= ( ( power_power @ nat @ B2 @ W )
= X2 ) ) ) ).
% of_nat_eq_of_nat_power_cancel_iff
thf(fact_256_of__nat__power__eq__of__nat__cancel__iff,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [X2: nat,B2: nat,W: nat] :
( ( ( semiring_1_of_nat @ A @ X2 )
= ( power_power @ A @ ( semiring_1_of_nat @ A @ B2 ) @ W ) )
= ( X2
= ( power_power @ nat @ B2 @ W ) ) ) ) ).
% of_nat_power_eq_of_nat_cancel_iff
thf(fact_257_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_258_numeral__power__eq__of__nat__cancel__iff,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [X2: num,N: nat,Y2: nat] :
( ( ( power_power @ A @ ( numeral_numeral @ A @ X2 ) @ N )
= ( semiring_1_of_nat @ A @ Y2 ) )
= ( ( power_power @ nat @ ( numeral_numeral @ nat @ X2 ) @ N )
= Y2 ) ) ) ).
% numeral_power_eq_of_nat_cancel_iff
thf(fact_259_real__of__nat__eq__numeral__power__cancel__iff,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [Y2: nat,X2: num,N: nat] :
( ( ( semiring_1_of_nat @ A @ Y2 )
= ( power_power @ A @ ( numeral_numeral @ A @ X2 ) @ N ) )
= ( Y2
= ( power_power @ nat @ ( numeral_numeral @ nat @ X2 ) @ N ) ) ) ) ).
% real_of_nat_eq_numeral_power_cancel_iff
thf(fact_260_of__nat__le__of__nat__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [B2: nat,W: nat,X2: nat] :
( ( ord_less_eq @ A @ ( power_power @ A @ ( semiring_1_of_nat @ A @ B2 ) @ W ) @ ( semiring_1_of_nat @ A @ X2 ) )
= ( ord_less_eq @ nat @ ( power_power @ nat @ B2 @ W ) @ X2 ) ) ) ).
% of_nat_le_of_nat_power_cancel_iff
thf(fact_261_of__nat__power__le__of__nat__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X2: nat,B2: nat,W: nat] :
( ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ X2 ) @ ( power_power @ A @ ( semiring_1_of_nat @ A @ B2 ) @ W ) )
= ( ord_less_eq @ nat @ X2 @ ( power_power @ nat @ B2 @ W ) ) ) ) ).
% of_nat_power_le_of_nat_cancel_iff
thf(fact_262_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_263_numeral__power__le__of__nat__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [I: num,N: nat,X2: nat] :
( ( ord_less_eq @ A @ ( power_power @ A @ ( numeral_numeral @ A @ I ) @ N ) @ ( semiring_1_of_nat @ A @ X2 ) )
= ( ord_less_eq @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ I ) @ N ) @ X2 ) ) ) ).
% numeral_power_le_of_nat_cancel_iff
thf(fact_264_of__nat__le__numeral__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X2: nat,I: num,N: nat] :
( ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ X2 ) @ ( power_power @ A @ ( numeral_numeral @ A @ I ) @ N ) )
= ( ord_less_eq @ nat @ X2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ I ) @ N ) ) ) ) ).
% of_nat_le_numeral_power_cancel_iff
thf(fact_265_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_266_mult__of__nat__commute,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [X2: nat,Y2: A] :
( ( times_times @ A @ ( semiring_1_of_nat @ A @ X2 ) @ Y2 )
= ( times_times @ A @ Y2 @ ( semiring_1_of_nat @ A @ X2 ) ) ) ) ).
% mult_of_nat_commute
thf(fact_267_real__of__nat__div4,axiom,
! [N: nat,X2: nat] : ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ ( divide_divide @ nat @ N @ X2 ) ) @ ( divide_divide @ real @ ( semiring_1_of_nat @ real @ N ) @ ( semiring_1_of_nat @ real @ X2 ) ) ) ).
% real_of_nat_div4
thf(fact_268_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_269_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_270_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_271_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_272_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_273_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_274_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_275_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_276_ab__semigroup__add__class_Oadd_Ocommute,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A )
=> ( ( plus_plus @ A )
= ( ^ [A4: A,B3: A] : ( plus_plus @ A @ B3 @ A4 ) ) ) ) ).
% ab_semigroup_add_class.add.commute
thf(fact_277_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_278_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_279_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_280_group__cancel_Oadd2,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [B5: A,K: A,B2: A,A2: A] :
( ( B5
= ( plus_plus @ A @ K @ B2 ) )
=> ( ( plus_plus @ A @ A2 @ B5 )
= ( plus_plus @ A @ K @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ) ).
% group_cancel.add2
thf(fact_281_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_282_add__mono__thms__linordered__semiring_I4_J,axiom,
! [A: $tType] :
( ( ordere6658533253407199908up_add @ A )
=> ! [I: A,J: A,K: A,L2: A] :
( ( ( I = J )
& ( K = L2 ) )
=> ( ( plus_plus @ A @ I @ K )
= ( plus_plus @ A @ J @ L2 ) ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_283_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_284_ab__semigroup__mult__class_Omult_Ocommute,axiom,
! [A: $tType] :
( ( ab_semigroup_mult @ A )
=> ( ( times_times @ A )
= ( ^ [A4: A,B3: A] : ( times_times @ A @ B3 @ A4 ) ) ) ) ).
% ab_semigroup_mult_class.mult.commute
thf(fact_285_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_286_complete__real,axiom,
! [S: set @ real] :
( ? [X4: real] : ( member @ real @ X4 @ S )
=> ( ? [Z4: real] :
! [X3: real] :
( ( member @ real @ X3 @ S )
=> ( ord_less_eq @ real @ X3 @ Z4 ) )
=> ? [Y3: real] :
( ! [X4: real] :
( ( member @ real @ X4 @ S )
=> ( ord_less_eq @ real @ X4 @ Y3 ) )
& ! [Z4: real] :
( ! [X3: real] :
( ( member @ real @ X3 @ S )
=> ( ord_less_eq @ real @ X3 @ Z4 ) )
=> ( ord_less_eq @ real @ Y3 @ Z4 ) ) ) ) ) ).
% complete_real
thf(fact_287_add__mono__thms__linordered__semiring_I3_J,axiom,
! [A: $tType] :
( ( ordere6658533253407199908up_add @ A )
=> ! [I: A,J: A,K: A,L2: A] :
( ( ( ord_less_eq @ A @ I @ J )
& ( K = L2 ) )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L2 ) ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_288_add__mono__thms__linordered__semiring_I2_J,axiom,
! [A: $tType] :
( ( ordere6658533253407199908up_add @ A )
=> ! [I: A,J: A,K: A,L2: A] :
( ( ( I = J )
& ( ord_less_eq @ A @ K @ L2 ) )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L2 ) ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_289_add__mono__thms__linordered__semiring_I1_J,axiom,
! [A: $tType] :
( ( ordere6658533253407199908up_add @ A )
=> ! [I: A,J: A,K: A,L2: A] :
( ( ( ord_less_eq @ A @ I @ J )
& ( ord_less_eq @ A @ K @ L2 ) )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L2 ) ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_290_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_291_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_292_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_293_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_294_le__iff__add,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A4: A,B3: A] :
? [C3: A] :
( B3
= ( plus_plus @ A @ A4 @ C3 ) ) ) ) ) ).
% le_iff_add
thf(fact_295_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_296_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_297_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_298_times__divide__times__eq,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [X2: A,Y2: A,Z: A,W: A] :
( ( times_times @ A @ ( divide_divide @ A @ X2 @ Y2 ) @ ( divide_divide @ A @ Z @ W ) )
= ( divide_divide @ A @ ( times_times @ A @ X2 @ Z ) @ ( times_times @ A @ Y2 @ W ) ) ) ) ).
% times_divide_times_eq
thf(fact_299_divide__divide__times__eq,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [X2: A,Y2: A,Z: A,W: A] :
( ( divide_divide @ A @ ( divide_divide @ A @ X2 @ Y2 ) @ ( divide_divide @ A @ Z @ W ) )
= ( divide_divide @ A @ ( times_times @ A @ X2 @ W ) @ ( times_times @ A @ Y2 @ Z ) ) ) ) ).
% divide_divide_times_eq
thf(fact_300_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_301_space__cnt,axiom,
! [T2: vEBT_VEBT] : ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ ( vEBT_VEBT_space2 @ T2 ) ) @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ ( bit1 @ one2 ) ) ) @ ( vEBT_VEBT_cnt @ T2 ) ) ) ).
% space_cnt
thf(fact_302_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_303_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_304_VEBT__internal_OTb_H_Oelims,axiom,
! [X2: nat,Y2: nat] :
( ( ( vEBT_VEBT_Tb2 @ X2 )
= Y2 )
=> ( ( ( X2
= ( zero_zero @ nat ) )
=> ( Y2
!= ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) ) )
=> ( ( ( X2
= ( suc @ ( zero_zero @ nat ) ) )
=> ( Y2
!= ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) ) )
=> ~ ! [N2: nat] :
( ( X2
= ( suc @ ( suc @ N2 ) ) )
=> ~ ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
=> ( Y2
= ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ one2 ) ) ) @ ( vEBT_VEBT_Tb2 @ ( suc @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( times_times @ nat @ ( vEBT_VEBT_Tb2 @ ( suc @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) )
& ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
=> ( Y2
= ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ one2 ) ) ) @ ( vEBT_VEBT_Tb2 @ ( suc @ ( suc @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) @ ( times_times @ nat @ ( vEBT_VEBT_Tb2 @ ( suc @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.Tb'.elims
thf(fact_305_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_306_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_307_VEBT__internal_OT__vebt__buildupi_H_Osimps_I3_J,axiom,
! [N: nat] :
( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( vEBT_V9176841429113362141ildupi @ ( suc @ ( suc @ N ) ) )
= ( plus_plus @ int @ ( numeral_numeral @ int @ ( bit1 @ one2 ) ) @ ( plus_plus @ int @ ( vEBT_V9176841429113362141ildupi @ ( suc @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( plus_plus @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ ( bit0 @ one2 ) ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( times_times @ int @ ( vEBT_V9176841429113362141ildupi @ ( suc @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) )
& ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( vEBT_V9176841429113362141ildupi @ ( suc @ ( suc @ N ) ) )
= ( plus_plus @ int @ ( numeral_numeral @ int @ ( bit1 @ one2 ) ) @ ( plus_plus @ int @ ( vEBT_V9176841429113362141ildupi @ ( suc @ ( suc @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( plus_plus @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ ( bit0 @ one2 ) ) ) @ ( times_times @ int @ ( vEBT_V9176841429113362141ildupi @ ( suc @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.T_vebt_buildupi'.simps(3)
thf(fact_308_high__def,axiom,
( vEBT_VEBT_high
= ( ^ [X: nat,N3: nat] : ( divide_divide @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) ) ) ) ).
% high_def
thf(fact_309_VEBT__internal_OTb_Osimps_I3_J,axiom,
! [N: nat] :
( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( vEBT_VEBT_Tb @ ( suc @ ( suc @ N ) ) )
= ( plus_plus @ int @ ( plus_plus @ int @ ( numeral_numeral @ int @ ( bit1 @ ( bit0 @ one2 ) ) ) @ ( vEBT_VEBT_Tb @ ( suc @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( times_times @ int @ ( vEBT_VEBT_Tb @ ( suc @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( suc @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) )
& ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( vEBT_VEBT_Tb @ ( suc @ ( suc @ N ) ) )
= ( plus_plus @ int @ ( plus_plus @ int @ ( numeral_numeral @ int @ ( bit1 @ ( bit0 @ one2 ) ) ) @ ( vEBT_VEBT_Tb @ ( suc @ ( suc @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) @ ( times_times @ int @ ( vEBT_VEBT_Tb @ ( suc @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.Tb.simps(3)
thf(fact_310_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_311_bezout__lemma__nat,axiom,
! [D2: nat,A2: nat,B2: nat,X2: nat,Y2: nat] :
( ( dvd_dvd @ nat @ D2 @ A2 )
=> ( ( dvd_dvd @ nat @ D2 @ B2 )
=> ( ( ( ( times_times @ nat @ A2 @ X2 )
= ( plus_plus @ nat @ ( times_times @ nat @ B2 @ Y2 ) @ D2 ) )
| ( ( times_times @ nat @ B2 @ X2 )
= ( plus_plus @ nat @ ( times_times @ nat @ A2 @ Y2 ) @ 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_312_Tb__Tb_H,axiom,
( vEBT_VEBT_Tb
= ( ^ [T: nat] : ( semiring_1_of_nat @ int @ ( vEBT_VEBT_Tb2 @ T ) ) ) ) ).
% Tb_Tb'
thf(fact_313_Tbuildupi__buildupi_H,axiom,
! [N: nat] :
( ( semiring_1_of_nat @ int @ ( vEBT_V441764108873111860ildupi @ N ) )
= ( vEBT_V9176841429113362141ildupi @ N ) ) ).
% Tbuildupi_buildupi'
thf(fact_314_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_315_add_Oright__neutral,axiom,
! [A: $tType] :
( ( monoid_add @ A )
=> ! [A2: A] :
( ( plus_plus @ A @ A2 @ ( zero_zero @ A ) )
= A2 ) ) ).
% add.right_neutral
thf(fact_316_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_317_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_318_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_319_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_320_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_321_add__eq__0__iff__both__eq__0,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [X2: A,Y2: A] :
( ( ( plus_plus @ A @ X2 @ Y2 )
= ( zero_zero @ A ) )
= ( ( X2
= ( zero_zero @ A ) )
& ( Y2
= ( zero_zero @ A ) ) ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_322_zero__eq__add__iff__both__eq__0,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [X2: A,Y2: A] :
( ( ( zero_zero @ A )
= ( plus_plus @ A @ X2 @ Y2 ) )
= ( ( X2
= ( zero_zero @ A ) )
& ( Y2
= ( zero_zero @ A ) ) ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_323_add__0,axiom,
! [A: $tType] :
( ( monoid_add @ A )
=> ! [A2: A] :
( ( plus_plus @ A @ ( zero_zero @ A ) @ A2 )
= A2 ) ) ).
% add_0
thf(fact_324_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_325_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_326_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_327_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_328_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_329_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_330_div__0,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [A2: A] :
( ( divide_divide @ A @ ( zero_zero @ A ) @ A2 )
= ( zero_zero @ A ) ) ) ).
% div_0
thf(fact_331_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_332_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_333_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_334_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_335_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_336_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_337_mult__1,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A2: A] :
( ( times_times @ A @ ( one_one @ A ) @ A2 )
= A2 ) ) ).
% mult_1
thf(fact_338_mult_Oright__neutral,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A2: A] :
( ( times_times @ A @ A2 @ ( one_one @ A ) )
= A2 ) ) ).
% mult.right_neutral
thf(fact_339_div__by__1,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [A2: A] :
( ( divide_divide @ A @ A2 @ ( one_one @ A ) )
= A2 ) ) ).
% div_by_1
thf(fact_340_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_341_power__one,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [N: nat] :
( ( power_power @ A @ ( one_one @ A ) @ N )
= ( one_one @ A ) ) ) ).
% power_one
thf(fact_342_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_343_dvd__0__right,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A2: A] : ( dvd_dvd @ A @ A2 @ ( zero_zero @ A ) ) ) ).
% dvd_0_right
thf(fact_344_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_345_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_346_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_347_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_348_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus @ nat @ M @ ( zero_zero @ nat ) )
= M ) ).
% Nat.add_0_right
thf(fact_349_le0,axiom,
! [N: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N ) ).
% le0
thf(fact_350_bot__nat__0_Oextremum,axiom,
! [A2: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ A2 ) ).
% bot_nat_0.extremum
thf(fact_351_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_352_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_353_mult__0__right,axiom,
! [M: nat] :
( ( times_times @ nat @ M @ ( zero_zero @ nat ) )
= ( zero_zero @ nat ) ) ).
% mult_0_right
thf(fact_354_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_355_triangle__0,axiom,
( ( nat_triangle @ ( zero_zero @ nat ) )
= ( zero_zero @ nat ) ) ).
% triangle_0
thf(fact_356_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_357_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_358_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_359_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_360_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_361_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_362_sum__squares__eq__zero__iff,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [X2: A,Y2: A] :
( ( ( plus_plus @ A @ ( times_times @ A @ X2 @ X2 ) @ ( times_times @ A @ Y2 @ Y2 ) )
= ( zero_zero @ A ) )
= ( ( X2
= ( zero_zero @ A ) )
& ( Y2
= ( zero_zero @ A ) ) ) ) ) ).
% sum_squares_eq_zero_iff
thf(fact_363_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_364_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_365_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_366_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_367_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_368_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_369_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_370_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_371_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_372_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_373_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_374_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_375_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_376_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_377_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_378_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_379_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_380_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_381_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_382_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_383_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_384_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_385_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_386_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_387_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_388_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_389_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_390_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_391_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_392_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_393_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_394_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_395_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_396_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_397_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_398_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_399_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_400_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_401_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_402_div__by__Suc__0,axiom,
! [M: nat] :
( ( divide_divide @ nat @ M @ ( suc @ ( zero_zero @ nat ) ) )
= M ) ).
% div_by_Suc_0
thf(fact_403_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_404_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_405_nat__power__eq__Suc__0__iff,axiom,
! [X2: nat,M: nat] :
( ( ( power_power @ nat @ X2 @ M )
= ( suc @ ( zero_zero @ nat ) ) )
= ( ( M
= ( zero_zero @ nat ) )
| ( X2
= ( suc @ ( zero_zero @ nat ) ) ) ) ) ).
% nat_power_eq_Suc_0_iff
thf(fact_406_power__Suc__0,axiom,
! [N: nat] :
( ( power_power @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% power_Suc_0
thf(fact_407_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_408_dvd__1__left,axiom,
! [K: nat] : ( dvd_dvd @ nat @ ( suc @ ( zero_zero @ nat ) ) @ K ) ).
% dvd_1_left
thf(fact_409_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_410_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_411_triangle__Suc,axiom,
! [N: nat] :
( ( nat_triangle @ ( suc @ N ) )
= ( plus_plus @ nat @ ( nat_triangle @ N ) @ ( suc @ N ) ) ) ).
% triangle_Suc
thf(fact_412_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_413_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_414_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_415_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_416_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_417_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_418_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_419_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_420_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_421_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_422_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_423_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_424_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_425_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_426_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_427_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_428_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_429_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_430_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_431_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_432_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_433_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_434_power2__eq__iff__nonneg,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X2: A,Y2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y2 )
=> ( ( ( power_power @ A @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( power_power @ A @ Y2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( X2 = Y2 ) ) ) ) ) ).
% power2_eq_iff_nonneg
thf(fact_435_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_436_sum__power2__eq__zero__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X2: A,Y2: A] :
( ( ( plus_plus @ A @ ( power_power @ A @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( zero_zero @ A ) )
= ( ( X2
= ( zero_zero @ A ) )
& ( Y2
= ( zero_zero @ A ) ) ) ) ) ).
% sum_power2_eq_zero_iff
thf(fact_437_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_438_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_439_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_440_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_441_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_442_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_443_zero__reorient,axiom,
! [A: $tType] :
( ( zero @ A )
=> ! [X2: A] :
( ( ( zero_zero @ A )
= X2 )
= ( X2
= ( zero_zero @ A ) ) ) ) ).
% zero_reorient
thf(fact_444_one__reorient,axiom,
! [A: $tType] :
( ( one @ A )
=> ! [X2: A] :
( ( ( one_one @ A )
= X2 )
= ( X2
= ( one_one @ A ) ) ) ) ).
% one_reorient
thf(fact_445_zero__neq__one,axiom,
! [A: $tType] :
( ( zero_neq_one @ A )
=> ( ( zero_zero @ A )
!= ( one_one @ A ) ) ) ).
% zero_neq_one
thf(fact_446_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_447_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_448_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_449_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_450_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_451_power__0,axiom,
! [A: $tType] :
( ( power @ A )
=> ! [A2: A] :
( ( power_power @ A @ A2 @ ( zero_zero @ nat ) )
= ( one_one @ A ) ) ) ).
% power_0
thf(fact_452_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_453_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_454_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_455_Euclid__induct,axiom,
! [P: nat > nat > $o,A2: nat,B2: nat] :
( ! [A5: nat,B4: nat] :
( ( P @ A5 @ B4 )
= ( P @ B4 @ A5 ) )
=> ( ! [A5: nat] : ( P @ A5 @ ( zero_zero @ nat ) )
=> ( ! [A5: nat,B4: nat] :
( ( P @ A5 @ B4 )
=> ( P @ A5 @ ( plus_plus @ nat @ A5 @ B4 ) ) )
=> ( P @ A2 @ B2 ) ) ) ) ).
% Euclid_induct
thf(fact_456_gcd__nat_Oextremum__uniqueI,axiom,
! [A2: nat] :
( ( dvd_dvd @ nat @ ( zero_zero @ nat ) @ A2 )
=> ( A2
= ( zero_zero @ nat ) ) ) ).
% gcd_nat.extremum_uniqueI
thf(fact_457_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_458_gcd__nat_Oextremum__unique,axiom,
! [A2: nat] :
( ( dvd_dvd @ nat @ ( zero_zero @ nat ) @ A2 )
= ( A2
= ( zero_zero @ nat ) ) ) ).
% gcd_nat.extremum_unique
thf(fact_459_gcd__nat_Oextremum__strict,axiom,
! [A2: nat] :
~ ( ( dvd_dvd @ nat @ ( zero_zero @ nat ) @ A2 )
& ( ( zero_zero @ nat )
!= A2 ) ) ).
% gcd_nat.extremum_strict
thf(fact_460_gcd__nat_Oextremum,axiom,
! [A2: nat] : ( dvd_dvd @ nat @ A2 @ ( zero_zero @ nat ) ) ).
% gcd_nat.extremum
thf(fact_461_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_462_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_463_mult__right__le__one__le,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X2: A,Y2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y2 )
=> ( ( ord_less_eq @ A @ Y2 @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ X2 @ Y2 ) @ X2 ) ) ) ) ) ).
% mult_right_le_one_le
thf(fact_464_mult__left__le__one__le,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X2: A,Y2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y2 )
=> ( ( ord_less_eq @ A @ Y2 @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ Y2 @ X2 ) @ X2 ) ) ) ) ) ).
% mult_left_le_one_le
thf(fact_465_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_466_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_467_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_468_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_469_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_470_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_471_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_472_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_473_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_474_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_475_zero__le,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [X2: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ X2 ) ) ).
% zero_le
thf(fact_476_zero__neq__numeral,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [N: num] :
( ( zero_zero @ A )
!= ( numeral_numeral @ A @ N ) ) ) ).
% zero_neq_numeral
thf(fact_477_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_478_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_479_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_480_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_481_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_482_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_483_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_484_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_485_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_486_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_487_one__dvd,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A2: A] : ( dvd_dvd @ A @ ( one_one @ A ) @ A2 ) ) ).
% one_dvd
thf(fact_488_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_489_not0__implies__Suc,axiom,
! [N: nat] :
( ( N
!= ( zero_zero @ nat ) )
=> ? [M3: nat] :
( N
= ( suc @ M3 ) ) ) ).
% not0_implies_Suc
thf(fact_490_Zero__not__Suc,axiom,
! [M: nat] :
( ( zero_zero @ nat )
!= ( suc @ M ) ) ).
% Zero_not_Suc
thf(fact_491_Zero__neq__Suc,axiom,
! [M: nat] :
( ( zero_zero @ nat )
!= ( suc @ M ) ) ).
% Zero_neq_Suc
thf(fact_492_Suc__neq__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= ( zero_zero @ nat ) ) ).
% Suc_neq_Zero
thf(fact_493_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_494_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_495_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_496_list__decode_Ocases,axiom,
! [X2: nat] :
( ( X2
!= ( zero_zero @ nat ) )
=> ~ ! [N2: nat] :
( X2
!= ( suc @ N2 ) ) ) ).
% list_decode.cases
thf(fact_497_old_Onat_Oexhaust,axiom,
! [Y2: nat] :
( ( Y2
!= ( zero_zero @ nat ) )
=> ~ ! [Nat3: nat] :
( Y2
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_498_nat_OdiscI,axiom,
! [Nat: nat,X23: nat] :
( ( Nat
= ( suc @ X23 ) )
=> ( Nat
!= ( zero_zero @ nat ) ) ) ).
% nat.discI
thf(fact_499_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( ( zero_zero @ nat )
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_500_old_Onat_Odistinct_I2_J,axiom,
! [Nat4: nat] :
( ( suc @ Nat4 )
!= ( zero_zero @ nat ) ) ).
% old.nat.distinct(2)
thf(fact_501_nat_Odistinct_I1_J,axiom,
! [X23: nat] :
( ( zero_zero @ nat )
!= ( suc @ X23 ) ) ).
% nat.distinct(1)
thf(fact_502_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_503_dvd__field__iff,axiom,
! [A: $tType] :
( ( field @ A )
=> ( ( dvd_dvd @ A )
= ( ^ [A4: A,B3: A] :
( ( A4
= ( zero_zero @ A ) )
=> ( B3
= ( zero_zero @ A ) ) ) ) ) ) ).
% dvd_field_iff
thf(fact_504_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus @ nat @ ( zero_zero @ nat ) @ N )
= N ) ).
% plus_nat.add_0
thf(fact_505_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_506_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq @ nat @ N @ ( zero_zero @ nat ) )
= ( N
= ( zero_zero @ nat ) ) ) ).
% le_0_eq
thf(fact_507_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_508_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_509_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N ) ).
% less_eq_nat.simps(1)
thf(fact_510_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_511_mult__0,axiom,
! [N: nat] :
( ( times_times @ nat @ ( zero_zero @ nat ) @ N )
= ( zero_zero @ nat ) ) ).
% mult_0
thf(fact_512_convex__bound__le,axiom,
! [A: $tType] :
( ( linord6961819062388156250ring_1 @ A )
=> ! [X2: A,A2: A,Y2: A,U: A,V: A] :
( ( ord_less_eq @ A @ X2 @ A2 )
=> ( ( ord_less_eq @ A @ Y2 @ 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 @ X2 ) @ ( times_times @ A @ V @ Y2 ) ) @ A2 ) ) ) ) ) ) ) ).
% convex_bound_le
thf(fact_513_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_514_power__decreasing,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N: nat,N5: nat,A2: A] :
( ( ord_less_eq @ nat @ N @ N5 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ A2 @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( power_power @ A @ A2 @ N5 ) @ ( power_power @ A @ A2 @ N ) ) ) ) ) ) ).
% power_decreasing
thf(fact_515_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_516_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_517_is__unitE,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A2: A,C2: A] :
( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) )
=> ~ ( ( A2
!= ( zero_zero @ A ) )
=> ! [B4: A] :
( ( B4
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ B4 @ ( one_one @ A ) )
=> ( ( ( divide_divide @ A @ ( one_one @ A ) @ A2 )
= B4 )
=> ( ( ( divide_divide @ A @ ( one_one @ A ) @ B4 )
= A2 )
=> ( ( ( times_times @ A @ A2 @ B4 )
= ( one_one @ A ) )
=> ( ( divide_divide @ A @ C2 @ A2 )
!= ( times_times @ A @ C2 @ B4 ) ) ) ) ) ) ) ) ) ) ).
% is_unitE
thf(fact_518_dvd__power__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [X2: A,M: nat,N: nat] :
( ( X2
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ ( power_power @ A @ X2 @ M ) @ ( power_power @ A @ X2 @ N ) )
= ( ( dvd_dvd @ A @ X2 @ ( one_one @ A ) )
| ( ord_less_eq @ nat @ M @ N ) ) ) ) ) ).
% dvd_power_iff
thf(fact_519_lambda__one,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ( ( ^ [X: A] : X )
= ( times_times @ A @ ( one_one @ A ) ) ) ) ).
% lambda_one
thf(fact_520_lambda__zero,axiom,
! [A: $tType] :
( ( mult_zero @ A )
=> ( ( ^ [H: A] : ( zero_zero @ A ) )
= ( times_times @ A @ ( zero_zero @ A ) ) ) ) ).
% lambda_zero
thf(fact_521_VEBT__internal_OTb_Osimps_I1_J,axiom,
( ( vEBT_VEBT_Tb @ ( zero_zero @ nat ) )
= ( numeral_numeral @ int @ ( bit1 @ one2 ) ) ) ).
% VEBT_internal.Tb.simps(1)
thf(fact_522_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_523_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_524_one__plus__numeral__commute,axiom,
! [A: $tType] :
( ( numeral @ A )
=> ! [X2: num] :
( ( plus_plus @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ X2 ) )
= ( plus_plus @ A @ ( numeral_numeral @ A @ X2 ) @ ( one_one @ A ) ) ) ) ).
% one_plus_numeral_commute
thf(fact_525_numeral__One,axiom,
! [A: $tType] :
( ( numeral @ A )
=> ( ( numeral_numeral @ A @ one2 )
= ( one_one @ A ) ) ) ).
% numeral_One
thf(fact_526_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_527_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_528_left__right__inverse__power,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [X2: A,Y2: A,N: nat] :
( ( ( times_times @ A @ X2 @ Y2 )
= ( one_one @ A ) )
=> ( ( times_times @ A @ ( power_power @ A @ X2 @ N ) @ ( power_power @ A @ Y2 @ N ) )
= ( one_one @ A ) ) ) ) ).
% left_right_inverse_power
thf(fact_529_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_530_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_531_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_532_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_533_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_534_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_535_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_536_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_537_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_538_add__nonneg__eq__0__iff,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [X2: A,Y2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y2 )
=> ( ( ( plus_plus @ A @ X2 @ Y2 )
= ( zero_zero @ A ) )
= ( ( X2
= ( zero_zero @ A ) )
& ( Y2
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_539_add__nonpos__eq__0__iff,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [X2: A,Y2: A] :
( ( ord_less_eq @ A @ X2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ Y2 @ ( zero_zero @ A ) )
=> ( ( ( plus_plus @ A @ X2 @ Y2 )
= ( zero_zero @ A ) )
= ( ( X2
= ( zero_zero @ A ) )
& ( Y2
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_540_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_541_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_542_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_543_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_544_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_545_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_546_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_547_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_548_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_549_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_550_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_551_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_552_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_553_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_554_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_555_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_556_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_557_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_558_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_559_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_560_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_561_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_562_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_563_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_564_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_565_divide__nonpos__nonpos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X2: A,Y2: A] :
( ( ord_less_eq @ A @ X2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ Y2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X2 @ Y2 ) ) ) ) ) ).
% divide_nonpos_nonpos
thf(fact_566_divide__nonpos__nonneg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X2: A,Y2: A] :
( ( ord_less_eq @ A @ X2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y2 )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ X2 @ Y2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% divide_nonpos_nonneg
thf(fact_567_divide__nonneg__nonpos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X2: A,Y2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X2 )
=> ( ( ord_less_eq @ A @ Y2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ X2 @ Y2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% divide_nonneg_nonpos
thf(fact_568_divide__nonneg__nonneg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X2: A,Y2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y2 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X2 @ Y2 ) ) ) ) ) ).
% divide_nonneg_nonneg
thf(fact_569_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_570_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_571_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_572_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_573_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_574_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_575_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_576_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_577_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_578_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_579_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_580_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_581_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_582_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_583_frac__eq__eq,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [Y2: A,Z: A,X2: A,W: A] :
( ( Y2
!= ( zero_zero @ A ) )
=> ( ( Z
!= ( zero_zero @ A ) )
=> ( ( ( divide_divide @ A @ X2 @ Y2 )
= ( divide_divide @ A @ W @ Z ) )
= ( ( times_times @ A @ X2 @ Z )
= ( times_times @ A @ W @ Y2 ) ) ) ) ) ) ).
% frac_eq_eq
thf(fact_584_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_585_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_586_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_587_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_588_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_589_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_590_VEBT__internal_OTb_Osimps_I2_J,axiom,
( ( vEBT_VEBT_Tb @ ( suc @ ( zero_zero @ nat ) ) )
= ( numeral_numeral @ int @ ( bit1 @ one2 ) ) ) ).
% VEBT_internal.Tb.simps(2)
thf(fact_591_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_592_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_593_power__increasing,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N: nat,N5: nat,A2: A] :
( ( ord_less_eq @ nat @ N @ N5 )
=> ( ( ord_less_eq @ A @ ( one_one @ A ) @ A2 )
=> ( ord_less_eq @ A @ ( power_power @ A @ A2 @ N ) @ ( power_power @ A @ A2 @ N5 ) ) ) ) ) ).
% power_increasing
thf(fact_594_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_595_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_596_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_597_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_598_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_599_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_600_sum__squares__ge__zero,axiom,
! [A: $tType] :
( ( linordered_ring @ A )
=> ! [X2: A,Y2: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ ( times_times @ A @ X2 @ X2 ) @ ( times_times @ A @ Y2 @ Y2 ) ) ) ) ).
% sum_squares_ge_zero
thf(fact_601_sum__squares__le__zero__iff,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [X2: A,Y2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ ( times_times @ A @ X2 @ X2 ) @ ( times_times @ A @ Y2 @ Y2 ) ) @ ( zero_zero @ A ) )
= ( ( X2
= ( zero_zero @ A ) )
& ( Y2
= ( zero_zero @ A ) ) ) ) ) ).
% sum_squares_le_zero_iff
thf(fact_602_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_603_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_604_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_605_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_606_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_607_divide__add__eq__iff,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z: A,X2: A,Y2: A] :
( ( Z
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( divide_divide @ A @ X2 @ Z ) @ Y2 )
= ( divide_divide @ A @ ( plus_plus @ A @ X2 @ ( times_times @ A @ Y2 @ Z ) ) @ Z ) ) ) ) ).
% divide_add_eq_iff
thf(fact_608_add__divide__eq__iff,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z: A,X2: A,Y2: A] :
( ( Z
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ X2 @ ( divide_divide @ A @ Y2 @ Z ) )
= ( divide_divide @ A @ ( plus_plus @ A @ ( times_times @ A @ X2 @ Z ) @ Y2 ) @ Z ) ) ) ) ).
% add_divide_eq_iff
thf(fact_609_add__num__frac,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [Y2: A,Z: A,X2: A] :
( ( Y2
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ Z @ ( divide_divide @ A @ X2 @ Y2 ) )
= ( divide_divide @ A @ ( plus_plus @ A @ X2 @ ( times_times @ A @ Z @ Y2 ) ) @ Y2 ) ) ) ) ).
% add_num_frac
thf(fact_610_add__frac__num,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [Y2: A,X2: A,Z: A] :
( ( Y2
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( divide_divide @ A @ X2 @ Y2 ) @ Z )
= ( divide_divide @ A @ ( plus_plus @ A @ X2 @ ( times_times @ A @ Z @ Y2 ) ) @ Y2 ) ) ) ) ).
% add_frac_num
thf(fact_611_add__frac__eq,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [Y2: A,Z: A,X2: A,W: A] :
( ( Y2
!= ( zero_zero @ A ) )
=> ( ( Z
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( divide_divide @ A @ X2 @ Y2 ) @ ( divide_divide @ A @ W @ Z ) )
= ( divide_divide @ A @ ( plus_plus @ A @ ( times_times @ A @ X2 @ Z ) @ ( times_times @ A @ W @ Y2 ) ) @ ( times_times @ A @ Y2 @ Z ) ) ) ) ) ) ).
% add_frac_eq
thf(fact_612_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_613_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_614_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_615_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_616_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_617_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_618_numeral__1__eq__Suc__0,axiom,
( ( numeral_numeral @ nat @ one2 )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% numeral_1_eq_Suc_0
thf(fact_619_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_620_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_621_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_622_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_623_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_624_gcd__nat_Oorder__iff__strict,axiom,
( ( dvd_dvd @ nat )
= ( ^ [A4: nat,B3: nat] :
( ( ( dvd_dvd @ nat @ A4 @ B3 )
& ( A4 != B3 ) )
| ( A4 = B3 ) ) ) ) ).
% gcd_nat.order_iff_strict
thf(fact_625_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_626_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_627_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_628_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_629_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_630_gcd__nat_Oirrefl,axiom,
! [A2: nat] :
~ ( ( dvd_dvd @ nat @ A2 @ A2 )
& ( A2 != A2 ) ) ).
% gcd_nat.irrefl
thf(fact_631_gcd__nat_Oeq__iff,axiom,
( ( ^ [Y5: nat,Z3: nat] : Y5 = Z3 )
= ( ^ [A4: nat,B3: nat] :
( ( dvd_dvd @ nat @ A4 @ B3 )
& ( dvd_dvd @ nat @ B3 @ A4 ) ) ) ) ).
% gcd_nat.eq_iff
thf(fact_632_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_633_gcd__nat_Orefl,axiom,
! [A2: nat] : ( dvd_dvd @ nat @ A2 @ A2 ) ).
% gcd_nat.refl
thf(fact_634_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_635_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_636_odd__one,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( one_one @ A ) ) ) ).
% odd_one
thf(fact_637_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_638_even__zero,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( zero_zero @ A ) ) ) ).
% even_zero
thf(fact_639_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_640_numeral__2__eq__2,axiom,
( ( numeral_numeral @ nat @ ( bit0 @ one2 ) )
= ( suc @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% numeral_2_eq_2
thf(fact_641_numeral__3__eq__3,axiom,
( ( numeral_numeral @ nat @ ( bit1 @ one2 ) )
= ( suc @ ( suc @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ).
% numeral_3_eq_3
thf(fact_642_VEBT__internal_OTb_H_Osimps_I1_J,axiom,
( ( vEBT_VEBT_Tb2 @ ( zero_zero @ nat ) )
= ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) ) ).
% VEBT_internal.Tb'.simps(1)
thf(fact_643_VEBT__internal_OTb_Oelims,axiom,
! [X2: nat,Y2: int] :
( ( ( vEBT_VEBT_Tb @ X2 )
= Y2 )
=> ( ( ( X2
= ( zero_zero @ nat ) )
=> ( Y2
!= ( numeral_numeral @ int @ ( bit1 @ one2 ) ) ) )
=> ( ( ( X2
= ( suc @ ( zero_zero @ nat ) ) )
=> ( Y2
!= ( numeral_numeral @ int @ ( bit1 @ one2 ) ) ) )
=> ~ ! [N2: nat] :
( ( X2
= ( suc @ ( suc @ N2 ) ) )
=> ~ ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
=> ( Y2
= ( plus_plus @ int @ ( plus_plus @ int @ ( numeral_numeral @ int @ ( bit1 @ ( bit0 @ one2 ) ) ) @ ( vEBT_VEBT_Tb @ ( suc @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( times_times @ int @ ( vEBT_VEBT_Tb @ ( suc @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( suc @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) )
& ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
=> ( Y2
= ( plus_plus @ int @ ( plus_plus @ int @ ( numeral_numeral @ int @ ( bit1 @ ( bit0 @ one2 ) ) ) @ ( vEBT_VEBT_Tb @ ( suc @ ( suc @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) @ ( times_times @ int @ ( vEBT_VEBT_Tb @ ( suc @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.Tb.elims
thf(fact_644_VEBT__internal_OT__vebt__buildupi_Oelims,axiom,
! [X2: nat,Y2: nat] :
( ( ( vEBT_V441764108873111860ildupi @ X2 )
= Y2 )
=> ( ( ( X2
= ( zero_zero @ nat ) )
=> ( Y2
!= ( suc @ ( zero_zero @ nat ) ) ) )
=> ( ( ( X2
= ( suc @ ( zero_zero @ nat ) ) )
=> ( Y2
!= ( suc @ ( zero_zero @ nat ) ) ) )
=> ~ ! [N2: nat] :
( ( X2
= ( suc @ ( suc @ N2 ) ) )
=> ~ ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
=> ( Y2
= ( 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 )
=> ( Y2
= ( 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_645_power2__le__imp__le,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X2: A,Y2: A] :
( ( ord_less_eq @ A @ ( power_power @ A @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y2 )
=> ( ord_less_eq @ A @ X2 @ Y2 ) ) ) ) ).
% power2_le_imp_le
thf(fact_646_power2__eq__imp__eq,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X2: A,Y2: A] :
( ( ( power_power @ A @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( power_power @ A @ Y2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y2 )
=> ( X2 = Y2 ) ) ) ) ) ).
% power2_eq_imp_eq
thf(fact_647_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_648_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_649_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_650_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_651_VEBT__internal_OTb_H_Osimps_I2_J,axiom,
( ( vEBT_VEBT_Tb2 @ ( suc @ ( zero_zero @ nat ) ) )
= ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) ) ).
% VEBT_internal.Tb'.simps(2)
thf(fact_652_oddE,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A2: A] :
( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
=> ~ ! [B4: A] :
( A2
!= ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B4 ) @ ( one_one @ A ) ) ) ) ) ).
% oddE
thf(fact_653_sum__power2__le__zero__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X2: A,Y2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ ( power_power @ A @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( zero_zero @ A ) )
= ( ( X2
= ( zero_zero @ A ) )
& ( Y2
= ( zero_zero @ A ) ) ) ) ) ).
% sum_power2_le_zero_iff
thf(fact_654_sum__power2__ge__zero,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X2: A,Y2: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ ( power_power @ A @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% sum_power2_ge_zero
thf(fact_655_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_656_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_657_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_658_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_659_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,C5: A] :
( ( A2
= ( times_times @ A @ B6 @ C5 ) )
& ( dvd_dvd @ A @ B6 @ B2 )
& ( dvd_dvd @ A @ C5 @ C2 ) ) ) ) ).
% division_decomp
thf(fact_660_dvd__productE,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [P2: A,A2: A,B2: A] :
( ( dvd_dvd @ A @ P2 @ ( times_times @ A @ A2 @ B2 ) )
=> ~ ! [X3: A,Y3: A] :
( ( P2
= ( times_times @ A @ X3 @ Y3 ) )
=> ( ( dvd_dvd @ A @ X3 @ A2 )
=> ~ ( dvd_dvd @ A @ Y3 @ B2 ) ) ) ) ) ).
% dvd_productE
thf(fact_661_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_662_arith__geo__mean,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [U: A,X2: A,Y2: A] :
( ( ( power_power @ A @ U @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( times_times @ A @ X2 @ Y2 ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y2 )
=> ( ord_less_eq @ A @ U @ ( divide_divide @ A @ ( plus_plus @ A @ X2 @ Y2 ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% arith_geo_mean
thf(fact_663_Bernoulli__inequality__even,axiom,
! [N: nat,X2: real] :
( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ord_less_eq @ real @ ( plus_plus @ real @ ( one_one @ real ) @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ X2 ) ) @ ( power_power @ real @ ( plus_plus @ real @ ( one_one @ real ) @ X2 ) @ N ) ) ) ).
% Bernoulli_inequality_even
thf(fact_664_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_665_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_666_high__inv,axiom,
! [X2: nat,N: nat,Y2: nat] :
( ( ord_less @ nat @ X2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
=> ( ( vEBT_VEBT_high @ ( plus_plus @ nat @ ( times_times @ nat @ Y2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) @ X2 ) @ N )
= Y2 ) ) ).
% high_inv
thf(fact_667_replicate__eq__replicate,axiom,
! [A: $tType,M: nat,X2: A,N: nat,Y2: A] :
( ( ( replicate @ A @ M @ X2 )
= ( replicate @ A @ N @ Y2 ) )
= ( ( M = N )
& ( ( M
!= ( zero_zero @ nat ) )
=> ( X2 = Y2 ) ) ) ) ).
% replicate_eq_replicate
thf(fact_668_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_669_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_670_Tb__T__vebt__buildupi_H_H,axiom,
! [N: nat] : ( ord_less_eq @ nat @ ( vEBT_V441764108873111860ildupi @ N ) @ ( minus_minus @ nat @ ( vEBT_VEBT_Tb2 @ N ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% Tb_T_vebt_buildupi''
thf(fact_671_space_H__bound,axiom,
! [T2: vEBT_VEBT,N: nat,U: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( U
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
=> ( ord_less_eq @ nat @ ( vEBT_VEBT_space2 @ T2 ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) @ U ) ) ) ) ).
% space'_bound
thf(fact_672_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_673_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_674_valid__tree__deg__neq__0,axiom,
! [T2: vEBT_VEBT] :
~ ( vEBT_invar_vebt @ T2 @ ( zero_zero @ nat ) ) ).
% valid_tree_deg_neq_0
thf(fact_675_valid__0__not,axiom,
! [T2: vEBT_VEBT] :
~ ( vEBT_invar_vebt @ T2 @ ( zero_zero @ nat ) ) ).
% valid_0_not
thf(fact_676_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_677_cnt__non__neg,axiom,
! [T2: vEBT_VEBT] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( vEBT_VEBT_cnt @ T2 ) ) ).
% cnt_non_neg
thf(fact_678_deg__not__0,axiom,
! [T2: vEBT_VEBT,N: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ).
% deg_not_0
thf(fact_679_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_680_T__vebt__buildupi__gq__0,axiom,
! [N: nat] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( vEBT_V441764108873111860ildupi @ N ) ) ).
% T_vebt_buildupi_gq_0
thf(fact_681_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_682_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_683_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_684_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_685_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_686_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_687_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_688_diff__self,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A] :
( ( minus_minus @ A @ A2 @ A2 )
= ( zero_zero @ A ) ) ) ).
% diff_self
thf(fact_689_diff__0__right,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A] :
( ( minus_minus @ A @ A2 @ ( zero_zero @ A ) )
= A2 ) ) ).
% diff_0_right
thf(fact_690_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_691_diff__zero,axiom,
! [A: $tType] :
( ( cancel1802427076303600483id_add @ A )
=> ! [A2: A] :
( ( minus_minus @ A @ A2 @ ( zero_zero @ A ) )
= A2 ) ) ).
% diff_zero
thf(fact_692_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_693_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_694_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_695_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_696_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_697_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_698_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_699_lessI,axiom,
! [N: nat] : ( ord_less @ nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_700_Suc__mono,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ N )
=> ( ord_less @ nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_701_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_702_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_703_neq0__conv,axiom,
! [N: nat] :
( ( N
!= ( zero_zero @ nat ) )
= ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ).
% neq0_conv
thf(fact_704_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% less_nat_zero_code
thf(fact_705_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_706_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_707_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_708_power__one__right,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A2: A] :
( ( power_power @ A @ A2 @ ( one_one @ nat ) )
= A2 ) ) ).
% power_one_right
thf(fact_709_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus @ nat @ ( zero_zero @ nat ) @ N )
= ( zero_zero @ nat ) ) ).
% diff_0_eq_0
thf(fact_710_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus @ nat @ M @ M )
= ( zero_zero @ nat ) ) ).
% diff_self_eq_0
thf(fact_711_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_712_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_713_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_714_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_715_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_716_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_717_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_718_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_719_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_720_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_721_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_722_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_723_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_724_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_725_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_726_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_727_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_728_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_729_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_730_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_731_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_732_zero__less__Suc,axiom,
! [N: nat] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_733_less__Suc0,axiom,
! [N: nat] :
( ( ord_less @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) )
= ( N
= ( zero_zero @ nat ) ) ) ).
% less_Suc0
thf(fact_734_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_735_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_736_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_737_less__one,axiom,
! [N: nat] :
( ( ord_less @ nat @ N @ ( one_one @ nat ) )
= ( N
= ( zero_zero @ nat ) ) ) ).
% less_one
thf(fact_738_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_739_diff__Suc__1,axiom,
! [N: nat] :
( ( minus_minus @ nat @ ( suc @ N ) @ ( one_one @ nat ) )
= N ) ).
% diff_Suc_1
thf(fact_740_div__less,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ N )
=> ( ( divide_divide @ nat @ M @ N )
= ( zero_zero @ nat ) ) ) ).
% div_less
thf(fact_741_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_742_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_743_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_744_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_745_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_746_nat__zero__less__power__iff,axiom,
! [X2: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( power_power @ nat @ X2 @ N ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ X2 )
| ( N
= ( zero_zero @ nat ) ) ) ) ).
% nat_zero_less_power_iff
thf(fact_747_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_748_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_749_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_750_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_751_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_752_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_753_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_754_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_755_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_756_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_757_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_758_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_759_power__strict__increasing__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [B2: A,X2: nat,Y2: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ B2 )
=> ( ( ord_less @ A @ ( power_power @ A @ B2 @ X2 ) @ ( power_power @ A @ B2 @ Y2 ) )
= ( ord_less @ nat @ X2 @ Y2 ) ) ) ) ).
% power_strict_increasing_iff
thf(fact_760_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_761_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_762_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_763_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_764_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_765_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_766_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_767_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_768_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_769_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_770_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_771_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_772_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_773_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_774_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_775_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_776_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_777_power__increasing__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [B2: A,X2: nat,Y2: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ B2 )
=> ( ( ord_less_eq @ A @ ( power_power @ A @ B2 @ X2 ) @ ( power_power @ A @ B2 @ Y2 ) )
= ( ord_less_eq @ nat @ X2 @ Y2 ) ) ) ) ).
% power_increasing_iff
thf(fact_778_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_779_Suc__1,axiom,
( ( suc @ ( one_one @ nat ) )
= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ).
% Suc_1
thf(fact_780_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_781_of__nat__less__of__nat__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [B2: nat,W: nat,X2: nat] :
( ( ord_less @ A @ ( power_power @ A @ ( semiring_1_of_nat @ A @ B2 ) @ W ) @ ( semiring_1_of_nat @ A @ X2 ) )
= ( ord_less @ nat @ ( power_power @ nat @ B2 @ W ) @ X2 ) ) ) ).
% of_nat_less_of_nat_power_cancel_iff
thf(fact_782_of__nat__power__less__of__nat__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X2: nat,B2: nat,W: nat] :
( ( ord_less @ A @ ( semiring_1_of_nat @ A @ X2 ) @ ( power_power @ A @ ( semiring_1_of_nat @ A @ B2 ) @ W ) )
= ( ord_less @ nat @ X2 @ ( power_power @ nat @ B2 @ W ) ) ) ) ).
% of_nat_power_less_of_nat_cancel_iff
thf(fact_783_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_784_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_785_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_786_of__nat__zero__less__power__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X2: nat,N: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( power_power @ A @ ( semiring_1_of_nat @ A @ X2 ) @ N ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ X2 )
| ( N
= ( zero_zero @ nat ) ) ) ) ) ).
% of_nat_zero_less_power_iff
thf(fact_787_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_788_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_789_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_790_numeral__power__less__of__nat__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [I: num,N: nat,X2: nat] :
( ( ord_less @ A @ ( power_power @ A @ ( numeral_numeral @ A @ I ) @ N ) @ ( semiring_1_of_nat @ A @ X2 ) )
= ( ord_less @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ I ) @ N ) @ X2 ) ) ) ).
% numeral_power_less_of_nat_cancel_iff
thf(fact_791_of__nat__less__numeral__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X2: nat,I: num,N: nat] :
( ( ord_less @ A @ ( semiring_1_of_nat @ A @ X2 ) @ ( power_power @ A @ ( numeral_numeral @ A @ I ) @ N ) )
= ( ord_less @ nat @ X2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ I ) @ N ) ) ) ) ).
% of_nat_less_numeral_power_cancel_iff
thf(fact_792_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_793_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_794_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_795_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_796_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_797_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_798_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_799_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_800_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_801_int__int__eq,axiom,
! [M: nat,N: nat] :
( ( ( semiring_1_of_nat @ int @ M )
= ( semiring_1_of_nat @ int @ N ) )
= ( M = N ) ) ).
% int_int_eq
thf(fact_802_odd__nonzero,axiom,
! [Z: int] :
( ( plus_plus @ int @ ( plus_plus @ int @ ( one_one @ int ) @ Z ) @ Z )
!= ( zero_zero @ int ) ) ).
% odd_nonzero
thf(fact_803_linorder__neqE__linordered__idom,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X2: A,Y2: A] :
( ( X2 != Y2 )
=> ( ~ ( ord_less @ A @ X2 @ Y2 )
=> ( ord_less @ A @ Y2 @ X2 ) ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_804_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_805_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_806_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ N ) ).
% less_not_refl
thf(fact_807_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less @ nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_808_less__not__refl3,axiom,
! [S3: nat,T2: nat] :
( ( ord_less @ nat @ S3 @ T2 )
=> ( S3 != T2 ) ) ).
% less_not_refl3
thf(fact_809_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_810_diff__less__mono2,axiom,
! [M: nat,N: nat,L2: nat] :
( ( ord_less @ nat @ M @ N )
=> ( ( ord_less @ nat @ M @ L2 )
=> ( ord_less @ nat @ ( minus_minus @ nat @ L2 @ N ) @ ( minus_minus @ nat @ L2 @ M ) ) ) ) ).
% diff_less_mono2
thf(fact_811_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_812_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M4: nat] :
( ( ord_less @ nat @ M4 @ N2 )
=> ( P @ M4 ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_813_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ~ ( P @ N2 )
=> ? [M4: nat] :
( ( ord_less @ nat @ M4 @ N2 )
& ~ ( P @ M4 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_814_linorder__neqE__nat,axiom,
! [X2: nat,Y2: nat] :
( ( X2 != Y2 )
=> ( ~ ( ord_less @ nat @ X2 @ Y2 )
=> ( ord_less @ nat @ Y2 @ X2 ) ) ) ).
% linorder_neqE_nat
thf(fact_815_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_816_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_817_infinite__descent__measure,axiom,
! [A: $tType,P: A > $o,V2: A > nat,X2: A] :
( ! [X3: A] :
( ~ ( P @ X3 )
=> ? [Y4: A] :
( ( ord_less @ nat @ ( V2 @ Y4 ) @ ( V2 @ X3 ) )
& ~ ( P @ Y4 ) ) )
=> ( P @ X2 ) ) ).
% infinite_descent_measure
thf(fact_818_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_819_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_820_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_821_linordered__field__no__lb,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X4: A] :
? [Y3: A] : ( ord_less @ A @ Y3 @ X4 ) ) ).
% linordered_field_no_lb
thf(fact_822_linordered__field__no__ub,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X4: A] :
? [X_1: A] : ( ord_less @ A @ X4 @ X_1 ) ) ).
% linordered_field_no_ub
thf(fact_823_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_824_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_825_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_826_less__iff__diff__less__0,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ( ( ord_less @ A )
= ( ^ [A4: A,B3: A] : ( ord_less @ A @ ( minus_minus @ A @ A4 @ B3 ) @ ( zero_zero @ A ) ) ) ) ) ).
% less_iff_diff_less_0
thf(fact_827_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_828_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_829_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_830_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_831_lift__Suc__mono__less,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [F2: nat > A,N: nat,N4: nat] :
( ! [N2: nat] : ( ord_less @ A @ ( F2 @ N2 ) @ ( F2 @ ( suc @ N2 ) ) )
=> ( ( ord_less @ nat @ N @ N4 )
=> ( ord_less @ A @ ( F2 @ N ) @ ( F2 @ N4 ) ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_832_diff__less__Suc,axiom,
! [M: nat,N: nat] : ( ord_less @ nat @ ( minus_minus @ nat @ M @ N ) @ ( suc @ M ) ) ).
% diff_less_Suc
thf(fact_833_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_834_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_835_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_836_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_837_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_838_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_839_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_840_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_841_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_842_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_843_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_844_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_845_int__ge__induct,axiom,
! [K: int,I: int,P: int > $o] :
( ( ord_less_eq @ int @ K @ I )
=> ( ( P @ K )
=> ( ! [I2: int] :
( ( ord_less_eq @ int @ K @ I2 )
=> ( ( P @ I2 )
=> ( P @ ( plus_plus @ int @ I2 @ ( one_one @ int ) ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_ge_induct
thf(fact_846_plus__int__code_I2_J,axiom,
! [L2: int] :
( ( plus_plus @ int @ ( zero_zero @ int ) @ L2 )
= L2 ) ).
% plus_int_code(2)
thf(fact_847_plus__int__code_I1_J,axiom,
! [K: int] :
( ( plus_plus @ int @ K @ ( zero_zero @ int ) )
= K ) ).
% plus_int_code(1)
thf(fact_848_times__int__code_I2_J,axiom,
! [L2: int] :
( ( times_times @ int @ ( zero_zero @ int ) @ L2 )
= ( zero_zero @ int ) ) ).
% times_int_code(2)
thf(fact_849_times__int__code_I1_J,axiom,
! [K: int] :
( ( times_times @ int @ K @ ( zero_zero @ int ) )
= ( zero_zero @ int ) ) ).
% times_int_code(1)
thf(fact_850_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_851_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_852_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_853_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_854_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_855_power__strict__increasing,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N: nat,N5: nat,A2: A] :
( ( ord_less @ nat @ N @ N5 )
=> ( ( ord_less @ A @ ( one_one @ A ) @ A2 )
=> ( ord_less @ A @ ( power_power @ A @ A2 @ N ) @ ( power_power @ A @ A2 @ N5 ) ) ) ) ) ).
% power_strict_increasing
thf(fact_856_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_857_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_858_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_859_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_860_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_861_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_862_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_863_eq__iff__diff__eq__0,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ( ( ^ [Y5: A,Z3: A] : Y5 = Z3 )
= ( ^ [A4: A,B3: A] :
( ( minus_minus @ A @ A4 @ B3 )
= ( zero_zero @ A ) ) ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_864_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_865_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_866_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_867_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_868_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_869_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_870_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_871_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_872_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_873_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_874_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_875_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_876_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_877_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_878_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_879_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_880_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_881_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_882_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_883_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_884_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_885_not__less__zero,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [N: A] :
~ ( ord_less @ A @ N @ ( zero_zero @ A ) ) ) ).
% not_less_zero
thf(fact_886_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_887_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_888_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_889_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_890_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_891_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_892_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_893_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_894_add__mono__thms__linordered__field_I1_J,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [I: A,J: A,K: A,L2: A] :
( ( ( ord_less @ A @ I @ J )
& ( K = L2 ) )
=> ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L2 ) ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_895_add__mono__thms__linordered__field_I2_J,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [I: A,J: A,K: A,L2: A] :
( ( ( I = J )
& ( ord_less @ A @ K @ L2 ) )
=> ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L2 ) ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_896_add__mono__thms__linordered__field_I5_J,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [I: A,J: A,K: A,L2: A] :
( ( ( ord_less @ A @ I @ J )
& ( ord_less @ A @ K @ L2 ) )
=> ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L2 ) ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_897_dvd__diff,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [X2: A,Y2: A,Z: A] :
( ( dvd_dvd @ A @ X2 @ Y2 )
=> ( ( dvd_dvd @ A @ X2 @ Z )
=> ( dvd_dvd @ A @ X2 @ ( minus_minus @ A @ Y2 @ Z ) ) ) ) ) ).
% dvd_diff
thf(fact_898_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_899_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_900_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_901_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus @ nat @ M @ ( zero_zero @ nat ) )
= M ) ).
% minus_nat.diff_0
thf(fact_902_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_903_diff__add__inverse2,axiom,
! [M: nat,N: nat] :
( ( minus_minus @ nat @ ( plus_plus @ nat @ M @ N ) @ N )
= M ) ).
% diff_add_inverse2
thf(fact_904_diff__add__inverse,axiom,
! [N: nat,M: nat] :
( ( minus_minus @ nat @ ( plus_plus @ nat @ N @ M ) @ N )
= M ) ).
% diff_add_inverse
thf(fact_905_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_906_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_907_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_908_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_909_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_910_diff__le__mono,axiom,
! [M: nat,N: nat,L2: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ord_less_eq @ nat @ ( minus_minus @ nat @ M @ L2 ) @ ( minus_minus @ nat @ N @ L2 ) ) ) ).
% diff_le_mono
thf(fact_911_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq @ nat @ ( minus_minus @ nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_912_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_913_diff__le__mono2,axiom,
! [M: nat,N: nat,L2: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ord_less_eq @ nat @ ( minus_minus @ nat @ L2 @ N ) @ ( minus_minus @ nat @ L2 @ M ) ) ) ).
% diff_le_mono2
thf(fact_914_Nat_OlessE,axiom,
! [I: nat,K: nat] :
( ( ord_less @ nat @ I @ K )
=> ( ( K
!= ( suc @ I ) )
=> ~ ! [J2: nat] :
( ( ord_less @ nat @ I @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ) ).
% Nat.lessE
thf(fact_915_Suc__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ ( suc @ M ) @ N )
=> ( ord_less @ nat @ M @ N ) ) ).
% Suc_lessD
thf(fact_916_Suc__lessE,axiom,
! [I: nat,K: nat] :
( ( ord_less @ nat @ ( suc @ I ) @ K )
=> ~ ! [J2: nat] :
( ( ord_less @ nat @ I @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ).
% Suc_lessE
thf(fact_917_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_918_less__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less @ nat @ M @ N )
=> ( M = N ) ) ) ).
% less_SucE
thf(fact_919_less__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ N )
=> ( ord_less @ nat @ M @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_920_Ex__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I3: nat] :
( ( ord_less @ nat @ I3 @ ( suc @ N ) )
& ( P @ I3 ) ) )
= ( ( P @ N )
| ? [I3: nat] :
( ( ord_less @ nat @ I3 @ N )
& ( P @ I3 ) ) ) ) ).
% Ex_less_Suc
thf(fact_921_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_922_not__less__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less @ nat @ M @ N ) )
= ( ord_less @ nat @ N @ ( suc @ M ) ) ) ).
% not_less_eq
thf(fact_923_Nat_OAll__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( suc @ N ) )
=> ( P @ I3 ) ) )
= ( ( P @ N )
& ! [I3: nat] :
( ( ord_less @ nat @ I3 @ N )
=> ( P @ I3 ) ) ) ) ).
% Nat.All_less_Suc
thf(fact_924_Suc__less__eq2,axiom,
! [N: nat,M: nat] :
( ( ord_less @ nat @ ( suc @ N ) @ M )
= ( ? [M6: nat] :
( ( M
= ( suc @ M6 ) )
& ( ord_less @ nat @ N @ M6 ) ) ) ) ).
% Suc_less_eq2
thf(fact_925_less__antisym,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less @ nat @ N @ M )
=> ( ( ord_less @ nat @ N @ ( suc @ M ) )
=> ( M = N ) ) ) ).
% less_antisym
thf(fact_926_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_927_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_928_less__Suc__induct,axiom,
! [I: nat,J: nat,P: nat > nat > $o] :
( ( ord_less @ nat @ I @ J )
=> ( ! [I2: nat] : ( P @ I2 @ ( suc @ I2 ) )
=> ( ! [I2: nat,J2: nat,K3: nat] :
( ( ord_less @ nat @ I2 @ J2 )
=> ( ( ord_less @ nat @ J2 @ K3 )
=> ( ( P @ I2 @ J2 )
=> ( ( P @ J2 @ K3 )
=> ( P @ I2 @ K3 ) ) ) ) )
=> ( P @ I @ J ) ) ) ) ).
% less_Suc_induct
thf(fact_929_strict__inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less @ nat @ I @ J )
=> ( ! [I2: nat] :
( ( J
= ( suc @ I2 ) )
=> ( P @ I2 ) )
=> ( ! [I2: nat] :
( ( ord_less @ nat @ I2 @ J )
=> ( ( P @ ( suc @ I2 ) )
=> ( P @ I2 ) ) )
=> ( P @ I ) ) ) ) ).
% strict_inc_induct
thf(fact_930_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_931_bot__nat__0_Oextremum__strict,axiom,
! [A2: nat] :
~ ( ord_less @ nat @ A2 @ ( zero_zero @ nat ) ) ).
% bot_nat_0.extremum_strict
thf(fact_932_gr0I,axiom,
! [N: nat] :
( ( N
!= ( zero_zero @ nat ) )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ).
% gr0I
thf(fact_933_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) )
= ( N
= ( zero_zero @ nat ) ) ) ).
% not_gr0
thf(fact_934_not__less0,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% not_less0
thf(fact_935_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% less_zeroE
thf(fact_936_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ N )
=> ( N
!= ( zero_zero @ nat ) ) ) ).
% gr_implies_not0
thf(fact_937_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ ( zero_zero @ nat ) )
=> ( ! [N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ~ ( P @ N2 )
=> ? [M4: nat] :
( ( ord_less @ nat @ M4 @ N2 )
& ~ ( P @ M4 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_938_infinite__descent0__measure,axiom,
! [A: $tType,V2: A > nat,P: A > $o,X2: 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 @ X2 ) ) ) ).
% infinite_descent0_measure
thf(fact_939_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_940_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_941_less__add__eq__less,axiom,
! [K: nat,L2: nat,M: nat,N: nat] :
( ( ord_less @ nat @ K @ L2 )
=> ( ( ( plus_plus @ nat @ M @ L2 )
= ( plus_plus @ nat @ K @ N ) )
=> ( ord_less @ nat @ M @ N ) ) ) ).
% less_add_eq_less
thf(fact_942_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_943_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_944_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_945_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less @ nat @ ( plus_plus @ nat @ J @ I ) @ I ) ).
% not_add_less2
thf(fact_946_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less @ nat @ ( plus_plus @ nat @ I @ J ) @ I ) ).
% not_add_less1
thf(fact_947_add__less__mono,axiom,
! [I: nat,J: nat,K: nat,L2: nat] :
( ( ord_less @ nat @ I @ J )
=> ( ( ord_less @ nat @ K @ L2 )
=> ( ord_less @ nat @ ( plus_plus @ nat @ I @ K ) @ ( plus_plus @ nat @ J @ L2 ) ) ) ) ).
% add_less_mono
thf(fact_948_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_949_nat__less__le,axiom,
( ( ord_less @ nat )
= ( ^ [M5: nat,N3: nat] :
( ( ord_less_eq @ nat @ M5 @ N3 )
& ( M5 != N3 ) ) ) ) ).
% nat_less_le
thf(fact_950_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_951_le__eq__less__or__eq,axiom,
( ( ord_less_eq @ nat )
= ( ^ [M5: nat,N3: nat] :
( ( ord_less @ nat @ M5 @ N3 )
| ( M5 = N3 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_952_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_953_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_954_less__mono__imp__le__mono,axiom,
! [F2: nat > nat,I: nat,J: nat] :
( ! [I2: nat,J2: nat] :
( ( ord_less @ nat @ I2 @ J2 )
=> ( ord_less @ nat @ ( F2 @ I2 ) @ ( F2 @ J2 ) ) )
=> ( ( ord_less_eq @ nat @ I @ J )
=> ( ord_less_eq @ nat @ ( F2 @ I ) @ ( F2 @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_955_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_956_nat__mult__1,axiom,
! [N: nat] :
( ( times_times @ nat @ ( one_one @ nat ) @ N )
= N ) ).
% nat_mult_1
thf(fact_957_nat__mult__1__right,axiom,
! [N: nat] :
( ( times_times @ nat @ N @ ( one_one @ nat ) )
= N ) ).
% nat_mult_1_right
thf(fact_958_frac__less__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [Y2: A,Z: A,X2: A,W: A] :
( ( Y2
!= ( zero_zero @ A ) )
=> ( ( Z
!= ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( divide_divide @ A @ X2 @ Y2 ) @ ( divide_divide @ A @ W @ Z ) )
= ( ord_less @ A @ ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ X2 @ Z ) @ ( times_times @ A @ W @ Y2 ) ) @ ( times_times @ A @ Y2 @ Z ) ) @ ( zero_zero @ A ) ) ) ) ) ) ).
% frac_less_eq
thf(fact_959_power__strict__decreasing,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N: nat,N5: nat,A2: A] :
( ( ord_less @ nat @ N @ N5 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ A @ A2 @ ( one_one @ A ) )
=> ( ord_less @ A @ ( power_power @ A @ A2 @ N5 ) @ ( power_power @ A @ A2 @ N ) ) ) ) ) ) ).
% power_strict_decreasing
thf(fact_960_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_961_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_962_div__if,axiom,
( ( divide_divide @ nat )
= ( ^ [M5: nat,N3: nat] :
( if @ nat
@ ( ( ord_less @ nat @ M5 @ N3 )
| ( N3
= ( zero_zero @ nat ) ) )
@ ( zero_zero @ nat )
@ ( suc @ ( divide_divide @ nat @ ( minus_minus @ nat @ M5 @ N3 ) @ N3 ) ) ) ) ) ).
% div_if
thf(fact_963_add__eq__if,axiom,
( ( plus_plus @ nat )
= ( ^ [M5: nat,N3: nat] :
( if @ nat
@ ( M5
= ( zero_zero @ nat ) )
@ N3
@ ( suc @ ( plus_plus @ nat @ ( minus_minus @ nat @ M5 @ ( one_one @ nat ) ) @ N3 ) ) ) ) ) ).
% add_eq_if
thf(fact_964_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_965_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_966_mult__eq__if,axiom,
( ( times_times @ nat )
= ( ^ [M5: nat,N3: nat] :
( if @ nat
@ ( M5
= ( zero_zero @ nat ) )
@ ( zero_zero @ nat )
@ ( plus_plus @ nat @ N3 @ ( times_times @ nat @ ( minus_minus @ nat @ M5 @ ( one_one @ nat ) ) @ N3 ) ) ) ) ) ).
% mult_eq_if
thf(fact_967_i0__lb,axiom,
! [N: extended_enat] : ( ord_less_eq @ extended_enat @ ( zero_zero @ extended_enat ) @ N ) ).
% i0_lb
thf(fact_968_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_969_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_970_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_971_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_972_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_973_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_974_le__iff__diff__le__0,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A4: A,B3: A] : ( ord_less_eq @ A @ ( minus_minus @ A @ A4 @ B3 ) @ ( zero_zero @ A ) ) ) ) ) ).
% le_iff_diff_le_0
thf(fact_975_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_976_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_977_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_978_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_979_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_980_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_981_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_982_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_983_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_984_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_985_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_986_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_987_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_988_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_989_mult__diff__mult,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [X2: A,Y2: A,A2: A,B2: A] :
( ( minus_minus @ A @ ( times_times @ A @ X2 @ Y2 ) @ ( times_times @ A @ A2 @ B2 ) )
= ( plus_plus @ A @ ( times_times @ A @ X2 @ ( minus_minus @ A @ Y2 @ B2 ) ) @ ( times_times @ A @ ( minus_minus @ A @ X2 @ A2 ) @ B2 ) ) ) ) ).
% mult_diff_mult
thf(fact_990_square__diff__square__factored,axiom,
! [A: $tType] :
( ( comm_ring @ A )
=> ! [X2: A,Y2: A] :
( ( minus_minus @ A @ ( times_times @ A @ X2 @ X2 ) @ ( times_times @ A @ Y2 @ Y2 ) )
= ( times_times @ A @ ( plus_plus @ A @ X2 @ Y2 ) @ ( minus_minus @ A @ X2 @ Y2 ) ) ) ) ).
% square_diff_square_factored
thf(fact_991_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_992_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_993_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_994_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_995_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_996_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_997_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_998_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_999_add__less__zeroD,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X2: A,Y2: A] :
( ( ord_less @ A @ ( plus_plus @ A @ X2 @ Y2 ) @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ X2 @ ( zero_zero @ A ) )
| ( ord_less @ A @ Y2 @ ( zero_zero @ A ) ) ) ) ) ).
% add_less_zeroD
thf(fact_1000_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_1001_zero__less__one,axiom,
! [A: $tType] :
( ( zero_less_one @ A )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( one_one @ A ) ) ) ).
% zero_less_one
thf(fact_1002_not__one__less__zero,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ~ ( ord_less @ A @ ( one_one @ A ) @ ( zero_zero @ A ) ) ) ).
% not_one_less_zero
thf(fact_1003_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_1004_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_1005_add__mono__thms__linordered__field_I3_J,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [I: A,J: A,K: A,L2: A] :
( ( ( ord_less @ A @ I @ J )
& ( ord_less_eq @ A @ K @ L2 ) )
=> ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L2 ) ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_1006_add__mono__thms__linordered__field_I4_J,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [I: A,J: A,K: A,L2: A] :
( ( ( ord_less_eq @ A @ I @ J )
& ( ord_less @ A @ K @ L2 ) )
=> ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L2 ) ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_1007_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_1008_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_1009_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_1010_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_1011_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_1012_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_1013_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_1014_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_1015_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_1016_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_1017_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_1018_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_1019_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_1020_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_1021_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_1022_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_1023_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_1024_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_1025_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_1026_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_1027_divide__neg__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X2: A,Y2: A] :
( ( ord_less @ A @ X2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ Y2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X2 @ Y2 ) ) ) ) ) ).
% divide_neg_neg
thf(fact_1028_divide__neg__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X2: A,Y2: A] :
( ( ord_less @ A @ X2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ Y2 )
=> ( ord_less @ A @ ( divide_divide @ A @ X2 @ Y2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% divide_neg_pos
thf(fact_1029_divide__pos__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X2: A,Y2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ X2 )
=> ( ( ord_less @ A @ Y2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( divide_divide @ A @ X2 @ Y2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% divide_pos_neg
thf(fact_1030_divide__pos__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X2: A,Y2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ X2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ Y2 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X2 @ Y2 ) ) ) ) ) ).
% divide_pos_pos
thf(fact_1031_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_1032_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_1033_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_1034_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_1035_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_1036_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_1037_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_1038_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_1039_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_1040_real__archimedian__rdiv__eq__0,axiom,
! [X2: real,C2: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ C2 )
=> ( ! [M3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ M3 )
=> ( ord_less_eq @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ M3 ) @ X2 ) @ C2 ) )
=> ( X2
= ( zero_zero @ real ) ) ) ) ) ).
% real_archimedian_rdiv_eq_0
thf(fact_1041_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_1042_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_1043_diff__add__0,axiom,
! [N: nat,M: nat] :
( ( minus_minus @ nat @ N @ ( plus_plus @ nat @ N @ M ) )
= ( zero_zero @ nat ) ) ).
% diff_add_0
thf(fact_1044_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_1045_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_1046_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_1047_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_1048_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_1049_less__Suc__eq__0__disj,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ ( suc @ N ) )
= ( ( M
= ( zero_zero @ nat ) )
| ? [J3: nat] :
( ( M
= ( suc @ J3 ) )
& ( ord_less @ nat @ J3 @ N ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_1050_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ? [M3: nat] :
( N
= ( suc @ M3 ) ) ) ).
% gr0_implies_Suc
thf(fact_1051_All__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( suc @ N ) )
=> ( P @ I3 ) ) )
= ( ( P @ ( zero_zero @ nat ) )
& ! [I3: nat] :
( ( ord_less @ nat @ I3 @ N )
=> ( P @ ( suc @ I3 ) ) ) ) ) ).
% All_less_Suc2
thf(fact_1052_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
= ( ? [M5: nat] :
( N
= ( suc @ M5 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_1053_Ex__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I3: nat] :
( ( ord_less @ nat @ I3 @ ( suc @ N ) )
& ( P @ I3 ) ) )
= ( ( P @ ( zero_zero @ nat ) )
| ? [I3: nat] :
( ( ord_less @ nat @ I3 @ N )
& ( P @ ( suc @ I3 ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_1054_less__imp__Suc__add,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ N )
=> ? [K3: nat] :
( N
= ( suc @ ( plus_plus @ nat @ M @ K3 ) ) ) ) ).
% less_imp_Suc_add
thf(fact_1055_less__iff__Suc__add,axiom,
( ( ord_less @ nat )
= ( ^ [M5: nat,N3: nat] :
? [K2: nat] :
( N3
= ( suc @ ( plus_plus @ nat @ M5 @ K2 ) ) ) ) ) ).
% less_iff_Suc_add
thf(fact_1056_less__add__Suc2,axiom,
! [I: nat,M: nat] : ( ord_less @ nat @ I @ ( suc @ ( plus_plus @ nat @ M @ I ) ) ) ).
% less_add_Suc2
thf(fact_1057_less__add__Suc1,axiom,
! [I: nat,M: nat] : ( ord_less @ nat @ I @ ( suc @ ( plus_plus @ nat @ I @ M ) ) ) ).
% less_add_Suc1
thf(fact_1058_less__natE,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ N )
=> ~ ! [Q3: nat] :
( N
!= ( suc @ ( plus_plus @ nat @ M @ Q3 ) ) ) ) ).
% less_natE
thf(fact_1059_Suc__leI,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ N )
=> ( ord_less_eq @ nat @ ( suc @ M ) @ N ) ) ).
% Suc_leI
thf(fact_1060_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_1061_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_1062_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_1063_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_1064_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_1065_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_1066_less__eq__Suc__le,axiom,
( ( ord_less @ nat )
= ( ^ [N3: nat] : ( ord_less_eq @ nat @ ( suc @ N3 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_1067_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_1068_less__imp__add__positive,axiom,
! [I: nat,J: nat] :
( ( ord_less @ nat @ I @ J )
=> ? [K3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ K3 )
& ( ( plus_plus @ nat @ I @ K3 )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_1069_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ ( zero_zero @ nat ) )
=> ? [K3: nat] :
( ( ord_less_eq @ nat @ K3 @ N )
& ! [I4: nat] :
( ( ord_less @ nat @ I4 @ K3 )
=> ~ ( P @ I4 ) )
& ( P @ K3 ) ) ) ) ).
% ex_least_nat_le
thf(fact_1070_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_1071_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_1072_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_1073_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_1074_mono__nat__linear__lb,axiom,
! [F2: nat > nat,M: nat,K: nat] :
( ! [M3: nat,N2: nat] :
( ( ord_less @ nat @ M3 @ N2 )
=> ( ord_less @ nat @ ( F2 @ M3 ) @ ( F2 @ N2 ) ) )
=> ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( F2 @ M ) @ K ) @ ( F2 @ ( plus_plus @ nat @ M @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_1075_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_1076_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_1077_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_1078_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_1079_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_1080_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_1081_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_1082_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_1083_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_1084_numerals_I1_J,axiom,
( ( numeral_numeral @ nat @ one2 )
= ( one_one @ nat ) ) ).
% numerals(1)
thf(fact_1085_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_1086_One__nat__def,axiom,
( ( one_one @ nat )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% One_nat_def
thf(fact_1087_Suc__eq__plus1__left,axiom,
( suc
= ( plus_plus @ nat @ ( one_one @ nat ) ) ) ).
% Suc_eq_plus1_left
thf(fact_1088_plus__1__eq__Suc,axiom,
( ( plus_plus @ nat @ ( one_one @ nat ) )
= suc ) ).
% plus_1_eq_Suc
thf(fact_1089_Suc__eq__plus1,axiom,
( suc
= ( ^ [N3: nat] : ( plus_plus @ nat @ N3 @ ( one_one @ nat ) ) ) ) ).
% Suc_eq_plus1
thf(fact_1090_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_1091_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_1092_VEBT__internal_OT__vebt__buildupi_H_Osimps_I1_J,axiom,
( ( vEBT_V9176841429113362141ildupi @ ( zero_zero @ nat ) )
= ( one_one @ int ) ) ).
% VEBT_internal.T_vebt_buildupi'.simps(1)
thf(fact_1093_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_1094_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_1095_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_1096_diff__frac__eq,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [Y2: A,Z: A,X2: A,W: A] :
( ( Y2
!= ( zero_zero @ A ) )
=> ( ( Z
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( divide_divide @ A @ X2 @ Y2 ) @ ( divide_divide @ A @ W @ Z ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ X2 @ Z ) @ ( times_times @ A @ W @ Y2 ) ) @ ( times_times @ A @ Y2 @ Z ) ) ) ) ) ) ).
% diff_frac_eq
thf(fact_1097_diff__divide__eq__iff,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z: A,X2: A,Y2: A] :
( ( Z
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ X2 @ ( divide_divide @ A @ Y2 @ Z ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ X2 @ Z ) @ Y2 ) @ Z ) ) ) ) ).
% diff_divide_eq_iff
thf(fact_1098_divide__diff__eq__iff,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z: A,X2: A,Y2: A] :
( ( Z
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( divide_divide @ A @ X2 @ Z ) @ Y2 )
= ( divide_divide @ A @ ( minus_minus @ A @ X2 @ ( times_times @ A @ Y2 @ Z ) ) @ Z ) ) ) ) ).
% divide_diff_eq_iff
thf(fact_1099_square__diff__one__factored,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [X2: A] :
( ( minus_minus @ A @ ( times_times @ A @ X2 @ X2 ) @ ( one_one @ A ) )
= ( times_times @ A @ ( plus_plus @ A @ X2 @ ( one_one @ A ) ) @ ( minus_minus @ A @ X2 @ ( one_one @ A ) ) ) ) ) ).
% square_diff_one_factored
thf(fact_1100_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_1101_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_1102_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_1103_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_1104_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_1105_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_1106_field__le__epsilon,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X2: A,Y2: A] :
( ! [E2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ E2 )
=> ( ord_less_eq @ A @ X2 @ ( plus_plus @ A @ Y2 @ E2 ) ) )
=> ( ord_less_eq @ A @ X2 @ Y2 ) ) ) ).
% field_le_epsilon
thf(fact_1107_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_1108_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_1109_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_1110_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_1111_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_1112_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_1113_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_1114_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_1115_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_1116_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_1117_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_1118_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_1119_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_1120_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_1121_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_1122_sum__squares__gt__zero__iff,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [X2: A,Y2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ ( times_times @ A @ X2 @ X2 ) @ ( times_times @ A @ Y2 @ Y2 ) ) )
= ( ( X2
!= ( zero_zero @ A ) )
| ( Y2
!= ( zero_zero @ A ) ) ) ) ) ).
% sum_squares_gt_zero_iff
thf(fact_1123_not__sum__squares__lt__zero,axiom,
! [A: $tType] :
( ( linordered_ring @ A )
=> ! [X2: A,Y2: A] :
~ ( ord_less @ A @ ( plus_plus @ A @ ( times_times @ A @ X2 @ X2 ) @ ( times_times @ A @ Y2 @ Y2 ) ) @ ( zero_zero @ A ) ) ) ).
% not_sum_squares_lt_zero
thf(fact_1124_frac__le,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [Y2: A,X2: A,W: A,Z: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y2 )
=> ( ( ord_less_eq @ A @ X2 @ Y2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ W )
=> ( ( ord_less_eq @ A @ W @ Z )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ X2 @ Z ) @ ( divide_divide @ A @ Y2 @ W ) ) ) ) ) ) ) ).
% frac_le
thf(fact_1125_frac__less,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X2: A,Y2: A,W: A,Z: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X2 )
=> ( ( ord_less @ A @ X2 @ Y2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ W )
=> ( ( ord_less_eq @ A @ W @ Z )
=> ( ord_less @ A @ ( divide_divide @ A @ X2 @ Z ) @ ( divide_divide @ A @ Y2 @ W ) ) ) ) ) ) ) ).
% frac_less
thf(fact_1126_frac__less2,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X2: A,Y2: A,W: A,Z: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ X2 )
=> ( ( ord_less_eq @ A @ X2 @ Y2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ W )
=> ( ( ord_less @ A @ W @ Z )
=> ( ord_less @ A @ ( divide_divide @ A @ X2 @ Z ) @ ( divide_divide @ A @ Y2 @ W ) ) ) ) ) ) ) ).
% frac_less2
thf(fact_1127_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_1128_divide__nonneg__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X2: A,Y2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X2 )
=> ( ( ord_less @ A @ Y2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ X2 @ Y2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% divide_nonneg_neg
thf(fact_1129_divide__nonneg__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X2: A,Y2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ Y2 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X2 @ Y2 ) ) ) ) ) ).
% divide_nonneg_pos
thf(fact_1130_divide__nonpos__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X2: A,Y2: A] :
( ( ord_less_eq @ A @ X2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ Y2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X2 @ Y2 ) ) ) ) ) ).
% divide_nonpos_neg
thf(fact_1131_divide__nonpos__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X2: A,Y2: A] :
( ( ord_less_eq @ A @ X2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ Y2 )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ X2 @ Y2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% divide_nonpos_pos
thf(fact_1132_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_1133_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_1134_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_1135_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_1136_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_1137_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_1138_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_1139_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_1140_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_1141_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_1142_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_1143_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_1144_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_1145_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_1146_mult__imp__div__pos__less,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [Y2: A,X2: A,Z: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Y2 )
=> ( ( ord_less @ A @ X2 @ ( times_times @ A @ Z @ Y2 ) )
=> ( ord_less @ A @ ( divide_divide @ A @ X2 @ Y2 ) @ Z ) ) ) ) ).
% mult_imp_div_pos_less
thf(fact_1147_mult__imp__less__div__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [Y2: A,Z: A,X2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Y2 )
=> ( ( ord_less @ A @ ( times_times @ A @ Z @ Y2 ) @ X2 )
=> ( ord_less @ A @ Z @ ( divide_divide @ A @ X2 @ Y2 ) ) ) ) ) ).
% mult_imp_less_div_pos
thf(fact_1148_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_1149_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_1150_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_1151_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_1152_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_1153_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_1154_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_1155_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_1156_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_1157_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_1158_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_1159_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_1160_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_1161_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_1162_ex__least__nat__less,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ ( zero_zero @ nat ) )
=> ? [K3: nat] :
( ( ord_less @ nat @ K3 @ N )
& ! [I4: nat] :
( ( ord_less_eq @ nat @ I4 @ K3 )
=> ~ ( P @ I4 ) )
& ( P @ ( suc @ K3 ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_1163_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_1164_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_1165_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_1166_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_1167_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_1168_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_1169_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_1170_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_1171_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_1172_div__less__iff__less__mult,axiom,
! [Q2: nat,M: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ Q2 )
=> ( ( ord_less @ nat @ ( divide_divide @ nat @ M @ Q2 ) @ N )
= ( ord_less @ nat @ M @ ( times_times @ nat @ N @ Q2 ) ) ) ) ).
% div_less_iff_less_mult
thf(fact_1173_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_1174_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_1175_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_1176_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_1177_zle__int,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ M ) @ ( semiring_1_of_nat @ int @ N ) )
= ( ord_less_eq @ nat @ M @ N ) ) ).
% zle_int
thf(fact_1178_zle__iff__zadd,axiom,
( ( ord_less_eq @ int )
= ( ^ [W2: int,Z5: int] :
? [N3: nat] :
( Z5
= ( plus_plus @ int @ W2 @ ( semiring_1_of_nat @ int @ N3 ) ) ) ) ) ).
% zle_iff_zadd
thf(fact_1179_VEBT__internal_OT__vebt__buildupi_H_Osimps_I2_J,axiom,
( ( vEBT_V9176841429113362141ildupi @ ( suc @ ( zero_zero @ nat ) ) )
= ( one_one @ int ) ) ).
% VEBT_internal.T_vebt_buildupi'.simps(2)
thf(fact_1180_frac__le__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [Y2: A,Z: A,X2: A,W: A] :
( ( Y2
!= ( zero_zero @ A ) )
=> ( ( Z
!= ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( divide_divide @ A @ X2 @ Y2 ) @ ( divide_divide @ A @ W @ Z ) )
= ( ord_less_eq @ A @ ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ X2 @ Z ) @ ( times_times @ A @ W @ Y2 ) ) @ ( times_times @ A @ Y2 @ Z ) ) @ ( zero_zero @ A ) ) ) ) ) ) ).
% frac_le_eq
thf(fact_1181_power2__commute,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [X2: A,Y2: A] :
( ( power_power @ A @ ( minus_minus @ A @ X2 @ Y2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( power_power @ A @ ( minus_minus @ A @ Y2 @ X2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% power2_commute
thf(fact_1182_field__le__mult__one__interval,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X2: A,Y2: A] :
( ! [Z2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Z2 )
=> ( ( ord_less @ A @ Z2 @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ Z2 @ X2 ) @ Y2 ) ) )
=> ( ord_less_eq @ A @ X2 @ Y2 ) ) ) ).
% field_le_mult_one_interval
thf(fact_1183_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_1184_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_1185_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_1186_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_1187_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_1188_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_1189_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_1190_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_1191_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_1192_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_1193_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_1194_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_1195_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_1196_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_1197_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_1198_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_1199_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_1200_mult__imp__div__pos__le,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [Y2: A,X2: A,Z: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Y2 )
=> ( ( ord_less_eq @ A @ X2 @ ( times_times @ A @ Z @ Y2 ) )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ X2 @ Y2 ) @ Z ) ) ) ) ).
% mult_imp_div_pos_le
thf(fact_1201_mult__imp__le__div__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [Y2: A,Z: A,X2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Y2 )
=> ( ( ord_less_eq @ A @ ( times_times @ A @ Z @ Y2 ) @ X2 )
=> ( ord_less_eq @ A @ Z @ ( divide_divide @ A @ X2 @ Y2 ) ) ) ) ) ).
% mult_imp_le_div_pos
thf(fact_1202_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_1203_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_1204_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_1205_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_1206_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_1207_power__eq__if,axiom,
! [A: $tType] :
( ( power @ A )
=> ( ( power_power @ A )
= ( ^ [P3: A,M5: nat] :
( if @ A
@ ( M5
= ( zero_zero @ nat ) )
@ ( one_one @ A )
@ ( times_times @ A @ P3 @ ( power_power @ A @ P3 @ ( minus_minus @ nat @ M5 @ ( one_one @ nat ) ) ) ) ) ) ) ) ).
% power_eq_if
thf(fact_1208_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_1209_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_1210_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_1211_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_1212_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_1213_pos2,axiom,
ord_less @ nat @ ( zero_zero @ nat ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ).
% pos2
thf(fact_1214_dvd__power,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [N: nat,X2: A] :
( ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
| ( X2
= ( one_one @ A ) ) )
=> ( dvd_dvd @ A @ X2 @ ( power_power @ A @ X2 @ N ) ) ) ) ).
% dvd_power
thf(fact_1215_dvd__minus__add,axiom,
! [Q2: nat,N: nat,R2: nat,M: nat] :
( ( ord_less_eq @ nat @ Q2 @ N )
=> ( ( ord_less_eq @ nat @ Q2 @ ( times_times @ nat @ R2 @ M ) )
=> ( ( dvd_dvd @ nat @ M @ ( minus_minus @ nat @ N @ Q2 ) )
= ( dvd_dvd @ nat @ M @ ( plus_plus @ nat @ N @ ( minus_minus @ nat @ ( times_times @ nat @ R2 @ M ) @ Q2 ) ) ) ) ) ) ).
% dvd_minus_add
thf(fact_1216_div__nat__eqI,axiom,
! [N: nat,Q2: nat,M: nat] :
( ( ord_less_eq @ nat @ ( times_times @ nat @ N @ Q2 ) @ M )
=> ( ( ord_less @ nat @ M @ ( times_times @ nat @ N @ ( suc @ Q2 ) ) )
=> ( ( divide_divide @ nat @ M @ N )
= Q2 ) ) ) ).
% div_nat_eqI
thf(fact_1217_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 ) )
=> ! [I3: nat,J3: nat] :
( ( ord_less @ nat @ J3 @ N )
=> ( ( M
= ( plus_plus @ nat @ ( times_times @ nat @ N @ I3 ) @ J3 ) )
=> ( P @ I3 ) ) ) ) ) ) ).
% split_div
thf(fact_1218_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_1219_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_1220_less__eq__div__iff__mult__less__eq,axiom,
! [Q2: nat,M: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ Q2 )
=> ( ( ord_less_eq @ nat @ M @ ( divide_divide @ nat @ N @ Q2 ) )
= ( ord_less_eq @ nat @ ( times_times @ nat @ M @ Q2 ) @ N ) ) ) ).
% less_eq_div_iff_mult_less_eq
thf(fact_1221_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_1222_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_1223_nat__less__real__le,axiom,
( ( ord_less @ nat )
= ( ^ [N3: nat,M5: nat] : ( ord_less_eq @ real @ ( plus_plus @ real @ ( semiring_1_of_nat @ real @ N3 ) @ ( one_one @ real ) ) @ ( semiring_1_of_nat @ real @ M5 ) ) ) ) ).
% nat_less_real_le
thf(fact_1224_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_1225_scaling__mono,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [U: A,V: A,R2: A,S3: A] :
( ( ord_less_eq @ A @ U @ V )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ R2 )
=> ( ( ord_less_eq @ A @ R2 @ S3 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ U @ ( divide_divide @ A @ ( times_times @ A @ R2 @ ( minus_minus @ A @ V @ U ) ) @ S3 ) ) @ V ) ) ) ) ) ).
% scaling_mono
thf(fact_1226_convex__bound__lt,axiom,
! [A: $tType] :
( ( linord715952674999750819strict @ A )
=> ! [X2: A,A2: A,Y2: A,U: A,V: A] :
( ( ord_less @ A @ X2 @ A2 )
=> ( ( ord_less @ A @ Y2 @ 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 @ X2 ) @ ( times_times @ A @ V @ Y2 ) ) @ A2 ) ) ) ) ) ) ) ).
% convex_bound_lt
thf(fact_1227_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_1228_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_1229_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_1230_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_1231_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_1232_field__less__half__sum,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X2: A,Y2: A] :
( ( ord_less @ A @ X2 @ Y2 )
=> ( ord_less @ A @ X2 @ ( divide_divide @ A @ ( plus_plus @ A @ X2 @ Y2 ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% field_less_half_sum
thf(fact_1233_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_1234_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_1235_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_1236_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_1237_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_1238_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_1239_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_1240_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 ) ) )
| ? [Q4: nat] :
( ( ord_less_eq @ nat @ ( times_times @ nat @ N @ Q4 ) @ M )
& ( ord_less @ nat @ M @ ( times_times @ nat @ N @ ( suc @ Q4 ) ) )
& ( P @ Q4 ) ) ) ) ).
% split_div'
thf(fact_1241_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_1242_power2__less__imp__less,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X2: A,Y2: A] :
( ( ord_less @ A @ ( power_power @ A @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y2 )
=> ( ord_less @ A @ X2 @ Y2 ) ) ) ) ).
% power2_less_imp_less
thf(fact_1243_not__sum__power2__lt__zero,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X2: A,Y2: A] :
~ ( ord_less @ A @ ( plus_plus @ A @ ( power_power @ A @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( zero_zero @ A ) ) ) ).
% not_sum_power2_lt_zero
thf(fact_1244_sum__power2__gt__zero__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X2: A,Y2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ ( power_power @ A @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
= ( ( X2
!= ( zero_zero @ A ) )
| ( Y2
!= ( zero_zero @ A ) ) ) ) ) ).
% sum_power2_gt_zero_iff
thf(fact_1245_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_1246_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_1247_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_1248_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p_Ocases,axiom,
! [X2: nat] :
( ( X2
!= ( zero_zero @ nat ) )
=> ( ( X2
!= ( suc @ ( zero_zero @ nat ) ) )
=> ~ ! [Va2: nat] :
( X2
!= ( suc @ ( suc @ Va2 ) ) ) ) ) ).
% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d\<^sub>u\<^sub>p.cases
thf(fact_1249_power2__diff,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [X2: A,Y2: A] :
( ( power_power @ A @ ( minus_minus @ A @ X2 @ Y2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( minus_minus @ A @ ( plus_plus @ A @ ( power_power @ A @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( times_times @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X2 ) @ Y2 ) ) ) ) ).
% power2_diff
thf(fact_1250_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_1251_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_1252_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_1253_int__distrib_I2_J,axiom,
! [W: int,Z1: int,Z22: int] :
( ( times_times @ int @ W @ ( plus_plus @ int @ Z1 @ Z22 ) )
= ( plus_plus @ int @ ( times_times @ int @ W @ Z1 ) @ ( times_times @ int @ W @ Z22 ) ) ) ).
% int_distrib(2)
thf(fact_1254_int__distrib_I1_J,axiom,
! [Z1: int,Z22: int,W: int] :
( ( times_times @ int @ ( plus_plus @ int @ Z1 @ Z22 ) @ W )
= ( plus_plus @ int @ ( times_times @ int @ Z1 @ W ) @ ( times_times @ int @ Z22 @ W ) ) ) ).
% int_distrib(1)
thf(fact_1255_zdvd__period,axiom,
! [A2: int,D2: int,X2: int,T2: int,C2: int] :
( ( dvd_dvd @ int @ A2 @ D2 )
=> ( ( dvd_dvd @ int @ A2 @ ( plus_plus @ int @ X2 @ T2 ) )
= ( dvd_dvd @ int @ A2 @ ( plus_plus @ int @ ( plus_plus @ int @ X2 @ ( times_times @ int @ C2 @ D2 ) ) @ T2 ) ) ) ) ).
% zdvd_period
thf(fact_1256_zdvd__reduce,axiom,
! [K: int,N: int,M: int] :
( ( dvd_dvd @ int @ K @ ( plus_plus @ int @ N @ ( times_times @ int @ K @ M ) ) )
= ( dvd_dvd @ int @ K @ N ) ) ).
% zdvd_reduce
thf(fact_1257_VEBT__internal_Oexp__split__high__low_I1_J,axiom,
! [X2: nat,N: nat,M: nat] :
( ( ord_less @ nat @ X2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( plus_plus @ nat @ N @ M ) ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
=> ( ord_less @ nat @ ( vEBT_VEBT_high @ X2 @ N ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) ) ) ) ) ).
% VEBT_internal.exp_split_high_low(1)
thf(fact_1258_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_1259_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_1260_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_1261_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p_Osimps_I3_J,axiom,
! [Va: nat] :
( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ Va ) ) )
=> ( ( vEBT_V8346862874174094_d_u_p @ ( suc @ ( suc @ Va ) ) )
= ( plus_plus @ nat @ ( one_one @ nat ) @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_V8346862874174094_d_u_p @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( plus_plus @ nat @ ( vEBT_V8346862874174094_d_u_p @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( one_one @ nat ) ) ) ) ) ) )
& ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ Va ) ) )
=> ( ( vEBT_V8346862874174094_d_u_p @ ( suc @ ( suc @ Va ) ) )
= ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_V8346862874174094_d_u_p @ ( suc @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( plus_plus @ nat @ ( vEBT_V8346862874174094_d_u_p @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( one_one @ nat ) ) ) ) ) ) ) ).
% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d\<^sub>u\<^sub>p.simps(3)
thf(fact_1262_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_Osimps_I3_J,axiom,
! [Va: nat] :
( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ Va ) ) )
=> ( ( vEBT_V8646137997579335489_i_l_d @ ( suc @ ( suc @ Va ) ) )
= ( plus_plus @ nat @ ( one_one @ nat ) @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_V8646137997579335489_i_l_d @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_V8646137997579335489_i_l_d @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) )
& ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ Va ) ) )
=> ( ( vEBT_V8646137997579335489_i_l_d @ ( suc @ ( suc @ Va ) ) )
= ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) @ ( vEBT_V8646137997579335489_i_l_d @ ( suc @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_V8646137997579335489_i_l_d @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d.simps(3)
thf(fact_1263_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p_Oelims,axiom,
! [X2: nat,Y2: nat] :
( ( ( vEBT_V8346862874174094_d_u_p @ X2 )
= Y2 )
=> ( ( ( X2
= ( zero_zero @ nat ) )
=> ( Y2
!= ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) ) )
=> ( ( ( X2
= ( suc @ ( zero_zero @ nat ) ) )
=> ( Y2
!= ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) ) )
=> ~ ! [Va2: nat] :
( ( X2
= ( suc @ ( suc @ Va2 ) ) )
=> ~ ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ Va2 ) ) )
=> ( Y2
= ( 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 ) ) )
=> ( Y2
= ( 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.elims
thf(fact_1264_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_Oelims,axiom,
! [X2: nat,Y2: nat] :
( ( ( vEBT_V8646137997579335489_i_l_d @ X2 )
= Y2 )
=> ( ( ( X2
= ( zero_zero @ nat ) )
=> ( Y2
!= ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) ) )
=> ( ( ( X2
= ( suc @ ( zero_zero @ nat ) ) )
=> ( Y2
!= ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) ) )
=> ~ ! [Va2: nat] :
( ( X2
= ( suc @ ( suc @ Va2 ) ) )
=> ~ ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ Va2 ) ) )
=> ( Y2
= ( 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 ) ) )
=> ( Y2
= ( 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.elims
thf(fact_1265_VEBT__internal_OT__vebt__buildupi_H_Oelims,axiom,
! [X2: nat,Y2: int] :
( ( ( vEBT_V9176841429113362141ildupi @ X2 )
= Y2 )
=> ( ( ( X2
= ( zero_zero @ nat ) )
=> ( Y2
!= ( one_one @ int ) ) )
=> ( ( ( X2
= ( suc @ ( zero_zero @ nat ) ) )
=> ( Y2
!= ( one_one @ int ) ) )
=> ~ ! [N2: nat] :
( ( X2
= ( suc @ ( suc @ N2 ) ) )
=> ~ ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
=> ( Y2
= ( plus_plus @ int @ ( numeral_numeral @ int @ ( bit1 @ one2 ) ) @ ( plus_plus @ int @ ( vEBT_V9176841429113362141ildupi @ ( suc @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( plus_plus @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ ( bit0 @ one2 ) ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( times_times @ int @ ( vEBT_V9176841429113362141ildupi @ ( suc @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) )
& ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
=> ( Y2
= ( plus_plus @ int @ ( numeral_numeral @ int @ ( bit1 @ one2 ) ) @ ( plus_plus @ int @ ( vEBT_V9176841429113362141ildupi @ ( suc @ ( suc @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( plus_plus @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ ( bit0 @ one2 ) ) ) @ ( times_times @ int @ ( vEBT_V9176841429113362141ildupi @ ( suc @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.T_vebt_buildupi'.elims
thf(fact_1266_space__bound,axiom,
! [T2: vEBT_VEBT,N: nat,U: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( U
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
=> ( ord_less_eq @ nat @ ( vEBT_VEBT_space @ T2 ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) @ U ) ) ) ) ).
% space_bound
thf(fact_1267_space__space_H,axiom,
! [T2: vEBT_VEBT] : ( ord_less @ nat @ ( vEBT_VEBT_space @ T2 ) @ ( vEBT_VEBT_space2 @ T2 ) ) ).
% space_space'
thf(fact_1268_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_1269_power__2__mult__step__le,axiom,
! [N4: nat,N: nat,K4: nat,K: nat] :
( ( ord_less_eq @ nat @ N4 @ N )
=> ( ( ord_less @ nat @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N4 ) @ K4 ) @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ K ) )
=> ( ord_less_eq @ nat @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N4 ) @ ( plus_plus @ nat @ K4 @ ( one_one @ nat ) ) ) @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ K ) ) ) ) ).
% power_2_mult_step_le
thf(fact_1270_small__powers__of__2,axiom,
! [X2: nat] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) @ X2 )
=> ( ord_less @ nat @ X2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ X2 @ ( one_one @ nat ) ) ) ) ) ).
% small_powers_of_2
thf(fact_1271_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_1272_space__2__pow__bound,axiom,
! [T2: vEBT_VEBT,N: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ ( vEBT_VEBT_space2 @ T2 ) ) @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) @ ( minus_minus @ real @ ( power_power @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ N ) @ ( one_one @ real ) ) ) ) ) ).
% space_2_pow_bound
thf(fact_1273_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_1274_less__two__pow__divD,axiom,
! [X2: nat,N: nat,M: nat] :
( ( ord_less @ nat @ X2 @ ( divide_divide @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) ) )
=> ( ( ord_less_eq @ nat @ M @ N )
& ( ord_less @ nat @ X2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ N @ M ) ) ) ) ) ).
% less_two_pow_divD
thf(fact_1275_less__two__pow__divI,axiom,
! [X2: nat,N: nat,M: nat] :
( ( ord_less @ nat @ X2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ N @ M ) ) )
=> ( ( ord_less_eq @ nat @ M @ N )
=> ( ord_less @ nat @ X2 @ ( divide_divide @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) ) ) ) ) ).
% less_two_pow_divI
thf(fact_1276_idiff__0__right,axiom,
! [N: extended_enat] :
( ( minus_minus @ extended_enat @ N @ ( zero_zero @ extended_enat ) )
= N ) ).
% idiff_0_right
thf(fact_1277_idiff__0,axiom,
! [N: extended_enat] :
( ( minus_minus @ extended_enat @ ( zero_zero @ extended_enat ) @ N )
= ( zero_zero @ extended_enat ) ) ).
% idiff_0
thf(fact_1278_i0__less,axiom,
! [N: extended_enat] :
( ( ord_less @ extended_enat @ ( zero_zero @ extended_enat ) @ N )
= ( N
!= ( zero_zero @ extended_enat ) ) ) ).
% i0_less
thf(fact_1279_Tb__T__vebt__buildupi_H,axiom,
! [N: nat] : ( ord_less_eq @ int @ ( vEBT_V9176841429113362141ildupi @ N ) @ ( minus_minus @ int @ ( vEBT_VEBT_Tb @ N ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ).
% Tb_T_vebt_buildupi'
thf(fact_1280_Tb__T__vebt__buildupi,axiom,
! [N: nat] : ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ ( vEBT_V441764108873111860ildupi @ N ) ) @ ( minus_minus @ int @ ( vEBT_VEBT_Tb @ N ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ).
% Tb_T_vebt_buildupi
thf(fact_1281_not__real__square__gt__zero,axiom,
! [X2: real] :
( ( ~ ( ord_less @ real @ ( zero_zero @ real ) @ ( times_times @ real @ X2 @ X2 ) ) )
= ( X2
= ( zero_zero @ real ) ) ) ).
% not_real_square_gt_zero
thf(fact_1282_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_1283_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_1284_semiring__norm_I75_J,axiom,
! [M: num] :
~ ( ord_less @ num @ M @ one2 ) ).
% semiring_norm(75)
thf(fact_1285_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_1286_cnt__bound_H,axiom,
! [T2: vEBT_VEBT,N: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ord_less_eq @ real @ ( vEBT_VEBT_cnt @ T2 ) @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( minus_minus @ real @ ( power_power @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ N ) @ ( one_one @ real ) ) ) ) ) ).
% cnt_bound'
thf(fact_1287_cnt__bound,axiom,
! [T2: vEBT_VEBT,N: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ord_less_eq @ real @ ( vEBT_VEBT_cnt @ T2 ) @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( minus_minus @ real @ ( power_power @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ N ) @ ( divide_divide @ real @ ( numeral_numeral @ real @ ( bit1 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) @ ( numeral_numeral @ real @ ( bit0 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ).
% cnt_bound
thf(fact_1288_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_1289_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_1290_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_1291_div__pos__pos__trivial,axiom,
! [K: int,L2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
=> ( ( ord_less @ int @ K @ L2 )
=> ( ( divide_divide @ int @ K @ L2 )
= ( zero_zero @ int ) ) ) ) ).
% div_pos_pos_trivial
thf(fact_1292_div__neg__neg__trivial,axiom,
! [K: int,L2: int] :
( ( ord_less_eq @ int @ K @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ L2 @ K )
=> ( ( divide_divide @ int @ K @ L2 )
= ( zero_zero @ int ) ) ) ) ).
% div_neg_neg_trivial
thf(fact_1293_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_1294_semiring__norm_I76_J,axiom,
! [N: num] : ( ord_less @ num @ one2 @ ( bit0 @ N ) ) ).
% semiring_norm(76)
thf(fact_1295_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_1296_semiring__norm_I77_J,axiom,
! [N: num] : ( ord_less @ num @ one2 @ ( bit1 @ N ) ) ).
% semiring_norm(77)
thf(fact_1297_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_1298_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_1299_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_1300_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_1301_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_1302_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_1303_not__iless0,axiom,
! [N: extended_enat] :
~ ( ord_less @ extended_enat @ N @ ( zero_zero @ extended_enat ) ) ).
% not_iless0
thf(fact_1304_zero__one__enat__neq_I1_J,axiom,
( ( zero_zero @ extended_enat )
!= ( one_one @ extended_enat ) ) ).
% zero_one_enat_neq(1)
thf(fact_1305_less__int__code_I1_J,axiom,
~ ( ord_less @ int @ ( zero_zero @ int ) @ ( zero_zero @ int ) ) ).
% less_int_code(1)
thf(fact_1306_minus__int__code_I1_J,axiom,
! [K: int] :
( ( minus_minus @ int @ K @ ( zero_zero @ int ) )
= K ) ).
% minus_int_code(1)
thf(fact_1307_int__less__induct,axiom,
! [I: int,K: int,P: int > $o] :
( ( ord_less @ int @ I @ K )
=> ( ( P @ ( minus_minus @ int @ K @ ( one_one @ int ) ) )
=> ( ! [I2: int] :
( ( ord_less @ int @ I2 @ K )
=> ( ( P @ I2 )
=> ( P @ ( minus_minus @ int @ I2 @ ( one_one @ int ) ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_less_induct
thf(fact_1308_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_1309_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_1310_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_1311_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_1312_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_1313_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_1314_int__le__induct,axiom,
! [I: int,K: int,P: int > $o] :
( ( ord_less_eq @ int @ I @ K )
=> ( ( P @ K )
=> ( ! [I2: int] :
( ( ord_less_eq @ int @ I2 @ K )
=> ( ( P @ I2 )
=> ( P @ ( minus_minus @ int @ I2 @ ( one_one @ int ) ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_le_induct
thf(fact_1315_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_1316_less__eq__int__code_I1_J,axiom,
ord_less_eq @ int @ ( zero_zero @ int ) @ ( zero_zero @ int ) ).
% less_eq_int_code(1)
thf(fact_1317_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_1318_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_1319_div__pos__geq,axiom,
! [L2: int,K: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ L2 )
=> ( ( ord_less_eq @ int @ L2 @ K )
=> ( ( divide_divide @ int @ K @ L2 )
= ( plus_plus @ int @ ( divide_divide @ int @ ( minus_minus @ int @ K @ L2 ) @ L2 ) @ ( one_one @ int ) ) ) ) ) ).
% div_pos_geq
thf(fact_1320_int__induct,axiom,
! [P: int > $o,K: int,I: int] :
( ( P @ K )
=> ( ! [I2: int] :
( ( ord_less_eq @ int @ K @ I2 )
=> ( ( P @ I2 )
=> ( P @ ( plus_plus @ int @ I2 @ ( one_one @ int ) ) ) ) )
=> ( ! [I2: int] :
( ( ord_less_eq @ int @ I2 @ K )
=> ( ( P @ I2 )
=> ( P @ ( minus_minus @ int @ I2 @ ( one_one @ int ) ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_induct
thf(fact_1321_int__distrib_I3_J,axiom,
! [Z1: int,Z22: int,W: int] :
( ( times_times @ int @ ( minus_minus @ int @ Z1 @ Z22 ) @ W )
= ( minus_minus @ int @ ( times_times @ int @ Z1 @ W ) @ ( times_times @ int @ Z22 @ W ) ) ) ).
% int_distrib(3)
thf(fact_1322_int__distrib_I4_J,axiom,
! [W: int,Z1: int,Z22: int] :
( ( times_times @ int @ W @ ( minus_minus @ int @ Z1 @ Z22 ) )
= ( minus_minus @ int @ ( times_times @ int @ W @ Z1 ) @ ( times_times @ int @ W @ Z22 ) ) ) ).
% int_distrib(4)
thf(fact_1323_int__diff__cases,axiom,
! [Z: int] :
~ ! [M3: nat,N2: nat] :
( Z
!= ( minus_minus @ int @ ( semiring_1_of_nat @ int @ M3 ) @ ( semiring_1_of_nat @ int @ N2 ) ) ) ).
% int_diff_cases
thf(fact_1324_less__eq__real__def,axiom,
( ( ord_less_eq @ real )
= ( ^ [X: real,Y: real] :
( ( ord_less @ real @ X @ Y )
| ( X = Y ) ) ) ) ).
% less_eq_real_def
thf(fact_1325_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_1326_int__gr__induct,axiom,
! [K: int,I: int,P: int > $o] :
( ( ord_less @ int @ K @ I )
=> ( ( P @ ( plus_plus @ int @ K @ ( one_one @ int ) ) )
=> ( ! [I2: int] :
( ( ord_less @ int @ K @ I2 )
=> ( ( P @ I2 )
=> ( P @ ( plus_plus @ int @ I2 @ ( one_one @ int ) ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_gr_induct
thf(fact_1327_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_1328_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_1329_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_1330_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_1331_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_1332_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_1333_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_1334_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_1335_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_1336_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_1337_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_1338_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_1339_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_1340_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_1341_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_1342_div__positive__int,axiom,
! [L2: int,K: int] :
( ( ord_less_eq @ int @ L2 @ K )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ L2 )
=> ( ord_less @ int @ ( zero_zero @ int ) @ ( divide_divide @ int @ K @ L2 ) ) ) ) ).
% div_positive_int
thf(fact_1343_div__int__pos__iff,axiom,
! [K: int,L2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( divide_divide @ int @ K @ L2 ) )
= ( ( K
= ( zero_zero @ int ) )
| ( L2
= ( zero_zero @ int ) )
| ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
& ( ord_less_eq @ int @ ( zero_zero @ int ) @ L2 ) )
| ( ( ord_less @ int @ K @ ( zero_zero @ int ) )
& ( ord_less @ int @ L2 @ ( zero_zero @ int ) ) ) ) ) ).
% div_int_pos_iff
thf(fact_1344_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_1345_zdiv__mono1__neg,axiom,
! [A2: int,A6: int,B2: int] :
( ( ord_less_eq @ int @ A2 @ A6 )
=> ( ( ord_less @ int @ B2 @ ( zero_zero @ int ) )
=> ( ord_less_eq @ int @ ( divide_divide @ int @ A6 @ B2 ) @ ( divide_divide @ int @ A2 @ B2 ) ) ) ) ).
% zdiv_mono1_neg
thf(fact_1346_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_1347_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_1348_zdiv__mono1,axiom,
! [A2: int,A6: int,B2: int] :
( ( ord_less_eq @ int @ A2 @ A6 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
=> ( ord_less_eq @ int @ ( divide_divide @ int @ A2 @ B2 ) @ ( divide_divide @ int @ A6 @ B2 ) ) ) ) ).
% zdiv_mono1
thf(fact_1349_int__div__less__self,axiom,
! [X2: int,K: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ X2 )
=> ( ( ord_less @ int @ ( one_one @ int ) @ K )
=> ( ord_less @ int @ ( divide_divide @ int @ X2 @ K ) @ X2 ) ) ) ).
% int_div_less_self
thf(fact_1350_add__diff__assoc__enat,axiom,
! [Z: extended_enat,Y2: extended_enat,X2: extended_enat] :
( ( ord_less_eq @ extended_enat @ Z @ Y2 )
=> ( ( plus_plus @ extended_enat @ X2 @ ( minus_minus @ extended_enat @ Y2 @ Z ) )
= ( minus_minus @ extended_enat @ ( plus_plus @ extended_enat @ X2 @ Y2 ) @ Z ) ) ) ).
% add_diff_assoc_enat
thf(fact_1351_strict__subset__divisors__dvd,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ ( set @ A )
@ ( collect @ A
@ ^ [C3: A] : ( dvd_dvd @ A @ C3 @ A2 ) )
@ ( collect @ A
@ ^ [C3: A] : ( dvd_dvd @ A @ C3 @ B2 ) ) )
= ( ( dvd_dvd @ A @ A2 @ B2 )
& ~ ( dvd_dvd @ A @ B2 @ A2 ) ) ) ) ).
% strict_subset_divisors_dvd
thf(fact_1352_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_1353_q__pos__lemma,axiom,
! [B7: int,Q5: int,R4: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( plus_plus @ int @ ( times_times @ int @ B7 @ Q5 ) @ R4 ) )
=> ( ( ord_less @ int @ R4 @ B7 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ B7 )
=> ( ord_less_eq @ int @ ( zero_zero @ int ) @ Q5 ) ) ) ) ).
% q_pos_lemma
thf(fact_1354_zdiv__mono2__lemma,axiom,
! [B2: int,Q2: int,R2: int,B7: int,Q5: int,R4: int] :
( ( ( plus_plus @ int @ ( times_times @ int @ B2 @ Q2 ) @ R2 )
= ( plus_plus @ int @ ( times_times @ int @ B7 @ Q5 ) @ R4 ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( plus_plus @ int @ ( times_times @ int @ B7 @ Q5 ) @ 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 @ Q2 @ Q5 ) ) ) ) ) ) ) ).
% zdiv_mono2_lemma
thf(fact_1355_zdiv__mono2__neg__lemma,axiom,
! [B2: int,Q2: int,R2: int,B7: int,Q5: int,R4: int] :
( ( ( plus_plus @ int @ ( times_times @ int @ B2 @ Q2 ) @ R2 )
= ( plus_plus @ int @ ( times_times @ int @ B7 @ Q5 ) @ R4 ) )
=> ( ( ord_less @ int @ ( plus_plus @ int @ ( times_times @ int @ B7 @ Q5 ) @ 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 @ Q5 @ Q2 ) ) ) ) ) ) ) ).
% zdiv_mono2_neg_lemma
thf(fact_1356_unique__quotient__lemma,axiom,
! [B2: int,Q5: int,R4: int,Q2: int,R2: int] :
( ( ord_less_eq @ int @ ( plus_plus @ int @ ( times_times @ int @ B2 @ Q5 ) @ R4 ) @ ( plus_plus @ int @ ( times_times @ int @ B2 @ Q2 ) @ R2 ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ R4 )
=> ( ( ord_less @ int @ R4 @ B2 )
=> ( ( ord_less @ int @ R2 @ B2 )
=> ( ord_less_eq @ int @ Q5 @ Q2 ) ) ) ) ) ).
% unique_quotient_lemma
thf(fact_1357_unique__quotient__lemma__neg,axiom,
! [B2: int,Q5: int,R4: int,Q2: int,R2: int] :
( ( ord_less_eq @ int @ ( plus_plus @ int @ ( times_times @ int @ B2 @ Q5 ) @ R4 ) @ ( plus_plus @ int @ ( times_times @ int @ B2 @ Q2 ) @ R2 ) )
=> ( ( ord_less_eq @ int @ R2 @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ B2 @ R2 )
=> ( ( ord_less @ int @ B2 @ R4 )
=> ( ord_less_eq @ int @ Q2 @ Q5 ) ) ) ) ) ).
% unique_quotient_lemma_neg
thf(fact_1358_real__arch__pow,axiom,
! [X2: real,Y2: real] :
( ( ord_less @ real @ ( one_one @ real ) @ X2 )
=> ? [N2: nat] : ( ord_less @ real @ Y2 @ ( power_power @ real @ X2 @ N2 ) ) ) ).
% real_arch_pow
thf(fact_1359_real__arch__pow__inv,axiom,
! [Y2: real,X2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ Y2 )
=> ( ( ord_less @ real @ X2 @ ( one_one @ real ) )
=> ? [N2: nat] : ( ord_less @ real @ ( power_power @ real @ X2 @ N2 ) @ Y2 ) ) ) ).
% real_arch_pow_inv
thf(fact_1360_reals__Archimedean3,axiom,
! [X2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ! [Y4: real] :
? [N2: nat] : ( ord_less @ real @ Y4 @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N2 ) @ X2 ) ) ) ).
% reals_Archimedean3
thf(fact_1361_ord__eq__le__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A2: A,B2: A,C2: A,D2: A] :
( ( A2 = B2 )
=> ( ( ord_less_eq @ A @ B2 @ C2 )
=> ( ( C2 = D2 )
=> ( ord_less_eq @ A @ A2 @ D2 ) ) ) ) ) ).
% ord_eq_le_eq_trans
thf(fact_1362_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_1363_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_1364_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_1365_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_1366_zless__iff__Suc__zadd,axiom,
( ( ord_less @ int )
= ( ^ [W2: int,Z5: int] :
? [N3: nat] :
( Z5
= ( plus_plus @ int @ W2 @ ( semiring_1_of_nat @ int @ ( suc @ N3 ) ) ) ) ) ) ).
% zless_iff_Suc_zadd
thf(fact_1367_subset__Collect__conv,axiom,
! [A: $tType,S: set @ A,P: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ S @ ( collect @ A @ P ) )
= ( ! [X: A] :
( ( member @ A @ X @ S )
=> ( P @ X ) ) ) ) ).
% subset_Collect_conv
thf(fact_1368_int__div__pos__eq,axiom,
! [A2: int,B2: int,Q2: int,R2: int] :
( ( A2
= ( plus_plus @ int @ ( times_times @ int @ B2 @ Q2 ) @ R2 ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ R2 )
=> ( ( ord_less @ int @ R2 @ B2 )
=> ( ( divide_divide @ int @ A2 @ B2 )
= Q2 ) ) ) ) ).
% int_div_pos_eq
thf(fact_1369_int__div__neg__eq,axiom,
! [A2: int,B2: int,Q2: int,R2: int] :
( ( A2
= ( plus_plus @ int @ ( times_times @ int @ B2 @ Q2 ) @ R2 ) )
=> ( ( ord_less_eq @ int @ R2 @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ B2 @ R2 )
=> ( ( divide_divide @ int @ A2 @ B2 )
= Q2 ) ) ) ) ).
% int_div_neg_eq
thf(fact_1370_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 )
=> ! [I3: int,J3: int] :
( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ J3 )
& ( ord_less @ int @ J3 @ K )
& ( N
= ( plus_plus @ int @ ( times_times @ int @ K @ I3 ) @ J3 ) ) )
=> ( P @ I3 ) ) )
& ( ( ord_less @ int @ K @ ( zero_zero @ int ) )
=> ! [I3: int,J3: int] :
( ( ( ord_less @ int @ K @ J3 )
& ( ord_less_eq @ int @ J3 @ ( zero_zero @ int ) )
& ( N
= ( plus_plus @ int @ ( times_times @ int @ K @ I3 ) @ J3 ) ) )
=> ( P @ I3 ) ) ) ) ) ).
% split_zdiv
thf(fact_1371_even__diff__iff,axiom,
! [K: int,L2: int] :
( ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( minus_minus @ int @ K @ L2 ) )
= ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( plus_plus @ int @ K @ L2 ) ) ) ).
% even_diff_iff
thf(fact_1372_real__of__nat__div2,axiom,
! [N: nat,X2: nat] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( minus_minus @ real @ ( divide_divide @ real @ ( semiring_1_of_nat @ real @ N ) @ ( semiring_1_of_nat @ real @ X2 ) ) @ ( semiring_1_of_nat @ real @ ( divide_divide @ nat @ N @ X2 ) ) ) ) ).
% real_of_nat_div2
thf(fact_1373_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_1374_nat__le__real__less,axiom,
( ( ord_less_eq @ nat )
= ( ^ [N3: nat,M5: nat] : ( ord_less @ real @ ( semiring_1_of_nat @ real @ N3 ) @ ( plus_plus @ real @ ( semiring_1_of_nat @ real @ M5 ) @ ( one_one @ real ) ) ) ) ) ).
% nat_le_real_less
thf(fact_1375_real__of__nat__div3,axiom,
! [N: nat,X2: nat] : ( ord_less_eq @ real @ ( minus_minus @ real @ ( divide_divide @ real @ ( semiring_1_of_nat @ real @ N ) @ ( semiring_1_of_nat @ real @ X2 ) ) @ ( semiring_1_of_nat @ real @ ( divide_divide @ nat @ N @ X2 ) ) ) @ ( one_one @ real ) ) ).
% real_of_nat_div3
thf(fact_1376_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_1377_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_1378_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_1379_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_1380_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_1381_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_1382_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_1383_nat__geq__1__eq__neqz,axiom,
! [X2: nat] :
( ( ord_less_eq @ nat @ ( one_one @ nat ) @ X2 )
= ( X2
!= ( zero_zero @ nat ) ) ) ).
% nat_geq_1_eq_neqz
thf(fact_1384_mlex__bound,axiom,
! [A2: nat,A3: nat,B2: nat,N5: nat] :
( ( ord_less @ nat @ A2 @ A3 )
=> ( ( ord_less @ nat @ B2 @ N5 )
=> ( ord_less @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ A2 @ N5 ) @ B2 ) @ ( times_times @ nat @ A3 @ N5 ) ) ) ) ).
% mlex_bound
thf(fact_1385_mlex__fst__decrI,axiom,
! [A2: nat,A6: nat,B2: nat,N5: nat,B7: nat] :
( ( ord_less @ nat @ A2 @ A6 )
=> ( ( ord_less @ nat @ B2 @ N5 )
=> ( ( ord_less @ nat @ B7 @ N5 )
=> ( ord_less @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ A2 @ N5 ) @ B2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ A6 @ N5 ) @ B7 ) ) ) ) ) ).
% mlex_fst_decrI
thf(fact_1386_mlex__snd__decrI,axiom,
! [A2: nat,A6: nat,B2: nat,B7: nat,N5: nat] :
( ( A2 = A6 )
=> ( ( ord_less @ nat @ B2 @ B7 )
=> ( ord_less @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ A2 @ N5 ) @ B2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ A6 @ N5 ) @ B7 ) ) ) ) ).
% mlex_snd_decrI
thf(fact_1387_mlex__leI,axiom,
! [A2: nat,A6: nat,B2: nat,B7: nat,N5: nat] :
( ( ord_less_eq @ nat @ A2 @ A6 )
=> ( ( ord_less_eq @ nat @ B2 @ B7 )
=> ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ A2 @ N5 ) @ B2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ A6 @ N5 ) @ B7 ) ) ) ) ).
% mlex_leI
thf(fact_1388_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_1389_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_1390_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_1391_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_1392_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_1393_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_1394_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_1395_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_1396_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_1397_axxdiv2,axiom,
! [X2: int] :
( ( ( divide_divide @ int @ ( plus_plus @ int @ ( plus_plus @ int @ ( one_one @ int ) @ X2 ) @ X2 ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) )
= X2 )
& ( ( divide_divide @ int @ ( plus_plus @ int @ ( plus_plus @ int @ ( zero_zero @ int ) @ X2 ) @ X2 ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) )
= X2 ) ) ).
% axxdiv2
thf(fact_1398_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_1399_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_1400_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_1401_nat__add__offset__less,axiom,
! [Y2: nat,N: nat,X2: nat,M: nat,Sz: nat] :
( ( ord_less @ nat @ Y2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
=> ( ( ord_less @ nat @ X2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) )
=> ( ( Sz
= ( plus_plus @ nat @ M @ N ) )
=> ( ord_less @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ X2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) @ Y2 ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Sz ) ) ) ) ) ).
% nat_add_offset_less
thf(fact_1402_nat__power__less__diff,axiom,
! [N: nat,Q2: nat,M: nat] :
( ( ord_less @ nat @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ Q2 ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) )
=> ( ord_less @ nat @ Q2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ M @ N ) ) ) ) ).
% nat_power_less_diff
thf(fact_1403_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_1404_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_1405_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_1406_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_1407_two__powr__height__bound__deg,axiom,
! [T2: vEBT_VEBT,N: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ord_less_eq @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( vEBT_VEBT_height @ T2 ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ).
% two_powr_height_bound_deg
thf(fact_1408_linear__plus__1__le__power,axiom,
! [X2: real,N: nat] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
=> ( ord_less_eq @ real @ ( plus_plus @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ X2 ) @ ( one_one @ real ) ) @ ( power_power @ real @ ( plus_plus @ real @ X2 @ ( one_one @ real ) ) @ N ) ) ) ).
% linear_plus_1_le_power
thf(fact_1409_member__bound,axiom,
! [Tree: vEBT_VEBT,X2: nat,N: nat] :
( ( vEBT_vebt_member @ Tree @ X2 )
=> ( ( vEBT_invar_vebt @ Tree @ N )
=> ( ord_less @ nat @ X2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% member_bound
thf(fact_1410_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_1411_zdiff__int__split,axiom,
! [P: int > $o,X2: nat,Y2: nat] :
( ( P @ ( semiring_1_of_nat @ int @ ( minus_minus @ nat @ X2 @ Y2 ) ) )
= ( ( ( ord_less_eq @ nat @ Y2 @ X2 )
=> ( P @ ( minus_minus @ int @ ( semiring_1_of_nat @ int @ X2 ) @ ( semiring_1_of_nat @ int @ Y2 ) ) ) )
& ( ( ord_less @ nat @ X2 @ Y2 )
=> ( P @ ( zero_zero @ int ) ) ) ) ) ).
% zdiff_int_split
thf(fact_1412_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_1413_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_1414_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_1415_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_1416_member__correct,axiom,
! [T2: vEBT_VEBT,N: nat,X2: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( vEBT_vebt_member @ T2 @ X2 )
= ( member @ nat @ X2 @ ( vEBT_set_vebt @ T2 ) ) ) ) ).
% member_correct
thf(fact_1417_enat__less__induct,axiom,
! [P: extended_enat > $o,N: extended_enat] :
( ! [N2: extended_enat] :
( ! [M4: extended_enat] :
( ( ord_less @ extended_enat @ M4 @ N2 )
=> ( P @ M4 ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% enat_less_induct
thf(fact_1418_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_1419_mult__commute__abs,axiom,
! [A: $tType] :
( ( ab_semigroup_mult @ A )
=> ! [C2: A] :
( ( ^ [X: A] : ( times_times @ A @ X @ C2 ) )
= ( times_times @ A @ C2 ) ) ) ).
% mult_commute_abs
thf(fact_1420_minf_I8_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z2: A] :
! [X4: A] :
( ( ord_less @ A @ X4 @ Z2 )
=> ~ ( ord_less_eq @ A @ T2 @ X4 ) ) ) ).
% minf(8)
thf(fact_1421_minf_I6_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z2: A] :
! [X4: A] :
( ( ord_less @ A @ X4 @ Z2 )
=> ( ord_less_eq @ A @ X4 @ T2 ) ) ) ).
% minf(6)
thf(fact_1422_pinf_I8_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z2: A] :
! [X4: A] :
( ( ord_less @ A @ Z2 @ X4 )
=> ( ord_less_eq @ A @ T2 @ X4 ) ) ) ).
% pinf(8)
thf(fact_1423_pinf_I6_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z2: A] :
! [X4: A] :
( ( ord_less @ A @ Z2 @ X4 )
=> ~ ( ord_less_eq @ A @ X4 @ T2 ) ) ) ).
% pinf(6)
thf(fact_1424_inf__period_I2_J,axiom,
! [A: $tType] :
( ( ( comm_ring @ A )
& ( dvd @ A ) )
=> ! [P: A > $o,D4: A,Q: A > $o] :
( ! [X3: A,K3: A] :
( ( P @ X3 )
= ( P @ ( minus_minus @ A @ X3 @ ( times_times @ A @ K3 @ D4 ) ) ) )
=> ( ! [X3: A,K3: A] :
( ( Q @ X3 )
= ( Q @ ( minus_minus @ A @ X3 @ ( times_times @ A @ K3 @ D4 ) ) ) )
=> ! [X4: A,K5: A] :
( ( ( P @ X4 )
| ( Q @ X4 ) )
= ( ( P @ ( minus_minus @ A @ X4 @ ( times_times @ A @ K5 @ D4 ) ) )
| ( Q @ ( minus_minus @ A @ X4 @ ( times_times @ A @ K5 @ D4 ) ) ) ) ) ) ) ) ).
% inf_period(2)
thf(fact_1425_inf__period_I1_J,axiom,
! [A: $tType] :
( ( ( comm_ring @ A )
& ( dvd @ A ) )
=> ! [P: A > $o,D4: A,Q: A > $o] :
( ! [X3: A,K3: A] :
( ( P @ X3 )
= ( P @ ( minus_minus @ A @ X3 @ ( times_times @ A @ K3 @ D4 ) ) ) )
=> ( ! [X3: A,K3: A] :
( ( Q @ X3 )
= ( Q @ ( minus_minus @ A @ X3 @ ( times_times @ A @ K3 @ D4 ) ) ) )
=> ! [X4: A,K5: A] :
( ( ( P @ X4 )
& ( Q @ X4 ) )
= ( ( P @ ( minus_minus @ A @ X4 @ ( times_times @ A @ K5 @ D4 ) ) )
& ( Q @ ( minus_minus @ A @ X4 @ ( times_times @ A @ K5 @ D4 ) ) ) ) ) ) ) ) ).
% inf_period(1)
thf(fact_1426_real__arch__simple,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X2: A] :
? [N2: nat] : ( ord_less_eq @ A @ X2 @ ( semiring_1_of_nat @ A @ N2 ) ) ) ).
% real_arch_simple
thf(fact_1427_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_1428_reals__Archimedean2,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X2: A] :
? [N2: nat] : ( ord_less @ A @ X2 @ ( semiring_1_of_nat @ A @ N2 ) ) ) ).
% reals_Archimedean2
thf(fact_1429_imp__le__cong,axiom,
! [X2: int,X5: int,P: $o,P4: $o] :
( ( X2 = X5 )
=> ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X5 )
=> ( P = P4 ) )
=> ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X2 )
=> P )
= ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X5 )
=> P4 ) ) ) ) ).
% imp_le_cong
thf(fact_1430_conj__le__cong,axiom,
! [X2: int,X5: int,P: $o,P4: $o] :
( ( X2 = X5 )
=> ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X5 )
=> ( P = P4 ) )
=> ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X2 )
& P )
= ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X5 )
& P4 ) ) ) ) ).
% conj_le_cong
thf(fact_1431_plusinfinity,axiom,
! [D2: int,P4: int > $o,P: int > $o] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D2 )
=> ( ! [X3: int,K3: int] :
( ( P4 @ X3 )
= ( P4 @ ( minus_minus @ int @ X3 @ ( times_times @ int @ K3 @ D2 ) ) ) )
=> ( ? [Z4: int] :
! [X3: int] :
( ( ord_less @ int @ Z4 @ X3 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [X_12: int] : ( P4 @ X_12 )
=> ? [X_1: int] : ( P @ X_1 ) ) ) ) ) ).
% plusinfinity
thf(fact_1432_minusinfinity,axiom,
! [D2: int,P1: int > $o,P: int > $o] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D2 )
=> ( ! [X3: int,K3: int] :
( ( P1 @ X3 )
= ( P1 @ ( minus_minus @ int @ X3 @ ( times_times @ int @ K3 @ 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_1433_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_1434_pinf_I9_J,axiom,
! [B: $tType] :
( ( ( plus @ B )
& ( linorder @ B )
& ( dvd @ B ) )
=> ! [D2: B,S3: B] :
? [Z2: B] :
! [X4: B] :
( ( ord_less @ B @ Z2 @ X4 )
=> ( ( dvd_dvd @ B @ D2 @ ( plus_plus @ B @ X4 @ S3 ) )
= ( dvd_dvd @ B @ D2 @ ( plus_plus @ B @ X4 @ S3 ) ) ) ) ) ).
% pinf(9)
thf(fact_1435_pinf_I10_J,axiom,
! [B: $tType] :
( ( ( plus @ B )
& ( linorder @ B )
& ( dvd @ B ) )
=> ! [D2: B,S3: B] :
? [Z2: B] :
! [X4: B] :
( ( ord_less @ B @ Z2 @ X4 )
=> ( ( ~ ( dvd_dvd @ B @ D2 @ ( plus_plus @ B @ X4 @ S3 ) ) )
= ( ~ ( dvd_dvd @ B @ D2 @ ( plus_plus @ B @ X4 @ S3 ) ) ) ) ) ) ).
% pinf(10)
thf(fact_1436_minf_I9_J,axiom,
! [B: $tType] :
( ( ( plus @ B )
& ( linorder @ B )
& ( dvd @ B ) )
=> ! [D2: B,S3: B] :
? [Z2: B] :
! [X4: B] :
( ( ord_less @ B @ X4 @ Z2 )
=> ( ( dvd_dvd @ B @ D2 @ ( plus_plus @ B @ X4 @ S3 ) )
= ( dvd_dvd @ B @ D2 @ ( plus_plus @ B @ X4 @ S3 ) ) ) ) ) ).
% minf(9)
thf(fact_1437_minf_I10_J,axiom,
! [B: $tType] :
( ( ( plus @ B )
& ( linorder @ B )
& ( dvd @ B ) )
=> ! [D2: B,S3: B] :
? [Z2: B] :
! [X4: B] :
( ( ord_less @ B @ X4 @ Z2 )
=> ( ( ~ ( dvd_dvd @ B @ D2 @ ( plus_plus @ B @ X4 @ S3 ) ) )
= ( ~ ( dvd_dvd @ B @ D2 @ ( plus_plus @ B @ X4 @ S3 ) ) ) ) ) ) ).
% minf(10)
thf(fact_1438_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 )
=> ! [X4: int] :
( ( P @ X4 )
=> ( P @ ( minus_minus @ int @ X4 @ ( times_times @ int @ K @ D2 ) ) ) ) ) ) ) ).
% decr_mult_lemma
thf(fact_1439_Bolzano,axiom,
! [A2: real,B2: real,P: real > real > $o] :
( ( ord_less_eq @ real @ A2 @ B2 )
=> ( ! [A5: real,B4: real,C4: real] :
( ( P @ A5 @ B4 )
=> ( ( P @ B4 @ C4 )
=> ( ( ord_less_eq @ real @ A5 @ B4 )
=> ( ( ord_less_eq @ real @ B4 @ C4 )
=> ( P @ A5 @ C4 ) ) ) ) )
=> ( ! [X3: real] :
( ( ord_less_eq @ real @ A2 @ X3 )
=> ( ( ord_less_eq @ real @ X3 @ B2 )
=> ? [D5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D5 )
& ! [A5: real,B4: real] :
( ( ( ord_less_eq @ real @ A5 @ X3 )
& ( ord_less_eq @ real @ X3 @ B4 )
& ( ord_less @ real @ ( minus_minus @ real @ B4 @ A5 ) @ D5 ) )
=> ( P @ A5 @ B4 ) ) ) ) )
=> ( P @ A2 @ B2 ) ) ) ) ).
% Bolzano
thf(fact_1440_zdvd__mono,axiom,
! [K: int,M: int,T2: int] :
( ( K
!= ( zero_zero @ int ) )
=> ( ( dvd_dvd @ int @ M @ T2 )
= ( dvd_dvd @ int @ ( times_times @ int @ K @ M ) @ ( times_times @ int @ K @ T2 ) ) ) ) ).
% zdvd_mono
thf(fact_1441_ex__less__of__nat__mult,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X2: A,Y2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ X2 )
=> ? [N2: nat] : ( ord_less @ A @ Y2 @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N2 ) @ X2 ) ) ) ) ).
% ex_less_of_nat_mult
thf(fact_1442_unity__coeff__ex,axiom,
! [A: $tType] :
( ( ( dvd @ A )
& ( semiring_0 @ A ) )
=> ! [P: A > $o,L2: A] :
( ( ? [X: A] : ( P @ ( times_times @ A @ L2 @ X ) ) )
= ( ? [X: A] :
( ( dvd_dvd @ A @ L2 @ ( plus_plus @ A @ X @ ( zero_zero @ A ) ) )
& ( P @ X ) ) ) ) ) ).
% unity_coeff_ex
thf(fact_1443_inf__period_I4_J,axiom,
! [A: $tType] :
( ( ( comm_ring @ A )
& ( dvd @ A ) )
=> ! [D2: A,D4: A,T2: A] :
( ( dvd_dvd @ A @ D2 @ D4 )
=> ! [X4: A,K5: A] :
( ( ~ ( dvd_dvd @ A @ D2 @ ( plus_plus @ A @ X4 @ T2 ) ) )
= ( ~ ( dvd_dvd @ A @ D2 @ ( plus_plus @ A @ ( minus_minus @ A @ X4 @ ( times_times @ A @ K5 @ D4 ) ) @ T2 ) ) ) ) ) ) ).
% inf_period(4)
thf(fact_1444_inf__period_I3_J,axiom,
! [A: $tType] :
( ( ( comm_ring @ A )
& ( dvd @ A ) )
=> ! [D2: A,D4: A,T2: A] :
( ( dvd_dvd @ A @ D2 @ D4 )
=> ! [X4: A,K5: A] :
( ( dvd_dvd @ A @ D2 @ ( plus_plus @ A @ X4 @ T2 ) )
= ( dvd_dvd @ A @ D2 @ ( plus_plus @ A @ ( minus_minus @ A @ X4 @ ( times_times @ A @ K5 @ D4 ) ) @ T2 ) ) ) ) ) ).
% inf_period(3)
thf(fact_1445_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 )
=> ! [X4: int] :
( ( P @ X4 )
=> ( P @ ( plus_plus @ int @ X4 @ ( times_times @ int @ K @ D2 ) ) ) ) ) ) ) ).
% incr_mult_lemma
thf(fact_1446_delete__bound__height,axiom,
! [T2: vEBT_VEBT,N: nat,X2: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ord_less_eq @ nat @ ( vEBT_T_d_e_l_e_t_e @ T2 @ X2 ) @ ( times_times @ nat @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_VEBT_height @ T2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% delete_bound_height
thf(fact_1447_delete__bound__height_H,axiom,
! [T2: vEBT_VEBT,N: nat,X2: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ord_less_eq @ nat @ ( vEBT_V1232361888498592333_e_t_e @ T2 @ X2 ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_VEBT_height @ T2 ) ) ) ) ).
% delete_bound_height'
thf(fact_1448_post__member__pre__member,axiom,
! [T2: vEBT_VEBT,N: nat,X2: nat,Y2: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( ord_less @ nat @ X2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
=> ( ( ord_less @ nat @ Y2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
=> ( ( vEBT_vebt_member @ ( vEBT_vebt_insert @ T2 @ X2 ) @ Y2 )
=> ( ( vEBT_vebt_member @ T2 @ Y2 )
| ( X2 = Y2 ) ) ) ) ) ) ).
% post_member_pre_member
thf(fact_1449_height__double__log__univ__size,axiom,
! [U: real,Deg: nat,T2: vEBT_VEBT] :
( ( U
= ( power_power @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ Deg ) )
=> ( ( vEBT_invar_vebt @ T2 @ Deg )
=> ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ ( vEBT_VEBT_height @ T2 ) ) @ ( plus_plus @ real @ ( one_one @ real ) @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ U ) ) ) ) ) ) ).
% height_double_log_univ_size
thf(fact_1450_set__vebt_H__def,axiom,
( vEBT_VEBT_set_vebt
= ( ^ [T: vEBT_VEBT] : ( collect @ nat @ ( vEBT_vebt_member @ T ) ) ) ) ).
% set_vebt'_def
thf(fact_1451_member__valid__both__member__options,axiom,
! [Tree: vEBT_VEBT,N: nat,X2: nat] :
( ( vEBT_invar_vebt @ Tree @ N )
=> ( ( vEBT_vebt_member @ Tree @ X2 )
=> ( ( vEBT_V5719532721284313246member @ Tree @ X2 )
| ( vEBT_VEBT_membermima @ Tree @ X2 ) ) ) ) ).
% member_valid_both_member_options
thf(fact_1452_psubsetI,axiom,
! [A: $tType,A3: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B5 )
=> ( ( A3 != B5 )
=> ( ord_less @ ( set @ A ) @ A3 @ B5 ) ) ) ).
% psubsetI
thf(fact_1453_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_1454_buildup__nothing__in__min__max,axiom,
! [N: nat,X2: nat] :
~ ( vEBT_VEBT_membermima @ ( vEBT_vebt_buildup @ N ) @ X2 ) ).
% buildup_nothing_in_min_max
thf(fact_1455_set__vebt__set__vebt_H__valid,axiom,
! [T2: vEBT_VEBT,N: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( vEBT_set_vebt @ T2 )
= ( vEBT_VEBT_set_vebt @ T2 ) ) ) ).
% set_vebt_set_vebt'_valid
thf(fact_1456_verit__eq__simplify_I8_J,axiom,
! [X23: num,Y22: num] :
( ( ( bit0 @ X23 )
= ( bit0 @ Y22 ) )
= ( X23 = Y22 ) ) ).
% verit_eq_simplify(8)
thf(fact_1457_verit__eq__simplify_I9_J,axiom,
! [X33: num,Y32: num] :
( ( ( bit1 @ X33 )
= ( bit1 @ Y32 ) )
= ( X33 = Y32 ) ) ).
% verit_eq_simplify(9)
thf(fact_1458_subset__antisym,axiom,
! [A: $tType,A3: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B5 )
=> ( ( ord_less_eq @ ( set @ A ) @ B5 @ A3 )
=> ( A3 = B5 ) ) ) ).
% subset_antisym
thf(fact_1459_subsetI,axiom,
! [A: $tType,A3: set @ A,B5: set @ A] :
( ! [X3: A] :
( ( member @ A @ X3 @ A3 )
=> ( member @ A @ X3 @ B5 ) )
=> ( ord_less_eq @ ( set @ A ) @ A3 @ B5 ) ) ).
% subsetI
thf(fact_1460_delete__bound__size__univ_H,axiom,
! [T2: vEBT_VEBT,N: nat,U: real,X2: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( U
= ( power_power @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ N ) )
=> ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ ( vEBT_V1232361888498592333_e_t_e @ T2 @ X2 ) ) @ ( plus_plus @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ U ) ) ) ) ) ) ).
% delete_bound_size_univ'
thf(fact_1461_delete__bound__size__univ,axiom,
! [T2: vEBT_VEBT,N: nat,U: real,X2: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( U
= ( power_power @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ N ) )
=> ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ ( vEBT_T_d_e_l_e_t_e @ T2 @ X2 ) ) @ ( plus_plus @ real @ ( numeral_numeral @ real @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ U ) ) ) ) ) ) ) ).
% delete_bound_size_univ
thf(fact_1462_succ__member,axiom,
! [T2: vEBT_VEBT,X2: nat,Y2: nat] :
( ( vEBT_is_succ_in_set @ ( vEBT_VEBT_set_vebt @ T2 ) @ X2 @ Y2 )
= ( ( vEBT_vebt_member @ T2 @ Y2 )
& ( ord_less @ nat @ X2 @ Y2 )
& ! [Z5: nat] :
( ( ( vEBT_vebt_member @ T2 @ Z5 )
& ( ord_less @ nat @ X2 @ Z5 ) )
=> ( ord_less_eq @ nat @ Y2 @ Z5 ) ) ) ) ).
% succ_member
thf(fact_1463_pred__member,axiom,
! [T2: vEBT_VEBT,X2: nat,Y2: nat] :
( ( vEBT_is_pred_in_set @ ( vEBT_VEBT_set_vebt @ T2 ) @ X2 @ Y2 )
= ( ( vEBT_vebt_member @ T2 @ Y2 )
& ( ord_less @ nat @ Y2 @ X2 )
& ! [Z5: nat] :
( ( ( vEBT_vebt_member @ T2 @ Z5 )
& ( ord_less @ nat @ Z5 @ X2 ) )
=> ( ord_less_eq @ nat @ Z5 @ Y2 ) ) ) ) ).
% pred_member
thf(fact_1464_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_1465_verit__comp__simplify1_I2_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A] : ( ord_less_eq @ A @ A2 @ A2 ) ) ).
% verit_comp_simplify1(2)
thf(fact_1466_double__diff,axiom,
! [A: $tType,A3: set @ A,B5: set @ A,C6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B5 )
=> ( ( ord_less_eq @ ( set @ A ) @ B5 @ C6 )
=> ( ( minus_minus @ ( set @ A ) @ B5 @ ( minus_minus @ ( set @ A ) @ C6 @ A3 ) )
= A3 ) ) ) ).
% double_diff
thf(fact_1467_Diff__subset,axiom,
! [A: $tType,A3: set @ A,B5: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ B5 ) @ A3 ) ).
% Diff_subset
thf(fact_1468_Diff__mono,axiom,
! [A: $tType,A3: set @ A,C6: set @ A,D4: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ C6 )
=> ( ( ord_less_eq @ ( set @ A ) @ D4 @ B5 )
=> ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ B5 ) @ ( minus_minus @ ( set @ A ) @ C6 @ D4 ) ) ) ) ).
% Diff_mono
thf(fact_1469_Collect__mono__iff,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) )
= ( ! [X: A] :
( ( P @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_1470_set__eq__subset,axiom,
! [A: $tType] :
( ( ^ [Y5: set @ A,Z3: set @ A] : Y5 = Z3 )
= ( ^ [A7: set @ A,B8: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A7 @ B8 )
& ( ord_less_eq @ ( set @ A ) @ B8 @ A7 ) ) ) ) ).
% set_eq_subset
thf(fact_1471_subset__trans,axiom,
! [A: $tType,A3: set @ A,B5: set @ A,C6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B5 )
=> ( ( ord_less_eq @ ( set @ A ) @ B5 @ C6 )
=> ( ord_less_eq @ ( set @ A ) @ A3 @ C6 ) ) ) ).
% subset_trans
thf(fact_1472_Collect__mono,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X3: A] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% Collect_mono
thf(fact_1473_subset__refl,axiom,
! [A: $tType,A3: set @ A] : ( ord_less_eq @ ( set @ A ) @ A3 @ A3 ) ).
% subset_refl
thf(fact_1474_subset__iff,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A7: set @ A,B8: set @ A] :
! [T: A] :
( ( member @ A @ T @ A7 )
=> ( member @ A @ T @ B8 ) ) ) ) ).
% subset_iff
thf(fact_1475_Set_OequalityD2,axiom,
! [A: $tType,A3: set @ A,B5: set @ A] :
( ( A3 = B5 )
=> ( ord_less_eq @ ( set @ A ) @ B5 @ A3 ) ) ).
% Set.equalityD2
thf(fact_1476_equalityD1,axiom,
! [A: $tType,A3: set @ A,B5: set @ A] :
( ( A3 = B5 )
=> ( ord_less_eq @ ( set @ A ) @ A3 @ B5 ) ) ).
% equalityD1
thf(fact_1477_subset__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A7: set @ A,B8: set @ A] :
! [X: A] :
( ( member @ A @ X @ A7 )
=> ( member @ A @ X @ B8 ) ) ) ) ).
% subset_eq
thf(fact_1478_equalityE,axiom,
! [A: $tType,A3: set @ A,B5: set @ A] :
( ( A3 = B5 )
=> ~ ( ( ord_less_eq @ ( set @ A ) @ A3 @ B5 )
=> ~ ( ord_less_eq @ ( set @ A ) @ B5 @ A3 ) ) ) ).
% equalityE
thf(fact_1479_subsetD,axiom,
! [A: $tType,A3: set @ A,B5: set @ A,C2: A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B5 )
=> ( ( member @ A @ C2 @ A3 )
=> ( member @ A @ C2 @ B5 ) ) ) ).
% subsetD
thf(fact_1480_in__mono,axiom,
! [A: $tType,A3: set @ A,B5: set @ A,X2: A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B5 )
=> ( ( member @ A @ X2 @ A3 )
=> ( member @ A @ X2 @ B5 ) ) ) ).
% in_mono
thf(fact_1481_nat__int__comparison_I1_J,axiom,
( ( ^ [Y5: nat,Z3: nat] : Y5 = Z3 )
= ( ^ [A4: nat,B3: nat] :
( ( semiring_1_of_nat @ int @ A4 )
= ( semiring_1_of_nat @ int @ B3 ) ) ) ) ).
% nat_int_comparison(1)
thf(fact_1482_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_1483_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_1484_less__eq__set__def,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A7: set @ A,B8: set @ A] :
( ord_less_eq @ ( A > $o )
@ ^ [X: A] : ( member @ A @ X @ A7 )
@ ^ [X: A] : ( member @ A @ X @ B8 ) ) ) ) ).
% less_eq_set_def
thf(fact_1485_Collect__subset,axiom,
! [A: $tType,A3: set @ A,P: A > $o] :
( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [X: A] :
( ( member @ A @ X @ A3 )
& ( P @ X ) ) )
@ A3 ) ).
% Collect_subset
thf(fact_1486_less__set__def,axiom,
! [A: $tType] :
( ( ord_less @ ( set @ A ) )
= ( ^ [A7: set @ A,B8: set @ A] :
( ord_less @ ( A > $o )
@ ^ [X: A] : ( member @ A @ X @ A7 )
@ ^ [X: A] : ( member @ A @ X @ B8 ) ) ) ) ).
% less_set_def
thf(fact_1487_verit__comp__simplify1_I3_J,axiom,
! [B: $tType] :
( ( linorder @ B )
=> ! [B7: B,A6: B] :
( ( ~ ( ord_less_eq @ B @ B7 @ A6 ) )
= ( ord_less @ B @ A6 @ B7 ) ) ) ).
% verit_comp_simplify1(3)
thf(fact_1488_verit__sum__simplify,axiom,
! [A: $tType] :
( ( cancel1802427076303600483id_add @ A )
=> ! [A2: A] :
( ( plus_plus @ A @ A2 @ ( zero_zero @ A ) )
= A2 ) ) ).
% verit_sum_simplify
thf(fact_1489_verit__eq__simplify_I10_J,axiom,
! [X23: num] :
( one2
!= ( bit0 @ X23 ) ) ).
% verit_eq_simplify(10)
thf(fact_1490_verit__eq__simplify_I14_J,axiom,
! [X23: num,X33: num] :
( ( bit0 @ X23 )
!= ( bit1 @ X33 ) ) ).
% verit_eq_simplify(14)
thf(fact_1491_verit__eq__simplify_I12_J,axiom,
! [X33: num] :
( one2
!= ( bit1 @ X33 ) ) ).
% verit_eq_simplify(12)
thf(fact_1492_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_1493_subset__psubset__trans,axiom,
! [A: $tType,A3: set @ A,B5: set @ A,C6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B5 )
=> ( ( ord_less @ ( set @ A ) @ B5 @ C6 )
=> ( ord_less @ ( set @ A ) @ A3 @ C6 ) ) ) ).
% subset_psubset_trans
thf(fact_1494_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_1495_psubset__subset__trans,axiom,
! [A: $tType,A3: set @ A,B5: set @ A,C6: set @ A] :
( ( ord_less @ ( set @ A ) @ A3 @ B5 )
=> ( ( ord_less_eq @ ( set @ A ) @ B5 @ C6 )
=> ( ord_less @ ( set @ A ) @ A3 @ C6 ) ) ) ).
% psubset_subset_trans
thf(fact_1496_psubset__imp__subset,axiom,
! [A: $tType,A3: set @ A,B5: set @ A] :
( ( ord_less @ ( set @ A ) @ A3 @ B5 )
=> ( ord_less_eq @ ( set @ A ) @ A3 @ B5 ) ) ).
% psubset_imp_subset
thf(fact_1497_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_1498_psubsetE,axiom,
! [A: $tType,A3: set @ A,B5: set @ A] :
( ( ord_less @ ( set @ A ) @ A3 @ B5 )
=> ~ ( ( ord_less_eq @ ( set @ A ) @ A3 @ B5 )
=> ( ord_less_eq @ ( set @ A ) @ B5 @ A3 ) ) ) ).
% psubsetE
thf(fact_1499_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_1500_int__ops_I1_J,axiom,
( ( semiring_1_of_nat @ int @ ( zero_zero @ nat ) )
= ( zero_zero @ int ) ) ).
% int_ops(1)
thf(fact_1501_nat__int__comparison_I2_J,axiom,
( ( ord_less @ nat )
= ( ^ [A4: nat,B3: nat] : ( ord_less @ int @ ( semiring_1_of_nat @ int @ A4 ) @ ( semiring_1_of_nat @ int @ B3 ) ) ) ) ).
% nat_int_comparison(2)
thf(fact_1502_nat__int__comparison_I3_J,axiom,
( ( ord_less_eq @ nat )
= ( ^ [A4: nat,B3: nat] : ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ A4 ) @ ( semiring_1_of_nat @ int @ B3 ) ) ) ) ).
% nat_int_comparison(3)
thf(fact_1503_int__ops_I2_J,axiom,
( ( semiring_1_of_nat @ int @ ( one_one @ nat ) )
= ( one_one @ int ) ) ).
% int_ops(2)
thf(fact_1504_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_1505_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_1506_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_1507_nat__less__as__int,axiom,
( ( ord_less @ nat )
= ( ^ [A4: nat,B3: nat] : ( ord_less @ int @ ( semiring_1_of_nat @ int @ A4 ) @ ( semiring_1_of_nat @ int @ B3 ) ) ) ) ).
% nat_less_as_int
thf(fact_1508_nat__leq__as__int,axiom,
( ( ord_less_eq @ nat )
= ( ^ [A4: nat,B3: nat] : ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ A4 ) @ ( semiring_1_of_nat @ int @ B3 ) ) ) ) ).
% nat_leq_as_int
thf(fact_1509_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_1510_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_1511_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_1512_zero__le__log__cancel__iff,axiom,
! [A2: real,X2: real] :
( ( ord_less @ real @ ( one_one @ real ) @ A2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( log @ A2 @ X2 ) )
= ( ord_less_eq @ real @ ( one_one @ real ) @ X2 ) ) ) ) ).
% zero_le_log_cancel_iff
thf(fact_1513_log__le__zero__cancel__iff,axiom,
! [A2: real,X2: real] :
( ( ord_less @ real @ ( one_one @ real ) @ A2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ord_less_eq @ real @ ( log @ A2 @ X2 ) @ ( zero_zero @ real ) )
= ( ord_less_eq @ real @ X2 @ ( one_one @ real ) ) ) ) ) ).
% log_le_zero_cancel_iff
thf(fact_1514_one__le__log__cancel__iff,axiom,
! [A2: real,X2: real] :
( ( ord_less @ real @ ( one_one @ real ) @ A2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ord_less_eq @ real @ ( one_one @ real ) @ ( log @ A2 @ X2 ) )
= ( ord_less_eq @ real @ A2 @ X2 ) ) ) ) ).
% one_le_log_cancel_iff
thf(fact_1515_log__le__one__cancel__iff,axiom,
! [A2: real,X2: real] :
( ( ord_less @ real @ ( one_one @ real ) @ A2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ord_less_eq @ real @ ( log @ A2 @ X2 ) @ ( one_one @ real ) )
= ( ord_less_eq @ real @ X2 @ A2 ) ) ) ) ).
% log_le_one_cancel_iff
thf(fact_1516_log__le__cancel__iff,axiom,
! [A2: real,X2: real,Y2: real] :
( ( ord_less @ real @ ( one_one @ real ) @ A2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ Y2 )
=> ( ( ord_less_eq @ real @ ( log @ A2 @ X2 ) @ ( log @ A2 @ Y2 ) )
= ( ord_less_eq @ real @ X2 @ Y2 ) ) ) ) ) ).
% log_le_cancel_iff
thf(fact_1517_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_1518_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_1519_log__one,axiom,
! [A2: real] :
( ( log @ A2 @ ( one_one @ real ) )
= ( zero_zero @ real ) ) ).
% log_one
thf(fact_1520_zero__less__log__cancel__iff,axiom,
! [A2: real,X2: real] :
( ( ord_less @ real @ ( one_one @ real ) @ A2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( log @ A2 @ X2 ) )
= ( ord_less @ real @ ( one_one @ real ) @ X2 ) ) ) ) ).
% zero_less_log_cancel_iff
thf(fact_1521_log__less__zero__cancel__iff,axiom,
! [A2: real,X2: real] :
( ( ord_less @ real @ ( one_one @ real ) @ A2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ord_less @ real @ ( log @ A2 @ X2 ) @ ( zero_zero @ real ) )
= ( ord_less @ real @ X2 @ ( one_one @ real ) ) ) ) ) ).
% log_less_zero_cancel_iff
thf(fact_1522_one__less__log__cancel__iff,axiom,
! [A2: real,X2: real] :
( ( ord_less @ real @ ( one_one @ real ) @ A2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ord_less @ real @ ( one_one @ real ) @ ( log @ A2 @ X2 ) )
= ( ord_less @ real @ A2 @ X2 ) ) ) ) ).
% one_less_log_cancel_iff
thf(fact_1523_log__less__one__cancel__iff,axiom,
! [A2: real,X2: real] :
( ( ord_less @ real @ ( one_one @ real ) @ A2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ord_less @ real @ ( log @ A2 @ X2 ) @ ( one_one @ real ) )
= ( ord_less @ real @ X2 @ A2 ) ) ) ) ).
% log_less_one_cancel_iff
thf(fact_1524_log__less__cancel__iff,axiom,
! [A2: real,X2: real,Y2: real] :
( ( ord_less @ real @ ( one_one @ real ) @ A2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ Y2 )
=> ( ( ord_less @ real @ ( log @ A2 @ X2 ) @ ( log @ A2 @ Y2 ) )
= ( ord_less @ real @ X2 @ Y2 ) ) ) ) ) ).
% log_less_cancel_iff
thf(fact_1525_minus__set__def,axiom,
! [A: $tType] :
( ( minus_minus @ ( set @ A ) )
= ( ^ [A7: set @ A,B8: set @ A] :
( collect @ A
@ ( minus_minus @ ( A > $o )
@ ^ [X: A] : ( member @ A @ X @ A7 )
@ ^ [X: A] : ( member @ A @ X @ B8 ) ) ) ) ) ).
% minus_set_def
thf(fact_1526_set__diff__eq,axiom,
! [A: $tType] :
( ( minus_minus @ ( set @ A ) )
= ( ^ [A7: set @ A,B8: set @ A] :
( collect @ A
@ ^ [X: A] :
( ( member @ A @ X @ A7 )
& ~ ( member @ A @ X @ B8 ) ) ) ) ) ).
% set_diff_eq
thf(fact_1527_log__base__change,axiom,
! [A2: real,B2: real,X2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
=> ( ( A2
!= ( one_one @ real ) )
=> ( ( log @ B2 @ X2 )
= ( divide_divide @ real @ ( log @ A2 @ X2 ) @ ( log @ A2 @ B2 ) ) ) ) ) ).
% log_base_change
thf(fact_1528_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_1529_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_1530_log__mult,axiom,
! [A2: real,X2: real,Y2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
=> ( ( A2
!= ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ Y2 )
=> ( ( log @ A2 @ ( times_times @ real @ X2 @ Y2 ) )
= ( plus_plus @ real @ ( log @ A2 @ X2 ) @ ( log @ A2 @ Y2 ) ) ) ) ) ) ) ).
% log_mult
thf(fact_1531_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_1532_log__divide,axiom,
! [A2: real,X2: real,Y2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
=> ( ( A2
!= ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ Y2 )
=> ( ( log @ A2 @ ( divide_divide @ real @ X2 @ Y2 ) )
= ( minus_minus @ real @ ( log @ A2 @ X2 ) @ ( log @ A2 @ Y2 ) ) ) ) ) ) ) ).
% log_divide
thf(fact_1533_log__base__pow,axiom,
! [A2: real,N: nat,X2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
=> ( ( log @ ( power_power @ real @ A2 @ N ) @ X2 )
= ( divide_divide @ real @ ( log @ A2 @ X2 ) @ ( semiring_1_of_nat @ real @ N ) ) ) ) ).
% log_base_pow
thf(fact_1534_log__nat__power,axiom,
! [X2: real,B2: real,N: nat] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( log @ B2 @ ( power_power @ real @ X2 @ N ) )
= ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ ( log @ B2 @ X2 ) ) ) ) ).
% log_nat_power
thf(fact_1535_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_1536_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_1537_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_1538_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_1539_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_1540_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_1541_arcosh__1,axiom,
! [A: $tType] :
( ( ln @ A )
=> ( ( arcosh @ A @ ( one_one @ A ) )
= ( zero_zero @ A ) ) ) ).
% arcosh_1
thf(fact_1542_inrange,axiom,
! [T2: vEBT_VEBT,N: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ord_less_eq @ ( set @ nat ) @ ( vEBT_VEBT_set_vebt @ T2 ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ ( one_one @ nat ) ) ) ) ) ).
% inrange
thf(fact_1543_pos__mult__pos__ge,axiom,
! [X2: int,N: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ X2 )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ N )
=> ( ord_less_eq @ int @ ( times_times @ int @ N @ ( one_one @ int ) ) @ ( times_times @ int @ N @ X2 ) ) ) ) ).
% pos_mult_pos_ge
thf(fact_1544_heigt__uplog__rel,axiom,
! [T2: vEBT_VEBT,N: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( semiring_1_of_nat @ int @ ( vEBT_VEBT_height @ T2 ) )
= ( archimedean_ceiling @ real @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ N ) ) ) ) ) ).
% heigt_uplog_rel
thf(fact_1545_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_1546_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_1547_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_1548_valid__insert__both__member__options__add,axiom,
! [T2: vEBT_VEBT,N: nat,X2: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( ord_less @ nat @ X2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
=> ( vEBT_V8194947554948674370ptions @ ( vEBT_vebt_insert @ T2 @ X2 ) @ X2 ) ) ) ).
% valid_insert_both_member_options_add
thf(fact_1549_both__member__options__equiv__member,axiom,
! [T2: vEBT_VEBT,N: nat,X2: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( vEBT_V8194947554948674370ptions @ T2 @ X2 )
= ( vEBT_vebt_member @ T2 @ X2 ) ) ) ).
% both_member_options_equiv_member
thf(fact_1550_valid__member__both__member__options,axiom,
! [T2: vEBT_VEBT,N: nat,X2: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( vEBT_V8194947554948674370ptions @ T2 @ X2 )
=> ( vEBT_vebt_member @ T2 @ X2 ) ) ) ).
% valid_member_both_member_options
thf(fact_1551_both__member__options__def,axiom,
( vEBT_V8194947554948674370ptions
= ( ^ [T: vEBT_VEBT,X: nat] :
( ( vEBT_V5719532721284313246member @ T @ X )
| ( vEBT_VEBT_membermima @ T @ X ) ) ) ) ).
% both_member_options_def
thf(fact_1552_valid__insert__both__member__options__pres,axiom,
! [T2: vEBT_VEBT,N: nat,X2: nat,Y2: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( ord_less @ nat @ X2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
=> ( ( ord_less @ nat @ Y2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
=> ( ( vEBT_V8194947554948674370ptions @ T2 @ X2 )
=> ( vEBT_V8194947554948674370ptions @ ( vEBT_vebt_insert @ T2 @ Y2 ) @ X2 ) ) ) ) ) ).
% valid_insert_both_member_options_pres
thf(fact_1553_ceiling__zero,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ( ( archimedean_ceiling @ A @ ( zero_zero @ A ) )
= ( zero_zero @ int ) ) ) ).
% ceiling_zero
thf(fact_1554_ceiling__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V: num] :
( ( archimedean_ceiling @ A @ ( numeral_numeral @ A @ V ) )
= ( numeral_numeral @ int @ V ) ) ) ).
% ceiling_numeral
thf(fact_1555_ceiling__one,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ( ( archimedean_ceiling @ A @ ( one_one @ A ) )
= ( one_one @ int ) ) ) ).
% ceiling_one
thf(fact_1556_ceiling__of__nat,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [N: nat] :
( ( archimedean_ceiling @ A @ ( semiring_1_of_nat @ A @ N ) )
= ( semiring_1_of_nat @ int @ N ) ) ) ).
% ceiling_of_nat
thf(fact_1557_ceiling__le__zero,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A] :
( ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ X2 ) @ ( zero_zero @ int ) )
= ( ord_less_eq @ A @ X2 @ ( zero_zero @ A ) ) ) ) ).
% ceiling_le_zero
thf(fact_1558_ceiling__le__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A,V: num] :
( ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ X2 ) @ ( numeral_numeral @ int @ V ) )
= ( ord_less_eq @ A @ X2 @ ( numeral_numeral @ A @ V ) ) ) ) ).
% ceiling_le_numeral
thf(fact_1559_zero__less__ceiling,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A] :
( ( ord_less @ int @ ( zero_zero @ int ) @ ( archimedean_ceiling @ A @ X2 ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ X2 ) ) ) ).
% zero_less_ceiling
thf(fact_1560_numeral__less__ceiling,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V: num,X2: A] :
( ( ord_less @ int @ ( numeral_numeral @ int @ V ) @ ( archimedean_ceiling @ A @ X2 ) )
= ( ord_less @ A @ ( numeral_numeral @ A @ V ) @ X2 ) ) ) ).
% numeral_less_ceiling
thf(fact_1561_ceiling__less__one,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A] :
( ( ord_less @ int @ ( archimedean_ceiling @ A @ X2 ) @ ( one_one @ int ) )
= ( ord_less_eq @ A @ X2 @ ( zero_zero @ A ) ) ) ) ).
% ceiling_less_one
thf(fact_1562_one__le__ceiling,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A] :
( ( ord_less_eq @ int @ ( one_one @ int ) @ ( archimedean_ceiling @ A @ X2 ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ X2 ) ) ) ).
% one_le_ceiling
thf(fact_1563_ceiling__add__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A,V: num] :
( ( archimedean_ceiling @ A @ ( plus_plus @ A @ X2 @ ( numeral_numeral @ A @ V ) ) )
= ( plus_plus @ int @ ( archimedean_ceiling @ A @ X2 ) @ ( numeral_numeral @ int @ V ) ) ) ) ).
% ceiling_add_numeral
thf(fact_1564_ceiling__le__one,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A] :
( ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ X2 ) @ ( one_one @ int ) )
= ( ord_less_eq @ A @ X2 @ ( one_one @ A ) ) ) ) ).
% ceiling_le_one
thf(fact_1565_one__less__ceiling,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A] :
( ( ord_less @ int @ ( one_one @ int ) @ ( archimedean_ceiling @ A @ X2 ) )
= ( ord_less @ A @ ( one_one @ A ) @ X2 ) ) ) ).
% one_less_ceiling
thf(fact_1566_ceiling__add__one,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A] :
( ( archimedean_ceiling @ A @ ( plus_plus @ A @ X2 @ ( one_one @ A ) ) )
= ( plus_plus @ int @ ( archimedean_ceiling @ A @ X2 ) @ ( one_one @ int ) ) ) ) ).
% ceiling_add_one
thf(fact_1567_ceiling__diff__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A,V: num] :
( ( archimedean_ceiling @ A @ ( minus_minus @ A @ X2 @ ( numeral_numeral @ A @ V ) ) )
= ( minus_minus @ int @ ( archimedean_ceiling @ A @ X2 ) @ ( numeral_numeral @ int @ V ) ) ) ) ).
% ceiling_diff_numeral
thf(fact_1568_ceiling__numeral__power,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: num,N: nat] :
( ( archimedean_ceiling @ A @ ( power_power @ A @ ( numeral_numeral @ A @ X2 ) @ N ) )
= ( power_power @ int @ ( numeral_numeral @ int @ X2 ) @ N ) ) ) ).
% ceiling_numeral_power
thf(fact_1569_ceiling__diff__one,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A] :
( ( archimedean_ceiling @ A @ ( minus_minus @ A @ X2 @ ( one_one @ A ) ) )
= ( minus_minus @ int @ ( archimedean_ceiling @ A @ X2 ) @ ( one_one @ int ) ) ) ) ).
% ceiling_diff_one
thf(fact_1570_ceiling__less__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A,V: num] :
( ( ord_less @ int @ ( archimedean_ceiling @ A @ X2 ) @ ( numeral_numeral @ int @ V ) )
= ( ord_less_eq @ A @ X2 @ ( minus_minus @ A @ ( numeral_numeral @ A @ V ) @ ( one_one @ A ) ) ) ) ) ).
% ceiling_less_numeral
thf(fact_1571_numeral__le__ceiling,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V: num,X2: A] :
( ( ord_less_eq @ int @ ( numeral_numeral @ int @ V ) @ ( archimedean_ceiling @ A @ X2 ) )
= ( ord_less @ A @ ( minus_minus @ A @ ( numeral_numeral @ A @ V ) @ ( one_one @ A ) ) @ X2 ) ) ) ).
% numeral_le_ceiling
thf(fact_1572_set__vebt__def,axiom,
( vEBT_set_vebt
= ( ^ [T: vEBT_VEBT] : ( collect @ nat @ ( vEBT_V8194947554948674370ptions @ T ) ) ) ) ).
% set_vebt_def
thf(fact_1573_ceiling__mono,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Y2: A,X2: A] :
( ( ord_less_eq @ A @ Y2 @ X2 )
=> ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ Y2 ) @ ( archimedean_ceiling @ A @ X2 ) ) ) ) ).
% ceiling_mono
thf(fact_1574_all__nat__less,axiom,
! [N: nat,P: nat > $o] :
( ( ! [M5: nat] :
( ( ord_less_eq @ nat @ M5 @ N )
=> ( P @ M5 ) ) )
= ( ! [X: nat] :
( ( member @ nat @ X @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
=> ( P @ X ) ) ) ) ).
% all_nat_less
thf(fact_1575_ex__nat__less,axiom,
! [N: nat,P: nat > $o] :
( ( ? [M5: nat] :
( ( ord_less_eq @ nat @ M5 @ N )
& ( P @ M5 ) ) )
= ( ? [X: nat] :
( ( member @ nat @ X @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
& ( P @ X ) ) ) ) ).
% ex_nat_less
thf(fact_1576_ceiling__add__le,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A,Y2: A] : ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ ( plus_plus @ A @ X2 @ Y2 ) ) @ ( plus_plus @ int @ ( archimedean_ceiling @ A @ X2 ) @ ( archimedean_ceiling @ A @ Y2 ) ) ) ) ).
% ceiling_add_le
thf(fact_1577_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_1578_lemma__NBseq__def2,axiom,
! [A: $tType,B: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [X6: A > B] :
( ( ? [K7: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K7 )
& ! [N3: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( X6 @ N3 ) ) @ K7 ) ) )
= ( ? [N8: nat] :
! [N3: A] : ( ord_less @ real @ ( real_V7770717601297561774m_norm @ B @ ( X6 @ N3 ) ) @ ( semiring_1_of_nat @ real @ ( suc @ N8 ) ) ) ) ) ) ).
% lemma_NBseq_def2
thf(fact_1579_lemma__NBseq__def,axiom,
! [A: $tType,B: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [X6: A > B] :
( ( ? [K7: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K7 )
& ! [N3: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( X6 @ N3 ) ) @ K7 ) ) )
= ( ? [N8: nat] :
! [N3: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( X6 @ N3 ) ) @ ( semiring_1_of_nat @ real @ ( suc @ N8 ) ) ) ) ) ) ).
% lemma_NBseq_def
thf(fact_1580_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_1581_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_1582_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_1583_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_1584_word__less__two__pow__divD,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A,N: nat,M: nat] :
( ( ord_less @ ( word @ A ) @ X2 @ ( divide_divide @ ( word @ A ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ M ) ) )
=> ( ( ord_less_eq @ nat @ M @ N )
& ( ord_less @ ( word @ A ) @ X2 @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ N @ M ) ) ) ) ) ) ).
% word_less_two_pow_divD
thf(fact_1585_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_1586_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_1587_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_1588_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_1589_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_1590_norm__le__zero__iff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X2: A] :
( ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ X2 ) @ ( zero_zero @ real ) )
= ( X2
= ( zero_zero @ A ) ) ) ) ).
% norm_le_zero_iff
thf(fact_1591_zero__less__norm__iff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X2: A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( real_V7770717601297561774m_norm @ A @ X2 ) )
= ( X2
!= ( zero_zero @ A ) ) ) ) ).
% zero_less_norm_iff
thf(fact_1592_norm__of__nat,axiom,
! [A: $tType] :
( ( real_V2822296259951069270ebra_1 @ A )
=> ! [N: nat] :
( ( real_V7770717601297561774m_norm @ A @ ( semiring_1_of_nat @ A @ N ) )
= ( semiring_1_of_nat @ real @ N ) ) ) ).
% norm_of_nat
thf(fact_1593_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_1594_norm__one,axiom,
! [A: $tType] :
( ( real_V2822296259951069270ebra_1 @ A )
=> ( ( real_V7770717601297561774m_norm @ A @ ( one_one @ A ) )
= ( one_one @ real ) ) ) ).
% norm_one
thf(fact_1595_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_1596_norm__eq__zero,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X2: A] :
( ( ( real_V7770717601297561774m_norm @ A @ X2 )
= ( zero_zero @ real ) )
= ( X2
= ( zero_zero @ A ) ) ) ) ).
% norm_eq_zero
thf(fact_1597_norm__zero,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ( ( real_V7770717601297561774m_norm @ A @ ( zero_zero @ A ) )
= ( zero_zero @ real ) ) ) ).
% norm_zero
thf(fact_1598_div__to__mult__word__lt,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A,Y2: word @ A,Z: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ X2 @ ( divide_divide @ ( word @ A ) @ Y2 @ Z ) )
=> ( ord_less_eq @ ( word @ A ) @ ( times_times @ ( word @ A ) @ X2 @ Z ) @ Y2 ) ) ) ).
% div_to_mult_word_lt
thf(fact_1599_More__Word_Oword__l__diffs_I2_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A,X2: word @ A,Z: word @ A] :
( ( ord_less @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W @ X2 ) @ Z )
=> ( ( ord_less_eq @ ( word @ A ) @ X2 @ ( plus_plus @ ( word @ A ) @ W @ X2 ) )
=> ( ord_less @ ( word @ A ) @ W @ ( minus_minus @ ( word @ A ) @ Z @ X2 ) ) ) ) ) ).
% More_Word.word_l_diffs(2)
thf(fact_1600_More__Word_Oword__l__diffs_I1_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A,Z: word @ A,X2: word @ A] :
( ( ord_less @ ( word @ A ) @ W @ ( minus_minus @ ( word @ A ) @ Z @ X2 ) )
=> ( ( ord_less_eq @ ( word @ A ) @ X2 @ Z )
=> ( ord_less @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W @ X2 ) @ Z ) ) ) ) ).
% More_Word.word_l_diffs(1)
thf(fact_1601_word__diff__ls_H_I2_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y2: word @ A,W: word @ A,X2: word @ A] :
( ( ord_less @ ( word @ A ) @ Y2 @ ( plus_plus @ ( word @ A ) @ W @ X2 ) )
=> ( ( ord_less_eq @ ( word @ A ) @ X2 @ Y2 )
=> ( ord_less @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ Y2 @ X2 ) @ W ) ) ) ) ).
% word_diff_ls'(2)
thf(fact_1602_word__diff__ls_H_I1_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y2: word @ A,X2: word @ A,W: word @ A] :
( ( ord_less @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ Y2 @ X2 ) @ W )
=> ( ( ord_less_eq @ ( word @ A ) @ X2 @ ( plus_plus @ ( word @ A ) @ W @ X2 ) )
=> ( ord_less @ ( word @ A ) @ Y2 @ ( plus_plus @ ( word @ A ) @ W @ X2 ) ) ) ) ) ).
% word_diff_ls'(1)
thf(fact_1603_word__l__diffs_H_I2_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A,X2: word @ A,Z: word @ A] :
( ( ord_less @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W @ X2 ) @ Z )
=> ( ( ord_less_eq @ ( word @ A ) @ X2 @ ( plus_plus @ ( word @ A ) @ W @ X2 ) )
=> ( ord_less @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W @ X2 ) @ X2 ) @ ( minus_minus @ ( word @ A ) @ Z @ X2 ) ) ) ) ) ).
% word_l_diffs'(2)
thf(fact_1604_word__l__diffs_H_I1_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A,X2: word @ A,Z: word @ A] :
( ( ord_less @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W @ X2 ) @ X2 ) @ ( minus_minus @ ( word @ A ) @ Z @ X2 ) )
=> ( ( ord_less_eq @ ( word @ A ) @ X2 @ Z )
=> ( ord_less @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W @ X2 ) @ Z ) ) ) ) ).
% word_l_diffs'(1)
thf(fact_1605_word__diff__ls_H_H_I2_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y2: word @ A,W: word @ A,X2: word @ A] :
( ( ord_less @ ( word @ A ) @ Y2 @ ( plus_plus @ ( word @ A ) @ W @ X2 ) )
=> ( ( ord_less_eq @ ( word @ A ) @ X2 @ Y2 )
=> ( ord_less @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ Y2 @ X2 ) @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W @ X2 ) @ X2 ) ) ) ) ) ).
% word_diff_ls''(2)
thf(fact_1606_word__diff__ls_H_H_I1_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y2: word @ A,X2: word @ A,W: word @ A] :
( ( ord_less @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ Y2 @ X2 ) @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W @ X2 ) @ X2 ) )
=> ( ( ord_less_eq @ ( word @ A ) @ X2 @ ( plus_plus @ ( word @ A ) @ W @ X2 ) )
=> ( ord_less @ ( word @ A ) @ Y2 @ ( plus_plus @ ( word @ A ) @ W @ X2 ) ) ) ) ) ).
% word_diff_ls''(1)
thf(fact_1607_lt1__neq0,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ ( one_one @ ( word @ A ) ) @ X2 )
= ( X2
!= ( zero_zero @ ( word @ A ) ) ) ) ) ).
% lt1_neq0
thf(fact_1608_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_1609_le__m1__iff__lt,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A,Y2: word @ A] :
( ( ord_less @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ X2 )
= ( ( ord_less_eq @ ( word @ A ) @ Y2 @ ( minus_minus @ ( word @ A ) @ X2 @ ( one_one @ ( word @ A ) ) ) )
= ( ord_less @ ( word @ A ) @ Y2 @ X2 ) ) ) ) ).
% le_m1_iff_lt
thf(fact_1610_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_1611_word__overflow,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A] :
( ( ord_less @ ( word @ A ) @ X2 @ ( plus_plus @ ( word @ A ) @ X2 @ ( one_one @ ( word @ A ) ) ) )
| ( ( plus_plus @ ( word @ A ) @ X2 @ ( one_one @ ( word @ A ) ) )
= ( zero_zero @ ( word @ A ) ) ) ) ) ).
% word_overflow
thf(fact_1612_word__sub__1__le,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A] :
( ( X2
!= ( zero_zero @ ( word @ A ) ) )
=> ( ord_less_eq @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ X2 @ ( one_one @ ( word @ A ) ) ) @ X2 ) ) ) ).
% word_sub_1_le
thf(fact_1613_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_1614_word__le__plus__1,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y2: word @ A,N: word @ A,A2: word @ A] :
( ( ord_less @ ( word @ A ) @ Y2 @ ( plus_plus @ ( word @ A ) @ Y2 @ N ) )
=> ( ( ord_less @ ( word @ A ) @ A2 @ N )
=> ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ Y2 @ A2 ) @ ( plus_plus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ Y2 @ A2 ) @ ( one_one @ ( word @ A ) ) ) ) ) ) ) ).
% word_le_plus_1
thf(fact_1615_word__must__wrap,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A,N: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ X2 @ ( minus_minus @ ( word @ A ) @ N @ ( one_one @ ( word @ A ) ) ) )
=> ( ( ord_less_eq @ ( word @ A ) @ N @ X2 )
=> ( N
= ( zero_zero @ ( word @ A ) ) ) ) ) ) ).
% word_must_wrap
thf(fact_1616_plus__one__helper,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A,N: word @ A] :
( ( ord_less @ ( word @ A ) @ X2 @ ( plus_plus @ ( word @ A ) @ N @ ( one_one @ ( word @ A ) ) ) )
=> ( ord_less_eq @ ( word @ A ) @ X2 @ N ) ) ) ).
% plus_one_helper
thf(fact_1617_plus__one__helper2,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A,N: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ X2 @ N )
=> ( ( ( plus_plus @ ( word @ A ) @ N @ ( one_one @ ( word @ A ) ) )
!= ( zero_zero @ ( word @ A ) ) )
=> ( ord_less @ ( word @ A ) @ X2 @ ( plus_plus @ ( word @ A ) @ N @ ( one_one @ ( word @ A ) ) ) ) ) ) ) ).
% plus_one_helper2
thf(fact_1618_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_1619_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_1620_word__less__nowrapI_H,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A,Z: word @ A,K: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ X2 @ ( minus_minus @ ( word @ A ) @ Z @ K ) )
=> ( ( ord_less_eq @ ( word @ A ) @ K @ Z )
=> ( ( ord_less @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ K )
=> ( ord_less @ ( word @ A ) @ X2 @ ( plus_plus @ ( word @ A ) @ X2 @ K ) ) ) ) ) ) ).
% word_less_nowrapI'
thf(fact_1621_le__step__down__word__2,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A,Y2: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ X2 @ Y2 )
=> ( ( X2 != Y2 )
=> ( ord_less_eq @ ( word @ A ) @ X2 @ ( minus_minus @ ( word @ A ) @ Y2 @ ( one_one @ ( word @ A ) ) ) ) ) ) ) ).
% le_step_down_word_2
thf(fact_1622_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_1623_word__le__minus__one__leq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A,Y2: word @ A] :
( ( ord_less @ ( word @ A ) @ X2 @ Y2 )
=> ( ord_less_eq @ ( word @ A ) @ X2 @ ( minus_minus @ ( word @ A ) @ Y2 @ ( one_one @ ( word @ A ) ) ) ) ) ) ).
% word_le_minus_one_leq
thf(fact_1624_word__leq__le__minus__one,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A,Y2: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ X2 @ Y2 )
=> ( ( X2
!= ( zero_zero @ ( word @ A ) ) )
=> ( ord_less @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ X2 @ ( one_one @ ( word @ A ) ) ) @ Y2 ) ) ) ) ).
% word_leq_le_minus_one
thf(fact_1625_word__leq__minus__one__le,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y2: word @ A,X2: word @ A] :
( ( Y2
!= ( zero_zero @ ( word @ A ) ) )
=> ( ( ord_less_eq @ ( word @ A ) @ X2 @ ( minus_minus @ ( word @ A ) @ Y2 @ ( one_one @ ( word @ A ) ) ) )
=> ( ord_less @ ( word @ A ) @ X2 @ Y2 ) ) ) ) ).
% word_leq_minus_one_le
thf(fact_1626_word__minus__one__le__leq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A,Y2: word @ A] :
( ( ord_less @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ X2 @ ( one_one @ ( word @ A ) ) ) @ Y2 )
=> ( ord_less_eq @ ( word @ A ) @ X2 @ Y2 ) ) ) ).
% word_minus_one_le_leq
thf(fact_1627_word__plus__one__nonzero,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A,Y2: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ X2 @ ( plus_plus @ ( word @ A ) @ X2 @ Y2 ) )
=> ( ( Y2
!= ( zero_zero @ ( word @ A ) ) )
=> ( ( plus_plus @ ( word @ A ) @ X2 @ ( one_one @ ( word @ A ) ) )
!= ( zero_zero @ ( word @ A ) ) ) ) ) ) ).
% word_plus_one_nonzero
thf(fact_1628_word__sub__plus__one__nonzero,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N4: word @ A,N: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ N4 @ N )
=> ( ( N4
!= ( zero_zero @ ( word @ A ) ) )
=> ( ( plus_plus @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ N @ N4 ) @ ( one_one @ ( word @ A ) ) )
!= ( zero_zero @ ( word @ A ) ) ) ) ) ) ).
% word_sub_plus_one_nonzero
thf(fact_1629_word__plus__strict__mono__right,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y2: word @ A,Z: word @ A,X2: word @ A] :
( ( ord_less @ ( word @ A ) @ Y2 @ Z )
=> ( ( ord_less_eq @ ( word @ A ) @ X2 @ ( plus_plus @ ( word @ A ) @ X2 @ Z ) )
=> ( ord_less @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ X2 @ Y2 ) @ ( plus_plus @ ( word @ A ) @ X2 @ Z ) ) ) ) ) ).
% word_plus_strict_mono_right
thf(fact_1630_word__less__cases,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A,Y2: word @ A] :
( ( ord_less @ ( word @ A ) @ X2 @ Y2 )
=> ( ( X2
= ( minus_minus @ ( word @ A ) @ Y2 @ ( one_one @ ( word @ A ) ) ) )
| ( ord_less @ ( word @ A ) @ X2 @ ( minus_minus @ ( word @ A ) @ Y2 @ ( one_one @ ( word @ A ) ) ) ) ) ) ) ).
% word_less_cases
thf(fact_1631_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_1632_gt0__iff__gem1,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A] :
( ( ord_less @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ X2 )
= ( ord_less @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ X2 @ ( one_one @ ( word @ A ) ) ) @ X2 ) ) ) ).
% gt0_iff_gem1
thf(fact_1633_div__less__dividend__word,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A,N: word @ A] :
( ( X2
!= ( zero_zero @ ( word @ A ) ) )
=> ( ( N
!= ( one_one @ ( word @ A ) ) )
=> ( ord_less @ ( word @ A ) @ ( divide_divide @ ( word @ A ) @ X2 @ N ) @ X2 ) ) ) ) ).
% div_less_dividend_word
thf(fact_1634_word__div__lt__eq__0,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A,Y2: word @ A] :
( ( ord_less @ ( word @ A ) @ X2 @ Y2 )
=> ( ( divide_divide @ ( word @ A ) @ X2 @ Y2 )
= ( zero_zero @ ( word @ A ) ) ) ) ) ).
% word_div_lt_eq_0
thf(fact_1635_word__less__div,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A,Y2: word @ A] :
( ( ( divide_divide @ ( word @ A ) @ X2 @ Y2 )
= ( zero_zero @ ( word @ A ) ) )
=> ( ( Y2
= ( zero_zero @ ( word @ A ) ) )
| ( ord_less @ ( word @ A ) @ X2 @ Y2 ) ) ) ) ).
% word_less_div
thf(fact_1636_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_1637_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_1638_word__div__sub,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y2: word @ A,X2: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ Y2 @ X2 )
=> ( ( ord_less @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ Y2 )
=> ( ( divide_divide @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ X2 @ Y2 ) @ Y2 )
= ( minus_minus @ ( word @ A ) @ ( divide_divide @ ( word @ A ) @ X2 @ Y2 ) @ ( one_one @ ( word @ A ) ) ) ) ) ) ) ).
% word_div_sub
thf(fact_1639_word__le__add,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A,Y2: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ X2 @ Y2 )
=> ? [N2: nat] :
( Y2
= ( plus_plus @ ( word @ A ) @ X2 @ ( semiring_1_of_nat @ ( word @ A ) @ N2 ) ) ) ) ) ).
% word_le_add
thf(fact_1640_word__1__0,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: word @ A,B2: word @ A,X2: 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 ) @ X2 ) )
=> ( ord_less @ ( word @ A ) @ A2 @ B2 ) ) ) ) ).
% word_1_0
thf(fact_1641_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_1642_div__by__0__word,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A] :
( ( divide_divide @ ( word @ A ) @ X2 @ ( zero_zero @ ( word @ A ) ) )
= ( zero_zero @ ( word @ A ) ) ) ) ).
% div_by_0_word
thf(fact_1643_norm__not__less__zero,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X2: A] :
~ ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X2 ) @ ( zero_zero @ real ) ) ) ).
% norm_not_less_zero
thf(fact_1644_norm__mult,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [X2: A,Y2: A] :
( ( real_V7770717601297561774m_norm @ A @ ( times_times @ A @ X2 @ Y2 ) )
= ( times_times @ real @ ( real_V7770717601297561774m_norm @ A @ X2 ) @ ( real_V7770717601297561774m_norm @ A @ Y2 ) ) ) ) ).
% norm_mult
thf(fact_1645_norm__ge__zero,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X2: A] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( real_V7770717601297561774m_norm @ A @ X2 ) ) ) ).
% norm_ge_zero
thf(fact_1646_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_1647_norm__power,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [X2: A,N: nat] :
( ( real_V7770717601297561774m_norm @ A @ ( power_power @ A @ X2 @ N ) )
= ( power_power @ real @ ( real_V7770717601297561774m_norm @ A @ X2 ) @ N ) ) ) ).
% norm_power
thf(fact_1648_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_1649_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_1650_norm__mult__less,axiom,
! [A: $tType] :
( ( real_V4412858255891104859lgebra @ A )
=> ! [X2: A,R2: real,Y2: A,S3: real] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X2 ) @ R2 )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ Y2 ) @ S3 )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( times_times @ A @ X2 @ Y2 ) ) @ ( times_times @ real @ R2 @ S3 ) ) ) ) ) ).
% norm_mult_less
thf(fact_1651_norm__mult__ineq,axiom,
! [A: $tType] :
( ( real_V4412858255891104859lgebra @ A )
=> ! [X2: A,Y2: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( times_times @ A @ X2 @ Y2 ) ) @ ( times_times @ real @ ( real_V7770717601297561774m_norm @ A @ X2 ) @ ( real_V7770717601297561774m_norm @ A @ Y2 ) ) ) ) ).
% norm_mult_ineq
thf(fact_1652_norm__triangle__lt,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X2: A,Y2: A,E: real] :
( ( ord_less @ real @ ( plus_plus @ real @ ( real_V7770717601297561774m_norm @ A @ X2 ) @ ( real_V7770717601297561774m_norm @ A @ Y2 ) ) @ E )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ X2 @ Y2 ) ) @ E ) ) ) ).
% norm_triangle_lt
thf(fact_1653_norm__add__less,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X2: A,R2: real,Y2: A,S3: real] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X2 ) @ R2 )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ Y2 ) @ S3 )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ X2 @ Y2 ) ) @ ( plus_plus @ real @ R2 @ S3 ) ) ) ) ) ).
% norm_add_less
thf(fact_1654_norm__triangle__mono,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [A2: A,R2: real,B2: A,S3: real] :
( ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ A2 ) @ R2 )
=> ( ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ B2 ) @ S3 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ A2 @ B2 ) ) @ ( plus_plus @ real @ R2 @ S3 ) ) ) ) ) ).
% norm_triangle_mono
thf(fact_1655_norm__triangle__ineq,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X2: A,Y2: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ X2 @ Y2 ) ) @ ( plus_plus @ real @ ( real_V7770717601297561774m_norm @ A @ X2 ) @ ( real_V7770717601297561774m_norm @ A @ Y2 ) ) ) ) ).
% norm_triangle_ineq
thf(fact_1656_norm__triangle__le,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X2: A,Y2: A,E: real] :
( ( ord_less_eq @ real @ ( plus_plus @ real @ ( real_V7770717601297561774m_norm @ A @ X2 ) @ ( real_V7770717601297561774m_norm @ A @ Y2 ) ) @ E )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ X2 @ Y2 ) ) @ E ) ) ) ).
% norm_triangle_le
thf(fact_1657_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_1658_norm__diff__triangle__less,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X2: A,Y2: A,E1: real,Z: A,E22: real] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ X2 @ Y2 ) ) @ E1 )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ Y2 @ Z ) ) @ E22 )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ X2 @ Z ) ) @ ( plus_plus @ real @ E1 @ E22 ) ) ) ) ) ).
% norm_diff_triangle_less
thf(fact_1659_norm__power__ineq,axiom,
! [A: $tType] :
( ( real_V2822296259951069270ebra_1 @ A )
=> ! [X2: A,N: nat] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( power_power @ A @ X2 @ N ) ) @ ( power_power @ real @ ( real_V7770717601297561774m_norm @ A @ X2 ) @ N ) ) ) ).
% norm_power_ineq
thf(fact_1660_norm__triangle__le__diff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X2: A,Y2: A,E: real] :
( ( ord_less_eq @ real @ ( plus_plus @ real @ ( real_V7770717601297561774m_norm @ A @ X2 ) @ ( real_V7770717601297561774m_norm @ A @ Y2 ) ) @ E )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ X2 @ Y2 ) ) @ E ) ) ) ).
% norm_triangle_le_diff
thf(fact_1661_norm__diff__triangle__le,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X2: A,Y2: A,E1: real,Z: A,E22: real] :
( ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ X2 @ Y2 ) ) @ E1 )
=> ( ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ Y2 @ Z ) ) @ E22 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ X2 @ Z ) ) @ ( plus_plus @ real @ E1 @ E22 ) ) ) ) ) ).
% norm_diff_triangle_le
thf(fact_1662_norm__triangle__ineq4,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [A2: A,B2: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ A2 @ B2 ) ) @ ( plus_plus @ real @ ( real_V7770717601297561774m_norm @ A @ A2 ) @ ( real_V7770717601297561774m_norm @ A @ B2 ) ) ) ) ).
% norm_triangle_ineq4
thf(fact_1663_norm__triangle__sub,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X2: A,Y2: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ X2 ) @ ( plus_plus @ real @ ( real_V7770717601297561774m_norm @ A @ Y2 ) @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ X2 @ Y2 ) ) ) ) ) ).
% norm_triangle_sub
thf(fact_1664_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_1665_norm__triangle__ineq2,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [A2: A,B2: A] : ( ord_less_eq @ real @ ( minus_minus @ real @ ( real_V7770717601297561774m_norm @ A @ A2 ) @ ( real_V7770717601297561774m_norm @ A @ B2 ) ) @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ A2 @ B2 ) ) ) ) ).
% norm_triangle_ineq2
thf(fact_1666_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_1667_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_1668_square__norm__one,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [X2: A] :
( ( ( power_power @ A @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( one_one @ A ) )
=> ( ( real_V7770717601297561774m_norm @ A @ X2 )
= ( one_one @ real ) ) ) ) ).
% square_norm_one
thf(fact_1669_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_1670_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_1671_atLeastatMost__subset__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_or1337092689740270186AtMost @ A @ C2 @ D2 ) )
= ( ~ ( ord_less_eq @ A @ A2 @ B2 )
| ( ( ord_less_eq @ A @ C2 @ A2 )
& ( ord_less_eq @ A @ B2 @ D2 ) ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_1672_word__less__1,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A] :
( ( ord_less @ ( word @ A ) @ X2 @ ( one_one @ ( word @ A ) ) )
= ( X2
= ( zero_zero @ ( word @ A ) ) ) ) ) ).
% word_less_1
thf(fact_1673_word__gt__0__no,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y2: num] :
( ( ord_less @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ ( numeral_numeral @ ( word @ A ) @ Y2 ) )
= ( ( zero_zero @ ( word @ A ) )
!= ( numeral_numeral @ ( word @ A ) @ Y2 ) ) ) ) ).
% word_gt_0_no
thf(fact_1674_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_1675_log__ceil__idem,axiom,
! [X2: real] :
( ( ord_less_eq @ real @ ( one_one @ real ) @ X2 )
=> ( ( archimedean_ceiling @ real @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ X2 ) )
= ( archimedean_ceiling @ real @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( ring_1_of_int @ real @ ( archimedean_ceiling @ real @ X2 ) ) ) ) ) ) ).
% log_ceil_idem
thf(fact_1676_Icc__eq__Icc,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L2: A,H2: A,L3: A,H3: A] :
( ( ( set_or1337092689740270186AtMost @ A @ L2 @ H2 )
= ( set_or1337092689740270186AtMost @ A @ L3 @ H3 ) )
= ( ( ( L2 = L3 )
& ( H2 = H3 ) )
| ( ~ ( ord_less_eq @ A @ L2 @ H2 )
& ~ ( ord_less_eq @ A @ L3 @ H3 ) ) ) ) ) ).
% Icc_eq_Icc
thf(fact_1677_atLeastAtMost__iff,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [I: A,L2: A,U: A] :
( ( member @ A @ I @ ( set_or1337092689740270186AtMost @ A @ L2 @ U ) )
= ( ( ord_less_eq @ A @ L2 @ I )
& ( ord_less_eq @ A @ I @ U ) ) ) ) ).
% atLeastAtMost_iff
thf(fact_1678_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_1679_word__le__0__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ X2 @ ( zero_zero @ ( word @ A ) ) )
= ( X2
= ( zero_zero @ ( word @ A ) ) ) ) ) ).
% word_le_0_iff
thf(fact_1680_of__int__ceiling__cancel,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A] :
( ( ( ring_1_of_int @ A @ ( archimedean_ceiling @ A @ X2 ) )
= X2 )
= ( ? [N3: int] :
( X2
= ( ring_1_of_int @ A @ N3 ) ) ) ) ) ).
% of_int_ceiling_cancel
thf(fact_1681_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_1682_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_1683_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_1684_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_1685_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_1686_of__int__le__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [W: int,Z: int] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ W ) @ ( ring_1_of_int @ A @ Z ) )
= ( ord_less_eq @ int @ W @ Z ) ) ) ).
% of_int_le_iff
thf(fact_1687_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_1688_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_1689_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_1690_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_1691_of__int__of__nat__eq,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [N: nat] :
( ( ring_1_of_int @ A @ ( semiring_1_of_nat @ int @ N ) )
= ( semiring_1_of_nat @ A @ N ) ) ) ).
% of_int_of_nat_eq
thf(fact_1692_of__int__power__eq__of__int__cancel__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [X2: int,B2: int,W: nat] :
( ( ( ring_1_of_int @ A @ X2 )
= ( power_power @ A @ ( ring_1_of_int @ A @ B2 ) @ W ) )
= ( X2
= ( power_power @ int @ B2 @ W ) ) ) ) ).
% of_int_power_eq_of_int_cancel_iff
thf(fact_1693_of__int__eq__of__int__power__cancel__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [B2: int,W: nat,X2: int] :
( ( ( power_power @ A @ ( ring_1_of_int @ A @ B2 ) @ W )
= ( ring_1_of_int @ A @ X2 ) )
= ( ( power_power @ int @ B2 @ W )
= X2 ) ) ) ).
% of_int_eq_of_int_power_cancel_iff
thf(fact_1694_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_1695_ceiling__add__of__int,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A,Z: int] :
( ( archimedean_ceiling @ A @ ( plus_plus @ A @ X2 @ ( ring_1_of_int @ A @ Z ) ) )
= ( plus_plus @ int @ ( archimedean_ceiling @ A @ X2 ) @ Z ) ) ) ).
% ceiling_add_of_int
thf(fact_1696_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_1697_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_1698_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_1699_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_1700_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_1701_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_1702_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_1703_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_1704_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_1705_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_1706_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_1707_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_1708_of__int__eq__numeral__power__cancel__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [Y2: int,X2: num,N: nat] :
( ( ( ring_1_of_int @ A @ Y2 )
= ( power_power @ A @ ( numeral_numeral @ A @ X2 ) @ N ) )
= ( Y2
= ( power_power @ int @ ( numeral_numeral @ int @ X2 ) @ N ) ) ) ) ).
% of_int_eq_numeral_power_cancel_iff
thf(fact_1709_numeral__power__eq__of__int__cancel__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [X2: num,N: nat,Y2: int] :
( ( ( power_power @ A @ ( numeral_numeral @ A @ X2 ) @ N )
= ( ring_1_of_int @ A @ Y2 ) )
= ( ( power_power @ int @ ( numeral_numeral @ int @ X2 ) @ N )
= Y2 ) ) ) ).
% numeral_power_eq_of_int_cancel_iff
thf(fact_1710_of__int__power__le__of__int__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X2: int,B2: int,W: nat] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ X2 ) @ ( power_power @ A @ ( ring_1_of_int @ A @ B2 ) @ W ) )
= ( ord_less_eq @ int @ X2 @ ( power_power @ int @ B2 @ W ) ) ) ) ).
% of_int_power_le_of_int_cancel_iff
thf(fact_1711_of__int__le__of__int__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [B2: int,W: nat,X2: int] :
( ( ord_less_eq @ A @ ( power_power @ A @ ( ring_1_of_int @ A @ B2 ) @ W ) @ ( ring_1_of_int @ A @ X2 ) )
= ( ord_less_eq @ int @ ( power_power @ int @ B2 @ W ) @ X2 ) ) ) ).
% of_int_le_of_int_power_cancel_iff
thf(fact_1712_of__int__less__of__int__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [B2: int,W: nat,X2: int] :
( ( ord_less @ A @ ( power_power @ A @ ( ring_1_of_int @ A @ B2 ) @ W ) @ ( ring_1_of_int @ A @ X2 ) )
= ( ord_less @ int @ ( power_power @ int @ B2 @ W ) @ X2 ) ) ) ).
% of_int_less_of_int_power_cancel_iff
thf(fact_1713_of__int__power__less__of__int__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X2: int,B2: int,W: nat] :
( ( ord_less @ A @ ( ring_1_of_int @ A @ X2 ) @ ( power_power @ A @ ( ring_1_of_int @ A @ B2 ) @ W ) )
= ( ord_less @ int @ X2 @ ( power_power @ int @ B2 @ W ) ) ) ) ).
% of_int_power_less_of_int_cancel_iff
thf(fact_1714_of__int__le__numeral__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: int,X2: num,N: nat] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ A2 ) @ ( power_power @ A @ ( numeral_numeral @ A @ X2 ) @ N ) )
= ( ord_less_eq @ int @ A2 @ ( power_power @ int @ ( numeral_numeral @ int @ X2 ) @ N ) ) ) ) ).
% of_int_le_numeral_power_cancel_iff
thf(fact_1715_numeral__power__le__of__int__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X2: num,N: nat,A2: int] :
( ( ord_less_eq @ A @ ( power_power @ A @ ( numeral_numeral @ A @ X2 ) @ N ) @ ( ring_1_of_int @ A @ A2 ) )
= ( ord_less_eq @ int @ ( power_power @ int @ ( numeral_numeral @ int @ X2 ) @ N ) @ A2 ) ) ) ).
% numeral_power_le_of_int_cancel_iff
thf(fact_1716_of__int__less__numeral__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: int,X2: num,N: nat] :
( ( ord_less @ A @ ( ring_1_of_int @ A @ A2 ) @ ( power_power @ A @ ( numeral_numeral @ A @ X2 ) @ N ) )
= ( ord_less @ int @ A2 @ ( power_power @ int @ ( numeral_numeral @ int @ X2 ) @ N ) ) ) ) ).
% of_int_less_numeral_power_cancel_iff
thf(fact_1717_numeral__power__less__of__int__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X2: num,N: nat,A2: int] :
( ( ord_less @ A @ ( power_power @ A @ ( numeral_numeral @ A @ X2 ) @ N ) @ ( ring_1_of_int @ A @ A2 ) )
= ( ord_less @ int @ ( power_power @ int @ ( numeral_numeral @ int @ X2 ) @ N ) @ A2 ) ) ) ).
% numeral_power_less_of_int_cancel_iff
thf(fact_1718_plus__minus__no__overflow__ab,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A,Ab: word @ A,C2: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ X2 @ ( minus_minus @ ( word @ A ) @ Ab @ C2 ) )
=> ( ( ord_less_eq @ ( word @ A ) @ C2 @ Ab )
=> ( ord_less_eq @ ( word @ A ) @ X2 @ ( plus_plus @ ( word @ A ) @ X2 @ C2 ) ) ) ) ) ).
% plus_minus_no_overflow_ab
thf(fact_1719_plus__minus__not__NULL__ab,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A,Ab: word @ A,C2: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ X2 @ ( minus_minus @ ( word @ A ) @ Ab @ C2 ) )
=> ( ( ord_less_eq @ ( word @ A ) @ C2 @ Ab )
=> ( ( C2
!= ( zero_zero @ ( word @ A ) ) )
=> ( ( plus_plus @ ( word @ A ) @ X2 @ C2 )
!= ( zero_zero @ ( word @ A ) ) ) ) ) ) ) ).
% plus_minus_not_NULL_ab
thf(fact_1720_neq__0__no__wrap,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A,Y2: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ X2 @ ( plus_plus @ ( word @ A ) @ X2 @ Y2 ) )
=> ( ( X2
!= ( zero_zero @ ( word @ A ) ) )
=> ( ( plus_plus @ ( word @ A ) @ X2 @ Y2 )
!= ( zero_zero @ ( word @ A ) ) ) ) ) ) ).
% neq_0_no_wrap
thf(fact_1721_le__minus_H,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: word @ A,C2: word @ A,B2: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ A2 @ C2 ) @ B2 )
=> ( ( ord_less_eq @ ( word @ A ) @ A2 @ ( plus_plus @ ( word @ A ) @ A2 @ C2 ) )
=> ( ord_less_eq @ ( word @ A ) @ C2 @ ( minus_minus @ ( word @ A ) @ B2 @ A2 ) ) ) ) ) ).
% le_minus'
thf(fact_1722_le__plus_H,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: word @ A,B2: word @ A,C2: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ A2 @ B2 )
=> ( ( ord_less_eq @ ( word @ A ) @ C2 @ ( minus_minus @ ( word @ A ) @ B2 @ A2 ) )
=> ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ A2 @ C2 ) @ B2 ) ) ) ) ).
% le_plus'
thf(fact_1723_le__plus,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [C2: word @ A,B2: word @ A,A2: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ C2 @ ( minus_minus @ ( word @ A ) @ B2 @ A2 ) )
=> ( ( ord_less_eq @ ( word @ A ) @ A2 @ B2 )
=> ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ A2 @ C2 ) @ B2 ) ) ) ) ).
% le_plus
thf(fact_1724_word__diff__ls_H_H_I3_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y2: word @ A,X2: word @ A,W: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ Y2 @ X2 ) @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W @ X2 ) @ X2 ) )
=> ( ( ord_less_eq @ ( word @ A ) @ X2 @ ( plus_plus @ ( word @ A ) @ W @ X2 ) )
=> ( ord_less_eq @ ( word @ A ) @ Y2 @ ( plus_plus @ ( word @ A ) @ W @ X2 ) ) ) ) ) ).
% word_diff_ls''(3)
thf(fact_1725_word__diff__ls_H_H_I4_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y2: word @ A,W: word @ A,X2: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ Y2 @ ( plus_plus @ ( word @ A ) @ W @ X2 ) )
=> ( ( ord_less_eq @ ( word @ A ) @ X2 @ Y2 )
=> ( ord_less_eq @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ Y2 @ X2 ) @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W @ X2 ) @ X2 ) ) ) ) ) ).
% word_diff_ls''(4)
thf(fact_1726_word__l__diffs_H_I3_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A,X2: word @ A,Z: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W @ X2 ) @ X2 ) @ ( minus_minus @ ( word @ A ) @ Z @ X2 ) )
=> ( ( ord_less_eq @ ( word @ A ) @ X2 @ Z )
=> ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W @ X2 ) @ Z ) ) ) ) ).
% word_l_diffs'(3)
thf(fact_1727_word__l__diffs_H_I4_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A,X2: word @ A,Z: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W @ X2 ) @ Z )
=> ( ( ord_less_eq @ ( word @ A ) @ X2 @ ( plus_plus @ ( word @ A ) @ W @ X2 ) )
=> ( ord_less_eq @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W @ X2 ) @ X2 ) @ ( minus_minus @ ( word @ A ) @ Z @ X2 ) ) ) ) ) ).
% word_l_diffs'(4)
thf(fact_1728_word__diff__ls_H_I3_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y2: word @ A,X2: word @ A,W: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ Y2 @ X2 ) @ W )
=> ( ( ord_less_eq @ ( word @ A ) @ X2 @ ( plus_plus @ ( word @ A ) @ W @ X2 ) )
=> ( ord_less_eq @ ( word @ A ) @ Y2 @ ( plus_plus @ ( word @ A ) @ W @ X2 ) ) ) ) ) ).
% word_diff_ls'(3)
thf(fact_1729_word__diff__ls_H_I4_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y2: word @ A,W: word @ A,X2: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ Y2 @ ( plus_plus @ ( word @ A ) @ W @ X2 ) )
=> ( ( ord_less_eq @ ( word @ A ) @ X2 @ Y2 )
=> ( ord_less_eq @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ Y2 @ X2 ) @ W ) ) ) ) ).
% word_diff_ls'(4)
thf(fact_1730_More__Word_Oword__l__diffs_I3_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A,Z: word @ A,X2: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ W @ ( minus_minus @ ( word @ A ) @ Z @ X2 ) )
=> ( ( ord_less_eq @ ( word @ A ) @ X2 @ Z )
=> ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W @ X2 ) @ Z ) ) ) ) ).
% More_Word.word_l_diffs(3)
thf(fact_1731_More__Word_Oword__l__diffs_I4_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A,X2: word @ A,Z: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W @ X2 ) @ Z )
=> ( ( ord_less_eq @ ( word @ A ) @ X2 @ ( plus_plus @ ( word @ A ) @ W @ X2 ) )
=> ( ord_less_eq @ ( word @ A ) @ W @ ( minus_minus @ ( word @ A ) @ Z @ X2 ) ) ) ) ) ).
% More_Word.word_l_diffs(4)
thf(fact_1732_word__plus__mcs_I3_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [V: word @ A,Xb: word @ A,X2: word @ A,W: word @ A,Xa: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ V @ Xb ) @ X2 ) @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W @ Xa ) @ X2 ) )
=> ( ( ord_less_eq @ ( word @ A ) @ X2 @ ( plus_plus @ ( word @ A ) @ W @ Xa ) )
=> ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ V @ Xb ) @ ( plus_plus @ ( word @ A ) @ W @ Xa ) ) ) ) ) ).
% word_plus_mcs(3)
thf(fact_1733_word__plus__mcs_I4_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [V: word @ A,Xb: word @ A,W: word @ A,Xa: word @ A,X2: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ V @ Xb ) @ ( plus_plus @ ( word @ A ) @ W @ Xa ) )
=> ( ( ord_less_eq @ ( word @ A ) @ X2 @ ( plus_plus @ ( word @ A ) @ V @ Xb ) )
=> ( ord_less_eq @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ V @ Xb ) @ X2 ) @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W @ Xa ) @ X2 ) ) ) ) ) ).
% word_plus_mcs(4)
thf(fact_1734_Word_Oword__l__diffs_I3_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A,Xa: word @ A,X2: word @ A,Z: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W @ Xa ) @ X2 ) @ ( minus_minus @ ( word @ A ) @ Z @ X2 ) )
=> ( ( ord_less_eq @ ( word @ A ) @ X2 @ Z )
=> ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W @ Xa ) @ Z ) ) ) ) ).
% Word.word_l_diffs(3)
thf(fact_1735_Word_Oword__l__diffs_I4_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A,Xa: word @ A,Z: word @ A,X2: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W @ Xa ) @ Z )
=> ( ( ord_less_eq @ ( word @ A ) @ X2 @ ( plus_plus @ ( word @ A ) @ W @ Xa ) )
=> ( ord_less_eq @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W @ Xa ) @ X2 ) @ ( minus_minus @ ( word @ A ) @ Z @ X2 ) ) ) ) ) ).
% Word.word_l_diffs(4)
thf(fact_1736_word__diff__ls_I3_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y2: word @ A,X2: word @ A,W: word @ A,Xa: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ Y2 @ X2 ) @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W @ Xa ) @ X2 ) )
=> ( ( ord_less_eq @ ( word @ A ) @ X2 @ ( plus_plus @ ( word @ A ) @ W @ Xa ) )
=> ( ord_less_eq @ ( word @ A ) @ Y2 @ ( plus_plus @ ( word @ A ) @ W @ Xa ) ) ) ) ) ).
% word_diff_ls(3)
thf(fact_1737_word__diff__ls_I4_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y2: word @ A,W: word @ A,Xa: word @ A,X2: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ Y2 @ ( plus_plus @ ( word @ A ) @ W @ Xa ) )
=> ( ( ord_less_eq @ ( word @ A ) @ X2 @ Y2 )
=> ( ord_less_eq @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ Y2 @ X2 ) @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W @ Xa ) @ X2 ) ) ) ) ) ).
% word_diff_ls(4)
thf(fact_1738_olen__add__eqv,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y2: word @ A,X2: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ Y2 @ ( plus_plus @ ( word @ A ) @ Y2 @ X2 ) )
= ( ord_less_eq @ ( word @ A ) @ X2 @ ( plus_plus @ ( word @ A ) @ Y2 @ X2 ) ) ) ) ).
% olen_add_eqv
thf(fact_1739_le__no__overflow,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A,B2: word @ A,A2: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ X2 @ B2 )
=> ( ( ord_less_eq @ ( word @ A ) @ A2 @ ( plus_plus @ ( word @ A ) @ A2 @ B2 ) )
=> ( ord_less_eq @ ( word @ A ) @ X2 @ ( plus_plus @ ( word @ A ) @ A2 @ B2 ) ) ) ) ) ).
% le_no_overflow
thf(fact_1740_word__random,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [P2: word @ A,X5: word @ A,X2: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ P2 @ ( plus_plus @ ( word @ A ) @ P2 @ X5 ) )
=> ( ( ord_less_eq @ ( word @ A ) @ X2 @ X5 )
=> ( ord_less_eq @ ( word @ A ) @ P2 @ ( plus_plus @ ( word @ A ) @ P2 @ X2 ) ) ) ) ) ).
% word_random
thf(fact_1741_word__sub__mono2,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: word @ A,B2: word @ A,C2: word @ A,D2: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ A2 @ B2 ) @ ( plus_plus @ ( word @ A ) @ C2 @ D2 ) )
=> ( ( ord_less_eq @ ( word @ A ) @ C2 @ A2 )
=> ( ( ord_less_eq @ ( word @ A ) @ B2 @ ( plus_plus @ ( word @ A ) @ A2 @ B2 ) )
=> ( ( ord_less_eq @ ( word @ A ) @ D2 @ ( plus_plus @ ( word @ A ) @ C2 @ D2 ) )
=> ( ord_less_eq @ ( word @ A ) @ B2 @ D2 ) ) ) ) ) ) ).
% word_sub_mono2
thf(fact_1742_word__le__plus__either,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A,Y2: word @ A,Z: word @ A] :
( ( ( ord_less_eq @ ( word @ A ) @ X2 @ Y2 )
| ( ord_less_eq @ ( word @ A ) @ X2 @ Z ) )
=> ( ( ord_less_eq @ ( word @ A ) @ Y2 @ ( plus_plus @ ( word @ A ) @ Y2 @ Z ) )
=> ( ord_less_eq @ ( word @ A ) @ X2 @ ( plus_plus @ ( word @ A ) @ Y2 @ Z ) ) ) ) ) ).
% word_le_plus_either
thf(fact_1743_word__plus__mcs__3,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [V: word @ A,W: word @ A,X2: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ V @ W )
=> ( ( ord_less_eq @ ( word @ A ) @ X2 @ ( plus_plus @ ( word @ A ) @ W @ X2 ) )
=> ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ V @ X2 ) @ ( plus_plus @ ( word @ A ) @ W @ X2 ) ) ) ) ) ).
% word_plus_mcs_3
thf(fact_1744_word__plus__mcs__4,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [V: word @ A,X2: word @ A,W: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ V @ X2 ) @ ( plus_plus @ ( word @ A ) @ W @ X2 ) )
=> ( ( ord_less_eq @ ( word @ A ) @ X2 @ ( plus_plus @ ( word @ A ) @ V @ X2 ) )
=> ( ord_less_eq @ ( word @ A ) @ V @ W ) ) ) ) ).
% word_plus_mcs_4
thf(fact_1745_word__plus__mono__right,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y2: word @ A,Z: word @ A,X2: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ Y2 @ Z )
=> ( ( ord_less_eq @ ( word @ A ) @ X2 @ ( plus_plus @ ( word @ A ) @ X2 @ Z ) )
=> ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ X2 @ Y2 ) @ ( plus_plus @ ( word @ A ) @ X2 @ Z ) ) ) ) ) ).
% word_plus_mono_right
thf(fact_1746_word__add__le__iff2,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [I: word @ A,K: word @ A,J: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ I @ ( plus_plus @ ( word @ A ) @ I @ K ) )
=> ( ( ord_less_eq @ ( word @ A ) @ J @ ( plus_plus @ ( word @ A ) @ J @ K ) )
=> ( ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ I @ K ) @ ( plus_plus @ ( word @ A ) @ J @ K ) )
= ( ord_less_eq @ ( word @ A ) @ I @ J ) ) ) ) ) ).
% word_add_le_iff2
thf(fact_1747_word__plus__mcs__4_H,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A,V: word @ A,W: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ X2 @ V ) @ ( plus_plus @ ( word @ A ) @ X2 @ W ) )
=> ( ( ord_less_eq @ ( word @ A ) @ X2 @ ( plus_plus @ ( word @ A ) @ X2 @ V ) )
=> ( ord_less_eq @ ( word @ A ) @ V @ W ) ) ) ) ).
% word_plus_mcs_4'
thf(fact_1748_word__plus__mono__right2,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: word @ A,B2: word @ A,C2: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ A2 @ ( plus_plus @ ( word @ A ) @ A2 @ B2 ) )
=> ( ( ord_less_eq @ ( word @ A ) @ C2 @ B2 )
=> ( ord_less_eq @ ( word @ A ) @ A2 @ ( plus_plus @ ( word @ A ) @ A2 @ C2 ) ) ) ) ) ).
% word_plus_mono_right2
thf(fact_1749_word__add__increasing,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [P2: word @ A,W: word @ A,X2: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ P2 @ W ) @ X2 )
=> ( ( ord_less_eq @ ( word @ A ) @ P2 @ ( plus_plus @ ( word @ A ) @ P2 @ W ) )
=> ( ord_less_eq @ ( word @ A ) @ P2 @ X2 ) ) ) ) ).
% word_add_increasing
thf(fact_1750_word__plus__mono__left,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y2: word @ A,Z: word @ A,X2: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ Y2 @ Z )
=> ( ( ord_less_eq @ ( word @ A ) @ X2 @ ( plus_plus @ ( word @ A ) @ X2 @ Z ) )
=> ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ Y2 @ X2 ) @ ( plus_plus @ ( word @ A ) @ Z @ X2 ) ) ) ) ) ).
% word_plus_mono_left
thf(fact_1751_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_1752_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_1753_word__zero__le,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y2: word @ A] : ( ord_less_eq @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ Y2 ) ) ).
% word_zero_le
thf(fact_1754_ex__le__of__int,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X2: A] :
? [Z2: int] : ( ord_less_eq @ A @ X2 @ ( ring_1_of_int @ A @ Z2 ) ) ) ).
% ex_le_of_int
thf(fact_1755_mult__of__int__commute,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [X2: int,Y2: A] :
( ( times_times @ A @ ( ring_1_of_int @ A @ X2 ) @ Y2 )
= ( times_times @ A @ Y2 @ ( ring_1_of_int @ A @ X2 ) ) ) ) ).
% mult_of_int_commute
thf(fact_1756_word__subset__less,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A,R2: word @ A,Y2: word @ A,S3: word @ A] :
( ( ord_less_eq @ ( set @ ( word @ A ) ) @ ( set_or1337092689740270186AtMost @ ( word @ A ) @ X2 @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ X2 @ R2 ) @ ( one_one @ ( word @ A ) ) ) ) @ ( set_or1337092689740270186AtMost @ ( word @ A ) @ Y2 @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ Y2 @ S3 ) @ ( one_one @ ( word @ A ) ) ) ) )
=> ( ( ord_less_eq @ ( word @ A ) @ X2 @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ X2 @ R2 ) @ ( one_one @ ( word @ A ) ) ) )
=> ( ( ord_less_eq @ ( word @ A ) @ Y2 @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ Y2 @ S3 ) @ ( one_one @ ( word @ A ) ) ) )
=> ( ( S3
!= ( zero_zero @ ( word @ A ) ) )
=> ( ord_less_eq @ ( word @ A ) @ R2 @ S3 ) ) ) ) ) ) ).
% word_subset_less
thf(fact_1757_le__of__int__ceiling,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A] : ( ord_less_eq @ A @ X2 @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ A @ X2 ) ) ) ) ).
% le_of_int_ceiling
thf(fact_1758_bset_I1_J,axiom,
! [D4: int,B5: set @ int,P: int > $o,Q: int > $o] :
( ! [X3: int] :
( ! [Xa2: int] :
( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ B5 )
=> ( X3
!= ( plus_plus @ int @ Xb2 @ Xa2 ) ) ) )
=> ( ( P @ X3 )
=> ( P @ ( minus_minus @ int @ X3 @ D4 ) ) ) )
=> ( ! [X3: int] :
( ! [Xa2: int] :
( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ B5 )
=> ( X3
!= ( plus_plus @ int @ Xb2 @ Xa2 ) ) ) )
=> ( ( Q @ X3 )
=> ( Q @ ( minus_minus @ int @ X3 @ D4 ) ) ) )
=> ! [X4: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ B5 )
=> ( X4
!= ( plus_plus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ( P @ X4 )
& ( Q @ X4 ) )
=> ( ( P @ ( minus_minus @ int @ X4 @ D4 ) )
& ( Q @ ( minus_minus @ int @ X4 @ D4 ) ) ) ) ) ) ) ).
% bset(1)
thf(fact_1759_bset_I2_J,axiom,
! [D4: int,B5: set @ int,P: int > $o,Q: int > $o] :
( ! [X3: int] :
( ! [Xa2: int] :
( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ B5 )
=> ( X3
!= ( plus_plus @ int @ Xb2 @ Xa2 ) ) ) )
=> ( ( P @ X3 )
=> ( P @ ( minus_minus @ int @ X3 @ D4 ) ) ) )
=> ( ! [X3: int] :
( ! [Xa2: int] :
( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ B5 )
=> ( X3
!= ( plus_plus @ int @ Xb2 @ Xa2 ) ) ) )
=> ( ( Q @ X3 )
=> ( Q @ ( minus_minus @ int @ X3 @ D4 ) ) ) )
=> ! [X4: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ B5 )
=> ( X4
!= ( plus_plus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ( P @ X4 )
| ( Q @ X4 ) )
=> ( ( P @ ( minus_minus @ int @ X4 @ D4 ) )
| ( Q @ ( minus_minus @ int @ X4 @ D4 ) ) ) ) ) ) ) ).
% bset(2)
thf(fact_1760_aset_I1_J,axiom,
! [D4: int,A3: set @ int,P: int > $o,Q: int > $o] :
( ! [X3: int] :
( ! [Xa2: int] :
( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ A3 )
=> ( X3
!= ( minus_minus @ int @ Xb2 @ Xa2 ) ) ) )
=> ( ( P @ X3 )
=> ( P @ ( plus_plus @ int @ X3 @ D4 ) ) ) )
=> ( ! [X3: int] :
( ! [Xa2: int] :
( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ A3 )
=> ( X3
!= ( minus_minus @ int @ Xb2 @ Xa2 ) ) ) )
=> ( ( Q @ X3 )
=> ( Q @ ( plus_plus @ int @ X3 @ D4 ) ) ) )
=> ! [X4: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ A3 )
=> ( X4
!= ( minus_minus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ( P @ X4 )
& ( Q @ X4 ) )
=> ( ( P @ ( plus_plus @ int @ X4 @ D4 ) )
& ( Q @ ( plus_plus @ int @ X4 @ D4 ) ) ) ) ) ) ) ).
% aset(1)
thf(fact_1761_aset_I2_J,axiom,
! [D4: int,A3: set @ int,P: int > $o,Q: int > $o] :
( ! [X3: int] :
( ! [Xa2: int] :
( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ A3 )
=> ( X3
!= ( minus_minus @ int @ Xb2 @ Xa2 ) ) ) )
=> ( ( P @ X3 )
=> ( P @ ( plus_plus @ int @ X3 @ D4 ) ) ) )
=> ( ! [X3: int] :
( ! [Xa2: int] :
( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ A3 )
=> ( X3
!= ( minus_minus @ int @ Xb2 @ Xa2 ) ) ) )
=> ( ( Q @ X3 )
=> ( Q @ ( plus_plus @ int @ X3 @ D4 ) ) ) )
=> ! [X4: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ A3 )
=> ( X4
!= ( minus_minus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ( P @ X4 )
| ( Q @ X4 ) )
=> ( ( P @ ( plus_plus @ int @ X4 @ D4 ) )
| ( Q @ ( plus_plus @ int @ X4 @ D4 ) ) ) ) ) ) ) ).
% aset(2)
thf(fact_1762_ceiling__le,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A,A2: int] :
( ( ord_less_eq @ A @ X2 @ ( ring_1_of_int @ A @ A2 ) )
=> ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ X2 ) @ A2 ) ) ) ).
% ceiling_le
thf(fact_1763_ceiling__le__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A,Z: int] :
( ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ X2 ) @ Z )
= ( ord_less_eq @ A @ X2 @ ( ring_1_of_int @ A @ Z ) ) ) ) ).
% ceiling_le_iff
thf(fact_1764_real__of__int__div4,axiom,
! [N: int,X2: int] : ( ord_less_eq @ real @ ( ring_1_of_int @ real @ ( divide_divide @ int @ N @ X2 ) ) @ ( divide_divide @ real @ ( ring_1_of_int @ real @ N ) @ ( ring_1_of_int @ real @ X2 ) ) ) ).
% real_of_int_div4
thf(fact_1765_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_1766_le__minus,axiom,
! [Aa: $tType,A: $tType] :
( ( ( type_len @ A )
& ( order @ Aa ) )
=> ! [Y2: Aa,X2: Aa,A2: word @ A,C2: word @ A,B2: word @ A] :
( ( ord_less_eq @ Aa @ Y2 @ X2 )
=> ( ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ A2 @ C2 ) @ B2 )
=> ( ( ord_less_eq @ ( word @ A ) @ A2 @ ( plus_plus @ ( word @ A ) @ A2 @ C2 ) )
=> ( ord_less_eq @ ( word @ A ) @ C2 @ ( minus_minus @ ( word @ A ) @ B2 @ A2 ) ) ) ) ) ) ).
% le_minus
thf(fact_1767_bounded__Max__nat,axiom,
! [P: nat > $o,X2: nat,M7: nat] :
( ( P @ X2 )
=> ( ! [X3: nat] :
( ( P @ X3 )
=> ( ord_less_eq @ nat @ X3 @ M7 ) )
=> ~ ! [M3: nat] :
( ( P @ M3 )
=> ~ ! [X4: nat] :
( ( P @ X4 )
=> ( ord_less_eq @ nat @ X4 @ M3 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_1768_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_1769_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_1770_word__le__sub1,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A] :
( ( X2
!= ( zero_zero @ ( word @ A ) ) )
=> ( ( ord_less_eq @ ( word @ A ) @ ( one_one @ ( word @ A ) ) @ X2 )
= ( ord_less_eq @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ ( minus_minus @ ( word @ A ) @ X2 @ ( one_one @ ( word @ A ) ) ) ) ) ) ) ).
% word_le_sub1
thf(fact_1771_plus__le__left__cancel__wrap,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A,Y6: word @ A,Y2: word @ A] :
( ( ord_less @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ X2 @ Y6 ) @ X2 )
=> ( ( ord_less @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ X2 @ Y2 ) @ X2 )
=> ( ( ord_less @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ X2 @ Y6 ) @ ( plus_plus @ ( word @ A ) @ X2 @ Y2 ) )
= ( ord_less @ ( word @ A ) @ Y6 @ Y2 ) ) ) ) ) ).
% plus_le_left_cancel_wrap
thf(fact_1772_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_1773_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_1774_word__gt__0,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y2: word @ A] :
( ( ord_less @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ Y2 )
= ( ( zero_zero @ ( word @ A ) )
!= Y2 ) ) ) ).
% word_gt_0
thf(fact_1775_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_1776_word__not__simps_I1_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A] :
~ ( ord_less @ ( word @ A ) @ X2 @ ( zero_zero @ ( word @ A ) ) ) ) ).
% word_not_simps(1)
thf(fact_1777_plus__le__left__cancel__nowrap,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A,Y6: word @ A,Y2: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ X2 @ ( plus_plus @ ( word @ A ) @ X2 @ Y6 ) )
=> ( ( ord_less_eq @ ( word @ A ) @ X2 @ ( plus_plus @ ( word @ A ) @ X2 @ Y2 ) )
=> ( ( ord_less @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ X2 @ Y6 ) @ ( plus_plus @ ( word @ A ) @ X2 @ Y2 ) )
= ( ord_less @ ( word @ A ) @ Y6 @ Y2 ) ) ) ) ) ).
% plus_le_left_cancel_nowrap
thf(fact_1778_plus__minus__no__overflow,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A,Ab: word @ A,C2: word @ A] :
( ( ord_less @ ( word @ A ) @ X2 @ ( minus_minus @ ( word @ A ) @ Ab @ C2 ) )
=> ( ( ord_less_eq @ ( word @ A ) @ C2 @ Ab )
=> ( ord_less_eq @ ( word @ A ) @ X2 @ ( plus_plus @ ( word @ A ) @ X2 @ C2 ) ) ) ) ) ).
% plus_minus_no_overflow
thf(fact_1779_word__less__sub__right,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A,Y2: word @ A,Z: word @ A] :
( ( ord_less @ ( word @ A ) @ X2 @ ( plus_plus @ ( word @ A ) @ Y2 @ Z ) )
=> ( ( ord_less_eq @ ( word @ A ) @ Y2 @ X2 )
=> ( ord_less @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ X2 @ Y2 ) @ Z ) ) ) ) ).
% word_less_sub_right
thf(fact_1780_word__less__add__right,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A,Y2: word @ A,Z: word @ A] :
( ( ord_less @ ( word @ A ) @ X2 @ ( minus_minus @ ( word @ A ) @ Y2 @ Z ) )
=> ( ( ord_less_eq @ ( word @ A ) @ Z @ Y2 )
=> ( ord_less @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ X2 @ Z ) @ Y2 ) ) ) ) ).
% word_less_add_right
thf(fact_1781_plus__minus__not__NULL,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A,Ab: word @ A,C2: word @ A] :
( ( ord_less @ ( word @ A ) @ X2 @ ( minus_minus @ ( word @ A ) @ Ab @ C2 ) )
=> ( ( ord_less_eq @ ( word @ A ) @ C2 @ Ab )
=> ( ( C2
!= ( zero_zero @ ( word @ A ) ) )
=> ( ( plus_plus @ ( word @ A ) @ X2 @ C2 )
!= ( zero_zero @ ( word @ A ) ) ) ) ) ) ) ).
% plus_minus_not_NULL
thf(fact_1782_word__less__nowrapI,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A,Z: word @ A,K: word @ A] :
( ( ord_less @ ( word @ A ) @ X2 @ ( minus_minus @ ( word @ A ) @ Z @ K ) )
=> ( ( ord_less_eq @ ( word @ A ) @ K @ Z )
=> ( ( ord_less @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ K )
=> ( ord_less @ ( word @ A ) @ X2 @ ( plus_plus @ ( word @ A ) @ X2 @ K ) ) ) ) ) ) ).
% word_less_nowrapI
thf(fact_1783_sub__wrap,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A,Z: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ X2 @ ( minus_minus @ ( word @ A ) @ X2 @ Z ) )
= ( ( Z
= ( zero_zero @ ( word @ A ) ) )
| ( ord_less @ ( word @ A ) @ X2 @ Z ) ) ) ) ).
% sub_wrap
thf(fact_1784_word__plus__mcs_I1_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [V: word @ A,Xb: word @ A,X2: word @ A,W: word @ A,Xa: word @ A] :
( ( ord_less @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ V @ Xb ) @ X2 ) @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W @ Xa ) @ X2 ) )
=> ( ( ord_less_eq @ ( word @ A ) @ X2 @ ( plus_plus @ ( word @ A ) @ W @ Xa ) )
=> ( ord_less @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ V @ Xb ) @ ( plus_plus @ ( word @ A ) @ W @ Xa ) ) ) ) ) ).
% word_plus_mcs(1)
thf(fact_1785_word__plus__mcs_I2_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [V: word @ A,Xb: word @ A,W: word @ A,Xa: word @ A,X2: word @ A] :
( ( ord_less @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ V @ Xb ) @ ( plus_plus @ ( word @ A ) @ W @ Xa ) )
=> ( ( ord_less_eq @ ( word @ A ) @ X2 @ ( plus_plus @ ( word @ A ) @ V @ Xb ) )
=> ( ord_less @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ V @ Xb ) @ X2 ) @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W @ Xa ) @ X2 ) ) ) ) ) ).
% word_plus_mcs(2)
thf(fact_1786_Word_Oword__l__diffs_I1_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A,Xa: word @ A,X2: word @ A,Z: word @ A] :
( ( ord_less @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W @ Xa ) @ X2 ) @ ( minus_minus @ ( word @ A ) @ Z @ X2 ) )
=> ( ( ord_less_eq @ ( word @ A ) @ X2 @ Z )
=> ( ord_less @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W @ Xa ) @ Z ) ) ) ) ).
% Word.word_l_diffs(1)
thf(fact_1787_Word_Oword__l__diffs_I2_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A,Xa: word @ A,Z: word @ A,X2: word @ A] :
( ( ord_less @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W @ Xa ) @ Z )
=> ( ( ord_less_eq @ ( word @ A ) @ X2 @ ( plus_plus @ ( word @ A ) @ W @ Xa ) )
=> ( ord_less @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W @ Xa ) @ X2 ) @ ( minus_minus @ ( word @ A ) @ Z @ X2 ) ) ) ) ) ).
% Word.word_l_diffs(2)
thf(fact_1788_word__diff__ls_I1_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y2: word @ A,X2: word @ A,W: word @ A,Xa: word @ A] :
( ( ord_less @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ Y2 @ X2 ) @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W @ Xa ) @ X2 ) )
=> ( ( ord_less_eq @ ( word @ A ) @ X2 @ ( plus_plus @ ( word @ A ) @ W @ Xa ) )
=> ( ord_less @ ( word @ A ) @ Y2 @ ( plus_plus @ ( word @ A ) @ W @ Xa ) ) ) ) ) ).
% word_diff_ls(1)
thf(fact_1789_word__diff__ls_I2_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y2: word @ A,W: word @ A,Xa: word @ A,X2: word @ A] :
( ( ord_less @ ( word @ A ) @ Y2 @ ( plus_plus @ ( word @ A ) @ W @ Xa ) )
=> ( ( ord_less_eq @ ( word @ A ) @ X2 @ Y2 )
=> ( ord_less @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ Y2 @ X2 ) @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W @ Xa ) @ X2 ) ) ) ) ) ).
% word_diff_ls(2)
thf(fact_1790_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_1791_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_1792_word__div__mult__le,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: word @ A,B2: word @ A] : ( ord_less_eq @ ( word @ A ) @ ( times_times @ ( word @ A ) @ ( divide_divide @ ( word @ A ) @ A2 @ B2 ) @ B2 ) @ A2 ) ) ).
% word_div_mult_le
thf(fact_1793_floor__exists,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X2: A] :
? [Z2: int] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Z2 ) @ X2 )
& ( ord_less @ A @ X2 @ ( ring_1_of_int @ A @ ( plus_plus @ int @ Z2 @ ( one_one @ int ) ) ) ) ) ) ).
% floor_exists
thf(fact_1794_floor__exists1,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X2: A] :
? [X3: int] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ X3 ) @ X2 )
& ( ord_less @ A @ X2 @ ( ring_1_of_int @ A @ ( plus_plus @ int @ X3 @ ( one_one @ int ) ) ) )
& ! [Y4: int] :
( ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Y4 ) @ X2 )
& ( ord_less @ A @ X2 @ ( ring_1_of_int @ A @ ( plus_plus @ int @ Y4 @ ( one_one @ int ) ) ) ) )
=> ( Y4 = X3 ) ) ) ) ).
% floor_exists1
thf(fact_1795_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_1796_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_1797_of__nat__less__of__int__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: nat,X2: int] :
( ( ord_less @ A @ ( semiring_1_of_nat @ A @ N ) @ ( ring_1_of_int @ A @ X2 ) )
= ( ord_less @ int @ ( semiring_1_of_nat @ int @ N ) @ X2 ) ) ) ).
% of_nat_less_of_int_iff
thf(fact_1798_bset_I9_J,axiom,
! [D2: int,D4: int,B5: set @ int,T2: int] :
( ( dvd_dvd @ int @ D2 @ D4 )
=> ! [X4: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ B5 )
=> ( X4
!= ( plus_plus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( dvd_dvd @ int @ D2 @ ( plus_plus @ int @ X4 @ T2 ) )
=> ( dvd_dvd @ int @ D2 @ ( plus_plus @ int @ ( minus_minus @ int @ X4 @ D4 ) @ T2 ) ) ) ) ) ).
% bset(9)
thf(fact_1799_bset_I10_J,axiom,
! [D2: int,D4: int,B5: set @ int,T2: int] :
( ( dvd_dvd @ int @ D2 @ D4 )
=> ! [X4: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ B5 )
=> ( X4
!= ( plus_plus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ~ ( dvd_dvd @ int @ D2 @ ( plus_plus @ int @ X4 @ T2 ) )
=> ~ ( dvd_dvd @ int @ D2 @ ( plus_plus @ int @ ( minus_minus @ int @ X4 @ D4 ) @ T2 ) ) ) ) ) ).
% bset(10)
thf(fact_1800_aset_I9_J,axiom,
! [D2: int,D4: int,A3: set @ int,T2: int] :
( ( dvd_dvd @ int @ D2 @ D4 )
=> ! [X4: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ A3 )
=> ( X4
!= ( minus_minus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( dvd_dvd @ int @ D2 @ ( plus_plus @ int @ X4 @ T2 ) )
=> ( dvd_dvd @ int @ D2 @ ( plus_plus @ int @ ( plus_plus @ int @ X4 @ D4 ) @ T2 ) ) ) ) ) ).
% aset(9)
thf(fact_1801_aset_I10_J,axiom,
! [D2: int,D4: int,A3: set @ int,T2: int] :
( ( dvd_dvd @ int @ D2 @ D4 )
=> ! [X4: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ A3 )
=> ( X4
!= ( minus_minus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ~ ( dvd_dvd @ int @ D2 @ ( plus_plus @ int @ X4 @ T2 ) )
=> ~ ( dvd_dvd @ int @ D2 @ ( plus_plus @ int @ ( plus_plus @ int @ X4 @ D4 ) @ T2 ) ) ) ) ) ).
% aset(10)
thf(fact_1802_int__le__real__less,axiom,
( ( ord_less_eq @ int )
= ( ^ [N3: int,M5: int] : ( ord_less @ real @ ( ring_1_of_int @ real @ N3 ) @ ( plus_plus @ real @ ( ring_1_of_int @ real @ M5 ) @ ( one_one @ real ) ) ) ) ) ).
% int_le_real_less
thf(fact_1803_int__less__real__le,axiom,
( ( ord_less @ int )
= ( ^ [N3: int,M5: int] : ( ord_less_eq @ real @ ( plus_plus @ real @ ( ring_1_of_int @ real @ N3 ) @ ( one_one @ real ) ) @ ( ring_1_of_int @ real @ M5 ) ) ) ) ).
% int_less_real_le
thf(fact_1804_ceiling__split,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [P: int > $o,T2: A] :
( ( P @ ( archimedean_ceiling @ A @ T2 ) )
= ( ! [I3: int] :
( ( ( ord_less @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ I3 ) @ ( one_one @ A ) ) @ T2 )
& ( ord_less_eq @ A @ T2 @ ( ring_1_of_int @ A @ I3 ) ) )
=> ( P @ I3 ) ) ) ) ) ).
% ceiling_split
thf(fact_1805_ceiling__eq__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A,A2: int] :
( ( ( archimedean_ceiling @ A @ X2 )
= A2 )
= ( ( ord_less @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ A2 ) @ ( one_one @ A ) ) @ X2 )
& ( ord_less_eq @ A @ X2 @ ( ring_1_of_int @ A @ A2 ) ) ) ) ) ).
% ceiling_eq_iff
thf(fact_1806_ceiling__unique,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z: int,X2: A] :
( ( ord_less @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ Z ) @ ( one_one @ A ) ) @ X2 )
=> ( ( ord_less_eq @ A @ X2 @ ( ring_1_of_int @ A @ Z ) )
=> ( ( archimedean_ceiling @ A @ X2 )
= Z ) ) ) ) ).
% ceiling_unique
thf(fact_1807_ceiling__correct,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A] :
( ( ord_less @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ A @ X2 ) ) @ ( one_one @ A ) ) @ X2 )
& ( ord_less_eq @ A @ X2 @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ A @ X2 ) ) ) ) ) ).
% ceiling_correct
thf(fact_1808_ceiling__less__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A,Z: int] :
( ( ord_less @ int @ ( archimedean_ceiling @ A @ X2 ) @ Z )
= ( ord_less_eq @ A @ X2 @ ( minus_minus @ A @ ( ring_1_of_int @ A @ Z ) @ ( one_one @ A ) ) ) ) ) ).
% ceiling_less_iff
thf(fact_1809_le__ceiling__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z: int,X2: A] :
( ( ord_less_eq @ int @ Z @ ( archimedean_ceiling @ A @ X2 ) )
= ( ord_less @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ Z ) @ ( one_one @ A ) ) @ X2 ) ) ) ).
% le_ceiling_iff
thf(fact_1810_bset_I3_J,axiom,
! [D4: int,T2: int,B5: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D4 )
=> ( ( member @ int @ ( minus_minus @ int @ T2 @ ( one_one @ int ) ) @ B5 )
=> ! [X4: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ B5 )
=> ( X4
!= ( plus_plus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( X4 = T2 )
=> ( ( minus_minus @ int @ X4 @ D4 )
= T2 ) ) ) ) ) ).
% bset(3)
thf(fact_1811_bset_I4_J,axiom,
! [D4: int,T2: int,B5: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D4 )
=> ( ( member @ int @ T2 @ B5 )
=> ! [X4: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ B5 )
=> ( X4
!= ( plus_plus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( X4 != T2 )
=> ( ( minus_minus @ int @ X4 @ D4 )
!= T2 ) ) ) ) ) ).
% bset(4)
thf(fact_1812_bset_I5_J,axiom,
! [D4: int,B5: set @ int,T2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D4 )
=> ! [X4: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ B5 )
=> ( X4
!= ( plus_plus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ord_less @ int @ X4 @ T2 )
=> ( ord_less @ int @ ( minus_minus @ int @ X4 @ D4 ) @ T2 ) ) ) ) ).
% bset(5)
thf(fact_1813_bset_I7_J,axiom,
! [D4: int,T2: int,B5: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D4 )
=> ( ( member @ int @ T2 @ B5 )
=> ! [X4: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ B5 )
=> ( X4
!= ( plus_plus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ord_less @ int @ T2 @ X4 )
=> ( ord_less @ int @ T2 @ ( minus_minus @ int @ X4 @ D4 ) ) ) ) ) ) ).
% bset(7)
thf(fact_1814_aset_I3_J,axiom,
! [D4: int,T2: int,A3: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D4 )
=> ( ( member @ int @ ( plus_plus @ int @ T2 @ ( one_one @ int ) ) @ A3 )
=> ! [X4: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ A3 )
=> ( X4
!= ( minus_minus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( X4 = T2 )
=> ( ( plus_plus @ int @ X4 @ D4 )
= T2 ) ) ) ) ) ).
% aset(3)
thf(fact_1815_aset_I4_J,axiom,
! [D4: int,T2: int,A3: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D4 )
=> ( ( member @ int @ T2 @ A3 )
=> ! [X4: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ A3 )
=> ( X4
!= ( minus_minus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( X4 != T2 )
=> ( ( plus_plus @ int @ X4 @ D4 )
!= T2 ) ) ) ) ) ).
% aset(4)
thf(fact_1816_aset_I5_J,axiom,
! [D4: int,T2: int,A3: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D4 )
=> ( ( member @ int @ T2 @ A3 )
=> ! [X4: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ A3 )
=> ( X4
!= ( minus_minus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ord_less @ int @ X4 @ T2 )
=> ( ord_less @ int @ ( plus_plus @ int @ X4 @ D4 ) @ T2 ) ) ) ) ) ).
% aset(5)
thf(fact_1817_aset_I7_J,axiom,
! [D4: int,A3: set @ int,T2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D4 )
=> ! [X4: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ A3 )
=> ( X4
!= ( minus_minus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ord_less @ int @ T2 @ X4 )
=> ( ord_less @ int @ T2 @ ( plus_plus @ int @ X4 @ D4 ) ) ) ) ) ).
% aset(7)
thf(fact_1818_periodic__finite__ex,axiom,
! [D2: int,P: int > $o] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D2 )
=> ( ! [X3: int,K3: int] :
( ( P @ X3 )
= ( P @ ( minus_minus @ int @ X3 @ ( times_times @ int @ K3 @ D2 ) ) ) )
=> ( ( ? [X7: int] : ( P @ X7 ) )
= ( ? [X: int] :
( ( member @ int @ X @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D2 ) )
& ( P @ X ) ) ) ) ) ) ).
% periodic_finite_ex
thf(fact_1819_real__of__int__div2,axiom,
! [N: int,X2: int] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( minus_minus @ real @ ( divide_divide @ real @ ( ring_1_of_int @ real @ N ) @ ( ring_1_of_int @ real @ X2 ) ) @ ( ring_1_of_int @ real @ ( divide_divide @ int @ N @ X2 ) ) ) ) ).
% real_of_int_div2
thf(fact_1820_real__of__int__div3,axiom,
! [N: int,X2: int] : ( ord_less_eq @ real @ ( minus_minus @ real @ ( divide_divide @ real @ ( ring_1_of_int @ real @ N ) @ ( ring_1_of_int @ real @ X2 ) ) @ ( ring_1_of_int @ real @ ( divide_divide @ int @ N @ X2 ) ) ) @ ( one_one @ real ) ) ).
% real_of_int_div3
thf(fact_1821_ceiling__divide__upper,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Q2: A,P2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Q2 )
=> ( ord_less_eq @ A @ P2 @ ( times_times @ A @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ A @ ( divide_divide @ A @ P2 @ Q2 ) ) ) @ Q2 ) ) ) ) ).
% ceiling_divide_upper
thf(fact_1822_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_1823_bset_I6_J,axiom,
! [D4: int,B5: set @ int,T2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D4 )
=> ! [X4: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ B5 )
=> ( X4
!= ( plus_plus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ord_less_eq @ int @ X4 @ T2 )
=> ( ord_less_eq @ int @ ( minus_minus @ int @ X4 @ D4 ) @ T2 ) ) ) ) ).
% bset(6)
thf(fact_1824_bset_I8_J,axiom,
! [D4: int,T2: int,B5: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D4 )
=> ( ( member @ int @ ( minus_minus @ int @ T2 @ ( one_one @ int ) ) @ B5 )
=> ! [X4: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ B5 )
=> ( X4
!= ( plus_plus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ord_less_eq @ int @ T2 @ X4 )
=> ( ord_less_eq @ int @ T2 @ ( minus_minus @ int @ X4 @ D4 ) ) ) ) ) ) ).
% bset(8)
thf(fact_1825_aset_I6_J,axiom,
! [D4: int,T2: int,A3: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D4 )
=> ( ( member @ int @ ( plus_plus @ int @ T2 @ ( one_one @ int ) ) @ A3 )
=> ! [X4: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ A3 )
=> ( X4
!= ( minus_minus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ord_less_eq @ int @ X4 @ T2 )
=> ( ord_less_eq @ int @ ( plus_plus @ int @ X4 @ D4 ) @ T2 ) ) ) ) ) ).
% aset(6)
thf(fact_1826_aset_I8_J,axiom,
! [D4: int,A3: set @ int,T2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D4 )
=> ! [X4: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ A3 )
=> ( X4
!= ( minus_minus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ord_less_eq @ int @ T2 @ X4 )
=> ( ord_less_eq @ int @ T2 @ ( plus_plus @ int @ X4 @ D4 ) ) ) ) ) ).
% aset(8)
thf(fact_1827_cppi,axiom,
! [D4: int,P: int > $o,P4: int > $o,A3: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D4 )
=> ( ? [Z4: int] :
! [X3: int] :
( ( ord_less @ int @ Z4 @ X3 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ! [X3: int] :
( ! [Xa2: int] :
( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ A3 )
=> ( X3
!= ( minus_minus @ int @ Xb2 @ Xa2 ) ) ) )
=> ( ( P @ X3 )
=> ( P @ ( plus_plus @ int @ X3 @ D4 ) ) ) )
=> ( ! [X3: int,K3: int] :
( ( P4 @ X3 )
= ( P4 @ ( minus_minus @ int @ X3 @ ( times_times @ int @ K3 @ D4 ) ) ) )
=> ( ( ? [X7: int] : ( P @ X7 ) )
= ( ? [X: int] :
( ( member @ int @ X @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
& ( P4 @ X ) )
| ? [X: int] :
( ( member @ int @ X @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
& ? [Y: int] :
( ( member @ int @ Y @ A3 )
& ( P @ ( minus_minus @ int @ Y @ X ) ) ) ) ) ) ) ) ) ) ).
% cppi
thf(fact_1828_cpmi,axiom,
! [D4: int,P: int > $o,P4: int > $o,B5: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D4 )
=> ( ? [Z4: int] :
! [X3: int] :
( ( ord_less @ int @ X3 @ Z4 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ! [X3: int] :
( ! [Xa2: int] :
( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ B5 )
=> ( X3
!= ( plus_plus @ int @ Xb2 @ Xa2 ) ) ) )
=> ( ( P @ X3 )
=> ( P @ ( minus_minus @ int @ X3 @ D4 ) ) ) )
=> ( ! [X3: int,K3: int] :
( ( P4 @ X3 )
= ( P4 @ ( minus_minus @ int @ X3 @ ( times_times @ int @ K3 @ D4 ) ) ) )
=> ( ( ? [X7: int] : ( P @ X7 ) )
= ( ? [X: int] :
( ( member @ int @ X @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
& ( P4 @ X ) )
| ? [X: int] :
( ( member @ int @ X @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
& ? [Y: int] :
( ( member @ int @ Y @ B5 )
& ( P @ ( plus_plus @ int @ Y @ X ) ) ) ) ) ) ) ) ) ) ).
% cpmi
thf(fact_1829_ceiling__divide__lower,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Q2: A,P2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Q2 )
=> ( ord_less @ A @ ( times_times @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ A @ ( divide_divide @ A @ P2 @ Q2 ) ) ) @ ( one_one @ A ) ) @ Q2 ) @ P2 ) ) ) ).
% ceiling_divide_lower
thf(fact_1830_ceiling__eq,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [N: int,X2: A] :
( ( ord_less @ A @ ( ring_1_of_int @ A @ N ) @ X2 )
=> ( ( ord_less_eq @ A @ X2 @ ( plus_plus @ A @ ( ring_1_of_int @ A @ N ) @ ( one_one @ A ) ) )
=> ( ( archimedean_ceiling @ A @ X2 )
= ( plus_plus @ int @ N @ ( one_one @ int ) ) ) ) ) ) ).
% ceiling_eq
thf(fact_1831_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_1832_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_1833_word__less__sub1,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A] :
( ( X2
!= ( zero_zero @ ( word @ A ) ) )
=> ( ( ord_less @ ( word @ A ) @ ( one_one @ ( word @ A ) ) @ X2 )
= ( ord_less @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ ( minus_minus @ ( word @ A ) @ X2 @ ( one_one @ ( word @ A ) ) ) ) ) ) ) ).
% word_less_sub1
thf(fact_1834_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_1835_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_1836_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_1837_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_1838_div__lt__mult,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [I: word @ A,K: word @ A,X2: word @ A] :
( ( ord_less @ ( word @ A ) @ I @ ( divide_divide @ ( word @ A ) @ K @ X2 ) )
=> ( ( ord_less @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ X2 )
=> ( ord_less @ ( word @ A ) @ ( times_times @ ( word @ A ) @ I @ X2 ) @ K ) ) ) ) ).
% div_lt_mult
thf(fact_1839_div__le__mult,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [I: word @ A,K: word @ A,X2: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ I @ ( divide_divide @ ( word @ A ) @ K @ X2 ) )
=> ( ( ord_less @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ X2 )
=> ( ord_less_eq @ ( word @ A ) @ ( times_times @ ( word @ A ) @ I @ X2 ) @ K ) ) ) ) ).
% div_le_mult
thf(fact_1840_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_1841_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_1842_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_1843_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_1844_vebt__insert_Osimps_I3_J,axiom,
! [Info: option @ ( product_prod @ nat @ nat ),Ts: list @ vEBT_VEBT,S3: vEBT_VEBT,X2: nat] :
( ( vEBT_vebt_insert @ ( vEBT_Node @ Info @ ( suc @ ( zero_zero @ nat ) ) @ Ts @ S3 ) @ X2 )
= ( vEBT_Node @ Info @ ( suc @ ( zero_zero @ nat ) ) @ Ts @ S3 ) ) ).
% vebt_insert.simps(3)
thf(fact_1845_low__inv,axiom,
! [X2: nat,N: nat,Y2: nat] :
( ( ord_less @ nat @ X2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
=> ( ( vEBT_VEBT_low @ ( plus_plus @ nat @ ( times_times @ nat @ Y2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) @ X2 ) @ N )
= X2 ) ) ).
% low_inv
thf(fact_1846_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_1847_height__compose__child,axiom,
! [T2: vEBT_VEBT,TreeList: list @ vEBT_VEBT,Info: option @ ( product_prod @ nat @ nat ),Deg: nat,Summary: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ T2 @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_VEBT_height @ T2 ) ) @ ( vEBT_VEBT_height @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) ) ) ) ).
% height_compose_child
thf(fact_1848_round__unique,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A,Y2: int] :
( ( ord_less @ A @ ( minus_minus @ A @ X2 @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ ( ring_1_of_int @ A @ Y2 ) )
=> ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Y2 ) @ ( plus_plus @ A @ X2 @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) )
=> ( ( archimedean_round @ A @ X2 )
= Y2 ) ) ) ) ).
% round_unique
thf(fact_1849_mult__le__cancel__iff2,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z: A,X2: A,Y2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Z )
=> ( ( ord_less_eq @ A @ ( times_times @ A @ Z @ X2 ) @ ( times_times @ A @ Z @ Y2 ) )
= ( ord_less_eq @ A @ X2 @ Y2 ) ) ) ) ).
% mult_le_cancel_iff2
thf(fact_1850_mult__le__cancel__iff1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z: A,X2: A,Y2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Z )
=> ( ( ord_less_eq @ A @ ( times_times @ A @ X2 @ Z ) @ ( times_times @ A @ Y2 @ Z ) )
= ( ord_less_eq @ A @ X2 @ Y2 ) ) ) ) ).
% mult_le_cancel_iff1
thf(fact_1851_bit__split__inv,axiom,
! [X2: nat,D2: nat] :
( ( vEBT_VEBT_bit_concat @ ( vEBT_VEBT_high @ X2 @ D2 ) @ ( vEBT_VEBT_low @ X2 @ D2 ) @ D2 )
= X2 ) ).
% bit_split_inv
thf(fact_1852_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_1853_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_1854_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_1855_Ball__set__replicate,axiom,
! [A: $tType,N: nat,A2: A,P: A > $o] :
( ( ! [X: A] :
( ( member @ A @ X @ ( set2 @ A @ ( replicate @ A @ N @ A2 ) ) )
=> ( P @ X ) ) )
= ( ( P @ A2 )
| ( N
= ( zero_zero @ nat ) ) ) ) ).
% Ball_set_replicate
thf(fact_1856_Bex__set__replicate,axiom,
! [A: $tType,N: nat,A2: A,P: A > $o] :
( ( ? [X: A] :
( ( member @ A @ X @ ( set2 @ A @ ( replicate @ A @ N @ A2 ) ) )
& ( P @ X ) ) )
= ( ( P @ A2 )
& ( N
!= ( zero_zero @ nat ) ) ) ) ).
% Bex_set_replicate
thf(fact_1857_in__set__replicate,axiom,
! [A: $tType,X2: A,N: nat,Y2: A] :
( ( member @ A @ X2 @ ( set2 @ A @ ( replicate @ A @ N @ Y2 ) ) )
= ( ( X2 = Y2 )
& ( N
!= ( zero_zero @ nat ) ) ) ) ).
% in_set_replicate
thf(fact_1858_round__0,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ( ( archimedean_round @ A @ ( zero_zero @ A ) )
= ( zero_zero @ int ) ) ) ).
% round_0
thf(fact_1859_round__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [N: num] :
( ( archimedean_round @ A @ ( numeral_numeral @ A @ N ) )
= ( numeral_numeral @ int @ N ) ) ) ).
% round_numeral
thf(fact_1860_round__1,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ( ( archimedean_round @ A @ ( one_one @ A ) )
= ( one_one @ int ) ) ) ).
% round_1
thf(fact_1861_round__of__nat,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [N: nat] :
( ( archimedean_round @ A @ ( semiring_1_of_nat @ A @ N ) )
= ( semiring_1_of_nat @ int @ N ) ) ) ).
% round_of_nat
thf(fact_1862_subset__code_I1_J,axiom,
! [A: $tType,Xs2: list @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs2 ) @ B5 )
= ( ! [X: A] :
( ( member @ A @ X @ ( set2 @ A @ Xs2 ) )
=> ( member @ A @ X @ B5 ) ) ) ) ).
% subset_code(1)
thf(fact_1863_word__numeral__alt,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( numeral_numeral @ ( word @ A ) )
= ( ^ [B3: num] : ( ring_1_of_int @ ( word @ A ) @ ( numeral_numeral @ int @ B3 ) ) ) ) ) ).
% word_numeral_alt
thf(fact_1864_wi__hom__add,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: int,B2: int] :
( ( plus_plus @ ( word @ A ) @ ( ring_1_of_int @ ( word @ A ) @ A2 ) @ ( ring_1_of_int @ ( word @ A ) @ B2 ) )
= ( ring_1_of_int @ ( word @ A ) @ ( plus_plus @ int @ A2 @ B2 ) ) ) ) ).
% wi_hom_add
thf(fact_1865_word__of__int__power__hom,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: int,N: nat] :
( ( power_power @ ( word @ A ) @ ( ring_1_of_int @ ( word @ A ) @ A2 ) @ N )
= ( ring_1_of_int @ ( word @ A ) @ ( power_power @ int @ A2 @ N ) ) ) ) ).
% word_of_int_power_hom
thf(fact_1866_wi__hom__mult,axiom,
! [C: $tType] :
( ( type_len @ C )
=> ! [A2: int,B2: int] :
( ( times_times @ ( word @ C ) @ ( ring_1_of_int @ ( word @ C ) @ A2 ) @ ( ring_1_of_int @ ( word @ C ) @ B2 ) )
= ( ring_1_of_int @ ( word @ C ) @ ( times_times @ int @ A2 @ B2 ) ) ) ) ).
% wi_hom_mult
thf(fact_1867_round__mono,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A,Y2: A] :
( ( ord_less_eq @ A @ X2 @ Y2 )
=> ( ord_less_eq @ int @ ( archimedean_round @ A @ X2 ) @ ( archimedean_round @ A @ Y2 ) ) ) ) ).
% round_mono
thf(fact_1868_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_1869_VEBT__internal_Oexp__split__high__low_I2_J,axiom,
! [X2: nat,N: nat,M: nat] :
( ( ord_less @ nat @ X2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( plus_plus @ nat @ N @ M ) ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
=> ( ord_less @ nat @ ( vEBT_VEBT_low @ X2 @ N ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ) ) ).
% VEBT_internal.exp_split_high_low(2)
thf(fact_1870_mult__less__iff1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z: A,X2: A,Y2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Z )
=> ( ( ord_less @ A @ ( times_times @ A @ X2 @ Z ) @ ( times_times @ A @ Y2 @ Z ) )
= ( ord_less @ A @ X2 @ Y2 ) ) ) ) ).
% mult_less_iff1
thf(fact_1871_of__int__round__le,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A] : ( ord_less_eq @ A @ ( ring_1_of_int @ A @ ( archimedean_round @ A @ X2 ) ) @ ( plus_plus @ A @ X2 @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% of_int_round_le
thf(fact_1872_of__int__round__ge,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A] : ( ord_less_eq @ A @ ( minus_minus @ A @ X2 @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ ( ring_1_of_int @ A @ ( archimedean_round @ A @ X2 ) ) ) ) ).
% of_int_round_ge
thf(fact_1873_of__int__round__gt,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A] : ( ord_less @ A @ ( minus_minus @ A @ X2 @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ ( ring_1_of_int @ A @ ( archimedean_round @ A @ X2 ) ) ) ) ).
% of_int_round_gt
thf(fact_1874_vebt__insert_Osimps_I2_J,axiom,
! [Info: option @ ( product_prod @ nat @ nat ),Ts: list @ vEBT_VEBT,S3: vEBT_VEBT,X2: nat] :
( ( vEBT_vebt_insert @ ( vEBT_Node @ Info @ ( zero_zero @ nat ) @ Ts @ S3 ) @ X2 )
= ( vEBT_Node @ Info @ ( zero_zero @ nat ) @ Ts @ S3 ) ) ).
% vebt_insert.simps(2)
thf(fact_1875_both__member__options__ding,axiom,
! [Info: option @ ( product_prod @ nat @ nat ),Deg: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,N: nat,X2: nat] :
( ( vEBT_invar_vebt @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) @ N )
=> ( ( ord_less @ nat @ X2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg ) )
=> ( ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
=> ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) @ X2 ) ) ) ) ).
% both_member_options_ding
thf(fact_1876_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_1877_artanh__0,axiom,
! [A: $tType] :
( ( ( real_V3459762299906320749_field @ A )
& ( ln @ A ) )
=> ( ( artanh @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% artanh_0
thf(fact_1878_arsinh__0,axiom,
! [A: $tType] :
( ( ln @ A )
=> ( ( arsinh @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% arsinh_0
thf(fact_1879_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_1880_low__def,axiom,
( vEBT_VEBT_low
= ( ^ [X: nat,N3: nat] : ( modulo_modulo @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) ) ) ) ).
% low_def
thf(fact_1881_vebt__buildup_Oelims,axiom,
! [X2: nat,Y2: vEBT_VEBT] :
( ( ( vEBT_vebt_buildup @ X2 )
= Y2 )
=> ( ( ( X2
= ( zero_zero @ nat ) )
=> ( Y2
!= ( vEBT_Leaf @ $false @ $false ) ) )
=> ( ( ( X2
= ( suc @ ( zero_zero @ nat ) ) )
=> ( Y2
!= ( vEBT_Leaf @ $false @ $false ) ) )
=> ~ ! [Va2: nat] :
( ( X2
= ( suc @ ( suc @ Va2 ) ) )
=> ~ ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ Va2 ) ) )
=> ( Y2
= ( 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 ) ) )
=> ( Y2
= ( 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.elims
thf(fact_1882_intind,axiom,
! [A: $tType,I: nat,N: nat,P: A > $o,X2: A] :
( ( ord_less @ nat @ I @ N )
=> ( ( P @ X2 )
=> ( P @ ( nth @ A @ ( replicate @ A @ N @ X2 ) @ I ) ) ) ) ).
% intind
thf(fact_1883_Leaf__0__not,axiom,
! [A2: $o,B2: $o] :
~ ( vEBT_invar_vebt @ ( vEBT_Leaf @ A2 @ B2 ) @ ( zero_zero @ nat ) ) ).
% Leaf_0_not
thf(fact_1884_deg__1__Leafy,axiom,
! [T2: vEBT_VEBT,N: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( N
= ( one_one @ nat ) )
=> ? [A5: $o,B4: $o] :
( T2
= ( vEBT_Leaf @ A5 @ B4 ) ) ) ) ).
% deg_1_Leafy
thf(fact_1885_deg__1__Leaf,axiom,
! [T2: vEBT_VEBT] :
( ( vEBT_invar_vebt @ T2 @ ( one_one @ nat ) )
=> ? [A5: $o,B4: $o] :
( T2
= ( vEBT_Leaf @ A5 @ B4 ) ) ) ).
% deg_1_Leaf
thf(fact_1886_deg1Leaf,axiom,
! [T2: vEBT_VEBT] :
( ( vEBT_invar_vebt @ T2 @ ( one_one @ nat ) )
= ( ? [A4: $o,B3: $o] :
( T2
= ( vEBT_Leaf @ A4 @ B3 ) ) ) ) ).
% deg1Leaf
thf(fact_1887_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_1888_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_1889_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_1890_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_1891_mod__self,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ! [A2: A] :
( ( modulo_modulo @ A @ A2 @ A2 )
= ( zero_zero @ A ) ) ) ).
% mod_self
thf(fact_1892_mod__by__0,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ! [A2: A] :
( ( modulo_modulo @ A @ A2 @ ( zero_zero @ A ) )
= A2 ) ) ).
% mod_by_0
thf(fact_1893_mod__0,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ! [A2: A] :
( ( modulo_modulo @ A @ ( zero_zero @ A ) @ A2 )
= ( zero_zero @ A ) ) ) ).
% mod_0
thf(fact_1894_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_1895_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_1896_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_1897_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_1898_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_1899_mod__less,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ N )
=> ( ( modulo_modulo @ nat @ M @ N )
= M ) ) ).
% mod_less
thf(fact_1900_length__replicate,axiom,
! [A: $tType,N: nat,X2: A] :
( ( size_size @ ( list @ A ) @ ( replicate @ A @ N @ X2 ) )
= N ) ).
% length_replicate
thf(fact_1901_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_1902_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_1903_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_1904_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_1905_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_1906_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_1907_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_1908_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_1909_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_1910_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_1911_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_1912_mod__by__Suc__0,axiom,
! [M: nat] :
( ( modulo_modulo @ nat @ M @ ( suc @ ( zero_zero @ nat ) ) )
= ( zero_zero @ nat ) ) ).
% mod_by_Suc_0
thf(fact_1913_nth__replicate,axiom,
! [A: $tType,I: nat,N: nat,X2: A] :
( ( ord_less @ nat @ I @ N )
=> ( ( nth @ A @ ( replicate @ A @ N @ X2 ) @ I )
= X2 ) ) ).
% nth_replicate
thf(fact_1914_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_1915_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_1916_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_1917_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_1918_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_1919_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_1920_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_1921_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_1922_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_1923_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_1924_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_1925_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_1926_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_1927_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_1928_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_1929_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_1930_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_1931_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_1932_size__neq__size__imp__neq,axiom,
! [A: $tType] :
( ( size @ A )
=> ! [X2: A,Y2: A] :
( ( ( size_size @ A @ X2 )
!= ( size_size @ A @ Y2 ) )
=> ( X2 != Y2 ) ) ) ).
% size_neq_size_imp_neq
thf(fact_1933_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_1934_mod__add__cong,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A2: A,C2: A,A6: A,B2: A,B7: A] :
( ( ( modulo_modulo @ A @ A2 @ C2 )
= ( modulo_modulo @ A @ A6 @ 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 @ A6 @ B7 ) @ C2 ) ) ) ) ) ).
% mod_add_cong
thf(fact_1935_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_1936_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_1937_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_1938_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_1939_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_1940_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_1941_mod__mult__cong,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A2: A,C2: A,A6: A,B2: A,B7: A] :
( ( ( modulo_modulo @ A @ A2 @ C2 )
= ( modulo_modulo @ A @ A6 @ 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 @ A6 @ B7 ) @ C2 ) ) ) ) ) ).
% mod_mult_cong
thf(fact_1942_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_1943_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_1944_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_1945_mod__diff__cong,axiom,
! [A: $tType] :
( ( euclid8851590272496341667cancel @ A )
=> ! [A2: A,C2: A,A6: A,B2: A,B7: A] :
( ( ( modulo_modulo @ A @ A2 @ C2 )
= ( modulo_modulo @ A @ A6 @ 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 @ A6 @ B7 ) @ C2 ) ) ) ) ) ).
% mod_diff_cong
thf(fact_1946_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_1947_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_1948_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_1949_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_1950_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_1951_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_1952_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_1953_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_1954_VEBT_Oexhaust,axiom,
! [Y2: vEBT_VEBT] :
( ! [X112: option @ ( product_prod @ nat @ nat ),X122: nat,X132: list @ vEBT_VEBT,X142: vEBT_VEBT] :
( Y2
!= ( vEBT_Node @ X112 @ X122 @ X132 @ X142 ) )
=> ~ ! [X212: $o,X223: $o] :
( Y2
!= ( vEBT_Leaf @ X212 @ X223 ) ) ) ).
% VEBT.exhaust
thf(fact_1955_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_1956_mod__plus__right,axiom,
! [A2: nat,X2: nat,M: nat,B2: nat] :
( ( ( modulo_modulo @ nat @ ( plus_plus @ nat @ A2 @ X2 ) @ M )
= ( modulo_modulo @ nat @ ( plus_plus @ nat @ B2 @ X2 ) @ M ) )
= ( ( modulo_modulo @ nat @ A2 @ M )
= ( modulo_modulo @ nat @ B2 @ M ) ) ) ).
% mod_plus_right
thf(fact_1957_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_1958_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_1959_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_1960_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_1961_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_1962_cong__exp__iff__simps_I9_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M: num,Q2: num,N: num] :
( ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit0 @ M ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q2 ) ) )
= ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit0 @ N ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q2 ) ) ) )
= ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ Q2 ) )
= ( modulo_modulo @ A @ ( numeral_numeral @ A @ N ) @ ( numeral_numeral @ A @ Q2 ) ) ) ) ) ).
% cong_exp_iff_simps(9)
thf(fact_1963_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_1964_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_1965_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_1966_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_1967_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_1968_dvd__eq__mod__eq__0,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ( ( dvd_dvd @ A )
= ( ^ [A4: A,B3: A] :
( ( modulo_modulo @ A @ B3 @ A4 )
= ( zero_zero @ A ) ) ) ) ) ).
% dvd_eq_mod_eq_0
thf(fact_1969_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_1970_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_1971_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_1972_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_1973_mod__induct,axiom,
! [P: nat > $o,N: nat,P2: nat,M: nat] :
( ( P @ N )
=> ( ( ord_less @ nat @ N @ P2 )
=> ( ( ord_less @ nat @ M @ P2 )
=> ( ! [N2: nat] :
( ( ord_less @ nat @ N2 @ P2 )
=> ( ( P @ N2 )
=> ( P @ ( modulo_modulo @ nat @ ( suc @ N2 ) @ P2 ) ) ) )
=> ( P @ M ) ) ) ) ) ).
% mod_induct
thf(fact_1974_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_1975_gcd__nat__induct,axiom,
! [P: nat > nat > $o,M: nat,N: nat] :
( ! [M3: nat] : ( P @ M3 @ ( zero_zero @ nat ) )
=> ( ! [M3: nat,N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( P @ N2 @ ( modulo_modulo @ nat @ M3 @ N2 ) )
=> ( P @ M3 @ N2 ) ) )
=> ( P @ M @ N ) ) ) ).
% gcd_nat_induct
thf(fact_1976_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_1977_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_1978_word__rot__lem,axiom,
! [L2: nat,K: nat,D2: nat,N: nat] :
( ( ( plus_plus @ nat @ L2 @ K )
= ( plus_plus @ nat @ D2 @ ( modulo_modulo @ nat @ K @ L2 ) ) )
=> ( ( ord_less @ nat @ N @ L2 )
=> ( ( modulo_modulo @ nat @ ( plus_plus @ nat @ D2 @ N ) @ L2 )
= N ) ) ) ).
% word_rot_lem
thf(fact_1979_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_1980_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_1981_mod__if,axiom,
( ( modulo_modulo @ nat )
= ( ^ [M5: nat,N3: nat] : ( if @ nat @ ( ord_less @ nat @ M5 @ N3 ) @ M5 @ ( modulo_modulo @ nat @ ( minus_minus @ nat @ M5 @ N3 ) @ N3 ) ) ) ) ).
% mod_if
thf(fact_1982_mod__eq__0D,axiom,
! [M: nat,D2: nat] :
( ( ( modulo_modulo @ nat @ M @ D2 )
= ( zero_zero @ nat ) )
=> ? [Q3: nat] :
( M
= ( times_times @ nat @ D2 @ Q3 ) ) ) ).
% mod_eq_0D
thf(fact_1983_nat__minus__mod__plus__right,axiom,
! [N: nat,X2: nat,M: nat] :
( ( modulo_modulo @ nat @ ( minus_minus @ nat @ ( plus_plus @ nat @ N @ X2 ) @ ( modulo_modulo @ nat @ N @ M ) ) @ M )
= ( modulo_modulo @ nat @ X2 @ M ) ) ).
% nat_minus_mod_plus_right
thf(fact_1984_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_1985_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_1986_nat__mod__eq__iff,axiom,
! [X2: nat,N: nat,Y2: nat] :
( ( ( modulo_modulo @ nat @ X2 @ N )
= ( modulo_modulo @ nat @ Y2 @ N ) )
= ( ? [Q1: nat,Q22: nat] :
( ( plus_plus @ nat @ X2 @ ( times_times @ nat @ N @ Q1 ) )
= ( plus_plus @ nat @ Y2 @ ( times_times @ nat @ N @ Q22 ) ) ) ) ) ).
% nat_mod_eq_iff
thf(fact_1987_replicate__length__same,axiom,
! [A: $tType,Xs2: list @ A,X2: A] :
( ! [X3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs2 ) )
=> ( X3 = X2 ) )
=> ( ( replicate @ A @ ( size_size @ ( list @ A ) @ Xs2 ) @ X2 )
= Xs2 ) ) ).
% replicate_length_same
thf(fact_1988_replicate__eqI,axiom,
! [A: $tType,Xs2: list @ A,N: nat,X2: A] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
= N )
=> ( ! [Y3: A] :
( ( member @ A @ Y3 @ ( set2 @ A @ Xs2 ) )
=> ( Y3 = X2 ) )
=> ( Xs2
= ( replicate @ A @ N @ X2 ) ) ) ) ).
% replicate_eqI
thf(fact_1989_vebt__buildup_Osimps_I1_J,axiom,
( ( vEBT_vebt_buildup @ ( zero_zero @ nat ) )
= ( vEBT_Leaf @ $false @ $false ) ) ).
% vebt_buildup.simps(1)
thf(fact_1990_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_1991_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_1992_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_1993_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_1994_cong__exp__iff__simps_I2_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [N: num,Q2: num] :
( ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit0 @ N ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q2 ) ) )
= ( zero_zero @ A ) )
= ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ N ) @ ( numeral_numeral @ A @ Q2 ) )
= ( zero_zero @ A ) ) ) ) ).
% cong_exp_iff_simps(2)
thf(fact_1995_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_1996_cong__exp__iff__simps_I6_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [Q2: num,N: num] :
( ( modulo_modulo @ A @ ( numeral_numeral @ A @ one2 ) @ ( numeral_numeral @ A @ ( bit0 @ Q2 ) ) )
!= ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit0 @ N ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q2 ) ) ) ) ) ).
% cong_exp_iff_simps(6)
thf(fact_1997_cong__exp__iff__simps_I8_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M: num,Q2: num] :
( ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit0 @ M ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q2 ) ) )
!= ( modulo_modulo @ A @ ( numeral_numeral @ A @ one2 ) @ ( numeral_numeral @ A @ ( bit0 @ Q2 ) ) ) ) ) ).
% cong_exp_iff_simps(8)
thf(fact_1998_cong__exp__iff__simps_I10_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M: num,Q2: num,N: num] :
( ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit0 @ M ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q2 ) ) )
!= ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit1 @ N ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q2 ) ) ) ) ) ).
% cong_exp_iff_simps(10)
thf(fact_1999_cong__exp__iff__simps_I12_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M: num,Q2: num,N: num] :
( ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit1 @ M ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q2 ) ) )
!= ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit0 @ N ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q2 ) ) ) ) ) ).
% cong_exp_iff_simps(12)
thf(fact_2000_cong__exp__iff__simps_I13_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M: num,Q2: num,N: num] :
( ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit1 @ M ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q2 ) ) )
= ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit1 @ N ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q2 ) ) ) )
= ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ Q2 ) )
= ( modulo_modulo @ A @ ( numeral_numeral @ A @ N ) @ ( numeral_numeral @ A @ Q2 ) ) ) ) ) ).
% cong_exp_iff_simps(13)
thf(fact_2001_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_2002_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_2003_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_2004_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_2005_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_2006_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_2007_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_2008_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_2009_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_2010_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_2011_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_2012_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_2013_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_2014_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_2015_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_2016_div__less__mono,axiom,
! [A3: nat,B5: nat,N: nat] :
( ( ord_less @ nat @ A3 @ B5 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ( modulo_modulo @ nat @ A3 @ N )
= ( zero_zero @ nat ) )
=> ( ( ( modulo_modulo @ nat @ B5 @ N )
= ( zero_zero @ nat ) )
=> ( ord_less @ nat @ ( divide_divide @ nat @ A3 @ N ) @ ( divide_divide @ nat @ B5 @ N ) ) ) ) ) ) ).
% div_less_mono
thf(fact_2017_mod__nat__add,axiom,
! [X2: nat,Z: nat,Y2: nat] :
( ( ord_less @ nat @ X2 @ Z )
=> ( ( ord_less @ nat @ Y2 @ Z )
=> ( ( ( ord_less @ nat @ ( plus_plus @ nat @ X2 @ Y2 ) @ Z )
=> ( ( modulo_modulo @ nat @ ( plus_plus @ nat @ X2 @ Y2 ) @ Z )
= ( plus_plus @ nat @ X2 @ Y2 ) ) )
& ( ~ ( ord_less @ nat @ ( plus_plus @ nat @ X2 @ Y2 ) @ Z )
=> ( ( modulo_modulo @ nat @ ( plus_plus @ nat @ X2 @ Y2 ) @ Z )
= ( minus_minus @ nat @ ( plus_plus @ nat @ X2 @ Y2 ) @ Z ) ) ) ) ) ) ).
% mod_nat_add
thf(fact_2018_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_2019_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_2020_nat__mod__eq__lemma,axiom,
! [X2: nat,N: nat,Y2: nat] :
( ( ( modulo_modulo @ nat @ X2 @ N )
= ( modulo_modulo @ nat @ Y2 @ N ) )
=> ( ( ord_less_eq @ nat @ Y2 @ X2 )
=> ? [Q3: nat] :
( X2
= ( plus_plus @ nat @ Y2 @ ( times_times @ nat @ N @ Q3 ) ) ) ) ) ).
% nat_mod_eq_lemma
thf(fact_2021_mod__eq__nat2E,axiom,
! [M: nat,Q2: nat,N: nat] :
( ( ( modulo_modulo @ nat @ M @ Q2 )
= ( modulo_modulo @ nat @ N @ Q2 ) )
=> ( ( ord_less_eq @ nat @ M @ N )
=> ~ ! [S2: nat] :
( N
!= ( plus_plus @ nat @ M @ ( times_times @ nat @ Q2 @ S2 ) ) ) ) ) ).
% mod_eq_nat2E
thf(fact_2022_mod__eq__nat1E,axiom,
! [M: nat,Q2: nat,N: nat] :
( ( ( modulo_modulo @ nat @ M @ Q2 )
= ( modulo_modulo @ nat @ N @ Q2 ) )
=> ( ( ord_less_eq @ nat @ N @ M )
=> ~ ! [S2: nat] :
( M
!= ( plus_plus @ nat @ N @ ( times_times @ nat @ Q2 @ S2 ) ) ) ) ) ).
% mod_eq_nat1E
thf(fact_2023_length__pos__if__in__set,axiom,
! [A: $tType,X2: A,Xs2: list @ A] :
( ( member @ A @ X2 @ ( set2 @ A @ Xs2 ) )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Xs2 ) ) ) ).
% length_pos_if_in_set
thf(fact_2024_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_2025_mod__mult2__eq,axiom,
! [M: nat,N: nat,Q2: nat] :
( ( modulo_modulo @ nat @ M @ ( times_times @ nat @ N @ Q2 ) )
= ( plus_plus @ nat @ ( times_times @ nat @ N @ ( modulo_modulo @ nat @ ( divide_divide @ nat @ M @ N ) @ Q2 ) ) @ ( modulo_modulo @ nat @ M @ N ) ) ) ).
% mod_mult2_eq
thf(fact_2026_modulo__nat__def,axiom,
( ( modulo_modulo @ nat )
= ( ^ [M5: nat,N3: nat] : ( minus_minus @ nat @ M5 @ ( times_times @ nat @ ( divide_divide @ nat @ M5 @ N3 ) @ N3 ) ) ) ) ).
% modulo_nat_def
thf(fact_2027_mod__eq__dvd__iff__nat,axiom,
! [N: nat,M: nat,Q2: nat] :
( ( ord_less_eq @ nat @ N @ M )
=> ( ( ( modulo_modulo @ nat @ M @ Q2 )
= ( modulo_modulo @ nat @ N @ Q2 ) )
= ( dvd_dvd @ nat @ Q2 @ ( minus_minus @ nat @ M @ N ) ) ) ) ).
% mod_eq_dvd_iff_nat
thf(fact_2028_vebt__buildup_Osimps_I2_J,axiom,
( ( vEBT_vebt_buildup @ ( suc @ ( zero_zero @ nat ) ) )
= ( vEBT_Leaf @ $false @ $false ) ) ).
% vebt_buildup.simps(2)
thf(fact_2029_vebt__insert_Osimps_I1_J,axiom,
! [X2: nat,A2: $o,B2: $o] :
( ( ( X2
= ( zero_zero @ nat ) )
=> ( ( vEBT_vebt_insert @ ( vEBT_Leaf @ A2 @ B2 ) @ X2 )
= ( vEBT_Leaf @ $true @ B2 ) ) )
& ( ( X2
!= ( zero_zero @ nat ) )
=> ( ( ( X2
= ( one_one @ nat ) )
=> ( ( vEBT_vebt_insert @ ( vEBT_Leaf @ A2 @ B2 ) @ X2 )
= ( vEBT_Leaf @ A2 @ $true ) ) )
& ( ( X2
!= ( one_one @ nat ) )
=> ( ( vEBT_vebt_insert @ ( vEBT_Leaf @ A2 @ B2 ) @ X2 )
= ( vEBT_Leaf @ A2 @ B2 ) ) ) ) ) ) ).
% vebt_insert.simps(1)
thf(fact_2030_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_2031_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_2032_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_2033_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_2034_VEBT__internal_Onaive__member_Osimps_I1_J,axiom,
! [A2: $o,B2: $o,X2: nat] :
( ( vEBT_V5719532721284313246member @ ( vEBT_Leaf @ A2 @ B2 ) @ X2 )
= ( ( ( X2
= ( zero_zero @ nat ) )
=> A2 )
& ( ( X2
!= ( zero_zero @ nat ) )
=> ( ( ( X2
= ( one_one @ nat ) )
=> B2 )
& ( X2
= ( one_one @ nat ) ) ) ) ) ) ).
% VEBT_internal.naive_member.simps(1)
thf(fact_2035_cong__exp__iff__simps_I3_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [N: num,Q2: num] :
( ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit1 @ N ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q2 ) ) )
!= ( zero_zero @ A ) ) ) ).
% cong_exp_iff_simps(3)
thf(fact_2036_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_2037_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_2038_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_2039_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_2040_mod__lemma,axiom,
! [C2: nat,R2: nat,B2: nat,Q2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ C2 )
=> ( ( ord_less @ nat @ R2 @ B2 )
=> ( ord_less @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ B2 @ ( modulo_modulo @ nat @ Q2 @ C2 ) ) @ R2 ) @ ( times_times @ nat @ B2 @ C2 ) ) ) ) ).
% mod_lemma
thf(fact_2041_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 ) )
=> ! [I3: nat,J3: nat] :
( ( ord_less @ nat @ J3 @ N )
=> ( ( M
= ( plus_plus @ nat @ ( times_times @ nat @ N @ I3 ) @ J3 ) )
=> ( P @ J3 ) ) ) ) ) ) ).
% split_mod
thf(fact_2042_VEBT__internal_Onaive__member_Oelims_I1_J,axiom,
! [X2: vEBT_VEBT,Xa: nat,Y2: $o] :
( ( ( vEBT_V5719532721284313246member @ X2 @ Xa )
= Y2 )
=> ( ! [A5: $o,B4: $o] :
( ( X2
= ( vEBT_Leaf @ A5 @ B4 ) )
=> ( Y2
= ( ~ ( ( ( Xa
= ( zero_zero @ nat ) )
=> A5 )
& ( ( Xa
!= ( zero_zero @ nat ) )
=> ( ( ( Xa
= ( one_one @ nat ) )
=> B4 )
& ( Xa
= ( one_one @ nat ) ) ) ) ) ) ) )
=> ( ( ? [Uu2: option @ ( product_prod @ nat @ nat ),Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT] :
( X2
= ( vEBT_Node @ Uu2 @ ( zero_zero @ nat ) @ Uv2 @ Uw2 ) )
=> Y2 )
=> ~ ! [Uy2: option @ ( product_prod @ nat @ nat ),V3: nat,TreeList2: list @ vEBT_VEBT] :
( ? [S2: vEBT_VEBT] :
( X2
= ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList2 @ S2 ) )
=> ( Y2
= ( ~ ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_V5719532721284313246member @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.naive_member.elims(1)
thf(fact_2043_VEBT__internal_Onaive__member_Oelims_I2_J,axiom,
! [X2: vEBT_VEBT,Xa: nat] :
( ( vEBT_V5719532721284313246member @ X2 @ Xa )
=> ( ! [A5: $o,B4: $o] :
( ( X2
= ( vEBT_Leaf @ A5 @ B4 ) )
=> ~ ( ( ( Xa
= ( zero_zero @ nat ) )
=> A5 )
& ( ( Xa
!= ( zero_zero @ nat ) )
=> ( ( ( Xa
= ( one_one @ nat ) )
=> B4 )
& ( Xa
= ( one_one @ nat ) ) ) ) ) )
=> ~ ! [Uy2: option @ ( product_prod @ nat @ nat ),V3: nat,TreeList2: list @ vEBT_VEBT] :
( ? [S2: vEBT_VEBT] :
( X2
= ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList2 @ S2 ) )
=> ~ ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_V5719532721284313246member @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ).
% VEBT_internal.naive_member.elims(2)
thf(fact_2044_VEBT__internal_Onaive__member_Oelims_I3_J,axiom,
! [X2: vEBT_VEBT,Xa: nat] :
( ~ ( vEBT_V5719532721284313246member @ X2 @ Xa )
=> ( ! [A5: $o,B4: $o] :
( ( X2
= ( vEBT_Leaf @ A5 @ B4 ) )
=> ( ( ( Xa
= ( zero_zero @ nat ) )
=> A5 )
& ( ( Xa
!= ( zero_zero @ nat ) )
=> ( ( ( Xa
= ( one_one @ nat ) )
=> B4 )
& ( Xa
= ( one_one @ nat ) ) ) ) ) )
=> ( ! [Uu2: option @ ( product_prod @ nat @ nat ),Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT] :
( X2
!= ( vEBT_Node @ Uu2 @ ( zero_zero @ nat ) @ Uv2 @ Uw2 ) )
=> ~ ! [Uy2: option @ ( product_prod @ nat @ nat ),V3: nat,TreeList2: list @ vEBT_VEBT] :
( ? [S2: vEBT_VEBT] :
( X2
= ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList2 @ S2 ) )
=> ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_V5719532721284313246member @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ) ).
% VEBT_internal.naive_member.elims(3)
thf(fact_2045_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_2046_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_2047_real__of__nat__div__aux,axiom,
! [X2: nat,D2: nat] :
( ( divide_divide @ real @ ( semiring_1_of_nat @ real @ X2 ) @ ( semiring_1_of_nat @ real @ D2 ) )
= ( plus_plus @ real @ ( semiring_1_of_nat @ real @ ( divide_divide @ nat @ X2 @ D2 ) ) @ ( divide_divide @ real @ ( semiring_1_of_nat @ real @ ( modulo_modulo @ nat @ X2 @ D2 ) ) @ ( semiring_1_of_nat @ real @ D2 ) ) ) ) ).
% real_of_nat_div_aux
thf(fact_2048_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_2049_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_2050_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_2051_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_2052_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_2053_cong__exp__iff__simps_I7_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [Q2: num,N: num] :
( ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ one2 ) @ ( numeral_numeral @ A @ ( bit0 @ Q2 ) ) )
= ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit1 @ N ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q2 ) ) ) )
= ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ N ) @ ( numeral_numeral @ A @ Q2 ) )
= ( zero_zero @ A ) ) ) ) ).
% cong_exp_iff_simps(7)
thf(fact_2054_cong__exp__iff__simps_I11_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M: num,Q2: num] :
( ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit1 @ M ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q2 ) ) )
= ( modulo_modulo @ A @ ( numeral_numeral @ A @ one2 ) @ ( numeral_numeral @ A @ ( bit0 @ Q2 ) ) ) )
= ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ Q2 ) )
= ( zero_zero @ A ) ) ) ) ).
% cong_exp_iff_simps(11)
thf(fact_2055_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_2056_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_2057_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_2058_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_2059_VEBT__internal_Onaive__member_Osimps_I3_J,axiom,
! [Uy: option @ ( product_prod @ nat @ nat ),V: nat,TreeList: list @ vEBT_VEBT,S3: vEBT_VEBT,X2: nat] :
( ( vEBT_V5719532721284313246member @ ( vEBT_Node @ Uy @ ( suc @ V ) @ TreeList @ S3 ) @ X2 )
= ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
=> ( vEBT_V5719532721284313246member @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) ) ) ) ).
% VEBT_internal.naive_member.simps(3)
thf(fact_2060_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_2061_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_2062_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_2063_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_2064_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_2065_VEBT__internal_Omembermima_Osimps_I5_J,axiom,
! [V: nat,TreeList: list @ vEBT_VEBT,Vd: vEBT_VEBT,X2: nat] :
( ( vEBT_VEBT_membermima @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V ) @ TreeList @ Vd ) @ X2 )
= ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
=> ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) ) ) ) ).
% VEBT_internal.membermima.simps(5)
thf(fact_2066_power__mod__div,axiom,
! [X2: nat,N: nat,M: nat] :
( ( divide_divide @ nat @ ( modulo_modulo @ nat @ X2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) )
= ( modulo_modulo @ nat @ ( divide_divide @ nat @ X2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ N @ M ) ) ) ) ).
% power_mod_div
thf(fact_2067_verit__le__mono__div,axiom,
! [A3: nat,B5: nat,N: nat] :
( ( ord_less @ nat @ A3 @ B5 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ord_less_eq @ nat
@ ( plus_plus @ nat @ ( divide_divide @ nat @ A3 @ N )
@ ( if @ nat
@ ( ( modulo_modulo @ nat @ B5 @ N )
= ( zero_zero @ nat ) )
@ ( one_one @ nat )
@ ( zero_zero @ nat ) ) )
@ ( divide_divide @ nat @ B5 @ N ) ) ) ) ).
% verit_le_mono_div
thf(fact_2068_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_2069_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_2070_mod__double__modulus,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M: A,X2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ M )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X2 )
=> ( ( ( modulo_modulo @ A @ X2 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) )
= ( modulo_modulo @ A @ X2 @ M ) )
| ( ( modulo_modulo @ A @ X2 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) )
= ( plus_plus @ A @ ( modulo_modulo @ A @ X2 @ M ) @ M ) ) ) ) ) ) ).
% mod_double_modulus
thf(fact_2071_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_2072_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_2073_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_2074_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_2075_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_2076_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_2077_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_2078_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_2079_in__children__def,axiom,
( vEBT_V5917875025757280293ildren
= ( ^ [N3: nat,TreeList3: list @ vEBT_VEBT,X: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ X @ N3 ) ) @ ( vEBT_VEBT_low @ X @ N3 ) ) ) ) ).
% in_children_def
thf(fact_2080_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_2081_artanh__def,axiom,
! [A: $tType] :
( ( ( real_V3459762299906320749_field @ A )
& ( ln @ A ) )
=> ( ( artanh @ A )
= ( ^ [X: A] : ( divide_divide @ A @ ( ln_ln @ A @ ( divide_divide @ A @ ( plus_plus @ A @ ( one_one @ A ) @ X ) @ ( minus_minus @ A @ ( one_one @ A ) @ X ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% artanh_def
thf(fact_2082_vebt__member_Osimps_I1_J,axiom,
! [A2: $o,B2: $o,X2: nat] :
( ( vEBT_vebt_member @ ( vEBT_Leaf @ A2 @ B2 ) @ X2 )
= ( ( ( X2
= ( zero_zero @ nat ) )
=> A2 )
& ( ( X2
!= ( zero_zero @ nat ) )
=> ( ( ( X2
= ( one_one @ nat ) )
=> B2 )
& ( X2
= ( one_one @ nat ) ) ) ) ) ) ).
% vebt_member.simps(1)
thf(fact_2083_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_2084_is__succ__in__set__def,axiom,
( vEBT_is_succ_in_set
= ( ^ [Xs: set @ nat,X: nat,Y: nat] :
( ( member @ nat @ Y @ Xs )
& ( ord_less @ nat @ X @ Y )
& ! [Z5: nat] :
( ( member @ nat @ Z5 @ Xs )
=> ( ( ord_less @ nat @ X @ Z5 )
=> ( ord_less_eq @ nat @ Y @ Z5 ) ) ) ) ) ) ).
% is_succ_in_set_def
thf(fact_2085_is__pred__in__set__def,axiom,
( vEBT_is_pred_in_set
= ( ^ [Xs: set @ nat,X: nat,Y: nat] :
( ( member @ nat @ Y @ Xs )
& ( ord_less @ nat @ Y @ X )
& ! [Z5: nat] :
( ( member @ nat @ Z5 @ Xs )
=> ( ( ord_less @ nat @ Z5 @ X )
=> ( ord_less_eq @ nat @ Z5 @ Y ) ) ) ) ) ) ).
% is_pred_in_set_def
thf(fact_2086_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_2087_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_2088_ln__inj__iff,axiom,
! [X2: real,Y2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ Y2 )
=> ( ( ( ln_ln @ real @ X2 )
= ( ln_ln @ real @ Y2 ) )
= ( X2 = Y2 ) ) ) ) ).
% ln_inj_iff
thf(fact_2089_ln__less__cancel__iff,axiom,
! [X2: real,Y2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ Y2 )
=> ( ( ord_less @ real @ ( ln_ln @ real @ X2 ) @ ( ln_ln @ real @ Y2 ) )
= ( ord_less @ real @ X2 @ Y2 ) ) ) ) ).
% ln_less_cancel_iff
thf(fact_2090_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_2091_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_2092_ln__le__cancel__iff,axiom,
! [X2: real,Y2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ Y2 )
=> ( ( ord_less_eq @ real @ ( ln_ln @ real @ X2 ) @ ( ln_ln @ real @ Y2 ) )
= ( ord_less_eq @ real @ X2 @ Y2 ) ) ) ) ).
% ln_le_cancel_iff
thf(fact_2093_ln__less__zero__iff,axiom,
! [X2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ord_less @ real @ ( ln_ln @ real @ X2 ) @ ( zero_zero @ real ) )
= ( ord_less @ real @ X2 @ ( one_one @ real ) ) ) ) ).
% ln_less_zero_iff
thf(fact_2094_ln__gt__zero__iff,axiom,
! [X2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( ln_ln @ real @ X2 ) )
= ( ord_less @ real @ ( one_one @ real ) @ X2 ) ) ) ).
% ln_gt_zero_iff
thf(fact_2095_ln__eq__zero__iff,axiom,
! [X2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ( ln_ln @ real @ X2 )
= ( zero_zero @ real ) )
= ( X2
= ( one_one @ real ) ) ) ) ).
% ln_eq_zero_iff
thf(fact_2096_ln__one,axiom,
! [A: $tType] :
( ( ln @ A )
=> ( ( ln_ln @ A @ ( one_one @ A ) )
= ( zero_zero @ A ) ) ) ).
% ln_one
thf(fact_2097_mod__neg__neg__trivial,axiom,
! [K: int,L2: int] :
( ( ord_less_eq @ int @ K @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ L2 @ K )
=> ( ( modulo_modulo @ int @ K @ L2 )
= K ) ) ) ).
% mod_neg_neg_trivial
thf(fact_2098_mod__pos__pos__trivial,axiom,
! [K: int,L2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
=> ( ( ord_less @ int @ K @ L2 )
=> ( ( modulo_modulo @ int @ K @ L2 )
= K ) ) ) ).
% mod_pos_pos_trivial
thf(fact_2099_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_2100_ln__ge__zero__iff,axiom,
! [X2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( ln_ln @ real @ X2 ) )
= ( ord_less_eq @ real @ ( one_one @ real ) @ X2 ) ) ) ).
% ln_ge_zero_iff
thf(fact_2101_ln__le__zero__iff,axiom,
! [X2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ord_less_eq @ real @ ( ln_ln @ real @ X2 ) @ ( zero_zero @ real ) )
= ( ord_less_eq @ real @ X2 @ ( one_one @ real ) ) ) ) ).
% ln_le_zero_iff
thf(fact_2102_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_2103_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_2104_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_2105_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_2106_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_2107_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_2108_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_2109_mod__plus__cong,axiom,
! [B2: int,B7: int,X2: int,X5: int,Y2: int,Y6: int,Z6: int] :
( ( B2 = B7 )
=> ( ( ( modulo_modulo @ int @ X2 @ B7 )
= ( modulo_modulo @ int @ X5 @ B7 ) )
=> ( ( ( modulo_modulo @ int @ Y2 @ B7 )
= ( modulo_modulo @ int @ Y6 @ B7 ) )
=> ( ( ( plus_plus @ int @ X5 @ Y6 )
= Z6 )
=> ( ( modulo_modulo @ int @ ( plus_plus @ int @ X2 @ Y2 ) @ B2 )
= ( modulo_modulo @ int @ Z6 @ B7 ) ) ) ) ) ) ).
% mod_plus_cong
thf(fact_2110_ln__less__self,axiom,
! [X2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( ord_less @ real @ ( ln_ln @ real @ X2 ) @ X2 ) ) ).
% ln_less_self
thf(fact_2111_log__def,axiom,
( log
= ( ^ [A4: real,X: real] : ( divide_divide @ real @ ( ln_ln @ real @ X ) @ ( ln_ln @ real @ A4 ) ) ) ) ).
% log_def
thf(fact_2112_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_2113_neg__mod__bound,axiom,
! [L2: int,K: int] :
( ( ord_less @ int @ L2 @ ( zero_zero @ int ) )
=> ( ord_less @ int @ L2 @ ( modulo_modulo @ int @ K @ L2 ) ) ) ).
% neg_mod_bound
thf(fact_2114_Euclidean__Division_Opos__mod__bound,axiom,
! [L2: int,K: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ L2 )
=> ( ord_less @ int @ ( modulo_modulo @ int @ K @ L2 ) @ L2 ) ) ).
% Euclidean_Division.pos_mod_bound
thf(fact_2115_zmod__eq__0__iff,axiom,
! [M: int,D2: int] :
( ( ( modulo_modulo @ int @ M @ D2 )
= ( zero_zero @ int ) )
= ( ? [Q4: int] :
( M
= ( times_times @ int @ D2 @ Q4 ) ) ) ) ).
% zmod_eq_0_iff
thf(fact_2116_zmod__eq__0D,axiom,
! [M: int,D2: int] :
( ( ( modulo_modulo @ int @ M @ D2 )
= ( zero_zero @ int ) )
=> ? [Q3: int] :
( M
= ( times_times @ int @ D2 @ Q3 ) ) ) ).
% zmod_eq_0D
thf(fact_2117_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_2118_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_2119_ln__bound,axiom,
! [X2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( ord_less_eq @ real @ ( ln_ln @ real @ X2 ) @ X2 ) ) ).
% ln_bound
thf(fact_2120_ln__gt__zero__imp__gt__one,axiom,
! [X2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( ln_ln @ real @ X2 ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( ord_less @ real @ ( one_one @ real ) @ X2 ) ) ) ).
% ln_gt_zero_imp_gt_one
thf(fact_2121_ln__less__zero,axiom,
! [X2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ord_less @ real @ X2 @ ( one_one @ real ) )
=> ( ord_less @ real @ ( ln_ln @ real @ X2 ) @ ( zero_zero @ real ) ) ) ) ).
% ln_less_zero
thf(fact_2122_ln__gt__zero,axiom,
! [X2: real] :
( ( ord_less @ real @ ( one_one @ real ) @ X2 )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( ln_ln @ real @ X2 ) ) ) ).
% ln_gt_zero
thf(fact_2123_ln__ge__zero,axiom,
! [X2: real] :
( ( ord_less_eq @ real @ ( one_one @ real ) @ X2 )
=> ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( ln_ln @ real @ X2 ) ) ) ).
% ln_ge_zero
thf(fact_2124_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_2125_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_2126_neg__mod__sign,axiom,
! [L2: int,K: int] :
( ( ord_less @ int @ L2 @ ( zero_zero @ int ) )
=> ( ord_less_eq @ int @ ( modulo_modulo @ int @ K @ L2 ) @ ( zero_zero @ int ) ) ) ).
% neg_mod_sign
thf(fact_2127_Euclidean__Division_Opos__mod__sign,axiom,
! [L2: int,K: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ L2 )
=> ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( modulo_modulo @ int @ K @ L2 ) ) ) ).
% Euclidean_Division.pos_mod_sign
thf(fact_2128_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_2129_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_2130_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_2131_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_2132_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_2133_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_2134_zdiv__mono__strict,axiom,
! [A3: int,B5: int,N: int] :
( ( ord_less @ int @ A3 @ B5 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ N )
=> ( ( ( modulo_modulo @ int @ A3 @ N )
= ( zero_zero @ int ) )
=> ( ( ( modulo_modulo @ int @ B5 @ N )
= ( zero_zero @ int ) )
=> ( ord_less @ int @ ( divide_divide @ int @ A3 @ N ) @ ( divide_divide @ int @ B5 @ N ) ) ) ) ) ) ).
% zdiv_mono_strict
thf(fact_2135_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_2136_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_2137_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_2138_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_2139_num_Osize_I4_J,axiom,
( ( size_size @ num @ one2 )
= ( zero_zero @ nat ) ) ).
% num.size(4)
thf(fact_2140_ln__ge__zero__imp__ge__one,axiom,
! [X2: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( ln_ln @ real @ X2 ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( ord_less_eq @ real @ ( one_one @ real ) @ X2 ) ) ) ).
% ln_ge_zero_imp_ge_one
thf(fact_2141_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_2142_ln__add__one__self__le__self,axiom,
! [X2: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
=> ( ord_less_eq @ real @ ( ln_ln @ real @ ( plus_plus @ real @ ( one_one @ real ) @ X2 ) ) @ X2 ) ) ).
% ln_add_one_self_le_self
thf(fact_2143_ln__mult,axiom,
! [X2: real,Y2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ Y2 )
=> ( ( ln_ln @ real @ ( times_times @ real @ X2 @ Y2 ) )
= ( plus_plus @ real @ ( ln_ln @ real @ X2 ) @ ( ln_ln @ real @ Y2 ) ) ) ) ) ).
% ln_mult
thf(fact_2144_ln__eq__minus__one,axiom,
! [X2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ( ln_ln @ real @ X2 )
= ( minus_minus @ real @ X2 @ ( one_one @ real ) ) )
=> ( X2
= ( one_one @ real ) ) ) ) ).
% ln_eq_minus_one
thf(fact_2145_ln__div,axiom,
! [X2: real,Y2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ Y2 )
=> ( ( ln_ln @ real @ ( divide_divide @ real @ X2 @ Y2 ) )
= ( minus_minus @ real @ ( ln_ln @ real @ X2 ) @ ( ln_ln @ real @ Y2 ) ) ) ) ) ).
% ln_div
thf(fact_2146_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_2147_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_2148_mod__pos__neg__trivial,axiom,
! [K: int,L2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ K )
=> ( ( ord_less_eq @ int @ ( plus_plus @ int @ K @ L2 ) @ ( zero_zero @ int ) )
=> ( ( modulo_modulo @ int @ K @ L2 )
= ( plus_plus @ int @ K @ L2 ) ) ) ) ).
% mod_pos_neg_trivial
thf(fact_2149_mod__pos__geq,axiom,
! [L2: int,K: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ L2 )
=> ( ( ord_less_eq @ int @ L2 @ K )
=> ( ( modulo_modulo @ int @ K @ L2 )
= ( modulo_modulo @ int @ ( minus_minus @ int @ K @ L2 ) @ L2 ) ) ) ) ).
% mod_pos_geq
thf(fact_2150_mod__int__pos__iff,axiom,
! [K: int,L2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( modulo_modulo @ int @ K @ L2 ) )
= ( ( dvd_dvd @ int @ L2 @ K )
| ( ( L2
= ( zero_zero @ int ) )
& ( ord_less_eq @ int @ ( zero_zero @ int ) @ K ) )
| ( ord_less @ int @ ( zero_zero @ int ) @ L2 ) ) ) ).
% mod_int_pos_iff
thf(fact_2151_real__of__int__div__aux,axiom,
! [X2: int,D2: int] :
( ( divide_divide @ real @ ( ring_1_of_int @ real @ X2 ) @ ( ring_1_of_int @ real @ D2 ) )
= ( plus_plus @ real @ ( ring_1_of_int @ real @ ( divide_divide @ int @ X2 @ D2 ) ) @ ( divide_divide @ real @ ( ring_1_of_int @ real @ ( modulo_modulo @ int @ X2 @ D2 ) ) @ ( ring_1_of_int @ real @ D2 ) ) ) ) ).
% real_of_int_div_aux
thf(fact_2152_ln__le__minus__one,axiom,
! [X2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( ord_less_eq @ real @ ( ln_ln @ real @ X2 ) @ ( minus_minus @ real @ X2 @ ( one_one @ real ) ) ) ) ).
% ln_le_minus_one
thf(fact_2153_ln__diff__le,axiom,
! [X2: real,Y2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ Y2 )
=> ( ord_less_eq @ real @ ( minus_minus @ real @ ( ln_ln @ real @ X2 ) @ ( ln_ln @ real @ Y2 ) ) @ ( divide_divide @ real @ ( minus_minus @ real @ X2 @ Y2 ) @ Y2 ) ) ) ) ).
% ln_diff_le
thf(fact_2154_ln__realpow,axiom,
! [X2: real,N: nat] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ln_ln @ real @ ( power_power @ real @ X2 @ N ) )
= ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ ( ln_ln @ real @ X2 ) ) ) ) ).
% ln_realpow
thf(fact_2155_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_2156_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_2157_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_2158_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_2159_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_2160_int__mod__pos__eq,axiom,
! [A2: int,B2: int,Q2: int,R2: int] :
( ( A2
= ( plus_plus @ int @ ( times_times @ int @ B2 @ Q2 ) @ R2 ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ R2 )
=> ( ( ord_less @ int @ R2 @ B2 )
=> ( ( modulo_modulo @ int @ A2 @ B2 )
= R2 ) ) ) ) ).
% int_mod_pos_eq
thf(fact_2161_int__mod__neg__eq,axiom,
! [A2: int,B2: int,Q2: int,R2: int] :
( ( A2
= ( plus_plus @ int @ ( times_times @ int @ B2 @ Q2 ) @ R2 ) )
=> ( ( ord_less_eq @ int @ R2 @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ B2 @ R2 )
=> ( ( modulo_modulo @ int @ A2 @ B2 )
= R2 ) ) ) ) ).
% int_mod_neg_eq
thf(fact_2162_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 )
=> ! [I3: int,J3: int] :
( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ J3 )
& ( ord_less @ int @ J3 @ K )
& ( N
= ( plus_plus @ int @ ( times_times @ int @ K @ I3 ) @ J3 ) ) )
=> ( P @ J3 ) ) )
& ( ( ord_less @ int @ K @ ( zero_zero @ int ) )
=> ! [I3: int,J3: int] :
( ( ( ord_less @ int @ K @ J3 )
& ( ord_less_eq @ int @ J3 @ ( zero_zero @ int ) )
& ( N
= ( plus_plus @ int @ ( times_times @ int @ K @ I3 ) @ J3 ) ) )
=> ( P @ J3 ) ) ) ) ) ).
% split_zmod
thf(fact_2163_mod__add__if__z,axiom,
! [X2: int,Z: int,Y2: int] :
( ( ord_less @ int @ X2 @ Z )
=> ( ( ord_less @ int @ Y2 @ Z )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y2 )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X2 )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z )
=> ( ( ( ord_less @ int @ ( plus_plus @ int @ X2 @ Y2 ) @ Z )
=> ( ( modulo_modulo @ int @ ( plus_plus @ int @ X2 @ Y2 ) @ Z )
= ( plus_plus @ int @ X2 @ Y2 ) ) )
& ( ~ ( ord_less @ int @ ( plus_plus @ int @ X2 @ Y2 ) @ Z )
=> ( ( modulo_modulo @ int @ ( plus_plus @ int @ X2 @ Y2 ) @ Z )
= ( minus_minus @ int @ ( plus_plus @ int @ X2 @ Y2 ) @ Z ) ) ) ) ) ) ) ) ) ).
% mod_add_if_z
thf(fact_2164_mod__sub__if__z,axiom,
! [X2: int,Z: int,Y2: int] :
( ( ord_less @ int @ X2 @ Z )
=> ( ( ord_less @ int @ Y2 @ Z )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y2 )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X2 )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z )
=> ( ( ( ord_less_eq @ int @ Y2 @ X2 )
=> ( ( modulo_modulo @ int @ ( minus_minus @ int @ X2 @ Y2 ) @ Z )
= ( minus_minus @ int @ X2 @ Y2 ) ) )
& ( ~ ( ord_less_eq @ int @ Y2 @ X2 )
=> ( ( modulo_modulo @ int @ ( minus_minus @ int @ X2 @ Y2 ) @ Z )
= ( plus_plus @ int @ ( minus_minus @ int @ X2 @ Y2 ) @ Z ) ) ) ) ) ) ) ) ) ).
% mod_sub_if_z
thf(fact_2165_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_2166_log__eq__div__ln__mult__log,axiom,
! [A2: real,B2: real,X2: 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 ) @ X2 )
=> ( ( log @ A2 @ X2 )
= ( times_times @ real @ ( divide_divide @ real @ ( ln_ln @ real @ B2 ) @ ( ln_ln @ real @ A2 ) ) @ ( log @ B2 @ X2 ) ) ) ) ) ) ) ) ).
% log_eq_div_ln_mult_log
thf(fact_2167_axxmod2,axiom,
! [X2: int] :
( ( ( modulo_modulo @ int @ ( plus_plus @ int @ ( plus_plus @ int @ ( one_one @ int ) @ X2 ) @ X2 ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) )
= ( one_one @ int ) )
& ( ( modulo_modulo @ int @ ( plus_plus @ int @ ( plus_plus @ int @ ( zero_zero @ int ) @ X2 ) @ X2 ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) )
= ( zero_zero @ int ) ) ) ).
% axxmod2
thf(fact_2168_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_2169_verit__le__mono__div__int,axiom,
! [A3: int,B5: int,N: int] :
( ( ord_less @ int @ A3 @ B5 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ N )
=> ( ord_less_eq @ int
@ ( plus_plus @ int @ ( divide_divide @ int @ A3 @ N )
@ ( if @ int
@ ( ( modulo_modulo @ int @ B5 @ N )
= ( zero_zero @ int ) )
@ ( one_one @ int )
@ ( zero_zero @ int ) ) )
@ ( divide_divide @ int @ B5 @ N ) ) ) ) ).
% verit_le_mono_div_int
thf(fact_2170_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 ) )
= ( ! [I3: int,J3: int] :
( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ J3 )
& ( ord_less @ int @ J3 @ K )
& ( N
= ( plus_plus @ int @ ( times_times @ int @ K @ I3 ) @ J3 ) ) )
=> ( P @ I3 @ J3 ) ) ) ) ) ).
% split_pos_lemma
thf(fact_2171_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 ) )
= ( ! [I3: int,J3: int] :
( ( ( ord_less @ int @ K @ J3 )
& ( ord_less_eq @ int @ J3 @ ( zero_zero @ int ) )
& ( N
= ( plus_plus @ int @ ( times_times @ int @ K @ I3 ) @ J3 ) ) )
=> ( P @ I3 @ J3 ) ) ) ) ) ).
% split_neg_lemma
thf(fact_2172_num_Osize_I5_J,axiom,
! [X23: num] :
( ( size_size @ num @ ( bit0 @ X23 ) )
= ( plus_plus @ nat @ ( size_size @ num @ X23 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% num.size(5)
thf(fact_2173_num_Osize_I6_J,axiom,
! [X33: num] :
( ( size_size @ num @ ( bit1 @ X33 ) )
= ( plus_plus @ nat @ ( size_size @ num @ X33 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% num.size(6)
thf(fact_2174_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_2175_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_2176_eq__diff__eq_H,axiom,
! [X2: real,Y2: real,Z: real] :
( ( X2
= ( minus_minus @ real @ Y2 @ Z ) )
= ( Y2
= ( plus_plus @ real @ X2 @ Z ) ) ) ).
% eq_diff_eq'
thf(fact_2177_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_2178_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_2179_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_2180_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_2181_ln__one__plus__pos__lower__bound,axiom,
! [X2: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ord_less_eq @ real @ X2 @ ( one_one @ real ) )
=> ( ord_less_eq @ real @ ( minus_minus @ real @ X2 @ ( power_power @ real @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( ln_ln @ real @ ( plus_plus @ real @ ( one_one @ real ) @ X2 ) ) ) ) ) ).
% ln_one_plus_pos_lower_bound
thf(fact_2182_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_2183_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_2184_vebt__member_Osimps_I2_J,axiom,
! [Uu: nat,Uv: list @ vEBT_VEBT,Uw: vEBT_VEBT,X2: nat] :
~ ( vEBT_vebt_member @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu @ Uv @ Uw ) @ X2 ) ).
% vebt_member.simps(2)
thf(fact_2185_foldr__zero,axiom,
! [Xs2: list @ nat,D2: nat] :
( ! [I2: nat] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ nat ) @ Xs2 ) )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ ( nth @ nat @ Xs2 @ I2 ) ) )
=> ( ord_less_eq @ nat @ ( size_size @ ( list @ nat ) @ Xs2 ) @ ( minus_minus @ nat @ ( foldr @ nat @ nat @ ( plus_plus @ nat ) @ Xs2 @ D2 ) @ D2 ) ) ) ).
% foldr_zero
thf(fact_2186_foldr__same,axiom,
! [Xs2: list @ real,Y2: 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 = Y2 ) )
=> ( ( foldr @ real @ real @ ( plus_plus @ real ) @ Xs2 @ ( zero_zero @ real ) )
= ( times_times @ real @ ( semiring_1_of_nat @ real @ ( size_size @ ( list @ real ) @ Xs2 ) ) @ Y2 ) ) ) ) ).
% foldr_same
thf(fact_2187_member__bound__size__univ,axiom,
! [T2: vEBT_VEBT,N: nat,U: real,X2: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( U
= ( power_power @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ N ) )
=> ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ ( vEBT_T_m_e_m_b_e_r @ T2 @ X2 ) ) @ ( plus_plus @ real @ ( numeral_numeral @ real @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit1 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ U ) ) ) ) ) ) ) ).
% member_bound_size_univ
thf(fact_2188_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_2189_obtain__set__succ,axiom,
! [X2: nat,Z: nat,A3: set @ nat,B5: set @ nat] :
( ( ord_less @ nat @ X2 @ Z )
=> ( ( vEBT_VEBT_max_in_set @ A3 @ Z )
=> ( ( finite_finite2 @ nat @ B5 )
=> ( ( A3 = B5 )
=> ? [X_1: nat] : ( vEBT_is_succ_in_set @ A3 @ X2 @ X_1 ) ) ) ) ) ).
% obtain_set_succ
thf(fact_2190_obtain__set__pred,axiom,
! [Z: nat,X2: nat,A3: set @ nat] :
( ( ord_less @ nat @ Z @ X2 )
=> ( ( vEBT_VEBT_min_in_set @ A3 @ Z )
=> ( ( finite_finite2 @ nat @ A3 )
=> ? [X_1: nat] : ( vEBT_is_pred_in_set @ A3 @ X2 @ X_1 ) ) ) ) ).
% obtain_set_pred
thf(fact_2191_ceiling__log__eq__powr__iff,axiom,
! [X2: real,B2: real,K: nat] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ord_less @ real @ ( one_one @ real ) @ B2 )
=> ( ( ( archimedean_ceiling @ real @ ( log @ B2 @ X2 ) )
= ( plus_plus @ int @ ( semiring_1_of_nat @ int @ K ) @ ( one_one @ int ) ) )
= ( ( ord_less @ real @ ( powr @ real @ B2 @ ( semiring_1_of_nat @ real @ K ) ) @ X2 )
& ( ord_less_eq @ real @ X2 @ ( powr @ real @ B2 @ ( semiring_1_of_nat @ real @ ( plus_plus @ nat @ K @ ( one_one @ nat ) ) ) ) ) ) ) ) ) ).
% ceiling_log_eq_powr_iff
thf(fact_2192_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_2193_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_2194_set__vebt__finite,axiom,
! [T2: vEBT_VEBT,N: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( finite_finite2 @ nat @ ( vEBT_VEBT_set_vebt @ T2 ) ) ) ).
% set_vebt_finite
thf(fact_2195_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 )
=> ~ ? [X4: nat] :
( ( member @ nat @ X4 @ Xs2 )
& ( ord_less @ nat @ X4 @ A2 ) ) ) ) ).
% pred_none_empty
thf(fact_2196_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 )
=> ~ ? [X4: nat] :
( ( member @ nat @ X4 @ Xs2 )
& ( ord_less @ nat @ A2 @ X4 ) ) ) ) ).
% succ_none_empty
thf(fact_2197_foldr__same__int,axiom,
! [Xs2: list @ nat,Y2: 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 = Y2 ) )
=> ( ( foldr @ nat @ nat @ ( plus_plus @ nat ) @ Xs2 @ ( zero_zero @ nat ) )
= ( times_times @ nat @ ( size_size @ ( list @ nat ) @ Xs2 ) @ Y2 ) ) ) ) ).
% foldr_same_int
thf(fact_2198_foldr__mono,axiom,
! [Xs2: list @ nat,Ys: list @ nat,C2: nat,D2: nat] :
( ( ( size_size @ ( list @ nat ) @ Xs2 )
= ( size_size @ ( list @ nat ) @ Ys ) )
=> ( ! [I2: nat] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ nat ) @ Xs2 ) )
=> ( ord_less @ nat @ ( nth @ nat @ Xs2 @ I2 ) @ ( nth @ nat @ Ys @ I2 ) ) )
=> ( ( ord_less_eq @ nat @ C2 @ 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_2199_powr__0,axiom,
! [A: $tType] :
( ( ln @ A )
=> ! [Z: A] :
( ( powr @ A @ ( zero_zero @ A ) @ Z )
= ( zero_zero @ A ) ) ) ).
% powr_0
thf(fact_2200_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_2201_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_2202_powr__zero__eq__one,axiom,
! [A: $tType] :
( ( ln @ A )
=> ! [X2: A] :
( ( ( X2
= ( zero_zero @ A ) )
=> ( ( powr @ A @ X2 @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) )
& ( ( X2
!= ( zero_zero @ A ) )
=> ( ( powr @ A @ X2 @ ( zero_zero @ A ) )
= ( one_one @ A ) ) ) ) ) ).
% powr_zero_eq_one
thf(fact_2203_powr__gt__zero,axiom,
! [X2: real,A2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( powr @ real @ X2 @ A2 ) )
= ( X2
!= ( zero_zero @ real ) ) ) ).
% powr_gt_zero
thf(fact_2204_powr__nonneg__iff,axiom,
! [A2: real,X2: real] :
( ( ord_less_eq @ real @ ( powr @ real @ A2 @ X2 ) @ ( zero_zero @ real ) )
= ( A2
= ( zero_zero @ real ) ) ) ).
% powr_nonneg_iff
thf(fact_2205_powr__less__cancel__iff,axiom,
! [X2: real,A2: real,B2: real] :
( ( ord_less @ real @ ( one_one @ real ) @ X2 )
=> ( ( ord_less @ real @ ( powr @ real @ X2 @ A2 ) @ ( powr @ real @ X2 @ B2 ) )
= ( ord_less @ real @ A2 @ B2 ) ) ) ).
% powr_less_cancel_iff
thf(fact_2206_powr__eq__one__iff,axiom,
! [A2: real,X2: real] :
( ( ord_less @ real @ ( one_one @ real ) @ A2 )
=> ( ( ( powr @ real @ A2 @ X2 )
= ( one_one @ real ) )
= ( X2
= ( zero_zero @ real ) ) ) ) ).
% powr_eq_one_iff
thf(fact_2207_powr__one__gt__zero__iff,axiom,
! [X2: real] :
( ( ( powr @ real @ X2 @ ( one_one @ real ) )
= X2 )
= ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 ) ) ).
% powr_one_gt_zero_iff
thf(fact_2208_powr__one,axiom,
! [X2: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( powr @ real @ X2 @ ( one_one @ real ) )
= X2 ) ) ).
% powr_one
thf(fact_2209_powr__le__cancel__iff,axiom,
! [X2: real,A2: real,B2: real] :
( ( ord_less @ real @ ( one_one @ real ) @ X2 )
=> ( ( ord_less_eq @ real @ ( powr @ real @ X2 @ A2 ) @ ( powr @ real @ X2 @ B2 ) )
= ( ord_less_eq @ real @ A2 @ B2 ) ) ) ).
% powr_le_cancel_iff
thf(fact_2210_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_2211_foldr__length,axiom,
! [A: $tType,L2: list @ A] :
( ( foldr @ A @ nat
@ ^ [X: A] : suc
@ L2
@ ( zero_zero @ nat ) )
= ( size_size @ ( list @ A ) @ L2 ) ) ).
% foldr_length
thf(fact_2212_log__powr__cancel,axiom,
! [A2: real,Y2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
=> ( ( A2
!= ( one_one @ real ) )
=> ( ( log @ A2 @ ( powr @ real @ A2 @ Y2 ) )
= Y2 ) ) ) ).
% log_powr_cancel
thf(fact_2213_powr__log__cancel,axiom,
! [A2: real,X2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
=> ( ( A2
!= ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( powr @ real @ A2 @ ( log @ A2 @ X2 ) )
= X2 ) ) ) ) ).
% powr_log_cancel
thf(fact_2214_powr__numeral,axiom,
! [X2: real,N: num] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( powr @ real @ X2 @ ( numeral_numeral @ real @ N ) )
= ( power_power @ real @ X2 @ ( numeral_numeral @ nat @ N ) ) ) ) ).
% powr_numeral
thf(fact_2215_finite__if__eq__beyond__finite,axiom,
! [A: $tType,S: set @ A,S4: set @ A] :
( ( finite_finite2 @ A @ S )
=> ( finite_finite2 @ ( set @ A )
@ ( collect @ ( set @ A )
@ ^ [S5: set @ A] :
( ( minus_minus @ ( set @ A ) @ S5 @ S )
= ( minus_minus @ ( set @ A ) @ S4 @ S ) ) ) ) ) ).
% finite_if_eq_beyond_finite
thf(fact_2216_powr__powr,axiom,
! [X2: real,A2: real,B2: real] :
( ( powr @ real @ ( powr @ real @ X2 @ A2 ) @ B2 )
= ( powr @ real @ X2 @ ( times_times @ real @ A2 @ B2 ) ) ) ).
% powr_powr
thf(fact_2217_finite__nat__set__iff__bounded__le,axiom,
( ( finite_finite2 @ nat )
= ( ^ [N8: set @ nat] :
? [M5: nat] :
! [X: nat] :
( ( member @ nat @ X @ N8 )
=> ( ord_less_eq @ nat @ X @ M5 ) ) ) ) ).
% finite_nat_set_iff_bounded_le
thf(fact_2218_finite__M__bounded__by__nat,axiom,
! [P: nat > $o,I: nat] :
( finite_finite2 @ nat
@ ( collect @ nat
@ ^ [K2: nat] :
( ( P @ K2 )
& ( ord_less @ nat @ K2 @ I ) ) ) ) ).
% finite_M_bounded_by_nat
thf(fact_2219_finite__less__ub,axiom,
! [F2: nat > nat,U: nat] :
( ! [N2: nat] : ( ord_less_eq @ nat @ N2 @ ( F2 @ N2 ) )
=> ( finite_finite2 @ nat
@ ( collect @ nat
@ ^ [N3: nat] : ( ord_less_eq @ nat @ ( F2 @ N3 ) @ U ) ) ) ) ).
% finite_less_ub
thf(fact_2220_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_2221_powr__less__mono2__neg,axiom,
! [A2: real,X2: real,Y2: real] :
( ( ord_less @ real @ A2 @ ( zero_zero @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ord_less @ real @ X2 @ Y2 )
=> ( ord_less @ real @ ( powr @ real @ Y2 @ A2 ) @ ( powr @ real @ X2 @ A2 ) ) ) ) ) ).
% powr_less_mono2_neg
thf(fact_2222_powr__non__neg,axiom,
! [A2: real,X2: real] :
~ ( ord_less @ real @ ( powr @ real @ A2 @ X2 ) @ ( zero_zero @ real ) ) ).
% powr_non_neg
thf(fact_2223_powr__mono2,axiom,
! [A2: real,X2: real,Y2: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ A2 )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ord_less_eq @ real @ X2 @ Y2 )
=> ( ord_less_eq @ real @ ( powr @ real @ X2 @ A2 ) @ ( powr @ real @ Y2 @ A2 ) ) ) ) ) ).
% powr_mono2
thf(fact_2224_powr__ge__pzero,axiom,
! [X2: real,Y2: real] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( powr @ real @ X2 @ Y2 ) ) ).
% powr_ge_pzero
thf(fact_2225_powr__less__cancel,axiom,
! [X2: real,A2: real,B2: real] :
( ( ord_less @ real @ ( powr @ real @ X2 @ A2 ) @ ( powr @ real @ X2 @ B2 ) )
=> ( ( ord_less @ real @ ( one_one @ real ) @ X2 )
=> ( ord_less @ real @ A2 @ B2 ) ) ) ).
% powr_less_cancel
thf(fact_2226_powr__less__mono,axiom,
! [A2: real,B2: real,X2: real] :
( ( ord_less @ real @ A2 @ B2 )
=> ( ( ord_less @ real @ ( one_one @ real ) @ X2 )
=> ( ord_less @ real @ ( powr @ real @ X2 @ A2 ) @ ( powr @ real @ X2 @ B2 ) ) ) ) ).
% powr_less_mono
thf(fact_2227_powr__mono,axiom,
! [A2: real,B2: real,X2: real] :
( ( ord_less_eq @ real @ A2 @ B2 )
=> ( ( ord_less_eq @ real @ ( one_one @ real ) @ X2 )
=> ( ord_less_eq @ real @ ( powr @ real @ X2 @ A2 ) @ ( powr @ real @ X2 @ B2 ) ) ) ) ).
% powr_mono
thf(fact_2228_ln__powr,axiom,
! [X2: real,Y2: real] :
( ( X2
!= ( zero_zero @ real ) )
=> ( ( ln_ln @ real @ ( powr @ real @ X2 @ Y2 ) )
= ( times_times @ real @ Y2 @ ( ln_ln @ real @ X2 ) ) ) ) ).
% ln_powr
thf(fact_2229_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_2230_powr__less__mono2,axiom,
! [A2: real,X2: real,Y2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ord_less @ real @ X2 @ Y2 )
=> ( ord_less @ real @ ( powr @ real @ X2 @ A2 ) @ ( powr @ real @ Y2 @ A2 ) ) ) ) ) ).
% powr_less_mono2
thf(fact_2231_powr__mono2_H,axiom,
! [A2: real,X2: real,Y2: real] :
( ( ord_less_eq @ real @ A2 @ ( zero_zero @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ord_less_eq @ real @ X2 @ Y2 )
=> ( ord_less_eq @ real @ ( powr @ real @ Y2 @ A2 ) @ ( powr @ real @ X2 @ A2 ) ) ) ) ) ).
% powr_mono2'
thf(fact_2232_powr__inj,axiom,
! [A2: real,X2: real,Y2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
=> ( ( A2
!= ( one_one @ real ) )
=> ( ( ( powr @ real @ A2 @ X2 )
= ( powr @ real @ A2 @ Y2 ) )
= ( X2 = Y2 ) ) ) ) ).
% powr_inj
thf(fact_2233_gr__one__powr,axiom,
! [X2: real,Y2: real] :
( ( ord_less @ real @ ( one_one @ real ) @ X2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ Y2 )
=> ( ord_less @ real @ ( one_one @ real ) @ ( powr @ real @ X2 @ Y2 ) ) ) ) ).
% gr_one_powr
thf(fact_2234_ge__one__powr__ge__zero,axiom,
! [X2: real,A2: real] :
( ( ord_less_eq @ real @ ( one_one @ real ) @ X2 )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ A2 )
=> ( ord_less_eq @ real @ ( one_one @ real ) @ ( powr @ real @ X2 @ A2 ) ) ) ) ).
% ge_one_powr_ge_zero
thf(fact_2235_powr__mono__both,axiom,
! [A2: real,B2: real,X2: real,Y2: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ A2 )
=> ( ( ord_less_eq @ real @ A2 @ B2 )
=> ( ( ord_less_eq @ real @ ( one_one @ real ) @ X2 )
=> ( ( ord_less_eq @ real @ X2 @ Y2 )
=> ( ord_less_eq @ real @ ( powr @ real @ X2 @ A2 ) @ ( powr @ real @ Y2 @ B2 ) ) ) ) ) ) ).
% powr_mono_both
thf(fact_2236_powr__le1,axiom,
! [A2: real,X2: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ A2 )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ord_less_eq @ real @ X2 @ ( one_one @ real ) )
=> ( ord_less_eq @ real @ ( powr @ real @ X2 @ A2 ) @ ( one_one @ real ) ) ) ) ) ).
% powr_le1
thf(fact_2237_powr__divide,axiom,
! [X2: real,Y2: real,A2: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y2 )
=> ( ( powr @ real @ ( divide_divide @ real @ X2 @ Y2 ) @ A2 )
= ( divide_divide @ real @ ( powr @ real @ X2 @ A2 ) @ ( powr @ real @ Y2 @ A2 ) ) ) ) ) ).
% powr_divide
thf(fact_2238_powr__mult,axiom,
! [X2: real,Y2: real,A2: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y2 )
=> ( ( powr @ real @ ( times_times @ real @ X2 @ Y2 ) @ A2 )
= ( times_times @ real @ ( powr @ real @ X2 @ A2 ) @ ( powr @ real @ Y2 @ A2 ) ) ) ) ) ).
% powr_mult
thf(fact_2239_log__base__powr,axiom,
! [A2: real,B2: real,X2: real] :
( ( A2
!= ( zero_zero @ real ) )
=> ( ( log @ ( powr @ real @ A2 @ B2 ) @ X2 )
= ( divide_divide @ real @ ( log @ A2 @ X2 ) @ B2 ) ) ) ).
% log_base_powr
thf(fact_2240_log__powr,axiom,
! [X2: real,B2: real,Y2: real] :
( ( X2
!= ( zero_zero @ real ) )
=> ( ( log @ B2 @ ( powr @ real @ X2 @ Y2 ) )
= ( times_times @ real @ Y2 @ ( log @ B2 @ X2 ) ) ) ) ).
% log_powr
thf(fact_2241_powr__add,axiom,
! [A: $tType] :
( ( ( real_V3459762299906320749_field @ A )
& ( ln @ A ) )
=> ! [X2: A,A2: A,B2: A] :
( ( powr @ A @ X2 @ ( plus_plus @ A @ A2 @ B2 ) )
= ( times_times @ A @ ( powr @ A @ X2 @ A2 ) @ ( powr @ A @ X2 @ B2 ) ) ) ) ).
% powr_add
thf(fact_2242_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_2243_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_2244_subset__eq__atLeast0__atMost__finite,axiom,
! [N5: set @ nat,N: nat] :
( ( ord_less_eq @ ( set @ nat ) @ N5 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
=> ( finite_finite2 @ nat @ N5 ) ) ).
% subset_eq_atLeast0_atMost_finite
thf(fact_2245_powr__realpow,axiom,
! [X2: real,N: nat] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( powr @ real @ X2 @ ( semiring_1_of_nat @ real @ N ) )
= ( power_power @ real @ X2 @ N ) ) ) ).
% powr_realpow
thf(fact_2246_powr__less__iff,axiom,
! [B2: real,X2: real,Y2: real] :
( ( ord_less @ real @ ( one_one @ real ) @ B2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ord_less @ real @ ( powr @ real @ B2 @ Y2 ) @ X2 )
= ( ord_less @ real @ Y2 @ ( log @ B2 @ X2 ) ) ) ) ) ).
% powr_less_iff
thf(fact_2247_less__powr__iff,axiom,
! [B2: real,X2: real,Y2: real] :
( ( ord_less @ real @ ( one_one @ real ) @ B2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ord_less @ real @ X2 @ ( powr @ real @ B2 @ Y2 ) )
= ( ord_less @ real @ ( log @ B2 @ X2 ) @ Y2 ) ) ) ) ).
% less_powr_iff
thf(fact_2248_log__less__iff,axiom,
! [B2: real,X2: real,Y2: real] :
( ( ord_less @ real @ ( one_one @ real ) @ B2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ord_less @ real @ ( log @ B2 @ X2 ) @ Y2 )
= ( ord_less @ real @ X2 @ ( powr @ real @ B2 @ Y2 ) ) ) ) ) ).
% log_less_iff
thf(fact_2249_less__log__iff,axiom,
! [B2: real,X2: real,Y2: real] :
( ( ord_less @ real @ ( one_one @ real ) @ B2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ord_less @ real @ Y2 @ ( log @ B2 @ X2 ) )
= ( ord_less @ real @ ( powr @ real @ B2 @ Y2 ) @ X2 ) ) ) ) ).
% less_log_iff
thf(fact_2250_foldr__length__aux,axiom,
! [A: $tType,L2: list @ A,A2: nat] :
( ( foldr @ A @ nat
@ ^ [X: A] : suc
@ L2
@ A2 )
= ( plus_plus @ nat @ A2 @ ( size_size @ ( list @ A ) @ L2 ) ) ) ).
% foldr_length_aux
thf(fact_2251_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
@ ^ [Z5: A] :
( ( power_power @ A @ Z5 @ N )
= ( one_one @ A ) ) ) ) ) ) ).
% finite_roots_unity
thf(fact_2252_ln__powr__bound,axiom,
! [X2: real,A2: real] :
( ( ord_less_eq @ real @ ( one_one @ real ) @ X2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
=> ( ord_less_eq @ real @ ( ln_ln @ real @ X2 ) @ ( divide_divide @ real @ ( powr @ real @ X2 @ A2 ) @ A2 ) ) ) ) ).
% ln_powr_bound
thf(fact_2253_ln__powr__bound2,axiom,
! [X2: real,A2: real] :
( ( ord_less @ real @ ( one_one @ real ) @ X2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
=> ( ord_less_eq @ real @ ( powr @ real @ ( ln_ln @ real @ X2 ) @ A2 ) @ ( times_times @ real @ ( powr @ real @ A2 @ A2 ) @ X2 ) ) ) ) ).
% ln_powr_bound2
thf(fact_2254_powr__mult__base,axiom,
! [X2: real,Y2: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( times_times @ real @ X2 @ ( powr @ real @ X2 @ Y2 ) )
= ( powr @ real @ X2 @ ( plus_plus @ real @ ( one_one @ real ) @ Y2 ) ) ) ) ).
% powr_mult_base
thf(fact_2255_powr__le__iff,axiom,
! [B2: real,X2: real,Y2: real] :
( ( ord_less @ real @ ( one_one @ real ) @ B2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ord_less_eq @ real @ ( powr @ real @ B2 @ Y2 ) @ X2 )
= ( ord_less_eq @ real @ Y2 @ ( log @ B2 @ X2 ) ) ) ) ) ).
% powr_le_iff
thf(fact_2256_le__powr__iff,axiom,
! [B2: real,X2: real,Y2: real] :
( ( ord_less @ real @ ( one_one @ real ) @ B2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ord_less_eq @ real @ X2 @ ( powr @ real @ B2 @ Y2 ) )
= ( ord_less_eq @ real @ ( log @ B2 @ X2 ) @ Y2 ) ) ) ) ).
% le_powr_iff
thf(fact_2257_log__le__iff,axiom,
! [B2: real,X2: real,Y2: real] :
( ( ord_less @ real @ ( one_one @ real ) @ B2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ord_less_eq @ real @ ( log @ B2 @ X2 ) @ Y2 )
= ( ord_less_eq @ real @ X2 @ ( powr @ real @ B2 @ Y2 ) ) ) ) ) ).
% log_le_iff
thf(fact_2258_le__log__iff,axiom,
! [B2: real,X2: real,Y2: real] :
( ( ord_less @ real @ ( one_one @ real ) @ B2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ord_less_eq @ real @ Y2 @ ( log @ B2 @ X2 ) )
= ( ord_less_eq @ real @ ( powr @ real @ B2 @ Y2 ) @ X2 ) ) ) ) ).
% le_log_iff
thf(fact_2259_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Osimps_I2_J,axiom,
! [Uu: nat,Uv: list @ vEBT_VEBT,Uw: vEBT_VEBT,X2: nat] :
( ( vEBT_T_m_e_m_b_e_r @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu @ Uv @ Uw ) @ X2 )
= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ).
% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.simps(2)
thf(fact_2260_log__add__eq__powr,axiom,
! [B2: real,X2: real,Y2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ B2 )
=> ( ( B2
!= ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( plus_plus @ real @ ( log @ B2 @ X2 ) @ Y2 )
= ( log @ B2 @ ( times_times @ real @ X2 @ ( powr @ real @ B2 @ Y2 ) ) ) ) ) ) ) ).
% log_add_eq_powr
thf(fact_2261_add__log__eq__powr,axiom,
! [B2: real,X2: real,Y2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ B2 )
=> ( ( B2
!= ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( plus_plus @ real @ Y2 @ ( log @ B2 @ X2 ) )
= ( log @ B2 @ ( times_times @ real @ ( powr @ real @ B2 @ Y2 ) @ X2 ) ) ) ) ) ) ).
% add_log_eq_powr
thf(fact_2262_minus__log__eq__powr,axiom,
! [B2: real,X2: real,Y2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ B2 )
=> ( ( B2
!= ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( minus_minus @ real @ Y2 @ ( log @ B2 @ X2 ) )
= ( log @ B2 @ ( divide_divide @ real @ ( powr @ real @ B2 @ Y2 ) @ X2 ) ) ) ) ) ) ).
% minus_log_eq_powr
thf(fact_2263_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,X2: nat] :
( ( vEBT_T_m_e_m_b_e_r @ ( vEBT_Leaf @ A2 @ B2 ) @ X2 )
= ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) )
@ ( if @ nat
@ ( X2
= ( zero_zero @ nat ) )
@ ( one_one @ nat )
@ ( plus_plus @ nat @ ( one_one @ nat ) @ ( one_one @ nat ) ) ) ) ) ).
% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.simps(1)
thf(fact_2264_member__bound__height,axiom,
! [T2: vEBT_VEBT,N: nat,X2: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ord_less_eq @ nat @ ( vEBT_T_m_e_m_b_e_r @ T2 @ X2 ) @ ( times_times @ nat @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_VEBT_height @ T2 ) ) @ ( numeral_numeral @ nat @ ( bit1 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) ) ) ).
% member_bound_height
thf(fact_2265_finite__Collect__le__nat,axiom,
! [K: nat] :
( finite_finite2 @ nat
@ ( collect @ nat
@ ^ [N3: nat] : ( ord_less_eq @ nat @ N3 @ K ) ) ) ).
% finite_Collect_le_nat
thf(fact_2266_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_2267_finite__Collect__subsets,axiom,
! [A: $tType,A3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( finite_finite2 @ ( set @ A )
@ ( collect @ ( set @ A )
@ ^ [B8: set @ A] : ( ord_less_eq @ ( set @ A ) @ B8 @ A3 ) ) ) ) ).
% finite_Collect_subsets
thf(fact_2268_succ__bound__size__univ,axiom,
! [T2: vEBT_VEBT,N: nat,U: real,X2: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( U
= ( power_power @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ N ) )
=> ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ ( vEBT_T_s_u_c_c @ T2 @ X2 ) ) @ ( plus_plus @ real @ ( numeral_numeral @ real @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) ) ) @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) ) @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ U ) ) ) ) ) ) ) ).
% succ_bound_size_univ
thf(fact_2269_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_2270_insert__bound__size__univ,axiom,
! [T2: vEBT_VEBT,N: nat,U: real,X2: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( U
= ( power_power @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ N ) )
=> ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ ( vEBT_T_i_n_s_e_r_t @ T2 @ X2 ) ) @ ( plus_plus @ real @ ( numeral_numeral @ real @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) ) ) @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) ) @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ U ) ) ) ) ) ) ) ).
% insert_bound_size_univ
thf(fact_2271_pred__bound__size__univ,axiom,
! [T2: vEBT_VEBT,N: nat,U: real,X2: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( U
= ( power_power @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ N ) )
=> ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ ( vEBT_T_p_r_e_d @ T2 @ X2 ) ) @ ( plus_plus @ real @ ( numeral_numeral @ real @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) ) @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ U ) ) ) ) ) ) ) ).
% pred_bound_size_univ
thf(fact_2272_finite__Collect__disjI,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X: A] :
( ( P @ X )
| ( Q @ X ) ) ) )
= ( ( finite_finite2 @ A @ ( collect @ A @ P ) )
& ( finite_finite2 @ A @ ( collect @ A @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_2273_finite__Collect__conjI,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ( ( finite_finite2 @ A @ ( collect @ A @ P ) )
| ( finite_finite2 @ A @ ( collect @ A @ Q ) ) )
=> ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X: A] :
( ( P @ X )
& ( Q @ X ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_2274_map__ident,axiom,
! [A: $tType] :
( ( map @ A @ A
@ ^ [X: A] : X )
= ( ^ [Xs: list @ A] : Xs ) ) ).
% map_ident
thf(fact_2275_finite__interval__int1,axiom,
! [A2: int,B2: int] :
( finite_finite2 @ int
@ ( collect @ int
@ ^ [I3: int] :
( ( ord_less_eq @ int @ A2 @ I3 )
& ( ord_less_eq @ int @ I3 @ B2 ) ) ) ) ).
% finite_interval_int1
thf(fact_2276_finite__interval__int4,axiom,
! [A2: int,B2: int] :
( finite_finite2 @ int
@ ( collect @ int
@ ^ [I3: int] :
( ( ord_less @ int @ A2 @ I3 )
& ( ord_less @ int @ I3 @ B2 ) ) ) ) ).
% finite_interval_int4
thf(fact_2277_listsum__bound,axiom,
! [A: $tType,Xs2: list @ A,F2: A > real,Y2: real] :
( ! [X3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs2 ) )
=> ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( F2 @ X3 ) ) )
=> ( ord_less_eq @ real @ Y2 @ ( foldr @ real @ real @ ( plus_plus @ real ) @ ( map @ A @ real @ F2 @ Xs2 ) @ Y2 ) ) ) ).
% listsum_bound
thf(fact_2278_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_2279_map__replicate,axiom,
! [A: $tType,B: $tType,F2: B > A,N: nat,X2: B] :
( ( map @ B @ A @ F2 @ ( replicate @ B @ N @ X2 ) )
= ( replicate @ A @ N @ ( F2 @ X2 ) ) ) ).
% map_replicate
thf(fact_2280_finite__interval__int3,axiom,
! [A2: int,B2: int] :
( finite_finite2 @ int
@ ( collect @ int
@ ^ [I3: int] :
( ( ord_less @ int @ A2 @ I3 )
& ( ord_less_eq @ int @ I3 @ B2 ) ) ) ) ).
% finite_interval_int3
thf(fact_2281_finite__interval__int2,axiom,
! [A2: int,B2: int] :
( finite_finite2 @ int
@ ( collect @ int
@ ^ [I3: int] :
( ( ord_less_eq @ int @ A2 @ I3 )
& ( ord_less @ int @ I3 @ B2 ) ) ) ) ).
% finite_interval_int2
thf(fact_2282_list_Omap__ident,axiom,
! [A: $tType,T2: list @ A] :
( ( map @ A @ A
@ ^ [X: A] : X
@ T2 )
= T2 ) ).
% list.map_ident
thf(fact_2283_map__replicate__const,axiom,
! [B: $tType,A: $tType,K: A,Lst: list @ B] :
( ( map @ B @ A
@ ^ [X: B] : K
@ Lst )
= ( replicate @ A @ ( size_size @ ( list @ B ) @ Lst ) @ K ) ) ).
% map_replicate_const
thf(fact_2284_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_2285_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,S3: vEBT_VEBT,X2: nat] :
( ( vEBT_T_i_n_s_e_r_t @ ( vEBT_Node @ Info @ ( zero_zero @ nat ) @ Ts @ S3 ) @ X2 )
= ( one_one @ nat ) ) ).
% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.simps(2)
thf(fact_2286_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_2287_pigeonhole__infinite__rel,axiom,
! [B: $tType,A: $tType,A3: set @ A,B5: set @ B,R: A > B > $o] :
( ~ ( finite_finite2 @ A @ A3 )
=> ( ( finite_finite2 @ B @ B5 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ A3 )
=> ? [Xa2: B] :
( ( member @ B @ Xa2 @ B5 )
& ( R @ X3 @ Xa2 ) ) )
=> ? [X3: B] :
( ( member @ B @ X3 @ B5 )
& ~ ( finite_finite2 @ A
@ ( collect @ A
@ ^ [A4: A] :
( ( member @ A @ A4 @ A3 )
& ( R @ A4 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_2288_not__finite__existsD,axiom,
! [A: $tType,P: A > $o] :
( ~ ( finite_finite2 @ A @ ( collect @ A @ P ) )
=> ? [X_1: A] : ( P @ X_1 ) ) ).
% not_finite_existsD
thf(fact_2289_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_2290_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I4_J,axiom,
! [Uy: nat,Uz: list @ vEBT_VEBT,Va: vEBT_VEBT,Vb: nat] :
( ( vEBT_T_p_r_e_d @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uy @ Uz @ Va ) @ Vb )
= ( one_one @ nat ) ) ).
% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(4)
thf(fact_2291_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Osimps_I3_J,axiom,
! [Ux: nat,Uy: list @ vEBT_VEBT,Uz: vEBT_VEBT,Va: nat] :
( ( vEBT_T_s_u_c_c @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Ux @ Uy @ Uz ) @ Va )
= ( one_one @ nat ) ) ).
% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.simps(3)
thf(fact_2292_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,S3: vEBT_VEBT,X2: nat] :
( ( vEBT_T_i_n_s_e_r_t @ ( vEBT_Node @ Info @ ( suc @ ( zero_zero @ nat ) ) @ Ts @ S3 ) @ X2 )
= ( one_one @ nat ) ) ).
% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.simps(3)
thf(fact_2293_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I3_J,axiom,
! [A2: $o,B2: $o,Va: nat] :
( ( vEBT_T_p_r_e_d @ ( vEBT_Leaf @ A2 @ B2 ) @ ( suc @ ( suc @ Va ) ) )
= ( plus_plus @ nat @ ( one_one @ nat ) @ ( if @ nat @ B2 @ ( one_one @ nat ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( one_one @ nat ) ) ) ) ) ).
% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(3)
thf(fact_2294_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,X2: nat] :
( ( vEBT_T_i_n_s_e_r_t @ ( vEBT_Leaf @ A2 @ B2 ) @ X2 )
= ( plus_plus @ nat @ ( one_one @ nat )
@ ( if @ nat
@ ( X2
= ( zero_zero @ nat ) )
@ ( one_one @ nat )
@ ( plus_plus @ nat @ ( one_one @ nat ) @ ( one_one @ nat ) ) ) ) ) ).
% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.simps(1)
thf(fact_2295_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_2296_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_2297_finite__has__minimal2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: set @ A,A2: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( member @ A @ A2 @ A3 )
=> ? [X3: A] :
( ( member @ A @ X3 @ A3 )
& ( ord_less_eq @ A @ X3 @ A2 )
& ! [Xa2: A] :
( ( member @ A @ Xa2 @ A3 )
=> ( ( ord_less_eq @ A @ Xa2 @ X3 )
=> ( X3 = Xa2 ) ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_2298_finite__has__maximal2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: set @ A,A2: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( member @ A @ A2 @ A3 )
=> ? [X3: A] :
( ( member @ A @ X3 @ A3 )
& ( ord_less_eq @ A @ A2 @ X3 )
& ! [Xa2: A] :
( ( member @ A @ Xa2 @ A3 )
=> ( ( ord_less_eq @ A @ X3 @ Xa2 )
=> ( X3 = Xa2 ) ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_2299_rev__finite__subset,axiom,
! [A: $tType,B5: set @ A,A3: set @ A] :
( ( finite_finite2 @ A @ B5 )
=> ( ( ord_less_eq @ ( set @ A ) @ A3 @ B5 )
=> ( finite_finite2 @ A @ A3 ) ) ) ).
% rev_finite_subset
thf(fact_2300_infinite__super,axiom,
! [A: $tType,S: set @ A,T3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ S @ T3 )
=> ( ~ ( finite_finite2 @ A @ S )
=> ~ ( finite_finite2 @ A @ T3 ) ) ) ).
% infinite_super
thf(fact_2301_finite__subset,axiom,
! [A: $tType,A3: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B5 )
=> ( ( finite_finite2 @ A @ B5 )
=> ( finite_finite2 @ A @ A3 ) ) ) ).
% finite_subset
thf(fact_2302_VEBT__internal_Ocnt_Oelims,axiom,
! [X2: vEBT_VEBT,Y2: real] :
( ( ( vEBT_VEBT_cnt @ X2 )
= Y2 )
=> ( ( ? [A5: $o,B4: $o] :
( X2
= ( vEBT_Leaf @ A5 @ B4 ) )
=> ( Y2
!= ( one_one @ real ) ) )
=> ~ ! [Info2: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( Y2
!= ( 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_2303_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_2304_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Osimps_I4_J,axiom,
! [V: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,X2: nat] :
( ( vEBT_T_i_n_s_e_r_t @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ V ) ) @ TreeList @ Summary ) @ X2 )
= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ).
% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.simps(4)
thf(fact_2305_insersimp,axiom,
! [T2: vEBT_VEBT,N: nat,Y2: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ T2 @ X_1 )
=> ( ord_less_eq @ nat @ ( vEBT_T_i_n_s_e_r_t @ T2 @ Y2 ) @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) ) ) ) ).
% insersimp
thf(fact_2306_pred__bound__height,axiom,
! [T2: vEBT_VEBT,N: nat,X2: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ord_less_eq @ nat @ ( vEBT_T_p_r_e_d @ T2 @ X2 ) @ ( times_times @ nat @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_VEBT_height @ T2 ) ) @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ).
% pred_bound_height
thf(fact_2307_insert__bound__height,axiom,
! [T2: vEBT_VEBT,N: nat,X2: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ord_less_eq @ nat @ ( vEBT_T_i_n_s_e_r_t @ T2 @ X2 ) @ ( times_times @ nat @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_VEBT_height @ T2 ) ) @ ( numeral_numeral @ nat @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% insert_bound_height
thf(fact_2308_succ__bound__height,axiom,
! [T2: vEBT_VEBT,N: nat,X2: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ord_less_eq @ nat @ ( vEBT_T_s_u_c_c @ T2 @ X2 ) @ ( times_times @ nat @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_VEBT_height @ T2 ) ) @ ( numeral_numeral @ nat @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ).
% succ_bound_height
thf(fact_2309_finite__nth__roots,axiom,
! [N: nat,C2: complex] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( finite_finite2 @ complex
@ ( collect @ complex
@ ^ [Z5: complex] :
( ( power_power @ complex @ Z5 @ N )
= C2 ) ) ) ) ).
% finite_nth_roots
thf(fact_2310_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
@ ^ [X: A] : ( semiring_1_of_nat @ real @ ( F2 @ X ) )
@ Xs2 )
@ ( semiring_1_of_nat @ real @ C2 ) ) ) ).
% real_nat_list
thf(fact_2311_diff__preserves__multiset,axiom,
! [A: $tType,M7: A > nat,N5: A > nat] :
( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( M7 @ X ) ) ) )
=> ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ ( M7 @ X ) @ ( N5 @ X ) ) ) ) ) ) ).
% diff_preserves_multiset
thf(fact_2312_add__mset__in__multiset,axiom,
! [A: $tType,M7: A > nat,A2: A] :
( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( M7 @ X ) ) ) )
=> ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( if @ nat @ ( X = A2 ) @ ( suc @ ( M7 @ X ) ) @ ( M7 @ X ) ) ) ) ) ) ).
% add_mset_in_multiset
thf(fact_2313_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_2314_succ__bound__size__univ_H,axiom,
! [T2: vEBT_VEBT,N: nat,U: real,X2: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( U
= ( power_power @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ N ) )
=> ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ ( vEBT_T_s_u_c_c2 @ T2 @ X2 ) ) @ ( plus_plus @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ U ) ) ) ) ) ) ).
% succ_bound_size_univ'
thf(fact_2315_size__empty,axiom,
! [A: $tType] :
( ( size_size @ ( multiset @ A ) @ ( zero_zero @ ( multiset @ A ) ) )
= ( zero_zero @ nat ) ) ).
% size_empty
thf(fact_2316_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_2317_size__union,axiom,
! [A: $tType,M7: multiset @ A,N5: multiset @ A] :
( ( size_size @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ M7 @ N5 ) )
= ( plus_plus @ nat @ ( size_size @ ( multiset @ A ) @ M7 ) @ ( size_size @ ( multiset @ A ) @ N5 ) ) ) ).
% size_union
thf(fact_2318_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_2319_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_2320_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Osimps_I3_J,axiom,
! [Ux: nat,Uy: list @ vEBT_VEBT,Uz: vEBT_VEBT,Va: nat] :
( ( vEBT_T_s_u_c_c2 @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Ux @ Uy @ Uz ) @ Va )
= ( one_one @ nat ) ) ).
% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(3)
thf(fact_2321_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_2322_nonempty__has__size,axiom,
! [A: $tType,S: multiset @ A] :
( ( S
!= ( zero_zero @ ( multiset @ A ) ) )
= ( ord_less @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( multiset @ A ) @ S ) ) ) ).
% nonempty_has_size
thf(fact_2323_diff__size__le__size__Diff,axiom,
! [A: $tType,M7: multiset @ A,M8: multiset @ A] : ( ord_less_eq @ nat @ ( minus_minus @ nat @ ( size_size @ ( multiset @ A ) @ M7 ) @ ( size_size @ ( multiset @ A ) @ M8 ) ) @ ( size_size @ ( multiset @ A ) @ ( minus_minus @ ( multiset @ A ) @ M7 @ M8 ) ) ) ).
% diff_size_le_size_Diff
thf(fact_2324_VEBT__internal_Ocnt_H_Oelims,axiom,
! [X2: vEBT_VEBT,Y2: nat] :
( ( ( vEBT_VEBT_cnt2 @ X2 )
= Y2 )
=> ( ( ? [A5: $o,B4: $o] :
( X2
= ( vEBT_Leaf @ A5 @ B4 ) )
=> ( Y2
!= ( one_one @ nat ) ) )
=> ~ ! [Info2: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( Y2
!= ( 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_2325_complex__mod__triangle__ineq2,axiom,
! [B2: complex,A2: complex] : ( ord_less_eq @ real @ ( minus_minus @ real @ ( real_V7770717601297561774m_norm @ complex @ ( plus_plus @ complex @ B2 @ A2 ) ) @ ( real_V7770717601297561774m_norm @ complex @ B2 ) ) @ ( real_V7770717601297561774m_norm @ complex @ A2 ) ) ).
% complex_mod_triangle_ineq2
thf(fact_2326_succ_H__bound__height,axiom,
! [T2: vEBT_VEBT,N: nat,X2: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ord_less_eq @ nat @ ( vEBT_T_s_u_c_c2 @ T2 @ X2 ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_VEBT_height @ T2 ) ) ) ) ).
% succ'_bound_height
thf(fact_2327_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_2328_VEBT__internal_Ospace_H_Oelims,axiom,
! [X2: vEBT_VEBT,Y2: nat] :
( ( ( vEBT_VEBT_space2 @ X2 )
= Y2 )
=> ( ( ? [A5: $o,B4: $o] :
( X2
= ( vEBT_Leaf @ A5 @ B4 ) )
=> ( Y2
!= ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) ) )
=> ~ ! [Info2: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( Y2
!= ( 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_2329_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_2330_VEBT__internal_Ospace_Oelims,axiom,
! [X2: vEBT_VEBT,Y2: nat] :
( ( ( vEBT_VEBT_space @ X2 )
= Y2 )
=> ( ( ? [A5: $o,B4: $o] :
( X2
= ( vEBT_Leaf @ A5 @ B4 ) )
=> ( Y2
!= ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) ) )
=> ~ ! [Info2: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( Y2
!= ( 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_2331_filter__preserves__multiset,axiom,
! [A: $tType,M7: A > nat,P: A > $o] :
( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( M7 @ X ) ) ) )
=> ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( if @ nat @ ( P @ X ) @ ( M7 @ X ) @ ( zero_zero @ nat ) ) ) ) ) ) ).
% filter_preserves_multiset
thf(fact_2332_pred__bound__size__univ_H,axiom,
! [T2: vEBT_VEBT,N: nat,U: real,X2: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( U
= ( power_power @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ N ) )
=> ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ ( vEBT_T_p_r_e_d2 @ T2 @ X2 ) ) @ ( plus_plus @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ U ) ) ) ) ) ) ).
% pred_bound_size_univ'
thf(fact_2333_tanh__ln__real,axiom,
! [X2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( tanh @ real @ ( ln_ln @ real @ X2 ) )
= ( divide_divide @ real @ ( minus_minus @ real @ ( power_power @ real @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ real ) ) @ ( plus_plus @ real @ ( power_power @ real @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ real ) ) ) ) ) ).
% tanh_ln_real
thf(fact_2334_ln__one__minus__pos__lower__bound,axiom,
! [X2: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ord_less_eq @ real @ X2 @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less_eq @ real @ ( minus_minus @ real @ ( uminus_uminus @ real @ X2 ) @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( power_power @ real @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( ln_ln @ real @ ( minus_minus @ real @ ( one_one @ real ) @ X2 ) ) ) ) ) ).
% ln_one_minus_pos_lower_bound
thf(fact_2335_arcosh__def,axiom,
! [A: $tType] :
( ( ln @ A )
=> ( ( arcosh @ A )
= ( ^ [X: A] : ( ln_ln @ A @ ( plus_plus @ A @ X @ ( powr @ A @ ( minus_minus @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) ) @ ( real_Vector_of_real @ A @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% arcosh_def
thf(fact_2336_abs__ln__one__plus__x__minus__x__bound,axiom,
! [X2: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ X2 ) @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less_eq @ real @ ( abs_abs @ real @ ( minus_minus @ real @ ( ln_ln @ real @ ( plus_plus @ real @ ( one_one @ real ) @ X2 ) ) @ X2 ) ) @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( power_power @ real @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% abs_ln_one_plus_x_minus_x_bound
thf(fact_2337_prod_Ofinite__Collect__op,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [I5: set @ B,X2: B > A,Y2: B > A] :
( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [I3: B] :
( ( member @ B @ I3 @ I5 )
& ( ( X2 @ I3 )
!= ( one_one @ A ) ) ) ) )
=> ( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [I3: B] :
( ( member @ B @ I3 @ I5 )
& ( ( Y2 @ I3 )
!= ( one_one @ A ) ) ) ) )
=> ( finite_finite2 @ B
@ ( collect @ B
@ ^ [I3: B] :
( ( member @ B @ I3 @ I5 )
& ( ( times_times @ A @ ( X2 @ I3 ) @ ( Y2 @ I3 ) )
!= ( one_one @ A ) ) ) ) ) ) ) ) ).
% prod.finite_Collect_op
thf(fact_2338_add_Oinverse__inverse,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A] :
( ( uminus_uminus @ A @ ( uminus_uminus @ A @ A2 ) )
= A2 ) ) ).
% add.inverse_inverse
thf(fact_2339_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_2340_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_2341_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_2342_add_Oinverse__neutral,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ( ( uminus_uminus @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% add.inverse_neutral
thf(fact_2343_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_2344_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_2345_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_2346_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_2347_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_2348_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_2349_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_2350_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_2351_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_2352_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_2353_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_2354_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_2355_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_2356_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_2357_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_2358_abs__0,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ( ( abs_abs @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% abs_0
thf(fact_2359_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_2360_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_2361_abs__zero,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ( ( abs_abs @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% abs_zero
thf(fact_2362_dvd__minus__iff,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [X2: A,Y2: A] :
( ( dvd_dvd @ A @ X2 @ ( uminus_uminus @ A @ Y2 ) )
= ( dvd_dvd @ A @ X2 @ Y2 ) ) ) ).
% dvd_minus_iff
thf(fact_2363_minus__dvd__iff,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [X2: A,Y2: A] :
( ( dvd_dvd @ A @ ( uminus_uminus @ A @ X2 ) @ Y2 )
= ( dvd_dvd @ A @ X2 @ Y2 ) ) ) ).
% minus_dvd_iff
thf(fact_2364_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_2365_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_2366_abs__1,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ( ( abs_abs @ A @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% abs_1
thf(fact_2367_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_2368_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_2369_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_2370_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_2371_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_2372_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_2373_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_2374_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_2375_subset__mset_Oadd__eq__0__iff__both__eq__0,axiom,
! [A: $tType,X2: multiset @ A,Y2: multiset @ A] :
( ( ( plus_plus @ ( multiset @ A ) @ X2 @ Y2 )
= ( zero_zero @ ( multiset @ A ) ) )
= ( ( X2
= ( zero_zero @ ( multiset @ A ) ) )
& ( Y2
= ( zero_zero @ ( multiset @ A ) ) ) ) ) ).
% subset_mset.add_eq_0_iff_both_eq_0
thf(fact_2376_subset__mset_Ozero__eq__add__iff__both__eq__0,axiom,
! [A: $tType,X2: multiset @ A,Y2: multiset @ A] :
( ( ( zero_zero @ ( multiset @ A ) )
= ( plus_plus @ ( multiset @ A ) @ X2 @ Y2 ) )
= ( ( X2
= ( zero_zero @ ( multiset @ A ) ) )
& ( Y2
= ( zero_zero @ ( multiset @ A ) ) ) ) ) ).
% subset_mset.zero_eq_add_iff_both_eq_0
thf(fact_2377_empty__eq__union,axiom,
! [A: $tType,M7: multiset @ A,N5: multiset @ A] :
( ( ( zero_zero @ ( multiset @ A ) )
= ( plus_plus @ ( multiset @ A ) @ M7 @ N5 ) )
= ( ( M7
= ( zero_zero @ ( multiset @ A ) ) )
& ( N5
= ( zero_zero @ ( multiset @ A ) ) ) ) ) ).
% empty_eq_union
thf(fact_2378_union__eq__empty,axiom,
! [A: $tType,M7: multiset @ A,N5: multiset @ A] :
( ( ( plus_plus @ ( multiset @ A ) @ M7 @ N5 )
= ( zero_zero @ ( multiset @ A ) ) )
= ( ( M7
= ( zero_zero @ ( multiset @ A ) ) )
& ( N5
= ( zero_zero @ ( multiset @ A ) ) ) ) ) ).
% union_eq_empty
thf(fact_2379_diff__diff__add__mset,axiom,
! [A: $tType,M7: multiset @ A,N5: multiset @ A,P: multiset @ A] :
( ( minus_minus @ ( multiset @ A ) @ ( minus_minus @ ( multiset @ A ) @ M7 @ N5 ) @ P )
= ( minus_minus @ ( multiset @ A ) @ M7 @ ( plus_plus @ ( multiset @ A ) @ N5 @ P ) ) ) ).
% diff_diff_add_mset
thf(fact_2380_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_2381_tanh__real__zero__iff,axiom,
! [X2: real] :
( ( ( tanh @ real @ X2 )
= ( zero_zero @ real ) )
= ( X2
= ( zero_zero @ real ) ) ) ).
% tanh_real_zero_iff
thf(fact_2382_tanh__real__le__iff,axiom,
! [X2: real,Y2: real] :
( ( ord_less_eq @ real @ ( tanh @ real @ X2 ) @ ( tanh @ real @ Y2 ) )
= ( ord_less_eq @ real @ X2 @ Y2 ) ) ).
% tanh_real_le_iff
thf(fact_2383_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_2384_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_2385_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_2386_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_2387_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_2388_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_2389_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_2390_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_2391_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_2392_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_2393_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_2394_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_2395_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_2396_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_2397_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_2398_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_2399_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_2400_divide__minus1,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [X2: A] :
( ( divide_divide @ A @ X2 @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( uminus_uminus @ A @ X2 ) ) ) ).
% divide_minus1
thf(fact_2401_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_2402_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_2403_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_2404_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_2405_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_2406_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_2407_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_2408_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_2409_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_2410_of__real__eq__0__iff,axiom,
! [A: $tType] :
( ( real_V2191834092415804123ebra_1 @ A )
=> ! [X2: real] :
( ( ( real_Vector_of_real @ A @ X2 )
= ( zero_zero @ A ) )
= ( X2
= ( zero_zero @ real ) ) ) ) ).
% of_real_eq_0_iff
thf(fact_2411_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_2412_of__real__eq__1__iff,axiom,
! [A: $tType] :
( ( real_V2191834092415804123ebra_1 @ A )
=> ! [X2: real] :
( ( ( real_Vector_of_real @ A @ X2 )
= ( one_one @ A ) )
= ( X2
= ( one_one @ real ) ) ) ) ).
% of_real_eq_1_iff
thf(fact_2413_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_2414_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_2415_of__real__mult,axiom,
! [A: $tType] :
( ( real_V2191834092415804123ebra_1 @ A )
=> ! [X2: real,Y2: real] :
( ( real_Vector_of_real @ A @ ( times_times @ real @ X2 @ Y2 ) )
= ( times_times @ A @ ( real_Vector_of_real @ A @ X2 ) @ ( real_Vector_of_real @ A @ Y2 ) ) ) ) ).
% of_real_mult
thf(fact_2416_of__real__divide,axiom,
! [A: $tType] :
( ( real_V5047593784448816457lgebra @ A )
=> ! [X2: real,Y2: real] :
( ( real_Vector_of_real @ A @ ( divide_divide @ real @ X2 @ Y2 ) )
= ( divide_divide @ A @ ( real_Vector_of_real @ A @ X2 ) @ ( real_Vector_of_real @ A @ Y2 ) ) ) ) ).
% of_real_divide
thf(fact_2417_of__real__add,axiom,
! [A: $tType] :
( ( real_V2191834092415804123ebra_1 @ A )
=> ! [X2: real,Y2: real] :
( ( real_Vector_of_real @ A @ ( plus_plus @ real @ X2 @ Y2 ) )
= ( plus_plus @ A @ ( real_Vector_of_real @ A @ X2 ) @ ( real_Vector_of_real @ A @ Y2 ) ) ) ) ).
% of_real_add
thf(fact_2418_of__real__power,axiom,
! [A: $tType] :
( ( real_V2191834092415804123ebra_1 @ A )
=> ! [X2: real,N: nat] :
( ( real_Vector_of_real @ A @ ( power_power @ real @ X2 @ N ) )
= ( power_power @ A @ ( real_Vector_of_real @ A @ X2 ) @ N ) ) ) ).
% of_real_power
thf(fact_2419_real__add__minus__iff,axiom,
! [X2: real,A2: real] :
( ( ( plus_plus @ real @ X2 @ ( uminus_uminus @ real @ A2 ) )
= ( zero_zero @ real ) )
= ( X2 = A2 ) ) ).
% real_add_minus_iff
thf(fact_2420_of__real__of__nat__eq,axiom,
! [A: $tType] :
( ( real_V2191834092415804123ebra_1 @ A )
=> ! [N: nat] :
( ( real_Vector_of_real @ A @ ( semiring_1_of_nat @ real @ N ) )
= ( semiring_1_of_nat @ A @ N ) ) ) ).
% of_real_of_nat_eq
thf(fact_2421_tanh__real__pos__iff,axiom,
! [X2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( tanh @ real @ X2 ) )
= ( ord_less @ real @ ( zero_zero @ real ) @ X2 ) ) ).
% tanh_real_pos_iff
thf(fact_2422_tanh__real__neg__iff,axiom,
! [X2: real] :
( ( ord_less @ real @ ( tanh @ real @ X2 ) @ ( zero_zero @ real ) )
= ( ord_less @ real @ X2 @ ( zero_zero @ real ) ) ) ).
% tanh_real_neg_iff
thf(fact_2423_tanh__real__nonpos__iff,axiom,
! [X2: real] :
( ( ord_less_eq @ real @ ( tanh @ real @ X2 ) @ ( zero_zero @ real ) )
= ( ord_less_eq @ real @ X2 @ ( zero_zero @ real ) ) ) ).
% tanh_real_nonpos_iff
thf(fact_2424_tanh__real__nonneg__iff,axiom,
! [X2: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( tanh @ real @ X2 ) )
= ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 ) ) ).
% tanh_real_nonneg_iff
thf(fact_2425_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_2426_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_2427_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_2428_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_2429_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_2430_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_2431_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_2432_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_2433_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_2434_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_2435_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_2436_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_2437_semiring__norm_I168_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [V: num,W: num,Y2: A] :
( ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) @ ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) @ Y2 ) )
= ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( plus_plus @ num @ V @ W ) ) ) @ Y2 ) ) ) ).
% semiring_norm(168)
thf(fact_2438_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_2439_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_2440_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_2441_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_2442_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_2443_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_2444_semiring__norm_I172_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [V: num,W: num,Y2: A] :
( ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) @ Y2 ) )
= ( times_times @ A @ ( numeral_numeral @ A @ ( times_times @ num @ V @ W ) ) @ Y2 ) ) ) ).
% semiring_norm(172)
thf(fact_2445_semiring__norm_I171_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [V: num,W: num,Y2: A] :
( ( times_times @ A @ ( numeral_numeral @ A @ V ) @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) @ Y2 ) )
= ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( times_times @ num @ V @ W ) ) ) @ Y2 ) ) ) ).
% semiring_norm(171)
thf(fact_2446_semiring__norm_I170_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [V: num,W: num,Y2: A] :
( ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ W ) @ Y2 ) )
= ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( times_times @ num @ V @ W ) ) ) @ Y2 ) ) ) ).
% semiring_norm(170)
thf(fact_2447_artanh__minus__real,axiom,
! [X2: real] :
( ( ord_less @ real @ ( abs_abs @ real @ X2 ) @ ( one_one @ real ) )
=> ( ( artanh @ real @ ( uminus_uminus @ real @ X2 ) )
= ( uminus_uminus @ real @ ( artanh @ real @ X2 ) ) ) ) ).
% artanh_minus_real
thf(fact_2448_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_2449_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_2450_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_2451_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_2452_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_2453_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_2454_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_2455_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_2456_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_2457_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_2458_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_2459_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_2460_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_2461_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_2462_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_2463_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_2464_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_2465_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_2466_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_2467_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_2468_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_2469_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_2470_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_2471_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_2472_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_2473_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_2474_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_2475_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_2476_of__int__eq__neg__numeral__power__cancel__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [Y2: int,X2: num,N: nat] :
( ( ( ring_1_of_int @ A @ Y2 )
= ( power_power @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ X2 ) ) @ N ) )
= ( Y2
= ( power_power @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ X2 ) ) @ N ) ) ) ) ).
% of_int_eq_neg_numeral_power_cancel_iff
thf(fact_2477_neg__numeral__power__eq__of__int__cancel__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [X2: num,N: nat,Y2: int] :
( ( ( power_power @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ X2 ) ) @ N )
= ( ring_1_of_int @ A @ Y2 ) )
= ( ( power_power @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ X2 ) ) @ N )
= Y2 ) ) ) ).
% neg_numeral_power_eq_of_int_cancel_iff
thf(fact_2478_ceiling__le__neg__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A,V: num] :
( ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ X2 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V ) ) )
= ( ord_less_eq @ A @ X2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) ) ) ) ).
% ceiling_le_neg_numeral
thf(fact_2479_neg__numeral__less__ceiling,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V: num,X2: A] :
( ( ord_less @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V ) ) @ ( archimedean_ceiling @ A @ X2 ) )
= ( ord_less @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) @ X2 ) ) ) ).
% neg_numeral_less_ceiling
thf(fact_2480_ceiling__less__zero,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A] :
( ( ord_less @ int @ ( archimedean_ceiling @ A @ X2 ) @ ( zero_zero @ int ) )
= ( ord_less_eq @ A @ X2 @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ).
% ceiling_less_zero
thf(fact_2481_zero__le__ceiling,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( archimedean_ceiling @ A @ X2 ) )
= ( ord_less @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ X2 ) ) ) ).
% zero_le_ceiling
thf(fact_2482_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_2483_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_2484_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_2485_norm__of__real__add1,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [X2: real] :
( ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ ( real_Vector_of_real @ A @ X2 ) @ ( one_one @ A ) ) )
= ( abs_abs @ real @ ( plus_plus @ real @ X2 @ ( one_one @ real ) ) ) ) ) ).
% norm_of_real_add1
thf(fact_2486_norm__of__real__addn,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [X2: real,B2: num] :
( ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ ( real_Vector_of_real @ A @ X2 ) @ ( numeral_numeral @ A @ B2 ) ) )
= ( abs_abs @ real @ ( plus_plus @ real @ X2 @ ( numeral_numeral @ real @ B2 ) ) ) ) ) ).
% norm_of_real_addn
thf(fact_2487_ceiling__less__neg__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A,V: num] :
( ( ord_less @ int @ ( archimedean_ceiling @ A @ X2 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V ) ) )
= ( ord_less_eq @ A @ X2 @ ( minus_minus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) @ ( one_one @ A ) ) ) ) ) ).
% ceiling_less_neg_numeral
thf(fact_2488_neg__numeral__le__ceiling,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V: num,X2: A] :
( ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V ) ) @ ( archimedean_ceiling @ A @ X2 ) )
= ( ord_less @ A @ ( minus_minus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) @ ( one_one @ A ) ) @ X2 ) ) ) ).
% neg_numeral_le_ceiling
thf(fact_2489_neg__numeral__power__le__of__int__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X2: num,N: nat,A2: int] :
( ( ord_less_eq @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ X2 ) ) @ N ) @ ( ring_1_of_int @ A @ A2 ) )
= ( ord_less_eq @ int @ ( power_power @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ X2 ) ) @ N ) @ A2 ) ) ) ).
% neg_numeral_power_le_of_int_cancel_iff
thf(fact_2490_of__int__le__neg__numeral__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: int,X2: num,N: nat] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ A2 ) @ ( power_power @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ X2 ) ) @ N ) )
= ( ord_less_eq @ int @ A2 @ ( power_power @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ X2 ) ) @ N ) ) ) ) ).
% of_int_le_neg_numeral_power_cancel_iff
thf(fact_2491_of__int__less__neg__numeral__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: int,X2: num,N: nat] :
( ( ord_less @ A @ ( ring_1_of_int @ A @ A2 ) @ ( power_power @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ X2 ) ) @ N ) )
= ( ord_less @ int @ A2 @ ( power_power @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ X2 ) ) @ N ) ) ) ) ).
% of_int_less_neg_numeral_power_cancel_iff
thf(fact_2492_neg__numeral__power__less__of__int__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X2: num,N: nat,A2: int] :
( ( ord_less @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ X2 ) ) @ N ) @ ( ring_1_of_int @ A @ A2 ) )
= ( ord_less @ int @ ( power_power @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ X2 ) ) @ N ) @ A2 ) ) ) ).
% neg_numeral_power_less_of_int_cancel_iff
thf(fact_2493_square__powr__half,axiom,
! [X2: real] :
( ( powr @ real @ ( power_power @ real @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
= ( abs_abs @ real @ X2 ) ) ).
% square_powr_half
thf(fact_2494_Multiset_Odiff__add,axiom,
! [A: $tType,M7: multiset @ A,N5: multiset @ A,Q: multiset @ A] :
( ( minus_minus @ ( multiset @ A ) @ M7 @ ( plus_plus @ ( multiset @ A ) @ N5 @ Q ) )
= ( minus_minus @ ( multiset @ A ) @ ( minus_minus @ ( multiset @ A ) @ M7 @ N5 ) @ Q ) ) ).
% Multiset.diff_add
thf(fact_2495_diff__union__cancelL,axiom,
! [A: $tType,N5: multiset @ A,M7: multiset @ A] :
( ( minus_minus @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ N5 @ M7 ) @ N5 )
= M7 ) ).
% diff_union_cancelL
thf(fact_2496_diff__union__cancelR,axiom,
! [A: $tType,M7: multiset @ A,N5: multiset @ A] :
( ( minus_minus @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ M7 @ N5 ) @ N5 )
= M7 ) ).
% diff_union_cancelR
thf(fact_2497_Multiset_Odiff__cancel,axiom,
! [A: $tType,A3: multiset @ A] :
( ( minus_minus @ ( multiset @ A ) @ A3 @ A3 )
= ( zero_zero @ ( multiset @ A ) ) ) ).
% Multiset.diff_cancel
thf(fact_2498_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_2499_union__assoc,axiom,
! [A: $tType,M7: multiset @ A,N5: multiset @ A,K6: multiset @ A] :
( ( plus_plus @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ M7 @ N5 ) @ K6 )
= ( plus_plus @ ( multiset @ A ) @ M7 @ ( plus_plus @ ( multiset @ A ) @ N5 @ K6 ) ) ) ).
% union_assoc
thf(fact_2500_union__lcomm,axiom,
! [A: $tType,M7: multiset @ A,N5: multiset @ A,K6: multiset @ A] :
( ( plus_plus @ ( multiset @ A ) @ M7 @ ( plus_plus @ ( multiset @ A ) @ N5 @ K6 ) )
= ( plus_plus @ ( multiset @ A ) @ N5 @ ( plus_plus @ ( multiset @ A ) @ M7 @ K6 ) ) ) ).
% union_lcomm
thf(fact_2501_union__commute,axiom,
! [A: $tType] :
( ( plus_plus @ ( multiset @ A ) )
= ( ^ [M9: multiset @ A,N8: multiset @ A] : ( plus_plus @ ( multiset @ A ) @ N8 @ M9 ) ) ) ).
% union_commute
thf(fact_2502_union__left__cancel,axiom,
! [A: $tType,K6: multiset @ A,M7: multiset @ A,N5: multiset @ A] :
( ( ( plus_plus @ ( multiset @ A ) @ K6 @ M7 )
= ( plus_plus @ ( multiset @ A ) @ K6 @ N5 ) )
= ( M7 = N5 ) ) ).
% union_left_cancel
thf(fact_2503_union__right__cancel,axiom,
! [A: $tType,M7: multiset @ A,K6: multiset @ A,N5: multiset @ A] :
( ( ( plus_plus @ ( multiset @ A ) @ M7 @ K6 )
= ( plus_plus @ ( multiset @ A ) @ N5 @ K6 ) )
= ( M7 = N5 ) ) ).
% union_right_cancel
thf(fact_2504_multi__union__self__other__eq,axiom,
! [A: $tType,A3: multiset @ A,X6: multiset @ A,Y7: multiset @ A] :
( ( ( plus_plus @ ( multiset @ A ) @ A3 @ X6 )
= ( plus_plus @ ( multiset @ A ) @ A3 @ Y7 ) )
=> ( X6 = Y7 ) ) ).
% multi_union_self_other_eq
thf(fact_2505_empty__neutral_I2_J,axiom,
! [A: $tType,X2: multiset @ A] :
( ( plus_plus @ ( multiset @ A ) @ X2 @ ( zero_zero @ ( multiset @ A ) ) )
= X2 ) ).
% empty_neutral(2)
thf(fact_2506_empty__neutral_I1_J,axiom,
! [A: $tType,X2: multiset @ A] :
( ( plus_plus @ ( multiset @ A ) @ ( zero_zero @ ( multiset @ A ) ) @ X2 )
= X2 ) ).
% empty_neutral(1)
thf(fact_2507_union__le__mono1,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [B5: multiset @ A,D4: multiset @ A,C6: multiset @ A] :
( ( ord_less @ ( multiset @ A ) @ B5 @ D4 )
=> ( ord_less @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ B5 @ C6 ) @ ( plus_plus @ ( multiset @ A ) @ D4 @ C6 ) ) ) ) ).
% union_le_mono1
thf(fact_2508_union__le__mono2,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [B5: multiset @ A,D4: multiset @ A,C6: multiset @ A] :
( ( ord_less @ ( multiset @ A ) @ B5 @ D4 )
=> ( ord_less @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ C6 @ B5 ) @ ( plus_plus @ ( multiset @ A ) @ C6 @ D4 ) ) ) ) ).
% union_le_mono2
thf(fact_2509_union__less__mono,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: multiset @ A,C6: multiset @ A,B5: multiset @ A,D4: multiset @ A] :
( ( ord_less @ ( multiset @ A ) @ A3 @ C6 )
=> ( ( ord_less @ ( multiset @ A ) @ B5 @ D4 )
=> ( ord_less @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ A3 @ B5 ) @ ( plus_plus @ ( multiset @ A ) @ C6 @ D4 ) ) ) ) ) ).
% union_less_mono
thf(fact_2510_mset__distrib,axiom,
! [A: $tType,A3: multiset @ A,B5: multiset @ A,M7: multiset @ A,N5: multiset @ A] :
( ( ( plus_plus @ ( multiset @ A ) @ A3 @ B5 )
= ( plus_plus @ ( multiset @ A ) @ M7 @ N5 ) )
=> ~ ! [Am: multiset @ A,An: multiset @ A] :
( ( A3
= ( plus_plus @ ( multiset @ A ) @ Am @ An ) )
=> ! [Bm: multiset @ A,Bn: multiset @ A] :
( ( B5
= ( plus_plus @ ( multiset @ A ) @ Bm @ Bn ) )
=> ( ( M7
= ( plus_plus @ ( multiset @ A ) @ Am @ Bm ) )
=> ( N5
!= ( plus_plus @ ( multiset @ A ) @ An @ Bn ) ) ) ) ) ) ).
% mset_distrib
thf(fact_2511_union__diff__assoc,axiom,
! [A: $tType,C6: multiset @ A,B5: multiset @ A,A3: multiset @ A] :
( ( ( minus_minus @ ( multiset @ A ) @ C6 @ B5 )
= ( zero_zero @ ( multiset @ A ) ) )
=> ( ( minus_minus @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ A3 @ B5 ) @ C6 )
= ( plus_plus @ ( multiset @ A ) @ A3 @ ( minus_minus @ ( multiset @ A ) @ B5 @ C6 ) ) ) ) ).
% union_diff_assoc
thf(fact_2512_abs__eq__iff,axiom,
! [A: $tType] :
( ( linordered_ring @ A )
=> ! [X2: A,Y2: A] :
( ( ( abs_abs @ A @ X2 )
= ( abs_abs @ A @ Y2 ) )
= ( ( X2 = Y2 )
| ( X2
= ( uminus_uminus @ A @ Y2 ) ) ) ) ) ).
% abs_eq_iff
thf(fact_2513_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_2514_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_2515_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_2516_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_2517_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_2518_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_2519_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_2520_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_2521_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_2522_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_2523_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_2524_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_2525_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_2526_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_2527_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_2528_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_2529_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_2530_abs__one,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( ( abs_abs @ A @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% abs_one
thf(fact_2531_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_2532_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_2533_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_2534_tanh__real__gt__neg1,axiom,
! [X2: real] : ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ ( tanh @ real @ X2 ) ) ).
% tanh_real_gt_neg1
thf(fact_2535_dvd__if__abs__eq,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [L2: A,K: A] :
( ( ( abs_abs @ A @ L2 )
= ( abs_abs @ A @ K ) )
=> ( dvd_dvd @ A @ L2 @ K ) ) ) ).
% dvd_if_abs_eq
thf(fact_2536_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_2537_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_2538_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_2539_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_2540_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_2541_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_2542_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_2543_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_2544_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_2545_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_2546_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_2547_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_2548_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_2549_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_2550_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_2551_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_2552_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_2553_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_2554_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_2555_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_2556_euclidean__ring__cancel__class_Omod__minus__cong,axiom,
! [A: $tType] :
( ( euclid8851590272496341667cancel @ A )
=> ! [A2: A,B2: A,A6: A] :
( ( ( modulo_modulo @ A @ A2 @ B2 )
= ( modulo_modulo @ A @ A6 @ B2 ) )
=> ( ( modulo_modulo @ A @ ( uminus_uminus @ A @ A2 ) @ B2 )
= ( modulo_modulo @ A @ ( uminus_uminus @ A @ A6 ) @ B2 ) ) ) ) ).
% euclidean_ring_cancel_class.mod_minus_cong
thf(fact_2557_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_2558_complex__mod__minus__le__complex__mod,axiom,
! [X2: complex] : ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( real_V7770717601297561774m_norm @ complex @ X2 ) ) @ ( real_V7770717601297561774m_norm @ complex @ X2 ) ) ).
% complex_mod_minus_le_complex_mod
thf(fact_2559_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_2560_norm__of__real__diff,axiom,
! [A: $tType] :
( ( real_V2822296259951069270ebra_1 @ A )
=> ! [B2: real,A2: real] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( real_Vector_of_real @ A @ B2 ) @ ( real_Vector_of_real @ A @ A2 ) ) ) @ ( abs_abs @ real @ ( minus_minus @ real @ B2 @ A2 ) ) ) ) ).
% norm_of_real_diff
thf(fact_2561_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_2562_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_2563_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_2564_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_2565_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_2566_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_2567_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_2568_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_2569_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_2570_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_2571_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_2572_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_2573_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_2574_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_2575_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_2576_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_2577_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_2578_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_2579_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_2580_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_2581_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_2582_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_2583_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_2584_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_2585_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_2586_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_2587_numeral__times__minus__swap,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [W: num,X2: A] :
( ( times_times @ A @ ( numeral_numeral @ A @ W ) @ ( uminus_uminus @ A @ X2 ) )
= ( times_times @ A @ X2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) ) ) ) ).
% numeral_times_minus_swap
thf(fact_2588_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_2589_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_2590_square__eq__1__iff,axiom,
! [A: $tType] :
( ( ring_15535105094025558882visors @ A )
=> ! [X2: A] :
( ( ( times_times @ A @ X2 @ X2 )
= ( one_one @ A ) )
= ( ( X2
= ( one_one @ A ) )
| ( X2
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ) ).
% square_eq_1_iff
thf(fact_2591_group__cancel_Osub2,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [B5: A,K: A,B2: A,A2: A] :
( ( B5
= ( plus_plus @ A @ K @ B2 ) )
=> ( ( minus_minus @ A @ A2 @ B5 )
= ( plus_plus @ A @ ( uminus_uminus @ A @ K ) @ ( minus_minus @ A @ A2 @ B2 ) ) ) ) ) ).
% group_cancel.sub2
thf(fact_2592_diff__conv__add__uminus,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ( ( minus_minus @ A )
= ( ^ [A4: A,B3: A] : ( plus_plus @ A @ A4 @ ( uminus_uminus @ A @ B3 ) ) ) ) ) ).
% diff_conv_add_uminus
thf(fact_2593_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ( ( minus_minus @ A )
= ( ^ [A4: A,B3: A] : ( plus_plus @ A @ A4 @ ( uminus_uminus @ A @ B3 ) ) ) ) ) ).
% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_2594_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_2595_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_2596_real__minus__mult__self__le,axiom,
! [U: real,X2: real] : ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( times_times @ real @ U @ U ) ) @ ( times_times @ real @ X2 @ X2 ) ) ).
% real_minus_mult_self_le
thf(fact_2597_minus__real__def,axiom,
( ( minus_minus @ real )
= ( ^ [X: real,Y: real] : ( plus_plus @ real @ X @ ( uminus_uminus @ real @ Y ) ) ) ) ).
% minus_real_def
thf(fact_2598_tanh__real__lt__1,axiom,
! [X2: real] : ( ord_less @ real @ ( tanh @ real @ X2 ) @ ( one_one @ real ) ) ).
% tanh_real_lt_1
thf(fact_2599_dense__eq0__I,axiom,
! [A: $tType] :
( ( ( ordere166539214618696060dd_abs @ A )
& ( dense_linorder @ A ) )
=> ! [X2: A] :
( ! [E2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ E2 )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ X2 ) @ E2 ) )
=> ( X2
= ( zero_zero @ A ) ) ) ) ).
% dense_eq0_I
thf(fact_2600_abs__mult__pos,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X2: A,Y2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X2 )
=> ( ( times_times @ A @ ( abs_abs @ A @ Y2 ) @ X2 )
= ( abs_abs @ A @ ( times_times @ A @ Y2 @ X2 ) ) ) ) ) ).
% abs_mult_pos
thf(fact_2601_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_2602_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_2603_abs__div__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [Y2: A,X2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Y2 )
=> ( ( divide_divide @ A @ ( abs_abs @ A @ X2 ) @ Y2 )
= ( abs_abs @ A @ ( divide_divide @ A @ X2 @ Y2 ) ) ) ) ) ).
% abs_div_pos
thf(fact_2604_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_2605_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_2606_abs__diff__le__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X2: A,A2: A,R2: A] :
( ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ X2 @ A2 ) ) @ R2 )
= ( ( ord_less_eq @ A @ ( minus_minus @ A @ A2 @ R2 ) @ X2 )
& ( ord_less_eq @ A @ X2 @ ( plus_plus @ A @ A2 @ R2 ) ) ) ) ) ).
% abs_diff_le_iff
thf(fact_2607_abs__diff__less__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X2: A,A2: A,R2: A] :
( ( ord_less @ A @ ( abs_abs @ A @ ( minus_minus @ A @ X2 @ A2 ) ) @ R2 )
= ( ( ord_less @ A @ ( minus_minus @ A @ A2 @ R2 ) @ X2 )
& ( ord_less @ A @ X2 @ ( plus_plus @ A @ A2 @ R2 ) ) ) ) ) ).
% abs_diff_less_iff
thf(fact_2608_nonzero__of__real__divide,axiom,
! [A: $tType] :
( ( real_V7773925162809079976_field @ A )
=> ! [Y2: real,X2: real] :
( ( Y2
!= ( zero_zero @ real ) )
=> ( ( real_Vector_of_real @ A @ ( divide_divide @ real @ X2 @ Y2 ) )
= ( divide_divide @ A @ ( real_Vector_of_real @ A @ X2 ) @ ( real_Vector_of_real @ A @ Y2 ) ) ) ) ) ).
% nonzero_of_real_divide
thf(fact_2609_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_2610_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_2611_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_2612_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_2613_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_2614_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_2615_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_2616_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_2617_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_2618_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_2619_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_2620_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_2621_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_2622_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_2623_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_2624_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_2625_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_2626_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_2627_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_2628_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_2629_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_2630_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_2631_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_2632_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_2633_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_2634_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_2635_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_2636_power__minus__Bit0,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [X2: A,K: num] :
( ( power_power @ A @ ( uminus_uminus @ A @ X2 ) @ ( numeral_numeral @ nat @ ( bit0 @ K ) ) )
= ( power_power @ A @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ K ) ) ) ) ) ).
% power_minus_Bit0
thf(fact_2637_power__minus__Bit1,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [X2: A,K: num] :
( ( power_power @ A @ ( uminus_uminus @ A @ X2 ) @ ( numeral_numeral @ nat @ ( bit1 @ K ) ) )
= ( uminus_uminus @ A @ ( power_power @ A @ X2 @ ( numeral_numeral @ nat @ ( bit1 @ K ) ) ) ) ) ) ).
% power_minus_Bit1
thf(fact_2638_lemma__interval__lt,axiom,
! [A2: real,X2: real,B2: real] :
( ( ord_less @ real @ A2 @ X2 )
=> ( ( ord_less @ real @ X2 @ B2 )
=> ? [D3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D3 )
& ! [Y4: real] :
( ( ord_less @ real @ ( abs_abs @ real @ ( minus_minus @ real @ X2 @ Y4 ) ) @ D3 )
=> ( ( ord_less @ real @ A2 @ Y4 )
& ( ord_less @ real @ Y4 @ B2 ) ) ) ) ) ) ).
% lemma_interval_lt
thf(fact_2639_norm__uminus__minus,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X2: A,Y2: A] :
( ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( uminus_uminus @ A @ X2 ) @ Y2 ) )
= ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ X2 @ Y2 ) ) ) ) ).
% norm_uminus_minus
thf(fact_2640_sin__bound__lemma,axiom,
! [X2: real,Y2: real,U: real,V: real] :
( ( X2 = Y2 )
=> ( ( ord_less_eq @ real @ ( abs_abs @ real @ U ) @ V )
=> ( ord_less_eq @ real @ ( abs_abs @ real @ ( minus_minus @ real @ ( plus_plus @ real @ X2 @ U ) @ Y2 ) ) @ V ) ) ) ).
% sin_bound_lemma
thf(fact_2641_powr__minus__divide,axiom,
! [A: $tType] :
( ( ( real_V3459762299906320749_field @ A )
& ( ln @ A ) )
=> ! [X2: A,A2: A] :
( ( powr @ A @ X2 @ ( uminus_uminus @ A @ A2 ) )
= ( divide_divide @ A @ ( one_one @ A ) @ ( powr @ A @ X2 @ A2 ) ) ) ) ).
% powr_minus_divide
thf(fact_2642_real__0__less__add__iff,axiom,
! [X2: real,Y2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( plus_plus @ real @ X2 @ Y2 ) )
= ( ord_less @ real @ ( uminus_uminus @ real @ X2 ) @ Y2 ) ) ).
% real_0_less_add_iff
thf(fact_2643_real__add__less__0__iff,axiom,
! [X2: real,Y2: real] :
( ( ord_less @ real @ ( plus_plus @ real @ X2 @ Y2 ) @ ( zero_zero @ real ) )
= ( ord_less @ real @ Y2 @ ( uminus_uminus @ real @ X2 ) ) ) ).
% real_add_less_0_iff
thf(fact_2644_real__add__le__0__iff,axiom,
! [X2: real,Y2: real] :
( ( ord_less_eq @ real @ ( plus_plus @ real @ X2 @ Y2 ) @ ( zero_zero @ real ) )
= ( ord_less_eq @ real @ Y2 @ ( uminus_uminus @ real @ X2 ) ) ) ).
% real_add_le_0_iff
thf(fact_2645_real__0__le__add__iff,axiom,
! [X2: real,Y2: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( plus_plus @ real @ X2 @ Y2 ) )
= ( ord_less_eq @ real @ ( uminus_uminus @ real @ X2 ) @ Y2 ) ) ).
% real_0_le_add_iff
thf(fact_2646_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_2647_abs__add__one__gt__zero,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X2: A] : ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ ( one_one @ A ) @ ( abs_abs @ A @ X2 ) ) ) ) ).
% abs_add_one_gt_zero
thf(fact_2648_norm__less__p1,axiom,
! [A: $tType] :
( ( real_V2822296259951069270ebra_1 @ A )
=> ! [X2: A] : ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X2 ) @ ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ ( real_Vector_of_real @ A @ ( real_V7770717601297561774m_norm @ A @ X2 ) ) @ ( one_one @ A ) ) ) ) ) ).
% norm_less_p1
thf(fact_2649_of__int__leD,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: int,X2: A] :
( ( ord_less_eq @ A @ ( abs_abs @ A @ ( ring_1_of_int @ A @ N ) ) @ X2 )
=> ( ( N
= ( zero_zero @ int ) )
| ( ord_less_eq @ A @ ( one_one @ A ) @ X2 ) ) ) ) ).
% of_int_leD
thf(fact_2650_of__int__lessD,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: int,X2: A] :
( ( ord_less @ A @ ( abs_abs @ A @ ( ring_1_of_int @ A @ N ) ) @ X2 )
=> ( ( N
= ( zero_zero @ int ) )
| ( ord_less @ A @ ( one_one @ A ) @ X2 ) ) ) ) ).
% of_int_lessD
thf(fact_2651_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_2652_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_2653_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_2654_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_2655_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_2656_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_2657_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_2658_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_2659_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_2660_minus__divide__add__eq__iff,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z: A,X2: A,Y2: A] :
( ( Z
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ X2 @ Z ) ) @ Y2 )
= ( divide_divide @ A @ ( plus_plus @ A @ ( uminus_uminus @ A @ X2 ) @ ( times_times @ A @ Y2 @ Z ) ) @ Z ) ) ) ) ).
% minus_divide_add_eq_iff
thf(fact_2661_minus__divide__diff__eq__iff,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z: A,X2: A,Y2: A] :
( ( Z
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ X2 @ Z ) ) @ Y2 )
= ( divide_divide @ A @ ( minus_minus @ A @ ( uminus_uminus @ A @ X2 ) @ ( times_times @ A @ Y2 @ Z ) ) @ Z ) ) ) ) ).
% minus_divide_diff_eq_iff
thf(fact_2662_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_2663_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_2664_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_2665_power2__eq__iff,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [X2: A,Y2: A] :
( ( ( power_power @ A @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( power_power @ A @ Y2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( ( X2 = Y2 )
| ( X2
= ( uminus_uminus @ A @ Y2 ) ) ) ) ) ).
% power2_eq_iff
thf(fact_2666_lemma__interval,axiom,
! [A2: real,X2: real,B2: real] :
( ( ord_less @ real @ A2 @ X2 )
=> ( ( ord_less @ real @ X2 @ B2 )
=> ? [D3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D3 )
& ! [Y4: real] :
( ( ord_less @ real @ ( abs_abs @ real @ ( minus_minus @ real @ X2 @ Y4 ) ) @ D3 )
=> ( ( ord_less_eq @ real @ A2 @ Y4 )
& ( ord_less_eq @ real @ Y4 @ B2 ) ) ) ) ) ) ).
% lemma_interval
thf(fact_2667_round__diff__minimal,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z: A,M: int] : ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ Z @ ( ring_1_of_int @ A @ ( archimedean_round @ A @ Z ) ) ) ) @ ( abs_abs @ A @ ( minus_minus @ A @ Z @ ( ring_1_of_int @ A @ M ) ) ) ) ) ).
% round_diff_minimal
thf(fact_2668_norm__triangle__ineq3,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [A2: A,B2: A] : ( ord_less_eq @ real @ ( abs_abs @ real @ ( minus_minus @ real @ ( real_V7770717601297561774m_norm @ A @ A2 ) @ ( real_V7770717601297561774m_norm @ A @ B2 ) ) ) @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ A2 @ B2 ) ) ) ) ).
% norm_triangle_ineq3
thf(fact_2669_abs__le__square__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X2: A,Y2: A] :
( ( ord_less_eq @ A @ ( abs_abs @ A @ X2 ) @ ( abs_abs @ A @ Y2 ) )
= ( ord_less_eq @ A @ ( power_power @ A @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% abs_le_square_iff
thf(fact_2670_abs__square__eq__1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X2: A] :
( ( ( power_power @ A @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( one_one @ A ) )
= ( ( abs_abs @ A @ X2 )
= ( one_one @ A ) ) ) ) ).
% abs_square_eq_1
thf(fact_2671_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I3_J,axiom,
! [A2: $o,B2: $o,Va: nat] :
( ( vEBT_T_p_r_e_d2 @ ( vEBT_Leaf @ A2 @ B2 ) @ ( suc @ ( suc @ Va ) ) )
= ( one_one @ nat ) ) ).
% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(3)
thf(fact_2672_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_2673_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_2674_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I4_J,axiom,
! [Uy: nat,Uz: list @ vEBT_VEBT,Va: vEBT_VEBT,Vb: nat] :
( ( vEBT_T_p_r_e_d2 @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uy @ Uz @ Va ) @ Vb )
= ( one_one @ nat ) ) ).
% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(4)
thf(fact_2675_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_2676_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_2677_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_2678_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_2679_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_2680_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_2681_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_2682_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_2683_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_2684_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_2685_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_2686_realpow__square__minus__le,axiom,
! [U: real,X2: real] : ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( power_power @ real @ U @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( power_power @ real @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% realpow_square_minus_le
thf(fact_2687_powr__neg__one,axiom,
! [X2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( powr @ real @ X2 @ ( uminus_uminus @ real @ ( one_one @ real ) ) )
= ( divide_divide @ real @ ( one_one @ real ) @ X2 ) ) ) ).
% powr_neg_one
thf(fact_2688_ln__add__one__self__le__self2,axiom,
! [X2: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X2 )
=> ( ord_less_eq @ real @ ( ln_ln @ real @ ( plus_plus @ real @ ( one_one @ real ) @ X2 ) ) @ X2 ) ) ).
% ln_add_one_self_le_self2
thf(fact_2689_power2__le__iff__abs__le,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Y2: A,X2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y2 )
=> ( ( ord_less_eq @ A @ ( power_power @ A @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( ord_less_eq @ A @ ( abs_abs @ A @ X2 ) @ Y2 ) ) ) ) ).
% power2_le_iff_abs_le
thf(fact_2690_abs__sqrt__wlog,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [P: A > A > $o,X2: 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 @ X2 ) @ ( power_power @ A @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% abs_sqrt_wlog
thf(fact_2691_abs__square__le__1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X2: A] :
( ( ord_less_eq @ A @ ( power_power @ A @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) )
= ( ord_less_eq @ A @ ( abs_abs @ A @ X2 ) @ ( one_one @ A ) ) ) ) ).
% abs_square_le_1
thf(fact_2692_abs__square__less__1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X2: A] :
( ( ord_less @ A @ ( power_power @ A @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) )
= ( ord_less @ A @ ( abs_abs @ A @ X2 ) @ ( one_one @ A ) ) ) ) ).
% abs_square_less_1
thf(fact_2693_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_2694_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_2695_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_2696_square__le__1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X2: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ X2 )
=> ( ( ord_less_eq @ A @ X2 @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( power_power @ A @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) ) ) ) ) ).
% square_le_1
thf(fact_2697_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_2698_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_2699_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_2700_Bernoulli__inequality,axiom,
! [X2: real,N: nat] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X2 )
=> ( ord_less_eq @ real @ ( plus_plus @ real @ ( one_one @ real ) @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ X2 ) ) @ ( power_power @ real @ ( plus_plus @ real @ ( one_one @ real ) @ X2 ) @ N ) ) ) ).
% Bernoulli_inequality
thf(fact_2701_ln__one__minus__pos__upper__bound,axiom,
! [X2: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ord_less @ real @ X2 @ ( one_one @ real ) )
=> ( ord_less_eq @ real @ ( ln_ln @ real @ ( minus_minus @ real @ ( one_one @ real ) @ X2 ) ) @ ( uminus_uminus @ real @ X2 ) ) ) ) ).
% ln_one_minus_pos_upper_bound
thf(fact_2702_abs__ln__one__plus__x__minus__x__bound__nonpos,axiom,
! [X2: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X2 )
=> ( ( ord_less_eq @ real @ X2 @ ( zero_zero @ real ) )
=> ( ord_less_eq @ real @ ( abs_abs @ real @ ( minus_minus @ real @ ( ln_ln @ real @ ( plus_plus @ real @ ( one_one @ real ) @ X2 ) ) @ X2 ) ) @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( power_power @ real @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% abs_ln_one_plus_x_minus_x_bound_nonpos
thf(fact_2703_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_2704_log__minus__eq__powr,axiom,
! [B2: real,X2: real,Y2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ B2 )
=> ( ( B2
!= ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( minus_minus @ real @ ( log @ B2 @ X2 ) @ Y2 )
= ( log @ B2 @ ( times_times @ real @ X2 @ ( powr @ real @ B2 @ ( uminus_uminus @ real @ Y2 ) ) ) ) ) ) ) ) ).
% log_minus_eq_powr
thf(fact_2705_pred__bound__height_H,axiom,
! [T2: vEBT_VEBT,N: nat,X2: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ord_less_eq @ nat @ ( vEBT_T_p_r_e_d2 @ T2 @ X2 ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_VEBT_height @ T2 ) ) ) ) ).
% pred_bound_height'
thf(fact_2706_powr__neg__numeral,axiom,
! [X2: real,N: num] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( powr @ real @ X2 @ ( uminus_uminus @ real @ ( numeral_numeral @ real @ N ) ) )
= ( divide_divide @ real @ ( one_one @ real ) @ ( power_power @ real @ X2 @ ( numeral_numeral @ nat @ N ) ) ) ) ) ).
% powr_neg_numeral
thf(fact_2707_of__int__round__abs__le,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A] : ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ ( archimedean_round @ A @ X2 ) ) @ X2 ) ) @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% of_int_round_abs_le
thf(fact_2708_round__unique_H,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A,N: int] :
( ( ord_less @ A @ ( abs_abs @ A @ ( minus_minus @ A @ X2 @ ( ring_1_of_int @ A @ N ) ) ) @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
=> ( ( archimedean_round @ A @ X2 )
= N ) ) ) ).
% round_unique'
thf(fact_2709_abs__ln__one__plus__x__minus__x__bound__nonneg,axiom,
! [X2: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ord_less_eq @ real @ X2 @ ( one_one @ real ) )
=> ( ord_less_eq @ real @ ( abs_abs @ real @ ( minus_minus @ real @ ( ln_ln @ real @ ( plus_plus @ real @ ( one_one @ real ) @ X2 ) ) @ X2 ) ) @ ( power_power @ real @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% abs_ln_one_plus_x_minus_x_bound_nonneg
thf(fact_2710_arsinh__def,axiom,
! [A: $tType] :
( ( ln @ A )
=> ( ( arsinh @ A )
= ( ^ [X: A] : ( ln_ln @ A @ ( plus_plus @ A @ X @ ( powr @ A @ ( plus_plus @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) ) @ ( real_Vector_of_real @ A @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% arsinh_def
thf(fact_2711_sum_Ofinite__Collect__op,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [I5: set @ B,X2: B > A,Y2: B > A] :
( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [I3: B] :
( ( member @ B @ I3 @ I5 )
& ( ( X2 @ I3 )
!= ( zero_zero @ A ) ) ) ) )
=> ( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [I3: B] :
( ( member @ B @ I3 @ I5 )
& ( ( Y2 @ I3 )
!= ( zero_zero @ A ) ) ) ) )
=> ( finite_finite2 @ B
@ ( collect @ B
@ ^ [I3: B] :
( ( member @ B @ I3 @ I5 )
& ( ( plus_plus @ A @ ( X2 @ I3 ) @ ( Y2 @ I3 ) )
!= ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% sum.finite_Collect_op
thf(fact_2712_of__int__code__if,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ( ( ring_1_of_int @ A )
= ( ^ [K2: int] :
( if @ A
@ ( K2
= ( zero_zero @ int ) )
@ ( zero_zero @ A )
@ ( if @ A @ ( ord_less @ int @ K2 @ ( zero_zero @ int ) ) @ ( uminus_uminus @ A @ ( ring_1_of_int @ A @ ( uminus_uminus @ int @ K2 ) ) )
@ ( if @ A
@ ( ( modulo_modulo @ int @ K2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) )
= ( zero_zero @ int ) )
@ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( ring_1_of_int @ A @ ( divide_divide @ int @ K2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) )
@ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( ring_1_of_int @ A @ ( divide_divide @ int @ K2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) @ ( one_one @ A ) ) ) ) ) ) ) ) ).
% of_int_code_if
thf(fact_2713_compl__le__compl__iff,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X2: A,Y2: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ X2 ) @ ( uminus_uminus @ A @ Y2 ) )
= ( ord_less_eq @ A @ Y2 @ X2 ) ) ) ).
% compl_le_compl_iff
thf(fact_2714_monoseq__arctan__series,axiom,
! [X2: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ X2 ) @ ( 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 @ X2 @ ( plus_plus @ nat @ ( times_times @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) ) ) ).
% monoseq_arctan_series
thf(fact_2715_ln__series,axiom,
! [X2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ord_less @ real @ X2 @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) )
=> ( ( ln_ln @ real @ X2 )
= ( suminf @ real
@ ^ [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 @ X2 @ ( one_one @ real ) ) @ ( suc @ N3 ) ) ) ) ) ) ) ).
% ln_series
thf(fact_2716_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_2717_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_2718_Compl__subset__Compl__iff,axiom,
! [A: $tType,A3: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A3 ) @ ( uminus_uminus @ ( set @ A ) @ B5 ) )
= ( ord_less_eq @ ( set @ A ) @ B5 @ A3 ) ) ).
% Compl_subset_Compl_iff
thf(fact_2719_Compl__anti__mono,axiom,
! [A: $tType,A3: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B5 )
=> ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ B5 ) @ ( uminus_uminus @ ( set @ A ) @ A3 ) ) ) ).
% Compl_anti_mono
thf(fact_2720_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_2721_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_2722_word__n1__ge,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y2: word @ A] : ( ord_less_eq @ ( word @ A ) @ Y2 @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) ) ) ).
% word_n1_ge
thf(fact_2723_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_2724_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_2725_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_2726_word__minus__one__le,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) @ X2 )
= ( X2
= ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) ) ) ) ).
% word_minus_one_le
thf(fact_2727_zdvd1__eq,axiom,
! [X2: int] :
( ( dvd_dvd @ int @ X2 @ ( one_one @ int ) )
= ( ( abs_abs @ int @ X2 )
= ( one_one @ int ) ) ) ).
% zdvd1_eq
thf(fact_2728_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_2729_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_2730_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_2731_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_2732_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_2733_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_2734_negative__zle,axiom,
! [N: nat,M: nat] : ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N ) ) @ ( semiring_1_of_nat @ int @ M ) ) ).
% negative_zle
thf(fact_2735_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_2736_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_2737_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_2738_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_2739_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_2740_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_2741_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_2742_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_2743_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_2744_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_2745_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_2746_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_2747_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_2748_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_2749_zabs__def,axiom,
( ( abs_abs @ int )
= ( ^ [I3: int] : ( if @ int @ ( ord_less @ int @ I3 @ ( zero_zero @ int ) ) @ ( uminus_uminus @ int @ I3 ) @ I3 ) ) ) ).
% zabs_def
thf(fact_2750_signed__take__bit__add,axiom,
! [N: nat,K: int,L2: int] :
( ( bit_ri4674362597316999326ke_bit @ int @ N @ ( plus_plus @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N @ K ) @ ( bit_ri4674362597316999326ke_bit @ int @ N @ L2 ) ) )
= ( bit_ri4674362597316999326ke_bit @ int @ N @ ( plus_plus @ int @ K @ L2 ) ) ) ).
% signed_take_bit_add
thf(fact_2751_signed__take__bit__mult,axiom,
! [N: nat,K: int,L2: int] :
( ( bit_ri4674362597316999326ke_bit @ int @ N @ ( times_times @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N @ K ) @ ( bit_ri4674362597316999326ke_bit @ int @ N @ L2 ) ) )
= ( bit_ri4674362597316999326ke_bit @ int @ N @ ( times_times @ int @ K @ L2 ) ) ) ).
% signed_take_bit_mult
thf(fact_2752_signed__take__bit__diff,axiom,
! [N: nat,K: int,L2: int] :
( ( bit_ri4674362597316999326ke_bit @ int @ N @ ( minus_minus @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N @ K ) @ ( bit_ri4674362597316999326ke_bit @ int @ N @ L2 ) ) )
= ( bit_ri4674362597316999326ke_bit @ int @ N @ ( minus_minus @ int @ K @ L2 ) ) ) ).
% signed_take_bit_diff
thf(fact_2753_uminus__int__code_I1_J,axiom,
( ( uminus_uminus @ int @ ( zero_zero @ int ) )
= ( zero_zero @ int ) ) ).
% uminus_int_code(1)
thf(fact_2754_int__cases2,axiom,
! [Z: int] :
( ! [N2: nat] :
( Z
!= ( semiring_1_of_nat @ int @ N2 ) )
=> ~ ! [N2: nat] :
( Z
!= ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N2 ) ) ) ) ).
% int_cases2
thf(fact_2755_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_2756_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_2757_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_2758_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_2759_abs__div,axiom,
! [Y2: int,X2: int] :
( ( dvd_dvd @ int @ Y2 @ X2 )
=> ( ( abs_abs @ int @ ( divide_divide @ int @ X2 @ Y2 ) )
= ( divide_divide @ int @ ( abs_abs @ int @ X2 ) @ ( abs_abs @ int @ Y2 ) ) ) ) ).
% abs_div
thf(fact_2760_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_2761_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_2762_minus__int__code_I2_J,axiom,
! [L2: int] :
( ( minus_minus @ int @ ( zero_zero @ int ) @ L2 )
= ( uminus_uminus @ int @ L2 ) ) ).
% minus_int_code(2)
thf(fact_2763_zmod__zminus1__not__zero,axiom,
! [K: int,L2: int] :
( ( ( modulo_modulo @ int @ ( uminus_uminus @ int @ K ) @ L2 )
!= ( zero_zero @ int ) )
=> ( ( modulo_modulo @ int @ K @ L2 )
!= ( zero_zero @ int ) ) ) ).
% zmod_zminus1_not_zero
thf(fact_2764_zmod__zminus2__not__zero,axiom,
! [K: int,L2: int] :
( ( ( modulo_modulo @ int @ K @ ( uminus_uminus @ int @ L2 ) )
!= ( zero_zero @ int ) )
=> ( ( modulo_modulo @ int @ K @ L2 )
!= ( zero_zero @ int ) ) ) ).
% zmod_zminus2_not_zero
thf(fact_2765_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_2766_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_2767_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_2768_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_2769_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_2770_word__not__simps_I3_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y2: word @ A] :
~ ( ord_less @ ( word @ A ) @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) @ Y2 ) ) ).
% word_not_simps(3)
thf(fact_2771_max__word__not__less,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A] :
~ ( ord_less @ ( word @ A ) @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) @ X2 ) ) ).
% max_word_not_less
thf(fact_2772_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_2773_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_2774_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_2775_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_2776_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_2777_abs__mod__less,axiom,
! [L2: int,K: int] :
( ( L2
!= ( zero_zero @ int ) )
=> ( ord_less @ int @ ( abs_abs @ int @ ( modulo_modulo @ int @ K @ L2 ) ) @ ( abs_abs @ int @ L2 ) ) ) ).
% abs_mod_less
thf(fact_2778_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_2779_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_2780_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_2781_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_2782_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_2783_overflow__plus__one__self,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [P2: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) @ P2 ) @ P2 )
= ( P2
= ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) ) ) ) ).
% overflow_plus_one_self
thf(fact_2784_plus__1__less,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ X2 @ ( one_one @ ( word @ A ) ) ) @ X2 )
= ( X2
= ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) ) ) ) ).
% plus_1_less
thf(fact_2785_max__word__wrap,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A] :
( ( ( plus_plus @ ( word @ A ) @ X2 @ ( one_one @ ( word @ A ) ) )
= ( zero_zero @ ( word @ A ) ) )
=> ( X2
= ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) ) ) ) ).
% max_word_wrap
thf(fact_2786_word__add__no__overflow,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A] :
( ( ord_less @ ( word @ A ) @ X2 @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) )
=> ( ord_less @ ( word @ A ) @ X2 @ ( plus_plus @ ( word @ A ) @ X2 @ ( one_one @ ( word @ A ) ) ) ) ) ) ).
% word_add_no_overflow
thf(fact_2787_less__x__plus__1,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A,Y2: word @ A] :
( ( X2
!= ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) )
=> ( ( ord_less @ ( word @ A ) @ Y2 @ ( plus_plus @ ( word @ A ) @ X2 @ ( one_one @ ( word @ A ) ) ) )
= ( ( ord_less @ ( word @ A ) @ Y2 @ X2 )
| ( Y2 = X2 ) ) ) ) ) ).
% less_x_plus_1
thf(fact_2788_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_2789_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_2790_no__plus__overflow__neg,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A,Y2: word @ A] :
( ( ord_less @ ( word @ A ) @ X2 @ ( uminus_uminus @ ( word @ A ) @ Y2 ) )
=> ( ord_less_eq @ ( word @ A ) @ X2 @ ( plus_plus @ ( word @ A ) @ X2 @ Y2 ) ) ) ) ).
% no_plus_overflow_neg
thf(fact_2791_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_2792_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_2793_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_2794_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_2795_negD,axiom,
! [X2: int] :
( ( ord_less @ int @ X2 @ ( zero_zero @ int ) )
=> ? [N2: nat] :
( X2
= ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ ( suc @ N2 ) ) ) ) ) ).
% negD
thf(fact_2796_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_2797_verit__less__mono__div__int2,axiom,
! [A3: int,B5: int,N: int] :
( ( ord_less_eq @ int @ A3 @ B5 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ ( uminus_uminus @ int @ N ) )
=> ( ord_less_eq @ int @ ( divide_divide @ int @ B5 @ N ) @ ( divide_divide @ int @ A3 @ N ) ) ) ) ).
% verit_less_mono_div_int2
thf(fact_2798_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_2799_word__le__make__less,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y2: word @ A,X2: word @ A] :
( ( Y2
!= ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) )
=> ( ( ord_less_eq @ ( word @ A ) @ X2 @ Y2 )
= ( ord_less @ ( word @ A ) @ X2 @ ( plus_plus @ ( word @ A ) @ Y2 @ ( one_one @ ( word @ A ) ) ) ) ) ) ) ).
% word_le_make_less
thf(fact_2800_word__Suc__leq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [K: word @ A,X2: word @ A] :
( ( K
!= ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) )
=> ( ( ord_less @ ( word @ A ) @ X2 @ ( plus_plus @ ( word @ A ) @ K @ ( one_one @ ( word @ A ) ) ) )
= ( ord_less_eq @ ( word @ A ) @ X2 @ K ) ) ) ) ).
% word_Suc_leq
thf(fact_2801_word__Suc__le,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A,K: word @ A] :
( ( X2
!= ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) )
=> ( ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ X2 @ ( one_one @ ( word @ A ) ) ) @ K )
= ( ord_less @ ( word @ A ) @ X2 @ K ) ) ) ) ).
% word_Suc_le
thf(fact_2802_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_2803_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_2804_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_2805_even__abs__add__iff,axiom,
! [K: int,L2: int] :
( ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( plus_plus @ int @ ( abs_abs @ int @ K ) @ L2 ) )
= ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( plus_plus @ int @ K @ L2 ) ) ) ).
% even_abs_add_iff
thf(fact_2806_even__add__abs__iff,axiom,
! [K: int,L2: int] :
( ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( plus_plus @ int @ K @ ( abs_abs @ int @ L2 ) ) )
= ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( plus_plus @ int @ K @ L2 ) ) ) ).
% even_add_abs_iff
thf(fact_2807_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_2808_minus__mod__int__eq,axiom,
! [L2: int,K: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ L2 )
=> ( ( modulo_modulo @ int @ ( uminus_uminus @ int @ K ) @ L2 )
= ( minus_minus @ int @ ( minus_minus @ int @ L2 @ ( one_one @ int ) ) @ ( modulo_modulo @ int @ ( minus_minus @ int @ K @ ( one_one @ int ) ) @ L2 ) ) ) ) ).
% minus_mod_int_eq
thf(fact_2809_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_2810_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_2811_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_2812_monoseq__realpow,axiom,
! [X2: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ord_less_eq @ real @ X2 @ ( one_one @ real ) )
=> ( topological_monoseq @ real @ ( power_power @ real @ X2 ) ) ) ) ).
% monoseq_realpow
thf(fact_2813_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_2814_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_2815_nat__intermed__int__val,axiom,
! [M: nat,N: nat,F2: nat > int,K: int] :
( ! [I2: nat] :
( ( ( ord_less_eq @ nat @ M @ I2 )
& ( ord_less @ nat @ I2 @ N ) )
=> ( ord_less_eq @ int @ ( abs_abs @ int @ ( minus_minus @ int @ ( F2 @ ( suc @ I2 ) ) @ ( F2 @ I2 ) ) ) @ ( one_one @ int ) ) )
=> ( ( ord_less_eq @ nat @ M @ N )
=> ( ( ord_less_eq @ int @ ( F2 @ M ) @ K )
=> ( ( ord_less_eq @ int @ K @ ( F2 @ N ) )
=> ? [I2: nat] :
( ( ord_less_eq @ nat @ M @ I2 )
& ( ord_less_eq @ nat @ I2 @ N )
& ( ( F2 @ I2 )
= K ) ) ) ) ) ) ).
% nat_intermed_int_val
thf(fact_2816_dbl__dec__def,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( ( neg_numeral_dbl_dec @ A )
= ( ^ [X: A] : ( minus_minus @ A @ ( plus_plus @ A @ X @ X ) @ ( one_one @ A ) ) ) ) ) ).
% dbl_dec_def
thf(fact_2817_incr__lemma,axiom,
! [D2: int,Z: int,X2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D2 )
=> ( ord_less @ int @ Z @ ( plus_plus @ int @ X2 @ ( times_times @ int @ ( plus_plus @ int @ ( abs_abs @ int @ ( minus_minus @ int @ X2 @ Z ) ) @ ( one_one @ int ) ) @ D2 ) ) ) ) ).
% incr_lemma
thf(fact_2818_decr__lemma,axiom,
! [D2: int,X2: int,Z: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D2 )
=> ( ord_less @ int @ ( minus_minus @ int @ X2 @ ( times_times @ int @ ( plus_plus @ int @ ( abs_abs @ int @ ( minus_minus @ int @ X2 @ Z ) ) @ ( one_one @ int ) ) @ D2 ) ) @ Z ) ) ).
% decr_lemma
thf(fact_2819_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_2820_div__pos__neg__trivial,axiom,
! [K: int,L2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ K )
=> ( ( ord_less_eq @ int @ ( plus_plus @ int @ K @ L2 ) @ ( zero_zero @ int ) )
=> ( ( divide_divide @ int @ K @ L2 )
= ( uminus_uminus @ int @ ( one_one @ int ) ) ) ) ) ).
% div_pos_neg_trivial
thf(fact_2821_compl__mono,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X2: A,Y2: A] :
( ( ord_less_eq @ A @ X2 @ Y2 )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ Y2 ) @ ( uminus_uminus @ A @ X2 ) ) ) ) ).
% compl_mono
thf(fact_2822_compl__le__swap1,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [Y2: A,X2: A] :
( ( ord_less_eq @ A @ Y2 @ ( uminus_uminus @ A @ X2 ) )
=> ( ord_less_eq @ A @ X2 @ ( uminus_uminus @ A @ Y2 ) ) ) ) ).
% compl_le_swap1
thf(fact_2823_compl__le__swap2,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [Y2: A,X2: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ Y2 ) @ X2 )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ X2 ) @ Y2 ) ) ) ).
% compl_le_swap2
thf(fact_2824_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_2825_nat__ivt__aux,axiom,
! [N: nat,F2: nat > int,K: int] :
( ! [I2: nat] :
( ( ord_less @ nat @ I2 @ N )
=> ( ord_less_eq @ int @ ( abs_abs @ int @ ( minus_minus @ int @ ( F2 @ ( suc @ I2 ) ) @ ( F2 @ I2 ) ) ) @ ( one_one @ int ) ) )
=> ( ( ord_less_eq @ int @ ( F2 @ ( zero_zero @ nat ) ) @ K )
=> ( ( ord_less_eq @ int @ K @ ( F2 @ N ) )
=> ? [I2: nat] :
( ( ord_less_eq @ nat @ I2 @ N )
& ( ( F2 @ I2 )
= K ) ) ) ) ) ).
% nat_ivt_aux
thf(fact_2826_int__bit__induct,axiom,
! [P: int > $o,K: int] :
( ( P @ ( zero_zero @ int ) )
=> ( ( P @ ( uminus_uminus @ int @ ( one_one @ int ) ) )
=> ( ! [K3: int] :
( ( P @ K3 )
=> ( ( K3
!= ( zero_zero @ int ) )
=> ( P @ ( times_times @ int @ K3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) )
=> ( ! [K3: int] :
( ( P @ K3 )
=> ( ( K3
!= ( uminus_uminus @ int @ ( one_one @ int ) ) )
=> ( P @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ K3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) )
=> ( P @ K ) ) ) ) ) ).
% int_bit_induct
thf(fact_2827_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_2828_nat0__intermed__int__val,axiom,
! [N: nat,F2: nat > int,K: int] :
( ! [I2: nat] :
( ( ord_less @ nat @ I2 @ N )
=> ( ord_less_eq @ int @ ( abs_abs @ int @ ( minus_minus @ int @ ( F2 @ ( plus_plus @ nat @ I2 @ ( one_one @ nat ) ) ) @ ( F2 @ I2 ) ) ) @ ( one_one @ int ) ) )
=> ( ( ord_less_eq @ int @ ( F2 @ ( zero_zero @ nat ) ) @ K )
=> ( ( ord_less_eq @ int @ K @ ( F2 @ N ) )
=> ? [I2: nat] :
( ( ord_less_eq @ nat @ I2 @ N )
& ( ( F2 @ I2 )
= K ) ) ) ) ) ).
% nat0_intermed_int_val
thf(fact_2829_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_2830_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_2831_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_2832_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_2833_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_2834_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_2835_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_2836_arctan__series,axiom,
! [X2: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ X2 ) @ ( one_one @ real ) )
=> ( ( arctan @ X2 )
= ( suminf @ real
@ ^ [K2: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ K2 ) @ ( times_times @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( semiring_1_of_nat @ real @ ( plus_plus @ nat @ ( times_times @ nat @ K2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) @ ( power_power @ real @ X2 @ ( plus_plus @ nat @ ( times_times @ nat @ K2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) ) ) ) ) ).
% arctan_series
thf(fact_2837_pi__series,axiom,
( ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ ( bit0 @ one2 ) ) ) )
= ( suminf @ real
@ ^ [K2: nat] : ( divide_divide @ real @ ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ K2 ) @ ( one_one @ real ) ) @ ( semiring_1_of_nat @ real @ ( plus_plus @ nat @ ( times_times @ nat @ K2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) ) ) ).
% pi_series
thf(fact_2838_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_2839_summable__arctan__series,axiom,
! [X2: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ X2 ) @ ( one_one @ real ) )
=> ( summable @ real
@ ^ [K2: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ K2 ) @ ( times_times @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( semiring_1_of_nat @ real @ ( plus_plus @ nat @ ( times_times @ nat @ K2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) @ ( power_power @ real @ X2 @ ( plus_plus @ nat @ ( times_times @ nat @ K2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) ) ) ) ).
% summable_arctan_series
thf(fact_2840_arctan__eq__zero__iff,axiom,
! [X2: real] :
( ( ( arctan @ X2 )
= ( zero_zero @ real ) )
= ( X2
= ( zero_zero @ real ) ) ) ).
% arctan_eq_zero_iff
thf(fact_2841_arctan__zero__zero,axiom,
( ( arctan @ ( zero_zero @ real ) )
= ( zero_zero @ real ) ) ).
% arctan_zero_zero
thf(fact_2842_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_2843_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_2844_summable__zero,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ( summable @ A
@ ^ [N3: nat] : ( zero_zero @ A ) ) ) ).
% summable_zero
thf(fact_2845_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_2846_zero__less__arctan__iff,axiom,
! [X2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( arctan @ X2 ) )
= ( ord_less @ real @ ( zero_zero @ real ) @ X2 ) ) ).
% zero_less_arctan_iff
thf(fact_2847_arctan__less__zero__iff,axiom,
! [X2: real] :
( ( ord_less @ real @ ( arctan @ X2 ) @ ( zero_zero @ real ) )
= ( ord_less @ real @ X2 @ ( zero_zero @ real ) ) ) ).
% arctan_less_zero_iff
thf(fact_2848_zero__le__arctan__iff,axiom,
! [X2: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( arctan @ X2 ) )
= ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 ) ) ).
% zero_le_arctan_iff
thf(fact_2849_arctan__le__zero__iff,axiom,
! [X2: real] :
( ( ord_less_eq @ real @ ( arctan @ X2 ) @ ( zero_zero @ real ) )
= ( ord_less_eq @ real @ X2 @ ( zero_zero @ real ) ) ) ).
% arctan_le_zero_iff
thf(fact_2850_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_2851_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_2852_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_2853_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_2854_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_2855_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_2856_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_2857_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_2858_Compl__eq,axiom,
! [A: $tType] :
( ( uminus_uminus @ ( set @ A ) )
= ( ^ [A7: set @ A] :
( collect @ A
@ ^ [X: A] :
~ ( member @ A @ X @ A7 ) ) ) ) ).
% Compl_eq
thf(fact_2859_Collect__neg__eq,axiom,
! [A: $tType,P: A > $o] :
( ( collect @ A
@ ^ [X: A] :
~ ( P @ X ) )
= ( uminus_uminus @ ( set @ A ) @ ( collect @ A @ P ) ) ) ).
% Collect_neg_eq
thf(fact_2860_uminus__set__def,axiom,
! [A: $tType] :
( ( uminus_uminus @ ( set @ A ) )
= ( ^ [A7: set @ A] :
( collect @ A
@ ( uminus_uminus @ ( A > $o )
@ ^ [X: A] : ( member @ A @ X @ A7 ) ) ) ) ) ).
% uminus_set_def
thf(fact_2861_summable__rabs__cancel,axiom,
! [F2: nat > real] :
( ( summable @ real
@ ^ [N3: nat] : ( abs_abs @ real @ ( F2 @ N3 ) ) )
=> ( summable @ real @ F2 ) ) ).
% summable_rabs_cancel
thf(fact_2862_summable__minus__iff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A] :
( ( summable @ A
@ ^ [N3: nat] : ( uminus_uminus @ A @ ( F2 @ N3 ) ) )
= ( summable @ A @ F2 ) ) ) ).
% summable_minus_iff
thf(fact_2863_summable__minus,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A] :
( ( summable @ A @ F2 )
=> ( summable @ A
@ ^ [N3: nat] : ( uminus_uminus @ A @ ( F2 @ N3 ) ) ) ) ) ).
% summable_minus
thf(fact_2864_summable__comparison__test_H,axiom,
! [A: $tType] :
( ( real_Vector_banach @ A )
=> ! [G: nat > real,N5: nat,F2: nat > A] :
( ( summable @ real @ G )
=> ( ! [N2: nat] :
( ( ord_less_eq @ nat @ N5 @ N2 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( F2 @ N2 ) ) @ ( G @ N2 ) ) )
=> ( summable @ A @ F2 ) ) ) ) ).
% summable_comparison_test'
thf(fact_2865_summable__comparison__test,axiom,
! [A: $tType] :
( ( real_Vector_banach @ A )
=> ! [F2: nat > A,G: nat > real] :
( ? [N9: nat] :
! [N2: nat] :
( ( ord_less_eq @ nat @ N9 @ N2 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( F2 @ N2 ) ) @ ( G @ N2 ) ) )
=> ( ( summable @ real @ G )
=> ( summable @ A @ F2 ) ) ) ) ).
% summable_comparison_test
thf(fact_2866_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_2867_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_2868_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_2869_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_2870_summable__diff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A,G: nat > A] :
( ( summable @ A @ F2 )
=> ( ( summable @ A @ G )
=> ( summable @ A
@ ^ [N3: nat] : ( minus_minus @ A @ ( F2 @ N3 ) @ ( G @ N3 ) ) ) ) ) ) ).
% summable_diff
thf(fact_2871_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_2872_summable__of__real,axiom,
! [A: $tType] :
( ( ( real_V2191834092415804123ebra_1 @ A )
& ( real_V822414075346904944vector @ A ) )
=> ! [X6: nat > real] :
( ( summable @ real @ X6 )
=> ( summable @ A
@ ^ [N3: nat] : ( real_Vector_of_real @ A @ ( X6 @ N3 ) ) ) ) ) ).
% summable_of_real
thf(fact_2873_summable__norm__cancel,axiom,
! [A: $tType] :
( ( real_Vector_banach @ A )
=> ! [F2: nat > A] :
( ( summable @ real
@ ^ [N3: nat] : ( real_V7770717601297561774m_norm @ A @ ( F2 @ N3 ) ) )
=> ( summable @ A @ F2 ) ) ) ).
% summable_norm_cancel
thf(fact_2874_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_2875_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_2876_pi__neq__zero,axiom,
( pi
!= ( zero_zero @ real ) ) ).
% pi_neq_zero
thf(fact_2877_arctan__le__iff,axiom,
! [X2: real,Y2: real] :
( ( ord_less_eq @ real @ ( arctan @ X2 ) @ ( arctan @ Y2 ) )
= ( ord_less_eq @ real @ X2 @ Y2 ) ) ).
% arctan_le_iff
thf(fact_2878_arctan__monotone_H,axiom,
! [X2: real,Y2: real] :
( ( ord_less_eq @ real @ X2 @ Y2 )
=> ( ord_less_eq @ real @ ( arctan @ X2 ) @ ( arctan @ Y2 ) ) ) ).
% arctan_monotone'
thf(fact_2879_suminf__le,axiom,
! [A: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F2: nat > A,G: nat > A] :
( ! [N2: nat] : ( ord_less_eq @ A @ ( F2 @ N2 ) @ ( G @ N2 ) )
=> ( ( summable @ A @ F2 )
=> ( ( summable @ A @ G )
=> ( ord_less_eq @ A @ ( suminf @ A @ F2 ) @ ( suminf @ A @ G ) ) ) ) ) ) ).
% suminf_le
thf(fact_2880_summable__finite,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [N5: set @ nat,F2: nat > A] :
( ( finite_finite2 @ nat @ N5 )
=> ( ! [N2: nat] :
( ~ ( member @ nat @ N2 @ N5 )
=> ( ( F2 @ N2 )
= ( zero_zero @ A ) ) )
=> ( summable @ A @ F2 ) ) ) ) ).
% summable_finite
thf(fact_2881_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_2882_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_2883_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_2884_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_2885_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_2886_suminf__diff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A,G: nat > A] :
( ( summable @ A @ F2 )
=> ( ( summable @ A @ G )
=> ( ( minus_minus @ A @ ( suminf @ A @ F2 ) @ ( suminf @ A @ G ) )
= ( suminf @ A
@ ^ [N3: nat] : ( minus_minus @ A @ ( F2 @ N3 ) @ ( G @ N3 ) ) ) ) ) ) ) ).
% suminf_diff
thf(fact_2887_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_2888_suminf__minus,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A] :
( ( summable @ A @ F2 )
=> ( ( suminf @ A
@ ^ [N3: nat] : ( uminus_uminus @ A @ ( F2 @ N3 ) ) )
= ( uminus_uminus @ A @ ( suminf @ A @ F2 ) ) ) ) ) ).
% suminf_minus
thf(fact_2889_powser__insidea,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [F2: nat > A,X2: A,Z: A] :
( ( summable @ A
@ ^ [N3: nat] : ( times_times @ A @ ( F2 @ N3 ) @ ( power_power @ A @ X2 @ N3 ) ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ Z ) @ ( real_V7770717601297561774m_norm @ A @ X2 ) )
=> ( summable @ real
@ ^ [N3: nat] : ( real_V7770717601297561774m_norm @ A @ ( times_times @ A @ ( F2 @ N3 ) @ ( power_power @ A @ Z @ N3 ) ) ) ) ) ) ) ).
% powser_insidea
thf(fact_2890_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_2891_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_2892_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_2893_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_2894_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_2895_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_2896_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_2897_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_2898_pi__gt__zero,axiom,
ord_less @ real @ ( zero_zero @ real ) @ pi ).
% pi_gt_zero
thf(fact_2899_pi__not__less__zero,axiom,
~ ( ord_less @ real @ pi @ ( zero_zero @ real ) ) ).
% pi_not_less_zero
thf(fact_2900_pi__ge__zero,axiom,
ord_less_eq @ real @ ( zero_zero @ real ) @ pi ).
% pi_ge_zero
thf(fact_2901_dbl__def,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( ( neg_numeral_dbl @ A )
= ( ^ [X: A] : ( plus_plus @ A @ X @ X ) ) ) ) ).
% dbl_def
thf(fact_2902_summable__norm__comparison__test,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A,G: nat > real] :
( ? [N9: nat] :
! [N2: nat] :
( ( ord_less_eq @ nat @ N9 @ N2 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( F2 @ N2 ) ) @ ( G @ N2 ) ) )
=> ( ( summable @ real @ G )
=> ( summable @ real
@ ^ [N3: nat] : ( real_V7770717601297561774m_norm @ A @ ( F2 @ N3 ) ) ) ) ) ) ).
% summable_norm_comparison_test
thf(fact_2903_summable__rabs__comparison__test,axiom,
! [F2: nat > real,G: nat > real] :
( ? [N9: nat] :
! [N2: nat] :
( ( ord_less_eq @ nat @ N9 @ N2 )
=> ( ord_less_eq @ real @ ( abs_abs @ real @ ( F2 @ N2 ) ) @ ( G @ N2 ) ) )
=> ( ( summable @ real @ G )
=> ( summable @ real
@ ^ [N3: nat] : ( abs_abs @ real @ ( F2 @ N3 ) ) ) ) ) ).
% summable_rabs_comparison_test
thf(fact_2904_summable__rabs,axiom,
! [F2: nat > real] :
( ( summable @ real
@ ^ [N3: nat] : ( abs_abs @ real @ ( F2 @ N3 ) ) )
=> ( ord_less_eq @ real @ ( abs_abs @ real @ ( suminf @ real @ F2 ) )
@ ( suminf @ real
@ ^ [N3: nat] : ( abs_abs @ real @ ( F2 @ N3 ) ) ) ) ) ).
% summable_rabs
thf(fact_2905_suminf__of__real,axiom,
! [A: $tType] :
( ( ( real_V2191834092415804123ebra_1 @ A )
& ( real_V822414075346904944vector @ A ) )
=> ! [X6: nat > real] :
( ( summable @ real @ X6 )
=> ( ( real_Vector_of_real @ A @ ( suminf @ real @ X6 ) )
= ( suminf @ A
@ ^ [N3: nat] : ( real_Vector_of_real @ A @ ( X6 @ N3 ) ) ) ) ) ) ).
% suminf_of_real
thf(fact_2906_arctan__ubound,axiom,
! [Y2: real] : ( ord_less @ real @ ( arctan @ Y2 ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ).
% arctan_ubound
thf(fact_2907_arctan__one,axiom,
( ( arctan @ ( one_one @ real ) )
= ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ).
% arctan_one
thf(fact_2908_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 ) )
= ( ? [I3: nat] : ( ord_less @ A @ ( zero_zero @ A ) @ ( F2 @ I3 ) ) ) ) ) ) ) ).
% suminf_pos_iff
thf(fact_2909_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_2910_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_2911_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_2912_complete__algebra__summable__geometric,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X2: A] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X2 ) @ ( one_one @ real ) )
=> ( summable @ A @ ( power_power @ A @ X2 ) ) ) ) ).
% complete_algebra_summable_geometric
thf(fact_2913_arctan__bounded,axiom,
! [Y2: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ ( arctan @ Y2 ) )
& ( ord_less @ real @ ( arctan @ Y2 ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ).
% arctan_bounded
thf(fact_2914_arctan__lbound,axiom,
! [Y2: real] : ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ ( arctan @ Y2 ) ) ).
% arctan_lbound
thf(fact_2915_summable__norm,axiom,
! [A: $tType] :
( ( real_Vector_banach @ A )
=> ! [F2: nat > A] :
( ( summable @ real
@ ^ [N3: nat] : ( real_V7770717601297561774m_norm @ A @ ( F2 @ N3 ) ) )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( suminf @ A @ F2 ) )
@ ( suminf @ real
@ ^ [N3: nat] : ( real_V7770717601297561774m_norm @ A @ ( F2 @ N3 ) ) ) ) ) ) ).
% summable_norm
thf(fact_2916_powser__inside,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V8999393235501362500lgebra @ A ) )
=> ! [F2: nat > A,X2: A,Z: A] :
( ( summable @ A
@ ^ [N3: nat] : ( times_times @ A @ ( F2 @ N3 ) @ ( power_power @ A @ X2 @ N3 ) ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ Z ) @ ( real_V7770717601297561774m_norm @ A @ X2 ) )
=> ( summable @ A
@ ^ [N3: nat] : ( times_times @ A @ ( F2 @ N3 ) @ ( power_power @ A @ Z @ N3 ) ) ) ) ) ) ).
% powser_inside
thf(fact_2917_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_2918_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_2919_pi__less__4,axiom,
ord_less @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ ( bit0 @ one2 ) ) ) ).
% pi_less_4
thf(fact_2920_pi__ge__two,axiom,
ord_less_eq @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ).
% pi_ge_two
thf(fact_2921_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_2922_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_2923_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_2924_suminf__exist__split,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [R2: real,F2: nat > A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R2 )
=> ( ( summable @ A @ F2 )
=> ? [N10: nat] :
! [N11: nat] :
( ( ord_less_eq @ nat @ N10 @ N11 )
=> ( ord_less @ real
@ ( real_V7770717601297561774m_norm @ A
@ ( suminf @ A
@ ^ [I3: nat] : ( F2 @ ( plus_plus @ nat @ I3 @ N11 ) ) ) )
@ R2 ) ) ) ) ) ).
% suminf_exist_split
thf(fact_2925_summable__power__series,axiom,
! [F2: nat > real,Z: real] :
( ! [I2: nat] : ( ord_less_eq @ real @ ( F2 @ I2 ) @ ( one_one @ real ) )
=> ( ! [I2: nat] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( F2 @ I2 ) )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Z )
=> ( ( ord_less @ real @ Z @ ( one_one @ real ) )
=> ( summable @ real
@ ^ [I3: nat] : ( times_times @ real @ ( F2 @ I3 ) @ ( power_power @ real @ Z @ I3 ) ) ) ) ) ) ) ).
% summable_power_series
thf(fact_2926_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_2927_summable__ratio__test,axiom,
! [A: $tType] :
( ( real_Vector_banach @ A )
=> ! [C2: real,N5: nat,F2: nat > A] :
( ( ord_less @ real @ C2 @ ( one_one @ real ) )
=> ( ! [N2: nat] :
( ( ord_less_eq @ nat @ N5 @ 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_2928_pi__half__neq__zero,axiom,
( ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) )
!= ( zero_zero @ real ) ) ).
% pi_half_neq_zero
thf(fact_2929_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_2930_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_2931_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_2932_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_2933_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_2934_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_2935_arctan__add,axiom,
! [X2: real,Y2: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ X2 ) @ ( one_one @ real ) )
=> ( ( ord_less @ real @ ( abs_abs @ real @ Y2 ) @ ( one_one @ real ) )
=> ( ( plus_plus @ real @ ( arctan @ X2 ) @ ( arctan @ Y2 ) )
= ( arctan @ ( divide_divide @ real @ ( plus_plus @ real @ X2 @ Y2 ) @ ( minus_minus @ real @ ( one_one @ real ) @ ( times_times @ real @ X2 @ Y2 ) ) ) ) ) ) ) ).
% arctan_add
thf(fact_2936_arctan__double,axiom,
! [X2: real] :
( ( ord_less @ real @ ( abs_abs @ real @ X2 ) @ ( one_one @ real ) )
=> ( ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( arctan @ X2 ) )
= ( arctan @ ( divide_divide @ real @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ X2 ) @ ( minus_minus @ real @ ( one_one @ real ) @ ( power_power @ real @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% arctan_double
thf(fact_2937_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_2938_signed__take__bit__numeral__minus__bit1,axiom,
! [L2: num,K: num] :
( ( bit_ri4674362597316999326ke_bit @ int @ ( numeral_numeral @ nat @ L2 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ K ) ) ) )
= ( plus_plus @ int @ ( times_times @ int @ ( bit_ri4674362597316999326ke_bit @ int @ ( pred_numeral @ L2 ) @ ( minus_minus @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) @ ( one_one @ int ) ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( one_one @ int ) ) ) ).
% signed_take_bit_numeral_minus_bit1
thf(fact_2939_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_2940_sum__gp,axiom,
! [A: $tType] :
( ( ( division_ring @ A )
& ( comm_ring @ A ) )
=> ! [N: nat,M: nat,X2: A] :
( ( ( ord_less @ nat @ N @ M )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X2 ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( zero_zero @ A ) ) )
& ( ~ ( ord_less @ nat @ N @ M )
=> ( ( ( X2
= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X2 ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) @ M ) ) ) )
& ( ( X2
!= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X2 ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( power_power @ A @ X2 @ M ) @ ( power_power @ A @ X2 @ ( suc @ N ) ) ) @ ( minus_minus @ A @ ( one_one @ A ) @ X2 ) ) ) ) ) ) ) ) ).
% sum_gp
thf(fact_2941_signed__take__bit__numeral__bit1,axiom,
! [L2: num,K: num] :
( ( bit_ri4674362597316999326ke_bit @ int @ ( numeral_numeral @ nat @ L2 ) @ ( numeral_numeral @ int @ ( bit1 @ K ) ) )
= ( plus_plus @ int @ ( times_times @ int @ ( bit_ri4674362597316999326ke_bit @ int @ ( pred_numeral @ L2 ) @ ( numeral_numeral @ int @ K ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( one_one @ int ) ) ) ).
% signed_take_bit_numeral_bit1
thf(fact_2942_summable__complex__of__real,axiom,
! [F2: nat > real] :
( ( summable @ complex
@ ^ [N3: nat] : ( real_Vector_of_real @ complex @ ( F2 @ N3 ) ) )
= ( summable @ real @ F2 ) ) ).
% summable_complex_of_real
thf(fact_2943_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_2944_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_2945_of__nat__sum,axiom,
! [A: $tType,B: $tType] :
( ( semiring_1 @ A )
=> ! [F2: B > nat,A3: set @ B] :
( ( semiring_1_of_nat @ A @ ( groups7311177749621191930dd_sum @ B @ nat @ F2 @ A3 ) )
= ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X: B] : ( semiring_1_of_nat @ A @ ( F2 @ X ) )
@ A3 ) ) ) ).
% of_nat_sum
thf(fact_2946_of__int__sum,axiom,
! [A: $tType,B: $tType] :
( ( ring_1 @ A )
=> ! [F2: B > int,A3: set @ B] :
( ( ring_1_of_int @ A @ ( groups7311177749621191930dd_sum @ B @ int @ F2 @ A3 ) )
= ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X: B] : ( ring_1_of_int @ A @ ( F2 @ X ) )
@ A3 ) ) ) ).
% of_int_sum
thf(fact_2947_abs__sum__abs,axiom,
! [B: $tType,A: $tType] :
( ( ordere166539214618696060dd_abs @ B )
=> ! [F2: A > B,A3: set @ A] :
( ( abs_abs @ B
@ ( groups7311177749621191930dd_sum @ A @ B
@ ^ [A4: A] : ( abs_abs @ B @ ( F2 @ A4 ) )
@ A3 ) )
= ( groups7311177749621191930dd_sum @ A @ B
@ ^ [A4: A] : ( abs_abs @ B @ ( F2 @ A4 ) )
@ A3 ) ) ) ).
% abs_sum_abs
thf(fact_2948_of__real__sum,axiom,
! [A: $tType,B: $tType] :
( ( real_V2191834092415804123ebra_1 @ A )
=> ! [F2: B > real,S3: set @ B] :
( ( real_Vector_of_real @ A @ ( groups7311177749621191930dd_sum @ B @ real @ F2 @ S3 ) )
= ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X: B] : ( real_Vector_of_real @ A @ ( F2 @ X ) )
@ S3 ) ) ) ).
% of_real_sum
thf(fact_2949_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_2950_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 ) )
= ( ! [X: B] :
( ( member @ B @ X @ F3 )
=> ( ( F2 @ X )
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% sum_eq_0_iff
thf(fact_2951_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_2952_pred__numeral__simps_I1_J,axiom,
( ( pred_numeral @ one2 )
= ( zero_zero @ nat ) ) ).
% pred_numeral_simps(1)
thf(fact_2953_eq__numeral__Suc,axiom,
! [K: num,N: nat] :
( ( ( numeral_numeral @ nat @ K )
= ( suc @ N ) )
= ( ( pred_numeral @ K )
= N ) ) ).
% eq_numeral_Suc
thf(fact_2954_Suc__eq__numeral,axiom,
! [N: nat,K: num] :
( ( ( suc @ N )
= ( numeral_numeral @ nat @ K ) )
= ( N
= ( pred_numeral @ K ) ) ) ).
% Suc_eq_numeral
thf(fact_2955_sin__pi,axiom,
( ( sin @ real @ pi )
= ( zero_zero @ real ) ) ).
% sin_pi
thf(fact_2956_sum_Odelta,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add @ A )
=> ! [S: set @ B,A2: B,B2: B > A] :
( ( finite_finite2 @ B @ S )
=> ( ( ( member @ B @ A2 @ S )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [K2: B] : ( if @ A @ ( K2 = A2 ) @ ( B2 @ K2 ) @ ( zero_zero @ A ) )
@ S )
= ( B2 @ A2 ) ) )
& ( ~ ( member @ B @ A2 @ S )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [K2: B] : ( if @ A @ ( K2 = A2 ) @ ( B2 @ K2 ) @ ( zero_zero @ A ) )
@ S )
= ( zero_zero @ A ) ) ) ) ) ) ).
% sum.delta
thf(fact_2957_sum_Odelta_H,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add @ A )
=> ! [S: set @ B,A2: B,B2: B > A] :
( ( finite_finite2 @ B @ S )
=> ( ( ( member @ B @ A2 @ S )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [K2: B] : ( if @ A @ ( A2 = K2 ) @ ( B2 @ K2 ) @ ( zero_zero @ A ) )
@ S )
= ( B2 @ A2 ) ) )
& ( ~ ( member @ B @ A2 @ S )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [K2: B] : ( if @ A @ ( A2 = K2 ) @ ( B2 @ K2 ) @ ( zero_zero @ A ) )
@ S )
= ( zero_zero @ A ) ) ) ) ) ) ).
% sum.delta'
thf(fact_2958_sum__abs,axiom,
! [B: $tType,A: $tType] :
( ( ordere166539214618696060dd_abs @ B )
=> ! [F2: A > B,A3: set @ A] :
( ord_less_eq @ B @ ( abs_abs @ B @ ( groups7311177749621191930dd_sum @ A @ B @ F2 @ A3 ) )
@ ( groups7311177749621191930dd_sum @ A @ B
@ ^ [I3: A] : ( abs_abs @ B @ ( F2 @ I3 ) )
@ A3 ) ) ) ).
% sum_abs
thf(fact_2959_pred__numeral__simps_I3_J,axiom,
! [K: num] :
( ( pred_numeral @ ( bit1 @ K ) )
= ( numeral_numeral @ nat @ ( bit0 @ K ) ) ) ).
% pred_numeral_simps(3)
thf(fact_2960_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_2961_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_2962_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_2963_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_2964_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_2965_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_2966_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_2967_cos__pi,axiom,
( ( cos @ real @ pi )
= ( uminus_uminus @ real @ ( one_one @ real ) ) ) ).
% cos_pi
thf(fact_2968_cos__periodic__pi2,axiom,
! [X2: real] :
( ( cos @ real @ ( plus_plus @ real @ pi @ X2 ) )
= ( uminus_uminus @ real @ ( cos @ real @ X2 ) ) ) ).
% cos_periodic_pi2
thf(fact_2969_cos__periodic__pi,axiom,
! [X2: real] :
( ( cos @ real @ ( plus_plus @ real @ X2 @ pi ) )
= ( uminus_uminus @ real @ ( cos @ real @ X2 ) ) ) ).
% cos_periodic_pi
thf(fact_2970_sin__periodic__pi2,axiom,
! [X2: real] :
( ( sin @ real @ ( plus_plus @ real @ pi @ X2 ) )
= ( uminus_uminus @ real @ ( sin @ real @ X2 ) ) ) ).
% sin_periodic_pi2
thf(fact_2971_sin__periodic__pi,axiom,
! [X2: real] :
( ( sin @ real @ ( plus_plus @ real @ X2 @ pi ) )
= ( uminus_uminus @ real @ ( sin @ real @ X2 ) ) ) ).
% sin_periodic_pi
thf(fact_2972_sum__abs__ge__zero,axiom,
! [B: $tType,A: $tType] :
( ( ordere166539214618696060dd_abs @ B )
=> ! [F2: A > B,A3: set @ A] :
( ord_less_eq @ B @ ( zero_zero @ B )
@ ( groups7311177749621191930dd_sum @ A @ B
@ ^ [I3: A] : ( abs_abs @ B @ ( F2 @ I3 ) )
@ A3 ) ) ) ).
% sum_abs_ge_zero
thf(fact_2973_sin__cos__squared__add3,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X2: A] :
( ( plus_plus @ A @ ( times_times @ A @ ( cos @ A @ X2 ) @ ( cos @ A @ X2 ) ) @ ( times_times @ A @ ( sin @ A @ X2 ) @ ( sin @ A @ X2 ) ) )
= ( one_one @ A ) ) ) ).
% sin_cos_squared_add3
thf(fact_2974_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_2975_sin__npi2,axiom,
! [N: nat] :
( ( sin @ real @ ( times_times @ real @ pi @ ( semiring_1_of_nat @ real @ N ) ) )
= ( zero_zero @ real ) ) ).
% sin_npi2
thf(fact_2976_sin__npi,axiom,
! [N: nat] :
( ( sin @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ pi ) )
= ( zero_zero @ real ) ) ).
% sin_npi
thf(fact_2977_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_2978_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_2979_sum__zero__power,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: set @ nat,C2: nat > A] :
( ( ( ( finite_finite2 @ nat @ A3 )
& ( member @ nat @ ( zero_zero @ nat ) @ A3 ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( C2 @ I3 ) @ ( power_power @ A @ ( zero_zero @ A ) @ I3 ) )
@ A3 )
= ( C2 @ ( zero_zero @ nat ) ) ) )
& ( ~ ( ( finite_finite2 @ nat @ A3 )
& ( member @ nat @ ( zero_zero @ nat ) @ A3 ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( C2 @ I3 ) @ ( power_power @ A @ ( zero_zero @ A ) @ I3 ) )
@ A3 )
= ( zero_zero @ A ) ) ) ) ) ).
% sum_zero_power
thf(fact_2980_cos__pi__half,axiom,
( ( cos @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
= ( zero_zero @ real ) ) ).
% cos_pi_half
thf(fact_2981_sin__two__pi,axiom,
( ( sin @ real @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) )
= ( zero_zero @ real ) ) ).
% sin_two_pi
thf(fact_2982_cos__two__pi,axiom,
( ( cos @ real @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) )
= ( one_one @ real ) ) ).
% cos_two_pi
thf(fact_2983_sin__pi__half,axiom,
( ( sin @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
= ( one_one @ real ) ) ).
% sin_pi_half
thf(fact_2984_signed__take__bit__numeral__bit0,axiom,
! [L2: num,K: num] :
( ( bit_ri4674362597316999326ke_bit @ int @ ( numeral_numeral @ nat @ L2 ) @ ( numeral_numeral @ int @ ( bit0 @ K ) ) )
= ( times_times @ int @ ( bit_ri4674362597316999326ke_bit @ int @ ( pred_numeral @ L2 ) @ ( numeral_numeral @ int @ K ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ).
% signed_take_bit_numeral_bit0
thf(fact_2985_cos__periodic,axiom,
! [X2: real] :
( ( cos @ real @ ( plus_plus @ real @ X2 @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) ) )
= ( cos @ real @ X2 ) ) ).
% cos_periodic
thf(fact_2986_sin__periodic,axiom,
! [X2: real] :
( ( sin @ real @ ( plus_plus @ real @ X2 @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) ) )
= ( sin @ real @ X2 ) ) ).
% sin_periodic
thf(fact_2987_cos__2pi__minus,axiom,
! [X2: real] :
( ( cos @ real @ ( minus_minus @ real @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) @ X2 ) )
= ( cos @ real @ X2 ) ) ).
% cos_2pi_minus
thf(fact_2988_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_2989_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_2990_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
@ ^ [I3: nat] : ( divide_divide @ A @ ( times_times @ A @ ( C2 @ I3 ) @ ( power_power @ A @ ( zero_zero @ A ) @ I3 ) ) @ ( D2 @ I3 ) )
@ 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
@ ^ [I3: nat] : ( divide_divide @ A @ ( times_times @ A @ ( C2 @ I3 ) @ ( power_power @ A @ ( zero_zero @ A ) @ I3 ) ) @ ( D2 @ I3 ) )
@ A3 )
= ( zero_zero @ A ) ) ) ) ) ).
% sum_zero_power'
thf(fact_2991_sin__cos__squared__add2,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X2: A] :
( ( plus_plus @ A @ ( power_power @ A @ ( cos @ A @ X2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ ( sin @ A @ X2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( one_one @ A ) ) ) ).
% sin_cos_squared_add2
thf(fact_2992_sin__cos__squared__add,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X2: A] :
( ( plus_plus @ A @ ( power_power @ A @ ( sin @ A @ X2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ ( cos @ A @ X2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( one_one @ A ) ) ) ).
% sin_cos_squared_add
thf(fact_2993_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_2994_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_2995_signed__take__bit__numeral__minus__bit0,axiom,
! [L2: num,K: num] :
( ( bit_ri4674362597316999326ke_bit @ int @ ( numeral_numeral @ nat @ L2 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ K ) ) ) )
= ( times_times @ int @ ( bit_ri4674362597316999326ke_bit @ int @ ( pred_numeral @ L2 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ).
% signed_take_bit_numeral_minus_bit0
thf(fact_2996_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_2997_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_2998_sin__2pi__minus,axiom,
! [X2: real] :
( ( sin @ real @ ( minus_minus @ real @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) @ X2 ) )
= ( uminus_uminus @ real @ ( sin @ real @ X2 ) ) ) ).
% sin_2pi_minus
thf(fact_2999_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_3000_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_3001_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_3002_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_3003_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_3004_dvd__sum,axiom,
! [A: $tType,B: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A3: set @ B,D2: A,F2: B > A] :
( ! [A5: B] :
( ( member @ B @ A5 @ A3 )
=> ( dvd_dvd @ A @ D2 @ ( F2 @ A5 ) ) )
=> ( dvd_dvd @ A @ D2 @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A3 ) ) ) ) ).
% dvd_sum
thf(fact_3005_sin__add,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X2: A,Y2: A] :
( ( sin @ A @ ( plus_plus @ A @ X2 @ Y2 ) )
= ( plus_plus @ A @ ( times_times @ A @ ( sin @ A @ X2 ) @ ( cos @ A @ Y2 ) ) @ ( times_times @ A @ ( cos @ A @ X2 ) @ ( sin @ A @ Y2 ) ) ) ) ) ).
% sin_add
thf(fact_3006_sin__diff,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X2: A,Y2: A] :
( ( sin @ A @ ( minus_minus @ A @ X2 @ Y2 ) )
= ( minus_minus @ A @ ( times_times @ A @ ( sin @ A @ X2 ) @ ( cos @ A @ Y2 ) ) @ ( times_times @ A @ ( cos @ A @ X2 ) @ ( sin @ A @ Y2 ) ) ) ) ) ).
% sin_diff
thf(fact_3007_sum_Oswap,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: B > C > A,B5: set @ C,A3: set @ B] :
( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [I3: B] : ( groups7311177749621191930dd_sum @ C @ A @ ( G @ I3 ) @ B5 )
@ A3 )
= ( groups7311177749621191930dd_sum @ C @ A
@ ^ [J3: C] :
( groups7311177749621191930dd_sum @ B @ A
@ ^ [I3: B] : ( G @ I3 @ J3 )
@ A3 )
@ B5 ) ) ) ).
% sum.swap
thf(fact_3008_cos__one__sin__zero,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X2: A] :
( ( ( cos @ A @ X2 )
= ( one_one @ A ) )
=> ( ( sin @ A @ X2 )
= ( zero_zero @ A ) ) ) ) ).
% cos_one_sin_zero
thf(fact_3009_sum__norm__le,axiom,
! [A: $tType,B: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [S: set @ B,F2: B > A,G: B > real] :
( ! [X3: B] :
( ( member @ B @ X3 @ S )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( F2 @ X3 ) ) @ ( G @ X3 ) ) )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ S ) ) @ ( groups7311177749621191930dd_sum @ B @ real @ G @ S ) ) ) ) ).
% sum_norm_le
thf(fact_3010_polar__Ex,axiom,
! [X2: real,Y2: real] :
? [R3: real,A5: real] :
( ( X2
= ( times_times @ real @ R3 @ ( cos @ real @ A5 ) ) )
& ( Y2
= ( times_times @ real @ R3 @ ( sin @ real @ A5 ) ) ) ) ).
% polar_Ex
thf(fact_3011_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_3012_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 ) )
=> ~ ! [A5: B] :
( ( member @ B @ A5 @ A3 )
=> ( ( G @ A5 )
= ( zero_zero @ A ) ) ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_3013_norm__sum,axiom,
! [A: $tType,B: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: B > A,A3: set @ B] :
( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A3 ) )
@ ( groups7311177749621191930dd_sum @ B @ real
@ ^ [I3: B] : ( real_V7770717601297561774m_norm @ A @ ( F2 @ I3 ) )
@ A3 ) ) ) ).
% norm_sum
thf(fact_3014_cos__add,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X2: A,Y2: A] :
( ( cos @ A @ ( plus_plus @ A @ X2 @ Y2 ) )
= ( minus_minus @ A @ ( times_times @ A @ ( cos @ A @ X2 ) @ ( cos @ A @ Y2 ) ) @ ( times_times @ A @ ( sin @ A @ X2 ) @ ( sin @ A @ Y2 ) ) ) ) ) ).
% cos_add
thf(fact_3015_cos__diff,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X2: A,Y2: A] :
( ( cos @ A @ ( minus_minus @ A @ X2 @ Y2 ) )
= ( plus_plus @ A @ ( times_times @ A @ ( cos @ A @ X2 ) @ ( cos @ A @ Y2 ) ) @ ( times_times @ A @ ( sin @ A @ X2 ) @ ( sin @ A @ Y2 ) ) ) ) ) ).
% cos_diff
thf(fact_3016_sin__zero__norm__cos__one,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X2: A] :
( ( ( sin @ A @ X2 )
= ( zero_zero @ A ) )
=> ( ( real_V7770717601297561774m_norm @ A @ ( cos @ A @ X2 ) )
= ( one_one @ real ) ) ) ) ).
% sin_zero_norm_cos_one
thf(fact_3017_sum__mono,axiom,
! [A: $tType,B: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [K6: set @ B,F2: B > A,G: B > A] :
( ! [I2: B] :
( ( member @ B @ I2 @ K6 )
=> ( ord_less_eq @ A @ ( F2 @ I2 ) @ ( G @ I2 ) ) )
=> ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ K6 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ K6 ) ) ) ) ).
% sum_mono
thf(fact_3018_sum_Odistrib,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: B > A,H2: B > A,A3: set @ B] :
( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X: B] : ( plus_plus @ A @ ( G @ X ) @ ( H2 @ X ) )
@ A3 )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ G @ A3 ) @ ( groups7311177749621191930dd_sum @ B @ A @ H2 @ A3 ) ) ) ) ).
% sum.distrib
thf(fact_3019_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_3020_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_3021_sum__product,axiom,
! [B: $tType,C: $tType,A: $tType] :
( ( semiring_0 @ B )
=> ! [F2: A > B,A3: set @ A,G: C > B,B5: set @ C] :
( ( times_times @ B @ ( groups7311177749621191930dd_sum @ A @ B @ F2 @ A3 ) @ ( groups7311177749621191930dd_sum @ C @ B @ G @ B5 ) )
= ( groups7311177749621191930dd_sum @ A @ B
@ ^ [I3: A] :
( groups7311177749621191930dd_sum @ C @ B
@ ^ [J3: C] : ( times_times @ B @ ( F2 @ I3 ) @ ( G @ J3 ) )
@ B5 )
@ A3 ) ) ) ).
% sum_product
thf(fact_3022_sum__subtractf,axiom,
! [A: $tType,B: $tType] :
( ( ab_group_add @ A )
=> ! [F2: B > A,G: B > A,A3: set @ B] :
( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X: B] : ( minus_minus @ A @ ( F2 @ X ) @ ( G @ X ) )
@ A3 )
= ( minus_minus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A3 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ A3 ) ) ) ) ).
% sum_subtractf
thf(fact_3023_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_3024_sum__negf,axiom,
! [A: $tType,B: $tType] :
( ( ab_group_add @ A )
=> ! [F2: B > A,A3: set @ B] :
( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X: B] : ( uminus_uminus @ A @ ( F2 @ X ) )
@ A3 )
= ( uminus_uminus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A3 ) ) ) ) ).
% sum_negf
thf(fact_3025_sum_Oswap__restrict,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( comm_monoid_add @ A )
=> ! [A3: set @ B,B5: set @ C,G: B > C > A,R: B > C > $o] :
( ( finite_finite2 @ B @ A3 )
=> ( ( finite_finite2 @ C @ B5 )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X: B] :
( groups7311177749621191930dd_sum @ C @ A @ ( G @ X )
@ ( collect @ C
@ ^ [Y: C] :
( ( member @ C @ Y @ B5 )
& ( R @ X @ Y ) ) ) )
@ A3 )
= ( groups7311177749621191930dd_sum @ C @ A
@ ^ [Y: C] :
( groups7311177749621191930dd_sum @ B @ A
@ ^ [X: B] : ( G @ X @ Y )
@ ( collect @ B
@ ^ [X: B] :
( ( member @ B @ X @ A3 )
& ( R @ X @ Y ) ) ) )
@ B5 ) ) ) ) ) ).
% sum.swap_restrict
thf(fact_3026_mod__sum__eq,axiom,
! [B: $tType,A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [F2: B > A,A2: A,A3: set @ B] :
( ( modulo_modulo @ A
@ ( groups7311177749621191930dd_sum @ B @ A
@ ^ [I3: B] : ( modulo_modulo @ A @ ( F2 @ I3 ) @ A2 )
@ A3 )
@ A2 )
= ( modulo_modulo @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A3 ) @ A2 ) ) ) ).
% mod_sum_eq
thf(fact_3027_sin__zero__abs__cos__one,axiom,
! [X2: real] :
( ( ( sin @ real @ X2 )
= ( zero_zero @ real ) )
=> ( ( abs_abs @ real @ ( cos @ real @ X2 ) )
= ( one_one @ real ) ) ) ).
% sin_zero_abs_cos_one
thf(fact_3028_summable__sum,axiom,
! [I6: $tType,A: $tType] :
( ( ( topolo5987344860129210374id_add @ A )
& ( topological_t2_space @ A ) )
=> ! [I5: set @ I6,F2: I6 > nat > A] :
( ! [I2: I6] :
( ( member @ I6 @ I2 @ I5 )
=> ( summable @ A @ ( F2 @ I2 ) ) )
=> ( summable @ A
@ ^ [N3: nat] :
( groups7311177749621191930dd_sum @ I6 @ A
@ ^ [I3: I6] : ( F2 @ I3 @ N3 )
@ I5 ) ) ) ) ).
% summable_sum
thf(fact_3029_sin__double,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X2: A] :
( ( sin @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X2 ) )
= ( times_times @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( sin @ A @ X2 ) ) @ ( cos @ A @ X2 ) ) ) ) ).
% sin_double
thf(fact_3030_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_3031_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_3032_sincos__principal__value,axiom,
! [X2: real] :
? [Y3: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ pi ) @ Y3 )
& ( ord_less_eq @ real @ Y3 @ pi )
& ( ( sin @ real @ Y3 )
= ( sin @ real @ X2 ) )
& ( ( cos @ real @ Y3 )
= ( cos @ real @ X2 ) ) ) ).
% sincos_principal_value
thf(fact_3033_sum__mono__inv,axiom,
! [A: $tType,I6: $tType] :
( ( ordere8940638589300402666id_add @ A )
=> ! [F2: I6 > A,I5: set @ I6,G: I6 > A,I: I6] :
( ( ( groups7311177749621191930dd_sum @ I6 @ A @ F2 @ I5 )
= ( groups7311177749621191930dd_sum @ I6 @ A @ G @ I5 ) )
=> ( ! [I2: I6] :
( ( member @ I6 @ I2 @ I5 )
=> ( ord_less_eq @ A @ ( F2 @ I2 ) @ ( G @ I2 ) ) )
=> ( ( member @ I6 @ I @ I5 )
=> ( ( finite_finite2 @ I6 @ I5 )
=> ( ( F2 @ I )
= ( G @ I ) ) ) ) ) ) ) ).
% sum_mono_inv
thf(fact_3034_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_3035_sin__x__le__x,axiom,
! [X2: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
=> ( ord_less_eq @ real @ ( sin @ real @ X2 ) @ X2 ) ) ).
% sin_x_le_x
thf(fact_3036_sin__le__one,axiom,
! [X2: real] : ( ord_less_eq @ real @ ( sin @ real @ X2 ) @ ( one_one @ real ) ) ).
% sin_le_one
thf(fact_3037_cos__le__one,axiom,
! [X2: real] : ( ord_less_eq @ real @ ( cos @ real @ X2 ) @ ( one_one @ real ) ) ).
% cos_le_one
thf(fact_3038_abs__sin__x__le__abs__x,axiom,
! [X2: real] : ( ord_less_eq @ real @ ( abs_abs @ real @ ( sin @ real @ X2 ) ) @ ( abs_abs @ real @ X2 ) ) ).
% abs_sin_x_le_abs_x
thf(fact_3039_cos__arctan__not__zero,axiom,
! [X2: real] :
( ( cos @ real @ ( arctan @ X2 ) )
!= ( zero_zero @ real ) ) ).
% cos_arctan_not_zero
thf(fact_3040_numeral__eq__Suc,axiom,
( ( numeral_numeral @ nat )
= ( ^ [K2: num] : ( suc @ ( pred_numeral @ K2 ) ) ) ) ).
% numeral_eq_Suc
thf(fact_3041_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
@ ^ [X: B] :
( ( member @ B @ X @ A3 )
& ( P @ X ) ) ) )
= ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X: B] : ( if @ A @ ( P @ X ) @ ( G @ X ) @ ( zero_zero @ A ) )
@ A3 ) ) ) ) ).
% sum.inter_filter
thf(fact_3042_cos__int__times__real,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [M: int,X2: real] :
( ( cos @ A @ ( times_times @ A @ ( ring_1_of_int @ A @ M ) @ ( real_Vector_of_real @ A @ X2 ) ) )
= ( real_Vector_of_real @ A @ ( cos @ real @ ( times_times @ real @ ( ring_1_of_int @ real @ M ) @ X2 ) ) ) ) ) ).
% cos_int_times_real
thf(fact_3043_sin__cos__le1,axiom,
! [X2: real,Y2: real] : ( ord_less_eq @ real @ ( abs_abs @ real @ ( plus_plus @ real @ ( times_times @ real @ ( sin @ real @ X2 ) @ ( sin @ real @ Y2 ) ) @ ( times_times @ real @ ( cos @ real @ X2 ) @ ( cos @ real @ Y2 ) ) ) ) @ ( one_one @ real ) ) ).
% sin_cos_le1
thf(fact_3044_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
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ).
% sum.shift_bounds_cl_Suc_ivl
thf(fact_3045_sin__int__times__real,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [M: int,X2: real] :
( ( sin @ A @ ( times_times @ A @ ( ring_1_of_int @ A @ M ) @ ( real_Vector_of_real @ A @ X2 ) ) )
= ( real_Vector_of_real @ A @ ( sin @ real @ ( times_times @ real @ ( ring_1_of_int @ real @ M ) @ X2 ) ) ) ) ) ).
% sin_int_times_real
thf(fact_3046_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
@ ^ [I3: nat] : ( G @ ( plus_plus @ nat @ I3 @ K ) )
@ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ).
% sum.shift_bounds_cl_nat_ivl
thf(fact_3047_suminf__sum,axiom,
! [A: $tType,I6: $tType] :
( ( ( topolo5987344860129210374id_add @ A )
& ( topological_t2_space @ A ) )
=> ! [I5: set @ I6,F2: I6 > nat > A] :
( ! [I2: I6] :
( ( member @ I6 @ I2 @ I5 )
=> ( summable @ A @ ( F2 @ I2 ) ) )
=> ( ( suminf @ A
@ ^ [N3: nat] :
( groups7311177749621191930dd_sum @ I6 @ A
@ ^ [I3: I6] : ( F2 @ I3 @ N3 )
@ I5 ) )
= ( groups7311177749621191930dd_sum @ I6 @ A
@ ^ [I3: I6] : ( suminf @ A @ ( F2 @ I3 ) )
@ I5 ) ) ) ) ).
% suminf_sum
thf(fact_3048_sin__squared__eq,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X2: A] :
( ( power_power @ A @ ( sin @ A @ X2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ ( cos @ A @ X2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% sin_squared_eq
thf(fact_3049_cos__squared__eq,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X2: A] :
( ( power_power @ A @ ( cos @ A @ X2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ ( sin @ A @ X2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% cos_squared_eq
thf(fact_3050_sum__le__included,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [S3: set @ B,T2: set @ C,G: C > A,I: C > B,F2: B > A] :
( ( finite_finite2 @ B @ S3 )
=> ( ( finite_finite2 @ C @ T2 )
=> ( ! [X3: C] :
( ( member @ C @ X3 @ T2 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( G @ X3 ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ S3 )
=> ? [Xa2: C] :
( ( member @ C @ Xa2 @ T2 )
& ( ( I @ Xa2 )
= X3 )
& ( ord_less_eq @ A @ ( F2 @ X3 ) @ ( G @ Xa2 ) ) ) )
=> ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ S3 ) @ ( groups7311177749621191930dd_sum @ C @ A @ G @ T2 ) ) ) ) ) ) ) ).
% sum_le_included
thf(fact_3051_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 ) )
= ( ! [X: B] :
( ( member @ B @ X @ A3 )
=> ( ( F2 @ X )
= ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
thf(fact_3052_sum__strict__mono__ex1,axiom,
! [A: $tType,I6: $tType] :
( ( ordere8940638589300402666id_add @ A )
=> ! [A3: set @ I6,F2: I6 > A,G: I6 > A] :
( ( finite_finite2 @ I6 @ A3 )
=> ( ! [X3: I6] :
( ( member @ I6 @ X3 @ A3 )
=> ( ord_less_eq @ A @ ( F2 @ X3 ) @ ( G @ X3 ) ) )
=> ( ? [X4: I6] :
( ( member @ I6 @ X4 @ A3 )
& ( ord_less @ A @ ( F2 @ X4 ) @ ( G @ X4 ) ) )
=> ( ord_less @ A @ ( groups7311177749621191930dd_sum @ I6 @ A @ F2 @ A3 ) @ ( groups7311177749621191930dd_sum @ I6 @ A @ G @ A3 ) ) ) ) ) ) ).
% sum_strict_mono_ex1
thf(fact_3053_sum_Orelated,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [R: A > A > $o,S: set @ B,H2: B > A,G: B > A] :
( ( R @ ( zero_zero @ A ) @ ( zero_zero @ A ) )
=> ( ! [X1: A,Y1: A,X22: A,Y23: A] :
( ( ( R @ X1 @ X22 )
& ( R @ Y1 @ Y23 ) )
=> ( R @ ( plus_plus @ A @ X1 @ Y1 ) @ ( plus_plus @ A @ X22 @ Y23 ) ) )
=> ( ( finite_finite2 @ B @ S )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ S )
=> ( R @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
=> ( R @ ( groups7311177749621191930dd_sum @ B @ A @ H2 @ S ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ S ) ) ) ) ) ) ) ).
% sum.related
thf(fact_3054_sum_Oreindex__bij__witness__not__neutral,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comm_monoid_add @ A )
=> ! [S6: set @ B,T4: set @ C,S: set @ B,I: C > B,J: B > C,T3: set @ C,G: B > A,H2: C > A] :
( ( finite_finite2 @ B @ S6 )
=> ( ( finite_finite2 @ C @ T4 )
=> ( ! [A5: B] :
( ( member @ B @ A5 @ ( minus_minus @ ( set @ B ) @ S @ S6 ) )
=> ( ( I @ ( J @ A5 ) )
= A5 ) )
=> ( ! [A5: B] :
( ( member @ B @ A5 @ ( minus_minus @ ( set @ B ) @ S @ S6 ) )
=> ( member @ C @ ( J @ A5 ) @ ( minus_minus @ ( set @ C ) @ T3 @ T4 ) ) )
=> ( ! [B4: C] :
( ( member @ C @ B4 @ ( minus_minus @ ( set @ C ) @ T3 @ T4 ) )
=> ( ( J @ ( I @ B4 ) )
= B4 ) )
=> ( ! [B4: C] :
( ( member @ C @ B4 @ ( minus_minus @ ( set @ C ) @ T3 @ T4 ) )
=> ( member @ B @ ( I @ B4 ) @ ( minus_minus @ ( set @ B ) @ S @ S6 ) ) )
=> ( ! [A5: B] :
( ( member @ B @ A5 @ S6 )
=> ( ( G @ A5 )
= ( zero_zero @ A ) ) )
=> ( ! [B4: C] :
( ( member @ C @ B4 @ T4 )
=> ( ( H2 @ B4 )
= ( zero_zero @ A ) ) )
=> ( ! [A5: B] :
( ( member @ B @ A5 @ S )
=> ( ( H2 @ ( J @ A5 ) )
= ( G @ A5 ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ S )
= ( groups7311177749621191930dd_sum @ C @ A @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness_not_neutral
thf(fact_3055_sin__gt__zero,axiom,
! [X2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ord_less @ real @ X2 @ pi )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( sin @ real @ X2 ) ) ) ) ).
% sin_gt_zero
thf(fact_3056_sin__x__ge__neg__x,axiom,
! [X2: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
=> ( ord_less_eq @ real @ ( uminus_uminus @ real @ X2 ) @ ( sin @ real @ X2 ) ) ) ).
% sin_x_ge_neg_x
thf(fact_3057_sin__ge__zero,axiom,
! [X2: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ord_less_eq @ real @ X2 @ pi )
=> ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( sin @ real @ X2 ) ) ) ) ).
% sin_ge_zero
thf(fact_3058_sin__ge__minus__one,axiom,
! [X2: real] : ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ ( sin @ real @ X2 ) ) ).
% sin_ge_minus_one
thf(fact_3059_cos__monotone__0__pi__le,axiom,
! [Y2: real,X2: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y2 )
=> ( ( ord_less_eq @ real @ Y2 @ X2 )
=> ( ( ord_less_eq @ real @ X2 @ pi )
=> ( ord_less_eq @ real @ ( cos @ real @ X2 ) @ ( cos @ real @ Y2 ) ) ) ) ) ).
% cos_monotone_0_pi_le
thf(fact_3060_cos__mono__le__eq,axiom,
! [X2: real,Y2: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ord_less_eq @ real @ X2 @ pi )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y2 )
=> ( ( ord_less_eq @ real @ Y2 @ pi )
=> ( ( ord_less_eq @ real @ ( cos @ real @ X2 ) @ ( cos @ real @ Y2 ) )
= ( ord_less_eq @ real @ Y2 @ X2 ) ) ) ) ) ) ).
% cos_mono_le_eq
thf(fact_3061_cos__inj__pi,axiom,
! [X2: real,Y2: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ord_less_eq @ real @ X2 @ pi )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y2 )
=> ( ( ord_less_eq @ real @ Y2 @ pi )
=> ( ( ( cos @ real @ X2 )
= ( cos @ real @ Y2 ) )
=> ( X2 = Y2 ) ) ) ) ) ) ).
% cos_inj_pi
thf(fact_3062_cos__ge__minus__one,axiom,
! [X2: real] : ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ ( cos @ real @ X2 ) ) ).
% cos_ge_minus_one
thf(fact_3063_abs__sin__le__one,axiom,
! [X2: real] : ( ord_less_eq @ real @ ( abs_abs @ real @ ( sin @ real @ X2 ) ) @ ( one_one @ real ) ) ).
% abs_sin_le_one
thf(fact_3064_abs__cos__le__one,axiom,
! [X2: real] : ( ord_less_eq @ real @ ( abs_abs @ real @ ( cos @ real @ X2 ) ) @ ( one_one @ real ) ) ).
% abs_cos_le_one
thf(fact_3065_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_3066_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_3067_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_3068_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_3069_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_3070_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_3071_cos__double,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X2: A] :
( ( cos @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X2 ) )
= ( minus_minus @ A @ ( power_power @ A @ ( cos @ A @ X2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ ( sin @ A @ X2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% cos_double
thf(fact_3072_sum__nonneg__0,axiom,
! [B: $tType,A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [S3: set @ B,F2: B > A,I: B] :
( ( finite_finite2 @ B @ S3 )
=> ( ! [I2: B] :
( ( member @ B @ I2 @ S3 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ I2 ) ) )
=> ( ( ( groups7311177749621191930dd_sum @ B @ A @ F2 @ S3 )
= ( zero_zero @ A ) )
=> ( ( member @ B @ I @ S3 )
=> ( ( F2 @ I )
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% sum_nonneg_0
thf(fact_3073_sum__nonneg__leq__bound,axiom,
! [B: $tType,A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [S3: set @ B,F2: B > A,B5: A,I: B] :
( ( finite_finite2 @ B @ S3 )
=> ( ! [I2: B] :
( ( member @ B @ I2 @ S3 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ I2 ) ) )
=> ( ( ( groups7311177749621191930dd_sum @ B @ A @ F2 @ S3 )
= B5 )
=> ( ( member @ B @ I @ S3 )
=> ( ord_less_eq @ A @ ( F2 @ I ) @ B5 ) ) ) ) ) ) ).
% sum_nonneg_leq_bound
thf(fact_3074_sin__cos__eq,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ( ( sin @ A )
= ( ^ [X: A] : ( cos @ A @ ( minus_minus @ A @ ( divide_divide @ A @ ( real_Vector_of_real @ A @ pi ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ X ) ) ) ) ) ).
% sin_cos_eq
thf(fact_3075_cos__sin__eq,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ( ( cos @ A )
= ( ^ [X: A] : ( sin @ A @ ( minus_minus @ A @ ( divide_divide @ A @ ( real_Vector_of_real @ A @ pi ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ X ) ) ) ) ) ).
% cos_sin_eq
thf(fact_3076_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
@ ^ [X: B] :
( ( G @ X )
= ( zero_zero @ A ) ) ) ) )
= ( groups7311177749621191930dd_sum @ B @ A @ G @ A3 ) ) ) ) ).
% sum.setdiff_irrelevant
thf(fact_3077_pred__numeral__def,axiom,
( pred_numeral
= ( ^ [K2: num] : ( minus_minus @ nat @ ( numeral_numeral @ nat @ K2 ) @ ( one_one @ nat ) ) ) ) ).
% pred_numeral_def
thf(fact_3078_sum__power__add,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [X2: A,M: nat,I5: set @ nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( power_power @ A @ X2 @ ( plus_plus @ nat @ M @ I3 ) )
@ I5 )
= ( times_times @ A @ ( power_power @ A @ X2 @ M ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X2 ) @ I5 ) ) ) ) ).
% sum_power_add
thf(fact_3079_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
@ ^ [I3: nat] : ( G @ ( minus_minus @ nat @ ( plus_plus @ nat @ M @ N ) @ I3 ) )
@ ( set_or1337092689740270186AtMost @ nat @ N @ M ) ) ) ) ).
% sum.atLeastAtMost_rev
thf(fact_3080_suminf__finite,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topological_t2_space @ A ) )
=> ! [N5: set @ nat,F2: nat > A] :
( ( finite_finite2 @ nat @ N5 )
=> ( ! [N2: nat] :
( ~ ( member @ nat @ N2 @ N5 )
=> ( ( F2 @ N2 )
= ( zero_zero @ A ) ) )
=> ( ( suminf @ A @ F2 )
= ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ N5 ) ) ) ) ) ).
% suminf_finite
thf(fact_3081_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_3082_minus__sin__cos__eq,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X2: A] :
( ( uminus_uminus @ A @ ( sin @ A @ X2 ) )
= ( cos @ A @ ( plus_plus @ A @ X2 @ ( divide_divide @ A @ ( real_Vector_of_real @ A @ pi ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% minus_sin_cos_eq
thf(fact_3083_sum__pos2,axiom,
! [A: $tType,B: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [I5: set @ B,I: B,F2: B > A] :
( ( finite_finite2 @ B @ I5 )
=> ( ( member @ B @ I @ I5 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ ( F2 @ I ) )
=> ( ! [I2: B] :
( ( member @ B @ I2 @ I5 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ I2 ) ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ I5 ) ) ) ) ) ) ) ).
% sum_pos2
thf(fact_3084_sum_Omono__neutral__cong__right,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [T3: set @ B,S: set @ B,G: B > A,H2: B > A] :
( ( finite_finite2 @ B @ T3 )
=> ( ( ord_less_eq @ ( set @ B ) @ S @ T3 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T3 @ S ) )
=> ( ( G @ X3 )
= ( zero_zero @ A ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ S )
=> ( ( G @ X3 )
= ( H2 @ X3 ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ T3 )
= ( groups7311177749621191930dd_sum @ B @ A @ H2 @ S ) ) ) ) ) ) ) ).
% sum.mono_neutral_cong_right
thf(fact_3085_sum_Omono__neutral__cong__left,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [T3: set @ B,S: set @ B,H2: B > A,G: B > A] :
( ( finite_finite2 @ B @ T3 )
=> ( ( ord_less_eq @ ( set @ B ) @ S @ T3 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T3 @ S ) )
=> ( ( H2 @ X3 )
= ( zero_zero @ A ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ S )
=> ( ( G @ X3 )
= ( H2 @ X3 ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ S )
= ( groups7311177749621191930dd_sum @ B @ A @ H2 @ T3 ) ) ) ) ) ) ) ).
% sum.mono_neutral_cong_left
thf(fact_3086_sum_Omono__neutral__right,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [T3: set @ B,S: set @ B,G: B > A] :
( ( finite_finite2 @ B @ T3 )
=> ( ( ord_less_eq @ ( set @ B ) @ S @ T3 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T3 @ S ) )
=> ( ( G @ X3 )
= ( zero_zero @ A ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ T3 )
= ( groups7311177749621191930dd_sum @ B @ A @ G @ S ) ) ) ) ) ) ).
% sum.mono_neutral_right
thf(fact_3087_sum_Omono__neutral__left,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [T3: set @ B,S: set @ B,G: B > A] :
( ( finite_finite2 @ B @ T3 )
=> ( ( ord_less_eq @ ( set @ B ) @ S @ T3 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T3 @ S ) )
=> ( ( G @ X3 )
= ( zero_zero @ A ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ S )
= ( groups7311177749621191930dd_sum @ B @ A @ G @ T3 ) ) ) ) ) ) ).
% sum.mono_neutral_left
thf(fact_3088_sum_Osame__carrierI,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [C6: set @ B,A3: set @ B,B5: set @ B,G: B > A,H2: B > A] :
( ( finite_finite2 @ B @ C6 )
=> ( ( ord_less_eq @ ( set @ B ) @ A3 @ C6 )
=> ( ( ord_less_eq @ ( set @ B ) @ B5 @ C6 )
=> ( ! [A5: B] :
( ( member @ B @ A5 @ ( minus_minus @ ( set @ B ) @ C6 @ A3 ) )
=> ( ( G @ A5 )
= ( zero_zero @ A ) ) )
=> ( ! [B4: B] :
( ( member @ B @ B4 @ ( minus_minus @ ( set @ B ) @ C6 @ B5 ) )
=> ( ( H2 @ B4 )
= ( 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 @ B5 ) ) ) ) ) ) ) ) ) ).
% sum.same_carrierI
thf(fact_3089_sum_Osame__carrier,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [C6: set @ B,A3: set @ B,B5: set @ B,G: B > A,H2: B > A] :
( ( finite_finite2 @ B @ C6 )
=> ( ( ord_less_eq @ ( set @ B ) @ A3 @ C6 )
=> ( ( ord_less_eq @ ( set @ B ) @ B5 @ C6 )
=> ( ! [A5: B] :
( ( member @ B @ A5 @ ( minus_minus @ ( set @ B ) @ C6 @ A3 ) )
=> ( ( G @ A5 )
= ( zero_zero @ A ) ) )
=> ( ! [B4: B] :
( ( member @ B @ B4 @ ( minus_minus @ ( set @ B ) @ C6 @ B5 ) )
=> ( ( H2 @ B4 )
= ( zero_zero @ A ) ) )
=> ( ( ( groups7311177749621191930dd_sum @ B @ A @ G @ A3 )
= ( groups7311177749621191930dd_sum @ B @ A @ H2 @ B5 ) )
= ( ( groups7311177749621191930dd_sum @ B @ A @ G @ C6 )
= ( groups7311177749621191930dd_sum @ B @ A @ H2 @ C6 ) ) ) ) ) ) ) ) ) ).
% sum.same_carrier
thf(fact_3090_sum_Osubset__diff,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [B5: set @ B,A3: set @ B,G: B > A] :
( ( ord_less_eq @ ( set @ B ) @ B5 @ A3 )
=> ( ( finite_finite2 @ B @ A3 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ A3 )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ A3 @ B5 ) ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ B5 ) ) ) ) ) ) ).
% sum.subset_diff
thf(fact_3091_sum__diff,axiom,
! [A: $tType,B: $tType] :
( ( ab_group_add @ A )
=> ! [A3: set @ B,B5: set @ B,F2: B > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ( ord_less_eq @ ( set @ B ) @ B5 @ A3 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ F2 @ ( minus_minus @ ( set @ B ) @ A3 @ B5 ) )
= ( minus_minus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A3 ) @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ B5 ) ) ) ) ) ) ).
% sum_diff
thf(fact_3092_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_3093_cos__two__neq__zero,axiom,
( ( cos @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) )
!= ( zero_zero @ real ) ) ).
% cos_two_neq_zero
thf(fact_3094_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_3095_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_3096_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_3097_cos__monotone__0__pi,axiom,
! [Y2: real,X2: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y2 )
=> ( ( ord_less @ real @ Y2 @ X2 )
=> ( ( ord_less_eq @ real @ X2 @ pi )
=> ( ord_less @ real @ ( cos @ real @ X2 ) @ ( cos @ real @ Y2 ) ) ) ) ) ).
% cos_monotone_0_pi
thf(fact_3098_cos__mono__less__eq,axiom,
! [X2: real,Y2: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ord_less_eq @ real @ X2 @ pi )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y2 )
=> ( ( ord_less_eq @ real @ Y2 @ pi )
=> ( ( ord_less @ real @ ( cos @ real @ X2 ) @ ( cos @ real @ Y2 ) )
= ( ord_less @ real @ Y2 @ X2 ) ) ) ) ) ) ).
% cos_mono_less_eq
thf(fact_3099_sin__eq__0__pi,axiom,
! [X2: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ pi ) @ X2 )
=> ( ( ord_less @ real @ X2 @ pi )
=> ( ( ( sin @ real @ X2 )
= ( zero_zero @ real ) )
=> ( X2
= ( zero_zero @ real ) ) ) ) ) ).
% sin_eq_0_pi
thf(fact_3100_sin__zero__pi__iff,axiom,
! [X2: real] :
( ( ord_less @ real @ ( abs_abs @ real @ X2 ) @ pi )
=> ( ( ( sin @ real @ X2 )
= ( zero_zero @ real ) )
= ( X2
= ( zero_zero @ real ) ) ) ) ).
% sin_zero_pi_iff
thf(fact_3101_cos__monotone__minus__pi__0_H,axiom,
! [Y2: real,X2: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ pi ) @ Y2 )
=> ( ( ord_less_eq @ real @ Y2 @ X2 )
=> ( ( ord_less_eq @ real @ X2 @ ( zero_zero @ real ) )
=> ( ord_less_eq @ real @ ( cos @ real @ Y2 ) @ ( cos @ real @ X2 ) ) ) ) ) ).
% cos_monotone_minus_pi_0'
thf(fact_3102_sin__zero__iff__int2,axiom,
! [X2: real] :
( ( ( sin @ real @ X2 )
= ( zero_zero @ real ) )
= ( ? [I3: int] :
( X2
= ( times_times @ real @ ( ring_1_of_int @ real @ I3 ) @ pi ) ) ) ) ).
% sin_zero_iff_int2
thf(fact_3103_sincos__total__pi,axiom,
! [Y2: real,X2: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y2 )
=> ( ( ( plus_plus @ real @ ( power_power @ real @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( one_one @ real ) )
=> ? [T5: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ T5 )
& ( ord_less_eq @ real @ T5 @ pi )
& ( X2
= ( cos @ real @ T5 ) )
& ( Y2
= ( sin @ real @ T5 ) ) ) ) ) ).
% sincos_total_pi
thf(fact_3104_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
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ) ) ).
% sum.Suc_reindex_ivl
thf(fact_3105_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
@ ^ [I3: nat] : ( minus_minus @ A @ ( F2 @ ( suc @ I3 ) ) @ ( F2 @ I3 ) )
@ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( minus_minus @ A @ ( F2 @ ( suc @ N ) ) @ ( F2 @ M ) ) ) ) ) ).
% sum_Suc_diff
thf(fact_3106_sin__expansion__lemma,axiom,
! [X2: real,M: nat] :
( ( sin @ real @ ( plus_plus @ real @ X2 @ ( divide_divide @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ ( suc @ M ) ) @ pi ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) )
= ( cos @ real @ ( plus_plus @ real @ X2 @ ( divide_divide @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ M ) @ pi ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ).
% sin_expansion_lemma
thf(fact_3107_sum__mono2,axiom,
! [A: $tType,B: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [B5: set @ B,A3: set @ B,F2: B > A] :
( ( finite_finite2 @ B @ B5 )
=> ( ( ord_less_eq @ ( set @ B ) @ A3 @ B5 )
=> ( ! [B4: B] :
( ( member @ B @ B4 @ ( minus_minus @ ( set @ B ) @ B5 @ A3 ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ B4 ) ) )
=> ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A3 ) @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ B5 ) ) ) ) ) ) ).
% sum_mono2
thf(fact_3108_cos__expansion__lemma,axiom,
! [X2: real,M: nat] :
( ( cos @ real @ ( plus_plus @ real @ X2 @ ( divide_divide @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ ( suc @ M ) ) @ pi ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) )
= ( uminus_uminus @ real @ ( sin @ real @ ( plus_plus @ real @ X2 @ ( divide_divide @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ M ) @ pi ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% cos_expansion_lemma
thf(fact_3109_sin__gt__zero__02,axiom,
! [X2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ord_less @ real @ X2 @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( sin @ real @ X2 ) ) ) ) ).
% sin_gt_zero_02
thf(fact_3110_cos__two__less__zero,axiom,
ord_less @ real @ ( cos @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( zero_zero @ real ) ).
% cos_two_less_zero
thf(fact_3111_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_3112_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_3113_sum_Oub__add__nat,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [M: nat,N: nat,G: nat > A,P2: 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 @ P2 ) ) )
= ( 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 @ P2 ) ) ) ) ) ) ) ).
% sum.ub_add_nat
thf(fact_3114_cos__monotone__minus__pi__0,axiom,
! [Y2: real,X2: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ pi ) @ Y2 )
=> ( ( ord_less @ real @ Y2 @ X2 )
=> ( ( ord_less_eq @ real @ X2 @ ( zero_zero @ real ) )
=> ( ord_less @ real @ ( cos @ real @ Y2 ) @ ( cos @ real @ X2 ) ) ) ) ) ).
% cos_monotone_minus_pi_0
thf(fact_3115_cos__total,axiom,
! [Y2: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ Y2 )
=> ( ( ord_less_eq @ real @ Y2 @ ( one_one @ real ) )
=> ? [X3: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 )
& ( ord_less_eq @ real @ X3 @ pi )
& ( ( cos @ real @ X3 )
= Y2 )
& ! [Y4: real] :
( ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y4 )
& ( ord_less_eq @ real @ Y4 @ pi )
& ( ( cos @ real @ Y4 )
= Y2 ) )
=> ( Y4 = X3 ) ) ) ) ) ).
% cos_total
thf(fact_3116_sum__le__suminf,axiom,
! [A: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F2: nat > A,I5: set @ nat] :
( ( summable @ A @ F2 )
=> ( ( finite_finite2 @ nat @ I5 )
=> ( ! [N2: nat] :
( ( member @ nat @ N2 @ ( uminus_uminus @ ( set @ nat ) @ I5 ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ N2 ) ) )
=> ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ I5 ) @ ( suminf @ A @ F2 ) ) ) ) ) ) ).
% sum_le_suminf
thf(fact_3117_sincos__total__pi__half,axiom,
! [X2: real,Y2: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y2 )
=> ( ( ( plus_plus @ real @ ( power_power @ real @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( one_one @ real ) )
=> ? [T5: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ T5 )
& ( ord_less_eq @ real @ T5 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
& ( X2
= ( cos @ real @ T5 ) )
& ( Y2
= ( sin @ real @ T5 ) ) ) ) ) ) ).
% sincos_total_pi_half
thf(fact_3118_sincos__total__2pi__le,axiom,
! [X2: real,Y2: real] :
( ( ( plus_plus @ real @ ( power_power @ real @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( one_one @ real ) )
=> ? [T5: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ T5 )
& ( ord_less_eq @ real @ T5 @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) )
& ( X2
= ( cos @ real @ T5 ) )
& ( Y2
= ( sin @ real @ T5 ) ) ) ) ).
% sincos_total_2pi_le
thf(fact_3119_sum__strict__mono2,axiom,
! [B: $tType,A: $tType] :
( ( ordere8940638589300402666id_add @ B )
=> ! [B5: set @ A,A3: set @ A,B2: A,F2: A > B] :
( ( finite_finite2 @ A @ B5 )
=> ( ( ord_less_eq @ ( set @ A ) @ A3 @ B5 )
=> ( ( member @ A @ B2 @ ( minus_minus @ ( set @ A ) @ B5 @ A3 ) )
=> ( ( ord_less @ B @ ( zero_zero @ B ) @ ( F2 @ B2 ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ B5 )
=> ( ord_less_eq @ B @ ( zero_zero @ B ) @ ( F2 @ X3 ) ) )
=> ( ord_less @ B @ ( groups7311177749621191930dd_sum @ A @ B @ F2 @ A3 ) @ ( groups7311177749621191930dd_sum @ A @ B @ F2 @ B5 ) ) ) ) ) ) ) ) ).
% sum_strict_mono2
thf(fact_3120_sincos__total__2pi,axiom,
! [X2: real,Y2: real] :
( ( ( plus_plus @ real @ ( power_power @ real @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( one_one @ real ) )
=> ~ ! [T5: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ T5 )
=> ( ( ord_less @ real @ T5 @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) )
=> ( ( X2
= ( cos @ real @ T5 ) )
=> ( Y2
!= ( sin @ real @ T5 ) ) ) ) ) ) ).
% sincos_total_2pi
thf(fact_3121_convex__sum__bound__le,axiom,
! [A: $tType,B: $tType] :
( ( linordered_idom @ B )
=> ! [I5: set @ A,X2: A > B,A2: A > B,B2: B,Delta: B] :
( ! [I2: A] :
( ( member @ A @ I2 @ I5 )
=> ( ord_less_eq @ B @ ( zero_zero @ B ) @ ( X2 @ I2 ) ) )
=> ( ( ( groups7311177749621191930dd_sum @ A @ B @ X2 @ I5 )
= ( one_one @ B ) )
=> ( ! [I2: A] :
( ( member @ A @ I2 @ I5 )
=> ( ord_less_eq @ B @ ( abs_abs @ B @ ( minus_minus @ B @ ( A2 @ I2 ) @ B2 ) ) @ Delta ) )
=> ( ord_less_eq @ B
@ ( abs_abs @ B
@ ( minus_minus @ B
@ ( groups7311177749621191930dd_sum @ A @ B
@ ^ [I3: A] : ( times_times @ B @ ( A2 @ I3 ) @ ( X2 @ I3 ) )
@ I5 )
@ B2 ) )
@ Delta ) ) ) ) ) ).
% convex_sum_bound_le
thf(fact_3122_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_3123_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
@ ^ [K2: nat] : ( minus_minus @ A @ ( F2 @ K2 ) @ ( F2 @ ( plus_plus @ nat @ K2 @ ( 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
@ ^ [K2: nat] : ( minus_minus @ A @ ( F2 @ K2 ) @ ( F2 @ ( plus_plus @ nat @ K2 @ ( one_one @ nat ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( zero_zero @ A ) ) ) ) ) ).
% sum_natinterval_diff
thf(fact_3124_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
@ ^ [K2: nat] : ( minus_minus @ A @ ( F2 @ K2 ) @ ( F2 @ ( minus_minus @ nat @ K2 @ ( one_one @ nat ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ N ) )
= ( minus_minus @ A @ ( F2 @ N ) @ ( F2 @ M ) ) ) ) ) ).
% sum_telescope''
thf(fact_3125_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_3126_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_3127_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 )
=> ~ ! [N10: nat] :
~ ! [M4: nat] :
( ( ord_less_eq @ nat @ N10 @ M4 )
=> ! [N11: nat] : ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_or1337092689740270186AtMost @ nat @ M4 @ N11 ) ) ) @ E ) ) ) ) ) ).
% summable_partial_sum_bound
thf(fact_3128_sin__gt__zero2,axiom,
! [X2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ord_less @ real @ X2 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( sin @ real @ X2 ) ) ) ) ).
% sin_gt_zero2
thf(fact_3129_sin__lt__zero,axiom,
! [X2: real] :
( ( ord_less @ real @ pi @ X2 )
=> ( ( ord_less @ real @ X2 @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) )
=> ( ord_less @ real @ ( sin @ real @ X2 ) @ ( zero_zero @ real ) ) ) ) ).
% sin_lt_zero
thf(fact_3130_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_3131_cos__double__less__one,axiom,
! [X2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ord_less @ real @ X2 @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) )
=> ( ord_less @ real @ ( cos @ real @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ X2 ) ) @ ( one_one @ real ) ) ) ) ).
% cos_double_less_one
thf(fact_3132_cos__gt__zero,axiom,
! [X2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ord_less @ real @ X2 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( cos @ real @ X2 ) ) ) ) ).
% cos_gt_zero
thf(fact_3133_sin__monotone__2pi__le,axiom,
! [Y2: real,X2: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ Y2 )
=> ( ( ord_less_eq @ real @ Y2 @ X2 )
=> ( ( ord_less_eq @ real @ X2 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less_eq @ real @ ( sin @ real @ Y2 ) @ ( sin @ real @ X2 ) ) ) ) ) ).
% sin_monotone_2pi_le
thf(fact_3134_sin__mono__le__eq,axiom,
! [X2: real,Y2: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X2 )
=> ( ( ord_less_eq @ real @ X2 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ Y2 )
=> ( ( ord_less_eq @ real @ Y2 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less_eq @ real @ ( sin @ real @ X2 ) @ ( sin @ real @ Y2 ) )
= ( ord_less_eq @ real @ X2 @ Y2 ) ) ) ) ) ) ).
% sin_mono_le_eq
thf(fact_3135_sin__inj__pi,axiom,
! [X2: real,Y2: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X2 )
=> ( ( ord_less_eq @ real @ X2 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ Y2 )
=> ( ( ord_less_eq @ real @ Y2 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ( sin @ real @ X2 )
= ( sin @ real @ Y2 ) )
=> ( X2 = Y2 ) ) ) ) ) ) ).
% sin_inj_pi
thf(fact_3136_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_3137_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
@ ^ [Q4: nat] : ( ord_less @ nat @ Q4 @ N ) ) ) ) ) ).
% mask_eq_sum_exp
thf(fact_3138_cos__one__2pi__int,axiom,
! [X2: real] :
( ( ( cos @ real @ X2 )
= ( one_one @ real ) )
= ( ? [X: int] :
( X2
= ( times_times @ real @ ( times_times @ real @ ( ring_1_of_int @ real @ X ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ pi ) ) ) ) ).
% cos_one_2pi_int
thf(fact_3139_sum__gp__multiplied,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [M: nat,N: nat,X2: A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( times_times @ A @ ( minus_minus @ A @ ( one_one @ A ) @ X2 ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X2 ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) )
= ( minus_minus @ A @ ( power_power @ A @ X2 @ M ) @ ( power_power @ A @ X2 @ ( suc @ N ) ) ) ) ) ) ).
% sum_gp_multiplied
thf(fact_3140_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
@ ^ [I3: nat] : ( plus_plus @ A @ ( G @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I3 ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ).
% sum.in_pairs
thf(fact_3141_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_3142_cos__treble__cos,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X2: A] :
( ( cos @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit1 @ one2 ) ) @ X2 ) )
= ( minus_minus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ ( cos @ A @ X2 ) @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit1 @ one2 ) ) @ ( cos @ A @ X2 ) ) ) ) ) ).
% cos_treble_cos
thf(fact_3143_sin__le__zero,axiom,
! [X2: real] :
( ( ord_less_eq @ real @ pi @ X2 )
=> ( ( ord_less @ real @ X2 @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) )
=> ( ord_less_eq @ real @ ( sin @ real @ X2 ) @ ( zero_zero @ real ) ) ) ) ).
% sin_le_zero
thf(fact_3144_sin__less__zero,axiom,
! [X2: real] :
( ( ord_less @ real @ ( divide_divide @ real @ ( uminus_uminus @ real @ pi ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ X2 )
=> ( ( ord_less @ real @ X2 @ ( zero_zero @ real ) )
=> ( ord_less @ real @ ( sin @ real @ X2 ) @ ( zero_zero @ real ) ) ) ) ).
% sin_less_zero
thf(fact_3145_sin__mono__less__eq,axiom,
! [X2: real,Y2: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X2 )
=> ( ( ord_less_eq @ real @ X2 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ Y2 )
=> ( ( ord_less_eq @ real @ Y2 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less @ real @ ( sin @ real @ X2 ) @ ( sin @ real @ Y2 ) )
= ( ord_less @ real @ X2 @ Y2 ) ) ) ) ) ) ).
% sin_mono_less_eq
thf(fact_3146_sin__monotone__2pi,axiom,
! [Y2: real,X2: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ Y2 )
=> ( ( ord_less @ real @ Y2 @ X2 )
=> ( ( ord_less_eq @ real @ X2 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less @ real @ ( sin @ real @ Y2 ) @ ( sin @ real @ X2 ) ) ) ) ) ).
% sin_monotone_2pi
thf(fact_3147_sin__total,axiom,
! [Y2: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ Y2 )
=> ( ( ord_less_eq @ real @ Y2 @ ( 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 )
= Y2 )
& ! [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 )
= Y2 ) )
=> ( Y4 = X3 ) ) ) ) ) ).
% sin_total
thf(fact_3148_cos__gt__zero__pi,axiom,
! [X2: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X2 )
=> ( ( ord_less @ real @ X2 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( cos @ real @ X2 ) ) ) ) ).
% cos_gt_zero_pi
thf(fact_3149_cos__ge__zero,axiom,
! [X2: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X2 )
=> ( ( ord_less_eq @ real @ X2 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( cos @ real @ X2 ) ) ) ) ).
% cos_ge_zero
thf(fact_3150_cos__one__2pi,axiom,
! [X2: real] :
( ( ( cos @ real @ X2 )
= ( one_one @ real ) )
= ( ? [X: nat] :
( X2
= ( times_times @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ X ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ pi ) )
| ? [X: nat] :
( X2
= ( uminus_uminus @ real @ ( times_times @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ X ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ pi ) ) ) ) ) ).
% cos_one_2pi
thf(fact_3151_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
@ ^ [Q4: nat] : ( ord_less @ nat @ Q4 @ N ) ) ) ) ).
% mask_eq_sum_exp_nat
thf(fact_3152_gauss__sum__nat,axiom,
! [N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [X: nat] : X
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
= ( divide_divide @ nat @ ( times_times @ nat @ N @ ( suc @ N ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% gauss_sum_nat
thf(fact_3153_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_3154_double__arith__series,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A2: A,D2: A,N: nat] :
( ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( plus_plus @ A @ A2 @ ( times_times @ A @ ( semiring_1_of_nat @ A @ I3 ) @ D2 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) )
= ( times_times @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N ) @ ( one_one @ A ) ) @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ D2 ) ) ) ) ) ).
% double_arith_series
thf(fact_3155_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_3156_arith__series__nat,axiom,
! [A2: nat,D2: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [I3: nat] : ( plus_plus @ nat @ A2 @ ( times_times @ nat @ I3 @ D2 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
= ( divide_divide @ nat @ ( times_times @ nat @ ( suc @ N ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ A2 ) @ ( times_times @ nat @ N @ D2 ) ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% arith_series_nat
thf(fact_3157_Sum__Icc__nat,axiom,
! [M: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [X: nat] : X
@ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( divide_divide @ nat @ ( minus_minus @ nat @ ( times_times @ nat @ N @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) ) @ ( times_times @ nat @ M @ ( minus_minus @ nat @ M @ ( one_one @ nat ) ) ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% Sum_Icc_nat
thf(fact_3158_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_3159_arith__series,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [A2: A,D2: A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( plus_plus @ A @ A2 @ ( times_times @ A @ ( semiring_1_of_nat @ A @ I3 ) @ D2 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
= ( divide_divide @ A @ ( times_times @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N ) @ ( one_one @ A ) ) @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ D2 ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% arith_series
thf(fact_3160_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_3161_sum__gp__offset,axiom,
! [A: $tType] :
( ( ( division_ring @ A )
& ( comm_ring @ A ) )
=> ! [X2: A,M: nat,N: nat] :
( ( ( X2
= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X2 ) @ ( set_or1337092689740270186AtMost @ nat @ M @ ( plus_plus @ nat @ M @ N ) ) )
= ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N ) @ ( one_one @ A ) ) ) )
& ( ( X2
!= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X2 ) @ ( set_or1337092689740270186AtMost @ nat @ M @ ( plus_plus @ nat @ M @ N ) ) )
= ( divide_divide @ A @ ( times_times @ A @ ( power_power @ A @ X2 @ M ) @ ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ X2 @ ( suc @ N ) ) ) ) @ ( minus_minus @ A @ ( one_one @ A ) @ X2 ) ) ) ) ) ) ).
% sum_gp_offset
thf(fact_3162_sin__zero__iff__int,axiom,
! [X2: real] :
( ( ( sin @ real @ X2 )
= ( zero_zero @ real ) )
= ( ? [I3: int] :
( ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ I3 )
& ( X2
= ( times_times @ real @ ( ring_1_of_int @ real @ I3 ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% sin_zero_iff_int
thf(fact_3163_cos__zero__iff__int,axiom,
! [X2: real] :
( ( ( cos @ real @ X2 )
= ( zero_zero @ real ) )
= ( ? [I3: int] :
( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ I3 )
& ( X2
= ( times_times @ real @ ( ring_1_of_int @ real @ I3 ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% cos_zero_iff_int
thf(fact_3164_sin__zero__lemma,axiom,
! [X2: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ( sin @ real @ X2 )
= ( zero_zero @ real ) )
=> ? [N2: nat] :
( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
& ( X2
= ( times_times @ real @ ( semiring_1_of_nat @ real @ N2 ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% sin_zero_lemma
thf(fact_3165_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_3166_sin__zero__iff,axiom,
! [X2: real] :
( ( ( sin @ real @ X2 )
= ( zero_zero @ real ) )
= ( ? [N3: nat] :
( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 )
& ( X2
= ( times_times @ real @ ( semiring_1_of_nat @ real @ N3 ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) )
| ? [N3: nat] :
( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 )
& ( X2
= ( 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_3167_cos__zero__lemma,axiom,
! [X2: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ( cos @ real @ X2 )
= ( zero_zero @ real ) )
=> ? [N2: nat] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
& ( X2
= ( times_times @ real @ ( semiring_1_of_nat @ real @ N2 ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% cos_zero_lemma
thf(fact_3168_cos__zero__iff,axiom,
! [X2: real] :
( ( ( cos @ real @ X2 )
= ( zero_zero @ real ) )
= ( ? [N3: nat] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 )
& ( X2
= ( times_times @ real @ ( semiring_1_of_nat @ real @ N3 ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) )
| ? [N3: nat] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 )
& ( X2
= ( 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_3169_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
@ ^ [P3: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [Q4: nat] : ( times_times @ A @ ( power_power @ A @ ( plus_plus @ A @ Z @ H2 ) @ Q4 ) @ ( power_power @ A @ Z @ ( minus_minus @ nat @ ( minus_minus @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ Q4 ) ) )
@ ( set_ord_lessThan @ nat @ ( minus_minus @ nat @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) @ P3 ) ) )
@ ( set_ord_lessThan @ nat @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ) ) ).
% lemma_termdiff2
thf(fact_3170_tan__double,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X2: A] :
( ( ( cos @ A @ X2 )
!= ( zero_zero @ A ) )
=> ( ( ( cos @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X2 ) )
!= ( zero_zero @ A ) )
=> ( ( tan @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X2 ) )
= ( divide_divide @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( tan @ A @ X2 ) ) @ ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ ( tan @ A @ X2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ).
% tan_double
thf(fact_3171_complex__unimodular__polar,axiom,
! [Z: complex] :
( ( ( real_V7770717601297561774m_norm @ complex @ Z )
= ( one_one @ real ) )
=> ~ ! [T5: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ T5 )
=> ( ( ord_less @ real @ T5 @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) )
=> ( Z
!= ( complex2 @ ( cos @ real @ T5 ) @ ( sin @ real @ T5 ) ) ) ) ) ) ).
% complex_unimodular_polar
thf(fact_3172_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_3173_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_3174_tan__pi,axiom,
( ( tan @ real @ pi )
= ( zero_zero @ real ) ) ).
% tan_pi
thf(fact_3175_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_3176_tan__periodic__pi,axiom,
! [X2: real] :
( ( tan @ real @ ( plus_plus @ real @ X2 @ pi ) )
= ( tan @ real @ X2 ) ) ).
% tan_periodic_pi
thf(fact_3177_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_3178_lessThan__subset__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: A,Y2: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_ord_lessThan @ A @ X2 ) @ ( set_ord_lessThan @ A @ Y2 ) )
= ( ord_less_eq @ A @ X2 @ Y2 ) ) ) ).
% lessThan_subset_iff
thf(fact_3179_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_3180_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_3181_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_3182_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_3183_tan__npi,axiom,
! [N: nat] :
( ( tan @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ pi ) )
= ( zero_zero @ real ) ) ).
% tan_npi
thf(fact_3184_tan__periodic__n,axiom,
! [X2: real,N: num] :
( ( tan @ real @ ( plus_plus @ real @ X2 @ ( times_times @ real @ ( numeral_numeral @ real @ N ) @ pi ) ) )
= ( tan @ real @ X2 ) ) ).
% tan_periodic_n
thf(fact_3185_tan__periodic__nat,axiom,
! [X2: real,N: nat] :
( ( tan @ real @ ( plus_plus @ real @ X2 @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ pi ) ) )
= ( tan @ real @ X2 ) ) ).
% tan_periodic_nat
thf(fact_3186_tan__periodic__int,axiom,
! [X2: real,I: int] :
( ( tan @ real @ ( plus_plus @ real @ X2 @ ( times_times @ real @ ( ring_1_of_int @ real @ I ) @ pi ) ) )
= ( tan @ real @ X2 ) ) ).
% tan_periodic_int
thf(fact_3187_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_3188_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_3189_powser__sums__zero__iff,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [A2: nat > A,X2: A] :
( ( sums @ A
@ ^ [N3: nat] : ( times_times @ A @ ( A2 @ N3 ) @ ( power_power @ A @ ( zero_zero @ A ) @ N3 ) )
@ X2 )
= ( ( A2 @ ( zero_zero @ nat ) )
= X2 ) ) ) ).
% powser_sums_zero_iff
thf(fact_3190_norm__cos__sin,axiom,
! [T2: real] :
( ( real_V7770717601297561774m_norm @ complex @ ( complex2 @ ( cos @ real @ T2 ) @ ( sin @ real @ T2 ) ) )
= ( one_one @ real ) ) ).
% norm_cos_sin
thf(fact_3191_tan__periodic,axiom,
! [X2: real] :
( ( tan @ real @ ( plus_plus @ real @ X2 @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) ) )
= ( tan @ real @ X2 ) ) ).
% tan_periodic
thf(fact_3192_Complex__sum_H,axiom,
! [A: $tType,F2: A > real,S3: set @ A] :
( ( groups7311177749621191930dd_sum @ A @ complex
@ ^ [X: A] : ( complex2 @ ( F2 @ X ) @ ( zero_zero @ real ) )
@ S3 )
= ( complex2 @ ( groups7311177749621191930dd_sum @ A @ real @ F2 @ S3 ) @ ( zero_zero @ real ) ) ) ).
% Complex_sum'
thf(fact_3193_int__sum,axiom,
! [B: $tType,F2: B > nat,A3: set @ B] :
( ( semiring_1_of_nat @ int @ ( groups7311177749621191930dd_sum @ B @ nat @ F2 @ A3 ) )
= ( groups7311177749621191930dd_sum @ B @ int
@ ^ [X: B] : ( semiring_1_of_nat @ int @ ( F2 @ X ) )
@ A3 ) ) ).
% int_sum
thf(fact_3194_sum__diff__distrib,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [Q: A > nat,P: A > nat,N: A] :
( ! [X3: A] : ( ord_less_eq @ nat @ ( Q @ X3 ) @ ( P @ X3 ) )
=> ( ( minus_minus @ nat @ ( groups7311177749621191930dd_sum @ A @ nat @ P @ ( set_ord_lessThan @ A @ N ) ) @ ( groups7311177749621191930dd_sum @ A @ nat @ Q @ ( set_ord_lessThan @ A @ N ) ) )
= ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [X: A] : ( minus_minus @ nat @ ( P @ X ) @ ( Q @ X ) )
@ ( set_ord_lessThan @ A @ N ) ) ) ) ) ).
% sum_diff_distrib
thf(fact_3195_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_3196_sums__of__real__iff,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [F2: nat > real,C2: real] :
( ( sums @ A
@ ^ [N3: nat] : ( real_Vector_of_real @ A @ ( F2 @ N3 ) )
@ ( real_Vector_of_real @ A @ C2 ) )
= ( sums @ real @ F2 @ C2 ) ) ) ).
% sums_of_real_iff
thf(fact_3197_sums__of__real,axiom,
! [A: $tType] :
( ( ( real_V2191834092415804123ebra_1 @ A )
& ( real_V822414075346904944vector @ A ) )
=> ! [X6: nat > real,A2: real] :
( ( sums @ real @ X6 @ A2 )
=> ( sums @ A
@ ^ [N3: nat] : ( real_Vector_of_real @ A @ ( X6 @ N3 ) )
@ ( real_Vector_of_real @ A @ A2 ) ) ) ) ).
% sums_of_real
thf(fact_3198_sums__le,axiom,
! [A: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F2: nat > A,G: nat > A,S3: A,T2: A] :
( ! [N2: nat] : ( ord_less_eq @ A @ ( F2 @ N2 ) @ ( G @ N2 ) )
=> ( ( sums @ A @ F2 @ S3 )
=> ( ( sums @ A @ G @ T2 )
=> ( ord_less_eq @ A @ S3 @ T2 ) ) ) ) ) ).
% sums_le
thf(fact_3199_complex__of__real__def,axiom,
( ( real_Vector_of_real @ complex )
= ( ^ [R5: real] : ( complex2 @ R5 @ ( zero_zero @ real ) ) ) ) ).
% complex_of_real_def
thf(fact_3200_complex__of__real__code,axiom,
( ( real_Vector_of_real @ complex )
= ( ^ [X: real] : ( complex2 @ X @ ( zero_zero @ real ) ) ) ) ).
% complex_of_real_code
thf(fact_3201_complex__eq__cancel__iff2,axiom,
! [X2: real,Y2: real,Xa: real] :
( ( ( complex2 @ X2 @ Y2 )
= ( real_Vector_of_real @ complex @ Xa ) )
= ( ( X2 = Xa )
& ( Y2
= ( zero_zero @ real ) ) ) ) ).
% complex_eq_cancel_iff2
thf(fact_3202_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_3203_zero__complex_Ocode,axiom,
( ( zero_zero @ complex )
= ( complex2 @ ( zero_zero @ real ) @ ( zero_zero @ real ) ) ) ).
% zero_complex.code
thf(fact_3204_Complex__mult__complex__of__real,axiom,
! [X2: real,Y2: real,R2: real] :
( ( times_times @ complex @ ( complex2 @ X2 @ Y2 ) @ ( real_Vector_of_real @ complex @ R2 ) )
= ( complex2 @ ( times_times @ real @ X2 @ R2 ) @ ( times_times @ real @ Y2 @ R2 ) ) ) ).
% Complex_mult_complex_of_real
thf(fact_3205_complex__of__real__mult__Complex,axiom,
! [R2: real,X2: real,Y2: real] :
( ( times_times @ complex @ ( real_Vector_of_real @ complex @ R2 ) @ ( complex2 @ X2 @ Y2 ) )
= ( complex2 @ ( times_times @ real @ R2 @ X2 ) @ ( times_times @ real @ R2 @ Y2 ) ) ) ).
% complex_of_real_mult_Complex
thf(fact_3206_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_3207_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_3208_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_3209_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_3210_sums__diff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A,A2: A,G: nat > A,B2: A] :
( ( sums @ A @ F2 @ A2 )
=> ( ( sums @ A @ G @ B2 )
=> ( sums @ A
@ ^ [N3: nat] : ( minus_minus @ A @ ( F2 @ N3 ) @ ( G @ N3 ) )
@ ( minus_minus @ A @ A2 @ B2 ) ) ) ) ) ).
% sums_diff
thf(fact_3211_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_3212_sums__minus,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A,A2: A] :
( ( sums @ A @ F2 @ A2 )
=> ( sums @ A
@ ^ [N3: nat] : ( uminus_uminus @ A @ ( F2 @ N3 ) )
@ ( uminus_uminus @ A @ A2 ) ) ) ) ).
% sums_minus
thf(fact_3213_sums__sum,axiom,
! [A: $tType,I6: $tType] :
( ( ( topolo5987344860129210374id_add @ A )
& ( topological_t2_space @ A ) )
=> ! [I5: set @ I6,F2: I6 > nat > A,X2: I6 > A] :
( ! [I2: I6] :
( ( member @ I6 @ I2 @ I5 )
=> ( sums @ A @ ( F2 @ I2 ) @ ( X2 @ I2 ) ) )
=> ( sums @ A
@ ^ [N3: nat] :
( groups7311177749621191930dd_sum @ I6 @ A
@ ^ [I3: I6] : ( F2 @ I3 @ N3 )
@ I5 )
@ ( groups7311177749621191930dd_sum @ I6 @ A @ X2 @ I5 ) ) ) ) ).
% sums_sum
thf(fact_3214_lessThan__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( set_ord_lessThan @ A )
= ( ^ [U2: A] :
( collect @ A
@ ^ [X: A] : ( ord_less @ A @ X @ U2 ) ) ) ) ) ).
% lessThan_def
thf(fact_3215_sums__iff__shift,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A,N: nat,S3: A] :
( ( sums @ A
@ ^ [I3: nat] : ( F2 @ ( plus_plus @ nat @ I3 @ N ) )
@ S3 )
= ( sums @ A @ F2 @ ( plus_plus @ A @ S3 @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ) ).
% sums_iff_shift
thf(fact_3216_sums__split__initial__segment,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A,S3: A,N: nat] :
( ( sums @ A @ F2 @ S3 )
=> ( sums @ A
@ ^ [I3: nat] : ( F2 @ ( plus_plus @ nat @ I3 @ N ) )
@ ( minus_minus @ A @ S3 @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ) ).
% sums_split_initial_segment
thf(fact_3217_sums__iff__shift_H,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A,N: nat,S3: A] :
( ( sums @ A
@ ^ [I3: nat] : ( F2 @ ( plus_plus @ nat @ I3 @ N ) )
@ ( minus_minus @ A @ S3 @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_ord_lessThan @ nat @ N ) ) ) )
= ( sums @ A @ F2 @ S3 ) ) ) ).
% sums_iff_shift'
thf(fact_3218_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 ) ) )
= ( ! [X: A] :
( ( member @ A @ X @ A3 )
=> ( ( F2 @ X )
= ( zero_zero @ ( multiset @ B ) ) ) ) ) ) ) ).
% sum_eq_empty_iff
thf(fact_3219_one__complex_Ocode,axiom,
( ( one_one @ complex )
= ( complex2 @ ( one_one @ real ) @ ( zero_zero @ real ) ) ) ).
% one_complex.code
thf(fact_3220_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_3221_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_3222_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_3223_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_3224_sum__subtractf__nat,axiom,
! [A: $tType,A3: set @ A,G: A > nat,F2: A > nat] :
( ! [X3: A] :
( ( member @ A @ X3 @ A3 )
=> ( ord_less_eq @ nat @ ( G @ X3 ) @ ( F2 @ X3 ) ) )
=> ( ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [X: A] : ( minus_minus @ nat @ ( F2 @ X ) @ ( G @ X ) )
@ A3 )
= ( minus_minus @ nat @ ( groups7311177749621191930dd_sum @ A @ nat @ F2 @ A3 ) @ ( groups7311177749621191930dd_sum @ A @ nat @ G @ A3 ) ) ) ) ).
% sum_subtractf_nat
thf(fact_3225_Complex__add__complex__of__real,axiom,
! [X2: real,Y2: real,R2: real] :
( ( plus_plus @ complex @ ( complex2 @ X2 @ Y2 ) @ ( real_Vector_of_real @ complex @ R2 ) )
= ( complex2 @ ( plus_plus @ real @ X2 @ R2 ) @ Y2 ) ) ).
% Complex_add_complex_of_real
thf(fact_3226_complex__of__real__add__Complex,axiom,
! [R2: real,X2: real,Y2: real] :
( ( plus_plus @ complex @ ( real_Vector_of_real @ complex @ R2 ) @ ( complex2 @ X2 @ Y2 ) )
= ( complex2 @ ( plus_plus @ real @ R2 @ X2 ) @ Y2 ) ) ).
% complex_of_real_add_Complex
thf(fact_3227_complex__add,axiom,
! [A2: real,B2: real,C2: real,D2: real] :
( ( plus_plus @ complex @ ( complex2 @ A2 @ B2 ) @ ( complex2 @ C2 @ D2 ) )
= ( complex2 @ ( plus_plus @ real @ A2 @ C2 ) @ ( plus_plus @ real @ B2 @ D2 ) ) ) ).
% complex_add
thf(fact_3228_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_3229_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 ) ) )
= ( ? [X: A] :
( ( member @ A @ X @ A3 )
& ( ( F2 @ X )
= ( suc @ ( zero_zero @ nat ) ) )
& ! [Y: A] :
( ( member @ A @ Y @ A3 )
=> ( ( X != Y )
=> ( ( F2 @ Y )
= ( zero_zero @ nat ) ) ) ) ) ) ) ) ).
% sum_eq_Suc0_iff
thf(fact_3230_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 ) )
= ( ? [X: A] :
( ( member @ A @ X @ A3 )
& ( ( F2 @ X )
= ( one_one @ nat ) )
& ! [Y: A] :
( ( member @ A @ Y @ A3 )
=> ( ( X != Y )
=> ( ( F2 @ Y )
= ( zero_zero @ nat ) ) ) ) ) ) ) ) ).
% sum_eq_1_iff
thf(fact_3231_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_3232_sums__Suc__imp,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A,S3: A] :
( ( ( F2 @ ( zero_zero @ nat ) )
= ( zero_zero @ A ) )
=> ( ( sums @ A
@ ^ [N3: nat] : ( F2 @ ( suc @ N3 ) )
@ S3 )
=> ( sums @ A @ F2 @ S3 ) ) ) ) ).
% sums_Suc_imp
thf(fact_3233_sums__Suc__iff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A,S3: A] :
( ( sums @ A
@ ^ [N3: nat] : ( F2 @ ( suc @ N3 ) )
@ S3 )
= ( sums @ A @ F2 @ ( plus_plus @ A @ S3 @ ( F2 @ ( zero_zero @ nat ) ) ) ) ) ) ).
% sums_Suc_iff
thf(fact_3234_sums__Suc,axiom,
! [A: $tType] :
( ( ( topolo5987344860129210374id_add @ A )
& ( topological_t2_space @ A ) )
=> ! [F2: nat > A,L2: A] :
( ( sums @ A
@ ^ [N3: nat] : ( F2 @ ( suc @ N3 ) )
@ L2 )
=> ( sums @ A @ F2 @ ( plus_plus @ A @ L2 @ ( F2 @ ( zero_zero @ nat ) ) ) ) ) ) ).
% sums_Suc
thf(fact_3235_sums__zero__iff__shift,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [N: nat,F2: nat > A,S3: A] :
( ! [I2: nat] :
( ( ord_less @ nat @ I2 @ N )
=> ( ( F2 @ I2 )
= ( zero_zero @ A ) ) )
=> ( ( sums @ A
@ ^ [I3: nat] : ( F2 @ ( plus_plus @ nat @ I3 @ N ) )
@ S3 )
= ( sums @ A @ F2 @ S3 ) ) ) ) ).
% sums_zero_iff_shift
thf(fact_3236_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_3237_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_3238_sums__finite,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [N5: set @ nat,F2: nat > A] :
( ( finite_finite2 @ nat @ N5 )
=> ( ! [N2: nat] :
( ~ ( member @ nat @ N2 @ N5 )
=> ( ( F2 @ N2 )
= ( zero_zero @ A ) ) )
=> ( sums @ A @ F2 @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ N5 ) ) ) ) ) ).
% sums_finite
thf(fact_3239_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_3240_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_3241_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_3242_sum_Onat__diff__reindex,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( G @ ( minus_minus @ nat @ N @ ( suc @ I3 ) ) )
@ ( set_ord_lessThan @ nat @ N ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% sum.nat_diff_reindex
thf(fact_3243_sum__nth__roots,axiom,
! [N: nat,C2: complex] :
( ( ord_less @ nat @ ( one_one @ nat ) @ N )
=> ( ( groups7311177749621191930dd_sum @ complex @ complex
@ ^ [X: complex] : X
@ ( collect @ complex
@ ^ [Z5: complex] :
( ( power_power @ complex @ Z5 @ N )
= C2 ) ) )
= ( zero_zero @ complex ) ) ) ).
% sum_nth_roots
thf(fact_3244_sum__roots__unity,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( one_one @ nat ) @ N )
=> ( ( groups7311177749621191930dd_sum @ complex @ complex
@ ^ [X: complex] : X
@ ( collect @ complex
@ ^ [Z5: complex] :
( ( power_power @ complex @ Z5 @ N )
= ( one_one @ complex ) ) ) )
= ( zero_zero @ complex ) ) ) ).
% sum_roots_unity
thf(fact_3245_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_3246_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_3247_sum__diff__nat,axiom,
! [A: $tType,B5: set @ A,A3: set @ A,F2: A > nat] :
( ( finite_finite2 @ A @ B5 )
=> ( ( ord_less_eq @ ( set @ A ) @ B5 @ A3 )
=> ( ( groups7311177749621191930dd_sum @ A @ nat @ F2 @ ( minus_minus @ ( set @ A ) @ A3 @ B5 ) )
= ( minus_minus @ nat @ ( groups7311177749621191930dd_sum @ A @ nat @ F2 @ A3 ) @ ( groups7311177749621191930dd_sum @ A @ nat @ F2 @ B5 ) ) ) ) ) ).
% sum_diff_nat
thf(fact_3248_tan__def,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ( ( tan @ A )
= ( ^ [X: A] : ( divide_divide @ A @ ( sin @ A @ X ) @ ( cos @ A @ X ) ) ) ) ) ).
% tan_def
thf(fact_3249_sums__If__finite__set_H,axiom,
! [A: $tType] :
( ( ( topolo1287966508704411220up_add @ A )
& ( topological_t2_space @ A ) )
=> ! [G: nat > A,S: A,A3: set @ nat,S6: A,F2: nat > A] :
( ( sums @ A @ G @ S )
=> ( ( finite_finite2 @ nat @ A3 )
=> ( ( S6
= ( plus_plus @ A @ S
@ ( 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 ) )
@ S6 ) ) ) ) ) ).
% sums_If_finite_set'
thf(fact_3250_suminf__le__const,axiom,
! [A: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F2: nat > A,X2: A] :
( ( summable @ A @ F2 )
=> ( ! [N2: nat] : ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_ord_lessThan @ nat @ N2 ) ) @ X2 )
=> ( ord_less_eq @ A @ ( suminf @ A @ F2 ) @ X2 ) ) ) ) ).
% suminf_le_const
thf(fact_3251_dbl__inc__def,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( ( neg_numeral_dbl_inc @ A )
= ( ^ [X: A] : ( plus_plus @ A @ ( plus_plus @ A @ X @ X ) @ ( one_one @ A ) ) ) ) ) ).
% dbl_inc_def
thf(fact_3252_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
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% sum.lessThan_Suc_shift
thf(fact_3253_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_3254_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_3255_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
@ ^ [I3: nat] : ( minus_minus @ A @ ( F2 @ I3 ) @ R2 )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% sumr_diff_mult_const2
thf(fact_3256_summableI__nonneg__bounded,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( ordere6911136660526730532id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F2: nat > A,X2: 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 ) ) @ X2 )
=> ( summable @ A @ F2 ) ) ) ) ).
% summableI_nonneg_bounded
thf(fact_3257_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
@ ^ [K2: nat] : ( G @ ( suc @ K2 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% sum.atLeast1_atMost_eq
thf(fact_3258_power__diff__1__eq,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [X2: A,N: nat] :
( ( minus_minus @ A @ ( power_power @ A @ X2 @ N ) @ ( one_one @ A ) )
= ( times_times @ A @ ( minus_minus @ A @ X2 @ ( one_one @ A ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X2 ) @ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% power_diff_1_eq
thf(fact_3259_one__diff__power__eq,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [X2: A,N: nat] :
( ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ X2 @ N ) )
= ( times_times @ A @ ( minus_minus @ A @ ( one_one @ A ) @ X2 ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X2 ) @ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% one_diff_power_eq
thf(fact_3260_geometric__sum,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [X2: A,N: nat] :
( ( X2
!= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X2 ) @ ( set_ord_lessThan @ nat @ N ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( power_power @ A @ X2 @ N ) @ ( one_one @ A ) ) @ ( minus_minus @ A @ X2 @ ( one_one @ A ) ) ) ) ) ) ).
% geometric_sum
thf(fact_3261_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_3262_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_3263_tan__45,axiom,
( ( tan @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ ( bit0 @ one2 ) ) ) ) )
= ( one_one @ real ) ) ).
% tan_45
thf(fact_3264_sum__less__suminf,axiom,
! [A: $tType] :
( ( ( ordere8940638589300402666id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F2: nat > A,N: nat] :
( ( summable @ A @ F2 )
=> ( ! [M3: nat] :
( ( ord_less_eq @ nat @ N @ M3 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( F2 @ M3 ) ) )
=> ( ord_less @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_ord_lessThan @ nat @ N ) ) @ ( suminf @ A @ F2 ) ) ) ) ) ).
% sum_less_suminf
thf(fact_3265_lemma__termdiff1,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [Z: A,H2: A,M: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [P3: nat] : ( minus_minus @ A @ ( times_times @ A @ ( power_power @ A @ ( plus_plus @ A @ Z @ H2 ) @ ( minus_minus @ nat @ M @ P3 ) ) @ ( power_power @ A @ Z @ P3 ) ) @ ( power_power @ A @ Z @ M ) )
@ ( set_ord_lessThan @ nat @ M ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [P3: nat] : ( times_times @ A @ ( power_power @ A @ Z @ P3 ) @ ( minus_minus @ A @ ( power_power @ A @ ( plus_plus @ A @ Z @ H2 ) @ ( minus_minus @ nat @ M @ P3 ) ) @ ( power_power @ A @ Z @ ( minus_minus @ nat @ M @ P3 ) ) ) )
@ ( set_ord_lessThan @ nat @ M ) ) ) ) ).
% lemma_termdiff1
thf(fact_3266_sum__gp__strict,axiom,
! [A: $tType] :
( ( ( division_ring @ A )
& ( comm_ring @ A ) )
=> ! [X2: A,N: nat] :
( ( ( X2
= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X2 ) @ ( set_ord_lessThan @ nat @ N ) )
= ( semiring_1_of_nat @ A @ N ) ) )
& ( ( X2
!= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X2 ) @ ( set_ord_lessThan @ nat @ N ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ X2 @ N ) ) @ ( minus_minus @ A @ ( one_one @ A ) @ X2 ) ) ) ) ) ) ).
% sum_gp_strict
thf(fact_3267_power__diff__sumr2,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [X2: A,N: nat,Y2: A] :
( ( minus_minus @ A @ ( power_power @ A @ X2 @ N ) @ ( power_power @ A @ Y2 @ N ) )
= ( times_times @ A @ ( minus_minus @ A @ X2 @ Y2 )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( power_power @ A @ Y2 @ ( minus_minus @ nat @ N @ ( suc @ I3 ) ) ) @ ( power_power @ A @ X2 @ I3 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% power_diff_sumr2
thf(fact_3268_diff__power__eq__sum,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [X2: A,N: nat,Y2: A] :
( ( minus_minus @ A @ ( power_power @ A @ X2 @ ( suc @ N ) ) @ ( power_power @ A @ Y2 @ ( suc @ N ) ) )
= ( times_times @ A @ ( minus_minus @ A @ X2 @ Y2 )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [P3: nat] : ( times_times @ A @ ( power_power @ A @ X2 @ P3 ) @ ( power_power @ A @ Y2 @ ( minus_minus @ nat @ N @ P3 ) ) )
@ ( set_ord_lessThan @ nat @ ( suc @ N ) ) ) ) ) ) ).
% diff_power_eq_sum
thf(fact_3269_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_3270_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_3271_lemma__tan__total,axiom,
! [Y2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ Y2 )
=> ? [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 @ Y2 @ ( tan @ real @ X3 ) ) ) ) ).
% lemma_tan_total
thf(fact_3272_tan__gt__zero,axiom,
! [X2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ord_less @ real @ X2 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( tan @ real @ X2 ) ) ) ) ).
% tan_gt_zero
thf(fact_3273_real__sum__nat__ivl__bounded2,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N: nat,F2: nat > A,K6: A,K: nat] :
( ! [P5: nat] :
( ( ord_less @ nat @ P5 @ N )
=> ( ord_less_eq @ A @ ( F2 @ P5 ) @ 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_3274_lemma__tan__total1,axiom,
! [Y2: 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 )
= Y2 ) ) ).
% lemma_tan_total1
thf(fact_3275_tan__mono__lt__eq,axiom,
! [X2: real,Y2: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X2 )
=> ( ( ord_less @ real @ X2 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ Y2 )
=> ( ( ord_less @ real @ Y2 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less @ real @ ( tan @ real @ X2 ) @ ( tan @ real @ Y2 ) )
= ( ord_less @ real @ X2 @ Y2 ) ) ) ) ) ) ).
% tan_mono_lt_eq
thf(fact_3276_tan__monotone_H,axiom,
! [Y2: real,X2: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ Y2 )
=> ( ( ord_less @ real @ Y2 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X2 )
=> ( ( ord_less @ real @ X2 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less @ real @ Y2 @ X2 )
= ( ord_less @ real @ ( tan @ real @ Y2 ) @ ( tan @ real @ X2 ) ) ) ) ) ) ) ).
% tan_monotone'
thf(fact_3277_tan__monotone,axiom,
! [Y2: real,X2: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ Y2 )
=> ( ( ord_less @ real @ Y2 @ X2 )
=> ( ( ord_less @ real @ X2 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less @ real @ ( tan @ real @ Y2 ) @ ( tan @ real @ X2 ) ) ) ) ) ).
% tan_monotone
thf(fact_3278_tan__total,axiom,
! [Y2: 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 )
= Y2 )
& ! [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 )
= Y2 ) )
=> ( Y4 = X3 ) ) ) ).
% tan_total
thf(fact_3279_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_3280_tan__inverse,axiom,
! [Y2: real] :
( ( divide_divide @ real @ ( one_one @ real ) @ ( tan @ real @ Y2 ) )
= ( tan @ real @ ( minus_minus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ Y2 ) ) ) ).
% tan_inverse
thf(fact_3281_sum__less__suminf2,axiom,
! [A: $tType] :
( ( ( ordere8940638589300402666id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F2: nat > A,N: nat,I: nat] :
( ( summable @ A @ F2 )
=> ( ! [M3: nat] :
( ( ord_less_eq @ nat @ N @ M3 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ M3 ) ) )
=> ( ( 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_3282_one__diff__power__eq_H,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [X2: A,N: nat] :
( ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ X2 @ N ) )
= ( times_times @ A @ ( minus_minus @ A @ ( one_one @ A ) @ X2 )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( power_power @ A @ X2 @ ( minus_minus @ nat @ N @ ( suc @ I3 ) ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% one_diff_power_eq'
thf(fact_3283_add__tan__eq,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X2: A,Y2: A] :
( ( ( cos @ A @ X2 )
!= ( zero_zero @ A ) )
=> ( ( ( cos @ A @ Y2 )
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( tan @ A @ X2 ) @ ( tan @ A @ Y2 ) )
= ( divide_divide @ A @ ( sin @ A @ ( plus_plus @ A @ X2 @ Y2 ) ) @ ( times_times @ A @ ( cos @ A @ X2 ) @ ( cos @ A @ Y2 ) ) ) ) ) ) ) ).
% add_tan_eq
thf(fact_3284_sums__if_H,axiom,
! [G: nat > real,X2: real] :
( ( sums @ real @ G @ X2 )
=> ( 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 ) ) ) ) )
@ X2 ) ) ).
% sums_if'
thf(fact_3285_sums__if,axiom,
! [G: nat > real,X2: real,F2: nat > real,Y2: real] :
( ( sums @ real @ G @ X2 )
=> ( ( sums @ real @ F2 @ Y2 )
=> ( 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 @ X2 @ Y2 ) ) ) ) ).
% sums_if
thf(fact_3286_tan__pos__pi2__le,axiom,
! [X2: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ord_less @ real @ X2 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( tan @ real @ X2 ) ) ) ) ).
% tan_pos_pi2_le
thf(fact_3287_tan__total__pos,axiom,
! [Y2: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y2 )
=> ? [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 )
= Y2 ) ) ) ).
% tan_total_pos
thf(fact_3288_tan__less__zero,axiom,
! [X2: real] :
( ( ord_less @ real @ ( divide_divide @ real @ ( uminus_uminus @ real @ pi ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ X2 )
=> ( ( ord_less @ real @ X2 @ ( zero_zero @ real ) )
=> ( ord_less @ real @ ( tan @ real @ X2 ) @ ( zero_zero @ real ) ) ) ) ).
% tan_less_zero
thf(fact_3289_tan__mono__le__eq,axiom,
! [X2: real,Y2: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X2 )
=> ( ( ord_less @ real @ X2 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ Y2 )
=> ( ( ord_less @ real @ Y2 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less_eq @ real @ ( tan @ real @ X2 ) @ ( tan @ real @ Y2 ) )
= ( ord_less_eq @ real @ X2 @ Y2 ) ) ) ) ) ) ).
% tan_mono_le_eq
thf(fact_3290_tan__mono__le,axiom,
! [X2: real,Y2: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X2 )
=> ( ( ord_less_eq @ real @ X2 @ Y2 )
=> ( ( ord_less @ real @ Y2 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less_eq @ real @ ( tan @ real @ X2 ) @ ( tan @ real @ Y2 ) ) ) ) ) ).
% tan_mono_le
thf(fact_3291_tan__bound__pi2,axiom,
! [X2: real] :
( ( ord_less @ real @ ( abs_abs @ real @ X2 ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ ( bit0 @ one2 ) ) ) ) )
=> ( ord_less @ real @ ( abs_abs @ real @ ( tan @ real @ X2 ) ) @ ( one_one @ real ) ) ) ).
% tan_bound_pi2
thf(fact_3292_arctan,axiom,
! [Y2: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ ( arctan @ Y2 ) )
& ( ord_less @ real @ ( arctan @ Y2 ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
& ( ( tan @ real @ ( arctan @ Y2 ) )
= Y2 ) ) ).
% arctan
thf(fact_3293_arctan__tan,axiom,
! [X2: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X2 )
=> ( ( ord_less @ real @ X2 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( arctan @ ( tan @ real @ X2 ) )
= X2 ) ) ) ).
% arctan_tan
thf(fact_3294_arctan__unique,axiom,
! [X2: real,Y2: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X2 )
=> ( ( ord_less @ real @ X2 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ( tan @ real @ X2 )
= Y2 )
=> ( ( arctan @ Y2 )
= X2 ) ) ) ) ).
% arctan_unique
thf(fact_3295_sum__split__even__odd,axiom,
! [F2: nat > real,G: nat > real,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I3: nat] : ( if @ real @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I3 ) @ ( F2 @ I3 ) @ ( G @ I3 ) )
@ ( set_ord_lessThan @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I3: nat] : ( F2 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I3 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I3: nat] : ( G @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I3 ) @ ( one_one @ nat ) ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% sum_split_even_odd
thf(fact_3296_lemma__tan__add1,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X2: A,Y2: A] :
( ( ( cos @ A @ X2 )
!= ( zero_zero @ A ) )
=> ( ( ( cos @ A @ Y2 )
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( tan @ A @ X2 ) @ ( tan @ A @ Y2 ) ) )
= ( divide_divide @ A @ ( cos @ A @ ( plus_plus @ A @ X2 @ Y2 ) ) @ ( times_times @ A @ ( cos @ A @ X2 ) @ ( cos @ A @ Y2 ) ) ) ) ) ) ) ).
% lemma_tan_add1
thf(fact_3297_tan__diff,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X2: A,Y2: A] :
( ( ( cos @ A @ X2 )
!= ( zero_zero @ A ) )
=> ( ( ( cos @ A @ Y2 )
!= ( zero_zero @ A ) )
=> ( ( ( cos @ A @ ( minus_minus @ A @ X2 @ Y2 ) )
!= ( zero_zero @ A ) )
=> ( ( tan @ A @ ( minus_minus @ A @ X2 @ Y2 ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( tan @ A @ X2 ) @ ( tan @ A @ Y2 ) ) @ ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( tan @ A @ X2 ) @ ( tan @ A @ Y2 ) ) ) ) ) ) ) ) ) ).
% tan_diff
thf(fact_3298_tan__add,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X2: A,Y2: A] :
( ( ( cos @ A @ X2 )
!= ( zero_zero @ A ) )
=> ( ( ( cos @ A @ Y2 )
!= ( zero_zero @ A ) )
=> ( ( ( cos @ A @ ( plus_plus @ A @ X2 @ Y2 ) )
!= ( zero_zero @ A ) )
=> ( ( tan @ A @ ( plus_plus @ A @ X2 @ Y2 ) )
= ( divide_divide @ A @ ( plus_plus @ A @ ( tan @ A @ X2 ) @ ( tan @ A @ Y2 ) ) @ ( minus_minus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( tan @ A @ X2 ) @ ( tan @ A @ Y2 ) ) ) ) ) ) ) ) ) ).
% tan_add
thf(fact_3299_tan__total__pi4,axiom,
! [X2: real] :
( ( ord_less @ real @ ( abs_abs @ real @ X2 ) @ ( one_one @ real ) )
=> ? [Z2: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) @ Z2 )
& ( ord_less @ real @ Z2 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ ( bit0 @ one2 ) ) ) ) )
& ( ( tan @ real @ Z2 )
= X2 ) ) ) ).
% tan_total_pi4
thf(fact_3300_Sum__Icc__int,axiom,
! [M: int,N: int] :
( ( ord_less_eq @ int @ M @ N )
=> ( ( groups7311177749621191930dd_sum @ int @ int
@ ^ [X: int] : X
@ ( set_or1337092689740270186AtMost @ int @ M @ N ) )
= ( divide_divide @ int @ ( minus_minus @ int @ ( times_times @ int @ N @ ( plus_plus @ int @ N @ ( one_one @ int ) ) ) @ ( times_times @ int @ M @ ( minus_minus @ int @ M @ ( one_one @ int ) ) ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ).
% Sum_Icc_int
thf(fact_3301_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_3302_tan__half,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ( ( tan @ A )
= ( ^ [X: A] : ( divide_divide @ A @ ( sin @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X ) ) @ ( plus_plus @ A @ ( cos @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X ) ) @ ( one_one @ A ) ) ) ) ) ) ).
% tan_half
thf(fact_3303_sum__bounds__lt__plus1,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [F2: nat > A,Mm: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K2: nat] : ( F2 @ ( suc @ K2 ) )
@ ( set_ord_lessThan @ nat @ Mm ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_or1337092689740270186AtMost @ nat @ ( one_one @ nat ) @ Mm ) ) ) ) ).
% sum_bounds_lt_plus1
thf(fact_3304_sumr__cos__zero__one,axiom,
! [N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M5: nat] : ( times_times @ real @ ( cos_coeff @ M5 ) @ ( power_power @ real @ ( zero_zero @ real ) @ M5 ) )
@ ( set_ord_lessThan @ nat @ ( suc @ N ) ) )
= ( one_one @ real ) ) ).
% sumr_cos_zero_one
thf(fact_3305_diffs__equiv,axiom,
! [A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( ring_1 @ A ) )
=> ! [C2: nat > A,X2: A] :
( ( summable @ A
@ ^ [N3: nat] : ( times_times @ A @ ( diffs @ A @ C2 @ N3 ) @ ( power_power @ A @ X2 @ N3 ) ) )
=> ( sums @ A
@ ^ [N3: nat] : ( times_times @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N3 ) @ ( C2 @ N3 ) ) @ ( power_power @ A @ X2 @ ( minus_minus @ nat @ N3 @ ( suc @ ( zero_zero @ nat ) ) ) ) )
@ ( suminf @ A
@ ^ [N3: nat] : ( times_times @ A @ ( diffs @ A @ C2 @ N3 ) @ ( power_power @ A @ X2 @ N3 ) ) ) ) ) ) ).
% diffs_equiv
thf(fact_3306_sin__paired,axiom,
! [X2: 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 @ X2 @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) @ ( one_one @ nat ) ) ) )
@ ( sin @ real @ X2 ) ) ).
% sin_paired
thf(fact_3307_sin__tan,axiom,
! [X2: real] :
( ( ord_less @ real @ ( abs_abs @ real @ X2 ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( sin @ real @ X2 )
= ( divide_divide @ real @ ( tan @ real @ X2 ) @ ( sqrt @ ( plus_plus @ real @ ( one_one @ real ) @ ( power_power @ real @ ( tan @ real @ X2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% sin_tan
thf(fact_3308_real__sqrt__eq__iff,axiom,
! [X2: real,Y2: real] :
( ( ( sqrt @ X2 )
= ( sqrt @ Y2 ) )
= ( X2 = Y2 ) ) ).
% real_sqrt_eq_iff
thf(fact_3309_of__nat__fact,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [N: nat] :
( ( semiring_1_of_nat @ A @ ( semiring_char_0_fact @ nat @ N ) )
= ( semiring_char_0_fact @ A @ N ) ) ) ).
% of_nat_fact
thf(fact_3310_real__sqrt__eq__zero__cancel__iff,axiom,
! [X2: real] :
( ( ( sqrt @ X2 )
= ( zero_zero @ real ) )
= ( X2
= ( zero_zero @ real ) ) ) ).
% real_sqrt_eq_zero_cancel_iff
thf(fact_3311_real__sqrt__zero,axiom,
( ( sqrt @ ( zero_zero @ real ) )
= ( zero_zero @ real ) ) ).
% real_sqrt_zero
thf(fact_3312_real__sqrt__less__iff,axiom,
! [X2: real,Y2: real] :
( ( ord_less @ real @ ( sqrt @ X2 ) @ ( sqrt @ Y2 ) )
= ( ord_less @ real @ X2 @ Y2 ) ) ).
% real_sqrt_less_iff
thf(fact_3313_real__sqrt__le__iff,axiom,
! [X2: real,Y2: real] :
( ( ord_less_eq @ real @ ( sqrt @ X2 ) @ ( sqrt @ Y2 ) )
= ( ord_less_eq @ real @ X2 @ Y2 ) ) ).
% real_sqrt_le_iff
thf(fact_3314_real__sqrt__eq__1__iff,axiom,
! [X2: real] :
( ( ( sqrt @ X2 )
= ( one_one @ real ) )
= ( X2
= ( one_one @ real ) ) ) ).
% real_sqrt_eq_1_iff
thf(fact_3315_real__sqrt__one,axiom,
( ( sqrt @ ( one_one @ real ) )
= ( one_one @ real ) ) ).
% real_sqrt_one
thf(fact_3316_fact__0,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ( ( semiring_char_0_fact @ A @ ( zero_zero @ nat ) )
= ( one_one @ A ) ) ) ).
% fact_0
thf(fact_3317_fact__1,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ( ( semiring_char_0_fact @ A @ ( one_one @ nat ) )
= ( one_one @ A ) ) ) ).
% fact_1
thf(fact_3318_real__sqrt__lt__0__iff,axiom,
! [X2: real] :
( ( ord_less @ real @ ( sqrt @ X2 ) @ ( zero_zero @ real ) )
= ( ord_less @ real @ X2 @ ( zero_zero @ real ) ) ) ).
% real_sqrt_lt_0_iff
thf(fact_3319_real__sqrt__gt__0__iff,axiom,
! [Y2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( sqrt @ Y2 ) )
= ( ord_less @ real @ ( zero_zero @ real ) @ Y2 ) ) ).
% real_sqrt_gt_0_iff
thf(fact_3320_real__sqrt__ge__0__iff,axiom,
! [Y2: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( sqrt @ Y2 ) )
= ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y2 ) ) ).
% real_sqrt_ge_0_iff
thf(fact_3321_real__sqrt__le__0__iff,axiom,
! [X2: real] :
( ( ord_less_eq @ real @ ( sqrt @ X2 ) @ ( zero_zero @ real ) )
= ( ord_less_eq @ real @ X2 @ ( zero_zero @ real ) ) ) ).
% real_sqrt_le_0_iff
thf(fact_3322_real__sqrt__lt__1__iff,axiom,
! [X2: real] :
( ( ord_less @ real @ ( sqrt @ X2 ) @ ( one_one @ real ) )
= ( ord_less @ real @ X2 @ ( one_one @ real ) ) ) ).
% real_sqrt_lt_1_iff
thf(fact_3323_real__sqrt__gt__1__iff,axiom,
! [Y2: real] :
( ( ord_less @ real @ ( one_one @ real ) @ ( sqrt @ Y2 ) )
= ( ord_less @ real @ ( one_one @ real ) @ Y2 ) ) ).
% real_sqrt_gt_1_iff
thf(fact_3324_real__sqrt__ge__1__iff,axiom,
! [Y2: real] :
( ( ord_less_eq @ real @ ( one_one @ real ) @ ( sqrt @ Y2 ) )
= ( ord_less_eq @ real @ ( one_one @ real ) @ Y2 ) ) ).
% real_sqrt_ge_1_iff
thf(fact_3325_real__sqrt__le__1__iff,axiom,
! [X2: real] :
( ( ord_less_eq @ real @ ( sqrt @ X2 ) @ ( one_one @ real ) )
= ( ord_less_eq @ real @ X2 @ ( one_one @ real ) ) ) ).
% real_sqrt_le_1_iff
thf(fact_3326_real__sqrt__mult__self,axiom,
! [A2: real] :
( ( times_times @ real @ ( sqrt @ A2 ) @ ( sqrt @ A2 ) )
= ( abs_abs @ real @ A2 ) ) ).
% real_sqrt_mult_self
thf(fact_3327_real__sqrt__abs2,axiom,
! [X2: real] :
( ( sqrt @ ( times_times @ real @ X2 @ X2 ) )
= ( abs_abs @ real @ X2 ) ) ).
% real_sqrt_abs2
thf(fact_3328_cos__coeff__0,axiom,
( ( cos_coeff @ ( zero_zero @ nat ) )
= ( one_one @ real ) ) ).
% cos_coeff_0
thf(fact_3329_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_3330_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_3331_real__sqrt__four,axiom,
( ( sqrt @ ( numeral_numeral @ real @ ( bit0 @ ( bit0 @ one2 ) ) ) )
= ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ).
% real_sqrt_four
thf(fact_3332_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_3333_real__sqrt__abs,axiom,
! [X2: real] :
( ( sqrt @ ( power_power @ real @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( abs_abs @ real @ X2 ) ) ).
% real_sqrt_abs
thf(fact_3334_real__sqrt__pow2__iff,axiom,
! [X2: real] :
( ( ( power_power @ real @ ( sqrt @ X2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= X2 )
= ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 ) ) ).
% real_sqrt_pow2_iff
thf(fact_3335_real__sqrt__pow2,axiom,
! [X2: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( power_power @ real @ ( sqrt @ X2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= X2 ) ) ).
% real_sqrt_pow2
thf(fact_3336_real__sqrt__sum__squares__mult__squared__eq,axiom,
! [X2: real,Y2: real,Xa: real,Ya: real] :
( ( power_power @ real @ ( sqrt @ ( times_times @ real @ ( plus_plus @ real @ ( power_power @ real @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( plus_plus @ real @ ( power_power @ real @ Xa @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Ya @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( times_times @ real @ ( plus_plus @ real @ ( power_power @ real @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y2 @ ( 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_3337_diffs__of__real,axiom,
! [A: $tType] :
( ( real_V2191834092415804123ebra_1 @ A )
=> ! [F2: nat > real] :
( ( diffs @ A
@ ^ [N3: nat] : ( real_Vector_of_real @ A @ ( F2 @ N3 ) ) )
= ( ^ [N3: nat] : ( real_Vector_of_real @ A @ ( diffs @ real @ F2 @ N3 ) ) ) ) ) ).
% diffs_of_real
thf(fact_3338_real__sqrt__minus,axiom,
! [X2: real] :
( ( sqrt @ ( uminus_uminus @ real @ X2 ) )
= ( uminus_uminus @ real @ ( sqrt @ X2 ) ) ) ).
% real_sqrt_minus
thf(fact_3339_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_3340_real__sqrt__less__mono,axiom,
! [X2: real,Y2: real] :
( ( ord_less @ real @ X2 @ Y2 )
=> ( ord_less @ real @ ( sqrt @ X2 ) @ ( sqrt @ Y2 ) ) ) ).
% real_sqrt_less_mono
thf(fact_3341_real__sqrt__le__mono,axiom,
! [X2: real,Y2: real] :
( ( ord_less_eq @ real @ X2 @ Y2 )
=> ( ord_less_eq @ real @ ( sqrt @ X2 ) @ ( sqrt @ Y2 ) ) ) ).
% real_sqrt_le_mono
thf(fact_3342_real__sqrt__power,axiom,
! [X2: real,K: nat] :
( ( sqrt @ ( power_power @ real @ X2 @ K ) )
= ( power_power @ real @ ( sqrt @ X2 ) @ K ) ) ).
% real_sqrt_power
thf(fact_3343_real__sqrt__mult,axiom,
! [X2: real,Y2: real] :
( ( sqrt @ ( times_times @ real @ X2 @ Y2 ) )
= ( times_times @ real @ ( sqrt @ X2 ) @ ( sqrt @ Y2 ) ) ) ).
% real_sqrt_mult
thf(fact_3344_real__sqrt__divide,axiom,
! [X2: real,Y2: real] :
( ( sqrt @ ( divide_divide @ real @ X2 @ Y2 ) )
= ( divide_divide @ real @ ( sqrt @ X2 ) @ ( sqrt @ Y2 ) ) ) ).
% real_sqrt_divide
thf(fact_3345_real__sqrt__gt__zero,axiom,
! [X2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( sqrt @ X2 ) ) ) ).
% real_sqrt_gt_zero
thf(fact_3346_real__sqrt__ge__zero,axiom,
! [X2: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
=> ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( sqrt @ X2 ) ) ) ).
% real_sqrt_ge_zero
thf(fact_3347_real__sqrt__eq__zero__cancel,axiom,
! [X2: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ( sqrt @ X2 )
= ( zero_zero @ real ) )
=> ( X2
= ( zero_zero @ real ) ) ) ) ).
% real_sqrt_eq_zero_cancel
thf(fact_3348_real__sqrt__ge__one,axiom,
! [X2: real] :
( ( ord_less_eq @ real @ ( one_one @ real ) @ X2 )
=> ( ord_less_eq @ real @ ( one_one @ real ) @ ( sqrt @ X2 ) ) ) ).
% real_sqrt_ge_one
thf(fact_3349_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_3350_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_3351_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_3352_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_3353_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_3354_fact__mono,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ord_less_eq @ A @ ( semiring_char_0_fact @ A @ M ) @ ( semiring_char_0_fact @ A @ N ) ) ) ) ).
% fact_mono
thf(fact_3355_diffs__minus,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [C2: nat > A] :
( ( diffs @ A
@ ^ [N3: nat] : ( uminus_uminus @ A @ ( C2 @ N3 ) ) )
= ( ^ [N3: nat] : ( uminus_uminus @ A @ ( diffs @ A @ C2 @ N3 ) ) ) ) ) ).
% diffs_minus
thf(fact_3356_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_3357_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_3358_real__div__sqrt,axiom,
! [X2: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( divide_divide @ real @ X2 @ ( sqrt @ X2 ) )
= ( sqrt @ X2 ) ) ) ).
% real_div_sqrt
thf(fact_3359_sqrt__add__le__add__sqrt,axiom,
! [X2: real,Y2: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y2 )
=> ( ord_less_eq @ real @ ( sqrt @ ( plus_plus @ real @ X2 @ Y2 ) ) @ ( plus_plus @ real @ ( sqrt @ X2 ) @ ( sqrt @ Y2 ) ) ) ) ) ).
% sqrt_add_le_add_sqrt
thf(fact_3360_le__real__sqrt__sumsq,axiom,
! [X2: real,Y2: real] : ( ord_less_eq @ real @ X2 @ ( sqrt @ ( plus_plus @ real @ ( times_times @ real @ X2 @ X2 ) @ ( times_times @ real @ Y2 @ Y2 ) ) ) ) ).
% le_real_sqrt_sumsq
thf(fact_3361_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_3362_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_3363_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_3364_sqrt2__less__2,axiom,
ord_less @ real @ ( sqrt @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ).
% sqrt2_less_2
thf(fact_3365_diffs__def,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ( ( diffs @ A )
= ( ^ [C3: nat > A,N3: nat] : ( times_times @ A @ ( semiring_1_of_nat @ A @ ( suc @ N3 ) ) @ ( C3 @ ( suc @ N3 ) ) ) ) ) ) ).
% diffs_def
thf(fact_3366_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_3367_real__less__rsqrt,axiom,
! [X2: real,Y2: real] :
( ( ord_less @ real @ ( power_power @ real @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ Y2 )
=> ( ord_less @ real @ X2 @ ( sqrt @ Y2 ) ) ) ).
% real_less_rsqrt
thf(fact_3368_real__le__rsqrt,axiom,
! [X2: real,Y2: real] :
( ( ord_less_eq @ real @ ( power_power @ real @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ Y2 )
=> ( ord_less_eq @ real @ X2 @ ( sqrt @ Y2 ) ) ) ).
% real_le_rsqrt
thf(fact_3369_sqrt__le__D,axiom,
! [X2: real,Y2: real] :
( ( ord_less_eq @ real @ ( sqrt @ X2 ) @ Y2 )
=> ( ord_less_eq @ real @ X2 @ ( power_power @ real @ Y2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% sqrt_le_D
thf(fact_3370_termdiff__converges__all,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [C2: nat > A,X2: 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 @ X2 @ N3 ) ) ) ) ) ).
% termdiff_converges_all
thf(fact_3371_tan__60,axiom,
( ( tan @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit1 @ one2 ) ) ) )
= ( sqrt @ ( numeral_numeral @ real @ ( bit1 @ one2 ) ) ) ) ).
% tan_60
thf(fact_3372_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_3373_real__sqrt__unique,axiom,
! [Y2: real,X2: real] :
( ( ( power_power @ real @ Y2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= X2 )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y2 )
=> ( ( sqrt @ X2 )
= Y2 ) ) ) ).
% real_sqrt_unique
thf(fact_3374_real__le__lsqrt,axiom,
! [X2: real,Y2: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y2 )
=> ( ( ord_less_eq @ real @ X2 @ ( power_power @ real @ Y2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
=> ( ord_less_eq @ real @ ( sqrt @ X2 ) @ Y2 ) ) ) ) ).
% real_le_lsqrt
thf(fact_3375_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_3376_real__sqrt__sum__squares__eq__cancel2,axiom,
! [X2: real,Y2: real] :
( ( ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
= Y2 )
=> ( X2
= ( zero_zero @ real ) ) ) ).
% real_sqrt_sum_squares_eq_cancel2
thf(fact_3377_real__sqrt__sum__squares__eq__cancel,axiom,
! [X2: real,Y2: real] :
( ( ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
= X2 )
=> ( Y2
= ( zero_zero @ real ) ) ) ).
% real_sqrt_sum_squares_eq_cancel
thf(fact_3378_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_3379_real__sqrt__sum__squares__ge2,axiom,
! [Y2: real,X2: real] : ( ord_less_eq @ real @ Y2 @ ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% real_sqrt_sum_squares_ge2
thf(fact_3380_real__sqrt__sum__squares__ge1,axiom,
! [X2: real,Y2: real] : ( ord_less_eq @ real @ X2 @ ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% real_sqrt_sum_squares_ge1
thf(fact_3381_sqrt__ge__absD,axiom,
! [X2: real,Y2: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ X2 ) @ ( sqrt @ Y2 ) )
=> ( ord_less_eq @ real @ ( power_power @ real @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ Y2 ) ) ).
% sqrt_ge_absD
thf(fact_3382_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_3383_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_3384_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_3385_fact__num__eq__if,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ( ( semiring_char_0_fact @ A )
= ( ^ [M5: nat] :
( if @ A
@ ( M5
= ( zero_zero @ nat ) )
@ ( one_one @ A )
@ ( times_times @ A @ ( semiring_1_of_nat @ A @ M5 ) @ ( semiring_char_0_fact @ A @ ( minus_minus @ nat @ M5 @ ( one_one @ nat ) ) ) ) ) ) ) ) ).
% fact_num_eq_if
thf(fact_3386_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_3387_real__less__lsqrt,axiom,
! [X2: real,Y2: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y2 )
=> ( ( ord_less @ real @ X2 @ ( power_power @ real @ Y2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
=> ( ord_less @ real @ ( sqrt @ X2 ) @ Y2 ) ) ) ) ).
% real_less_lsqrt
thf(fact_3388_sqrt__sum__squares__le__sum,axiom,
! [X2: real,Y2: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y2 )
=> ( ord_less_eq @ real @ ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( plus_plus @ real @ X2 @ Y2 ) ) ) ) ).
% sqrt_sum_squares_le_sum
thf(fact_3389_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_3390_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_3391_real__sqrt__ge__abs1,axiom,
! [X2: real,Y2: real] : ( ord_less_eq @ real @ ( abs_abs @ real @ X2 ) @ ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% real_sqrt_ge_abs1
thf(fact_3392_real__sqrt__ge__abs2,axiom,
! [Y2: real,X2: real] : ( ord_less_eq @ real @ ( abs_abs @ real @ Y2 ) @ ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% real_sqrt_ge_abs2
thf(fact_3393_sqrt__sum__squares__le__sum__abs,axiom,
! [X2: real,Y2: real] : ( ord_less_eq @ real @ ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( plus_plus @ real @ ( abs_abs @ real @ X2 ) @ ( abs_abs @ real @ Y2 ) ) ) ).
% sqrt_sum_squares_le_sum_abs
thf(fact_3394_ln__sqrt,axiom,
! [X2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ln_ln @ real @ ( sqrt @ X2 ) )
= ( divide_divide @ real @ ( ln_ln @ real @ X2 ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ).
% ln_sqrt
thf(fact_3395_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_3396_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_3397_arsinh__real__def,axiom,
( ( arsinh @ real )
= ( ^ [X: real] : ( ln_ln @ real @ ( plus_plus @ real @ X @ ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ real ) ) ) ) ) ) ) ).
% arsinh_real_def
thf(fact_3398_complex__norm,axiom,
! [X2: real,Y2: real] :
( ( real_V7770717601297561774m_norm @ complex @ ( complex2 @ X2 @ Y2 ) )
= ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% complex_norm
thf(fact_3399_Maclaurin__zero,axiom,
! [A: $tType] :
( ( zero @ A )
=> ! [X2: real,N: nat,Diff: nat > A > real] :
( ( X2
= ( zero_zero @ real ) )
=> ( ( N
!= ( zero_zero @ nat ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M5: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M5 @ ( zero_zero @ A ) ) @ ( semiring_char_0_fact @ real @ M5 ) ) @ ( power_power @ real @ X2 @ M5 ) )
@ ( set_ord_lessThan @ nat @ N ) )
= ( Diff @ ( zero_zero @ nat ) @ ( zero_zero @ A ) ) ) ) ) ) ).
% Maclaurin_zero
thf(fact_3400_Maclaurin__lemma,axiom,
! [H2: real,F2: real > real,J: nat > real,N: nat] :
( ( ord_less @ real @ ( zero_zero @ real ) @ H2 )
=> ? [B9: real] :
( ( F2 @ H2 )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M5: nat] : ( times_times @ real @ ( divide_divide @ real @ ( J @ M5 ) @ ( semiring_char_0_fact @ real @ M5 ) ) @ ( power_power @ real @ H2 @ M5 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ B9 @ ( divide_divide @ real @ ( power_power @ real @ H2 @ N ) @ ( semiring_char_0_fact @ real @ N ) ) ) ) ) ) ).
% Maclaurin_lemma
thf(fact_3401_termdiff__converges,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X2: A,K6: real,C2: nat > A] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X2 ) @ 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 @ X2 @ N3 ) ) ) ) ) ) ).
% termdiff_converges
thf(fact_3402_real__sqrt__power__even,axiom,
! [N: nat,X2: real] :
( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( power_power @ real @ ( sqrt @ X2 ) @ N )
= ( power_power @ real @ X2 @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% real_sqrt_power_even
thf(fact_3403_arsinh__real__aux,axiom,
! [X2: real] : ( ord_less @ real @ ( zero_zero @ real ) @ ( plus_plus @ real @ X2 @ ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ real ) ) ) ) ) ).
% arsinh_real_aux
thf(fact_3404_real__sqrt__sum__squares__mult__ge__zero,axiom,
! [X2: real,Y2: real,Xa: real,Ya: real] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( sqrt @ ( times_times @ real @ ( plus_plus @ real @ ( power_power @ real @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y2 @ ( 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_3405_arith__geo__mean__sqrt,axiom,
! [X2: real,Y2: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y2 )
=> ( ord_less_eq @ real @ ( sqrt @ ( times_times @ real @ X2 @ Y2 ) ) @ ( divide_divide @ real @ ( plus_plus @ real @ X2 @ Y2 ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ).
% arith_geo_mean_sqrt
thf(fact_3406_powr__half__sqrt,axiom,
! [X2: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( powr @ real @ X2 @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
= ( sqrt @ X2 ) ) ) ).
% powr_half_sqrt
thf(fact_3407_cos__x__y__le__one,axiom,
! [X2: real,Y2: real] : ( ord_less_eq @ real @ ( abs_abs @ real @ ( divide_divide @ real @ X2 @ ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) @ ( one_one @ real ) ) ).
% cos_x_y_le_one
thf(fact_3408_real__sqrt__sum__squares__less,axiom,
! [X2: real,U: real,Y2: real] :
( ( ord_less @ real @ ( abs_abs @ real @ X2 ) @ ( divide_divide @ real @ U @ ( sqrt @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) )
=> ( ( ord_less @ real @ ( abs_abs @ real @ Y2 ) @ ( divide_divide @ real @ U @ ( sqrt @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) )
=> ( ord_less @ real @ ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ U ) ) ) ).
% real_sqrt_sum_squares_less
thf(fact_3409_Maclaurin__cos__expansion,axiom,
! [X2: real,N: nat] :
? [T5: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ T5 ) @ ( abs_abs @ real @ X2 ) )
& ( ( cos @ real @ X2 )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M5: nat] : ( times_times @ real @ ( cos_coeff @ M5 ) @ ( power_power @ real @ X2 @ M5 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( cos @ real @ ( plus_plus @ real @ T5 @ ( times_times @ real @ ( times_times @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( semiring_1_of_nat @ real @ N ) ) @ pi ) ) ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ X2 @ N ) ) ) ) ) ).
% Maclaurin_cos_expansion
thf(fact_3410_arcosh__real__def,axiom,
! [X2: real] :
( ( ord_less_eq @ real @ ( one_one @ real ) @ X2 )
=> ( ( arcosh @ real @ X2 )
= ( ln_ln @ real @ ( plus_plus @ real @ X2 @ ( sqrt @ ( minus_minus @ real @ ( power_power @ real @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ real ) ) ) ) ) ) ) ).
% arcosh_real_def
thf(fact_3411_cos__arctan,axiom,
! [X2: real] :
( ( cos @ real @ ( arctan @ X2 ) )
= ( divide_divide @ real @ ( one_one @ real ) @ ( sqrt @ ( plus_plus @ real @ ( one_one @ real ) @ ( power_power @ real @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% cos_arctan
thf(fact_3412_sin__arctan,axiom,
! [X2: real] :
( ( sin @ real @ ( arctan @ X2 ) )
= ( divide_divide @ real @ X2 @ ( sqrt @ ( plus_plus @ real @ ( one_one @ real ) @ ( power_power @ real @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% sin_arctan
thf(fact_3413_sqrt__sum__squares__half__less,axiom,
! [X2: real,U: real,Y2: real] :
( ( ord_less @ real @ X2 @ ( divide_divide @ real @ U @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less @ real @ Y2 @ ( divide_divide @ real @ U @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y2 )
=> ( ord_less @ real @ ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ U ) ) ) ) ) ).
% sqrt_sum_squares_half_less
thf(fact_3414_sin__cos__sqrt,axiom,
! [X2: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( sin @ real @ X2 ) )
=> ( ( sin @ real @ X2 )
= ( sqrt @ ( minus_minus @ real @ ( one_one @ real ) @ ( power_power @ real @ ( cos @ real @ X2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% sin_cos_sqrt
thf(fact_3415_arctan__half,axiom,
( arctan
= ( ^ [X: real] : ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( arctan @ ( divide_divide @ real @ X @ ( plus_plus @ real @ ( one_one @ real ) @ ( sqrt @ ( plus_plus @ real @ ( one_one @ real ) @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ).
% arctan_half
thf(fact_3416_Maclaurin__minus__cos__expansion,axiom,
! [N: nat,X2: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ X2 @ ( zero_zero @ real ) )
=> ? [T5: real] :
( ( ord_less @ real @ X2 @ T5 )
& ( ord_less @ real @ T5 @ ( zero_zero @ real ) )
& ( ( cos @ real @ X2 )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M5: nat] : ( times_times @ real @ ( cos_coeff @ M5 ) @ ( power_power @ real @ X2 @ M5 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( cos @ real @ ( plus_plus @ real @ T5 @ ( times_times @ real @ ( times_times @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( semiring_1_of_nat @ real @ N ) ) @ pi ) ) ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ X2 @ N ) ) ) ) ) ) ) ).
% Maclaurin_minus_cos_expansion
thf(fact_3417_Maclaurin__cos__expansion2,axiom,
! [X2: real,N: nat] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ? [T5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ T5 )
& ( ord_less @ real @ T5 @ X2 )
& ( ( cos @ real @ X2 )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M5: nat] : ( times_times @ real @ ( cos_coeff @ M5 ) @ ( power_power @ real @ X2 @ M5 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( cos @ real @ ( plus_plus @ real @ T5 @ ( times_times @ real @ ( times_times @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( semiring_1_of_nat @ real @ N ) ) @ pi ) ) ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ X2 @ N ) ) ) ) ) ) ) ).
% Maclaurin_cos_expansion2
thf(fact_3418_cos__paired,axiom,
! [X2: 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 @ X2 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) ) )
@ ( cos @ real @ X2 ) ) ).
% cos_paired
thf(fact_3419_cos__tan,axiom,
! [X2: real] :
( ( ord_less @ real @ ( abs_abs @ real @ X2 ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( cos @ real @ X2 )
= ( divide_divide @ real @ ( one_one @ real ) @ ( sqrt @ ( plus_plus @ real @ ( one_one @ real ) @ ( power_power @ real @ ( tan @ real @ X2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% cos_tan
thf(fact_3420_Maclaurin__sin__expansion3,axiom,
! [N: nat,X2: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ? [T5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ T5 )
& ( ord_less @ real @ T5 @ X2 )
& ( ( sin @ real @ X2 )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M5: nat] : ( times_times @ real @ ( sin_coeff @ M5 ) @ ( power_power @ real @ X2 @ M5 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( sin @ real @ ( plus_plus @ real @ T5 @ ( times_times @ real @ ( times_times @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( semiring_1_of_nat @ real @ N ) ) @ pi ) ) ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ X2 @ N ) ) ) ) ) ) ) ).
% Maclaurin_sin_expansion3
thf(fact_3421_Maclaurin__sin__expansion4,axiom,
! [X2: real,N: nat] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ? [T5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ T5 )
& ( ord_less_eq @ real @ T5 @ X2 )
& ( ( sin @ real @ X2 )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M5: nat] : ( times_times @ real @ ( sin_coeff @ M5 ) @ ( power_power @ real @ X2 @ M5 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( sin @ real @ ( plus_plus @ real @ T5 @ ( times_times @ real @ ( times_times @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( semiring_1_of_nat @ real @ N ) ) @ pi ) ) ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ X2 @ N ) ) ) ) ) ) ).
% Maclaurin_sin_expansion4
thf(fact_3422_Maclaurin__sin__expansion2,axiom,
! [X2: real,N: nat] :
? [T5: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ T5 ) @ ( abs_abs @ real @ X2 ) )
& ( ( sin @ real @ X2 )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M5: nat] : ( times_times @ real @ ( sin_coeff @ M5 ) @ ( power_power @ real @ X2 @ M5 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( sin @ real @ ( plus_plus @ real @ T5 @ ( times_times @ real @ ( times_times @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( semiring_1_of_nat @ real @ N ) ) @ pi ) ) ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ X2 @ N ) ) ) ) ) ).
% Maclaurin_sin_expansion2
thf(fact_3423_Maclaurin__sin__expansion,axiom,
! [X2: real,N: nat] :
? [T5: real] :
( ( sin @ real @ X2 )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M5: nat] : ( times_times @ real @ ( sin_coeff @ M5 ) @ ( power_power @ real @ X2 @ M5 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( sin @ real @ ( plus_plus @ real @ T5 @ ( times_times @ real @ ( times_times @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( semiring_1_of_nat @ real @ N ) ) @ pi ) ) ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ X2 @ N ) ) ) ) ).
% Maclaurin_sin_expansion
thf(fact_3424_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_3425_sin__coeff__0,axiom,
( ( sin_coeff @ ( zero_zero @ nat ) )
= ( zero_zero @ real ) ) ).
% sin_coeff_0
thf(fact_3426_fact__ge__self,axiom,
! [N: nat] : ( ord_less_eq @ nat @ N @ ( semiring_char_0_fact @ nat @ N ) ) ).
% fact_ge_self
thf(fact_3427_fact__mono__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ord_less_eq @ nat @ ( semiring_char_0_fact @ nat @ M ) @ ( semiring_char_0_fact @ nat @ N ) ) ) ).
% fact_mono_nat
thf(fact_3428_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_3429_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_3430_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_3431_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_3432_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_3433_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_3434_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_3435_Maclaurin__exp__lt,axiom,
! [X2: real,N: nat] :
( ( X2
!= ( zero_zero @ real ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ? [T5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( abs_abs @ real @ T5 ) )
& ( ord_less @ real @ ( abs_abs @ real @ T5 ) @ ( abs_abs @ real @ X2 ) )
& ( ( exp @ real @ X2 )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M5: nat] : ( divide_divide @ real @ ( power_power @ real @ X2 @ M5 ) @ ( semiring_char_0_fact @ real @ M5 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( exp @ real @ T5 ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ X2 @ N ) ) ) ) ) ) ) ).
% Maclaurin_exp_lt
thf(fact_3436_cos__arcsin,axiom,
! [X2: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X2 )
=> ( ( ord_less_eq @ real @ X2 @ ( one_one @ real ) )
=> ( ( cos @ real @ ( arcsin @ X2 ) )
= ( sqrt @ ( minus_minus @ real @ ( one_one @ real ) @ ( power_power @ real @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% cos_arcsin
thf(fact_3437_sin__arccos__abs,axiom,
! [Y2: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ Y2 ) @ ( one_one @ real ) )
=> ( ( sin @ real @ ( arccos @ Y2 ) )
= ( sqrt @ ( minus_minus @ real @ ( one_one @ real ) @ ( power_power @ real @ Y2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% sin_arccos_abs
thf(fact_3438_sin__arccos,axiom,
! [X2: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X2 )
=> ( ( ord_less_eq @ real @ X2 @ ( one_one @ real ) )
=> ( ( sin @ real @ ( arccos @ X2 ) )
= ( sqrt @ ( minus_minus @ real @ ( one_one @ real ) @ ( power_power @ real @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% sin_arccos
thf(fact_3439_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_3440_exp__le__cancel__iff,axiom,
! [X2: real,Y2: real] :
( ( ord_less_eq @ real @ ( exp @ real @ X2 ) @ ( exp @ real @ Y2 ) )
= ( ord_less_eq @ real @ X2 @ Y2 ) ) ).
% exp_le_cancel_iff
thf(fact_3441_arcsin__0,axiom,
( ( arcsin @ ( zero_zero @ real ) )
= ( zero_zero @ real ) ) ).
% arcsin_0
thf(fact_3442_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_3443_exp__eq__one__iff,axiom,
! [X2: real] :
( ( ( exp @ real @ X2 )
= ( one_one @ real ) )
= ( X2
= ( zero_zero @ real ) ) ) ).
% exp_eq_one_iff
thf(fact_3444_arccos__1,axiom,
( ( arccos @ ( one_one @ real ) )
= ( zero_zero @ real ) ) ).
% arccos_1
thf(fact_3445_exp__less__one__iff,axiom,
! [X2: real] :
( ( ord_less @ real @ ( exp @ real @ X2 ) @ ( one_one @ real ) )
= ( ord_less @ real @ X2 @ ( zero_zero @ real ) ) ) ).
% exp_less_one_iff
thf(fact_3446_one__less__exp__iff,axiom,
! [X2: real] :
( ( ord_less @ real @ ( one_one @ real ) @ ( exp @ real @ X2 ) )
= ( ord_less @ real @ ( zero_zero @ real ) @ X2 ) ) ).
% one_less_exp_iff
thf(fact_3447_one__le__exp__iff,axiom,
! [X2: real] :
( ( ord_less_eq @ real @ ( one_one @ real ) @ ( exp @ real @ X2 ) )
= ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 ) ) ).
% one_le_exp_iff
thf(fact_3448_exp__le__one__iff,axiom,
! [X2: real] :
( ( ord_less_eq @ real @ ( exp @ real @ X2 ) @ ( one_one @ real ) )
= ( ord_less_eq @ real @ X2 @ ( zero_zero @ real ) ) ) ).
% exp_le_one_iff
thf(fact_3449_exp__ln,axiom,
! [X2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( exp @ real @ ( ln_ln @ real @ X2 ) )
= X2 ) ) ).
% exp_ln
thf(fact_3450_exp__ln__iff,axiom,
! [X2: real] :
( ( ( exp @ real @ ( ln_ln @ real @ X2 ) )
= X2 )
= ( ord_less @ real @ ( zero_zero @ real ) @ X2 ) ) ).
% exp_ln_iff
thf(fact_3451_arccos__minus__1,axiom,
( ( arccos @ ( uminus_uminus @ real @ ( one_one @ real ) ) )
= pi ) ).
% arccos_minus_1
thf(fact_3452_cos__arccos,axiom,
! [Y2: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ Y2 )
=> ( ( ord_less_eq @ real @ Y2 @ ( one_one @ real ) )
=> ( ( cos @ real @ ( arccos @ Y2 ) )
= Y2 ) ) ) ).
% cos_arccos
thf(fact_3453_sin__arcsin,axiom,
! [Y2: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ Y2 )
=> ( ( ord_less_eq @ real @ Y2 @ ( one_one @ real ) )
=> ( ( sin @ real @ ( arcsin @ Y2 ) )
= Y2 ) ) ) ).
% sin_arcsin
thf(fact_3454_arccos__0,axiom,
( ( arccos @ ( zero_zero @ real ) )
= ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ).
% arccos_0
thf(fact_3455_arcsin__1,axiom,
( ( arcsin @ ( one_one @ real ) )
= ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ).
% arcsin_1
thf(fact_3456_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_3457_norm__exp,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X2: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( exp @ A @ X2 ) ) @ ( exp @ real @ ( real_V7770717601297561774m_norm @ A @ X2 ) ) ) ) ).
% norm_exp
thf(fact_3458_exp__not__eq__zero,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X2: A] :
( ( exp @ A @ X2 )
!= ( zero_zero @ A ) ) ) ).
% exp_not_eq_zero
thf(fact_3459_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_3460_not__exp__less__zero,axiom,
! [X2: real] :
~ ( ord_less @ real @ ( exp @ real @ X2 ) @ ( zero_zero @ real ) ) ).
% not_exp_less_zero
thf(fact_3461_exp__gt__zero,axiom,
! [X2: real] : ( ord_less @ real @ ( zero_zero @ real ) @ ( exp @ real @ X2 ) ) ).
% exp_gt_zero
thf(fact_3462_exp__total,axiom,
! [Y2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ Y2 )
=> ? [X3: real] :
( ( exp @ real @ X3 )
= Y2 ) ) ).
% exp_total
thf(fact_3463_exp__ge__zero,axiom,
! [X2: real] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( exp @ real @ X2 ) ) ).
% exp_ge_zero
thf(fact_3464_not__exp__le__zero,axiom,
! [X2: real] :
~ ( ord_less_eq @ real @ ( exp @ real @ X2 ) @ ( zero_zero @ real ) ) ).
% not_exp_le_zero
thf(fact_3465_mult__exp__exp,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X2: A,Y2: A] :
( ( times_times @ A @ ( exp @ A @ X2 ) @ ( exp @ A @ Y2 ) )
= ( exp @ A @ ( plus_plus @ A @ X2 @ Y2 ) ) ) ) ).
% mult_exp_exp
thf(fact_3466_exp__add__commuting,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X2: A,Y2: A] :
( ( ( times_times @ A @ X2 @ Y2 )
= ( times_times @ A @ Y2 @ X2 ) )
=> ( ( exp @ A @ ( plus_plus @ A @ X2 @ Y2 ) )
= ( times_times @ A @ ( exp @ A @ X2 ) @ ( exp @ A @ Y2 ) ) ) ) ) ).
% exp_add_commuting
thf(fact_3467_exp__diff,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X2: A,Y2: A] :
( ( exp @ A @ ( minus_minus @ A @ X2 @ Y2 ) )
= ( divide_divide @ A @ ( exp @ A @ X2 ) @ ( exp @ A @ Y2 ) ) ) ) ).
% exp_diff
thf(fact_3468_exp__gt__one,axiom,
! [X2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( ord_less @ real @ ( one_one @ real ) @ ( exp @ real @ X2 ) ) ) ).
% exp_gt_one
thf(fact_3469_exp__ge__add__one__self,axiom,
! [X2: real] : ( ord_less_eq @ real @ ( plus_plus @ real @ ( one_one @ real ) @ X2 ) @ ( exp @ real @ X2 ) ) ).
% exp_ge_add_one_self
thf(fact_3470_exp__minus__inverse,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X2: A] :
( ( times_times @ A @ ( exp @ A @ X2 ) @ ( exp @ A @ ( uminus_uminus @ A @ X2 ) ) )
= ( one_one @ A ) ) ) ).
% exp_minus_inverse
thf(fact_3471_exp__of__nat2__mult,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X2: A,N: nat] :
( ( exp @ A @ ( times_times @ A @ X2 @ ( semiring_1_of_nat @ A @ N ) ) )
= ( power_power @ A @ ( exp @ A @ X2 ) @ N ) ) ) ).
% exp_of_nat2_mult
thf(fact_3472_exp__of__nat__mult,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [N: nat,X2: A] :
( ( exp @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ X2 ) )
= ( power_power @ A @ ( exp @ A @ X2 ) @ N ) ) ) ).
% exp_of_nat_mult
thf(fact_3473_log__ln,axiom,
( ( ln_ln @ real )
= ( log @ ( exp @ real @ ( one_one @ real ) ) ) ) ).
% log_ln
thf(fact_3474_arccos__le__arccos,axiom,
! [X2: real,Y2: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X2 )
=> ( ( ord_less_eq @ real @ X2 @ Y2 )
=> ( ( ord_less_eq @ real @ Y2 @ ( one_one @ real ) )
=> ( ord_less_eq @ real @ ( arccos @ Y2 ) @ ( arccos @ X2 ) ) ) ) ) ).
% arccos_le_arccos
thf(fact_3475_arccos__le__mono,axiom,
! [X2: real,Y2: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ X2 ) @ ( one_one @ real ) )
=> ( ( ord_less_eq @ real @ ( abs_abs @ real @ Y2 ) @ ( one_one @ real ) )
=> ( ( ord_less_eq @ real @ ( arccos @ X2 ) @ ( arccos @ Y2 ) )
= ( ord_less_eq @ real @ Y2 @ X2 ) ) ) ) ).
% arccos_le_mono
thf(fact_3476_arccos__eq__iff,axiom,
! [X2: real,Y2: real] :
( ( ( ord_less_eq @ real @ ( abs_abs @ real @ X2 ) @ ( one_one @ real ) )
& ( ord_less_eq @ real @ ( abs_abs @ real @ Y2 ) @ ( one_one @ real ) ) )
=> ( ( ( arccos @ X2 )
= ( arccos @ Y2 ) )
= ( X2 = Y2 ) ) ) ).
% arccos_eq_iff
thf(fact_3477_arcsin__le__arcsin,axiom,
! [X2: real,Y2: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X2 )
=> ( ( ord_less_eq @ real @ X2 @ Y2 )
=> ( ( ord_less_eq @ real @ Y2 @ ( one_one @ real ) )
=> ( ord_less_eq @ real @ ( arcsin @ X2 ) @ ( arcsin @ Y2 ) ) ) ) ) ).
% arcsin_le_arcsin
thf(fact_3478_arcsin__minus,axiom,
! [X2: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X2 )
=> ( ( ord_less_eq @ real @ X2 @ ( one_one @ real ) )
=> ( ( arcsin @ ( uminus_uminus @ real @ X2 ) )
= ( uminus_uminus @ real @ ( arcsin @ X2 ) ) ) ) ) ).
% arcsin_minus
thf(fact_3479_arcsin__le__mono,axiom,
! [X2: real,Y2: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ X2 ) @ ( one_one @ real ) )
=> ( ( ord_less_eq @ real @ ( abs_abs @ real @ Y2 ) @ ( one_one @ real ) )
=> ( ( ord_less_eq @ real @ ( arcsin @ X2 ) @ ( arcsin @ Y2 ) )
= ( ord_less_eq @ real @ X2 @ Y2 ) ) ) ) ).
% arcsin_le_mono
thf(fact_3480_arcsin__eq__iff,axiom,
! [X2: real,Y2: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ X2 ) @ ( one_one @ real ) )
=> ( ( ord_less_eq @ real @ ( abs_abs @ real @ Y2 ) @ ( one_one @ real ) )
=> ( ( ( arcsin @ X2 )
= ( arcsin @ Y2 ) )
= ( X2 = Y2 ) ) ) ) ).
% arcsin_eq_iff
thf(fact_3481_exp__ge__add__one__self__aux,axiom,
! [X2: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
=> ( ord_less_eq @ real @ ( plus_plus @ real @ ( one_one @ real ) @ X2 ) @ ( exp @ real @ X2 ) ) ) ).
% exp_ge_add_one_self_aux
thf(fact_3482_lemma__exp__total,axiom,
! [Y2: real] :
( ( ord_less_eq @ real @ ( one_one @ real ) @ Y2 )
=> ? [X3: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 )
& ( ord_less_eq @ real @ X3 @ ( minus_minus @ real @ Y2 @ ( one_one @ real ) ) )
& ( ( exp @ real @ X3 )
= Y2 ) ) ) ).
% lemma_exp_total
thf(fact_3483_ln__ge__iff,axiom,
! [X2: real,Y2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ord_less_eq @ real @ Y2 @ ( ln_ln @ real @ X2 ) )
= ( ord_less_eq @ real @ ( exp @ real @ Y2 ) @ X2 ) ) ) ).
% ln_ge_iff
thf(fact_3484_ln__x__over__x__mono,axiom,
! [X2: real,Y2: real] :
( ( ord_less_eq @ real @ ( exp @ real @ ( one_one @ real ) ) @ X2 )
=> ( ( ord_less_eq @ real @ X2 @ Y2 )
=> ( ord_less_eq @ real @ ( divide_divide @ real @ ( ln_ln @ real @ Y2 ) @ Y2 ) @ ( divide_divide @ real @ ( ln_ln @ real @ X2 ) @ X2 ) ) ) ) ).
% ln_x_over_x_mono
thf(fact_3485_powr__def,axiom,
! [A: $tType] :
( ( ln @ A )
=> ( ( powr @ A )
= ( ^ [X: A,A4: A] :
( if @ A
@ ( X
= ( zero_zero @ A ) )
@ ( zero_zero @ A )
@ ( exp @ A @ ( times_times @ A @ A4 @ ( ln_ln @ A @ X ) ) ) ) ) ) ) ).
% powr_def
thf(fact_3486_arccos__lbound,axiom,
! [Y2: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ Y2 )
=> ( ( ord_less_eq @ real @ Y2 @ ( one_one @ real ) )
=> ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( arccos @ Y2 ) ) ) ) ).
% arccos_lbound
thf(fact_3487_arccos__less__arccos,axiom,
! [X2: real,Y2: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X2 )
=> ( ( ord_less @ real @ X2 @ Y2 )
=> ( ( ord_less_eq @ real @ Y2 @ ( one_one @ real ) )
=> ( ord_less @ real @ ( arccos @ Y2 ) @ ( arccos @ X2 ) ) ) ) ) ).
% arccos_less_arccos
thf(fact_3488_arccos__less__mono,axiom,
! [X2: real,Y2: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ X2 ) @ ( one_one @ real ) )
=> ( ( ord_less_eq @ real @ ( abs_abs @ real @ Y2 ) @ ( one_one @ real ) )
=> ( ( ord_less @ real @ ( arccos @ X2 ) @ ( arccos @ Y2 ) )
= ( ord_less @ real @ Y2 @ X2 ) ) ) ) ).
% arccos_less_mono
thf(fact_3489_exp__le,axiom,
ord_less_eq @ real @ ( exp @ real @ ( one_one @ real ) ) @ ( numeral_numeral @ real @ ( bit1 @ one2 ) ) ).
% exp_le
thf(fact_3490_arccos__ubound,axiom,
! [Y2: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ Y2 )
=> ( ( ord_less_eq @ real @ Y2 @ ( one_one @ real ) )
=> ( ord_less_eq @ real @ ( arccos @ Y2 ) @ pi ) ) ) ).
% arccos_ubound
thf(fact_3491_arccos__cos,axiom,
! [X2: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ord_less_eq @ real @ X2 @ pi )
=> ( ( arccos @ ( cos @ real @ X2 ) )
= X2 ) ) ) ).
% arccos_cos
thf(fact_3492_arcsin__less__arcsin,axiom,
! [X2: real,Y2: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X2 )
=> ( ( ord_less @ real @ X2 @ Y2 )
=> ( ( ord_less_eq @ real @ Y2 @ ( one_one @ real ) )
=> ( ord_less @ real @ ( arcsin @ X2 ) @ ( arcsin @ Y2 ) ) ) ) ) ).
% arcsin_less_arcsin
thf(fact_3493_arcsin__less__mono,axiom,
! [X2: real,Y2: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ X2 ) @ ( one_one @ real ) )
=> ( ( ord_less_eq @ real @ ( abs_abs @ real @ Y2 ) @ ( one_one @ real ) )
=> ( ( ord_less @ real @ ( arcsin @ X2 ) @ ( arcsin @ Y2 ) )
= ( ord_less @ real @ X2 @ Y2 ) ) ) ) ).
% arcsin_less_mono
thf(fact_3494_cos__arccos__abs,axiom,
! [Y2: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ Y2 ) @ ( one_one @ real ) )
=> ( ( cos @ real @ ( arccos @ Y2 ) )
= Y2 ) ) ).
% cos_arccos_abs
thf(fact_3495_arccos__cos__eq__abs,axiom,
! [Theta: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ Theta ) @ pi )
=> ( ( arccos @ ( cos @ real @ Theta ) )
= ( abs_abs @ real @ Theta ) ) ) ).
% arccos_cos_eq_abs
thf(fact_3496_exp__divide__power__eq,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [N: nat,X2: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( power_power @ A @ ( exp @ A @ ( divide_divide @ A @ X2 @ ( semiring_1_of_nat @ A @ N ) ) ) @ N )
= ( exp @ A @ X2 ) ) ) ) ).
% exp_divide_power_eq
thf(fact_3497_tanh__altdef,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ( ( tanh @ A )
= ( ^ [X: A] : ( divide_divide @ A @ ( minus_minus @ A @ ( exp @ A @ X ) @ ( exp @ A @ ( uminus_uminus @ A @ X ) ) ) @ ( plus_plus @ A @ ( exp @ A @ X ) @ ( exp @ A @ ( uminus_uminus @ A @ X ) ) ) ) ) ) ) ).
% tanh_altdef
thf(fact_3498_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_3499_arccos__lt__bounded,axiom,
! [Y2: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ Y2 )
=> ( ( ord_less @ real @ Y2 @ ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( arccos @ Y2 ) )
& ( ord_less @ real @ ( arccos @ Y2 ) @ pi ) ) ) ) ).
% arccos_lt_bounded
thf(fact_3500_arccos__bounded,axiom,
! [Y2: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ Y2 )
=> ( ( ord_less_eq @ real @ Y2 @ ( one_one @ real ) )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( arccos @ Y2 ) )
& ( ord_less_eq @ real @ ( arccos @ Y2 ) @ pi ) ) ) ) ).
% arccos_bounded
thf(fact_3501_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_3502_sin__arccos__nonzero,axiom,
! [X2: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X2 )
=> ( ( ord_less @ real @ X2 @ ( one_one @ real ) )
=> ( ( sin @ real @ ( arccos @ X2 ) )
!= ( zero_zero @ real ) ) ) ) ).
% sin_arccos_nonzero
thf(fact_3503_arccos__cos2,axiom,
! [X2: real] :
( ( ord_less_eq @ real @ X2 @ ( zero_zero @ real ) )
=> ( ( ord_less_eq @ real @ ( uminus_uminus @ real @ pi ) @ X2 )
=> ( ( arccos @ ( cos @ real @ X2 ) )
= ( uminus_uminus @ real @ X2 ) ) ) ) ).
% arccos_cos2
thf(fact_3504_arccos__minus,axiom,
! [X2: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X2 )
=> ( ( ord_less_eq @ real @ X2 @ ( one_one @ real ) )
=> ( ( arccos @ ( uminus_uminus @ real @ X2 ) )
= ( minus_minus @ real @ pi @ ( arccos @ X2 ) ) ) ) ) ).
% arccos_minus
thf(fact_3505_cos__arcsin__nonzero,axiom,
! [X2: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X2 )
=> ( ( ord_less @ real @ X2 @ ( one_one @ real ) )
=> ( ( cos @ real @ ( arcsin @ X2 ) )
!= ( zero_zero @ real ) ) ) ) ).
% cos_arcsin_nonzero
thf(fact_3506_arccos,axiom,
! [Y2: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ Y2 )
=> ( ( ord_less_eq @ real @ Y2 @ ( one_one @ real ) )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( arccos @ Y2 ) )
& ( ord_less_eq @ real @ ( arccos @ Y2 ) @ pi )
& ( ( cos @ real @ ( arccos @ Y2 ) )
= Y2 ) ) ) ) ).
% arccos
thf(fact_3507_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_3508_arccos__minus__abs,axiom,
! [X2: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ X2 ) @ ( one_one @ real ) )
=> ( ( arccos @ ( uminus_uminus @ real @ X2 ) )
= ( minus_minus @ real @ pi @ ( arccos @ X2 ) ) ) ) ).
% arccos_minus_abs
thf(fact_3509_VEBT__internal_Oheight_Ocases,axiom,
! [X2: vEBT_VEBT] :
( ! [A5: $o,B4: $o] :
( X2
!= ( vEBT_Leaf @ A5 @ B4 ) )
=> ~ ! [Uu2: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( X2
!= ( vEBT_Node @ Uu2 @ Deg2 @ TreeList2 @ Summary2 ) ) ) ).
% VEBT_internal.height.cases
thf(fact_3510_exp__bound,axiom,
! [X2: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ord_less_eq @ real @ X2 @ ( one_one @ real ) )
=> ( ord_less_eq @ real @ ( exp @ real @ X2 ) @ ( plus_plus @ real @ ( plus_plus @ real @ ( one_one @ real ) @ X2 ) @ ( power_power @ real @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% exp_bound
thf(fact_3511_real__exp__bound__lemma,axiom,
! [X2: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ord_less_eq @ real @ X2 @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less_eq @ real @ ( exp @ real @ X2 ) @ ( plus_plus @ real @ ( one_one @ real ) @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ X2 ) ) ) ) ) ).
% real_exp_bound_lemma
thf(fact_3512_exp__ge__one__plus__x__over__n__power__n,axiom,
! [N: nat,X2: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( semiring_1_of_nat @ real @ N ) ) @ X2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ord_less_eq @ real @ ( power_power @ real @ ( plus_plus @ real @ ( one_one @ real ) @ ( divide_divide @ real @ X2 @ ( semiring_1_of_nat @ real @ N ) ) ) @ N ) @ ( exp @ real @ X2 ) ) ) ) ).
% exp_ge_one_plus_x_over_n_power_n
thf(fact_3513_exp__ge__one__minus__x__over__n__power__n,axiom,
! [X2: real,N: nat] :
( ( ord_less_eq @ real @ X2 @ ( semiring_1_of_nat @ real @ N ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ord_less_eq @ real @ ( power_power @ real @ ( minus_minus @ real @ ( one_one @ real ) @ ( divide_divide @ real @ X2 @ ( semiring_1_of_nat @ real @ N ) ) ) @ N ) @ ( exp @ real @ ( uminus_uminus @ real @ X2 ) ) ) ) ) ).
% exp_ge_one_minus_x_over_n_power_n
thf(fact_3514_arccos__le__pi2,axiom,
! [Y2: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y2 )
=> ( ( ord_less_eq @ real @ Y2 @ ( one_one @ real ) )
=> ( ord_less_eq @ real @ ( arccos @ Y2 ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ).
% arccos_le_pi2
thf(fact_3515_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_3516_arcsin__lt__bounded,axiom,
! [Y2: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ Y2 )
=> ( ( ord_less @ real @ Y2 @ ( one_one @ real ) )
=> ( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ ( arcsin @ Y2 ) )
& ( ord_less @ real @ ( arcsin @ Y2 ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% arcsin_lt_bounded
thf(fact_3517_arcsin__lbound,axiom,
! [Y2: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ Y2 )
=> ( ( ord_less_eq @ real @ Y2 @ ( one_one @ real ) )
=> ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ ( arcsin @ Y2 ) ) ) ) ).
% arcsin_lbound
thf(fact_3518_arcsin__ubound,axiom,
! [Y2: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ Y2 )
=> ( ( ord_less_eq @ real @ Y2 @ ( one_one @ real ) )
=> ( ord_less_eq @ real @ ( arcsin @ Y2 ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ).
% arcsin_ubound
thf(fact_3519_arcsin__bounded,axiom,
! [Y2: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ Y2 )
=> ( ( ord_less_eq @ real @ Y2 @ ( one_one @ real ) )
=> ( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ ( arcsin @ Y2 ) )
& ( ord_less_eq @ real @ ( arcsin @ Y2 ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% arcsin_bounded
thf(fact_3520_Maclaurin__exp__le,axiom,
! [X2: real,N: nat] :
? [T5: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ T5 ) @ ( abs_abs @ real @ X2 ) )
& ( ( exp @ real @ X2 )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M5: nat] : ( divide_divide @ real @ ( power_power @ real @ X2 @ M5 ) @ ( semiring_char_0_fact @ real @ M5 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( exp @ real @ T5 ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ X2 @ N ) ) ) ) ) ).
% Maclaurin_exp_le
thf(fact_3521_exp__lower__Taylor__quadratic,axiom,
! [X2: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
=> ( ord_less_eq @ real @ ( plus_plus @ real @ ( plus_plus @ real @ ( one_one @ real ) @ X2 ) @ ( divide_divide @ real @ ( power_power @ real @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ ( exp @ real @ X2 ) ) ) ).
% exp_lower_Taylor_quadratic
thf(fact_3522_arcsin__sin,axiom,
! [X2: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X2 )
=> ( ( ord_less_eq @ real @ X2 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( arcsin @ ( sin @ real @ X2 ) )
= X2 ) ) ) ).
% arcsin_sin
thf(fact_3523_log__base__10__eq2,axiom,
! [X2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( log @ ( numeral_numeral @ real @ ( bit0 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) @ X2 )
= ( times_times @ real @ ( log @ ( numeral_numeral @ real @ ( bit0 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) @ ( exp @ real @ ( one_one @ real ) ) ) @ ( ln_ln @ real @ X2 ) ) ) ) ).
% log_base_10_eq2
thf(fact_3524_tanh__real__altdef,axiom,
( ( tanh @ real )
= ( ^ [X: real] : ( divide_divide @ real @ ( minus_minus @ real @ ( one_one @ real ) @ ( exp @ real @ ( times_times @ real @ ( uminus_uminus @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ X ) ) ) @ ( plus_plus @ real @ ( one_one @ real ) @ ( exp @ real @ ( times_times @ real @ ( uminus_uminus @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ X ) ) ) ) ) ) ).
% tanh_real_altdef
thf(fact_3525_log__base__10__eq1,axiom,
! [X2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( log @ ( numeral_numeral @ real @ ( bit0 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) @ X2 )
= ( times_times @ real @ ( divide_divide @ real @ ( ln_ln @ real @ ( exp @ real @ ( one_one @ real ) ) ) @ ( ln_ln @ real @ ( numeral_numeral @ real @ ( bit0 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) ) ) @ ( ln_ln @ real @ X2 ) ) ) ) ).
% log_base_10_eq1
thf(fact_3526_arcsin,axiom,
! [Y2: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ Y2 )
=> ( ( ord_less_eq @ real @ Y2 @ ( one_one @ real ) )
=> ( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ ( arcsin @ Y2 ) )
& ( ord_less_eq @ real @ ( arcsin @ Y2 ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
& ( ( sin @ real @ ( arcsin @ Y2 ) )
= Y2 ) ) ) ) ).
% arcsin
thf(fact_3527_arcsin__pi,axiom,
! [Y2: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ Y2 )
=> ( ( ord_less_eq @ real @ Y2 @ ( one_one @ real ) )
=> ( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ ( arcsin @ Y2 ) )
& ( ord_less_eq @ real @ ( arcsin @ Y2 ) @ pi )
& ( ( sin @ real @ ( arcsin @ Y2 ) )
= Y2 ) ) ) ) ).
% arcsin_pi
thf(fact_3528_arcsin__le__iff,axiom,
! [X2: real,Y2: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X2 )
=> ( ( ord_less_eq @ real @ X2 @ ( one_one @ real ) )
=> ( ( ord_less_eq @ real @ ( divide_divide @ real @ ( uminus_uminus @ real @ pi ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ Y2 )
=> ( ( ord_less_eq @ real @ Y2 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less_eq @ real @ ( arcsin @ X2 ) @ Y2 )
= ( ord_less_eq @ real @ X2 @ ( sin @ real @ Y2 ) ) ) ) ) ) ) ).
% arcsin_le_iff
thf(fact_3529_le__arcsin__iff,axiom,
! [X2: real,Y2: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X2 )
=> ( ( ord_less_eq @ real @ X2 @ ( one_one @ real ) )
=> ( ( ord_less_eq @ real @ ( divide_divide @ real @ ( uminus_uminus @ real @ pi ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ Y2 )
=> ( ( ord_less_eq @ real @ Y2 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less_eq @ real @ Y2 @ ( arcsin @ X2 ) )
= ( ord_less_eq @ real @ ( sin @ real @ Y2 ) @ X2 ) ) ) ) ) ) ).
% le_arcsin_iff
thf(fact_3530_arccos__cos__eq__abs__2pi,axiom,
! [Theta: real] :
~ ! [K3: int] :
( ( arccos @ ( cos @ real @ Theta ) )
!= ( abs_abs @ real @ ( minus_minus @ real @ Theta @ ( times_times @ real @ ( ring_1_of_int @ real @ K3 ) @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) ) ) ) ) ).
% arccos_cos_eq_abs_2pi
thf(fact_3531_monoseq__def,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( topological_monoseq @ A )
= ( ^ [X7: nat > A] :
( ! [M5: nat,N3: nat] :
( ( ord_less_eq @ nat @ M5 @ N3 )
=> ( ord_less_eq @ A @ ( X7 @ M5 ) @ ( X7 @ N3 ) ) )
| ! [M5: nat,N3: nat] :
( ( ord_less_eq @ nat @ M5 @ N3 )
=> ( ord_less_eq @ A @ ( X7 @ N3 ) @ ( X7 @ M5 ) ) ) ) ) ) ) ).
% monoseq_def
thf(fact_3532_monoI2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X6: nat > A] :
( ! [M3: nat,N2: nat] :
( ( ord_less_eq @ nat @ M3 @ N2 )
=> ( ord_less_eq @ A @ ( X6 @ N2 ) @ ( X6 @ M3 ) ) )
=> ( topological_monoseq @ A @ X6 ) ) ) ).
% monoI2
thf(fact_3533_monoI1,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X6: nat > A] :
( ! [M3: nat,N2: nat] :
( ( ord_less_eq @ nat @ M3 @ N2 )
=> ( ord_less_eq @ A @ ( X6 @ M3 ) @ ( X6 @ N2 ) ) )
=> ( topological_monoseq @ A @ X6 ) ) ) ).
% monoI1
thf(fact_3534_monoseq__Suc,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( topological_monoseq @ A )
= ( ^ [X7: nat > A] :
( ! [N3: nat] : ( ord_less_eq @ A @ ( X7 @ N3 ) @ ( X7 @ ( suc @ N3 ) ) )
| ! [N3: nat] : ( ord_less_eq @ A @ ( X7 @ ( suc @ N3 ) ) @ ( X7 @ N3 ) ) ) ) ) ) ).
% monoseq_Suc
thf(fact_3535_mono__SucI2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X6: nat > A] :
( ! [N2: nat] : ( ord_less_eq @ A @ ( X6 @ ( suc @ N2 ) ) @ ( X6 @ N2 ) )
=> ( topological_monoseq @ A @ X6 ) ) ) ).
% mono_SucI2
thf(fact_3536_complex__exp__exists,axiom,
! [Z: complex] :
? [A5: complex,R3: real] :
( Z
= ( times_times @ complex @ ( real_Vector_of_real @ complex @ R3 ) @ ( exp @ complex @ A5 ) ) ) ).
% complex_exp_exists
thf(fact_3537_monoseq__minus,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: nat > A] :
( ( topological_monoseq @ A @ A2 )
=> ( topological_monoseq @ A
@ ^ [N3: nat] : ( uminus_uminus @ A @ ( A2 @ N3 ) ) ) ) ) ).
% monoseq_minus
thf(fact_3538_mono__SucI1,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X6: nat > A] :
( ! [N2: nat] : ( ord_less_eq @ A @ ( X6 @ N2 ) @ ( X6 @ ( suc @ N2 ) ) )
=> ( topological_monoseq @ A @ X6 ) ) ) ).
% mono_SucI1
thf(fact_3539_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_3540_of__nat__code,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ( ( semiring_1_of_nat @ A )
= ( ^ [N3: nat] :
( semiri8178284476397505188at_aux @ A
@ ^ [I3: A] : ( plus_plus @ A @ I3 @ ( one_one @ A ) )
@ N3
@ ( zero_zero @ A ) ) ) ) ) ).
% of_nat_code
thf(fact_3541_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_3542_Maclaurin__sin__bound,axiom,
! [X2: real,N: nat] :
( ord_less_eq @ real
@ ( abs_abs @ real
@ ( minus_minus @ real @ ( sin @ real @ X2 )
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M5: nat] : ( times_times @ real @ ( sin_coeff @ M5 ) @ ( power_power @ real @ X2 @ M5 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) )
@ ( times_times @ real @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ ( abs_abs @ real @ X2 ) @ N ) ) ) ).
% Maclaurin_sin_bound
thf(fact_3543_gchoose__row__sum__weighted,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [R2: A,M: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K2: nat] : ( times_times @ A @ ( gbinomial @ A @ R2 @ K2 ) @ ( minus_minus @ A @ ( divide_divide @ A @ R2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( semiring_1_of_nat @ A @ K2 ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ M ) )
= ( times_times @ A @ ( divide_divide @ A @ ( semiring_1_of_nat @ A @ ( suc @ M ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( gbinomial @ A @ R2 @ ( suc @ M ) ) ) ) ) ).
% gchoose_row_sum_weighted
thf(fact_3544_inverse__inverse__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A] :
( ( inverse_inverse @ A @ ( inverse_inverse @ A @ A2 ) )
= A2 ) ) ).
% inverse_inverse_eq
thf(fact_3545_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_3546_inverse__zero,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ( ( inverse_inverse @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% inverse_zero
thf(fact_3547_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_3548_inverse__eq__1__iff,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [X2: A] :
( ( ( inverse_inverse @ A @ X2 )
= ( one_one @ A ) )
= ( X2
= ( one_one @ A ) ) ) ) ).
% inverse_eq_1_iff
thf(fact_3549_inverse__1,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ( ( inverse_inverse @ A @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% inverse_1
thf(fact_3550_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_3551_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_3552_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_3553_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_3554_of__int__floor__cancel,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A] :
( ( ( ring_1_of_int @ A @ ( archim6421214686448440834_floor @ A @ X2 ) )
= X2 )
= ( ? [N3: int] :
( X2
= ( ring_1_of_int @ A @ N3 ) ) ) ) ) ).
% of_int_floor_cancel
thf(fact_3555_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_3556_pochhammer__1,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A2: A] :
( ( comm_s3205402744901411588hammer @ A @ A2 @ ( one_one @ nat ) )
= A2 ) ) ).
% pochhammer_1
thf(fact_3557_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_3558_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_3559_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_3560_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_3561_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_3562_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_3563_floor__zero,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ( ( archim6421214686448440834_floor @ A @ ( zero_zero @ A ) )
= ( zero_zero @ int ) ) ) ).
% floor_zero
thf(fact_3564_floor__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V: num] :
( ( archim6421214686448440834_floor @ A @ ( numeral_numeral @ A @ V ) )
= ( numeral_numeral @ int @ V ) ) ) ).
% floor_numeral
thf(fact_3565_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_3566_floor__one,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ( ( archim6421214686448440834_floor @ A @ ( one_one @ A ) )
= ( one_one @ int ) ) ) ).
% floor_one
thf(fact_3567_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_3568_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_3569_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_3570_pochhammer__Suc0,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A2: A] :
( ( comm_s3205402744901411588hammer @ A @ A2 @ ( suc @ ( zero_zero @ nat ) ) )
= A2 ) ) ).
% pochhammer_Suc0
thf(fact_3571_floor__of__nat,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [N: nat] :
( ( archim6421214686448440834_floor @ A @ ( semiring_1_of_nat @ A @ N ) )
= ( semiring_1_of_nat @ int @ N ) ) ) ).
% floor_of_nat
thf(fact_3572_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_3573_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_3574_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_3575_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_3576_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_3577_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_3578_zero__le__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( archim6421214686448440834_floor @ A @ X2 ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ X2 ) ) ) ).
% zero_le_floor
thf(fact_3579_numeral__le__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V: num,X2: A] :
( ( ord_less_eq @ int @ ( numeral_numeral @ int @ V ) @ ( archim6421214686448440834_floor @ A @ X2 ) )
= ( ord_less_eq @ A @ ( numeral_numeral @ A @ V ) @ X2 ) ) ) ).
% numeral_le_floor
thf(fact_3580_floor__less__zero,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A] :
( ( ord_less @ int @ ( archim6421214686448440834_floor @ A @ X2 ) @ ( zero_zero @ int ) )
= ( ord_less @ A @ X2 @ ( zero_zero @ A ) ) ) ) ).
% floor_less_zero
thf(fact_3581_floor__less__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A,V: num] :
( ( ord_less @ int @ ( archim6421214686448440834_floor @ A @ X2 ) @ ( numeral_numeral @ int @ V ) )
= ( ord_less @ A @ X2 @ ( numeral_numeral @ A @ V ) ) ) ) ).
% floor_less_numeral
thf(fact_3582_zero__less__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A] :
( ( ord_less @ int @ ( zero_zero @ int ) @ ( archim6421214686448440834_floor @ A @ X2 ) )
= ( ord_less_eq @ A @ ( one_one @ A ) @ X2 ) ) ) ).
% zero_less_floor
thf(fact_3583_floor__le__zero,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A] :
( ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X2 ) @ ( zero_zero @ int ) )
= ( ord_less @ A @ X2 @ ( one_one @ A ) ) ) ) ).
% floor_le_zero
thf(fact_3584_one__le__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A] :
( ( ord_less_eq @ int @ ( one_one @ int ) @ ( archim6421214686448440834_floor @ A @ X2 ) )
= ( ord_less_eq @ A @ ( one_one @ A ) @ X2 ) ) ) ).
% one_le_floor
thf(fact_3585_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_3586_floor__less__one,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A] :
( ( ord_less @ int @ ( archim6421214686448440834_floor @ A @ X2 ) @ ( one_one @ int ) )
= ( ord_less @ A @ X2 @ ( one_one @ A ) ) ) ) ).
% floor_less_one
thf(fact_3587_floor__diff__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A,V: num] :
( ( archim6421214686448440834_floor @ A @ ( minus_minus @ A @ X2 @ ( numeral_numeral @ A @ V ) ) )
= ( minus_minus @ int @ ( archim6421214686448440834_floor @ A @ X2 ) @ ( numeral_numeral @ int @ V ) ) ) ) ).
% floor_diff_numeral
thf(fact_3588_floor__numeral__power,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: num,N: nat] :
( ( archim6421214686448440834_floor @ A @ ( power_power @ A @ ( numeral_numeral @ A @ X2 ) @ N ) )
= ( power_power @ int @ ( numeral_numeral @ int @ X2 ) @ N ) ) ) ).
% floor_numeral_power
thf(fact_3589_floor__diff__one,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A] :
( ( archim6421214686448440834_floor @ A @ ( minus_minus @ A @ X2 @ ( one_one @ A ) ) )
= ( minus_minus @ int @ ( archim6421214686448440834_floor @ A @ X2 ) @ ( one_one @ int ) ) ) ) ).
% floor_diff_one
thf(fact_3590_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_3591_numeral__less__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V: num,X2: A] :
( ( ord_less @ int @ ( numeral_numeral @ int @ V ) @ ( archim6421214686448440834_floor @ A @ X2 ) )
= ( ord_less_eq @ A @ ( plus_plus @ A @ ( numeral_numeral @ A @ V ) @ ( one_one @ A ) ) @ X2 ) ) ) ).
% numeral_less_floor
thf(fact_3592_floor__le__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A,V: num] :
( ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X2 ) @ ( numeral_numeral @ int @ V ) )
= ( ord_less @ A @ X2 @ ( plus_plus @ A @ ( numeral_numeral @ A @ V ) @ ( one_one @ A ) ) ) ) ) ).
% floor_le_numeral
thf(fact_3593_one__less__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A] :
( ( ord_less @ int @ ( one_one @ int ) @ ( archim6421214686448440834_floor @ A @ X2 ) )
= ( ord_less_eq @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X2 ) ) ) ).
% one_less_floor
thf(fact_3594_floor__le__one,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A] :
( ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X2 ) @ ( one_one @ int ) )
= ( ord_less @ A @ X2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% floor_le_one
thf(fact_3595_neg__numeral__le__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V: num,X2: A] :
( ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V ) ) @ ( archim6421214686448440834_floor @ A @ X2 ) )
= ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) @ X2 ) ) ) ).
% neg_numeral_le_floor
thf(fact_3596_floor__less__neg__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A,V: num] :
( ( ord_less @ int @ ( archim6421214686448440834_floor @ A @ X2 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V ) ) )
= ( ord_less @ A @ X2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) ) ) ) ).
% floor_less_neg_numeral
thf(fact_3597_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_3598_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_3599_neg__numeral__less__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V: num,X2: A] :
( ( ord_less @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V ) ) @ ( archim6421214686448440834_floor @ A @ X2 ) )
= ( ord_less_eq @ A @ ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) @ ( one_one @ A ) ) @ X2 ) ) ) ).
% neg_numeral_less_floor
thf(fact_3600_floor__le__neg__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A,V: num] :
( ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X2 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V ) ) )
= ( ord_less @ A @ X2 @ ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) @ ( one_one @ A ) ) ) ) ) ).
% floor_le_neg_numeral
thf(fact_3601_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_3602_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_3603_pochhammer__of__nat,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [X2: nat,N: nat] :
( ( comm_s3205402744901411588hammer @ A @ ( semiring_1_of_nat @ A @ X2 ) @ N )
= ( semiring_1_of_nat @ A @ ( comm_s3205402744901411588hammer @ nat @ X2 @ N ) ) ) ) ).
% pochhammer_of_nat
thf(fact_3604_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_3605_real__sqrt__inverse,axiom,
! [X2: real] :
( ( sqrt @ ( inverse_inverse @ real @ X2 ) )
= ( inverse_inverse @ real @ ( sqrt @ X2 ) ) ) ).
% real_sqrt_inverse
thf(fact_3606_of__nat__gbinomial,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [N: nat,K: nat] :
( ( semiring_1_of_nat @ A @ ( gbinomial @ nat @ N @ K ) )
= ( gbinomial @ A @ ( semiring_1_of_nat @ A @ N ) @ K ) ) ) ).
% of_nat_gbinomial
thf(fact_3607_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_3608_mult__commute__imp__mult__inverse__commute,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Y2: A,X2: A] :
( ( ( times_times @ A @ Y2 @ X2 )
= ( times_times @ A @ X2 @ Y2 ) )
=> ( ( times_times @ A @ ( inverse_inverse @ A @ Y2 ) @ X2 )
= ( times_times @ A @ X2 @ ( inverse_inverse @ A @ Y2 ) ) ) ) ) ).
% mult_commute_imp_mult_inverse_commute
thf(fact_3609_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_3610_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_3611_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_3612_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_3613_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_3614_nonzero__of__real__inverse,axiom,
! [A: $tType] :
( ( real_V5047593784448816457lgebra @ A )
=> ! [X2: real] :
( ( X2
!= ( zero_zero @ real ) )
=> ( ( real_Vector_of_real @ A @ ( inverse_inverse @ real @ X2 ) )
= ( inverse_inverse @ A @ ( real_Vector_of_real @ A @ X2 ) ) ) ) ) ).
% nonzero_of_real_inverse
thf(fact_3615_norm__inverse__le__norm,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [R2: real,X2: A] :
( ( ord_less_eq @ real @ R2 @ ( real_V7770717601297561774m_norm @ A @ X2 ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ R2 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( inverse_inverse @ A @ X2 ) ) @ ( inverse_inverse @ real @ R2 ) ) ) ) ) ).
% norm_inverse_le_norm
thf(fact_3616_of__int__floor__le,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A] : ( ord_less_eq @ A @ ( ring_1_of_int @ A @ ( archim6421214686448440834_floor @ A @ X2 ) ) @ X2 ) ) ).
% of_int_floor_le
thf(fact_3617_floor__mono,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A,Y2: A] :
( ( ord_less_eq @ A @ X2 @ Y2 )
=> ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X2 ) @ ( archim6421214686448440834_floor @ A @ Y2 ) ) ) ) ).
% floor_mono
thf(fact_3618_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_3619_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_3620_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_3621_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_3622_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_3623_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_3624_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_3625_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_3626_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_3627_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_3628_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_3629_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_3630_inverse__eq__divide,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ( ( inverse_inverse @ A )
= ( divide_divide @ A @ ( one_one @ A ) ) ) ) ).
% inverse_eq_divide
thf(fact_3631_divide__inverse__commute,axiom,
! [A: $tType] :
( ( field @ A )
=> ( ( divide_divide @ A )
= ( ^ [A4: A,B3: A] : ( times_times @ A @ ( inverse_inverse @ A @ B3 ) @ A4 ) ) ) ) ).
% divide_inverse_commute
thf(fact_3632_divide__inverse,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ( ( divide_divide @ A )
= ( ^ [A4: A,B3: A] : ( times_times @ A @ A4 @ ( inverse_inverse @ A @ B3 ) ) ) ) ) ).
% divide_inverse
thf(fact_3633_field__class_Ofield__divide__inverse,axiom,
! [A: $tType] :
( ( field @ A )
=> ( ( divide_divide @ A )
= ( ^ [A4: A,B3: A] : ( times_times @ A @ A4 @ ( inverse_inverse @ A @ B3 ) ) ) ) ) ).
% field_class.field_divide_inverse
thf(fact_3634_power__mult__power__inverse__commute,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X2: A,M: nat,N: nat] :
( ( times_times @ A @ ( power_power @ A @ X2 @ M ) @ ( power_power @ A @ ( inverse_inverse @ A @ X2 ) @ N ) )
= ( times_times @ A @ ( power_power @ A @ ( inverse_inverse @ A @ X2 ) @ N ) @ ( power_power @ A @ X2 @ M ) ) ) ) ).
% power_mult_power_inverse_commute
thf(fact_3635_power__mult__inverse__distrib,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X2: A,M: nat] :
( ( times_times @ A @ ( power_power @ A @ X2 @ M ) @ ( inverse_inverse @ A @ X2 ) )
= ( times_times @ A @ ( inverse_inverse @ A @ X2 ) @ ( power_power @ A @ X2 @ M ) ) ) ) ).
% power_mult_inverse_distrib
thf(fact_3636_mult__inverse__of__nat__commute,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Xa: nat,X2: A] :
( ( times_times @ A @ ( inverse_inverse @ A @ ( semiring_1_of_nat @ A @ Xa ) ) @ X2 )
= ( times_times @ A @ X2 @ ( inverse_inverse @ A @ ( semiring_1_of_nat @ A @ Xa ) ) ) ) ) ).
% mult_inverse_of_nat_commute
thf(fact_3637_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_3638_mult__inverse__of__int__commute,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Xa: int,X2: A] :
( ( times_times @ A @ ( inverse_inverse @ A @ ( ring_1_of_int @ A @ Xa ) ) @ X2 )
= ( times_times @ A @ X2 @ ( inverse_inverse @ A @ ( ring_1_of_int @ A @ Xa ) ) ) ) ) ).
% mult_inverse_of_int_commute
thf(fact_3639_divide__real__def,axiom,
( ( divide_divide @ real )
= ( ^ [X: real,Y: real] : ( times_times @ real @ X @ ( inverse_inverse @ real @ Y ) ) ) ) ).
% divide_real_def
thf(fact_3640_pochhammer__pos,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X2: A,N: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ X2 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( comm_s3205402744901411588hammer @ A @ X2 @ N ) ) ) ) ).
% pochhammer_pos
thf(fact_3641_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_3642_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_3643_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_3644_exp__fdiffs,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ( ( diffs @ A
@ ^ [N3: nat] : ( inverse_inverse @ A @ ( semiring_char_0_fact @ A @ N3 ) ) )
= ( ^ [N3: nat] : ( inverse_inverse @ A @ ( semiring_char_0_fact @ A @ N3 ) ) ) ) ) ).
% exp_fdiffs
thf(fact_3645_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_3646_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_3647_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_3648_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_3649_inverse__le__1__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X2: A] :
( ( ord_less_eq @ A @ ( inverse_inverse @ A @ X2 ) @ ( one_one @ A ) )
= ( ( ord_less_eq @ A @ X2 @ ( zero_zero @ A ) )
| ( ord_less_eq @ A @ ( one_one @ A ) @ X2 ) ) ) ) ).
% inverse_le_1_iff
thf(fact_3650_one__less__inverse__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X2: A] :
( ( ord_less @ A @ ( one_one @ A ) @ ( inverse_inverse @ A @ X2 ) )
= ( ( ord_less @ A @ ( zero_zero @ A ) @ X2 )
& ( ord_less @ A @ X2 @ ( one_one @ A ) ) ) ) ) ).
% one_less_inverse_iff
thf(fact_3651_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_3652_le__floor__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z: int,X2: A] :
( ( ord_less_eq @ int @ Z @ ( archim6421214686448440834_floor @ A @ X2 ) )
= ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Z ) @ X2 ) ) ) ).
% le_floor_iff
thf(fact_3653_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_3654_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_3655_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_3656_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_3657_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_3658_floor__add__int,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A,Z: int] :
( ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X2 ) @ Z )
= ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ X2 @ ( ring_1_of_int @ A @ Z ) ) ) ) ) ).
% floor_add_int
thf(fact_3659_int__add__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z: int,X2: A] :
( ( plus_plus @ int @ Z @ ( archim6421214686448440834_floor @ A @ X2 ) )
= ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ ( ring_1_of_int @ A @ Z ) @ X2 ) ) ) ) ).
% int_add_floor
thf(fact_3660_le__floor__add,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A,Y2: A] : ( ord_less_eq @ int @ ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X2 ) @ ( archim6421214686448440834_floor @ A @ Y2 ) ) @ ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ X2 @ Y2 ) ) ) ) ).
% le_floor_add
thf(fact_3661_floor__power,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A,N: nat] :
( ( X2
= ( ring_1_of_int @ A @ ( archim6421214686448440834_floor @ A @ X2 ) ) )
=> ( ( archim6421214686448440834_floor @ A @ ( power_power @ A @ X2 @ N ) )
= ( power_power @ int @ ( archim6421214686448440834_floor @ A @ X2 ) @ N ) ) ) ) ).
% floor_power
thf(fact_3662_floor__divide__of__int__eq,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [K: int,L2: int] :
( ( archim6421214686448440834_floor @ A @ ( divide_divide @ A @ ( ring_1_of_int @ A @ K ) @ ( ring_1_of_int @ A @ L2 ) ) )
= ( divide_divide @ int @ K @ L2 ) ) ) ).
% floor_divide_of_int_eq
thf(fact_3663_gbinomial__pochhammer,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( ( gbinomial @ A )
= ( ^ [A4: A,K2: nat] : ( divide_divide @ A @ ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ K2 ) @ ( comm_s3205402744901411588hammer @ A @ ( uminus_uminus @ A @ A4 ) @ K2 ) ) @ ( semiring_char_0_fact @ A @ K2 ) ) ) ) ) ).
% gbinomial_pochhammer
thf(fact_3664_gbinomial__pochhammer_H,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( ( gbinomial @ A )
= ( ^ [A4: A,K2: nat] : ( divide_divide @ A @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ ( minus_minus @ A @ A4 @ ( semiring_1_of_nat @ A @ K2 ) ) @ ( one_one @ A ) ) @ K2 ) @ ( semiring_char_0_fact @ A @ K2 ) ) ) ) ) ).
% gbinomial_pochhammer'
thf(fact_3665_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_3666_gbinomial__of__nat__symmetric,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K: nat,N: nat] :
( ( ord_less_eq @ nat @ K @ N )
=> ( ( gbinomial @ A @ ( semiring_1_of_nat @ A @ N ) @ K )
= ( gbinomial @ A @ ( semiring_1_of_nat @ A @ N ) @ ( minus_minus @ nat @ N @ K ) ) ) ) ) ).
% gbinomial_of_nat_symmetric
thf(fact_3667_pochhammer__nonneg,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X2: A,N: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ X2 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( comm_s3205402744901411588hammer @ A @ X2 @ N ) ) ) ) ).
% pochhammer_nonneg
thf(fact_3668_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_3669_inverse__powr,axiom,
! [Y2: real,A2: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y2 )
=> ( ( powr @ real @ ( inverse_inverse @ real @ Y2 ) @ A2 )
= ( inverse_inverse @ real @ ( powr @ real @ Y2 @ A2 ) ) ) ) ).
% inverse_powr
thf(fact_3670_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_3671_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_3672_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_3673_inverse__less__1__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X2: A] :
( ( ord_less @ A @ ( inverse_inverse @ A @ X2 ) @ ( one_one @ A ) )
= ( ( ord_less_eq @ A @ X2 @ ( zero_zero @ A ) )
| ( ord_less @ A @ ( one_one @ A ) @ X2 ) ) ) ) ).
% inverse_less_1_iff
thf(fact_3674_one__le__inverse__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X2: A] :
( ( ord_less_eq @ A @ ( one_one @ A ) @ ( inverse_inverse @ A @ X2 ) )
= ( ( ord_less @ A @ ( zero_zero @ A ) @ X2 )
& ( ord_less_eq @ A @ X2 @ ( one_one @ A ) ) ) ) ) ).
% one_le_inverse_iff
thf(fact_3675_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_3676_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_3677_one__add__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A] :
( ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X2 ) @ ( one_one @ int ) )
= ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ X2 @ ( one_one @ A ) ) ) ) ) ).
% one_add_floor
thf(fact_3678_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_3679_reals__Archimedean,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ X2 )
=> ? [N2: nat] : ( ord_less @ A @ ( inverse_inverse @ A @ ( semiring_1_of_nat @ A @ ( suc @ N2 ) ) ) @ X2 ) ) ) ).
% reals_Archimedean
thf(fact_3680_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_3681_ceiling__altdef,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ( ( archimedean_ceiling @ A )
= ( ^ [X: A] :
( if @ int
@ ( X
= ( ring_1_of_int @ A @ ( archim6421214686448440834_floor @ A @ X ) ) )
@ ( archim6421214686448440834_floor @ A @ X )
@ ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( one_one @ int ) ) ) ) ) ) ).
% ceiling_altdef
thf(fact_3682_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_3683_ceiling__diff__floor__le__1,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A] : ( ord_less_eq @ int @ ( minus_minus @ int @ ( archimedean_ceiling @ A @ X2 ) @ ( archim6421214686448440834_floor @ A @ X2 ) ) @ ( one_one @ int ) ) ) ).
% ceiling_diff_floor_le_1
thf(fact_3684_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_3685_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_3686_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_3687_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_3688_floor__eq,axiom,
! [N: int,X2: real] :
( ( ord_less @ real @ ( ring_1_of_int @ real @ N ) @ X2 )
=> ( ( ord_less @ real @ X2 @ ( plus_plus @ real @ ( ring_1_of_int @ real @ N ) @ ( one_one @ real ) ) )
=> ( ( archim6421214686448440834_floor @ real @ X2 )
= N ) ) ) ).
% floor_eq
thf(fact_3689_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_3690_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_3691_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_3692_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_3693_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_3694_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_3695_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_3696_sqrt__divide__self__eq,axiom,
! [X2: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( divide_divide @ real @ ( sqrt @ X2 ) @ X2 )
= ( inverse_inverse @ real @ ( sqrt @ X2 ) ) ) ) ).
% sqrt_divide_self_eq
thf(fact_3697_ln__inverse,axiom,
! [X2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ln_ln @ real @ ( inverse_inverse @ real @ X2 ) )
= ( uminus_uminus @ real @ ( ln_ln @ real @ X2 ) ) ) ) ).
% ln_inverse
thf(fact_3698_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_3699_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_3700_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_3701_pochhammer__eq__0__iff,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A2: A,N: nat] :
( ( ( comm_s3205402744901411588hammer @ A @ A2 @ N )
= ( zero_zero @ A ) )
= ( ? [K2: nat] :
( ( ord_less @ nat @ K2 @ N )
& ( A2
= ( uminus_uminus @ A @ ( semiring_1_of_nat @ A @ K2 ) ) ) ) ) ) ) ).
% pochhammer_eq_0_iff
thf(fact_3702_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_3703_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_3704_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_3705_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_3706_summable__exp,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X2: A] :
( summable @ A
@ ^ [N3: nat] : ( times_times @ A @ ( inverse_inverse @ A @ ( semiring_char_0_fact @ A @ N3 ) ) @ ( power_power @ A @ X2 @ N3 ) ) ) ) ).
% summable_exp
thf(fact_3707_floor__unique,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z: int,X2: A] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Z ) @ X2 )
=> ( ( ord_less @ A @ X2 @ ( plus_plus @ A @ ( ring_1_of_int @ A @ Z ) @ ( one_one @ A ) ) )
=> ( ( archim6421214686448440834_floor @ A @ X2 )
= Z ) ) ) ) ).
% floor_unique
thf(fact_3708_floor__eq__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A,A2: int] :
( ( ( archim6421214686448440834_floor @ A @ X2 )
= A2 )
= ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ A2 ) @ X2 )
& ( ord_less @ A @ X2 @ ( plus_plus @ A @ ( ring_1_of_int @ A @ A2 ) @ ( one_one @ A ) ) ) ) ) ) ).
% floor_eq_iff
thf(fact_3709_floor__split,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [P: int > $o,T2: A] :
( ( P @ ( archim6421214686448440834_floor @ A @ T2 ) )
= ( ! [I3: int] :
( ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ I3 ) @ T2 )
& ( ord_less @ A @ T2 @ ( plus_plus @ A @ ( ring_1_of_int @ A @ I3 ) @ ( one_one @ A ) ) ) )
=> ( P @ I3 ) ) ) ) ) ).
% floor_split
thf(fact_3710_less__floor__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z: int,X2: A] :
( ( ord_less @ int @ Z @ ( archim6421214686448440834_floor @ A @ X2 ) )
= ( ord_less_eq @ A @ ( plus_plus @ A @ ( ring_1_of_int @ A @ Z ) @ ( one_one @ A ) ) @ X2 ) ) ) ).
% less_floor_iff
thf(fact_3711_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_3712_floor__le__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A,Z: int] :
( ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X2 ) @ Z )
= ( ord_less @ A @ X2 @ ( plus_plus @ A @ ( ring_1_of_int @ A @ Z ) @ ( one_one @ A ) ) ) ) ) ).
% floor_le_iff
thf(fact_3713_floor__correct,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ ( archim6421214686448440834_floor @ A @ X2 ) ) @ X2 )
& ( ord_less @ A @ X2 @ ( ring_1_of_int @ A @ ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X2 ) @ ( one_one @ int ) ) ) ) ) ) ).
% floor_correct
thf(fact_3714_ex__inverse__of__nat__less,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ X2 )
=> ? [N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
& ( ord_less @ A @ ( inverse_inverse @ A @ ( semiring_1_of_nat @ A @ N2 ) ) @ X2 ) ) ) ) ).
% ex_inverse_of_nat_less
thf(fact_3715_power__diff__conv__inverse,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X2: A,M: nat,N: nat] :
( ( X2
!= ( zero_zero @ A ) )
=> ( ( ord_less_eq @ nat @ M @ N )
=> ( ( power_power @ A @ X2 @ ( minus_minus @ nat @ N @ M ) )
= ( times_times @ A @ ( power_power @ A @ X2 @ N ) @ ( power_power @ A @ ( inverse_inverse @ A @ X2 ) @ M ) ) ) ) ) ) ).
% power_diff_conv_inverse
thf(fact_3716_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_3717_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_3718_floor__eq2,axiom,
! [N: int,X2: real] :
( ( ord_less_eq @ real @ ( ring_1_of_int @ real @ N ) @ X2 )
=> ( ( ord_less @ real @ X2 @ ( plus_plus @ real @ ( ring_1_of_int @ real @ N ) @ ( one_one @ real ) ) )
=> ( ( archim6421214686448440834_floor @ real @ X2 )
= N ) ) ) ).
% floor_eq2
thf(fact_3719_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_3720_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_3721_log__inverse,axiom,
! [A2: real,X2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
=> ( ( A2
!= ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( log @ A2 @ ( inverse_inverse @ real @ X2 ) )
= ( uminus_uminus @ real @ ( log @ A2 @ X2 ) ) ) ) ) ) ).
% log_inverse
thf(fact_3722_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_3723_floor__divide__lower,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Q2: A,P2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Q2 )
=> ( ord_less_eq @ A @ ( times_times @ A @ ( ring_1_of_int @ A @ ( archim6421214686448440834_floor @ A @ ( divide_divide @ A @ P2 @ Q2 ) ) ) @ Q2 ) @ P2 ) ) ) ).
% floor_divide_lower
thf(fact_3724_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_3725_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_3726_gbinomial__negated__upper,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( ( gbinomial @ A )
= ( ^ [A4: A,K2: nat] : ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ K2 ) @ ( gbinomial @ A @ ( minus_minus @ A @ ( minus_minus @ A @ ( semiring_1_of_nat @ A @ K2 ) @ A4 ) @ ( one_one @ A ) ) @ K2 ) ) ) ) ) ).
% gbinomial_negated_upper
thf(fact_3727_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_3728_exp__plus__inverse__exp,axiom,
! [X2: real] : ( ord_less_eq @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( plus_plus @ real @ ( exp @ real @ X2 ) @ ( inverse_inverse @ real @ ( exp @ real @ X2 ) ) ) ) ).
% exp_plus_inverse_exp
thf(fact_3729_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_3730_floor__divide__upper,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Q2: A,P2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Q2 )
=> ( ord_less @ A @ P2 @ ( times_times @ A @ ( plus_plus @ A @ ( ring_1_of_int @ A @ ( archim6421214686448440834_floor @ A @ ( divide_divide @ A @ P2 @ Q2 ) ) ) @ ( one_one @ A ) ) @ Q2 ) ) ) ) ).
% floor_divide_upper
thf(fact_3731_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_3732_round__def,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ( ( archimedean_round @ A )
= ( ^ [X: A] : ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ X @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% round_def
thf(fact_3733_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_3734_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_3735_plus__inverse__ge__2,axiom,
! [X2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( ord_less_eq @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( plus_plus @ real @ X2 @ ( inverse_inverse @ real @ X2 ) ) ) ) ).
% plus_inverse_ge_2
thf(fact_3736_real__inv__sqrt__pow2,axiom,
! [X2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( power_power @ real @ ( inverse_inverse @ real @ ( sqrt @ X2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( inverse_inverse @ real @ X2 ) ) ) ).
% real_inv_sqrt_pow2
thf(fact_3737_tan__cot,axiom,
! [X2: real] :
( ( tan @ real @ ( minus_minus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ X2 ) )
= ( inverse_inverse @ real @ ( tan @ real @ X2 ) ) ) ).
% tan_cot
thf(fact_3738_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_3739_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_3740_real__le__x__sinh,axiom,
! [X2: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
=> ( ord_less_eq @ real @ X2 @ ( divide_divide @ real @ ( minus_minus @ real @ ( exp @ real @ X2 ) @ ( inverse_inverse @ real @ ( exp @ real @ X2 ) ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ).
% real_le_x_sinh
thf(fact_3741_real__le__abs__sinh,axiom,
! [X2: real] : ( ord_less_eq @ real @ ( abs_abs @ real @ X2 ) @ ( abs_abs @ real @ ( divide_divide @ real @ ( minus_minus @ real @ ( exp @ real @ X2 ) @ ( inverse_inverse @ real @ ( exp @ real @ X2 ) ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ).
% real_le_abs_sinh
thf(fact_3742_gbinomial__sum__up__index,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [J3: nat] : ( gbinomial @ A @ ( semiring_1_of_nat @ A @ J3 ) @ K )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
= ( gbinomial @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N ) @ ( one_one @ A ) ) @ ( plus_plus @ nat @ K @ ( one_one @ nat ) ) ) ) ) ).
% gbinomial_sum_up_index
thf(fact_3743_tan__sec,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X2: A] :
( ( ( cos @ A @ X2 )
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( one_one @ A ) @ ( power_power @ A @ ( tan @ A @ X2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( power_power @ A @ ( inverse_inverse @ A @ ( cos @ A @ X2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% tan_sec
thf(fact_3744_floor__log__eq__powr__iff,axiom,
! [X2: real,B2: real,K: int] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ord_less @ real @ ( one_one @ real ) @ B2 )
=> ( ( ( archim6421214686448440834_floor @ real @ ( log @ B2 @ X2 ) )
= K )
= ( ( ord_less_eq @ real @ ( powr @ real @ B2 @ ( ring_1_of_int @ real @ K ) ) @ X2 )
& ( ord_less @ real @ X2 @ ( powr @ real @ B2 @ ( ring_1_of_int @ real @ ( plus_plus @ int @ K @ ( one_one @ int ) ) ) ) ) ) ) ) ) ).
% floor_log_eq_powr_iff
thf(fact_3745_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_3746_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_3747_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_3748_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_3749_gbinomial__code,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( ( gbinomial @ A )
= ( ^ [A4: A,K2: nat] :
( if @ A
@ ( K2
= ( zero_zero @ nat ) )
@ ( one_one @ A )
@ ( divide_divide @ A
@ ( set_fo6178422350223883121st_nat @ A
@ ^ [L: nat] : ( times_times @ A @ ( minus_minus @ A @ A4 @ ( semiring_1_of_nat @ A @ L ) ) )
@ ( zero_zero @ nat )
@ ( minus_minus @ nat @ K2 @ ( one_one @ nat ) )
@ ( one_one @ A ) )
@ ( semiring_char_0_fact @ A @ K2 ) ) ) ) ) ) ).
% gbinomial_code
thf(fact_3750_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
@ ^ [K2: nat] : ( plus_plus @ A @ Z @ ( divide_divide @ A @ ( semiring_1_of_nat @ A @ K2 ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ ( one_one @ nat ) ) ) ) ) ) ).
% pochhammer_times_pochhammer_half
thf(fact_3751_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_3752_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_3753_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_3754_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_3755_of__nat__prod,axiom,
! [A: $tType,B: $tType] :
( ( comm_semiring_1 @ A )
=> ! [F2: B > nat,A3: set @ B] :
( ( semiring_1_of_nat @ A @ ( groups7121269368397514597t_prod @ B @ nat @ F2 @ A3 ) )
= ( groups7121269368397514597t_prod @ B @ A
@ ^ [X: B] : ( semiring_1_of_nat @ A @ ( F2 @ X ) )
@ A3 ) ) ) ).
% of_nat_prod
thf(fact_3756_of__int__prod,axiom,
! [A: $tType,B: $tType] :
( ( comm_ring_1 @ A )
=> ! [F2: B > int,A3: set @ B] :
( ( ring_1_of_int @ A @ ( groups7121269368397514597t_prod @ B @ int @ F2 @ A3 ) )
= ( groups7121269368397514597t_prod @ B @ A
@ ^ [X: B] : ( ring_1_of_int @ A @ ( F2 @ X ) )
@ A3 ) ) ) ).
% of_int_prod
thf(fact_3757_of__real__prod,axiom,
! [A: $tType,B: $tType] :
( ( ( comm_monoid_mult @ A )
& ( real_V2191834092415804123ebra_1 @ A ) )
=> ! [F2: B > real,S3: set @ B] :
( ( real_Vector_of_real @ A @ ( groups7121269368397514597t_prod @ B @ real @ F2 @ S3 ) )
= ( groups7121269368397514597t_prod @ B @ A
@ ^ [X: B] : ( real_Vector_of_real @ A @ ( F2 @ X ) )
@ S3 ) ) ) ).
% of_real_prod
thf(fact_3758_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 ) )
= ( ? [X: B] :
( ( member @ B @ X @ A3 )
& ( ( F2 @ X )
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% prod_zero_iff
thf(fact_3759_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_3760_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_3761_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_3762_norm__ii,axiom,
( ( real_V7770717601297561774m_norm @ complex @ imaginary_unit )
= ( one_one @ real ) ) ).
% norm_ii
thf(fact_3763_prod_Odelta,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S: set @ B,A2: B,B2: B > A] :
( ( finite_finite2 @ B @ S )
=> ( ( ( member @ B @ A2 @ S )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K2: B] : ( if @ A @ ( K2 = A2 ) @ ( B2 @ K2 ) @ ( one_one @ A ) )
@ S )
= ( B2 @ A2 ) ) )
& ( ~ ( member @ B @ A2 @ S )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K2: B] : ( if @ A @ ( K2 = A2 ) @ ( B2 @ K2 ) @ ( one_one @ A ) )
@ S )
= ( one_one @ A ) ) ) ) ) ) ).
% prod.delta
thf(fact_3764_prod_Odelta_H,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S: set @ B,A2: B,B2: B > A] :
( ( finite_finite2 @ B @ S )
=> ( ( ( member @ B @ A2 @ S )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K2: B] : ( if @ A @ ( A2 = K2 ) @ ( B2 @ K2 ) @ ( one_one @ A ) )
@ S )
= ( B2 @ A2 ) ) )
& ( ~ ( member @ B @ A2 @ S )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K2: B] : ( if @ A @ ( A2 = K2 ) @ ( B2 @ K2 ) @ ( one_one @ A ) )
@ S )
= ( one_one @ A ) ) ) ) ) ) ).
% prod.delta'
thf(fact_3765_divide__i,axiom,
! [X2: complex] :
( ( divide_divide @ complex @ X2 @ imaginary_unit )
= ( times_times @ complex @ ( uminus_uminus @ complex @ imaginary_unit ) @ X2 ) ) ).
% divide_i
thf(fact_3766_complex__i__mult__minus,axiom,
! [X2: complex] :
( ( times_times @ complex @ imaginary_unit @ ( times_times @ complex @ imaginary_unit @ X2 ) )
= ( uminus_uminus @ complex @ X2 ) ) ).
% complex_i_mult_minus
thf(fact_3767_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_3768_i__squared,axiom,
( ( times_times @ complex @ imaginary_unit @ imaginary_unit )
= ( uminus_uminus @ complex @ ( one_one @ complex ) ) ) ).
% i_squared
thf(fact_3769_divide__numeral__i,axiom,
! [Z: complex,N: num] :
( ( divide_divide @ complex @ Z @ ( times_times @ complex @ ( numeral_numeral @ complex @ N ) @ imaginary_unit ) )
= ( divide_divide @ complex @ ( uminus_uminus @ complex @ ( times_times @ complex @ imaginary_unit @ Z ) ) @ ( numeral_numeral @ complex @ N ) ) ) ).
% divide_numeral_i
thf(fact_3770_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_3771_power2__i,axiom,
( ( power_power @ complex @ imaginary_unit @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( uminus_uminus @ complex @ ( one_one @ complex ) ) ) ).
% power2_i
thf(fact_3772_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_3773_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_3774_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_3775_divide__complex__def,axiom,
( ( divide_divide @ complex )
= ( ^ [X: complex,Y: complex] : ( times_times @ complex @ X @ ( inverse_inverse @ complex @ Y ) ) ) ) ).
% divide_complex_def
thf(fact_3776_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 ) )
=> ~ ! [A5: B] :
( ( member @ B @ A5 @ A3 )
=> ( ( G @ A5 )
= ( one_one @ A ) ) ) ) ) ).
% prod.not_neutral_contains_not_neutral
thf(fact_3777_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_3778_prod__dvd__prod,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: set @ B,F2: B > A,G: B > A] :
( ! [A5: B] :
( ( member @ B @ A5 @ A3 )
=> ( dvd_dvd @ A @ ( F2 @ A5 ) @ ( G @ A5 ) ) )
=> ( dvd_dvd @ A @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ A3 ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ A3 ) ) ) ) ).
% prod_dvd_prod
thf(fact_3779_prod__norm,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V8999393235501362500lgebra @ B )
& ( comm_semiring_1 @ B ) )
=> ! [F2: A > B,A3: set @ A] :
( ( groups7121269368397514597t_prod @ A @ real
@ ^ [X: A] : ( real_V7770717601297561774m_norm @ B @ ( F2 @ X ) )
@ A3 )
= ( real_V7770717601297561774m_norm @ B @ ( groups7121269368397514597t_prod @ A @ B @ F2 @ A3 ) ) ) ) ).
% prod_norm
thf(fact_3780_prod_Oswap,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: B > C > A,B5: set @ C,A3: set @ B] :
( ( groups7121269368397514597t_prod @ B @ A
@ ^ [I3: B] : ( groups7121269368397514597t_prod @ C @ A @ ( G @ I3 ) @ B5 )
@ A3 )
= ( groups7121269368397514597t_prod @ C @ A
@ ^ [J3: C] :
( groups7121269368397514597t_prod @ B @ A
@ ^ [I3: B] : ( G @ I3 @ J3 )
@ A3 )
@ B5 ) ) ) ).
% prod.swap
thf(fact_3781_norm__prod__le,axiom,
! [A: $tType,B: $tType] :
( ( ( comm_monoid_mult @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [F2: B > A,A3: set @ B] :
( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ A3 ) )
@ ( groups7121269368397514597t_prod @ B @ real
@ ^ [A4: B] : ( real_V7770717601297561774m_norm @ A @ ( F2 @ A4 ) )
@ A3 ) ) ) ).
% norm_prod_le
thf(fact_3782_prod_Odistrib,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: B > A,H2: B > A,A3: set @ B] :
( ( groups7121269368397514597t_prod @ B @ A
@ ^ [X: B] : ( times_times @ A @ ( G @ X ) @ ( H2 @ X ) )
@ A3 )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ G @ A3 ) @ ( groups7121269368397514597t_prod @ B @ A @ H2 @ A3 ) ) ) ) ).
% prod.distrib
thf(fact_3783_prod__dividef,axiom,
! [A: $tType,B: $tType] :
( ( field @ A )
=> ! [F2: B > A,G: B > A,A3: set @ B] :
( ( groups7121269368397514597t_prod @ B @ A
@ ^ [X: B] : ( divide_divide @ A @ ( F2 @ X ) @ ( G @ X ) )
@ A3 )
= ( divide_divide @ A @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ A3 ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ A3 ) ) ) ) ).
% prod_dividef
thf(fact_3784_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
@ ^ [X: A] : ( power_power @ B @ ( F2 @ X ) @ N )
@ A3 ) ) ) ).
% prod_power_distrib
thf(fact_3785_prod_Oswap__restrict,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: set @ B,B5: set @ C,G: B > C > A,R: B > C > $o] :
( ( finite_finite2 @ B @ A3 )
=> ( ( finite_finite2 @ C @ B5 )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [X: B] :
( groups7121269368397514597t_prod @ C @ A @ ( G @ X )
@ ( collect @ C
@ ^ [Y: C] :
( ( member @ C @ Y @ B5 )
& ( R @ X @ Y ) ) ) )
@ A3 )
= ( groups7121269368397514597t_prod @ C @ A
@ ^ [Y: C] :
( groups7121269368397514597t_prod @ B @ A
@ ^ [X: B] : ( G @ X @ Y )
@ ( collect @ B
@ ^ [X: B] :
( ( member @ B @ X @ A3 )
& ( R @ X @ Y ) ) ) )
@ B5 ) ) ) ) ) ).
% prod.swap_restrict
thf(fact_3786_mod__prod__eq,axiom,
! [B: $tType,A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [F2: B > A,A2: A,A3: set @ B] :
( ( modulo_modulo @ A
@ ( groups7121269368397514597t_prod @ B @ A
@ ^ [I3: B] : ( modulo_modulo @ A @ ( F2 @ I3 ) @ A2 )
@ A3 )
@ A2 )
= ( modulo_modulo @ A @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ A3 ) @ A2 ) ) ) ).
% mod_prod_eq
thf(fact_3787_abs__prod,axiom,
! [A: $tType,B: $tType] :
( ( linordered_field @ A )
=> ! [F2: B > A,A3: set @ B] :
( ( abs_abs @ A @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ A3 ) )
= ( groups7121269368397514597t_prod @ B @ A
@ ^ [X: B] : ( abs_abs @ A @ ( F2 @ X ) )
@ A3 ) ) ) ).
% abs_prod
thf(fact_3788_complex__i__not__zero,axiom,
( imaginary_unit
!= ( zero_zero @ complex ) ) ).
% complex_i_not_zero
thf(fact_3789_complex__i__not__one,axiom,
( imaginary_unit
!= ( one_one @ complex ) ) ).
% complex_i_not_one
thf(fact_3790_prod__mono,axiom,
! [A: $tType,B: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: set @ B,F2: B > A,G: B > A] :
( ! [I2: B] :
( ( member @ B @ I2 @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ I2 ) )
& ( ord_less_eq @ A @ ( F2 @ I2 ) @ ( G @ I2 ) ) ) )
=> ( ord_less_eq @ A @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ A3 ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ A3 ) ) ) ) ).
% prod_mono
thf(fact_3791_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_3792_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_3793_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_3794_prod__zero,axiom,
! [B: $tType,A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A3: set @ B,F2: B > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ? [X4: B] :
( ( member @ B @ X4 @ A3 )
& ( ( F2 @ X4 )
= ( zero_zero @ A ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ F2 @ A3 )
= ( zero_zero @ A ) ) ) ) ) ).
% prod_zero
thf(fact_3795_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_3796_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
@ ^ [X: B] :
( ( member @ B @ X @ A3 )
& ( P @ X ) ) ) )
= ( groups7121269368397514597t_prod @ B @ A
@ ^ [X: B] : ( if @ A @ ( P @ X ) @ ( G @ X ) @ ( one_one @ A ) )
@ A3 ) ) ) ) ).
% prod.inter_filter
thf(fact_3797_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
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ).
% prod.shift_bounds_cl_Suc_ivl
thf(fact_3798_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_3799_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
@ ^ [I3: nat] : ( G @ ( plus_plus @ nat @ I3 @ K ) )
@ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ).
% prod.shift_bounds_cl_nat_ivl
thf(fact_3800_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_3801_prod_Orelated,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [R: A > A > $o,S: set @ B,H2: B > A,G: B > A] :
( ( R @ ( one_one @ A ) @ ( one_one @ A ) )
=> ( ! [X1: A,Y1: A,X22: A,Y23: A] :
( ( ( R @ X1 @ X22 )
& ( R @ Y1 @ Y23 ) )
=> ( R @ ( times_times @ A @ X1 @ Y1 ) @ ( times_times @ A @ X22 @ Y23 ) ) )
=> ( ( finite_finite2 @ B @ S )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ S )
=> ( R @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
=> ( R @ ( groups7121269368397514597t_prod @ B @ A @ H2 @ S ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ S ) ) ) ) ) ) ) ).
% prod.related
thf(fact_3802_prod__dvd__prod__subset2,axiom,
! [A: $tType,B: $tType] :
( ( comm_semiring_1 @ A )
=> ! [B5: set @ B,A3: set @ B,F2: B > A,G: B > A] :
( ( finite_finite2 @ B @ B5 )
=> ( ( ord_less_eq @ ( set @ B ) @ A3 @ B5 )
=> ( ! [A5: B] :
( ( member @ B @ A5 @ A3 )
=> ( dvd_dvd @ A @ ( F2 @ A5 ) @ ( G @ A5 ) ) )
=> ( dvd_dvd @ A @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ A3 ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ B5 ) ) ) ) ) ) ).
% prod_dvd_prod_subset2
thf(fact_3803_prod__dvd__prod__subset,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [B5: set @ B,A3: set @ B,F2: B > A] :
( ( finite_finite2 @ B @ B5 )
=> ( ( ord_less_eq @ ( set @ B ) @ A3 @ B5 )
=> ( dvd_dvd @ A @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ A3 ) @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ B5 ) ) ) ) ) ).
% prod_dvd_prod_subset
thf(fact_3804_prod_Oreindex__bij__witness__not__neutral,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S6: set @ B,T4: set @ C,S: set @ B,I: C > B,J: B > C,T3: set @ C,G: B > A,H2: C > A] :
( ( finite_finite2 @ B @ S6 )
=> ( ( finite_finite2 @ C @ T4 )
=> ( ! [A5: B] :
( ( member @ B @ A5 @ ( minus_minus @ ( set @ B ) @ S @ S6 ) )
=> ( ( I @ ( J @ A5 ) )
= A5 ) )
=> ( ! [A5: B] :
( ( member @ B @ A5 @ ( minus_minus @ ( set @ B ) @ S @ S6 ) )
=> ( member @ C @ ( J @ A5 ) @ ( minus_minus @ ( set @ C ) @ T3 @ T4 ) ) )
=> ( ! [B4: C] :
( ( member @ C @ B4 @ ( minus_minus @ ( set @ C ) @ T3 @ T4 ) )
=> ( ( J @ ( I @ B4 ) )
= B4 ) )
=> ( ! [B4: C] :
( ( member @ C @ B4 @ ( minus_minus @ ( set @ C ) @ T3 @ T4 ) )
=> ( member @ B @ ( I @ B4 ) @ ( minus_minus @ ( set @ B ) @ S @ S6 ) ) )
=> ( ! [A5: B] :
( ( member @ B @ A5 @ S6 )
=> ( ( G @ A5 )
= ( one_one @ A ) ) )
=> ( ! [B4: C] :
( ( member @ C @ B4 @ T4 )
=> ( ( H2 @ B4 )
= ( one_one @ A ) ) )
=> ( ! [A5: B] :
( ( member @ B @ A5 @ S )
=> ( ( H2 @ ( J @ A5 ) )
= ( G @ A5 ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ S )
= ( groups7121269368397514597t_prod @ C @ A @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ) ).
% prod.reindex_bij_witness_not_neutral
thf(fact_3805_i__times__eq__iff,axiom,
! [W: complex,Z: complex] :
( ( ( times_times @ complex @ imaginary_unit @ W )
= Z )
= ( W
= ( uminus_uminus @ complex @ ( times_times @ complex @ imaginary_unit @ Z ) ) ) ) ).
% i_times_eq_iff
thf(fact_3806_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
@ ^ [X: B] :
( ( G @ X )
= ( one_one @ A ) ) ) ) )
= ( groups7121269368397514597t_prod @ B @ A @ G @ A3 ) ) ) ) ).
% prod.setdiff_irrelevant
thf(fact_3807_exp__sum,axiom,
! [B: $tType,A: $tType] :
( ( ( comm_monoid_mult @ B )
& ( real_Vector_banach @ B )
& ( real_V2822296259951069270ebra_1 @ B ) )
=> ! [I5: set @ A,F2: A > B] :
( ( finite_finite2 @ A @ I5 )
=> ( ( exp @ B @ ( groups7311177749621191930dd_sum @ A @ B @ F2 @ I5 ) )
= ( groups7121269368397514597t_prod @ A @ B
@ ^ [X: A] : ( exp @ B @ ( F2 @ X ) )
@ I5 ) ) ) ) ).
% exp_sum
thf(fact_3808_prod_Onat__diff__reindex,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( G @ ( minus_minus @ nat @ N @ ( suc @ I3 ) ) )
@ ( set_ord_lessThan @ nat @ N ) )
= ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% prod.nat_diff_reindex
thf(fact_3809_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
@ ^ [I3: nat] : ( G @ ( minus_minus @ nat @ ( plus_plus @ nat @ M @ N ) @ I3 ) )
@ ( set_or1337092689740270186AtMost @ nat @ N @ M ) ) ) ) ).
% prod.atLeastAtMost_rev
thf(fact_3810_less__1__prod2,axiom,
! [B: $tType,A: $tType] :
( ( linordered_idom @ B )
=> ! [I5: set @ A,I: A,F2: A > B] :
( ( finite_finite2 @ A @ I5 )
=> ( ( member @ A @ I @ I5 )
=> ( ( ord_less @ B @ ( one_one @ B ) @ ( F2 @ I ) )
=> ( ! [I2: A] :
( ( member @ A @ I2 @ I5 )
=> ( ord_less_eq @ B @ ( one_one @ B ) @ ( F2 @ I2 ) ) )
=> ( ord_less @ B @ ( one_one @ B ) @ ( groups7121269368397514597t_prod @ A @ B @ F2 @ I5 ) ) ) ) ) ) ) ).
% less_1_prod2
thf(fact_3811_Complex__eq__i,axiom,
! [X2: real,Y2: real] :
( ( ( complex2 @ X2 @ Y2 )
= imaginary_unit )
= ( ( X2
= ( zero_zero @ real ) )
& ( Y2
= ( one_one @ real ) ) ) ) ).
% Complex_eq_i
thf(fact_3812_imaginary__unit_Ocode,axiom,
( imaginary_unit
= ( complex2 @ ( zero_zero @ real ) @ ( one_one @ real ) ) ) ).
% imaginary_unit.code
thf(fact_3813_prod_Osame__carrier,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [C6: set @ B,A3: set @ B,B5: set @ B,G: B > A,H2: B > A] :
( ( finite_finite2 @ B @ C6 )
=> ( ( ord_less_eq @ ( set @ B ) @ A3 @ C6 )
=> ( ( ord_less_eq @ ( set @ B ) @ B5 @ C6 )
=> ( ! [A5: B] :
( ( member @ B @ A5 @ ( minus_minus @ ( set @ B ) @ C6 @ A3 ) )
=> ( ( G @ A5 )
= ( one_one @ A ) ) )
=> ( ! [B4: B] :
( ( member @ B @ B4 @ ( minus_minus @ ( set @ B ) @ C6 @ B5 ) )
=> ( ( H2 @ B4 )
= ( one_one @ A ) ) )
=> ( ( ( groups7121269368397514597t_prod @ B @ A @ G @ A3 )
= ( groups7121269368397514597t_prod @ B @ A @ H2 @ B5 ) )
= ( ( groups7121269368397514597t_prod @ B @ A @ G @ C6 )
= ( groups7121269368397514597t_prod @ B @ A @ H2 @ C6 ) ) ) ) ) ) ) ) ) ).
% prod.same_carrier
thf(fact_3814_prod_Osame__carrierI,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [C6: set @ B,A3: set @ B,B5: set @ B,G: B > A,H2: B > A] :
( ( finite_finite2 @ B @ C6 )
=> ( ( ord_less_eq @ ( set @ B ) @ A3 @ C6 )
=> ( ( ord_less_eq @ ( set @ B ) @ B5 @ C6 )
=> ( ! [A5: B] :
( ( member @ B @ A5 @ ( minus_minus @ ( set @ B ) @ C6 @ A3 ) )
=> ( ( G @ A5 )
= ( one_one @ A ) ) )
=> ( ! [B4: B] :
( ( member @ B @ B4 @ ( minus_minus @ ( set @ B ) @ C6 @ B5 ) )
=> ( ( H2 @ B4 )
= ( 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 @ B5 ) ) ) ) ) ) ) ) ) ).
% prod.same_carrierI
thf(fact_3815_prod_Omono__neutral__left,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [T3: set @ B,S: set @ B,G: B > A] :
( ( finite_finite2 @ B @ T3 )
=> ( ( ord_less_eq @ ( set @ B ) @ S @ T3 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T3 @ S ) )
=> ( ( G @ X3 )
= ( one_one @ A ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ S )
= ( groups7121269368397514597t_prod @ B @ A @ G @ T3 ) ) ) ) ) ) ).
% prod.mono_neutral_left
thf(fact_3816_prod_Omono__neutral__right,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [T3: set @ B,S: set @ B,G: B > A] :
( ( finite_finite2 @ B @ T3 )
=> ( ( ord_less_eq @ ( set @ B ) @ S @ T3 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T3 @ S ) )
=> ( ( G @ X3 )
= ( one_one @ A ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ T3 )
= ( groups7121269368397514597t_prod @ B @ A @ G @ S ) ) ) ) ) ) ).
% prod.mono_neutral_right
thf(fact_3817_prod_Omono__neutral__cong__left,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [T3: set @ B,S: set @ B,H2: B > A,G: B > A] :
( ( finite_finite2 @ B @ T3 )
=> ( ( ord_less_eq @ ( set @ B ) @ S @ T3 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T3 @ S ) )
=> ( ( H2 @ X3 )
= ( one_one @ A ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ S )
=> ( ( G @ X3 )
= ( H2 @ X3 ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ S )
= ( groups7121269368397514597t_prod @ B @ A @ H2 @ T3 ) ) ) ) ) ) ) ).
% prod.mono_neutral_cong_left
thf(fact_3818_prod_Omono__neutral__cong__right,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [T3: set @ B,S: set @ B,G: B > A,H2: B > A] :
( ( finite_finite2 @ B @ T3 )
=> ( ( ord_less_eq @ ( set @ B ) @ S @ T3 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T3 @ S ) )
=> ( ( G @ X3 )
= ( one_one @ A ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ S )
=> ( ( G @ X3 )
= ( H2 @ X3 ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ T3 )
= ( groups7121269368397514597t_prod @ B @ A @ H2 @ S ) ) ) ) ) ) ) ).
% prod.mono_neutral_cong_right
thf(fact_3819_prod_Osubset__diff,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [B5: set @ B,A3: set @ B,G: B > A] :
( ( ord_less_eq @ ( set @ B ) @ B5 @ A3 )
=> ( ( finite_finite2 @ B @ A3 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ A3 )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ A3 @ B5 ) ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ B5 ) ) ) ) ) ) ).
% prod.subset_diff
thf(fact_3820_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_3821_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_3822_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_3823_Complex__mult__i,axiom,
! [A2: real,B2: real] :
( ( times_times @ complex @ ( complex2 @ A2 @ B2 ) @ imaginary_unit )
= ( complex2 @ ( uminus_uminus @ real @ B2 ) @ A2 ) ) ).
% Complex_mult_i
thf(fact_3824_i__mult__Complex,axiom,
! [A2: real,B2: real] :
( ( times_times @ complex @ imaginary_unit @ ( complex2 @ A2 @ B2 ) )
= ( complex2 @ ( uminus_uminus @ real @ B2 ) @ A2 ) ) ).
% i_mult_Complex
thf(fact_3825_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
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% prod.lessThan_Suc_shift
thf(fact_3826_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
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ) ) ).
% prod.Suc_reindex_ivl
thf(fact_3827_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
@ ^ [K2: nat] : ( G @ ( suc @ K2 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% prod.atLeast1_atMost_eq
thf(fact_3828_fact__prod,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ( ( semiring_char_0_fact @ A )
= ( ^ [N3: nat] :
( semiring_1_of_nat @ A
@ ( groups7121269368397514597t_prod @ nat @ nat
@ ^ [X: nat] : X
@ ( set_or1337092689740270186AtMost @ nat @ ( one_one @ nat ) @ N3 ) ) ) ) ) ) ).
% fact_prod
thf(fact_3829_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 ) )
= ( ? [X: B] :
( ( member @ B @ X @ A3 )
& ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( F2 @ X ) ) ) ) ) ) ) ).
% even_prod_iff
thf(fact_3830_prod_Oub__add__nat,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M: nat,N: nat,G: nat > A,P2: 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 @ P2 ) ) )
= ( 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 @ P2 ) ) ) ) ) ) ) ).
% prod.ub_add_nat
thf(fact_3831_fold__atLeastAtMost__nat_Osimps,axiom,
! [A: $tType] :
( ( set_fo6178422350223883121st_nat @ A )
= ( ^ [F4: nat > A > A,A4: nat,B3: nat,Acc: A] : ( if @ A @ ( ord_less @ nat @ B3 @ A4 ) @ Acc @ ( set_fo6178422350223883121st_nat @ A @ F4 @ ( plus_plus @ nat @ A4 @ ( one_one @ nat ) ) @ B3 @ ( F4 @ A4 @ Acc ) ) ) ) ) ).
% fold_atLeastAtMost_nat.simps
thf(fact_3832_fold__atLeastAtMost__nat_Oelims,axiom,
! [A: $tType,X2: nat > A > A,Xa: nat,Xb: nat,Xc: A,Y2: A] :
( ( ( set_fo6178422350223883121st_nat @ A @ X2 @ Xa @ Xb @ Xc )
= Y2 )
=> ( ( ( ord_less @ nat @ Xb @ Xa )
=> ( Y2 = Xc ) )
& ( ~ ( ord_less @ nat @ Xb @ Xa )
=> ( Y2
= ( set_fo6178422350223883121st_nat @ A @ X2 @ ( plus_plus @ nat @ Xa @ ( one_one @ nat ) ) @ Xb @ ( X2 @ Xa @ Xc ) ) ) ) ) ) ).
% fold_atLeastAtMost_nat.elims
thf(fact_3833_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_3834_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_3835_norm__prod__diff,axiom,
! [A: $tType,I6: $tType] :
( ( ( comm_monoid_mult @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [I5: set @ I6,Z: I6 > A,W: I6 > A] :
( ! [I2: I6] :
( ( member @ I6 @ I2 @ I5 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( Z @ I2 ) ) @ ( one_one @ real ) ) )
=> ( ! [I2: I6] :
( ( member @ I6 @ I2 @ I5 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( W @ I2 ) ) @ ( one_one @ real ) ) )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( groups7121269368397514597t_prod @ I6 @ A @ Z @ I5 ) @ ( groups7121269368397514597t_prod @ I6 @ A @ W @ I5 ) ) )
@ ( groups7311177749621191930dd_sum @ I6 @ real
@ ^ [I3: I6] : ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( Z @ I3 ) @ ( W @ I3 ) ) )
@ I5 ) ) ) ) ) ).
% norm_prod_diff
thf(fact_3836_fact__eq__fact__times,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq @ nat @ N @ M )
=> ( ( semiring_char_0_fact @ nat @ M )
= ( times_times @ nat @ ( semiring_char_0_fact @ nat @ N )
@ ( groups7121269368397514597t_prod @ nat @ nat
@ ^ [X: nat] : X
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ N ) @ M ) ) ) ) ) ).
% fact_eq_fact_times
thf(fact_3837_Complex__eq,axiom,
( complex2
= ( ^ [A4: real,B3: real] : ( plus_plus @ complex @ ( real_Vector_of_real @ complex @ A4 ) @ ( times_times @ complex @ imaginary_unit @ ( real_Vector_of_real @ complex @ B3 ) ) ) ) ) ).
% Complex_eq
thf(fact_3838_prod__mono2,axiom,
! [B: $tType,A: $tType] :
( ( linordered_idom @ B )
=> ! [B5: set @ A,A3: set @ A,F2: A > B] :
( ( finite_finite2 @ A @ B5 )
=> ( ( ord_less_eq @ ( set @ A ) @ A3 @ B5 )
=> ( ! [B4: A] :
( ( member @ A @ B4 @ ( minus_minus @ ( set @ A ) @ B5 @ A3 ) )
=> ( ord_less_eq @ B @ ( one_one @ B ) @ ( F2 @ B4 ) ) )
=> ( ! [A5: A] :
( ( member @ A @ A5 @ A3 )
=> ( ord_less_eq @ B @ ( zero_zero @ B ) @ ( F2 @ A5 ) ) )
=> ( ord_less_eq @ B @ ( groups7121269368397514597t_prod @ A @ B @ F2 @ A3 ) @ ( groups7121269368397514597t_prod @ A @ B @ F2 @ B5 ) ) ) ) ) ) ) ).
% prod_mono2
thf(fact_3839_pochhammer__Suc__prod,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A2: A,N: nat] :
( ( comm_s3205402744901411588hammer @ A @ A2 @ ( suc @ N ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( plus_plus @ A @ A2 @ ( semiring_1_of_nat @ A @ I3 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% pochhammer_Suc_prod
thf(fact_3840_pochhammer__prod__rev,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ( ( comm_s3205402744901411588hammer @ A )
= ( ^ [A4: A,N3: nat] :
( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( plus_plus @ A @ A4 @ ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ N3 @ I3 ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( one_one @ nat ) @ N3 ) ) ) ) ) ).
% pochhammer_prod_rev
thf(fact_3841_fact__div__fact,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq @ nat @ N @ M )
=> ( ( divide_divide @ nat @ ( semiring_char_0_fact @ nat @ M ) @ ( semiring_char_0_fact @ nat @ N ) )
= ( groups7121269368397514597t_prod @ nat @ nat
@ ^ [X: nat] : X
@ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) @ M ) ) ) ) ).
% fact_div_fact
thf(fact_3842_complex__split__polar,axiom,
! [Z: complex] :
? [R3: real,A5: real] :
( Z
= ( times_times @ complex @ ( real_Vector_of_real @ complex @ R3 ) @ ( plus_plus @ complex @ ( real_Vector_of_real @ complex @ ( cos @ real @ A5 ) ) @ ( times_times @ complex @ imaginary_unit @ ( real_Vector_of_real @ complex @ ( sin @ real @ A5 ) ) ) ) ) ) ).
% complex_split_polar
thf(fact_3843_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
@ ^ [I3: nat] : ( times_times @ A @ ( G @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I3 ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ).
% prod.in_pairs
thf(fact_3844_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_3845_pochhammer__Suc__prod__rev,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A2: A,N: nat] :
( ( comm_s3205402744901411588hammer @ A @ A2 @ ( suc @ N ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( plus_plus @ A @ A2 @ ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ N @ I3 ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% pochhammer_Suc_prod_rev
thf(fact_3846_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_3847_gbinomial__Suc,axiom,
! [A: $tType] :
( ( ( semiring_char_0 @ A )
& ( semidom_divide @ A ) )
=> ! [A2: A,K: nat] :
( ( gbinomial @ A @ A2 @ ( suc @ K ) )
= ( divide_divide @ A
@ ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( minus_minus @ A @ A2 @ ( semiring_1_of_nat @ A @ I3 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ K ) )
@ ( semiring_char_0_fact @ A @ ( suc @ K ) ) ) ) ) ).
% gbinomial_Suc
thf(fact_3848_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_3849_cmod__complex__polar,axiom,
! [R2: real,A2: real] :
( ( real_V7770717601297561774m_norm @ complex @ ( times_times @ complex @ ( real_Vector_of_real @ complex @ R2 ) @ ( plus_plus @ complex @ ( real_Vector_of_real @ complex @ ( cos @ real @ A2 ) ) @ ( times_times @ complex @ imaginary_unit @ ( real_Vector_of_real @ complex @ ( sin @ real @ A2 ) ) ) ) ) )
= ( abs_abs @ real @ R2 ) ) ).
% cmod_complex_polar
thf(fact_3850_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_3851_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_3852_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_3853_Arg__ii,axiom,
( ( arg @ imaginary_unit )
= ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ).
% Arg_ii
thf(fact_3854_gbinomial__partial__row__sum,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A2: A,M: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K2: nat] : ( times_times @ A @ ( gbinomial @ A @ A2 @ K2 ) @ ( minus_minus @ A @ ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( semiring_1_of_nat @ A @ K2 ) ) )
@ ( set_ord_atMost @ nat @ M ) )
= ( times_times @ A @ ( divide_divide @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ M ) @ ( one_one @ A ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( gbinomial @ A @ A2 @ ( plus_plus @ nat @ M @ ( one_one @ nat ) ) ) ) ) ) ).
% gbinomial_partial_row_sum
thf(fact_3855_atMost__iff,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [I: A,K: A] :
( ( member @ A @ I @ ( set_ord_atMost @ A @ K ) )
= ( ord_less_eq @ A @ I @ K ) ) ) ).
% atMost_iff
thf(fact_3856_csqrt__0,axiom,
( ( csqrt @ ( zero_zero @ complex ) )
= ( zero_zero @ complex ) ) ).
% csqrt_0
thf(fact_3857_csqrt__eq__0,axiom,
! [Z: complex] :
( ( ( csqrt @ Z )
= ( zero_zero @ complex ) )
= ( Z
= ( zero_zero @ complex ) ) ) ).
% csqrt_eq_0
thf(fact_3858_csqrt__1,axiom,
( ( csqrt @ ( one_one @ complex ) )
= ( one_one @ complex ) ) ).
% csqrt_1
thf(fact_3859_csqrt__eq__1,axiom,
! [Z: complex] :
( ( ( csqrt @ Z )
= ( one_one @ complex ) )
= ( Z
= ( one_one @ complex ) ) ) ).
% csqrt_eq_1
thf(fact_3860_atMost__subset__iff,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X2: A,Y2: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_ord_atMost @ A @ X2 ) @ ( set_ord_atMost @ A @ Y2 ) )
= ( ord_less_eq @ A @ X2 @ Y2 ) ) ) ).
% atMost_subset_iff
thf(fact_3861_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 ) )
= ( ! [X: A] :
( ( member @ A @ X @ A3 )
=> ( ( F2 @ X )
= ( one_one @ nat ) ) ) ) ) ) ).
% prod_eq_1_iff
thf(fact_3862_Icc__subset__Iic__iff,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [L2: A,H2: A,H3: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ L2 @ H2 ) @ ( set_ord_atMost @ A @ H3 ) )
= ( ~ ( ord_less_eq @ A @ L2 @ H2 )
| ( ord_less_eq @ A @ H2 @ H3 ) ) ) ) ).
% Icc_subset_Iic_iff
thf(fact_3863_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_3864_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_3865_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 ) )
= ( ! [X: A] :
( ( member @ A @ X @ A3 )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F2 @ X ) ) ) ) ) ) ).
% prod_pos_nat_iff
thf(fact_3866_power2__csqrt,axiom,
! [Z: complex] :
( ( power_power @ complex @ ( csqrt @ Z ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= Z ) ).
% power2_csqrt
thf(fact_3867_int__prod,axiom,
! [B: $tType,F2: B > nat,A3: set @ B] :
( ( semiring_1_of_nat @ int @ ( groups7121269368397514597t_prod @ B @ nat @ F2 @ A3 ) )
= ( groups7121269368397514597t_prod @ B @ int
@ ^ [X: B] : ( semiring_1_of_nat @ int @ ( F2 @ X ) )
@ A3 ) ) ).
% int_prod
thf(fact_3868_atMost__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( set_ord_atMost @ A )
= ( ^ [U2: A] :
( collect @ A
@ ^ [X: A] : ( ord_less_eq @ A @ X @ U2 ) ) ) ) ) ).
% atMost_def
thf(fact_3869_atMost__atLeast0,axiom,
( ( set_ord_atMost @ nat )
= ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) ) ) ).
% atMost_atLeast0
thf(fact_3870_lessThan__Suc__atMost,axiom,
! [K: nat] :
( ( set_ord_lessThan @ nat @ ( suc @ K ) )
= ( set_ord_atMost @ nat @ K ) ) ).
% lessThan_Suc_atMost
thf(fact_3871_not__Iic__le__Icc,axiom,
! [A: $tType] :
( ( no_bot @ A )
=> ! [H2: A,L3: A,H3: A] :
~ ( ord_less_eq @ ( set @ A ) @ ( set_ord_atMost @ A @ H2 ) @ ( set_or1337092689740270186AtMost @ A @ L3 @ H3 ) ) ) ).
% not_Iic_le_Icc
thf(fact_3872_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_3873_prod__int__eq,axiom,
! [I: nat,J: nat] :
( ( groups7121269368397514597t_prod @ nat @ int @ ( semiring_1_of_nat @ int ) @ ( set_or1337092689740270186AtMost @ nat @ I @ J ) )
= ( groups7121269368397514597t_prod @ int @ int
@ ^ [X: int] : X
@ ( set_or1337092689740270186AtMost @ int @ ( semiring_1_of_nat @ int @ I ) @ ( semiring_1_of_nat @ int @ J ) ) ) ) ).
% prod_int_eq
thf(fact_3874_Arg__zero,axiom,
( ( arg @ ( zero_zero @ complex ) )
= ( zero_zero @ real ) ) ).
% Arg_zero
thf(fact_3875_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
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ) ).
% sum.atMost_Suc_shift
thf(fact_3876_sum__telescope,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [F2: nat > A,I: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( minus_minus @ A @ ( F2 @ I3 ) @ ( F2 @ ( suc @ I3 ) ) )
@ ( set_ord_atMost @ nat @ I ) )
= ( minus_minus @ A @ ( F2 @ ( zero_zero @ nat ) ) @ ( F2 @ ( suc @ I ) ) ) ) ) ).
% sum_telescope
thf(fact_3877_polyfun__eq__coeffs,axiom,
! [A: $tType] :
( ( ( real_V8999393235501362500lgebra @ A )
& ( idom @ A ) )
=> ! [C2: nat > A,N: nat,D2: nat > A] :
( ( ! [X: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( C2 @ I3 ) @ ( power_power @ A @ X @ I3 ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( D2 @ I3 ) @ ( power_power @ A @ X @ I3 ) )
@ ( set_ord_atMost @ nat @ N ) ) ) )
= ( ! [I3: nat] :
( ( ord_less_eq @ nat @ I3 @ N )
=> ( ( C2 @ I3 )
= ( D2 @ I3 ) ) ) ) ) ) ).
% polyfun_eq_coeffs
thf(fact_3878_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
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ) ).
% prod.atMost_Suc_shift
thf(fact_3879_bounded__imp__summable,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( linord2810124833399127020strict @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [A2: nat > A,B5: A] :
( ! [N2: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( A2 @ N2 ) )
=> ( ! [N2: nat] : ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ A2 @ ( set_ord_atMost @ nat @ N2 ) ) @ B5 )
=> ( summable @ A @ A2 ) ) ) ) ).
% bounded_imp_summable
thf(fact_3880_prod__int__plus__eq,axiom,
! [I: nat,J: nat] :
( ( groups7121269368397514597t_prod @ nat @ int @ ( semiring_1_of_nat @ int ) @ ( set_or1337092689740270186AtMost @ nat @ I @ ( plus_plus @ nat @ I @ J ) ) )
= ( groups7121269368397514597t_prod @ int @ int
@ ^ [X: int] : X
@ ( set_or1337092689740270186AtMost @ int @ ( semiring_1_of_nat @ int @ I ) @ ( semiring_1_of_nat @ int @ ( plus_plus @ nat @ I @ J ) ) ) ) ) ).
% prod_int_plus_eq
thf(fact_3881_sum_Onested__swap_H,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A2: nat > nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( groups7311177749621191930dd_sum @ nat @ A @ ( A2 @ I3 ) @ ( set_ord_lessThan @ nat @ I3 ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [J3: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( A2 @ I3 @ J3 )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ J3 ) @ N ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% sum.nested_swap'
thf(fact_3882_prod_Onested__swap_H,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A2: nat > nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( groups7121269368397514597t_prod @ nat @ A @ ( A2 @ I3 ) @ ( set_ord_lessThan @ nat @ I3 ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [J3: nat] :
( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( A2 @ I3 @ J3 )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ J3 ) @ N ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% prod.nested_swap'
thf(fact_3883_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
@ ^ [I3: nat] : ( times_times @ A @ ( C2 @ I3 ) @ ( power_power @ A @ W3 @ I3 ) )
@ ( 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_3884_polyfun__eq__0,axiom,
! [A: $tType] :
( ( ( real_V8999393235501362500lgebra @ A )
& ( idom @ A ) )
=> ! [C2: nat > A,N: nat] :
( ( ! [X: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( C2 @ I3 ) @ ( power_power @ A @ X @ I3 ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( zero_zero @ A ) ) )
= ( ! [I3: nat] :
( ( ord_less_eq @ nat @ I3 @ N )
=> ( ( C2 @ I3 )
= ( zero_zero @ A ) ) ) ) ) ) ).
% polyfun_eq_0
thf(fact_3885_ln__prod,axiom,
! [A: $tType,I5: set @ A,F2: A > real] :
( ( finite_finite2 @ A @ I5 )
=> ( ! [I2: A] :
( ( member @ A @ I2 @ I5 )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( F2 @ I2 ) ) )
=> ( ( ln_ln @ real @ ( groups7121269368397514597t_prod @ A @ real @ F2 @ I5 ) )
= ( groups7311177749621191930dd_sum @ A @ real
@ ^ [X: A] : ( ln_ln @ real @ ( F2 @ X ) )
@ I5 ) ) ) ) ).
% ln_prod
thf(fact_3886_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
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% sum.atMost_shift
thf(fact_3887_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_3888_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
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% prod.atMost_shift
thf(fact_3889_gbinomial__parallel__sum,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A2: A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K2: nat] : ( gbinomial @ A @ ( plus_plus @ A @ A2 @ ( semiring_1_of_nat @ A @ K2 ) ) @ K2 )
@ ( 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_3890_sum__gp__basic,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [X2: A,N: nat] :
( ( times_times @ A @ ( minus_minus @ A @ ( one_one @ A ) @ X2 ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X2 ) @ ( set_ord_atMost @ nat @ N ) ) )
= ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ X2 @ ( suc @ N ) ) ) ) ) ).
% sum_gp_basic
thf(fact_3891_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
@ ^ [Z5: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( C2 @ I3 ) @ ( power_power @ A @ Z5 @ I3 ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% polyfun_roots_finite
thf(fact_3892_polyfun__finite__roots,axiom,
! [A: $tType] :
( ( ( real_V8999393235501362500lgebra @ A )
& ( idom @ A ) )
=> ! [C2: nat > A,N: nat] :
( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( C2 @ I3 ) @ ( power_power @ A @ X @ I3 ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( zero_zero @ A ) ) ) )
= ( ? [I3: nat] :
( ( ord_less_eq @ nat @ I3 @ N )
& ( ( C2 @ I3 )
!= ( zero_zero @ A ) ) ) ) ) ) ).
% polyfun_finite_roots
thf(fact_3893_of__real__sqrt,axiom,
! [X2: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( real_Vector_of_real @ complex @ ( sqrt @ X2 ) )
= ( csqrt @ ( real_Vector_of_real @ complex @ X2 ) ) ) ) ).
% of_real_sqrt
thf(fact_3894_polyfun__linear__factor__root,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [C2: nat > A,A2: A,N: nat] :
( ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( C2 @ I3 ) @ ( power_power @ A @ A2 @ I3 ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( zero_zero @ A ) )
=> ~ ! [B4: nat > A] :
~ ! [Z4: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( C2 @ I3 ) @ ( power_power @ A @ Z4 @ I3 ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( times_times @ A @ ( minus_minus @ A @ Z4 @ A2 )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( B4 @ I3 ) @ ( power_power @ A @ Z4 @ I3 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ) ).
% polyfun_linear_factor_root
thf(fact_3895_polyfun__linear__factor,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [C2: nat > A,N: nat,A2: A] :
? [B4: nat > A] :
! [Z4: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( C2 @ I3 ) @ ( power_power @ A @ Z4 @ I3 ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( plus_plus @ A
@ ( times_times @ A @ ( minus_minus @ A @ Z4 @ A2 )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( B4 @ I3 ) @ ( power_power @ A @ Z4 @ I3 ) )
@ ( set_ord_lessThan @ nat @ N ) ) )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( C2 @ I3 ) @ ( power_power @ A @ A2 @ I3 ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ) ).
% polyfun_linear_factor
thf(fact_3896_sum__power__shift,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [M: nat,N: nat,X2: A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X2 ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( times_times @ A @ ( power_power @ A @ X2 @ M ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X2 ) @ ( set_ord_atMost @ nat @ ( minus_minus @ nat @ N @ M ) ) ) ) ) ) ) ).
% sum_power_shift
thf(fact_3897_summable__Cauchy__product,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V4412858255891104859lgebra @ A ) )
=> ! [A2: nat > A,B2: nat > A] :
( ( summable @ real
@ ^ [K2: nat] : ( real_V7770717601297561774m_norm @ A @ ( A2 @ K2 ) ) )
=> ( ( summable @ real
@ ^ [K2: nat] : ( real_V7770717601297561774m_norm @ A @ ( B2 @ K2 ) ) )
=> ( summable @ A
@ ^ [K2: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( A2 @ I3 ) @ ( B2 @ ( minus_minus @ nat @ K2 @ I3 ) ) )
@ ( set_ord_atMost @ nat @ K2 ) ) ) ) ) ) ).
% summable_Cauchy_product
thf(fact_3898_Cauchy__product,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V4412858255891104859lgebra @ A ) )
=> ! [A2: nat > A,B2: nat > A] :
( ( summable @ real
@ ^ [K2: nat] : ( real_V7770717601297561774m_norm @ A @ ( A2 @ K2 ) ) )
=> ( ( summable @ real
@ ^ [K2: nat] : ( real_V7770717601297561774m_norm @ A @ ( B2 @ K2 ) ) )
=> ( ( times_times @ A @ ( suminf @ A @ A2 ) @ ( suminf @ A @ B2 ) )
= ( suminf @ A
@ ^ [K2: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( A2 @ I3 ) @ ( B2 @ ( minus_minus @ nat @ K2 @ I3 ) ) )
@ ( set_ord_atMost @ nat @ K2 ) ) ) ) ) ) ) ).
% Cauchy_product
thf(fact_3899_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_3900_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
@ ^ [I3: nat] : ( plus_plus @ A @ ( G @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I3 ) ) ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% sum.in_pairs_0
thf(fact_3901_polynomial__product,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [M: nat,A2: nat > A,N: nat,B2: nat > A,X2: A] :
( ! [I2: nat] :
( ( ord_less @ nat @ M @ I2 )
=> ( ( A2 @ I2 )
= ( zero_zero @ A ) ) )
=> ( ! [J2: nat] :
( ( ord_less @ nat @ N @ J2 )
=> ( ( B2 @ J2 )
= ( zero_zero @ A ) ) )
=> ( ( times_times @ A
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( A2 @ I3 ) @ ( power_power @ A @ X2 @ I3 ) )
@ ( set_ord_atMost @ nat @ M ) )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [J3: nat] : ( times_times @ A @ ( B2 @ J3 ) @ ( power_power @ A @ X2 @ J3 ) )
@ ( set_ord_atMost @ nat @ N ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [R5: nat] :
( times_times @ A
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K2: nat] : ( times_times @ A @ ( A2 @ K2 ) @ ( B2 @ ( minus_minus @ nat @ R5 @ K2 ) ) )
@ ( set_ord_atMost @ nat @ R5 ) )
@ ( power_power @ A @ X2 @ R5 ) )
@ ( set_ord_atMost @ nat @ ( plus_plus @ nat @ M @ N ) ) ) ) ) ) ) ).
% polynomial_product
thf(fact_3902_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
@ ^ [I3: nat] : ( times_times @ A @ ( G @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I3 ) ) ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% prod.in_pairs_0
thf(fact_3903_polyfun__eq__const,axiom,
! [A: $tType] :
( ( ( real_V8999393235501362500lgebra @ A )
& ( idom @ A ) )
=> ! [C2: nat > A,N: nat,K: A] :
( ( ! [X: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( C2 @ I3 ) @ ( power_power @ A @ X @ I3 ) )
@ ( set_ord_atMost @ nat @ N ) )
= K ) )
= ( ( ( C2 @ ( zero_zero @ nat ) )
= K )
& ! [X: nat] :
( ( member @ nat @ X @ ( set_or1337092689740270186AtMost @ nat @ ( one_one @ nat ) @ N ) )
=> ( ( C2 @ X )
= ( zero_zero @ A ) ) ) ) ) ) ).
% polyfun_eq_const
thf(fact_3904_gbinomial__sum__lower__neg,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A2: A,M: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K2: nat] : ( times_times @ A @ ( gbinomial @ A @ A2 @ K2 ) @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ K2 ) )
@ ( set_ord_atMost @ nat @ M ) )
= ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ M ) @ ( gbinomial @ A @ ( minus_minus @ A @ A2 @ ( one_one @ A ) ) @ M ) ) ) ) ).
% gbinomial_sum_lower_neg
thf(fact_3905_polynomial__product__nat,axiom,
! [M: nat,A2: nat > nat,N: nat,B2: nat > nat,X2: nat] :
( ! [I2: nat] :
( ( ord_less @ nat @ M @ I2 )
=> ( ( A2 @ I2 )
= ( zero_zero @ nat ) ) )
=> ( ! [J2: nat] :
( ( ord_less @ nat @ N @ J2 )
=> ( ( B2 @ J2 )
= ( zero_zero @ nat ) ) )
=> ( ( times_times @ nat
@ ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [I3: nat] : ( times_times @ nat @ ( A2 @ I3 ) @ ( power_power @ nat @ X2 @ I3 ) )
@ ( set_ord_atMost @ nat @ M ) )
@ ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [J3: nat] : ( times_times @ nat @ ( B2 @ J3 ) @ ( power_power @ nat @ X2 @ J3 ) )
@ ( set_ord_atMost @ nat @ N ) ) )
= ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [R5: nat] :
( times_times @ nat
@ ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [K2: nat] : ( times_times @ nat @ ( A2 @ K2 ) @ ( B2 @ ( minus_minus @ nat @ R5 @ K2 ) ) )
@ ( set_ord_atMost @ nat @ R5 ) )
@ ( power_power @ nat @ X2 @ R5 ) )
@ ( set_ord_atMost @ nat @ ( plus_plus @ nat @ M @ N ) ) ) ) ) ) ).
% polynomial_product_nat
thf(fact_3906_Cauchy__product__sums,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V4412858255891104859lgebra @ A ) )
=> ! [A2: nat > A,B2: nat > A] :
( ( summable @ real
@ ^ [K2: nat] : ( real_V7770717601297561774m_norm @ A @ ( A2 @ K2 ) ) )
=> ( ( summable @ real
@ ^ [K2: nat] : ( real_V7770717601297561774m_norm @ A @ ( B2 @ K2 ) ) )
=> ( sums @ A
@ ^ [K2: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( A2 @ I3 ) @ ( B2 @ ( minus_minus @ nat @ K2 @ I3 ) ) )
@ ( set_ord_atMost @ nat @ K2 ) )
@ ( times_times @ A @ ( suminf @ A @ A2 ) @ ( suminf @ A @ B2 ) ) ) ) ) ) ).
% Cauchy_product_sums
thf(fact_3907_sum_Ozero__middle,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [P2: nat,K: nat,G: nat > A,H2: nat > A] :
( ( ord_less_eq @ nat @ ( one_one @ nat ) @ P2 )
=> ( ( ord_less_eq @ nat @ K @ P2 )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [J3: nat] : ( if @ A @ ( ord_less @ nat @ J3 @ K ) @ ( G @ J3 ) @ ( if @ A @ ( J3 = K ) @ ( zero_zero @ A ) @ ( H2 @ ( minus_minus @ nat @ J3 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) )
@ ( set_ord_atMost @ nat @ P2 ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [J3: nat] : ( if @ A @ ( ord_less @ nat @ J3 @ K ) @ ( G @ J3 ) @ ( H2 @ J3 ) )
@ ( set_ord_atMost @ nat @ ( minus_minus @ nat @ P2 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ) ) ).
% sum.zero_middle
thf(fact_3908_prod_Ozero__middle,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [P2: nat,K: nat,G: nat > A,H2: nat > A] :
( ( ord_less_eq @ nat @ ( one_one @ nat ) @ P2 )
=> ( ( ord_less_eq @ nat @ K @ P2 )
=> ( ( groups7121269368397514597t_prod @ nat @ A
@ ^ [J3: nat] : ( if @ A @ ( ord_less @ nat @ J3 @ K ) @ ( G @ J3 ) @ ( if @ A @ ( J3 = K ) @ ( one_one @ A ) @ ( H2 @ ( minus_minus @ nat @ J3 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) )
@ ( set_ord_atMost @ nat @ P2 ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [J3: nat] : ( if @ A @ ( ord_less @ nat @ J3 @ K ) @ ( G @ J3 ) @ ( H2 @ J3 ) )
@ ( set_ord_atMost @ nat @ ( minus_minus @ nat @ P2 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ) ) ).
% prod.zero_middle
thf(fact_3909_gbinomial__partial__sum__poly,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [M: nat,A2: A,X2: A,Y2: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K2: nat] : ( times_times @ A @ ( times_times @ A @ ( gbinomial @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ M ) @ A2 ) @ K2 ) @ ( power_power @ A @ X2 @ K2 ) ) @ ( power_power @ A @ Y2 @ ( minus_minus @ nat @ M @ K2 ) ) )
@ ( set_ord_atMost @ nat @ M ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K2: nat] : ( times_times @ A @ ( times_times @ A @ ( gbinomial @ A @ ( uminus_uminus @ A @ A2 ) @ K2 ) @ ( power_power @ A @ ( uminus_uminus @ A @ X2 ) @ K2 ) ) @ ( power_power @ A @ ( plus_plus @ A @ X2 @ Y2 ) @ ( minus_minus @ nat @ M @ K2 ) ) )
@ ( set_ord_atMost @ nat @ M ) ) ) ) ).
% gbinomial_partial_sum_poly
thf(fact_3910_root__polyfun,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [N: nat,Z: A,A2: A] :
( ( ord_less_eq @ nat @ ( one_one @ nat ) @ N )
=> ( ( ( power_power @ A @ Z @ N )
= A2 )
= ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] :
( times_times @ A
@ ( if @ A
@ ( I3
= ( zero_zero @ nat ) )
@ ( uminus_uminus @ A @ A2 )
@ ( if @ A @ ( I3 = N ) @ ( one_one @ A ) @ ( zero_zero @ A ) ) )
@ ( power_power @ A @ Z @ I3 ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( zero_zero @ A ) ) ) ) ) ).
% root_polyfun
thf(fact_3911_sum__gp0,axiom,
! [A: $tType] :
( ( ( division_ring @ A )
& ( comm_ring @ A ) )
=> ! [X2: A,N: nat] :
( ( ( X2
= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X2 ) @ ( set_ord_atMost @ nat @ N ) )
= ( semiring_1_of_nat @ A @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) ) ) )
& ( ( X2
!= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X2 ) @ ( set_ord_atMost @ nat @ N ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ X2 @ ( suc @ N ) ) ) @ ( minus_minus @ A @ ( one_one @ A ) @ X2 ) ) ) ) ) ) ).
% sum_gp0
thf(fact_3912_gbinomial__sum__nat__pow2,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [M: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K2: nat] : ( divide_divide @ A @ ( gbinomial @ A @ ( semiring_1_of_nat @ A @ ( plus_plus @ nat @ M @ K2 ) ) @ K2 ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ K2 ) )
@ ( set_ord_atMost @ nat @ M ) )
= ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) ) ) ).
% gbinomial_sum_nat_pow2
thf(fact_3913_gbinomial__partial__sum__poly__xpos,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [M: nat,A2: A,X2: A,Y2: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K2: nat] : ( times_times @ A @ ( times_times @ A @ ( gbinomial @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ M ) @ A2 ) @ K2 ) @ ( power_power @ A @ X2 @ K2 ) ) @ ( power_power @ A @ Y2 @ ( minus_minus @ nat @ M @ K2 ) ) )
@ ( set_ord_atMost @ nat @ M ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K2: nat] : ( times_times @ A @ ( times_times @ A @ ( gbinomial @ A @ ( minus_minus @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ K2 ) @ A2 ) @ ( one_one @ A ) ) @ K2 ) @ ( power_power @ A @ X2 @ K2 ) ) @ ( power_power @ A @ ( plus_plus @ A @ X2 @ Y2 ) @ ( minus_minus @ nat @ M @ K2 ) ) )
@ ( set_ord_atMost @ nat @ M ) ) ) ) ).
% gbinomial_partial_sum_poly_xpos
thf(fact_3914_polyfun__diff__alt,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [N: nat,A2: nat > A,X2: A,Y2: A] :
( ( ord_less_eq @ nat @ ( one_one @ nat ) @ N )
=> ( ( minus_minus @ A
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( A2 @ I3 ) @ ( power_power @ A @ X2 @ I3 ) )
@ ( set_ord_atMost @ nat @ N ) )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( A2 @ I3 ) @ ( power_power @ A @ Y2 @ I3 ) )
@ ( set_ord_atMost @ nat @ N ) ) )
= ( times_times @ A @ ( minus_minus @ A @ X2 @ Y2 )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [J3: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K2: nat] : ( times_times @ A @ ( times_times @ A @ ( A2 @ ( plus_plus @ nat @ ( plus_plus @ nat @ J3 @ K2 ) @ ( one_one @ nat ) ) ) @ ( power_power @ A @ Y2 @ K2 ) ) @ ( power_power @ A @ X2 @ J3 ) )
@ ( set_ord_lessThan @ nat @ ( minus_minus @ nat @ N @ J3 ) ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ) ).
% polyfun_diff_alt
thf(fact_3915_polyfun__extremal__lemma,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [E: real,C2: nat > A,N: nat] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E )
=> ? [M10: real] :
! [Z4: A] :
( ( ord_less_eq @ real @ M10 @ ( real_V7770717601297561774m_norm @ A @ Z4 ) )
=> ( ord_less_eq @ real
@ ( real_V7770717601297561774m_norm @ A
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( C2 @ I3 ) @ ( power_power @ A @ Z4 @ I3 ) )
@ ( set_ord_atMost @ nat @ N ) ) )
@ ( times_times @ real @ E @ ( power_power @ real @ ( real_V7770717601297561774m_norm @ A @ Z4 ) @ ( suc @ N ) ) ) ) ) ) ) ).
% polyfun_extremal_lemma
thf(fact_3916_polyfun__diff,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [N: nat,A2: nat > A,X2: A,Y2: A] :
( ( ord_less_eq @ nat @ ( one_one @ nat ) @ N )
=> ( ( minus_minus @ A
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( A2 @ I3 ) @ ( power_power @ A @ X2 @ I3 ) )
@ ( set_ord_atMost @ nat @ N ) )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( A2 @ I3 ) @ ( power_power @ A @ Y2 @ I3 ) )
@ ( set_ord_atMost @ nat @ N ) ) )
= ( times_times @ A @ ( minus_minus @ A @ X2 @ Y2 )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [J3: nat] :
( times_times @ A
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( A2 @ I3 ) @ ( power_power @ A @ Y2 @ ( minus_minus @ nat @ ( minus_minus @ nat @ I3 @ J3 ) @ ( one_one @ nat ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ J3 ) @ N ) )
@ ( power_power @ A @ X2 @ J3 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ) ).
% polyfun_diff
thf(fact_3917_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_3918_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_3919_binomial__code,axiom,
( binomial
= ( ^ [N3: nat,K2: nat] : ( if @ nat @ ( ord_less @ nat @ N3 @ K2 ) @ ( zero_zero @ nat ) @ ( if @ nat @ ( ord_less @ nat @ N3 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K2 ) ) @ ( binomial @ N3 @ ( minus_minus @ nat @ N3 @ K2 ) ) @ ( divide_divide @ nat @ ( set_fo6178422350223883121st_nat @ nat @ ( times_times @ nat ) @ ( plus_plus @ nat @ ( minus_minus @ nat @ N3 @ K2 ) @ ( one_one @ nat ) ) @ N3 @ ( one_one @ nat ) ) @ ( semiring_char_0_fact @ nat @ K2 ) ) ) ) ) ) ).
% binomial_code
thf(fact_3920_choose__even__sum,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( if @ A @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I3 ) @ ( semiring_1_of_nat @ A @ ( binomial @ N @ I3 ) ) @ ( zero_zero @ A ) )
@ ( set_ord_atMost @ nat @ N ) ) )
= ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% choose_even_sum
thf(fact_3921_choose__odd__sum,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] :
( if @ A
@ ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I3 )
@ ( semiring_1_of_nat @ A @ ( binomial @ N @ I3 ) )
@ ( zero_zero @ A ) )
@ ( set_ord_atMost @ nat @ N ) ) )
= ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% choose_odd_sum
thf(fact_3922_binomial__Suc__n,axiom,
! [N: nat] :
( ( binomial @ ( suc @ N ) @ N )
= ( suc @ N ) ) ).
% binomial_Suc_n
thf(fact_3923_binomial__n__n,axiom,
! [N: nat] :
( ( binomial @ N @ N )
= ( one_one @ nat ) ) ).
% binomial_n_n
thf(fact_3924_binomial__0__Suc,axiom,
! [K: nat] :
( ( binomial @ ( zero_zero @ nat ) @ ( suc @ K ) )
= ( zero_zero @ nat ) ) ).
% binomial_0_Suc
thf(fact_3925_binomial__1,axiom,
! [N: nat] :
( ( binomial @ N @ ( suc @ ( zero_zero @ nat ) ) )
= N ) ).
% binomial_1
thf(fact_3926_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_3927_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_3928_binomial__n__0,axiom,
! [N: nat] :
( ( binomial @ N @ ( zero_zero @ nat ) )
= ( one_one @ nat ) ) ).
% binomial_n_0
thf(fact_3929_norm__cis,axiom,
! [A2: real] :
( ( real_V7770717601297561774m_norm @ complex @ ( cis @ A2 ) )
= ( one_one @ real ) ) ).
% norm_cis
thf(fact_3930_cis__zero,axiom,
( ( cis @ ( zero_zero @ real ) )
= ( one_one @ complex ) ) ).
% cis_zero
thf(fact_3931_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_3932_cis__pi,axiom,
( ( cis @ pi )
= ( uminus_uminus @ complex @ ( one_one @ complex ) ) ) ).
% cis_pi
thf(fact_3933_cis__pi__half,axiom,
( ( cis @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
= imaginary_unit ) ).
% cis_pi_half
thf(fact_3934_cis__2pi,axiom,
( ( cis @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) )
= ( one_one @ complex ) ) ).
% cis_2pi
thf(fact_3935_choose__one,axiom,
! [N: nat] :
( ( binomial @ N @ ( one_one @ nat ) )
= N ) ).
% choose_one
thf(fact_3936_cis__neq__zero,axiom,
! [A2: real] :
( ( cis @ A2 )
!= ( zero_zero @ complex ) ) ).
% cis_neq_zero
thf(fact_3937_binomial__eq__0,axiom,
! [N: nat,K: nat] :
( ( ord_less @ nat @ N @ K )
=> ( ( binomial @ N @ K )
= ( zero_zero @ nat ) ) ) ).
% binomial_eq_0
thf(fact_3938_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_3939_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_3940_binomial__symmetric,axiom,
! [K: nat,N: nat] :
( ( ord_less_eq @ nat @ K @ N )
=> ( ( binomial @ N @ K )
= ( binomial @ N @ ( minus_minus @ nat @ N @ K ) ) ) ) ).
% binomial_symmetric
thf(fact_3941_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_3942_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_3943_binomial__gbinomial,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [N: nat,K: nat] :
( ( semiring_1_of_nat @ A @ ( binomial @ N @ K ) )
= ( gbinomial @ A @ ( semiring_1_of_nat @ A @ N ) @ K ) ) ) ).
% binomial_gbinomial
thf(fact_3944_cis__mult,axiom,
! [A2: real,B2: real] :
( ( times_times @ complex @ ( cis @ A2 ) @ ( cis @ B2 ) )
= ( cis @ ( plus_plus @ real @ A2 @ B2 ) ) ) ).
% cis_mult
thf(fact_3945_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_3946_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_3947_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_3948_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_3949_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_3950_sum__choose__upper,axiom,
! [M: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [K2: nat] : ( binomial @ K2 @ M )
@ ( set_ord_atMost @ nat @ N ) )
= ( binomial @ ( suc @ N ) @ ( suc @ M ) ) ) ).
% sum_choose_upper
thf(fact_3951_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_3952_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_3953_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_3954_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_3955_sum__choose__lower,axiom,
! [R2: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [K2: nat] : ( binomial @ ( plus_plus @ nat @ R2 @ K2 ) @ K2 )
@ ( set_ord_atMost @ nat @ N ) )
= ( binomial @ ( suc @ ( plus_plus @ nat @ R2 @ N ) ) @ N ) ) ).
% sum_choose_lower
thf(fact_3956_choose__rising__sum_I2_J,axiom,
! [N: nat,M: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [J3: nat] : ( binomial @ ( plus_plus @ nat @ N @ J3 ) @ N )
@ ( set_ord_atMost @ nat @ M ) )
= ( binomial @ ( plus_plus @ nat @ ( plus_plus @ nat @ N @ M ) @ ( one_one @ nat ) ) @ M ) ) ).
% choose_rising_sum(2)
thf(fact_3957_choose__rising__sum_I1_J,axiom,
! [N: nat,M: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [J3: nat] : ( binomial @ ( plus_plus @ nat @ N @ J3 ) @ N )
@ ( set_ord_atMost @ nat @ M ) )
= ( binomial @ ( plus_plus @ nat @ ( plus_plus @ nat @ N @ M ) @ ( one_one @ nat ) ) @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) ) ) ).
% choose_rising_sum(1)
thf(fact_3958_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_3959_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_3960_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_3961_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_3962_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_3963_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_3964_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_3965_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_3966_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_3967_sum__choose__diagonal,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [K2: nat] : ( binomial @ ( minus_minus @ nat @ N @ K2 ) @ ( minus_minus @ nat @ M @ K2 ) )
@ ( set_ord_atMost @ nat @ M ) )
= ( binomial @ ( suc @ N ) @ M ) ) ) ).
% sum_choose_diagonal
thf(fact_3968_vandermonde,axiom,
! [M: nat,N: nat,R2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [K2: nat] : ( times_times @ nat @ ( binomial @ M @ K2 ) @ ( binomial @ N @ ( minus_minus @ nat @ R2 @ K2 ) ) )
@ ( set_ord_atMost @ nat @ R2 ) )
= ( binomial @ ( plus_plus @ nat @ M @ N ) @ R2 ) ) ).
% vandermonde
thf(fact_3969_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_3970_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_3971_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_3972_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_3973_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_3974_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_3975_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_3976_cis__conv__exp,axiom,
( cis
= ( ^ [B3: real] : ( exp @ complex @ ( times_times @ complex @ imaginary_unit @ ( real_Vector_of_real @ complex @ B3 ) ) ) ) ) ).
% cis_conv_exp
thf(fact_3977_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_3978_binomial,axiom,
! [A2: nat,B2: nat,N: nat] :
( ( power_power @ nat @ ( plus_plus @ nat @ A2 @ B2 ) @ N )
= ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [K2: nat] : ( times_times @ nat @ ( times_times @ nat @ ( semiring_1_of_nat @ nat @ ( binomial @ N @ K2 ) ) @ ( power_power @ nat @ A2 @ K2 ) ) @ ( power_power @ nat @ B2 @ ( minus_minus @ nat @ N @ K2 ) ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ).
% binomial
thf(fact_3979_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_3980_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
@ ^ [K2: nat] : ( times_times @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( binomial @ N @ K2 ) ) @ ( power_power @ A @ A2 @ K2 ) ) @ ( power_power @ A @ B2 @ ( minus_minus @ nat @ N @ K2 ) ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% binomial_ring
thf(fact_3981_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
@ ^ [K2: nat] : ( times_times @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( binomial @ N @ K2 ) ) @ ( comm_s3205402744901411588hammer @ A @ A2 @ K2 ) ) @ ( comm_s3205402744901411588hammer @ A @ B2 @ ( minus_minus @ nat @ N @ K2 ) ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% pochhammer_binomial_sum
thf(fact_3982_choose__square__sum,axiom,
! [N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [K2: nat] : ( power_power @ nat @ ( binomial @ N @ K2 ) @ ( 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_3983_choose__alternating__linear__sum,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [N: nat] :
( ( N
!= ( one_one @ nat ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ I3 ) @ ( semiring_1_of_nat @ A @ I3 ) ) @ ( semiring_1_of_nat @ A @ ( binomial @ N @ I3 ) ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( zero_zero @ A ) ) ) ) ).
% choose_alternating_linear_sum
thf(fact_3984_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_3985_choose__linear__sum,axiom,
! [N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [I3: nat] : ( times_times @ nat @ I3 @ ( binomial @ N @ I3 ) )
@ ( 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_3986_choose__alternating__sum,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ I3 ) @ ( semiring_1_of_nat @ A @ ( binomial @ N @ I3 ) ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( zero_zero @ A ) ) ) ) ).
% choose_alternating_sum
thf(fact_3987_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_3988_sin__x__sin__y,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X2: A,Y2: A] :
( sums @ A
@ ^ [P3: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [N3: nat] :
( if @ A
@ ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ P3 )
& ~ ( 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 @ P3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( semiring_1_of_nat @ int @ ( binomial @ P3 @ N3 ) ) ) ) @ ( semiring_char_0_fact @ real @ P3 ) ) ) @ ( power_power @ A @ X2 @ N3 ) ) @ ( power_power @ A @ Y2 @ ( minus_minus @ nat @ P3 @ N3 ) ) )
@ ( zero_zero @ A ) )
@ ( set_ord_atMost @ nat @ P3 ) )
@ ( times_times @ A @ ( sin @ A @ X2 ) @ ( sin @ A @ Y2 ) ) ) ) ).
% sin_x_sin_y
thf(fact_3989_sums__cos__x__plus__y,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X2: A,Y2: A] :
( sums @ A
@ ^ [P3: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [N3: nat] : ( if @ A @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ P3 ) @ ( 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 @ P3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( semiring_1_of_nat @ int @ ( binomial @ P3 @ N3 ) ) ) ) @ ( semiring_char_0_fact @ real @ P3 ) ) @ ( power_power @ A @ X2 @ N3 ) ) @ ( power_power @ A @ Y2 @ ( minus_minus @ nat @ P3 @ N3 ) ) ) @ ( zero_zero @ A ) )
@ ( set_ord_atMost @ nat @ P3 ) )
@ ( cos @ A @ ( plus_plus @ A @ X2 @ Y2 ) ) ) ) ).
% sums_cos_x_plus_y
thf(fact_3990_cos__x__cos__y,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X2: A,Y2: A] :
( sums @ A
@ ^ [P3: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [N3: nat] :
( if @ A
@ ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ P3 )
& ( 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 @ P3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( semiring_1_of_nat @ int @ ( binomial @ P3 @ N3 ) ) ) ) @ ( semiring_char_0_fact @ real @ P3 ) ) @ ( power_power @ A @ X2 @ N3 ) ) @ ( power_power @ A @ Y2 @ ( minus_minus @ nat @ P3 @ N3 ) ) )
@ ( zero_zero @ A ) )
@ ( set_ord_atMost @ nat @ P3 ) )
@ ( times_times @ A @ ( cos @ A @ X2 ) @ ( cos @ A @ Y2 ) ) ) ) ).
% cos_x_cos_y
thf(fact_3991_bij__betw__roots__unity,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( bij_betw @ nat @ complex
@ ^ [K2: nat] : ( cis @ ( divide_divide @ real @ ( times_times @ real @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) @ ( semiring_1_of_nat @ real @ K2 ) ) @ ( semiring_1_of_nat @ real @ N ) ) )
@ ( set_ord_lessThan @ nat @ N )
@ ( collect @ complex
@ ^ [Z5: complex] :
( ( power_power @ complex @ Z5 @ N )
= ( one_one @ complex ) ) ) ) ) ).
% bij_betw_roots_unity
thf(fact_3992_of__nat__id,axiom,
( ( semiring_1_of_nat @ nat )
= ( ^ [N3: nat] : N3 ) ) ).
% of_nat_id
thf(fact_3993_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_3994_scaleR__cancel__right,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [A2: real,X2: A,B2: real] :
( ( ( real_V8093663219630862766scaleR @ A @ A2 @ X2 )
= ( real_V8093663219630862766scaleR @ A @ B2 @ X2 ) )
= ( ( A2 = B2 )
| ( X2
= ( zero_zero @ A ) ) ) ) ) ).
% scaleR_cancel_right
thf(fact_3995_mult__scaleR__left,axiom,
! [A: $tType] :
( ( real_V6157519004096292374lgebra @ A )
=> ! [A2: real,X2: A,Y2: A] :
( ( times_times @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ X2 ) @ Y2 )
= ( real_V8093663219630862766scaleR @ A @ A2 @ ( times_times @ A @ X2 @ Y2 ) ) ) ) ).
% mult_scaleR_left
thf(fact_3996_mult__scaleR__right,axiom,
! [A: $tType] :
( ( real_V6157519004096292374lgebra @ A )
=> ! [X2: A,A2: real,Y2: A] :
( ( times_times @ A @ X2 @ ( real_V8093663219630862766scaleR @ A @ A2 @ Y2 ) )
= ( real_V8093663219630862766scaleR @ A @ A2 @ ( times_times @ A @ X2 @ Y2 ) ) ) ) ).
% mult_scaleR_right
thf(fact_3997_scaleR__cancel__left,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [A2: real,X2: A,Y2: A] :
( ( ( real_V8093663219630862766scaleR @ A @ A2 @ X2 )
= ( real_V8093663219630862766scaleR @ A @ A2 @ Y2 ) )
= ( ( X2 = Y2 )
| ( A2
= ( zero_zero @ real ) ) ) ) ) ).
% scaleR_cancel_left
thf(fact_3998_scaleR__one,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [X2: A] :
( ( real_V8093663219630862766scaleR @ A @ ( one_one @ real ) @ X2 )
= X2 ) ) ).
% scaleR_one
thf(fact_3999_scaleR__scaleR,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [A2: real,B2: real,X2: A] :
( ( real_V8093663219630862766scaleR @ A @ A2 @ ( real_V8093663219630862766scaleR @ A @ B2 @ X2 ) )
= ( real_V8093663219630862766scaleR @ A @ ( times_times @ real @ A2 @ B2 ) @ X2 ) ) ) ).
% scaleR_scaleR
thf(fact_4000_scaleR__eq__0__iff,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [A2: real,X2: A] :
( ( ( real_V8093663219630862766scaleR @ A @ A2 @ X2 )
= ( zero_zero @ A ) )
= ( ( A2
= ( zero_zero @ real ) )
| ( X2
= ( zero_zero @ A ) ) ) ) ) ).
% scaleR_eq_0_iff
thf(fact_4001_scaleR__zero__left,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [X2: A] :
( ( real_V8093663219630862766scaleR @ A @ ( zero_zero @ real ) @ X2 )
= ( zero_zero @ A ) ) ) ).
% scaleR_zero_left
thf(fact_4002_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_4003_scaleR__power,axiom,
! [A: $tType] :
( ( real_V2822296259951069270ebra_1 @ A )
=> ! [X2: real,Y2: A,N: nat] :
( ( power_power @ A @ ( real_V8093663219630862766scaleR @ A @ X2 @ Y2 ) @ N )
= ( real_V8093663219630862766scaleR @ A @ ( power_power @ real @ X2 @ N ) @ ( power_power @ A @ Y2 @ N ) ) ) ) ).
% scaleR_power
thf(fact_4004_scaleR__minus1__left,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [X2: A] :
( ( real_V8093663219630862766scaleR @ A @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X2 )
= ( uminus_uminus @ A @ X2 ) ) ) ).
% scaleR_minus1_left
thf(fact_4005_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_4006_norm__scaleR,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [A2: real,X2: A] :
( ( real_V7770717601297561774m_norm @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ X2 ) )
= ( times_times @ real @ ( abs_abs @ real @ A2 ) @ ( real_V7770717601297561774m_norm @ A @ X2 ) ) ) ) ).
% norm_scaleR
thf(fact_4007_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_4008_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_4009_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_4010_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_4011_prod_Oreindex__bij__betw,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comm_monoid_mult @ A )
=> ! [H2: B > C,S: set @ B,T3: set @ C,G: C > A] :
( ( bij_betw @ B @ C @ H2 @ S @ T3 )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [X: B] : ( G @ ( H2 @ X ) )
@ S )
= ( groups7121269368397514597t_prod @ C @ A @ G @ T3 ) ) ) ) ).
% prod.reindex_bij_betw
thf(fact_4012_sum_Oreindex__bij__betw,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comm_monoid_add @ A )
=> ! [H2: B > C,S: set @ B,T3: set @ C,G: C > A] :
( ( bij_betw @ B @ C @ H2 @ S @ T3 )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X: B] : ( G @ ( H2 @ X ) )
@ S )
= ( groups7311177749621191930dd_sum @ C @ A @ G @ T3 ) ) ) ) ).
% sum.reindex_bij_betw
thf(fact_4013_scaleR__right__distrib,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [A2: real,X2: A,Y2: A] :
( ( real_V8093663219630862766scaleR @ A @ A2 @ ( plus_plus @ A @ X2 @ Y2 ) )
= ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ X2 ) @ ( real_V8093663219630862766scaleR @ A @ A2 @ Y2 ) ) ) ) ).
% scaleR_right_distrib
thf(fact_4014_scaleR__right__imp__eq,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [X2: A,A2: real,B2: real] :
( ( X2
!= ( zero_zero @ A ) )
=> ( ( ( real_V8093663219630862766scaleR @ A @ A2 @ X2 )
= ( real_V8093663219630862766scaleR @ A @ B2 @ X2 ) )
=> ( A2 = B2 ) ) ) ) ).
% scaleR_right_imp_eq
thf(fact_4015_real__scaleR__def,axiom,
( ( real_V8093663219630862766scaleR @ real )
= ( times_times @ real ) ) ).
% real_scaleR_def
thf(fact_4016_scaleR__left__imp__eq,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [A2: real,X2: A,Y2: A] :
( ( A2
!= ( zero_zero @ real ) )
=> ( ( ( real_V8093663219630862766scaleR @ A @ A2 @ X2 )
= ( real_V8093663219630862766scaleR @ A @ A2 @ Y2 ) )
=> ( X2 = Y2 ) ) ) ) ).
% scaleR_left_imp_eq
thf(fact_4017_scaleR__sum__right,axiom,
! [A: $tType,C: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [A2: real,F2: C > A,A3: set @ C] :
( ( real_V8093663219630862766scaleR @ A @ A2 @ ( groups7311177749621191930dd_sum @ C @ A @ F2 @ A3 ) )
= ( groups7311177749621191930dd_sum @ C @ A
@ ^ [X: C] : ( real_V8093663219630862766scaleR @ A @ A2 @ ( F2 @ X ) )
@ A3 ) ) ) ).
% scaleR_sum_right
thf(fact_4018_scaleR__right_Osum,axiom,
! [A: $tType,C: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [A2: real,G: C > A,A3: set @ C] :
( ( real_V8093663219630862766scaleR @ A @ A2 @ ( groups7311177749621191930dd_sum @ C @ A @ G @ A3 ) )
= ( groups7311177749621191930dd_sum @ C @ A
@ ^ [X: C] : ( real_V8093663219630862766scaleR @ A @ A2 @ ( G @ X ) )
@ A3 ) ) ) ).
% scaleR_right.sum
thf(fact_4019_summable__scaleR__right,axiom,
! [B: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [X6: nat > B,R2: real] :
( ( summable @ B @ X6 )
=> ( summable @ B
@ ^ [N3: nat] : ( real_V8093663219630862766scaleR @ B @ R2 @ ( X6 @ N3 ) ) ) ) ) ).
% summable_scaleR_right
thf(fact_4020_sums__scaleR__right,axiom,
! [B: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [X6: nat > B,A2: B,R2: real] :
( ( sums @ B @ X6 @ A2 )
=> ( sums @ B
@ ^ [N3: nat] : ( real_V8093663219630862766scaleR @ B @ R2 @ ( X6 @ N3 ) )
@ ( real_V8093663219630862766scaleR @ B @ R2 @ A2 ) ) ) ) ).
% sums_scaleR_right
thf(fact_4021_scaleR__left_Oadd,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [X2: real,Y2: real,Xa: A] :
( ( real_V8093663219630862766scaleR @ A @ ( plus_plus @ real @ X2 @ Y2 ) @ Xa )
= ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ X2 @ Xa ) @ ( real_V8093663219630862766scaleR @ A @ Y2 @ Xa ) ) ) ) ).
% scaleR_left.add
thf(fact_4022_scaleR__left__distrib,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [A2: real,B2: real,X2: A] :
( ( real_V8093663219630862766scaleR @ A @ ( plus_plus @ real @ A2 @ B2 ) @ X2 )
= ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ X2 ) @ ( real_V8093663219630862766scaleR @ A @ B2 @ X2 ) ) ) ) ).
% scaleR_left_distrib
thf(fact_4023_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_4024_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_4025_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_4026_scaleR__sum__left,axiom,
! [A: $tType,C: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [F2: C > real,A3: set @ C,X2: A] :
( ( real_V8093663219630862766scaleR @ A @ ( groups7311177749621191930dd_sum @ C @ real @ F2 @ A3 ) @ X2 )
= ( groups7311177749621191930dd_sum @ C @ A
@ ^ [A4: C] : ( real_V8093663219630862766scaleR @ A @ ( F2 @ A4 ) @ X2 )
@ A3 ) ) ) ).
% scaleR_sum_left
thf(fact_4027_scaleR__left_Osum,axiom,
! [A: $tType,C: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [G: C > real,A3: set @ C,X2: A] :
( ( real_V8093663219630862766scaleR @ A @ ( groups7311177749621191930dd_sum @ C @ real @ G @ A3 ) @ X2 )
= ( groups7311177749621191930dd_sum @ C @ A
@ ^ [X: C] : ( real_V8093663219630862766scaleR @ A @ ( G @ X ) @ X2 )
@ A3 ) ) ) ).
% scaleR_left.sum
thf(fact_4028_suminf__scaleR__right,axiom,
! [B: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [X6: nat > B,R2: real] :
( ( summable @ B @ X6 )
=> ( ( real_V8093663219630862766scaleR @ B @ R2 @ ( suminf @ B @ X6 ) )
= ( suminf @ B
@ ^ [N3: nat] : ( real_V8093663219630862766scaleR @ B @ R2 @ ( X6 @ N3 ) ) ) ) ) ) ).
% suminf_scaleR_right
thf(fact_4029_summable__scaleR__left,axiom,
! [B: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [X6: nat > real,X2: B] :
( ( summable @ real @ X6 )
=> ( summable @ B
@ ^ [N3: nat] : ( real_V8093663219630862766scaleR @ B @ ( X6 @ N3 ) @ X2 ) ) ) ) ).
% summable_scaleR_left
thf(fact_4030_sums__scaleR__left,axiom,
! [B: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [X6: nat > real,A2: real,X2: B] :
( ( sums @ real @ X6 @ A2 )
=> ( sums @ B
@ ^ [N3: nat] : ( real_V8093663219630862766scaleR @ B @ ( X6 @ N3 ) @ X2 )
@ ( real_V8093663219630862766scaleR @ B @ A2 @ X2 ) ) ) ) ).
% sums_scaleR_left
thf(fact_4031_scaleR__right__mono,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [A2: real,B2: real,X2: A] :
( ( ord_less_eq @ real @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X2 )
=> ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ X2 ) @ ( real_V8093663219630862766scaleR @ A @ B2 @ X2 ) ) ) ) ) ).
% scaleR_right_mono
thf(fact_4032_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_4033_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_4034_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_4035_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_4036_scaleR__left__mono,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [X2: A,Y2: A,A2: real] :
( ( ord_less_eq @ A @ X2 @ Y2 )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ A2 )
=> ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ X2 ) @ ( real_V8093663219630862766scaleR @ A @ A2 @ Y2 ) ) ) ) ) ).
% scaleR_left_mono
thf(fact_4037_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_4038_eq__vector__fraction__iff,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [X2: A,U: real,V: real,A2: A] :
( ( X2
= ( real_V8093663219630862766scaleR @ A @ ( divide_divide @ real @ U @ V ) @ A2 ) )
= ( ( ( V
= ( zero_zero @ real ) )
=> ( X2
= ( zero_zero @ A ) ) )
& ( ( V
!= ( zero_zero @ real ) )
=> ( ( real_V8093663219630862766scaleR @ A @ V @ X2 )
= ( real_V8093663219630862766scaleR @ A @ U @ A2 ) ) ) ) ) ) ).
% eq_vector_fraction_iff
thf(fact_4039_vector__fraction__eq__iff,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [U: real,V: real,A2: A,X2: A] :
( ( ( real_V8093663219630862766scaleR @ A @ ( divide_divide @ real @ U @ V ) @ A2 )
= X2 )
= ( ( ( V
= ( zero_zero @ real ) )
=> ( X2
= ( zero_zero @ A ) ) )
& ( ( V
!= ( zero_zero @ real ) )
=> ( ( real_V8093663219630862766scaleR @ A @ U @ A2 )
= ( real_V8093663219630862766scaleR @ A @ V @ X2 ) ) ) ) ) ) ).
% vector_fraction_eq_iff
thf(fact_4040_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_4041_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_4042_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_4043_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_4044_suminf__scaleR__left,axiom,
! [B: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [X6: nat > real,X2: B] :
( ( summable @ real @ X6 )
=> ( ( real_V8093663219630862766scaleR @ B @ ( suminf @ real @ X6 ) @ X2 )
= ( suminf @ B
@ ^ [N3: nat] : ( real_V8093663219630862766scaleR @ B @ ( X6 @ N3 ) @ X2 ) ) ) ) ) ).
% suminf_scaleR_left
thf(fact_4045_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_4046_scaleR__nonpos__nonneg,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [A2: real,X2: A] :
( ( ord_less_eq @ real @ A2 @ ( zero_zero @ real ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X2 )
=> ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ X2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% scaleR_nonpos_nonneg
thf(fact_4047_scaleR__nonneg__nonpos,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [A2: real,X2: A] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ A2 )
=> ( ( ord_less_eq @ A @ X2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ X2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% scaleR_nonneg_nonpos
thf(fact_4048_scaleR__nonneg__nonneg,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [A2: real,X2: A] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ A2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X2 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( real_V8093663219630862766scaleR @ A @ A2 @ X2 ) ) ) ) ) ).
% scaleR_nonneg_nonneg
thf(fact_4049_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_4050_split__scaleR__neg__le,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [A2: real,X2: A] :
( ( ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ A2 )
& ( ord_less_eq @ A @ X2 @ ( zero_zero @ A ) ) )
| ( ( ord_less_eq @ real @ A2 @ ( zero_zero @ real ) )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ X2 ) ) )
=> ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ X2 ) @ ( zero_zero @ A ) ) ) ) ).
% split_scaleR_neg_le
thf(fact_4051_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_4052_scaleR__mono,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [A2: real,B2: real,X2: A,Y2: A] :
( ( ord_less_eq @ real @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ X2 @ Y2 )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ B2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X2 )
=> ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ X2 ) @ ( real_V8093663219630862766scaleR @ A @ B2 @ Y2 ) ) ) ) ) ) ) ).
% scaleR_mono
thf(fact_4053_scaleR__2,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [X2: A] :
( ( real_V8093663219630862766scaleR @ A @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ X2 )
= ( plus_plus @ A @ X2 @ X2 ) ) ) ).
% scaleR_2
thf(fact_4054_scaleR__left__le__one__le,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [X2: A,A2: real] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X2 )
=> ( ( ord_less_eq @ real @ A2 @ ( one_one @ real ) )
=> ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ X2 ) @ X2 ) ) ) ) ).
% scaleR_left_le_one_le
thf(fact_4055_sum_Oreindex__bij__betw__not__neutral,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comm_monoid_add @ A )
=> ! [S6: set @ B,T4: set @ C,H2: B > C,S: set @ B,T3: set @ C,G: C > A] :
( ( finite_finite2 @ B @ S6 )
=> ( ( finite_finite2 @ C @ T4 )
=> ( ( bij_betw @ B @ C @ H2 @ ( minus_minus @ ( set @ B ) @ S @ S6 ) @ ( minus_minus @ ( set @ C ) @ T3 @ T4 ) )
=> ( ! [A5: B] :
( ( member @ B @ A5 @ S6 )
=> ( ( G @ ( H2 @ A5 ) )
= ( zero_zero @ A ) ) )
=> ( ! [B4: C] :
( ( member @ C @ B4 @ T4 )
=> ( ( G @ B4 )
= ( zero_zero @ A ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X: B] : ( G @ ( H2 @ X ) )
@ S )
= ( groups7311177749621191930dd_sum @ C @ A @ G @ T3 ) ) ) ) ) ) ) ) ).
% sum.reindex_bij_betw_not_neutral
thf(fact_4056_prod_Oreindex__bij__betw__not__neutral,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S6: set @ B,T4: set @ C,H2: B > C,S: set @ B,T3: set @ C,G: C > A] :
( ( finite_finite2 @ B @ S6 )
=> ( ( finite_finite2 @ C @ T4 )
=> ( ( bij_betw @ B @ C @ H2 @ ( minus_minus @ ( set @ B ) @ S @ S6 ) @ ( minus_minus @ ( set @ C ) @ T3 @ T4 ) )
=> ( ! [A5: B] :
( ( member @ B @ A5 @ S6 )
=> ( ( G @ ( H2 @ A5 ) )
= ( one_one @ A ) ) )
=> ( ! [B4: C] :
( ( member @ C @ B4 @ T4 )
=> ( ( G @ B4 )
= ( one_one @ A ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [X: B] : ( G @ ( H2 @ X ) )
@ S )
= ( groups7121269368397514597t_prod @ C @ A @ G @ T3 ) ) ) ) ) ) ) ) ).
% prod.reindex_bij_betw_not_neutral
thf(fact_4057_real__vector__affinity__eq,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [M: real,X2: A,C2: A,Y2: A] :
( ( M
!= ( zero_zero @ real ) )
=> ( ( ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ M @ X2 ) @ C2 )
= Y2 )
= ( X2
= ( minus_minus @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ M ) @ Y2 ) @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ M ) @ C2 ) ) ) ) ) ) ).
% real_vector_affinity_eq
thf(fact_4058_real__vector__eq__affinity,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [M: real,Y2: A,X2: A,C2: A] :
( ( M
!= ( zero_zero @ real ) )
=> ( ( Y2
= ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ M @ X2 ) @ C2 ) )
= ( ( minus_minus @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ M ) @ Y2 ) @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ M ) @ C2 ) )
= X2 ) ) ) ) ).
% real_vector_eq_affinity
thf(fact_4059_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_4060_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_4061_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_4062_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_4063_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_4064_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_4065_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_4066_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_4067_nonzero__inverse__scaleR__distrib,axiom,
! [A: $tType] :
( ( real_V5047593784448816457lgebra @ A )
=> ! [A2: real,X2: A] :
( ( A2
!= ( zero_zero @ real ) )
=> ( ( X2
!= ( zero_zero @ A ) )
=> ( ( inverse_inverse @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ X2 ) )
= ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ A2 ) @ ( inverse_inverse @ A @ X2 ) ) ) ) ) ) ).
% nonzero_inverse_scaleR_distrib
thf(fact_4068_summable__exp__generic,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X2: A] :
( summable @ A
@ ^ [N3: nat] : ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ N3 ) ) @ ( power_power @ A @ X2 @ N3 ) ) ) ) ).
% summable_exp_generic
thf(fact_4069_sin__converges,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X2: A] :
( sums @ A
@ ^ [N3: nat] : ( real_V8093663219630862766scaleR @ A @ ( sin_coeff @ N3 ) @ ( power_power @ A @ X2 @ N3 ) )
@ ( sin @ A @ X2 ) ) ) ).
% sin_converges
thf(fact_4070_sin__def,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ( ( sin @ A )
= ( ^ [X: A] :
( suminf @ A
@ ^ [N3: nat] : ( real_V8093663219630862766scaleR @ A @ ( sin_coeff @ N3 ) @ ( power_power @ A @ X @ N3 ) ) ) ) ) ) ).
% sin_def
thf(fact_4071_cos__converges,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X2: A] :
( sums @ A
@ ^ [N3: nat] : ( real_V8093663219630862766scaleR @ A @ ( cos_coeff @ N3 ) @ ( power_power @ A @ X2 @ N3 ) )
@ ( cos @ A @ X2 ) ) ) ).
% cos_converges
thf(fact_4072_cos__def,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ( ( cos @ A )
= ( ^ [X: A] :
( suminf @ A
@ ^ [N3: nat] : ( real_V8093663219630862766scaleR @ A @ ( cos_coeff @ N3 ) @ ( power_power @ A @ X @ N3 ) ) ) ) ) ) ).
% cos_def
thf(fact_4073_summable__norm__sin,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X2: A] :
( summable @ real
@ ^ [N3: nat] : ( real_V7770717601297561774m_norm @ A @ ( real_V8093663219630862766scaleR @ A @ ( sin_coeff @ N3 ) @ ( power_power @ A @ X2 @ N3 ) ) ) ) ) ).
% summable_norm_sin
thf(fact_4074_summable__norm__cos,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X2: A] :
( summable @ real
@ ^ [N3: nat] : ( real_V7770717601297561774m_norm @ A @ ( real_V8093663219630862766scaleR @ A @ ( cos_coeff @ N3 ) @ ( power_power @ A @ X2 @ N3 ) ) ) ) ) ).
% summable_norm_cos
thf(fact_4075_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_4076_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_4077_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_4078_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_4079_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_4080_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_4081_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_4082_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_4083_exp__converges,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X2: A] :
( sums @ A
@ ^ [N3: nat] : ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ N3 ) ) @ ( power_power @ A @ X2 @ N3 ) )
@ ( exp @ A @ X2 ) ) ) ).
% exp_converges
thf(fact_4084_exp__def,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ( ( exp @ A )
= ( ^ [X: A] :
( suminf @ A
@ ^ [N3: nat] : ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ N3 ) ) @ ( power_power @ A @ X @ N3 ) ) ) ) ) ) ).
% exp_def
thf(fact_4085_summable__norm__exp,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X2: A] :
( summable @ real
@ ^ [N3: nat] : ( real_V7770717601297561774m_norm @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ N3 ) ) @ ( power_power @ A @ X2 @ N3 ) ) ) ) ) ).
% summable_norm_exp
thf(fact_4086_sin__minus__converges,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X2: A] :
( sums @ A
@ ^ [N3: nat] : ( uminus_uminus @ A @ ( real_V8093663219630862766scaleR @ A @ ( sin_coeff @ N3 ) @ ( power_power @ A @ ( uminus_uminus @ A @ X2 ) @ N3 ) ) )
@ ( sin @ A @ X2 ) ) ) ).
% sin_minus_converges
thf(fact_4087_cos__minus__converges,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X2: A] :
( sums @ A
@ ^ [N3: nat] : ( real_V8093663219630862766scaleR @ A @ ( cos_coeff @ N3 ) @ ( power_power @ A @ ( uminus_uminus @ A @ X2 ) @ N3 ) )
@ ( cos @ A @ X2 ) ) ) ).
% cos_minus_converges
thf(fact_4088_exp__series__add__commuting,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X2: A,Y2: A,N: nat] :
( ( ( times_times @ A @ X2 @ Y2 )
= ( times_times @ A @ Y2 @ X2 ) )
=> ( ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ A @ ( plus_plus @ A @ X2 @ Y2 ) @ N ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ I3 ) ) @ ( power_power @ A @ X2 @ I3 ) ) @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ ( minus_minus @ nat @ N @ I3 ) ) ) @ ( power_power @ A @ Y2 @ ( minus_minus @ nat @ N @ I3 ) ) ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ) ).
% exp_series_add_commuting
thf(fact_4089_exp__first__term,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ( ( exp @ A )
= ( ^ [X: A] :
( plus_plus @ A @ ( one_one @ A )
@ ( suminf @ A
@ ^ [N3: nat] : ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ ( suc @ N3 ) ) ) @ ( power_power @ A @ X @ ( suc @ N3 ) ) ) ) ) ) ) ) ).
% exp_first_term
thf(fact_4090_exp__first__terms,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [K: nat] :
( ( exp @ A )
= ( ^ [X: A] :
( plus_plus @ A
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [N3: nat] : ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ N3 ) ) @ ( power_power @ A @ X @ N3 ) )
@ ( set_ord_lessThan @ nat @ K ) )
@ ( suminf @ A
@ ^ [N3: nat] : ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ ( plus_plus @ nat @ N3 @ K ) ) ) @ ( power_power @ A @ X @ ( plus_plus @ nat @ N3 @ K ) ) ) ) ) ) ) ) ).
% exp_first_terms
thf(fact_4091_exp__first__two__terms,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ( ( exp @ A )
= ( ^ [X: A] :
( plus_plus @ A @ ( plus_plus @ A @ ( one_one @ A ) @ X )
@ ( suminf @ A
@ ^ [N3: nat] : ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ ( plus_plus @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( power_power @ A @ X @ ( plus_plus @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% exp_first_two_terms
thf(fact_4092_sinh__converges,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X2: 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 @ X2 @ N3 ) ) )
@ ( sinh @ A @ X2 ) ) ) ).
% sinh_converges
thf(fact_4093_round__altdef,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ( ( archimedean_round @ A )
= ( ^ [X: A] : ( if @ int @ ( ord_less_eq @ A @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( archimedean_frac @ A @ X ) ) @ ( archimedean_ceiling @ A @ X ) @ ( archim6421214686448440834_floor @ A @ X ) ) ) ) ) ).
% round_altdef
thf(fact_4094_cosh__converges,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X2: 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 @ X2 @ N3 ) ) @ ( zero_zero @ A ) )
@ ( cosh @ A @ X2 ) ) ) ).
% cosh_converges
thf(fact_4095_sinh__real__zero__iff,axiom,
! [X2: real] :
( ( ( sinh @ real @ X2 )
= ( zero_zero @ real ) )
= ( X2
= ( zero_zero @ real ) ) ) ).
% sinh_real_zero_iff
thf(fact_4096_sinh__real__le__iff,axiom,
! [X2: real,Y2: real] :
( ( ord_less_eq @ real @ ( sinh @ real @ X2 ) @ ( sinh @ real @ Y2 ) )
= ( ord_less_eq @ real @ X2 @ Y2 ) ) ).
% sinh_real_le_iff
thf(fact_4097_sinh__real__pos__iff,axiom,
! [X2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( sinh @ real @ X2 ) )
= ( ord_less @ real @ ( zero_zero @ real ) @ X2 ) ) ).
% sinh_real_pos_iff
thf(fact_4098_sinh__real__neg__iff,axiom,
! [X2: real] :
( ( ord_less @ real @ ( sinh @ real @ X2 ) @ ( zero_zero @ real ) )
= ( ord_less @ real @ X2 @ ( zero_zero @ real ) ) ) ).
% sinh_real_neg_iff
thf(fact_4099_sinh__real__nonpos__iff,axiom,
! [X2: real] :
( ( ord_less_eq @ real @ ( sinh @ real @ X2 ) @ ( zero_zero @ real ) )
= ( ord_less_eq @ real @ X2 @ ( zero_zero @ real ) ) ) ).
% sinh_real_nonpos_iff
thf(fact_4100_sinh__real__nonneg__iff,axiom,
! [X2: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( sinh @ real @ X2 ) )
= ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 ) ) ).
% sinh_real_nonneg_iff
thf(fact_4101_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_4102_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_4103_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_4104_tanh__def,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ( ( tanh @ A )
= ( ^ [X: A] : ( divide_divide @ A @ ( sinh @ A @ X ) @ ( cosh @ A @ X ) ) ) ) ) ).
% tanh_def
thf(fact_4105_cosh__plus__sinh,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X2: A] :
( ( plus_plus @ A @ ( cosh @ A @ X2 ) @ ( sinh @ A @ X2 ) )
= ( exp @ A @ X2 ) ) ) ).
% cosh_plus_sinh
thf(fact_4106_sinh__plus__cosh,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X2: A] :
( ( plus_plus @ A @ ( sinh @ A @ X2 ) @ ( cosh @ A @ X2 ) )
= ( exp @ A @ X2 ) ) ) ).
% sinh_plus_cosh
thf(fact_4107_cosh__real__nonzero,axiom,
! [X2: real] :
( ( cosh @ real @ X2 )
!= ( zero_zero @ real ) ) ).
% cosh_real_nonzero
thf(fact_4108_sinh__diff,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X2: A,Y2: A] :
( ( sinh @ A @ ( minus_minus @ A @ X2 @ Y2 ) )
= ( minus_minus @ A @ ( times_times @ A @ ( sinh @ A @ X2 ) @ ( cosh @ A @ Y2 ) ) @ ( times_times @ A @ ( cosh @ A @ X2 ) @ ( sinh @ A @ Y2 ) ) ) ) ) ).
% sinh_diff
thf(fact_4109_cosh__diff,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X2: A,Y2: A] :
( ( cosh @ A @ ( minus_minus @ A @ X2 @ Y2 ) )
= ( minus_minus @ A @ ( times_times @ A @ ( cosh @ A @ X2 ) @ ( cosh @ A @ Y2 ) ) @ ( times_times @ A @ ( sinh @ A @ X2 ) @ ( sinh @ A @ Y2 ) ) ) ) ) ).
% cosh_diff
thf(fact_4110_sinh__add,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X2: A,Y2: A] :
( ( sinh @ A @ ( plus_plus @ A @ X2 @ Y2 ) )
= ( plus_plus @ A @ ( times_times @ A @ ( sinh @ A @ X2 ) @ ( cosh @ A @ Y2 ) ) @ ( times_times @ A @ ( cosh @ A @ X2 ) @ ( sinh @ A @ Y2 ) ) ) ) ) ).
% sinh_add
thf(fact_4111_cosh__add,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X2: A,Y2: A] :
( ( cosh @ A @ ( plus_plus @ A @ X2 @ Y2 ) )
= ( plus_plus @ A @ ( times_times @ A @ ( cosh @ A @ X2 ) @ ( cosh @ A @ Y2 ) ) @ ( times_times @ A @ ( sinh @ A @ X2 ) @ ( sinh @ A @ Y2 ) ) ) ) ) ).
% cosh_add
thf(fact_4112_sinh__le__cosh__real,axiom,
! [X2: real] : ( ord_less_eq @ real @ ( sinh @ real @ X2 ) @ ( cosh @ real @ X2 ) ) ).
% sinh_le_cosh_real
thf(fact_4113_cosh__real__pos,axiom,
! [X2: real] : ( ord_less @ real @ ( zero_zero @ real ) @ ( cosh @ real @ X2 ) ) ).
% cosh_real_pos
thf(fact_4114_cosh__real__nonpos__le__iff,axiom,
! [X2: real,Y2: real] :
( ( ord_less_eq @ real @ X2 @ ( zero_zero @ real ) )
=> ( ( ord_less_eq @ real @ Y2 @ ( zero_zero @ real ) )
=> ( ( ord_less_eq @ real @ ( cosh @ real @ X2 ) @ ( cosh @ real @ Y2 ) )
= ( ord_less_eq @ real @ Y2 @ X2 ) ) ) ) ).
% cosh_real_nonpos_le_iff
thf(fact_4115_cosh__real__nonneg__le__iff,axiom,
! [X2: real,Y2: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y2 )
=> ( ( ord_less_eq @ real @ ( cosh @ real @ X2 ) @ ( cosh @ real @ Y2 ) )
= ( ord_less_eq @ real @ X2 @ Y2 ) ) ) ) ).
% cosh_real_nonneg_le_iff
thf(fact_4116_cosh__real__nonneg,axiom,
! [X2: real] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( cosh @ real @ X2 ) ) ).
% cosh_real_nonneg
thf(fact_4117_sinh__double,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X2: A] :
( ( sinh @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X2 ) )
= ( times_times @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( sinh @ A @ X2 ) ) @ ( cosh @ A @ X2 ) ) ) ) ).
% sinh_double
thf(fact_4118_cosh__real__ge__1,axiom,
! [X2: real] : ( ord_less_eq @ real @ ( one_one @ real ) @ ( cosh @ real @ X2 ) ) ).
% cosh_real_ge_1
thf(fact_4119_frac__ge__0,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( archimedean_frac @ A @ X2 ) ) ) ).
% frac_ge_0
thf(fact_4120_frac__lt__1,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A] : ( ord_less @ A @ ( archimedean_frac @ A @ X2 ) @ ( one_one @ A ) ) ) ).
% frac_lt_1
thf(fact_4121_frac__1__eq,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A] :
( ( archimedean_frac @ A @ ( plus_plus @ A @ X2 @ ( one_one @ A ) ) )
= ( archimedean_frac @ A @ X2 ) ) ) ).
% frac_1_eq
thf(fact_4122_cosh__real__strict__mono,axiom,
! [X2: real,Y2: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ord_less @ real @ X2 @ Y2 )
=> ( ord_less @ real @ ( cosh @ real @ X2 ) @ ( cosh @ real @ Y2 ) ) ) ) ).
% cosh_real_strict_mono
thf(fact_4123_cosh__real__nonneg__less__iff,axiom,
! [X2: real,Y2: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y2 )
=> ( ( ord_less @ real @ ( cosh @ real @ X2 ) @ ( cosh @ real @ Y2 ) )
= ( ord_less @ real @ X2 @ Y2 ) ) ) ) ).
% cosh_real_nonneg_less_iff
thf(fact_4124_cosh__real__nonpos__less__iff,axiom,
! [X2: real,Y2: real] :
( ( ord_less_eq @ real @ X2 @ ( zero_zero @ real ) )
=> ( ( ord_less_eq @ real @ Y2 @ ( zero_zero @ real ) )
=> ( ( ord_less @ real @ ( cosh @ real @ X2 ) @ ( cosh @ real @ Y2 ) )
= ( ord_less @ real @ Y2 @ X2 ) ) ) ) ).
% cosh_real_nonpos_less_iff
thf(fact_4125_cosh__square__eq,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X2: A] :
( ( power_power @ A @ ( cosh @ A @ X2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( plus_plus @ A @ ( power_power @ A @ ( sinh @ A @ X2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) ) ) ) ).
% cosh_square_eq
thf(fact_4126_sinh__square__eq,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X2: A] :
( ( power_power @ A @ ( sinh @ A @ X2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( minus_minus @ A @ ( power_power @ A @ ( cosh @ A @ X2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) ) ) ) ).
% sinh_square_eq
thf(fact_4127_hyperbolic__pythagoras,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X2: A] :
( ( minus_minus @ A @ ( power_power @ A @ ( cosh @ A @ X2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ ( sinh @ A @ X2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( one_one @ A ) ) ) ).
% hyperbolic_pythagoras
thf(fact_4128_arcosh__cosh__real,axiom,
! [X2: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( arcosh @ real @ ( cosh @ real @ X2 ) )
= X2 ) ) ).
% arcosh_cosh_real
thf(fact_4129_cosh__double,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X2: A] :
( ( cosh @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X2 ) )
= ( plus_plus @ A @ ( power_power @ A @ ( cosh @ A @ X2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ ( sinh @ A @ X2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% cosh_double
thf(fact_4130_frac__eq,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A] :
( ( ( archimedean_frac @ A @ X2 )
= X2 )
= ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X2 )
& ( ord_less @ A @ X2 @ ( one_one @ A ) ) ) ) ) ).
% frac_eq
thf(fact_4131_frac__add,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A,Y2: A] :
( ( ( ord_less @ A @ ( plus_plus @ A @ ( archimedean_frac @ A @ X2 ) @ ( archimedean_frac @ A @ Y2 ) ) @ ( one_one @ A ) )
=> ( ( archimedean_frac @ A @ ( plus_plus @ A @ X2 @ Y2 ) )
= ( plus_plus @ A @ ( archimedean_frac @ A @ X2 ) @ ( archimedean_frac @ A @ Y2 ) ) ) )
& ( ~ ( ord_less @ A @ ( plus_plus @ A @ ( archimedean_frac @ A @ X2 ) @ ( archimedean_frac @ A @ Y2 ) ) @ ( one_one @ A ) )
=> ( ( archimedean_frac @ A @ ( plus_plus @ A @ X2 @ Y2 ) )
= ( minus_minus @ A @ ( plus_plus @ A @ ( archimedean_frac @ A @ X2 ) @ ( archimedean_frac @ A @ Y2 ) ) @ ( one_one @ A ) ) ) ) ) ) ).
% frac_add
thf(fact_4132_tanh__add,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X2: A,Y2: A] :
( ( ( cosh @ A @ X2 )
!= ( zero_zero @ A ) )
=> ( ( ( cosh @ A @ Y2 )
!= ( zero_zero @ A ) )
=> ( ( tanh @ A @ ( plus_plus @ A @ X2 @ Y2 ) )
= ( divide_divide @ A @ ( plus_plus @ A @ ( tanh @ A @ X2 ) @ ( tanh @ A @ Y2 ) ) @ ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( tanh @ A @ X2 ) @ ( tanh @ A @ Y2 ) ) ) ) ) ) ) ) ).
% tanh_add
thf(fact_4133_cosh__field__def,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ( ( cosh @ A )
= ( ^ [Z5: A] : ( divide_divide @ A @ ( plus_plus @ A @ ( exp @ A @ Z5 ) @ ( exp @ A @ ( uminus_uminus @ A @ Z5 ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% cosh_field_def
thf(fact_4134_floor__add,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A,Y2: A] :
( ( ( ord_less @ A @ ( plus_plus @ A @ ( archimedean_frac @ A @ X2 ) @ ( archimedean_frac @ A @ Y2 ) ) @ ( one_one @ A ) )
=> ( ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ X2 @ Y2 ) )
= ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X2 ) @ ( archim6421214686448440834_floor @ A @ Y2 ) ) ) )
& ( ~ ( ord_less @ A @ ( plus_plus @ A @ ( archimedean_frac @ A @ X2 ) @ ( archimedean_frac @ A @ Y2 ) ) @ ( one_one @ A ) )
=> ( ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ X2 @ Y2 ) )
= ( plus_plus @ int @ ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X2 ) @ ( archim6421214686448440834_floor @ A @ Y2 ) ) @ ( one_one @ int ) ) ) ) ) ) ).
% floor_add
thf(fact_4135_sinh__field__def,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ( ( sinh @ A )
= ( ^ [Z5: A] : ( divide_divide @ A @ ( minus_minus @ A @ ( exp @ A @ Z5 ) @ ( exp @ A @ ( uminus_uminus @ A @ Z5 ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% sinh_field_def
thf(fact_4136_cosh__zero__iff,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X2: A] :
( ( ( cosh @ A @ X2 )
= ( zero_zero @ A ) )
= ( ( power_power @ A @ ( exp @ A @ X2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ).
% cosh_zero_iff
thf(fact_4137_cosh__def,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ( ( cosh @ A )
= ( ^ [X: A] : ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( plus_plus @ A @ ( exp @ A @ X ) @ ( exp @ A @ ( uminus_uminus @ A @ X ) ) ) ) ) ) ) ).
% cosh_def
thf(fact_4138_cosh__ln__real,axiom,
! [X2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( cosh @ real @ ( ln_ln @ real @ X2 ) )
= ( divide_divide @ real @ ( plus_plus @ real @ X2 @ ( inverse_inverse @ real @ X2 ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ).
% cosh_ln_real
thf(fact_4139_sinh__def,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ( ( sinh @ A )
= ( ^ [X: A] : ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( minus_minus @ A @ ( exp @ A @ X ) @ ( exp @ A @ ( uminus_uminus @ A @ X ) ) ) ) ) ) ) ).
% sinh_def
thf(fact_4140_sinh__ln__real,axiom,
! [X2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( sinh @ real @ ( ln_ln @ real @ X2 ) )
= ( divide_divide @ real @ ( minus_minus @ real @ X2 @ ( inverse_inverse @ real @ X2 ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ).
% sinh_ln_real
thf(fact_4141_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
@ ^ [Z5: complex] :
( ( power_power @ complex @ Z5 @ N )
= ( one_one @ complex ) ) )
@ ( collect @ complex
@ ^ [Z5: complex] :
( ( power_power @ complex @ Z5 @ N )
= C2 ) ) ) ) ) ).
% bij_betw_nth_root_unity
thf(fact_4142_cot__less__zero,axiom,
! [X2: real] :
( ( ord_less @ real @ ( divide_divide @ real @ ( uminus_uminus @ real @ pi ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ X2 )
=> ( ( ord_less @ real @ X2 @ ( zero_zero @ real ) )
=> ( ord_less @ real @ ( cot @ real @ X2 ) @ ( zero_zero @ real ) ) ) ) ).
% cot_less_zero
thf(fact_4143_real__root__zero,axiom,
! [N: nat] :
( ( root @ N @ ( zero_zero @ real ) )
= ( zero_zero @ real ) ) ).
% real_root_zero
thf(fact_4144_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_4145_real__root__Suc__0,axiom,
! [X2: real] :
( ( root @ ( suc @ ( zero_zero @ nat ) ) @ X2 )
= X2 ) ).
% real_root_Suc_0
thf(fact_4146_real__root__eq__iff,axiom,
! [N: nat,X2: real,Y2: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ( root @ N @ X2 )
= ( root @ N @ Y2 ) )
= ( X2 = Y2 ) ) ) ).
% real_root_eq_iff
thf(fact_4147_root__0,axiom,
! [X2: real] :
( ( root @ ( zero_zero @ nat ) @ X2 )
= ( zero_zero @ real ) ) ).
% root_0
thf(fact_4148_cot__pi,axiom,
( ( cot @ real @ pi )
= ( zero_zero @ real ) ) ).
% cot_pi
thf(fact_4149_real__root__eq__0__iff,axiom,
! [N: nat,X2: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ( root @ N @ X2 )
= ( zero_zero @ real ) )
= ( X2
= ( zero_zero @ real ) ) ) ) ).
% real_root_eq_0_iff
thf(fact_4150_real__root__less__iff,axiom,
! [N: nat,X2: real,Y2: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ ( root @ N @ X2 ) @ ( root @ N @ Y2 ) )
= ( ord_less @ real @ X2 @ Y2 ) ) ) ).
% real_root_less_iff
thf(fact_4151_real__root__le__iff,axiom,
! [N: nat,X2: real,Y2: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less_eq @ real @ ( root @ N @ X2 ) @ ( root @ N @ Y2 ) )
= ( ord_less_eq @ real @ X2 @ Y2 ) ) ) ).
% real_root_le_iff
thf(fact_4152_real__root__eq__1__iff,axiom,
! [N: nat,X2: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ( root @ N @ X2 )
= ( one_one @ real ) )
= ( X2
= ( one_one @ real ) ) ) ) ).
% real_root_eq_1_iff
thf(fact_4153_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_4154_real__root__gt__0__iff,axiom,
! [N: nat,Y2: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( root @ N @ Y2 ) )
= ( ord_less @ real @ ( zero_zero @ real ) @ Y2 ) ) ) ).
% real_root_gt_0_iff
thf(fact_4155_real__root__lt__0__iff,axiom,
! [N: nat,X2: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ ( root @ N @ X2 ) @ ( zero_zero @ real ) )
= ( ord_less @ real @ X2 @ ( zero_zero @ real ) ) ) ) ).
% real_root_lt_0_iff
thf(fact_4156_real__root__le__0__iff,axiom,
! [N: nat,X2: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less_eq @ real @ ( root @ N @ X2 ) @ ( zero_zero @ real ) )
= ( ord_less_eq @ real @ X2 @ ( zero_zero @ real ) ) ) ) ).
% real_root_le_0_iff
thf(fact_4157_real__root__ge__0__iff,axiom,
! [N: nat,Y2: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( root @ N @ Y2 ) )
= ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y2 ) ) ) ).
% real_root_ge_0_iff
thf(fact_4158_real__root__lt__1__iff,axiom,
! [N: nat,X2: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ ( root @ N @ X2 ) @ ( one_one @ real ) )
= ( ord_less @ real @ X2 @ ( one_one @ real ) ) ) ) ).
% real_root_lt_1_iff
thf(fact_4159_real__root__gt__1__iff,axiom,
! [N: nat,Y2: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ ( one_one @ real ) @ ( root @ N @ Y2 ) )
= ( ord_less @ real @ ( one_one @ real ) @ Y2 ) ) ) ).
% real_root_gt_1_iff
thf(fact_4160_real__root__le__1__iff,axiom,
! [N: nat,X2: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less_eq @ real @ ( root @ N @ X2 ) @ ( one_one @ real ) )
= ( ord_less_eq @ real @ X2 @ ( one_one @ real ) ) ) ) ).
% real_root_le_1_iff
thf(fact_4161_real__root__ge__1__iff,axiom,
! [N: nat,Y2: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less_eq @ real @ ( one_one @ real ) @ ( root @ N @ Y2 ) )
= ( ord_less_eq @ real @ ( one_one @ real ) @ Y2 ) ) ) ).
% real_root_ge_1_iff
thf(fact_4162_cot__npi,axiom,
! [N: nat] :
( ( cot @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ pi ) )
= ( zero_zero @ real ) ) ).
% cot_npi
thf(fact_4163_real__root__pow__pos2,axiom,
! [N: nat,X2: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( power_power @ real @ ( root @ N @ X2 ) @ N )
= X2 ) ) ) ).
% real_root_pow_pos2
thf(fact_4164_cot__periodic,axiom,
! [X2: real] :
( ( cot @ real @ ( plus_plus @ real @ X2 @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) ) )
= ( cot @ real @ X2 ) ) ).
% cot_periodic
thf(fact_4165_real__root__commute,axiom,
! [M: nat,N: nat,X2: real] :
( ( root @ M @ ( root @ N @ X2 ) )
= ( root @ N @ ( root @ M @ X2 ) ) ) ).
% real_root_commute
thf(fact_4166_real__root__mult,axiom,
! [N: nat,X2: real,Y2: real] :
( ( root @ N @ ( times_times @ real @ X2 @ Y2 ) )
= ( times_times @ real @ ( root @ N @ X2 ) @ ( root @ N @ Y2 ) ) ) ).
% real_root_mult
thf(fact_4167_real__root__mult__exp,axiom,
! [M: nat,N: nat,X2: real] :
( ( root @ ( times_times @ nat @ M @ N ) @ X2 )
= ( root @ M @ ( root @ N @ X2 ) ) ) ).
% real_root_mult_exp
thf(fact_4168_real__root__minus,axiom,
! [N: nat,X2: real] :
( ( root @ N @ ( uminus_uminus @ real @ X2 ) )
= ( uminus_uminus @ real @ ( root @ N @ X2 ) ) ) ).
% real_root_minus
thf(fact_4169_real__root__divide,axiom,
! [N: nat,X2: real,Y2: real] :
( ( root @ N @ ( divide_divide @ real @ X2 @ Y2 ) )
= ( divide_divide @ real @ ( root @ N @ X2 ) @ ( root @ N @ Y2 ) ) ) ).
% real_root_divide
thf(fact_4170_real__root__inverse,axiom,
! [N: nat,X2: real] :
( ( root @ N @ ( inverse_inverse @ real @ X2 ) )
= ( inverse_inverse @ real @ ( root @ N @ X2 ) ) ) ).
% real_root_inverse
thf(fact_4171_real__root__pos__pos__le,axiom,
! [X2: real,N: nat] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
=> ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( root @ N @ X2 ) ) ) ).
% real_root_pos_pos_le
thf(fact_4172_real__root__less__mono,axiom,
! [N: nat,X2: real,Y2: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ X2 @ Y2 )
=> ( ord_less @ real @ ( root @ N @ X2 ) @ ( root @ N @ Y2 ) ) ) ) ).
% real_root_less_mono
thf(fact_4173_real__root__le__mono,axiom,
! [N: nat,X2: real,Y2: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less_eq @ real @ X2 @ Y2 )
=> ( ord_less_eq @ real @ ( root @ N @ X2 ) @ ( root @ N @ Y2 ) ) ) ) ).
% real_root_le_mono
thf(fact_4174_real__root__power,axiom,
! [N: nat,X2: real,K: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( root @ N @ ( power_power @ real @ X2 @ K ) )
= ( power_power @ real @ ( root @ N @ X2 ) @ K ) ) ) ).
% real_root_power
thf(fact_4175_real__root__abs,axiom,
! [N: nat,X2: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( root @ N @ ( abs_abs @ real @ X2 ) )
= ( abs_abs @ real @ ( root @ N @ X2 ) ) ) ) ).
% real_root_abs
thf(fact_4176_real__root__gt__zero,axiom,
! [N: nat,X2: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( root @ N @ X2 ) ) ) ) ).
% real_root_gt_zero
thf(fact_4177_real__root__strict__decreasing,axiom,
! [N: nat,N5: nat,X2: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ nat @ N @ N5 )
=> ( ( ord_less @ real @ ( one_one @ real ) @ X2 )
=> ( ord_less @ real @ ( root @ N5 @ X2 ) @ ( root @ N @ X2 ) ) ) ) ) ).
% real_root_strict_decreasing
thf(fact_4178_sqrt__def,axiom,
( sqrt
= ( root @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% sqrt_def
thf(fact_4179_root__abs__power,axiom,
! [N: nat,Y2: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( abs_abs @ real @ ( root @ N @ ( power_power @ real @ Y2 @ N ) ) )
= ( abs_abs @ real @ Y2 ) ) ) ).
% root_abs_power
thf(fact_4180_real__root__pos__pos,axiom,
! [N: nat,X2: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( root @ N @ X2 ) ) ) ) ).
% real_root_pos_pos
thf(fact_4181_real__root__strict__increasing,axiom,
! [N: nat,N5: nat,X2: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ nat @ N @ N5 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ord_less @ real @ X2 @ ( one_one @ real ) )
=> ( ord_less @ real @ ( root @ N @ X2 ) @ ( root @ N5 @ X2 ) ) ) ) ) ) ).
% real_root_strict_increasing
thf(fact_4182_real__root__decreasing,axiom,
! [N: nat,N5: nat,X2: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less_eq @ nat @ N @ N5 )
=> ( ( ord_less_eq @ real @ ( one_one @ real ) @ X2 )
=> ( ord_less_eq @ real @ ( root @ N5 @ X2 ) @ ( root @ N @ X2 ) ) ) ) ) ).
% real_root_decreasing
thf(fact_4183_odd__real__root__power__cancel,axiom,
! [N: nat,X2: real] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( root @ N @ ( power_power @ real @ X2 @ N ) )
= X2 ) ) ).
% odd_real_root_power_cancel
thf(fact_4184_odd__real__root__unique,axiom,
! [N: nat,Y2: real,X2: real] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( ( power_power @ real @ Y2 @ N )
= X2 )
=> ( ( root @ N @ X2 )
= Y2 ) ) ) ).
% odd_real_root_unique
thf(fact_4185_odd__real__root__pow,axiom,
! [N: nat,X2: real] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( power_power @ real @ ( root @ N @ X2 ) @ N )
= X2 ) ) ).
% odd_real_root_pow
thf(fact_4186_real__root__pow__pos,axiom,
! [N: nat,X2: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( power_power @ real @ ( root @ N @ X2 ) @ N )
= X2 ) ) ) ).
% real_root_pow_pos
thf(fact_4187_real__root__pos__unique,axiom,
! [N: nat,Y2: real,X2: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y2 )
=> ( ( ( power_power @ real @ Y2 @ N )
= X2 )
=> ( ( root @ N @ X2 )
= Y2 ) ) ) ) ).
% real_root_pos_unique
thf(fact_4188_real__root__power__cancel,axiom,
! [N: nat,X2: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( root @ N @ ( power_power @ real @ X2 @ N ) )
= X2 ) ) ) ).
% real_root_power_cancel
thf(fact_4189_cot__def,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ( ( cot @ A )
= ( ^ [X: A] : ( divide_divide @ A @ ( cos @ A @ X ) @ ( sin @ A @ X ) ) ) ) ) ).
% cot_def
thf(fact_4190_real__root__increasing,axiom,
! [N: nat,N5: nat,X2: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less_eq @ nat @ N @ N5 )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ord_less_eq @ real @ X2 @ ( one_one @ real ) )
=> ( ord_less_eq @ real @ ( root @ N @ X2 ) @ ( root @ N5 @ X2 ) ) ) ) ) ) ).
% real_root_increasing
thf(fact_4191_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_4192_log__base__root,axiom,
! [N: nat,B2: real,X2: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ B2 )
=> ( ( log @ ( root @ N @ B2 ) @ X2 )
= ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ ( log @ B2 @ X2 ) ) ) ) ) ).
% log_base_root
thf(fact_4193_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_4194_root__powr__inverse,axiom,
! [N: nat,X2: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( root @ N @ X2 )
= ( powr @ real @ X2 @ ( divide_divide @ real @ ( one_one @ real ) @ ( semiring_1_of_nat @ real @ N ) ) ) ) ) ) ).
% root_powr_inverse
thf(fact_4195_cot__gt__zero,axiom,
! [X2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ord_less @ real @ X2 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( cot @ real @ X2 ) ) ) ) ).
% cot_gt_zero
thf(fact_4196_tan__cot_H,axiom,
! [X2: real] :
( ( tan @ real @ ( minus_minus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ X2 ) )
= ( cot @ real @ X2 ) ) ).
% tan_cot'
thf(fact_4197_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_4198_lowi__def,axiom,
( vEBT_VEBT_lowi
= ( ^ [X: nat,N3: nat] : ( heap_Time_return @ nat @ ( modulo_modulo @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) ) ) ) ) ).
% lowi_def
thf(fact_4199_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_4200_option_Osize_I3_J,axiom,
! [A: $tType] :
( ( size_size @ ( option @ A ) @ ( none @ A ) )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% option.size(3)
thf(fact_4201_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_4202_More__Word_Osint__0,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A] :
( ( ( ring_1_signed @ A @ int @ X2 )
= ( zero_zero @ int ) )
= ( X2
= ( zero_zero @ ( word @ A ) ) ) ) ) ).
% More_Word.sint_0
thf(fact_4203_highi__def,axiom,
( vEBT_VEBT_highi
= ( ^ [X: nat,N3: nat] : ( heap_Time_return @ nat @ ( divide_divide @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) ) ) ) ) ).
% highi_def
thf(fact_4204_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_4205_sint__minus1,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A] :
( ( ( ring_1_signed @ A @ int @ X2 )
= ( uminus_uminus @ int @ ( one_one @ int ) ) )
= ( X2
= ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) ) ) ) ).
% sint_minus1
thf(fact_4206_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_4207_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_4208_option_Osize__neq,axiom,
! [A: $tType,X2: option @ A] :
( ( size_size @ ( option @ A ) @ X2 )
!= ( zero_zero @ nat ) ) ).
% option.size_neq
thf(fact_4209_ex__has__least__nat,axiom,
! [A: $tType,P: A > $o,K: A,M: A > nat] :
( ( P @ K )
=> ? [X3: A] :
( ( P @ X3 )
& ! [Y4: A] :
( ( P @ Y4 )
=> ( ord_less_eq @ nat @ ( M @ X3 ) @ ( M @ Y4 ) ) ) ) ) ).
% ex_has_least_nat
thf(fact_4210_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_4211_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_4212_sint__above__size,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A,X2: int] :
( ( ord_less_eq @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ ( size_size @ ( word @ A ) @ W ) @ ( one_one @ nat ) ) ) @ X2 )
=> ( ord_less @ int @ ( ring_1_signed @ A @ int @ W ) @ X2 ) ) ) ).
% sint_above_size
thf(fact_4213_sint__below__size,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: int,W: word @ A] :
( ( ord_less_eq @ int @ X2 @ ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ ( size_size @ ( word @ A ) @ W ) @ ( one_one @ nat ) ) ) ) )
=> ( ord_less_eq @ int @ X2 @ ( ring_1_signed @ A @ int @ W ) ) ) ) ).
% sint_below_size
thf(fact_4214_set__decode__0,axiom,
! [X2: nat] :
( ( member @ nat @ ( zero_zero @ nat ) @ ( nat_set_decode @ X2 ) )
= ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ X2 ) ) ) ).
% set_decode_0
thf(fact_4215_Comparator__Generator_OAll__less__Suc,axiom,
! [X2: nat,P: nat > $o] :
( ( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( suc @ X2 ) )
=> ( P @ I3 ) ) )
= ( ( P @ ( zero_zero @ nat ) )
& ! [I3: nat] :
( ( ord_less @ nat @ I3 @ X2 )
=> ( P @ ( suc @ I3 ) ) ) ) ) ).
% Comparator_Generator.All_less_Suc
thf(fact_4216_forall__finite_I2_J,axiom,
! [P: nat > $o] :
( ( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( suc @ ( zero_zero @ nat ) ) )
=> ( P @ I3 ) ) )
= ( P @ ( zero_zero @ nat ) ) ) ).
% forall_finite(2)
thf(fact_4217_forall__finite_I3_J,axiom,
! [X2: nat,P: nat > $o] :
( ( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( suc @ ( suc @ X2 ) ) )
=> ( P @ I3 ) ) )
= ( ( P @ ( zero_zero @ nat ) )
& ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( suc @ X2 ) )
=> ( P @ ( suc @ I3 ) ) ) ) ) ).
% forall_finite(3)
thf(fact_4218_set__decode__Suc,axiom,
! [N: nat,X2: nat] :
( ( member @ nat @ ( suc @ N ) @ ( nat_set_decode @ X2 ) )
= ( member @ nat @ N @ ( nat_set_decode @ ( divide_divide @ nat @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% set_decode_Suc
thf(fact_4219_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_4220_finite__set__decode,axiom,
! [N: nat] : ( finite_finite2 @ nat @ ( nat_set_decode @ N ) ) ).
% finite_set_decode
thf(fact_4221_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_4222_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_4223_forall__finite_I1_J,axiom,
! [P: nat > $o,I4: nat] :
( ( ord_less @ nat @ I4 @ ( zero_zero @ nat ) )
=> ( P @ I4 ) ) ).
% forall_finite(1)
thf(fact_4224_set__decode__def,axiom,
( nat_set_decode
= ( ^ [X: nat] :
( collect @ nat
@ ^ [N3: nat] :
~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) ) ) ) ) ) ).
% set_decode_def
thf(fact_4225_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_4226_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_4227_uint32_Osize__eq,axiom,
( ( size_size @ uint32 )
= ( ^ [P3: uint32] : ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% uint32.size_eq
thf(fact_4228_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_4229_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_4230_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_4231_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_4232_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_4233_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_4234_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_4235_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_4236_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_4237_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_4238_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_4239_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_4240_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_4241_take__bit__numeral__1,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [L2: num] :
( ( bit_se2584673776208193580ke_bit @ A @ ( numeral_numeral @ nat @ L2 ) @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% take_bit_numeral_1
thf(fact_4242_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_4243_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_4244_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_4245_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_4246_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_4247_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_4248_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_4249_odd__of__bool__self,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [P2: $o] :
( ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( zero_neq_one_of_bool @ A @ P2 ) ) )
= P2 ) ) ).
% odd_of_bool_self
thf(fact_4250_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_4251_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_4252_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_4253_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_4254_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_4255_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_4256_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_4257_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_4258_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_4259_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_4260_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_4261_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_4262_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_4263_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_4264_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_4265_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_4266_of__bool__eq__iff,axiom,
! [A: $tType] :
( ( zero_neq_one @ A )
=> ! [P2: $o,Q2: $o] :
( ( ( zero_neq_one_of_bool @ A @ P2 )
= ( zero_neq_one_of_bool @ A @ Q2 ) )
= ( P2 = Q2 ) ) ) ).
% of_bool_eq_iff
thf(fact_4267_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_4268_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_4269_of__bool__def,axiom,
! [A: $tType] :
( ( zero_neq_one @ A )
=> ( ( zero_neq_one_of_bool @ A )
= ( ^ [P3: $o] : ( if @ A @ P3 @ ( one_one @ A ) @ ( zero_zero @ A ) ) ) ) ) ).
% of_bool_def
thf(fact_4270_split__of__bool,axiom,
! [A: $tType] :
( ( zero_neq_one @ A )
=> ! [P: A > $o,P2: $o] :
( ( P @ ( zero_neq_one_of_bool @ A @ P2 ) )
= ( ( P2
=> ( P @ ( one_one @ A ) ) )
& ( ~ P2
=> ( P @ ( zero_zero @ A ) ) ) ) ) ) ).
% split_of_bool
thf(fact_4271_split__of__bool__asm,axiom,
! [A: $tType] :
( ( zero_neq_one @ A )
=> ! [P: A > $o,P2: $o] :
( ( P @ ( zero_neq_one_of_bool @ A @ P2 ) )
= ( ~ ( ( P2
& ~ ( P @ ( one_one @ A ) ) )
| ( ~ P2
& ~ ( P @ ( zero_zero @ A ) ) ) ) ) ) ) ).
% split_of_bool_asm
thf(fact_4272_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_4273_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_4274_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_4275_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_4276_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_4277_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_4278_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_4279_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_4280_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_4281_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_4282_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_4283_bin__last__bintrunc,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [L2: nat,N: A] :
( ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se2584673776208193580ke_bit @ A @ L2 @ N ) ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ L2 )
& ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% bin_last_bintrunc
thf(fact_4284_take__bit__numeral__bit0,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [L2: num,K: num] :
( ( bit_se2584673776208193580ke_bit @ A @ ( numeral_numeral @ nat @ L2 ) @ ( numeral_numeral @ A @ ( bit0 @ K ) ) )
= ( times_times @ A @ ( bit_se2584673776208193580ke_bit @ A @ ( pred_numeral @ L2 ) @ ( numeral_numeral @ A @ K ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% take_bit_numeral_bit0
thf(fact_4285_bits__induct,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [P: A > $o,A2: A] :
( ! [A5: A] :
( ( ( divide_divide @ A @ A5 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= A5 )
=> ( P @ A5 ) )
=> ( ! [A5: A,B4: $o] :
( ( P @ A5 )
=> ( ( ( divide_divide @ A @ ( plus_plus @ A @ ( zero_neq_one_of_bool @ A @ B4 ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A5 ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= A5 )
=> ( P @ ( plus_plus @ A @ ( zero_neq_one_of_bool @ A @ B4 ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A5 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% bits_induct
thf(fact_4286_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_4287_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_4288_crossproduct__eq,axiom,
! [A: $tType] :
( ( semiri1453513574482234551roduct @ A )
=> ! [W: A,Y2: A,X2: A,Z: A] :
( ( ( plus_plus @ A @ ( times_times @ A @ W @ Y2 ) @ ( times_times @ A @ X2 @ Z ) )
= ( plus_plus @ A @ ( times_times @ A @ W @ Z ) @ ( times_times @ A @ X2 @ Y2 ) ) )
= ( ( W = X2 )
| ( Y2 = Z ) ) ) ) ).
% crossproduct_eq
thf(fact_4289_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_4290_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_4291_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_4292_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_4293_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_4294_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_4295_take__bit__numeral__bit1,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [L2: num,K: num] :
( ( bit_se2584673776208193580ke_bit @ A @ ( numeral_numeral @ nat @ L2 ) @ ( numeral_numeral @ A @ ( bit1 @ K ) ) )
= ( plus_plus @ A @ ( times_times @ A @ ( bit_se2584673776208193580ke_bit @ A @ ( pred_numeral @ L2 ) @ ( numeral_numeral @ A @ K ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) ) ) ) ).
% take_bit_numeral_bit1
thf(fact_4296_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_4297_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_4298_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_4299_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_4300_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_4301_powr__int,axiom,
! [X2: real,I: int] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ I )
=> ( ( powr @ real @ X2 @ ( ring_1_of_int @ real @ I ) )
= ( power_power @ real @ X2 @ ( nat2 @ I ) ) ) )
& ( ~ ( ord_less_eq @ int @ ( zero_zero @ int ) @ I )
=> ( ( powr @ real @ X2 @ ( ring_1_of_int @ real @ I ) )
= ( divide_divide @ real @ ( one_one @ real ) @ ( power_power @ real @ X2 @ ( nat2 @ ( uminus_uminus @ int @ I ) ) ) ) ) ) ) ) ).
% powr_int
thf(fact_4302_Word_Oof__nat__unat,axiom,
! [A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( semiring_1 @ A ) )
=> ! [W: word @ B] :
( ( semiring_1_of_nat @ A @ ( semiring_1_unsigned @ B @ nat @ W ) )
= ( semiring_1_unsigned @ B @ A @ W ) ) ) ).
% Word.of_nat_unat
thf(fact_4303_nat__int,axiom,
! [N: nat] :
( ( nat2 @ ( semiring_1_of_nat @ int @ N ) )
= N ) ).
% nat_int
thf(fact_4304_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_4305_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_4306_nat__numeral,axiom,
! [K: num] :
( ( nat2 @ ( numeral_numeral @ int @ K ) )
= ( numeral_numeral @ nat @ K ) ) ).
% nat_numeral
thf(fact_4307_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_4308_uint__le__0__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A] :
( ( ord_less_eq @ int @ ( semiring_1_unsigned @ A @ int @ X2 ) @ ( zero_zero @ int ) )
= ( ( semiring_1_unsigned @ A @ int @ X2 )
= ( zero_zero @ int ) ) ) ) ).
% uint_le_0_iff
thf(fact_4309_uint__ge__0,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A] : ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( semiring_1_unsigned @ A @ int @ X2 ) ) ) ).
% uint_ge_0
thf(fact_4310_uint__lt__0,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A] :
~ ( ord_less @ int @ ( semiring_1_unsigned @ A @ int @ X2 ) @ ( zero_zero @ int ) ) ) ).
% uint_lt_0
thf(fact_4311_pred__numeral__inc,axiom,
! [K: num] :
( ( pred_numeral @ ( inc @ K ) )
= ( numeral_numeral @ nat @ K ) ) ).
% pred_numeral_inc
thf(fact_4312_slice__complete,axiom,
! [A: $tType,Xs2: list @ A] :
( ( slice @ A @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Xs2 ) @ Xs2 )
= Xs2 ) ).
% slice_complete
thf(fact_4313_nat__1,axiom,
( ( nat2 @ ( one_one @ int ) )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% nat_1
thf(fact_4314_nat__le__0,axiom,
! [Z: int] :
( ( ord_less_eq @ int @ Z @ ( zero_zero @ int ) )
=> ( ( nat2 @ Z )
= ( zero_zero @ nat ) ) ) ).
% nat_le_0
thf(fact_4315_nat__0__iff,axiom,
! [I: int] :
( ( ( nat2 @ I )
= ( zero_zero @ nat ) )
= ( ord_less_eq @ int @ I @ ( zero_zero @ int ) ) ) ).
% nat_0_iff
thf(fact_4316_nat__neg__numeral,axiom,
! [K: num] :
( ( nat2 @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) )
= ( zero_zero @ nat ) ) ).
% nat_neg_numeral
thf(fact_4317_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_4318_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_4319_nat__zminus__int,axiom,
! [N: nat] :
( ( nat2 @ ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N ) ) )
= ( zero_zero @ nat ) ) ).
% nat_zminus_int
thf(fact_4320_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_4321_slice__len,axiom,
! [A: $tType,From: nat,To: nat,Xs2: list @ A] :
( ( ord_less_eq @ nat @ From @ To )
=> ( ( ord_less_eq @ nat @ To @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( size_size @ ( list @ A ) @ ( slice @ A @ From @ To @ Xs2 ) )
= ( minus_minus @ nat @ To @ From ) ) ) ) ).
% slice_len
thf(fact_4322_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_4323_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_4324_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_4325_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_4326_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_4327_diff__nat__numeral,axiom,
! [V: num,V4: num] :
( ( minus_minus @ nat @ ( numeral_numeral @ nat @ V ) @ ( numeral_numeral @ nat @ V4 ) )
= ( nat2 @ ( minus_minus @ int @ ( numeral_numeral @ int @ V ) @ ( numeral_numeral @ int @ V4 ) ) ) ) ).
% diff_nat_numeral
thf(fact_4328_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_4329_numeral__power__eq__nat__cancel__iff,axiom,
! [X2: num,N: nat,Y2: int] :
( ( ( power_power @ nat @ ( numeral_numeral @ nat @ X2 ) @ N )
= ( nat2 @ Y2 ) )
= ( ( power_power @ int @ ( numeral_numeral @ int @ X2 ) @ N )
= Y2 ) ) ).
% numeral_power_eq_nat_cancel_iff
thf(fact_4330_nat__eq__numeral__power__cancel__iff,axiom,
! [Y2: int,X2: num,N: nat] :
( ( ( nat2 @ Y2 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ X2 ) @ N ) )
= ( Y2
= ( power_power @ int @ ( numeral_numeral @ int @ X2 ) @ N ) ) ) ).
% nat_eq_numeral_power_cancel_iff
thf(fact_4331_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_4332_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_4333_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_4334_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_4335_nat__ceiling__le__eq,axiom,
! [X2: real,A2: nat] :
( ( ord_less_eq @ nat @ ( nat2 @ ( archimedean_ceiling @ real @ X2 ) ) @ A2 )
= ( ord_less_eq @ real @ X2 @ ( semiring_1_of_nat @ real @ A2 ) ) ) ).
% nat_ceiling_le_eq
thf(fact_4336_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_4337_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_4338_numeral__power__less__nat__cancel__iff,axiom,
! [X2: num,N: nat,A2: int] :
( ( ord_less @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ X2 ) @ N ) @ ( nat2 @ A2 ) )
= ( ord_less @ int @ ( power_power @ int @ ( numeral_numeral @ int @ X2 ) @ N ) @ A2 ) ) ).
% numeral_power_less_nat_cancel_iff
thf(fact_4339_nat__less__numeral__power__cancel__iff,axiom,
! [A2: int,X2: num,N: nat] :
( ( ord_less @ nat @ ( nat2 @ A2 ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ X2 ) @ N ) )
= ( ord_less @ int @ A2 @ ( power_power @ int @ ( numeral_numeral @ int @ X2 ) @ N ) ) ) ).
% nat_less_numeral_power_cancel_iff
thf(fact_4340_numeral__power__le__nat__cancel__iff,axiom,
! [X2: num,N: nat,A2: int] :
( ( ord_less_eq @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ X2 ) @ N ) @ ( nat2 @ A2 ) )
= ( ord_less_eq @ int @ ( power_power @ int @ ( numeral_numeral @ int @ X2 ) @ N ) @ A2 ) ) ).
% numeral_power_le_nat_cancel_iff
thf(fact_4341_nat__le__numeral__power__cancel__iff,axiom,
! [A2: int,X2: num,N: nat] :
( ( ord_less_eq @ nat @ ( nat2 @ A2 ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ X2 ) @ N ) )
= ( ord_less_eq @ int @ A2 @ ( power_power @ int @ ( numeral_numeral @ int @ X2 ) @ N ) ) ) ).
% nat_le_numeral_power_cancel_iff
thf(fact_4342_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_4343_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_4344_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_4345_take__bit__mult,axiom,
! [N: nat,K: int,L2: int] :
( ( bit_se2584673776208193580ke_bit @ int @ N @ ( times_times @ int @ ( bit_se2584673776208193580ke_bit @ int @ N @ K ) @ ( bit_se2584673776208193580ke_bit @ int @ N @ L2 ) ) )
= ( bit_se2584673776208193580ke_bit @ int @ N @ ( times_times @ int @ K @ L2 ) ) ) ).
% take_bit_mult
thf(fact_4346_take__bit__diff,axiom,
! [N: nat,K: int,L2: int] :
( ( bit_se2584673776208193580ke_bit @ int @ N @ ( minus_minus @ int @ ( bit_se2584673776208193580ke_bit @ int @ N @ K ) @ ( bit_se2584673776208193580ke_bit @ int @ N @ L2 ) ) )
= ( bit_se2584673776208193580ke_bit @ int @ N @ ( minus_minus @ int @ K @ L2 ) ) ) ).
% take_bit_diff
thf(fact_4347_take__bit__tightened__less__eq__nat,axiom,
! [M: nat,N: nat,Q2: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ord_less_eq @ nat @ ( bit_se2584673776208193580ke_bit @ nat @ M @ Q2 ) @ ( bit_se2584673776208193580ke_bit @ nat @ N @ Q2 ) ) ) ).
% take_bit_tightened_less_eq_nat
thf(fact_4348_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_4349_num__induct,axiom,
! [P: num > $o,X2: num] :
( ( P @ one2 )
=> ( ! [X3: num] :
( ( P @ X3 )
=> ( P @ ( inc @ X3 ) ) )
=> ( P @ X2 ) ) ) ).
% num_induct
thf(fact_4350_add__inc,axiom,
! [X2: num,Y2: num] :
( ( plus_plus @ num @ X2 @ ( inc @ Y2 ) )
= ( inc @ ( plus_plus @ num @ X2 @ Y2 ) ) ) ).
% add_inc
thf(fact_4351_nat__numeral__as__int,axiom,
( ( numeral_numeral @ nat )
= ( ^ [I3: num] : ( nat2 @ ( numeral_numeral @ int @ I3 ) ) ) ) ).
% nat_numeral_as_int
thf(fact_4352_nat__zero__as__int,axiom,
( ( zero_zero @ nat )
= ( nat2 @ ( zero_zero @ int ) ) ) ).
% nat_zero_as_int
thf(fact_4353_nat__mono,axiom,
! [X2: int,Y2: int] :
( ( ord_less_eq @ int @ X2 @ Y2 )
=> ( ord_less_eq @ nat @ ( nat2 @ X2 ) @ ( nat2 @ Y2 ) ) ) ).
% nat_mono
thf(fact_4354_nat__one__as__int,axiom,
( ( one_one @ nat )
= ( nat2 @ ( one_one @ int ) ) ) ).
% nat_one_as_int
thf(fact_4355_eq__nat__nat__iff,axiom,
! [Z: int,Z6: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z6 )
=> ( ( ( nat2 @ Z )
= ( nat2 @ Z6 ) )
= ( Z = Z6 ) ) ) ) ).
% eq_nat_nat_iff
thf(fact_4356_all__nat,axiom,
( ( ^ [P6: nat > $o] :
! [X8: nat] : ( P6 @ X8 ) )
= ( ^ [P7: nat > $o] :
! [X: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
=> ( P7 @ ( nat2 @ X ) ) ) ) ) ).
% all_nat
thf(fact_4357_ex__nat,axiom,
( ( ^ [P6: nat > $o] :
? [X8: nat] : ( P6 @ X8 ) )
= ( ^ [P7: nat > $o] :
? [X: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
& ( P7 @ ( nat2 @ X ) ) ) ) ) ).
% ex_nat
thf(fact_4358_uint__0__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A] :
( ( ( semiring_1_unsigned @ A @ int @ X2 )
= ( zero_zero @ int ) )
= ( X2
= ( zero_zero @ ( word @ A ) ) ) ) ) ).
% uint_0_iff
thf(fact_4359_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_4360_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_4361_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_4362_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_4363_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_4364_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_4365_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_4366_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_4367_uint__div,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A,Y2: word @ A] :
( ( semiring_1_unsigned @ A @ int @ ( divide_divide @ ( word @ A ) @ X2 @ Y2 ) )
= ( divide_divide @ int @ ( semiring_1_unsigned @ A @ int @ X2 ) @ ( semiring_1_unsigned @ A @ int @ Y2 ) ) ) ) ).
% uint_div
thf(fact_4368_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_4369_word__less__eq__iff__unsigned,axiom,
! [A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( linordered_semidom @ A ) )
=> ( ( ord_less_eq @ ( word @ B ) )
= ( ^ [A4: word @ B,B3: word @ B] : ( ord_less_eq @ A @ ( semiring_1_unsigned @ B @ A @ A4 ) @ ( semiring_1_unsigned @ B @ A @ B3 ) ) ) ) ) ).
% word_less_eq_iff_unsigned
thf(fact_4370_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_4371_unset__bit__nat__def,axiom,
( ( bit_se2638667681897837118et_bit @ nat )
= ( ^ [M5: nat,N3: nat] : ( nat2 @ ( bit_se2638667681897837118et_bit @ int @ M5 @ ( semiring_1_of_nat @ int @ N3 ) ) ) ) ) ).
% unset_bit_nat_def
thf(fact_4372_inc_Osimps_I1_J,axiom,
( ( inc @ one2 )
= ( bit0 @ one2 ) ) ).
% inc.simps(1)
thf(fact_4373_inc_Osimps_I2_J,axiom,
! [X2: num] :
( ( inc @ ( bit0 @ X2 ) )
= ( bit1 @ X2 ) ) ).
% inc.simps(2)
thf(fact_4374_inc_Osimps_I3_J,axiom,
! [X2: num] :
( ( inc @ ( bit1 @ X2 ) )
= ( bit0 @ ( inc @ X2 ) ) ) ).
% inc.simps(3)
thf(fact_4375_add__One,axiom,
! [X2: num] :
( ( plus_plus @ num @ X2 @ one2 )
= ( inc @ X2 ) ) ).
% add_One
thf(fact_4376_mult__inc,axiom,
! [X2: num,Y2: num] :
( ( times_times @ num @ X2 @ ( inc @ Y2 ) )
= ( plus_plus @ num @ ( times_times @ num @ X2 @ Y2 ) @ X2 ) ) ).
% mult_inc
thf(fact_4377_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_4378_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_4379_nat__le__iff,axiom,
! [X2: int,N: nat] :
( ( ord_less_eq @ nat @ ( nat2 @ X2 ) @ N )
= ( ord_less_eq @ int @ X2 @ ( semiring_1_of_nat @ int @ N ) ) ) ).
% nat_le_iff
thf(fact_4380_of__nat__ceiling,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [R2: A] : ( ord_less_eq @ A @ R2 @ ( semiring_1_of_nat @ A @ ( nat2 @ ( archimedean_ceiling @ A @ R2 ) ) ) ) ) ).
% of_nat_ceiling
thf(fact_4381_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_4382_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_4383_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_4384_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_4385_uint__add__ge0,axiom,
! [Aa: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ Aa ) )
=> ! [Xa: word @ Aa,X2: word @ A] : ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( plus_plus @ int @ ( semiring_1_unsigned @ Aa @ int @ Xa ) @ ( semiring_1_unsigned @ A @ int @ X2 ) ) ) ) ).
% uint_add_ge0
thf(fact_4386_uint__mult__ge0,axiom,
! [Aa: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ Aa ) )
=> ! [Xa: word @ Aa,X2: word @ A] : ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( times_times @ int @ ( semiring_1_unsigned @ Aa @ int @ Xa ) @ ( semiring_1_unsigned @ A @ int @ X2 ) ) ) ) ).
% uint_mult_ge0
thf(fact_4387_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_4388_uint__add__le,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A,Y2: word @ A] : ( ord_less_eq @ int @ ( semiring_1_unsigned @ A @ int @ ( plus_plus @ ( word @ A ) @ X2 @ Y2 ) ) @ ( plus_plus @ int @ ( semiring_1_unsigned @ A @ int @ X2 ) @ ( semiring_1_unsigned @ A @ int @ Y2 ) ) ) ) ).
% uint_add_le
thf(fact_4389_uint__plus__simple__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A,Y2: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ X2 @ ( plus_plus @ ( word @ A ) @ X2 @ Y2 ) )
= ( ( semiring_1_unsigned @ A @ int @ ( plus_plus @ ( word @ A ) @ X2 @ Y2 ) )
= ( plus_plus @ int @ ( semiring_1_unsigned @ A @ int @ X2 ) @ ( semiring_1_unsigned @ A @ int @ Y2 ) ) ) ) ) ).
% uint_plus_simple_iff
thf(fact_4390_uint__plus__simple,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A,Y2: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ X2 @ ( plus_plus @ ( word @ A ) @ X2 @ Y2 ) )
=> ( ( semiring_1_unsigned @ A @ int @ ( plus_plus @ ( word @ A ) @ X2 @ Y2 ) )
= ( plus_plus @ int @ ( semiring_1_unsigned @ A @ int @ X2 ) @ ( semiring_1_unsigned @ A @ int @ Y2 ) ) ) ) ) ).
% uint_plus_simple
thf(fact_4391_word__add__def,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( plus_plus @ ( word @ A ) )
= ( ^ [A4: word @ A,B3: word @ A] : ( ring_1_of_int @ ( word @ A ) @ ( plus_plus @ int @ ( semiring_1_unsigned @ A @ int @ A4 ) @ ( semiring_1_unsigned @ A @ int @ B3 ) ) ) ) ) ) ).
% word_add_def
thf(fact_4392_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_4393_word__arith__power__alt,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( power_power @ ( word @ A ) )
= ( ^ [A4: word @ A,N3: nat] : ( ring_1_of_int @ ( word @ A ) @ ( power_power @ int @ ( semiring_1_unsigned @ A @ int @ A4 ) @ N3 ) ) ) ) ) ).
% word_arith_power_alt
thf(fact_4394_word__mult__def,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( times_times @ ( word @ A ) )
= ( ^ [A4: word @ A,B3: word @ A] : ( ring_1_of_int @ ( word @ A ) @ ( times_times @ int @ ( semiring_1_unsigned @ A @ int @ A4 ) @ ( semiring_1_unsigned @ A @ int @ B3 ) ) ) ) ) ) ).
% word_mult_def
thf(fact_4395_word__div__def,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( divide_divide @ ( word @ A ) )
= ( ^ [A4: word @ A,B3: word @ A] : ( ring_1_of_int @ ( word @ A ) @ ( divide_divide @ int @ ( semiring_1_unsigned @ A @ int @ A4 ) @ ( semiring_1_unsigned @ A @ int @ B3 ) ) ) ) ) ) ).
% word_div_def
thf(fact_4396_real__nat__ceiling__ge,axiom,
! [X2: real] : ( ord_less_eq @ real @ X2 @ ( semiring_1_of_nat @ real @ ( nat2 @ ( archimedean_ceiling @ real @ X2 ) ) ) ) ).
% real_nat_ceiling_ge
thf(fact_4397_nat__plus__as__int,axiom,
( ( plus_plus @ nat )
= ( ^ [A4: nat,B3: nat] : ( nat2 @ ( plus_plus @ int @ ( semiring_1_of_nat @ int @ A4 ) @ ( semiring_1_of_nat @ int @ B3 ) ) ) ) ) ).
% nat_plus_as_int
thf(fact_4398_nat__times__as__int,axiom,
( ( times_times @ nat )
= ( ^ [A4: nat,B3: nat] : ( nat2 @ ( times_times @ int @ ( semiring_1_of_nat @ int @ A4 ) @ ( semiring_1_of_nat @ int @ B3 ) ) ) ) ) ).
% nat_times_as_int
thf(fact_4399_nat__minus__as__int,axiom,
( ( minus_minus @ nat )
= ( ^ [A4: nat,B3: nat] : ( nat2 @ ( minus_minus @ int @ ( semiring_1_of_nat @ int @ A4 ) @ ( semiring_1_of_nat @ int @ B3 ) ) ) ) ) ).
% nat_minus_as_int
thf(fact_4400_nat__div__as__int,axiom,
( ( divide_divide @ nat )
= ( ^ [A4: nat,B3: nat] : ( nat2 @ ( divide_divide @ int @ ( semiring_1_of_nat @ int @ A4 ) @ ( semiring_1_of_nat @ int @ B3 ) ) ) ) ) ).
% nat_div_as_int
thf(fact_4401_nat__mod__as__int,axiom,
( ( modulo_modulo @ nat )
= ( ^ [A4: nat,B3: nat] : ( nat2 @ ( modulo_modulo @ int @ ( semiring_1_of_nat @ int @ A4 ) @ ( semiring_1_of_nat @ int @ B3 ) ) ) ) ) ).
% nat_mod_as_int
thf(fact_4402_numeral__inc,axiom,
! [A: $tType] :
( ( numeral @ A )
=> ! [X2: num] :
( ( numeral_numeral @ A @ ( inc @ X2 ) )
= ( plus_plus @ A @ ( numeral_numeral @ A @ X2 ) @ ( one_one @ A ) ) ) ) ).
% numeral_inc
thf(fact_4403_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_4404_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_4405_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_4406_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_4407_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_4408_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_4409_nat__add__distrib,axiom,
! [Z: int,Z6: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z6 )
=> ( ( nat2 @ ( plus_plus @ int @ Z @ Z6 ) )
= ( plus_plus @ nat @ ( nat2 @ Z ) @ ( nat2 @ Z6 ) ) ) ) ) ).
% nat_add_distrib
thf(fact_4410_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_4411_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_4412_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_4413_nat__mult__distrib,axiom,
! [Z: int,Z6: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z )
=> ( ( nat2 @ ( times_times @ int @ Z @ Z6 ) )
= ( times_times @ nat @ ( nat2 @ Z ) @ ( nat2 @ Z6 ) ) ) ) ).
% nat_mult_distrib
thf(fact_4414_nat__diff__distrib_H,axiom,
! [X2: int,Y2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X2 )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y2 )
=> ( ( nat2 @ ( minus_minus @ int @ X2 @ Y2 ) )
= ( minus_minus @ nat @ ( nat2 @ X2 ) @ ( nat2 @ Y2 ) ) ) ) ) ).
% nat_diff_distrib'
thf(fact_4415_nat__diff__distrib,axiom,
! [Z6: int,Z: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z6 )
=> ( ( ord_less_eq @ int @ Z6 @ Z )
=> ( ( nat2 @ ( minus_minus @ int @ Z @ Z6 ) )
= ( minus_minus @ nat @ ( nat2 @ Z ) @ ( nat2 @ Z6 ) ) ) ) ) ).
% nat_diff_distrib
thf(fact_4416_nat__abs__triangle__ineq,axiom,
! [K: int,L2: int] : ( ord_less_eq @ nat @ ( nat2 @ ( abs_abs @ int @ ( plus_plus @ int @ K @ L2 ) ) ) @ ( plus_plus @ nat @ ( nat2 @ ( abs_abs @ int @ K ) ) @ ( nat2 @ ( abs_abs @ int @ L2 ) ) ) ) ).
% nat_abs_triangle_ineq
thf(fact_4417_nat__div__distrib,axiom,
! [X2: int,Y2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X2 )
=> ( ( nat2 @ ( divide_divide @ int @ X2 @ Y2 ) )
= ( divide_divide @ nat @ ( nat2 @ X2 ) @ ( nat2 @ Y2 ) ) ) ) ).
% nat_div_distrib
thf(fact_4418_nat__div__distrib_H,axiom,
! [Y2: int,X2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y2 )
=> ( ( nat2 @ ( divide_divide @ int @ X2 @ Y2 ) )
= ( divide_divide @ nat @ ( nat2 @ X2 ) @ ( nat2 @ Y2 ) ) ) ) ).
% nat_div_distrib'
thf(fact_4419_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_4420_nat__mod__distrib,axiom,
! [X2: int,Y2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X2 )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y2 )
=> ( ( nat2 @ ( modulo_modulo @ int @ X2 @ Y2 ) )
= ( modulo_modulo @ nat @ ( nat2 @ X2 ) @ ( nat2 @ Y2 ) ) ) ) ) ).
% nat_mod_distrib
thf(fact_4421_udvd__incr__lem0,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Up: int,Uq: int,N: int,K6: word @ A,N4: int] :
( ( ord_less @ int @ Up @ Uq )
=> ( ( Up
= ( times_times @ int @ N @ ( semiring_1_unsigned @ A @ int @ K6 ) ) )
=> ( ( Uq
= ( times_times @ int @ N4 @ ( semiring_1_unsigned @ A @ int @ K6 ) ) )
=> ( ord_less_eq @ int @ ( plus_plus @ int @ Up @ ( semiring_1_unsigned @ A @ int @ K6 ) ) @ Uq ) ) ) ) ) ).
% udvd_incr_lem0
thf(fact_4422_udvd__incr__lem,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Up: int,Uq: int,Ua: int,N: int,K6: word @ A,N4: 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 @ N4 @ ( semiring_1_unsigned @ A @ int @ K6 ) ) ) )
=> ( ord_less_eq @ int @ ( plus_plus @ int @ Up @ ( semiring_1_unsigned @ A @ int @ K6 ) ) @ Uq ) ) ) ) ) ).
% udvd_incr_lem
thf(fact_4423_word__of__int__nat,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X2 )
=> ( ( ring_1_of_int @ ( word @ A ) @ X2 )
= ( semiring_1_of_nat @ ( word @ A ) @ ( nat2 @ X2 ) ) ) ) ) ).
% word_of_int_nat
thf(fact_4424_div__abs__eq__div__nat,axiom,
! [K: int,L2: int] :
( ( divide_divide @ int @ ( abs_abs @ int @ K ) @ ( abs_abs @ int @ L2 ) )
= ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ ( nat2 @ ( abs_abs @ int @ K ) ) @ ( nat2 @ ( abs_abs @ int @ L2 ) ) ) ) ) ).
% div_abs_eq_div_nat
thf(fact_4425_nat__floor__neg,axiom,
! [X2: real] :
( ( ord_less_eq @ real @ X2 @ ( zero_zero @ real ) )
=> ( ( nat2 @ ( archim6421214686448440834_floor @ real @ X2 ) )
= ( zero_zero @ nat ) ) ) ).
% nat_floor_neg
thf(fact_4426_mod__abs__eq__div__nat,axiom,
! [K: int,L2: int] :
( ( modulo_modulo @ int @ ( abs_abs @ int @ K ) @ ( abs_abs @ int @ L2 ) )
= ( semiring_1_of_nat @ int @ ( modulo_modulo @ nat @ ( nat2 @ ( abs_abs @ int @ K ) ) @ ( nat2 @ ( abs_abs @ int @ L2 ) ) ) ) ) ).
% mod_abs_eq_div_nat
thf(fact_4427_udvd__incr0,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [P2: word @ A,Q2: word @ A,N: int,K6: word @ A,N4: int] :
( ( ord_less @ ( word @ A ) @ P2 @ Q2 )
=> ( ( ( semiring_1_unsigned @ A @ int @ P2 )
= ( times_times @ int @ N @ ( semiring_1_unsigned @ A @ int @ K6 ) ) )
=> ( ( ( semiring_1_unsigned @ A @ int @ Q2 )
= ( times_times @ int @ N4 @ ( semiring_1_unsigned @ A @ int @ K6 ) ) )
=> ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ P2 @ K6 ) @ Q2 ) ) ) ) ) ).
% udvd_incr0
thf(fact_4428_udvd__decr0,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [P2: word @ A,Q2: word @ A,N: int,K6: word @ A,N4: int] :
( ( ord_less @ ( word @ A ) @ P2 @ Q2 )
=> ( ( ( semiring_1_unsigned @ A @ int @ P2 )
= ( times_times @ int @ N @ ( semiring_1_unsigned @ A @ int @ K6 ) ) )
=> ( ( ( semiring_1_unsigned @ A @ int @ Q2 )
= ( times_times @ int @ N4 @ ( semiring_1_unsigned @ A @ int @ K6 ) ) )
=> ( ( ( semiring_1_unsigned @ A @ int @ Q2 )
= ( times_times @ int @ N4 @ ( semiring_1_unsigned @ A @ int @ K6 ) ) )
=> ( ord_less_eq @ ( word @ A ) @ P2 @ ( minus_minus @ ( word @ A ) @ Q2 @ K6 ) ) ) ) ) ) ) ).
% udvd_decr0
thf(fact_4429_floor__eq3,axiom,
! [N: nat,X2: real] :
( ( ord_less @ real @ ( semiring_1_of_nat @ real @ N ) @ X2 )
=> ( ( ord_less @ real @ X2 @ ( semiring_1_of_nat @ real @ ( suc @ N ) ) )
=> ( ( nat2 @ ( archim6421214686448440834_floor @ real @ X2 ) )
= N ) ) ) ).
% floor_eq3
thf(fact_4430_le__nat__floor,axiom,
! [X2: nat,A2: real] :
( ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ X2 ) @ A2 )
=> ( ord_less_eq @ nat @ X2 @ ( nat2 @ ( archim6421214686448440834_floor @ real @ A2 ) ) ) ) ).
% le_nat_floor
thf(fact_4431_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_4432_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_4433_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_4434_take__bit__nat__def,axiom,
( ( bit_se2584673776208193580ke_bit @ nat )
= ( ^ [N3: nat,M5: nat] : ( modulo_modulo @ nat @ M5 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) ) ) ) ).
% take_bit_nat_def
thf(fact_4435_nat__2,axiom,
( ( nat2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) )
= ( suc @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% nat_2
thf(fact_4436_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_4437_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_4438_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_4439_nat__mult__distrib__neg,axiom,
! [Z: int,Z6: int] :
( ( ord_less_eq @ int @ Z @ ( zero_zero @ int ) )
=> ( ( nat2 @ ( times_times @ int @ Z @ Z6 ) )
= ( times_times @ nat @ ( nat2 @ ( uminus_uminus @ int @ Z ) ) @ ( nat2 @ ( uminus_uminus @ int @ Z6 ) ) ) ) ) ).
% nat_mult_distrib_neg
thf(fact_4440_nat__abs__int__diff,axiom,
! [A2: nat,B2: nat] :
( ( ( ord_less_eq @ nat @ A2 @ B2 )
=> ( ( nat2 @ ( abs_abs @ int @ ( minus_minus @ int @ ( semiring_1_of_nat @ int @ A2 ) @ ( semiring_1_of_nat @ int @ B2 ) ) ) )
= ( minus_minus @ nat @ B2 @ A2 ) ) )
& ( ~ ( ord_less_eq @ nat @ A2 @ B2 )
=> ( ( nat2 @ ( abs_abs @ int @ ( minus_minus @ int @ ( semiring_1_of_nat @ int @ A2 ) @ ( semiring_1_of_nat @ int @ B2 ) ) ) )
= ( minus_minus @ nat @ A2 @ B2 ) ) ) ) ).
% nat_abs_int_diff
thf(fact_4441_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_4442_take__bit__int__def,axiom,
( ( bit_se2584673776208193580ke_bit @ int )
= ( ^ [N3: nat,K2: int] : ( modulo_modulo @ int @ K2 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N3 ) ) ) ) ).
% take_bit_int_def
thf(fact_4443_floor__eq4,axiom,
! [N: nat,X2: real] :
( ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ N ) @ X2 )
=> ( ( ord_less @ real @ X2 @ ( semiring_1_of_nat @ real @ ( suc @ N ) ) )
=> ( ( nat2 @ ( archim6421214686448440834_floor @ real @ X2 ) )
= N ) ) ) ).
% floor_eq4
thf(fact_4444_udvd__incr_H,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [P2: word @ A,Q2: word @ A,Ua: int,N: int,K6: word @ A,N4: int] :
( ( ord_less @ ( word @ A ) @ P2 @ Q2 )
=> ( ( ( semiring_1_unsigned @ A @ int @ P2 )
= ( plus_plus @ int @ Ua @ ( times_times @ int @ N @ ( semiring_1_unsigned @ A @ int @ K6 ) ) ) )
=> ( ( ( semiring_1_unsigned @ A @ int @ Q2 )
= ( plus_plus @ int @ Ua @ ( times_times @ int @ N4 @ ( semiring_1_unsigned @ A @ int @ K6 ) ) ) )
=> ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ P2 @ K6 ) @ Q2 ) ) ) ) ) ).
% udvd_incr'
thf(fact_4445_udvd__decr_H,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [P2: word @ A,Q2: word @ A,Ua: int,N: int,K6: word @ A,N4: int] :
( ( ord_less @ ( word @ A ) @ P2 @ Q2 )
=> ( ( ( semiring_1_unsigned @ A @ int @ P2 )
= ( plus_plus @ int @ Ua @ ( times_times @ int @ N @ ( semiring_1_unsigned @ A @ int @ K6 ) ) ) )
=> ( ( ( semiring_1_unsigned @ A @ int @ Q2 )
= ( plus_plus @ int @ Ua @ ( times_times @ int @ N4 @ ( semiring_1_unsigned @ A @ int @ K6 ) ) ) )
=> ( ( ( semiring_1_unsigned @ A @ int @ Q2 )
= ( plus_plus @ int @ Ua @ ( times_times @ int @ N4 @ ( semiring_1_unsigned @ A @ int @ K6 ) ) ) )
=> ( ord_less_eq @ ( word @ A ) @ P2 @ ( minus_minus @ ( word @ A ) @ Q2 @ K6 ) ) ) ) ) ) ) ).
% udvd_decr'
thf(fact_4446_diff__nat__eq__if,axiom,
! [Z6: int,Z: int] :
( ( ( ord_less @ int @ Z6 @ ( zero_zero @ int ) )
=> ( ( minus_minus @ nat @ ( nat2 @ Z ) @ ( nat2 @ Z6 ) )
= ( nat2 @ Z ) ) )
& ( ~ ( ord_less @ int @ Z6 @ ( zero_zero @ int ) )
=> ( ( minus_minus @ nat @ ( nat2 @ Z ) @ ( nat2 @ Z6 ) )
= ( if @ nat @ ( ord_less @ int @ ( minus_minus @ int @ Z @ Z6 ) @ ( zero_zero @ int ) ) @ ( zero_zero @ nat ) @ ( nat2 @ ( minus_minus @ int @ Z @ Z6 ) ) ) ) ) ) ).
% diff_nat_eq_if
thf(fact_4447_take__bit__numeral__minus__bit1,axiom,
! [L2: num,K: num] :
( ( bit_se2584673776208193580ke_bit @ int @ ( numeral_numeral @ nat @ L2 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ K ) ) ) )
= ( plus_plus @ int @ ( times_times @ int @ ( bit_se2584673776208193580ke_bit @ int @ ( pred_numeral @ L2 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( inc @ K ) ) ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( one_one @ int ) ) ) ).
% take_bit_numeral_minus_bit1
thf(fact_4448_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_4449_of__int__of__nat,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ( ( ring_1_of_int @ A )
= ( ^ [K2: int] : ( if @ A @ ( ord_less @ int @ K2 @ ( zero_zero @ int ) ) @ ( uminus_uminus @ A @ ( semiring_1_of_nat @ A @ ( nat2 @ ( uminus_uminus @ int @ K2 ) ) ) ) @ ( semiring_1_of_nat @ A @ ( nat2 @ K2 ) ) ) ) ) ) ).
% of_int_of_nat
thf(fact_4450_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_4451_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_4452_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_4453_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_4454_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_4455_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_4456_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_4457_take__bit__numeral__minus__bit0,axiom,
! [L2: num,K: num] :
( ( bit_se2584673776208193580ke_bit @ int @ ( numeral_numeral @ nat @ L2 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ K ) ) ) )
= ( times_times @ int @ ( bit_se2584673776208193580ke_bit @ int @ ( pred_numeral @ L2 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ).
% take_bit_numeral_minus_bit0
thf(fact_4458_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_4459_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_4460_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_4461_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_4462_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_4463_signed__take__bit__eq__take__bit__shift,axiom,
( ( bit_ri4674362597316999326ke_bit @ int )
= ( ^ [N3: nat,K2: int] : ( minus_minus @ int @ ( bit_se2584673776208193580ke_bit @ int @ ( suc @ N3 ) @ ( plus_plus @ int @ K2 @ ( 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_4464_no__plus__overflow__uint__size,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A,Y2: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ X2 @ ( plus_plus @ ( word @ A ) @ X2 @ Y2 ) )
= ( ord_less @ int @ ( plus_plus @ int @ ( semiring_1_unsigned @ A @ int @ X2 ) @ ( semiring_1_unsigned @ A @ int @ Y2 ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( size_size @ ( word @ A ) @ X2 ) ) ) ) ) ).
% no_plus_overflow_uint_size
thf(fact_4465_uint__plus__if__size,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A,Y2: word @ A] :
( ( ( ord_less @ int @ ( plus_plus @ int @ ( semiring_1_unsigned @ A @ int @ X2 ) @ ( semiring_1_unsigned @ A @ int @ Y2 ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( size_size @ ( word @ A ) @ X2 ) ) )
=> ( ( semiring_1_unsigned @ A @ int @ ( plus_plus @ ( word @ A ) @ X2 @ Y2 ) )
= ( plus_plus @ int @ ( semiring_1_unsigned @ A @ int @ X2 ) @ ( semiring_1_unsigned @ A @ int @ Y2 ) ) ) )
& ( ~ ( ord_less @ int @ ( plus_plus @ int @ ( semiring_1_unsigned @ A @ int @ X2 ) @ ( semiring_1_unsigned @ A @ int @ Y2 ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( size_size @ ( word @ A ) @ X2 ) ) )
=> ( ( semiring_1_unsigned @ A @ int @ ( plus_plus @ ( word @ A ) @ X2 @ Y2 ) )
= ( minus_minus @ int @ ( plus_plus @ int @ ( semiring_1_unsigned @ A @ int @ X2 ) @ ( semiring_1_unsigned @ A @ int @ Y2 ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( size_size @ ( word @ A ) @ X2 ) ) ) ) ) ) ) ).
% uint_plus_if_size
thf(fact_4466_uint__sub__if__size,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y2: word @ A,X2: word @ A] :
( ( ( ord_less_eq @ int @ ( semiring_1_unsigned @ A @ int @ Y2 ) @ ( semiring_1_unsigned @ A @ int @ X2 ) )
=> ( ( semiring_1_unsigned @ A @ int @ ( minus_minus @ ( word @ A ) @ X2 @ Y2 ) )
= ( minus_minus @ int @ ( semiring_1_unsigned @ A @ int @ X2 ) @ ( semiring_1_unsigned @ A @ int @ Y2 ) ) ) )
& ( ~ ( ord_less_eq @ int @ ( semiring_1_unsigned @ A @ int @ Y2 ) @ ( semiring_1_unsigned @ A @ int @ X2 ) )
=> ( ( semiring_1_unsigned @ A @ int @ ( minus_minus @ ( word @ A ) @ X2 @ Y2 ) )
= ( plus_plus @ int @ ( minus_minus @ int @ ( semiring_1_unsigned @ A @ int @ X2 ) @ ( semiring_1_unsigned @ A @ int @ Y2 ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( size_size @ ( word @ A ) @ X2 ) ) ) ) ) ) ) ).
% uint_sub_if_size
thf(fact_4467_powr__real__of__int,axiom,
! [X2: real,N: int] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ N )
=> ( ( powr @ real @ X2 @ ( ring_1_of_int @ real @ N ) )
= ( power_power @ real @ X2 @ ( nat2 @ N ) ) ) )
& ( ~ ( ord_less_eq @ int @ ( zero_zero @ int ) @ N )
=> ( ( powr @ real @ X2 @ ( ring_1_of_int @ real @ N ) )
= ( inverse_inverse @ real @ ( power_power @ real @ X2 @ ( nat2 @ ( uminus_uminus @ int @ N ) ) ) ) ) ) ) ) ).
% powr_real_of_int
thf(fact_4468_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_4469_arctan__def,axiom,
( arctan
= ( ^ [Y: real] :
( the @ real
@ ^ [X: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X )
& ( ord_less @ real @ X @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
& ( ( tan @ real @ X )
= Y ) ) ) ) ) ).
% arctan_def
thf(fact_4470_arcsin__def,axiom,
( arcsin
= ( ^ [Y: real] :
( the @ real
@ ^ [X: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X )
& ( ord_less_eq @ real @ X @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
& ( ( sin @ real @ X )
= Y ) ) ) ) ) ).
% arcsin_def
thf(fact_4471_divide__int__def,axiom,
( ( divide_divide @ int )
= ( ^ [K2: int,L: int] :
( if @ int
@ ( L
= ( zero_zero @ int ) )
@ ( zero_zero @ int )
@ ( if @ int
@ ( ( sgn_sgn @ int @ K2 )
= ( sgn_sgn @ int @ L ) )
@ ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ ( nat2 @ ( abs_abs @ int @ K2 ) ) @ ( nat2 @ ( abs_abs @ int @ L ) ) ) )
@ ( uminus_uminus @ int
@ ( semiring_1_of_nat @ int
@ ( plus_plus @ nat @ ( divide_divide @ nat @ ( nat2 @ ( abs_abs @ int @ K2 ) ) @ ( nat2 @ ( abs_abs @ int @ L ) ) )
@ ( zero_neq_one_of_bool @ nat
@ ~ ( dvd_dvd @ int @ L @ K2 ) ) ) ) ) ) ) ) ) ).
% divide_int_def
thf(fact_4472_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_4473_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_4474_sgn__0,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ( ( sgn_sgn @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% sgn_0
thf(fact_4475_sgn__zero,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ( ( sgn_sgn @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% sgn_zero
thf(fact_4476_sgn__1,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ( ( sgn_sgn @ A @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% sgn_1
thf(fact_4477_sgn__one,axiom,
! [A: $tType] :
( ( real_V2822296259951069270ebra_1 @ A )
=> ( ( sgn_sgn @ A @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% sgn_one
thf(fact_4478_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_4479_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_4480_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_4481_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_4482_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_4483_set__decode__inverse,axiom,
! [N: nat] :
( ( nat_set_encode @ ( nat_set_decode @ N ) )
= N ) ).
% set_decode_inverse
thf(fact_4484_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_4485_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_4486_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_4487_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_4488_set__encode__inverse,axiom,
! [A3: set @ nat] :
( ( finite_finite2 @ nat @ A3 )
=> ( ( nat_set_decode @ ( nat_set_encode @ A3 ) )
= A3 ) ) ).
% set_encode_inverse
thf(fact_4489_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_4490_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_4491_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_4492_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_4493_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_4494_sgn__mult__dvd__iff,axiom,
! [R2: int,L2: int,K: int] :
( ( dvd_dvd @ int @ ( times_times @ int @ ( sgn_sgn @ int @ R2 ) @ L2 ) @ K )
= ( ( dvd_dvd @ int @ L2 @ K )
& ( ( R2
= ( zero_zero @ int ) )
=> ( K
= ( zero_zero @ int ) ) ) ) ) ).
% sgn_mult_dvd_iff
thf(fact_4495_mult__sgn__dvd__iff,axiom,
! [L2: int,R2: int,K: int] :
( ( dvd_dvd @ int @ ( times_times @ int @ L2 @ ( sgn_sgn @ int @ R2 ) ) @ K )
= ( ( dvd_dvd @ int @ L2 @ K )
& ( ( R2
= ( zero_zero @ int ) )
=> ( K
= ( zero_zero @ int ) ) ) ) ) ).
% mult_sgn_dvd_iff
thf(fact_4496_dvd__sgn__mult__iff,axiom,
! [L2: int,R2: int,K: int] :
( ( dvd_dvd @ int @ L2 @ ( times_times @ int @ ( sgn_sgn @ int @ R2 ) @ K ) )
= ( ( dvd_dvd @ int @ L2 @ K )
| ( R2
= ( zero_zero @ int ) ) ) ) ).
% dvd_sgn_mult_iff
thf(fact_4497_dvd__mult__sgn__iff,axiom,
! [L2: int,K: int,R2: int] :
( ( dvd_dvd @ int @ L2 @ ( times_times @ int @ K @ ( sgn_sgn @ int @ R2 ) ) )
= ( ( dvd_dvd @ int @ L2 @ K )
| ( R2
= ( zero_zero @ int ) ) ) ) ).
% dvd_mult_sgn_iff
thf(fact_4498_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_4499_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_4500_Suc__unat__minus__one,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A] :
( ( X2
!= ( zero_zero @ ( word @ A ) ) )
=> ( ( suc @ ( semiring_1_unsigned @ A @ nat @ ( minus_minus @ ( word @ A ) @ X2 @ ( one_one @ ( word @ A ) ) ) ) )
= ( semiring_1_unsigned @ A @ nat @ X2 ) ) ) ) ).
% Suc_unat_minus_one
thf(fact_4501_word__unat__Rep__inject1,axiom,
! [B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B ) )
=> ! [X2: word @ A] :
( ( ( semiring_1_unsigned @ A @ nat @ X2 )
= ( semiring_1_unsigned @ B @ nat @ ( one_one @ ( word @ B ) ) ) )
= ( X2
= ( one_one @ ( word @ A ) ) ) ) ) ).
% word_unat_Rep_inject1
thf(fact_4502_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_4503_ucast__0__I,axiom,
! [A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( type_len @ A ) )
=> ! [X2: word @ A] :
( ( X2
= ( zero_zero @ ( word @ A ) ) )
=> ( ( semiring_1_unsigned @ A @ ( word @ B ) @ X2 )
= ( zero_zero @ ( word @ B ) ) ) ) ) ).
% ucast_0_I
thf(fact_4504_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_4505_ucast__nat__def,axiom,
! [B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B ) )
=> ! [X2: word @ A] :
( ( semiring_1_of_nat @ ( word @ B ) @ ( semiring_1_unsigned @ A @ nat @ X2 ) )
= ( semiring_1_unsigned @ A @ ( word @ B ) @ X2 ) ) ) ).
% ucast_nat_def
thf(fact_4506_sgn__zero__iff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X2: A] :
( ( ( sgn_sgn @ A @ X2 )
= ( zero_zero @ A ) )
= ( X2
= ( zero_zero @ A ) ) ) ) ).
% sgn_zero_iff
thf(fact_4507_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_4508_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_4509_Real__Vector__Spaces_Osgn__mult,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [X2: A,Y2: A] :
( ( sgn_sgn @ A @ ( times_times @ A @ X2 @ Y2 ) )
= ( times_times @ A @ ( sgn_sgn @ A @ X2 ) @ ( sgn_sgn @ A @ Y2 ) ) ) ) ).
% Real_Vector_Spaces.sgn_mult
thf(fact_4510_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_4511_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_4512_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_4513_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_4514_int__sgnE,axiom,
! [K: int] :
~ ! [N2: nat,L4: int] :
( K
!= ( times_times @ int @ ( sgn_sgn @ int @ L4 ) @ ( semiring_1_of_nat @ int @ N2 ) ) ) ).
% int_sgnE
thf(fact_4515_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_4516_linordered__idom__class_Oabs__sgn,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( ( abs_abs @ A )
= ( ^ [K2: A] : ( times_times @ A @ K2 @ ( sgn_sgn @ A @ K2 ) ) ) ) ) ).
% linordered_idom_class.abs_sgn
thf(fact_4517_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_4518_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_4519_mult__sgn__abs,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X2: A] :
( ( times_times @ A @ ( sgn_sgn @ A @ X2 ) @ ( abs_abs @ A @ X2 ) )
= X2 ) ) ).
% mult_sgn_abs
thf(fact_4520_unat__eq__zero,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A] :
( ( ( semiring_1_unsigned @ A @ nat @ X2 )
= ( zero_zero @ nat ) )
= ( X2
= ( zero_zero @ ( word @ A ) ) ) ) ) ).
% unat_eq_zero
thf(fact_4521_unat__0,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( semiring_1_unsigned @ A @ nat @ ( zero_zero @ ( word @ A ) ) )
= ( zero_zero @ nat ) ) ) ).
% unat_0
thf(fact_4522_word__le__nat__alt,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( ord_less_eq @ ( word @ A ) )
= ( ^ [A4: word @ A,B3: word @ A] : ( ord_less_eq @ nat @ ( semiring_1_unsigned @ A @ nat @ A4 ) @ ( semiring_1_unsigned @ A @ nat @ B3 ) ) ) ) ) ).
% word_le_nat_alt
thf(fact_4523_unat__1,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( semiring_1_unsigned @ A @ nat @ ( one_one @ ( word @ A ) ) )
= ( one_one @ nat ) ) ) ).
% unat_1
thf(fact_4524_div__eq__sgn__abs,axiom,
! [K: int,L2: int] :
( ( ( sgn_sgn @ int @ K )
= ( sgn_sgn @ int @ L2 ) )
=> ( ( divide_divide @ int @ K @ L2 )
= ( divide_divide @ int @ ( abs_abs @ int @ K ) @ ( abs_abs @ int @ L2 ) ) ) ) ).
% div_eq_sgn_abs
thf(fact_4525_uint__nat,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( semiring_1_unsigned @ A @ int )
= ( ^ [W2: word @ A] : ( semiring_1_of_nat @ int @ ( semiring_1_unsigned @ A @ nat @ W2 ) ) ) ) ) ).
% uint_nat
thf(fact_4526_le__unat__uoi,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y2: nat,Z: word @ A] :
( ( ord_less_eq @ nat @ Y2 @ ( semiring_1_unsigned @ A @ nat @ Z ) )
=> ( ( semiring_1_unsigned @ A @ nat @ ( semiring_1_of_nat @ ( word @ A ) @ Y2 ) )
= Y2 ) ) ) ).
% le_unat_uoi
thf(fact_4527_set__encode__eq,axiom,
! [A3: set @ nat,B5: set @ nat] :
( ( finite_finite2 @ nat @ A3 )
=> ( ( finite_finite2 @ nat @ B5 )
=> ( ( ( nat_set_encode @ A3 )
= ( nat_set_encode @ B5 ) )
= ( A3 = B5 ) ) ) ) ).
% set_encode_eq
thf(fact_4528_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_4529_unat__div,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A,Y2: word @ A] :
( ( semiring_1_unsigned @ A @ nat @ ( divide_divide @ ( word @ A ) @ X2 @ Y2 ) )
= ( divide_divide @ nat @ ( semiring_1_unsigned @ A @ nat @ X2 ) @ ( semiring_1_unsigned @ A @ nat @ Y2 ) ) ) ) ).
% unat_div
thf(fact_4530_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_4531_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_4532_ln__real__def,axiom,
( ( ln_ln @ real )
= ( ^ [X: real] :
( the @ real
@ ^ [U2: real] :
( ( exp @ real @ U2 )
= X ) ) ) ) ).
% ln_real_def
thf(fact_4533_suminf__def,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ( ( suminf @ A )
= ( ^ [F4: nat > A] : ( the @ A @ ( sums @ A @ F4 ) ) ) ) ) ).
% suminf_def
thf(fact_4534_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_4535_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_4536_unat__eq__1,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A] :
( ( ( semiring_1_unsigned @ A @ nat @ X2 )
= ( suc @ ( zero_zero @ nat ) ) )
= ( X2
= ( one_one @ ( word @ A ) ) ) ) ) ).
% unat_eq_1
thf(fact_4537_sgn__mod,axiom,
! [L2: int,K: int] :
( ( L2
!= ( zero_zero @ int ) )
=> ( ~ ( dvd_dvd @ int @ L2 @ K )
=> ( ( sgn_sgn @ int @ ( modulo_modulo @ int @ K @ L2 ) )
= ( sgn_sgn @ int @ L2 ) ) ) ) ).
% sgn_mod
thf(fact_4538_unat__gt__0,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( semiring_1_unsigned @ A @ nat @ X2 ) )
= ( X2
!= ( zero_zero @ ( word @ A ) ) ) ) ) ).
% unat_gt_0
thf(fact_4539_un__ui__le,axiom,
! [A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( type_len @ A ) )
=> ! [A2: word @ A,B2: word @ B] :
( ( ord_less_eq @ nat @ ( semiring_1_unsigned @ A @ nat @ A2 ) @ ( semiring_1_unsigned @ B @ nat @ B2 ) )
= ( ord_less_eq @ int @ ( semiring_1_unsigned @ A @ int @ A2 ) @ ( semiring_1_unsigned @ B @ int @ B2 ) ) ) ) ).
% un_ui_le
thf(fact_4540_unat__plus__simple,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A,Y2: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ X2 @ ( plus_plus @ ( word @ A ) @ X2 @ Y2 ) )
= ( ( semiring_1_unsigned @ A @ nat @ ( plus_plus @ ( word @ A ) @ X2 @ Y2 ) )
= ( plus_plus @ nat @ ( semiring_1_unsigned @ A @ nat @ X2 ) @ ( semiring_1_unsigned @ A @ nat @ Y2 ) ) ) ) ) ).
% unat_plus_simple
thf(fact_4541_word__of__nat__less,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,X2: word @ A] :
( ( ord_less @ nat @ N @ ( semiring_1_unsigned @ A @ nat @ X2 ) )
=> ( ord_less @ ( word @ A ) @ ( semiring_1_of_nat @ ( word @ A ) @ N ) @ X2 ) ) ) ).
% word_of_nat_less
thf(fact_4542_unat__less__helper,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A,N: nat] :
( ( ord_less @ ( word @ A ) @ X2 @ ( semiring_1_of_nat @ ( word @ A ) @ N ) )
=> ( ord_less @ nat @ ( semiring_1_unsigned @ A @ nat @ X2 ) @ N ) ) ) ).
% unat_less_helper
thf(fact_4543_word__unat__less__le,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: word @ A,B2: nat] :
( ( ord_less_eq @ ( word @ A ) @ A2 @ ( semiring_1_of_nat @ ( word @ A ) @ B2 ) )
=> ( ord_less_eq @ nat @ ( semiring_1_unsigned @ A @ nat @ A2 ) @ B2 ) ) ) ).
% word_unat_less_le
thf(fact_4544_word__of__nat__le,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,X2: word @ A] :
( ( ord_less_eq @ nat @ N @ ( semiring_1_unsigned @ A @ nat @ X2 ) )
=> ( ord_less_eq @ ( word @ A ) @ ( semiring_1_of_nat @ ( word @ A ) @ N ) @ X2 ) ) ) ).
% word_of_nat_le
thf(fact_4545_word__arith__nat__add,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( plus_plus @ ( word @ A ) )
= ( ^ [A4: word @ A,B3: word @ A] : ( semiring_1_of_nat @ ( word @ A ) @ ( plus_plus @ nat @ ( semiring_1_unsigned @ A @ nat @ A4 ) @ ( semiring_1_unsigned @ A @ nat @ B3 ) ) ) ) ) ) ).
% word_arith_nat_add
thf(fact_4546_word__arith__nat__mult,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( times_times @ ( word @ A ) )
= ( ^ [A4: word @ A,B3: word @ A] : ( semiring_1_of_nat @ ( word @ A ) @ ( times_times @ nat @ ( semiring_1_unsigned @ A @ nat @ A4 ) @ ( semiring_1_unsigned @ A @ nat @ B3 ) ) ) ) ) ) ).
% word_arith_nat_mult
thf(fact_4547_word__arith__nat__div,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( divide_divide @ ( word @ A ) )
= ( ^ [A4: word @ A,B3: word @ A] : ( semiring_1_of_nat @ ( word @ A ) @ ( divide_divide @ nat @ ( semiring_1_unsigned @ A @ nat @ A4 ) @ ( semiring_1_unsigned @ A @ nat @ B3 ) ) ) ) ) ) ).
% word_arith_nat_div
thf(fact_4548_word__arith__nat__mod,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( modulo_modulo @ ( word @ A ) )
= ( ^ [A4: word @ A,B3: word @ A] : ( semiring_1_of_nat @ ( word @ A ) @ ( modulo_modulo @ nat @ ( semiring_1_unsigned @ A @ nat @ A4 ) @ ( semiring_1_unsigned @ A @ nat @ B3 ) ) ) ) ) ) ).
% word_arith_nat_mod
thf(fact_4549_ln__neg__is__const,axiom,
! [X2: real] :
( ( ord_less_eq @ real @ X2 @ ( zero_zero @ real ) )
=> ( ( ln_ln @ real @ X2 )
= ( the @ real
@ ^ [X: real] : $false ) ) ) ).
% ln_neg_is_const
thf(fact_4550_set__encode__inf,axiom,
! [A3: set @ nat] :
( ~ ( finite_finite2 @ nat @ A3 )
=> ( ( nat_set_encode @ A3 )
= ( zero_zero @ nat ) ) ) ).
% set_encode_inf
thf(fact_4551_sgn__if,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( ( sgn_sgn @ A )
= ( ^ [X: A] :
( if @ A
@ ( X
= ( zero_zero @ A ) )
@ ( zero_zero @ A )
@ ( if @ A @ ( ord_less @ A @ ( zero_zero @ A ) @ X ) @ ( one_one @ A ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ) ) ).
% sgn_if
thf(fact_4552_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_4553_zsgn__def,axiom,
( ( sgn_sgn @ int )
= ( ^ [I3: int] :
( if @ int
@ ( I3
= ( zero_zero @ int ) )
@ ( zero_zero @ int )
@ ( if @ int @ ( ord_less @ int @ ( zero_zero @ int ) @ I3 ) @ ( one_one @ int ) @ ( uminus_uminus @ int @ ( one_one @ int ) ) ) ) ) ) ).
% zsgn_def
thf(fact_4554_norm__sgn,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X2: A] :
( ( ( X2
= ( zero_zero @ A ) )
=> ( ( real_V7770717601297561774m_norm @ A @ ( sgn_sgn @ A @ X2 ) )
= ( zero_zero @ real ) ) )
& ( ( X2
!= ( zero_zero @ A ) )
=> ( ( real_V7770717601297561774m_norm @ A @ ( sgn_sgn @ A @ X2 ) )
= ( one_one @ real ) ) ) ) ) ).
% norm_sgn
thf(fact_4555_div__sgn__abs__cancel,axiom,
! [V: int,K: int,L2: 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 @ L2 ) ) )
= ( divide_divide @ int @ ( abs_abs @ int @ K ) @ ( abs_abs @ int @ L2 ) ) ) ) ).
% div_sgn_abs_cancel
thf(fact_4556_unat__1__0,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ ( one_one @ ( word @ A ) ) @ X2 )
= ( ord_less @ nat @ ( zero_zero @ nat ) @ ( semiring_1_unsigned @ A @ nat @ X2 ) ) ) ) ).
% unat_1_0
thf(fact_4557_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_4558_div__dvd__sgn__abs,axiom,
! [L2: int,K: int] :
( ( dvd_dvd @ int @ L2 @ K )
=> ( ( divide_divide @ int @ K @ L2 )
= ( times_times @ int @ ( times_times @ int @ ( sgn_sgn @ int @ K ) @ ( sgn_sgn @ int @ L2 ) ) @ ( divide_divide @ int @ ( abs_abs @ int @ K ) @ ( abs_abs @ int @ L2 ) ) ) ) ) ).
% div_dvd_sgn_abs
thf(fact_4559_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_4560_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_4561_Suc__unat__diff__1,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ ( one_one @ ( word @ A ) ) @ X2 )
=> ( ( suc @ ( semiring_1_unsigned @ A @ nat @ ( minus_minus @ ( word @ A ) @ X2 @ ( one_one @ ( word @ A ) ) ) ) )
= ( semiring_1_unsigned @ A @ nat @ X2 ) ) ) ) ).
% Suc_unat_diff_1
thf(fact_4562_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_4563_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_4564_measure__unat,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [P2: word @ A] :
( ( P2
!= ( zero_zero @ ( word @ A ) ) )
=> ( ord_less @ nat @ ( semiring_1_unsigned @ A @ nat @ ( minus_minus @ ( word @ A ) @ P2 @ ( one_one @ ( word @ A ) ) ) ) @ ( semiring_1_unsigned @ A @ nat @ P2 ) ) ) ) ).
% measure_unat
thf(fact_4565_word__overflow__unat,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A] :
( ( ( semiring_1_unsigned @ A @ nat @ ( plus_plus @ ( word @ A ) @ X2 @ ( one_one @ ( word @ A ) ) ) )
= ( plus_plus @ nat @ ( semiring_1_unsigned @ A @ nat @ X2 ) @ ( one_one @ nat ) ) )
| ( ( plus_plus @ ( word @ A ) @ X2 @ ( one_one @ ( word @ A ) ) )
= ( zero_zero @ ( word @ A ) ) ) ) ) ).
% word_overflow_unat
thf(fact_4566_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_4567_arccos__def,axiom,
( arccos
= ( ^ [Y: real] :
( the @ real
@ ^ [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
& ( ord_less_eq @ real @ X @ pi )
& ( ( cos @ real @ X )
= Y ) ) ) ) ) ).
% arccos_def
thf(fact_4568_div__noneq__sgn__abs,axiom,
! [L2: int,K: int] :
( ( L2
!= ( zero_zero @ int ) )
=> ( ( ( sgn_sgn @ int @ K )
!= ( sgn_sgn @ int @ L2 ) )
=> ( ( divide_divide @ int @ K @ L2 )
= ( minus_minus @ int @ ( uminus_uminus @ int @ ( divide_divide @ int @ ( abs_abs @ int @ K ) @ ( abs_abs @ int @ L2 ) ) )
@ ( zero_neq_one_of_bool @ int
@ ~ ( dvd_dvd @ int @ L2 @ K ) ) ) ) ) ) ).
% div_noneq_sgn_abs
thf(fact_4569_lt__plus__1__le__word,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,MaxBound: word @ A,X2: word @ A] :
( ( ord_less @ nat @ N @ ( semiring_1_unsigned @ A @ nat @ MaxBound ) )
=> ( ( ord_less @ ( word @ A ) @ X2 @ ( plus_plus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) @ ( semiring_1_of_nat @ ( word @ A ) @ N ) ) )
= ( ord_less_eq @ ( word @ A ) @ X2 @ ( semiring_1_of_nat @ ( word @ A ) @ N ) ) ) ) ) ).
% lt_plus_1_le_word
thf(fact_4570_even__word__imp__odd__next,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A] :
( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( semiring_1_unsigned @ A @ nat @ X2 ) )
=> ( ( ( plus_plus @ ( word @ A ) @ X2 @ ( one_one @ ( word @ A ) ) )
= ( zero_zero @ ( word @ A ) ) )
| ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( semiring_1_unsigned @ A @ nat @ ( plus_plus @ ( word @ A ) @ X2 @ ( one_one @ ( word @ A ) ) ) ) ) ) ) ) ).
% even_word_imp_odd_next
thf(fact_4571_odd__word__imp__even__next,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( semiring_1_unsigned @ A @ nat @ X2 ) )
=> ( ( ( plus_plus @ ( word @ A ) @ X2 @ ( one_one @ ( word @ A ) ) )
= ( zero_zero @ ( word @ A ) ) )
| ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( semiring_1_unsigned @ A @ nat @ ( plus_plus @ ( word @ A ) @ X2 @ ( one_one @ ( word @ A ) ) ) ) ) ) ) ) ).
% odd_word_imp_even_next
thf(fact_4572_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_4573_of__nat__eq__size,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,W: word @ A] :
( ( ( semiring_1_of_nat @ ( word @ A ) @ N )
= W )
= ( ? [Q4: nat] :
( N
= ( plus_plus @ nat @ ( semiring_1_unsigned @ A @ nat @ W ) @ ( times_times @ nat @ Q4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( size_size @ ( word @ A ) @ W ) ) ) ) ) ) ) ) ).
% of_nat_eq_size
thf(fact_4574_set__encode__def,axiom,
( nat_set_encode
= ( groups7311177749621191930dd_sum @ nat @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% set_encode_def
thf(fact_4575_pi__half,axiom,
( ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) )
= ( the @ real
@ ^ [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
& ( ord_less_eq @ real @ X @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) )
& ( ( cos @ real @ X )
= ( zero_zero @ real ) ) ) ) ) ).
% pi_half
thf(fact_4576_pi__def,axiom,
( pi
= ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) )
@ ( the @ real
@ ^ [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
& ( ord_less_eq @ real @ X @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) )
& ( ( cos @ real @ X )
= ( zero_zero @ real ) ) ) ) ) ) ).
% pi_def
thf(fact_4577_unat__plus__if__size,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A,Y2: word @ A] :
( ( ( ord_less @ nat @ ( plus_plus @ nat @ ( semiring_1_unsigned @ A @ nat @ X2 ) @ ( semiring_1_unsigned @ A @ nat @ Y2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( size_size @ ( word @ A ) @ X2 ) ) )
=> ( ( semiring_1_unsigned @ A @ nat @ ( plus_plus @ ( word @ A ) @ X2 @ Y2 ) )
= ( plus_plus @ nat @ ( semiring_1_unsigned @ A @ nat @ X2 ) @ ( semiring_1_unsigned @ A @ nat @ Y2 ) ) ) )
& ( ~ ( ord_less @ nat @ ( plus_plus @ nat @ ( semiring_1_unsigned @ A @ nat @ X2 ) @ ( semiring_1_unsigned @ A @ nat @ Y2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( size_size @ ( word @ A ) @ X2 ) ) )
=> ( ( semiring_1_unsigned @ A @ nat @ ( plus_plus @ ( word @ A ) @ X2 @ Y2 ) )
= ( minus_minus @ nat @ ( plus_plus @ nat @ ( semiring_1_unsigned @ A @ nat @ X2 ) @ ( semiring_1_unsigned @ A @ nat @ Y2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( size_size @ ( word @ A ) @ X2 ) ) ) ) ) ) ) ).
% unat_plus_if_size
thf(fact_4578_modulo__int__def,axiom,
( ( modulo_modulo @ int )
= ( ^ [K2: int,L: int] :
( if @ int
@ ( L
= ( zero_zero @ int ) )
@ K2
@ ( if @ int
@ ( ( sgn_sgn @ int @ K2 )
= ( sgn_sgn @ int @ L ) )
@ ( times_times @ int @ ( sgn_sgn @ int @ L ) @ ( semiring_1_of_nat @ int @ ( modulo_modulo @ nat @ ( nat2 @ ( abs_abs @ int @ K2 ) ) @ ( nat2 @ ( abs_abs @ int @ L ) ) ) ) )
@ ( times_times @ int @ ( sgn_sgn @ int @ L )
@ ( minus_minus @ int
@ ( times_times @ int @ ( abs_abs @ int @ L )
@ ( zero_neq_one_of_bool @ int
@ ~ ( dvd_dvd @ int @ L @ K2 ) ) )
@ ( semiring_1_of_nat @ int @ ( modulo_modulo @ nat @ ( nat2 @ ( abs_abs @ int @ K2 ) ) @ ( nat2 @ ( abs_abs @ int @ L ) ) ) ) ) ) ) ) ) ) ).
% modulo_int_def
thf(fact_4579_unat__sub__if__size,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y2: word @ A,X2: word @ A] :
( ( ( ord_less_eq @ nat @ ( semiring_1_unsigned @ A @ nat @ Y2 ) @ ( semiring_1_unsigned @ A @ nat @ X2 ) )
=> ( ( semiring_1_unsigned @ A @ nat @ ( minus_minus @ ( word @ A ) @ X2 @ Y2 ) )
= ( minus_minus @ nat @ ( semiring_1_unsigned @ A @ nat @ X2 ) @ ( semiring_1_unsigned @ A @ nat @ Y2 ) ) ) )
& ( ~ ( ord_less_eq @ nat @ ( semiring_1_unsigned @ A @ nat @ Y2 ) @ ( semiring_1_unsigned @ A @ nat @ X2 ) )
=> ( ( semiring_1_unsigned @ A @ nat @ ( minus_minus @ ( word @ A ) @ X2 @ Y2 ) )
= ( minus_minus @ nat @ ( plus_plus @ nat @ ( semiring_1_unsigned @ A @ nat @ X2 ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( size_size @ ( word @ A ) @ X2 ) ) ) @ ( semiring_1_unsigned @ A @ nat @ Y2 ) ) ) ) ) ) ).
% unat_sub_if_size
thf(fact_4580_no__plus__overflow__unat__size,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A,Y2: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ X2 @ ( plus_plus @ ( word @ A ) @ X2 @ Y2 ) )
= ( ord_less @ nat @ ( plus_plus @ nat @ ( semiring_1_unsigned @ A @ nat @ X2 ) @ ( semiring_1_unsigned @ A @ nat @ Y2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( size_size @ ( word @ A ) @ X2 ) ) ) ) ) ).
% no_plus_overflow_unat_size
thf(fact_4581_divide__int__unfold,axiom,
! [L2: int,K: int,N: nat,M: nat] :
( ( ( ( ( sgn_sgn @ int @ L2 )
= ( 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 @ L2 ) @ ( semiring_1_of_nat @ int @ N ) ) )
= ( zero_zero @ int ) ) )
& ( ~ ( ( ( sgn_sgn @ int @ L2 )
= ( zero_zero @ int ) )
| ( ( sgn_sgn @ int @ K )
= ( zero_zero @ int ) )
| ( N
= ( zero_zero @ nat ) ) )
=> ( ( ( ( sgn_sgn @ int @ K )
= ( sgn_sgn @ int @ L2 ) )
=> ( ( divide_divide @ int @ ( times_times @ int @ ( sgn_sgn @ int @ K ) @ ( semiring_1_of_nat @ int @ M ) ) @ ( times_times @ int @ ( sgn_sgn @ int @ L2 ) @ ( semiring_1_of_nat @ int @ N ) ) )
= ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ M @ N ) ) ) )
& ( ( ( sgn_sgn @ int @ K )
!= ( sgn_sgn @ int @ L2 ) )
=> ( ( divide_divide @ int @ ( times_times @ int @ ( sgn_sgn @ int @ K ) @ ( semiring_1_of_nat @ int @ M ) ) @ ( times_times @ int @ ( sgn_sgn @ int @ L2 ) @ ( semiring_1_of_nat @ int @ 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_4582_modulo__int__unfold,axiom,
! [L2: int,K: int,N: nat,M: nat] :
( ( ( ( ( sgn_sgn @ int @ L2 )
= ( 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 @ L2 ) @ ( semiring_1_of_nat @ int @ N ) ) )
= ( times_times @ int @ ( sgn_sgn @ int @ K ) @ ( semiring_1_of_nat @ int @ M ) ) ) )
& ( ~ ( ( ( sgn_sgn @ int @ L2 )
= ( zero_zero @ int ) )
| ( ( sgn_sgn @ int @ K )
= ( zero_zero @ int ) )
| ( N
= ( zero_zero @ nat ) ) )
=> ( ( ( ( sgn_sgn @ int @ K )
= ( sgn_sgn @ int @ L2 ) )
=> ( ( modulo_modulo @ int @ ( times_times @ int @ ( sgn_sgn @ int @ K ) @ ( semiring_1_of_nat @ int @ M ) ) @ ( times_times @ int @ ( sgn_sgn @ int @ L2 ) @ ( semiring_1_of_nat @ int @ N ) ) )
= ( times_times @ int @ ( sgn_sgn @ int @ L2 ) @ ( semiring_1_of_nat @ int @ ( modulo_modulo @ nat @ M @ N ) ) ) ) )
& ( ( ( sgn_sgn @ int @ K )
!= ( sgn_sgn @ int @ L2 ) )
=> ( ( modulo_modulo @ int @ ( times_times @ int @ ( sgn_sgn @ int @ K ) @ ( semiring_1_of_nat @ int @ M ) ) @ ( times_times @ int @ ( sgn_sgn @ int @ L2 ) @ ( semiring_1_of_nat @ int @ N ) ) )
= ( times_times @ int @ ( sgn_sgn @ int @ L2 )
@ ( 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_4583_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_4584_the__sym__eq__trivial,axiom,
! [A: $tType,X2: A] :
( ( the @ A
@ ( ^ [Y5: A,Z3: A] : Y5 = Z3
@ X2 ) )
= X2 ) ).
% the_sym_eq_trivial
thf(fact_4585_the__eq__trivial,axiom,
! [A: $tType,A2: A] :
( ( the @ A
@ ^ [X: A] : X = A2 )
= A2 ) ).
% the_eq_trivial
thf(fact_4586_the__equality,axiom,
! [A: $tType,P: A > $o,A2: A] :
( ( P @ A2 )
=> ( ! [X3: A] :
( ( P @ X3 )
=> ( X3 = A2 ) )
=> ( ( the @ A @ P )
= A2 ) ) ) ).
% the_equality
thf(fact_4587_sgn__le__0__iff,axiom,
! [X2: real] :
( ( ord_less_eq @ real @ ( sgn_sgn @ real @ X2 ) @ ( zero_zero @ real ) )
= ( ord_less_eq @ real @ X2 @ ( zero_zero @ real ) ) ) ).
% sgn_le_0_iff
thf(fact_4588_zero__le__sgn__iff,axiom,
! [X2: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( sgn_sgn @ real @ X2 ) )
= ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 ) ) ).
% zero_le_sgn_iff
thf(fact_4589_real__sgn__eq,axiom,
( ( sgn_sgn @ real )
= ( ^ [X: real] : ( divide_divide @ real @ X @ ( abs_abs @ real @ X ) ) ) ) ).
% real_sgn_eq
thf(fact_4590_sgn__root,axiom,
! [N: nat,X2: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( sgn_sgn @ real @ ( root @ N @ X2 ) )
= ( sgn_sgn @ real @ X2 ) ) ) ).
% sgn_root
thf(fact_4591_cis__Arg,axiom,
! [Z: complex] :
( ( Z
!= ( zero_zero @ complex ) )
=> ( ( cis @ ( arg @ Z ) )
= ( sgn_sgn @ complex @ Z ) ) ) ).
% cis_Arg
thf(fact_4592_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_4593_sgn__power__injE,axiom,
! [A2: real,N: nat,X2: real,B2: real] :
( ( ( times_times @ real @ ( sgn_sgn @ real @ A2 ) @ ( power_power @ real @ ( abs_abs @ real @ A2 ) @ N ) )
= X2 )
=> ( ( X2
= ( 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_4594_theI,axiom,
! [A: $tType,P: A > $o,A2: A] :
( ( P @ A2 )
=> ( ! [X3: A] :
( ( P @ X3 )
=> ( X3 = A2 ) )
=> ( P @ ( the @ A @ P ) ) ) ) ).
% theI
thf(fact_4595_theI_H,axiom,
! [A: $tType,P: A > $o] :
( ? [X4: A] :
( ( P @ X4 )
& ! [Y3: A] :
( ( P @ Y3 )
=> ( Y3 = X4 ) ) )
=> ( P @ ( the @ A @ P ) ) ) ).
% theI'
thf(fact_4596_theI2,axiom,
! [A: $tType,P: A > $o,A2: A,Q: A > $o] :
( ( P @ A2 )
=> ( ! [X3: A] :
( ( P @ X3 )
=> ( X3 = A2 ) )
=> ( ! [X3: A] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( Q @ ( the @ A @ P ) ) ) ) ) ).
% theI2
thf(fact_4597_If__def,axiom,
! [A: $tType] :
( ( if @ A )
= ( ^ [P7: $o,X: A,Y: A] :
( the @ A
@ ^ [Z5: A] :
( ( P7
=> ( Z5 = X ) )
& ( ~ P7
=> ( Z5 = Y ) ) ) ) ) ) ).
% If_def
thf(fact_4598_the1I2,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ? [X4: A] :
( ( P @ X4 )
& ! [Y3: A] :
( ( P @ Y3 )
=> ( Y3 = X4 ) ) )
=> ( ! [X3: A] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( Q @ ( the @ A @ P ) ) ) ) ).
% the1I2
thf(fact_4599_the1__equality,axiom,
! [A: $tType,P: A > $o,A2: A] :
( ? [X4: A] :
( ( P @ X4 )
& ! [Y3: A] :
( ( P @ Y3 )
=> ( Y3 = X4 ) ) )
=> ( ( P @ A2 )
=> ( ( the @ A @ P )
= A2 ) ) ) ).
% the1_equality
thf(fact_4600_sgn__power__root,axiom,
! [N: nat,X2: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( times_times @ real @ ( sgn_sgn @ real @ ( root @ N @ X2 ) ) @ ( power_power @ real @ ( abs_abs @ real @ ( root @ N @ X2 ) ) @ N ) )
= X2 ) ) ).
% sgn_power_root
thf(fact_4601_root__sgn__power,axiom,
! [N: nat,Y2: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( root @ N @ ( times_times @ real @ ( sgn_sgn @ real @ Y2 ) @ ( power_power @ real @ ( abs_abs @ real @ Y2 ) @ N ) ) )
= Y2 ) ) ).
% root_sgn_power
thf(fact_4602_cis__Arg__unique,axiom,
! [Z: complex,X2: real] :
( ( ( sgn_sgn @ complex @ Z )
= ( cis @ X2 ) )
=> ( ( ord_less @ real @ ( uminus_uminus @ real @ pi ) @ X2 )
=> ( ( ord_less_eq @ real @ X2 @ pi )
=> ( ( arg @ Z )
= X2 ) ) ) ) ).
% cis_Arg_unique
thf(fact_4603_split__root,axiom,
! [P: real > $o,N: nat,X2: real] :
( ( P @ ( root @ N @ X2 ) )
= ( ( ( N
= ( zero_zero @ nat ) )
=> ( P @ ( zero_zero @ real ) ) )
& ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ! [Y: real] :
( ( ( times_times @ real @ ( sgn_sgn @ real @ Y ) @ ( power_power @ real @ ( abs_abs @ real @ Y ) @ N ) )
= X2 )
=> ( P @ Y ) ) ) ) ) ).
% split_root
thf(fact_4604_floor__real__def,axiom,
( ( archim6421214686448440834_floor @ real )
= ( ^ [X: real] :
( the @ int
@ ^ [Z5: int] :
( ( ord_less_eq @ real @ ( ring_1_of_int @ real @ Z5 ) @ X )
& ( ord_less @ real @ X @ ( ring_1_of_int @ real @ ( plus_plus @ int @ Z5 @ ( one_one @ int ) ) ) ) ) ) ) ) ).
% floor_real_def
thf(fact_4605_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_4606_arctan__inverse,axiom,
! [X2: real] :
( ( X2
!= ( zero_zero @ real ) )
=> ( ( arctan @ ( divide_divide @ real @ ( one_one @ real ) @ X2 ) )
= ( minus_minus @ real @ ( divide_divide @ real @ ( times_times @ real @ ( sgn_sgn @ real @ X2 ) @ pi ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( arctan @ X2 ) ) ) ) ).
% arctan_inverse
thf(fact_4607_signed__take__bit__eq__take__bit__minus,axiom,
( ( bit_ri4674362597316999326ke_bit @ int )
= ( ^ [N3: nat,K2: int] : ( minus_minus @ int @ ( bit_se2584673776208193580ke_bit @ int @ ( suc @ N3 ) @ K2 ) @ ( times_times @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( suc @ N3 ) ) @ ( zero_neq_one_of_bool @ int @ ( bit_se5641148757651400278ts_bit @ int @ K2 @ N3 ) ) ) ) ) ) ).
% signed_take_bit_eq_take_bit_minus
thf(fact_4608_num_Osize__gen_I3_J,axiom,
! [X33: num] :
( ( size_num @ ( bit1 @ X33 ) )
= ( plus_plus @ nat @ ( size_num @ X33 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% num.size_gen(3)
thf(fact_4609_mi__eq__ma__no__ch,axiom,
! [Mi: nat,Ma: nat,Deg: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ Deg )
=> ( ( Mi = Ma )
=> ( ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X_12 ) )
& ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_12 ) ) ) ) ).
% mi_eq_ma_no_ch
thf(fact_4610_insert__simp__mima,axiom,
! [X2: nat,Mi: nat,Ma: nat,Deg: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( ( X2 = Mi )
| ( X2 = Ma ) )
=> ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X2 )
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) ) ) ) ).
% insert_simp_mima
thf(fact_4611_not__None__eq,axiom,
! [A: $tType,X2: option @ A] :
( ( X2
!= ( none @ A ) )
= ( ? [Y: A] :
( X2
= ( some @ A @ Y ) ) ) ) ).
% not_None_eq
thf(fact_4612_not__Some__eq,axiom,
! [A: $tType,X2: option @ A] :
( ( ! [Y: A] :
( X2
!= ( some @ A @ Y ) ) )
= ( X2
= ( none @ A ) ) ) ).
% not_Some_eq
thf(fact_4613_tdeletemimi,axiom,
! [Deg: nat,Mi: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,X2: nat] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ord_less_eq @ nat @ ( vEBT_T_d_e_l_e_t_e @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Mi ) ) @ Deg @ TreeList @ Summary ) @ X2 ) @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ).
% tdeletemimi
thf(fact_4614_tdeletemimi_H,axiom,
! [Deg: nat,Mi: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,X2: nat] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ord_less_eq @ nat @ ( vEBT_V1232361888498592333_e_t_e @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Mi ) ) @ Deg @ TreeList @ Summary ) @ X2 ) @ ( one_one @ nat ) ) ) ).
% tdeletemimi'
thf(fact_4615_mi__ma__2__deg,axiom,
! [Mi: nat,Ma: nat,Deg: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ N )
=> ( ( ord_less_eq @ nat @ Mi @ Ma )
& ( ord_less @ nat @ Ma @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg ) ) ) ) ).
% mi_ma_2_deg
thf(fact_4616_not__Some__eq2,axiom,
! [B: $tType,A: $tType,V: option @ ( product_prod @ A @ B )] :
( ( ! [X: A,Y: B] :
( V
!= ( some @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) ) ) )
= ( V
= ( none @ ( product_prod @ A @ B ) ) ) ) ).
% not_Some_eq2
thf(fact_4617_both__member__options__from__complete__tree__to__child,axiom,
! [Deg: nat,Mi: nat,Ma: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,X2: nat] :
( ( ord_less_eq @ nat @ ( one_one @ nat ) @ Deg )
=> ( ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X2 )
=> ( ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
| ( X2 = Mi )
| ( X2 = Ma ) ) ) ) ).
% both_member_options_from_complete_tree_to_child
thf(fact_4618_member__inv,axiom,
! [Mi: nat,Ma: nat,Deg: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,X2: nat] :
( ( vEBT_vebt_member @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X2 )
=> ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
& ( ( X2 = Mi )
| ( X2 = Ma )
| ( ( ord_less @ nat @ X2 @ Ma )
& ( ord_less @ nat @ Mi @ X2 )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
& ( vEBT_vebt_member @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ).
% member_inv
thf(fact_4619_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_4620_both__member__options__from__chilf__to__complete__tree,axiom,
! [X2: nat,Deg: nat,TreeList: list @ vEBT_VEBT,Mi: nat,Ma: nat,Summary: vEBT_VEBT] :
( ( ord_less @ nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
=> ( ( ord_less_eq @ nat @ ( one_one @ nat ) @ Deg )
=> ( ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
=> ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X2 ) ) ) ) ).
% both_member_options_from_chilf_to_complete_tree
thf(fact_4621_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_4622_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_4623_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_4624_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_4625_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_4626_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_4627_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_4628_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_4629_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_4630_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_4631_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_4632_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_4633_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_4634_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_4635_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_4636_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_4637_prod__decode__aux_Ocases,axiom,
! [X2: product_prod @ nat @ nat] :
~ ! [K3: nat,M3: nat] :
( X2
!= ( product_Pair @ nat @ nat @ K3 @ M3 ) ) ).
% prod_decode_aux.cases
thf(fact_4638_le__some__optE,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [M: A,X2: option @ A] :
( ( ord_less_eq @ ( option @ A ) @ ( some @ A @ M ) @ X2 )
=> ~ ! [M11: A] :
( ( X2
= ( some @ A @ M11 ) )
=> ~ ( ord_less_eq @ A @ M @ M11 ) ) ) ) ).
% le_some_optE
thf(fact_4639_option_Odistinct_I1_J,axiom,
! [A: $tType,X23: A] :
( ( none @ A )
!= ( some @ A @ X23 ) ) ).
% option.distinct(1)
thf(fact_4640_option_OdiscI,axiom,
! [A: $tType,Option: option @ A,X23: A] :
( ( Option
= ( some @ A @ X23 ) )
=> ( Option
!= ( none @ A ) ) ) ).
% option.discI
thf(fact_4641_option_Oexhaust,axiom,
! [A: $tType,Y2: option @ A] :
( ( Y2
!= ( none @ A ) )
=> ~ ! [X22: A] :
( Y2
!= ( some @ A @ X22 ) ) ) ).
% option.exhaust
thf(fact_4642_split__option__ex,axiom,
! [A: $tType] :
( ( ^ [P6: ( option @ A ) > $o] :
? [X8: option @ A] : ( P6 @ X8 ) )
= ( ^ [P7: ( option @ A ) > $o] :
( ( P7 @ ( none @ A ) )
| ? [X: A] : ( P7 @ ( some @ A @ X ) ) ) ) ) ).
% split_option_ex
thf(fact_4643_split__option__all,axiom,
! [A: $tType] :
( ( ^ [P6: ( option @ A ) > $o] :
! [X8: option @ A] : ( P6 @ X8 ) )
= ( ^ [P7: ( option @ A ) > $o] :
( ( P7 @ ( none @ A ) )
& ! [X: A] : ( P7 @ ( some @ A @ X ) ) ) ) ) ).
% split_option_all
thf(fact_4644_combine__options__cases,axiom,
! [A: $tType,B: $tType,X2: option @ A,P: ( option @ A ) > ( option @ B ) > $o,Y2: option @ B] :
( ( ( X2
= ( none @ A ) )
=> ( P @ X2 @ Y2 ) )
=> ( ( ( Y2
= ( none @ B ) )
=> ( P @ X2 @ Y2 ) )
=> ( ! [A5: A,B4: B] :
( ( X2
= ( some @ A @ A5 ) )
=> ( ( Y2
= ( some @ B @ B4 ) )
=> ( P @ X2 @ Y2 ) ) )
=> ( P @ X2 @ Y2 ) ) ) ) ).
% combine_options_cases
thf(fact_4645_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Ocases,axiom,
! [X2: vEBT_VEBT] :
( ! [A5: $o,B4: $o] :
( X2
!= ( vEBT_Leaf @ A5 @ B4 ) )
=> ( ! [Uu2: nat,Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT] :
( X2
!= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) )
=> ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( X2
!= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) ) ) ) ).
% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.cases
thf(fact_4646_VEBT__internal_Omembermima_Osimps_I3_J,axiom,
! [Mi: nat,Ma: nat,Va: list @ vEBT_VEBT,Vb: vEBT_VEBT,X2: nat] :
( ( vEBT_VEBT_membermima @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( zero_zero @ nat ) @ Va @ Vb ) @ X2 )
= ( ( X2 = Mi )
| ( X2 = Ma ) ) ) ).
% VEBT_internal.membermima.simps(3)
thf(fact_4647_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_4648_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_4649_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_4650_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_4651_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_4652_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_4653_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_4654_T_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_Osimps_I5_J,axiom,
! [Mi: nat,Ma: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,X2: nat] :
( ( vEBT_T_d_e_l_e_t_e @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( zero_zero @ nat ) @ TreeList @ Summary ) @ X2 )
= ( one_one @ nat ) ) ).
% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(5)
thf(fact_4655_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Osimps_I5_J,axiom,
! [Mi: nat,Ma: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,X2: nat] :
( ( vEBT_V1232361888498592333_e_t_e @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( zero_zero @ nat ) @ TreeList @ Summary ) @ X2 )
= ( one_one @ nat ) ) ).
% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.simps(5)
thf(fact_4656_vebt__insert_Osimps_I4_J,axiom,
! [V: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,X2: nat] :
( ( vEBT_vebt_insert @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ V ) ) @ TreeList @ Summary ) @ X2 )
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ X2 @ X2 ) ) @ ( suc @ ( suc @ V ) ) @ TreeList @ Summary ) ) ).
% vebt_insert.simps(4)
thf(fact_4657_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Osimps_I6_J,axiom,
! [Mi: nat,Ma: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,X2: nat] :
( ( vEBT_V1232361888498592333_e_t_e @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( zero_zero @ nat ) ) @ TreeList @ Summary ) @ X2 )
= ( one_one @ nat ) ) ).
% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.simps(6)
thf(fact_4658_T_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_Osimps_I6_J,axiom,
! [Mi: nat,Ma: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,X2: nat] :
( ( vEBT_T_d_e_l_e_t_e @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( zero_zero @ nat ) ) @ TreeList @ Summary ) @ X2 )
= ( one_one @ nat ) ) ).
% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(6)
thf(fact_4659_height__node,axiom,
! [Mi: nat,Ma: nat,Deg: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ N )
=> ( ord_less_eq @ nat @ ( one_one @ nat ) @ ( vEBT_VEBT_height @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) ) ) ) ).
% height_node
thf(fact_4660_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_4661_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l_Ocases,axiom,
! [X2: vEBT_VEBT] :
( ( X2
!= ( vEBT_Leaf @ $false @ $false ) )
=> ( ! [Uv2: $o] :
( X2
!= ( vEBT_Leaf @ $true @ Uv2 ) )
=> ( ! [Uu2: $o] :
( X2
!= ( vEBT_Leaf @ Uu2 @ $true ) )
=> ( ! [Uw2: nat,Ux2: list @ vEBT_VEBT,Uy2: vEBT_VEBT] :
( X2
!= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uw2 @ Ux2 @ Uy2 ) )
=> ~ ! [Uz2: product_prod @ nat @ nat,Va3: nat,Vb2: list @ vEBT_VEBT,Vc: vEBT_VEBT] :
( X2
!= ( 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_4662_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_4663_vebt__member_Osimps_I3_J,axiom,
! [V: product_prod @ nat @ nat,Uy: list @ vEBT_VEBT,Uz: vEBT_VEBT,X2: nat] :
~ ( vEBT_vebt_member @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V ) @ ( zero_zero @ nat ) @ Uy @ Uz ) @ X2 ) ).
% vebt_member.simps(3)
thf(fact_4664_option_Osize_I4_J,axiom,
! [A: $tType,X23: A] :
( ( size_size @ ( option @ A ) @ ( some @ A @ X23 ) )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% option.size(4)
thf(fact_4665_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_4666_vebt__member_Osimps_I4_J,axiom,
! [V: product_prod @ nat @ nat,Vb: list @ vEBT_VEBT,Vc2: vEBT_VEBT,X2: nat] :
~ ( vEBT_vebt_member @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb @ Vc2 ) @ X2 ) ).
% vebt_member.simps(4)
thf(fact_4667_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_4668_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_4669_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_4670_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_4671_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_4672_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_4673_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_4674_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_4675_int__bit__bound,axiom,
! [K: int] :
~ ! [N2: nat] :
( ! [M4: nat] :
( ( ord_less_eq @ nat @ N2 @ M4 )
=> ( ( bit_se5641148757651400278ts_bit @ int @ K @ M4 )
= ( 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_4676_num_Osize__gen_I1_J,axiom,
( ( size_num @ one2 )
= ( zero_zero @ nat ) ) ).
% num.size_gen(1)
thf(fact_4677_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_4678_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_4679_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_4680_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_4681_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_4682_bit__int__def,axiom,
( ( bit_se5641148757651400278ts_bit @ int )
= ( ^ [K2: int,N3: nat] :
~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( divide_divide @ int @ K2 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N3 ) ) ) ) ) ).
% bit_int_def
thf(fact_4683_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_4684_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_4685_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Osimps_I3_J,axiom,
! [V: product_prod @ nat @ nat,Uy: list @ vEBT_VEBT,Uz: vEBT_VEBT,X2: nat] :
( ( vEBT_T_m_e_m_b_e_r @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V ) @ ( zero_zero @ nat ) @ Uy @ Uz ) @ X2 )
= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ).
% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.simps(3)
thf(fact_4686_vebt__member_Osimps_I5_J,axiom,
! [Mi: nat,Ma: nat,Va: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,X2: nat] :
( ( vEBT_vebt_member @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X2 )
= ( ( X2 != Mi )
=> ( ( X2 != Ma )
=> ( ~ ( ord_less @ nat @ X2 @ Mi )
& ( ~ ( ord_less @ nat @ X2 @ Mi )
=> ( ~ ( ord_less @ nat @ Ma @ X2 )
& ( ~ ( ord_less @ nat @ Ma @ X2 )
=> ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
=> ( vEBT_vebt_member @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) ) ) ) ) ) ) ) ) ) ).
% vebt_member.simps(5)
thf(fact_4687_VEBT__internal_Omembermima_Osimps_I4_J,axiom,
! [Mi: nat,Ma: nat,V: nat,TreeList: list @ vEBT_VEBT,Vc2: vEBT_VEBT,X2: nat] :
( ( vEBT_VEBT_membermima @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ V ) @ TreeList @ Vc2 ) @ X2 )
= ( ( X2 = Mi )
| ( X2 = Ma )
| ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
=> ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) ) ) ) ) ).
% VEBT_internal.membermima.simps(4)
thf(fact_4688_bit__sum__mult__2__cases,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A,B2: A,N: nat] :
( ! [J2: nat] :
~ ( bit_se5641148757651400278ts_bit @ A @ A2 @ ( suc @ J2 ) )
=> ( ( bit_se5641148757651400278ts_bit @ A @ ( plus_plus @ A @ A2 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) @ N )
= ( ( ( N
= ( zero_zero @ nat ) )
=> ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) )
& ( ( N
!= ( zero_zero @ nat ) )
=> ( bit_se5641148757651400278ts_bit @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) @ N ) ) ) ) ) ) ).
% bit_sum_mult_2_cases
thf(fact_4689_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_4690_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,X2: nat] :
( ( vEBT_T_m_e_m_b_e_r @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb @ Vc2 ) @ X2 )
= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ).
% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.simps(4)
thf(fact_4691_VEBT__internal_Omembermima_Oelims_I2_J,axiom,
! [X2: vEBT_VEBT,Xa: nat] :
( ( vEBT_VEBT_membermima @ X2 @ Xa )
=> ( ! [Mi2: nat,Ma2: nat] :
( ? [Va3: list @ vEBT_VEBT,Vb2: vEBT_VEBT] :
( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ Va3 @ Vb2 ) )
=> ~ ( ( Xa = Mi2 )
| ( Xa = Ma2 ) ) )
=> ( ! [Mi2: nat,Ma2: nat,V3: nat,TreeList2: list @ vEBT_VEBT] :
( ? [Vc: vEBT_VEBT] :
( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList2 @ Vc ) )
=> ~ ( ( Xa = Mi2 )
| ( Xa = Ma2 )
| ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) )
=> ~ ! [V3: nat,TreeList2: list @ vEBT_VEBT] :
( ? [Vd2: vEBT_VEBT] :
( X2
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V3 ) @ TreeList2 @ Vd2 ) )
=> ~ ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ) ).
% VEBT_internal.membermima.elims(2)
thf(fact_4692_invar__vebt_Ointros_I4_J,axiom,
! [TreeList: list @ vEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat,Mi: nat,Ma: nat] :
( ! [X3: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X3 @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ( vEBT_invar_vebt @ X3 @ N ) )
=> ( ( vEBT_invar_vebt @ Summary @ M )
=> ( ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) )
=> ( ( M = N )
=> ( ( Deg
= ( plus_plus @ nat @ N @ M ) )
=> ( ! [I2: nat] :
( ( ord_less @ nat @ I2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) )
=> ( ( ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList @ I2 ) @ X7 ) )
= ( vEBT_V8194947554948674370ptions @ Summary @ I2 ) ) )
=> ( ( ( Mi = Ma )
=> ! [X3: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X3 @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X_1 ) ) )
=> ( ( ord_less_eq @ nat @ Mi @ Ma )
=> ( ( ord_less @ nat @ Ma @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg ) )
=> ( ( ( Mi != Ma )
=> ! [I2: nat] :
( ( ord_less @ nat @ I2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) )
=> ( ( ( ( vEBT_VEBT_high @ Ma @ N )
= I2 )
=> ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList @ I2 ) @ ( vEBT_VEBT_low @ Ma @ N ) ) )
& ! [X3: nat] :
( ( ( ( vEBT_VEBT_high @ X3 @ N )
= I2 )
& ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList @ I2 ) @ ( vEBT_VEBT_low @ X3 @ N ) ) )
=> ( ( ord_less @ nat @ Mi @ X3 )
& ( ord_less_eq @ nat @ X3 @ Ma ) ) ) ) ) )
=> ( vEBT_invar_vebt @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ Deg ) ) ) ) ) ) ) ) ) ) ) ).
% invar_vebt.intros(4)
thf(fact_4693_vebt__member_Oelims_I2_J,axiom,
! [X2: vEBT_VEBT,Xa: nat] :
( ( vEBT_vebt_member @ X2 @ Xa )
=> ( ! [A5: $o,B4: $o] :
( ( X2
= ( vEBT_Leaf @ A5 @ B4 ) )
=> ~ ( ( ( Xa
= ( zero_zero @ nat ) )
=> A5 )
& ( ( Xa
!= ( zero_zero @ nat ) )
=> ( ( ( Xa
= ( one_one @ nat ) )
=> B4 )
& ( Xa
= ( one_one @ nat ) ) ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list @ vEBT_VEBT] :
( ? [Summary2: vEBT_VEBT] :
( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary2 ) )
=> ~ ( ( Xa != Mi2 )
=> ( ( Xa != Ma2 )
=> ( ~ ( ord_less @ nat @ Xa @ Mi2 )
& ( ~ ( ord_less @ nat @ Xa @ Mi2 )
=> ( ~ ( ord_less @ nat @ Ma2 @ Xa )
& ( ~ ( ord_less @ nat @ Ma2 @ Xa )
=> ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_member.elims(2)
thf(fact_4694_VEBT__internal_Omembermima_Oelims_I3_J,axiom,
! [X2: vEBT_VEBT,Xa: nat] :
( ~ ( vEBT_VEBT_membermima @ X2 @ Xa )
=> ( ! [Uu2: $o,Uv2: $o] :
( X2
!= ( vEBT_Leaf @ Uu2 @ Uv2 ) )
=> ( ! [Ux2: list @ vEBT_VEBT,Uy2: vEBT_VEBT] :
( X2
!= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( zero_zero @ nat ) @ Ux2 @ Uy2 ) )
=> ( ! [Mi2: nat,Ma2: nat] :
( ? [Va3: list @ vEBT_VEBT,Vb2: vEBT_VEBT] :
( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ Va3 @ Vb2 ) )
=> ( ( Xa = Mi2 )
| ( Xa = Ma2 ) ) )
=> ( ! [Mi2: nat,Ma2: nat,V3: nat,TreeList2: list @ vEBT_VEBT] :
( ? [Vc: vEBT_VEBT] :
( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList2 @ Vc ) )
=> ( ( Xa = Mi2 )
| ( Xa = Ma2 )
| ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) )
=> ~ ! [V3: nat,TreeList2: list @ vEBT_VEBT] :
( ? [Vd2: vEBT_VEBT] :
( X2
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V3 ) @ TreeList2 @ Vd2 ) )
=> ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.membermima.elims(3)
thf(fact_4695_VEBT__internal_Omembermima_Oelims_I1_J,axiom,
! [X2: vEBT_VEBT,Xa: nat,Y2: $o] :
( ( ( vEBT_VEBT_membermima @ X2 @ Xa )
= Y2 )
=> ( ( ? [Uu2: $o,Uv2: $o] :
( X2
= ( vEBT_Leaf @ Uu2 @ Uv2 ) )
=> Y2 )
=> ( ( ? [Ux2: list @ vEBT_VEBT,Uy2: vEBT_VEBT] :
( X2
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( zero_zero @ nat ) @ Ux2 @ Uy2 ) )
=> Y2 )
=> ( ! [Mi2: nat,Ma2: nat] :
( ? [Va3: list @ vEBT_VEBT,Vb2: vEBT_VEBT] :
( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ Va3 @ Vb2 ) )
=> ( Y2
= ( ~ ( ( Xa = Mi2 )
| ( Xa = Ma2 ) ) ) ) )
=> ( ! [Mi2: nat,Ma2: nat,V3: nat,TreeList2: list @ vEBT_VEBT] :
( ? [Vc: vEBT_VEBT] :
( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList2 @ Vc ) )
=> ( Y2
= ( ~ ( ( Xa = Mi2 )
| ( Xa = Ma2 )
| ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) )
=> ~ ! [V3: nat,TreeList2: list @ vEBT_VEBT] :
( ? [Vd2: vEBT_VEBT] :
( X2
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V3 ) @ TreeList2 @ Vd2 ) )
=> ( Y2
= ( ~ ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.membermima.elims(1)
thf(fact_4696_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Osimps_I5_J,axiom,
! [Mi: nat,Ma: nat,Va: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,X2: nat] :
( ( vEBT_T_m_e_m_b_e_r @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X2 )
= ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( if @ nat @ ( X2 = Mi ) @ ( one_one @ nat ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( if @ nat @ ( X2 = Ma ) @ ( one_one @ nat ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( if @ nat @ ( ord_less @ nat @ X2 @ Mi ) @ ( one_one @ nat ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( if @ nat @ ( ord_less @ nat @ Ma @ X2 ) @ ( one_one @ nat ) @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) @ ( if @ nat @ ( ord_less @ nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_T_m_e_m_b_e_r @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( one_one @ nat ) ) ) ) ) ) ) ) ) ) ) ) ).
% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.simps(5)
thf(fact_4697_invar__vebt_Ointros_I5_J,axiom,
! [TreeList: list @ vEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat,Mi: nat,Ma: nat] :
( ! [X3: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X3 @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ( vEBT_invar_vebt @ X3 @ N ) )
=> ( ( vEBT_invar_vebt @ Summary @ M )
=> ( ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) )
=> ( ( M
= ( suc @ N ) )
=> ( ( Deg
= ( plus_plus @ nat @ N @ M ) )
=> ( ! [I2: nat] :
( ( ord_less @ nat @ I2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) )
=> ( ( ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList @ I2 ) @ X7 ) )
= ( vEBT_V8194947554948674370ptions @ Summary @ I2 ) ) )
=> ( ( ( Mi = Ma )
=> ! [X3: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X3 @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X_1 ) ) )
=> ( ( ord_less_eq @ nat @ Mi @ Ma )
=> ( ( ord_less @ nat @ Ma @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg ) )
=> ( ( ( Mi != Ma )
=> ! [I2: nat] :
( ( ord_less @ nat @ I2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) )
=> ( ( ( ( vEBT_VEBT_high @ Ma @ N )
= I2 )
=> ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList @ I2 ) @ ( vEBT_VEBT_low @ Ma @ N ) ) )
& ! [X3: nat] :
( ( ( ( vEBT_VEBT_high @ X3 @ N )
= I2 )
& ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList @ I2 ) @ ( vEBT_VEBT_low @ X3 @ N ) ) )
=> ( ( ord_less @ nat @ Mi @ X3 )
& ( ord_less_eq @ nat @ X3 @ Ma ) ) ) ) ) )
=> ( vEBT_invar_vebt @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ Deg ) ) ) ) ) ) ) ) ) ) ) ).
% invar_vebt.intros(5)
thf(fact_4698_Bit__Operations_Oset__bit__eq,axiom,
( ( bit_se5668285175392031749et_bit @ int )
= ( ^ [N3: nat,K2: int] :
( plus_plus @ int @ K2
@ ( times_times @ int
@ ( zero_neq_one_of_bool @ int
@ ~ ( bit_se5641148757651400278ts_bit @ int @ K2 @ N3 ) )
@ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N3 ) ) ) ) ) ).
% Bit_Operations.set_bit_eq
thf(fact_4699_unset__bit__eq,axiom,
( ( bit_se2638667681897837118et_bit @ int )
= ( ^ [N3: nat,K2: int] : ( minus_minus @ int @ K2 @ ( times_times @ int @ ( zero_neq_one_of_bool @ int @ ( bit_se5641148757651400278ts_bit @ int @ K2 @ N3 ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N3 ) ) ) ) ) ).
% unset_bit_eq
thf(fact_4700_vebt__member_Oelims_I3_J,axiom,
! [X2: vEBT_VEBT,Xa: nat] :
( ~ ( vEBT_vebt_member @ X2 @ Xa )
=> ( ! [A5: $o,B4: $o] :
( ( X2
= ( vEBT_Leaf @ A5 @ B4 ) )
=> ( ( ( Xa
= ( zero_zero @ nat ) )
=> A5 )
& ( ( Xa
!= ( zero_zero @ nat ) )
=> ( ( ( Xa
= ( one_one @ nat ) )
=> B4 )
& ( Xa
= ( one_one @ nat ) ) ) ) ) )
=> ( ! [Uu2: nat,Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT] :
( X2
!= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) )
=> ( ! [V3: product_prod @ nat @ nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( X2
!= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Uy2 @ Uz2 ) )
=> ( ! [V3: product_prod @ nat @ nat,Vb2: list @ vEBT_VEBT,Vc: vEBT_VEBT] :
( X2
!= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb2 @ Vc ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list @ vEBT_VEBT] :
( ? [Summary2: vEBT_VEBT] :
( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary2 ) )
=> ( ( Xa != Mi2 )
=> ( ( Xa != Ma2 )
=> ( ~ ( ord_less @ nat @ Xa @ Mi2 )
& ( ~ ( ord_less @ nat @ Xa @ Mi2 )
=> ( ~ ( ord_less @ nat @ Ma2 @ Xa )
& ( ~ ( ord_less @ nat @ Ma2 @ Xa )
=> ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_member.elims(3)
thf(fact_4701_vebt__member_Oelims_I1_J,axiom,
! [X2: vEBT_VEBT,Xa: nat,Y2: $o] :
( ( ( vEBT_vebt_member @ X2 @ Xa )
= Y2 )
=> ( ! [A5: $o,B4: $o] :
( ( X2
= ( vEBT_Leaf @ A5 @ B4 ) )
=> ( Y2
= ( ~ ( ( ( Xa
= ( zero_zero @ nat ) )
=> A5 )
& ( ( Xa
!= ( zero_zero @ nat ) )
=> ( ( ( Xa
= ( one_one @ nat ) )
=> B4 )
& ( Xa
= ( one_one @ nat ) ) ) ) ) ) ) )
=> ( ( ? [Uu2: nat,Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT] :
( X2
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) )
=> Y2 )
=> ( ( ? [V3: product_prod @ nat @ nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Uy2 @ Uz2 ) )
=> Y2 )
=> ( ( ? [V3: product_prod @ nat @ nat,Vb2: list @ vEBT_VEBT,Vc: vEBT_VEBT] :
( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb2 @ Vc ) )
=> Y2 )
=> ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list @ vEBT_VEBT] :
( ? [Summary2: vEBT_VEBT] :
( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary2 ) )
=> ( Y2
= ( ~ ( ( Xa != Mi2 )
=> ( ( Xa != Ma2 )
=> ( ~ ( ord_less @ nat @ Xa @ Mi2 )
& ( ~ ( ord_less @ nat @ Xa @ Mi2 )
=> ( ~ ( ord_less @ nat @ Ma2 @ Xa )
& ( ~ ( ord_less @ nat @ Ma2 @ Xa )
=> ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_member.elims(1)
thf(fact_4702_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_4703_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Oelims,axiom,
! [X2: vEBT_VEBT,Xa: nat,Y2: nat] :
( ( ( vEBT_T_m_e_m_b_e_r @ X2 @ Xa )
= Y2 )
=> ( ( ? [A5: $o,B4: $o] :
( X2
= ( vEBT_Leaf @ A5 @ B4 ) )
=> ( Y2
!= ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) )
@ ( if @ nat
@ ( Xa
= ( zero_zero @ nat ) )
@ ( one_one @ nat )
@ ( plus_plus @ nat @ ( one_one @ nat ) @ ( one_one @ nat ) ) ) ) ) )
=> ( ( ? [Uu2: nat,Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT] :
( X2
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) )
=> ( Y2
!= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
=> ( ( ? [V3: product_prod @ nat @ nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Uy2 @ Uz2 ) )
=> ( Y2
!= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
=> ( ( ? [V3: product_prod @ nat @ nat,Vb2: list @ vEBT_VEBT,Vc: vEBT_VEBT] :
( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb2 @ Vc ) )
=> ( Y2
!= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list @ vEBT_VEBT] :
( ? [Summary2: vEBT_VEBT] :
( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary2 ) )
=> ( Y2
!= ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( if @ nat @ ( Xa = Mi2 ) @ ( one_one @ nat ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( if @ nat @ ( Xa = Ma2 ) @ ( one_one @ nat ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ ( one_one @ nat ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( if @ nat @ ( ord_less @ nat @ Ma2 @ Xa ) @ ( one_one @ nat ) @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) @ ( if @ nat @ ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_T_m_e_m_b_e_r @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( 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.elims
thf(fact_4704_num_Osize__gen_I2_J,axiom,
! [X23: num] :
( ( size_num @ ( bit0 @ X23 ) )
= ( plus_plus @ nat @ ( size_num @ X23 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% num.size_gen(2)
thf(fact_4705_invar__vebt_Ocases,axiom,
! [A1: vEBT_VEBT,A22: nat] :
( ( vEBT_invar_vebt @ A1 @ A22 )
=> ( ( ? [A5: $o,B4: $o] :
( A1
= ( vEBT_Leaf @ A5 @ B4 ) )
=> ( A22
!= ( suc @ ( zero_zero @ nat ) ) ) )
=> ( ! [TreeList2: list @ vEBT_VEBT,N2: nat,Summary2: vEBT_VEBT,M3: nat,Deg2: nat] :
( ( A1
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( ( A22 = Deg2 )
=> ( ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ( vEBT_invar_vebt @ X4 @ N2 ) )
=> ( ( vEBT_invar_vebt @ Summary2 @ M3 )
=> ( ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M3 ) )
=> ( ( M3 = N2 )
=> ( ( Deg2
= ( plus_plus @ nat @ N2 @ M3 ) )
=> ( ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X_12 )
=> ~ ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X_12 ) ) ) ) ) ) ) ) ) )
=> ( ! [TreeList2: list @ vEBT_VEBT,N2: nat,Summary2: vEBT_VEBT,M3: nat,Deg2: nat] :
( ( A1
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( ( A22 = Deg2 )
=> ( ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ( vEBT_invar_vebt @ X4 @ N2 ) )
=> ( ( vEBT_invar_vebt @ Summary2 @ M3 )
=> ( ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M3 ) )
=> ( ( M3
= ( suc @ N2 ) )
=> ( ( Deg2
= ( plus_plus @ nat @ N2 @ M3 ) )
=> ( ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X_12 )
=> ~ ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X_12 ) ) ) ) ) ) ) ) ) )
=> ( ! [TreeList2: list @ vEBT_VEBT,N2: nat,Summary2: vEBT_VEBT,M3: nat,Deg2: nat,Mi2: nat,Ma2: nat] :
( ( A1
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( ( A22 = Deg2 )
=> ( ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ( vEBT_invar_vebt @ X4 @ N2 ) )
=> ( ( vEBT_invar_vebt @ Summary2 @ M3 )
=> ( ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M3 ) )
=> ( ( M3 = N2 )
=> ( ( Deg2
= ( plus_plus @ nat @ N2 @ M3 ) )
=> ( ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M3 ) )
=> ( ( ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I4 ) @ X7 ) )
= ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
=> ( ( ( Mi2 = Ma2 )
=> ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X_12 ) ) )
=> ( ( ord_less_eq @ nat @ Mi2 @ Ma2 )
=> ( ( ord_less @ nat @ Ma2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
=> ~ ( ( Mi2 != Ma2 )
=> ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M3 ) )
=> ( ( ( ( vEBT_VEBT_high @ Ma2 @ N2 )
= I4 )
=> ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I4 ) @ ( vEBT_VEBT_low @ Ma2 @ N2 ) ) )
& ! [X4: nat] :
( ( ( ( vEBT_VEBT_high @ X4 @ N2 )
= I4 )
& ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I4 ) @ ( vEBT_VEBT_low @ X4 @ N2 ) ) )
=> ( ( ord_less @ nat @ Mi2 @ X4 )
& ( ord_less_eq @ nat @ X4 @ Ma2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
=> ~ ! [TreeList2: list @ vEBT_VEBT,N2: nat,Summary2: vEBT_VEBT,M3: nat,Deg2: nat,Mi2: nat,Ma2: nat] :
( ( A1
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( ( A22 = Deg2 )
=> ( ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ( vEBT_invar_vebt @ X4 @ N2 ) )
=> ( ( vEBT_invar_vebt @ Summary2 @ M3 )
=> ( ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M3 ) )
=> ( ( M3
= ( suc @ N2 ) )
=> ( ( Deg2
= ( plus_plus @ nat @ N2 @ M3 ) )
=> ( ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M3 ) )
=> ( ( ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I4 ) @ X7 ) )
= ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
=> ( ( ( Mi2 = Ma2 )
=> ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X_12 ) ) )
=> ( ( ord_less_eq @ nat @ Mi2 @ Ma2 )
=> ( ( ord_less @ nat @ Ma2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
=> ~ ( ( Mi2 != Ma2 )
=> ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M3 ) )
=> ( ( ( ( vEBT_VEBT_high @ Ma2 @ N2 )
= I4 )
=> ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I4 ) @ ( vEBT_VEBT_low @ Ma2 @ N2 ) ) )
& ! [X4: nat] :
( ( ( ( vEBT_VEBT_high @ X4 @ N2 )
= I4 )
& ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I4 ) @ ( vEBT_VEBT_low @ X4 @ N2 ) ) )
=> ( ( ord_less @ nat @ Mi2 @ X4 )
& ( ord_less_eq @ nat @ X4 @ Ma2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% invar_vebt.cases
thf(fact_4706_invar__vebt_Osimps,axiom,
( vEBT_invar_vebt
= ( ^ [A12: vEBT_VEBT,A23: nat] :
( ( ? [A4: $o,B3: $o] :
( A12
= ( vEBT_Leaf @ A4 @ B3 ) )
& ( 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 ) )
& ! [X: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ TreeList3 ) )
=> ( vEBT_invar_vebt @ X @ 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 ) )
& ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X7 )
& ! [X: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ TreeList3 ) )
=> ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X @ X7 ) ) )
| ? [TreeList3: list @ vEBT_VEBT,N3: nat,Summary3: vEBT_VEBT] :
( ( A12
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ A23 @ TreeList3 @ Summary3 ) )
& ! [X: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ TreeList3 ) )
=> ( vEBT_invar_vebt @ X @ 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 ) ) )
& ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X7 )
& ! [X: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ TreeList3 ) )
=> ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X @ X7 ) ) )
| ? [TreeList3: list @ vEBT_VEBT,N3: nat,Summary3: vEBT_VEBT,Mi3: nat,Ma3: nat] :
( ( A12
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi3 @ Ma3 ) ) @ A23 @ TreeList3 @ Summary3 ) )
& ! [X: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ TreeList3 ) )
=> ( vEBT_invar_vebt @ X @ 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 ) )
& ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) )
=> ( ( ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList3 @ I3 ) @ X7 ) )
= ( vEBT_V8194947554948674370ptions @ Summary3 @ I3 ) ) )
& ( ( Mi3 = Ma3 )
=> ! [X: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ TreeList3 ) )
=> ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X @ X7 ) ) )
& ( ord_less_eq @ nat @ Mi3 @ Ma3 )
& ( ord_less @ nat @ Ma3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ A23 ) )
& ( ( Mi3 != Ma3 )
=> ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) )
=> ( ( ( ( vEBT_VEBT_high @ Ma3 @ N3 )
= I3 )
=> ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList3 @ I3 ) @ ( vEBT_VEBT_low @ Ma3 @ N3 ) ) )
& ! [X: nat] :
( ( ( ( vEBT_VEBT_high @ X @ N3 )
= I3 )
& ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList3 @ I3 ) @ ( vEBT_VEBT_low @ X @ N3 ) ) )
=> ( ( ord_less @ nat @ Mi3 @ X )
& ( ord_less_eq @ nat @ X @ Ma3 ) ) ) ) ) ) )
| ? [TreeList3: list @ vEBT_VEBT,N3: nat,Summary3: vEBT_VEBT,Mi3: nat,Ma3: nat] :
( ( A12
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi3 @ Ma3 ) ) @ A23 @ TreeList3 @ Summary3 ) )
& ! [X: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ TreeList3 ) )
=> ( vEBT_invar_vebt @ X @ 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 ) ) )
& ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ N3 ) ) )
=> ( ( ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList3 @ I3 ) @ X7 ) )
= ( vEBT_V8194947554948674370ptions @ Summary3 @ I3 ) ) )
& ( ( Mi3 = Ma3 )
=> ! [X: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ TreeList3 ) )
=> ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X @ X7 ) ) )
& ( ord_less_eq @ nat @ Mi3 @ Ma3 )
& ( ord_less @ nat @ Ma3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ A23 ) )
& ( ( Mi3 != Ma3 )
=> ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ N3 ) ) )
=> ( ( ( ( vEBT_VEBT_high @ Ma3 @ N3 )
= I3 )
=> ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList3 @ I3 ) @ ( vEBT_VEBT_low @ Ma3 @ N3 ) ) )
& ! [X: nat] :
( ( ( ( vEBT_VEBT_high @ X @ N3 )
= I3 )
& ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList3 @ I3 ) @ ( vEBT_VEBT_low @ X @ N3 ) ) )
=> ( ( ord_less @ nat @ Mi3 @ X )
& ( ord_less_eq @ nat @ X @ Ma3 ) ) ) ) ) ) ) ) ) ) ).
% invar_vebt.simps
thf(fact_4707_div__half__nat,axiom,
! [Y2: nat,X2: nat] :
( ( Y2
!= ( zero_zero @ nat ) )
=> ( ( product_Pair @ nat @ nat @ ( divide_divide @ nat @ X2 @ Y2 ) @ ( modulo_modulo @ nat @ X2 @ Y2 ) )
= ( if @ ( product_prod @ nat @ nat ) @ ( ord_less_eq @ nat @ Y2 @ ( minus_minus @ nat @ X2 @ ( times_times @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( divide_divide @ nat @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ Y2 ) ) @ Y2 ) ) ) @ ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( divide_divide @ nat @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ Y2 ) ) @ ( one_one @ nat ) ) @ ( minus_minus @ nat @ ( minus_minus @ nat @ X2 @ ( times_times @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( divide_divide @ nat @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ Y2 ) ) @ Y2 ) ) @ Y2 ) ) @ ( product_Pair @ nat @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( divide_divide @ nat @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ Y2 ) ) @ ( minus_minus @ nat @ X2 @ ( times_times @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( divide_divide @ nat @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ Y2 ) ) @ Y2 ) ) ) ) ) ) ).
% div_half_nat
thf(fact_4708_pred__list__to__short,axiom,
! [Deg: nat,X2: nat,Ma: nat,TreeList: list @ vEBT_VEBT,Mi: nat,Summary: vEBT_VEBT] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ( ord_less_eq @ nat @ X2 @ Ma )
=> ( ( ord_less_eq @ nat @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
=> ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X2 )
= ( none @ nat ) ) ) ) ) ).
% pred_list_to_short
thf(fact_4709_succ__list__to__short,axiom,
! [Deg: nat,Mi: nat,X2: nat,TreeList: list @ vEBT_VEBT,Ma: nat,Summary: vEBT_VEBT] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ( ord_less_eq @ nat @ Mi @ X2 )
=> ( ( ord_less_eq @ nat @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
=> ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X2 )
= ( none @ nat ) ) ) ) ) ).
% succ_list_to_short
thf(fact_4710_power__shift,axiom,
! [X2: nat,Y2: nat,Z: nat] :
( ( ( power_power @ nat @ X2 @ Y2 )
= Z )
= ( ( vEBT_VEBT_power @ ( some @ nat @ X2 ) @ ( some @ nat @ Y2 ) )
= ( some @ nat @ Z ) ) ) ).
% power_shift
thf(fact_4711_succ__corr,axiom,
! [T2: vEBT_VEBT,N: nat,X2: nat,Sx: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( ( vEBT_vebt_succ @ T2 @ X2 )
= ( some @ nat @ Sx ) )
= ( vEBT_is_succ_in_set @ ( vEBT_VEBT_set_vebt @ T2 ) @ X2 @ Sx ) ) ) ).
% succ_corr
thf(fact_4712_pred__corr,axiom,
! [T2: vEBT_VEBT,N: nat,X2: nat,Px: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( ( vEBT_vebt_pred @ T2 @ X2 )
= ( some @ nat @ Px ) )
= ( vEBT_is_pred_in_set @ ( vEBT_VEBT_set_vebt @ T2 ) @ X2 @ Px ) ) ) ).
% pred_corr
thf(fact_4713_succ__correct,axiom,
! [T2: vEBT_VEBT,N: nat,X2: nat,Sx: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( ( vEBT_vebt_succ @ T2 @ X2 )
= ( some @ nat @ Sx ) )
= ( vEBT_is_succ_in_set @ ( vEBT_set_vebt @ T2 ) @ X2 @ Sx ) ) ) ).
% succ_correct
thf(fact_4714_pred__correct,axiom,
! [T2: vEBT_VEBT,N: nat,X2: nat,Sx: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( ( vEBT_vebt_pred @ T2 @ X2 )
= ( some @ nat @ Sx ) )
= ( vEBT_is_pred_in_set @ ( vEBT_set_vebt @ T2 ) @ X2 @ Sx ) ) ) ).
% pred_correct
thf(fact_4715_helpypredd,axiom,
! [T2: vEBT_VEBT,N: nat,X2: nat,Y2: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( ( vEBT_vebt_pred @ T2 @ X2 )
= ( some @ nat @ Y2 ) )
=> ( ord_less @ nat @ Y2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% helpypredd
thf(fact_4716_helpyd,axiom,
! [T2: vEBT_VEBT,N: nat,X2: nat,Y2: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( ( vEBT_vebt_succ @ T2 @ X2 )
= ( some @ nat @ Y2 ) )
=> ( ord_less @ nat @ Y2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% helpyd
thf(fact_4717_geqmaxNone,axiom,
! [Mi: nat,Ma: nat,Deg: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,N: nat,X2: nat] :
( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ N )
=> ( ( ord_less_eq @ nat @ Ma @ X2 )
=> ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X2 )
= ( none @ nat ) ) ) ) ).
% geqmaxNone
thf(fact_4718_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_4719_succ__min,axiom,
! [Deg: nat,X2: nat,Mi: nat,Ma: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ( ord_less @ nat @ X2 @ Mi )
=> ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X2 )
= ( some @ nat @ Mi ) ) ) ) ).
% succ_min
thf(fact_4720_pred__max,axiom,
! [Deg: nat,Ma: nat,X2: nat,Mi: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ( ord_less @ nat @ Ma @ X2 )
=> ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X2 )
= ( some @ nat @ Ma ) ) ) ) ).
% pred_max
thf(fact_4721_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_4722_lesseq__shift,axiom,
( ( ord_less_eq @ nat )
= ( ^ [X: nat,Y: nat] : ( vEBT_VEBT_lesseq @ ( some @ nat @ X ) @ ( some @ nat @ Y ) ) ) ) ).
% lesseq_shift
thf(fact_4723_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_4724_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_4725_nth__0,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat] :
~ ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ N ) ) ).
% nth_0
thf(fact_4726_finite__bit__word,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A] : ( finite_finite2 @ nat @ ( collect @ nat @ ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ W ) ) ) ) ).
% finite_bit_word
thf(fact_4727_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_4728_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_4729_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_4730_vebt__pred_Osimps_I3_J,axiom,
! [B2: $o,A2: $o,Va: nat] :
( ( B2
=> ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A2 @ B2 ) @ ( suc @ ( suc @ Va ) ) )
= ( some @ nat @ ( one_one @ nat ) ) ) )
& ( ~ B2
=> ( ( A2
=> ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A2 @ B2 ) @ ( suc @ ( suc @ Va ) ) )
= ( some @ nat @ ( zero_zero @ nat ) ) ) )
& ( ~ A2
=> ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A2 @ B2 ) @ ( suc @ ( suc @ Va ) ) )
= ( none @ nat ) ) ) ) ) ) ).
% vebt_pred.simps(3)
thf(fact_4731_VEBT__internal_Oreplicatei_Ocases,axiom,
! [A: $tType,X2: product_prod @ nat @ ( heap_Time_Heap @ A )] :
( ! [X3: heap_Time_Heap @ A] :
( X2
!= ( product_Pair @ nat @ ( heap_Time_Heap @ A ) @ ( zero_zero @ nat ) @ X3 ) )
=> ~ ! [N2: nat,X3: heap_Time_Heap @ A] :
( X2
!= ( product_Pair @ nat @ ( heap_Time_Heap @ A ) @ ( suc @ N2 ) @ X3 ) ) ) ).
% VEBT_internal.replicatei.cases
thf(fact_4732_test__bit__over,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A,N: nat] :
( ( ord_less_eq @ nat @ ( size_size @ ( word @ A ) @ X2 ) @ N )
=> ~ ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ X2 @ N ) ) ) ).
% test_bit_over
thf(fact_4733_VEBT__internal_Ovalid_H_Ocases,axiom,
! [X2: product_prod @ vEBT_VEBT @ nat] :
( ! [Uu2: $o,Uv2: $o,D3: nat] :
( X2
!= ( 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] :
( X2
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList2 @ Summary2 ) @ Deg3 ) ) ) ).
% VEBT_internal.valid'.cases
thf(fact_4734_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_4735_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_4736_vebt__succ_Osimps_I3_J,axiom,
! [Ux: nat,Uy: list @ vEBT_VEBT,Uz: vEBT_VEBT,Va: nat] :
( ( vEBT_vebt_succ @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Ux @ Uy @ Uz ) @ Va )
= ( none @ nat ) ) ).
% vebt_succ.simps(3)
thf(fact_4737_vebt__pred_Osimps_I4_J,axiom,
! [Uy: nat,Uz: list @ vEBT_VEBT,Va: vEBT_VEBT,Vb: nat] :
( ( vEBT_vebt_pred @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uy @ Uz @ Va ) @ Vb )
= ( none @ nat ) ) ).
% vebt_pred.simps(4)
thf(fact_4738_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_4739_xor__num_Ocases,axiom,
! [X2: product_prod @ num @ num] :
( ( X2
!= ( product_Pair @ num @ num @ one2 @ one2 ) )
=> ( ! [N2: num] :
( X2
!= ( product_Pair @ num @ num @ one2 @ ( bit0 @ N2 ) ) )
=> ( ! [N2: num] :
( X2
!= ( product_Pair @ num @ num @ one2 @ ( bit1 @ N2 ) ) )
=> ( ! [M3: num] :
( X2
!= ( product_Pair @ num @ num @ ( bit0 @ M3 ) @ one2 ) )
=> ( ! [M3: num,N2: num] :
( X2
!= ( product_Pair @ num @ num @ ( bit0 @ M3 ) @ ( bit0 @ N2 ) ) )
=> ( ! [M3: num,N2: num] :
( X2
!= ( product_Pair @ num @ num @ ( bit0 @ M3 ) @ ( bit1 @ N2 ) ) )
=> ( ! [M3: num] :
( X2
!= ( product_Pair @ num @ num @ ( bit1 @ M3 ) @ one2 ) )
=> ( ! [M3: num,N2: num] :
( X2
!= ( product_Pair @ num @ num @ ( bit1 @ M3 ) @ ( bit0 @ N2 ) ) )
=> ~ ! [M3: num,N2: num] :
( X2
!= ( product_Pair @ num @ num @ ( bit1 @ M3 ) @ ( bit1 @ N2 ) ) ) ) ) ) ) ) ) ) ) ).
% xor_num.cases
thf(fact_4740_lsb__this__or__next,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A] :
( ~ ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ X2 @ ( one_one @ ( word @ A ) ) ) @ ( zero_zero @ nat ) )
=> ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ X2 @ ( zero_zero @ nat ) ) ) ) ).
% lsb_this_or_next
thf(fact_4741_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_4742_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_4743_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_4744_overflow__imp__lsb,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A] :
( ( ( plus_plus @ ( word @ A ) @ X2 @ ( one_one @ ( word @ A ) ) )
= ( zero_zero @ ( word @ A ) ) )
=> ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ X2 @ ( zero_zero @ nat ) ) ) ) ).
% overflow_imp_lsb
thf(fact_4745_VEBT__internal_Onaive__member_Ocases,axiom,
! [X2: product_prod @ vEBT_VEBT @ nat] :
( ! [A5: $o,B4: $o,X3: nat] :
( X2
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A5 @ B4 ) @ X3 ) )
=> ( ! [Uu2: option @ ( product_prod @ nat @ nat ),Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT,Ux2: nat] :
( X2
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Uu2 @ ( zero_zero @ nat ) @ Uv2 @ Uw2 ) @ Ux2 ) )
=> ~ ! [Uy2: option @ ( product_prod @ nat @ nat ),V3: nat,TreeList2: list @ vEBT_VEBT,S2: vEBT_VEBT,X3: nat] :
( X2
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList2 @ S2 ) @ X3 ) ) ) ) ).
% VEBT_internal.naive_member.cases
thf(fact_4746_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_4747_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_4748_bang__is__le,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A,M: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ X2 @ M )
=> ( ord_less_eq @ ( word @ A ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ M ) @ X2 ) ) ) ).
% bang_is_le
thf(fact_4749_bit__nat__def,axiom,
( ( bit_se5641148757651400278ts_bit @ nat )
= ( ^ [M5: nat,N3: nat] :
~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ M5 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) ) ) ) ) ).
% bit_nat_def
thf(fact_4750_odd__iff__lsb,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A] :
( ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( semiring_1_unsigned @ A @ nat @ X2 ) ) )
= ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ X2 @ ( zero_zero @ nat ) ) ) ) ).
% odd_iff_lsb
thf(fact_4751_VEBT__internal_Omembermima_Ocases,axiom,
! [X2: product_prod @ vEBT_VEBT @ nat] :
( ! [Uu2: $o,Uv2: $o,Uw2: nat] :
( X2
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Uw2 ) )
=> ( ! [Ux2: list @ vEBT_VEBT,Uy2: vEBT_VEBT,Uz2: nat] :
( X2
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( zero_zero @ nat ) @ Ux2 @ Uy2 ) @ Uz2 ) )
=> ( ! [Mi2: nat,Ma2: nat,Va3: list @ vEBT_VEBT,Vb2: vEBT_VEBT,X3: nat] :
( X2
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ Va3 @ Vb2 ) @ X3 ) )
=> ( ! [Mi2: nat,Ma2: nat,V3: nat,TreeList2: list @ vEBT_VEBT,Vc: vEBT_VEBT,X3: nat] :
( X2
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList2 @ Vc ) @ X3 ) )
=> ~ ! [V3: nat,TreeList2: list @ vEBT_VEBT,Vd2: vEBT_VEBT,X3: nat] :
( X2
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V3 ) @ TreeList2 @ Vd2 ) @ X3 ) ) ) ) ) ) ).
% VEBT_internal.membermima.cases
thf(fact_4752_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Ocases,axiom,
! [X2: product_prod @ vEBT_VEBT @ nat] :
( ! [A5: $o,B4: $o] :
( X2
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A5 @ B4 ) @ ( zero_zero @ nat ) ) )
=> ( ! [A5: $o,B4: $o] :
( X2
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A5 @ B4 ) @ ( suc @ ( zero_zero @ nat ) ) ) )
=> ( ! [A5: $o,B4: $o,N2: nat] :
( X2
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A5 @ B4 ) @ ( suc @ ( suc @ N2 ) ) ) )
=> ( ! [Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT,Uu2: nat] :
( X2
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg2 @ TreeList2 @ Summary2 ) @ Uu2 ) )
=> ( ! [Mi2: nat,Ma2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT,X3: nat] :
( X2
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ TreeList2 @ Summary2 ) @ X3 ) )
=> ( ! [Mi2: nat,Ma2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT,X3: nat] :
( X2
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( zero_zero @ nat ) ) @ TreeList2 @ Summary2 ) @ X3 ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT,X3: nat] :
( X2
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary2 ) @ X3 ) ) ) ) ) ) ) ) ).
% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.cases
thf(fact_4753_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Ocases,axiom,
! [X2: product_prod @ vEBT_VEBT @ nat] :
( ! [Uu2: $o,Uv2: $o] :
( X2
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ ( zero_zero @ nat ) ) )
=> ( ! [A5: $o,Uw2: $o] :
( X2
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A5 @ Uw2 ) @ ( suc @ ( zero_zero @ nat ) ) ) )
=> ( ! [A5: $o,B4: $o,Va2: nat] :
( X2
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A5 @ B4 ) @ ( suc @ ( suc @ Va2 ) ) ) )
=> ( ! [Uy2: nat,Uz2: list @ vEBT_VEBT,Va3: vEBT_VEBT,Vb2: nat] :
( X2
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uy2 @ Uz2 @ Va3 ) @ Vb2 ) )
=> ( ! [V3: product_prod @ nat @ nat,Vd2: list @ vEBT_VEBT,Ve2: vEBT_VEBT,Vf2: nat] :
( X2
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Vd2 @ Ve2 ) @ Vf2 ) )
=> ( ! [V3: product_prod @ nat @ nat,Vh2: list @ vEBT_VEBT,Vi2: vEBT_VEBT,Vj2: nat] :
( X2
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vh2 @ Vi2 ) @ Vj2 ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT,X3: nat] :
( X2
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary2 ) @ X3 ) ) ) ) ) ) ) ) ).
% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.cases
thf(fact_4754_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Ocases,axiom,
! [X2: product_prod @ vEBT_VEBT @ nat] :
( ! [Uu2: $o,B4: $o] :
( X2
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uu2 @ B4 ) @ ( zero_zero @ nat ) ) )
=> ( ! [Uv2: $o,Uw2: $o,N2: nat] :
( X2
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uv2 @ Uw2 ) @ ( suc @ N2 ) ) )
=> ( ! [Ux2: nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT,Va3: nat] :
( X2
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Ux2 @ Uy2 @ Uz2 ) @ Va3 ) )
=> ( ! [V3: product_prod @ nat @ nat,Vc: list @ vEBT_VEBT,Vd2: vEBT_VEBT,Ve2: nat] :
( X2
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Vc @ Vd2 ) @ Ve2 ) )
=> ( ! [V3: product_prod @ nat @ nat,Vg2: list @ vEBT_VEBT,Vh2: vEBT_VEBT,Vi2: nat] :
( X2
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vg2 @ Vh2 ) @ Vi2 ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT,X3: nat] :
( X2
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary2 ) @ X3 ) ) ) ) ) ) ) ).
% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.cases
thf(fact_4755_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Ocases,axiom,
! [X2: product_prod @ vEBT_VEBT @ nat] :
( ! [A5: $o,B4: $o,X3: nat] :
( X2
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A5 @ B4 ) @ X3 ) )
=> ( ! [Info2: option @ ( product_prod @ nat @ nat ),Ts2: list @ vEBT_VEBT,S2: vEBT_VEBT,X3: nat] :
( X2
!= ( 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] :
( X2
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Info2 @ ( suc @ ( zero_zero @ nat ) ) @ Ts2 @ S2 ) @ X3 ) )
=> ( ! [V3: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT,X3: nat] :
( X2
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ V3 ) ) @ TreeList2 @ Summary2 ) @ X3 ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT,X3: nat] :
( X2
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary2 ) @ X3 ) ) ) ) ) ) ).
% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.cases
thf(fact_4756_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Ocases,axiom,
! [X2: product_prod @ vEBT_VEBT @ nat] :
( ! [A5: $o,B4: $o,X3: nat] :
( X2
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A5 @ B4 ) @ X3 ) )
=> ( ! [Uu2: nat,Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT,X3: nat] :
( X2
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) @ X3 ) )
=> ( ! [V3: product_prod @ nat @ nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT,X3: nat] :
( X2
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Uy2 @ Uz2 ) @ X3 ) )
=> ( ! [V3: product_prod @ nat @ nat,Vb2: list @ vEBT_VEBT,Vc: vEBT_VEBT,X3: nat] :
( X2
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb2 @ Vc ) @ X3 ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT,X3: nat] :
( X2
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary2 ) @ X3 ) ) ) ) ) ) ).
% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.cases
thf(fact_4757_divmod__step__eq,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [L2: num,R2: A,Q2: A] :
( ( ( ord_less_eq @ A @ ( numeral_numeral @ A @ L2 ) @ R2 )
=> ( ( unique1321980374590559556d_step @ A @ L2 @ ( product_Pair @ A @ A @ Q2 @ R2 ) )
= ( product_Pair @ A @ A @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Q2 ) @ ( one_one @ A ) ) @ ( minus_minus @ A @ R2 @ ( numeral_numeral @ A @ L2 ) ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( numeral_numeral @ A @ L2 ) @ R2 )
=> ( ( unique1321980374590559556d_step @ A @ L2 @ ( product_Pair @ A @ A @ Q2 @ R2 ) )
= ( product_Pair @ A @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Q2 ) @ R2 ) ) ) ) ) ).
% divmod_step_eq
thf(fact_4758_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Oelims,axiom,
! [X2: vEBT_VEBT,Xa: nat,Y2: nat] :
( ( ( vEBT_T_m_e_m_b_e_r2 @ X2 @ Xa )
= Y2 )
=> ( ( ? [A5: $o,B4: $o] :
( X2
= ( vEBT_Leaf @ A5 @ B4 ) )
=> ( Y2
!= ( one_one @ nat ) ) )
=> ( ( ? [Uu2: nat,Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT] :
( X2
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) )
=> ( Y2
!= ( one_one @ nat ) ) )
=> ( ( ? [V3: product_prod @ nat @ nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Uy2 @ Uz2 ) )
=> ( Y2
!= ( one_one @ nat ) ) )
=> ( ( ? [V3: product_prod @ nat @ nat,Vb2: list @ vEBT_VEBT,Vc: vEBT_VEBT] :
( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb2 @ Vc ) )
=> ( Y2
!= ( one_one @ nat ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list @ vEBT_VEBT] :
( ? [Summary2: vEBT_VEBT] :
( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary2 ) )
=> ( Y2
!= ( plus_plus @ nat @ ( one_one @ nat )
@ ( if @ nat @ ( Xa = Mi2 ) @ ( zero_zero @ nat )
@ ( if @ nat @ ( Xa = Ma2 ) @ ( zero_zero @ nat )
@ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ ( zero_zero @ nat )
@ ( if @ nat @ ( ord_less @ nat @ Ma2 @ Xa ) @ ( zero_zero @ nat )
@ ( if @ nat
@ ( ( ord_less @ nat @ Mi2 @ Xa )
& ( ord_less @ nat @ Xa @ Ma2 ) )
@ ( if @ nat @ ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) @ ( vEBT_T_m_e_m_b_e_r2 @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( 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'.elims
thf(fact_4759_less__eq__option__Some__None,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X2: A] :
~ ( ord_less_eq @ ( option @ A ) @ ( some @ A @ X2 ) @ ( none @ A ) ) ) ).
% less_eq_option_Some_None
thf(fact_4760_less__option__None,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X2: option @ A] :
~ ( ord_less @ ( option @ A ) @ X2 @ ( none @ A ) ) ) ).
% less_option_None
thf(fact_4761_less__option__None__Some__code,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X2: A] : ( ord_less @ ( option @ A ) @ ( none @ A ) @ ( some @ A @ X2 ) ) ) ).
% less_option_None_Some_code
thf(fact_4762_less__eq__option__None__code,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X2: option @ A] : ( ord_less_eq @ ( option @ A ) @ ( none @ A ) @ X2 ) ) ).
% less_eq_option_None_code
thf(fact_4763_less__eq__option__Some,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X2: A,Y2: A] :
( ( ord_less_eq @ ( option @ A ) @ ( some @ A @ X2 ) @ ( some @ A @ Y2 ) )
= ( ord_less_eq @ A @ X2 @ Y2 ) ) ) ).
% less_eq_option_Some
thf(fact_4764_local_Opower__def,axiom,
( vEBT_VEBT_power
= ( vEBT_V2048590022279873568_shift @ nat @ ( power_power @ nat ) ) ) ).
% local.power_def
thf(fact_4765_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,X2: nat] :
( ( vEBT_T_m_e_m_b_e_r2 @ ( vEBT_Leaf @ A2 @ B2 ) @ X2 )
= ( one_one @ nat ) ) ).
% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.simps(1)
thf(fact_4766_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Osimps_I2_J,axiom,
! [Uu: nat,Uv: list @ vEBT_VEBT,Uw: vEBT_VEBT,X2: nat] :
( ( vEBT_T_m_e_m_b_e_r2 @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu @ Uv @ Uw ) @ X2 )
= ( one_one @ nat ) ) ).
% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.simps(2)
thf(fact_4767_less__option__None__Some,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X2: A] : ( ord_less @ ( option @ A ) @ ( none @ A ) @ ( some @ A @ X2 ) ) ) ).
% less_option_None_Some
thf(fact_4768_less__option__None__is__Some,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X2: option @ A] :
( ( ord_less @ ( option @ A ) @ ( none @ A ) @ X2 )
=> ? [Z2: A] :
( X2
= ( some @ A @ Z2 ) ) ) ) ).
% less_option_None_is_Some
thf(fact_4769_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Osimps_I3_J,axiom,
! [V: product_prod @ nat @ nat,Uy: list @ vEBT_VEBT,Uz: vEBT_VEBT,X2: nat] :
( ( vEBT_T_m_e_m_b_e_r2 @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V ) @ ( zero_zero @ nat ) @ Uy @ Uz ) @ X2 )
= ( one_one @ nat ) ) ).
% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.simps(3)
thf(fact_4770_less__eq__option__None,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X2: option @ A] : ( ord_less_eq @ ( option @ A ) @ ( none @ A ) @ X2 ) ) ).
% less_eq_option_None
thf(fact_4771_less__eq__option__None__is__None,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X2: option @ A] :
( ( ord_less_eq @ ( option @ A ) @ X2 @ ( none @ A ) )
=> ( X2
= ( none @ A ) ) ) ) ).
% less_eq_option_None_is_None
thf(fact_4772_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,X2: nat] :
( ( vEBT_T_m_e_m_b_e_r2 @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb @ Vc2 ) @ X2 )
= ( one_one @ nat ) ) ).
% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.simps(4)
thf(fact_4773_member__bound__height_H,axiom,
! [T2: vEBT_VEBT,N: nat,X2: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ord_less_eq @ nat @ ( vEBT_T_m_e_m_b_e_r2 @ T2 @ X2 ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_VEBT_height @ T2 ) ) ) ) ).
% member_bound_height'
thf(fact_4774_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Osimps_I5_J,axiom,
! [Mi: nat,Ma: nat,Va: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,X2: nat] :
( ( vEBT_T_m_e_m_b_e_r2 @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X2 )
= ( plus_plus @ nat @ ( one_one @ nat )
@ ( if @ nat @ ( X2 = Mi ) @ ( zero_zero @ nat )
@ ( if @ nat @ ( X2 = Ma ) @ ( zero_zero @ nat )
@ ( if @ nat @ ( ord_less @ nat @ X2 @ Mi ) @ ( zero_zero @ nat )
@ ( if @ nat @ ( ord_less @ nat @ Ma @ X2 ) @ ( zero_zero @ nat )
@ ( if @ nat
@ ( ( ord_less @ nat @ Mi @ X2 )
& ( ord_less @ nat @ X2 @ Ma ) )
@ ( if @ nat @ ( ord_less @ nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) ) @ ( vEBT_T_m_e_m_b_e_r2 @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( zero_zero @ nat ) )
@ ( zero_zero @ nat ) ) ) ) ) ) ) ) ).
% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.simps(5)
thf(fact_4775_mintlistlength,axiom,
! [Mi: nat,Ma: nat,Deg: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ N )
=> ( ( Mi != Ma )
=> ( ( ord_less @ nat @ Mi @ Ma )
& ? [M3: nat] :
( ( ( some @ nat @ M3 )
= ( vEBT_vebt_mint @ Summary ) )
& ( ord_less @ nat @ M3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ N @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% mintlistlength
thf(fact_4776_del__single__cont,axiom,
! [X2: nat,Mi: nat,Ma: nat,Deg: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( ( X2 = Mi )
& ( X2 = Ma ) )
=> ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X2 )
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg @ TreeList @ Summary ) ) ) ) ).
% del_single_cont
thf(fact_4777_delt__out__of__range,axiom,
! [X2: nat,Mi: nat,Ma: nat,Deg: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( ( ord_less @ nat @ X2 @ Mi )
| ( ord_less @ nat @ Ma @ X2 ) )
=> ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X2 )
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) ) ) ) ).
% delt_out_of_range
thf(fact_4778_delete__pres__valid,axiom,
! [T2: vEBT_VEBT,N: nat,X2: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( vEBT_invar_vebt @ ( vEBT_vebt_delete @ T2 @ X2 ) @ N ) ) ).
% delete_pres_valid
thf(fact_4779_dele__bmo__cont__corr,axiom,
! [T2: vEBT_VEBT,N: nat,X2: nat,Y2: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( vEBT_V8194947554948674370ptions @ ( vEBT_vebt_delete @ T2 @ X2 ) @ Y2 )
= ( ( X2 != Y2 )
& ( vEBT_V8194947554948674370ptions @ T2 @ Y2 ) ) ) ) ).
% dele_bmo_cont_corr
thf(fact_4780_dele__member__cont__corr,axiom,
! [T2: vEBT_VEBT,N: nat,X2: nat,Y2: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( vEBT_vebt_member @ ( vEBT_vebt_delete @ T2 @ X2 ) @ Y2 )
= ( ( X2 != Y2 )
& ( vEBT_vebt_member @ T2 @ Y2 ) ) ) ) ).
% dele_member_cont_corr
thf(fact_4781_mint__member,axiom,
! [T2: vEBT_VEBT,N: nat,Maxi: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( ( vEBT_vebt_mint @ T2 )
= ( some @ nat @ Maxi ) )
=> ( vEBT_vebt_member @ T2 @ Maxi ) ) ) ).
% mint_member
thf(fact_4782_mint__corr__help,axiom,
! [T2: vEBT_VEBT,N: nat,Mini: nat,X2: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( ( vEBT_vebt_mint @ T2 )
= ( some @ nat @ Mini ) )
=> ( ( vEBT_vebt_member @ T2 @ X2 )
=> ( ord_less_eq @ nat @ Mini @ X2 ) ) ) ) ).
% mint_corr_help
thf(fact_4783_mint__corr,axiom,
! [T2: vEBT_VEBT,N: nat,X2: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( ( vEBT_vebt_mint @ T2 )
= ( some @ nat @ X2 ) )
=> ( vEBT_VEBT_min_in_set @ ( vEBT_VEBT_set_vebt @ T2 ) @ X2 ) ) ) ).
% mint_corr
thf(fact_4784_mint__sound,axiom,
! [T2: vEBT_VEBT,N: nat,X2: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( vEBT_VEBT_min_in_set @ ( vEBT_VEBT_set_vebt @ T2 ) @ X2 )
=> ( ( vEBT_vebt_mint @ T2 )
= ( some @ nat @ X2 ) ) ) ) ).
% mint_sound
thf(fact_4785_misiz,axiom,
! [T2: vEBT_VEBT,N: nat,M: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( ( some @ nat @ M )
= ( vEBT_vebt_mint @ T2 ) )
=> ( ord_less @ nat @ M @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% misiz
thf(fact_4786_nested__mint,axiom,
! [Mi: nat,Ma: nat,Deg: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,N: nat,Va: nat] :
( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ N )
=> ( ( N
= ( suc @ ( suc @ Va ) ) )
=> ( ~ ( ord_less @ nat @ Ma @ Mi )
=> ( ( Ma != Mi )
=> ( ord_less @ nat @ ( vEBT_VEBT_high @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Va @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( suc @ ( divide_divide @ nat @ Va @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) ) ) ) ) ) ).
% nested_mint
thf(fact_4787_vebt__mint_Oelims,axiom,
! [X2: vEBT_VEBT,Y2: option @ nat] :
( ( ( vEBT_vebt_mint @ X2 )
= Y2 )
=> ( ! [A5: $o,B4: $o] :
( ( X2
= ( vEBT_Leaf @ A5 @ B4 ) )
=> ~ ( ( A5
=> ( Y2
= ( some @ nat @ ( zero_zero @ nat ) ) ) )
& ( ~ A5
=> ( ( B4
=> ( Y2
= ( some @ nat @ ( one_one @ nat ) ) ) )
& ( ~ B4
=> ( Y2
= ( none @ nat ) ) ) ) ) ) )
=> ( ( ? [Uu2: nat,Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT] :
( X2
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) )
=> ( Y2
!= ( none @ nat ) ) )
=> ~ ! [Mi2: nat] :
( ? [Ma2: nat,Ux2: nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
=> ( Y2
!= ( some @ nat @ Mi2 ) ) ) ) ) ) ).
% vebt_mint.elims
thf(fact_4788_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Oelims,axiom,
! [X2: vEBT_VEBT,Xa: nat,Y2: nat] :
( ( ( vEBT_T_p_r_e_d2 @ X2 @ Xa )
= Y2 )
=> ( ( ? [Uu2: $o,Uv2: $o] :
( X2
= ( vEBT_Leaf @ Uu2 @ Uv2 ) )
=> ( ( Xa
= ( zero_zero @ nat ) )
=> ( Y2
!= ( one_one @ nat ) ) ) )
=> ( ( ? [A5: $o,Uw2: $o] :
( X2
= ( vEBT_Leaf @ A5 @ Uw2 ) )
=> ( ( Xa
= ( suc @ ( zero_zero @ nat ) ) )
=> ( Y2
!= ( one_one @ nat ) ) ) )
=> ( ( ? [A5: $o,B4: $o] :
( X2
= ( vEBT_Leaf @ A5 @ B4 ) )
=> ( ? [Va2: nat] :
( Xa
= ( suc @ ( suc @ Va2 ) ) )
=> ( Y2
!= ( one_one @ nat ) ) ) )
=> ( ( ? [Uy2: nat,Uz2: list @ vEBT_VEBT,Va3: vEBT_VEBT] :
( X2
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uy2 @ Uz2 @ Va3 ) )
=> ( Y2
!= ( one_one @ nat ) ) )
=> ( ( ? [V3: product_prod @ nat @ nat,Vd2: list @ vEBT_VEBT,Ve2: vEBT_VEBT] :
( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Vd2 @ Ve2 ) )
=> ( Y2
!= ( one_one @ nat ) ) )
=> ( ( ? [V3: product_prod @ nat @ nat,Vh2: list @ vEBT_VEBT,Vi2: vEBT_VEBT] :
( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vh2 @ Vi2 ) )
=> ( Y2
!= ( one_one @ nat ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary2 ) )
=> ~ ( ( ( ord_less @ nat @ Ma2 @ Xa )
=> ( Y2
= ( one_one @ nat ) ) )
& ( ~ ( ord_less @ nat @ Ma2 @ Xa )
=> ( Y2
= ( if @ nat @ ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
@ ( if @ nat
@ ( ( ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
!= ( none @ nat ) )
& ( vEBT_VEBT_greater @ ( some @ nat @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
@ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_T_p_r_e_d2 @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( plus_plus @ nat @ ( vEBT_T_p_r_e_d2 @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( 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'.elims
thf(fact_4789_greater__shift,axiom,
( ( ord_less @ nat )
= ( ^ [Y: nat,X: nat] : ( vEBT_VEBT_greater @ ( some @ nat @ X ) @ ( some @ nat @ Y ) ) ) ) ).
% greater_shift
thf(fact_4790_option_Ocollapse,axiom,
! [A: $tType,Option: option @ A] :
( ( Option
!= ( none @ A ) )
=> ( ( some @ A @ ( the2 @ A @ Option ) )
= Option ) ) ).
% option.collapse
thf(fact_4791_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_4792_option_Oexhaust__sel,axiom,
! [A: $tType,Option: option @ A] :
( ( Option
!= ( none @ A ) )
=> ( Option
= ( some @ A @ ( the2 @ A @ Option ) ) ) ) ).
% option.exhaust_sel
thf(fact_4793_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
! [A: $tType,X2: product_prod @ ( A > A > $o ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) )] :
( ! [Uu2: A > A > $o,Uv2: option @ A] :
( X2
!= ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) ) @ Uu2 @ ( product_Pair @ ( option @ A ) @ ( option @ A ) @ ( none @ A ) @ Uv2 ) ) )
=> ( ! [Uw2: A > A > $o,V3: A] :
( X2
!= ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) ) @ Uw2 @ ( product_Pair @ ( option @ A ) @ ( option @ A ) @ ( some @ A @ V3 ) @ ( none @ A ) ) ) )
=> ~ ! [F5: A > A > $o,X3: A,Y3: A] :
( X2
!= ( 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_4794_VEBT__internal_Ooption__shift_Ocases,axiom,
! [A: $tType,X2: product_prod @ ( A > A > A ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) )] :
( ! [Uu2: A > A > A,Uv2: option @ A] :
( X2
!= ( product_Pair @ ( A > A > A ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) ) @ Uu2 @ ( product_Pair @ ( option @ A ) @ ( option @ A ) @ ( none @ A ) @ Uv2 ) ) )
=> ( ! [Uw2: A > A > A,V3: A] :
( X2
!= ( product_Pair @ ( A > A > A ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) ) @ Uw2 @ ( product_Pair @ ( option @ A ) @ ( option @ A ) @ ( some @ A @ V3 ) @ ( none @ A ) ) ) )
=> ~ ! [F5: A > A > A,A5: A,B4: A] :
( X2
!= ( product_Pair @ ( A > A > A ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) ) @ F5 @ ( product_Pair @ ( option @ A ) @ ( option @ A ) @ ( some @ A @ A5 ) @ ( some @ A @ B4 ) ) ) ) ) ) ).
% VEBT_internal.option_shift.cases
thf(fact_4795_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_4796_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_4797_VEBT__internal_Ooption__shift_Oelims,axiom,
! [A: $tType,X2: A > A > A,Xa: option @ A,Xb: option @ A,Y2: option @ A] :
( ( ( vEBT_V2048590022279873568_shift @ A @ X2 @ Xa @ Xb )
= Y2 )
=> ( ( ( Xa
= ( none @ A ) )
=> ( Y2
!= ( none @ A ) ) )
=> ( ( ? [V3: A] :
( Xa
= ( some @ A @ V3 ) )
=> ( ( Xb
= ( none @ A ) )
=> ( Y2
!= ( none @ A ) ) ) )
=> ~ ! [A5: A] :
( ( Xa
= ( some @ A @ A5 ) )
=> ! [B4: A] :
( ( Xb
= ( some @ A @ B4 ) )
=> ( Y2
!= ( some @ A @ ( X2 @ A5 @ B4 ) ) ) ) ) ) ) ) ).
% VEBT_internal.option_shift.elims
thf(fact_4798_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_4799_vebt__mint_Osimps_I3_J,axiom,
! [Mi: nat,Ma: nat,Ux: nat,Uy: list @ vEBT_VEBT,Uz: vEBT_VEBT] :
( ( vEBT_vebt_mint @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Ux @ Uy @ Uz ) )
= ( some @ nat @ Mi ) ) ).
% vebt_mint.simps(3)
thf(fact_4800_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_4801_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I7_J,axiom,
! [Ma: nat,X2: nat,Mi: nat,Va: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( ( ord_less @ nat @ Ma @ X2 )
=> ( ( vEBT_T_p_r_e_d2 @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X2 )
= ( one_one @ nat ) ) )
& ( ~ ( ord_less @ nat @ Ma @ X2 )
=> ( ( vEBT_T_p_r_e_d2 @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X2 )
= ( if @ nat @ ( ord_less @ nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
@ ( if @ nat
@ ( ( ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
!= ( none @ nat ) )
& ( vEBT_VEBT_greater @ ( some @ nat @ ( vEBT_VEBT_low @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
@ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_T_p_r_e_d2 @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( plus_plus @ nat @ ( vEBT_T_p_r_e_d2 @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( one_one @ nat ) ) )
@ ( one_one @ nat ) ) ) ) ) ).
% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(7)
thf(fact_4802_summaxma,axiom,
! [Mi: nat,Ma: nat,Deg: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ Deg )
=> ( ( Mi != Ma )
=> ( ( the2 @ nat @ ( vEBT_vebt_maxt @ Summary ) )
= ( vEBT_VEBT_high @ Ma @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% summaxma
thf(fact_4803_minNull__delete__time__bound,axiom,
! [T2: vEBT_VEBT,N: nat,X2: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ T2 @ X2 ) )
=> ( ord_less_eq @ nat @ ( vEBT_T_d_e_l_e_t_e @ T2 @ X2 ) @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% minNull_delete_time_bound
thf(fact_4804_mul__def,axiom,
( vEBT_VEBT_mul
= ( vEBT_V2048590022279873568_shift @ nat @ ( times_times @ nat ) ) ) ).
% mul_def
thf(fact_4805_not__min__Null__member,axiom,
! [T2: vEBT_VEBT] :
( ~ ( vEBT_VEBT_minNull @ T2 )
=> ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ T2 @ X_1 ) ) ).
% not_min_Null_member
thf(fact_4806_min__Null__member,axiom,
! [T2: vEBT_VEBT,X2: nat] :
( ( vEBT_VEBT_minNull @ T2 )
=> ~ ( vEBT_vebt_member @ T2 @ X2 ) ) ).
% min_Null_member
thf(fact_4807_maxbmo,axiom,
! [T2: vEBT_VEBT,X2: nat] :
( ( ( vEBT_vebt_maxt @ T2 )
= ( some @ nat @ X2 ) )
=> ( vEBT_V8194947554948674370ptions @ T2 @ X2 ) ) ).
% maxbmo
thf(fact_4808_minNullmin,axiom,
! [T2: vEBT_VEBT] :
( ( vEBT_VEBT_minNull @ T2 )
=> ( ( vEBT_vebt_mint @ T2 )
= ( none @ nat ) ) ) ).
% minNullmin
thf(fact_4809_minminNull,axiom,
! [T2: vEBT_VEBT] :
( ( ( vEBT_vebt_mint @ T2 )
= ( none @ nat ) )
=> ( vEBT_VEBT_minNull @ T2 ) ) ).
% minminNull
thf(fact_4810_mul__shift,axiom,
! [X2: nat,Y2: nat,Z: nat] :
( ( ( times_times @ nat @ X2 @ Y2 )
= Z )
= ( ( vEBT_VEBT_mul @ ( some @ nat @ X2 ) @ ( some @ nat @ Y2 ) )
= ( some @ nat @ Z ) ) ) ).
% mul_shift
thf(fact_4811_maxt__member,axiom,
! [T2: vEBT_VEBT,N: nat,Maxi: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( ( vEBT_vebt_maxt @ T2 )
= ( some @ nat @ Maxi ) )
=> ( vEBT_vebt_member @ T2 @ Maxi ) ) ) ).
% maxt_member
thf(fact_4812_maxt__corr__help,axiom,
! [T2: vEBT_VEBT,N: nat,Maxi: nat,X2: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( ( vEBT_vebt_maxt @ T2 )
= ( some @ nat @ Maxi ) )
=> ( ( vEBT_vebt_member @ T2 @ X2 )
=> ( ord_less_eq @ nat @ X2 @ Maxi ) ) ) ) ).
% maxt_corr_help
thf(fact_4813_maxt__corr,axiom,
! [T2: vEBT_VEBT,N: nat,X2: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( ( vEBT_vebt_maxt @ T2 )
= ( some @ nat @ X2 ) )
=> ( vEBT_VEBT_max_in_set @ ( vEBT_VEBT_set_vebt @ T2 ) @ X2 ) ) ) ).
% maxt_corr
thf(fact_4814_maxt__sound,axiom,
! [T2: vEBT_VEBT,N: nat,X2: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( vEBT_VEBT_max_in_set @ ( vEBT_VEBT_set_vebt @ T2 ) @ X2 )
=> ( ( vEBT_vebt_maxt @ T2 )
= ( some @ nat @ X2 ) ) ) ) ).
% maxt_sound
thf(fact_4815_minNull__delete__time__bound_H,axiom,
! [T2: vEBT_VEBT,N: nat,X2: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ T2 @ X2 ) )
=> ( ord_less_eq @ nat @ ( vEBT_V1232361888498592333_e_t_e @ T2 @ X2 ) @ ( one_one @ nat ) ) ) ) ).
% minNull_delete_time_bound'
thf(fact_4816_VEBT__internal_OminNull_Osimps_I5_J,axiom,
! [Uz: product_prod @ nat @ nat,Va: nat,Vb: list @ vEBT_VEBT,Vc2: vEBT_VEBT] :
~ ( vEBT_VEBT_minNull @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ Uz ) @ Va @ Vb @ Vc2 ) ) ).
% VEBT_internal.minNull.simps(5)
thf(fact_4817_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_4818_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_4819_VEBT__internal_OminNull_Oelims_I3_J,axiom,
! [X2: vEBT_VEBT] :
( ~ ( vEBT_VEBT_minNull @ X2 )
=> ( ! [Uv2: $o] :
( X2
!= ( vEBT_Leaf @ $true @ Uv2 ) )
=> ( ! [Uu2: $o] :
( X2
!= ( vEBT_Leaf @ Uu2 @ $true ) )
=> ~ ! [Uz2: product_prod @ nat @ nat,Va3: nat,Vb2: list @ vEBT_VEBT,Vc: vEBT_VEBT] :
( X2
!= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ Uz2 ) @ Va3 @ Vb2 @ Vc ) ) ) ) ) ).
% VEBT_internal.minNull.elims(3)
thf(fact_4820_VEBT__internal_OminNull_Oelims_I2_J,axiom,
! [X2: vEBT_VEBT] :
( ( vEBT_VEBT_minNull @ X2 )
=> ( ( X2
!= ( vEBT_Leaf @ $false @ $false ) )
=> ~ ! [Uw2: nat,Ux2: list @ vEBT_VEBT,Uy2: vEBT_VEBT] :
( X2
!= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uw2 @ Ux2 @ Uy2 ) ) ) ) ).
% VEBT_internal.minNull.elims(2)
thf(fact_4821_vebt__maxt_Osimps_I3_J,axiom,
! [Mi: nat,Ma: nat,Ux: nat,Uy: list @ vEBT_VEBT,Uz: vEBT_VEBT] :
( ( vEBT_vebt_maxt @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Ux @ Uy @ Uz ) )
= ( some @ nat @ Ma ) ) ).
% vebt_maxt.simps(3)
thf(fact_4822_VEBT__internal_OminNull_Oelims_I1_J,axiom,
! [X2: vEBT_VEBT,Y2: $o] :
( ( ( vEBT_VEBT_minNull @ X2 )
= Y2 )
=> ( ( ( X2
= ( vEBT_Leaf @ $false @ $false ) )
=> ~ Y2 )
=> ( ( ? [Uv2: $o] :
( X2
= ( vEBT_Leaf @ $true @ Uv2 ) )
=> Y2 )
=> ( ( ? [Uu2: $o] :
( X2
= ( vEBT_Leaf @ Uu2 @ $true ) )
=> Y2 )
=> ( ( ? [Uw2: nat,Ux2: list @ vEBT_VEBT,Uy2: vEBT_VEBT] :
( X2
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uw2 @ Ux2 @ Uy2 ) )
=> ~ Y2 )
=> ~ ( ? [Uz2: product_prod @ nat @ nat,Va3: nat,Vb2: list @ vEBT_VEBT,Vc: vEBT_VEBT] :
( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ Uz2 ) @ Va3 @ Vb2 @ Vc ) )
=> Y2 ) ) ) ) ) ) ).
% VEBT_internal.minNull.elims(1)
thf(fact_4823_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_4824_insertsimp,axiom,
! [T2: vEBT_VEBT,N: nat,L2: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( vEBT_VEBT_minNull @ T2 )
=> ( ord_less_eq @ nat @ ( vEBT_T_i_n_s_e_r_t @ T2 @ L2 ) @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) ) ) ) ).
% insertsimp
thf(fact_4825_vebt__maxt_Oelims,axiom,
! [X2: vEBT_VEBT,Y2: option @ nat] :
( ( ( vEBT_vebt_maxt @ X2 )
= Y2 )
=> ( ! [A5: $o,B4: $o] :
( ( X2
= ( vEBT_Leaf @ A5 @ B4 ) )
=> ~ ( ( B4
=> ( Y2
= ( some @ nat @ ( one_one @ nat ) ) ) )
& ( ~ B4
=> ( ( A5
=> ( Y2
= ( some @ nat @ ( zero_zero @ nat ) ) ) )
& ( ~ A5
=> ( Y2
= ( none @ nat ) ) ) ) ) ) )
=> ( ( ? [Uu2: nat,Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT] :
( X2
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) )
=> ( Y2
!= ( none @ nat ) ) )
=> ~ ! [Mi2: nat,Ma2: nat] :
( ? [Ux2: nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
=> ( Y2
!= ( some @ nat @ Ma2 ) ) ) ) ) ) ).
% vebt_maxt.elims
thf(fact_4826_pred__less__length__list,axiom,
! [Deg: nat,X2: nat,Ma: nat,TreeList: list @ vEBT_VEBT,Mi: nat,Summary: vEBT_VEBT] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ( ord_less_eq @ nat @ X2 @ Ma )
=> ( ( ord_less @ nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
=> ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X2 )
= ( if @ ( option @ nat )
@ ( ( ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
!= ( none @ nat ) )
& ( vEBT_VEBT_greater @ ( some @ nat @ ( vEBT_VEBT_low @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
@ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( some @ nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( if @ ( option @ nat )
@ ( ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
= ( none @ nat ) )
@ ( if @ ( option @ nat ) @ ( ord_less @ nat @ Mi @ X2 ) @ ( some @ nat @ Mi ) @ ( none @ nat ) )
@ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% pred_less_length_list
thf(fact_4827_pred__lesseq__max,axiom,
! [Deg: nat,X2: nat,Ma: nat,Mi: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ( ord_less_eq @ nat @ X2 @ Ma )
=> ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X2 )
= ( if @ ( option @ nat ) @ ( ord_less @ nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
@ ( if @ ( option @ nat )
@ ( ( ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
!= ( none @ nat ) )
& ( vEBT_VEBT_greater @ ( some @ nat @ ( vEBT_VEBT_low @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
@ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( some @ nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( if @ ( option @ nat )
@ ( ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
= ( none @ nat ) )
@ ( if @ ( option @ nat ) @ ( ord_less @ nat @ Mi @ X2 ) @ ( some @ nat @ Mi ) @ ( none @ nat ) )
@ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) )
@ ( none @ nat ) ) ) ) ) ).
% pred_lesseq_max
thf(fact_4828_del__x__mia,axiom,
! [X2: nat,Mi: nat,Ma: nat,Deg: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( ( X2 = Mi )
& ( ord_less @ nat @ X2 @ Ma ) )
=> ( ( Mi != Ma )
=> ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X2 )
= ( if @ vEBT_VEBT @ ( ord_less @ nat @ ( vEBT_VEBT_high @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
@ ( if @ vEBT_VEBT @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( vEBT_Node
@ ( some @ ( product_prod @ nat @ nat )
@ ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
@ ( if @ nat
@ ( ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
= Ma )
@ ( if @ nat
@ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
= ( none @ nat ) )
@ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
@ ( plus_plus @ nat @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ ( list_update @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) )
@ Ma ) ) )
@ Deg
@ ( list_update @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( vEBT_Node
@ ( some @ ( product_prod @ nat @ nat )
@ ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
@ ( if @ nat
@ ( ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
= Ma )
@ ( plus_plus @ nat @ ( times_times @ nat @ ( vEBT_VEBT_high @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ ( list_update @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
@ Ma ) ) )
@ Deg
@ ( list_update @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ Summary ) )
@ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) ) ) ) ) ) ).
% del_x_mia
thf(fact_4829_add__shift,axiom,
! [X2: nat,Y2: nat,Z: nat] :
( ( ( plus_plus @ nat @ X2 @ Y2 )
= Z )
= ( ( vEBT_VEBT_add @ ( some @ nat @ X2 ) @ ( some @ nat @ Y2 ) )
= ( some @ nat @ Z ) ) ) ).
% add_shift
thf(fact_4830_add__def,axiom,
( vEBT_VEBT_add
= ( vEBT_V2048590022279873568_shift @ nat @ ( plus_plus @ nat ) ) ) ).
% add_def
thf(fact_4831_list__update__beyond,axiom,
! [A: $tType,Xs2: list @ A,I: nat,X2: A] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ I )
=> ( ( list_update @ A @ Xs2 @ I @ X2 )
= Xs2 ) ) ).
% list_update_beyond
thf(fact_4832_del__x__not__mi__newnode__not__nil,axiom,
! [Mi: nat,X2: nat,Ma: nat,Deg: nat,H2: nat,L2: nat,Newnode: vEBT_VEBT,TreeList: list @ vEBT_VEBT,Newlist: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( ( ord_less @ nat @ Mi @ X2 )
& ( ord_less_eq @ nat @ X2 @ Ma ) )
=> ( ( Mi != Ma )
=> ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ( ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= H2 )
=> ( ( ( vEBT_VEBT_low @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= L2 )
=> ( ( Newnode
= ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ H2 ) @ L2 ) )
=> ( ~ ( vEBT_VEBT_minNull @ Newnode )
=> ( ( Newlist
= ( list_update @ vEBT_VEBT @ TreeList @ H2 @ Newnode ) )
=> ( ( ord_less @ nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
=> ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X2 )
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ ( if @ nat @ ( X2 = Ma ) @ ( plus_plus @ nat @ ( times_times @ nat @ H2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ Newlist @ H2 ) ) ) ) @ Ma ) ) ) @ Deg @ Newlist @ Summary ) ) ) ) ) ) ) ) ) ) ) ).
% del_x_not_mi_newnode_not_nil
thf(fact_4833_del__x__mi__lets__in__not__minNull,axiom,
! [X2: nat,Mi: nat,Ma: nat,Deg: nat,Xn: nat,H2: nat,Summary: vEBT_VEBT,TreeList: list @ vEBT_VEBT,L2: nat,Newnode: vEBT_VEBT,Newlist: list @ vEBT_VEBT] :
( ( ( X2 = Mi )
& ( ord_less @ nat @ X2 @ Ma ) )
=> ( ( Mi != Ma )
=> ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ( ( vEBT_VEBT_high @ Xn @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= H2 )
=> ( ( Xn
= ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) )
=> ( ( ( vEBT_VEBT_low @ Xn @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= L2 )
=> ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xn @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
=> ( ( Newnode
= ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ H2 ) @ L2 ) )
=> ( ( Newlist
= ( list_update @ vEBT_VEBT @ TreeList @ H2 @ Newnode ) )
=> ( ~ ( vEBT_VEBT_minNull @ Newnode )
=> ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X2 )
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Xn @ ( if @ nat @ ( Xn = Ma ) @ ( plus_plus @ nat @ ( times_times @ nat @ H2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ Newlist @ H2 ) ) ) ) @ Ma ) ) ) @ Deg @ Newlist @ Summary ) ) ) ) ) ) ) ) ) ) ) ) ).
% del_x_mi_lets_in_not_minNull
thf(fact_4834_del__x__not__mi,axiom,
! [Mi: nat,X2: nat,Ma: nat,Deg: nat,H2: nat,L2: nat,Newnode: vEBT_VEBT,TreeList: list @ vEBT_VEBT,Newlist: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( ( ord_less @ nat @ Mi @ X2 )
& ( ord_less_eq @ nat @ X2 @ Ma ) )
=> ( ( Mi != Ma )
=> ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ( ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= H2 )
=> ( ( ( vEBT_VEBT_low @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= L2 )
=> ( ( Newnode
= ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ H2 ) @ L2 ) )
=> ( ( Newlist
= ( list_update @ vEBT_VEBT @ TreeList @ H2 @ Newnode ) )
=> ( ( ord_less @ nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
=> ( ( ( vEBT_VEBT_minNull @ Newnode )
=> ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X2 )
= ( vEBT_Node
@ ( some @ ( product_prod @ nat @ nat )
@ ( product_Pair @ nat @ nat @ Mi
@ ( if @ nat @ ( X2 = Ma )
@ ( if @ nat
@ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) )
= ( none @ nat ) )
@ Mi
@ ( plus_plus @ nat @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ Newlist @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) ) ) ) ) )
@ Ma ) ) )
@ Deg
@ Newlist
@ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) )
& ( ~ ( vEBT_VEBT_minNull @ Newnode )
=> ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X2 )
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ ( if @ nat @ ( X2 = Ma ) @ ( plus_plus @ nat @ ( times_times @ nat @ H2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ Newlist @ H2 ) ) ) ) @ Ma ) ) ) @ Deg @ Newlist @ Summary ) ) ) ) ) ) ) ) ) ) ) ) ).
% del_x_not_mi
thf(fact_4835_del__x__not__mi__new__node__nil,axiom,
! [Mi: nat,X2: nat,Ma: nat,Deg: nat,H2: nat,L2: nat,Newnode: vEBT_VEBT,TreeList: list @ vEBT_VEBT,Sn: vEBT_VEBT,Summary: vEBT_VEBT,Newlist: list @ vEBT_VEBT] :
( ( ( ord_less @ nat @ Mi @ X2 )
& ( ord_less_eq @ nat @ X2 @ Ma ) )
=> ( ( Mi != Ma )
=> ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ( ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= H2 )
=> ( ( ( vEBT_VEBT_low @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= L2 )
=> ( ( Newnode
= ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ H2 ) @ L2 ) )
=> ( ( vEBT_VEBT_minNull @ Newnode )
=> ( ( Sn
= ( vEBT_vebt_delete @ Summary @ H2 ) )
=> ( ( Newlist
= ( list_update @ vEBT_VEBT @ TreeList @ H2 @ Newnode ) )
=> ( ( ord_less @ nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
=> ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X2 )
= ( vEBT_Node
@ ( some @ ( product_prod @ nat @ nat )
@ ( product_Pair @ nat @ nat @ Mi
@ ( if @ nat @ ( X2 = Ma )
@ ( if @ nat
@ ( ( vEBT_vebt_maxt @ Sn )
= ( none @ nat ) )
@ Mi
@ ( plus_plus @ nat @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ Sn ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ Newlist @ ( the2 @ nat @ ( vEBT_vebt_maxt @ Sn ) ) ) ) ) ) )
@ Ma ) ) )
@ Deg
@ Newlist
@ Sn ) ) ) ) ) ) ) ) ) ) ) ) ).
% del_x_not_mi_new_node_nil
thf(fact_4836_del__x__not__mia,axiom,
! [Mi: nat,X2: nat,Ma: nat,Deg: nat,H2: nat,L2: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( ( ord_less @ nat @ Mi @ X2 )
& ( ord_less_eq @ nat @ X2 @ Ma ) )
=> ( ( Mi != Ma )
=> ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ( ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= H2 )
=> ( ( ( vEBT_VEBT_low @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= L2 )
=> ( ( ord_less @ nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
=> ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X2 )
= ( if @ vEBT_VEBT @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ H2 ) @ L2 ) )
@ ( vEBT_Node
@ ( some @ ( product_prod @ nat @ nat )
@ ( product_Pair @ nat @ nat @ Mi
@ ( if @ nat @ ( X2 = Ma )
@ ( if @ nat
@ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) )
= ( none @ nat ) )
@ Mi
@ ( plus_plus @ nat @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ ( list_update @ vEBT_VEBT @ TreeList @ H2 @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ H2 ) @ L2 ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) ) ) ) ) )
@ Ma ) ) )
@ Deg
@ ( list_update @ vEBT_VEBT @ TreeList @ H2 @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ H2 ) @ L2 ) )
@ ( vEBT_vebt_delete @ Summary @ H2 ) )
@ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ ( if @ nat @ ( X2 = Ma ) @ ( plus_plus @ nat @ ( times_times @ nat @ H2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ ( list_update @ vEBT_VEBT @ TreeList @ H2 @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ H2 ) @ L2 ) ) @ H2 ) ) ) ) @ Ma ) ) ) @ Deg @ ( list_update @ vEBT_VEBT @ TreeList @ H2 @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ H2 ) @ L2 ) ) @ Summary ) ) ) ) ) ) ) ) ) ).
% del_x_not_mia
thf(fact_4837_del__in__range,axiom,
! [Mi: nat,X2: nat,Ma: nat,Deg: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( ( ord_less_eq @ nat @ Mi @ X2 )
& ( ord_less_eq @ nat @ X2 @ Ma ) )
=> ( ( Mi != Ma )
=> ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X2 )
= ( if @ vEBT_VEBT @ ( ord_less @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
@ ( if @ vEBT_VEBT @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( vEBT_Node
@ ( some @ ( product_prod @ nat @ nat )
@ ( product_Pair @ nat @ nat @ ( if @ nat @ ( X2 = Mi ) @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ Mi )
@ ( if @ nat
@ ( ( ( X2 = Mi )
=> ( ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
= Ma ) )
& ( ( X2 != Mi )
=> ( X2 = Ma ) ) )
@ ( if @ nat
@ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
= ( none @ nat ) )
@ ( if @ nat @ ( X2 = Mi ) @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ Mi )
@ ( plus_plus @ nat @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ ( list_update @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) )
@ Ma ) ) )
@ Deg
@ ( list_update @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( vEBT_Node
@ ( some @ ( product_prod @ nat @ nat )
@ ( product_Pair @ nat @ nat @ ( if @ nat @ ( X2 = Mi ) @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ Mi )
@ ( if @ nat
@ ( ( ( X2 = Mi )
=> ( ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
= Ma ) )
& ( ( X2 != Mi )
=> ( X2 = Ma ) ) )
@ ( plus_plus @ nat @ ( times_times @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ ( list_update @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
@ Ma ) ) )
@ Deg
@ ( list_update @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ Summary ) )
@ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) ) ) ) ) ) ).
% del_in_range
thf(fact_4838_del__x__mi,axiom,
! [X2: nat,Mi: nat,Ma: nat,Deg: nat,Xn: nat,H2: nat,Summary: vEBT_VEBT,TreeList: list @ vEBT_VEBT,L2: nat] :
( ( ( X2 = Mi )
& ( ord_less @ nat @ X2 @ Ma ) )
=> ( ( Mi != Ma )
=> ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ( ( vEBT_VEBT_high @ Xn @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= H2 )
=> ( ( Xn
= ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) )
=> ( ( ( vEBT_VEBT_low @ Xn @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= L2 )
=> ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xn @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
=> ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X2 )
= ( if @ vEBT_VEBT @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ H2 ) @ L2 ) )
@ ( vEBT_Node
@ ( some @ ( product_prod @ nat @ nat )
@ ( product_Pair @ nat @ nat @ Xn
@ ( if @ nat @ ( Xn = Ma )
@ ( if @ nat
@ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) )
= ( none @ nat ) )
@ Xn
@ ( plus_plus @ nat @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ ( list_update @ vEBT_VEBT @ TreeList @ H2 @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ H2 ) @ L2 ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) ) ) ) ) )
@ Ma ) ) )
@ Deg
@ ( list_update @ vEBT_VEBT @ TreeList @ H2 @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ H2 ) @ L2 ) )
@ ( vEBT_vebt_delete @ Summary @ H2 ) )
@ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Xn @ ( if @ nat @ ( Xn = Ma ) @ ( plus_plus @ nat @ ( times_times @ nat @ H2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ ( list_update @ vEBT_VEBT @ TreeList @ H2 @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ H2 ) @ L2 ) ) @ H2 ) ) ) ) @ Ma ) ) ) @ Deg @ ( list_update @ vEBT_VEBT @ TreeList @ H2 @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ H2 ) @ L2 ) ) @ Summary ) ) ) ) ) ) ) ) ) ) ).
% del_x_mi
thf(fact_4839_del__x__mi__lets__in,axiom,
! [X2: nat,Mi: nat,Ma: nat,Deg: nat,Xn: nat,H2: nat,Summary: vEBT_VEBT,TreeList: list @ vEBT_VEBT,L2: nat,Newnode: vEBT_VEBT,Newlist: list @ vEBT_VEBT] :
( ( ( X2 = Mi )
& ( ord_less @ nat @ X2 @ Ma ) )
=> ( ( Mi != Ma )
=> ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ( ( vEBT_VEBT_high @ Xn @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= H2 )
=> ( ( Xn
= ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) )
=> ( ( ( vEBT_VEBT_low @ Xn @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= L2 )
=> ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xn @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
=> ( ( Newnode
= ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ H2 ) @ L2 ) )
=> ( ( Newlist
= ( list_update @ vEBT_VEBT @ TreeList @ H2 @ Newnode ) )
=> ( ( ( vEBT_VEBT_minNull @ Newnode )
=> ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X2 )
= ( vEBT_Node
@ ( some @ ( product_prod @ nat @ nat )
@ ( product_Pair @ nat @ nat @ Xn
@ ( if @ nat @ ( Xn = Ma )
@ ( if @ nat
@ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) )
= ( none @ nat ) )
@ Xn
@ ( plus_plus @ nat @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ Newlist @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) ) ) ) ) )
@ Ma ) ) )
@ Deg
@ Newlist
@ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) )
& ( ~ ( vEBT_VEBT_minNull @ Newnode )
=> ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X2 )
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Xn @ ( if @ nat @ ( Xn = Ma ) @ ( plus_plus @ nat @ ( times_times @ nat @ H2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ Newlist @ H2 ) ) ) ) @ Ma ) ) ) @ Deg @ Newlist @ Summary ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% del_x_mi_lets_in
thf(fact_4840_del__x__mi__lets__in__minNull,axiom,
! [X2: nat,Mi: nat,Ma: nat,Deg: nat,Xn: nat,H2: nat,Summary: vEBT_VEBT,TreeList: list @ vEBT_VEBT,L2: nat,Newnode: vEBT_VEBT,Newlist: list @ vEBT_VEBT,Sn: vEBT_VEBT] :
( ( ( X2 = Mi )
& ( ord_less @ nat @ X2 @ Ma ) )
=> ( ( Mi != Ma )
=> ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ( ( vEBT_VEBT_high @ Xn @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= H2 )
=> ( ( Xn
= ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) )
=> ( ( ( vEBT_VEBT_low @ Xn @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= L2 )
=> ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xn @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
=> ( ( Newnode
= ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ H2 ) @ L2 ) )
=> ( ( Newlist
= ( list_update @ vEBT_VEBT @ TreeList @ H2 @ Newnode ) )
=> ( ( vEBT_VEBT_minNull @ Newnode )
=> ( ( Sn
= ( vEBT_vebt_delete @ Summary @ H2 ) )
=> ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X2 )
= ( vEBT_Node
@ ( some @ ( product_prod @ nat @ nat )
@ ( product_Pair @ nat @ nat @ Xn
@ ( if @ nat @ ( Xn = Ma )
@ ( if @ nat
@ ( ( vEBT_vebt_maxt @ Sn )
= ( none @ nat ) )
@ Xn
@ ( plus_plus @ nat @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ Sn ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ Newlist @ ( the2 @ nat @ ( vEBT_vebt_maxt @ Sn ) ) ) ) ) ) )
@ Ma ) ) )
@ Deg
@ Newlist
@ Sn ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% del_x_mi_lets_in_minNull
thf(fact_4841_set__update__subsetI,axiom,
! [A: $tType,Xs2: list @ A,A3: set @ A,X2: A,I: nat] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs2 ) @ A3 )
=> ( ( member @ A @ X2 @ A3 )
=> ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ ( list_update @ A @ Xs2 @ I @ X2 ) ) @ A3 ) ) ) ).
% set_update_subsetI
thf(fact_4842_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Osimps_I7_J,axiom,
! [X2: nat,Mi: nat,Ma: nat,Va: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( ( ( ord_less @ nat @ X2 @ Mi )
| ( ord_less @ nat @ Ma @ X2 ) )
=> ( ( vEBT_V1232361888498592333_e_t_e @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X2 )
= ( one_one @ nat ) ) )
& ( ~ ( ( ord_less @ nat @ X2 @ Mi )
| ( ord_less @ nat @ Ma @ X2 ) )
=> ( ( ( ( X2 = Mi )
& ( X2 = Ma ) )
=> ( ( vEBT_V1232361888498592333_e_t_e @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X2 )
= ( one_one @ nat ) ) )
& ( ~ ( ( X2 = Mi )
& ( X2 = Ma ) )
=> ( ( vEBT_V1232361888498592333_e_t_e @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X2 )
= ( if @ nat @ ( ord_less @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) ) @ ( plus_plus @ nat @ ( vEBT_V1232361888498592333_e_t_e @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( if @ nat @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_V1232361888498592333_e_t_e @ Summary @ ( vEBT_VEBT_high @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( one_one @ nat ) ) ) @ ( one_one @ nat ) ) ) ) ) ) ) ).
% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.simps(7)
thf(fact_4843_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Oelims,axiom,
! [X2: vEBT_VEBT,Xa: nat,Y2: nat] :
( ( ( vEBT_V1232361888498592333_e_t_e @ X2 @ Xa )
= Y2 )
=> ( ( ? [A5: $o,B4: $o] :
( X2
= ( vEBT_Leaf @ A5 @ B4 ) )
=> ( ( Xa
= ( zero_zero @ nat ) )
=> ( Y2
!= ( one_one @ nat ) ) ) )
=> ( ( ? [A5: $o,B4: $o] :
( X2
= ( vEBT_Leaf @ A5 @ B4 ) )
=> ( ( Xa
= ( suc @ ( zero_zero @ nat ) ) )
=> ( Y2
!= ( one_one @ nat ) ) ) )
=> ( ( ? [A5: $o,B4: $o] :
( X2
= ( vEBT_Leaf @ A5 @ B4 ) )
=> ( ? [N2: nat] :
( Xa
= ( suc @ ( suc @ N2 ) ) )
=> ( Y2
!= ( one_one @ nat ) ) ) )
=> ( ( ? [Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( X2
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( Y2
!= ( one_one @ nat ) ) )
=> ( ( ? [Mi2: nat,Ma2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ TreeList2 @ Summary2 ) )
=> ( Y2
!= ( one_one @ nat ) ) )
=> ( ( ? [Mi2: nat,Ma2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( zero_zero @ nat ) ) @ TreeList2 @ Summary2 ) )
=> ( Y2
!= ( one_one @ nat ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary2 ) )
=> ~ ( ( ( ( ord_less @ nat @ Xa @ Mi2 )
| ( ord_less @ nat @ Ma2 @ Xa ) )
=> ( Y2
= ( one_one @ nat ) ) )
& ( ~ ( ( ord_less @ nat @ Xa @ Mi2 )
| ( ord_less @ nat @ Ma2 @ Xa ) )
=> ( ( ( ( Xa = Mi2 )
& ( Xa = Ma2 ) )
=> ( Y2
= ( one_one @ nat ) ) )
& ( ~ ( ( Xa = Mi2 )
& ( Xa = Ma2 ) )
=> ( Y2
= ( if @ nat @ ( ord_less @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) @ ( plus_plus @ nat @ ( vEBT_V1232361888498592333_e_t_e @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( if @ nat @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_V1232361888498592333_e_t_e @ Summary2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( 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'.elims
thf(fact_4844_vebt__pred_Osimps_I7_J,axiom,
! [Ma: nat,X2: nat,Mi: nat,Va: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( ( ord_less @ nat @ Ma @ X2 )
=> ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X2 )
= ( some @ nat @ Ma ) ) )
& ( ~ ( ord_less @ nat @ Ma @ X2 )
=> ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X2 )
= ( if @ ( option @ nat ) @ ( ord_less @ nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
@ ( if @ ( option @ nat )
@ ( ( ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
!= ( none @ nat ) )
& ( vEBT_VEBT_greater @ ( some @ nat @ ( vEBT_VEBT_low @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
@ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( some @ nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( if @ ( option @ nat )
@ ( ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
= ( none @ nat ) )
@ ( if @ ( option @ nat ) @ ( ord_less @ nat @ Mi @ X2 ) @ ( some @ nat @ Mi ) @ ( none @ nat ) )
@ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) )
@ ( none @ nat ) ) ) ) ) ).
% vebt_pred.simps(7)
thf(fact_4845_vebt__pred_Oelims,axiom,
! [X2: vEBT_VEBT,Xa: nat,Y2: option @ nat] :
( ( ( vEBT_vebt_pred @ X2 @ Xa )
= Y2 )
=> ( ( ? [Uu2: $o,Uv2: $o] :
( X2
= ( vEBT_Leaf @ Uu2 @ Uv2 ) )
=> ( ( Xa
= ( zero_zero @ nat ) )
=> ( Y2
!= ( none @ nat ) ) ) )
=> ( ! [A5: $o] :
( ? [Uw2: $o] :
( X2
= ( vEBT_Leaf @ A5 @ Uw2 ) )
=> ( ( Xa
= ( suc @ ( zero_zero @ nat ) ) )
=> ~ ( ( A5
=> ( Y2
= ( some @ nat @ ( zero_zero @ nat ) ) ) )
& ( ~ A5
=> ( Y2
= ( none @ nat ) ) ) ) ) )
=> ( ! [A5: $o,B4: $o] :
( ( X2
= ( vEBT_Leaf @ A5 @ B4 ) )
=> ( ? [Va2: nat] :
( Xa
= ( suc @ ( suc @ Va2 ) ) )
=> ~ ( ( B4
=> ( Y2
= ( some @ nat @ ( one_one @ nat ) ) ) )
& ( ~ B4
=> ( ( A5
=> ( Y2
= ( some @ nat @ ( zero_zero @ nat ) ) ) )
& ( ~ A5
=> ( Y2
= ( none @ nat ) ) ) ) ) ) ) )
=> ( ( ? [Uy2: nat,Uz2: list @ vEBT_VEBT,Va3: vEBT_VEBT] :
( X2
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uy2 @ Uz2 @ Va3 ) )
=> ( Y2
!= ( none @ nat ) ) )
=> ( ( ? [V3: product_prod @ nat @ nat,Vd2: list @ vEBT_VEBT,Ve2: vEBT_VEBT] :
( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Vd2 @ Ve2 ) )
=> ( Y2
!= ( none @ nat ) ) )
=> ( ( ? [V3: product_prod @ nat @ nat,Vh2: list @ vEBT_VEBT,Vi2: vEBT_VEBT] :
( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vh2 @ Vi2 ) )
=> ( Y2
!= ( none @ nat ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary2 ) )
=> ~ ( ( ( ord_less @ nat @ Ma2 @ Xa )
=> ( Y2
= ( some @ nat @ Ma2 ) ) )
& ( ~ ( ord_less @ nat @ Ma2 @ Xa )
=> ( Y2
= ( if @ ( option @ nat ) @ ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
@ ( if @ ( option @ nat )
@ ( ( ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
!= ( none @ nat ) )
& ( vEBT_VEBT_greater @ ( some @ nat @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( 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 @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( if @ ( option @ nat )
@ ( ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
= ( none @ nat ) )
@ ( if @ ( option @ nat ) @ ( ord_less @ nat @ Mi2 @ Xa ) @ ( some @ nat @ Mi2 ) @ ( none @ nat ) )
@ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) )
@ ( none @ nat ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_pred.elims
thf(fact_4846_succ__less__length__list,axiom,
! [Deg: nat,Mi: nat,X2: nat,TreeList: list @ vEBT_VEBT,Ma: nat,Summary: vEBT_VEBT] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ( ord_less_eq @ nat @ Mi @ X2 )
=> ( ( ord_less @ nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
=> ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X2 )
= ( if @ ( option @ nat )
@ ( ( ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
!= ( none @ nat ) )
& ( vEBT_VEBT_less @ ( some @ nat @ ( vEBT_VEBT_low @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
@ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( some @ nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( if @ ( option @ nat )
@ ( ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
= ( none @ nat ) )
@ ( none @ nat )
@ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% succ_less_length_list
thf(fact_4847_succ__greatereq__min,axiom,
! [Deg: nat,Mi: nat,X2: nat,Ma: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ( ord_less_eq @ nat @ Mi @ X2 )
=> ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X2 )
= ( if @ ( option @ nat ) @ ( ord_less @ nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
@ ( if @ ( option @ nat )
@ ( ( ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
!= ( none @ nat ) )
& ( vEBT_VEBT_less @ ( some @ nat @ ( vEBT_VEBT_low @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
@ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( some @ nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( if @ ( option @ nat )
@ ( ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
= ( none @ nat ) )
@ ( none @ nat )
@ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) )
@ ( none @ nat ) ) ) ) ) ).
% succ_greatereq_min
thf(fact_4848_vebt__delete_Oelims,axiom,
! [X2: vEBT_VEBT,Xa: nat,Y2: vEBT_VEBT] :
( ( ( vEBT_vebt_delete @ X2 @ Xa )
= Y2 )
=> ( ! [A5: $o,B4: $o] :
( ( X2
= ( vEBT_Leaf @ A5 @ B4 ) )
=> ( ( Xa
= ( zero_zero @ nat ) )
=> ( Y2
!= ( vEBT_Leaf @ $false @ B4 ) ) ) )
=> ( ! [A5: $o] :
( ? [B4: $o] :
( X2
= ( vEBT_Leaf @ A5 @ B4 ) )
=> ( ( Xa
= ( suc @ ( zero_zero @ nat ) ) )
=> ( Y2
!= ( vEBT_Leaf @ A5 @ $false ) ) ) )
=> ( ! [A5: $o,B4: $o] :
( ( X2
= ( vEBT_Leaf @ A5 @ B4 ) )
=> ( ? [N2: nat] :
( Xa
= ( suc @ ( suc @ N2 ) ) )
=> ( Y2
!= ( vEBT_Leaf @ A5 @ B4 ) ) ) )
=> ( ! [Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( Y2
!= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg2 @ TreeList2 @ Summary2 ) ) )
=> ( ! [Mi2: nat,Ma2: nat,TrLst: list @ vEBT_VEBT,Smry: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ TrLst @ Smry ) )
=> ( Y2
!= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ TrLst @ Smry ) ) )
=> ( ! [Mi2: nat,Ma2: nat,Tr: list @ vEBT_VEBT,Sm: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( zero_zero @ nat ) ) @ Tr @ Sm ) )
=> ( Y2
!= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( zero_zero @ nat ) ) @ Tr @ Sm ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary2 ) )
=> ~ ( ( ( ( ord_less @ nat @ Xa @ Mi2 )
| ( ord_less @ nat @ Ma2 @ Xa ) )
=> ( Y2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary2 ) ) )
& ( ~ ( ( ord_less @ nat @ Xa @ Mi2 )
| ( ord_less @ nat @ Ma2 @ Xa ) )
=> ( ( ( ( Xa = Mi2 )
& ( Xa = Ma2 ) )
=> ( Y2
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary2 ) ) )
& ( ~ ( ( Xa = Mi2 )
& ( Xa = Ma2 ) )
=> ( Y2
= ( if @ vEBT_VEBT @ ( ord_less @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
@ ( if @ vEBT_VEBT @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( vEBT_Node
@ ( some @ ( product_prod @ nat @ nat )
@ ( product_Pair @ nat @ nat @ ( if @ nat @ ( Xa = Mi2 ) @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ Mi2 )
@ ( if @ nat
@ ( ( ( Xa = Mi2 )
=> ( ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) )
= Ma2 ) )
& ( ( Xa != Mi2 )
=> ( Xa = Ma2 ) ) )
@ ( if @ nat
@ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
= ( none @ nat ) )
@ ( if @ nat @ ( Xa = Mi2 ) @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ Mi2 )
@ ( plus_plus @ nat @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ ( list_update @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) )
@ Ma2 ) ) )
@ ( suc @ ( suc @ Va2 ) )
@ ( list_update @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( vEBT_Node
@ ( some @ ( product_prod @ nat @ nat )
@ ( product_Pair @ nat @ nat @ ( if @ nat @ ( Xa = Mi2 ) @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ Mi2 )
@ ( if @ nat
@ ( ( ( Xa = Mi2 )
=> ( ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) )
= Ma2 ) )
& ( ( Xa != Mi2 )
=> ( Xa = Ma2 ) ) )
@ ( plus_plus @ nat @ ( times_times @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ ( list_update @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
@ Ma2 ) ) )
@ ( suc @ ( suc @ Va2 ) )
@ ( list_update @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ Summary2 ) )
@ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_delete.elims
thf(fact_4849_less__shift,axiom,
( ( ord_less @ nat )
= ( ^ [X: nat,Y: nat] : ( vEBT_VEBT_less @ ( some @ nat @ X ) @ ( some @ nat @ Y ) ) ) ) ).
% less_shift
thf(fact_4850_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_4851_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_4852_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_4853_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_4854_vebt__delete_Osimps_I5_J,axiom,
! [Mi: nat,Ma: nat,TrLst2: list @ vEBT_VEBT,Smry2: vEBT_VEBT,X2: nat] :
( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( zero_zero @ nat ) @ TrLst2 @ Smry2 ) @ X2 )
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( zero_zero @ nat ) @ TrLst2 @ Smry2 ) ) ).
% vebt_delete.simps(5)
thf(fact_4855_vebt__delete_Osimps_I6_J,axiom,
! [Mi: nat,Ma: nat,Tr2: list @ vEBT_VEBT,Sm2: vEBT_VEBT,X2: nat] :
( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( zero_zero @ nat ) ) @ Tr2 @ Sm2 ) @ X2 )
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( zero_zero @ nat ) ) @ Tr2 @ Sm2 ) ) ).
% vebt_delete.simps(6)
thf(fact_4856_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Osimps_I6_J,axiom,
! [X2: nat,Mi: nat,Ma: nat,Va: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( ( ord_less @ nat @ X2 @ Mi )
=> ( ( vEBT_T_s_u_c_c2 @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X2 )
= ( one_one @ nat ) ) )
& ( ~ ( ord_less @ nat @ X2 @ Mi )
=> ( ( vEBT_T_s_u_c_c2 @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X2 )
= ( if @ nat @ ( ord_less @ nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
@ ( if @ nat
@ ( ( ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
!= ( none @ nat ) )
& ( vEBT_VEBT_less @ ( some @ nat @ ( vEBT_VEBT_low @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
@ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_T_s_u_c_c2 @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( plus_plus @ nat @ ( vEBT_T_s_u_c_c2 @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( one_one @ nat ) ) )
@ ( one_one @ nat ) ) ) ) ) ).
% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(6)
thf(fact_4857_vebt__succ_Osimps_I6_J,axiom,
! [X2: nat,Mi: nat,Ma: nat,Va: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( ( ord_less @ nat @ X2 @ Mi )
=> ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X2 )
= ( some @ nat @ Mi ) ) )
& ( ~ ( ord_less @ nat @ X2 @ Mi )
=> ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X2 )
= ( if @ ( option @ nat ) @ ( ord_less @ nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
@ ( if @ ( option @ nat )
@ ( ( ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
!= ( none @ nat ) )
& ( vEBT_VEBT_less @ ( some @ nat @ ( vEBT_VEBT_low @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
@ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( some @ nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( if @ ( option @ nat )
@ ( ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
= ( none @ nat ) )
@ ( none @ nat )
@ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) )
@ ( none @ nat ) ) ) ) ) ).
% vebt_succ.simps(6)
thf(fact_4858_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Oelims,axiom,
! [X2: vEBT_VEBT,Xa: nat,Y2: nat] :
( ( ( vEBT_T_s_u_c_c2 @ X2 @ Xa )
= Y2 )
=> ( ( ? [Uu2: $o,B4: $o] :
( X2
= ( vEBT_Leaf @ Uu2 @ B4 ) )
=> ( ( Xa
= ( zero_zero @ nat ) )
=> ( Y2
!= ( one_one @ nat ) ) ) )
=> ( ( ? [Uv2: $o,Uw2: $o] :
( X2
= ( vEBT_Leaf @ Uv2 @ Uw2 ) )
=> ( ? [N2: nat] :
( Xa
= ( suc @ N2 ) )
=> ( Y2
!= ( one_one @ nat ) ) ) )
=> ( ( ? [Ux2: nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( X2
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Ux2 @ Uy2 @ Uz2 ) )
=> ( Y2
!= ( one_one @ nat ) ) )
=> ( ( ? [V3: product_prod @ nat @ nat,Vc: list @ vEBT_VEBT,Vd2: vEBT_VEBT] :
( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Vc @ Vd2 ) )
=> ( Y2
!= ( one_one @ nat ) ) )
=> ( ( ? [V3: product_prod @ nat @ nat,Vg2: list @ vEBT_VEBT,Vh2: vEBT_VEBT] :
( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vg2 @ Vh2 ) )
=> ( Y2
!= ( one_one @ nat ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary2 ) )
=> ~ ( ( ( ord_less @ nat @ Xa @ Mi2 )
=> ( Y2
= ( one_one @ nat ) ) )
& ( ~ ( ord_less @ nat @ Xa @ Mi2 )
=> ( Y2
= ( if @ nat @ ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
@ ( if @ nat
@ ( ( ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
!= ( none @ nat ) )
& ( vEBT_VEBT_less @ ( some @ nat @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
@ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_T_s_u_c_c2 @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( plus_plus @ nat @ ( vEBT_T_s_u_c_c2 @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( 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'.elims
thf(fact_4859_vebt__succ_Oelims,axiom,
! [X2: vEBT_VEBT,Xa: nat,Y2: option @ nat] :
( ( ( vEBT_vebt_succ @ X2 @ Xa )
= Y2 )
=> ( ! [Uu2: $o,B4: $o] :
( ( X2
= ( vEBT_Leaf @ Uu2 @ B4 ) )
=> ( ( Xa
= ( zero_zero @ nat ) )
=> ~ ( ( B4
=> ( Y2
= ( some @ nat @ ( one_one @ nat ) ) ) )
& ( ~ B4
=> ( Y2
= ( none @ nat ) ) ) ) ) )
=> ( ( ? [Uv2: $o,Uw2: $o] :
( X2
= ( vEBT_Leaf @ Uv2 @ Uw2 ) )
=> ( ? [N2: nat] :
( Xa
= ( suc @ N2 ) )
=> ( Y2
!= ( none @ nat ) ) ) )
=> ( ( ? [Ux2: nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( X2
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Ux2 @ Uy2 @ Uz2 ) )
=> ( Y2
!= ( none @ nat ) ) )
=> ( ( ? [V3: product_prod @ nat @ nat,Vc: list @ vEBT_VEBT,Vd2: vEBT_VEBT] :
( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Vc @ Vd2 ) )
=> ( Y2
!= ( none @ nat ) ) )
=> ( ( ? [V3: product_prod @ nat @ nat,Vg2: list @ vEBT_VEBT,Vh2: vEBT_VEBT] :
( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vg2 @ Vh2 ) )
=> ( Y2
!= ( none @ nat ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary2 ) )
=> ~ ( ( ( ord_less @ nat @ Xa @ Mi2 )
=> ( Y2
= ( some @ nat @ Mi2 ) ) )
& ( ~ ( ord_less @ nat @ Xa @ Mi2 )
=> ( Y2
= ( if @ ( option @ nat ) @ ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
@ ( if @ ( option @ nat )
@ ( ( ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
!= ( none @ nat ) )
& ( vEBT_VEBT_less @ ( some @ nat @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( 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 @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( if @ ( option @ nat )
@ ( ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( 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 @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) )
@ ( none @ nat ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_succ.elims
thf(fact_4860_vebt__delete_Osimps_I7_J,axiom,
! [X2: nat,Mi: nat,Ma: nat,Va: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( ( ( ord_less @ nat @ X2 @ Mi )
| ( ord_less @ nat @ Ma @ X2 ) )
=> ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X2 )
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) ) )
& ( ~ ( ( ord_less @ nat @ X2 @ Mi )
| ( ord_less @ nat @ Ma @ X2 ) )
=> ( ( ( ( X2 = Mi )
& ( X2 = Ma ) )
=> ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X2 )
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) ) )
& ( ~ ( ( X2 = Mi )
& ( X2 = Ma ) )
=> ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X2 )
= ( if @ vEBT_VEBT @ ( ord_less @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
@ ( if @ vEBT_VEBT @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( vEBT_Node
@ ( some @ ( product_prod @ nat @ nat )
@ ( product_Pair @ nat @ nat @ ( if @ nat @ ( X2 = Mi ) @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ Mi )
@ ( if @ nat
@ ( ( ( X2 = Mi )
=> ( ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
= Ma ) )
& ( ( X2 != Mi )
=> ( X2 = Ma ) ) )
@ ( if @ nat
@ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
= ( none @ nat ) )
@ ( if @ nat @ ( X2 = Mi ) @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ Mi )
@ ( plus_plus @ nat @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ ( list_update @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) )
@ Ma ) ) )
@ ( suc @ ( suc @ Va ) )
@ ( list_update @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( vEBT_Node
@ ( some @ ( product_prod @ nat @ nat )
@ ( product_Pair @ nat @ nat @ ( if @ nat @ ( X2 = Mi ) @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ Mi )
@ ( if @ nat
@ ( ( ( X2 = Mi )
=> ( ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
= Ma ) )
& ( ( X2 != Mi )
=> ( X2 = Ma ) ) )
@ ( plus_plus @ nat @ ( times_times @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ ( list_update @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
@ Ma ) ) )
@ ( suc @ ( suc @ Va ) )
@ ( list_update @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ Summary ) )
@ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) ) ) ) ) ) ) ).
% vebt_delete.simps(7)
thf(fact_4861_insert__simp__norm,axiom,
! [X2: nat,Deg: nat,TreeList: list @ vEBT_VEBT,Mi: nat,Ma: nat,Summary: vEBT_VEBT] :
( ( ord_less @ nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
=> ( ( ord_less @ nat @ Mi @ X2 )
=> ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ( X2 != Ma )
=> ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X2 )
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ ( ord_max @ nat @ X2 @ Ma ) ) ) @ Deg @ ( list_update @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_insert @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( if @ vEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ Summary ) ) ) ) ) ) ) ).
% insert_simp_norm
thf(fact_4862_insert__simp__excp,axiom,
! [Mi: nat,Deg: nat,TreeList: list @ vEBT_VEBT,X2: nat,Ma: nat,Summary: vEBT_VEBT] :
( ( ord_less @ nat @ ( vEBT_VEBT_high @ Mi @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
=> ( ( ord_less @ nat @ X2 @ Mi )
=> ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ( X2 != Ma )
=> ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X2 )
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ X2 @ ( ord_max @ nat @ Mi @ Ma ) ) ) @ Deg @ ( list_update @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Mi @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_insert @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Mi @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Mi @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( if @ vEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Mi @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary @ ( vEBT_VEBT_high @ Mi @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ Summary ) ) ) ) ) ) ) ).
% insert_simp_excp
thf(fact_4863_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Oelims,axiom,
! [X2: vEBT_VEBT,Xa: nat,Y2: nat] :
( ( ( vEBT_T_i_n_s_e_r_t @ X2 @ Xa )
= Y2 )
=> ( ( ? [A5: $o,B4: $o] :
( X2
= ( vEBT_Leaf @ A5 @ B4 ) )
=> ( Y2
!= ( 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] :
( X2
= ( vEBT_Node @ Info2 @ ( zero_zero @ nat ) @ Ts2 @ S2 ) )
=> ( Y2
!= ( one_one @ nat ) ) )
=> ( ( ? [Info2: option @ ( product_prod @ nat @ nat ),Ts2: list @ vEBT_VEBT,S2: vEBT_VEBT] :
( X2
= ( vEBT_Node @ Info2 @ ( suc @ ( zero_zero @ nat ) ) @ Ts2 @ S2 ) )
=> ( Y2
!= ( one_one @ nat ) ) )
=> ( ( ? [V3: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( X2
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ V3 ) ) @ TreeList2 @ Summary2 ) )
=> ( Y2
!= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary2 ) )
=> ( Y2
!= ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) )
@ ( if @ nat
@ ( ( ord_less @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
& ~ ( ( Xa = Mi2 )
| ( Xa = Ma2 ) ) )
@ ( plus_plus @ nat @ ( plus_plus @ nat @ ( vEBT_T_i_n_s_e_r_t @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_T_m_i_n_N_u_l_l @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) @ ( if @ nat @ ( vEBT_VEBT_minNull @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_T_i_n_s_e_r_t @ Summary2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ 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.elims
thf(fact_4864_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_4865_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_4866_max__nat_Oleft__neutral,axiom,
! [A2: nat] :
( ( ord_max @ nat @ ( zero_zero @ nat ) @ A2 )
= A2 ) ).
% max_nat.left_neutral
thf(fact_4867_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_4868_max__nat_Oright__neutral,axiom,
! [A2: nat] :
( ( ord_max @ nat @ A2 @ ( zero_zero @ nat ) )
= A2 ) ).
% max_nat.right_neutral
thf(fact_4869_max__0L,axiom,
! [N: nat] :
( ( ord_max @ nat @ ( zero_zero @ nat ) @ N )
= N ) ).
% max_0L
thf(fact_4870_max__0R,axiom,
! [N: nat] :
( ( ord_max @ nat @ N @ ( zero_zero @ nat ) )
= N ) ).
% max_0R
thf(fact_4871_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_4872_max__0__1_I3_J,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [X2: num] :
( ( ord_max @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ X2 ) )
= ( numeral_numeral @ A @ X2 ) ) ) ).
% max_0_1(3)
thf(fact_4873_max__0__1_I4_J,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [X2: num] :
( ( ord_max @ A @ ( numeral_numeral @ A @ X2 ) @ ( zero_zero @ A ) )
= ( numeral_numeral @ A @ X2 ) ) ) ).
% max_0_1(4)
thf(fact_4874_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_4875_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_4876_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_4877_max__0__1_I5_J,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [X2: num] :
( ( ord_max @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ X2 ) )
= ( numeral_numeral @ A @ X2 ) ) ) ).
% max_0_1(5)
thf(fact_4878_max__0__1_I6_J,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [X2: num] :
( ( ord_max @ A @ ( numeral_numeral @ A @ X2 ) @ ( one_one @ A ) )
= ( numeral_numeral @ A @ X2 ) ) ) ).
% max_0_1(6)
thf(fact_4879_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_4880_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_4881_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_4882_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_4883_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_4884_max__add__distrib__right,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [X2: A,Y2: A,Z: A] :
( ( plus_plus @ A @ X2 @ ( ord_max @ A @ Y2 @ Z ) )
= ( ord_max @ A @ ( plus_plus @ A @ X2 @ Y2 ) @ ( plus_plus @ A @ X2 @ Z ) ) ) ) ).
% max_add_distrib_right
thf(fact_4885_max__add__distrib__left,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [X2: A,Y2: A,Z: A] :
( ( plus_plus @ A @ ( ord_max @ A @ X2 @ Y2 ) @ Z )
= ( ord_max @ A @ ( plus_plus @ A @ X2 @ Z ) @ ( plus_plus @ A @ Y2 @ Z ) ) ) ) ).
% max_add_distrib_left
thf(fact_4886_nat__add__max__right,axiom,
! [M: nat,N: nat,Q2: nat] :
( ( plus_plus @ nat @ M @ ( ord_max @ nat @ N @ Q2 ) )
= ( ord_max @ nat @ ( plus_plus @ nat @ M @ N ) @ ( plus_plus @ nat @ M @ Q2 ) ) ) ).
% nat_add_max_right
thf(fact_4887_nat__add__max__left,axiom,
! [M: nat,N: nat,Q2: nat] :
( ( plus_plus @ nat @ ( ord_max @ nat @ M @ N ) @ Q2 )
= ( ord_max @ nat @ ( plus_plus @ nat @ M @ Q2 ) @ ( plus_plus @ nat @ N @ Q2 ) ) ) ).
% nat_add_max_left
thf(fact_4888_max__def__raw,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( ord_max @ A )
= ( ^ [A4: A,B3: A] : ( if @ A @ ( ord_less_eq @ A @ A4 @ B3 ) @ B3 @ A4 ) ) ) ) ).
% max_def_raw
thf(fact_4889_of__nat__max,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [X2: nat,Y2: nat] :
( ( semiring_1_of_nat @ A @ ( ord_max @ nat @ X2 @ Y2 ) )
= ( ord_max @ A @ ( semiring_1_of_nat @ A @ X2 ) @ ( semiring_1_of_nat @ A @ Y2 ) ) ) ) ).
% of_nat_max
thf(fact_4890_nat__mult__max__left,axiom,
! [M: nat,N: nat,Q2: nat] :
( ( times_times @ nat @ ( ord_max @ nat @ M @ N ) @ Q2 )
= ( ord_max @ nat @ ( times_times @ nat @ M @ Q2 ) @ ( times_times @ nat @ N @ Q2 ) ) ) ).
% nat_mult_max_left
thf(fact_4891_nat__mult__max__right,axiom,
! [M: nat,N: nat,Q2: nat] :
( ( times_times @ nat @ M @ ( ord_max @ nat @ N @ Q2 ) )
= ( ord_max @ nat @ ( times_times @ nat @ M @ N ) @ ( times_times @ nat @ M @ Q2 ) ) ) ).
% nat_mult_max_right
thf(fact_4892_max__diff__distrib__left,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [X2: A,Y2: A,Z: A] :
( ( minus_minus @ A @ ( ord_max @ A @ X2 @ Y2 ) @ Z )
= ( ord_max @ A @ ( minus_minus @ A @ X2 @ Z ) @ ( minus_minus @ A @ Y2 @ Z ) ) ) ) ).
% max_diff_distrib_left
thf(fact_4893_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_4894_minNull__bound,axiom,
! [T2: vEBT_VEBT] : ( ord_less_eq @ nat @ ( vEBT_T_m_i_n_N_u_l_l @ T2 ) @ ( one_one @ nat ) ) ).
% minNull_bound
thf(fact_4895_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_4896_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_4897_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_4898_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l_Osimps_I5_J,axiom,
! [Uz: product_prod @ nat @ nat,Va: nat,Vb: list @ vEBT_VEBT,Vc2: vEBT_VEBT] :
( ( vEBT_T_m_i_n_N_u_l_l @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ Uz ) @ Va @ Vb @ Vc2 ) )
= ( one_one @ nat ) ) ).
% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.simps(5)
thf(fact_4899_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_4900_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l_Oelims,axiom,
! [X2: vEBT_VEBT,Y2: nat] :
( ( ( vEBT_T_m_i_n_N_u_l_l @ X2 )
= Y2 )
=> ( ( ( X2
= ( vEBT_Leaf @ $false @ $false ) )
=> ( Y2
!= ( one_one @ nat ) ) )
=> ( ( ? [Uv2: $o] :
( X2
= ( vEBT_Leaf @ $true @ Uv2 ) )
=> ( Y2
!= ( one_one @ nat ) ) )
=> ( ( ? [Uu2: $o] :
( X2
= ( vEBT_Leaf @ Uu2 @ $true ) )
=> ( Y2
!= ( one_one @ nat ) ) )
=> ( ( ? [Uw2: nat,Ux2: list @ vEBT_VEBT,Uy2: vEBT_VEBT] :
( X2
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uw2 @ Ux2 @ Uy2 ) )
=> ( Y2
!= ( one_one @ nat ) ) )
=> ~ ( ? [Uz2: product_prod @ nat @ nat,Va3: nat,Vb2: list @ vEBT_VEBT,Vc: vEBT_VEBT] :
( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ Uz2 ) @ Va3 @ Vb2 @ Vc ) )
=> ( Y2
!= ( one_one @ nat ) ) ) ) ) ) ) ) ).
% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.elims
thf(fact_4901_vebt__insert_Osimps_I5_J,axiom,
! [Mi: nat,Ma: nat,Va: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,X2: nat] :
( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X2 )
= ( if @ vEBT_VEBT
@ ( ( ord_less @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ X2 @ Mi ) @ Mi @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
& ~ ( ( X2 = Mi )
| ( X2 = Ma ) ) )
@ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ ( if @ nat @ ( ord_less @ nat @ X2 @ Mi ) @ X2 @ Mi ) @ ( ord_max @ nat @ ( if @ nat @ ( ord_less @ nat @ X2 @ Mi ) @ Mi @ X2 ) @ Ma ) ) ) @ ( suc @ ( suc @ Va ) ) @ ( list_update @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ X2 @ Mi ) @ Mi @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_insert @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ X2 @ Mi ) @ Mi @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( ord_less @ nat @ X2 @ Mi ) @ Mi @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( if @ vEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ X2 @ Mi ) @ Mi @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ X2 @ Mi ) @ Mi @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ Summary ) )
@ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) ) ) ).
% vebt_insert.simps(5)
thf(fact_4902_vebt__insert_Oelims,axiom,
! [X2: vEBT_VEBT,Xa: nat,Y2: vEBT_VEBT] :
( ( ( vEBT_vebt_insert @ X2 @ Xa )
= Y2 )
=> ( ! [A5: $o,B4: $o] :
( ( X2
= ( vEBT_Leaf @ A5 @ B4 ) )
=> ~ ( ( ( Xa
= ( zero_zero @ nat ) )
=> ( Y2
= ( vEBT_Leaf @ $true @ B4 ) ) )
& ( ( Xa
!= ( zero_zero @ nat ) )
=> ( ( ( Xa
= ( one_one @ nat ) )
=> ( Y2
= ( vEBT_Leaf @ A5 @ $true ) ) )
& ( ( Xa
!= ( one_one @ nat ) )
=> ( Y2
= ( vEBT_Leaf @ A5 @ B4 ) ) ) ) ) ) )
=> ( ! [Info2: option @ ( product_prod @ nat @ nat ),Ts2: list @ vEBT_VEBT,S2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ Info2 @ ( zero_zero @ nat ) @ Ts2 @ S2 ) )
=> ( Y2
!= ( vEBT_Node @ Info2 @ ( zero_zero @ nat ) @ Ts2 @ S2 ) ) )
=> ( ! [Info2: option @ ( product_prod @ nat @ nat ),Ts2: list @ vEBT_VEBT,S2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ Info2 @ ( suc @ ( zero_zero @ nat ) ) @ Ts2 @ S2 ) )
=> ( Y2
!= ( vEBT_Node @ Info2 @ ( suc @ ( zero_zero @ nat ) ) @ Ts2 @ S2 ) ) )
=> ( ! [V3: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ V3 ) ) @ TreeList2 @ Summary2 ) )
=> ( Y2
!= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Xa @ Xa ) ) @ ( suc @ ( suc @ V3 ) ) @ TreeList2 @ Summary2 ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary2 ) )
=> ( Y2
!= ( if @ vEBT_VEBT
@ ( ( ord_less @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
& ~ ( ( Xa = Mi2 )
| ( Xa = Ma2 ) ) )
@ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ Xa @ Mi2 ) @ ( ord_max @ nat @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ Ma2 ) ) ) @ ( suc @ ( suc @ Va2 ) ) @ ( list_update @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_insert @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( if @ vEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ Summary2 ) )
@ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary2 ) ) ) ) ) ) ) ) ) ).
% vebt_insert.elims
thf(fact_4903_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Osimps_I5_J,axiom,
! [Mi: nat,Ma: nat,Va: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,X2: nat] :
( ( vEBT_T_i_n_s_e_r_t @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X2 )
= ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) )
@ ( if @ nat
@ ( ( ord_less @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ X2 @ Mi ) @ Mi @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
& ~ ( ( X2 = Mi )
| ( X2 = Ma ) ) )
@ ( plus_plus @ nat @ ( plus_plus @ nat @ ( vEBT_T_i_n_s_e_r_t @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ X2 @ Mi ) @ Mi @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( ord_less @ nat @ X2 @ Mi ) @ Mi @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_T_m_i_n_N_u_l_l @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ X2 @ Mi ) @ Mi @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) @ ( if @ nat @ ( vEBT_VEBT_minNull @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ X2 @ Mi ) @ Mi @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_T_i_n_s_e_r_t @ Summary @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ X2 @ Mi ) @ Mi @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( one_one @ nat ) ) )
@ ( one_one @ nat ) ) ) ) ).
% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.simps(5)
thf(fact_4904_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Oelims,axiom,
! [X2: vEBT_VEBT,Xa: nat,Y2: nat] :
( ( ( vEBT_T_i_n_s_e_r_t2 @ X2 @ Xa )
= Y2 )
=> ( ( ? [A5: $o,B4: $o] :
( X2
= ( vEBT_Leaf @ A5 @ B4 ) )
=> ( Y2
!= ( one_one @ nat ) ) )
=> ( ( ? [Info2: option @ ( product_prod @ nat @ nat ),Ts2: list @ vEBT_VEBT,S2: vEBT_VEBT] :
( X2
= ( vEBT_Node @ Info2 @ ( zero_zero @ nat ) @ Ts2 @ S2 ) )
=> ( Y2
!= ( one_one @ nat ) ) )
=> ( ( ? [Info2: option @ ( product_prod @ nat @ nat ),Ts2: list @ vEBT_VEBT,S2: vEBT_VEBT] :
( X2
= ( vEBT_Node @ Info2 @ ( suc @ ( zero_zero @ nat ) ) @ Ts2 @ S2 ) )
=> ( Y2
!= ( one_one @ nat ) ) )
=> ( ( ? [V3: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( X2
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ V3 ) ) @ TreeList2 @ Summary2 ) )
=> ( Y2
!= ( one_one @ nat ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary2 ) )
=> ( Y2
!= ( if @ nat
@ ( ( ord_less @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
& ~ ( ( Xa = Mi2 )
| ( Xa = Ma2 ) ) )
@ ( plus_plus @ nat @ ( vEBT_T_i_n_s_e_r_t2 @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( if @ nat @ ( vEBT_VEBT_minNull @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_T_i_n_s_e_r_t2 @ Summary2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ 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'.elims
thf(fact_4905_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Osimps_I5_J,axiom,
! [Mi: nat,Ma: nat,Va: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,X2: nat] :
( ( vEBT_T_i_n_s_e_r_t2 @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X2 )
= ( if @ nat
@ ( ( ord_less @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ X2 @ Mi ) @ Mi @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
& ~ ( ( X2 = Mi )
| ( X2 = Ma ) ) )
@ ( plus_plus @ nat @ ( vEBT_T_i_n_s_e_r_t2 @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ X2 @ Mi ) @ Mi @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( ord_less @ nat @ X2 @ Mi ) @ Mi @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( if @ nat @ ( vEBT_VEBT_minNull @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ X2 @ Mi ) @ Mi @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_T_i_n_s_e_r_t2 @ Summary @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ X2 @ Mi ) @ Mi @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( one_one @ nat ) ) )
@ ( one_one @ nat ) ) ) ).
% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.simps(5)
thf(fact_4906_T_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_Oelims,axiom,
! [X2: vEBT_VEBT,Xa: nat,Y2: nat] :
( ( ( vEBT_T_d_e_l_e_t_e @ X2 @ Xa )
= Y2 )
=> ( ( ? [A5: $o,B4: $o] :
( X2
= ( vEBT_Leaf @ A5 @ B4 ) )
=> ( ( Xa
= ( zero_zero @ nat ) )
=> ( Y2
!= ( one_one @ nat ) ) ) )
=> ( ( ? [A5: $o,B4: $o] :
( X2
= ( vEBT_Leaf @ A5 @ B4 ) )
=> ( ( Xa
= ( suc @ ( zero_zero @ nat ) ) )
=> ( Y2
!= ( one_one @ nat ) ) ) )
=> ( ( ? [A5: $o,B4: $o] :
( X2
= ( vEBT_Leaf @ A5 @ B4 ) )
=> ( ? [N2: nat] :
( Xa
= ( suc @ ( suc @ N2 ) ) )
=> ( Y2
!= ( one_one @ nat ) ) ) )
=> ( ( ? [Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( X2
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( Y2
!= ( one_one @ nat ) ) )
=> ( ( ? [Mi2: nat,Ma2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ TreeList2 @ Summary2 ) )
=> ( Y2
!= ( one_one @ nat ) ) )
=> ( ( ? [Mi2: nat,Ma2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( zero_zero @ nat ) ) @ TreeList2 @ Summary2 ) )
=> ( Y2
!= ( one_one @ nat ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary2 ) )
=> ( Y2
!= ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) )
@ ( if @ nat
@ ( ( ord_less @ nat @ Xa @ Mi2 )
| ( ord_less @ nat @ Ma2 @ Xa ) )
@ ( one_one @ nat )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) )
@ ( if @ nat
@ ( ( Xa = Mi2 )
& ( Xa = Ma2 ) )
@ ( numeral_numeral @ nat @ ( bit1 @ one2 ) )
@ ( plus_plus @ nat @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( vEBT_T_m_i_n_t @ Summary2 ) @ ( vEBT_T_m_i_n_t @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) @ ( numeral_numeral @ nat @ ( bit1 @ ( bit1 @ one2 ) ) ) ) @ ( one_one @ nat ) ) ) @ ( one_one @ nat ) )
@ ( if @ nat @ ( ord_less @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
@ ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) @ ( vEBT_T_d_e_l_e_t_e @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( plus_plus @ nat @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_T_m_i_n_N_u_l_l @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) )
@ ( if @ nat @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( 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 @ Summary2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) )
@ ( if @ nat
@ ( ( ( Xa = Mi2 )
=> ( ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) )
= Ma2 ) )
& ( ( Xa != Mi2 )
=> ( Xa = Ma2 ) ) )
@ ( plus_plus @ nat @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_T_m_a_x_t @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( 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 @ Summary2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
= ( none @ nat ) )
@ ( one_one @ nat )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_T_m_a_x_t @ ( nth @ vEBT_VEBT @ ( list_update @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) )
@ ( one_one @ nat ) ) ) )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) )
@ ( if @ nat
@ ( ( ( Xa = Mi2 )
=> ( ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) )
= Ma2 ) )
& ( ( Xa != Mi2 )
=> ( Xa = Ma2 ) ) )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit1 @ one2 ) ) ) @ ( vEBT_T_m_a_x_t @ ( nth @ vEBT_VEBT @ ( list_update @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( 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.elims
thf(fact_4907_max__enat__simps_I3_J,axiom,
! [Q2: extended_enat] :
( ( ord_max @ extended_enat @ ( zero_zero @ extended_enat ) @ Q2 )
= Q2 ) ).
% max_enat_simps(3)
thf(fact_4908_max__enat__simps_I2_J,axiom,
! [Q2: extended_enat] :
( ( ord_max @ extended_enat @ Q2 @ ( zero_zero @ extended_enat ) )
= Q2 ) ).
% max_enat_simps(2)
thf(fact_4909_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_4910_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,X2: nat] :
( ( vEBT_T_i_n_s_e_r_t2 @ ( vEBT_Leaf @ A2 @ B2 ) @ X2 )
= ( one_one @ nat ) ) ).
% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.simps(1)
thf(fact_4911_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_4912_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_4913_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_4914_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,S3: vEBT_VEBT,X2: nat] :
( ( vEBT_T_i_n_s_e_r_t2 @ ( vEBT_Node @ Info @ ( zero_zero @ nat ) @ Ts @ S3 ) @ X2 )
= ( one_one @ nat ) ) ).
% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.simps(2)
thf(fact_4915_maxt__bound,axiom,
! [T2: vEBT_VEBT] : ( ord_less_eq @ nat @ ( vEBT_T_m_a_x_t @ T2 ) @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) ) ).
% maxt_bound
thf(fact_4916_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t_Osimps_I3_J,axiom,
! [Mi: nat,Ma: nat,Ux: nat,Uy: list @ vEBT_VEBT,Uz: vEBT_VEBT] :
( ( vEBT_T_m_a_x_t @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Ux @ Uy @ Uz ) )
= ( one_one @ nat ) ) ).
% T\<^sub>m\<^sub>a\<^sub>x\<^sub>t.simps(3)
thf(fact_4917_mint__bound,axiom,
! [T2: vEBT_VEBT] : ( ord_less_eq @ nat @ ( vEBT_T_m_i_n_t @ T2 ) @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) ) ).
% mint_bound
thf(fact_4918_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_4919_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Osimps_I3_J,axiom,
! [Mi: nat,Ma: nat,Ux: nat,Uy: list @ vEBT_VEBT,Uz: vEBT_VEBT] :
( ( vEBT_T_m_i_n_t @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Ux @ Uy @ Uz ) )
= ( one_one @ nat ) ) ).
% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.simps(3)
thf(fact_4920_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,S3: vEBT_VEBT,X2: nat] :
( ( vEBT_T_i_n_s_e_r_t2 @ ( vEBT_Node @ Info @ ( suc @ ( zero_zero @ nat ) ) @ Ts @ S3 ) @ X2 )
= ( one_one @ nat ) ) ).
% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.simps(3)
thf(fact_4921_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Osimps_I4_J,axiom,
! [V: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,X2: nat] :
( ( vEBT_T_i_n_s_e_r_t2 @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ V ) ) @ TreeList @ Summary ) @ X2 )
= ( one_one @ nat ) ) ).
% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.simps(4)
thf(fact_4922_insersimp_H,axiom,
! [T2: vEBT_VEBT,N: nat,Y2: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ T2 @ X_1 )
=> ( ord_less_eq @ nat @ ( vEBT_T_i_n_s_e_r_t2 @ T2 @ Y2 ) @ ( one_one @ nat ) ) ) ) ).
% insersimp'
thf(fact_4923_insertsimp_H,axiom,
! [T2: vEBT_VEBT,N: nat,L2: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( vEBT_VEBT_minNull @ T2 )
=> ( ord_less_eq @ nat @ ( vEBT_T_i_n_s_e_r_t2 @ T2 @ L2 ) @ ( one_one @ nat ) ) ) ) ).
% insertsimp'
thf(fact_4924_insert_H__bound__height,axiom,
! [T2: vEBT_VEBT,N: nat,X2: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ord_less_eq @ nat @ ( vEBT_T_i_n_s_e_r_t2 @ T2 @ X2 ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_VEBT_height @ T2 ) ) ) ) ).
% insert'_bound_height
thf(fact_4925_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t_Oelims,axiom,
! [X2: vEBT_VEBT,Y2: nat] :
( ( ( vEBT_T_m_a_x_t @ X2 )
= Y2 )
=> ( ! [A5: $o,B4: $o] :
( ( X2
= ( vEBT_Leaf @ A5 @ B4 ) )
=> ( Y2
!= ( plus_plus @ nat @ ( one_one @ nat ) @ ( if @ nat @ B4 @ ( one_one @ nat ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( one_one @ nat ) ) ) ) ) )
=> ( ( ? [Uu2: nat,Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT] :
( X2
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) )
=> ( Y2
!= ( one_one @ nat ) ) )
=> ~ ( ? [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
=> ( Y2
!= ( one_one @ nat ) ) ) ) ) ) ).
% T\<^sub>m\<^sub>a\<^sub>x\<^sub>t.elims
thf(fact_4926_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Oelims,axiom,
! [X2: vEBT_VEBT,Y2: nat] :
( ( ( vEBT_T_m_i_n_t @ X2 )
= Y2 )
=> ( ! [A5: $o] :
( ? [B4: $o] :
( X2
= ( vEBT_Leaf @ A5 @ B4 ) )
=> ( Y2
!= ( plus_plus @ nat @ ( one_one @ nat ) @ ( if @ nat @ A5 @ ( zero_zero @ nat ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( one_one @ nat ) ) ) ) ) )
=> ( ( ? [Uu2: nat,Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT] :
( X2
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) )
=> ( Y2
!= ( one_one @ nat ) ) )
=> ~ ( ? [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
=> ( Y2
!= ( one_one @ nat ) ) ) ) ) ) ).
% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.elims
thf(fact_4927_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I7_J,axiom,
! [Mi: nat,Ma: nat,Va: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,X2: nat] :
( ( vEBT_T_p_r_e_d @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X2 )
= ( plus_plus @ nat @ ( one_one @ nat )
@ ( if @ nat @ ( ord_less @ nat @ Ma @ X2 ) @ ( one_one @ nat )
@ ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) @ ( one_one @ nat ) )
@ ( if @ nat @ ( ord_less @ nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
@ ( plus_plus @ nat @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( vEBT_T_m_i_n_t @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) )
@ ( if @ nat
@ ( ( ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
!= ( none @ nat ) )
& ( vEBT_VEBT_greater @ ( some @ nat @ ( vEBT_VEBT_low @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) @ ( vEBT_T_p_r_e_d @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( plus_plus @ nat @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_T_p_r_e_d @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( one_one @ nat ) )
@ ( if @ nat
@ ( ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
= ( none @ nat ) )
@ ( plus_plus @ nat @ ( one_one @ nat ) @ ( one_one @ nat ) )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) @ ( vEBT_T_m_a_x_t @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) )
@ ( one_one @ nat ) ) ) ) ) ) ).
% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(7)
thf(fact_4928_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Osimps_I6_J,axiom,
! [Mi: nat,Ma: nat,Va: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,X2: nat] :
( ( vEBT_T_s_u_c_c @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X2 )
= ( plus_plus @ nat @ ( one_one @ nat )
@ ( if @ nat @ ( ord_less @ nat @ X2 @ Mi ) @ ( one_one @ nat )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit1 @ ( bit0 @ one2 ) ) ) )
@ ( if @ nat @ ( ord_less @ nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
@ ( plus_plus @ nat @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_T_m_a_x_t @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) )
@ ( if @ nat
@ ( ( ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
!= ( none @ nat ) )
& ( vEBT_VEBT_less @ ( some @ nat @ ( vEBT_VEBT_low @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) @ ( vEBT_T_s_u_c_c @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( plus_plus @ nat @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_T_s_u_c_c @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( one_one @ nat ) )
@ ( if @ nat
@ ( ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
= ( none @ nat ) )
@ ( one_one @ nat )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) @ ( vEBT_T_m_i_n_t @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) )
@ ( one_one @ nat ) ) ) ) ) ) ).
% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.simps(6)
thf(fact_4929_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Oelims,axiom,
! [X2: vEBT_VEBT,Xa: nat,Y2: nat] :
( ( ( vEBT_T_p_r_e_d @ X2 @ Xa )
= Y2 )
=> ( ( ? [Uu2: $o,Uv2: $o] :
( X2
= ( vEBT_Leaf @ Uu2 @ Uv2 ) )
=> ( ( Xa
= ( zero_zero @ nat ) )
=> ( Y2
!= ( one_one @ nat ) ) ) )
=> ( ( ? [A5: $o,Uw2: $o] :
( X2
= ( vEBT_Leaf @ A5 @ Uw2 ) )
=> ( ( Xa
= ( suc @ ( zero_zero @ nat ) ) )
=> ( Y2
!= ( plus_plus @ nat @ ( one_one @ nat ) @ ( one_one @ nat ) ) ) ) )
=> ( ! [A5: $o,B4: $o] :
( ( X2
= ( vEBT_Leaf @ A5 @ B4 ) )
=> ( ? [Va2: nat] :
( Xa
= ( suc @ ( suc @ Va2 ) ) )
=> ( Y2
!= ( plus_plus @ nat @ ( one_one @ nat ) @ ( if @ nat @ B4 @ ( one_one @ nat ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( one_one @ nat ) ) ) ) ) ) )
=> ( ( ? [Uy2: nat,Uz2: list @ vEBT_VEBT,Va3: vEBT_VEBT] :
( X2
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uy2 @ Uz2 @ Va3 ) )
=> ( Y2
!= ( one_one @ nat ) ) )
=> ( ( ? [V3: product_prod @ nat @ nat,Vd2: list @ vEBT_VEBT,Ve2: vEBT_VEBT] :
( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Vd2 @ Ve2 ) )
=> ( Y2
!= ( one_one @ nat ) ) )
=> ( ( ? [V3: product_prod @ nat @ nat,Vh2: list @ vEBT_VEBT,Vi2: vEBT_VEBT] :
( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vh2 @ Vi2 ) )
=> ( Y2
!= ( one_one @ nat ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary2 ) )
=> ( Y2
!= ( 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 @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
@ ( plus_plus @ nat @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( vEBT_T_m_i_n_t @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) )
@ ( if @ nat
@ ( ( ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
!= ( none @ nat ) )
& ( vEBT_VEBT_greater @ ( some @ nat @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) @ ( vEBT_T_p_r_e_d @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( 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 @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( one_one @ nat ) )
@ ( if @ nat
@ ( ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
= ( none @ nat ) )
@ ( plus_plus @ nat @ ( one_one @ nat ) @ ( one_one @ nat ) )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) @ ( vEBT_T_m_a_x_t @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) )
@ ( one_one @ nat ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.elims
thf(fact_4930_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Oelims,axiom,
! [X2: vEBT_VEBT,Xa: nat,Y2: nat] :
( ( ( vEBT_T_s_u_c_c @ X2 @ Xa )
= Y2 )
=> ( ( ? [Uu2: $o,B4: $o] :
( X2
= ( vEBT_Leaf @ Uu2 @ B4 ) )
=> ( ( Xa
= ( zero_zero @ nat ) )
=> ( Y2
!= ( plus_plus @ nat @ ( one_one @ nat ) @ ( one_one @ nat ) ) ) ) )
=> ( ( ? [Uv2: $o,Uw2: $o] :
( X2
= ( vEBT_Leaf @ Uv2 @ Uw2 ) )
=> ( ? [N2: nat] :
( Xa
= ( suc @ N2 ) )
=> ( Y2
!= ( one_one @ nat ) ) ) )
=> ( ( ? [Ux2: nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( X2
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Ux2 @ Uy2 @ Uz2 ) )
=> ( Y2
!= ( one_one @ nat ) ) )
=> ( ( ? [V3: product_prod @ nat @ nat,Vc: list @ vEBT_VEBT,Vd2: vEBT_VEBT] :
( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Vc @ Vd2 ) )
=> ( Y2
!= ( one_one @ nat ) ) )
=> ( ( ? [V3: product_prod @ nat @ nat,Vg2: list @ vEBT_VEBT,Vh2: vEBT_VEBT] :
( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vg2 @ Vh2 ) )
=> ( Y2
!= ( one_one @ nat ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary2 ) )
=> ( Y2
!= ( plus_plus @ nat @ ( one_one @ nat )
@ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ ( one_one @ nat )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit1 @ ( bit0 @ one2 ) ) ) )
@ ( if @ nat @ ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
@ ( plus_plus @ nat @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_T_m_a_x_t @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) )
@ ( if @ nat
@ ( ( ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
!= ( none @ nat ) )
& ( vEBT_VEBT_less @ ( some @ nat @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) @ ( vEBT_T_s_u_c_c @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( 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 @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( one_one @ nat ) )
@ ( if @ nat
@ ( ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
= ( none @ nat ) )
@ ( one_one @ nat )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) @ ( vEBT_T_m_i_n_t @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) )
@ ( one_one @ nat ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.elims
thf(fact_4931_T_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_Osimps_I7_J,axiom,
! [Mi: nat,Ma: nat,Va: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,X2: nat] :
( ( vEBT_T_d_e_l_e_t_e @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X2 )
= ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) )
@ ( if @ nat
@ ( ( ord_less @ nat @ X2 @ Mi )
| ( ord_less @ nat @ Ma @ X2 ) )
@ ( one_one @ nat )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) )
@ ( if @ nat
@ ( ( X2 = Mi )
& ( X2 = Ma ) )
@ ( numeral_numeral @ nat @ ( bit1 @ one2 ) )
@ ( plus_plus @ nat @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( vEBT_T_m_i_n_t @ Summary ) @ ( vEBT_T_m_i_n_t @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) @ ( numeral_numeral @ nat @ ( bit1 @ ( bit1 @ one2 ) ) ) ) @ ( one_one @ nat ) ) ) @ ( one_one @ nat ) )
@ ( if @ nat @ ( ord_less @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
@ ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) @ ( vEBT_T_d_e_l_e_t_e @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( plus_plus @ nat @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_T_m_i_n_N_u_l_l @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) )
@ ( if @ nat @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( plus_plus @ nat @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_T_d_e_l_e_t_e @ Summary @ ( vEBT_VEBT_high @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) )
@ ( if @ nat
@ ( ( ( X2 = Mi )
=> ( ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
= Ma ) )
& ( ( X2 != Mi )
=> ( X2 = Ma ) ) )
@ ( plus_plus @ nat @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_T_m_a_x_t @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) )
@ ( plus_plus @ nat @ ( one_one @ nat )
@ ( if @ nat
@ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
= ( none @ nat ) )
@ ( one_one @ nat )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_T_m_a_x_t @ ( nth @ vEBT_VEBT @ ( list_update @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) )
@ ( one_one @ nat ) ) ) )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) )
@ ( if @ nat
@ ( ( ( X2 = Mi )
=> ( ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
= Ma ) )
& ( ( X2 != Mi )
=> ( X2 = Ma ) ) )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit1 @ one2 ) ) ) @ ( vEBT_T_m_a_x_t @ ( nth @ vEBT_VEBT @ ( list_update @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) )
@ ( one_one @ nat ) ) ) ) ) )
@ ( one_one @ nat ) ) ) ) ) ) ) ) ).
% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(7)
thf(fact_4932_T_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_Opelims,axiom,
! [X2: vEBT_VEBT,Xa: nat,Y2: nat] :
( ( ( vEBT_T_d_e_l_e_t_e @ X2 @ Xa )
= Y2 )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T8441311223069195367_e_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X2 @ Xa ) )
=> ( ! [A5: $o,B4: $o] :
( ( X2
= ( vEBT_Leaf @ A5 @ B4 ) )
=> ( ( Xa
= ( zero_zero @ nat ) )
=> ( ( Y2
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T8441311223069195367_e_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A5 @ B4 ) @ ( zero_zero @ nat ) ) ) ) ) )
=> ( ! [A5: $o,B4: $o] :
( ( X2
= ( vEBT_Leaf @ A5 @ B4 ) )
=> ( ( Xa
= ( suc @ ( zero_zero @ nat ) ) )
=> ( ( Y2
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T8441311223069195367_e_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A5 @ B4 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) )
=> ( ! [A5: $o,B4: $o] :
( ( X2
= ( vEBT_Leaf @ A5 @ B4 ) )
=> ! [N2: nat] :
( ( Xa
= ( suc @ ( suc @ N2 ) ) )
=> ( ( Y2
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T8441311223069195367_e_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A5 @ B4 ) @ ( suc @ ( suc @ N2 ) ) ) ) ) ) )
=> ( ! [Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( ( Y2
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T8441311223069195367_e_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg2 @ TreeList2 @ Summary2 ) @ Xa ) ) ) )
=> ( ! [Mi2: nat,Ma2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ TreeList2 @ Summary2 ) )
=> ( ( Y2
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T8441311223069195367_e_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ TreeList2 @ Summary2 ) @ Xa ) ) ) )
=> ( ! [Mi2: nat,Ma2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( zero_zero @ nat ) ) @ TreeList2 @ Summary2 ) )
=> ( ( Y2
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T8441311223069195367_e_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( zero_zero @ nat ) ) @ TreeList2 @ Summary2 ) @ Xa ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary2 ) )
=> ( ( Y2
= ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) )
@ ( if @ nat
@ ( ( ord_less @ nat @ Xa @ Mi2 )
| ( ord_less @ nat @ Ma2 @ Xa ) )
@ ( one_one @ nat )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) )
@ ( if @ nat
@ ( ( Xa = Mi2 )
& ( Xa = Ma2 ) )
@ ( numeral_numeral @ nat @ ( bit1 @ one2 ) )
@ ( plus_plus @ nat @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( vEBT_T_m_i_n_t @ Summary2 ) @ ( vEBT_T_m_i_n_t @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) @ ( numeral_numeral @ nat @ ( bit1 @ ( bit1 @ one2 ) ) ) ) @ ( one_one @ nat ) ) ) @ ( one_one @ nat ) )
@ ( if @ nat @ ( ord_less @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
@ ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) @ ( vEBT_T_d_e_l_e_t_e @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( plus_plus @ nat @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_T_m_i_n_N_u_l_l @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) )
@ ( if @ nat @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( 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 @ Summary2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) )
@ ( if @ nat
@ ( ( ( Xa = Mi2 )
=> ( ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) )
= Ma2 ) )
& ( ( Xa != Mi2 )
=> ( Xa = Ma2 ) ) )
@ ( plus_plus @ nat @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_T_m_a_x_t @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( 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 @ Summary2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
= ( none @ nat ) )
@ ( one_one @ nat )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_T_m_a_x_t @ ( nth @ vEBT_VEBT @ ( list_update @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) )
@ ( one_one @ nat ) ) ) )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) )
@ ( if @ nat
@ ( ( ( Xa = Mi2 )
=> ( ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) )
= Ma2 ) )
& ( ( Xa != Mi2 )
=> ( Xa = Ma2 ) ) )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit1 @ one2 ) ) ) @ ( vEBT_T_m_a_x_t @ ( nth @ vEBT_VEBT @ ( list_update @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) )
@ ( one_one @ nat ) ) ) ) ) )
@ ( one_one @ nat ) ) ) ) ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T8441311223069195367_e_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ) ) ).
% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.pelims
thf(fact_4933_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Opelims,axiom,
! [X2: vEBT_VEBT,Xa: nat,Y2: nat] :
( ( ( vEBT_T_s_u_c_c @ X2 @ Xa )
= Y2 )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_s_u_c_c_rel2 @ ( product_Pair @ vEBT_VEBT @ nat @ X2 @ Xa ) )
=> ( ! [Uu2: $o,B4: $o] :
( ( X2
= ( vEBT_Leaf @ Uu2 @ B4 ) )
=> ( ( Xa
= ( zero_zero @ nat ) )
=> ( ( Y2
= ( plus_plus @ nat @ ( one_one @ nat ) @ ( one_one @ nat ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_s_u_c_c_rel2 @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uu2 @ B4 ) @ ( zero_zero @ nat ) ) ) ) ) )
=> ( ! [Uv2: $o,Uw2: $o] :
( ( X2
= ( vEBT_Leaf @ Uv2 @ Uw2 ) )
=> ! [N2: nat] :
( ( Xa
= ( suc @ N2 ) )
=> ( ( Y2
= ( 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] :
( ( X2
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Ux2 @ Uy2 @ Uz2 ) )
=> ( ( Y2
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_s_u_c_c_rel2 @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Ux2 @ Uy2 @ Uz2 ) @ Xa ) ) ) )
=> ( ! [V3: product_prod @ nat @ nat,Vc: list @ vEBT_VEBT,Vd2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Vc @ Vd2 ) )
=> ( ( Y2
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_s_u_c_c_rel2 @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Vc @ Vd2 ) @ Xa ) ) ) )
=> ( ! [V3: product_prod @ nat @ nat,Vg2: list @ vEBT_VEBT,Vh2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vg2 @ Vh2 ) )
=> ( ( Y2
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_s_u_c_c_rel2 @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vg2 @ Vh2 ) @ Xa ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary2 ) )
=> ( ( Y2
= ( plus_plus @ nat @ ( one_one @ nat )
@ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ ( one_one @ nat )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit1 @ ( bit0 @ one2 ) ) ) )
@ ( if @ nat @ ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
@ ( plus_plus @ nat @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_T_m_a_x_t @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) )
@ ( if @ nat
@ ( ( ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
!= ( none @ nat ) )
& ( vEBT_VEBT_less @ ( some @ nat @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) @ ( vEBT_T_s_u_c_c @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( 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 @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( one_one @ nat ) )
@ ( if @ nat
@ ( ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
= ( none @ nat ) )
@ ( one_one @ nat )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) @ ( vEBT_T_m_i_n_t @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) )
@ ( one_one @ nat ) ) ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_s_u_c_c_rel2 @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ) ).
% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.pelims
thf(fact_4934_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Opelims,axiom,
! [X2: vEBT_VEBT,Xa: nat,Y2: nat] :
( ( ( vEBT_T_p_r_e_d @ X2 @ Xa )
= Y2 )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_p_r_e_d_rel2 @ ( product_Pair @ vEBT_VEBT @ nat @ X2 @ Xa ) )
=> ( ! [Uu2: $o,Uv2: $o] :
( ( X2
= ( vEBT_Leaf @ Uu2 @ Uv2 ) )
=> ( ( Xa
= ( zero_zero @ nat ) )
=> ( ( Y2
= ( 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 ) ) ) ) ) )
=> ( ! [A5: $o,Uw2: $o] :
( ( X2
= ( vEBT_Leaf @ A5 @ Uw2 ) )
=> ( ( Xa
= ( suc @ ( zero_zero @ nat ) ) )
=> ( ( Y2
= ( 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 @ A5 @ Uw2 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) )
=> ( ! [A5: $o,B4: $o] :
( ( X2
= ( vEBT_Leaf @ A5 @ B4 ) )
=> ! [Va2: nat] :
( ( Xa
= ( suc @ ( suc @ Va2 ) ) )
=> ( ( Y2
= ( plus_plus @ nat @ ( one_one @ nat ) @ ( if @ nat @ B4 @ ( one_one @ nat ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( one_one @ nat ) ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_p_r_e_d_rel2 @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A5 @ B4 ) @ ( suc @ ( suc @ Va2 ) ) ) ) ) ) )
=> ( ! [Uy2: nat,Uz2: list @ vEBT_VEBT,Va3: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uy2 @ Uz2 @ Va3 ) )
=> ( ( Y2
= ( 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 ) ) ) )
=> ( ! [V3: product_prod @ nat @ nat,Vd2: list @ vEBT_VEBT,Ve2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Vd2 @ Ve2 ) )
=> ( ( Y2
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_p_r_e_d_rel2 @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Vd2 @ Ve2 ) @ Xa ) ) ) )
=> ( ! [V3: product_prod @ nat @ nat,Vh2: list @ vEBT_VEBT,Vi2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vh2 @ Vi2 ) )
=> ( ( Y2
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_p_r_e_d_rel2 @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vh2 @ Vi2 ) @ Xa ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary2 ) )
=> ( ( Y2
= ( 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 @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
@ ( plus_plus @ nat @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( vEBT_T_m_i_n_t @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) )
@ ( if @ nat
@ ( ( ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
!= ( none @ nat ) )
& ( vEBT_VEBT_greater @ ( some @ nat @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) @ ( vEBT_T_p_r_e_d @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( 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 @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( one_one @ nat ) )
@ ( if @ nat
@ ( ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
= ( none @ nat ) )
@ ( plus_plus @ nat @ ( one_one @ nat ) @ ( one_one @ nat ) )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) @ ( vEBT_T_m_a_x_t @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) )
@ ( one_one @ nat ) ) ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_p_r_e_d_rel2 @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ) ) ).
% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.pelims
thf(fact_4935_vebt__succ_Opelims,axiom,
! [X2: vEBT_VEBT,Xa: nat,Y2: option @ nat] :
( ( ( vEBT_vebt_succ @ X2 @ Xa )
= Y2 )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_succ_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X2 @ Xa ) )
=> ( ! [Uu2: $o,B4: $o] :
( ( X2
= ( vEBT_Leaf @ Uu2 @ B4 ) )
=> ( ( Xa
= ( zero_zero @ nat ) )
=> ( ( ( B4
=> ( Y2
= ( some @ nat @ ( one_one @ nat ) ) ) )
& ( ~ B4
=> ( Y2
= ( none @ nat ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_succ_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uu2 @ B4 ) @ ( zero_zero @ nat ) ) ) ) ) )
=> ( ! [Uv2: $o,Uw2: $o] :
( ( X2
= ( vEBT_Leaf @ Uv2 @ Uw2 ) )
=> ! [N2: nat] :
( ( Xa
= ( suc @ N2 ) )
=> ( ( Y2
= ( 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] :
( ( X2
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Ux2 @ Uy2 @ Uz2 ) )
=> ( ( Y2
= ( none @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_succ_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Ux2 @ Uy2 @ Uz2 ) @ Xa ) ) ) )
=> ( ! [V3: product_prod @ nat @ nat,Vc: list @ vEBT_VEBT,Vd2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Vc @ Vd2 ) )
=> ( ( Y2
= ( none @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_succ_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Vc @ Vd2 ) @ Xa ) ) ) )
=> ( ! [V3: product_prod @ nat @ nat,Vg2: list @ vEBT_VEBT,Vh2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vg2 @ Vh2 ) )
=> ( ( Y2
= ( none @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_succ_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vg2 @ Vh2 ) @ Xa ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary2 ) )
=> ( ( ( ( ord_less @ nat @ Xa @ Mi2 )
=> ( Y2
= ( some @ nat @ Mi2 ) ) )
& ( ~ ( ord_less @ nat @ Xa @ Mi2 )
=> ( Y2
= ( if @ ( option @ nat ) @ ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
@ ( if @ ( option @ nat )
@ ( ( ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
!= ( none @ nat ) )
& ( vEBT_VEBT_less @ ( some @ nat @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( 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 @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( if @ ( option @ nat )
@ ( ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( 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 @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) )
@ ( none @ nat ) ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_succ_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_succ.pelims
thf(fact_4936_vebt__pred_Opelims,axiom,
! [X2: vEBT_VEBT,Xa: nat,Y2: option @ nat] :
( ( ( vEBT_vebt_pred @ X2 @ Xa )
= Y2 )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_pred_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X2 @ Xa ) )
=> ( ! [Uu2: $o,Uv2: $o] :
( ( X2
= ( vEBT_Leaf @ Uu2 @ Uv2 ) )
=> ( ( Xa
= ( zero_zero @ nat ) )
=> ( ( Y2
= ( none @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_pred_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ ( zero_zero @ nat ) ) ) ) ) )
=> ( ! [A5: $o,Uw2: $o] :
( ( X2
= ( vEBT_Leaf @ A5 @ Uw2 ) )
=> ( ( Xa
= ( suc @ ( zero_zero @ nat ) ) )
=> ( ( ( A5
=> ( Y2
= ( some @ nat @ ( zero_zero @ nat ) ) ) )
& ( ~ A5
=> ( Y2
= ( none @ nat ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_pred_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A5 @ Uw2 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) )
=> ( ! [A5: $o,B4: $o] :
( ( X2
= ( vEBT_Leaf @ A5 @ B4 ) )
=> ! [Va2: nat] :
( ( Xa
= ( suc @ ( suc @ Va2 ) ) )
=> ( ( ( B4
=> ( Y2
= ( some @ nat @ ( one_one @ nat ) ) ) )
& ( ~ B4
=> ( ( A5
=> ( Y2
= ( some @ nat @ ( zero_zero @ nat ) ) ) )
& ( ~ A5
=> ( Y2
= ( none @ nat ) ) ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_pred_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A5 @ B4 ) @ ( suc @ ( suc @ Va2 ) ) ) ) ) ) )
=> ( ! [Uy2: nat,Uz2: list @ vEBT_VEBT,Va3: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uy2 @ Uz2 @ Va3 ) )
=> ( ( Y2
= ( 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 ) ) ) )
=> ( ! [V3: product_prod @ nat @ nat,Vd2: list @ vEBT_VEBT,Ve2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Vd2 @ Ve2 ) )
=> ( ( Y2
= ( none @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_pred_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Vd2 @ Ve2 ) @ Xa ) ) ) )
=> ( ! [V3: product_prod @ nat @ nat,Vh2: list @ vEBT_VEBT,Vi2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vh2 @ Vi2 ) )
=> ( ( Y2
= ( none @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_pred_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vh2 @ Vi2 ) @ Xa ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary2 ) )
=> ( ( ( ( ord_less @ nat @ Ma2 @ Xa )
=> ( Y2
= ( some @ nat @ Ma2 ) ) )
& ( ~ ( ord_less @ nat @ Ma2 @ Xa )
=> ( Y2
= ( if @ ( option @ nat ) @ ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
@ ( if @ ( option @ nat )
@ ( ( ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
!= ( none @ nat ) )
& ( vEBT_VEBT_greater @ ( some @ nat @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( 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 @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( if @ ( option @ nat )
@ ( ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
= ( none @ nat ) )
@ ( if @ ( option @ nat ) @ ( ord_less @ nat @ Mi2 @ Xa ) @ ( some @ nat @ Mi2 ) @ ( none @ nat ) )
@ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) )
@ ( none @ nat ) ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_pred_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_pred.pelims
thf(fact_4937_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Opelims,axiom,
! [X2: vEBT_VEBT,Xa: nat,Y2: nat] :
( ( ( vEBT_V1232361888498592333_e_t_e @ X2 @ Xa )
= Y2 )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V6368547301243506412_e_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X2 @ Xa ) )
=> ( ! [A5: $o,B4: $o] :
( ( X2
= ( vEBT_Leaf @ A5 @ B4 ) )
=> ( ( Xa
= ( zero_zero @ nat ) )
=> ( ( Y2
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V6368547301243506412_e_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A5 @ B4 ) @ ( zero_zero @ nat ) ) ) ) ) )
=> ( ! [A5: $o,B4: $o] :
( ( X2
= ( vEBT_Leaf @ A5 @ B4 ) )
=> ( ( Xa
= ( suc @ ( zero_zero @ nat ) ) )
=> ( ( Y2
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V6368547301243506412_e_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A5 @ B4 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) )
=> ( ! [A5: $o,B4: $o] :
( ( X2
= ( vEBT_Leaf @ A5 @ B4 ) )
=> ! [N2: nat] :
( ( Xa
= ( suc @ ( suc @ N2 ) ) )
=> ( ( Y2
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V6368547301243506412_e_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A5 @ B4 ) @ ( suc @ ( suc @ N2 ) ) ) ) ) ) )
=> ( ! [Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( ( Y2
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V6368547301243506412_e_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg2 @ TreeList2 @ Summary2 ) @ Xa ) ) ) )
=> ( ! [Mi2: nat,Ma2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ TreeList2 @ Summary2 ) )
=> ( ( Y2
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V6368547301243506412_e_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ TreeList2 @ Summary2 ) @ Xa ) ) ) )
=> ( ! [Mi2: nat,Ma2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( zero_zero @ nat ) ) @ TreeList2 @ Summary2 ) )
=> ( ( Y2
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V6368547301243506412_e_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( zero_zero @ nat ) ) @ TreeList2 @ Summary2 ) @ Xa ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary2 ) )
=> ( ( ( ( ( ord_less @ nat @ Xa @ Mi2 )
| ( ord_less @ nat @ Ma2 @ Xa ) )
=> ( Y2
= ( one_one @ nat ) ) )
& ( ~ ( ( ord_less @ nat @ Xa @ Mi2 )
| ( ord_less @ nat @ Ma2 @ Xa ) )
=> ( ( ( ( Xa = Mi2 )
& ( Xa = Ma2 ) )
=> ( Y2
= ( one_one @ nat ) ) )
& ( ~ ( ( Xa = Mi2 )
& ( Xa = Ma2 ) )
=> ( Y2
= ( if @ nat @ ( ord_less @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) @ ( plus_plus @ nat @ ( vEBT_V1232361888498592333_e_t_e @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( if @ nat @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_V1232361888498592333_e_t_e @ Summary2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( one_one @ nat ) ) ) @ ( one_one @ nat ) ) ) ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V6368547301243506412_e_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.pelims
thf(fact_4938_vebt__delete_Opelims,axiom,
! [X2: vEBT_VEBT,Xa: nat,Y2: vEBT_VEBT] :
( ( ( vEBT_vebt_delete @ X2 @ Xa )
= Y2 )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_delete_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X2 @ Xa ) )
=> ( ! [A5: $o,B4: $o] :
( ( X2
= ( vEBT_Leaf @ A5 @ B4 ) )
=> ( ( Xa
= ( zero_zero @ nat ) )
=> ( ( Y2
= ( vEBT_Leaf @ $false @ B4 ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_delete_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A5 @ B4 ) @ ( zero_zero @ nat ) ) ) ) ) )
=> ( ! [A5: $o,B4: $o] :
( ( X2
= ( vEBT_Leaf @ A5 @ B4 ) )
=> ( ( Xa
= ( suc @ ( zero_zero @ nat ) ) )
=> ( ( Y2
= ( vEBT_Leaf @ A5 @ $false ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_delete_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A5 @ B4 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) )
=> ( ! [A5: $o,B4: $o] :
( ( X2
= ( vEBT_Leaf @ A5 @ B4 ) )
=> ! [N2: nat] :
( ( Xa
= ( suc @ ( suc @ N2 ) ) )
=> ( ( Y2
= ( vEBT_Leaf @ A5 @ B4 ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_delete_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A5 @ B4 ) @ ( suc @ ( suc @ N2 ) ) ) ) ) ) )
=> ( ! [Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( ( Y2
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg2 @ TreeList2 @ Summary2 ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_delete_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg2 @ TreeList2 @ Summary2 ) @ Xa ) ) ) )
=> ( ! [Mi2: nat,Ma2: nat,TrLst: list @ vEBT_VEBT,Smry: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ TrLst @ Smry ) )
=> ( ( Y2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ TrLst @ Smry ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_delete_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ TrLst @ Smry ) @ Xa ) ) ) )
=> ( ! [Mi2: nat,Ma2: nat,Tr: list @ vEBT_VEBT,Sm: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( zero_zero @ nat ) ) @ Tr @ Sm ) )
=> ( ( Y2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( zero_zero @ nat ) ) @ Tr @ Sm ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_delete_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( zero_zero @ nat ) ) @ Tr @ Sm ) @ Xa ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary2 ) )
=> ( ( ( ( ( ord_less @ nat @ Xa @ Mi2 )
| ( ord_less @ nat @ Ma2 @ Xa ) )
=> ( Y2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary2 ) ) )
& ( ~ ( ( ord_less @ nat @ Xa @ Mi2 )
| ( ord_less @ nat @ Ma2 @ Xa ) )
=> ( ( ( ( Xa = Mi2 )
& ( Xa = Ma2 ) )
=> ( Y2
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary2 ) ) )
& ( ~ ( ( Xa = Mi2 )
& ( Xa = Ma2 ) )
=> ( Y2
= ( if @ vEBT_VEBT @ ( ord_less @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
@ ( if @ vEBT_VEBT @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( vEBT_Node
@ ( some @ ( product_prod @ nat @ nat )
@ ( product_Pair @ nat @ nat @ ( if @ nat @ ( Xa = Mi2 ) @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ Mi2 )
@ ( if @ nat
@ ( ( ( Xa = Mi2 )
=> ( ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) )
= Ma2 ) )
& ( ( Xa != Mi2 )
=> ( Xa = Ma2 ) ) )
@ ( if @ nat
@ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
= ( none @ nat ) )
@ ( if @ nat @ ( Xa = Mi2 ) @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ Mi2 )
@ ( plus_plus @ nat @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ ( list_update @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) )
@ Ma2 ) ) )
@ ( suc @ ( suc @ Va2 ) )
@ ( list_update @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( vEBT_Node
@ ( some @ ( product_prod @ nat @ nat )
@ ( product_Pair @ nat @ nat @ ( if @ nat @ ( Xa = Mi2 ) @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ Mi2 )
@ ( if @ nat
@ ( ( ( Xa = Mi2 )
=> ( ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) )
= Ma2 ) )
& ( ( Xa != Mi2 )
=> ( Xa = Ma2 ) ) )
@ ( plus_plus @ nat @ ( times_times @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ ( list_update @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
@ Ma2 ) ) )
@ ( suc @ ( suc @ Va2 ) )
@ ( list_update @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ Summary2 ) )
@ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary2 ) ) ) ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_delete_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_delete.pelims
thf(fact_4939_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Opelims,axiom,
! [X2: vEBT_VEBT,Xa: nat,Y2: nat] :
( ( ( vEBT_T_s_u_c_c2 @ X2 @ Xa )
= Y2 )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_s_u_c_c_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X2 @ Xa ) )
=> ( ! [Uu2: $o,B4: $o] :
( ( X2
= ( vEBT_Leaf @ Uu2 @ B4 ) )
=> ( ( Xa
= ( zero_zero @ nat ) )
=> ( ( Y2
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_s_u_c_c_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uu2 @ B4 ) @ ( zero_zero @ nat ) ) ) ) ) )
=> ( ! [Uv2: $o,Uw2: $o] :
( ( X2
= ( vEBT_Leaf @ Uv2 @ Uw2 ) )
=> ! [N2: nat] :
( ( Xa
= ( suc @ N2 ) )
=> ( ( Y2
= ( 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] :
( ( X2
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Ux2 @ Uy2 @ Uz2 ) )
=> ( ( Y2
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_s_u_c_c_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Ux2 @ Uy2 @ Uz2 ) @ Xa ) ) ) )
=> ( ! [V3: product_prod @ nat @ nat,Vc: list @ vEBT_VEBT,Vd2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Vc @ Vd2 ) )
=> ( ( Y2
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_s_u_c_c_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Vc @ Vd2 ) @ Xa ) ) ) )
=> ( ! [V3: product_prod @ nat @ nat,Vg2: list @ vEBT_VEBT,Vh2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vg2 @ Vh2 ) )
=> ( ( Y2
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_s_u_c_c_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vg2 @ Vh2 ) @ Xa ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary2 ) )
=> ( ( ( ( ord_less @ nat @ Xa @ Mi2 )
=> ( Y2
= ( one_one @ nat ) ) )
& ( ~ ( ord_less @ nat @ Xa @ Mi2 )
=> ( Y2
= ( if @ nat @ ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
@ ( if @ nat
@ ( ( ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
!= ( none @ nat ) )
& ( vEBT_VEBT_less @ ( some @ nat @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
@ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_T_s_u_c_c2 @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( plus_plus @ nat @ ( vEBT_T_s_u_c_c2 @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( one_one @ nat ) ) )
@ ( one_one @ nat ) ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_s_u_c_c_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ) ).
% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.pelims
thf(fact_4940_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Opelims,axiom,
! [X2: vEBT_VEBT,Xa: nat,Y2: nat] :
( ( ( vEBT_T_i_n_s_e_r_t @ X2 @ Xa )
= Y2 )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T9217963907923527482_t_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X2 @ Xa ) )
=> ( ! [A5: $o,B4: $o] :
( ( X2
= ( vEBT_Leaf @ A5 @ B4 ) )
=> ( ( Y2
= ( 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 @ A5 @ B4 ) @ Xa ) ) ) )
=> ( ! [Info2: option @ ( product_prod @ nat @ nat ),Ts2: list @ vEBT_VEBT,S2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ Info2 @ ( zero_zero @ nat ) @ Ts2 @ S2 ) )
=> ( ( Y2
= ( 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] :
( ( X2
= ( vEBT_Node @ Info2 @ ( suc @ ( zero_zero @ nat ) ) @ Ts2 @ S2 ) )
=> ( ( Y2
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T9217963907923527482_t_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Info2 @ ( suc @ ( zero_zero @ nat ) ) @ Ts2 @ S2 ) @ Xa ) ) ) )
=> ( ! [V3: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ V3 ) ) @ TreeList2 @ Summary2 ) )
=> ( ( Y2
= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T9217963907923527482_t_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ V3 ) ) @ TreeList2 @ Summary2 ) @ Xa ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary2 ) )
=> ( ( Y2
= ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) )
@ ( if @ nat
@ ( ( ord_less @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
& ~ ( ( Xa = Mi2 )
| ( Xa = Ma2 ) ) )
@ ( plus_plus @ nat @ ( plus_plus @ nat @ ( vEBT_T_i_n_s_e_r_t @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_T_m_i_n_N_u_l_l @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) @ ( if @ nat @ ( vEBT_VEBT_minNull @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_T_i_n_s_e_r_t @ Summary2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( one_one @ nat ) ) )
@ ( one_one @ nat ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T9217963907923527482_t_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ).
% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.pelims
thf(fact_4941_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Opelims,axiom,
! [X2: vEBT_VEBT,Xa: nat,Y2: nat] :
( ( ( vEBT_T_p_r_e_d2 @ X2 @ Xa )
= Y2 )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_p_r_e_d_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X2 @ Xa ) )
=> ( ! [Uu2: $o,Uv2: $o] :
( ( X2
= ( vEBT_Leaf @ Uu2 @ Uv2 ) )
=> ( ( Xa
= ( zero_zero @ nat ) )
=> ( ( Y2
= ( 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 ) ) ) ) ) )
=> ( ! [A5: $o,Uw2: $o] :
( ( X2
= ( vEBT_Leaf @ A5 @ Uw2 ) )
=> ( ( Xa
= ( suc @ ( zero_zero @ nat ) ) )
=> ( ( Y2
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_p_r_e_d_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A5 @ Uw2 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) )
=> ( ! [A5: $o,B4: $o] :
( ( X2
= ( vEBT_Leaf @ A5 @ B4 ) )
=> ! [Va2: nat] :
( ( Xa
= ( suc @ ( suc @ Va2 ) ) )
=> ( ( Y2
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_p_r_e_d_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A5 @ B4 ) @ ( suc @ ( suc @ Va2 ) ) ) ) ) ) )
=> ( ! [Uy2: nat,Uz2: list @ vEBT_VEBT,Va3: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uy2 @ Uz2 @ Va3 ) )
=> ( ( Y2
= ( 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 ) ) ) )
=> ( ! [V3: product_prod @ nat @ nat,Vd2: list @ vEBT_VEBT,Ve2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Vd2 @ Ve2 ) )
=> ( ( Y2
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_p_r_e_d_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Vd2 @ Ve2 ) @ Xa ) ) ) )
=> ( ! [V3: product_prod @ nat @ nat,Vh2: list @ vEBT_VEBT,Vi2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vh2 @ Vi2 ) )
=> ( ( Y2
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_p_r_e_d_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vh2 @ Vi2 ) @ Xa ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary2 ) )
=> ( ( ( ( ord_less @ nat @ Ma2 @ Xa )
=> ( Y2
= ( one_one @ nat ) ) )
& ( ~ ( ord_less @ nat @ Ma2 @ Xa )
=> ( Y2
= ( if @ nat @ ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
@ ( if @ nat
@ ( ( ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
!= ( none @ nat ) )
& ( vEBT_VEBT_greater @ ( some @ nat @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
@ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_T_p_r_e_d2 @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( plus_plus @ nat @ ( vEBT_T_p_r_e_d2 @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( one_one @ nat ) ) )
@ ( one_one @ nat ) ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_p_r_e_d_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ) ) ).
% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.pelims
thf(fact_4942_vebt__insert_Opelims,axiom,
! [X2: vEBT_VEBT,Xa: nat,Y2: vEBT_VEBT] :
( ( ( vEBT_vebt_insert @ X2 @ Xa )
= Y2 )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_insert_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X2 @ Xa ) )
=> ( ! [A5: $o,B4: $o] :
( ( X2
= ( vEBT_Leaf @ A5 @ B4 ) )
=> ( ( ( ( Xa
= ( zero_zero @ nat ) )
=> ( Y2
= ( vEBT_Leaf @ $true @ B4 ) ) )
& ( ( Xa
!= ( zero_zero @ nat ) )
=> ( ( ( Xa
= ( one_one @ nat ) )
=> ( Y2
= ( vEBT_Leaf @ A5 @ $true ) ) )
& ( ( Xa
!= ( one_one @ nat ) )
=> ( Y2
= ( vEBT_Leaf @ A5 @ B4 ) ) ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_insert_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A5 @ B4 ) @ Xa ) ) ) )
=> ( ! [Info2: option @ ( product_prod @ nat @ nat ),Ts2: list @ vEBT_VEBT,S2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ Info2 @ ( zero_zero @ nat ) @ Ts2 @ S2 ) )
=> ( ( Y2
= ( 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] :
( ( X2
= ( vEBT_Node @ Info2 @ ( suc @ ( zero_zero @ nat ) ) @ Ts2 @ S2 ) )
=> ( ( Y2
= ( vEBT_Node @ Info2 @ ( suc @ ( zero_zero @ nat ) ) @ Ts2 @ S2 ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_insert_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Info2 @ ( suc @ ( zero_zero @ nat ) ) @ Ts2 @ S2 ) @ Xa ) ) ) )
=> ( ! [V3: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ V3 ) ) @ TreeList2 @ Summary2 ) )
=> ( ( Y2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Xa @ Xa ) ) @ ( suc @ ( suc @ V3 ) ) @ TreeList2 @ Summary2 ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_insert_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ V3 ) ) @ TreeList2 @ Summary2 ) @ Xa ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary2 ) )
=> ( ( Y2
= ( if @ vEBT_VEBT
@ ( ( ord_less @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
& ~ ( ( Xa = Mi2 )
| ( Xa = Ma2 ) ) )
@ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ Xa @ Mi2 ) @ ( ord_max @ nat @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ Ma2 ) ) ) @ ( suc @ ( suc @ Va2 ) ) @ ( list_update @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_insert @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( if @ vEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ Summary2 ) )
@ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary2 ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_insert_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ).
% vebt_insert.pelims
thf(fact_4943_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Opelims,axiom,
! [X2: vEBT_VEBT,Xa: nat,Y2: nat] :
( ( ( vEBT_T_i_n_s_e_r_t2 @ X2 @ Xa )
= Y2 )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T5076183648494686801_t_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X2 @ Xa ) )
=> ( ! [A5: $o,B4: $o] :
( ( X2
= ( vEBT_Leaf @ A5 @ B4 ) )
=> ( ( Y2
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T5076183648494686801_t_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A5 @ B4 ) @ Xa ) ) ) )
=> ( ! [Info2: option @ ( product_prod @ nat @ nat ),Ts2: list @ vEBT_VEBT,S2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ Info2 @ ( zero_zero @ nat ) @ Ts2 @ S2 ) )
=> ( ( Y2
= ( 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] :
( ( X2
= ( vEBT_Node @ Info2 @ ( suc @ ( zero_zero @ nat ) ) @ Ts2 @ S2 ) )
=> ( ( Y2
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T5076183648494686801_t_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Info2 @ ( suc @ ( zero_zero @ nat ) ) @ Ts2 @ S2 ) @ Xa ) ) ) )
=> ( ! [V3: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ V3 ) ) @ TreeList2 @ Summary2 ) )
=> ( ( Y2
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T5076183648494686801_t_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ V3 ) ) @ TreeList2 @ Summary2 ) @ Xa ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary2 ) )
=> ( ( Y2
= ( if @ nat
@ ( ( ord_less @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
& ~ ( ( Xa = Mi2 )
| ( Xa = Ma2 ) ) )
@ ( plus_plus @ nat @ ( vEBT_T_i_n_s_e_r_t2 @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( if @ nat @ ( vEBT_VEBT_minNull @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_T_i_n_s_e_r_t2 @ Summary2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( one_one @ nat ) ) )
@ ( one_one @ nat ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T5076183648494686801_t_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ).
% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.pelims
thf(fact_4944_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Opelims,axiom,
! [X2: vEBT_VEBT,Xa: nat,Y2: nat] :
( ( ( vEBT_T_m_e_m_b_e_r @ X2 @ Xa )
= Y2 )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T5837161174952499735_r_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X2 @ Xa ) )
=> ( ! [A5: $o,B4: $o] :
( ( X2
= ( vEBT_Leaf @ A5 @ B4 ) )
=> ( ( Y2
= ( 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 @ A5 @ B4 ) @ Xa ) ) ) )
=> ( ! [Uu2: nat,Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) )
=> ( ( Y2
= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T5837161174952499735_r_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) @ Xa ) ) ) )
=> ( ! [V3: product_prod @ nat @ nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Uy2 @ Uz2 ) )
=> ( ( Y2
= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T5837161174952499735_r_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Uy2 @ Uz2 ) @ Xa ) ) ) )
=> ( ! [V3: product_prod @ nat @ nat,Vb2: list @ vEBT_VEBT,Vc: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb2 @ Vc ) )
=> ( ( Y2
= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T5837161174952499735_r_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb2 @ Vc ) @ Xa ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary2 ) )
=> ( ( Y2
= ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( if @ nat @ ( Xa = Mi2 ) @ ( one_one @ nat ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( if @ nat @ ( Xa = Ma2 ) @ ( one_one @ nat ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ ( one_one @ nat ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( if @ nat @ ( ord_less @ nat @ Ma2 @ Xa ) @ ( one_one @ nat ) @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) @ ( if @ nat @ ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_T_m_e_m_b_e_r @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( one_one @ nat ) ) ) ) ) ) ) ) ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T5837161174952499735_r_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ).
% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.pelims
thf(fact_4945_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Opelims,axiom,
! [X2: vEBT_VEBT,Xa: nat,Y2: nat] :
( ( ( vEBT_T_m_e_m_b_e_r2 @ X2 @ Xa )
= Y2 )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T8099345112685741742_r_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X2 @ Xa ) )
=> ( ! [A5: $o,B4: $o] :
( ( X2
= ( vEBT_Leaf @ A5 @ B4 ) )
=> ( ( Y2
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T8099345112685741742_r_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A5 @ B4 ) @ Xa ) ) ) )
=> ( ! [Uu2: nat,Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) )
=> ( ( Y2
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T8099345112685741742_r_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) @ Xa ) ) ) )
=> ( ! [V3: product_prod @ nat @ nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Uy2 @ Uz2 ) )
=> ( ( Y2
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T8099345112685741742_r_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Uy2 @ Uz2 ) @ Xa ) ) ) )
=> ( ! [V3: product_prod @ nat @ nat,Vb2: list @ vEBT_VEBT,Vc: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb2 @ Vc ) )
=> ( ( Y2
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T8099345112685741742_r_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb2 @ Vc ) @ Xa ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary2 ) )
=> ( ( Y2
= ( plus_plus @ nat @ ( one_one @ nat )
@ ( if @ nat @ ( Xa = Mi2 ) @ ( zero_zero @ nat )
@ ( if @ nat @ ( Xa = Ma2 ) @ ( zero_zero @ nat )
@ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ ( zero_zero @ nat )
@ ( if @ nat @ ( ord_less @ nat @ Ma2 @ Xa ) @ ( zero_zero @ nat )
@ ( if @ nat
@ ( ( ord_less @ nat @ Mi2 @ Xa )
& ( ord_less @ nat @ Xa @ Ma2 ) )
@ ( if @ nat @ ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) @ ( vEBT_T_m_e_m_b_e_r2 @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( zero_zero @ nat ) )
@ ( zero_zero @ nat ) ) ) ) ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T8099345112685741742_r_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ).
% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.pelims
thf(fact_4946_vebt__member_Opelims_I3_J,axiom,
! [X2: vEBT_VEBT,Xa: nat] :
( ~ ( vEBT_vebt_member @ X2 @ Xa )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X2 @ Xa ) )
=> ( ! [A5: $o,B4: $o] :
( ( X2
= ( vEBT_Leaf @ A5 @ B4 ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A5 @ B4 ) @ Xa ) )
=> ( ( ( Xa
= ( zero_zero @ nat ) )
=> A5 )
& ( ( Xa
!= ( zero_zero @ nat ) )
=> ( ( ( Xa
= ( one_one @ nat ) )
=> B4 )
& ( Xa
= ( one_one @ nat ) ) ) ) ) ) )
=> ( ! [Uu2: nat,Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) @ Xa ) ) )
=> ( ! [V3: product_prod @ nat @ nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Uy2 @ Uz2 ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Uy2 @ Uz2 ) @ Xa ) ) )
=> ( ! [V3: product_prod @ nat @ nat,Vb2: list @ vEBT_VEBT,Vc: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( 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 ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb2 @ Vc ) @ Xa ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary2 ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary2 ) @ Xa ) )
=> ( ( Xa != Mi2 )
=> ( ( Xa != Ma2 )
=> ( ~ ( ord_less @ nat @ Xa @ Mi2 )
& ( ~ ( ord_less @ nat @ Xa @ Mi2 )
=> ( ~ ( ord_less @ nat @ Ma2 @ Xa )
& ( ~ ( ord_less @ nat @ Ma2 @ Xa )
=> ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_member.pelims(3)
thf(fact_4947_vebt__member_Opelims_I2_J,axiom,
! [X2: vEBT_VEBT,Xa: nat] :
( ( vEBT_vebt_member @ X2 @ Xa )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X2 @ Xa ) )
=> ( ! [A5: $o,B4: $o] :
( ( X2
= ( vEBT_Leaf @ A5 @ B4 ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A5 @ B4 ) @ Xa ) )
=> ~ ( ( ( Xa
= ( zero_zero @ nat ) )
=> A5 )
& ( ( Xa
!= ( zero_zero @ nat ) )
=> ( ( ( Xa
= ( one_one @ nat ) )
=> B4 )
& ( Xa
= ( one_one @ nat ) ) ) ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary2 ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary2 ) @ Xa ) )
=> ~ ( ( Xa != Mi2 )
=> ( ( Xa != Ma2 )
=> ( ~ ( ord_less @ nat @ Xa @ Mi2 )
& ( ~ ( ord_less @ nat @ Xa @ Mi2 )
=> ( ~ ( ord_less @ nat @ Ma2 @ Xa )
& ( ~ ( ord_less @ nat @ Ma2 @ Xa )
=> ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_member.pelims(2)
thf(fact_4948_vebt__member_Opelims_I1_J,axiom,
! [X2: vEBT_VEBT,Xa: nat,Y2: $o] :
( ( ( vEBT_vebt_member @ X2 @ Xa )
= Y2 )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X2 @ Xa ) )
=> ( ! [A5: $o,B4: $o] :
( ( X2
= ( vEBT_Leaf @ A5 @ B4 ) )
=> ( ( Y2
= ( ( ( Xa
= ( zero_zero @ nat ) )
=> A5 )
& ( ( Xa
!= ( zero_zero @ nat ) )
=> ( ( ( Xa
= ( one_one @ nat ) )
=> B4 )
& ( Xa
= ( one_one @ nat ) ) ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A5 @ B4 ) @ Xa ) ) ) )
=> ( ! [Uu2: nat,Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) )
=> ( ~ Y2
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) @ Xa ) ) ) )
=> ( ! [V3: product_prod @ nat @ nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Uy2 @ Uz2 ) )
=> ( ~ Y2
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Uy2 @ Uz2 ) @ Xa ) ) ) )
=> ( ! [V3: product_prod @ nat @ nat,Vb2: list @ vEBT_VEBT,Vc: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb2 @ Vc ) )
=> ( ~ Y2
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb2 @ Vc ) @ Xa ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary2 ) )
=> ( ( Y2
= ( ( Xa != Mi2 )
=> ( ( Xa != Ma2 )
=> ( ~ ( ord_less @ nat @ Xa @ Mi2 )
& ( ~ ( ord_less @ nat @ Xa @ Mi2 )
=> ( ~ ( ord_less @ nat @ Ma2 @ Xa )
& ( ~ ( ord_less @ nat @ Ma2 @ Xa )
=> ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ).
% vebt_member.pelims(1)
thf(fact_4949_VEBT__internal_Onaive__member_Opelims_I1_J,axiom,
! [X2: vEBT_VEBT,Xa: nat,Y2: $o] :
( ( ( vEBT_V5719532721284313246member @ X2 @ Xa )
= Y2 )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X2 @ Xa ) )
=> ( ! [A5: $o,B4: $o] :
( ( X2
= ( vEBT_Leaf @ A5 @ B4 ) )
=> ( ( Y2
= ( ( ( Xa
= ( zero_zero @ nat ) )
=> A5 )
& ( ( Xa
!= ( zero_zero @ nat ) )
=> ( ( ( Xa
= ( one_one @ nat ) )
=> B4 )
& ( Xa
= ( one_one @ nat ) ) ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A5 @ B4 ) @ Xa ) ) ) )
=> ( ! [Uu2: option @ ( product_prod @ nat @ nat ),Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ Uu2 @ ( zero_zero @ nat ) @ Uv2 @ Uw2 ) )
=> ( ~ Y2
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Uu2 @ ( zero_zero @ nat ) @ Uv2 @ Uw2 ) @ Xa ) ) ) )
=> ~ ! [Uy2: option @ ( product_prod @ nat @ nat ),V3: nat,TreeList2: list @ vEBT_VEBT,S2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList2 @ S2 ) )
=> ( ( Y2
= ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_V5719532721284313246member @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList2 @ S2 ) @ Xa ) ) ) ) ) ) ) ) ).
% VEBT_internal.naive_member.pelims(1)
thf(fact_4950_VEBT__internal_Onaive__member_Opelims_I2_J,axiom,
! [X2: vEBT_VEBT,Xa: nat] :
( ( vEBT_V5719532721284313246member @ X2 @ Xa )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X2 @ Xa ) )
=> ( ! [A5: $o,B4: $o] :
( ( X2
= ( vEBT_Leaf @ A5 @ B4 ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A5 @ B4 ) @ Xa ) )
=> ~ ( ( ( Xa
= ( zero_zero @ nat ) )
=> A5 )
& ( ( Xa
!= ( zero_zero @ nat ) )
=> ( ( ( Xa
= ( one_one @ nat ) )
=> B4 )
& ( Xa
= ( one_one @ nat ) ) ) ) ) ) )
=> ~ ! [Uy2: option @ ( product_prod @ nat @ nat ),V3: nat,TreeList2: list @ vEBT_VEBT,S2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList2 @ S2 ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList2 @ S2 ) @ Xa ) )
=> ~ ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_V5719532721284313246member @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ) ) ).
% VEBT_internal.naive_member.pelims(2)
thf(fact_4951_VEBT__internal_Onaive__member_Opelims_I3_J,axiom,
! [X2: vEBT_VEBT,Xa: nat] :
( ~ ( vEBT_V5719532721284313246member @ X2 @ Xa )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X2 @ Xa ) )
=> ( ! [A5: $o,B4: $o] :
( ( X2
= ( vEBT_Leaf @ A5 @ B4 ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A5 @ B4 ) @ Xa ) )
=> ( ( ( Xa
= ( zero_zero @ nat ) )
=> A5 )
& ( ( Xa
!= ( zero_zero @ nat ) )
=> ( ( ( Xa
= ( one_one @ nat ) )
=> B4 )
& ( Xa
= ( one_one @ nat ) ) ) ) ) ) )
=> ( ! [Uu2: option @ ( product_prod @ nat @ nat ),Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ Uu2 @ ( zero_zero @ nat ) @ Uv2 @ Uw2 ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Uu2 @ ( zero_zero @ nat ) @ Uv2 @ Uw2 ) @ Xa ) ) )
=> ~ ! [Uy2: option @ ( product_prod @ nat @ nat ),V3: nat,TreeList2: list @ vEBT_VEBT,S2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList2 @ S2 ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList2 @ S2 ) @ Xa ) )
=> ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_V5719532721284313246member @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.naive_member.pelims(3)
thf(fact_4952_VEBT__internal_Omembermima_Opelims_I3_J,axiom,
! [X2: vEBT_VEBT,Xa: nat] :
( ~ ( vEBT_VEBT_membermima @ X2 @ Xa )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X2 @ Xa ) )
=> ( ! [Uu2: $o,Uv2: $o] :
( ( X2
= ( vEBT_Leaf @ Uu2 @ Uv2 ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa ) ) )
=> ( ! [Ux2: list @ vEBT_VEBT,Uy2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( zero_zero @ nat ) @ Ux2 @ Uy2 ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( zero_zero @ nat ) @ Ux2 @ Uy2 ) @ Xa ) ) )
=> ( ! [Mi2: nat,Ma2: nat,Va3: list @ vEBT_VEBT,Vb2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ Va3 @ Vb2 ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ Va3 @ Vb2 ) @ Xa ) )
=> ( ( Xa = Mi2 )
| ( Xa = Ma2 ) ) ) )
=> ( ! [Mi2: nat,Ma2: nat,V3: nat,TreeList2: list @ vEBT_VEBT,Vc: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ V3 ) @ 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 @ Mi2 @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList2 @ Vc ) @ Xa ) )
=> ( ( Xa = Mi2 )
| ( Xa = Ma2 )
| ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) )
=> ~ ! [V3: nat,TreeList2: list @ vEBT_VEBT,Vd2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V3 ) @ TreeList2 @ Vd2 ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V3 ) @ TreeList2 @ Vd2 ) @ Xa ) )
=> ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.membermima.pelims(3)
thf(fact_4953_VEBT__internal_Omembermima_Opelims_I1_J,axiom,
! [X2: vEBT_VEBT,Xa: nat,Y2: $o] :
( ( ( vEBT_VEBT_membermima @ X2 @ Xa )
= Y2 )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X2 @ Xa ) )
=> ( ! [Uu2: $o,Uv2: $o] :
( ( X2
= ( vEBT_Leaf @ Uu2 @ Uv2 ) )
=> ( ~ Y2
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa ) ) ) )
=> ( ! [Ux2: list @ vEBT_VEBT,Uy2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( zero_zero @ nat ) @ Ux2 @ Uy2 ) )
=> ( ~ Y2
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( zero_zero @ nat ) @ Ux2 @ Uy2 ) @ Xa ) ) ) )
=> ( ! [Mi2: nat,Ma2: nat,Va3: list @ vEBT_VEBT,Vb2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ Va3 @ Vb2 ) )
=> ( ( Y2
= ( ( Xa = Mi2 )
| ( Xa = Ma2 ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ Va3 @ Vb2 ) @ Xa ) ) ) )
=> ( ! [Mi2: nat,Ma2: nat,V3: nat,TreeList2: list @ vEBT_VEBT,Vc: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList2 @ Vc ) )
=> ( ( Y2
= ( ( Xa = Mi2 )
| ( Xa = Ma2 )
| ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList2 @ Vc ) @ Xa ) ) ) )
=> ~ ! [V3: nat,TreeList2: list @ vEBT_VEBT,Vd2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V3 ) @ TreeList2 @ Vd2 ) )
=> ( ( Y2
= ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V3 ) @ TreeList2 @ Vd2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.membermima.pelims(1)
thf(fact_4954_VEBT__internal_Omembermima_Opelims_I2_J,axiom,
! [X2: vEBT_VEBT,Xa: nat] :
( ( vEBT_VEBT_membermima @ X2 @ Xa )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X2 @ Xa ) )
=> ( ! [Mi2: nat,Ma2: nat,Va3: list @ vEBT_VEBT,Vb2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ Va3 @ Vb2 ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ Va3 @ Vb2 ) @ Xa ) )
=> ~ ( ( Xa = Mi2 )
| ( Xa = Ma2 ) ) ) )
=> ( ! [Mi2: nat,Ma2: nat,V3: nat,TreeList2: list @ vEBT_VEBT,Vc: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ V3 ) @ 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 @ Mi2 @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList2 @ Vc ) @ Xa ) )
=> ~ ( ( Xa = Mi2 )
| ( Xa = Ma2 )
| ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) )
=> ~ ! [V3: nat,TreeList2: list @ vEBT_VEBT,Vd2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V3 ) @ TreeList2 @ Vd2 ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V3 ) @ TreeList2 @ Vd2 ) @ Xa ) )
=> ~ ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.membermima.pelims(2)
thf(fact_4955_succ__empty,axiom,
! [T2: vEBT_VEBT,N: nat,X2: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( ( vEBT_vebt_succ @ T2 @ X2 )
= ( none @ nat ) )
= ( ( collect @ nat
@ ^ [Y: nat] :
( ( vEBT_vebt_member @ T2 @ Y )
& ( ord_less @ nat @ X2 @ Y ) ) )
= ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% succ_empty
thf(fact_4956_pred__empty,axiom,
! [T2: vEBT_VEBT,N: nat,X2: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( ( vEBT_vebt_pred @ T2 @ X2 )
= ( none @ nat ) )
= ( ( collect @ nat
@ ^ [Y: nat] :
( ( vEBT_vebt_member @ T2 @ Y )
& ( ord_less @ nat @ Y @ X2 ) ) )
= ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% pred_empty
thf(fact_4957_max_Oabsorb1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,A2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( ord_max @ A @ A2 @ B2 )
= A2 ) ) ) ).
% max.absorb1
thf(fact_4958_buildup__gives__empty,axiom,
! [N: nat] :
( ( vEBT_VEBT_set_vebt @ ( vEBT_vebt_buildup @ N ) )
= ( bot_bot @ ( set @ nat ) ) ) ).
% buildup_gives_empty
thf(fact_4959_mint__corr__help__empty,axiom,
! [T2: vEBT_VEBT,N: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( ( vEBT_vebt_mint @ T2 )
= ( none @ nat ) )
=> ( ( vEBT_VEBT_set_vebt @ T2 )
= ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% mint_corr_help_empty
thf(fact_4960_maxt__corr__help__empty,axiom,
! [T2: vEBT_VEBT,N: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( ( vEBT_vebt_maxt @ T2 )
= ( none @ nat ) )
=> ( ( vEBT_VEBT_set_vebt @ T2 )
= ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% maxt_corr_help_empty
thf(fact_4961_empty__subsetI,axiom,
! [A: $tType,A3: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A3 ) ).
% empty_subsetI
thf(fact_4962_subset__empty,axiom,
! [A: $tType,A3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ ( bot_bot @ ( set @ A ) ) )
= ( A3
= ( bot_bot @ ( set @ A ) ) ) ) ).
% subset_empty
thf(fact_4963_max_Obounded__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,C2: A,A2: A] :
( ( ord_less_eq @ A @ ( ord_max @ A @ B2 @ C2 ) @ A2 )
= ( ( ord_less_eq @ A @ B2 @ A2 )
& ( ord_less_eq @ A @ C2 @ A2 ) ) ) ) ).
% max.bounded_iff
thf(fact_4964_max_Oabsorb2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_max @ A @ A2 @ B2 )
= B2 ) ) ) ).
% max.absorb2
thf(fact_4965_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_4966_atLeastatMost__empty__iff2,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A2: A,B2: A] :
( ( ( bot_bot @ ( set @ A ) )
= ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
= ( ~ ( ord_less_eq @ A @ A2 @ B2 ) ) ) ) ).
% atLeastatMost_empty_iff2
thf(fact_4967_atLeastatMost__empty__iff,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A2: A,B2: A] :
( ( ( set_or1337092689740270186AtMost @ A @ A2 @ B2 )
= ( bot_bot @ ( set @ A ) ) )
= ( ~ ( ord_less_eq @ A @ A2 @ B2 ) ) ) ) ).
% atLeastatMost_empty_iff
thf(fact_4968_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_4969_Diff__eq__empty__iff,axiom,
! [A: $tType,A3: set @ A,B5: set @ A] :
( ( ( minus_minus @ ( set @ A ) @ A3 @ B5 )
= ( bot_bot @ ( set @ A ) ) )
= ( ord_less_eq @ ( set @ A ) @ A3 @ B5 ) ) ).
% Diff_eq_empty_iff
thf(fact_4970_lessThan__0,axiom,
( ( set_ord_lessThan @ nat @ ( zero_zero @ nat ) )
= ( bot_bot @ ( set @ nat ) ) ) ).
% lessThan_0
thf(fact_4971_set__decode__zero,axiom,
( ( nat_set_decode @ ( zero_zero @ nat ) )
= ( bot_bot @ ( set @ nat ) ) ) ).
% set_decode_zero
thf(fact_4972_set__encode__empty,axiom,
( ( nat_set_encode @ ( bot_bot @ ( set @ nat ) ) )
= ( zero_zero @ nat ) ) ).
% set_encode_empty
thf(fact_4973_Set_Oempty__def,axiom,
! [A: $tType] :
( ( bot_bot @ ( set @ A ) )
= ( collect @ A
@ ^ [X: A] : $false ) ) ).
% Set.empty_def
thf(fact_4974_diff__shunt__var,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X2: A,Y2: A] :
( ( ( minus_minus @ A @ X2 @ Y2 )
= ( bot_bot @ A ) )
= ( ord_less_eq @ A @ X2 @ Y2 ) ) ) ).
% diff_shunt_var
thf(fact_4975_finite__has__maximal,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ? [X3: A] :
( ( member @ A @ X3 @ A3 )
& ! [Xa2: A] :
( ( member @ A @ Xa2 @ A3 )
=> ( ( ord_less_eq @ A @ X3 @ Xa2 )
=> ( X3 = Xa2 ) ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_4976_finite__has__minimal,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ? [X3: A] :
( ( member @ A @ X3 @ A3 )
& ! [Xa2: A] :
( ( member @ A @ Xa2 @ A3 )
=> ( ( ord_less_eq @ A @ Xa2 @ X3 )
=> ( X3 = Xa2 ) ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_4977_subset__minus__empty,axiom,
! [A: $tType,A3: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B5 )
=> ( ( minus_minus @ ( set @ A ) @ A3 @ B5 )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% subset_minus_empty
thf(fact_4978_lessThan__empty__iff,axiom,
! [N: nat] :
( ( ( set_ord_lessThan @ nat @ N )
= ( bot_bot @ ( set @ nat ) ) )
= ( N
= ( zero_zero @ nat ) ) ) ).
% lessThan_empty_iff
thf(fact_4979_subset__Compl__self__eq,axiom,
! [A: $tType,A3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ ( uminus_uminus @ ( set @ A ) @ A3 ) )
= ( A3
= ( bot_bot @ ( set @ A ) ) ) ) ).
% subset_Compl_self_eq
thf(fact_4980_sum__pos,axiom,
! [A: $tType,B: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [I5: set @ B,F2: B > A] :
( ( finite_finite2 @ B @ I5 )
=> ( ( I5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [I2: B] :
( ( member @ B @ I2 @ I5 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( F2 @ I2 ) ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ I5 ) ) ) ) ) ) ).
% sum_pos
thf(fact_4981_less__1__prod,axiom,
! [B: $tType,A: $tType] :
( ( linordered_idom @ B )
=> ! [I5: set @ A,F2: A > B] :
( ( finite_finite2 @ A @ I5 )
=> ( ( I5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [I2: A] :
( ( member @ A @ I2 @ I5 )
=> ( ord_less @ B @ ( one_one @ B ) @ ( F2 @ I2 ) ) )
=> ( ord_less @ B @ ( one_one @ B ) @ ( groups7121269368397514597t_prod @ A @ B @ F2 @ I5 ) ) ) ) ) ) ).
% less_1_prod
thf(fact_4982_prod__mono__strict,axiom,
! [A: $tType,B: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: set @ B,F2: B > A,G: B > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ! [I2: B] :
( ( member @ B @ I2 @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ I2 ) )
& ( ord_less @ A @ ( F2 @ I2 ) @ ( G @ I2 ) ) ) )
=> ( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ord_less @ A @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ A3 ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ A3 ) ) ) ) ) ) ).
% prod_mono_strict
thf(fact_4983_max_Omono,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [C2: A,A2: A,D2: A,B2: A] :
( ( ord_less_eq @ A @ C2 @ A2 )
=> ( ( ord_less_eq @ A @ D2 @ B2 )
=> ( ord_less_eq @ A @ ( ord_max @ A @ C2 @ D2 ) @ ( ord_max @ A @ A2 @ B2 ) ) ) ) ) ).
% max.mono
thf(fact_4984_max_OorderE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,A2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( A2
= ( ord_max @ A @ A2 @ B2 ) ) ) ) ).
% max.orderE
thf(fact_4985_max_OorderI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A] :
( ( A2
= ( ord_max @ A @ A2 @ B2 ) )
=> ( ord_less_eq @ A @ B2 @ A2 ) ) ) ).
% max.orderI
thf(fact_4986_max_OboundedE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,C2: A,A2: A] :
( ( ord_less_eq @ A @ ( ord_max @ A @ B2 @ C2 ) @ A2 )
=> ~ ( ( ord_less_eq @ A @ B2 @ A2 )
=> ~ ( ord_less_eq @ A @ C2 @ A2 ) ) ) ) ).
% max.boundedE
thf(fact_4987_max_OboundedI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( ord_less_eq @ A @ C2 @ A2 )
=> ( ord_less_eq @ A @ ( ord_max @ A @ B2 @ C2 ) @ A2 ) ) ) ) ).
% max.boundedI
thf(fact_4988_max_Oorder__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B3: A,A4: A] :
( A4
= ( ord_max @ A @ A4 @ B3 ) ) ) ) ) ).
% max.order_iff
thf(fact_4989_max_Ocobounded1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A] : ( ord_less_eq @ A @ A2 @ ( ord_max @ A @ A2 @ B2 ) ) ) ).
% max.cobounded1
thf(fact_4990_max_Ocobounded2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,A2: A] : ( ord_less_eq @ A @ B2 @ ( ord_max @ A @ A2 @ B2 ) ) ) ).
% max.cobounded2
thf(fact_4991_le__max__iff__disj,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Z: A,X2: A,Y2: A] :
( ( ord_less_eq @ A @ Z @ ( ord_max @ A @ X2 @ Y2 ) )
= ( ( ord_less_eq @ A @ Z @ X2 )
| ( ord_less_eq @ A @ Z @ Y2 ) ) ) ) ).
% le_max_iff_disj
thf(fact_4992_max_Oabsorb__iff1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B3: A,A4: A] :
( ( ord_max @ A @ A4 @ B3 )
= A4 ) ) ) ) ).
% max.absorb_iff1
thf(fact_4993_max_Oabsorb__iff2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A4: A,B3: A] :
( ( ord_max @ A @ A4 @ B3 )
= B3 ) ) ) ) ).
% max.absorb_iff2
thf(fact_4994_max_OcoboundedI1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ord_less_eq @ A @ C2 @ A2 )
=> ( ord_less_eq @ A @ C2 @ ( ord_max @ A @ A2 @ B2 ) ) ) ) ).
% max.coboundedI1
thf(fact_4995_max_OcoboundedI2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [C2: A,B2: A,A2: A] :
( ( ord_less_eq @ A @ C2 @ B2 )
=> ( ord_less_eq @ A @ C2 @ ( ord_max @ A @ A2 @ B2 ) ) ) ) ).
% max.coboundedI2
thf(fact_4996_delete__correct,axiom,
! [T2: vEBT_VEBT,N: nat,X2: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( vEBT_VEBT_set_vebt @ ( vEBT_vebt_delete @ T2 @ X2 ) )
= ( minus_minus @ ( set @ nat ) @ ( vEBT_set_vebt @ T2 ) @ ( insert @ nat @ X2 @ ( bot_bot @ ( set @ nat ) ) ) ) ) ) ).
% delete_correct
thf(fact_4997_floor__rat__def,axiom,
( ( archim6421214686448440834_floor @ rat )
= ( ^ [X: rat] :
( the @ int
@ ^ [Z5: int] :
( ( ord_less_eq @ rat @ ( ring_1_of_int @ rat @ Z5 ) @ X )
& ( ord_less @ rat @ X @ ( ring_1_of_int @ rat @ ( plus_plus @ int @ Z5 @ ( one_one @ int ) ) ) ) ) ) ) ) ).
% floor_rat_def
thf(fact_4998_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_4999_delete__correct_H,axiom,
! [T2: vEBT_VEBT,N: nat,X2: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( vEBT_VEBT_set_vebt @ ( vEBT_vebt_delete @ T2 @ X2 ) )
= ( minus_minus @ ( set @ nat ) @ ( vEBT_VEBT_set_vebt @ T2 ) @ ( insert @ nat @ X2 @ ( bot_bot @ ( set @ nat ) ) ) ) ) ) ).
% delete_correct'
thf(fact_5000_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_5001_insert__subset,axiom,
! [A: $tType,X2: A,A3: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X2 @ A3 ) @ B5 )
= ( ( member @ A @ X2 @ B5 )
& ( ord_less_eq @ ( set @ A ) @ A3 @ B5 ) ) ) ).
% insert_subset
thf(fact_5002_singleton__conv,axiom,
! [A: $tType,A2: A] :
( ( collect @ A
@ ^ [X: A] : X = A2 )
= ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% singleton_conv
thf(fact_5003_singleton__conv2,axiom,
! [A: $tType,A2: A] :
( ( collect @ A
@ ( ^ [Y5: A,Z3: A] : Y5 = Z3
@ A2 ) )
= ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% singleton_conv2
thf(fact_5004_singleton__insert__inj__eq_H,axiom,
! [A: $tType,A2: A,A3: set @ A,B2: A] :
( ( ( insert @ A @ A2 @ A3 )
= ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) )
= ( ( A2 = B2 )
& ( ord_less_eq @ ( set @ A ) @ A3 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_5005_singleton__insert__inj__eq,axiom,
! [A: $tType,B2: A,A2: A,A3: set @ A] :
( ( ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) )
= ( insert @ A @ A2 @ A3 ) )
= ( ( A2 = B2 )
& ( ord_less_eq @ ( set @ A ) @ A3 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_5006_sum_Oinsert,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A3: set @ B,X2: B,G: B > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ~ ( member @ B @ X2 @ A3 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( insert @ B @ X2 @ A3 ) )
= ( plus_plus @ A @ ( G @ X2 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ A3 ) ) ) ) ) ) ).
% sum.insert
thf(fact_5007_prod_Oinsert,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: set @ B,X2: B,G: B > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ~ ( member @ B @ X2 @ A3 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( insert @ B @ X2 @ A3 ) )
= ( times_times @ A @ ( G @ X2 ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ A3 ) ) ) ) ) ) ).
% prod.insert
thf(fact_5008_subset__Compl__singleton,axiom,
! [A: $tType,A3: set @ A,B2: A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ ( uminus_uminus @ ( set @ A ) @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( ~ ( member @ A @ B2 @ A3 ) ) ) ).
% subset_Compl_singleton
thf(fact_5009_atMost__0,axiom,
( ( set_ord_atMost @ nat @ ( zero_zero @ nat ) )
= ( insert @ nat @ ( zero_zero @ nat ) @ ( bot_bot @ ( set @ nat ) ) ) ) ).
% atMost_0
thf(fact_5010_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_5011_set__replicate,axiom,
! [A: $tType,N: nat,X2: A] :
( ( N
!= ( zero_zero @ nat ) )
=> ( ( set2 @ A @ ( replicate @ A @ N @ X2 ) )
= ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% set_replicate
thf(fact_5012_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_5013_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_5014_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_5015_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_5016_bot__option__def,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( bot_bot @ ( option @ A ) )
= ( none @ A ) ) ) ).
% bot_option_def
thf(fact_5017_bot__nat__def,axiom,
( ( bot_bot @ nat )
= ( zero_zero @ nat ) ) ).
% bot_nat_def
thf(fact_5018_bot__enat__def,axiom,
( ( bot_bot @ extended_enat )
= ( zero_zero @ extended_enat ) ) ).
% bot_enat_def
thf(fact_5019_Collect__conv__if2,axiom,
! [A: $tType,P: A > $o,A2: A] :
( ( ( P @ A2 )
=> ( ( collect @ A
@ ^ [X: A] :
( ( A2 = X )
& ( P @ X ) ) )
= ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) )
& ( ~ ( P @ A2 )
=> ( ( collect @ A
@ ^ [X: A] :
( ( A2 = X )
& ( P @ X ) ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Collect_conv_if2
thf(fact_5020_Collect__conv__if,axiom,
! [A: $tType,P: A > $o,A2: A] :
( ( ( P @ A2 )
=> ( ( collect @ A
@ ^ [X: A] :
( ( X = A2 )
& ( P @ X ) ) )
= ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) )
& ( ~ ( P @ A2 )
=> ( ( collect @ A
@ ^ [X: A] :
( ( X = A2 )
& ( P @ X ) ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Collect_conv_if
thf(fact_5021_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_5022_obtain__pos__sum,axiom,
! [R2: rat] :
( ( ord_less @ rat @ ( zero_zero @ rat ) @ R2 )
=> ~ ! [S2: rat] :
( ( ord_less @ rat @ ( zero_zero @ rat ) @ S2 )
=> ! [T5: rat] :
( ( ord_less @ rat @ ( zero_zero @ rat ) @ T5 )
=> ( R2
!= ( plus_plus @ rat @ S2 @ T5 ) ) ) ) ) ).
% obtain_pos_sum
thf(fact_5023_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_5024_subset__insertI2,axiom,
! [A: $tType,A3: set @ A,B5: set @ A,B2: A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B5 )
=> ( ord_less_eq @ ( set @ A ) @ A3 @ ( insert @ A @ B2 @ B5 ) ) ) ).
% subset_insertI2
thf(fact_5025_subset__insertI,axiom,
! [A: $tType,B5: set @ A,A2: A] : ( ord_less_eq @ ( set @ A ) @ B5 @ ( insert @ A @ A2 @ B5 ) ) ).
% subset_insertI
thf(fact_5026_subset__insert,axiom,
! [A: $tType,X2: A,A3: set @ A,B5: set @ A] :
( ~ ( member @ A @ X2 @ A3 )
=> ( ( ord_less_eq @ ( set @ A ) @ A3 @ ( insert @ A @ X2 @ B5 ) )
= ( ord_less_eq @ ( set @ A ) @ A3 @ B5 ) ) ) ).
% subset_insert
thf(fact_5027_insert__mono,axiom,
! [A: $tType,C6: set @ A,D4: set @ A,A2: A] :
( ( ord_less_eq @ ( set @ A ) @ C6 @ D4 )
=> ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ A2 @ C6 ) @ ( insert @ A @ A2 @ D4 ) ) ) ).
% insert_mono
thf(fact_5028_insert__compr,axiom,
! [A: $tType] :
( ( insert @ A )
= ( ^ [A4: A,B8: set @ A] :
( collect @ A
@ ^ [X: A] :
( ( X = A4 )
| ( member @ A @ X @ B8 ) ) ) ) ) ).
% insert_compr
thf(fact_5029_insert__Collect,axiom,
! [A: $tType,A2: A,P: A > $o] :
( ( insert @ A @ A2 @ ( collect @ A @ P ) )
= ( collect @ A
@ ^ [U2: A] :
( ( U2 != A2 )
=> ( P @ U2 ) ) ) ) ).
% insert_Collect
thf(fact_5030_subset__singleton__iff,axiom,
! [A: $tType,X6: set @ A,A2: A] :
( ( ord_less_eq @ ( set @ A ) @ X6 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) )
= ( ( X6
= ( bot_bot @ ( set @ A ) ) )
| ( X6
= ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% subset_singleton_iff
thf(fact_5031_subset__singletonD,axiom,
! [A: $tType,A3: set @ A,X2: A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) )
=> ( ( A3
= ( bot_bot @ ( set @ A ) ) )
| ( A3
= ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% subset_singletonD
thf(fact_5032_subset__Diff__insert,axiom,
! [A: $tType,A3: set @ A,B5: set @ A,X2: A,C6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ ( minus_minus @ ( set @ A ) @ B5 @ ( insert @ A @ X2 @ C6 ) ) )
= ( ( ord_less_eq @ ( set @ A ) @ A3 @ ( minus_minus @ ( set @ A ) @ B5 @ C6 ) )
& ~ ( member @ A @ X2 @ A3 ) ) ) ).
% subset_Diff_insert
thf(fact_5033_lessThan__Suc,axiom,
! [K: nat] :
( ( set_ord_lessThan @ nat @ ( suc @ K ) )
= ( insert @ nat @ K @ ( set_ord_lessThan @ nat @ K ) ) ) ).
% lessThan_Suc
thf(fact_5034_atMost__Suc,axiom,
! [K: nat] :
( ( set_ord_atMost @ nat @ ( suc @ K ) )
= ( insert @ nat @ ( suc @ K ) @ ( set_ord_atMost @ nat @ K ) ) ) ).
% atMost_Suc
thf(fact_5035_finite__ranking__induct,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [S: set @ B,P: ( set @ B ) > $o,F2: B > A] :
( ( finite_finite2 @ B @ S )
=> ( ( P @ ( bot_bot @ ( set @ B ) ) )
=> ( ! [X3: B,S7: set @ B] :
( ( finite_finite2 @ B @ S7 )
=> ( ! [Y4: B] :
( ( member @ B @ Y4 @ S7 )
=> ( ord_less_eq @ A @ ( F2 @ Y4 ) @ ( F2 @ X3 ) ) )
=> ( ( P @ S7 )
=> ( P @ ( insert @ B @ X3 @ S7 ) ) ) ) )
=> ( P @ S ) ) ) ) ) ).
% finite_ranking_induct
thf(fact_5036_sum_Oinsert__if,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A3: set @ B,X2: B,G: B > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ( ( member @ B @ X2 @ A3 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( insert @ B @ X2 @ A3 ) )
= ( groups7311177749621191930dd_sum @ B @ A @ G @ A3 ) ) )
& ( ~ ( member @ B @ X2 @ A3 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( insert @ B @ X2 @ A3 ) )
= ( plus_plus @ A @ ( G @ X2 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ A3 ) ) ) ) ) ) ) ).
% sum.insert_if
thf(fact_5037_finite__subset__induct,axiom,
! [A: $tType,F3: set @ A,A3: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ F3 )
=> ( ( ord_less_eq @ ( set @ A ) @ F3 @ A3 )
=> ( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [A5: A,F6: set @ A] :
( ( finite_finite2 @ A @ F6 )
=> ( ( member @ A @ A5 @ A3 )
=> ( ~ ( member @ A @ A5 @ F6 )
=> ( ( P @ F6 )
=> ( P @ ( insert @ A @ A5 @ F6 ) ) ) ) ) )
=> ( P @ F3 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_5038_finite__subset__induct_H,axiom,
! [A: $tType,F3: set @ A,A3: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ F3 )
=> ( ( ord_less_eq @ ( set @ A ) @ F3 @ A3 )
=> ( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [A5: A,F6: set @ A] :
( ( finite_finite2 @ A @ F6 )
=> ( ( member @ A @ A5 @ A3 )
=> ( ( ord_less_eq @ ( set @ A ) @ F6 @ A3 )
=> ( ~ ( member @ A @ A5 @ F6 )
=> ( ( P @ F6 )
=> ( P @ ( insert @ A @ A5 @ F6 ) ) ) ) ) ) )
=> ( P @ F3 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_5039_prod_Oinsert__if,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: set @ B,X2: B,G: B > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ( ( member @ B @ X2 @ A3 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( insert @ B @ X2 @ A3 ) )
= ( groups7121269368397514597t_prod @ B @ A @ G @ A3 ) ) )
& ( ~ ( member @ B @ X2 @ A3 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( insert @ B @ X2 @ A3 ) )
= ( times_times @ A @ ( G @ X2 ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ A3 ) ) ) ) ) ) ) ).
% prod.insert_if
thf(fact_5040_Diff__single__insert,axiom,
! [A: $tType,A3: set @ A,X2: A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) @ B5 )
=> ( ord_less_eq @ ( set @ A ) @ A3 @ ( insert @ A @ X2 @ B5 ) ) ) ).
% Diff_single_insert
thf(fact_5041_subset__insert__iff,axiom,
! [A: $tType,A3: set @ A,X2: A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ ( insert @ A @ X2 @ B5 ) )
= ( ( ( member @ A @ X2 @ A3 )
=> ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) @ B5 ) )
& ( ~ ( member @ A @ X2 @ A3 )
=> ( ord_less_eq @ ( set @ A ) @ A3 @ B5 ) ) ) ) ).
% subset_insert_iff
thf(fact_5042_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_5043_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_5044_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_5045_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_5046_set__update__subset__insert,axiom,
! [A: $tType,Xs2: list @ A,I: nat,X2: A] : ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ ( list_update @ A @ Xs2 @ I @ X2 ) ) @ ( insert @ A @ X2 @ ( set2 @ A @ Xs2 ) ) ) ).
% set_update_subset_insert
thf(fact_5047_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_5048_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_5049_divmod__int__def,axiom,
( ( unique8689654367752047608divmod @ int )
= ( ^ [M5: num,N3: num] : ( product_Pair @ int @ int @ ( divide_divide @ int @ ( numeral_numeral @ int @ M5 ) @ ( numeral_numeral @ int @ N3 ) ) @ ( modulo_modulo @ int @ ( numeral_numeral @ int @ M5 ) @ ( numeral_numeral @ int @ N3 ) ) ) ) ) ).
% divmod_int_def
thf(fact_5050_remove__induct,axiom,
! [A: $tType,P: ( set @ A ) > $o,B5: set @ A] :
( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ( ~ ( finite_finite2 @ A @ B5 )
=> ( P @ B5 ) )
=> ( ! [A8: set @ A] :
( ( finite_finite2 @ A @ A8 )
=> ( ( A8
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ A ) @ A8 @ B5 )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ A8 )
=> ( P @ ( minus_minus @ ( set @ A ) @ A8 @ ( insert @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) ) ) )
=> ( P @ A8 ) ) ) ) )
=> ( P @ B5 ) ) ) ) ).
% remove_induct
thf(fact_5051_finite__remove__induct,axiom,
! [A: $tType,B5: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ B5 )
=> ( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [A8: set @ A] :
( ( finite_finite2 @ A @ A8 )
=> ( ( A8
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ A ) @ A8 @ B5 )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ A8 )
=> ( P @ ( minus_minus @ ( set @ A ) @ A8 @ ( insert @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) ) ) )
=> ( P @ A8 ) ) ) ) )
=> ( P @ B5 ) ) ) ) ).
% finite_remove_induct
thf(fact_5052_set__replicate__Suc,axiom,
! [A: $tType,N: nat,X2: A] :
( ( set2 @ A @ ( replicate @ A @ ( suc @ N ) @ X2 ) )
= ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% set_replicate_Suc
thf(fact_5053_psubset__insert__iff,axiom,
! [A: $tType,A3: set @ A,X2: A,B5: set @ A] :
( ( ord_less @ ( set @ A ) @ A3 @ ( insert @ A @ X2 @ B5 ) )
= ( ( ( member @ A @ X2 @ B5 )
=> ( ord_less @ ( set @ A ) @ A3 @ B5 ) )
& ( ~ ( member @ A @ X2 @ B5 )
=> ( ( ( member @ A @ X2 @ A3 )
=> ( ord_less @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) @ B5 ) )
& ( ~ ( member @ A @ X2 @ A3 )
=> ( ord_less_eq @ ( set @ A ) @ A3 @ B5 ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_5054_set__replicate__conv__if,axiom,
! [A: $tType,N: nat,X2: A] :
( ( ( N
= ( zero_zero @ nat ) )
=> ( ( set2 @ A @ ( replicate @ A @ N @ X2 ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( N
!= ( zero_zero @ nat ) )
=> ( ( set2 @ A @ ( replicate @ A @ N @ X2 ) )
= ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% set_replicate_conv_if
thf(fact_5055_sum_Oinsert__remove,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A3: set @ B,G: B > A,X2: B] :
( ( finite_finite2 @ B @ A3 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( insert @ B @ X2 @ A3 ) )
= ( plus_plus @ A @ ( G @ X2 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ A3 @ ( insert @ B @ X2 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ).
% sum.insert_remove
thf(fact_5056_sum_Oremove,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A3: set @ B,X2: B,G: B > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ( member @ B @ X2 @ A3 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ A3 )
= ( plus_plus @ A @ ( G @ X2 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ A3 @ ( insert @ B @ X2 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ) ).
% sum.remove
thf(fact_5057_prod_Oremove,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: set @ B,X2: B,G: B > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ( member @ B @ X2 @ A3 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ A3 )
= ( times_times @ A @ ( G @ X2 ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ A3 @ ( insert @ B @ X2 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ) ).
% prod.remove
thf(fact_5058_prod_Oinsert__remove,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: set @ B,G: B > A,X2: B] :
( ( finite_finite2 @ B @ A3 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( insert @ B @ X2 @ A3 ) )
= ( times_times @ A @ ( G @ X2 ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ A3 @ ( insert @ B @ X2 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ).
% prod.insert_remove
thf(fact_5059_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_5060_sum_Odelta__remove,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [S: set @ B,A2: B,B2: B > A,C2: B > A] :
( ( finite_finite2 @ B @ S )
=> ( ( ( member @ B @ A2 @ S )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [K2: B] : ( if @ A @ ( K2 = A2 ) @ ( B2 @ K2 ) @ ( C2 @ K2 ) )
@ S )
= ( plus_plus @ A @ ( B2 @ A2 ) @ ( groups7311177749621191930dd_sum @ B @ A @ C2 @ ( minus_minus @ ( set @ B ) @ S @ ( insert @ B @ A2 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) )
& ( ~ ( member @ B @ A2 @ S )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [K2: B] : ( if @ A @ ( K2 = A2 ) @ ( B2 @ K2 ) @ ( C2 @ K2 ) )
@ S )
= ( groups7311177749621191930dd_sum @ B @ A @ C2 @ ( minus_minus @ ( set @ B ) @ S @ ( insert @ B @ A2 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ) ).
% sum.delta_remove
thf(fact_5061_prod_Odelta__remove,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S: set @ B,A2: B,B2: B > A,C2: B > A] :
( ( finite_finite2 @ B @ S )
=> ( ( ( member @ B @ A2 @ S )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K2: B] : ( if @ A @ ( K2 = A2 ) @ ( B2 @ K2 ) @ ( C2 @ K2 ) )
@ S )
= ( times_times @ A @ ( B2 @ A2 ) @ ( groups7121269368397514597t_prod @ B @ A @ C2 @ ( minus_minus @ ( set @ B ) @ S @ ( insert @ B @ A2 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) )
& ( ~ ( member @ B @ A2 @ S )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K2: B] : ( if @ A @ ( K2 = A2 ) @ ( B2 @ K2 ) @ ( C2 @ K2 ) )
@ S )
= ( groups7121269368397514597t_prod @ B @ A @ C2 @ ( minus_minus @ ( set @ B ) @ S @ ( insert @ B @ A2 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ) ).
% prod.delta_remove
thf(fact_5062_divmod__def,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ( ( unique8689654367752047608divmod @ A )
= ( ^ [M5: num,N3: num] : ( product_Pair @ A @ A @ ( divide_divide @ A @ ( numeral_numeral @ A @ M5 ) @ ( numeral_numeral @ A @ N3 ) ) @ ( modulo_modulo @ A @ ( numeral_numeral @ A @ M5 ) @ ( numeral_numeral @ A @ N3 ) ) ) ) ) ) ).
% divmod_def
thf(fact_5063_divmod_H__nat__def,axiom,
( ( unique8689654367752047608divmod @ nat )
= ( ^ [M5: num,N3: num] : ( product_Pair @ nat @ nat @ ( divide_divide @ nat @ ( numeral_numeral @ nat @ M5 ) @ ( numeral_numeral @ nat @ N3 ) ) @ ( modulo_modulo @ nat @ ( numeral_numeral @ nat @ M5 ) @ ( numeral_numeral @ nat @ N3 ) ) ) ) ) ).
% divmod'_nat_def
thf(fact_5064_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_5065_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_5066_sinh__zero__iff,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X2: A] :
( ( ( sinh @ A @ X2 )
= ( zero_zero @ A ) )
= ( member @ A @ ( exp @ A @ X2 ) @ ( insert @ A @ ( one_one @ A ) @ ( insert @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% sinh_zero_iff
thf(fact_5067_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_5068_divmod__divmod__step,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ( ( unique8689654367752047608divmod @ A )
= ( ^ [M5: num,N3: num] : ( if @ ( product_prod @ A @ A ) @ ( ord_less @ num @ M5 @ N3 ) @ ( product_Pair @ A @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ M5 ) ) @ ( unique1321980374590559556d_step @ A @ N3 @ ( unique8689654367752047608divmod @ A @ M5 @ ( bit0 @ N3 ) ) ) ) ) ) ) ).
% divmod_divmod_step
thf(fact_5069_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_5070_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_5071_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_5072_divides__aux__eq,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [Q2: A,R2: A] :
( ( unique5940410009612947441es_aux @ A @ ( product_Pair @ A @ A @ Q2 @ R2 ) )
= ( R2
= ( zero_zero @ A ) ) ) ) ).
% divides_aux_eq
thf(fact_5073_Divides_Oadjust__div__eq,axiom,
! [Q2: int,R2: int] :
( ( adjust_div @ ( product_Pair @ int @ int @ Q2 @ R2 ) )
= ( plus_plus @ int @ Q2
@ ( zero_neq_one_of_bool @ int
@ ( R2
!= ( zero_zero @ int ) ) ) ) ) ).
% Divides.adjust_div_eq
thf(fact_5074_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_5075_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_5076_diff__rat__def,axiom,
( ( minus_minus @ rat )
= ( ^ [Q4: rat,R5: rat] : ( plus_plus @ rat @ Q4 @ ( uminus_uminus @ rat @ R5 ) ) ) ) ).
% diff_rat_def
thf(fact_5077_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_5078_simp__from__to,axiom,
( ( set_or1337092689740270186AtMost @ int )
= ( ^ [I3: int,J3: int] : ( if @ ( set @ int ) @ ( ord_less @ int @ J3 @ I3 ) @ ( bot_bot @ ( set @ int ) ) @ ( insert @ int @ I3 @ ( set_or1337092689740270186AtMost @ int @ ( plus_plus @ int @ I3 @ ( one_one @ int ) ) @ J3 ) ) ) ) ) ).
% simp_from_to
thf(fact_5079_rel__of__empty,axiom,
! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o] :
( ( rel_of @ A @ B
@ ^ [X: A] : ( none @ B )
@ P )
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% rel_of_empty
thf(fact_5080_neg__eucl__rel__int__mult__2,axiom,
! [B2: int,A2: int,Q2: int,R2: int] :
( ( ord_less_eq @ int @ B2 @ ( zero_zero @ int ) )
=> ( ( eucl_rel_int @ ( plus_plus @ int @ A2 @ ( one_one @ int ) ) @ B2 @ ( product_Pair @ int @ int @ Q2 @ R2 ) )
=> ( eucl_rel_int @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ A2 ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ B2 ) @ ( product_Pair @ int @ int @ Q2 @ ( minus_minus @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ R2 ) @ ( one_one @ int ) ) ) ) ) ) ).
% neg_eucl_rel_int_mult_2
thf(fact_5081_pos__eucl__rel__int__mult__2,axiom,
! [B2: int,A2: int,Q2: int,R2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ B2 )
=> ( ( eucl_rel_int @ A2 @ B2 @ ( product_Pair @ int @ int @ Q2 @ R2 ) )
=> ( eucl_rel_int @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ A2 ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ B2 ) @ ( product_Pair @ int @ int @ Q2 @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ R2 ) ) ) ) ) ) ).
% pos_eucl_rel_int_mult_2
thf(fact_5082_unique__remainder,axiom,
! [A2: int,B2: int,Q2: int,R2: int,Q5: int,R4: int] :
( ( eucl_rel_int @ A2 @ B2 @ ( product_Pair @ int @ int @ Q2 @ R2 ) )
=> ( ( eucl_rel_int @ A2 @ B2 @ ( product_Pair @ int @ int @ Q5 @ R4 ) )
=> ( R2 = R4 ) ) ) ).
% unique_remainder
thf(fact_5083_unique__quotient,axiom,
! [A2: int,B2: int,Q2: int,R2: int,Q5: int,R4: int] :
( ( eucl_rel_int @ A2 @ B2 @ ( product_Pair @ int @ int @ Q2 @ R2 ) )
=> ( ( eucl_rel_int @ A2 @ B2 @ ( product_Pair @ int @ int @ Q5 @ R4 ) )
=> ( Q2 = Q5 ) ) ) ).
% unique_quotient
thf(fact_5084_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_5085_mod__int__unique,axiom,
! [K: int,L2: int,Q2: int,R2: int] :
( ( eucl_rel_int @ K @ L2 @ ( product_Pair @ int @ int @ Q2 @ R2 ) )
=> ( ( modulo_modulo @ int @ K @ L2 )
= R2 ) ) ).
% mod_int_unique
thf(fact_5086_div__int__unique,axiom,
! [K: int,L2: int,Q2: int,R2: int] :
( ( eucl_rel_int @ K @ L2 @ ( product_Pair @ int @ int @ Q2 @ R2 ) )
=> ( ( divide_divide @ int @ K @ L2 )
= Q2 ) ) ).
% div_int_unique
thf(fact_5087_eucl__rel__int__dividesI,axiom,
! [L2: int,K: int,Q2: int] :
( ( L2
!= ( zero_zero @ int ) )
=> ( ( K
= ( times_times @ int @ Q2 @ L2 ) )
=> ( eucl_rel_int @ K @ L2 @ ( product_Pair @ int @ int @ Q2 @ ( zero_zero @ int ) ) ) ) ) ).
% eucl_rel_int_dividesI
thf(fact_5088_eucl__rel__int,axiom,
! [K: int,L2: int] : ( eucl_rel_int @ K @ L2 @ ( product_Pair @ int @ int @ ( divide_divide @ int @ K @ L2 ) @ ( modulo_modulo @ int @ K @ L2 ) ) ) ).
% eucl_rel_int
thf(fact_5089_zminus1__lemma,axiom,
! [A2: int,B2: int,Q2: int,R2: int] :
( ( eucl_rel_int @ A2 @ B2 @ ( product_Pair @ int @ int @ Q2 @ R2 ) )
=> ( ( B2
!= ( zero_zero @ int ) )
=> ( eucl_rel_int @ ( uminus_uminus @ int @ A2 ) @ B2
@ ( product_Pair @ int @ int
@ ( if @ int
@ ( R2
= ( zero_zero @ int ) )
@ ( uminus_uminus @ int @ Q2 )
@ ( minus_minus @ int @ ( uminus_uminus @ int @ Q2 ) @ ( one_one @ int ) ) )
@ ( if @ int
@ ( R2
= ( zero_zero @ int ) )
@ ( zero_zero @ int )
@ ( minus_minus @ int @ B2 @ R2 ) ) ) ) ) ) ).
% zminus1_lemma
thf(fact_5090_eucl__rel__int__iff,axiom,
! [K: int,L2: int,Q2: int,R2: int] :
( ( eucl_rel_int @ K @ L2 @ ( product_Pair @ int @ int @ Q2 @ R2 ) )
= ( ( K
= ( plus_plus @ int @ ( times_times @ int @ L2 @ Q2 ) @ R2 ) )
& ( ( ord_less @ int @ ( zero_zero @ int ) @ L2 )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ R2 )
& ( ord_less @ int @ R2 @ L2 ) ) )
& ( ~ ( ord_less @ int @ ( zero_zero @ int ) @ L2 )
=> ( ( ( ord_less @ int @ L2 @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ L2 @ R2 )
& ( ord_less_eq @ int @ R2 @ ( zero_zero @ int ) ) ) )
& ( ~ ( ord_less @ int @ L2 @ ( zero_zero @ int ) )
=> ( Q2
= ( zero_zero @ int ) ) ) ) ) ) ) ).
% eucl_rel_int_iff
thf(fact_5091_eucl__rel__int__remainderI,axiom,
! [R2: int,L2: int,K: int,Q2: int] :
( ( ( sgn_sgn @ int @ R2 )
= ( sgn_sgn @ int @ L2 ) )
=> ( ( ord_less @ int @ ( abs_abs @ int @ R2 ) @ ( abs_abs @ int @ L2 ) )
=> ( ( K
= ( plus_plus @ int @ ( times_times @ int @ Q2 @ L2 ) @ R2 ) )
=> ( eucl_rel_int @ K @ L2 @ ( product_Pair @ int @ int @ Q2 @ R2 ) ) ) ) ) ).
% eucl_rel_int_remainderI
thf(fact_5092_eucl__rel__int_Osimps,axiom,
( eucl_rel_int
= ( ^ [A12: int,A23: int,A32: product_prod @ int @ int] :
( ? [K2: int] :
( ( A12 = K2 )
& ( A23
= ( zero_zero @ int ) )
& ( A32
= ( product_Pair @ int @ int @ ( zero_zero @ int ) @ K2 ) ) )
| ? [L: int,K2: int,Q4: int] :
( ( A12 = K2 )
& ( A23 = L )
& ( A32
= ( product_Pair @ int @ int @ Q4 @ ( zero_zero @ int ) ) )
& ( L
!= ( zero_zero @ int ) )
& ( K2
= ( times_times @ int @ Q4 @ L ) ) )
| ? [R5: int,L: int,K2: int,Q4: int] :
( ( A12 = K2 )
& ( A23 = L )
& ( A32
= ( product_Pair @ int @ int @ Q4 @ R5 ) )
& ( ( sgn_sgn @ int @ R5 )
= ( sgn_sgn @ int @ L ) )
& ( ord_less @ int @ ( abs_abs @ int @ R5 ) @ ( abs_abs @ int @ L ) )
& ( K2
= ( plus_plus @ int @ ( times_times @ int @ Q4 @ L ) @ R5 ) ) ) ) ) ) ).
% eucl_rel_int.simps
thf(fact_5093_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 ) ) )
=> ( ! [Q3: int] :
( ( A33
= ( product_Pair @ int @ int @ Q3 @ ( zero_zero @ int ) ) )
=> ( ( A22
!= ( zero_zero @ int ) )
=> ( A1
!= ( times_times @ int @ Q3 @ A22 ) ) ) )
=> ~ ! [R3: int,Q3: int] :
( ( A33
= ( product_Pair @ int @ int @ Q3 @ 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 @ Q3 @ A22 ) @ R3 ) ) ) ) ) ) ) ) ).
% eucl_rel_int.cases
thf(fact_5094_and__int_Oelims,axiom,
! [X2: int,Xa: int,Y2: int] :
( ( ( bit_se5824344872417868541ns_and @ int @ X2 @ Xa )
= Y2 )
=> ( ( ( ( member @ int @ X2 @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) )
& ( member @ int @ Xa @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) )
=> ( Y2
= ( uminus_uminus @ int
@ ( zero_neq_one_of_bool @ int
@ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ X2 )
& ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ Xa ) ) ) ) ) )
& ( ~ ( ( member @ int @ X2 @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) )
& ( member @ int @ Xa @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) )
=> ( Y2
= ( plus_plus @ int
@ ( zero_neq_one_of_bool @ int
@ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ X2 )
& ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ Xa ) ) )
@ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( divide_divide @ int @ X2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ Xa @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% and_int.elims
thf(fact_5095_and__int_Osimps,axiom,
( ( bit_se5824344872417868541ns_and @ int )
= ( ^ [K2: int,L: int] :
( if @ int
@ ( ( member @ int @ K2 @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) )
& ( member @ int @ L @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) )
@ ( uminus_uminus @ int
@ ( zero_neq_one_of_bool @ int
@ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K2 )
& ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ L ) ) ) )
@ ( plus_plus @ int
@ ( zero_neq_one_of_bool @ int
@ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K2 )
& ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ L ) ) )
@ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( divide_divide @ int @ K2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ).
% and_int.simps
thf(fact_5096_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_5097_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_5098_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_5099_and_Oidem,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A] :
( ( bit_se5824344872417868541ns_and @ A @ A2 @ A2 )
= A2 ) ) ).
% and.idem
thf(fact_5100_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_5101_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_5102_bit_Oconj__zero__left,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X2: A] :
( ( bit_se5824344872417868541ns_and @ A @ ( zero_zero @ A ) @ X2 )
= ( zero_zero @ A ) ) ) ).
% bit.conj_zero_left
thf(fact_5103_bit_Oconj__zero__right,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X2: A] :
( ( bit_se5824344872417868541ns_and @ A @ X2 @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% bit.conj_zero_right
thf(fact_5104_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_5105_bit_Oconj__one__right,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X2: A] :
( ( bit_se5824344872417868541ns_and @ A @ X2 @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= X2 ) ) ).
% bit.conj_one_right
thf(fact_5106_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_5107_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_5108_and__nonnegative__int__iff,axiom,
! [K: int,L2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se5824344872417868541ns_and @ int @ K @ L2 ) )
= ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
| ( ord_less_eq @ int @ ( zero_zero @ int ) @ L2 ) ) ) ).
% and_nonnegative_int_iff
thf(fact_5109_and__negative__int__iff,axiom,
! [K: int,L2: int] :
( ( ord_less @ int @ ( bit_se5824344872417868541ns_and @ int @ K @ L2 ) @ ( zero_zero @ int ) )
= ( ( ord_less @ int @ K @ ( zero_zero @ int ) )
& ( ord_less @ int @ L2 @ ( zero_zero @ int ) ) ) ) ).
% and_negative_int_iff
thf(fact_5110_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_5111_and__numerals_I2_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [Y2: num] :
( ( bit_se5824344872417868541ns_and @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit1 @ Y2 ) ) )
= ( one_one @ A ) ) ) ).
% and_numerals(2)
thf(fact_5112_and__numerals_I8_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X2: num] :
( ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ ( bit1 @ X2 ) ) @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% and_numerals(8)
thf(fact_5113_and__numerals_I1_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [Y2: num] :
( ( bit_se5824344872417868541ns_and @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ Y2 ) ) )
= ( zero_zero @ A ) ) ) ).
% and_numerals(1)
thf(fact_5114_and__numerals_I5_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X2: num] :
( ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ ( bit0 @ X2 ) ) @ ( one_one @ A ) )
= ( zero_zero @ A ) ) ) ).
% and_numerals(5)
thf(fact_5115_and__numerals_I3_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X2: num,Y2: num] :
( ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ ( bit0 @ X2 ) ) @ ( numeral_numeral @ A @ ( bit0 @ Y2 ) ) )
= ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ X2 ) @ ( numeral_numeral @ A @ Y2 ) ) ) ) ) ).
% and_numerals(3)
thf(fact_5116_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_5117_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_5118_and__numerals_I4_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X2: num,Y2: num] :
( ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ ( bit0 @ X2 ) ) @ ( numeral_numeral @ A @ ( bit1 @ Y2 ) ) )
= ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ X2 ) @ ( numeral_numeral @ A @ Y2 ) ) ) ) ) ).
% and_numerals(4)
thf(fact_5119_and__numerals_I6_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X2: num,Y2: num] :
( ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ ( bit1 @ X2 ) ) @ ( numeral_numeral @ A @ ( bit0 @ Y2 ) ) )
= ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ X2 ) @ ( numeral_numeral @ A @ Y2 ) ) ) ) ) ).
% and_numerals(6)
thf(fact_5120_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_5121_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_5122_and__numerals_I7_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X2: num,Y2: num] :
( ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ ( bit1 @ X2 ) ) @ ( numeral_numeral @ A @ ( bit1 @ Y2 ) ) )
= ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ X2 ) @ ( numeral_numeral @ A @ Y2 ) ) ) ) ) ) ).
% and_numerals(7)
thf(fact_5123_of__int__and__eq,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [K: int,L2: int] :
( ( ring_1_of_int @ A @ ( bit_se5824344872417868541ns_and @ int @ K @ L2 ) )
= ( bit_se5824344872417868541ns_and @ A @ ( ring_1_of_int @ A @ K ) @ ( ring_1_of_int @ A @ L2 ) ) ) ) ).
% of_int_and_eq
thf(fact_5124_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_5125_bit__and__int__iff,axiom,
! [K: int,L2: int,N: nat] :
( ( bit_se5641148757651400278ts_bit @ int @ ( bit_se5824344872417868541ns_and @ int @ K @ L2 ) @ N )
= ( ( bit_se5641148757651400278ts_bit @ int @ K @ N )
& ( bit_se5641148757651400278ts_bit @ int @ L2 @ N ) ) ) ).
% bit_and_int_iff
thf(fact_5126_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_5127_and_Ocommute,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se5824344872417868541ns_and @ A )
= ( ^ [A4: A,B3: A] : ( bit_se5824344872417868541ns_and @ A @ B3 @ A4 ) ) ) ) ).
% and.commute
thf(fact_5128_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_5129_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_5130_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_5131_AND__upper2_H,axiom,
! [Y2: int,Z: int,X2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y2 )
=> ( ( ord_less_eq @ int @ Y2 @ Z )
=> ( ord_less_eq @ int @ ( bit_se5824344872417868541ns_and @ int @ X2 @ Y2 ) @ Z ) ) ) ).
% AND_upper2'
thf(fact_5132_AND__upper1_H,axiom,
! [Y2: int,Z: int,Ya: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y2 )
=> ( ( ord_less_eq @ int @ Y2 @ Z )
=> ( ord_less_eq @ int @ ( bit_se5824344872417868541ns_and @ int @ Y2 @ Ya ) @ Z ) ) ) ).
% AND_upper1'
thf(fact_5133_AND__upper2,axiom,
! [Y2: int,X2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y2 )
=> ( ord_less_eq @ int @ ( bit_se5824344872417868541ns_and @ int @ X2 @ Y2 ) @ Y2 ) ) ).
% AND_upper2
thf(fact_5134_AND__upper1,axiom,
! [X2: int,Y2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X2 )
=> ( ord_less_eq @ int @ ( bit_se5824344872417868541ns_and @ int @ X2 @ Y2 ) @ X2 ) ) ).
% AND_upper1
thf(fact_5135_AND__lower,axiom,
! [X2: int,Y2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X2 )
=> ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se5824344872417868541ns_and @ int @ X2 @ Y2 ) ) ) ).
% AND_lower
thf(fact_5136_AND__upper2_H_H,axiom,
! [Y2: int,Z: int,X2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y2 )
=> ( ( ord_less @ int @ Y2 @ Z )
=> ( ord_less @ int @ ( bit_se5824344872417868541ns_and @ int @ X2 @ Y2 ) @ Z ) ) ) ).
% AND_upper2''
thf(fact_5137_AND__upper1_H_H,axiom,
! [Y2: int,Z: int,Ya: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y2 )
=> ( ( ord_less @ int @ Y2 @ Z )
=> ( ord_less @ int @ ( bit_se5824344872417868541ns_and @ int @ Y2 @ Ya ) @ Z ) ) ) ).
% AND_upper1''
thf(fact_5138_and__less__eq,axiom,
! [L2: int,K: int] :
( ( ord_less @ int @ L2 @ ( zero_zero @ int ) )
=> ( ord_less_eq @ int @ ( bit_se5824344872417868541ns_and @ int @ K @ L2 ) @ K ) ) ).
% and_less_eq
thf(fact_5139_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_5140_even__and__iff__int,axiom,
! [K: int,L2: int] :
( ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ K @ L2 ) )
= ( ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K )
| ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ L2 ) ) ) ).
% even_and_iff_int
thf(fact_5141_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_5142_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_5143_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_5144_and__int__rec,axiom,
( ( bit_se5824344872417868541ns_and @ int )
= ( ^ [K2: int,L: int] :
( plus_plus @ int
@ ( zero_neq_one_of_bool @ int
@ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K2 )
& ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ L ) ) )
@ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( divide_divide @ int @ K2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% and_int_rec
thf(fact_5145_and__int__unfold,axiom,
( ( bit_se5824344872417868541ns_and @ int )
= ( ^ [K2: int,L: int] :
( if @ int
@ ( ( K2
= ( zero_zero @ int ) )
| ( L
= ( zero_zero @ int ) ) )
@ ( zero_zero @ int )
@ ( if @ int
@ ( K2
= ( uminus_uminus @ int @ ( one_one @ int ) ) )
@ L
@ ( if @ int
@ ( L
= ( uminus_uminus @ int @ ( one_one @ int ) ) )
@ K2
@ ( plus_plus @ int @ ( times_times @ int @ ( modulo_modulo @ int @ K2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ int @ L @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( divide_divide @ int @ K2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ).
% and_int_unfold
thf(fact_5146_sgn__integer__code,axiom,
( ( sgn_sgn @ code_integer )
= ( ^ [K2: code_integer] :
( if @ code_integer
@ ( K2
= ( zero_zero @ code_integer ) )
@ ( zero_zero @ code_integer )
@ ( if @ code_integer @ ( ord_less @ code_integer @ K2 @ ( zero_zero @ code_integer ) ) @ ( uminus_uminus @ code_integer @ ( one_one @ code_integer ) ) @ ( one_one @ code_integer ) ) ) ) ) ).
% sgn_integer_code
thf(fact_5147_and__int_Opsimps,axiom,
! [K: int,L2: int] :
( ( accp @ ( product_prod @ int @ int ) @ bit_and_int_rel @ ( product_Pair @ int @ int @ K @ L2 ) )
=> ( ( ( ( member @ int @ K @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) )
& ( member @ int @ L2 @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) )
=> ( ( bit_se5824344872417868541ns_and @ int @ K @ L2 )
= ( uminus_uminus @ int
@ ( zero_neq_one_of_bool @ int
@ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K )
& ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ L2 ) ) ) ) ) )
& ( ~ ( ( member @ int @ K @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) )
& ( member @ int @ L2 @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) )
=> ( ( bit_se5824344872417868541ns_and @ int @ K @ L2 )
= ( plus_plus @ int
@ ( zero_neq_one_of_bool @ int
@ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K )
& ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ L2 ) ) )
@ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( divide_divide @ int @ K @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% and_int.psimps
thf(fact_5148_and__int_Opelims,axiom,
! [X2: int,Xa: int,Y2: int] :
( ( ( bit_se5824344872417868541ns_and @ int @ X2 @ Xa )
= Y2 )
=> ( ( accp @ ( product_prod @ int @ int ) @ bit_and_int_rel @ ( product_Pair @ int @ int @ X2 @ Xa ) )
=> ~ ( ( ( ( ( member @ int @ X2 @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) )
& ( member @ int @ Xa @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) )
=> ( Y2
= ( uminus_uminus @ int
@ ( zero_neq_one_of_bool @ int
@ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ X2 )
& ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ Xa ) ) ) ) ) )
& ( ~ ( ( member @ int @ X2 @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) )
& ( member @ int @ Xa @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) )
=> ( Y2
= ( plus_plus @ int
@ ( zero_neq_one_of_bool @ int
@ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ X2 )
& ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ Xa ) ) )
@ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( divide_divide @ int @ X2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ Xa @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) )
=> ~ ( accp @ ( product_prod @ int @ int ) @ bit_and_int_rel @ ( product_Pair @ int @ int @ X2 @ Xa ) ) ) ) ) ).
% and_int.pelims
thf(fact_5149_word__bitwise__m1__simps_I2_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A] :
( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) @ X2 )
= X2 ) ) ).
% word_bitwise_m1_simps(2)
thf(fact_5150_word__and__max,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A] :
( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X2 @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) )
= X2 ) ) ).
% word_and_max
thf(fact_5151_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_5152_and__nat__numerals_I3_J,axiom,
! [X2: num] :
( ( bit_se5824344872417868541ns_and @ nat @ ( numeral_numeral @ nat @ ( bit0 @ X2 ) ) @ ( suc @ ( zero_zero @ nat ) ) )
= ( zero_zero @ nat ) ) ).
% and_nat_numerals(3)
thf(fact_5153_and__nat__numerals_I1_J,axiom,
! [Y2: num] :
( ( bit_se5824344872417868541ns_and @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( numeral_numeral @ nat @ ( bit0 @ Y2 ) ) )
= ( zero_zero @ nat ) ) ).
% and_nat_numerals(1)
thf(fact_5154_and__nat__numerals_I4_J,axiom,
! [X2: num] :
( ( bit_se5824344872417868541ns_and @ nat @ ( numeral_numeral @ nat @ ( bit1 @ X2 ) ) @ ( suc @ ( zero_zero @ nat ) ) )
= ( one_one @ nat ) ) ).
% and_nat_numerals(4)
thf(fact_5155_and__nat__numerals_I2_J,axiom,
! [Y2: num] :
( ( bit_se5824344872417868541ns_and @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( numeral_numeral @ nat @ ( bit1 @ Y2 ) ) )
= ( one_one @ nat ) ) ).
% and_nat_numerals(2)
thf(fact_5156_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_5157_word__bitwise__1__simps_I4_J,axiom,
! [D6: $tType] :
( ( type_len @ D6 )
=> ! [A2: num] :
( ( bit_se5824344872417868541ns_and @ ( word @ D6 ) @ ( numeral_numeral @ ( word @ D6 ) @ A2 ) @ ( one_one @ ( word @ D6 ) ) )
= ( ring_1_of_int @ ( word @ D6 ) @ ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ A2 ) @ ( one_one @ int ) ) ) ) ) ).
% word_bitwise_1_simps(4)
thf(fact_5158_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_5159_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_5160_word__no__log__defs_I4_J,axiom,
! [D6: $tType] :
( ( type_len @ D6 )
=> ! [A2: num,B2: num] :
( ( bit_se5824344872417868541ns_and @ ( word @ D6 ) @ ( numeral_numeral @ ( word @ D6 ) @ A2 ) @ ( uminus_uminus @ ( word @ D6 ) @ ( numeral_numeral @ ( word @ D6 ) @ B2 ) ) )
= ( ring_1_of_int @ ( word @ D6 ) @ ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ A2 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ B2 ) ) ) ) ) ) ).
% word_no_log_defs(4)
thf(fact_5161_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_5162_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_5163_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_5164_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_5165_word__log__esimps_I1_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A] :
( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X2 @ ( zero_zero @ ( word @ A ) ) )
= ( zero_zero @ ( word @ A ) ) ) ) ).
% word_log_esimps(1)
thf(fact_5166_word__log__esimps_I7_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A] :
( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ X2 )
= ( zero_zero @ ( word @ A ) ) ) ) ).
% word_log_esimps(7)
thf(fact_5167_word__and__max__word,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A,A2: word @ A] :
( ( X2
= ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) )
=> ( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ A2 @ X2 )
= A2 ) ) ) ).
% word_and_max_word
thf(fact_5168_and__nat__def,axiom,
( ( bit_se5824344872417868541ns_and @ nat )
= ( ^ [M5: nat,N3: nat] : ( nat2 @ ( bit_se5824344872417868541ns_and @ int @ ( semiring_1_of_nat @ int @ M5 ) @ ( semiring_1_of_nat @ int @ N3 ) ) ) ) ) ).
% and_nat_def
thf(fact_5169_zero__natural_Orsp,axiom,
( ( zero_zero @ nat )
= ( zero_zero @ nat ) ) ).
% zero_natural.rsp
thf(fact_5170_one__natural_Orsp,axiom,
( ( one_one @ nat )
= ( one_one @ nat ) ) ).
% one_natural.rsp
thf(fact_5171_zero__integer_Orsp,axiom,
( ( zero_zero @ int )
= ( zero_zero @ int ) ) ).
% zero_integer.rsp
thf(fact_5172_one__integer_Orsp,axiom,
( ( one_one @ int )
= ( one_one @ int ) ) ).
% one_integer.rsp
thf(fact_5173_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_5174_uminus__integer__code_I1_J,axiom,
( ( uminus_uminus @ code_integer @ ( zero_zero @ code_integer ) )
= ( zero_zero @ code_integer ) ) ).
% uminus_integer_code(1)
thf(fact_5175_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_5176_times__integer__code_I2_J,axiom,
! [L2: code_integer] :
( ( times_times @ code_integer @ ( zero_zero @ code_integer ) @ L2 )
= ( zero_zero @ code_integer ) ) ).
% times_integer_code(2)
thf(fact_5177_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_5178_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_5179_nth__is__and__neq__0,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) )
= ( ^ [X: word @ A,N3: nat] :
( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N3 ) )
!= ( zero_zero @ ( word @ A ) ) ) ) ) ) ).
% nth_is_and_neq_0
thf(fact_5180_and__neq__0__is__nth,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y2: word @ A,N: nat,X2: word @ A] :
( ( Y2
= ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) )
=> ( ( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X2 @ Y2 )
!= ( zero_zero @ ( word @ A ) ) )
= ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ X2 @ N ) ) ) ) ).
% and_neq_0_is_nth
thf(fact_5181_and__nat__unfold,axiom,
( ( bit_se5824344872417868541ns_and @ nat )
= ( ^ [M5: nat,N3: nat] :
( if @ nat
@ ( ( M5
= ( zero_zero @ nat ) )
| ( N3
= ( zero_zero @ nat ) ) )
@ ( zero_zero @ nat )
@ ( plus_plus @ nat @ ( times_times @ nat @ ( modulo_modulo @ nat @ M5 @ ( 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 @ M5 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ).
% and_nat_unfold
thf(fact_5182_and__nat__rec,axiom,
( ( bit_se5824344872417868541ns_and @ nat )
= ( ^ [M5: nat,N3: nat] :
( plus_plus @ nat
@ ( zero_neq_one_of_bool @ nat
@ ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M5 )
& ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) ) )
@ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ nat @ ( divide_divide @ nat @ M5 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% and_nat_rec
thf(fact_5183_abs__integer__code,axiom,
( ( abs_abs @ code_integer )
= ( ^ [K2: code_integer] : ( if @ code_integer @ ( ord_less @ code_integer @ K2 @ ( zero_zero @ code_integer ) ) @ ( uminus_uminus @ code_integer @ K2 ) @ K2 ) ) ) ).
% abs_integer_code
thf(fact_5184_less__integer__code_I1_J,axiom,
~ ( ord_less @ code_integer @ ( zero_zero @ code_integer ) @ ( zero_zero @ code_integer ) ) ).
% less_integer_code(1)
thf(fact_5185_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 ) )
=> ( ! [K3: int,L4: int] :
( ( accp @ ( product_prod @ int @ int ) @ bit_and_int_rel @ ( product_Pair @ int @ int @ K3 @ L4 ) )
=> ( ( ~ ( ( member @ int @ K3 @ ( 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 @ K3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L4 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) )
=> ( P @ K3 @ L4 ) ) )
=> ( P @ A0 @ A1 ) ) ) ).
% and_int.pinduct
thf(fact_5186_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_5187_plus__integer__code_I2_J,axiom,
! [L2: code_integer] :
( ( plus_plus @ code_integer @ ( zero_zero @ code_integer ) @ L2 )
= L2 ) ).
% plus_integer_code(2)
thf(fact_5188_minus__integer__code_I2_J,axiom,
! [L2: code_integer] :
( ( minus_minus @ code_integer @ ( zero_zero @ code_integer ) @ L2 )
= ( uminus_uminus @ code_integer @ L2 ) ) ).
% minus_integer_code(2)
thf(fact_5189_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_5190_upto_Opinduct,axiom,
! [A0: int,A1: int,P: int > int > $o] :
( ( accp @ ( product_prod @ int @ int ) @ upto_rel @ ( product_Pair @ int @ int @ A0 @ A1 ) )
=> ( ! [I2: int,J2: int] :
( ( accp @ ( product_prod @ int @ int ) @ upto_rel @ ( product_Pair @ int @ int @ I2 @ J2 ) )
=> ( ( ( ord_less_eq @ int @ I2 @ J2 )
=> ( P @ ( plus_plus @ int @ I2 @ ( one_one @ int ) ) @ J2 ) )
=> ( P @ I2 @ J2 ) ) )
=> ( P @ A0 @ A1 ) ) ) ).
% upto.pinduct
thf(fact_5191_integer__of__int__code,axiom,
( code_integer_of_int
= ( ^ [K2: int] :
( if @ code_integer @ ( ord_less @ int @ K2 @ ( zero_zero @ int ) ) @ ( uminus_uminus @ code_integer @ ( code_integer_of_int @ ( uminus_uminus @ int @ K2 ) ) )
@ ( if @ code_integer
@ ( K2
= ( zero_zero @ int ) )
@ ( zero_zero @ code_integer )
@ ( if @ code_integer
@ ( ( modulo_modulo @ int @ K2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) )
= ( zero_zero @ int ) )
@ ( times_times @ code_integer @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) @ ( code_integer_of_int @ ( divide_divide @ int @ K2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) )
@ ( plus_plus @ code_integer @ ( times_times @ code_integer @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) @ ( code_integer_of_int @ ( divide_divide @ int @ K2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) @ ( one_one @ code_integer ) ) ) ) ) ) ) ).
% integer_of_int_code
thf(fact_5192_Arg__def,axiom,
( arg
= ( ^ [Z5: complex] :
( if @ real
@ ( Z5
= ( zero_zero @ complex ) )
@ ( zero_zero @ real )
@ ( fChoice @ real
@ ^ [A4: real] :
( ( ( sgn_sgn @ complex @ Z5 )
= ( cis @ A4 ) )
& ( ord_less @ real @ ( uminus_uminus @ real @ pi ) @ A4 )
& ( ord_less_eq @ real @ A4 @ pi ) ) ) ) ) ) ).
% Arg_def
thf(fact_5193_some__insert__self,axiom,
! [A: $tType,S: set @ A] :
( ( S
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( insert @ A
@ ( fChoice @ A
@ ^ [X: A] : ( member @ A @ X @ S ) )
@ S )
= S ) ) ).
% some_insert_self
thf(fact_5194_divide__integer_Oabs__eq,axiom,
! [Xa: int,X2: int] :
( ( divide_divide @ code_integer @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X2 ) )
= ( code_integer_of_int @ ( divide_divide @ int @ Xa @ X2 ) ) ) ).
% divide_integer.abs_eq
thf(fact_5195_times__integer_Oabs__eq,axiom,
! [Xa: int,X2: int] :
( ( times_times @ code_integer @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X2 ) )
= ( code_integer_of_int @ ( times_times @ int @ Xa @ X2 ) ) ) ).
% times_integer.abs_eq
thf(fact_5196_some__elem,axiom,
! [A: $tType,S: set @ A] :
( ( S
!= ( bot_bot @ ( set @ A ) ) )
=> ( member @ A
@ ( fChoice @ A
@ ^ [X: A] : ( member @ A @ X @ S ) )
@ S ) ) ).
% some_elem
thf(fact_5197_verit__sko__forall__indirect2,axiom,
! [A: $tType,X2: A,P: A > $o,P4: A > $o] :
( ( X2
= ( fChoice @ A
@ ^ [X: A] :
~ ( P @ X ) ) )
=> ( ! [X3: A] :
( ( P @ X3 )
= ( P4 @ X3 ) )
=> ( ( ! [X7: A] : ( P4 @ X7 ) )
= ( P @ X2 ) ) ) ) ).
% verit_sko_forall_indirect2
thf(fact_5198_verit__sko__forall__indirect,axiom,
! [A: $tType,X2: A,P: A > $o] :
( ( X2
= ( fChoice @ A
@ ^ [X: A] :
~ ( P @ X ) ) )
=> ( ( ! [X7: A] : ( P @ X7 ) )
= ( P @ X2 ) ) ) ).
% verit_sko_forall_indirect
thf(fact_5199_verit__sko__ex__indirect2,axiom,
! [A: $tType,X2: A,P: A > $o,P4: A > $o] :
( ( X2
= ( fChoice @ A @ P ) )
=> ( ! [X3: A] :
( ( P @ X3 )
= ( P4 @ X3 ) )
=> ( ( ? [X7: A] : ( P4 @ X7 ) )
= ( P @ X2 ) ) ) ) ).
% verit_sko_ex_indirect2
thf(fact_5200_verit__sko__ex__indirect,axiom,
! [A: $tType,X2: A,P: A > $o] :
( ( X2
= ( fChoice @ A @ P ) )
=> ( ( ? [X7: A] : ( P @ X7 ) )
= ( P @ X2 ) ) ) ).
% verit_sko_ex_indirect
thf(fact_5201_verit__sko__forall_H_H,axiom,
! [A: $tType,B5: A,A3: A,P: A > $o] :
( ( B5 = A3 )
=> ( ( ( fChoice @ A @ P )
= A3 )
= ( ( fChoice @ A @ P )
= B5 ) ) ) ).
% verit_sko_forall''
thf(fact_5202_verit__sko__forall_H,axiom,
! [A: $tType,P: A > $o,A3: $o] :
( ( ( P
@ ( fChoice @ A
@ ^ [X: A] :
~ ( P @ X ) ) )
= A3 )
=> ( ( ! [X7: A] : ( P @ X7 ) )
= A3 ) ) ).
% verit_sko_forall'
thf(fact_5203_verit__sko__forall,axiom,
! [A: $tType] :
( ( ^ [P6: A > $o] :
! [X8: A] : ( P6 @ X8 ) )
= ( ^ [P7: A > $o] :
( P7
@ ( fChoice @ A
@ ^ [X: A] :
~ ( P7 @ X ) ) ) ) ) ).
% verit_sko_forall
thf(fact_5204_verit__sko__ex_H,axiom,
! [A: $tType,P: A > $o,A3: $o] :
( ( ( P @ ( fChoice @ A @ P ) )
= A3 )
=> ( ( ? [X7: A] : ( P @ X7 ) )
= A3 ) ) ).
% verit_sko_ex'
thf(fact_5205_zero__integer__def,axiom,
( ( zero_zero @ code_integer )
= ( code_integer_of_int @ ( zero_zero @ int ) ) ) ).
% zero_integer_def
thf(fact_5206_one__integer__def,axiom,
( ( one_one @ code_integer )
= ( code_integer_of_int @ ( one_one @ int ) ) ) ).
% one_integer_def
thf(fact_5207_plus__integer_Oabs__eq,axiom,
! [Xa: int,X2: int] :
( ( plus_plus @ code_integer @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X2 ) )
= ( code_integer_of_int @ ( plus_plus @ int @ Xa @ X2 ) ) ) ).
% plus_integer.abs_eq
thf(fact_5208_some__equality,axiom,
! [A: $tType,P: A > $o,A2: A] :
( ( P @ A2 )
=> ( ! [X3: A] :
( ( P @ X3 )
=> ( X3 = A2 ) )
=> ( ( fChoice @ A @ P )
= A2 ) ) ) ).
% some_equality
thf(fact_5209_some__eq__trivial,axiom,
! [A: $tType,X2: A] :
( ( fChoice @ A
@ ^ [Y: A] : Y = X2 )
= X2 ) ).
% some_eq_trivial
thf(fact_5210_some__sym__eq__trivial,axiom,
! [A: $tType,X2: A] :
( ( fChoice @ A
@ ( ^ [Y5: A,Z3: A] : Y5 = Z3
@ X2 ) )
= X2 ) ).
% some_sym_eq_trivial
thf(fact_5211_some1__equality,axiom,
! [A: $tType,P: A > $o,A2: A] :
( ? [X4: A] :
( ( P @ X4 )
& ! [Y3: A] :
( ( P @ Y3 )
=> ( Y3 = X4 ) ) )
=> ( ( P @ A2 )
=> ( ( fChoice @ A @ P )
= A2 ) ) ) ).
% some1_equality
thf(fact_5212_some__eq__ex,axiom,
! [A: $tType,P: A > $o] :
( ( P @ ( fChoice @ A @ P ) )
= ( ? [X7: A] : ( P @ X7 ) ) ) ).
% some_eq_ex
thf(fact_5213_someI2__bex,axiom,
! [A: $tType,A3: set @ A,P: A > $o,Q: A > $o] :
( ? [X4: A] :
( ( member @ A @ X4 @ A3 )
& ( P @ X4 ) )
=> ( ! [X3: A] :
( ( ( member @ A @ X3 @ A3 )
& ( P @ X3 ) )
=> ( Q @ X3 ) )
=> ( Q
@ ( fChoice @ A
@ ^ [X: A] :
( ( member @ A @ X @ A3 )
& ( P @ X ) ) ) ) ) ) ).
% someI2_bex
thf(fact_5214_someI2__ex,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ? [X_12: A] : ( P @ X_12 )
=> ( ! [X3: A] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( Q @ ( fChoice @ A @ P ) ) ) ) ).
% someI2_ex
thf(fact_5215_someI__ex,axiom,
! [A: $tType,P: A > $o] :
( ? [X_12: A] : ( P @ X_12 )
=> ( P @ ( fChoice @ A @ P ) ) ) ).
% someI_ex
thf(fact_5216_someI2,axiom,
! [A: $tType,P: A > $o,A2: A,Q: A > $o] :
( ( P @ A2 )
=> ( ! [X3: A] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( Q @ ( fChoice @ A @ P ) ) ) ) ).
% someI2
thf(fact_5217_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] :
! [N11: nat] :
( ( P @ N11 @ ( F5 @ N11 ) )
& ( Q @ N11 @ ( F5 @ N11 ) @ ( F5 @ ( suc @ N11 ) ) ) ) ) ) ).
% dependent_nat_choice
thf(fact_5218_some__in__eq,axiom,
! [A: $tType,A3: set @ A] :
( ( member @ A
@ ( fChoice @ A
@ ^ [X: A] : ( member @ A @ X @ A3 ) )
@ A3 )
= ( A3
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% some_in_eq
thf(fact_5219_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_5220_fun__of__rel__def,axiom,
! [A: $tType,B: $tType] :
( ( fun_of_rel @ B @ A )
= ( ^ [R6: set @ ( product_prod @ B @ A ),X: B] :
( fChoice @ A
@ ^ [Y: A] : ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X @ Y ) @ R6 ) ) ) ) ).
% fun_of_rel_def
thf(fact_5221_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_5222_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_5223_mask__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se2239418461657761734s_mask @ A @ ( zero_zero @ nat ) )
= ( zero_zero @ A ) ) ) ).
% mask_0
thf(fact_5224_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_5225_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_5226_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_5227_pred__numeral__simps_I2_J,axiom,
! [K: num] :
( ( pred_numeral @ ( bit0 @ K ) )
= ( numeral_numeral @ nat @ ( bitM @ K ) ) ) ).
% pred_numeral_simps(2)
thf(fact_5228_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_5229_semiring__norm_I26_J,axiom,
( ( bitM @ one2 )
= one2 ) ).
% semiring_norm(26)
thf(fact_5230_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_5231_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_5232_mask__twice2,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,M: nat,X2: word @ A] :
( ( ord_less_eq @ nat @ N @ M )
=> ( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X2 @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ M ) ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) )
= ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X2 @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) ) ) ) ).
% mask_twice2
thf(fact_5233_mask__eqs_I1_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: word @ A,N: nat,B2: word @ A] :
( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ A2 @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) @ B2 ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) )
= ( bit_se5824344872417868541ns_and @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ A2 @ B2 ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) ) ) ).
% mask_eqs(1)
thf(fact_5234_mask__eqs_I2_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: word @ A,B2: word @ A,N: nat] :
( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ A2 @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ B2 @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) )
= ( bit_se5824344872417868541ns_and @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ A2 @ B2 ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) ) ) ).
% mask_eqs(2)
thf(fact_5235_mask__eqs_I7_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: word @ A,N: nat,B2: word @ A] :
( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ A2 @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ B2 @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) )
= ( bit_se5824344872417868541ns_and @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ A2 @ B2 ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) ) ) ).
% mask_eqs(7)
thf(fact_5236_mask__power__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A,N: nat,K: nat] :
( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ ( power_power @ ( word @ A ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X2 @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) @ K ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) )
= ( bit_se5824344872417868541ns_and @ ( word @ A ) @ ( power_power @ ( word @ A ) @ X2 @ K ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) ) ) ).
% mask_power_eq
thf(fact_5237_mask__eqs_I5_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: word @ A,B2: word @ A,N: nat] :
( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ ( times_times @ ( word @ A ) @ A2 @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ B2 @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) )
= ( bit_se5824344872417868541ns_and @ ( word @ A ) @ ( times_times @ ( word @ A ) @ A2 @ B2 ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) ) ) ).
% mask_eqs(5)
thf(fact_5238_mask__eqs_I6_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [B2: word @ A,N: nat,A2: word @ A] :
( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ ( times_times @ ( word @ A ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ B2 @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) @ A2 ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) )
= ( bit_se5824344872417868541ns_and @ ( word @ A ) @ ( times_times @ ( word @ A ) @ B2 @ A2 ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) ) ) ).
% mask_eqs(6)
thf(fact_5239_mask__eqs_I9_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: word @ A,N: nat,B2: word @ A] :
( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ ( times_times @ ( word @ A ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ A2 @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ B2 @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) )
= ( bit_se5824344872417868541ns_and @ ( word @ A ) @ ( times_times @ ( word @ A ) @ A2 @ B2 ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) ) ) ).
% mask_eqs(9)
thf(fact_5240_semiring__norm_I27_J,axiom,
! [N: num] :
( ( bitM @ ( bit0 @ N ) )
= ( bit1 @ ( bitM @ N ) ) ) ).
% semiring_norm(27)
thf(fact_5241_semiring__norm_I28_J,axiom,
! [N: num] :
( ( bitM @ ( bit1 @ N ) )
= ( bit1 @ ( bit0 @ N ) ) ) ).
% semiring_norm(28)
thf(fact_5242_mask__1,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se2239418461657761734s_mask @ A @ ( one_one @ nat ) )
= ( one_one @ A ) ) ) ).
% mask_1
thf(fact_5243_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_5244_inc__BitM__eq,axiom,
! [N: num] :
( ( inc @ ( bitM @ N ) )
= ( bit0 @ N ) ) ).
% inc_BitM_eq
thf(fact_5245_BitM__inc__eq,axiom,
! [N: num] :
( ( bitM @ ( inc @ N ) )
= ( bit1 @ N ) ) ).
% BitM_inc_eq
thf(fact_5246_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_5247_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_5248_one__plus__BitM,axiom,
! [N: num] :
( ( plus_plus @ num @ one2 @ ( bitM @ N ) )
= ( bit0 @ N ) ) ).
% one_plus_BitM
thf(fact_5249_BitM__plus__one,axiom,
! [N: num] :
( ( plus_plus @ num @ ( bitM @ N ) @ one2 )
= ( bit0 @ N ) ) ).
% BitM_plus_one
thf(fact_5250_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_5251_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_5252_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_5253_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_5254_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_5255_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_5256_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_5257_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_5258_less__mask__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A,N: nat] :
( ( ord_less @ ( word @ A ) @ X2 @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) )
=> ( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X2 @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) )
= X2 ) ) ) ).
% less_mask_eq
thf(fact_5259_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_5260_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_5261_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_5262_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_5263_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_5264_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_5265_add__mask__fold,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A,N: nat] :
( ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ X2 @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) ) @ ( one_one @ ( word @ A ) ) )
= ( plus_plus @ ( word @ A ) @ X2 @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) ) ) ).
% add_mask_fold
thf(fact_5266_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_5267_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_5268_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_5269_and__mask__less__size,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,X2: word @ A] :
( ( ord_less @ nat @ N @ ( size_size @ ( word @ A ) @ X2 ) )
=> ( ord_less @ ( word @ A ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X2 @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% and_mask_less_size
thf(fact_5270_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_5271_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_5272_word__mod__2p__is__mask,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,X2: word @ A] :
( ( ord_less @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) )
=> ( ( modulo_modulo @ ( word @ A ) @ X2 @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) )
= ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X2 @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) ) ) ) ).
% word_mod_2p_is_mask
thf(fact_5273_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_5274_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_5275_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_5276_option_Osize__gen_I2_J,axiom,
! [A: $tType,X2: A > nat,X23: A] :
( ( size_option @ A @ X2 @ ( some @ A @ X23 ) )
= ( plus_plus @ nat @ ( X2 @ X23 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% option.size_gen(2)
thf(fact_5277_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 )
@ ^ [Q4: A,R5: A] : ( product_Pair @ A @ A @ Q4 @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ R5 ) @ ( one_one @ A ) ) )
@ ( unique8689654367752047608divmod @ A @ M @ N ) ) ) ) ).
% divmod_algorithm_code(6)
thf(fact_5278_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_5279_bit_Odouble__compl,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X2: A] :
( ( bit_ri4277139882892585799ns_not @ A @ ( bit_ri4277139882892585799ns_not @ A @ X2 ) )
= X2 ) ) ).
% bit.double_compl
thf(fact_5280_bit_Ocompl__eq__compl__iff,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X2: A,Y2: A] :
( ( ( bit_ri4277139882892585799ns_not @ A @ X2 )
= ( bit_ri4277139882892585799ns_not @ A @ Y2 ) )
= ( X2 = Y2 ) ) ) ).
% bit.compl_eq_compl_iff
thf(fact_5281_bit_Oconj__cancel__right,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X2: A] :
( ( bit_se5824344872417868541ns_and @ A @ X2 @ ( bit_ri4277139882892585799ns_not @ A @ X2 ) )
= ( zero_zero @ A ) ) ) ).
% bit.conj_cancel_right
thf(fact_5282_bit_Oconj__cancel__left,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X2: A] :
( ( bit_se5824344872417868541ns_and @ A @ ( bit_ri4277139882892585799ns_not @ A @ X2 ) @ X2 )
= ( zero_zero @ A ) ) ) ).
% bit.conj_cancel_left
thf(fact_5283_word__and__not,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A] :
( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X2 @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ X2 ) )
= ( zero_zero @ ( word @ A ) ) ) ) ).
% word_and_not
thf(fact_5284_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_5285_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_5286_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_5287_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_5288_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_5289_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_5290_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_5291_word__add__not,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A] :
( ( plus_plus @ ( word @ A ) @ X2 @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ X2 ) )
= ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) ) ) ).
% word_add_not
thf(fact_5292_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_5293_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_5294_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_5295_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 )
@ ^ [Q4: A,R5: A] : ( product_Pair @ A @ A @ Q4 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ R5 ) )
@ ( unique8689654367752047608divmod @ A @ M @ N ) ) ) ) ).
% divmod_algorithm_code(5)
thf(fact_5296_nat__mask__eq,axiom,
! [N: nat] :
( ( nat2 @ ( bit_se2239418461657761734s_mask @ int @ N ) )
= ( bit_se2239418461657761734s_mask @ nat @ N ) ) ).
% nat_mask_eq
thf(fact_5297_less__eq__mask,axiom,
! [N: nat] : ( ord_less_eq @ nat @ N @ ( bit_se2239418461657761734s_mask @ nat @ N ) ) ).
% less_eq_mask
thf(fact_5298_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_5299_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_5300_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_5301_nested__case__prod__simp,axiom,
! [A: $tType,D6: $tType,C: $tType,B: $tType] :
( ( product_case_prod @ B @ C @ ( D6 > A ) )
= ( ^ [F4: B > C > D6 > A,X: product_prod @ B @ C,Y: D6] :
( product_case_prod @ B @ C @ A
@ ^ [A4: B,B3: C] : ( F4 @ A4 @ B3 @ Y )
@ X ) ) ) ).
% nested_case_prod_simp
thf(fact_5302_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_5303_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_5304_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_5305_mask__eq__0__eq__x,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A,W: word @ A] :
( ( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X2 @ W )
= ( zero_zero @ ( word @ A ) ) )
= ( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X2 @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ W ) )
= X2 ) ) ) ).
% mask_eq_0_eq_x
thf(fact_5306_mask__eq__x__eq__0,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A,W: word @ A] :
( ( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X2 @ W )
= X2 )
= ( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X2 @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ W ) )
= ( zero_zero @ ( word @ A ) ) ) ) ) ).
% mask_eq_x_eq_0
thf(fact_5307_word__plus__and__or__coroll2,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A,W: word @ A] :
( ( plus_plus @ ( word @ A ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X2 @ W ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X2 @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ W ) ) )
= X2 ) ) ).
% word_plus_and_or_coroll2
thf(fact_5308_mask__nonnegative__int,axiom,
! [N: nat] : ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se2239418461657761734s_mask @ int @ N ) ) ).
% mask_nonnegative_int
thf(fact_5309_not__mask__negative__int,axiom,
! [N: nat] :
~ ( ord_less @ int @ ( bit_se2239418461657761734s_mask @ int @ N ) @ ( zero_zero @ int ) ) ).
% not_mask_negative_int
thf(fact_5310_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_5311_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_5312_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_5313_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_5314_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_5315_mask__lower__twice,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,M: nat,X2: word @ A] :
( ( ord_less_eq @ nat @ N @ M )
=> ( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X2 @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) ) @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ M ) ) )
= ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X2 @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ M ) ) ) ) ) ) ).
% mask_lower_twice
thf(fact_5316_mask__out__first__mask__some,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A,N: nat,Y2: word @ A,M: nat] :
( ( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X2 @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) )
= Y2 )
=> ( ( ord_less_eq @ nat @ N @ M )
=> ( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X2 @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ M ) ) )
= ( bit_se5824344872417868541ns_and @ ( word @ A ) @ Y2 @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ M ) ) ) ) ) ) ) ).
% mask_out_first_mask_some
thf(fact_5317_NOT__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( bit_ri4277139882892585799ns_not @ ( word @ A ) )
= ( ^ [X: word @ A] : ( minus_minus @ ( word @ A ) @ ( uminus_uminus @ ( word @ A ) @ X ) @ ( one_one @ ( word @ A ) ) ) ) ) ) ).
% NOT_eq
thf(fact_5318_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_5319_AND__NOT__mask__plus__AND__mask__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A,N: nat] :
( ( plus_plus @ ( word @ A ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ W @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ W @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) )
= W ) ) ).
% AND_NOT_mask_plus_AND_mask_eq
thf(fact_5320_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_5321_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_5322_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_5323_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_5324_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_5325_multiple__mask__trivia,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [M: nat,N: nat,X2: word @ A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( plus_plus @ ( word @ A ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X2 @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X2 @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ M ) ) ) ) )
= ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X2 @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ M ) ) ) ) ) ) ).
% multiple_mask_trivia
thf(fact_5326_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_5327_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_5328_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_5329_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_5330_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_5331_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_5332_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_5333_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_5334_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_5335_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_5336_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_5337_option_Osize__gen_I1_J,axiom,
! [A: $tType,X2: A > nat] :
( ( size_option @ A @ X2 @ ( none @ A ) )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% option.size_gen(1)
thf(fact_5338_divmod__step__nat__def,axiom,
( ( unique1321980374590559556d_step @ nat )
= ( ^ [L: num] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [Q4: nat,R5: nat] : ( if @ ( product_prod @ nat @ nat ) @ ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ L ) @ R5 ) @ ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Q4 ) @ ( one_one @ nat ) ) @ ( minus_minus @ nat @ R5 @ ( numeral_numeral @ nat @ L ) ) ) @ ( product_Pair @ nat @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Q4 ) @ R5 ) ) ) ) ) ).
% divmod_step_nat_def
thf(fact_5339_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_5340_divmod__step__int__def,axiom,
( ( unique1321980374590559556d_step @ int )
= ( ^ [L: num] :
( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [Q4: int,R5: int] : ( if @ ( product_prod @ int @ int ) @ ( ord_less_eq @ int @ ( numeral_numeral @ int @ L ) @ R5 ) @ ( product_Pair @ int @ int @ ( plus_plus @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ Q4 ) @ ( one_one @ int ) ) @ ( minus_minus @ int @ R5 @ ( numeral_numeral @ int @ L ) ) ) @ ( product_Pair @ int @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ Q4 ) @ R5 ) ) ) ) ) ).
% divmod_step_int_def
thf(fact_5341_divmod__step__integer__def,axiom,
( ( unique1321980374590559556d_step @ code_integer )
= ( ^ [L: num] :
( product_case_prod @ code_integer @ code_integer @ ( product_prod @ code_integer @ code_integer )
@ ^ [Q4: code_integer,R5: code_integer] : ( if @ ( product_prod @ code_integer @ code_integer ) @ ( ord_less_eq @ code_integer @ ( numeral_numeral @ code_integer @ L ) @ R5 ) @ ( product_Pair @ code_integer @ code_integer @ ( plus_plus @ code_integer @ ( times_times @ code_integer @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) @ Q4 ) @ ( one_one @ code_integer ) ) @ ( minus_minus @ code_integer @ R5 @ ( numeral_numeral @ code_integer @ L ) ) ) @ ( product_Pair @ code_integer @ code_integer @ ( times_times @ code_integer @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) @ Q4 ) @ R5 ) ) ) ) ) ).
% divmod_step_integer_def
thf(fact_5342_divmod__step__def,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ( ( unique1321980374590559556d_step @ A )
= ( ^ [L: num] :
( product_case_prod @ A @ A @ ( product_prod @ A @ A )
@ ^ [Q4: A,R5: A] : ( if @ ( product_prod @ A @ A ) @ ( ord_less_eq @ A @ ( numeral_numeral @ A @ L ) @ R5 ) @ ( product_Pair @ A @ A @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Q4 ) @ ( one_one @ A ) ) @ ( minus_minus @ A @ R5 @ ( numeral_numeral @ A @ L ) ) ) @ ( product_Pair @ A @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Q4 ) @ R5 ) ) ) ) ) ) ).
% divmod_step_def
thf(fact_5343_case__prod__conv,axiom,
! [B: $tType,A: $tType,C: $tType,F2: B > C > A,A2: B,B2: C] :
( ( product_case_prod @ B @ C @ A @ F2 @ ( product_Pair @ B @ C @ A2 @ B2 ) )
= ( F2 @ A2 @ B2 ) ) ).
% case_prod_conv
thf(fact_5344_divmod__nat__if,axiom,
( divmod_nat
= ( ^ [M5: nat,N3: nat] :
( if @ ( product_prod @ nat @ nat )
@ ( ( N3
= ( zero_zero @ nat ) )
| ( ord_less @ nat @ M5 @ N3 ) )
@ ( product_Pair @ nat @ nat @ ( zero_zero @ nat ) @ M5 )
@ ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [Q4: nat] : ( product_Pair @ nat @ nat @ ( suc @ Q4 ) )
@ ( divmod_nat @ ( minus_minus @ nat @ M5 @ N3 ) @ N3 ) ) ) ) ) ).
% divmod_nat_if
thf(fact_5345_neg__mask__add__mask,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A,N: nat] :
( ( plus_plus @ ( word @ A ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X2 @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) ) @ ( minus_minus @ ( word @ A ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) @ ( one_one @ ( word @ A ) ) ) )
= ( bit_se1065995026697491101ons_or @ ( word @ A ) @ X2 @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) ) ) ).
% neg_mask_add_mask
thf(fact_5346_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_5347_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_5348_or_Oidem,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A] :
( ( bit_se1065995026697491101ons_or @ A @ A2 @ A2 )
= A2 ) ) ).
% or.idem
thf(fact_5349_case__prodI,axiom,
! [A: $tType,B: $tType,F2: A > B > $o,A2: A,B2: B] :
( ( F2 @ A2 @ B2 )
=> ( product_case_prod @ A @ B @ $o @ F2 @ ( product_Pair @ A @ B @ A2 @ B2 ) ) ) ).
% case_prodI
thf(fact_5350_case__prodI2,axiom,
! [B: $tType,A: $tType,P2: product_prod @ A @ B,C2: A > B > $o] :
( ! [A5: A,B4: B] :
( ( P2
= ( product_Pair @ A @ B @ A5 @ B4 ) )
=> ( C2 @ A5 @ B4 ) )
=> ( product_case_prod @ A @ B @ $o @ C2 @ P2 ) ) ).
% case_prodI2
thf(fact_5351_case__prodI2_H,axiom,
! [A: $tType,B: $tType,C: $tType,P2: product_prod @ A @ B,C2: A > B > C > $o,X2: C] :
( ! [A5: A,B4: B] :
( ( ( product_Pair @ A @ B @ A5 @ B4 )
= P2 )
=> ( C2 @ A5 @ B4 @ X2 ) )
=> ( product_case_prod @ A @ B @ ( C > $o ) @ C2 @ P2 @ X2 ) ) ).
% case_prodI2'
thf(fact_5352_mem__case__prodI,axiom,
! [A: $tType,B: $tType,C: $tType,Z: A,C2: B > C > ( set @ A ),A2: B,B2: C] :
( ( member @ A @ Z @ ( C2 @ A2 @ B2 ) )
=> ( member @ A @ Z @ ( product_case_prod @ B @ C @ ( set @ A ) @ C2 @ ( product_Pair @ B @ C @ A2 @ B2 ) ) ) ) ).
% mem_case_prodI
thf(fact_5353_mem__case__prodI2,axiom,
! [C: $tType,B: $tType,A: $tType,P2: product_prod @ A @ B,Z: C,C2: A > B > ( set @ C )] :
( ! [A5: A,B4: B] :
( ( P2
= ( product_Pair @ A @ B @ A5 @ B4 ) )
=> ( member @ C @ Z @ ( C2 @ A5 @ B4 ) ) )
=> ( member @ C @ Z @ ( product_case_prod @ A @ B @ ( set @ C ) @ C2 @ P2 ) ) ) ).
% mem_case_prodI2
thf(fact_5354_The__split__eq,axiom,
! [A: $tType,B: $tType,X2: A,Y2: B] :
( ( the @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [X9: A,Y8: B] :
( ( X2 = X9 )
& ( Y2 = Y8 ) ) ) )
= ( product_Pair @ A @ B @ X2 @ Y2 ) ) ).
% The_split_eq
thf(fact_5355_Eps__case__prod__eq,axiom,
! [A: $tType,B: $tType,X2: A,Y2: B] :
( ( fChoice @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [X9: A,Y8: B] :
( ( X2 = X9 )
& ( Y2 = Y8 ) ) ) )
= ( product_Pair @ A @ B @ X2 @ Y2 ) ) ).
% Eps_case_prod_eq
thf(fact_5356_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_5357_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_5358_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_5359_bit_Odisj__one__right,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X2: A] :
( ( bit_se1065995026697491101ons_or @ A @ X2 @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% bit.disj_one_right
thf(fact_5360_bit_Odisj__one__left,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X2: A] :
( ( bit_se1065995026697491101ons_or @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ X2 )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% bit.disj_one_left
thf(fact_5361_bit_Ode__Morgan__conj,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X2: A,Y2: A] :
( ( bit_ri4277139882892585799ns_not @ A @ ( bit_se5824344872417868541ns_and @ A @ X2 @ Y2 ) )
= ( bit_se1065995026697491101ons_or @ A @ ( bit_ri4277139882892585799ns_not @ A @ X2 ) @ ( bit_ri4277139882892585799ns_not @ A @ Y2 ) ) ) ) ).
% bit.de_Morgan_conj
thf(fact_5362_bit_Ode__Morgan__disj,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X2: A,Y2: A] :
( ( bit_ri4277139882892585799ns_not @ A @ ( bit_se1065995026697491101ons_or @ A @ X2 @ Y2 ) )
= ( bit_se5824344872417868541ns_and @ A @ ( bit_ri4277139882892585799ns_not @ A @ X2 ) @ ( bit_ri4277139882892585799ns_not @ A @ Y2 ) ) ) ) ).
% bit.de_Morgan_disj
thf(fact_5363_word__plus__and__or,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A,Y2: word @ A] :
( ( plus_plus @ ( word @ A ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X2 @ Y2 ) @ ( bit_se1065995026697491101ons_or @ ( word @ A ) @ X2 @ Y2 ) )
= ( plus_plus @ ( word @ A ) @ X2 @ Y2 ) ) ) ).
% word_plus_and_or
thf(fact_5364_word__or__max,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A] :
( ( bit_se1065995026697491101ons_or @ ( word @ A ) @ X2 @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) )
= ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) ) ) ).
% word_or_max
thf(fact_5365_word__bitwise__m1__simps_I4_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A] :
( ( bit_se1065995026697491101ons_or @ ( word @ A ) @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) @ X2 )
= ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) ) ) ).
% word_bitwise_m1_simps(4)
thf(fact_5366_or__numerals_I2_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [Y2: num] :
( ( bit_se1065995026697491101ons_or @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit1 @ Y2 ) ) )
= ( numeral_numeral @ A @ ( bit1 @ Y2 ) ) ) ) ).
% or_numerals(2)
thf(fact_5367_or__numerals_I8_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X2: num] :
( ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ ( bit1 @ X2 ) ) @ ( one_one @ A ) )
= ( numeral_numeral @ A @ ( bit1 @ X2 ) ) ) ) ).
% or_numerals(8)
thf(fact_5368_bit_Odisj__cancel__left,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X2: A] :
( ( bit_se1065995026697491101ons_or @ A @ ( bit_ri4277139882892585799ns_not @ A @ X2 ) @ X2 )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% bit.disj_cancel_left
thf(fact_5369_bit_Odisj__cancel__right,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X2: A] :
( ( bit_se1065995026697491101ons_or @ A @ X2 @ ( bit_ri4277139882892585799ns_not @ A @ X2 ) )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% bit.disj_cancel_right
thf(fact_5370_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_5371_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_5372_word__or__not,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A] :
( ( bit_se1065995026697491101ons_or @ ( word @ A ) @ X2 @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ X2 ) )
= ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) ) ) ).
% word_or_not
thf(fact_5373_or__numerals_I3_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X2: num,Y2: num] :
( ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ ( bit0 @ X2 ) ) @ ( numeral_numeral @ A @ ( bit0 @ Y2 ) ) )
= ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ X2 ) @ ( numeral_numeral @ A @ Y2 ) ) ) ) ) ).
% or_numerals(3)
thf(fact_5374_or__numerals_I1_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [Y2: num] :
( ( bit_se1065995026697491101ons_or @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ Y2 ) ) )
= ( numeral_numeral @ A @ ( bit1 @ Y2 ) ) ) ) ).
% or_numerals(1)
thf(fact_5375_or__numerals_I5_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X2: num] :
( ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ ( bit0 @ X2 ) ) @ ( one_one @ A ) )
= ( numeral_numeral @ A @ ( bit1 @ X2 ) ) ) ) ).
% or_numerals(5)
thf(fact_5376_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_5377_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_5378_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_5379_or__numerals_I4_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X2: num,Y2: num] :
( ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ ( bit0 @ X2 ) ) @ ( numeral_numeral @ A @ ( bit1 @ Y2 ) ) )
= ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ X2 ) @ ( numeral_numeral @ A @ Y2 ) ) ) ) ) ) ).
% or_numerals(4)
thf(fact_5380_or__numerals_I6_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X2: num,Y2: num] :
( ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ ( bit1 @ X2 ) ) @ ( numeral_numeral @ A @ ( bit0 @ Y2 ) ) )
= ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ X2 ) @ ( numeral_numeral @ A @ Y2 ) ) ) ) ) ) ).
% or_numerals(6)
thf(fact_5381_or__numerals_I7_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X2: num,Y2: num] :
( ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ ( bit1 @ X2 ) ) @ ( numeral_numeral @ A @ ( bit1 @ Y2 ) ) )
= ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ X2 ) @ ( numeral_numeral @ A @ Y2 ) ) ) ) ) ) ).
% or_numerals(7)
thf(fact_5382_case__prodD,axiom,
! [A: $tType,B: $tType,F2: A > B > $o,A2: A,B2: B] :
( ( product_case_prod @ A @ B @ $o @ F2 @ ( product_Pair @ A @ B @ A2 @ B2 ) )
=> ( F2 @ A2 @ B2 ) ) ).
% case_prodD
thf(fact_5383_case__prodE,axiom,
! [A: $tType,B: $tType,C2: A > B > $o,P2: product_prod @ A @ B] :
( ( product_case_prod @ A @ B @ $o @ C2 @ P2 )
=> ~ ! [X3: A,Y3: B] :
( ( P2
= ( product_Pair @ A @ B @ X3 @ Y3 ) )
=> ~ ( C2 @ X3 @ Y3 ) ) ) ).
% case_prodE
thf(fact_5384_case__prodD_H,axiom,
! [B: $tType,A: $tType,C: $tType,R: A > B > C > $o,A2: A,B2: B,C2: C] :
( ( product_case_prod @ A @ B @ ( C > $o ) @ R @ ( product_Pair @ A @ B @ A2 @ B2 ) @ C2 )
=> ( R @ A2 @ B2 @ C2 ) ) ).
% case_prodD'
thf(fact_5385_case__prodE_H,axiom,
! [B: $tType,A: $tType,C: $tType,C2: A > B > C > $o,P2: product_prod @ A @ B,Z: C] :
( ( product_case_prod @ A @ B @ ( C > $o ) @ C2 @ P2 @ Z )
=> ~ ! [X3: A,Y3: B] :
( ( P2
= ( product_Pair @ A @ B @ X3 @ Y3 ) )
=> ~ ( C2 @ X3 @ Y3 @ Z ) ) ) ).
% case_prodE'
thf(fact_5386_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_5387_split__paired__Eps,axiom,
! [B: $tType,A: $tType] :
( ( fChoice @ ( product_prod @ A @ B ) )
= ( ^ [P7: ( product_prod @ A @ B ) > $o] :
( fChoice @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [A4: A,B3: B] : ( P7 @ ( product_Pair @ A @ B @ A4 @ B3 ) ) ) ) ) ) ).
% split_paired_Eps
thf(fact_5388_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_5389_of__int__or__eq,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [K: int,L2: int] :
( ( ring_1_of_int @ A @ ( bit_se1065995026697491101ons_or @ int @ K @ L2 ) )
= ( bit_se1065995026697491101ons_or @ A @ ( ring_1_of_int @ A @ K ) @ ( ring_1_of_int @ A @ L2 ) ) ) ) ).
% of_int_or_eq
thf(fact_5390_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_5391_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_5392_or_Ocommute,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se1065995026697491101ons_or @ A )
= ( ^ [A4: A,B3: A] : ( bit_se1065995026697491101ons_or @ A @ B3 @ A4 ) ) ) ) ).
% or.commute
thf(fact_5393_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_5394_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_5395_bit_Odisj__zero__right,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X2: A] :
( ( bit_se1065995026697491101ons_or @ A @ X2 @ ( zero_zero @ A ) )
= X2 ) ) ).
% bit.disj_zero_right
thf(fact_5396_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_5397_word__log__esimps_I3_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A] :
( ( bit_se1065995026697491101ons_or @ ( word @ A ) @ X2 @ ( zero_zero @ ( word @ A ) ) )
= X2 ) ) ).
% word_log_esimps(3)
thf(fact_5398_word__log__esimps_I9_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A] :
( ( bit_se1065995026697491101ons_or @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ X2 )
= X2 ) ) ).
% word_log_esimps(9)
thf(fact_5399_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_5400_bit_Oconj__disj__distrib,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X2: A,Y2: A,Z: A] :
( ( bit_se5824344872417868541ns_and @ A @ X2 @ ( bit_se1065995026697491101ons_or @ A @ Y2 @ Z ) )
= ( bit_se1065995026697491101ons_or @ A @ ( bit_se5824344872417868541ns_and @ A @ X2 @ Y2 ) @ ( bit_se5824344872417868541ns_and @ A @ X2 @ Z ) ) ) ) ).
% bit.conj_disj_distrib
thf(fact_5401_bit_Odisj__conj__distrib,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X2: A,Y2: A,Z: A] :
( ( bit_se1065995026697491101ons_or @ A @ X2 @ ( bit_se5824344872417868541ns_and @ A @ Y2 @ Z ) )
= ( bit_se5824344872417868541ns_and @ A @ ( bit_se1065995026697491101ons_or @ A @ X2 @ Y2 ) @ ( bit_se1065995026697491101ons_or @ A @ X2 @ Z ) ) ) ) ).
% bit.disj_conj_distrib
thf(fact_5402_bit_Oconj__disj__distrib2,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [Y2: A,Z: A,X2: A] :
( ( bit_se5824344872417868541ns_and @ A @ ( bit_se1065995026697491101ons_or @ A @ Y2 @ Z ) @ X2 )
= ( bit_se1065995026697491101ons_or @ A @ ( bit_se5824344872417868541ns_and @ A @ Y2 @ X2 ) @ ( bit_se5824344872417868541ns_and @ A @ Z @ X2 ) ) ) ) ).
% bit.conj_disj_distrib2
thf(fact_5403_bit_Odisj__conj__distrib2,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [Y2: A,Z: A,X2: A] :
( ( bit_se1065995026697491101ons_or @ A @ ( bit_se5824344872417868541ns_and @ A @ Y2 @ Z ) @ X2 )
= ( bit_se5824344872417868541ns_and @ A @ ( bit_se1065995026697491101ons_or @ A @ Y2 @ X2 ) @ ( bit_se1065995026697491101ons_or @ A @ Z @ X2 ) ) ) ) ).
% bit.disj_conj_distrib2
thf(fact_5404_and__eq__not__not__or,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( ( bit_se5824344872417868541ns_and @ A )
= ( ^ [A4: A,B3: A] : ( bit_ri4277139882892585799ns_not @ A @ ( bit_se1065995026697491101ons_or @ A @ ( bit_ri4277139882892585799ns_not @ A @ A4 ) @ ( bit_ri4277139882892585799ns_not @ A @ B3 ) ) ) ) ) ) ).
% and_eq_not_not_or
thf(fact_5405_or__eq__not__not__and,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( ( bit_se1065995026697491101ons_or @ A )
= ( ^ [A4: A,B3: A] : ( bit_ri4277139882892585799ns_not @ A @ ( bit_se5824344872417868541ns_and @ A @ ( bit_ri4277139882892585799ns_not @ A @ A4 ) @ ( bit_ri4277139882892585799ns_not @ A @ B3 ) ) ) ) ) ) ).
% or_eq_not_not_and
thf(fact_5406_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
@ ^ [I3: nat,J3: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ I3 @ J3 ) @ N ) ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K2: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( G @ I3 @ ( minus_minus @ nat @ K2 @ I3 ) )
@ ( set_ord_atMost @ nat @ K2 ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% sum.triangle_reindex_eq
thf(fact_5407_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
@ ^ [I3: nat,J3: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ I3 @ J3 ) @ N ) ) ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [K2: nat] :
( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( G @ I3 @ ( minus_minus @ nat @ K2 @ I3 ) )
@ ( set_ord_atMost @ nat @ K2 ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% prod.triangle_reindex_eq
thf(fact_5408_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
@ ^ [I3: nat,J3: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ I3 @ J3 ) @ N ) ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K2: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( G @ I3 @ ( minus_minus @ nat @ K2 @ I3 ) )
@ ( set_ord_atMost @ nat @ K2 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% sum.triangle_reindex
thf(fact_5409_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
@ ^ [I3: nat,J3: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ I3 @ J3 ) @ N ) ) ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [K2: nat] :
( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( G @ I3 @ ( minus_minus @ nat @ K2 @ I3 ) )
@ ( set_ord_atMost @ nat @ K2 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% prod.triangle_reindex
thf(fact_5410_not__int__def,axiom,
( ( bit_ri4277139882892585799ns_not @ int )
= ( ^ [K2: int] : ( minus_minus @ int @ ( uminus_uminus @ int @ K2 ) @ ( one_one @ int ) ) ) ) ).
% not_int_def
thf(fact_5411_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_5412_word__plus__and__or__coroll,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A,Y2: word @ A] :
( ( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X2 @ Y2 )
= ( zero_zero @ ( word @ A ) ) )
=> ( ( plus_plus @ ( word @ A ) @ X2 @ Y2 )
= ( bit_se1065995026697491101ons_or @ ( word @ A ) @ X2 @ Y2 ) ) ) ) ).
% word_plus_and_or_coroll
thf(fact_5413_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_5414_bit_Ocomplement__unique,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [A2: A,X2: A,Y2: A] :
( ( ( bit_se5824344872417868541ns_and @ A @ A2 @ X2 )
= ( zero_zero @ A ) )
=> ( ( ( bit_se1065995026697491101ons_or @ A @ A2 @ X2 )
= ( uminus_uminus @ A @ ( one_one @ A ) ) )
=> ( ( ( bit_se5824344872417868541ns_and @ A @ A2 @ Y2 )
= ( zero_zero @ A ) )
=> ( ( ( bit_se1065995026697491101ons_or @ A @ A2 @ Y2 )
= ( uminus_uminus @ A @ ( one_one @ A ) ) )
=> ( X2 = Y2 ) ) ) ) ) ) ).
% bit.complement_unique
thf(fact_5415_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_5416_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_5417_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_5418_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_5419_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_5420_mask__subsume,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,M: nat,X2: word @ A,Y2: word @ A] :
( ( ord_less_eq @ nat @ N @ M )
=> ( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ ( bit_se1065995026697491101ons_or @ ( word @ A ) @ X2 @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ Y2 @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) ) @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ M ) ) )
= ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X2 @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ M ) ) ) ) ) ) ).
% mask_subsume
thf(fact_5421_rel__of__def,axiom,
! [B: $tType,A: $tType] :
( ( rel_of @ A @ B )
= ( ^ [M5: A > ( option @ B ),P7: ( product_prod @ A @ B ) > $o] :
( collect @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [K2: A,V5: B] :
( ( ( M5 @ K2 )
= ( some @ B @ V5 ) )
& ( P7 @ ( product_Pair @ A @ B @ K2 @ V5 ) ) ) ) ) ) ) ).
% rel_of_def
thf(fact_5422_bit_Ocompl__unique,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X2: A,Y2: A] :
( ( ( bit_se5824344872417868541ns_and @ A @ X2 @ Y2 )
= ( zero_zero @ A ) )
=> ( ( ( bit_se1065995026697491101ons_or @ A @ X2 @ Y2 )
= ( uminus_uminus @ A @ ( one_one @ A ) ) )
=> ( ( bit_ri4277139882892585799ns_not @ A @ X2 )
= Y2 ) ) ) ) ).
% bit.compl_unique
thf(fact_5423_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_5424_prod_Ocase__distrib,axiom,
! [C: $tType,D6: $tType,B: $tType,A: $tType,H2: C > D6,F2: A > B > C,Prod: product_prod @ A @ B] :
( ( H2 @ ( product_case_prod @ A @ B @ C @ F2 @ Prod ) )
= ( product_case_prod @ A @ B @ D6
@ ^ [X15: A,X24: B] : ( H2 @ ( F2 @ X15 @ X24 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_5425_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_5426_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_5427_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_5428_Divides_Oadjust__div__def,axiom,
( adjust_div
= ( product_case_prod @ int @ int @ int
@ ^ [Q4: int,R5: int] :
( plus_plus @ int @ Q4
@ ( zero_neq_one_of_bool @ int
@ ( R5
!= ( zero_zero @ int ) ) ) ) ) ) ).
% Divides.adjust_div_def
thf(fact_5429_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_5430_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_5431_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_5432_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_5433_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_5434_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_5435_divmod__nat__def,axiom,
( divmod_nat
= ( ^ [M5: nat,N3: nat] : ( product_Pair @ nat @ nat @ ( divide_divide @ nat @ M5 @ N3 ) @ ( modulo_modulo @ nat @ M5 @ N3 ) ) ) ) ).
% divmod_nat_def
thf(fact_5436_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_5437_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_5438_not__int__rec,axiom,
( ( bit_ri4277139882892585799ns_not @ int )
= ( ^ [K2: int] : ( plus_plus @ int @ ( zero_neq_one_of_bool @ int @ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K2 ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_ri4277139882892585799ns_not @ int @ ( divide_divide @ int @ K2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% not_int_rec
thf(fact_5439_cond__case__prod__eta,axiom,
! [C: $tType,B: $tType,A: $tType,F2: A > B > C,G: ( product_prod @ A @ B ) > C] :
( ! [X3: A,Y3: B] :
( ( F2 @ X3 @ Y3 )
= ( G @ ( product_Pair @ A @ B @ X3 @ Y3 ) ) )
=> ( ( product_case_prod @ A @ B @ C @ F2 )
= G ) ) ).
% cond_case_prod_eta
thf(fact_5440_case__prod__eta,axiom,
! [C: $tType,B: $tType,A: $tType,F2: ( product_prod @ A @ B ) > C] :
( ( product_case_prod @ A @ B @ C
@ ^ [X: A,Y: B] : ( F2 @ ( product_Pair @ A @ B @ X @ Y ) ) )
= F2 ) ).
% case_prod_eta
thf(fact_5441_case__prodE2,axiom,
! [B: $tType,A: $tType,C: $tType,Q: A > $o,P: B > C > A,Z: product_prod @ B @ C] :
( ( Q @ ( product_case_prod @ B @ C @ A @ P @ Z ) )
=> ~ ! [X3: B,Y3: C] :
( ( Z
= ( product_Pair @ B @ C @ X3 @ Y3 ) )
=> ~ ( Q @ ( P @ X3 @ Y3 ) ) ) ) ).
% case_prodE2
thf(fact_5442_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
@ ^ [M5: nat,Q4: nat] :
( if @ A
@ ( Q4
= ( zero_zero @ nat ) )
@ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ A @ M5 ) )
@ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ A @ M5 ) ) @ ( one_one @ A ) ) )
@ ( divmod_nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% of_nat_code_if
thf(fact_5443_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_5444_bitNOT__integer__code,axiom,
( ( bit_ri4277139882892585799ns_not @ code_integer )
= ( ^ [I3: code_integer] : ( minus_minus @ code_integer @ ( uminus_uminus @ code_integer @ I3 ) @ ( one_one @ code_integer ) ) ) ) ).
% bitNOT_integer_code
thf(fact_5445_split__part,axiom,
! [B: $tType,A: $tType,P: $o,Q: A > B > $o] :
( ( product_case_prod @ A @ B @ $o
@ ^ [A4: A,B3: B] :
( P
& ( Q @ A4 @ B3 ) ) )
= ( ^ [Ab2: product_prod @ A @ B] :
( P
& ( product_case_prod @ A @ B @ $o @ Q @ Ab2 ) ) ) ) ).
% split_part
thf(fact_5446_or__nonnegative__int__iff,axiom,
! [K: int,L2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se1065995026697491101ons_or @ int @ K @ L2 ) )
= ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
& ( ord_less_eq @ int @ ( zero_zero @ int ) @ L2 ) ) ) ).
% or_nonnegative_int_iff
thf(fact_5447_or__negative__int__iff,axiom,
! [K: int,L2: int] :
( ( ord_less @ int @ ( bit_se1065995026697491101ons_or @ int @ K @ L2 ) @ ( zero_zero @ int ) )
= ( ( ord_less @ int @ K @ ( zero_zero @ int ) )
| ( ord_less @ int @ L2 @ ( zero_zero @ int ) ) ) ) ).
% or_negative_int_iff
thf(fact_5448_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_5449_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_5450_or__nat__numerals_I4_J,axiom,
! [X2: num] :
( ( bit_se1065995026697491101ons_or @ nat @ ( numeral_numeral @ nat @ ( bit1 @ X2 ) ) @ ( suc @ ( zero_zero @ nat ) ) )
= ( numeral_numeral @ nat @ ( bit1 @ X2 ) ) ) ).
% or_nat_numerals(4)
thf(fact_5451_or__nat__numerals_I2_J,axiom,
! [Y2: num] :
( ( bit_se1065995026697491101ons_or @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( numeral_numeral @ nat @ ( bit1 @ Y2 ) ) )
= ( numeral_numeral @ nat @ ( bit1 @ Y2 ) ) ) ).
% or_nat_numerals(2)
thf(fact_5452_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_5453_or__nat__numerals_I3_J,axiom,
! [X2: num] :
( ( bit_se1065995026697491101ons_or @ nat @ ( numeral_numeral @ nat @ ( bit0 @ X2 ) ) @ ( suc @ ( zero_zero @ nat ) ) )
= ( numeral_numeral @ nat @ ( bit1 @ X2 ) ) ) ).
% or_nat_numerals(3)
thf(fact_5454_or__nat__numerals_I1_J,axiom,
! [Y2: num] :
( ( bit_se1065995026697491101ons_or @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( numeral_numeral @ nat @ ( bit0 @ Y2 ) ) )
= ( numeral_numeral @ nat @ ( bit1 @ Y2 ) ) ) ).
% or_nat_numerals(1)
thf(fact_5455_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_5456_word__bitwise__1__simps_I8_J,axiom,
! [H4: $tType] :
( ( type_len @ H4 )
=> ! [A2: num] :
( ( bit_se1065995026697491101ons_or @ ( word @ H4 ) @ ( numeral_numeral @ ( word @ H4 ) @ A2 ) @ ( one_one @ ( word @ H4 ) ) )
= ( ring_1_of_int @ ( word @ H4 ) @ ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ A2 ) @ ( one_one @ int ) ) ) ) ) ).
% word_bitwise_1_simps(8)
thf(fact_5457_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_5458_word__no__log__defs_I9_J,axiom,
! [I6: $tType] :
( ( type_len @ I6 )
=> ! [A2: num,B2: num] :
( ( bit_se1065995026697491101ons_or @ ( word @ I6 ) @ ( uminus_uminus @ ( word @ I6 ) @ ( numeral_numeral @ ( word @ I6 ) @ A2 ) ) @ ( numeral_numeral @ ( word @ I6 ) @ B2 ) )
= ( ring_1_of_int @ ( word @ I6 ) @ ( bit_se1065995026697491101ons_or @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ A2 ) ) @ ( numeral_numeral @ int @ B2 ) ) ) ) ) ).
% word_no_log_defs(9)
thf(fact_5459_word__no__log__defs_I8_J,axiom,
! [H4: $tType] :
( ( type_len @ H4 )
=> ! [A2: num,B2: num] :
( ( bit_se1065995026697491101ons_or @ ( word @ H4 ) @ ( numeral_numeral @ ( word @ H4 ) @ A2 ) @ ( uminus_uminus @ ( word @ H4 ) @ ( numeral_numeral @ ( word @ H4 ) @ B2 ) ) )
= ( ring_1_of_int @ ( word @ H4 ) @ ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ A2 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ B2 ) ) ) ) ) ) ).
% word_no_log_defs(8)
thf(fact_5460_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_5461_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_5462_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_5463_word__bitwise__1__simps_I9_J,axiom,
! [I6: $tType] :
( ( type_len @ I6 )
=> ! [A2: num] :
( ( bit_se1065995026697491101ons_or @ ( word @ I6 ) @ ( uminus_uminus @ ( word @ I6 ) @ ( numeral_numeral @ ( word @ I6 ) @ A2 ) ) @ ( one_one @ ( word @ I6 ) ) )
= ( ring_1_of_int @ ( word @ I6 ) @ ( bit_se1065995026697491101ons_or @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ A2 ) ) @ ( one_one @ int ) ) ) ) ) ).
% word_bitwise_1_simps(9)
thf(fact_5464_prod_Odisc__eq__case,axiom,
! [B: $tType,A: $tType,Prod: product_prod @ A @ B] :
( product_case_prod @ A @ B @ $o
@ ^ [Uu3: A,Uv3: B] : $true
@ Prod ) ).
% prod.disc_eq_case
thf(fact_5465_bit__or__int__iff,axiom,
! [K: int,L2: int,N: nat] :
( ( bit_se5641148757651400278ts_bit @ int @ ( bit_se1065995026697491101ons_or @ int @ K @ L2 ) @ N )
= ( ( bit_se5641148757651400278ts_bit @ int @ K @ N )
| ( bit_se5641148757651400278ts_bit @ int @ L2 @ N ) ) ) ).
% bit_or_int_iff
thf(fact_5466_or__nat__def,axiom,
( ( bit_se1065995026697491101ons_or @ nat )
= ( ^ [M5: nat,N3: nat] : ( nat2 @ ( bit_se1065995026697491101ons_or @ int @ ( semiring_1_of_nat @ int @ M5 ) @ ( semiring_1_of_nat @ int @ N3 ) ) ) ) ) ).
% or_nat_def
thf(fact_5467_int__or__code_I2_J,axiom,
! [I: int] :
( ( bit_se1065995026697491101ons_or @ int @ I @ ( zero_zero @ int ) )
= I ) ).
% int_or_code(2)
thf(fact_5468_int__or__code_I1_J,axiom,
! [J: int] :
( ( bit_se1065995026697491101ons_or @ int @ ( zero_zero @ int ) @ J )
= J ) ).
% int_or_code(1)
thf(fact_5469_OR__lower,axiom,
! [X2: int,Y2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X2 )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y2 )
=> ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se1065995026697491101ons_or @ int @ X2 @ Y2 ) ) ) ) ).
% OR_lower
thf(fact_5470_or__greater__eq,axiom,
! [L2: int,K: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ L2 )
=> ( ord_less_eq @ int @ K @ ( bit_se1065995026697491101ons_or @ int @ K @ L2 ) ) ) ).
% or_greater_eq
thf(fact_5471_plus__and__or,axiom,
! [X2: int,Y2: int] :
( ( plus_plus @ int @ ( bit_se5824344872417868541ns_and @ int @ X2 @ Y2 ) @ ( bit_se1065995026697491101ons_or @ int @ X2 @ Y2 ) )
= ( plus_plus @ int @ X2 @ Y2 ) ) ).
% plus_and_or
thf(fact_5472_or__int__def,axiom,
( ( bit_se1065995026697491101ons_or @ int )
= ( ^ [K2: int,L: int] : ( bit_ri4277139882892585799ns_not @ int @ ( bit_se5824344872417868541ns_and @ int @ ( bit_ri4277139882892585799ns_not @ int @ K2 ) @ ( bit_ri4277139882892585799ns_not @ int @ L ) ) ) ) ) ).
% or_int_def
thf(fact_5473_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_5474_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_5475_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_5476_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_5477_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_5478_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_5479_OR__upper,axiom,
! [X2: int,N: nat,Y2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X2 )
=> ( ( ord_less @ int @ X2 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) )
=> ( ( ord_less @ int @ Y2 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) )
=> ( ord_less @ int @ ( bit_se1065995026697491101ons_or @ int @ X2 @ Y2 ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ) ) ) ).
% OR_upper
thf(fact_5480_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_5481_test__bit__int__code_I1_J,axiom,
! [N: nat] :
~ ( bit_se5641148757651400278ts_bit @ int @ ( zero_zero @ int ) @ N ) ).
% test_bit_int_code(1)
thf(fact_5482_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_5483_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_5484_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_5485_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_5486_or__nat__rec,axiom,
( ( bit_se1065995026697491101ons_or @ nat )
= ( ^ [M5: nat,N3: nat] :
( plus_plus @ nat
@ ( zero_neq_one_of_bool @ nat
@ ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M5 )
| ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) ) )
@ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ nat @ ( divide_divide @ nat @ M5 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% or_nat_rec
thf(fact_5487_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_5488_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_5489_or__int__rec,axiom,
( ( bit_se1065995026697491101ons_or @ int )
= ( ^ [K2: int,L: int] :
( plus_plus @ int
@ ( zero_neq_one_of_bool @ int
@ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K2 )
| ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ L ) ) )
@ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ int @ ( divide_divide @ int @ K2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% or_int_rec
thf(fact_5490_or__nat__unfold,axiom,
( ( bit_se1065995026697491101ons_or @ nat )
= ( ^ [M5: nat,N3: nat] :
( if @ nat
@ ( M5
= ( zero_zero @ nat ) )
@ N3
@ ( if @ nat
@ ( N3
= ( zero_zero @ nat ) )
@ M5
@ ( plus_plus @ nat @ ( ord_max @ nat @ ( modulo_modulo @ nat @ M5 @ ( 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 @ M5 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% or_nat_unfold
thf(fact_5491_or__int__unfold,axiom,
( ( bit_se1065995026697491101ons_or @ int )
= ( ^ [K2: int,L: int] :
( if @ int
@ ( ( K2
= ( uminus_uminus @ int @ ( one_one @ int ) ) )
| ( L
= ( uminus_uminus @ int @ ( one_one @ int ) ) ) )
@ ( uminus_uminus @ int @ ( one_one @ int ) )
@ ( if @ int
@ ( K2
= ( zero_zero @ int ) )
@ L
@ ( if @ int
@ ( L
= ( zero_zero @ int ) )
@ K2
@ ( plus_plus @ int @ ( ord_max @ int @ ( modulo_modulo @ int @ K2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ int @ L @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ int @ ( divide_divide @ int @ K2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ).
% or_int_unfold
thf(fact_5492_Bit__integer_Oabs__eq,axiom,
! [Xa: int,X2: $o] :
( ( bits_Bit_integer @ ( code_integer_of_int @ Xa ) @ X2 )
= ( code_integer_of_int @ ( plus_plus @ int @ ( zero_neq_one_of_bool @ int @ X2 ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ Xa ) ) ) ) ).
% Bit_integer.abs_eq
thf(fact_5493_xor__int__unfold,axiom,
( ( bit_se5824344971392196577ns_xor @ int )
= ( ^ [K2: int,L: int] :
( if @ int
@ ( K2
= ( uminus_uminus @ int @ ( one_one @ int ) ) )
@ ( bit_ri4277139882892585799ns_not @ int @ L )
@ ( if @ int
@ ( L
= ( uminus_uminus @ int @ ( one_one @ int ) ) )
@ ( bit_ri4277139882892585799ns_not @ int @ K2 )
@ ( if @ int
@ ( K2
= ( zero_zero @ int ) )
@ L
@ ( if @ int
@ ( L
= ( zero_zero @ int ) )
@ K2
@ ( plus_plus @ int @ ( abs_abs @ int @ ( minus_minus @ int @ ( modulo_modulo @ int @ K2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ int @ L @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344971392196577ns_xor @ int @ ( divide_divide @ int @ K2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ).
% xor_int_unfold
thf(fact_5494_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_5495_bit_Oxor__left__self,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X2: A,Y2: A] :
( ( bit_se5824344971392196577ns_xor @ A @ X2 @ ( bit_se5824344971392196577ns_xor @ A @ X2 @ Y2 ) )
= Y2 ) ) ).
% bit.xor_left_self
thf(fact_5496_bit_Oxor__self,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X2: A] :
( ( bit_se5824344971392196577ns_xor @ A @ X2 @ X2 )
= ( zero_zero @ A ) ) ) ).
% bit.xor_self
thf(fact_5497_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_5498_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_5499_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_5500_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_5501_bit_Oxor__compl__left,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X2: A,Y2: A] :
( ( bit_se5824344971392196577ns_xor @ A @ ( bit_ri4277139882892585799ns_not @ A @ X2 ) @ Y2 )
= ( bit_ri4277139882892585799ns_not @ A @ ( bit_se5824344971392196577ns_xor @ A @ X2 @ Y2 ) ) ) ) ).
% bit.xor_compl_left
thf(fact_5502_bit_Oxor__compl__right,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X2: A,Y2: A] :
( ( bit_se5824344971392196577ns_xor @ A @ X2 @ ( bit_ri4277139882892585799ns_not @ A @ Y2 ) )
= ( bit_ri4277139882892585799ns_not @ A @ ( bit_se5824344971392196577ns_xor @ A @ X2 @ Y2 ) ) ) ) ).
% bit.xor_compl_right
thf(fact_5503_xor__nonnegative__int__iff,axiom,
! [K: int,L2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se5824344971392196577ns_xor @ int @ K @ L2 ) )
= ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
= ( ord_less_eq @ int @ ( zero_zero @ int ) @ L2 ) ) ) ).
% xor_nonnegative_int_iff
thf(fact_5504_xor__negative__int__iff,axiom,
! [K: int,L2: int] :
( ( ord_less @ int @ ( bit_se5824344971392196577ns_xor @ int @ K @ L2 ) @ ( zero_zero @ int ) )
= ( ( ord_less @ int @ K @ ( zero_zero @ int ) )
!= ( ord_less @ int @ L2 @ ( zero_zero @ int ) ) ) ) ).
% xor_negative_int_iff
thf(fact_5505_bit_Oxor__one__left,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X2: A] :
( ( bit_se5824344971392196577ns_xor @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ X2 )
= ( bit_ri4277139882892585799ns_not @ A @ X2 ) ) ) ).
% bit.xor_one_left
thf(fact_5506_bit_Oxor__one__right,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X2: A] :
( ( bit_se5824344971392196577ns_xor @ A @ X2 @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( bit_ri4277139882892585799ns_not @ A @ X2 ) ) ) ).
% bit.xor_one_right
thf(fact_5507_bit_Oxor__cancel__left,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X2: A] :
( ( bit_se5824344971392196577ns_xor @ A @ ( bit_ri4277139882892585799ns_not @ A @ X2 ) @ X2 )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% bit.xor_cancel_left
thf(fact_5508_bit_Oxor__cancel__right,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X2: A] :
( ( bit_se5824344971392196577ns_xor @ A @ X2 @ ( bit_ri4277139882892585799ns_not @ A @ X2 ) )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% bit.xor_cancel_right
thf(fact_5509_xor__numerals_I3_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X2: num,Y2: num] :
( ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ ( bit0 @ X2 ) ) @ ( numeral_numeral @ A @ ( bit0 @ Y2 ) ) )
= ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ X2 ) @ ( numeral_numeral @ A @ Y2 ) ) ) ) ) ).
% xor_numerals(3)
thf(fact_5510_xor__numerals_I1_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [Y2: num] :
( ( bit_se5824344971392196577ns_xor @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ Y2 ) ) )
= ( numeral_numeral @ A @ ( bit1 @ Y2 ) ) ) ) ).
% xor_numerals(1)
thf(fact_5511_xor__numerals_I2_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [Y2: num] :
( ( bit_se5824344971392196577ns_xor @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit1 @ Y2 ) ) )
= ( numeral_numeral @ A @ ( bit0 @ Y2 ) ) ) ) ).
% xor_numerals(2)
thf(fact_5512_xor__numerals_I5_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X2: num] :
( ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ ( bit0 @ X2 ) ) @ ( one_one @ A ) )
= ( numeral_numeral @ A @ ( bit1 @ X2 ) ) ) ) ).
% xor_numerals(5)
thf(fact_5513_xor__numerals_I8_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X2: num] :
( ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ ( bit1 @ X2 ) ) @ ( one_one @ A ) )
= ( numeral_numeral @ A @ ( bit0 @ X2 ) ) ) ) ).
% xor_numerals(8)
thf(fact_5514_xor__numerals_I7_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X2: num,Y2: num] :
( ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ ( bit1 @ X2 ) ) @ ( numeral_numeral @ A @ ( bit1 @ Y2 ) ) )
= ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ X2 ) @ ( numeral_numeral @ A @ Y2 ) ) ) ) ) ).
% xor_numerals(7)
thf(fact_5515_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_5516_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_5517_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_5518_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_5519_xor__numerals_I4_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X2: num,Y2: num] :
( ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ ( bit0 @ X2 ) ) @ ( numeral_numeral @ A @ ( bit1 @ Y2 ) ) )
= ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ X2 ) @ ( numeral_numeral @ A @ Y2 ) ) ) ) ) ) ).
% xor_numerals(4)
thf(fact_5520_xor__numerals_I6_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X2: num,Y2: num] :
( ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ ( bit1 @ X2 ) ) @ ( numeral_numeral @ A @ ( bit0 @ Y2 ) ) )
= ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ X2 ) @ ( numeral_numeral @ A @ Y2 ) ) ) ) ) ) ).
% xor_numerals(6)
thf(fact_5521_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_5522_int__xor__code_I1_J,axiom,
! [J: int] :
( ( bit_se5824344971392196577ns_xor @ int @ ( zero_zero @ int ) @ J )
= J ) ).
% int_xor_code(1)
thf(fact_5523_int__xor__code_I2_J,axiom,
! [I: int] :
( ( bit_se5824344971392196577ns_xor @ int @ I @ ( zero_zero @ int ) )
= I ) ).
% int_xor_code(2)
thf(fact_5524_XOR__lower,axiom,
! [X2: int,Y2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X2 )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y2 )
=> ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se5824344971392196577ns_xor @ int @ X2 @ Y2 ) ) ) ) ).
% XOR_lower
thf(fact_5525_of__int__xor__eq,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [K: int,L2: int] :
( ( ring_1_of_int @ A @ ( bit_se5824344971392196577ns_xor @ int @ K @ L2 ) )
= ( bit_se5824344971392196577ns_xor @ A @ ( ring_1_of_int @ A @ K ) @ ( ring_1_of_int @ A @ L2 ) ) ) ) ).
% of_int_xor_eq
thf(fact_5526_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_5527_bit__xor__int__iff,axiom,
! [K: int,L2: int,N: nat] :
( ( bit_se5641148757651400278ts_bit @ int @ ( bit_se5824344971392196577ns_xor @ int @ K @ L2 ) @ N )
= ( ( bit_se5641148757651400278ts_bit @ int @ K @ N )
!= ( bit_se5641148757651400278ts_bit @ int @ L2 @ N ) ) ) ).
% bit_xor_int_iff
thf(fact_5528_or__not__num__neg_Osimps_I1_J,axiom,
( ( bit_or_not_num_neg @ one2 @ one2 )
= one2 ) ).
% or_not_num_neg.simps(1)
thf(fact_5529_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_5530_xor_Ocommute,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se5824344971392196577ns_xor @ A )
= ( ^ [A4: A,B3: A] : ( bit_se5824344971392196577ns_xor @ A @ B3 @ A4 ) ) ) ) ).
% xor.commute
thf(fact_5531_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_5532_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_5533_bit_Oconj__xor__distrib,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X2: A,Y2: A,Z: A] :
( ( bit_se5824344872417868541ns_and @ A @ X2 @ ( bit_se5824344971392196577ns_xor @ A @ Y2 @ Z ) )
= ( bit_se5824344971392196577ns_xor @ A @ ( bit_se5824344872417868541ns_and @ A @ X2 @ Y2 ) @ ( bit_se5824344872417868541ns_and @ A @ X2 @ Z ) ) ) ) ).
% bit.conj_xor_distrib
thf(fact_5534_bit_Oconj__xor__distrib2,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [Y2: A,Z: A,X2: A] :
( ( bit_se5824344872417868541ns_and @ A @ ( bit_se5824344971392196577ns_xor @ A @ Y2 @ Z ) @ X2 )
= ( bit_se5824344971392196577ns_xor @ A @ ( bit_se5824344872417868541ns_and @ A @ Y2 @ X2 ) @ ( bit_se5824344872417868541ns_and @ A @ Z @ X2 ) ) ) ) ).
% bit.conj_xor_distrib2
thf(fact_5535_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_5536_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_5537_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_5538_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_5539_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_5540_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_5541_xor__int__def,axiom,
( ( bit_se5824344971392196577ns_xor @ int )
= ( ^ [K2: int,L: int] : ( bit_se1065995026697491101ons_or @ int @ ( bit_se5824344872417868541ns_and @ int @ K2 @ ( bit_ri4277139882892585799ns_not @ int @ L ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( bit_ri4277139882892585799ns_not @ int @ K2 ) @ L ) ) ) ) ).
% xor_int_def
thf(fact_5542_bit_Oxor__def2,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( ( bit_se5824344971392196577ns_xor @ A )
= ( ^ [X: A,Y: A] : ( bit_se5824344872417868541ns_and @ A @ ( bit_se1065995026697491101ons_or @ A @ X @ Y ) @ ( bit_se1065995026697491101ons_or @ A @ ( bit_ri4277139882892585799ns_not @ A @ X ) @ ( bit_ri4277139882892585799ns_not @ A @ Y ) ) ) ) ) ) ).
% bit.xor_def2
thf(fact_5543_bit_Oxor__def,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( ( bit_se5824344971392196577ns_xor @ A )
= ( ^ [X: A,Y: A] : ( bit_se1065995026697491101ons_or @ A @ ( bit_se5824344872417868541ns_and @ A @ X @ ( bit_ri4277139882892585799ns_not @ A @ Y ) ) @ ( bit_se5824344872417868541ns_and @ A @ ( bit_ri4277139882892585799ns_not @ A @ X ) @ Y ) ) ) ) ) ).
% bit.xor_def
thf(fact_5544_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_5545_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_5546_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_5547_or__not__num__neg_Oelims,axiom,
! [X2: num,Xa: num,Y2: num] :
( ( ( bit_or_not_num_neg @ X2 @ Xa )
= Y2 )
=> ( ( ( X2 = one2 )
=> ( ( Xa = one2 )
=> ( Y2 != one2 ) ) )
=> ( ( ( X2 = one2 )
=> ! [M3: num] :
( ( Xa
= ( bit0 @ M3 ) )
=> ( Y2
!= ( bit1 @ M3 ) ) ) )
=> ( ( ( X2 = one2 )
=> ! [M3: num] :
( ( Xa
= ( bit1 @ M3 ) )
=> ( Y2
!= ( bit1 @ M3 ) ) ) )
=> ( ( ? [N2: num] :
( X2
= ( bit0 @ N2 ) )
=> ( ( Xa = one2 )
=> ( Y2
!= ( bit0 @ one2 ) ) ) )
=> ( ! [N2: num] :
( ( X2
= ( bit0 @ N2 ) )
=> ! [M3: num] :
( ( Xa
= ( bit0 @ M3 ) )
=> ( Y2
!= ( bitM @ ( bit_or_not_num_neg @ N2 @ M3 ) ) ) ) )
=> ( ! [N2: num] :
( ( X2
= ( bit0 @ N2 ) )
=> ! [M3: num] :
( ( Xa
= ( bit1 @ M3 ) )
=> ( Y2
!= ( bit0 @ ( bit_or_not_num_neg @ N2 @ M3 ) ) ) ) )
=> ( ( ? [N2: num] :
( X2
= ( bit1 @ N2 ) )
=> ( ( Xa = one2 )
=> ( Y2 != one2 ) ) )
=> ( ! [N2: num] :
( ( X2
= ( bit1 @ N2 ) )
=> ! [M3: num] :
( ( Xa
= ( bit0 @ M3 ) )
=> ( Y2
!= ( bitM @ ( bit_or_not_num_neg @ N2 @ M3 ) ) ) ) )
=> ~ ! [N2: num] :
( ( X2
= ( bit1 @ N2 ) )
=> ! [M3: num] :
( ( Xa
= ( bit1 @ M3 ) )
=> ( Y2
!= ( bitM @ ( bit_or_not_num_neg @ N2 @ M3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% or_not_num_neg.elims
thf(fact_5548_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_5549_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_5550_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_5551_XOR__upper,axiom,
! [X2: int,N: nat,Y2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X2 )
=> ( ( ord_less @ int @ X2 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) )
=> ( ( ord_less @ int @ Y2 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) )
=> ( ord_less @ int @ ( bit_se5824344971392196577ns_xor @ int @ X2 @ Y2 ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ) ) ) ).
% XOR_upper
thf(fact_5552_xor__int__rec,axiom,
( ( bit_se5824344971392196577ns_xor @ int )
= ( ^ [K2: int,L: int] :
( plus_plus @ int
@ ( zero_neq_one_of_bool @ int
@ ( ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K2 ) )
!= ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ L ) ) ) )
@ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344971392196577ns_xor @ int @ ( divide_divide @ int @ K2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% xor_int_rec
thf(fact_5553_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_5554_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_5555_int__ge__less__than__def,axiom,
( int_ge_less_than
= ( ^ [D: int] :
( collect @ ( product_prod @ int @ int )
@ ( product_case_prod @ int @ int @ $o
@ ^ [Z7: int,Z5: int] :
( ( ord_less_eq @ int @ D @ Z7 )
& ( ord_less @ int @ Z7 @ Z5 ) ) ) ) ) ) ).
% int_ge_less_than_def
thf(fact_5556_int__ge__less__than2__def,axiom,
( int_ge_less_than2
= ( ^ [D: int] :
( collect @ ( product_prod @ int @ int )
@ ( product_case_prod @ int @ int @ $o
@ ^ [Z7: int,Z5: int] :
( ( ord_less_eq @ int @ D @ Z5 )
& ( ord_less @ int @ Z7 @ Z5 ) ) ) ) ) ) ).
% int_ge_less_than2_def
thf(fact_5557_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_5558_word__bitwise__m1__simps_I7_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A] :
( ( bit_se5824344971392196577ns_xor @ ( word @ A ) @ X2 @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) )
= ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ X2 ) ) ) ).
% word_bitwise_m1_simps(7)
thf(fact_5559_word__bitwise__m1__simps_I6_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A] :
( ( bit_se5824344971392196577ns_xor @ ( word @ A ) @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) @ X2 )
= ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ X2 ) ) ) ).
% word_bitwise_m1_simps(6)
thf(fact_5560_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_5561_xor__nat__numerals_I4_J,axiom,
! [X2: num] :
( ( bit_se5824344971392196577ns_xor @ nat @ ( numeral_numeral @ nat @ ( bit1 @ X2 ) ) @ ( suc @ ( zero_zero @ nat ) ) )
= ( numeral_numeral @ nat @ ( bit0 @ X2 ) ) ) ).
% xor_nat_numerals(4)
thf(fact_5562_xor__nat__numerals_I3_J,axiom,
! [X2: num] :
( ( bit_se5824344971392196577ns_xor @ nat @ ( numeral_numeral @ nat @ ( bit0 @ X2 ) ) @ ( suc @ ( zero_zero @ nat ) ) )
= ( numeral_numeral @ nat @ ( bit1 @ X2 ) ) ) ).
% xor_nat_numerals(3)
thf(fact_5563_xor__nat__numerals_I2_J,axiom,
! [Y2: num] :
( ( bit_se5824344971392196577ns_xor @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( numeral_numeral @ nat @ ( bit1 @ Y2 ) ) )
= ( numeral_numeral @ nat @ ( bit0 @ Y2 ) ) ) ).
% xor_nat_numerals(2)
thf(fact_5564_xor__nat__numerals_I1_J,axiom,
! [Y2: num] :
( ( bit_se5824344971392196577ns_xor @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( numeral_numeral @ nat @ ( bit0 @ Y2 ) ) )
= ( numeral_numeral @ nat @ ( bit1 @ Y2 ) ) ) ).
% xor_nat_numerals(1)
thf(fact_5565_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_5566_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_5567_word__no__log__defs_I14_J,axiom,
! [N12: $tType] :
( ( type_len @ N12 )
=> ! [A2: num,B2: num] :
( ( bit_se5824344971392196577ns_xor @ ( word @ N12 ) @ ( uminus_uminus @ ( word @ N12 ) @ ( numeral_numeral @ ( word @ N12 ) @ A2 ) ) @ ( uminus_uminus @ ( word @ N12 ) @ ( numeral_numeral @ ( word @ N12 ) @ B2 ) ) )
= ( ring_1_of_int @ ( word @ N12 ) @ ( 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_5568_word__no__log__defs_I13_J,axiom,
! [M12: $tType] :
( ( type_len @ M12 )
=> ! [A2: num,B2: num] :
( ( bit_se5824344971392196577ns_xor @ ( word @ M12 ) @ ( uminus_uminus @ ( word @ M12 ) @ ( numeral_numeral @ ( word @ M12 ) @ A2 ) ) @ ( numeral_numeral @ ( word @ M12 ) @ B2 ) )
= ( ring_1_of_int @ ( word @ M12 ) @ ( bit_se5824344971392196577ns_xor @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ A2 ) ) @ ( numeral_numeral @ int @ B2 ) ) ) ) ) ).
% word_no_log_defs(13)
thf(fact_5569_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_5570_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_5571_word__bitwise__1__simps_I13_J,axiom,
! [M12: $tType] :
( ( type_len @ M12 )
=> ! [A2: num] :
( ( bit_se5824344971392196577ns_xor @ ( word @ M12 ) @ ( uminus_uminus @ ( word @ M12 ) @ ( numeral_numeral @ ( word @ M12 ) @ A2 ) ) @ ( one_one @ ( word @ M12 ) ) )
= ( ring_1_of_int @ ( word @ M12 ) @ ( bit_se5824344971392196577ns_xor @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ A2 ) ) @ ( one_one @ int ) ) ) ) ) ).
% word_bitwise_1_simps(13)
thf(fact_5572_word__log__esimps_I5_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A] :
( ( bit_se5824344971392196577ns_xor @ ( word @ A ) @ X2 @ ( zero_zero @ ( word @ A ) ) )
= X2 ) ) ).
% word_log_esimps(5)
thf(fact_5573_word__log__esimps_I11_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A] :
( ( bit_se5824344971392196577ns_xor @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ X2 )
= X2 ) ) ).
% word_log_esimps(11)
thf(fact_5574_word__bw__same_I3_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A] :
( ( bit_se5824344971392196577ns_xor @ ( word @ A ) @ X2 @ X2 )
= ( zero_zero @ ( word @ A ) ) ) ) ).
% word_bw_same(3)
thf(fact_5575_xor__nat__def,axiom,
( ( bit_se5824344971392196577ns_xor @ nat )
= ( ^ [M5: nat,N3: nat] : ( nat2 @ ( bit_se5824344971392196577ns_xor @ int @ ( semiring_1_of_nat @ int @ M5 ) @ ( semiring_1_of_nat @ int @ N3 ) ) ) ) ) ).
% xor_nat_def
thf(fact_5576_bit__twiddle__max,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A,Y2: word @ A] :
( ( bit_se5824344971392196577ns_xor @ ( word @ A ) @ X2 @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ ( bit_se5824344971392196577ns_xor @ ( word @ A ) @ X2 @ Y2 ) @ ( if @ ( word @ A ) @ ( ord_less @ ( word @ A ) @ X2 @ Y2 ) @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) @ ( zero_zero @ ( word @ A ) ) ) ) )
= ( ord_max @ ( word @ A ) @ X2 @ Y2 ) ) ) ).
% bit_twiddle_max
thf(fact_5577_xor__nat__unfold,axiom,
( ( bit_se5824344971392196577ns_xor @ nat )
= ( ^ [M5: nat,N3: nat] :
( if @ nat
@ ( M5
= ( zero_zero @ nat ) )
@ N3
@ ( if @ nat
@ ( N3
= ( zero_zero @ nat ) )
@ M5
@ ( plus_plus @ nat @ ( modulo_modulo @ nat @ ( plus_plus @ nat @ ( modulo_modulo @ nat @ M5 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ nat @ 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 @ M5 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% xor_nat_unfold
thf(fact_5578_xor__nat__rec,axiom,
( ( bit_se5824344971392196577ns_xor @ nat )
= ( ^ [M5: nat,N3: nat] :
( plus_plus @ nat
@ ( zero_neq_one_of_bool @ nat
@ ( ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M5 ) )
!= ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) ) ) )
@ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( bit_se5824344971392196577ns_xor @ nat @ ( divide_divide @ nat @ M5 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% xor_nat_rec
thf(fact_5579_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_5580_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_5581_word__ops__lsb,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A,Y2: word @ A] :
( ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( bit_se1065995026697491101ons_or @ ( word @ A ) @ X2 @ Y2 ) @ ( zero_zero @ nat ) )
= ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ X2 @ ( zero_zero @ nat ) )
| ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ Y2 @ ( zero_zero @ nat ) ) ) )
& ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X2 @ Y2 ) @ ( zero_zero @ nat ) )
= ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ X2 @ ( zero_zero @ nat ) )
& ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ Y2 @ ( zero_zero @ nat ) ) ) )
& ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( bit_se5824344971392196577ns_xor @ ( word @ A ) @ X2 @ Y2 ) @ ( zero_zero @ nat ) )
= ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ X2 @ ( zero_zero @ nat ) )
!= ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ Y2 @ ( zero_zero @ nat ) ) ) )
& ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ X2 ) @ ( zero_zero @ nat ) )
= ( ~ ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ X2 @ ( zero_zero @ nat ) ) ) ) ) ) ).
% word_ops_lsb
thf(fact_5582_hash__code__prod__simps,axiom,
! [A: $tType,B: $tType,H_a: A > uint32,H_b: B > uint32,X2: A,Xa: B] :
( ( hash_hash_code_prod @ A @ B @ H_a @ H_b @ ( product_Pair @ A @ B @ X2 @ Xa ) )
= ( plus_plus @ uint32 @ ( times_times @ uint32 @ ( H_a @ X2 ) @ ( numeral_numeral @ uint32 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( plus_plus @ uint32 @ ( times_times @ uint32 @ ( H_b @ Xa ) @ ( numeral_numeral @ uint32 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numeral_numeral @ uint32 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% hash_code_prod_simps
thf(fact_5583_sum__count__set,axiom,
! [A: $tType,Xs2: list @ A,X6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs2 ) @ X6 )
=> ( ( finite_finite2 @ A @ X6 )
=> ( ( groups7311177749621191930dd_sum @ A @ nat @ ( count_list @ A @ Xs2 ) @ X6 )
= ( size_size @ ( list @ A ) @ Xs2 ) ) ) ) ).
% sum_count_set
thf(fact_5584_count__notin,axiom,
! [A: $tType,X2: A,Xs2: list @ A] :
( ~ ( member @ A @ X2 @ ( set2 @ A @ Xs2 ) )
=> ( ( count_list @ A @ Xs2 @ X2 )
= ( zero_zero @ nat ) ) ) ).
% count_notin
thf(fact_5585_count__le__length,axiom,
! [A: $tType,Xs2: list @ A,X2: A] : ( ord_less_eq @ nat @ ( count_list @ A @ Xs2 @ X2 ) @ ( size_size @ ( list @ A ) @ Xs2 ) ) ).
% count_le_length
thf(fact_5586_hash__code__option__simps_I2_J,axiom,
! [A: $tType,H_a: A > uint32,X2: A] :
( ( hash_h1887023736457453652option @ A @ H_a @ ( some @ A @ X2 ) )
= ( plus_plus @ uint32 @ ( times_times @ uint32 @ ( H_a @ X2 ) @ ( numeral_numeral @ uint32 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numeral_numeral @ uint32 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% hash_code_option_simps(2)
thf(fact_5587_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_5588_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_5589_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_5590_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_5591_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_5592_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_5593_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_5594_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_5595_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_5596_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_5597_dup__1,axiom,
( ( code_dup @ ( one_one @ code_integer ) )
= ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) ).
% dup_1
thf(fact_5598_power__numeral,axiom,
! [A: $tType] :
( ( semiring_numeral @ A )
=> ! [K: num,L2: num] :
( ( power_power @ A @ ( numeral_numeral @ A @ K ) @ ( numeral_numeral @ nat @ L2 ) )
= ( numeral_numeral @ A @ ( pow @ K @ L2 ) ) ) ) ).
% power_numeral
thf(fact_5599_VEBT__internal_OT__vebt__buildupi_H_Opelims,axiom,
! [X2: nat,Y2: int] :
( ( ( vEBT_V9176841429113362141ildupi @ X2 )
= Y2 )
=> ( ( accp @ nat @ vEBT_V3352910403632780892pi_rel @ X2 )
=> ( ( ( X2
= ( zero_zero @ nat ) )
=> ( ( Y2
= ( one_one @ int ) )
=> ~ ( accp @ nat @ vEBT_V3352910403632780892pi_rel @ ( zero_zero @ nat ) ) ) )
=> ( ( ( X2
= ( suc @ ( zero_zero @ nat ) ) )
=> ( ( Y2
= ( one_one @ int ) )
=> ~ ( accp @ nat @ vEBT_V3352910403632780892pi_rel @ ( suc @ ( zero_zero @ nat ) ) ) ) )
=> ~ ! [N2: nat] :
( ( X2
= ( suc @ ( suc @ N2 ) ) )
=> ( ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
=> ( Y2
= ( plus_plus @ int @ ( numeral_numeral @ int @ ( bit1 @ one2 ) ) @ ( plus_plus @ int @ ( vEBT_V9176841429113362141ildupi @ ( suc @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( plus_plus @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ ( bit0 @ one2 ) ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( times_times @ int @ ( vEBT_V9176841429113362141ildupi @ ( suc @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) )
& ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
=> ( Y2
= ( plus_plus @ int @ ( numeral_numeral @ int @ ( bit1 @ one2 ) ) @ ( plus_plus @ int @ ( vEBT_V9176841429113362141ildupi @ ( suc @ ( suc @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( plus_plus @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ ( bit0 @ one2 ) ) ) @ ( times_times @ int @ ( vEBT_V9176841429113362141ildupi @ ( suc @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) )
=> ~ ( accp @ nat @ vEBT_V3352910403632780892pi_rel @ ( suc @ ( suc @ N2 ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.T_vebt_buildupi'.pelims
thf(fact_5600_int__lsb__numeral_I1_J,axiom,
~ ( least_8051144512741203767sb_lsb @ int @ ( zero_zero @ int ) ) ).
% int_lsb_numeral(1)
thf(fact_5601_int__lsb__numeral_I2_J,axiom,
least_8051144512741203767sb_lsb @ int @ ( one_one @ int ) ).
% int_lsb_numeral(2)
thf(fact_5602_int__lsb__numeral_I6_J,axiom,
! [W: num] :
~ ( least_8051144512741203767sb_lsb @ int @ ( numeral_numeral @ int @ ( bit0 @ W ) ) ) ).
% int_lsb_numeral(6)
thf(fact_5603_int__lsb__numeral_I3_J,axiom,
least_8051144512741203767sb_lsb @ int @ ( numeral_numeral @ int @ one2 ) ).
% int_lsb_numeral(3)
thf(fact_5604_int__lsb__numeral_I7_J,axiom,
! [W: num] : ( least_8051144512741203767sb_lsb @ int @ ( numeral_numeral @ int @ ( bit1 @ W ) ) ) ).
% int_lsb_numeral(7)
thf(fact_5605_int__lsb__numeral_I4_J,axiom,
least_8051144512741203767sb_lsb @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) ).
% int_lsb_numeral(4)
thf(fact_5606_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_5607_int__lsb__numeral_I5_J,axiom,
least_8051144512741203767sb_lsb @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ one2 ) ) ).
% int_lsb_numeral(5)
thf(fact_5608_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_5609_Code__Numeral_Odup__code_I1_J,axiom,
( ( code_dup @ ( zero_zero @ code_integer ) )
= ( zero_zero @ code_integer ) ) ).
% Code_Numeral.dup_code(1)
thf(fact_5610_lsb__integer__code,axiom,
( ( least_8051144512741203767sb_lsb @ code_integer )
= ( ^ [X: code_integer] : ( bit_se5641148757651400278ts_bit @ code_integer @ X @ ( zero_zero @ nat ) ) ) ) ).
% lsb_integer_code
thf(fact_5611_pow_Osimps_I1_J,axiom,
! [X2: num] :
( ( pow @ X2 @ one2 )
= X2 ) ).
% pow.simps(1)
thf(fact_5612_lsb__int__def,axiom,
( ( least_8051144512741203767sb_lsb @ int )
= ( ^ [I3: int] : ( bit_se5641148757651400278ts_bit @ int @ I3 @ ( zero_zero @ nat ) ) ) ) ).
% lsb_int_def
thf(fact_5613_dup_Oabs__eq,axiom,
! [X2: int] :
( ( code_dup @ ( code_integer_of_int @ X2 ) )
= ( code_integer_of_int @ ( plus_plus @ int @ X2 @ X2 ) ) ) ).
% dup.abs_eq
thf(fact_5614_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_5615_vebt__buildup_Opelims,axiom,
! [X2: nat,Y2: vEBT_VEBT] :
( ( ( vEBT_vebt_buildup @ X2 )
= Y2 )
=> ( ( accp @ nat @ vEBT_v4011308405150292612up_rel @ X2 )
=> ( ( ( X2
= ( zero_zero @ nat ) )
=> ( ( Y2
= ( vEBT_Leaf @ $false @ $false ) )
=> ~ ( accp @ nat @ vEBT_v4011308405150292612up_rel @ ( zero_zero @ nat ) ) ) )
=> ( ( ( X2
= ( suc @ ( zero_zero @ nat ) ) )
=> ( ( Y2
= ( vEBT_Leaf @ $false @ $false ) )
=> ~ ( accp @ nat @ vEBT_v4011308405150292612up_rel @ ( suc @ ( zero_zero @ nat ) ) ) ) )
=> ~ ! [Va2: nat] :
( ( X2
= ( suc @ ( suc @ Va2 ) ) )
=> ( ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ Va2 ) ) )
=> ( Y2
= ( 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 ) ) )
=> ( Y2
= ( 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 ) ) ) ) ) ) ) ) )
=> ~ ( accp @ nat @ vEBT_v4011308405150292612up_rel @ ( suc @ ( suc @ Va2 ) ) ) ) ) ) ) ) ) ).
% vebt_buildup.pelims
thf(fact_5616_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_Opelims,axiom,
! [X2: nat,Y2: nat] :
( ( ( vEBT_V8646137997579335489_i_l_d @ X2 )
= Y2 )
=> ( ( accp @ nat @ vEBT_V5144397997797733112_d_rel @ X2 )
=> ( ( ( X2
= ( zero_zero @ nat ) )
=> ( ( Y2
= ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) )
=> ~ ( accp @ nat @ vEBT_V5144397997797733112_d_rel @ ( zero_zero @ nat ) ) ) )
=> ( ( ( X2
= ( suc @ ( zero_zero @ nat ) ) )
=> ( ( Y2
= ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) )
=> ~ ( accp @ nat @ vEBT_V5144397997797733112_d_rel @ ( suc @ ( zero_zero @ nat ) ) ) ) )
=> ~ ! [Va2: nat] :
( ( X2
= ( suc @ ( suc @ Va2 ) ) )
=> ( ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ Va2 ) ) )
=> ( Y2
= ( 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 ) ) )
=> ( Y2
= ( 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 ) ) ) ) ) ) ) ) )
=> ~ ( accp @ nat @ vEBT_V5144397997797733112_d_rel @ ( suc @ ( suc @ Va2 ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d.pelims
thf(fact_5617_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p_Opelims,axiom,
! [X2: nat,Y2: nat] :
( ( ( vEBT_V8346862874174094_d_u_p @ X2 )
= Y2 )
=> ( ( accp @ nat @ vEBT_V1247956027447740395_p_rel @ X2 )
=> ( ( ( X2
= ( zero_zero @ nat ) )
=> ( ( Y2
= ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) )
=> ~ ( accp @ nat @ vEBT_V1247956027447740395_p_rel @ ( zero_zero @ nat ) ) ) )
=> ( ( ( X2
= ( suc @ ( zero_zero @ nat ) ) )
=> ( ( Y2
= ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) )
=> ~ ( accp @ nat @ vEBT_V1247956027447740395_p_rel @ ( suc @ ( zero_zero @ nat ) ) ) ) )
=> ~ ! [Va2: nat] :
( ( X2
= ( suc @ ( suc @ Va2 ) ) )
=> ( ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ Va2 ) ) )
=> ( Y2
= ( 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 ) ) )
=> ( Y2
= ( 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 ) ) ) ) ) ) )
=> ~ ( accp @ nat @ vEBT_V1247956027447740395_p_rel @ ( suc @ ( suc @ Va2 ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d\<^sub>u\<^sub>p.pelims
thf(fact_5618_VEBT__internal_OTb_Opelims,axiom,
! [X2: nat,Y2: int] :
( ( ( vEBT_VEBT_Tb @ X2 )
= Y2 )
=> ( ( accp @ nat @ vEBT_VEBT_Tb_rel2 @ X2 )
=> ( ( ( X2
= ( zero_zero @ nat ) )
=> ( ( Y2
= ( numeral_numeral @ int @ ( bit1 @ one2 ) ) )
=> ~ ( accp @ nat @ vEBT_VEBT_Tb_rel2 @ ( zero_zero @ nat ) ) ) )
=> ( ( ( X2
= ( suc @ ( zero_zero @ nat ) ) )
=> ( ( Y2
= ( numeral_numeral @ int @ ( bit1 @ one2 ) ) )
=> ~ ( accp @ nat @ vEBT_VEBT_Tb_rel2 @ ( suc @ ( zero_zero @ nat ) ) ) ) )
=> ~ ! [N2: nat] :
( ( X2
= ( suc @ ( suc @ N2 ) ) )
=> ( ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
=> ( Y2
= ( plus_plus @ int @ ( plus_plus @ int @ ( numeral_numeral @ int @ ( bit1 @ ( bit0 @ one2 ) ) ) @ ( vEBT_VEBT_Tb @ ( suc @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( times_times @ int @ ( vEBT_VEBT_Tb @ ( suc @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( suc @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) )
& ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
=> ( Y2
= ( plus_plus @ int @ ( plus_plus @ int @ ( numeral_numeral @ int @ ( bit1 @ ( bit0 @ one2 ) ) ) @ ( vEBT_VEBT_Tb @ ( suc @ ( suc @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) @ ( times_times @ int @ ( vEBT_VEBT_Tb @ ( suc @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) )
=> ~ ( accp @ nat @ vEBT_VEBT_Tb_rel2 @ ( suc @ ( suc @ N2 ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.Tb.pelims
thf(fact_5619_VEBT__internal_OTb_H_Opelims,axiom,
! [X2: nat,Y2: nat] :
( ( ( vEBT_VEBT_Tb2 @ X2 )
= Y2 )
=> ( ( accp @ nat @ vEBT_VEBT_Tb_rel @ X2 )
=> ( ( ( X2
= ( zero_zero @ nat ) )
=> ( ( Y2
= ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) )
=> ~ ( accp @ nat @ vEBT_VEBT_Tb_rel @ ( zero_zero @ nat ) ) ) )
=> ( ( ( X2
= ( suc @ ( zero_zero @ nat ) ) )
=> ( ( Y2
= ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) )
=> ~ ( accp @ nat @ vEBT_VEBT_Tb_rel @ ( suc @ ( zero_zero @ nat ) ) ) ) )
=> ~ ! [N2: nat] :
( ( X2
= ( suc @ ( suc @ N2 ) ) )
=> ( ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
=> ( Y2
= ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ one2 ) ) ) @ ( vEBT_VEBT_Tb2 @ ( suc @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( times_times @ nat @ ( vEBT_VEBT_Tb2 @ ( suc @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) )
& ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
=> ( Y2
= ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ one2 ) ) ) @ ( vEBT_VEBT_Tb2 @ ( suc @ ( suc @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) @ ( times_times @ nat @ ( vEBT_VEBT_Tb2 @ ( suc @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) )
=> ~ ( accp @ nat @ vEBT_VEBT_Tb_rel @ ( suc @ ( suc @ N2 ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.Tb'.pelims
thf(fact_5620_VEBT__internal_OT__vebt__buildupi_Opelims,axiom,
! [X2: nat,Y2: nat] :
( ( ( vEBT_V441764108873111860ildupi @ X2 )
= Y2 )
=> ( ( accp @ nat @ vEBT_V2957053500504383685pi_rel @ X2 )
=> ( ( ( X2
= ( zero_zero @ nat ) )
=> ( ( Y2
= ( suc @ ( zero_zero @ nat ) ) )
=> ~ ( accp @ nat @ vEBT_V2957053500504383685pi_rel @ ( zero_zero @ nat ) ) ) )
=> ( ( ( X2
= ( suc @ ( zero_zero @ nat ) ) )
=> ( ( Y2
= ( suc @ ( zero_zero @ nat ) ) )
=> ~ ( accp @ nat @ vEBT_V2957053500504383685pi_rel @ ( suc @ ( zero_zero @ nat ) ) ) ) )
=> ~ ! [N2: nat] :
( ( X2
= ( suc @ ( suc @ N2 ) ) )
=> ( ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
=> ( Y2
= ( 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 )
=> ( Y2
= ( 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_5621_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_5622_ran__nth__set__encoding__conv,axiom,
! [A: $tType,L2: list @ A] :
( ( ran @ nat @ A
@ ^ [I3: nat] : ( if @ ( option @ A ) @ ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ L2 ) ) @ ( some @ A @ ( nth @ A @ L2 @ I3 ) ) @ ( none @ A ) ) )
= ( set2 @ A @ L2 ) ) ).
% ran_nth_set_encoding_conv
thf(fact_5623_rat__inverse__code,axiom,
! [P2: rat] :
( ( quotient_of @ ( inverse_inverse @ rat @ P2 ) )
= ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [A4: int,B3: 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 ) @ B3 ) @ ( abs_abs @ int @ A4 ) ) )
@ ( quotient_of @ P2 ) ) ) ).
% rat_inverse_code
thf(fact_5624_frac__eq__0__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A] :
( ( ( archimedean_frac @ A @ X2 )
= ( zero_zero @ A ) )
= ( member @ A @ X2 @ ( ring_1_Ints @ A ) ) ) ) ).
% frac_eq_0_iff
thf(fact_5625_floor__add2,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A,Y2: A] :
( ( ( member @ A @ X2 @ ( ring_1_Ints @ A ) )
| ( member @ A @ Y2 @ ( ring_1_Ints @ A ) ) )
=> ( ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ X2 @ Y2 ) )
= ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X2 ) @ ( archim6421214686448440834_floor @ A @ Y2 ) ) ) ) ) ).
% floor_add2
thf(fact_5626_frac__gt__0__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( archimedean_frac @ A @ X2 ) )
= ( ~ ( member @ A @ X2 @ ( ring_1_Ints @ A ) ) ) ) ) ).
% frac_gt_0_iff
thf(fact_5627_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_5628_rat__one__code,axiom,
( ( quotient_of @ ( one_one @ rat ) )
= ( product_Pair @ int @ int @ ( one_one @ int ) @ ( one_one @ int ) ) ) ).
% rat_one_code
thf(fact_5629_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
@ ^ [X: A] : ( if @ ( option @ B ) @ ( X = K ) @ ( some @ B @ V ) @ ( M @ X ) ) )
= ( insert @ B @ V @ ( ran @ A @ B @ M ) ) ) ) ).
% map_update_eta_repair(2)
thf(fact_5630_rat__zero__code,axiom,
( ( quotient_of @ ( zero_zero @ rat ) )
= ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) ) ) ).
% rat_zero_code
thf(fact_5631_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_5632_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_5633_divide__rat__def,axiom,
( ( divide_divide @ rat )
= ( ^ [Q4: rat,R5: rat] : ( times_times @ rat @ Q4 @ ( inverse_inverse @ rat @ R5 ) ) ) ) ).
% divide_rat_def
thf(fact_5634_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_5635_Ints__numeral,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [N: num] : ( member @ A @ ( numeral_numeral @ A @ N ) @ ( ring_1_Ints @ A ) ) ) ).
% Ints_numeral
thf(fact_5636_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_5637_Ints__0,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ( member @ A @ ( zero_zero @ A ) @ ( ring_1_Ints @ A ) ) ) ).
% Ints_0
thf(fact_5638_Ints__1,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ( member @ A @ ( one_one @ A ) @ ( ring_1_Ints @ A ) ) ) ).
% Ints_1
thf(fact_5639_Ints__of__nat,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [N: nat] : ( member @ A @ ( semiring_1_of_nat @ A @ N ) @ ( ring_1_Ints @ A ) ) ) ).
% Ints_of_nat
thf(fact_5640_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_5641_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_5642_finite__int__segment,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [A2: A,B2: A] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X: A] :
( ( member @ A @ X @ ( ring_1_Ints @ A ) )
& ( ord_less_eq @ A @ A2 @ X )
& ( ord_less_eq @ A @ X @ B2 ) ) ) ) ) ).
% finite_int_segment
thf(fact_5643_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_5644_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_5645_quotient__of__denom__pos,axiom,
! [R2: rat,P2: int,Q2: int] :
( ( ( quotient_of @ R2 )
= ( product_Pair @ int @ int @ P2 @ Q2 ) )
=> ( ord_less @ int @ ( zero_zero @ int ) @ Q2 ) ) ).
% quotient_of_denom_pos
thf(fact_5646_finite__abs__int__segment,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [A2: A] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [K2: A] :
( ( member @ A @ K2 @ ( ring_1_Ints @ A ) )
& ( ord_less_eq @ A @ ( abs_abs @ A @ K2 ) @ A2 ) ) ) ) ) ).
% finite_abs_int_segment
thf(fact_5647_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_5648_Ints__nonzero__abs__ge1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X2: A] :
( ( member @ A @ X2 @ ( ring_1_Ints @ A ) )
=> ( ( X2
!= ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ ( abs_abs @ A @ X2 ) ) ) ) ) ).
% Ints_nonzero_abs_ge1
thf(fact_5649_Ints__nonzero__abs__less1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X2: A] :
( ( member @ A @ X2 @ ( ring_1_Ints @ A ) )
=> ( ( ord_less @ A @ ( abs_abs @ A @ X2 ) @ ( one_one @ A ) )
=> ( X2
= ( zero_zero @ A ) ) ) ) ) ).
% Ints_nonzero_abs_less1
thf(fact_5650_Ints__eq__abs__less1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X2: A,Y2: A] :
( ( member @ A @ X2 @ ( ring_1_Ints @ A ) )
=> ( ( member @ A @ Y2 @ ( ring_1_Ints @ A ) )
=> ( ( X2 = Y2 )
= ( ord_less @ A @ ( abs_abs @ A @ ( minus_minus @ A @ X2 @ Y2 ) ) @ ( one_one @ A ) ) ) ) ) ) ).
% Ints_eq_abs_less1
thf(fact_5651_sin__times__pi__eq__0,axiom,
! [X2: real] :
( ( ( sin @ real @ ( times_times @ real @ X2 @ pi ) )
= ( zero_zero @ real ) )
= ( member @ real @ X2 @ ( ring_1_Ints @ real ) ) ) ).
% sin_times_pi_eq_0
thf(fact_5652_frac__neg,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A] :
( ( ( member @ A @ X2 @ ( ring_1_Ints @ A ) )
=> ( ( archimedean_frac @ A @ ( uminus_uminus @ A @ X2 ) )
= ( zero_zero @ A ) ) )
& ( ~ ( member @ A @ X2 @ ( ring_1_Ints @ A ) )
=> ( ( archimedean_frac @ A @ ( uminus_uminus @ A @ X2 ) )
= ( minus_minus @ A @ ( one_one @ A ) @ ( archimedean_frac @ A @ X2 ) ) ) ) ) ) ).
% frac_neg
thf(fact_5653_rat__uminus__code,axiom,
! [P2: rat] :
( ( quotient_of @ ( uminus_uminus @ rat @ P2 ) )
= ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [A4: int] : ( product_Pair @ int @ int @ ( uminus_uminus @ int @ A4 ) )
@ ( quotient_of @ P2 ) ) ) ).
% rat_uminus_code
thf(fact_5654_rat__less__code,axiom,
( ( ord_less @ rat )
= ( ^ [P3: rat,Q4: rat] :
( product_case_prod @ int @ int @ $o
@ ^ [A4: int,C3: int] :
( product_case_prod @ int @ int @ $o
@ ^ [B3: int,D: int] : ( ord_less @ int @ ( times_times @ int @ A4 @ D ) @ ( times_times @ int @ C3 @ B3 ) )
@ ( quotient_of @ Q4 ) )
@ ( quotient_of @ P3 ) ) ) ) ).
% rat_less_code
thf(fact_5655_rat__floor__code,axiom,
( ( archim6421214686448440834_floor @ rat )
= ( ^ [P3: rat] : ( product_case_prod @ int @ int @ int @ ( divide_divide @ int ) @ ( quotient_of @ P3 ) ) ) ) ).
% rat_floor_code
thf(fact_5656_rat__abs__code,axiom,
! [P2: rat] :
( ( quotient_of @ ( abs_abs @ rat @ P2 ) )
= ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [A4: int] : ( product_Pair @ int @ int @ ( abs_abs @ int @ A4 ) )
@ ( quotient_of @ P2 ) ) ) ).
% rat_abs_code
thf(fact_5657_rat__less__eq__code,axiom,
( ( ord_less_eq @ rat )
= ( ^ [P3: rat,Q4: rat] :
( product_case_prod @ int @ int @ $o
@ ^ [A4: int,C3: int] :
( product_case_prod @ int @ int @ $o
@ ^ [B3: int,D: int] : ( ord_less_eq @ int @ ( times_times @ int @ A4 @ D ) @ ( times_times @ int @ C3 @ B3 ) )
@ ( quotient_of @ Q4 ) )
@ ( quotient_of @ P3 ) ) ) ) ).
% rat_less_eq_code
thf(fact_5658_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_5659_frac__unique__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A,A2: A] :
( ( ( archimedean_frac @ A @ X2 )
= A2 )
= ( ( member @ A @ ( minus_minus @ A @ X2 @ 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_5660_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_5661_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_5662_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_5663_ran__empty,axiom,
! [B: $tType,A: $tType] :
( ( ran @ B @ A
@ ^ [X: B] : ( none @ A ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% ran_empty
thf(fact_5664_quotient__of__int,axiom,
! [A2: int] :
( ( quotient_of @ ( of_int @ A2 ) )
= ( product_Pair @ int @ int @ A2 @ ( one_one @ int ) ) ) ).
% quotient_of_int
thf(fact_5665_rat__minus__code,axiom,
! [P2: rat,Q2: rat] :
( ( quotient_of @ ( minus_minus @ rat @ P2 @ Q2 ) )
= ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [A4: int,C3: int] :
( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [B3: int,D: int] : ( normalize @ ( product_Pair @ int @ int @ ( minus_minus @ int @ ( times_times @ int @ A4 @ D ) @ ( times_times @ int @ B3 @ C3 ) ) @ ( times_times @ int @ C3 @ D ) ) )
@ ( quotient_of @ Q2 ) )
@ ( quotient_of @ P2 ) ) ) ).
% rat_minus_code
thf(fact_5666_normalize__denom__zero,axiom,
! [P2: int] :
( ( normalize @ ( product_Pair @ int @ int @ P2 @ ( zero_zero @ int ) ) )
= ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) ) ) ).
% normalize_denom_zero
thf(fact_5667_normalize__negative,axiom,
! [Q2: int,P2: int] :
( ( ord_less @ int @ Q2 @ ( zero_zero @ int ) )
=> ( ( normalize @ ( product_Pair @ int @ int @ P2 @ Q2 ) )
= ( normalize @ ( product_Pair @ int @ int @ ( uminus_uminus @ int @ P2 ) @ ( uminus_uminus @ int @ Q2 ) ) ) ) ) ).
% normalize_negative
thf(fact_5668_normalize__denom__pos,axiom,
! [R2: product_prod @ int @ int,P2: int,Q2: int] :
( ( ( normalize @ R2 )
= ( product_Pair @ int @ int @ P2 @ Q2 ) )
=> ( ord_less @ int @ ( zero_zero @ int ) @ Q2 ) ) ).
% normalize_denom_pos
thf(fact_5669_normalize__crossproduct,axiom,
! [Q2: int,S3: int,P2: int,R2: int] :
( ( Q2
!= ( zero_zero @ int ) )
=> ( ( S3
!= ( zero_zero @ int ) )
=> ( ( ( normalize @ ( product_Pair @ int @ int @ P2 @ Q2 ) )
= ( normalize @ ( product_Pair @ int @ int @ R2 @ S3 ) ) )
=> ( ( times_times @ int @ P2 @ S3 )
= ( times_times @ int @ R2 @ Q2 ) ) ) ) ) ).
% normalize_crossproduct
thf(fact_5670_rat__times__code,axiom,
! [P2: rat,Q2: rat] :
( ( quotient_of @ ( times_times @ rat @ P2 @ Q2 ) )
= ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [A4: int,C3: int] :
( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [B3: int,D: int] : ( normalize @ ( product_Pair @ int @ int @ ( times_times @ int @ A4 @ B3 ) @ ( times_times @ int @ C3 @ D ) ) )
@ ( quotient_of @ Q2 ) )
@ ( quotient_of @ P2 ) ) ) ).
% rat_times_code
thf(fact_5671_rat__divide__code,axiom,
! [P2: rat,Q2: rat] :
( ( quotient_of @ ( divide_divide @ rat @ P2 @ Q2 ) )
= ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [A4: int,C3: int] :
( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [B3: int,D: int] : ( normalize @ ( product_Pair @ int @ int @ ( times_times @ int @ A4 @ D ) @ ( times_times @ int @ C3 @ B3 ) ) )
@ ( quotient_of @ Q2 ) )
@ ( quotient_of @ P2 ) ) ) ).
% rat_divide_code
thf(fact_5672_rat__plus__code,axiom,
! [P2: rat,Q2: rat] :
( ( quotient_of @ ( plus_plus @ rat @ P2 @ Q2 ) )
= ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [A4: int,C3: int] :
( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [B3: int,D: int] : ( normalize @ ( product_Pair @ int @ int @ ( plus_plus @ int @ ( times_times @ int @ A4 @ D ) @ ( times_times @ int @ B3 @ C3 ) ) @ ( times_times @ int @ C3 @ D ) ) )
@ ( quotient_of @ Q2 ) )
@ ( quotient_of @ P2 ) ) ) ).
% rat_plus_code
thf(fact_5673_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_5674_setceilmax,axiom,
! [S3: vEBT_VEBT,M: nat,Listy: list @ vEBT_VEBT,N: nat] :
( ( vEBT_invar_vebt @ S3 @ 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 @ S3 ) )
= ( 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 @ S3 @ ( set2 @ vEBT_VEBT @ Listy ) ) ) ) )
= ( archimedean_ceiling @ real @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ M ) ) ) ) ) ) ) ) ) ).
% setceilmax
thf(fact_5675_VEBT__internal_Ospace_Opelims,axiom,
! [X2: vEBT_VEBT,Y2: nat] :
( ( ( vEBT_VEBT_space @ X2 )
= Y2 )
=> ( ( accp @ vEBT_VEBT @ vEBT_VEBT_space_rel2 @ X2 )
=> ( ! [A5: $o,B4: $o] :
( ( X2
= ( vEBT_Leaf @ A5 @ B4 ) )
=> ( ( Y2
= ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_VEBT_space_rel2 @ ( vEBT_Leaf @ A5 @ B4 ) ) ) )
=> ~ ! [Info2: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( ( Y2
= ( 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_5676_image__ident,axiom,
! [A: $tType,Y7: set @ A] :
( ( image @ A @ A
@ ^ [X: A] : X
@ Y7 )
= Y7 ) ).
% image_ident
thf(fact_5677_height__compose__list,axiom,
! [T2: vEBT_VEBT,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ T2 @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ( ord_less_eq @ nat @ ( vEBT_VEBT_height @ T2 ) @ ( lattic643756798349783984er_Max @ nat @ ( image @ vEBT_VEBT @ nat @ vEBT_VEBT_height @ ( insert @ vEBT_VEBT @ Summary @ ( set2 @ vEBT_VEBT @ TreeList ) ) ) ) ) ) ).
% height_compose_list
thf(fact_5678_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_5679_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_5680_image__add__0,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [S: set @ A] :
( ( image @ A @ A @ ( plus_plus @ A @ ( zero_zero @ A ) ) @ S )
= S ) ) ).
% image_add_0
thf(fact_5681_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_5682_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_5683_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_5684_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_5685_image__minus__const__atLeastAtMost_H,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [D2: A,A2: A,B2: A] :
( ( image @ A @ A
@ ^ [T: A] : ( minus_minus @ A @ T @ D2 )
@ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( minus_minus @ A @ A2 @ D2 ) @ ( minus_minus @ A @ B2 @ D2 ) ) ) ) ).
% image_minus_const_atLeastAtMost'
thf(fact_5686_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_5687_Max_Obounded__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,X2: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ ( lattic643756798349783984er_Max @ A @ A3 ) @ X2 )
= ( ! [X: A] :
( ( member @ A @ X @ A3 )
=> ( ord_less_eq @ A @ X @ X2 ) ) ) ) ) ) ) ).
% Max.bounded_iff
thf(fact_5688_Max__const,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ B,C2: A] :
( ( finite_finite2 @ B @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( lattic643756798349783984er_Max @ A
@ ( image @ B @ A
@ ^ [Uu3: B] : C2
@ A3 ) )
= C2 ) ) ) ) ).
% Max_const
thf(fact_5689_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_5690_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
@ ^ [C3: A] : ( divide_divide @ A @ C3 @ D2 )
@ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( divide_divide @ A @ A2 @ D2 ) @ ( divide_divide @ A @ B2 @ D2 ) ) ) ) ) ).
% image_divide_atLeastAtMost
thf(fact_5691_Max_OcoboundedI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,A2: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( member @ A @ A2 @ A3 )
=> ( ord_less_eq @ A @ A2 @ ( lattic643756798349783984er_Max @ A @ A3 ) ) ) ) ) ).
% Max.coboundedI
thf(fact_5692_Max__eq__if,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,B5: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( finite_finite2 @ A @ B5 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ A3 )
=> ? [Xa2: A] :
( ( member @ A @ Xa2 @ B5 )
& ( ord_less_eq @ A @ X3 @ Xa2 ) ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ B5 )
=> ? [Xa2: A] :
( ( member @ A @ Xa2 @ A3 )
& ( ord_less_eq @ A @ X3 @ Xa2 ) ) )
=> ( ( lattic643756798349783984er_Max @ A @ A3 )
= ( lattic643756798349783984er_Max @ A @ B5 ) ) ) ) ) ) ) ).
% Max_eq_if
thf(fact_5693_Max__eqI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,X2: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ! [Y3: A] :
( ( member @ A @ Y3 @ A3 )
=> ( ord_less_eq @ A @ Y3 @ X2 ) )
=> ( ( member @ A @ X2 @ A3 )
=> ( ( lattic643756798349783984er_Max @ A @ A3 )
= X2 ) ) ) ) ) ).
% Max_eqI
thf(fact_5694_Max__ge,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,X2: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( member @ A @ X2 @ A3 )
=> ( ord_less_eq @ A @ X2 @ ( lattic643756798349783984er_Max @ A @ A3 ) ) ) ) ) ).
% Max_ge
thf(fact_5695_Max__add__commute,axiom,
! [B: $tType,A: $tType] :
( ( linord4140545234300271783up_add @ A )
=> ! [S: set @ B,F2: B > A,K: A] :
( ( finite_finite2 @ B @ S )
=> ( ( S
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( lattic643756798349783984er_Max @ A
@ ( image @ B @ A
@ ^ [X: B] : ( plus_plus @ A @ ( F2 @ X ) @ K )
@ S ) )
= ( plus_plus @ A @ ( lattic643756798349783984er_Max @ A @ ( image @ B @ A @ F2 @ S ) ) @ K ) ) ) ) ) ).
% Max_add_commute
thf(fact_5696_image__diff__subset,axiom,
! [A: $tType,B: $tType,F2: B > A,A3: set @ B,B5: set @ B] : ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ ( image @ B @ A @ F2 @ A3 ) @ ( image @ B @ A @ F2 @ B5 ) ) @ ( image @ B @ A @ F2 @ ( minus_minus @ ( set @ B ) @ A3 @ B5 ) ) ) ).
% image_diff_subset
thf(fact_5697_all__subset__image,axiom,
! [A: $tType,B: $tType,F2: B > A,A3: set @ B,P: ( set @ A ) > $o] :
( ( ! [B8: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ B8 @ ( image @ B @ A @ F2 @ A3 ) )
=> ( P @ B8 ) ) )
= ( ! [B8: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ B8 @ A3 )
=> ( P @ ( image @ B @ A @ F2 @ B8 ) ) ) ) ) ).
% all_subset_image
thf(fact_5698_subset__image__iff,axiom,
! [A: $tType,B: $tType,B5: set @ A,F2: B > A,A3: set @ B] :
( ( ord_less_eq @ ( set @ A ) @ B5 @ ( image @ B @ A @ F2 @ A3 ) )
= ( ? [AA: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ AA @ A3 )
& ( B5
= ( image @ B @ A @ F2 @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_5699_image__subset__iff,axiom,
! [A: $tType,B: $tType,F2: B > A,A3: set @ B,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ F2 @ A3 ) @ B5 )
= ( ! [X: B] :
( ( member @ B @ X @ A3 )
=> ( member @ A @ ( F2 @ X ) @ B5 ) ) ) ) ).
% image_subset_iff
thf(fact_5700_subset__imageE,axiom,
! [A: $tType,B: $tType,B5: set @ A,F2: B > A,A3: set @ B] :
( ( ord_less_eq @ ( set @ A ) @ B5 @ ( image @ B @ A @ F2 @ A3 ) )
=> ~ ! [C7: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ C7 @ A3 )
=> ( B5
!= ( image @ B @ A @ F2 @ C7 ) ) ) ) ).
% subset_imageE
thf(fact_5701_image__subsetI,axiom,
! [A: $tType,B: $tType,A3: set @ A,F2: A > B,B5: set @ B] :
( ! [X3: A] :
( ( member @ A @ X3 @ A3 )
=> ( member @ B @ ( F2 @ X3 ) @ B5 ) )
=> ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F2 @ A3 ) @ B5 ) ) ).
% image_subsetI
thf(fact_5702_image__mono,axiom,
! [B: $tType,A: $tType,A3: set @ A,B5: set @ A,F2: A > B] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B5 )
=> ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F2 @ A3 ) @ ( image @ A @ B @ F2 @ B5 ) ) ) ).
% image_mono
thf(fact_5703_imageE,axiom,
! [A: $tType,B: $tType,B2: A,F2: B > A,A3: set @ B] :
( ( member @ A @ B2 @ ( image @ B @ A @ F2 @ A3 ) )
=> ~ ! [X3: B] :
( ( B2
= ( F2 @ X3 ) )
=> ~ ( member @ B @ X3 @ A3 ) ) ) ).
% imageE
thf(fact_5704_image__image,axiom,
! [A: $tType,B: $tType,C: $tType,F2: B > A,G: C > B,A3: set @ C] :
( ( image @ B @ A @ F2 @ ( image @ C @ B @ G @ A3 ) )
= ( image @ C @ A
@ ^ [X: C] : ( F2 @ ( G @ X ) )
@ A3 ) ) ).
% image_image
thf(fact_5705_Compr__image__eq,axiom,
! [A: $tType,B: $tType,F2: B > A,A3: set @ B,P: A > $o] :
( ( collect @ A
@ ^ [X: A] :
( ( member @ A @ X @ ( image @ B @ A @ F2 @ A3 ) )
& ( P @ X ) ) )
= ( image @ B @ A @ F2
@ ( collect @ B
@ ^ [X: B] :
( ( member @ B @ X @ A3 )
& ( P @ ( F2 @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_5706_pigeonhole__infinite,axiom,
! [B: $tType,A: $tType,A3: set @ A,F2: A > B] :
( ~ ( finite_finite2 @ A @ A3 )
=> ( ( finite_finite2 @ B @ ( image @ A @ B @ F2 @ A3 ) )
=> ? [X3: A] :
( ( member @ A @ X3 @ A3 )
& ~ ( finite_finite2 @ A
@ ( collect @ A
@ ^ [A4: A] :
( ( member @ A @ A4 @ A3 )
& ( ( F2 @ A4 )
= ( F2 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_5707_finite__surj,axiom,
! [A: $tType,B: $tType,A3: set @ A,B5: set @ B,F2: A > B] :
( ( finite_finite2 @ A @ A3 )
=> ( ( ord_less_eq @ ( set @ B ) @ B5 @ ( image @ A @ B @ F2 @ A3 ) )
=> ( finite_finite2 @ B @ B5 ) ) ) ).
% finite_surj
thf(fact_5708_finite__subset__image,axiom,
! [A: $tType,B: $tType,B5: set @ A,F2: B > A,A3: set @ B] :
( ( finite_finite2 @ A @ B5 )
=> ( ( ord_less_eq @ ( set @ A ) @ B5 @ ( image @ B @ A @ F2 @ A3 ) )
=> ? [C7: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ C7 @ A3 )
& ( finite_finite2 @ B @ C7 )
& ( B5
= ( image @ B @ A @ F2 @ C7 ) ) ) ) ) ).
% finite_subset_image
thf(fact_5709_ex__finite__subset__image,axiom,
! [A: $tType,B: $tType,F2: B > A,A3: set @ B,P: ( set @ A ) > $o] :
( ( ? [B8: set @ A] :
( ( finite_finite2 @ A @ B8 )
& ( ord_less_eq @ ( set @ A ) @ B8 @ ( image @ B @ A @ F2 @ A3 ) )
& ( P @ B8 ) ) )
= ( ? [B8: set @ B] :
( ( finite_finite2 @ B @ B8 )
& ( ord_less_eq @ ( set @ B ) @ B8 @ A3 )
& ( P @ ( image @ B @ A @ F2 @ B8 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_5710_all__finite__subset__image,axiom,
! [A: $tType,B: $tType,F2: B > A,A3: set @ B,P: ( set @ A ) > $o] :
( ( ! [B8: set @ A] :
( ( ( finite_finite2 @ A @ B8 )
& ( ord_less_eq @ ( set @ A ) @ B8 @ ( image @ B @ A @ F2 @ A3 ) ) )
=> ( P @ B8 ) ) )
= ( ! [B8: set @ B] :
( ( ( finite_finite2 @ B @ B8 )
& ( ord_less_eq @ ( set @ B ) @ B8 @ A3 ) )
=> ( P @ ( image @ B @ A @ F2 @ B8 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_5711_image__constant__conv,axiom,
! [B: $tType,A: $tType,A3: set @ B,C2: A] :
( ( ( A3
= ( bot_bot @ ( set @ B ) ) )
=> ( ( image @ B @ A
@ ^ [X: B] : C2
@ A3 )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( image @ B @ A
@ ^ [X: B] : C2
@ A3 )
= ( insert @ A @ C2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% image_constant_conv
thf(fact_5712_image__constant,axiom,
! [A: $tType,B: $tType,X2: A,A3: set @ A,C2: B] :
( ( member @ A @ X2 @ A3 )
=> ( ( image @ A @ B
@ ^ [X: A] : C2
@ A3 )
= ( insert @ B @ C2 @ ( bot_bot @ ( set @ B ) ) ) ) ) ).
% image_constant
thf(fact_5713_sum_Oimage__gen,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [S: set @ B,H2: B > A,G: B > C] :
( ( finite_finite2 @ B @ S )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ H2 @ S )
= ( groups7311177749621191930dd_sum @ C @ A
@ ^ [Y: C] :
( groups7311177749621191930dd_sum @ B @ A @ H2
@ ( collect @ B
@ ^ [X: B] :
( ( member @ B @ X @ S )
& ( ( G @ X )
= Y ) ) ) )
@ ( image @ B @ C @ G @ S ) ) ) ) ) ).
% sum.image_gen
thf(fact_5714_prod_Oimage__gen,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S: set @ B,H2: B > A,G: B > C] :
( ( finite_finite2 @ B @ S )
=> ( ( groups7121269368397514597t_prod @ B @ A @ H2 @ S )
= ( groups7121269368397514597t_prod @ C @ A
@ ^ [Y: C] :
( groups7121269368397514597t_prod @ B @ A @ H2
@ ( collect @ B
@ ^ [X: B] :
( ( member @ B @ X @ S )
& ( ( G @ X )
= Y ) ) ) )
@ ( image @ B @ C @ G @ S ) ) ) ) ) ).
% prod.image_gen
thf(fact_5715_Max__eq__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,M: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ( lattic643756798349783984er_Max @ A @ A3 )
= M )
= ( ( member @ A @ M @ A3 )
& ! [X: A] :
( ( member @ A @ X @ A3 )
=> ( ord_less_eq @ A @ X @ M ) ) ) ) ) ) ) ).
% Max_eq_iff
thf(fact_5716_Max__ge__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,X2: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ X2 @ ( lattic643756798349783984er_Max @ A @ A3 ) )
= ( ? [X: A] :
( ( member @ A @ X @ A3 )
& ( ord_less_eq @ A @ X2 @ X ) ) ) ) ) ) ) ).
% Max_ge_iff
thf(fact_5717_eq__Max__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,M: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( M
= ( lattic643756798349783984er_Max @ A @ A3 ) )
= ( ( member @ A @ M @ A3 )
& ! [X: A] :
( ( member @ A @ X @ A3 )
=> ( ord_less_eq @ A @ X @ M ) ) ) ) ) ) ) ).
% eq_Max_iff
thf(fact_5718_Max_OboundedE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,X2: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ ( lattic643756798349783984er_Max @ A @ A3 ) @ X2 )
=> ! [A9: A] :
( ( member @ A @ A9 @ A3 )
=> ( ord_less_eq @ A @ A9 @ X2 ) ) ) ) ) ) ).
% Max.boundedE
thf(fact_5719_Max_OboundedI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,X2: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [A5: A] :
( ( member @ A @ A5 @ A3 )
=> ( ord_less_eq @ A @ A5 @ X2 ) )
=> ( ord_less_eq @ A @ ( lattic643756798349783984er_Max @ A @ A3 ) @ X2 ) ) ) ) ) ).
% Max.boundedI
thf(fact_5720_Max__insert2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,A2: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ! [B4: A] :
( ( member @ A @ B4 @ A3 )
=> ( ord_less_eq @ A @ B4 @ A2 ) )
=> ( ( lattic643756798349783984er_Max @ A @ ( insert @ A @ A2 @ A3 ) )
= A2 ) ) ) ) ).
% Max_insert2
thf(fact_5721_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_5722_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_5723_sum_Ogroup,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [S: set @ B,T3: set @ C,G: B > C,H2: B > A] :
( ( finite_finite2 @ B @ S )
=> ( ( finite_finite2 @ C @ T3 )
=> ( ( ord_less_eq @ ( set @ C ) @ ( image @ B @ C @ G @ S ) @ T3 )
=> ( ( groups7311177749621191930dd_sum @ C @ A
@ ^ [Y: C] :
( groups7311177749621191930dd_sum @ B @ A @ H2
@ ( collect @ B
@ ^ [X: B] :
( ( member @ B @ X @ S )
& ( ( G @ X )
= Y ) ) ) )
@ T3 )
= ( groups7311177749621191930dd_sum @ B @ A @ H2 @ S ) ) ) ) ) ) ).
% sum.group
thf(fact_5724_prod_Ogroup,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S: set @ B,T3: set @ C,G: B > C,H2: B > A] :
( ( finite_finite2 @ B @ S )
=> ( ( finite_finite2 @ C @ T3 )
=> ( ( ord_less_eq @ ( set @ C ) @ ( image @ B @ C @ G @ S ) @ T3 )
=> ( ( groups7121269368397514597t_prod @ C @ A
@ ^ [Y: C] :
( groups7121269368397514597t_prod @ B @ A @ H2
@ ( collect @ B
@ ^ [X: B] :
( ( member @ B @ X @ S )
& ( ( G @ X )
= Y ) ) ) )
@ T3 )
= ( groups7121269368397514597t_prod @ B @ A @ H2 @ S ) ) ) ) ) ) ).
% prod.group
thf(fact_5725_scaleR__image__atLeastAtMost,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C2: real,X2: A,Y2: A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ C2 )
=> ( ( image @ A @ A @ ( real_V8093663219630862766scaleR @ A @ C2 ) @ ( set_or1337092689740270186AtMost @ A @ X2 @ Y2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( real_V8093663219630862766scaleR @ A @ C2 @ X2 ) @ ( real_V8093663219630862766scaleR @ A @ C2 @ Y2 ) ) ) ) ) ).
% scaleR_image_atLeastAtMost
thf(fact_5726_Max_Osubset__imp,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B5 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B5 )
=> ( ord_less_eq @ A @ ( lattic643756798349783984er_Max @ A @ A3 ) @ ( lattic643756798349783984er_Max @ A @ B5 ) ) ) ) ) ) ).
% Max.subset_imp
thf(fact_5727_Max__mono,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [M7: set @ A,N5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ M7 @ N5 )
=> ( ( M7
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ N5 )
=> ( ord_less_eq @ A @ ( lattic643756798349783984er_Max @ A @ M7 ) @ ( lattic643756798349783984er_Max @ A @ N5 ) ) ) ) ) ) ).
% Max_mono
thf(fact_5728_VEBT__internal_Oheight_Oelims,axiom,
! [X2: vEBT_VEBT,Y2: nat] :
( ( ( vEBT_VEBT_height @ X2 )
= Y2 )
=> ( ( ? [A5: $o,B4: $o] :
( X2
= ( vEBT_Leaf @ A5 @ B4 ) )
=> ( Y2
!= ( zero_zero @ nat ) ) )
=> ~ ! [Uu2: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ Uu2 @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( Y2
!= ( 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_5729_Max_Osubset,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,B5: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( B5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ A ) @ B5 @ A3 )
=> ( ( ord_max @ A @ ( lattic643756798349783984er_Max @ A @ B5 ) @ ( lattic643756798349783984er_Max @ A @ A3 ) )
= ( lattic643756798349783984er_Max @ A @ A3 ) ) ) ) ) ) ).
% Max.subset
thf(fact_5730_divide__nat__def,axiom,
( ( divide_divide @ nat )
= ( ^ [M5: nat,N3: nat] :
( if @ nat
@ ( N3
= ( zero_zero @ nat ) )
@ ( zero_zero @ nat )
@ ( lattic643756798349783984er_Max @ nat
@ ( collect @ nat
@ ^ [K2: nat] : ( ord_less_eq @ nat @ ( times_times @ nat @ K2 @ N3 ) @ M5 ) ) ) ) ) ) ).
% divide_nat_def
thf(fact_5731_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_5732_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_5733_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_5734_image__mult__atLeastAtMost__if,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,X2: A,Y2: A] :
( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( image @ A @ A @ ( times_times @ A @ C2 ) @ ( set_or1337092689740270186AtMost @ A @ X2 @ Y2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( times_times @ A @ C2 @ X2 ) @ ( times_times @ A @ C2 @ Y2 ) ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less_eq @ A @ X2 @ Y2 )
=> ( ( image @ A @ A @ ( times_times @ A @ C2 ) @ ( set_or1337092689740270186AtMost @ A @ X2 @ Y2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( times_times @ A @ C2 @ Y2 ) @ ( times_times @ A @ C2 @ X2 ) ) ) )
& ( ~ ( ord_less_eq @ A @ X2 @ Y2 )
=> ( ( image @ A @ A @ ( times_times @ A @ C2 ) @ ( set_or1337092689740270186AtMost @ A @ X2 @ Y2 ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% image_mult_atLeastAtMost_if
thf(fact_5735_image__mult__atLeastAtMost__if_H,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X2: A,Y2: A,C2: A] :
( ( ( ord_less_eq @ A @ X2 @ Y2 )
=> ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( image @ A @ A
@ ^ [X: A] : ( times_times @ A @ X @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ X2 @ Y2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( times_times @ A @ X2 @ C2 ) @ ( times_times @ A @ Y2 @ C2 ) ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( image @ A @ A
@ ^ [X: A] : ( times_times @ A @ X @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ X2 @ Y2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( times_times @ A @ Y2 @ C2 ) @ ( times_times @ A @ X2 @ C2 ) ) ) ) ) )
& ( ~ ( ord_less_eq @ A @ X2 @ Y2 )
=> ( ( image @ A @ A
@ ^ [X: A] : ( times_times @ A @ X @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ X2 @ Y2 ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% image_mult_atLeastAtMost_if'
thf(fact_5736_image__affinity__atLeastAtMost,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A,M: A,C2: A] :
( ( ( ( set_or1337092689740270186AtMost @ A @ A2 @ B2 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( image @ A @ A
@ ^ [X: A] : ( plus_plus @ A @ ( times_times @ A @ M @ X ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( ( set_or1337092689740270186AtMost @ A @ A2 @ B2 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ M )
=> ( ( image @ A @ A
@ ^ [X: A] : ( plus_plus @ A @ ( times_times @ A @ M @ X ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( plus_plus @ A @ ( times_times @ A @ M @ A2 ) @ C2 ) @ ( plus_plus @ A @ ( times_times @ A @ M @ B2 ) @ C2 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ M )
=> ( ( image @ A @ A
@ ^ [X: A] : ( plus_plus @ A @ ( times_times @ A @ M @ X ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( plus_plus @ A @ ( times_times @ A @ M @ B2 ) @ C2 ) @ ( plus_plus @ A @ ( times_times @ A @ M @ A2 ) @ C2 ) ) ) ) ) ) ) ) ).
% image_affinity_atLeastAtMost
thf(fact_5737_image__affinity__atLeastAtMost__diff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A,M: A,C2: A] :
( ( ( ( set_or1337092689740270186AtMost @ A @ A2 @ B2 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( image @ A @ A
@ ^ [X: A] : ( minus_minus @ A @ ( times_times @ A @ M @ X ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( ( set_or1337092689740270186AtMost @ A @ A2 @ B2 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ M )
=> ( ( image @ A @ A
@ ^ [X: A] : ( minus_minus @ A @ ( times_times @ A @ M @ X ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( minus_minus @ A @ ( times_times @ A @ M @ A2 ) @ C2 ) @ ( minus_minus @ A @ ( times_times @ A @ M @ B2 ) @ C2 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ M )
=> ( ( image @ A @ A
@ ^ [X: A] : ( minus_minus @ A @ ( times_times @ A @ M @ X ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( minus_minus @ A @ ( times_times @ A @ M @ B2 ) @ C2 ) @ ( minus_minus @ A @ ( times_times @ A @ M @ A2 ) @ C2 ) ) ) ) ) ) ) ) ).
% image_affinity_atLeastAtMost_diff
thf(fact_5738_image__affinity__atLeastAtMost__div,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A,M: A,C2: A] :
( ( ( ( set_or1337092689740270186AtMost @ A @ A2 @ B2 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( image @ A @ A
@ ^ [X: A] : ( plus_plus @ A @ ( divide_divide @ A @ X @ M ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( ( set_or1337092689740270186AtMost @ A @ A2 @ B2 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ M )
=> ( ( image @ A @ A
@ ^ [X: A] : ( plus_plus @ A @ ( divide_divide @ A @ X @ M ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( plus_plus @ A @ ( divide_divide @ A @ A2 @ M ) @ C2 ) @ ( plus_plus @ A @ ( divide_divide @ A @ B2 @ M ) @ C2 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ M )
=> ( ( image @ A @ A
@ ^ [X: A] : ( plus_plus @ A @ ( divide_divide @ A @ X @ M ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( plus_plus @ A @ ( divide_divide @ A @ B2 @ M ) @ C2 ) @ ( plus_plus @ A @ ( divide_divide @ A @ A2 @ M ) @ C2 ) ) ) ) ) ) ) ) ).
% image_affinity_atLeastAtMost_div
thf(fact_5739_image__affinity__atLeastAtMost__div__diff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A,M: A,C2: A] :
( ( ( ( set_or1337092689740270186AtMost @ A @ A2 @ B2 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( image @ A @ A
@ ^ [X: A] : ( minus_minus @ A @ ( divide_divide @ A @ X @ M ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( ( set_or1337092689740270186AtMost @ A @ A2 @ B2 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ M )
=> ( ( image @ A @ A
@ ^ [X: A] : ( minus_minus @ A @ ( divide_divide @ A @ X @ M ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( minus_minus @ A @ ( divide_divide @ A @ A2 @ M ) @ C2 ) @ ( minus_minus @ A @ ( divide_divide @ A @ B2 @ M ) @ C2 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ M )
=> ( ( image @ A @ A
@ ^ [X: A] : ( minus_minus @ A @ ( divide_divide @ A @ X @ M ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( minus_minus @ A @ ( divide_divide @ A @ B2 @ M ) @ C2 ) @ ( minus_minus @ A @ ( divide_divide @ A @ A2 @ M ) @ C2 ) ) ) ) ) ) ) ) ).
% image_affinity_atLeastAtMost_div_diff
thf(fact_5740_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_5741_Frct__code__post_I6_J,axiom,
! [K: num,L2: num] :
( ( frct @ ( product_Pair @ int @ int @ ( numeral_numeral @ int @ K ) @ ( numeral_numeral @ int @ L2 ) ) )
= ( divide_divide @ rat @ ( numeral_numeral @ rat @ K ) @ ( numeral_numeral @ rat @ L2 ) ) ) ).
% Frct_code_post(6)
thf(fact_5742_VEBT__internal_Ospace_H_Opelims,axiom,
! [X2: vEBT_VEBT,Y2: nat] :
( ( ( vEBT_VEBT_space2 @ X2 )
= Y2 )
=> ( ( accp @ vEBT_VEBT @ vEBT_VEBT_space_rel @ X2 )
=> ( ! [A5: $o,B4: $o] :
( ( X2
= ( vEBT_Leaf @ A5 @ B4 ) )
=> ( ( Y2
= ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_VEBT_space_rel @ ( vEBT_Leaf @ A5 @ B4 ) ) ) )
=> ~ ! [Info2: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( ( Y2
= ( 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_5743_VEBT__internal_Oheight_Opelims,axiom,
! [X2: vEBT_VEBT,Y2: nat] :
( ( ( vEBT_VEBT_height @ X2 )
= Y2 )
=> ( ( accp @ vEBT_VEBT @ vEBT_VEBT_height_rel @ X2 )
=> ( ! [A5: $o,B4: $o] :
( ( X2
= ( vEBT_Leaf @ A5 @ B4 ) )
=> ( ( Y2
= ( zero_zero @ nat ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_VEBT_height_rel @ ( vEBT_Leaf @ A5 @ B4 ) ) ) )
=> ~ ! [Uu2: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ Uu2 @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( ( Y2
= ( 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_5744_VEBT__internal_Ocnt_Opelims,axiom,
! [X2: vEBT_VEBT,Y2: real] :
( ( ( vEBT_VEBT_cnt @ X2 )
= Y2 )
=> ( ( accp @ vEBT_VEBT @ vEBT_VEBT_cnt_rel2 @ X2 )
=> ( ! [A5: $o,B4: $o] :
( ( X2
= ( vEBT_Leaf @ A5 @ B4 ) )
=> ( ( Y2
= ( one_one @ real ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_VEBT_cnt_rel2 @ ( vEBT_Leaf @ A5 @ B4 ) ) ) )
=> ~ ! [Info2: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( ( Y2
= ( 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_5745_bij__betw__Suc,axiom,
! [M7: set @ nat,N5: set @ nat] :
( ( bij_betw @ nat @ nat @ suc @ M7 @ N5 )
= ( ( image @ nat @ nat @ suc @ M7 )
= N5 ) ) ).
% bij_betw_Suc
thf(fact_5746_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_5747_bij__betw__of__nat,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [N5: set @ nat,A3: set @ A] :
( ( bij_betw @ nat @ A @ ( semiring_1_of_nat @ A ) @ N5 @ A3 )
= ( ( image @ nat @ A @ ( semiring_1_of_nat @ A ) @ N5 )
= A3 ) ) ) ).
% bij_betw_of_nat
thf(fact_5748_pair__imageI,axiom,
! [C: $tType,B: $tType,A: $tType,A2: A,B2: B,A3: set @ ( product_prod @ A @ B ),F2: A > B > C] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ A3 )
=> ( member @ C @ ( F2 @ A2 @ B2 ) @ ( image @ ( product_prod @ A @ B ) @ C @ ( product_case_prod @ A @ B @ C @ F2 ) @ A3 ) ) ) ).
% pair_imageI
thf(fact_5749_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_5750_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_5751_zero__notin__Suc__image,axiom,
! [A3: set @ nat] :
~ ( member @ nat @ ( zero_zero @ nat ) @ ( image @ nat @ nat @ suc @ A3 ) ) ).
% zero_notin_Suc_image
thf(fact_5752_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
@ ^ [I3: nat] : ( ord_less @ nat @ I3 @ N3 ) ) ) ) ) ) ).
% finite_conv_nat_seg_image
thf(fact_5753_nat__seg__image__imp__finite,axiom,
! [A: $tType,A3: set @ A,F2: nat > A,N: nat] :
( ( A3
= ( image @ nat @ A @ F2
@ ( collect @ nat
@ ^ [I3: nat] : ( ord_less @ nat @ I3 @ N ) ) ) )
=> ( finite_finite2 @ A @ A3 ) ) ).
% nat_seg_image_imp_finite
thf(fact_5754_image__int__atLeastAtMost,axiom,
! [A2: nat,B2: nat] :
( ( image @ nat @ int @ ( semiring_1_of_nat @ int ) @ ( set_or1337092689740270186AtMost @ nat @ A2 @ B2 ) )
= ( set_or1337092689740270186AtMost @ int @ ( semiring_1_of_nat @ int @ A2 ) @ ( semiring_1_of_nat @ int @ B2 ) ) ) ).
% image_int_atLeastAtMost
thf(fact_5755_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_5756_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_5757_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_5758_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_5759_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_5760_VEBT__internal_Ocnt_H_Opelims,axiom,
! [X2: vEBT_VEBT,Y2: nat] :
( ( ( vEBT_VEBT_cnt2 @ X2 )
= Y2 )
=> ( ( accp @ vEBT_VEBT @ vEBT_VEBT_cnt_rel @ X2 )
=> ( ! [A5: $o,B4: $o] :
( ( X2
= ( vEBT_Leaf @ A5 @ B4 ) )
=> ( ( Y2
= ( one_one @ nat ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_VEBT_cnt_rel @ ( vEBT_Leaf @ A5 @ B4 ) ) ) )
=> ~ ! [Info2: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( ( Y2
= ( 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_5761_vebt__maxt_Opelims,axiom,
! [X2: vEBT_VEBT,Y2: option @ nat] :
( ( ( vEBT_vebt_maxt @ X2 )
= Y2 )
=> ( ( accp @ vEBT_VEBT @ vEBT_vebt_maxt_rel @ X2 )
=> ( ! [A5: $o,B4: $o] :
( ( X2
= ( vEBT_Leaf @ A5 @ B4 ) )
=> ( ( ( B4
=> ( Y2
= ( some @ nat @ ( one_one @ nat ) ) ) )
& ( ~ B4
=> ( ( A5
=> ( Y2
= ( some @ nat @ ( zero_zero @ nat ) ) ) )
& ( ~ A5
=> ( Y2
= ( none @ nat ) ) ) ) ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_vebt_maxt_rel @ ( vEBT_Leaf @ A5 @ B4 ) ) ) )
=> ( ! [Uu2: nat,Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) )
=> ( ( Y2
= ( none @ nat ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_vebt_maxt_rel @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
=> ( ( Y2
= ( some @ nat @ Ma2 ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_vebt_maxt_rel @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) ) ) ) ) ) ) ) ).
% vebt_maxt.pelims
thf(fact_5762_vebt__mint_Opelims,axiom,
! [X2: vEBT_VEBT,Y2: option @ nat] :
( ( ( vEBT_vebt_mint @ X2 )
= Y2 )
=> ( ( accp @ vEBT_VEBT @ vEBT_vebt_mint_rel @ X2 )
=> ( ! [A5: $o,B4: $o] :
( ( X2
= ( vEBT_Leaf @ A5 @ B4 ) )
=> ( ( ( A5
=> ( Y2
= ( some @ nat @ ( zero_zero @ nat ) ) ) )
& ( ~ A5
=> ( ( B4
=> ( Y2
= ( some @ nat @ ( one_one @ nat ) ) ) )
& ( ~ B4
=> ( Y2
= ( none @ nat ) ) ) ) ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_vebt_mint_rel @ ( vEBT_Leaf @ A5 @ B4 ) ) ) )
=> ( ! [Uu2: nat,Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) )
=> ( ( Y2
= ( none @ nat ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_vebt_mint_rel @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
=> ( ( Y2
= ( some @ nat @ Mi2 ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_vebt_mint_rel @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) ) ) ) ) ) ) ) ).
% vebt_mint.pelims
thf(fact_5763_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Opelims,axiom,
! [X2: vEBT_VEBT,Y2: nat] :
( ( ( vEBT_T_m_i_n_t @ X2 )
= Y2 )
=> ( ( accp @ vEBT_VEBT @ vEBT_T_m_i_n_t_rel @ X2 )
=> ( ! [A5: $o,B4: $o] :
( ( X2
= ( vEBT_Leaf @ A5 @ B4 ) )
=> ( ( Y2
= ( plus_plus @ nat @ ( one_one @ nat ) @ ( if @ nat @ A5 @ ( zero_zero @ nat ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( one_one @ nat ) ) ) ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_T_m_i_n_t_rel @ ( vEBT_Leaf @ A5 @ B4 ) ) ) )
=> ( ! [Uu2: nat,Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) )
=> ( ( Y2
= ( one_one @ nat ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_T_m_i_n_t_rel @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
=> ( ( Y2
= ( one_one @ nat ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_T_m_i_n_t_rel @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) ) ) ) ) ) ) ) ).
% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.pelims
thf(fact_5764_bij__betw__add,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [A2: A,A3: set @ A,B5: set @ A] :
( ( bij_betw @ A @ A @ ( plus_plus @ A @ A2 ) @ A3 @ B5 )
= ( ( image @ A @ A @ ( plus_plus @ A @ A2 ) @ A3 )
= B5 ) ) ) ).
% bij_betw_add
thf(fact_5765_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t_Opelims,axiom,
! [X2: vEBT_VEBT,Y2: nat] :
( ( ( vEBT_T_m_a_x_t @ X2 )
= Y2 )
=> ( ( accp @ vEBT_VEBT @ vEBT_T_m_a_x_t_rel @ X2 )
=> ( ! [A5: $o,B4: $o] :
( ( X2
= ( vEBT_Leaf @ A5 @ B4 ) )
=> ( ( Y2
= ( plus_plus @ nat @ ( one_one @ nat ) @ ( if @ nat @ B4 @ ( one_one @ nat ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( one_one @ nat ) ) ) ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_T_m_a_x_t_rel @ ( vEBT_Leaf @ A5 @ B4 ) ) ) )
=> ( ! [Uu2: nat,Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) )
=> ( ( Y2
= ( one_one @ nat ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_T_m_a_x_t_rel @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
=> ( ( Y2
= ( one_one @ nat ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_T_m_a_x_t_rel @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) ) ) ) ) ) ) ) ).
% T\<^sub>m\<^sub>a\<^sub>x\<^sub>t.pelims
thf(fact_5766_translation__diff,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A2: A,S3: set @ A,T2: set @ A] :
( ( image @ A @ A @ ( plus_plus @ A @ A2 ) @ ( minus_minus @ ( set @ A ) @ S3 @ T2 ) )
= ( minus_minus @ ( set @ A ) @ ( image @ A @ A @ ( plus_plus @ A @ A2 ) @ S3 ) @ ( image @ A @ A @ ( plus_plus @ A @ A2 ) @ T2 ) ) ) ) ).
% translation_diff
thf(fact_5767_bij__betw__subset,axiom,
! [A: $tType,B: $tType,F2: A > B,A3: set @ A,A10: set @ B,B5: set @ A,B10: set @ B] :
( ( bij_betw @ A @ B @ F2 @ A3 @ A10 )
=> ( ( ord_less_eq @ ( set @ A ) @ B5 @ A3 )
=> ( ( ( image @ A @ B @ F2 @ B5 )
= B10 )
=> ( bij_betw @ A @ B @ F2 @ B5 @ B10 ) ) ) ) ).
% bij_betw_subset
thf(fact_5768_bij__betw__byWitness,axiom,
! [A: $tType,B: $tType,A3: set @ A,F7: B > A,F2: A > B,A10: set @ B] :
( ! [X3: A] :
( ( member @ A @ X3 @ A3 )
=> ( ( F7 @ ( F2 @ X3 ) )
= X3 ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ A10 )
=> ( ( F2 @ ( F7 @ X3 ) )
= X3 ) )
=> ( ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F2 @ A3 ) @ A10 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ F7 @ A10 ) @ A3 )
=> ( bij_betw @ A @ B @ F2 @ A3 @ A10 ) ) ) ) ) ).
% bij_betw_byWitness
thf(fact_5769_translation__Compl,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A2: A,T2: set @ A] :
( ( image @ A @ A @ ( plus_plus @ A @ A2 ) @ ( uminus_uminus @ ( set @ A ) @ T2 ) )
= ( uminus_uminus @ ( set @ A ) @ ( image @ A @ A @ ( plus_plus @ A @ A2 ) @ T2 ) ) ) ) ).
% translation_Compl
thf(fact_5770_translation__subtract__diff,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A2: A,S3: set @ A,T2: set @ A] :
( ( image @ A @ A
@ ^ [X: A] : ( minus_minus @ A @ X @ A2 )
@ ( minus_minus @ ( set @ A ) @ S3 @ T2 ) )
= ( minus_minus @ ( set @ A )
@ ( image @ A @ A
@ ^ [X: A] : ( minus_minus @ A @ X @ A2 )
@ S3 )
@ ( image @ A @ A
@ ^ [X: A] : ( minus_minus @ A @ X @ A2 )
@ T2 ) ) ) ) ).
% translation_subtract_diff
thf(fact_5771_translation__subtract__Compl,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A2: A,T2: set @ A] :
( ( image @ A @ A
@ ^ [X: A] : ( minus_minus @ A @ X @ A2 )
@ ( uminus_uminus @ ( set @ A ) @ T2 ) )
= ( uminus_uminus @ ( set @ A )
@ ( image @ A @ A
@ ^ [X: A] : ( minus_minus @ A @ X @ A2 )
@ T2 ) ) ) ) ).
% translation_subtract_Compl
thf(fact_5772_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l_Opelims,axiom,
! [X2: vEBT_VEBT,Y2: nat] :
( ( ( vEBT_T_m_i_n_N_u_l_l @ X2 )
= Y2 )
=> ( ( accp @ vEBT_VEBT @ vEBT_T5462971552011256508_l_rel @ X2 )
=> ( ( ( X2
= ( vEBT_Leaf @ $false @ $false ) )
=> ( ( Y2
= ( one_one @ nat ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_T5462971552011256508_l_rel @ ( vEBT_Leaf @ $false @ $false ) ) ) )
=> ( ! [Uv2: $o] :
( ( X2
= ( vEBT_Leaf @ $true @ Uv2 ) )
=> ( ( Y2
= ( one_one @ nat ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_T5462971552011256508_l_rel @ ( vEBT_Leaf @ $true @ Uv2 ) ) ) )
=> ( ! [Uu2: $o] :
( ( X2
= ( vEBT_Leaf @ Uu2 @ $true ) )
=> ( ( Y2
= ( one_one @ nat ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_T5462971552011256508_l_rel @ ( vEBT_Leaf @ Uu2 @ $true ) ) ) )
=> ( ! [Uw2: nat,Ux2: list @ vEBT_VEBT,Uy2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uw2 @ Ux2 @ Uy2 ) )
=> ( ( Y2
= ( 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] :
( ( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ Uz2 ) @ Va3 @ Vb2 @ Vc ) )
=> ( ( Y2
= ( 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_5773_VEBT__internal_OminNull_Opelims_I1_J,axiom,
! [X2: vEBT_VEBT,Y2: $o] :
( ( ( vEBT_VEBT_minNull @ X2 )
= Y2 )
=> ( ( accp @ vEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ X2 )
=> ( ( ( X2
= ( vEBT_Leaf @ $false @ $false ) )
=> ( Y2
=> ~ ( accp @ vEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $false @ $false ) ) ) )
=> ( ! [Uv2: $o] :
( ( X2
= ( vEBT_Leaf @ $true @ Uv2 ) )
=> ( ~ Y2
=> ~ ( accp @ vEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $true @ Uv2 ) ) ) )
=> ( ! [Uu2: $o] :
( ( X2
= ( vEBT_Leaf @ Uu2 @ $true ) )
=> ( ~ Y2
=> ~ ( accp @ vEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ Uu2 @ $true ) ) ) )
=> ( ! [Uw2: nat,Ux2: list @ vEBT_VEBT,Uy2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uw2 @ Ux2 @ Uy2 ) )
=> ( Y2
=> ~ ( 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] :
( ( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ Uz2 ) @ Va3 @ Vb2 @ Vc ) )
=> ( ~ Y2
=> ~ ( accp @ vEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ Uz2 ) @ Va3 @ Vb2 @ Vc ) ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.minNull.pelims(1)
thf(fact_5774_sofl__test,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A,Y2: word @ A] :
( ( ( plus_plus @ int @ ( ring_1_signed @ A @ int @ X2 ) @ ( ring_1_signed @ A @ int @ Y2 ) )
= ( ring_1_signed @ A @ int @ ( plus_plus @ ( word @ A ) @ X2 @ Y2 ) ) )
= ( ( bit_se4197421643247451524op_bit @ ( word @ A ) @ ( minus_minus @ nat @ ( size_size @ ( word @ A ) @ X2 ) @ ( one_one @ nat ) ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ ( bit_se5824344971392196577ns_xor @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ X2 @ Y2 ) @ X2 ) @ ( bit_se5824344971392196577ns_xor @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ X2 @ Y2 ) @ Y2 ) ) )
= ( zero_zero @ ( word @ A ) ) ) ) ) ).
% sofl_test
thf(fact_5775_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_5776_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_5777_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_5778_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_5779_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_5780_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_5781_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_5782_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_5783_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_5784_drop__bit__numeral__bit0,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [L2: num,K: num] :
( ( bit_se4197421643247451524op_bit @ A @ ( numeral_numeral @ nat @ L2 ) @ ( numeral_numeral @ A @ ( bit0 @ K ) ) )
= ( bit_se4197421643247451524op_bit @ A @ ( pred_numeral @ L2 ) @ ( numeral_numeral @ A @ K ) ) ) ) ).
% drop_bit_numeral_bit0
thf(fact_5785_drop__bit__numeral__bit1,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [L2: num,K: num] :
( ( bit_se4197421643247451524op_bit @ A @ ( numeral_numeral @ nat @ L2 ) @ ( numeral_numeral @ A @ ( bit1 @ K ) ) )
= ( bit_se4197421643247451524op_bit @ A @ ( pred_numeral @ L2 ) @ ( numeral_numeral @ A @ K ) ) ) ) ).
% drop_bit_numeral_bit1
thf(fact_5786_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_5787_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_5788_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_5789_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_5790_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_5791_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_5792_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_5793_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_5794_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_5795_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_5796_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_5797_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_5798_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_5799_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_5800_VEBT__internal_OminNull_Opelims_I3_J,axiom,
! [X2: vEBT_VEBT] :
( ~ ( vEBT_VEBT_minNull @ X2 )
=> ( ( accp @ vEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ X2 )
=> ( ! [Uv2: $o] :
( ( X2
= ( vEBT_Leaf @ $true @ Uv2 ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $true @ Uv2 ) ) )
=> ( ! [Uu2: $o] :
( ( X2
= ( vEBT_Leaf @ Uu2 @ $true ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ Uu2 @ $true ) ) )
=> ~ ! [Uz2: product_prod @ nat @ nat,Va3: nat,Vb2: list @ vEBT_VEBT,Vc: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ Uz2 ) @ Va3 @ Vb2 @ Vc ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ Uz2 ) @ Va3 @ Vb2 @ Vc ) ) ) ) ) ) ) ).
% VEBT_internal.minNull.pelims(3)
thf(fact_5801_VEBT__internal_OminNull_Opelims_I2_J,axiom,
! [X2: vEBT_VEBT] :
( ( vEBT_VEBT_minNull @ X2 )
=> ( ( accp @ vEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ X2 )
=> ( ( ( X2
= ( vEBT_Leaf @ $false @ $false ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $false @ $false ) ) )
=> ~ ! [Uw2: nat,Ux2: list @ vEBT_VEBT,Uy2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uw2 @ Ux2 @ Uy2 ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uw2 @ Ux2 @ Uy2 ) ) ) ) ) ) ).
% VEBT_internal.minNull.pelims(2)
thf(fact_5802_div__half__word,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y2: word @ A,X2: word @ A] :
( ( Y2
!= ( zero_zero @ ( word @ A ) ) )
=> ( ( product_Pair @ ( word @ A ) @ ( word @ A ) @ ( divide_divide @ ( word @ A ) @ X2 @ Y2 ) @ ( modulo_modulo @ ( word @ A ) @ X2 @ Y2 ) )
= ( if @ ( product_prod @ ( word @ A ) @ ( word @ A ) ) @ ( ord_less_eq @ ( word @ A ) @ Y2 @ ( minus_minus @ ( word @ A ) @ X2 @ ( times_times @ ( word @ A ) @ ( bit_se4730199178511100633sh_bit @ ( word @ A ) @ ( one_one @ nat ) @ ( divide_divide @ ( word @ A ) @ ( bit_se4197421643247451524op_bit @ ( word @ A ) @ ( one_one @ nat ) @ X2 ) @ Y2 ) ) @ Y2 ) ) ) @ ( product_Pair @ ( word @ A ) @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ ( bit_se4730199178511100633sh_bit @ ( word @ A ) @ ( one_one @ nat ) @ ( divide_divide @ ( word @ A ) @ ( bit_se4197421643247451524op_bit @ ( word @ A ) @ ( one_one @ nat ) @ X2 ) @ Y2 ) ) @ ( one_one @ ( word @ A ) ) ) @ ( minus_minus @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ X2 @ ( times_times @ ( word @ A ) @ ( bit_se4730199178511100633sh_bit @ ( word @ A ) @ ( one_one @ nat ) @ ( divide_divide @ ( word @ A ) @ ( bit_se4197421643247451524op_bit @ ( word @ A ) @ ( one_one @ nat ) @ X2 ) @ Y2 ) ) @ Y2 ) ) @ Y2 ) ) @ ( product_Pair @ ( word @ A ) @ ( word @ A ) @ ( bit_se4730199178511100633sh_bit @ ( word @ A ) @ ( one_one @ nat ) @ ( divide_divide @ ( word @ A ) @ ( bit_se4197421643247451524op_bit @ ( word @ A ) @ ( one_one @ nat ) @ X2 ) @ Y2 ) ) @ ( minus_minus @ ( word @ A ) @ X2 @ ( times_times @ ( word @ A ) @ ( bit_se4730199178511100633sh_bit @ ( word @ A ) @ ( one_one @ nat ) @ ( divide_divide @ ( word @ A ) @ ( bit_se4197421643247451524op_bit @ ( word @ A ) @ ( one_one @ nat ) @ X2 ) @ Y2 ) ) @ Y2 ) ) ) ) ) ) ) ).
% div_half_word
thf(fact_5803_pred__subset__eq,axiom,
! [A: $tType,R: set @ A,S: set @ A] :
( ( ord_less_eq @ ( A > $o )
@ ^ [X: A] : ( member @ A @ X @ R )
@ ^ [X: A] : ( member @ A @ X @ S ) )
= ( ord_less_eq @ ( set @ A ) @ R @ S ) ) ).
% pred_subset_eq
thf(fact_5804_case__prod__app,axiom,
! [A: $tType,D6: $tType,C: $tType,B: $tType] :
( ( product_case_prod @ B @ C @ ( D6 > A ) )
= ( ^ [F4: B > C > D6 > A,X: product_prod @ B @ C,Y: D6] :
( product_case_prod @ B @ C @ A
@ ^ [L: B,R5: C] : ( F4 @ L @ R5 @ Y )
@ X ) ) ) ).
% case_prod_app
thf(fact_5805_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_5806_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_5807_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_5808_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_5809_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_5810_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_5811_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_5812_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_5813_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_5814_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_5815_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_5816_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_5817_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_5818_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_5819_drop__bit__numeral__minus__bit0,axiom,
! [L2: num,K: num] :
( ( bit_se4197421643247451524op_bit @ int @ ( numeral_numeral @ nat @ L2 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ K ) ) ) )
= ( bit_se4197421643247451524op_bit @ int @ ( pred_numeral @ L2 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) ) ) ).
% drop_bit_numeral_minus_bit0
thf(fact_5820_push__bit__numeral,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [L2: num,K: num] :
( ( bit_se4730199178511100633sh_bit @ A @ ( numeral_numeral @ nat @ L2 ) @ ( numeral_numeral @ A @ K ) )
= ( bit_se4730199178511100633sh_bit @ A @ ( pred_numeral @ L2 ) @ ( numeral_numeral @ A @ ( bit0 @ K ) ) ) ) ) ).
% push_bit_numeral
thf(fact_5821_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_5822_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_5823_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_5824_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_5825_push__bit__minus__numeral,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [L2: num,K: num] :
( ( bit_se4730199178511100633sh_bit @ A @ ( numeral_numeral @ nat @ L2 ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ K ) ) )
= ( bit_se4730199178511100633sh_bit @ A @ ( pred_numeral @ L2 ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( bit0 @ K ) ) ) ) ) ) ).
% push_bit_minus_numeral
thf(fact_5826_drop__bit__numeral__minus__bit1,axiom,
! [L2: num,K: num] :
( ( bit_se4197421643247451524op_bit @ int @ ( numeral_numeral @ nat @ L2 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ K ) ) ) )
= ( bit_se4197421643247451524op_bit @ int @ ( pred_numeral @ L2 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( inc @ K ) ) ) ) ) ).
% drop_bit_numeral_minus_bit1
thf(fact_5827_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_5828_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_5829_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_5830_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_5831_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_5832_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_5833_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_5834_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_5835_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_5836_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_5837_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_5838_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_5839_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_5840_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_5841_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_5842_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_5843_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_5844_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_5845_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_5846_shiftr__integer__conv__div__pow2,axiom,
( ( bit_se4197421643247451524op_bit @ code_integer )
= ( ^ [N3: nat,X: code_integer] : ( divide_divide @ code_integer @ X @ ( power_power @ code_integer @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) @ N3 ) ) ) ) ).
% shiftr_integer_conv_div_pow2
thf(fact_5847_fun__cong__unused__0,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( zero @ B )
=> ! [F2: ( A > B ) > C,G: C] :
( ( F2
= ( ^ [X: A > B] : G ) )
=> ( ( F2
@ ^ [X: A] : ( zero_zero @ B ) )
= G ) ) ) ).
% fun_cong_unused_0
thf(fact_5848_pred__subset__eq2,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ A @ B )] :
( ( ord_less_eq @ ( A > B > $o )
@ ^ [X: A,Y: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ R )
@ ^ [X: A,Y: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ S ) )
= ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R @ S ) ) ).
% pred_subset_eq2
thf(fact_5849_pred__equals__eq2,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ A @ B )] :
( ( ( ^ [X: A,Y: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ R ) )
= ( ^ [X: A,Y: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ S ) ) )
= ( R = S ) ) ).
% pred_equals_eq2
thf(fact_5850_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_5851_drop__bit__int__def,axiom,
( ( bit_se4197421643247451524op_bit @ int )
= ( ^ [N3: nat,K2: int] : ( divide_divide @ int @ K2 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N3 ) ) ) ) ).
% drop_bit_int_def
thf(fact_5852_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_5853_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 )
=> ~ ! [B4: A] :
( A2
!= ( bit_se4730199178511100633sh_bit @ A @ N @ B4 ) ) ) ) ).
% exp_dvdE
thf(fact_5854_drop__bit__nat__def,axiom,
( ( bit_se4197421643247451524op_bit @ nat )
= ( ^ [N3: nat,M5: nat] : ( divide_divide @ nat @ M5 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) ) ) ) ).
% drop_bit_nat_def
thf(fact_5855_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_5856_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_5857_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_5858_bot__empty__eq2,axiom,
! [B: $tType,A: $tType] :
( ( bot_bot @ ( A > B > $o ) )
= ( ^ [X: A,Y: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ) ) ).
% bot_empty_eq2
thf(fact_5859_case__prod__Pair__iden,axiom,
! [B: $tType,A: $tType,P2: product_prod @ A @ B] :
( ( product_case_prod @ A @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B ) @ P2 )
= P2 ) ).
% case_prod_Pair_iden
thf(fact_5860_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_5861_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_5862_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 ) ) )
& ! [M5: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ A2 @ M5 )
= ( ( ord_less @ nat @ M5 @ ( size_size @ ( word @ A ) @ A2 ) )
& ( bit_se5641148757651400278ts_bit @ ( word @ C ) @ C2 @ ( plus_plus @ nat @ M5 @ ( size_size @ ( word @ B ) @ B2 ) ) ) ) ) ) ) ) ).
% test_bit_split_eq
thf(fact_5863_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_5864_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_5865_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_5866_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_5867_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_5868_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_5869_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_5870_flip__bit__nat__def,axiom,
( ( bit_se8732182000553998342ip_bit @ nat )
= ( ^ [M5: nat,N3: nat] : ( bit_se5824344971392196577ns_xor @ nat @ N3 @ ( bit_se4730199178511100633sh_bit @ nat @ M5 @ ( one_one @ nat ) ) ) ) ) ).
% flip_bit_nat_def
thf(fact_5871_set__bit__nat__def,axiom,
( ( bit_se5668285175392031749et_bit @ nat )
= ( ^ [M5: nat,N3: nat] : ( bit_se1065995026697491101ons_or @ nat @ N3 @ ( bit_se4730199178511100633sh_bit @ nat @ M5 @ ( one_one @ nat ) ) ) ) ) ).
% set_bit_nat_def
thf(fact_5872_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_5873_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_5874_Bit__Operations_Oset__bit__int__def,axiom,
( ( bit_se5668285175392031749et_bit @ int )
= ( ^ [N3: nat,K2: int] : ( bit_se1065995026697491101ons_or @ int @ K2 @ ( bit_se4730199178511100633sh_bit @ int @ N3 @ ( one_one @ int ) ) ) ) ) ).
% Bit_Operations.set_bit_int_def
thf(fact_5875_bit__push__bit__iff__nat,axiom,
! [M: nat,Q2: nat,N: nat] :
( ( bit_se5641148757651400278ts_bit @ nat @ ( bit_se4730199178511100633sh_bit @ nat @ M @ Q2 ) @ N )
= ( ( ord_less_eq @ nat @ M @ N )
& ( bit_se5641148757651400278ts_bit @ nat @ Q2 @ ( minus_minus @ nat @ N @ M ) ) ) ) ).
% bit_push_bit_iff_nat
thf(fact_5876_flip__bit__int__def,axiom,
( ( bit_se8732182000553998342ip_bit @ int )
= ( ^ [N3: nat,K2: int] : ( bit_se5824344971392196577ns_xor @ int @ K2 @ ( bit_se4730199178511100633sh_bit @ int @ N3 @ ( one_one @ int ) ) ) ) ) ).
% flip_bit_int_def
thf(fact_5877_shiftl__integer__conv__mult__pow2,axiom,
( ( bit_se4730199178511100633sh_bit @ code_integer )
= ( ^ [N3: nat,X: code_integer] : ( times_times @ code_integer @ X @ ( power_power @ code_integer @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) @ N3 ) ) ) ) ).
% shiftl_integer_conv_mult_pow2
thf(fact_5878_unset__bit__int__def,axiom,
( ( bit_se2638667681897837118et_bit @ int )
= ( ^ [N3: nat,K2: int] : ( bit_se5824344872417868541ns_and @ int @ K2 @ ( bit_ri4277139882892585799ns_not @ int @ ( bit_se4730199178511100633sh_bit @ int @ N3 @ ( one_one @ int ) ) ) ) ) ) ).
% unset_bit_int_def
thf(fact_5879_push__bit__int__def,axiom,
( ( bit_se4730199178511100633sh_bit @ int )
= ( ^ [N3: nat,K2: int] : ( times_times @ int @ K2 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N3 ) ) ) ) ).
% push_bit_int_def
thf(fact_5880_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_5881_push__bit__nat__def,axiom,
( ( bit_se4730199178511100633sh_bit @ nat )
= ( ^ [N3: nat,M5: nat] : ( times_times @ nat @ M5 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) ) ) ) ).
% push_bit_nat_def
thf(fact_5882_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_5883_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 ) )
=> ( ! [N11: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ B ) @ B2 @ N11 )
= ( ( ord_less @ nat @ N11 @ ( size_size @ ( word @ B ) @ B2 ) )
& ( bit_se5641148757651400278ts_bit @ ( word @ C ) @ C2 @ N11 ) ) )
& ! [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
thf(fact_5884_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 ) )
=> ! [N11: nat,M4: nat] :
( ( ( bit_se5641148757651400278ts_bit @ ( word @ B ) @ B2 @ N11 )
= ( ( ord_less @ nat @ N11 @ ( size_size @ ( word @ B ) @ B2 ) )
& ( bit_se5641148757651400278ts_bit @ ( word @ C ) @ C2 @ N11 ) ) )
& ( ( 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'
thf(fact_5885_bin__rest__integer_Oabs__eq,axiom,
! [X2: int] :
( ( bits_b2549910563261871055nteger @ ( code_integer_of_int @ X2 ) )
= ( code_integer_of_int @ ( divide_divide @ int @ X2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ).
% bin_rest_integer.abs_eq
thf(fact_5886_the__elem__def,axiom,
! [A: $tType] :
( ( the_elem @ A )
= ( ^ [X7: set @ A] :
( the @ A
@ ^ [X: A] :
( X7
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% the_elem_def
thf(fact_5887_concat__bit__Suc,axiom,
! [N: nat,K: int,L2: int] :
( ( bit_concat_bit @ ( suc @ N ) @ K @ L2 )
= ( plus_plus @ int @ ( modulo_modulo @ int @ K @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_concat_bit @ N @ ( divide_divide @ int @ K @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ L2 ) ) ) ) ).
% concat_bit_Suc
thf(fact_5888_concat__bit__0,axiom,
! [K: int,L2: int] :
( ( bit_concat_bit @ ( zero_zero @ nat ) @ K @ L2 )
= L2 ) ).
% concat_bit_0
thf(fact_5889_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_5890_concat__bit__nonnegative__iff,axiom,
! [N: nat,K: int,L2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_concat_bit @ N @ K @ L2 ) )
= ( ord_less_eq @ int @ ( zero_zero @ int ) @ L2 ) ) ).
% concat_bit_nonnegative_iff
thf(fact_5891_concat__bit__negative__iff,axiom,
! [N: nat,K: int,L2: int] :
( ( ord_less @ int @ ( bit_concat_bit @ N @ K @ L2 ) @ ( zero_zero @ int ) )
= ( ord_less @ int @ L2 @ ( zero_zero @ int ) ) ) ).
% concat_bit_negative_iff
thf(fact_5892_concat__bit__of__zero__1,axiom,
! [N: nat,L2: int] :
( ( bit_concat_bit @ N @ ( zero_zero @ int ) @ L2 )
= ( bit_se4730199178511100633sh_bit @ int @ N @ L2 ) ) ).
% concat_bit_of_zero_1
thf(fact_5893_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_5894_concat__bit__eq__iff,axiom,
! [N: nat,K: int,L2: int,R2: int,S3: int] :
( ( ( bit_concat_bit @ N @ K @ L2 )
= ( bit_concat_bit @ N @ R2 @ S3 ) )
= ( ( ( bit_se2584673776208193580ke_bit @ int @ N @ K )
= ( bit_se2584673776208193580ke_bit @ int @ N @ R2 ) )
& ( L2 = S3 ) ) ) ).
% concat_bit_eq_iff
thf(fact_5895_concat__bit__assoc,axiom,
! [N: nat,K: int,M: nat,L2: int,R2: int] :
( ( bit_concat_bit @ N @ K @ ( bit_concat_bit @ M @ L2 @ R2 ) )
= ( bit_concat_bit @ ( plus_plus @ nat @ M @ N ) @ ( bit_concat_bit @ N @ K @ L2 ) @ R2 ) ) ).
% concat_bit_assoc
thf(fact_5896_concat__bit__eq,axiom,
( bit_concat_bit
= ( ^ [N3: nat,K2: int,L: int] : ( plus_plus @ int @ ( bit_se2584673776208193580ke_bit @ int @ N3 @ K2 ) @ ( bit_se4730199178511100633sh_bit @ int @ N3 @ L ) ) ) ) ).
% concat_bit_eq
thf(fact_5897_concat__bit__def,axiom,
( bit_concat_bit
= ( ^ [N3: nat,K2: int,L: int] : ( bit_se1065995026697491101ons_or @ int @ ( bit_se2584673776208193580ke_bit @ int @ N3 @ K2 ) @ ( bit_se4730199178511100633sh_bit @ int @ N3 @ L ) ) ) ) ).
% concat_bit_def
thf(fact_5898_bit__concat__bit__iff,axiom,
! [M: nat,K: int,L2: int,N: nat] :
( ( bit_se5641148757651400278ts_bit @ int @ ( bit_concat_bit @ M @ K @ L2 ) @ N )
= ( ( ( ord_less @ nat @ N @ M )
& ( bit_se5641148757651400278ts_bit @ int @ K @ N ) )
| ( ( ord_less_eq @ nat @ M @ N )
& ( bit_se5641148757651400278ts_bit @ int @ L2 @ ( minus_minus @ nat @ N @ M ) ) ) ) ) ).
% bit_concat_bit_iff
thf(fact_5899_signed__take__bit__eq__concat__bit,axiom,
( ( bit_ri4674362597316999326ke_bit @ int )
= ( ^ [N3: nat,K2: int] : ( bit_concat_bit @ N3 @ K2 @ ( uminus_uminus @ int @ ( zero_neq_one_of_bool @ int @ ( bit_se5641148757651400278ts_bit @ int @ K2 @ N3 ) ) ) ) ) ) ).
% signed_take_bit_eq_concat_bit
thf(fact_5900_bin__rest__integer__code,axiom,
( bits_b2549910563261871055nteger
= ( ^ [I3: code_integer] : ( divide_divide @ code_integer @ I3 @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) ) ) ).
% bin_rest_integer_code
thf(fact_5901_bitXOR__integer__unfold,axiom,
( ( bit_se5824344971392196577ns_xor @ code_integer )
= ( ^ [X: code_integer,Y: code_integer] :
( if @ code_integer
@ ( X
= ( zero_zero @ code_integer ) )
@ Y
@ ( if @ code_integer
@ ( X
= ( uminus_uminus @ code_integer @ ( one_one @ code_integer ) ) )
@ ( bit_ri4277139882892585799ns_not @ code_integer @ Y )
@ ( bits_Bit_integer @ ( bit_se5824344971392196577ns_xor @ code_integer @ ( bits_b2549910563261871055nteger @ X ) @ ( bits_b2549910563261871055nteger @ Y ) )
@ ( ( ~ ( bits_b8758750999018896077nteger @ X ) )
= ( bits_b8758750999018896077nteger @ Y ) ) ) ) ) ) ) ).
% bitXOR_integer_unfold
thf(fact_5902_bitOR__integer__unfold,axiom,
( ( bit_se1065995026697491101ons_or @ code_integer )
= ( ^ [X: code_integer,Y: code_integer] :
( if @ code_integer
@ ( X
= ( zero_zero @ code_integer ) )
@ Y
@ ( if @ code_integer
@ ( X
= ( 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 @ X ) @ ( bits_b2549910563261871055nteger @ Y ) )
@ ( ( bits_b8758750999018896077nteger @ X )
| ( bits_b8758750999018896077nteger @ Y ) ) ) ) ) ) ) ).
% bitOR_integer_unfold
thf(fact_5903_bitAND__integer__unfold,axiom,
( ( bit_se5824344872417868541ns_and @ code_integer )
= ( ^ [X: code_integer,Y: code_integer] :
( if @ code_integer
@ ( X
= ( zero_zero @ code_integer ) )
@ ( zero_zero @ code_integer )
@ ( if @ code_integer
@ ( X
= ( uminus_uminus @ code_integer @ ( one_one @ code_integer ) ) )
@ Y
@ ( bits_Bit_integer @ ( bit_se5824344872417868541ns_and @ code_integer @ ( bits_b2549910563261871055nteger @ X ) @ ( bits_b2549910563261871055nteger @ Y ) )
@ ( ( bits_b8758750999018896077nteger @ X )
& ( bits_b8758750999018896077nteger @ Y ) ) ) ) ) ) ) ).
% bitAND_integer_unfold
thf(fact_5904_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_5905_bin__last__integer__code,axiom,
( bits_b8758750999018896077nteger
= ( ^ [I3: code_integer] :
( ( bit_se5824344872417868541ns_and @ code_integer @ I3 @ ( one_one @ code_integer ) )
!= ( zero_zero @ code_integer ) ) ) ) ).
% bin_last_integer_code
thf(fact_5906_bin__last__integer__nbe,axiom,
( bits_b8758750999018896077nteger
= ( ^ [I3: code_integer] :
( ( modulo_modulo @ code_integer @ I3 @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) )
!= ( zero_zero @ code_integer ) ) ) ) ).
% bin_last_integer_nbe
thf(fact_5907_bin__last__integer_Oabs__eq,axiom,
! [X2: int] :
( ( bits_b8758750999018896077nteger @ ( code_integer_of_int @ X2 ) )
= ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ X2 ) ) ) ).
% bin_last_integer.abs_eq
thf(fact_5908_set__bit__integer__conv__masks,axiom,
( ( generi7602027413899671122et_bit @ code_integer )
= ( ^ [X: code_integer,I3: nat,B3: $o] : ( if @ code_integer @ B3 @ ( bit_se1065995026697491101ons_or @ code_integer @ X @ ( bit_se4730199178511100633sh_bit @ code_integer @ I3 @ ( one_one @ code_integer ) ) ) @ ( bit_se5824344872417868541ns_and @ code_integer @ X @ ( bit_ri4277139882892585799ns_not @ code_integer @ ( bit_se4730199178511100633sh_bit @ code_integer @ I3 @ ( one_one @ code_integer ) ) ) ) ) ) ) ).
% set_bit_integer_conv_masks
thf(fact_5909_Uint32__code,axiom,
( uint322
= ( ^ [I3: code_integer] : ( if @ uint32 @ ( bit_se5641148757651400278ts_bit @ code_integer @ ( bit_se5824344872417868541ns_and @ code_integer @ I3 @ ( 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 @ I3 @ ( 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 @ I3 @ ( 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_5910_word__cat__split__size,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( ( type_len @ C )
& ( type_len @ A )
& ( type_len @ B ) )
=> ! [T2: word @ A,U: word @ B,V: word @ C] :
( ( ord_less_eq @ nat @ ( size_size @ ( word @ A ) @ T2 ) @ ( plus_plus @ nat @ ( size_size @ ( word @ B ) @ U ) @ ( size_size @ ( word @ C ) @ V ) ) )
=> ( ( ( product_Pair @ ( word @ B ) @ ( word @ C ) @ U @ V )
= ( word_split @ A @ B @ C @ T2 ) )
=> ( T2
= ( word_cat @ B @ C @ A @ U @ V ) ) ) ) ) ).
% word_cat_split_size
thf(fact_5911_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_5912_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_5913_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_5914_Uint32__signed__def,axiom,
( uint32_signed
= ( ^ [I3: code_integer] :
( if @ uint32
@ ( ( ord_less @ code_integer @ I3 @ ( 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 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ I3 ) )
@ ( undefined @ ( ( code_integer > uint32 ) > code_integer > uint32 ) @ uint322 @ I3 )
@ ( uint322 @ I3 ) ) ) ) ).
% Uint32_signed_def
thf(fact_5915_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_5916_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_5917_set__bit__beyond,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A,N: nat,B2: $o] :
( ( ord_less_eq @ nat @ ( size_size @ ( word @ A ) @ X2 ) @ N )
=> ( ( generi7602027413899671122et_bit @ ( word @ A ) @ X2 @ N @ B2 )
= X2 ) ) ) ).
% set_bit_beyond
thf(fact_5918_slice__id,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [T2: word @ A] :
( ( slice2 @ A @ A @ ( zero_zero @ nat ) @ T2 )
= T2 ) ) ).
% slice_id
thf(fact_5919_word__set__nth__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A,N: nat,B2: $o] :
( ( ( generi7602027413899671122et_bit @ ( word @ A ) @ W @ N @ B2 )
= W )
= ( ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ W @ N )
= B2 )
| ( ord_less_eq @ nat @ ( size_size @ ( word @ A ) @ W ) @ N ) ) ) ) ).
% word_set_nth_iff
thf(fact_5920_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_5921_slice__cat2,axiom,
! [B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B ) )
=> ! [A2: word @ B,T2: word @ A] :
( ( slice2 @ A @ A @ ( zero_zero @ nat ) @ ( word_cat @ B @ A @ A @ A2 @ T2 ) )
= T2 ) ) ).
% slice_cat2
thf(fact_5922_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_5923_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_5924_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_5925_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_5926_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_5927_int__set__bit__conv__ops,axiom,
( ( generi7602027413899671122et_bit @ int )
= ( ^ [I3: int,N3: nat,B3: $o] : ( if @ int @ B3 @ ( bit_se1065995026697491101ons_or @ int @ I3 @ ( bit_se4730199178511100633sh_bit @ int @ N3 @ ( one_one @ int ) ) ) @ ( bit_se5824344872417868541ns_and @ int @ I3 @ ( bit_ri4277139882892585799ns_not @ int @ ( bit_se4730199178511100633sh_bit @ int @ N3 @ ( one_one @ int ) ) ) ) ) ) ) ).
% int_set_bit_conv_ops
thf(fact_5928_sum__diff1_H__aux,axiom,
! [B: $tType,A: $tType] :
( ( ab_group_add @ B )
=> ! [F3: set @ A,I5: set @ A,F2: A > B,I: A] :
( ( finite_finite2 @ A @ F3 )
=> ( ( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [I3: A] :
( ( member @ A @ I3 @ I5 )
& ( ( F2 @ I3 )
!= ( zero_zero @ B ) ) ) )
@ F3 )
=> ( ( ( member @ A @ I @ I5 )
=> ( ( groups1027152243600224163dd_sum @ A @ B @ F2 @ ( minus_minus @ ( set @ A ) @ I5 @ ( insert @ A @ I @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( minus_minus @ B @ ( groups1027152243600224163dd_sum @ A @ B @ F2 @ I5 ) @ ( F2 @ I ) ) ) )
& ( ~ ( member @ A @ I @ I5 )
=> ( ( groups1027152243600224163dd_sum @ A @ B @ F2 @ ( minus_minus @ ( set @ A ) @ I5 @ ( insert @ A @ I @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( groups1027152243600224163dd_sum @ A @ B @ F2 @ I5 ) ) ) ) ) ) ) ).
% sum_diff1'_aux
thf(fact_5929_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_5930_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_5931_int__sdiv__simps_I2_J,axiom,
! [A2: int] :
( ( signed7115095781618012415divide @ int @ A2 @ ( zero_zero @ int ) )
= ( zero_zero @ int ) ) ).
% int_sdiv_simps(2)
thf(fact_5932_sdiv__int__0__div,axiom,
! [X2: int] :
( ( signed7115095781618012415divide @ int @ ( zero_zero @ int ) @ X2 )
= ( zero_zero @ int ) ) ).
% sdiv_int_0_div
thf(fact_5933_sdiv__int__div__0,axiom,
! [X2: int] :
( ( signed7115095781618012415divide @ int @ X2 @ ( zero_zero @ int ) )
= ( zero_zero @ int ) ) ).
% sdiv_int_div_0
thf(fact_5934_int__sdiv__simps_I1_J,axiom,
! [A2: int] :
( ( signed7115095781618012415divide @ int @ A2 @ ( one_one @ int ) )
= A2 ) ).
% int_sdiv_simps(1)
thf(fact_5935_sum_Oempty_H,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add @ A )
=> ! [P2: B > A] :
( ( groups1027152243600224163dd_sum @ B @ A @ P2 @ ( bot_bot @ ( set @ B ) ) )
= ( zero_zero @ A ) ) ) ).
% sum.empty'
thf(fact_5936_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_5937_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_5938_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_5939_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_5940_sum_Oinsert_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [I5: set @ B,P2: B > A,I: B] :
( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X: B] :
( ( member @ B @ X @ I5 )
& ( ( P2 @ X )
!= ( zero_zero @ A ) ) ) ) )
=> ( ( ( member @ B @ I @ I5 )
=> ( ( groups1027152243600224163dd_sum @ B @ A @ P2 @ ( insert @ B @ I @ I5 ) )
= ( groups1027152243600224163dd_sum @ B @ A @ P2 @ I5 ) ) )
& ( ~ ( member @ B @ I @ I5 )
=> ( ( groups1027152243600224163dd_sum @ B @ A @ P2 @ ( insert @ B @ I @ I5 ) )
= ( plus_plus @ A @ ( P2 @ I ) @ ( groups1027152243600224163dd_sum @ B @ A @ P2 @ I5 ) ) ) ) ) ) ) ).
% sum.insert'
thf(fact_5941_sum_Onon__neutral_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: B > A,I5: set @ B] :
( ( groups1027152243600224163dd_sum @ B @ A @ G
@ ( collect @ B
@ ^ [X: B] :
( ( member @ B @ X @ I5 )
& ( ( G @ X )
!= ( zero_zero @ A ) ) ) ) )
= ( groups1027152243600224163dd_sum @ B @ A @ G @ I5 ) ) ) ).
% sum.non_neutral'
thf(fact_5942_sum_Odistrib__triv_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [I5: set @ B,G: B > A,H2: B > A] :
( ( finite_finite2 @ B @ I5 )
=> ( ( groups1027152243600224163dd_sum @ B @ A
@ ^ [I3: B] : ( plus_plus @ A @ ( G @ I3 ) @ ( H2 @ I3 ) )
@ I5 )
= ( plus_plus @ A @ ( groups1027152243600224163dd_sum @ B @ A @ G @ I5 ) @ ( groups1027152243600224163dd_sum @ B @ A @ H2 @ I5 ) ) ) ) ) ).
% sum.distrib_triv'
thf(fact_5943_sum_Omono__neutral__cong__right_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [S: set @ B,T3: set @ B,G: B > A,H2: B > A] :
( ( ord_less_eq @ ( set @ B ) @ S @ T3 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T3 @ S ) )
=> ( ( G @ X3 )
= ( zero_zero @ A ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ S )
=> ( ( G @ X3 )
= ( H2 @ X3 ) ) )
=> ( ( groups1027152243600224163dd_sum @ B @ A @ G @ T3 )
= ( groups1027152243600224163dd_sum @ B @ A @ H2 @ S ) ) ) ) ) ) ).
% sum.mono_neutral_cong_right'
thf(fact_5944_sum_Omono__neutral__cong__left_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [S: set @ B,T3: set @ B,H2: B > A,G: B > A] :
( ( ord_less_eq @ ( set @ B ) @ S @ T3 )
=> ( ! [I2: B] :
( ( member @ B @ I2 @ ( minus_minus @ ( set @ B ) @ T3 @ S ) )
=> ( ( H2 @ I2 )
= ( zero_zero @ A ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ S )
=> ( ( G @ X3 )
= ( H2 @ X3 ) ) )
=> ( ( groups1027152243600224163dd_sum @ B @ A @ G @ S )
= ( groups1027152243600224163dd_sum @ B @ A @ H2 @ T3 ) ) ) ) ) ) ).
% sum.mono_neutral_cong_left'
thf(fact_5945_sum_Omono__neutral__right_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [S: set @ B,T3: set @ B,G: B > A] :
( ( ord_less_eq @ ( set @ B ) @ S @ T3 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T3 @ S ) )
=> ( ( G @ X3 )
= ( zero_zero @ A ) ) )
=> ( ( groups1027152243600224163dd_sum @ B @ A @ G @ T3 )
= ( groups1027152243600224163dd_sum @ B @ A @ G @ S ) ) ) ) ) ).
% sum.mono_neutral_right'
thf(fact_5946_sum_Omono__neutral__left_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [S: set @ B,T3: set @ B,G: B > A] :
( ( ord_less_eq @ ( set @ B ) @ S @ T3 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T3 @ S ) )
=> ( ( G @ X3 )
= ( zero_zero @ A ) ) )
=> ( ( groups1027152243600224163dd_sum @ B @ A @ G @ S )
= ( groups1027152243600224163dd_sum @ B @ A @ G @ T3 ) ) ) ) ) ).
% sum.mono_neutral_left'
thf(fact_5947_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_5948_sum_Odistrib_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [I5: set @ B,G: B > A,H2: B > A] :
( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X: B] :
( ( member @ B @ X @ I5 )
& ( ( G @ X )
!= ( zero_zero @ A ) ) ) ) )
=> ( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X: B] :
( ( member @ B @ X @ I5 )
& ( ( H2 @ X )
!= ( zero_zero @ A ) ) ) ) )
=> ( ( groups1027152243600224163dd_sum @ B @ A
@ ^ [I3: B] : ( plus_plus @ A @ ( G @ I3 ) @ ( H2 @ I3 ) )
@ I5 )
= ( plus_plus @ A @ ( groups1027152243600224163dd_sum @ B @ A @ G @ I5 ) @ ( groups1027152243600224163dd_sum @ B @ A @ H2 @ I5 ) ) ) ) ) ) ).
% sum.distrib'
thf(fact_5949_sum_OG__def,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ( ( groups1027152243600224163dd_sum @ B @ A )
= ( ^ [P3: B > A,I7: set @ B] :
( if @ A
@ ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X: B] :
( ( member @ B @ X @ I7 )
& ( ( P3 @ X )
!= ( zero_zero @ A ) ) ) ) )
@ ( groups7311177749621191930dd_sum @ B @ A @ P3
@ ( collect @ B
@ ^ [X: B] :
( ( member @ B @ X @ I7 )
& ( ( P3 @ X )
!= ( zero_zero @ A ) ) ) ) )
@ ( zero_zero @ A ) ) ) ) ) ).
% sum.G_def
thf(fact_5950_signed__divide__int__def,axiom,
( ( signed7115095781618012415divide @ int )
= ( ^ [K2: int,L: int] : ( times_times @ int @ ( times_times @ int @ ( sgn_sgn @ int @ K2 ) @ ( sgn_sgn @ int @ L ) ) @ ( divide_divide @ int @ ( abs_abs @ int @ K2 ) @ ( abs_abs @ int @ L ) ) ) ) ) ).
% signed_divide_int_def
thf(fact_5951_sum__diff1_H,axiom,
! [B: $tType,A: $tType] :
( ( ab_group_add @ B )
=> ! [I5: set @ A,F2: A > B,I: A] :
( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [I3: A] :
( ( member @ A @ I3 @ I5 )
& ( ( F2 @ I3 )
!= ( zero_zero @ B ) ) ) ) )
=> ( ( ( member @ A @ I @ I5 )
=> ( ( groups1027152243600224163dd_sum @ A @ B @ F2 @ ( minus_minus @ ( set @ A ) @ I5 @ ( insert @ A @ I @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( minus_minus @ B @ ( groups1027152243600224163dd_sum @ A @ B @ F2 @ I5 ) @ ( F2 @ I ) ) ) )
& ( ~ ( member @ A @ I @ I5 )
=> ( ( groups1027152243600224163dd_sum @ A @ B @ F2 @ ( minus_minus @ ( set @ A ) @ I5 @ ( insert @ A @ I @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( groups1027152243600224163dd_sum @ A @ B @ F2 @ I5 ) ) ) ) ) ) ).
% sum_diff1'
thf(fact_5952_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_5953_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_5954_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
@ ^ [X: B,B3: A] : ( plus_plus @ A @ ( F4 @ X ) @ ( times_times @ A @ A4 @ B3 ) )
@ Xs
@ ( zero_zero @ A ) ) ) ) ) ).
% horner_sum_foldr
thf(fact_5955_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_5956_Cauchy__iff2,axiom,
( ( topolo3814608138187158403Cauchy @ real )
= ( ^ [X7: nat > real] :
! [J3: nat] :
? [M9: nat] :
! [M5: nat] :
( ( ord_less_eq @ nat @ M9 @ M5 )
=> ! [N3: nat] :
( ( ord_less_eq @ nat @ M9 @ N3 )
=> ( ord_less @ real @ ( abs_abs @ real @ ( minus_minus @ real @ ( X7 @ M5 ) @ ( X7 @ N3 ) ) ) @ ( inverse_inverse @ real @ ( semiring_1_of_nat @ real @ ( suc @ J3 ) ) ) ) ) ) ) ) ).
% Cauchy_iff2
thf(fact_5957_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_5958_map__add__upt,axiom,
! [N: nat,M: nat] :
( ( map @ nat @ nat
@ ^ [I3: nat] : ( plus_plus @ nat @ I3 @ N )
@ ( upt @ ( zero_zero @ nat ) @ M ) )
= ( upt @ N @ ( plus_plus @ nat @ M @ N ) ) ) ).
% map_add_upt
thf(fact_5959_map__add__upt_H,axiom,
! [Ofs: nat,A2: nat,B2: nat] :
( ( map @ nat @ nat
@ ^ [I3: nat] : ( plus_plus @ nat @ I3 @ Ofs )
@ ( upt @ A2 @ B2 ) )
= ( upt @ ( plus_plus @ nat @ A2 @ Ofs ) @ ( plus_plus @ nat @ B2 @ Ofs ) ) ) ).
% map_add_upt'
thf(fact_5960_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_5961_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_5962_atLeastAtMost__upt,axiom,
( ( set_or1337092689740270186AtMost @ nat )
= ( ^ [N3: nat,M5: nat] : ( set2 @ nat @ ( upt @ N3 @ ( suc @ M5 ) ) ) ) ) ).
% atLeastAtMost_upt
thf(fact_5963_atLeast__upt,axiom,
( ( set_ord_lessThan @ nat )
= ( ^ [N3: nat] : ( set2 @ nat @ ( upt @ ( zero_zero @ nat ) @ N3 ) ) ) ) ).
% atLeast_upt
thf(fact_5964_map__replicate__trivial,axiom,
! [A: $tType,X2: A,I: nat] :
( ( map @ nat @ A
@ ^ [I3: nat] : X2
@ ( upt @ ( zero_zero @ nat ) @ I ) )
= ( replicate @ A @ I @ X2 ) ) ).
% map_replicate_trivial
thf(fact_5965_atMost__upto,axiom,
( ( set_ord_atMost @ nat )
= ( ^ [N3: nat] : ( set2 @ nat @ ( upt @ ( zero_zero @ nat ) @ ( suc @ N3 ) ) ) ) ) ).
% atMost_upto
thf(fact_5966_map__bit__range__eq__if__take__bit__eq,axiom,
! [N: nat,K: int,L2: int] :
( ( ( bit_se2584673776208193580ke_bit @ int @ N @ K )
= ( bit_se2584673776208193580ke_bit @ int @ N @ L2 ) )
=> ( ( map @ nat @ $o @ ( bit_se5641148757651400278ts_bit @ int @ K ) @ ( upt @ ( zero_zero @ nat ) @ N ) )
= ( map @ nat @ $o @ ( bit_se5641148757651400278ts_bit @ int @ L2 ) @ ( upt @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% map_bit_range_eq_if_take_bit_eq
thf(fact_5967_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_5968_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_5969_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 ) )
=> ( ! [I2: nat] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( nth @ A @ Xs2 @ I2 )
= ( F2 @ ( plus_plus @ nat @ M @ I2 ) ) ) )
=> ( ( map @ nat @ A @ F2 @ ( upt @ M @ N ) )
= Xs2 ) ) ) ).
% map_upt_eqI
thf(fact_5970_CauchyD,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X6: nat > A,E: real] :
( ( topolo3814608138187158403Cauchy @ A @ X6 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ E )
=> ? [M10: nat] :
! [M4: nat] :
( ( ord_less_eq @ nat @ M10 @ M4 )
=> ! [N11: nat] :
( ( ord_less_eq @ nat @ M10 @ N11 )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( X6 @ M4 ) @ ( X6 @ N11 ) ) ) @ E ) ) ) ) ) ) ).
% CauchyD
thf(fact_5971_CauchyI,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X6: nat > A] :
( ! [E2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E2 )
=> ? [M13: nat] :
! [M3: nat] :
( ( ord_less_eq @ nat @ M13 @ M3 )
=> ! [N2: nat] :
( ( ord_less_eq @ nat @ M13 @ N2 )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( X6 @ M3 ) @ ( X6 @ N2 ) ) ) @ E2 ) ) ) )
=> ( topolo3814608138187158403Cauchy @ A @ X6 ) ) ) ).
% CauchyI
thf(fact_5972_Cauchy__iff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ( ( topolo3814608138187158403Cauchy @ A )
= ( ^ [X7: nat > A] :
! [E4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E4 )
=> ? [M9: nat] :
! [M5: nat] :
( ( ord_less_eq @ nat @ M9 @ M5 )
=> ! [N3: nat] :
( ( ord_less_eq @ nat @ M9 @ N3 )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( X7 @ M5 ) @ ( X7 @ N3 ) ) ) @ E4 ) ) ) ) ) ) ) ).
% Cauchy_iff
thf(fact_5973_order__refl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X2: A] : ( ord_less_eq @ A @ X2 @ X2 ) ) ).
% order_refl
thf(fact_5974_dual__order_Orefl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A2: A] : ( ord_less_eq @ A @ A2 @ A2 ) ) ).
% dual_order.refl
thf(fact_5975_order__antisym__conv,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [Y2: A,X2: A] :
( ( ord_less_eq @ A @ Y2 @ X2 )
=> ( ( ord_less_eq @ A @ X2 @ Y2 )
= ( X2 = Y2 ) ) ) ) ).
% order_antisym_conv
thf(fact_5976_linorder__le__cases,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: A,Y2: A] :
( ~ ( ord_less_eq @ A @ X2 @ Y2 )
=> ( ord_less_eq @ A @ Y2 @ X2 ) ) ) ).
% linorder_le_cases
thf(fact_5977_ord__le__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A2: A,B2: A,F2: A > B,C2: B] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ( F2 @ B2 )
= C2 )
=> ( ! [X3: A,Y3: A] :
( ( ord_less_eq @ A @ X3 @ Y3 )
=> ( ord_less_eq @ B @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq @ B @ ( F2 @ A2 ) @ C2 ) ) ) ) ) ).
% ord_le_eq_subst
thf(fact_5978_ord__eq__le__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A2: A,F2: B > A,B2: B,C2: B] :
( ( A2
= ( F2 @ B2 ) )
=> ( ( ord_less_eq @ B @ B2 @ C2 )
=> ( ! [X3: B,Y3: B] :
( ( ord_less_eq @ B @ X3 @ Y3 )
=> ( ord_less_eq @ A @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq @ A @ A2 @ ( F2 @ C2 ) ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_5979_linorder__linear,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: A,Y2: A] :
( ( ord_less_eq @ A @ X2 @ Y2 )
| ( ord_less_eq @ A @ Y2 @ X2 ) ) ) ).
% linorder_linear
thf(fact_5980_order__eq__refl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X2: A,Y2: A] :
( ( X2 = Y2 )
=> ( ord_less_eq @ A @ X2 @ Y2 ) ) ) ).
% order_eq_refl
thf(fact_5981_order__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C )
& ( order @ A ) )
=> ! [A2: A,B2: A,F2: A > C,C2: C] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ C @ ( F2 @ B2 ) @ C2 )
=> ( ! [X3: A,Y3: A] :
( ( ord_less_eq @ A @ X3 @ Y3 )
=> ( ord_less_eq @ C @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq @ C @ ( F2 @ A2 ) @ C2 ) ) ) ) ) ).
% order_subst2
thf(fact_5982_order__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A2: A,F2: B > A,B2: B,C2: B] :
( ( ord_less_eq @ A @ A2 @ ( F2 @ B2 ) )
=> ( ( ord_less_eq @ B @ B2 @ C2 )
=> ( ! [X3: B,Y3: B] :
( ( ord_less_eq @ B @ X3 @ Y3 )
=> ( ord_less_eq @ A @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq @ A @ A2 @ ( F2 @ C2 ) ) ) ) ) ) ).
% order_subst1
thf(fact_5983_Orderings_Oorder__eq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y5: A,Z3: A] : Y5 = Z3 )
= ( ^ [A4: A,B3: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
& ( ord_less_eq @ A @ B3 @ A4 ) ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_5984_antisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ) ).
% antisym
thf(fact_5985_dual__order_Otrans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( ord_less_eq @ A @ C2 @ B2 )
=> ( ord_less_eq @ A @ C2 @ A2 ) ) ) ) ).
% dual_order.trans
thf(fact_5986_dual__order_Oantisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B2: A,A2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( ord_less_eq @ A @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ) ).
% dual_order.antisym
thf(fact_5987_dual__order_Oeq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y5: A,Z3: A] : Y5 = Z3 )
= ( ^ [A4: A,B3: A] :
( ( ord_less_eq @ A @ B3 @ A4 )
& ( ord_less_eq @ A @ A4 @ B3 ) ) ) ) ) ).
% dual_order.eq_iff
thf(fact_5988_linorder__wlog,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > A > $o,A2: A,B2: A] :
( ! [A5: A,B4: A] :
( ( ord_less_eq @ A @ A5 @ B4 )
=> ( P @ A5 @ B4 ) )
=> ( ! [A5: A,B4: A] :
( ( P @ B4 @ A5 )
=> ( P @ A5 @ B4 ) )
=> ( P @ A2 @ B2 ) ) ) ) ).
% linorder_wlog
thf(fact_5989_order__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X2: A,Y2: A,Z: A] :
( ( ord_less_eq @ A @ X2 @ Y2 )
=> ( ( ord_less_eq @ A @ Y2 @ Z )
=> ( ord_less_eq @ A @ X2 @ Z ) ) ) ) ).
% order_trans
thf(fact_5990_order_Otrans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ C2 )
=> ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% order.trans
thf(fact_5991_order__antisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X2: A,Y2: A] :
( ( ord_less_eq @ A @ X2 @ Y2 )
=> ( ( ord_less_eq @ A @ Y2 @ X2 )
=> ( X2 = Y2 ) ) ) ) ).
% order_antisym
thf(fact_5992_ord__le__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( B2 = C2 )
=> ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% ord_le_eq_trans
thf(fact_5993_ord__eq__le__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( A2 = B2 )
=> ( ( ord_less_eq @ A @ B2 @ C2 )
=> ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% ord_eq_le_trans
thf(fact_5994_order__class_Oorder__eq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y5: A,Z3: A] : Y5 = Z3 )
= ( ^ [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
& ( ord_less_eq @ A @ Y @ X ) ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_5995_le__cases3,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: A,Y2: A,Z: A] :
( ( ( ord_less_eq @ A @ X2 @ Y2 )
=> ~ ( ord_less_eq @ A @ Y2 @ Z ) )
=> ( ( ( ord_less_eq @ A @ Y2 @ X2 )
=> ~ ( ord_less_eq @ A @ X2 @ Z ) )
=> ( ( ( ord_less_eq @ A @ X2 @ Z )
=> ~ ( ord_less_eq @ A @ Z @ Y2 ) )
=> ( ( ( ord_less_eq @ A @ Z @ Y2 )
=> ~ ( ord_less_eq @ A @ Y2 @ X2 ) )
=> ( ( ( ord_less_eq @ A @ Y2 @ Z )
=> ~ ( ord_less_eq @ A @ Z @ X2 ) )
=> ~ ( ( ord_less_eq @ A @ Z @ X2 )
=> ~ ( ord_less_eq @ A @ X2 @ Y2 ) ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_5996_nle__le,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A] :
( ( ~ ( ord_less_eq @ A @ A2 @ B2 ) )
= ( ( ord_less_eq @ A @ B2 @ A2 )
& ( B2 != A2 ) ) ) ) ).
% nle_le
thf(fact_5997_order__le__imp__less__or__eq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X2: A,Y2: A] :
( ( ord_less_eq @ A @ X2 @ Y2 )
=> ( ( ord_less @ A @ X2 @ Y2 )
| ( X2 = Y2 ) ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_5998_linorder__le__less__linear,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: A,Y2: A] :
( ( ord_less_eq @ A @ X2 @ Y2 )
| ( ord_less @ A @ Y2 @ X2 ) ) ) ).
% linorder_le_less_linear
thf(fact_5999_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_6000_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_6001_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_6002_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_6003_order__less__le__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X2: A,Y2: A,Z: A] :
( ( ord_less @ A @ X2 @ Y2 )
=> ( ( ord_less_eq @ A @ Y2 @ Z )
=> ( ord_less @ A @ X2 @ Z ) ) ) ) ).
% order_less_le_trans
thf(fact_6004_order__le__less__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X2: A,Y2: A,Z: A] :
( ( ord_less_eq @ A @ X2 @ Y2 )
=> ( ( ord_less @ A @ Y2 @ Z )
=> ( ord_less @ A @ X2 @ Z ) ) ) ) ).
% order_le_less_trans
thf(fact_6005_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_6006_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_6007_order__less__imp__le,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X2: A,Y2: A] :
( ( ord_less @ A @ X2 @ Y2 )
=> ( ord_less_eq @ A @ X2 @ Y2 ) ) ) ).
% order_less_imp_le
thf(fact_6008_linorder__not__less,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: A,Y2: A] :
( ( ~ ( ord_less @ A @ X2 @ Y2 ) )
= ( ord_less_eq @ A @ Y2 @ X2 ) ) ) ).
% linorder_not_less
thf(fact_6009_linorder__not__le,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: A,Y2: A] :
( ( ~ ( ord_less_eq @ A @ X2 @ Y2 ) )
= ( ord_less @ A @ Y2 @ X2 ) ) ) ).
% linorder_not_le
thf(fact_6010_order__less__le,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less @ A )
= ( ^ [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
& ( X != Y ) ) ) ) ) ).
% order_less_le
thf(fact_6011_order__le__less,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
| ( X = Y ) ) ) ) ) ).
% order_le_less
thf(fact_6012_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_6013_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_6014_dual__order_Ostrict__iff__not,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( ( ord_less @ A )
= ( ^ [B3: A,A4: A] :
( ( ord_less_eq @ A @ B3 @ A4 )
& ~ ( ord_less_eq @ A @ A4 @ B3 ) ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_6015_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_6016_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_6017_dual__order_Ostrict__iff__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less @ A )
= ( ^ [B3: A,A4: A] :
( ( ord_less_eq @ A @ B3 @ A4 )
& ( A4 != B3 ) ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_6018_dual__order_Oorder__iff__strict,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B3: A,A4: A] :
( ( ord_less @ A @ B3 @ A4 )
| ( A4 = B3 ) ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_6019_dense__le__bounded,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [X2: A,Y2: A,Z: A] :
( ( ord_less @ A @ X2 @ Y2 )
=> ( ! [W3: A] :
( ( ord_less @ A @ X2 @ W3 )
=> ( ( ord_less @ A @ W3 @ Y2 )
=> ( ord_less_eq @ A @ W3 @ Z ) ) )
=> ( ord_less_eq @ A @ Y2 @ Z ) ) ) ) ).
% dense_le_bounded
thf(fact_6020_dense__ge__bounded,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [Z: A,X2: A,Y2: A] :
( ( ord_less @ A @ Z @ X2 )
=> ( ! [W3: A] :
( ( ord_less @ A @ Z @ W3 )
=> ( ( ord_less @ A @ W3 @ X2 )
=> ( ord_less_eq @ A @ Y2 @ W3 ) ) )
=> ( ord_less_eq @ A @ Y2 @ Z ) ) ) ) ).
% dense_ge_bounded
thf(fact_6021_order_Ostrict__iff__not,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( ( ord_less @ A )
= ( ^ [A4: A,B3: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
& ~ ( ord_less_eq @ A @ B3 @ A4 ) ) ) ) ) ).
% order.strict_iff_not
thf(fact_6022_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_6023_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_6024_order_Ostrict__iff__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less @ A )
= ( ^ [A4: A,B3: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
& ( A4 != B3 ) ) ) ) ) ).
% order.strict_iff_order
thf(fact_6025_order_Oorder__iff__strict,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A4: A,B3: A] :
( ( ord_less @ A @ A4 @ B3 )
| ( A4 = B3 ) ) ) ) ) ).
% order.order_iff_strict
thf(fact_6026_not__le__imp__less,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Y2: A,X2: A] :
( ~ ( ord_less_eq @ A @ Y2 @ X2 )
=> ( ord_less @ A @ X2 @ Y2 ) ) ) ).
% not_le_imp_less
thf(fact_6027_less__le__not__le,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( ( ord_less @ A )
= ( ^ [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
& ~ ( ord_less_eq @ A @ Y @ X ) ) ) ) ) ).
% less_le_not_le
thf(fact_6028_dense__le,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [Y2: A,Z: A] :
( ! [X3: A] :
( ( ord_less @ A @ X3 @ Y2 )
=> ( ord_less_eq @ A @ X3 @ Z ) )
=> ( ord_less_eq @ A @ Y2 @ Z ) ) ) ).
% dense_le
thf(fact_6029_dense__ge,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [Z: A,Y2: A] :
( ! [X3: A] :
( ( ord_less @ A @ Z @ X3 )
=> ( ord_less_eq @ A @ Y2 @ X3 ) )
=> ( ord_less_eq @ A @ Y2 @ Z ) ) ) ).
% dense_ge
thf(fact_6030_antisym__conv2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X2: A,Y2: A] :
( ( ord_less_eq @ A @ X2 @ Y2 )
=> ( ( ~ ( ord_less @ A @ X2 @ Y2 ) )
= ( X2 = Y2 ) ) ) ) ).
% antisym_conv2
thf(fact_6031_antisym__conv1,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X2: A,Y2: A] :
( ~ ( ord_less @ A @ X2 @ Y2 )
=> ( ( ord_less_eq @ A @ X2 @ Y2 )
= ( X2 = Y2 ) ) ) ) ).
% antisym_conv1
thf(fact_6032_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_6033_leI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: A,Y2: A] :
( ~ ( ord_less @ A @ X2 @ Y2 )
=> ( ord_less_eq @ A @ Y2 @ X2 ) ) ) ).
% leI
thf(fact_6034_leD,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [Y2: A,X2: A] :
( ( ord_less_eq @ A @ Y2 @ X2 )
=> ~ ( ord_less @ A @ X2 @ Y2 ) ) ) ).
% leD
thf(fact_6035_bot_Oextremum__uniqueI,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [A2: A] :
( ( ord_less_eq @ A @ A2 @ ( bot_bot @ A ) )
=> ( A2
= ( bot_bot @ A ) ) ) ) ).
% bot.extremum_uniqueI
thf(fact_6036_bot_Oextremum__unique,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [A2: A] :
( ( ord_less_eq @ A @ A2 @ ( bot_bot @ A ) )
= ( A2
= ( bot_bot @ A ) ) ) ) ).
% bot.extremum_unique
thf(fact_6037_bot_Oextremum,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [A2: A] : ( ord_less_eq @ A @ ( bot_bot @ A ) @ A2 ) ) ).
% bot.extremum
thf(fact_6038_max__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( ord_max @ A )
= ( ^ [A4: A,B3: A] : ( if @ A @ ( ord_less_eq @ A @ A4 @ B3 ) @ B3 @ A4 ) ) ) ) ).
% max_def
thf(fact_6039_max__absorb1,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [Y2: A,X2: A] :
( ( ord_less_eq @ A @ Y2 @ X2 )
=> ( ( ord_max @ A @ X2 @ Y2 )
= X2 ) ) ) ).
% max_absorb1
thf(fact_6040_max__absorb2,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X2: A,Y2: A] :
( ( ord_less_eq @ A @ X2 @ Y2 )
=> ( ( ord_max @ A @ X2 @ Y2 )
= Y2 ) ) ) ).
% max_absorb2
thf(fact_6041_le__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ( ( ord_less_eq @ ( A > B ) )
= ( ^ [F4: A > B,G3: A > B] :
! [X: A] : ( ord_less_eq @ B @ ( F4 @ X ) @ ( G3 @ X ) ) ) ) ) ).
% le_fun_def
thf(fact_6042_le__funI,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F2: A > B,G: A > B] :
( ! [X3: A] : ( ord_less_eq @ B @ ( F2 @ X3 ) @ ( G @ X3 ) )
=> ( ord_less_eq @ ( A > B ) @ F2 @ G ) ) ) ).
% le_funI
thf(fact_6043_le__funE,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F2: A > B,G: A > B,X2: A] :
( ( ord_less_eq @ ( A > B ) @ F2 @ G )
=> ( ord_less_eq @ B @ ( F2 @ X2 ) @ ( G @ X2 ) ) ) ) ).
% le_funE
thf(fact_6044_le__funD,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F2: A > B,G: A > B,X2: A] :
( ( ord_less_eq @ ( A > B ) @ F2 @ G )
=> ( ord_less_eq @ B @ ( F2 @ X2 ) @ ( G @ X2 ) ) ) ) ).
% le_funD
thf(fact_6045_Sum__Ico__nat,axiom,
! [M: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [X: nat] : X
@ ( set_or7035219750837199246ssThan @ nat @ M @ N ) )
= ( divide_divide @ nat @ ( minus_minus @ nat @ ( times_times @ nat @ N @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) @ ( times_times @ nat @ M @ ( minus_minus @ nat @ M @ ( one_one @ nat ) ) ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% Sum_Ico_nat
thf(fact_6046_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_6047_atLeastLessThan__iff,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [I: A,L2: A,U: A] :
( ( member @ A @ I @ ( set_or7035219750837199246ssThan @ A @ L2 @ U ) )
= ( ( ord_less_eq @ A @ L2 @ I )
& ( ord_less @ A @ I @ U ) ) ) ) ).
% atLeastLessThan_iff
thf(fact_6048_atLeastLessThan__empty,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B2: A,A2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( set_or7035219750837199246ssThan @ A @ A2 @ B2 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% atLeastLessThan_empty
thf(fact_6049_ivl__subset,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [I: A,J: A,M: A,N: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ I @ J ) @ ( set_or7035219750837199246ssThan @ A @ M @ N ) )
= ( ( ord_less_eq @ A @ J @ I )
| ( ( ord_less_eq @ A @ M @ I )
& ( ord_less_eq @ A @ J @ N ) ) ) ) ) ).
% ivl_subset
thf(fact_6050_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_6051_ivl__diff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [I: A,N: A,M: A] :
( ( ord_less_eq @ A @ I @ N )
=> ( ( minus_minus @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ I @ M ) @ ( set_or7035219750837199246ssThan @ A @ I @ N ) )
= ( set_or7035219750837199246ssThan @ A @ N @ M ) ) ) ) ).
% ivl_diff
thf(fact_6052_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_6053_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_6054_atLeastLessThan__singleton,axiom,
! [M: nat] :
( ( set_or7035219750837199246ssThan @ nat @ M @ ( suc @ M ) )
= ( insert @ nat @ M @ ( bot_bot @ ( set @ nat ) ) ) ) ).
% atLeastLessThan_singleton
thf(fact_6055_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_6056_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_6057_nth__image__indices,axiom,
! [A: $tType,L2: list @ A] :
( ( image @ nat @ A @ ( nth @ A @ L2 ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ L2 ) ) )
= ( set2 @ A @ L2 ) ) ).
% nth_image_indices
thf(fact_6058_all__nat__less__eq,axiom,
! [N: nat,P: nat > $o] :
( ( ! [M5: nat] :
( ( ord_less @ nat @ M5 @ N )
=> ( P @ M5 ) ) )
= ( ! [X: nat] :
( ( member @ nat @ X @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) )
=> ( P @ X ) ) ) ) ).
% all_nat_less_eq
thf(fact_6059_ex__nat__less__eq,axiom,
! [N: nat,P: nat > $o] :
( ( ? [M5: nat] :
( ( ord_less @ nat @ M5 @ N )
& ( P @ M5 ) ) )
= ( ? [X: nat] :
( ( member @ nat @ X @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) )
& ( P @ X ) ) ) ) ).
% ex_nat_less_eq
thf(fact_6060_lessThan__atLeast0,axiom,
( ( set_ord_lessThan @ nat )
= ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) ) ) ).
% lessThan_atLeast0
thf(fact_6061_atLeastLessThanSuc__atLeastAtMost,axiom,
! [L2: nat,U: nat] :
( ( set_or7035219750837199246ssThan @ nat @ L2 @ ( suc @ U ) )
= ( set_or1337092689740270186AtMost @ nat @ L2 @ U ) ) ).
% atLeastLessThanSuc_atLeastAtMost
thf(fact_6062_atLeastLessThan0,axiom,
! [M: nat] :
( ( set_or7035219750837199246ssThan @ nat @ M @ ( zero_zero @ nat ) )
= ( bot_bot @ ( set @ nat ) ) ) ).
% atLeastLessThan0
thf(fact_6063_atLeastLessThan__subset__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A,C2: A,D2: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ A2 @ B2 ) @ ( set_or7035219750837199246ssThan @ A @ C2 @ D2 ) )
=> ( ( ord_less_eq @ A @ B2 @ A2 )
| ( ( ord_less_eq @ A @ C2 @ A2 )
& ( ord_less_eq @ A @ B2 @ D2 ) ) ) ) ) ).
% atLeastLessThan_subset_iff
thf(fact_6064_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
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) ) ) ).
% sum.shift_bounds_Suc_ivl
thf(fact_6065_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
@ ^ [I3: nat] : ( G @ ( plus_plus @ nat @ I3 @ K ) )
@ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) ) ) ).
% sum.shift_bounds_nat_ivl
thf(fact_6066_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
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) ) ) ).
% prod.shift_bounds_Suc_ivl
thf(fact_6067_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
@ ^ [I3: nat] : ( G @ ( plus_plus @ nat @ I3 @ K ) )
@ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) ) ) ).
% prod.shift_bounds_nat_ivl
thf(fact_6068_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_6069_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_6070_sum_OatLeastLessThan__concat,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [M: nat,N: nat,P2: nat,G: nat > A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( ord_less_eq @ nat @ N @ P2 )
=> ( ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ N @ P2 ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ P2 ) ) ) ) ) ) ).
% sum.atLeastLessThan_concat
thf(fact_6071_sum__diff__nat__ivl,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [M: nat,N: nat,P2: nat,F2: nat > A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( ord_less_eq @ nat @ N @ P2 )
=> ( ( minus_minus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_or7035219750837199246ssThan @ nat @ M @ P2 ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_or7035219750837199246ssThan @ nat @ N @ P2 ) ) ) ) ) ) ).
% sum_diff_nat_ivl
thf(fact_6072_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_6073_size__list__estimation_H,axiom,
! [A: $tType,X2: A,Xs2: list @ A,Y2: nat,F2: A > nat] :
( ( member @ A @ X2 @ ( set2 @ A @ Xs2 ) )
=> ( ( ord_less_eq @ nat @ Y2 @ ( F2 @ X2 ) )
=> ( ord_less_eq @ nat @ Y2 @ ( size_list @ A @ F2 @ Xs2 ) ) ) ) ).
% size_list_estimation'
thf(fact_6074_prod_OatLeastLessThan__concat,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M: nat,N: nat,P2: nat,G: nat > A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( ord_less_eq @ nat @ N @ P2 )
=> ( ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ N @ P2 ) ) )
= ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ P2 ) ) ) ) ) ) ).
% prod.atLeastLessThan_concat
thf(fact_6075_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_6076_subset__eq__atLeast0__lessThan__finite,axiom,
! [N5: set @ nat,N: nat] :
( ( ord_less_eq @ ( set @ nat ) @ N5 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) )
=> ( finite_finite2 @ nat @ N5 ) ) ).
% subset_eq_atLeast0_lessThan_finite
thf(fact_6077_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_6078_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_6079_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_6080_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_6081_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_6082_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_6083_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_6084_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_6085_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_6086_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_6087_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_6088_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_6089_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
@ ^ [I3: nat] : ( minus_minus @ A @ ( F2 @ ( suc @ I3 ) ) @ ( F2 @ I3 ) )
@ ( set_or7035219750837199246ssThan @ nat @ M @ N ) )
= ( minus_minus @ A @ ( F2 @ N ) @ ( F2 @ M ) ) ) ) ) ).
% sum_Suc_diff'
thf(fact_6090_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_6091_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
@ ^ [I3: nat] : ( G @ ( minus_minus @ nat @ ( plus_plus @ nat @ M @ N ) @ ( suc @ I3 ) ) )
@ ( set_or7035219750837199246ssThan @ nat @ N @ M ) ) ) ) ).
% sum.atLeastLessThan_rev
thf(fact_6092_sum_Onested__swap,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A2: nat > nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( groups7311177749621191930dd_sum @ nat @ A @ ( A2 @ I3 ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ I3 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [J3: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( A2 @ I3 @ J3 )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ J3 ) @ N ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% sum.nested_swap
thf(fact_6093_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
@ ^ [I3: nat] : ( G @ ( minus_minus @ nat @ ( plus_plus @ nat @ M @ N ) @ ( suc @ I3 ) ) )
@ ( set_or7035219750837199246ssThan @ nat @ N @ M ) ) ) ) ).
% prod.atLeastLessThan_rev
thf(fact_6094_prod_Onested__swap,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A2: nat > nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( groups7121269368397514597t_prod @ nat @ A @ ( A2 @ I3 ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ I3 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [J3: nat] :
( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( A2 @ I3 @ J3 )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ J3 ) @ N ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% prod.nested_swap
thf(fact_6095_sum_Onat__group,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,K: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [M5: nat] : ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( times_times @ nat @ M5 @ K ) @ ( plus_plus @ nat @ ( times_times @ nat @ M5 @ K ) @ K ) ) )
@ ( set_ord_lessThan @ nat @ N ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_ord_lessThan @ nat @ ( times_times @ nat @ N @ K ) ) ) ) ) ).
% sum.nat_group
thf(fact_6096_prod_Onat__group,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,K: nat,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A
@ ^ [M5: nat] : ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( times_times @ nat @ M5 @ K ) @ ( plus_plus @ nat @ ( times_times @ nat @ M5 @ K ) @ K ) ) )
@ ( set_ord_lessThan @ nat @ N ) )
= ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_lessThan @ nat @ ( times_times @ nat @ N @ K ) ) ) ) ) ).
% prod.nat_group
thf(fact_6097_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_6098_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_6099_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_6100_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_6101_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_6102_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_6103_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
@ ^ [I3: nat] : ( G @ ( minus_minus @ nat @ ( plus_plus @ nat @ M @ N ) @ I3 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ N ) @ M ) ) ) ) ).
% sum.atLeastLessThan_rev_at_least_Suc_atMost
thf(fact_6104_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
@ ^ [I3: nat] : ( G @ ( minus_minus @ nat @ ( plus_plus @ nat @ M @ N ) @ I3 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ N ) @ M ) ) ) ) ).
% prod.atLeastLessThan_rev_at_least_Suc_atMost
thf(fact_6105_pochhammer__prod,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ( ( comm_s3205402744901411588hammer @ A )
= ( ^ [A4: A,N3: nat] :
( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( plus_plus @ A @ A4 @ ( semiring_1_of_nat @ A @ I3 ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N3 ) ) ) ) ) ).
% pochhammer_prod
thf(fact_6106_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_6107_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] :
! [M5: nat] :
( ( ord_less_eq @ nat @ N8 @ M5 )
=> ! [N3: nat] : ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F4 @ ( set_or7035219750837199246ssThan @ nat @ M5 @ N3 ) ) ) @ E4 ) ) ) ) ) ) ).
% summable_Cauchy
thf(fact_6108_sums__group,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [F2: nat > A,S3: A,K: nat] :
( ( sums @ A @ F2 @ S3 )
=> ( ( 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 ) ) )
@ S3 ) ) ) ) ).
% sums_group
thf(fact_6109_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_6110_take__bit__sum,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ( ( bit_se2584673776208193580ke_bit @ A )
= ( ^ [N3: nat,A4: A] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K2: nat] : ( bit_se4730199178511100633sh_bit @ A @ K2 @ ( zero_neq_one_of_bool @ A @ ( bit_se5641148757651400278ts_bit @ A @ A4 @ K2 ) ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N3 ) ) ) ) ) ).
% take_bit_sum
thf(fact_6111_image__minus__const__atLeastLessThan__nat,axiom,
! [C2: nat,Y2: nat,X2: nat] :
( ( ( ord_less @ nat @ C2 @ Y2 )
=> ( ( image @ nat @ nat
@ ^ [I3: nat] : ( minus_minus @ nat @ I3 @ C2 )
@ ( set_or7035219750837199246ssThan @ nat @ X2 @ Y2 ) )
= ( set_or7035219750837199246ssThan @ nat @ ( minus_minus @ nat @ X2 @ C2 ) @ ( minus_minus @ nat @ Y2 @ C2 ) ) ) )
& ( ~ ( ord_less @ nat @ C2 @ Y2 )
=> ( ( ( ord_less @ nat @ X2 @ Y2 )
=> ( ( image @ nat @ nat
@ ^ [I3: nat] : ( minus_minus @ nat @ I3 @ C2 )
@ ( set_or7035219750837199246ssThan @ nat @ X2 @ Y2 ) )
= ( insert @ nat @ ( zero_zero @ nat ) @ ( bot_bot @ ( set @ nat ) ) ) ) )
& ( ~ ( ord_less @ nat @ X2 @ Y2 )
=> ( ( image @ nat @ nat
@ ^ [I3: nat] : ( minus_minus @ nat @ I3 @ C2 )
@ ( set_or7035219750837199246ssThan @ nat @ X2 @ Y2 ) )
= ( bot_bot @ ( set @ nat ) ) ) ) ) ) ) ).
% image_minus_const_atLeastLessThan_nat
thf(fact_6112_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_6113_binomial__altdef__of__nat,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K: nat,N: nat] :
( ( ord_less_eq @ nat @ K @ N )
=> ( ( semiring_1_of_nat @ A @ ( binomial @ N @ K ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( divide_divide @ A @ ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ N @ I3 ) ) @ ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ K @ I3 ) ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ K ) ) ) ) ) ).
% binomial_altdef_of_nat
thf(fact_6114_gbinomial__altdef__of__nat,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( ( gbinomial @ A )
= ( ^ [A4: A,K2: nat] :
( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( divide_divide @ A @ ( minus_minus @ A @ A4 @ ( semiring_1_of_nat @ A @ I3 ) ) @ ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ K2 @ I3 ) ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ K2 ) ) ) ) ) ).
% gbinomial_altdef_of_nat
thf(fact_6115_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_6116_gbinomial__mult__fact,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K: nat,A2: A] :
( ( times_times @ A @ ( semiring_char_0_fact @ A @ K ) @ ( gbinomial @ A @ A2 @ K ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( minus_minus @ A @ A2 @ ( semiring_1_of_nat @ A @ I3 ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ K ) ) ) ) ).
% gbinomial_mult_fact
thf(fact_6117_gbinomial__mult__fact_H,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A2: A,K: nat] :
( ( times_times @ A @ ( gbinomial @ A @ A2 @ K ) @ ( semiring_char_0_fact @ A @ K ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( minus_minus @ A @ A2 @ ( semiring_1_of_nat @ A @ I3 ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ K ) ) ) ) ).
% gbinomial_mult_fact'
thf(fact_6118_gbinomial__prod__rev,axiom,
! [A: $tType] :
( ( ( semiring_char_0 @ A )
& ( semidom_divide @ A ) )
=> ( ( gbinomial @ A )
= ( ^ [A4: A,K2: nat] :
( divide_divide @ A
@ ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( minus_minus @ A @ A4 @ ( semiring_1_of_nat @ A @ I3 ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ K2 ) )
@ ( semiring_char_0_fact @ A @ K2 ) ) ) ) ) ).
% gbinomial_prod_rev
thf(fact_6119_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_6120_Chebyshev__sum__upper,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: nat,A2: nat > A,B2: nat > A] :
( ! [I2: nat,J2: nat] :
( ( ord_less_eq @ nat @ I2 @ J2 )
=> ( ( ord_less @ nat @ J2 @ N )
=> ( ord_less_eq @ A @ ( A2 @ I2 ) @ ( A2 @ J2 ) ) ) )
=> ( ! [I2: nat,J2: nat] :
( ( ord_less_eq @ nat @ I2 @ J2 )
=> ( ( ord_less @ nat @ J2 @ N )
=> ( ord_less_eq @ A @ ( B2 @ J2 ) @ ( B2 @ I2 ) ) ) )
=> ( ord_less_eq @ A
@ ( times_times @ A @ ( semiring_1_of_nat @ A @ N )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K2: nat] : ( times_times @ A @ ( A2 @ K2 ) @ ( B2 @ K2 ) )
@ ( 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_6121_Chebyshev__sum__upper__nat,axiom,
! [N: nat,A2: nat > nat,B2: nat > nat] :
( ! [I2: nat,J2: nat] :
( ( ord_less_eq @ nat @ I2 @ J2 )
=> ( ( ord_less @ nat @ J2 @ N )
=> ( ord_less_eq @ nat @ ( A2 @ I2 ) @ ( A2 @ J2 ) ) ) )
=> ( ! [I2: nat,J2: nat] :
( ( ord_less_eq @ nat @ I2 @ J2 )
=> ( ( ord_less @ nat @ J2 @ N )
=> ( ord_less_eq @ nat @ ( B2 @ J2 ) @ ( B2 @ I2 ) ) ) )
=> ( ord_less_eq @ nat
@ ( times_times @ nat @ N
@ ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [I3: nat] : ( times_times @ nat @ ( A2 @ I3 ) @ ( B2 @ I3 ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) )
@ ( times_times @ nat @ ( groups7311177749621191930dd_sum @ nat @ nat @ A2 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) @ ( groups7311177749621191930dd_sum @ nat @ nat @ B2 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ) ) ).
% Chebyshev_sum_upper_nat
thf(fact_6122_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_6123_finite__atLeastZeroLessThan__int,axiom,
! [U: int] : ( finite_finite2 @ int @ ( set_or7035219750837199246ssThan @ int @ ( zero_zero @ int ) @ U ) ) ).
% finite_atLeastZeroLessThan_int
thf(fact_6124_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_6125_atLeastLessThanPlusOne__atLeastAtMost__integer,axiom,
! [L2: code_integer,U: code_integer] :
( ( set_or7035219750837199246ssThan @ code_integer @ L2 @ ( plus_plus @ code_integer @ U @ ( one_one @ code_integer ) ) )
= ( set_or1337092689740270186AtMost @ code_integer @ L2 @ U ) ) ).
% atLeastLessThanPlusOne_atLeastAtMost_integer
thf(fact_6126_atLeastLessThanPlusOne__atLeastAtMost__int,axiom,
! [L2: int,U: int] :
( ( set_or7035219750837199246ssThan @ int @ L2 @ ( plus_plus @ int @ U @ ( one_one @ int ) ) )
= ( set_or1337092689740270186AtMost @ int @ L2 @ U ) ) ).
% atLeastLessThanPlusOne_atLeastAtMost_int
thf(fact_6127_image__add__int__atLeastLessThan,axiom,
! [L2: int,U: int] :
( ( image @ int @ int
@ ^ [X: int] : ( plus_plus @ int @ X @ L2 )
@ ( set_or7035219750837199246ssThan @ int @ ( zero_zero @ int ) @ ( minus_minus @ int @ U @ L2 ) ) )
= ( set_or7035219750837199246ssThan @ int @ L2 @ U ) ) ).
% image_add_int_atLeastLessThan
thf(fact_6128_image__int__atLeastLessThan,axiom,
! [A2: nat,B2: nat] :
( ( image @ nat @ int @ ( semiring_1_of_nat @ int ) @ ( set_or7035219750837199246ssThan @ nat @ A2 @ B2 ) )
= ( set_or7035219750837199246ssThan @ int @ ( semiring_1_of_nat @ int @ A2 ) @ ( semiring_1_of_nat @ int @ B2 ) ) ) ).
% image_int_atLeastLessThan
thf(fact_6129_image__add__integer__atLeastLessThan,axiom,
! [L2: code_integer,U: code_integer] :
( ( image @ code_integer @ code_integer
@ ^ [X: code_integer] : ( plus_plus @ code_integer @ X @ L2 )
@ ( set_or7035219750837199246ssThan @ code_integer @ ( zero_zero @ code_integer ) @ ( minus_minus @ code_integer @ U @ L2 ) ) )
= ( set_or7035219750837199246ssThan @ code_integer @ L2 @ U ) ) ).
% image_add_integer_atLeastLessThan
thf(fact_6130_word__atLeastLessThan__Suc__atLeastAtMost,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [U: word @ A,L2: word @ A] :
( ( U
!= ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) )
=> ( ( set_or7035219750837199246ssThan @ ( word @ A ) @ L2 @ ( plus_plus @ ( word @ A ) @ U @ ( one_one @ ( word @ A ) ) ) )
= ( set_or1337092689740270186AtMost @ ( word @ A ) @ L2 @ U ) ) ) ) ).
% word_atLeastLessThan_Suc_atLeastAtMost
thf(fact_6131_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_6132_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_6133_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_6134_int__of__nat__def,axiom,
( code_T6385005292777649522of_nat
= ( semiring_1_of_nat @ int ) ) ).
% int_of_nat_def
thf(fact_6135_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_6136_image__Collect__subsetI,axiom,
! [A: $tType,B: $tType,P: A > $o,F2: A > B,B5: set @ B] :
( ! [X3: A] :
( ( P @ X3 )
=> ( member @ B @ ( F2 @ X3 ) @ B5 ) )
=> ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F2 @ ( collect @ A @ P ) ) @ B5 ) ) ).
% image_Collect_subsetI
thf(fact_6137_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_6138_Collect__restrict,axiom,
! [A: $tType,X6: set @ A,P: A > $o] :
( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [X: A] :
( ( member @ A @ X @ X6 )
& ( P @ X ) ) )
@ X6 ) ).
% Collect_restrict
thf(fact_6139_prop__restrict,axiom,
! [A: $tType,X2: A,Z8: set @ A,X6: set @ A,P: A > $o] :
( ( member @ A @ X2 @ Z8 )
=> ( ( ord_less_eq @ ( set @ A ) @ Z8
@ ( collect @ A
@ ^ [X: A] :
( ( member @ A @ X @ X6 )
& ( P @ X ) ) ) )
=> ( P @ X2 ) ) ) ).
% prop_restrict
thf(fact_6140_subset__emptyI,axiom,
! [A: $tType,A3: set @ A] :
( ! [X3: A] :
~ ( member @ A @ X3 @ A3 )
=> ( ord_less_eq @ ( set @ A ) @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% subset_emptyI
thf(fact_6141_insert__subsetI,axiom,
! [A: $tType,X2: A,A3: set @ A,X6: set @ A] :
( ( member @ A @ X2 @ A3 )
=> ( ( ord_less_eq @ ( set @ A ) @ X6 @ A3 )
=> ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X2 @ X6 ) @ A3 ) ) ) ).
% insert_subsetI
thf(fact_6142_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_6143_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_6144_product__concat__map,axiom,
! [B: $tType,A: $tType] :
( ( product @ A @ B )
= ( ^ [Xs: list @ A,Ys2: list @ B] :
( concat @ ( product_prod @ A @ B )
@ ( map @ A @ ( list @ ( product_prod @ A @ B ) )
@ ^ [X: A] : ( map @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X ) @ Ys2 )
@ Xs ) ) ) ) ).
% product_concat_map
thf(fact_6145_subset__subseqs,axiom,
! [A: $tType,X6: set @ A,Xs2: list @ A] :
( ( ord_less_eq @ ( set @ A ) @ X6 @ ( set2 @ A @ Xs2 ) )
=> ( member @ ( set @ A ) @ X6 @ ( image @ ( list @ A ) @ ( set @ A ) @ ( set2 @ A ) @ ( set2 @ ( list @ A ) @ ( subseqs @ A @ Xs2 ) ) ) ) ) ).
% subset_subseqs
thf(fact_6146_set__n__lists,axiom,
! [A: $tType,N: nat,Xs2: list @ A] :
( ( set2 @ ( list @ A ) @ ( n_lists @ A @ N @ Xs2 ) )
= ( collect @ ( list @ A )
@ ^ [Ys2: list @ A] :
( ( ( size_size @ ( list @ A ) @ Ys2 )
= N )
& ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Ys2 ) @ ( set2 @ A @ Xs2 ) ) ) ) ) ).
% set_n_lists
thf(fact_6147_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_6148_smod__int__0__mod,axiom,
! [X2: int] :
( ( signed6721504322012087516modulo @ int @ ( zero_zero @ int ) @ X2 )
= ( zero_zero @ int ) ) ).
% smod_int_0_mod
thf(fact_6149_smod__int__mod__0,axiom,
! [X2: int] :
( ( signed6721504322012087516modulo @ int @ X2 @ ( zero_zero @ int ) )
= X2 ) ).
% smod_int_mod_0
thf(fact_6150_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_6151_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_6152_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_6153_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_6154_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_6155_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_6156_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_6157_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_6158_signed__modulo__int__def,axiom,
( ( signed6721504322012087516modulo @ int )
= ( ^ [K2: int,L: int] : ( minus_minus @ int @ K2 @ ( times_times @ int @ ( signed7115095781618012415divide @ int @ K2 @ L ) @ L ) ) ) ) ).
% signed_modulo_int_def
thf(fact_6159_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_6160_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_6161_smod__int__alt__def,axiom,
( ( signed6721504322012087516modulo @ int )
= ( ^ [A4: int,B3: int] : ( times_times @ int @ ( sgn_sgn @ int @ A4 ) @ ( modulo_modulo @ int @ ( abs_abs @ int @ A4 ) @ ( abs_abs @ int @ B3 ) ) ) ) ) ).
% smod_int_alt_def
thf(fact_6162_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_6163_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 ) )
@ ^ [X: A] : ( map @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X ) @ Ys )
@ Xs2 ) ) ) ) ).
% product_code
thf(fact_6164_valid__eq,axiom,
vEBT_VEBT_valid = vEBT_invar_vebt ).
% valid_eq
thf(fact_6165_valid__eq2,axiom,
! [T2: vEBT_VEBT,D2: nat] :
( ( vEBT_VEBT_valid @ T2 @ D2 )
=> ( vEBT_invar_vebt @ T2 @ D2 ) ) ).
% valid_eq2
thf(fact_6166_valid__eq1,axiom,
! [T2: vEBT_VEBT,D2: nat] :
( ( vEBT_invar_vebt @ T2 @ D2 )
=> ( vEBT_VEBT_valid @ T2 @ D2 ) ) ).
% valid_eq1
thf(fact_6167_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_6168_Multiset_Ois__empty__def,axiom,
! [A: $tType] :
( ( is_empty @ A )
= ( ^ [A7: multiset @ A] :
( A7
= ( zero_zero @ ( multiset @ A ) ) ) ) ) ).
% Multiset.is_empty_def
thf(fact_6169_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_6170_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_6171_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_6172_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_6173_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_6174_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_6175_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_6176_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_6177_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_6178_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_6179_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_6180_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_6181_unat__lt2p,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A] : ( ord_less @ nat @ ( semiring_1_unsigned @ A @ nat @ X2 ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ).
% unat_lt2p
thf(fact_6182_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_6183_uint__lt2p,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A] : ( ord_less @ int @ ( semiring_1_unsigned @ A @ int @ X2 ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ).
% uint_lt2p
thf(fact_6184_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_6185_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_6186_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_6187_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_6188_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_6189_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_6190_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_6191_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_6192_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_6193_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_6194_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_6195_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_6196_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_6197_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_6198_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_6199_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_6200_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_6201_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_6202_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_6203_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_6204_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_6205_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_6206_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_6207_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_6208_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_6209_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_6210_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_6211_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_6212_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_6213_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_6214_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_6215_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_6216_word__less__sub__le,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,X2: word @ A] :
( ( ord_less @ nat @ N @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ( ord_less_eq @ ( word @ A ) @ X2 @ ( minus_minus @ ( word @ A ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) @ ( one_one @ ( word @ A ) ) ) )
= ( ord_less @ ( word @ A ) @ X2 @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) ) ) ) ) ).
% word_less_sub_le
thf(fact_6217_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_6218_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_6219_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_6220_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_6221_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_6222_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_6223_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_6224_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_6225_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_6226_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_6227_bintr__uint,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,W: word @ A] :
( ( ord_less_eq @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ N )
=> ( ( bit_se2584673776208193580ke_bit @ int @ N @ ( semiring_1_unsigned @ A @ int @ W ) )
= ( semiring_1_unsigned @ A @ int @ W ) ) ) ) ).
% bintr_uint
thf(fact_6228_up__ucast__inj,axiom,
! [B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B ) )
=> ! [X2: word @ A,Y2: word @ A] :
( ( ( semiring_1_unsigned @ A @ ( word @ B ) @ X2 )
= ( semiring_1_unsigned @ A @ ( word @ B ) @ Y2 ) )
=> ( ( ord_less_eq @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( type_len0_len_of @ B @ ( type2 @ B ) ) )
=> ( X2 = Y2 ) ) ) ) ).
% up_ucast_inj
thf(fact_6229_ucast__le__ucast,axiom,
! [B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B ) )
=> ! [X2: word @ A,Y2: word @ A] :
( ( ord_less_eq @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( type_len0_len_of @ B @ ( type2 @ B ) ) )
=> ( ( ord_less_eq @ ( word @ B ) @ ( semiring_1_unsigned @ A @ ( word @ B ) @ X2 ) @ ( semiring_1_unsigned @ A @ ( word @ B ) @ Y2 ) )
= ( ord_less_eq @ ( word @ A ) @ X2 @ Y2 ) ) ) ) ).
% ucast_le_ucast
thf(fact_6230_ucast__ucast__eq,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( type_len @ A )
& ( type_len @ C ) )
=> ! [X2: word @ A,Y2: word @ B] :
( ( ( semiring_1_unsigned @ A @ ( word @ C ) @ X2 )
= ( semiring_1_unsigned @ A @ ( word @ C ) @ ( semiring_1_unsigned @ B @ ( word @ A ) @ Y2 ) ) )
=> ( ( ord_less_eq @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( type_len0_len_of @ B @ ( type2 @ B ) ) )
=> ( ( ord_less_eq @ nat @ ( type_len0_len_of @ B @ ( type2 @ B ) ) @ ( type_len0_len_of @ C @ ( type2 @ C ) ) )
=> ( X2
= ( semiring_1_unsigned @ B @ ( word @ A ) @ Y2 ) ) ) ) ) ) ).
% ucast_ucast_eq
thf(fact_6231_up__ucast__inj__eq,axiom,
! [B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B ) )
=> ! [X2: word @ A,Y2: word @ A] :
( ( ord_less_eq @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( type_len0_len_of @ B @ ( type2 @ B ) ) )
=> ( ( ( semiring_1_unsigned @ A @ ( word @ B ) @ X2 )
= ( semiring_1_unsigned @ A @ ( word @ B ) @ Y2 ) )
= ( X2 = Y2 ) ) ) ) ).
% up_ucast_inj_eq
thf(fact_6232_ucast__up__mono__le,axiom,
! [B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B ) )
=> ! [X2: word @ A,Y2: word @ A] :
( ( ord_less_eq @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( type_len0_len_of @ B @ ( type2 @ B ) ) )
=> ( ( ord_less_eq @ ( word @ A ) @ X2 @ Y2 )
=> ( ord_less_eq @ ( word @ B ) @ ( semiring_1_unsigned @ A @ ( word @ B ) @ X2 ) @ ( semiring_1_unsigned @ A @ ( word @ B ) @ Y2 ) ) ) ) ) ).
% ucast_up_mono_le
thf(fact_6233_eq__ucast__ucast__eq,axiom,
! [A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( type_len @ A ) )
=> ! [X2: word @ A,Y2: word @ B] :
( ( ord_less_eq @ nat @ ( type_len0_len_of @ B @ ( type2 @ B ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ( X2
= ( semiring_1_unsigned @ B @ ( word @ A ) @ Y2 ) )
=> ( ( semiring_1_unsigned @ A @ ( word @ B ) @ X2 )
= Y2 ) ) ) ) ).
% eq_ucast_ucast_eq
thf(fact_6234_unat__ucast__up__simp,axiom,
! [B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B ) )
=> ! [X2: word @ A] :
( ( ord_less_eq @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( type_len0_len_of @ B @ ( type2 @ B ) ) )
=> ( ( semiring_1_unsigned @ B @ nat @ ( semiring_1_unsigned @ A @ ( word @ B ) @ X2 ) )
= ( semiring_1_unsigned @ A @ nat @ X2 ) ) ) ) ).
% unat_ucast_up_simp
thf(fact_6235_wi__bintr,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,W: int] :
( ( ord_less_eq @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ N )
=> ( ( ring_1_of_int @ ( word @ A ) @ ( bit_se2584673776208193580ke_bit @ int @ N @ W ) )
= ( ring_1_of_int @ ( word @ A ) @ W ) ) ) ) ).
% wi_bintr
thf(fact_6236_take__bit__word__beyond__length__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,W: word @ A] :
( ( ord_less_eq @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ N )
=> ( ( bit_se2584673776208193580ke_bit @ ( word @ A ) @ N @ W )
= W ) ) ) ).
% take_bit_word_beyond_length_eq
thf(fact_6237_up__scast__inj__eq,axiom,
! [B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B ) )
=> ! [X2: word @ A,Y2: word @ A] :
( ( ord_less_eq @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( type_len0_len_of @ B @ ( type2 @ B ) ) )
=> ( ( ( ring_1_signed @ A @ ( word @ B ) @ X2 )
= ( ring_1_signed @ A @ ( word @ B ) @ Y2 ) )
= ( X2 = Y2 ) ) ) ) ).
% up_scast_inj_eq
thf(fact_6238_ucast__up__mono,axiom,
! [B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B ) )
=> ! [X2: word @ A,Y2: word @ A] :
( ( ord_less_eq @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( type_len0_len_of @ B @ ( type2 @ B ) ) )
=> ( ( ord_less @ ( word @ A ) @ X2 @ Y2 )
=> ( ord_less @ ( word @ B ) @ ( semiring_1_unsigned @ A @ ( word @ B ) @ X2 ) @ ( semiring_1_unsigned @ A @ ( word @ B ) @ Y2 ) ) ) ) ) ).
% ucast_up_mono
thf(fact_6239_ucast__less__ucast,axiom,
! [B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B ) )
=> ! [X2: word @ A,Y2: word @ A] :
( ( ord_less_eq @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( type_len0_len_of @ B @ ( type2 @ B ) ) )
=> ( ( ord_less @ ( word @ B ) @ ( semiring_1_unsigned @ A @ ( word @ B ) @ X2 ) @ ( semiring_1_unsigned @ A @ ( word @ B ) @ Y2 ) )
= ( ord_less @ ( word @ A ) @ X2 @ Y2 ) ) ) ) ).
% ucast_less_ucast
thf(fact_6240_less__ucast__ucast__less,axiom,
! [A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( type_len @ A ) )
=> ! [X2: word @ A,Y2: word @ B] :
( ( ord_less_eq @ nat @ ( type_len0_len_of @ B @ ( type2 @ B ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ( ord_less @ ( word @ A ) @ X2 @ ( semiring_1_unsigned @ B @ ( word @ A ) @ Y2 ) )
=> ( ord_less @ ( word @ B ) @ ( semiring_1_unsigned @ A @ ( word @ B ) @ X2 ) @ Y2 ) ) ) ) ).
% less_ucast_ucast_less
thf(fact_6241_ucast__ucast__add,axiom,
! [A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( type_len @ A ) )
=> ! [X2: word @ A,Y2: word @ B] :
( ( ord_less_eq @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( type_len0_len_of @ B @ ( type2 @ B ) ) )
=> ( ( semiring_1_unsigned @ B @ ( word @ A ) @ ( plus_plus @ ( word @ B ) @ ( semiring_1_unsigned @ A @ ( word @ B ) @ X2 ) @ Y2 ) )
= ( plus_plus @ ( word @ A ) @ X2 @ ( semiring_1_unsigned @ B @ ( word @ A ) @ Y2 ) ) ) ) ) ).
% ucast_ucast_add
thf(fact_6242_ucast__sub__ucast,axiom,
! [B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B ) )
=> ! [Y2: word @ A,X2: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ Y2 @ X2 )
=> ( ( ord_less_eq @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( type_len0_len_of @ B @ ( type2 @ B ) ) )
=> ( ( semiring_1_unsigned @ A @ ( word @ B ) @ ( minus_minus @ ( word @ A ) @ X2 @ Y2 ) )
= ( minus_minus @ ( word @ B ) @ ( semiring_1_unsigned @ A @ ( word @ B ) @ X2 ) @ ( semiring_1_unsigned @ A @ ( word @ B ) @ Y2 ) ) ) ) ) ) ).
% ucast_sub_ucast
thf(fact_6243_word__of__nat__less__eq__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [M: nat,N: nat] :
( ( ord_less_eq @ ( word @ A ) @ ( semiring_1_of_nat @ ( word @ A ) @ M ) @ ( semiring_1_of_nat @ ( word @ A ) @ N ) )
= ( ord_less_eq @ nat @ ( bit_se2584673776208193580ke_bit @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ M ) @ ( bit_se2584673776208193580ke_bit @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ N ) ) ) ) ).
% word_of_nat_less_eq_iff
thf(fact_6244_ucast__drop__bit__eq,axiom,
! [B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B ) )
=> ! [N: nat,W: word @ A] :
( ( ord_less_eq @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( type_len0_len_of @ B @ ( type2 @ B ) ) )
=> ( ( semiring_1_unsigned @ A @ ( word @ B ) @ ( bit_se4197421643247451524op_bit @ ( word @ A ) @ N @ W ) )
= ( bit_se4197421643247451524op_bit @ ( word @ B ) @ N @ ( semiring_1_unsigned @ A @ ( word @ B ) @ W ) ) ) ) ) ).
% ucast_drop_bit_eq
thf(fact_6245_ucast__mask__drop,axiom,
! [A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( type_len @ A ) )
=> ! [N: nat,X2: word @ B] :
( ( ord_less_eq @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ N )
=> ( ( semiring_1_unsigned @ B @ ( word @ A ) @ ( bit_se5824344872417868541ns_and @ ( word @ B ) @ X2 @ ( bit_se2239418461657761734s_mask @ ( word @ B ) @ N ) ) )
= ( semiring_1_unsigned @ B @ ( word @ A ) @ X2 ) ) ) ) ).
% ucast_mask_drop
thf(fact_6246_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_6247_unsigned__of__nat,axiom,
! [A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( semiring_1 @ A ) )
=> ! [N: nat] :
( ( semiring_1_unsigned @ B @ A @ ( semiring_1_of_nat @ ( word @ B ) @ N ) )
= ( semiring_1_of_nat @ A @ ( bit_se2584673776208193580ke_bit @ nat @ ( type_len0_len_of @ B @ ( type2 @ B ) ) @ N ) ) ) ) ).
% unsigned_of_nat
thf(fact_6248_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_6249_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_6250_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_6251_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_6252_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_6253_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_6254_word__of__nat__eq__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [M: nat,N: nat] :
( ( ( semiring_1_of_nat @ ( word @ A ) @ M )
= ( semiring_1_of_nat @ ( word @ A ) @ N ) )
= ( ( 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_eq_iff
thf(fact_6255_degenerate__word,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A] :
( ( ( type_len0_len_of @ A @ ( type2 @ A ) )
= ( one_one @ nat ) )
=> ( ( X2
= ( zero_zero @ ( word @ A ) ) )
| ( X2
= ( one_one @ ( word @ A ) ) ) ) ) ) ).
% degenerate_word
thf(fact_6256_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_6257_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_6258_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_6259_uint__word__arith__bintrs_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 ) )
= ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( plus_plus @ int @ ( semiring_1_unsigned @ A @ int @ A2 ) @ ( semiring_1_unsigned @ A @ int @ B2 ) ) ) ) ) ).
% uint_word_arith_bintrs(1)
thf(fact_6260_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_6261_uint__word__arith__bintrs_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 ) )
= ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( times_times @ int @ ( semiring_1_unsigned @ A @ int @ A2 ) @ ( semiring_1_unsigned @ A @ int @ B2 ) ) ) ) ) ).
% uint_word_arith_bintrs(3)
thf(fact_6262_num__abs__bintr,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( numeral_numeral @ ( word @ A ) )
= ( ^ [X: num] : ( ring_1_of_int @ ( word @ A ) @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( numeral_numeral @ int @ X ) ) ) ) ) ) ).
% num_abs_bintr
thf(fact_6263_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_6264_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_6265_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_6266_word__cat__bin_H,axiom,
! [B: $tType,C: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ C )
& ( type_len @ B ) )
=> ( ( word_cat @ A @ B @ C )
= ( ^ [V5: word @ A,W2: word @ B] : ( plus_plus @ ( word @ C ) @ ( bit_se4730199178511100633sh_bit @ ( word @ C ) @ ( type_len0_len_of @ B @ ( type2 @ B ) ) @ ( semiring_1_unsigned @ A @ ( word @ C ) @ V5 ) ) @ ( semiring_1_unsigned @ B @ ( word @ C ) @ W2 ) ) ) ) ) ).
% word_cat_bin'
thf(fact_6267_up__scast__inj,axiom,
! [B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B ) )
=> ! [X2: word @ A,Y2: word @ A] :
( ( ( ring_1_signed @ A @ ( word @ B ) @ X2 )
= ( ring_1_signed @ A @ ( word @ B ) @ Y2 ) )
=> ( ( ord_less_eq @ nat @ ( size_size @ ( word @ A ) @ X2 ) @ ( type_len0_len_of @ B @ ( type2 @ B ) ) )
=> ( X2 = Y2 ) ) ) ) ).
% up_scast_inj
thf(fact_6268_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_6269_signed__take__bit__decr__length__iff,axiom,
! [A: $tType,B: $tType] :
( ( ( bit_ri3973907225187159222ations @ B )
& ( type_len @ A ) )
=> ! [K: B,L2: B] :
( ( ( bit_ri4674362597316999326ke_bit @ B @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) @ K )
= ( bit_ri4674362597316999326ke_bit @ B @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) @ L2 ) )
= ( ( bit_se2584673776208193580ke_bit @ B @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ K )
= ( bit_se2584673776208193580ke_bit @ B @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ L2 ) ) ) ) ).
% signed_take_bit_decr_length_iff
thf(fact_6270_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_6271_unsigned__of__int,axiom,
! [A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( semiring_1 @ A ) )
=> ! [K: int] :
( ( semiring_1_unsigned @ B @ A @ ( ring_1_of_int @ ( word @ B ) @ K ) )
= ( semiring_1_of_nat @ A @ ( nat2 @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ B @ ( type2 @ B ) ) @ K ) ) ) ) ) ).
% unsigned_of_int
thf(fact_6272_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_6273_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_6274_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_6275_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_6276_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_6277_mask__exceed,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,X2: word @ A] :
( ( ord_less_eq @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ N )
=> ( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X2 @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) )
= ( zero_zero @ ( word @ A ) ) ) ) ) ).
% mask_exceed
thf(fact_6278_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_6279_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_6280_word__nat__cases,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A] :
~ ! [N2: nat] :
( ( X2
= ( 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_6281_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_6282_word__of__nat__inj,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: nat,Y2: nat] :
( ( ord_less @ nat @ X2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
=> ( ( ord_less @ nat @ Y2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
=> ( ( ( semiring_1_of_nat @ ( word @ A ) @ X2 )
= ( semiring_1_of_nat @ ( word @ A ) @ Y2 ) )
=> ( X2 = Y2 ) ) ) ) ) ).
% word_of_nat_inj
thf(fact_6283_of__nat__inj,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: nat,Y2: nat] :
( ( ord_less @ nat @ X2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
=> ( ( ord_less @ nat @ Y2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
=> ( ( ( semiring_1_of_nat @ ( word @ A ) @ X2 )
= ( semiring_1_of_nat @ ( word @ A ) @ Y2 ) )
= ( X2 = Y2 ) ) ) ) ) ).
% of_nat_inj
thf(fact_6284_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_6285_word__power__increasing,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: nat,Y2: nat] :
( ( ord_less @ ( word @ A ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ X2 ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ Y2 ) )
=> ( ( ord_less @ nat @ X2 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ( ord_less @ nat @ Y2 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ord_less @ nat @ X2 @ Y2 ) ) ) ) ) ).
% word_power_increasing
thf(fact_6286_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_6287_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_6288_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_6289_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_6290_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_6291_of__nat__neq__iff__word,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: nat,Y2: nat] :
( ( ( modulo_modulo @ nat @ X2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
!= ( modulo_modulo @ nat @ Y2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) )
=> ( ( ( semiring_1_of_nat @ ( word @ A ) @ X2 )
!= ( semiring_1_of_nat @ ( word @ A ) @ Y2 ) )
= ( X2 != Y2 ) ) ) ) ).
% of_nat_neq_iff_word
thf(fact_6292_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_6293_num__abs__sbintr,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( numeral_numeral @ ( word @ A ) )
= ( ^ [X: 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 @ X ) ) ) ) ) ) ).
% num_abs_sbintr
thf(fact_6294_ucast__mono,axiom,
! [A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( type_len @ A ) )
=> ! [X2: word @ B,Y2: word @ B] :
( ( ord_less @ ( word @ B ) @ X2 @ Y2 )
=> ( ( ord_less @ ( word @ B ) @ Y2 @ ( power_power @ ( word @ B ) @ ( numeral_numeral @ ( word @ B ) @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
=> ( ord_less @ ( word @ A ) @ ( semiring_1_unsigned @ B @ ( word @ A ) @ X2 ) @ ( semiring_1_unsigned @ B @ ( word @ A ) @ Y2 ) ) ) ) ) ).
% ucast_mono
thf(fact_6295_ucast__ucast__len,axiom,
! [B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B ) )
=> ! [X2: word @ A] :
( ( ord_less @ ( word @ A ) @ X2 @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ B @ ( type2 @ B ) ) ) )
=> ( ( semiring_1_unsigned @ B @ ( word @ A ) @ ( semiring_1_unsigned @ A @ ( word @ B ) @ X2 ) )
= X2 ) ) ) ).
% ucast_ucast_len
thf(fact_6296_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_6297_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_6298_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_6299_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_6300_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_6301_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_6302_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_6303_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_6304_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_6305_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_6306_less__Suc__unat__less__bound,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,X2: word @ A] :
( ( ord_less @ nat @ N @ ( suc @ ( semiring_1_unsigned @ A @ nat @ X2 ) ) )
=> ( ord_less @ nat @ N @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).
% less_Suc_unat_less_bound
thf(fact_6307_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_6308_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_6309_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_6310_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_6311_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_6312_of__nat__mono__maybe_H,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: nat,Y2: nat] :
( ( ord_less @ nat @ X2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
=> ( ( ord_less @ nat @ Y2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
=> ( ( ord_less @ nat @ Y2 @ X2 )
= ( ord_less @ ( word @ A ) @ ( semiring_1_of_nat @ ( word @ A ) @ Y2 ) @ ( semiring_1_of_nat @ ( word @ A ) @ X2 ) ) ) ) ) ) ).
% of_nat_mono_maybe'
thf(fact_6313_of__nat__mono__maybe,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: nat,Y2: nat] :
( ( ord_less @ nat @ X2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
=> ( ( ord_less @ nat @ Y2 @ X2 )
=> ( ord_less @ ( word @ A ) @ ( semiring_1_of_nat @ ( word @ A ) @ Y2 ) @ ( semiring_1_of_nat @ ( word @ A ) @ X2 ) ) ) ) ) ).
% of_nat_mono_maybe
thf(fact_6314_unat__split__asm,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [P: nat > $o,X2: word @ A] :
( ( P @ ( semiring_1_unsigned @ A @ nat @ X2 ) )
= ( ~ ? [N3: nat] :
( ( ( semiring_1_of_nat @ ( word @ A ) @ N3 )
= X2 )
& ( 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_6315_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_6316_unat__split,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [P: nat > $o,X2: word @ A] :
( ( P @ ( semiring_1_unsigned @ A @ nat @ X2 ) )
= ( ! [N3: nat] :
( ( ( ( semiring_1_of_nat @ ( word @ A ) @ N3 )
= X2 )
& ( 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_6317_unat__of__nat__len,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: nat] :
( ( ord_less @ nat @ X2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
=> ( ( semiring_1_unsigned @ A @ nat @ ( semiring_1_of_nat @ ( word @ A ) @ X2 ) )
= X2 ) ) ) ).
% unat_of_nat_len
thf(fact_6318_unat__eq__of__nat,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,X2: word @ A] :
( ( ord_less @ nat @ N @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
=> ( ( ( semiring_1_unsigned @ A @ nat @ X2 )
= N )
= ( X2
= ( semiring_1_of_nat @ ( word @ A ) @ N ) ) ) ) ) ).
% unat_eq_of_nat
thf(fact_6319_x__less__2__0__1_H,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A] :
( ( ( type_len0_len_of @ A @ ( type2 @ A ) )
!= ( one_one @ nat ) )
=> ( ( ord_less @ ( word @ A ) @ X2 @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) )
=> ( ( X2
= ( zero_zero @ ( word @ A ) ) )
| ( X2
= ( one_one @ ( word @ A ) ) ) ) ) ) ) ).
% x_less_2_0_1'
thf(fact_6320_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_6321_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_6322_Word_Oof__nat__0,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [M: nat] :
( ( ( semiring_1_of_nat @ ( word @ A ) @ M )
= ( zero_zero @ ( word @ A ) ) )
= ( ? [Q4: nat] :
( M
= ( times_times @ nat @ Q4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ) ) ).
% Word.of_nat_0
thf(fact_6323_uint__sub__lt2p,axiom,
! [B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B ) )
=> ! [X2: word @ A,Y2: word @ B] : ( ord_less @ int @ ( minus_minus @ int @ ( semiring_1_unsigned @ A @ int @ X2 ) @ ( semiring_1_unsigned @ B @ int @ Y2 ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ).
% uint_sub_lt2p
thf(fact_6324_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_6325_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_6326_ucast__of__nat__small,axiom,
! [A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( type_len @ A ) )
=> ! [X2: nat] :
( ( ord_less @ nat @ X2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
=> ( ( semiring_1_unsigned @ A @ ( word @ B ) @ ( semiring_1_of_nat @ ( word @ A ) @ X2 ) )
= ( semiring_1_of_nat @ ( word @ B ) @ X2 ) ) ) ) ).
% ucast_of_nat_small
thf(fact_6327_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_6328_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_6329_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_6330_unat__ucast,axiom,
! [B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B ) )
=> ! [X2: word @ B] :
( ( semiring_1_unsigned @ A @ nat @ ( semiring_1_unsigned @ B @ ( word @ A ) @ X2 ) )
= ( modulo_modulo @ nat @ ( semiring_1_unsigned @ B @ nat @ X2 ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).
% unat_ucast
thf(fact_6331_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_6332_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_6333_unat__of__nat,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: nat] :
( ( semiring_1_unsigned @ A @ nat @ ( semiring_1_of_nat @ ( word @ A ) @ X2 ) )
= ( modulo_modulo @ nat @ X2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).
% unat_of_nat
thf(fact_6334_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_6335_ucast__less,axiom,
! [A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( type_len @ A ) )
=> ! [X2: word @ B] :
( ( ord_less @ nat @ ( type_len0_len_of @ B @ ( type2 @ B ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ord_less @ ( word @ A ) @ ( semiring_1_unsigned @ B @ ( word @ A ) @ X2 ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ B @ ( type2 @ B ) ) ) ) ) ) ).
% ucast_less
thf(fact_6336_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_6337_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_6338_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_6339_upper__trivial,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A] :
( ( X2
!= ( minus_minus @ ( word @ A ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) @ ( one_one @ ( word @ A ) ) ) )
=> ( ord_less @ ( word @ A ) @ X2 @ ( minus_minus @ ( word @ A ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) @ ( one_one @ ( word @ A ) ) ) ) ) ) ).
% upper_trivial
thf(fact_6340_complement__nth__w2p,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N4: nat,N: nat] :
( ( ord_less @ nat @ N4 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) ) @ N4 )
= ( N4 != N ) ) ) ) ).
% complement_nth_w2p
thf(fact_6341_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_6342_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_6343_word__power__less__diff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,Q2: word @ A,M: nat] :
( ( ord_less @ ( word @ A ) @ ( times_times @ ( word @ A ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) @ Q2 ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ M ) )
=> ( ( ord_less @ ( word @ A ) @ Q2 @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ N ) ) )
=> ( ord_less @ ( word @ A ) @ Q2 @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ M @ N ) ) ) ) ) ) ).
% word_power_less_diff
thf(fact_6344_ucast__mono__le,axiom,
! [B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B ) )
=> ! [X2: word @ A,Y2: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ X2 @ Y2 )
=> ( ( ord_less @ ( word @ A ) @ Y2 @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ B @ ( type2 @ B ) ) ) )
=> ( ord_less_eq @ ( word @ B ) @ ( semiring_1_unsigned @ A @ ( word @ B ) @ X2 ) @ ( semiring_1_unsigned @ A @ ( word @ B ) @ Y2 ) ) ) ) ) ).
% ucast_mono_le
thf(fact_6345_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_6346_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_6347_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_6348_msb0,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A,Y2: word @ A,I: nat] :
( ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( bit_se1065995026697491101ons_or @ ( word @ A ) @ X2 @ Y2 ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ I ) ) )
= ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ X2 @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ I ) ) )
| ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ Y2 @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ I ) ) ) ) )
& ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X2 @ Y2 ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ I ) ) )
= ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ X2 @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ I ) ) )
& ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ Y2 @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ I ) ) ) ) )
& ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( bit_se5824344971392196577ns_xor @ ( word @ A ) @ X2 @ Y2 ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ I ) ) )
= ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ X2 @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ I ) ) )
!= ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ Y2 @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ I ) ) ) ) )
& ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ X2 ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ I ) ) )
= ( ~ ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ X2 @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ I ) ) ) ) ) ) ) ).
% msb0
thf(fact_6349_unat__add__lem,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A,Y2: word @ A] :
( ( ord_less @ nat @ ( plus_plus @ nat @ ( semiring_1_unsigned @ A @ nat @ X2 ) @ ( semiring_1_unsigned @ A @ nat @ Y2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
= ( ( semiring_1_unsigned @ A @ nat @ ( plus_plus @ ( word @ A ) @ X2 @ Y2 ) )
= ( plus_plus @ nat @ ( semiring_1_unsigned @ A @ nat @ X2 ) @ ( semiring_1_unsigned @ A @ nat @ Y2 ) ) ) ) ) ).
% unat_add_lem
thf(fact_6350_unat__add__lem_H,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A,Y2: word @ A] :
( ( ord_less @ nat @ ( plus_plus @ nat @ ( semiring_1_unsigned @ A @ nat @ X2 ) @ ( semiring_1_unsigned @ A @ nat @ Y2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
=> ( ( semiring_1_unsigned @ A @ nat @ ( plus_plus @ ( word @ A ) @ X2 @ Y2 ) )
= ( plus_plus @ nat @ ( semiring_1_unsigned @ A @ nat @ X2 ) @ ( semiring_1_unsigned @ A @ nat @ Y2 ) ) ) ) ) ).
% unat_add_lem'
thf(fact_6351_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_6352_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_6353_of__nat__mono__maybe__le,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: nat,Y2: nat] :
( ( ord_less @ nat @ X2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
=> ( ( ord_less @ nat @ Y2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
=> ( ( ord_less_eq @ nat @ Y2 @ X2 )
= ( ord_less_eq @ ( word @ A ) @ ( semiring_1_of_nat @ ( word @ A ) @ Y2 ) @ ( semiring_1_of_nat @ ( word @ A ) @ X2 ) ) ) ) ) ) ).
% of_nat_mono_maybe_le
thf(fact_6354_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_6355_uint__range_H,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( semiring_1_unsigned @ A @ int @ X2 ) )
& ( ord_less @ int @ ( semiring_1_unsigned @ A @ int @ X2 ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).
% uint_range'
thf(fact_6356_bool__mask_H,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A] :
( ( ord_less @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ( ord_less @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X2 @ ( one_one @ ( word @ A ) ) ) )
= ( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X2 @ ( one_one @ ( word @ A ) ) )
= ( one_one @ ( word @ A ) ) ) ) ) ) ).
% bool_mask'
thf(fact_6357_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_6358_sint__lt,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A] : ( ord_less @ int @ ( ring_1_signed @ A @ int @ X2 ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) ) ) ) ).
% sint_lt
thf(fact_6359_word__int__cases,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A] :
~ ! [N2: int] :
( ( X2
= ( 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_6360_word__of__int__inj,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: int,Y2: int] :
( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X2 )
& ( ord_less @ int @ X2 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) )
=> ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y2 )
& ( ord_less @ int @ Y2 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) )
=> ( ( ( ring_1_of_int @ ( word @ A ) @ X2 )
= ( ring_1_of_int @ ( word @ A ) @ Y2 ) )
= ( X2 = Y2 ) ) ) ) ) ).
% word_of_int_inj
thf(fact_6361_ucast__mono__le_H,axiom,
! [B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B ) )
=> ! [Y2: word @ A,X2: word @ A] :
( ( ord_less @ nat @ ( semiring_1_unsigned @ A @ nat @ Y2 ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ B @ ( type2 @ B ) ) ) )
=> ( ( ord_less @ nat @ ( type_len0_len_of @ B @ ( type2 @ B ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ( ord_less_eq @ ( word @ A ) @ X2 @ Y2 )
=> ( ord_less_eq @ ( word @ B ) @ ( semiring_1_unsigned @ A @ ( word @ B ) @ X2 ) @ ( semiring_1_unsigned @ A @ ( word @ B ) @ Y2 ) ) ) ) ) ) ).
% ucast_mono_le'
thf(fact_6362_of__nat__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,W: word @ A] :
( ( ( semiring_1_of_nat @ ( word @ A ) @ N )
= W )
= ( ? [Q4: nat] :
( N
= ( plus_plus @ nat @ ( semiring_1_unsigned @ A @ nat @ W ) @ ( times_times @ nat @ Q4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ) ) ) ).
% of_nat_eq
thf(fact_6363_unat__mult__lem,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A,Y2: word @ A] :
( ( ord_less @ nat @ ( times_times @ nat @ ( semiring_1_unsigned @ A @ nat @ X2 ) @ ( semiring_1_unsigned @ A @ nat @ Y2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
= ( ( semiring_1_unsigned @ A @ nat @ ( times_times @ ( word @ A ) @ X2 @ Y2 ) )
= ( times_times @ nat @ ( semiring_1_unsigned @ A @ nat @ X2 ) @ ( semiring_1_unsigned @ A @ nat @ Y2 ) ) ) ) ) ).
% unat_mult_lem
thf(fact_6364_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_6365_uint__m2p__not__non__neg,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A] :
~ ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( minus_minus @ int @ ( semiring_1_unsigned @ A @ int @ X2 ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).
% uint_m2p_not_non_neg
thf(fact_6366_uint__m2p__neg,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A] : ( ord_less @ int @ ( minus_minus @ int @ ( semiring_1_unsigned @ A @ int @ X2 ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) @ ( zero_zero @ int ) ) ) ).
% uint_m2p_neg
thf(fact_6367_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_6368_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_6369_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_6370_less__2p__is__upper__bits__unset,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [P2: word @ A,N: nat] :
( ( ord_less @ ( word @ A ) @ P2 @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) )
= ( ( ord_less @ nat @ N @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
& ! [N13: nat] :
( ( ord_less_eq @ nat @ N @ N13 )
=> ( ( ord_less @ nat @ N13 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ~ ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ P2 @ N13 ) ) ) ) ) ) ).
% less_2p_is_upper_bits_unset
thf(fact_6371_upper__bits__unset__is__l2p,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,P2: word @ A] :
( ( ord_less @ nat @ N @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ( ! [N13: nat] :
( ( ord_less_eq @ nat @ N @ N13 )
=> ( ( ord_less @ nat @ N13 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ~ ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ P2 @ N13 ) ) ) )
= ( ord_less @ ( word @ A ) @ P2 @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) ) ) ) ) ).
% upper_bits_unset_is_l2p
thf(fact_6372_nth__bounded,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A,N: nat,M: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ X2 @ N )
=> ( ( ord_less @ ( word @ A ) @ X2 @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ M ) )
=> ( ( ord_less_eq @ nat @ M @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ord_less @ nat @ N @ M ) ) ) ) ) ).
% nth_bounded
thf(fact_6373_uint__add__lem,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A,Y2: word @ A] :
( ( ord_less @ int @ ( plus_plus @ int @ ( semiring_1_unsigned @ A @ int @ X2 ) @ ( semiring_1_unsigned @ A @ int @ Y2 ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
= ( ( semiring_1_unsigned @ A @ int @ ( plus_plus @ ( word @ A ) @ X2 @ Y2 ) )
= ( plus_plus @ int @ ( semiring_1_unsigned @ A @ int @ X2 ) @ ( semiring_1_unsigned @ A @ int @ Y2 ) ) ) ) ) ).
% uint_add_lem
thf(fact_6374_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_6375_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_6376_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_6377_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_6378_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_6379_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_6380_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_6381_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_6382_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_6383_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_6384_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_6385_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_6386_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_6387_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_6388_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_6389_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_6390_uint__mult__lem,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A,Y2: word @ A] :
( ( ord_less @ int @ ( times_times @ int @ ( semiring_1_unsigned @ A @ int @ X2 ) @ ( semiring_1_unsigned @ A @ int @ Y2 ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
= ( ( semiring_1_unsigned @ A @ int @ ( times_times @ ( word @ A ) @ X2 @ Y2 ) )
= ( times_times @ int @ ( semiring_1_unsigned @ A @ int @ X2 ) @ ( semiring_1_unsigned @ A @ int @ Y2 ) ) ) ) ) ).
% uint_mult_lem
thf(fact_6391_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_6392_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_6393_word__power__mod__div,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,M: nat,X2: word @ A] :
( ( ord_less @ nat @ N @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ( ord_less @ nat @ M @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ( divide_divide @ ( word @ A ) @ ( modulo_modulo @ ( word @ A ) @ X2 @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ M ) )
= ( modulo_modulo @ ( word @ A ) @ ( divide_divide @ ( word @ A ) @ X2 @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ M ) ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ N @ M ) ) ) ) ) ) ) ).
% word_power_mod_div
thf(fact_6394_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_6395_msb1,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A,Y2: word @ A] :
( ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( bit_se1065995026697491101ons_or @ ( word @ A ) @ X2 @ Y2 ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) )
= ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ X2 @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) )
| ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ Y2 @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ) )
& ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X2 @ Y2 ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) )
= ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ X2 @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) )
& ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ Y2 @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ) )
& ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( bit_se5824344971392196577ns_xor @ ( word @ A ) @ X2 @ Y2 ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) )
= ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ X2 @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) )
!= ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ Y2 @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ) )
& ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ X2 ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) )
= ( ~ ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ X2 @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ) ).
% msb1
thf(fact_6396_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_6397_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_6398_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_6399_unat__sub__if_H,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y2: word @ A,X2: word @ A] :
( ( ( ord_less_eq @ nat @ ( semiring_1_unsigned @ A @ nat @ Y2 ) @ ( semiring_1_unsigned @ A @ nat @ X2 ) )
=> ( ( semiring_1_unsigned @ A @ nat @ ( minus_minus @ ( word @ A ) @ X2 @ Y2 ) )
= ( minus_minus @ nat @ ( semiring_1_unsigned @ A @ nat @ X2 ) @ ( semiring_1_unsigned @ A @ nat @ Y2 ) ) ) )
& ( ~ ( ord_less_eq @ nat @ ( semiring_1_unsigned @ A @ nat @ Y2 ) @ ( semiring_1_unsigned @ A @ nat @ X2 ) )
=> ( ( semiring_1_unsigned @ A @ nat @ ( minus_minus @ ( word @ A ) @ X2 @ Y2 ) )
= ( minus_minus @ nat @ ( plus_plus @ nat @ ( semiring_1_unsigned @ A @ nat @ X2 ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) @ ( semiring_1_unsigned @ A @ nat @ Y2 ) ) ) ) ) ) ).
% unat_sub_if'
thf(fact_6400_no__olen__add__nat,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A,Y2: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ X2 @ ( plus_plus @ ( word @ A ) @ X2 @ Y2 ) )
= ( ord_less @ nat @ ( plus_plus @ nat @ ( semiring_1_unsigned @ A @ nat @ X2 ) @ ( semiring_1_unsigned @ A @ nat @ Y2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).
% no_olen_add_nat
thf(fact_6401_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_6402_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_6403_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_6404_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_6405_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_6406_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_6407_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_6408_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_6409_sint__ge,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A] : ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) ) ) @ ( ring_1_signed @ A @ int @ X2 ) ) ) ).
% sint_ge
thf(fact_6410_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_6411_uint__split,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [P: int > $o,X2: word @ A] :
( ( P @ ( semiring_1_unsigned @ A @ int @ X2 ) )
= ( ! [I3: int] :
( ( ( ( ring_1_of_int @ ( word @ A ) @ I3 )
= X2 )
& ( ord_less_eq @ int @ ( zero_zero @ int ) @ I3 )
& ( ord_less @ int @ I3 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) )
=> ( P @ I3 ) ) ) ) ) ).
% uint_split
thf(fact_6412_uint__split__asm,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [P: int > $o,X2: word @ A] :
( ( P @ ( semiring_1_unsigned @ A @ int @ X2 ) )
= ( ~ ? [I3: int] :
( ( ( ring_1_of_int @ ( word @ A ) @ I3 )
= X2 )
& ( ord_less_eq @ int @ ( zero_zero @ int ) @ I3 )
& ( ord_less @ int @ I3 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
& ~ ( P @ I3 ) ) ) ) ) ).
% uint_split_asm
thf(fact_6413_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_6414_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_6415_div__lt_H,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [I: word @ A,K: word @ A,X2: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ I @ ( divide_divide @ ( word @ A ) @ K @ X2 ) )
=> ( ord_less @ nat @ ( times_times @ nat @ ( semiring_1_unsigned @ A @ nat @ I ) @ ( semiring_1_unsigned @ A @ nat @ X2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).
% div_lt'
thf(fact_6416_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_6417_div__lt_H_H,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [I: word @ A,K: word @ A,X2: word @ A] :
( ( ord_less @ ( word @ A ) @ I @ ( divide_divide @ ( word @ A ) @ K @ X2 ) )
=> ( ord_less @ nat @ ( times_times @ nat @ ( semiring_1_unsigned @ A @ nat @ I ) @ ( semiring_1_unsigned @ A @ nat @ X2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).
% div_lt''
thf(fact_6418_word__le__exists_H,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A,Y2: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ X2 @ Y2 )
=> ? [Z2: word @ A] :
( ( Y2
= ( plus_plus @ ( word @ A ) @ X2 @ Z2 ) )
& ( ord_less @ int @ ( plus_plus @ int @ ( semiring_1_unsigned @ A @ int @ X2 ) @ ( semiring_1_unsigned @ A @ int @ Z2 ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ) ).
% word_le_exists'
thf(fact_6419_no__olen__add_H,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A,Y2: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ X2 @ ( plus_plus @ ( word @ A ) @ Y2 @ X2 ) )
= ( ord_less @ int @ ( plus_plus @ int @ ( semiring_1_unsigned @ A @ int @ Y2 ) @ ( semiring_1_unsigned @ A @ int @ X2 ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).
% no_olen_add'
thf(fact_6420_no__olen__add,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A,Y2: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ X2 @ ( plus_plus @ ( word @ A ) @ X2 @ Y2 ) )
= ( ord_less @ int @ ( plus_plus @ int @ ( semiring_1_unsigned @ A @ int @ X2 ) @ ( semiring_1_unsigned @ A @ int @ Y2 ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).
% no_olen_add
thf(fact_6421_More__Word_Oof__nat__power,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [P2: nat,X2: nat] :
( ( ord_less @ nat @ P2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ X2 ) )
=> ( ( ord_less @ nat @ X2 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ord_less @ ( word @ A ) @ ( semiring_1_of_nat @ ( word @ A ) @ P2 ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ X2 ) ) ) ) ) ).
% More_Word.of_nat_power
thf(fact_6422_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_6423_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_6424_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_6425_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_6426_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_6427_word__less__two__pow__divI,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A,N: nat,M: nat] :
( ( ord_less @ ( word @ A ) @ X2 @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ N @ M ) ) )
=> ( ( ord_less_eq @ nat @ M @ N )
=> ( ( ord_less @ nat @ N @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ord_less @ ( word @ A ) @ X2 @ ( divide_divide @ ( word @ A ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ M ) ) ) ) ) ) ) ).
% word_less_two_pow_divI
thf(fact_6428_word__power__nonzero,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A,N: nat] :
( ( ord_less @ ( word @ A ) @ X2 @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ N ) ) )
=> ( ( ord_less @ nat @ N @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ( X2
!= ( zero_zero @ ( word @ A ) ) )
=> ( ( times_times @ ( word @ A ) @ X2 @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) )
!= ( zero_zero @ ( word @ A ) ) ) ) ) ) ) ).
% word_power_nonzero
thf(fact_6429_div__lt__uint_H,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [I: word @ A,K: word @ A,X2: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ I @ ( divide_divide @ ( word @ A ) @ K @ X2 ) )
=> ( ord_less @ int @ ( times_times @ int @ ( semiring_1_unsigned @ A @ int @ I ) @ ( semiring_1_unsigned @ A @ int @ X2 ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).
% div_lt_uint'
thf(fact_6430_mult__pow2__inj,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [M: nat,N: nat,X2: word @ A,Y2: word @ A] :
( ( ord_less_eq @ nat @ ( plus_plus @ nat @ M @ N ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ( ord_less_eq @ ( word @ A ) @ X2 @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ M ) )
=> ( ( ord_less_eq @ ( word @ A ) @ Y2 @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ M ) )
=> ( ( ( times_times @ ( word @ A ) @ X2 @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) )
= ( times_times @ ( word @ A ) @ Y2 @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) ) )
=> ( X2 = Y2 ) ) ) ) ) ) ).
% mult_pow2_inj
thf(fact_6431_div__lt__uint_H_H,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [I: word @ A,K: word @ A,X2: word @ A] :
( ( ord_less @ ( word @ A ) @ I @ ( divide_divide @ ( word @ A ) @ K @ X2 ) )
=> ( ord_less @ int @ ( times_times @ int @ ( semiring_1_unsigned @ A @ int @ I ) @ ( semiring_1_unsigned @ A @ int @ X2 ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).
% div_lt_uint''
thf(fact_6432_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_6433_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_6434_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_6435_int__eq__sint,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: nat] :
( ( ord_less @ nat @ X2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) ) )
=> ( ( ring_1_signed @ A @ int @ ( semiring_1_of_nat @ ( word @ A ) @ X2 ) )
= ( semiring_1_of_nat @ int @ X2 ) ) ) ) ).
% int_eq_sint
thf(fact_6436_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_6437_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_6438_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_6439_le__2p__upper__bits,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [P2: word @ A,N: nat] :
( ( ord_less_eq @ ( word @ A ) @ P2 @ ( 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 ) @ P2 @ N6 ) ) ) ) ) ) ).
% le_2p_upper_bits
thf(fact_6440_le2p__bits__unset,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [P2: word @ A,N: nat] :
( ( ord_less_eq @ ( word @ A ) @ P2 @ ( 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 ) @ P2 @ N6 ) ) ) ) ) ).
% le2p_bits_unset
thf(fact_6441_word__add__offset__less,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y2: word @ A,N: nat,X2: word @ A,M: nat,Sz: nat] :
( ( ord_less @ ( word @ A ) @ Y2 @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) )
=> ( ( ord_less @ ( word @ A ) @ X2 @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ M ) )
=> ( ( ord_less @ nat @ Sz @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ( ord_less @ ( word @ A ) @ X2 @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ N ) ) )
=> ( ( Sz
= ( plus_plus @ nat @ M @ N ) )
=> ( ord_less @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ ( times_times @ ( word @ A ) @ X2 @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) ) @ Y2 ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ Sz ) ) ) ) ) ) ) ) ).
% word_add_offset_less
thf(fact_6442_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_6443_div__power__helper,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: nat,Y2: nat] :
( ( ord_less_eq @ nat @ X2 @ Y2 )
=> ( ( ord_less @ nat @ Y2 @ ( 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 ) ) @ Y2 ) @ ( one_one @ ( word @ A ) ) ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ X2 ) )
= ( minus_minus @ ( word @ A ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Y2 @ X2 ) ) @ ( one_one @ ( word @ A ) ) ) ) ) ) ) ).
% div_power_helper
thf(fact_6444_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_6445_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_6446_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_6447_sint__of__nat__ge__zero,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: nat] :
( ( ord_less @ nat @ X2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) ) )
=> ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( ring_1_signed @ A @ int @ ( semiring_1_of_nat @ ( word @ A ) @ X2 ) ) ) ) ) ).
% sint_of_nat_ge_zero
thf(fact_6448_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_6449_sint__of__int__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: int] :
( ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) ) ) @ X2 )
=> ( ( ord_less @ int @ X2 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) ) )
=> ( ( ring_1_signed @ A @ int @ ( ring_1_of_int @ ( word @ A ) @ X2 ) )
= X2 ) ) ) ) ).
% sint_of_int_eq
thf(fact_6450_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_6451_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_6452_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_6453_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_6454_int__word__sint,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: int] :
( ( ring_1_signed @ A @ int @ ( ring_1_of_int @ ( word @ A ) @ X2 ) )
= ( minus_minus @ int @ ( modulo_modulo @ int @ ( plus_plus @ int @ X2 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) ) ) ) ) ).
% int_word_sint
thf(fact_6455_Word_Oword__div__mult,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y2: word @ A,X2: word @ A] :
( ( ord_less @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ Y2 )
=> ( ( ord_less @ nat @ ( times_times @ nat @ ( semiring_1_unsigned @ A @ nat @ X2 ) @ ( semiring_1_unsigned @ A @ nat @ Y2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
=> ( ( divide_divide @ ( word @ A ) @ ( times_times @ ( word @ A ) @ X2 @ Y2 ) @ Y2 )
= X2 ) ) ) ) ).
% Word.word_div_mult
thf(fact_6456_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 )
@ ^ [X: word @ 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 ) ) ) )
= ( image @ nat @ ( word @ A ) @ ( semiring_1_of_nat @ ( word @ A ) )
@ ( collect @ nat
@ ^ [K2: nat] : ( ord_less @ nat @ K2 @ ( divide_divide @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) ) ) ) ) ) ) ) ).
% of_nat_less_two_pow_div_set
thf(fact_6457_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_6458_word__bit__induct,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [P: ( word @ A ) > $o,A2: word @ A] :
( ( P @ ( zero_zero @ ( word @ A ) ) )
=> ( ! [A5: word @ A] :
( ( P @ A5 )
=> ( ( ord_less @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ A5 )
=> ( ( ord_less @ ( word @ A ) @ A5 @ ( 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 ) ) @ A5 ) ) ) ) )
=> ( ! [A5: word @ A] :
( ( P @ A5 )
=> ( ( ord_less @ ( word @ A ) @ A5 @ ( 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 ) ) @ A5 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ) ).
% word_bit_induct
thf(fact_6459_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_6460_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_6461_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_6462_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_6463_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_6464_alignUp__div__helper,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [K: nat,N: nat,X2: 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 ) ) )
=> ( ( X2
= ( times_times @ ( word @ A ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) @ ( semiring_1_of_nat @ ( word @ A ) @ K ) ) )
=> ( ( ord_less_eq @ ( word @ A ) @ A2 @ X2 )
=> ( ( 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_6465_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_6466_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_6467_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_6468_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_6469_len__num0,axiom,
( ( type_len0_len_of @ numeral_num0 )
= ( ^ [Uu4: itself @ numeral_num0] : ( zero_zero @ nat ) ) ) ).
% len_num0
thf(fact_6470_len__num1,axiom,
( ( type_len0_len_of @ numeral_num1 )
= ( ^ [Uu4: itself @ numeral_num1] : ( one_one @ nat ) ) ) ).
% len_num1
thf(fact_6471_len__of__finite__1__def,axiom,
( ( type_len0_len_of @ finite_1 )
= ( ^ [X: itself @ finite_1] : ( one_one @ nat ) ) ) ).
% len_of_finite_1_def
thf(fact_6472_len__of__finite__3__def,axiom,
( ( type_len0_len_of @ finite_3 )
= ( ^ [X: itself @ finite_3] : ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ).
% len_of_finite_3_def
thf(fact_6473_len__of__finite__2__def,axiom,
( ( type_len0_len_of @ finite_2 )
= ( ^ [X: itself @ finite_2] : ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% len_of_finite_2_def
thf(fact_6474_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_6475_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_6476_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_6477_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_6478_divmod__via__sdivmod,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y2: word @ A,X2: word @ A] :
( ( Y2
!= ( 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 ) ) ) @ Y2 )
=> ( ( ( ord_less @ ( word @ A ) @ X2 @ Y2 )
=> ( ( product_Pair @ ( word @ A ) @ ( word @ A ) @ ( divide_divide @ ( word @ A ) @ X2 @ Y2 ) @ ( modulo_modulo @ ( word @ A ) @ X2 @ Y2 ) )
= ( product_Pair @ ( word @ A ) @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ X2 ) ) )
& ( ~ ( ord_less @ ( word @ A ) @ X2 @ Y2 )
=> ( ( product_Pair @ ( word @ A ) @ ( word @ A ) @ ( divide_divide @ ( word @ A ) @ X2 @ Y2 ) @ ( modulo_modulo @ ( word @ A ) @ X2 @ Y2 ) )
= ( product_Pair @ ( word @ A ) @ ( word @ A ) @ ( one_one @ ( word @ A ) ) @ ( minus_minus @ ( word @ A ) @ X2 @ Y2 ) ) ) ) ) )
& ( ~ ( 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 ) ) ) @ Y2 )
=> ( ( product_Pair @ ( word @ A ) @ ( word @ A ) @ ( divide_divide @ ( word @ A ) @ X2 @ Y2 ) @ ( modulo_modulo @ ( word @ A ) @ X2 @ Y2 ) )
= ( if @ ( product_prod @ ( word @ A ) @ ( word @ A ) ) @ ( ord_less_eq @ ( word @ A ) @ Y2 @ ( minus_minus @ ( word @ A ) @ X2 @ ( times_times @ ( word @ A ) @ ( bit_se4730199178511100633sh_bit @ ( word @ A ) @ ( one_one @ nat ) @ ( signed7115095781618012415divide @ ( word @ A ) @ ( bit_se4197421643247451524op_bit @ ( word @ A ) @ ( one_one @ nat ) @ X2 ) @ Y2 ) ) @ Y2 ) ) ) @ ( product_Pair @ ( word @ A ) @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ ( bit_se4730199178511100633sh_bit @ ( word @ A ) @ ( one_one @ nat ) @ ( signed7115095781618012415divide @ ( word @ A ) @ ( bit_se4197421643247451524op_bit @ ( word @ A ) @ ( one_one @ nat ) @ X2 ) @ Y2 ) ) @ ( one_one @ ( word @ A ) ) ) @ ( minus_minus @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ X2 @ ( times_times @ ( word @ A ) @ ( bit_se4730199178511100633sh_bit @ ( word @ A ) @ ( one_one @ nat ) @ ( signed7115095781618012415divide @ ( word @ A ) @ ( bit_se4197421643247451524op_bit @ ( word @ A ) @ ( one_one @ nat ) @ X2 ) @ Y2 ) ) @ Y2 ) ) @ Y2 ) ) @ ( product_Pair @ ( word @ A ) @ ( word @ A ) @ ( bit_se4730199178511100633sh_bit @ ( word @ A ) @ ( one_one @ nat ) @ ( signed7115095781618012415divide @ ( word @ A ) @ ( bit_se4197421643247451524op_bit @ ( word @ A ) @ ( one_one @ nat ) @ X2 ) @ Y2 ) ) @ ( minus_minus @ ( word @ A ) @ X2 @ ( times_times @ ( word @ A ) @ ( bit_se4730199178511100633sh_bit @ ( word @ A ) @ ( one_one @ nat ) @ ( signed7115095781618012415divide @ ( word @ A ) @ ( bit_se4197421643247451524op_bit @ ( word @ A ) @ ( one_one @ nat ) @ X2 ) @ Y2 ) ) @ Y2 ) ) ) ) ) ) ) ) ) ).
% divmod_via_sdivmod
thf(fact_6479_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_6480_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_6481_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_6482_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_6483_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_6484_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_6485_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_6486_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_6487_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_6488_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_6489_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_6490_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_6491_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_6492_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_6493_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_6494_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_6495_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_6496_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_6497_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_6498_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_6499_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_6500_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_6501_word__sdiv__numerals_I3_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y2: num] :
( ( signed7115095781618012415divide @ ( word @ A ) @ ( one_one @ ( word @ A ) ) @ ( numeral_numeral @ ( word @ A ) @ Y2 ) )
= ( 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 ) @ Y2 ) ) ) ) ) ) ).
% word_sdiv_numerals(3)
thf(fact_6502_word__sdiv__numerals_I7_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: num] :
( ( signed7115095781618012415divide @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ X2 ) @ ( one_one @ ( word @ A ) ) )
= ( ring_1_of_int @ ( word @ A ) @ ( signed7115095781618012415divide @ int @ ( ring_1_signed @ A @ int @ ( numeral_numeral @ ( word @ A ) @ X2 ) ) @ ( ring_1_signed @ A @ int @ ( one_one @ ( word @ A ) ) ) ) ) ) ) ).
% word_sdiv_numerals(7)
thf(fact_6503_word__sdiv__numerals_I2_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y2: num] :
( ( signed7115095781618012415divide @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ ( numeral_numeral @ ( word @ A ) @ Y2 ) )
= ( 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 ) @ Y2 ) ) ) ) ) ) ).
% word_sdiv_numerals(2)
thf(fact_6504_word__sdiv__numerals_I4_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: num] :
( ( signed7115095781618012415divide @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ X2 ) @ ( zero_zero @ ( word @ A ) ) )
= ( ring_1_of_int @ ( word @ A ) @ ( signed7115095781618012415divide @ int @ ( ring_1_signed @ A @ int @ ( numeral_numeral @ ( word @ A ) @ X2 ) ) @ ( ring_1_signed @ A @ int @ ( zero_zero @ ( word @ A ) ) ) ) ) ) ) ).
% word_sdiv_numerals(4)
thf(fact_6505_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_6506_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_6507_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_6508_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_6509_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_6510_word__int__split,axiom,
! [A: $tType,B: $tType] :
( ( type_len @ B )
=> ! [P: A > $o,F2: int > A,X2: word @ B] :
( ( P @ ( word_int_case @ A @ B @ F2 @ X2 ) )
= ( ! [I3: int] :
( ( ( X2
= ( ring_1_of_int @ ( word @ B ) @ I3 ) )
& ( ord_less_eq @ int @ ( zero_zero @ int ) @ I3 )
& ( ord_less @ int @ I3 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ B @ ( type2 @ B ) ) ) ) )
=> ( P @ ( F2 @ I3 ) ) ) ) ) ) ).
% word_int_split
thf(fact_6511_word__int__split__asm,axiom,
! [A: $tType,B: $tType] :
( ( type_len @ B )
=> ! [P: A > $o,F2: int > A,X2: word @ B] :
( ( P @ ( word_int_case @ A @ B @ F2 @ X2 ) )
= ( ~ ? [N3: int] :
( ( X2
= ( 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_6512_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_6513_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_6514_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_6515_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_6516_smod__word__0__mod,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A] :
( ( signed6721504322012087516modulo @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ X2 )
= ( zero_zero @ ( word @ A ) ) ) ) ).
% smod_word_0_mod
thf(fact_6517_smod__word__mod__0,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A] :
( ( signed6721504322012087516modulo @ ( word @ A ) @ X2 @ ( zero_zero @ ( word @ A ) ) )
= X2 ) ) ).
% smod_word_mod_0
thf(fact_6518_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_6519_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_6520_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_6521_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_6522_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_6523_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_6524_sdiv__smod__id,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: word @ A,B2: word @ A] :
( ( plus_plus @ ( word @ A ) @ ( times_times @ ( word @ A ) @ ( signed7115095781618012415divide @ ( word @ A ) @ A2 @ B2 ) @ B2 ) @ ( signed6721504322012087516modulo @ ( word @ A ) @ A2 @ B2 ) )
= A2 ) ) ).
% sdiv_smod_id
thf(fact_6525_smod__word__alt__def,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( signed6721504322012087516modulo @ ( word @ A ) )
= ( ^ [A4: word @ A,B3: word @ A] : ( minus_minus @ ( word @ A ) @ A4 @ ( times_times @ ( word @ A ) @ ( signed7115095781618012415divide @ ( word @ A ) @ A4 @ B3 ) @ B3 ) ) ) ) ) ).
% smod_word_alt_def
thf(fact_6526_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_6527_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_6528_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_6529_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_6530_word__smod__numerals_I3_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y2: num] :
( ( signed6721504322012087516modulo @ ( word @ A ) @ ( one_one @ ( word @ A ) ) @ ( numeral_numeral @ ( word @ A ) @ Y2 ) )
= ( 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 ) @ Y2 ) ) ) ) ) ) ).
% word_smod_numerals(3)
thf(fact_6531_word__smod__numerals_I7_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: num] :
( ( signed6721504322012087516modulo @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ X2 ) @ ( one_one @ ( word @ A ) ) )
= ( ring_1_of_int @ ( word @ A ) @ ( signed6721504322012087516modulo @ int @ ( ring_1_signed @ A @ int @ ( numeral_numeral @ ( word @ A ) @ X2 ) ) @ ( ring_1_signed @ A @ int @ ( one_one @ ( word @ A ) ) ) ) ) ) ) ).
% word_smod_numerals(7)
thf(fact_6532_word__smod__numerals_I2_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y2: num] :
( ( signed6721504322012087516modulo @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ ( numeral_numeral @ ( word @ A ) @ Y2 ) )
= ( 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 ) @ Y2 ) ) ) ) ) ) ).
% word_smod_numerals(2)
thf(fact_6533_word__smod__numerals_I4_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: num] :
( ( signed6721504322012087516modulo @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ X2 ) @ ( zero_zero @ ( word @ A ) ) )
= ( ring_1_of_int @ ( word @ A ) @ ( signed6721504322012087516modulo @ int @ ( ring_1_signed @ A @ int @ ( numeral_numeral @ ( word @ A ) @ X2 ) ) @ ( ring_1_signed @ A @ int @ ( zero_zero @ ( word @ A ) ) ) ) ) ) ) ).
% word_smod_numerals(4)
thf(fact_6534_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_6535_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_6536_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_6537_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_6538_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_6539_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_6540_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_6541_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_6542_succ__pred__no_I4_J,axiom,
! [D6: $tType] :
( ( type_len @ D6 )
=> ! [W: num] :
( ( word_pred @ D6 @ ( uminus_uminus @ ( word @ D6 ) @ ( numeral_numeral @ ( word @ D6 ) @ W ) ) )
= ( minus_minus @ ( word @ D6 ) @ ( uminus_uminus @ ( word @ D6 ) @ ( numeral_numeral @ ( word @ D6 ) @ W ) ) @ ( one_one @ ( word @ D6 ) ) ) ) ) ).
% succ_pred_no(4)
thf(fact_6543_word__sp__01,axiom,
! [C: $tType,A: $tType,B: $tType,D6: $tType] :
( ( ( type_len @ D6 )
& ( type_len @ B )
& ( type_len @ A )
& ( type_len @ C ) )
=> ( ( ( word_succ @ A @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) )
= ( zero_zero @ ( word @ A ) ) )
& ( ( word_succ @ B @ ( zero_zero @ ( word @ B ) ) )
= ( one_one @ ( word @ B ) ) )
& ( ( word_pred @ C @ ( zero_zero @ ( word @ C ) ) )
= ( uminus_uminus @ ( word @ C ) @ ( one_one @ ( word @ C ) ) ) )
& ( ( word_pred @ D6 @ ( one_one @ ( word @ D6 ) ) )
= ( zero_zero @ ( word @ D6 ) ) ) ) ) ).
% word_sp_01
thf(fact_6544_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_6545_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_6546_word__m1__ge,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y2: word @ A] : ( ord_less_eq @ ( word @ A ) @ Y2 @ ( word_pred @ A @ ( zero_zero @ ( word @ A ) ) ) ) ) ).
% word_m1_ge
thf(fact_6547_word__not__simps_I2_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y2: word @ A] :
~ ( ord_less @ ( word @ A ) @ ( word_pred @ A @ ( zero_zero @ ( word @ A ) ) ) @ Y2 ) ) ).
% word_not_simps(2)
thf(fact_6548_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_6549_word__mult__succ,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: word @ A,B2: word @ A] :
( ( times_times @ ( word @ A ) @ ( word_succ @ A @ A2 ) @ B2 )
= ( plus_plus @ ( word @ A ) @ B2 @ ( times_times @ ( word @ A ) @ A2 @ B2 ) ) ) ) ).
% word_mult_succ
thf(fact_6550_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_6551_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_6552_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_6553_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_6554_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_6555_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_6556_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_6557_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_6558_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_6559_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_6560_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_6561_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_6562_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_6563_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_6564_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_6565_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_6566_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_6567_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_6568_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_6569_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_6570_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_6571_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_6572_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_6573_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_6574_signed_Olift__Suc__mono__le,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [F2: nat > ( word @ A ),N: nat,N4: nat] :
( ! [N2: nat] : ( word_sle @ A @ ( F2 @ N2 ) @ ( F2 @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq @ nat @ N @ N4 )
=> ( word_sle @ A @ ( F2 @ N ) @ ( F2 @ N4 ) ) ) ) ) ).
% signed.lift_Suc_mono_le
thf(fact_6575_signed_Olift__Suc__antimono__le,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [F2: nat > ( word @ A ),N: nat,N4: nat] :
( ! [N2: nat] : ( word_sle @ A @ ( F2 @ ( suc @ N2 ) ) @ ( F2 @ N2 ) )
=> ( ( ord_less_eq @ nat @ N @ N4 )
=> ( word_sle @ A @ ( F2 @ N4 ) @ ( F2 @ N ) ) ) ) ) ).
% signed.lift_Suc_antimono_le
thf(fact_6576_word__roti__add,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [M: int,N: int,W: word @ A] :
( ( word_roti @ A @ ( plus_plus @ int @ M @ N ) @ W )
= ( word_roti @ A @ M @ ( word_roti @ A @ N @ W ) ) ) ) ).
% word_roti_add
thf(fact_6577_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_6578_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_6579_word__0__sle__from__less,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A] :
( ( ord_less @ ( word @ A ) @ X2 @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) ) )
=> ( word_sle @ A @ ( zero_zero @ ( word @ A ) ) @ X2 ) ) ) ).
% word_0_sle_from_less
thf(fact_6580_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_6581_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_6582_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_6583_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_6584_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_6585_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_6586_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_6587_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_6588_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_6589_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_6590_signed_Olift__Suc__mono__less,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [F2: nat > ( word @ A ),N: nat,N4: nat] :
( ! [N2: nat] : ( word_sless @ A @ ( F2 @ N2 ) @ ( F2 @ ( suc @ N2 ) ) )
=> ( ( ord_less @ nat @ N @ N4 )
=> ( word_sless @ A @ ( F2 @ N ) @ ( F2 @ N4 ) ) ) ) ) ).
% signed.lift_Suc_mono_less
thf(fact_6591_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_6592_word__sless__sint__le,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: word @ A,Y2: word @ A] :
( ( word_sless @ A @ X2 @ Y2 )
=> ( ord_less_eq @ int @ ( ring_1_signed @ A @ int @ X2 ) @ ( minus_minus @ int @ ( ring_1_signed @ A @ int @ Y2 ) @ ( one_one @ int ) ) ) ) ) ).
% word_sless_sint_le
thf(fact_6593_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_6594_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_6595_min__eq__arg_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [M: A,N: A] :
( ( ( ord_min @ A @ M @ N )
= M )
= ( ord_less_eq @ A @ M @ N ) ) ) ).
% min_eq_arg(1)
thf(fact_6596_min__eq__arg_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [M: A,N: A] :
( ( ( ord_min @ A @ M @ N )
= N )
= ( ord_less_eq @ A @ N @ M ) ) ) ).
% min_eq_arg(2)
thf(fact_6597_min__arg__le_I1_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [N: A,M: A] :
( ( ord_less_eq @ A @ N @ ( ord_min @ A @ M @ N ) )
= ( ( ord_min @ A @ M @ N )
= N ) ) ) ).
% min_arg_le(1)
thf(fact_6598_min__arg__le_I2_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [M: A,N: A] :
( ( ord_less_eq @ A @ M @ ( ord_min @ A @ M @ N ) )
= ( ( ord_min @ A @ M @ N )
= M ) ) ) ).
% min_arg_le(2)
thf(fact_6599_min_Oabsorb1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_min @ A @ A2 @ B2 )
= A2 ) ) ) ).
% min.absorb1
thf(fact_6600_min_Oabsorb2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,A2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( ord_min @ A @ A2 @ B2 )
= B2 ) ) ) ).
% min.absorb2
thf(fact_6601_min_Obounded__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ ( ord_min @ A @ B2 @ C2 ) )
= ( ( ord_less_eq @ A @ A2 @ B2 )
& ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% min.bounded_iff
thf(fact_6602_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_6603_min__0L,axiom,
! [N: nat] :
( ( ord_min @ nat @ ( zero_zero @ nat ) @ N )
= ( zero_zero @ nat ) ) ).
% min_0L
thf(fact_6604_min__0R,axiom,
! [N: nat] :
( ( ord_min @ nat @ N @ ( zero_zero @ nat ) )
= ( zero_zero @ nat ) ) ).
% min_0R
thf(fact_6605_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_6606_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_6607_min__0__1_I3_J,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [X2: num] :
( ( ord_min @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ X2 ) )
= ( zero_zero @ A ) ) ) ).
% min_0_1(3)
thf(fact_6608_min__0__1_I4_J,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [X2: num] :
( ( ord_min @ A @ ( numeral_numeral @ A @ X2 ) @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% min_0_1(4)
thf(fact_6609_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_6610_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_6611_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_6612_min__0__1_I5_J,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [X2: num] :
( ( ord_min @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ X2 ) )
= ( one_one @ A ) ) ) ).
% min_0_1(5)
thf(fact_6613_min__0__1_I6_J,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [X2: num] :
( ( ord_min @ A @ ( numeral_numeral @ A @ X2 ) @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% min_0_1(6)
thf(fact_6614_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_6615_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_6616_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_6617_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_6618_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_6619_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_6620_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_6621_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_6622_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_6623_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_6624_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_6625_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_6626_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_6627_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_6628_min__absorb2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [Y2: A,X2: A] :
( ( ord_less_eq @ A @ Y2 @ X2 )
=> ( ( ord_min @ A @ X2 @ Y2 )
= Y2 ) ) ) ).
% min_absorb2
thf(fact_6629_min__absorb1,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X2: A,Y2: A] :
( ( ord_less_eq @ A @ X2 @ Y2 )
=> ( ( ord_min @ A @ X2 @ Y2 )
= X2 ) ) ) ).
% min_absorb1
thf(fact_6630_min__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( ord_min @ A )
= ( ^ [A4: A,B3: A] : ( if @ A @ ( ord_less_eq @ A @ A4 @ B3 ) @ A4 @ B3 ) ) ) ) ).
% min_def
thf(fact_6631_min__def__raw,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( ord_min @ A )
= ( ^ [A4: A,B3: A] : ( if @ A @ ( ord_less_eq @ A @ A4 @ B3 ) @ A4 @ B3 ) ) ) ) ).
% min_def_raw
thf(fact_6632_min__le__iff__disj,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: A,Y2: A,Z: A] :
( ( ord_less_eq @ A @ ( ord_min @ A @ X2 @ Y2 ) @ Z )
= ( ( ord_less_eq @ A @ X2 @ Z )
| ( ord_less_eq @ A @ Y2 @ Z ) ) ) ) ).
% min_le_iff_disj
thf(fact_6633_min_OcoboundedI2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,C2: A,A2: A] :
( ( ord_less_eq @ A @ B2 @ C2 )
=> ( ord_less_eq @ A @ ( ord_min @ A @ A2 @ B2 ) @ C2 ) ) ) ).
% min.coboundedI2
thf(fact_6634_min_OcoboundedI1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ C2 )
=> ( ord_less_eq @ A @ ( ord_min @ A @ A2 @ B2 ) @ C2 ) ) ) ).
% min.coboundedI1
thf(fact_6635_min_Oabsorb__iff2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B3: A,A4: A] :
( ( ord_min @ A @ A4 @ B3 )
= B3 ) ) ) ) ).
% min.absorb_iff2
thf(fact_6636_min_Oabsorb__iff1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A4: A,B3: A] :
( ( ord_min @ A @ A4 @ B3 )
= A4 ) ) ) ) ).
% min.absorb_iff1
thf(fact_6637_min_Ocobounded2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A] : ( ord_less_eq @ A @ ( ord_min @ A @ A2 @ B2 ) @ B2 ) ) ).
% min.cobounded2
thf(fact_6638_min_Ocobounded1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A] : ( ord_less_eq @ A @ ( ord_min @ A @ A2 @ B2 ) @ A2 ) ) ).
% min.cobounded1
thf(fact_6639_min_Oorder__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A4: A,B3: A] :
( A4
= ( ord_min @ A @ A4 @ B3 ) ) ) ) ) ).
% min.order_iff
thf(fact_6640_min_OboundedI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ A2 @ C2 )
=> ( ord_less_eq @ A @ A2 @ ( ord_min @ A @ B2 @ C2 ) ) ) ) ) ).
% min.boundedI
thf(fact_6641_min_OboundedE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ ( ord_min @ A @ B2 @ C2 ) )
=> ~ ( ( ord_less_eq @ A @ A2 @ B2 )
=> ~ ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% min.boundedE
thf(fact_6642_min_OorderI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A] :
( ( A2
= ( ord_min @ A @ A2 @ B2 ) )
=> ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% min.orderI
thf(fact_6643_min_OorderE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( A2
= ( ord_min @ A @ A2 @ B2 ) ) ) ) ).
% min.orderE
thf(fact_6644_min_Omono,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,C2: A,B2: A,D2: A] :
( ( ord_less_eq @ A @ A2 @ C2 )
=> ( ( ord_less_eq @ A @ B2 @ D2 )
=> ( ord_less_eq @ A @ ( ord_min @ A @ A2 @ B2 ) @ ( ord_min @ A @ C2 @ D2 ) ) ) ) ) ).
% min.mono
thf(fact_6645_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_6646_min__diff__distrib__left,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [X2: A,Y2: A,Z: A] :
( ( minus_minus @ A @ ( ord_min @ A @ X2 @ Y2 ) @ Z )
= ( ord_min @ A @ ( minus_minus @ A @ X2 @ Z ) @ ( minus_minus @ A @ Y2 @ Z ) ) ) ) ).
% min_diff_distrib_left
thf(fact_6647_of__nat__min,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [X2: nat,Y2: nat] :
( ( semiring_1_of_nat @ A @ ( ord_min @ nat @ X2 @ Y2 ) )
= ( ord_min @ A @ ( semiring_1_of_nat @ A @ X2 ) @ ( semiring_1_of_nat @ A @ Y2 ) ) ) ) ).
% of_nat_min
thf(fact_6648_minus__min__eq__max,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [X2: A,Y2: A] :
( ( uminus_uminus @ A @ ( ord_min @ A @ X2 @ Y2 ) )
= ( ord_max @ A @ ( uminus_uminus @ A @ X2 ) @ ( uminus_uminus @ A @ Y2 ) ) ) ) ).
% minus_min_eq_max
thf(fact_6649_minus__max__eq__min,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [X2: A,Y2: A] :
( ( uminus_uminus @ A @ ( ord_max @ A @ X2 @ Y2 ) )
= ( ord_min @ A @ ( uminus_uminus @ A @ X2 ) @ ( uminus_uminus @ A @ Y2 ) ) ) ) ).
% minus_max_eq_min
thf(fact_6650_min__add__distrib__left,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [X2: A,Y2: A,Z: A] :
( ( plus_plus @ A @ ( ord_min @ A @ X2 @ Y2 ) @ Z )
= ( ord_min @ A @ ( plus_plus @ A @ X2 @ Z ) @ ( plus_plus @ A @ Y2 @ Z ) ) ) ) ).
% min_add_distrib_left
thf(fact_6651_min__add__distrib__right,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [X2: A,Y2: A,Z: A] :
( ( plus_plus @ A @ X2 @ ( ord_min @ A @ Y2 @ Z ) )
= ( ord_min @ A @ ( plus_plus @ A @ X2 @ Y2 ) @ ( plus_plus @ A @ X2 @ Z ) ) ) ) ).
% min_add_distrib_right
thf(fact_6652_nat__mult__min__left,axiom,
! [M: nat,N: nat,Q2: nat] :
( ( times_times @ nat @ ( ord_min @ nat @ M @ N ) @ Q2 )
= ( ord_min @ nat @ ( times_times @ nat @ M @ Q2 ) @ ( times_times @ nat @ N @ Q2 ) ) ) ).
% nat_mult_min_left
thf(fact_6653_nat__mult__min__right,axiom,
! [M: nat,N: nat,Q2: nat] :
( ( times_times @ nat @ M @ ( ord_min @ nat @ N @ Q2 ) )
= ( ord_min @ nat @ ( times_times @ nat @ M @ N ) @ ( times_times @ nat @ M @ Q2 ) ) ) ).
% nat_mult_min_right
thf(fact_6654_concat__bit__assoc__sym,axiom,
! [M: nat,N: nat,K: int,L2: int,R2: int] :
( ( bit_concat_bit @ M @ ( bit_concat_bit @ N @ K @ L2 ) @ R2 )
= ( bit_concat_bit @ ( ord_min @ nat @ M @ N ) @ K @ ( bit_concat_bit @ ( minus_minus @ nat @ M @ N ) @ L2 @ R2 ) ) ) ).
% concat_bit_assoc_sym
thf(fact_6655_take__bit__concat__bit__eq,axiom,
! [M: nat,N: nat,K: int,L2: int] :
( ( bit_se2584673776208193580ke_bit @ int @ M @ ( bit_concat_bit @ N @ K @ L2 ) )
= ( bit_concat_bit @ ( ord_min @ nat @ M @ N ) @ K @ ( bit_se2584673776208193580ke_bit @ int @ ( minus_minus @ nat @ M @ N ) @ L2 ) ) ) ).
% take_bit_concat_bit_eq
thf(fact_6656_max__mult__distrib__left,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [P2: A,X2: A,Y2: A] :
( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ P2 )
=> ( ( times_times @ A @ P2 @ ( ord_max @ A @ X2 @ Y2 ) )
= ( ord_max @ A @ ( times_times @ A @ P2 @ X2 ) @ ( times_times @ A @ P2 @ Y2 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ P2 )
=> ( ( times_times @ A @ P2 @ ( ord_max @ A @ X2 @ Y2 ) )
= ( ord_min @ A @ ( times_times @ A @ P2 @ X2 ) @ ( times_times @ A @ P2 @ Y2 ) ) ) ) ) ) ).
% max_mult_distrib_left
thf(fact_6657_min__mult__distrib__left,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [P2: A,X2: A,Y2: A] :
( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ P2 )
=> ( ( times_times @ A @ P2 @ ( ord_min @ A @ X2 @ Y2 ) )
= ( ord_min @ A @ ( times_times @ A @ P2 @ X2 ) @ ( times_times @ A @ P2 @ Y2 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ P2 )
=> ( ( times_times @ A @ P2 @ ( ord_min @ A @ X2 @ Y2 ) )
= ( ord_max @ A @ ( times_times @ A @ P2 @ X2 ) @ ( times_times @ A @ P2 @ Y2 ) ) ) ) ) ) ).
% min_mult_distrib_left
thf(fact_6658_max__mult__distrib__right,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [P2: A,X2: A,Y2: A] :
( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ P2 )
=> ( ( times_times @ A @ ( ord_max @ A @ X2 @ Y2 ) @ P2 )
= ( ord_max @ A @ ( times_times @ A @ X2 @ P2 ) @ ( times_times @ A @ Y2 @ P2 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ P2 )
=> ( ( times_times @ A @ ( ord_max @ A @ X2 @ Y2 ) @ P2 )
= ( ord_min @ A @ ( times_times @ A @ X2 @ P2 ) @ ( times_times @ A @ Y2 @ P2 ) ) ) ) ) ) ).
% max_mult_distrib_right
thf(fact_6659_min__mult__distrib__right,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [P2: A,X2: A,Y2: A] :
( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ P2 )
=> ( ( times_times @ A @ ( ord_min @ A @ X2 @ Y2 ) @ P2 )
= ( ord_min @ A @ ( times_times @ A @ X2 @ P2 ) @ ( times_times @ A @ Y2 @ P2 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ P2 )
=> ( ( times_times @ A @ ( ord_min @ A @ X2 @ Y2 ) @ P2 )
= ( ord_max @ A @ ( times_times @ A @ X2 @ P2 ) @ ( times_times @ A @ Y2 @ P2 ) ) ) ) ) ) ).
% min_mult_distrib_right
thf(fact_6660_min__divide__distrib__right,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [P2: A,X2: A,Y2: A] :
( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ P2 )
=> ( ( divide_divide @ A @ ( ord_min @ A @ X2 @ Y2 ) @ P2 )
= ( ord_min @ A @ ( divide_divide @ A @ X2 @ P2 ) @ ( divide_divide @ A @ Y2 @ P2 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ P2 )
=> ( ( divide_divide @ A @ ( ord_min @ A @ X2 @ Y2 ) @ P2 )
= ( ord_max @ A @ ( divide_divide @ A @ X2 @ P2 ) @ ( divide_divide @ A @ Y2 @ P2 ) ) ) ) ) ) ).
% min_divide_distrib_right
thf(fact_6661_max__divide__distrib__right,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [P2: A,X2: A,Y2: A] :
( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ P2 )
=> ( ( divide_divide @ A @ ( ord_max @ A @ X2 @ Y2 ) @ P2 )
= ( ord_max @ A @ ( divide_divide @ A @ X2 @ P2 ) @ ( divide_divide @ A @ Y2 @ P2 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ P2 )
=> ( ( divide_divide @ A @ ( ord_max @ A @ X2 @ Y2 ) @ P2 )
= ( ord_min @ A @ ( divide_divide @ A @ X2 @ P2 ) @ ( divide_divide @ A @ Y2 @ P2 ) ) ) ) ) ) ).
% max_divide_distrib_right
thf(fact_6662_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_6663_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_6664_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_6665_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_6666_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_6667_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_6668_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_6669_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_6670_min__enat__simps_I3_J,axiom,
! [Q2: extended_enat] :
( ( ord_min @ extended_enat @ ( zero_zero @ extended_enat ) @ Q2 )
= ( zero_zero @ extended_enat ) ) ).
% min_enat_simps(3)
thf(fact_6671_min__enat__simps_I2_J,axiom,
! [Q2: extended_enat] :
( ( ord_min @ extended_enat @ Q2 @ ( zero_zero @ extended_enat ) )
= ( zero_zero @ extended_enat ) ) ).
% min_enat_simps(2)
thf(fact_6672_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_6673_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_6674_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_6675_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_6676_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_6677_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_6678_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_6679_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_6680_bit__eq__iff,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ( ( ^ [Y5: A,Z3: A] : Y5 = Z3 )
= ( ^ [A4: A,B3: A] :
! [N3: nat] :
( ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N3 )
=> ( ( bit_se5641148757651400278ts_bit @ A @ A4 @ N3 )
= ( bit_se5641148757651400278ts_bit @ A @ B3 @ N3 ) ) ) ) ) ) ).
% bit_eq_iff
thf(fact_6681_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_6682_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_6683_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_6684_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_6685_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_6686_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_6687_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_6688_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_6689_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_6690_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_6691_bit__twiddle__min,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y2: word @ A,X2: word @ A] :
( ( bit_se5824344971392196577ns_xor @ ( word @ A ) @ Y2 @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ ( bit_se5824344971392196577ns_xor @ ( word @ A ) @ X2 @ Y2 ) @ ( if @ ( word @ A ) @ ( ord_less @ ( word @ A ) @ X2 @ Y2 ) @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) @ ( zero_zero @ ( word @ A ) ) ) ) )
= ( ord_min @ ( word @ A ) @ X2 @ Y2 ) ) ) ).
% bit_twiddle_min
thf(fact_6692_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_6693_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_6694_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_6695_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_6696_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_6697_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_6698_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_6699_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_6700_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_6701_word__set__bits__def,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( bit_bi4170147762399595738t_bits @ ( word @ A ) )
= ( ^ [P7: nat > $o] : ( groups4207007520872428315er_sum @ $o @ ( word @ A ) @ ( zero_neq_one_of_bool @ ( word @ A ) ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( map @ nat @ $o @ P7 @ ( upt @ ( zero_zero @ nat ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ) ) ).
% word_set_bits_def
thf(fact_6702_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_6703_Collect__const,axiom,
! [A: $tType,P: $o] :
( ( P
=> ( ( collect @ A
@ ^ [S5: A] : P )
= ( top_top @ ( set @ A ) ) ) )
& ( ~ P
=> ( ( collect @ A
@ ^ [S5: A] : P )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Collect_const
thf(fact_6704_finite__Collect__not,axiom,
! [A: $tType,P: A > $o] :
( ( finite_finite2 @ A @ ( collect @ A @ P ) )
=> ( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X: A] :
~ ( P @ X ) ) )
= ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) ) ) ) ).
% finite_Collect_not
thf(fact_6705_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_6706_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_6707_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_6708_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_6709_surj__diff__right,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A2: A] :
( ( image @ A @ A
@ ^ [X: A] : ( minus_minus @ A @ X @ A2 )
@ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ A ) ) ) ) ).
% surj_diff_right
thf(fact_6710_range__constant,axiom,
! [B: $tType,A: $tType,X2: A] :
( ( image @ B @ A
@ ^ [Uu3: B] : X2
@ ( top_top @ ( set @ B ) ) )
= ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% range_constant
thf(fact_6711_finite__range__imageI,axiom,
! [C: $tType,A: $tType,B: $tType,G: B > A,F2: A > C] :
( ( finite_finite2 @ A @ ( image @ B @ A @ G @ ( top_top @ ( set @ B ) ) ) )
=> ( finite_finite2 @ C
@ ( image @ B @ C
@ ^ [X: B] : ( F2 @ ( G @ X ) )
@ ( top_top @ ( set @ B ) ) ) ) ) ).
% finite_range_imageI
thf(fact_6712_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_6713_range__composition,axiom,
! [A: $tType,C: $tType,B: $tType,F2: C > A,G: B > C] :
( ( image @ B @ A
@ ^ [X: B] : ( F2 @ ( G @ X ) )
@ ( top_top @ ( set @ B ) ) )
= ( image @ C @ A @ F2 @ ( image @ B @ C @ G @ ( top_top @ ( set @ B ) ) ) ) ) ).
% range_composition
thf(fact_6714_rangeE,axiom,
! [A: $tType,B: $tType,B2: A,F2: B > A] :
( ( member @ A @ B2 @ ( image @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) ) )
=> ~ ! [X3: B] :
( B2
!= ( F2 @ X3 ) ) ) ).
% rangeE
thf(fact_6715_range__subsetD,axiom,
! [B: $tType,A: $tType,F2: B > A,B5: set @ A,I: B] :
( ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) ) @ B5 )
=> ( member @ A @ ( F2 @ I ) @ B5 ) ) ).
% range_subsetD
thf(fact_6716_not__UNIV__le__Icc,axiom,
! [A: $tType] :
( ( no_top @ A )
=> ! [L2: A,H2: A] :
~ ( ord_less_eq @ ( set @ A ) @ ( top_top @ ( set @ A ) ) @ ( set_or1337092689740270186AtMost @ A @ L2 @ H2 ) ) ) ).
% not_UNIV_le_Icc
thf(fact_6717_top__greatest,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [A2: A] : ( ord_less_eq @ A @ A2 @ ( top_top @ A ) ) ) ).
% top_greatest
thf(fact_6718_top_Oextremum__unique,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [A2: A] :
( ( ord_less_eq @ A @ ( top_top @ A ) @ A2 )
= ( A2
= ( top_top @ A ) ) ) ) ).
% top.extremum_unique
thf(fact_6719_top_Oextremum__uniqueI,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [A2: A] :
( ( ord_less_eq @ A @ ( top_top @ A ) @ A2 )
=> ( A2
= ( top_top @ A ) ) ) ) ).
% top.extremum_uniqueI
thf(fact_6720_subset__UNIV,axiom,
! [A: $tType,A3: set @ A] : ( ord_less_eq @ ( set @ A ) @ A3 @ ( top_top @ ( set @ A ) ) ) ).
% subset_UNIV
thf(fact_6721_UNIV__def,axiom,
! [A: $tType] :
( ( top_top @ ( set @ A ) )
= ( collect @ A
@ ^ [X: A] : $true ) ) ).
% UNIV_def
thf(fact_6722_not__UNIV__le__Iic,axiom,
! [A: $tType] :
( ( no_top @ A )
=> ! [H2: A] :
~ ( ord_less_eq @ ( set @ A ) @ ( top_top @ ( set @ A ) ) @ ( set_ord_atMost @ A @ H2 ) ) ) ).
% not_UNIV_le_Iic
thf(fact_6723_surj__Compl__image__subset,axiom,
! [A: $tType,B: $tType,F2: B > A,A3: set @ B] :
( ( ( image @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
=> ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ ( image @ B @ A @ F2 @ A3 ) ) @ ( image @ B @ A @ F2 @ ( uminus_uminus @ ( set @ B ) @ A3 ) ) ) ) ).
% surj_Compl_image_subset
thf(fact_6724_notin__range__Some,axiom,
! [A: $tType,X2: option @ A] :
( ( ~ ( member @ ( option @ A ) @ X2 @ ( image @ A @ ( option @ A ) @ ( some @ A ) @ ( top_top @ ( set @ A ) ) ) ) )
= ( X2
= ( none @ A ) ) ) ).
% notin_range_Some
thf(fact_6725_word__of__int__conv__set__bits,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( ring_1_of_int @ ( word @ A ) )
= ( ^ [I3: int] : ( bit_bi4170147762399595738t_bits @ ( word @ A ) @ ( bit_se5641148757651400278ts_bit @ int @ I3 ) ) ) ) ) ).
% word_of_int_conv_set_bits
thf(fact_6726_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_6727_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_6728_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_6729_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_6730_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 )
@ ^ [X: word @ B] : ( ord_less @ ( word @ B ) @ X @ ( power_power @ ( word @ B ) @ ( numeral_numeral @ ( word @ B ) @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ) ) ).
% ucast_range_less
thf(fact_6731_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_6732_word__roti_Oabs__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Xa: int,X2: int] :
( ( word_roti @ A @ Xa @ ( word2 @ A @ X2 ) )
= ( word2 @ A @ ( bit_concat_bit @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( nat2 @ ( modulo_modulo @ int @ Xa @ ( semiring_1_of_nat @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) @ ( bit_se4197421643247451524op_bit @ int @ ( nat2 @ ( modulo_modulo @ int @ Xa @ ( semiring_1_of_nat @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ X2 ) ) @ ( bit_se2584673776208193580ke_bit @ int @ ( nat2 @ ( modulo_modulo @ int @ Xa @ ( semiring_1_of_nat @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) @ X2 ) ) ) ) ) ).
% word_roti.abs_eq
thf(fact_6733_int__set__bits__K__False,axiom,
( ( bit_bi4170147762399595738t_bits @ int
@ ^ [Uu3: nat] : $false )
= ( zero_zero @ int ) ) ).
% int_set_bits_K_False
thf(fact_6734_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_6735_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_6736_Collect__const__case__prod,axiom,
! [B: $tType,A: $tType,P: $o] :
( ( P
=> ( ( collect @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [A4: A,B3: B] : P ) )
= ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) )
& ( ~ P
=> ( ( collect @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [A4: A,B3: B] : P ) )
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ) ) ).
% Collect_const_case_prod
thf(fact_6737_length__nth__simps_I4_J,axiom,
! [B: $tType,X2: B,Xs2: list @ B,N: nat] :
( ( nth @ B @ ( cons @ B @ X2 @ Xs2 ) @ ( suc @ N ) )
= ( nth @ B @ Xs2 @ N ) ) ).
% length_nth_simps(4)
thf(fact_6738_nth__Cons__Suc,axiom,
! [A: $tType,X2: A,Xs2: list @ A,N: nat] :
( ( nth @ A @ ( cons @ A @ X2 @ Xs2 ) @ ( suc @ N ) )
= ( nth @ A @ Xs2 @ N ) ) ).
% nth_Cons_Suc
thf(fact_6739_length__nth__simps_I3_J,axiom,
! [B: $tType,X2: B,Xs2: list @ B] :
( ( nth @ B @ ( cons @ B @ X2 @ Xs2 ) @ ( zero_zero @ nat ) )
= X2 ) ).
% length_nth_simps(3)
thf(fact_6740_nth__Cons__0,axiom,
! [A: $tType,X2: A,Xs2: list @ A] :
( ( nth @ A @ ( cons @ A @ X2 @ Xs2 ) @ ( zero_zero @ nat ) )
= X2 ) ).
% nth_Cons_0
thf(fact_6741_horner__sum__simps_I2_J,axiom,
! [A: $tType,B: $tType] :
( ( comm_semiring_0 @ A )
=> ! [F2: B > A,A2: A,X2: B,Xs2: list @ B] :
( ( groups4207007520872428315er_sum @ B @ A @ F2 @ A2 @ ( cons @ B @ X2 @ Xs2 ) )
= ( plus_plus @ A @ ( F2 @ X2 ) @ ( times_times @ A @ A2 @ ( groups4207007520872428315er_sum @ B @ A @ F2 @ A2 @ Xs2 ) ) ) ) ) ).
% horner_sum_simps(2)
thf(fact_6742_nth__Cons__numeral,axiom,
! [A: $tType,X2: A,Xs2: list @ A,V: num] :
( ( nth @ A @ ( cons @ A @ X2 @ Xs2 ) @ ( numeral_numeral @ nat @ V ) )
= ( nth @ A @ Xs2 @ ( minus_minus @ nat @ ( numeral_numeral @ nat @ V ) @ ( one_one @ nat ) ) ) ) ).
% nth_Cons_numeral
thf(fact_6743_nth__Cons__pos,axiom,
! [A: $tType,N: nat,X2: A,Xs2: list @ A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( nth @ A @ ( cons @ A @ X2 @ Xs2 ) @ N )
= ( nth @ A @ Xs2 @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ).
% nth_Cons_pos
thf(fact_6744_top__empty__eq2,axiom,
! [B: $tType,A: $tType] :
( ( top_top @ ( A > B > $o ) )
= ( ^ [X: A,Y: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) ) ) ).
% top_empty_eq2
thf(fact_6745_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 ) )
@ ^ [X: A] : ( map @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X ) @ ( product_lists @ A @ Xss ) )
@ Xs2 ) ) ) ).
% product_lists.simps(2)
thf(fact_6746_times__word_Oabs__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Xa: int,X2: int] :
( ( times_times @ ( word @ A ) @ ( word2 @ A @ Xa ) @ ( word2 @ A @ X2 ) )
= ( word2 @ A @ ( times_times @ int @ Xa @ X2 ) ) ) ) ).
% times_word.abs_eq
thf(fact_6747_plus__word_Oabs__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Xa: int,X2: int] :
( ( plus_plus @ ( word @ A ) @ ( word2 @ A @ Xa ) @ ( word2 @ A @ X2 ) )
= ( word2 @ A @ ( plus_plus @ int @ Xa @ X2 ) ) ) ) ).
% plus_word.abs_eq
thf(fact_6748_zero__word__def,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( zero_zero @ ( word @ A ) )
= ( word2 @ A @ ( zero_zero @ int ) ) ) ) ).
% zero_word_def
thf(fact_6749_one__word__def,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( one_one @ ( word @ A ) )
= ( word2 @ A @ ( one_one @ int ) ) ) ) ).
% one_word_def
thf(fact_6750_replicate__Suc,axiom,
! [A: $tType,N: nat,X2: A] :
( ( replicate @ A @ ( suc @ N ) @ X2 )
= ( cons @ A @ X2 @ ( replicate @ A @ N @ X2 ) ) ) ).
% replicate_Suc
thf(fact_6751_list__update__code_I2_J,axiom,
! [A: $tType,X2: A,Xs2: list @ A,Y2: A] :
( ( list_update @ A @ ( cons @ A @ X2 @ Xs2 ) @ ( zero_zero @ nat ) @ Y2 )
= ( cons @ A @ Y2 @ Xs2 ) ) ).
% list_update_code(2)
thf(fact_6752_list__update__code_I3_J,axiom,
! [A: $tType,X2: A,Xs2: list @ A,I: nat,Y2: A] :
( ( list_update @ A @ ( cons @ A @ X2 @ Xs2 ) @ ( suc @ I ) @ Y2 )
= ( cons @ A @ X2 @ ( list_update @ A @ Xs2 @ I @ Y2 ) ) ) ).
% list_update_code(3)
thf(fact_6753_set__subset__Cons,axiom,
! [A: $tType,Xs2: list @ A,X2: A] : ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs2 ) @ ( set2 @ A @ ( cons @ A @ X2 @ Xs2 ) ) ) ).
% set_subset_Cons
thf(fact_6754_Suc__length__conv,axiom,
! [A: $tType,N: nat,Xs2: list @ A] :
( ( ( suc @ N )
= ( size_size @ ( list @ A ) @ Xs2 ) )
= ( ? [Y: A,Ys2: list @ A] :
( ( Xs2
= ( cons @ A @ Y @ Ys2 ) )
& ( ( size_size @ ( list @ A ) @ Ys2 )
= N ) ) ) ) ).
% Suc_length_conv
thf(fact_6755_length__Suc__conv,axiom,
! [A: $tType,Xs2: list @ A,N: nat] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
= ( suc @ N ) )
= ( ? [Y: A,Ys2: list @ A] :
( ( Xs2
= ( cons @ A @ Y @ Ys2 ) )
& ( ( size_size @ ( list @ A ) @ Ys2 )
= N ) ) ) ) ).
% length_Suc_conv
thf(fact_6756_length__nth__simps_I2_J,axiom,
! [B: $tType,X2: B,Xs2: list @ B] :
( ( size_size @ ( list @ B ) @ ( cons @ B @ X2 @ Xs2 ) )
= ( suc @ ( size_size @ ( list @ B ) @ Xs2 ) ) ) ).
% length_nth_simps(2)
thf(fact_6757_impossible__Cons,axiom,
! [A: $tType,Xs2: list @ A,Ys: list @ A,X2: A] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ ( size_size @ ( list @ A ) @ Ys ) )
=> ( Xs2
!= ( cons @ A @ X2 @ Ys ) ) ) ).
% impossible_Cons
thf(fact_6758_Suc__le__length__iff,axiom,
! [A: $tType,N: nat,Xs2: list @ A] :
( ( ord_less_eq @ nat @ ( suc @ N ) @ ( size_size @ ( list @ A ) @ Xs2 ) )
= ( ? [X: A,Ys2: list @ A] :
( ( Xs2
= ( cons @ A @ X @ Ys2 ) )
& ( ord_less_eq @ nat @ N @ ( size_size @ ( list @ A ) @ Ys2 ) ) ) ) ) ).
% Suc_le_length_iff
thf(fact_6759_word__succ_Oabs__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: int] :
( ( word_succ @ A @ ( word2 @ A @ X2 ) )
= ( word2 @ A @ ( plus_plus @ int @ X2 @ ( one_one @ int ) ) ) ) ) ).
% word_succ.abs_eq
thf(fact_6760_word__pred_Oabs__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: int] :
( ( word_pred @ A @ ( word2 @ A @ X2 ) )
= ( word2 @ A @ ( minus_minus @ int @ X2 @ ( one_one @ int ) ) ) ) ) ).
% word_pred.abs_eq
thf(fact_6761_count__list_Osimps_I2_J,axiom,
! [A: $tType,X2: A,Y2: A,Xs2: list @ A] :
( ( ( X2 = Y2 )
=> ( ( count_list @ A @ ( cons @ A @ X2 @ Xs2 ) @ Y2 )
= ( plus_plus @ nat @ ( count_list @ A @ Xs2 @ Y2 ) @ ( one_one @ nat ) ) ) )
& ( ( X2 != Y2 )
=> ( ( count_list @ A @ ( cons @ A @ X2 @ Xs2 ) @ Y2 )
= ( count_list @ A @ Xs2 @ Y2 ) ) ) ) ).
% count_list.simps(2)
thf(fact_6762_int__in__range__abs,axiom,
! [N: nat] : ( member @ int @ ( semiring_1_of_nat @ int @ N ) @ ( image @ int @ int @ ( abs_abs @ int ) @ ( top_top @ ( set @ int ) ) ) ) ).
% int_in_range_abs
thf(fact_6763_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_6764_nth__Cons_H,axiom,
! [A: $tType,N: nat,X2: A,Xs2: list @ A] :
( ( ( N
= ( zero_zero @ nat ) )
=> ( ( nth @ A @ ( cons @ A @ X2 @ Xs2 ) @ N )
= X2 ) )
& ( ( N
!= ( zero_zero @ nat ) )
=> ( ( nth @ A @ ( cons @ A @ X2 @ Xs2 ) @ N )
= ( nth @ A @ Xs2 @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ) ).
% nth_Cons'
thf(fact_6765_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 ) )
@ ^ [Ys2: list @ A] :
( map @ A @ ( list @ A )
@ ^ [Y: A] : ( cons @ A @ Y @ Ys2 )
@ Xs2 )
@ ( n_lists @ A @ N @ Xs2 ) ) ) ) ).
% n_lists.simps(2)
thf(fact_6766_list_Osize__gen_I2_J,axiom,
! [A: $tType,X2: A > nat,X21: A,X222: list @ A] :
( ( size_list @ A @ X2 @ ( cons @ A @ X21 @ X222 ) )
= ( plus_plus @ nat @ ( plus_plus @ nat @ ( X2 @ X21 ) @ ( size_list @ A @ X2 @ X222 ) ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% list.size_gen(2)
thf(fact_6767_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
@ ^ [I3: nat] : ( F2 @ ( suc @ I3 ) )
@ ( upt @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% map_upt_Suc
thf(fact_6768_divide__word_Oabs__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Xa: int,X2: int] :
( ( divide_divide @ ( word @ A ) @ ( word2 @ A @ Xa ) @ ( word2 @ A @ X2 ) )
= ( 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 ) ) @ X2 ) ) ) ) ) ).
% divide_word.abs_eq
thf(fact_6769_nth__non__equal__first__eq,axiom,
! [A: $tType,X2: A,Y2: A,Xs2: list @ A,N: nat] :
( ( X2 != Y2 )
=> ( ( ( nth @ A @ ( cons @ A @ X2 @ Xs2 ) @ N )
= Y2 )
= ( ( ( nth @ A @ Xs2 @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) )
= Y2 )
& ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% nth_non_equal_first_eq
thf(fact_6770_nth__equal__first__eq,axiom,
! [A: $tType,X2: A,Xs2: list @ A,N: nat] :
( ~ ( member @ A @ X2 @ ( set2 @ A @ Xs2 ) )
=> ( ( ord_less_eq @ nat @ N @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( ( nth @ A @ ( cons @ A @ X2 @ Xs2 ) @ N )
= X2 )
= ( N
= ( zero_zero @ nat ) ) ) ) ) ).
% nth_equal_first_eq
thf(fact_6771_Cons__replicate__eq,axiom,
! [A: $tType,X2: A,Xs2: list @ A,N: nat,Y2: A] :
( ( ( cons @ A @ X2 @ Xs2 )
= ( replicate @ A @ N @ Y2 ) )
= ( ( X2 = Y2 )
& ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
& ( Xs2
= ( replicate @ A @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ X2 ) ) ) ) ).
% Cons_replicate_eq
thf(fact_6772_slice__Cons,axiom,
! [A: $tType,Begin: nat,End: nat,X2: A,Xs2: list @ A] :
( ( ( ( Begin
= ( zero_zero @ nat ) )
& ( ord_less @ nat @ ( zero_zero @ nat ) @ End ) )
=> ( ( slice @ A @ Begin @ End @ ( cons @ A @ X2 @ Xs2 ) )
= ( cons @ A @ X2 @ ( 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 @ X2 @ Xs2 ) )
= ( slice @ A @ ( minus_minus @ nat @ Begin @ ( one_one @ nat ) ) @ ( minus_minus @ nat @ End @ ( one_one @ nat ) ) @ Xs2 ) ) ) ) ).
% slice_Cons
thf(fact_6773_unsigned_Oabs__eq,axiom,
! [A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( semiring_1 @ A ) )
=> ! [X2: int] :
( ( semiring_1_unsigned @ B @ A @ ( word2 @ B @ X2 ) )
= ( semiring_1_of_nat @ A @ ( nat2 @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ B @ ( type2 @ B ) ) @ X2 ) ) ) ) ) ).
% unsigned.abs_eq
thf(fact_6774_range__mod,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( image @ nat @ nat
@ ^ [M5: nat] : ( modulo_modulo @ nat @ M5 @ N )
@ ( top_top @ ( set @ nat ) ) )
= ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ).
% range_mod
thf(fact_6775_word__sle_Oabs__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Xa: int,X2: int] :
( ( word_sle @ A @ ( word2 @ A @ Xa ) @ ( word2 @ A @ X2 ) )
= ( ord_less_eq @ int @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) @ Xa ) @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) @ X2 ) ) ) ) ).
% word_sle.abs_eq
thf(fact_6776_word__sless_Oabs__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Xa: int,X2: int] :
( ( word_sless @ A @ ( word2 @ A @ Xa ) @ ( word2 @ A @ X2 ) )
= ( ord_less @ int @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) @ Xa ) @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) @ X2 ) ) ) ) ).
% word_sless.abs_eq
thf(fact_6777_signed_Oabs__eq,axiom,
! [A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( ring_1 @ A ) )
=> ! [X2: int] :
( ( ring_1_signed @ B @ A @ ( word2 @ B @ X2 ) )
= ( ring_1_of_int @ A @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ B @ ( type2 @ B ) ) @ ( suc @ ( zero_zero @ nat ) ) ) @ X2 ) ) ) ) ).
% signed.abs_eq
thf(fact_6778_signed__drop__bit_Oabs__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Xa: nat,X2: int] :
( ( signed_drop_bit @ A @ Xa @ ( word2 @ A @ X2 ) )
= ( word2 @ A @ ( bit_se4197421643247451524op_bit @ int @ Xa @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) @ X2 ) ) ) ) ) ).
% signed_drop_bit.abs_eq
thf(fact_6779_signed__modulo__word_Oabs__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Xa: int,X2: int] :
( ( signed6721504322012087516modulo @ ( word @ A ) @ ( word2 @ A @ Xa ) @ ( word2 @ A @ X2 ) )
= ( word2 @ A @ ( signed6721504322012087516modulo @ int @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) @ Xa ) @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) @ X2 ) ) ) ) ) ).
% signed_modulo_word.abs_eq
thf(fact_6780_signed__divide__word_Oabs__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Xa: int,X2: int] :
( ( signed7115095781618012415divide @ ( word @ A ) @ ( word2 @ A @ Xa ) @ ( word2 @ A @ X2 ) )
= ( word2 @ A @ ( signed7115095781618012415divide @ int @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) @ Xa ) @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) @ X2 ) ) ) ) ) ).
% signed_divide_word.abs_eq
thf(fact_6781_hash__code__list__simps_I2_J,axiom,
! [A: $tType,H_a: A > uint32,X2: A,Xa: list @ A] :
( ( hash_hash_code_list @ A @ H_a @ ( cons @ A @ X2 @ Xa ) )
= ( plus_plus @ uint32 @ ( times_times @ uint32 @ ( H_a @ X2 ) @ ( numeral_numeral @ uint32 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( plus_plus @ uint32 @ ( times_times @ uint32 @ ( hash_hash_code_list @ A @ H_a @ Xa ) @ ( numeral_numeral @ uint32 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numeral_numeral @ uint32 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% hash_code_list_simps(2)
thf(fact_6782_signed__cast_Oabs__eq,axiom,
! [B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B ) )
=> ! [X2: int] :
( ( signed_cast @ A @ B @ ( word2 @ A @ X2 ) )
= ( word2 @ B @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) @ X2 ) ) ) ) ).
% signed_cast.abs_eq
thf(fact_6783_upt__conv__Cons__Cons,axiom,
! [M: nat,N: nat,Ns: list @ nat,Q2: nat] :
( ( ( cons @ nat @ M @ ( cons @ nat @ N @ Ns ) )
= ( upt @ M @ Q2 ) )
= ( ( cons @ nat @ N @ Ns )
= ( upt @ ( suc @ M ) @ Q2 ) ) ) ).
% upt_conv_Cons_Cons
thf(fact_6784_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_6785_upt__eq__Cons__conv,axiom,
! [I: nat,J: nat,X2: nat,Xs2: list @ nat] :
( ( ( upt @ I @ J )
= ( cons @ nat @ X2 @ Xs2 ) )
= ( ( ord_less @ nat @ I @ J )
& ( I = X2 )
& ( ( upt @ ( plus_plus @ nat @ I @ ( one_one @ nat ) ) @ J )
= Xs2 ) ) ) ).
% upt_eq_Cons_conv
thf(fact_6786_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_6787_the__signed__int_Oabs__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: int] :
( ( the_signed_int @ A @ ( word2 @ A @ X2 ) )
= ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) @ X2 ) ) ) ).
% the_signed_int.abs_eq
thf(fact_6788_of__nat_Oabs__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( of_nat @ A )
= ( ^ [X: nat] : ( word2 @ A @ ( semiring_1_of_nat @ int @ ( bit_se2584673776208193580ke_bit @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ X ) ) ) ) ) ) ).
% of_nat.abs_eq
thf(fact_6789_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_6790_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_6791_wf__set__bits__int__const,axiom,
! [B2: $o] :
( bit_wf_set_bits_int
@ ^ [Uu3: nat] : B2 ) ).
% wf_set_bits_int_const
thf(fact_6792_ones,axiom,
! [N: nat,F2: nat > $o] :
( ! [N7: nat] :
( ( ord_less_eq @ nat @ N @ N7 )
=> ( F2 @ N7 ) )
=> ( bit_wf_set_bits_int @ F2 ) ) ).
% ones
thf(fact_6793_wf__set__bits__int_Ocases,axiom,
! [F2: nat > $o] :
( ( bit_wf_set_bits_int @ F2 )
=> ( ! [N2: nat] :
~ ! [N6: nat] :
( ( ord_less_eq @ nat @ N2 @ N6 )
=> ~ ( F2 @ N6 ) )
=> ~ ! [N2: nat] :
~ ! [N6: nat] :
( ( ord_less_eq @ nat @ N2 @ N6 )
=> ( F2 @ N6 ) ) ) ) ).
% wf_set_bits_int.cases
thf(fact_6794_wf__set__bits__int_Osimps,axiom,
( bit_wf_set_bits_int
= ( ^ [F4: nat > $o] :
( ? [N3: nat] :
! [N13: nat] :
( ( ord_less_eq @ nat @ N3 @ N13 )
=> ~ ( F4 @ N13 ) )
| ? [N3: nat] :
! [N13: nat] :
( ( ord_less_eq @ nat @ N3 @ N13 )
=> ( F4 @ N13 ) ) ) ) ) ).
% wf_set_bits_int.simps
thf(fact_6795_zeros,axiom,
! [N: nat,F2: nat > $o] :
( ! [N7: nat] :
( ( ord_less_eq @ nat @ N @ N7 )
=> ~ ( F2 @ N7 ) )
=> ( bit_wf_set_bits_int @ F2 ) ) ).
% zeros
thf(fact_6796_wf__set__bits__int__simps,axiom,
( bit_wf_set_bits_int
= ( ^ [F4: nat > $o] :
? [N3: nat] :
( ! [N13: nat] :
( ( ord_less_eq @ nat @ N3 @ N13 )
=> ~ ( F4 @ N13 ) )
| ! [N13: nat] :
( ( ord_less_eq @ nat @ N3 @ N13 )
=> ( F4 @ N13 ) ) ) ) ) ).
% wf_set_bits_int_simps
thf(fact_6797_root__def,axiom,
( root
= ( ^ [N3: nat,X: real] :
( if @ real
@ ( N3
= ( zero_zero @ nat ) )
@ ( zero_zero @ real )
@ ( the_inv_into @ real @ real @ ( top_top @ ( set @ real ) )
@ ^ [Y: real] : ( times_times @ real @ ( sgn_sgn @ real @ Y ) @ ( power_power @ real @ ( abs_abs @ real @ Y ) @ N3 ) )
@ X ) ) ) ) ).
% root_def
thf(fact_6798_length__Cons,axiom,
! [A: $tType,X2: A,Xs2: list @ A] :
( ( size_size @ ( list @ A ) @ ( cons @ A @ X2 @ Xs2 ) )
= ( suc @ ( size_size @ ( list @ A ) @ Xs2 ) ) ) ).
% length_Cons
thf(fact_6799_the__inv__into__def,axiom,
! [B: $tType,A: $tType] :
( ( the_inv_into @ A @ B )
= ( ^ [A7: set @ A,F4: A > B,X: B] :
( the @ A
@ ^ [Y: A] :
( ( member @ A @ Y @ A7 )
& ( ( F4 @ Y )
= X ) ) ) ) ) ).
% the_inv_into_def
thf(fact_6800_subset__CollectI,axiom,
! [A: $tType,B5: set @ A,A3: set @ A,Q: A > $o,P: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ B5 @ A3 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ B5 )
=> ( ( Q @ X3 )
=> ( P @ X3 ) ) )
=> ( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [X: A] :
( ( member @ A @ X @ B5 )
& ( Q @ X ) ) )
@ ( collect @ A
@ ^ [X: A] :
( ( member @ A @ X @ A3 )
& ( P @ X ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_6801_subset__Collect__iff,axiom,
! [A: $tType,B5: set @ A,A3: set @ A,P: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ B5 @ A3 )
=> ( ( ord_less_eq @ ( set @ A ) @ B5
@ ( collect @ A
@ ^ [X: A] :
( ( member @ A @ X @ A3 )
& ( P @ X ) ) ) )
= ( ! [X: A] :
( ( member @ A @ X @ B5 )
=> ( P @ X ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_6802_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_6803_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_6804_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_6805_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_6806_bij__betw__comp__iff2,axiom,
! [C: $tType,A: $tType,B: $tType,F7: A > B,A10: set @ A,A11: set @ B,F2: C > A,A3: set @ C] :
( ( bij_betw @ A @ B @ F7 @ A10 @ A11 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image @ C @ A @ F2 @ A3 ) @ A10 )
=> ( ( bij_betw @ C @ A @ F2 @ A3 @ A10 )
= ( bij_betw @ C @ B @ ( comp @ A @ B @ C @ F7 @ F2 ) @ A3 @ A11 ) ) ) ) ).
% bij_betw_comp_iff2
thf(fact_6807_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_6808_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_6809_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_6810_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_6811_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_6812_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_6813_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_6814_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_6815_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_6816_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_6817_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_6818_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_6819_sum__image__le,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( ordere6911136660526730532id_add @ B )
=> ! [I5: set @ C,G: A > B,F2: C > A] :
( ( finite_finite2 @ C @ I5 )
=> ( ! [I2: C] :
( ( member @ C @ I2 @ I5 )
=> ( ord_less_eq @ B @ ( zero_zero @ B ) @ ( G @ ( F2 @ I2 ) ) ) )
=> ( ord_less_eq @ B @ ( groups7311177749621191930dd_sum @ A @ B @ G @ ( image @ C @ A @ F2 @ I5 ) ) @ ( groups7311177749621191930dd_sum @ C @ B @ ( comp @ A @ B @ C @ G @ F2 ) @ I5 ) ) ) ) ) ).
% sum_image_le
thf(fact_6820_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_6821_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_6822_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_6823_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_6824_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_6825_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_6826_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_6827_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_6828_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_6829_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_6830_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_6831_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_6832_word__roti__eq__word__rotr__word__rotl,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( word_roti @ A )
= ( ^ [I3: int,W2: word @ A] : ( if @ ( word @ A ) @ ( ord_less_eq @ int @ ( zero_zero @ int ) @ I3 ) @ ( word_rotr @ A @ ( nat2 @ I3 ) @ W2 ) @ ( word_rotl @ A @ ( nat2 @ ( uminus_uminus @ int @ I3 ) ) @ W2 ) ) ) ) ) ).
% word_roti_eq_word_rotr_word_rotl
thf(fact_6833_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_6834_CHAR__eq__0,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ( ( semiri4206861660011772517g_char @ A @ ( type2 @ A ) )
= ( zero_zero @ nat ) ) ) ).
% CHAR_eq_0
thf(fact_6835_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_6836_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_6837_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_6838_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_6839_sum_OatLeast__int__atMost__int__shift,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: int > A,M: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ int @ A @ G @ ( set_or1337092689740270186AtMost @ int @ ( semiring_1_of_nat @ int @ M ) @ ( semiring_1_of_nat @ int @ N ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ int @ A @ nat @ G @ ( semiring_1_of_nat @ int ) ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ).
% sum.atLeast_int_atMost_int_shift
thf(fact_6840_sum_OatLeast__int__lessThan__int__shift,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: int > A,M: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ int @ A @ G @ ( set_or7035219750837199246ssThan @ int @ ( semiring_1_of_nat @ int @ M ) @ ( semiring_1_of_nat @ int @ N ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ int @ A @ nat @ G @ ( semiring_1_of_nat @ int ) ) @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) ) ) ).
% sum.atLeast_int_lessThan_int_shift
thf(fact_6841_prod_OatLeast__int__atMost__int__shift,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: int > A,M: nat,N: nat] :
( ( groups7121269368397514597t_prod @ int @ A @ G @ ( set_or1337092689740270186AtMost @ int @ ( semiring_1_of_nat @ int @ M ) @ ( semiring_1_of_nat @ int @ N ) ) )
= ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ int @ A @ nat @ G @ ( semiring_1_of_nat @ int ) ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ).
% prod.atLeast_int_atMost_int_shift
thf(fact_6842_prod_OatLeast__int__lessThan__int__shift,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: int > A,M: nat,N: nat] :
( ( groups7121269368397514597t_prod @ int @ A @ G @ ( set_or7035219750837199246ssThan @ int @ ( semiring_1_of_nat @ int @ M ) @ ( semiring_1_of_nat @ int @ N ) ) )
= ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ int @ A @ nat @ G @ ( semiring_1_of_nat @ int ) ) @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) ) ) ).
% prod.atLeast_int_lessThan_int_shift
thf(fact_6843_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_6844_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_6845_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_6846_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_6847_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_6848_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_6849_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_6850_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_6851_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_6852_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_6853_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_6854_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_6855_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_6856_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_6857_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_6858_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_6859_shiftl__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat] :
( ( bit_Sh4282982442137083160shiftl @ A @ ( zero_zero @ A ) @ N )
= ( zero_zero @ A ) ) ) ).
% shiftl_0
thf(fact_6860_shiftl__of__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A] :
( ( bit_Sh4282982442137083160shiftl @ A @ A2 @ ( zero_zero @ nat ) )
= A2 ) ) ).
% shiftl_of_0
thf(fact_6861_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_6862_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_6863_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_6864_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_6865_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_6866_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_6867_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_6868_shiftl__eq__mult,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_Sh4282982442137083160shiftl @ A )
= ( ^ [X: A,N3: nat] : ( times_times @ A @ X @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N3 ) ) ) ) ) ).
% shiftl_eq_mult
thf(fact_6869_bit__shiftl__iff,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A,M: nat,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( bit_Sh4282982442137083160shiftl @ A @ A2 @ M ) @ N )
= ( ( ord_less_eq @ nat @ M @ N )
& ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N )
& ( bit_se5641148757651400278ts_bit @ A @ A2 @ ( minus_minus @ nat @ N @ M ) ) ) ) ) ).
% bit_shiftl_iff
thf(fact_6870_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_6871_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_6872_shiftr__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat] :
( ( bit_Sh4282982442137083166shiftr @ A @ ( zero_zero @ A ) @ N )
= ( zero_zero @ A ) ) ) ).
% shiftr_0
thf(fact_6873_shiftr__of__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A] :
( ( bit_Sh4282982442137083166shiftr @ A @ A2 @ ( zero_zero @ nat ) )
= A2 ) ) ).
% shiftr_of_0
thf(fact_6874_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_6875_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_6876_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_6877_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_6878_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_6879_shiftr__eq__div,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_Sh4282982442137083166shiftr @ A )
= ( ^ [X: A,N3: nat] : ( divide_divide @ A @ X @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N3 ) ) ) ) ) ).
% shiftr_eq_div
thf(fact_6880_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_6881_upto__aux__rec,axiom,
( upto_aux
= ( ^ [I3: int,J3: int,Js: list @ int] : ( if @ ( list @ int ) @ ( ord_less @ int @ J3 @ I3 ) @ Js @ ( upto_aux @ I3 @ ( minus_minus @ int @ J3 @ ( one_one @ int ) ) @ ( cons @ int @ J3 @ Js ) ) ) ) ) ).
% upto_aux_rec
thf(fact_6882_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_6883_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_6884_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_6885_inj__divide__right,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A2: A] :
( ( inj_on @ A @ A
@ ^ [B3: A] : ( divide_divide @ A @ B3 @ A2 )
@ ( top_top @ ( set @ A ) ) )
= ( A2
!= ( zero_zero @ A ) ) ) ) ).
% inj_divide_right
thf(fact_6886_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_6887_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_6888_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_6889_finite__inverse__image__gen,axiom,
! [A: $tType,B: $tType,A3: set @ A,F2: B > A,D4: set @ B] :
( ( finite_finite2 @ A @ A3 )
=> ( ( inj_on @ B @ A @ F2 @ D4 )
=> ( finite_finite2 @ B
@ ( collect @ B
@ ^ [J3: B] :
( ( member @ B @ J3 @ D4 )
& ( member @ A @ ( F2 @ J3 ) @ A3 ) ) ) ) ) ) ).
% finite_inverse_image_gen
thf(fact_6890_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_6891_inj__on__subset,axiom,
! [B: $tType,A: $tType,F2: A > B,A3: set @ A,B5: set @ A] :
( ( inj_on @ A @ B @ F2 @ A3 )
=> ( ( ord_less_eq @ ( set @ A ) @ B5 @ A3 )
=> ( inj_on @ A @ B @ F2 @ B5 ) ) ) ).
% inj_on_subset
thf(fact_6892_subset__inj__on,axiom,
! [B: $tType,A: $tType,F2: A > B,B5: set @ A,A3: set @ A] :
( ( inj_on @ A @ B @ F2 @ B5 )
=> ( ( ord_less_eq @ ( set @ A ) @ A3 @ B5 )
=> ( inj_on @ A @ B @ F2 @ A3 ) ) ) ).
% subset_inj_on
thf(fact_6893_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_6894_word__msb__0,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ~ ( most_s684356279273892711sb_msb @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) ) ) ).
% word_msb_0
thf(fact_6895_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_6896_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_6897_inj__on__add_H,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [A2: A,A3: set @ A] :
( inj_on @ A @ A
@ ^ [B3: A] : ( plus_plus @ A @ B3 @ A2 )
@ A3 ) ) ).
% inj_on_add'
thf(fact_6898_inj__on__id2,axiom,
! [A: $tType,A3: set @ A] :
( inj_on @ A @ A
@ ^ [X: A] : X
@ A3 ) ).
% inj_on_id2
thf(fact_6899_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_6900_inj__on__image__eq__iff,axiom,
! [B: $tType,A: $tType,F2: A > B,C6: set @ A,A3: set @ A,B5: set @ A] :
( ( inj_on @ A @ B @ F2 @ C6 )
=> ( ( ord_less_eq @ ( set @ A ) @ A3 @ C6 )
=> ( ( ord_less_eq @ ( set @ A ) @ B5 @ C6 )
=> ( ( ( image @ A @ B @ F2 @ A3 )
= ( image @ A @ B @ F2 @ B5 ) )
= ( A3 = B5 ) ) ) ) ) ).
% inj_on_image_eq_iff
thf(fact_6901_inj__on__image__mem__iff,axiom,
! [B: $tType,A: $tType,F2: A > B,B5: set @ A,A2: A,A3: set @ A] :
( ( inj_on @ A @ B @ F2 @ B5 )
=> ( ( member @ A @ A2 @ B5 )
=> ( ( ord_less_eq @ ( set @ A ) @ A3 @ B5 )
=> ( ( member @ B @ ( F2 @ A2 ) @ ( image @ A @ B @ F2 @ A3 ) )
= ( member @ A @ A2 @ A3 ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_6902_subset__image__inj,axiom,
! [A: $tType,B: $tType,S: set @ A,F2: B > A,T3: set @ B] :
( ( ord_less_eq @ ( set @ A ) @ S @ ( image @ B @ A @ F2 @ T3 ) )
= ( ? [U3: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ U3 @ T3 )
& ( inj_on @ B @ A @ F2 @ U3 )
& ( S
= ( image @ B @ A @ F2 @ U3 ) ) ) ) ) ).
% subset_image_inj
thf(fact_6903_inj__diff__right,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A2: A] :
( inj_on @ A @ A
@ ^ [B3: A] : ( minus_minus @ A @ B3 @ A2 )
@ ( top_top @ ( set @ A ) ) ) ) ).
% inj_diff_right
thf(fact_6904_inj__fun,axiom,
! [B: $tType,C: $tType,A: $tType,F2: A > B] :
( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
=> ( inj_on @ A @ ( C > B )
@ ^ [X: A,Y: C] : ( F2 @ X )
@ ( top_top @ ( set @ A ) ) ) ) ).
% inj_fun
thf(fact_6905_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_6906_sorted__list__of__set_Oinj__on,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( inj_on @ A @ A
@ ^ [X: A] : X
@ ( top_top @ ( set @ A ) ) ) ) ).
% sorted_list_of_set.inj_on
thf(fact_6907_finite__Collect,axiom,
! [A: $tType,B: $tType,S: set @ A,F2: B > A] :
( ( finite_finite2 @ A @ S )
=> ( ( inj_on @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) )
=> ( finite_finite2 @ B
@ ( collect @ B
@ ^ [A4: B] : ( member @ A @ ( F2 @ A4 ) @ S ) ) ) ) ) ).
% finite_Collect
thf(fact_6908_finite__inverse__image,axiom,
! [A: $tType,B: $tType,A3: set @ A,F2: B > A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( inj_on @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) )
=> ( finite_finite2 @ B
@ ( collect @ B
@ ^ [J3: B] : ( member @ A @ ( F2 @ J3 ) @ A3 ) ) ) ) ) ).
% finite_inverse_image
thf(fact_6909_prod_Oimage__eq,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: B > A,A3: set @ B] :
( ( inj_on @ B @ A @ G @ A3 )
=> ( ( groups7121269368397514597t_prod @ A @ A
@ ^ [X: A] : X
@ ( image @ B @ A @ G @ A3 ) )
= ( groups7121269368397514597t_prod @ B @ A @ G @ A3 ) ) ) ) ).
% prod.image_eq
thf(fact_6910_sum_Oimage__eq,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: B > A,A3: set @ B] :
( ( inj_on @ B @ A @ G @ A3 )
=> ( ( groups7311177749621191930dd_sum @ A @ A
@ ^ [X: A] : X
@ ( image @ B @ A @ G @ A3 ) )
= ( groups7311177749621191930dd_sum @ B @ A @ G @ A3 ) ) ) ) ).
% sum.image_eq
thf(fact_6911_inj__on__iff__surj,axiom,
! [A: $tType,B: $tType,A3: set @ A,A10: set @ B] :
( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ? [F4: A > B] :
( ( inj_on @ A @ B @ F4 @ A3 )
& ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F4 @ A3 ) @ A10 ) ) )
= ( ? [G3: B > A] :
( ( image @ B @ A @ G3 @ A10 )
= A3 ) ) ) ) ).
% inj_on_iff_surj
thf(fact_6912_inj__image__subset__iff,axiom,
! [B: $tType,A: $tType,F2: A > B,A3: set @ A,B5: set @ A] :
( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F2 @ A3 ) @ ( image @ A @ B @ F2 @ B5 ) )
= ( ord_less_eq @ ( set @ A ) @ A3 @ B5 ) ) ) ).
% inj_image_subset_iff
thf(fact_6913_endo__inj__surj,axiom,
! [A: $tType,A3: set @ A,F2: A > A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ F2 @ A3 ) @ A3 )
=> ( ( inj_on @ A @ A @ F2 @ A3 )
=> ( ( image @ A @ A @ F2 @ A3 )
= A3 ) ) ) ) ).
% endo_inj_surj
thf(fact_6914_inj__on__finite,axiom,
! [B: $tType,A: $tType,F2: A > B,A3: set @ A,B5: set @ B] :
( ( inj_on @ A @ B @ F2 @ A3 )
=> ( ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F2 @ A3 ) @ B5 )
=> ( ( finite_finite2 @ B @ B5 )
=> ( finite_finite2 @ A @ A3 ) ) ) ) ).
% inj_on_finite
thf(fact_6915_finite__surj__inj,axiom,
! [A: $tType,A3: set @ A,F2: A > A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( ord_less_eq @ ( set @ A ) @ A3 @ ( image @ A @ A @ F2 @ A3 ) )
=> ( inj_on @ A @ A @ F2 @ A3 ) ) ) ).
% finite_surj_inj
thf(fact_6916_inj__on__image__set__diff,axiom,
! [B: $tType,A: $tType,F2: A > B,C6: set @ A,A3: set @ A,B5: set @ A] :
( ( inj_on @ A @ B @ F2 @ C6 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ B5 ) @ C6 )
=> ( ( ord_less_eq @ ( set @ A ) @ B5 @ C6 )
=> ( ( image @ A @ B @ F2 @ ( minus_minus @ ( set @ A ) @ A3 @ B5 ) )
= ( minus_minus @ ( set @ B ) @ ( image @ A @ B @ F2 @ A3 ) @ ( image @ A @ B @ F2 @ B5 ) ) ) ) ) ) ).
% inj_on_image_set_diff
thf(fact_6917_the__inv__into__into,axiom,
! [B: $tType,A: $tType,F2: A > B,A3: set @ A,X2: B,B5: set @ A] :
( ( inj_on @ A @ B @ F2 @ A3 )
=> ( ( member @ B @ X2 @ ( image @ A @ B @ F2 @ A3 ) )
=> ( ( ord_less_eq @ ( set @ A ) @ A3 @ B5 )
=> ( member @ A @ ( the_inv_into @ A @ B @ A3 @ F2 @ X2 ) @ B5 ) ) ) ) ).
% the_inv_into_into
thf(fact_6918_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_6919_inj__image__Compl__subset,axiom,
! [B: $tType,A: $tType,F2: A > B,A3: set @ A] :
( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
=> ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F2 @ ( uminus_uminus @ ( set @ A ) @ A3 ) ) @ ( uminus_uminus @ ( set @ B ) @ ( image @ A @ B @ F2 @ A3 ) ) ) ) ).
% inj_image_Compl_subset
thf(fact_6920_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_6921_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_6922_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_6923_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_6924_msb__word__of__int,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X2: int] :
( ( most_s684356279273892711sb_msb @ ( word @ A ) @ ( ring_1_of_int @ ( word @ A ) @ X2 ) )
= ( bit_se5641148757651400278ts_bit @ int @ X2 @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) ) ) ) ).
% msb_word_of_int
thf(fact_6925_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_6926_word__sint__msb__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( ring_1_signed @ A @ int )
= ( ^ [X: word @ A] : ( minus_minus @ int @ ( semiring_1_unsigned @ A @ int @ X ) @ ( if @ int @ ( most_s684356279273892711sb_msb @ ( word @ A ) @ X ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( size_size @ ( word @ A ) @ X ) ) @ ( zero_zero @ int ) ) ) ) ) ) ).
% word_sint_msb_eq
thf(fact_6927_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_6928_inj__on__map__inv__f,axiom,
! [B: $tType,A: $tType,L2: list @ A,A3: set @ A,F2: A > B] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ L2 ) @ A3 )
=> ( ( inj_on @ A @ B @ F2 @ A3 )
=> ( ( map @ B @ A @ ( inv_on @ A @ B @ F2 @ A3 ) @ ( map @ A @ B @ F2 @ L2 ) )
= L2 ) ) ) ).
% inj_on_map_inv_f
thf(fact_6929_Schroeder__Bernstein,axiom,
! [A: $tType,B: $tType,F2: A > B,A3: set @ A,B5: set @ B,G: B > A] :
( ( inj_on @ A @ B @ F2 @ A3 )
=> ( ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F2 @ A3 ) @ B5 )
=> ( ( inj_on @ B @ A @ G @ B5 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ G @ B5 ) @ A3 )
=> ? [H5: A > B] : ( bij_betw @ A @ B @ H5 @ A3 @ B5 ) ) ) ) ) ).
% Schroeder_Bernstein
thf(fact_6930_inj__split__Cons,axiom,
! [A: $tType,X6: set @ ( product_prod @ ( list @ A ) @ A )] :
( inj_on @ ( product_prod @ ( list @ A ) @ A ) @ ( list @ A )
@ ( product_case_prod @ ( list @ A ) @ A @ ( list @ A )
@ ^ [Xs: list @ A,N3: A] : ( cons @ A @ N3 @ Xs ) )
@ X6 ) ).
% inj_split_Cons
thf(fact_6931_swap__inj__on,axiom,
! [B: $tType,A: $tType,A3: set @ ( product_prod @ A @ B )] :
( inj_on @ ( product_prod @ A @ B ) @ ( product_prod @ B @ A )
@ ( product_case_prod @ A @ B @ ( product_prod @ B @ A )
@ ^ [I3: A,J3: B] : ( product_Pair @ B @ A @ J3 @ I3 ) )
@ A3 ) ).
% swap_inj_on
thf(fact_6932_inj__on__diff__nat,axiom,
! [N5: set @ nat,K: nat] :
( ! [N2: nat] :
( ( member @ nat @ N2 @ N5 )
=> ( ord_less_eq @ nat @ K @ N2 ) )
=> ( inj_on @ nat @ nat
@ ^ [N3: nat] : ( minus_minus @ nat @ N3 @ K )
@ N5 ) ) ).
% inj_on_diff_nat
thf(fact_6933_inj__Suc,axiom,
! [N5: set @ nat] : ( inj_on @ nat @ nat @ suc @ N5 ) ).
% inj_Suc
thf(fact_6934_inj__on__of__nat,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [N5: set @ nat] : ( inj_on @ nat @ A @ ( semiring_1_of_nat @ A ) @ N5 ) ) ).
% inj_on_of_nat
thf(fact_6935_inj__Pair_I2_J,axiom,
! [B: $tType,A: $tType,C2: A > B,S: set @ A] :
( inj_on @ A @ ( product_prod @ B @ A )
@ ^ [X: A] : ( product_Pair @ B @ A @ ( C2 @ X ) @ X )
@ S ) ).
% inj_Pair(2)
thf(fact_6936_inj__Pair_I1_J,axiom,
! [B: $tType,A: $tType,C2: A > B,S: set @ A] :
( inj_on @ A @ ( product_prod @ A @ B )
@ ^ [X: A] : ( product_Pair @ A @ B @ X @ ( C2 @ X ) )
@ S ) ).
% inj_Pair(1)
thf(fact_6937_inj__on__convol__ident,axiom,
! [B: $tType,A: $tType,F2: A > B,X6: set @ A] :
( inj_on @ A @ ( product_prod @ A @ B )
@ ^ [X: A] : ( product_Pair @ A @ B @ X @ ( F2 @ X ) )
@ X6 ) ).
% inj_on_convol_ident
thf(fact_6938_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_6939_inj__singleton,axiom,
! [A: $tType,A3: set @ A] :
( inj_on @ A @ ( set @ A )
@ ^ [X: A] : ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) )
@ A3 ) ).
% inj_singleton
thf(fact_6940_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_6941_signed_Osorted__list__of__set_Oinj__on,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( inj_on @ ( word @ A ) @ ( word @ A )
@ ^ [X: word @ A] : X
@ ( top_top @ ( set @ ( word @ A ) ) ) ) ) ).
% signed.sorted_list_of_set.inj_on
thf(fact_6942_inj__graph,axiom,
! [B: $tType,A: $tType] :
( inj_on @ ( A > B ) @ ( set @ ( product_prod @ A @ B ) )
@ ^ [F4: A > B] :
( collect @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [X: A,Y: B] :
( Y
= ( F4 @ X ) ) ) )
@ ( top_top @ ( set @ ( A > B ) ) ) ) ).
% inj_graph
thf(fact_6943_le__rel__bool__arg__iff,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( ord_less_eq @ ( $o > A ) )
= ( ^ [X7: $o > A,Y9: $o > A] :
( ( ord_less_eq @ A @ ( X7 @ $false ) @ ( Y9 @ $false ) )
& ( ord_less_eq @ A @ ( X7 @ $true ) @ ( Y9 @ $true ) ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_6944_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
@ ^ [I3: nat] : ( ord_less @ nat @ I3 @ N2 ) ) ) )
& ( inj_on @ nat @ A @ F5
@ ( collect @ nat
@ ^ [I3: nat] : ( ord_less @ nat @ I3 @ N2 ) ) ) ) ) ).
% finite_imp_nat_seg_image_inj_on
thf(fact_6945_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
@ ^ [I3: nat] : ( ord_less @ nat @ I3 @ N2 ) ) )
& ( inj_on @ A @ nat @ F5 @ A3 ) ) ) ).
% finite_imp_inj_to_nat_seg
thf(fact_6946_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
@ ^ [I3: nat] : ( ord_less @ nat @ I3 @ N2 ) ) )
=> ~ ( inj_on @ A @ nat @ F5 @ A3 ) ) ) ).
% finite_imp_inj_to_nat_seg'
thf(fact_6947_infinite__iff__countable__subset,axiom,
! [A: $tType,S: set @ A] :
( ( ~ ( finite_finite2 @ A @ S ) )
= ( ? [F4: nat > A] :
( ( inj_on @ nat @ A @ F4 @ ( top_top @ ( set @ nat ) ) )
& ( ord_less_eq @ ( set @ A ) @ ( image @ nat @ A @ F4 @ ( top_top @ ( set @ nat ) ) ) @ S ) ) ) ) ).
% infinite_iff_countable_subset
thf(fact_6948_infinite__countable__subset,axiom,
! [A: $tType,S: set @ A] :
( ~ ( finite_finite2 @ A @ S )
=> ? [F5: nat > A] :
( ( inj_on @ nat @ A @ F5 @ ( top_top @ ( set @ nat ) ) )
& ( ord_less_eq @ ( set @ A ) @ ( image @ nat @ A @ F5 @ ( top_top @ ( set @ nat ) ) ) @ S ) ) ) ).
% infinite_countable_subset
thf(fact_6949_inv__on__def,axiom,
! [B: $tType,A: $tType] :
( ( inv_on @ A @ B )
= ( ^ [F4: A > B,A7: set @ A,X: B] :
( fChoice @ A
@ ^ [Y: A] :
( ( member @ A @ Y @ A7 )
& ( ( F4 @ Y )
= X ) ) ) ) ) ).
% inv_on_def
thf(fact_6950_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_6951_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_6952_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_6953_inj__sgn__power,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( inj_on @ real @ real
@ ^ [Y: real] : ( times_times @ real @ ( sgn_sgn @ real @ Y ) @ ( power_power @ real @ ( abs_abs @ real @ Y ) @ N ) )
@ ( top_top @ ( set @ real ) ) ) ) ).
% inj_sgn_power
thf(fact_6954_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_6955_ex__subset__image__inj,axiom,
! [A: $tType,B: $tType,F2: B > A,S: set @ B,P: ( set @ A ) > $o] :
( ( ? [T6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ T6 @ ( image @ B @ A @ F2 @ S ) )
& ( P @ T6 ) ) )
= ( ? [T6: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ T6 @ S )
& ( inj_on @ B @ A @ F2 @ T6 )
& ( P @ ( image @ B @ A @ F2 @ T6 ) ) ) ) ) ).
% ex_subset_image_inj
thf(fact_6956_all__subset__image__inj,axiom,
! [A: $tType,B: $tType,F2: B > A,S: set @ B,P: ( set @ A ) > $o] :
( ( ! [T6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ T6 @ ( image @ B @ A @ F2 @ S ) )
=> ( P @ T6 ) ) )
= ( ! [T6: set @ B] :
( ( ( ord_less_eq @ ( set @ B ) @ T6 @ S )
& ( inj_on @ B @ A @ F2 @ T6 ) )
=> ( P @ ( image @ B @ A @ F2 @ T6 ) ) ) ) ) ).
% all_subset_image_inj
thf(fact_6957_empty__natural,axiom,
! [C: $tType,B: $tType,D6: $tType,A: $tType,F2: A > C,G: D6 > B] :
( ( comp @ C @ ( set @ B ) @ A
@ ^ [Uu3: C] : ( bot_bot @ ( set @ B ) )
@ F2 )
= ( comp @ ( set @ D6 ) @ ( set @ B ) @ A @ ( image @ D6 @ B @ G )
@ ^ [Uu3: A] : ( bot_bot @ ( set @ D6 ) ) ) ) ).
% empty_natural
thf(fact_6958_DERIV__even__real__root,axiom,
! [N: nat,X2: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( ord_less @ real @ X2 @ ( zero_zero @ real ) )
=> ( has_field_derivative @ real @ ( root @ N ) @ ( inverse_inverse @ real @ ( times_times @ real @ ( uminus_uminus @ real @ ( semiring_1_of_nat @ real @ N ) ) @ ( power_power @ real @ ( root @ N @ X2 ) @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ).
% DERIV_even_real_root
thf(fact_6959_DERIV__subset,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,F7: A,X2: A,S3: set @ A,T2: set @ A] :
( ( has_field_derivative @ A @ F2 @ F7 @ ( topolo174197925503356063within @ A @ X2 @ S3 ) )
=> ( ( ord_less_eq @ ( set @ A ) @ T2 @ S3 )
=> ( has_field_derivative @ A @ F2 @ F7 @ ( topolo174197925503356063within @ A @ X2 @ T2 ) ) ) ) ) ).
% DERIV_subset
thf(fact_6960_has__field__derivative__subset,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,Y2: A,X2: A,S3: set @ A,T2: set @ A] :
( ( has_field_derivative @ A @ F2 @ Y2 @ ( topolo174197925503356063within @ A @ X2 @ S3 ) )
=> ( ( ord_less_eq @ ( set @ A ) @ T2 @ S3 )
=> ( has_field_derivative @ A @ F2 @ Y2 @ ( topolo174197925503356063within @ A @ X2 @ T2 ) ) ) ) ) ).
% has_field_derivative_subset
thf(fact_6961_at__le,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [S3: set @ A,T2: set @ A,X2: A] :
( ( ord_less_eq @ ( set @ A ) @ S3 @ T2 )
=> ( ord_less_eq @ ( filter @ A ) @ ( topolo174197925503356063within @ A @ X2 @ S3 ) @ ( topolo174197925503356063within @ A @ X2 @ T2 ) ) ) ) ).
% at_le
thf(fact_6962_field__differentiable__minus,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,F7: A,F3: filter @ A] :
( ( has_field_derivative @ A @ F2 @ F7 @ F3 )
=> ( has_field_derivative @ A
@ ^ [Z5: A] : ( uminus_uminus @ A @ ( F2 @ Z5 ) )
@ ( uminus_uminus @ A @ F7 )
@ F3 ) ) ) ).
% field_differentiable_minus
thf(fact_6963_DERIV__minus,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,D4: A,X2: A,S3: set @ A] :
( ( has_field_derivative @ A @ F2 @ D4 @ ( topolo174197925503356063within @ A @ X2 @ S3 ) )
=> ( has_field_derivative @ A
@ ^ [X: A] : ( uminus_uminus @ A @ ( F2 @ X ) )
@ ( uminus_uminus @ A @ D4 )
@ ( topolo174197925503356063within @ A @ X2 @ S3 ) ) ) ) ).
% DERIV_minus
thf(fact_6964_DERIV__diff,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,D4: A,X2: A,S3: set @ A,G: A > A,E5: A] :
( ( has_field_derivative @ A @ F2 @ D4 @ ( topolo174197925503356063within @ A @ X2 @ S3 ) )
=> ( ( has_field_derivative @ A @ G @ E5 @ ( topolo174197925503356063within @ A @ X2 @ S3 ) )
=> ( has_field_derivative @ A
@ ^ [X: A] : ( minus_minus @ A @ ( F2 @ X ) @ ( G @ X ) )
@ ( minus_minus @ A @ D4 @ E5 )
@ ( topolo174197925503356063within @ A @ X2 @ S3 ) ) ) ) ) ).
% DERIV_diff
thf(fact_6965_field__differentiable__diff,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,F7: A,F3: filter @ A,G: A > A,G4: A] :
( ( has_field_derivative @ A @ F2 @ F7 @ F3 )
=> ( ( has_field_derivative @ A @ G @ G4 @ F3 )
=> ( has_field_derivative @ A
@ ^ [Z5: A] : ( minus_minus @ A @ ( F2 @ Z5 ) @ ( G @ Z5 ) )
@ ( minus_minus @ A @ F7 @ G4 )
@ F3 ) ) ) ) ).
% field_differentiable_diff
thf(fact_6966_has__field__derivative__scaleR__right,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,D4: A,F3: filter @ A,C2: real] :
( ( has_field_derivative @ A @ F2 @ D4 @ F3 )
=> ( has_field_derivative @ A
@ ^ [X: A] : ( real_V8093663219630862766scaleR @ A @ C2 @ ( F2 @ X ) )
@ ( real_V8093663219630862766scaleR @ A @ C2 @ D4 )
@ F3 ) ) ) ).
% has_field_derivative_scaleR_right
thf(fact_6967_DERIV__ident,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: filter @ A] :
( has_field_derivative @ A
@ ^ [X: A] : X
@ ( one_one @ A )
@ F3 ) ) ).
% DERIV_ident
thf(fact_6968_DERIV__const,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [K: A,F3: filter @ A] :
( has_field_derivative @ A
@ ^ [X: A] : K
@ ( zero_zero @ A )
@ F3 ) ) ).
% DERIV_const
thf(fact_6969_DERIV__add,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,D4: A,X2: A,S3: set @ A,G: A > A,E5: A] :
( ( has_field_derivative @ A @ F2 @ D4 @ ( topolo174197925503356063within @ A @ X2 @ S3 ) )
=> ( ( has_field_derivative @ A @ G @ E5 @ ( topolo174197925503356063within @ A @ X2 @ S3 ) )
=> ( has_field_derivative @ A
@ ^ [X: A] : ( plus_plus @ A @ ( F2 @ X ) @ ( G @ X ) )
@ ( plus_plus @ A @ D4 @ E5 )
@ ( topolo174197925503356063within @ A @ X2 @ S3 ) ) ) ) ) ).
% DERIV_add
thf(fact_6970_field__differentiable__add,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,F7: A,F3: filter @ A,G: A > A,G4: A] :
( ( has_field_derivative @ A @ F2 @ F7 @ F3 )
=> ( ( has_field_derivative @ A @ G @ G4 @ F3 )
=> ( has_field_derivative @ A
@ ^ [Z5: A] : ( plus_plus @ A @ ( F2 @ Z5 ) @ ( G @ Z5 ) )
@ ( plus_plus @ A @ F7 @ G4 )
@ F3 ) ) ) ) ).
% field_differentiable_add
thf(fact_6971_DERIV__divide,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,D4: A,X2: A,S3: set @ A,G: A > A,E5: A] :
( ( has_field_derivative @ A @ F2 @ D4 @ ( topolo174197925503356063within @ A @ X2 @ S3 ) )
=> ( ( has_field_derivative @ A @ G @ E5 @ ( topolo174197925503356063within @ A @ X2 @ S3 ) )
=> ( ( ( G @ X2 )
!= ( zero_zero @ A ) )
=> ( has_field_derivative @ A
@ ^ [X: A] : ( divide_divide @ A @ ( F2 @ X ) @ ( G @ X ) )
@ ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ D4 @ ( G @ X2 ) ) @ ( times_times @ A @ ( F2 @ X2 ) @ E5 ) ) @ ( times_times @ A @ ( G @ X2 ) @ ( G @ X2 ) ) )
@ ( topolo174197925503356063within @ A @ X2 @ S3 ) ) ) ) ) ) ).
% DERIV_divide
thf(fact_6972_DERIV__mult_H,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,D4: A,X2: A,S3: set @ A,G: A > A,E5: A] :
( ( has_field_derivative @ A @ F2 @ D4 @ ( topolo174197925503356063within @ A @ X2 @ S3 ) )
=> ( ( has_field_derivative @ A @ G @ E5 @ ( topolo174197925503356063within @ A @ X2 @ S3 ) )
=> ( has_field_derivative @ A
@ ^ [X: A] : ( times_times @ A @ ( F2 @ X ) @ ( G @ X ) )
@ ( plus_plus @ A @ ( times_times @ A @ ( F2 @ X2 ) @ E5 ) @ ( times_times @ A @ D4 @ ( G @ X2 ) ) )
@ ( topolo174197925503356063within @ A @ X2 @ S3 ) ) ) ) ) ).
% DERIV_mult'
thf(fact_6973_DERIV__mult,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,Da: A,X2: A,S3: set @ A,G: A > A,Db: A] :
( ( has_field_derivative @ A @ F2 @ Da @ ( topolo174197925503356063within @ A @ X2 @ S3 ) )
=> ( ( has_field_derivative @ A @ G @ Db @ ( topolo174197925503356063within @ A @ X2 @ S3 ) )
=> ( has_field_derivative @ A
@ ^ [X: A] : ( times_times @ A @ ( F2 @ X ) @ ( G @ X ) )
@ ( plus_plus @ A @ ( times_times @ A @ Da @ ( G @ X2 ) ) @ ( times_times @ A @ Db @ ( F2 @ X2 ) ) )
@ ( topolo174197925503356063within @ A @ X2 @ S3 ) ) ) ) ) ).
% DERIV_mult
thf(fact_6974_has__field__derivative__cosh,axiom,
! [A13: $tType] :
( ( ( real_Vector_banach @ A13 )
& ( real_V3459762299906320749_field @ A13 ) )
=> ! [G: A13 > A13,Db: A13,X2: A13,S3: set @ A13] :
( ( has_field_derivative @ A13 @ G @ Db @ ( topolo174197925503356063within @ A13 @ X2 @ S3 ) )
=> ( has_field_derivative @ A13
@ ^ [X: A13] : ( cosh @ A13 @ ( G @ X ) )
@ ( times_times @ A13 @ ( sinh @ A13 @ ( G @ X2 ) ) @ Db )
@ ( topolo174197925503356063within @ A13 @ X2 @ S3 ) ) ) ) ).
% has_field_derivative_cosh
thf(fact_6975_has__field__derivative__sinh,axiom,
! [A13: $tType] :
( ( ( real_Vector_banach @ A13 )
& ( real_V3459762299906320749_field @ A13 ) )
=> ! [G: A13 > A13,Db: A13,X2: A13,S3: set @ A13] :
( ( has_field_derivative @ A13 @ G @ Db @ ( topolo174197925503356063within @ A13 @ X2 @ S3 ) )
=> ( has_field_derivative @ A13
@ ^ [X: A13] : ( sinh @ A13 @ ( G @ X ) )
@ ( times_times @ A13 @ ( cosh @ A13 @ ( G @ X2 ) ) @ Db )
@ ( topolo174197925503356063within @ A13 @ X2 @ S3 ) ) ) ) ).
% has_field_derivative_sinh
thf(fact_6976_DERIV__cmult,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,D4: A,X2: A,S3: set @ A,C2: A] :
( ( has_field_derivative @ A @ F2 @ D4 @ ( topolo174197925503356063within @ A @ X2 @ S3 ) )
=> ( has_field_derivative @ A
@ ^ [X: A] : ( times_times @ A @ C2 @ ( F2 @ X ) )
@ ( times_times @ A @ C2 @ D4 )
@ ( topolo174197925503356063within @ A @ X2 @ S3 ) ) ) ) ).
% DERIV_cmult
thf(fact_6977_DERIV__cmult__right,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,D4: A,X2: A,S3: set @ A,C2: A] :
( ( has_field_derivative @ A @ F2 @ D4 @ ( topolo174197925503356063within @ A @ X2 @ S3 ) )
=> ( has_field_derivative @ A
@ ^ [X: A] : ( times_times @ A @ ( F2 @ X ) @ C2 )
@ ( times_times @ A @ D4 @ C2 )
@ ( topolo174197925503356063within @ A @ X2 @ S3 ) ) ) ) ).
% DERIV_cmult_right
thf(fact_6978_DERIV__cdivide,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,D4: A,X2: A,S3: set @ A,C2: A] :
( ( has_field_derivative @ A @ F2 @ D4 @ ( topolo174197925503356063within @ A @ X2 @ S3 ) )
=> ( has_field_derivative @ A
@ ^ [X: A] : ( divide_divide @ A @ ( F2 @ X ) @ C2 )
@ ( divide_divide @ A @ D4 @ C2 )
@ ( topolo174197925503356063within @ A @ X2 @ S3 ) ) ) ) ).
% DERIV_cdivide
thf(fact_6979_DERIV__inverse_H,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,D4: A,X2: A,S3: set @ A] :
( ( has_field_derivative @ A @ F2 @ D4 @ ( topolo174197925503356063within @ A @ X2 @ S3 ) )
=> ( ( ( F2 @ X2 )
!= ( zero_zero @ A ) )
=> ( has_field_derivative @ A
@ ^ [X: A] : ( inverse_inverse @ A @ ( F2 @ X ) )
@ ( uminus_uminus @ A @ ( times_times @ A @ ( times_times @ A @ ( inverse_inverse @ A @ ( F2 @ X2 ) ) @ D4 ) @ ( inverse_inverse @ A @ ( F2 @ X2 ) ) ) )
@ ( topolo174197925503356063within @ A @ X2 @ S3 ) ) ) ) ) ).
% DERIV_inverse'
thf(fact_6980_DERIV__at__within__shift,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,Y2: A,Z: A,X2: A,S: set @ A] :
( ( has_field_derivative @ A @ F2 @ Y2 @ ( topolo174197925503356063within @ A @ ( plus_plus @ A @ Z @ X2 ) @ ( image @ A @ A @ ( plus_plus @ A @ Z ) @ S ) ) )
= ( has_field_derivative @ A
@ ^ [X: A] : ( F2 @ ( plus_plus @ A @ Z @ X ) )
@ Y2
@ ( topolo174197925503356063within @ A @ X2 @ S ) ) ) ) ).
% DERIV_at_within_shift
thf(fact_6981_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_6982_DERIV__fun__cos,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [G: A > A,M: A,X2: A] :
( ( has_field_derivative @ A @ G @ M @ ( topolo174197925503356063within @ A @ X2 @ ( top_top @ ( set @ A ) ) ) )
=> ( has_field_derivative @ A
@ ^ [X: A] : ( cos @ A @ ( G @ X ) )
@ ( times_times @ A @ ( uminus_uminus @ A @ ( sin @ A @ ( G @ X2 ) ) ) @ M )
@ ( topolo174197925503356063within @ A @ X2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% DERIV_fun_cos
thf(fact_6983_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_6984_DERIV__chain__s,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [S3: set @ A,G: A > A,G4: A > A,F2: A > A,F7: A,X2: A] :
( ! [X3: A] :
( ( member @ A @ X3 @ S3 )
=> ( has_field_derivative @ A @ G @ ( G4 @ X3 ) @ ( topolo174197925503356063within @ A @ X3 @ ( top_top @ ( set @ A ) ) ) ) )
=> ( ( has_field_derivative @ A @ F2 @ F7 @ ( topolo174197925503356063within @ A @ X2 @ ( top_top @ ( set @ A ) ) ) )
=> ( ( member @ A @ ( F2 @ X2 ) @ S3 )
=> ( has_field_derivative @ A
@ ^ [X: A] : ( G @ ( F2 @ X ) )
@ ( times_times @ A @ F7 @ ( G4 @ ( F2 @ X2 ) ) )
@ ( topolo174197925503356063within @ A @ X2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ).
% DERIV_chain_s
thf(fact_6985_DERIV__chain3,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [G: A > A,G4: A > A,F2: A > A,F7: A,X2: A] :
( ! [X3: A] : ( has_field_derivative @ A @ G @ ( G4 @ X3 ) @ ( topolo174197925503356063within @ A @ X3 @ ( top_top @ ( set @ A ) ) ) )
=> ( ( has_field_derivative @ A @ F2 @ F7 @ ( topolo174197925503356063within @ A @ X2 @ ( top_top @ ( set @ A ) ) ) )
=> ( has_field_derivative @ A
@ ^ [X: A] : ( G @ ( F2 @ X ) )
@ ( times_times @ A @ F7 @ ( G4 @ ( F2 @ X2 ) ) )
@ ( topolo174197925503356063within @ A @ X2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% DERIV_chain3
thf(fact_6986_DERIV__chain2,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,Da: A,G: A > A,X2: A,Db: A,S3: set @ A] :
( ( has_field_derivative @ A @ F2 @ Da @ ( topolo174197925503356063within @ A @ ( G @ X2 ) @ ( top_top @ ( set @ A ) ) ) )
=> ( ( has_field_derivative @ A @ G @ Db @ ( topolo174197925503356063within @ A @ X2 @ S3 ) )
=> ( has_field_derivative @ A
@ ^ [X: A] : ( F2 @ ( G @ X ) )
@ ( times_times @ A @ Da @ Db )
@ ( topolo174197925503356063within @ A @ X2 @ S3 ) ) ) ) ) ).
% DERIV_chain2
thf(fact_6987_DERIV__chain_H,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,D4: A,X2: A,S3: set @ A,G: A > A,E5: A] :
( ( has_field_derivative @ A @ F2 @ D4 @ ( topolo174197925503356063within @ A @ X2 @ S3 ) )
=> ( ( has_field_derivative @ A @ G @ E5 @ ( topolo174197925503356063within @ A @ ( F2 @ X2 ) @ ( top_top @ ( set @ A ) ) ) )
=> ( has_field_derivative @ A
@ ^ [X: A] : ( G @ ( F2 @ X ) )
@ ( times_times @ A @ E5 @ D4 )
@ ( topolo174197925503356063within @ A @ X2 @ S3 ) ) ) ) ) ).
% DERIV_chain'
thf(fact_6988_DERIV__fun__sin,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [G: A > A,M: A,X2: A] :
( ( has_field_derivative @ A @ G @ M @ ( topolo174197925503356063within @ A @ X2 @ ( top_top @ ( set @ A ) ) ) )
=> ( has_field_derivative @ A
@ ^ [X: A] : ( sin @ A @ ( G @ X ) )
@ ( times_times @ A @ ( cos @ A @ ( G @ X2 ) ) @ M )
@ ( topolo174197925503356063within @ A @ X2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% DERIV_fun_sin
thf(fact_6989_DERIV__fun__exp,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [G: A > A,M: A,X2: A] :
( ( has_field_derivative @ A @ G @ M @ ( topolo174197925503356063within @ A @ X2 @ ( top_top @ ( set @ A ) ) ) )
=> ( has_field_derivative @ A
@ ^ [X: A] : ( exp @ A @ ( G @ X ) )
@ ( times_times @ A @ ( exp @ A @ ( G @ X2 ) ) @ M )
@ ( topolo174197925503356063within @ A @ X2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% DERIV_fun_exp
thf(fact_6990_DERIV__cos__add,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [K: A,Xa: A] :
( has_field_derivative @ A
@ ^ [X: A] : ( cos @ A @ ( plus_plus @ A @ X @ K ) )
@ ( uminus_uminus @ A @ ( sin @ A @ ( plus_plus @ A @ Xa @ K ) ) )
@ ( topolo174197925503356063within @ A @ Xa @ ( top_top @ ( set @ A ) ) ) ) ) ).
% DERIV_cos_add
thf(fact_6991_DERIV__shift,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,Y2: A,X2: A,Z: A] :
( ( has_field_derivative @ A @ F2 @ Y2 @ ( topolo174197925503356063within @ A @ ( plus_plus @ A @ X2 @ Z ) @ ( top_top @ ( set @ A ) ) ) )
= ( has_field_derivative @ A
@ ^ [X: A] : ( F2 @ ( plus_plus @ A @ X @ Z ) )
@ Y2
@ ( topolo174197925503356063within @ A @ X2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% DERIV_shift
thf(fact_6992_DERIV__neg__dec__right,axiom,
! [F2: real > real,L2: real,X2: real] :
( ( has_field_derivative @ real @ F2 @ L2 @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( ord_less @ real @ L2 @ ( zero_zero @ real ) )
=> ? [D3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D3 )
& ! [H6: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ H6 )
=> ( ( ord_less @ real @ H6 @ D3 )
=> ( ord_less @ real @ ( F2 @ ( plus_plus @ real @ X2 @ H6 ) ) @ ( F2 @ X2 ) ) ) ) ) ) ) ).
% DERIV_neg_dec_right
thf(fact_6993_DERIV__pos__inc__right,axiom,
! [F2: real > real,L2: real,X2: real] :
( ( has_field_derivative @ real @ F2 @ L2 @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ L2 )
=> ? [D3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D3 )
& ! [H6: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ H6 )
=> ( ( ord_less @ real @ H6 @ D3 )
=> ( ord_less @ real @ ( F2 @ X2 ) @ ( F2 @ ( plus_plus @ real @ X2 @ H6 ) ) ) ) ) ) ) ) ).
% DERIV_pos_inc_right
thf(fact_6994_DERIV__local__const,axiom,
! [F2: real > real,L2: real,X2: real,D2: real] :
( ( has_field_derivative @ real @ F2 @ L2 @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ D2 )
=> ( ! [Y3: real] :
( ( ord_less @ real @ ( abs_abs @ real @ ( minus_minus @ real @ X2 @ Y3 ) ) @ D2 )
=> ( ( F2 @ X2 )
= ( F2 @ Y3 ) ) )
=> ( L2
= ( zero_zero @ real ) ) ) ) ) ).
% DERIV_local_const
thf(fact_6995_DERIV__isconst__all,axiom,
! [F2: real > real,X2: real,Y2: real] :
( ! [X3: real] : ( has_field_derivative @ real @ F2 @ ( zero_zero @ real ) @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( F2 @ X2 )
= ( F2 @ Y2 ) ) ) ).
% DERIV_isconst_all
thf(fact_6996_DERIV__pos__inc__left,axiom,
! [F2: real > real,L2: real,X2: real] :
( ( has_field_derivative @ real @ F2 @ L2 @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ L2 )
=> ? [D3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D3 )
& ! [H6: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ H6 )
=> ( ( ord_less @ real @ H6 @ D3 )
=> ( ord_less @ real @ ( F2 @ ( minus_minus @ real @ X2 @ H6 ) ) @ ( F2 @ X2 ) ) ) ) ) ) ) ).
% DERIV_pos_inc_left
thf(fact_6997_DERIV__neg__dec__left,axiom,
! [F2: real > real,L2: real,X2: real] :
( ( has_field_derivative @ real @ F2 @ L2 @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( ord_less @ real @ L2 @ ( zero_zero @ real ) )
=> ? [D3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D3 )
& ! [H6: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ H6 )
=> ( ( ord_less @ real @ H6 @ D3 )
=> ( ord_less @ real @ ( F2 @ X2 ) @ ( F2 @ ( minus_minus @ real @ X2 @ H6 ) ) ) ) ) ) ) ) ).
% DERIV_neg_dec_left
thf(fact_6998_DERIV__mirror,axiom,
! [F2: real > real,Y2: real,X2: real] :
( ( has_field_derivative @ real @ F2 @ Y2 @ ( topolo174197925503356063within @ real @ ( uminus_uminus @ real @ X2 ) @ ( top_top @ ( set @ real ) ) ) )
= ( has_field_derivative @ real
@ ^ [X: real] : ( F2 @ ( uminus_uminus @ real @ X ) )
@ ( uminus_uminus @ real @ Y2 )
@ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) ) ) ).
% DERIV_mirror
thf(fact_6999_DERIV__ln,axiom,
! [X2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( has_field_derivative @ real @ ( ln_ln @ real ) @ ( inverse_inverse @ real @ X2 ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) ) ) ).
% DERIV_ln
thf(fact_7000_MVT2,axiom,
! [A2: real,B2: real,F2: real > real,F7: 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 @ ( F7 @ X3 ) @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) ) ) )
=> ? [Z2: real] :
( ( ord_less @ real @ A2 @ Z2 )
& ( ord_less @ real @ Z2 @ B2 )
& ( ( minus_minus @ real @ ( F2 @ B2 ) @ ( F2 @ A2 ) )
= ( times_times @ real @ ( minus_minus @ real @ B2 @ A2 ) @ ( F7 @ Z2 ) ) ) ) ) ) ).
% MVT2
thf(fact_7001_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_7002_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_7003_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_7004_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_7005_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_7006_DERIV__sum,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( real_V3459762299906320749_field @ B )
=> ! [S: set @ A,F2: B > A > B,F7: C > A > B,X2: C,F3: filter @ B] :
( ! [N2: A] :
( ( member @ A @ N2 @ S )
=> ( has_field_derivative @ B
@ ^ [X: B] : ( F2 @ X @ N2 )
@ ( F7 @ X2 @ N2 )
@ F3 ) )
=> ( has_field_derivative @ B
@ ^ [X: B] : ( groups7311177749621191930dd_sum @ A @ B @ ( F2 @ X ) @ S )
@ ( groups7311177749621191930dd_sum @ A @ B @ ( F7 @ X2 ) @ S )
@ F3 ) ) ) ).
% DERIV_sum
thf(fact_7007_DERIV__chain,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,Da: A,G: A > A,X2: A,Db: A,S3: set @ A] :
( ( has_field_derivative @ A @ F2 @ Da @ ( topolo174197925503356063within @ A @ ( G @ X2 ) @ ( top_top @ ( set @ A ) ) ) )
=> ( ( has_field_derivative @ A @ G @ Db @ ( topolo174197925503356063within @ A @ X2 @ S3 ) )
=> ( has_field_derivative @ A @ ( comp @ A @ A @ A @ F2 @ G ) @ ( times_times @ A @ Da @ Db ) @ ( topolo174197925503356063within @ A @ X2 @ S3 ) ) ) ) ) ).
% DERIV_chain
thf(fact_7008_DERIV__image__chain,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,Da: A,G: A > A,X2: A,S3: set @ A,Db: A] :
( ( has_field_derivative @ A @ F2 @ Da @ ( topolo174197925503356063within @ A @ ( G @ X2 ) @ ( image @ A @ A @ G @ S3 ) ) )
=> ( ( has_field_derivative @ A @ G @ Db @ ( topolo174197925503356063within @ A @ X2 @ S3 ) )
=> ( has_field_derivative @ A @ ( comp @ A @ A @ A @ F2 @ G ) @ ( times_times @ A @ Da @ Db ) @ ( topolo174197925503356063within @ A @ X2 @ S3 ) ) ) ) ) ).
% DERIV_image_chain
thf(fact_7009_DERIV__at__within__shift__lemma,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,Y2: A,Z: A,X2: A,S: set @ A] :
( ( has_field_derivative @ A @ F2 @ Y2 @ ( topolo174197925503356063within @ A @ ( plus_plus @ A @ Z @ X2 ) @ ( image @ A @ A @ ( plus_plus @ A @ Z ) @ S ) ) )
=> ( has_field_derivative @ A @ ( comp @ A @ A @ A @ F2 @ ( plus_plus @ A @ Z ) ) @ Y2 @ ( topolo174197925503356063within @ A @ X2 @ S ) ) ) ) ).
% DERIV_at_within_shift_lemma
thf(fact_7010_DERIV__power__Suc,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,D4: A,X2: A,S3: set @ A,N: nat] :
( ( has_field_derivative @ A @ F2 @ D4 @ ( topolo174197925503356063within @ A @ X2 @ S3 ) )
=> ( has_field_derivative @ A
@ ^ [X: A] : ( power_power @ A @ ( F2 @ X ) @ ( suc @ N ) )
@ ( times_times @ A @ ( plus_plus @ A @ ( one_one @ A ) @ ( semiring_1_of_nat @ A @ N ) ) @ ( times_times @ A @ D4 @ ( power_power @ A @ ( F2 @ X2 ) @ N ) ) )
@ ( topolo174197925503356063within @ A @ X2 @ S3 ) ) ) ) ).
% DERIV_power_Suc
thf(fact_7011_DERIV__cmult__Id,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [C2: A,X2: A,S3: set @ A] : ( has_field_derivative @ A @ ( times_times @ A @ C2 ) @ C2 @ ( topolo174197925503356063within @ A @ X2 @ S3 ) ) ) ).
% DERIV_cmult_Id
thf(fact_7012_has__real__derivative__neg__dec__right,axiom,
! [F2: real > real,L2: real,X2: real,S: set @ real] :
( ( has_field_derivative @ real @ F2 @ L2 @ ( topolo174197925503356063within @ real @ X2 @ S ) )
=> ( ( ord_less @ real @ L2 @ ( zero_zero @ real ) )
=> ? [D3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D3 )
& ! [H6: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ H6 )
=> ( ( member @ real @ ( plus_plus @ real @ X2 @ H6 ) @ S )
=> ( ( ord_less @ real @ H6 @ D3 )
=> ( ord_less @ real @ ( F2 @ ( plus_plus @ real @ X2 @ H6 ) ) @ ( F2 @ X2 ) ) ) ) ) ) ) ) ).
% has_real_derivative_neg_dec_right
thf(fact_7013_has__real__derivative__pos__inc__right,axiom,
! [F2: real > real,L2: real,X2: real,S: set @ real] :
( ( has_field_derivative @ real @ F2 @ L2 @ ( topolo174197925503356063within @ real @ X2 @ S ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ L2 )
=> ? [D3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D3 )
& ! [H6: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ H6 )
=> ( ( member @ real @ ( plus_plus @ real @ X2 @ H6 ) @ S )
=> ( ( ord_less @ real @ H6 @ D3 )
=> ( ord_less @ real @ ( F2 @ X2 ) @ ( F2 @ ( plus_plus @ real @ X2 @ H6 ) ) ) ) ) ) ) ) ) ).
% has_real_derivative_pos_inc_right
thf(fact_7014_has__real__derivative__pos__inc__left,axiom,
! [F2: real > real,L2: real,X2: real,S: set @ real] :
( ( has_field_derivative @ real @ F2 @ L2 @ ( topolo174197925503356063within @ real @ X2 @ S ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ L2 )
=> ? [D3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D3 )
& ! [H6: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ H6 )
=> ( ( member @ real @ ( minus_minus @ real @ X2 @ H6 ) @ S )
=> ( ( ord_less @ real @ H6 @ D3 )
=> ( ord_less @ real @ ( F2 @ ( minus_minus @ real @ X2 @ H6 ) ) @ ( F2 @ X2 ) ) ) ) ) ) ) ) ).
% has_real_derivative_pos_inc_left
thf(fact_7015_has__real__derivative__neg__dec__left,axiom,
! [F2: real > real,L2: real,X2: real,S: set @ real] :
( ( has_field_derivative @ real @ F2 @ L2 @ ( topolo174197925503356063within @ real @ X2 @ S ) )
=> ( ( ord_less @ real @ L2 @ ( zero_zero @ real ) )
=> ? [D3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D3 )
& ! [H6: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ H6 )
=> ( ( member @ real @ ( minus_minus @ real @ X2 @ H6 ) @ S )
=> ( ( ord_less @ real @ H6 @ D3 )
=> ( ord_less @ real @ ( F2 @ X2 ) @ ( F2 @ ( minus_minus @ real @ X2 @ H6 ) ) ) ) ) ) ) ) ) ).
% has_real_derivative_neg_dec_left
thf(fact_7016_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_7017_DERIV__power,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,D4: A,X2: A,S3: set @ A,N: nat] :
( ( has_field_derivative @ A @ F2 @ D4 @ ( topolo174197925503356063within @ A @ X2 @ S3 ) )
=> ( has_field_derivative @ A
@ ^ [X: A] : ( power_power @ A @ ( F2 @ X ) @ N )
@ ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ ( times_times @ A @ D4 @ ( power_power @ A @ ( F2 @ X2 ) @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) ) ) )
@ ( topolo174197925503356063within @ A @ X2 @ S3 ) ) ) ) ).
% DERIV_power
thf(fact_7018_DERIV__inverse,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [X2: A,S3: set @ A] :
( ( X2
!= ( zero_zero @ A ) )
=> ( has_field_derivative @ A @ ( inverse_inverse @ A ) @ ( uminus_uminus @ A @ ( power_power @ A @ ( inverse_inverse @ A @ X2 ) @ ( suc @ ( suc @ ( zero_zero @ nat ) ) ) ) ) @ ( topolo174197925503356063within @ A @ X2 @ S3 ) ) ) ) ).
% DERIV_inverse
thf(fact_7019_DERIV__local__max,axiom,
! [F2: real > real,L2: real,X2: real,D2: real] :
( ( has_field_derivative @ real @ F2 @ L2 @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ D2 )
=> ( ! [Y3: real] :
( ( ord_less @ real @ ( abs_abs @ real @ ( minus_minus @ real @ X2 @ Y3 ) ) @ D2 )
=> ( ord_less_eq @ real @ ( F2 @ Y3 ) @ ( F2 @ X2 ) ) )
=> ( L2
= ( zero_zero @ real ) ) ) ) ) ).
% DERIV_local_max
thf(fact_7020_DERIV__local__min,axiom,
! [F2: real > real,L2: real,X2: real,D2: real] :
( ( has_field_derivative @ real @ F2 @ L2 @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ D2 )
=> ( ! [Y3: real] :
( ( ord_less @ real @ ( abs_abs @ real @ ( minus_minus @ real @ X2 @ Y3 ) ) @ D2 )
=> ( ord_less_eq @ real @ ( F2 @ X2 ) @ ( F2 @ Y3 ) ) )
=> ( L2
= ( zero_zero @ real ) ) ) ) ) ).
% DERIV_local_min
thf(fact_7021_DERIV__ln__divide,axiom,
! [X2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( has_field_derivative @ real @ ( ln_ln @ real ) @ ( divide_divide @ real @ ( one_one @ real ) @ X2 ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) ) ) ).
% DERIV_ln_divide
thf(fact_7022_DERIV__pow,axiom,
! [N: nat,X2: real,S3: set @ real] :
( has_field_derivative @ real
@ ^ [X: real] : ( power_power @ real @ X @ N )
@ ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ ( power_power @ real @ X2 @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) ) )
@ ( topolo174197925503356063within @ real @ X2 @ S3 ) ) ).
% DERIV_pow
thf(fact_7023_termdiffs__strong__converges__everywhere,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [C2: nat > A,X2: A] :
( ! [Y3: A] :
( summable @ A
@ ^ [N3: nat] : ( times_times @ A @ ( C2 @ N3 ) @ ( power_power @ A @ Y3 @ N3 ) ) )
=> ( has_field_derivative @ A
@ ^ [X: A] :
( suminf @ A
@ ^ [N3: nat] : ( times_times @ A @ ( C2 @ N3 ) @ ( power_power @ A @ X @ N3 ) ) )
@ ( suminf @ A
@ ^ [N3: nat] : ( times_times @ A @ ( diffs @ A @ C2 @ N3 ) @ ( power_power @ A @ X2 @ N3 ) ) )
@ ( topolo174197925503356063within @ A @ X2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% termdiffs_strong_converges_everywhere
thf(fact_7024_DERIV__fun__pow,axiom,
! [G: real > real,M: real,X2: real,N: nat] :
( ( has_field_derivative @ real @ G @ M @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
=> ( has_field_derivative @ real
@ ^ [X: real] : ( power_power @ real @ ( G @ X ) @ N )
@ ( times_times @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ ( power_power @ real @ ( G @ X2 ) @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) @ M )
@ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) ) ) ).
% DERIV_fun_pow
thf(fact_7025_DERIV__quotient,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,D2: A,X2: A,S3: set @ A,G: A > A,E: A] :
( ( has_field_derivative @ A @ F2 @ D2 @ ( topolo174197925503356063within @ A @ X2 @ S3 ) )
=> ( ( has_field_derivative @ A @ G @ E @ ( topolo174197925503356063within @ A @ X2 @ S3 ) )
=> ( ( ( G @ X2 )
!= ( zero_zero @ A ) )
=> ( has_field_derivative @ A
@ ^ [Y: A] : ( divide_divide @ A @ ( F2 @ Y ) @ ( G @ Y ) )
@ ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ D2 @ ( G @ X2 ) ) @ ( times_times @ A @ E @ ( F2 @ X2 ) ) ) @ ( power_power @ A @ ( G @ X2 ) @ ( suc @ ( suc @ ( zero_zero @ nat ) ) ) ) )
@ ( topolo174197925503356063within @ A @ X2 @ S3 ) ) ) ) ) ) ).
% DERIV_quotient
thf(fact_7026_DERIV__inverse__fun,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,D2: A,X2: A,S3: set @ A] :
( ( has_field_derivative @ A @ F2 @ D2 @ ( topolo174197925503356063within @ A @ X2 @ S3 ) )
=> ( ( ( F2 @ X2 )
!= ( zero_zero @ A ) )
=> ( has_field_derivative @ A
@ ^ [X: A] : ( inverse_inverse @ A @ ( F2 @ X ) )
@ ( uminus_uminus @ A @ ( times_times @ A @ D2 @ ( inverse_inverse @ A @ ( power_power @ A @ ( F2 @ X2 ) @ ( suc @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) )
@ ( topolo174197925503356063within @ A @ X2 @ S3 ) ) ) ) ) ).
% DERIV_inverse_fun
thf(fact_7027_termdiffs__sums__strong,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [K6: real,C2: nat > A,F2: A > A,F7: A,Z: A] :
( ! [Z2: A] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ Z2 ) @ K6 )
=> ( sums @ A
@ ^ [N3: nat] : ( times_times @ A @ ( C2 @ N3 ) @ ( power_power @ A @ Z2 @ N3 ) )
@ ( F2 @ Z2 ) ) )
=> ( ( has_field_derivative @ A @ F2 @ F7 @ ( topolo174197925503356063within @ A @ Z @ ( top_top @ ( set @ A ) ) ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ Z ) @ K6 )
=> ( sums @ A
@ ^ [N3: nat] : ( times_times @ A @ ( diffs @ A @ C2 @ N3 ) @ ( power_power @ A @ Z @ N3 ) )
@ F7 ) ) ) ) ) ).
% termdiffs_sums_strong
thf(fact_7028_has__real__derivative__powr,axiom,
! [Z: real,R2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ Z )
=> ( has_field_derivative @ real
@ ^ [Z5: real] : ( powr @ real @ Z5 @ 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_7029_termdiffs,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [C2: nat > A,K6: A,X2: 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 @ X2 ) @ ( real_V7770717601297561774m_norm @ A @ K6 ) )
=> ( has_field_derivative @ A
@ ^ [X: A] :
( suminf @ A
@ ^ [N3: nat] : ( times_times @ A @ ( C2 @ N3 ) @ ( power_power @ A @ X @ N3 ) ) )
@ ( suminf @ A
@ ^ [N3: nat] : ( times_times @ A @ ( diffs @ A @ C2 @ N3 ) @ ( power_power @ A @ X2 @ N3 ) ) )
@ ( topolo174197925503356063within @ A @ X2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ) ).
% termdiffs
thf(fact_7030_termdiffs__strong,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [C2: nat > A,K6: A,X2: A] :
( ( summable @ A
@ ^ [N3: nat] : ( times_times @ A @ ( C2 @ N3 ) @ ( power_power @ A @ K6 @ N3 ) ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X2 ) @ ( real_V7770717601297561774m_norm @ A @ K6 ) )
=> ( has_field_derivative @ A
@ ^ [X: A] :
( suminf @ A
@ ^ [N3: nat] : ( times_times @ A @ ( C2 @ N3 ) @ ( power_power @ A @ X @ N3 ) ) )
@ ( suminf @ A
@ ^ [N3: nat] : ( times_times @ A @ ( diffs @ A @ C2 @ N3 ) @ ( power_power @ A @ X2 @ N3 ) ) )
@ ( topolo174197925503356063within @ A @ X2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% termdiffs_strong
thf(fact_7031_termdiffs__strong_H,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [K6: real,C2: nat > A,Z: A] :
( ! [Z2: A] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ Z2 ) @ K6 )
=> ( summable @ A
@ ^ [N3: nat] : ( times_times @ A @ ( C2 @ N3 ) @ ( power_power @ A @ Z2 @ N3 ) ) ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ Z ) @ K6 )
=> ( has_field_derivative @ A
@ ^ [Z5: A] :
( suminf @ A
@ ^ [N3: nat] : ( times_times @ A @ ( C2 @ N3 ) @ ( power_power @ A @ Z5 @ 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_7032_DERIV__fun__powr,axiom,
! [G: real > real,M: real,X2: real,R2: real] :
( ( has_field_derivative @ real @ G @ M @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( G @ X2 ) )
=> ( has_field_derivative @ real
@ ^ [X: real] : ( powr @ real @ ( G @ X ) @ R2 )
@ ( times_times @ real @ ( times_times @ real @ R2 @ ( powr @ real @ ( G @ X2 ) @ ( minus_minus @ real @ R2 @ ( semiring_1_of_nat @ real @ ( one_one @ nat ) ) ) ) ) @ M )
@ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) ) ) ) ).
% DERIV_fun_powr
thf(fact_7033_DERIV__log,axiom,
! [X2: real,B2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( has_field_derivative @ real @ ( log @ B2 ) @ ( divide_divide @ real @ ( one_one @ real ) @ ( times_times @ real @ ( ln_ln @ real @ B2 ) @ X2 ) ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) ) ) ).
% DERIV_log
thf(fact_7034_DERIV__powr,axiom,
! [G: real > real,M: real,X2: real,F2: real > real,R2: real] :
( ( has_field_derivative @ real @ G @ M @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( G @ X2 ) )
=> ( ( has_field_derivative @ real @ F2 @ R2 @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
=> ( has_field_derivative @ real
@ ^ [X: real] : ( powr @ real @ ( G @ X ) @ ( F2 @ X ) )
@ ( times_times @ real @ ( powr @ real @ ( G @ X2 ) @ ( F2 @ X2 ) ) @ ( plus_plus @ real @ ( times_times @ real @ R2 @ ( ln_ln @ real @ ( G @ X2 ) ) ) @ ( divide_divide @ real @ ( times_times @ real @ M @ ( F2 @ X2 ) ) @ ( G @ X2 ) ) ) )
@ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ).
% DERIV_powr
thf(fact_7035_DERIV__tan,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X2: A] :
( ( ( cos @ A @ X2 )
!= ( zero_zero @ A ) )
=> ( has_field_derivative @ A @ ( tan @ A ) @ ( inverse_inverse @ A @ ( power_power @ A @ ( cos @ A @ X2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( topolo174197925503356063within @ A @ X2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% DERIV_tan
thf(fact_7036_DERIV__real__sqrt,axiom,
! [X2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( has_field_derivative @ real @ sqrt @ ( divide_divide @ real @ ( inverse_inverse @ real @ ( sqrt @ X2 ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) ) ) ).
% DERIV_real_sqrt
thf(fact_7037_DERIV__arctan,axiom,
! [X2: real] : ( has_field_derivative @ real @ arctan @ ( inverse_inverse @ real @ ( plus_plus @ real @ ( one_one @ real ) @ ( power_power @ real @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) ) ).
% DERIV_arctan
thf(fact_7038_arsinh__real__has__field__derivative,axiom,
! [X2: real,A3: set @ real] : ( has_field_derivative @ real @ ( arsinh @ real ) @ ( divide_divide @ real @ ( one_one @ real ) @ ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ real ) ) ) ) @ ( topolo174197925503356063within @ real @ X2 @ A3 ) ) ).
% arsinh_real_has_field_derivative
thf(fact_7039_DERIV__cot,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X2: A] :
( ( ( sin @ A @ X2 )
!= ( zero_zero @ A ) )
=> ( has_field_derivative @ A @ ( cot @ A ) @ ( uminus_uminus @ A @ ( inverse_inverse @ A @ ( power_power @ A @ ( sin @ A @ X2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( topolo174197925503356063within @ A @ X2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% DERIV_cot
thf(fact_7040_conj__subset__def,axiom,
! [A: $tType,A3: set @ A,P: A > $o,Q: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ A3
@ ( collect @ A
@ ^ [X: A] :
( ( P @ X )
& ( Q @ X ) ) ) )
= ( ( ord_less_eq @ ( set @ A ) @ A3 @ ( collect @ A @ P ) )
& ( ord_less_eq @ ( set @ A ) @ A3 @ ( collect @ A @ Q ) ) ) ) ).
% conj_subset_def
thf(fact_7041_has__field__derivative__tanh,axiom,
! [A13: $tType] :
( ( ( real_Vector_banach @ A13 )
& ( real_V3459762299906320749_field @ A13 ) )
=> ! [G: A13 > A13,X2: A13,Db: A13,S3: set @ A13] :
( ( ( cosh @ A13 @ ( G @ X2 ) )
!= ( zero_zero @ A13 ) )
=> ( ( has_field_derivative @ A13 @ G @ Db @ ( topolo174197925503356063within @ A13 @ X2 @ S3 ) )
=> ( has_field_derivative @ A13
@ ^ [X: A13] : ( tanh @ A13 @ ( G @ X ) )
@ ( times_times @ A13 @ ( minus_minus @ A13 @ ( one_one @ A13 ) @ ( power_power @ A13 @ ( tanh @ A13 @ ( G @ X2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ Db )
@ ( topolo174197925503356063within @ A13 @ X2 @ S3 ) ) ) ) ) ).
% has_field_derivative_tanh
thf(fact_7042_DERIV__real__sqrt__generic,axiom,
! [X2: real,D4: real] :
( ( X2
!= ( zero_zero @ real ) )
=> ( ( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( D4
= ( divide_divide @ real @ ( inverse_inverse @ real @ ( sqrt @ X2 ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) )
=> ( ( ( ord_less @ real @ X2 @ ( zero_zero @ real ) )
=> ( D4
= ( divide_divide @ real @ ( uminus_uminus @ real @ ( inverse_inverse @ real @ ( sqrt @ X2 ) ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) )
=> ( has_field_derivative @ real @ sqrt @ D4 @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ).
% DERIV_real_sqrt_generic
thf(fact_7043_arcosh__real__has__field__derivative,axiom,
! [X2: real,A3: set @ real] :
( ( ord_less @ real @ ( one_one @ real ) @ X2 )
=> ( has_field_derivative @ real @ ( arcosh @ real ) @ ( divide_divide @ real @ ( one_one @ real ) @ ( sqrt @ ( minus_minus @ real @ ( power_power @ real @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ real ) ) ) ) @ ( topolo174197925503356063within @ real @ X2 @ A3 ) ) ) ).
% arcosh_real_has_field_derivative
thf(fact_7044_artanh__real__has__field__derivative,axiom,
! [X2: real,A3: set @ real] :
( ( ord_less @ real @ ( abs_abs @ real @ X2 ) @ ( one_one @ real ) )
=> ( has_field_derivative @ real @ ( artanh @ real ) @ ( divide_divide @ real @ ( one_one @ real ) @ ( minus_minus @ real @ ( one_one @ real ) @ ( power_power @ real @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( topolo174197925503356063within @ real @ X2 @ A3 ) ) ) ).
% artanh_real_has_field_derivative
thf(fact_7045_conj__comp__iff,axiom,
! [B: $tType,A: $tType,P: B > $o,Q: B > $o,G: A > B] :
( ( comp @ B @ $o @ A
@ ^ [X: B] :
( ( P @ X )
& ( Q @ X ) )
@ G )
= ( ^ [X: A] :
( ( comp @ B @ $o @ A @ P @ G @ X )
& ( comp @ B @ $o @ A @ Q @ G @ X ) ) ) ) ).
% conj_comp_iff
thf(fact_7046_DERIV__real__root,axiom,
! [N: nat,X2: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( has_field_derivative @ real @ ( root @ N ) @ ( inverse_inverse @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ ( power_power @ real @ ( root @ N @ X2 ) @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) ) ) ) ).
% DERIV_real_root
thf(fact_7047_DERIV__arccos,axiom,
! [X2: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X2 )
=> ( ( ord_less @ real @ X2 @ ( one_one @ real ) )
=> ( has_field_derivative @ real @ arccos @ ( inverse_inverse @ real @ ( uminus_uminus @ real @ ( sqrt @ ( minus_minus @ real @ ( one_one @ real ) @ ( power_power @ real @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) ) ) ) ).
% DERIV_arccos
thf(fact_7048_DERIV__arcsin,axiom,
! [X2: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X2 )
=> ( ( ord_less @ real @ X2 @ ( one_one @ real ) )
=> ( has_field_derivative @ real @ arcsin @ ( inverse_inverse @ real @ ( sqrt @ ( minus_minus @ real @ ( one_one @ real ) @ ( power_power @ real @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) ) ) ) ).
% DERIV_arcsin
thf(fact_7049_Maclaurin__all__le,axiom,
! [Diff: nat > real > real,F2: real > real,X2: real,N: nat] :
( ( ( Diff @ ( zero_zero @ nat ) )
= F2 )
=> ( ! [M3: nat,X3: real] : ( has_field_derivative @ real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ X3 ) @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) )
=> ? [T5: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ T5 ) @ ( abs_abs @ real @ X2 ) )
& ( ( F2 @ X2 )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M5: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M5 @ ( zero_zero @ real ) ) @ ( semiring_char_0_fact @ real @ M5 ) ) @ ( power_power @ real @ X2 @ M5 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N @ T5 ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ X2 @ N ) ) ) ) ) ) ) ).
% Maclaurin_all_le
thf(fact_7050_Maclaurin__all__le__objl,axiom,
! [Diff: nat > real > real,F2: real > real,X2: real,N: nat] :
( ( ( ( Diff @ ( zero_zero @ nat ) )
= F2 )
& ! [M3: nat,X3: real] : ( has_field_derivative @ real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ X3 ) @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) ) )
=> ? [T5: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ T5 ) @ ( abs_abs @ real @ X2 ) )
& ( ( F2 @ X2 )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M5: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M5 @ ( zero_zero @ real ) ) @ ( semiring_char_0_fact @ real @ M5 ) ) @ ( power_power @ real @ X2 @ M5 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N @ T5 ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ X2 @ N ) ) ) ) ) ) ).
% Maclaurin_all_le_objl
thf(fact_7051_DERIV__odd__real__root,axiom,
! [N: nat,X2: real] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( X2
!= ( zero_zero @ real ) )
=> ( has_field_derivative @ real @ ( root @ N ) @ ( inverse_inverse @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ ( power_power @ real @ ( root @ N @ X2 ) @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) ) ) ) ).
% DERIV_odd_real_root
thf(fact_7052_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 )
=> ( ! [M3: nat,T5: real] :
( ( ( ord_less @ nat @ M3 @ N )
& ( ord_less_eq @ real @ ( zero_zero @ real ) @ T5 )
& ( ord_less_eq @ real @ T5 @ H2 ) )
=> ( has_field_derivative @ real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ T5 ) @ ( topolo174197925503356063within @ real @ T5 @ ( top_top @ ( set @ real ) ) ) ) )
=> ? [T5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ T5 )
& ( ord_less @ real @ T5 @ H2 )
& ( ( F2 @ H2 )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M5: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M5 @ ( zero_zero @ real ) ) @ ( semiring_char_0_fact @ real @ M5 ) ) @ ( power_power @ real @ H2 @ M5 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N @ T5 ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ H2 @ N ) ) ) ) ) ) ) ) ) ).
% Maclaurin
thf(fact_7053_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 )
=> ( ! [M3: nat,T5: real] :
( ( ( ord_less @ nat @ M3 @ N )
& ( ord_less_eq @ real @ ( zero_zero @ real ) @ T5 )
& ( ord_less_eq @ real @ T5 @ H2 ) )
=> ( has_field_derivative @ real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ T5 ) @ ( topolo174197925503356063within @ real @ T5 @ ( top_top @ ( set @ real ) ) ) ) )
=> ? [T5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ T5 )
& ( ord_less_eq @ real @ T5 @ H2 )
& ( ( F2 @ H2 )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M5: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M5 @ ( zero_zero @ real ) ) @ ( semiring_char_0_fact @ real @ M5 ) ) @ ( power_power @ real @ H2 @ M5 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N @ T5 ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ H2 @ N ) ) ) ) ) ) ) ) ).
% Maclaurin2
thf(fact_7054_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 )
=> ( ! [M3: nat,T5: real] :
( ( ( ord_less @ nat @ M3 @ N )
& ( ord_less_eq @ real @ H2 @ T5 )
& ( ord_less_eq @ real @ T5 @ ( zero_zero @ real ) ) )
=> ( has_field_derivative @ real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ T5 ) @ ( topolo174197925503356063within @ real @ T5 @ ( top_top @ ( set @ real ) ) ) ) )
=> ? [T5: real] :
( ( ord_less @ real @ H2 @ T5 )
& ( ord_less @ real @ T5 @ ( zero_zero @ real ) )
& ( ( F2 @ H2 )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M5: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M5 @ ( zero_zero @ real ) ) @ ( semiring_char_0_fact @ real @ M5 ) ) @ ( power_power @ real @ H2 @ M5 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N @ T5 ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ H2 @ N ) ) ) ) ) ) ) ) ) ).
% Maclaurin_minus
thf(fact_7055_Maclaurin__all__lt,axiom,
! [Diff: nat > real > real,F2: real > real,N: nat,X2: real] :
( ( ( Diff @ ( zero_zero @ nat ) )
= F2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( X2
!= ( zero_zero @ real ) )
=> ( ! [M3: nat,X3: real] : ( has_field_derivative @ real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ X3 ) @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) )
=> ? [T5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( abs_abs @ real @ T5 ) )
& ( ord_less @ real @ ( abs_abs @ real @ T5 ) @ ( abs_abs @ real @ X2 ) )
& ( ( F2 @ X2 )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M5: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M5 @ ( zero_zero @ real ) ) @ ( semiring_char_0_fact @ real @ M5 ) ) @ ( power_power @ real @ X2 @ M5 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N @ T5 ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ X2 @ N ) ) ) ) ) ) ) ) ) ).
% Maclaurin_all_lt
thf(fact_7056_Maclaurin__bi__le,axiom,
! [Diff: nat > real > real,F2: real > real,N: nat,X2: real] :
( ( ( Diff @ ( zero_zero @ nat ) )
= F2 )
=> ( ! [M3: nat,T5: real] :
( ( ( ord_less @ nat @ M3 @ N )
& ( ord_less_eq @ real @ ( abs_abs @ real @ T5 ) @ ( abs_abs @ real @ X2 ) ) )
=> ( has_field_derivative @ real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ T5 ) @ ( topolo174197925503356063within @ real @ T5 @ ( top_top @ ( set @ real ) ) ) ) )
=> ? [T5: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ T5 ) @ ( abs_abs @ real @ X2 ) )
& ( ( F2 @ X2 )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M5: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M5 @ ( zero_zero @ real ) ) @ ( semiring_char_0_fact @ real @ M5 ) ) @ ( power_power @ real @ X2 @ M5 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N @ T5 ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ X2 @ N ) ) ) ) ) ) ) ).
% Maclaurin_bi_le
thf(fact_7057_Taylor,axiom,
! [N: nat,Diff: nat > real > real,F2: real > real,A2: real,B2: real,C2: real,X2: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ( Diff @ ( zero_zero @ nat ) )
= F2 )
=> ( ! [M3: nat,T5: real] :
( ( ( ord_less @ nat @ M3 @ N )
& ( ord_less_eq @ real @ A2 @ T5 )
& ( ord_less_eq @ real @ T5 @ B2 ) )
=> ( has_field_derivative @ real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ T5 ) @ ( topolo174197925503356063within @ real @ T5 @ ( top_top @ ( set @ real ) ) ) ) )
=> ( ( ord_less_eq @ real @ A2 @ C2 )
=> ( ( ord_less_eq @ real @ C2 @ B2 )
=> ( ( ord_less_eq @ real @ A2 @ X2 )
=> ( ( ord_less_eq @ real @ X2 @ B2 )
=> ( ( X2 != C2 )
=> ? [T5: real] :
( ( ( ord_less @ real @ X2 @ C2 )
=> ( ( ord_less @ real @ X2 @ T5 )
& ( ord_less @ real @ T5 @ C2 ) ) )
& ( ~ ( ord_less @ real @ X2 @ C2 )
=> ( ( ord_less @ real @ C2 @ T5 )
& ( ord_less @ real @ T5 @ X2 ) ) )
& ( ( F2 @ X2 )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M5: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M5 @ C2 ) @ ( semiring_char_0_fact @ real @ M5 ) ) @ ( power_power @ real @ ( minus_minus @ real @ X2 @ C2 ) @ M5 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N @ T5 ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ ( minus_minus @ real @ X2 @ C2 ) @ N ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% Taylor
thf(fact_7058_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 )
=> ( ! [M3: nat,T5: real] :
( ( ( ord_less @ nat @ M3 @ N )
& ( ord_less_eq @ real @ A2 @ T5 )
& ( ord_less_eq @ real @ T5 @ B2 ) )
=> ( has_field_derivative @ real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ T5 ) @ ( topolo174197925503356063within @ real @ T5 @ ( top_top @ ( set @ real ) ) ) ) )
=> ( ( ord_less_eq @ real @ A2 @ C2 )
=> ( ( ord_less @ real @ C2 @ B2 )
=> ? [T5: real] :
( ( ord_less @ real @ C2 @ T5 )
& ( ord_less @ real @ T5 @ B2 )
& ( ( F2 @ B2 )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M5: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M5 @ C2 ) @ ( semiring_char_0_fact @ real @ M5 ) ) @ ( power_power @ real @ ( minus_minus @ real @ B2 @ C2 ) @ M5 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N @ T5 ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ ( minus_minus @ real @ B2 @ C2 ) @ N ) ) ) ) ) ) ) ) ) ) ).
% Taylor_up
thf(fact_7059_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 )
=> ( ! [M3: nat,T5: real] :
( ( ( ord_less @ nat @ M3 @ N )
& ( ord_less_eq @ real @ A2 @ T5 )
& ( ord_less_eq @ real @ T5 @ B2 ) )
=> ( has_field_derivative @ real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ T5 ) @ ( topolo174197925503356063within @ real @ T5 @ ( top_top @ ( set @ real ) ) ) ) )
=> ( ( ord_less @ real @ A2 @ C2 )
=> ( ( ord_less_eq @ real @ C2 @ B2 )
=> ? [T5: real] :
( ( ord_less @ real @ A2 @ T5 )
& ( ord_less @ real @ T5 @ C2 )
& ( ( F2 @ A2 )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M5: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M5 @ C2 ) @ ( semiring_char_0_fact @ real @ M5 ) ) @ ( power_power @ real @ ( minus_minus @ real @ A2 @ C2 ) @ M5 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N @ T5 ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ ( minus_minus @ real @ A2 @ C2 ) @ N ) ) ) ) ) ) ) ) ) ) ).
% Taylor_down
thf(fact_7060_Maclaurin__lemma2,axiom,
! [N: nat,H2: real,Diff: nat > real > real,K: nat,B5: real] :
( ! [M3: nat,T5: real] :
( ( ( ord_less @ nat @ M3 @ N )
& ( ord_less_eq @ real @ ( zero_zero @ real ) @ T5 )
& ( ord_less_eq @ real @ T5 @ H2 ) )
=> ( has_field_derivative @ real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ T5 ) @ ( topolo174197925503356063within @ real @ T5 @ ( top_top @ ( set @ real ) ) ) ) )
=> ( ( N
= ( suc @ K ) )
=> ! [M4: nat,T7: real] :
( ( ( ord_less @ nat @ M4 @ N )
& ( ord_less_eq @ real @ ( zero_zero @ real ) @ T7 )
& ( ord_less_eq @ real @ T7 @ H2 ) )
=> ( has_field_derivative @ real
@ ^ [U2: real] :
( minus_minus @ real @ ( Diff @ M4 @ U2 )
@ ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [P3: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ ( plus_plus @ nat @ M4 @ P3 ) @ ( zero_zero @ real ) ) @ ( semiring_char_0_fact @ real @ P3 ) ) @ ( power_power @ real @ U2 @ P3 ) )
@ ( set_ord_lessThan @ nat @ ( minus_minus @ nat @ N @ M4 ) ) )
@ ( times_times @ real @ B5 @ ( divide_divide @ real @ ( power_power @ real @ U2 @ ( minus_minus @ nat @ N @ M4 ) ) @ ( semiring_char_0_fact @ real @ ( minus_minus @ nat @ N @ M4 ) ) ) ) ) )
@ ( minus_minus @ real @ ( Diff @ ( suc @ M4 ) @ T7 )
@ ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [P3: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ ( plus_plus @ nat @ ( suc @ M4 ) @ P3 ) @ ( zero_zero @ real ) ) @ ( semiring_char_0_fact @ real @ P3 ) ) @ ( power_power @ real @ T7 @ P3 ) )
@ ( set_ord_lessThan @ nat @ ( minus_minus @ nat @ N @ ( suc @ M4 ) ) ) )
@ ( times_times @ real @ B5 @ ( divide_divide @ real @ ( power_power @ real @ T7 @ ( minus_minus @ nat @ N @ ( suc @ M4 ) ) ) @ ( semiring_char_0_fact @ real @ ( minus_minus @ nat @ N @ ( suc @ M4 ) ) ) ) ) ) )
@ ( topolo174197925503356063within @ real @ T7 @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ).
% Maclaurin_lemma2
thf(fact_7061_DERIV__arctan__series,axiom,
! [X2: real] :
( ( ord_less @ real @ ( abs_abs @ real @ X2 ) @ ( one_one @ real ) )
=> ( has_field_derivative @ real
@ ^ [X9: real] :
( suminf @ real
@ ^ [K2: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ K2 ) @ ( times_times @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( semiring_1_of_nat @ real @ ( plus_plus @ nat @ ( times_times @ nat @ K2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) @ ( power_power @ real @ X9 @ ( plus_plus @ nat @ ( times_times @ nat @ K2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) ) )
@ ( suminf @ real
@ ^ [K2: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ K2 ) @ ( power_power @ real @ X2 @ ( times_times @ nat @ K2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) ) ) ).
% DERIV_arctan_series
thf(fact_7062_DERIV__real__root__generic,axiom,
! [N: nat,X2: real,D4: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( X2
!= ( zero_zero @ real ) )
=> ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
=> ( D4
= ( inverse_inverse @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ ( power_power @ real @ ( root @ N @ X2 ) @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ) )
=> ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( ord_less @ real @ X2 @ ( zero_zero @ real ) )
=> ( D4
= ( uminus_uminus @ real @ ( inverse_inverse @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ ( power_power @ real @ ( root @ N @ X2 ) @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ) ) )
=> ( ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( D4
= ( inverse_inverse @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ ( power_power @ real @ ( root @ N @ X2 ) @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) )
=> ( has_field_derivative @ real @ ( root @ N ) @ D4 @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ) ) ).
% DERIV_real_root_generic
thf(fact_7063_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
@ ^ [X: real] :
( suminf @ real
@ ^ [N3: nat] : ( times_times @ real @ ( F2 @ N3 ) @ ( power_power @ real @ X @ ( suc @ N3 ) ) ) )
@ ( suminf @ real
@ ^ [N3: nat] : ( times_times @ real @ ( times_times @ real @ ( F2 @ N3 ) @ ( semiring_1_of_nat @ real @ ( suc @ N3 ) ) ) @ ( power_power @ real @ X0 @ N3 ) ) )
@ ( topolo174197925503356063within @ real @ X0 @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ).
% DERIV_power_series'
thf(fact_7064_has__derivative__arcsin,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [G: A > real,X2: A,G4: A > real,S3: set @ A] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ ( G @ X2 ) )
=> ( ( ord_less @ real @ ( G @ X2 ) @ ( one_one @ real ) )
=> ( ( has_derivative @ A @ real @ G @ G4 @ ( topolo174197925503356063within @ A @ X2 @ S3 ) )
=> ( has_derivative @ A @ real
@ ^ [X: A] : ( arcsin @ ( G @ X ) )
@ ^ [X: A] : ( times_times @ real @ ( G4 @ X ) @ ( inverse_inverse @ real @ ( sqrt @ ( minus_minus @ real @ ( one_one @ real ) @ ( power_power @ real @ ( G @ X2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) )
@ ( topolo174197925503356063within @ A @ X2 @ S3 ) ) ) ) ) ) ).
% has_derivative_arcsin
thf(fact_7065_greaterThanLessThan__empty__iff2,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A2: A,B2: A] :
( ( ( bot_bot @ ( set @ A ) )
= ( set_or5935395276787703475ssThan @ A @ A2 @ B2 ) )
= ( ord_less_eq @ A @ B2 @ A2 ) ) ) ).
% greaterThanLessThan_empty_iff2
thf(fact_7066_greaterThanLessThan__empty__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A2: A,B2: A] :
( ( ( set_or5935395276787703475ssThan @ A @ A2 @ B2 )
= ( bot_bot @ ( set @ A ) ) )
= ( ord_less_eq @ A @ B2 @ A2 ) ) ) ).
% greaterThanLessThan_empty_iff
thf(fact_7067_greaterThanLessThan__empty,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L2: A,K: A] :
( ( ord_less_eq @ A @ L2 @ K )
=> ( ( set_or5935395276787703475ssThan @ A @ K @ L2 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% greaterThanLessThan_empty
thf(fact_7068_has__derivative__subset,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,F7: A > B,X2: A,S3: set @ A,T2: set @ A] :
( ( has_derivative @ A @ B @ F2 @ F7 @ ( topolo174197925503356063within @ A @ X2 @ S3 ) )
=> ( ( ord_less_eq @ ( set @ A ) @ T2 @ S3 )
=> ( has_derivative @ A @ B @ F2 @ F7 @ ( topolo174197925503356063within @ A @ X2 @ T2 ) ) ) ) ) ).
% has_derivative_subset
thf(fact_7069_has__derivative__scaleR,axiom,
! [C: $tType,D6: $tType] :
( ( ( real_V822414075346904944vector @ D6 )
& ( real_V822414075346904944vector @ C ) )
=> ! [F2: D6 > real,F7: D6 > real,X2: D6,S3: set @ D6,G: D6 > C,G4: D6 > C] :
( ( has_derivative @ D6 @ real @ F2 @ F7 @ ( topolo174197925503356063within @ D6 @ X2 @ S3 ) )
=> ( ( has_derivative @ D6 @ C @ G @ G4 @ ( topolo174197925503356063within @ D6 @ X2 @ S3 ) )
=> ( has_derivative @ D6 @ C
@ ^ [X: D6] : ( real_V8093663219630862766scaleR @ C @ ( F2 @ X ) @ ( G @ X ) )
@ ^ [H: D6] : ( plus_plus @ C @ ( real_V8093663219630862766scaleR @ C @ ( F2 @ X2 ) @ ( G4 @ H ) ) @ ( real_V8093663219630862766scaleR @ C @ ( F7 @ H ) @ ( G @ X2 ) ) )
@ ( topolo174197925503356063within @ D6 @ X2 @ S3 ) ) ) ) ) ).
% has_derivative_scaleR
thf(fact_7070_has__derivative__in__compose,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B )
& ( real_V822414075346904944vector @ C ) )
=> ! [F2: A > B,F7: A > B,X2: A,S3: set @ A,G: B > C,G4: B > C] :
( ( has_derivative @ A @ B @ F2 @ F7 @ ( topolo174197925503356063within @ A @ X2 @ S3 ) )
=> ( ( has_derivative @ B @ C @ G @ G4 @ ( topolo174197925503356063within @ B @ ( F2 @ X2 ) @ ( image @ A @ B @ F2 @ S3 ) ) )
=> ( has_derivative @ A @ C
@ ^ [X: A] : ( G @ ( F2 @ X ) )
@ ^ [X: A] : ( G4 @ ( F7 @ X ) )
@ ( topolo174197925503356063within @ A @ X2 @ S3 ) ) ) ) ) ).
% has_derivative_in_compose
thf(fact_7071_has__derivative__compose,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B )
& ( real_V822414075346904944vector @ C ) )
=> ! [F2: A > B,F7: A > B,X2: A,S3: set @ A,G: B > C,G4: B > C] :
( ( has_derivative @ A @ B @ F2 @ F7 @ ( topolo174197925503356063within @ A @ X2 @ S3 ) )
=> ( ( has_derivative @ B @ C @ G @ G4 @ ( topolo174197925503356063within @ B @ ( F2 @ X2 ) @ ( top_top @ ( set @ B ) ) ) )
=> ( has_derivative @ A @ C
@ ^ [X: A] : ( G @ ( F2 @ X ) )
@ ^ [X: A] : ( G4 @ ( F7 @ X ) )
@ ( topolo174197925503356063within @ A @ X2 @ S3 ) ) ) ) ) ).
% has_derivative_compose
thf(fact_7072_has__field__derivative__imp__has__derivative,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,D4: A,F3: filter @ A] :
( ( has_field_derivative @ A @ F2 @ D4 @ F3 )
=> ( has_derivative @ A @ A @ F2 @ ( times_times @ A @ D4 ) @ F3 ) ) ) ).
% has_field_derivative_imp_has_derivative
thf(fact_7073_has__derivative__imp__has__field__derivative,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,D4: A > A,F3: filter @ A,D7: A] :
( ( has_derivative @ A @ A @ F2 @ D4 @ F3 )
=> ( ! [X3: A] :
( ( times_times @ A @ X3 @ D7 )
= ( D4 @ X3 ) )
=> ( has_field_derivative @ A @ F2 @ D7 @ F3 ) ) ) ) ).
% has_derivative_imp_has_field_derivative
thf(fact_7074_has__field__derivative__def,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ( ( has_field_derivative @ A )
= ( ^ [F4: A > A,D8: A] : ( has_derivative @ A @ A @ F4 @ ( times_times @ A @ D8 ) ) ) ) ) ).
% has_field_derivative_def
thf(fact_7075_has__derivative__sum,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( ( real_V822414075346904944vector @ B )
& ( real_V822414075346904944vector @ C ) )
=> ! [I5: set @ A,F2: A > B > C,F7: A > B > C,F3: filter @ B] :
( ! [I2: A] :
( ( member @ A @ I2 @ I5 )
=> ( has_derivative @ B @ C @ ( F2 @ I2 ) @ ( F7 @ I2 ) @ F3 ) )
=> ( has_derivative @ B @ C
@ ^ [X: B] :
( groups7311177749621191930dd_sum @ A @ C
@ ^ [I3: A] : ( F2 @ I3 @ X )
@ I5 )
@ ^ [X: B] :
( groups7311177749621191930dd_sum @ A @ C
@ ^ [I3: A] : ( F7 @ I3 @ X )
@ I5 )
@ F3 ) ) ) ).
% has_derivative_sum
thf(fact_7076_has__derivative__mult__left,axiom,
! [A: $tType,C: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V4412858255891104859lgebra @ A ) )
=> ! [G: C > A,G4: C > A,F3: filter @ C,Y2: A] :
( ( has_derivative @ C @ A @ G @ G4 @ F3 )
=> ( has_derivative @ C @ A
@ ^ [X: C] : ( times_times @ A @ ( G @ X ) @ Y2 )
@ ^ [X: C] : ( times_times @ A @ ( G4 @ X ) @ Y2 )
@ F3 ) ) ) ).
% has_derivative_mult_left
thf(fact_7077_has__derivative__mult__right,axiom,
! [A: $tType,C: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V4412858255891104859lgebra @ A ) )
=> ! [G: C > A,G4: C > A,F3: filter @ C,X2: A] :
( ( has_derivative @ C @ A @ G @ G4 @ F3 )
=> ( has_derivative @ C @ A
@ ^ [X: C] : ( times_times @ A @ X2 @ ( G @ X ) )
@ ^ [X: C] : ( times_times @ A @ X2 @ ( G4 @ X ) )
@ F3 ) ) ) ).
% has_derivative_mult_right
thf(fact_7078_has__derivative__add,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,F7: A > B,F3: filter @ A,G: A > B,G4: A > B] :
( ( has_derivative @ A @ B @ F2 @ F7 @ F3 )
=> ( ( has_derivative @ A @ B @ G @ G4 @ F3 )
=> ( has_derivative @ A @ B
@ ^ [X: A] : ( plus_plus @ B @ ( F2 @ X ) @ ( G @ X ) )
@ ^ [X: A] : ( plus_plus @ B @ ( F7 @ X ) @ ( G4 @ X ) )
@ F3 ) ) ) ) ).
% has_derivative_add
thf(fact_7079_has__derivative__const,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [C2: B,F3: filter @ A] :
( has_derivative @ A @ B
@ ^ [X: A] : C2
@ ^ [X: A] : ( zero_zero @ B )
@ F3 ) ) ).
% has_derivative_const
thf(fact_7080_has__derivative__ident,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: filter @ A] :
( has_derivative @ A @ A
@ ^ [X: A] : X
@ ^ [X: A] : X
@ F3 ) ) ).
% has_derivative_ident
thf(fact_7081_has__derivative__of__real,axiom,
! [A: $tType,C: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V2191834092415804123ebra_1 @ A )
& ( real_V822414075346904944vector @ A ) )
=> ! [G: C > real,G4: C > real,F3: filter @ C] :
( ( has_derivative @ C @ real @ G @ G4 @ F3 )
=> ( has_derivative @ C @ A
@ ^ [X: C] : ( real_Vector_of_real @ A @ ( G @ X ) )
@ ^ [X: C] : ( real_Vector_of_real @ A @ ( G4 @ X ) )
@ F3 ) ) ) ).
% has_derivative_of_real
thf(fact_7082_has__derivative__scaleR__left,axiom,
! [B: $tType,C: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V822414075346904944vector @ B ) )
=> ! [G: C > real,G4: C > real,F3: filter @ C,X2: B] :
( ( has_derivative @ C @ real @ G @ G4 @ F3 )
=> ( has_derivative @ C @ B
@ ^ [X: C] : ( real_V8093663219630862766scaleR @ B @ ( G @ X ) @ X2 )
@ ^ [X: C] : ( real_V8093663219630862766scaleR @ B @ ( G4 @ X ) @ X2 )
@ F3 ) ) ) ).
% has_derivative_scaleR_left
thf(fact_7083_has__derivative__scaleR__right,axiom,
! [B: $tType,C: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V822414075346904944vector @ B ) )
=> ! [G: C > B,G4: C > B,F3: filter @ C,R2: real] :
( ( has_derivative @ C @ B @ G @ G4 @ F3 )
=> ( has_derivative @ C @ B
@ ^ [X: C] : ( real_V8093663219630862766scaleR @ B @ R2 @ ( G @ X ) )
@ ^ [X: C] : ( real_V8093663219630862766scaleR @ B @ R2 @ ( G4 @ X ) )
@ F3 ) ) ) ).
% has_derivative_scaleR_right
thf(fact_7084_has__derivative__diff,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,F7: A > B,F3: filter @ A,G: A > B,G4: A > B] :
( ( has_derivative @ A @ B @ F2 @ F7 @ F3 )
=> ( ( has_derivative @ A @ B @ G @ G4 @ F3 )
=> ( has_derivative @ A @ B
@ ^ [X: A] : ( minus_minus @ B @ ( F2 @ X ) @ ( G @ X ) )
@ ^ [X: A] : ( minus_minus @ B @ ( F7 @ X ) @ ( G4 @ X ) )
@ F3 ) ) ) ) ).
% has_derivative_diff
thf(fact_7085_has__derivative__minus,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,F7: A > B,F3: filter @ A] :
( ( has_derivative @ A @ B @ F2 @ F7 @ F3 )
=> ( has_derivative @ A @ B
@ ^ [X: A] : ( uminus_uminus @ B @ ( F2 @ X ) )
@ ^ [X: A] : ( uminus_uminus @ B @ ( F7 @ X ) )
@ F3 ) ) ) ).
% has_derivative_minus
thf(fact_7086_has__derivative__mult,axiom,
! [A: $tType,D6: $tType] :
( ( ( real_V822414075346904944vector @ D6 )
& ( real_V4412858255891104859lgebra @ A ) )
=> ! [F2: D6 > A,F7: D6 > A,X2: D6,S3: set @ D6,G: D6 > A,G4: D6 > A] :
( ( has_derivative @ D6 @ A @ F2 @ F7 @ ( topolo174197925503356063within @ D6 @ X2 @ S3 ) )
=> ( ( has_derivative @ D6 @ A @ G @ G4 @ ( topolo174197925503356063within @ D6 @ X2 @ S3 ) )
=> ( has_derivative @ D6 @ A
@ ^ [X: D6] : ( times_times @ A @ ( F2 @ X ) @ ( G @ X ) )
@ ^ [H: D6] : ( plus_plus @ A @ ( times_times @ A @ ( F2 @ X2 ) @ ( G4 @ H ) ) @ ( times_times @ A @ ( F7 @ H ) @ ( G @ X2 ) ) )
@ ( topolo174197925503356063within @ D6 @ X2 @ S3 ) ) ) ) ) ).
% has_derivative_mult
thf(fact_7087_has__derivative__zero__unique,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F3: A > B,X2: A] :
( ( has_derivative @ A @ B
@ ^ [X: A] : ( zero_zero @ B )
@ F3
@ ( topolo174197925503356063within @ A @ X2 @ ( top_top @ ( set @ A ) ) ) )
=> ( F3
= ( ^ [H: A] : ( zero_zero @ B ) ) ) ) ) ).
% has_derivative_zero_unique
thf(fact_7088_has__derivative__in__compose2,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [T2: set @ A,G: A > B,G4: A > A > B,F2: C > A,S3: set @ C,X2: C,F7: C > A] :
( ! [X3: A] :
( ( member @ A @ X3 @ T2 )
=> ( has_derivative @ A @ B @ G @ ( G4 @ X3 ) @ ( topolo174197925503356063within @ A @ X3 @ T2 ) ) )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image @ C @ A @ F2 @ S3 ) @ T2 )
=> ( ( member @ C @ X2 @ S3 )
=> ( ( has_derivative @ C @ A @ F2 @ F7 @ ( topolo174197925503356063within @ C @ X2 @ S3 ) )
=> ( has_derivative @ C @ B
@ ^ [X: C] : ( G @ ( F2 @ X ) )
@ ^ [Y: C] : ( G4 @ ( F2 @ X2 ) @ ( F7 @ Y ) )
@ ( topolo174197925503356063within @ C @ X2 @ S3 ) ) ) ) ) ) ) ).
% has_derivative_in_compose2
thf(fact_7089_has__derivative__exp,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [G: A > real,G4: A > real,X2: A,S3: set @ A] :
( ( has_derivative @ A @ real @ G @ G4 @ ( topolo174197925503356063within @ A @ X2 @ S3 ) )
=> ( has_derivative @ A @ real
@ ^ [X: A] : ( exp @ real @ ( G @ X ) )
@ ^ [X: A] : ( times_times @ real @ ( G4 @ X ) @ ( exp @ real @ ( G @ X2 ) ) )
@ ( topolo174197925503356063within @ A @ X2 @ S3 ) ) ) ) ).
% has_derivative_exp
thf(fact_7090_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_7091_tanh__real__bounds,axiom,
! [X2: real] : ( member @ real @ ( tanh @ real @ X2 ) @ ( set_or5935395276787703475ssThan @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ ( one_one @ real ) ) ) ).
% tanh_real_bounds
thf(fact_7092_has__derivative__sinh,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [G: A > A,Db: A,X2: A,S3: set @ A] :
( ( has_derivative @ A @ A @ G @ ( times_times @ A @ Db ) @ ( topolo174197925503356063within @ A @ X2 @ S3 ) )
=> ( has_derivative @ A @ A
@ ^ [X: A] : ( sinh @ A @ ( G @ X ) )
@ ( times_times @ A @ ( times_times @ A @ ( cosh @ A @ ( G @ X2 ) ) @ Db ) )
@ ( topolo174197925503356063within @ A @ X2 @ S3 ) ) ) ) ).
% has_derivative_sinh
thf(fact_7093_has__derivative__cosh,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [G: A > A,Db: A,X2: A,S3: set @ A] :
( ( has_derivative @ A @ A @ G @ ( times_times @ A @ Db ) @ ( topolo174197925503356063within @ A @ X2 @ S3 ) )
=> ( has_derivative @ A @ A
@ ^ [X: A] : ( cosh @ A @ ( G @ X ) )
@ ( times_times @ A @ ( times_times @ A @ ( sinh @ A @ ( G @ X2 ) ) @ Db ) )
@ ( topolo174197925503356063within @ A @ X2 @ S3 ) ) ) ) ).
% has_derivative_cosh
thf(fact_7094_has__derivative__sin,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [G: A > real,G4: A > real,X2: A,S3: set @ A] :
( ( has_derivative @ A @ real @ G @ G4 @ ( topolo174197925503356063within @ A @ X2 @ S3 ) )
=> ( has_derivative @ A @ real
@ ^ [X: A] : ( sin @ real @ ( G @ X ) )
@ ^ [X: A] : ( times_times @ real @ ( G4 @ X ) @ ( cos @ real @ ( G @ X2 ) ) )
@ ( topolo174197925503356063within @ A @ X2 @ S3 ) ) ) ) ).
% has_derivative_sin
thf(fact_7095_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_7096_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_7097_has__derivative__divide_H,axiom,
! [A: $tType,C: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V3459762299906320749_field @ A ) )
=> ! [F2: C > A,F7: C > A,X2: C,S: set @ C,G: C > A,G4: C > A] :
( ( has_derivative @ C @ A @ F2 @ F7 @ ( topolo174197925503356063within @ C @ X2 @ S ) )
=> ( ( has_derivative @ C @ A @ G @ G4 @ ( topolo174197925503356063within @ C @ X2 @ S ) )
=> ( ( ( G @ X2 )
!= ( zero_zero @ A ) )
=> ( has_derivative @ C @ A
@ ^ [X: C] : ( divide_divide @ A @ ( F2 @ X ) @ ( G @ X ) )
@ ^ [H: C] : ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ ( F7 @ H ) @ ( G @ X2 ) ) @ ( times_times @ A @ ( F2 @ X2 ) @ ( G4 @ H ) ) ) @ ( times_times @ A @ ( G @ X2 ) @ ( G @ X2 ) ) )
@ ( topolo174197925503356063within @ C @ X2 @ S ) ) ) ) ) ) ).
% has_derivative_divide'
thf(fact_7098_has__derivative__inverse_H,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [X2: A,S: set @ A] :
( ( X2
!= ( zero_zero @ A ) )
=> ( has_derivative @ A @ A @ ( inverse_inverse @ A )
@ ^ [H: A] : ( uminus_uminus @ A @ ( times_times @ A @ ( times_times @ A @ ( inverse_inverse @ A @ X2 ) @ H ) @ ( inverse_inverse @ A @ X2 ) ) )
@ ( topolo174197925503356063within @ A @ X2 @ S ) ) ) ) ).
% has_derivative_inverse'
thf(fact_7099_has__derivative__inverse,axiom,
! [A: $tType,C: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V8999393235501362500lgebra @ A ) )
=> ! [F2: C > A,X2: C,F7: C > A,S: set @ C] :
( ( ( F2 @ X2 )
!= ( zero_zero @ A ) )
=> ( ( has_derivative @ C @ A @ F2 @ F7 @ ( topolo174197925503356063within @ C @ X2 @ S ) )
=> ( has_derivative @ C @ A
@ ^ [X: C] : ( inverse_inverse @ A @ ( F2 @ X ) )
@ ^ [H: C] : ( uminus_uminus @ A @ ( times_times @ A @ ( times_times @ A @ ( inverse_inverse @ A @ ( F2 @ X2 ) ) @ ( F7 @ H ) ) @ ( inverse_inverse @ A @ ( F2 @ X2 ) ) ) )
@ ( topolo174197925503356063within @ C @ X2 @ S ) ) ) ) ) ).
% has_derivative_inverse
thf(fact_7100_DERIV__compose__FDERIV,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: real > real,F7: real,G: A > real,X2: A,G4: A > real,S3: set @ A] :
( ( has_field_derivative @ real @ F2 @ F7 @ ( topolo174197925503356063within @ real @ ( G @ X2 ) @ ( top_top @ ( set @ real ) ) ) )
=> ( ( has_derivative @ A @ real @ G @ G4 @ ( topolo174197925503356063within @ A @ X2 @ S3 ) )
=> ( has_derivative @ A @ real
@ ^ [X: A] : ( F2 @ ( G @ X ) )
@ ^ [X: A] : ( times_times @ real @ ( G4 @ X ) @ F7 )
@ ( topolo174197925503356063within @ A @ X2 @ S3 ) ) ) ) ) ).
% DERIV_compose_FDERIV
thf(fact_7101_has__derivative__cos,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [G: A > real,G4: A > real,X2: A,S3: set @ A] :
( ( has_derivative @ A @ real @ G @ G4 @ ( topolo174197925503356063within @ A @ X2 @ S3 ) )
=> ( has_derivative @ A @ real
@ ^ [X: A] : ( cos @ real @ ( G @ X ) )
@ ^ [X: A] : ( times_times @ real @ ( G4 @ X ) @ ( uminus_uminus @ real @ ( sin @ real @ ( G @ X2 ) ) ) )
@ ( topolo174197925503356063within @ A @ X2 @ S3 ) ) ) ) ).
% has_derivative_cos
thf(fact_7102_DERIV__isconst3,axiom,
! [A2: real,B2: real,X2: real,Y2: real,F2: real > real] :
( ( ord_less @ real @ A2 @ B2 )
=> ( ( member @ real @ X2 @ ( set_or5935395276787703475ssThan @ real @ A2 @ B2 ) )
=> ( ( member @ real @ Y2 @ ( 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 @ X2 )
= ( F2 @ Y2 ) ) ) ) ) ) ).
% DERIV_isconst3
thf(fact_7103_has__derivative__power,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V3459762299906320749_field @ B ) )
=> ! [F2: A > B,F7: A > B,X2: A,S: set @ A,N: nat] :
( ( has_derivative @ A @ B @ F2 @ F7 @ ( topolo174197925503356063within @ A @ X2 @ S ) )
=> ( has_derivative @ A @ B
@ ^ [X: A] : ( power_power @ B @ ( F2 @ X ) @ N )
@ ^ [Y: A] : ( times_times @ B @ ( times_times @ B @ ( semiring_1_of_nat @ B @ N ) @ ( F7 @ Y ) ) @ ( power_power @ B @ ( F2 @ X2 ) @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) )
@ ( topolo174197925503356063within @ A @ X2 @ S ) ) ) ) ).
% has_derivative_power
thf(fact_7104_has__derivative__ln,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [G: A > real,X2: A,G4: A > real,S3: set @ A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( G @ X2 ) )
=> ( ( has_derivative @ A @ real @ G @ G4 @ ( topolo174197925503356063within @ A @ X2 @ S3 ) )
=> ( has_derivative @ A @ real
@ ^ [X: A] : ( ln_ln @ real @ ( G @ X ) )
@ ^ [X: A] : ( times_times @ real @ ( G4 @ X ) @ ( inverse_inverse @ real @ ( G @ X2 ) ) )
@ ( topolo174197925503356063within @ A @ X2 @ S3 ) ) ) ) ) ).
% has_derivative_ln
thf(fact_7105_has__derivative__divide,axiom,
! [A: $tType,C: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V8999393235501362500lgebra @ A ) )
=> ! [F2: C > A,F7: C > A,X2: C,S: set @ C,G: C > A,G4: C > A] :
( ( has_derivative @ C @ A @ F2 @ F7 @ ( topolo174197925503356063within @ C @ X2 @ S ) )
=> ( ( has_derivative @ C @ A @ G @ G4 @ ( topolo174197925503356063within @ C @ X2 @ S ) )
=> ( ( ( G @ X2 )
!= ( zero_zero @ A ) )
=> ( has_derivative @ C @ A
@ ^ [X: C] : ( divide_divide @ A @ ( F2 @ X ) @ ( G @ X ) )
@ ^ [H: C] : ( plus_plus @ A @ ( times_times @ A @ ( uminus_uminus @ A @ ( F2 @ X2 ) ) @ ( times_times @ A @ ( times_times @ A @ ( inverse_inverse @ A @ ( G @ X2 ) ) @ ( G4 @ H ) ) @ ( inverse_inverse @ A @ ( G @ X2 ) ) ) ) @ ( divide_divide @ A @ ( F7 @ H ) @ ( G @ X2 ) ) )
@ ( topolo174197925503356063within @ C @ X2 @ S ) ) ) ) ) ) ).
% has_derivative_divide
thf(fact_7106_has__derivative__prod,axiom,
! [B: $tType,I6: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V3459762299906320749_field @ B ) )
=> ! [I5: set @ I6,F2: I6 > A > B,F7: I6 > A > B,X2: A,S: set @ A] :
( ! [I2: I6] :
( ( member @ I6 @ I2 @ I5 )
=> ( has_derivative @ A @ B @ ( F2 @ I2 ) @ ( F7 @ I2 ) @ ( topolo174197925503356063within @ A @ X2 @ S ) ) )
=> ( has_derivative @ A @ B
@ ^ [X: A] :
( groups7121269368397514597t_prod @ I6 @ B
@ ^ [I3: I6] : ( F2 @ I3 @ X )
@ I5 )
@ ^ [Y: A] :
( groups7311177749621191930dd_sum @ I6 @ B
@ ^ [I3: I6] :
( times_times @ B @ ( F7 @ I3 @ Y )
@ ( groups7121269368397514597t_prod @ I6 @ B
@ ^ [J3: I6] : ( F2 @ J3 @ X2 )
@ ( minus_minus @ ( set @ I6 ) @ I5 @ ( insert @ I6 @ I3 @ ( bot_bot @ ( set @ I6 ) ) ) ) ) )
@ I5 )
@ ( topolo174197925503356063within @ A @ X2 @ S ) ) ) ) ).
% has_derivative_prod
thf(fact_7107_has__derivative__powr,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [G: A > real,G4: A > real,X2: A,X6: set @ A,F2: A > real,F7: A > real] :
( ( has_derivative @ A @ real @ G @ G4 @ ( topolo174197925503356063within @ A @ X2 @ X6 ) )
=> ( ( has_derivative @ A @ real @ F2 @ F7 @ ( topolo174197925503356063within @ A @ X2 @ X6 ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( G @ X2 ) )
=> ( ( member @ A @ X2 @ X6 )
=> ( has_derivative @ A @ real
@ ^ [X: A] : ( powr @ real @ ( G @ X ) @ ( F2 @ X ) )
@ ^ [H: A] : ( times_times @ real @ ( powr @ real @ ( G @ X2 ) @ ( F2 @ X2 ) ) @ ( plus_plus @ real @ ( times_times @ real @ ( F7 @ H ) @ ( ln_ln @ real @ ( G @ X2 ) ) ) @ ( divide_divide @ real @ ( times_times @ real @ ( G4 @ H ) @ ( F2 @ X2 ) ) @ ( G @ X2 ) ) ) )
@ ( topolo174197925503356063within @ A @ X2 @ X6 ) ) ) ) ) ) ) ).
% has_derivative_powr
thf(fact_7108_has__derivative__real__sqrt,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [G: A > real,X2: A,G4: A > real,S3: set @ A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( G @ X2 ) )
=> ( ( has_derivative @ A @ real @ G @ G4 @ ( topolo174197925503356063within @ A @ X2 @ S3 ) )
=> ( has_derivative @ A @ real
@ ^ [X: A] : ( sqrt @ ( G @ X ) )
@ ^ [X: A] : ( times_times @ real @ ( G4 @ X ) @ ( divide_divide @ real @ ( inverse_inverse @ real @ ( sqrt @ ( G @ X2 ) ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
@ ( topolo174197925503356063within @ A @ X2 @ S3 ) ) ) ) ) ).
% has_derivative_real_sqrt
thf(fact_7109_has__derivative__arctan,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [G: A > real,G4: A > real,X2: A,S3: set @ A] :
( ( has_derivative @ A @ real @ G @ G4 @ ( topolo174197925503356063within @ A @ X2 @ S3 ) )
=> ( has_derivative @ A @ real
@ ^ [X: A] : ( arctan @ ( G @ X ) )
@ ^ [X: A] : ( times_times @ real @ ( G4 @ X ) @ ( inverse_inverse @ real @ ( plus_plus @ real @ ( one_one @ real ) @ ( power_power @ real @ ( G @ X2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( topolo174197925503356063within @ A @ X2 @ S3 ) ) ) ) ).
% has_derivative_arctan
thf(fact_7110_has__derivative__tan,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [G: A > real,X2: A,G4: A > real,S3: set @ A] :
( ( ( cos @ real @ ( G @ X2 ) )
!= ( zero_zero @ real ) )
=> ( ( has_derivative @ A @ real @ G @ G4 @ ( topolo174197925503356063within @ A @ X2 @ S3 ) )
=> ( has_derivative @ A @ real
@ ^ [X: A] : ( tan @ real @ ( G @ X ) )
@ ^ [X: A] : ( times_times @ real @ ( G4 @ X ) @ ( inverse_inverse @ real @ ( power_power @ real @ ( cos @ real @ ( G @ X2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
@ ( topolo174197925503356063within @ A @ X2 @ S3 ) ) ) ) ) ).
% has_derivative_tan
thf(fact_7111_DERIV__series_H,axiom,
! [F2: real > nat > real,F7: real > nat > real,X0: real,A2: real,B2: real,L6: nat > real] :
( ! [N2: nat] :
( has_field_derivative @ real
@ ^ [X: real] : ( F2 @ X @ N2 )
@ ( F7 @ 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 @ ( F7 @ 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
@ ^ [X: real] : ( suminf @ real @ ( F2 @ X ) )
@ ( suminf @ real @ ( F7 @ X0 ) )
@ ( topolo174197925503356063within @ real @ X0 @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ) ) ) ).
% DERIV_series'
thf(fact_7112_has__derivative__arccos,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [G: A > real,X2: A,G4: A > real,S3: set @ A] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ ( G @ X2 ) )
=> ( ( ord_less @ real @ ( G @ X2 ) @ ( one_one @ real ) )
=> ( ( has_derivative @ A @ real @ G @ G4 @ ( topolo174197925503356063within @ A @ X2 @ S3 ) )
=> ( has_derivative @ A @ real
@ ^ [X: A] : ( arccos @ ( G @ X ) )
@ ^ [X: A] : ( times_times @ real @ ( G4 @ X ) @ ( inverse_inverse @ real @ ( uminus_uminus @ real @ ( sqrt @ ( minus_minus @ real @ ( one_one @ real ) @ ( power_power @ real @ ( G @ X2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
@ ( topolo174197925503356063within @ A @ X2 @ S3 ) ) ) ) ) ) ).
% has_derivative_arccos
thf(fact_7113_has__derivative__floor,axiom,
! [Aa: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( archim2362893244070406136eiling @ Aa )
& ( topolo2564578578187576103pology @ Aa ) )
=> ! [G: A > real,X2: A,F2: real > Aa,G4: A > real,S3: set @ A] :
( ( topolo3448309680560233919inuous @ real @ Aa @ ( topolo174197925503356063within @ real @ ( G @ X2 ) @ ( top_top @ ( set @ real ) ) ) @ F2 )
=> ( ~ ( member @ Aa @ ( F2 @ ( G @ X2 ) ) @ ( ring_1_Ints @ Aa ) )
=> ( ( has_derivative @ A @ real @ G @ G4 @ ( topolo174197925503356063within @ A @ X2 @ S3 ) )
=> ( has_derivative @ A @ real
@ ^ [X: A] : ( ring_1_of_int @ real @ ( archim6421214686448440834_floor @ Aa @ ( F2 @ ( G @ X ) ) ) )
@ ^ [X: A] : ( times_times @ real @ ( G4 @ X ) @ ( zero_zero @ real ) )
@ ( topolo174197925503356063within @ A @ X2 @ S3 ) ) ) ) ) ) ).
% has_derivative_floor
thf(fact_7114_termdiffs__aux,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [C2: nat > A,K6: A,X2: 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 @ X2 ) @ ( 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 @ X2 @ H ) @ N3 ) @ ( power_power @ A @ X2 @ N3 ) ) @ H ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N3 ) @ ( power_power @ A @ X2 @ ( 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_7115_tendsto__const,axiom,
! [A: $tType,B: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [K: A,F3: filter @ B] :
( filterlim @ B @ A
@ ^ [X: B] : K
@ ( topolo7230453075368039082e_nhds @ A @ K )
@ F3 ) ) ).
% tendsto_const
thf(fact_7116_tendsto__ident__at,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [A2: A,S3: set @ A] :
( filterlim @ A @ A
@ ^ [X: A] : X
@ ( topolo7230453075368039082e_nhds @ A @ A2 )
@ ( topolo174197925503356063within @ A @ A2 @ S3 ) ) ) ).
% tendsto_ident_at
thf(fact_7117_tendsto__mult__left__iff,axiom,
! [A: $tType,B: $tType] :
( ( ( field @ A )
& ( topolo4211221413907600880p_mult @ A ) )
=> ! [C2: A,F2: B > A,L2: A,F3: filter @ B] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( filterlim @ B @ A
@ ^ [X: B] : ( times_times @ A @ C2 @ ( F2 @ X ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( times_times @ A @ C2 @ L2 ) )
@ F3 )
= ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L2 ) @ F3 ) ) ) ) ).
% tendsto_mult_left_iff
thf(fact_7118_tendsto__mult__right__iff,axiom,
! [A: $tType,B: $tType] :
( ( ( field @ A )
& ( topolo4211221413907600880p_mult @ A ) )
=> ! [C2: A,F2: B > A,L2: A,F3: filter @ B] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( filterlim @ B @ A
@ ^ [X: B] : ( times_times @ A @ ( F2 @ X ) @ C2 )
@ ( topolo7230453075368039082e_nhds @ A @ ( times_times @ A @ L2 @ C2 ) )
@ F3 )
= ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L2 ) @ F3 ) ) ) ) ).
% tendsto_mult_right_iff
thf(fact_7119_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
@ ^ [X: A] : ( power_power @ real @ ( F2 @ X ) @ 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_7120_tendsto__within__subset,axiom,
! [B: $tType,A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [F2: A > B,L2: filter @ B,X2: A,S: set @ A,T3: set @ A] :
( ( filterlim @ A @ B @ F2 @ L2 @ ( topolo174197925503356063within @ A @ X2 @ S ) )
=> ( ( ord_less_eq @ ( set @ A ) @ T3 @ S )
=> ( filterlim @ A @ B @ F2 @ L2 @ ( topolo174197925503356063within @ A @ X2 @ T3 ) ) ) ) ) ).
% tendsto_within_subset
thf(fact_7121_continuous__ident,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [X2: A,S: set @ A] :
( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ X2 @ S )
@ ^ [X: A] : X ) ) ).
% continuous_ident
thf(fact_7122_has__field__derivativeD,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,D4: A,X2: A,S: set @ A] :
( ( has_field_derivative @ A @ F2 @ D4 @ ( topolo174197925503356063within @ A @ X2 @ S ) )
=> ( filterlim @ A @ A
@ ^ [Y: A] : ( divide_divide @ A @ ( minus_minus @ A @ ( F2 @ Y ) @ ( F2 @ X2 ) ) @ ( minus_minus @ A @ Y @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ D4 )
@ ( topolo174197925503356063within @ A @ X2 @ S ) ) ) ) ).
% has_field_derivativeD
thf(fact_7123_has__field__derivative__iff,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,D4: A,X2: A,S: set @ A] :
( ( has_field_derivative @ A @ F2 @ D4 @ ( topolo174197925503356063within @ A @ X2 @ S ) )
= ( filterlim @ A @ A
@ ^ [Y: A] : ( divide_divide @ A @ ( minus_minus @ A @ ( F2 @ Y ) @ ( F2 @ X2 ) ) @ ( minus_minus @ A @ Y @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ D4 )
@ ( topolo174197925503356063within @ A @ X2 @ S ) ) ) ) ).
% has_field_derivative_iff
thf(fact_7124_continuous__within__compose2,axiom,
! [B: $tType,C: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( topolo4958980785337419405_space @ C )
& ( topological_t2_space @ B ) )
=> ! [X2: A,S3: set @ A,F2: A > B,G: B > C] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ X2 @ S3 ) @ F2 )
=> ( ( topolo3448309680560233919inuous @ B @ C @ ( topolo174197925503356063within @ B @ ( F2 @ X2 ) @ ( image @ A @ B @ F2 @ S3 ) ) @ G )
=> ( topolo3448309680560233919inuous @ A @ C @ ( topolo174197925503356063within @ A @ X2 @ S3 )
@ ^ [X: A] : ( G @ ( F2 @ X ) ) ) ) ) ) ).
% continuous_within_compose2
thf(fact_7125_isCont__Pair,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( topolo4958980785337419405_space @ B )
& ( topolo4958980785337419405_space @ C ) )
=> ! [A2: A,F2: A > B,G: A > C] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) @ F2 )
=> ( ( topolo3448309680560233919inuous @ A @ C @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) @ G )
=> ( topolo3448309680560233919inuous @ A @ ( product_prod @ B @ C ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) )
@ ^ [X: A] : ( product_Pair @ B @ C @ ( F2 @ X ) @ ( G @ X ) ) ) ) ) ) ).
% isCont_Pair
thf(fact_7126_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
@ ^ [X: A] : ( F2 @ ( plus_plus @ A @ X @ K ) )
@ ( topolo7230453075368039082e_nhds @ B @ L6 )
@ ( topolo174197925503356063within @ A @ ( minus_minus @ A @ A2 @ K ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% LIM_offset
thf(fact_7127_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_7128_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_7129_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_7130_isCont__iff,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [X2: A,F2: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ X2 @ ( top_top @ ( set @ A ) ) ) @ F2 )
= ( filterlim @ A @ B
@ ^ [H: A] : ( F2 @ ( plus_plus @ A @ X2 @ H ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( F2 @ X2 ) )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% isCont_iff
thf(fact_7131_LIM__not__zero,axiom,
! [Aa: $tType,A: $tType] :
( ( ( topolo8386298272705272623_space @ A )
& ( zero @ Aa )
& ( topological_t2_space @ Aa ) )
=> ! [K: Aa,A2: A] :
( ( K
!= ( zero_zero @ Aa ) )
=> ~ ( filterlim @ A @ Aa
@ ^ [X: A] : K
@ ( topolo7230453075368039082e_nhds @ Aa @ ( zero_zero @ Aa ) )
@ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% LIM_not_zero
thf(fact_7132_isCont__norm,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 )
=> ( topolo3448309680560233919inuous @ A @ real @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) )
@ ^ [X: A] : ( real_V7770717601297561774m_norm @ B @ ( F2 @ X ) ) ) ) ) ).
% isCont_norm
thf(fact_7133_isCont__of__real,axiom,
! [A: $tType,C: $tType] :
( ( ( topological_t2_space @ C )
& ( real_V2191834092415804123ebra_1 @ A )
& ( real_V822414075346904944vector @ A ) )
=> ! [A2: C,G: C > real] :
( ( topolo3448309680560233919inuous @ C @ real @ ( topolo174197925503356063within @ C @ A2 @ ( top_top @ ( set @ C ) ) ) @ G )
=> ( topolo3448309680560233919inuous @ C @ A @ ( topolo174197925503356063within @ C @ A2 @ ( top_top @ ( set @ C ) ) )
@ ^ [X: C] : ( real_Vector_of_real @ A @ ( G @ X ) ) ) ) ) ).
% isCont_of_real
thf(fact_7134_filterlim__at__If,axiom,
! [B: $tType,A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [F2: A > B,G5: filter @ B,X2: A,P: A > $o,G: A > B] :
( ( filterlim @ A @ B @ F2 @ G5 @ ( topolo174197925503356063within @ A @ X2 @ ( collect @ A @ P ) ) )
=> ( ( filterlim @ A @ B @ G @ G5
@ ( topolo174197925503356063within @ A @ X2
@ ( collect @ A
@ ^ [X: A] :
~ ( P @ X ) ) ) )
=> ( filterlim @ A @ B
@ ^ [X: A] : ( if @ B @ ( P @ X ) @ ( F2 @ X ) @ ( G @ X ) )
@ G5
@ ( topolo174197925503356063within @ A @ X2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% filterlim_at_If
thf(fact_7135_isCont__o2,axiom,
! [B: $tType,C: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( topolo4958980785337419405_space @ C )
& ( topological_t2_space @ B ) )
=> ! [A2: A,F2: A > B,G: B > C] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) @ F2 )
=> ( ( topolo3448309680560233919inuous @ B @ C @ ( topolo174197925503356063within @ B @ ( F2 @ A2 ) @ ( top_top @ ( set @ B ) ) ) @ G )
=> ( topolo3448309680560233919inuous @ A @ C @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) )
@ ^ [X: A] : ( G @ ( F2 @ X ) ) ) ) ) ) ).
% isCont_o2
thf(fact_7136_LIM__const__eq,axiom,
! [A: $tType,B: $tType] :
( ( ( topological_t2_space @ B )
& ( topolo8386298272705272623_space @ A ) )
=> ! [K: B,L6: B,A2: A] :
( ( filterlim @ A @ B
@ ^ [X: A] : K
@ ( topolo7230453075368039082e_nhds @ B @ L6 )
@ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) )
=> ( K = L6 ) ) ) ).
% LIM_const_eq
thf(fact_7137_tendsto__compose,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( ( topolo4958980785337419405_space @ B )
& ( topolo4958980785337419405_space @ A ) )
=> ! [G: A > B,L2: A,F2: C > A,F3: filter @ C] :
( ( filterlim @ A @ B @ G @ ( topolo7230453075368039082e_nhds @ B @ ( G @ L2 ) ) @ ( topolo174197925503356063within @ A @ L2 @ ( top_top @ ( set @ A ) ) ) )
=> ( ( filterlim @ C @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L2 ) @ F3 )
=> ( filterlim @ C @ B
@ ^ [X: C] : ( G @ ( F2 @ X ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( G @ L2 ) )
@ F3 ) ) ) ) ).
% tendsto_compose
thf(fact_7138_LIM__const__not__eq,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo8386298272705272623_space @ A )
& ( topological_t2_space @ B ) )
=> ! [K: B,L6: B,A2: A] :
( ( K != L6 )
=> ~ ( filterlim @ A @ B
@ ^ [X: A] : K
@ ( topolo7230453075368039082e_nhds @ B @ L6 )
@ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% LIM_const_not_eq
thf(fact_7139_isCont__tendsto__compose,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( ( topolo4958980785337419405_space @ B )
& ( topological_t2_space @ A ) )
=> ! [L2: A,G: A > B,F2: C > A,F3: filter @ C] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ L2 @ ( top_top @ ( set @ A ) ) ) @ G )
=> ( ( filterlim @ C @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L2 ) @ F3 )
=> ( filterlim @ C @ B
@ ^ [X: C] : ( G @ ( F2 @ X ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( G @ L2 ) )
@ F3 ) ) ) ) ).
% isCont_tendsto_compose
thf(fact_7140_continuous__within__compose3,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( ( topological_t2_space @ C )
& ( topolo4958980785337419405_space @ B )
& ( topological_t2_space @ A ) )
=> ! [F2: C > A,X2: C,G: A > B,S3: set @ C] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ ( F2 @ X2 ) @ ( top_top @ ( set @ A ) ) ) @ G )
=> ( ( topolo3448309680560233919inuous @ C @ A @ ( topolo174197925503356063within @ C @ X2 @ S3 ) @ F2 )
=> ( topolo3448309680560233919inuous @ C @ B @ ( topolo174197925503356063within @ C @ X2 @ S3 )
@ ^ [X: C] : ( G @ ( F2 @ X ) ) ) ) ) ) ).
% continuous_within_compose3
thf(fact_7141_isCont__scaleR,axiom,
! [C: $tType,D6: $tType] :
( ( ( topological_t2_space @ D6 )
& ( real_V822414075346904944vector @ C ) )
=> ! [A2: D6,F2: D6 > real,G: D6 > C] :
( ( topolo3448309680560233919inuous @ D6 @ real @ ( topolo174197925503356063within @ D6 @ A2 @ ( top_top @ ( set @ D6 ) ) ) @ F2 )
=> ( ( topolo3448309680560233919inuous @ D6 @ C @ ( topolo174197925503356063within @ D6 @ A2 @ ( top_top @ ( set @ D6 ) ) ) @ G )
=> ( topolo3448309680560233919inuous @ D6 @ C @ ( topolo174197925503356063within @ D6 @ A2 @ ( top_top @ ( set @ D6 ) ) )
@ ^ [X: D6] : ( real_V8093663219630862766scaleR @ C @ ( F2 @ X ) @ ( G @ X ) ) ) ) ) ) ).
% isCont_scaleR
thf(fact_7142_LIM__equal2,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [R: real,A2: A,F2: A > B,G: A > B,L2: B] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R )
=> ( ! [X3: A] :
( ( X3 != A2 )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ X3 @ A2 ) ) @ R )
=> ( ( F2 @ X3 )
= ( G @ X3 ) ) ) )
=> ( ( filterlim @ A @ B @ G @ ( topolo7230453075368039082e_nhds @ B @ L2 ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) )
=> ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L2 ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ).
% LIM_equal2
thf(fact_7143_LIM__eq,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,L6: B,A2: A] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L6 ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) )
= ( ! [R5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R5 )
=> ? [S5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ S5 )
& ! [X: A] :
( ( ( X != A2 )
& ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ X @ A2 ) ) @ S5 ) )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ B @ ( minus_minus @ B @ ( F2 @ X ) @ L6 ) ) @ R5 ) ) ) ) ) ) ) ).
% LIM_eq
thf(fact_7144_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 )
=> ? [S8: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ S8 )
& ! [X3: A] :
( ( ( X3 != A2 )
& ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ X3 @ A2 ) ) @ S8 ) )
=> ( 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_7145_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 )
& ! [X4: A] :
( ( ( X4 != A2 )
& ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ X4 @ A2 ) ) @ S2 ) )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ B @ ( minus_minus @ B @ ( F2 @ X4 ) @ L6 ) ) @ R2 ) ) ) ) ) ) ).
% LIM_D
thf(fact_7146_isCont__LIM__compose2,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( topolo4958980785337419405_space @ B )
& ( topolo4958980785337419405_space @ C ) )
=> ! [A2: A,F2: A > B,G: B > C,L2: C] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) @ F2 )
=> ( ( filterlim @ B @ C @ G @ ( topolo7230453075368039082e_nhds @ C @ L2 ) @ ( topolo174197925503356063within @ B @ ( F2 @ A2 ) @ ( top_top @ ( set @ B ) ) ) )
=> ( ? [D5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D5 )
& ! [X3: A] :
( ( ( X3 != A2 )
& ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ X3 @ A2 ) ) @ D5 ) )
=> ( ( F2 @ X3 )
!= ( F2 @ A2 ) ) ) )
=> ( filterlim @ A @ C
@ ^ [X: A] : ( G @ ( F2 @ X ) )
@ ( topolo7230453075368039082e_nhds @ C @ L2 )
@ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ).
% isCont_LIM_compose2
thf(fact_7147_LIM__imp__LIM,axiom,
! [B: $tType,C: $tType,A: $tType] :
( ( ( topolo4958980785337419405_space @ A )
& ( real_V822414075346904944vector @ C )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,L2: B,A2: A,G: A > C,M: C] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L2 ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) )
=> ( ! [X3: A] :
( ( X3 != A2 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ C @ ( minus_minus @ C @ ( G @ X3 ) @ M ) ) @ ( real_V7770717601297561774m_norm @ B @ ( minus_minus @ B @ ( F2 @ X3 ) @ L2 ) ) ) )
=> ( filterlim @ A @ C @ G @ ( topolo7230453075368039082e_nhds @ C @ M ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% LIM_imp_LIM
thf(fact_7148_IVT2,axiom,
! [A: $tType,B: $tType] :
( ( ( topolo1944317154257567458pology @ B )
& ( topolo8458572112393995274pology @ A ) )
=> ! [F2: A > B,B2: A,Y2: B,A2: A] :
( ( ord_less_eq @ B @ ( F2 @ B2 ) @ Y2 )
=> ( ( ord_less_eq @ B @ Y2 @ ( F2 @ A2 ) )
=> ( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ! [X3: A] :
( ( ( ord_less_eq @ A @ A2 @ X3 )
& ( ord_less_eq @ A @ X3 @ B2 ) )
=> ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ X3 @ ( top_top @ ( set @ A ) ) ) @ F2 ) )
=> ? [X3: A] :
( ( ord_less_eq @ A @ A2 @ X3 )
& ( ord_less_eq @ A @ X3 @ B2 )
& ( ( F2 @ X3 )
= Y2 ) ) ) ) ) ) ) ).
% IVT2
thf(fact_7149_IVT,axiom,
! [A: $tType,B: $tType] :
( ( ( topolo1944317154257567458pology @ B )
& ( topolo8458572112393995274pology @ A ) )
=> ! [F2: A > B,A2: A,Y2: B,B2: A] :
( ( ord_less_eq @ B @ ( F2 @ A2 ) @ Y2 )
=> ( ( ord_less_eq @ B @ Y2 @ ( F2 @ B2 ) )
=> ( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ! [X3: A] :
( ( ( ord_less_eq @ A @ A2 @ X3 )
& ( ord_less_eq @ A @ X3 @ B2 ) )
=> ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ X3 @ ( top_top @ ( set @ A ) ) ) @ F2 ) )
=> ? [X3: A] :
( ( ord_less_eq @ A @ A2 @ X3 )
& ( ord_less_eq @ A @ X3 @ B2 )
& ( ( F2 @ X3 )
= Y2 ) ) ) ) ) ) ) ).
% IVT
thf(fact_7150_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_7151_atLeastSucLessThan__greaterThanLessThan,axiom,
! [L2: nat,U: nat] :
( ( set_or7035219750837199246ssThan @ nat @ ( suc @ L2 ) @ U )
= ( set_or5935395276787703475ssThan @ nat @ L2 @ U ) ) ).
% atLeastSucLessThan_greaterThanLessThan
thf(fact_7152_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
@ ^ [X: A] : ( power_power @ B @ ( F2 @ X ) @ N )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ F3 ) ) ) ) ).
% tendsto_null_power
thf(fact_7153_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
@ ^ [X: A] : ( power_power @ B @ ( F2 @ X ) @ N )
@ ( topolo7230453075368039082e_nhds @ B @ ( power_power @ B @ A2 @ N ) )
@ F3 ) ) ) ).
% tendsto_power
thf(fact_7154_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
@ ^ [X: C] : ( power_power @ B @ ( F2 @ X ) @ ( G @ X ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( power_power @ B @ A2 @ B2 ) )
@ F3 ) ) ) ) ).
% tendsto_power_strong
thf(fact_7155_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
@ ^ [X: C] : ( power_power @ B @ ( F2 @ X ) @ ( G @ X ) ) ) ) ) ) ).
% continuous_power'
thf(fact_7156_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
@ ^ [X: A] : ( power_power @ B @ ( F2 @ X ) @ N ) ) ) ) ).
% continuous_power
thf(fact_7157_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
@ ^ [X: B] : ( inverse_inverse @ A @ ( F2 @ X ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( inverse_inverse @ A @ A2 ) )
@ F3 ) ) ) ) ).
% tendsto_inverse
thf(fact_7158_tendsto__sgn,axiom,
! [A: $tType,B: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: B > A,L2: A,F3: filter @ B] :
( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L2 ) @ F3 )
=> ( ( L2
!= ( zero_zero @ A ) )
=> ( filterlim @ B @ A
@ ^ [X: B] : ( sgn_sgn @ A @ ( F2 @ X ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( sgn_sgn @ A @ L2 ) )
@ F3 ) ) ) ) ).
% tendsto_sgn
thf(fact_7159_continuous__minus,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( topolo1633459387980952147up_add @ B ) )
=> ! [F3: filter @ A,F2: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ F3 @ F2 )
=> ( topolo3448309680560233919inuous @ A @ B @ F3
@ ^ [X: A] : ( uminus_uminus @ B @ ( F2 @ X ) ) ) ) ) ).
% continuous_minus
thf(fact_7160_tendsto__minus__cancel__left,axiom,
! [B: $tType,A: $tType] :
( ( topolo1633459387980952147up_add @ B )
=> ! [F2: A > B,Y2: B,F3: filter @ A] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ ( uminus_uminus @ B @ Y2 ) ) @ F3 )
= ( filterlim @ A @ B
@ ^ [X: A] : ( uminus_uminus @ B @ ( F2 @ X ) )
@ ( topolo7230453075368039082e_nhds @ B @ Y2 )
@ F3 ) ) ) ).
% tendsto_minus_cancel_left
thf(fact_7161_tendsto__minus__cancel,axiom,
! [A: $tType,B: $tType] :
( ( topolo1633459387980952147up_add @ A )
=> ! [F2: B > A,A2: A,F3: filter @ B] :
( ( filterlim @ B @ A
@ ^ [X: B] : ( uminus_uminus @ A @ ( F2 @ X ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( uminus_uminus @ A @ A2 ) )
@ F3 )
=> ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ F3 ) ) ) ).
% tendsto_minus_cancel
thf(fact_7162_tendsto__minus,axiom,
! [A: $tType,B: $tType] :
( ( topolo1633459387980952147up_add @ A )
=> ! [F2: B > A,A2: A,F3: filter @ B] :
( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ F3 )
=> ( filterlim @ B @ A
@ ^ [X: B] : ( uminus_uminus @ A @ ( F2 @ X ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( uminus_uminus @ A @ A2 ) )
@ F3 ) ) ) ).
% tendsto_minus
thf(fact_7163_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
@ ^ [X: B] : ( minus_minus @ A @ ( F2 @ X ) @ ( G @ X ) )
@ ( 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_7164_LIM__zero__cancel,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [F2: A > B,L2: B,F3: filter @ A] :
( ( filterlim @ A @ B
@ ^ [X: A] : ( minus_minus @ B @ ( F2 @ X ) @ L2 )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ F3 )
=> ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L2 ) @ F3 ) ) ) ).
% LIM_zero_cancel
thf(fact_7165_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
@ ^ [X: B] : ( minus_minus @ A @ ( F2 @ X ) @ ( G @ X ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ F3 )
=> ( filterlim @ B @ A @ G @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ F3 ) ) ) ) ).
% Lim_transform2
thf(fact_7166_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
@ ^ [X: B] : ( minus_minus @ A @ ( F2 @ X ) @ ( G @ X ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ F3 )
=> ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ F3 ) ) ) ) ).
% Lim_transform
thf(fact_7167_LIM__zero__iff,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [F2: A > B,L2: B,F3: filter @ A] :
( ( filterlim @ A @ B
@ ^ [X: A] : ( minus_minus @ B @ ( F2 @ X ) @ L2 )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ F3 )
= ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L2 ) @ F3 ) ) ) ).
% LIM_zero_iff
thf(fact_7168_LIM__zero,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [F2: A > B,L2: B,F3: filter @ A] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L2 ) @ F3 )
=> ( filterlim @ A @ B
@ ^ [X: A] : ( minus_minus @ B @ ( F2 @ X ) @ L2 )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ F3 ) ) ) ).
% LIM_zero
thf(fact_7169_tendsto__diff,axiom,
! [A: $tType,B: $tType] :
( ( topolo1633459387980952147up_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
@ ^ [X: B] : ( minus_minus @ A @ ( F2 @ X ) @ ( G @ X ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( minus_minus @ A @ A2 @ B2 ) )
@ F3 ) ) ) ) ).
% tendsto_diff
thf(fact_7170_continuous__diff,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( topolo1633459387980952147up_add @ B ) )
=> ! [F3: filter @ A,F2: A > B,G: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ F3 @ F2 )
=> ( ( topolo3448309680560233919inuous @ A @ B @ F3 @ G )
=> ( topolo3448309680560233919inuous @ A @ B @ F3
@ ^ [X: A] : ( minus_minus @ B @ ( F2 @ X ) @ ( G @ X ) ) ) ) ) ) ).
% continuous_diff
thf(fact_7171_continuous__scaleR,axiom,
! [C: $tType,D6: $tType] :
( ( ( topological_t2_space @ D6 )
& ( real_V822414075346904944vector @ C ) )
=> ! [F3: filter @ D6,F2: D6 > real,G: D6 > C] :
( ( topolo3448309680560233919inuous @ D6 @ real @ F3 @ F2 )
=> ( ( topolo3448309680560233919inuous @ D6 @ C @ F3 @ G )
=> ( topolo3448309680560233919inuous @ D6 @ C @ F3
@ ^ [X: D6] : ( real_V8093663219630862766scaleR @ C @ ( F2 @ X ) @ ( G @ X ) ) ) ) ) ) ).
% continuous_scaleR
thf(fact_7172_tendsto__scaleR,axiom,
! [C: $tType,D6: $tType] :
( ( real_V822414075346904944vector @ C )
=> ! [F2: D6 > real,A2: real,F3: filter @ D6,G: D6 > C,B2: C] :
( ( filterlim @ D6 @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ A2 ) @ F3 )
=> ( ( filterlim @ D6 @ C @ G @ ( topolo7230453075368039082e_nhds @ C @ B2 ) @ F3 )
=> ( filterlim @ D6 @ C
@ ^ [X: D6] : ( real_V8093663219630862766scaleR @ C @ ( F2 @ X ) @ ( G @ X ) )
@ ( topolo7230453075368039082e_nhds @ C @ ( real_V8093663219630862766scaleR @ C @ A2 @ B2 ) )
@ F3 ) ) ) ) ).
% tendsto_scaleR
thf(fact_7173_continuous__cos,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_Vector_banach @ B )
& ( real_V3459762299906320749_field @ B ) )
=> ! [F3: filter @ A,F2: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ F3 @ F2 )
=> ( topolo3448309680560233919inuous @ A @ B @ F3
@ ^ [X: A] : ( cos @ B @ ( F2 @ X ) ) ) ) ) ).
% continuous_cos
thf(fact_7174_continuous__const,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [F3: filter @ A,C2: B] :
( topolo3448309680560233919inuous @ A @ B @ F3
@ ^ [X: A] : C2 ) ) ).
% continuous_const
thf(fact_7175_tendsto__arsinh,axiom,
! [B: $tType,F2: B > real,A2: real,F3: filter @ B] :
( ( filterlim @ B @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ A2 ) @ F3 )
=> ( filterlim @ B @ real
@ ^ [X: B] : ( arsinh @ real @ ( F2 @ X ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( arsinh @ real @ A2 ) )
@ F3 ) ) ).
% tendsto_arsinh
thf(fact_7176_tendsto__sin,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_Vector_banach @ B )
& ( real_V3459762299906320749_field @ B ) )
=> ! [F2: A > B,A2: B,F3: filter @ A] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ A2 ) @ F3 )
=> ( filterlim @ A @ B
@ ^ [X: A] : ( sin @ B @ ( F2 @ X ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( sin @ B @ A2 ) )
@ F3 ) ) ) ).
% tendsto_sin
thf(fact_7177_tendsto__cos,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_Vector_banach @ B )
& ( real_V3459762299906320749_field @ B ) )
=> ! [F2: A > B,A2: B,F3: filter @ A] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ A2 ) @ F3 )
=> ( filterlim @ A @ B
@ ^ [X: A] : ( cos @ B @ ( F2 @ X ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( cos @ B @ A2 ) )
@ F3 ) ) ) ).
% tendsto_cos
thf(fact_7178_tendsto__Complex,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 )
=> ( filterlim @ A @ complex
@ ^ [X: A] : ( complex2 @ ( F2 @ X ) @ ( G @ X ) )
@ ( topolo7230453075368039082e_nhds @ complex @ ( complex2 @ A2 @ B2 ) )
@ F3 ) ) ) ).
% tendsto_Complex
thf(fact_7179_tendsto__of__real__iff,axiom,
! [A: $tType,B: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [F2: B > real,C2: real,F3: filter @ B] :
( ( filterlim @ B @ A
@ ^ [X: B] : ( real_Vector_of_real @ A @ ( F2 @ X ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( real_Vector_of_real @ A @ C2 ) )
@ F3 )
= ( filterlim @ B @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ C2 ) @ F3 ) ) ) ).
% tendsto_of_real_iff
thf(fact_7180_continuous__of__real,axiom,
! [A: $tType,C: $tType] :
( ( ( topological_t2_space @ C )
& ( real_V2191834092415804123ebra_1 @ A )
& ( real_V822414075346904944vector @ A ) )
=> ! [F3: filter @ C,G: C > real] :
( ( topolo3448309680560233919inuous @ C @ real @ F3 @ G )
=> ( topolo3448309680560233919inuous @ C @ A @ F3
@ ^ [X: C] : ( real_Vector_of_real @ A @ ( G @ X ) ) ) ) ) ).
% continuous_of_real
thf(fact_7181_tendsto__of__real,axiom,
! [A: $tType,C: $tType] :
( ( ( real_V2191834092415804123ebra_1 @ A )
& ( real_V822414075346904944vector @ A ) )
=> ! [G: C > real,A2: real,F3: filter @ C] :
( ( filterlim @ C @ real @ G @ ( topolo7230453075368039082e_nhds @ real @ A2 ) @ F3 )
=> ( filterlim @ C @ A
@ ^ [X: C] : ( real_Vector_of_real @ A @ ( G @ X ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( real_Vector_of_real @ A @ A2 ) )
@ F3 ) ) ) ).
% tendsto_of_real
thf(fact_7182_tendsto__arctan,axiom,
! [A: $tType,F2: A > real,X2: real,F3: filter @ A] :
( ( filterlim @ A @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ X2 ) @ F3 )
=> ( filterlim @ A @ real
@ ^ [X: A] : ( arctan @ ( F2 @ X ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( arctan @ X2 ) )
@ F3 ) ) ).
% tendsto_arctan
thf(fact_7183_continuous__norm,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F3: filter @ A,F2: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ F3 @ F2 )
=> ( topolo3448309680560233919inuous @ A @ real @ F3
@ ^ [X: A] : ( real_V7770717601297561774m_norm @ B @ ( F2 @ X ) ) ) ) ) ).
% continuous_norm
thf(fact_7184_tendsto__norm,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [F2: A > B,A2: B,F3: filter @ A] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ A2 ) @ F3 )
=> ( filterlim @ A @ real
@ ^ [X: A] : ( real_V7770717601297561774m_norm @ B @ ( F2 @ X ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( real_V7770717601297561774m_norm @ B @ A2 ) )
@ F3 ) ) ) ).
% tendsto_norm
thf(fact_7185_tendsto__real__sqrt,axiom,
! [A: $tType,F2: A > real,X2: real,F3: filter @ A] :
( ( filterlim @ A @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ X2 ) @ F3 )
=> ( filterlim @ A @ real
@ ^ [X: A] : ( sqrt @ ( F2 @ X ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( sqrt @ X2 ) )
@ F3 ) ) ).
% tendsto_real_sqrt
thf(fact_7186_tendsto__exp,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 )
=> ( filterlim @ C @ A
@ ^ [X: C] : ( exp @ A @ ( F2 @ X ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( exp @ A @ A2 ) )
@ F3 ) ) ) ).
% tendsto_exp
thf(fact_7187_continuous__exp,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 )
=> ( topolo3448309680560233919inuous @ C @ A @ F3
@ ^ [X: C] : ( exp @ A @ ( F2 @ X ) ) ) ) ) ).
% continuous_exp
thf(fact_7188_continuous__sin,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_Vector_banach @ B )
& ( real_V3459762299906320749_field @ B ) )
=> ! [F3: filter @ A,F2: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ F3 @ F2 )
=> ( topolo3448309680560233919inuous @ A @ B @ F3
@ ^ [X: A] : ( sin @ B @ ( F2 @ X ) ) ) ) ) ).
% continuous_sin
thf(fact_7189_continuous__sinh,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 )
=> ( topolo3448309680560233919inuous @ C @ A @ F3
@ ^ [X: C] : ( sinh @ A @ ( F2 @ X ) ) ) ) ) ).
% continuous_sinh
thf(fact_7190_continuous__cosh,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 )
=> ( topolo3448309680560233919inuous @ C @ A @ F3
@ ^ [X: C] : ( cosh @ A @ ( F2 @ X ) ) ) ) ) ).
% continuous_cosh
thf(fact_7191_tendsto__real__root,axiom,
! [A: $tType,F2: A > real,X2: real,F3: filter @ A,N: nat] :
( ( filterlim @ A @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ X2 ) @ F3 )
=> ( filterlim @ A @ real
@ ^ [X: A] : ( root @ N @ ( F2 @ X ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( root @ N @ X2 ) )
@ F3 ) ) ).
% tendsto_real_root
thf(fact_7192_tendsto__sinh,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 )
=> ( filterlim @ C @ A
@ ^ [X: C] : ( sinh @ A @ ( F2 @ X ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( sinh @ A @ A2 ) )
@ F3 ) ) ) ).
% tendsto_sinh
thf(fact_7193_tendsto__cosh,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 )
=> ( filterlim @ C @ A
@ ^ [X: C] : ( cosh @ A @ ( F2 @ X ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( cosh @ A @ A2 ) )
@ F3 ) ) ) ).
% tendsto_cosh
thf(fact_7194_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
@ ^ [X: A] : ( tan @ A @ ( F2 @ X ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( tan @ A @ A2 ) )
@ F3 ) ) ) ) ).
% tendsto_tan
thf(fact_7195_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
@ ^ [X: A] : ( real_V7770717601297561774m_norm @ B @ ( F2 @ X ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ F3 ) ) ) ).
% tendsto_norm_zero
thf(fact_7196_tendsto__norm__zero__iff,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [F2: A > B,F3: filter @ A] :
( ( filterlim @ A @ real
@ ^ [X: A] : ( real_V7770717601297561774m_norm @ B @ ( F2 @ X ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ F3 )
= ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) ) @ F3 ) ) ) ).
% tendsto_norm_zero_iff
thf(fact_7197_tendsto__norm__zero__cancel,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [F2: A > B,F3: filter @ A] :
( ( filterlim @ A @ real
@ ^ [X: A] : ( real_V7770717601297561774m_norm @ B @ ( F2 @ X ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ F3 )
=> ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) ) @ F3 ) ) ) ).
% tendsto_norm_zero_cancel
thf(fact_7198_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
@ ^ [X: A] : ( ln_ln @ real @ ( F2 @ X ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( ln_ln @ real @ A2 ) )
@ F3 ) ) ) ).
% tendsto_ln
thf(fact_7199_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
@ ^ [X: A] : ( powr @ real @ ( F2 @ X ) @ ( G @ X ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( powr @ real @ A2 @ B2 ) )
@ F3 ) ) ) ) ).
% tendsto_powr
thf(fact_7200_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
@ ^ [X: C] : ( tanh @ A @ ( F2 @ X ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( tanh @ A @ A2 ) )
@ F3 ) ) ) ) ).
% tendsto_tanh
thf(fact_7201_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
@ ^ [X: A] : ( cot @ A @ ( F2 @ X ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( cot @ A @ A2 ) )
@ F3 ) ) ) ) ).
% tendsto_cot
thf(fact_7202_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
@ ^ [X: B] : ( arcosh @ real @ ( F2 @ X ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( arcosh @ real @ A2 ) )
@ F3 ) ) ) ).
% tendsto_arcosh
thf(fact_7203_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
@ ^ [X: A] : ( abs_abs @ real @ ( F2 @ X ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ F3 ) ) ).
% tendsto_rabs_zero
thf(fact_7204_tendsto__rabs__zero__iff,axiom,
! [A: $tType,F2: A > real,F3: filter @ A] :
( ( filterlim @ A @ real
@ ^ [X: A] : ( abs_abs @ real @ ( F2 @ X ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ F3 )
= ( filterlim @ A @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ F3 ) ) ).
% tendsto_rabs_zero_iff
thf(fact_7205_tendsto__rabs__zero__cancel,axiom,
! [A: $tType,F2: A > real,F3: filter @ A] :
( ( filterlim @ A @ real
@ ^ [X: A] : ( abs_abs @ real @ ( F2 @ X ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ F3 )
=> ( filterlim @ A @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ F3 ) ) ).
% tendsto_rabs_zero_cancel
thf(fact_7206_tendsto__rabs,axiom,
! [A: $tType,F2: A > real,L2: real,F3: filter @ A] :
( ( filterlim @ A @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ L2 ) @ F3 )
=> ( filterlim @ A @ real
@ ^ [X: A] : ( abs_abs @ real @ ( F2 @ X ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( abs_abs @ real @ L2 ) )
@ F3 ) ) ).
% tendsto_rabs
thf(fact_7207_continuous__max,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( topolo1944317154257567458pology @ B ) )
=> ! [F3: filter @ A,F2: A > B,G: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ F3 @ F2 )
=> ( ( topolo3448309680560233919inuous @ A @ B @ F3 @ G )
=> ( topolo3448309680560233919inuous @ A @ B @ F3
@ ^ [X: A] : ( ord_max @ B @ ( F2 @ X ) @ ( G @ X ) ) ) ) ) ) ).
% continuous_max
thf(fact_7208_tendsto__max,axiom,
! [A: $tType,B: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [X6: B > A,X2: A,Net: filter @ B,Y7: B > A,Y2: A] :
( ( filterlim @ B @ A @ X6 @ ( topolo7230453075368039082e_nhds @ A @ X2 ) @ Net )
=> ( ( filterlim @ B @ A @ Y7 @ ( topolo7230453075368039082e_nhds @ A @ Y2 ) @ Net )
=> ( filterlim @ B @ A
@ ^ [X: B] : ( ord_max @ A @ ( X6 @ X ) @ ( Y7 @ X ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( ord_max @ A @ X2 @ Y2 ) )
@ Net ) ) ) ) ).
% tendsto_max
thf(fact_7209_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
@ ^ [X: A] : ( log @ ( F2 @ X ) @ ( G @ X ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( log @ A2 @ B2 ) )
@ F3 ) ) ) ) ) ) ).
% tendsto_log
thf(fact_7210_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
@ ^ [X: B] : ( plus_plus @ A @ ( F2 @ X ) @ ( G @ X ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( plus_plus @ A @ A2 @ B2 ) )
@ F3 ) ) ) ) ).
% tendsto_add
thf(fact_7211_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
@ ^ [X: B] : ( plus_plus @ A @ C2 @ ( F2 @ X ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( plus_plus @ A @ C2 @ D2 ) )
@ F3 )
= ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ D2 ) @ F3 ) ) ) ).
% tendsto_add_const_iff
thf(fact_7212_continuous__add,axiom,
! [B: $tType,D6: $tType] :
( ( ( topological_t2_space @ D6 )
& ( topolo6943815403480290642id_add @ B ) )
=> ! [F3: filter @ D6,F2: D6 > B,G: D6 > B] :
( ( topolo3448309680560233919inuous @ D6 @ B @ F3 @ F2 )
=> ( ( topolo3448309680560233919inuous @ D6 @ B @ F3 @ G )
=> ( topolo3448309680560233919inuous @ D6 @ B @ F3
@ ^ [X: D6] : ( plus_plus @ B @ ( F2 @ X ) @ ( G @ X ) ) ) ) ) ) ).
% continuous_add
thf(fact_7213_tendsto__add__zero,axiom,
! [B: $tType,D6: $tType] :
( ( topolo6943815403480290642id_add @ B )
=> ! [F2: D6 > B,F3: filter @ D6,G: D6 > B] :
( ( filterlim @ D6 @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) ) @ F3 )
=> ( ( filterlim @ D6 @ B @ G @ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) ) @ F3 )
=> ( filterlim @ D6 @ B
@ ^ [X: D6] : ( plus_plus @ B @ ( F2 @ X ) @ ( G @ X ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ F3 ) ) ) ) ).
% tendsto_add_zero
thf(fact_7214_tendsto__mult__zero,axiom,
! [A: $tType,D6: $tType] :
( ( real_V4412858255891104859lgebra @ A )
=> ! [F2: D6 > A,F3: filter @ D6,G: D6 > A] :
( ( filterlim @ D6 @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ F3 )
=> ( ( filterlim @ D6 @ A @ G @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ F3 )
=> ( filterlim @ D6 @ A
@ ^ [X: D6] : ( times_times @ A @ ( F2 @ X ) @ ( G @ X ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ F3 ) ) ) ) ).
% tendsto_mult_zero
thf(fact_7215_tendsto__mult__left__zero,axiom,
! [A: $tType,D6: $tType] :
( ( real_V4412858255891104859lgebra @ A )
=> ! [F2: D6 > A,F3: filter @ D6,C2: A] :
( ( filterlim @ D6 @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ F3 )
=> ( filterlim @ D6 @ A
@ ^ [X: D6] : ( times_times @ A @ ( F2 @ X ) @ C2 )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ F3 ) ) ) ).
% tendsto_mult_left_zero
thf(fact_7216_tendsto__mult__right__zero,axiom,
! [A: $tType,D6: $tType] :
( ( real_V4412858255891104859lgebra @ A )
=> ! [F2: D6 > A,F3: filter @ D6,C2: A] :
( ( filterlim @ D6 @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ F3 )
=> ( filterlim @ D6 @ A
@ ^ [X: D6] : ( times_times @ A @ C2 @ ( F2 @ X ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ F3 ) ) ) ).
% tendsto_mult_right_zero
thf(fact_7217_tendsto__mult__one,axiom,
! [B: $tType,D6: $tType] :
( ( topolo1898628316856586783d_mult @ B )
=> ! [F2: D6 > B,F3: filter @ D6,G: D6 > B] :
( ( filterlim @ D6 @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ ( one_one @ B ) ) @ F3 )
=> ( ( filterlim @ D6 @ B @ G @ ( topolo7230453075368039082e_nhds @ B @ ( one_one @ B ) ) @ F3 )
=> ( filterlim @ D6 @ B
@ ^ [X: D6] : ( times_times @ B @ ( F2 @ X ) @ ( G @ X ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( one_one @ B ) )
@ F3 ) ) ) ) ).
% tendsto_mult_one
thf(fact_7218_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
@ ^ [X: B] : ( times_times @ A @ ( F2 @ X ) @ ( G @ X ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( times_times @ A @ A2 @ B2 ) )
@ F3 ) ) ) ) ).
% tendsto_mult
thf(fact_7219_tendsto__mult__left,axiom,
! [A: $tType,B: $tType] :
( ( topolo4211221413907600880p_mult @ A )
=> ! [F2: B > A,L2: A,F3: filter @ B,C2: A] :
( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L2 ) @ F3 )
=> ( filterlim @ B @ A
@ ^ [X: B] : ( times_times @ A @ C2 @ ( F2 @ X ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( times_times @ A @ C2 @ L2 ) )
@ F3 ) ) ) ).
% tendsto_mult_left
thf(fact_7220_tendsto__mult__right,axiom,
! [A: $tType,B: $tType] :
( ( topolo4211221413907600880p_mult @ A )
=> ! [F2: B > A,L2: A,F3: filter @ B,C2: A] :
( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L2 ) @ F3 )
=> ( filterlim @ B @ A
@ ^ [X: B] : ( times_times @ A @ ( F2 @ X ) @ C2 )
@ ( topolo7230453075368039082e_nhds @ A @ ( times_times @ A @ L2 @ C2 ) )
@ F3 ) ) ) ).
% tendsto_mult_right
thf(fact_7221_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
@ ^ [X: B] : ( times_times @ A @ ( F2 @ X ) @ C2 ) ) ) ) ).
% continuous_mult_right
thf(fact_7222_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
@ ^ [X: B] : ( times_times @ A @ C2 @ ( F2 @ X ) ) ) ) ) ).
% continuous_mult_left
thf(fact_7223_continuous__mult_H,axiom,
! [B: $tType,D6: $tType] :
( ( ( topological_t2_space @ D6 )
& ( topolo4211221413907600880p_mult @ B ) )
=> ! [F3: filter @ D6,F2: D6 > B,G: D6 > B] :
( ( topolo3448309680560233919inuous @ D6 @ B @ F3 @ F2 )
=> ( ( topolo3448309680560233919inuous @ D6 @ B @ F3 @ G )
=> ( topolo3448309680560233919inuous @ D6 @ B @ F3
@ ^ [X: D6] : ( times_times @ B @ ( F2 @ X ) @ ( G @ X ) ) ) ) ) ) ).
% continuous_mult'
thf(fact_7224_continuous__mult,axiom,
! [A: $tType,D6: $tType] :
( ( ( topological_t2_space @ D6 )
& ( real_V4412858255891104859lgebra @ A ) )
=> ! [F3: filter @ D6,F2: D6 > A,G: D6 > A] :
( ( topolo3448309680560233919inuous @ D6 @ A @ F3 @ F2 )
=> ( ( topolo3448309680560233919inuous @ D6 @ A @ F3 @ G )
=> ( topolo3448309680560233919inuous @ D6 @ A @ F3
@ ^ [X: D6] : ( times_times @ A @ ( F2 @ X ) @ ( G @ X ) ) ) ) ) ) ).
% continuous_mult
thf(fact_7225_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
@ ^ [X: B] : ( divide_divide @ A @ ( F2 @ X ) @ C2 )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ F3 ) ) ) ).
% tendsto_divide_zero
thf(fact_7226_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
@ ^ [X: B] : ( divide_divide @ A @ ( F2 @ X ) @ ( G @ X ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( divide_divide @ A @ A2 @ B2 ) )
@ F3 ) ) ) ) ) ).
% tendsto_divide
thf(fact_7227_tendsto__of__int__ceiling,axiom,
! [B: $tType,C: $tType,A: $tType] :
( ( ( ring_1 @ C )
& ( topolo4958980785337419405_space @ C )
& ( archim2362893244070406136eiling @ B )
& ( topolo2564578578187576103pology @ B ) )
=> ! [F2: A > B,L2: B,F3: filter @ A] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L2 ) @ F3 )
=> ( ~ ( member @ B @ L2 @ ( ring_1_Ints @ B ) )
=> ( filterlim @ A @ C
@ ^ [X: A] : ( ring_1_of_int @ C @ ( archimedean_ceiling @ B @ ( F2 @ X ) ) )
@ ( topolo7230453075368039082e_nhds @ C @ ( ring_1_of_int @ C @ ( archimedean_ceiling @ B @ L2 ) ) )
@ F3 ) ) ) ) ).
% tendsto_of_int_ceiling
thf(fact_7228_tendsto__of__int__floor,axiom,
! [B: $tType,C: $tType,A: $tType] :
( ( ( ring_1 @ C )
& ( topolo4958980785337419405_space @ C )
& ( archim2362893244070406136eiling @ B )
& ( topolo2564578578187576103pology @ B ) )
=> ! [F2: A > B,L2: B,F3: filter @ A] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L2 ) @ F3 )
=> ( ~ ( member @ B @ L2 @ ( ring_1_Ints @ B ) )
=> ( filterlim @ A @ C
@ ^ [X: A] : ( ring_1_of_int @ C @ ( archim6421214686448440834_floor @ B @ ( F2 @ X ) ) )
@ ( topolo7230453075368039082e_nhds @ C @ ( ring_1_of_int @ C @ ( archim6421214686448440834_floor @ B @ L2 ) ) )
@ F3 ) ) ) ) ).
% tendsto_of_int_floor
thf(fact_7229_tendsto__Pair,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( topolo4958980785337419405_space @ B )
& ( topolo4958980785337419405_space @ C ) )
=> ! [F2: A > B,A2: B,F3: filter @ A,G: A > C,B2: C] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ A2 ) @ F3 )
=> ( ( filterlim @ A @ C @ G @ ( topolo7230453075368039082e_nhds @ C @ B2 ) @ F3 )
=> ( filterlim @ A @ ( product_prod @ B @ C )
@ ^ [X: A] : ( product_Pair @ B @ C @ ( F2 @ X ) @ ( G @ X ) )
@ ( topolo7230453075368039082e_nhds @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ A2 @ B2 ) )
@ F3 ) ) ) ) ).
% tendsto_Pair
thf(fact_7230_continuous__Pair,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( topolo4958980785337419405_space @ B )
& ( topolo4958980785337419405_space @ C ) )
=> ! [F3: filter @ A,F2: A > B,G: A > C] :
( ( topolo3448309680560233919inuous @ A @ B @ F3 @ F2 )
=> ( ( topolo3448309680560233919inuous @ A @ C @ F3 @ G )
=> ( topolo3448309680560233919inuous @ A @ ( product_prod @ B @ C ) @ F3
@ ^ [X: A] : ( product_Pair @ B @ C @ ( F2 @ X ) @ ( G @ X ) ) ) ) ) ) ).
% continuous_Pair
thf(fact_7231_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
@ ^ [X: A] : ( artanh @ real @ ( F2 @ X ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( artanh @ real @ A2 ) )
@ F3 ) ) ) ) ).
% tendsto_artanh
thf(fact_7232_tendsto__min,axiom,
! [A: $tType,B: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [X6: B > A,X2: A,Net: filter @ B,Y7: B > A,Y2: A] :
( ( filterlim @ B @ A @ X6 @ ( topolo7230453075368039082e_nhds @ A @ X2 ) @ Net )
=> ( ( filterlim @ B @ A @ Y7 @ ( topolo7230453075368039082e_nhds @ A @ Y2 ) @ Net )
=> ( filterlim @ B @ A
@ ^ [X: B] : ( ord_min @ A @ ( X6 @ X ) @ ( Y7 @ X ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( ord_min @ A @ X2 @ Y2 ) )
@ Net ) ) ) ) ).
% tendsto_min
thf(fact_7233_continuous__min,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( topolo1944317154257567458pology @ B ) )
=> ! [F3: filter @ A,F2: A > B,G: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ F3 @ F2 )
=> ( ( topolo3448309680560233919inuous @ A @ B @ F3 @ G )
=> ( topolo3448309680560233919inuous @ A @ B @ F3
@ ^ [X: A] : ( ord_min @ B @ ( F2 @ X ) @ ( G @ X ) ) ) ) ) ) ).
% continuous_min
thf(fact_7234_tendsto__const__iff,axiom,
! [B: $tType,A: $tType] :
( ( topological_t2_space @ A )
=> ! [F3: filter @ B,A2: A,B2: A] :
( ( F3
!= ( bot_bot @ ( filter @ B ) ) )
=> ( ( filterlim @ B @ A
@ ^ [X: B] : A2
@ ( topolo7230453075368039082e_nhds @ A @ B2 )
@ F3 )
= ( A2 = B2 ) ) ) ) ).
% tendsto_const_iff
thf(fact_7235_tendsto__prod,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( ( real_V4412858255891104859lgebra @ C )
& ( comm_ring_1 @ C ) )
=> ! [S: set @ A,F2: A > B > C,L6: A > C,F3: filter @ B] :
( ! [I2: A] :
( ( member @ A @ I2 @ S )
=> ( filterlim @ B @ C @ ( F2 @ I2 ) @ ( topolo7230453075368039082e_nhds @ C @ ( L6 @ I2 ) ) @ F3 ) )
=> ( filterlim @ B @ C
@ ^ [X: B] :
( groups7121269368397514597t_prod @ A @ C
@ ^ [I3: A] : ( F2 @ I3 @ X )
@ S )
@ ( topolo7230453075368039082e_nhds @ C @ ( groups7121269368397514597t_prod @ A @ C @ L6 @ S ) )
@ F3 ) ) ) ).
% tendsto_prod
thf(fact_7236_tendsto__prod_H,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( topolo4987421752381908075d_mult @ C )
=> ! [I5: set @ A,F2: A > B > C,A2: A > C,F3: filter @ B] :
( ! [I2: A] :
( ( member @ A @ I2 @ I5 )
=> ( filterlim @ B @ C @ ( F2 @ I2 ) @ ( topolo7230453075368039082e_nhds @ C @ ( A2 @ I2 ) ) @ F3 ) )
=> ( filterlim @ B @ C
@ ^ [X: B] :
( groups7121269368397514597t_prod @ A @ C
@ ^ [I3: A] : ( F2 @ I3 @ X )
@ I5 )
@ ( topolo7230453075368039082e_nhds @ C @ ( groups7121269368397514597t_prod @ A @ C @ A2 @ I5 ) )
@ F3 ) ) ) ).
% tendsto_prod'
thf(fact_7237_continuous__prod_H,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( ( topological_t2_space @ B )
& ( topolo4987421752381908075d_mult @ C ) )
=> ! [I5: set @ A,F3: filter @ B,F2: A > B > C] :
( ! [I2: A] :
( ( member @ A @ I2 @ I5 )
=> ( topolo3448309680560233919inuous @ B @ C @ F3 @ ( F2 @ I2 ) ) )
=> ( topolo3448309680560233919inuous @ B @ C @ F3
@ ^ [X: B] :
( groups7121269368397514597t_prod @ A @ C
@ ^ [I3: A] : ( F2 @ I3 @ X )
@ I5 ) ) ) ) ).
% continuous_prod'
thf(fact_7238_continuous__prod,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( ( topological_t2_space @ B )
& ( real_V4412858255891104859lgebra @ C )
& ( comm_ring_1 @ C ) )
=> ! [S: set @ A,F3: filter @ B,F2: A > B > C] :
( ! [I2: A] :
( ( member @ A @ I2 @ S )
=> ( topolo3448309680560233919inuous @ B @ C @ F3 @ ( F2 @ I2 ) ) )
=> ( topolo3448309680560233919inuous @ B @ C @ F3
@ ^ [X: B] :
( groups7121269368397514597t_prod @ A @ C
@ ^ [I3: A] : ( F2 @ I3 @ X )
@ S ) ) ) ) ).
% continuous_prod
thf(fact_7239_tendsto__sum,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( topolo5987344860129210374id_add @ C )
=> ! [I5: set @ A,F2: A > B > C,A2: A > C,F3: filter @ B] :
( ! [I2: A] :
( ( member @ A @ I2 @ I5 )
=> ( filterlim @ B @ C @ ( F2 @ I2 ) @ ( topolo7230453075368039082e_nhds @ C @ ( A2 @ I2 ) ) @ F3 ) )
=> ( filterlim @ B @ C
@ ^ [X: B] :
( groups7311177749621191930dd_sum @ A @ C
@ ^ [I3: A] : ( F2 @ I3 @ X )
@ I5 )
@ ( topolo7230453075368039082e_nhds @ C @ ( groups7311177749621191930dd_sum @ A @ C @ A2 @ I5 ) )
@ F3 ) ) ) ).
% tendsto_sum
thf(fact_7240_continuous__sum,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( ( topological_t2_space @ B )
& ( topolo5987344860129210374id_add @ C ) )
=> ! [I5: set @ A,F3: filter @ B,F2: A > B > C] :
( ! [I2: A] :
( ( member @ A @ I2 @ I5 )
=> ( topolo3448309680560233919inuous @ B @ C @ F3 @ ( F2 @ I2 ) ) )
=> ( topolo3448309680560233919inuous @ B @ C @ F3
@ ^ [X: B] :
( groups7311177749621191930dd_sum @ A @ C
@ ^ [I3: A] : ( F2 @ I3 @ X )
@ I5 ) ) ) ) ).
% continuous_sum
thf(fact_7241_tendsto__one__prod_H,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( topolo4987421752381908075d_mult @ C )
=> ! [I5: set @ B,F2: A > B > C,F3: filter @ A] :
( ! [I2: B] :
( ( member @ B @ I2 @ I5 )
=> ( filterlim @ A @ C
@ ^ [X: A] : ( F2 @ X @ I2 )
@ ( topolo7230453075368039082e_nhds @ C @ ( one_one @ C ) )
@ F3 ) )
=> ( filterlim @ A @ C
@ ^ [I3: A] : ( groups7121269368397514597t_prod @ B @ C @ ( F2 @ I3 ) @ I5 )
@ ( topolo7230453075368039082e_nhds @ C @ ( one_one @ C ) )
@ F3 ) ) ) ).
% tendsto_one_prod'
thf(fact_7242_tendsto__null__sum,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( topolo5987344860129210374id_add @ C )
=> ! [I5: set @ B,F2: A > B > C,F3: filter @ A] :
( ! [I2: B] :
( ( member @ B @ I2 @ I5 )
=> ( filterlim @ A @ C
@ ^ [X: A] : ( F2 @ X @ I2 )
@ ( topolo7230453075368039082e_nhds @ C @ ( zero_zero @ C ) )
@ F3 ) )
=> ( filterlim @ A @ C
@ ^ [I3: A] : ( groups7311177749621191930dd_sum @ B @ C @ ( F2 @ I3 ) @ I5 )
@ ( topolo7230453075368039082e_nhds @ C @ ( zero_zero @ C ) )
@ F3 ) ) ) ).
% tendsto_null_sum
thf(fact_7243_DERIV__LIM__iff,axiom,
! [A: $tType] :
( ( ( inverse @ A )
& ( real_V822414075346904944vector @ A ) )
=> ! [F2: A > A,A2: A,D4: A] :
( ( filterlim @ A @ A
@ ^ [H: A] : ( divide_divide @ A @ ( minus_minus @ A @ ( F2 @ ( plus_plus @ A @ A2 @ H ) ) @ ( F2 @ A2 ) ) @ H )
@ ( topolo7230453075368039082e_nhds @ A @ D4 )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) )
= ( filterlim @ A @ A
@ ^ [X: A] : ( divide_divide @ A @ ( minus_minus @ A @ ( F2 @ X ) @ ( F2 @ A2 ) ) @ ( minus_minus @ A @ X @ A2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ D4 )
@ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% DERIV_LIM_iff
thf(fact_7244_isCont__Lb__Ub,axiom,
! [A2: real,B2: real,F2: real > real] :
( ( ord_less_eq @ real @ A2 @ B2 )
=> ( ! [X3: real] :
( ( ( ord_less_eq @ real @ A2 @ X3 )
& ( ord_less_eq @ real @ X3 @ B2 ) )
=> ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) @ F2 ) )
=> ? [L7: real,M10: real] :
( ! [X4: real] :
( ( ( ord_less_eq @ real @ A2 @ X4 )
& ( ord_less_eq @ real @ X4 @ B2 ) )
=> ( ( ord_less_eq @ real @ L7 @ ( F2 @ X4 ) )
& ( ord_less_eq @ real @ ( F2 @ X4 ) @ M10 ) ) )
& ! [Y4: real] :
( ( ( ord_less_eq @ real @ L7 @ Y4 )
& ( ord_less_eq @ real @ Y4 @ M10 ) )
=> ? [X3: real] :
( ( ord_less_eq @ real @ A2 @ X3 )
& ( ord_less_eq @ real @ X3 @ B2 )
& ( ( F2 @ X3 )
= Y4 ) ) ) ) ) ) ).
% isCont_Lb_Ub
thf(fact_7245_LIM__fun__less__zero,axiom,
! [F2: real > real,L2: real,C2: real] :
( ( filterlim @ real @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ L2 ) @ ( topolo174197925503356063within @ real @ C2 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( ord_less @ real @ L2 @ ( zero_zero @ real ) )
=> ? [R3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R3 )
& ! [X4: real] :
( ( ( X4 != C2 )
& ( ord_less @ real @ ( abs_abs @ real @ ( minus_minus @ real @ C2 @ X4 ) ) @ R3 ) )
=> ( ord_less @ real @ ( F2 @ X4 ) @ ( zero_zero @ real ) ) ) ) ) ) ).
% LIM_fun_less_zero
thf(fact_7246_LIM__fun__not__zero,axiom,
! [F2: real > real,L2: real,C2: real] :
( ( filterlim @ real @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ L2 ) @ ( topolo174197925503356063within @ real @ C2 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( L2
!= ( zero_zero @ real ) )
=> ? [R3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R3 )
& ! [X4: real] :
( ( ( X4 != C2 )
& ( ord_less @ real @ ( abs_abs @ real @ ( minus_minus @ real @ C2 @ X4 ) ) @ R3 ) )
=> ( ( F2 @ X4 )
!= ( zero_zero @ real ) ) ) ) ) ) ).
% LIM_fun_not_zero
thf(fact_7247_LIM__fun__gt__zero,axiom,
! [F2: real > real,L2: real,C2: real] :
( ( filterlim @ real @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ L2 ) @ ( topolo174197925503356063within @ real @ C2 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ L2 )
=> ? [R3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R3 )
& ! [X4: real] :
( ( ( X4 != C2 )
& ( ord_less @ real @ ( abs_abs @ real @ ( minus_minus @ real @ C2 @ X4 ) ) @ R3 ) )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( F2 @ X4 ) ) ) ) ) ) ).
% LIM_fun_gt_zero
thf(fact_7248_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 ) ) ) )
=> ( ? [D5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D5 )
& ! [X3: A] :
( ( ( X3 != A2 )
& ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ X3 @ A2 ) ) @ D5 ) )
=> ( ( F2 @ X3 )
!= B2 ) ) )
=> ( filterlim @ A @ C
@ ^ [X: A] : ( G @ ( F2 @ X ) )
@ ( topolo7230453075368039082e_nhds @ C @ C2 )
@ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ).
% LIM_compose2
thf(fact_7249_isCont__real__sqrt,axiom,
! [X2: real] : ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) @ sqrt ) ).
% isCont_real_sqrt
thf(fact_7250_isCont__real__root,axiom,
! [X2: real,N: nat] : ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) @ ( root @ N ) ) ).
% isCont_real_root
thf(fact_7251_continuous__at__within__divide,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_V3459762299906320749_field @ B ) )
=> ! [A2: A,S3: set @ A,F2: A > B,G: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ S3 ) @ F2 )
=> ( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ S3 ) @ G )
=> ( ( ( G @ A2 )
!= ( zero_zero @ B ) )
=> ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ S3 )
@ ^ [X: A] : ( divide_divide @ B @ ( F2 @ X ) @ ( G @ X ) ) ) ) ) ) ) ).
% continuous_at_within_divide
thf(fact_7252_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 ) ) )
@ ^ [X: A] : ( plus_plus @ B @ ( F2 @ X ) @ ( G @ X ) ) ) ) ) ) ).
% isCont_add
thf(fact_7253_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 ) ) )
@ ^ [X: A] : ( times_times @ B @ ( F2 @ X ) @ ( G @ X ) ) ) ) ) ) ).
% isCont_mult
thf(fact_7254_isCont__diff,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [A2: A,F2: A > B,G: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) @ F2 )
=> ( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) @ G )
=> ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) )
@ ^ [X: A] : ( minus_minus @ B @ ( F2 @ X ) @ ( G @ X ) ) ) ) ) ) ).
% isCont_diff
thf(fact_7255_isCont__minus,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 )
=> ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) )
@ ^ [X: A] : ( uminus_uminus @ B @ ( F2 @ X ) ) ) ) ) ).
% isCont_minus
thf(fact_7256_isCont__power,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( power @ B )
& ( real_V4412858255891104859lgebra @ B ) )
=> ! [A2: A,F2: A > B,N: nat] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) @ F2 )
=> ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) )
@ ^ [X: A] : ( power_power @ B @ ( F2 @ X ) @ N ) ) ) ) ).
% isCont_power
thf(fact_7257_continuous__at__within__inverse,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_V8999393235501362500lgebra @ B ) )
=> ! [A2: A,S3: set @ A,F2: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ S3 ) @ F2 )
=> ( ( ( F2 @ A2 )
!= ( zero_zero @ B ) )
=> ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ S3 )
@ ^ [X: A] : ( inverse_inverse @ B @ ( F2 @ X ) ) ) ) ) ) ).
% continuous_at_within_inverse
thf(fact_7258_isCont__sum,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( ( topological_t2_space @ B )
& ( topolo5987344860129210374id_add @ C ) )
=> ! [A3: set @ A,A2: B,F2: A > B > C] :
( ! [X3: A] :
( ( member @ A @ X3 @ A3 )
=> ( topolo3448309680560233919inuous @ B @ C @ ( topolo174197925503356063within @ B @ A2 @ ( top_top @ ( set @ B ) ) ) @ ( F2 @ X3 ) ) )
=> ( topolo3448309680560233919inuous @ B @ C @ ( topolo174197925503356063within @ B @ A2 @ ( top_top @ ( set @ B ) ) )
@ ^ [X: B] :
( groups7311177749621191930dd_sum @ A @ C
@ ^ [I3: A] : ( F2 @ I3 @ X )
@ A3 ) ) ) ) ).
% isCont_sum
thf(fact_7259_continuous__at__within__sgn,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [A2: A,S3: set @ A,F2: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ S3 ) @ F2 )
=> ( ( ( F2 @ A2 )
!= ( zero_zero @ B ) )
=> ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ S3 )
@ ^ [X: A] : ( sgn_sgn @ B @ ( F2 @ X ) ) ) ) ) ) ).
% continuous_at_within_sgn
thf(fact_7260_isCont__cos_H,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_Vector_banach @ B )
& ( real_V3459762299906320749_field @ B ) )
=> ! [A2: A,F2: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) @ F2 )
=> ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) )
@ ^ [X: A] : ( cos @ B @ ( F2 @ X ) ) ) ) ) ).
% isCont_cos'
thf(fact_7261_isCont__sin_H,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_Vector_banach @ B )
& ( real_V3459762299906320749_field @ B ) )
=> ! [A2: A,F2: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) @ F2 )
=> ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) )
@ ^ [X: A] : ( sin @ B @ ( F2 @ X ) ) ) ) ) ).
% isCont_sin'
thf(fact_7262_isCont__exp_H,axiom,
! [A: $tType,C: $tType] :
( ( ( topological_t2_space @ C )
& ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [A2: C,F2: C > A] :
( ( topolo3448309680560233919inuous @ C @ A @ ( topolo174197925503356063within @ C @ A2 @ ( top_top @ ( set @ C ) ) ) @ F2 )
=> ( topolo3448309680560233919inuous @ C @ A @ ( topolo174197925503356063within @ C @ A2 @ ( top_top @ ( set @ C ) ) )
@ ^ [X: C] : ( exp @ A @ ( F2 @ X ) ) ) ) ) ).
% isCont_exp'
thf(fact_7263_isCont__pochhammer,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [Z: A,N: nat] :
( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ Z @ ( top_top @ ( set @ A ) ) )
@ ^ [Z5: A] : ( comm_s3205402744901411588hammer @ A @ Z5 @ N ) ) ) ).
% isCont_pochhammer
thf(fact_7264_DERIV__def,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,D4: A,X2: A] :
( ( has_field_derivative @ A @ F2 @ D4 @ ( topolo174197925503356063within @ A @ X2 @ ( top_top @ ( set @ A ) ) ) )
= ( filterlim @ A @ A
@ ^ [H: A] : ( divide_divide @ A @ ( minus_minus @ A @ ( F2 @ ( plus_plus @ A @ X2 @ H ) ) @ ( F2 @ X2 ) ) @ H )
@ ( topolo7230453075368039082e_nhds @ A @ D4 )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% DERIV_def
thf(fact_7265_DERIV__D,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,D4: A,X2: A] :
( ( has_field_derivative @ A @ F2 @ D4 @ ( topolo174197925503356063within @ A @ X2 @ ( top_top @ ( set @ A ) ) ) )
=> ( filterlim @ A @ A
@ ^ [H: A] : ( divide_divide @ A @ ( minus_minus @ A @ ( F2 @ ( plus_plus @ A @ X2 @ H ) ) @ ( F2 @ X2 ) ) @ H )
@ ( topolo7230453075368039082e_nhds @ A @ D4 )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% DERIV_D
thf(fact_7266_lim__exp__minus__1,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ( filterlim @ A @ A
@ ^ [Z5: A] : ( divide_divide @ A @ ( minus_minus @ A @ ( exp @ A @ Z5 ) @ ( one_one @ A ) ) @ Z5 )
@ ( topolo7230453075368039082e_nhds @ A @ ( one_one @ A ) )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ).
% lim_exp_minus_1
thf(fact_7267_atLeastPlusOneLessThan__greaterThanLessThan__int,axiom,
! [L2: int,U: int] :
( ( set_or7035219750837199246ssThan @ int @ ( plus_plus @ int @ L2 @ ( one_one @ int ) ) @ U )
= ( set_or5935395276787703475ssThan @ int @ L2 @ U ) ) ).
% atLeastPlusOneLessThan_greaterThanLessThan_int
thf(fact_7268_greaterThanLessThan__upt,axiom,
( ( set_or5935395276787703475ssThan @ nat )
= ( ^ [N3: nat,M5: nat] : ( set2 @ nat @ ( upt @ ( suc @ N3 ) @ M5 ) ) ) ) ).
% greaterThanLessThan_upt
thf(fact_7269_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 )
=> ( ! [H5: A] :
( ( H5
!= ( zero_zero @ A ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ H5 ) @ K )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( F2 @ H5 ) ) @ ( times_times @ real @ K6 @ ( real_V7770717601297561774m_norm @ A @ H5 ) ) ) ) )
=> ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) ) @ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% lemma_termdiff4
thf(fact_7270_isCont__bounded,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [A2: real,B2: real,F2: real > A] :
( ( ord_less_eq @ real @ A2 @ B2 )
=> ( ! [X3: real] :
( ( ( ord_less_eq @ real @ A2 @ X3 )
& ( ord_less_eq @ real @ X3 @ B2 ) )
=> ( topolo3448309680560233919inuous @ real @ A @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) @ F2 ) )
=> ? [M10: A] :
! [X4: real] :
( ( ( ord_less_eq @ real @ A2 @ X4 )
& ( ord_less_eq @ real @ X4 @ B2 ) )
=> ( ord_less_eq @ A @ ( F2 @ X4 ) @ M10 ) ) ) ) ) ).
% isCont_bounded
thf(fact_7271_isCont__eq__Ub,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [A2: real,B2: real,F2: real > A] :
( ( ord_less_eq @ real @ A2 @ B2 )
=> ( ! [X3: real] :
( ( ( ord_less_eq @ real @ A2 @ X3 )
& ( ord_less_eq @ real @ X3 @ B2 ) )
=> ( topolo3448309680560233919inuous @ real @ A @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) @ F2 ) )
=> ? [M10: A] :
( ! [X4: real] :
( ( ( ord_less_eq @ real @ A2 @ X4 )
& ( ord_less_eq @ real @ X4 @ B2 ) )
=> ( ord_less_eq @ A @ ( F2 @ X4 ) @ M10 ) )
& ? [X3: real] :
( ( ord_less_eq @ real @ A2 @ X3 )
& ( ord_less_eq @ real @ X3 @ B2 )
& ( ( F2 @ X3 )
= M10 ) ) ) ) ) ) ).
% isCont_eq_Ub
thf(fact_7272_isCont__eq__Lb,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [A2: real,B2: real,F2: real > A] :
( ( ord_less_eq @ real @ A2 @ B2 )
=> ( ! [X3: real] :
( ( ( ord_less_eq @ real @ A2 @ X3 )
& ( ord_less_eq @ real @ X3 @ B2 ) )
=> ( topolo3448309680560233919inuous @ real @ A @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) @ F2 ) )
=> ? [M10: A] :
( ! [X4: real] :
( ( ( ord_less_eq @ real @ A2 @ X4 )
& ( ord_less_eq @ real @ X4 @ B2 ) )
=> ( ord_less_eq @ A @ M10 @ ( F2 @ X4 ) ) )
& ? [X3: real] :
( ( ord_less_eq @ real @ A2 @ X3 )
& ( ord_less_eq @ real @ X3 @ B2 )
& ( ( F2 @ X3 )
= M10 ) ) ) ) ) ) ).
% isCont_eq_Lb
thf(fact_7273_isCont__inverse__function2,axiom,
! [A2: real,X2: real,B2: real,G: real > real,F2: real > real] :
( ( ord_less @ real @ A2 @ X2 )
=> ( ( ord_less @ real @ X2 @ B2 )
=> ( ! [Z2: real] :
( ( ord_less_eq @ real @ A2 @ Z2 )
=> ( ( ord_less_eq @ real @ Z2 @ B2 )
=> ( ( G @ ( F2 @ Z2 ) )
= Z2 ) ) )
=> ( ! [Z2: real] :
( ( ord_less_eq @ real @ A2 @ Z2 )
=> ( ( ord_less_eq @ real @ Z2 @ B2 )
=> ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ Z2 @ ( top_top @ ( set @ real ) ) ) @ F2 ) ) )
=> ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ ( F2 @ X2 ) @ ( top_top @ ( set @ real ) ) ) @ G ) ) ) ) ) ).
% isCont_inverse_function2
thf(fact_7274_field__has__derivative__at,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,D4: A,X2: A] :
( ( has_derivative @ A @ A @ F2 @ ( times_times @ A @ D4 ) @ ( topolo174197925503356063within @ A @ X2 @ ( top_top @ ( set @ A ) ) ) )
= ( filterlim @ A @ A
@ ^ [H: A] : ( divide_divide @ A @ ( minus_minus @ A @ ( F2 @ ( plus_plus @ A @ X2 @ H ) ) @ ( F2 @ X2 ) ) @ H )
@ ( topolo7230453075368039082e_nhds @ A @ D4 )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% field_has_derivative_at
thf(fact_7275_isCont__divide,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_V3459762299906320749_field @ B ) )
=> ! [A2: A,F2: A > B,G: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) @ F2 )
=> ( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) @ G )
=> ( ( ( G @ A2 )
!= ( zero_zero @ B ) )
=> ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) )
@ ^ [X: A] : ( divide_divide @ B @ ( F2 @ X ) @ ( G @ X ) ) ) ) ) ) ) ).
% isCont_divide
thf(fact_7276_isCont__ln,axiom,
! [X2: real] :
( ( X2
!= ( zero_zero @ real ) )
=> ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) @ ( ln_ln @ real ) ) ) ).
% isCont_ln
thf(fact_7277_isCont__sgn,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [A2: A,F2: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) @ F2 )
=> ( ( ( F2 @ A2 )
!= ( zero_zero @ B ) )
=> ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) )
@ ^ [X: A] : ( sgn_sgn @ B @ ( F2 @ X ) ) ) ) ) ) ).
% isCont_sgn
thf(fact_7278_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
@ ^ [X: A] : ( F2 @ ( plus_plus @ A @ X @ A2 ) )
@ F3
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% filterlim_at_to_0
thf(fact_7279_continuous__within__tan,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X2: A,S3: set @ A,F2: A > A] :
( ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ X2 @ S3 ) @ F2 )
=> ( ( ( cos @ A @ ( F2 @ X2 ) )
!= ( zero_zero @ A ) )
=> ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ X2 @ S3 )
@ ^ [X: A] : ( tan @ A @ ( F2 @ X ) ) ) ) ) ) ).
% continuous_within_tan
thf(fact_7280_atLeastPlusOneLessThan__greaterThanLessThan__integer,axiom,
! [L2: code_integer,U: code_integer] :
( ( set_or7035219750837199246ssThan @ code_integer @ ( plus_plus @ code_integer @ L2 @ ( one_one @ code_integer ) ) @ U )
= ( set_or5935395276787703475ssThan @ code_integer @ L2 @ U ) ) ).
% atLeastPlusOneLessThan_greaterThanLessThan_integer
thf(fact_7281_continuous__within__cot,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X2: A,S3: set @ A,F2: A > A] :
( ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ X2 @ S3 ) @ F2 )
=> ( ( ( sin @ A @ ( F2 @ X2 ) )
!= ( zero_zero @ A ) )
=> ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ X2 @ S3 )
@ ^ [X: A] : ( cot @ A @ ( F2 @ X ) ) ) ) ) ) ).
% continuous_within_cot
thf(fact_7282_continuous__at__within__tanh,axiom,
! [A: $tType,C: $tType] :
( ( ( topological_t2_space @ C )
& ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X2: C,A3: set @ C,F2: C > A] :
( ( topolo3448309680560233919inuous @ C @ A @ ( topolo174197925503356063within @ C @ X2 @ A3 ) @ F2 )
=> ( ( ( cosh @ A @ ( F2 @ X2 ) )
!= ( zero_zero @ A ) )
=> ( topolo3448309680560233919inuous @ C @ A @ ( topolo174197925503356063within @ C @ X2 @ A3 )
@ ^ [X: C] : ( tanh @ A @ ( F2 @ X ) ) ) ) ) ) ).
% continuous_at_within_tanh
thf(fact_7283_CARAT__DERIV,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,L2: A,X2: A] :
( ( has_field_derivative @ A @ F2 @ L2 @ ( topolo174197925503356063within @ A @ X2 @ ( top_top @ ( set @ A ) ) ) )
= ( ? [G3: A > A] :
( ! [Z5: A] :
( ( minus_minus @ A @ ( F2 @ Z5 ) @ ( F2 @ X2 ) )
= ( times_times @ A @ ( G3 @ Z5 ) @ ( minus_minus @ A @ Z5 @ X2 ) ) )
& ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ X2 @ ( top_top @ ( set @ A ) ) ) @ G3 )
& ( ( G3 @ X2 )
= L2 ) ) ) ) ) ).
% CARAT_DERIV
thf(fact_7284_isCont__has__Ub,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [A2: real,B2: real,F2: real > A] :
( ( ord_less_eq @ real @ A2 @ B2 )
=> ( ! [X3: real] :
( ( ( ord_less_eq @ real @ A2 @ X3 )
& ( ord_less_eq @ real @ X3 @ B2 ) )
=> ( topolo3448309680560233919inuous @ real @ A @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) @ F2 ) )
=> ? [M10: A] :
( ! [X4: real] :
( ( ( ord_less_eq @ real @ A2 @ X4 )
& ( ord_less_eq @ real @ X4 @ B2 ) )
=> ( ord_less_eq @ A @ ( F2 @ X4 ) @ M10 ) )
& ! [N9: A] :
( ( ord_less @ A @ N9 @ M10 )
=> ? [X3: real] :
( ( ord_less_eq @ real @ A2 @ X3 )
& ( ord_less_eq @ real @ X3 @ B2 )
& ( ord_less @ A @ N9 @ ( F2 @ X3 ) ) ) ) ) ) ) ) ).
% isCont_has_Ub
thf(fact_7285_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_7286_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_7287_isCont__tan,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X2: A] :
( ( ( cos @ A @ X2 )
!= ( zero_zero @ A ) )
=> ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ X2 @ ( top_top @ ( set @ A ) ) ) @ ( tan @ A ) ) ) ) ).
% isCont_tan
thf(fact_7288_isCont__cot,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X2: A] :
( ( ( sin @ A @ X2 )
!= ( zero_zero @ A ) )
=> ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ X2 @ ( top_top @ ( set @ A ) ) ) @ ( cot @ A ) ) ) ) ).
% isCont_cot
thf(fact_7289_isCont__tanh,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X2: A] :
( ( ( cosh @ A @ X2 )
!= ( zero_zero @ A ) )
=> ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ X2 @ ( top_top @ ( set @ A ) ) ) @ ( tanh @ A ) ) ) ) ).
% isCont_tanh
thf(fact_7290_powser__limit__0__strong,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [S3: real,A2: nat > A,F2: A > A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ S3 )
=> ( ! [X3: A] :
( ( X3
!= ( zero_zero @ A ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X3 ) @ S3 )
=> ( 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_7291_powser__limit__0,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [S3: real,A2: nat > A,F2: A > A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ S3 )
=> ( ! [X3: A] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X3 ) @ S3 )
=> ( 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_7292_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 )
=> ( ! [H5: A,N2: nat] :
( ( H5
!= ( zero_zero @ A ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ H5 ) @ K )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( G @ H5 @ N2 ) ) @ ( times_times @ real @ ( F2 @ N2 ) @ ( real_V7770717601297561774m_norm @ A @ H5 ) ) ) ) )
=> ( 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_7293_isCont__tan_H,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [A2: A,F2: A > A] :
( ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) @ F2 )
=> ( ( ( cos @ A @ ( F2 @ A2 ) )
!= ( zero_zero @ A ) )
=> ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) )
@ ^ [X: A] : ( tan @ A @ ( F2 @ X ) ) ) ) ) ) ).
% isCont_tan'
thf(fact_7294_isCont__arcosh,axiom,
! [X2: real] :
( ( ord_less @ real @ ( one_one @ real ) @ X2 )
=> ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) @ ( arcosh @ real ) ) ) ).
% isCont_arcosh
thf(fact_7295_LIM__cos__div__sin,axiom,
( filterlim @ real @ real
@ ^ [X: real] : ( divide_divide @ real @ ( cos @ real @ X ) @ ( sin @ real @ X ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ ( topolo174197925503356063within @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( top_top @ ( set @ real ) ) ) ) ).
% LIM_cos_div_sin
thf(fact_7296_isCont__cot_H,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [A2: A,F2: A > A] :
( ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) @ F2 )
=> ( ( ( sin @ A @ ( F2 @ A2 ) )
!= ( zero_zero @ A ) )
=> ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) )
@ ^ [X: A] : ( cot @ A @ ( F2 @ X ) ) ) ) ) ) ).
% isCont_cot'
thf(fact_7297_DERIV__inverse__function,axiom,
! [F2: real > real,D4: real,G: real > real,X2: real,A2: real,B2: real] :
( ( has_field_derivative @ real @ F2 @ D4 @ ( topolo174197925503356063within @ real @ ( G @ X2 ) @ ( top_top @ ( set @ real ) ) ) )
=> ( ( D4
!= ( zero_zero @ real ) )
=> ( ( ord_less @ real @ A2 @ X2 )
=> ( ( ord_less @ real @ X2 @ B2 )
=> ( ! [Y3: real] :
( ( ord_less @ real @ A2 @ Y3 )
=> ( ( ord_less @ real @ Y3 @ B2 )
=> ( ( F2 @ ( G @ Y3 ) )
= Y3 ) ) )
=> ( ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) @ G )
=> ( has_field_derivative @ real @ G @ ( inverse_inverse @ real @ D4 ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ) ) ) ).
% DERIV_inverse_function
thf(fact_7298_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
@ ^ [I3: nat] : ( times_times @ A @ ( C2 @ I3 ) @ ( power_power @ A @ W2 @ I3 ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% isCont_polynom
thf(fact_7299_isCont__powser__converges__everywhere,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [C2: nat > A,X2: A] :
( ! [Y3: A] :
( summable @ A
@ ^ [N3: nat] : ( times_times @ A @ ( C2 @ N3 ) @ ( power_power @ A @ Y3 @ N3 ) ) )
=> ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ X2 @ ( top_top @ ( set @ A ) ) )
@ ^ [X: A] :
( suminf @ A
@ ^ [N3: nat] : ( times_times @ A @ ( C2 @ N3 ) @ ( power_power @ A @ X @ N3 ) ) ) ) ) ) ).
% isCont_powser_converges_everywhere
thf(fact_7300_isCont__arccos,axiom,
! [X2: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X2 )
=> ( ( ord_less @ real @ X2 @ ( one_one @ real ) )
=> ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) @ arccos ) ) ) ).
% isCont_arccos
thf(fact_7301_isCont__arcsin,axiom,
! [X2: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X2 )
=> ( ( ord_less @ real @ X2 @ ( one_one @ real ) )
=> ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) @ arcsin ) ) ) ).
% isCont_arcsin
thf(fact_7302_LIM__less__bound,axiom,
! [B2: real,X2: real,F2: real > real] :
( ( ord_less @ real @ B2 @ X2 )
=> ( ! [X3: real] :
( ( member @ real @ X3 @ ( set_or5935395276787703475ssThan @ real @ B2 @ X2 ) )
=> ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( F2 @ X3 ) ) )
=> ( ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) @ F2 )
=> ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( F2 @ X2 ) ) ) ) ) ).
% LIM_less_bound
thf(fact_7303_isCont__artanh,axiom,
! [X2: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X2 )
=> ( ( ord_less @ real @ X2 @ ( one_one @ real ) )
=> ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) @ ( artanh @ real ) ) ) ) ).
% isCont_artanh
thf(fact_7304_isCont__inverse__function,axiom,
! [D2: real,X2: real,G: real > real,F2: real > real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D2 )
=> ( ! [Z2: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ ( minus_minus @ real @ Z2 @ X2 ) ) @ D2 )
=> ( ( G @ ( F2 @ Z2 ) )
= Z2 ) )
=> ( ! [Z2: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ ( minus_minus @ real @ Z2 @ X2 ) ) @ D2 )
=> ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ Z2 @ ( top_top @ ( set @ real ) ) ) @ F2 ) )
=> ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ ( F2 @ X2 ) @ ( top_top @ ( set @ real ) ) ) @ G ) ) ) ) ).
% isCont_inverse_function
thf(fact_7305_GMVT_H,axiom,
! [A2: real,B2: real,F2: real > real,G: real > real,G4: real > real,F7: real > real] :
( ( ord_less @ real @ A2 @ B2 )
=> ( ! [Z2: real] :
( ( ord_less_eq @ real @ A2 @ Z2 )
=> ( ( ord_less_eq @ real @ Z2 @ B2 )
=> ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ Z2 @ ( top_top @ ( set @ real ) ) ) @ F2 ) ) )
=> ( ! [Z2: real] :
( ( ord_less_eq @ real @ A2 @ Z2 )
=> ( ( ord_less_eq @ real @ Z2 @ B2 )
=> ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ Z2 @ ( top_top @ ( set @ real ) ) ) @ G ) ) )
=> ( ! [Z2: real] :
( ( ord_less @ real @ A2 @ Z2 )
=> ( ( ord_less @ real @ Z2 @ B2 )
=> ( has_field_derivative @ real @ G @ ( G4 @ Z2 ) @ ( topolo174197925503356063within @ real @ Z2 @ ( top_top @ ( set @ real ) ) ) ) ) )
=> ( ! [Z2: real] :
( ( ord_less @ real @ A2 @ Z2 )
=> ( ( ord_less @ real @ Z2 @ B2 )
=> ( has_field_derivative @ real @ F2 @ ( F7 @ Z2 ) @ ( topolo174197925503356063within @ real @ Z2 @ ( top_top @ ( set @ real ) ) ) ) ) )
=> ? [C4: real] :
( ( ord_less @ real @ A2 @ C4 )
& ( ord_less @ real @ C4 @ B2 )
& ( ( times_times @ real @ ( minus_minus @ real @ ( F2 @ B2 ) @ ( F2 @ A2 ) ) @ ( G4 @ C4 ) )
= ( times_times @ real @ ( minus_minus @ real @ ( G @ B2 ) @ ( G @ A2 ) ) @ ( F7 @ C4 ) ) ) ) ) ) ) ) ) ).
% GMVT'
thf(fact_7306_floor__has__real__derivative,axiom,
! [A: $tType] :
( ( ( archim2362893244070406136eiling @ A )
& ( topolo2564578578187576103pology @ A ) )
=> ! [X2: real,F2: real > A] :
( ( topolo3448309680560233919inuous @ real @ A @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) @ F2 )
=> ( ~ ( member @ A @ ( F2 @ X2 ) @ ( ring_1_Ints @ A ) )
=> ( has_field_derivative @ real
@ ^ [X: real] : ( ring_1_of_int @ real @ ( archim6421214686448440834_floor @ A @ ( F2 @ X ) ) )
@ ( zero_zero @ real )
@ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ).
% floor_has_real_derivative
thf(fact_7307_isCont__powser,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [C2: nat > A,K6: A,X2: A] :
( ( summable @ A
@ ^ [N3: nat] : ( times_times @ A @ ( C2 @ N3 ) @ ( power_power @ A @ K6 @ N3 ) ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X2 ) @ ( real_V7770717601297561774m_norm @ A @ K6 ) )
=> ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ X2 @ ( top_top @ ( set @ A ) ) )
@ ^ [X: A] :
( suminf @ A
@ ^ [N3: nat] : ( times_times @ A @ ( C2 @ N3 ) @ ( power_power @ A @ X @ N3 ) ) ) ) ) ) ) ).
% isCont_powser
thf(fact_7308_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 ) ) )
@ ^ [X: A] :
( suminf @ Aa
@ ^ [N3: nat] : ( times_times @ Aa @ ( C2 @ N3 ) @ ( power_power @ Aa @ ( F2 @ X ) @ N3 ) ) ) ) ) ) ) ) ).
% isCont_powser'
thf(fact_7309_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 ) ) )
=> ! [N11: nat] :
( member @ real
@ ( suminf @ real
@ ^ [I3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I3 ) @ ( A2 @ I3 ) ) )
@ ( set_or1337092689740270186AtMost @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I3 ) @ ( A2 @ I3 ) )
@ ( set_ord_lessThan @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N11 ) ) )
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I3 ) @ ( A2 @ I3 ) )
@ ( set_ord_lessThan @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N11 ) @ ( one_one @ nat ) ) ) ) ) ) ) ) ) ).
% summable_Leibniz(2)
thf(fact_7310_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 ) )
=> ! [N11: nat] :
( member @ real
@ ( suminf @ real
@ ^ [I3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I3 ) @ ( A2 @ I3 ) ) )
@ ( set_or1337092689740270186AtMost @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I3 ) @ ( A2 @ I3 ) )
@ ( set_ord_lessThan @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N11 ) @ ( one_one @ nat ) ) ) )
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I3 ) @ ( A2 @ I3 ) )
@ ( set_ord_lessThan @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N11 ) ) ) ) ) ) ) ) ).
% summable_Leibniz(3)
thf(fact_7311_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_7312_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_7313_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_7314_seq__offset__neg,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [F2: nat > A,L2: A,K: nat] :
( ( filterlim @ nat @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L2 ) @ ( at_top @ nat ) )
=> ( filterlim @ nat @ A
@ ^ [I3: nat] : ( F2 @ ( minus_minus @ nat @ I3 @ K ) )
@ ( topolo7230453075368039082e_nhds @ A @ L2 )
@ ( at_top @ nat ) ) ) ) ).
% seq_offset_neg
thf(fact_7315_continuous__arctan,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [F3: filter @ A,F2: A > real] :
( ( topolo3448309680560233919inuous @ A @ real @ F3 @ F2 )
=> ( topolo3448309680560233919inuous @ A @ real @ F3
@ ^ [X: A] : ( arctan @ ( F2 @ X ) ) ) ) ) ).
% continuous_arctan
thf(fact_7316_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
@ ^ [X: A] : ( sqrt @ ( F2 @ X ) ) ) ) ) ).
% continuous_real_sqrt
thf(fact_7317_LIMSEQ__const__iff,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [K: A,L2: A] :
( ( filterlim @ nat @ A
@ ^ [N3: nat] : K
@ ( topolo7230453075368039082e_nhds @ A @ L2 )
@ ( at_top @ nat ) )
= ( K = L2 ) ) ) ).
% LIMSEQ_const_iff
thf(fact_7318_LIMSEQ__imp__Suc,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [F2: nat > A,L2: A] :
( ( filterlim @ nat @ A
@ ^ [N3: nat] : ( F2 @ ( suc @ N3 ) )
@ ( topolo7230453075368039082e_nhds @ A @ L2 )
@ ( at_top @ nat ) )
=> ( filterlim @ nat @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L2 ) @ ( at_top @ nat ) ) ) ) ).
% LIMSEQ_imp_Suc
thf(fact_7319_LIMSEQ__Suc,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [F2: nat > A,L2: A] :
( ( filterlim @ nat @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L2 ) @ ( at_top @ nat ) )
=> ( filterlim @ nat @ A
@ ^ [N3: nat] : ( F2 @ ( suc @ N3 ) )
@ ( topolo7230453075368039082e_nhds @ A @ L2 )
@ ( at_top @ nat ) ) ) ) ).
% LIMSEQ_Suc
thf(fact_7320_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
@ ^ [X: A] : ( root @ N @ ( F2 @ X ) ) ) ) ) ).
% continuous_real_root
thf(fact_7321_continuous__arsinh,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [F3: filter @ A,F2: A > real] :
( ( topolo3448309680560233919inuous @ A @ real @ F3 @ F2 )
=> ( topolo3448309680560233919inuous @ A @ real @ F3
@ ^ [X: A] : ( arsinh @ real @ ( F2 @ X ) ) ) ) ) ).
% continuous_arsinh
thf(fact_7322_continuous__rabs,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [F3: filter @ A,F2: A > real] :
( ( topolo3448309680560233919inuous @ A @ real @ F3 @ F2 )
=> ( topolo3448309680560233919inuous @ A @ real @ F3
@ ^ [X: A] : ( abs_abs @ real @ ( F2 @ X ) ) ) ) ) ).
% continuous_rabs
thf(fact_7323_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_7324_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_7325_lim__mono,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [N5: nat,X6: nat > A,Y7: nat > A,X2: A,Y2: A] :
( ! [N2: nat] :
( ( ord_less_eq @ nat @ N5 @ N2 )
=> ( ord_less_eq @ A @ ( X6 @ N2 ) @ ( Y7 @ N2 ) ) )
=> ( ( filterlim @ nat @ A @ X6 @ ( topolo7230453075368039082e_nhds @ A @ X2 ) @ ( at_top @ nat ) )
=> ( ( filterlim @ nat @ A @ Y7 @ ( topolo7230453075368039082e_nhds @ A @ Y2 ) @ ( at_top @ nat ) )
=> ( ord_less_eq @ A @ X2 @ Y2 ) ) ) ) ) ).
% lim_mono
thf(fact_7326_LIMSEQ__le,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [X6: nat > A,X2: A,Y7: nat > A,Y2: A] :
( ( filterlim @ nat @ A @ X6 @ ( topolo7230453075368039082e_nhds @ A @ X2 ) @ ( at_top @ nat ) )
=> ( ( filterlim @ nat @ A @ Y7 @ ( topolo7230453075368039082e_nhds @ A @ Y2 ) @ ( at_top @ nat ) )
=> ( ? [N9: nat] :
! [N2: nat] :
( ( ord_less_eq @ nat @ N9 @ N2 )
=> ( ord_less_eq @ A @ ( X6 @ N2 ) @ ( Y7 @ N2 ) ) )
=> ( ord_less_eq @ A @ X2 @ Y2 ) ) ) ) ) ).
% LIMSEQ_le
thf(fact_7327_Lim__bounded,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [F2: nat > A,L2: A,M7: nat,C6: A] :
( ( filterlim @ nat @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L2 ) @ ( at_top @ nat ) )
=> ( ! [N2: nat] :
( ( ord_less_eq @ nat @ M7 @ N2 )
=> ( ord_less_eq @ A @ ( F2 @ N2 ) @ C6 ) )
=> ( ord_less_eq @ A @ L2 @ C6 ) ) ) ) ).
% Lim_bounded
thf(fact_7328_Lim__bounded2,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [F2: nat > A,L2: A,N5: nat,C6: A] :
( ( filterlim @ nat @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L2 ) @ ( at_top @ nat ) )
=> ( ! [N2: nat] :
( ( ord_less_eq @ nat @ N5 @ N2 )
=> ( ord_less_eq @ A @ C6 @ ( F2 @ N2 ) ) )
=> ( ord_less_eq @ A @ C6 @ L2 ) ) ) ) ).
% Lim_bounded2
thf(fact_7329_LIMSEQ__le__const,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [X6: nat > A,X2: A,A2: A] :
( ( filterlim @ nat @ A @ X6 @ ( topolo7230453075368039082e_nhds @ A @ X2 ) @ ( at_top @ nat ) )
=> ( ? [N9: nat] :
! [N2: nat] :
( ( ord_less_eq @ nat @ N9 @ N2 )
=> ( ord_less_eq @ A @ A2 @ ( X6 @ N2 ) ) )
=> ( ord_less_eq @ A @ A2 @ X2 ) ) ) ) ).
% LIMSEQ_le_const
thf(fact_7330_LIMSEQ__le__const2,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [X6: nat > A,X2: A,A2: A] :
( ( filterlim @ nat @ A @ X6 @ ( topolo7230453075368039082e_nhds @ A @ X2 ) @ ( at_top @ nat ) )
=> ( ? [N9: nat] :
! [N2: nat] :
( ( ord_less_eq @ nat @ N9 @ N2 )
=> ( ord_less_eq @ A @ ( X6 @ N2 ) @ A2 ) )
=> ( ord_less_eq @ A @ X2 @ A2 ) ) ) ) ).
% LIMSEQ_le_const2
thf(fact_7331_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_7332_isCont__rabs,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [A2: A,F2: A > real] :
( ( topolo3448309680560233919inuous @ A @ real @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) @ F2 )
=> ( topolo3448309680560233919inuous @ A @ real @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) )
@ ^ [X: A] : ( abs_abs @ real @ ( F2 @ X ) ) ) ) ) ).
% isCont_rabs
thf(fact_7333_continuous__at__within__powr,axiom,
! [C: $tType] :
( ( topological_t2_space @ C )
=> ! [A2: C,S3: set @ C,F2: C > real,G: C > real] :
( ( topolo3448309680560233919inuous @ C @ real @ ( topolo174197925503356063within @ C @ A2 @ S3 ) @ F2 )
=> ( ( topolo3448309680560233919inuous @ C @ real @ ( topolo174197925503356063within @ C @ A2 @ S3 ) @ G )
=> ( ( ( F2 @ A2 )
!= ( zero_zero @ real ) )
=> ( topolo3448309680560233919inuous @ C @ real @ ( topolo174197925503356063within @ C @ A2 @ S3 )
@ ^ [X: C] : ( powr @ real @ ( F2 @ X ) @ ( G @ X ) ) ) ) ) ) ) ).
% continuous_at_within_powr
thf(fact_7334_continuous__within__ln,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [X2: A,S3: set @ A,F2: A > real] :
( ( topolo3448309680560233919inuous @ A @ real @ ( topolo174197925503356063within @ A @ X2 @ S3 ) @ F2 )
=> ( ( ( F2 @ X2 )
!= ( zero_zero @ real ) )
=> ( topolo3448309680560233919inuous @ A @ real @ ( topolo174197925503356063within @ A @ X2 @ S3 )
@ ^ [X: A] : ( ln_ln @ real @ ( F2 @ X ) ) ) ) ) ) ).
% continuous_within_ln
thf(fact_7335_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_7336_mult__nat__right__at__top,axiom,
! [C2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ C2 )
=> ( filterlim @ nat @ nat
@ ^ [X: nat] : ( times_times @ nat @ X @ C2 )
@ ( at_top @ nat )
@ ( at_top @ nat ) ) ) ).
% mult_nat_right_at_top
thf(fact_7337_monoseq__convergent,axiom,
! [X6: nat > real,B5: real] :
( ( topological_monoseq @ real @ X6 )
=> ( ! [I2: nat] : ( ord_less_eq @ real @ ( abs_abs @ real @ ( X6 @ I2 ) ) @ B5 )
=> ~ ! [L7: real] :
~ ( filterlim @ nat @ real @ X6 @ ( topolo7230453075368039082e_nhds @ real @ L7 ) @ ( at_top @ nat ) ) ) ) ).
% monoseq_convergent
thf(fact_7338_LIMSEQ__lessThan__iff__atMost,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [F2: ( set @ nat ) > A,X2: A] :
( ( filterlim @ nat @ A
@ ^ [N3: nat] : ( F2 @ ( set_ord_lessThan @ nat @ N3 ) )
@ ( topolo7230453075368039082e_nhds @ A @ X2 )
@ ( at_top @ nat ) )
= ( filterlim @ nat @ A
@ ^ [N3: nat] : ( F2 @ ( set_ord_atMost @ nat @ N3 ) )
@ ( topolo7230453075368039082e_nhds @ A @ X2 )
@ ( at_top @ nat ) ) ) ) ).
% LIMSEQ_lessThan_iff_atMost
thf(fact_7339_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_7340_isCont__powr,axiom,
! [C: $tType] :
( ( topological_t2_space @ C )
=> ! [A2: C,F2: C > real,G: C > real] :
( ( topolo3448309680560233919inuous @ C @ real @ ( topolo174197925503356063within @ C @ A2 @ ( top_top @ ( set @ C ) ) ) @ F2 )
=> ( ( topolo3448309680560233919inuous @ C @ real @ ( topolo174197925503356063within @ C @ A2 @ ( top_top @ ( set @ C ) ) ) @ G )
=> ( ( ( F2 @ A2 )
!= ( zero_zero @ real ) )
=> ( topolo3448309680560233919inuous @ C @ real @ ( topolo174197925503356063within @ C @ A2 @ ( top_top @ ( set @ C ) ) )
@ ^ [X: C] : ( powr @ real @ ( F2 @ X ) @ ( G @ X ) ) ) ) ) ) ) ).
% isCont_powr
thf(fact_7341_isCont__ln_H,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [X2: A,F2: A > real] :
( ( topolo3448309680560233919inuous @ A @ real @ ( topolo174197925503356063within @ A @ X2 @ ( top_top @ ( set @ A ) ) ) @ F2 )
=> ( ( ( F2 @ X2 )
!= ( zero_zero @ real ) )
=> ( topolo3448309680560233919inuous @ A @ real @ ( topolo174197925503356063within @ A @ X2 @ ( top_top @ ( set @ A ) ) )
@ ^ [X: A] : ( ln_ln @ real @ ( F2 @ X ) ) ) ) ) ) ).
% isCont_ln'
thf(fact_7342_monoseq__le,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [A2: nat > A,X2: A] :
( ( topological_monoseq @ A @ A2 )
=> ( ( filterlim @ nat @ A @ A2 @ ( topolo7230453075368039082e_nhds @ A @ X2 ) @ ( at_top @ nat ) )
=> ( ( ! [N11: nat] : ( ord_less_eq @ A @ ( A2 @ N11 ) @ X2 )
& ! [M4: nat,N11: nat] :
( ( ord_less_eq @ nat @ M4 @ N11 )
=> ( ord_less_eq @ A @ ( A2 @ M4 ) @ ( A2 @ N11 ) ) ) )
| ( ! [N11: nat] : ( ord_less_eq @ A @ X2 @ ( A2 @ N11 ) )
& ! [M4: nat,N11: nat] :
( ( ord_less_eq @ nat @ M4 @ N11 )
=> ( ord_less_eq @ A @ ( A2 @ N11 ) @ ( A2 @ M4 ) ) ) ) ) ) ) ) ).
% monoseq_le
thf(fact_7343_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_7344_LIMSEQ__linear,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [X6: nat > A,X2: A,L2: nat] :
( ( filterlim @ nat @ A @ X6 @ ( topolo7230453075368039082e_nhds @ A @ X2 ) @ ( at_top @ nat ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ L2 )
=> ( filterlim @ nat @ A
@ ^ [N3: nat] : ( X6 @ ( times_times @ nat @ N3 @ L2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ X2 )
@ ( at_top @ nat ) ) ) ) ) ).
% LIMSEQ_linear
thf(fact_7345_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_7346_LIMSEQ__SEQ__conv2,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo3112930676232923870pology @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [A2: A,F2: A > B,L2: B] :
( ! [S7: nat > A] :
( ( ! [N11: nat] :
( ( S7 @ N11 )
!= A2 )
& ( filterlim @ nat @ A @ S7 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ ( at_top @ nat ) ) )
=> ( filterlim @ nat @ B
@ ^ [N3: nat] : ( F2 @ ( S7 @ N3 ) )
@ ( topolo7230453075368039082e_nhds @ B @ L2 )
@ ( at_top @ nat ) ) )
=> ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L2 ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% LIMSEQ_SEQ_conv2
thf(fact_7347_LIMSEQ__SEQ__conv1,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo4958980785337419405_space @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [F2: A > B,L2: B,A2: A] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L2 ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) )
=> ! [S9: nat > A] :
( ( ! [N2: nat] :
( ( S9 @ N2 )
!= A2 )
& ( filterlim @ nat @ A @ S9 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ ( at_top @ nat ) ) )
=> ( filterlim @ nat @ B
@ ^ [N3: nat] : ( F2 @ ( S9 @ N3 ) )
@ ( topolo7230453075368039082e_nhds @ B @ L2 )
@ ( at_top @ nat ) ) ) ) ) ).
% LIMSEQ_SEQ_conv1
thf(fact_7348_LIMSEQ__SEQ__conv,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo3112930676232923870pology @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [A2: A,X6: A > B,L6: B] :
( ( ! [S10: nat > A] :
( ( ! [N3: nat] :
( ( S10 @ N3 )
!= A2 )
& ( filterlim @ nat @ A @ S10 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ ( at_top @ nat ) ) )
=> ( filterlim @ nat @ B
@ ^ [N3: nat] : ( X6 @ ( S10 @ N3 ) )
@ ( topolo7230453075368039082e_nhds @ B @ L6 )
@ ( at_top @ nat ) ) ) )
= ( filterlim @ A @ B @ X6 @ ( topolo7230453075368039082e_nhds @ B @ L6 ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% LIMSEQ_SEQ_conv
thf(fact_7349_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_7350_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_7351_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] :
( ! [N11: nat] : ( ord_less_eq @ real @ ( F2 @ N11 ) @ L4 )
& ( filterlim @ nat @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ L4 ) @ ( at_top @ nat ) )
& ! [N11: nat] : ( ord_less_eq @ real @ L4 @ ( G @ N11 ) )
& ( filterlim @ nat @ real @ G @ ( topolo7230453075368039082e_nhds @ real @ L4 ) @ ( at_top @ nat ) ) ) ) ) ) ) ).
% nested_sequence_unique
thf(fact_7352_LIMSEQ__inverse__zero,axiom,
! [X6: nat > real] :
( ! [R3: real] :
? [N9: nat] :
! [N2: nat] :
( ( ord_less_eq @ nat @ N9 @ N2 )
=> ( ord_less @ real @ R3 @ ( X6 @ N2 ) ) )
=> ( filterlim @ nat @ real
@ ^ [N3: nat] : ( inverse_inverse @ real @ ( X6 @ N3 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ ( at_top @ nat ) ) ) ).
% LIMSEQ_inverse_zero
thf(fact_7353_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_7354_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_7355_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_7356_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_7357_sums__def,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ( ( sums @ A )
= ( ^ [F4: nat > A,S5: A] :
( filterlim @ nat @ A
@ ^ [N3: nat] : ( groups7311177749621191930dd_sum @ nat @ A @ F4 @ ( set_ord_lessThan @ nat @ N3 ) )
@ ( topolo7230453075368039082e_nhds @ A @ S5 )
@ ( at_top @ nat ) ) ) ) ) ).
% sums_def
thf(fact_7358_sums__def__le,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ( ( sums @ A )
= ( ^ [F4: nat > A,S5: A] :
( filterlim @ nat @ A
@ ^ [N3: nat] : ( groups7311177749621191930dd_sum @ nat @ A @ F4 @ ( set_ord_atMost @ nat @ N3 ) )
@ ( topolo7230453075368039082e_nhds @ A @ S5 )
@ ( at_top @ nat ) ) ) ) ) ).
% sums_def_le
thf(fact_7359_continuous__at__within__log,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [A2: A,S3: set @ A,F2: A > real,G: A > real] :
( ( topolo3448309680560233919inuous @ A @ real @ ( topolo174197925503356063within @ A @ A2 @ S3 ) @ F2 )
=> ( ( topolo3448309680560233919inuous @ A @ real @ ( topolo174197925503356063within @ A @ A2 @ S3 ) @ 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 @ S3 )
@ ^ [X: A] : ( log @ ( F2 @ X ) @ ( G @ X ) ) ) ) ) ) ) ) ) ).
% continuous_at_within_log
thf(fact_7360_increasing__LIMSEQ,axiom,
! [F2: nat > real,L2: real] :
( ! [N2: nat] : ( ord_less_eq @ real @ ( F2 @ N2 ) @ ( F2 @ ( suc @ N2 ) ) )
=> ( ! [N2: nat] : ( ord_less_eq @ real @ ( F2 @ N2 ) @ L2 )
=> ( ! [E2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E2 )
=> ? [N11: nat] : ( ord_less_eq @ real @ L2 @ ( plus_plus @ real @ ( F2 @ N11 ) @ E2 ) ) )
=> ( filterlim @ nat @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ L2 ) @ ( at_top @ nat ) ) ) ) ) ).
% increasing_LIMSEQ
thf(fact_7361_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_7362_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_7363_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_7364_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_7365_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_7366_LIMSEQ__realpow__zero,axiom,
! [X2: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ord_less @ real @ X2 @ ( one_one @ real ) )
=> ( filterlim @ nat @ real @ ( power_power @ real @ X2 ) @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) ) ) ) ).
% LIMSEQ_realpow_zero
thf(fact_7367_LIMSEQ__divide__realpow__zero,axiom,
! [X2: real,A2: real] :
( ( ord_less @ real @ ( one_one @ real ) @ X2 )
=> ( filterlim @ nat @ real
@ ^ [N3: nat] : ( divide_divide @ real @ A2 @ ( power_power @ real @ X2 @ N3 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ ( at_top @ nat ) ) ) ).
% LIMSEQ_divide_realpow_zero
thf(fact_7368_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_7369_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_7370_LIMSEQ__inverse__realpow__zero,axiom,
! [X2: real] :
( ( ord_less @ real @ ( one_one @ real ) @ X2 )
=> ( filterlim @ nat @ real
@ ^ [N3: nat] : ( inverse_inverse @ real @ ( power_power @ real @ X2 @ N3 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ ( at_top @ nat ) ) ) ).
% LIMSEQ_inverse_realpow_zero
thf(fact_7371_sums__def_H,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ( ( sums @ A )
= ( ^ [F4: nat > A,S5: A] :
( filterlim @ nat @ A
@ ^ [N3: nat] : ( groups7311177749621191930dd_sum @ nat @ A @ F4 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N3 ) )
@ ( topolo7230453075368039082e_nhds @ A @ S5 )
@ ( at_top @ nat ) ) ) ) ) ).
% sums_def'
thf(fact_7372_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_7373_root__test__convergence,axiom,
! [A: $tType] :
( ( real_Vector_banach @ A )
=> ! [F2: nat > A,X2: real] :
( ( filterlim @ nat @ real
@ ^ [N3: nat] : ( root @ N3 @ ( real_V7770717601297561774m_norm @ A @ ( F2 @ N3 ) ) )
@ ( topolo7230453075368039082e_nhds @ real @ X2 )
@ ( at_top @ nat ) )
=> ( ( ord_less @ real @ X2 @ ( one_one @ real ) )
=> ( summable @ A @ F2 ) ) ) ) ).
% root_test_convergence
thf(fact_7374_summable__LIMSEQ,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topological_t2_space @ A ) )
=> ! [F2: nat > A] :
( ( summable @ A @ F2 )
=> ( filterlim @ nat @ A
@ ^ [N3: nat] : ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_ord_lessThan @ nat @ N3 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( suminf @ A @ F2 ) )
@ ( at_top @ nat ) ) ) ) ).
% summable_LIMSEQ
thf(fact_7375_summable__LIMSEQ_H,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topological_t2_space @ A ) )
=> ! [F2: nat > A] :
( ( summable @ A @ F2 )
=> ( filterlim @ nat @ A
@ ^ [N3: nat] : ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_ord_atMost @ nat @ N3 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( suminf @ A @ F2 ) )
@ ( at_top @ nat ) ) ) ) ).
% summable_LIMSEQ'
thf(fact_7376_isCont__log,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [A2: A,F2: A > real,G: A > real] :
( ( topolo3448309680560233919inuous @ A @ real @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) @ F2 )
=> ( ( topolo3448309680560233919inuous @ A @ real @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) @ G )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( F2 @ A2 ) )
=> ( ( ( F2 @ A2 )
!= ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( G @ A2 ) )
=> ( topolo3448309680560233919inuous @ A @ real @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) )
@ ^ [X: A] : ( log @ ( F2 @ X ) @ ( G @ X ) ) ) ) ) ) ) ) ) ).
% isCont_log
thf(fact_7377_LIMSEQ__D,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X6: nat > A,L6: A,R2: real] :
( ( filterlim @ nat @ A @ X6 @ ( topolo7230453075368039082e_nhds @ A @ L6 ) @ ( at_top @ nat ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ R2 )
=> ? [No: nat] :
! [N11: nat] :
( ( ord_less_eq @ nat @ No @ N11 )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( X6 @ N11 ) @ L6 ) ) @ R2 ) ) ) ) ) ).
% LIMSEQ_D
thf(fact_7378_LIMSEQ__I,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X6: 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 @ ( X6 @ N2 ) @ L6 ) ) @ R3 ) ) )
=> ( filterlim @ nat @ A @ X6 @ ( topolo7230453075368039082e_nhds @ A @ L6 ) @ ( at_top @ nat ) ) ) ) ).
% LIMSEQ_I
thf(fact_7379_LIMSEQ__iff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X6: nat > A,L6: A] :
( ( filterlim @ nat @ A @ X6 @ ( topolo7230453075368039082e_nhds @ A @ L6 ) @ ( at_top @ nat ) )
= ( ! [R5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R5 )
=> ? [No3: nat] :
! [N3: nat] :
( ( ord_less_eq @ nat @ No3 @ N3 )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( X6 @ N3 ) @ L6 ) ) @ R5 ) ) ) ) ) ) ).
% LIMSEQ_iff
thf(fact_7380_LIMSEQ__power__zero,axiom,
! [A: $tType] :
( ( real_V2822296259951069270ebra_1 @ A )
=> ! [X2: A] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X2 ) @ ( one_one @ real ) )
=> ( filterlim @ nat @ A @ ( power_power @ A @ X2 ) @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ ( at_top @ nat ) ) ) ) ).
% LIMSEQ_power_zero
thf(fact_7381_tendsto__exp__limit__sequentially,axiom,
! [X2: real] :
( filterlim @ nat @ real
@ ^ [N3: nat] : ( power_power @ real @ ( plus_plus @ real @ ( one_one @ real ) @ ( divide_divide @ real @ X2 @ ( semiring_1_of_nat @ real @ N3 ) ) ) @ N3 )
@ ( topolo7230453075368039082e_nhds @ real @ ( exp @ real @ X2 ) )
@ ( at_top @ nat ) ) ).
% tendsto_exp_limit_sequentially
thf(fact_7382_tendsto__power__zero,axiom,
! [A: $tType,B: $tType] :
( ( real_V2822296259951069270ebra_1 @ A )
=> ! [F2: B > nat,F3: filter @ B,X2: A] :
( ( filterlim @ B @ nat @ F2 @ ( at_top @ nat ) @ F3 )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X2 ) @ ( one_one @ real ) )
=> ( filterlim @ B @ A
@ ^ [Y: B] : ( power_power @ A @ X2 @ ( F2 @ Y ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ F3 ) ) ) ) ).
% tendsto_power_zero
thf(fact_7383_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_7384_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_7385_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_7386_field__derivative__lim__unique,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,Df: A,Z: A,S3: nat > A,A2: A] :
( ( has_field_derivative @ A @ F2 @ Df @ ( topolo174197925503356063within @ A @ Z @ ( top_top @ ( set @ A ) ) ) )
=> ( ( filterlim @ nat @ A @ S3 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ ( at_top @ nat ) )
=> ( ! [N2: nat] :
( ( S3 @ N2 )
!= ( zero_zero @ A ) )
=> ( ( filterlim @ nat @ A
@ ^ [N3: nat] : ( divide_divide @ A @ ( minus_minus @ A @ ( F2 @ ( plus_plus @ A @ Z @ ( S3 @ N3 ) ) ) @ ( F2 @ Z ) ) @ ( S3 @ N3 ) )
@ ( topolo7230453075368039082e_nhds @ A @ A2 )
@ ( at_top @ nat ) )
=> ( Df = A2 ) ) ) ) ) ) ).
% field_derivative_lim_unique
thf(fact_7387_powser__times__n__limit__0,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V8999393235501362500lgebra @ A ) )
=> ! [X2: A] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X2 ) @ ( one_one @ real ) )
=> ( filterlim @ nat @ A
@ ^ [N3: nat] : ( times_times @ A @ ( semiring_1_of_nat @ A @ N3 ) @ ( power_power @ A @ X2 @ N3 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ ( at_top @ nat ) ) ) ) ).
% powser_times_n_limit_0
thf(fact_7388_lim__n__over__pown,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X2: A] :
( ( ord_less @ real @ ( one_one @ real ) @ ( real_V7770717601297561774m_norm @ A @ X2 ) )
=> ( filterlim @ nat @ A
@ ^ [N3: nat] : ( divide_divide @ A @ ( semiring_1_of_nat @ A @ N3 ) @ ( power_power @ A @ X2 @ N3 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ ( at_top @ nat ) ) ) ) ).
% lim_n_over_pown
thf(fact_7389_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_7390_cos__diff__limit__1,axiom,
! [Theta: nat > real,Theta2: real] :
( ( filterlim @ nat @ real
@ ^ [J3: nat] : ( cos @ real @ ( minus_minus @ real @ ( Theta @ J3 ) @ Theta2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( one_one @ real ) )
@ ( at_top @ nat ) )
=> ~ ! [K3: nat > int] :
~ ( filterlim @ nat @ real
@ ^ [J3: nat] : ( minus_minus @ real @ ( Theta @ J3 ) @ ( times_times @ real @ ( ring_1_of_int @ real @ ( K3 @ J3 ) ) @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) ) )
@ ( topolo7230453075368039082e_nhds @ real @ Theta2 )
@ ( at_top @ nat ) ) ) ).
% cos_diff_limit_1
thf(fact_7391_cos__limit__1,axiom,
! [Theta: nat > real] :
( ( filterlim @ nat @ real
@ ^ [J3: nat] : ( cos @ real @ ( Theta @ J3 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( one_one @ real ) )
@ ( at_top @ nat ) )
=> ? [K3: nat > int] :
( filterlim @ nat @ real
@ ^ [J3: nat] : ( minus_minus @ real @ ( Theta @ J3 ) @ ( times_times @ real @ ( ring_1_of_int @ real @ ( K3 @ J3 ) ) @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ ( at_top @ nat ) ) ) ).
% cos_limit_1
thf(fact_7392_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
@ ^ [I3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I3 ) @ ( A2 @ I3 ) )
@ ( set_ord_lessThan @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) ) )
@ ( topolo7230453075368039082e_nhds @ real
@ ( suminf @ real
@ ^ [I3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I3 ) @ ( A2 @ I3 ) ) ) )
@ ( at_top @ nat ) ) ) ) ).
% summable_Leibniz(4)
thf(fact_7393_zeroseq__arctan__series,axiom,
! [X2: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ X2 ) @ ( 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 @ X2 @ ( 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_7394_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
@ ^ [I3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I3 ) @ ( A2 @ I3 ) )
@ ( set_ord_lessThan @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) ) )
@ ( topolo7230453075368039082e_nhds @ real
@ ( suminf @ real
@ ^ [I3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I3 ) @ ( A2 @ I3 ) ) ) )
@ ( at_top @ nat ) ) ) ) ) ).
% summable_Leibniz'(3)
thf(fact_7395_summable__Leibniz_H_I2_J,axiom,
! [A2: nat > real,N: nat] :
( ( filterlim @ nat @ real @ A2 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) )
=> ( ! [N2: nat] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( A2 @ N2 ) )
=> ( ! [N2: nat] : ( ord_less_eq @ real @ ( A2 @ ( suc @ N2 ) ) @ ( A2 @ N2 ) )
=> ( ord_less_eq @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I3 ) @ ( A2 @ I3 ) )
@ ( set_ord_lessThan @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) )
@ ( suminf @ real
@ ^ [I3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I3 ) @ ( A2 @ I3 ) ) ) ) ) ) ) ).
% summable_Leibniz'(2)
thf(fact_7396_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] :
( ! [N11: nat] :
( ord_less_eq @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I3 ) @ ( A2 @ I3 ) )
@ ( set_ord_lessThan @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N11 ) ) )
@ L4 )
& ( filterlim @ nat @ real
@ ^ [N3: nat] :
( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I3 ) @ ( A2 @ I3 ) )
@ ( set_ord_lessThan @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) ) )
@ ( topolo7230453075368039082e_nhds @ real @ L4 )
@ ( at_top @ nat ) )
& ! [N11: nat] :
( ord_less_eq @ real @ L4
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I3 ) @ ( A2 @ I3 ) )
@ ( set_ord_lessThan @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N11 ) @ ( one_one @ nat ) ) ) ) )
& ( filterlim @ nat @ real
@ ^ [N3: nat] :
( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I3 ) @ ( A2 @ I3 ) )
@ ( set_ord_lessThan @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) @ ( one_one @ nat ) ) ) )
@ ( topolo7230453075368039082e_nhds @ real @ L4 )
@ ( at_top @ nat ) ) ) ) ) ) ).
% sums_alternating_upper_lower
thf(fact_7397_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
@ ^ [I3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I3 ) @ ( A2 @ I3 ) )
@ ( set_ord_lessThan @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) @ ( one_one @ nat ) ) ) )
@ ( topolo7230453075368039082e_nhds @ real
@ ( suminf @ real
@ ^ [I3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I3 ) @ ( A2 @ I3 ) ) ) )
@ ( at_top @ nat ) ) ) ) ).
% summable_Leibniz(5)
thf(fact_7398_summable__Leibniz_H_I4_J,axiom,
! [A2: nat > real,N: nat] :
( ( filterlim @ nat @ real @ A2 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) )
=> ( ! [N2: nat] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( A2 @ N2 ) )
=> ( ! [N2: nat] : ( ord_less_eq @ real @ ( A2 @ ( suc @ N2 ) ) @ ( A2 @ N2 ) )
=> ( ord_less_eq @ real
@ ( suminf @ real
@ ^ [I3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I3 ) @ ( A2 @ I3 ) ) )
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I3 ) @ ( A2 @ I3 ) )
@ ( set_ord_lessThan @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ ( one_one @ nat ) ) ) ) ) ) ) ) ).
% summable_Leibniz'(4)
thf(fact_7399_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
@ ^ [I3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I3 ) @ ( A2 @ I3 ) )
@ ( set_ord_lessThan @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) @ ( one_one @ nat ) ) ) )
@ ( topolo7230453075368039082e_nhds @ real
@ ( suminf @ real
@ ^ [I3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I3 ) @ ( A2 @ I3 ) ) ) )
@ ( at_top @ nat ) ) ) ) ) ).
% summable_Leibniz'(5)
thf(fact_7400_filterlim__sequentially__Suc,axiom,
! [A: $tType,F2: nat > A,F3: filter @ A] :
( ( filterlim @ nat @ A
@ ^ [X: nat] : ( F2 @ ( suc @ X ) )
@ F3
@ ( at_top @ nat ) )
= ( filterlim @ nat @ A @ F2 @ F3 @ ( at_top @ nat ) ) ) ).
% filterlim_sequentially_Suc
thf(fact_7401_has__derivative__at2,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,F7: A > B,X2: A] :
( ( has_derivative @ A @ B @ F2 @ F7 @ ( topolo174197925503356063within @ A @ X2 @ ( top_top @ ( set @ A ) ) ) )
= ( ( real_V3181309239436604168linear @ A @ B @ F7 )
& ( filterlim @ A @ B
@ ^ [Y: A] : ( real_V8093663219630862766scaleR @ B @ ( divide_divide @ real @ ( one_one @ real ) @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ Y @ X2 ) ) ) @ ( minus_minus @ B @ ( F2 @ Y ) @ ( plus_plus @ B @ ( F2 @ X2 ) @ ( F7 @ ( minus_minus @ A @ Y @ X2 ) ) ) ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ ( topolo174197925503356063within @ A @ X2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% has_derivative_at2
thf(fact_7402_bounded__linear_Ocontinuous,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( ( topological_t2_space @ C )
& ( real_V822414075346904944vector @ B )
& ( real_V822414075346904944vector @ A ) )
=> ! [F2: A > B,F3: filter @ C,G: C > A] :
( ( real_V3181309239436604168linear @ A @ B @ F2 )
=> ( ( topolo3448309680560233919inuous @ C @ A @ F3 @ G )
=> ( topolo3448309680560233919inuous @ C @ B @ F3
@ ^ [X: C] : ( F2 @ ( G @ X ) ) ) ) ) ) ).
% bounded_linear.continuous
thf(fact_7403_bounded__linear_Otendsto,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( ( real_V822414075346904944vector @ B )
& ( real_V822414075346904944vector @ A ) )
=> ! [F2: A > B,G: C > A,A2: A,F3: filter @ C] :
( ( real_V3181309239436604168linear @ A @ B @ F2 )
=> ( ( filterlim @ C @ A @ G @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ F3 )
=> ( filterlim @ C @ B
@ ^ [X: C] : ( F2 @ ( G @ X ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( F2 @ A2 ) )
@ F3 ) ) ) ) ).
% bounded_linear.tendsto
thf(fact_7404_bounded__linear_Obounded,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B] :
( ( real_V3181309239436604168linear @ A @ B @ F2 )
=> ? [K9: real] :
! [X4: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( F2 @ X4 ) ) @ ( times_times @ real @ ( real_V7770717601297561774m_norm @ A @ X4 ) @ K9 ) ) ) ) ).
% bounded_linear.bounded
thf(fact_7405_bounded__linear_Osuminf,axiom,
! [A: $tType,B: $tType] :
( ( ( real_V822414075346904944vector @ B )
& ( real_V822414075346904944vector @ A ) )
=> ! [F2: A > B,X6: nat > A] :
( ( real_V3181309239436604168linear @ A @ B @ F2 )
=> ( ( summable @ A @ X6 )
=> ( ( F2 @ ( suminf @ A @ X6 ) )
= ( suminf @ B
@ ^ [N3: nat] : ( F2 @ ( X6 @ N3 ) ) ) ) ) ) ) ).
% bounded_linear.suminf
thf(fact_7406_bounded__linear__minus,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B] :
( ( real_V3181309239436604168linear @ A @ B @ F2 )
=> ( real_V3181309239436604168linear @ A @ B
@ ^ [X: A] : ( uminus_uminus @ B @ ( F2 @ X ) ) ) ) ) ).
% bounded_linear_minus
thf(fact_7407_bounded__linear__sub,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,G: A > B] :
( ( real_V3181309239436604168linear @ A @ B @ F2 )
=> ( ( real_V3181309239436604168linear @ A @ B @ G )
=> ( real_V3181309239436604168linear @ A @ B
@ ^ [X: A] : ( minus_minus @ B @ ( F2 @ X ) @ ( G @ X ) ) ) ) ) ) ).
% bounded_linear_sub
thf(fact_7408_bounded__linear__scaleR__const,axiom,
! [B: $tType,C: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V822414075346904944vector @ B ) )
=> ! [G: C > real,X2: B] :
( ( real_V3181309239436604168linear @ C @ real @ G )
=> ( real_V3181309239436604168linear @ C @ B
@ ^ [X: C] : ( real_V8093663219630862766scaleR @ B @ ( G @ X ) @ X2 ) ) ) ) ).
% bounded_linear_scaleR_const
thf(fact_7409_bounded__linear__scaleR__left,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X2: A] :
( real_V3181309239436604168linear @ real @ A
@ ^ [R5: real] : ( real_V8093663219630862766scaleR @ A @ R5 @ X2 ) ) ) ).
% bounded_linear_scaleR_left
thf(fact_7410_bounded__linear__const__scaleR,axiom,
! [B: $tType,C: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V822414075346904944vector @ B ) )
=> ! [G: C > B,R2: real] :
( ( real_V3181309239436604168linear @ C @ B @ G )
=> ( real_V3181309239436604168linear @ C @ B
@ ^ [X: C] : ( real_V8093663219630862766scaleR @ B @ R2 @ ( G @ X ) ) ) ) ) ).
% bounded_linear_const_scaleR
thf(fact_7411_bounded__linear__scaleR__right,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [R2: real] : ( real_V3181309239436604168linear @ A @ A @ ( real_V8093663219630862766scaleR @ A @ R2 ) ) ) ).
% bounded_linear_scaleR_right
thf(fact_7412_bounded__linear__compose,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V822414075346904944vector @ B )
& ( real_V822414075346904944vector @ A ) )
=> ! [F2: A > B,G: C > A] :
( ( real_V3181309239436604168linear @ A @ B @ F2 )
=> ( ( real_V3181309239436604168linear @ C @ A @ G )
=> ( real_V3181309239436604168linear @ C @ B
@ ^ [X: C] : ( F2 @ ( G @ X ) ) ) ) ) ) ).
% bounded_linear_compose
thf(fact_7413_bounded__linear__ident,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ( real_V3181309239436604168linear @ A @ A
@ ^ [X: A] : X ) ) ).
% bounded_linear_ident
thf(fact_7414_bounded__linear_Osummable,axiom,
! [A: $tType,B: $tType] :
( ( ( real_V822414075346904944vector @ B )
& ( real_V822414075346904944vector @ A ) )
=> ! [F2: A > B,X6: nat > A] :
( ( real_V3181309239436604168linear @ A @ B @ F2 )
=> ( ( summable @ A @ X6 )
=> ( summable @ B
@ ^ [N3: nat] : ( F2 @ ( X6 @ N3 ) ) ) ) ) ) ).
% bounded_linear.summable
thf(fact_7415_bounded__linear__of__real,axiom,
! [A: $tType] :
( ( ( real_V2191834092415804123ebra_1 @ A )
& ( real_V822414075346904944vector @ A ) )
=> ( real_V3181309239436604168linear @ real @ A @ ( real_Vector_of_real @ A ) ) ) ).
% bounded_linear_of_real
thf(fact_7416_bounded__linear_Osums,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,X6: nat > A,A2: A] :
( ( real_V3181309239436604168linear @ A @ B @ F2 )
=> ( ( sums @ A @ X6 @ A2 )
=> ( sums @ B
@ ^ [N3: nat] : ( F2 @ ( X6 @ N3 ) )
@ ( F2 @ A2 ) ) ) ) ) ).
% bounded_linear.sums
thf(fact_7417_bounded__linear__zero,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ( real_V3181309239436604168linear @ A @ B
@ ^ [X: A] : ( zero_zero @ B ) ) ) ).
% bounded_linear_zero
thf(fact_7418_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
@ ^ [X: A] : ( plus_plus @ B @ ( F2 @ X ) @ ( G @ X ) ) ) ) ) ) ).
% bounded_linear_add
thf(fact_7419_bounded__linear__mult__right,axiom,
! [A: $tType] :
( ( real_V4412858255891104859lgebra @ A )
=> ! [X2: A] : ( real_V3181309239436604168linear @ A @ A @ ( times_times @ A @ X2 ) ) ) ).
% bounded_linear_mult_right
thf(fact_7420_bounded__linear__mult__const,axiom,
! [A: $tType,C: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V4412858255891104859lgebra @ A ) )
=> ! [G: C > A,Y2: A] :
( ( real_V3181309239436604168linear @ C @ A @ G )
=> ( real_V3181309239436604168linear @ C @ A
@ ^ [X: C] : ( times_times @ A @ ( G @ X ) @ Y2 ) ) ) ) ).
% bounded_linear_mult_const
thf(fact_7421_bounded__linear__const__mult,axiom,
! [A: $tType,C: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V4412858255891104859lgebra @ A ) )
=> ! [G: C > A,X2: A] :
( ( real_V3181309239436604168linear @ C @ A @ G )
=> ( real_V3181309239436604168linear @ C @ A
@ ^ [X: C] : ( times_times @ A @ X2 @ ( G @ X ) ) ) ) ) ).
% bounded_linear_const_mult
thf(fact_7422_bounded__linear__mult__left,axiom,
! [A: $tType] :
( ( real_V4412858255891104859lgebra @ A )
=> ! [Y2: A] :
( real_V3181309239436604168linear @ A @ A
@ ^ [X: A] : ( times_times @ A @ X @ Y2 ) ) ) ).
% bounded_linear_mult_left
thf(fact_7423_bounded__linear__divide,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [Y2: A] :
( real_V3181309239436604168linear @ A @ A
@ ^ [X: A] : ( divide_divide @ A @ X @ Y2 ) ) ) ).
% bounded_linear_divide
thf(fact_7424_bounded__linear_OCauchy,axiom,
! [A: $tType,B: $tType] :
( ( ( real_V822414075346904944vector @ B )
& ( real_V822414075346904944vector @ A ) )
=> ! [F2: A > B,X6: nat > A] :
( ( real_V3181309239436604168linear @ A @ B @ F2 )
=> ( ( topolo3814608138187158403Cauchy @ A @ X6 )
=> ( topolo3814608138187158403Cauchy @ B
@ ^ [N3: nat] : ( F2 @ ( X6 @ N3 ) ) ) ) ) ) ).
% bounded_linear.Cauchy
thf(fact_7425_bounded__linear__sum,axiom,
! [I6: $tType,B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [I5: set @ I6,F2: I6 > A > B] :
( ! [I2: I6] :
( ( member @ I6 @ I2 @ I5 )
=> ( real_V3181309239436604168linear @ A @ B @ ( F2 @ I2 ) ) )
=> ( real_V3181309239436604168linear @ A @ B
@ ^ [X: A] :
( groups7311177749621191930dd_sum @ I6 @ B
@ ^ [I3: I6] : ( F2 @ I3 @ X )
@ I5 ) ) ) ) ).
% bounded_linear_sum
thf(fact_7426_real__bounded__linear,axiom,
( ( real_V3181309239436604168linear @ real @ real )
= ( ^ [F4: real > real] :
? [C3: real] :
( F4
= ( ^ [X: real] : ( times_times @ real @ X @ C3 ) ) ) ) ) ).
% real_bounded_linear
thf(fact_7427_bounded__linear_Ohas__derivative,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,G: C > A,G4: C > A,F3: filter @ C] :
( ( real_V3181309239436604168linear @ A @ B @ F2 )
=> ( ( has_derivative @ C @ A @ G @ G4 @ F3 )
=> ( has_derivative @ C @ B
@ ^ [X: C] : ( F2 @ ( G @ X ) )
@ ^ [X: C] : ( F2 @ ( G4 @ X ) )
@ F3 ) ) ) ) ).
% bounded_linear.has_derivative
thf(fact_7428_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
@ ^ [X: C] : ( F2 @ ( G @ X ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ F3 ) ) ) ) ).
% bounded_linear.tendsto_zero
thf(fact_7429_bounded__linear_OisCont,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( ( topological_t2_space @ C )
& ( real_V822414075346904944vector @ B )
& ( real_V822414075346904944vector @ A ) )
=> ! [F2: A > B,A2: C,G: C > A] :
( ( real_V3181309239436604168linear @ A @ B @ F2 )
=> ( ( topolo3448309680560233919inuous @ C @ A @ ( topolo174197925503356063within @ C @ A2 @ ( top_top @ ( set @ C ) ) ) @ G )
=> ( topolo3448309680560233919inuous @ C @ B @ ( topolo174197925503356063within @ C @ A2 @ ( top_top @ ( set @ C ) ) )
@ ^ [X: C] : ( F2 @ ( G @ X ) ) ) ) ) ) ).
% bounded_linear.isCont
thf(fact_7430_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 )
& ! [X4: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( F2 @ X4 ) ) @ ( times_times @ real @ ( real_V7770717601297561774m_norm @ A @ X4 ) @ K9 ) ) ) ) ) ).
% bounded_linear.nonneg_bounded
thf(fact_7431_filterlim__ident,axiom,
! [A: $tType,F3: filter @ A] :
( filterlim @ A @ A
@ ^ [X: A] : X
@ F3
@ F3 ) ).
% filterlim_ident
thf(fact_7432_filterlim__compose,axiom,
! [B: $tType,A: $tType,C: $tType,G: A > B,F32: filter @ B,F22: filter @ A,F2: C > A,F1: filter @ C] :
( ( filterlim @ A @ B @ G @ F32 @ F22 )
=> ( ( filterlim @ C @ A @ F2 @ F22 @ F1 )
=> ( filterlim @ C @ B
@ ^ [X: C] : ( G @ ( F2 @ X ) )
@ F32
@ F1 ) ) ) ).
% filterlim_compose
thf(fact_7433_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 )
& ! [X4: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( F2 @ X4 ) ) @ ( times_times @ real @ ( real_V7770717601297561774m_norm @ A @ X4 ) @ K9 ) ) ) ) ) ).
% bounded_linear.pos_bounded
thf(fact_7434_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_7435_filterlim__Suc,axiom,
filterlim @ nat @ nat @ suc @ ( at_top @ nat ) @ ( at_top @ nat ) ).
% filterlim_Suc
thf(fact_7436_has__derivative__iff__norm,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,F7: A > B,X2: A,S3: set @ A] :
( ( has_derivative @ A @ B @ F2 @ F7 @ ( topolo174197925503356063within @ A @ X2 @ S3 ) )
= ( ( real_V3181309239436604168linear @ A @ B @ F7 )
& ( filterlim @ A @ real
@ ^ [Y: A] : ( divide_divide @ real @ ( real_V7770717601297561774m_norm @ B @ ( minus_minus @ B @ ( minus_minus @ B @ ( F2 @ Y ) @ ( F2 @ X2 ) ) @ ( F7 @ ( minus_minus @ A @ Y @ X2 ) ) ) ) @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ Y @ X2 ) ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ ( topolo174197925503356063within @ A @ X2 @ S3 ) ) ) ) ) ).
% has_derivative_iff_norm
thf(fact_7437_has__derivative__at__within,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,F7: A > B,X2: A,S3: set @ A] :
( ( has_derivative @ A @ B @ F2 @ F7 @ ( topolo174197925503356063within @ A @ X2 @ S3 ) )
= ( ( real_V3181309239436604168linear @ A @ B @ F7 )
& ( filterlim @ A @ B
@ ^ [Y: A] : ( real_V8093663219630862766scaleR @ B @ ( inverse_inverse @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ Y @ X2 ) ) ) @ ( minus_minus @ B @ ( minus_minus @ B @ ( F2 @ Y ) @ ( F2 @ X2 ) ) @ ( F7 @ ( minus_minus @ A @ Y @ X2 ) ) ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ ( topolo174197925503356063within @ A @ X2 @ S3 ) ) ) ) ) ).
% has_derivative_at_within
thf(fact_7438_has__derivativeI,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F7: A > B,X2: A,F2: A > B,S3: set @ A] :
( ( real_V3181309239436604168linear @ A @ B @ F7 )
=> ( ( filterlim @ A @ B
@ ^ [Y: A] : ( real_V8093663219630862766scaleR @ B @ ( inverse_inverse @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ Y @ X2 ) ) ) @ ( minus_minus @ B @ ( minus_minus @ B @ ( F2 @ Y ) @ ( F2 @ X2 ) ) @ ( F7 @ ( minus_minus @ A @ Y @ X2 ) ) ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ ( topolo174197925503356063within @ A @ X2 @ S3 ) )
=> ( has_derivative @ A @ B @ F2 @ F7 @ ( topolo174197925503356063within @ A @ X2 @ S3 ) ) ) ) ) ).
% has_derivativeI
thf(fact_7439_has__derivative__iff__Ex,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,F7: A > B,X2: A] :
( ( has_derivative @ A @ B @ F2 @ F7 @ ( topolo174197925503356063within @ A @ X2 @ ( top_top @ ( set @ A ) ) ) )
= ( ( real_V3181309239436604168linear @ A @ B @ F7 )
& ? [E4: A > B] :
( ! [H: A] :
( ( F2 @ ( plus_plus @ A @ X2 @ H ) )
= ( plus_plus @ B @ ( plus_plus @ B @ ( F2 @ X2 ) @ ( F7 @ H ) ) @ ( E4 @ H ) ) )
& ( filterlim @ A @ real
@ ^ [H: A] : ( divide_divide @ real @ ( real_V7770717601297561774m_norm @ B @ ( E4 @ H ) ) @ ( real_V7770717601297561774m_norm @ A @ H ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ).
% has_derivative_iff_Ex
thf(fact_7440_has__derivative__within,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,F7: A > B,X2: A,S3: set @ A] :
( ( has_derivative @ A @ B @ F2 @ F7 @ ( topolo174197925503356063within @ A @ X2 @ S3 ) )
= ( ( real_V3181309239436604168linear @ A @ B @ F7 )
& ( filterlim @ A @ B
@ ^ [Y: A] : ( real_V8093663219630862766scaleR @ B @ ( divide_divide @ real @ ( one_one @ real ) @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ Y @ X2 ) ) ) @ ( minus_minus @ B @ ( F2 @ Y ) @ ( plus_plus @ B @ ( F2 @ X2 ) @ ( F7 @ ( minus_minus @ A @ Y @ X2 ) ) ) ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ ( topolo174197925503356063within @ A @ X2 @ S3 ) ) ) ) ) ).
% has_derivative_within
thf(fact_7441_has__derivative__at,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,D4: A > B,X2: A] :
( ( has_derivative @ A @ B @ F2 @ D4 @ ( topolo174197925503356063within @ A @ X2 @ ( top_top @ ( set @ A ) ) ) )
= ( ( real_V3181309239436604168linear @ A @ B @ D4 )
& ( filterlim @ A @ real
@ ^ [H: A] : ( divide_divide @ real @ ( real_V7770717601297561774m_norm @ B @ ( minus_minus @ B @ ( minus_minus @ B @ ( F2 @ ( plus_plus @ A @ X2 @ H ) ) @ ( F2 @ X2 ) ) @ ( D4 @ H ) ) ) @ ( real_V7770717601297561774m_norm @ A @ H ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% has_derivative_at
thf(fact_7442_has__derivative__def,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ( ( has_derivative @ A @ B )
= ( ^ [F4: A > B,F8: A > B,F9: filter @ A] :
( ( real_V3181309239436604168linear @ A @ B @ F8 )
& ( filterlim @ A @ B
@ ^ [Y: A] :
( real_V8093663219630862766scaleR @ B
@ ( inverse_inverse @ real
@ ( real_V7770717601297561774m_norm @ A
@ ( minus_minus @ A @ Y
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F9
@ ^ [X: A] : X ) ) ) )
@ ( minus_minus @ B
@ ( minus_minus @ B @ ( F4 @ Y )
@ ( F4
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F9
@ ^ [X: A] : X ) ) )
@ ( F8
@ ( minus_minus @ A @ Y
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F9
@ ^ [X: A] : X ) ) ) ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ F9 ) ) ) ) ) ).
% has_derivative_def
thf(fact_7443_has__derivative__at__within__iff__Ex,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [X2: A,S: set @ A,F2: A > B,F7: A > B] :
( ( member @ A @ X2 @ S )
=> ( ( topolo1002775350975398744n_open @ A @ S )
=> ( ( has_derivative @ A @ B @ F2 @ F7 @ ( topolo174197925503356063within @ A @ X2 @ S ) )
= ( ( real_V3181309239436604168linear @ A @ B @ F7 )
& ? [E4: A > B] :
( ! [H: A] :
( ( member @ A @ ( plus_plus @ A @ X2 @ H ) @ S )
=> ( ( F2 @ ( plus_plus @ A @ X2 @ H ) )
= ( plus_plus @ B @ ( plus_plus @ B @ ( F2 @ X2 ) @ ( F7 @ H ) ) @ ( E4 @ H ) ) ) )
& ( filterlim @ A @ real
@ ^ [H: A] : ( divide_divide @ real @ ( real_V7770717601297561774m_norm @ B @ ( E4 @ H ) ) @ ( real_V7770717601297561774m_norm @ A @ H ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ) ) ).
% has_derivative_at_within_iff_Ex
thf(fact_7444_lim__const,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [A2: A] :
( ( topolo3827282254853284352ce_Lim @ nat @ A @ ( at_top @ nat )
@ ^ [M5: nat] : A2 )
= A2 ) ) ).
% lim_const
thf(fact_7445_open__Collect__conj,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [P: A > $o,Q: A > $o] :
( ( topolo1002775350975398744n_open @ A @ ( collect @ A @ P ) )
=> ( ( topolo1002775350975398744n_open @ A @ ( collect @ A @ Q ) )
=> ( topolo1002775350975398744n_open @ A
@ ( collect @ A
@ ^ [X: A] :
( ( P @ X )
& ( Q @ X ) ) ) ) ) ) ) ).
% open_Collect_conj
thf(fact_7446_open__Collect__disj,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [P: A > $o,Q: A > $o] :
( ( topolo1002775350975398744n_open @ A @ ( collect @ A @ P ) )
=> ( ( topolo1002775350975398744n_open @ A @ ( collect @ A @ Q ) )
=> ( topolo1002775350975398744n_open @ A
@ ( collect @ A
@ ^ [X: A] :
( ( P @ X )
| ( Q @ X ) ) ) ) ) ) ) ).
% open_Collect_disj
thf(fact_7447_open__Collect__const,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [P: $o] :
( topolo1002775350975398744n_open @ A
@ ( collect @ A
@ ^ [X: A] : P ) ) ) ).
% open_Collect_const
thf(fact_7448_open__right,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [S: set @ A,X2: A,Y2: A] :
( ( topolo1002775350975398744n_open @ A @ S )
=> ( ( member @ A @ X2 @ S )
=> ( ( ord_less @ A @ X2 @ Y2 )
=> ? [B4: A] :
( ( ord_less @ A @ X2 @ B4 )
& ( ord_less_eq @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ X2 @ B4 ) @ S ) ) ) ) ) ) ).
% open_right
thf(fact_7449_at__within__open__subset,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [A2: A,S: set @ A,T3: set @ A] :
( ( member @ A @ A2 @ S )
=> ( ( topolo1002775350975398744n_open @ A @ S )
=> ( ( ord_less_eq @ ( set @ A ) @ S @ T3 )
=> ( ( topolo174197925503356063within @ A @ A2 @ T3 )
= ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ).
% at_within_open_subset
thf(fact_7450_openI,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [S: set @ A] :
( ! [X3: A] :
( ( member @ A @ X3 @ S )
=> ? [T8: set @ A] :
( ( topolo1002775350975398744n_open @ A @ T8 )
& ( member @ A @ X3 @ T8 )
& ( ord_less_eq @ ( set @ A ) @ T8 @ S ) ) )
=> ( topolo1002775350975398744n_open @ A @ S ) ) ) ).
% openI
thf(fact_7451_open__subopen,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ( ( topolo1002775350975398744n_open @ A )
= ( ^ [S10: set @ A] :
! [X: A] :
( ( member @ A @ X @ S10 )
=> ? [T6: set @ A] :
( ( topolo1002775350975398744n_open @ A @ T6 )
& ( member @ A @ X @ T6 )
& ( ord_less_eq @ ( set @ A ) @ T6 @ S10 ) ) ) ) ) ) ).
% open_subopen
thf(fact_7452_first__countable__basis,axiom,
! [A: $tType] :
( ( topolo3112930676232923870pology @ A )
=> ! [X2: A] :
? [A8: nat > ( set @ A )] :
( ! [I4: nat] :
( ( member @ A @ X2 @ ( A8 @ I4 ) )
& ( topolo1002775350975398744n_open @ A @ ( A8 @ I4 ) ) )
& ! [S9: set @ A] :
( ( ( topolo1002775350975398744n_open @ A @ S9 )
& ( member @ A @ X2 @ S9 ) )
=> ? [I2: nat] : ( ord_less_eq @ ( set @ A ) @ ( A8 @ I2 ) @ S9 ) ) ) ) ).
% first_countable_basis
thf(fact_7453_Lim__ident__at,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [X2: A,S3: set @ A] :
( ( ( topolo174197925503356063within @ A @ X2 @ S3 )
!= ( bot_bot @ ( filter @ A ) ) )
=> ( ( topolo3827282254853284352ce_Lim @ A @ A @ ( topolo174197925503356063within @ A @ X2 @ S3 )
@ ^ [X: A] : X )
= X2 ) ) ) ).
% Lim_ident_at
thf(fact_7454_lim__explicit,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [F2: nat > A,F0: A] :
( ( filterlim @ nat @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ F0 ) @ ( at_top @ nat ) )
= ( ! [S10: set @ A] :
( ( topolo1002775350975398744n_open @ A @ S10 )
=> ( ( member @ A @ F0 @ S10 )
=> ? [N8: nat] :
! [N3: nat] :
( ( ord_less_eq @ nat @ N8 @ N3 )
=> ( member @ A @ ( F2 @ N3 ) @ S10 ) ) ) ) ) ) ) ).
% lim_explicit
thf(fact_7455_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
@ ^ [X: A] : X ) )
!= ( zero_zero @ B ) )
=> ( topolo3448309680560233919inuous @ A @ B @ F3
@ ^ [X: A] : ( divide_divide @ B @ ( F2 @ X ) @ ( G @ X ) ) ) ) ) ) ) ).
% continuous_divide
thf(fact_7456_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
@ ^ [X: A] : X ) )
!= ( zero_zero @ B ) )
=> ( topolo3448309680560233919inuous @ A @ B @ F3
@ ^ [X: A] : ( inverse_inverse @ B @ ( F2 @ X ) ) ) ) ) ) ).
% continuous_inverse
thf(fact_7457_continuous__def,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ( ( topolo3448309680560233919inuous @ A @ B )
= ( ^ [F9: filter @ A,F4: A > B] :
( filterlim @ A @ B @ F4
@ ( topolo7230453075368039082e_nhds @ B
@ ( F4
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F9
@ ^ [X: A] : X ) ) )
@ F9 ) ) ) ) ).
% continuous_def
thf(fact_7458_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
@ ^ [X: A] : X ) )
!= ( zero_zero @ B ) )
=> ( topolo3448309680560233919inuous @ A @ B @ F3
@ ^ [X: A] : ( sgn_sgn @ B @ ( F2 @ X ) ) ) ) ) ) ).
% continuous_sgn
thf(fact_7459_t2__space__class_OLim__def,axiom,
! [A: $tType,F: $tType] :
( ( topological_t2_space @ A )
=> ( ( topolo3827282254853284352ce_Lim @ F @ A )
= ( ^ [A7: filter @ F,F4: F > A] :
( the @ A
@ ^ [L: A] : ( filterlim @ F @ A @ F4 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ A7 ) ) ) ) ) ).
% t2_space_class.Lim_def
thf(fact_7460_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
@ ^ [X: A] : X ) )
!= ( zero_zero @ real ) )
=> ( topolo3448309680560233919inuous @ A @ real @ F3
@ ^ [X: A] : ( powr @ real @ ( F2 @ X ) @ ( G @ X ) ) ) ) ) ) ) ).
% continuous_powr
thf(fact_7461_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
@ ^ [X: A] : X ) )
!= ( zero_zero @ real ) )
=> ( topolo3448309680560233919inuous @ A @ real @ F3
@ ^ [X: A] : ( ln_ln @ real @ ( F2 @ X ) ) ) ) ) ) ).
% continuous_ln
thf(fact_7462_suminf__eq__lim,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topological_t2_space @ A ) )
=> ( ( suminf @ A )
= ( ^ [F4: nat > A] :
( topolo3827282254853284352ce_Lim @ nat @ A @ ( at_top @ nat )
@ ^ [N3: nat] : ( groups7311177749621191930dd_sum @ nat @ A @ F4 @ ( set_ord_lessThan @ nat @ N3 ) ) ) ) ) ) ).
% suminf_eq_lim
thf(fact_7463_Topological__Spaces_Olim__def,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [X6: nat > A] :
( ( topolo3827282254853284352ce_Lim @ nat @ A @ ( at_top @ nat ) @ X6 )
= ( the @ A
@ ^ [L8: A] : ( filterlim @ nat @ A @ X6 @ ( topolo7230453075368039082e_nhds @ A @ L8 ) @ ( at_top @ nat ) ) ) ) ) ).
% Topological_Spaces.lim_def
thf(fact_7464_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
@ ^ [X: A] : X ) ) )
!= ( zero_zero @ A ) )
=> ( topolo3448309680560233919inuous @ A @ A @ F3
@ ^ [X: A] : ( tan @ A @ ( F2 @ X ) ) ) ) ) ) ).
% continuous_tan
thf(fact_7465_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
@ ^ [X: A] : X ) ) )
!= ( zero_zero @ A ) )
=> ( topolo3448309680560233919inuous @ A @ A @ F3
@ ^ [X: A] : ( cot @ A @ ( F2 @ X ) ) ) ) ) ) ).
% continuous_cot
thf(fact_7466_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
@ ^ [X: C] : X ) ) )
!= ( zero_zero @ A ) )
=> ( topolo3448309680560233919inuous @ C @ A @ F3
@ ^ [X: C] : ( tanh @ A @ ( F2 @ X ) ) ) ) ) ) ).
% continuous_tanh
thf(fact_7467_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
@ ^ [X: A] : X ) ) )
=> ( topolo3448309680560233919inuous @ A @ real @ F3
@ ^ [X: A] : ( arcosh @ real @ ( F2 @ X ) ) ) ) ) ) ).
% continuous_arcosh
thf(fact_7468_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
@ ^ [X: A] : X ) ) )
=> ( ( ( F2
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F3
@ ^ [X: A] : X ) )
!= ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real )
@ ( G
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F3
@ ^ [X: A] : X ) ) )
=> ( topolo3448309680560233919inuous @ A @ real @ F3
@ ^ [X: A] : ( log @ ( F2 @ X ) @ ( G @ X ) ) ) ) ) ) ) ) ) ).
% continuous_log
thf(fact_7469_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
@ ^ [X: A] : X ) )
@ ( set_or5935395276787703475ssThan @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ ( one_one @ real ) ) )
=> ( topolo3448309680560233919inuous @ A @ real @ F3
@ ^ [X: A] : ( artanh @ real @ ( F2 @ X ) ) ) ) ) ) ).
% continuous_artanh
thf(fact_7470_tendsto__offset__zero__iff,axiom,
! [C: $tType,D6: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( topolo4958980785337419405_space @ D6 )
& ( zero @ C ) )
=> ! [A2: A,S: set @ A,F2: A > D6,L6: D6] :
( ( nO_MATCH @ C @ A @ ( zero_zero @ C ) @ A2 )
=> ( ( member @ A @ A2 @ S )
=> ( ( topolo1002775350975398744n_open @ A @ S )
=> ( ( filterlim @ A @ D6 @ F2 @ ( topolo7230453075368039082e_nhds @ D6 @ L6 ) @ ( topolo174197925503356063within @ A @ A2 @ S ) )
= ( filterlim @ A @ D6
@ ^ [H: A] : ( F2 @ ( plus_plus @ A @ A2 @ H ) )
@ ( topolo7230453075368039082e_nhds @ D6 @ L6 )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ) ).
% tendsto_offset_zero_iff
thf(fact_7471_has__derivativeI__sandwich,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [E: real,F7: A > B,S3: set @ A,X2: A,F2: A > B,H7: A > real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E )
=> ( ( real_V3181309239436604168linear @ A @ B @ F7 )
=> ( ! [Y3: A] :
( ( member @ A @ Y3 @ S3 )
=> ( ( Y3 != X2 )
=> ( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ Y3 @ X2 ) @ E )
=> ( ord_less_eq @ real @ ( divide_divide @ real @ ( real_V7770717601297561774m_norm @ B @ ( minus_minus @ B @ ( minus_minus @ B @ ( F2 @ Y3 ) @ ( F2 @ X2 ) ) @ ( F7 @ ( minus_minus @ A @ Y3 @ X2 ) ) ) ) @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ Y3 @ X2 ) ) ) @ ( H7 @ Y3 ) ) ) ) )
=> ( ( filterlim @ A @ real @ H7 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( topolo174197925503356063within @ A @ X2 @ S3 ) )
=> ( has_derivative @ A @ B @ F2 @ F7 @ ( topolo174197925503356063within @ A @ X2 @ S3 ) ) ) ) ) ) ) ).
% has_derivativeI_sandwich
thf(fact_7472_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_7473_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_7474_dist__eq__0__iff,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X2: A,Y2: A] :
( ( ( real_V557655796197034286t_dist @ A @ X2 @ Y2 )
= ( zero_zero @ real ) )
= ( X2 = Y2 ) ) ) ).
% dist_eq_0_iff
thf(fact_7475_dist__self,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X2: A] :
( ( real_V557655796197034286t_dist @ A @ X2 @ X2 )
= ( zero_zero @ real ) ) ) ).
% dist_self
thf(fact_7476_dist__0__norm,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X2: A] :
( ( real_V557655796197034286t_dist @ A @ ( zero_zero @ A ) @ X2 )
= ( real_V7770717601297561774m_norm @ A @ X2 ) ) ) ).
% dist_0_norm
thf(fact_7477_zero__less__dist__iff,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X2: A,Y2: A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( real_V557655796197034286t_dist @ A @ X2 @ Y2 ) )
= ( X2 != Y2 ) ) ) ).
% zero_less_dist_iff
thf(fact_7478_dist__le__zero__iff,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X2: A,Y2: A] :
( ( ord_less_eq @ real @ ( real_V557655796197034286t_dist @ A @ X2 @ Y2 ) @ ( zero_zero @ real ) )
= ( X2 = Y2 ) ) ) ).
% dist_le_zero_iff
thf(fact_7479_div__add__self2__no__field,axiom,
! [B: $tType,A: $tType] :
( ( ( euclid4440199948858584721cancel @ A )
& ( field @ B ) )
=> ! [X2: B,B2: A,A2: A] :
( ( nO_MATCH @ B @ A @ X2 @ 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_7480_div__add__self1__no__field,axiom,
! [B: $tType,A: $tType] :
( ( ( euclid4440199948858584721cancel @ A )
& ( field @ B ) )
=> ! [X2: B,B2: A,A2: A] :
( ( nO_MATCH @ B @ A @ X2 @ 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_7481_dist__scaleR,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X2: real,A2: A,Y2: real] :
( ( real_V557655796197034286t_dist @ A @ ( real_V8093663219630862766scaleR @ A @ X2 @ A2 ) @ ( real_V8093663219630862766scaleR @ A @ Y2 @ A2 ) )
= ( times_times @ real @ ( abs_abs @ real @ ( minus_minus @ real @ X2 @ Y2 ) ) @ ( real_V7770717601297561774m_norm @ A @ A2 ) ) ) ) ).
% dist_scaleR
thf(fact_7482_open__dist,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ( ( topolo1002775350975398744n_open @ A )
= ( ^ [S10: set @ A] :
! [X: A] :
( ( member @ A @ X @ S10 )
=> ? [E4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E4 )
& ! [Y: A] :
( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ Y @ X ) @ E4 )
=> ( member @ A @ Y @ S10 ) ) ) ) ) ) ) ).
% open_dist
thf(fact_7483_open__ball,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X2: A,D2: real] :
( topolo1002775350975398744n_open @ A
@ ( collect @ A
@ ^ [Y: A] : ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X2 @ Y ) @ D2 ) ) ) ) ).
% open_ball
thf(fact_7484_continuous__dist,axiom,
! [A: $tType,D6: $tType] :
( ( ( topological_t2_space @ D6 )
& ( real_V7819770556892013058_space @ A ) )
=> ! [F3: filter @ D6,F2: D6 > A,G: D6 > A] :
( ( topolo3448309680560233919inuous @ D6 @ A @ F3 @ F2 )
=> ( ( topolo3448309680560233919inuous @ D6 @ A @ F3 @ G )
=> ( topolo3448309680560233919inuous @ D6 @ real @ F3
@ ^ [X: D6] : ( real_V557655796197034286t_dist @ A @ ( F2 @ X ) @ ( G @ X ) ) ) ) ) ) ).
% continuous_dist
thf(fact_7485_abs__dist__diff__le,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [A2: A,B2: A,C2: A] : ( ord_less_eq @ real @ ( abs_abs @ real @ ( minus_minus @ real @ ( real_V557655796197034286t_dist @ A @ A2 @ B2 ) @ ( real_V557655796197034286t_dist @ A @ B2 @ C2 ) ) ) @ ( real_V557655796197034286t_dist @ A @ A2 @ C2 ) ) ) ).
% abs_dist_diff_le
thf(fact_7486_dist__triangle__le,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X2: A,Z: A,Y2: A,E: real] :
( ( ord_less_eq @ real @ ( plus_plus @ real @ ( real_V557655796197034286t_dist @ A @ X2 @ Z ) @ ( real_V557655796197034286t_dist @ A @ Y2 @ Z ) ) @ E )
=> ( ord_less_eq @ real @ ( real_V557655796197034286t_dist @ A @ X2 @ Y2 ) @ E ) ) ) ).
% dist_triangle_le
thf(fact_7487_dist__triangle3,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X2: A,Y2: A,A2: A] : ( ord_less_eq @ real @ ( real_V557655796197034286t_dist @ A @ X2 @ Y2 ) @ ( plus_plus @ real @ ( real_V557655796197034286t_dist @ A @ A2 @ X2 ) @ ( real_V557655796197034286t_dist @ A @ A2 @ Y2 ) ) ) ) ).
% dist_triangle3
thf(fact_7488_dist__triangle2,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X2: A,Y2: A,Z: A] : ( ord_less_eq @ real @ ( real_V557655796197034286t_dist @ A @ X2 @ Y2 ) @ ( plus_plus @ real @ ( real_V557655796197034286t_dist @ A @ X2 @ Z ) @ ( real_V557655796197034286t_dist @ A @ Y2 @ Z ) ) ) ) ).
% dist_triangle2
thf(fact_7489_dist__triangle,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X2: A,Z: A,Y2: A] : ( ord_less_eq @ real @ ( real_V557655796197034286t_dist @ A @ X2 @ Z ) @ ( plus_plus @ real @ ( real_V557655796197034286t_dist @ A @ X2 @ Y2 ) @ ( real_V557655796197034286t_dist @ A @ Y2 @ Z ) ) ) ) ).
% dist_triangle
thf(fact_7490_zero__le__dist,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X2: A,Y2: A] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( real_V557655796197034286t_dist @ A @ X2 @ Y2 ) ) ) ).
% zero_le_dist
thf(fact_7491_dist__triangle__less__add,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X16: A,Y2: A,E1: real,X23: A,E22: real] :
( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X16 @ Y2 ) @ E1 )
=> ( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X23 @ Y2 ) @ E22 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X16 @ X23 ) @ ( plus_plus @ real @ E1 @ E22 ) ) ) ) ) ).
% dist_triangle_less_add
thf(fact_7492_dist__triangle__lt,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X2: A,Z: A,Y2: A,E: real] :
( ( ord_less @ real @ ( plus_plus @ real @ ( real_V557655796197034286t_dist @ A @ X2 @ Z ) @ ( real_V557655796197034286t_dist @ A @ Y2 @ Z ) ) @ E )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X2 @ Y2 ) @ E ) ) ) ).
% dist_triangle_lt
thf(fact_7493_norm__conv__dist,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ( ( real_V7770717601297561774m_norm @ A )
= ( ^ [X: A] : ( real_V557655796197034286t_dist @ A @ X @ ( zero_zero @ A ) ) ) ) ) ).
% norm_conv_dist
thf(fact_7494_dist__not__less__zero,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X2: A,Y2: A] :
~ ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X2 @ Y2 ) @ ( zero_zero @ real ) ) ) ).
% dist_not_less_zero
thf(fact_7495_dist__pos__lt,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X2: A,Y2: A] :
( ( X2 != Y2 )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( real_V557655796197034286t_dist @ A @ X2 @ Y2 ) ) ) ) ).
% dist_pos_lt
thf(fact_7496_has__field__derivative__transform__within,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,F7: A,A2: A,S: set @ A,D2: real,G: A > A] :
( ( has_field_derivative @ A @ F2 @ F7 @ ( topolo174197925503356063within @ A @ A2 @ S ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ D2 )
=> ( ( member @ A @ A2 @ S )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ S )
=> ( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X3 @ A2 ) @ D2 )
=> ( ( F2 @ X3 )
= ( G @ X3 ) ) ) )
=> ( has_field_derivative @ A @ G @ F7 @ ( topolo174197925503356063within @ A @ A2 @ S ) ) ) ) ) ) ) ).
% has_field_derivative_transform_within
thf(fact_7497_has__derivative__transform__within,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,F7: A > B,X2: A,S3: set @ A,D2: real,G: A > B] :
( ( has_derivative @ A @ B @ F2 @ F7 @ ( topolo174197925503356063within @ A @ X2 @ S3 ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ D2 )
=> ( ( member @ A @ X2 @ S3 )
=> ( ! [X10: A] :
( ( member @ A @ X10 @ S3 )
=> ( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X10 @ X2 ) @ D2 )
=> ( ( F2 @ X10 )
= ( G @ X10 ) ) ) )
=> ( has_derivative @ A @ B @ G @ F7 @ ( topolo174197925503356063within @ A @ X2 @ S3 ) ) ) ) ) ) ) ).
% has_derivative_transform_within
thf(fact_7498_Cauchy__def,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ( ( topolo3814608138187158403Cauchy @ A )
= ( ^ [X7: nat > A] :
! [E4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E4 )
=> ? [M9: nat] :
! [M5: nat] :
( ( ord_less_eq @ nat @ M9 @ M5 )
=> ! [N3: nat] :
( ( ord_less_eq @ nat @ M9 @ N3 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( X7 @ M5 ) @ ( X7 @ N3 ) ) @ E4 ) ) ) ) ) ) ) ).
% Cauchy_def
thf(fact_7499_Cauchy__altdef2,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ( ( topolo3814608138187158403Cauchy @ A )
= ( ^ [S5: nat > A] :
! [E4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E4 )
=> ? [N8: nat] :
! [N3: nat] :
( ( ord_less_eq @ nat @ N8 @ N3 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( S5 @ N3 ) @ ( S5 @ N8 ) ) @ E4 ) ) ) ) ) ) ).
% Cauchy_altdef2
thf(fact_7500_metric__CauchyD,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X6: nat > A,E: real] :
( ( topolo3814608138187158403Cauchy @ A @ X6 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ E )
=> ? [M10: nat] :
! [M4: nat] :
( ( ord_less_eq @ nat @ M10 @ M4 )
=> ! [N11: nat] :
( ( ord_less_eq @ nat @ M10 @ N11 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( X6 @ M4 ) @ ( X6 @ N11 ) ) @ E ) ) ) ) ) ) ).
% metric_CauchyD
thf(fact_7501_metric__CauchyI,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X6: nat > A] :
( ! [E2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E2 )
=> ? [M13: nat] :
! [M3: nat] :
( ( ord_less_eq @ nat @ M13 @ M3 )
=> ! [N2: nat] :
( ( ord_less_eq @ nat @ M13 @ N2 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( X6 @ M3 ) @ ( X6 @ N2 ) ) @ E2 ) ) ) )
=> ( topolo3814608138187158403Cauchy @ A @ X6 ) ) ) ).
% metric_CauchyI
thf(fact_7502_tendsto__dist,axiom,
! [A: $tType,B: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [F2: B > A,L2: A,F3: filter @ B,G: B > A,M: A] :
( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L2 ) @ F3 )
=> ( ( filterlim @ B @ A @ G @ ( topolo7230453075368039082e_nhds @ A @ M ) @ F3 )
=> ( filterlim @ B @ real
@ ^ [X: B] : ( real_V557655796197034286t_dist @ A @ ( F2 @ X ) @ ( G @ X ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( real_V557655796197034286t_dist @ A @ L2 @ M ) )
@ F3 ) ) ) ) ).
% tendsto_dist
thf(fact_7503_scale__right__distrib__NO__MATCH,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [X2: A,Y2: A,A2: real] :
( ( nO_MATCH @ A @ real @ ( divide_divide @ A @ X2 @ Y2 ) @ A2 )
=> ( ( real_V8093663219630862766scaleR @ A @ A2 @ ( plus_plus @ A @ X2 @ Y2 ) )
= ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ X2 ) @ ( real_V8093663219630862766scaleR @ A @ A2 @ Y2 ) ) ) ) ) ).
% scale_right_distrib_NO_MATCH
thf(fact_7504_scale__right__diff__distrib__NO__MATCH,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [X2: A,Y2: A,A2: real] :
( ( nO_MATCH @ A @ real @ ( divide_divide @ A @ X2 @ Y2 ) @ A2 )
=> ( ( real_V8093663219630862766scaleR @ A @ A2 @ ( minus_minus @ A @ X2 @ Y2 ) )
= ( minus_minus @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ X2 ) @ ( real_V8093663219630862766scaleR @ A @ A2 @ Y2 ) ) ) ) ) ).
% scale_right_diff_distrib_NO_MATCH
thf(fact_7505_distrib__left__NO__MATCH,axiom,
! [B: $tType,A: $tType] :
( ( semiring @ A )
=> ! [X2: B,Y2: B,A2: A,B2: A,C2: A] :
( ( nO_MATCH @ B @ A @ ( divide_divide @ B @ X2 @ Y2 ) @ 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_7506_distrib__right__NO__MATCH,axiom,
! [B: $tType,A: $tType] :
( ( semiring @ A )
=> ! [X2: B,Y2: B,C2: A,A2: A,B2: A] :
( ( nO_MATCH @ B @ A @ ( divide_divide @ B @ X2 @ Y2 ) @ 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_7507_left__diff__distrib__NO__MATCH,axiom,
! [B: $tType,A: $tType] :
( ( ring @ A )
=> ! [X2: B,Y2: B,C2: A,A2: A,B2: A] :
( ( nO_MATCH @ B @ A @ ( divide_divide @ B @ X2 @ Y2 ) @ 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_7508_right__diff__distrib__NO__MATCH,axiom,
! [B: $tType,A: $tType] :
( ( ring @ A )
=> ! [X2: B,Y2: B,A2: A,B2: A,C2: A] :
( ( nO_MATCH @ B @ A @ ( divide_divide @ B @ X2 @ Y2 ) @ 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_7509_dist__triangle__half__r,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [Y2: A,X16: A,E: real,X23: A] :
( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ Y2 @ X16 ) @ ( divide_divide @ real @ E @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ Y2 @ X23 ) @ ( divide_divide @ real @ E @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X16 @ X23 ) @ E ) ) ) ) ).
% dist_triangle_half_r
thf(fact_7510_dist__triangle__half__l,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X16: A,Y2: A,E: real,X23: A] :
( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X16 @ Y2 ) @ ( divide_divide @ real @ E @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X23 @ Y2 ) @ ( divide_divide @ real @ E @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X16 @ X23 ) @ E ) ) ) ) ).
% dist_triangle_half_l
thf(fact_7511_metric__LIM__imp__LIM,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( ( topolo4958980785337419405_space @ C )
& ( real_V7819770556892013058_space @ B )
& ( real_V7819770556892013058_space @ A ) )
=> ! [F2: C > A,L2: A,A2: C,G: C > B,M: B] :
( ( filterlim @ C @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L2 ) @ ( topolo174197925503356063within @ C @ A2 @ ( top_top @ ( set @ C ) ) ) )
=> ( ! [X3: C] :
( ( X3 != A2 )
=> ( ord_less_eq @ real @ ( real_V557655796197034286t_dist @ B @ ( G @ X3 ) @ M ) @ ( real_V557655796197034286t_dist @ A @ ( F2 @ X3 ) @ L2 ) ) )
=> ( filterlim @ C @ B @ G @ ( topolo7230453075368039082e_nhds @ B @ M ) @ ( topolo174197925503356063within @ C @ A2 @ ( top_top @ ( set @ C ) ) ) ) ) ) ) ).
% metric_LIM_imp_LIM
thf(fact_7512_dist__triangle__third,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X16: A,X23: A,E: real,X33: A,X42: A] :
( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X16 @ X23 ) @ ( divide_divide @ real @ E @ ( numeral_numeral @ real @ ( bit1 @ one2 ) ) ) )
=> ( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X23 @ X33 ) @ ( divide_divide @ real @ E @ ( numeral_numeral @ real @ ( bit1 @ one2 ) ) ) )
=> ( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X33 @ X42 ) @ ( divide_divide @ real @ E @ ( numeral_numeral @ real @ ( bit1 @ one2 ) ) ) )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X16 @ X42 ) @ E ) ) ) ) ) ).
% dist_triangle_third
thf(fact_7513_Lim__transform__within,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V7819770556892013058_space @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [F2: A > B,L2: B,X2: A,S: set @ A,D2: real,G: A > B] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L2 ) @ ( topolo174197925503356063within @ A @ X2 @ S ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ D2 )
=> ( ! [X10: A] :
( ( member @ A @ X10 @ S )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( real_V557655796197034286t_dist @ A @ X10 @ X2 ) )
=> ( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X10 @ X2 ) @ D2 )
=> ( ( F2 @ X10 )
= ( G @ X10 ) ) ) ) )
=> ( filterlim @ A @ B @ G @ ( topolo7230453075368039082e_nhds @ B @ L2 ) @ ( topolo174197925503356063within @ A @ X2 @ S ) ) ) ) ) ) ).
% Lim_transform_within
thf(fact_7514_filterlim__transform__within,axiom,
! [B: $tType,A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [G: A > B,G5: filter @ B,X2: A,S: set @ A,F3: filter @ B,D2: real,F2: A > B] :
( ( filterlim @ A @ B @ G @ G5 @ ( topolo174197925503356063within @ A @ X2 @ S ) )
=> ( ( ord_less_eq @ ( filter @ B ) @ G5 @ F3 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ D2 )
=> ( ! [X10: A] :
( ( member @ A @ X10 @ S )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( real_V557655796197034286t_dist @ A @ X10 @ X2 ) )
=> ( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X10 @ X2 ) @ D2 )
=> ( ( F2 @ X10 )
= ( G @ X10 ) ) ) ) )
=> ( filterlim @ A @ B @ F2 @ F3 @ ( topolo174197925503356063within @ A @ X2 @ S ) ) ) ) ) ) ) ).
% filterlim_transform_within
thf(fact_7515_CauchyI_H,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X6: nat > A] :
( ! [E2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E2 )
=> ? [M13: nat] :
! [M3: nat] :
( ( ord_less_eq @ nat @ M13 @ M3 )
=> ! [N2: nat] :
( ( ord_less @ nat @ M3 @ N2 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( X6 @ M3 ) @ ( X6 @ N2 ) ) @ E2 ) ) ) )
=> ( topolo3814608138187158403Cauchy @ A @ X6 ) ) ) ).
% CauchyI'
thf(fact_7516_Cauchy__altdef,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ( ( topolo3814608138187158403Cauchy @ A )
= ( ^ [F4: nat > A] :
! [E4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E4 )
=> ? [M9: nat] :
! [M5: nat] :
( ( ord_less_eq @ nat @ M9 @ M5 )
=> ! [N3: nat] :
( ( ord_less @ nat @ M5 @ N3 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( F4 @ M5 ) @ ( F4 @ N3 ) ) @ E4 ) ) ) ) ) ) ) ).
% Cauchy_altdef
thf(fact_7517_dist__of__nat,axiom,
! [A: $tType] :
( ( real_V2822296259951069270ebra_1 @ A )
=> ! [M: nat,N: nat] :
( ( real_V557655796197034286t_dist @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) )
= ( ring_1_of_int @ real @ ( abs_abs @ int @ ( minus_minus @ int @ ( semiring_1_of_nat @ int @ M ) @ ( semiring_1_of_nat @ int @ N ) ) ) ) ) ) ).
% dist_of_nat
thf(fact_7518_tendsto__dist__iff,axiom,
! [B: $tType,A: $tType] :
( ( real_V7819770556892013058_space @ B )
=> ! [F2: A > B,L2: B,F3: filter @ A] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L2 ) @ F3 )
= ( filterlim @ A @ real
@ ^ [X: A] : ( real_V557655796197034286t_dist @ B @ ( F2 @ X ) @ L2 )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ F3 ) ) ) ).
% tendsto_dist_iff
thf(fact_7519_metric__LIM__equal2,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V7819770556892013058_space @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [G: A > B,L2: B,A2: A,R: real,F2: A > B] :
( ( filterlim @ A @ B @ G @ ( topolo7230453075368039082e_nhds @ B @ L2 ) @ ( 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 @ L2 ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ).
% metric_LIM_equal2
thf(fact_7520_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 )
=> ? [S8: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ S8 )
& ! [X3: A] :
( ( ( X3 != A2 )
& ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X3 @ A2 ) @ S8 ) )
=> ( 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_7521_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 )
& ! [X4: A] :
( ( ( X4 != A2 )
& ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X4 @ A2 ) @ S2 ) )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ B @ ( F2 @ X4 ) @ L6 ) @ R2 ) ) ) ) ) ) ).
% metric_LIM_D
thf(fact_7522_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 )
=> ? [S5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ S5 )
& ! [X: A] :
( ( ( X != A2 )
& ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X @ A2 ) @ S5 ) )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ B @ ( F2 @ X ) @ L6 ) @ R5 ) ) ) ) ) ) ) ).
% LIM_def
thf(fact_7523_lim__sequentially,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X6: nat > A,L6: A] :
( ( filterlim @ nat @ A @ X6 @ ( topolo7230453075368039082e_nhds @ A @ L6 ) @ ( at_top @ nat ) )
= ( ! [R5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R5 )
=> ? [No3: nat] :
! [N3: nat] :
( ( ord_less_eq @ nat @ No3 @ N3 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( X6 @ N3 ) @ L6 ) @ R5 ) ) ) ) ) ) ).
% lim_sequentially
thf(fact_7524_metric__LIMSEQ__I,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X6: 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 @ ( X6 @ N2 ) @ L6 ) @ R3 ) ) )
=> ( filterlim @ nat @ A @ X6 @ ( topolo7230453075368039082e_nhds @ A @ L6 ) @ ( at_top @ nat ) ) ) ) ).
% metric_LIMSEQ_I
thf(fact_7525_metric__LIMSEQ__D,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X6: nat > A,L6: A,R2: real] :
( ( filterlim @ nat @ A @ X6 @ ( topolo7230453075368039082e_nhds @ A @ L6 ) @ ( at_top @ nat ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ R2 )
=> ? [No: nat] :
! [N11: nat] :
( ( ord_less_eq @ nat @ No @ N11 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( X6 @ N11 ) @ L6 ) @ R2 ) ) ) ) ) ).
% metric_LIMSEQ_D
thf(fact_7526_power__minus_H,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [X2: A,N: nat] :
( ( nO_MATCH @ A @ A @ ( one_one @ A ) @ X2 )
=> ( ( power_power @ A @ ( uminus_uminus @ A @ X2 ) @ N )
= ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N ) @ ( power_power @ A @ X2 @ N ) ) ) ) ) ).
% power_minus'
thf(fact_7527_metric__Cauchy__iff2,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ( ( topolo3814608138187158403Cauchy @ A )
= ( ^ [X7: nat > A] :
! [J3: nat] :
? [M9: nat] :
! [M5: nat] :
( ( ord_less_eq @ nat @ M9 @ M5 )
=> ! [N3: nat] :
( ( ord_less_eq @ nat @ M9 @ N3 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( X7 @ M5 ) @ ( X7 @ N3 ) ) @ ( inverse_inverse @ real @ ( semiring_1_of_nat @ real @ ( suc @ J3 ) ) ) ) ) ) ) ) ) ).
% metric_Cauchy_iff2
thf(fact_7528_scale__left__distrib__NO__MATCH,axiom,
! [C: $tType,A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [X2: A,Y2: A,C2: C,A2: real,B2: real] :
( ( nO_MATCH @ A @ C @ ( divide_divide @ A @ X2 @ Y2 ) @ C2 )
=> ( ( real_V8093663219630862766scaleR @ A @ ( plus_plus @ real @ A2 @ B2 ) @ X2 )
= ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ X2 ) @ ( real_V8093663219630862766scaleR @ A @ B2 @ X2 ) ) ) ) ) ).
% scale_left_distrib_NO_MATCH
thf(fact_7529_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 ) ) ) )
=> ( ? [D5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D5 )
& ! [X3: A] :
( ( ( X3 != A2 )
& ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X3 @ A2 ) @ D5 ) )
=> ( ( F2 @ X3 )
!= B2 ) ) )
=> ( filterlim @ A @ C
@ ^ [X: A] : ( G @ ( F2 @ X ) )
@ ( topolo7230453075368039082e_nhds @ C @ C2 )
@ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ).
% metric_LIM_compose2
thf(fact_7530_scale__left__diff__distrib__NO__MATCH,axiom,
! [C: $tType,A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [X2: A,Y2: A,C2: C,A2: real,B2: real] :
( ( nO_MATCH @ A @ C @ ( divide_divide @ A @ X2 @ Y2 ) @ C2 )
=> ( ( real_V8093663219630862766scaleR @ A @ ( minus_minus @ real @ A2 @ B2 ) @ X2 )
= ( minus_minus @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ X2 ) @ ( real_V8093663219630862766scaleR @ A @ B2 @ X2 ) ) ) ) ) ).
% scale_left_diff_distrib_NO_MATCH
thf(fact_7531_metric__isCont__LIM__compose2,axiom,
! [D6: $tType,C: $tType,A: $tType] :
( ( ( real_V7819770556892013058_space @ A )
& ( topolo4958980785337419405_space @ C )
& ( topolo4958980785337419405_space @ D6 ) )
=> ! [A2: A,F2: A > C,G: C > D6,L2: D6] :
( ( topolo3448309680560233919inuous @ A @ C @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) @ F2 )
=> ( ( filterlim @ C @ D6 @ G @ ( topolo7230453075368039082e_nhds @ D6 @ L2 ) @ ( topolo174197925503356063within @ C @ ( F2 @ A2 ) @ ( top_top @ ( set @ C ) ) ) )
=> ( ? [D5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D5 )
& ! [X3: A] :
( ( ( X3 != A2 )
& ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X3 @ A2 ) @ D5 ) )
=> ( ( F2 @ X3 )
!= ( F2 @ A2 ) ) ) )
=> ( filterlim @ A @ D6
@ ^ [X: A] : ( G @ ( F2 @ X ) )
@ ( topolo7230453075368039082e_nhds @ D6 @ L2 )
@ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ).
% metric_isCont_LIM_compose2
thf(fact_7532_LIMSEQ__iff__nz,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X6: nat > A,L6: A] :
( ( filterlim @ nat @ A @ X6 @ ( topolo7230453075368039082e_nhds @ A @ L6 ) @ ( at_top @ nat ) )
= ( ! [R5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R5 )
=> ? [No3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ No3 )
& ! [N3: nat] :
( ( ord_less_eq @ nat @ No3 @ N3 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( X6 @ N3 ) @ L6 ) @ R5 ) ) ) ) ) ) ) ).
% LIMSEQ_iff_nz
thf(fact_7533_LIM__offset__zero__iff,axiom,
! [C: $tType,D6: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( topolo4958980785337419405_space @ D6 )
& ( zero @ C ) )
=> ! [A2: A,F2: A > D6,L6: D6] :
( ( nO_MATCH @ C @ A @ ( zero_zero @ C ) @ A2 )
=> ( ( filterlim @ A @ D6 @ F2 @ ( topolo7230453075368039082e_nhds @ D6 @ L6 ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) )
= ( filterlim @ A @ D6
@ ^ [H: A] : ( F2 @ ( plus_plus @ A @ A2 @ H ) )
@ ( topolo7230453075368039082e_nhds @ D6 @ L6 )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% LIM_offset_zero_iff
thf(fact_7534_tendsto__exp__limit__at__right,axiom,
! [X2: real] :
( filterlim @ real @ real
@ ^ [Y: real] : ( powr @ real @ ( plus_plus @ real @ ( one_one @ real ) @ ( times_times @ real @ X2 @ Y ) ) @ ( divide_divide @ real @ ( one_one @ real ) @ Y ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( exp @ real @ X2 ) )
@ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) ) ).
% tendsto_exp_limit_at_right
thf(fact_7535_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_7536_greaterThan__subset__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: A,Y2: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_ord_greaterThan @ A @ X2 ) @ ( set_ord_greaterThan @ A @ Y2 ) )
= ( ord_less_eq @ A @ Y2 @ X2 ) ) ) ).
% greaterThan_subset_iff
thf(fact_7537_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_7538_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_7539_greaterThan__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( set_ord_greaterThan @ A )
= ( ^ [L: A] : ( collect @ A @ ( ord_less @ A @ L ) ) ) ) ) ).
% greaterThan_def
thf(fact_7540_filterlim__at__left__to__right,axiom,
! [A: $tType,F2: real > A,F3: filter @ A,A2: real] :
( ( filterlim @ real @ A @ F2 @ F3 @ ( topolo174197925503356063within @ real @ A2 @ ( set_ord_lessThan @ real @ A2 ) ) )
= ( filterlim @ real @ A
@ ^ [X: real] : ( F2 @ ( uminus_uminus @ real @ X ) )
@ F3
@ ( topolo174197925503356063within @ real @ ( uminus_uminus @ real @ A2 ) @ ( set_ord_greaterThan @ real @ ( uminus_uminus @ real @ A2 ) ) ) ) ) ).
% filterlim_at_left_to_right
thf(fact_7541_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
@ ^ [X: real] : ( F2 @ ( plus_plus @ real @ X @ A2 ) )
@ F3
@ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) ) ) ).
% filterlim_at_right_to_0
thf(fact_7542_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_7543_exp__at__bot,axiom,
filterlim @ real @ real @ ( exp @ real ) @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_bot @ real ) ).
% exp_at_bot
thf(fact_7544_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_7545_filterlim__times__pos,axiom,
! [A: $tType,B: $tType] :
( ( ( linordered_field @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F2: B > A,P2: A,F1: filter @ B,C2: A,L2: A] :
( ( filterlim @ B @ A @ F2 @ ( topolo174197925503356063within @ A @ P2 @ ( set_ord_greaterThan @ A @ P2 ) ) @ F1 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( L2
= ( times_times @ A @ C2 @ P2 ) )
=> ( filterlim @ B @ A
@ ^ [X: B] : ( times_times @ A @ C2 @ ( F2 @ X ) )
@ ( topolo174197925503356063within @ A @ L2 @ ( set_ord_greaterThan @ A @ L2 ) )
@ F1 ) ) ) ) ) ).
% filterlim_times_pos
thf(fact_7546_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_7547_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_7548_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
@ ^ [X: A] : ( times_times @ real @ ( F2 @ X ) @ ( G @ X ) )
@ ( at_bot @ real )
@ F3 ) ) ) ) ).
% filterlim_tendsto_pos_mult_at_bot
thf(fact_7549_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_7550_tendsto__at__botI__sequentially,axiom,
! [B: $tType] :
( ( topolo3112930676232923870pology @ B )
=> ! [F2: real > B,Y2: B] :
( ! [X17: nat > real] :
( ( filterlim @ nat @ real @ X17 @ ( at_bot @ real ) @ ( at_top @ nat ) )
=> ( filterlim @ nat @ B
@ ^ [N3: nat] : ( F2 @ ( X17 @ N3 ) )
@ ( topolo7230453075368039082e_nhds @ B @ Y2 )
@ ( at_top @ nat ) ) )
=> ( filterlim @ real @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ Y2 ) @ ( at_bot @ real ) ) ) ) ).
% tendsto_at_botI_sequentially
thf(fact_7551_isCont__If__ge,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo1944317154257567458pology @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [A2: A,G: A > B,F2: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ ( set_ord_lessThan @ A @ A2 ) ) @ G )
=> ( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ ( G @ A2 ) ) @ ( topolo174197925503356063within @ A @ A2 @ ( set_ord_greaterThan @ A @ A2 ) ) )
=> ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) )
@ ^ [X: A] : ( if @ B @ ( ord_less_eq @ A @ X @ A2 ) @ ( G @ X ) @ ( F2 @ X ) ) ) ) ) ) ).
% isCont_If_ge
thf(fact_7552_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_7553_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
@ ^ [X: real] : ( power_power @ real @ ( F2 @ X ) @ N )
@ ( at_bot @ real )
@ F3 ) ) ) ) ).
% filterlim_pow_at_bot_odd
thf(fact_7554_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
@ ^ [X: real] : ( power_power @ real @ ( F2 @ X ) @ N )
@ ( at_top @ real )
@ F3 ) ) ) ) ).
% filterlim_pow_at_bot_even
thf(fact_7555_lim__zero__infinity,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,L2: A] :
( ( filterlim @ A @ A
@ ^ [X: A] : ( F2 @ ( divide_divide @ A @ ( one_one @ A ) @ X ) )
@ ( topolo7230453075368039082e_nhds @ A @ L2 )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) )
=> ( filterlim @ A @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L2 ) @ ( at_infinity @ A ) ) ) ) ).
% lim_zero_infinity
thf(fact_7556_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
@ ^ [X: A] : ( times_times @ real @ ( F2 @ X ) @ ( G @ X ) )
@ ( at_top @ real )
@ F3 ) ) ) ).
% filterlim_at_top_mult_at_top
thf(fact_7557_filterlim__at__top__add__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
@ ^ [X: A] : ( plus_plus @ real @ ( F2 @ X ) @ ( G @ X ) )
@ ( at_top @ real )
@ F3 ) ) ) ).
% filterlim_at_top_add_at_top
thf(fact_7558_exp__at__top,axiom,
filterlim @ real @ real @ ( exp @ real ) @ ( at_top @ real ) @ ( at_top @ real ) ).
% exp_at_top
thf(fact_7559_sqrt__at__top,axiom,
filterlim @ real @ real @ sqrt @ ( at_top @ real ) @ ( at_top @ real ) ).
% sqrt_at_top
thf(fact_7560_ln__at__top,axiom,
filterlim @ real @ real @ ( ln_ln @ real ) @ ( at_top @ real ) @ ( at_top @ real ) ).
% ln_at_top
thf(fact_7561_filterlim__at__infinity__imp__norm__at__top,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [F2: A > B,F3: filter @ A] :
( ( filterlim @ A @ B @ F2 @ ( at_infinity @ B ) @ F3 )
=> ( filterlim @ A @ real
@ ^ [X: A] : ( real_V7770717601297561774m_norm @ B @ ( F2 @ X ) )
@ ( at_top @ real )
@ F3 ) ) ) ).
% filterlim_at_infinity_imp_norm_at_top
thf(fact_7562_filterlim__norm__at__top__imp__at__infinity,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [F2: A > B,F3: filter @ A] :
( ( filterlim @ A @ real
@ ^ [X: A] : ( real_V7770717601297561774m_norm @ B @ ( F2 @ X ) )
@ ( at_top @ real )
@ F3 )
=> ( filterlim @ A @ B @ F2 @ ( at_infinity @ B ) @ F3 ) ) ) ).
% filterlim_norm_at_top_imp_at_infinity
thf(fact_7563_filterlim__at__infinity__conv__norm__at__top,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [F2: A > B,G5: filter @ A] :
( ( filterlim @ A @ B @ F2 @ ( at_infinity @ B ) @ G5 )
= ( filterlim @ A @ real
@ ^ [X: A] : ( real_V7770717601297561774m_norm @ B @ ( F2 @ X ) )
@ ( at_top @ real )
@ G5 ) ) ) ).
% filterlim_at_infinity_conv_norm_at_top
thf(fact_7564_filterlim__tendsto__add__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
@ ^ [X: A] : ( plus_plus @ real @ ( F2 @ X ) @ ( G @ X ) )
@ ( at_top @ real )
@ F3 ) ) ) ).
% filterlim_tendsto_add_at_top
thf(fact_7565_filterlim__real__sequentially,axiom,
filterlim @ nat @ real @ ( semiring_1_of_nat @ real ) @ ( at_top @ real ) @ ( at_top @ nat ) ).
% filterlim_real_sequentially
thf(fact_7566_filterlim__uminus__at__bot,axiom,
! [A: $tType,F2: A > real,F3: filter @ A] :
( ( filterlim @ A @ real @ F2 @ ( at_bot @ real ) @ F3 )
= ( filterlim @ A @ real
@ ^ [X: A] : ( uminus_uminus @ real @ ( F2 @ X ) )
@ ( at_top @ real )
@ F3 ) ) ).
% filterlim_uminus_at_bot
thf(fact_7567_filterlim__uminus__at__top,axiom,
! [A: $tType,F2: A > real,F3: filter @ A] :
( ( filterlim @ A @ real @ F2 @ ( at_top @ real ) @ F3 )
= ( filterlim @ A @ real
@ ^ [X: A] : ( uminus_uminus @ real @ ( F2 @ X ) )
@ ( at_bot @ real )
@ F3 ) ) ).
% filterlim_uminus_at_top
thf(fact_7568_filterlim__at__bot__mirror,axiom,
! [A: $tType,F2: real > A,F3: filter @ A] :
( ( filterlim @ real @ A @ F2 @ F3 @ ( at_bot @ real ) )
= ( filterlim @ real @ A
@ ^ [X: real] : ( F2 @ ( uminus_uminus @ real @ X ) )
@ F3
@ ( at_top @ real ) ) ) ).
% filterlim_at_bot_mirror
thf(fact_7569_filterlim__at__top__mirror,axiom,
! [A: $tType,F2: real > A,F3: filter @ A] :
( ( filterlim @ real @ A @ F2 @ F3 @ ( at_top @ real ) )
= ( filterlim @ real @ A
@ ^ [X: real] : ( F2 @ ( uminus_uminus @ real @ X ) )
@ F3
@ ( at_bot @ real ) ) ) ).
% filterlim_at_top_mirror
thf(fact_7570_filterlim__uminus__at__bot__at__top,axiom,
filterlim @ real @ real @ ( uminus_uminus @ real ) @ ( at_bot @ real ) @ ( at_top @ real ) ).
% filterlim_uminus_at_bot_at_top
thf(fact_7571_filterlim__uminus__at__top__at__bot,axiom,
filterlim @ real @ real @ ( uminus_uminus @ real ) @ ( at_top @ real ) @ ( at_bot @ real ) ).
% filterlim_uminus_at_top_at_bot
thf(fact_7572_filterlim__real__at__infinity__sequentially,axiom,
filterlim @ nat @ real @ ( semiring_1_of_nat @ real ) @ ( at_infinity @ real ) @ ( at_top @ nat ) ).
% filterlim_real_at_infinity_sequentially
thf(fact_7573_tendsto__of__nat,axiom,
! [A: $tType] :
( ( real_V2822296259951069270ebra_1 @ A )
=> ( filterlim @ nat @ A @ ( semiring_1_of_nat @ A ) @ ( at_infinity @ A ) @ ( at_top @ nat ) ) ) ).
% tendsto_of_nat
thf(fact_7574_greaterThan__0,axiom,
( ( set_ord_greaterThan @ nat @ ( zero_zero @ nat ) )
= ( image @ nat @ nat @ suc @ ( top_top @ ( set @ nat ) ) ) ) ).
% greaterThan_0
thf(fact_7575_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
@ ^ [X: A] : ( power_power @ real @ ( F2 @ X ) @ N )
@ ( at_top @ real )
@ F3 ) ) ) ).
% filterlim_pow_at_top
thf(fact_7576_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_7577_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
@ ^ [X: A] : ( plus_plus @ B @ ( F2 @ X ) @ ( G @ X ) )
@ ( at_infinity @ B )
@ F3 ) ) ) ) ).
% tendsto_add_filterlim_at_infinity
thf(fact_7578_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
@ ^ [X: A] : ( plus_plus @ B @ ( F2 @ X ) @ ( G @ X ) )
@ ( at_infinity @ B )
@ F3 ) ) ) ) ).
% tendsto_add_filterlim_at_infinity'
thf(fact_7579_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_7580_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
@ ^ [X: A] : ( divide_divide @ real @ ( F2 @ X ) @ ( G @ X ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ F3 ) ) ) ).
% real_tendsto_divide_at_top
thf(fact_7581_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_7582_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
@ ^ [X: A] : ( inverse_inverse @ real @ ( F2 @ X ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ F3 ) ) ).
% tendsto_inverse_0_at_top
thf(fact_7583_filterlim__sequentially__iff__filterlim__real,axiom,
! [A: $tType,F2: A > nat,F3: filter @ A] :
( ( filterlim @ A @ nat @ F2 @ ( at_top @ nat ) @ F3 )
= ( filterlim @ A @ real
@ ^ [X: A] : ( semiring_1_of_nat @ real @ ( F2 @ X ) )
@ ( at_top @ real )
@ F3 ) ) ).
% filterlim_sequentially_iff_filterlim_real
thf(fact_7584_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_7585_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
@ ^ [X: A] : ( times_times @ real @ ( F2 @ X ) @ ( G @ X ) )
@ ( at_top @ real )
@ F3 ) ) ) ) ).
% filterlim_tendsto_pos_mult_at_top
thf(fact_7586_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
@ ^ [X: A] : ( times_times @ real @ ( G @ X ) @ ( F2 @ X ) )
@ ( at_top @ real )
@ F3 ) ) ) ) ).
% filterlim_at_top_mult_tendsto_pos
thf(fact_7587_tendsto__neg__powr,axiom,
! [A: $tType,S3: real,F2: A > real,F3: filter @ A] :
( ( ord_less @ real @ S3 @ ( zero_zero @ real ) )
=> ( ( filterlim @ A @ real @ F2 @ ( at_top @ real ) @ F3 )
=> ( filterlim @ A @ real
@ ^ [X: A] : ( powr @ real @ ( F2 @ X ) @ S3 )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ F3 ) ) ) ).
% tendsto_neg_powr
thf(fact_7588_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
@ ^ [X: B] : ( times_times @ A @ ( F2 @ X ) @ ( G @ X ) )
@ ( at_infinity @ A )
@ F3 ) ) ) ) ) ).
% tendsto_mult_filterlim_at_infinity
thf(fact_7589_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
@ ^ [X: C] : ( divide_divide @ A @ ( F2 @ X ) @ ( G @ X ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ F3 ) ) ) ) ).
% tendsto_divide_0
thf(fact_7590_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
@ ^ [X: A] : ( power_power @ B @ ( F2 @ X ) @ N )
@ ( at_infinity @ B )
@ F3 ) ) ) ) ).
% filterlim_power_at_infinity
thf(fact_7591_ln__x__over__x__tendsto__0,axiom,
( filterlim @ real @ real
@ ^ [X: real] : ( divide_divide @ real @ ( ln_ln @ real @ X ) @ X )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ ( at_top @ real ) ) ).
% ln_x_over_x_tendsto_0
thf(fact_7592_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
@ ^ [X: real] : ( F2 @ ( inverse_inverse @ real @ X ) )
@ F3
@ ( at_top @ real ) ) ) ).
% filterlim_at_right_to_top
thf(fact_7593_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
@ ^ [X: real] : ( F2 @ ( inverse_inverse @ real @ X ) )
@ F3
@ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) ) ) ).
% filterlim_at_top_to_right
thf(fact_7594_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_7595_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_7596_tendsto__at__topI__sequentially,axiom,
! [B: $tType] :
( ( topolo3112930676232923870pology @ B )
=> ! [F2: real > B,Y2: B] :
( ! [X17: nat > real] :
( ( filterlim @ nat @ real @ X17 @ ( at_top @ real ) @ ( at_top @ nat ) )
=> ( filterlim @ nat @ B
@ ^ [N3: nat] : ( F2 @ ( X17 @ N3 ) )
@ ( topolo7230453075368039082e_nhds @ B @ Y2 )
@ ( at_top @ nat ) ) )
=> ( filterlim @ real @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ Y2 ) @ ( at_top @ real ) ) ) ) ).
% tendsto_at_topI_sequentially
thf(fact_7597_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_7598_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
@ ^ [X: A] : ( times_times @ real @ ( F2 @ X ) @ ( G @ X ) )
@ ( at_bot @ real )
@ F3 ) ) ) ) ).
% filterlim_tendsto_neg_mult_at_bot
thf(fact_7599_tendsto__power__div__exp__0,axiom,
! [K: nat] :
( filterlim @ real @ real
@ ^ [X: real] : ( divide_divide @ real @ ( power_power @ real @ X @ K ) @ ( exp @ real @ X ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ ( at_top @ real ) ) ).
% tendsto_power_div_exp_0
thf(fact_7600_lim__infinity__imp__sequentially,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [F2: real > A,L2: A] :
( ( filterlim @ real @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L2 ) @ ( at_infinity @ real ) )
=> ( filterlim @ nat @ A
@ ^ [N3: nat] : ( F2 @ ( semiring_1_of_nat @ real @ N3 ) )
@ ( topolo7230453075368039082e_nhds @ A @ L2 )
@ ( at_top @ nat ) ) ) ) ).
% lim_infinity_imp_sequentially
thf(fact_7601_filterlim__inverse__at__iff,axiom,
! [B: $tType,A: $tType] :
( ( real_V8999393235501362500lgebra @ B )
=> ! [G: A > B,F3: filter @ A] :
( ( filterlim @ A @ B
@ ^ [X: A] : ( inverse_inverse @ B @ ( G @ X ) )
@ ( topolo174197925503356063within @ B @ ( zero_zero @ B ) @ ( top_top @ ( set @ B ) ) )
@ F3 )
= ( filterlim @ A @ B @ G @ ( at_infinity @ B ) @ F3 ) ) ) ).
% filterlim_inverse_at_iff
thf(fact_7602_tendsto__exp__limit__at__top,axiom,
! [X2: real] :
( filterlim @ real @ real
@ ^ [Y: real] : ( powr @ real @ ( plus_plus @ real @ ( one_one @ real ) @ ( divide_divide @ real @ X2 @ Y ) ) @ Y )
@ ( topolo7230453075368039082e_nhds @ real @ ( exp @ real @ X2 ) )
@ ( at_top @ real ) ) ).
% tendsto_exp_limit_at_top
thf(fact_7603_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
@ ^ [X: A] : ( divide_divide @ A @ ( F2 @ X ) @ ( G @ X ) )
@ ( at_infinity @ A )
@ F3 ) ) ) ) ) ).
% filterlim_divide_at_infinity
thf(fact_7604_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_7605_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_7606_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_7607_filterlim__realpow__sequentially__gt1,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [X2: A] :
( ( ord_less @ real @ ( one_one @ real ) @ ( real_V7770717601297561774m_norm @ A @ X2 ) )
=> ( filterlim @ nat @ A @ ( power_power @ A @ X2 ) @ ( at_infinity @ A ) @ ( at_top @ nat ) ) ) ) ).
% filterlim_realpow_sequentially_gt1
thf(fact_7608_lim__at__infinity__0,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,L2: A] :
( ( filterlim @ A @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L2 ) @ ( at_infinity @ A ) )
= ( filterlim @ A @ A @ ( comp @ A @ A @ A @ F2 @ ( inverse_inverse @ A ) ) @ ( topolo7230453075368039082e_nhds @ A @ L2 ) @ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% lim_at_infinity_0
thf(fact_7609_polyfun__extremal,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [C2: nat > A,K: nat,N: nat,B5: real] :
( ( ( C2 @ K )
!= ( zero_zero @ A ) )
=> ( ( ord_less_eq @ nat @ ( one_one @ nat ) @ K )
=> ( ( ord_less_eq @ nat @ K @ N )
=> ( eventually @ A
@ ^ [Z5: A] :
( ord_less_eq @ real @ B5
@ ( real_V7770717601297561774m_norm @ A
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( C2 @ I3 ) @ ( power_power @ A @ Z5 @ I3 ) )
@ ( set_ord_atMost @ nat @ N ) ) ) )
@ ( at_infinity @ A ) ) ) ) ) ) ).
% polyfun_extremal
thf(fact_7610_lhopital__left__at__top,axiom,
! [G: real > real,X2: real,G4: real > real,F2: real > real,F7: real > real,Y2: real] :
( ( filterlim @ real @ real @ G @ ( at_top @ real ) @ ( topolo174197925503356063within @ real @ X2 @ ( set_ord_lessThan @ real @ X2 ) ) )
=> ( ( eventually @ real
@ ^ [X: real] :
( ( G4 @ X )
!= ( zero_zero @ real ) )
@ ( topolo174197925503356063within @ real @ X2 @ ( set_ord_lessThan @ real @ X2 ) ) )
=> ( ( eventually @ real
@ ^ [X: real] : ( has_field_derivative @ real @ F2 @ ( F7 @ X ) @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ X2 @ ( set_ord_lessThan @ real @ X2 ) ) )
=> ( ( eventually @ real
@ ^ [X: real] : ( has_field_derivative @ real @ G @ ( G4 @ X ) @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ X2 @ ( set_ord_lessThan @ real @ X2 ) ) )
=> ( ( filterlim @ real @ real
@ ^ [X: real] : ( divide_divide @ real @ ( F7 @ X ) @ ( G4 @ X ) )
@ ( topolo7230453075368039082e_nhds @ real @ Y2 )
@ ( topolo174197925503356063within @ real @ X2 @ ( set_ord_lessThan @ real @ X2 ) ) )
=> ( filterlim @ real @ real
@ ^ [X: real] : ( divide_divide @ real @ ( F2 @ X ) @ ( G @ X ) )
@ ( topolo7230453075368039082e_nhds @ real @ Y2 )
@ ( topolo174197925503356063within @ real @ X2 @ ( set_ord_lessThan @ real @ X2 ) ) ) ) ) ) ) ) ).
% lhopital_left_at_top
thf(fact_7611_eventually__sequentially__Suc,axiom,
! [P: nat > $o] :
( ( eventually @ nat
@ ^ [I3: nat] : ( P @ ( suc @ I3 ) )
@ ( at_top @ nat ) )
= ( eventually @ nat @ P @ ( at_top @ nat ) ) ) ).
% eventually_sequentially_Suc
thf(fact_7612_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_7613_eventually__const,axiom,
! [A: $tType,F3: filter @ A,P: $o] :
( ( F3
!= ( bot_bot @ ( filter @ A ) ) )
=> ( ( eventually @ A
@ ^ [X: A] : P
@ F3 )
= P ) ) ).
% eventually_const
thf(fact_7614_tendsto__sandwich,axiom,
! [A: $tType,B: $tType] :
( ( topolo2564578578187576103pology @ A )
=> ! [F2: B > A,G: B > A,Net: filter @ B,H2: B > A,C2: A] :
( ( eventually @ B
@ ^ [N3: B] : ( ord_less_eq @ A @ ( F2 @ N3 ) @ ( G @ N3 ) )
@ Net )
=> ( ( eventually @ B
@ ^ [N3: B] : ( ord_less_eq @ A @ ( G @ N3 ) @ ( H2 @ N3 ) )
@ Net )
=> ( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ C2 ) @ Net )
=> ( ( filterlim @ B @ A @ H2 @ ( topolo7230453075368039082e_nhds @ A @ C2 ) @ Net )
=> ( filterlim @ B @ A @ G @ ( topolo7230453075368039082e_nhds @ A @ C2 ) @ Net ) ) ) ) ) ) ).
% tendsto_sandwich
thf(fact_7615_order__tendsto__iff,axiom,
! [B: $tType,A: $tType] :
( ( topolo2564578578187576103pology @ A )
=> ! [F2: B > A,X2: A,F3: filter @ B] :
( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ X2 ) @ F3 )
= ( ! [L: A] :
( ( ord_less @ A @ L @ X2 )
=> ( eventually @ B
@ ^ [X: B] : ( ord_less @ A @ L @ ( F2 @ X ) )
@ F3 ) )
& ! [U2: A] :
( ( ord_less @ A @ X2 @ U2 )
=> ( eventually @ B
@ ^ [X: B] : ( ord_less @ A @ ( F2 @ X ) @ U2 )
@ F3 ) ) ) ) ) ).
% order_tendsto_iff
thf(fact_7616_order__tendstoI,axiom,
! [A: $tType,B: $tType] :
( ( topolo2564578578187576103pology @ A )
=> ! [Y2: A,F2: B > A,F3: filter @ B] :
( ! [A5: A] :
( ( ord_less @ A @ A5 @ Y2 )
=> ( eventually @ B
@ ^ [X: B] : ( ord_less @ A @ A5 @ ( F2 @ X ) )
@ F3 ) )
=> ( ! [A5: A] :
( ( ord_less @ A @ Y2 @ A5 )
=> ( eventually @ B
@ ^ [X: B] : ( ord_less @ A @ ( F2 @ X ) @ A5 )
@ F3 ) )
=> ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ Y2 ) @ F3 ) ) ) ) ).
% order_tendstoI
thf(fact_7617_order__tendstoD_I1_J,axiom,
! [A: $tType,B: $tType] :
( ( topolo2564578578187576103pology @ A )
=> ! [F2: B > A,Y2: A,F3: filter @ B,A2: A] :
( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ Y2 ) @ F3 )
=> ( ( ord_less @ A @ A2 @ Y2 )
=> ( eventually @ B
@ ^ [X: B] : ( ord_less @ A @ A2 @ ( F2 @ X ) )
@ F3 ) ) ) ) ).
% order_tendstoD(1)
thf(fact_7618_order__tendstoD_I2_J,axiom,
! [A: $tType,B: $tType] :
( ( topolo2564578578187576103pology @ A )
=> ! [F2: B > A,Y2: A,F3: filter @ B,A2: A] :
( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ Y2 ) @ F3 )
=> ( ( ord_less @ A @ Y2 @ A2 )
=> ( eventually @ B
@ ^ [X: B] : ( ord_less @ A @ ( F2 @ X ) @ A2 )
@ F3 ) ) ) ) ).
% order_tendstoD(2)
thf(fact_7619_sequentially__offset,axiom,
! [P: nat > $o,K: nat] :
( ( eventually @ nat @ P @ ( at_top @ nat ) )
=> ( eventually @ nat
@ ^ [I3: nat] : ( P @ ( plus_plus @ nat @ I3 @ K ) )
@ ( at_top @ nat ) ) ) ).
% sequentially_offset
thf(fact_7620_summable__cong,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A,G: nat > A] :
( ( eventually @ nat
@ ^ [X: nat] :
( ( F2 @ X )
= ( G @ X ) )
@ ( at_top @ nat ) )
=> ( ( summable @ A @ F2 )
= ( summable @ A @ G ) ) ) ) ).
% summable_cong
thf(fact_7621_eventually__at__infinity,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [P: A > $o] :
( ( eventually @ A @ P @ ( at_infinity @ A ) )
= ( ? [B3: real] :
! [X: A] :
( ( ord_less_eq @ real @ B3 @ ( real_V7770717601297561774m_norm @ A @ X ) )
=> ( P @ X ) ) ) ) ) ).
% eventually_at_infinity
thf(fact_7622_eventually__not__equal__at__infinity,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [A2: A] :
( eventually @ A
@ ^ [X: A] : X != A2
@ ( at_infinity @ A ) ) ) ).
% eventually_not_equal_at_infinity
thf(fact_7623_eventually__ge__at__top,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [C2: A] : ( eventually @ A @ ( ord_less_eq @ A @ C2 ) @ ( at_top @ A ) ) ) ).
% eventually_ge_at_top
thf(fact_7624_eventually__sequentially,axiom,
! [P: nat > $o] :
( ( eventually @ nat @ P @ ( at_top @ nat ) )
= ( ? [N8: nat] :
! [N3: nat] :
( ( ord_less_eq @ nat @ N8 @ N3 )
=> ( P @ N3 ) ) ) ) ).
% eventually_sequentially
thf(fact_7625_eventually__sequentiallyI,axiom,
! [C2: nat,P: nat > $o] :
( ! [X3: nat] :
( ( ord_less_eq @ nat @ C2 @ X3 )
=> ( P @ X3 ) )
=> ( eventually @ nat @ P @ ( at_top @ nat ) ) ) ).
% eventually_sequentiallyI
thf(fact_7626_eventually__at__top__linorder,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > $o] :
( ( eventually @ A @ P @ ( at_top @ A ) )
= ( ? [N8: A] :
! [N3: A] :
( ( ord_less_eq @ A @ N8 @ N3 )
=> ( P @ N3 ) ) ) ) ) ).
% eventually_at_top_linorder
thf(fact_7627_eventually__at__top__linorderI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [C2: A,P: A > $o] :
( ! [X3: A] :
( ( ord_less_eq @ A @ C2 @ X3 )
=> ( P @ X3 ) )
=> ( eventually @ A @ P @ ( at_top @ A ) ) ) ) ).
% eventually_at_top_linorderI
thf(fact_7628_le__sequentially,axiom,
! [F3: filter @ nat] :
( ( ord_less_eq @ ( filter @ nat ) @ F3 @ ( at_top @ nat ) )
= ( ! [N8: nat] : ( eventually @ nat @ ( ord_less_eq @ nat @ N8 ) @ F3 ) ) ) ).
% le_sequentially
thf(fact_7629_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_7630_eventually__at__top__not__equal,axiom,
! [A: $tType] :
( ( ( linorder @ A )
& ( no_top @ A ) )
=> ! [C2: A] :
( eventually @ A
@ ^ [X: A] : X != C2
@ ( at_top @ A ) ) ) ).
% eventually_at_top_not_equal
thf(fact_7631_eventually__False__sequentially,axiom,
~ ( eventually @ nat
@ ^ [N3: nat] : $false
@ ( at_top @ nat ) ) ).
% eventually_False_sequentially
thf(fact_7632_eventually__compose__filterlim,axiom,
! [A: $tType,B: $tType,P: A > $o,F3: filter @ A,F2: B > A,G5: filter @ B] :
( ( eventually @ A @ P @ F3 )
=> ( ( filterlim @ B @ A @ F2 @ F3 @ G5 )
=> ( eventually @ B
@ ^ [X: B] : ( P @ ( F2 @ X ) )
@ G5 ) ) ) ).
% eventually_compose_filterlim
thf(fact_7633_filterlim__cong,axiom,
! [A: $tType,B: $tType,F1: filter @ A,F12: filter @ A,F22: filter @ B,F23: filter @ B,F2: B > A,G: B > A] :
( ( F1 = F12 )
=> ( ( F22 = F23 )
=> ( ( eventually @ B
@ ^ [X: B] :
( ( F2 @ X )
= ( G @ X ) )
@ F22 )
=> ( ( filterlim @ B @ A @ F2 @ F1 @ F22 )
= ( filterlim @ B @ A @ G @ F12 @ F23 ) ) ) ) ) ).
% filterlim_cong
thf(fact_7634_filterlim__iff,axiom,
! [B: $tType,A: $tType] :
( ( filterlim @ A @ B )
= ( ^ [F4: A > B,F24: filter @ B,F13: filter @ A] :
! [P7: B > $o] :
( ( eventually @ B @ P7 @ F24 )
=> ( eventually @ A
@ ^ [X: A] : ( P7 @ ( F4 @ X ) )
@ F13 ) ) ) ) ).
% filterlim_iff
thf(fact_7635_Lim__transform__eventually,axiom,
! [B: $tType,A: $tType] :
( ( topolo4958980785337419405_space @ B )
=> ! [F2: A > B,L2: B,F3: filter @ A,G: A > B] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L2 ) @ F3 )
=> ( ( eventually @ A
@ ^ [X: A] :
( ( F2 @ X )
= ( G @ X ) )
@ F3 )
=> ( filterlim @ A @ B @ G @ ( topolo7230453075368039082e_nhds @ B @ L2 ) @ F3 ) ) ) ) ).
% Lim_transform_eventually
thf(fact_7636_filterlim__at__infinity__imp__eventually__ne,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [F2: A > B,F3: filter @ A,C2: B] :
( ( filterlim @ A @ B @ F2 @ ( at_infinity @ B ) @ F3 )
=> ( eventually @ A
@ ^ [Z5: A] :
( ( F2 @ Z5 )
!= C2 )
@ F3 ) ) ) ).
% filterlim_at_infinity_imp_eventually_ne
thf(fact_7637_tendsto__cong,axiom,
! [A: $tType,B: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [F2: B > A,G: B > A,F3: filter @ B,C2: A] :
( ( eventually @ B
@ ^ [X: B] :
( ( F2 @ X )
= ( G @ X ) )
@ F3 )
=> ( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ C2 ) @ F3 )
= ( filterlim @ B @ A @ G @ ( topolo7230453075368039082e_nhds @ A @ C2 ) @ F3 ) ) ) ) ).
% tendsto_cong
thf(fact_7638_tendsto__discrete,axiom,
! [A: $tType,B: $tType] :
( ( topolo8865339358273720382pology @ A )
=> ! [F2: B > A,Y2: A,F3: filter @ B] :
( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ Y2 ) @ F3 )
= ( eventually @ B
@ ^ [X: B] :
( ( F2 @ X )
= Y2 )
@ F3 ) ) ) ).
% tendsto_discrete
thf(fact_7639_tendsto__imp__eventually__ne,axiom,
! [A: $tType,B: $tType] :
( ( topological_t1_space @ A )
=> ! [F2: B > A,C2: A,F3: filter @ B,C8: A] :
( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ C2 ) @ F3 )
=> ( ( C2 != C8 )
=> ( eventually @ B
@ ^ [Z5: B] :
( ( F2 @ Z5 )
!= C8 )
@ F3 ) ) ) ) ).
% tendsto_imp_eventually_ne
thf(fact_7640_tendsto__eventually,axiom,
! [A: $tType,B: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [F2: B > A,L2: A,Net: filter @ B] :
( ( eventually @ B
@ ^ [X: B] :
( ( F2 @ X )
= L2 )
@ Net )
=> ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L2 ) @ Net ) ) ) ).
% tendsto_eventually
thf(fact_7641_eventually__nhds__iff__sequentially,axiom,
! [A: $tType] :
( ( topolo3112930676232923870pology @ A )
=> ! [P: A > $o,A2: A] :
( ( eventually @ A @ P @ ( topolo7230453075368039082e_nhds @ A @ A2 ) )
= ( ! [F4: nat > A] :
( ( filterlim @ nat @ A @ F4 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ ( at_top @ nat ) )
=> ( eventually @ nat
@ ^ [N3: nat] : ( P @ ( F4 @ N3 ) )
@ ( at_top @ nat ) ) ) ) ) ) ).
% eventually_nhds_iff_sequentially
thf(fact_7642_filterlim__mono__eventually,axiom,
! [B: $tType,A: $tType,F2: A > B,F3: filter @ B,G5: filter @ A,F10: filter @ B,G6: filter @ A,F7: A > B] :
( ( filterlim @ A @ B @ F2 @ F3 @ G5 )
=> ( ( ord_less_eq @ ( filter @ B ) @ F3 @ F10 )
=> ( ( ord_less_eq @ ( filter @ A ) @ G6 @ G5 )
=> ( ( eventually @ A
@ ^ [X: A] :
( ( F2 @ X )
= ( F7 @ X ) )
@ G6 )
=> ( filterlim @ A @ B @ F7 @ F10 @ G6 ) ) ) ) ) ).
% filterlim_mono_eventually
thf(fact_7643_filterlim__at__top__at__top,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( linorder @ B ) )
=> ! [Q: A > $o,F2: A > B,P: B > $o,G: B > A] :
( ! [X3: A,Y3: A] :
( ( Q @ X3 )
=> ( ( Q @ Y3 )
=> ( ( ord_less_eq @ A @ X3 @ Y3 )
=> ( ord_less_eq @ B @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) ) ) )
=> ( ! [X3: B] :
( ( P @ X3 )
=> ( ( F2 @ ( G @ X3 ) )
= X3 ) )
=> ( ! [X3: B] :
( ( P @ X3 )
=> ( Q @ ( G @ X3 ) ) )
=> ( ( eventually @ A @ Q @ ( at_top @ A ) )
=> ( ( eventually @ B @ P @ ( at_top @ B ) )
=> ( filterlim @ A @ B @ F2 @ ( at_top @ B ) @ ( at_top @ A ) ) ) ) ) ) ) ) ).
% filterlim_at_top_at_top
thf(fact_7644_sequentially__imp__eventually__at,axiom,
! [A: $tType] :
( ( topolo3112930676232923870pology @ A )
=> ! [A2: A,P: A > $o] :
( ! [F5: nat > A] :
( ( ! [N11: nat] :
( ( F5 @ N11 )
!= A2 )
& ( 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 @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% sequentially_imp_eventually_at
thf(fact_7645_sequentially__imp__eventually__within,axiom,
! [A: $tType] :
( ( topolo3112930676232923870pology @ A )
=> ! [S3: set @ A,A2: A,P: A > $o] :
( ! [F5: nat > A] :
( ( ! [N11: nat] :
( ( member @ A @ ( F5 @ N11 ) @ S3 )
& ( ( F5 @ N11 )
!= A2 ) )
& ( 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 @ S3 ) ) ) ) ).
% sequentially_imp_eventually_within
thf(fact_7646_filterlim__at__within__not__equal,axiom,
! [B: $tType,A: $tType] :
( ( topological_t2_space @ B )
=> ! [F2: A > B,A2: B,S3: set @ B,F3: filter @ A,B2: B] :
( ( filterlim @ A @ B @ F2 @ ( topolo174197925503356063within @ B @ A2 @ S3 ) @ F3 )
=> ( eventually @ A
@ ^ [W2: A] :
( ( member @ B @ ( F2 @ W2 ) @ S3 )
& ( ( F2 @ W2 )
!= B2 ) )
@ F3 ) ) ) ).
% filterlim_at_within_not_equal
thf(fact_7647_filterlim__at__top,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ B )
=> ! [F2: A > B,F3: filter @ A] :
( ( filterlim @ A @ B @ F2 @ ( at_top @ B ) @ F3 )
= ( ! [Z9: B] :
( eventually @ A
@ ^ [X: A] : ( ord_less_eq @ B @ Z9 @ ( F2 @ X ) )
@ F3 ) ) ) ) ).
% filterlim_at_top
thf(fact_7648_filterlim__at__top__ge,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ B )
=> ! [F2: A > B,F3: filter @ A,C2: B] :
( ( filterlim @ A @ B @ F2 @ ( at_top @ B ) @ F3 )
= ( ! [Z9: B] :
( ( ord_less_eq @ B @ C2 @ Z9 )
=> ( eventually @ A
@ ^ [X: A] : ( ord_less_eq @ B @ Z9 @ ( F2 @ X ) )
@ F3 ) ) ) ) ) ).
% filterlim_at_top_ge
thf(fact_7649_filterlim__at__top__mono,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F2: B > A,F3: filter @ B,G: B > A] :
( ( filterlim @ B @ A @ F2 @ ( at_top @ A ) @ F3 )
=> ( ( eventually @ B
@ ^ [X: B] : ( ord_less_eq @ A @ ( F2 @ X ) @ ( G @ X ) )
@ F3 )
=> ( filterlim @ B @ A @ G @ ( at_top @ A ) @ F3 ) ) ) ) ).
% filterlim_at_top_mono
thf(fact_7650_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
@ ^ [X: A] : ( ord_less @ B @ Z9 @ ( F2 @ X ) )
@ F3 ) ) ) ) ).
% filterlim_at_top_dense
thf(fact_7651_eventually__Lim__ident__at,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [P: A > A > $o,X2: A,X6: set @ A] :
( ( eventually @ A
@ ( P
@ ( topolo3827282254853284352ce_Lim @ A @ A @ ( topolo174197925503356063within @ A @ X2 @ X6 )
@ ^ [X: A] : X ) )
@ ( topolo174197925503356063within @ A @ X2 @ X6 ) )
= ( eventually @ A @ ( P @ X2 ) @ ( topolo174197925503356063within @ A @ X2 @ X6 ) ) ) ) ).
% eventually_Lim_ident_at
thf(fact_7652_trivial__limit__def,axiom,
! [A: $tType,F3: filter @ A] :
( ( F3
= ( bot_bot @ ( filter @ A ) ) )
= ( eventually @ A
@ ^ [X: A] : $false
@ F3 ) ) ).
% trivial_limit_def
thf(fact_7653_eventually__const__iff,axiom,
! [A: $tType,P: $o,F3: filter @ A] :
( ( eventually @ A
@ ^ [X: A] : P
@ F3 )
= ( P
| ( F3
= ( bot_bot @ ( filter @ A ) ) ) ) ) ).
% eventually_const_iff
thf(fact_7654_False__imp__not__eventually,axiom,
! [A: $tType,P: A > $o,Net: filter @ A] :
( ! [X3: A] :
~ ( P @ X3 )
=> ( ( Net
!= ( bot_bot @ ( filter @ A ) ) )
=> ~ ( eventually @ A @ P @ Net ) ) ) ).
% False_imp_not_eventually
thf(fact_7655_eventually__frequently__const__simps_I6_J,axiom,
! [A: $tType,C6: $o,P: A > $o,F3: filter @ A] :
( ( eventually @ A
@ ^ [X: A] :
( C6
=> ( P @ X ) )
@ F3 )
= ( C6
=> ( eventually @ A @ P @ F3 ) ) ) ).
% eventually_frequently_const_simps(6)
thf(fact_7656_eventually__frequently__const__simps_I4_J,axiom,
! [A: $tType,C6: $o,P: A > $o,F3: filter @ A] :
( ( eventually @ A
@ ^ [X: A] :
( C6
| ( P @ X ) )
@ F3 )
= ( C6
| ( eventually @ A @ P @ F3 ) ) ) ).
% eventually_frequently_const_simps(4)
thf(fact_7657_eventually__frequently__const__simps_I3_J,axiom,
! [A: $tType,P: A > $o,C6: $o,F3: filter @ A] :
( ( eventually @ A
@ ^ [X: A] :
( ( P @ X )
| C6 )
@ F3 )
= ( ( eventually @ A @ P @ F3 )
| C6 ) ) ).
% eventually_frequently_const_simps(3)
thf(fact_7658_eventually__mp,axiom,
! [A: $tType,P: A > $o,Q: A > $o,F3: filter @ A] :
( ( eventually @ A
@ ^ [X: A] :
( ( P @ X )
=> ( Q @ X ) )
@ F3 )
=> ( ( eventually @ A @ P @ F3 )
=> ( eventually @ A @ Q @ F3 ) ) ) ).
% eventually_mp
thf(fact_7659_eventually__True,axiom,
! [A: $tType,F3: filter @ A] :
( eventually @ A
@ ^ [X: A] : $true
@ F3 ) ).
% eventually_True
thf(fact_7660_eventually__conj,axiom,
! [A: $tType,P: A > $o,F3: filter @ A,Q: A > $o] :
( ( eventually @ A @ P @ F3 )
=> ( ( eventually @ A @ Q @ F3 )
=> ( eventually @ A
@ ^ [X: A] :
( ( P @ X )
& ( Q @ X ) )
@ F3 ) ) ) ).
% eventually_conj
thf(fact_7661_eventually__elim2,axiom,
! [A: $tType,P: A > $o,F3: filter @ A,Q: A > $o,R: A > $o] :
( ( eventually @ A @ P @ F3 )
=> ( ( eventually @ A @ Q @ F3 )
=> ( ! [I2: A] :
( ( P @ I2 )
=> ( ( Q @ I2 )
=> ( R @ I2 ) ) )
=> ( eventually @ A @ R @ F3 ) ) ) ) ).
% eventually_elim2
thf(fact_7662_eventually__subst,axiom,
! [A: $tType,P: A > $o,Q: A > $o,F3: filter @ A] :
( ( eventually @ A
@ ^ [N3: A] :
( ( P @ N3 )
= ( Q @ N3 ) )
@ F3 )
=> ( ( eventually @ A @ P @ F3 )
= ( eventually @ A @ Q @ F3 ) ) ) ).
% eventually_subst
thf(fact_7663_eventually__rev__mp,axiom,
! [A: $tType,P: A > $o,F3: filter @ A,Q: A > $o] :
( ( eventually @ A @ P @ F3 )
=> ( ( eventually @ A
@ ^ [X: A] :
( ( P @ X )
=> ( Q @ X ) )
@ F3 )
=> ( eventually @ A @ Q @ F3 ) ) ) ).
% eventually_rev_mp
thf(fact_7664_eventually__conj__iff,axiom,
! [A: $tType,P: A > $o,Q: A > $o,F3: filter @ A] :
( ( eventually @ A
@ ^ [X: A] :
( ( P @ X )
& ( Q @ X ) )
@ F3 )
= ( ( eventually @ A @ P @ F3 )
& ( eventually @ A @ Q @ F3 ) ) ) ).
% eventually_conj_iff
thf(fact_7665_not__eventually__impI,axiom,
! [A: $tType,P: A > $o,F3: filter @ A,Q: A > $o] :
( ( eventually @ A @ P @ F3 )
=> ( ~ ( eventually @ A @ Q @ F3 )
=> ~ ( eventually @ A
@ ^ [X: A] :
( ( P @ X )
=> ( Q @ X ) )
@ F3 ) ) ) ).
% not_eventually_impI
thf(fact_7666_has__derivative__transform__eventually,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,F7: A > B,X2: A,S3: set @ A,G: A > B] :
( ( has_derivative @ A @ B @ F2 @ F7 @ ( topolo174197925503356063within @ A @ X2 @ S3 ) )
=> ( ( eventually @ A
@ ^ [X9: A] :
( ( F2 @ X9 )
= ( G @ X9 ) )
@ ( topolo174197925503356063within @ A @ X2 @ S3 ) )
=> ( ( ( F2 @ X2 )
= ( G @ X2 ) )
=> ( ( member @ A @ X2 @ S3 )
=> ( has_derivative @ A @ B @ G @ F7 @ ( topolo174197925503356063within @ A @ X2 @ S3 ) ) ) ) ) ) ) ).
% has_derivative_transform_eventually
thf(fact_7667_has__field__derivative__cong__eventually,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,G: A > A,X2: A,S: set @ A,U: A] :
( ( eventually @ A
@ ^ [X: A] :
( ( F2 @ X )
= ( G @ X ) )
@ ( topolo174197925503356063within @ A @ X2 @ S ) )
=> ( ( ( F2 @ X2 )
= ( G @ X2 ) )
=> ( ( has_field_derivative @ A @ F2 @ U @ ( topolo174197925503356063within @ A @ X2 @ S ) )
= ( has_field_derivative @ A @ G @ U @ ( topolo174197925503356063within @ A @ X2 @ S ) ) ) ) ) ) ).
% has_field_derivative_cong_eventually
thf(fact_7668_t1__space__nhds,axiom,
! [A: $tType] :
( ( topological_t1_space @ A )
=> ! [X2: A,Y2: A] :
( ( X2 != Y2 )
=> ( eventually @ A
@ ^ [X: A] : X != Y2
@ ( topolo7230453075368039082e_nhds @ A @ X2 ) ) ) ) ).
% t1_space_nhds
thf(fact_7669_eventually__eventually,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [P: A > $o,X2: A] :
( ( eventually @ A
@ ^ [Y: A] : ( eventually @ A @ P @ ( topolo7230453075368039082e_nhds @ A @ Y ) )
@ ( topolo7230453075368039082e_nhds @ A @ X2 ) )
= ( eventually @ A @ P @ ( topolo7230453075368039082e_nhds @ A @ X2 ) ) ) ) ).
% eventually_eventually
thf(fact_7670_eventually__at__filter,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [P: A > $o,A2: A,S3: set @ A] :
( ( eventually @ A @ P @ ( topolo174197925503356063within @ A @ A2 @ S3 ) )
= ( eventually @ A
@ ^ [X: A] :
( ( X != A2 )
=> ( ( member @ A @ X @ S3 )
=> ( P @ X ) ) )
@ ( topolo7230453075368039082e_nhds @ A @ A2 ) ) ) ) ).
% eventually_at_filter
thf(fact_7671_eventually__nhds__in__open,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [S3: set @ A,X2: A] :
( ( topolo1002775350975398744n_open @ A @ S3 )
=> ( ( member @ A @ X2 @ S3 )
=> ( eventually @ A
@ ^ [Y: A] : ( member @ A @ Y @ S3 )
@ ( topolo7230453075368039082e_nhds @ A @ X2 ) ) ) ) ) ).
% eventually_nhds_in_open
thf(fact_7672_eventually__at__bot__not__equal,axiom,
! [A: $tType] :
( ( ( linorder @ A )
& ( no_bot @ A ) )
=> ! [C2: A] :
( eventually @ A
@ ^ [X: A] : X != C2
@ ( at_bot @ A ) ) ) ).
% eventually_at_bot_not_equal
thf(fact_7673_eventually__gt__at__bot,axiom,
! [A: $tType] :
( ( unboun7993243217541854897norder @ A )
=> ! [C2: A] :
( eventually @ A
@ ^ [X: A] : ( ord_less @ A @ X @ C2 )
@ ( at_bot @ A ) ) ) ).
% eventually_gt_at_bot
thf(fact_7674_eventually__le__at__bot,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [C2: A] :
( eventually @ A
@ ^ [X: A] : ( ord_less_eq @ A @ X @ C2 )
@ ( at_bot @ A ) ) ) ).
% eventually_le_at_bot
thf(fact_7675_eventually__at__bot__linorder,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > $o] :
( ( eventually @ A @ P @ ( at_bot @ A ) )
= ( ? [N8: A] :
! [N3: A] :
( ( ord_less_eq @ A @ N3 @ N8 )
=> ( P @ N3 ) ) ) ) ) ).
% eventually_at_bot_linorder
thf(fact_7676_filterlim__at,axiom,
! [A: $tType,B: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [F2: B > A,B2: A,S3: set @ A,F3: filter @ B] :
( ( filterlim @ B @ A @ F2 @ ( topolo174197925503356063within @ A @ B2 @ S3 ) @ F3 )
= ( ( eventually @ B
@ ^ [X: B] :
( ( member @ A @ ( F2 @ X ) @ S3 )
& ( ( F2 @ X )
!= B2 ) )
@ F3 )
& ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ B2 ) @ F3 ) ) ) ) ).
% filterlim_at
thf(fact_7677_eventually__at__right__less,axiom,
! [A: $tType] :
( ( ( no_top @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [X2: A] : ( eventually @ A @ ( ord_less @ A @ X2 ) @ ( topolo174197925503356063within @ A @ X2 @ ( set_ord_greaterThan @ A @ X2 ) ) ) ) ).
% eventually_at_right_less
thf(fact_7678_has__field__derivative__cong__ev,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [X2: A,Y2: A,S: set @ A,F2: A > A,G: A > A,U: A,V: A,T2: set @ A] :
( ( X2 = Y2 )
=> ( ( eventually @ A
@ ^ [X: A] :
( ( member @ A @ X @ S )
=> ( ( F2 @ X )
= ( G @ X ) ) )
@ ( topolo7230453075368039082e_nhds @ A @ X2 ) )
=> ( ( U = V )
=> ( ( S = T2 )
=> ( ( member @ A @ X2 @ S )
=> ( ( has_field_derivative @ A @ F2 @ U @ ( topolo174197925503356063within @ A @ X2 @ S ) )
= ( has_field_derivative @ A @ G @ V @ ( topolo174197925503356063within @ A @ Y2 @ T2 ) ) ) ) ) ) ) ) ) ).
% has_field_derivative_cong_ev
thf(fact_7679_topological__tendstoI,axiom,
! [A: $tType,B: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [L2: A,F2: B > A,F3: filter @ B] :
( ! [S7: set @ A] :
( ( topolo1002775350975398744n_open @ A @ S7 )
=> ( ( member @ A @ L2 @ S7 )
=> ( eventually @ B
@ ^ [X: B] : ( member @ A @ ( F2 @ X ) @ S7 )
@ F3 ) ) )
=> ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L2 ) @ F3 ) ) ) ).
% topological_tendstoI
thf(fact_7680_topological__tendstoD,axiom,
! [A: $tType,B: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [F2: B > A,L2: A,F3: filter @ B,S: set @ A] :
( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L2 ) @ F3 )
=> ( ( topolo1002775350975398744n_open @ A @ S )
=> ( ( member @ A @ L2 @ S )
=> ( eventually @ B
@ ^ [X: B] : ( member @ A @ ( F2 @ X ) @ S )
@ F3 ) ) ) ) ) ).
% topological_tendstoD
thf(fact_7681_tendsto__def,axiom,
! [B: $tType,A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [F2: B > A,L2: A,F3: filter @ B] :
( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L2 ) @ F3 )
= ( ! [S10: set @ A] :
( ( topolo1002775350975398744n_open @ A @ S10 )
=> ( ( member @ A @ L2 @ S10 )
=> ( eventually @ B
@ ^ [X: B] : ( member @ A @ ( F2 @ X ) @ S10 )
@ F3 ) ) ) ) ) ) ).
% tendsto_def
thf(fact_7682_filterlim__at__bot,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ B )
=> ! [F2: A > B,F3: filter @ A] :
( ( filterlim @ A @ B @ F2 @ ( at_bot @ B ) @ F3 )
= ( ! [Z9: B] :
( eventually @ A
@ ^ [X: A] : ( ord_less_eq @ B @ ( F2 @ X ) @ Z9 )
@ F3 ) ) ) ) ).
% filterlim_at_bot
thf(fact_7683_filterlim__at__bot__le,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ B )
=> ! [F2: A > B,F3: filter @ A,C2: B] :
( ( filterlim @ A @ B @ F2 @ ( at_bot @ B ) @ F3 )
= ( ! [Z9: B] :
( ( ord_less_eq @ B @ Z9 @ C2 )
=> ( eventually @ A
@ ^ [X: A] : ( ord_less_eq @ B @ ( F2 @ X ) @ Z9 )
@ F3 ) ) ) ) ) ).
% filterlim_at_bot_le
thf(fact_7684_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
@ ^ [X: A] : ( ord_less @ B @ ( F2 @ X ) @ Z9 )
@ F3 ) ) ) ) ).
% filterlim_at_bot_dense
thf(fact_7685_real__tendsto__sandwich,axiom,
! [B: $tType,F2: B > real,G: B > real,Net: filter @ B,H2: B > real,C2: real] :
( ( eventually @ B
@ ^ [N3: B] : ( ord_less_eq @ real @ ( F2 @ N3 ) @ ( G @ N3 ) )
@ Net )
=> ( ( eventually @ B
@ ^ [N3: B] : ( ord_less_eq @ real @ ( G @ N3 ) @ ( H2 @ N3 ) )
@ Net )
=> ( ( filterlim @ B @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ C2 ) @ Net )
=> ( ( filterlim @ B @ real @ H2 @ ( topolo7230453075368039082e_nhds @ real @ C2 ) @ Net )
=> ( filterlim @ B @ real @ G @ ( topolo7230453075368039082e_nhds @ real @ C2 ) @ Net ) ) ) ) ) ).
% real_tendsto_sandwich
thf(fact_7686_countable__basis__at__decseq,axiom,
! [A: $tType] :
( ( topolo3112930676232923870pology @ A )
=> ! [X2: A] :
~ ! [A8: nat > ( set @ A )] :
( ! [I4: nat] : ( topolo1002775350975398744n_open @ A @ ( A8 @ I4 ) )
=> ( ! [I4: nat] : ( member @ A @ X2 @ ( A8 @ I4 ) )
=> ~ ! [S9: set @ A] :
( ( topolo1002775350975398744n_open @ A @ S9 )
=> ( ( member @ A @ X2 @ S9 )
=> ( eventually @ nat
@ ^ [I3: nat] : ( ord_less_eq @ ( set @ A ) @ ( A8 @ I3 ) @ S9 )
@ ( at_top @ nat ) ) ) ) ) ) ) ).
% countable_basis_at_decseq
thf(fact_7687_eventually__at,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [P: A > $o,A2: A,S: set @ A] :
( ( eventually @ A @ P @ ( topolo174197925503356063within @ A @ A2 @ S ) )
= ( ? [D: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D )
& ! [X: A] :
( ( member @ A @ X @ S )
=> ( ( ( X != A2 )
& ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X @ A2 ) @ D ) )
=> ( P @ X ) ) ) ) ) ) ) ).
% eventually_at
thf(fact_7688_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 )
& ! [X: A] :
( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X @ A2 ) @ D )
=> ( P @ X ) ) ) ) ) ) ).
% eventually_nhds_metric
thf(fact_7689_eventually__at__to__0,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [P: A > $o,A2: A] :
( ( eventually @ A @ P @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) )
= ( eventually @ A
@ ^ [X: A] : ( P @ ( plus_plus @ A @ X @ A2 ) )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% eventually_at_to_0
thf(fact_7690_decreasing__tendsto,axiom,
! [A: $tType,B: $tType] :
( ( topolo2564578578187576103pology @ A )
=> ! [L2: A,F2: B > A,F3: filter @ B] :
( ( eventually @ B
@ ^ [N3: B] : ( ord_less_eq @ A @ L2 @ ( F2 @ N3 ) )
@ F3 )
=> ( ! [X3: A] :
( ( ord_less @ A @ L2 @ X3 )
=> ( eventually @ B
@ ^ [N3: B] : ( ord_less @ A @ ( F2 @ N3 ) @ X3 )
@ F3 ) )
=> ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L2 ) @ F3 ) ) ) ) ).
% decreasing_tendsto
thf(fact_7691_increasing__tendsto,axiom,
! [A: $tType,B: $tType] :
( ( topolo2564578578187576103pology @ A )
=> ! [F2: B > A,L2: A,F3: filter @ B] :
( ( eventually @ B
@ ^ [N3: B] : ( ord_less_eq @ A @ ( F2 @ N3 ) @ L2 )
@ F3 )
=> ( ! [X3: A] :
( ( ord_less @ A @ X3 @ L2 )
=> ( eventually @ B
@ ^ [N3: B] : ( ord_less @ A @ X3 @ ( F2 @ N3 ) )
@ F3 ) )
=> ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L2 ) @ F3 ) ) ) ) ).
% increasing_tendsto
thf(fact_7692_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
@ ^ [X: A] : ( ord_less_eq @ B @ Z9 @ ( F2 @ X ) )
@ F3 ) ) ) ) ) ).
% filterlim_at_top_gt
thf(fact_7693_tendsto__compose__eventually,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( ( topolo4958980785337419405_space @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [G: A > B,M: B,L2: A,F2: C > A,F3: filter @ C] :
( ( filterlim @ A @ B @ G @ ( topolo7230453075368039082e_nhds @ B @ M ) @ ( topolo174197925503356063within @ A @ L2 @ ( top_top @ ( set @ A ) ) ) )
=> ( ( filterlim @ C @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L2 ) @ F3 )
=> ( ( eventually @ C
@ ^ [X: C] :
( ( F2 @ X )
!= L2 )
@ F3 )
=> ( filterlim @ C @ B
@ ^ [X: C] : ( G @ ( F2 @ X ) )
@ ( topolo7230453075368039082e_nhds @ B @ M )
@ F3 ) ) ) ) ) ).
% tendsto_compose_eventually
thf(fact_7694_LIM__compose__eventually,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( topolo4958980785337419405_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 ) ) ) )
=> ( ( eventually @ A
@ ^ [X: A] :
( ( F2 @ X )
!= B2 )
@ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) )
=> ( filterlim @ A @ C
@ ^ [X: A] : ( G @ ( F2 @ X ) )
@ ( topolo7230453075368039082e_nhds @ C @ C2 )
@ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ).
% LIM_compose_eventually
thf(fact_7695_filterlim__atI,axiom,
! [A: $tType,B: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [F2: B > A,C2: A,F3: filter @ B] :
( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ C2 ) @ F3 )
=> ( ( eventually @ B
@ ^ [X: B] :
( ( F2 @ X )
!= C2 )
@ F3 )
=> ( filterlim @ B @ A @ F2 @ ( topolo174197925503356063within @ A @ C2 @ ( top_top @ ( set @ A ) ) ) @ F3 ) ) ) ) ).
% filterlim_atI
thf(fact_7696_isCont__cong,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [F2: A > B,G: A > B,X2: A] :
( ( eventually @ A
@ ^ [X: A] :
( ( F2 @ X )
= ( G @ X ) )
@ ( topolo7230453075368039082e_nhds @ A @ X2 ) )
=> ( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ X2 @ ( top_top @ ( set @ A ) ) ) @ F2 )
= ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ X2 @ ( top_top @ ( set @ A ) ) ) @ G ) ) ) ) ).
% isCont_cong
thf(fact_7697_DERIV__cong__ev,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [X2: A,Y2: A,F2: A > A,G: A > A,U: A,V: A] :
( ( X2 = Y2 )
=> ( ( eventually @ A
@ ^ [X: A] :
( ( F2 @ X )
= ( G @ X ) )
@ ( topolo7230453075368039082e_nhds @ A @ X2 ) )
=> ( ( U = V )
=> ( ( has_field_derivative @ A @ F2 @ U @ ( topolo174197925503356063within @ A @ X2 @ ( top_top @ ( set @ A ) ) ) )
= ( has_field_derivative @ A @ G @ V @ ( topolo174197925503356063within @ A @ Y2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ) ).
% DERIV_cong_ev
thf(fact_7698_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
@ ^ [X: A] : ( ord_less_eq @ B @ ( F2 @ X ) @ Z9 )
@ F3 ) ) ) ) ) ).
% filterlim_at_bot_lt
thf(fact_7699_tendsto__upperbound,axiom,
! [B: $tType,A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [F2: B > A,X2: A,F3: filter @ B,A2: A] :
( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ X2 ) @ F3 )
=> ( ( eventually @ B
@ ^ [I3: B] : ( ord_less_eq @ A @ ( F2 @ I3 ) @ A2 )
@ F3 )
=> ( ( F3
!= ( bot_bot @ ( filter @ B ) ) )
=> ( ord_less_eq @ A @ X2 @ A2 ) ) ) ) ) ).
% tendsto_upperbound
thf(fact_7700_tendsto__lowerbound,axiom,
! [B: $tType,A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [F2: B > A,X2: A,F3: filter @ B,A2: A] :
( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ X2 ) @ F3 )
=> ( ( eventually @ B
@ ^ [I3: B] : ( ord_less_eq @ A @ A2 @ ( F2 @ I3 ) )
@ F3 )
=> ( ( F3
!= ( bot_bot @ ( filter @ B ) ) )
=> ( ord_less_eq @ A @ A2 @ X2 ) ) ) ) ) ).
% tendsto_lowerbound
thf(fact_7701_tendsto__le,axiom,
! [B: $tType,A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [F3: filter @ B,F2: B > A,X2: A,G: B > A,Y2: A] :
( ( F3
!= ( bot_bot @ ( filter @ B ) ) )
=> ( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ X2 ) @ F3 )
=> ( ( filterlim @ B @ A @ G @ ( topolo7230453075368039082e_nhds @ A @ Y2 ) @ F3 )
=> ( ( eventually @ B
@ ^ [X: B] : ( ord_less_eq @ A @ ( G @ X ) @ ( F2 @ X ) )
@ F3 )
=> ( ord_less_eq @ A @ Y2 @ X2 ) ) ) ) ) ) ).
% tendsto_le
thf(fact_7702_metric__tendsto__imp__tendsto,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( ( real_V7819770556892013058_space @ B )
& ( real_V7819770556892013058_space @ A ) )
=> ! [F2: C > A,A2: A,F3: filter @ C,G: C > B,B2: B] :
( ( filterlim @ C @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ F3 )
=> ( ( eventually @ C
@ ^ [X: C] : ( ord_less_eq @ real @ ( real_V557655796197034286t_dist @ B @ ( G @ X ) @ B2 ) @ ( real_V557655796197034286t_dist @ A @ ( F2 @ X ) @ A2 ) )
@ F3 )
=> ( filterlim @ C @ B @ G @ ( topolo7230453075368039082e_nhds @ B @ B2 ) @ F3 ) ) ) ) ).
% metric_tendsto_imp_tendsto
thf(fact_7703_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
@ ^ [X: A] : ( ord_less @ real @ ( zero_zero @ real ) @ ( F2 @ X ) )
@ F3 )
=> ( filterlim @ A @ real @ F2 @ ( at_top @ real ) @ F3 ) ) ) ).
% filterlim_at_infinity_imp_filterlim_at_top
thf(fact_7704_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
@ ^ [X: A] : ( ord_less @ real @ ( F2 @ X ) @ ( zero_zero @ real ) )
@ F3 )
=> ( filterlim @ A @ real @ F2 @ ( at_bot @ real ) @ F3 ) ) ) ).
% filterlim_at_infinity_imp_filterlim_at_bot
thf(fact_7705_eventually__floor__eq,axiom,
! [B: $tType,A: $tType] :
( ( ( archim2362893244070406136eiling @ B )
& ( topolo2564578578187576103pology @ B ) )
=> ! [F2: A > B,L2: B,F3: filter @ A] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L2 ) @ F3 )
=> ( ~ ( member @ B @ L2 @ ( ring_1_Ints @ B ) )
=> ( eventually @ A
@ ^ [X: A] :
( ( archim6421214686448440834_floor @ B @ ( F2 @ X ) )
= ( archim6421214686448440834_floor @ B @ L2 ) )
@ F3 ) ) ) ) ).
% eventually_floor_eq
thf(fact_7706_eventually__ceiling__eq,axiom,
! [B: $tType,A: $tType] :
( ( ( archim2362893244070406136eiling @ B )
& ( topolo2564578578187576103pology @ B ) )
=> ! [F2: A > B,L2: B,F3: filter @ A] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L2 ) @ F3 )
=> ( ~ ( member @ B @ L2 @ ( ring_1_Ints @ B ) )
=> ( eventually @ A
@ ^ [X: A] :
( ( archimedean_ceiling @ B @ ( F2 @ X ) )
= ( archimedean_ceiling @ B @ L2 ) )
@ F3 ) ) ) ) ).
% eventually_ceiling_eq
thf(fact_7707_eventually__at__right__to__0,axiom,
! [P: real > $o,A2: real] :
( ( eventually @ real @ P @ ( topolo174197925503356063within @ real @ A2 @ ( set_ord_greaterThan @ real @ A2 ) ) )
= ( eventually @ real
@ ^ [X: real] : ( P @ ( plus_plus @ real @ X @ A2 ) )
@ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) ) ) ).
% eventually_at_right_to_0
thf(fact_7708_eventually__at__left__to__right,axiom,
! [P: real > $o,A2: real] :
( ( eventually @ real @ P @ ( topolo174197925503356063within @ real @ A2 @ ( set_ord_lessThan @ real @ A2 ) ) )
= ( eventually @ real
@ ^ [X: real] : ( P @ ( uminus_uminus @ real @ X ) )
@ ( topolo174197925503356063within @ real @ ( uminus_uminus @ real @ A2 ) @ ( set_ord_greaterThan @ real @ ( uminus_uminus @ real @ A2 ) ) ) ) ) ).
% eventually_at_left_to_right
thf(fact_7709_continuous__arcosh__strong,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [F3: filter @ A,F2: A > real] :
( ( topolo3448309680560233919inuous @ A @ real @ F3 @ F2 )
=> ( ( eventually @ A
@ ^ [X: A] : ( ord_less_eq @ real @ ( one_one @ real ) @ ( F2 @ X ) )
@ F3 )
=> ( topolo3448309680560233919inuous @ A @ real @ F3
@ ^ [X: A] : ( arcosh @ real @ ( F2 @ X ) ) ) ) ) ) ).
% continuous_arcosh_strong
thf(fact_7710_eventually__at__right__real,axiom,
! [A2: real,B2: real] :
( ( ord_less @ real @ A2 @ B2 )
=> ( eventually @ real
@ ^ [X: real] : ( member @ real @ X @ ( set_or5935395276787703475ssThan @ real @ A2 @ B2 ) )
@ ( topolo174197925503356063within @ real @ A2 @ ( set_ord_greaterThan @ real @ A2 ) ) ) ) ).
% eventually_at_right_real
thf(fact_7711_eventually__at__left__real,axiom,
! [B2: real,A2: real] :
( ( ord_less @ real @ B2 @ A2 )
=> ( eventually @ real
@ ^ [X: real] : ( member @ real @ X @ ( set_or5935395276787703475ssThan @ real @ B2 @ A2 ) )
@ ( topolo174197925503356063within @ real @ A2 @ ( set_ord_lessThan @ real @ A2 ) ) ) ) ).
% eventually_at_left_real
thf(fact_7712_eventually__at__le,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [P: A > $o,A2: A,S: set @ A] :
( ( eventually @ A @ P @ ( topolo174197925503356063within @ A @ A2 @ S ) )
= ( ? [D: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D )
& ! [X: A] :
( ( member @ A @ X @ S )
=> ( ( ( X != A2 )
& ( ord_less_eq @ real @ ( real_V557655796197034286t_dist @ A @ X @ A2 ) @ D ) )
=> ( P @ X ) ) ) ) ) ) ) ).
% eventually_at_le
thf(fact_7713_eventually__at__infinity__pos,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [P2: A > $o] :
( ( eventually @ A @ P2 @ ( at_infinity @ A ) )
= ( ? [B3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ B3 )
& ! [X: A] :
( ( ord_less_eq @ real @ B3 @ ( real_V7770717601297561774m_norm @ A @ X ) )
=> ( P2 @ X ) ) ) ) ) ) ).
% eventually_at_infinity_pos
thf(fact_7714_tendsto__compose__at,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( topolo4958980785337419405_space @ B )
& ( topolo4958980785337419405_space @ C ) )
=> ! [F2: A > B,Y2: B,F3: filter @ A,G: B > C,Z: C] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ Y2 ) @ F3 )
=> ( ( filterlim @ B @ C @ G @ ( topolo7230453075368039082e_nhds @ C @ Z ) @ ( topolo174197925503356063within @ B @ Y2 @ ( top_top @ ( set @ B ) ) ) )
=> ( ( eventually @ A
@ ^ [W2: A] :
( ( ( F2 @ W2 )
= Y2 )
=> ( ( G @ Y2 )
= Z ) )
@ F3 )
=> ( filterlim @ A @ C @ ( comp @ B @ C @ A @ G @ F2 ) @ ( topolo7230453075368039082e_nhds @ C @ Z ) @ F3 ) ) ) ) ) ).
% tendsto_compose_at
thf(fact_7715_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
@ ^ [X: A] : ( ord_less @ B @ ( F2 @ X ) @ L6 )
@ F3 )
=> ( filterlim @ A @ B @ F2 @ ( topolo174197925503356063within @ B @ L6 @ ( set_ord_lessThan @ B @ L6 ) ) @ F3 ) ) ) ) ).
% tendsto_imp_filterlim_at_left
thf(fact_7716_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
@ ^ [X: A] : ( ord_less @ B @ L6 @ ( F2 @ X ) )
@ F3 )
=> ( filterlim @ A @ B @ F2 @ ( topolo174197925503356063within @ B @ L6 @ ( set_ord_greaterThan @ B @ L6 ) ) @ F3 ) ) ) ) ).
% tendsto_imp_filterlim_at_right
thf(fact_7717_tendstoD,axiom,
! [A: $tType,B: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [F2: B > A,L2: A,F3: filter @ B,E: real] :
( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L2 ) @ F3 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ E )
=> ( eventually @ B
@ ^ [X: B] : ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( F2 @ X ) @ L2 ) @ E )
@ F3 ) ) ) ) ).
% tendstoD
thf(fact_7718_tendstoI,axiom,
! [A: $tType,B: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [F2: B > A,L2: A,F3: filter @ B] :
( ! [E2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E2 )
=> ( eventually @ B
@ ^ [X: B] : ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( F2 @ X ) @ L2 ) @ E2 )
@ F3 ) )
=> ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L2 ) @ F3 ) ) ) ).
% tendstoI
thf(fact_7719_tendsto__iff,axiom,
! [B: $tType,A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [F2: B > A,L2: A,F3: filter @ B] :
( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L2 ) @ F3 )
= ( ! [E4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E4 )
=> ( eventually @ B
@ ^ [X: B] : ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( F2 @ X ) @ L2 ) @ E4 )
@ F3 ) ) ) ) ) ).
% tendsto_iff
thf(fact_7720_summable__comparison__test__ev,axiom,
! [A: $tType] :
( ( real_Vector_banach @ A )
=> ! [F2: nat > A,G: nat > real] :
( ( eventually @ nat
@ ^ [N3: nat] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( F2 @ N3 ) ) @ ( G @ N3 ) )
@ ( at_top @ nat ) )
=> ( ( summable @ real @ G )
=> ( summable @ A @ F2 ) ) ) ) ).
% summable_comparison_test_ev
thf(fact_7721_eventually__at__right__to__top,axiom,
! [P: real > $o] :
( ( eventually @ real @ P @ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) )
= ( eventually @ real
@ ^ [X: real] : ( P @ ( inverse_inverse @ real @ X ) )
@ ( at_top @ real ) ) ) ).
% eventually_at_right_to_top
thf(fact_7722_eventually__at__top__to__right,axiom,
! [P: real > $o] :
( ( eventually @ real @ P @ ( at_top @ real ) )
= ( eventually @ real
@ ^ [X: real] : ( P @ ( inverse_inverse @ real @ X ) )
@ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) ) ) ).
% eventually_at_top_to_right
thf(fact_7723_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
@ ^ [X: B] : ( ord_less_eq @ real @ ( one_one @ real ) @ ( F2 @ X ) )
@ F3 )
=> ( filterlim @ B @ real
@ ^ [X: B] : ( arcosh @ real @ ( F2 @ X ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( arcosh @ real @ A2 ) )
@ F3 ) ) ) ) ).
% tendsto_arcosh_strong
thf(fact_7724_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 ) ) )
=> ( ! [B4: A] :
( ( Q @ B4 )
=> ( ord_less @ A @ B4 @ 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_7725_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 ) ) )
=> ( ! [B4: A] :
( ( Q @ B4 )
=> ( ord_less @ A @ A2 @ B4 ) )
=> ( ( 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_7726_filterlim__at__withinI,axiom,
! [A: $tType,B: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [F2: B > A,C2: A,F3: filter @ B,A3: set @ A] :
( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ C2 ) @ F3 )
=> ( ( eventually @ B
@ ^ [X: B] : ( member @ A @ ( F2 @ X ) @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ C2 @ ( bot_bot @ ( set @ A ) ) ) ) )
@ F3 )
=> ( filterlim @ B @ A @ F2 @ ( topolo174197925503356063within @ A @ C2 @ A3 ) @ F3 ) ) ) ) ).
% filterlim_at_withinI
thf(fact_7727_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
@ ^ [X: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ C @ ( G @ X ) ) @ ( times_times @ real @ ( real_V7770717601297561774m_norm @ B @ ( F2 @ X ) ) @ K6 ) )
@ F3 )
=> ( filterlim @ A @ C @ G @ ( topolo7230453075368039082e_nhds @ C @ ( zero_zero @ C ) ) @ F3 ) ) ) ) ).
% tendsto_0_le
thf(fact_7728_eventually__floor__less,axiom,
! [B: $tType,A: $tType] :
( ( ( archim2362893244070406136eiling @ B )
& ( topolo2564578578187576103pology @ B ) )
=> ! [F2: A > B,L2: B,F3: filter @ A] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L2 ) @ F3 )
=> ( ~ ( member @ B @ L2 @ ( ring_1_Ints @ B ) )
=> ( eventually @ A
@ ^ [X: A] : ( ord_less @ B @ ( ring_1_of_int @ B @ ( archim6421214686448440834_floor @ B @ L2 ) ) @ ( F2 @ X ) )
@ F3 ) ) ) ) ).
% eventually_floor_less
thf(fact_7729_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
@ ^ [X: C] : ( ord_less_eq @ real @ R5 @ ( real_V7770717601297561774m_norm @ A @ ( F2 @ X ) ) )
@ F3 ) ) ) ) ) ) ).
% filterlim_at_infinity
thf(fact_7730_eventually__less__ceiling,axiom,
! [B: $tType,A: $tType] :
( ( ( archim2362893244070406136eiling @ B )
& ( topolo2564578578187576103pology @ B ) )
=> ! [F2: A > B,L2: B,F3: filter @ A] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L2 ) @ F3 )
=> ( ~ ( member @ B @ L2 @ ( ring_1_Ints @ B ) )
=> ( eventually @ A
@ ^ [X: A] : ( ord_less @ B @ ( F2 @ X ) @ ( ring_1_of_int @ B @ ( archimedean_ceiling @ B @ L2 ) ) )
@ F3 ) ) ) ) ).
% eventually_less_ceiling
thf(fact_7731_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
@ ^ [X: A] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( F2 @ X ) )
@ F3 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ B2 )
=> ( filterlim @ A @ real
@ ^ [X: A] : ( powr @ real @ ( F2 @ X ) @ ( G @ X ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ F3 ) ) ) ) ) ).
% tendsto_zero_powrI
thf(fact_7732_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
@ ^ [X: A] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( F2 @ X ) )
@ F3 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ B2 )
=> ( filterlim @ A @ real
@ ^ [X: A] : ( powr @ real @ ( F2 @ X ) @ ( G @ X ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( powr @ real @ A2 @ B2 ) )
@ F3 ) ) ) ) ) ).
% tendsto_powr2
thf(fact_7733_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
@ ^ [X: A] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( F2 @ X ) )
@ F3 ) ) )
=> ( filterlim @ A @ real
@ ^ [X: A] : ( powr @ real @ ( F2 @ X ) @ ( G @ X ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( powr @ real @ A2 @ B2 ) )
@ F3 ) ) ) ) ).
% tendsto_powr'
thf(fact_7734_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
@ ^ [X: A] : ( ord_less @ real @ ( zero_zero @ real ) @ ( G @ X ) )
@ F3 )
=> ( filterlim @ A @ real
@ ^ [X: A] : ( divide_divide @ real @ ( F2 @ X ) @ ( G @ X ) )
@ ( at_top @ real )
@ F3 ) ) ) ) ) ).
% LIM_at_top_divide
thf(fact_7735_filterlim__at__top__iff__inverse__0,axiom,
! [A: $tType,F2: A > real,F3: filter @ A] :
( ( eventually @ A
@ ^ [X: A] : ( ord_less @ real @ ( zero_zero @ real ) @ ( F2 @ X ) )
@ 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_7736_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
@ ^ [X: A] : ( ord_less @ real @ ( zero_zero @ real ) @ ( F2 @ X ) )
@ F3 )
=> ( filterlim @ A @ real
@ ^ [X: A] : ( inverse_inverse @ real @ ( F2 @ X ) )
@ ( at_top @ real )
@ F3 ) ) ) ).
% filterlim_inverse_at_top
thf(fact_7737_filterlim__inverse__at__top__iff,axiom,
! [A: $tType,F2: A > real,F3: filter @ A] :
( ( eventually @ A
@ ^ [X: A] : ( ord_less @ real @ ( zero_zero @ real ) @ ( F2 @ X ) )
@ F3 )
=> ( ( filterlim @ A @ real
@ ^ [X: A] : ( inverse_inverse @ real @ ( F2 @ X ) )
@ ( at_top @ real )
@ F3 )
= ( filterlim @ A @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ F3 ) ) ) ).
% filterlim_inverse_at_top_iff
thf(fact_7738_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
@ ^ [X: A] : ( ord_less @ real @ ( F2 @ X ) @ ( zero_zero @ real ) )
@ F3 )
=> ( filterlim @ A @ real
@ ^ [X: A] : ( inverse_inverse @ real @ ( F2 @ X ) )
@ ( at_bot @ real )
@ F3 ) ) ) ).
% filterlim_inverse_at_bot
thf(fact_7739_lhopital__at__top__at__top,axiom,
! [F2: real > real,A2: real,G: real > real,F7: 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
@ ^ [X: real] : ( has_field_derivative @ real @ F2 @ ( F7 @ X ) @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ A2 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( eventually @ real
@ ^ [X: real] : ( has_field_derivative @ real @ G @ ( G4 @ X ) @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ A2 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( filterlim @ real @ real
@ ^ [X: real] : ( divide_divide @ real @ ( F7 @ X ) @ ( G4 @ X ) )
@ ( at_top @ real )
@ ( topolo174197925503356063within @ real @ A2 @ ( top_top @ ( set @ real ) ) ) )
=> ( filterlim @ real @ real
@ ^ [X: real] : ( divide_divide @ real @ ( F2 @ X ) @ ( G @ X ) )
@ ( at_top @ real )
@ ( topolo174197925503356063within @ real @ A2 @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ) ) ).
% lhopital_at_top_at_top
thf(fact_7740_lhopital,axiom,
! [F2: real > real,X2: real,G: real > real,G4: real > real,F7: real > real,F3: filter @ real] :
( ( filterlim @ real @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( filterlim @ real @ real @ G @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( eventually @ real
@ ^ [X: real] :
( ( G @ X )
!= ( zero_zero @ real ) )
@ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( eventually @ real
@ ^ [X: real] :
( ( G4 @ X )
!= ( zero_zero @ real ) )
@ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( eventually @ real
@ ^ [X: real] : ( has_field_derivative @ real @ F2 @ ( F7 @ X ) @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( eventually @ real
@ ^ [X: real] : ( has_field_derivative @ real @ G @ ( G4 @ X ) @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( filterlim @ real @ real
@ ^ [X: real] : ( divide_divide @ real @ ( F7 @ X ) @ ( G4 @ X ) )
@ F3
@ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
=> ( filterlim @ real @ real
@ ^ [X: real] : ( divide_divide @ real @ ( F2 @ X ) @ ( G @ X ) )
@ F3
@ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ) ) ) ) ).
% lhopital
thf(fact_7741_lhopital__right__at__top__at__top,axiom,
! [F2: real > real,A2: real,G: real > real,F7: 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
@ ^ [X: real] : ( has_field_derivative @ real @ F2 @ ( F7 @ X ) @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ A2 @ ( set_ord_greaterThan @ real @ A2 ) ) )
=> ( ( eventually @ real
@ ^ [X: real] : ( has_field_derivative @ real @ G @ ( G4 @ X ) @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ A2 @ ( set_ord_greaterThan @ real @ A2 ) ) )
=> ( ( filterlim @ real @ real
@ ^ [X: real] : ( divide_divide @ real @ ( F7 @ X ) @ ( G4 @ X ) )
@ ( at_top @ real )
@ ( topolo174197925503356063within @ real @ A2 @ ( set_ord_greaterThan @ real @ A2 ) ) )
=> ( filterlim @ real @ real
@ ^ [X: real] : ( divide_divide @ real @ ( F2 @ X ) @ ( G @ X ) )
@ ( at_top @ real )
@ ( topolo174197925503356063within @ real @ A2 @ ( set_ord_greaterThan @ real @ A2 ) ) ) ) ) ) ) ) ).
% lhopital_right_at_top_at_top
thf(fact_7742_lhopital__at__top__at__bot,axiom,
! [F2: real > real,A2: real,G: real > real,F7: 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
@ ^ [X: real] : ( has_field_derivative @ real @ F2 @ ( F7 @ X ) @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ A2 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( eventually @ real
@ ^ [X: real] : ( has_field_derivative @ real @ G @ ( G4 @ X ) @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ A2 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( filterlim @ real @ real
@ ^ [X: real] : ( divide_divide @ real @ ( F7 @ X ) @ ( G4 @ X ) )
@ ( at_bot @ real )
@ ( topolo174197925503356063within @ real @ A2 @ ( top_top @ ( set @ real ) ) ) )
=> ( filterlim @ real @ real
@ ^ [X: real] : ( divide_divide @ real @ ( F2 @ X ) @ ( G @ X ) )
@ ( at_bot @ real )
@ ( topolo174197925503356063within @ real @ A2 @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ) ) ).
% lhopital_at_top_at_bot
thf(fact_7743_lhopital__left__at__top__at__top,axiom,
! [F2: real > real,A2: real,G: real > real,F7: 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
@ ^ [X: real] : ( has_field_derivative @ real @ F2 @ ( F7 @ X ) @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ A2 @ ( set_ord_lessThan @ real @ A2 ) ) )
=> ( ( eventually @ real
@ ^ [X: real] : ( has_field_derivative @ real @ G @ ( G4 @ X ) @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ A2 @ ( set_ord_lessThan @ real @ A2 ) ) )
=> ( ( filterlim @ real @ real
@ ^ [X: real] : ( divide_divide @ real @ ( F7 @ X ) @ ( G4 @ X ) )
@ ( at_top @ real )
@ ( topolo174197925503356063within @ real @ A2 @ ( set_ord_lessThan @ real @ A2 ) ) )
=> ( filterlim @ real @ real
@ ^ [X: real] : ( divide_divide @ real @ ( F2 @ X ) @ ( G @ X ) )
@ ( at_top @ real )
@ ( topolo174197925503356063within @ real @ A2 @ ( set_ord_lessThan @ real @ A2 ) ) ) ) ) ) ) ) ).
% lhopital_left_at_top_at_top
thf(fact_7744_lhopital__at__top,axiom,
! [G: real > real,X2: real,G4: real > real,F2: real > real,F7: real > real,Y2: real] :
( ( filterlim @ real @ real @ G @ ( at_top @ real ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( eventually @ real
@ ^ [X: real] :
( ( G4 @ X )
!= ( zero_zero @ real ) )
@ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( eventually @ real
@ ^ [X: real] : ( has_field_derivative @ real @ F2 @ ( F7 @ X ) @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( eventually @ real
@ ^ [X: real] : ( has_field_derivative @ real @ G @ ( G4 @ X ) @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( filterlim @ real @ real
@ ^ [X: real] : ( divide_divide @ real @ ( F7 @ X ) @ ( G4 @ X ) )
@ ( topolo7230453075368039082e_nhds @ real @ Y2 )
@ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
=> ( filterlim @ real @ real
@ ^ [X: real] : ( divide_divide @ real @ ( F2 @ X ) @ ( G @ X ) )
@ ( topolo7230453075368039082e_nhds @ real @ Y2 )
@ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ) ) ).
% lhopital_at_top
thf(fact_7745_lhospital__at__top__at__top,axiom,
! [G: real > real,G4: real > real,F2: real > real,F7: real > real,X2: real] :
( ( filterlim @ real @ real @ G @ ( at_top @ real ) @ ( at_top @ real ) )
=> ( ( eventually @ real
@ ^ [X: real] :
( ( G4 @ X )
!= ( zero_zero @ real ) )
@ ( at_top @ real ) )
=> ( ( eventually @ real
@ ^ [X: real] : ( has_field_derivative @ real @ F2 @ ( F7 @ X ) @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
@ ( at_top @ real ) )
=> ( ( eventually @ real
@ ^ [X: real] : ( has_field_derivative @ real @ G @ ( G4 @ X ) @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
@ ( at_top @ real ) )
=> ( ( filterlim @ real @ real
@ ^ [X: real] : ( divide_divide @ real @ ( F7 @ X ) @ ( G4 @ X ) )
@ ( topolo7230453075368039082e_nhds @ real @ X2 )
@ ( at_top @ real ) )
=> ( filterlim @ real @ real
@ ^ [X: real] : ( divide_divide @ real @ ( F2 @ X ) @ ( G @ X ) )
@ ( topolo7230453075368039082e_nhds @ real @ X2 )
@ ( at_top @ real ) ) ) ) ) ) ) ).
% lhospital_at_top_at_top
thf(fact_7746_lhopital__right,axiom,
! [F2: real > real,X2: real,G: real > real,G4: real > real,F7: real > real,F3: filter @ real] :
( ( filterlim @ real @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( topolo174197925503356063within @ real @ X2 @ ( set_ord_greaterThan @ real @ X2 ) ) )
=> ( ( filterlim @ real @ real @ G @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( topolo174197925503356063within @ real @ X2 @ ( set_ord_greaterThan @ real @ X2 ) ) )
=> ( ( eventually @ real
@ ^ [X: real] :
( ( G @ X )
!= ( zero_zero @ real ) )
@ ( topolo174197925503356063within @ real @ X2 @ ( set_ord_greaterThan @ real @ X2 ) ) )
=> ( ( eventually @ real
@ ^ [X: real] :
( ( G4 @ X )
!= ( zero_zero @ real ) )
@ ( topolo174197925503356063within @ real @ X2 @ ( set_ord_greaterThan @ real @ X2 ) ) )
=> ( ( eventually @ real
@ ^ [X: real] : ( has_field_derivative @ real @ F2 @ ( F7 @ X ) @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ X2 @ ( set_ord_greaterThan @ real @ X2 ) ) )
=> ( ( eventually @ real
@ ^ [X: real] : ( has_field_derivative @ real @ G @ ( G4 @ X ) @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ X2 @ ( set_ord_greaterThan @ real @ X2 ) ) )
=> ( ( filterlim @ real @ real
@ ^ [X: real] : ( divide_divide @ real @ ( F7 @ X ) @ ( G4 @ X ) )
@ F3
@ ( topolo174197925503356063within @ real @ X2 @ ( set_ord_greaterThan @ real @ X2 ) ) )
=> ( filterlim @ real @ real
@ ^ [X: real] : ( divide_divide @ real @ ( F2 @ X ) @ ( G @ X ) )
@ F3
@ ( topolo174197925503356063within @ real @ X2 @ ( set_ord_greaterThan @ real @ X2 ) ) ) ) ) ) ) ) ) ) ).
% lhopital_right
thf(fact_7747_lhopital__right__0,axiom,
! [F0: real > real,G0: real > real,G4: real > real,F7: 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
@ ^ [X: real] :
( ( G0 @ X )
!= ( zero_zero @ real ) )
@ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) )
=> ( ( eventually @ real
@ ^ [X: real] :
( ( G4 @ X )
!= ( zero_zero @ real ) )
@ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) )
=> ( ( eventually @ real
@ ^ [X: real] : ( has_field_derivative @ real @ F0 @ ( F7 @ X ) @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) )
=> ( ( eventually @ real
@ ^ [X: real] : ( has_field_derivative @ real @ G0 @ ( G4 @ X ) @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) )
=> ( ( filterlim @ real @ real
@ ^ [X: real] : ( divide_divide @ real @ ( F7 @ X ) @ ( G4 @ X ) )
@ F3
@ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) )
=> ( filterlim @ real @ real
@ ^ [X: real] : ( divide_divide @ real @ ( F0 @ X ) @ ( G0 @ X ) )
@ F3
@ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) ) ) ) ) ) ) ) ) ).
% lhopital_right_0
thf(fact_7748_lhopital__left,axiom,
! [F2: real > real,X2: real,G: real > real,G4: real > real,F7: real > real,F3: filter @ real] :
( ( filterlim @ real @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( topolo174197925503356063within @ real @ X2 @ ( set_ord_lessThan @ real @ X2 ) ) )
=> ( ( filterlim @ real @ real @ G @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( topolo174197925503356063within @ real @ X2 @ ( set_ord_lessThan @ real @ X2 ) ) )
=> ( ( eventually @ real
@ ^ [X: real] :
( ( G @ X )
!= ( zero_zero @ real ) )
@ ( topolo174197925503356063within @ real @ X2 @ ( set_ord_lessThan @ real @ X2 ) ) )
=> ( ( eventually @ real
@ ^ [X: real] :
( ( G4 @ X )
!= ( zero_zero @ real ) )
@ ( topolo174197925503356063within @ real @ X2 @ ( set_ord_lessThan @ real @ X2 ) ) )
=> ( ( eventually @ real
@ ^ [X: real] : ( has_field_derivative @ real @ F2 @ ( F7 @ X ) @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ X2 @ ( set_ord_lessThan @ real @ X2 ) ) )
=> ( ( eventually @ real
@ ^ [X: real] : ( has_field_derivative @ real @ G @ ( G4 @ X ) @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ X2 @ ( set_ord_lessThan @ real @ X2 ) ) )
=> ( ( filterlim @ real @ real
@ ^ [X: real] : ( divide_divide @ real @ ( F7 @ X ) @ ( G4 @ X ) )
@ F3
@ ( topolo174197925503356063within @ real @ X2 @ ( set_ord_lessThan @ real @ X2 ) ) )
=> ( filterlim @ real @ real
@ ^ [X: real] : ( divide_divide @ real @ ( F2 @ X ) @ ( G @ X ) )
@ F3
@ ( topolo174197925503356063within @ real @ X2 @ ( set_ord_lessThan @ real @ X2 ) ) ) ) ) ) ) ) ) ) ).
% lhopital_left
thf(fact_7749_lhopital__right__at__top__at__bot,axiom,
! [F2: real > real,A2: real,G: real > real,F7: 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
@ ^ [X: real] : ( has_field_derivative @ real @ F2 @ ( F7 @ X ) @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ A2 @ ( set_ord_greaterThan @ real @ A2 ) ) )
=> ( ( eventually @ real
@ ^ [X: real] : ( has_field_derivative @ real @ G @ ( G4 @ X ) @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ A2 @ ( set_ord_greaterThan @ real @ A2 ) ) )
=> ( ( filterlim @ real @ real
@ ^ [X: real] : ( divide_divide @ real @ ( F7 @ X ) @ ( G4 @ X ) )
@ ( at_bot @ real )
@ ( topolo174197925503356063within @ real @ A2 @ ( set_ord_greaterThan @ real @ A2 ) ) )
=> ( filterlim @ real @ real
@ ^ [X: real] : ( divide_divide @ real @ ( F2 @ X ) @ ( G @ X ) )
@ ( at_bot @ real )
@ ( topolo174197925503356063within @ real @ A2 @ ( set_ord_greaterThan @ real @ A2 ) ) ) ) ) ) ) ) ).
% lhopital_right_at_top_at_bot
thf(fact_7750_lhopital__left__at__top__at__bot,axiom,
! [F2: real > real,A2: real,G: real > real,F7: 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
@ ^ [X: real] : ( has_field_derivative @ real @ F2 @ ( F7 @ X ) @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ A2 @ ( set_ord_lessThan @ real @ A2 ) ) )
=> ( ( eventually @ real
@ ^ [X: real] : ( has_field_derivative @ real @ G @ ( G4 @ X ) @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ A2 @ ( set_ord_lessThan @ real @ A2 ) ) )
=> ( ( filterlim @ real @ real
@ ^ [X: real] : ( divide_divide @ real @ ( F7 @ X ) @ ( G4 @ X ) )
@ ( at_bot @ real )
@ ( topolo174197925503356063within @ real @ A2 @ ( set_ord_lessThan @ real @ A2 ) ) )
=> ( filterlim @ real @ real
@ ^ [X: real] : ( divide_divide @ real @ ( F2 @ X ) @ ( G @ X ) )
@ ( at_bot @ real )
@ ( topolo174197925503356063within @ real @ A2 @ ( set_ord_lessThan @ real @ A2 ) ) ) ) ) ) ) ) ).
% lhopital_left_at_top_at_bot
thf(fact_7751_lhopital__right__at__top,axiom,
! [G: real > real,X2: real,G4: real > real,F2: real > real,F7: real > real,Y2: real] :
( ( filterlim @ real @ real @ G @ ( at_top @ real ) @ ( topolo174197925503356063within @ real @ X2 @ ( set_ord_greaterThan @ real @ X2 ) ) )
=> ( ( eventually @ real
@ ^ [X: real] :
( ( G4 @ X )
!= ( zero_zero @ real ) )
@ ( topolo174197925503356063within @ real @ X2 @ ( set_ord_greaterThan @ real @ X2 ) ) )
=> ( ( eventually @ real
@ ^ [X: real] : ( has_field_derivative @ real @ F2 @ ( F7 @ X ) @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ X2 @ ( set_ord_greaterThan @ real @ X2 ) ) )
=> ( ( eventually @ real
@ ^ [X: real] : ( has_field_derivative @ real @ G @ ( G4 @ X ) @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ X2 @ ( set_ord_greaterThan @ real @ X2 ) ) )
=> ( ( filterlim @ real @ real
@ ^ [X: real] : ( divide_divide @ real @ ( F7 @ X ) @ ( G4 @ X ) )
@ ( topolo7230453075368039082e_nhds @ real @ Y2 )
@ ( topolo174197925503356063within @ real @ X2 @ ( set_ord_greaterThan @ real @ X2 ) ) )
=> ( filterlim @ real @ real
@ ^ [X: real] : ( divide_divide @ real @ ( F2 @ X ) @ ( G @ X ) )
@ ( topolo7230453075368039082e_nhds @ real @ Y2 )
@ ( topolo174197925503356063within @ real @ X2 @ ( set_ord_greaterThan @ real @ X2 ) ) ) ) ) ) ) ) ).
% lhopital_right_at_top
thf(fact_7752_lhopital__right__0__at__top,axiom,
! [G: real > real,G4: real > real,F2: real > real,F7: real > real,X2: real] :
( ( filterlim @ real @ real @ G @ ( at_top @ real ) @ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) )
=> ( ( eventually @ real
@ ^ [X: real] :
( ( G4 @ X )
!= ( zero_zero @ real ) )
@ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) )
=> ( ( eventually @ real
@ ^ [X: real] : ( has_field_derivative @ real @ F2 @ ( F7 @ X ) @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) )
=> ( ( eventually @ real
@ ^ [X: real] : ( has_field_derivative @ real @ G @ ( G4 @ X ) @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) )
=> ( ( filterlim @ real @ real
@ ^ [X: real] : ( divide_divide @ real @ ( F7 @ X ) @ ( G4 @ X ) )
@ ( topolo7230453075368039082e_nhds @ real @ X2 )
@ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) )
=> ( filterlim @ real @ real
@ ^ [X: real] : ( divide_divide @ real @ ( F2 @ X ) @ ( G @ X ) )
@ ( topolo7230453075368039082e_nhds @ real @ X2 )
@ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) ) ) ) ) ) ) ).
% lhopital_right_0_at_top
thf(fact_7753_summable__Cauchy_H,axiom,
! [A: $tType] :
( ( real_Vector_banach @ A )
=> ! [F2: nat > A,G: nat > real] :
( ( eventually @ nat
@ ^ [M5: nat] :
! [N3: nat] :
( ( ord_less_eq @ nat @ M5 @ N3 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_or7035219750837199246ssThan @ nat @ M5 @ N3 ) ) ) @ ( G @ M5 ) ) )
@ ( at_top @ nat ) )
=> ( ( filterlim @ nat @ real @ G @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) )
=> ( summable @ A @ F2 ) ) ) ) ).
% summable_Cauchy'
thf(fact_7754_Bfun__metric__def,axiom,
! [B: $tType,A: $tType] :
( ( real_V7819770556892013058_space @ B )
=> ( ( bfun @ A @ B )
= ( ^ [F4: A > B,F9: filter @ A] :
? [Y: B,K7: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K7 )
& ( eventually @ A
@ ^ [X: A] : ( ord_less_eq @ real @ ( real_V557655796197034286t_dist @ B @ ( F4 @ X ) @ Y ) @ K7 )
@ F9 ) ) ) ) ) ).
% Bfun_metric_def
thf(fact_7755_eventually__all__finite,axiom,
! [B: $tType,A: $tType] :
( ( finite_finite @ B )
=> ! [P: A > B > $o,Net: filter @ A] :
( ! [Y3: B] :
( eventually @ A
@ ^ [X: A] : ( P @ X @ Y3 )
@ Net )
=> ( eventually @ A
@ ^ [X: A] :
! [X7: B] : ( P @ X @ X7 )
@ Net ) ) ) ).
% eventually_all_finite
thf(fact_7756_finite__set__of__finite__funs,axiom,
! [B: $tType,A: $tType,A3: set @ A,B5: set @ B,D2: B] :
( ( finite_finite2 @ A @ A3 )
=> ( ( finite_finite2 @ B @ B5 )
=> ( finite_finite2 @ ( A > B )
@ ( collect @ ( A > B )
@ ^ [F4: A > B] :
! [X: A] :
( ( ( member @ A @ X @ A3 )
=> ( member @ B @ ( F4 @ X ) @ B5 ) )
& ( ~ ( member @ A @ X @ A3 )
=> ( ( F4 @ X )
= D2 ) ) ) ) ) ) ) ).
% finite_set_of_finite_funs
thf(fact_7757_Bfun__const,axiom,
! [B: $tType,A: $tType] :
( ( real_V7819770556892013058_space @ B )
=> ! [C2: B,F3: filter @ A] :
( bfun @ A @ B
@ ^ [Uu3: A] : C2
@ F3 ) ) ).
% Bfun_const
thf(fact_7758_filterlim__int__sequentially,axiom,
filterlim @ nat @ int @ ( semiring_1_of_nat @ int ) @ ( at_top @ int ) @ ( at_top @ nat ) ).
% filterlim_int_sequentially
thf(fact_7759_Bseq__minus__iff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X6: nat > A] :
( ( bfun @ nat @ A
@ ^ [N3: nat] : ( uminus_uminus @ A @ ( X6 @ N3 ) )
@ ( at_top @ nat ) )
= ( bfun @ nat @ A @ X6 @ ( at_top @ nat ) ) ) ) ).
% Bseq_minus_iff
thf(fact_7760_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_7761_Bseq__subseq,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A,G: nat > nat] :
( ( bfun @ nat @ A @ F2 @ ( at_top @ nat ) )
=> ( bfun @ nat @ A
@ ^ [X: nat] : ( F2 @ ( G @ X ) )
@ ( at_top @ nat ) ) ) ) ).
% Bseq_subseq
thf(fact_7762_Bseq__add,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A,C2: A] :
( ( bfun @ nat @ A @ F2 @ ( at_top @ nat ) )
=> ( bfun @ nat @ A
@ ^ [X: nat] : ( plus_plus @ A @ ( F2 @ X ) @ C2 )
@ ( at_top @ nat ) ) ) ) ).
% Bseq_add
thf(fact_7763_Bseq__add__iff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A,C2: A] :
( ( bfun @ nat @ A
@ ^ [X: nat] : ( plus_plus @ A @ ( F2 @ X ) @ C2 )
@ ( at_top @ nat ) )
= ( bfun @ nat @ A @ F2 @ ( at_top @ nat ) ) ) ) ).
% Bseq_add_iff
thf(fact_7764_Bseq__offset,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X6: nat > A,K: nat] :
( ( bfun @ nat @ A
@ ^ [N3: nat] : ( X6 @ ( plus_plus @ nat @ N3 @ K ) )
@ ( at_top @ nat ) )
=> ( bfun @ nat @ A @ X6 @ ( at_top @ nat ) ) ) ) ).
% Bseq_offset
thf(fact_7765_Bseq__ignore__initial__segment,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X6: nat > A,K: nat] :
( ( bfun @ nat @ A @ X6 @ ( at_top @ nat ) )
=> ( bfun @ nat @ A
@ ^ [N3: nat] : ( X6 @ ( plus_plus @ nat @ N3 @ K ) )
@ ( at_top @ nat ) ) ) ) ).
% Bseq_ignore_initial_segment
thf(fact_7766_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
@ ^ [X: nat] : ( times_times @ A @ ( F2 @ X ) @ ( G @ X ) )
@ ( at_top @ nat ) ) ) ) ) ).
% Bseq_mult
thf(fact_7767_BseqI_H,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X6: nat > A,K6: real] :
( ! [N2: nat] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( X6 @ N2 ) ) @ K6 )
=> ( bfun @ nat @ A @ X6 @ ( at_top @ nat ) ) ) ) ).
% BseqI'
thf(fact_7768_eventually__all__ge__at__top,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > $o] :
( ( eventually @ A @ P @ ( at_top @ A ) )
=> ( eventually @ A
@ ^ [X: A] :
! [Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( P @ Y ) )
@ ( at_top @ A ) ) ) ) ).
% eventually_all_ge_at_top
thf(fact_7769_Bseq__cmult__iff,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [C2: A,F2: nat > A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( bfun @ nat @ A
@ ^ [X: nat] : ( times_times @ A @ C2 @ ( F2 @ X ) )
@ ( at_top @ nat ) )
= ( bfun @ nat @ A @ F2 @ ( at_top @ nat ) ) ) ) ) ).
% Bseq_cmult_iff
thf(fact_7770_Bseq__eventually__mono,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: nat > A,G: nat > B] :
( ( eventually @ nat
@ ^ [N3: nat] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( F2 @ N3 ) ) @ ( real_V7770717601297561774m_norm @ B @ ( G @ N3 ) ) )
@ ( at_top @ nat ) )
=> ( ( bfun @ nat @ B @ G @ ( at_top @ nat ) )
=> ( bfun @ nat @ A @ F2 @ ( at_top @ nat ) ) ) ) ) ).
% Bseq_eventually_mono
thf(fact_7771_filterlim__int__of__nat__at__topD,axiom,
! [A: $tType,F2: int > A,F3: filter @ A] :
( ( filterlim @ nat @ A
@ ^ [X: nat] : ( F2 @ ( semiring_1_of_nat @ int @ X ) )
@ F3
@ ( at_top @ nat ) )
=> ( filterlim @ int @ A @ F2 @ F3 @ ( at_top @ int ) ) ) ).
% filterlim_int_of_nat_at_topD
thf(fact_7772_BseqD,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X6: nat > A] :
( ( bfun @ nat @ A @ X6 @ ( at_top @ nat ) )
=> ? [K9: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K9 )
& ! [N11: nat] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( X6 @ N11 ) ) @ K9 ) ) ) ) ).
% BseqD
thf(fact_7773_BseqE,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X6: nat > A] :
( ( bfun @ nat @ A @ X6 @ ( at_top @ nat ) )
=> ~ ! [K9: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K9 )
=> ~ ! [N11: nat] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( X6 @ N11 ) ) @ K9 ) ) ) ) ).
% BseqE
thf(fact_7774_BseqI,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [K6: real,X6: nat > A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K6 )
=> ( ! [N2: nat] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( X6 @ N2 ) ) @ K6 )
=> ( bfun @ nat @ A @ X6 @ ( at_top @ nat ) ) ) ) ) ).
% BseqI
thf(fact_7775_Bseq__def,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X6: nat > A] :
( ( bfun @ nat @ A @ X6 @ ( at_top @ nat ) )
= ( ? [K7: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K7 )
& ! [N3: nat] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( X6 @ N3 ) ) @ K7 ) ) ) ) ) ).
% Bseq_def
thf(fact_7776_Bseq__iff1a,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X6: nat > A] :
( ( bfun @ nat @ A @ X6 @ ( at_top @ nat ) )
= ( ? [N8: nat] :
! [N3: nat] : ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( X6 @ N3 ) ) @ ( semiring_1_of_nat @ real @ ( suc @ N8 ) ) ) ) ) ) ).
% Bseq_iff1a
thf(fact_7777_Bseq__iff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X6: nat > A] :
( ( bfun @ nat @ A @ X6 @ ( at_top @ nat ) )
= ( ? [N8: nat] :
! [N3: nat] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( X6 @ N3 ) ) @ ( semiring_1_of_nat @ real @ ( suc @ N8 ) ) ) ) ) ) ).
% Bseq_iff
thf(fact_7778_Bseq__realpow,axiom,
! [X2: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
=> ( ( ord_less_eq @ real @ X2 @ ( one_one @ real ) )
=> ( bfun @ nat @ real @ ( power_power @ real @ X2 ) @ ( at_top @ nat ) ) ) ) ).
% Bseq_realpow
thf(fact_7779_BfunI,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [F2: A > B,K6: real,F3: filter @ A] :
( ( eventually @ A
@ ^ [X: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( F2 @ X ) ) @ K6 )
@ F3 )
=> ( bfun @ A @ B @ F2 @ F3 ) ) ) ).
% BfunI
thf(fact_7780_Bseq__iff3,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X6: nat > A] :
( ( bfun @ nat @ A @ X6 @ ( at_top @ nat ) )
= ( ? [K2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K2 )
& ? [N8: nat] :
! [N3: nat] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ ( X6 @ N3 ) @ ( uminus_uminus @ A @ ( X6 @ N8 ) ) ) ) @ K2 ) ) ) ) ) ).
% Bseq_iff3
thf(fact_7781_Bseq__iff2,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X6: nat > A] :
( ( bfun @ nat @ A @ X6 @ ( at_top @ nat ) )
= ( ? [K2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K2 )
& ? [X: A] :
! [N3: nat] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ ( X6 @ N3 ) @ ( uminus_uminus @ A @ X ) ) ) @ K2 ) ) ) ) ) ).
% Bseq_iff2
thf(fact_7782_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
@ ^ [X: B] : ( inverse_inverse @ A @ ( F2 @ X ) )
@ F3 ) ) ) ) ).
% Bfun_inverse
thf(fact_7783_BfunE,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [F2: A > B,F3: filter @ A] :
( ( bfun @ A @ B @ F2 @ F3 )
=> ~ ! [B9: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ B9 )
=> ~ ( eventually @ A
@ ^ [X: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( F2 @ X ) ) @ B9 )
@ F3 ) ) ) ) ).
% BfunE
thf(fact_7784_Bfun__def,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ( ( bfun @ A @ B )
= ( ^ [F4: A > B,F9: filter @ A] :
? [K7: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K7 )
& ( eventually @ A
@ ^ [X: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( F4 @ X ) ) @ K7 )
@ F9 ) ) ) ) ) ).
% Bfun_def
thf(fact_7785_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 )
=> ! [B3: nat] :
( ( ord_less @ nat @ A4 @ B3 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_or3652927894154168847AtMost @ nat @ A4 @ B3 ) ) ) @ ( 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_7786_Greatest__def,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( order_Greatest @ A )
= ( ^ [P7: A > $o] :
( the @ A
@ ^ [X: A] :
( ( P7 @ X )
& ! [Y: A] :
( ( P7 @ Y )
=> ( ord_less_eq @ A @ Y @ X ) ) ) ) ) ) ) ).
% Greatest_def
thf(fact_7787_greaterThanAtMost__iff,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [I: A,L2: A,U: A] :
( ( member @ A @ I @ ( set_or3652927894154168847AtMost @ A @ L2 @ U ) )
= ( ( ord_less @ A @ L2 @ I )
& ( ord_less_eq @ A @ I @ U ) ) ) ) ).
% greaterThanAtMost_iff
thf(fact_7788_greaterThanAtMost__empty,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L2: A,K: A] :
( ( ord_less_eq @ A @ L2 @ K )
=> ( ( set_or3652927894154168847AtMost @ A @ K @ L2 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% greaterThanAtMost_empty
thf(fact_7789_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_7790_Ioc__subset__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A,C2: A,D2: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ A2 @ B2 ) @ ( set_or3652927894154168847AtMost @ A @ C2 @ D2 ) )
= ( ( ord_less_eq @ A @ B2 @ A2 )
| ( ( ord_less_eq @ A @ C2 @ A2 )
& ( ord_less_eq @ A @ B2 @ D2 ) ) ) ) ) ).
% Ioc_subset_iff
thf(fact_7791_Ioc__inj,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A,C2: A,D2: A] :
( ( ( set_or3652927894154168847AtMost @ A @ A2 @ B2 )
= ( set_or3652927894154168847AtMost @ A @ C2 @ D2 ) )
= ( ( ( ord_less_eq @ A @ B2 @ A2 )
& ( ord_less_eq @ A @ D2 @ C2 ) )
| ( ( A2 = C2 )
& ( B2 = D2 ) ) ) ) ) ).
% Ioc_inj
thf(fact_7792_GreatestI__ex__nat,axiom,
! [P: nat > $o,B2: nat] :
( ? [X_12: nat] : ( P @ X_12 )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq @ nat @ Y3 @ B2 ) )
=> ( P @ ( order_Greatest @ nat @ P ) ) ) ) ).
% GreatestI_ex_nat
thf(fact_7793_Greatest__le__nat,axiom,
! [P: nat > $o,K: nat,B2: nat] :
( ( P @ K )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq @ nat @ Y3 @ B2 ) )
=> ( ord_less_eq @ nat @ K @ ( order_Greatest @ nat @ P ) ) ) ) ).
% Greatest_le_nat
thf(fact_7794_GreatestI__nat,axiom,
! [P: nat > $o,K: nat,B2: nat] :
( ( P @ K )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq @ nat @ Y3 @ B2 ) )
=> ( P @ ( order_Greatest @ nat @ P ) ) ) ) ).
% GreatestI_nat
thf(fact_7795_open__left,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [S: set @ A,X2: A,Y2: A] :
( ( topolo1002775350975398744n_open @ A @ S )
=> ( ( member @ A @ X2 @ S )
=> ( ( ord_less @ A @ Y2 @ X2 )
=> ? [B4: A] :
( ( ord_less @ A @ B4 @ X2 )
& ( ord_less_eq @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ B4 @ X2 ) @ S ) ) ) ) ) ) ).
% open_left
thf(fact_7796_atLeastSucAtMost__greaterThanAtMost,axiom,
! [L2: nat,U: nat] :
( ( set_or1337092689740270186AtMost @ nat @ ( suc @ L2 ) @ U )
= ( set_or3652927894154168847AtMost @ nat @ L2 @ U ) ) ).
% atLeastSucAtMost_greaterThanAtMost
thf(fact_7797_greaterThanAtMost__upt,axiom,
( ( set_or3652927894154168847AtMost @ nat )
= ( ^ [N3: nat,M5: nat] : ( set2 @ nat @ ( upt @ ( suc @ N3 ) @ ( suc @ M5 ) ) ) ) ) ).
% greaterThanAtMost_upt
thf(fact_7798_GreatestI2__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [P: A > $o,X2: A,Q: A > $o] :
( ( P @ X2 )
=> ( ! [Y3: A] :
( ( P @ Y3 )
=> ( ord_less_eq @ A @ Y3 @ X2 ) )
=> ( ! [X3: A] :
( ( P @ X3 )
=> ( ! [Y4: A] :
( ( P @ Y4 )
=> ( ord_less_eq @ A @ Y4 @ X3 ) )
=> ( Q @ X3 ) ) )
=> ( Q @ ( order_Greatest @ A @ P ) ) ) ) ) ) ).
% GreatestI2_order
thf(fact_7799_Greatest__equality,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [P: A > $o,X2: A] :
( ( P @ X2 )
=> ( ! [Y3: A] :
( ( P @ Y3 )
=> ( ord_less_eq @ A @ Y3 @ X2 ) )
=> ( ( order_Greatest @ A @ P )
= X2 ) ) ) ) ).
% Greatest_equality
thf(fact_7800_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_7801_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_7802_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_7803_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_7804_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_7805_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] :
( ! [N11: nat] : ( ord_less @ A @ A2 @ ( F5 @ N11 ) )
=> ( ! [N11: nat] : ( ord_less @ A @ ( F5 @ N11 ) @ 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_7806_map__filter__on__comp,axiom,
! [A: $tType,C: $tType,B: $tType,G: B > A,Y7: set @ B,X6: set @ A,F3: filter @ B,F2: A > C] :
( ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ G @ Y7 ) @ X6 )
=> ( ( eventually @ B
@ ^ [X: B] : ( member @ B @ X @ Y7 )
@ F3 )
=> ( ( map_filter_on @ A @ C @ X6 @ F2 @ ( map_filter_on @ B @ A @ Y7 @ G @ F3 ) )
= ( map_filter_on @ B @ C @ Y7 @ ( comp @ A @ C @ B @ F2 @ G ) @ F3 ) ) ) ) ).
% map_filter_on_comp
thf(fact_7807_decseq__const,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [K: A] :
( order_antimono @ nat @ A
@ ^ [X: nat] : K ) ) ).
% decseq_const
thf(fact_7808_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_7809_decseq__SucI,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X6: nat > A] :
( ! [N2: nat] : ( ord_less_eq @ A @ ( X6 @ ( suc @ N2 ) ) @ ( X6 @ N2 ) )
=> ( order_antimono @ nat @ A @ X6 ) ) ) ).
% decseq_SucI
thf(fact_7810_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_7811_decseqD,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [F2: nat > A,I: nat,J: nat] :
( ( order_antimono @ nat @ A @ F2 )
=> ( ( ord_less_eq @ nat @ I @ J )
=> ( ord_less_eq @ A @ ( F2 @ J ) @ ( F2 @ I ) ) ) ) ) ).
% decseqD
thf(fact_7812_decseq__def,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( order_antimono @ nat @ A )
= ( ^ [X7: nat > A] :
! [M5: nat,N3: nat] :
( ( ord_less_eq @ nat @ M5 @ N3 )
=> ( ord_less_eq @ A @ ( X7 @ N3 ) @ ( X7 @ M5 ) ) ) ) ) ) ).
% decseq_def
thf(fact_7813_antimonoD,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [F2: A > B,X2: A,Y2: A] :
( ( order_antimono @ A @ B @ F2 )
=> ( ( ord_less_eq @ A @ X2 @ Y2 )
=> ( ord_less_eq @ B @ ( F2 @ Y2 ) @ ( F2 @ X2 ) ) ) ) ) ).
% antimonoD
thf(fact_7814_antimonoE,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [F2: A > B,X2: A,Y2: A] :
( ( order_antimono @ A @ B @ F2 )
=> ( ( ord_less_eq @ A @ X2 @ Y2 )
=> ( ord_less_eq @ B @ ( F2 @ Y2 ) @ ( F2 @ X2 ) ) ) ) ) ).
% antimonoE
thf(fact_7815_antimonoI,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [F2: A > B] :
( ! [X3: A,Y3: A] :
( ( ord_less_eq @ A @ X3 @ Y3 )
=> ( ord_less_eq @ B @ ( F2 @ Y3 ) @ ( F2 @ X3 ) ) )
=> ( order_antimono @ A @ B @ F2 ) ) ) ).
% antimonoI
thf(fact_7816_antimono__def,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ( ( order_antimono @ A @ B )
= ( ^ [F4: A > B] :
! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ord_less_eq @ B @ ( F4 @ Y ) @ ( F4 @ X ) ) ) ) ) ) ).
% antimono_def
thf(fact_7817_atLeastPlusOneAtMost__greaterThanAtMost__int,axiom,
! [L2: int,U: int] :
( ( set_or1337092689740270186AtMost @ int @ ( plus_plus @ int @ L2 @ ( one_one @ int ) ) @ U )
= ( set_or3652927894154168847AtMost @ int @ L2 @ U ) ) ).
% atLeastPlusOneAtMost_greaterThanAtMost_int
thf(fact_7818_eventually__map__filter__on,axiom,
! [B: $tType,A: $tType,X6: set @ A,F3: filter @ A,P: B > $o,F2: A > B] :
( ( eventually @ A
@ ^ [X: A] : ( member @ A @ X @ X6 )
@ F3 )
=> ( ( eventually @ B @ P @ ( map_filter_on @ A @ B @ X6 @ F2 @ F3 ) )
= ( eventually @ A
@ ^ [X: A] :
( ( P @ ( F2 @ X ) )
& ( member @ A @ X @ X6 ) )
@ F3 ) ) ) ).
% eventually_map_filter_on
thf(fact_7819_decseq__bounded,axiom,
! [X6: nat > real,B5: real] :
( ( order_antimono @ nat @ real @ X6 )
=> ( ! [I2: nat] : ( ord_less_eq @ real @ B5 @ ( X6 @ I2 ) )
=> ( bfun @ nat @ real @ X6 @ ( at_top @ nat ) ) ) ) ).
% decseq_bounded
thf(fact_7820_atLeastPlusOneAtMost__greaterThanAtMost__integer,axiom,
! [L2: code_integer,U: code_integer] :
( ( set_or1337092689740270186AtMost @ code_integer @ ( plus_plus @ code_integer @ L2 @ ( one_one @ code_integer ) ) @ U )
= ( set_or3652927894154168847AtMost @ code_integer @ L2 @ U ) ) ).
% atLeastPlusOneAtMost_greaterThanAtMost_integer
thf(fact_7821_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_7822_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_7823_decseq__ge,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [X6: nat > A,L6: A,N: nat] :
( ( order_antimono @ nat @ A @ X6 )
=> ( ( filterlim @ nat @ A @ X6 @ ( topolo7230453075368039082e_nhds @ A @ L6 ) @ ( at_top @ nat ) )
=> ( ord_less_eq @ A @ L6 @ ( X6 @ N ) ) ) ) ) ).
% decseq_ge
thf(fact_7824_decseq__convergent,axiom,
! [X6: nat > real,B5: real] :
( ( order_antimono @ nat @ real @ X6 )
=> ( ! [I2: nat] : ( ord_less_eq @ real @ B5 @ ( X6 @ I2 ) )
=> ~ ! [L7: real] :
( ( filterlim @ nat @ real @ X6 @ ( topolo7230453075368039082e_nhds @ real @ L7 ) @ ( at_top @ nat ) )
=> ~ ! [I4: nat] : ( ord_less_eq @ real @ L7 @ ( X6 @ I4 ) ) ) ) ) ).
% decseq_convergent
thf(fact_7825_tendsto__at__right__sequentially,axiom,
! [C: $tType,B: $tType] :
( ( ( topolo3112930676232923870pology @ B )
& ( topolo1944317154257567458pology @ B )
& ( topolo4958980785337419405_space @ C ) )
=> ! [A2: B,B2: B,X6: B > C,L6: C] :
( ( ord_less @ B @ A2 @ B2 )
=> ( ! [S7: nat > B] :
( ! [N11: nat] : ( ord_less @ B @ A2 @ ( S7 @ N11 ) )
=> ( ! [N11: nat] : ( ord_less @ B @ ( S7 @ N11 ) @ B2 )
=> ( ( order_antimono @ nat @ B @ S7 )
=> ( ( filterlim @ nat @ B @ S7 @ ( topolo7230453075368039082e_nhds @ B @ A2 ) @ ( at_top @ nat ) )
=> ( filterlim @ nat @ C
@ ^ [N3: nat] : ( X6 @ ( S7 @ N3 ) )
@ ( topolo7230453075368039082e_nhds @ C @ L6 )
@ ( at_top @ nat ) ) ) ) ) )
=> ( filterlim @ B @ C @ X6 @ ( topolo7230453075368039082e_nhds @ C @ L6 ) @ ( topolo174197925503356063within @ B @ A2 @ ( set_ord_greaterThan @ B @ A2 ) ) ) ) ) ) ).
% tendsto_at_right_sequentially
thf(fact_7826_cauchy__filter__metric,axiom,
! [A: $tType] :
( ( ( real_V768167426530841204y_dist @ A )
& ( topolo7287701948861334536_space @ A ) )
=> ( ( topolo6773858410816713723filter @ A )
= ( ^ [F9: filter @ A] :
! [E4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E4 )
=> ? [P7: A > $o] :
( ( eventually @ A @ P7 @ F9 )
& ! [X: A,Y: A] :
( ( ( P7 @ X )
& ( P7 @ Y ) )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X @ Y ) @ E4 ) ) ) ) ) ) ) ).
% cauchy_filter_metric
thf(fact_7827_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_7828_differentiable__cmult__left__iff,axiom,
! [A: $tType,B: $tType] :
( ( ( real_V822414075346904944vector @ B )
& ( real_V3459762299906320749_field @ A ) )
=> ! [C2: A,Q2: B > A,T2: B] :
( ( differentiable @ B @ A
@ ^ [T: B] : ( times_times @ A @ C2 @ ( Q2 @ T ) )
@ ( topolo174197925503356063within @ B @ T2 @ ( top_top @ ( set @ B ) ) ) )
= ( ( C2
= ( zero_zero @ A ) )
| ( differentiable @ B @ A @ Q2 @ ( topolo174197925503356063within @ B @ T2 @ ( top_top @ ( set @ B ) ) ) ) ) ) ) ).
% differentiable_cmult_left_iff
thf(fact_7829_differentiable__cmult__right__iff,axiom,
! [A: $tType,B: $tType] :
( ( ( real_V822414075346904944vector @ B )
& ( real_V3459762299906320749_field @ A ) )
=> ! [Q2: B > A,C2: A,T2: B] :
( ( differentiable @ B @ A
@ ^ [T: B] : ( times_times @ A @ ( Q2 @ T ) @ C2 )
@ ( topolo174197925503356063within @ B @ T2 @ ( top_top @ ( set @ B ) ) ) )
= ( ( C2
= ( zero_zero @ A ) )
| ( differentiable @ B @ A @ Q2 @ ( topolo174197925503356063within @ B @ T2 @ ( top_top @ ( set @ B ) ) ) ) ) ) ) ).
% differentiable_cmult_right_iff
thf(fact_7830_differentiable__within__subset,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,X2: A,S3: set @ A,T2: set @ A] :
( ( differentiable @ A @ B @ F2 @ ( topolo174197925503356063within @ A @ X2 @ S3 ) )
=> ( ( ord_less_eq @ ( set @ A ) @ T2 @ S3 )
=> ( differentiable @ A @ B @ F2 @ ( topolo174197925503356063within @ A @ X2 @ T2 ) ) ) ) ) ).
% differentiable_within_subset
thf(fact_7831_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
@ ^ [X: A] : ( plus_plus @ B @ ( F2 @ X ) @ ( G @ X ) )
@ F3 ) ) ) ) ).
% differentiable_add
thf(fact_7832_differentiable__const,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [A2: B,F3: filter @ A] :
( differentiable @ A @ B
@ ^ [Z5: A] : A2
@ F3 ) ) ).
% differentiable_const
thf(fact_7833_differentiable__ident,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: filter @ A] :
( differentiable @ A @ A
@ ^ [X: A] : X
@ F3 ) ) ).
% differentiable_ident
thf(fact_7834_differentiable__diff,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
@ ^ [X: A] : ( minus_minus @ B @ ( F2 @ X ) @ ( G @ X ) )
@ F3 ) ) ) ) ).
% differentiable_diff
thf(fact_7835_differentiable__minus,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,F3: filter @ A] :
( ( differentiable @ A @ B @ F2 @ F3 )
=> ( differentiable @ A @ B
@ ^ [X: A] : ( uminus_uminus @ B @ ( F2 @ X ) )
@ F3 ) ) ) ).
% differentiable_minus
thf(fact_7836_differentiable__compose,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,G: C > A,X2: C,S3: set @ C] :
( ( differentiable @ A @ B @ F2 @ ( topolo174197925503356063within @ A @ ( G @ X2 ) @ ( top_top @ ( set @ A ) ) ) )
=> ( ( differentiable @ C @ A @ G @ ( topolo174197925503356063within @ C @ X2 @ S3 ) )
=> ( differentiable @ C @ B
@ ^ [X: C] : ( F2 @ ( G @ X ) )
@ ( topolo174197925503356063within @ C @ X2 @ S3 ) ) ) ) ) ).
% differentiable_compose
thf(fact_7837_differentiable__in__compose,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,G: C > A,X2: C,S3: set @ C] :
( ( differentiable @ A @ B @ F2 @ ( topolo174197925503356063within @ A @ ( G @ X2 ) @ ( image @ C @ A @ G @ S3 ) ) )
=> ( ( differentiable @ C @ A @ G @ ( topolo174197925503356063within @ C @ X2 @ S3 ) )
=> ( differentiable @ C @ B
@ ^ [X: C] : ( F2 @ ( G @ X ) )
@ ( topolo174197925503356063within @ C @ X2 @ S3 ) ) ) ) ) ).
% differentiable_in_compose
thf(fact_7838_differentiable__mult,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V4412858255891104859lgebra @ B ) )
=> ! [F2: A > B,X2: A,S3: set @ A,G: A > B] :
( ( differentiable @ A @ B @ F2 @ ( topolo174197925503356063within @ A @ X2 @ S3 ) )
=> ( ( differentiable @ A @ B @ G @ ( topolo174197925503356063within @ A @ X2 @ S3 ) )
=> ( differentiable @ A @ B
@ ^ [X: A] : ( times_times @ B @ ( F2 @ X ) @ ( G @ X ) )
@ ( topolo174197925503356063within @ A @ X2 @ S3 ) ) ) ) ) ).
% differentiable_mult
thf(fact_7839_differentiable__scaleR,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > real,X2: A,S3: set @ A,G: A > B] :
( ( differentiable @ A @ real @ F2 @ ( topolo174197925503356063within @ A @ X2 @ S3 ) )
=> ( ( differentiable @ A @ B @ G @ ( topolo174197925503356063within @ A @ X2 @ S3 ) )
=> ( differentiable @ A @ B
@ ^ [X: A] : ( real_V8093663219630862766scaleR @ B @ ( F2 @ X ) @ ( G @ X ) )
@ ( topolo174197925503356063within @ A @ X2 @ S3 ) ) ) ) ) ).
% differentiable_scaleR
thf(fact_7840_differentiable__power,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V3459762299906320749_field @ B ) )
=> ! [F2: A > B,X2: A,S3: set @ A,N: nat] :
( ( differentiable @ A @ B @ F2 @ ( topolo174197925503356063within @ A @ X2 @ S3 ) )
=> ( differentiable @ A @ B
@ ^ [X: A] : ( power_power @ B @ ( F2 @ X ) @ N )
@ ( topolo174197925503356063within @ A @ X2 @ S3 ) ) ) ) ).
% differentiable_power
thf(fact_7841_differentiable__sum,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( ( real_V822414075346904944vector @ B )
& ( real_V822414075346904944vector @ C ) )
=> ! [S3: set @ A,F2: A > B > C,Net: filter @ B] :
( ( finite_finite2 @ A @ S3 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ S3 )
=> ( differentiable @ B @ C @ ( F2 @ X3 ) @ Net ) )
=> ( differentiable @ B @ C
@ ^ [X: B] :
( groups7311177749621191930dd_sum @ A @ C
@ ^ [A4: A] : ( F2 @ A4 @ X )
@ S3 )
@ Net ) ) ) ) ).
% differentiable_sum
thf(fact_7842_differentiable__divide,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V3459762299906320749_field @ B ) )
=> ! [F2: A > B,X2: A,S3: set @ A,G: A > B] :
( ( differentiable @ A @ B @ F2 @ ( topolo174197925503356063within @ A @ X2 @ S3 ) )
=> ( ( differentiable @ A @ B @ G @ ( topolo174197925503356063within @ A @ X2 @ S3 ) )
=> ( ( ( G @ X2 )
!= ( zero_zero @ B ) )
=> ( differentiable @ A @ B
@ ^ [X: A] : ( divide_divide @ B @ ( F2 @ X ) @ ( G @ X ) )
@ ( topolo174197925503356063within @ A @ X2 @ S3 ) ) ) ) ) ) ).
% differentiable_divide
thf(fact_7843_differentiable__inverse,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V3459762299906320749_field @ B ) )
=> ! [F2: A > B,X2: A,S3: set @ A] :
( ( differentiable @ A @ B @ F2 @ ( topolo174197925503356063within @ A @ X2 @ S3 ) )
=> ( ( ( F2 @ X2 )
!= ( zero_zero @ B ) )
=> ( differentiable @ A @ B
@ ^ [X: A] : ( inverse_inverse @ B @ ( F2 @ X ) )
@ ( topolo174197925503356063within @ A @ X2 @ S3 ) ) ) ) ) ).
% differentiable_inverse
thf(fact_7844_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,Z2: real] :
( ( ord_less @ real @ A2 @ Z2 )
& ( ord_less @ real @ Z2 @ B2 )
& ( has_field_derivative @ real @ F2 @ L4 @ ( topolo174197925503356063within @ real @ Z2 @ ( top_top @ ( set @ real ) ) ) )
& ( ( minus_minus @ real @ ( F2 @ B2 ) @ ( F2 @ A2 ) )
= ( times_times @ real @ ( minus_minus @ real @ B2 @ A2 ) @ L4 ) ) ) ) ) ) ).
% MVT
thf(fact_7845_interval__cases,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [S: set @ A] :
( ! [A5: A,B4: A,X3: A] :
( ( member @ A @ A5 @ S )
=> ( ( member @ A @ B4 @ S )
=> ( ( ord_less_eq @ A @ A5 @ X3 )
=> ( ( ord_less_eq @ A @ X3 @ B4 )
=> ( member @ A @ X3 @ S ) ) ) ) )
=> ? [A5: A,B4: A] :
( ( S
= ( bot_bot @ ( set @ A ) ) )
| ( S
= ( top_top @ ( set @ A ) ) )
| ( S
= ( set_ord_lessThan @ A @ B4 ) )
| ( S
= ( set_ord_atMost @ A @ B4 ) )
| ( S
= ( set_ord_greaterThan @ A @ A5 ) )
| ( S
= ( set_ord_atLeast @ A @ A5 ) )
| ( S
= ( set_or5935395276787703475ssThan @ A @ A5 @ B4 ) )
| ( S
= ( set_or3652927894154168847AtMost @ A @ A5 @ B4 ) )
| ( S
= ( set_or7035219750837199246ssThan @ A @ A5 @ B4 ) )
| ( S
= ( set_or1337092689740270186AtMost @ A @ A5 @ B4 ) ) ) ) ) ).
% interval_cases
thf(fact_7846_atLeast__0,axiom,
( ( set_ord_atLeast @ nat @ ( zero_zero @ nat ) )
= ( top_top @ ( set @ nat ) ) ) ).
% atLeast_0
thf(fact_7847_atLeast__iff,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [I: A,K: A] :
( ( member @ A @ I @ ( set_ord_atLeast @ A @ K ) )
= ( ord_less_eq @ A @ K @ I ) ) ) ).
% atLeast_iff
thf(fact_7848_atLeast__subset__iff,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X2: A,Y2: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_ord_atLeast @ A @ X2 ) @ ( set_ord_atLeast @ A @ Y2 ) )
= ( ord_less_eq @ A @ Y2 @ X2 ) ) ) ).
% atLeast_subset_iff
thf(fact_7849_image__add__atLeast,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [K: A,I: A] :
( ( image @ A @ A @ ( plus_plus @ A @ K ) @ ( set_ord_atLeast @ A @ I ) )
= ( set_ord_atLeast @ A @ ( plus_plus @ A @ K @ I ) ) ) ) ).
% image_add_atLeast
thf(fact_7850_Icc__subset__Ici__iff,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [L2: A,H2: A,L3: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ L2 @ H2 ) @ ( set_ord_atLeast @ A @ L3 ) )
= ( ~ ( ord_less_eq @ A @ L2 @ H2 )
| ( ord_less_eq @ A @ L3 @ L2 ) ) ) ) ).
% Icc_subset_Ici_iff
thf(fact_7851_continuous__on__tendsto__compose,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( ( topolo4958980785337419405_space @ B )
& ( topolo4958980785337419405_space @ A ) )
=> ! [S3: set @ A,F2: A > B,G: C > A,L2: A,F3: filter @ C] :
( ( topolo81223032696312382ous_on @ A @ B @ S3 @ F2 )
=> ( ( filterlim @ C @ A @ G @ ( topolo7230453075368039082e_nhds @ A @ L2 ) @ F3 )
=> ( ( member @ A @ L2 @ S3 )
=> ( ( eventually @ C
@ ^ [X: C] : ( member @ A @ ( G @ X ) @ S3 )
@ F3 )
=> ( filterlim @ C @ B
@ ^ [X: C] : ( F2 @ ( G @ X ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( F2 @ L2 ) )
@ F3 ) ) ) ) ) ) ).
% continuous_on_tendsto_compose
thf(fact_7852_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
@ ^ [X: C] : ( tanh @ A @ ( F2 @ X ) ) ) ) ) ) ).
% continuous_on_tanh
thf(fact_7853_atLeast__Suc__greaterThan,axiom,
! [K: nat] :
( ( set_ord_atLeast @ nat @ ( suc @ K ) )
= ( set_ord_greaterThan @ nat @ K ) ) ).
% atLeast_Suc_greaterThan
thf(fact_7854_continuous__on__tan,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [S3: set @ A,F2: A > A] :
( ( topolo81223032696312382ous_on @ A @ A @ S3 @ F2 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ S3 )
=> ( ( cos @ A @ ( F2 @ X3 ) )
!= ( zero_zero @ A ) ) )
=> ( topolo81223032696312382ous_on @ A @ A @ S3
@ ^ [X: A] : ( tan @ A @ ( F2 @ X ) ) ) ) ) ) ).
% continuous_on_tan
thf(fact_7855_bounded__linear_Ocontinuous__on,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( ( topolo4958980785337419405_space @ C )
& ( real_V822414075346904944vector @ B )
& ( real_V822414075346904944vector @ A ) )
=> ! [F2: A > B,S3: set @ C,G: C > A] :
( ( real_V3181309239436604168linear @ A @ B @ F2 )
=> ( ( topolo81223032696312382ous_on @ C @ A @ S3 @ G )
=> ( topolo81223032696312382ous_on @ C @ B @ S3
@ ^ [X: C] : ( F2 @ ( G @ X ) ) ) ) ) ) ).
% bounded_linear.continuous_on
thf(fact_7856_open__Collect__less,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo4958980785337419405_space @ A )
& ( topolo1944317154257567458pology @ B ) )
=> ! [F2: A > B,G: A > B] :
( ( topolo81223032696312382ous_on @ A @ B @ ( top_top @ ( set @ A ) ) @ F2 )
=> ( ( topolo81223032696312382ous_on @ A @ B @ ( top_top @ ( set @ A ) ) @ G )
=> ( topolo1002775350975398744n_open @ A
@ ( collect @ A
@ ^ [X: A] : ( ord_less @ B @ ( F2 @ X ) @ ( G @ X ) ) ) ) ) ) ) ).
% open_Collect_less
thf(fact_7857_open__Collect__neq,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo4958980785337419405_space @ A )
& ( topological_t2_space @ B ) )
=> ! [F2: A > B,G: A > B] :
( ( topolo81223032696312382ous_on @ A @ B @ ( top_top @ ( set @ A ) ) @ F2 )
=> ( ( topolo81223032696312382ous_on @ A @ B @ ( top_top @ ( set @ A ) ) @ G )
=> ( topolo1002775350975398744n_open @ A
@ ( collect @ A
@ ^ [X: A] :
( ( F2 @ X )
!= ( G @ X ) ) ) ) ) ) ) ).
% open_Collect_neq
thf(fact_7858_continuous__on__powr,axiom,
! [C: $tType] :
( ( topolo4958980785337419405_space @ C )
=> ! [S3: set @ C,F2: C > real,G: C > real] :
( ( topolo81223032696312382ous_on @ C @ real @ S3 @ F2 )
=> ( ( topolo81223032696312382ous_on @ C @ real @ S3 @ G )
=> ( ! [X3: C] :
( ( member @ C @ X3 @ S3 )
=> ( ( F2 @ X3 )
!= ( zero_zero @ real ) ) )
=> ( topolo81223032696312382ous_on @ C @ real @ S3
@ ^ [X: C] : ( powr @ real @ ( F2 @ X ) @ ( G @ X ) ) ) ) ) ) ) ).
% continuous_on_powr
thf(fact_7859_continuous__on__ln,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [S3: set @ A,F2: A > real] :
( ( topolo81223032696312382ous_on @ A @ real @ S3 @ F2 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ S3 )
=> ( ( F2 @ X3 )
!= ( zero_zero @ real ) ) )
=> ( topolo81223032696312382ous_on @ A @ real @ S3
@ ^ [X: A] : ( ln_ln @ real @ ( F2 @ X ) ) ) ) ) ) ).
% continuous_on_ln
thf(fact_7860_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
@ ^ [X: C] : ( power_power @ B @ ( F2 @ X ) @ ( G @ X ) ) ) ) ) ) ).
% continuous_on_power'
thf(fact_7861_continuous__on__power,axiom,
! [B: $tType,C: $tType] :
( ( ( topolo4958980785337419405_space @ C )
& ( power @ B )
& ( real_V4412858255891104859lgebra @ B ) )
=> ! [S3: set @ C,F2: C > B,N: nat] :
( ( topolo81223032696312382ous_on @ C @ B @ S3 @ F2 )
=> ( topolo81223032696312382ous_on @ C @ B @ S3
@ ^ [X: C] : ( power_power @ B @ ( F2 @ X ) @ N ) ) ) ) ).
% continuous_on_power
thf(fact_7862_continuous__on__sgn,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo4958980785337419405_space @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [S3: set @ A,F2: A > B] :
( ( topolo81223032696312382ous_on @ A @ B @ S3 @ F2 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ S3 )
=> ( ( F2 @ X3 )
!= ( zero_zero @ B ) ) )
=> ( topolo81223032696312382ous_on @ A @ B @ S3
@ ^ [X: A] : ( sgn_sgn @ B @ ( F2 @ X ) ) ) ) ) ) ).
% continuous_on_sgn
thf(fact_7863_continuous__on__inverse,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo4958980785337419405_space @ A )
& ( real_V8999393235501362500lgebra @ B ) )
=> ! [S3: set @ A,F2: A > B] :
( ( topolo81223032696312382ous_on @ A @ B @ S3 @ F2 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ S3 )
=> ( ( F2 @ X3 )
!= ( zero_zero @ B ) ) )
=> ( topolo81223032696312382ous_on @ A @ B @ S3
@ ^ [X: A] : ( inverse_inverse @ B @ ( F2 @ X ) ) ) ) ) ) ).
% continuous_on_inverse
thf(fact_7864_continuous__on__minus,axiom,
! [B: $tType,C: $tType] :
( ( ( topolo4958980785337419405_space @ C )
& ( topolo1633459387980952147up_add @ B ) )
=> ! [S3: set @ C,F2: C > B] :
( ( topolo81223032696312382ous_on @ C @ B @ S3 @ F2 )
=> ( topolo81223032696312382ous_on @ C @ B @ S3
@ ^ [X: C] : ( uminus_uminus @ B @ ( F2 @ X ) ) ) ) ) ).
% continuous_on_minus
thf(fact_7865_continuous__on__diff,axiom,
! [B: $tType,D6: $tType] :
( ( ( topolo4958980785337419405_space @ D6 )
& ( topolo1633459387980952147up_add @ B ) )
=> ! [S3: set @ D6,F2: D6 > B,G: D6 > B] :
( ( topolo81223032696312382ous_on @ D6 @ B @ S3 @ F2 )
=> ( ( topolo81223032696312382ous_on @ D6 @ B @ S3 @ G )
=> ( topolo81223032696312382ous_on @ D6 @ B @ S3
@ ^ [X: D6] : ( minus_minus @ B @ ( F2 @ X ) @ ( G @ X ) ) ) ) ) ) ).
% continuous_on_diff
thf(fact_7866_continuous__on__scaleR,axiom,
! [C: $tType,D6: $tType] :
( ( ( topolo4958980785337419405_space @ D6 )
& ( real_V822414075346904944vector @ C ) )
=> ! [S3: set @ D6,F2: D6 > real,G: D6 > C] :
( ( topolo81223032696312382ous_on @ D6 @ real @ S3 @ F2 )
=> ( ( topolo81223032696312382ous_on @ D6 @ C @ S3 @ G )
=> ( topolo81223032696312382ous_on @ D6 @ C @ S3
@ ^ [X: D6] : ( real_V8093663219630862766scaleR @ C @ ( F2 @ X ) @ ( G @ X ) ) ) ) ) ) ).
% continuous_on_scaleR
thf(fact_7867_continuous__on__pochhammer,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [A3: set @ A,N: nat] :
( topolo81223032696312382ous_on @ A @ A @ A3
@ ^ [Z5: A] : ( comm_s3205402744901411588hammer @ A @ Z5 @ N ) ) ) ).
% continuous_on_pochhammer
thf(fact_7868_continuous__on__pochhammer_H,axiom,
! [A: $tType,C: $tType] :
( ( ( topolo4958980785337419405_space @ C )
& ( real_V3459762299906320749_field @ A ) )
=> ! [S3: set @ C,F2: C > A,N: nat] :
( ( topolo81223032696312382ous_on @ C @ A @ S3 @ F2 )
=> ( topolo81223032696312382ous_on @ C @ A @ S3
@ ^ [X: C] : ( comm_s3205402744901411588hammer @ A @ ( F2 @ X ) @ N ) ) ) ) ).
% continuous_on_pochhammer'
thf(fact_7869_continuous__on__sin,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_Vector_banach @ B )
& ( real_V3459762299906320749_field @ B ) )
=> ! [S3: set @ A,F2: A > B] :
( ( topolo81223032696312382ous_on @ A @ B @ S3 @ F2 )
=> ( topolo81223032696312382ous_on @ A @ B @ S3
@ ^ [X: A] : ( sin @ B @ ( F2 @ X ) ) ) ) ) ).
% continuous_on_sin
thf(fact_7870_continuous__on__exp,axiom,
! [A: $tType,C: $tType] :
( ( ( topolo4958980785337419405_space @ C )
& ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [S3: set @ C,F2: C > A] :
( ( topolo81223032696312382ous_on @ C @ A @ S3 @ F2 )
=> ( topolo81223032696312382ous_on @ C @ A @ S3
@ ^ [X: C] : ( exp @ A @ ( F2 @ X ) ) ) ) ) ).
% continuous_on_exp
thf(fact_7871_continuous__on__const,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo4958980785337419405_space @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [S3: set @ A,C2: B] :
( topolo81223032696312382ous_on @ A @ B @ S3
@ ^ [X: A] : C2 ) ) ).
% continuous_on_const
thf(fact_7872_continuous__on__id,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [S3: set @ A] :
( topolo81223032696312382ous_on @ A @ A @ S3
@ ^ [X: A] : X ) ) ).
% continuous_on_id
thf(fact_7873_continuous__on__of__real,axiom,
! [A: $tType,C: $tType] :
( ( ( topolo4958980785337419405_space @ C )
& ( real_V2191834092415804123ebra_1 @ A )
& ( real_V822414075346904944vector @ A ) )
=> ! [S3: set @ C,G: C > real] :
( ( topolo81223032696312382ous_on @ C @ real @ S3 @ G )
=> ( topolo81223032696312382ous_on @ C @ A @ S3
@ ^ [X: C] : ( real_Vector_of_real @ A @ ( G @ X ) ) ) ) ) ).
% continuous_on_of_real
thf(fact_7874_continuous__on__norm,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo4958980785337419405_space @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [S3: set @ A,F2: A > B] :
( ( topolo81223032696312382ous_on @ A @ B @ S3 @ F2 )
=> ( topolo81223032696312382ous_on @ A @ real @ S3
@ ^ [X: A] : ( real_V7770717601297561774m_norm @ B @ ( F2 @ X ) ) ) ) ) ).
% continuous_on_norm
thf(fact_7875_continuous__on__cos,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_Vector_banach @ B )
& ( real_V3459762299906320749_field @ B ) )
=> ! [S3: set @ A,F2: A > B] :
( ( topolo81223032696312382ous_on @ A @ B @ S3 @ F2 )
=> ( topolo81223032696312382ous_on @ A @ B @ S3
@ ^ [X: A] : ( cos @ B @ ( F2 @ X ) ) ) ) ) ).
% continuous_on_cos
thf(fact_7876_continuous__on__real__sqrt,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [S3: set @ A,F2: A > real] :
( ( topolo81223032696312382ous_on @ A @ real @ S3 @ F2 )
=> ( topolo81223032696312382ous_on @ A @ real @ S3
@ ^ [X: A] : ( sqrt @ ( F2 @ X ) ) ) ) ) ).
% continuous_on_real_sqrt
thf(fact_7877_continuous__on__arctan,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [S3: set @ A,F2: A > real] :
( ( topolo81223032696312382ous_on @ A @ real @ S3 @ F2 )
=> ( topolo81223032696312382ous_on @ A @ real @ S3
@ ^ [X: A] : ( arctan @ ( F2 @ X ) ) ) ) ) ).
% continuous_on_arctan
thf(fact_7878_continuous__on__real__root,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [S3: set @ A,F2: A > real,N: nat] :
( ( topolo81223032696312382ous_on @ A @ real @ S3 @ F2 )
=> ( topolo81223032696312382ous_on @ A @ real @ S3
@ ^ [X: A] : ( root @ N @ ( F2 @ X ) ) ) ) ) ).
% continuous_on_real_root
thf(fact_7879_continuous__on__sinh,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 )
=> ( topolo81223032696312382ous_on @ C @ A @ A3
@ ^ [X: C] : ( sinh @ A @ ( F2 @ X ) ) ) ) ) ).
% continuous_on_sinh
thf(fact_7880_continuous__on__cosh,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 )
=> ( topolo81223032696312382ous_on @ C @ A @ A3
@ ^ [X: C] : ( cosh @ A @ ( F2 @ X ) ) ) ) ) ).
% continuous_on_cosh
thf(fact_7881_continuous__on__arsinh_H,axiom,
! [A3: set @ real,F2: real > real] :
( ( topolo81223032696312382ous_on @ real @ real @ A3 @ F2 )
=> ( topolo81223032696312382ous_on @ real @ real @ A3
@ ^ [X: real] : ( arsinh @ real @ ( F2 @ X ) ) ) ) ).
% continuous_on_arsinh'
thf(fact_7882_continuous__on__rabs,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [S3: set @ A,F2: A > real] :
( ( topolo81223032696312382ous_on @ A @ real @ S3 @ F2 )
=> ( topolo81223032696312382ous_on @ A @ real @ S3
@ ^ [X: A] : ( abs_abs @ real @ ( F2 @ X ) ) ) ) ) ).
% continuous_on_rabs
thf(fact_7883_continuous__on__max,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo4958980785337419405_space @ A )
& ( topolo1944317154257567458pology @ B ) )
=> ! [A3: set @ A,F2: A > B,G: A > B] :
( ( topolo81223032696312382ous_on @ A @ B @ A3 @ F2 )
=> ( ( topolo81223032696312382ous_on @ A @ B @ A3 @ G )
=> ( topolo81223032696312382ous_on @ A @ B @ A3
@ ^ [X: A] : ( ord_max @ B @ ( F2 @ X ) @ ( G @ X ) ) ) ) ) ) ).
% continuous_on_max
thf(fact_7884_continuous__on__add,axiom,
! [B: $tType,D6: $tType] :
( ( ( topolo4958980785337419405_space @ D6 )
& ( topolo6943815403480290642id_add @ B ) )
=> ! [S3: set @ D6,F2: D6 > B,G: D6 > B] :
( ( topolo81223032696312382ous_on @ D6 @ B @ S3 @ F2 )
=> ( ( topolo81223032696312382ous_on @ D6 @ B @ S3 @ G )
=> ( topolo81223032696312382ous_on @ D6 @ B @ S3
@ ^ [X: D6] : ( plus_plus @ B @ ( F2 @ X ) @ ( G @ X ) ) ) ) ) ) ).
% continuous_on_add
thf(fact_7885_continuous__on__divide,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo4958980785337419405_space @ A )
& ( real_V3459762299906320749_field @ B ) )
=> ! [S3: set @ A,F2: A > B,G: A > B] :
( ( topolo81223032696312382ous_on @ A @ B @ S3 @ F2 )
=> ( ( topolo81223032696312382ous_on @ A @ B @ S3 @ G )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ S3 )
=> ( ( G @ X3 )
!= ( zero_zero @ B ) ) )
=> ( topolo81223032696312382ous_on @ A @ B @ S3
@ ^ [X: A] : ( divide_divide @ B @ ( F2 @ X ) @ ( G @ X ) ) ) ) ) ) ) ).
% continuous_on_divide
thf(fact_7886_continuous__on__mult__right,axiom,
! [A: $tType,B: $tType] :
( ( ( topolo4958980785337419405_space @ B )
& ( real_V4412858255891104859lgebra @ A ) )
=> ! [S3: set @ B,F2: B > A,C2: A] :
( ( topolo81223032696312382ous_on @ B @ A @ S3 @ F2 )
=> ( topolo81223032696312382ous_on @ B @ A @ S3
@ ^ [X: B] : ( times_times @ A @ ( F2 @ X ) @ C2 ) ) ) ) ).
% continuous_on_mult_right
thf(fact_7887_continuous__on__mult__left,axiom,
! [A: $tType,B: $tType] :
( ( ( topolo4958980785337419405_space @ B )
& ( real_V4412858255891104859lgebra @ A ) )
=> ! [S3: set @ B,F2: B > A,C2: A] :
( ( topolo81223032696312382ous_on @ B @ A @ S3 @ F2 )
=> ( topolo81223032696312382ous_on @ B @ A @ S3
@ ^ [X: B] : ( times_times @ A @ C2 @ ( F2 @ X ) ) ) ) ) ).
% continuous_on_mult_left
thf(fact_7888_continuous__on__mult_H,axiom,
! [B: $tType,D6: $tType] :
( ( ( topolo4958980785337419405_space @ D6 )
& ( topolo4211221413907600880p_mult @ B ) )
=> ! [A3: set @ D6,F2: D6 > B,G: D6 > B] :
( ( topolo81223032696312382ous_on @ D6 @ B @ A3 @ F2 )
=> ( ( topolo81223032696312382ous_on @ D6 @ B @ A3 @ G )
=> ( topolo81223032696312382ous_on @ D6 @ B @ A3
@ ^ [X: D6] : ( times_times @ B @ ( F2 @ X ) @ ( G @ X ) ) ) ) ) ) ).
% continuous_on_mult'
thf(fact_7889_continuous__on__mult,axiom,
! [A: $tType,D6: $tType] :
( ( ( topolo4958980785337419405_space @ D6 )
& ( real_V4412858255891104859lgebra @ A ) )
=> ! [S3: set @ D6,F2: D6 > A,G: D6 > A] :
( ( topolo81223032696312382ous_on @ D6 @ A @ S3 @ F2 )
=> ( ( topolo81223032696312382ous_on @ D6 @ A @ S3 @ G )
=> ( topolo81223032696312382ous_on @ D6 @ A @ S3
@ ^ [X: D6] : ( times_times @ A @ ( F2 @ X ) @ ( G @ X ) ) ) ) ) ) ).
% continuous_on_mult
thf(fact_7890_continuous__on__Pair,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( topolo4958980785337419405_space @ A )
& ( topolo4958980785337419405_space @ B )
& ( topolo4958980785337419405_space @ C ) )
=> ! [S3: set @ A,F2: A > B,G: A > C] :
( ( topolo81223032696312382ous_on @ A @ B @ S3 @ F2 )
=> ( ( topolo81223032696312382ous_on @ A @ C @ S3 @ G )
=> ( topolo81223032696312382ous_on @ A @ ( product_prod @ B @ C ) @ S3
@ ^ [X: A] : ( product_Pair @ B @ C @ ( F2 @ X ) @ ( G @ X ) ) ) ) ) ) ).
% continuous_on_Pair
thf(fact_7891_continuous__on__min,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo4958980785337419405_space @ A )
& ( topolo1944317154257567458pology @ B ) )
=> ! [A3: set @ A,F2: A > B,G: A > B] :
( ( topolo81223032696312382ous_on @ A @ B @ A3 @ F2 )
=> ( ( topolo81223032696312382ous_on @ A @ B @ A3 @ G )
=> ( topolo81223032696312382ous_on @ A @ B @ A3
@ ^ [X: A] : ( ord_min @ B @ ( F2 @ X ) @ ( G @ X ) ) ) ) ) ) ).
% continuous_on_min
thf(fact_7892_continuous__on__prod_H,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( ( topolo4958980785337419405_space @ B )
& ( topolo4987421752381908075d_mult @ C ) )
=> ! [I5: set @ A,S: set @ B,F2: A > B > C] :
( ! [I2: A] :
( ( member @ A @ I2 @ I5 )
=> ( topolo81223032696312382ous_on @ B @ C @ S @ ( F2 @ I2 ) ) )
=> ( topolo81223032696312382ous_on @ B @ C @ S
@ ^ [X: B] :
( groups7121269368397514597t_prod @ A @ C
@ ^ [I3: A] : ( F2 @ I3 @ X )
@ I5 ) ) ) ) ).
% continuous_on_prod'
thf(fact_7893_continuous__on__prod,axiom,
! [A: $tType,C: $tType,D6: $tType] :
( ( ( topolo4958980785337419405_space @ D6 )
& ( real_V4412858255891104859lgebra @ C )
& ( comm_ring_1 @ C ) )
=> ! [S: set @ A,S3: set @ D6,F2: A > D6 > C] :
( ! [I2: A] :
( ( member @ A @ I2 @ S )
=> ( topolo81223032696312382ous_on @ D6 @ C @ S3 @ ( F2 @ I2 ) ) )
=> ( topolo81223032696312382ous_on @ D6 @ C @ S3
@ ^ [X: D6] :
( groups7121269368397514597t_prod @ A @ C
@ ^ [I3: A] : ( F2 @ I3 @ X )
@ S ) ) ) ) ).
% continuous_on_prod
thf(fact_7894_continuous__on__sum,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( ( topolo4958980785337419405_space @ B )
& ( topolo5987344860129210374id_add @ C ) )
=> ! [I5: set @ A,S: set @ B,F2: A > B > C] :
( ! [I2: A] :
( ( member @ A @ I2 @ I5 )
=> ( topolo81223032696312382ous_on @ B @ C @ S @ ( F2 @ I2 ) ) )
=> ( topolo81223032696312382ous_on @ B @ C @ S
@ ^ [X: B] :
( groups7311177749621191930dd_sum @ A @ C
@ ^ [I3: A] : ( F2 @ I3 @ X )
@ I5 ) ) ) ) ).
% continuous_on_sum
thf(fact_7895_continuous__on__mult__const,axiom,
! [A: $tType] :
( ( real_V4412858255891104859lgebra @ A )
=> ! [S3: set @ A,C2: A] : ( topolo81223032696312382ous_on @ A @ A @ S3 @ ( times_times @ A @ C2 ) ) ) ).
% continuous_on_mult_const
thf(fact_7896_continuous__on__dist,axiom,
! [A: $tType,D6: $tType] :
( ( ( topolo4958980785337419405_space @ D6 )
& ( real_V7819770556892013058_space @ A ) )
=> ! [S3: set @ D6,F2: D6 > A,G: D6 > A] :
( ( topolo81223032696312382ous_on @ D6 @ A @ S3 @ F2 )
=> ( ( topolo81223032696312382ous_on @ D6 @ A @ S3 @ G )
=> ( topolo81223032696312382ous_on @ D6 @ real @ S3
@ ^ [X: D6] : ( real_V557655796197034286t_dist @ A @ ( F2 @ X ) @ ( G @ X ) ) ) ) ) ) ).
% continuous_on_dist
thf(fact_7897_continuous__on__cot,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [S3: set @ A,F2: A > A] :
( ( topolo81223032696312382ous_on @ A @ A @ S3 @ F2 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ S3 )
=> ( ( sin @ A @ ( F2 @ X3 ) )
!= ( zero_zero @ A ) ) )
=> ( topolo81223032696312382ous_on @ A @ A @ S3
@ ^ [X: A] : ( cot @ A @ ( F2 @ X ) ) ) ) ) ) ).
% continuous_on_cot
thf(fact_7898_not__Ici__le__Icc,axiom,
! [A: $tType] :
( ( no_top @ A )
=> ! [L2: A,L3: A,H3: A] :
~ ( ord_less_eq @ ( set @ A ) @ ( set_ord_atLeast @ A @ L2 ) @ ( set_or1337092689740270186AtMost @ A @ L3 @ H3 ) ) ) ).
% not_Ici_le_Icc
thf(fact_7899_IVT2_H,axiom,
! [A: $tType,B: $tType] :
( ( ( topolo1944317154257567458pology @ B )
& ( topolo8458572112393995274pology @ A ) )
=> ! [F2: A > B,B2: A,Y2: B,A2: A] :
( ( ord_less_eq @ B @ ( F2 @ B2 ) @ Y2 )
=> ( ( ord_less_eq @ B @ Y2 @ ( F2 @ A2 ) )
=> ( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( topolo81223032696312382ous_on @ A @ B @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) @ F2 )
=> ? [X3: A] :
( ( ord_less_eq @ A @ A2 @ X3 )
& ( ord_less_eq @ A @ X3 @ B2 )
& ( ( F2 @ X3 )
= Y2 ) ) ) ) ) ) ) ).
% IVT2'
thf(fact_7900_IVT_H,axiom,
! [A: $tType,B: $tType] :
( ( ( topolo1944317154257567458pology @ B )
& ( topolo8458572112393995274pology @ A ) )
=> ! [F2: A > B,A2: A,Y2: B,B2: A] :
( ( ord_less_eq @ B @ ( F2 @ A2 ) @ Y2 )
=> ( ( ord_less_eq @ B @ Y2 @ ( F2 @ B2 ) )
=> ( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( topolo81223032696312382ous_on @ A @ B @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) @ F2 )
=> ? [X3: A] :
( ( ord_less_eq @ A @ A2 @ X3 )
& ( ord_less_eq @ A @ X3 @ B2 )
& ( ( F2 @ X3 )
= Y2 ) ) ) ) ) ) ) ).
% IVT'
thf(fact_7901_continuous__on__arcosh,axiom,
! [A3: set @ real] :
( ( ord_less_eq @ ( set @ real ) @ A3 @ ( set_ord_atLeast @ real @ ( one_one @ real ) ) )
=> ( topolo81223032696312382ous_on @ real @ real @ A3 @ ( arcosh @ real ) ) ) ).
% continuous_on_arcosh
thf(fact_7902_continuous__on__subset,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo4958980785337419405_space @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [S3: set @ A,F2: A > B,T2: set @ A] :
( ( topolo81223032696312382ous_on @ A @ B @ S3 @ F2 )
=> ( ( ord_less_eq @ ( set @ A ) @ T2 @ S3 )
=> ( topolo81223032696312382ous_on @ A @ B @ T2 @ F2 ) ) ) ) ).
% continuous_on_subset
thf(fact_7903_atLeast__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( set_ord_atLeast @ A )
= ( ^ [L: A] : ( collect @ A @ ( ord_less_eq @ A @ L ) ) ) ) ) ).
% atLeast_def
thf(fact_7904_not__Ici__le__Iic,axiom,
! [A: $tType] :
( ( no_top @ A )
=> ! [L2: A,H3: A] :
~ ( ord_less_eq @ ( set @ A ) @ ( set_ord_atLeast @ A @ L2 ) @ ( set_ord_atMost @ A @ H3 ) ) ) ).
% not_Ici_le_Iic
thf(fact_7905_not__Iic__le__Ici,axiom,
! [A: $tType] :
( ( no_bot @ A )
=> ! [H2: A,L3: A] :
~ ( ord_less_eq @ ( set @ A ) @ ( set_ord_atMost @ A @ H2 ) @ ( set_ord_atLeast @ A @ L3 ) ) ) ).
% not_Iic_le_Ici
thf(fact_7906_continuous__on__compose2,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( ( topolo4958980785337419405_space @ C )
& ( topolo4958980785337419405_space @ B )
& ( topolo4958980785337419405_space @ A ) )
=> ! [T2: set @ A,G: A > B,S3: set @ C,F2: C > A] :
( ( topolo81223032696312382ous_on @ A @ B @ T2 @ G )
=> ( ( topolo81223032696312382ous_on @ C @ A @ S3 @ F2 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image @ C @ A @ F2 @ S3 ) @ T2 )
=> ( topolo81223032696312382ous_on @ C @ B @ S3
@ ^ [X: C] : ( G @ ( F2 @ X ) ) ) ) ) ) ) ).
% continuous_on_compose2
thf(fact_7907_not__UNIV__le__Ici,axiom,
! [A: $tType] :
( ( no_bot @ A )
=> ! [L2: A] :
~ ( ord_less_eq @ ( set @ A ) @ ( top_top @ ( set @ A ) ) @ ( set_ord_atLeast @ A @ L2 ) ) ) ).
% not_UNIV_le_Ici
thf(fact_7908_continuous__onI__mono,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo1944317154257567458pology @ A )
& ( dense_order @ B )
& ( topolo1944317154257567458pology @ B ) )
=> ! [F2: A > B,A3: set @ A] :
( ( topolo1002775350975398744n_open @ B @ ( image @ A @ B @ F2 @ A3 ) )
=> ( ! [X3: A,Y3: A] :
( ( member @ A @ X3 @ A3 )
=> ( ( member @ A @ Y3 @ A3 )
=> ( ( ord_less_eq @ A @ X3 @ Y3 )
=> ( ord_less_eq @ B @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) ) ) )
=> ( topolo81223032696312382ous_on @ A @ B @ A3 @ F2 ) ) ) ) ).
% continuous_onI_mono
thf(fact_7909_Ioi__le__Ico,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A2: A] : ( ord_less_eq @ ( set @ A ) @ ( set_ord_greaterThan @ A @ A2 ) @ ( set_ord_atLeast @ A @ A2 ) ) ) ).
% Ioi_le_Ico
thf(fact_7910_Ici__subset__Ioi__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_ord_atLeast @ A @ A2 ) @ ( set_ord_greaterThan @ A @ B2 ) )
= ( ord_less @ A @ B2 @ A2 ) ) ) ).
% Ici_subset_Ioi_iff
thf(fact_7911_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
@ ^ [X: real] : ( arcosh @ real @ ( F2 @ X ) ) ) ) ) ).
% continuous_on_arcosh'
thf(fact_7912_continuous__image__closed__interval,axiom,
! [A2: real,B2: real,F2: real > real] :
( ( ord_less_eq @ real @ A2 @ B2 )
=> ( ( topolo81223032696312382ous_on @ real @ real @ ( set_or1337092689740270186AtMost @ real @ A2 @ B2 ) @ F2 )
=> ? [C4: real,D3: real] :
( ( ( image @ real @ real @ F2 @ ( set_or1337092689740270186AtMost @ real @ A2 @ B2 ) )
= ( set_or1337092689740270186AtMost @ real @ C4 @ D3 ) )
& ( ord_less_eq @ real @ C4 @ D3 ) ) ) ) ).
% continuous_image_closed_interval
thf(fact_7913_continuous__on__powr_H,axiom,
! [C: $tType] :
( ( topolo4958980785337419405_space @ C )
=> ! [S3: set @ C,F2: C > real,G: C > real] :
( ( topolo81223032696312382ous_on @ C @ real @ S3 @ F2 )
=> ( ( topolo81223032696312382ous_on @ C @ real @ S3 @ G )
=> ( ! [X3: C] :
( ( member @ C @ X3 @ S3 )
=> ( ( 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 @ S3
@ ^ [X: C] : ( powr @ real @ ( F2 @ X ) @ ( G @ X ) ) ) ) ) ) ) ).
% continuous_on_powr'
thf(fact_7914_continuous__on__log,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [S3: set @ A,F2: A > real,G: A > real] :
( ( topolo81223032696312382ous_on @ A @ real @ S3 @ F2 )
=> ( ( topolo81223032696312382ous_on @ A @ real @ S3 @ G )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ S3 )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( F2 @ X3 ) ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ S3 )
=> ( ( F2 @ X3 )
!= ( one_one @ real ) ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ S3 )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( G @ X3 ) ) )
=> ( topolo81223032696312382ous_on @ A @ real @ S3
@ ^ [X: A] : ( log @ ( F2 @ X ) @ ( G @ X ) ) ) ) ) ) ) ) ) ).
% continuous_on_log
thf(fact_7915_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_7916_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_7917_continuous__on__arccos,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [S3: set @ A,F2: A > real] :
( ( topolo81223032696312382ous_on @ A @ real @ S3 @ F2 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ S3 )
=> ( ( 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 @ S3
@ ^ [X: A] : ( arccos @ ( F2 @ X ) ) ) ) ) ) ).
% continuous_on_arccos
thf(fact_7918_continuous__on__arcsin,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [S3: set @ A,F2: A > real] :
( ( topolo81223032696312382ous_on @ A @ real @ S3 @ F2 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ S3 )
=> ( ( 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 @ S3
@ ^ [X: A] : ( arcsin @ ( F2 @ X ) ) ) ) ) ) ).
% continuous_on_arcsin
thf(fact_7919_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_7920_DERIV__atLeastAtMost__imp__continuous__on,axiom,
! [A: $tType] :
( ( ( ord @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [A2: A,B2: A,F2: A > A] :
( ! [X3: A] :
( ( ord_less_eq @ A @ A2 @ X3 )
=> ( ( ord_less_eq @ A @ X3 @ B2 )
=> ? [Y4: A] : ( has_field_derivative @ A @ F2 @ Y4 @ ( topolo174197925503356063within @ A @ X3 @ ( top_top @ ( set @ A ) ) ) ) ) )
=> ( topolo81223032696312382ous_on @ A @ A @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) @ F2 ) ) ) ).
% DERIV_atLeastAtMost_imp_continuous_on
thf(fact_7921_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
@ ^ [X: real] : ( artanh @ real @ ( F2 @ X ) ) ) ) ) ).
% continuous_on_artanh'
thf(fact_7922_atLeast__Suc,axiom,
! [K: nat] :
( ( set_ord_atLeast @ nat @ ( suc @ K ) )
= ( minus_minus @ ( set @ nat ) @ ( set_ord_atLeast @ nat @ K ) @ ( insert @ nat @ K @ ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% atLeast_Suc
thf(fact_7923_Rolle__deriv,axiom,
! [A2: real,B2: real,F2: real > real,F7: 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 @ ( F7 @ X3 ) @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) ) ) )
=> ? [Z2: real] :
( ( ord_less @ real @ A2 @ Z2 )
& ( ord_less @ real @ Z2 @ B2 )
& ( ( F7 @ Z2 )
= ( ^ [V5: real] : ( zero_zero @ real ) ) ) ) ) ) ) ) ).
% Rolle_deriv
thf(fact_7924_continuous__on__of__int__floor,axiom,
! [B: $tType,A: $tType] :
( ( ( archim2362893244070406136eiling @ A )
& ( topolo2564578578187576103pology @ A )
& ( ring_1 @ B )
& ( topolo4958980785337419405_space @ B ) )
=> ( topolo81223032696312382ous_on @ A @ B @ ( minus_minus @ ( set @ A ) @ ( top_top @ ( set @ A ) ) @ ( ring_1_Ints @ A ) )
@ ^ [X: A] : ( ring_1_of_int @ B @ ( archim6421214686448440834_floor @ A @ X ) ) ) ) ).
% continuous_on_of_int_floor
thf(fact_7925_continuous__on__of__int__ceiling,axiom,
! [B: $tType,A: $tType] :
( ( ( archim2362893244070406136eiling @ A )
& ( topolo2564578578187576103pology @ A )
& ( ring_1 @ B )
& ( topolo4958980785337419405_space @ B ) )
=> ( topolo81223032696312382ous_on @ A @ B @ ( minus_minus @ ( set @ A ) @ ( top_top @ ( set @ A ) ) @ ( ring_1_Ints @ A ) )
@ ^ [X: A] : ( ring_1_of_int @ B @ ( archimedean_ceiling @ A @ X ) ) ) ) ).
% continuous_on_of_int_ceiling
thf(fact_7926_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_7927_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_7928_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_7929_complete__interval,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [A2: A,B2: A,P: A > $o] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( P @ A2 )
=> ( ~ ( P @ B2 )
=> ? [C4: A] :
( ( ord_less_eq @ A @ A2 @ C4 )
& ( ord_less_eq @ A @ C4 @ B2 )
& ! [X4: A] :
( ( ( ord_less_eq @ A @ A2 @ X4 )
& ( ord_less @ A @ X4 @ C4 ) )
=> ( P @ X4 ) )
& ! [D5: A] :
( ! [X3: A] :
( ( ( ord_less_eq @ A @ A2 @ X3 )
& ( ord_less @ A @ X3 @ D5 ) )
=> ( P @ X3 ) )
=> ( ord_less_eq @ A @ D5 @ C4 ) ) ) ) ) ) ) ).
% complete_interval
thf(fact_7930_DERIV__isconst2,axiom,
! [A2: real,B2: real,F2: real > real,X2: 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 @ X2 )
=> ( ( ord_less_eq @ real @ X2 @ B2 )
=> ( ( F2 @ X2 )
= ( F2 @ A2 ) ) ) ) ) ) ) ).
% DERIV_isconst2
thf(fact_7931_Rolle,axiom,
! [A2: real,B2: real,F2: real > real] :
( ( ord_less @ real @ A2 @ B2 )
=> ( ( ( F2 @ A2 )
= ( F2 @ B2 ) )
=> ( ( topolo81223032696312382ous_on @ real @ real @ ( set_or1337092689740270186AtMost @ real @ A2 @ B2 ) @ F2 )
=> ( ! [X3: real] :
( ( ord_less @ real @ A2 @ X3 )
=> ( ( ord_less @ real @ X3 @ B2 )
=> ( differentiable @ real @ real @ F2 @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) ) ) )
=> ? [Z2: real] :
( ( ord_less @ real @ A2 @ Z2 )
& ( ord_less @ real @ Z2 @ B2 )
& ( has_field_derivative @ real @ F2 @ ( zero_zero @ real ) @ ( topolo174197925503356063within @ real @ Z2 @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ) ) ).
% Rolle
thf(fact_7932_set__bits__int__def,axiom,
( ( bit_bi4170147762399595738t_bits @ int )
= ( ^ [F4: nat > $o] :
( if @ int
@ ? [N3: nat] :
! [M5: nat] :
( ( ord_less_eq @ nat @ N3 @ M5 )
=> ( ( F4 @ M5 )
= ( F4 @ N3 ) ) )
@ ( bit_ri4674362597316999326ke_bit @ int
@ ( ord_Least @ nat
@ ^ [N3: nat] :
! [M5: nat] :
( ( ord_less_eq @ nat @ N3 @ M5 )
=> ( ( F4 @ M5 )
= ( 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] :
! [M5: nat] :
( ( ord_less_eq @ nat @ N3 @ M5 )
=> ( ( F4 @ M5 )
= ( F4 @ N3 ) ) ) ) ) ) ) ) )
@ ( zero_zero @ int ) ) ) ) ).
% set_bits_int_def
thf(fact_7933_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_7934_finite__Collect__bounded__ex,axiom,
! [B: $tType,A: $tType,P: A > $o,Q: B > A > $o] :
( ( finite_finite2 @ A @ ( collect @ A @ P ) )
=> ( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X: B] :
? [Y: A] :
( ( P @ Y )
& ( Q @ X @ Y ) ) ) )
= ( ! [Y: A] :
( ( P @ Y )
=> ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X: B] : ( Q @ X @ Y ) ) ) ) ) ) ) ).
% finite_Collect_bounded_ex
thf(fact_7935_eq__or__mem__image__simp,axiom,
! [B: $tType,A: $tType,F2: B > A,A2: B,B5: set @ B] :
( ( collect @ A
@ ^ [Uu3: A] :
? [L: B] :
( ( Uu3
= ( F2 @ L ) )
& ( ( L = A2 )
| ( member @ B @ L @ B5 ) ) ) )
= ( insert @ A @ ( F2 @ A2 )
@ ( collect @ A
@ ^ [Uu3: A] :
? [L: B] :
( ( Uu3
= ( F2 @ L ) )
& ( member @ B @ L @ B5 ) ) ) ) ) ).
% eq_or_mem_image_simp
thf(fact_7936_Least__eq__0,axiom,
! [P: nat > $o] :
( ( P @ ( zero_zero @ nat ) )
=> ( ( ord_Least @ nat @ P )
= ( zero_zero @ nat ) ) ) ).
% Least_eq_0
thf(fact_7937_continuous__on__of__real__o__iff,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [S: set @ A,G: A > real] :
( ( topolo81223032696312382ous_on @ A @ complex @ S
@ ^ [X: A] : ( real_Vector_of_real @ complex @ ( G @ X ) ) )
= ( topolo81223032696312382ous_on @ A @ real @ S @ G ) ) ) ).
% continuous_on_of_real_o_iff
thf(fact_7938_continuous__on__cis,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [A3: set @ A,F2: A > real] :
( ( topolo81223032696312382ous_on @ A @ real @ A3 @ F2 )
=> ( topolo81223032696312382ous_on @ A @ complex @ A3
@ ^ [X: A] : ( cis @ ( F2 @ X ) ) ) ) ) ).
% continuous_on_cis
thf(fact_7939_eventually__ex,axiom,
! [B: $tType,A: $tType,P: A > B > $o,F3: filter @ A] :
( ( eventually @ A
@ ^ [X: A] :
? [X7: B] : ( P @ X @ X7 )
@ F3 )
= ( ? [Y9: A > B] :
( eventually @ A
@ ^ [X: A] : ( P @ X @ ( Y9 @ X ) )
@ F3 ) ) ) ).
% eventually_ex
thf(fact_7940_LeastI2__wellorder__ex,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [P: A > $o,Q: A > $o] :
( ? [X_12: A] : ( P @ X_12 )
=> ( ! [A5: A] :
( ( P @ A5 )
=> ( ! [B11: A] :
( ( P @ B11 )
=> ( ord_less_eq @ A @ A5 @ B11 ) )
=> ( Q @ A5 ) ) )
=> ( Q @ ( ord_Least @ A @ P ) ) ) ) ) ).
% LeastI2_wellorder_ex
thf(fact_7941_LeastI2__wellorder,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [P: A > $o,A2: A,Q: A > $o] :
( ( P @ A2 )
=> ( ! [A5: A] :
( ( P @ A5 )
=> ( ! [B11: A] :
( ( P @ B11 )
=> ( ord_less_eq @ A @ A5 @ B11 ) )
=> ( Q @ A5 ) ) )
=> ( Q @ ( ord_Least @ A @ P ) ) ) ) ) ).
% LeastI2_wellorder
thf(fact_7942_Least__equality,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [P: A > $o,X2: A] :
( ( P @ X2 )
=> ( ! [Y3: A] :
( ( P @ Y3 )
=> ( ord_less_eq @ A @ X2 @ Y3 ) )
=> ( ( ord_Least @ A @ P )
= X2 ) ) ) ) ).
% Least_equality
thf(fact_7943_LeastI2__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [P: A > $o,X2: A,Q: A > $o] :
( ( P @ X2 )
=> ( ! [Y3: A] :
( ( P @ Y3 )
=> ( ord_less_eq @ A @ X2 @ Y3 ) )
=> ( ! [X3: A] :
( ( P @ X3 )
=> ( ! [Y4: A] :
( ( P @ Y4 )
=> ( ord_less_eq @ A @ X3 @ Y4 ) )
=> ( Q @ X3 ) ) )
=> ( Q @ ( ord_Least @ A @ P ) ) ) ) ) ) ).
% LeastI2_order
thf(fact_7944_Least1__le,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [P: A > $o,Z: A] :
( ? [X4: A] :
( ( P @ X4 )
& ! [Y3: A] :
( ( P @ Y3 )
=> ( ord_less_eq @ A @ X4 @ Y3 ) )
& ! [Y3: A] :
( ( ( P @ Y3 )
& ! [Ya2: A] :
( ( P @ Ya2 )
=> ( ord_less_eq @ A @ Y3 @ Ya2 ) ) )
=> ( Y3 = X4 ) ) )
=> ( ( P @ Z )
=> ( ord_less_eq @ A @ ( ord_Least @ A @ P ) @ Z ) ) ) ) ).
% Least1_le
thf(fact_7945_Least1I,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [P: A > $o] :
( ? [X4: A] :
( ( P @ X4 )
& ! [Y3: A] :
( ( P @ Y3 )
=> ( ord_less_eq @ A @ X4 @ Y3 ) )
& ! [Y3: A] :
( ( ( P @ Y3 )
& ! [Ya2: A] :
( ( P @ Ya2 )
=> ( ord_less_eq @ A @ Y3 @ Ya2 ) ) )
=> ( Y3 = X4 ) ) )
=> ( P @ ( ord_Least @ A @ P ) ) ) ) ).
% Least1I
thf(fact_7946_Least__le,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [P: A > $o,K: A] :
( ( P @ K )
=> ( ord_less_eq @ A @ ( ord_Least @ A @ P ) @ K ) ) ) ).
% Least_le
thf(fact_7947_Least__Suc2,axiom,
! [P: nat > $o,N: nat,Q: nat > $o,M: nat] :
( ( P @ N )
=> ( ( Q @ M )
=> ( ~ ( P @ ( zero_zero @ nat ) )
=> ( ! [K3: nat] :
( ( P @ ( suc @ K3 ) )
= ( Q @ K3 ) )
=> ( ( ord_Least @ nat @ P )
= ( suc @ ( ord_Least @ nat @ Q ) ) ) ) ) ) ) ).
% Least_Suc2
thf(fact_7948_full__SetCompr__eq,axiom,
! [A: $tType,B: $tType,F2: B > A] :
( ( collect @ A
@ ^ [U2: A] :
? [X: B] :
( U2
= ( F2 @ X ) ) )
= ( image @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) ) ) ).
% full_SetCompr_eq
thf(fact_7949_Setcompr__eq__image,axiom,
! [A: $tType,B: $tType,F2: B > A,A3: set @ B] :
( ( collect @ A
@ ^ [Uu3: A] :
? [X: B] :
( ( Uu3
= ( F2 @ X ) )
& ( member @ B @ X @ A3 ) ) )
= ( image @ B @ A @ F2 @ A3 ) ) ).
% Setcompr_eq_image
thf(fact_7950_setcompr__eq__image,axiom,
! [A: $tType,B: $tType,F2: B > A,P: B > $o] :
( ( collect @ A
@ ^ [Uu3: A] :
? [X: B] :
( ( Uu3
= ( F2 @ X ) )
& ( P @ X ) ) )
= ( image @ B @ A @ F2 @ ( collect @ B @ P ) ) ) ).
% setcompr_eq_image
thf(fact_7951_some__theI,axiom,
! [A: $tType,B: $tType,P: A > B > $o] :
( ? [A9: A,X_12: B] : ( P @ A9 @ X_12 )
=> ( ! [B1: B,B22: B] :
( ? [A5: A] : ( P @ A5 @ B1 )
=> ( ? [A5: A] : ( P @ A5 @ B22 )
=> ( B1 = B22 ) ) )
=> ( P
@ ( fChoice @ A
@ ^ [A4: A] :
? [X7: B] : ( P @ A4 @ X7 ) )
@ ( the @ B
@ ^ [B3: B] :
? [A4: A] : ( P @ A4 @ B3 ) ) ) ) ) ).
% some_theI
thf(fact_7952_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_7953_LeastI,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [P: A > $o,K: A] :
( ( P @ K )
=> ( P @ ( ord_Least @ A @ P ) ) ) ) ).
% LeastI
thf(fact_7954_ran__def,axiom,
! [B: $tType,A: $tType] :
( ( ran @ A @ B )
= ( ^ [M5: A > ( option @ B )] :
( collect @ B
@ ^ [B3: B] :
? [A4: A] :
( ( M5 @ A4 )
= ( some @ B @ B3 ) ) ) ) ) ).
% ran_def
thf(fact_7955_Least__Suc,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ ( zero_zero @ nat ) )
=> ( ( ord_Least @ nat @ P )
= ( suc
@ ( ord_Least @ nat
@ ^ [M5: nat] : ( P @ ( suc @ M5 ) ) ) ) ) ) ) ).
% Least_Suc
thf(fact_7956_open__Collect__ex,axiom,
! [B: $tType,A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [P: B > A > $o] :
( ! [I2: B] : ( topolo1002775350975398744n_open @ A @ ( collect @ A @ ( P @ I2 ) ) )
=> ( topolo1002775350975398744n_open @ A
@ ( collect @ A
@ ^ [X: A] :
? [I3: B] : ( P @ I3 @ X ) ) ) ) ) ).
% open_Collect_ex
thf(fact_7957_open__diagonal__complement,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ( topolo1002775350975398744n_open @ ( product_prod @ A @ A )
@ ( collect @ ( product_prod @ A @ A )
@ ^ [Uu3: product_prod @ A @ A] :
? [X: A,Y: A] :
( ( Uu3
= ( product_Pair @ A @ A @ X @ Y ) )
& ( X != Y ) ) ) ) ) ).
% open_diagonal_complement
thf(fact_7958_open__subdiagonal,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ( topolo1002775350975398744n_open @ ( product_prod @ A @ A )
@ ( collect @ ( product_prod @ A @ A )
@ ^ [Uu3: product_prod @ A @ A] :
? [X: A,Y: A] :
( ( Uu3
= ( product_Pair @ A @ A @ X @ Y ) )
& ( ord_less @ A @ X @ Y ) ) ) ) ) ).
% open_subdiagonal
thf(fact_7959_open__superdiagonal,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ( topolo1002775350975398744n_open @ ( product_prod @ A @ A )
@ ( collect @ ( product_prod @ A @ A )
@ ^ [Uu3: product_prod @ A @ A] :
? [X: A,Y: A] :
( ( Uu3
= ( product_Pair @ A @ A @ X @ Y ) )
& ( ord_less @ A @ Y @ X ) ) ) ) ) ).
% open_superdiagonal
thf(fact_7960_finite__image__set2,axiom,
! [A: $tType,B: $tType,C: $tType,P: A > $o,Q: B > $o,F2: A > B > C] :
( ( finite_finite2 @ A @ ( collect @ A @ P ) )
=> ( ( finite_finite2 @ B @ ( collect @ B @ Q ) )
=> ( finite_finite2 @ C
@ ( collect @ C
@ ^ [Uu3: C] :
? [X: A,Y: B] :
( ( Uu3
= ( F2 @ X @ Y ) )
& ( P @ X )
& ( Q @ Y ) ) ) ) ) ) ).
% finite_image_set2
thf(fact_7961_finite__image__set,axiom,
! [A: $tType,B: $tType,P: A > $o,F2: A > B] :
( ( finite_finite2 @ A @ ( collect @ A @ P ) )
=> ( finite_finite2 @ B
@ ( collect @ B
@ ^ [Uu3: B] :
? [X: A] :
( ( Uu3
= ( F2 @ X ) )
& ( P @ X ) ) ) ) ) ).
% finite_image_set
thf(fact_7962_set__conv__nth,axiom,
! [A: $tType] :
( ( set2 @ A )
= ( ^ [Xs: list @ A] :
( collect @ A
@ ^ [Uu3: A] :
? [I3: nat] :
( ( Uu3
= ( nth @ A @ Xs @ I3 ) )
& ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ) ) ).
% set_conv_nth
thf(fact_7963_Least__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( ord_Least @ A )
= ( ^ [P7: A > $o] :
( the @ A
@ ^ [X: A] :
( ( P7 @ X )
& ! [Y: A] :
( ( P7 @ Y )
=> ( ord_less_eq @ A @ X @ Y ) ) ) ) ) ) ) ).
% Least_def
thf(fact_7964_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_7965_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_7966_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_7967_pairself__image__eq,axiom,
! [B: $tType,A: $tType,F2: B > A,P: B > B > $o] :
( ( image @ ( product_prod @ B @ B ) @ ( product_prod @ A @ A ) @ ( pairself @ B @ A @ F2 ) @ ( collect @ ( product_prod @ B @ B ) @ ( product_case_prod @ B @ B @ $o @ P ) ) )
= ( collect @ ( product_prod @ A @ A )
@ ^ [Uu3: product_prod @ A @ A] :
? [A4: B,B3: B] :
( ( Uu3
= ( product_Pair @ A @ A @ ( F2 @ A4 ) @ ( F2 @ B3 ) ) )
& ( P @ A4 @ B3 ) ) ) ) ).
% pairself_image_eq
thf(fact_7968_VEBT__internal_Ovalid_H_Oelims_I3_J,axiom,
! [X2: vEBT_VEBT,Xa: nat] :
( ~ ( vEBT_VEBT_valid @ X2 @ Xa )
=> ( ( ? [Uu2: $o,Uv2: $o] :
( X2
= ( vEBT_Leaf @ Uu2 @ Uv2 ) )
=> ( Xa
= ( one_one @ nat ) ) )
=> ~ ! [Mima: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X2
= ( 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 )
@ ( ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X7 )
& ! [X: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X @ X7 ) ) )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [Mi3: nat,Ma3: nat] :
( ( ord_less_eq @ nat @ Mi3 @ Ma3 )
& ( ord_less @ nat @ Ma3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
& ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
=> ( ( ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I3 ) @ X7 ) )
= ( vEBT_V8194947554948674370ptions @ Summary2 @ I3 ) ) )
& ( ( Mi3 = Ma3 )
=> ! [X: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X @ X7 ) ) )
& ( ( Mi3 != Ma3 )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList2 @ Ma3 )
& ! [X: nat] :
( ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList2 @ X )
=> ( ( ord_less @ nat @ Mi3 @ X )
& ( ord_less_eq @ nat @ X @ Ma3 ) ) ) ) ) ) ) )
@ Mima ) ) ) ) ) ).
% VEBT_internal.valid'.elims(3)
thf(fact_7969_less__eq__option__def,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( ( ord_less_eq @ ( option @ A ) )
= ( ^ [X: option @ A,Y: option @ A] :
( case_option @ $o @ A @ $true
@ ^ [Z5: A] : ( case_option @ $o @ A @ $false @ ( ord_less_eq @ A @ Z5 ) @ Y )
@ X ) ) ) ) ).
% less_eq_option_def
thf(fact_7970_eventually__ball__finite,axiom,
! [A: $tType,B: $tType,A3: set @ A,P: B > A > $o,Net: filter @ B] :
( ( finite_finite2 @ A @ A3 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ A3 )
=> ( eventually @ B
@ ^ [Y: B] : ( P @ Y @ X3 )
@ Net ) )
=> ( eventually @ B
@ ^ [X: B] :
! [Y: A] :
( ( member @ A @ Y @ A3 )
=> ( P @ X @ Y ) )
@ Net ) ) ) ).
% eventually_ball_finite
thf(fact_7971_eventually__ball__finite__distrib,axiom,
! [B: $tType,A: $tType,A3: set @ A,P: B > A > $o,Net: filter @ B] :
( ( finite_finite2 @ A @ A3 )
=> ( ( eventually @ B
@ ^ [X: B] :
! [Y: A] :
( ( member @ A @ Y @ A3 )
=> ( P @ X @ Y ) )
@ Net )
= ( ! [X: A] :
( ( member @ A @ X @ A3 )
=> ( eventually @ B
@ ^ [Y: B] : ( P @ Y @ X )
@ Net ) ) ) ) ) ).
% eventually_ball_finite_distrib
thf(fact_7972_Ball__def__raw,axiom,
! [A: $tType] :
( ( ball @ A )
= ( ^ [A7: set @ A,P7: A > $o] :
! [X: A] :
( ( member @ A @ X @ A7 )
=> ( P7 @ X ) ) ) ) ).
% Ball_def_raw
thf(fact_7973_case__optionE,axiom,
! [A: $tType,P: $o,Q: A > $o,X2: option @ A] :
( ( case_option @ $o @ A @ P @ Q @ X2 )
=> ( ( ( X2
= ( none @ A ) )
=> ~ P )
=> ~ ! [Y3: A] :
( ( X2
= ( some @ A @ Y3 ) )
=> ~ ( Q @ Y3 ) ) ) ) ).
% case_optionE
thf(fact_7974_option_Ocase__distrib,axiom,
! [C: $tType,B: $tType,A: $tType,H2: B > C,F14: B,F25: A > B,Option: option @ A] :
( ( H2 @ ( case_option @ B @ A @ F14 @ F25 @ Option ) )
= ( case_option @ C @ A @ ( H2 @ F14 )
@ ^ [X: A] : ( H2 @ ( F25 @ X ) )
@ Option ) ) ).
% option.case_distrib
thf(fact_7975_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_7976_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_7977_option_Osimps_I4_J,axiom,
! [A: $tType,B: $tType,F14: B,F25: A > B] :
( ( case_option @ B @ A @ F14 @ F25 @ ( none @ A ) )
= F14 ) ).
% option.simps(4)
thf(fact_7978_option_Ocase__eq__if,axiom,
! [A: $tType,B: $tType] :
( ( case_option @ B @ A )
= ( ^ [F15: B,F26: A > B,Option3: option @ A] :
( if @ B
@ ( Option3
= ( none @ A ) )
@ F15
@ ( F26 @ ( the2 @ A @ Option3 ) ) ) ) ) ).
% option.case_eq_if
thf(fact_7979_Ball__comp__iff,axiom,
! [C: $tType,B: $tType,A: $tType,A3: B > ( set @ C ),F2: C > $o,G: A > B] :
( ( comp @ B @ $o @ A
@ ^ [X: B] :
! [Y: C] :
( ( member @ C @ Y @ ( A3 @ X ) )
=> ( F2 @ Y ) )
@ G )
= ( ^ [X: A] :
! [Y: C] :
( ( member @ C @ Y @ ( comp @ B @ ( set @ C ) @ A @ A3 @ G @ X ) )
=> ( F2 @ Y ) ) ) ) ).
% Ball_comp_iff
thf(fact_7980_less__option__def,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( ( ord_less @ ( option @ A ) )
= ( ^ [X: option @ A] :
( case_option @ $o @ A @ $false
@ ^ [Y: A] :
( case_option @ $o @ A @ $true
@ ^ [Z5: A] : ( ord_less @ A @ Z5 @ Y )
@ X ) ) ) ) ) ).
% less_option_def
thf(fact_7981_option_Othe__def,axiom,
! [A: $tType] :
( ( the2 @ A )
= ( case_option @ A @ A @ ( undefined @ A )
@ ^ [X24: A] : X24 ) ) ).
% option.the_def
thf(fact_7982_option_Osplit__sel__asm,axiom,
! [B: $tType,A: $tType,P: B > $o,F14: B,F25: A > B,Option: option @ A] :
( ( P @ ( case_option @ B @ A @ F14 @ F25 @ Option ) )
= ( ~ ( ( ( Option
= ( none @ A ) )
& ~ ( P @ F14 ) )
| ( ( Option
= ( some @ A @ ( the2 @ A @ Option ) ) )
& ~ ( P @ ( F25 @ ( the2 @ A @ Option ) ) ) ) ) ) ) ).
% option.split_sel_asm
thf(fact_7983_option_Osplit__sel,axiom,
! [B: $tType,A: $tType,P: B > $o,F14: B,F25: A > B,Option: option @ A] :
( ( P @ ( case_option @ B @ A @ F14 @ F25 @ Option ) )
= ( ( ( Option
= ( none @ A ) )
=> ( P @ F14 ) )
& ( ( Option
= ( some @ A @ ( the2 @ A @ Option ) ) )
=> ( P @ ( F25 @ ( the2 @ A @ Option ) ) ) ) ) ) ).
% option.split_sel
thf(fact_7984_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 )
& ! [X: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ( vEBT_VEBT_valid @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
& ( vEBT_VEBT_valid @ Summary @ ( minus_minus @ nat @ Deg @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
& ( ( size_size @ ( list @ vEBT_VEBT ) @ 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 )
@ ( ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X7 )
& ! [X: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X @ X7 ) ) )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [Mi3: nat,Ma3: nat] :
( ( ord_less_eq @ nat @ Mi3 @ Ma3 )
& ( ord_less @ nat @ Ma3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg ) )
& ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
=> ( ( ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList @ I3 ) @ X7 ) )
= ( vEBT_V8194947554948674370ptions @ Summary @ I3 ) ) )
& ( ( Mi3 = Ma3 )
=> ! [X: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X @ X7 ) ) )
& ( ( Mi3 != Ma3 )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList @ Ma3 )
& ! [X: nat] :
( ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg ) )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList @ X )
=> ( ( ord_less @ nat @ Mi3 @ X )
& ( ord_less_eq @ nat @ X @ Ma3 ) ) ) ) ) ) ) )
@ Mima2 ) ) ) ).
% VEBT_internal.valid'.simps(2)
thf(fact_7985_VEBT__internal_Ovalid_H_Oelims_I1_J,axiom,
! [X2: vEBT_VEBT,Xa: nat,Y2: $o] :
( ( ( vEBT_VEBT_valid @ X2 @ Xa )
= Y2 )
=> ( ( ? [Uu2: $o,Uv2: $o] :
( X2
= ( vEBT_Leaf @ Uu2 @ Uv2 ) )
=> ( Y2
= ( Xa
!= ( one_one @ nat ) ) ) )
=> ~ ! [Mima: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ Mima @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( Y2
= ( ~ ( ( Deg2 = Xa )
& ! [X: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ( vEBT_VEBT_valid @ X @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
& ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
& ( ( size_size @ ( list @ vEBT_VEBT ) @ 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 )
@ ( ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X7 )
& ! [X: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X @ X7 ) ) )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [Mi3: nat,Ma3: nat] :
( ( ord_less_eq @ nat @ Mi3 @ Ma3 )
& ( ord_less @ nat @ Ma3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
& ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
=> ( ( ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I3 ) @ X7 ) )
= ( vEBT_V8194947554948674370ptions @ Summary2 @ I3 ) ) )
& ( ( Mi3 = Ma3 )
=> ! [X: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X @ X7 ) ) )
& ( ( Mi3 != Ma3 )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList2 @ Ma3 )
& ! [X: nat] :
( ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList2 @ X )
=> ( ( ord_less @ nat @ Mi3 @ X )
& ( ord_less_eq @ nat @ X @ Ma3 ) ) ) ) ) ) ) )
@ Mima ) ) ) ) ) ) ) ).
% VEBT_internal.valid'.elims(1)
thf(fact_7986_VEBT__internal_Ovalid_H_Oelims_I2_J,axiom,
! [X2: vEBT_VEBT,Xa: nat] :
( ( vEBT_VEBT_valid @ X2 @ Xa )
=> ( ( ? [Uu2: $o,Uv2: $o] :
( X2
= ( vEBT_Leaf @ Uu2 @ Uv2 ) )
=> ( Xa
!= ( one_one @ nat ) ) )
=> ~ ! [Mima: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X2
= ( vEBT_Node @ Mima @ Deg2 @ TreeList2 @ Summary2 ) )
=> ~ ( ( Deg2 = Xa )
& ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ( vEBT_VEBT_valid @ X4 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
& ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
& ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( case_option @ $o @ ( product_prod @ nat @ nat )
@ ( ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X7 )
& ! [X: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X @ X7 ) ) )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [Mi3: nat,Ma3: nat] :
( ( ord_less_eq @ nat @ Mi3 @ Ma3 )
& ( ord_less @ nat @ Ma3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
& ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
=> ( ( ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I3 ) @ X7 ) )
= ( vEBT_V8194947554948674370ptions @ Summary2 @ I3 ) ) )
& ( ( Mi3 = Ma3 )
=> ! [X: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X @ X7 ) ) )
& ( ( Mi3 != Ma3 )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList2 @ Ma3 )
& ! [X: nat] :
( ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList2 @ X )
=> ( ( ord_less @ nat @ Mi3 @ X )
& ( ord_less_eq @ nat @ X @ Ma3 ) ) ) ) ) ) ) )
@ Mima ) ) ) ) ) ).
% VEBT_internal.valid'.elims(2)
thf(fact_7987_VEBT__internal_Ovalid_H_Opelims_I3_J,axiom,
! [X2: vEBT_VEBT,Xa: nat] :
( ~ ( vEBT_VEBT_valid @ X2 @ Xa )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_VEBT_valid_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X2 @ Xa ) )
=> ( ! [Uu2: $o,Uv2: $o] :
( ( X2
= ( 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] :
( ( X2
= ( 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 )
@ ( ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X7 )
& ! [X: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X @ X7 ) ) )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [Mi3: nat,Ma3: nat] :
( ( ord_less_eq @ nat @ Mi3 @ Ma3 )
& ( ord_less @ nat @ Ma3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
& ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
=> ( ( ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I3 ) @ X7 ) )
= ( vEBT_V8194947554948674370ptions @ Summary2 @ I3 ) ) )
& ( ( Mi3 = Ma3 )
=> ! [X: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X @ X7 ) ) )
& ( ( Mi3 != Ma3 )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList2 @ Ma3 )
& ! [X: nat] :
( ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList2 @ X )
=> ( ( ord_less @ nat @ Mi3 @ X )
& ( ord_less_eq @ nat @ X @ Ma3 ) ) ) ) ) ) ) )
@ Mima ) ) ) ) ) ) ) ).
% VEBT_internal.valid'.pelims(3)
thf(fact_7988_VEBT__internal_Ovalid_H_Opelims_I2_J,axiom,
! [X2: vEBT_VEBT,Xa: nat] :
( ( vEBT_VEBT_valid @ X2 @ Xa )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_VEBT_valid_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X2 @ Xa ) )
=> ( ! [Uu2: $o,Uv2: $o] :
( ( X2
= ( 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] :
( ( X2
= ( 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 )
& ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ( vEBT_VEBT_valid @ X4 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
& ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
& ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( case_option @ $o @ ( product_prod @ nat @ nat )
@ ( ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X7 )
& ! [X: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X @ X7 ) ) )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [Mi3: nat,Ma3: nat] :
( ( ord_less_eq @ nat @ Mi3 @ Ma3 )
& ( ord_less @ nat @ Ma3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
& ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
=> ( ( ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I3 ) @ X7 ) )
= ( vEBT_V8194947554948674370ptions @ Summary2 @ I3 ) ) )
& ( ( Mi3 = Ma3 )
=> ! [X: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X @ X7 ) ) )
& ( ( Mi3 != Ma3 )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList2 @ Ma3 )
& ! [X: nat] :
( ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList2 @ X )
=> ( ( ord_less @ nat @ Mi3 @ X )
& ( ord_less_eq @ nat @ X @ Ma3 ) ) ) ) ) ) ) )
@ Mima ) ) ) ) ) ) ) ).
% VEBT_internal.valid'.pelims(2)
thf(fact_7989_VEBT__internal_Ovalid_H_Opelims_I1_J,axiom,
! [X2: vEBT_VEBT,Xa: nat,Y2: $o] :
( ( ( vEBT_VEBT_valid @ X2 @ Xa )
= Y2 )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_VEBT_valid_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X2 @ Xa ) )
=> ( ! [Uu2: $o,Uv2: $o] :
( ( X2
= ( vEBT_Leaf @ Uu2 @ Uv2 ) )
=> ( ( Y2
= ( 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] :
( ( X2
= ( vEBT_Node @ Mima @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( ( Y2
= ( ( Deg2 = Xa )
& ! [X: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ( vEBT_VEBT_valid @ X @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
& ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
& ( ( size_size @ ( list @ vEBT_VEBT ) @ 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 )
@ ( ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X7 )
& ! [X: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X @ X7 ) ) )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [Mi3: nat,Ma3: nat] :
( ( ord_less_eq @ nat @ Mi3 @ Ma3 )
& ( ord_less @ nat @ Ma3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
& ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
=> ( ( ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I3 ) @ X7 ) )
= ( vEBT_V8194947554948674370ptions @ Summary2 @ I3 ) ) )
& ( ( Mi3 = Ma3 )
=> ! [X: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X @ X7 ) ) )
& ( ( Mi3 != Ma3 )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList2 @ Ma3 )
& ! [X: nat] :
( ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList2 @ X )
=> ( ( ord_less @ nat @ Mi3 @ X )
& ( ord_less_eq @ nat @ X @ Ma3 ) ) ) ) ) ) ) )
@ Mima ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_VEBT_valid_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList2 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ).
% VEBT_internal.valid'.pelims(1)
thf(fact_7990_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 ) )
@ ^ [Q4: 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 ) @ Q4 ) ) )
@ ( bit_take_bit_num @ ( numeral_numeral @ nat @ M ) @ N ) ) ) ) ).
% take_bit_numeral_minus_numeral_word
thf(fact_7991_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 )
@ ^ [Q4: 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 @ Q4 ) ) )
@ ( bit_take_bit_num @ ( numeral_numeral @ nat @ M ) @ N ) ) ) ).
% take_bit_numeral_minus_numeral_int
thf(fact_7992_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_7993_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_7994_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_7995_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 )
@ ^ [Q4: num] : ( some @ num @ ( bit0 @ Q4 ) )
@ ( bit_take_bit_num @ N @ M ) ) ) ).
% take_bit_num_simps(3)
thf(fact_7996_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_7997_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 )
@ ^ [Q4: num] : ( some @ num @ ( bit0 @ Q4 ) )
@ ( bit_take_bit_num @ ( pred_numeral @ R2 ) @ M ) ) ) ).
% take_bit_num_simps(6)
thf(fact_7998_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_7999_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_8000_take__bit__num__eq__Some__imp,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M: nat,N: num,Q2: num] :
( ( ( bit_take_bit_num @ M @ N )
= ( some @ num @ Q2 ) )
=> ( ( bit_se2584673776208193580ke_bit @ A @ M @ ( numeral_numeral @ A @ N ) )
= ( numeral_numeral @ A @ Q2 ) ) ) ) ).
% take_bit_num_eq_Some_imp
thf(fact_8001_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_8002_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_8003_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_8004_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_8005_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_8006_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_8007_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_8008_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_8009_and__not__num_Osimps_I1_J,axiom,
( ( bit_and_not_num @ one2 @ one2 )
= ( none @ num ) ) ).
% and_not_num.simps(1)
thf(fact_8010_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_8011_and__not__num__eq__Some__iff,axiom,
! [M: num,N: num,Q2: num] :
( ( ( bit_and_not_num @ M @ N )
= ( some @ num @ Q2 ) )
= ( ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ M ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ N ) ) )
= ( numeral_numeral @ int @ Q2 ) ) ) ).
% and_not_num_eq_Some_iff
thf(fact_8012_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 )
@ ^ [N13: num] : ( some @ num @ ( bit1 @ N13 ) )
@ ( bit_and_not_num @ M @ N ) ) ) ).
% and_not_num.simps(8)
thf(fact_8013_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_8014_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_8015_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_8016_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_8017_disjE__realizer2,axiom,
! [B: $tType,A: $tType,P: $o,Q: A > $o,X2: option @ A,R: B > $o,F2: B,G: A > B] :
( ( case_option @ $o @ A @ P @ Q @ X2 )
=> ( ( P
=> ( R @ F2 ) )
=> ( ! [Q3: A] :
( ( Q @ Q3 )
=> ( R @ ( G @ Q3 ) ) )
=> ( R @ ( case_option @ B @ A @ F2 @ G @ X2 ) ) ) ) ) ).
% disjE_realizer2
thf(fact_8018_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_8019_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_8020_nat_Ocase__distrib,axiom,
! [B: $tType,A: $tType,H2: A > B,F14: A,F25: nat > A,Nat: nat] :
( ( H2 @ ( case_nat @ A @ F14 @ F25 @ Nat ) )
= ( case_nat @ B @ ( H2 @ F14 )
@ ^ [X: nat] : ( H2 @ ( F25 @ X ) )
@ Nat ) ) ).
% nat.case_distrib
thf(fact_8021_old_Onat_Osimps_I5_J,axiom,
! [A: $tType,F14: A,F25: nat > A,X23: nat] :
( ( case_nat @ A @ F14 @ F25 @ ( suc @ X23 ) )
= ( F25 @ X23 ) ) ).
% old.nat.simps(5)
thf(fact_8022_old_Onat_Osimps_I4_J,axiom,
! [A: $tType,F14: A,F25: nat > A] :
( ( case_nat @ A @ F14 @ F25 @ ( zero_zero @ nat ) )
= F14 ) ).
% old.nat.simps(4)
thf(fact_8023_nth__Cons,axiom,
! [A: $tType,X2: A,Xs2: list @ A,N: nat] :
( ( nth @ A @ ( cons @ A @ X2 @ Xs2 ) @ N )
= ( case_nat @ A @ X2 @ ( nth @ A @ Xs2 ) @ N ) ) ).
% nth_Cons
thf(fact_8024_Nitpick_Ocase__nat__unfold,axiom,
! [A: $tType] :
( ( case_nat @ A )
= ( ^ [X: A,F4: nat > A,N3: nat] :
( if @ A
@ ( N3
= ( zero_zero @ nat ) )
@ X
@ ( F4 @ ( minus_minus @ nat @ N3 @ ( one_one @ nat ) ) ) ) ) ) ).
% Nitpick.case_nat_unfold
thf(fact_8025_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_8026_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 )
@ ^ [Q4: num] : ( some @ num @ ( bit0 @ Q4 ) )
@ ( bit_take_bit_num @ N3 @ M ) )
@ N ) ) ).
% Code_Abstract_Nat.take_bit_num_code(2)
thf(fact_8027_Bit__Operations_Otake__bit__num__code,axiom,
( bit_take_bit_num
= ( ^ [N3: nat,M5: num] :
( product_case_prod @ nat @ num @ ( option @ num )
@ ^ [A4: nat,X: num] :
( case_nat @ ( option @ num ) @ ( none @ num )
@ ^ [O: nat] :
( case_num @ ( option @ num ) @ ( some @ num @ one2 )
@ ^ [P3: num] :
( case_option @ ( option @ num ) @ num @ ( none @ num )
@ ^ [Q4: num] : ( some @ num @ ( bit0 @ Q4 ) )
@ ( bit_take_bit_num @ O @ P3 ) )
@ ^ [P3: num] : ( some @ num @ ( case_option @ num @ num @ one2 @ bit1 @ ( bit_take_bit_num @ O @ P3 ) ) )
@ X )
@ A4 )
@ ( product_Pair @ nat @ num @ N3 @ M5 ) ) ) ) ).
% Bit_Operations.take_bit_num_code
thf(fact_8028_take__bit__num__def,axiom,
( bit_take_bit_num
= ( ^ [N3: nat,M5: num] :
( if @ ( option @ num )
@ ( ( bit_se2584673776208193580ke_bit @ nat @ N3 @ ( numeral_numeral @ nat @ M5 ) )
= ( zero_zero @ nat ) )
@ ( none @ num )
@ ( some @ num @ ( num_of_nat @ ( bit_se2584673776208193580ke_bit @ nat @ N3 @ ( numeral_numeral @ nat @ M5 ) ) ) ) ) ) ) ).
% take_bit_num_def
thf(fact_8029_num__of__nat__numeral__eq,axiom,
! [Q2: num] :
( ( num_of_nat @ ( numeral_numeral @ nat @ Q2 ) )
= Q2 ) ).
% num_of_nat_numeral_eq
thf(fact_8030_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_8031_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_8032_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_8033_num__of__nat_Osimps_I1_J,axiom,
( ( num_of_nat @ ( zero_zero @ nat ) )
= one2 ) ).
% num_of_nat.simps(1)
thf(fact_8034_verit__eq__simplify_I17_J,axiom,
! [A: $tType,F14: A,F25: num > A,F33: num > A,X23: num] :
( ( case_num @ A @ F14 @ F25 @ F33 @ ( bit0 @ X23 ) )
= ( F25 @ X23 ) ) ).
% verit_eq_simplify(17)
thf(fact_8035_verit__eq__simplify_I16_J,axiom,
! [A: $tType,F14: A,F25: num > A,F33: num > A] :
( ( case_num @ A @ F14 @ F25 @ F33 @ one2 )
= F14 ) ).
% verit_eq_simplify(16)
thf(fact_8036_verit__eq__simplify_I18_J,axiom,
! [A: $tType,F14: A,F25: num > A,F33: num > A,X33: num] :
( ( case_num @ A @ F14 @ F25 @ F33 @ ( bit1 @ X33 ) )
= ( F33 @ X33 ) ) ).
% verit_eq_simplify(18)
thf(fact_8037_num_Ocase__distrib,axiom,
! [B: $tType,A: $tType,H2: A > B,F14: A,F25: num > A,F33: num > A,Num: num] :
( ( H2 @ ( case_num @ A @ F14 @ F25 @ F33 @ Num ) )
= ( case_num @ B @ ( H2 @ F14 )
@ ^ [X: num] : ( H2 @ ( F25 @ X ) )
@ ^ [X: num] : ( H2 @ ( F33 @ X ) )
@ Num ) ) ).
% num.case_distrib
thf(fact_8038_max__Suc1,axiom,
! [N: nat,M: nat] :
( ( ord_max @ nat @ ( suc @ N ) @ M )
= ( case_nat @ nat @ ( suc @ N )
@ ^ [M6: nat] : ( suc @ ( ord_max @ nat @ N @ M6 ) )
@ M ) ) ).
% max_Suc1
thf(fact_8039_max__Suc2,axiom,
! [M: nat,N: nat] :
( ( ord_max @ nat @ M @ ( suc @ N ) )
= ( case_nat @ nat @ ( suc @ N )
@ ^ [M6: nat] : ( suc @ ( ord_max @ nat @ M6 @ N ) )
@ M ) ) ).
% max_Suc2
thf(fact_8040_list__update_Osimps_I2_J,axiom,
! [A: $tType,X2: A,Xs2: list @ A,I: nat,V: A] :
( ( list_update @ A @ ( cons @ A @ X2 @ Xs2 ) @ I @ V )
= ( case_nat @ ( list @ A ) @ ( cons @ A @ V @ Xs2 )
@ ^ [J3: nat] : ( cons @ A @ X2 @ ( list_update @ A @ Xs2 @ J3 @ V ) )
@ I ) ) ).
% list_update.simps(2)
thf(fact_8041_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_8042_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_8043_diff__Suc,axiom,
! [M: nat,N: nat] :
( ( minus_minus @ nat @ M @ ( suc @ N ) )
= ( case_nat @ nat @ ( zero_zero @ nat )
@ ^ [K2: nat] : K2
@ ( minus_minus @ nat @ M @ N ) ) ) ).
% diff_Suc
thf(fact_8044_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_8045_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_8046_min__Suc2,axiom,
! [M: nat,N: nat] :
( ( ord_min @ nat @ M @ ( suc @ N ) )
= ( case_nat @ nat @ ( zero_zero @ nat )
@ ^ [M6: nat] : ( suc @ ( ord_min @ nat @ M6 @ N ) )
@ M ) ) ).
% min_Suc2
thf(fact_8047_min__Suc1,axiom,
! [N: nat,M: nat] :
( ( ord_min @ nat @ ( suc @ N ) @ M )
= ( case_nat @ nat @ ( zero_zero @ nat )
@ ^ [M6: nat] : ( suc @ ( ord_min @ nat @ N @ M6 ) )
@ M ) ) ).
% min_Suc1
thf(fact_8048_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_8049_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_8050_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_8051_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_8052_set__bits__int__unfold_H,axiom,
( ( bit_bi4170147762399595738t_bits @ int )
= ( ^ [F4: nat > $o] :
( if @ int
@ ? [N3: nat] :
! [N13: nat] :
( ( ord_less_eq @ nat @ N3 @ N13 )
=> ~ ( F4 @ N13 ) )
@ ( 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] :
! [N13: nat] :
( ( ord_less_eq @ nat @ N3 @ N13 )
=> ~ ( F4 @ N13 ) ) ) ) ) )
@ ( if @ int
@ ? [N3: nat] :
! [N13: nat] :
( ( ord_less_eq @ nat @ N3 @ N13 )
=> ( F4 @ N13 ) )
@ ( bit_ri4674362597316999326ke_bit @ int
@ ( ord_Least @ nat
@ ^ [N3: nat] :
! [N13: nat] :
( ( ord_less_eq @ nat @ N3 @ N13 )
=> ( F4 @ N13 ) ) )
@ ( 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] :
! [N13: nat] :
( ( ord_less_eq @ nat @ N3 @ N13 )
=> ( F4 @ N13 ) ) ) ) )
@ ( cons @ $o @ $true @ ( nil @ $o ) ) ) ) )
@ ( zero_zero @ int ) ) ) ) ) ).
% set_bits_int_unfold'
thf(fact_8053_Eps__Opt__def,axiom,
! [A: $tType] :
( ( eps_Opt @ A )
= ( ^ [P7: A > $o] :
( if @ ( option @ A )
@ ? [X7: A] : ( P7 @ X7 )
@ ( some @ A @ ( fChoice @ A @ P7 ) )
@ ( none @ A ) ) ) ) ).
% Eps_Opt_def
thf(fact_8054_concat__inth,axiom,
! [A: $tType,Xs2: list @ A,X2: A,Ys: list @ A] :
( ( nth @ A @ ( append @ A @ Xs2 @ ( append @ A @ ( cons @ A @ X2 @ ( nil @ A ) ) @ Ys ) ) @ ( size_size @ ( list @ A ) @ Xs2 ) )
= X2 ) ).
% concat_inth
thf(fact_8055_some__opt__eq__trivial,axiom,
! [A: $tType,X2: A] :
( ( eps_Opt @ A
@ ^ [Y: A] : Y = X2 )
= ( some @ A @ X2 ) ) ).
% some_opt_eq_trivial
thf(fact_8056_some__opt__false__trivial,axiom,
! [A: $tType] :
( ( eps_Opt @ A
@ ^ [Uu3: A] : $false )
= ( none @ A ) ) ).
% some_opt_false_trivial
thf(fact_8057_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_8058_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_8059_empty__replicate,axiom,
! [A: $tType,N: nat,X2: A] :
( ( ( nil @ A )
= ( replicate @ A @ N @ X2 ) )
= ( N
= ( zero_zero @ nat ) ) ) ).
% empty_replicate
thf(fact_8060_replicate__empty,axiom,
! [A: $tType,N: nat,X2: A] :
( ( ( replicate @ A @ N @ X2 )
= ( nil @ A ) )
= ( N
= ( zero_zero @ nat ) ) ) ).
% replicate_empty
thf(fact_8061_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_8062_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_8063_Eps__Opt__eq__None,axiom,
! [A: $tType,P: A > $o] :
( ( ( eps_Opt @ A @ P )
= ( none @ A ) )
= ( ~ ? [X7: A] : ( P @ X7 ) ) ) ).
% Eps_Opt_eq_None
thf(fact_8064_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_8065_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_8066_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_8067_length__ge__1__conv,axiom,
! [A: $tType,L2: list @ A] :
( ( ord_less_eq @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( size_size @ ( list @ A ) @ L2 ) )
= ( L2
!= ( nil @ A ) ) ) ).
% length_ge_1_conv
thf(fact_8068_concat__map__singleton,axiom,
! [A: $tType,B: $tType,F2: B > A,Xs2: list @ B] :
( ( concat @ A
@ ( map @ B @ ( list @ A )
@ ^ [X: B] : ( cons @ A @ ( F2 @ X ) @ ( nil @ A ) )
@ Xs2 ) )
= ( map @ B @ A @ F2 @ Xs2 ) ) ).
% concat_map_singleton
thf(fact_8069_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_8070_list__rest__coinc,axiom,
! [A: $tType,S22: list @ A,S1: list @ A,R1: list @ A,R22: list @ A] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ S22 ) @ ( size_size @ ( list @ A ) @ S1 ) )
=> ( ( ( append @ A @ S1 @ R1 )
= ( append @ A @ S22 @ R22 ) )
=> ? [R1p: list @ A] :
( R22
= ( append @ A @ R1p @ R1 ) ) ) ) ).
% list_rest_coinc
thf(fact_8071_length__compl__rev__induct,axiom,
! [A: $tType,P: ( list @ A ) > $o,L2: list @ A] :
( ( P @ ( nil @ A ) )
=> ( ! [L4: list @ A,E2: A] :
( ! [Ll: list @ A] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Ll ) @ ( size_size @ ( list @ A ) @ L4 ) )
=> ( P @ Ll ) )
=> ( P @ ( append @ A @ L4 @ ( cons @ A @ E2 @ ( nil @ A ) ) ) ) )
=> ( P @ L2 ) ) ) ).
% length_compl_rev_induct
thf(fact_8072_length__compl__induct,axiom,
! [A: $tType,P: ( list @ A ) > $o,L2: list @ A] :
( ( P @ ( nil @ A ) )
=> ( ! [E2: A,L4: list @ A] :
( ! [Ll: list @ A] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Ll ) @ ( size_size @ ( list @ A ) @ L4 ) )
=> ( P @ Ll ) )
=> ( P @ ( cons @ A @ E2 @ L4 ) ) )
=> ( P @ L2 ) ) ) ).
% length_compl_induct
thf(fact_8073_comm__append__is__replicate,axiom,
! [A: $tType,Xs2: list @ A,Ys: list @ A] :
( ( Xs2
!= ( nil @ A ) )
=> ( ( Ys
!= ( nil @ A ) )
=> ( ( ( append @ A @ Xs2 @ Ys )
= ( append @ A @ Ys @ Xs2 ) )
=> ? [N2: nat,Zs: list @ A] :
( ( ord_less @ nat @ ( one_one @ nat ) @ N2 )
& ( ( concat @ A @ ( replicate @ ( list @ A ) @ N2 @ Zs ) )
= ( append @ A @ Xs2 @ Ys ) ) ) ) ) ) ).
% comm_append_is_replicate
thf(fact_8074_count__list_Osimps_I1_J,axiom,
! [A: $tType,Y2: A] :
( ( count_list @ A @ ( nil @ A ) @ Y2 )
= ( zero_zero @ nat ) ) ).
% count_list.simps(1)
thf(fact_8075_replicate__0,axiom,
! [A: $tType,X2: A] :
( ( replicate @ A @ ( zero_zero @ nat ) @ X2 )
= ( nil @ A ) ) ).
% replicate_0
thf(fact_8076_replicate__add,axiom,
! [A: $tType,N: nat,M: nat,X2: A] :
( ( replicate @ A @ ( plus_plus @ nat @ N @ M ) @ X2 )
= ( append @ A @ ( replicate @ A @ N @ X2 ) @ ( replicate @ A @ M @ X2 ) ) ) ).
% replicate_add
thf(fact_8077_append__replicate__commute,axiom,
! [A: $tType,N: nat,X2: A,K: nat] :
( ( append @ A @ ( replicate @ A @ N @ X2 ) @ ( replicate @ A @ K @ X2 ) )
= ( append @ A @ ( replicate @ A @ K @ X2 ) @ ( replicate @ A @ N @ X2 ) ) ) ).
% append_replicate_commute
thf(fact_8078_list_Osize__gen_I1_J,axiom,
! [A: $tType,X2: A > nat] :
( ( size_list @ A @ X2 @ ( nil @ A ) )
= ( zero_zero @ nat ) ) ).
% list.size_gen(1)
thf(fact_8079_length__nth__simps_I1_J,axiom,
! [A: $tType] :
( ( size_size @ ( list @ A ) @ ( nil @ A ) )
= ( zero_zero @ nat ) ) ).
% length_nth_simps(1)
thf(fact_8080_length__append__singleton,axiom,
! [A: $tType,Xs2: list @ A,X2: A] :
( ( size_size @ ( list @ A ) @ ( append @ A @ Xs2 @ ( cons @ A @ X2 @ ( nil @ A ) ) ) )
= ( suc @ ( size_size @ ( list @ A ) @ Xs2 ) ) ) ).
% length_append_singleton
thf(fact_8081_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_8082_replicate__append__same,axiom,
! [A: $tType,I: nat,X2: A] :
( ( append @ A @ ( replicate @ A @ I @ X2 ) @ ( cons @ A @ X2 @ ( nil @ A ) ) )
= ( cons @ A @ X2 @ ( replicate @ A @ I @ X2 ) ) ) ).
% replicate_append_same
thf(fact_8083_replicate__app__Cons__same,axiom,
! [A: $tType,N: nat,X2: A,Xs2: list @ A] :
( ( append @ A @ ( replicate @ A @ N @ X2 ) @ ( cons @ A @ X2 @ Xs2 ) )
= ( cons @ A @ X2 @ ( append @ A @ ( replicate @ A @ N @ X2 ) @ Xs2 ) ) ) ).
% replicate_app_Cons_same
thf(fact_8084_replicate__Suc__conv__snoc,axiom,
! [A: $tType,N: nat,X2: A] :
( ( replicate @ A @ ( suc @ N ) @ X2 )
= ( append @ A @ ( replicate @ A @ N @ X2 ) @ ( cons @ A @ X2 @ ( nil @ A ) ) ) ) ).
% replicate_Suc_conv_snoc
thf(fact_8085_length__Suc__rev__conv,axiom,
! [A: $tType,Xs2: list @ A,N: nat] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
= ( suc @ N ) )
= ( ? [Ys2: list @ A,Y: A] :
( ( Xs2
= ( append @ A @ Ys2 @ ( cons @ A @ Y @ ( nil @ A ) ) ) )
& ( ( size_size @ ( list @ A ) @ Ys2 )
= N ) ) ) ) ).
% length_Suc_rev_conv
thf(fact_8086_length__Suc__conv__rev,axiom,
! [A: $tType,Xs2: list @ A,N: nat] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
= ( suc @ N ) )
= ( ? [Y: A,Ys2: list @ A] :
( ( Xs2
= ( append @ A @ Ys2 @ ( cons @ A @ Y @ ( nil @ A ) ) ) )
& ( ( size_size @ ( list @ A ) @ Ys2 )
= N ) ) ) ) ).
% length_Suc_conv_rev
thf(fact_8087_list__decomp__1,axiom,
! [A: $tType,L2: list @ A] :
( ( ( size_size @ ( list @ A ) @ L2 )
= ( one_one @ nat ) )
=> ? [A5: A] :
( L2
= ( cons @ A @ A5 @ ( nil @ A ) ) ) ) ).
% list_decomp_1
thf(fact_8088_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_8089_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_8090_list__decomp__2,axiom,
! [A: $tType,L2: list @ A] :
( ( ( size_size @ ( list @ A ) @ L2 )
= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
=> ? [A5: A,B4: A] :
( L2
= ( cons @ A @ A5 @ ( cons @ A @ B4 @ ( nil @ A ) ) ) ) ) ).
% list_decomp_2
thf(fact_8091_quickcheck__narrowing__samples_Onarrowing__samples_Oelims,axiom,
! [A: $tType] :
( ( ( code_term_of @ A )
& ( quickc6926020345158392990erm_of @ A ) )
=> ! [A_of_integer: code_integer > ( product_prod @ A @ A ),Zero: A,X2: code_integer,Y2: list @ A] :
( ( ( code_T4080844693773952564amples @ A @ A_of_integer @ Zero @ X2 )
= Y2 )
=> ( ( ( ord_less @ code_integer @ ( zero_zero @ code_integer ) @ X2 )
=> ( Y2
= ( product_case_prod @ A @ A @ ( list @ A )
@ ^ [A4: A,A14: A] : ( append @ A @ ( code_T4080844693773952564amples @ A @ A_of_integer @ Zero @ ( minus_minus @ code_integer @ X2 @ ( one_one @ code_integer ) ) ) @ ( cons @ A @ A4 @ ( cons @ A @ A14 @ ( nil @ A ) ) ) )
@ ( A_of_integer @ X2 ) ) ) )
& ( ~ ( ord_less @ code_integer @ ( zero_zero @ code_integer ) @ X2 )
=> ( Y2
= ( cons @ A @ Zero @ ( nil @ A ) ) ) ) ) ) ) ).
% quickcheck_narrowing_samples.narrowing_samples.elims
thf(fact_8092_quickcheck__narrowing__samples_Onarrowing__samples_Osimps,axiom,
! [A: $tType] :
( ( ( code_term_of @ A )
& ( quickc6926020345158392990erm_of @ A ) )
=> ( ( code_T4080844693773952564amples @ A )
= ( ^ [A_of_integer2: code_integer > ( product_prod @ A @ A ),Zero2: A,I3: code_integer] :
( if @ ( list @ A ) @ ( ord_less @ code_integer @ ( zero_zero @ code_integer ) @ I3 )
@ ( product_case_prod @ A @ A @ ( list @ A )
@ ^ [A4: A,A14: A] : ( append @ A @ ( code_T4080844693773952564amples @ A @ A_of_integer2 @ Zero2 @ ( minus_minus @ code_integer @ I3 @ ( one_one @ code_integer ) ) ) @ ( cons @ A @ A4 @ ( cons @ A @ A14 @ ( nil @ A ) ) ) )
@ ( A_of_integer2 @ I3 ) )
@ ( cons @ A @ Zero2 @ ( nil @ A ) ) ) ) ) ) ).
% quickcheck_narrowing_samples.narrowing_samples.simps
thf(fact_8093_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_8094_upt__conv__Nil,axiom,
! [J: nat,I: nat] :
( ( ord_less_eq @ nat @ J @ I )
=> ( ( upt @ I @ J )
= ( nil @ nat ) ) ) ).
% upt_conv_Nil
thf(fact_8095_upt__merge,axiom,
! [I: nat,J: nat,K: nat] :
( ( ( ord_less_eq @ nat @ I @ J )
& ( ord_less_eq @ nat @ J @ K ) )
=> ( ( append @ nat @ ( upt @ I @ J ) @ ( upt @ J @ K ) )
= ( upt @ I @ K ) ) ) ).
% upt_merge
thf(fact_8096_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_8097_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_8098_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_8099_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_8100_upt__eq__append__conv,axiom,
! [I: nat,J: nat,Xs2: list @ nat,Ys: list @ nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( ( upt @ I @ J )
= ( append @ nat @ Xs2 @ Ys ) )
= ( ? [K2: nat] :
( ( ord_less_eq @ nat @ I @ K2 )
& ( ord_less_eq @ nat @ K2 @ J )
& ( ( upt @ I @ K2 )
= Xs2 )
& ( ( upt @ K2 @ J )
= Ys ) ) ) ) ) ).
% upt_eq_append_conv
thf(fact_8101_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_8102_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_8103_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_8104_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_8105_upt__0,axiom,
! [I: nat] :
( ( upt @ I @ ( zero_zero @ nat ) )
= ( nil @ nat ) ) ).
% upt_0
thf(fact_8106_list__encode_Ocases,axiom,
! [X2: list @ nat] :
( ( X2
!= ( nil @ nat ) )
=> ~ ! [X3: nat,Xs3: list @ nat] :
( X2
!= ( cons @ nat @ X3 @ Xs3 ) ) ) ).
% list_encode.cases
thf(fact_8107_subseqs_Osimps_I2_J,axiom,
! [A: $tType,X2: A,Xs2: list @ A] :
( ( subseqs @ A @ ( cons @ A @ X2 @ Xs2 ) )
= ( append @ ( list @ A ) @ ( map @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X2 ) @ ( subseqs @ A @ Xs2 ) ) @ ( subseqs @ A @ Xs2 ) ) ) ).
% subseqs.simps(2)
thf(fact_8108_upt__rec,axiom,
( upt
= ( ^ [I3: nat,J3: nat] : ( if @ ( list @ nat ) @ ( ord_less @ nat @ I3 @ J3 ) @ ( cons @ nat @ I3 @ ( upt @ ( suc @ I3 ) @ J3 ) ) @ ( nil @ nat ) ) ) ) ).
% upt_rec
thf(fact_8109_upt__eq__lel__conv,axiom,
! [L2: nat,H2: nat,Is1: list @ nat,I: nat,Is2: list @ nat] :
( ( ( upt @ L2 @ H2 )
= ( append @ nat @ Is1 @ ( cons @ nat @ I @ Is2 ) ) )
= ( ( Is1
= ( upt @ L2 @ I ) )
& ( Is2
= ( upt @ ( suc @ I ) @ H2 ) )
& ( ord_less_eq @ nat @ L2 @ I )
& ( ord_less @ nat @ I @ H2 ) ) ) ).
% upt_eq_lel_conv
thf(fact_8110_transpose__rectangle,axiom,
! [A: $tType,Xs2: list @ ( list @ A ),N: nat] :
( ( ( Xs2
= ( nil @ ( list @ A ) ) )
=> ( N
= ( zero_zero @ nat ) ) )
=> ( ! [I2: nat] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ ( list @ A ) ) @ Xs2 ) )
=> ( ( size_size @ ( list @ A ) @ ( nth @ ( list @ A ) @ Xs2 @ I2 ) )
= N ) )
=> ( ( transpose @ A @ Xs2 )
= ( map @ nat @ ( list @ A )
@ ^ [I3: nat] :
( map @ nat @ A
@ ^ [J3: nat] : ( nth @ A @ ( nth @ ( list @ A ) @ Xs2 @ J3 ) @ I3 )
@ ( upt @ ( zero_zero @ nat ) @ ( size_size @ ( list @ ( list @ A ) ) @ Xs2 ) ) )
@ ( upt @ ( zero_zero @ nat ) @ N ) ) ) ) ) ).
% transpose_rectangle
thf(fact_8111_lexn__conv,axiom,
! [A: $tType] :
( ( lexn @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A ),N3: nat] :
( collect @ ( product_prod @ ( list @ A ) @ ( list @ A ) )
@ ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ $o
@ ^ [Xs: list @ A,Ys2: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs )
= N3 )
& ( ( size_size @ ( list @ A ) @ Ys2 )
= N3 )
& ? [Xys: list @ A,X: A,Y: A,Xs4: list @ A,Ys3: list @ A] :
( ( Xs
= ( append @ A @ Xys @ ( cons @ A @ X @ Xs4 ) ) )
& ( Ys2
= ( append @ A @ Xys @ ( cons @ A @ Y @ Ys3 ) ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R5 ) ) ) ) ) ) ) ).
% lexn_conv
thf(fact_8112_lexn_Osimps_I1_J,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( lexn @ A @ R2 @ ( zero_zero @ nat ) )
= ( bot_bot @ ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) ) ) ).
% lexn.simps(1)
thf(fact_8113_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_8114_lex__conv,axiom,
! [A: $tType] :
( ( lex @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A )] :
( collect @ ( product_prod @ ( list @ A ) @ ( list @ A ) )
@ ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ $o
@ ^ [Xs: list @ A,Ys2: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ A ) @ Ys2 ) )
& ? [Xys: list @ A,X: A,Y: A,Xs4: list @ A,Ys3: list @ A] :
( ( Xs
= ( append @ A @ Xys @ ( cons @ A @ X @ Xs4 ) ) )
& ( Ys2
= ( append @ A @ Xys @ ( cons @ A @ Y @ Ys3 ) ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R5 ) ) ) ) ) ) ) ).
% lex_conv
thf(fact_8115_less__eq__char__simp,axiom,
! [B0: $o,B12: $o,B23: $o,B32: $o,B42: $o,B52: $o,B62: $o,B72: $o,C0: $o,C1: $o,C22: $o,C32: $o,C42: $o,C52: $o,C62: $o,C72: $o] :
( ( ord_less_eq @ char @ ( char2 @ B0 @ B12 @ B23 @ B32 @ B42 @ B52 @ B62 @ B72 ) @ ( char2 @ C0 @ C1 @ C22 @ C32 @ C42 @ C52 @ C62 @ C72 ) )
= ( ord_less_eq @ nat
@ ( foldr @ $o @ nat
@ ^ [B3: $o,K2: nat] : ( plus_plus @ nat @ ( zero_neq_one_of_bool @ nat @ B3 ) @ ( times_times @ nat @ K2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
@ ( cons @ $o @ B0 @ ( cons @ $o @ B12 @ ( cons @ $o @ B23 @ ( cons @ $o @ B32 @ ( cons @ $o @ B42 @ ( cons @ $o @ B52 @ ( cons @ $o @ B62 @ ( cons @ $o @ B72 @ ( nil @ $o ) ) ) ) ) ) ) ) )
@ ( zero_zero @ nat ) )
@ ( foldr @ $o @ nat
@ ^ [B3: $o,K2: nat] : ( plus_plus @ nat @ ( zero_neq_one_of_bool @ nat @ B3 ) @ ( times_times @ nat @ K2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
@ ( cons @ $o @ C0 @ ( cons @ $o @ C1 @ ( cons @ $o @ C22 @ ( cons @ $o @ C32 @ ( cons @ $o @ C42 @ ( cons @ $o @ C52 @ ( cons @ $o @ C62 @ ( cons @ $o @ C72 @ ( nil @ $o ) ) ) ) ) ) ) ) )
@ ( zero_zero @ nat ) ) ) ) ).
% less_eq_char_simp
thf(fact_8116_less__char__simp,axiom,
! [B0: $o,B12: $o,B23: $o,B32: $o,B42: $o,B52: $o,B62: $o,B72: $o,C0: $o,C1: $o,C22: $o,C32: $o,C42: $o,C52: $o,C62: $o,C72: $o] :
( ( ord_less @ char @ ( char2 @ B0 @ B12 @ B23 @ B32 @ B42 @ B52 @ B62 @ B72 ) @ ( char2 @ C0 @ C1 @ C22 @ C32 @ C42 @ C52 @ C62 @ C72 ) )
= ( ord_less @ nat
@ ( foldr @ $o @ nat
@ ^ [B3: $o,K2: nat] : ( plus_plus @ nat @ ( zero_neq_one_of_bool @ nat @ B3 ) @ ( times_times @ nat @ K2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
@ ( cons @ $o @ B0 @ ( cons @ $o @ B12 @ ( cons @ $o @ B23 @ ( cons @ $o @ B32 @ ( cons @ $o @ B42 @ ( cons @ $o @ B52 @ ( cons @ $o @ B62 @ ( cons @ $o @ B72 @ ( nil @ $o ) ) ) ) ) ) ) ) )
@ ( zero_zero @ nat ) )
@ ( foldr @ $o @ nat
@ ^ [B3: $o,K2: nat] : ( plus_plus @ nat @ ( zero_neq_one_of_bool @ nat @ B3 ) @ ( times_times @ nat @ K2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
@ ( cons @ $o @ C0 @ ( cons @ $o @ C1 @ ( cons @ $o @ C22 @ ( cons @ $o @ C32 @ ( cons @ $o @ C42 @ ( cons @ $o @ C52 @ ( cons @ $o @ C62 @ ( cons @ $o @ C72 @ ( nil @ $o ) ) ) ) ) ) ) ) )
@ ( zero_zero @ nat ) ) ) ) ).
% less_char_simp
thf(fact_8117_integer__of__char__code,axiom,
! [B0: $o,B12: $o,B23: $o,B32: $o,B42: $o,B52: $o,B62: $o,B72: $o] :
( ( integer_of_char @ ( char2 @ B0 @ B12 @ B23 @ 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 @ B23 ) ) @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) @ ( zero_neq_one_of_bool @ code_integer @ B12 ) ) @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) @ ( zero_neq_one_of_bool @ code_integer @ B0 ) ) ) ).
% integer_of_char_code
thf(fact_8118_of__char__Char,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [B0: $o,B12: $o,B23: $o,B32: $o,B42: $o,B52: $o,B62: $o,B72: $o] :
( ( comm_s6883823935334413003f_char @ A @ ( char2 @ B0 @ B12 @ B23 @ 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 @ B12 @ ( cons @ $o @ B23 @ ( cons @ $o @ B32 @ ( cons @ $o @ B42 @ ( cons @ $o @ B52 @ ( cons @ $o @ B62 @ ( cons @ $o @ B72 @ ( nil @ $o ) ) ) ) ) ) ) ) ) ) ) ) ).
% of_char_Char
thf(fact_8119_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_8120_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_8121_less__eq__char__def,axiom,
( ( ord_less_eq @ char )
= ( ^ [C12: char,C23: char] : ( ord_less_eq @ nat @ ( comm_s6883823935334413003f_char @ nat @ C12 ) @ ( comm_s6883823935334413003f_char @ nat @ C23 ) ) ) ) ).
% less_eq_char_def
thf(fact_8122_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_8123_char_Osize_I2_J,axiom,
! [X16: $o,X23: $o,X33: $o,X42: $o,X52: $o,X62: $o,X72: $o,X82: $o] :
( ( size_size @ char @ ( char2 @ X16 @ X23 @ X33 @ X42 @ X52 @ X62 @ X72 @ X82 ) )
= ( zero_zero @ nat ) ) ).
% char.size(2)
thf(fact_8124_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_8125_char_Osize__gen,axiom,
! [X16: $o,X23: $o,X33: $o,X42: $o,X52: $o,X62: $o,X72: $o,X82: $o] :
( ( size_char @ ( char2 @ X16 @ X23 @ X33 @ X42 @ X52 @ X62 @ X72 @ X82 ) )
= ( zero_zero @ nat ) ) ).
% char.size_gen
thf(fact_8126_char__of__nat,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [N: nat] :
( ( unique5772411509450598832har_of @ A @ ( semiring_1_of_nat @ A @ N ) )
= ( unique5772411509450598832har_of @ nat @ N ) ) ) ).
% char_of_nat
thf(fact_8127_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_8128_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_8129_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_8130_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_8131_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_8132_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_8133_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_8134_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_8135_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_8136_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_8137_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_8138_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_8139_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_8140_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_8141_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_8142_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_8143_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_8144_atLeastLessThan__upto,axiom,
( ( set_or7035219750837199246ssThan @ int )
= ( ^ [I3: int,J3: int] : ( set2 @ int @ ( upto @ I3 @ ( minus_minus @ int @ J3 @ ( one_one @ int ) ) ) ) ) ) ).
% atLeastLessThan_upto
thf(fact_8145_greaterThanAtMost__upto,axiom,
( ( set_or3652927894154168847AtMost @ int )
= ( ^ [I3: int,J3: int] : ( set2 @ int @ ( upto @ ( plus_plus @ int @ I3 @ ( one_one @ int ) ) @ J3 ) ) ) ) ).
% greaterThanAtMost_upto
thf(fact_8146_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_8147_upto_Oelims,axiom,
! [X2: int,Xa: int,Y2: list @ int] :
( ( ( upto @ X2 @ Xa )
= Y2 )
=> ( ( ( ord_less_eq @ int @ X2 @ Xa )
=> ( Y2
= ( cons @ int @ X2 @ ( upto @ ( plus_plus @ int @ X2 @ ( one_one @ int ) ) @ Xa ) ) ) )
& ( ~ ( ord_less_eq @ int @ X2 @ Xa )
=> ( Y2
= ( nil @ int ) ) ) ) ) ).
% upto.elims
thf(fact_8148_upto_Osimps,axiom,
( upto
= ( ^ [I3: int,J3: int] : ( if @ ( list @ int ) @ ( ord_less_eq @ int @ I3 @ J3 ) @ ( cons @ int @ I3 @ ( upto @ ( plus_plus @ int @ I3 @ ( one_one @ int ) ) @ J3 ) ) @ ( nil @ int ) ) ) ) ).
% upto.simps
thf(fact_8149_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_8150_greaterThanLessThan__upto,axiom,
( ( set_or5935395276787703475ssThan @ int )
= ( ^ [I3: int,J3: int] : ( set2 @ int @ ( upto @ ( plus_plus @ int @ I3 @ ( one_one @ int ) ) @ ( minus_minus @ int @ J3 @ ( one_one @ int ) ) ) ) ) ) ).
% greaterThanLessThan_upto
thf(fact_8151_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_8152_upto_Opelims,axiom,
! [X2: int,Xa: int,Y2: list @ int] :
( ( ( upto @ X2 @ Xa )
= Y2 )
=> ( ( accp @ ( product_prod @ int @ int ) @ upto_rel @ ( product_Pair @ int @ int @ X2 @ Xa ) )
=> ~ ( ( ( ( ord_less_eq @ int @ X2 @ Xa )
=> ( Y2
= ( cons @ int @ X2 @ ( upto @ ( plus_plus @ int @ X2 @ ( one_one @ int ) ) @ Xa ) ) ) )
& ( ~ ( ord_less_eq @ int @ X2 @ Xa )
=> ( Y2
= ( nil @ int ) ) ) )
=> ~ ( accp @ ( product_prod @ int @ int ) @ upto_rel @ ( product_Pair @ int @ int @ X2 @ Xa ) ) ) ) ) ).
% upto.pelims
thf(fact_8153_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 )
@ ^ [X: A] : ( cons @ A @ X @ ( nil @ A ) )
@ A3 ) ) ).
% set_Cons_sing_Nil
thf(fact_8154_list_Ocase__distrib,axiom,
! [B: $tType,C: $tType,A: $tType,H2: B > C,F14: B,F25: A > ( list @ A ) > B,List: list @ A] :
( ( H2 @ ( case_list @ B @ A @ F14 @ F25 @ List ) )
= ( case_list @ C @ A @ ( H2 @ F14 )
@ ^ [X15: A,X24: list @ A] : ( H2 @ ( F25 @ X15 @ X24 ) )
@ List ) ) ).
% list.case_distrib
thf(fact_8155_transpose_Oelims,axiom,
! [A: $tType,X2: list @ ( list @ A ),Y2: list @ ( list @ A )] :
( ( ( transpose @ A @ X2 )
= Y2 )
=> ( ( ( X2
= ( nil @ ( list @ A ) ) )
=> ( Y2
!= ( nil @ ( list @ A ) ) ) )
=> ( ! [Xss2: list @ ( list @ A )] :
( ( X2
= ( cons @ ( list @ A ) @ ( nil @ A ) @ Xss2 ) )
=> ( Y2
!= ( transpose @ A @ Xss2 ) ) )
=> ~ ! [X3: A,Xs3: list @ A,Xss2: list @ ( list @ A )] :
( ( X2
= ( cons @ ( list @ A ) @ ( cons @ A @ X3 @ Xs3 ) @ Xss2 ) )
=> ( Y2
!= ( cons @ ( list @ A )
@ ( cons @ A @ X3
@ ( concat @ A
@ ( map @ ( list @ A ) @ ( list @ A )
@ ( case_list @ ( list @ A ) @ A @ ( nil @ A )
@ ^ [H: A,T: list @ A] : ( cons @ A @ H @ ( nil @ A ) ) )
@ Xss2 ) ) )
@ ( 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,T: list @ A] : ( cons @ ( list @ A ) @ T @ ( nil @ ( list @ A ) ) ) )
@ Xss2 ) ) ) ) ) ) ) ) ) ) ).
% transpose.elims
thf(fact_8156_transpose_Osimps_I3_J,axiom,
! [A: $tType,X2: A,Xs2: list @ A,Xss: list @ ( list @ A )] :
( ( transpose @ A @ ( cons @ ( list @ A ) @ ( cons @ A @ X2 @ Xs2 ) @ Xss ) )
= ( cons @ ( list @ A )
@ ( cons @ A @ X2
@ ( concat @ A
@ ( map @ ( list @ A ) @ ( list @ A )
@ ( case_list @ ( list @ A ) @ A @ ( nil @ A )
@ ^ [H: A,T: 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,T: list @ A] : ( cons @ ( list @ A ) @ T @ ( nil @ ( list @ A ) ) ) )
@ Xss ) ) ) ) ) ) ).
% transpose.simps(3)
thf(fact_8157_set__Cons__def,axiom,
! [A: $tType] :
( ( set_Cons @ A )
= ( ^ [A7: set @ A,XS: set @ ( list @ A )] :
( collect @ ( list @ A )
@ ^ [Z5: list @ A] :
? [X: A,Xs: list @ A] :
( ( Z5
= ( cons @ A @ X @ Xs ) )
& ( member @ A @ X @ A7 )
& ( member @ ( list @ A ) @ Xs @ XS ) ) ) ) ) ).
% set_Cons_def
thf(fact_8158_min__list_Oelims,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X2: list @ A,Y2: A] :
( ( ( min_list @ A @ X2 )
= Y2 )
=> ( ! [X3: A,Xs3: list @ A] :
( ( X2
= ( cons @ A @ X3 @ Xs3 ) )
=> ( Y2
!= ( case_list @ A @ A @ X3
@ ^ [A4: A,List2: list @ A] : ( ord_min @ A @ X3 @ ( min_list @ A @ Xs3 ) )
@ Xs3 ) ) )
=> ~ ( ( X2
= ( nil @ A ) )
=> ( Y2
!= ( undefined @ A ) ) ) ) ) ) ).
% min_list.elims
thf(fact_8159_transpose_Opsimps_I3_J,axiom,
! [A: $tType,X2: A,Xs2: list @ A,Xss: list @ ( list @ A )] :
( ( accp @ ( list @ ( list @ A ) ) @ ( transpose_rel @ A ) @ ( cons @ ( list @ A ) @ ( cons @ A @ X2 @ Xs2 ) @ Xss ) )
=> ( ( transpose @ A @ ( cons @ ( list @ A ) @ ( cons @ A @ X2 @ Xs2 ) @ Xss ) )
= ( cons @ ( list @ A )
@ ( cons @ A @ X2
@ ( concat @ A
@ ( map @ ( list @ A ) @ ( list @ A )
@ ( case_list @ ( list @ A ) @ A @ ( nil @ A )
@ ^ [H: A,T: 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,T: list @ A] : ( cons @ ( list @ A ) @ T @ ( nil @ ( list @ A ) ) ) )
@ Xss ) ) ) ) ) ) ) ).
% transpose.psimps(3)
thf(fact_8160_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_8161_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_8162_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 ) ) ) )
=> ( ! [Xss2: list @ ( list @ A )] :
( ( accp @ ( list @ ( list @ A ) ) @ ( transpose_rel @ A ) @ ( cons @ ( list @ A ) @ ( nil @ A ) @ Xss2 ) )
=> ( ( P @ Xss2 )
=> ( P @ ( cons @ ( list @ A ) @ ( nil @ A ) @ Xss2 ) ) ) )
=> ( ! [X3: A,Xs3: list @ A,Xss2: list @ ( list @ A )] :
( ( accp @ ( list @ ( list @ A ) ) @ ( transpose_rel @ A ) @ ( cons @ ( list @ A ) @ ( cons @ A @ X3 @ Xs3 ) @ Xss2 ) )
=> ( ( 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,T: list @ A] : ( cons @ ( list @ A ) @ T @ ( nil @ ( list @ A ) ) ) )
@ Xss2 ) ) ) )
=> ( P @ ( cons @ ( list @ A ) @ ( cons @ A @ X3 @ Xs3 ) @ Xss2 ) ) ) )
=> ( P @ A0 ) ) ) ) ) ).
% transpose.pinduct
thf(fact_8163_min__list_Osimps,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X2: A,Xs2: list @ A] :
( ( min_list @ A @ ( cons @ A @ X2 @ Xs2 ) )
= ( case_list @ A @ A @ X2
@ ^ [A4: A,List2: list @ A] : ( ord_min @ A @ X2 @ ( min_list @ A @ Xs2 ) )
@ Xs2 ) ) ) ).
% min_list.simps
thf(fact_8164_transpose_Opelims,axiom,
! [A: $tType,X2: list @ ( list @ A ),Y2: list @ ( list @ A )] :
( ( ( transpose @ A @ X2 )
= Y2 )
=> ( ( accp @ ( list @ ( list @ A ) ) @ ( transpose_rel @ A ) @ X2 )
=> ( ( ( X2
= ( nil @ ( list @ A ) ) )
=> ( ( Y2
= ( nil @ ( list @ A ) ) )
=> ~ ( accp @ ( list @ ( list @ A ) ) @ ( transpose_rel @ A ) @ ( nil @ ( list @ A ) ) ) ) )
=> ( ! [Xss2: list @ ( list @ A )] :
( ( X2
= ( cons @ ( list @ A ) @ ( nil @ A ) @ Xss2 ) )
=> ( ( Y2
= ( transpose @ A @ Xss2 ) )
=> ~ ( accp @ ( list @ ( list @ A ) ) @ ( transpose_rel @ A ) @ ( cons @ ( list @ A ) @ ( nil @ A ) @ Xss2 ) ) ) )
=> ~ ! [X3: A,Xs3: list @ A,Xss2: list @ ( list @ A )] :
( ( X2
= ( cons @ ( list @ A ) @ ( cons @ A @ X3 @ Xs3 ) @ Xss2 ) )
=> ( ( Y2
= ( cons @ ( list @ A )
@ ( cons @ A @ X3
@ ( concat @ A
@ ( map @ ( list @ A ) @ ( list @ A )
@ ( case_list @ ( list @ A ) @ A @ ( nil @ A )
@ ^ [H: A,T: list @ A] : ( cons @ A @ H @ ( nil @ A ) ) )
@ Xss2 ) ) )
@ ( 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,T: list @ A] : ( cons @ ( list @ A ) @ T @ ( nil @ ( list @ A ) ) ) )
@ Xss2 ) ) ) ) ) )
=> ~ ( accp @ ( list @ ( list @ A ) ) @ ( transpose_rel @ A ) @ ( cons @ ( list @ A ) @ ( cons @ A @ X3 @ Xs3 ) @ Xss2 ) ) ) ) ) ) ) ) ).
% transpose.pelims
thf(fact_8165_min__list_Opelims,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X2: list @ A,Y2: A] :
( ( ( min_list @ A @ X2 )
= Y2 )
=> ( ( accp @ ( list @ A ) @ ( min_list_rel @ A ) @ X2 )
=> ( ! [X3: A,Xs3: list @ A] :
( ( X2
= ( cons @ A @ X3 @ Xs3 ) )
=> ( ( Y2
= ( 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 ) ) ) )
=> ~ ( ( X2
= ( nil @ A ) )
=> ( ( Y2
= ( undefined @ A ) )
=> ~ ( accp @ ( list @ A ) @ ( min_list_rel @ A ) @ ( nil @ A ) ) ) ) ) ) ) ) ).
% min_list.pelims
thf(fact_8166_filter__upt__last,axiom,
! [A: $tType,P: A > $o,L2: list @ A,Js2: list @ nat,J: nat,I: nat] :
( ( ( filter2 @ nat
@ ^ [K2: nat] : ( P @ ( nth @ A @ L2 @ K2 ) )
@ ( upt @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ L2 ) ) )
= ( append @ nat @ Js2 @ ( cons @ nat @ J @ ( nil @ nat ) ) ) )
=> ( ( ord_less @ nat @ J @ I )
=> ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ L2 ) )
=> ~ ( P @ ( nth @ A @ L2 @ I ) ) ) ) ) ).
% filter_upt_last
thf(fact_8167_filter__filter,axiom,
! [A: $tType,P: A > $o,Q: A > $o,Xs2: list @ A] :
( ( filter2 @ A @ P @ ( filter2 @ A @ Q @ Xs2 ) )
= ( filter2 @ A
@ ^ [X: A] :
( ( Q @ X )
& ( P @ X ) )
@ Xs2 ) ) ).
% filter_filter
thf(fact_8168_set__filter,axiom,
! [A: $tType,P: A > $o,Xs2: list @ A] :
( ( set2 @ A @ ( filter2 @ A @ P @ Xs2 ) )
= ( collect @ A
@ ^ [X: A] :
( ( member @ A @ X @ ( set2 @ A @ Xs2 ) )
& ( P @ X ) ) ) ) ).
% set_filter
thf(fact_8169_filter__conv__foldr,axiom,
! [A: $tType] :
( ( filter2 @ A )
= ( ^ [P7: A > $o,Xs: list @ A] :
( foldr @ A @ ( list @ A )
@ ^ [X: A,Xt: list @ A] : ( if @ ( list @ A ) @ ( P7 @ X ) @ ( cons @ A @ X @ Xt ) @ Xt )
@ Xs
@ ( nil @ A ) ) ) ) ).
% filter_conv_foldr
thf(fact_8170_filter__replicate,axiom,
! [A: $tType,P: A > $o,X2: A,N: nat] :
( ( ( P @ X2 )
=> ( ( filter2 @ A @ P @ ( replicate @ A @ N @ X2 ) )
= ( replicate @ A @ N @ X2 ) ) )
& ( ~ ( P @ X2 )
=> ( ( filter2 @ A @ P @ ( replicate @ A @ N @ X2 ) )
= ( nil @ A ) ) ) ) ).
% filter_replicate
thf(fact_8171_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
@ ^ [X: A] :
~ ( P @ X )
@ Xs2 ) ) )
= ( size_size @ ( list @ A ) @ Xs2 ) ) ).
% sum_length_filter_compl
thf(fact_8172_replicate__length__filter,axiom,
! [A: $tType,X2: A,Xs2: list @ A] :
( ( replicate @ A
@ ( size_size @ ( list @ A )
@ ( filter2 @ A
@ ( ^ [Y5: A,Z3: A] : Y5 = Z3
@ X2 )
@ Xs2 ) )
@ X2 )
= ( filter2 @ A
@ ( ^ [Y5: A,Z3: A] : Y5 = Z3
@ X2 )
@ Xs2 ) ) ).
% replicate_length_filter
thf(fact_8173_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_8174_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_8175_inj__on__filter__key__eq,axiom,
! [B: $tType,A: $tType,F2: A > B,Y2: A,Xs2: list @ A] :
( ( inj_on @ A @ B @ F2 @ ( insert @ A @ Y2 @ ( set2 @ A @ Xs2 ) ) )
=> ( ( filter2 @ A
@ ^ [X: A] :
( ( F2 @ Y2 )
= ( F2 @ X ) )
@ Xs2 )
= ( filter2 @ A
@ ( ^ [Y5: A,Z3: A] : Y5 = Z3
@ Y2 )
@ Xs2 ) ) ) ).
% inj_on_filter_key_eq
thf(fact_8176_set__minus__filter__out,axiom,
! [A: $tType,Xs2: list @ A,Y2: A] :
( ( minus_minus @ ( set @ A ) @ ( set2 @ A @ Xs2 ) @ ( insert @ A @ Y2 @ ( bot_bot @ ( set @ A ) ) ) )
= ( set2 @ A
@ ( filter2 @ A
@ ^ [X: A] : X != Y2
@ Xs2 ) ) ) ).
% set_minus_filter_out
thf(fact_8177_upt__filter__extend,axiom,
! [U: nat,U4: nat,P: nat > $o] :
( ( ord_less_eq @ nat @ U @ U4 )
=> ( ! [I2: nat] :
( ( ( ord_less_eq @ nat @ U @ I2 )
& ( ord_less @ nat @ I2 @ U4 ) )
=> ~ ( P @ I2 ) )
=> ( ( filter2 @ nat @ P @ ( upt @ ( zero_zero @ nat ) @ U ) )
= ( filter2 @ nat @ P @ ( upt @ ( zero_zero @ nat ) @ U4 ) ) ) ) ) ).
% upt_filter_extend
thf(fact_8178_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
@ ^ [X: B] : ( ord_less @ A @ ( F4 @ X ) @ ( F4 @ ( nth @ B @ Xs @ ( divide_divide @ nat @ ( size_size @ ( list @ B ) @ Xs ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ Xs ) )
@ ( append @ B
@ ( filter2 @ B
@ ^ [X: B] :
( ( F4 @ X )
= ( 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
@ ^ [X: B] : ( ord_less @ A @ ( F4 @ ( nth @ B @ Xs @ ( divide_divide @ nat @ ( size_size @ ( list @ B ) @ Xs ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( F4 @ X ) )
@ Xs ) ) ) ) ) ) ) ).
% sort_key_by_quicksort
thf(fact_8179_sort__upt,axiom,
! [M: nat,N: nat] :
( ( linorder_sort_key @ nat @ nat
@ ^ [X: nat] : X
@ ( upt @ M @ N ) )
= ( upt @ M @ N ) ) ).
% sort_upt
% Type constructors (1243)
thf(tcon_VEBT__Definitions_OVEBT___Typerep_Otyperep,axiom,
typerep @ vEBT_VEBT ).
thf(tcon_Heap__Time__Monad_OHeap___Typerep_Otyperep_1,axiom,
! [A13: $tType] :
( ( typerep @ A13 )
=> ( typerep @ ( heap_Time_Heap @ A13 ) ) ) ).
thf(tcon_Code__Numeral_Ointeger___Typerep_Otyperep_2,axiom,
typerep @ code_integer ).
thf(tcon_Product__Type_Oprod___Typerep_Otyperep_3,axiom,
! [A13: $tType,A15: $tType] :
( ( ( typerep @ A13 )
& ( typerep @ A15 ) )
=> ( typerep @ ( product_prod @ A13 @ A15 ) ) ) ).
thf(tcon_Numeral__Type_Onum1___Typerep_Otyperep_4,axiom,
typerep @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum0___Typerep_Otyperep_5,axiom,
typerep @ numeral_num0 ).
thf(tcon_Numeral__Type_Obit1___Typerep_Otyperep_6,axiom,
! [A13: $tType] :
( ( typerep @ A13 )
=> ( typerep @ ( numeral_bit1 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Typerep_Otyperep_7,axiom,
! [A13: $tType] :
( ( typerep @ A13 )
=> ( typerep @ ( numeral_bit0 @ A13 ) ) ) ).
thf(tcon_Multiset_Omultiset___Typerep_Otyperep_8,axiom,
! [A13: $tType] :
( ( typerep @ A13 )
=> ( typerep @ ( multiset @ A13 ) ) ) ).
thf(tcon_Extended__Nat_Oenat___Typerep_Otyperep_9,axiom,
typerep @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Complete__Lattices_Ocomplete__linorder,axiom,
comple5582772986160207858norder @ extended_enat ).
thf(tcon_Complex_Ocomplex___Typerep_Otyperep_10,axiom,
typerep @ complex ).
thf(tcon_Uint32_Ouint32___Typerep_Otyperep_11,axiom,
typerep @ uint32 ).
thf(tcon_Option_Ooption___Typerep_Otyperep_12,axiom,
! [A13: $tType] :
( ( typerep @ A13 )
=> ( typerep @ ( option @ A13 ) ) ) ).
thf(tcon_Option_Ooption___Complete__Lattices_Ocomplete__linorder_13,axiom,
! [A13: $tType] :
( ( comple5582772986160207858norder @ A13 )
=> ( comple5582772986160207858norder @ ( option @ A13 ) ) ) ).
thf(tcon_Filter_Ofilter___Typerep_Otyperep_14,axiom,
! [A13: $tType] :
( ( typerep @ A13 )
=> ( typerep @ ( filter @ A13 ) ) ) ).
thf(tcon_Enum_Ofinite__3___Typerep_Otyperep_15,axiom,
typerep @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Complete__Lattices_Ocomplete__linorder_16,axiom,
comple5582772986160207858norder @ finite_3 ).
thf(tcon_Enum_Ofinite__2___Typerep_Otyperep_17,axiom,
typerep @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Complete__Lattices_Ocomplete__linorder_18,axiom,
comple5582772986160207858norder @ finite_2 ).
thf(tcon_Enum_Ofinite__1___Typerep_Otyperep_19,axiom,
typerep @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Complete__Lattices_Ocomplete__linorder_20,axiom,
comple5582772986160207858norder @ finite_1 ).
thf(tcon_String_Ochar___Typerep_Otyperep_21,axiom,
typerep @ char ).
thf(tcon_Word_Oword___Typerep_Otyperep_22,axiom,
! [A13: $tType] :
( ( typerep @ A13 )
=> ( typerep @ ( word @ A13 ) ) ) ).
thf(tcon_Real_Oreal___Typerep_Otyperep_23,axiom,
typerep @ real ).
thf(tcon_List_Olist___Typerep_Otyperep_24,axiom,
! [A13: $tType] :
( ( typerep @ A13 )
=> ( typerep @ ( list @ A13 ) ) ) ).
thf(tcon_HOL_Obool___Typerep_Otyperep_25,axiom,
typerep @ $o ).
thf(tcon_Set_Oset___Typerep_Otyperep_26,axiom,
! [A13: $tType] :
( ( typerep @ A13 )
=> ( typerep @ ( set @ A13 ) ) ) ).
thf(tcon_Rat_Orat___Typerep_Otyperep_27,axiom,
typerep @ rat ).
thf(tcon_Num_Onum___Typerep_Otyperep_28,axiom,
typerep @ num ).
thf(tcon_Nat_Onat___Typerep_Otyperep_29,axiom,
typerep @ nat ).
thf(tcon_Int_Oint___Typerep_Otyperep_30,axiom,
typerep @ int ).
thf(tcon_fun___Typerep_Otyperep_31,axiom,
! [A13: $tType,A15: $tType] :
( ( ( typerep @ A13 )
& ( typerep @ A15 ) )
=> ( typerep @ ( A13 > A15 ) ) ) ).
thf(tcon_fun___Boolean__Algebras_Oboolean__algebra,axiom,
! [A13: $tType,A15: $tType] :
( ( boolea8198339166811842893lgebra @ A15 )
=> ( boolea8198339166811842893lgebra @ ( A13 > A15 ) ) ) ).
thf(tcon_fun___Code__Evaluation_Oterm__of,axiom,
! [A13: $tType,A15: $tType] :
( ( ( typerep @ A13 )
& ( typerep @ A15 ) )
=> ( code_term_of @ ( A13 > A15 ) ) ) ).
thf(tcon_fun___Orderings_Oorder__top,axiom,
! [A13: $tType,A15: $tType] :
( ( order_top @ A15 )
=> ( order_top @ ( A13 > A15 ) ) ) ).
thf(tcon_fun___Orderings_Oorder__bot,axiom,
! [A13: $tType,A15: $tType] :
( ( order_bot @ A15 )
=> ( order_bot @ ( A13 > A15 ) ) ) ).
thf(tcon_fun___Orderings_Opreorder,axiom,
! [A13: $tType,A15: $tType] :
( ( preorder @ A15 )
=> ( preorder @ ( A13 > A15 ) ) ) ).
thf(tcon_fun___Finite__Set_Ofinite,axiom,
! [A13: $tType,A15: $tType] :
( ( ( finite_finite @ A13 )
& ( finite_finite @ A15 ) )
=> ( finite_finite @ ( A13 > A15 ) ) ) ).
thf(tcon_fun___Orderings_Oorder,axiom,
! [A13: $tType,A15: $tType] :
( ( order @ A15 )
=> ( order @ ( A13 > A15 ) ) ) ).
thf(tcon_fun___Orderings_Oord,axiom,
! [A13: $tType,A15: $tType] :
( ( ord @ A15 )
=> ( ord @ ( A13 > A15 ) ) ) ).
thf(tcon_fun___Groups_Ouminus,axiom,
! [A13: $tType,A15: $tType] :
( ( uminus @ A15 )
=> ( uminus @ ( A13 > A15 ) ) ) ).
thf(tcon_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___Topological__Spaces_Odiscrete__topology,axiom,
topolo8865339358273720382pology @ int ).
thf(tcon_Int_Oint___Quickcheck__Narrowing_Opartial__term__of,axiom,
quickc6926020345158392990erm_of @ int ).
thf(tcon_Int_Oint___Limits_Otopological__comm__monoid__mult,axiom,
topolo4987421752381908075d_mult @ int ).
thf(tcon_Int_Oint___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___Topological__Spaces_Ot1__space,axiom,
topological_t1_space @ int ).
thf(tcon_Int_Oint___Rings_Oordered__comm__semiring,axiom,
ordere2520102378445227354miring @ int ).
thf(tcon_Int_Oint___Rings_Olinordered__semiring__1,axiom,
linord6961819062388156250ring_1 @ int ).
thf(tcon_Int_Oint___Groups_Oordered__ab__group__add,axiom,
ordered_ab_group_add @ int ).
thf(tcon_Int_Oint___Groups_Ocancel__semigroup__add,axiom,
cancel_semigroup_add @ int ).
thf(tcon_Int_Oint___Rings_Olinordered__semiring,axiom,
linordered_semiring @ int ).
thf(tcon_Int_Oint___Least__significant__bit_Olsb,axiom,
least_6119777620449941438nt_lsb @ int ).
thf(tcon_Int_Oint___Rings_Oordered__semiring__0,axiom,
ordered_semiring_0 @ int ).
thf(tcon_Int_Oint___Rings_Olinordered__semidom,axiom,
linordered_semidom @ int ).
thf(tcon_Int_Oint___Groups_Oab__semigroup__mult,axiom,
ab_semigroup_mult @ int ).
thf(tcon_Int_Oint___Rings_Osemiring__1__cancel,axiom,
semiring_1_cancel @ int ).
thf(tcon_Int_Oint___Rings_Oalgebraic__semidom,axiom,
algebraic_semidom @ int ).
thf(tcon_Int_Oint___Groups_Ocomm__monoid__mult,axiom,
comm_monoid_mult @ int ).
thf(tcon_Int_Oint___Groups_Oab__semigroup__add,axiom,
ab_semigroup_add @ int ).
thf(tcon_Int_Oint___Generic__set__bit_Oset__bit,axiom,
generic_set_set_bit @ int ).
thf(tcon_Int_Oint___Code__Evaluation_Oterm__of_32,axiom,
code_term_of @ int ).
thf(tcon_Int_Oint___Rings_Oordered__semiring,axiom,
ordered_semiring @ int ).
thf(tcon_Int_Oint___Rings_Oordered__ring__abs,axiom,
ordered_ring_abs @ int ).
thf(tcon_Int_Oint___Parity_Osemiring__parity,axiom,
semiring_parity @ int ).
thf(tcon_Int_Oint___Groups_Ocomm__monoid__add,axiom,
comm_monoid_add @ int ).
thf(tcon_Int_Oint___Rings_Osemiring__modulo,axiom,
semiring_modulo @ int ).
thf(tcon_Int_Oint___Rings_Olinordered__ring,axiom,
linordered_ring @ int ).
thf(tcon_Int_Oint___Rings_Olinordered__idom,axiom,
linordered_idom @ int ).
thf(tcon_Int_Oint___Rings_Ocomm__semiring__1,axiom,
comm_semiring_1 @ int ).
thf(tcon_Int_Oint___Rings_Ocomm__semiring__0,axiom,
comm_semiring_0 @ int ).
thf(tcon_Int_Oint___Groups_Osemigroup__mult,axiom,
semigroup_mult @ int ).
thf(tcon_Int_Oint___Rings_Osemidom__modulo,axiom,
semidom_modulo @ int ).
thf(tcon_Int_Oint___Rings_Osemidom__divide,axiom,
semidom_divide @ int ).
thf(tcon_Int_Oint___Num_Osemiring__numeral,axiom,
semiring_numeral @ int ).
thf(tcon_Int_Oint___Groups_Osemigroup__add,axiom,
semigroup_add @ int ).
thf(tcon_Int_Oint___Rings_Ozero__less__one,axiom,
zero_less_one @ int ).
thf(tcon_Int_Oint___Rings_Ocomm__semiring,axiom,
comm_semiring @ int ).
thf(tcon_Int_Oint___Nat_Osemiring__char__0,axiom,
semiring_char_0 @ int ).
thf(tcon_Int_Oint___Groups_Oab__group__add,axiom,
ab_group_add @ int ).
thf(tcon_Int_Oint___Rings_Ozero__neq__one,axiom,
zero_neq_one @ int ).
thf(tcon_Int_Oint___Rings_Oordered__ring,axiom,
ordered_ring @ int ).
thf(tcon_Int_Oint___Rings_Oidom__abs__sgn,axiom,
idom_abs_sgn @ int ).
thf(tcon_Int_Oint___Parity_Oring__parity,axiom,
ring_parity @ int ).
thf(tcon_Int_Oint___Orderings_Opreorder_33,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_34,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_35,axiom,
ord @ int ).
thf(tcon_Int_Oint___Groups_Ouminus_36,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___Power_Opower,axiom,
power @ int ).
thf(tcon_Int_Oint___Num_Onumeral,axiom,
numeral @ int ).
thf(tcon_Int_Oint___Groups_Ozero,axiom,
zero @ int ).
thf(tcon_Int_Oint___Groups_Oplus,axiom,
plus @ int ).
thf(tcon_Int_Oint___Rings_Oring,axiom,
ring @ int ).
thf(tcon_Int_Oint___Rings_Oidom,axiom,
idom @ int ).
thf(tcon_Int_Oint___Groups_Oone,axiom,
one @ int ).
thf(tcon_Int_Oint___Rings_Odvd,axiom,
dvd @ int ).
thf(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_37,axiom,
condit6923001295902523014norder @ nat ).
thf(tcon_Nat_Onat___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_38,axiom,
bit_un5681908812861735899ations @ nat ).
thf(tcon_Nat_Onat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_39,axiom,
semiri1453513574482234551roduct @ nat ).
thf(tcon_Nat_Onat___Euclidean__Division_Ounique__euclidean__semiring__with__nat_40,axiom,
euclid5411537665997757685th_nat @ nat ).
thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__monoid__add__imp__le_41,axiom,
ordere1937475149494474687imp_le @ nat ).
thf(tcon_Nat_Onat___Euclidean__Division_Ounique__euclidean__semiring_42,axiom,
euclid3128863361964157862miring @ nat ).
thf(tcon_Nat_Onat___Euclidean__Division_Oeuclidean__semiring__cancel_43,axiom,
euclid4440199948858584721cancel @ nat ).
thf(tcon_Nat_Onat___Divides_Ounique__euclidean__semiring__numeral_44,axiom,
unique1627219031080169319umeral @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors__cancel_45,axiom,
semiri6575147826004484403cancel @ nat ).
thf(tcon_Nat_Onat___Groups_Ostrict__ordered__ab__semigroup__add_46,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_47,axiom,
ordere580206878836729694up_add @ nat ).
thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le_48,axiom,
ordere2412721322843649153imp_le @ nat ).
thf(tcon_Nat_Onat___Bit__Operations_Osemiring__bit__operations_49,axiom,
bit_se359711467146920520ations @ nat ).
thf(tcon_Nat_Onat___Rings_Olinordered__comm__semiring__strict_50,axiom,
linord2810124833399127020strict @ nat ).
thf(tcon_Nat_Onat___Groups_Ostrict__ordered__comm__monoid__add_51,axiom,
strict7427464778891057005id_add @ nat ).
thf(tcon_Nat_Onat___Groups_Oordered__cancel__comm__monoid__add_52,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_53,axiom,
euclid3725896446679973847miring @ nat ).
thf(tcon_Nat_Onat___Topological__Spaces_Otopological__space_54,axiom,
topolo4958980785337419405_space @ nat ).
thf(tcon_Nat_Onat___Topological__Spaces_Olinorder__topology_55,axiom,
topolo1944317154257567458pology @ nat ).
thf(tcon_Nat_Onat___Topological__Spaces_Odiscrete__topology_56,axiom,
topolo8865339358273720382pology @ nat ).
thf(tcon_Nat_Onat___Quickcheck__Narrowing_Opartial__term__of_57,axiom,
quickc6926020345158392990erm_of @ nat ).
thf(tcon_Nat_Onat___Limits_Otopological__comm__monoid__mult_58,axiom,
topolo4987421752381908075d_mult @ nat ).
thf(tcon_Nat_Onat___Limits_Otopological__comm__monoid__add_59,axiom,
topolo5987344860129210374id_add @ nat ).
thf(tcon_Nat_Onat___Groups_Olinordered__ab__semigroup__add_60,axiom,
linord4140545234300271783up_add @ nat ).
thf(tcon_Nat_Onat___Topological__Spaces_Oorder__topology_61,axiom,
topolo2564578578187576103pology @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__1__no__zero__divisors_62,axiom,
semiri2026040879449505780visors @ nat ).
thf(tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring_63,axiom,
linord181362715937106298miring @ nat ).
thf(tcon_Nat_Onat___Limits_Otopological__semigroup__mult_64,axiom,
topolo4211221413907600880p_mult @ nat ).
thf(tcon_Nat_Onat___Rings_Olinordered__semiring__strict_65,axiom,
linord8928482502909563296strict @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors_66,axiom,
semiri3467727345109120633visors @ nat ).
thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add_67,axiom,
ordere6658533253407199908up_add @ nat ).
thf(tcon_Nat_Onat___Limits_Otopological__monoid__mult_68,axiom,
topolo1898628316856586783d_mult @ nat ).
thf(tcon_Nat_Onat___Groups_Oordered__comm__monoid__add_69,axiom,
ordere6911136660526730532id_add @ nat ).
thf(tcon_Nat_Onat___Groups_Ocancel__ab__semigroup__add_70,axiom,
cancel2418104881723323429up_add @ nat ).
thf(tcon_Nat_Onat___Limits_Otopological__monoid__add_71,axiom,
topolo6943815403480290642id_add @ nat ).
thf(tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add_72,axiom,
cancel1802427076303600483id_add @ nat ).
thf(tcon_Nat_Onat___Rings_Ocomm__semiring__1__cancel_73,axiom,
comm_s4317794764714335236cancel @ nat ).
thf(tcon_Nat_Onat___Bit__Operations_Osemiring__bits_74,axiom,
bit_semiring_bits @ nat ).
thf(tcon_Nat_Onat___Topological__Spaces_Ot2__space_75,axiom,
topological_t2_space @ nat ).
thf(tcon_Nat_Onat___Topological__Spaces_Ot1__space_76,axiom,
topological_t1_space @ nat ).
thf(tcon_Nat_Onat___Rings_Oordered__comm__semiring_77,axiom,
ordere2520102378445227354miring @ nat ).
thf(tcon_Nat_Onat___Groups_Ocancel__semigroup__add_78,axiom,
cancel_semigroup_add @ nat ).
thf(tcon_Nat_Onat___Rings_Olinordered__semiring_79,axiom,
linordered_semiring @ nat ).
thf(tcon_Nat_Onat___Rings_Oordered__semiring__0_80,axiom,
ordered_semiring_0 @ nat ).
thf(tcon_Nat_Onat___Rings_Olinordered__semidom_81,axiom,
linordered_semidom @ nat ).
thf(tcon_Nat_Onat___Groups_Oab__semigroup__mult_82,axiom,
ab_semigroup_mult @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__1__cancel_83,axiom,
semiring_1_cancel @ nat ).
thf(tcon_Nat_Onat___Rings_Oalgebraic__semidom_84,axiom,
algebraic_semidom @ nat ).
thf(tcon_Nat_Onat___Groups_Ocomm__monoid__mult_85,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_86,axiom,
ab_semigroup_add @ nat ).
thf(tcon_Nat_Onat___Code__Evaluation_Oterm__of_87,axiom,
code_term_of @ nat ).
thf(tcon_Nat_Onat___Rings_Oordered__semiring_88,axiom,
ordered_semiring @ nat ).
thf(tcon_Nat_Onat___Parity_Osemiring__parity_89,axiom,
semiring_parity @ nat ).
thf(tcon_Nat_Onat___Groups_Ocomm__monoid__add_90,axiom,
comm_monoid_add @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__modulo_91,axiom,
semiring_modulo @ nat ).
thf(tcon_Nat_Onat___Rings_Ocomm__semiring__1_92,axiom,
comm_semiring_1 @ nat ).
thf(tcon_Nat_Onat___Rings_Ocomm__semiring__0_93,axiom,
comm_semiring_0 @ nat ).
thf(tcon_Nat_Onat___Groups_Osemigroup__mult_94,axiom,
semigroup_mult @ nat ).
thf(tcon_Nat_Onat___Rings_Osemidom__modulo_95,axiom,
semidom_modulo @ nat ).
thf(tcon_Nat_Onat___Rings_Osemidom__divide_96,axiom,
semidom_divide @ nat ).
thf(tcon_Nat_Onat___Num_Osemiring__numeral_97,axiom,
semiring_numeral @ nat ).
thf(tcon_Nat_Onat___Groups_Osemigroup__add_98,axiom,
semigroup_add @ nat ).
thf(tcon_Nat_Onat___Rings_Ozero__less__one_99,axiom,
zero_less_one @ nat ).
thf(tcon_Nat_Onat___Rings_Ocomm__semiring_100,axiom,
comm_semiring @ nat ).
thf(tcon_Nat_Onat___Orderings_Owellorder,axiom,
wellorder @ nat ).
thf(tcon_Nat_Onat___Orderings_Oorder__bot_101,axiom,
order_bot @ nat ).
thf(tcon_Nat_Onat___Nat_Osemiring__char__0_102,axiom,
semiring_char_0 @ nat ).
thf(tcon_Nat_Onat___Rings_Ozero__neq__one_103,axiom,
zero_neq_one @ nat ).
thf(tcon_Nat_Onat___Orderings_Opreorder_104,axiom,
preorder @ nat ).
thf(tcon_Nat_Onat___Orderings_Olinorder_105,axiom,
linorder @ nat ).
thf(tcon_Nat_Onat___Groups_Omonoid__mult_106,axiom,
monoid_mult @ nat ).
thf(tcon_Nat_Onat___Groups_Omonoid__add_107,axiom,
monoid_add @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__1_108,axiom,
semiring_1 @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__0_109,axiom,
semiring_0 @ nat ).
thf(tcon_Nat_Onat___Orderings_Ono__top_110,axiom,
no_top @ nat ).
thf(tcon_Nat_Onat___GCD_Osemiring__gcd_111,axiom,
semiring_gcd @ nat ).
thf(tcon_Nat_Onat___Rings_Omult__zero_112,axiom,
mult_zero @ nat ).
thf(tcon_Nat_Onat___Orderings_Oorder_113,axiom,
order @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring_114,axiom,
semiring @ nat ).
thf(tcon_Nat_Onat___Rings_Osemidom_115,axiom,
semidom @ nat ).
thf(tcon_Nat_Onat___Orderings_Oord_116,axiom,
ord @ nat ).
thf(tcon_Nat_Onat___Power_Opower_117,axiom,
power @ nat ).
thf(tcon_Nat_Onat___Num_Onumeral_118,axiom,
numeral @ nat ).
thf(tcon_Nat_Onat___Groups_Ozero_119,axiom,
zero @ nat ).
thf(tcon_Nat_Onat___Groups_Oplus_120,axiom,
plus @ nat ).
thf(tcon_Nat_Onat___Groups_Oone_121,axiom,
one @ nat ).
thf(tcon_Nat_Onat___Rings_Odvd_122,axiom,
dvd @ nat ).
thf(tcon_Nat_Onat___Nat_Osize,axiom,
size @ nat ).
thf(tcon_Num_Onum___Quickcheck__Narrowing_Opartial__term__of_123,axiom,
quickc6926020345158392990erm_of @ num ).
thf(tcon_Num_Onum___Code__Evaluation_Oterm__of_124,axiom,
code_term_of @ num ).
thf(tcon_Num_Onum___Orderings_Opreorder_125,axiom,
preorder @ num ).
thf(tcon_Num_Onum___Orderings_Olinorder_126,axiom,
linorder @ num ).
thf(tcon_Num_Onum___Orderings_Oorder_127,axiom,
order @ num ).
thf(tcon_Num_Onum___Orderings_Oord_128,axiom,
ord @ num ).
thf(tcon_Num_Onum___Groups_Oplus_129,axiom,
plus @ num ).
thf(tcon_Num_Onum___Nat_Osize_130,axiom,
size @ num ).
thf(tcon_Rat_Orat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_131,axiom,
semiri1453513574482234551roduct @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__monoid__add__imp__le_132,axiom,
ordere1937475149494474687imp_le @ rat ).
thf(tcon_Rat_Orat___Rings_Osemiring__no__zero__divisors__cancel_133,axiom,
semiri6575147826004484403cancel @ rat ).
thf(tcon_Rat_Orat___Groups_Ostrict__ordered__ab__semigroup__add_134,axiom,
strict9044650504122735259up_add @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__cancel__ab__semigroup__add_135,axiom,
ordere580206878836729694up_add @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__add__imp__le_136,axiom,
ordere2412721322843649153imp_le @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__comm__semiring__strict_137,axiom,
linord2810124833399127020strict @ rat ).
thf(tcon_Rat_Orat___Groups_Ostrict__ordered__comm__monoid__add_138,axiom,
strict7427464778891057005id_add @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__cancel__comm__monoid__add_139,axiom,
ordere8940638589300402666id_add @ rat ).
thf(tcon_Rat_Orat___Quickcheck__Narrowing_Opartial__term__of_140,axiom,
quickc6926020345158392990erm_of @ rat ).
thf(tcon_Rat_Orat___Archimedean__Field_Oarchimedean__field,axiom,
archim462609752435547400_field @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__semiring__1__strict_141,axiom,
linord715952674999750819strict @ rat ).
thf(tcon_Rat_Orat___Orderings_Ounbounded__dense__linorder,axiom,
unboun7993243217541854897norder @ rat ).
thf(tcon_Rat_Orat___Groups_Olinordered__ab__semigroup__add_142,axiom,
linord4140545234300271783up_add @ rat ).
thf(tcon_Rat_Orat___Rings_Osemiring__1__no__zero__divisors_143,axiom,
semiri2026040879449505780visors @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__nonzero__semiring_144,axiom,
linord181362715937106298miring @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__semiring__strict_145,axiom,
linord8928482502909563296strict @ rat ).
thf(tcon_Rat_Orat___Rings_Osemiring__no__zero__divisors_146,axiom,
semiri3467727345109120633visors @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__add_147,axiom,
ordere6658533253407199908up_add @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__ab__group__add__abs_148,axiom,
ordere166539214618696060dd_abs @ rat ).
thf(tcon_Rat_Orat___Archimedean__Field_Ofloor__ceiling,axiom,
archim2362893244070406136eiling @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__comm__monoid__add_149,axiom,
ordere6911136660526730532id_add @ rat ).
thf(tcon_Rat_Orat___Groups_Olinordered__ab__group__add_150,axiom,
linord5086331880401160121up_add @ rat ).
thf(tcon_Rat_Orat___Groups_Ocancel__ab__semigroup__add_151,axiom,
cancel2418104881723323429up_add @ rat ).
thf(tcon_Rat_Orat___Rings_Oring__1__no__zero__divisors_152,axiom,
ring_15535105094025558882visors @ rat ).
thf(tcon_Rat_Orat___Groups_Ocancel__comm__monoid__add_153,axiom,
cancel1802427076303600483id_add @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__ring__strict_154,axiom,
linord4710134922213307826strict @ rat ).
thf(tcon_Rat_Orat___Rings_Ocomm__semiring__1__cancel_155,axiom,
comm_s4317794764714335236cancel @ rat ).
thf(tcon_Rat_Orat___Rings_Oordered__comm__semiring_156,axiom,
ordere2520102378445227354miring @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__semiring__1_157,axiom,
linord6961819062388156250ring_1 @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__ab__group__add_158,axiom,
ordered_ab_group_add @ rat ).
thf(tcon_Rat_Orat___Groups_Ocancel__semigroup__add_159,axiom,
cancel_semigroup_add @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__semiring_160,axiom,
linordered_semiring @ rat ).
thf(tcon_Rat_Orat___Rings_Oordered__semiring__0_161,axiom,
ordered_semiring_0 @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__semidom_162,axiom,
linordered_semidom @ rat ).
thf(tcon_Rat_Orat___Orderings_Odense__linorder,axiom,
dense_linorder @ rat ).
thf(tcon_Rat_Orat___Groups_Oab__semigroup__mult_163,axiom,
ab_semigroup_mult @ rat ).
thf(tcon_Rat_Orat___Rings_Osemiring__1__cancel_164,axiom,
semiring_1_cancel @ rat ).
thf(tcon_Rat_Orat___Groups_Ocomm__monoid__mult_165,axiom,
comm_monoid_mult @ rat ).
thf(tcon_Rat_Orat___Groups_Oab__semigroup__add_166,axiom,
ab_semigroup_add @ rat ).
thf(tcon_Rat_Orat___Fields_Olinordered__field,axiom,
linordered_field @ rat ).
thf(tcon_Rat_Orat___Code__Evaluation_Oterm__of_167,axiom,
code_term_of @ rat ).
thf(tcon_Rat_Orat___Rings_Oordered__semiring_168,axiom,
ordered_semiring @ rat ).
thf(tcon_Rat_Orat___Rings_Oordered__ring__abs_169,axiom,
ordered_ring_abs @ rat ).
thf(tcon_Rat_Orat___Groups_Ocomm__monoid__add_170,axiom,
comm_monoid_add @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__ring_171,axiom,
linordered_ring @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__idom_172,axiom,
linordered_idom @ rat ).
thf(tcon_Rat_Orat___Rings_Ocomm__semiring__1_173,axiom,
comm_semiring_1 @ rat ).
thf(tcon_Rat_Orat___Rings_Ocomm__semiring__0_174,axiom,
comm_semiring_0 @ rat ).
thf(tcon_Rat_Orat___Orderings_Odense__order,axiom,
dense_order @ rat ).
thf(tcon_Rat_Orat___Groups_Osemigroup__mult_175,axiom,
semigroup_mult @ rat ).
thf(tcon_Rat_Orat___Rings_Osemidom__divide_176,axiom,
semidom_divide @ rat ).
thf(tcon_Rat_Orat___Num_Osemiring__numeral_177,axiom,
semiring_numeral @ rat ).
thf(tcon_Rat_Orat___Groups_Osemigroup__add_178,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_179,axiom,
zero_less_one @ rat ).
thf(tcon_Rat_Orat___Rings_Ocomm__semiring_180,axiom,
comm_semiring @ rat ).
thf(tcon_Rat_Orat___Nat_Osemiring__char__0_181,axiom,
semiring_char_0 @ rat ).
thf(tcon_Rat_Orat___Groups_Oab__group__add_182,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_183,axiom,
zero_neq_one @ rat ).
thf(tcon_Rat_Orat___Rings_Oordered__ring_184,axiom,
ordered_ring @ rat ).
thf(tcon_Rat_Orat___Rings_Oidom__abs__sgn_185,axiom,
idom_abs_sgn @ rat ).
thf(tcon_Rat_Orat___Orderings_Opreorder_186,axiom,
preorder @ rat ).
thf(tcon_Rat_Orat___Orderings_Olinorder_187,axiom,
linorder @ rat ).
thf(tcon_Rat_Orat___Groups_Omonoid__mult_188,axiom,
monoid_mult @ rat ).
thf(tcon_Rat_Orat___Rings_Oidom__divide_189,axiom,
idom_divide @ rat ).
thf(tcon_Rat_Orat___Rings_Ocomm__ring__1_190,axiom,
comm_ring_1 @ rat ).
thf(tcon_Rat_Orat___Groups_Omonoid__add_191,axiom,
monoid_add @ rat ).
thf(tcon_Rat_Orat___Rings_Osemiring__1_192,axiom,
semiring_1 @ rat ).
thf(tcon_Rat_Orat___Rings_Osemiring__0_193,axiom,
semiring_0 @ rat ).
thf(tcon_Rat_Orat___Orderings_Ono__top_194,axiom,
no_top @ rat ).
thf(tcon_Rat_Orat___Orderings_Ono__bot_195,axiom,
no_bot @ rat ).
thf(tcon_Rat_Orat___Groups_Ogroup__add_196,axiom,
group_add @ rat ).
thf(tcon_Rat_Orat___Rings_Omult__zero_197,axiom,
mult_zero @ rat ).
thf(tcon_Rat_Orat___Rings_Ocomm__ring_198,axiom,
comm_ring @ rat ).
thf(tcon_Rat_Orat___Orderings_Oorder_199,axiom,
order @ rat ).
thf(tcon_Rat_Orat___Num_Oneg__numeral_200,axiom,
neg_numeral @ rat ).
thf(tcon_Rat_Orat___Nat_Oring__char__0_201,axiom,
ring_char_0 @ rat ).
thf(tcon_Rat_Orat___Rings_Osemiring_202,axiom,
semiring @ rat ).
thf(tcon_Rat_Orat___Fields_Oinverse,axiom,
inverse @ rat ).
thf(tcon_Rat_Orat___Rings_Osemidom_203,axiom,
semidom @ rat ).
thf(tcon_Rat_Orat___Orderings_Oord_204,axiom,
ord @ rat ).
thf(tcon_Rat_Orat___Groups_Ouminus_205,axiom,
uminus @ rat ).
thf(tcon_Rat_Orat___Rings_Oring__1_206,axiom,
ring_1 @ rat ).
thf(tcon_Rat_Orat___Rings_Oabs__if_207,axiom,
abs_if @ rat ).
thf(tcon_Rat_Orat___Fields_Ofield,axiom,
field @ rat ).
thf(tcon_Rat_Orat___Power_Opower_208,axiom,
power @ rat ).
thf(tcon_Rat_Orat___Num_Onumeral_209,axiom,
numeral @ rat ).
thf(tcon_Rat_Orat___Groups_Ozero_210,axiom,
zero @ rat ).
thf(tcon_Rat_Orat___Groups_Oplus_211,axiom,
plus @ rat ).
thf(tcon_Rat_Orat___Rings_Oring_212,axiom,
ring @ rat ).
thf(tcon_Rat_Orat___Rings_Oidom_213,axiom,
idom @ rat ).
thf(tcon_Rat_Orat___Groups_Oone_214,axiom,
one @ rat ).
thf(tcon_Rat_Orat___Rings_Odvd_215,axiom,
dvd @ rat ).
thf(tcon_Set_Oset___Quickcheck__Narrowing_Opartial__term__of_216,axiom,
! [A13: $tType] :
( ( typerep @ A13 )
=> ( quickc6926020345158392990erm_of @ ( set @ A13 ) ) ) ).
thf(tcon_Set_Oset___Boolean__Algebras_Oboolean__algebra_217,axiom,
! [A13: $tType] : ( boolea8198339166811842893lgebra @ ( set @ A13 ) ) ).
thf(tcon_Set_Oset___Code__Evaluation_Oterm__of_218,axiom,
! [A13: $tType] :
( ( typerep @ A13 )
=> ( code_term_of @ ( set @ A13 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder__top_219,axiom,
! [A13: $tType] : ( order_top @ ( set @ A13 ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder__bot_220,axiom,
! [A13: $tType] : ( order_bot @ ( set @ A13 ) ) ).
thf(tcon_Set_Oset___Orderings_Opreorder_221,axiom,
! [A13: $tType] : ( preorder @ ( set @ A13 ) ) ).
thf(tcon_Set_Oset___Finite__Set_Ofinite_222,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( finite_finite @ ( set @ A13 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder_223,axiom,
! [A13: $tType] : ( order @ ( set @ A13 ) ) ).
thf(tcon_Set_Oset___Orderings_Oord_224,axiom,
! [A13: $tType] : ( ord @ ( set @ A13 ) ) ).
thf(tcon_Set_Oset___Groups_Ouminus_225,axiom,
! [A13: $tType] : ( uminus @ ( set @ A13 ) ) ).
thf(tcon_HOL_Obool___Topological__Spaces_Otopological__space_226,axiom,
topolo4958980785337419405_space @ $o ).
thf(tcon_HOL_Obool___Topological__Spaces_Olinorder__topology_227,axiom,
topolo1944317154257567458pology @ $o ).
thf(tcon_HOL_Obool___Topological__Spaces_Odiscrete__topology_228,axiom,
topolo8865339358273720382pology @ $o ).
thf(tcon_HOL_Obool___Quickcheck__Narrowing_Opartial__term__of_229,axiom,
quickc6926020345158392990erm_of @ $o ).
thf(tcon_HOL_Obool___Topological__Spaces_Oorder__topology_230,axiom,
topolo2564578578187576103pology @ $o ).
thf(tcon_HOL_Obool___Boolean__Algebras_Oboolean__algebra_231,axiom,
boolea8198339166811842893lgebra @ $o ).
thf(tcon_HOL_Obool___Topological__Spaces_Ot2__space_232,axiom,
topological_t2_space @ $o ).
thf(tcon_HOL_Obool___Topological__Spaces_Ot1__space_233,axiom,
topological_t1_space @ $o ).
thf(tcon_HOL_Obool___Code__Evaluation_Oterm__of_234,axiom,
code_term_of @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder__top_235,axiom,
order_top @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder__bot_236,axiom,
order_bot @ $o ).
thf(tcon_HOL_Obool___Orderings_Opreorder_237,axiom,
preorder @ $o ).
thf(tcon_HOL_Obool___Orderings_Olinorder_238,axiom,
linorder @ $o ).
thf(tcon_HOL_Obool___Finite__Set_Ofinite_239,axiom,
finite_finite @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder_240,axiom,
order @ $o ).
thf(tcon_HOL_Obool___Orderings_Oord_241,axiom,
ord @ $o ).
thf(tcon_HOL_Obool___Groups_Ouminus_242,axiom,
uminus @ $o ).
thf(tcon_List_Olist___Quickcheck__Narrowing_Opartial__term__of_243,axiom,
! [A13: $tType] :
( ( typerep @ A13 )
=> ( quickc6926020345158392990erm_of @ ( list @ A13 ) ) ) ).
thf(tcon_List_Olist___Code__Evaluation_Oterm__of_244,axiom,
! [A13: $tType] :
( ( typerep @ A13 )
=> ( code_term_of @ ( list @ A13 ) ) ) ).
thf(tcon_List_Olist___Nat_Osize_245,axiom,
! [A13: $tType] : ( size @ ( list @ A13 ) ) ).
thf(tcon_Real_Oreal___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_246,axiom,
condit6923001295902523014norder @ real ).
thf(tcon_Real_Oreal___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_247,axiom,
semiri1453513574482234551roduct @ real ).
thf(tcon_Real_Oreal___Groups_Oordered__ab__semigroup__monoid__add__imp__le_248,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_249,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_250,axiom,
strict9044650504122735259up_add @ real ).
thf(tcon_Real_Oreal___Groups_Oordered__cancel__ab__semigroup__add_251,axiom,
ordere580206878836729694up_add @ real ).
thf(tcon_Real_Oreal___Groups_Oordered__ab__semigroup__add__imp__le_252,axiom,
ordere2412721322843649153imp_le @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__comm__semiring__strict_253,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_254,axiom,
strict7427464778891057005id_add @ real ).
thf(tcon_Real_Oreal___Groups_Oordered__cancel__comm__monoid__add_255,axiom,
ordere8940638589300402666id_add @ real ).
thf(tcon_Real_Oreal___Topological__Spaces_Otopological__space_256,axiom,
topolo4958980785337419405_space @ real ).
thf(tcon_Real_Oreal___Topological__Spaces_Olinorder__topology_257,axiom,
topolo1944317154257567458pology @ real ).
thf(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__normed__field,axiom,
real_V3459762299906320749_field @ real ).
thf(tcon_Real_Oreal___Quickcheck__Narrowing_Opartial__term__of_258,axiom,
quickc6926020345158392990erm_of @ real ).
thf(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__div__algebra,axiom,
real_V5047593784448816457lgebra @ real ).
thf(tcon_Real_Oreal___Archimedean__Field_Oarchimedean__field_259,axiom,
archim462609752435547400_field @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__semiring__1__strict_260,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_261,axiom,
unboun7993243217541854897norder @ real ).
thf(tcon_Real_Oreal___Limits_Otopological__comm__monoid__add_262,axiom,
topolo5987344860129210374id_add @ real ).
thf(tcon_Real_Oreal___Groups_Olinordered__ab__semigroup__add_263,axiom,
linord4140545234300271783up_add @ real ).
thf(tcon_Real_Oreal___Topological__Spaces_Oorder__topology_264,axiom,
topolo2564578578187576103pology @ real ).
thf(tcon_Real_Oreal___Rings_Osemiring__1__no__zero__divisors_265,axiom,
semiri2026040879449505780visors @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__nonzero__semiring_266,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_267,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_268,axiom,
linord8928482502909563296strict @ real ).
thf(tcon_Real_Oreal___Rings_Osemiring__no__zero__divisors_269,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_270,axiom,
ordere6658533253407199908up_add @ real ).
thf(tcon_Real_Oreal___Groups_Oordered__ab__group__add__abs_271,axiom,
ordere166539214618696060dd_abs @ real ).
thf(tcon_Real_Oreal___Archimedean__Field_Ofloor__ceiling_272,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_273,axiom,
ordere6911136660526730532id_add @ real ).
thf(tcon_Real_Oreal___Groups_Olinordered__ab__group__add_274,axiom,
linord5086331880401160121up_add @ real ).
thf(tcon_Real_Oreal___Groups_Ocancel__ab__semigroup__add_275,axiom,
cancel2418104881723323429up_add @ real ).
thf(tcon_Real_Oreal___Rings_Oring__1__no__zero__divisors_276,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_277,axiom,
topolo6943815403480290642id_add @ real ).
thf(tcon_Real_Oreal___Groups_Ocancel__comm__monoid__add_278,axiom,
cancel1802427076303600483id_add @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__ring__strict_279,axiom,
linord4710134922213307826strict @ real ).
thf(tcon_Real_Oreal___Rings_Ocomm__semiring__1__cancel_280,axiom,
comm_s4317794764714335236cancel @ real ).
thf(tcon_Real_Oreal___Limits_Otopological__group__add,axiom,
topolo1633459387980952147up_add @ real ).
thf(tcon_Real_Oreal___Topological__Spaces_Ot2__space_281,axiom,
topological_t2_space @ real ).
thf(tcon_Real_Oreal___Topological__Spaces_Ot1__space_282,axiom,
topological_t1_space @ real ).
thf(tcon_Real_Oreal___Rings_Oordered__comm__semiring_283,axiom,
ordere2520102378445227354miring @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__semiring__1_284,axiom,
linord6961819062388156250ring_1 @ real ).
thf(tcon_Real_Oreal___Groups_Oordered__ab__group__add_285,axiom,
ordered_ab_group_add @ real ).
thf(tcon_Real_Oreal___Groups_Ocancel__semigroup__add_286,axiom,
cancel_semigroup_add @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__semiring_287,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_288,axiom,
ordered_semiring_0 @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__semidom_289,axiom,
linordered_semidom @ real ).
thf(tcon_Real_Oreal___Orderings_Odense__linorder_290,axiom,
dense_linorder @ real ).
thf(tcon_Real_Oreal___Groups_Oab__semigroup__mult_291,axiom,
ab_semigroup_mult @ real ).
thf(tcon_Real_Oreal___Rings_Osemiring__1__cancel_292,axiom,
semiring_1_cancel @ real ).
thf(tcon_Real_Oreal___Groups_Ocomm__monoid__mult_293,axiom,
comm_monoid_mult @ real ).
thf(tcon_Real_Oreal___Groups_Oab__semigroup__add_294,axiom,
ab_semigroup_add @ real ).
thf(tcon_Real_Oreal___Fields_Olinordered__field_295,axiom,
linordered_field @ real ).
thf(tcon_Real_Oreal___Code__Evaluation_Oterm__of_296,axiom,
code_term_of @ real ).
thf(tcon_Real_Oreal___Rings_Oordered__semiring_297,axiom,
ordered_semiring @ real ).
thf(tcon_Real_Oreal___Rings_Oordered__ring__abs_298,axiom,
ordered_ring_abs @ real ).
thf(tcon_Real_Oreal___Groups_Ocomm__monoid__add_299,axiom,
comm_monoid_add @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__ring_300,axiom,
linordered_ring @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__idom_301,axiom,
linordered_idom @ real ).
thf(tcon_Real_Oreal___Rings_Ocomm__semiring__1_302,axiom,
comm_semiring_1 @ real ).
thf(tcon_Real_Oreal___Rings_Ocomm__semiring__0_303,axiom,
comm_semiring_0 @ real ).
thf(tcon_Real_Oreal___Orderings_Odense__order_304,axiom,
dense_order @ real ).
thf(tcon_Real_Oreal___Groups_Osemigroup__mult_305,axiom,
semigroup_mult @ real ).
thf(tcon_Real_Oreal___Rings_Osemidom__divide_306,axiom,
semidom_divide @ real ).
thf(tcon_Real_Oreal___Num_Osemiring__numeral_307,axiom,
semiring_numeral @ real ).
thf(tcon_Real_Oreal___Groups_Osemigroup__add_308,axiom,
semigroup_add @ real ).
thf(tcon_Real_Oreal___Fields_Ofield__abs__sgn_309,axiom,
field_abs_sgn @ real ).
thf(tcon_Real_Oreal___Fields_Odivision__ring_310,axiom,
division_ring @ real ).
thf(tcon_Real_Oreal___Rings_Ozero__less__one_311,axiom,
zero_less_one @ real ).
thf(tcon_Real_Oreal___Rings_Ocomm__semiring_312,axiom,
comm_semiring @ real ).
thf(tcon_Real_Oreal___Nat_Osemiring__char__0_313,axiom,
semiring_char_0 @ real ).
thf(tcon_Real_Oreal___Groups_Oab__group__add_314,axiom,
ab_group_add @ real ).
thf(tcon_Real_Oreal___Fields_Ofield__char__0_315,axiom,
field_char_0 @ real ).
thf(tcon_Real_Oreal___Rings_Ozero__neq__one_316,axiom,
zero_neq_one @ real ).
thf(tcon_Real_Oreal___Rings_Oordered__ring_317,axiom,
ordered_ring @ real ).
thf(tcon_Real_Oreal___Rings_Oidom__abs__sgn_318,axiom,
idom_abs_sgn @ real ).
thf(tcon_Real_Oreal___Orderings_Opreorder_319,axiom,
preorder @ real ).
thf(tcon_Real_Oreal___Orderings_Olinorder_320,axiom,
linorder @ real ).
thf(tcon_Real_Oreal___Groups_Omonoid__mult_321,axiom,
monoid_mult @ real ).
thf(tcon_Real_Oreal___Transcendental_Oln,axiom,
ln @ real ).
thf(tcon_Real_Oreal___Rings_Oidom__divide_322,axiom,
idom_divide @ real ).
thf(tcon_Real_Oreal___Rings_Ocomm__ring__1_323,axiom,
comm_ring_1 @ real ).
thf(tcon_Real_Oreal___Groups_Omonoid__add_324,axiom,
monoid_add @ real ).
thf(tcon_Real_Oreal___Rings_Osemiring__1_325,axiom,
semiring_1 @ real ).
thf(tcon_Real_Oreal___Rings_Osemiring__0_326,axiom,
semiring_0 @ real ).
thf(tcon_Real_Oreal___Orderings_Ono__top_327,axiom,
no_top @ real ).
thf(tcon_Real_Oreal___Orderings_Ono__bot_328,axiom,
no_bot @ real ).
thf(tcon_Real_Oreal___Groups_Ogroup__add_329,axiom,
group_add @ real ).
thf(tcon_Real_Oreal___Rings_Omult__zero_330,axiom,
mult_zero @ real ).
thf(tcon_Real_Oreal___Rings_Ocomm__ring_331,axiom,
comm_ring @ real ).
thf(tcon_Real_Oreal___Orderings_Oorder_332,axiom,
order @ real ).
thf(tcon_Real_Oreal___Num_Oneg__numeral_333,axiom,
neg_numeral @ real ).
thf(tcon_Real_Oreal___Nat_Oring__char__0_334,axiom,
ring_char_0 @ real ).
thf(tcon_Real_Oreal___Rings_Osemiring_335,axiom,
semiring @ real ).
thf(tcon_Real_Oreal___Fields_Oinverse_336,axiom,
inverse @ real ).
thf(tcon_Real_Oreal___Rings_Osemidom_337,axiom,
semidom @ real ).
thf(tcon_Real_Oreal___Orderings_Oord_338,axiom,
ord @ real ).
thf(tcon_Real_Oreal___Groups_Ouminus_339,axiom,
uminus @ real ).
thf(tcon_Real_Oreal___Rings_Oring__1_340,axiom,
ring_1 @ real ).
thf(tcon_Real_Oreal___Rings_Oabs__if_341,axiom,
abs_if @ real ).
thf(tcon_Real_Oreal___Fields_Ofield_342,axiom,
field @ real ).
thf(tcon_Real_Oreal___Power_Opower_343,axiom,
power @ real ).
thf(tcon_Real_Oreal___Num_Onumeral_344,axiom,
numeral @ real ).
thf(tcon_Real_Oreal___Groups_Ozero_345,axiom,
zero @ real ).
thf(tcon_Real_Oreal___Groups_Oplus_346,axiom,
plus @ real ).
thf(tcon_Real_Oreal___Rings_Oring_347,axiom,
ring @ real ).
thf(tcon_Real_Oreal___Rings_Oidom_348,axiom,
idom @ real ).
thf(tcon_Real_Oreal___Groups_Oone_349,axiom,
one @ real ).
thf(tcon_Real_Oreal___Rings_Odvd_350,axiom,
dvd @ real ).
thf(tcon_Word_Oword___Bit__Operations_Osemiring__bit__operations_351,axiom,
! [A13: $tType] :
( ( type_len @ A13 )
=> ( bit_se359711467146920520ations @ ( word @ A13 ) ) ) ).
thf(tcon_Word_Oword___Quickcheck__Narrowing_Opartial__term__of_352,axiom,
! [A13: $tType] :
( ( typerep @ A13 )
=> ( quickc6926020345158392990erm_of @ ( word @ A13 ) ) ) ).
thf(tcon_Word_Oword___Bit__Comprehension_Obit__comprehension_353,axiom,
! [A13: $tType] :
( ( type_len @ A13 )
=> ( bit_bi6583157726757044596ension @ ( word @ A13 ) ) ) ).
thf(tcon_Word_Oword___Bit__Operations_Oring__bit__operations_354,axiom,
! [A13: $tType] :
( ( type_len @ A13 )
=> ( bit_ri3973907225187159222ations @ ( word @ A13 ) ) ) ).
thf(tcon_Word_Oword___Groups_Ocancel__ab__semigroup__add_355,axiom,
! [A13: $tType] :
( ( type_len @ A13 )
=> ( cancel2418104881723323429up_add @ ( word @ A13 ) ) ) ).
thf(tcon_Word_Oword___Groups_Ocancel__comm__monoid__add_356,axiom,
! [A13: $tType] :
( ( type_len @ A13 )
=> ( cancel1802427076303600483id_add @ ( word @ A13 ) ) ) ).
thf(tcon_Word_Oword___Rings_Ocomm__semiring__1__cancel_357,axiom,
! [A13: $tType] :
( ( type_len @ A13 )
=> ( comm_s4317794764714335236cancel @ ( word @ A13 ) ) ) ).
thf(tcon_Word_Oword___Bit__Operations_Osemiring__bits_358,axiom,
! [A13: $tType] :
( ( type_len @ A13 )
=> ( bit_semiring_bits @ ( word @ A13 ) ) ) ).
thf(tcon_Word_Oword___Groups_Ocancel__semigroup__add_359,axiom,
! [A13: $tType] :
( ( type_len @ A13 )
=> ( cancel_semigroup_add @ ( word @ A13 ) ) ) ).
thf(tcon_Word_Oword___Least__significant__bit_Olsb_360,axiom,
! [A13: $tType] :
( ( type_len @ A13 )
=> ( least_6119777620449941438nt_lsb @ ( word @ A13 ) ) ) ).
thf(tcon_Word_Oword___Groups_Oab__semigroup__mult_361,axiom,
! [A13: $tType] :
( ( type_len @ A13 )
=> ( ab_semigroup_mult @ ( word @ A13 ) ) ) ).
thf(tcon_Word_Oword___Rings_Osemiring__1__cancel_362,axiom,
! [A13: $tType] :
( ( type_len @ A13 )
=> ( semiring_1_cancel @ ( word @ A13 ) ) ) ).
thf(tcon_Word_Oword___Groups_Ocomm__monoid__mult_363,axiom,
! [A13: $tType] :
( ( type_len @ A13 )
=> ( comm_monoid_mult @ ( word @ A13 ) ) ) ).
thf(tcon_Word_Oword___Groups_Oab__semigroup__add_364,axiom,
! [A13: $tType] :
( ( type_len @ A13 )
=> ( ab_semigroup_add @ ( word @ A13 ) ) ) ).
thf(tcon_Word_Oword___Generic__set__bit_Oset__bit_365,axiom,
! [A13: $tType] :
( ( type_len @ A13 )
=> ( generic_set_set_bit @ ( word @ A13 ) ) ) ).
thf(tcon_Word_Oword___Code__Evaluation_Oterm__of_366,axiom,
! [A13: $tType] :
( ( typerep @ A13 )
=> ( code_term_of @ ( word @ A13 ) ) ) ).
thf(tcon_Word_Oword___Parity_Osemiring__parity_367,axiom,
! [A13: $tType] :
( ( type_len @ A13 )
=> ( semiring_parity @ ( word @ A13 ) ) ) ).
thf(tcon_Word_Oword___Groups_Ocomm__monoid__add_368,axiom,
! [A13: $tType] :
( ( type_len @ A13 )
=> ( comm_monoid_add @ ( word @ A13 ) ) ) ).
thf(tcon_Word_Oword___Rings_Osemiring__modulo_369,axiom,
! [A13: $tType] :
( ( type_len @ A13 )
=> ( semiring_modulo @ ( word @ A13 ) ) ) ).
thf(tcon_Word_Oword___Rings_Ocomm__semiring__1_370,axiom,
! [A13: $tType] :
( ( type_len @ A13 )
=> ( comm_semiring_1 @ ( word @ A13 ) ) ) ).
thf(tcon_Word_Oword___Rings_Ocomm__semiring__0_371,axiom,
! [A13: $tType] :
( ( type_len @ A13 )
=> ( comm_semiring_0 @ ( word @ A13 ) ) ) ).
thf(tcon_Word_Oword___Groups_Osemigroup__mult_372,axiom,
! [A13: $tType] :
( ( type_len @ A13 )
=> ( semigroup_mult @ ( word @ A13 ) ) ) ).
thf(tcon_Word_Oword___Num_Osemiring__numeral_373,axiom,
! [A13: $tType] :
( ( type_len @ A13 )
=> ( semiring_numeral @ ( word @ A13 ) ) ) ).
thf(tcon_Word_Oword___Groups_Osemigroup__add_374,axiom,
! [A13: $tType] :
( ( type_len @ A13 )
=> ( semigroup_add @ ( word @ A13 ) ) ) ).
thf(tcon_Word_Oword___Rings_Ocomm__semiring_375,axiom,
! [A13: $tType] :
( ( type_len @ A13 )
=> ( comm_semiring @ ( word @ A13 ) ) ) ).
thf(tcon_Word_Oword___Orderings_Owellorder_376,axiom,
! [A13: $tType] :
( ( type_len @ A13 )
=> ( wellorder @ ( word @ A13 ) ) ) ).
thf(tcon_Word_Oword___Groups_Oab__group__add_377,axiom,
! [A13: $tType] :
( ( type_len @ A13 )
=> ( ab_group_add @ ( word @ A13 ) ) ) ).
thf(tcon_Word_Oword___Rings_Ozero__neq__one_378,axiom,
! [A13: $tType] :
( ( type_len @ A13 )
=> ( zero_neq_one @ ( word @ A13 ) ) ) ).
thf(tcon_Word_Oword___Parity_Oring__parity_379,axiom,
! [A13: $tType] :
( ( type_len @ A13 )
=> ( ring_parity @ ( word @ A13 ) ) ) ).
thf(tcon_Word_Oword___Orderings_Opreorder_380,axiom,
! [A13: $tType] :
( ( type_len @ A13 )
=> ( preorder @ ( word @ A13 ) ) ) ).
thf(tcon_Word_Oword___Orderings_Olinorder_381,axiom,
! [A13: $tType] :
( ( type_len @ A13 )
=> ( linorder @ ( word @ A13 ) ) ) ).
thf(tcon_Word_Oword___Groups_Omonoid__mult_382,axiom,
! [A13: $tType] :
( ( type_len @ A13 )
=> ( monoid_mult @ ( word @ A13 ) ) ) ).
thf(tcon_Word_Oword___Rings_Ocomm__ring__1_383,axiom,
! [A13: $tType] :
( ( type_len @ A13 )
=> ( comm_ring_1 @ ( word @ A13 ) ) ) ).
thf(tcon_Word_Oword___Groups_Omonoid__add_384,axiom,
! [A13: $tType] :
( ( type_len @ A13 )
=> ( monoid_add @ ( word @ A13 ) ) ) ).
thf(tcon_Word_Oword___Finite__Set_Ofinite_385,axiom,
! [A13: $tType] :
( ( type_len @ A13 )
=> ( finite_finite @ ( word @ A13 ) ) ) ).
thf(tcon_Word_Oword___Rings_Osemiring__1_386,axiom,
! [A13: $tType] :
( ( type_len @ A13 )
=> ( semiring_1 @ ( word @ A13 ) ) ) ).
thf(tcon_Word_Oword___Rings_Osemiring__0_387,axiom,
! [A13: $tType] :
( ( type_len @ A13 )
=> ( semiring_0 @ ( word @ A13 ) ) ) ).
thf(tcon_Word_Oword___Groups_Ogroup__add_388,axiom,
! [A13: $tType] :
( ( type_len @ A13 )
=> ( group_add @ ( word @ A13 ) ) ) ).
thf(tcon_Word_Oword___Rings_Omult__zero_389,axiom,
! [A13: $tType] :
( ( type_len @ A13 )
=> ( mult_zero @ ( word @ A13 ) ) ) ).
thf(tcon_Word_Oword___Rings_Ocomm__ring_390,axiom,
! [A13: $tType] :
( ( type_len @ A13 )
=> ( comm_ring @ ( word @ A13 ) ) ) ).
thf(tcon_Word_Oword___Orderings_Oorder_391,axiom,
! [A13: $tType] :
( ( type_len @ A13 )
=> ( order @ ( word @ A13 ) ) ) ).
thf(tcon_Word_Oword___Num_Oneg__numeral_392,axiom,
! [A13: $tType] :
( ( type_len @ A13 )
=> ( neg_numeral @ ( word @ A13 ) ) ) ).
thf(tcon_Word_Oword___Rings_Osemiring_393,axiom,
! [A13: $tType] :
( ( type_len @ A13 )
=> ( semiring @ ( word @ A13 ) ) ) ).
thf(tcon_Word_Oword___Orderings_Oord_394,axiom,
! [A13: $tType] :
( ( type_len @ A13 )
=> ( ord @ ( word @ A13 ) ) ) ).
thf(tcon_Word_Oword___Groups_Ouminus_395,axiom,
! [A13: $tType] :
( ( type_len @ A13 )
=> ( uminus @ ( word @ A13 ) ) ) ).
thf(tcon_Word_Oword___Rings_Oring__1_396,axiom,
! [A13: $tType] :
( ( type_len @ A13 )
=> ( ring_1 @ ( word @ A13 ) ) ) ).
thf(tcon_Word_Oword___Power_Opower_397,axiom,
! [A13: $tType] :
( ( type_len @ A13 )
=> ( power @ ( word @ A13 ) ) ) ).
thf(tcon_Word_Oword___Num_Onumeral_398,axiom,
! [A13: $tType] :
( ( type_len @ A13 )
=> ( numeral @ ( word @ A13 ) ) ) ).
thf(tcon_Word_Oword___Groups_Ozero_399,axiom,
! [A13: $tType] :
( ( type_len @ A13 )
=> ( zero @ ( word @ A13 ) ) ) ).
thf(tcon_Word_Oword___Groups_Oplus_400,axiom,
! [A13: $tType] :
( ( type_len @ A13 )
=> ( plus @ ( word @ A13 ) ) ) ).
thf(tcon_Word_Oword___Rings_Oring_401,axiom,
! [A13: $tType] :
( ( type_len @ A13 )
=> ( ring @ ( word @ A13 ) ) ) ).
thf(tcon_Word_Oword___Groups_Oone_402,axiom,
! [A13: $tType] :
( ( type_len @ A13 )
=> ( one @ ( word @ A13 ) ) ) ).
thf(tcon_Word_Oword___Rings_Odvd_403,axiom,
! [A13: $tType] :
( ( type_len @ A13 )
=> ( dvd @ ( word @ A13 ) ) ) ).
thf(tcon_Word_Oword___Nat_Osize_404,axiom,
! [A13: $tType] :
( ( type_len @ A13 )
=> ( size @ ( word @ A13 ) ) ) ).
thf(tcon_String_Ochar___Quickcheck__Narrowing_Opartial__term__of_405,axiom,
quickc6926020345158392990erm_of @ char ).
thf(tcon_String_Ochar___Code__Evaluation_Oterm__of_406,axiom,
code_term_of @ char ).
thf(tcon_String_Ochar___Orderings_Opreorder_407,axiom,
preorder @ char ).
thf(tcon_String_Ochar___Orderings_Olinorder_408,axiom,
linorder @ char ).
thf(tcon_String_Ochar___Finite__Set_Ofinite_409,axiom,
finite_finite @ char ).
thf(tcon_String_Ochar___Orderings_Oorder_410,axiom,
order @ char ).
thf(tcon_String_Ochar___Orderings_Oord_411,axiom,
ord @ char ).
thf(tcon_String_Ochar___Nat_Osize_412,axiom,
size @ char ).
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___Quickcheck__Narrowing_Opartial__term__of_424,axiom,
quickc6926020345158392990erm_of @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Groups_Olinordered__ab__semigroup__add_425,axiom,
linord4140545234300271783up_add @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Rings_Olinordered__semiring__strict_426,axiom,
linord8928482502909563296strict @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Boolean__Algebras_Oboolean__algebra_427,axiom,
boolea8198339166811842893lgebra @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Rings_Osemiring__no__zero__divisors_428,axiom,
semiri3467727345109120633visors @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Groups_Oordered__ab__semigroup__add_429,axiom,
ordere6658533253407199908up_add @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Groups_Oordered__ab__group__add__abs_430,axiom,
ordere166539214618696060dd_abs @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Groups_Oordered__comm__monoid__add_431,axiom,
ordere6911136660526730532id_add @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Groups_Olinordered__ab__group__add_432,axiom,
linord5086331880401160121up_add @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Groups_Ocancel__ab__semigroup__add_433,axiom,
cancel2418104881723323429up_add @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Groups_Ocancel__comm__monoid__add_434,axiom,
cancel1802427076303600483id_add @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Rings_Olinordered__ring__strict_435,axiom,
linord4710134922213307826strict @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Rings_Oordered__comm__semiring_436,axiom,
ordere2520102378445227354miring @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Groups_Oordered__ab__group__add_437,axiom,
ordered_ab_group_add @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Groups_Ocancel__semigroup__add_438,axiom,
cancel_semigroup_add @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Rings_Olinordered__semiring_439,axiom,
linordered_semiring @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Rings_Oordered__semiring__0_440,axiom,
ordered_semiring_0 @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Orderings_Odense__linorder_441,axiom,
dense_linorder @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Groups_Oab__semigroup__mult_442,axiom,
ab_semigroup_mult @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Groups_Ocomm__monoid__mult_443,axiom,
comm_monoid_mult @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Groups_Ocomm__monoid__diff_444,axiom,
comm_monoid_diff @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Groups_Oab__semigroup__add_445,axiom,
ab_semigroup_add @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Code__Evaluation_Oterm__of_446,axiom,
code_term_of @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Rings_Oordered__semiring_447,axiom,
ordered_semiring @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Rings_Oordered__ring__abs_448,axiom,
ordered_ring_abs @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Groups_Ocomm__monoid__add_449,axiom,
comm_monoid_add @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Rings_Olinordered__ring_450,axiom,
linordered_ring @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Rings_Ocomm__semiring__0_451,axiom,
comm_semiring_0 @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Orderings_Odense__order_452,axiom,
dense_order @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Groups_Osemigroup__mult_453,axiom,
semigroup_mult @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Groups_Osemigroup__add_454,axiom,
semigroup_add @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Rings_Ocomm__semiring_455,axiom,
comm_semiring @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Orderings_Owellorder_456,axiom,
wellorder @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Orderings_Oorder__top_457,axiom,
order_top @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Orderings_Oorder__bot_458,axiom,
order_bot @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Groups_Oab__group__add_459,axiom,
ab_group_add @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Rings_Oordered__ring_460,axiom,
ordered_ring @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Orderings_Opreorder_461,axiom,
preorder @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Orderings_Olinorder_462,axiom,
linorder @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Groups_Omonoid__mult_463,axiom,
monoid_mult @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Groups_Omonoid__add_464,axiom,
monoid_add @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Finite__Set_Ofinite_465,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_466,axiom,
semiring_0 @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Groups_Ogroup__add_467,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_468,axiom,
mult_zero @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Rings_Ocomm__ring_469,axiom,
comm_ring @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Orderings_Oorder_470,axiom,
order @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Rings_Osemiring_471,axiom,
semiring @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Fields_Oinverse_472,axiom,
inverse @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Orderings_Oord_473,axiom,
ord @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Groups_Ouminus_474,axiom,
uminus @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Rings_Oabs__if_475,axiom,
abs_if @ 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__2___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_482,axiom,
condit6923001295902523014norder @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_483,axiom,
semiri1453513574482234551roduct @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Euclidean__Division_Ounique__euclidean__semiring_484,axiom,
euclid3128863361964157862miring @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Euclidean__Division_Oeuclidean__semiring__cancel_485,axiom,
euclid4440199948858584721cancel @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Rings_Osemiring__no__zero__divisors__cancel_486,axiom,
semiri6575147826004484403cancel @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Euclidean__Division_Oeuclidean__semiring_487,axiom,
euclid3725896446679973847miring @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Quickcheck__Narrowing_Opartial__term__of_488,axiom,
quickc6926020345158392990erm_of @ 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___Code__Evaluation_Oterm__of_501,axiom,
code_term_of @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Groups_Ocomm__monoid__add_502,axiom,
comm_monoid_add @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Rings_Osemiring__modulo_503,axiom,
semiring_modulo @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Rings_Ocomm__semiring__1_504,axiom,
comm_semiring_1 @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Rings_Ocomm__semiring__0_505,axiom,
comm_semiring_0 @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Groups_Osemigroup__mult_506,axiom,
semigroup_mult @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Rings_Osemidom__modulo_507,axiom,
semidom_modulo @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Rings_Osemidom__divide_508,axiom,
semidom_divide @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Num_Osemiring__numeral_509,axiom,
semiring_numeral @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Groups_Osemigroup__add_510,axiom,
semigroup_add @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Fields_Odivision__ring_511,axiom,
division_ring @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Rings_Ocomm__semiring_512,axiom,
comm_semiring @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Orderings_Owellorder_513,axiom,
wellorder @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Orderings_Oorder__top_514,axiom,
order_top @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Orderings_Oorder__bot_515,axiom,
order_bot @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Groups_Oab__group__add_516,axiom,
ab_group_add @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Rings_Ozero__neq__one_517,axiom,
zero_neq_one @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Rings_Oidom__abs__sgn_518,axiom,
idom_abs_sgn @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Orderings_Opreorder_519,axiom,
preorder @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Orderings_Olinorder_520,axiom,
linorder @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Groups_Omonoid__mult_521,axiom,
monoid_mult @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Rings_Oidom__modulo_522,axiom,
idom_modulo @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Rings_Oidom__divide_523,axiom,
idom_divide @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Rings_Ocomm__ring__1_524,axiom,
comm_ring_1 @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Groups_Omonoid__add_525,axiom,
monoid_add @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Finite__Set_Ofinite_526,axiom,
finite_finite @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Type__Length_Olen0_527,axiom,
type_len0 @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Rings_Osemiring__1_528,axiom,
semiring_1 @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Rings_Osemiring__0_529,axiom,
semiring_0 @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Groups_Ogroup__add_530,axiom,
group_add @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Type__Length_Olen_531,axiom,
type_len @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Rings_Omult__zero_532,axiom,
mult_zero @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Rings_Ocomm__ring_533,axiom,
comm_ring @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Orderings_Oorder_534,axiom,
order @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Num_Oneg__numeral_535,axiom,
neg_numeral @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Rings_Osemiring_536,axiom,
semiring @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Fields_Oinverse_537,axiom,
inverse @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Rings_Osemidom_538,axiom,
semidom @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Orderings_Oord_539,axiom,
ord @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Groups_Ouminus_540,axiom,
uminus @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Rings_Oring__1_541,axiom,
ring_1 @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Fields_Ofield_542,axiom,
field @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Power_Opower_543,axiom,
power @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Num_Onumeral_544,axiom,
numeral @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Groups_Ozero_545,axiom,
zero @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Groups_Oplus_546,axiom,
plus @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Rings_Oring_547,axiom,
ring @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Rings_Oidom_548,axiom,
idom @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Groups_Oone_549,axiom,
one @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Rings_Odvd_550,axiom,
dvd @ finite_2 ).
thf(tcon_Enum_Ofinite__3___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_551,axiom,
condit6923001295902523014norder @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_552,axiom,
semiri1453513574482234551roduct @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Euclidean__Division_Ounique__euclidean__semiring_553,axiom,
euclid3128863361964157862miring @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Euclidean__Division_Oeuclidean__semiring__cancel_554,axiom,
euclid4440199948858584721cancel @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Rings_Osemiring__no__zero__divisors__cancel_555,axiom,
semiri6575147826004484403cancel @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Euclidean__Division_Oeuclidean__semiring_556,axiom,
euclid3725896446679973847miring @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Quickcheck__Narrowing_Opartial__term__of_557,axiom,
quickc6926020345158392990erm_of @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Rings_Osemiring__1__no__zero__divisors_558,axiom,
semiri2026040879449505780visors @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Rings_Osemiring__no__zero__divisors_559,axiom,
semiri3467727345109120633visors @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Groups_Ocancel__ab__semigroup__add_560,axiom,
cancel2418104881723323429up_add @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Rings_Oring__1__no__zero__divisors_561,axiom,
ring_15535105094025558882visors @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Groups_Ocancel__comm__monoid__add_562,axiom,
cancel1802427076303600483id_add @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Rings_Ocomm__semiring__1__cancel_563,axiom,
comm_s4317794764714335236cancel @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Groups_Ocancel__semigroup__add_564,axiom,
cancel_semigroup_add @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Groups_Oab__semigroup__mult_565,axiom,
ab_semigroup_mult @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Rings_Osemiring__1__cancel_566,axiom,
semiring_1_cancel @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Rings_Oalgebraic__semidom_567,axiom,
algebraic_semidom @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Groups_Ocomm__monoid__mult_568,axiom,
comm_monoid_mult @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Groups_Oab__semigroup__add_569,axiom,
ab_semigroup_add @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Code__Evaluation_Oterm__of_570,axiom,
code_term_of @ 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___Fields_Ofield_611,axiom,
field @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Power_Opower_612,axiom,
power @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Num_Onumeral_613,axiom,
numeral @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Groups_Ozero_614,axiom,
zero @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Groups_Oplus_615,axiom,
plus @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Rings_Oring_616,axiom,
ring @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Rings_Oidom_617,axiom,
idom @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Groups_Oone_618,axiom,
one @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Rings_Odvd_619,axiom,
dvd @ finite_3 ).
thf(tcon_Filter_Ofilter___Quickcheck__Narrowing_Opartial__term__of_620,axiom,
! [A13: $tType] :
( ( typerep @ A13 )
=> ( quickc6926020345158392990erm_of @ ( filter @ A13 ) ) ) ).
thf(tcon_Filter_Ofilter___Code__Evaluation_Oterm__of_621,axiom,
! [A13: $tType] :
( ( typerep @ A13 )
=> ( code_term_of @ ( filter @ A13 ) ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Oorder__top_622,axiom,
! [A13: $tType] : ( order_top @ ( filter @ A13 ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Oorder__bot_623,axiom,
! [A13: $tType] : ( order_bot @ ( filter @ A13 ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Opreorder_624,axiom,
! [A13: $tType] : ( preorder @ ( filter @ A13 ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Oorder_625,axiom,
! [A13: $tType] : ( order @ ( filter @ A13 ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Oord_626,axiom,
! [A13: $tType] : ( ord @ ( filter @ A13 ) ) ).
thf(tcon_Option_Ooption___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_627,axiom,
! [A13: $tType] :
( ( comple5582772986160207858norder @ A13 )
=> ( condit6923001295902523014norder @ ( option @ A13 ) ) ) ).
thf(tcon_Option_Ooption___Quickcheck__Narrowing_Opartial__term__of_628,axiom,
! [A13: $tType] :
( ( typerep @ A13 )
=> ( quickc6926020345158392990erm_of @ ( option @ A13 ) ) ) ).
thf(tcon_Option_Ooption___Code__Evaluation_Oterm__of_629,axiom,
! [A13: $tType] :
( ( typerep @ A13 )
=> ( code_term_of @ ( option @ A13 ) ) ) ).
thf(tcon_Option_Ooption___Orderings_Owellorder_630,axiom,
! [A13: $tType] :
( ( wellorder @ A13 )
=> ( wellorder @ ( option @ A13 ) ) ) ).
thf(tcon_Option_Ooption___Orderings_Oorder__top_631,axiom,
! [A13: $tType] :
( ( order_top @ A13 )
=> ( order_top @ ( option @ A13 ) ) ) ).
thf(tcon_Option_Ooption___Orderings_Oorder__bot_632,axiom,
! [A13: $tType] :
( ( order @ A13 )
=> ( order_bot @ ( option @ A13 ) ) ) ).
thf(tcon_Option_Ooption___Orderings_Opreorder_633,axiom,
! [A13: $tType] :
( ( preorder @ A13 )
=> ( preorder @ ( option @ A13 ) ) ) ).
thf(tcon_Option_Ooption___Orderings_Olinorder_634,axiom,
! [A13: $tType] :
( ( linorder @ A13 )
=> ( linorder @ ( option @ A13 ) ) ) ).
thf(tcon_Option_Ooption___Finite__Set_Ofinite_635,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( finite_finite @ ( option @ A13 ) ) ) ).
thf(tcon_Option_Ooption___Orderings_Oorder_636,axiom,
! [A13: $tType] :
( ( order @ A13 )
=> ( order @ ( option @ A13 ) ) ) ).
thf(tcon_Option_Ooption___Orderings_Oord_637,axiom,
! [A13: $tType] :
( ( preorder @ A13 )
=> ( ord @ ( option @ A13 ) ) ) ).
thf(tcon_Option_Ooption___Nat_Osize_638,axiom,
! [A13: $tType] : ( size @ ( option @ A13 ) ) ).
thf(tcon_Uint32_Ouint32___Bit__Operations_Osemiring__bit__operations_639,axiom,
bit_se359711467146920520ations @ uint32 ).
thf(tcon_Uint32_Ouint32___Quickcheck__Narrowing_Opartial__term__of_640,axiom,
quickc6926020345158392990erm_of @ uint32 ).
thf(tcon_Uint32_Ouint32___Bit__Comprehension_Obit__comprehension_641,axiom,
bit_bi6583157726757044596ension @ uint32 ).
thf(tcon_Uint32_Ouint32___Bit__Operations_Oring__bit__operations_642,axiom,
bit_ri3973907225187159222ations @ uint32 ).
thf(tcon_Uint32_Ouint32___Groups_Ocancel__ab__semigroup__add_643,axiom,
cancel2418104881723323429up_add @ uint32 ).
thf(tcon_Uint32_Ouint32___Groups_Ocancel__comm__monoid__add_644,axiom,
cancel1802427076303600483id_add @ uint32 ).
thf(tcon_Uint32_Ouint32___Rings_Ocomm__semiring__1__cancel_645,axiom,
comm_s4317794764714335236cancel @ uint32 ).
thf(tcon_Uint32_Ouint32___Bit__Operations_Osemiring__bits_646,axiom,
bit_semiring_bits @ uint32 ).
thf(tcon_Uint32_Ouint32___Groups_Ocancel__semigroup__add_647,axiom,
cancel_semigroup_add @ uint32 ).
thf(tcon_Uint32_Ouint32___Least__significant__bit_Olsb_648,axiom,
least_6119777620449941438nt_lsb @ uint32 ).
thf(tcon_Uint32_Ouint32___Groups_Oab__semigroup__mult_649,axiom,
ab_semigroup_mult @ uint32 ).
thf(tcon_Uint32_Ouint32___Rings_Osemiring__1__cancel_650,axiom,
semiring_1_cancel @ uint32 ).
thf(tcon_Uint32_Ouint32___Groups_Ocomm__monoid__mult_651,axiom,
comm_monoid_mult @ uint32 ).
thf(tcon_Uint32_Ouint32___Groups_Oab__semigroup__add_652,axiom,
ab_semigroup_add @ uint32 ).
thf(tcon_Uint32_Ouint32___Generic__set__bit_Oset__bit_653,axiom,
generic_set_set_bit @ uint32 ).
thf(tcon_Uint32_Ouint32___Code__Evaluation_Oterm__of_654,axiom,
code_term_of @ 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___Power_Opower_683,axiom,
power @ uint32 ).
thf(tcon_Uint32_Ouint32___Num_Onumeral_684,axiom,
numeral @ uint32 ).
thf(tcon_Uint32_Ouint32___Groups_Ozero_685,axiom,
zero @ uint32 ).
thf(tcon_Uint32_Ouint32___Groups_Oplus_686,axiom,
plus @ uint32 ).
thf(tcon_Uint32_Ouint32___Rings_Oring_687,axiom,
ring @ uint32 ).
thf(tcon_Uint32_Ouint32___Groups_Oone_688,axiom,
one @ uint32 ).
thf(tcon_Uint32_Ouint32___Rings_Odvd_689,axiom,
dvd @ uint32 ).
thf(tcon_Uint32_Ouint32___Nat_Osize_690,axiom,
size @ uint32 ).
thf(tcon_Complex_Ocomplex___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_691,axiom,
semiri1453513574482234551roduct @ complex ).
thf(tcon_Complex_Ocomplex___Topological__Spaces_Ofirst__countable__topology_692,axiom,
topolo3112930676232923870pology @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__div__algebra_693,axiom,
real_V8999393235501362500lgebra @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__algebra__1_694,axiom,
real_V2822296259951069270ebra_1 @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Osemiring__no__zero__divisors__cancel_695,axiom,
semiri6575147826004484403cancel @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__algebra_696,axiom,
real_V4412858255891104859lgebra @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__vector_697,axiom,
real_V822414075346904944vector @ complex ).
thf(tcon_Complex_Ocomplex___Topological__Spaces_Otopological__space_698,axiom,
topolo4958980785337419405_space @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__field_699,axiom,
real_V3459762299906320749_field @ complex ).
thf(tcon_Complex_Ocomplex___Quickcheck__Narrowing_Opartial__term__of_700,axiom,
quickc6926020345158392990erm_of @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__div__algebra_701,axiom,
real_V5047593784448816457lgebra @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Ouniformity__dist_702,axiom,
real_V768167426530841204y_dist @ complex ).
thf(tcon_Complex_Ocomplex___Limits_Otopological__comm__monoid__add_703,axiom,
topolo5987344860129210374id_add @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Osemiring__1__no__zero__divisors_704,axiom,
semiri2026040879449505780visors @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__algebra__1_705,axiom,
real_V2191834092415804123ebra_1 @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Ocomplete__space_706,axiom,
real_V8037385150606011577_space @ complex ).
thf(tcon_Complex_Ocomplex___Limits_Otopological__semigroup__mult_707,axiom,
topolo4211221413907600880p_mult @ complex ).
thf(tcon_Complex_Ocomplex___Topological__Spaces_Ouniform__space_708,axiom,
topolo7287701948861334536_space @ complex ).
thf(tcon_Complex_Ocomplex___Topological__Spaces_Operfect__space_709,axiom,
topolo8386298272705272623_space @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Osemiring__no__zero__divisors_710,axiom,
semiri3467727345109120633visors @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__algebra_711,axiom,
real_V6157519004096292374lgebra @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Ometric__space_712,axiom,
real_V7819770556892013058_space @ complex ).
thf(tcon_Complex_Ocomplex___Limits_Otopological__ab__group__add_713,axiom,
topolo1287966508704411220up_add @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__vector_714,axiom,
real_V4867850818363320053vector @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Ocancel__ab__semigroup__add_715,axiom,
cancel2418104881723323429up_add @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Oring__1__no__zero__divisors_716,axiom,
ring_15535105094025558882visors @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__field_717,axiom,
real_V7773925162809079976_field @ complex ).
thf(tcon_Complex_Ocomplex___Limits_Otopological__monoid__add_718,axiom,
topolo6943815403480290642id_add @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Ocancel__comm__monoid__add_719,axiom,
cancel1802427076303600483id_add @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Ocomm__semiring__1__cancel_720,axiom,
comm_s4317794764714335236cancel @ complex ).
thf(tcon_Complex_Ocomplex___Limits_Otopological__group__add_721,axiom,
topolo1633459387980952147up_add @ complex ).
thf(tcon_Complex_Ocomplex___Topological__Spaces_Ot2__space_722,axiom,
topological_t2_space @ complex ).
thf(tcon_Complex_Ocomplex___Topological__Spaces_Ot1__space_723,axiom,
topological_t1_space @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Ocancel__semigroup__add_724,axiom,
cancel_semigroup_add @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Obanach_725,axiom,
real_Vector_banach @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Oab__semigroup__mult_726,axiom,
ab_semigroup_mult @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Osemiring__1__cancel_727,axiom,
semiring_1_cancel @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Ocomm__monoid__mult_728,axiom,
comm_monoid_mult @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Oab__semigroup__add_729,axiom,
ab_semigroup_add @ complex ).
thf(tcon_Complex_Ocomplex___Code__Evaluation_Oterm__of_730,axiom,
code_term_of @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Ocomm__monoid__add_731,axiom,
comm_monoid_add @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Ocomm__semiring__1_732,axiom,
comm_semiring_1 @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Ocomm__semiring__0_733,axiom,
comm_semiring_0 @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Osemigroup__mult_734,axiom,
semigroup_mult @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Osemidom__divide_735,axiom,
semidom_divide @ complex ).
thf(tcon_Complex_Ocomplex___Num_Osemiring__numeral_736,axiom,
semiring_numeral @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Osemigroup__add_737,axiom,
semigroup_add @ complex ).
thf(tcon_Complex_Ocomplex___Fields_Ofield__abs__sgn_738,axiom,
field_abs_sgn @ complex ).
thf(tcon_Complex_Ocomplex___Fields_Odivision__ring_739,axiom,
division_ring @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Ocomm__semiring_740,axiom,
comm_semiring @ complex ).
thf(tcon_Complex_Ocomplex___Nat_Osemiring__char__0_741,axiom,
semiring_char_0 @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Oab__group__add_742,axiom,
ab_group_add @ complex ).
thf(tcon_Complex_Ocomplex___Fields_Ofield__char__0_743,axiom,
field_char_0 @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Ozero__neq__one_744,axiom,
zero_neq_one @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Oidom__abs__sgn_745,axiom,
idom_abs_sgn @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Omonoid__mult_746,axiom,
monoid_mult @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Oidom__divide_747,axiom,
idom_divide @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Ocomm__ring__1_748,axiom,
comm_ring_1 @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Omonoid__add_749,axiom,
monoid_add @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Osemiring__1_750,axiom,
semiring_1 @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Osemiring__0_751,axiom,
semiring_0 @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Ogroup__add_752,axiom,
group_add @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Omult__zero_753,axiom,
mult_zero @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Ocomm__ring_754,axiom,
comm_ring @ complex ).
thf(tcon_Complex_Ocomplex___Num_Oneg__numeral_755,axiom,
neg_numeral @ complex ).
thf(tcon_Complex_Ocomplex___Nat_Oring__char__0_756,axiom,
ring_char_0 @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Osemiring_757,axiom,
semiring @ complex ).
thf(tcon_Complex_Ocomplex___Fields_Oinverse_758,axiom,
inverse @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Osemidom_759,axiom,
semidom @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Ouminus_760,axiom,
uminus @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Oring__1_761,axiom,
ring_1 @ complex ).
thf(tcon_Complex_Ocomplex___Fields_Ofield_762,axiom,
field @ complex ).
thf(tcon_Complex_Ocomplex___Power_Opower_763,axiom,
power @ complex ).
thf(tcon_Complex_Ocomplex___Num_Onumeral_764,axiom,
numeral @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Ozero_765,axiom,
zero @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Oplus_766,axiom,
plus @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Oring_767,axiom,
ring @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Oidom_768,axiom,
idom @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Oone_769,axiom,
one @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Odvd_770,axiom,
dvd @ complex ).
thf(tcon_Extended__Nat_Oenat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_771,axiom,
condit6923001295902523014norder @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Ostrict__ordered__ab__semigroup__add_772,axiom,
strict9044650504122735259up_add @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Ostrict__ordered__comm__monoid__add_773,axiom,
strict7427464778891057005id_add @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Ocanonically__ordered__monoid__add_774,axiom,
canoni5634975068530333245id_add @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Quickcheck__Narrowing_Opartial__term__of_775,axiom,
quickc6926020345158392990erm_of @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Olinordered__ab__semigroup__add_776,axiom,
linord4140545234300271783up_add @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Olinordered__nonzero__semiring_777,axiom,
linord181362715937106298miring @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Osemiring__no__zero__divisors_778,axiom,
semiri3467727345109120633visors @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Oordered__ab__semigroup__add_779,axiom,
ordere6658533253407199908up_add @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Oordered__comm__monoid__add_780,axiom,
ordere6911136660526730532id_add @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Oordered__comm__semiring_781,axiom,
ordere2520102378445227354miring @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Oab__semigroup__mult_782,axiom,
ab_semigroup_mult @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Ocomm__monoid__mult_783,axiom,
comm_monoid_mult @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Oab__semigroup__add_784,axiom,
ab_semigroup_add @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Code__Evaluation_Oterm__of_785,axiom,
code_term_of @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Oordered__semiring_786,axiom,
ordered_semiring @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Ocomm__monoid__add_787,axiom,
comm_monoid_add @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Ocomm__semiring__1_788,axiom,
comm_semiring_1 @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Ocomm__semiring__0_789,axiom,
comm_semiring_0 @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Osemigroup__mult_790,axiom,
semigroup_mult @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Num_Osemiring__numeral_791,axiom,
semiring_numeral @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Osemigroup__add_792,axiom,
semigroup_add @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Ozero__less__one_793,axiom,
zero_less_one @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Ocomm__semiring_794,axiom,
comm_semiring @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Orderings_Owellorder_795,axiom,
wellorder @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Orderings_Oorder__top_796,axiom,
order_top @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Orderings_Oorder__bot_797,axiom,
order_bot @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Nat_Osemiring__char__0_798,axiom,
semiring_char_0 @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Ozero__neq__one_799,axiom,
zero_neq_one @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Orderings_Opreorder_800,axiom,
preorder @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Orderings_Olinorder_801,axiom,
linorder @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Omonoid__mult_802,axiom,
monoid_mult @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Omonoid__add_803,axiom,
monoid_add @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Osemiring__1_804,axiom,
semiring_1 @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Osemiring__0_805,axiom,
semiring_0 @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Omult__zero_806,axiom,
mult_zero @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Orderings_Oorder_807,axiom,
order @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Osemiring_808,axiom,
semiring @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Orderings_Oord_809,axiom,
ord @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Power_Opower_810,axiom,
power @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Num_Onumeral_811,axiom,
numeral @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Ozero_812,axiom,
zero @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Oplus_813,axiom,
plus @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Oone_814,axiom,
one @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Odvd_815,axiom,
dvd @ extended_enat ).
thf(tcon_Multiset_Omultiset___Quickcheck__Narrowing_Opartial__term__of_816,axiom,
! [A13: $tType] :
( ( typerep @ A13 )
=> ( quickc6926020345158392990erm_of @ ( multiset @ A13 ) ) ) ).
thf(tcon_Multiset_Omultiset___Groups_Oordered__ab__semigroup__add_817,axiom,
! [A13: $tType] :
( ( preorder @ A13 )
=> ( ordere6658533253407199908up_add @ ( multiset @ A13 ) ) ) ).
thf(tcon_Multiset_Omultiset___Groups_Ocancel__ab__semigroup__add_818,axiom,
! [A13: $tType] : ( cancel2418104881723323429up_add @ ( multiset @ A13 ) ) ).
thf(tcon_Multiset_Omultiset___Groups_Ocancel__comm__monoid__add_819,axiom,
! [A13: $tType] : ( cancel1802427076303600483id_add @ ( multiset @ A13 ) ) ).
thf(tcon_Multiset_Omultiset___Groups_Ocancel__semigroup__add_820,axiom,
! [A13: $tType] : ( cancel_semigroup_add @ ( multiset @ A13 ) ) ).
thf(tcon_Multiset_Omultiset___Groups_Ocomm__monoid__diff_821,axiom,
! [A13: $tType] : ( comm_monoid_diff @ ( multiset @ A13 ) ) ).
thf(tcon_Multiset_Omultiset___Groups_Oab__semigroup__add_822,axiom,
! [A13: $tType] : ( ab_semigroup_add @ ( multiset @ A13 ) ) ).
thf(tcon_Multiset_Omultiset___Code__Evaluation_Oterm__of_823,axiom,
! [A13: $tType] :
( ( typerep @ A13 )
=> ( code_term_of @ ( multiset @ A13 ) ) ) ).
thf(tcon_Multiset_Omultiset___Groups_Ocomm__monoid__add_824,axiom,
! [A13: $tType] : ( comm_monoid_add @ ( multiset @ A13 ) ) ).
thf(tcon_Multiset_Omultiset___Groups_Osemigroup__add_825,axiom,
! [A13: $tType] : ( semigroup_add @ ( multiset @ A13 ) ) ).
thf(tcon_Multiset_Omultiset___Orderings_Opreorder_826,axiom,
! [A13: $tType] :
( ( preorder @ A13 )
=> ( preorder @ ( multiset @ A13 ) ) ) ).
thf(tcon_Multiset_Omultiset___Groups_Omonoid__add_827,axiom,
! [A13: $tType] : ( monoid_add @ ( multiset @ A13 ) ) ).
thf(tcon_Multiset_Omultiset___Orderings_Oorder_828,axiom,
! [A13: $tType] :
( ( preorder @ A13 )
=> ( order @ ( multiset @ A13 ) ) ) ).
thf(tcon_Multiset_Omultiset___Orderings_Oord_829,axiom,
! [A13: $tType] :
( ( preorder @ A13 )
=> ( ord @ ( multiset @ A13 ) ) ) ).
thf(tcon_Multiset_Omultiset___Groups_Ozero_830,axiom,
! [A13: $tType] : ( zero @ ( multiset @ A13 ) ) ).
thf(tcon_Multiset_Omultiset___Groups_Oplus_831,axiom,
! [A13: $tType] : ( plus @ ( multiset @ A13 ) ) ).
thf(tcon_Multiset_Omultiset___Nat_Osize_832,axiom,
! [A13: $tType] : ( size @ ( multiset @ A13 ) ) ).
thf(tcon_Numeral__Type_Obit0___Groups_Ocancel__ab__semigroup__add_833,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( cancel2418104881723323429up_add @ ( numeral_bit0 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Groups_Ocancel__comm__monoid__add_834,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( cancel1802427076303600483id_add @ ( numeral_bit0 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Rings_Ocomm__semiring__1__cancel_835,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( comm_s4317794764714335236cancel @ ( numeral_bit0 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Groups_Ocancel__semigroup__add_836,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( cancel_semigroup_add @ ( numeral_bit0 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Groups_Oab__semigroup__mult_837,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( ab_semigroup_mult @ ( numeral_bit0 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Rings_Osemiring__1__cancel_838,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( semiring_1_cancel @ ( numeral_bit0 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Groups_Ocomm__monoid__mult_839,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( comm_monoid_mult @ ( numeral_bit0 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Groups_Oab__semigroup__add_840,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( ab_semigroup_add @ ( numeral_bit0 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Code__Evaluation_Oterm__of_841,axiom,
! [A13: $tType] :
( ( typerep @ A13 )
=> ( code_term_of @ ( numeral_bit0 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Groups_Ocomm__monoid__add_842,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( comm_monoid_add @ ( numeral_bit0 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Rings_Ocomm__semiring__1_843,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( comm_semiring_1 @ ( numeral_bit0 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Rings_Ocomm__semiring__0_844,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( comm_semiring_0 @ ( numeral_bit0 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Groups_Osemigroup__mult_845,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( semigroup_mult @ ( numeral_bit0 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Num_Osemiring__numeral_846,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( semiring_numeral @ ( numeral_bit0 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Groups_Osemigroup__add_847,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( semigroup_add @ ( numeral_bit0 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Rings_Ocomm__semiring_848,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( comm_semiring @ ( numeral_bit0 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Orderings_Owellorder_849,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( wellorder @ ( numeral_bit0 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Groups_Oab__group__add_850,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( ab_group_add @ ( numeral_bit0 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Rings_Ozero__neq__one_851,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( zero_neq_one @ ( numeral_bit0 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Orderings_Opreorder_852,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( preorder @ ( numeral_bit0 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Orderings_Olinorder_853,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( linorder @ ( numeral_bit0 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Groups_Omonoid__mult_854,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( monoid_mult @ ( numeral_bit0 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Rings_Ocomm__ring__1_855,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( comm_ring_1 @ ( numeral_bit0 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Groups_Omonoid__add_856,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( monoid_add @ ( numeral_bit0 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Finite__Set_Ofinite_857,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( finite_finite @ ( numeral_bit0 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Type__Length_Olen0_858,axiom,
! [A13: $tType] :
( ( type_len0 @ A13 )
=> ( type_len0 @ ( numeral_bit0 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Rings_Osemiring__1_859,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( semiring_1 @ ( numeral_bit0 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Rings_Osemiring__0_860,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( semiring_0 @ ( numeral_bit0 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Groups_Ogroup__add_861,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( group_add @ ( numeral_bit0 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Type__Length_Olen_862,axiom,
! [A13: $tType] :
( ( type_len @ A13 )
=> ( type_len @ ( numeral_bit0 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Rings_Omult__zero_863,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( mult_zero @ ( numeral_bit0 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Rings_Ocomm__ring_864,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( comm_ring @ ( numeral_bit0 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Orderings_Oorder_865,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( order @ ( numeral_bit0 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Num_Oneg__numeral_866,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( neg_numeral @ ( numeral_bit0 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Rings_Osemiring_867,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( semiring @ ( numeral_bit0 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Orderings_Oord_868,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( ord @ ( numeral_bit0 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Groups_Ouminus_869,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( uminus @ ( numeral_bit0 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Rings_Oring__1_870,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( ring_1 @ ( numeral_bit0 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Power_Opower_871,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( power @ ( numeral_bit0 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Num_Onumeral_872,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( numeral @ ( numeral_bit0 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Groups_Ozero_873,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( zero @ ( numeral_bit0 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Groups_Oplus_874,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( plus @ ( numeral_bit0 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Rings_Oring_875,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( ring @ ( numeral_bit0 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Groups_Oone_876,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( one @ ( numeral_bit0 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Rings_Odvd_877,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( dvd @ ( numeral_bit0 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Groups_Ocancel__ab__semigroup__add_878,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( cancel2418104881723323429up_add @ ( numeral_bit1 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Groups_Ocancel__comm__monoid__add_879,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( cancel1802427076303600483id_add @ ( numeral_bit1 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Rings_Ocomm__semiring__1__cancel_880,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( comm_s4317794764714335236cancel @ ( numeral_bit1 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Groups_Ocancel__semigroup__add_881,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( cancel_semigroup_add @ ( numeral_bit1 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Groups_Oab__semigroup__mult_882,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( ab_semigroup_mult @ ( numeral_bit1 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Rings_Osemiring__1__cancel_883,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( semiring_1_cancel @ ( numeral_bit1 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Groups_Ocomm__monoid__mult_884,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( comm_monoid_mult @ ( numeral_bit1 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Groups_Oab__semigroup__add_885,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( ab_semigroup_add @ ( numeral_bit1 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Code__Evaluation_Oterm__of_886,axiom,
! [A13: $tType] :
( ( typerep @ A13 )
=> ( code_term_of @ ( numeral_bit1 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Groups_Ocomm__monoid__add_887,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( comm_monoid_add @ ( numeral_bit1 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Rings_Ocomm__semiring__1_888,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( comm_semiring_1 @ ( numeral_bit1 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Rings_Ocomm__semiring__0_889,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( comm_semiring_0 @ ( numeral_bit1 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Groups_Osemigroup__mult_890,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( semigroup_mult @ ( numeral_bit1 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Num_Osemiring__numeral_891,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( semiring_numeral @ ( numeral_bit1 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Groups_Osemigroup__add_892,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( semigroup_add @ ( numeral_bit1 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Rings_Ocomm__semiring_893,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( comm_semiring @ ( numeral_bit1 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Orderings_Owellorder_894,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( wellorder @ ( numeral_bit1 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Groups_Oab__group__add_895,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( ab_group_add @ ( numeral_bit1 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Rings_Ozero__neq__one_896,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( zero_neq_one @ ( numeral_bit1 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Orderings_Opreorder_897,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( preorder @ ( numeral_bit1 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Orderings_Olinorder_898,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( linorder @ ( numeral_bit1 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Groups_Omonoid__mult_899,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( monoid_mult @ ( numeral_bit1 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Rings_Ocomm__ring__1_900,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( comm_ring_1 @ ( numeral_bit1 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Groups_Omonoid__add_901,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( monoid_add @ ( numeral_bit1 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Finite__Set_Ofinite_902,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( finite_finite @ ( numeral_bit1 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Type__Length_Olen0_903,axiom,
! [A13: $tType] :
( ( type_len0 @ A13 )
=> ( type_len0 @ ( numeral_bit1 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Rings_Osemiring__1_904,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( semiring_1 @ ( numeral_bit1 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Rings_Osemiring__0_905,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( semiring_0 @ ( numeral_bit1 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Groups_Ogroup__add_906,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( group_add @ ( numeral_bit1 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Type__Length_Olen_907,axiom,
! [A13: $tType] :
( ( type_len0 @ A13 )
=> ( type_len @ ( numeral_bit1 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Rings_Omult__zero_908,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( mult_zero @ ( numeral_bit1 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Rings_Ocomm__ring_909,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( comm_ring @ ( numeral_bit1 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Orderings_Oorder_910,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( order @ ( numeral_bit1 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Num_Oneg__numeral_911,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( neg_numeral @ ( numeral_bit1 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Rings_Osemiring_912,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( semiring @ ( numeral_bit1 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Orderings_Oord_913,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( ord @ ( numeral_bit1 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Groups_Ouminus_914,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( uminus @ ( numeral_bit1 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Rings_Oring__1_915,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( ring_1 @ ( numeral_bit1 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Power_Opower_916,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( power @ ( numeral_bit1 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Num_Onumeral_917,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( numeral @ ( numeral_bit1 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Groups_Ozero_918,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( zero @ ( numeral_bit1 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Groups_Oplus_919,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( plus @ ( numeral_bit1 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Rings_Oring_920,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( ring @ ( numeral_bit1 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Groups_Oone_921,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( one @ ( numeral_bit1 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Rings_Odvd_922,axiom,
! [A13: $tType] :
( ( finite_finite @ A13 )
=> ( dvd @ ( numeral_bit1 @ A13 ) ) ) ).
thf(tcon_Numeral__Type_Onum0___Quickcheck__Narrowing_Opartial__term__of_923,axiom,
quickc6926020345158392990erm_of @ numeral_num0 ).
thf(tcon_Numeral__Type_Onum0___Code__Evaluation_Oterm__of_924,axiom,
code_term_of @ numeral_num0 ).
thf(tcon_Numeral__Type_Onum0___Type__Length_Olen0_925,axiom,
type_len0 @ numeral_num0 ).
thf(tcon_Numeral__Type_Onum1___Quickcheck__Narrowing_Opartial__term__of_926,axiom,
quickc6926020345158392990erm_of @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Groups_Ocancel__ab__semigroup__add_927,axiom,
cancel2418104881723323429up_add @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Groups_Ocancel__comm__monoid__add_928,axiom,
cancel1802427076303600483id_add @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Groups_Ocancel__semigroup__add_929,axiom,
cancel_semigroup_add @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Groups_Oab__semigroup__mult_930,axiom,
ab_semigroup_mult @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Groups_Ocomm__monoid__mult_931,axiom,
comm_monoid_mult @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Groups_Oab__semigroup__add_932,axiom,
ab_semigroup_add @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Code__Evaluation_Oterm__of_933,axiom,
code_term_of @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Groups_Ocomm__monoid__add_934,axiom,
comm_monoid_add @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Rings_Ocomm__semiring__0_935,axiom,
comm_semiring_0 @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Groups_Osemigroup__mult_936,axiom,
semigroup_mult @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Groups_Osemigroup__add_937,axiom,
semigroup_add @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Rings_Ocomm__semiring_938,axiom,
comm_semiring @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Orderings_Owellorder_939,axiom,
wellorder @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Groups_Oab__group__add_940,axiom,
ab_group_add @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Orderings_Opreorder_941,axiom,
preorder @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Orderings_Olinorder_942,axiom,
linorder @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Groups_Omonoid__mult_943,axiom,
monoid_mult @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Groups_Omonoid__add_944,axiom,
monoid_add @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Finite__Set_Ofinite_945,axiom,
finite_finite @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Type__Length_Olen0_946,axiom,
type_len0 @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Rings_Osemiring__0_947,axiom,
semiring_0 @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Groups_Ogroup__add_948,axiom,
group_add @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Type__Length_Olen_949,axiom,
type_len @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Rings_Omult__zero_950,axiom,
mult_zero @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Rings_Ocomm__ring_951,axiom,
comm_ring @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Orderings_Oorder_952,axiom,
order @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Rings_Osemiring_953,axiom,
semiring @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Orderings_Oord_954,axiom,
ord @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Groups_Ouminus_955,axiom,
uminus @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Power_Opower_956,axiom,
power @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Num_Onumeral_957,axiom,
numeral @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Groups_Ozero_958,axiom,
zero @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Groups_Oplus_959,axiom,
plus @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Rings_Oring_960,axiom,
ring @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Groups_Oone_961,axiom,
one @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Rings_Odvd_962,axiom,
dvd @ numeral_num1 ).
thf(tcon_Product__Type_Oprod___Topological__Spaces_Otopological__space_963,axiom,
! [A13: $tType,A15: $tType] :
( ( ( topolo4958980785337419405_space @ A13 )
& ( topolo4958980785337419405_space @ A15 ) )
=> ( topolo4958980785337419405_space @ ( product_prod @ A13 @ A15 ) ) ) ).
thf(tcon_Product__Type_Oprod___Quickcheck__Narrowing_Opartial__term__of_964,axiom,
! [A13: $tType,A15: $tType] :
( ( ( typerep @ A13 )
& ( typerep @ A15 ) )
=> ( quickc6926020345158392990erm_of @ ( product_prod @ A13 @ A15 ) ) ) ).
thf(tcon_Product__Type_Oprod___Topological__Spaces_Ot2__space_965,axiom,
! [A13: $tType,A15: $tType] :
( ( ( topological_t2_space @ A13 )
& ( topological_t2_space @ A15 ) )
=> ( topological_t2_space @ ( product_prod @ A13 @ A15 ) ) ) ).
thf(tcon_Product__Type_Oprod___Topological__Spaces_Ot1__space_966,axiom,
! [A13: $tType,A15: $tType] :
( ( ( topological_t1_space @ A13 )
& ( topological_t1_space @ A15 ) )
=> ( topological_t1_space @ ( product_prod @ A13 @ A15 ) ) ) ).
thf(tcon_Product__Type_Oprod___Code__Evaluation_Oterm__of_967,axiom,
! [A13: $tType,A15: $tType] :
( ( ( typerep @ A13 )
& ( typerep @ A15 ) )
=> ( code_term_of @ ( product_prod @ A13 @ A15 ) ) ) ).
thf(tcon_Product__Type_Oprod___Finite__Set_Ofinite_968,axiom,
! [A13: $tType,A15: $tType] :
( ( ( finite_finite @ A13 )
& ( finite_finite @ A15 ) )
=> ( finite_finite @ ( product_prod @ A13 @ A15 ) ) ) ).
thf(tcon_Product__Type_Oprod___Nat_Osize_969,axiom,
! [A13: $tType,A15: $tType] : ( size @ ( product_prod @ A13 @ A15 ) ) ).
thf(tcon_Code__Numeral_Ointeger___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_970,axiom,
bit_un5681908812861735899ations @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_971,axiom,
semiri1453513574482234551roduct @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__semiring__with__nat_972,axiom,
euclid5411537665997757685th_nat @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__ring__with__nat_973,axiom,
euclid8789492081693882211th_nat @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__monoid__add__imp__le_974,axiom,
ordere1937475149494474687imp_le @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__semiring_975,axiom,
euclid3128863361964157862miring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__semiring__cancel_976,axiom,
euclid4440199948858584721cancel @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Divides_Ounique__euclidean__semiring__numeral_977,axiom,
unique1627219031080169319umeral @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__ring__cancel_978,axiom,
euclid8851590272496341667cancel @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__no__zero__divisors__cancel_979,axiom,
semiri6575147826004484403cancel @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ostrict__ordered__ab__semigroup__add_980,axiom,
strict9044650504122735259up_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__cancel__ab__semigroup__add_981,axiom,
ordere580206878836729694up_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__add__imp__le_982,axiom,
ordere2412721322843649153imp_le @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Bit__Operations_Osemiring__bit__operations_983,axiom,
bit_se359711467146920520ations @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__comm__semiring__strict_984,axiom,
linord2810124833399127020strict @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ostrict__ordered__comm__monoid__add_985,axiom,
strict7427464778891057005id_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__cancel__comm__monoid__add_986,axiom,
ordere8940638589300402666id_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__semiring_987,axiom,
euclid3725896446679973847miring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Quickcheck__Narrowing_Opartial__term__of_988,axiom,
quickc6926020345158392990erm_of @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__1__strict_989,axiom,
linord715952674999750819strict @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Olinordered__ab__semigroup__add_990,axiom,
linord4140545234300271783up_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Bit__Operations_Oring__bit__operations_991,axiom,
bit_ri3973907225187159222ations @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1__no__zero__divisors_992,axiom,
semiri2026040879449505780visors @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__nonzero__semiring_993,axiom,
linord181362715937106298miring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__ring_994,axiom,
euclid5891614535332579305n_ring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__strict_995,axiom,
linord8928482502909563296strict @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__no__zero__divisors_996,axiom,
semiri3467727345109120633visors @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__add_997,axiom,
ordere6658533253407199908up_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__group__add__abs_998,axiom,
ordere166539214618696060dd_abs @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__comm__monoid__add_999,axiom,
ordere6911136660526730532id_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Olinordered__ab__group__add_1000,axiom,
linord5086331880401160121up_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ocancel__ab__semigroup__add_1001,axiom,
cancel2418104881723323429up_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oring__1__no__zero__divisors_1002,axiom,
ring_15535105094025558882visors @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ocancel__comm__monoid__add_1003,axiom,
cancel1802427076303600483id_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__ring__strict_1004,axiom,
linord4710134922213307826strict @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__1__cancel_1005,axiom,
comm_s4317794764714335236cancel @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Bit__Operations_Osemiring__bits_1006,axiom,
bit_semiring_bits @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oordered__comm__semiring_1007,axiom,
ordere2520102378445227354miring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__1_1008,axiom,
linord6961819062388156250ring_1 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__group__add_1009,axiom,
ordered_ab_group_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ocancel__semigroup__add_1010,axiom,
cancel_semigroup_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring_1011,axiom,
linordered_semiring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Least__significant__bit_Olsb_1012,axiom,
least_6119777620449941438nt_lsb @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oordered__semiring__0_1013,axiom,
ordered_semiring_0 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semidom_1014,axiom,
linordered_semidom @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oab__semigroup__mult_1015,axiom,
ab_semigroup_mult @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1__cancel_1016,axiom,
semiring_1_cancel @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oalgebraic__semidom_1017,axiom,
algebraic_semidom @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ocomm__monoid__mult_1018,axiom,
comm_monoid_mult @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oab__semigroup__add_1019,axiom,
ab_semigroup_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Generic__set__bit_Oset__bit_1020,axiom,
generic_set_set_bit @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Code__Evaluation_Oterm__of_1021,axiom,
code_term_of @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oordered__semiring_1022,axiom,
ordered_semiring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oordered__ring__abs_1023,axiom,
ordered_ring_abs @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Parity_Osemiring__parity_1024,axiom,
semiring_parity @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ocomm__monoid__add_1025,axiom,
comm_monoid_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__modulo_1026,axiom,
semiring_modulo @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__ring_1027,axiom,
linordered_ring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__idom_1028,axiom,
linordered_idom @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__1_1029,axiom,
comm_semiring_1 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__0_1030,axiom,
comm_semiring_0 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Osemigroup__mult_1031,axiom,
semigroup_mult @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemidom__modulo_1032,axiom,
semidom_modulo @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemidom__divide_1033,axiom,
semidom_divide @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Num_Osemiring__numeral_1034,axiom,
semiring_numeral @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Osemigroup__add_1035,axiom,
semigroup_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ozero__less__one_1036,axiom,
zero_less_one @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring_1037,axiom,
comm_semiring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Nat_Osemiring__char__0_1038,axiom,
semiring_char_0 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oab__group__add_1039,axiom,
ab_group_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ozero__neq__one_1040,axiom,
zero_neq_one @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oordered__ring_1041,axiom,
ordered_ring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oidom__abs__sgn_1042,axiom,
idom_abs_sgn @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Parity_Oring__parity_1043,axiom,
ring_parity @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Orderings_Opreorder_1044,axiom,
preorder @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Orderings_Olinorder_1045,axiom,
linorder @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Omonoid__mult_1046,axiom,
monoid_mult @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oidom__modulo_1047,axiom,
idom_modulo @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oidom__divide_1048,axiom,
idom_divide @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__ring__1_1049,axiom,
comm_ring_1 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Omonoid__add_1050,axiom,
monoid_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1_1051,axiom,
semiring_1 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__0_1052,axiom,
semiring_0 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ogroup__add_1053,axiom,
group_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Omult__zero_1054,axiom,
mult_zero @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__ring_1055,axiom,
comm_ring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Orderings_Oorder_1056,axiom,
order @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Num_Oneg__numeral_1057,axiom,
neg_numeral @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Nat_Oring__char__0_1058,axiom,
ring_char_0 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring_1059,axiom,
semiring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemidom_1060,axiom,
semidom @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Orderings_Oord_1061,axiom,
ord @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ouminus_1062,axiom,
uminus @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oring__1_1063,axiom,
ring_1 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oabs__if_1064,axiom,
abs_if @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Power_Opower_1065,axiom,
power @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Num_Onumeral_1066,axiom,
numeral @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ozero_1067,axiom,
zero @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oplus_1068,axiom,
plus @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oring_1069,axiom,
ring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oidom_1070,axiom,
idom @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oone_1071,axiom,
one @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Odvd_1072,axiom,
dvd @ code_integer ).
thf(tcon_Heap__Time__Monad_OHeap___Quickcheck__Narrowing_Opartial__term__of_1073,axiom,
! [A13: $tType] :
( ( typerep @ A13 )
=> ( quickc6926020345158392990erm_of @ ( heap_Time_Heap @ A13 ) ) ) ).
thf(tcon_Heap__Time__Monad_OHeap___Code__Evaluation_Oterm__of_1074,axiom,
! [A13: $tType] :
( ( typerep @ A13 )
=> ( code_term_of @ ( heap_Time_Heap @ A13 ) ) ) ).
thf(tcon_Heap__Time__Monad_OHeap___Nat_Osize_1075,axiom,
! [A13: $tType] : ( size @ ( heap_Time_Heap @ A13 ) ) ).
thf(tcon_VEBT__Definitions_OVEBT___Quickcheck__Narrowing_Opartial__term__of_1076,axiom,
quickc6926020345158392990erm_of @ vEBT_VEBT ).
thf(tcon_VEBT__Definitions_OVEBT___Code__Evaluation_Oterm__of_1077,axiom,
code_term_of @ vEBT_VEBT ).
thf(tcon_VEBT__Definitions_OVEBT___Nat_Osize_1078,axiom,
size @ vEBT_VEBT ).
% Helper facts (4)
thf(help_If_3_1_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_T,axiom,
! [A: $tType,X2: A,Y2: A] :
( ( if @ A @ $false @ X2 @ Y2 )
= Y2 ) ).
thf(help_If_1_1_T,axiom,
! [A: $tType,X2: A,Y2: A] :
( ( if @ A @ $true @ X2 @ Y2 )
= X2 ) ).
thf(help_fChoice_1_1_T,axiom,
! [A: $tType,P: A > $o] :
( ( P @ ( fChoice @ A @ P ) )
= ( ? [X7: A] : ( P @ X7 ) ) ) ).
% Conjectures (1)
thf(conj_0,conjecture,
ord_less_eq @ nat @ ( vEBT_VEBT_Tb2 @ ( suc @ ( suc @ na ) ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ one2 ) ) ) @ ( vEBT_VEBT_cnt2 @ ( if @ vEBT_VEBT @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ na ) ) ) @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ na ) ) @ ( replicate @ vEBT_VEBT @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ na ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide @ nat @ ( suc @ ( suc @ na ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide @ nat @ ( suc @ ( suc @ na ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ na ) ) @ ( replicate @ vEBT_VEBT @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( divide_divide @ nat @ ( suc @ ( suc @ na ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide @ nat @ ( suc @ ( suc @ na ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( suc @ ( divide_divide @ nat @ ( suc @ ( suc @ na ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ).
%------------------------------------------------------------------------------