TPTP Problem File: ITP293^4.p
View Solutions
- Solve Problem
%------------------------------------------------------------------------------
% File : ITP293^4 : TPTP v8.2.0. Released v8.0.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer problem VEBT_Example 00027_000692
% 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 : 0097_VEBT_Example_00027_000692 [Des22]
% Status : Theorem
% Rating : 1.00 v8.1.0
% Syntax : Number of formulae : 9952 (2859 unt; 498 typ; 0 def)
% Number of atoms : 28114 (8637 equ; 0 cnn)
% Maximal formula atoms : 28 ( 2 avg)
% Number of connectives : 167639 (1766 ~; 308 |;2054 &;150862 @)
% ( 0 <=>;12649 =>; 0 <=; 0 <~>)
% Maximal formula depth : 40 ( 7 avg)
% Number of types : 19 ( 18 usr)
% Number of type conns : 3021 (3021 >; 0 *; 0 +; 0 <<)
% Number of symbols : 483 ( 480 usr; 9 con; 0-9 aty)
% Number of variables : 26611 (2624 ^;23147 !; 432 ?;26611 :)
% ( 408 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% from the van Emde Boas Trees session in the Archive of Formal
% proofs -
% www.isa-afp.org/browser_info/current/AFP/Van_Emde_Boas_Trees
% 2022-02-18 23:43:26.191
%------------------------------------------------------------------------------
% Could-be-implicit typings (30)
thf(ty_t_VEBT__BuildupMemImp_OVEBTi,type,
vEBT_VEBTi: $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_Ounit,type,
product_unit: $tType ).
thf(ty_t_Product__Type_Oprod,type,
product_prod: $tType > $tType > $tType ).
thf(ty_t_Numeral__Type_Onum1,type,
numeral_num1: $tType ).
thf(ty_t_Numeral__Type_Onum0,type,
numeral_num0: $tType ).
thf(ty_t_Numeral__Type_Obit1,type,
numeral_bit1: $tType > $tType ).
thf(ty_t_Numeral__Type_Obit0,type,
numeral_bit0: $tType > $tType ).
thf(ty_t_Extended__Nat_Oenat,type,
extended_enat: $tType ).
thf(ty_t_Multiset_Omultiset,type,
multiset: $tType > $tType ).
thf(ty_t_Complex_Ocomplex,type,
complex: $tType ).
thf(ty_t_Assertions_Oassn,type,
assn: $tType ).
thf(ty_t_String_Oliteral,type,
literal: $tType ).
thf(ty_t_Enum_Ofinite__3,type,
finite_3: $tType ).
thf(ty_t_Enum_Ofinite__2,type,
finite_2: $tType ).
thf(ty_t_Enum_Ofinite__1,type,
finite_1: $tType ).
thf(ty_t_Uint32_Ouint32,type,
uint32: $tType ).
thf(ty_t_Sum__Type_Osum,type,
sum_sum: $tType > $tType > $tType ).
thf(ty_t_Option_Ooption,type,
option: $tType > $tType ).
thf(ty_t_Filter_Ofilter,type,
filter: $tType > $tType ).
thf(ty_t_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 (468)
thf(sy_cl_Complete__Lattices_Ocomplete__linorder,type,
comple5582772986160207858norder:
!>[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_Groups_Ominus,type,
minus:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oabs__if,type,
abs_if:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oring__1,type,
ring_1:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ouminus,type,
uminus:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Obot,type,
bot:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oord,type,
ord:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Otop,type,
top:
!>[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_Lattices_Olattice,type,
lattice:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Ono__bot,type,
no_bot:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Ono__top,type,
no_top:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemiring__0,type,
semiring_0:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemiring__1,type,
semiring_1:
!>[A: $tType] : $o ).
thf(sy_cl_Type__Length_Olen0,type,
type_len0:
!>[A: $tType] : $o ).
thf(sy_cl_Cardinality_Ocard2,type,
card2:
!>[A: $tType] : $o ).
thf(sy_cl_Finite__Set_Ofinite,type,
finite_finite:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Omonoid__add,type,
monoid_add:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Ocomm__ring__1,type,
comm_ring_1:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oidom__divide,type,
idom_divide:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oidom__modulo,type,
idom_modulo:
!>[A: $tType] : $o ).
thf(sy_cl_Transcendental_Oln,type,
ln:
!>[A: $tType] : $o ).
thf(sy_cl_Cardinality_OCARD__1,type,
cARD_1:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Omonoid__mult,type,
monoid_mult:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Olinorder,type,
linorder:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Opreorder,type,
preorder:
!>[A: $tType] : $o ).
thf(sy_cl_Parity_Oring__parity,type,
ring_parity:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oidom__abs__sgn,type,
idom_abs_sgn:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oordered__ring,type,
ordered_ring:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Ozero__neq__one,type,
zero_neq_one:
!>[A: $tType] : $o ).
thf(sy_cl_Fields_Ofield__char__0,type,
field_char_0:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oab__group__add,type,
ab_group_add:
!>[A: $tType] : $o ).
thf(sy_cl_Nat_Osemiring__char__0,type,
semiring_char_0:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oorder__bot,type,
order_bot:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oorder__top,type,
order_top:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Owellorder,type,
wellorder:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Ocomm__semiring,type,
comm_semiring:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Ozero__less__one,type,
zero_less_one:
!>[A: $tType] : $o ).
thf(sy_cl_Fields_Odivision__ring,type,
division_ring:
!>[A: $tType] : $o ).
thf(sy_cl_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__1,type,
comm_semiring_1:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__idom,type,
linordered_idom:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__ring,type,
linordered_ring:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemiring__modulo,type,
semiring_modulo:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocomm__monoid__add,type,
comm_monoid_add:
!>[A: $tType] : $o ).
thf(sy_cl_Parity_Osemiring__parity,type,
semiring_parity:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oordered__ring__abs,type,
ordered_ring_abs:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oordered__semiring,type,
ordered_semiring:
!>[A: $tType] : $o ).
thf(sy_cl_Fields_Olinordered__field,type,
linordered_field:
!>[A: $tType] : $o ).
thf(sy_cl_Generic__set__bit_Oset__bit,type,
generic_set_set_bit:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oab__semigroup__add,type,
ab_semigroup_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocomm__monoid__diff,type,
comm_monoid_diff:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocomm__monoid__mult,type,
comm_monoid_mult:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oalgebraic__semidom,type,
algebraic_semidom:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemiring__1__cancel,type,
semiring_1_cancel:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oab__semigroup__mult,type,
ab_semigroup_mult:
!>[A: $tType] : $o ).
thf(sy_cl_Lattices_Obounded__lattice,type,
bounded_lattice:
!>[A: $tType] : $o ).
thf(sy_cl_Lattices_Osemilattice__sup,type,
semilattice_sup:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Odense__linorder,type,
dense_linorder:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__semidom,type,
linordered_semidom:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oordered__semiring__0,type,
ordered_semiring_0:
!>[A: $tType] : $o ).
thf(sy_cl_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_Lattices_Obounded__lattice__top,type,
bounded_lattice_top:
!>[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_Rings_Osemiring__no__zero__divisors,type,
semiri3467727345109120633visors:
!>[A: $tType] : $o ).
thf(sy_cl_Boolean__Algebras_Oboolean__algebra,type,
boolea8198339166811842893lgebra:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__semiring__strict,type,
linord8928482502909563296strict:
!>[A: $tType] : $o ).
thf(sy_cl_Topological__Spaces_Operfect__space,type,
topolo8386298272705272623_space:
!>[A: $tType] : $o ).
thf(sy_cl_Topological__Spaces_Ouniform__space,type,
topolo7287701948861334536_space:
!>[A: $tType] : $o ).
thf(sy_cl_Limits_Otopological__semigroup__mult,type,
topolo4211221413907600880p_mult:
!>[A: $tType] : $o ).
thf(sy_cl_Real__Vector__Spaces_Ocomplete__space,type,
real_V8037385150606011577_space:
!>[A: $tType] : $o ).
thf(sy_cl_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_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_Lattices_Obounded__semilattice__sup__bot,type,
bounde4967611905675639751up_bot:
!>[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_Topological__Spaces_Olinear__continuum__topology,type,
topolo8458572112393995274pology:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__ab__semigroup__monoid__add__imp__le,type,
ordere1937475149494474687imp_le:
!>[A: $tType] : $o ).
thf(sy_cl_Conditionally__Complete__Lattices_Olinear__continuum,type,
condit5016429287641298734tinuum:
!>[A: $tType] : $o ).
thf(sy_cl_Euclidean__Division_Ounique__euclidean__ring__with__nat,type,
euclid8789492081693882211th_nat:
!>[A: $tType] : $o ).
thf(sy_cl_Euclidean__Division_Ounique__euclidean__semiring__with__nat,type,
euclid5411537665997757685th_nat:
!>[A: $tType] : $o ).
thf(sy_cl_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct,type,
semiri1453513574482234551roduct:
!>[A: $tType] : $o ).
thf(sy_cl_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations,type,
bit_un5681908812861735899ations:
!>[A: $tType] : $o ).
thf(sy_cl_Conditionally__Complete__Lattices_Oconditionally__complete__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_Assertions_Opure__assn,type,
pure_assn: $o > assn ).
thf(sy_c_BNF__Wellorder__Constructions_OFunc,type,
bNF_Wellorder_Func:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( set @ B ) > ( set @ ( A > B ) ) ) ).
thf(sy_c_BNF__Wellorder__Constructions_OFunc__map,type,
bNF_We4925052301507509544nc_map:
!>[B: $tType,C: $tType,A: $tType,D: $tType] : ( ( set @ B ) > ( C > A ) > ( B > D ) > ( D > C ) > B > A ) ).
thf(sy_c_Binomial_Obinomial,type,
binomial: nat > nat > nat ).
thf(sy_c_Binomial_Ogbinomial,type,
gbinomial:
!>[A: $tType] : ( A > nat > A ) ).
thf(sy_c_Bit__Comprehension_Obit__comprehension__class_Oset__bits,type,
bit_bi4170147762399595738t_bits:
!>[A: $tType] : ( ( nat > $o ) > A ) ).
thf(sy_c_Bit__Comprehension_Owf__set__bits__int,type,
bit_wf_set_bits_int: ( nat > $o ) > $o ).
thf(sy_c_Bit__Operations_Oand__int__rel,type,
bit_and_int_rel: ( product_prod @ int @ int ) > ( product_prod @ int @ int ) > $o ).
thf(sy_c_Bit__Operations_Oconcat__bit,type,
bit_concat_bit: nat > int > int > int ).
thf(sy_c_Bit__Operations_Oor__not__num__neg,type,
bit_or_not_num_neg: num > num > num ).
thf(sy_c_Bit__Operations_Oring__bit__operations__class_Onot,type,
bit_ri4277139882892585799ns_not:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Bit__Operations_Oring__bit__operations__class_Osigned__take__bit,type,
bit_ri4674362597316999326ke_bit:
!>[A: $tType] : ( nat > A > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oand,type,
bit_se5824344872417868541ns_and:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Odrop__bit,type,
bit_se4197421643247451524op_bit:
!>[A: $tType] : ( nat > A > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oflip__bit,type,
bit_se8732182000553998342ip_bit:
!>[A: $tType] : ( nat > A > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Omask,type,
bit_se2239418461657761734s_mask:
!>[A: $tType] : ( nat > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oor,type,
bit_se1065995026697491101ons_or:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Opush__bit,type,
bit_se4730199178511100633sh_bit:
!>[A: $tType] : ( nat > A > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oset__bit,type,
bit_se5668285175392031749et_bit:
!>[A: $tType] : ( nat > A > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Otake__bit,type,
bit_se2584673776208193580ke_bit:
!>[A: $tType] : ( nat > A > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Ounset__bit,type,
bit_se2638667681897837118et_bit:
!>[A: $tType] : ( nat > A > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oxor,type,
bit_se5824344971392196577ns_xor:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bits__class_Obit,type,
bit_se5641148757651400278ts_bit:
!>[A: $tType] : ( A > nat > $o ) ).
thf(sy_c_Bit__Operations_Osemiring__bits__class_Opossible__bit,type,
bit_se6407376104438227557le_bit:
!>[A: $tType] : ( ( itself @ A ) > nat > $o ) ).
thf(sy_c_Bit__Shifts__Infix__Syntax_Osemiring__bit__operations__class_Oshiftl,type,
bit_Sh4282982442137083160shiftl:
!>[A: $tType] : ( A > nat > A ) ).
thf(sy_c_Bit__Shifts__Infix__Syntax_Osemiring__bit__operations__class_Oshiftr,type,
bit_Sh4282982442137083166shiftr:
!>[A: $tType] : ( A > nat > A ) ).
thf(sy_c_Bit__Shifts__Infix__Syntax_Osshiftr,type,
bit_Sh8784991116023147202shiftr:
!>[A: $tType] : ( ( word @ A ) > nat > ( word @ A ) ) ).
thf(sy_c_Bits__Integer_OBit__integer,type,
bits_Bit_integer: code_integer > $o > code_integer ).
thf(sy_c_Bits__Integer_Obin__last__integer,type,
bits_b8758750999018896077nteger: code_integer > $o ).
thf(sy_c_Bits__Integer_Obin__rest__integer,type,
bits_b2549910563261871055nteger: code_integer > code_integer ).
thf(sy_c_Bits__Integer_Ointeger__set__bit,type,
bits_integer_set_bit: code_integer > code_integer > $o > code_integer ).
thf(sy_c_Bits__Integer_Ointeger__shiftl,type,
bits_integer_shiftl: code_integer > code_integer > code_integer ).
thf(sy_c_Bits__Integer_Ointeger__shiftr,type,
bits_integer_shiftr: code_integer > code_integer > code_integer ).
thf(sy_c_Boolean__Algebras_Oabstract__boolean__algebra__sym__diff,type,
boolea3799213064322606851m_diff:
!>[A: $tType] : ( ( A > A > A ) > ( A > A > A ) > ( A > A ) > A > A > ( A > A > A ) > $o ) ).
thf(sy_c_Code__Int__Integer__Conversion_Oint__of__integer__symbolic,type,
code_I935103866777955880mbolic: code_integer > int ).
thf(sy_c_Code__Numeral_Obit__cut__integer,type,
code_bit_cut_integer: code_integer > ( product_prod @ code_integer @ $o ) ).
thf(sy_c_Code__Numeral_Odivmod__abs,type,
code_divmod_abs: code_integer > code_integer > ( product_prod @ code_integer @ code_integer ) ).
thf(sy_c_Code__Numeral_Odivmod__integer,type,
code_divmod_integer: code_integer > code_integer > ( product_prod @ code_integer @ code_integer ) ).
thf(sy_c_Code__Numeral_Odup,type,
code_dup: code_integer > code_integer ).
thf(sy_c_Code__Numeral_Ointeger_Oint__of__integer,type,
code_int_of_integer: code_integer > int ).
thf(sy_c_Code__Numeral_Ointeger_Ointeger__of__int,type,
code_integer_of_int: int > code_integer ).
thf(sy_c_Code__Numeral_Ointeger__of__nat,type,
code_integer_of_nat: nat > code_integer ).
thf(sy_c_Code__Numeral_Onat__of__integer,type,
code_nat_of_integer: code_integer > nat ).
thf(sy_c_Code__Numeral_Onum__of__integer,type,
code_num_of_integer: code_integer > num ).
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_Oset__bits__aux,type,
code_T2661198915054445665ts_aux:
!>[A: $tType] : ( ( nat > $o ) > nat > ( word @ A ) > ( word @ A ) ) ).
thf(sy_c_Complex_OArg,type,
arg: complex > real ).
thf(sy_c_Complex_Ocis,type,
cis: real > complex ).
thf(sy_c_Complex_Ocnj,type,
cnj: complex > complex ).
thf(sy_c_Complex_Ocomplex_OComplex,type,
complex2: real > real > complex ).
thf(sy_c_Complex_Ocomplex_OIm,type,
im: complex > real ).
thf(sy_c_Complex_Ocomplex_ORe,type,
re: complex > real ).
thf(sy_c_Complex_Ocsqrt,type,
csqrt: complex > complex ).
thf(sy_c_Complex_Oimaginary__unit,type,
imaginary_unit: complex ).
thf(sy_c_Deriv_Odifferentiable,type,
differentiable:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( filter @ A ) > $o ) ).
thf(sy_c_Deriv_Ohas__derivative,type,
has_derivative:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( A > B ) > ( filter @ A ) > $o ) ).
thf(sy_c_Deriv_Ohas__field__derivative,type,
has_field_derivative:
!>[A: $tType] : ( ( A > A ) > A > ( filter @ A ) > $o ) ).
thf(sy_c_Divides_Oadjust__div,type,
adjust_div: ( product_prod @ int @ int ) > int ).
thf(sy_c_Divides_Odivmod__nat,type,
divmod_nat: nat > nat > ( product_prod @ nat @ nat ) ).
thf(sy_c_Divides_Oeucl__rel__int,type,
eucl_rel_int: int > int > ( product_prod @ int @ int ) > $o ).
thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivides__aux,type,
unique5940410009612947441es_aux:
!>[A: $tType] : ( ( product_prod @ A @ A ) > $o ) ).
thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod,type,
unique8689654367752047608divmod:
!>[A: $tType] : ( num > num > ( product_prod @ A @ A ) ) ).
thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod__step,type,
unique1321980374590559556d_step:
!>[A: $tType] : ( num > ( product_prod @ A @ A ) > ( product_prod @ A @ A ) ) ).
thf(sy_c_Factorial_Ocomm__semiring__1__class_Opochhammer,type,
comm_s3205402744901411588hammer:
!>[A: $tType] : ( A > nat > A ) ).
thf(sy_c_Factorial_Osemiring__char__0__class_Ofact,type,
semiring_char_0_fact:
!>[A: $tType] : ( nat > A ) ).
thf(sy_c_Fields_Oinverse__class_Oinverse,type,
inverse_inverse:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Filter_Oat__bot,type,
at_bot:
!>[A: $tType] : ( filter @ A ) ).
thf(sy_c_Filter_Oat__top,type,
at_top:
!>[A: $tType] : ( filter @ A ) ).
thf(sy_c_Filter_Oeventually,type,
eventually:
!>[A: $tType] : ( ( A > $o ) > ( filter @ A ) > $o ) ).
thf(sy_c_Filter_Ofilterlim,type,
filterlim:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( filter @ B ) > ( filter @ A ) > $o ) ).
thf(sy_c_Filter_Omap__filter__on,type,
map_filter_on:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B ) > ( filter @ A ) > ( filter @ B ) ) ).
thf(sy_c_Finite__Set_Ocard,type,
finite_card:
!>[B: $tType] : ( ( set @ B ) > nat ) ).
thf(sy_c_Finite__Set_Ofinite,type,
finite_finite2:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_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__prod,type,
hash_hash_code_prod:
!>[A: $tType,B: $tType] : ( ( A > uint32 ) > ( B > uint32 ) > ( product_prod @ A @ B ) > uint32 ) ).
thf(sy_c_Hoare__Triple_Ohoare__triple,type,
hoare_hoare_triple:
!>[A: $tType] : ( assn > ( heap_Time_Heap @ A ) > ( A > assn ) > $o ) ).
thf(sy_c_If,type,
if:
!>[A: $tType] : ( $o > A > A > A ) ).
thf(sy_c_Int_Oint__ge__less__than,type,
int_ge_less_than: int > ( set @ ( product_prod @ int @ int ) ) ).
thf(sy_c_Int_Oint__ge__less__than2,type,
int_ge_less_than2: int > ( set @ ( product_prod @ int @ int ) ) ).
thf(sy_c_Int_Onat,type,
nat2: int > nat ).
thf(sy_c_Int_Oring__1__class_OInts,type,
ring_1_Ints:
!>[A: $tType] : ( set @ A ) ).
thf(sy_c_Int_Oring__1__class_Oof__int,type,
ring_1_of_int:
!>[A: $tType] : ( int > A ) ).
thf(sy_c_Lattices_Osup__class_Osup,type,
sup_sup:
!>[A: $tType] : ( A > 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_Ocount__list,type,
count_list:
!>[A: $tType] : ( ( list @ A ) > A > nat ) ).
thf(sy_c_List_Odistinct,type,
distinct:
!>[A: $tType] : ( ( list @ A ) > $o ) ).
thf(sy_c_List_Olist_OCons,type,
cons:
!>[A: $tType] : ( A > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Olist_Oset,type,
set2:
!>[A: $tType] : ( ( list @ A ) > ( set @ 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_Ounion,type,
union:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Oupto__aux,type,
upto_aux: int > int > ( list @ int ) > ( list @ int ) ).
thf(sy_c_List_Oupto__rel,type,
upto_rel: ( product_prod @ int @ int ) > ( product_prod @ int @ int ) > $o ).
thf(sy_c_Misc_Ofun__of__rel,type,
fun_of_rel:
!>[B: $tType,A: $tType] : ( ( set @ ( product_prod @ B @ A ) ) > B > A ) ).
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_Osemiring__1__class_Oof__nat,type,
semiring_1_of_nat:
!>[A: $tType] : ( nat > A ) ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat__aux,type,
semiri8178284476397505188at_aux:
!>[A: $tType] : ( ( A > A ) > nat > A > A ) ).
thf(sy_c_Nat_Osize__class_Osize,type,
size_size:
!>[A: $tType] : ( A > nat ) ).
thf(sy_c_Nat__Bijection_Oset__decode,type,
nat_set_decode: nat > ( set @ nat ) ).
thf(sy_c_Nat__Bijection_Oset__encode,type,
nat_set_encode: ( set @ nat ) > nat ).
thf(sy_c_Nat__Bijection_Otriangle,type,
nat_triangle: nat > nat ).
thf(sy_c_NthRoot_Oroot,type,
root: nat > real > real ).
thf(sy_c_NthRoot_Osqrt,type,
sqrt: real > real ).
thf(sy_c_Num_OBitM,type,
bitM: num > num ).
thf(sy_c_Num_Oinc,type,
inc: num > num ).
thf(sy_c_Num_Oneg__numeral__class_Odbl,type,
neg_numeral_dbl:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__dec,type,
neg_numeral_dbl_dec:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__inc,type,
neg_numeral_dbl_inc:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Num_Onum_OBit0,type,
bit0: num > num ).
thf(sy_c_Num_Onum_OBit1,type,
bit1: num > num ).
thf(sy_c_Num_Onum_OOne,type,
one2: num ).
thf(sy_c_Num_Onum_Osize__num,type,
size_num: num > nat ).
thf(sy_c_Num_Onumeral__class_Onumeral,type,
numeral_numeral:
!>[A: $tType] : ( num > A ) ).
thf(sy_c_Num_Opow,type,
pow: num > num > num ).
thf(sy_c_Num_Opred__numeral,type,
pred_numeral: num > nat ).
thf(sy_c_Numeral__Type_OAbs__bit1_H,type,
numeral_Abs_bit1:
!>[A: $tType] : ( int > ( numeral_bit1 @ A ) ) ).
thf(sy_c_Numeral__Type_Obit1_OAbs__bit1,type,
numeral_Abs_bit12:
!>[A: $tType] : ( int > ( numeral_bit1 @ A ) ) ).
thf(sy_c_Numeral__Type_Obit1_ORep__bit1,type,
numeral_Rep_bit1:
!>[A: $tType] : ( ( numeral_bit1 @ A ) > int ) ).
thf(sy_c_Orderings_Obot__class_Obot,type,
bot_bot:
!>[A: $tType] : A ).
thf(sy_c_Orderings_Oord__class_Oless,type,
ord_less:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Orderings_Oord__class_Oless__eq,type,
ord_less_eq:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Orderings_Oord__class_Omax,type,
ord_max:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Orderings_Oord__class_Omin,type,
ord_min:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Orderings_Oorder__class_OGreatest,type,
order_Greatest:
!>[A: $tType] : ( ( A > $o ) > A ) ).
thf(sy_c_Orderings_Oorder__class_Oantimono,type,
order_antimono:
!>[A: $tType,B: $tType] : ( ( A > B ) > $o ) ).
thf(sy_c_Orderings_Otop__class_Otop,type,
top_top:
!>[A: $tType] : A ).
thf(sy_c_Power_Opower__class_Opower,type,
power_power:
!>[A: $tType] : ( A > nat > A ) ).
thf(sy_c_Product__Type_OPair,type,
product_Pair:
!>[A: $tType,B: $tType] : ( A > B > ( product_prod @ A @ B ) ) ).
thf(sy_c_Product__Type_Oapsnd,type,
product_apsnd:
!>[B: $tType,C: $tType,A: $tType] : ( ( B > C ) > ( product_prod @ A @ B ) > ( product_prod @ A @ C ) ) ).
thf(sy_c_Product__Type_Oprod_Ocase__prod,type,
product_case_prod:
!>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
thf(sy_c_Pure_Otype,type,
type2:
!>[A: $tType] : ( itself @ A ) ).
thf(sy_c_Rat_OFrct,type,
frct: ( product_prod @ int @ int ) > rat ).
thf(sy_c_Rat_Onormalize,type,
normalize: ( product_prod @ int @ int ) > ( product_prod @ int @ int ) ).
thf(sy_c_Rat_Oof__int,type,
of_int: int > rat ).
thf(sy_c_Rat_Oquotient__of,type,
quotient_of: rat > ( product_prod @ int @ int ) ).
thf(sy_c_Real__Vector__Spaces_OReals,type,
real_Vector_Reals:
!>[A: $tType] : ( set @ A ) ).
thf(sy_c_Real__Vector__Spaces_Obounded__linear,type,
real_V3181309239436604168linear:
!>[A: $tType,B: $tType] : ( ( A > B ) > $o ) ).
thf(sy_c_Real__Vector__Spaces_Odist__class_Odist,type,
real_V557655796197034286t_dist:
!>[A: $tType] : ( A > A > real ) ).
thf(sy_c_Real__Vector__Spaces_Onorm__class_Onorm,type,
real_V7770717601297561774m_norm:
!>[A: $tType] : ( A > real ) ).
thf(sy_c_Real__Vector__Spaces_Oof__real,type,
real_Vector_of_real:
!>[A: $tType] : ( real > A ) ).
thf(sy_c_Real__Vector__Spaces_OscaleR__class_OscaleR,type,
real_V8093663219630862766scaleR:
!>[A: $tType] : ( real > A > A ) ).
thf(sy_c_Rings_Odivide__class_Odivide,type,
divide_divide:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Rings_Odvd__class_Odvd,type,
dvd_dvd:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Rings_Omodulo__class_Omodulo,type,
modulo_modulo:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool,type,
zero_neq_one_of_bool:
!>[A: $tType] : ( $o > A ) ).
thf(sy_c_Series_Osuminf,type,
suminf:
!>[A: $tType] : ( ( nat > A ) > A ) ).
thf(sy_c_Series_Osummable,type,
summable:
!>[A: $tType] : ( ( nat > A ) > $o ) ).
thf(sy_c_Series_Osums,type,
sums:
!>[A: $tType] : ( ( nat > A ) > A > $o ) ).
thf(sy_c_Set_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_Ois__singleton,type,
is_singleton:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
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_Time__Reasoning_OTBOUND,type,
time_TBOUND:
!>[A: $tType] : ( ( heap_Time_Heap @ A ) > nat > $o ) ).
thf(sy_c_Time__Reasoning_Ohtt,type,
time_htt:
!>[A: $tType] : ( assn > ( heap_Time_Heap @ A ) > ( A > assn ) > nat > $o ) ).
thf(sy_c_Topological__Spaces_Ocontinuous,type,
topolo3448309680560233919inuous:
!>[A: $tType,B: $tType] : ( ( filter @ A ) > ( A > B ) > $o ) ).
thf(sy_c_Topological__Spaces_Ocontinuous__on,type,
topolo81223032696312382ous_on:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B ) > $o ) ).
thf(sy_c_Topological__Spaces_Omonoseq,type,
topological_monoseq:
!>[A: $tType] : ( ( nat > A ) > $o ) ).
thf(sy_c_Topological__Spaces_Oopen__class_Oopen,type,
topolo1002775350975398744n_open:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_Topological__Spaces_Ot2__space__class_OLim,type,
topolo3827282254853284352ce_Lim:
!>[F: $tType,A: $tType] : ( ( filter @ F ) > ( F > A ) > A ) ).
thf(sy_c_Topological__Spaces_Otopological__space__class_Oat__within,type,
topolo174197925503356063within:
!>[A: $tType] : ( A > ( set @ A ) > ( filter @ A ) ) ).
thf(sy_c_Topological__Spaces_Otopological__space__class_Onhds,type,
topolo7230453075368039082e_nhds:
!>[A: $tType] : ( A > ( filter @ A ) ) ).
thf(sy_c_Topological__Spaces_Ouniform__space__class_OCauchy,type,
topolo3814608138187158403Cauchy:
!>[A: $tType] : ( ( nat > A ) > $o ) ).
thf(sy_c_Topological__Spaces_Ouniform__space__class_Ocauchy__filter,type,
topolo6773858410816713723filter:
!>[A: $tType] : ( ( filter @ A ) > $o ) ).
thf(sy_c_Transcendental_Oarccos,type,
arccos: real > real ).
thf(sy_c_Transcendental_Oarcosh,type,
arcosh:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Transcendental_Oarcsin,type,
arcsin: real > real ).
thf(sy_c_Transcendental_Oarctan,type,
arctan: real > real ).
thf(sy_c_Transcendental_Oarsinh,type,
arsinh:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Transcendental_Oartanh,type,
artanh:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Transcendental_Ocos,type,
cos:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Transcendental_Ocos__coeff,type,
cos_coeff: nat > real ).
thf(sy_c_Transcendental_Ocosh,type,
cosh:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Transcendental_Ocot,type,
cot:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Transcendental_Odiffs,type,
diffs:
!>[A: $tType] : ( ( nat > A ) > nat > A ) ).
thf(sy_c_Transcendental_Oexp,type,
exp:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Transcendental_Oln__class_Oln,type,
ln_ln:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Transcendental_Olog,type,
log: real > real > real ).
thf(sy_c_Transcendental_Opi,type,
pi: real ).
thf(sy_c_Transcendental_Opowr,type,
powr:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Transcendental_Osin,type,
sin:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Transcendental_Osin__coeff,type,
sin_coeff: nat > real ).
thf(sy_c_Transcendental_Osinh,type,
sinh:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Transcendental_Otan,type,
tan:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Transcendental_Otanh,type,
tanh:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Type__Length_Olen0__class_Olen__of,type,
type_len0_len_of:
!>[A: $tType] : ( ( itself @ A ) > nat ) ).
thf(sy_c_Typedef_Otype__definition,type,
type_definition:
!>[B: $tType,A: $tType] : ( ( B > A ) > ( A > B ) > ( set @ A ) > $o ) ).
thf(sy_c_Uint32_ORep__uint32_H,type,
rep_uint32: uint32 > ( word @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ numeral_num1 ) ) ) ) ) ) ).
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_Uint32_Odiv0__uint32,type,
div0_uint32: uint32 > uint32 ).
thf(sy_c_Uint32_Ointeger__of__uint32,type,
integer_of_uint32: uint32 > code_integer ).
thf(sy_c_Uint32_Ointeger__of__uint32__signed,type,
intege5370686899274169573signed: uint32 > code_integer ).
thf(sy_c_Uint32_Omod0__uint32,type,
mod0_uint32: uint32 > uint32 ).
thf(sy_c_Uint32_Oset__bits__aux__uint32,type,
set_bits_aux_uint32: ( nat > $o ) > nat > uint32 > uint32 ).
thf(sy_c_Uint32_Osigned__drop__bit__uint32,type,
signed489701013188660438uint32: nat > uint32 > uint32 ).
thf(sy_c_Uint32_Ouint32_OAbs__uint32,type,
abs_uint32: ( word @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ numeral_num1 ) ) ) ) ) ) > uint32 ).
thf(sy_c_Uint32_Ouint32_ORep__uint32,type,
rep_uint322: uint32 > ( word @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ numeral_num1 ) ) ) ) ) ) ).
thf(sy_c_Uint32_Ouint32__div,type,
uint32_div: uint32 > uint32 > uint32 ).
thf(sy_c_Uint32_Ouint32__divmod,type,
uint32_divmod: uint32 > uint32 > ( product_prod @ uint32 @ uint32 ) ).
thf(sy_c_Uint32_Ouint32__mod,type,
uint32_mod: uint32 > uint32 > uint32 ).
thf(sy_c_Uint32_Ouint32__sdiv,type,
uint32_sdiv: uint32 > uint32 > uint32 ).
thf(sy_c_Uint32_Ouint32__set__bit,type,
uint32_set_bit: uint32 > code_integer > $o > uint32 ).
thf(sy_c_Uint32_Ouint32__shiftl,type,
uint32_shiftl: uint32 > code_integer > uint32 ).
thf(sy_c_Uint32_Ouint32__shiftr,type,
uint32_shiftr: uint32 > code_integer > uint32 ).
thf(sy_c_Uint32_Ouint32__sshiftr,type,
uint32_sshiftr: uint32 > code_integer > uint32 ).
thf(sy_c_Uint32_Ouint32__test__bit,type,
uint32_test_bit: uint32 > code_integer > $o ).
thf(sy_c_VEBT__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__buildupi,type,
vEBT_vebt_buildupi: nat > ( heap_Time_Heap @ vEBT_VEBTi ) ).
thf(sy_c_VEBT__BuildupMemImp_Ovebt__inserti,type,
vEBT_vebt_inserti: vEBT_VEBTi > nat > ( heap_Time_Heap @ vEBT_VEBTi ) ).
thf(sy_c_VEBT__BuildupMemImp_Ovebt__memberi,type,
vEBT_vebt_memberi: vEBT_VEBTi > nat > ( heap_Time_Heap @ $o ) ).
thf(sy_c_VEBT__DelImperative_Ovebt__deletei,type,
vEBT_vebt_deletei: vEBT_VEBTi > nat > ( heap_Time_Heap @ vEBT_VEBTi ) ).
thf(sy_c_VEBT__Example__Setup_Omfold,type,
vEBT_Example_mfold:
!>[A: $tType,B: $tType] : ( ( A > B > ( heap_Time_Heap @ B ) ) > ( list @ A ) > B > ( heap_Time_Heap @ B ) ) ).
thf(sy_c_VEBT__Intf__Imperative_Ovebt__assn,type,
vEBT_Intf_vebt_assn: nat > ( set @ nat ) > vEBT_VEBTi > assn ).
thf(sy_c_VEBT__Member_OVEBT__internal_Obit__concat,type,
vEBT_VEBT_bit_concat: nat > nat > 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,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_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_Ocast,type,
cast:
!>[A: $tType,B: $tType] : ( ( word @ A ) > ( word @ B ) ) ).
thf(sy_c_Word_Oeven__word,type,
even_word:
!>[A: $tType] : ( ( word @ A ) > $o ) ).
thf(sy_c_Word_Oof__int,type,
of_int2:
!>[A: $tType] : ( int > ( word @ A ) ) ).
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__int,type,
the_int:
!>[A: $tType] : ( ( word @ A ) > int ) ).
thf(sy_c_Word_Othe__nat,type,
the_nat:
!>[A: $tType] : ( ( word @ A ) > nat ) ).
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_n,type,
n: nat ).
thf(sy_v_s,type,
s: set @ nat ).
thf(sy_v_t,type,
t: vEBT_VEBTi ).
thf(sy_v_xs,type,
xs: list @ nat ).
% Relevant facts (8178)
thf(fact_0_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_1_semiring__norm_I85_J,axiom,
! [M: num] :
( ( bit0 @ M )
!= one2 ) ).
% semiring_norm(85)
thf(fact_2_semiring__norm_I83_J,axiom,
! [N: num] :
( one2
!= ( bit0 @ N ) ) ).
% semiring_norm(83)
thf(fact_3_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_4_UnCI,axiom,
! [A: $tType,C2: A,B2: set @ A,A2: set @ A] :
( ( ~ ( member @ A @ C2 @ B2 )
=> ( member @ A @ C2 @ A2 ) )
=> ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) ) ) ).
% UnCI
thf(fact_5_Un__iff,axiom,
! [A: $tType,C2: A,A2: set @ A,B2: set @ A] :
( ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) )
= ( ( member @ A @ C2 @ A2 )
| ( member @ A @ C2 @ B2 ) ) ) ).
% Un_iff
thf(fact_6_sup_Oidem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: A] :
( ( sup_sup @ A @ A3 @ A3 )
= A3 ) ) ).
% sup.idem
thf(fact_7_sup__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X: A] :
( ( sup_sup @ A @ X @ X )
= X ) ) ).
% sup_idem
thf(fact_8_sup_Oleft__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: A,B3: A] :
( ( sup_sup @ A @ A3 @ ( sup_sup @ A @ A3 @ B3 ) )
= ( sup_sup @ A @ A3 @ B3 ) ) ) ).
% sup.left_idem
thf(fact_9_sup__left__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X: A,Y: A] :
( ( sup_sup @ A @ X @ ( sup_sup @ A @ X @ Y ) )
= ( sup_sup @ A @ X @ Y ) ) ) ).
% sup_left_idem
thf(fact_10_sup_Oright__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: A,B3: A] :
( ( sup_sup @ A @ ( sup_sup @ A @ A3 @ B3 ) @ B3 )
= ( sup_sup @ A @ A3 @ B3 ) ) ) ).
% sup.right_idem
thf(fact_11_sup__apply,axiom,
! [B: $tType,A: $tType] :
( ( semilattice_sup @ B )
=> ( ( sup_sup @ ( A > B ) )
= ( ^ [F2: A > B,G: A > B,X2: A] : ( sup_sup @ B @ ( F2 @ X2 ) @ ( G @ X2 ) ) ) ) ) ).
% sup_apply
thf(fact_12_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_13_semiring__norm_I87_J,axiom,
! [M: num,N: num] :
( ( ( bit0 @ M )
= ( bit0 @ N ) )
= ( M = N ) ) ).
% semiring_norm(87)
thf(fact_14_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_15_semiring__norm_I75_J,axiom,
! [M: num] :
~ ( ord_less @ num @ M @ one2 ) ).
% semiring_norm(75)
thf(fact_16_semiring__norm_I76_J,axiom,
! [N: num] : ( ord_less @ num @ one2 @ ( bit0 @ N ) ) ).
% semiring_norm(76)
thf(fact_17_sup__set__def,axiom,
! [A: $tType] :
( ( sup_sup @ ( set @ A ) )
= ( ^ [A4: set @ A,B4: set @ A] :
( collect @ A
@ ( sup_sup @ ( A > $o )
@ ^ [X2: A] : ( member @ A @ X2 @ A4 )
@ ^ [X2: A] : ( member @ A @ X2 @ B4 ) ) ) ) ) ).
% sup_set_def
thf(fact_18_sup__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( semilattice_sup @ B )
=> ( ( sup_sup @ ( A > B ) )
= ( ^ [F2: A > B,G: A > B,X2: A] : ( sup_sup @ B @ ( F2 @ X2 ) @ ( G @ X2 ) ) ) ) ) ).
% sup_fun_def
thf(fact_19_sup__left__commute,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X: A,Y: A,Z: A] :
( ( sup_sup @ A @ X @ ( sup_sup @ A @ Y @ Z ) )
= ( sup_sup @ A @ Y @ ( sup_sup @ A @ X @ Z ) ) ) ) ).
% sup_left_commute
thf(fact_20_sup_Oleft__commute,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B3: A,A3: A,C2: A] :
( ( sup_sup @ A @ B3 @ ( sup_sup @ A @ A3 @ C2 ) )
= ( sup_sup @ A @ A3 @ ( sup_sup @ A @ B3 @ C2 ) ) ) ) ).
% sup.left_commute
thf(fact_21_sup__commute,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( sup_sup @ A )
= ( ^ [X2: A,Y2: A] : ( sup_sup @ A @ Y2 @ X2 ) ) ) ) ).
% sup_commute
thf(fact_22_sup_Ocommute,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( sup_sup @ A )
= ( ^ [A5: A,B5: A] : ( sup_sup @ A @ B5 @ A5 ) ) ) ) ).
% sup.commute
thf(fact_23_sup__assoc,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X: A,Y: A,Z: A] :
( ( sup_sup @ A @ ( sup_sup @ A @ X @ Y ) @ Z )
= ( sup_sup @ A @ X @ ( sup_sup @ A @ Y @ Z ) ) ) ) ).
% sup_assoc
thf(fact_24_sup_Oassoc,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: A,B3: A,C2: A] :
( ( sup_sup @ A @ ( sup_sup @ A @ A3 @ B3 ) @ C2 )
= ( sup_sup @ A @ A3 @ ( sup_sup @ A @ B3 @ C2 ) ) ) ) ).
% sup.assoc
thf(fact_25_inf__sup__aci_I5_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ( ( sup_sup @ A )
= ( ^ [X2: A,Y2: A] : ( sup_sup @ A @ Y2 @ X2 ) ) ) ) ).
% inf_sup_aci(5)
thf(fact_26_inf__sup__aci_I6_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X: A,Y: A,Z: A] :
( ( sup_sup @ A @ ( sup_sup @ A @ X @ Y ) @ Z )
= ( sup_sup @ A @ X @ ( sup_sup @ A @ Y @ Z ) ) ) ) ).
% inf_sup_aci(6)
thf(fact_27_inf__sup__aci_I7_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X: A,Y: A,Z: A] :
( ( sup_sup @ A @ X @ ( sup_sup @ A @ Y @ Z ) )
= ( sup_sup @ A @ Y @ ( sup_sup @ A @ X @ Z ) ) ) ) ).
% inf_sup_aci(7)
thf(fact_28_inf__sup__aci_I8_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X: A,Y: A] :
( ( sup_sup @ A @ X @ ( sup_sup @ A @ X @ Y ) )
= ( sup_sup @ A @ X @ Y ) ) ) ).
% inf_sup_aci(8)
thf(fact_29_Un__left__commute,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,C3: set @ A] :
( ( sup_sup @ ( set @ A ) @ A2 @ ( sup_sup @ ( set @ A ) @ B2 @ C3 ) )
= ( sup_sup @ ( set @ A ) @ B2 @ ( sup_sup @ ( set @ A ) @ A2 @ C3 ) ) ) ).
% Un_left_commute
thf(fact_30_Un__left__absorb,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( sup_sup @ ( set @ A ) @ A2 @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) )
= ( sup_sup @ ( set @ A ) @ A2 @ B2 ) ) ).
% Un_left_absorb
thf(fact_31_Un__commute,axiom,
! [A: $tType] :
( ( sup_sup @ ( set @ A ) )
= ( ^ [A4: set @ A,B4: set @ A] : ( sup_sup @ ( set @ A ) @ B4 @ A4 ) ) ) ).
% Un_commute
thf(fact_32_Un__absorb,axiom,
! [A: $tType,A2: set @ A] :
( ( sup_sup @ ( set @ A ) @ A2 @ A2 )
= A2 ) ).
% Un_absorb
thf(fact_33_Un__assoc,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,C3: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) @ C3 )
= ( sup_sup @ ( set @ A ) @ A2 @ ( sup_sup @ ( set @ A ) @ B2 @ C3 ) ) ) ).
% Un_assoc
thf(fact_34_ball__Un,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,P: A > $o] :
( ( ! [X2: A] :
( ( member @ A @ X2 @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) )
=> ( P @ X2 ) ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ A2 )
=> ( P @ X2 ) )
& ! [X2: A] :
( ( member @ A @ X2 @ B2 )
=> ( P @ X2 ) ) ) ) ).
% ball_Un
thf(fact_35_bex__Un,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,P: A > $o] :
( ( ? [X2: A] :
( ( member @ A @ X2 @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) )
& ( P @ X2 ) ) )
= ( ? [X2: A] :
( ( member @ A @ X2 @ A2 )
& ( P @ X2 ) )
| ? [X2: A] :
( ( member @ A @ X2 @ B2 )
& ( P @ X2 ) ) ) ) ).
% bex_Un
thf(fact_36_UnI2,axiom,
! [A: $tType,C2: A,B2: set @ A,A2: set @ A] :
( ( member @ A @ C2 @ B2 )
=> ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) ) ) ).
% UnI2
thf(fact_37_UnI1,axiom,
! [A: $tType,C2: A,A2: set @ A,B2: set @ A] :
( ( member @ A @ C2 @ A2 )
=> ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) ) ) ).
% UnI1
thf(fact_38_UnE,axiom,
! [A: $tType,C2: A,A2: set @ A,B2: set @ A] :
( ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) )
=> ( ~ ( member @ A @ C2 @ A2 )
=> ( member @ A @ C2 @ B2 ) ) ) ).
% UnE
thf(fact_39_Collect__disj__eq,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ( collect @ A
@ ^ [X2: A] :
( ( P @ X2 )
| ( Q @ X2 ) ) )
= ( sup_sup @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% Collect_disj_eq
thf(fact_40_Un__def,axiom,
! [A: $tType] :
( ( sup_sup @ ( set @ A ) )
= ( ^ [A4: set @ A,B4: set @ A] :
( collect @ A
@ ^ [X2: A] :
( ( member @ A @ X2 @ A4 )
| ( member @ A @ X2 @ B4 ) ) ) ) ) ).
% Un_def
thf(fact_41_sup_Ostrict__coboundedI2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [C2: A,B3: A,A3: A] :
( ( ord_less @ A @ C2 @ B3 )
=> ( ord_less @ A @ C2 @ ( sup_sup @ A @ A3 @ B3 ) ) ) ) ).
% sup.strict_coboundedI2
thf(fact_42_sup_Ostrict__coboundedI1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [C2: A,A3: A,B3: A] :
( ( ord_less @ A @ C2 @ A3 )
=> ( ord_less @ A @ C2 @ ( sup_sup @ A @ A3 @ B3 ) ) ) ) ).
% sup.strict_coboundedI1
thf(fact_43_mem__Collect__eq,axiom,
! [A: $tType,A3: A,P: A > $o] :
( ( member @ A @ A3 @ ( collect @ A @ P ) )
= ( P @ A3 ) ) ).
% mem_Collect_eq
thf(fact_44_Collect__mem__eq,axiom,
! [A: $tType,A2: set @ A] :
( ( collect @ A
@ ^ [X2: A] : ( member @ A @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_45_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_46_ext,axiom,
! [B: $tType,A: $tType,F3: A > B,G2: A > B] :
( ! [X3: A] :
( ( F3 @ X3 )
= ( G2 @ X3 ) )
=> ( F3 = G2 ) ) ).
% ext
thf(fact_47_sup_Ostrict__order__iff,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( ord_less @ A )
= ( ^ [B5: A,A5: A] :
( ( A5
= ( sup_sup @ A @ A5 @ B5 ) )
& ( A5 != B5 ) ) ) ) ) ).
% sup.strict_order_iff
thf(fact_48_sup_Ostrict__boundedE,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B3: A,C2: A,A3: A] :
( ( ord_less @ A @ ( sup_sup @ A @ B3 @ C2 ) @ A3 )
=> ~ ( ( ord_less @ A @ B3 @ A3 )
=> ~ ( ord_less @ A @ C2 @ A3 ) ) ) ) ).
% sup.strict_boundedE
thf(fact_49_sup_Oabsorb4,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ A3 @ B3 )
=> ( ( sup_sup @ A @ A3 @ B3 )
= B3 ) ) ) ).
% sup.absorb4
thf(fact_50_sup_Oabsorb3,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B3: A,A3: A] :
( ( ord_less @ A @ B3 @ A3 )
=> ( ( sup_sup @ A @ A3 @ B3 )
= A3 ) ) ) ).
% sup.absorb3
thf(fact_51_less__supI2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X: A,B3: A,A3: A] :
( ( ord_less @ A @ X @ B3 )
=> ( ord_less @ A @ X @ ( sup_sup @ A @ A3 @ B3 ) ) ) ) ).
% less_supI2
thf(fact_52_less__supI1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X: A,A3: A,B3: A] :
( ( ord_less @ A @ X @ A3 )
=> ( ord_less @ A @ X @ ( sup_sup @ A @ A3 @ B3 ) ) ) ) ).
% less_supI1
thf(fact_53_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_54_verit__eq__simplify_I8_J,axiom,
! [X22: num,Y22: num] :
( ( ( bit0 @ X22 )
= ( bit0 @ Y22 ) )
= ( X22 = Y22 ) ) ).
% verit_eq_simplify(8)
thf(fact_55_verit__eq__simplify_I10_J,axiom,
! [X22: num] :
( one2
!= ( bit0 @ X22 ) ) ).
% verit_eq_simplify(10)
thf(fact_56_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_57_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_58_of__nat__less__numeral__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X: nat,I: num,N: nat] :
( ( ord_less @ A @ ( semiring_1_of_nat @ A @ X ) @ ( power_power @ A @ ( numeral_numeral @ A @ I ) @ N ) )
= ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ I ) @ N ) ) ) ) ).
% of_nat_less_numeral_power_cancel_iff
thf(fact_59_numeral__power__less__of__nat__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [I: num,N: nat,X: nat] :
( ( ord_less @ A @ ( power_power @ A @ ( numeral_numeral @ A @ I ) @ N ) @ ( semiring_1_of_nat @ A @ X ) )
= ( ord_less @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ I ) @ N ) @ X ) ) ) ).
% numeral_power_less_of_nat_cancel_iff
thf(fact_60_sup__Un__eq,axiom,
! [A: $tType,R: set @ A,S: set @ A] :
( ( sup_sup @ ( A > $o )
@ ^ [X2: A] : ( member @ A @ X2 @ R )
@ ^ [X2: A] : ( member @ A @ X2 @ S ) )
= ( ^ [X2: A] : ( member @ A @ X2 @ ( sup_sup @ ( set @ A ) @ R @ S ) ) ) ) ).
% sup_Un_eq
thf(fact_61_vebt__heap__rules_I3_J,axiom,
! [X: nat,N: nat,S2: set @ nat,Ti: vEBT_VEBTi] :
( ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
=> ( hoare_hoare_triple @ vEBT_VEBTi @ ( vEBT_Intf_vebt_assn @ N @ S2 @ Ti ) @ ( vEBT_vebt_inserti @ Ti @ X ) @ ( vEBT_Intf_vebt_assn @ N @ ( sup_sup @ ( set @ nat ) @ S2 @ ( insert @ nat @ X @ ( bot_bot @ ( set @ nat ) ) ) ) ) ) ) ).
% vebt_heap_rules(3)
thf(fact_62_power__numeral,axiom,
! [A: $tType] :
( ( semiring_numeral @ A )
=> ! [K: num,L: num] :
( ( power_power @ A @ ( numeral_numeral @ A @ K ) @ ( numeral_numeral @ nat @ L ) )
= ( numeral_numeral @ A @ ( pow @ K @ L ) ) ) ) ).
% power_numeral
thf(fact_63_zero__less__power2,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( A3
!= ( zero_zero @ A ) ) ) ) ).
% zero_less_power2
thf(fact_64_set__union,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( set2 @ A @ ( union @ A @ Xs @ Ys ) )
= ( sup_sup @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ Ys ) ) ) ).
% set_union
thf(fact_65_power__one__right,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A3: A] :
( ( power_power @ A @ A3 @ ( one_one @ nat ) )
= A3 ) ) ).
% power_one_right
thf(fact_66_empty__iff,axiom,
! [A: $tType,C2: A] :
~ ( member @ A @ C2 @ ( bot_bot @ ( set @ A ) ) ) ).
% empty_iff
thf(fact_67_all__not__in__conv,axiom,
! [A: $tType,A2: set @ A] :
( ( ! [X2: A] :
~ ( member @ A @ X2 @ A2 ) )
= ( A2
= ( bot_bot @ ( set @ A ) ) ) ) ).
% all_not_in_conv
thf(fact_68_Collect__empty__eq,axiom,
! [A: $tType,P: A > $o] :
( ( ( collect @ A @ P )
= ( bot_bot @ ( set @ A ) ) )
= ( ! [X2: A] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_69_empty__Collect__eq,axiom,
! [A: $tType,P: A > $o] :
( ( ( bot_bot @ ( set @ A ) )
= ( collect @ A @ P ) )
= ( ! [X2: A] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_70_insertCI,axiom,
! [A: $tType,A3: A,B2: set @ A,B3: A] :
( ( ~ ( member @ A @ A3 @ B2 )
=> ( A3 = B3 ) )
=> ( member @ A @ A3 @ ( insert @ A @ B3 @ B2 ) ) ) ).
% insertCI
thf(fact_71_insert__iff,axiom,
! [A: $tType,A3: A,B3: A,A2: set @ A] :
( ( member @ A @ A3 @ ( insert @ A @ B3 @ A2 ) )
= ( ( A3 = B3 )
| ( member @ A @ A3 @ A2 ) ) ) ).
% insert_iff
thf(fact_72_insert__absorb2,axiom,
! [A: $tType,X: A,A2: set @ A] :
( ( insert @ A @ X @ ( insert @ A @ X @ A2 ) )
= ( insert @ A @ X @ A2 ) ) ).
% insert_absorb2
thf(fact_73_i0__less,axiom,
! [N: extended_enat] :
( ( ord_less @ extended_enat @ ( zero_zero @ extended_enat ) @ N )
= ( N
!= ( zero_zero @ extended_enat ) ) ) ).
% i0_less
thf(fact_74_power__one,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [N: nat] :
( ( power_power @ A @ ( one_one @ A ) @ N )
= ( one_one @ A ) ) ) ).
% power_one
thf(fact_75_singletonI,axiom,
! [A: $tType,A3: A] : ( member @ A @ A3 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% singletonI
thf(fact_76_sup__bot__left,axiom,
! [A: $tType] :
( ( bounde4967611905675639751up_bot @ A )
=> ! [X: A] :
( ( sup_sup @ A @ ( bot_bot @ A ) @ X )
= X ) ) ).
% sup_bot_left
thf(fact_77_sup__bot__right,axiom,
! [A: $tType] :
( ( bounde4967611905675639751up_bot @ A )
=> ! [X: A] :
( ( sup_sup @ A @ X @ ( bot_bot @ A ) )
= X ) ) ).
% sup_bot_right
thf(fact_78_bot__eq__sup__iff,axiom,
! [A: $tType] :
( ( bounde4967611905675639751up_bot @ A )
=> ! [X: A,Y: A] :
( ( ( bot_bot @ A )
= ( sup_sup @ A @ X @ Y ) )
= ( ( X
= ( bot_bot @ A ) )
& ( Y
= ( bot_bot @ A ) ) ) ) ) ).
% bot_eq_sup_iff
thf(fact_79_sup__eq__bot__iff,axiom,
! [A: $tType] :
( ( bounde4967611905675639751up_bot @ A )
=> ! [X: A,Y: A] :
( ( ( sup_sup @ A @ X @ Y )
= ( bot_bot @ A ) )
= ( ( X
= ( bot_bot @ A ) )
& ( Y
= ( bot_bot @ A ) ) ) ) ) ).
% sup_eq_bot_iff
thf(fact_80_sup__bot_Oeq__neutr__iff,axiom,
! [A: $tType] :
( ( bounde4967611905675639751up_bot @ A )
=> ! [A3: A,B3: A] :
( ( ( sup_sup @ A @ A3 @ B3 )
= ( bot_bot @ A ) )
= ( ( A3
= ( bot_bot @ A ) )
& ( B3
= ( bot_bot @ A ) ) ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_81_sup__bot_Oleft__neutral,axiom,
! [A: $tType] :
( ( bounde4967611905675639751up_bot @ A )
=> ! [A3: A] :
( ( sup_sup @ A @ ( bot_bot @ A ) @ A3 )
= A3 ) ) ).
% sup_bot.left_neutral
thf(fact_82_sup__bot_Oneutr__eq__iff,axiom,
! [A: $tType] :
( ( bounde4967611905675639751up_bot @ A )
=> ! [A3: A,B3: A] :
( ( ( bot_bot @ A )
= ( sup_sup @ A @ A3 @ B3 ) )
= ( ( A3
= ( bot_bot @ A ) )
& ( B3
= ( bot_bot @ A ) ) ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_83_sup__bot_Oright__neutral,axiom,
! [A: $tType] :
( ( bounde4967611905675639751up_bot @ A )
=> ! [A3: A] :
( ( sup_sup @ A @ A3 @ ( bot_bot @ A ) )
= A3 ) ) ).
% sup_bot.right_neutral
thf(fact_84_nat__zero__less__power__iff,axiom,
! [X: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( power_power @ nat @ X @ N ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ X )
| ( N
= ( zero_zero @ nat ) ) ) ) ).
% nat_zero_less_power_iff
thf(fact_85_Un__empty,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( ( sup_sup @ ( set @ A ) @ A2 @ B2 )
= ( bot_bot @ ( set @ A ) ) )
= ( ( A2
= ( bot_bot @ ( set @ A ) ) )
& ( B2
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Un_empty
thf(fact_86_Un__insert__left,axiom,
! [A: $tType,A3: A,B2: set @ A,C3: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( insert @ A @ A3 @ B2 ) @ C3 )
= ( insert @ A @ A3 @ ( sup_sup @ ( set @ A ) @ B2 @ C3 ) ) ) ).
% Un_insert_left
thf(fact_87_Un__insert__right,axiom,
! [A: $tType,A2: set @ A,A3: A,B2: set @ A] :
( ( sup_sup @ ( set @ A ) @ A2 @ ( insert @ A @ A3 @ B2 ) )
= ( insert @ A @ A3 @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) ) ) ).
% Un_insert_right
thf(fact_88_singleton__conv,axiom,
! [A: $tType,A3: A] :
( ( collect @ A
@ ^ [X2: A] : X2 = A3 )
= ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% singleton_conv
thf(fact_89_singleton__conv2,axiom,
! [A: $tType,A3: A] :
( ( collect @ A
@ ( ^ [Y3: A,Z2: A] : Y3 = Z2
@ A3 ) )
= ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% singleton_conv2
thf(fact_90_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_91_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_92_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_93_power__inject__exp,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,M: nat,N: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ A3 )
=> ( ( ( power_power @ A @ A3 @ M )
= ( power_power @ A @ A3 @ N ) )
= ( M = N ) ) ) ) ).
% power_inject_exp
thf(fact_94_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_95_power__eq__0__iff,axiom,
! [A: $tType] :
( ( semiri2026040879449505780visors @ A )
=> ! [A3: A,N: nat] :
( ( ( power_power @ A @ A3 @ N )
= ( zero_zero @ A ) )
= ( ( A3
= ( zero_zero @ A ) )
& ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% power_eq_0_iff
thf(fact_96_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_97_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_98_of__nat__eq__of__nat__power__cancel__iff,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [B3: nat,W: nat,X: nat] :
( ( ( power_power @ A @ ( semiring_1_of_nat @ A @ B3 ) @ W )
= ( semiring_1_of_nat @ A @ X ) )
= ( ( power_power @ nat @ B3 @ W )
= X ) ) ) ).
% of_nat_eq_of_nat_power_cancel_iff
thf(fact_99_of__nat__power__eq__of__nat__cancel__iff,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [X: nat,B3: nat,W: nat] :
( ( ( semiring_1_of_nat @ A @ X )
= ( power_power @ A @ ( semiring_1_of_nat @ A @ B3 ) @ W ) )
= ( X
= ( power_power @ nat @ B3 @ W ) ) ) ) ).
% of_nat_power_eq_of_nat_cancel_iff
thf(fact_100_power__strict__increasing__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [B3: A,X: nat,Y: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ B3 )
=> ( ( ord_less @ A @ ( power_power @ A @ B3 @ X ) @ ( power_power @ A @ B3 @ Y ) )
= ( ord_less @ nat @ X @ Y ) ) ) ) ).
% power_strict_increasing_iff
thf(fact_101_zero__eq__power2,axiom,
! [A: $tType] :
( ( semiri2026040879449505780visors @ A )
=> ! [A3: A] :
( ( ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( zero_zero @ A ) )
= ( A3
= ( zero_zero @ A ) ) ) ) ).
% zero_eq_power2
thf(fact_102_power__strict__decreasing__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [B3: A,M: nat,N: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ B3 )
=> ( ( ord_less @ A @ B3 @ ( one_one @ A ) )
=> ( ( ord_less @ A @ ( power_power @ A @ B3 @ M ) @ ( power_power @ A @ B3 @ N ) )
= ( ord_less @ nat @ N @ M ) ) ) ) ) ).
% power_strict_decreasing_iff
thf(fact_103_of__nat__zero__less__power__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X: nat,N: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( power_power @ A @ ( semiring_1_of_nat @ A @ X ) @ N ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ X )
| ( N
= ( zero_zero @ nat ) ) ) ) ) ).
% of_nat_zero_less_power_iff
thf(fact_104_numeral__power__eq__of__nat__cancel__iff,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [X: num,N: nat,Y: nat] :
( ( ( power_power @ A @ ( numeral_numeral @ A @ X ) @ N )
= ( semiring_1_of_nat @ A @ Y ) )
= ( ( power_power @ nat @ ( numeral_numeral @ nat @ X ) @ N )
= Y ) ) ) ).
% numeral_power_eq_of_nat_cancel_iff
thf(fact_105_real__of__nat__eq__numeral__power__cancel__iff,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [Y: nat,X: num,N: nat] :
( ( ( semiring_1_of_nat @ A @ Y )
= ( power_power @ A @ ( numeral_numeral @ A @ X ) @ N ) )
= ( Y
= ( power_power @ nat @ ( numeral_numeral @ nat @ X ) @ N ) ) ) ) ).
% real_of_nat_eq_numeral_power_cancel_iff
thf(fact_106_of__nat__less__of__nat__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [B3: nat,W: nat,X: nat] :
( ( ord_less @ A @ ( power_power @ A @ ( semiring_1_of_nat @ A @ B3 ) @ W ) @ ( semiring_1_of_nat @ A @ X ) )
= ( ord_less @ nat @ ( power_power @ nat @ B3 @ W ) @ X ) ) ) ).
% of_nat_less_of_nat_power_cancel_iff
thf(fact_107_of__nat__power__less__of__nat__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X: nat,B3: nat,W: nat] :
( ( ord_less @ A @ ( semiring_1_of_nat @ A @ X ) @ ( power_power @ A @ ( semiring_1_of_nat @ A @ B3 ) @ W ) )
= ( ord_less @ nat @ X @ ( power_power @ nat @ B3 @ W ) ) ) ) ).
% of_nat_power_less_of_nat_cancel_iff
thf(fact_108_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_109_emptyE,axiom,
! [A: $tType,A3: A] :
~ ( member @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ).
% emptyE
thf(fact_110_insertE,axiom,
! [A: $tType,A3: A,B3: A,A2: set @ A] :
( ( member @ A @ A3 @ ( insert @ A @ B3 @ A2 ) )
=> ( ( A3 != B3 )
=> ( member @ A @ A3 @ A2 ) ) ) ).
% insertE
thf(fact_111_equals0D,axiom,
! [A: $tType,A2: set @ A,A3: A] :
( ( A2
= ( bot_bot @ ( set @ A ) ) )
=> ~ ( member @ A @ A3 @ A2 ) ) ).
% equals0D
thf(fact_112_equals0I,axiom,
! [A: $tType,A2: set @ A] :
( ! [Y4: A] :
~ ( member @ A @ Y4 @ A2 )
=> ( A2
= ( bot_bot @ ( set @ A ) ) ) ) ).
% equals0I
thf(fact_113_insertI1,axiom,
! [A: $tType,A3: A,B2: set @ A] : ( member @ A @ A3 @ ( insert @ A @ A3 @ B2 ) ) ).
% insertI1
thf(fact_114_insertI2,axiom,
! [A: $tType,A3: A,B2: set @ A,B3: A] :
( ( member @ A @ A3 @ B2 )
=> ( member @ A @ A3 @ ( insert @ A @ B3 @ B2 ) ) ) ).
% insertI2
thf(fact_115_ex__in__conv,axiom,
! [A: $tType,A2: set @ A] :
( ( ? [X2: A] : ( member @ A @ X2 @ A2 ) )
= ( A2
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% ex_in_conv
thf(fact_116_Set_Oset__insert,axiom,
! [A: $tType,X: A,A2: set @ A] :
( ( member @ A @ X @ A2 )
=> ~ ! [B6: set @ A] :
( ( A2
= ( insert @ A @ X @ B6 ) )
=> ( member @ A @ X @ B6 ) ) ) ).
% Set.set_insert
thf(fact_117_singletonD,axiom,
! [A: $tType,B3: A,A3: A] :
( ( member @ A @ B3 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) )
=> ( B3 = A3 ) ) ).
% singletonD
thf(fact_118_insert__ident,axiom,
! [A: $tType,X: A,A2: set @ A,B2: set @ A] :
( ~ ( member @ A @ X @ A2 )
=> ( ~ ( member @ A @ X @ B2 )
=> ( ( ( insert @ A @ X @ A2 )
= ( insert @ A @ X @ B2 ) )
= ( A2 = B2 ) ) ) ) ).
% insert_ident
thf(fact_119_insert__absorb,axiom,
! [A: $tType,A3: A,A2: set @ A] :
( ( member @ A @ A3 @ A2 )
=> ( ( insert @ A @ A3 @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_120_insert__eq__iff,axiom,
! [A: $tType,A3: A,A2: set @ A,B3: A,B2: set @ A] :
( ~ ( member @ A @ A3 @ A2 )
=> ( ~ ( member @ A @ B3 @ B2 )
=> ( ( ( insert @ A @ A3 @ A2 )
= ( insert @ A @ B3 @ B2 ) )
= ( ( ( A3 = B3 )
=> ( A2 = B2 ) )
& ( ( A3 != B3 )
=> ? [C4: set @ A] :
( ( A2
= ( insert @ A @ B3 @ C4 ) )
& ~ ( member @ A @ B3 @ C4 )
& ( B2
= ( insert @ A @ A3 @ C4 ) )
& ~ ( member @ A @ A3 @ C4 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_121_singleton__iff,axiom,
! [A: $tType,B3: A,A3: A] :
( ( member @ A @ B3 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( B3 = A3 ) ) ).
% singleton_iff
thf(fact_122_insert__commute,axiom,
! [A: $tType,X: A,Y: A,A2: set @ A] :
( ( insert @ A @ X @ ( insert @ A @ Y @ A2 ) )
= ( insert @ A @ Y @ ( insert @ A @ X @ A2 ) ) ) ).
% insert_commute
thf(fact_123_Collect__conv__if,axiom,
! [A: $tType,P: A > $o,A3: A] :
( ( ( P @ A3 )
=> ( ( collect @ A
@ ^ [X2: A] :
( ( X2 = A3 )
& ( P @ X2 ) ) )
= ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) )
& ( ~ ( P @ A3 )
=> ( ( collect @ A
@ ^ [X2: A] :
( ( X2 = A3 )
& ( P @ X2 ) ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Collect_conv_if
thf(fact_124_Collect__conv__if2,axiom,
! [A: $tType,P: A > $o,A3: A] :
( ( ( P @ A3 )
=> ( ( collect @ A
@ ^ [X2: A] :
( ( A3 = X2 )
& ( P @ X2 ) ) )
= ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) )
& ( ~ ( P @ A3 )
=> ( ( collect @ A
@ ^ [X2: A] :
( ( A3 = X2 )
& ( P @ X2 ) ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Collect_conv_if2
thf(fact_125_doubleton__eq__iff,axiom,
! [A: $tType,A3: A,B3: A,C2: A,D2: A] :
( ( ( insert @ A @ A3 @ ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( insert @ A @ C2 @ ( insert @ A @ D2 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( ( ( A3 = C2 )
& ( B3 = D2 ) )
| ( ( A3 = D2 )
& ( B3 = C2 ) ) ) ) ).
% doubleton_eq_iff
thf(fact_126_insert__not__empty,axiom,
! [A: $tType,A3: A,A2: set @ A] :
( ( insert @ A @ A3 @ A2 )
!= ( bot_bot @ ( set @ A ) ) ) ).
% insert_not_empty
thf(fact_127_singleton__inject,axiom,
! [A: $tType,A3: A,B3: A] :
( ( ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) )
= ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) )
=> ( A3 = B3 ) ) ).
% singleton_inject
thf(fact_128_not__psubset__empty,axiom,
! [A: $tType,A2: set @ A] :
~ ( ord_less @ ( set @ A ) @ A2 @ ( bot_bot @ ( set @ A ) ) ) ).
% not_psubset_empty
thf(fact_129_mk__disjoint__insert,axiom,
! [A: $tType,A3: A,A2: set @ A] :
( ( member @ A @ A3 @ A2 )
=> ? [B6: set @ A] :
( ( A2
= ( insert @ A @ A3 @ B6 ) )
& ~ ( member @ A @ A3 @ B6 ) ) ) ).
% mk_disjoint_insert
thf(fact_130_not__iless0,axiom,
! [N: extended_enat] :
~ ( ord_less @ extended_enat @ N @ ( zero_zero @ extended_enat ) ) ).
% not_iless0
thf(fact_131_enat__less__induct,axiom,
! [P: extended_enat > $o,N: extended_enat] :
( ! [N2: extended_enat] :
( ! [M2: extended_enat] :
( ( ord_less @ extended_enat @ M2 @ N2 )
=> ( P @ M2 ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% enat_less_induct
thf(fact_132_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_133_power__0,axiom,
! [A: $tType] :
( ( power @ A )
=> ! [A3: A] :
( ( power_power @ A @ A3 @ ( zero_zero @ nat ) )
= ( one_one @ A ) ) ) ).
% power_0
thf(fact_134_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_135_singleton__Un__iff,axiom,
! [A: $tType,X: A,A2: set @ A,B2: set @ A] :
( ( ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) )
= ( sup_sup @ ( set @ A ) @ A2 @ B2 ) )
= ( ( ( A2
= ( bot_bot @ ( set @ A ) ) )
& ( B2
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
| ( ( A2
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
& ( B2
= ( bot_bot @ ( set @ A ) ) ) )
| ( ( A2
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
& ( B2
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_136_Un__singleton__iff,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,X: A] :
( ( ( sup_sup @ ( set @ A ) @ A2 @ B2 )
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( ( ( A2
= ( bot_bot @ ( set @ A ) ) )
& ( B2
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
| ( ( A2
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
& ( B2
= ( bot_bot @ ( set @ A ) ) ) )
| ( ( A2
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
& ( B2
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_137_insert__is__Un,axiom,
! [A: $tType] :
( ( insert @ A )
= ( ^ [A5: A] : ( sup_sup @ ( set @ A ) @ ( insert @ A @ A5 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% insert_is_Un
thf(fact_138_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_139_nat__int__comparison_I2_J,axiom,
( ( ord_less @ nat )
= ( ^ [A5: nat,B5: nat] : ( ord_less @ int @ ( semiring_1_of_nat @ int @ A5 ) @ ( semiring_1_of_nat @ int @ B5 ) ) ) ) ).
% nat_int_comparison(2)
thf(fact_140_insert__Collect,axiom,
! [A: $tType,A3: A,P: A > $o] :
( ( insert @ A @ A3 @ ( collect @ A @ P ) )
= ( collect @ A
@ ^ [U: A] :
( ( U != A3 )
=> ( P @ U ) ) ) ) ).
% insert_Collect
thf(fact_141_insert__compr,axiom,
! [A: $tType] :
( ( insert @ A )
= ( ^ [A5: A,B4: set @ A] :
( collect @ A
@ ^ [X2: A] :
( ( X2 = A5 )
| ( member @ A @ X2 @ B4 ) ) ) ) ) ).
% insert_compr
thf(fact_142_semiring__1__no__zero__divisors__class_Opower__not__zero,axiom,
! [A: $tType] :
( ( semiri2026040879449505780visors @ A )
=> ! [A3: A,N: nat] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( power_power @ A @ A3 @ N )
!= ( zero_zero @ A ) ) ) ) ).
% semiring_1_no_zero_divisors_class.power_not_zero
thf(fact_143_Set_Oempty__def,axiom,
! [A: $tType] :
( ( bot_bot @ ( set @ A ) )
= ( collect @ A
@ ^ [X2: A] : $false ) ) ).
% Set.empty_def
thf(fact_144_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_145_nat__less__as__int,axiom,
( ( ord_less @ nat )
= ( ^ [A5: nat,B5: nat] : ( ord_less @ int @ ( semiring_1_of_nat @ int @ A5 ) @ ( semiring_1_of_nat @ int @ B5 ) ) ) ) ).
% nat_less_as_int
thf(fact_146_power__strict__decreasing,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N: nat,N3: nat,A3: A] :
( ( ord_less @ nat @ N @ N3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ A3 @ ( one_one @ A ) )
=> ( ord_less @ A @ ( power_power @ A @ A3 @ N3 ) @ ( power_power @ A @ A3 @ N ) ) ) ) ) ) ).
% power_strict_decreasing
thf(fact_147_one__less__power,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,N: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ A3 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ord_less @ A @ ( one_one @ A ) @ ( power_power @ A @ A3 @ N ) ) ) ) ) ).
% one_less_power
thf(fact_148_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_149_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_150_zero__neq__numeral,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [N: num] :
( ( zero_zero @ A )
!= ( numeral_numeral @ A @ N ) ) ) ).
% zero_neq_numeral
thf(fact_151_Un__empty__left,axiom,
! [A: $tType,B2: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ B2 )
= B2 ) ).
% Un_empty_left
thf(fact_152_Un__empty__right,axiom,
! [A: $tType,A2: set @ A] :
( ( sup_sup @ ( set @ A ) @ A2 @ ( bot_bot @ ( set @ A ) ) )
= A2 ) ).
% Un_empty_right
thf(fact_153_zero__less__power,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,N: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A3 @ N ) ) ) ) ).
% zero_less_power
thf(fact_154_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_155_insert__def,axiom,
! [A: $tType] :
( ( insert @ A )
= ( ^ [A5: A] :
( sup_sup @ ( set @ A )
@ ( collect @ A
@ ^ [X2: A] : X2 = A5 ) ) ) ) ).
% insert_def
thf(fact_156_power__less__imp__less__exp,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,M: nat,N: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ A3 )
=> ( ( ord_less @ A @ ( power_power @ A @ A3 @ M ) @ ( power_power @ A @ A3 @ N ) )
=> ( ord_less @ nat @ M @ N ) ) ) ) ).
% power_less_imp_less_exp
thf(fact_157_power__strict__increasing,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N: nat,N3: nat,A3: A] :
( ( ord_less @ nat @ N @ N3 )
=> ( ( ord_less @ A @ ( one_one @ A ) @ A3 )
=> ( ord_less @ A @ ( power_power @ A @ A3 @ N ) @ ( power_power @ A @ A3 @ N3 ) ) ) ) ) ).
% power_strict_increasing
thf(fact_158_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_159_numeral__One,axiom,
! [A: $tType] :
( ( numeral @ A )
=> ( ( numeral_numeral @ A @ one2 )
= ( one_one @ A ) ) ) ).
% numeral_One
thf(fact_160_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_161_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_162_numerals_I1_J,axiom,
( ( numeral_numeral @ nat @ one2 )
= ( one_one @ nat ) ) ).
% numerals(1)
thf(fact_163_pow_Osimps_I1_J,axiom,
! [X: num] :
( ( pow @ X @ one2 )
= X ) ).
% pow.simps(1)
thf(fact_164_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_165_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_166_verit__comp__simplify1_I1_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A] :
~ ( ord_less @ A @ A3 @ A3 ) ) ).
% verit_comp_simplify1(1)
thf(fact_167_power2__less__0,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A] :
~ ( ord_less @ A @ ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( zero_zero @ A ) ) ) ).
% power2_less_0
thf(fact_168_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_169_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_170_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_171_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_172_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_173_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_174_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_175_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_176_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_177_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_178_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_179_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_180_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_181_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% less_nat_zero_code
thf(fact_182_neq0__conv,axiom,
! [N: nat] :
( ( N
!= ( zero_zero @ nat ) )
= ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ).
% neq0_conv
thf(fact_183_bot__nat__0_Onot__eq__extremum,axiom,
! [A3: nat] :
( ( A3
!= ( zero_zero @ nat ) )
= ( ord_less @ nat @ ( zero_zero @ nat ) @ A3 ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_184_less__one,axiom,
! [N: nat] :
( ( ord_less @ nat @ N @ ( one_one @ nat ) )
= ( N
= ( zero_zero @ nat ) ) ) ).
% less_one
thf(fact_185_psubsetD,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,C2: A] :
( ( ord_less @ ( set @ A ) @ A2 @ B2 )
=> ( ( member @ A @ C2 @ A2 )
=> ( member @ A @ C2 @ B2 ) ) ) ).
% psubsetD
thf(fact_186_less__set__def,axiom,
! [A: $tType] :
( ( ord_less @ ( set @ A ) )
= ( ^ [A4: set @ A,B4: set @ A] :
( ord_less @ ( A > $o )
@ ^ [X2: A] : ( member @ A @ X2 @ A4 )
@ ^ [X2: A] : ( member @ A @ X2 @ B4 ) ) ) ) ).
% less_set_def
thf(fact_187_psubset__trans,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,C3: set @ A] :
( ( ord_less @ ( set @ A ) @ A2 @ B2 )
=> ( ( ord_less @ ( set @ A ) @ B2 @ C3 )
=> ( ord_less @ ( set @ A ) @ A2 @ C3 ) ) ) ).
% psubset_trans
thf(fact_188_real__arch__pow,axiom,
! [X: real,Y: real] :
( ( ord_less @ real @ ( one_one @ real ) @ X )
=> ? [N2: nat] : ( ord_less @ real @ Y @ ( power_power @ real @ X @ N2 ) ) ) ).
% real_arch_pow
thf(fact_189_real__arch__pow__inv,axiom,
! [Y: real,X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ Y )
=> ( ( ord_less @ real @ X @ ( one_one @ real ) )
=> ? [N2: nat] : ( ord_less @ real @ ( power_power @ real @ X @ N2 ) @ Y ) ) ) ).
% real_arch_pow_inv
thf(fact_190_bot__enat__def,axiom,
( ( bot_bot @ extended_enat )
= ( zero_zero @ extended_enat ) ) ).
% bot_enat_def
thf(fact_191_bot__nat__def,axiom,
( ( bot_bot @ nat )
= ( zero_zero @ nat ) ) ).
% bot_nat_def
thf(fact_192_int__if,axiom,
! [P: $o,A3: nat,B3: nat] :
( ( P
=> ( ( semiring_1_of_nat @ int @ ( if @ nat @ P @ A3 @ B3 ) )
= ( semiring_1_of_nat @ int @ A3 ) ) )
& ( ~ P
=> ( ( semiring_1_of_nat @ int @ ( if @ nat @ P @ A3 @ B3 ) )
= ( semiring_1_of_nat @ int @ B3 ) ) ) ) ).
% int_if
thf(fact_193_zero__one__enat__neq_I1_J,axiom,
( ( zero_zero @ extended_enat )
!= ( one_one @ extended_enat ) ) ).
% zero_one_enat_neq(1)
thf(fact_194_nat__int__comparison_I1_J,axiom,
( ( ^ [Y3: nat,Z2: nat] : Y3 = Z2 )
= ( ^ [A5: nat,B5: nat] :
( ( semiring_1_of_nat @ int @ A5 )
= ( semiring_1_of_nat @ int @ B5 ) ) ) ) ).
% nat_int_comparison(1)
thf(fact_195_int__ops_I1_J,axiom,
( ( semiring_1_of_nat @ int @ ( zero_zero @ nat ) )
= ( zero_zero @ int ) ) ).
% int_ops(1)
thf(fact_196_int__ops_I2_J,axiom,
( ( semiring_1_of_nat @ int @ ( one_one @ nat ) )
= ( one_one @ int ) ) ).
% int_ops(2)
thf(fact_197_bot__empty__eq,axiom,
! [A: $tType] :
( ( bot_bot @ ( A > $o ) )
= ( ^ [X2: A] : ( member @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% bot_empty_eq
thf(fact_198_bot__set__def,axiom,
! [A: $tType] :
( ( bot_bot @ ( set @ A ) )
= ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) ) ).
% bot_set_def
thf(fact_199_zero__reorient,axiom,
! [A: $tType] :
( ( zero @ A )
=> ! [X: A] :
( ( ( zero_zero @ A )
= X )
= ( X
= ( zero_zero @ A ) ) ) ) ).
% zero_reorient
thf(fact_200_measure__induct__rule,axiom,
! [B: $tType,A: $tType] :
( ( wellorder @ B )
=> ! [F3: A > B,P: A > $o,A3: A] :
( ! [X3: A] :
( ! [Y5: A] :
( ( ord_less @ B @ ( F3 @ Y5 ) @ ( F3 @ X3 ) )
=> ( P @ Y5 ) )
=> ( P @ X3 ) )
=> ( P @ A3 ) ) ) ).
% measure_induct_rule
thf(fact_201_measure__induct,axiom,
! [B: $tType,A: $tType] :
( ( wellorder @ B )
=> ! [F3: A > B,P: A > $o,A3: A] :
( ! [X3: A] :
( ! [Y5: A] :
( ( ord_less @ B @ ( F3 @ Y5 ) @ ( F3 @ X3 ) )
=> ( P @ Y5 ) )
=> ( P @ X3 ) )
=> ( P @ A3 ) ) ) ).
% measure_induct
thf(fact_202_one__reorient,axiom,
! [A: $tType] :
( ( one @ A )
=> ! [X: A] :
( ( ( one_one @ A )
= X )
= ( X
= ( one_one @ A ) ) ) ) ).
% one_reorient
thf(fact_203_infinite__descent__measure,axiom,
! [A: $tType,P: A > $o,V: A > nat,X: A] :
( ! [X3: A] :
( ~ ( P @ X3 )
=> ? [Y5: A] :
( ( ord_less @ nat @ ( V @ Y5 ) @ ( V @ X3 ) )
& ~ ( P @ Y5 ) ) )
=> ( P @ X ) ) ).
% infinite_descent_measure
thf(fact_204_linorder__neqE__nat,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less @ nat @ X @ Y )
=> ( ord_less @ nat @ Y @ X ) ) ) ).
% linorder_neqE_nat
thf(fact_205_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ~ ( P @ N2 )
=> ? [M2: nat] :
( ( ord_less @ nat @ M2 @ N2 )
& ~ ( P @ M2 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_206_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M2: nat] :
( ( ord_less @ nat @ M2 @ N2 )
=> ( P @ M2 ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_207_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_208_less__not__refl3,axiom,
! [S2: nat,T: nat] :
( ( ord_less @ nat @ S2 @ T )
=> ( S2 != T ) ) ).
% less_not_refl3
thf(fact_209_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less @ nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_210_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ N ) ).
% less_not_refl
thf(fact_211_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_212_infinite__descent0__measure,axiom,
! [A: $tType,V: A > nat,P: A > $o,X: A] :
( ! [X3: A] :
( ( ( V @ X3 )
= ( zero_zero @ nat ) )
=> ( P @ X3 ) )
=> ( ! [X3: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( V @ X3 ) )
=> ( ~ ( P @ X3 )
=> ? [Y5: A] :
( ( ord_less @ nat @ ( V @ Y5 ) @ ( V @ X3 ) )
& ~ ( P @ Y5 ) ) ) )
=> ( P @ X ) ) ) ).
% infinite_descent0_measure
thf(fact_213_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ ( zero_zero @ nat ) )
=> ( ! [N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ~ ( P @ N2 )
=> ? [M2: nat] :
( ( ord_less @ nat @ M2 @ N2 )
& ~ ( P @ M2 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_214_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ N )
=> ( N
!= ( zero_zero @ nat ) ) ) ).
% gr_implies_not0
thf(fact_215_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% less_zeroE
thf(fact_216_not__less0,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% not_less0
thf(fact_217_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) )
= ( N
= ( zero_zero @ nat ) ) ) ).
% not_gr0
thf(fact_218_gr0I,axiom,
! [N: nat] :
( ( N
!= ( zero_zero @ nat ) )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ).
% gr0I
thf(fact_219_bot__nat__0_Oextremum__strict,axiom,
! [A3: nat] :
~ ( ord_less @ nat @ A3 @ ( zero_zero @ nat ) ) ).
% bot_nat_0.extremum_strict
thf(fact_220_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 )
=> ? [E: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ E )
& ( ord_less @ A @ E @ D1 )
& ( ord_less @ A @ E @ D22 ) ) ) ) ) ).
% field_lbound_gt_zero
thf(fact_221_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_222_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_223_not__less__zero,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [N: A] :
~ ( ord_less @ A @ N @ ( zero_zero @ A ) ) ) ).
% not_less_zero
thf(fact_224_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_225_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_226_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_227_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_228_int__eq__iff__numeral,axiom,
! [M: nat,V2: num] :
( ( ( semiring_1_of_nat @ int @ M )
= ( numeral_numeral @ int @ V2 ) )
= ( M
= ( numeral_numeral @ nat @ V2 ) ) ) ).
% int_eq_iff_numeral
thf(fact_229_pos2,axiom,
ord_less @ nat @ ( zero_zero @ nat ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ).
% pos2
thf(fact_230_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_231_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_232_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_233_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_234_sup1CI,axiom,
! [A: $tType,B2: A > $o,X: A,A2: A > $o] :
( ( ~ ( B2 @ X )
=> ( A2 @ X ) )
=> ( sup_sup @ ( A > $o ) @ A2 @ B2 @ X ) ) ).
% sup1CI
thf(fact_235_realpow__pos__nth__unique,axiom,
! [N: nat,A3: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ A3 )
=> ? [X3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
& ( ( power_power @ real @ X3 @ N )
= A3 )
& ! [Y5: real] :
( ( ( ord_less @ real @ ( zero_zero @ real ) @ Y5 )
& ( ( power_power @ real @ Y5 @ N )
= A3 ) )
=> ( Y5 = X3 ) ) ) ) ) ).
% realpow_pos_nth_unique
thf(fact_236_realpow__pos__nth,axiom,
! [N: nat,A3: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ A3 )
=> ? [R2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R2 )
& ( ( power_power @ real @ R2 @ N )
= A3 ) ) ) ) ).
% realpow_pos_nth
thf(fact_237_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_238_not__one__less__zero,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ~ ( ord_less @ A @ ( one_one @ A ) @ ( zero_zero @ A ) ) ) ).
% not_one_less_zero
thf(fact_239_word__gt__0__no,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y: num] :
( ( ord_less @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ ( numeral_numeral @ ( word @ A ) @ Y ) )
= ( ( zero_zero @ ( word @ A ) )
!= ( numeral_numeral @ ( word @ A ) @ Y ) ) ) ) ).
% word_gt_0_no
thf(fact_240_word__less__1,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] :
( ( ord_less @ ( word @ A ) @ X @ ( one_one @ ( word @ A ) ) )
= ( X
= ( zero_zero @ ( word @ A ) ) ) ) ) ).
% word_less_1
thf(fact_241_linorder__neqE__linordered__idom,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A,Y: A] :
( ( X != Y )
=> ( ~ ( ord_less @ A @ X @ Y )
=> ( ord_less @ A @ Y @ X ) ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_242_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_243_boolean__algebra__cancel_Osup1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A2: A,K: A,A3: A,B3: A] :
( ( A2
= ( sup_sup @ A @ K @ A3 ) )
=> ( ( sup_sup @ A @ A2 @ B3 )
= ( sup_sup @ A @ K @ ( sup_sup @ A @ A3 @ B3 ) ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_244_boolean__algebra__cancel_Osup2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B2: A,K: A,B3: A,A3: A] :
( ( B2
= ( sup_sup @ A @ K @ B3 ) )
=> ( ( sup_sup @ A @ A3 @ B2 )
= ( sup_sup @ A @ K @ ( sup_sup @ A @ A3 @ B3 ) ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_245_less__int__code_I1_J,axiom,
~ ( ord_less @ int @ ( zero_zero @ int ) @ ( zero_zero @ int ) ) ).
% less_int_code(1)
thf(fact_246_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_247_sup1E,axiom,
! [A: $tType,A2: A > $o,B2: A > $o,X: A] :
( ( sup_sup @ ( A > $o ) @ A2 @ B2 @ X )
=> ( ~ ( A2 @ X )
=> ( B2 @ X ) ) ) ).
% sup1E
thf(fact_248_sup1I1,axiom,
! [A: $tType,A2: A > $o,X: A,B2: A > $o] :
( ( A2 @ X )
=> ( sup_sup @ ( A > $o ) @ A2 @ B2 @ X ) ) ).
% sup1I1
thf(fact_249_sup1I2,axiom,
! [A: $tType,B2: A > $o,X: A,A2: A > $o] :
( ( B2 @ X )
=> ( sup_sup @ ( A > $o ) @ A2 @ B2 @ X ) ) ).
% sup1I2
thf(fact_250_zero__neq__one,axiom,
! [A: $tType] :
( ( zero_neq_one @ A )
=> ( ( zero_zero @ A )
!= ( one_one @ A ) ) ) ).
% zero_neq_one
thf(fact_251_boolean__algebra_Odisj__zero__right,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A] :
( ( sup_sup @ A @ X @ ( bot_bot @ A ) )
= X ) ) ).
% boolean_algebra.disj_zero_right
thf(fact_252_zero__less__one,axiom,
! [A: $tType] :
( ( zero_less_one @ A )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( one_one @ A ) ) ) ).
% zero_less_one
thf(fact_253_bot__apply,axiom,
! [C: $tType,D: $tType] :
( ( bot @ C )
=> ( ( bot_bot @ ( D > C ) )
= ( ^ [X2: D] : ( bot_bot @ C ) ) ) ) ).
% bot_apply
thf(fact_254_false__rule,axiom,
! [A: $tType,C2: heap_Time_Heap @ A,Q: A > assn] : ( hoare_hoare_triple @ A @ ( bot_bot @ assn ) @ C2 @ Q ) ).
% false_rule
thf(fact_255_the__elem__eq,axiom,
! [A: $tType,X: A] :
( ( the_elem @ A @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= X ) ).
% the_elem_eq
thf(fact_256_Collect__empty__eq__bot,axiom,
! [A: $tType,P: A > $o] :
( ( ( collect @ A @ P )
= ( bot_bot @ ( set @ A ) ) )
= ( P
= ( bot_bot @ ( A > $o ) ) ) ) ).
% Collect_empty_eq_bot
thf(fact_257_vebt__buildupi__rule__basic,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( hoare_hoare_triple @ vEBT_VEBTi @ ( one_one @ assn ) @ ( vEBT_vebt_buildupi @ N ) @ ( vEBT_Intf_vebt_assn @ N @ ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% vebt_buildupi_rule_basic
thf(fact_258_bot__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( bot @ B )
=> ( ( bot_bot @ ( A > B ) )
= ( ^ [X2: A] : ( bot_bot @ B ) ) ) ) ).
% bot_fun_def
thf(fact_259_is__singletonI,axiom,
! [A: $tType,X: A] : ( is_singleton @ A @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ).
% is_singletonI
thf(fact_260_forall__finite_I1_J,axiom,
! [P: nat > $o,I2: nat] :
( ( ord_less @ nat @ I2 @ ( zero_zero @ nat ) )
=> ( P @ I2 ) ) ).
% forall_finite(1)
thf(fact_261_reals__Archimedean2,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X: A] :
? [N2: nat] : ( ord_less @ A @ X @ ( semiring_1_of_nat @ A @ N2 ) ) ) ).
% reals_Archimedean2
thf(fact_262_bot_Oextremum__strict,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [A3: A] :
~ ( ord_less @ A @ A3 @ ( bot_bot @ A ) ) ) ).
% bot.extremum_strict
thf(fact_263_bot_Onot__eq__extremum,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [A3: A] :
( ( A3
!= ( bot_bot @ A ) )
= ( ord_less @ A @ ( bot_bot @ A ) @ A3 ) ) ) ).
% bot.not_eq_extremum
thf(fact_264_word__not__simps_I1_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] :
~ ( ord_less @ ( word @ A ) @ X @ ( zero_zero @ ( word @ A ) ) ) ) ).
% word_not_simps(1)
thf(fact_265_word__coorder_Oextremum__strict,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A] :
~ ( ord_less @ ( word @ A ) @ A3 @ ( zero_zero @ ( word @ A ) ) ) ) ).
% word_coorder.extremum_strict
thf(fact_266_word__gt__0,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y: word @ A] :
( ( ord_less @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ Y )
= ( ( zero_zero @ ( word @ A ) )
!= Y ) ) ) ).
% word_gt_0
thf(fact_267_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_268_word__gt__a__gt__0,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A,N: word @ A] :
( ( ord_less @ ( word @ A ) @ A3 @ N )
=> ( ord_less @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ N ) ) ) ).
% word_gt_a_gt_0
thf(fact_269_word__greater__zero__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A] :
( ( ord_less @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ A3 )
= ( A3
!= ( zero_zero @ ( word @ A ) ) ) ) ) ).
% word_greater_zero_iff
thf(fact_270_is__singleton__the__elem,axiom,
! [A: $tType] :
( ( is_singleton @ A )
= ( ^ [A4: set @ A] :
( A4
= ( insert @ A @ ( the_elem @ A @ A4 ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% is_singleton_the_elem
thf(fact_271_is__singletonI_H,axiom,
! [A: $tType,A2: set @ A] :
( ( A2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A,Y4: A] :
( ( member @ A @ X3 @ A2 )
=> ( ( member @ A @ Y4 @ A2 )
=> ( X3 = Y4 ) ) )
=> ( is_singleton @ A @ A2 ) ) ) ).
% is_singletonI'
thf(fact_272_order__less__imp__not__less,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ~ ( ord_less @ A @ Y @ X ) ) ) ).
% order_less_imp_not_less
thf(fact_273_order__less__imp__not__eq2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ( Y != X ) ) ) ).
% order_less_imp_not_eq2
thf(fact_274_order__less__imp__not__eq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ( X != Y ) ) ) ).
% order_less_imp_not_eq
thf(fact_275_linorder__less__linear,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
| ( X = Y )
| ( ord_less @ A @ Y @ X ) ) ) ).
% linorder_less_linear
thf(fact_276_order__less__imp__triv,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A,P: $o] :
( ( ord_less @ A @ X @ Y )
=> ( ( ord_less @ A @ Y @ X )
=> P ) ) ) ).
% order_less_imp_triv
thf(fact_277_order__less__not__sym,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ~ ( ord_less @ A @ Y @ X ) ) ) ).
% order_less_not_sym
thf(fact_278_order__less__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C )
& ( order @ A ) )
=> ! [A3: A,B3: A,F3: A > C,C2: C] :
( ( ord_less @ A @ A3 @ B3 )
=> ( ( ord_less @ C @ ( F3 @ B3 ) @ C2 )
=> ( ! [X3: A,Y4: A] :
( ( ord_less @ A @ X3 @ Y4 )
=> ( ord_less @ C @ ( F3 @ X3 ) @ ( F3 @ Y4 ) ) )
=> ( ord_less @ C @ ( F3 @ A3 ) @ C2 ) ) ) ) ) ).
% order_less_subst2
thf(fact_279_order__less__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A3: A,F3: B > A,B3: B,C2: B] :
( ( ord_less @ A @ A3 @ ( F3 @ B3 ) )
=> ( ( ord_less @ B @ B3 @ C2 )
=> ( ! [X3: B,Y4: B] :
( ( ord_less @ B @ X3 @ Y4 )
=> ( ord_less @ A @ ( F3 @ X3 ) @ ( F3 @ Y4 ) ) )
=> ( ord_less @ A @ A3 @ ( F3 @ C2 ) ) ) ) ) ) ).
% order_less_subst1
thf(fact_280_order__less__irrefl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A] :
~ ( ord_less @ A @ X @ X ) ) ).
% order_less_irrefl
thf(fact_281_ord__less__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A3: A,B3: A,F3: A > B,C2: B] :
( ( ord_less @ A @ A3 @ B3 )
=> ( ( ( F3 @ B3 )
= C2 )
=> ( ! [X3: A,Y4: A] :
( ( ord_less @ A @ X3 @ Y4 )
=> ( ord_less @ B @ ( F3 @ X3 ) @ ( F3 @ Y4 ) ) )
=> ( ord_less @ B @ ( F3 @ A3 ) @ C2 ) ) ) ) ) ).
% ord_less_eq_subst
thf(fact_282_ord__eq__less__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A3: A,F3: B > A,B3: B,C2: B] :
( ( A3
= ( F3 @ B3 ) )
=> ( ( ord_less @ B @ B3 @ C2 )
=> ( ! [X3: B,Y4: B] :
( ( ord_less @ B @ X3 @ Y4 )
=> ( ord_less @ A @ ( F3 @ X3 ) @ ( F3 @ Y4 ) ) )
=> ( ord_less @ A @ A3 @ ( F3 @ C2 ) ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_283_order__less__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A,Z: A] :
( ( ord_less @ A @ X @ Y )
=> ( ( ord_less @ A @ Y @ Z )
=> ( ord_less @ A @ X @ Z ) ) ) ) ).
% order_less_trans
thf(fact_284_order__less__asym_H,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ A3 @ B3 )
=> ~ ( ord_less @ A @ B3 @ A3 ) ) ) ).
% order_less_asym'
thf(fact_285_linorder__neq__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( X != Y )
= ( ( ord_less @ A @ X @ Y )
| ( ord_less @ A @ Y @ X ) ) ) ) ).
% linorder_neq_iff
thf(fact_286_order__less__asym,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ~ ( ord_less @ A @ Y @ X ) ) ) ).
% order_less_asym
thf(fact_287_linorder__neqE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( X != Y )
=> ( ~ ( ord_less @ A @ X @ Y )
=> ( ord_less @ A @ Y @ X ) ) ) ) ).
% linorder_neqE
thf(fact_288_dual__order_Ostrict__implies__not__eq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B3: A,A3: A] :
( ( ord_less @ A @ B3 @ A3 )
=> ( A3 != B3 ) ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_289_order_Ostrict__implies__not__eq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ A3 @ B3 )
=> ( A3 != B3 ) ) ) ).
% order.strict_implies_not_eq
thf(fact_290_dual__order_Ostrict__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [B3: A,A3: A,C2: A] :
( ( ord_less @ A @ B3 @ A3 )
=> ( ( ord_less @ A @ C2 @ B3 )
=> ( ord_less @ A @ C2 @ A3 ) ) ) ) ).
% dual_order.strict_trans
thf(fact_291_not__less__iff__gr__or__eq,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( ~ ( ord_less @ A @ X @ Y ) )
= ( ( ord_less @ A @ Y @ X )
| ( X = Y ) ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_292_order_Ostrict__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: A,B3: A,C2: A] :
( ( ord_less @ A @ A3 @ B3 )
=> ( ( ord_less @ A @ B3 @ C2 )
=> ( ord_less @ A @ A3 @ C2 ) ) ) ) ).
% order.strict_trans
thf(fact_293_linorder__less__wlog,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > A > $o,A3: A,B3: A] :
( ! [A6: A,B7: A] :
( ( ord_less @ A @ A6 @ B7 )
=> ( P @ A6 @ B7 ) )
=> ( ! [A6: A] : ( P @ A6 @ A6 )
=> ( ! [A6: A,B7: A] :
( ( P @ B7 @ A6 )
=> ( P @ A6 @ B7 ) )
=> ( P @ A3 @ B3 ) ) ) ) ) ).
% linorder_less_wlog
thf(fact_294_exists__least__iff,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ( ( ^ [P2: A > $o] :
? [X4: A] : ( P2 @ X4 ) )
= ( ^ [P3: A > $o] :
? [N4: A] :
( ( P3 @ N4 )
& ! [M3: A] :
( ( ord_less @ A @ M3 @ N4 )
=> ~ ( P3 @ M3 ) ) ) ) ) ) ).
% exists_least_iff
thf(fact_295_dual__order_Oirrefl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: A] :
~ ( ord_less @ A @ A3 @ A3 ) ) ).
% dual_order.irrefl
thf(fact_296_dual__order_Oasym,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [B3: A,A3: A] :
( ( ord_less @ A @ B3 @ A3 )
=> ~ ( ord_less @ A @ A3 @ B3 ) ) ) ).
% dual_order.asym
thf(fact_297_linorder__cases,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ~ ( ord_less @ A @ X @ Y )
=> ( ( X != Y )
=> ( ord_less @ A @ Y @ X ) ) ) ) ).
% linorder_cases
thf(fact_298_antisym__conv3,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Y: A,X: A] :
( ~ ( ord_less @ A @ Y @ X )
=> ( ( ~ ( ord_less @ A @ X @ Y ) )
= ( X = Y ) ) ) ) ).
% antisym_conv3
thf(fact_299_less__induct,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [P: A > $o,A3: A] :
( ! [X3: A] :
( ! [Y5: A] :
( ( ord_less @ A @ Y5 @ X3 )
=> ( P @ Y5 ) )
=> ( P @ X3 ) )
=> ( P @ A3 ) ) ) ).
% less_induct
thf(fact_300_ord__less__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A3: A,B3: A,C2: A] :
( ( ord_less @ A @ A3 @ B3 )
=> ( ( B3 = C2 )
=> ( ord_less @ A @ A3 @ C2 ) ) ) ) ).
% ord_less_eq_trans
thf(fact_301_ord__eq__less__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A3: A,B3: A,C2: A] :
( ( A3 = B3 )
=> ( ( ord_less @ A @ B3 @ C2 )
=> ( ord_less @ A @ A3 @ C2 ) ) ) ) ).
% ord_eq_less_trans
thf(fact_302_order_Oasym,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ A3 @ B3 )
=> ~ ( ord_less @ A @ B3 @ A3 ) ) ) ).
% order.asym
thf(fact_303_less__imp__neq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ( X != Y ) ) ) ).
% less_imp_neq
thf(fact_304_dense,axiom,
! [A: $tType] :
( ( dense_order @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ? [Z3: A] :
( ( ord_less @ A @ X @ Z3 )
& ( ord_less @ A @ Z3 @ Y ) ) ) ) ).
% dense
thf(fact_305_gt__ex,axiom,
! [A: $tType] :
( ( no_top @ A )
=> ! [X: A] :
? [X_1: A] : ( ord_less @ A @ X @ X_1 ) ) ).
% gt_ex
thf(fact_306_lt__ex,axiom,
! [A: $tType] :
( ( no_bot @ A )
=> ! [X: A] :
? [Y4: A] : ( ord_less @ A @ Y4 @ X ) ) ).
% lt_ex
thf(fact_307_split__rule,axiom,
! [A: $tType,P: assn,C2: heap_Time_Heap @ A,R: A > assn,Q: assn] :
( ( hoare_hoare_triple @ A @ P @ C2 @ R )
=> ( ( hoare_hoare_triple @ A @ Q @ C2 @ R )
=> ( hoare_hoare_triple @ A @ ( sup_sup @ assn @ P @ Q ) @ C2 @ R ) ) ) ).
% split_rule
thf(fact_308_if__rule,axiom,
! [A: $tType,B3: $o,P: assn,F3: heap_Time_Heap @ A,Q: A > assn,G2: heap_Time_Heap @ A] :
( ( B3
=> ( hoare_hoare_triple @ A @ P @ F3 @ Q ) )
=> ( ( ~ B3
=> ( hoare_hoare_triple @ A @ P @ G2 @ Q ) )
=> ( hoare_hoare_triple @ A @ P @ ( if @ ( heap_Time_Heap @ A ) @ B3 @ F3 @ G2 ) @ Q ) ) ) ).
% if_rule
thf(fact_309_is__singleton__def,axiom,
! [A: $tType] :
( ( is_singleton @ A )
= ( ^ [A4: set @ A] :
? [X2: A] :
( A4
= ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% is_singleton_def
thf(fact_310_is__singletonE,axiom,
! [A: $tType,A2: set @ A] :
( ( is_singleton @ A @ A2 )
=> ~ ! [X3: A] :
( A2
!= ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% is_singletonE
thf(fact_311_assn__basic__inequalities_I3_J,axiom,
( ( bot_bot @ assn )
!= ( one_one @ assn ) ) ).
% assn_basic_inequalities(3)
thf(fact_312_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_313_vebt__heap__rules_I1_J,axiom,
! [N: nat] : ( hoare_hoare_triple @ vEBT_VEBTi @ ( pure_assn @ ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) @ ( vEBT_vebt_buildupi @ N ) @ ( vEBT_Intf_vebt_assn @ N @ ( bot_bot @ ( set @ nat ) ) ) ) ).
% vebt_heap_rules(1)
thf(fact_314_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_315_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_316_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_317_power2__eq__iff__nonneg,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
=> ( ( ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( X = Y ) ) ) ) ) ).
% power2_eq_iff_nonneg
thf(fact_318_power2__less__eq__zero__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( zero_zero @ A ) )
= ( A3
= ( zero_zero @ A ) ) ) ) ).
% power2_less_eq_zero_iff
thf(fact_319_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_320_power2__less__imp__less,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
=> ( ord_less @ A @ X @ Y ) ) ) ) ).
% power2_less_imp_less
thf(fact_321_sum__power2__eq__zero__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A,Y: A] :
( ( ( plus_plus @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( zero_zero @ A ) )
= ( ( X
= ( zero_zero @ A ) )
& ( Y
= ( zero_zero @ A ) ) ) ) ) ).
% sum_power2_eq_zero_iff
thf(fact_322_order__refl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% order_refl
thf(fact_323_dual__order_Orefl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: A] : ( ord_less_eq @ A @ A3 @ A3 ) ) ).
% dual_order.refl
thf(fact_324_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_325_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_326_semiring__norm_I68_J,axiom,
! [N: num] : ( ord_less_eq @ num @ one2 @ N ) ).
% semiring_norm(68)
thf(fact_327_add__left__cancel,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [A3: A,B3: A,C2: A] :
( ( ( plus_plus @ A @ A3 @ B3 )
= ( plus_plus @ A @ A3 @ C2 ) )
= ( B3 = C2 ) ) ) ).
% add_left_cancel
thf(fact_328_add__right__cancel,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [B3: A,A3: A,C2: A] :
( ( ( plus_plus @ A @ B3 @ A3 )
= ( plus_plus @ A @ C2 @ A3 ) )
= ( B3 = C2 ) ) ) ).
% add_right_cancel
thf(fact_329_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_330_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus @ nat @ M @ ( zero_zero @ nat ) )
= M ) ).
% Nat.add_0_right
thf(fact_331_bot__nat__0_Oextremum,axiom,
! [A3: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ A3 ) ).
% bot_nat_0.extremum
thf(fact_332_le0,axiom,
! [N: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N ) ).
% le0
thf(fact_333_add_Oinverse__inverse,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A] :
( ( uminus_uminus @ A @ ( uminus_uminus @ A @ A3 ) )
= A3 ) ) ).
% add.inverse_inverse
thf(fact_334_neg__equal__iff__equal,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B3: A] :
( ( ( uminus_uminus @ A @ A3 )
= ( uminus_uminus @ A @ B3 ) )
= ( A3 = B3 ) ) ) ).
% neg_equal_iff_equal
thf(fact_335_uminus__apply,axiom,
! [B: $tType,A: $tType] :
( ( uminus @ B )
=> ( ( uminus_uminus @ ( A > B ) )
= ( ^ [A4: A > B,X2: A] : ( uminus_uminus @ B @ ( A4 @ X2 ) ) ) ) ) ).
% uminus_apply
thf(fact_336_verit__minus__simplify_I4_J,axiom,
! [B: $tType] :
( ( group_add @ B )
=> ! [B3: B] :
( ( uminus_uminus @ B @ ( uminus_uminus @ B @ B3 ) )
= B3 ) ) ).
% verit_minus_simplify(4)
thf(fact_337_boolean__algebra__class_Oboolean__algebra_Odouble__compl,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A] :
( ( uminus_uminus @ A @ ( uminus_uminus @ A @ X ) )
= X ) ) ).
% boolean_algebra_class.boolean_algebra.double_compl
thf(fact_338_boolean__algebra__class_Oboolean__algebra_Ocompl__eq__compl__iff,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,Y: A] :
( ( ( uminus_uminus @ A @ X )
= ( uminus_uminus @ A @ Y ) )
= ( X = Y ) ) ) ).
% boolean_algebra_class.boolean_algebra.compl_eq_compl_iff
thf(fact_339_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_340_subset__empty,axiom,
! [A: $tType,A2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ ( bot_bot @ ( set @ A ) ) )
= ( A2
= ( bot_bot @ ( set @ A ) ) ) ) ).
% subset_empty
thf(fact_341_empty__subsetI,axiom,
! [A: $tType,A2: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A2 ) ).
% empty_subsetI
thf(fact_342_real__add__minus__iff,axiom,
! [X: real,A3: real] :
( ( ( plus_plus @ real @ X @ ( uminus_uminus @ real @ A3 ) )
= ( zero_zero @ real ) )
= ( X = A3 ) ) ).
% real_add_minus_iff
thf(fact_343_insert__subset,axiom,
! [A: $tType,X: A,A2: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X @ A2 ) @ B2 )
= ( ( member @ A @ X @ B2 )
& ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ) ).
% insert_subset
thf(fact_344_Un__subset__iff,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,C3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) @ C3 )
= ( ( ord_less_eq @ ( set @ A ) @ A2 @ C3 )
& ( ord_less_eq @ ( set @ A ) @ B2 @ C3 ) ) ) ).
% Un_subset_iff
thf(fact_345_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_346_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_347_word__minus__one__le,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) @ X )
= ( X
= ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) ) ) ) ).
% word_minus_one_le
thf(fact_348_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_349_word__n1__ge,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y: word @ A] : ( ord_less_eq @ ( word @ A ) @ Y @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) ) ) ).
% word_n1_ge
thf(fact_350_word__order_Oextremum__unique,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) @ A3 )
= ( A3
= ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) ) ) ) ).
% word_order.extremum_unique
thf(fact_351_word__order_Oextremum,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A] : ( ord_less_eq @ ( word @ A ) @ A3 @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) ) ) ).
% word_order.extremum
thf(fact_352_word__coorder_Oextremum__unique,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ A3 @ ( zero_zero @ ( word @ A ) ) )
= ( A3
= ( zero_zero @ ( word @ A ) ) ) ) ) ).
% word_coorder.extremum_unique
thf(fact_353_word__le__0__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ X @ ( zero_zero @ ( word @ A ) ) )
= ( X
= ( zero_zero @ ( word @ A ) ) ) ) ) ).
% word_le_0_iff
thf(fact_354_psubsetI,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_less @ ( set @ A ) @ A2 @ B2 ) ) ) ).
% psubsetI
thf(fact_355_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_356_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_357_double__eq__0__iff,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A3: A] :
( ( ( plus_plus @ A @ A3 @ A3 )
= ( zero_zero @ A ) )
= ( A3
= ( zero_zero @ A ) ) ) ) ).
% double_eq_0_iff
thf(fact_358_add__0,axiom,
! [A: $tType] :
( ( monoid_add @ A )
=> ! [A3: A] :
( ( plus_plus @ A @ ( zero_zero @ A ) @ A3 )
= A3 ) ) ).
% add_0
thf(fact_359_zero__eq__add__iff__both__eq__0,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [X: A,Y: A] :
( ( ( zero_zero @ A )
= ( plus_plus @ A @ X @ Y ) )
= ( ( X
= ( zero_zero @ A ) )
& ( Y
= ( zero_zero @ A ) ) ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_360_add__eq__0__iff__both__eq__0,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [X: A,Y: A] :
( ( ( plus_plus @ A @ X @ Y )
= ( zero_zero @ A ) )
= ( ( X
= ( zero_zero @ A ) )
& ( Y
= ( zero_zero @ A ) ) ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_361_add__cancel__right__right,axiom,
! [A: $tType] :
( ( cancel1802427076303600483id_add @ A )
=> ! [A3: A,B3: A] :
( ( A3
= ( plus_plus @ A @ A3 @ B3 ) )
= ( B3
= ( zero_zero @ A ) ) ) ) ).
% add_cancel_right_right
thf(fact_362_add__cancel__right__left,axiom,
! [A: $tType] :
( ( cancel1802427076303600483id_add @ A )
=> ! [A3: A,B3: A] :
( ( A3
= ( plus_plus @ A @ B3 @ A3 ) )
= ( B3
= ( zero_zero @ A ) ) ) ) ).
% add_cancel_right_left
thf(fact_363_add__cancel__left__right,axiom,
! [A: $tType] :
( ( cancel1802427076303600483id_add @ A )
=> ! [A3: A,B3: A] :
( ( ( plus_plus @ A @ A3 @ B3 )
= A3 )
= ( B3
= ( zero_zero @ A ) ) ) ) ).
% add_cancel_left_right
thf(fact_364_add__cancel__left__left,axiom,
! [A: $tType] :
( ( cancel1802427076303600483id_add @ A )
=> ! [B3: A,A3: A] :
( ( ( plus_plus @ A @ B3 @ A3 )
= A3 )
= ( B3
= ( zero_zero @ A ) ) ) ) ).
% add_cancel_left_left
thf(fact_365_double__zero__sym,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A3: A] :
( ( ( zero_zero @ A )
= ( plus_plus @ A @ A3 @ A3 ) )
= ( A3
= ( zero_zero @ A ) ) ) ) ).
% double_zero_sym
thf(fact_366_add_Oright__neutral,axiom,
! [A: $tType] :
( ( monoid_add @ A )
=> ! [A3: A] :
( ( plus_plus @ A @ A3 @ ( zero_zero @ A ) )
= A3 ) ) ).
% add.right_neutral
thf(fact_367_add__le__cancel__left,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [C2: A,A3: A,B3: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ C2 @ A3 ) @ ( plus_plus @ A @ C2 @ B3 ) )
= ( ord_less_eq @ A @ A3 @ B3 ) ) ) ).
% add_le_cancel_left
thf(fact_368_add__le__cancel__right,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [A3: A,C2: A,B3: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ A3 @ C2 ) @ ( plus_plus @ A @ B3 @ C2 ) )
= ( ord_less_eq @ A @ A3 @ B3 ) ) ) ).
% add_le_cancel_right
thf(fact_369_add__less__cancel__left,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [C2: A,A3: A,B3: A] :
( ( ord_less @ A @ ( plus_plus @ A @ C2 @ A3 ) @ ( plus_plus @ A @ C2 @ B3 ) )
= ( ord_less @ A @ A3 @ B3 ) ) ) ).
% add_less_cancel_left
thf(fact_370_add__less__cancel__right,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [A3: A,C2: A,B3: A] :
( ( ord_less @ A @ ( plus_plus @ A @ A3 @ C2 ) @ ( plus_plus @ A @ B3 @ C2 ) )
= ( ord_less @ A @ A3 @ B3 ) ) ) ).
% add_less_cancel_right
thf(fact_371_add__numeral__left,axiom,
! [A: $tType] :
( ( numeral @ A )
=> ! [V2: num,W: num,Z: A] :
( ( plus_plus @ A @ ( numeral_numeral @ A @ V2 ) @ ( plus_plus @ A @ ( numeral_numeral @ A @ W ) @ Z ) )
= ( plus_plus @ A @ ( numeral_numeral @ A @ ( plus_plus @ num @ V2 @ W ) ) @ Z ) ) ) ).
% add_numeral_left
thf(fact_372_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_373_semiring__norm_I2_J,axiom,
( ( plus_plus @ num @ one2 @ one2 )
= ( bit0 @ one2 ) ) ).
% semiring_norm(2)
thf(fact_374_semiring__norm_I69_J,axiom,
! [M: num] :
~ ( ord_less_eq @ num @ ( bit0 @ M ) @ one2 ) ).
% semiring_norm(69)
thf(fact_375_neg__equal__zero,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A3: A] :
( ( ( uminus_uminus @ A @ A3 )
= A3 )
= ( A3
= ( zero_zero @ A ) ) ) ) ).
% neg_equal_zero
thf(fact_376_equal__neg__zero,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A3: A] :
( ( A3
= ( uminus_uminus @ A @ A3 ) )
= ( A3
= ( zero_zero @ A ) ) ) ) ).
% equal_neg_zero
thf(fact_377_neg__equal__0__iff__equal,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A] :
( ( ( uminus_uminus @ A @ A3 )
= ( zero_zero @ A ) )
= ( A3
= ( zero_zero @ A ) ) ) ) ).
% neg_equal_0_iff_equal
thf(fact_378_neg__0__equal__iff__equal,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A] :
( ( ( zero_zero @ A )
= ( uminus_uminus @ A @ A3 ) )
= ( ( zero_zero @ A )
= A3 ) ) ) ).
% neg_0_equal_iff_equal
thf(fact_379_add_Oinverse__neutral,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ( ( uminus_uminus @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% add.inverse_neutral
thf(fact_380_neg__le__iff__le,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [B3: A,A3: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ B3 ) @ ( uminus_uminus @ A @ A3 ) )
= ( ord_less_eq @ A @ A3 @ B3 ) ) ) ).
% neg_le_iff_le
thf(fact_381_compl__le__compl__iff,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ X ) @ ( uminus_uminus @ A @ Y ) )
= ( ord_less_eq @ A @ Y @ X ) ) ) ).
% compl_le_compl_iff
thf(fact_382_compl__less__compl__iff,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ X ) @ ( uminus_uminus @ A @ Y ) )
= ( ord_less @ A @ Y @ X ) ) ) ).
% compl_less_compl_iff
thf(fact_383_neg__less__iff__less,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [B3: A,A3: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ B3 ) @ ( uminus_uminus @ A @ A3 ) )
= ( ord_less @ A @ A3 @ B3 ) ) ) ).
% neg_less_iff_less
thf(fact_384_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_385_add__minus__cancel,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B3: A] :
( ( plus_plus @ A @ A3 @ ( plus_plus @ A @ ( uminus_uminus @ A @ A3 ) @ B3 ) )
= B3 ) ) ).
% add_minus_cancel
thf(fact_386_minus__add__cancel,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B3: A] :
( ( plus_plus @ A @ ( uminus_uminus @ A @ A3 ) @ ( plus_plus @ A @ A3 @ B3 ) )
= B3 ) ) ).
% minus_add_cancel
thf(fact_387_minus__add__distrib,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A3: A,B3: A] :
( ( uminus_uminus @ A @ ( plus_plus @ A @ A3 @ B3 ) )
= ( plus_plus @ A @ ( uminus_uminus @ A @ A3 ) @ ( uminus_uminus @ A @ B3 ) ) ) ) ).
% minus_add_distrib
thf(fact_388_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_389_le__sup__iff,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X: A,Y: A,Z: A] :
( ( ord_less_eq @ A @ ( sup_sup @ A @ X @ Y ) @ Z )
= ( ( ord_less_eq @ A @ X @ Z )
& ( ord_less_eq @ A @ Y @ Z ) ) ) ) ).
% le_sup_iff
thf(fact_390_sup_Obounded__iff,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B3: A,C2: A,A3: A] :
( ( ord_less_eq @ A @ ( sup_sup @ A @ B3 @ C2 ) @ A3 )
= ( ( ord_less_eq @ A @ B3 @ A3 )
& ( ord_less_eq @ A @ C2 @ A3 ) ) ) ) ).
% sup.bounded_iff
thf(fact_391_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_392_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_393_subset__Compl__singleton,axiom,
! [A: $tType,A2: set @ A,B3: A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ ( uminus_uminus @ ( set @ A ) @ ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( ~ ( member @ A @ B3 @ A2 ) ) ) ).
% subset_Compl_singleton
thf(fact_394_singleton__insert__inj__eq,axiom,
! [A: $tType,B3: A,A3: A,A2: set @ A] :
( ( ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) )
= ( insert @ A @ A3 @ A2 ) )
= ( ( A3 = B3 )
& ( ord_less_eq @ ( set @ A ) @ A2 @ ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_395_singleton__insert__inj__eq_H,axiom,
! [A: $tType,A3: A,A2: set @ A,B3: A] :
( ( ( insert @ A @ A3 @ A2 )
= ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( ( A3 = B3 )
& ( ord_less_eq @ ( set @ A ) @ A2 @ ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_396_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_397_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_398_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_399_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_400_uint__le__0__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] :
( ( ord_less_eq @ int @ ( semiring_1_unsigned @ A @ int @ X ) @ ( zero_zero @ int ) )
= ( ( semiring_1_unsigned @ A @ int @ X )
= ( zero_zero @ int ) ) ) ) ).
% uint_le_0_iff
thf(fact_401_uint__ge__0,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] : ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( semiring_1_unsigned @ A @ int @ X ) ) ) ).
% uint_ge_0
thf(fact_402_word__le__no,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: num,B3: num] :
( ( ord_less_eq @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ A3 ) @ ( numeral_numeral @ ( word @ A ) @ B3 ) )
= ( ord_less_eq @ int @ ( semiring_1_unsigned @ A @ int @ ( numeral_numeral @ ( word @ A ) @ A3 ) ) @ ( semiring_1_unsigned @ A @ int @ ( numeral_numeral @ ( word @ A ) @ B3 ) ) ) ) ) ).
% word_le_no
thf(fact_403_shiftl__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat] :
( ( bit_Sh4282982442137083160shiftl @ A @ ( zero_zero @ A ) @ N )
= ( zero_zero @ A ) ) ) ).
% shiftl_0
thf(fact_404_pure__true,axiom,
( ( pure_assn @ $true )
= ( one_one @ assn ) ) ).
% pure_true
thf(fact_405_pure__assn__eq__emp__iff,axiom,
! [P: $o] :
( ( ( pure_assn @ P )
= ( one_one @ assn ) )
= P ) ).
% pure_assn_eq_emp_iff
thf(fact_406_shiftl__of__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A] :
( ( bit_Sh4282982442137083160shiftl @ A @ A3 @ ( zero_zero @ nat ) )
= A3 ) ) ).
% shiftl_of_0
thf(fact_407_pure__false,axiom,
( ( pure_assn @ $false )
= ( bot_bot @ assn ) ) ).
% pure_false
thf(fact_408_pure__assn__eq__false__iff,axiom,
! [P: $o] :
( ( ( pure_assn @ P )
= ( bot_bot @ assn ) )
= ~ P ) ).
% pure_assn_eq_false_iff
thf(fact_409_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_410_add__le__same__cancel1,axiom,
! [A: $tType] :
( ( ordere1937475149494474687imp_le @ A )
=> ! [B3: A,A3: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ B3 @ A3 ) @ B3 )
= ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% add_le_same_cancel1
thf(fact_411_add__le__same__cancel2,axiom,
! [A: $tType] :
( ( ordere1937475149494474687imp_le @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ A3 @ B3 ) @ B3 )
= ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% add_le_same_cancel2
thf(fact_412_le__add__same__cancel1,axiom,
! [A: $tType] :
( ( ordere1937475149494474687imp_le @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ A3 @ ( plus_plus @ A @ A3 @ B3 ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ B3 ) ) ) ).
% le_add_same_cancel1
thf(fact_413_le__add__same__cancel2,axiom,
! [A: $tType] :
( ( ordere1937475149494474687imp_le @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ A3 @ ( plus_plus @ A @ B3 @ A3 ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ B3 ) ) ) ).
% le_add_same_cancel2
thf(fact_414_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ A3 @ A3 ) @ ( zero_zero @ A ) )
= ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% double_add_le_zero_iff_single_add_le_zero
thf(fact_415_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A3 @ A3 ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% zero_le_double_add_iff_zero_le_single_add
thf(fact_416_add__less__same__cancel1,axiom,
! [A: $tType] :
( ( ordere1937475149494474687imp_le @ A )
=> ! [B3: A,A3: A] :
( ( ord_less @ A @ ( plus_plus @ A @ B3 @ A3 ) @ B3 )
= ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% add_less_same_cancel1
thf(fact_417_add__less__same__cancel2,axiom,
! [A: $tType] :
( ( ordere1937475149494474687imp_le @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ ( plus_plus @ A @ A3 @ B3 ) @ B3 )
= ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% add_less_same_cancel2
thf(fact_418_less__add__same__cancel1,axiom,
! [A: $tType] :
( ( ordere1937475149494474687imp_le @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ A3 @ ( plus_plus @ A @ A3 @ B3 ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ B3 ) ) ) ).
% less_add_same_cancel1
thf(fact_419_less__add__same__cancel2,axiom,
! [A: $tType] :
( ( ordere1937475149494474687imp_le @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ A3 @ ( plus_plus @ A @ B3 @ A3 ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ B3 ) ) ) ).
% less_add_same_cancel2
thf(fact_420_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( plus_plus @ A @ A3 @ A3 ) @ ( zero_zero @ A ) )
= ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% double_add_less_zero_iff_single_add_less_zero
thf(fact_421_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A3 @ A3 ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% zero_less_double_add_iff_zero_less_single_add
thf(fact_422_neg__0__le__iff__le,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ A3 ) )
= ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% neg_0_le_iff_le
thf(fact_423_neg__le__0__iff__le,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A3 ) @ ( zero_zero @ A ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% neg_le_0_iff_le
thf(fact_424_less__eq__neg__nonpos,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ A3 @ ( uminus_uminus @ A @ A3 ) )
= ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% less_eq_neg_nonpos
thf(fact_425_neg__less__eq__nonneg,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A3 ) @ A3 )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% neg_less_eq_nonneg
thf(fact_426_less__neg__neg,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A3: A] :
( ( ord_less @ A @ A3 @ ( uminus_uminus @ A @ A3 ) )
= ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% less_neg_neg
thf(fact_427_neg__less__pos,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ A3 ) @ A3 )
= ( ord_less @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% neg_less_pos
thf(fact_428_neg__0__less__iff__less,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ A3 ) )
= ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% neg_0_less_iff_less
thf(fact_429_neg__less__0__iff__less,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ A3 ) @ ( zero_zero @ A ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% neg_less_0_iff_less
thf(fact_430_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_431_add_Oright__inverse,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A] :
( ( plus_plus @ A @ A3 @ ( uminus_uminus @ A @ A3 ) )
= ( zero_zero @ A ) ) ) ).
% add.right_inverse
thf(fact_432_ab__left__minus,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A] :
( ( plus_plus @ A @ ( uminus_uminus @ A @ A3 ) @ A3 )
= ( zero_zero @ A ) ) ) ).
% ab_left_minus
thf(fact_433_semiring__norm_I168_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [V2: num,W: num,Y: A] :
( ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) @ ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) @ Y ) )
= ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( plus_plus @ num @ V2 @ W ) ) ) @ Y ) ) ) ).
% semiring_norm(168)
thf(fact_434_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_435_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_436_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_437_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_438_uint__lt__0,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] :
~ ( ord_less @ int @ ( semiring_1_unsigned @ A @ int @ X ) @ ( zero_zero @ int ) ) ) ).
% uint_lt_0
thf(fact_439_log__one,axiom,
! [A3: real] :
( ( log @ A3 @ ( one_one @ real ) )
= ( zero_zero @ real ) ) ).
% log_one
thf(fact_440_norm__pre__pure__iff__sng,axiom,
! [A: $tType,B3: $o,F3: heap_Time_Heap @ A,Q: A > assn] :
( ( hoare_hoare_triple @ A @ ( pure_assn @ B3 ) @ F3 @ Q )
= ( B3
=> ( hoare_hoare_triple @ A @ ( one_one @ assn ) @ F3 @ Q ) ) ) ).
% norm_pre_pure_iff_sng
thf(fact_441_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_442_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_443_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_444_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_445_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_446_power__increasing__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [B3: A,X: nat,Y: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ B3 )
=> ( ( ord_less_eq @ A @ ( power_power @ A @ B3 @ X ) @ ( power_power @ A @ B3 @ Y ) )
= ( ord_less_eq @ nat @ X @ Y ) ) ) ) ).
% power_increasing_iff
thf(fact_447_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_448_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_449_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_450_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_451_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_452_of__nat__le__of__nat__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [B3: nat,W: nat,X: nat] :
( ( ord_less_eq @ A @ ( power_power @ A @ ( semiring_1_of_nat @ A @ B3 ) @ W ) @ ( semiring_1_of_nat @ A @ X ) )
= ( ord_less_eq @ nat @ ( power_power @ nat @ B3 @ W ) @ X ) ) ) ).
% of_nat_le_of_nat_power_cancel_iff
thf(fact_453_of__nat__power__le__of__nat__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X: nat,B3: nat,W: nat] :
( ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ X ) @ ( power_power @ A @ ( semiring_1_of_nat @ A @ B3 ) @ W ) )
= ( ord_less_eq @ nat @ X @ ( power_power @ nat @ B3 @ W ) ) ) ) ).
% of_nat_power_le_of_nat_cancel_iff
thf(fact_454_log__le__cancel__iff,axiom,
! [A3: real,X: real,Y: real] :
( ( ord_less @ real @ ( one_one @ real ) @ A3 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ Y )
=> ( ( ord_less_eq @ real @ ( log @ A3 @ X ) @ ( log @ A3 @ Y ) )
= ( ord_less_eq @ real @ X @ Y ) ) ) ) ) ).
% log_le_cancel_iff
thf(fact_455_log__le__one__cancel__iff,axiom,
! [A3: real,X: real] :
( ( ord_less @ real @ ( one_one @ real ) @ A3 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ ( log @ A3 @ X ) @ ( one_one @ real ) )
= ( ord_less_eq @ real @ X @ A3 ) ) ) ) ).
% log_le_one_cancel_iff
thf(fact_456_one__le__log__cancel__iff,axiom,
! [A3: real,X: real] :
( ( ord_less @ real @ ( one_one @ real ) @ A3 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ ( one_one @ real ) @ ( log @ A3 @ X ) )
= ( ord_less_eq @ real @ A3 @ X ) ) ) ) ).
% one_le_log_cancel_iff
thf(fact_457_log__le__zero__cancel__iff,axiom,
! [A3: real,X: real] :
( ( ord_less @ real @ ( one_one @ real ) @ A3 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ ( log @ A3 @ X ) @ ( zero_zero @ real ) )
= ( ord_less_eq @ real @ X @ ( one_one @ real ) ) ) ) ) ).
% log_le_zero_cancel_iff
thf(fact_458_zero__le__log__cancel__iff,axiom,
! [A3: real,X: real] :
( ( ord_less @ real @ ( one_one @ real ) @ A3 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( log @ A3 @ X ) )
= ( ord_less_eq @ real @ ( one_one @ real ) @ X ) ) ) ) ).
% zero_le_log_cancel_iff
thf(fact_459_zero__less__log__cancel__iff,axiom,
! [A3: real,X: real] :
( ( ord_less @ real @ ( one_one @ real ) @ A3 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( log @ A3 @ X ) )
= ( ord_less @ real @ ( one_one @ real ) @ X ) ) ) ) ).
% zero_less_log_cancel_iff
thf(fact_460_log__less__zero__cancel__iff,axiom,
! [A3: real,X: real] :
( ( ord_less @ real @ ( one_one @ real ) @ A3 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ ( log @ A3 @ X ) @ ( zero_zero @ real ) )
= ( ord_less @ real @ X @ ( one_one @ real ) ) ) ) ) ).
% log_less_zero_cancel_iff
thf(fact_461_one__less__log__cancel__iff,axiom,
! [A3: real,X: real] :
( ( ord_less @ real @ ( one_one @ real ) @ A3 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ ( one_one @ real ) @ ( log @ A3 @ X ) )
= ( ord_less @ real @ A3 @ X ) ) ) ) ).
% one_less_log_cancel_iff
thf(fact_462_log__less__one__cancel__iff,axiom,
! [A3: real,X: real] :
( ( ord_less @ real @ ( one_one @ real ) @ A3 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ ( log @ A3 @ X ) @ ( one_one @ real ) )
= ( ord_less @ real @ X @ A3 ) ) ) ) ).
% log_less_one_cancel_iff
thf(fact_463_log__less__cancel__iff,axiom,
! [A3: real,X: real,Y: real] :
( ( ord_less @ real @ ( one_one @ real ) @ A3 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ Y )
=> ( ( ord_less @ real @ ( log @ A3 @ X ) @ ( log @ A3 @ Y ) )
= ( ord_less @ real @ X @ Y ) ) ) ) ) ).
% log_less_cancel_iff
thf(fact_464_log__eq__one,axiom,
! [A3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ A3 )
=> ( ( A3
!= ( one_one @ real ) )
=> ( ( log @ A3 @ A3 )
= ( one_one @ real ) ) ) ) ).
% log_eq_one
thf(fact_465_word__less__no,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: num,B3: num] :
( ( ord_less @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ A3 ) @ ( numeral_numeral @ ( word @ A ) @ B3 ) )
= ( ord_less @ int @ ( semiring_1_unsigned @ A @ int @ ( numeral_numeral @ ( word @ A ) @ A3 ) ) @ ( semiring_1_unsigned @ A @ int @ ( numeral_numeral @ ( word @ A ) @ B3 ) ) ) ) ) ).
% word_less_no
thf(fact_466_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_467_power__decreasing__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [B3: A,M: nat,N: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ B3 )
=> ( ( ord_less @ A @ B3 @ ( one_one @ A ) )
=> ( ( ord_less_eq @ A @ ( power_power @ A @ B3 @ M ) @ ( power_power @ A @ B3 @ N ) )
= ( ord_less_eq @ nat @ N @ M ) ) ) ) ) ).
% power_decreasing_iff
thf(fact_468_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_469_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_470_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_471_power__mono__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,B3: A,N: nat] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B3 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less_eq @ A @ ( power_power @ A @ A3 @ N ) @ ( power_power @ A @ B3 @ N ) )
= ( ord_less_eq @ A @ A3 @ B3 ) ) ) ) ) ) ).
% power_mono_iff
thf(fact_472_power2__minus,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [A3: A] :
( ( power_power @ A @ ( uminus_uminus @ A @ A3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% power2_minus
thf(fact_473_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_474_numeral__power__le__of__nat__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [I: num,N: nat,X: nat] :
( ( ord_less_eq @ A @ ( power_power @ A @ ( numeral_numeral @ A @ I ) @ N ) @ ( semiring_1_of_nat @ A @ X ) )
= ( ord_less_eq @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ I ) @ N ) @ X ) ) ) ).
% numeral_power_le_of_nat_cancel_iff
thf(fact_475_of__nat__le__numeral__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X: nat,I: num,N: nat] :
( ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ X ) @ ( power_power @ A @ ( numeral_numeral @ A @ I ) @ N ) )
= ( ord_less_eq @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ I ) @ N ) ) ) ) ).
% of_nat_le_numeral_power_cancel_iff
thf(fact_476_log__pow__cancel,axiom,
! [A3: real,B3: nat] :
( ( ord_less @ real @ ( zero_zero @ real ) @ A3 )
=> ( ( A3
!= ( one_one @ real ) )
=> ( ( log @ A3 @ ( power_power @ real @ A3 @ B3 ) )
= ( semiring_1_of_nat @ real @ B3 ) ) ) ) ).
% log_pow_cancel
thf(fact_477_no__plus__overflow__neg,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] :
( ( ord_less @ ( word @ A ) @ X @ ( uminus_uminus @ ( word @ A ) @ Y ) )
=> ( ord_less_eq @ ( word @ A ) @ X @ ( plus_plus @ ( word @ A ) @ X @ Y ) ) ) ) ).
% no_plus_overflow_neg
thf(fact_478_overflow__plus__one__self,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [P4: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) @ P4 ) @ P4 )
= ( P4
= ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) ) ) ) ).
% overflow_plus_one_self
thf(fact_479_real__add__le__0__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( plus_plus @ real @ X @ Y ) @ ( zero_zero @ real ) )
= ( ord_less_eq @ real @ Y @ ( uminus_uminus @ real @ X ) ) ) ).
% real_add_le_0_iff
thf(fact_480_real__0__le__add__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( plus_plus @ real @ X @ Y ) )
= ( ord_less_eq @ real @ ( uminus_uminus @ real @ X ) @ Y ) ) ).
% real_0_le_add_iff
thf(fact_481_plus__1__less,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ X @ ( one_one @ ( word @ A ) ) ) @ X )
= ( X
= ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) ) ) ) ).
% plus_1_less
thf(fact_482_uint__add__ge0,axiom,
! [Aa: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ Aa ) )
=> ! [Xa: word @ Aa,X: word @ A] : ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( plus_plus @ int @ ( semiring_1_unsigned @ Aa @ int @ Xa ) @ ( semiring_1_unsigned @ A @ int @ X ) ) ) ) ).
% uint_add_ge0
thf(fact_483_nle__le,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B3: A] :
( ( ~ ( ord_less_eq @ A @ A3 @ B3 ) )
= ( ( ord_less_eq @ A @ B3 @ A3 )
& ( B3 != A3 ) ) ) ) ).
% nle_le
thf(fact_484_le__cases3,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A,Z: A] :
( ( ( ord_less_eq @ A @ X @ Y )
=> ~ ( ord_less_eq @ A @ Y @ Z ) )
=> ( ( ( ord_less_eq @ A @ Y @ X )
=> ~ ( ord_less_eq @ A @ X @ Z ) )
=> ( ( ( ord_less_eq @ A @ X @ Z )
=> ~ ( ord_less_eq @ A @ Z @ Y ) )
=> ( ( ( ord_less_eq @ A @ Z @ Y )
=> ~ ( ord_less_eq @ A @ Y @ X ) )
=> ( ( ( ord_less_eq @ A @ Y @ Z )
=> ~ ( ord_less_eq @ A @ Z @ X ) )
=> ~ ( ( ord_less_eq @ A @ Z @ X )
=> ~ ( ord_less_eq @ A @ X @ Y ) ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_485_order__class_Oorder__eq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y3: A,Z2: A] : Y3 = Z2 )
= ( ^ [X2: A,Y2: A] :
( ( ord_less_eq @ A @ X2 @ Y2 )
& ( ord_less_eq @ A @ Y2 @ X2 ) ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_486_ord__eq__le__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A3: A,B3: A,C2: A] :
( ( A3 = B3 )
=> ( ( ord_less_eq @ A @ B3 @ C2 )
=> ( ord_less_eq @ A @ A3 @ C2 ) ) ) ) ).
% ord_eq_le_trans
thf(fact_487_ord__le__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A3: A,B3: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( B3 = C2 )
=> ( ord_less_eq @ A @ A3 @ C2 ) ) ) ) ).
% ord_le_eq_trans
thf(fact_488_order__antisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less_eq @ A @ Y @ X )
=> ( X = Y ) ) ) ) ).
% order_antisym
thf(fact_489_order_Otrans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: A,B3: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ A @ B3 @ C2 )
=> ( ord_less_eq @ A @ A3 @ C2 ) ) ) ) ).
% order.trans
thf(fact_490_order__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A,Z: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less_eq @ A @ Y @ Z )
=> ( ord_less_eq @ A @ X @ Z ) ) ) ) ).
% order_trans
thf(fact_491_linorder__wlog,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > A > $o,A3: A,B3: A] :
( ! [A6: A,B7: A] :
( ( ord_less_eq @ A @ A6 @ B7 )
=> ( P @ A6 @ B7 ) )
=> ( ! [A6: A,B7: A] :
( ( P @ B7 @ A6 )
=> ( P @ A6 @ B7 ) )
=> ( P @ A3 @ B3 ) ) ) ) ).
% linorder_wlog
thf(fact_492_dual__order_Oeq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y3: A,Z2: A] : Y3 = Z2 )
= ( ^ [A5: A,B5: A] :
( ( ord_less_eq @ A @ B5 @ A5 )
& ( ord_less_eq @ A @ A5 @ B5 ) ) ) ) ) ).
% dual_order.eq_iff
thf(fact_493_dual__order_Oantisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B3: A,A3: A] :
( ( ord_less_eq @ A @ B3 @ A3 )
=> ( ( ord_less_eq @ A @ A3 @ B3 )
=> ( A3 = B3 ) ) ) ) ).
% dual_order.antisym
thf(fact_494_dual__order_Otrans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [B3: A,A3: A,C2: A] :
( ( ord_less_eq @ A @ B3 @ A3 )
=> ( ( ord_less_eq @ A @ C2 @ B3 )
=> ( ord_less_eq @ A @ C2 @ A3 ) ) ) ) ).
% dual_order.trans
thf(fact_495_antisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ A @ B3 @ A3 )
=> ( A3 = B3 ) ) ) ) ).
% antisym
thf(fact_496_le__funD,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F3: A > B,G2: A > B,X: A] :
( ( ord_less_eq @ ( A > B ) @ F3 @ G2 )
=> ( ord_less_eq @ B @ ( F3 @ X ) @ ( G2 @ X ) ) ) ) ).
% le_funD
thf(fact_497_le__funE,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F3: A > B,G2: A > B,X: A] :
( ( ord_less_eq @ ( A > B ) @ F3 @ G2 )
=> ( ord_less_eq @ B @ ( F3 @ X ) @ ( G2 @ X ) ) ) ) ).
% le_funE
thf(fact_498_le__funI,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F3: A > B,G2: A > B] :
( ! [X3: A] : ( ord_less_eq @ B @ ( F3 @ X3 ) @ ( G2 @ X3 ) )
=> ( ord_less_eq @ ( A > B ) @ F3 @ G2 ) ) ) ).
% le_funI
thf(fact_499_le__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ( ( ord_less_eq @ ( A > B ) )
= ( ^ [F2: A > B,G: A > B] :
! [X2: A] : ( ord_less_eq @ B @ ( F2 @ X2 ) @ ( G @ X2 ) ) ) ) ) ).
% le_fun_def
thf(fact_500_Orderings_Oorder__eq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y3: A,Z2: A] : Y3 = Z2 )
= ( ^ [A5: A,B5: A] :
( ( ord_less_eq @ A @ A5 @ B5 )
& ( ord_less_eq @ A @ B5 @ A5 ) ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_501_order__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A3: A,F3: B > A,B3: B,C2: B] :
( ( ord_less_eq @ A @ A3 @ ( F3 @ B3 ) )
=> ( ( ord_less_eq @ B @ B3 @ C2 )
=> ( ! [X3: B,Y4: B] :
( ( ord_less_eq @ B @ X3 @ Y4 )
=> ( ord_less_eq @ A @ ( F3 @ X3 ) @ ( F3 @ Y4 ) ) )
=> ( ord_less_eq @ A @ A3 @ ( F3 @ C2 ) ) ) ) ) ) ).
% order_subst1
thf(fact_502_order__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C )
& ( order @ A ) )
=> ! [A3: A,B3: A,F3: A > C,C2: C] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ C @ ( F3 @ B3 ) @ C2 )
=> ( ! [X3: A,Y4: A] :
( ( ord_less_eq @ A @ X3 @ Y4 )
=> ( ord_less_eq @ C @ ( F3 @ X3 ) @ ( F3 @ Y4 ) ) )
=> ( ord_less_eq @ C @ ( F3 @ A3 ) @ C2 ) ) ) ) ) ).
% order_subst2
thf(fact_503_order__eq__refl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A] :
( ( X = Y )
=> ( ord_less_eq @ A @ X @ Y ) ) ) ).
% order_eq_refl
thf(fact_504_linorder__linear,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
| ( ord_less_eq @ A @ Y @ X ) ) ) ).
% linorder_linear
thf(fact_505_ord__eq__le__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A3: A,F3: B > A,B3: B,C2: B] :
( ( A3
= ( F3 @ B3 ) )
=> ( ( ord_less_eq @ B @ B3 @ C2 )
=> ( ! [X3: B,Y4: B] :
( ( ord_less_eq @ B @ X3 @ Y4 )
=> ( ord_less_eq @ A @ ( F3 @ X3 ) @ ( F3 @ Y4 ) ) )
=> ( ord_less_eq @ A @ A3 @ ( F3 @ C2 ) ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_506_ord__le__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A3: A,B3: A,F3: A > B,C2: B] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ( F3 @ B3 )
= C2 )
=> ( ! [X3: A,Y4: A] :
( ( ord_less_eq @ A @ X3 @ Y4 )
=> ( ord_less_eq @ B @ ( F3 @ X3 ) @ ( F3 @ Y4 ) ) )
=> ( ord_less_eq @ B @ ( F3 @ A3 ) @ C2 ) ) ) ) ) ).
% ord_le_eq_subst
thf(fact_507_linorder__le__cases,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ~ ( ord_less_eq @ A @ X @ Y )
=> ( ord_less_eq @ A @ Y @ X ) ) ) ).
% linorder_le_cases
thf(fact_508_order__antisym__conv,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ Y @ X )
=> ( ( ord_less_eq @ A @ X @ Y )
= ( X = Y ) ) ) ) ).
% order_antisym_conv
thf(fact_509_add__mono__thms__linordered__semiring_I4_J,axiom,
! [A: $tType] :
( ( ordere6658533253407199908up_add @ A )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus @ A @ I @ K )
= ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_510_add__mono__thms__linordered__semiring_I3_J,axiom,
! [A: $tType] :
( ( ordere6658533253407199908up_add @ A )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( ord_less_eq @ A @ I @ J )
& ( K = L ) )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_511_add__mono__thms__linordered__semiring_I2_J,axiom,
! [A: $tType] :
( ( ordere6658533253407199908up_add @ A )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( I = J )
& ( ord_less_eq @ A @ K @ L ) )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_512_add__mono__thms__linordered__semiring_I1_J,axiom,
! [A: $tType] :
( ( ordere6658533253407199908up_add @ A )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( ord_less_eq @ A @ I @ J )
& ( ord_less_eq @ A @ K @ L ) )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_513_group__cancel_Oadd1,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A2: A,K: A,A3: A,B3: A] :
( ( A2
= ( plus_plus @ A @ K @ A3 ) )
=> ( ( plus_plus @ A @ A2 @ B3 )
= ( plus_plus @ A @ K @ ( plus_plus @ A @ A3 @ B3 ) ) ) ) ) ).
% group_cancel.add1
thf(fact_514_group__cancel_Oadd2,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [B2: A,K: A,B3: A,A3: A] :
( ( B2
= ( plus_plus @ A @ K @ B3 ) )
=> ( ( plus_plus @ A @ A3 @ B2 )
= ( plus_plus @ A @ K @ ( plus_plus @ A @ A3 @ B3 ) ) ) ) ) ).
% group_cancel.add2
thf(fact_515_group__cancel_Oneg1,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A2: A,K: A,A3: A] :
( ( A2
= ( plus_plus @ A @ K @ A3 ) )
=> ( ( uminus_uminus @ A @ A2 )
= ( plus_plus @ A @ ( uminus_uminus @ A @ K ) @ ( uminus_uminus @ A @ A3 ) ) ) ) ) ).
% group_cancel.neg1
thf(fact_516_add_Oassoc,axiom,
! [A: $tType] :
( ( semigroup_add @ A )
=> ! [A3: A,B3: A,C2: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ A3 @ B3 ) @ C2 )
= ( plus_plus @ A @ A3 @ ( plus_plus @ A @ B3 @ C2 ) ) ) ) ).
% add.assoc
thf(fact_517_add_Oleft__cancel,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B3: A,C2: A] :
( ( ( plus_plus @ A @ A3 @ B3 )
= ( plus_plus @ A @ A3 @ C2 ) )
= ( B3 = C2 ) ) ) ).
% add.left_cancel
thf(fact_518_add_Oright__cancel,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [B3: A,A3: A,C2: A] :
( ( ( plus_plus @ A @ B3 @ A3 )
= ( plus_plus @ A @ C2 @ A3 ) )
= ( B3 = C2 ) ) ) ).
% add.right_cancel
thf(fact_519_ab__semigroup__add__class_Oadd_Ocommute,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A )
=> ( ( plus_plus @ A )
= ( ^ [A5: A,B5: A] : ( plus_plus @ A @ B5 @ A5 ) ) ) ) ).
% ab_semigroup_add_class.add.commute
thf(fact_520_equation__minus__iff,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B3: A] :
( ( A3
= ( uminus_uminus @ A @ B3 ) )
= ( B3
= ( uminus_uminus @ A @ A3 ) ) ) ) ).
% equation_minus_iff
thf(fact_521_minus__equation__iff,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B3: A] :
( ( ( uminus_uminus @ A @ A3 )
= B3 )
= ( ( uminus_uminus @ A @ B3 )
= A3 ) ) ) ).
% minus_equation_iff
thf(fact_522_ab__semigroup__add__class_Oadd_Oleft__commute,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A )
=> ! [B3: A,A3: A,C2: A] :
( ( plus_plus @ A @ B3 @ ( plus_plus @ A @ A3 @ C2 ) )
= ( plus_plus @ A @ A3 @ ( plus_plus @ A @ B3 @ C2 ) ) ) ) ).
% ab_semigroup_add_class.add.left_commute
thf(fact_523_le__minus__iff,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ A3 @ ( uminus_uminus @ A @ B3 ) )
= ( ord_less_eq @ A @ B3 @ ( uminus_uminus @ A @ A3 ) ) ) ) ).
% le_minus_iff
thf(fact_524_minus__le__iff,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A3 ) @ B3 )
= ( ord_less_eq @ A @ ( uminus_uminus @ A @ B3 ) @ A3 ) ) ) ).
% minus_le_iff
thf(fact_525_add__mono,axiom,
! [A: $tType] :
( ( ordere6658533253407199908up_add @ A )
=> ! [A3: A,B3: A,C2: A,D2: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ A @ C2 @ D2 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A3 @ C2 ) @ ( plus_plus @ A @ B3 @ D2 ) ) ) ) ) ).
% add_mono
thf(fact_526_add_Oinverse__distrib__swap,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B3: A] :
( ( uminus_uminus @ A @ ( plus_plus @ A @ A3 @ B3 ) )
= ( plus_plus @ A @ ( uminus_uminus @ A @ B3 ) @ ( uminus_uminus @ A @ A3 ) ) ) ) ).
% add.inverse_distrib_swap
thf(fact_527_le__imp__neg__le,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ B3 ) @ ( uminus_uminus @ A @ A3 ) ) ) ) ).
% le_imp_neg_le
thf(fact_528_add__left__imp__eq,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [A3: A,B3: A,C2: A] :
( ( ( plus_plus @ A @ A3 @ B3 )
= ( plus_plus @ A @ A3 @ C2 ) )
=> ( B3 = C2 ) ) ) ).
% add_left_imp_eq
thf(fact_529_add__right__imp__eq,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [B3: A,A3: A,C2: A] :
( ( ( plus_plus @ A @ B3 @ A3 )
= ( plus_plus @ A @ C2 @ A3 ) )
=> ( B3 = C2 ) ) ) ).
% add_right_imp_eq
thf(fact_530_add__left__mono,axiom,
! [A: $tType] :
( ( ordere6658533253407199908up_add @ A )
=> ! [A3: A,B3: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ C2 @ A3 ) @ ( plus_plus @ A @ C2 @ B3 ) ) ) ) ).
% add_left_mono
thf(fact_531_less__eqE,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ~ ! [C5: A] :
( B3
!= ( plus_plus @ A @ A3 @ C5 ) ) ) ) ).
% less_eqE
thf(fact_532_add__right__mono,axiom,
! [A: $tType] :
( ( ordere6658533253407199908up_add @ A )
=> ! [A3: A,B3: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A3 @ C2 ) @ ( plus_plus @ A @ B3 @ C2 ) ) ) ) ).
% add_right_mono
thf(fact_533_le__iff__add,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A5: A,B5: A] :
? [C6: A] :
( B5
= ( plus_plus @ A @ A5 @ C6 ) ) ) ) ) ).
% le_iff_add
thf(fact_534_add__le__imp__le__left,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [C2: A,A3: A,B3: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ C2 @ A3 ) @ ( plus_plus @ A @ C2 @ B3 ) )
=> ( ord_less_eq @ A @ A3 @ B3 ) ) ) ).
% add_le_imp_le_left
thf(fact_535_add__le__imp__le__right,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [A3: A,C2: A,B3: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ A3 @ C2 ) @ ( plus_plus @ A @ B3 @ C2 ) )
=> ( ord_less_eq @ A @ A3 @ B3 ) ) ) ).
% add_le_imp_le_right
thf(fact_536_is__num__normalize_I8_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [A3: A,B3: A] :
( ( uminus_uminus @ A @ ( plus_plus @ A @ A3 @ B3 ) )
= ( plus_plus @ A @ ( uminus_uminus @ A @ B3 ) @ ( uminus_uminus @ A @ A3 ) ) ) ) ).
% is_num_normalize(8)
thf(fact_537_is__num__normalize_I1_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [A3: A,B3: A,C2: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ A3 @ B3 ) @ C2 )
= ( plus_plus @ A @ A3 @ ( plus_plus @ A @ B3 @ C2 ) ) ) ) ).
% is_num_normalize(1)
thf(fact_538_fun__Compl__def,axiom,
! [B: $tType,A: $tType] :
( ( uminus @ B )
=> ( ( uminus_uminus @ ( A > B ) )
= ( ^ [A4: A > B,X2: A] : ( uminus_uminus @ B @ ( A4 @ X2 ) ) ) ) ) ).
% fun_Compl_def
thf(fact_539_verit__comp__simplify1_I2_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A] : ( ord_less_eq @ A @ A3 @ A3 ) ) ).
% verit_comp_simplify1(2)
thf(fact_540_verit__negate__coefficient_I3_J,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B3: A] :
( ( A3 = B3 )
=> ( ( uminus_uminus @ A @ A3 )
= ( uminus_uminus @ A @ B3 ) ) ) ) ).
% verit_negate_coefficient(3)
thf(fact_541_verit__la__disequality,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B3: A] :
( ( A3 = B3 )
| ~ ( ord_less_eq @ A @ A3 @ B3 )
| ~ ( ord_less_eq @ A @ B3 @ A3 ) ) ) ).
% verit_la_disequality
thf(fact_542_unsigned__word__eqI,axiom,
! [A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( semiring_char_0 @ A ) )
=> ! [V2: word @ B,W: word @ B] :
( ( ( semiring_1_unsigned @ B @ A @ V2 )
= ( semiring_1_unsigned @ B @ A @ W ) )
=> ( V2 = W ) ) ) ).
% unsigned_word_eqI
thf(fact_543_word__eq__iff__unsigned,axiom,
! [A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( semiring_char_0 @ A ) )
=> ( ( ^ [Y3: word @ B,Z2: word @ B] : Y3 = Z2 )
= ( ^ [V3: word @ B,W2: word @ B] :
( ( semiring_1_unsigned @ B @ A @ V3 )
= ( semiring_1_unsigned @ B @ A @ W2 ) ) ) ) ) ).
% word_eq_iff_unsigned
thf(fact_544_compl__mono,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ Y ) @ ( uminus_uminus @ A @ X ) ) ) ) ).
% compl_mono
thf(fact_545_compl__le__swap1,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ Y @ ( uminus_uminus @ A @ X ) )
=> ( ord_less_eq @ A @ X @ ( uminus_uminus @ A @ Y ) ) ) ) ).
% compl_le_swap1
thf(fact_546_compl__le__swap2,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ Y ) @ X )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ X ) @ Y ) ) ) ).
% compl_le_swap2
thf(fact_547_word__less__eq__iff__unsigned,axiom,
! [A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( linordered_semidom @ A ) )
=> ( ( ord_less_eq @ ( word @ B ) )
= ( ^ [A5: word @ B,B5: word @ B] : ( ord_less_eq @ A @ ( semiring_1_unsigned @ B @ A @ A5 ) @ ( semiring_1_unsigned @ B @ A @ B5 ) ) ) ) ) ).
% word_less_eq_iff_unsigned
thf(fact_548_uint__cong,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] :
( ( X = Y )
=> ( ( semiring_1_unsigned @ A @ int @ X )
= ( semiring_1_unsigned @ A @ int @ Y ) ) ) ) ).
% uint_cong
thf(fact_549_uint__add__le,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] : ( ord_less_eq @ int @ ( semiring_1_unsigned @ A @ int @ ( plus_plus @ ( word @ A ) @ X @ Y ) ) @ ( plus_plus @ int @ ( semiring_1_unsigned @ A @ int @ X ) @ ( semiring_1_unsigned @ A @ int @ Y ) ) ) ) ).
% uint_add_le
thf(fact_550_word__le__def,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( ord_less_eq @ ( word @ A ) )
= ( ^ [A5: word @ A,B5: word @ A] : ( ord_less_eq @ int @ ( semiring_1_unsigned @ A @ int @ A5 ) @ ( semiring_1_unsigned @ A @ int @ B5 ) ) ) ) ) ).
% word_le_def
thf(fact_551_word__uint__eqI,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A,B3: word @ A] :
( ( ( semiring_1_unsigned @ A @ int @ A3 )
= ( semiring_1_unsigned @ A @ int @ B3 ) )
=> ( A3 = B3 ) ) ) ).
% word_uint_eqI
thf(fact_552_uint__plus__simple,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ X @ ( plus_plus @ ( word @ A ) @ X @ Y ) )
=> ( ( semiring_1_unsigned @ A @ int @ ( plus_plus @ ( word @ A ) @ X @ Y ) )
= ( plus_plus @ int @ ( semiring_1_unsigned @ A @ int @ X ) @ ( semiring_1_unsigned @ A @ int @ Y ) ) ) ) ) ).
% uint_plus_simple
thf(fact_553_word__uint__eq__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( ^ [Y3: word @ A,Z2: word @ A] : Y3 = Z2 )
= ( ^ [A5: word @ A,B5: word @ A] :
( ( semiring_1_unsigned @ A @ int @ A5 )
= ( semiring_1_unsigned @ A @ int @ B5 ) ) ) ) ) ).
% word_uint_eq_iff
thf(fact_554_uint__plus__simple__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ X @ ( plus_plus @ ( word @ A ) @ X @ Y ) )
= ( ( semiring_1_unsigned @ A @ int @ ( plus_plus @ ( word @ A ) @ X @ Y ) )
= ( plus_plus @ int @ ( semiring_1_unsigned @ A @ int @ X ) @ ( semiring_1_unsigned @ A @ int @ Y ) ) ) ) ) ).
% uint_plus_simple_iff
thf(fact_555_add__eq__0__iff,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B3: A] :
( ( ( plus_plus @ A @ A3 @ B3 )
= ( zero_zero @ A ) )
= ( B3
= ( uminus_uminus @ A @ A3 ) ) ) ) ).
% add_eq_0_iff
thf(fact_556_ab__group__add__class_Oab__left__minus,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A3: A] :
( ( plus_plus @ A @ ( uminus_uminus @ A @ A3 ) @ A3 )
= ( zero_zero @ A ) ) ) ).
% ab_group_add_class.ab_left_minus
thf(fact_557_add_Oinverse__unique,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B3: A] :
( ( ( plus_plus @ A @ A3 @ B3 )
= ( zero_zero @ A ) )
=> ( ( uminus_uminus @ A @ A3 )
= B3 ) ) ) ).
% add.inverse_unique
thf(fact_558_eq__neg__iff__add__eq__0,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B3: A] :
( ( A3
= ( uminus_uminus @ A @ B3 ) )
= ( ( plus_plus @ A @ A3 @ B3 )
= ( zero_zero @ A ) ) ) ) ).
% eq_neg_iff_add_eq_0
thf(fact_559_neg__eq__iff__add__eq__0,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B3: A] :
( ( ( uminus_uminus @ A @ A3 )
= B3 )
= ( ( plus_plus @ A @ A3 @ B3 )
= ( zero_zero @ A ) ) ) ) ).
% neg_eq_iff_add_eq_0
thf(fact_560_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_561_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_562_add__decreasing,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A3: A,C2: A,B3: A] :
( ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ C2 @ B3 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A3 @ C2 ) @ B3 ) ) ) ) ).
% add_decreasing
thf(fact_563_add__increasing,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A3: A,B3: A,C2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ B3 @ C2 )
=> ( ord_less_eq @ A @ B3 @ ( plus_plus @ A @ A3 @ C2 ) ) ) ) ) ).
% add_increasing
thf(fact_564_add__decreasing2,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [C2: A,A3: A,B3: A] :
( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A3 @ C2 ) @ B3 ) ) ) ) ).
% add_decreasing2
thf(fact_565_add__increasing2,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [C2: A,B3: A,A3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ord_less_eq @ A @ B3 @ A3 )
=> ( ord_less_eq @ A @ B3 @ ( plus_plus @ A @ A3 @ C2 ) ) ) ) ) ).
% add_increasing2
thf(fact_566_add__nonneg__nonneg,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B3 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A3 @ B3 ) ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_567_add__nonpos__nonpos,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ B3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A3 @ B3 ) @ ( zero_zero @ A ) ) ) ) ) ).
% add_nonpos_nonpos
thf(fact_568_add__nonneg__eq__0__iff,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
=> ( ( ( plus_plus @ A @ X @ Y )
= ( zero_zero @ A ) )
= ( ( X
= ( zero_zero @ A ) )
& ( Y
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_569_add__nonpos__eq__0__iff,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ Y @ ( zero_zero @ A ) )
=> ( ( ( plus_plus @ A @ X @ Y )
= ( zero_zero @ A ) )
= ( ( X
= ( zero_zero @ A ) )
& ( Y
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_570_add__less__le__mono,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [A3: A,B3: A,C2: A,D2: A] :
( ( ord_less @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ A @ C2 @ D2 )
=> ( ord_less @ A @ ( plus_plus @ A @ A3 @ C2 ) @ ( plus_plus @ A @ B3 @ D2 ) ) ) ) ) ).
% add_less_le_mono
thf(fact_571_add__le__less__mono,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [A3: A,B3: A,C2: A,D2: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ord_less @ A @ C2 @ D2 )
=> ( ord_less @ A @ ( plus_plus @ A @ A3 @ C2 ) @ ( plus_plus @ A @ B3 @ D2 ) ) ) ) ) ).
% add_le_less_mono
thf(fact_572_add__mono__thms__linordered__field_I3_J,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( ord_less @ A @ I @ J )
& ( ord_less_eq @ A @ K @ L ) )
=> ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_573_add__mono__thms__linordered__field_I4_J,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( ord_less_eq @ A @ I @ J )
& ( ord_less @ A @ K @ L ) )
=> ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_574_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_575_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_576_subset__Compl__self__eq,axiom,
! [A: $tType,A2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ ( uminus_uminus @ ( set @ A ) @ A2 ) )
= ( A2
= ( bot_bot @ ( set @ A ) ) ) ) ).
% subset_Compl_self_eq
thf(fact_577_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_578_Abs__fnat__hom__add,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [A3: nat,B3: nat] :
( ( plus_plus @ A @ ( semiring_1_of_nat @ A @ A3 ) @ ( semiring_1_of_nat @ A @ B3 ) )
= ( semiring_1_of_nat @ A @ ( plus_plus @ nat @ A3 @ B3 ) ) ) ) ).
% Abs_fnat_hom_add
thf(fact_579_mono__nat__linear__lb,axiom,
! [F3: nat > nat,M: nat,K: nat] :
( ! [M4: nat,N2: nat] :
( ( ord_less @ nat @ M4 @ N2 )
=> ( ord_less @ nat @ ( F3 @ M4 ) @ ( F3 @ N2 ) ) )
=> ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( F3 @ M ) @ K ) @ ( F3 @ ( plus_plus @ nat @ M @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_580_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_581_int__ge__induct,axiom,
! [K: int,I: int,P: int > $o] :
( ( ord_less_eq @ int @ K @ I )
=> ( ( P @ K )
=> ( ! [I3: int] :
( ( ord_less_eq @ int @ K @ I3 )
=> ( ( P @ I3 )
=> ( P @ ( plus_plus @ int @ I3 @ ( one_one @ int ) ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_ge_induct
thf(fact_582_word__order_Oextremum__uniqueI,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) @ A3 )
=> ( A3
= ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) ) ) ) ).
% word_order.extremum_uniqueI
thf(fact_583_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_584_int__ops_I5_J,axiom,
! [A3: nat,B3: nat] :
( ( semiring_1_of_nat @ int @ ( plus_plus @ nat @ A3 @ B3 ) )
= ( plus_plus @ int @ ( semiring_1_of_nat @ int @ A3 ) @ ( semiring_1_of_nat @ int @ B3 ) ) ) ).
% int_ops(5)
thf(fact_585_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_586_zle__iff__zadd,axiom,
( ( ord_less_eq @ int )
= ( ^ [W2: int,Z4: int] :
? [N4: nat] :
( Z4
= ( plus_plus @ int @ W2 @ ( semiring_1_of_nat @ int @ N4 ) ) ) ) ) ).
% zle_iff_zadd
thf(fact_587_neq__0__no__wrap,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ X @ ( plus_plus @ ( word @ A ) @ X @ Y ) )
=> ( ( X
!= ( zero_zero @ ( word @ A ) ) )
=> ( ( plus_plus @ ( word @ A ) @ X @ Y )
!= ( zero_zero @ ( word @ A ) ) ) ) ) ) ).
% neq_0_no_wrap
thf(fact_588_word__plus__strict__mono__right,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y: word @ A,Z: word @ A,X: word @ A] :
( ( ord_less @ ( word @ A ) @ Y @ Z )
=> ( ( ord_less_eq @ ( word @ A ) @ X @ ( plus_plus @ ( word @ A ) @ X @ Z ) )
=> ( ord_less @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ X @ Y ) @ ( plus_plus @ ( word @ A ) @ X @ Z ) ) ) ) ) ).
% word_plus_strict_mono_right
thf(fact_589_plus__le__left__cancel__nowrap,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y6: word @ A,Y: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ X @ ( plus_plus @ ( word @ A ) @ X @ Y6 ) )
=> ( ( ord_less_eq @ ( word @ A ) @ X @ ( plus_plus @ ( word @ A ) @ X @ Y ) )
=> ( ( ord_less @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ X @ Y6 ) @ ( plus_plus @ ( word @ A ) @ X @ Y ) )
= ( ord_less @ ( word @ A ) @ Y6 @ Y ) ) ) ) ) ).
% plus_le_left_cancel_nowrap
thf(fact_590_word__le__plus,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A,B3: word @ A,C2: word @ A] :
( ( ord_less @ ( word @ A ) @ A3 @ ( plus_plus @ ( word @ A ) @ A3 @ B3 ) )
=> ( ( ord_less @ ( word @ A ) @ C2 @ B3 )
=> ( ord_less_eq @ ( word @ A ) @ A3 @ ( plus_plus @ ( word @ A ) @ A3 @ C2 ) ) ) ) ) ).
% word_le_plus
thf(fact_591_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_592_nat__int__comparison_I3_J,axiom,
( ( ord_less_eq @ nat )
= ( ^ [A5: nat,B5: nat] : ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ A5 ) @ ( semiring_1_of_nat @ int @ B5 ) ) ) ) ).
% nat_int_comparison(3)
thf(fact_593_word__le__add,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ X @ Y )
=> ? [N2: nat] :
( Y
= ( plus_plus @ ( word @ A ) @ X @ ( semiring_1_of_nat @ ( word @ A ) @ N2 ) ) ) ) ) ).
% word_le_add
thf(fact_594_word__le__make__less,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y: word @ A,X: word @ A] :
( ( Y
!= ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) )
=> ( ( ord_less_eq @ ( word @ A ) @ X @ Y )
= ( ord_less @ ( word @ A ) @ X @ ( plus_plus @ ( word @ A ) @ Y @ ( one_one @ ( word @ A ) ) ) ) ) ) ) ).
% word_le_make_less
thf(fact_595_word__Suc__leq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [K: word @ A,X: word @ A] :
( ( K
!= ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) )
=> ( ( ord_less @ ( word @ A ) @ X @ ( plus_plus @ ( word @ A ) @ K @ ( one_one @ ( word @ A ) ) ) )
= ( ord_less_eq @ ( word @ A ) @ X @ K ) ) ) ) ).
% word_Suc_leq
thf(fact_596_word__Suc__le,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,K: word @ A] :
( ( X
!= ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) )
=> ( ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ X @ ( one_one @ ( word @ A ) ) ) @ K )
= ( ord_less @ ( word @ A ) @ X @ K ) ) ) ) ).
% word_Suc_le
thf(fact_597_dbl__inc__def,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( ( neg_numeral_dbl_inc @ A )
= ( ^ [X2: A] : ( plus_plus @ A @ ( plus_plus @ A @ X2 @ X2 ) @ ( one_one @ A ) ) ) ) ) ).
% dbl_inc_def
thf(fact_598_nat__leq__as__int,axiom,
( ( ord_less_eq @ nat )
= ( ^ [A5: nat,B5: nat] : ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ A5 ) @ ( semiring_1_of_nat @ int @ B5 ) ) ) ) ).
% nat_leq_as_int
thf(fact_599_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_600_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_601_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_602_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_603_add__neg__nonpos,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ B3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( plus_plus @ A @ A3 @ B3 ) @ ( zero_zero @ A ) ) ) ) ) ).
% add_neg_nonpos
thf(fact_604_add__nonneg__pos,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B3 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A3 @ B3 ) ) ) ) ) ).
% add_nonneg_pos
thf(fact_605_add__nonpos__neg,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ B3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( plus_plus @ A @ A3 @ B3 ) @ ( zero_zero @ A ) ) ) ) ) ).
% add_nonpos_neg
thf(fact_606_add__pos__nonneg,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B3 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A3 @ B3 ) ) ) ) ) ).
% add_pos_nonneg
thf(fact_607_add__strict__increasing,axiom,
! [A: $tType] :
( ( ordere8940638589300402666id_add @ A )
=> ! [A3: A,B3: A,C2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ B3 @ C2 )
=> ( ord_less @ A @ B3 @ ( plus_plus @ A @ A3 @ C2 ) ) ) ) ) ).
% add_strict_increasing
thf(fact_608_add__strict__increasing2,axiom,
! [A: $tType] :
( ( ordere8940638589300402666id_add @ A )
=> ! [A3: A,B3: A,C2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ B3 @ C2 )
=> ( ord_less @ A @ B3 @ ( plus_plus @ A @ A3 @ C2 ) ) ) ) ) ).
% add_strict_increasing2
thf(fact_609_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_610_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_611_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_612_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_613_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_614_power__increasing,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N: nat,N3: nat,A3: A] :
( ( ord_less_eq @ nat @ N @ N3 )
=> ( ( ord_less_eq @ A @ ( one_one @ A ) @ A3 )
=> ( ord_less_eq @ A @ ( power_power @ A @ A3 @ N ) @ ( power_power @ A @ A3 @ N3 ) ) ) ) ) ).
% power_increasing
thf(fact_615_real__add__less__0__iff,axiom,
! [X: real,Y: real] :
( ( ord_less @ real @ ( plus_plus @ real @ X @ Y ) @ ( zero_zero @ real ) )
= ( ord_less @ real @ Y @ ( uminus_uminus @ real @ X ) ) ) ).
% real_add_less_0_iff
thf(fact_616_real__0__less__add__iff,axiom,
! [X: real,Y: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( plus_plus @ real @ X @ Y ) )
= ( ord_less @ real @ ( uminus_uminus @ real @ X ) @ Y ) ) ).
% real_0_less_add_iff
thf(fact_617_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_618_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_619_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_620_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_621_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_622_max__word__wrap,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] :
( ( ( plus_plus @ ( word @ A ) @ X @ ( one_one @ ( word @ A ) ) )
= ( zero_zero @ ( word @ A ) ) )
=> ( X
= ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) ) ) ) ).
% max_word_wrap
thf(fact_623_less__x__plus__1,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] :
( ( X
!= ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) )
=> ( ( ord_less @ ( word @ A ) @ Y @ ( plus_plus @ ( word @ A ) @ X @ ( one_one @ ( word @ A ) ) ) )
= ( ( ord_less @ ( word @ A ) @ Y @ X )
| ( Y = X ) ) ) ) ) ).
% less_x_plus_1
thf(fact_624_word__add__no__overflow,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] :
( ( ord_less @ ( word @ A ) @ X @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) )
=> ( ord_less @ ( word @ A ) @ X @ ( plus_plus @ ( word @ A ) @ X @ ( one_one @ ( word @ A ) ) ) ) ) ) ).
% word_add_no_overflow
thf(fact_625_word__plus__one__nonzero,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ X @ ( plus_plus @ ( word @ A ) @ X @ Y ) )
=> ( ( Y
!= ( zero_zero @ ( word @ A ) ) )
=> ( ( plus_plus @ ( word @ A ) @ X @ ( one_one @ ( word @ A ) ) )
!= ( zero_zero @ ( word @ A ) ) ) ) ) ) ).
% word_plus_one_nonzero
thf(fact_626_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_627_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_628_word__le__plus__1,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y: word @ A,N: word @ A,A3: word @ A] :
( ( ord_less @ ( word @ A ) @ Y @ ( plus_plus @ ( word @ A ) @ Y @ N ) )
=> ( ( ord_less @ ( word @ A ) @ A3 @ N )
=> ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ Y @ A3 ) @ ( plus_plus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ Y @ A3 ) @ ( one_one @ ( word @ A ) ) ) ) ) ) ) ).
% word_le_plus_1
thf(fact_629_plus__one__helper,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,N: word @ A] :
( ( ord_less @ ( word @ A ) @ X @ ( plus_plus @ ( word @ A ) @ N @ ( one_one @ ( word @ A ) ) ) )
=> ( ord_less_eq @ ( word @ A ) @ X @ N ) ) ) ).
% plus_one_helper
thf(fact_630_compl__less__swap1,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [Y: A,X: A] :
( ( ord_less @ A @ Y @ ( uminus_uminus @ A @ X ) )
=> ( ord_less @ A @ X @ ( uminus_uminus @ A @ Y ) ) ) ) ).
% compl_less_swap1
thf(fact_631_compl__less__swap2,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [Y: A,X: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ Y ) @ X )
=> ( ord_less @ A @ ( uminus_uminus @ A @ X ) @ Y ) ) ) ).
% compl_less_swap2
thf(fact_632_verit__negate__coefficient_I2_J,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ A3 @ B3 )
=> ( ord_less @ A @ ( uminus_uminus @ A @ B3 ) @ ( uminus_uminus @ A @ A3 ) ) ) ) ).
% verit_negate_coefficient(2)
thf(fact_633_less__minus__iff,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ A3 @ ( uminus_uminus @ A @ B3 ) )
= ( ord_less @ A @ B3 @ ( uminus_uminus @ A @ A3 ) ) ) ) ).
% less_minus_iff
thf(fact_634_minus__less__iff,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ A3 ) @ B3 )
= ( ord_less @ A @ ( uminus_uminus @ A @ B3 ) @ A3 ) ) ) ).
% minus_less_iff
thf(fact_635_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_636_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_637_verit__sum__simplify,axiom,
! [A: $tType] :
( ( cancel1802427076303600483id_add @ A )
=> ! [A3: A] :
( ( plus_plus @ A @ A3 @ ( zero_zero @ A ) )
= A3 ) ) ).
% verit_sum_simplify
thf(fact_638_add_Ogroup__left__neutral,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A] :
( ( plus_plus @ A @ ( zero_zero @ A ) @ A3 )
= A3 ) ) ).
% add.group_left_neutral
thf(fact_639_add_Ocomm__neutral,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A3: A] :
( ( plus_plus @ A @ A3 @ ( zero_zero @ A ) )
= A3 ) ) ).
% add.comm_neutral
thf(fact_640_comm__monoid__add__class_Oadd__0,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A3: A] :
( ( plus_plus @ A @ ( zero_zero @ A ) @ A3 )
= A3 ) ) ).
% comm_monoid_add_class.add_0
thf(fact_641_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_642_add__mono__thms__linordered__field_I5_J,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( ord_less @ A @ I @ J )
& ( ord_less @ A @ K @ L ) )
=> ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_643_add__mono__thms__linordered__field_I2_J,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( I = J )
& ( ord_less @ A @ K @ L ) )
=> ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_644_add__mono__thms__linordered__field_I1_J,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( ord_less @ A @ I @ J )
& ( K = L ) )
=> ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_645_add__strict__mono,axiom,
! [A: $tType] :
( ( strict9044650504122735259up_add @ A )
=> ! [A3: A,B3: A,C2: A,D2: A] :
( ( ord_less @ A @ A3 @ B3 )
=> ( ( ord_less @ A @ C2 @ D2 )
=> ( ord_less @ A @ ( plus_plus @ A @ A3 @ C2 ) @ ( plus_plus @ A @ B3 @ D2 ) ) ) ) ) ).
% add_strict_mono
thf(fact_646_add__strict__left__mono,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [A3: A,B3: A,C2: A] :
( ( ord_less @ A @ A3 @ B3 )
=> ( ord_less @ A @ ( plus_plus @ A @ C2 @ A3 ) @ ( plus_plus @ A @ C2 @ B3 ) ) ) ) ).
% add_strict_left_mono
thf(fact_647_add__strict__right__mono,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [A3: A,B3: A,C2: A] :
( ( ord_less @ A @ A3 @ B3 )
=> ( ord_less @ A @ ( plus_plus @ A @ A3 @ C2 ) @ ( plus_plus @ A @ B3 @ C2 ) ) ) ) ).
% add_strict_right_mono
thf(fact_648_add__less__imp__less__left,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [C2: A,A3: A,B3: A] :
( ( ord_less @ A @ ( plus_plus @ A @ C2 @ A3 ) @ ( plus_plus @ A @ C2 @ B3 ) )
=> ( ord_less @ A @ A3 @ B3 ) ) ) ).
% add_less_imp_less_left
thf(fact_649_add__less__imp__less__right,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [A3: A,C2: A,B3: A] :
( ( ord_less @ A @ ( plus_plus @ A @ A3 @ C2 ) @ ( plus_plus @ A @ B3 @ C2 ) )
=> ( ord_less @ A @ A3 @ B3 ) ) ) ).
% add_less_imp_less_right
thf(fact_650_add__One__commute,axiom,
! [N: num] :
( ( plus_plus @ num @ one2 @ N )
= ( plus_plus @ num @ N @ one2 ) ) ).
% add_One_commute
thf(fact_651_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_652_zero__le,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [X: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ X ) ) ).
% zero_le
thf(fact_653_verit__comp__simplify1_I3_J,axiom,
! [B: $tType] :
( ( linorder @ B )
=> ! [B8: B,A7: B] :
( ( ~ ( ord_less_eq @ B @ B8 @ A7 ) )
= ( ord_less @ B @ A7 @ B8 ) ) ) ).
% verit_comp_simplify1(3)
thf(fact_654_leD,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ Y @ X )
=> ~ ( ord_less @ A @ X @ Y ) ) ) ).
% leD
thf(fact_655_leI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ~ ( ord_less @ A @ X @ Y )
=> ( ord_less_eq @ A @ Y @ X ) ) ) ).
% leI
thf(fact_656_nless__le,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A,B3: A] :
( ( ~ ( ord_less @ A @ A3 @ B3 ) )
= ( ~ ( ord_less_eq @ A @ A3 @ B3 )
| ( A3 = B3 ) ) ) ) ).
% nless_le
thf(fact_657_antisym__conv1,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X: A,Y: A] :
( ~ ( ord_less @ A @ X @ Y )
=> ( ( ord_less_eq @ A @ X @ Y )
= ( X = Y ) ) ) ) ).
% antisym_conv1
thf(fact_658_antisym__conv2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ~ ( ord_less @ A @ X @ Y ) )
= ( X = Y ) ) ) ) ).
% antisym_conv2
thf(fact_659_dense__ge,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [Z: A,Y: A] :
( ! [X3: A] :
( ( ord_less @ A @ Z @ X3 )
=> ( ord_less_eq @ A @ Y @ X3 ) )
=> ( ord_less_eq @ A @ Y @ Z ) ) ) ).
% dense_ge
thf(fact_660_dense__le,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [Y: A,Z: A] :
( ! [X3: A] :
( ( ord_less @ A @ X3 @ Y )
=> ( ord_less_eq @ A @ X3 @ Z ) )
=> ( ord_less_eq @ A @ Y @ Z ) ) ) ).
% dense_le
thf(fact_661_less__le__not__le,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( ( ord_less @ A )
= ( ^ [X2: A,Y2: A] :
( ( ord_less_eq @ A @ X2 @ Y2 )
& ~ ( ord_less_eq @ A @ Y2 @ X2 ) ) ) ) ) ).
% less_le_not_le
thf(fact_662_not__le__imp__less,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Y: A,X: A] :
( ~ ( ord_less_eq @ A @ Y @ X )
=> ( ord_less @ A @ X @ Y ) ) ) ).
% not_le_imp_less
thf(fact_663_order_Oorder__iff__strict,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A5: A,B5: A] :
( ( ord_less @ A @ A5 @ B5 )
| ( A5 = B5 ) ) ) ) ) ).
% order.order_iff_strict
thf(fact_664_order_Ostrict__iff__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less @ A )
= ( ^ [A5: A,B5: A] :
( ( ord_less_eq @ A @ A5 @ B5 )
& ( A5 != B5 ) ) ) ) ) ).
% order.strict_iff_order
thf(fact_665_order_Ostrict__trans1,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: A,B3: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ord_less @ A @ B3 @ C2 )
=> ( ord_less @ A @ A3 @ C2 ) ) ) ) ).
% order.strict_trans1
thf(fact_666_order_Ostrict__trans2,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: A,B3: A,C2: A] :
( ( ord_less @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ A @ B3 @ C2 )
=> ( ord_less @ A @ A3 @ C2 ) ) ) ) ).
% order.strict_trans2
thf(fact_667_order_Ostrict__iff__not,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( ( ord_less @ A )
= ( ^ [A5: A,B5: A] :
( ( ord_less_eq @ A @ A5 @ B5 )
& ~ ( ord_less_eq @ A @ B5 @ A5 ) ) ) ) ) ).
% order.strict_iff_not
thf(fact_668_dense__ge__bounded,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [Z: A,X: A,Y: A] :
( ( ord_less @ A @ Z @ X )
=> ( ! [W3: A] :
( ( ord_less @ A @ Z @ W3 )
=> ( ( ord_less @ A @ W3 @ X )
=> ( ord_less_eq @ A @ Y @ W3 ) ) )
=> ( ord_less_eq @ A @ Y @ Z ) ) ) ) ).
% dense_ge_bounded
thf(fact_669_dense__le__bounded,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [X: A,Y: A,Z: A] :
( ( ord_less @ A @ X @ Y )
=> ( ! [W3: A] :
( ( ord_less @ A @ X @ W3 )
=> ( ( ord_less @ A @ W3 @ Y )
=> ( ord_less_eq @ A @ W3 @ Z ) ) )
=> ( ord_less_eq @ A @ Y @ Z ) ) ) ) ).
% dense_le_bounded
thf(fact_670_dual__order_Oorder__iff__strict,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B5: A,A5: A] :
( ( ord_less @ A @ B5 @ A5 )
| ( A5 = B5 ) ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_671_dual__order_Ostrict__iff__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less @ A )
= ( ^ [B5: A,A5: A] :
( ( ord_less_eq @ A @ B5 @ A5 )
& ( A5 != B5 ) ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_672_dual__order_Ostrict__trans1,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [B3: A,A3: A,C2: A] :
( ( ord_less_eq @ A @ B3 @ A3 )
=> ( ( ord_less @ A @ C2 @ B3 )
=> ( ord_less @ A @ C2 @ A3 ) ) ) ) ).
% dual_order.strict_trans1
thf(fact_673_dual__order_Ostrict__trans2,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [B3: A,A3: A,C2: A] :
( ( ord_less @ A @ B3 @ A3 )
=> ( ( ord_less_eq @ A @ C2 @ B3 )
=> ( ord_less @ A @ C2 @ A3 ) ) ) ) ).
% dual_order.strict_trans2
thf(fact_674_dual__order_Ostrict__iff__not,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( ( ord_less @ A )
= ( ^ [B5: A,A5: A] :
( ( ord_less_eq @ A @ B5 @ A5 )
& ~ ( ord_less_eq @ A @ A5 @ B5 ) ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_675_order_Ostrict__implies__order,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ A3 @ B3 )
=> ( ord_less_eq @ A @ A3 @ B3 ) ) ) ).
% order.strict_implies_order
thf(fact_676_dual__order_Ostrict__implies__order,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [B3: A,A3: A] :
( ( ord_less @ A @ B3 @ A3 )
=> ( ord_less_eq @ A @ B3 @ A3 ) ) ) ).
% dual_order.strict_implies_order
thf(fact_677_order__le__less,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [X2: A,Y2: A] :
( ( ord_less @ A @ X2 @ Y2 )
| ( X2 = Y2 ) ) ) ) ) ).
% order_le_less
thf(fact_678_order__less__le,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less @ A )
= ( ^ [X2: A,Y2: A] :
( ( ord_less_eq @ A @ X2 @ Y2 )
& ( X2 != Y2 ) ) ) ) ) ).
% order_less_le
thf(fact_679_linorder__not__le,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( ~ ( ord_less_eq @ A @ X @ Y ) )
= ( ord_less @ A @ Y @ X ) ) ) ).
% linorder_not_le
thf(fact_680_linorder__not__less,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( ~ ( ord_less @ A @ X @ Y ) )
= ( ord_less_eq @ A @ Y @ X ) ) ) ).
% linorder_not_less
thf(fact_681_order__less__imp__le,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ( ord_less_eq @ A @ X @ Y ) ) ) ).
% order_less_imp_le
thf(fact_682_order__le__neq__trans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( A3 != B3 )
=> ( ord_less @ A @ A3 @ B3 ) ) ) ) ).
% order_le_neq_trans
thf(fact_683_order__neq__le__trans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A,B3: A] :
( ( A3 != B3 )
=> ( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ord_less @ A @ A3 @ B3 ) ) ) ) ).
% order_neq_le_trans
thf(fact_684_order__le__less__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A,Z: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less @ A @ Y @ Z )
=> ( ord_less @ A @ X @ Z ) ) ) ) ).
% order_le_less_trans
thf(fact_685_order__less__le__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A,Z: A] :
( ( ord_less @ A @ X @ Y )
=> ( ( ord_less_eq @ A @ Y @ Z )
=> ( ord_less @ A @ X @ Z ) ) ) ) ).
% order_less_le_trans
thf(fact_686_order__le__less__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A3: A,F3: B > A,B3: B,C2: B] :
( ( ord_less_eq @ A @ A3 @ ( F3 @ B3 ) )
=> ( ( ord_less @ B @ B3 @ C2 )
=> ( ! [X3: B,Y4: B] :
( ( ord_less @ B @ X3 @ Y4 )
=> ( ord_less @ A @ ( F3 @ X3 ) @ ( F3 @ Y4 ) ) )
=> ( ord_less @ A @ A3 @ ( F3 @ C2 ) ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_687_order__le__less__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C )
& ( order @ A ) )
=> ! [A3: A,B3: A,F3: A > C,C2: C] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ord_less @ C @ ( F3 @ B3 ) @ C2 )
=> ( ! [X3: A,Y4: A] :
( ( ord_less_eq @ A @ X3 @ Y4 )
=> ( ord_less_eq @ C @ ( F3 @ X3 ) @ ( F3 @ Y4 ) ) )
=> ( ord_less @ C @ ( F3 @ A3 ) @ C2 ) ) ) ) ) ).
% order_le_less_subst2
thf(fact_688_order__less__le__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A3: A,F3: B > A,B3: B,C2: B] :
( ( ord_less @ A @ A3 @ ( F3 @ B3 ) )
=> ( ( ord_less_eq @ B @ B3 @ C2 )
=> ( ! [X3: B,Y4: B] :
( ( ord_less_eq @ B @ X3 @ Y4 )
=> ( ord_less_eq @ A @ ( F3 @ X3 ) @ ( F3 @ Y4 ) ) )
=> ( ord_less @ A @ A3 @ ( F3 @ C2 ) ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_689_order__less__le__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C )
& ( order @ A ) )
=> ! [A3: A,B3: A,F3: A > C,C2: C] :
( ( ord_less @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ C @ ( F3 @ B3 ) @ C2 )
=> ( ! [X3: A,Y4: A] :
( ( ord_less @ A @ X3 @ Y4 )
=> ( ord_less @ C @ ( F3 @ X3 ) @ ( F3 @ Y4 ) ) )
=> ( ord_less @ C @ ( F3 @ A3 ) @ C2 ) ) ) ) ) ).
% order_less_le_subst2
thf(fact_690_linorder__le__less__linear,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
| ( ord_less @ A @ Y @ X ) ) ) ).
% linorder_le_less_linear
thf(fact_691_order__le__imp__less__or__eq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less @ A @ X @ Y )
| ( X = Y ) ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_692_le__num__One__iff,axiom,
! [X: num] :
( ( ord_less_eq @ num @ X @ one2 )
= ( X = one2 ) ) ).
% le_num_One_iff
thf(fact_693_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_694_bot_Oextremum__uniqueI,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ A3 @ ( bot_bot @ A ) )
=> ( A3
= ( bot_bot @ A ) ) ) ) ).
% bot.extremum_uniqueI
thf(fact_695_bot_Oextremum__unique,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ A3 @ ( bot_bot @ A ) )
= ( A3
= ( bot_bot @ A ) ) ) ) ).
% bot.extremum_unique
thf(fact_696_bot_Oextremum,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [A3: A] : ( ord_less_eq @ A @ ( bot_bot @ A ) @ A3 ) ) ).
% bot.extremum
thf(fact_697_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus @ nat @ ( zero_zero @ nat ) @ N )
= N ) ).
% plus_nat.add_0
thf(fact_698_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_699_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_700_add__less__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less @ nat @ I @ J )
=> ( ( ord_less @ nat @ K @ L )
=> ( ord_less @ nat @ ( plus_plus @ nat @ I @ K ) @ ( plus_plus @ nat @ J @ L ) ) ) ) ).
% add_less_mono
thf(fact_701_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less @ nat @ ( plus_plus @ nat @ I @ J ) @ I ) ).
% not_add_less1
thf(fact_702_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less @ nat @ ( plus_plus @ nat @ J @ I ) @ I ) ).
% not_add_less2
thf(fact_703_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_704_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_705_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_706_less__add__eq__less,axiom,
! [K: nat,L: nat,M: nat,N: nat] :
( ( ord_less @ nat @ K @ L )
=> ( ( ( plus_plus @ nat @ M @ L )
= ( plus_plus @ nat @ K @ N ) )
=> ( ord_less @ nat @ M @ N ) ) ) ).
% less_add_eq_less
thf(fact_707_uminus__int__code_I1_J,axiom,
( ( uminus_uminus @ int @ ( zero_zero @ int ) )
= ( zero_zero @ int ) ) ).
% uminus_int_code(1)
thf(fact_708_plus__int__code_I1_J,axiom,
! [K: int] :
( ( plus_plus @ int @ K @ ( zero_zero @ int ) )
= K ) ).
% plus_int_code(1)
thf(fact_709_plus__int__code_I2_J,axiom,
! [L: int] :
( ( plus_plus @ int @ ( zero_zero @ int ) @ L )
= L ) ).
% plus_int_code(2)
thf(fact_710_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N ) ).
% less_eq_nat.simps(1)
thf(fact_711_bot__nat__0_Oextremum__unique,axiom,
! [A3: nat] :
( ( ord_less_eq @ nat @ A3 @ ( zero_zero @ nat ) )
= ( A3
= ( zero_zero @ nat ) ) ) ).
% bot_nat_0.extremum_unique
thf(fact_712_bot__nat__0_Oextremum__uniqueI,axiom,
! [A3: nat] :
( ( ord_less_eq @ nat @ A3 @ ( zero_zero @ nat ) )
=> ( A3
= ( zero_zero @ nat ) ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_713_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq @ nat @ N @ ( zero_zero @ nat ) )
= ( N
= ( zero_zero @ nat ) ) ) ).
% le_0_eq
thf(fact_714_nat__less__le,axiom,
( ( ord_less @ nat )
= ( ^ [M3: nat,N4: nat] :
( ( ord_less_eq @ nat @ M3 @ N4 )
& ( M3 != N4 ) ) ) ) ).
% nat_less_le
thf(fact_715_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_716_le__eq__less__or__eq,axiom,
( ( ord_less_eq @ nat )
= ( ^ [M3: nat,N4: nat] :
( ( ord_less @ nat @ M3 @ N4 )
| ( M3 = N4 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_717_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_718_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_719_less__mono__imp__le__mono,axiom,
! [F3: nat > nat,I: nat,J: nat] :
( ! [I3: nat,J2: nat] :
( ( ord_less @ nat @ I3 @ J2 )
=> ( ord_less @ nat @ ( F3 @ I3 ) @ ( F3 @ J2 ) ) )
=> ( ( ord_less_eq @ nat @ I @ J )
=> ( ord_less_eq @ nat @ ( F3 @ I ) @ ( F3 @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_720_real__arch__simple,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X: A] :
? [N2: nat] : ( ord_less_eq @ A @ X @ ( semiring_1_of_nat @ A @ N2 ) ) ) ).
% real_arch_simple
thf(fact_721_inf__sup__ord_I4_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [Y: A,X: A] : ( ord_less_eq @ A @ Y @ ( sup_sup @ A @ X @ Y ) ) ) ).
% inf_sup_ord(4)
thf(fact_722_inf__sup__ord_I3_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X: A,Y: A] : ( ord_less_eq @ A @ X @ ( sup_sup @ A @ X @ Y ) ) ) ).
% inf_sup_ord(3)
thf(fact_723_le__supE,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: A,B3: A,X: A] :
( ( ord_less_eq @ A @ ( sup_sup @ A @ A3 @ B3 ) @ X )
=> ~ ( ( ord_less_eq @ A @ A3 @ X )
=> ~ ( ord_less_eq @ A @ B3 @ X ) ) ) ) ).
% le_supE
thf(fact_724_le__supI,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: A,X: A,B3: A] :
( ( ord_less_eq @ A @ A3 @ X )
=> ( ( ord_less_eq @ A @ B3 @ X )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ A3 @ B3 ) @ X ) ) ) ) ).
% le_supI
thf(fact_725_sup__ge1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X: A,Y: A] : ( ord_less_eq @ A @ X @ ( sup_sup @ A @ X @ Y ) ) ) ).
% sup_ge1
thf(fact_726_sup__ge2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [Y: A,X: A] : ( ord_less_eq @ A @ Y @ ( sup_sup @ A @ X @ Y ) ) ) ).
% sup_ge2
thf(fact_727_le__supI1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X: A,A3: A,B3: A] :
( ( ord_less_eq @ A @ X @ A3 )
=> ( ord_less_eq @ A @ X @ ( sup_sup @ A @ A3 @ B3 ) ) ) ) ).
% le_supI1
thf(fact_728_le__supI2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X: A,B3: A,A3: A] :
( ( ord_less_eq @ A @ X @ B3 )
=> ( ord_less_eq @ A @ X @ ( sup_sup @ A @ A3 @ B3 ) ) ) ) ).
% le_supI2
thf(fact_729_sup_Omono,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [C2: A,A3: A,D2: A,B3: A] :
( ( ord_less_eq @ A @ C2 @ A3 )
=> ( ( ord_less_eq @ A @ D2 @ B3 )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ C2 @ D2 ) @ ( sup_sup @ A @ A3 @ B3 ) ) ) ) ) ).
% sup.mono
thf(fact_730_sup__mono,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: A,C2: A,B3: A,D2: A] :
( ( ord_less_eq @ A @ A3 @ C2 )
=> ( ( ord_less_eq @ A @ B3 @ D2 )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ A3 @ B3 ) @ ( sup_sup @ A @ C2 @ D2 ) ) ) ) ) ).
% sup_mono
thf(fact_731_sup__least,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [Y: A,X: A,Z: A] :
( ( ord_less_eq @ A @ Y @ X )
=> ( ( ord_less_eq @ A @ Z @ X )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ Y @ Z ) @ X ) ) ) ) ).
% sup_least
thf(fact_732_le__iff__sup,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [X2: A,Y2: A] :
( ( sup_sup @ A @ X2 @ Y2 )
= Y2 ) ) ) ) ).
% le_iff_sup
thf(fact_733_sup_OorderE,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B3: A,A3: A] :
( ( ord_less_eq @ A @ B3 @ A3 )
=> ( A3
= ( sup_sup @ A @ A3 @ B3 ) ) ) ) ).
% sup.orderE
thf(fact_734_sup_OorderI,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: A,B3: A] :
( ( A3
= ( sup_sup @ A @ A3 @ B3 ) )
=> ( ord_less_eq @ A @ B3 @ A3 ) ) ) ).
% sup.orderI
thf(fact_735_sup__unique,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [F3: A > A > A,X: A,Y: A] :
( ! [X3: A,Y4: A] : ( ord_less_eq @ A @ X3 @ ( F3 @ X3 @ Y4 ) )
=> ( ! [X3: A,Y4: A] : ( ord_less_eq @ A @ Y4 @ ( F3 @ X3 @ Y4 ) )
=> ( ! [X3: A,Y4: A,Z3: A] :
( ( ord_less_eq @ A @ Y4 @ X3 )
=> ( ( ord_less_eq @ A @ Z3 @ X3 )
=> ( ord_less_eq @ A @ ( F3 @ Y4 @ Z3 ) @ X3 ) ) )
=> ( ( sup_sup @ A @ X @ Y )
= ( F3 @ X @ Y ) ) ) ) ) ) ).
% sup_unique
thf(fact_736_sup_Oabsorb1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B3: A,A3: A] :
( ( ord_less_eq @ A @ B3 @ A3 )
=> ( ( sup_sup @ A @ A3 @ B3 )
= A3 ) ) ) ).
% sup.absorb1
thf(fact_737_sup_Oabsorb2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( sup_sup @ A @ A3 @ B3 )
= B3 ) ) ) ).
% sup.absorb2
thf(fact_738_sup__absorb1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ Y @ X )
=> ( ( sup_sup @ A @ X @ Y )
= X ) ) ) ).
% sup_absorb1
thf(fact_739_sup__absorb2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( sup_sup @ A @ X @ Y )
= Y ) ) ) ).
% sup_absorb2
thf(fact_740_sup_OboundedE,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B3: A,C2: A,A3: A] :
( ( ord_less_eq @ A @ ( sup_sup @ A @ B3 @ C2 ) @ A3 )
=> ~ ( ( ord_less_eq @ A @ B3 @ A3 )
=> ~ ( ord_less_eq @ A @ C2 @ A3 ) ) ) ) ).
% sup.boundedE
thf(fact_741_sup_OboundedI,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B3: A,A3: A,C2: A] :
( ( ord_less_eq @ A @ B3 @ A3 )
=> ( ( ord_less_eq @ A @ C2 @ A3 )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ B3 @ C2 ) @ A3 ) ) ) ) ).
% sup.boundedI
thf(fact_742_sup_Oorder__iff,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B5: A,A5: A] :
( A5
= ( sup_sup @ A @ A5 @ B5 ) ) ) ) ) ).
% sup.order_iff
thf(fact_743_sup_Ocobounded1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: A,B3: A] : ( ord_less_eq @ A @ A3 @ ( sup_sup @ A @ A3 @ B3 ) ) ) ).
% sup.cobounded1
thf(fact_744_sup_Ocobounded2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B3: A,A3: A] : ( ord_less_eq @ A @ B3 @ ( sup_sup @ A @ A3 @ B3 ) ) ) ).
% sup.cobounded2
thf(fact_745_sup_Oabsorb__iff1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B5: A,A5: A] :
( ( sup_sup @ A @ A5 @ B5 )
= A5 ) ) ) ) ).
% sup.absorb_iff1
thf(fact_746_sup_Oabsorb__iff2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A5: A,B5: A] :
( ( sup_sup @ A @ A5 @ B5 )
= B5 ) ) ) ) ).
% sup.absorb_iff2
thf(fact_747_sup_OcoboundedI1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [C2: A,A3: A,B3: A] :
( ( ord_less_eq @ A @ C2 @ A3 )
=> ( ord_less_eq @ A @ C2 @ ( sup_sup @ A @ A3 @ B3 ) ) ) ) ).
% sup.coboundedI1
thf(fact_748_sup_OcoboundedI2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [C2: A,B3: A,A3: A] :
( ( ord_less_eq @ A @ C2 @ B3 )
=> ( ord_less_eq @ A @ C2 @ ( sup_sup @ A @ A3 @ B3 ) ) ) ) ).
% sup.coboundedI2
thf(fact_749_insert__mono,axiom,
! [A: $tType,C3: set @ A,D3: set @ A,A3: A] :
( ( ord_less_eq @ ( set @ A ) @ C3 @ D3 )
=> ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ A3 @ C3 ) @ ( insert @ A @ A3 @ D3 ) ) ) ).
% insert_mono
thf(fact_750_subset__insert,axiom,
! [A: $tType,X: A,A2: set @ A,B2: set @ A] :
( ~ ( member @ A @ X @ A2 )
=> ( ( ord_less_eq @ ( set @ A ) @ A2 @ ( insert @ A @ X @ B2 ) )
= ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ) ).
% subset_insert
thf(fact_751_subset__insertI,axiom,
! [A: $tType,B2: set @ A,A3: A] : ( ord_less_eq @ ( set @ A ) @ B2 @ ( insert @ A @ A3 @ B2 ) ) ).
% subset_insertI
thf(fact_752_subset__insertI2,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,B3: A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ord_less_eq @ ( set @ A ) @ A2 @ ( insert @ A @ B3 @ B2 ) ) ) ).
% subset_insertI2
thf(fact_753_less__eq__real__def,axiom,
( ( ord_less_eq @ real )
= ( ^ [X2: real,Y2: real] :
( ( ord_less @ real @ X2 @ Y2 )
| ( X2 = Y2 ) ) ) ) ).
% less_eq_real_def
thf(fact_754_less__eq__int__code_I1_J,axiom,
ord_less_eq @ int @ ( zero_zero @ int ) @ ( zero_zero @ int ) ).
% less_eq_int_code(1)
thf(fact_755_subset__code_I1_J,axiom,
! [A: $tType,Xs: list @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ B2 )
= ( ! [X2: A] :
( ( member @ A @ X2 @ ( set2 @ A @ Xs ) )
=> ( member @ A @ X2 @ B2 ) ) ) ) ).
% subset_code(1)
thf(fact_756_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_757_plus__le__left__cancel__wrap,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y6: word @ A,Y: word @ A] :
( ( ord_less @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ X @ Y6 ) @ X )
=> ( ( ord_less @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ X @ Y ) @ X )
=> ( ( ord_less @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ X @ Y6 ) @ ( plus_plus @ ( word @ A ) @ X @ Y ) )
= ( ord_less @ ( word @ A ) @ Y6 @ Y ) ) ) ) ) ).
% plus_le_left_cancel_wrap
thf(fact_758_Un__mono,axiom,
! [A: $tType,A2: set @ A,C3: set @ A,B2: set @ A,D3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ C3 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ D3 )
=> ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) @ ( sup_sup @ ( set @ A ) @ C3 @ D3 ) ) ) ) ).
% Un_mono
thf(fact_759_Un__least,axiom,
! [A: $tType,A2: set @ A,C3: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ C3 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ C3 )
=> ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) @ C3 ) ) ) ).
% Un_least
thf(fact_760_Un__upper1,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] : ( ord_less_eq @ ( set @ A ) @ A2 @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) ) ).
% Un_upper1
thf(fact_761_Un__upper2,axiom,
! [A: $tType,B2: set @ A,A2: set @ A] : ( ord_less_eq @ ( set @ A ) @ B2 @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) ) ).
% Un_upper2
thf(fact_762_Un__absorb1,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ( sup_sup @ ( set @ A ) @ A2 @ B2 )
= B2 ) ) ).
% Un_absorb1
thf(fact_763_Un__absorb2,axiom,
! [A: $tType,B2: set @ A,A2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ B2 @ A2 )
=> ( ( sup_sup @ ( set @ A ) @ A2 @ B2 )
= A2 ) ) ).
% Un_absorb2
thf(fact_764_subset__UnE,axiom,
! [A: $tType,C3: set @ A,A2: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ C3 @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) )
=> ~ ! [A8: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A8 @ A2 )
=> ! [B9: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ B9 @ B2 )
=> ( C3
!= ( sup_sup @ ( set @ A ) @ A8 @ B9 ) ) ) ) ) ).
% subset_UnE
thf(fact_765_subset__Un__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A4: set @ A,B4: set @ A] :
( ( sup_sup @ ( set @ A ) @ A4 @ B4 )
= B4 ) ) ) ).
% subset_Un_eq
thf(fact_766_word__coorder_Oextremum,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A] : ( ord_less_eq @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ A3 ) ) ).
% word_coorder.extremum
thf(fact_767_word__coorder_Oextremum__uniqueI,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ A3 @ ( zero_zero @ ( word @ A ) ) )
=> ( A3
= ( zero_zero @ ( word @ A ) ) ) ) ) ).
% word_coorder.extremum_uniqueI
thf(fact_768_word__zero__le,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y: word @ A] : ( ord_less_eq @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ Y ) ) ).
% word_zero_le
thf(fact_769_word__le__not__less,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( ord_less_eq @ ( word @ A ) )
= ( ^ [B5: word @ A,A5: word @ A] :
~ ( ord_less @ ( word @ A ) @ A5 @ B5 ) ) ) ) ).
% word_le_not_less
thf(fact_770_word__not__le,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] :
( ( ~ ( ord_less_eq @ ( word @ A ) @ X @ Y ) )
= ( ord_less @ ( word @ A ) @ Y @ X ) ) ) ).
% word_not_le
thf(fact_771_word__le__less__eq,axiom,
! [Z5: $tType] :
( ( type_len @ Z5 )
=> ( ( ord_less_eq @ ( word @ Z5 ) )
= ( ^ [X2: word @ Z5,Y2: word @ Z5] :
( ( X2 = Y2 )
| ( ord_less @ ( word @ Z5 ) @ X2 @ Y2 ) ) ) ) ) ).
% word_le_less_eq
thf(fact_772_subset__iff__psubset__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A4: set @ A,B4: set @ A] :
( ( ord_less @ ( set @ A ) @ A4 @ B4 )
| ( A4 = B4 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_773_subset__psubset__trans,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,C3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ( ord_less @ ( set @ A ) @ B2 @ C3 )
=> ( ord_less @ ( set @ A ) @ A2 @ C3 ) ) ) ).
% subset_psubset_trans
thf(fact_774_subset__not__subset__eq,axiom,
! [A: $tType] :
( ( ord_less @ ( set @ A ) )
= ( ^ [A4: set @ A,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
& ~ ( ord_less_eq @ ( set @ A ) @ B4 @ A4 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_775_psubset__subset__trans,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,C3: set @ A] :
( ( ord_less @ ( set @ A ) @ A2 @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ C3 )
=> ( ord_less @ ( set @ A ) @ A2 @ C3 ) ) ) ).
% psubset_subset_trans
thf(fact_776_psubset__imp__subset,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( ord_less @ ( set @ A ) @ A2 @ B2 )
=> ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ).
% psubset_imp_subset
thf(fact_777_psubset__eq,axiom,
! [A: $tType] :
( ( ord_less @ ( set @ A ) )
= ( ^ [A4: set @ A,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
& ( A4 != B4 ) ) ) ) ).
% psubset_eq
thf(fact_778_psubsetE,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( ord_less @ ( set @ A ) @ A2 @ B2 )
=> ~ ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ord_less_eq @ ( set @ A ) @ B2 @ A2 ) ) ) ).
% psubsetE
thf(fact_779_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_780_i0__lb,axiom,
! [N: extended_enat] : ( ord_less_eq @ extended_enat @ ( zero_zero @ extended_enat ) @ N ) ).
% i0_lb
thf(fact_781_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_782_power__decreasing,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N: nat,N3: nat,A3: A] :
( ( ord_less_eq @ nat @ N @ N3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ A3 @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( power_power @ A @ A3 @ N3 ) @ ( power_power @ A @ A3 @ N ) ) ) ) ) ) ).
% power_decreasing
thf(fact_783_power__le__imp__le__exp,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,M: nat,N: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ ( power_power @ A @ A3 @ M ) @ ( power_power @ A @ A3 @ N ) )
=> ( ord_less_eq @ nat @ M @ N ) ) ) ) ).
% power_le_imp_le_exp
thf(fact_784_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_785_le__log__of__power,axiom,
! [B3: real,N: nat,M: real] :
( ( ord_less_eq @ real @ ( power_power @ real @ B3 @ N ) @ M )
=> ( ( ord_less @ real @ ( one_one @ real ) @ B3 )
=> ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ N ) @ ( log @ B3 @ M ) ) ) ) ).
% le_log_of_power
thf(fact_786_nat__less__real__le,axiom,
( ( ord_less @ nat )
= ( ^ [N4: nat,M3: nat] : ( ord_less_eq @ real @ ( plus_plus @ real @ ( semiring_1_of_nat @ real @ N4 ) @ ( one_one @ real ) ) @ ( semiring_1_of_nat @ real @ M3 ) ) ) ) ).
% nat_less_real_le
thf(fact_787_nat__le__real__less,axiom,
( ( ord_less_eq @ nat )
= ( ^ [N4: nat,M3: nat] : ( ord_less @ real @ ( semiring_1_of_nat @ real @ N4 ) @ ( plus_plus @ real @ ( semiring_1_of_nat @ real @ M3 ) @ ( one_one @ real ) ) ) ) ) ).
% nat_le_real_less
thf(fact_788_plus__one__helper2,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,N: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ X @ N )
=> ( ( ( plus_plus @ ( word @ A ) @ N @ ( one_one @ ( word @ A ) ) )
!= ( zero_zero @ ( word @ A ) ) )
=> ( ord_less @ ( word @ A ) @ X @ ( plus_plus @ ( word @ A ) @ N @ ( one_one @ ( word @ A ) ) ) ) ) ) ) ).
% plus_one_helper2
thf(fact_789_Collect__imp__eq,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ( collect @ A
@ ^ [X2: A] :
( ( P @ X2 )
=> ( Q @ X2 ) ) )
= ( sup_sup @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ ( collect @ A @ P ) ) @ ( collect @ A @ Q ) ) ) ).
% Collect_imp_eq
thf(fact_790_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_791_word__1__0,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A,B3: word @ A,X: nat] :
( ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ A3 @ ( one_one @ ( word @ A ) ) ) @ B3 )
=> ( ( ord_less @ ( word @ A ) @ A3 @ ( semiring_1_of_nat @ ( word @ A ) @ X ) )
=> ( ord_less @ ( word @ A ) @ A3 @ B3 ) ) ) ) ).
% word_1_0
thf(fact_792_dbl__def,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( ( neg_numeral_dbl @ A )
= ( ^ [X2: A] : ( plus_plus @ A @ X2 @ X2 ) ) ) ) ).
% dbl_def
thf(fact_793_realpow__square__minus__le,axiom,
! [U2: real,X: real] : ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( power_power @ real @ U2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% realpow_square_minus_le
thf(fact_794_less__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ( ( ord_less @ ( A > B ) )
= ( ^ [F2: A > B,G: A > B] :
( ( ord_less_eq @ ( A > B ) @ F2 @ G )
& ~ ( ord_less_eq @ ( A > B ) @ G @ F2 ) ) ) ) ) ).
% less_fun_def
thf(fact_795_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_796_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_797_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_798_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_799_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_800_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_801_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_802_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_803_add__less__zeroD,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ ( plus_plus @ A @ X @ Y ) @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ X @ ( zero_zero @ A ) )
| ( ord_less @ A @ Y @ ( zero_zero @ A ) ) ) ) ) ).
% add_less_zeroD
thf(fact_804_add__neg__neg,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ B3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( plus_plus @ A @ A3 @ B3 ) @ ( zero_zero @ A ) ) ) ) ) ).
% add_neg_neg
thf(fact_805_add__pos__pos,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B3 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A3 @ B3 ) ) ) ) ) ).
% add_pos_pos
thf(fact_806_canonically__ordered__monoid__add__class_OlessE,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ A3 @ B3 )
=> ~ ! [C5: A] :
( ( B3
= ( plus_plus @ A @ A3 @ C5 ) )
=> ( C5
= ( zero_zero @ A ) ) ) ) ) ).
% canonically_ordered_monoid_add_class.lessE
thf(fact_807_pos__add__strict,axiom,
! [A: $tType] :
( ( strict7427464778891057005id_add @ A )
=> ! [A3: A,B3: A,C2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ B3 @ C2 )
=> ( ord_less @ A @ B3 @ ( plus_plus @ A @ A3 @ C2 ) ) ) ) ) ).
% pos_add_strict
thf(fact_808_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_809_less__add__one,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A] : ( ord_less @ A @ A3 @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) ) ) ).
% less_add_one
thf(fact_810_add__mono1,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ A3 @ B3 )
=> ( ord_less @ A @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) @ ( plus_plus @ A @ B3 @ ( one_one @ A ) ) ) ) ) ).
% add_mono1
thf(fact_811_one__plus__numeral__commute,axiom,
! [A: $tType] :
( ( numeral @ A )
=> ! [X: num] :
( ( plus_plus @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ X ) )
= ( plus_plus @ A @ ( numeral_numeral @ A @ X ) @ ( one_one @ A ) ) ) ) ).
% one_plus_numeral_commute
thf(fact_812_uint__0__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] :
( ( ( semiring_1_unsigned @ A @ int @ X )
= ( zero_zero @ int ) )
= ( X
= ( zero_zero @ ( word @ A ) ) ) ) ) ).
% uint_0_iff
thf(fact_813_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_814_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_815_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_816_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_817_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_818_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_819_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_820_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_821_word__less__def,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( ord_less @ ( word @ A ) )
= ( ^ [A5: word @ A,B5: word @ A] : ( ord_less @ int @ ( semiring_1_unsigned @ A @ int @ A5 ) @ ( semiring_1_unsigned @ A @ int @ B5 ) ) ) ) ) ).
% word_less_def
thf(fact_822_zero__le__power,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,N: nat] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A3 @ N ) ) ) ) ).
% zero_le_power
thf(fact_823_power__mono,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,B3: A,N: nat] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ord_less_eq @ A @ ( power_power @ A @ A3 @ N ) @ ( power_power @ A @ B3 @ N ) ) ) ) ) ).
% power_mono
thf(fact_824_norm__pre__pure__rule2,axiom,
! [A: $tType,B3: $o,F3: heap_Time_Heap @ A,Q: A > assn] :
( ( B3
=> ( hoare_hoare_triple @ A @ ( one_one @ assn ) @ F3 @ Q ) )
=> ( hoare_hoare_triple @ A @ ( pure_assn @ B3 ) @ F3 @ Q ) ) ).
% norm_pre_pure_rule2
thf(fact_825_one__le__power,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,N: nat] :
( ( ord_less_eq @ A @ ( one_one @ A ) @ A3 )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ ( power_power @ A @ A3 @ N ) ) ) ) ).
% one_le_power
thf(fact_826_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_827_less__imp__add__positive,axiom,
! [I: nat,J: nat] :
( ( ord_less @ nat @ I @ J )
=> ? [K2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ K2 )
& ( ( plus_plus @ nat @ I @ K2 )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_828_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_829_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ ( zero_zero @ nat ) )
=> ? [K2: nat] :
( ( ord_less_eq @ nat @ K2 @ N )
& ! [I2: nat] :
( ( ord_less @ nat @ I2 @ K2 )
=> ~ ( P @ I2 ) )
& ( P @ K2 ) ) ) ) ).
% ex_least_nat_le
thf(fact_830_subset__singletonD,axiom,
! [A: $tType,A2: set @ A,X: A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
=> ( ( A2
= ( bot_bot @ ( set @ A ) ) )
| ( A2
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% subset_singletonD
thf(fact_831_subset__singleton__iff,axiom,
! [A: $tType,X5: set @ A,A3: A] :
( ( ord_less_eq @ ( set @ A ) @ X5 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( ( X5
= ( bot_bot @ ( set @ A ) ) )
| ( X5
= ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% subset_singleton_iff
thf(fact_832_odd__nonzero,axiom,
! [Z: int] :
( ( plus_plus @ int @ ( plus_plus @ int @ ( one_one @ int ) @ Z ) @ Z )
!= ( zero_zero @ int ) ) ).
% odd_nonzero
thf(fact_833_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_834_int__gr__induct,axiom,
! [K: int,I: int,P: int > $o] :
( ( ord_less @ int @ K @ I )
=> ( ( P @ ( plus_plus @ int @ K @ ( one_one @ int ) ) )
=> ( ! [I3: int] :
( ( ord_less @ int @ K @ I3 )
=> ( ( P @ I3 )
=> ( P @ ( plus_plus @ int @ I3 @ ( one_one @ int ) ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_gr_induct
thf(fact_835_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_836_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_837_word__not__simps_I3_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y: word @ A] :
~ ( ord_less @ ( word @ A ) @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) @ Y ) ) ).
% word_not_simps(3)
thf(fact_838_word__order_Oextremum__strict,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A] :
~ ( ord_less @ ( word @ A ) @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) @ A3 ) ) ).
% word_order.extremum_strict
thf(fact_839_word__order_Onot__eq__extremum,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A] :
( ( A3
!= ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) )
= ( ord_less @ ( word @ A ) @ A3 @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) ) ) ) ).
% word_order.not_eq_extremum
thf(fact_840_max__word__not__less,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] :
~ ( ord_less @ ( word @ A ) @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) @ X ) ) ).
% max_word_not_less
thf(fact_841_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_842_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_843_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_844_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_845_lt1__neq0,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ ( one_one @ ( word @ A ) ) @ X )
= ( X
!= ( zero_zero @ ( word @ A ) ) ) ) ) ).
% lt1_neq0
thf(fact_846_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_847_word__less__iff__unsigned,axiom,
! [A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( linordered_semidom @ A ) )
=> ( ( ord_less @ ( word @ B ) )
= ( ^ [A5: word @ B,B5: word @ B] : ( ord_less @ A @ ( semiring_1_unsigned @ B @ A @ A5 ) @ ( semiring_1_unsigned @ B @ A @ B5 ) ) ) ) ) ).
% word_less_iff_unsigned
thf(fact_848_log__of__power__le,axiom,
! [M: nat,B3: real,N: nat] :
( ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ M ) @ ( power_power @ real @ B3 @ N ) )
=> ( ( ord_less @ real @ ( one_one @ real ) @ B3 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
=> ( ord_less_eq @ real @ ( log @ B3 @ ( semiring_1_of_nat @ real @ M ) ) @ ( semiring_1_of_nat @ real @ N ) ) ) ) ) ).
% log_of_power_le
thf(fact_849_square__le__1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ X )
=> ( ( ord_less_eq @ A @ X @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) ) ) ) ) ).
% square_le_1
thf(fact_850_sum__power2__ge__zero,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A,Y: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% sum_power2_ge_zero
thf(fact_851_sum__power2__le__zero__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( zero_zero @ A ) )
= ( ( X
= ( zero_zero @ A ) )
& ( Y
= ( zero_zero @ A ) ) ) ) ) ).
% sum_power2_le_zero_iff
thf(fact_852_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_853_less__log__of__power,axiom,
! [B3: real,N: nat,M: real] :
( ( ord_less @ real @ ( power_power @ real @ B3 @ N ) @ M )
=> ( ( ord_less @ real @ ( one_one @ real ) @ B3 )
=> ( ord_less @ real @ ( semiring_1_of_nat @ real @ N ) @ ( log @ B3 @ M ) ) ) ) ).
% less_log_of_power
thf(fact_854_log__of__power__eq,axiom,
! [M: nat,B3: real,N: nat] :
( ( ( semiring_1_of_nat @ real @ M )
= ( power_power @ real @ B3 @ N ) )
=> ( ( ord_less @ real @ ( one_one @ real ) @ B3 )
=> ( ( semiring_1_of_nat @ real @ N )
= ( log @ B3 @ ( semiring_1_of_nat @ real @ M ) ) ) ) ) ).
% log_of_power_eq
thf(fact_855_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_856_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_857_ex__power__ivl1,axiom,
! [B3: nat,K: nat] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ B3 )
=> ( ( ord_less_eq @ nat @ ( one_one @ nat ) @ K )
=> ? [N2: nat] :
( ( ord_less_eq @ nat @ ( power_power @ nat @ B3 @ N2 ) @ K )
& ( ord_less @ nat @ K @ ( power_power @ nat @ B3 @ ( plus_plus @ nat @ N2 @ ( one_one @ nat ) ) ) ) ) ) ) ).
% ex_power_ivl1
thf(fact_858_ex__power__ivl2,axiom,
! [B3: nat,K: nat] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ B3 )
=> ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K )
=> ? [N2: nat] :
( ( ord_less @ nat @ ( power_power @ nat @ B3 @ N2 ) @ K )
& ( ord_less_eq @ nat @ K @ ( power_power @ nat @ B3 @ ( plus_plus @ nat @ N2 @ ( one_one @ nat ) ) ) ) ) ) ) ).
% ex_power_ivl2
thf(fact_859_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_860_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_861_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_862_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_863_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_864_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_865_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_866_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_867_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_868_power__minus__Bit0,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [X: A,K: num] :
( ( power_power @ A @ ( uminus_uminus @ A @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ K ) ) )
= ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ K ) ) ) ) ) ).
% power_minus_Bit0
thf(fact_869_power__less__imp__less__base,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,N: nat,B3: A] :
( ( ord_less @ A @ ( power_power @ A @ A3 @ N ) @ ( power_power @ A @ B3 @ N ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B3 )
=> ( ord_less @ A @ A3 @ B3 ) ) ) ) ).
% power_less_imp_less_base
thf(fact_870_power__le__one,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,N: nat] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ A3 @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( power_power @ A @ A3 @ N ) @ ( one_one @ A ) ) ) ) ) ).
% power_le_one
thf(fact_871_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_872_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_873_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_874_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_875_word__overflow,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] :
( ( ord_less @ ( word @ A ) @ X @ ( plus_plus @ ( word @ A ) @ X @ ( one_one @ ( word @ A ) ) ) )
| ( ( plus_plus @ ( word @ A ) @ X @ ( one_one @ ( word @ A ) ) )
= ( zero_zero @ ( word @ A ) ) ) ) ) ).
% word_overflow
thf(fact_876_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_877_less__is__non__zero__p1,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A,K: word @ A] :
( ( ord_less @ ( word @ A ) @ A3 @ K )
=> ( ( plus_plus @ ( word @ A ) @ A3 @ ( one_one @ ( word @ A ) ) )
!= ( zero_zero @ ( word @ A ) ) ) ) ) ).
% less_is_non_zero_p1
thf(fact_878_power2__eq__iff,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [X: A,Y: A] :
( ( ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( ( X = Y )
| ( X
= ( uminus_uminus @ A @ Y ) ) ) ) ) ).
% power2_eq_iff
thf(fact_879_power__eq__iff__eq__base,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N: nat,A3: A,B3: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B3 )
=> ( ( ( power_power @ A @ A3 @ N )
= ( power_power @ A @ B3 @ N ) )
= ( A3 = B3 ) ) ) ) ) ) ).
% power_eq_iff_eq_base
thf(fact_880_power__eq__imp__eq__base,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,N: nat,B3: A] :
( ( ( power_power @ A @ A3 @ N )
= ( power_power @ A @ B3 @ N ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B3 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( A3 = B3 ) ) ) ) ) ) ).
% power_eq_imp_eq_base
thf(fact_881_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_882_self__le__power,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,N: nat] :
( ( ord_less_eq @ A @ ( one_one @ A ) @ A3 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ord_less_eq @ A @ A3 @ ( power_power @ A @ A3 @ N ) ) ) ) ) ).
% self_le_power
thf(fact_883_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_884_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_885_log__of__power__less,axiom,
! [M: nat,B3: real,N: nat] :
( ( ord_less @ real @ ( semiring_1_of_nat @ real @ M ) @ ( power_power @ real @ B3 @ N ) )
=> ( ( ord_less @ real @ ( one_one @ real ) @ B3 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
=> ( ord_less @ real @ ( log @ B3 @ ( semiring_1_of_nat @ real @ M ) ) @ ( semiring_1_of_nat @ real @ N ) ) ) ) ) ).
% log_of_power_less
thf(fact_886_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_887_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_888_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_889_power2__eq__1__iff,axiom,
! [A: $tType] :
( ( ring_15535105094025558882visors @ A )
=> ! [A3: A] :
( ( ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( one_one @ A ) )
= ( ( A3
= ( one_one @ A ) )
| ( A3
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ) ).
% power2_eq_1_iff
thf(fact_890_zero__le__power2,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% zero_le_power2
thf(fact_891_power2__eq__imp__eq,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X: A,Y: A] :
( ( ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
=> ( X = Y ) ) ) ) ) ).
% power2_eq_imp_eq
thf(fact_892_power2__le__imp__le,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
=> ( ord_less_eq @ A @ X @ Y ) ) ) ) ).
% power2_le_imp_le
thf(fact_893_power__strict__mono,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,B3: A,N: nat] :
( ( ord_less @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ord_less @ A @ ( power_power @ A @ A3 @ N ) @ ( power_power @ A @ B3 @ N ) ) ) ) ) ) ).
% power_strict_mono
thf(fact_894_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_895_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_896_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_897_not__sum__power2__lt__zero,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A,Y: A] :
~ ( ord_less @ A @ ( plus_plus @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( zero_zero @ A ) ) ) ).
% not_sum_power2_lt_zero
thf(fact_898_sum__power2__gt__zero__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
= ( ( X
!= ( zero_zero @ A ) )
| ( Y
!= ( zero_zero @ A ) ) ) ) ) ).
% sum_power2_gt_zero_iff
thf(fact_899_arcosh__1,axiom,
! [A: $tType] :
( ( ln @ A )
=> ( ( arcosh @ A @ ( one_one @ A ) )
= ( zero_zero @ A ) ) ) ).
% arcosh_1
thf(fact_900_VEBT__internal_Otwo__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 ) ) ) ).
% VEBT_internal.two_realpow_ge_two
thf(fact_901_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_902_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_903_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_904_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_905_discrete,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ( ( ord_less @ A )
= ( ^ [A5: A] : ( ord_less_eq @ A @ ( plus_plus @ A @ A5 @ ( one_one @ A ) ) ) ) ) ) ).
% discrete
thf(fact_906_field__le__epsilon,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y: A] :
( ! [E: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ E )
=> ( ord_less_eq @ A @ X @ ( plus_plus @ A @ Y @ E ) ) )
=> ( ord_less_eq @ A @ X @ Y ) ) ) ).
% field_le_epsilon
thf(fact_907_artanh__0,axiom,
! [A: $tType] :
( ( ( real_V3459762299906320749_field @ A )
& ( ln @ A ) )
=> ( ( artanh @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% artanh_0
thf(fact_908_arsinh__0,axiom,
! [A: $tType] :
( ( ln @ A )
=> ( ( arsinh @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% arsinh_0
thf(fact_909_of__nat__code,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ( ( semiring_1_of_nat @ A )
= ( ^ [N4: nat] :
( semiri8178284476397505188at_aux @ A
@ ^ [I4: A] : ( plus_plus @ A @ I4 @ ( one_one @ A ) )
@ N4
@ ( zero_zero @ A ) ) ) ) ) ).
% of_nat_code
thf(fact_910_subsetI,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ! [X3: A] :
( ( member @ A @ X3 @ A2 )
=> ( member @ A @ X3 @ B2 ) )
=> ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ).
% subsetI
thf(fact_911_subset__antisym,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% subset_antisym
thf(fact_912_ComplI,axiom,
! [A: $tType,C2: A,A2: set @ A] :
( ~ ( member @ A @ C2 @ A2 )
=> ( member @ A @ C2 @ ( uminus_uminus @ ( set @ A ) @ A2 ) ) ) ).
% ComplI
thf(fact_913_Compl__iff,axiom,
! [A: $tType,C2: A,A2: set @ A] :
( ( member @ A @ C2 @ ( uminus_uminus @ ( set @ A ) @ A2 ) )
= ( ~ ( member @ A @ C2 @ A2 ) ) ) ).
% Compl_iff
thf(fact_914_Compl__eq__Compl__iff,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( ( uminus_uminus @ ( set @ A ) @ A2 )
= ( uminus_uminus @ ( set @ A ) @ B2 ) )
= ( A2 = B2 ) ) ).
% Compl_eq_Compl_iff
thf(fact_915_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_916_Compl__anti__mono,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ B2 ) @ ( uminus_uminus @ ( set @ A ) @ A2 ) ) ) ).
% Compl_anti_mono
thf(fact_917_Compl__subset__Compl__iff,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A2 ) @ ( uminus_uminus @ ( set @ A ) @ B2 ) )
= ( ord_less_eq @ ( set @ A ) @ B2 @ A2 ) ) ).
% Compl_subset_Compl_iff
thf(fact_918_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_919_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq @ nat @ N @ ( plus_plus @ nat @ N @ M ) ) ).
% le_add1
thf(fact_920_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq @ nat @ N @ ( plus_plus @ nat @ M @ N ) ) ).
% le_add2
thf(fact_921_le__refl,axiom,
! [N: nat] : ( ord_less_eq @ nat @ N @ N ) ).
% le_refl
thf(fact_922_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_923_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_924_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_925_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq @ nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_926_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq @ nat @ K @ L )
=> ? [N2: nat] :
( L
= ( plus_plus @ nat @ K @ N2 ) ) ) ).
% le_Suc_ex
thf(fact_927_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_928_add__le__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( ord_less_eq @ nat @ K @ L )
=> ( ord_less_eq @ nat @ ( plus_plus @ nat @ I @ K ) @ ( plus_plus @ nat @ J @ L ) ) ) ) ).
% add_le_mono
thf(fact_929_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_930_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_931_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_932_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_933_nat__le__iff__add,axiom,
( ( ord_less_eq @ nat )
= ( ^ [M3: nat,N4: nat] :
? [K3: nat] :
( N4
= ( plus_plus @ nat @ M3 @ K3 ) ) ) ) ).
% nat_le_iff_add
thf(fact_934_complete__real,axiom,
! [S: set @ real] :
( ? [X6: real] : ( member @ real @ X6 @ S )
=> ( ? [Z6: real] :
! [X3: real] :
( ( member @ real @ X3 @ S )
=> ( ord_less_eq @ real @ X3 @ Z6 ) )
=> ? [Y4: real] :
( ! [X6: real] :
( ( member @ real @ X6 @ S )
=> ( ord_less_eq @ real @ X6 @ Y4 ) )
& ! [Z6: real] :
( ! [X3: real] :
( ( member @ real @ X3 @ S )
=> ( ord_less_eq @ real @ X3 @ Z6 ) )
=> ( ord_less_eq @ real @ Y4 @ Z6 ) ) ) ) ) ).
% complete_real
thf(fact_935_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B3: nat] :
( ( P @ K )
=> ( ! [Y4: nat] :
( ( P @ Y4 )
=> ( ord_less_eq @ nat @ Y4 @ B3 ) )
=> ? [X3: nat] :
( ( P @ X3 )
& ! [Y5: nat] :
( ( P @ Y5 )
=> ( ord_less_eq @ nat @ Y5 @ X3 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_936_ComplD,axiom,
! [A: $tType,C2: A,A2: set @ A] :
( ( member @ A @ C2 @ ( uminus_uminus @ ( set @ A ) @ A2 ) )
=> ~ ( member @ A @ C2 @ A2 ) ) ).
% ComplD
thf(fact_937_in__mono,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,X: A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ( member @ A @ X @ A2 )
=> ( member @ A @ X @ B2 ) ) ) ).
% in_mono
thf(fact_938_subsetD,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,C2: A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ( member @ A @ C2 @ A2 )
=> ( member @ A @ C2 @ B2 ) ) ) ).
% subsetD
thf(fact_939_equalityE,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( A2 = B2 )
=> ~ ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ~ ( ord_less_eq @ ( set @ A ) @ B2 @ A2 ) ) ) ).
% equalityE
thf(fact_940_subset__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A4: set @ A,B4: set @ A] :
! [X2: A] :
( ( member @ A @ X2 @ A4 )
=> ( member @ A @ X2 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_941_equalityD1,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( A2 = B2 )
=> ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ).
% equalityD1
thf(fact_942_Set_OequalityD2,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( A2 = B2 )
=> ( ord_less_eq @ ( set @ A ) @ B2 @ A2 ) ) ).
% Set.equalityD2
thf(fact_943_subset__iff,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A4: set @ A,B4: set @ A] :
! [T2: A] :
( ( member @ A @ T2 @ A4 )
=> ( member @ A @ T2 @ B4 ) ) ) ) ).
% subset_iff
thf(fact_944_subset__refl,axiom,
! [A: $tType,A2: set @ A] : ( ord_less_eq @ ( set @ A ) @ A2 @ A2 ) ).
% subset_refl
thf(fact_945_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_946_subset__trans,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,C3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ C3 )
=> ( ord_less_eq @ ( set @ A ) @ A2 @ C3 ) ) ) ).
% subset_trans
thf(fact_947_set__eq__subset,axiom,
! [A: $tType] :
( ( ^ [Y3: set @ A,Z2: set @ A] : Y3 = Z2 )
= ( ^ [A4: set @ A,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
& ( ord_less_eq @ ( set @ A ) @ B4 @ A4 ) ) ) ) ).
% set_eq_subset
thf(fact_948_uminus__set__def,axiom,
! [A: $tType] :
( ( uminus_uminus @ ( set @ A ) )
= ( ^ [A4: set @ A] :
( collect @ A
@ ( uminus_uminus @ ( A > $o )
@ ^ [X2: A] : ( member @ A @ X2 @ A4 ) ) ) ) ) ).
% uminus_set_def
thf(fact_949_less__eq__set__def,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A4: set @ A,B4: set @ A] :
( ord_less_eq @ ( A > $o )
@ ^ [X2: A] : ( member @ A @ X2 @ A4 )
@ ^ [X2: A] : ( member @ A @ X2 @ B4 ) ) ) ) ).
% less_eq_set_def
thf(fact_950_Collect__mono__iff,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) )
= ( ! [X2: A] :
( ( P @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_951_double__complement,axiom,
! [A: $tType,A2: set @ A] :
( ( uminus_uminus @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A2 ) )
= A2 ) ).
% double_complement
thf(fact_952_verit__la__generic,axiom,
! [A3: int,X: int] :
( ( ord_less_eq @ int @ A3 @ X )
| ( A3 = X )
| ( ord_less_eq @ int @ X @ A3 ) ) ).
% verit_la_generic
thf(fact_953_pred__subset__eq,axiom,
! [A: $tType,R: set @ A,S: set @ A] :
( ( ord_less_eq @ ( A > $o )
@ ^ [X2: A] : ( member @ A @ X2 @ R )
@ ^ [X2: A] : ( member @ A @ X2 @ S ) )
= ( ord_less_eq @ ( set @ A ) @ R @ S ) ) ).
% pred_subset_eq
thf(fact_954_unat__cong,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] :
( ( X = Y )
=> ( ( semiring_1_unsigned @ A @ nat @ X )
= ( semiring_1_unsigned @ A @ nat @ Y ) ) ) ) ).
% unat_cong
thf(fact_955_olen__add__eqv,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y: word @ A,X: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ Y @ ( plus_plus @ ( word @ A ) @ Y @ X ) )
= ( ord_less_eq @ ( word @ A ) @ X @ ( plus_plus @ ( word @ A ) @ Y @ X ) ) ) ) ).
% olen_add_eqv
thf(fact_956_le__no__overflow,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,B3: word @ A,A3: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ X @ B3 )
=> ( ( ord_less_eq @ ( word @ A ) @ A3 @ ( plus_plus @ ( word @ A ) @ A3 @ B3 ) )
=> ( ord_less_eq @ ( word @ A ) @ X @ ( plus_plus @ ( word @ A ) @ A3 @ B3 ) ) ) ) ) ).
% le_no_overflow
thf(fact_957_word__le__nat__alt,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( ord_less_eq @ ( word @ A ) )
= ( ^ [A5: word @ A,B5: word @ A] : ( ord_less_eq @ nat @ ( semiring_1_unsigned @ A @ nat @ A5 ) @ ( semiring_1_unsigned @ A @ nat @ B5 ) ) ) ) ) ).
% word_le_nat_alt
thf(fact_958_word__random,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [P4: word @ A,X7: word @ A,X: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ P4 @ ( plus_plus @ ( word @ A ) @ P4 @ X7 ) )
=> ( ( ord_less_eq @ ( word @ A ) @ X @ X7 )
=> ( ord_less_eq @ ( word @ A ) @ P4 @ ( plus_plus @ ( word @ A ) @ P4 @ X ) ) ) ) ) ).
% word_random
thf(fact_959_unat__plus__simple,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ X @ ( plus_plus @ ( word @ A ) @ X @ Y ) )
= ( ( semiring_1_unsigned @ A @ nat @ ( plus_plus @ ( word @ A ) @ X @ Y ) )
= ( plus_plus @ nat @ ( semiring_1_unsigned @ A @ nat @ X ) @ ( semiring_1_unsigned @ A @ nat @ Y ) ) ) ) ) ).
% unat_plus_simple
thf(fact_960_word__unat__eq__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( ^ [Y3: word @ A,Z2: word @ A] : Y3 = Z2 )
= ( ^ [V3: word @ A,W2: word @ A] :
( ( semiring_1_unsigned @ A @ nat @ V3 )
= ( semiring_1_unsigned @ A @ nat @ W2 ) ) ) ) ) ).
% word_unat_eq_iff
thf(fact_961_word__eq__unatI,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [V2: word @ A,W: word @ A] :
( ( ( semiring_1_unsigned @ A @ nat @ V2 )
= ( semiring_1_unsigned @ A @ nat @ W ) )
=> ( V2 = W ) ) ) ).
% word_eq_unatI
thf(fact_962_word__sub__mono2,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A,B3: word @ A,C2: word @ A,D2: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ A3 @ B3 ) @ ( plus_plus @ ( word @ A ) @ C2 @ D2 ) )
=> ( ( ord_less_eq @ ( word @ A ) @ C2 @ A3 )
=> ( ( ord_less_eq @ ( word @ A ) @ B3 @ ( plus_plus @ ( word @ A ) @ A3 @ B3 ) )
=> ( ( ord_less_eq @ ( word @ A ) @ D2 @ ( plus_plus @ ( word @ A ) @ C2 @ D2 ) )
=> ( ord_less_eq @ ( word @ A ) @ B3 @ D2 ) ) ) ) ) ) ).
% word_sub_mono2
thf(fact_963_word__le__plus__either,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A,Z: word @ A] :
( ( ( ord_less_eq @ ( word @ A ) @ X @ Y )
| ( ord_less_eq @ ( word @ A ) @ X @ Z ) )
=> ( ( ord_less_eq @ ( word @ A ) @ Y @ ( plus_plus @ ( word @ A ) @ Y @ Z ) )
=> ( ord_less_eq @ ( word @ A ) @ X @ ( plus_plus @ ( word @ A ) @ Y @ Z ) ) ) ) ) ).
% word_le_plus_either
thf(fact_964_word__plus__mcs__3,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [V2: word @ A,W: word @ A,X: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ V2 @ W )
=> ( ( ord_less_eq @ ( word @ A ) @ X @ ( plus_plus @ ( word @ A ) @ W @ X ) )
=> ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ V2 @ X ) @ ( plus_plus @ ( word @ A ) @ W @ X ) ) ) ) ) ).
% word_plus_mcs_3
thf(fact_965_word__plus__mcs__4,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [V2: word @ A,X: word @ A,W: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ V2 @ X ) @ ( plus_plus @ ( word @ A ) @ W @ X ) )
=> ( ( ord_less_eq @ ( word @ A ) @ X @ ( plus_plus @ ( word @ A ) @ V2 @ X ) )
=> ( ord_less_eq @ ( word @ A ) @ V2 @ W ) ) ) ) ).
% word_plus_mcs_4
thf(fact_966_word__plus__mono__right,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y: word @ A,Z: word @ A,X: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ Y @ Z )
=> ( ( ord_less_eq @ ( word @ A ) @ X @ ( plus_plus @ ( word @ A ) @ X @ Z ) )
=> ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ X @ Y ) @ ( plus_plus @ ( word @ A ) @ X @ Z ) ) ) ) ) ).
% word_plus_mono_right
thf(fact_967_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_968_word__plus__mcs__4_H,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,V2: word @ A,W: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ X @ V2 ) @ ( plus_plus @ ( word @ A ) @ X @ W ) )
=> ( ( ord_less_eq @ ( word @ A ) @ X @ ( plus_plus @ ( word @ A ) @ X @ V2 ) )
=> ( ord_less_eq @ ( word @ A ) @ V2 @ W ) ) ) ) ).
% word_plus_mcs_4'
thf(fact_969_word__plus__mono__right2,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A,B3: word @ A,C2: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ A3 @ ( plus_plus @ ( word @ A ) @ A3 @ B3 ) )
=> ( ( ord_less_eq @ ( word @ A ) @ C2 @ B3 )
=> ( ord_less_eq @ ( word @ A ) @ A3 @ ( plus_plus @ ( word @ A ) @ A3 @ C2 ) ) ) ) ) ).
% word_plus_mono_right2
thf(fact_970_le__ucast__ucast__le,axiom,
! [A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( type_len @ A ) )
=> ! [X: word @ A,Y: word @ B] :
( ( ord_less_eq @ ( word @ A ) @ X @ ( semiring_1_unsigned @ B @ ( word @ A ) @ Y ) )
=> ( ord_less_eq @ ( word @ B ) @ ( semiring_1_unsigned @ A @ ( word @ B ) @ X ) @ Y ) ) ) ).
% le_ucast_ucast_le
thf(fact_971_word__add__increasing,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [P4: word @ A,W: word @ A,X: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ P4 @ W ) @ X )
=> ( ( ord_less_eq @ ( word @ A ) @ P4 @ ( plus_plus @ ( word @ A ) @ P4 @ W ) )
=> ( ord_less_eq @ ( word @ A ) @ P4 @ X ) ) ) ) ).
% word_add_increasing
thf(fact_972_word__plus__mono__left,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y: word @ A,Z: word @ A,X: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ Y @ Z )
=> ( ( ord_less_eq @ ( word @ A ) @ X @ ( plus_plus @ ( word @ A ) @ X @ Z ) )
=> ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ Y @ X ) @ ( plus_plus @ ( word @ A ) @ Z @ X ) ) ) ) ) ).
% word_plus_mono_left
thf(fact_973_Collect__neg__eq,axiom,
! [A: $tType,P: A > $o] :
( ( collect @ A
@ ^ [X2: A] :
~ ( P @ X2 ) )
= ( uminus_uminus @ ( set @ A ) @ ( collect @ A @ P ) ) ) ).
% Collect_neg_eq
thf(fact_974_Compl__eq,axiom,
! [A: $tType] :
( ( uminus_uminus @ ( set @ A ) )
= ( ^ [A4: set @ A] :
( collect @ A
@ ^ [X2: A] :
~ ( member @ A @ X2 @ A4 ) ) ) ) ).
% Compl_eq
thf(fact_975_constraint__expand,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Lower: word @ A,Upper: word @ A] :
( ( member @ ( word @ A ) @ X
@ ( collect @ ( word @ A )
@ ^ [Y2: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ Lower @ Y2 )
& ( ord_less_eq @ ( word @ A ) @ Y2 @ Upper ) ) ) )
= ( ( ord_less_eq @ ( word @ A ) @ Lower @ X )
& ( ord_less_eq @ ( word @ A ) @ X @ Upper ) ) ) ) ).
% constraint_expand
thf(fact_976_Collect__subset,axiom,
! [A: $tType,A2: set @ A,P: A > $o] :
( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [X2: A] :
( ( member @ A @ X2 @ A2 )
& ( P @ X2 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_977_ucast__nat__def,axiom,
! [B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B ) )
=> ! [X: word @ A] :
( ( semiring_1_of_nat @ ( word @ B ) @ ( semiring_1_unsigned @ A @ nat @ X ) )
= ( semiring_1_unsigned @ A @ ( word @ B ) @ X ) ) ) ).
% ucast_nat_def
thf(fact_978_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_979_ucast__0__I,axiom,
! [A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( type_len @ A ) )
=> ! [X: word @ A] :
( ( X
= ( zero_zero @ ( word @ A ) ) )
=> ( ( semiring_1_unsigned @ A @ ( word @ B ) @ X )
= ( zero_zero @ ( word @ B ) ) ) ) ) ).
% ucast_0_I
thf(fact_980_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_981_word__unat__Rep__inject1,axiom,
! [B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B ) )
=> ! [X: word @ A] :
( ( ( semiring_1_unsigned @ A @ nat @ X )
= ( semiring_1_unsigned @ B @ nat @ ( one_one @ ( word @ B ) ) ) )
= ( X
= ( one_one @ ( word @ A ) ) ) ) ) ).
% word_unat_Rep_inject1
thf(fact_982_un__ui__le,axiom,
! [A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( type_len @ A ) )
=> ! [A3: word @ A,B3: word @ B] :
( ( ord_less_eq @ nat @ ( semiring_1_unsigned @ A @ nat @ A3 ) @ ( semiring_1_unsigned @ B @ nat @ B3 ) )
= ( ord_less_eq @ int @ ( semiring_1_unsigned @ A @ int @ A3 ) @ ( semiring_1_unsigned @ B @ int @ B3 ) ) ) ) ).
% un_ui_le
thf(fact_983_le__unat__uoi,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y: nat,Z: word @ A] :
( ( ord_less_eq @ nat @ Y @ ( semiring_1_unsigned @ A @ nat @ Z ) )
=> ( ( semiring_1_unsigned @ A @ nat @ ( semiring_1_of_nat @ ( word @ A ) @ Y ) )
= Y ) ) ) ).
% le_unat_uoi
thf(fact_984_word__unat__less__le,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A,B3: nat] :
( ( ord_less_eq @ ( word @ A ) @ A3 @ ( semiring_1_of_nat @ ( word @ A ) @ B3 ) )
=> ( ord_less_eq @ nat @ ( semiring_1_unsigned @ A @ nat @ A3 ) @ B3 ) ) ) ).
% word_unat_less_le
thf(fact_985_word__of__nat__le,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,X: word @ A] :
( ( ord_less_eq @ nat @ N @ ( semiring_1_unsigned @ A @ nat @ X ) )
=> ( ord_less_eq @ ( word @ A ) @ ( semiring_1_of_nat @ ( word @ A ) @ N ) @ X ) ) ) ).
% word_of_nat_le
thf(fact_986_word__arith__nat__add,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( plus_plus @ ( word @ A ) )
= ( ^ [A5: word @ A,B5: word @ A] : ( semiring_1_of_nat @ ( word @ A ) @ ( plus_plus @ nat @ ( semiring_1_unsigned @ A @ nat @ A5 ) @ ( semiring_1_unsigned @ A @ nat @ B5 ) ) ) ) ) ) ).
% word_arith_nat_add
thf(fact_987_unat__eq__zero,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] :
( ( ( semiring_1_unsigned @ A @ nat @ X )
= ( zero_zero @ nat ) )
= ( X
= ( zero_zero @ ( word @ A ) ) ) ) ) ).
% unat_eq_zero
thf(fact_988_unat__0,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( semiring_1_unsigned @ A @ nat @ ( zero_zero @ ( word @ A ) ) )
= ( zero_zero @ nat ) ) ) ).
% unat_0
thf(fact_989_unat__mono,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A,B3: word @ A] :
( ( ord_less @ ( word @ A ) @ A3 @ B3 )
=> ( ord_less @ nat @ ( semiring_1_unsigned @ A @ nat @ A3 ) @ ( semiring_1_unsigned @ A @ nat @ B3 ) ) ) ) ).
% unat_mono
thf(fact_990_word__less__nat__alt,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( ord_less @ ( word @ A ) )
= ( ^ [A5: word @ A,B5: word @ A] : ( ord_less @ nat @ ( semiring_1_unsigned @ A @ nat @ A5 ) @ ( semiring_1_unsigned @ A @ nat @ B5 ) ) ) ) ) ).
% word_less_nat_alt
thf(fact_991_unat__1,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( semiring_1_unsigned @ A @ nat @ ( one_one @ ( word @ A ) ) )
= ( one_one @ nat ) ) ) ).
% unat_1
thf(fact_992_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_993_unat__of__nat__less,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A,B3: word @ A,C2: nat] :
( ( ord_less @ ( word @ A ) @ A3 @ B3 )
=> ( ( ( semiring_1_unsigned @ A @ nat @ B3 )
= C2 )
=> ( ord_less @ ( word @ A ) @ A3 @ ( semiring_1_of_nat @ ( word @ A ) @ C2 ) ) ) ) ) ).
% unat_of_nat_less
thf(fact_994_unat__1__0,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ ( one_one @ ( word @ A ) ) @ X )
= ( ord_less @ nat @ ( zero_zero @ nat ) @ ( semiring_1_unsigned @ A @ nat @ X ) ) ) ) ).
% unat_1_0
thf(fact_995_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_996_unat__gt__0,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( semiring_1_unsigned @ A @ nat @ X ) )
= ( X
!= ( zero_zero @ ( word @ A ) ) ) ) ) ).
% unat_gt_0
thf(fact_997_word__overflow__unat,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] :
( ( ( semiring_1_unsigned @ A @ nat @ ( plus_plus @ ( word @ A ) @ X @ ( one_one @ ( word @ A ) ) ) )
= ( plus_plus @ nat @ ( semiring_1_unsigned @ A @ nat @ X ) @ ( one_one @ nat ) ) )
| ( ( plus_plus @ ( word @ A ) @ X @ ( one_one @ ( word @ A ) ) )
= ( zero_zero @ ( word @ A ) ) ) ) ) ).
% word_overflow_unat
thf(fact_998_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_999_word__of__nat__less,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,X: word @ A] :
( ( ord_less @ nat @ N @ ( semiring_1_unsigned @ A @ nat @ X ) )
=> ( ord_less @ ( word @ A ) @ ( semiring_1_of_nat @ ( word @ A ) @ N ) @ X ) ) ) ).
% word_of_nat_less
thf(fact_1000_unat__less__helper,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,N: nat] :
( ( ord_less @ ( word @ A ) @ X @ ( semiring_1_of_nat @ ( word @ A ) @ N ) )
=> ( ord_less @ nat @ ( semiring_1_unsigned @ A @ nat @ X ) @ N ) ) ) ).
% unat_less_helper
thf(fact_1001_linordered__field__no__ub,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X6: A] :
? [X_1: A] : ( ord_less @ A @ X6 @ X_1 ) ) ).
% linordered_field_no_ub
thf(fact_1002_linordered__field__no__lb,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X6: A] :
? [Y4: A] : ( ord_less @ A @ Y4 @ X6 ) ) ).
% linordered_field_no_lb
thf(fact_1003_lt__plus__1__le__word,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,MaxBound: word @ A,X: word @ A] :
( ( ord_less @ nat @ N @ ( semiring_1_unsigned @ A @ nat @ MaxBound ) )
=> ( ( ord_less @ ( word @ A ) @ X @ ( plus_plus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) @ ( semiring_1_of_nat @ ( word @ A ) @ N ) ) )
= ( ord_less_eq @ ( word @ A ) @ X @ ( semiring_1_of_nat @ ( word @ A ) @ N ) ) ) ) ) ).
% lt_plus_1_le_word
thf(fact_1004_ceiling__log__nat__eq__powr__iff,axiom,
! [B3: nat,K: nat,N: nat] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ B3 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ( ( archimedean_ceiling @ real @ ( log @ ( semiring_1_of_nat @ real @ B3 ) @ ( semiring_1_of_nat @ real @ K ) ) )
= ( plus_plus @ int @ ( semiring_1_of_nat @ int @ N ) @ ( one_one @ int ) ) )
= ( ( ord_less @ nat @ ( power_power @ nat @ B3 @ N ) @ K )
& ( ord_less_eq @ nat @ K @ ( power_power @ nat @ B3 @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ) ) ) ).
% ceiling_log_nat_eq_powr_iff
thf(fact_1005_ceiling__log__nat__eq__if,axiom,
! [B3: nat,N: nat,K: nat] :
( ( ord_less @ nat @ ( power_power @ nat @ B3 @ N ) @ K )
=> ( ( ord_less_eq @ nat @ K @ ( power_power @ nat @ B3 @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) ) )
=> ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ B3 )
=> ( ( archimedean_ceiling @ real @ ( log @ ( semiring_1_of_nat @ real @ B3 ) @ ( semiring_1_of_nat @ real @ K ) ) )
= ( plus_plus @ int @ ( semiring_1_of_nat @ int @ N ) @ ( one_one @ int ) ) ) ) ) ) ).
% ceiling_log_nat_eq_if
thf(fact_1006_floor__log__nat__eq__powr__iff,axiom,
! [B3: nat,K: nat,N: nat] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ B3 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ( ( archim6421214686448440834_floor @ real @ ( log @ ( semiring_1_of_nat @ real @ B3 ) @ ( semiring_1_of_nat @ real @ K ) ) )
= ( semiring_1_of_nat @ int @ N ) )
= ( ( ord_less_eq @ nat @ ( power_power @ nat @ B3 @ N ) @ K )
& ( ord_less @ nat @ K @ ( power_power @ nat @ B3 @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ) ) ) ).
% floor_log_nat_eq_powr_iff
thf(fact_1007_ex__has__greatest__nat__lemma,axiom,
! [A: $tType,P: A > $o,K: A,F3: A > nat,N: nat] :
( ( P @ K )
=> ( ! [X3: A] :
( ( P @ X3 )
=> ? [Y5: A] :
( ( P @ Y5 )
& ~ ( ord_less_eq @ nat @ ( F3 @ Y5 ) @ ( F3 @ X3 ) ) ) )
=> ? [Y4: A] :
( ( P @ Y4 )
& ~ ( ord_less @ nat @ ( F3 @ Y4 ) @ ( plus_plus @ nat @ ( F3 @ K ) @ N ) ) ) ) ) ).
% ex_has_greatest_nat_lemma
thf(fact_1008_nat__geq__1__eq__neqz,axiom,
! [X: nat] :
( ( ord_less_eq @ nat @ ( one_one @ nat ) @ X )
= ( X
!= ( zero_zero @ nat ) ) ) ).
% nat_geq_1_eq_neqz
thf(fact_1009_ucast__id,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( semiring_1_unsigned @ A @ ( word @ A ) )
= ( ^ [W2: word @ A] : W2 ) ) ) ).
% ucast_id
thf(fact_1010_predicate1I,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X3: A] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_less_eq @ ( A > $o ) @ P @ Q ) ) ).
% predicate1I
thf(fact_1011_floor__zero,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ( ( archim6421214686448440834_floor @ A @ ( zero_zero @ A ) )
= ( zero_zero @ int ) ) ) ).
% floor_zero
thf(fact_1012_floor__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V2: num] :
( ( archim6421214686448440834_floor @ A @ ( numeral_numeral @ A @ V2 ) )
= ( numeral_numeral @ int @ V2 ) ) ) ).
% floor_numeral
thf(fact_1013_ceiling__zero,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ( ( archimedean_ceiling @ A @ ( zero_zero @ A ) )
= ( zero_zero @ int ) ) ) ).
% ceiling_zero
thf(fact_1014_ceiling__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V2: num] :
( ( archimedean_ceiling @ A @ ( numeral_numeral @ A @ V2 ) )
= ( numeral_numeral @ int @ V2 ) ) ) ).
% ceiling_numeral
thf(fact_1015_floor__one,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ( ( archim6421214686448440834_floor @ A @ ( one_one @ A ) )
= ( one_one @ int ) ) ) ).
% floor_one
thf(fact_1016_ceiling__one,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ( ( archimedean_ceiling @ A @ ( one_one @ A ) )
= ( one_one @ int ) ) ) ).
% ceiling_one
thf(fact_1017_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_1018_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_1019_zero__le__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( archim6421214686448440834_floor @ A @ X ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ X ) ) ) ).
% zero_le_floor
thf(fact_1020_numeral__le__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V2: num,X: A] :
( ( ord_less_eq @ int @ ( numeral_numeral @ int @ V2 ) @ ( archim6421214686448440834_floor @ A @ X ) )
= ( ord_less_eq @ A @ ( numeral_numeral @ A @ V2 ) @ X ) ) ) ).
% numeral_le_floor
thf(fact_1021_floor__less__zero,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ord_less @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( zero_zero @ int ) )
= ( ord_less @ A @ X @ ( zero_zero @ A ) ) ) ) ).
% floor_less_zero
thf(fact_1022_floor__less__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,V2: num] :
( ( ord_less @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( numeral_numeral @ int @ V2 ) )
= ( ord_less @ A @ X @ ( numeral_numeral @ A @ V2 ) ) ) ) ).
% floor_less_numeral
thf(fact_1023_ceiling__le__zero,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ X ) @ ( zero_zero @ int ) )
= ( ord_less_eq @ A @ X @ ( zero_zero @ A ) ) ) ) ).
% ceiling_le_zero
thf(fact_1024_ceiling__le__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,V2: num] :
( ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ X ) @ ( numeral_numeral @ int @ V2 ) )
= ( ord_less_eq @ A @ X @ ( numeral_numeral @ A @ V2 ) ) ) ) ).
% ceiling_le_numeral
thf(fact_1025_zero__less__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ord_less @ int @ ( zero_zero @ int ) @ ( archim6421214686448440834_floor @ A @ X ) )
= ( ord_less_eq @ A @ ( one_one @ A ) @ X ) ) ) ).
% zero_less_floor
thf(fact_1026_floor__le__zero,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( zero_zero @ int ) )
= ( ord_less @ A @ X @ ( one_one @ A ) ) ) ) ).
% floor_le_zero
thf(fact_1027_zero__less__ceiling,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ord_less @ int @ ( zero_zero @ int ) @ ( archimedean_ceiling @ A @ X ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ X ) ) ) ).
% zero_less_ceiling
thf(fact_1028_numeral__less__ceiling,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V2: num,X: A] :
( ( ord_less @ int @ ( numeral_numeral @ int @ V2 ) @ ( archimedean_ceiling @ A @ X ) )
= ( ord_less @ A @ ( numeral_numeral @ A @ V2 ) @ X ) ) ) ).
% numeral_less_ceiling
thf(fact_1029_one__le__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ord_less_eq @ int @ ( one_one @ int ) @ ( archim6421214686448440834_floor @ A @ X ) )
= ( ord_less_eq @ A @ ( one_one @ A ) @ X ) ) ) ).
% one_le_floor
thf(fact_1030_ceiling__less__one,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ord_less @ int @ ( archimedean_ceiling @ A @ X ) @ ( one_one @ int ) )
= ( ord_less_eq @ A @ X @ ( zero_zero @ A ) ) ) ) ).
% ceiling_less_one
thf(fact_1031_one__le__ceiling,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ord_less_eq @ int @ ( one_one @ int ) @ ( archimedean_ceiling @ A @ X ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ X ) ) ) ).
% one_le_ceiling
thf(fact_1032_ceiling__add__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,V2: num] :
( ( archimedean_ceiling @ A @ ( plus_plus @ A @ X @ ( numeral_numeral @ A @ V2 ) ) )
= ( plus_plus @ int @ ( archimedean_ceiling @ A @ X ) @ ( numeral_numeral @ int @ V2 ) ) ) ) ).
% ceiling_add_numeral
thf(fact_1033_floor__less__one,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ord_less @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( one_one @ int ) )
= ( ord_less @ A @ X @ ( one_one @ A ) ) ) ) ).
% floor_less_one
thf(fact_1034_floor__neg__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V2: num] :
( ( archim6421214686448440834_floor @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) )
= ( uminus_uminus @ int @ ( numeral_numeral @ int @ V2 ) ) ) ) ).
% floor_neg_numeral
thf(fact_1035_ceiling__le__one,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ X ) @ ( one_one @ int ) )
= ( ord_less_eq @ A @ X @ ( one_one @ A ) ) ) ) ).
% ceiling_le_one
thf(fact_1036_one__less__ceiling,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ord_less @ int @ ( one_one @ int ) @ ( archimedean_ceiling @ A @ X ) )
= ( ord_less @ A @ ( one_one @ A ) @ X ) ) ) ).
% one_less_ceiling
thf(fact_1037_ceiling__neg__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V2: num] :
( ( archimedean_ceiling @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) )
= ( uminus_uminus @ int @ ( numeral_numeral @ int @ V2 ) ) ) ) ).
% ceiling_neg_numeral
thf(fact_1038_ceiling__add__one,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( archimedean_ceiling @ A @ ( plus_plus @ A @ X @ ( one_one @ A ) ) )
= ( plus_plus @ int @ ( archimedean_ceiling @ A @ X ) @ ( one_one @ int ) ) ) ) ).
% ceiling_add_one
thf(fact_1039_floor__numeral__power,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: num,N: nat] :
( ( archim6421214686448440834_floor @ A @ ( power_power @ A @ ( numeral_numeral @ A @ X ) @ N ) )
= ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N ) ) ) ).
% floor_numeral_power
thf(fact_1040_ceiling__numeral__power,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: num,N: nat] :
( ( archimedean_ceiling @ A @ ( power_power @ A @ ( numeral_numeral @ A @ X ) @ N ) )
= ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N ) ) ) ).
% ceiling_numeral_power
thf(fact_1041_ceiling__less__zero,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ord_less @ int @ ( archimedean_ceiling @ A @ X ) @ ( zero_zero @ int ) )
= ( ord_less_eq @ A @ X @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ).
% ceiling_less_zero
thf(fact_1042_zero__le__ceiling,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( archimedean_ceiling @ A @ X ) )
= ( ord_less @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ X ) ) ) ).
% zero_le_ceiling
thf(fact_1043_numeral__less__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V2: num,X: A] :
( ( ord_less @ int @ ( numeral_numeral @ int @ V2 ) @ ( archim6421214686448440834_floor @ A @ X ) )
= ( ord_less_eq @ A @ ( plus_plus @ A @ ( numeral_numeral @ A @ V2 ) @ ( one_one @ A ) ) @ X ) ) ) ).
% numeral_less_floor
thf(fact_1044_floor__le__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,V2: num] :
( ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( numeral_numeral @ int @ V2 ) )
= ( ord_less @ A @ X @ ( plus_plus @ A @ ( numeral_numeral @ A @ V2 ) @ ( one_one @ A ) ) ) ) ) ).
% floor_le_numeral
thf(fact_1045_one__less__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ord_less @ int @ ( one_one @ int ) @ ( archim6421214686448440834_floor @ A @ X ) )
= ( ord_less_eq @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X ) ) ) ).
% one_less_floor
thf(fact_1046_floor__le__one,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( one_one @ int ) )
= ( ord_less @ A @ X @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% floor_le_one
thf(fact_1047_neg__numeral__le__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V2: num,X: A] :
( ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V2 ) ) @ ( archim6421214686448440834_floor @ A @ X ) )
= ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) @ X ) ) ) ).
% neg_numeral_le_floor
thf(fact_1048_floor__less__neg__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,V2: num] :
( ( ord_less @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V2 ) ) )
= ( ord_less @ A @ X @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) ) ) ) ).
% floor_less_neg_numeral
thf(fact_1049_ceiling__le__neg__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,V2: num] :
( ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ X ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V2 ) ) )
= ( ord_less_eq @ A @ X @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) ) ) ) ).
% ceiling_le_neg_numeral
thf(fact_1050_neg__numeral__less__ceiling,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V2: num,X: A] :
( ( ord_less @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V2 ) ) @ ( archimedean_ceiling @ A @ X ) )
= ( ord_less @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) @ X ) ) ) ).
% neg_numeral_less_ceiling
thf(fact_1051_neg__numeral__less__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V2: num,X: A] :
( ( ord_less @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V2 ) ) @ ( archim6421214686448440834_floor @ A @ X ) )
= ( ord_less_eq @ A @ ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) @ ( one_one @ A ) ) @ X ) ) ) ).
% neg_numeral_less_floor
thf(fact_1052_floor__le__neg__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,V2: num] :
( ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V2 ) ) )
= ( ord_less @ A @ X @ ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) @ ( one_one @ A ) ) ) ) ) ).
% floor_le_neg_numeral
thf(fact_1053_predicate1D,axiom,
! [A: $tType,P: A > $o,Q: A > $o,X: A] :
( ( ord_less_eq @ ( A > $o ) @ P @ Q )
=> ( ( P @ X )
=> ( Q @ X ) ) ) ).
% predicate1D
thf(fact_1054_rev__predicate1D,axiom,
! [A: $tType,P: A > $o,X: A,Q: A > $o] :
( ( P @ X )
=> ( ( ord_less_eq @ ( A > $o ) @ P @ Q )
=> ( Q @ X ) ) ) ).
% rev_predicate1D
thf(fact_1055_floor__le__ceiling,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] : ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( archimedean_ceiling @ A @ X ) ) ) ).
% floor_le_ceiling
thf(fact_1056_ceiling__minus,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( archimedean_ceiling @ A @ ( uminus_uminus @ A @ X ) )
= ( uminus_uminus @ int @ ( archim6421214686448440834_floor @ A @ X ) ) ) ) ).
% ceiling_minus
thf(fact_1057_floor__minus,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( archim6421214686448440834_floor @ A @ ( uminus_uminus @ A @ X ) )
= ( uminus_uminus @ int @ ( archimedean_ceiling @ A @ X ) ) ) ) ).
% floor_minus
thf(fact_1058_ceiling__def,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ( ( archimedean_ceiling @ A )
= ( ^ [X2: A] : ( uminus_uminus @ int @ ( archim6421214686448440834_floor @ A @ ( uminus_uminus @ A @ X2 ) ) ) ) ) ) ).
% ceiling_def
thf(fact_1059_floor__mono,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( archim6421214686448440834_floor @ A @ Y ) ) ) ) ).
% floor_mono
thf(fact_1060_floor__less__cancel,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( archim6421214686448440834_floor @ A @ Y ) )
=> ( ord_less @ A @ X @ Y ) ) ) ).
% floor_less_cancel
thf(fact_1061_ceiling__mono,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ Y @ X )
=> ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ Y ) @ ( archimedean_ceiling @ A @ X ) ) ) ) ).
% ceiling_mono
thf(fact_1062_ceiling__less__cancel,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ int @ ( archimedean_ceiling @ A @ X ) @ ( archimedean_ceiling @ A @ Y ) )
=> ( ord_less @ A @ X @ Y ) ) ) ).
% ceiling_less_cancel
thf(fact_1063_le__floor__add,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Y: A] : ( ord_less_eq @ int @ ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( archim6421214686448440834_floor @ A @ Y ) ) @ ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ X @ Y ) ) ) ) ).
% le_floor_add
thf(fact_1064_ceiling__add__le,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Y: A] : ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ ( plus_plus @ A @ X @ Y ) ) @ ( plus_plus @ int @ ( archimedean_ceiling @ A @ X ) @ ( archimedean_ceiling @ A @ Y ) ) ) ) ).
% ceiling_add_le
thf(fact_1065_memb__imp__not__empty,axiom,
! [A: $tType,X: A,S: set @ A] :
( ( member @ A @ X @ S )
=> ( S
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% memb_imp_not_empty
thf(fact_1066_set__notEmptyE,axiom,
! [A: $tType,S: set @ A] :
( ( S
!= ( bot_bot @ ( set @ A ) ) )
=> ~ ! [X3: A] :
~ ( member @ A @ X3 @ S ) ) ).
% set_notEmptyE
thf(fact_1067_one__add__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( one_one @ int ) )
= ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ X @ ( one_one @ A ) ) ) ) ) ).
% one_add_floor
thf(fact_1068_exists__leI,axiom,
! [N: nat,P: nat > $o] :
( ( ! [N5: nat] :
( ( ord_less @ nat @ N5 @ N )
=> ~ ( P @ N5 ) )
=> ( P @ N ) )
=> ? [N6: nat] :
( ( ord_less_eq @ nat @ N6 @ N )
& ( P @ N6 ) ) ) ).
% exists_leI
thf(fact_1069_Lattices__Big_Oex__has__greatest__nat,axiom,
! [A: $tType,P: A > $o,K: A,F3: A > nat,B3: nat] :
( ( P @ K )
=> ( ! [Y4: A] :
( ( P @ Y4 )
=> ( ord_less @ nat @ ( F3 @ Y4 ) @ B3 ) )
=> ? [X3: A] :
( ( P @ X3 )
& ! [Y5: A] :
( ( P @ Y5 )
=> ( ord_less_eq @ nat @ ( F3 @ Y5 ) @ ( F3 @ X3 ) ) ) ) ) ) ).
% Lattices_Big.ex_has_greatest_nat
thf(fact_1070_floor__log__nat__eq__if,axiom,
! [B3: nat,N: nat,K: nat] :
( ( ord_less_eq @ nat @ ( power_power @ nat @ B3 @ N ) @ K )
=> ( ( ord_less @ nat @ K @ ( power_power @ nat @ B3 @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) ) )
=> ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ B3 )
=> ( ( archim6421214686448440834_floor @ real @ ( log @ ( semiring_1_of_nat @ real @ B3 ) @ ( semiring_1_of_nat @ real @ K ) ) )
= ( semiring_1_of_nat @ int @ N ) ) ) ) ) ).
% floor_log_nat_eq_if
thf(fact_1071_ceiling__log__eq__powr__iff,axiom,
! [X: real,B3: real,K: nat] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ ( one_one @ real ) @ B3 )
=> ( ( ( archimedean_ceiling @ real @ ( log @ B3 @ X ) )
= ( plus_plus @ int @ ( semiring_1_of_nat @ int @ K ) @ ( one_one @ int ) ) )
= ( ( ord_less @ real @ ( powr @ real @ B3 @ ( semiring_1_of_nat @ real @ K ) ) @ X )
& ( ord_less_eq @ real @ X @ ( powr @ real @ B3 @ ( semiring_1_of_nat @ real @ ( plus_plus @ nat @ K @ ( one_one @ nat ) ) ) ) ) ) ) ) ) ).
% ceiling_log_eq_powr_iff
thf(fact_1072_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_1073_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_1074_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_1075_neg__numeral__le__ceiling,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V2: num,X: A] :
( ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V2 ) ) @ ( archimedean_ceiling @ A @ X ) )
= ( ord_less @ A @ ( minus_minus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) @ ( one_one @ A ) ) @ X ) ) ) ).
% neg_numeral_le_ceiling
thf(fact_1076_ceiling__less__neg__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,V2: num] :
( ( ord_less @ int @ ( archimedean_ceiling @ A @ X ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V2 ) ) )
= ( ord_less_eq @ A @ X @ ( minus_minus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) @ ( one_one @ A ) ) ) ) ) ).
% ceiling_less_neg_numeral
thf(fact_1077_power__le__zero__eq__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,W: num] :
( ( ord_less_eq @ A @ ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ W ) ) @ ( zero_zero @ A ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( numeral_numeral @ nat @ W ) )
& ( ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ W ) )
& ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) ) )
| ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ W ) )
& ( A3
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% power_le_zero_eq_numeral
thf(fact_1078_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_1079_neg__numeral__power__less__of__int__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: num,N: nat,A3: int] :
( ( ord_less @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ X ) ) @ N ) @ ( ring_1_of_int @ A @ A3 ) )
= ( ord_less @ int @ ( power_power @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ X ) ) @ N ) @ A3 ) ) ) ).
% neg_numeral_power_less_of_int_cancel_iff
thf(fact_1080_minus__apply,axiom,
! [B: $tType,A: $tType] :
( ( minus @ B )
=> ( ( minus_minus @ ( A > B ) )
= ( ^ [A4: A > B,B4: A > B,X2: A] : ( minus_minus @ B @ ( A4 @ X2 ) @ ( B4 @ X2 ) ) ) ) ) ).
% minus_apply
thf(fact_1081_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus @ nat @ M @ M )
= ( zero_zero @ nat ) ) ).
% diff_self_eq_0
thf(fact_1082_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus @ nat @ ( zero_zero @ nat ) @ N )
= ( zero_zero @ nat ) ) ).
% diff_0_eq_0
thf(fact_1083_Diff__cancel,axiom,
! [A: $tType,A2: set @ A] :
( ( minus_minus @ ( set @ A ) @ A2 @ A2 )
= ( bot_bot @ ( set @ A ) ) ) ).
% Diff_cancel
thf(fact_1084_empty__Diff,axiom,
! [A: $tType,A2: set @ A] :
( ( minus_minus @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A2 )
= ( bot_bot @ ( set @ A ) ) ) ).
% empty_Diff
thf(fact_1085_Diff__empty,axiom,
! [A: $tType,A2: set @ A] :
( ( minus_minus @ ( set @ A ) @ A2 @ ( bot_bot @ ( set @ A ) ) )
= A2 ) ).
% Diff_empty
thf(fact_1086_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_1087_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_1088_insert__Diff1,axiom,
! [A: $tType,X: A,B2: set @ A,A2: set @ A] :
( ( member @ A @ X @ B2 )
=> ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X @ A2 ) @ B2 )
= ( minus_minus @ ( set @ A ) @ A2 @ B2 ) ) ) ).
% insert_Diff1
thf(fact_1089_Diff__insert0,axiom,
! [A: $tType,X: A,A2: set @ A,B2: set @ A] :
( ~ ( member @ A @ X @ A2 )
=> ( ( minus_minus @ ( set @ A ) @ A2 @ ( insert @ A @ X @ B2 ) )
= ( minus_minus @ ( set @ A ) @ A2 @ B2 ) ) ) ).
% Diff_insert0
thf(fact_1090_of__int__eq__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [W: int,Z: int] :
( ( ( ring_1_of_int @ A @ W )
= ( ring_1_of_int @ A @ Z ) )
= ( W = Z ) ) ) ).
% of_int_eq_iff
thf(fact_1091_Un__Diff__cancel2,axiom,
! [A: $tType,B2: set @ A,A2: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ B2 @ A2 ) @ A2 )
= ( sup_sup @ ( set @ A ) @ B2 @ A2 ) ) ).
% Un_Diff_cancel2
thf(fact_1092_Un__Diff__cancel,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( sup_sup @ ( set @ A ) @ A2 @ ( minus_minus @ ( set @ A ) @ B2 @ A2 ) )
= ( sup_sup @ ( set @ A ) @ A2 @ B2 ) ) ).
% Un_Diff_cancel
thf(fact_1093_idiff__0__right,axiom,
! [N: extended_enat] :
( ( minus_minus @ extended_enat @ N @ ( zero_zero @ extended_enat ) )
= N ) ).
% idiff_0_right
thf(fact_1094_idiff__0,axiom,
! [N: extended_enat] :
( ( minus_minus @ extended_enat @ ( zero_zero @ extended_enat ) @ N )
= ( zero_zero @ extended_enat ) ) ).
% idiff_0
thf(fact_1095_diff__self,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A] :
( ( minus_minus @ A @ A3 @ A3 )
= ( zero_zero @ A ) ) ) ).
% diff_self
thf(fact_1096_diff__0__right,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A] :
( ( minus_minus @ A @ A3 @ ( zero_zero @ A ) )
= A3 ) ) ).
% diff_0_right
thf(fact_1097_zero__diff,axiom,
! [A: $tType] :
( ( comm_monoid_diff @ A )
=> ! [A3: A] :
( ( minus_minus @ A @ ( zero_zero @ A ) @ A3 )
= ( zero_zero @ A ) ) ) ).
% zero_diff
thf(fact_1098_diff__zero,axiom,
! [A: $tType] :
( ( cancel1802427076303600483id_add @ A )
=> ! [A3: A] :
( ( minus_minus @ A @ A3 @ ( zero_zero @ A ) )
= A3 ) ) ).
% diff_zero
thf(fact_1099_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: $tType] :
( ( cancel1802427076303600483id_add @ A )
=> ! [A3: A] :
( ( minus_minus @ A @ A3 @ A3 )
= ( zero_zero @ A ) ) ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_1100_divide__eq__0__iff,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A3: A,B3: A] :
( ( ( divide_divide @ A @ A3 @ B3 )
= ( zero_zero @ A ) )
= ( ( A3
= ( zero_zero @ A ) )
| ( B3
= ( zero_zero @ A ) ) ) ) ) ).
% divide_eq_0_iff
thf(fact_1101_divide__cancel__left,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [C2: A,A3: A,B3: A] :
( ( ( divide_divide @ A @ C2 @ A3 )
= ( divide_divide @ A @ C2 @ B3 ) )
= ( ( C2
= ( zero_zero @ A ) )
| ( A3 = B3 ) ) ) ) ).
% divide_cancel_left
thf(fact_1102_divide__cancel__right,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A3: A,C2: A,B3: A] :
( ( ( divide_divide @ A @ A3 @ C2 )
= ( divide_divide @ A @ B3 @ C2 ) )
= ( ( C2
= ( zero_zero @ A ) )
| ( A3 = B3 ) ) ) ) ).
% divide_cancel_right
thf(fact_1103_bits__div__0,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A3: A] :
( ( divide_divide @ A @ ( zero_zero @ A ) @ A3 )
= ( zero_zero @ A ) ) ) ).
% bits_div_0
thf(fact_1104_bits__div__by__0,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A3: A] :
( ( divide_divide @ A @ A3 @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% bits_div_by_0
thf(fact_1105_division__ring__divide__zero,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A] :
( ( divide_divide @ A @ A3 @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% division_ring_divide_zero
thf(fact_1106_div__0,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [A3: A] :
( ( divide_divide @ A @ ( zero_zero @ A ) @ A3 )
= ( zero_zero @ A ) ) ) ).
% div_0
thf(fact_1107_div__by__0,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [A3: A] :
( ( divide_divide @ A @ A3 @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% div_by_0
thf(fact_1108_add__diff__cancel__right_H,axiom,
! [A: $tType] :
( ( cancel2418104881723323429up_add @ A )
=> ! [A3: A,B3: A] :
( ( minus_minus @ A @ ( plus_plus @ A @ A3 @ B3 ) @ B3 )
= A3 ) ) ).
% add_diff_cancel_right'
thf(fact_1109_add__diff__cancel__right,axiom,
! [A: $tType] :
( ( cancel2418104881723323429up_add @ A )
=> ! [A3: A,C2: A,B3: A] :
( ( minus_minus @ A @ ( plus_plus @ A @ A3 @ C2 ) @ ( plus_plus @ A @ B3 @ C2 ) )
= ( minus_minus @ A @ A3 @ B3 ) ) ) ).
% add_diff_cancel_right
thf(fact_1110_add__diff__cancel__left_H,axiom,
! [A: $tType] :
( ( cancel2418104881723323429up_add @ A )
=> ! [A3: A,B3: A] :
( ( minus_minus @ A @ ( plus_plus @ A @ A3 @ B3 ) @ A3 )
= B3 ) ) ).
% add_diff_cancel_left'
thf(fact_1111_add__diff__cancel__left,axiom,
! [A: $tType] :
( ( cancel2418104881723323429up_add @ A )
=> ! [C2: A,A3: A,B3: A] :
( ( minus_minus @ A @ ( plus_plus @ A @ C2 @ A3 ) @ ( plus_plus @ A @ C2 @ B3 ) )
= ( minus_minus @ A @ A3 @ B3 ) ) ) ).
% add_diff_cancel_left
thf(fact_1112_diff__add__cancel,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B3: A] :
( ( plus_plus @ A @ ( minus_minus @ A @ A3 @ B3 ) @ B3 )
= A3 ) ) ).
% diff_add_cancel
thf(fact_1113_add__diff__cancel,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B3: A] :
( ( minus_minus @ A @ ( plus_plus @ A @ A3 @ B3 ) @ B3 )
= A3 ) ) ).
% add_diff_cancel
thf(fact_1114_bits__div__by__1,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A3: A] :
( ( divide_divide @ A @ A3 @ ( one_one @ A ) )
= A3 ) ) ).
% bits_div_by_1
thf(fact_1115_div__by__1,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [A3: A] :
( ( divide_divide @ A @ A3 @ ( one_one @ A ) )
= A3 ) ) ).
% div_by_1
thf(fact_1116_dvd__0__right,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A3: A] : ( dvd_dvd @ A @ A3 @ ( zero_zero @ A ) ) ) ).
% dvd_0_right
thf(fact_1117_dvd__0__left__iff,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A3: A] :
( ( dvd_dvd @ A @ ( zero_zero @ A ) @ A3 )
= ( A3
= ( zero_zero @ A ) ) ) ) ).
% dvd_0_left_iff
thf(fact_1118_minus__diff__eq,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B3: A] :
( ( uminus_uminus @ A @ ( minus_minus @ A @ A3 @ B3 ) )
= ( minus_minus @ A @ B3 @ A3 ) ) ) ).
% minus_diff_eq
thf(fact_1119_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_1120_dvd__add__triv__right__iff,axiom,
! [A: $tType] :
( ( comm_s4317794764714335236cancel @ A )
=> ! [A3: A,B3: A] :
( ( dvd_dvd @ A @ A3 @ ( plus_plus @ A @ B3 @ A3 ) )
= ( dvd_dvd @ A @ A3 @ B3 ) ) ) ).
% dvd_add_triv_right_iff
thf(fact_1121_dvd__add__triv__left__iff,axiom,
! [A: $tType] :
( ( comm_s4317794764714335236cancel @ A )
=> ! [A3: A,B3: A] :
( ( dvd_dvd @ A @ A3 @ ( plus_plus @ A @ A3 @ B3 ) )
= ( dvd_dvd @ A @ A3 @ B3 ) ) ) ).
% dvd_add_triv_left_iff
thf(fact_1122_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_1123_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_1124_div__dvd__div,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B3: A,C2: A] :
( ( dvd_dvd @ A @ A3 @ B3 )
=> ( ( dvd_dvd @ A @ A3 @ C2 )
=> ( ( dvd_dvd @ A @ ( divide_divide @ A @ B3 @ A3 ) @ ( divide_divide @ A @ C2 @ A3 ) )
= ( dvd_dvd @ A @ B3 @ C2 ) ) ) ) ) ).
% div_dvd_div
thf(fact_1125_Diff__eq__empty__iff,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( ( minus_minus @ ( set @ A ) @ A2 @ B2 )
= ( bot_bot @ ( set @ A ) ) )
= ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ).
% Diff_eq_empty_iff
thf(fact_1126_minus__dvd__iff,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [X: A,Y: A] :
( ( dvd_dvd @ A @ ( uminus_uminus @ A @ X ) @ Y )
= ( dvd_dvd @ A @ X @ Y ) ) ) ).
% minus_dvd_iff
thf(fact_1127_dvd__minus__iff,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [X: A,Y: A] :
( ( dvd_dvd @ A @ X @ ( uminus_uminus @ A @ Y ) )
= ( dvd_dvd @ A @ X @ Y ) ) ) ).
% dvd_minus_iff
thf(fact_1128_insert__Diff__single,axiom,
! [A: $tType,A3: A,A2: set @ A] :
( ( insert @ A @ A3 @ ( minus_minus @ ( set @ A ) @ A2 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( insert @ A @ A3 @ A2 ) ) ).
% insert_Diff_single
thf(fact_1129_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_1130_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_1131_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_1132_of__int__diff,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [W: int,Z: int] :
( ( ring_1_of_int @ A @ ( minus_minus @ int @ W @ Z ) )
= ( minus_minus @ A @ ( ring_1_of_int @ A @ W ) @ ( ring_1_of_int @ A @ Z ) ) ) ) ).
% of_int_diff
thf(fact_1133_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_1134_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_1135_powr__0,axiom,
! [A: $tType] :
( ( ln @ A )
=> ! [Z: A] :
( ( powr @ A @ ( zero_zero @ A ) @ Z )
= ( zero_zero @ A ) ) ) ).
% powr_0
thf(fact_1136_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_1137_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_1138_powr__one__eq__one,axiom,
! [A: $tType] :
( ( ln @ A )
=> ! [A3: A] :
( ( powr @ A @ ( one_one @ A ) @ A3 )
= ( one_one @ A ) ) ) ).
% powr_one_eq_one
thf(fact_1139_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_1140_of__int__floor__cancel,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ( ring_1_of_int @ A @ ( archim6421214686448440834_floor @ A @ X ) )
= X )
= ( ? [N4: int] :
( X
= ( ring_1_of_int @ A @ N4 ) ) ) ) ) ).
% of_int_floor_cancel
thf(fact_1141_floor__of__int,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z: int] :
( ( archim6421214686448440834_floor @ A @ ( ring_1_of_int @ A @ Z ) )
= Z ) ) ).
% floor_of_int
thf(fact_1142_of__int__ceiling__cancel,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ( ring_1_of_int @ A @ ( archimedean_ceiling @ A @ X ) )
= X )
= ( ? [N4: int] :
( X
= ( ring_1_of_int @ A @ N4 ) ) ) ) ) ).
% of_int_ceiling_cancel
thf(fact_1143_ceiling__of__int,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z: int] :
( ( archimedean_ceiling @ A @ ( ring_1_of_int @ A @ Z ) )
= Z ) ) ).
% ceiling_of_int
thf(fact_1144_Compl__Diff__eq,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( uminus_uminus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A2 @ B2 ) )
= ( sup_sup @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A2 ) @ B2 ) ) ).
% Compl_Diff_eq
thf(fact_1145_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_1146_diff__ge__0__iff__ge,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( minus_minus @ A @ A3 @ B3 ) )
= ( ord_less_eq @ A @ B3 @ A3 ) ) ) ).
% diff_ge_0_iff_ge
thf(fact_1147_zero__comp__diff__simps_I1_J,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( minus_minus @ A @ A3 @ B3 ) )
= ( ord_less_eq @ A @ B3 @ A3 ) ) ) ).
% zero_comp_diff_simps(1)
thf(fact_1148_diff__gt__0__iff__gt,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( minus_minus @ A @ A3 @ B3 ) )
= ( ord_less @ A @ B3 @ A3 ) ) ) ).
% diff_gt_0_iff_gt
thf(fact_1149_zero__comp__diff__simps_I2_J,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( minus_minus @ A @ A3 @ B3 ) )
= ( ord_less @ A @ B3 @ A3 ) ) ) ).
% zero_comp_diff_simps(2)
thf(fact_1150_diff__add__zero,axiom,
! [A: $tType] :
( ( comm_monoid_diff @ A )
=> ! [A3: A,B3: A] :
( ( minus_minus @ A @ A3 @ ( plus_plus @ A @ A3 @ B3 ) )
= ( zero_zero @ A ) ) ) ).
% diff_add_zero
thf(fact_1151_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_1152_le__add__diff__inverse2,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [B3: A,A3: A] :
( ( ord_less_eq @ A @ B3 @ A3 )
=> ( ( plus_plus @ A @ ( minus_minus @ A @ A3 @ B3 ) @ B3 )
= A3 ) ) ) ).
% le_add_diff_inverse2
thf(fact_1153_le__add__diff__inverse,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [B3: A,A3: A] :
( ( ord_less_eq @ A @ B3 @ A3 )
=> ( ( plus_plus @ A @ B3 @ ( minus_minus @ A @ A3 @ B3 ) )
= A3 ) ) ) ).
% le_add_diff_inverse
thf(fact_1154_divide__eq__1__iff,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A3: A,B3: A] :
( ( ( divide_divide @ A @ A3 @ B3 )
= ( one_one @ A ) )
= ( ( B3
!= ( zero_zero @ A ) )
& ( A3 = B3 ) ) ) ) ).
% divide_eq_1_iff
thf(fact_1155_one__eq__divide__iff,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A3: A,B3: A] :
( ( ( one_one @ A )
= ( divide_divide @ A @ A3 @ B3 ) )
= ( ( B3
!= ( zero_zero @ A ) )
& ( A3 = B3 ) ) ) ) ).
% one_eq_divide_iff
thf(fact_1156_divide__self,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ A3 @ A3 )
= ( one_one @ A ) ) ) ) ).
% divide_self
thf(fact_1157_divide__self__if,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A] :
( ( ( A3
= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ A3 @ A3 )
= ( zero_zero @ A ) ) )
& ( ( A3
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ A3 @ A3 )
= ( one_one @ A ) ) ) ) ) ).
% divide_self_if
thf(fact_1158_divide__eq__eq__1,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B3: A,A3: A] :
( ( ( divide_divide @ A @ B3 @ A3 )
= ( one_one @ A ) )
= ( ( A3
!= ( zero_zero @ A ) )
& ( A3 = B3 ) ) ) ) ).
% divide_eq_eq_1
thf(fact_1159_eq__divide__eq__1,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B3: A,A3: A] :
( ( ( one_one @ A )
= ( divide_divide @ A @ B3 @ A3 ) )
= ( ( A3
!= ( zero_zero @ A ) )
& ( A3 = B3 ) ) ) ) ).
% eq_divide_eq_1
thf(fact_1160_one__divide__eq__0__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A] :
( ( ( divide_divide @ A @ ( one_one @ A ) @ A3 )
= ( zero_zero @ A ) )
= ( A3
= ( zero_zero @ A ) ) ) ) ).
% one_divide_eq_0_iff
thf(fact_1161_zero__eq__1__divide__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A] :
( ( ( zero_zero @ A )
= ( divide_divide @ A @ ( one_one @ A ) @ A3 ) )
= ( A3
= ( zero_zero @ A ) ) ) ) ).
% zero_eq_1_divide_iff
thf(fact_1162_div__self,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [A3: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ A3 @ A3 )
= ( one_one @ A ) ) ) ) ).
% div_self
thf(fact_1163_diff__0,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A] :
( ( minus_minus @ A @ ( zero_zero @ A ) @ A3 )
= ( uminus_uminus @ A @ A3 ) ) ) ).
% diff_0
thf(fact_1164_verit__minus__simplify_I3_J,axiom,
! [B: $tType] :
( ( group_add @ B )
=> ! [B3: B] :
( ( minus_minus @ B @ ( zero_zero @ B ) @ B3 )
= ( uminus_uminus @ B @ B3 ) ) ) ).
% verit_minus_simplify(3)
thf(fact_1165_uminus__add__conv__diff,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A3: A,B3: A] :
( ( plus_plus @ A @ ( uminus_uminus @ A @ A3 ) @ B3 )
= ( minus_minus @ A @ B3 @ A3 ) ) ) ).
% uminus_add_conv_diff
thf(fact_1166_diff__minus__eq__add,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B3: A] :
( ( minus_minus @ A @ A3 @ ( uminus_uminus @ A @ B3 ) )
= ( plus_plus @ A @ A3 @ B3 ) ) ) ).
% diff_minus_eq_add
thf(fact_1167_divide__minus1,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [X: A] :
( ( divide_divide @ A @ X @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( uminus_uminus @ A @ X ) ) ) ).
% divide_minus1
thf(fact_1168_div__add,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [C2: A,A3: A,B3: A] :
( ( dvd_dvd @ A @ C2 @ A3 )
=> ( ( dvd_dvd @ A @ C2 @ B3 )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ A3 @ B3 ) @ C2 )
= ( plus_plus @ A @ ( divide_divide @ A @ A3 @ C2 ) @ ( divide_divide @ A @ B3 @ C2 ) ) ) ) ) ) ).
% div_add
thf(fact_1169_unit__div__1__div__1,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ( divide_divide @ A @ ( one_one @ A ) @ ( divide_divide @ A @ ( one_one @ A ) @ A3 ) )
= A3 ) ) ) ).
% unit_div_1_div_1
thf(fact_1170_unit__div__1__unit,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( dvd_dvd @ A @ ( divide_divide @ A @ ( one_one @ A ) @ A3 ) @ ( one_one @ A ) ) ) ) ).
% unit_div_1_unit
thf(fact_1171_unit__div,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B3: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ B3 @ ( one_one @ A ) )
=> ( dvd_dvd @ A @ ( divide_divide @ A @ A3 @ B3 ) @ ( one_one @ A ) ) ) ) ) ).
% unit_div
thf(fact_1172_div__diff,axiom,
! [A: $tType] :
( ( idom_modulo @ A )
=> ! [C2: A,A3: A,B3: A] :
( ( dvd_dvd @ A @ C2 @ A3 )
=> ( ( dvd_dvd @ A @ C2 @ B3 )
=> ( ( divide_divide @ A @ ( minus_minus @ A @ A3 @ B3 ) @ C2 )
= ( minus_minus @ A @ ( divide_divide @ A @ A3 @ C2 ) @ ( divide_divide @ A @ B3 @ C2 ) ) ) ) ) ) ).
% div_diff
thf(fact_1173_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_1174_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_1175_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_1176_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_1177_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_1178_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_1179_of__int__less__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [W: int,Z: int] :
( ( ord_less @ A @ ( ring_1_of_int @ A @ W ) @ ( ring_1_of_int @ A @ Z ) )
= ( ord_less @ int @ W @ Z ) ) ) ).
% of_int_less_iff
thf(fact_1180_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_1181_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_1182_powr__zero__eq__one,axiom,
! [A: $tType] :
( ( ln @ A )
=> ! [X: A] :
( ( ( X
= ( zero_zero @ A ) )
=> ( ( powr @ A @ X @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) )
& ( ( X
!= ( zero_zero @ A ) )
=> ( ( powr @ A @ X @ ( zero_zero @ A ) )
= ( one_one @ A ) ) ) ) ) ).
% powr_zero_eq_one
thf(fact_1183_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_1184_of__int__minus,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [Z: int] :
( ( ring_1_of_int @ A @ ( uminus_uminus @ int @ Z ) )
= ( uminus_uminus @ A @ ( ring_1_of_int @ A @ Z ) ) ) ) ).
% of_int_minus
thf(fact_1185_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_1186_of__int__power__eq__of__int__cancel__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [X: int,B3: int,W: nat] :
( ( ( ring_1_of_int @ A @ X )
= ( power_power @ A @ ( ring_1_of_int @ A @ B3 ) @ W ) )
= ( X
= ( power_power @ int @ B3 @ W ) ) ) ) ).
% of_int_power_eq_of_int_cancel_iff
thf(fact_1187_of__int__eq__of__int__power__cancel__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [B3: int,W: nat,X: int] :
( ( ( power_power @ A @ ( ring_1_of_int @ A @ B3 ) @ W )
= ( ring_1_of_int @ A @ X ) )
= ( ( power_power @ int @ B3 @ W )
= X ) ) ) ).
% of_int_eq_of_int_power_cancel_iff
thf(fact_1188_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_1189_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_1190_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_1191_Word_Oof__int__uint,axiom,
! [A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( ring_1 @ A ) )
=> ! [W: word @ B] :
( ( ring_1_of_int @ A @ ( semiring_1_unsigned @ B @ int @ W ) )
= ( semiring_1_unsigned @ B @ A @ W ) ) ) ).
% Word.of_int_uint
thf(fact_1192_floor__diff__of__int,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Z: int] :
( ( archim6421214686448440834_floor @ A @ ( minus_minus @ A @ X @ ( ring_1_of_int @ A @ Z ) ) )
= ( minus_minus @ int @ ( archim6421214686448440834_floor @ A @ X ) @ Z ) ) ) ).
% floor_diff_of_int
thf(fact_1193_ceiling__diff__of__int,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Z: int] :
( ( archimedean_ceiling @ A @ ( minus_minus @ A @ X @ ( ring_1_of_int @ A @ Z ) ) )
= ( minus_minus @ int @ ( archimedean_ceiling @ A @ X ) @ Z ) ) ) ).
% ceiling_diff_of_int
thf(fact_1194_powr__gt__zero,axiom,
! [X: real,A3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( powr @ real @ X @ A3 ) )
= ( X
!= ( zero_zero @ real ) ) ) ).
% powr_gt_zero
thf(fact_1195_powr__nonneg__iff,axiom,
! [A3: real,X: real] :
( ( ord_less_eq @ real @ ( powr @ real @ A3 @ X ) @ ( zero_zero @ real ) )
= ( A3
= ( zero_zero @ real ) ) ) ).
% powr_nonneg_iff
thf(fact_1196_powr__less__cancel__iff,axiom,
! [X: real,A3: real,B3: real] :
( ( ord_less @ real @ ( one_one @ real ) @ X )
=> ( ( ord_less @ real @ ( powr @ real @ X @ A3 ) @ ( powr @ real @ X @ B3 ) )
= ( ord_less @ real @ A3 @ B3 ) ) ) ).
% powr_less_cancel_iff
thf(fact_1197_divide__le__0__1__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ ( one_one @ A ) @ A3 ) @ ( zero_zero @ A ) )
= ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% divide_le_0_1_iff
thf(fact_1198_zero__le__divide__1__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ ( one_one @ A ) @ A3 ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% zero_le_divide_1_iff
thf(fact_1199_zero__less__divide__1__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ ( one_one @ A ) @ A3 ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% zero_less_divide_1_iff
thf(fact_1200_less__divide__eq__1__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B3 @ A3 ) )
= ( ord_less @ A @ A3 @ B3 ) ) ) ) ).
% less_divide_eq_1_pos
thf(fact_1201_less__divide__eq__1__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B3 @ A3 ) )
= ( ord_less @ A @ B3 @ A3 ) ) ) ) ).
% less_divide_eq_1_neg
thf(fact_1202_divide__less__eq__1__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ ( divide_divide @ A @ B3 @ A3 ) @ ( one_one @ A ) )
= ( ord_less @ A @ B3 @ A3 ) ) ) ) ).
% divide_less_eq_1_pos
thf(fact_1203_divide__less__eq__1__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( divide_divide @ A @ B3 @ A3 ) @ ( one_one @ A ) )
= ( ord_less @ A @ A3 @ B3 ) ) ) ) ).
% divide_less_eq_1_neg
thf(fact_1204_divide__less__0__1__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( divide_divide @ A @ ( one_one @ A ) @ A3 ) @ ( zero_zero @ A ) )
= ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% divide_less_0_1_iff
thf(fact_1205_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_1206_floor__diff__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,V2: num] :
( ( archim6421214686448440834_floor @ A @ ( minus_minus @ A @ X @ ( numeral_numeral @ A @ V2 ) ) )
= ( minus_minus @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( numeral_numeral @ int @ V2 ) ) ) ) ).
% floor_diff_numeral
thf(fact_1207_ceiling__diff__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,V2: num] :
( ( archimedean_ceiling @ A @ ( minus_minus @ A @ X @ ( numeral_numeral @ A @ V2 ) ) )
= ( minus_minus @ int @ ( archimedean_ceiling @ A @ X ) @ ( numeral_numeral @ int @ V2 ) ) ) ) ).
% ceiling_diff_numeral
thf(fact_1208_floor__diff__one,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( archim6421214686448440834_floor @ A @ ( minus_minus @ A @ X @ ( one_one @ A ) ) )
= ( minus_minus @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( one_one @ int ) ) ) ) ).
% floor_diff_one
thf(fact_1209_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_1210_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_1211_ceiling__diff__one,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( archimedean_ceiling @ A @ ( minus_minus @ A @ X @ ( one_one @ A ) ) )
= ( minus_minus @ int @ ( archimedean_ceiling @ A @ X ) @ ( one_one @ int ) ) ) ) ).
% ceiling_diff_one
thf(fact_1212_ceiling__add__of__int,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Z: int] :
( ( archimedean_ceiling @ A @ ( plus_plus @ A @ X @ ( ring_1_of_int @ A @ Z ) ) )
= ( plus_plus @ int @ ( archimedean_ceiling @ A @ X ) @ Z ) ) ) ).
% ceiling_add_of_int
thf(fact_1213_floor__uminus__of__int,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z: int] :
( ( archim6421214686448440834_floor @ A @ ( uminus_uminus @ A @ ( ring_1_of_int @ A @ Z ) ) )
= ( uminus_uminus @ int @ Z ) ) ) ).
% floor_uminus_of_int
thf(fact_1214_powr__eq__one__iff,axiom,
! [A3: real,X: real] :
( ( ord_less @ real @ ( one_one @ real ) @ A3 )
=> ( ( ( powr @ real @ A3 @ X )
= ( one_one @ real ) )
= ( X
= ( zero_zero @ real ) ) ) ) ).
% powr_eq_one_iff
thf(fact_1215_powr__one,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( powr @ real @ X @ ( one_one @ real ) )
= X ) ) ).
% powr_one
thf(fact_1216_powr__one__gt__zero__iff,axiom,
! [X: real] :
( ( ( powr @ real @ X @ ( one_one @ real ) )
= X )
= ( ord_less_eq @ real @ ( zero_zero @ real ) @ X ) ) ).
% powr_one_gt_zero_iff
thf(fact_1217_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_1218_powr__le__cancel__iff,axiom,
! [X: real,A3: real,B3: real] :
( ( ord_less @ real @ ( one_one @ real ) @ X )
=> ( ( ord_less_eq @ real @ ( powr @ real @ X @ A3 ) @ ( powr @ real @ X @ B3 ) )
= ( ord_less_eq @ real @ A3 @ B3 ) ) ) ).
% powr_le_cancel_iff
thf(fact_1219_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_1220_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_1221_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_1222_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_1223_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_1224_le__divide__eq__1__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B3 @ A3 ) )
= ( ord_less_eq @ A @ A3 @ B3 ) ) ) ) ).
% le_divide_eq_1_pos
thf(fact_1225_le__divide__eq__1__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B3 @ A3 ) )
= ( ord_less_eq @ A @ B3 @ A3 ) ) ) ) ).
% le_divide_eq_1_neg
thf(fact_1226_divide__le__eq__1__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ ( divide_divide @ A @ B3 @ A3 ) @ ( one_one @ A ) )
= ( ord_less_eq @ A @ B3 @ A3 ) ) ) ) ).
% divide_le_eq_1_pos
thf(fact_1227_divide__le__eq__1__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( divide_divide @ A @ B3 @ A3 ) @ ( one_one @ A ) )
= ( ord_less_eq @ A @ A3 @ B3 ) ) ) ) ).
% divide_le_eq_1_neg
thf(fact_1228_even__add,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A3: A,B3: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( plus_plus @ A @ A3 @ B3 ) )
= ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
= ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B3 ) ) ) ) ).
% even_add
thf(fact_1229_odd__add,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A3: A,B3: A] :
( ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( plus_plus @ A @ A3 @ B3 ) ) )
= ( ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) )
!= ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B3 ) ) ) ) ) ).
% odd_add
thf(fact_1230_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_1231_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_1232_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_1233_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_1234_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_1235_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_1236_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_1237_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_1238_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_1239_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_1240_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_1241_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_1242_of__int__eq__numeral__power__cancel__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [Y: int,X: num,N: nat] :
( ( ( ring_1_of_int @ A @ Y )
= ( power_power @ A @ ( numeral_numeral @ A @ X ) @ N ) )
= ( Y
= ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N ) ) ) ) ).
% of_int_eq_numeral_power_cancel_iff
thf(fact_1243_numeral__power__eq__of__int__cancel__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [X: num,N: nat,Y: int] :
( ( ( power_power @ A @ ( numeral_numeral @ A @ X ) @ N )
= ( ring_1_of_int @ A @ Y ) )
= ( ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N )
= Y ) ) ) ).
% numeral_power_eq_of_int_cancel_iff
thf(fact_1244_of__int__le__of__int__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [B3: int,W: nat,X: int] :
( ( ord_less_eq @ A @ ( power_power @ A @ ( ring_1_of_int @ A @ B3 ) @ W ) @ ( ring_1_of_int @ A @ X ) )
= ( ord_less_eq @ int @ ( power_power @ int @ B3 @ W ) @ X ) ) ) ).
% of_int_le_of_int_power_cancel_iff
thf(fact_1245_of__int__power__le__of__int__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: int,B3: int,W: nat] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ X ) @ ( power_power @ A @ ( ring_1_of_int @ A @ B3 ) @ W ) )
= ( ord_less_eq @ int @ X @ ( power_power @ int @ B3 @ W ) ) ) ) ).
% of_int_power_le_of_int_cancel_iff
thf(fact_1246_of__int__power__less__of__int__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: int,B3: int,W: nat] :
( ( ord_less @ A @ ( ring_1_of_int @ A @ X ) @ ( power_power @ A @ ( ring_1_of_int @ A @ B3 ) @ W ) )
= ( ord_less @ int @ X @ ( power_power @ int @ B3 @ W ) ) ) ) ).
% of_int_power_less_of_int_cancel_iff
thf(fact_1247_of__int__less__of__int__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [B3: int,W: nat,X: int] :
( ( ord_less @ A @ ( power_power @ A @ ( ring_1_of_int @ A @ B3 ) @ W ) @ ( ring_1_of_int @ A @ X ) )
= ( ord_less @ int @ ( power_power @ int @ B3 @ W ) @ X ) ) ) ).
% of_int_less_of_int_power_cancel_iff
thf(fact_1248_powr__log__cancel,axiom,
! [A3: real,X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ A3 )
=> ( ( A3
!= ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( powr @ real @ A3 @ ( log @ A3 @ X ) )
= X ) ) ) ) ).
% powr_log_cancel
thf(fact_1249_log__powr__cancel,axiom,
! [A3: real,Y: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ A3 )
=> ( ( A3
!= ( one_one @ real ) )
=> ( ( log @ A3 @ ( powr @ real @ A3 @ Y ) )
= Y ) ) ) ).
% log_powr_cancel
thf(fact_1250_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_1251_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_1252_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_1253_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_1254_even__plus__one__iff,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A3: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) )
= ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) ) ) ) ).
% even_plus_one_iff
thf(fact_1255_even__diff,axiom,
! [A: $tType] :
( ( ring_parity @ A )
=> ! [A3: A,B3: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( minus_minus @ A @ A3 @ B3 ) )
= ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( plus_plus @ A @ A3 @ B3 ) ) ) ) ).
% even_diff
thf(fact_1256_Parity_Oring__1__class_Opower__minus__even,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [N: nat,A3: A] :
( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( power_power @ A @ ( uminus_uminus @ A @ A3 ) @ N )
= ( power_power @ A @ A3 @ N ) ) ) ) ).
% Parity.ring_1_class.power_minus_even
thf(fact_1257_power__minus__odd,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [N: nat,A3: A] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( power_power @ A @ ( uminus_uminus @ A @ A3 ) @ N )
= ( uminus_uminus @ A @ ( power_power @ A @ A3 @ N ) ) ) ) ) ).
% power_minus_odd
thf(fact_1258_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_1259_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_1260_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_1261_powr__numeral,axiom,
! [X: real,N: num] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( powr @ real @ X @ ( numeral_numeral @ real @ N ) )
= ( power_power @ real @ X @ ( numeral_numeral @ nat @ N ) ) ) ) ).
% powr_numeral
thf(fact_1262_even__succ__div__two,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [A3: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% even_succ_div_two
thf(fact_1263_odd__succ__div__two,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [A3: A] :
( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( plus_plus @ A @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) ) ) ) ) ).
% odd_succ_div_two
thf(fact_1264_even__succ__div__2,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A3: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ ( one_one @ A ) @ A3 ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% even_succ_div_2
thf(fact_1265_even__power,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A3: A,N: nat] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( power_power @ A @ A3 @ N ) )
= ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
& ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% even_power
thf(fact_1266_zero__le__power__eq__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,W: num] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ W ) ) )
= ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ W ) )
| ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ W ) )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 ) ) ) ) ) ).
% zero_le_power_eq_numeral
thf(fact_1267_power__less__zero__eq__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,W: num] :
( ( ord_less @ A @ ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ W ) ) @ ( zero_zero @ A ) )
= ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ W ) )
& ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ) ).
% power_less_zero_eq_numeral
thf(fact_1268_power__less__zero__eq,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,N: nat] :
( ( ord_less @ A @ ( power_power @ A @ A3 @ N ) @ ( zero_zero @ A ) )
= ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
& ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ) ).
% power_less_zero_eq
thf(fact_1269_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_1270_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_1271_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_1272_ceiling__less__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,V2: num] :
( ( ord_less @ int @ ( archimedean_ceiling @ A @ X ) @ ( numeral_numeral @ int @ V2 ) )
= ( ord_less_eq @ A @ X @ ( minus_minus @ A @ ( numeral_numeral @ A @ V2 ) @ ( one_one @ A ) ) ) ) ) ).
% ceiling_less_numeral
thf(fact_1273_numeral__le__ceiling,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V2: num,X: A] :
( ( ord_less_eq @ int @ ( numeral_numeral @ int @ V2 ) @ ( archimedean_ceiling @ A @ X ) )
= ( ord_less @ A @ ( minus_minus @ A @ ( numeral_numeral @ A @ V2 ) @ ( one_one @ A ) ) @ X ) ) ) ).
% numeral_le_ceiling
thf(fact_1274_of__int__le__numeral__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: int,X: num,N: nat] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ A3 ) @ ( power_power @ A @ ( numeral_numeral @ A @ X ) @ N ) )
= ( ord_less_eq @ int @ A3 @ ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N ) ) ) ) ).
% of_int_le_numeral_power_cancel_iff
thf(fact_1275_numeral__power__le__of__int__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: num,N: nat,A3: int] :
( ( ord_less_eq @ A @ ( power_power @ A @ ( numeral_numeral @ A @ X ) @ N ) @ ( ring_1_of_int @ A @ A3 ) )
= ( ord_less_eq @ int @ ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N ) @ A3 ) ) ) ).
% numeral_power_le_of_int_cancel_iff
thf(fact_1276_of__int__less__numeral__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: int,X: num,N: nat] :
( ( ord_less @ A @ ( ring_1_of_int @ A @ A3 ) @ ( power_power @ A @ ( numeral_numeral @ A @ X ) @ N ) )
= ( ord_less @ int @ A3 @ ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N ) ) ) ) ).
% of_int_less_numeral_power_cancel_iff
thf(fact_1277_numeral__power__less__of__int__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: num,N: nat,A3: int] :
( ( ord_less @ A @ ( power_power @ A @ ( numeral_numeral @ A @ X ) @ N ) @ ( ring_1_of_int @ A @ A3 ) )
= ( ord_less @ int @ ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N ) @ A3 ) ) ) ).
% numeral_power_less_of_int_cancel_iff
thf(fact_1278_of__int__eq__neg__numeral__power__cancel__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [Y: int,X: num,N: nat] :
( ( ( ring_1_of_int @ A @ Y )
= ( power_power @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ X ) ) @ N ) )
= ( Y
= ( power_power @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ X ) ) @ N ) ) ) ) ).
% of_int_eq_neg_numeral_power_cancel_iff
thf(fact_1279_neg__numeral__power__eq__of__int__cancel__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [X: num,N: nat,Y: int] :
( ( ( power_power @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ X ) ) @ N )
= ( ring_1_of_int @ A @ Y ) )
= ( ( power_power @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ X ) ) @ N )
= Y ) ) ) ).
% neg_numeral_power_eq_of_int_cancel_iff
thf(fact_1280_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_1281_zero__less__power__eq__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,W: num] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ W ) ) )
= ( ( ( numeral_numeral @ nat @ W )
= ( zero_zero @ nat ) )
| ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ W ) )
& ( A3
!= ( zero_zero @ A ) ) )
| ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ W ) )
& ( ord_less @ A @ ( zero_zero @ A ) @ A3 ) ) ) ) ) ).
% zero_less_power_eq_numeral
thf(fact_1282_of__int__le__neg__numeral__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: int,X: num,N: nat] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ A3 ) @ ( power_power @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ X ) ) @ N ) )
= ( ord_less_eq @ int @ A3 @ ( power_power @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ X ) ) @ N ) ) ) ) ).
% of_int_le_neg_numeral_power_cancel_iff
thf(fact_1283_neg__numeral__power__le__of__int__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: num,N: nat,A3: int] :
( ( ord_less_eq @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ X ) ) @ N ) @ ( ring_1_of_int @ A @ A3 ) )
= ( ord_less_eq @ int @ ( power_power @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ X ) ) @ N ) @ A3 ) ) ) ).
% neg_numeral_power_le_of_int_cancel_iff
thf(fact_1284_of__int__less__neg__numeral__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: int,X: num,N: nat] :
( ( ord_less @ A @ ( ring_1_of_int @ A @ A3 ) @ ( power_power @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ X ) ) @ N ) )
= ( ord_less @ int @ A3 @ ( power_power @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ X ) ) @ N ) ) ) ) ).
% of_int_less_neg_numeral_power_cancel_iff
thf(fact_1285_even__succ__div__exp,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A3: A,N: nat] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ ( one_one @ A ) @ A3 ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) )
= ( divide_divide @ A @ A3 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) ) ) ) ) ).
% even_succ_div_exp
thf(fact_1286_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_1287_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_1288_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_1289_word__sub__wi,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( minus_minus @ ( word @ A ) )
= ( ^ [A5: word @ A,B5: word @ A] : ( ring_1_of_int @ ( word @ A ) @ ( minus_minus @ int @ ( semiring_1_unsigned @ A @ int @ A5 ) @ ( semiring_1_unsigned @ A @ int @ B5 ) ) ) ) ) ) ).
% word_sub_wi
thf(fact_1290_dvd__neg__div,axiom,
! [A: $tType] :
( ( idom_divide @ A )
=> ! [B3: A,A3: A] :
( ( dvd_dvd @ A @ B3 @ A3 )
=> ( ( divide_divide @ A @ ( uminus_uminus @ A @ A3 ) @ B3 )
= ( uminus_uminus @ A @ ( divide_divide @ A @ A3 @ B3 ) ) ) ) ) ).
% dvd_neg_div
thf(fact_1291_dvd__div__neg,axiom,
! [A: $tType] :
( ( idom_divide @ A )
=> ! [B3: A,A3: A] :
( ( dvd_dvd @ A @ B3 @ A3 )
=> ( ( divide_divide @ A @ A3 @ ( uminus_uminus @ A @ B3 ) )
= ( uminus_uminus @ A @ ( divide_divide @ A @ A3 @ B3 ) ) ) ) ) ).
% dvd_div_neg
thf(fact_1292_floor__divide__of__int__eq,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [K: int,L: int] :
( ( archim6421214686448440834_floor @ A @ ( divide_divide @ A @ ( ring_1_of_int @ A @ K ) @ ( ring_1_of_int @ A @ L ) ) )
= ( divide_divide @ int @ K @ L ) ) ) ).
% floor_divide_of_int_eq
thf(fact_1293_size__neq__size__imp__neq,axiom,
! [A: $tType] :
( ( size @ A )
=> ! [X: A,Y: A] :
( ( ( size_size @ A @ X )
!= ( size_size @ A @ Y ) )
=> ( X != Y ) ) ) ).
% size_neq_size_imp_neq
thf(fact_1294_div__div__div__same,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [D2: A,B3: A,A3: A] :
( ( dvd_dvd @ A @ D2 @ B3 )
=> ( ( dvd_dvd @ A @ B3 @ A3 )
=> ( ( divide_divide @ A @ ( divide_divide @ A @ A3 @ D2 ) @ ( divide_divide @ A @ B3 @ D2 ) )
= ( divide_divide @ A @ A3 @ B3 ) ) ) ) ) ).
% div_div_div_same
thf(fact_1295_dvd__div__eq__cancel,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [A3: A,C2: A,B3: A] :
( ( ( divide_divide @ A @ A3 @ C2 )
= ( divide_divide @ A @ B3 @ C2 ) )
=> ( ( dvd_dvd @ A @ C2 @ A3 )
=> ( ( dvd_dvd @ A @ C2 @ B3 )
=> ( A3 = B3 ) ) ) ) ) ).
% dvd_div_eq_cancel
thf(fact_1296_dvd__div__eq__iff,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [C2: A,A3: A,B3: A] :
( ( dvd_dvd @ A @ C2 @ A3 )
=> ( ( dvd_dvd @ A @ C2 @ B3 )
=> ( ( ( divide_divide @ A @ A3 @ C2 )
= ( divide_divide @ A @ B3 @ C2 ) )
= ( A3 = B3 ) ) ) ) ) ).
% dvd_div_eq_iff
thf(fact_1297_dvd__trans,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: A,B3: A,C2: A] :
( ( dvd_dvd @ A @ A3 @ B3 )
=> ( ( dvd_dvd @ A @ B3 @ C2 )
=> ( dvd_dvd @ A @ A3 @ C2 ) ) ) ) ).
% dvd_trans
thf(fact_1298_dvd__refl,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: A] : ( dvd_dvd @ A @ A3 @ A3 ) ) ).
% dvd_refl
thf(fact_1299_dvd__diff,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [X: A,Y: A,Z: A] :
( ( dvd_dvd @ A @ X @ Y )
=> ( ( dvd_dvd @ A @ X @ Z )
=> ( dvd_dvd @ A @ X @ ( minus_minus @ A @ Y @ Z ) ) ) ) ) ).
% dvd_diff
thf(fact_1300_fun__diff__def,axiom,
! [B: $tType,A: $tType] :
( ( minus @ B )
=> ( ( minus_minus @ ( A > B ) )
= ( ^ [A4: A > B,B4: A > B,X2: A] : ( minus_minus @ B @ ( A4 @ X2 ) @ ( B4 @ X2 ) ) ) ) ) ).
% fun_diff_def
thf(fact_1301_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_1302_dvd__antisym,axiom,
! [M: nat,N: nat] :
( ( dvd_dvd @ nat @ M @ N )
=> ( ( dvd_dvd @ nat @ N @ M )
=> ( M = N ) ) ) ).
% dvd_antisym
thf(fact_1303_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A: $tType] :
( ( cancel2418104881723323429up_add @ A )
=> ! [A3: A,C2: A,B3: A] :
( ( minus_minus @ A @ ( minus_minus @ A @ A3 @ C2 ) @ B3 )
= ( minus_minus @ A @ ( minus_minus @ A @ A3 @ B3 ) @ C2 ) ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_1304_diff__eq__diff__eq,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B3: A,C2: A,D2: A] :
( ( ( minus_minus @ A @ A3 @ B3 )
= ( minus_minus @ A @ C2 @ D2 ) )
=> ( ( A3 = B3 )
= ( C2 = D2 ) ) ) ) ).
% diff_eq_diff_eq
thf(fact_1305_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_1306_div__power,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B3: A,A3: A,N: nat] :
( ( dvd_dvd @ A @ B3 @ A3 )
=> ( ( power_power @ A @ ( divide_divide @ A @ A3 @ B3 ) @ N )
= ( divide_divide @ A @ ( power_power @ A @ A3 @ N ) @ ( power_power @ A @ B3 @ N ) ) ) ) ) ).
% div_power
thf(fact_1307_dvd__div__unit__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B3: A,A3: A,C2: A] :
( ( dvd_dvd @ A @ B3 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ A3 @ ( divide_divide @ A @ C2 @ B3 ) )
= ( dvd_dvd @ A @ A3 @ C2 ) ) ) ) ).
% dvd_div_unit_iff
thf(fact_1308_div__unit__dvd__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B3: A,A3: A,C2: A] :
( ( dvd_dvd @ A @ B3 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ ( divide_divide @ A @ A3 @ B3 ) @ C2 )
= ( dvd_dvd @ A @ A3 @ C2 ) ) ) ) ).
% div_unit_dvd_iff
thf(fact_1309_unit__div__cancel,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B3: A,C2: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ( ( divide_divide @ A @ B3 @ A3 )
= ( divide_divide @ A @ C2 @ A3 ) )
= ( B3 = C2 ) ) ) ) ).
% unit_div_cancel
thf(fact_1310_dvd__div__eq__0__iff,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [B3: A,A3: A] :
( ( dvd_dvd @ A @ B3 @ A3 )
=> ( ( ( divide_divide @ A @ A3 @ B3 )
= ( zero_zero @ A ) )
= ( A3
= ( zero_zero @ A ) ) ) ) ) ).
% dvd_div_eq_0_iff
thf(fact_1311_ceiling__divide__eq__div,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [A3: int,B3: int] :
( ( archimedean_ceiling @ A @ ( divide_divide @ A @ ( ring_1_of_int @ A @ A3 ) @ ( ring_1_of_int @ A @ B3 ) ) )
= ( uminus_uminus @ int @ ( divide_divide @ int @ ( uminus_uminus @ int @ A3 ) @ B3 ) ) ) ) ).
% ceiling_divide_eq_div
thf(fact_1312_unit__div__eq__0__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B3: A,A3: A] :
( ( dvd_dvd @ A @ B3 @ ( one_one @ A ) )
=> ( ( ( divide_divide @ A @ A3 @ B3 )
= ( zero_zero @ A ) )
= ( A3
= ( zero_zero @ A ) ) ) ) ) ).
% unit_div_eq_0_iff
thf(fact_1313_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_1314_uint__sub__lem,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y: word @ A,X: word @ A] :
( ( ord_less_eq @ int @ ( semiring_1_unsigned @ A @ int @ Y ) @ ( semiring_1_unsigned @ A @ int @ X ) )
= ( ( semiring_1_unsigned @ A @ int @ ( minus_minus @ ( word @ A ) @ X @ Y ) )
= ( minus_minus @ int @ ( semiring_1_unsigned @ A @ int @ X ) @ ( semiring_1_unsigned @ A @ int @ Y ) ) ) ) ) ).
% uint_sub_lem
thf(fact_1315_uint__sub__ge,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] : ( ord_less_eq @ int @ ( minus_minus @ int @ ( semiring_1_unsigned @ A @ int @ X ) @ ( semiring_1_unsigned @ A @ int @ Y ) ) @ ( semiring_1_unsigned @ A @ int @ ( minus_minus @ ( word @ A ) @ X @ Y ) ) ) ) ).
% uint_sub_ge
thf(fact_1316_uint__minus__simple__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ X @ Y ) @ X )
= ( ( semiring_1_unsigned @ A @ int @ ( minus_minus @ ( word @ A ) @ X @ Y ) )
= ( minus_minus @ int @ ( semiring_1_unsigned @ A @ int @ X ) @ ( semiring_1_unsigned @ A @ int @ Y ) ) ) ) ) ).
% uint_minus_simple_iff
thf(fact_1317_uint__minus__simple__alt,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( ord_less_eq @ ( word @ A ) )
= ( ^ [Y2: word @ A,X2: word @ A] :
( ( semiring_1_unsigned @ A @ int @ ( minus_minus @ ( word @ A ) @ X2 @ Y2 ) )
= ( minus_minus @ int @ ( semiring_1_unsigned @ A @ int @ X2 ) @ ( semiring_1_unsigned @ A @ int @ Y2 ) ) ) ) ) ) ).
% uint_minus_simple_alt
thf(fact_1318_size__0__same_H,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A,V2: word @ A] :
( ( ( size_size @ ( word @ A ) @ W )
= ( zero_zero @ nat ) )
=> ( W = V2 ) ) ) ).
% size_0_same'
thf(fact_1319_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_1320_size__0__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A,V2: word @ A] :
( ( ( size_size @ ( word @ A ) @ W )
= ( zero_zero @ nat ) )
=> ( V2 = W ) ) ) ).
% size_0_eq
thf(fact_1321_unat__sub,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [B3: word @ A,A3: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ B3 @ A3 )
=> ( ( semiring_1_unsigned @ A @ nat @ ( minus_minus @ ( word @ A ) @ A3 @ B3 ) )
= ( minus_minus @ nat @ ( semiring_1_unsigned @ A @ nat @ A3 ) @ ( semiring_1_unsigned @ A @ nat @ B3 ) ) ) ) ) ).
% unat_sub
thf(fact_1322_ex__le__of__int,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X: A] :
? [Z3: int] : ( ord_less_eq @ A @ X @ ( ring_1_of_int @ A @ Z3 ) ) ) ).
% ex_le_of_int
thf(fact_1323_ex__of__int__less,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X: A] :
? [Z3: int] : ( ord_less @ A @ ( ring_1_of_int @ A @ Z3 ) @ X ) ) ).
% ex_of_int_less
thf(fact_1324_ex__less__of__int,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X: A] :
? [Z3: int] : ( ord_less @ A @ X @ ( ring_1_of_int @ A @ Z3 ) ) ) ).
% ex_less_of_int
thf(fact_1325_dvd__field__iff,axiom,
! [A: $tType] :
( ( field @ A )
=> ( ( dvd_dvd @ A )
= ( ^ [A5: A,B5: A] :
( ( A5
= ( zero_zero @ A ) )
=> ( B5
= ( zero_zero @ A ) ) ) ) ) ) ).
% dvd_field_iff
thf(fact_1326_dvd__0__left,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A3: A] :
( ( dvd_dvd @ A @ ( zero_zero @ A ) @ A3 )
=> ( A3
= ( zero_zero @ A ) ) ) ) ).
% dvd_0_left
thf(fact_1327_eq__iff__diff__eq__0,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ( ( ^ [Y3: A,Z2: A] : Y3 = Z2 )
= ( ^ [A5: A,B5: A] :
( ( minus_minus @ A @ A5 @ B5 )
= ( zero_zero @ A ) ) ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_1328_diff__eq__diff__less__eq,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B3: A,C2: A,D2: A] :
( ( ( minus_minus @ A @ A3 @ B3 )
= ( minus_minus @ A @ C2 @ D2 ) )
=> ( ( ord_less_eq @ A @ A3 @ B3 )
= ( ord_less_eq @ A @ C2 @ D2 ) ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_1329_diff__right__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B3: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ord_less_eq @ A @ ( minus_minus @ A @ A3 @ C2 ) @ ( minus_minus @ A @ B3 @ C2 ) ) ) ) ).
% diff_right_mono
thf(fact_1330_diff__left__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [B3: A,A3: A,C2: A] :
( ( ord_less_eq @ A @ B3 @ A3 )
=> ( ord_less_eq @ A @ ( minus_minus @ A @ C2 @ A3 ) @ ( minus_minus @ A @ C2 @ B3 ) ) ) ) ).
% diff_left_mono
thf(fact_1331_diff__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B3: A,D2: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ A @ D2 @ C2 )
=> ( ord_less_eq @ A @ ( minus_minus @ A @ A3 @ C2 ) @ ( minus_minus @ A @ B3 @ D2 ) ) ) ) ) ).
% diff_mono
thf(fact_1332_dvd__add__right__iff,axiom,
! [A: $tType] :
( ( comm_s4317794764714335236cancel @ A )
=> ! [A3: A,B3: A,C2: A] :
( ( dvd_dvd @ A @ A3 @ B3 )
=> ( ( dvd_dvd @ A @ A3 @ ( plus_plus @ A @ B3 @ C2 ) )
= ( dvd_dvd @ A @ A3 @ C2 ) ) ) ) ).
% dvd_add_right_iff
thf(fact_1333_dvd__add__left__iff,axiom,
! [A: $tType] :
( ( comm_s4317794764714335236cancel @ A )
=> ! [A3: A,C2: A,B3: A] :
( ( dvd_dvd @ A @ A3 @ C2 )
=> ( ( dvd_dvd @ A @ A3 @ ( plus_plus @ A @ B3 @ C2 ) )
= ( dvd_dvd @ A @ A3 @ B3 ) ) ) ) ).
% dvd_add_left_iff
thf(fact_1334_dvd__add,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A3: A,B3: A,C2: A] :
( ( dvd_dvd @ A @ A3 @ B3 )
=> ( ( dvd_dvd @ A @ A3 @ C2 )
=> ( dvd_dvd @ A @ A3 @ ( plus_plus @ A @ B3 @ C2 ) ) ) ) ) ).
% dvd_add
thf(fact_1335_one__dvd,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: A] : ( dvd_dvd @ A @ ( one_one @ A ) @ A3 ) ) ).
% one_dvd
thf(fact_1336_unit__imp__dvd,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B3: A,A3: A] :
( ( dvd_dvd @ A @ B3 @ ( one_one @ A ) )
=> ( dvd_dvd @ A @ B3 @ A3 ) ) ) ).
% unit_imp_dvd
thf(fact_1337_dvd__unit__imp__unit,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B3: A] :
( ( dvd_dvd @ A @ A3 @ B3 )
=> ( ( dvd_dvd @ A @ B3 @ ( one_one @ A ) )
=> ( dvd_dvd @ A @ A3 @ ( one_one @ A ) ) ) ) ) ).
% dvd_unit_imp_unit
thf(fact_1338_diff__strict__right__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B3: A,C2: A] :
( ( ord_less @ A @ A3 @ B3 )
=> ( ord_less @ A @ ( minus_minus @ A @ A3 @ C2 ) @ ( minus_minus @ A @ B3 @ C2 ) ) ) ) ).
% diff_strict_right_mono
thf(fact_1339_diff__strict__left__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [B3: A,A3: A,C2: A] :
( ( ord_less @ A @ B3 @ A3 )
=> ( ord_less @ A @ ( minus_minus @ A @ C2 @ A3 ) @ ( minus_minus @ A @ C2 @ B3 ) ) ) ) ).
% diff_strict_left_mono
thf(fact_1340_diff__eq__diff__less,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B3: A,C2: A,D2: A] :
( ( ( minus_minus @ A @ A3 @ B3 )
= ( minus_minus @ A @ C2 @ D2 ) )
=> ( ( ord_less @ A @ A3 @ B3 )
= ( ord_less @ A @ C2 @ D2 ) ) ) ) ).
% diff_eq_diff_less
thf(fact_1341_diff__strict__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B3: A,D2: A,C2: A] :
( ( ord_less @ A @ A3 @ B3 )
=> ( ( ord_less @ A @ D2 @ C2 )
=> ( ord_less @ A @ ( minus_minus @ A @ A3 @ C2 ) @ ( minus_minus @ A @ B3 @ D2 ) ) ) ) ) ).
% diff_strict_mono
thf(fact_1342_diff__diff__eq,axiom,
! [A: $tType] :
( ( cancel2418104881723323429up_add @ A )
=> ! [A3: A,B3: A,C2: A] :
( ( minus_minus @ A @ ( minus_minus @ A @ A3 @ B3 ) @ C2 )
= ( minus_minus @ A @ A3 @ ( plus_plus @ A @ B3 @ C2 ) ) ) ) ).
% diff_diff_eq
thf(fact_1343_add__diff__add,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A3: A,C2: A,B3: A,D2: A] :
( ( minus_minus @ A @ ( plus_plus @ A @ A3 @ C2 ) @ ( plus_plus @ A @ B3 @ D2 ) )
= ( plus_plus @ A @ ( minus_minus @ A @ A3 @ B3 ) @ ( minus_minus @ A @ C2 @ D2 ) ) ) ) ).
% add_diff_add
thf(fact_1344_add__implies__diff,axiom,
! [A: $tType] :
( ( cancel1802427076303600483id_add @ A )
=> ! [C2: A,B3: A,A3: A] :
( ( ( plus_plus @ A @ C2 @ B3 )
= A3 )
=> ( C2
= ( minus_minus @ A @ A3 @ B3 ) ) ) ) ).
% add_implies_diff
thf(fact_1345_diff__add__eq__diff__diff__swap,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B3: A,C2: A] :
( ( minus_minus @ A @ A3 @ ( plus_plus @ A @ B3 @ C2 ) )
= ( minus_minus @ A @ ( minus_minus @ A @ A3 @ C2 ) @ B3 ) ) ) ).
% diff_add_eq_diff_diff_swap
thf(fact_1346_diff__add__eq,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A3: A,B3: A,C2: A] :
( ( plus_plus @ A @ ( minus_minus @ A @ A3 @ B3 ) @ C2 )
= ( minus_minus @ A @ ( plus_plus @ A @ A3 @ C2 ) @ B3 ) ) ) ).
% diff_add_eq
thf(fact_1347_diff__diff__eq2,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B3: A,C2: A] :
( ( minus_minus @ A @ A3 @ ( minus_minus @ A @ B3 @ C2 ) )
= ( minus_minus @ A @ ( plus_plus @ A @ A3 @ C2 ) @ B3 ) ) ) ).
% diff_diff_eq2
thf(fact_1348_add__diff__eq,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B3: A,C2: A] :
( ( plus_plus @ A @ A3 @ ( minus_minus @ A @ B3 @ C2 ) )
= ( minus_minus @ A @ ( plus_plus @ A @ A3 @ B3 ) @ C2 ) ) ) ).
% add_diff_eq
thf(fact_1349_eq__diff__eq,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,C2: A,B3: A] :
( ( A3
= ( minus_minus @ A @ C2 @ B3 ) )
= ( ( plus_plus @ A @ A3 @ B3 )
= C2 ) ) ) ).
% eq_diff_eq
thf(fact_1350_diff__eq__eq,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B3: A,C2: A] :
( ( ( minus_minus @ A @ A3 @ B3 )
= C2 )
= ( A3
= ( plus_plus @ A @ C2 @ B3 ) ) ) ) ).
% diff_eq_eq
thf(fact_1351_group__cancel_Osub1,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A2: A,K: A,A3: A,B3: A] :
( ( A2
= ( plus_plus @ A @ K @ A3 ) )
=> ( ( minus_minus @ A @ A2 @ B3 )
= ( plus_plus @ A @ K @ ( minus_minus @ A @ A3 @ B3 ) ) ) ) ) ).
% group_cancel.sub1
thf(fact_1352_dvd__power__same,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [X: A,Y: A,N: nat] :
( ( dvd_dvd @ A @ X @ Y )
=> ( dvd_dvd @ A @ ( power_power @ A @ X @ N ) @ ( power_power @ A @ Y @ N ) ) ) ) ).
% dvd_power_same
thf(fact_1353_minus__diff__minus,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A3: A,B3: A] :
( ( minus_minus @ A @ ( uminus_uminus @ A @ A3 ) @ ( uminus_uminus @ A @ B3 ) )
= ( uminus_uminus @ A @ ( minus_minus @ A @ A3 @ B3 ) ) ) ) ).
% minus_diff_minus
thf(fact_1354_minus__diff__commute,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [B3: A,A3: A] :
( ( minus_minus @ A @ ( uminus_uminus @ A @ B3 ) @ A3 )
= ( minus_minus @ A @ ( uminus_uminus @ A @ A3 ) @ B3 ) ) ) ).
% minus_diff_commute
thf(fact_1355_power__divide,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A3: A,B3: A,N: nat] :
( ( power_power @ A @ ( divide_divide @ A @ A3 @ B3 ) @ N )
= ( divide_divide @ A @ ( power_power @ A @ A3 @ N ) @ ( power_power @ A @ B3 @ N ) ) ) ) ).
% power_divide
thf(fact_1356_powr__minus__divide,axiom,
! [A: $tType] :
( ( ( real_V3459762299906320749_field @ A )
& ( ln @ A ) )
=> ! [X: A,A3: A] :
( ( powr @ A @ X @ ( uminus_uminus @ A @ A3 ) )
= ( divide_divide @ A @ ( one_one @ A ) @ ( powr @ A @ X @ A3 ) ) ) ) ).
% powr_minus_divide
thf(fact_1357_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_1358_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_1359_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus @ nat @ M @ ( zero_zero @ nat ) )
= M ) ).
% minus_nat.diff_0
thf(fact_1360_Diff__mono,axiom,
! [A: $tType,A2: set @ A,C3: set @ A,D3: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ C3 )
=> ( ( ord_less_eq @ ( set @ A ) @ D3 @ B2 )
=> ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A2 @ B2 ) @ ( minus_minus @ ( set @ A ) @ C3 @ D3 ) ) ) ) ).
% Diff_mono
thf(fact_1361_Diff__subset,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A2 @ B2 ) @ A2 ) ).
% Diff_subset
thf(fact_1362_double__diff,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,C3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ C3 )
=> ( ( minus_minus @ ( set @ A ) @ B2 @ ( minus_minus @ ( set @ A ) @ C3 @ A2 ) )
= A2 ) ) ) ).
% double_diff
thf(fact_1363_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_1364_diff__less__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less @ nat @ M @ N )
=> ( ( ord_less @ nat @ M @ L )
=> ( ord_less @ nat @ ( minus_minus @ nat @ L @ N ) @ ( minus_minus @ nat @ L @ M ) ) ) ) ).
% diff_less_mono2
thf(fact_1365_diff__diff__less,axiom,
! [I: nat,M: nat,N: nat] :
( ( ord_less @ nat @ I @ ( minus_minus @ nat @ M @ ( minus_minus @ nat @ M @ N ) ) )
= ( ( ord_less @ nat @ I @ M )
& ( ord_less @ nat @ I @ N ) ) ) ).
% diff_diff_less
thf(fact_1366_insert__Diff__if,axiom,
! [A: $tType,X: A,B2: set @ A,A2: set @ A] :
( ( ( member @ A @ X @ B2 )
=> ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X @ A2 ) @ B2 )
= ( minus_minus @ ( set @ A ) @ A2 @ B2 ) ) )
& ( ~ ( member @ A @ X @ B2 )
=> ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X @ A2 ) @ B2 )
= ( insert @ A @ X @ ( minus_minus @ ( set @ A ) @ A2 @ B2 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_1367_minus__int__code_I1_J,axiom,
! [K: int] :
( ( minus_minus @ int @ K @ ( zero_zero @ int ) )
= K ) ).
% minus_int_code(1)
thf(fact_1368_word__of__int__uint,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A] :
( ( ring_1_of_int @ ( word @ A ) @ ( semiring_1_unsigned @ A @ int @ W ) )
= W ) ) ).
% word_of_int_uint
thf(fact_1369_More__Word_Oof__int__uint,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] :
( ( ring_1_of_int @ ( word @ A ) @ ( semiring_1_unsigned @ A @ int @ X ) )
= X ) ) ).
% More_Word.of_int_uint
thf(fact_1370_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_1371_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_1372_diff__add__inverse,axiom,
! [N: nat,M: nat] :
( ( minus_minus @ nat @ ( plus_plus @ nat @ N @ M ) @ N )
= M ) ).
% diff_add_inverse
thf(fact_1373_diff__add__inverse2,axiom,
! [M: nat,N: nat] :
( ( minus_minus @ nat @ ( plus_plus @ nat @ M @ N ) @ N )
= M ) ).
% diff_add_inverse2
thf(fact_1374_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_1375_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_1376_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_1377_diff__le__mono,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ord_less_eq @ nat @ ( minus_minus @ nat @ M @ L ) @ ( minus_minus @ nat @ N @ L ) ) ) ).
% diff_le_mono
thf(fact_1378_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq @ nat @ ( minus_minus @ nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_1379_le__diff__iff_H,axiom,
! [A3: nat,C2: nat,B3: nat] :
( ( ord_less_eq @ nat @ A3 @ C2 )
=> ( ( ord_less_eq @ nat @ B3 @ C2 )
=> ( ( ord_less_eq @ nat @ ( minus_minus @ nat @ C2 @ A3 ) @ ( minus_minus @ nat @ C2 @ B3 ) )
= ( ord_less_eq @ nat @ B3 @ A3 ) ) ) ) ).
% le_diff_iff'
thf(fact_1380_diff__le__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ord_less_eq @ nat @ ( minus_minus @ nat @ L @ N ) @ ( minus_minus @ nat @ L @ M ) ) ) ).
% diff_le_mono2
thf(fact_1381_Un__Diff,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,C3: set @ A] :
( ( minus_minus @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) @ C3 )
= ( sup_sup @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A2 @ C3 ) @ ( minus_minus @ ( set @ A ) @ B2 @ C3 ) ) ) ).
% Un_Diff
thf(fact_1382_set__diff__diff__left,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,C3: set @ A] :
( ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A2 @ B2 ) @ C3 )
= ( minus_minus @ ( set @ A ) @ A2 @ ( sup_sup @ ( set @ A ) @ B2 @ C3 ) ) ) ).
% set_diff_diff_left
thf(fact_1383_int__diff__cases,axiom,
! [Z: int] :
~ ! [M4: nat,N2: nat] :
( Z
!= ( minus_minus @ int @ ( semiring_1_of_nat @ int @ M4 ) @ ( semiring_1_of_nat @ int @ N2 ) ) ) ).
% int_diff_cases
thf(fact_1384_word__sub__le,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y: word @ A,X: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ Y @ X )
=> ( ord_less_eq @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ X @ Y ) @ X ) ) ) ).
% word_sub_le
thf(fact_1385_word__sub__le__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ X @ Y ) @ X )
= ( ord_less_eq @ ( word @ A ) @ Y @ X ) ) ) ).
% word_sub_le_iff
thf(fact_1386_word__le__minus__mono,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A,C2: word @ A,D2: word @ A,B3: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ A3 @ C2 )
=> ( ( ord_less_eq @ ( word @ A ) @ D2 @ B3 )
=> ( ( ord_less_eq @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ A3 @ B3 ) @ A3 )
=> ( ( ord_less_eq @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ C2 @ D2 ) @ C2 )
=> ( ord_less_eq @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ A3 @ B3 ) @ ( minus_minus @ ( word @ A ) @ C2 @ D2 ) ) ) ) ) ) ) ).
% word_le_minus_mono
thf(fact_1387_word__le__minus__cancel,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y: word @ A,X: word @ A,Z: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ Y @ X ) @ ( minus_minus @ ( word @ A ) @ Z @ X ) )
=> ( ( ord_less_eq @ ( word @ A ) @ X @ Z )
=> ( ord_less_eq @ ( word @ A ) @ Y @ Z ) ) ) ) ).
% word_le_minus_cancel
thf(fact_1388_word__le__minus__mono__left,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y: word @ A,Z: word @ A,X: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ Y @ Z )
=> ( ( ord_less_eq @ ( word @ A ) @ X @ Y )
=> ( ord_less_eq @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ Y @ X ) @ ( minus_minus @ ( word @ A ) @ Z @ X ) ) ) ) ) ).
% word_le_minus_mono_left
thf(fact_1389_word__le__imp__diff__le,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [K: word @ A,N: word @ A,M: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ K @ N )
=> ( ( ord_less_eq @ ( word @ A ) @ N @ M )
=> ( ord_less_eq @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ N @ K ) @ M ) ) ) ) ).
% word_le_imp_diff_le
thf(fact_1390_word__le__minus__mono__right,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Z: word @ A,Y: word @ A,X: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ Z @ Y )
=> ( ( ord_less_eq @ ( word @ A ) @ Y @ X )
=> ( ( ord_less_eq @ ( word @ A ) @ Z @ X )
=> ( ord_less_eq @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ X @ Y ) @ ( minus_minus @ ( word @ A ) @ X @ Z ) ) ) ) ) ) ).
% word_le_minus_mono_right
thf(fact_1391_sub__wrap__lt,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Z: word @ A] :
( ( ord_less @ ( word @ A ) @ X @ ( minus_minus @ ( word @ A ) @ X @ Z ) )
= ( ord_less @ ( word @ A ) @ X @ Z ) ) ) ).
% sub_wrap_lt
thf(fact_1392_word__sub__less__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] :
( ( ord_less @ ( word @ A ) @ X @ ( minus_minus @ ( word @ A ) @ X @ Y ) )
= ( ord_less @ ( word @ A ) @ X @ Y ) ) ) ).
% word_sub_less_iff
thf(fact_1393_word__less__minus__mono,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A,C2: word @ A,D2: word @ A,B3: word @ A] :
( ( ord_less @ ( word @ A ) @ A3 @ C2 )
=> ( ( ord_less @ ( word @ A ) @ D2 @ B3 )
=> ( ( ord_less @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ A3 @ B3 ) @ A3 )
=> ( ( ord_less @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ C2 @ D2 ) @ C2 )
=> ( ord_less @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ A3 @ B3 ) @ ( minus_minus @ ( word @ A ) @ C2 @ D2 ) ) ) ) ) ) ) ).
% word_less_minus_mono
thf(fact_1394_psubset__imp__ex__mem,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( ord_less @ ( set @ A ) @ A2 @ B2 )
=> ? [B7: A] : ( member @ A @ B7 @ ( minus_minus @ ( set @ A ) @ B2 @ A2 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_1395_dbl__dec__def,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( ( neg_numeral_dbl_dec @ A )
= ( ^ [X2: A] : ( minus_minus @ A @ ( plus_plus @ A @ X2 @ X2 ) @ ( one_one @ A ) ) ) ) ) ).
% dbl_dec_def
thf(fact_1396_bit__eq__rec,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ( ( ^ [Y3: A,Z2: A] : Y3 = Z2 )
= ( ^ [A5: A,B5: A] :
( ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A5 )
= ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B5 ) )
& ( ( divide_divide @ A @ A5 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( divide_divide @ A @ B5 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% bit_eq_rec
thf(fact_1397_of__int__ceiling__diff__one__le,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [R3: A] : ( ord_less_eq @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ A @ R3 ) ) @ ( one_one @ A ) ) @ R3 ) ) ).
% of_int_ceiling_diff_one_le
thf(fact_1398_power__diff,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [A3: A,N: nat,M: nat] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( ord_less_eq @ nat @ N @ M )
=> ( ( power_power @ A @ A3 @ ( minus_minus @ nat @ M @ N ) )
= ( divide_divide @ A @ ( power_power @ A @ A3 @ M ) @ ( power_power @ A @ A3 @ N ) ) ) ) ) ) ).
% power_diff
thf(fact_1399_subset__divisors__dvd,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [C6: A] : ( dvd_dvd @ A @ C6 @ A3 ) )
@ ( collect @ A
@ ^ [C6: A] : ( dvd_dvd @ A @ C6 @ B3 ) ) )
= ( dvd_dvd @ A @ A3 @ B3 ) ) ) ).
% subset_divisors_dvd
thf(fact_1400_int__ops_I6_J,axiom,
! [A3: nat,B3: nat] :
( ( ( ord_less @ int @ ( semiring_1_of_nat @ int @ A3 ) @ ( semiring_1_of_nat @ int @ B3 ) )
=> ( ( semiring_1_of_nat @ int @ ( minus_minus @ nat @ A3 @ B3 ) )
= ( zero_zero @ int ) ) )
& ( ~ ( ord_less @ int @ ( semiring_1_of_nat @ int @ A3 ) @ ( semiring_1_of_nat @ int @ B3 ) )
=> ( ( semiring_1_of_nat @ int @ ( minus_minus @ nat @ A3 @ B3 ) )
= ( minus_minus @ int @ ( semiring_1_of_nat @ int @ A3 ) @ ( semiring_1_of_nat @ int @ B3 ) ) ) ) ) ).
% int_ops(6)
thf(fact_1401_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_1402_real__of__int__floor__gt__diff__one,axiom,
! [R3: real] : ( ord_less @ real @ ( minus_minus @ real @ R3 @ ( one_one @ real ) ) @ ( ring_1_of_int @ real @ ( archim6421214686448440834_floor @ real @ R3 ) ) ) ).
% real_of_int_floor_gt_diff_one
thf(fact_1403_real__of__int__floor__ge__diff__one,axiom,
! [R3: real] : ( ord_less_eq @ real @ ( minus_minus @ real @ R3 @ ( one_one @ real ) ) @ ( ring_1_of_int @ real @ ( archim6421214686448440834_floor @ real @ R3 ) ) ) ).
% real_of_int_floor_ge_diff_one
thf(fact_1404_strict__subset__divisors__dvd,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ ( set @ A )
@ ( collect @ A
@ ^ [C6: A] : ( dvd_dvd @ A @ C6 @ A3 ) )
@ ( collect @ A
@ ^ [C6: A] : ( dvd_dvd @ A @ C6 @ B3 ) ) )
= ( ( dvd_dvd @ A @ A3 @ B3 )
& ~ ( dvd_dvd @ A @ B3 @ A3 ) ) ) ) ).
% strict_subset_divisors_dvd
thf(fact_1405_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_1406_ceiling__correct,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ord_less @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ A @ X ) ) @ ( one_one @ A ) ) @ X )
& ( ord_less_eq @ A @ X @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ A @ X ) ) ) ) ) ).
% ceiling_correct
thf(fact_1407_ceiling__unique,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z: int,X: A] :
( ( ord_less @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ Z ) @ ( one_one @ A ) ) @ X )
=> ( ( ord_less_eq @ A @ X @ ( ring_1_of_int @ A @ Z ) )
=> ( ( archimedean_ceiling @ A @ X )
= Z ) ) ) ) ).
% ceiling_unique
thf(fact_1408_ceiling__eq__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,A3: int] :
( ( ( archimedean_ceiling @ A @ X )
= A3 )
= ( ( ord_less @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ A3 ) @ ( one_one @ A ) ) @ X )
& ( ord_less_eq @ A @ X @ ( ring_1_of_int @ A @ A3 ) ) ) ) ) ).
% ceiling_eq_iff
thf(fact_1409_ceiling__split,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [P: int > $o,T: A] :
( ( P @ ( archimedean_ceiling @ A @ T ) )
= ( ! [I4: int] :
( ( ( ord_less @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ I4 ) @ ( one_one @ A ) ) @ T )
& ( ord_less_eq @ A @ T @ ( ring_1_of_int @ A @ I4 ) ) )
=> ( P @ I4 ) ) ) ) ) ).
% ceiling_split
thf(fact_1410_ceiling__less__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Z: int] :
( ( ord_less @ int @ ( archimedean_ceiling @ A @ X ) @ Z )
= ( ord_less_eq @ A @ X @ ( minus_minus @ A @ ( ring_1_of_int @ A @ Z ) @ ( one_one @ A ) ) ) ) ) ).
% ceiling_less_iff
thf(fact_1411_le__ceiling__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z: int,X: A] :
( ( ord_less_eq @ int @ Z @ ( archimedean_ceiling @ A @ X ) )
= ( ord_less @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ Z ) @ ( one_one @ A ) ) @ X ) ) ) ).
% le_ceiling_iff
thf(fact_1412_powr__non__neg,axiom,
! [A3: real,X: real] :
~ ( ord_less @ real @ ( powr @ real @ A3 @ X ) @ ( zero_zero @ real ) ) ).
% powr_non_neg
thf(fact_1413_powr__less__mono2__neg,axiom,
! [A3: real,X: real,Y: real] :
( ( ord_less @ real @ A3 @ ( zero_zero @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ X @ Y )
=> ( ord_less @ real @ ( powr @ real @ Y @ A3 ) @ ( powr @ real @ X @ A3 ) ) ) ) ) ).
% powr_less_mono2_neg
thf(fact_1414_powr__mono2,axiom,
! [A3: real,X: real,Y: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ A3 )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ X @ Y )
=> ( ord_less_eq @ real @ ( powr @ real @ X @ A3 ) @ ( powr @ real @ Y @ A3 ) ) ) ) ) ).
% powr_mono2
thf(fact_1415_powr__ge__pzero,axiom,
! [X: real,Y: real] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( powr @ real @ X @ Y ) ) ).
% powr_ge_pzero
thf(fact_1416_powr__less__mono,axiom,
! [A3: real,B3: real,X: real] :
( ( ord_less @ real @ A3 @ B3 )
=> ( ( ord_less @ real @ ( one_one @ real ) @ X )
=> ( ord_less @ real @ ( powr @ real @ X @ A3 ) @ ( powr @ real @ X @ B3 ) ) ) ) ).
% powr_less_mono
thf(fact_1417_powr__less__cancel,axiom,
! [X: real,A3: real,B3: real] :
( ( ord_less @ real @ ( powr @ real @ X @ A3 ) @ ( powr @ real @ X @ B3 ) )
=> ( ( ord_less @ real @ ( one_one @ real ) @ X )
=> ( ord_less @ real @ A3 @ B3 ) ) ) ).
% powr_less_cancel
thf(fact_1418_powr__mono,axiom,
! [A3: real,B3: real,X: real] :
( ( ord_less_eq @ real @ A3 @ B3 )
=> ( ( ord_less_eq @ real @ ( one_one @ real ) @ X )
=> ( ord_less_eq @ real @ ( powr @ real @ X @ A3 ) @ ( powr @ real @ X @ B3 ) ) ) ) ).
% powr_mono
thf(fact_1419_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_1420_le__iff__diff__le__0,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A5: A,B5: A] : ( ord_less_eq @ A @ ( minus_minus @ A @ A5 @ B5 ) @ ( zero_zero @ A ) ) ) ) ) ).
% le_iff_diff_le_0
thf(fact_1421_divide__le__0__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ A3 @ B3 ) @ ( zero_zero @ A ) )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
& ( ord_less_eq @ A @ B3 @ ( zero_zero @ A ) ) )
| ( ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ B3 ) ) ) ) ) ).
% divide_le_0_iff
thf(fact_1422_divide__right__mono,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B3: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ A3 @ C2 ) @ ( divide_divide @ A @ B3 @ C2 ) ) ) ) ) ).
% divide_right_mono
thf(fact_1423_zero__le__divide__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ A3 @ B3 ) )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ B3 ) )
| ( ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) )
& ( ord_less_eq @ A @ B3 @ ( zero_zero @ A ) ) ) ) ) ) ).
% zero_le_divide_iff
thf(fact_1424_divide__nonneg__nonneg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X @ Y ) ) ) ) ) ).
% divide_nonneg_nonneg
thf(fact_1425_divide__nonneg__nonpos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less_eq @ A @ Y @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ X @ Y ) @ ( zero_zero @ A ) ) ) ) ) ).
% divide_nonneg_nonpos
thf(fact_1426_divide__nonpos__nonneg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ X @ Y ) @ ( zero_zero @ A ) ) ) ) ) ).
% divide_nonpos_nonneg
thf(fact_1427_divide__nonpos__nonpos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ Y @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X @ Y ) ) ) ) ) ).
% divide_nonpos_nonpos
thf(fact_1428_divide__right__mono__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B3: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ B3 @ C2 ) @ ( divide_divide @ A @ A3 @ C2 ) ) ) ) ) ).
% divide_right_mono_neg
thf(fact_1429_less__iff__diff__less__0,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ( ( ord_less @ A )
= ( ^ [A5: A,B5: A] : ( ord_less @ A @ ( minus_minus @ A @ A5 @ B5 ) @ ( zero_zero @ A ) ) ) ) ) ).
% less_iff_diff_less_0
thf(fact_1430_divide__strict__right__mono__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B3: A,A3: A,C2: A] :
( ( ord_less @ A @ B3 @ A3 )
=> ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( divide_divide @ A @ A3 @ C2 ) @ ( divide_divide @ A @ B3 @ C2 ) ) ) ) ) ).
% divide_strict_right_mono_neg
thf(fact_1431_divide__strict__right__mono,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B3: A,C2: A] :
( ( ord_less @ A @ A3 @ B3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( divide_divide @ A @ A3 @ C2 ) @ ( divide_divide @ A @ B3 @ C2 ) ) ) ) ) ).
% divide_strict_right_mono
thf(fact_1432_zero__less__divide__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ A3 @ B3 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
& ( ord_less @ A @ ( zero_zero @ A ) @ B3 ) )
| ( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
& ( ord_less @ A @ B3 @ ( zero_zero @ A ) ) ) ) ) ) ).
% zero_less_divide_iff
thf(fact_1433_divide__less__cancel,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,C2: A,B3: A] :
( ( ord_less @ A @ ( divide_divide @ A @ A3 @ C2 ) @ ( divide_divide @ A @ B3 @ C2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ A3 @ B3 ) )
& ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ B3 @ A3 ) )
& ( C2
!= ( zero_zero @ A ) ) ) ) ) ).
% divide_less_cancel
thf(fact_1434_divide__less__0__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ ( divide_divide @ A @ A3 @ B3 ) @ ( zero_zero @ A ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
& ( ord_less @ A @ B3 @ ( zero_zero @ A ) ) )
| ( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
& ( ord_less @ A @ ( zero_zero @ A ) @ B3 ) ) ) ) ) ).
% divide_less_0_iff
thf(fact_1435_divide__pos__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ Y )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X @ Y ) ) ) ) ) ).
% divide_pos_pos
thf(fact_1436_divide__pos__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less @ A @ Y @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( divide_divide @ A @ X @ Y ) @ ( zero_zero @ A ) ) ) ) ) ).
% divide_pos_neg
thf(fact_1437_divide__neg__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ Y )
=> ( ord_less @ A @ ( divide_divide @ A @ X @ Y ) @ ( zero_zero @ A ) ) ) ) ) ).
% divide_neg_pos
thf(fact_1438_divide__neg__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ Y @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X @ Y ) ) ) ) ) ).
% divide_neg_neg
thf(fact_1439_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_1440_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_1441_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_1442_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A3: A,B3: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ( minus_minus @ A @ B3 @ A3 )
= C2 )
= ( B3
= ( plus_plus @ A @ C2 @ A3 ) ) ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
thf(fact_1443_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( plus_plus @ A @ A3 @ ( minus_minus @ A @ B3 @ A3 ) )
= B3 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
thf(fact_1444_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A3: A,B3: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( minus_minus @ A @ C2 @ ( minus_minus @ A @ B3 @ A3 ) )
= ( minus_minus @ A @ ( plus_plus @ A @ C2 @ A3 ) @ B3 ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_diff_right
thf(fact_1445_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A3: A,B3: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( minus_minus @ A @ ( plus_plus @ A @ B3 @ C2 ) @ A3 )
= ( plus_plus @ A @ ( minus_minus @ A @ B3 @ A3 ) @ C2 ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
thf(fact_1446_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A3: A,B3: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( plus_plus @ A @ ( minus_minus @ A @ B3 @ A3 ) @ C2 )
= ( minus_minus @ A @ ( plus_plus @ A @ B3 @ C2 ) @ A3 ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
thf(fact_1447_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A3: A,B3: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( minus_minus @ A @ ( plus_plus @ A @ C2 @ B3 ) @ A3 )
= ( plus_plus @ A @ C2 @ ( minus_minus @ A @ B3 @ A3 ) ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
thf(fact_1448_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A3: A,B3: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( plus_plus @ A @ C2 @ ( minus_minus @ A @ B3 @ A3 ) )
= ( minus_minus @ A @ ( plus_plus @ A @ C2 @ B3 ) @ A3 ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
thf(fact_1449_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A3: A,B3: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ A @ C2 @ ( minus_minus @ A @ B3 @ A3 ) )
= ( ord_less_eq @ A @ ( plus_plus @ A @ C2 @ A3 ) @ B3 ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
thf(fact_1450_le__add__diff,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A3: A,B3: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ord_less_eq @ A @ C2 @ ( minus_minus @ A @ ( plus_plus @ A @ B3 @ C2 ) @ A3 ) ) ) ) ).
% le_add_diff
thf(fact_1451_ordered__cancel__comm__monoid__diff__class_Odiff__add,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( plus_plus @ A @ ( minus_minus @ A @ B3 @ A3 ) @ A3 )
= B3 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add
thf(fact_1452_le__diff__eq,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,C2: A,B3: A] :
( ( ord_less_eq @ A @ A3 @ ( minus_minus @ A @ C2 @ B3 ) )
= ( ord_less_eq @ A @ ( plus_plus @ A @ A3 @ B3 ) @ C2 ) ) ) ).
% le_diff_eq
thf(fact_1453_diff__le__eq,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B3: A,C2: A] :
( ( ord_less_eq @ A @ ( minus_minus @ A @ A3 @ B3 ) @ C2 )
= ( ord_less_eq @ A @ A3 @ ( plus_plus @ A @ C2 @ B3 ) ) ) ) ).
% diff_le_eq
thf(fact_1454_right__inverse__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B3: A,A3: A] :
( ( B3
!= ( zero_zero @ A ) )
=> ( ( ( divide_divide @ A @ A3 @ B3 )
= ( one_one @ A ) )
= ( A3 = B3 ) ) ) ) ).
% right_inverse_eq
thf(fact_1455_diff__less__eq,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B3: A,C2: A] :
( ( ord_less @ A @ ( minus_minus @ A @ A3 @ B3 ) @ C2 )
= ( ord_less @ A @ A3 @ ( plus_plus @ A @ C2 @ B3 ) ) ) ) ).
% diff_less_eq
thf(fact_1456_less__diff__eq,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,C2: A,B3: A] :
( ( ord_less @ A @ A3 @ ( minus_minus @ A @ C2 @ B3 ) )
= ( ord_less @ A @ ( plus_plus @ A @ A3 @ B3 ) @ C2 ) ) ) ).
% less_diff_eq
thf(fact_1457_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,B3: A] :
( ~ ( ord_less @ A @ A3 @ B3 )
=> ( ( plus_plus @ A @ B3 @ ( minus_minus @ A @ A3 @ B3 ) )
= A3 ) ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_1458_divide__numeral__1,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A3: A] :
( ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ one2 ) )
= A3 ) ) ).
% divide_numeral_1
thf(fact_1459_diff__shunt__var,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,Y: A] :
( ( ( minus_minus @ A @ X @ Y )
= ( bot_bot @ A ) )
= ( ord_less_eq @ A @ X @ Y ) ) ) ).
% diff_shunt_var
thf(fact_1460_nonzero__minus__divide__right,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B3: A,A3: A] :
( ( B3
!= ( zero_zero @ A ) )
=> ( ( uminus_uminus @ A @ ( divide_divide @ A @ A3 @ B3 ) )
= ( divide_divide @ A @ A3 @ ( uminus_uminus @ A @ B3 ) ) ) ) ) ).
% nonzero_minus_divide_right
thf(fact_1461_nonzero__minus__divide__divide,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B3: A,A3: A] :
( ( B3
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( uminus_uminus @ A @ A3 ) @ ( uminus_uminus @ A @ B3 ) )
= ( divide_divide @ A @ A3 @ B3 ) ) ) ) ).
% nonzero_minus_divide_divide
thf(fact_1462_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ( ( minus_minus @ A )
= ( ^ [A5: A,B5: A] : ( plus_plus @ A @ A5 @ ( uminus_uminus @ A @ B5 ) ) ) ) ) ).
% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_1463_diff__conv__add__uminus,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ( ( minus_minus @ A )
= ( ^ [A5: A,B5: A] : ( plus_plus @ A @ A5 @ ( uminus_uminus @ A @ B5 ) ) ) ) ) ).
% diff_conv_add_uminus
thf(fact_1464_group__cancel_Osub2,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [B2: A,K: A,B3: A,A3: A] :
( ( B2
= ( plus_plus @ A @ K @ B3 ) )
=> ( ( minus_minus @ A @ A3 @ B2 )
= ( plus_plus @ A @ ( uminus_uminus @ A @ K ) @ ( minus_minus @ A @ A3 @ B3 ) ) ) ) ) ).
% group_cancel.sub2
thf(fact_1465_of__int__floor__le,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] : ( ord_less_eq @ A @ ( ring_1_of_int @ A @ ( archim6421214686448440834_floor @ A @ X ) ) @ X ) ) ).
% of_int_floor_le
thf(fact_1466_power__one__over,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A,N: nat] :
( ( power_power @ A @ ( divide_divide @ A @ ( one_one @ A ) @ A3 ) @ N )
= ( divide_divide @ A @ ( one_one @ A ) @ ( power_power @ A @ A3 @ N ) ) ) ) ).
% power_one_over
thf(fact_1467_dvd__power__le,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [X: A,Y: A,N: nat,M: nat] :
( ( dvd_dvd @ A @ X @ Y )
=> ( ( ord_less_eq @ nat @ N @ M )
=> ( dvd_dvd @ A @ ( power_power @ A @ X @ N ) @ ( power_power @ A @ Y @ M ) ) ) ) ) ).
% dvd_power_le
thf(fact_1468_power__le__dvd,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A3: A,N: nat,B3: A,M: nat] :
( ( dvd_dvd @ A @ ( power_power @ A @ A3 @ N ) @ B3 )
=> ( ( ord_less_eq @ nat @ M @ N )
=> ( dvd_dvd @ A @ ( power_power @ A @ A3 @ M ) @ B3 ) ) ) ) ).
% power_le_dvd
thf(fact_1469_le__imp__power__dvd,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [M: nat,N: nat,A3: A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( dvd_dvd @ A @ ( power_power @ A @ A3 @ M ) @ ( power_power @ A @ A3 @ N ) ) ) ) ).
% le_imp_power_dvd
thf(fact_1470_le__of__int__ceiling,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] : ( ord_less_eq @ A @ X @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ A @ X ) ) ) ) ).
% le_of_int_ceiling
thf(fact_1471_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_1472_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_1473_word__numeral__alt,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( numeral_numeral @ ( word @ A ) )
= ( ^ [B5: num] : ( ring_1_of_int @ ( word @ A ) @ ( numeral_numeral @ int @ B5 ) ) ) ) ) ).
% word_numeral_alt
thf(fact_1474_subset__minus__empty,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ( minus_minus @ ( set @ A ) @ A2 @ B2 )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% subset_minus_empty
thf(fact_1475_unat__sub__if__size,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y: word @ A,X: word @ A] :
( ( ( ord_less_eq @ nat @ ( semiring_1_unsigned @ A @ nat @ Y ) @ ( semiring_1_unsigned @ A @ nat @ X ) )
=> ( ( semiring_1_unsigned @ A @ nat @ ( minus_minus @ ( word @ A ) @ X @ Y ) )
= ( minus_minus @ nat @ ( semiring_1_unsigned @ A @ nat @ X ) @ ( semiring_1_unsigned @ A @ nat @ Y ) ) ) )
& ( ~ ( ord_less_eq @ nat @ ( semiring_1_unsigned @ A @ nat @ Y ) @ ( semiring_1_unsigned @ A @ nat @ X ) )
=> ( ( semiring_1_unsigned @ A @ nat @ ( minus_minus @ ( word @ A ) @ X @ Y ) )
= ( minus_minus @ nat @ ( plus_plus @ nat @ ( semiring_1_unsigned @ A @ nat @ X ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( size_size @ ( word @ A ) @ X ) ) ) @ ( semiring_1_unsigned @ A @ nat @ Y ) ) ) ) ) ) ).
% unat_sub_if_size
thf(fact_1476_Diff__insert__absorb,axiom,
! [A: $tType,X: A,A2: set @ A] :
( ~ ( member @ A @ X @ A2 )
=> ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X @ A2 ) @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_1477_Diff__insert2,axiom,
! [A: $tType,A2: set @ A,A3: A,B2: set @ A] :
( ( minus_minus @ ( set @ A ) @ A2 @ ( insert @ A @ A3 @ B2 ) )
= ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A2 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) @ B2 ) ) ).
% Diff_insert2
thf(fact_1478_insert__Diff,axiom,
! [A: $tType,A3: A,A2: set @ A] :
( ( member @ A @ A3 @ A2 )
=> ( ( insert @ A @ A3 @ ( minus_minus @ ( set @ A ) @ A2 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_1479_Diff__insert,axiom,
! [A: $tType,A2: set @ A,A3: A,B2: set @ A] :
( ( minus_minus @ ( set @ A ) @ A2 @ ( insert @ A @ A3 @ B2 ) )
= ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A2 @ B2 ) @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Diff_insert
thf(fact_1480_insert__minus__eq,axiom,
! [A: $tType,X: A,Y: A,A2: set @ A] :
( ( X != Y )
=> ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X @ A2 ) @ ( insert @ A @ Y @ ( bot_bot @ ( set @ A ) ) ) )
= ( insert @ A @ X @ ( minus_minus @ ( set @ A ) @ A2 @ ( insert @ A @ Y @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% insert_minus_eq
thf(fact_1481_set__minus__singleton__eq,axiom,
! [A: $tType,X: A,X5: set @ A] :
( ~ ( member @ A @ X @ X5 )
=> ( ( minus_minus @ ( set @ A ) @ X5 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= X5 ) ) ).
% set_minus_singleton_eq
thf(fact_1482_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_1483_subset__Diff__insert,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,X: A,C3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ ( minus_minus @ ( set @ A ) @ B2 @ ( insert @ A @ X @ C3 ) ) )
= ( ( ord_less_eq @ ( set @ A ) @ A2 @ ( minus_minus @ ( set @ A ) @ B2 @ C3 ) )
& ~ ( member @ A @ X @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_1484_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_1485_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_1486_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_1487_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_1488_diff__less__mono,axiom,
! [A3: nat,B3: nat,C2: nat] :
( ( ord_less @ nat @ A3 @ B3 )
=> ( ( ord_less_eq @ nat @ C2 @ A3 )
=> ( ord_less @ nat @ ( minus_minus @ nat @ A3 @ C2 ) @ ( minus_minus @ nat @ B3 @ C2 ) ) ) ) ).
% diff_less_mono
thf(fact_1489_uint__sub__if__size,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y: word @ A,X: word @ A] :
( ( ( ord_less_eq @ int @ ( semiring_1_unsigned @ A @ int @ Y ) @ ( semiring_1_unsigned @ A @ int @ X ) )
=> ( ( semiring_1_unsigned @ A @ int @ ( minus_minus @ ( word @ A ) @ X @ Y ) )
= ( minus_minus @ int @ ( semiring_1_unsigned @ A @ int @ X ) @ ( semiring_1_unsigned @ A @ int @ Y ) ) ) )
& ( ~ ( ord_less_eq @ int @ ( semiring_1_unsigned @ A @ int @ Y ) @ ( semiring_1_unsigned @ A @ int @ X ) )
=> ( ( semiring_1_unsigned @ A @ int @ ( minus_minus @ ( word @ A ) @ X @ Y ) )
= ( plus_plus @ int @ ( minus_minus @ int @ ( semiring_1_unsigned @ A @ int @ X ) @ ( semiring_1_unsigned @ A @ int @ Y ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( size_size @ ( word @ A ) @ X ) ) ) ) ) ) ) ).
% uint_sub_if_size
thf(fact_1490_wi__hom__add,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: int,B3: int] :
( ( plus_plus @ ( word @ A ) @ ( ring_1_of_int @ ( word @ A ) @ A3 ) @ ( ring_1_of_int @ ( word @ A ) @ B3 ) )
= ( ring_1_of_int @ ( word @ A ) @ ( plus_plus @ int @ A3 @ B3 ) ) ) ) ).
% wi_hom_add
thf(fact_1491_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_1492_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_1493_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_1494_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_1495_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_1496_Diff__partition,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ( sup_sup @ ( set @ A ) @ A2 @ ( minus_minus @ ( set @ A ) @ B2 @ A2 ) )
= B2 ) ) ).
% Diff_partition
thf(fact_1497_Diff__subset__conv,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,C3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A2 @ B2 ) @ C3 )
= ( ord_less_eq @ ( set @ A ) @ A2 @ ( sup_sup @ ( set @ A ) @ B2 @ C3 ) ) ) ).
% Diff_subset_conv
thf(fact_1498_minus__int__code_I2_J,axiom,
! [L: int] :
( ( minus_minus @ int @ ( zero_zero @ int ) @ L )
= ( uminus_uminus @ int @ L ) ) ).
% minus_int_code(2)
thf(fact_1499_wi__hom__neg,axiom,
! [D: $tType] :
( ( type_len @ D )
=> ! [A3: int] :
( ( uminus_uminus @ ( word @ D ) @ ( ring_1_of_int @ ( word @ D ) @ A3 ) )
= ( ring_1_of_int @ ( word @ D ) @ ( uminus_uminus @ int @ A3 ) ) ) ) ).
% wi_hom_neg
thf(fact_1500_int__le__induct,axiom,
! [I: int,K: int,P: int > $o] :
( ( ord_less_eq @ int @ I @ K )
=> ( ( P @ K )
=> ( ! [I3: int] :
( ( ord_less_eq @ int @ I3 @ K )
=> ( ( P @ I3 )
=> ( P @ ( minus_minus @ int @ I3 @ ( one_one @ int ) ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_le_induct
thf(fact_1501_int__less__induct,axiom,
! [I: int,K: int,P: int > $o] :
( ( ord_less @ int @ I @ K )
=> ( ( P @ ( minus_minus @ int @ K @ ( one_one @ int ) ) )
=> ( ! [I3: int] :
( ( ord_less @ int @ I3 @ K )
=> ( ( P @ I3 )
=> ( P @ ( minus_minus @ int @ I3 @ ( one_one @ int ) ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_less_induct
thf(fact_1502_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_1503_le__step__down__word__2,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_eq @ ( word @ A ) @ X @ ( minus_minus @ ( word @ A ) @ Y @ ( one_one @ ( word @ A ) ) ) ) ) ) ) ).
% le_step_down_word_2
thf(fact_1504_ucast__eq,axiom,
! [A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( type_len @ A ) )
=> ( ( semiring_1_unsigned @ B @ ( word @ A ) )
= ( ^ [W2: word @ B] : ( ring_1_of_int @ ( word @ A ) @ ( semiring_1_unsigned @ B @ int @ W2 ) ) ) ) ) ).
% ucast_eq
thf(fact_1505_word__of__int__power__hom,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: int,N: nat] :
( ( power_power @ ( word @ A ) @ ( ring_1_of_int @ ( word @ A ) @ A3 ) @ N )
= ( ring_1_of_int @ ( word @ A ) @ ( power_power @ int @ A3 @ N ) ) ) ) ).
% word_of_int_power_hom
thf(fact_1506_word__diff__ls_I4_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y: word @ A,W: word @ A,Xa: word @ A,X: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ Y @ ( plus_plus @ ( word @ A ) @ W @ Xa ) )
=> ( ( ord_less_eq @ ( word @ A ) @ X @ Y )
=> ( ord_less_eq @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ Y @ X ) @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W @ Xa ) @ X ) ) ) ) ) ).
% word_diff_ls(4)
thf(fact_1507_word__diff__ls_I3_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y: word @ A,X: word @ A,W: word @ A,Xa: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ Y @ X ) @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W @ Xa ) @ X ) )
=> ( ( ord_less_eq @ ( word @ A ) @ X @ ( plus_plus @ ( word @ A ) @ W @ Xa ) )
=> ( ord_less_eq @ ( word @ A ) @ Y @ ( plus_plus @ ( word @ A ) @ W @ Xa ) ) ) ) ) ).
% word_diff_ls(3)
thf(fact_1508_Word_Oword__l__diffs_I4_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A,Xa: word @ A,Z: word @ A,X: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W @ Xa ) @ Z )
=> ( ( ord_less_eq @ ( word @ A ) @ X @ ( plus_plus @ ( word @ A ) @ W @ Xa ) )
=> ( ord_less_eq @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W @ Xa ) @ X ) @ ( minus_minus @ ( word @ A ) @ Z @ X ) ) ) ) ) ).
% Word.word_l_diffs(4)
thf(fact_1509_Word_Oword__l__diffs_I3_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A,Xa: word @ A,X: word @ A,Z: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W @ Xa ) @ X ) @ ( minus_minus @ ( word @ A ) @ Z @ X ) )
=> ( ( ord_less_eq @ ( word @ A ) @ X @ Z )
=> ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W @ Xa ) @ Z ) ) ) ) ).
% Word.word_l_diffs(3)
thf(fact_1510_word__plus__mcs_I4_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [V2: word @ A,Xb: word @ A,W: word @ A,Xa: word @ A,X: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ V2 @ Xb ) @ ( plus_plus @ ( word @ A ) @ W @ Xa ) )
=> ( ( ord_less_eq @ ( word @ A ) @ X @ ( plus_plus @ ( word @ A ) @ V2 @ Xb ) )
=> ( ord_less_eq @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ V2 @ Xb ) @ X ) @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W @ Xa ) @ X ) ) ) ) ) ).
% word_plus_mcs(4)
thf(fact_1511_word__plus__mcs_I3_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [V2: word @ A,Xb: word @ A,X: word @ A,W: word @ A,Xa: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ V2 @ Xb ) @ X ) @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W @ Xa ) @ X ) )
=> ( ( ord_less_eq @ ( word @ A ) @ X @ ( plus_plus @ ( word @ A ) @ W @ Xa ) )
=> ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ V2 @ Xb ) @ ( plus_plus @ ( word @ A ) @ W @ Xa ) ) ) ) ) ).
% word_plus_mcs(3)
thf(fact_1512_More__Word_Oword__l__diffs_I4_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A,X: word @ A,Z: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W @ X ) @ Z )
=> ( ( ord_less_eq @ ( word @ A ) @ X @ ( plus_plus @ ( word @ A ) @ W @ X ) )
=> ( ord_less_eq @ ( word @ A ) @ W @ ( minus_minus @ ( word @ A ) @ Z @ X ) ) ) ) ) ).
% More_Word.word_l_diffs(4)
thf(fact_1513_More__Word_Oword__l__diffs_I3_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A,Z: word @ A,X: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ W @ ( minus_minus @ ( word @ A ) @ Z @ X ) )
=> ( ( ord_less_eq @ ( word @ A ) @ X @ Z )
=> ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W @ X ) @ Z ) ) ) ) ).
% More_Word.word_l_diffs(3)
thf(fact_1514_word__diff__ls_H_I4_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y: word @ A,W: word @ A,X: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ Y @ ( plus_plus @ ( word @ A ) @ W @ X ) )
=> ( ( ord_less_eq @ ( word @ A ) @ X @ Y )
=> ( ord_less_eq @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ Y @ X ) @ W ) ) ) ) ).
% word_diff_ls'(4)
thf(fact_1515_word__diff__ls_H_I3_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y: word @ A,X: word @ A,W: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ Y @ X ) @ W )
=> ( ( ord_less_eq @ ( word @ A ) @ X @ ( plus_plus @ ( word @ A ) @ W @ X ) )
=> ( ord_less_eq @ ( word @ A ) @ Y @ ( plus_plus @ ( word @ A ) @ W @ X ) ) ) ) ) ).
% word_diff_ls'(3)
thf(fact_1516_word__l__diffs_H_I4_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A,X: word @ A,Z: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W @ X ) @ Z )
=> ( ( ord_less_eq @ ( word @ A ) @ X @ ( plus_plus @ ( word @ A ) @ W @ X ) )
=> ( ord_less_eq @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W @ X ) @ X ) @ ( minus_minus @ ( word @ A ) @ Z @ X ) ) ) ) ) ).
% word_l_diffs'(4)
thf(fact_1517_word__l__diffs_H_I3_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A,X: word @ A,Z: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W @ X ) @ X ) @ ( minus_minus @ ( word @ A ) @ Z @ X ) )
=> ( ( ord_less_eq @ ( word @ A ) @ X @ Z )
=> ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W @ X ) @ Z ) ) ) ) ).
% word_l_diffs'(3)
thf(fact_1518_word__diff__ls_H_H_I4_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y: word @ A,W: word @ A,X: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ Y @ ( plus_plus @ ( word @ A ) @ W @ X ) )
=> ( ( ord_less_eq @ ( word @ A ) @ X @ Y )
=> ( ord_less_eq @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ Y @ X ) @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W @ X ) @ X ) ) ) ) ) ).
% word_diff_ls''(4)
thf(fact_1519_word__diff__ls_H_H_I3_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y: word @ A,X: word @ A,W: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ Y @ X ) @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W @ X ) @ X ) )
=> ( ( ord_less_eq @ ( word @ A ) @ X @ ( plus_plus @ ( word @ A ) @ W @ X ) )
=> ( ord_less_eq @ ( word @ A ) @ Y @ ( plus_plus @ ( word @ A ) @ W @ X ) ) ) ) ) ).
% word_diff_ls''(3)
thf(fact_1520_le__plus,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [C2: word @ A,B3: word @ A,A3: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ C2 @ ( minus_minus @ ( word @ A ) @ B3 @ A3 ) )
=> ( ( ord_less_eq @ ( word @ A ) @ A3 @ B3 )
=> ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ A3 @ C2 ) @ B3 ) ) ) ) ).
% le_plus
thf(fact_1521_le__plus_H,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A,B3: word @ A,C2: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ A3 @ B3 )
=> ( ( ord_less_eq @ ( word @ A ) @ C2 @ ( minus_minus @ ( word @ A ) @ B3 @ A3 ) )
=> ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ A3 @ C2 ) @ B3 ) ) ) ) ).
% le_plus'
thf(fact_1522_le__minus_H,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A,C2: word @ A,B3: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ A3 @ C2 ) @ B3 )
=> ( ( ord_less_eq @ ( word @ A ) @ A3 @ ( plus_plus @ ( word @ A ) @ A3 @ C2 ) )
=> ( ord_less_eq @ ( word @ A ) @ C2 @ ( minus_minus @ ( word @ A ) @ B3 @ A3 ) ) ) ) ) ).
% le_minus'
thf(fact_1523_plus__minus__no__overflow__ab,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Ab: word @ A,C2: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ X @ ( minus_minus @ ( word @ A ) @ Ab @ C2 ) )
=> ( ( ord_less_eq @ ( word @ A ) @ C2 @ Ab )
=> ( ord_less_eq @ ( word @ A ) @ X @ ( plus_plus @ ( word @ A ) @ X @ C2 ) ) ) ) ) ).
% plus_minus_no_overflow_ab
thf(fact_1524_word__less__cases,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] :
( ( ord_less @ ( word @ A ) @ X @ Y )
=> ( ( X
= ( minus_minus @ ( word @ A ) @ Y @ ( one_one @ ( word @ A ) ) ) )
| ( ord_less @ ( word @ A ) @ X @ ( minus_minus @ ( word @ A ) @ Y @ ( one_one @ ( word @ A ) ) ) ) ) ) ) ).
% word_less_cases
thf(fact_1525_word__less__minus__cancel,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y: word @ A,X: word @ A,Z: word @ A] :
( ( ord_less @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ Y @ X ) @ ( minus_minus @ ( word @ A ) @ Z @ X ) )
=> ( ( ord_less_eq @ ( word @ A ) @ X @ Z )
=> ( ord_less @ ( word @ A ) @ Y @ Z ) ) ) ) ).
% word_less_minus_cancel
thf(fact_1526_word__less__minus__mono__left,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y: word @ A,Z: word @ A,X: word @ A] :
( ( ord_less @ ( word @ A ) @ Y @ Z )
=> ( ( ord_less_eq @ ( word @ A ) @ X @ Y )
=> ( ord_less @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ Y @ X ) @ ( minus_minus @ ( word @ A ) @ Z @ X ) ) ) ) ) ).
% word_less_minus_mono_left
thf(fact_1527_word__less__imp__diff__less,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [K: word @ A,N: word @ A,M: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ K @ N )
=> ( ( ord_less @ ( word @ A ) @ N @ M )
=> ( ord_less @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ N @ K ) @ M ) ) ) ) ).
% word_less_imp_diff_less
thf(fact_1528_minus__real__def,axiom,
( ( minus_minus @ real )
= ( ^ [X2: real,Y2: real] : ( plus_plus @ real @ X2 @ ( uminus_uminus @ real @ Y2 ) ) ) ) ).
% minus_real_def
thf(fact_1529_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_1530_add__diff__assoc__enat,axiom,
! [Z: extended_enat,Y: extended_enat,X: extended_enat] :
( ( ord_less_eq @ extended_enat @ Z @ Y )
=> ( ( plus_plus @ extended_enat @ X @ ( minus_minus @ extended_enat @ Y @ Z ) )
= ( minus_minus @ extended_enat @ ( plus_plus @ extended_enat @ X @ Y ) @ Z ) ) ) ).
% add_diff_assoc_enat
thf(fact_1531_powr__less__mono2,axiom,
! [A3: real,X: real,Y: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ A3 )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ X @ Y )
=> ( ord_less @ real @ ( powr @ real @ X @ A3 ) @ ( powr @ real @ Y @ A3 ) ) ) ) ) ).
% powr_less_mono2
thf(fact_1532_powr__mono2_H,axiom,
! [A3: real,X: real,Y: real] :
( ( ord_less_eq @ real @ A3 @ ( zero_zero @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ X @ Y )
=> ( ord_less_eq @ real @ ( powr @ real @ Y @ A3 ) @ ( powr @ real @ X @ A3 ) ) ) ) ) ).
% powr_mono2'
thf(fact_1533_powr__inj,axiom,
! [A3: real,X: real,Y: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ A3 )
=> ( ( A3
!= ( one_one @ real ) )
=> ( ( ( powr @ real @ A3 @ X )
= ( powr @ real @ A3 @ Y ) )
= ( X = Y ) ) ) ) ).
% powr_inj
thf(fact_1534_gr__one__powr,axiom,
! [X: real,Y: real] :
( ( ord_less @ real @ ( one_one @ real ) @ X )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ Y )
=> ( ord_less @ real @ ( one_one @ real ) @ ( powr @ real @ X @ Y ) ) ) ) ).
% gr_one_powr
thf(fact_1535_powr__le1,axiom,
! [A3: real,X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ A3 )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ X @ ( one_one @ real ) )
=> ( ord_less_eq @ real @ ( powr @ real @ X @ A3 ) @ ( one_one @ real ) ) ) ) ) ).
% powr_le1
thf(fact_1536_powr__mono__both,axiom,
! [A3: real,B3: real,X: real,Y: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ A3 )
=> ( ( ord_less_eq @ real @ A3 @ B3 )
=> ( ( ord_less_eq @ real @ ( one_one @ real ) @ X )
=> ( ( ord_less_eq @ real @ X @ Y )
=> ( ord_less_eq @ real @ ( powr @ real @ X @ A3 ) @ ( powr @ real @ Y @ B3 ) ) ) ) ) ) ).
% powr_mono_both
thf(fact_1537_ge__one__powr__ge__zero,axiom,
! [X: real,A3: real] :
( ( ord_less_eq @ real @ ( one_one @ real ) @ X )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ A3 )
=> ( ord_less_eq @ real @ ( one_one @ real ) @ ( powr @ real @ X @ A3 ) ) ) ) ).
% ge_one_powr_ge_zero
thf(fact_1538_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_1539_div__positive,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [B3: A,A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ B3 )
=> ( ( ord_less_eq @ A @ B3 @ A3 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ A3 @ B3 ) ) ) ) ) ).
% div_positive
thf(fact_1540_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ A3 @ B3 )
=> ( ( divide_divide @ A @ A3 @ B3 )
= ( zero_zero @ A ) ) ) ) ) ).
% unique_euclidean_semiring_numeral_class.div_less
thf(fact_1541_divide__nonpos__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ Y )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ X @ Y ) @ ( zero_zero @ A ) ) ) ) ) ).
% divide_nonpos_pos
thf(fact_1542_divide__nonpos__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ Y @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X @ Y ) ) ) ) ) ).
% divide_nonpos_neg
thf(fact_1543_divide__nonneg__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ Y )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X @ Y ) ) ) ) ) ).
% divide_nonneg_pos
thf(fact_1544_divide__nonneg__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less @ A @ Y @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ X @ Y ) @ ( zero_zero @ A ) ) ) ) ) ).
% divide_nonneg_neg
thf(fact_1545_divide__le__cancel,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,C2: A,B3: A] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ A3 @ C2 ) @ ( divide_divide @ A @ B3 @ C2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ A3 @ B3 ) )
& ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ B3 @ A3 ) ) ) ) ) ).
% divide_le_cancel
thf(fact_1546_frac__less2,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y: A,W: A,Z: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ W )
=> ( ( ord_less @ A @ W @ Z )
=> ( ord_less @ A @ ( divide_divide @ A @ X @ Z ) @ ( divide_divide @ A @ Y @ W ) ) ) ) ) ) ) ).
% frac_less2
thf(fact_1547_frac__less,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y: A,W: A,Z: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less @ A @ X @ Y )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ W )
=> ( ( ord_less_eq @ A @ W @ Z )
=> ( ord_less @ A @ ( divide_divide @ A @ X @ Z ) @ ( divide_divide @ A @ Y @ W ) ) ) ) ) ) ) ).
% frac_less
thf(fact_1548_frac__le,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [Y: A,X: A,W: A,Z: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
=> ( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ W )
=> ( ( ord_less_eq @ A @ W @ Z )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ X @ Z ) @ ( divide_divide @ A @ Y @ W ) ) ) ) ) ) ) ).
% frac_le
thf(fact_1549_less__divide__eq__1,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B3: A,A3: A] :
( ( ord_less @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B3 @ A3 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
& ( ord_less @ A @ A3 @ B3 ) )
| ( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
& ( ord_less @ A @ B3 @ A3 ) ) ) ) ) ).
% less_divide_eq_1
thf(fact_1550_divide__less__eq__1,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B3: A,A3: A] :
( ( ord_less @ A @ ( divide_divide @ A @ B3 @ A3 ) @ ( one_one @ A ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
& ( ord_less @ A @ B3 @ A3 ) )
| ( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
& ( ord_less @ A @ A3 @ B3 ) )
| ( A3
= ( zero_zero @ A ) ) ) ) ) ).
% divide_less_eq_1
thf(fact_1551_less__half__sum,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ A3 @ B3 )
=> ( ord_less @ A @ A3 @ ( divide_divide @ A @ ( plus_plus @ A @ A3 @ B3 ) @ ( plus_plus @ A @ ( one_one @ A ) @ ( one_one @ A ) ) ) ) ) ) ).
% less_half_sum
thf(fact_1552_gt__half__sum,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ A3 @ B3 )
=> ( ord_less @ A @ ( divide_divide @ A @ ( plus_plus @ A @ A3 @ B3 ) @ ( plus_plus @ A @ ( one_one @ A ) @ ( one_one @ A ) ) ) @ B3 ) ) ) ).
% gt_half_sum
thf(fact_1553_is__unit__power__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,N: nat] :
( ( dvd_dvd @ A @ ( power_power @ A @ A3 @ N ) @ ( one_one @ A ) )
= ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
| ( N
= ( zero_zero @ nat ) ) ) ) ) ).
% is_unit_power_iff
thf(fact_1554_divide__eq__minus__1__iff,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A3: A,B3: A] :
( ( ( divide_divide @ A @ A3 @ B3 )
= ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( ( B3
!= ( zero_zero @ A ) )
& ( A3
= ( uminus_uminus @ A @ B3 ) ) ) ) ) ).
% divide_eq_minus_1_iff
thf(fact_1555_le__floor__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z: int,X: A] :
( ( ord_less_eq @ int @ Z @ ( archim6421214686448440834_floor @ A @ X ) )
= ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Z ) @ X ) ) ) ).
% le_floor_iff
thf(fact_1556_floor__less__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Z: int] :
( ( ord_less @ int @ ( archim6421214686448440834_floor @ A @ X ) @ Z )
= ( ord_less @ A @ X @ ( ring_1_of_int @ A @ Z ) ) ) ) ).
% floor_less_iff
thf(fact_1557_ceiling__le__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Z: int] :
( ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ X ) @ Z )
= ( ord_less_eq @ A @ X @ ( ring_1_of_int @ A @ Z ) ) ) ) ).
% ceiling_le_iff
thf(fact_1558_ceiling__le,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,A3: int] :
( ( ord_less_eq @ A @ X @ ( ring_1_of_int @ A @ A3 ) )
=> ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ X ) @ A3 ) ) ) ).
% ceiling_le
thf(fact_1559_less__ceiling__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z: int,X: A] :
( ( ord_less @ int @ Z @ ( archimedean_ceiling @ A @ X ) )
= ( ord_less @ A @ ( ring_1_of_int @ A @ Z ) @ X ) ) ) ).
% less_ceiling_iff
thf(fact_1560_floor__add__int,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Z: int] :
( ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X ) @ Z )
= ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ X @ ( ring_1_of_int @ A @ Z ) ) ) ) ) ).
% floor_add_int
thf(fact_1561_int__add__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z: int,X: A] :
( ( plus_plus @ int @ Z @ ( archim6421214686448440834_floor @ A @ X ) )
= ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ ( ring_1_of_int @ A @ Z ) @ X ) ) ) ) ).
% int_add_floor
thf(fact_1562_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_1563_floor__power,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,N: nat] :
( ( X
= ( ring_1_of_int @ A @ ( archim6421214686448440834_floor @ A @ X ) ) )
=> ( ( archim6421214686448440834_floor @ A @ ( power_power @ A @ X @ N ) )
= ( power_power @ int @ ( archim6421214686448440834_floor @ A @ X ) @ N ) ) ) ) ).
% floor_power
thf(fact_1564_Diff__single__insert,axiom,
! [A: $tType,A2: set @ A,X: A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A2 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) @ B2 )
=> ( ord_less_eq @ ( set @ A ) @ A2 @ ( insert @ A @ X @ B2 ) ) ) ).
% Diff_single_insert
thf(fact_1565_subset__insert__iff,axiom,
! [A: $tType,A2: set @ A,X: A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ ( insert @ A @ X @ B2 ) )
= ( ( ( member @ A @ X @ A2 )
=> ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A2 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) @ B2 ) )
& ( ~ ( member @ A @ X @ A2 )
=> ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ) ) ).
% subset_insert_iff
thf(fact_1566_nat__diff__split,axiom,
! [P: nat > $o,A3: nat,B3: nat] :
( ( P @ ( minus_minus @ nat @ A3 @ B3 ) )
= ( ( ( ord_less @ nat @ A3 @ B3 )
=> ( P @ ( zero_zero @ nat ) ) )
& ! [D4: nat] :
( ( A3
= ( plus_plus @ nat @ B3 @ D4 ) )
=> ( P @ D4 ) ) ) ) ).
% nat_diff_split
thf(fact_1567_nat__diff__split__asm,axiom,
! [P: nat > $o,A3: nat,B3: nat] :
( ( P @ ( minus_minus @ nat @ A3 @ B3 ) )
= ( ~ ( ( ( ord_less @ nat @ A3 @ B3 )
& ~ ( P @ ( zero_zero @ nat ) ) )
| ? [D4: nat] :
( ( A3
= ( plus_plus @ nat @ B3 @ D4 ) )
& ~ ( P @ D4 ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_1568_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_1569_le__minus,axiom,
! [Aa: $tType,A: $tType] :
( ( ( type_len @ A )
& ( order @ Aa ) )
=> ! [Y: Aa,X: Aa,A3: word @ A,C2: word @ A,B3: word @ A] :
( ( ord_less_eq @ Aa @ Y @ X )
=> ( ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ A3 @ C2 ) @ B3 )
=> ( ( ord_less_eq @ ( word @ A ) @ A3 @ ( plus_plus @ ( word @ A ) @ A3 @ C2 ) )
=> ( ord_less_eq @ ( word @ A ) @ C2 @ ( minus_minus @ ( word @ A ) @ B3 @ A3 ) ) ) ) ) ) ).
% le_minus
thf(fact_1570_word__add__def,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( plus_plus @ ( word @ A ) )
= ( ^ [A5: word @ A,B5: word @ A] : ( ring_1_of_int @ ( word @ A ) @ ( plus_plus @ int @ ( semiring_1_unsigned @ A @ int @ A5 ) @ ( semiring_1_unsigned @ A @ int @ B5 ) ) ) ) ) ) ).
% word_add_def
thf(fact_1571_word__minus__def,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( uminus_uminus @ ( word @ A ) )
= ( ^ [A5: word @ A] : ( ring_1_of_int @ ( word @ A ) @ ( uminus_uminus @ int @ ( semiring_1_unsigned @ A @ int @ A5 ) ) ) ) ) ) ).
% word_minus_def
thf(fact_1572_remove__subset,axiom,
! [A: $tType,X: A,S: set @ A] :
( ( member @ A @ X @ S )
=> ( ord_less @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ S @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) @ S ) ) ).
% remove_subset
thf(fact_1573_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_1574_int__induct,axiom,
! [P: int > $o,K: int,I: int] :
( ( P @ K )
=> ( ! [I3: int] :
( ( ord_less_eq @ int @ K @ I3 )
=> ( ( P @ I3 )
=> ( P @ ( plus_plus @ int @ I3 @ ( one_one @ int ) ) ) ) )
=> ( ! [I3: int] :
( ( ord_less_eq @ int @ I3 @ K )
=> ( ( P @ I3 )
=> ( P @ ( minus_minus @ int @ I3 @ ( one_one @ int ) ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_induct
thf(fact_1575_no__ulen__sub,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ X @ Y ) @ X )
= ( ord_less_eq @ int @ ( semiring_1_unsigned @ A @ int @ Y ) @ ( semiring_1_unsigned @ A @ int @ X ) ) ) ) ).
% no_ulen_sub
thf(fact_1576_word__arith__power__alt,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( power_power @ ( word @ A ) )
= ( ^ [A5: word @ A,N4: nat] : ( ring_1_of_int @ ( word @ A ) @ ( power_power @ int @ ( semiring_1_unsigned @ A @ int @ A5 ) @ N4 ) ) ) ) ) ).
% word_arith_power_alt
thf(fact_1577_Compl__insert,axiom,
! [A: $tType,X: A,A2: set @ A] :
( ( uminus_uminus @ ( set @ A ) @ ( insert @ A @ X @ A2 ) )
= ( minus_minus @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A2 ) @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Compl_insert
thf(fact_1578_word__le__sub1,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] :
( ( X
!= ( zero_zero @ ( word @ A ) ) )
=> ( ( ord_less_eq @ ( word @ A ) @ ( one_one @ ( word @ A ) ) @ X )
= ( ord_less_eq @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ ( minus_minus @ ( word @ A ) @ X @ ( one_one @ ( word @ A ) ) ) ) ) ) ) ).
% word_le_sub1
thf(fact_1579_word__sub__1__le,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] :
( ( X
!= ( zero_zero @ ( word @ A ) ) )
=> ( ord_less_eq @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ X @ ( one_one @ ( word @ A ) ) ) @ X ) ) ) ).
% word_sub_1_le
thf(fact_1580_word__must__wrap,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,N: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ X @ ( minus_minus @ ( word @ A ) @ N @ ( one_one @ ( word @ A ) ) ) )
=> ( ( ord_less_eq @ ( word @ A ) @ N @ X )
=> ( N
= ( zero_zero @ ( word @ A ) ) ) ) ) ) ).
% word_must_wrap
thf(fact_1581_word__minus__one__le__leq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] :
( ( ord_less @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ X @ ( one_one @ ( word @ A ) ) ) @ Y )
=> ( ord_less_eq @ ( word @ A ) @ X @ Y ) ) ) ).
% word_minus_one_le_leq
thf(fact_1582_word__le__minus__one__leq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] :
( ( ord_less @ ( word @ A ) @ X @ Y )
=> ( ord_less_eq @ ( word @ A ) @ X @ ( minus_minus @ ( word @ A ) @ Y @ ( one_one @ ( word @ A ) ) ) ) ) ) ).
% word_le_minus_one_leq
thf(fact_1583_plus__minus__not__NULL__ab,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Ab: word @ A,C2: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ X @ ( minus_minus @ ( word @ A ) @ Ab @ C2 ) )
=> ( ( ord_less_eq @ ( word @ A ) @ C2 @ Ab )
=> ( ( C2
!= ( zero_zero @ ( word @ A ) ) )
=> ( ( plus_plus @ ( word @ A ) @ X @ C2 )
!= ( zero_zero @ ( word @ A ) ) ) ) ) ) ) ).
% plus_minus_not_NULL_ab
thf(fact_1584_gt0__iff__gem1,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] :
( ( ord_less @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ X )
= ( ord_less @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ X @ ( one_one @ ( word @ A ) ) ) @ X ) ) ) ).
% gt0_iff_gem1
thf(fact_1585_word__less__sub1,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] :
( ( X
!= ( zero_zero @ ( word @ A ) ) )
=> ( ( ord_less @ ( word @ A ) @ ( one_one @ ( word @ A ) ) @ X )
= ( ord_less @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ ( minus_minus @ ( word @ A ) @ X @ ( one_one @ ( word @ A ) ) ) ) ) ) ) ).
% word_less_sub1
thf(fact_1586_sub__wrap,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Z: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ X @ ( minus_minus @ ( word @ A ) @ X @ Z ) )
= ( ( Z
= ( zero_zero @ ( word @ A ) ) )
| ( ord_less @ ( word @ A ) @ X @ Z ) ) ) ) ).
% sub_wrap
thf(fact_1587_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_1588_plus__minus__no__overflow,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Ab: word @ A,C2: word @ A] :
( ( ord_less @ ( word @ A ) @ X @ ( minus_minus @ ( word @ A ) @ Ab @ C2 ) )
=> ( ( ord_less_eq @ ( word @ A ) @ C2 @ Ab )
=> ( ord_less_eq @ ( word @ A ) @ X @ ( plus_plus @ ( word @ A ) @ X @ C2 ) ) ) ) ) ).
% plus_minus_no_overflow
thf(fact_1589_word__less__sub__right,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A,Z: word @ A] :
( ( ord_less @ ( word @ A ) @ X @ ( plus_plus @ ( word @ A ) @ Y @ Z ) )
=> ( ( ord_less_eq @ ( word @ A ) @ Y @ X )
=> ( ord_less @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ X @ Y ) @ Z ) ) ) ) ).
% word_less_sub_right
thf(fact_1590_word__less__add__right,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A,Z: word @ A] :
( ( ord_less @ ( word @ A ) @ X @ ( minus_minus @ ( word @ A ) @ Y @ Z ) )
=> ( ( ord_less_eq @ ( word @ A ) @ Z @ Y )
=> ( ord_less @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ X @ Z ) @ Y ) ) ) ) ).
% word_less_add_right
thf(fact_1591_word__diff__ls_H_H_I1_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y: word @ A,X: word @ A,W: word @ A] :
( ( ord_less @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ Y @ X ) @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W @ X ) @ X ) )
=> ( ( ord_less_eq @ ( word @ A ) @ X @ ( plus_plus @ ( word @ A ) @ W @ X ) )
=> ( ord_less @ ( word @ A ) @ Y @ ( plus_plus @ ( word @ A ) @ W @ X ) ) ) ) ) ).
% word_diff_ls''(1)
thf(fact_1592_word__diff__ls_H_H_I2_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y: word @ A,W: word @ A,X: word @ A] :
( ( ord_less @ ( word @ A ) @ Y @ ( plus_plus @ ( word @ A ) @ W @ X ) )
=> ( ( ord_less_eq @ ( word @ A ) @ X @ Y )
=> ( ord_less @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ Y @ X ) @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W @ X ) @ X ) ) ) ) ) ).
% word_diff_ls''(2)
thf(fact_1593_word__l__diffs_H_I1_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A,X: word @ A,Z: word @ A] :
( ( ord_less @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W @ X ) @ X ) @ ( minus_minus @ ( word @ A ) @ Z @ X ) )
=> ( ( ord_less_eq @ ( word @ A ) @ X @ Z )
=> ( ord_less @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W @ X ) @ Z ) ) ) ) ).
% word_l_diffs'(1)
thf(fact_1594_word__l__diffs_H_I2_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A,X: word @ A,Z: word @ A] :
( ( ord_less @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W @ X ) @ Z )
=> ( ( ord_less_eq @ ( word @ A ) @ X @ ( plus_plus @ ( word @ A ) @ W @ X ) )
=> ( ord_less @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W @ X ) @ X ) @ ( minus_minus @ ( word @ A ) @ Z @ X ) ) ) ) ) ).
% word_l_diffs'(2)
thf(fact_1595_word__diff__ls_H_I1_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y: word @ A,X: word @ A,W: word @ A] :
( ( ord_less @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ Y @ X ) @ W )
=> ( ( ord_less_eq @ ( word @ A ) @ X @ ( plus_plus @ ( word @ A ) @ W @ X ) )
=> ( ord_less @ ( word @ A ) @ Y @ ( plus_plus @ ( word @ A ) @ W @ X ) ) ) ) ) ).
% word_diff_ls'(1)
thf(fact_1596_word__diff__ls_H_I2_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y: word @ A,W: word @ A,X: word @ A] :
( ( ord_less @ ( word @ A ) @ Y @ ( plus_plus @ ( word @ A ) @ W @ X ) )
=> ( ( ord_less_eq @ ( word @ A ) @ X @ Y )
=> ( ord_less @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ Y @ X ) @ W ) ) ) ) ).
% word_diff_ls'(2)
thf(fact_1597_More__Word_Oword__l__diffs_I1_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A,Z: word @ A,X: word @ A] :
( ( ord_less @ ( word @ A ) @ W @ ( minus_minus @ ( word @ A ) @ Z @ X ) )
=> ( ( ord_less_eq @ ( word @ A ) @ X @ Z )
=> ( ord_less @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W @ X ) @ Z ) ) ) ) ).
% More_Word.word_l_diffs(1)
thf(fact_1598_More__Word_Oword__l__diffs_I2_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A,X: word @ A,Z: word @ A] :
( ( ord_less @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W @ X ) @ Z )
=> ( ( ord_less_eq @ ( word @ A ) @ X @ ( plus_plus @ ( word @ A ) @ W @ X ) )
=> ( ord_less @ ( word @ A ) @ W @ ( minus_minus @ ( word @ A ) @ Z @ X ) ) ) ) ) ).
% More_Word.word_l_diffs(2)
thf(fact_1599_word__plus__mcs_I1_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [V2: word @ A,Xb: word @ A,X: word @ A,W: word @ A,Xa: word @ A] :
( ( ord_less @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ V2 @ Xb ) @ X ) @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W @ Xa ) @ X ) )
=> ( ( ord_less_eq @ ( word @ A ) @ X @ ( plus_plus @ ( word @ A ) @ W @ Xa ) )
=> ( ord_less @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ V2 @ Xb ) @ ( plus_plus @ ( word @ A ) @ W @ Xa ) ) ) ) ) ).
% word_plus_mcs(1)
thf(fact_1600_word__plus__mcs_I2_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [V2: word @ A,Xb: word @ A,W: word @ A,Xa: word @ A,X: word @ A] :
( ( ord_less @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ V2 @ Xb ) @ ( plus_plus @ ( word @ A ) @ W @ Xa ) )
=> ( ( ord_less_eq @ ( word @ A ) @ X @ ( plus_plus @ ( word @ A ) @ V2 @ Xb ) )
=> ( ord_less @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ V2 @ Xb ) @ X ) @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W @ Xa ) @ X ) ) ) ) ) ).
% word_plus_mcs(2)
thf(fact_1601_Word_Oword__l__diffs_I1_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A,Xa: word @ A,X: word @ A,Z: word @ A] :
( ( ord_less @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W @ Xa ) @ X ) @ ( minus_minus @ ( word @ A ) @ Z @ X ) )
=> ( ( ord_less_eq @ ( word @ A ) @ X @ Z )
=> ( ord_less @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W @ Xa ) @ Z ) ) ) ) ).
% Word.word_l_diffs(1)
thf(fact_1602_Word_Oword__l__diffs_I2_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A,Xa: word @ A,Z: word @ A,X: word @ A] :
( ( ord_less @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W @ Xa ) @ Z )
=> ( ( ord_less_eq @ ( word @ A ) @ X @ ( plus_plus @ ( word @ A ) @ W @ Xa ) )
=> ( ord_less @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W @ Xa ) @ X ) @ ( minus_minus @ ( word @ A ) @ Z @ X ) ) ) ) ) ).
% Word.word_l_diffs(2)
thf(fact_1603_word__diff__ls_I1_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y: word @ A,X: word @ A,W: word @ A,Xa: word @ A] :
( ( ord_less @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ Y @ X ) @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W @ Xa ) @ X ) )
=> ( ( ord_less_eq @ ( word @ A ) @ X @ ( plus_plus @ ( word @ A ) @ W @ Xa ) )
=> ( ord_less @ ( word @ A ) @ Y @ ( plus_plus @ ( word @ A ) @ W @ Xa ) ) ) ) ) ).
% word_diff_ls(1)
thf(fact_1604_word__diff__ls_I2_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y: word @ A,W: word @ A,Xa: word @ A,X: word @ A] :
( ( ord_less @ ( word @ A ) @ Y @ ( plus_plus @ ( word @ A ) @ W @ Xa ) )
=> ( ( ord_less_eq @ ( word @ A ) @ X @ Y )
=> ( ord_less @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ Y @ X ) @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W @ Xa ) @ X ) ) ) ) ) ).
% word_diff_ls(2)
thf(fact_1605_unat__plus__if__size,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] :
( ( ( ord_less @ nat @ ( plus_plus @ nat @ ( semiring_1_unsigned @ A @ nat @ X ) @ ( semiring_1_unsigned @ A @ nat @ Y ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( size_size @ ( word @ A ) @ X ) ) )
=> ( ( semiring_1_unsigned @ A @ nat @ ( plus_plus @ ( word @ A ) @ X @ Y ) )
= ( plus_plus @ nat @ ( semiring_1_unsigned @ A @ nat @ X ) @ ( semiring_1_unsigned @ A @ nat @ Y ) ) ) )
& ( ~ ( ord_less @ nat @ ( plus_plus @ nat @ ( semiring_1_unsigned @ A @ nat @ X ) @ ( semiring_1_unsigned @ A @ nat @ Y ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( size_size @ ( word @ A ) @ X ) ) )
=> ( ( semiring_1_unsigned @ A @ nat @ ( plus_plus @ ( word @ A ) @ X @ Y ) )
= ( minus_minus @ nat @ ( plus_plus @ nat @ ( semiring_1_unsigned @ A @ nat @ X ) @ ( semiring_1_unsigned @ A @ nat @ Y ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( size_size @ ( word @ A ) @ X ) ) ) ) ) ) ) ).
% unat_plus_if_size
thf(fact_1606_uint__plus__if__size,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] :
( ( ( ord_less @ int @ ( plus_plus @ int @ ( semiring_1_unsigned @ A @ int @ X ) @ ( semiring_1_unsigned @ A @ int @ Y ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( size_size @ ( word @ A ) @ X ) ) )
=> ( ( semiring_1_unsigned @ A @ int @ ( plus_plus @ ( word @ A ) @ X @ Y ) )
= ( plus_plus @ int @ ( semiring_1_unsigned @ A @ int @ X ) @ ( semiring_1_unsigned @ A @ int @ Y ) ) ) )
& ( ~ ( ord_less @ int @ ( plus_plus @ int @ ( semiring_1_unsigned @ A @ int @ X ) @ ( semiring_1_unsigned @ A @ int @ Y ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( size_size @ ( word @ A ) @ X ) ) )
=> ( ( semiring_1_unsigned @ A @ int @ ( plus_plus @ ( word @ A ) @ X @ Y ) )
= ( minus_minus @ int @ ( plus_plus @ int @ ( semiring_1_unsigned @ A @ int @ X ) @ ( semiring_1_unsigned @ A @ int @ Y ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( size_size @ ( word @ A ) @ X ) ) ) ) ) ) ) ).
% uint_plus_if_size
thf(fact_1607_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_1608_powr__realpow,axiom,
! [X: real,N: nat] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( powr @ real @ X @ ( semiring_1_of_nat @ real @ N ) )
= ( power_power @ real @ X @ N ) ) ) ).
% powr_realpow
thf(fact_1609_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_1610_powr__less__iff,axiom,
! [B3: real,X: real,Y: real] :
( ( ord_less @ real @ ( one_one @ real ) @ B3 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ ( powr @ real @ B3 @ Y ) @ X )
= ( ord_less @ real @ Y @ ( log @ B3 @ X ) ) ) ) ) ).
% powr_less_iff
thf(fact_1611_less__powr__iff,axiom,
! [B3: real,X: real,Y: real] :
( ( ord_less @ real @ ( one_one @ real ) @ B3 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ X @ ( powr @ real @ B3 @ Y ) )
= ( ord_less @ real @ ( log @ B3 @ X ) @ Y ) ) ) ) ).
% less_powr_iff
thf(fact_1612_log__less__iff,axiom,
! [B3: real,X: real,Y: real] :
( ( ord_less @ real @ ( one_one @ real ) @ B3 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ ( log @ B3 @ X ) @ Y )
= ( ord_less @ real @ X @ ( powr @ real @ B3 @ Y ) ) ) ) ) ).
% log_less_iff
thf(fact_1613_less__log__iff,axiom,
! [B3: real,X: real,Y: real] :
( ( ord_less @ real @ ( one_one @ real ) @ B3 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ Y @ ( log @ B3 @ X ) )
= ( ord_less @ real @ ( powr @ real @ B3 @ Y ) @ X ) ) ) ) ).
% less_log_iff
thf(fact_1614_floor__log__eq__powr__iff,axiom,
! [X: real,B3: real,K: int] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ ( one_one @ real ) @ B3 )
=> ( ( ( archim6421214686448440834_floor @ real @ ( log @ B3 @ X ) )
= K )
= ( ( ord_less_eq @ real @ ( powr @ real @ B3 @ ( ring_1_of_int @ real @ K ) ) @ X )
& ( ord_less @ real @ X @ ( powr @ real @ B3 @ ( ring_1_of_int @ real @ ( plus_plus @ int @ K @ ( one_one @ int ) ) ) ) ) ) ) ) ) ).
% floor_log_eq_powr_iff
thf(fact_1615_floor__exists,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X: A] :
? [Z3: int] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Z3 ) @ X )
& ( ord_less @ A @ X @ ( ring_1_of_int @ A @ ( plus_plus @ int @ Z3 @ ( one_one @ int ) ) ) ) ) ) ).
% floor_exists
thf(fact_1616_floor__exists1,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X: A] :
? [X3: int] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ X3 ) @ X )
& ( ord_less @ A @ X @ ( ring_1_of_int @ A @ ( plus_plus @ int @ X3 @ ( one_one @ int ) ) ) )
& ! [Y5: int] :
( ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Y5 ) @ X )
& ( ord_less @ A @ X @ ( ring_1_of_int @ A @ ( plus_plus @ int @ Y5 @ ( one_one @ int ) ) ) ) )
=> ( Y5 = X3 ) ) ) ) ).
% floor_exists1
thf(fact_1617_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_1618_even__zero,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( zero_zero @ A ) ) ) ).
% even_zero
thf(fact_1619_of__int__ceiling__le__add__one,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [R3: A] : ( ord_less_eq @ A @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ A @ R3 ) ) @ ( plus_plus @ A @ R3 @ ( one_one @ A ) ) ) ) ).
% of_int_ceiling_le_add_one
thf(fact_1620_odd__even__add,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A3: A,B3: A] :
( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
=> ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B3 )
=> ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( plus_plus @ A @ A3 @ B3 ) ) ) ) ) ).
% odd_even_add
thf(fact_1621_odd__one,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( one_one @ A ) ) ) ).
% odd_one
thf(fact_1622_even__minus,axiom,
! [A: $tType] :
( ( ring_parity @ A )
=> ! [A3: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( uminus_uminus @ A @ A3 ) )
= ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) ) ) ).
% even_minus
thf(fact_1623_le__divide__eq__1,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B3: A,A3: A] :
( ( ord_less_eq @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B3 @ A3 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
& ( ord_less_eq @ A @ A3 @ B3 ) )
| ( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
& ( ord_less_eq @ A @ B3 @ A3 ) ) ) ) ) ).
% le_divide_eq_1
thf(fact_1624_divide__le__eq__1,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B3: A,A3: A] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ B3 @ A3 ) @ ( one_one @ A ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
& ( ord_less_eq @ A @ B3 @ A3 ) )
| ( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
& ( ord_less_eq @ A @ A3 @ B3 ) )
| ( A3
= ( zero_zero @ A ) ) ) ) ) ).
% divide_le_eq_1
thf(fact_1625_of__nat__less__of__int__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: nat,X: int] :
( ( ord_less @ A @ ( semiring_1_of_nat @ A @ N ) @ ( ring_1_of_int @ A @ X ) )
= ( ord_less @ int @ ( semiring_1_of_nat @ int @ N ) @ X ) ) ) ).
% of_nat_less_of_int_iff
thf(fact_1626_field__sum__of__halves,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A] :
( ( plus_plus @ A @ ( divide_divide @ A @ X @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ A @ X @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
= X ) ) ).
% field_sum_of_halves
thf(fact_1627_dvd__power__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [X: A,M: nat,N: nat] :
( ( X
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ ( power_power @ A @ X @ M ) @ ( power_power @ A @ X @ N ) )
= ( ( dvd_dvd @ A @ X @ ( one_one @ A ) )
| ( ord_less_eq @ nat @ M @ N ) ) ) ) ) ).
% dvd_power_iff
thf(fact_1628_dvd__power,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [N: nat,X: A] :
( ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
| ( X
= ( one_one @ A ) ) )
=> ( dvd_dvd @ A @ X @ ( power_power @ A @ X @ N ) ) ) ) ).
% dvd_power
thf(fact_1629_power2__commute,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [X: A,Y: A] :
( ( power_power @ A @ ( minus_minus @ A @ X @ Y ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( power_power @ A @ ( minus_minus @ A @ Y @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% power2_commute
thf(fact_1630_int__le__real__less,axiom,
( ( ord_less_eq @ int )
= ( ^ [N4: int,M3: int] : ( ord_less @ real @ ( ring_1_of_int @ real @ N4 ) @ ( plus_plus @ real @ ( ring_1_of_int @ real @ M3 ) @ ( one_one @ real ) ) ) ) ) ).
% int_le_real_less
thf(fact_1631_int__less__real__le,axiom,
( ( ord_less @ int )
= ( ^ [N4: int,M3: int] : ( ord_less_eq @ real @ ( plus_plus @ real @ ( ring_1_of_int @ real @ N4 ) @ ( one_one @ real ) ) @ ( ring_1_of_int @ real @ M3 ) ) ) ) ).
% int_less_real_le
thf(fact_1632_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_1633_word__neg__numeral__alt,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [B3: num] :
( ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ B3 ) )
= ( ring_1_of_int @ ( word @ A ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ B3 ) ) ) ) ) ).
% word_neg_numeral_alt
thf(fact_1634_ceiling__altdef,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ( ( archimedean_ceiling @ A )
= ( ^ [X2: A] :
( if @ int
@ ( X2
= ( ring_1_of_int @ A @ ( archim6421214686448440834_floor @ A @ X2 ) ) )
@ ( archim6421214686448440834_floor @ A @ X2 )
@ ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X2 ) @ ( one_one @ int ) ) ) ) ) ) ).
% ceiling_altdef
thf(fact_1635_psubset__insert__iff,axiom,
! [A: $tType,A2: set @ A,X: A,B2: set @ A] :
( ( ord_less @ ( set @ A ) @ A2 @ ( insert @ A @ X @ B2 ) )
= ( ( ( member @ A @ X @ B2 )
=> ( ord_less @ ( set @ A ) @ A2 @ B2 ) )
& ( ~ ( member @ A @ X @ B2 )
=> ( ( ( member @ A @ X @ A2 )
=> ( ord_less @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A2 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) @ B2 ) )
& ( ~ ( member @ A @ X @ A2 )
=> ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_1636_measure__unat,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [P4: word @ A] :
( ( P4
!= ( zero_zero @ ( word @ A ) ) )
=> ( ord_less @ nat @ ( semiring_1_unsigned @ A @ nat @ ( minus_minus @ ( word @ A ) @ P4 @ ( one_one @ ( word @ A ) ) ) ) @ ( semiring_1_unsigned @ A @ nat @ P4 ) ) ) ) ).
% measure_unat
thf(fact_1637_real__of__int__floor__add__one__gt,axiom,
! [R3: real] : ( ord_less @ real @ R3 @ ( plus_plus @ real @ ( ring_1_of_int @ real @ ( archim6421214686448440834_floor @ real @ R3 ) ) @ ( one_one @ real ) ) ) ).
% real_of_int_floor_add_one_gt
thf(fact_1638_floor__eq,axiom,
! [N: int,X: real] :
( ( ord_less @ real @ ( ring_1_of_int @ real @ N ) @ X )
=> ( ( ord_less @ real @ X @ ( plus_plus @ real @ ( ring_1_of_int @ real @ N ) @ ( one_one @ real ) ) )
=> ( ( archim6421214686448440834_floor @ real @ X )
= N ) ) ) ).
% floor_eq
thf(fact_1639_real__of__int__floor__add__one__ge,axiom,
! [R3: real] : ( ord_less_eq @ real @ R3 @ ( plus_plus @ real @ ( ring_1_of_int @ real @ ( archim6421214686448440834_floor @ real @ R3 ) ) @ ( one_one @ real ) ) ) ).
% real_of_int_floor_add_one_ge
thf(fact_1640_word__sub__plus__one__nonzero,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N7: word @ A,N: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ N7 @ N )
=> ( ( N7
!= ( zero_zero @ ( word @ A ) ) )
=> ( ( plus_plus @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ N @ N7 ) @ ( one_one @ ( word @ A ) ) )
!= ( zero_zero @ ( word @ A ) ) ) ) ) ) ).
% word_sub_plus_one_nonzero
thf(fact_1641_word__leq__minus__one__le,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y: word @ A,X: word @ A] :
( ( Y
!= ( zero_zero @ ( word @ A ) ) )
=> ( ( ord_less_eq @ ( word @ A ) @ X @ ( minus_minus @ ( word @ A ) @ Y @ ( one_one @ ( word @ A ) ) ) )
=> ( ord_less @ ( word @ A ) @ X @ Y ) ) ) ) ).
% word_leq_minus_one_le
thf(fact_1642_word__leq__le__minus__one,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ X @ Y )
=> ( ( X
!= ( zero_zero @ ( word @ A ) ) )
=> ( ord_less @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ X @ ( one_one @ ( word @ A ) ) ) @ Y ) ) ) ) ).
% word_leq_le_minus_one
thf(fact_1643_le__m1__iff__lt,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] :
( ( ord_less @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ X )
= ( ( ord_less_eq @ ( word @ A ) @ Y @ ( minus_minus @ ( word @ A ) @ X @ ( one_one @ ( word @ A ) ) ) )
= ( ord_less @ ( word @ A ) @ Y @ X ) ) ) ) ).
% le_m1_iff_lt
thf(fact_1644_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_1645_word__less__nowrapI,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Z: word @ A,K: word @ A] :
( ( ord_less @ ( word @ A ) @ X @ ( minus_minus @ ( word @ A ) @ Z @ K ) )
=> ( ( ord_less_eq @ ( word @ A ) @ K @ Z )
=> ( ( ord_less @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ K )
=> ( ord_less @ ( word @ A ) @ X @ ( plus_plus @ ( word @ A ) @ X @ K ) ) ) ) ) ) ).
% word_less_nowrapI
thf(fact_1646_plus__minus__not__NULL,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Ab: word @ A,C2: word @ A] :
( ( ord_less @ ( word @ A ) @ X @ ( minus_minus @ ( word @ A ) @ Ab @ C2 ) )
=> ( ( ord_less_eq @ ( word @ A ) @ C2 @ Ab )
=> ( ( C2
!= ( zero_zero @ ( word @ A ) ) )
=> ( ( plus_plus @ ( word @ A ) @ X @ C2 )
!= ( zero_zero @ ( word @ A ) ) ) ) ) ) ) ).
% plus_minus_not_NULL
thf(fact_1647_word__less__nowrapI_H,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Z: word @ A,K: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ X @ ( minus_minus @ ( word @ A ) @ Z @ K ) )
=> ( ( ord_less_eq @ ( word @ A ) @ K @ Z )
=> ( ( ord_less @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ K )
=> ( ord_less @ ( word @ A ) @ X @ ( plus_plus @ ( word @ A ) @ X @ K ) ) ) ) ) ) ).
% word_less_nowrapI'
thf(fact_1648_ceiling__diff__floor__le__1,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] : ( ord_less_eq @ int @ ( minus_minus @ int @ ( archimedean_ceiling @ A @ X ) @ ( archim6421214686448440834_floor @ A @ X ) ) @ ( one_one @ int ) ) ) ).
% ceiling_diff_floor_le_1
thf(fact_1649_powr__le__iff,axiom,
! [B3: real,X: real,Y: real] :
( ( ord_less @ real @ ( one_one @ real ) @ B3 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ ( powr @ real @ B3 @ Y ) @ X )
= ( ord_less_eq @ real @ Y @ ( log @ B3 @ X ) ) ) ) ) ).
% powr_le_iff
thf(fact_1650_le__powr__iff,axiom,
! [B3: real,X: real,Y: real] :
( ( ord_less @ real @ ( one_one @ real ) @ B3 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ X @ ( powr @ real @ B3 @ Y ) )
= ( ord_less_eq @ real @ ( log @ B3 @ X ) @ Y ) ) ) ) ).
% le_powr_iff
thf(fact_1651_log__le__iff,axiom,
! [B3: real,X: real,Y: real] :
( ( ord_less @ real @ ( one_one @ real ) @ B3 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ ( log @ B3 @ X ) @ Y )
= ( ord_less_eq @ real @ X @ ( powr @ real @ B3 @ Y ) ) ) ) ) ).
% log_le_iff
thf(fact_1652_le__log__iff,axiom,
! [B3: real,X: real,Y: real] :
( ( ord_less @ real @ ( one_one @ real ) @ B3 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ Y @ ( log @ B3 @ X ) )
= ( ord_less_eq @ real @ ( powr @ real @ B3 @ Y ) @ X ) ) ) ) ).
% le_log_iff
thf(fact_1653_floor__unique,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z: int,X: A] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Z ) @ X )
=> ( ( ord_less @ A @ X @ ( plus_plus @ A @ ( ring_1_of_int @ A @ Z ) @ ( one_one @ A ) ) )
=> ( ( archim6421214686448440834_floor @ A @ X )
= Z ) ) ) ) ).
% floor_unique
thf(fact_1654_floor__eq__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,A3: int] :
( ( ( archim6421214686448440834_floor @ A @ X )
= A3 )
= ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ A3 ) @ X )
& ( ord_less @ A @ X @ ( plus_plus @ A @ ( ring_1_of_int @ A @ A3 ) @ ( one_one @ A ) ) ) ) ) ) ).
% floor_eq_iff
thf(fact_1655_floor__split,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [P: int > $o,T: A] :
( ( P @ ( archim6421214686448440834_floor @ A @ T ) )
= ( ! [I4: int] :
( ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ I4 ) @ T )
& ( ord_less @ A @ T @ ( plus_plus @ A @ ( ring_1_of_int @ A @ I4 ) @ ( one_one @ A ) ) ) )
=> ( P @ I4 ) ) ) ) ) ).
% floor_split
thf(fact_1656_less__floor__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z: int,X: A] :
( ( ord_less @ int @ Z @ ( archim6421214686448440834_floor @ A @ X ) )
= ( ord_less_eq @ A @ ( plus_plus @ A @ ( ring_1_of_int @ A @ Z ) @ ( one_one @ A ) ) @ X ) ) ) ).
% less_floor_iff
thf(fact_1657_floor__le__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Z: int] :
( ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X ) @ Z )
= ( ord_less @ A @ X @ ( plus_plus @ A @ ( ring_1_of_int @ A @ Z ) @ ( one_one @ A ) ) ) ) ) ).
% floor_le_iff
thf(fact_1658_half__gt__zero__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% half_gt_zero_iff
thf(fact_1659_half__gt__zero,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% half_gt_zero
thf(fact_1660_field__less__half__sum,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ( ord_less @ A @ X @ ( divide_divide @ A @ ( plus_plus @ A @ X @ Y ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% field_less_half_sum
thf(fact_1661_floor__correct,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ ( archim6421214686448440834_floor @ A @ X ) ) @ X )
& ( ord_less @ A @ X @ ( ring_1_of_int @ A @ ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( one_one @ int ) ) ) ) ) ) ).
% floor_correct
thf(fact_1662_div__exp__eq,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A3: A,M: nat,N: nat] :
( ( divide_divide @ A @ ( divide_divide @ A @ A3 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) )
= ( divide_divide @ A @ A3 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( plus_plus @ nat @ M @ N ) ) ) ) ) ).
% div_exp_eq
thf(fact_1663_power__mono__odd,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: nat,A3: A,B3: A] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ord_less_eq @ A @ ( power_power @ A @ A3 @ N ) @ ( power_power @ A @ B3 @ N ) ) ) ) ) ).
% power_mono_odd
thf(fact_1664_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_1665_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_1666_uminus__power__if,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [N: nat,A3: A] :
( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( power_power @ A @ ( uminus_uminus @ A @ A3 ) @ N )
= ( power_power @ A @ A3 @ N ) ) )
& ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( power_power @ A @ ( uminus_uminus @ A @ A3 ) @ N )
= ( uminus_uminus @ A @ ( power_power @ A @ A3 @ N ) ) ) ) ) ) ).
% uminus_power_if
thf(fact_1667_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_1668_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_1669_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_1670_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_1671_floor__eq2,axiom,
! [N: int,X: real] :
( ( ord_less_eq @ real @ ( ring_1_of_int @ real @ N ) @ X )
=> ( ( ord_less @ real @ X @ ( plus_plus @ real @ ( ring_1_of_int @ real @ N ) @ ( one_one @ real ) ) )
=> ( ( archim6421214686448440834_floor @ real @ X )
= N ) ) ) ).
% floor_eq2
thf(fact_1672_zero__le__even__power,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: nat,A3: A] :
( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A3 @ N ) ) ) ) ).
% zero_le_even_power
thf(fact_1673_zero__le__odd__power,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: nat,A3: A] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A3 @ N ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 ) ) ) ) ).
% zero_le_odd_power
thf(fact_1674_zero__le__power__eq,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,N: nat] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A3 @ N ) )
= ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
| ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 ) ) ) ) ) ).
% zero_le_power_eq
thf(fact_1675_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_1676_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_1677_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_1678_ceiling__eq,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [N: int,X: A] :
( ( ord_less @ A @ ( ring_1_of_int @ A @ N ) @ X )
=> ( ( ord_less_eq @ A @ X @ ( plus_plus @ A @ ( ring_1_of_int @ A @ N ) @ ( one_one @ A ) ) )
=> ( ( archimedean_ceiling @ A @ X )
= ( plus_plus @ int @ N @ ( one_one @ int ) ) ) ) ) ) ).
% ceiling_eq
thf(fact_1679_zero__less__power__eq,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,N: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A3 @ N ) )
= ( ( N
= ( zero_zero @ nat ) )
| ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
& ( A3
!= ( zero_zero @ A ) ) )
| ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
& ( ord_less @ A @ ( zero_zero @ A ) @ A3 ) ) ) ) ) ).
% zero_less_power_eq
thf(fact_1680_odd__word__imp__even__next,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( semiring_1_unsigned @ A @ nat @ X ) )
=> ( ( ( plus_plus @ ( word @ A ) @ X @ ( one_one @ ( word @ A ) ) )
= ( zero_zero @ ( word @ A ) ) )
| ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( semiring_1_unsigned @ A @ nat @ ( plus_plus @ ( word @ A ) @ X @ ( one_one @ ( word @ A ) ) ) ) ) ) ) ) ).
% odd_word_imp_even_next
thf(fact_1681_even__word__imp__odd__next,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] :
( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( semiring_1_unsigned @ A @ nat @ X ) )
=> ( ( ( plus_plus @ ( word @ A ) @ X @ ( one_one @ ( word @ A ) ) )
= ( zero_zero @ ( word @ A ) ) )
| ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( semiring_1_unsigned @ A @ nat @ ( plus_plus @ ( word @ A ) @ X @ ( one_one @ ( word @ A ) ) ) ) ) ) ) ) ).
% even_word_imp_odd_next
thf(fact_1682_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_1683_power__le__zero__eq,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,N: nat] :
( ( ord_less_eq @ A @ ( power_power @ A @ A3 @ N ) @ ( zero_zero @ A ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
& ( ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
& ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) ) )
| ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
& ( A3
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% power_le_zero_eq
thf(fact_1684_no__plus__overflow__unat__size,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ X @ ( plus_plus @ ( word @ A ) @ X @ Y ) )
= ( ord_less @ nat @ ( plus_plus @ nat @ ( semiring_1_unsigned @ A @ nat @ X ) @ ( semiring_1_unsigned @ A @ nat @ Y ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( size_size @ ( word @ A ) @ X ) ) ) ) ) ).
% no_plus_overflow_unat_size
thf(fact_1685_no__plus__overflow__uint__size,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ X @ ( plus_plus @ ( word @ A ) @ X @ Y ) )
= ( ord_less @ int @ ( plus_plus @ int @ ( semiring_1_unsigned @ A @ int @ X ) @ ( semiring_1_unsigned @ A @ int @ Y ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( size_size @ ( word @ A ) @ X ) ) ) ) ) ).
% no_plus_overflow_uint_size
thf(fact_1686_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_1687_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_1688_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_1689_pow__divides__pow__iff,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [N: nat,A3: A,B3: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( dvd_dvd @ A @ ( power_power @ A @ A3 @ N ) @ ( power_power @ A @ B3 @ N ) )
= ( dvd_dvd @ A @ A3 @ B3 ) ) ) ) ).
% pow_divides_pow_iff
thf(fact_1690_div__less,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ N )
=> ( ( divide_divide @ nat @ M @ N )
= ( zero_zero @ nat ) ) ) ).
% div_less
thf(fact_1691_VEBT__internal_Olog__ceil__idem,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( one_one @ real ) @ X )
=> ( ( archimedean_ceiling @ real @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ X ) )
= ( archimedean_ceiling @ real @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( ring_1_of_int @ real @ ( archimedean_ceiling @ real @ X ) ) ) ) ) ) ).
% VEBT_internal.log_ceil_idem
thf(fact_1692_div__minus1__right,axiom,
! [A: $tType] :
( ( euclid8851590272496341667cancel @ A )
=> ! [A3: A] :
( ( divide_divide @ A @ A3 @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( uminus_uminus @ A @ A3 ) ) ) ).
% div_minus1_right
thf(fact_1693_less__two__pow__divI,axiom,
! [X: nat,N: nat,M: nat] :
( ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ N @ M ) ) )
=> ( ( ord_less_eq @ nat @ M @ N )
=> ( ord_less @ nat @ X @ ( divide_divide @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) ) ) ) ) ).
% less_two_pow_divI
thf(fact_1694_less__two__pow__divD,axiom,
! [X: nat,N: nat,M: nat] :
( ( ord_less @ nat @ X @ ( divide_divide @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) ) )
=> ( ( ord_less_eq @ nat @ M @ N )
& ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ N @ M ) ) ) ) ) ).
% less_two_pow_divD
thf(fact_1695_Diff__idemp,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A2 @ B2 ) @ B2 )
= ( minus_minus @ ( set @ A ) @ A2 @ B2 ) ) ).
% Diff_idemp
thf(fact_1696_Diff__iff,axiom,
! [A: $tType,C2: A,A2: set @ A,B2: set @ A] :
( ( member @ A @ C2 @ ( minus_minus @ ( set @ A ) @ A2 @ B2 ) )
= ( ( member @ A @ C2 @ A2 )
& ~ ( member @ A @ C2 @ B2 ) ) ) ).
% Diff_iff
thf(fact_1697_DiffI,axiom,
! [A: $tType,C2: A,A2: set @ A,B2: set @ A] :
( ( member @ A @ C2 @ A2 )
=> ( ~ ( member @ A @ C2 @ B2 )
=> ( member @ A @ C2 @ ( minus_minus @ ( set @ A ) @ A2 @ B2 ) ) ) ) ).
% DiffI
thf(fact_1698_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_1699_zdiv__numeral__Bit0,axiom,
! [V2: num,W: num] :
( ( divide_divide @ int @ ( numeral_numeral @ int @ ( bit0 @ V2 ) ) @ ( numeral_numeral @ int @ ( bit0 @ W ) ) )
= ( divide_divide @ int @ ( numeral_numeral @ int @ V2 ) @ ( numeral_numeral @ int @ W ) ) ) ).
% zdiv_numeral_Bit0
thf(fact_1700_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_1701_div__neg__neg__trivial,axiom,
! [K: int,L: int] :
( ( ord_less_eq @ int @ K @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ L @ K )
=> ( ( divide_divide @ int @ K @ L )
= ( zero_zero @ int ) ) ) ) ).
% div_neg_neg_trivial
thf(fact_1702_div__pos__pos__trivial,axiom,
! [K: int,L: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
=> ( ( ord_less @ int @ K @ L )
=> ( ( divide_divide @ int @ K @ L )
= ( zero_zero @ int ) ) ) ) ).
% div_pos_pos_trivial
thf(fact_1703_int__div__same__is__1,axiom,
! [A3: int,B3: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ A3 )
=> ( ( ( divide_divide @ int @ A3 @ B3 )
= A3 )
= ( B3
= ( one_one @ int ) ) ) ) ).
% int_div_same_is_1
thf(fact_1704_word__div__no,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: num,B3: num] :
( ( divide_divide @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ A3 ) @ ( numeral_numeral @ ( word @ A ) @ B3 ) )
= ( ring_1_of_int @ ( word @ A ) @ ( divide_divide @ int @ ( semiring_1_unsigned @ A @ int @ ( numeral_numeral @ ( word @ A ) @ A3 ) ) @ ( semiring_1_unsigned @ A @ int @ ( numeral_numeral @ ( word @ A ) @ B3 ) ) ) ) ) ) ).
% word_div_no
thf(fact_1705_floor__divide__eq__div__numeral,axiom,
! [A3: num,B3: num] :
( ( archim6421214686448440834_floor @ real @ ( divide_divide @ real @ ( numeral_numeral @ real @ A3 ) @ ( numeral_numeral @ real @ B3 ) ) )
= ( divide_divide @ int @ ( numeral_numeral @ int @ A3 ) @ ( numeral_numeral @ int @ B3 ) ) ) ).
% floor_divide_eq_div_numeral
thf(fact_1706_int__div__minus__is__minus1,axiom,
! [A3: int,B3: int] :
( ( ord_less @ int @ A3 @ ( zero_zero @ int ) )
=> ( ( ( divide_divide @ int @ A3 @ B3 )
= ( uminus_uminus @ int @ A3 ) )
= ( B3
= ( uminus_uminus @ int @ ( one_one @ int ) ) ) ) ) ).
% int_div_minus_is_minus1
thf(fact_1707_ceiling__divide__eq__div__numeral,axiom,
! [A3: num,B3: num] :
( ( archimedean_ceiling @ real @ ( divide_divide @ real @ ( numeral_numeral @ real @ A3 ) @ ( numeral_numeral @ real @ B3 ) ) )
= ( uminus_uminus @ int @ ( divide_divide @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ A3 ) ) @ ( numeral_numeral @ int @ B3 ) ) ) ) ).
% ceiling_divide_eq_div_numeral
thf(fact_1708_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_1709_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_1710_floor__one__divide__eq__div__numeral,axiom,
! [B3: num] :
( ( archim6421214686448440834_floor @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ B3 ) ) )
= ( divide_divide @ int @ ( one_one @ int ) @ ( numeral_numeral @ int @ B3 ) ) ) ).
% floor_one_divide_eq_div_numeral
thf(fact_1711_floor__minus__divide__eq__div__numeral,axiom,
! [A3: num,B3: num] :
( ( archim6421214686448440834_floor @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ ( numeral_numeral @ real @ A3 ) @ ( numeral_numeral @ real @ B3 ) ) ) )
= ( divide_divide @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ A3 ) ) @ ( numeral_numeral @ int @ B3 ) ) ) ).
% floor_minus_divide_eq_div_numeral
thf(fact_1712_ceiling__minus__divide__eq__div__numeral,axiom,
! [A3: num,B3: num] :
( ( archimedean_ceiling @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ ( numeral_numeral @ real @ A3 ) @ ( numeral_numeral @ real @ B3 ) ) ) )
= ( uminus_uminus @ int @ ( divide_divide @ int @ ( numeral_numeral @ int @ A3 ) @ ( numeral_numeral @ int @ B3 ) ) ) ) ).
% ceiling_minus_divide_eq_div_numeral
thf(fact_1713_floor__minus__one__divide__eq__div__numeral,axiom,
! [B3: num] :
( ( archim6421214686448440834_floor @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ B3 ) ) ) )
= ( divide_divide @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( numeral_numeral @ int @ B3 ) ) ) ).
% floor_minus_one_divide_eq_div_numeral
thf(fact_1714_real__of__int__div4,axiom,
! [N: int,X: int] : ( ord_less_eq @ real @ ( ring_1_of_int @ real @ ( divide_divide @ int @ N @ X ) ) @ ( divide_divide @ real @ ( ring_1_of_int @ real @ N ) @ ( ring_1_of_int @ real @ X ) ) ) ).
% real_of_int_div4
thf(fact_1715_word__div__def,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( divide_divide @ ( word @ A ) )
= ( ^ [A5: word @ A,B5: word @ A] : ( ring_1_of_int @ ( word @ A ) @ ( divide_divide @ int @ ( semiring_1_unsigned @ A @ int @ A5 ) @ ( semiring_1_unsigned @ A @ int @ B5 ) ) ) ) ) ) ).
% word_div_def
thf(fact_1716_uint__div__distrib,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [V2: word @ A,W: word @ A] :
( ( semiring_1_unsigned @ A @ int @ ( divide_divide @ ( word @ A ) @ V2 @ W ) )
= ( divide_divide @ int @ ( semiring_1_unsigned @ A @ int @ V2 ) @ ( semiring_1_unsigned @ A @ int @ W ) ) ) ) ).
% uint_div_distrib
thf(fact_1717_uint__div,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] :
( ( semiring_1_unsigned @ A @ int @ ( divide_divide @ ( word @ A ) @ X @ Y ) )
= ( divide_divide @ int @ ( semiring_1_unsigned @ A @ int @ X ) @ ( semiring_1_unsigned @ A @ int @ Y ) ) ) ) ).
% uint_div
thf(fact_1718_set__diff__eq,axiom,
! [A: $tType] :
( ( minus_minus @ ( set @ A ) )
= ( ^ [A4: set @ A,B4: set @ A] :
( collect @ A
@ ^ [X2: A] :
( ( member @ A @ X2 @ A4 )
& ~ ( member @ A @ X2 @ B4 ) ) ) ) ) ).
% set_diff_eq
thf(fact_1719_word__of__int__Ex,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] :
? [Y4: int] :
( X
= ( ring_1_of_int @ ( word @ A ) @ Y4 ) ) ) ).
% word_of_int_Ex
thf(fact_1720_wi__hom__sub,axiom,
! [B: $tType] :
( ( type_len @ B )
=> ! [A3: int,B3: int] :
( ( minus_minus @ ( word @ B ) @ ( ring_1_of_int @ ( word @ B ) @ A3 ) @ ( ring_1_of_int @ ( word @ B ) @ B3 ) )
= ( ring_1_of_int @ ( word @ B ) @ ( minus_minus @ int @ A3 @ B3 ) ) ) ) ).
% wi_hom_sub
thf(fact_1721_zdvd__zdiffD,axiom,
! [K: int,M: int,N: int] :
( ( dvd_dvd @ int @ K @ ( minus_minus @ int @ M @ N ) )
=> ( ( dvd_dvd @ int @ K @ N )
=> ( dvd_dvd @ int @ K @ M ) ) ) ).
% zdvd_zdiffD
thf(fact_1722_minus__set__def,axiom,
! [A: $tType] :
( ( minus_minus @ ( set @ A ) )
= ( ^ [A4: set @ A,B4: set @ A] :
( collect @ A
@ ( minus_minus @ ( A > $o )
@ ^ [X2: A] : ( member @ A @ X2 @ A4 )
@ ^ [X2: A] : ( member @ A @ X2 @ B4 ) ) ) ) ) ).
% minus_set_def
thf(fact_1723_DiffD2,axiom,
! [A: $tType,C2: A,A2: set @ A,B2: set @ A] :
( ( member @ A @ C2 @ ( minus_minus @ ( set @ A ) @ A2 @ B2 ) )
=> ~ ( member @ A @ C2 @ B2 ) ) ).
% DiffD2
thf(fact_1724_DiffD1,axiom,
! [A: $tType,C2: A,A2: set @ A,B2: set @ A] :
( ( member @ A @ C2 @ ( minus_minus @ ( set @ A ) @ A2 @ B2 ) )
=> ( member @ A @ C2 @ A2 ) ) ).
% DiffD1
thf(fact_1725_DiffE,axiom,
! [A: $tType,C2: A,A2: set @ A,B2: set @ A] :
( ( member @ A @ C2 @ ( minus_minus @ ( set @ A ) @ A2 @ B2 ) )
=> ~ ( ( member @ A @ C2 @ A2 )
=> ( member @ A @ C2 @ B2 ) ) ) ).
% DiffE
thf(fact_1726_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_1727_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_1728_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_1729_real__of__int__div2,axiom,
! [N: int,X: int] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( minus_minus @ real @ ( divide_divide @ real @ ( ring_1_of_int @ real @ N ) @ ( ring_1_of_int @ real @ X ) ) @ ( ring_1_of_int @ real @ ( divide_divide @ int @ N @ X ) ) ) ) ).
% real_of_int_div2
thf(fact_1730_real__of__int__div3,axiom,
! [N: int,X: int] : ( ord_less_eq @ real @ ( minus_minus @ real @ ( divide_divide @ real @ ( ring_1_of_int @ real @ N ) @ ( ring_1_of_int @ real @ X ) ) @ ( ring_1_of_int @ real @ ( divide_divide @ int @ N @ X ) ) ) @ ( one_one @ real ) ) ).
% real_of_int_div3
thf(fact_1731_length__induct,axiom,
! [A: $tType,P: ( list @ A ) > $o,Xs: list @ A] :
( ! [Xs2: list @ A] :
( ! [Ys2: list @ A] :
( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Ys2 ) @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( P @ Ys2 ) )
=> ( P @ Xs2 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_1732_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_1733_div__by__0__word,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] :
( ( divide_divide @ ( word @ A ) @ X @ ( zero_zero @ ( word @ A ) ) )
= ( zero_zero @ ( word @ A ) ) ) ) ).
% div_by_0_word
thf(fact_1734_floor__divide__real__eq__div,axiom,
! [B3: int,A3: real] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ B3 )
=> ( ( archim6421214686448440834_floor @ real @ ( divide_divide @ real @ A3 @ ( ring_1_of_int @ real @ B3 ) ) )
= ( divide_divide @ int @ ( archim6421214686448440834_floor @ real @ A3 ) @ B3 ) ) ) ).
% floor_divide_real_eq_div
thf(fact_1735_zdiv__le__dividend,axiom,
! [A3: int,B3: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ A3 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ B3 )
=> ( ord_less_eq @ int @ ( divide_divide @ int @ A3 @ B3 ) @ A3 ) ) ) ).
% zdiv_le_dividend
thf(fact_1736_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_1737_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_1738_pos__imp__zdiv__neg__iff,axiom,
! [B3: int,A3: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ B3 )
=> ( ( ord_less @ int @ ( divide_divide @ int @ A3 @ B3 ) @ ( zero_zero @ int ) )
= ( ord_less @ int @ A3 @ ( zero_zero @ int ) ) ) ) ).
% pos_imp_zdiv_neg_iff
thf(fact_1739_neg__imp__zdiv__neg__iff,axiom,
! [B3: int,A3: int] :
( ( ord_less @ int @ B3 @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ ( divide_divide @ int @ A3 @ B3 ) @ ( zero_zero @ int ) )
= ( ord_less @ int @ ( zero_zero @ int ) @ A3 ) ) ) ).
% neg_imp_zdiv_neg_iff
thf(fact_1740_div__neg__pos__less0,axiom,
! [A3: int,B3: int] :
( ( ord_less @ int @ A3 @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ B3 )
=> ( ord_less @ int @ ( divide_divide @ int @ A3 @ B3 ) @ ( zero_zero @ int ) ) ) ) ).
% div_neg_pos_less0
thf(fact_1741_zdiv__int,axiom,
! [A3: nat,B3: nat] :
( ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ A3 @ B3 ) )
= ( divide_divide @ int @ ( semiring_1_of_nat @ int @ A3 ) @ ( semiring_1_of_nat @ int @ B3 ) ) ) ).
% zdiv_int
thf(fact_1742_unat__div__distrib,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [V2: word @ A,W: word @ A] :
( ( semiring_1_unsigned @ A @ nat @ ( divide_divide @ ( word @ A ) @ V2 @ W ) )
= ( divide_divide @ nat @ ( semiring_1_unsigned @ A @ nat @ V2 ) @ ( semiring_1_unsigned @ A @ nat @ W ) ) ) ) ).
% unat_div_distrib
thf(fact_1743_unat__div,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] :
( ( semiring_1_unsigned @ A @ nat @ ( divide_divide @ ( word @ A ) @ X @ Y ) )
= ( divide_divide @ nat @ ( semiring_1_unsigned @ A @ nat @ X ) @ ( semiring_1_unsigned @ A @ nat @ Y ) ) ) ) ).
% unat_div
thf(fact_1744_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_1745_word__div__lt__eq__0,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] :
( ( ord_less @ ( word @ A ) @ X @ Y )
=> ( ( divide_divide @ ( word @ A ) @ X @ Y )
= ( zero_zero @ ( word @ A ) ) ) ) ) ).
% word_div_lt_eq_0
thf(fact_1746_word__less__div,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] :
( ( ( divide_divide @ ( word @ A ) @ X @ Y )
= ( zero_zero @ ( word @ A ) ) )
=> ( ( Y
= ( zero_zero @ ( word @ A ) ) )
| ( ord_less @ ( word @ A ) @ X @ Y ) ) ) ) ).
% word_less_div
thf(fact_1747_word__div__less,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A,V2: word @ A] :
( ( ord_less @ ( word @ A ) @ W @ V2 )
=> ( ( divide_divide @ ( word @ A ) @ W @ V2 )
= ( zero_zero @ ( word @ A ) ) ) ) ) ).
% word_div_less
thf(fact_1748_length__pos__if__in__set,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_1749_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_1750_zdiv__mono1,axiom,
! [A3: int,A7: int,B3: int] :
( ( ord_less_eq @ int @ A3 @ A7 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ B3 )
=> ( ord_less_eq @ int @ ( divide_divide @ int @ A3 @ B3 ) @ ( divide_divide @ int @ A7 @ B3 ) ) ) ) ).
% zdiv_mono1
thf(fact_1751_zdiv__mono2,axiom,
! [A3: int,B8: int,B3: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ A3 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ B8 )
=> ( ( ord_less_eq @ int @ B8 @ B3 )
=> ( ord_less_eq @ int @ ( divide_divide @ int @ A3 @ B3 ) @ ( divide_divide @ int @ A3 @ B8 ) ) ) ) ) ).
% zdiv_mono2
thf(fact_1752_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_1753_zdiv__mono1__neg,axiom,
! [A3: int,A7: int,B3: int] :
( ( ord_less_eq @ int @ A3 @ A7 )
=> ( ( ord_less @ int @ B3 @ ( zero_zero @ int ) )
=> ( ord_less_eq @ int @ ( divide_divide @ int @ A7 @ B3 ) @ ( divide_divide @ int @ A3 @ B3 ) ) ) ) ).
% zdiv_mono1_neg
thf(fact_1754_zdiv__mono2__neg,axiom,
! [A3: int,B8: int,B3: int] :
( ( ord_less @ int @ A3 @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ B8 )
=> ( ( ord_less_eq @ int @ B8 @ B3 )
=> ( ord_less_eq @ int @ ( divide_divide @ int @ A3 @ B8 ) @ ( divide_divide @ int @ A3 @ B3 ) ) ) ) ) ).
% zdiv_mono2_neg
thf(fact_1755_div__int__pos__iff,axiom,
! [K: int,L: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( divide_divide @ int @ K @ L ) )
= ( ( K
= ( zero_zero @ int ) )
| ( L
= ( zero_zero @ int ) )
| ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
& ( ord_less_eq @ int @ ( zero_zero @ int ) @ L ) )
| ( ( ord_less @ int @ K @ ( zero_zero @ int ) )
& ( ord_less @ int @ L @ ( zero_zero @ int ) ) ) ) ) ).
% div_int_pos_iff
thf(fact_1756_div__positive__int,axiom,
! [L: int,K: int] :
( ( ord_less_eq @ int @ L @ K )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ L )
=> ( ord_less @ int @ ( zero_zero @ int ) @ ( divide_divide @ int @ K @ L ) ) ) ) ).
% div_positive_int
thf(fact_1757_div__nonneg__neg__le0,axiom,
! [A3: int,B3: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ A3 )
=> ( ( ord_less @ int @ B3 @ ( zero_zero @ int ) )
=> ( ord_less_eq @ int @ ( divide_divide @ int @ A3 @ B3 ) @ ( zero_zero @ int ) ) ) ) ).
% div_nonneg_neg_le0
thf(fact_1758_div__nonpos__pos__le0,axiom,
! [A3: int,B3: int] :
( ( ord_less_eq @ int @ A3 @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ B3 )
=> ( ord_less_eq @ int @ ( divide_divide @ int @ A3 @ B3 ) @ ( zero_zero @ int ) ) ) ) ).
% div_nonpos_pos_le0
thf(fact_1759_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_1760_neg__imp__zdiv__nonneg__iff,axiom,
! [B3: int,A3: int] :
( ( ord_less @ int @ B3 @ ( zero_zero @ int ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( divide_divide @ int @ A3 @ B3 ) )
= ( ord_less_eq @ int @ A3 @ ( zero_zero @ int ) ) ) ) ).
% neg_imp_zdiv_nonneg_iff
thf(fact_1761_pos__imp__zdiv__nonneg__iff,axiom,
! [B3: int,A3: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ B3 )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( divide_divide @ int @ A3 @ B3 ) )
= ( ord_less_eq @ int @ ( zero_zero @ int ) @ A3 ) ) ) ).
% pos_imp_zdiv_nonneg_iff
thf(fact_1762_nonneg1__imp__zdiv__pos__iff,axiom,
! [A3: int,B3: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ A3 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ ( divide_divide @ int @ A3 @ B3 ) )
= ( ( ord_less_eq @ int @ B3 @ A3 )
& ( ord_less @ int @ ( zero_zero @ int ) @ B3 ) ) ) ) ).
% nonneg1_imp_zdiv_pos_iff
thf(fact_1763_int__div__less__self,axiom,
! [X: int,K: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ X )
=> ( ( ord_less @ int @ ( one_one @ int ) @ K )
=> ( ord_less @ int @ ( divide_divide @ int @ X @ K ) @ X ) ) ) ).
% int_div_less_self
thf(fact_1764_real__of__nat__div4,axiom,
! [N: nat,X: nat] : ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ ( divide_divide @ nat @ N @ X ) ) @ ( divide_divide @ real @ ( semiring_1_of_nat @ real @ N ) @ ( semiring_1_of_nat @ real @ X ) ) ) ).
% real_of_nat_div4
thf(fact_1765_div__less__dividend__word,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,N: word @ A] :
( ( X
!= ( zero_zero @ ( word @ A ) ) )
=> ( ( N
!= ( one_one @ ( word @ A ) ) )
=> ( ord_less @ ( word @ A ) @ ( divide_divide @ ( word @ A ) @ X @ N ) @ X ) ) ) ) ).
% div_less_dividend_word
thf(fact_1766_powr__divide,axiom,
! [X: real,Y: real,A3: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y )
=> ( ( powr @ real @ ( divide_divide @ real @ X @ Y ) @ A3 )
= ( divide_divide @ real @ ( powr @ real @ X @ A3 ) @ ( powr @ real @ Y @ A3 ) ) ) ) ) ).
% powr_divide
thf(fact_1767_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_1768_word__arith__nat__div,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( divide_divide @ ( word @ A ) )
= ( ^ [A5: word @ A,B5: word @ A] : ( semiring_1_of_nat @ ( word @ A ) @ ( divide_divide @ nat @ ( semiring_1_unsigned @ A @ nat @ A5 ) @ ( semiring_1_unsigned @ A @ nat @ B5 ) ) ) ) ) ) ).
% word_arith_nat_div
thf(fact_1769_log__base__powr,axiom,
! [A3: real,B3: real,X: real] :
( ( A3
!= ( zero_zero @ real ) )
=> ( ( log @ ( powr @ real @ A3 @ B3 ) @ X )
= ( divide_divide @ real @ ( log @ A3 @ X ) @ B3 ) ) ) ).
% log_base_powr
thf(fact_1770_num_Osize_I4_J,axiom,
( ( size_size @ num @ one2 )
= ( zero_zero @ nat ) ) ).
% num.size(4)
thf(fact_1771_even__diff__iff,axiom,
! [K: int,L: int] :
( ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( minus_minus @ int @ K @ L ) )
= ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( plus_plus @ int @ K @ L ) ) ) ).
% even_diff_iff
thf(fact_1772_verit__less__mono__div__int2,axiom,
! [A2: int,B2: int,N: int] :
( ( ord_less_eq @ int @ A2 @ B2 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ ( uminus_uminus @ int @ N ) )
=> ( ord_less_eq @ int @ ( divide_divide @ int @ B2 @ N ) @ ( divide_divide @ int @ A2 @ N ) ) ) ) ).
% verit_less_mono_div_int2
thf(fact_1773_div__eq__minus1,axiom,
! [B3: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ B3 )
=> ( ( divide_divide @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ B3 )
= ( uminus_uminus @ int @ ( one_one @ int ) ) ) ) ).
% div_eq_minus1
thf(fact_1774_real__of__nat__div2,axiom,
! [N: nat,X: nat] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( minus_minus @ real @ ( divide_divide @ real @ ( semiring_1_of_nat @ real @ N ) @ ( semiring_1_of_nat @ real @ X ) ) @ ( semiring_1_of_nat @ real @ ( divide_divide @ nat @ N @ X ) ) ) ) ).
% real_of_nat_div2
thf(fact_1775_real__of__nat__div3,axiom,
! [N: nat,X: nat] : ( ord_less_eq @ real @ ( minus_minus @ real @ ( divide_divide @ real @ ( semiring_1_of_nat @ real @ N ) @ ( semiring_1_of_nat @ real @ X ) ) @ ( semiring_1_of_nat @ real @ ( divide_divide @ nat @ N @ X ) ) ) @ ( one_one @ real ) ) ).
% real_of_nat_div3
thf(fact_1776_log__base__change,axiom,
! [A3: real,B3: real,X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ A3 )
=> ( ( A3
!= ( one_one @ real ) )
=> ( ( log @ B3 @ X )
= ( divide_divide @ real @ ( log @ A3 @ X ) @ ( log @ A3 @ B3 ) ) ) ) ) ).
% log_base_change
thf(fact_1777_log__divide,axiom,
! [A3: real,X: real,Y: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ A3 )
=> ( ( A3
!= ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ Y )
=> ( ( log @ A3 @ ( divide_divide @ real @ X @ Y ) )
= ( minus_minus @ real @ ( log @ A3 @ X ) @ ( log @ A3 @ Y ) ) ) ) ) ) ) ).
% log_divide
thf(fact_1778_word__div__sub,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y: word @ A,X: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ Y @ X )
=> ( ( ord_less @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ Y )
=> ( ( divide_divide @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ X @ Y ) @ Y )
= ( minus_minus @ ( word @ A ) @ ( divide_divide @ ( word @ A ) @ X @ Y ) @ ( one_one @ ( word @ A ) ) ) ) ) ) ) ).
% word_div_sub
thf(fact_1779_axxdiv2,axiom,
! [X: int] :
( ( ( divide_divide @ int @ ( plus_plus @ int @ ( plus_plus @ int @ ( one_one @ int ) @ X ) @ X ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) )
= X )
& ( ( divide_divide @ int @ ( plus_plus @ int @ ( plus_plus @ int @ ( zero_zero @ int ) @ X ) @ X ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) )
= X ) ) ).
% axxdiv2
thf(fact_1780_div__pos__neg__trivial,axiom,
! [K: int,L: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ K )
=> ( ( ord_less_eq @ int @ ( plus_plus @ int @ K @ L ) @ ( zero_zero @ int ) )
=> ( ( divide_divide @ int @ K @ L )
= ( uminus_uminus @ int @ ( one_one @ int ) ) ) ) ) ).
% div_pos_neg_trivial
thf(fact_1781_div__pos__geq,axiom,
! [L: int,K: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ L )
=> ( ( ord_less_eq @ int @ L @ K )
=> ( ( divide_divide @ int @ K @ L )
= ( plus_plus @ int @ ( divide_divide @ int @ ( minus_minus @ int @ K @ L ) @ L ) @ ( one_one @ int ) ) ) ) ) ).
% div_pos_geq
thf(fact_1782_powr__neg__one,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( powr @ real @ X @ ( uminus_uminus @ real @ ( one_one @ real ) ) )
= ( divide_divide @ real @ ( one_one @ real ) @ X ) ) ) ).
% powr_neg_one
thf(fact_1783_log__base__pow,axiom,
! [A3: real,N: nat,X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ A3 )
=> ( ( log @ ( power_power @ real @ A3 @ N ) @ X )
= ( divide_divide @ real @ ( log @ A3 @ X ) @ ( semiring_1_of_nat @ real @ N ) ) ) ) ).
% log_base_pow
thf(fact_1784_minus__log__eq__powr,axiom,
! [B3: real,X: real,Y: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ B3 )
=> ( ( B3
!= ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( minus_minus @ real @ Y @ ( log @ B3 @ X ) )
= ( log @ B3 @ ( divide_divide @ real @ ( powr @ real @ B3 @ Y ) @ X ) ) ) ) ) ) ).
% minus_log_eq_powr
thf(fact_1785_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_1786_word__less__two__pow__divD,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,N: nat,M: nat] :
( ( ord_less @ ( word @ A ) @ X @ ( divide_divide @ ( word @ A ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ M ) ) )
=> ( ( ord_less_eq @ nat @ M @ N )
& ( ord_less @ ( word @ A ) @ X @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ N @ M ) ) ) ) ) ) ).
% word_less_two_pow_divD
thf(fact_1787_Euclid__induct,axiom,
! [P: nat > nat > $o,A3: nat,B3: nat] :
( ! [A6: nat,B7: nat] :
( ( P @ A6 @ B7 )
= ( P @ B7 @ A6 ) )
=> ( ! [A6: nat] : ( P @ A6 @ ( zero_zero @ nat ) )
=> ( ! [A6: nat,B7: nat] :
( ( P @ A6 @ B7 )
=> ( P @ A6 @ ( plus_plus @ nat @ A6 @ B7 ) ) )
=> ( P @ A3 @ B3 ) ) ) ) ).
% Euclid_induct
thf(fact_1788_gcd__nat_Oextremum__uniqueI,axiom,
! [A3: nat] :
( ( dvd_dvd @ nat @ ( zero_zero @ nat ) @ A3 )
=> ( A3
= ( zero_zero @ nat ) ) ) ).
% gcd_nat.extremum_uniqueI
thf(fact_1789_gcd__nat_Onot__eq__extremum,axiom,
! [A3: nat] :
( ( A3
!= ( zero_zero @ nat ) )
= ( ( dvd_dvd @ nat @ A3 @ ( zero_zero @ nat ) )
& ( A3
!= ( zero_zero @ nat ) ) ) ) ).
% gcd_nat.not_eq_extremum
thf(fact_1790_gcd__nat_Oextremum__unique,axiom,
! [A3: nat] :
( ( dvd_dvd @ nat @ ( zero_zero @ nat ) @ A3 )
= ( A3
= ( zero_zero @ nat ) ) ) ).
% gcd_nat.extremum_unique
thf(fact_1791_gcd__nat_Oextremum__strict,axiom,
! [A3: nat] :
~ ( ( dvd_dvd @ nat @ ( zero_zero @ nat ) @ A3 )
& ( ( zero_zero @ nat )
!= A3 ) ) ).
% gcd_nat.extremum_strict
thf(fact_1792_gcd__nat_Oextremum,axiom,
! [A3: nat] : ( dvd_dvd @ nat @ A3 @ ( zero_zero @ nat ) ) ).
% gcd_nat.extremum
thf(fact_1793_powr__neg__numeral,axiom,
! [X: real,N: num] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( powr @ real @ X @ ( uminus_uminus @ real @ ( numeral_numeral @ real @ N ) ) )
= ( divide_divide @ real @ ( one_one @ real ) @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ N ) ) ) ) ) ).
% powr_neg_numeral
thf(fact_1794_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_1795_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_1796_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_1797_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_1798_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_1799_div__add__self2,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [B3: A,A3: A] :
( ( B3
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ A3 @ B3 ) @ B3 )
= ( plus_plus @ A @ ( divide_divide @ A @ A3 @ B3 ) @ ( one_one @ A ) ) ) ) ) ).
% div_add_self2
thf(fact_1800_div__add__self1,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [B3: A,A3: A] :
( ( B3
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ B3 @ A3 ) @ B3 )
= ( plus_plus @ A @ ( divide_divide @ A @ A3 @ B3 ) @ ( one_one @ A ) ) ) ) ) ).
% div_add_self1
thf(fact_1801_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_1802_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_1803_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_1804_power__diff__power__eq,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A3: A,N: nat,M: nat] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( ( ord_less_eq @ nat @ N @ M )
=> ( ( divide_divide @ A @ ( power_power @ A @ A3 @ M ) @ ( power_power @ A @ A3 @ N ) )
= ( power_power @ A @ A3 @ ( minus_minus @ nat @ M @ N ) ) ) )
& ( ~ ( ord_less_eq @ nat @ N @ M )
=> ( ( divide_divide @ A @ ( power_power @ A @ A3 @ M ) @ ( power_power @ A @ A3 @ N ) )
= ( divide_divide @ A @ ( one_one @ A ) @ ( power_power @ A @ A3 @ ( minus_minus @ nat @ N @ M ) ) ) ) ) ) ) ) ).
% power_diff_power_eq
thf(fact_1805_power__sub,axiom,
! [N: nat,M: nat,A3: nat] :
( ( ord_less_eq @ nat @ N @ M )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ A3 )
=> ( ( power_power @ nat @ A3 @ ( minus_minus @ nat @ M @ N ) )
= ( divide_divide @ nat @ ( power_power @ nat @ A3 @ M ) @ ( power_power @ nat @ A3 @ N ) ) ) ) ) ).
% power_sub
thf(fact_1806_power__minus__is__div,axiom,
! [B3: nat,A3: nat] :
( ( ord_less_eq @ nat @ B3 @ A3 )
=> ( ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ A3 @ B3 ) )
= ( divide_divide @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ A3 ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ B3 ) ) ) ) ).
% power_minus_is_div
thf(fact_1807_two__pow__div__gt__le,axiom,
! [V2: nat,N: nat,M: nat] :
( ( ord_less @ nat @ V2 @ ( 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_1808_real__average__minus__second,axiom,
! [B3: real,A3: real] :
( ( minus_minus @ real @ ( divide_divide @ real @ ( plus_plus @ real @ B3 @ A3 ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ A3 )
= ( divide_divide @ real @ ( minus_minus @ real @ B3 @ A3 ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ).
% real_average_minus_second
thf(fact_1809_real__average__minus__first,axiom,
! [A3: real,B3: real] :
( ( minus_minus @ real @ ( divide_divide @ real @ ( plus_plus @ real @ A3 @ B3 ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ A3 )
= ( divide_divide @ real @ ( minus_minus @ real @ B3 @ A3 ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ).
% real_average_minus_first
thf(fact_1810_zdiff__int__split,axiom,
! [P: int > $o,X: nat,Y: nat] :
( ( P @ ( semiring_1_of_nat @ int @ ( minus_minus @ nat @ X @ Y ) ) )
= ( ( ( ord_less_eq @ nat @ Y @ X )
=> ( P @ ( minus_minus @ int @ ( semiring_1_of_nat @ int @ X ) @ ( semiring_1_of_nat @ int @ Y ) ) ) )
& ( ( ord_less @ nat @ X @ Y )
=> ( P @ ( zero_zero @ int ) ) ) ) ) ).
% zdiff_int_split
thf(fact_1811_VEBT__internal_Opow__sum,axiom,
! [A3: nat,B3: nat] :
( ( divide_divide @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( plus_plus @ nat @ A3 @ B3 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ A3 ) )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ B3 ) ) ).
% VEBT_internal.pow_sum
thf(fact_1812_artanh__def,axiom,
! [A: $tType] :
( ( ( real_V3459762299906320749_field @ A )
& ( ln @ A ) )
=> ( ( artanh @ A )
= ( ^ [X2: A] : ( divide_divide @ A @ ( ln_ln @ A @ ( divide_divide @ A @ ( plus_plus @ A @ ( one_one @ A ) @ X2 ) @ ( minus_minus @ A @ ( one_one @ A ) @ X2 ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% artanh_def
thf(fact_1813_div2__even__ext__nat,axiom,
! [X: nat,Y: nat] :
( ( ( divide_divide @ nat @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( divide_divide @ nat @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
=> ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ X )
= ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Y ) )
=> ( X = Y ) ) ) ).
% div2_even_ext_nat
thf(fact_1814_ln__inj__iff,axiom,
! [X: real,Y: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ Y )
=> ( ( ( ln_ln @ real @ X )
= ( ln_ln @ real @ Y ) )
= ( X = Y ) ) ) ) ).
% ln_inj_iff
thf(fact_1815_ln__less__cancel__iff,axiom,
! [X: real,Y: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ Y )
=> ( ( ord_less @ real @ ( ln_ln @ real @ X ) @ ( ln_ln @ real @ Y ) )
= ( ord_less @ real @ X @ Y ) ) ) ) ).
% ln_less_cancel_iff
thf(fact_1816_ln__le__cancel__iff,axiom,
! [X: real,Y: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ Y )
=> ( ( ord_less_eq @ real @ ( ln_ln @ real @ X ) @ ( ln_ln @ real @ Y ) )
= ( ord_less_eq @ real @ X @ Y ) ) ) ) ).
% ln_le_cancel_iff
thf(fact_1817_ln__eq__zero__iff,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ( ln_ln @ real @ X )
= ( zero_zero @ real ) )
= ( X
= ( one_one @ real ) ) ) ) ).
% ln_eq_zero_iff
thf(fact_1818_ln__gt__zero__iff,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( ln_ln @ real @ X ) )
= ( ord_less @ real @ ( one_one @ real ) @ X ) ) ) ).
% ln_gt_zero_iff
thf(fact_1819_ln__less__zero__iff,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ ( ln_ln @ real @ X ) @ ( zero_zero @ real ) )
= ( ord_less @ real @ X @ ( one_one @ real ) ) ) ) ).
% ln_less_zero_iff
thf(fact_1820_ln__one,axiom,
! [A: $tType] :
( ( ln @ A )
=> ( ( ln_ln @ A @ ( one_one @ A ) )
= ( zero_zero @ A ) ) ) ).
% ln_one
thf(fact_1821_ln__le__zero__iff,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ ( ln_ln @ real @ X ) @ ( zero_zero @ real ) )
= ( ord_less_eq @ real @ X @ ( one_one @ real ) ) ) ) ).
% ln_le_zero_iff
thf(fact_1822_ln__ge__zero__iff,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( ln_ln @ real @ X ) )
= ( ord_less_eq @ real @ ( one_one @ real ) @ X ) ) ) ).
% ln_ge_zero_iff
thf(fact_1823_neq__if__length__neq,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs )
!= ( size_size @ ( list @ A ) @ Ys ) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
thf(fact_1824_Ex__list__of__length,axiom,
! [A: $tType,N: nat] :
? [Xs2: list @ A] :
( ( size_size @ ( list @ A ) @ Xs2 )
= N ) ).
% Ex_list_of_length
thf(fact_1825_ln__less__self,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ord_less @ real @ ( ln_ln @ real @ X ) @ X ) ) ).
% ln_less_self
thf(fact_1826_ln__bound,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ord_less_eq @ real @ ( ln_ln @ real @ X ) @ X ) ) ).
% ln_bound
thf(fact_1827_ln__gt__zero,axiom,
! [X: real] :
( ( ord_less @ real @ ( one_one @ real ) @ X )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( ln_ln @ real @ X ) ) ) ).
% ln_gt_zero
thf(fact_1828_ln__less__zero,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ X @ ( one_one @ real ) )
=> ( ord_less @ real @ ( ln_ln @ real @ X ) @ ( zero_zero @ real ) ) ) ) ).
% ln_less_zero
thf(fact_1829_ln__gt__zero__imp__gt__one,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( ln_ln @ real @ X ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ord_less @ real @ ( one_one @ real ) @ X ) ) ) ).
% ln_gt_zero_imp_gt_one
thf(fact_1830_ln__ge__zero,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( one_one @ real ) @ X )
=> ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( ln_ln @ real @ X ) ) ) ).
% ln_ge_zero
thf(fact_1831_ln__div,axiom,
! [X: real,Y: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ Y )
=> ( ( ln_ln @ real @ ( divide_divide @ real @ X @ Y ) )
= ( minus_minus @ real @ ( ln_ln @ real @ X ) @ ( ln_ln @ real @ Y ) ) ) ) ) ).
% ln_div
thf(fact_1832_pinf_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > $o,P5: A > $o,Q: A > $o,Q2: A > $o] :
( ? [Z6: A] :
! [X3: A] :
( ( ord_less @ A @ Z6 @ X3 )
=> ( ( P @ X3 )
= ( P5 @ X3 ) ) )
=> ( ? [Z6: A] :
! [X3: A] :
( ( ord_less @ A @ Z6 @ X3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z3: A] :
! [X6: A] :
( ( ord_less @ A @ Z3 @ X6 )
=> ( ( ( P @ X6 )
& ( Q @ X6 ) )
= ( ( P5 @ X6 )
& ( Q2 @ X6 ) ) ) ) ) ) ) ).
% pinf(1)
thf(fact_1833_pinf_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > $o,P5: A > $o,Q: A > $o,Q2: A > $o] :
( ? [Z6: A] :
! [X3: A] :
( ( ord_less @ A @ Z6 @ X3 )
=> ( ( P @ X3 )
= ( P5 @ X3 ) ) )
=> ( ? [Z6: A] :
! [X3: A] :
( ( ord_less @ A @ Z6 @ X3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z3: A] :
! [X6: A] :
( ( ord_less @ A @ Z3 @ X6 )
=> ( ( ( P @ X6 )
| ( Q @ X6 ) )
= ( ( P5 @ X6 )
| ( Q2 @ X6 ) ) ) ) ) ) ) ).
% pinf(2)
thf(fact_1834_pinf_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T: A] :
? [Z3: A] :
! [X6: A] :
( ( ord_less @ A @ Z3 @ X6 )
=> ( X6 != T ) ) ) ).
% pinf(3)
thf(fact_1835_pinf_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T: A] :
? [Z3: A] :
! [X6: A] :
( ( ord_less @ A @ Z3 @ X6 )
=> ( X6 != T ) ) ) ).
% pinf(4)
thf(fact_1836_pinf_I5_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T: A] :
? [Z3: A] :
! [X6: A] :
( ( ord_less @ A @ Z3 @ X6 )
=> ~ ( ord_less @ A @ X6 @ T ) ) ) ).
% pinf(5)
thf(fact_1837_pinf_I7_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T: A] :
? [Z3: A] :
! [X6: A] :
( ( ord_less @ A @ Z3 @ X6 )
=> ( ord_less @ A @ T @ X6 ) ) ) ).
% pinf(7)
thf(fact_1838_pinf_I11_J,axiom,
! [C: $tType,D: $tType] :
( ( ord @ C )
=> ! [F4: D] :
? [Z3: C] :
! [X6: C] :
( ( ord_less @ C @ Z3 @ X6 )
=> ( F4 = F4 ) ) ) ).
% pinf(11)
thf(fact_1839_minf_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > $o,P5: A > $o,Q: A > $o,Q2: A > $o] :
( ? [Z6: A] :
! [X3: A] :
( ( ord_less @ A @ X3 @ Z6 )
=> ( ( P @ X3 )
= ( P5 @ X3 ) ) )
=> ( ? [Z6: A] :
! [X3: A] :
( ( ord_less @ A @ X3 @ Z6 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z3: A] :
! [X6: A] :
( ( ord_less @ A @ X6 @ Z3 )
=> ( ( ( P @ X6 )
& ( Q @ X6 ) )
= ( ( P5 @ X6 )
& ( Q2 @ X6 ) ) ) ) ) ) ) ).
% minf(1)
thf(fact_1840_minf_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > $o,P5: A > $o,Q: A > $o,Q2: A > $o] :
( ? [Z6: A] :
! [X3: A] :
( ( ord_less @ A @ X3 @ Z6 )
=> ( ( P @ X3 )
= ( P5 @ X3 ) ) )
=> ( ? [Z6: A] :
! [X3: A] :
( ( ord_less @ A @ X3 @ Z6 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z3: A] :
! [X6: A] :
( ( ord_less @ A @ X6 @ Z3 )
=> ( ( ( P @ X6 )
| ( Q @ X6 ) )
= ( ( P5 @ X6 )
| ( Q2 @ X6 ) ) ) ) ) ) ) ).
% minf(2)
thf(fact_1841_minf_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T: A] :
? [Z3: A] :
! [X6: A] :
( ( ord_less @ A @ X6 @ Z3 )
=> ( X6 != T ) ) ) ).
% minf(3)
thf(fact_1842_minf_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T: A] :
? [Z3: A] :
! [X6: A] :
( ( ord_less @ A @ X6 @ Z3 )
=> ( X6 != T ) ) ) ).
% minf(4)
thf(fact_1843_minf_I5_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T: A] :
? [Z3: A] :
! [X6: A] :
( ( ord_less @ A @ X6 @ Z3 )
=> ( ord_less @ A @ X6 @ T ) ) ) ).
% minf(5)
thf(fact_1844_minf_I7_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T: A] :
? [Z3: A] :
! [X6: A] :
( ( ord_less @ A @ X6 @ Z3 )
=> ~ ( ord_less @ A @ T @ X6 ) ) ) ).
% minf(7)
thf(fact_1845_minf_I11_J,axiom,
! [C: $tType,D: $tType] :
( ( ord @ C )
=> ! [F4: D] :
? [Z3: C] :
! [X6: C] :
( ( ord_less @ C @ X6 @ Z3 )
=> ( F4 = F4 ) ) ) ).
% minf(11)
thf(fact_1846_ln__ge__zero__imp__ge__one,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( ln_ln @ real @ X ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ord_less_eq @ real @ ( one_one @ real ) @ X ) ) ) ).
% ln_ge_zero_imp_ge_one
thf(fact_1847_ln__add__one__self__le__self,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ord_less_eq @ real @ ( ln_ln @ real @ ( plus_plus @ real @ ( one_one @ real ) @ X ) ) @ X ) ) ).
% ln_add_one_self_le_self
thf(fact_1848_ln__diff__le,axiom,
! [X: real,Y: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ Y )
=> ( ord_less_eq @ real @ ( minus_minus @ real @ ( ln_ln @ real @ X ) @ ( ln_ln @ real @ Y ) ) @ ( divide_divide @ real @ ( minus_minus @ real @ X @ Y ) @ Y ) ) ) ) ).
% ln_diff_le
thf(fact_1849_ln__eq__minus__one,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ( ln_ln @ real @ X )
= ( minus_minus @ real @ X @ ( one_one @ real ) ) )
=> ( X
= ( one_one @ real ) ) ) ) ).
% ln_eq_minus_one
thf(fact_1850_ln__le__minus__one,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ord_less_eq @ real @ ( ln_ln @ real @ X ) @ ( minus_minus @ real @ X @ ( one_one @ real ) ) ) ) ).
% ln_le_minus_one
thf(fact_1851_ln__add__one__self__le__self2,axiom,
! [X: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X )
=> ( ord_less_eq @ real @ ( ln_ln @ real @ ( plus_plus @ real @ ( one_one @ real ) @ X ) ) @ X ) ) ).
% ln_add_one_self_le_self2
thf(fact_1852_ln__powr__bound,axiom,
! [X: real,A3: real] :
( ( ord_less_eq @ real @ ( one_one @ real ) @ X )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ A3 )
=> ( ord_less_eq @ real @ ( ln_ln @ real @ X ) @ ( divide_divide @ real @ ( powr @ real @ X @ A3 ) @ A3 ) ) ) ) ).
% ln_powr_bound
thf(fact_1853_ln__one__minus__pos__upper__bound,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ X @ ( one_one @ real ) )
=> ( ord_less_eq @ real @ ( ln_ln @ real @ ( minus_minus @ real @ ( one_one @ real ) @ X ) ) @ ( uminus_uminus @ real @ X ) ) ) ) ).
% ln_one_minus_pos_upper_bound
thf(fact_1854_pinf_I6_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T: A] :
? [Z3: A] :
! [X6: A] :
( ( ord_less @ A @ Z3 @ X6 )
=> ~ ( ord_less_eq @ A @ X6 @ T ) ) ) ).
% pinf(6)
thf(fact_1855_pinf_I8_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T: A] :
? [Z3: A] :
! [X6: A] :
( ( ord_less @ A @ Z3 @ X6 )
=> ( ord_less_eq @ A @ T @ X6 ) ) ) ).
% pinf(8)
thf(fact_1856_minf_I6_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T: A] :
? [Z3: A] :
! [X6: A] :
( ( ord_less @ A @ X6 @ Z3 )
=> ( ord_less_eq @ A @ X6 @ T ) ) ) ).
% minf(6)
thf(fact_1857_minf_I8_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T: A] :
? [Z3: A] :
! [X6: A] :
( ( ord_less @ A @ X6 @ Z3 )
=> ~ ( ord_less_eq @ A @ T @ X6 ) ) ) ).
% minf(8)
thf(fact_1858_imp__le__cong,axiom,
! [X: int,X7: int,P: $o,P5: $o] :
( ( X = X7 )
=> ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X7 )
=> ( P = P5 ) )
=> ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
=> P )
= ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X7 )
=> P5 ) ) ) ) ).
% imp_le_cong
thf(fact_1859_conj__le__cong,axiom,
! [X: int,X7: int,P: $o,P5: $o] :
( ( X = X7 )
=> ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X7 )
=> ( P = P5 ) )
=> ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
& P )
= ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X7 )
& P5 ) ) ) ) ).
% conj_le_cong
thf(fact_1860_ln__one__plus__pos__lower__bound,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ X @ ( one_one @ real ) )
=> ( ord_less_eq @ real @ ( minus_minus @ real @ X @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( ln_ln @ real @ ( plus_plus @ real @ ( one_one @ real ) @ X ) ) ) ) ) ).
% ln_one_plus_pos_lower_bound
thf(fact_1861_pinf_I9_J,axiom,
! [B: $tType] :
( ( ( plus @ B )
& ( linorder @ B )
& ( dvd @ B ) )
=> ! [D2: B,S2: B] :
? [Z3: B] :
! [X6: B] :
( ( ord_less @ B @ Z3 @ X6 )
=> ( ( dvd_dvd @ B @ D2 @ ( plus_plus @ B @ X6 @ S2 ) )
= ( dvd_dvd @ B @ D2 @ ( plus_plus @ B @ X6 @ S2 ) ) ) ) ) ).
% pinf(9)
thf(fact_1862_pinf_I10_J,axiom,
! [B: $tType] :
( ( ( plus @ B )
& ( linorder @ B )
& ( dvd @ B ) )
=> ! [D2: B,S2: B] :
? [Z3: B] :
! [X6: B] :
( ( ord_less @ B @ Z3 @ X6 )
=> ( ( ~ ( dvd_dvd @ B @ D2 @ ( plus_plus @ B @ X6 @ S2 ) ) )
= ( ~ ( dvd_dvd @ B @ D2 @ ( plus_plus @ B @ X6 @ S2 ) ) ) ) ) ) ).
% pinf(10)
thf(fact_1863_minf_I9_J,axiom,
! [B: $tType] :
( ( ( plus @ B )
& ( linorder @ B )
& ( dvd @ B ) )
=> ! [D2: B,S2: B] :
? [Z3: B] :
! [X6: B] :
( ( ord_less @ B @ X6 @ Z3 )
=> ( ( dvd_dvd @ B @ D2 @ ( plus_plus @ B @ X6 @ S2 ) )
= ( dvd_dvd @ B @ D2 @ ( plus_plus @ B @ X6 @ S2 ) ) ) ) ) ).
% minf(9)
thf(fact_1864_minf_I10_J,axiom,
! [B: $tType] :
( ( ( plus @ B )
& ( linorder @ B )
& ( dvd @ B ) )
=> ! [D2: B,S2: B] :
? [Z3: B] :
! [X6: B] :
( ( ord_less @ B @ X6 @ Z3 )
=> ( ( ~ ( dvd_dvd @ B @ D2 @ ( plus_plus @ B @ X6 @ S2 ) ) )
= ( ~ ( dvd_dvd @ B @ D2 @ ( plus_plus @ B @ X6 @ S2 ) ) ) ) ) ) ).
% minf(10)
thf(fact_1865_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_1866_Bolzano,axiom,
! [A3: real,B3: real,P: real > real > $o] :
( ( ord_less_eq @ real @ A3 @ B3 )
=> ( ! [A6: real,B7: real,C5: real] :
( ( P @ A6 @ B7 )
=> ( ( P @ B7 @ C5 )
=> ( ( ord_less_eq @ real @ A6 @ B7 )
=> ( ( ord_less_eq @ real @ B7 @ C5 )
=> ( P @ A6 @ C5 ) ) ) ) )
=> ( ! [X3: real] :
( ( ord_less_eq @ real @ A3 @ X3 )
=> ( ( ord_less_eq @ real @ X3 @ B3 )
=> ? [D5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D5 )
& ! [A6: real,B7: real] :
( ( ( ord_less_eq @ real @ A6 @ X3 )
& ( ord_less_eq @ real @ X3 @ B7 )
& ( ord_less @ real @ ( minus_minus @ real @ B7 @ A6 ) @ D5 ) )
=> ( P @ A6 @ B7 ) ) ) ) )
=> ( P @ A3 @ B3 ) ) ) ) ).
% Bolzano
thf(fact_1867_round__unique,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Y: int] :
( ( ord_less @ A @ ( minus_minus @ A @ X @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ ( ring_1_of_int @ A @ Y ) )
=> ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Y ) @ ( plus_plus @ A @ X @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) )
=> ( ( archimedean_round @ A @ X )
= Y ) ) ) ) ).
% round_unique
thf(fact_1868_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_1869_arcosh__def,axiom,
! [A: $tType] :
( ( ln @ A )
=> ( ( arcosh @ A )
= ( ^ [X2: A] : ( ln_ln @ A @ ( plus_plus @ A @ X2 @ ( powr @ A @ ( minus_minus @ A @ ( power_power @ A @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) ) @ ( real_Vector_of_real @ A @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% arcosh_def
thf(fact_1870_ln__one__minus__pos__lower__bound,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ X @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less_eq @ real @ ( minus_minus @ real @ ( uminus_uminus @ real @ X ) @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( ln_ln @ real @ ( minus_minus @ real @ ( one_one @ real ) @ X ) ) ) ) ) ).
% ln_one_minus_pos_lower_bound
thf(fact_1871_tanh__ln__real,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( tanh @ real @ ( ln_ln @ real @ X ) )
= ( divide_divide @ real @ ( minus_minus @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ real ) ) @ ( plus_plus @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ real ) ) ) ) ) ).
% tanh_ln_real
thf(fact_1872_mult__cancel__right,axiom,
! [A: $tType] :
( ( semiri6575147826004484403cancel @ A )
=> ! [A3: A,C2: A,B3: A] :
( ( ( times_times @ A @ A3 @ C2 )
= ( times_times @ A @ B3 @ C2 ) )
= ( ( C2
= ( zero_zero @ A ) )
| ( A3 = B3 ) ) ) ) ).
% mult_cancel_right
thf(fact_1873_mult__cancel__left,axiom,
! [A: $tType] :
( ( semiri6575147826004484403cancel @ A )
=> ! [C2: A,A3: A,B3: A] :
( ( ( times_times @ A @ C2 @ A3 )
= ( times_times @ A @ C2 @ B3 ) )
= ( ( C2
= ( zero_zero @ A ) )
| ( A3 = B3 ) ) ) ) ).
% mult_cancel_left
thf(fact_1874_mult__eq__0__iff,axiom,
! [A: $tType] :
( ( semiri3467727345109120633visors @ A )
=> ! [A3: A,B3: A] :
( ( ( times_times @ A @ A3 @ B3 )
= ( zero_zero @ A ) )
= ( ( A3
= ( zero_zero @ A ) )
| ( B3
= ( zero_zero @ A ) ) ) ) ) ).
% mult_eq_0_iff
thf(fact_1875_mult__zero__right,axiom,
! [A: $tType] :
( ( mult_zero @ A )
=> ! [A3: A] :
( ( times_times @ A @ A3 @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% mult_zero_right
thf(fact_1876_mult__zero__left,axiom,
! [A: $tType] :
( ( mult_zero @ A )
=> ! [A3: A] :
( ( times_times @ A @ ( zero_zero @ A ) @ A3 )
= ( zero_zero @ A ) ) ) ).
% mult_zero_left
thf(fact_1877_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_1878_mult__numeral__left__semiring__numeral,axiom,
! [A: $tType] :
( ( semiring_numeral @ A )
=> ! [V2: num,W: num,Z: A] :
( ( times_times @ A @ ( numeral_numeral @ A @ V2 ) @ ( times_times @ A @ ( numeral_numeral @ A @ W ) @ Z ) )
= ( times_times @ A @ ( numeral_numeral @ A @ ( times_times @ num @ V2 @ W ) ) @ Z ) ) ) ).
% mult_numeral_left_semiring_numeral
thf(fact_1879_mult__1,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A3: A] :
( ( times_times @ A @ ( one_one @ A ) @ A3 )
= A3 ) ) ).
% mult_1
thf(fact_1880_mult_Oright__neutral,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A3: A] :
( ( times_times @ A @ A3 @ ( one_one @ A ) )
= A3 ) ) ).
% mult.right_neutral
thf(fact_1881_mult__minus__left,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A3: A,B3: A] :
( ( times_times @ A @ ( uminus_uminus @ A @ A3 ) @ B3 )
= ( uminus_uminus @ A @ ( times_times @ A @ A3 @ B3 ) ) ) ) ).
% mult_minus_left
thf(fact_1882_minus__mult__minus,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A3: A,B3: A] :
( ( times_times @ A @ ( uminus_uminus @ A @ A3 ) @ ( uminus_uminus @ A @ B3 ) )
= ( times_times @ A @ A3 @ B3 ) ) ) ).
% minus_mult_minus
thf(fact_1883_mult__minus__right,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A3: A,B3: A] :
( ( times_times @ A @ A3 @ ( uminus_uminus @ A @ B3 ) )
= ( uminus_uminus @ A @ ( times_times @ A @ A3 @ B3 ) ) ) ) ).
% mult_minus_right
thf(fact_1884_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_1885_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_1886_real__divide__square__eq,axiom,
! [R3: real,A3: real] :
( ( divide_divide @ real @ ( times_times @ real @ R3 @ A3 ) @ ( times_times @ real @ R3 @ R3 ) )
= ( divide_divide @ real @ A3 @ R3 ) ) ).
% real_divide_square_eq
thf(fact_1887_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_1888_tanh__real__zero__iff,axiom,
! [X: real] :
( ( ( tanh @ real @ X )
= ( zero_zero @ real ) )
= ( X
= ( zero_zero @ real ) ) ) ).
% tanh_real_zero_iff
thf(fact_1889_tanh__real__less__iff,axiom,
! [X: real,Y: real] :
( ( ord_less @ real @ ( tanh @ real @ X ) @ ( tanh @ real @ Y ) )
= ( ord_less @ real @ X @ Y ) ) ).
% tanh_real_less_iff
thf(fact_1890_round__of__int,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [N: int] :
( ( archimedean_round @ A @ ( ring_1_of_int @ A @ N ) )
= N ) ) ).
% round_of_int
thf(fact_1891_sum__squares__eq__zero__iff,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [X: A,Y: A] :
( ( ( plus_plus @ A @ ( times_times @ A @ X @ X ) @ ( times_times @ A @ Y @ Y ) )
= ( zero_zero @ A ) )
= ( ( X
= ( zero_zero @ A ) )
& ( Y
= ( zero_zero @ A ) ) ) ) ) ).
% sum_squares_eq_zero_iff
thf(fact_1892_mult__cancel__right2,axiom,
! [A: $tType] :
( ( ring_15535105094025558882visors @ A )
=> ! [A3: A,C2: A] :
( ( ( times_times @ A @ A3 @ C2 )
= C2 )
= ( ( C2
= ( zero_zero @ A ) )
| ( A3
= ( one_one @ A ) ) ) ) ) ).
% mult_cancel_right2
thf(fact_1893_mult__cancel__right1,axiom,
! [A: $tType] :
( ( ring_15535105094025558882visors @ A )
=> ! [C2: A,B3: A] :
( ( C2
= ( times_times @ A @ B3 @ C2 ) )
= ( ( C2
= ( zero_zero @ A ) )
| ( B3
= ( one_one @ A ) ) ) ) ) ).
% mult_cancel_right1
thf(fact_1894_mult__cancel__left2,axiom,
! [A: $tType] :
( ( ring_15535105094025558882visors @ A )
=> ! [C2: A,A3: A] :
( ( ( times_times @ A @ C2 @ A3 )
= C2 )
= ( ( C2
= ( zero_zero @ A ) )
| ( A3
= ( one_one @ A ) ) ) ) ) ).
% mult_cancel_left2
thf(fact_1895_mult__cancel__left1,axiom,
! [A: $tType] :
( ( ring_15535105094025558882visors @ A )
=> ! [C2: A,B3: A] :
( ( C2
= ( times_times @ A @ C2 @ B3 ) )
= ( ( C2
= ( zero_zero @ A ) )
| ( B3
= ( one_one @ A ) ) ) ) ) ).
% mult_cancel_left1
thf(fact_1896_distrib__right__numeral,axiom,
! [A: $tType] :
( ( ( numeral @ A )
& ( semiring @ A ) )
=> ! [A3: A,B3: A,V2: num] :
( ( times_times @ A @ ( plus_plus @ A @ A3 @ B3 ) @ ( numeral_numeral @ A @ V2 ) )
= ( plus_plus @ A @ ( times_times @ A @ A3 @ ( numeral_numeral @ A @ V2 ) ) @ ( times_times @ A @ B3 @ ( numeral_numeral @ A @ V2 ) ) ) ) ) ).
% distrib_right_numeral
thf(fact_1897_distrib__left__numeral,axiom,
! [A: $tType] :
( ( ( numeral @ A )
& ( semiring @ A ) )
=> ! [V2: num,B3: A,C2: A] :
( ( times_times @ A @ ( numeral_numeral @ A @ V2 ) @ ( plus_plus @ A @ B3 @ C2 ) )
= ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ V2 ) @ B3 ) @ ( times_times @ A @ ( numeral_numeral @ A @ V2 ) @ C2 ) ) ) ) ).
% distrib_left_numeral
thf(fact_1898_div__mult__mult1__if,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [C2: A,A3: A,B3: A] :
( ( ( C2
= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B3 ) )
= ( zero_zero @ A ) ) )
& ( ( C2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B3 ) )
= ( divide_divide @ A @ A3 @ B3 ) ) ) ) ) ).
% div_mult_mult1_if
thf(fact_1899_div__mult__mult2,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [C2: A,A3: A,B3: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B3 @ C2 ) )
= ( divide_divide @ A @ A3 @ B3 ) ) ) ) ).
% div_mult_mult2
thf(fact_1900_div__mult__mult1,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [C2: A,A3: A,B3: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B3 ) )
= ( divide_divide @ A @ A3 @ B3 ) ) ) ) ).
% div_mult_mult1
thf(fact_1901_mult__divide__mult__cancel__left__if,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [C2: A,A3: A,B3: A] :
( ( ( C2
= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B3 ) )
= ( zero_zero @ A ) ) )
& ( ( C2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B3 ) )
= ( divide_divide @ A @ A3 @ B3 ) ) ) ) ) ).
% mult_divide_mult_cancel_left_if
thf(fact_1902_nonzero__mult__divide__mult__cancel__left,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [C2: A,A3: A,B3: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B3 ) )
= ( divide_divide @ A @ A3 @ B3 ) ) ) ) ).
% nonzero_mult_divide_mult_cancel_left
thf(fact_1903_nonzero__mult__divide__mult__cancel__left2,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [C2: A,A3: A,B3: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ B3 @ C2 ) )
= ( divide_divide @ A @ A3 @ B3 ) ) ) ) ).
% nonzero_mult_divide_mult_cancel_left2
thf(fact_1904_nonzero__mult__divide__mult__cancel__right,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [C2: A,A3: A,B3: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B3 @ C2 ) )
= ( divide_divide @ A @ A3 @ B3 ) ) ) ) ).
% nonzero_mult_divide_mult_cancel_right
thf(fact_1905_nonzero__mult__divide__mult__cancel__right2,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [C2: A,A3: A,B3: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ C2 @ B3 ) )
= ( divide_divide @ A @ A3 @ B3 ) ) ) ) ).
% nonzero_mult_divide_mult_cancel_right2
thf(fact_1906_nonzero__mult__div__cancel__left,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [A3: A,B3: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ A3 @ B3 ) @ A3 )
= B3 ) ) ) ).
% nonzero_mult_div_cancel_left
thf(fact_1907_nonzero__mult__div__cancel__right,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [B3: A,A3: A] :
( ( B3
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ A3 @ B3 ) @ B3 )
= A3 ) ) ) ).
% nonzero_mult_div_cancel_right
thf(fact_1908_left__diff__distrib__numeral,axiom,
! [A: $tType] :
( ( ( numeral @ A )
& ( ring @ A ) )
=> ! [A3: A,B3: A,V2: num] :
( ( times_times @ A @ ( minus_minus @ A @ A3 @ B3 ) @ ( numeral_numeral @ A @ V2 ) )
= ( minus_minus @ A @ ( times_times @ A @ A3 @ ( numeral_numeral @ A @ V2 ) ) @ ( times_times @ A @ B3 @ ( numeral_numeral @ A @ V2 ) ) ) ) ) ).
% left_diff_distrib_numeral
thf(fact_1909_right__diff__distrib__numeral,axiom,
! [A: $tType] :
( ( ( numeral @ A )
& ( ring @ A ) )
=> ! [V2: num,B3: A,C2: A] :
( ( times_times @ A @ ( numeral_numeral @ A @ V2 ) @ ( minus_minus @ A @ B3 @ C2 ) )
= ( minus_minus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ V2 ) @ B3 ) @ ( times_times @ A @ ( numeral_numeral @ A @ V2 ) @ C2 ) ) ) ) ).
% right_diff_distrib_numeral
thf(fact_1910_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_1911_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_1912_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_1913_semiring__norm_I170_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [V2: num,W: num,Y: A] :
( ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ W ) @ Y ) )
= ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( times_times @ num @ V2 @ W ) ) ) @ Y ) ) ) ).
% semiring_norm(170)
thf(fact_1914_semiring__norm_I171_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [V2: num,W: num,Y: A] :
( ( times_times @ A @ ( numeral_numeral @ A @ V2 ) @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) @ Y ) )
= ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( times_times @ num @ V2 @ W ) ) ) @ Y ) ) ) ).
% semiring_norm(171)
thf(fact_1915_semiring__norm_I172_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [V2: num,W: num,Y: A] :
( ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) @ Y ) )
= ( times_times @ A @ ( numeral_numeral @ A @ ( times_times @ num @ V2 @ W ) ) @ Y ) ) ) ).
% semiring_norm(172)
thf(fact_1916_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_1917_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_1918_dvd__mult__cancel__left,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [C2: A,A3: A,B3: A] :
( ( dvd_dvd @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B3 ) )
= ( ( C2
= ( zero_zero @ A ) )
| ( dvd_dvd @ A @ A3 @ B3 ) ) ) ) ).
% dvd_mult_cancel_left
thf(fact_1919_dvd__mult__cancel__right,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [A3: A,C2: A,B3: A] :
( ( dvd_dvd @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B3 @ C2 ) )
= ( ( C2
= ( zero_zero @ A ) )
| ( dvd_dvd @ A @ A3 @ B3 ) ) ) ) ).
% dvd_mult_cancel_right
thf(fact_1920_dvd__times__left__cancel__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B3: A,C2: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ ( times_times @ A @ A3 @ B3 ) @ ( times_times @ A @ A3 @ C2 ) )
= ( dvd_dvd @ A @ B3 @ C2 ) ) ) ) ).
% dvd_times_left_cancel_iff
thf(fact_1921_dvd__times__right__cancel__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B3: A,C2: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ ( times_times @ A @ B3 @ A3 ) @ ( times_times @ A @ C2 @ A3 ) )
= ( dvd_dvd @ A @ B3 @ C2 ) ) ) ) ).
% dvd_times_right_cancel_iff
thf(fact_1922_dvd__add__times__triv__right__iff,axiom,
! [A: $tType] :
( ( comm_s4317794764714335236cancel @ A )
=> ! [A3: A,B3: A,C2: A] :
( ( dvd_dvd @ A @ A3 @ ( plus_plus @ A @ B3 @ ( times_times @ A @ C2 @ A3 ) ) )
= ( dvd_dvd @ A @ A3 @ B3 ) ) ) ).
% dvd_add_times_triv_right_iff
thf(fact_1923_dvd__add__times__triv__left__iff,axiom,
! [A: $tType] :
( ( comm_s4317794764714335236cancel @ A )
=> ! [A3: A,C2: A,B3: A] :
( ( dvd_dvd @ A @ A3 @ ( plus_plus @ A @ ( times_times @ A @ C2 @ A3 ) @ B3 ) )
= ( dvd_dvd @ A @ A3 @ B3 ) ) ) ).
% dvd_add_times_triv_left_iff
thf(fact_1924_unit__prod,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B3: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ B3 @ ( one_one @ A ) )
=> ( dvd_dvd @ A @ ( times_times @ A @ A3 @ B3 ) @ ( one_one @ A ) ) ) ) ) ).
% unit_prod
thf(fact_1925_dvd__div__mult__self,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B3: A] :
( ( dvd_dvd @ A @ A3 @ B3 )
=> ( ( times_times @ A @ ( divide_divide @ A @ B3 @ A3 ) @ A3 )
= B3 ) ) ) ).
% dvd_div_mult_self
thf(fact_1926_dvd__mult__div__cancel,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B3: A] :
( ( dvd_dvd @ A @ A3 @ B3 )
=> ( ( times_times @ A @ A3 @ ( divide_divide @ A @ B3 @ A3 ) )
= B3 ) ) ) ).
% dvd_mult_div_cancel
thf(fact_1927_not__real__square__gt__zero,axiom,
! [X: real] :
( ( ~ ( ord_less @ real @ ( zero_zero @ real ) @ ( times_times @ real @ X @ X ) ) )
= ( X
= ( zero_zero @ real ) ) ) ).
% not_real_square_gt_zero
thf(fact_1928_round__0,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ( ( archimedean_round @ A @ ( zero_zero @ A ) )
= ( zero_zero @ int ) ) ) ).
% round_0
thf(fact_1929_round__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [N: num] :
( ( archimedean_round @ A @ ( numeral_numeral @ A @ N ) )
= ( numeral_numeral @ int @ N ) ) ) ).
% round_numeral
thf(fact_1930_tanh__real__pos__iff,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( tanh @ real @ X ) )
= ( ord_less @ real @ ( zero_zero @ real ) @ X ) ) ).
% tanh_real_pos_iff
thf(fact_1931_tanh__real__neg__iff,axiom,
! [X: real] :
( ( ord_less @ real @ ( tanh @ real @ X ) @ ( zero_zero @ real ) )
= ( ord_less @ real @ X @ ( zero_zero @ real ) ) ) ).
% tanh_real_neg_iff
thf(fact_1932_tanh__real__nonpos__iff,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( tanh @ real @ X ) @ ( zero_zero @ real ) )
= ( ord_less_eq @ real @ X @ ( zero_zero @ real ) ) ) ).
% tanh_real_nonpos_iff
thf(fact_1933_tanh__real__nonneg__iff,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( tanh @ real @ X ) )
= ( ord_less_eq @ real @ ( zero_zero @ real ) @ X ) ) ).
% tanh_real_nonneg_iff
thf(fact_1934_round__1,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ( ( archimedean_round @ A @ ( one_one @ A ) )
= ( one_one @ int ) ) ) ).
% round_1
thf(fact_1935_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_1936_divide__eq__eq__numeral1_I1_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B3: A,W: num,A3: A] :
( ( ( divide_divide @ A @ B3 @ ( numeral_numeral @ A @ W ) )
= A3 )
= ( ( ( ( numeral_numeral @ A @ W )
!= ( zero_zero @ A ) )
=> ( B3
= ( times_times @ A @ A3 @ ( numeral_numeral @ A @ W ) ) ) )
& ( ( ( numeral_numeral @ A @ W )
= ( zero_zero @ A ) )
=> ( A3
= ( zero_zero @ A ) ) ) ) ) ) ).
% divide_eq_eq_numeral1(1)
thf(fact_1937_eq__divide__eq__numeral1_I1_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A,B3: A,W: num] :
( ( A3
= ( divide_divide @ A @ B3 @ ( numeral_numeral @ A @ W ) ) )
= ( ( ( ( numeral_numeral @ A @ W )
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ A3 @ ( numeral_numeral @ A @ W ) )
= B3 ) )
& ( ( ( numeral_numeral @ A @ W )
= ( zero_zero @ A ) )
=> ( A3
= ( zero_zero @ A ) ) ) ) ) ) ).
% eq_divide_eq_numeral1(1)
thf(fact_1938_le__divide__eq__numeral1_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B3: A,W: num] :
( ( ord_less_eq @ A @ A3 @ ( divide_divide @ A @ B3 @ ( numeral_numeral @ A @ W ) ) )
= ( ord_less_eq @ A @ ( times_times @ A @ A3 @ ( numeral_numeral @ A @ W ) ) @ B3 ) ) ) ).
% le_divide_eq_numeral1(1)
thf(fact_1939_divide__le__eq__numeral1_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B3: A,W: num,A3: A] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ B3 @ ( numeral_numeral @ A @ W ) ) @ A3 )
= ( ord_less_eq @ A @ B3 @ ( times_times @ A @ A3 @ ( numeral_numeral @ A @ W ) ) ) ) ) ).
% divide_le_eq_numeral1(1)
thf(fact_1940_divide__less__eq__numeral1_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B3: A,W: num,A3: A] :
( ( ord_less @ A @ ( divide_divide @ A @ B3 @ ( numeral_numeral @ A @ W ) ) @ A3 )
= ( ord_less @ A @ B3 @ ( times_times @ A @ A3 @ ( numeral_numeral @ A @ W ) ) ) ) ) ).
% divide_less_eq_numeral1(1)
thf(fact_1941_less__divide__eq__numeral1_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B3: A,W: num] :
( ( ord_less @ A @ A3 @ ( divide_divide @ A @ B3 @ ( numeral_numeral @ A @ W ) ) )
= ( ord_less @ A @ ( times_times @ A @ A3 @ ( numeral_numeral @ A @ W ) ) @ B3 ) ) ) ).
% less_divide_eq_numeral1(1)
thf(fact_1942_div__mult__self4,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [B3: A,C2: A,A3: A] :
( ( B3
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ ( times_times @ A @ B3 @ C2 ) @ A3 ) @ B3 )
= ( plus_plus @ A @ C2 @ ( divide_divide @ A @ A3 @ B3 ) ) ) ) ) ).
% div_mult_self4
thf(fact_1943_div__mult__self3,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [B3: A,C2: A,A3: A] :
( ( B3
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ ( times_times @ A @ C2 @ B3 ) @ A3 ) @ B3 )
= ( plus_plus @ A @ C2 @ ( divide_divide @ A @ A3 @ B3 ) ) ) ) ) ).
% div_mult_self3
thf(fact_1944_div__mult__self2,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [B3: A,A3: A,C2: A] :
( ( B3
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ A3 @ ( times_times @ A @ B3 @ C2 ) ) @ B3 )
= ( plus_plus @ A @ C2 @ ( divide_divide @ A @ A3 @ B3 ) ) ) ) ) ).
% div_mult_self2
thf(fact_1945_div__mult__self1,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [B3: A,A3: A,C2: A] :
( ( B3
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ A3 @ ( times_times @ A @ C2 @ B3 ) ) @ B3 )
= ( plus_plus @ A @ C2 @ ( divide_divide @ A @ A3 @ B3 ) ) ) ) ) ).
% div_mult_self1
thf(fact_1946_nonzero__divide__mult__cancel__right,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [B3: A,A3: A] :
( ( B3
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ B3 @ ( times_times @ A @ A3 @ B3 ) )
= ( divide_divide @ A @ ( one_one @ A ) @ A3 ) ) ) ) ).
% nonzero_divide_mult_cancel_right
thf(fact_1947_nonzero__divide__mult__cancel__left,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A3: A,B3: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ A3 @ ( times_times @ A @ A3 @ B3 ) )
= ( divide_divide @ A @ ( one_one @ A ) @ B3 ) ) ) ) ).
% nonzero_divide_mult_cancel_left
thf(fact_1948_left__minus__one__mult__self,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [N: nat,A3: A] :
( ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N ) @ ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N ) @ A3 ) )
= A3 ) ) ).
% left_minus_one_mult_self
thf(fact_1949_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_1950_unit__div__mult__self,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B3: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ( times_times @ A @ ( divide_divide @ A @ B3 @ A3 ) @ A3 )
= B3 ) ) ) ).
% unit_div_mult_self
thf(fact_1951_unit__mult__div__div,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B3: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ( times_times @ A @ B3 @ ( divide_divide @ A @ ( one_one @ A ) @ A3 ) )
= ( divide_divide @ A @ B3 @ A3 ) ) ) ) ).
% unit_mult_div_div
thf(fact_1952_power__add__numeral,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A3: A,M: num,N: num] :
( ( times_times @ A @ ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ M ) ) @ ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ N ) ) )
= ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ ( plus_plus @ num @ M @ N ) ) ) ) ) ).
% power_add_numeral
thf(fact_1953_power__add__numeral2,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A3: A,M: num,N: num,B3: A] :
( ( times_times @ A @ ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ M ) ) @ ( times_times @ A @ ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ N ) ) @ B3 ) )
= ( times_times @ A @ ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ ( plus_plus @ num @ M @ N ) ) ) @ B3 ) ) ) ).
% power_add_numeral2
thf(fact_1954_divide__eq__eq__numeral1_I2_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B3: A,W: num,A3: A] :
( ( ( divide_divide @ A @ B3 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) )
= A3 )
= ( ( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) )
!= ( zero_zero @ A ) )
=> ( B3
= ( times_times @ A @ A3 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) ) ) )
& ( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) )
= ( zero_zero @ A ) )
=> ( A3
= ( zero_zero @ A ) ) ) ) ) ) ).
% divide_eq_eq_numeral1(2)
thf(fact_1955_eq__divide__eq__numeral1_I2_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A,B3: A,W: num] :
( ( A3
= ( divide_divide @ A @ B3 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) ) )
= ( ( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) )
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ A3 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) )
= B3 ) )
& ( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) )
= ( zero_zero @ A ) )
=> ( A3
= ( zero_zero @ A ) ) ) ) ) ) ).
% eq_divide_eq_numeral1(2)
thf(fact_1956_le__divide__eq__numeral1_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B3: A,W: num] :
( ( ord_less_eq @ A @ A3 @ ( divide_divide @ A @ B3 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) ) )
= ( ord_less_eq @ A @ B3 @ ( times_times @ A @ A3 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) ) ) ) ) ).
% le_divide_eq_numeral1(2)
thf(fact_1957_divide__le__eq__numeral1_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B3: A,W: num,A3: A] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ B3 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) ) @ A3 )
= ( ord_less_eq @ A @ ( times_times @ A @ A3 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) ) @ B3 ) ) ) ).
% divide_le_eq_numeral1(2)
thf(fact_1958_divide__less__eq__numeral1_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B3: A,W: num,A3: A] :
( ( ord_less @ A @ ( divide_divide @ A @ B3 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) ) @ A3 )
= ( ord_less @ A @ ( times_times @ A @ A3 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) ) @ B3 ) ) ) ).
% divide_less_eq_numeral1(2)
thf(fact_1959_less__divide__eq__numeral1_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B3: A,W: num] :
( ( ord_less @ A @ A3 @ ( divide_divide @ A @ B3 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) ) )
= ( ord_less @ A @ B3 @ ( times_times @ A @ A3 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) ) ) ) ) ).
% less_divide_eq_numeral1(2)
thf(fact_1960_even__mult__iff,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A3: A,B3: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( times_times @ A @ A3 @ B3 ) )
= ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
| ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B3 ) ) ) ) ).
% even_mult_iff
thf(fact_1961_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_1962_odd__two__times__div__two__succ,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [A3: A] :
( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
=> ( ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ ( one_one @ A ) )
= A3 ) ) ) ).
% odd_two_times_div_two_succ
thf(fact_1963_mult__commute__abs,axiom,
! [A: $tType] :
( ( ab_semigroup_mult @ A )
=> ! [C2: A] :
( ( ^ [X2: A] : ( times_times @ A @ X2 @ C2 ) )
= ( times_times @ A @ C2 ) ) ) ).
% mult_commute_abs
thf(fact_1964_mult_Oassoc,axiom,
! [A: $tType] :
( ( semigroup_mult @ A )
=> ! [A3: A,B3: A,C2: A] :
( ( times_times @ A @ ( times_times @ A @ A3 @ B3 ) @ C2 )
= ( times_times @ A @ A3 @ ( times_times @ A @ B3 @ C2 ) ) ) ) ).
% mult.assoc
thf(fact_1965_ab__semigroup__mult__class_Omult_Ocommute,axiom,
! [A: $tType] :
( ( ab_semigroup_mult @ A )
=> ( ( times_times @ A )
= ( ^ [A5: A,B5: A] : ( times_times @ A @ B5 @ A5 ) ) ) ) ).
% ab_semigroup_mult_class.mult.commute
thf(fact_1966_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
! [A: $tType] :
( ( ab_semigroup_mult @ A )
=> ! [B3: A,A3: A,C2: A] :
( ( times_times @ A @ B3 @ ( times_times @ A @ A3 @ C2 ) )
= ( times_times @ A @ A3 @ ( times_times @ A @ B3 @ C2 ) ) ) ) ).
% ab_semigroup_mult_class.mult.left_commute
thf(fact_1967_mult__right__cancel,axiom,
! [A: $tType] :
( ( semiri6575147826004484403cancel @ A )
=> ! [C2: A,A3: A,B3: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( ( times_times @ A @ A3 @ C2 )
= ( times_times @ A @ B3 @ C2 ) )
= ( A3 = B3 ) ) ) ) ).
% mult_right_cancel
thf(fact_1968_mult__left__cancel,axiom,
! [A: $tType] :
( ( semiri6575147826004484403cancel @ A )
=> ! [C2: A,A3: A,B3: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( ( times_times @ A @ C2 @ A3 )
= ( times_times @ A @ C2 @ B3 ) )
= ( A3 = B3 ) ) ) ) ).
% mult_left_cancel
thf(fact_1969_no__zero__divisors,axiom,
! [A: $tType] :
( ( semiri3467727345109120633visors @ A )
=> ! [A3: A,B3: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( B3
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ A3 @ B3 )
!= ( zero_zero @ A ) ) ) ) ) ).
% no_zero_divisors
thf(fact_1970_divisors__zero,axiom,
! [A: $tType] :
( ( semiri3467727345109120633visors @ A )
=> ! [A3: A,B3: A] :
( ( ( times_times @ A @ A3 @ B3 )
= ( zero_zero @ A ) )
=> ( ( A3
= ( zero_zero @ A ) )
| ( B3
= ( zero_zero @ A ) ) ) ) ) ).
% divisors_zero
thf(fact_1971_mult__not__zero,axiom,
! [A: $tType] :
( ( mult_zero @ A )
=> ! [A3: A,B3: A] :
( ( ( times_times @ A @ A3 @ B3 )
!= ( zero_zero @ A ) )
=> ( ( A3
!= ( zero_zero @ A ) )
& ( B3
!= ( zero_zero @ A ) ) ) ) ) ).
% mult_not_zero
thf(fact_1972_ring__class_Oring__distribs_I2_J,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A3: A,B3: A,C2: A] :
( ( times_times @ A @ ( plus_plus @ A @ A3 @ B3 ) @ C2 )
= ( plus_plus @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B3 @ C2 ) ) ) ) ).
% ring_class.ring_distribs(2)
thf(fact_1973_ring__class_Oring__distribs_I1_J,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A3: A,B3: A,C2: A] :
( ( times_times @ A @ A3 @ ( plus_plus @ A @ B3 @ C2 ) )
= ( plus_plus @ A @ ( times_times @ A @ A3 @ B3 ) @ ( times_times @ A @ A3 @ C2 ) ) ) ) ).
% ring_class.ring_distribs(1)
thf(fact_1974_comm__semiring__class_Odistrib,axiom,
! [A: $tType] :
( ( comm_semiring @ A )
=> ! [A3: A,B3: A,C2: A] :
( ( times_times @ A @ ( plus_plus @ A @ A3 @ B3 ) @ C2 )
= ( plus_plus @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B3 @ C2 ) ) ) ) ).
% comm_semiring_class.distrib
thf(fact_1975_distrib__left,axiom,
! [A: $tType] :
( ( semiring @ A )
=> ! [A3: A,B3: A,C2: A] :
( ( times_times @ A @ A3 @ ( plus_plus @ A @ B3 @ C2 ) )
= ( plus_plus @ A @ ( times_times @ A @ A3 @ B3 ) @ ( times_times @ A @ A3 @ C2 ) ) ) ) ).
% distrib_left
thf(fact_1976_distrib__right,axiom,
! [A: $tType] :
( ( semiring @ A )
=> ! [A3: A,B3: A,C2: A] :
( ( times_times @ A @ ( plus_plus @ A @ A3 @ B3 ) @ C2 )
= ( plus_plus @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B3 @ C2 ) ) ) ) ).
% distrib_right
thf(fact_1977_combine__common__factor,axiom,
! [A: $tType] :
( ( semiring @ A )
=> ! [A3: A,E2: A,B3: A,C2: A] :
( ( plus_plus @ A @ ( times_times @ A @ A3 @ E2 ) @ ( plus_plus @ A @ ( times_times @ A @ B3 @ E2 ) @ C2 ) )
= ( plus_plus @ A @ ( times_times @ A @ ( plus_plus @ A @ A3 @ B3 ) @ E2 ) @ C2 ) ) ) ).
% combine_common_factor
thf(fact_1978_mult_Ocomm__neutral,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: A] :
( ( times_times @ A @ A3 @ ( one_one @ A ) )
= A3 ) ) ).
% mult.comm_neutral
thf(fact_1979_comm__monoid__mult__class_Omult__1,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: A] :
( ( times_times @ A @ ( one_one @ A ) @ A3 )
= A3 ) ) ).
% comm_monoid_mult_class.mult_1
thf(fact_1980_left__diff__distrib,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A3: A,B3: A,C2: A] :
( ( times_times @ A @ ( minus_minus @ A @ A3 @ B3 ) @ C2 )
= ( minus_minus @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B3 @ C2 ) ) ) ) ).
% left_diff_distrib
thf(fact_1981_right__diff__distrib,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A3: A,B3: A,C2: A] :
( ( times_times @ A @ A3 @ ( minus_minus @ A @ B3 @ C2 ) )
= ( minus_minus @ A @ ( times_times @ A @ A3 @ B3 ) @ ( times_times @ A @ A3 @ C2 ) ) ) ) ).
% right_diff_distrib
thf(fact_1982_left__diff__distrib_H,axiom,
! [A: $tType] :
( ( comm_s4317794764714335236cancel @ A )
=> ! [B3: A,C2: A,A3: A] :
( ( times_times @ A @ ( minus_minus @ A @ B3 @ C2 ) @ A3 )
= ( minus_minus @ A @ ( times_times @ A @ B3 @ A3 ) @ ( times_times @ A @ C2 @ A3 ) ) ) ) ).
% left_diff_distrib'
thf(fact_1983_right__diff__distrib_H,axiom,
! [A: $tType] :
( ( comm_s4317794764714335236cancel @ A )
=> ! [A3: A,B3: A,C2: A] :
( ( times_times @ A @ A3 @ ( minus_minus @ A @ B3 @ C2 ) )
= ( minus_minus @ A @ ( times_times @ A @ A3 @ B3 ) @ ( times_times @ A @ A3 @ C2 ) ) ) ) ).
% right_diff_distrib'
thf(fact_1984_power__commutes,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A3: A,N: nat] :
( ( times_times @ A @ ( power_power @ A @ A3 @ N ) @ A3 )
= ( times_times @ A @ A3 @ ( power_power @ A @ A3 @ N ) ) ) ) ).
% power_commutes
thf(fact_1985_power__mult__distrib,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: A,B3: A,N: nat] :
( ( power_power @ A @ ( times_times @ A @ A3 @ B3 ) @ N )
= ( times_times @ A @ ( power_power @ A @ A3 @ N ) @ ( power_power @ A @ B3 @ N ) ) ) ) ).
% power_mult_distrib
thf(fact_1986_power__commuting__commutes,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [X: A,Y: A,N: nat] :
( ( ( times_times @ A @ X @ Y )
= ( times_times @ A @ Y @ X ) )
=> ( ( times_times @ A @ ( power_power @ A @ X @ N ) @ Y )
= ( times_times @ A @ Y @ ( power_power @ A @ X @ N ) ) ) ) ) ).
% power_commuting_commutes
thf(fact_1987_square__eq__iff,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [A3: A,B3: A] :
( ( ( times_times @ A @ A3 @ A3 )
= ( times_times @ A @ B3 @ B3 ) )
= ( ( A3 = B3 )
| ( A3
= ( uminus_uminus @ A @ B3 ) ) ) ) ) ).
% square_eq_iff
thf(fact_1988_minus__mult__commute,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A3: A,B3: A] :
( ( times_times @ A @ ( uminus_uminus @ A @ A3 ) @ B3 )
= ( times_times @ A @ A3 @ ( uminus_uminus @ A @ B3 ) ) ) ) ).
% minus_mult_commute
thf(fact_1989_mult__of__nat__commute,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [X: nat,Y: A] :
( ( times_times @ A @ ( semiring_1_of_nat @ A @ X ) @ Y )
= ( times_times @ A @ Y @ ( semiring_1_of_nat @ A @ X ) ) ) ) ).
% mult_of_nat_commute
thf(fact_1990_dvdE,axiom,
! [A: $tType] :
( ( dvd @ A )
=> ! [B3: A,A3: A] :
( ( dvd_dvd @ A @ B3 @ A3 )
=> ~ ! [K2: A] :
( A3
!= ( times_times @ A @ B3 @ K2 ) ) ) ) ).
% dvdE
thf(fact_1991_dvdI,axiom,
! [A: $tType] :
( ( dvd @ A )
=> ! [A3: A,B3: A,K: A] :
( ( A3
= ( times_times @ A @ B3 @ K ) )
=> ( dvd_dvd @ A @ B3 @ A3 ) ) ) ).
% dvdI
thf(fact_1992_dvd__def,axiom,
! [A: $tType] :
( ( dvd @ A )
=> ( ( dvd_dvd @ A )
= ( ^ [B5: A,A5: A] :
? [K3: A] :
( A5
= ( times_times @ A @ B5 @ K3 ) ) ) ) ) ).
% dvd_def
thf(fact_1993_dvd__mult,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: A,C2: A,B3: A] :
( ( dvd_dvd @ A @ A3 @ C2 )
=> ( dvd_dvd @ A @ A3 @ ( times_times @ A @ B3 @ C2 ) ) ) ) ).
% dvd_mult
thf(fact_1994_dvd__mult2,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: A,B3: A,C2: A] :
( ( dvd_dvd @ A @ A3 @ B3 )
=> ( dvd_dvd @ A @ A3 @ ( times_times @ A @ B3 @ C2 ) ) ) ) ).
% dvd_mult2
thf(fact_1995_dvd__mult__left,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: A,B3: A,C2: A] :
( ( dvd_dvd @ A @ ( times_times @ A @ A3 @ B3 ) @ C2 )
=> ( dvd_dvd @ A @ A3 @ C2 ) ) ) ).
% dvd_mult_left
thf(fact_1996_dvd__triv__left,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: A,B3: A] : ( dvd_dvd @ A @ A3 @ ( times_times @ A @ A3 @ B3 ) ) ) ).
% dvd_triv_left
thf(fact_1997_mult__dvd__mono,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: A,B3: A,C2: A,D2: A] :
( ( dvd_dvd @ A @ A3 @ B3 )
=> ( ( dvd_dvd @ A @ C2 @ D2 )
=> ( dvd_dvd @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B3 @ D2 ) ) ) ) ) ).
% mult_dvd_mono
thf(fact_1998_dvd__mult__right,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: A,B3: A,C2: A] :
( ( dvd_dvd @ A @ ( times_times @ A @ A3 @ B3 ) @ C2 )
=> ( dvd_dvd @ A @ B3 @ C2 ) ) ) ).
% dvd_mult_right
thf(fact_1999_dvd__triv__right,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: A,B3: A] : ( dvd_dvd @ A @ A3 @ ( times_times @ A @ B3 @ A3 ) ) ) ).
% dvd_triv_right
thf(fact_2000_mult__of__int__commute,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [X: int,Y: A] :
( ( times_times @ A @ ( ring_1_of_int @ A @ X ) @ Y )
= ( times_times @ A @ Y @ ( ring_1_of_int @ A @ X ) ) ) ) ).
% mult_of_int_commute
thf(fact_2001_lambda__zero,axiom,
! [A: $tType] :
( ( mult_zero @ A )
=> ( ( ^ [H: A] : ( zero_zero @ A ) )
= ( times_times @ A @ ( zero_zero @ A ) ) ) ) ).
% lambda_zero
thf(fact_2002_lambda__one,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ( ( ^ [X2: A] : X2 )
= ( times_times @ A @ ( one_one @ A ) ) ) ) ).
% lambda_one
thf(fact_2003_le__mult__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B3 )
=> ( ord_less_eq @ int @ ( times_times @ int @ ( archim6421214686448440834_floor @ A @ A3 ) @ ( archim6421214686448440834_floor @ A @ B3 ) ) @ ( archim6421214686448440834_floor @ A @ ( times_times @ A @ A3 @ B3 ) ) ) ) ) ) ).
% le_mult_floor
thf(fact_2004_mult__ceiling__le,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B3 )
=> ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ ( times_times @ A @ A3 @ B3 ) ) @ ( times_times @ int @ ( archimedean_ceiling @ A @ A3 ) @ ( archimedean_ceiling @ A @ B3 ) ) ) ) ) ) ).
% mult_ceiling_le
thf(fact_2005_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: $tType] :
( ( ordere2520102378445227354miring @ A )
=> ! [A3: A,B3: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B3 ) ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_2006_zero__le__mult__iff,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ B3 ) )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ B3 ) )
| ( ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) )
& ( ord_less_eq @ A @ B3 @ ( zero_zero @ A ) ) ) ) ) ) ).
% zero_le_mult_iff
thf(fact_2007_mult__nonneg__nonpos2,axiom,
! [A: $tType] :
( ( ordered_semiring_0 @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ B3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ B3 @ A3 ) @ ( zero_zero @ A ) ) ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_2008_mult__nonpos__nonneg,axiom,
! [A: $tType] :
( ( ordered_semiring_0 @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B3 )
=> ( ord_less_eq @ A @ ( times_times @ A @ A3 @ B3 ) @ ( zero_zero @ A ) ) ) ) ) ).
% mult_nonpos_nonneg
thf(fact_2009_mult__nonneg__nonpos,axiom,
! [A: $tType] :
( ( ordered_semiring_0 @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ B3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ A3 @ B3 ) @ ( zero_zero @ A ) ) ) ) ) ).
% mult_nonneg_nonpos
thf(fact_2010_mult__nonneg__nonneg,axiom,
! [A: $tType] :
( ( ordered_semiring_0 @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B3 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ B3 ) ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_2011_split__mult__neg__le,axiom,
! [A: $tType] :
( ( ordered_semiring_0 @ A )
=> ! [A3: A,B3: A] :
( ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
& ( ord_less_eq @ A @ B3 @ ( zero_zero @ A ) ) )
| ( ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ B3 ) ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ A3 @ B3 ) @ ( zero_zero @ A ) ) ) ) ).
% split_mult_neg_le
thf(fact_2012_mult__le__0__iff,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ ( times_times @ A @ A3 @ B3 ) @ ( zero_zero @ A ) )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
& ( ord_less_eq @ A @ B3 @ ( zero_zero @ A ) ) )
| ( ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ B3 ) ) ) ) ) ).
% mult_le_0_iff
thf(fact_2013_mult__right__mono,axiom,
! [A: $tType] :
( ( ordered_semiring @ A )
=> ! [A3: A,B3: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B3 @ C2 ) ) ) ) ) ).
% mult_right_mono
thf(fact_2014_mult__right__mono__neg,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [B3: A,A3: A,C2: A] :
( ( ord_less_eq @ A @ B3 @ A3 )
=> ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B3 @ C2 ) ) ) ) ) ).
% mult_right_mono_neg
thf(fact_2015_mult__left__mono,axiom,
! [A: $tType] :
( ( ordered_semiring @ A )
=> ! [A3: A,B3: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B3 ) ) ) ) ) ).
% mult_left_mono
thf(fact_2016_mult__nonpos__nonpos,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ B3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ B3 ) ) ) ) ) ).
% mult_nonpos_nonpos
thf(fact_2017_mult__left__mono__neg,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [B3: A,A3: A,C2: A] :
( ( ord_less_eq @ A @ B3 @ A3 )
=> ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B3 ) ) ) ) ) ).
% mult_left_mono_neg
thf(fact_2018_split__mult__pos__le,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [A3: A,B3: A] :
( ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ B3 ) )
| ( ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) )
& ( ord_less_eq @ A @ B3 @ ( zero_zero @ A ) ) ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ B3 ) ) ) ) ).
% split_mult_pos_le
thf(fact_2019_zero__le__square,axiom,
! [A: $tType] :
( ( linordered_ring @ A )
=> ! [A3: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ A3 ) ) ) ).
% zero_le_square
thf(fact_2020_mult__mono_H,axiom,
! [A: $tType] :
( ( ordered_semiring @ A )
=> ! [A3: A,B3: A,C2: A,D2: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ A @ C2 @ D2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B3 @ D2 ) ) ) ) ) ) ) ).
% mult_mono'
thf(fact_2021_mult__mono,axiom,
! [A: $tType] :
( ( ordered_semiring @ A )
=> ! [A3: A,B3: A,C2: A,D2: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ A @ C2 @ D2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B3 @ D2 ) ) ) ) ) ) ) ).
% mult_mono
thf(fact_2022_mult__neg__neg,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ B3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ B3 ) ) ) ) ) ).
% mult_neg_neg
thf(fact_2023_not__square__less__zero,axiom,
! [A: $tType] :
( ( linordered_ring @ A )
=> ! [A3: A] :
~ ( ord_less @ A @ ( times_times @ A @ A3 @ A3 ) @ ( zero_zero @ A ) ) ) ).
% not_square_less_zero
thf(fact_2024_mult__less__0__iff,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ ( times_times @ A @ A3 @ B3 ) @ ( zero_zero @ A ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
& ( ord_less @ A @ B3 @ ( zero_zero @ A ) ) )
| ( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
& ( ord_less @ A @ ( zero_zero @ A ) @ B3 ) ) ) ) ) ).
% mult_less_0_iff
thf(fact_2025_mult__neg__pos,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B3 )
=> ( ord_less @ A @ ( times_times @ A @ A3 @ B3 ) @ ( zero_zero @ A ) ) ) ) ) ).
% mult_neg_pos
thf(fact_2026_mult__pos__neg,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ B3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ A3 @ B3 ) @ ( zero_zero @ A ) ) ) ) ) ).
% mult_pos_neg
thf(fact_2027_mult__pos__pos,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B3 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ B3 ) ) ) ) ) ).
% mult_pos_pos
thf(fact_2028_mult__pos__neg2,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ B3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ B3 @ A3 ) @ ( zero_zero @ A ) ) ) ) ) ).
% mult_pos_neg2
thf(fact_2029_zero__less__mult__iff,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ B3 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
& ( ord_less @ A @ ( zero_zero @ A ) @ B3 ) )
| ( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
& ( ord_less @ A @ B3 @ ( zero_zero @ A ) ) ) ) ) ) ).
% zero_less_mult_iff
thf(fact_2030_zero__less__mult__pos,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ B3 ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ B3 ) ) ) ) ).
% zero_less_mult_pos
thf(fact_2031_zero__less__mult__pos2,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [B3: A,A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ B3 @ A3 ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ B3 ) ) ) ) ).
% zero_less_mult_pos2
thf(fact_2032_mult__less__cancel__left__neg,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [C2: A,A3: A,B3: A] :
( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B3 ) )
= ( ord_less @ A @ B3 @ A3 ) ) ) ) ).
% mult_less_cancel_left_neg
thf(fact_2033_mult__less__cancel__left__pos,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [C2: A,A3: A,B3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ord_less @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B3 ) )
= ( ord_less @ A @ A3 @ B3 ) ) ) ) ).
% mult_less_cancel_left_pos
thf(fact_2034_mult__strict__left__mono__neg,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [B3: A,A3: A,C2: A] :
( ( ord_less @ A @ B3 @ A3 )
=> ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B3 ) ) ) ) ) ).
% mult_strict_left_mono_neg
thf(fact_2035_mult__strict__left__mono,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A3: A,B3: A,C2: A] :
( ( ord_less @ A @ A3 @ B3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B3 ) ) ) ) ) ).
% mult_strict_left_mono
thf(fact_2036_mult__less__cancel__left__disj,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [C2: A,A3: A,B3: A] :
( ( ord_less @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B3 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
& ( ord_less @ A @ A3 @ B3 ) )
| ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
& ( ord_less @ A @ B3 @ A3 ) ) ) ) ) ).
% mult_less_cancel_left_disj
thf(fact_2037_mult__strict__right__mono__neg,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [B3: A,A3: A,C2: A] :
( ( ord_less @ A @ B3 @ A3 )
=> ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B3 @ C2 ) ) ) ) ) ).
% mult_strict_right_mono_neg
thf(fact_2038_mult__strict__right__mono,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A3: A,B3: A,C2: A] :
( ( ord_less @ A @ A3 @ B3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B3 @ C2 ) ) ) ) ) ).
% mult_strict_right_mono
thf(fact_2039_mult__less__cancel__right__disj,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [A3: A,C2: A,B3: A] :
( ( ord_less @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B3 @ C2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
& ( ord_less @ A @ A3 @ B3 ) )
| ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
& ( ord_less @ A @ B3 @ A3 ) ) ) ) ) ).
% mult_less_cancel_right_disj
thf(fact_2040_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: $tType] :
( ( linord2810124833399127020strict @ A )
=> ! [A3: A,B3: A,C2: A] :
( ( ord_less @ A @ A3 @ B3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B3 ) ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_2041_mult__numeral__1,axiom,
! [A: $tType] :
( ( semiring_numeral @ A )
=> ! [A3: A] :
( ( times_times @ A @ ( numeral_numeral @ A @ one2 ) @ A3 )
= A3 ) ) ).
% mult_numeral_1
thf(fact_2042_mult__numeral__1__right,axiom,
! [A: $tType] :
( ( semiring_numeral @ A )
=> ! [A3: A] :
( ( times_times @ A @ A3 @ ( numeral_numeral @ A @ one2 ) )
= A3 ) ) ).
% mult_numeral_1_right
thf(fact_2043_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_2044_frac__eq__eq,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [Y: A,Z: A,X: A,W: A] :
( ( Y
!= ( zero_zero @ A ) )
=> ( ( Z
!= ( zero_zero @ A ) )
=> ( ( ( divide_divide @ A @ X @ Y )
= ( divide_divide @ A @ W @ Z ) )
= ( ( times_times @ A @ X @ Z )
= ( times_times @ A @ W @ Y ) ) ) ) ) ) ).
% frac_eq_eq
thf(fact_2045_divide__eq__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B3: A,C2: A,A3: A] :
( ( ( divide_divide @ A @ B3 @ C2 )
= A3 )
= ( ( ( C2
!= ( zero_zero @ A ) )
=> ( B3
= ( times_times @ A @ A3 @ C2 ) ) )
& ( ( C2
= ( zero_zero @ A ) )
=> ( A3
= ( zero_zero @ A ) ) ) ) ) ) ).
% divide_eq_eq
thf(fact_2046_eq__divide__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A,B3: A,C2: A] :
( ( A3
= ( divide_divide @ A @ B3 @ C2 ) )
= ( ( ( C2
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ A3 @ C2 )
= B3 ) )
& ( ( C2
= ( zero_zero @ A ) )
=> ( A3
= ( zero_zero @ A ) ) ) ) ) ) ).
% eq_divide_eq
thf(fact_2047_divide__eq__imp,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [C2: A,B3: A,A3: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( B3
= ( times_times @ A @ A3 @ C2 ) )
=> ( ( divide_divide @ A @ B3 @ C2 )
= A3 ) ) ) ) ).
% divide_eq_imp
thf(fact_2048_eq__divide__imp,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [C2: A,A3: A,B3: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( ( times_times @ A @ A3 @ C2 )
= B3 )
=> ( A3
= ( divide_divide @ A @ B3 @ C2 ) ) ) ) ) ).
% eq_divide_imp
thf(fact_2049_nonzero__divide__eq__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [C2: A,B3: A,A3: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( ( divide_divide @ A @ B3 @ C2 )
= A3 )
= ( B3
= ( times_times @ A @ A3 @ C2 ) ) ) ) ) ).
% nonzero_divide_eq_eq
thf(fact_2050_nonzero__eq__divide__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [C2: A,A3: A,B3: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( A3
= ( divide_divide @ A @ B3 @ C2 ) )
= ( ( times_times @ A @ A3 @ C2 )
= B3 ) ) ) ) ).
% nonzero_eq_divide_eq
thf(fact_2051_square__diff__square__factored,axiom,
! [A: $tType] :
( ( comm_ring @ A )
=> ! [X: A,Y: A] :
( ( minus_minus @ A @ ( times_times @ A @ X @ X ) @ ( times_times @ A @ Y @ Y ) )
= ( times_times @ A @ ( plus_plus @ A @ X @ Y ) @ ( minus_minus @ A @ X @ Y ) ) ) ) ).
% square_diff_square_factored
thf(fact_2052_eq__add__iff2,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A3: A,E2: A,C2: A,B3: A,D2: A] :
( ( ( plus_plus @ A @ ( times_times @ A @ A3 @ E2 ) @ C2 )
= ( plus_plus @ A @ ( times_times @ A @ B3 @ E2 ) @ D2 ) )
= ( C2
= ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ B3 @ A3 ) @ E2 ) @ D2 ) ) ) ) ).
% eq_add_iff2
thf(fact_2053_eq__add__iff1,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A3: A,E2: A,C2: A,B3: A,D2: A] :
( ( ( plus_plus @ A @ ( times_times @ A @ A3 @ E2 ) @ C2 )
= ( plus_plus @ A @ ( times_times @ A @ B3 @ E2 ) @ D2 ) )
= ( ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ A3 @ B3 ) @ E2 ) @ C2 )
= D2 ) ) ) ).
% eq_add_iff1
thf(fact_2054_mult__diff__mult,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [X: A,Y: A,A3: A,B3: A] :
( ( minus_minus @ A @ ( times_times @ A @ X @ Y ) @ ( times_times @ A @ A3 @ B3 ) )
= ( plus_plus @ A @ ( times_times @ A @ X @ ( minus_minus @ A @ Y @ B3 ) ) @ ( times_times @ A @ ( minus_minus @ A @ X @ A3 ) @ B3 ) ) ) ) ).
% mult_diff_mult
thf(fact_2055_numeral__times__minus__swap,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [W: num,X: A] :
( ( times_times @ A @ ( numeral_numeral @ A @ W ) @ ( uminus_uminus @ A @ X ) )
= ( times_times @ A @ X @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) ) ) ) ).
% numeral_times_minus_swap
thf(fact_2056_left__right__inverse__power,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [X: A,Y: A,N: nat] :
( ( ( times_times @ A @ X @ Y )
= ( one_one @ A ) )
=> ( ( times_times @ A @ ( power_power @ A @ X @ N ) @ ( power_power @ A @ Y @ N ) )
= ( one_one @ A ) ) ) ) ).
% left_right_inverse_power
thf(fact_2057_square__eq__1__iff,axiom,
! [A: $tType] :
( ( ring_15535105094025558882visors @ A )
=> ! [X: A] :
( ( ( times_times @ A @ X @ X )
= ( one_one @ A ) )
= ( ( X
= ( one_one @ A ) )
| ( X
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ) ).
% square_eq_1_iff
thf(fact_2058_is__unit__mult__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B3: A] :
( ( dvd_dvd @ A @ ( times_times @ A @ A3 @ B3 ) @ ( one_one @ A ) )
= ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
& ( dvd_dvd @ A @ B3 @ ( one_one @ A ) ) ) ) ) ).
% is_unit_mult_iff
thf(fact_2059_dvd__mult__unit__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B3: A,A3: A,C2: A] :
( ( dvd_dvd @ A @ B3 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ A3 @ ( times_times @ A @ C2 @ B3 ) )
= ( dvd_dvd @ A @ A3 @ C2 ) ) ) ) ).
% dvd_mult_unit_iff
thf(fact_2060_mult__unit__dvd__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B3: A,A3: A,C2: A] :
( ( dvd_dvd @ A @ B3 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ ( times_times @ A @ A3 @ B3 ) @ C2 )
= ( dvd_dvd @ A @ A3 @ C2 ) ) ) ) ).
% mult_unit_dvd_iff
thf(fact_2061_dvd__mult__unit__iff_H,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B3: A,A3: A,C2: A] :
( ( dvd_dvd @ A @ B3 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ A3 @ ( times_times @ A @ B3 @ C2 ) )
= ( dvd_dvd @ A @ A3 @ C2 ) ) ) ) ).
% dvd_mult_unit_iff'
thf(fact_2062_mult__unit__dvd__iff_H,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B3: A,C2: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ ( times_times @ A @ A3 @ B3 ) @ C2 )
= ( dvd_dvd @ A @ B3 @ C2 ) ) ) ) ).
% mult_unit_dvd_iff'
thf(fact_2063_unit__mult__left__cancel,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B3: A,C2: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ( ( times_times @ A @ A3 @ B3 )
= ( times_times @ A @ A3 @ C2 ) )
= ( B3 = C2 ) ) ) ) ).
% unit_mult_left_cancel
thf(fact_2064_unit__mult__right__cancel,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B3: A,C2: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ( ( times_times @ A @ B3 @ A3 )
= ( times_times @ A @ C2 @ A3 ) )
= ( B3 = C2 ) ) ) ) ).
% unit_mult_right_cancel
thf(fact_2065_power__add,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A3: A,M: nat,N: nat] :
( ( power_power @ A @ A3 @ ( plus_plus @ nat @ M @ N ) )
= ( times_times @ A @ ( power_power @ A @ A3 @ M ) @ ( power_power @ A @ A3 @ N ) ) ) ) ).
% power_add
thf(fact_2066_div__mult2__eq_H,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [A3: A,M: nat,N: nat] :
( ( divide_divide @ A @ A3 @ ( times_times @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) ) )
= ( divide_divide @ A @ ( divide_divide @ A @ A3 @ ( semiring_1_of_nat @ A @ M ) ) @ ( semiring_1_of_nat @ A @ N ) ) ) ) ).
% div_mult2_eq'
thf(fact_2067_dvd__div__mult,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [C2: A,B3: A,A3: A] :
( ( dvd_dvd @ A @ C2 @ B3 )
=> ( ( times_times @ A @ ( divide_divide @ A @ B3 @ C2 ) @ A3 )
= ( divide_divide @ A @ ( times_times @ A @ B3 @ A3 ) @ C2 ) ) ) ) ).
% dvd_div_mult
thf(fact_2068_div__mult__swap,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [C2: A,B3: A,A3: A] :
( ( dvd_dvd @ A @ C2 @ B3 )
=> ( ( times_times @ A @ A3 @ ( divide_divide @ A @ B3 @ C2 ) )
= ( divide_divide @ A @ ( times_times @ A @ A3 @ B3 ) @ C2 ) ) ) ) ).
% div_mult_swap
thf(fact_2069_div__div__eq__right,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [C2: A,B3: A,A3: A] :
( ( dvd_dvd @ A @ C2 @ B3 )
=> ( ( dvd_dvd @ A @ B3 @ A3 )
=> ( ( divide_divide @ A @ A3 @ ( divide_divide @ A @ B3 @ C2 ) )
= ( times_times @ A @ ( divide_divide @ A @ A3 @ B3 ) @ C2 ) ) ) ) ) ).
% div_div_eq_right
thf(fact_2070_dvd__div__mult2__eq,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B3: A,C2: A,A3: A] :
( ( dvd_dvd @ A @ ( times_times @ A @ B3 @ C2 ) @ A3 )
=> ( ( divide_divide @ A @ A3 @ ( times_times @ A @ B3 @ C2 ) )
= ( divide_divide @ A @ ( divide_divide @ A @ A3 @ B3 ) @ C2 ) ) ) ) ).
% dvd_div_mult2_eq
thf(fact_2071_dvd__mult__imp__div,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,C2: A,B3: A] :
( ( dvd_dvd @ A @ ( times_times @ A @ A3 @ C2 ) @ B3 )
=> ( dvd_dvd @ A @ A3 @ ( divide_divide @ A @ B3 @ C2 ) ) ) ) ).
% dvd_mult_imp_div
thf(fact_2072_div__mult__div__if__dvd,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B3: A,A3: A,D2: A,C2: A] :
( ( dvd_dvd @ A @ B3 @ A3 )
=> ( ( dvd_dvd @ A @ D2 @ C2 )
=> ( ( times_times @ A @ ( divide_divide @ A @ A3 @ B3 ) @ ( divide_divide @ A @ C2 @ D2 ) )
= ( divide_divide @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B3 @ D2 ) ) ) ) ) ) ).
% div_mult_div_if_dvd
thf(fact_2073_real__minus__mult__self__le,axiom,
! [U2: real,X: real] : ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( times_times @ real @ U2 @ U2 ) ) @ ( times_times @ real @ X @ X ) ) ).
% real_minus_mult_self_le
thf(fact_2074_ln__powr,axiom,
! [X: real,Y: real] :
( ( X
!= ( zero_zero @ real ) )
=> ( ( ln_ln @ real @ ( powr @ real @ X @ Y ) )
= ( times_times @ real @ Y @ ( ln_ln @ real @ X ) ) ) ) ).
% ln_powr
thf(fact_2075_minus__power__mult__self,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [A3: A,N: nat] :
( ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ A3 ) @ N ) @ ( power_power @ A @ ( uminus_uminus @ A @ A3 ) @ N ) )
= ( power_power @ A @ A3 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% minus_power_mult_self
thf(fact_2076_tanh__real__lt__1,axiom,
! [X: real] : ( ord_less @ real @ ( tanh @ real @ X ) @ ( one_one @ real ) ) ).
% tanh_real_lt_1
thf(fact_2077_round__mono,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ord_less_eq @ int @ ( archimedean_round @ A @ X ) @ ( archimedean_round @ A @ Y ) ) ) ) ).
% round_mono
thf(fact_2078_mult__le__cancel__left,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [C2: A,A3: A,B3: A] :
( ( ord_less_eq @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B3 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ A3 @ B3 ) )
& ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ B3 @ A3 ) ) ) ) ) ).
% mult_le_cancel_left
thf(fact_2079_mult__le__cancel__right,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [A3: A,C2: A,B3: A] :
( ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B3 @ C2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ A3 @ B3 ) )
& ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ B3 @ A3 ) ) ) ) ) ).
% mult_le_cancel_right
thf(fact_2080_mult__left__less__imp__less,axiom,
! [A: $tType] :
( ( linordered_semiring @ A )
=> ! [C2: A,A3: A,B3: A] :
( ( ord_less @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B3 ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ A3 @ B3 ) ) ) ) ).
% mult_left_less_imp_less
thf(fact_2081_mult__strict__mono,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A3: A,B3: A,C2: A,D2: A] :
( ( ord_less @ A @ A3 @ B3 )
=> ( ( ord_less @ A @ C2 @ D2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B3 @ D2 ) ) ) ) ) ) ) ).
% mult_strict_mono
thf(fact_2082_mult__less__cancel__left,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [C2: A,A3: A,B3: A] :
( ( ord_less @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B3 ) )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ A3 @ B3 ) )
& ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ B3 @ A3 ) ) ) ) ) ).
% mult_less_cancel_left
thf(fact_2083_mult__right__less__imp__less,axiom,
! [A: $tType] :
( ( linordered_semiring @ A )
=> ! [A3: A,C2: A,B3: A] :
( ( ord_less @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B3 @ C2 ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ A3 @ B3 ) ) ) ) ).
% mult_right_less_imp_less
thf(fact_2084_mult__strict__mono_H,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A3: A,B3: A,C2: A,D2: A] :
( ( ord_less @ A @ A3 @ B3 )
=> ( ( ord_less @ A @ C2 @ D2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B3 @ D2 ) ) ) ) ) ) ) ).
% mult_strict_mono'
thf(fact_2085_mult__less__cancel__right,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [A3: A,C2: A,B3: A] :
( ( ord_less @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B3 @ C2 ) )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ A3 @ B3 ) )
& ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ B3 @ A3 ) ) ) ) ) ).
% mult_less_cancel_right
thf(fact_2086_mult__le__cancel__left__neg,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [C2: A,A3: A,B3: A] :
( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B3 ) )
= ( ord_less_eq @ A @ B3 @ A3 ) ) ) ) ).
% mult_le_cancel_left_neg
thf(fact_2087_mult__le__cancel__left__pos,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [C2: A,A3: A,B3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ord_less_eq @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B3 ) )
= ( ord_less_eq @ A @ A3 @ B3 ) ) ) ) ).
% mult_le_cancel_left_pos
thf(fact_2088_mult__left__le__imp__le,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [C2: A,A3: A,B3: A] :
( ( ord_less_eq @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B3 ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ A3 @ B3 ) ) ) ) ).
% mult_left_le_imp_le
thf(fact_2089_mult__right__le__imp__le,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A3: A,C2: A,B3: A] :
( ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B3 @ C2 ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ A3 @ B3 ) ) ) ) ).
% mult_right_le_imp_le
thf(fact_2090_mult__le__less__imp__less,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A3: A,B3: A,C2: A,D2: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ord_less @ A @ C2 @ D2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B3 @ D2 ) ) ) ) ) ) ) ).
% mult_le_less_imp_less
thf(fact_2091_mult__less__le__imp__less,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A3: A,B3: A,C2: A,D2: A] :
( ( ord_less @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ A @ C2 @ D2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B3 @ D2 ) ) ) ) ) ) ) ).
% mult_less_le_imp_less
thf(fact_2092_sum__squares__le__zero__iff,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ ( times_times @ A @ X @ X ) @ ( times_times @ A @ Y @ Y ) ) @ ( zero_zero @ A ) )
= ( ( X
= ( zero_zero @ A ) )
& ( Y
= ( zero_zero @ A ) ) ) ) ) ).
% sum_squares_le_zero_iff
thf(fact_2093_sum__squares__ge__zero,axiom,
! [A: $tType] :
( ( linordered_ring @ A )
=> ! [X: A,Y: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ ( times_times @ A @ X @ X ) @ ( times_times @ A @ Y @ Y ) ) ) ) ).
% sum_squares_ge_zero
thf(fact_2094_mult__left__le,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [C2: A,A3: A] :
( ( ord_less_eq @ A @ C2 @ ( one_one @ A ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C2 ) @ A3 ) ) ) ) ).
% mult_left_le
thf(fact_2095_mult__le__one,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ A3 @ ( one_one @ A ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B3 )
=> ( ( ord_less_eq @ A @ B3 @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ A3 @ B3 ) @ ( one_one @ A ) ) ) ) ) ) ).
% mult_le_one
thf(fact_2096_mult__right__le__one__le,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
=> ( ( ord_less_eq @ A @ Y @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ X @ Y ) @ X ) ) ) ) ) ).
% mult_right_le_one_le
thf(fact_2097_mult__left__le__one__le,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
=> ( ( ord_less_eq @ A @ Y @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ Y @ X ) @ X ) ) ) ) ) ).
% mult_left_le_one_le
thf(fact_2098_not__sum__squares__lt__zero,axiom,
! [A: $tType] :
( ( linordered_ring @ A )
=> ! [X: A,Y: A] :
~ ( ord_less @ A @ ( plus_plus @ A @ ( times_times @ A @ X @ X ) @ ( times_times @ A @ Y @ Y ) ) @ ( zero_zero @ A ) ) ) ).
% not_sum_squares_lt_zero
thf(fact_2099_sum__squares__gt__zero__iff,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ ( times_times @ A @ X @ X ) @ ( times_times @ A @ Y @ Y ) ) )
= ( ( X
!= ( zero_zero @ A ) )
| ( Y
!= ( zero_zero @ A ) ) ) ) ) ).
% sum_squares_gt_zero_iff
thf(fact_2100_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [C2: A,A3: A,B3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( divide_divide @ A @ A3 @ ( times_times @ A @ B3 @ C2 ) )
= ( divide_divide @ A @ ( divide_divide @ A @ A3 @ B3 ) @ C2 ) ) ) ) ).
% unique_euclidean_semiring_numeral_class.div_mult2_eq
thf(fact_2101_divide__less__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B3: A,C2: A,A3: A] :
( ( ord_less @ A @ ( divide_divide @ A @ B3 @ C2 ) @ A3 )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ B3 @ ( times_times @ A @ A3 @ C2 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ A3 @ C2 ) @ B3 ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ A3 ) ) ) ) ) ) ) ).
% divide_less_eq
thf(fact_2102_less__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B3: A,C2: A] :
( ( ord_less @ A @ A3 @ ( divide_divide @ A @ B3 @ C2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( times_times @ A @ A3 @ C2 ) @ B3 ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ B3 @ ( times_times @ A @ A3 @ C2 ) ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% less_divide_eq
thf(fact_2103_neg__divide__less__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,B3: A,A3: A] :
( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( divide_divide @ A @ B3 @ C2 ) @ A3 )
= ( ord_less @ A @ ( times_times @ A @ A3 @ C2 ) @ B3 ) ) ) ) ).
% neg_divide_less_eq
thf(fact_2104_neg__less__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,A3: A,B3: A] :
( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ A3 @ ( divide_divide @ A @ B3 @ C2 ) )
= ( ord_less @ A @ B3 @ ( times_times @ A @ A3 @ C2 ) ) ) ) ) ).
% neg_less_divide_eq
thf(fact_2105_pos__divide__less__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,B3: A,A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ord_less @ A @ ( divide_divide @ A @ B3 @ C2 ) @ A3 )
= ( ord_less @ A @ B3 @ ( times_times @ A @ A3 @ C2 ) ) ) ) ) ).
% pos_divide_less_eq
thf(fact_2106_pos__less__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,A3: A,B3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ord_less @ A @ A3 @ ( divide_divide @ A @ B3 @ C2 ) )
= ( ord_less @ A @ ( times_times @ A @ A3 @ C2 ) @ B3 ) ) ) ) ).
% pos_less_divide_eq
thf(fact_2107_mult__imp__div__pos__less,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [Y: A,X: A,Z: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Y )
=> ( ( ord_less @ A @ X @ ( times_times @ A @ Z @ Y ) )
=> ( ord_less @ A @ ( divide_divide @ A @ X @ Y ) @ Z ) ) ) ) ).
% mult_imp_div_pos_less
thf(fact_2108_mult__imp__less__div__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [Y: A,Z: A,X: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Y )
=> ( ( ord_less @ A @ ( times_times @ A @ Z @ Y ) @ X )
=> ( ord_less @ A @ Z @ ( divide_divide @ A @ X @ Y ) ) ) ) ) ).
% mult_imp_less_div_pos
thf(fact_2109_divide__strict__left__mono,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B3: A,A3: A,C2: A] :
( ( ord_less @ A @ B3 @ A3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ B3 ) )
=> ( ord_less @ A @ ( divide_divide @ A @ C2 @ A3 ) @ ( divide_divide @ A @ C2 @ B3 ) ) ) ) ) ) ).
% divide_strict_left_mono
thf(fact_2110_divide__strict__left__mono__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B3: A,C2: A] :
( ( ord_less @ A @ A3 @ B3 )
=> ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ B3 ) )
=> ( ord_less @ A @ ( divide_divide @ A @ C2 @ A3 ) @ ( divide_divide @ A @ C2 @ B3 ) ) ) ) ) ) ).
% divide_strict_left_mono_neg
thf(fact_2111_divide__eq__eq__numeral_I1_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B3: A,C2: A,W: num] :
( ( ( divide_divide @ A @ B3 @ C2 )
= ( numeral_numeral @ A @ W ) )
= ( ( ( C2
!= ( zero_zero @ A ) )
=> ( B3
= ( 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_2112_eq__divide__eq__numeral_I1_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [W: num,B3: A,C2: A] :
( ( ( numeral_numeral @ A @ W )
= ( divide_divide @ A @ B3 @ C2 ) )
= ( ( ( C2
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ ( numeral_numeral @ A @ W ) @ C2 )
= B3 ) )
& ( ( C2
= ( zero_zero @ A ) )
=> ( ( numeral_numeral @ A @ W )
= ( zero_zero @ A ) ) ) ) ) ) ).
% eq_divide_eq_numeral(1)
thf(fact_2113_ordered__ring__class_Ole__add__iff1,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [A3: A,E2: A,C2: A,B3: A,D2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ ( times_times @ A @ A3 @ E2 ) @ C2 ) @ ( plus_plus @ A @ ( times_times @ A @ B3 @ E2 ) @ D2 ) )
= ( ord_less_eq @ A @ ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ A3 @ B3 ) @ E2 ) @ C2 ) @ D2 ) ) ) ).
% ordered_ring_class.le_add_iff1
thf(fact_2114_ordered__ring__class_Ole__add__iff2,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [A3: A,E2: A,C2: A,B3: A,D2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ ( times_times @ A @ A3 @ E2 ) @ C2 ) @ ( plus_plus @ A @ ( times_times @ A @ B3 @ E2 ) @ D2 ) )
= ( ord_less_eq @ A @ C2 @ ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ B3 @ A3 ) @ E2 ) @ D2 ) ) ) ) ).
% ordered_ring_class.le_add_iff2
thf(fact_2115_divide__add__eq__iff,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z: A,X: A,Y: A] :
( ( Z
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( divide_divide @ A @ X @ Z ) @ Y )
= ( divide_divide @ A @ ( plus_plus @ A @ X @ ( times_times @ A @ Y @ Z ) ) @ Z ) ) ) ) ).
% divide_add_eq_iff
thf(fact_2116_add__divide__eq__iff,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z: A,X: A,Y: A] :
( ( Z
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ X @ ( divide_divide @ A @ Y @ Z ) )
= ( divide_divide @ A @ ( plus_plus @ A @ ( times_times @ A @ X @ Z ) @ Y ) @ Z ) ) ) ) ).
% add_divide_eq_iff
thf(fact_2117_add__num__frac,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [Y: A,Z: A,X: A] :
( ( Y
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ Z @ ( divide_divide @ A @ X @ Y ) )
= ( divide_divide @ A @ ( plus_plus @ A @ X @ ( times_times @ A @ Z @ Y ) ) @ Y ) ) ) ) ).
% add_num_frac
thf(fact_2118_add__frac__num,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [Y: A,X: A,Z: A] :
( ( Y
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( divide_divide @ A @ X @ Y ) @ Z )
= ( divide_divide @ A @ ( plus_plus @ A @ X @ ( times_times @ A @ Z @ Y ) ) @ Y ) ) ) ) ).
% add_frac_num
thf(fact_2119_add__frac__eq,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [Y: A,Z: A,X: A,W: A] :
( ( Y
!= ( zero_zero @ A ) )
=> ( ( Z
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( divide_divide @ A @ X @ Y ) @ ( divide_divide @ A @ W @ Z ) )
= ( divide_divide @ A @ ( plus_plus @ A @ ( times_times @ A @ X @ Z ) @ ( times_times @ A @ W @ Y ) ) @ ( times_times @ A @ Y @ Z ) ) ) ) ) ) ).
% add_frac_eq
thf(fact_2120_add__divide__eq__if__simps_I1_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z: A,A3: A,B3: A] :
( ( ( Z
= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ A3 @ ( divide_divide @ A @ B3 @ Z ) )
= A3 ) )
& ( ( Z
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ A3 @ ( divide_divide @ A @ B3 @ Z ) )
= ( divide_divide @ A @ ( plus_plus @ A @ ( times_times @ A @ A3 @ Z ) @ B3 ) @ Z ) ) ) ) ) ).
% add_divide_eq_if_simps(1)
thf(fact_2121_add__divide__eq__if__simps_I2_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z: A,A3: A,B3: A] :
( ( ( Z
= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( divide_divide @ A @ A3 @ Z ) @ B3 )
= B3 ) )
& ( ( Z
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( divide_divide @ A @ A3 @ Z ) @ B3 )
= ( divide_divide @ A @ ( plus_plus @ A @ A3 @ ( times_times @ A @ B3 @ Z ) ) @ Z ) ) ) ) ) ).
% add_divide_eq_if_simps(2)
thf(fact_2122_less__add__iff1,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [A3: A,E2: A,C2: A,B3: A,D2: A] :
( ( ord_less @ A @ ( plus_plus @ A @ ( times_times @ A @ A3 @ E2 ) @ C2 ) @ ( plus_plus @ A @ ( times_times @ A @ B3 @ E2 ) @ D2 ) )
= ( ord_less @ A @ ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ A3 @ B3 ) @ E2 ) @ C2 ) @ D2 ) ) ) ).
% less_add_iff1
thf(fact_2123_less__add__iff2,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [A3: A,E2: A,C2: A,B3: A,D2: A] :
( ( ord_less @ A @ ( plus_plus @ A @ ( times_times @ A @ A3 @ E2 ) @ C2 ) @ ( plus_plus @ A @ ( times_times @ A @ B3 @ E2 ) @ D2 ) )
= ( ord_less @ A @ C2 @ ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ B3 @ A3 ) @ E2 ) @ D2 ) ) ) ) ).
% less_add_iff2
thf(fact_2124_add__divide__eq__if__simps_I4_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z: A,A3: A,B3: A] :
( ( ( Z
= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ A3 @ ( divide_divide @ A @ B3 @ Z ) )
= A3 ) )
& ( ( Z
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ A3 @ ( divide_divide @ A @ B3 @ Z ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ A3 @ Z ) @ B3 ) @ Z ) ) ) ) ) ).
% add_divide_eq_if_simps(4)
thf(fact_2125_diff__frac__eq,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [Y: A,Z: A,X: A,W: A] :
( ( Y
!= ( zero_zero @ A ) )
=> ( ( Z
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( divide_divide @ A @ X @ Y ) @ ( divide_divide @ A @ W @ Z ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ X @ Z ) @ ( times_times @ A @ W @ Y ) ) @ ( times_times @ A @ Y @ Z ) ) ) ) ) ) ).
% diff_frac_eq
thf(fact_2126_diff__divide__eq__iff,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z: A,X: A,Y: A] :
( ( Z
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ X @ ( divide_divide @ A @ Y @ Z ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ X @ Z ) @ Y ) @ Z ) ) ) ) ).
% diff_divide_eq_iff
thf(fact_2127_divide__diff__eq__iff,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z: A,X: A,Y: A] :
( ( Z
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( divide_divide @ A @ X @ Z ) @ Y )
= ( divide_divide @ A @ ( minus_minus @ A @ X @ ( times_times @ A @ Y @ Z ) ) @ Z ) ) ) ) ).
% divide_diff_eq_iff
thf(fact_2128_square__diff__one__factored,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [X: A] :
( ( minus_minus @ A @ ( times_times @ A @ X @ X ) @ ( one_one @ A ) )
= ( times_times @ A @ ( plus_plus @ A @ X @ ( one_one @ A ) ) @ ( minus_minus @ A @ X @ ( one_one @ A ) ) ) ) ) ).
% square_diff_one_factored
thf(fact_2129_mult__1s__ring__1_I2_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [B3: A] :
( ( times_times @ A @ B3 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ one2 ) ) )
= ( uminus_uminus @ A @ B3 ) ) ) ).
% mult_1s_ring_1(2)
thf(fact_2130_mult__1s__ring__1_I1_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [B3: A] :
( ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ one2 ) ) @ B3 )
= ( uminus_uminus @ A @ B3 ) ) ) ).
% mult_1s_ring_1(1)
thf(fact_2131_power__gt1__lemma,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,N: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ A3 )
=> ( ord_less @ A @ ( one_one @ A ) @ ( times_times @ A @ A3 @ ( power_power @ A @ A3 @ N ) ) ) ) ) ).
% power_gt1_lemma
thf(fact_2132_power__less__power__Suc,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,N: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ A3 )
=> ( ord_less @ A @ ( power_power @ A @ A3 @ N ) @ ( times_times @ A @ A3 @ ( power_power @ A @ A3 @ N ) ) ) ) ) ).
% power_less_power_Suc
thf(fact_2133_ex__less__of__nat__mult,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ X )
=> ? [N2: nat] : ( ord_less @ A @ Y @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N2 ) @ X ) ) ) ) ).
% ex_less_of_nat_mult
thf(fact_2134_nonzero__neg__divide__eq__eq2,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B3: A,C2: A,A3: A] :
( ( B3
!= ( zero_zero @ A ) )
=> ( ( C2
= ( uminus_uminus @ A @ ( divide_divide @ A @ A3 @ B3 ) ) )
= ( ( times_times @ A @ C2 @ B3 )
= ( uminus_uminus @ A @ A3 ) ) ) ) ) ).
% nonzero_neg_divide_eq_eq2
thf(fact_2135_nonzero__neg__divide__eq__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B3: A,A3: A,C2: A] :
( ( B3
!= ( zero_zero @ A ) )
=> ( ( ( uminus_uminus @ A @ ( divide_divide @ A @ A3 @ B3 ) )
= C2 )
= ( ( uminus_uminus @ A @ A3 )
= ( times_times @ A @ C2 @ B3 ) ) ) ) ) ).
% nonzero_neg_divide_eq_eq
thf(fact_2136_minus__divide__eq__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B3: A,C2: A,A3: A] :
( ( ( uminus_uminus @ A @ ( divide_divide @ A @ B3 @ C2 ) )
= A3 )
= ( ( ( C2
!= ( zero_zero @ A ) )
=> ( ( uminus_uminus @ A @ B3 )
= ( times_times @ A @ A3 @ C2 ) ) )
& ( ( C2
= ( zero_zero @ A ) )
=> ( A3
= ( zero_zero @ A ) ) ) ) ) ) ).
% minus_divide_eq_eq
thf(fact_2137_eq__minus__divide__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A,B3: A,C2: A] :
( ( A3
= ( uminus_uminus @ A @ ( divide_divide @ A @ B3 @ C2 ) ) )
= ( ( ( C2
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ A3 @ C2 )
= ( uminus_uminus @ A @ B3 ) ) )
& ( ( C2
= ( zero_zero @ A ) )
=> ( A3
= ( zero_zero @ A ) ) ) ) ) ) ).
% eq_minus_divide_eq
thf(fact_2138_unity__coeff__ex,axiom,
! [A: $tType] :
( ( ( dvd @ A )
& ( semiring_0 @ A ) )
=> ! [P: A > $o,L: A] :
( ( ? [X2: A] : ( P @ ( times_times @ A @ L @ X2 ) ) )
= ( ? [X2: A] :
( ( dvd_dvd @ A @ L @ ( plus_plus @ A @ X2 @ ( zero_zero @ A ) ) )
& ( P @ X2 ) ) ) ) ) ).
% unity_coeff_ex
thf(fact_2139_unit__dvdE,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B3: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ~ ( ( A3
!= ( zero_zero @ A ) )
=> ! [C5: A] :
( B3
!= ( times_times @ A @ A3 @ C5 ) ) ) ) ) ).
% unit_dvdE
thf(fact_2140_power__minus,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [A3: A,N: nat] :
( ( power_power @ A @ ( uminus_uminus @ A @ A3 ) @ N )
= ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N ) @ ( power_power @ A @ A3 @ N ) ) ) ) ).
% power_minus
thf(fact_2141_dvd__div__div__eq__mult,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,C2: A,B3: A,D2: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( C2
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ A3 @ B3 )
=> ( ( dvd_dvd @ A @ C2 @ D2 )
=> ( ( ( divide_divide @ A @ B3 @ A3 )
= ( divide_divide @ A @ D2 @ C2 ) )
= ( ( times_times @ A @ B3 @ C2 )
= ( times_times @ A @ A3 @ D2 ) ) ) ) ) ) ) ) ).
% dvd_div_div_eq_mult
thf(fact_2142_dvd__div__iff__mult,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [C2: A,B3: A,A3: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ C2 @ B3 )
=> ( ( dvd_dvd @ A @ A3 @ ( divide_divide @ A @ B3 @ C2 ) )
= ( dvd_dvd @ A @ ( times_times @ A @ A3 @ C2 ) @ B3 ) ) ) ) ) ).
% dvd_div_iff_mult
thf(fact_2143_div__dvd__iff__mult,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B3: A,A3: A,C2: A] :
( ( B3
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ B3 @ A3 )
=> ( ( dvd_dvd @ A @ ( divide_divide @ A @ A3 @ B3 ) @ C2 )
= ( dvd_dvd @ A @ A3 @ ( times_times @ A @ C2 @ B3 ) ) ) ) ) ) ).
% div_dvd_iff_mult
thf(fact_2144_dvd__div__eq__mult,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B3: A,C2: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ A3 @ B3 )
=> ( ( ( divide_divide @ A @ B3 @ A3 )
= C2 )
= ( B3
= ( times_times @ A @ C2 @ A3 ) ) ) ) ) ) ).
% dvd_div_eq_mult
thf(fact_2145_is__unit__div__mult2__eq,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B3: A,C2: A,A3: A] :
( ( dvd_dvd @ A @ B3 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ C2 @ ( one_one @ A ) )
=> ( ( divide_divide @ A @ A3 @ ( times_times @ A @ B3 @ C2 ) )
= ( divide_divide @ A @ ( divide_divide @ A @ A3 @ B3 ) @ C2 ) ) ) ) ) ).
% is_unit_div_mult2_eq
thf(fact_2146_unit__div__mult__swap,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [C2: A,A3: A,B3: A] :
( ( dvd_dvd @ A @ C2 @ ( one_one @ A ) )
=> ( ( times_times @ A @ A3 @ ( divide_divide @ A @ B3 @ C2 ) )
= ( divide_divide @ A @ ( times_times @ A @ A3 @ B3 ) @ C2 ) ) ) ) ).
% unit_div_mult_swap
thf(fact_2147_unit__div__commute,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B3: A,A3: A,C2: A] :
( ( dvd_dvd @ A @ B3 @ ( one_one @ A ) )
=> ( ( times_times @ A @ ( divide_divide @ A @ A3 @ B3 ) @ C2 )
= ( divide_divide @ A @ ( times_times @ A @ A3 @ C2 ) @ B3 ) ) ) ) ).
% unit_div_commute
thf(fact_2148_div__mult__unit2,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [C2: A,B3: A,A3: A] :
( ( dvd_dvd @ A @ C2 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ B3 @ A3 )
=> ( ( divide_divide @ A @ A3 @ ( times_times @ A @ B3 @ C2 ) )
= ( divide_divide @ A @ ( divide_divide @ A @ A3 @ B3 ) @ C2 ) ) ) ) ) ).
% div_mult_unit2
thf(fact_2149_unit__eq__div2,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B3: A,A3: A,C2: A] :
( ( dvd_dvd @ A @ B3 @ ( one_one @ A ) )
=> ( ( A3
= ( divide_divide @ A @ C2 @ B3 ) )
= ( ( times_times @ A @ A3 @ B3 )
= C2 ) ) ) ) ).
% unit_eq_div2
thf(fact_2150_unit__eq__div1,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B3: A,A3: A,C2: A] :
( ( dvd_dvd @ A @ B3 @ ( one_one @ A ) )
=> ( ( ( divide_divide @ A @ A3 @ B3 )
= C2 )
= ( A3
= ( times_times @ A @ C2 @ B3 ) ) ) ) ) ).
% unit_eq_div1
thf(fact_2151_floor__le__round,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] : ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( archimedean_round @ A @ X ) ) ) ).
% floor_le_round
thf(fact_2152_ceiling__ge__round,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] : ( ord_less_eq @ int @ ( archimedean_round @ A @ X ) @ ( archimedean_ceiling @ A @ X ) ) ) ).
% ceiling_ge_round
thf(fact_2153_ln__mult,axiom,
! [X: real,Y: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ Y )
=> ( ( ln_ln @ real @ ( times_times @ real @ X @ Y ) )
= ( plus_plus @ real @ ( ln_ln @ real @ X ) @ ( ln_ln @ real @ Y ) ) ) ) ) ).
% ln_mult
thf(fact_2154_reals__Archimedean3,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ! [Y5: real] :
? [N2: nat] : ( ord_less @ real @ Y5 @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N2 ) @ X ) ) ) ).
% reals_Archimedean3
thf(fact_2155_powr__mult,axiom,
! [X: real,Y: real,A3: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y )
=> ( ( powr @ real @ ( times_times @ real @ X @ Y ) @ A3 )
= ( times_times @ real @ ( powr @ real @ X @ A3 ) @ ( powr @ real @ Y @ A3 ) ) ) ) ) ).
% powr_mult
thf(fact_2156_log__powr,axiom,
! [X: real,B3: real,Y: real] :
( ( X
!= ( zero_zero @ real ) )
=> ( ( log @ B3 @ ( powr @ real @ X @ Y ) )
= ( times_times @ real @ Y @ ( log @ B3 @ X ) ) ) ) ).
% log_powr
thf(fact_2157_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_2158_tanh__real__gt__neg1,axiom,
! [X: real] : ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ ( tanh @ real @ X ) ) ).
% tanh_real_gt_neg1
thf(fact_2159_mult__le__cancel__left1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [C2: A,B3: A] :
( ( ord_less_eq @ A @ C2 @ ( times_times @ A @ C2 @ B3 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ B3 ) )
& ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ B3 @ ( one_one @ A ) ) ) ) ) ) ).
% mult_le_cancel_left1
thf(fact_2160_mult__le__cancel__left2,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [C2: A,A3: A] :
( ( ord_less_eq @ A @ ( times_times @ A @ C2 @ A3 ) @ C2 )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ A3 @ ( one_one @ A ) ) )
& ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ A3 ) ) ) ) ) ).
% mult_le_cancel_left2
thf(fact_2161_mult__le__cancel__right1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [C2: A,B3: A] :
( ( ord_less_eq @ A @ C2 @ ( times_times @ A @ B3 @ C2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ B3 ) )
& ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ B3 @ ( one_one @ A ) ) ) ) ) ) ).
% mult_le_cancel_right1
thf(fact_2162_mult__le__cancel__right2,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,C2: A] :
( ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C2 ) @ C2 )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ A3 @ ( one_one @ A ) ) )
& ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ A3 ) ) ) ) ) ).
% mult_le_cancel_right2
thf(fact_2163_mult__less__cancel__left1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [C2: A,B3: A] :
( ( ord_less @ A @ C2 @ ( times_times @ A @ C2 @ B3 ) )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( one_one @ A ) @ B3 ) )
& ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ B3 @ ( one_one @ A ) ) ) ) ) ) ).
% mult_less_cancel_left1
thf(fact_2164_mult__less__cancel__left2,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [C2: A,A3: A] :
( ( ord_less @ A @ ( times_times @ A @ C2 @ A3 ) @ C2 )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ A3 @ ( one_one @ A ) ) )
& ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( one_one @ A ) @ A3 ) ) ) ) ) ).
% mult_less_cancel_left2
thf(fact_2165_mult__less__cancel__right1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [C2: A,B3: A] :
( ( ord_less @ A @ C2 @ ( times_times @ A @ B3 @ C2 ) )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( one_one @ A ) @ B3 ) )
& ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ B3 @ ( one_one @ A ) ) ) ) ) ) ).
% mult_less_cancel_right1
thf(fact_2166_mult__less__cancel__right2,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,C2: A] :
( ( ord_less @ A @ ( times_times @ A @ A3 @ C2 ) @ C2 )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ A3 @ ( one_one @ A ) ) )
& ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( one_one @ A ) @ A3 ) ) ) ) ) ).
% mult_less_cancel_right2
thf(fact_2167_field__le__mult__one__interval,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y: A] :
( ! [Z3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Z3 )
=> ( ( ord_less @ A @ Z3 @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ Z3 @ X ) @ Y ) ) )
=> ( ord_less_eq @ A @ X @ Y ) ) ) ).
% field_le_mult_one_interval
thf(fact_2168_convex__bound__le,axiom,
! [A: $tType] :
( ( linord6961819062388156250ring_1 @ A )
=> ! [X: A,A3: A,Y: A,U2: A,V2: A] :
( ( ord_less_eq @ A @ X @ A3 )
=> ( ( ord_less_eq @ A @ Y @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ U2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ V2 )
=> ( ( ( plus_plus @ A @ U2 @ V2 )
= ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ ( times_times @ A @ U2 @ X ) @ ( times_times @ A @ V2 @ Y ) ) @ A3 ) ) ) ) ) ) ) ).
% convex_bound_le
thf(fact_2169_divide__left__mono__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B3: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ B3 ) )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ C2 @ A3 ) @ ( divide_divide @ A @ C2 @ B3 ) ) ) ) ) ) ).
% divide_left_mono_neg
thf(fact_2170_mult__imp__le__div__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [Y: A,Z: A,X: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Y )
=> ( ( ord_less_eq @ A @ ( times_times @ A @ Z @ Y ) @ X )
=> ( ord_less_eq @ A @ Z @ ( divide_divide @ A @ X @ Y ) ) ) ) ) ).
% mult_imp_le_div_pos
thf(fact_2171_mult__imp__div__pos__le,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [Y: A,X: A,Z: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Y )
=> ( ( ord_less_eq @ A @ X @ ( times_times @ A @ Z @ Y ) )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ X @ Y ) @ Z ) ) ) ) ).
% mult_imp_div_pos_le
thf(fact_2172_pos__le__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,A3: A,B3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ord_less_eq @ A @ A3 @ ( divide_divide @ A @ B3 @ C2 ) )
= ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C2 ) @ B3 ) ) ) ) ).
% pos_le_divide_eq
thf(fact_2173_pos__divide__le__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,B3: A,A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ord_less_eq @ A @ ( divide_divide @ A @ B3 @ C2 ) @ A3 )
= ( ord_less_eq @ A @ B3 @ ( times_times @ A @ A3 @ C2 ) ) ) ) ) ).
% pos_divide_le_eq
thf(fact_2174_neg__le__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,A3: A,B3: A] :
( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ A3 @ ( divide_divide @ A @ B3 @ C2 ) )
= ( ord_less_eq @ A @ B3 @ ( times_times @ A @ A3 @ C2 ) ) ) ) ) ).
% neg_le_divide_eq
thf(fact_2175_neg__divide__le__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,B3: A,A3: A] :
( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( divide_divide @ A @ B3 @ C2 ) @ A3 )
= ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C2 ) @ B3 ) ) ) ) ).
% neg_divide_le_eq
thf(fact_2176_divide__left__mono,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B3: A,A3: A,C2: A] :
( ( ord_less_eq @ A @ B3 @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ B3 ) )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ C2 @ A3 ) @ ( divide_divide @ A @ C2 @ B3 ) ) ) ) ) ) ).
% divide_left_mono
thf(fact_2177_le__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B3: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ ( divide_divide @ A @ B3 @ C2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C2 ) @ B3 ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ B3 @ ( times_times @ A @ A3 @ C2 ) ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% le_divide_eq
thf(fact_2178_divide__le__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B3: A,C2: A,A3: A] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ B3 @ C2 ) @ A3 )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ B3 @ ( times_times @ A @ A3 @ C2 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C2 ) @ B3 ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 ) ) ) ) ) ) ) ).
% divide_le_eq
thf(fact_2179_divide__less__eq__numeral_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B3: A,C2: A,W: num] :
( ( ord_less @ A @ ( divide_divide @ A @ B3 @ C2 ) @ ( numeral_numeral @ A @ W ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ B3 @ ( 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 ) @ B3 ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ W ) ) ) ) ) ) ) ) ).
% divide_less_eq_numeral(1)
thf(fact_2180_less__divide__eq__numeral_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [W: num,B3: A,C2: A] :
( ( ord_less @ A @ ( numeral_numeral @ A @ W ) @ ( divide_divide @ A @ B3 @ C2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( times_times @ A @ ( numeral_numeral @ A @ W ) @ C2 ) @ B3 ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ B3 @ ( 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_2181_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_2182_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_2183_left__add__twice,axiom,
! [A: $tType] :
( ( semiring_numeral @ A )
=> ! [A3: A,B3: A] :
( ( plus_plus @ A @ A3 @ ( plus_plus @ A @ A3 @ B3 ) )
= ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) @ B3 ) ) ) ).
% left_add_twice
thf(fact_2184_frac__le__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [Y: A,Z: A,X: A,W: A] :
( ( Y
!= ( zero_zero @ A ) )
=> ( ( Z
!= ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( divide_divide @ A @ X @ Y ) @ ( divide_divide @ A @ W @ Z ) )
= ( ord_less_eq @ A @ ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ X @ Z ) @ ( times_times @ A @ W @ Y ) ) @ ( times_times @ A @ Y @ Z ) ) @ ( zero_zero @ A ) ) ) ) ) ) ).
% frac_le_eq
thf(fact_2185_power__Suc__less,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,N: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ A3 @ ( one_one @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ A3 @ ( power_power @ A @ A3 @ N ) ) @ ( power_power @ A @ A3 @ N ) ) ) ) ) ).
% power_Suc_less
thf(fact_2186_frac__less__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [Y: A,Z: A,X: A,W: A] :
( ( Y
!= ( zero_zero @ A ) )
=> ( ( Z
!= ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( divide_divide @ A @ X @ Y ) @ ( divide_divide @ A @ W @ Z ) )
= ( ord_less @ A @ ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ X @ Z ) @ ( times_times @ A @ W @ Y ) ) @ ( times_times @ A @ Y @ Z ) ) @ ( zero_zero @ A ) ) ) ) ) ) ).
% frac_less_eq
thf(fact_2187_less__minus__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B3: A,C2: A] :
( ( ord_less @ A @ A3 @ ( uminus_uminus @ A @ ( divide_divide @ A @ B3 @ C2 ) ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( times_times @ A @ A3 @ C2 ) @ ( uminus_uminus @ A @ B3 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( uminus_uminus @ A @ B3 ) @ ( times_times @ A @ A3 @ C2 ) ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% less_minus_divide_eq
thf(fact_2188_minus__divide__less__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B3: A,C2: A,A3: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ B3 @ C2 ) ) @ A3 )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( uminus_uminus @ A @ B3 ) @ ( times_times @ A @ A3 @ C2 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ A3 @ C2 ) @ ( uminus_uminus @ A @ B3 ) ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ A3 ) ) ) ) ) ) ) ).
% minus_divide_less_eq
thf(fact_2189_neg__less__minus__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,A3: A,B3: A] :
( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ A3 @ ( uminus_uminus @ A @ ( divide_divide @ A @ B3 @ C2 ) ) )
= ( ord_less @ A @ ( uminus_uminus @ A @ B3 ) @ ( times_times @ A @ A3 @ C2 ) ) ) ) ) ).
% neg_less_minus_divide_eq
thf(fact_2190_neg__minus__divide__less__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,B3: A,A3: A] :
( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ B3 @ C2 ) ) @ A3 )
= ( ord_less @ A @ ( times_times @ A @ A3 @ C2 ) @ ( uminus_uminus @ A @ B3 ) ) ) ) ) ).
% neg_minus_divide_less_eq
thf(fact_2191_pos__less__minus__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,A3: A,B3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ord_less @ A @ A3 @ ( uminus_uminus @ A @ ( divide_divide @ A @ B3 @ C2 ) ) )
= ( ord_less @ A @ ( times_times @ A @ A3 @ C2 ) @ ( uminus_uminus @ A @ B3 ) ) ) ) ) ).
% pos_less_minus_divide_eq
thf(fact_2192_pos__minus__divide__less__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,B3: A,A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ord_less @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ B3 @ C2 ) ) @ A3 )
= ( ord_less @ A @ ( uminus_uminus @ A @ B3 ) @ ( times_times @ A @ A3 @ C2 ) ) ) ) ) ).
% pos_minus_divide_less_eq
thf(fact_2193_divide__eq__eq__numeral_I2_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B3: A,C2: A,W: num] :
( ( ( divide_divide @ A @ B3 @ C2 )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) )
= ( ( ( C2
!= ( zero_zero @ A ) )
=> ( B3
= ( 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_2194_eq__divide__eq__numeral_I2_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [W: num,B3: A,C2: A] :
( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) )
= ( divide_divide @ A @ B3 @ C2 ) )
= ( ( ( C2
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) @ C2 )
= B3 ) )
& ( ( C2
= ( zero_zero @ A ) )
=> ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) )
= ( zero_zero @ A ) ) ) ) ) ) ).
% eq_divide_eq_numeral(2)
thf(fact_2195_minus__divide__add__eq__iff,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z: A,X: A,Y: A] :
( ( Z
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ X @ Z ) ) @ Y )
= ( divide_divide @ A @ ( plus_plus @ A @ ( uminus_uminus @ A @ X ) @ ( times_times @ A @ Y @ Z ) ) @ Z ) ) ) ) ).
% minus_divide_add_eq_iff
thf(fact_2196_add__divide__eq__if__simps_I3_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z: A,A3: A,B3: A] :
( ( ( Z
= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ A3 @ Z ) ) @ B3 )
= B3 ) )
& ( ( Z
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ A3 @ Z ) ) @ B3 )
= ( divide_divide @ A @ ( plus_plus @ A @ ( uminus_uminus @ A @ A3 ) @ ( times_times @ A @ B3 @ Z ) ) @ Z ) ) ) ) ) ).
% add_divide_eq_if_simps(3)
thf(fact_2197_evenE,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A3: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
=> ~ ! [B7: A] :
( A3
!= ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B7 ) ) ) ) ).
% evenE
thf(fact_2198_power4__eq__xxxx,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [X: A] :
( ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) )
= ( times_times @ A @ ( times_times @ A @ ( times_times @ A @ X @ X ) @ X ) @ X ) ) ) ).
% power4_eq_xxxx
thf(fact_2199_power2__eq__square,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A3: A] :
( ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( times_times @ A @ A3 @ A3 ) ) ) ).
% power2_eq_square
thf(fact_2200_minus__divide__diff__eq__iff,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z: A,X: A,Y: A] :
( ( Z
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ X @ Z ) ) @ Y )
= ( divide_divide @ A @ ( minus_minus @ A @ ( uminus_uminus @ A @ X ) @ ( times_times @ A @ Y @ Z ) ) @ Z ) ) ) ) ).
% minus_divide_diff_eq_iff
thf(fact_2201_add__divide__eq__if__simps_I5_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z: A,A3: A,B3: A] :
( ( ( Z
= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( divide_divide @ A @ A3 @ Z ) @ B3 )
= ( uminus_uminus @ A @ B3 ) ) )
& ( ( Z
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( divide_divide @ A @ A3 @ Z ) @ B3 )
= ( divide_divide @ A @ ( minus_minus @ A @ A3 @ ( times_times @ A @ B3 @ Z ) ) @ Z ) ) ) ) ) ).
% add_divide_eq_if_simps(5)
thf(fact_2202_add__divide__eq__if__simps_I6_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z: A,A3: A,B3: A] :
( ( ( Z
= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ A3 @ Z ) ) @ B3 )
= ( uminus_uminus @ A @ B3 ) ) )
& ( ( Z
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ A3 @ Z ) ) @ B3 )
= ( divide_divide @ A @ ( minus_minus @ A @ ( uminus_uminus @ A @ A3 ) @ ( times_times @ A @ B3 @ Z ) ) @ Z ) ) ) ) ) ).
% add_divide_eq_if_simps(6)
thf(fact_2203_is__unitE,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,C2: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ~ ( ( A3
!= ( zero_zero @ A ) )
=> ! [B7: A] :
( ( B7
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ B7 @ ( one_one @ A ) )
=> ( ( ( divide_divide @ A @ ( one_one @ A ) @ A3 )
= B7 )
=> ( ( ( divide_divide @ A @ ( one_one @ A ) @ B7 )
= A3 )
=> ( ( ( times_times @ A @ A3 @ B7 )
= ( one_one @ A ) )
=> ( ( divide_divide @ A @ C2 @ A3 )
!= ( times_times @ A @ C2 @ B7 ) ) ) ) ) ) ) ) ) ) ).
% is_unitE
thf(fact_2204_is__unit__div__mult__cancel__left,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B3: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ B3 @ ( one_one @ A ) )
=> ( ( divide_divide @ A @ A3 @ ( times_times @ A @ A3 @ B3 ) )
= ( divide_divide @ A @ ( one_one @ A ) @ B3 ) ) ) ) ) ).
% is_unit_div_mult_cancel_left
thf(fact_2205_is__unit__div__mult__cancel__right,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B3: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ B3 @ ( one_one @ A ) )
=> ( ( divide_divide @ A @ A3 @ ( times_times @ A @ B3 @ A3 ) )
= ( divide_divide @ A @ ( one_one @ A ) @ B3 ) ) ) ) ) ).
% is_unit_div_mult_cancel_right
thf(fact_2206_ln__realpow,axiom,
! [X: real,N: nat] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ln_ln @ real @ ( power_power @ real @ X @ N ) )
= ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ ( ln_ln @ real @ X ) ) ) ) ).
% ln_realpow
thf(fact_2207_linear__plus__1__le__power,axiom,
! [X: real,N: nat] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ord_less_eq @ real @ ( plus_plus @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ X ) @ ( one_one @ real ) ) @ ( power_power @ real @ ( plus_plus @ real @ X @ ( one_one @ real ) ) @ N ) ) ) ).
% linear_plus_1_le_power
thf(fact_2208_convex__bound__lt,axiom,
! [A: $tType] :
( ( linord715952674999750819strict @ A )
=> ! [X: A,A3: A,Y: A,U2: A,V2: A] :
( ( ord_less @ A @ X @ A3 )
=> ( ( ord_less @ A @ Y @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ U2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ V2 )
=> ( ( ( plus_plus @ A @ U2 @ V2 )
= ( one_one @ A ) )
=> ( ord_less @ A @ ( plus_plus @ A @ ( times_times @ A @ U2 @ X ) @ ( times_times @ A @ V2 @ Y ) ) @ A3 ) ) ) ) ) ) ) ).
% convex_bound_lt
thf(fact_2209_le__divide__eq__numeral_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [W: num,B3: A,C2: A] :
( ( ord_less_eq @ A @ ( numeral_numeral @ A @ W ) @ ( divide_divide @ A @ B3 @ C2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( times_times @ A @ ( numeral_numeral @ A @ W ) @ C2 ) @ B3 ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ B3 @ ( 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_2210_divide__le__eq__numeral_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B3: A,C2: A,W: num] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ B3 @ C2 ) @ ( numeral_numeral @ A @ W ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ B3 @ ( 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 ) @ B3 ) )
& ( ~ ( 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_2211_scaling__mono,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [U2: A,V2: A,R3: A,S2: A] :
( ( ord_less_eq @ A @ U2 @ V2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ R3 )
=> ( ( ord_less_eq @ A @ R3 @ S2 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ U2 @ ( divide_divide @ A @ ( times_times @ A @ R3 @ ( minus_minus @ A @ V2 @ U2 ) ) @ S2 ) ) @ V2 ) ) ) ) ) ).
% scaling_mono
thf(fact_2212_le__minus__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B3: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ ( uminus_uminus @ A @ ( divide_divide @ A @ B3 @ C2 ) ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C2 ) @ ( uminus_uminus @ A @ B3 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ B3 ) @ ( times_times @ A @ A3 @ C2 ) ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% le_minus_divide_eq
thf(fact_2213_minus__divide__le__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B3: A,C2: A,A3: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ B3 @ C2 ) ) @ A3 )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ B3 ) @ ( times_times @ A @ A3 @ C2 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C2 ) @ ( uminus_uminus @ A @ B3 ) ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 ) ) ) ) ) ) ) ).
% minus_divide_le_eq
thf(fact_2214_neg__le__minus__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,A3: A,B3: A] :
( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ A3 @ ( uminus_uminus @ A @ ( divide_divide @ A @ B3 @ C2 ) ) )
= ( ord_less_eq @ A @ ( uminus_uminus @ A @ B3 ) @ ( times_times @ A @ A3 @ C2 ) ) ) ) ) ).
% neg_le_minus_divide_eq
thf(fact_2215_neg__minus__divide__le__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,B3: A,A3: A] :
( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ B3 @ C2 ) ) @ A3 )
= ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C2 ) @ ( uminus_uminus @ A @ B3 ) ) ) ) ) ).
% neg_minus_divide_le_eq
thf(fact_2216_pos__le__minus__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,A3: A,B3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ord_less_eq @ A @ A3 @ ( uminus_uminus @ A @ ( divide_divide @ A @ B3 @ C2 ) ) )
= ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C2 ) @ ( uminus_uminus @ A @ B3 ) ) ) ) ) ).
% pos_le_minus_divide_eq
thf(fact_2217_pos__minus__divide__le__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,B3: A,A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ B3 @ C2 ) ) @ A3 )
= ( ord_less_eq @ A @ ( uminus_uminus @ A @ B3 ) @ ( times_times @ A @ A3 @ C2 ) ) ) ) ) ).
% pos_minus_divide_le_eq
thf(fact_2218_less__divide__eq__numeral_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [W: num,B3: A,C2: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) @ ( divide_divide @ A @ B3 @ C2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) @ C2 ) @ B3 ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ B3 @ ( 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_2219_divide__less__eq__numeral_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B3: A,C2: A,W: num] :
( ( ord_less @ A @ ( divide_divide @ A @ B3 @ C2 ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ B3 @ ( 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 ) @ B3 ) )
& ( ~ ( 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_2220_even__two__times__div__two,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [A3: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
=> ( ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
= A3 ) ) ) ).
% even_two_times_div_two
thf(fact_2221_power__eq__if,axiom,
! [A: $tType] :
( ( power @ A )
=> ( ( power_power @ A )
= ( ^ [P6: A,M3: nat] :
( if @ A
@ ( M3
= ( zero_zero @ nat ) )
@ ( one_one @ A )
@ ( times_times @ A @ P6 @ ( power_power @ A @ P6 @ ( minus_minus @ nat @ M3 @ ( one_one @ nat ) ) ) ) ) ) ) ) ).
% power_eq_if
thf(fact_2222_power__minus__mult,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [N: nat,A3: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( times_times @ A @ ( power_power @ A @ A3 @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) @ A3 )
= ( power_power @ A @ A3 @ N ) ) ) ) ).
% power_minus_mult
thf(fact_2223_four__x__squared,axiom,
! [X: real] :
( ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( power_power @ real @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% four_x_squared
thf(fact_2224_real__archimedian__rdiv__eq__0,axiom,
! [X: real,C2: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ C2 )
=> ( ! [M4: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ M4 )
=> ( ord_less_eq @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ M4 ) @ X ) @ C2 ) )
=> ( X
= ( zero_zero @ real ) ) ) ) ) ).
% real_archimedian_rdiv_eq_0
thf(fact_2225_ln__powr__bound2,axiom,
! [X: real,A3: real] :
( ( ord_less @ real @ ( one_one @ real ) @ X )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ A3 )
=> ( ord_less_eq @ real @ ( powr @ real @ ( ln_ln @ real @ X ) @ A3 ) @ ( times_times @ real @ ( powr @ real @ A3 @ A3 ) @ X ) ) ) ) ).
% ln_powr_bound2
thf(fact_2226_shiftl__eq__mult,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_Sh4282982442137083160shiftl @ A )
= ( ^ [X2: A,N4: nat] : ( times_times @ A @ X2 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N4 ) ) ) ) ) ).
% shiftl_eq_mult
thf(fact_2227_log__eq__div__ln__mult__log,axiom,
! [A3: real,B3: real,X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ A3 )
=> ( ( A3
!= ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ B3 )
=> ( ( B3
!= ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( log @ A3 @ X )
= ( times_times @ real @ ( divide_divide @ real @ ( ln_ln @ real @ B3 ) @ ( ln_ln @ real @ A3 ) ) @ ( log @ B3 @ X ) ) ) ) ) ) ) ) ).
% log_eq_div_ln_mult_log
thf(fact_2228_log__mult,axiom,
! [A3: real,X: real,Y: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ A3 )
=> ( ( A3
!= ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ Y )
=> ( ( log @ A3 @ ( times_times @ real @ X @ Y ) )
= ( plus_plus @ real @ ( log @ A3 @ X ) @ ( log @ A3 @ Y ) ) ) ) ) ) ) ).
% log_mult
thf(fact_2229_powr__mult__base,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( times_times @ real @ X @ ( powr @ real @ X @ Y ) )
= ( powr @ real @ X @ ( plus_plus @ real @ ( one_one @ real ) @ Y ) ) ) ) ).
% powr_mult_base
thf(fact_2230_log__nat__power,axiom,
! [X: real,B3: real,N: nat] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( log @ B3 @ ( power_power @ real @ X @ N ) )
= ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ ( log @ B3 @ X ) ) ) ) ).
% log_nat_power
thf(fact_2231_divide__le__eq__numeral_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B3: A,C2: A,W: num] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ B3 @ C2 ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ B3 @ ( 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 ) @ B3 ) )
& ( ~ ( 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_2232_le__divide__eq__numeral_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [W: num,B3: A,C2: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) @ ( divide_divide @ A @ B3 @ C2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) @ C2 ) @ B3 ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ B3 @ ( 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_2233_power2__sum,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [X: A,Y: A] :
( ( power_power @ A @ ( plus_plus @ A @ X @ Y ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( plus_plus @ A @ ( plus_plus @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( times_times @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X ) @ Y ) ) ) ) ).
% power2_sum
thf(fact_2234_oddE,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A3: A] :
( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
=> ~ ! [B7: A] :
( A3
!= ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B7 ) @ ( one_one @ A ) ) ) ) ) ).
% oddE
thf(fact_2235_floor__divide__lower,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Q3: A,P4: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Q3 )
=> ( ord_less_eq @ A @ ( times_times @ A @ ( ring_1_of_int @ A @ ( archim6421214686448440834_floor @ A @ ( divide_divide @ A @ P4 @ Q3 ) ) ) @ Q3 ) @ P4 ) ) ) ).
% floor_divide_lower
thf(fact_2236_ceiling__divide__upper,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Q3: A,P4: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Q3 )
=> ( ord_less_eq @ A @ P4 @ ( times_times @ A @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ A @ ( divide_divide @ A @ P4 @ Q3 ) ) ) @ Q3 ) ) ) ) ).
% ceiling_divide_upper
thf(fact_2237_Bernoulli__inequality,axiom,
! [X: real,N: nat] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X )
=> ( ord_less_eq @ real @ ( plus_plus @ real @ ( one_one @ real ) @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ X ) ) @ ( power_power @ real @ ( plus_plus @ real @ ( one_one @ real ) @ X ) @ N ) ) ) ).
% Bernoulli_inequality
thf(fact_2238_add__log__eq__powr,axiom,
! [B3: real,X: real,Y: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ B3 )
=> ( ( B3
!= ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( plus_plus @ real @ Y @ ( log @ B3 @ X ) )
= ( log @ B3 @ ( times_times @ real @ ( powr @ real @ B3 @ Y ) @ X ) ) ) ) ) ) ).
% add_log_eq_powr
thf(fact_2239_log__add__eq__powr,axiom,
! [B3: real,X: real,Y: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ B3 )
=> ( ( B3
!= ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( plus_plus @ real @ ( log @ B3 @ X ) @ Y )
= ( log @ B3 @ ( times_times @ real @ X @ ( powr @ real @ B3 @ Y ) ) ) ) ) ) ) ).
% log_add_eq_powr
thf(fact_2240_sum__squares__bound,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y: A] : ( ord_less_eq @ A @ ( times_times @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X ) @ Y ) @ ( plus_plus @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% sum_squares_bound
thf(fact_2241_power2__diff,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [X: A,Y: A] :
( ( power_power @ A @ ( minus_minus @ A @ X @ Y ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( minus_minus @ A @ ( plus_plus @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( times_times @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X ) @ Y ) ) ) ) ).
% power2_diff
thf(fact_2242_floor__divide__upper,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Q3: A,P4: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Q3 )
=> ( ord_less @ A @ P4 @ ( times_times @ A @ ( plus_plus @ A @ ( ring_1_of_int @ A @ ( archim6421214686448440834_floor @ A @ ( divide_divide @ A @ P4 @ Q3 ) ) ) @ ( one_one @ A ) ) @ Q3 ) ) ) ) ).
% floor_divide_upper
thf(fact_2243_ceiling__divide__lower,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Q3: A,P4: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Q3 )
=> ( ord_less @ A @ ( times_times @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ A @ ( divide_divide @ A @ P4 @ Q3 ) ) ) @ ( one_one @ A ) ) @ Q3 ) @ P4 ) ) ) ).
% ceiling_divide_lower
thf(fact_2244_L2__set__mult__ineq__lemma,axiom,
! [A3: real,C2: real,B3: real,D2: real] : ( ord_less_eq @ real @ ( times_times @ real @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( times_times @ real @ A3 @ C2 ) ) @ ( times_times @ real @ B3 @ D2 ) ) @ ( plus_plus @ real @ ( times_times @ real @ ( power_power @ real @ A3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ D2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( times_times @ real @ ( power_power @ real @ B3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ C2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% L2_set_mult_ineq_lemma
thf(fact_2245_log__minus__eq__powr,axiom,
! [B3: real,X: real,Y: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ B3 )
=> ( ( B3
!= ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( minus_minus @ real @ ( log @ B3 @ X ) @ Y )
= ( log @ B3 @ ( times_times @ real @ X @ ( powr @ real @ B3 @ ( uminus_uminus @ real @ Y ) ) ) ) ) ) ) ) ).
% log_minus_eq_powr
thf(fact_2246_arith__geo__mean,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [U2: A,X: A,Y: A] :
( ( ( power_power @ A @ U2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( times_times @ A @ X @ Y ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
=> ( ord_less_eq @ A @ U2 @ ( divide_divide @ A @ ( plus_plus @ A @ X @ Y ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% arith_geo_mean
thf(fact_2247_Bernoulli__inequality__even,axiom,
! [N: nat,X: real] :
( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ord_less_eq @ real @ ( plus_plus @ real @ ( one_one @ real ) @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ X ) ) @ ( power_power @ real @ ( plus_plus @ real @ ( one_one @ real ) @ X ) @ N ) ) ) ).
% Bernoulli_inequality_even
thf(fact_2248_round__def,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ( ( archimedean_round @ A )
= ( ^ [X2: A] : ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ X2 @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% round_def
thf(fact_2249_even__mult__exp__div__exp__iff,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A3: A,M: nat,N: nat] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ ( times_times @ A @ A3 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) )
= ( ( ord_less @ nat @ N @ M )
| ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N )
= ( zero_zero @ A ) )
| ( ( ord_less_eq @ nat @ M @ N )
& ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A3 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ N @ M ) ) ) ) ) ) ) ) ).
% even_mult_exp_div_exp_iff
thf(fact_2250_of__int__round__le,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] : ( ord_less_eq @ A @ ( ring_1_of_int @ A @ ( archimedean_round @ A @ X ) ) @ ( plus_plus @ A @ X @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% of_int_round_le
thf(fact_2251_of__int__round__ge,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] : ( ord_less_eq @ A @ ( minus_minus @ A @ X @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ ( ring_1_of_int @ A @ ( archimedean_round @ A @ X ) ) ) ) ).
% of_int_round_ge
thf(fact_2252_of__int__round__gt,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] : ( ord_less @ A @ ( minus_minus @ A @ X @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ ( ring_1_of_int @ A @ ( archimedean_round @ A @ X ) ) ) ) ).
% of_int_round_gt
thf(fact_2253_arsinh__def,axiom,
! [A: $tType] :
( ( ln @ A )
=> ( ( arsinh @ A )
= ( ^ [X2: A] : ( ln_ln @ A @ ( plus_plus @ A @ X2 @ ( powr @ A @ ( plus_plus @ A @ ( power_power @ A @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) ) @ ( real_Vector_of_real @ A @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% arsinh_def
thf(fact_2254_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_2255_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_2256_of__real__power,axiom,
! [A: $tType] :
( ( real_V2191834092415804123ebra_1 @ A )
=> ! [X: real,N: nat] :
( ( real_Vector_of_real @ A @ ( power_power @ real @ X @ N ) )
= ( power_power @ A @ ( real_Vector_of_real @ A @ X ) @ N ) ) ) ).
% of_real_power
thf(fact_2257_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_2258_of__real__eq__1__iff,axiom,
! [A: $tType] :
( ( real_V2191834092415804123ebra_1 @ A )
=> ! [X: real] :
( ( ( real_Vector_of_real @ A @ X )
= ( one_one @ A ) )
= ( X
= ( one_one @ real ) ) ) ) ).
% of_real_eq_1_iff
thf(fact_2259_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_2260_of__real__eq__0__iff,axiom,
! [A: $tType] :
( ( real_V2191834092415804123ebra_1 @ A )
=> ! [X: real] :
( ( ( real_Vector_of_real @ A @ X )
= ( zero_zero @ A ) )
= ( X
= ( zero_zero @ real ) ) ) ) ).
% of_real_eq_0_iff
thf(fact_2261_star__false__right,axiom,
! [P: assn] :
( ( times_times @ assn @ P @ ( bot_bot @ assn ) )
= ( bot_bot @ assn ) ) ).
% star_false_right
thf(fact_2262_star__false__left,axiom,
! [P: assn] :
( ( times_times @ assn @ ( bot_bot @ assn ) @ P )
= ( bot_bot @ assn ) ) ).
% star_false_left
thf(fact_2263_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_2264_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_2265_mult__0__right,axiom,
! [M: nat] :
( ( times_times @ nat @ M @ ( zero_zero @ nat ) )
= ( zero_zero @ nat ) ) ).
% mult_0_right
thf(fact_2266_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_2267_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_2268_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_2269_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_2270_semiring__norm_I11_J,axiom,
! [M: num] :
( ( times_times @ num @ M @ one2 )
= M ) ).
% semiring_norm(11)
thf(fact_2271_semiring__norm_I12_J,axiom,
! [N: num] :
( ( times_times @ num @ one2 @ N )
= N ) ).
% semiring_norm(12)
thf(fact_2272_norm__pre__pure__iff,axiom,
! [A: $tType,P: assn,B3: $o,F3: heap_Time_Heap @ A,Q: A > assn] :
( ( hoare_hoare_triple @ A @ ( times_times @ assn @ P @ ( pure_assn @ B3 ) ) @ F3 @ Q )
= ( B3
=> ( hoare_hoare_triple @ A @ P @ F3 @ Q ) ) ) ).
% norm_pre_pure_iff
thf(fact_2273_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_2274_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_2275_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_2276_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_2277_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_2278_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_2279_num__double,axiom,
! [N: num] :
( ( times_times @ num @ ( bit0 @ one2 ) @ N )
= ( bit0 @ N ) ) ).
% num_double
thf(fact_2280_power__mult__numeral,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A3: A,M: num,N: num] :
( ( power_power @ A @ ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ M ) ) @ ( numeral_numeral @ nat @ N ) )
= ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ ( times_times @ num @ M @ N ) ) ) ) ) ).
% power_mult_numeral
thf(fact_2281_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_2282_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_2283_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_2284_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_2285_Power_Oring__1__class_Opower__minus__even,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [A3: A,N: nat] :
( ( power_power @ A @ ( uminus_uminus @ A @ A3 ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
= ( power_power @ A @ A3 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% Power.ring_1_class.power_minus_even
thf(fact_2286_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_2287_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_2288_int__ops_I7_J,axiom,
! [A3: nat,B3: nat] :
( ( semiring_1_of_nat @ int @ ( times_times @ nat @ A3 @ B3 ) )
= ( times_times @ int @ ( semiring_1_of_nat @ int @ A3 ) @ ( semiring_1_of_nat @ int @ B3 ) ) ) ).
% int_ops(7)
thf(fact_2289_Abs__fnat__hom__mult,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: nat,B3: nat] :
( ( times_times @ ( word @ A ) @ ( semiring_1_of_nat @ ( word @ A ) @ A3 ) @ ( semiring_1_of_nat @ ( word @ A ) @ B3 ) )
= ( semiring_1_of_nat @ ( word @ A ) @ ( times_times @ nat @ A3 @ B3 ) ) ) ) ).
% Abs_fnat_hom_mult
thf(fact_2290_wi__hom__mult,axiom,
! [C: $tType] :
( ( type_len @ C )
=> ! [A3: int,B3: int] :
( ( times_times @ ( word @ C ) @ ( ring_1_of_int @ ( word @ C ) @ A3 ) @ ( ring_1_of_int @ ( word @ C ) @ B3 ) )
= ( ring_1_of_int @ ( word @ C ) @ ( times_times @ int @ A3 @ B3 ) ) ) ) ).
% wi_hom_mult
thf(fact_2291_word__mult__def,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( times_times @ ( word @ A ) )
= ( ^ [A5: word @ A,B5: word @ A] : ( ring_1_of_int @ ( word @ A ) @ ( times_times @ int @ ( semiring_1_unsigned @ A @ int @ A5 ) @ ( semiring_1_unsigned @ A @ int @ B5 ) ) ) ) ) ) ).
% word_mult_def
thf(fact_2292_word__arith__nat__mult,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( times_times @ ( word @ A ) )
= ( ^ [A5: word @ A,B5: word @ A] : ( semiring_1_of_nat @ ( word @ A ) @ ( times_times @ nat @ ( semiring_1_unsigned @ A @ nat @ A5 ) @ ( semiring_1_unsigned @ A @ nat @ B5 ) ) ) ) ) ) ).
% word_arith_nat_mult
thf(fact_2293_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_2294_mult__0,axiom,
! [N: nat] :
( ( times_times @ nat @ ( zero_zero @ nat ) @ N )
= ( zero_zero @ nat ) ) ).
% mult_0
thf(fact_2295_power__mult,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A3: A,M: nat,N: nat] :
( ( power_power @ A @ A3 @ ( times_times @ nat @ M @ N ) )
= ( power_power @ A @ ( power_power @ A @ A3 @ M ) @ N ) ) ) ).
% power_mult
thf(fact_2296_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_2297_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_2298_left__add__mult__distrib,axiom,
! [I: nat,U2: nat,J: nat,K: nat] :
( ( plus_plus @ nat @ ( times_times @ nat @ I @ U2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ J @ U2 ) @ K ) )
= ( plus_plus @ nat @ ( times_times @ nat @ ( plus_plus @ nat @ I @ J ) @ U2 ) @ K ) ) ).
% left_add_mult_distrib
thf(fact_2299_le__cube,axiom,
! [M: nat] : ( ord_less_eq @ nat @ M @ ( times_times @ nat @ M @ ( times_times @ nat @ M @ M ) ) ) ).
% le_cube
thf(fact_2300_le__square,axiom,
! [M: nat] : ( ord_less_eq @ nat @ M @ ( times_times @ nat @ M @ M ) ) ).
% le_square
thf(fact_2301_mult__le__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( ord_less_eq @ nat @ K @ L )
=> ( ord_less_eq @ nat @ ( times_times @ nat @ I @ K ) @ ( times_times @ nat @ J @ L ) ) ) ) ).
% mult_le_mono
thf(fact_2302_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_2303_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_2304_nat__mult__1__right,axiom,
! [N: nat] :
( ( times_times @ nat @ N @ ( one_one @ nat ) )
= N ) ).
% nat_mult_1_right
thf(fact_2305_nat__mult__1,axiom,
! [N: nat] :
( ( times_times @ nat @ ( one_one @ nat ) @ N )
= N ) ).
% nat_mult_1
thf(fact_2306_times__int__code_I2_J,axiom,
! [L: int] :
( ( times_times @ int @ ( zero_zero @ int ) @ L )
= ( zero_zero @ int ) ) ).
% times_int_code(2)
thf(fact_2307_times__int__code_I1_J,axiom,
! [K: int] :
( ( times_times @ int @ K @ ( zero_zero @ int ) )
= ( zero_zero @ int ) ) ).
% times_int_code(1)
thf(fact_2308_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_2309_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_2310_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_2311_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_2312_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_2313_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_2314_assn__one__left,axiom,
! [P: assn] :
( ( times_times @ assn @ ( one_one @ assn ) @ P )
= P ) ).
% assn_one_left
thf(fact_2315_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_2316_frame__rule,axiom,
! [A: $tType,P: assn,C2: heap_Time_Heap @ A,Q: A > assn,R: assn] :
( ( hoare_hoare_triple @ A @ P @ C2 @ Q )
=> ( hoare_hoare_triple @ A @ ( times_times @ assn @ P @ R ) @ C2
@ ^ [X2: A] : ( times_times @ assn @ ( Q @ X2 ) @ R ) ) ) ).
% frame_rule
thf(fact_2317_div__mult2__numeral__eq,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [A3: A,K: num,L: num] :
( ( divide_divide @ A @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ K ) ) @ ( numeral_numeral @ A @ L ) )
= ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( times_times @ num @ K @ L ) ) ) ) ) ).
% div_mult2_numeral_eq
thf(fact_2318_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_2319_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_2320_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_2321_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_2322_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_2323_mlex__snd__decrI,axiom,
! [A3: nat,A7: nat,B3: nat,B8: nat,N3: nat] :
( ( A3 = A7 )
=> ( ( ord_less @ nat @ B3 @ B8 )
=> ( ord_less @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ A3 @ N3 ) @ B3 ) @ ( plus_plus @ nat @ ( times_times @ nat @ A7 @ N3 ) @ B8 ) ) ) ) ).
% mlex_snd_decrI
thf(fact_2324_mlex__fst__decrI,axiom,
! [A3: nat,A7: nat,B3: nat,N3: nat,B8: nat] :
( ( ord_less @ nat @ A3 @ A7 )
=> ( ( ord_less @ nat @ B3 @ N3 )
=> ( ( ord_less @ nat @ B8 @ N3 )
=> ( ord_less @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ A3 @ N3 ) @ B3 ) @ ( plus_plus @ nat @ ( times_times @ nat @ A7 @ N3 ) @ B8 ) ) ) ) ) ).
% mlex_fst_decrI
thf(fact_2325_mlex__bound,axiom,
! [A3: nat,A2: nat,B3: nat,N3: nat] :
( ( ord_less @ nat @ A3 @ A2 )
=> ( ( ord_less @ nat @ B3 @ N3 )
=> ( ord_less @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ A3 @ N3 ) @ B3 ) @ ( times_times @ nat @ A2 @ N3 ) ) ) ) ).
% mlex_bound
thf(fact_2326_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_2327_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_2328_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_2329_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_2330_zdvd__mono,axiom,
! [K: int,M: int,T: int] :
( ( K
!= ( zero_zero @ int ) )
=> ( ( dvd_dvd @ int @ M @ T )
= ( dvd_dvd @ int @ ( times_times @ int @ K @ M ) @ ( times_times @ int @ K @ T ) ) ) ) ).
% zdvd_mono
thf(fact_2331_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_2332_zdvd__period,axiom,
! [A3: int,D2: int,X: int,T: int,C2: int] :
( ( dvd_dvd @ int @ A3 @ D2 )
=> ( ( dvd_dvd @ int @ A3 @ ( plus_plus @ int @ X @ T ) )
= ( dvd_dvd @ int @ A3 @ ( plus_plus @ int @ ( plus_plus @ int @ X @ ( times_times @ int @ C2 @ D2 ) ) @ T ) ) ) ) ).
% zdvd_period
thf(fact_2333_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_2334_div__to__mult__word__lt,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A,Z: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ X @ ( divide_divide @ ( word @ A ) @ Y @ Z ) )
=> ( ord_less_eq @ ( word @ A ) @ ( times_times @ ( word @ A ) @ X @ Z ) @ Y ) ) ) ).
% div_to_mult_word_lt
thf(fact_2335_word__div__mult__le,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A,B3: word @ A] : ( ord_less_eq @ ( word @ A ) @ ( times_times @ ( word @ A ) @ ( divide_divide @ ( word @ A ) @ A3 @ B3 ) @ B3 ) @ A3 ) ) ).
% word_div_mult_le
thf(fact_2336_norm__pre__pure__rule1,axiom,
! [A: $tType,B3: $o,P: assn,F3: heap_Time_Heap @ A,Q: A > assn] :
( ( B3
=> ( hoare_hoare_triple @ A @ P @ F3 @ Q ) )
=> ( hoare_hoare_triple @ A @ ( times_times @ assn @ P @ ( pure_assn @ B3 ) ) @ F3 @ Q ) ) ).
% norm_pre_pure_rule1
thf(fact_2337_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_2338_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_2339_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_2340_td__gal__lt,axiom,
! [C2: nat,A3: nat,B3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ C2 )
=> ( ( ord_less @ nat @ A3 @ ( times_times @ nat @ B3 @ C2 ) )
= ( ord_less @ nat @ ( divide_divide @ nat @ A3 @ C2 ) @ B3 ) ) ) ).
% td_gal_lt
thf(fact_2341_div__less__iff__less__mult,axiom,
! [Q3: nat,M: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ Q3 )
=> ( ( ord_less @ nat @ ( divide_divide @ nat @ M @ Q3 ) @ N )
= ( ord_less @ nat @ M @ ( times_times @ nat @ N @ Q3 ) ) ) ) ).
% div_less_iff_less_mult
thf(fact_2342_nat__eq__add__iff1,axiom,
! [J: nat,I: nat,U2: nat,M: nat,N: nat] :
( ( ord_less_eq @ nat @ J @ I )
=> ( ( ( plus_plus @ nat @ ( times_times @ nat @ I @ U2 ) @ M )
= ( plus_plus @ nat @ ( times_times @ nat @ J @ U2 ) @ N ) )
= ( ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ I @ J ) @ U2 ) @ M )
= N ) ) ) ).
% nat_eq_add_iff1
thf(fact_2343_nat__eq__add__iff2,axiom,
! [I: nat,J: nat,U2: nat,M: nat,N: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( ( plus_plus @ nat @ ( times_times @ nat @ I @ U2 ) @ M )
= ( plus_plus @ nat @ ( times_times @ nat @ J @ U2 ) @ N ) )
= ( M
= ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ J @ I ) @ U2 ) @ N ) ) ) ) ).
% nat_eq_add_iff2
thf(fact_2344_nat__le__add__iff1,axiom,
! [J: nat,I: nat,U2: nat,M: nat,N: nat] :
( ( ord_less_eq @ nat @ J @ I )
=> ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ I @ U2 ) @ M ) @ ( plus_plus @ nat @ ( times_times @ nat @ J @ U2 ) @ N ) )
= ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ I @ J ) @ U2 ) @ M ) @ N ) ) ) ).
% nat_le_add_iff1
thf(fact_2345_nat__le__add__iff2,axiom,
! [I: nat,J: nat,U2: nat,M: nat,N: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ I @ U2 ) @ M ) @ ( plus_plus @ nat @ ( times_times @ nat @ J @ U2 ) @ N ) )
= ( ord_less_eq @ nat @ M @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ J @ I ) @ U2 ) @ N ) ) ) ) ).
% nat_le_add_iff2
thf(fact_2346_nat__diff__add__eq1,axiom,
! [J: nat,I: nat,U2: nat,M: nat,N: nat] :
( ( ord_less_eq @ nat @ J @ I )
=> ( ( minus_minus @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ I @ U2 ) @ M ) @ ( plus_plus @ nat @ ( times_times @ nat @ J @ U2 ) @ N ) )
= ( minus_minus @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ I @ J ) @ U2 ) @ M ) @ N ) ) ) ).
% nat_diff_add_eq1
thf(fact_2347_nat__diff__add__eq2,axiom,
! [I: nat,J: nat,U2: nat,M: nat,N: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( minus_minus @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ I @ U2 ) @ M ) @ ( plus_plus @ nat @ ( times_times @ nat @ J @ U2 ) @ N ) )
= ( minus_minus @ nat @ M @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ J @ I ) @ U2 ) @ N ) ) ) ) ).
% nat_diff_add_eq2
thf(fact_2348_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_2349_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_2350_bezout__add__strong__nat,axiom,
! [A3: nat,B3: nat] :
( ( A3
!= ( zero_zero @ nat ) )
=> ? [D6: nat,X3: nat,Y4: nat] :
( ( dvd_dvd @ nat @ D6 @ A3 )
& ( dvd_dvd @ nat @ D6 @ B3 )
& ( ( times_times @ nat @ A3 @ X3 )
= ( plus_plus @ nat @ ( times_times @ nat @ B3 @ Y4 ) @ D6 ) ) ) ) ).
% bezout_add_strong_nat
thf(fact_2351_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_2352_uint__mult__ge0,axiom,
! [Aa: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ Aa ) )
=> ! [Xa: word @ Aa,X: word @ A] : ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( times_times @ int @ ( semiring_1_unsigned @ Aa @ int @ Xa ) @ ( semiring_1_unsigned @ A @ int @ X ) ) ) ) ).
% uint_mult_ge0
thf(fact_2353_minusinfinity,axiom,
! [D2: int,P1: int > $o,P: int > $o] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D2 )
=> ( ! [X3: int,K2: int] :
( ( P1 @ X3 )
= ( P1 @ ( minus_minus @ int @ X3 @ ( times_times @ int @ K2 @ D2 ) ) ) )
=> ( ? [Z6: int] :
! [X3: int] :
( ( ord_less @ int @ X3 @ Z6 )
=> ( ( P @ X3 )
= ( P1 @ X3 ) ) )
=> ( ? [X_12: int] : ( P1 @ X_12 )
=> ? [X_1: int] : ( P @ X_1 ) ) ) ) ) ).
% minusinfinity
thf(fact_2354_plusinfinity,axiom,
! [D2: int,P5: int > $o,P: int > $o] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D2 )
=> ( ! [X3: int,K2: int] :
( ( P5 @ X3 )
= ( P5 @ ( minus_minus @ int @ X3 @ ( times_times @ int @ K2 @ D2 ) ) ) )
=> ( ? [Z6: int] :
! [X3: int] :
( ( ord_less @ int @ Z6 @ X3 )
=> ( ( P @ X3 )
= ( P5 @ X3 ) ) )
=> ( ? [X_12: int] : ( P5 @ X_12 )
=> ? [X_1: int] : ( P @ X_1 ) ) ) ) ) ).
% plusinfinity
thf(fact_2355_zdiv__zmult2__eq,axiom,
! [C2: int,A3: int,B3: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ C2 )
=> ( ( divide_divide @ int @ A3 @ ( times_times @ int @ B3 @ C2 ) )
= ( divide_divide @ int @ ( divide_divide @ int @ A3 @ B3 ) @ C2 ) ) ) ).
% zdiv_zmult2_eq
thf(fact_2356_zdiv__mult__self,axiom,
! [M: int,A3: int,N: int] :
( ( M
!= ( zero_zero @ int ) )
=> ( ( divide_divide @ int @ ( plus_plus @ int @ A3 @ ( times_times @ int @ M @ N ) ) @ M )
= ( plus_plus @ int @ ( divide_divide @ int @ A3 @ M ) @ N ) ) ) ).
% zdiv_mult_self
thf(fact_2357_div__lt__mult,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [I: word @ A,K: word @ A,X: word @ A] :
( ( ord_less @ ( word @ A ) @ I @ ( divide_divide @ ( word @ A ) @ K @ X ) )
=> ( ( ord_less @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ X )
=> ( ord_less @ ( word @ A ) @ ( times_times @ ( word @ A ) @ I @ X ) @ K ) ) ) ) ).
% div_lt_mult
thf(fact_2358_More__Word_Oword__div__mult,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [C2: word @ A,A3: word @ A,B3: word @ A] :
( ( ord_less @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ C2 )
=> ( ( ord_less @ ( word @ A ) @ A3 @ ( times_times @ ( word @ A ) @ B3 @ C2 ) )
=> ( ord_less @ ( word @ A ) @ ( divide_divide @ ( word @ A ) @ A3 @ C2 ) @ B3 ) ) ) ) ).
% More_Word.word_div_mult
thf(fact_2359_power__even__eq,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A3: A,N: nat] :
( ( power_power @ A @ A3 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
= ( power_power @ A @ ( power_power @ A @ A3 @ N ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% power_even_eq
thf(fact_2360_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_2361_split__div,axiom,
! [P: nat > $o,M: nat,N: nat] :
( ( P @ ( divide_divide @ nat @ M @ N ) )
= ( ( ( N
= ( zero_zero @ nat ) )
=> ( P @ ( zero_zero @ nat ) ) )
& ( ( N
!= ( zero_zero @ nat ) )
=> ! [I4: nat,J3: nat] :
( ( ord_less @ nat @ J3 @ N )
=> ( ( M
= ( plus_plus @ nat @ ( times_times @ nat @ N @ I4 ) @ J3 ) )
=> ( P @ I4 ) ) ) ) ) ) ).
% split_div
thf(fact_2362_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_2363_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_2364_td__gal,axiom,
! [C2: nat,B3: nat,A3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ C2 )
=> ( ( ord_less_eq @ nat @ ( times_times @ nat @ B3 @ C2 ) @ A3 )
= ( ord_less_eq @ nat @ B3 @ ( divide_divide @ nat @ A3 @ C2 ) ) ) ) ).
% td_gal
thf(fact_2365_less__eq__div__iff__mult__less__eq,axiom,
! [Q3: nat,M: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ Q3 )
=> ( ( ord_less_eq @ nat @ M @ ( divide_divide @ nat @ N @ Q3 ) )
= ( ord_less_eq @ nat @ ( times_times @ nat @ M @ Q3 ) @ N ) ) ) ).
% less_eq_div_iff_mult_less_eq
thf(fact_2366_mult__eq__if,axiom,
( ( times_times @ nat )
= ( ^ [M3: nat,N4: nat] :
( if @ nat
@ ( M3
= ( zero_zero @ nat ) )
@ ( zero_zero @ nat )
@ ( plus_plus @ nat @ N4 @ ( times_times @ nat @ ( minus_minus @ nat @ M3 @ ( one_one @ nat ) ) @ N4 ) ) ) ) ) ).
% mult_eq_if
thf(fact_2367_nat__less__add__iff2,axiom,
! [I: nat,J: nat,U2: nat,M: nat,N: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( ord_less @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ I @ U2 ) @ M ) @ ( plus_plus @ nat @ ( times_times @ nat @ J @ U2 ) @ N ) )
= ( ord_less @ nat @ M @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ J @ I ) @ U2 ) @ N ) ) ) ) ).
% nat_less_add_iff2
thf(fact_2368_nat__less__add__iff1,axiom,
! [J: nat,I: nat,U2: nat,M: nat,N: nat] :
( ( ord_less_eq @ nat @ J @ I )
=> ( ( ord_less @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ I @ U2 ) @ M ) @ ( plus_plus @ nat @ ( times_times @ nat @ J @ U2 ) @ N ) )
= ( ord_less @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ I @ J ) @ U2 ) @ M ) @ N ) ) ) ).
% nat_less_add_iff1
thf(fact_2369_nat__mult__power__less__eq,axiom,
! [B3: nat,A3: nat,N: nat,M: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ B3 )
=> ( ( ord_less @ nat @ ( times_times @ nat @ A3 @ ( power_power @ nat @ B3 @ N ) ) @ ( power_power @ nat @ B3 @ M ) )
= ( ord_less @ nat @ A3 @ ( power_power @ nat @ B3 @ ( minus_minus @ nat @ M @ N ) ) ) ) ) ).
% nat_mult_power_less_eq
thf(fact_2370_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_2371_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_2372_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_2373_dvd__minus__add,axiom,
! [Q3: nat,N: nat,R3: nat,M: nat] :
( ( ord_less_eq @ nat @ Q3 @ N )
=> ( ( ord_less_eq @ nat @ Q3 @ ( times_times @ nat @ R3 @ M ) )
=> ( ( dvd_dvd @ nat @ M @ ( minus_minus @ nat @ N @ Q3 ) )
= ( dvd_dvd @ nat @ M @ ( plus_plus @ nat @ N @ ( minus_minus @ nat @ ( times_times @ nat @ R3 @ M ) @ Q3 ) ) ) ) ) ) ).
% dvd_minus_add
thf(fact_2374_pos__mult__pos__ge,axiom,
! [X: int,N: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ X )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ N )
=> ( ord_less_eq @ int @ ( times_times @ int @ N @ ( one_one @ int ) ) @ ( times_times @ int @ N @ X ) ) ) ) ).
% pos_mult_pos_ge
thf(fact_2375_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 )
=> ! [X6: int] :
( ( P @ X6 )
=> ( P @ ( plus_plus @ int @ X6 @ ( times_times @ int @ K @ D2 ) ) ) ) ) ) ) ).
% incr_mult_lemma
thf(fact_2376_unique__quotient__lemma__neg,axiom,
! [B3: int,Q4: int,R4: int,Q3: int,R3: int] :
( ( ord_less_eq @ int @ ( plus_plus @ int @ ( times_times @ int @ B3 @ Q4 ) @ R4 ) @ ( plus_plus @ int @ ( times_times @ int @ B3 @ Q3 ) @ R3 ) )
=> ( ( ord_less_eq @ int @ R3 @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ B3 @ R3 )
=> ( ( ord_less @ int @ B3 @ R4 )
=> ( ord_less_eq @ int @ Q3 @ Q4 ) ) ) ) ) ).
% unique_quotient_lemma_neg
thf(fact_2377_unique__quotient__lemma,axiom,
! [B3: int,Q4: int,R4: int,Q3: int,R3: int] :
( ( ord_less_eq @ int @ ( plus_plus @ int @ ( times_times @ int @ B3 @ Q4 ) @ R4 ) @ ( plus_plus @ int @ ( times_times @ int @ B3 @ Q3 ) @ R3 ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ R4 )
=> ( ( ord_less @ int @ R4 @ B3 )
=> ( ( ord_less @ int @ R3 @ B3 )
=> ( ord_less_eq @ int @ Q4 @ Q3 ) ) ) ) ) ).
% unique_quotient_lemma
thf(fact_2378_zdiv__mono2__neg__lemma,axiom,
! [B3: int,Q3: int,R3: int,B8: int,Q4: int,R4: int] :
( ( ( plus_plus @ int @ ( times_times @ int @ B3 @ Q3 ) @ R3 )
= ( plus_plus @ int @ ( times_times @ int @ B8 @ Q4 ) @ R4 ) )
=> ( ( ord_less @ int @ ( plus_plus @ int @ ( times_times @ int @ B8 @ Q4 ) @ R4 ) @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ R3 @ B3 )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ R4 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ B8 )
=> ( ( ord_less_eq @ int @ B8 @ B3 )
=> ( ord_less_eq @ int @ Q4 @ Q3 ) ) ) ) ) ) ) ).
% zdiv_mono2_neg_lemma
thf(fact_2379_zdiv__mono2__lemma,axiom,
! [B3: int,Q3: int,R3: int,B8: int,Q4: int,R4: int] :
( ( ( plus_plus @ int @ ( times_times @ int @ B3 @ Q3 ) @ R3 )
= ( plus_plus @ int @ ( times_times @ int @ B8 @ Q4 ) @ R4 ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( plus_plus @ int @ ( times_times @ int @ B8 @ Q4 ) @ R4 ) )
=> ( ( ord_less @ int @ R4 @ B8 )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ R3 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ B8 )
=> ( ( ord_less_eq @ int @ B8 @ B3 )
=> ( ord_less_eq @ int @ Q3 @ Q4 ) ) ) ) ) ) ) ).
% zdiv_mono2_lemma
thf(fact_2380_q__pos__lemma,axiom,
! [B8: int,Q4: int,R4: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( plus_plus @ int @ ( times_times @ int @ B8 @ Q4 ) @ R4 ) )
=> ( ( ord_less @ int @ R4 @ B8 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ B8 )
=> ( ord_less_eq @ int @ ( zero_zero @ int ) @ Q4 ) ) ) ) ).
% q_pos_lemma
thf(fact_2381_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 )
=> ! [X6: int] :
( ( P @ X6 )
=> ( P @ ( minus_minus @ int @ X6 @ ( times_times @ int @ K @ D2 ) ) ) ) ) ) ) ).
% decr_mult_lemma
thf(fact_2382_udvd__incr__lem0,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Up: int,Uq: int,N: int,K4: word @ A,N7: int] :
( ( ord_less @ int @ Up @ Uq )
=> ( ( Up
= ( times_times @ int @ N @ ( semiring_1_unsigned @ A @ int @ K4 ) ) )
=> ( ( Uq
= ( times_times @ int @ N7 @ ( semiring_1_unsigned @ A @ int @ K4 ) ) )
=> ( ord_less_eq @ int @ ( plus_plus @ int @ Up @ ( semiring_1_unsigned @ A @ int @ K4 ) ) @ Uq ) ) ) ) ) ).
% udvd_incr_lem0
thf(fact_2383_udvd__incr__lem,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Up: int,Uq: int,Ua: int,N: int,K4: word @ A,N7: int] :
( ( ord_less @ int @ Up @ Uq )
=> ( ( Up
= ( plus_plus @ int @ Ua @ ( times_times @ int @ N @ ( semiring_1_unsigned @ A @ int @ K4 ) ) ) )
=> ( ( Uq
= ( plus_plus @ int @ Ua @ ( times_times @ int @ N7 @ ( semiring_1_unsigned @ A @ int @ K4 ) ) ) )
=> ( ord_less_eq @ int @ ( plus_plus @ int @ Up @ ( semiring_1_unsigned @ A @ int @ K4 ) ) @ Uq ) ) ) ) ) ).
% udvd_incr_lem
thf(fact_2384_udvd__incr0,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [P4: word @ A,Q3: word @ A,N: int,K4: word @ A,N7: int] :
( ( ord_less @ ( word @ A ) @ P4 @ Q3 )
=> ( ( ( semiring_1_unsigned @ A @ int @ P4 )
= ( times_times @ int @ N @ ( semiring_1_unsigned @ A @ int @ K4 ) ) )
=> ( ( ( semiring_1_unsigned @ A @ int @ Q3 )
= ( times_times @ int @ N7 @ ( semiring_1_unsigned @ A @ int @ K4 ) ) )
=> ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ P4 @ K4 ) @ Q3 ) ) ) ) ) ).
% udvd_incr0
thf(fact_2385_udvd__decr0,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [P4: word @ A,Q3: word @ A,N: int,K4: word @ A,N7: int] :
( ( ord_less @ ( word @ A ) @ P4 @ Q3 )
=> ( ( ( semiring_1_unsigned @ A @ int @ P4 )
= ( times_times @ int @ N @ ( semiring_1_unsigned @ A @ int @ K4 ) ) )
=> ( ( ( semiring_1_unsigned @ A @ int @ Q3 )
= ( times_times @ int @ N7 @ ( semiring_1_unsigned @ A @ int @ K4 ) ) )
=> ( ( ( semiring_1_unsigned @ A @ int @ Q3 )
= ( times_times @ int @ N7 @ ( semiring_1_unsigned @ A @ int @ K4 ) ) )
=> ( ord_less_eq @ ( word @ A ) @ P4 @ ( minus_minus @ ( word @ A ) @ Q3 @ K4 ) ) ) ) ) ) ) ).
% udvd_decr0
thf(fact_2386_div__le__mult,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [I: word @ A,K: word @ A,X: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ I @ ( divide_divide @ ( word @ A ) @ K @ X ) )
=> ( ( ord_less @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ X )
=> ( ord_less_eq @ ( word @ A ) @ ( times_times @ ( word @ A ) @ I @ X ) @ K ) ) ) ) ).
% div_le_mult
thf(fact_2387_int__div__pos__eq,axiom,
! [A3: int,B3: int,Q3: int,R3: int] :
( ( A3
= ( plus_plus @ int @ ( times_times @ int @ B3 @ Q3 ) @ R3 ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ R3 )
=> ( ( ord_less @ int @ R3 @ B3 )
=> ( ( divide_divide @ int @ A3 @ B3 )
= Q3 ) ) ) ) ).
% int_div_pos_eq
thf(fact_2388_int__div__neg__eq,axiom,
! [A3: int,B3: int,Q3: int,R3: int] :
( ( A3
= ( plus_plus @ int @ ( times_times @ int @ B3 @ Q3 ) @ R3 ) )
=> ( ( ord_less_eq @ int @ R3 @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ B3 @ R3 )
=> ( ( divide_divide @ int @ A3 @ B3 )
= Q3 ) ) ) ) ).
% int_div_neg_eq
thf(fact_2389_split__zdiv,axiom,
! [P: int > $o,N: int,K: int] :
( ( P @ ( divide_divide @ int @ N @ K ) )
= ( ( ( K
= ( zero_zero @ int ) )
=> ( P @ ( zero_zero @ int ) ) )
& ( ( ord_less @ int @ ( zero_zero @ int ) @ K )
=> ! [I4: int,J3: int] :
( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ J3 )
& ( ord_less @ int @ J3 @ K )
& ( N
= ( plus_plus @ int @ ( times_times @ int @ K @ I4 ) @ J3 ) ) )
=> ( P @ I4 ) ) )
& ( ( ord_less @ int @ K @ ( zero_zero @ int ) )
=> ! [I4: int,J3: int] :
( ( ( ord_less @ int @ K @ J3 )
& ( ord_less_eq @ int @ J3 @ ( zero_zero @ int ) )
& ( N
= ( plus_plus @ int @ ( times_times @ int @ K @ I4 ) @ J3 ) ) )
=> ( P @ I4 ) ) ) ) ) ).
% split_zdiv
thf(fact_2390_udvd__incr_H,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [P4: word @ A,Q3: word @ A,Ua: int,N: int,K4: word @ A,N7: int] :
( ( ord_less @ ( word @ A ) @ P4 @ Q3 )
=> ( ( ( semiring_1_unsigned @ A @ int @ P4 )
= ( plus_plus @ int @ Ua @ ( times_times @ int @ N @ ( semiring_1_unsigned @ A @ int @ K4 ) ) ) )
=> ( ( ( semiring_1_unsigned @ A @ int @ Q3 )
= ( plus_plus @ int @ Ua @ ( times_times @ int @ N7 @ ( semiring_1_unsigned @ A @ int @ K4 ) ) ) )
=> ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ P4 @ K4 ) @ Q3 ) ) ) ) ) ).
% udvd_incr'
thf(fact_2391_udvd__decr_H,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [P4: word @ A,Q3: word @ A,Ua: int,N: int,K4: word @ A,N7: int] :
( ( ord_less @ ( word @ A ) @ P4 @ Q3 )
=> ( ( ( semiring_1_unsigned @ A @ int @ P4 )
= ( plus_plus @ int @ Ua @ ( times_times @ int @ N @ ( semiring_1_unsigned @ A @ int @ K4 ) ) ) )
=> ( ( ( semiring_1_unsigned @ A @ int @ Q3 )
= ( plus_plus @ int @ Ua @ ( times_times @ int @ N7 @ ( semiring_1_unsigned @ A @ int @ K4 ) ) ) )
=> ( ( ( semiring_1_unsigned @ A @ int @ Q3 )
= ( plus_plus @ int @ Ua @ ( times_times @ int @ N7 @ ( semiring_1_unsigned @ A @ int @ K4 ) ) ) )
=> ( ord_less_eq @ ( word @ A ) @ P4 @ ( minus_minus @ ( word @ A ) @ Q3 @ K4 ) ) ) ) ) ) ) ).
% udvd_decr'
thf(fact_2392_zero__le__even__power_H,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,N: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A3 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% zero_le_even_power'
thf(fact_2393_nat__add__offset__less,axiom,
! [Y: nat,N: nat,X: nat,M: nat,Sz: nat] :
( ( ord_less @ nat @ Y @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
=> ( ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) )
=> ( ( Sz
= ( plus_plus @ nat @ M @ N ) )
=> ( ord_less @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) @ Y ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Sz ) ) ) ) ) ).
% nat_add_offset_less
thf(fact_2394_nat__power__less__diff,axiom,
! [N: nat,Q3: nat,M: nat] :
( ( ord_less @ nat @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ Q3 ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) )
=> ( ord_less @ nat @ Q3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ M @ N ) ) ) ) ).
% nat_power_less_diff
thf(fact_2395_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_2396_z1pdiv2,axiom,
! [B3: int] :
( ( divide_divide @ int @ ( plus_plus @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ B3 ) @ ( one_one @ int ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) )
= B3 ) ).
% z1pdiv2
thf(fact_2397_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_2398_int__bit__induct,axiom,
! [P: int > $o,K: int] :
( ( P @ ( zero_zero @ int ) )
=> ( ( P @ ( uminus_uminus @ int @ ( one_one @ int ) ) )
=> ( ! [K2: int] :
( ( P @ K2 )
=> ( ( K2
!= ( zero_zero @ int ) )
=> ( P @ ( times_times @ int @ K2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) )
=> ( ! [K2: int] :
( ( P @ K2 )
=> ( ( K2
!= ( uminus_uminus @ int @ ( one_one @ int ) ) )
=> ( P @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ K2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) )
=> ( P @ K ) ) ) ) ) ).
% int_bit_induct
thf(fact_2399_power__2__mult__step__le,axiom,
! [N7: nat,N: nat,K5: nat,K: nat] :
( ( ord_less_eq @ nat @ N7 @ N )
=> ( ( ord_less @ nat @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N7 ) @ K5 ) @ ( 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 ) ) @ N7 ) @ ( plus_plus @ nat @ K5 @ ( one_one @ nat ) ) ) @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ K ) ) ) ) ).
% power_2_mult_step_le
thf(fact_2400_neg__zdiv__mult__2,axiom,
! [A3: int,B3: int] :
( ( ord_less_eq @ int @ A3 @ ( zero_zero @ int ) )
=> ( ( divide_divide @ int @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ B3 ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ A3 ) )
= ( divide_divide @ int @ ( plus_plus @ int @ B3 @ ( one_one @ int ) ) @ A3 ) ) ) ).
% neg_zdiv_mult_2
thf(fact_2401_pos__zdiv__mult__2,axiom,
! [A3: int,B3: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ A3 )
=> ( ( divide_divide @ int @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ B3 ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ A3 ) )
= ( divide_divide @ int @ B3 @ A3 ) ) ) ).
% pos_zdiv_mult_2
thf(fact_2402_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_2403_of__nat__eq__size,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,W: word @ A] :
( ( ( semiring_1_of_nat @ ( word @ A ) @ N )
= W )
= ( ? [Q5: nat] :
( N
= ( plus_plus @ nat @ ( semiring_1_unsigned @ A @ nat @ W ) @ ( times_times @ nat @ Q5 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( size_size @ ( word @ A ) @ W ) ) ) ) ) ) ) ) ).
% of_nat_eq_size
thf(fact_2404_nonzero__of__real__divide,axiom,
! [A: $tType] :
( ( real_V7773925162809079976_field @ A )
=> ! [Y: real,X: real] :
( ( Y
!= ( zero_zero @ real ) )
=> ( ( real_Vector_of_real @ A @ ( divide_divide @ real @ X @ Y ) )
= ( divide_divide @ A @ ( real_Vector_of_real @ A @ X ) @ ( real_Vector_of_real @ A @ Y ) ) ) ) ) ).
% nonzero_of_real_divide
thf(fact_2405_VEBT__internal_Omulcomm,axiom,
! [I: nat,Va: nat] :
( ( times_times @ nat @ I @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Va @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
= ( times_times @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Va @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ I ) ) ).
% VEBT_internal.mulcomm
thf(fact_2406_set__bit__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A] :
( ( bit_se5668285175392031749et_bit @ A @ ( zero_zero @ nat ) @ A3 )
= ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% set_bit_0
thf(fact_2407_abs__ln__one__plus__x__minus__x__bound__nonpos,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X )
=> ( ( ord_less_eq @ real @ X @ ( zero_zero @ real ) )
=> ( ord_less_eq @ real @ ( abs_abs @ real @ ( minus_minus @ real @ ( ln_ln @ real @ ( plus_plus @ real @ ( one_one @ real ) @ X ) ) @ X ) ) @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% abs_ln_one_plus_x_minus_x_bound_nonpos
thf(fact_2408_mult__le__cancel__iff2,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z: A,X: A,Y: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Z )
=> ( ( ord_less_eq @ A @ ( times_times @ A @ Z @ X ) @ ( times_times @ A @ Z @ Y ) )
= ( ord_less_eq @ A @ X @ Y ) ) ) ) ).
% mult_le_cancel_iff2
thf(fact_2409_mult__le__cancel__iff1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z: A,X: A,Y: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Z )
=> ( ( ord_less_eq @ A @ ( times_times @ A @ X @ Z ) @ ( times_times @ A @ Y @ Z ) )
= ( ord_less_eq @ A @ X @ Y ) ) ) ) ).
% mult_le_cancel_iff1
thf(fact_2410_unset__bit__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A] :
( ( bit_se2638667681897837118et_bit @ A @ ( zero_zero @ nat ) @ A3 )
= ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% unset_bit_0
thf(fact_2411_abs__idempotent,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A] :
( ( abs_abs @ A @ ( abs_abs @ A @ A3 ) )
= ( abs_abs @ A @ A3 ) ) ) ).
% abs_idempotent
thf(fact_2412_abs__abs,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ! [A3: A] :
( ( abs_abs @ A @ ( abs_abs @ A @ A3 ) )
= ( abs_abs @ A @ A3 ) ) ) ).
% abs_abs
thf(fact_2413_abs__0,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ( ( abs_abs @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% abs_0
thf(fact_2414_abs__zero,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ( ( abs_abs @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% abs_zero
thf(fact_2415_abs__eq__0,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A] :
( ( ( abs_abs @ A @ A3 )
= ( zero_zero @ A ) )
= ( A3
= ( zero_zero @ A ) ) ) ) ).
% abs_eq_0
thf(fact_2416_abs__0__eq,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A] :
( ( ( zero_zero @ A )
= ( abs_abs @ A @ A3 ) )
= ( A3
= ( zero_zero @ A ) ) ) ) ).
% abs_0_eq
thf(fact_2417_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_2418_abs__add__abs,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A,B3: A] :
( ( abs_abs @ A @ ( plus_plus @ A @ ( abs_abs @ A @ A3 ) @ ( abs_abs @ A @ B3 ) ) )
= ( plus_plus @ A @ ( abs_abs @ A @ A3 ) @ ( abs_abs @ A @ B3 ) ) ) ) ).
% abs_add_abs
thf(fact_2419_abs__1,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ( ( abs_abs @ A @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% abs_1
thf(fact_2420_abs__mult__self__eq,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A] :
( ( times_times @ A @ ( abs_abs @ A @ A3 ) @ ( abs_abs @ A @ A3 ) )
= ( times_times @ A @ A3 @ A3 ) ) ) ).
% abs_mult_self_eq
thf(fact_2421_abs__minus__cancel,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A] :
( ( abs_abs @ A @ ( uminus_uminus @ A @ A3 ) )
= ( abs_abs @ A @ A3 ) ) ) ).
% abs_minus_cancel
thf(fact_2422_abs__minus,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ! [A3: A] :
( ( abs_abs @ A @ ( uminus_uminus @ A @ A3 ) )
= ( abs_abs @ A @ A3 ) ) ) ).
% abs_minus
thf(fact_2423_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_2424_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_2425_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_2426_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_2427_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_2428_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_2429_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_2430_of__int__abs,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: int] :
( ( ring_1_of_int @ A @ ( abs_abs @ int @ X ) )
= ( abs_abs @ A @ ( ring_1_of_int @ A @ X ) ) ) ) ).
% of_int_abs
thf(fact_2431_abs__of__nonneg,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( abs_abs @ A @ A3 )
= A3 ) ) ) ).
% abs_of_nonneg
thf(fact_2432_abs__le__self__iff,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ ( abs_abs @ A @ A3 ) @ A3 )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% abs_le_self_iff
thf(fact_2433_abs__le__zero__iff,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ ( abs_abs @ A @ A3 ) @ ( zero_zero @ A ) )
= ( A3
= ( zero_zero @ A ) ) ) ) ).
% abs_le_zero_iff
thf(fact_2434_zero__less__abs__iff,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( abs_abs @ A @ A3 ) )
= ( A3
!= ( zero_zero @ A ) ) ) ) ).
% zero_less_abs_iff
thf(fact_2435_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_2436_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_2437_abs__power__minus,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,N: nat] :
( ( abs_abs @ A @ ( power_power @ A @ ( uminus_uminus @ A @ A3 ) @ N ) )
= ( abs_abs @ A @ ( power_power @ A @ A3 @ N ) ) ) ) ).
% abs_power_minus
thf(fact_2438_zero__le__divide__abs__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ A3 @ ( abs_abs @ A @ B3 ) ) )
= ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
| ( B3
= ( zero_zero @ A ) ) ) ) ) ).
% zero_le_divide_abs_iff
thf(fact_2439_divide__le__0__abs__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ A3 @ ( abs_abs @ A @ B3 ) ) @ ( zero_zero @ A ) )
= ( ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) )
| ( B3
= ( zero_zero @ A ) ) ) ) ) ).
% divide_le_0_abs_iff
thf(fact_2440_abs__of__nonpos,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( abs_abs @ A @ A3 )
= ( uminus_uminus @ A @ A3 ) ) ) ) ).
% abs_of_nonpos
thf(fact_2441_artanh__minus__real,axiom,
! [X: real] :
( ( ord_less @ real @ ( abs_abs @ real @ X ) @ ( one_one @ real ) )
=> ( ( artanh @ real @ ( uminus_uminus @ real @ X ) )
= ( uminus_uminus @ real @ ( artanh @ real @ X ) ) ) ) ).
% artanh_minus_real
thf(fact_2442_zero__less__power__abs__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,N: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( power_power @ A @ ( abs_abs @ A @ A3 ) @ N ) )
= ( ( A3
!= ( zero_zero @ A ) )
| ( N
= ( zero_zero @ nat ) ) ) ) ) ).
% zero_less_power_abs_iff
thf(fact_2443_abs__power2,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A] :
( ( abs_abs @ A @ ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% abs_power2
thf(fact_2444_power2__abs,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A] :
( ( power_power @ A @ ( abs_abs @ A @ A3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% power2_abs
thf(fact_2445_power__even__abs__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [W: num,A3: A] :
( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ W ) )
=> ( ( power_power @ A @ ( abs_abs @ A @ A3 ) @ ( numeral_numeral @ nat @ W ) )
= ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ W ) ) ) ) ) ).
% power_even_abs_numeral
thf(fact_2446_square__powr__half,axiom,
! [X: real] :
( ( powr @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
= ( abs_abs @ real @ X ) ) ).
% square_powr_half
thf(fact_2447_abs__eq__0__iff,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ! [A3: A] :
( ( ( abs_abs @ A @ A3 )
= ( zero_zero @ A ) )
= ( A3
= ( zero_zero @ A ) ) ) ) ).
% abs_eq_0_iff
thf(fact_2448_abs__le__D1,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ ( abs_abs @ A @ A3 ) @ B3 )
=> ( ord_less_eq @ A @ A3 @ B3 ) ) ) ).
% abs_le_D1
thf(fact_2449_abs__ge__self,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A] : ( ord_less_eq @ A @ A3 @ ( abs_abs @ A @ A3 ) ) ) ).
% abs_ge_self
thf(fact_2450_abs__one,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( ( abs_abs @ A @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% abs_one
thf(fact_2451_abs__mult,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ! [A3: A,B3: A] :
( ( abs_abs @ A @ ( times_times @ A @ A3 @ B3 ) )
= ( times_times @ A @ ( abs_abs @ A @ A3 ) @ ( abs_abs @ A @ B3 ) ) ) ) ).
% abs_mult
thf(fact_2452_abs__minus__commute,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A,B3: A] :
( ( abs_abs @ A @ ( minus_minus @ A @ A3 @ B3 ) )
= ( abs_abs @ A @ ( minus_minus @ A @ B3 @ A3 ) ) ) ) ).
% abs_minus_commute
thf(fact_2453_power__abs,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,N: nat] :
( ( abs_abs @ A @ ( power_power @ A @ A3 @ N ) )
= ( power_power @ A @ ( abs_abs @ A @ A3 ) @ N ) ) ) ).
% power_abs
thf(fact_2454_abs__eq__iff,axiom,
! [A: $tType] :
( ( linordered_ring @ A )
=> ! [X: A,Y: A] :
( ( ( abs_abs @ A @ X )
= ( abs_abs @ A @ Y ) )
= ( ( X = Y )
| ( X
= ( uminus_uminus @ A @ Y ) ) ) ) ) ).
% abs_eq_iff
thf(fact_2455_dvd__if__abs__eq,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [L: A,K: A] :
( ( ( abs_abs @ A @ L )
= ( abs_abs @ A @ K ) )
=> ( dvd_dvd @ A @ L @ K ) ) ) ).
% dvd_if_abs_eq
thf(fact_2456_abs__ge__zero,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( abs_abs @ A @ A3 ) ) ) ).
% abs_ge_zero
thf(fact_2457_abs__of__pos,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( abs_abs @ A @ A3 )
= A3 ) ) ) ).
% abs_of_pos
thf(fact_2458_abs__not__less__zero,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A] :
~ ( ord_less @ A @ ( abs_abs @ A @ A3 ) @ ( zero_zero @ A ) ) ) ).
% abs_not_less_zero
thf(fact_2459_abs__triangle__ineq,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A,B3: A] : ( ord_less_eq @ A @ ( abs_abs @ A @ ( plus_plus @ A @ A3 @ B3 ) ) @ ( plus_plus @ A @ ( abs_abs @ A @ A3 ) @ ( abs_abs @ A @ B3 ) ) ) ) ).
% abs_triangle_ineq
thf(fact_2460_abs__mult__less,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,C2: A,B3: A,D2: A] :
( ( ord_less @ A @ ( abs_abs @ A @ A3 ) @ C2 )
=> ( ( ord_less @ A @ ( abs_abs @ A @ B3 ) @ D2 )
=> ( ord_less @ A @ ( times_times @ A @ ( abs_abs @ A @ A3 ) @ ( abs_abs @ A @ B3 ) ) @ ( times_times @ A @ C2 @ D2 ) ) ) ) ) ).
% abs_mult_less
thf(fact_2461_abs__triangle__ineq2,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A,B3: A] : ( ord_less_eq @ A @ ( minus_minus @ A @ ( abs_abs @ A @ A3 ) @ ( abs_abs @ A @ B3 ) ) @ ( abs_abs @ A @ ( minus_minus @ A @ A3 @ B3 ) ) ) ) ).
% abs_triangle_ineq2
thf(fact_2462_abs__triangle__ineq3,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A,B3: A] : ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ ( abs_abs @ A @ A3 ) @ ( abs_abs @ A @ B3 ) ) ) @ ( abs_abs @ A @ ( minus_minus @ A @ A3 @ B3 ) ) ) ) ).
% abs_triangle_ineq3
thf(fact_2463_abs__triangle__ineq2__sym,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A,B3: A] : ( ord_less_eq @ A @ ( minus_minus @ A @ ( abs_abs @ A @ A3 ) @ ( abs_abs @ A @ B3 ) ) @ ( abs_abs @ A @ ( minus_minus @ A @ B3 @ A3 ) ) ) ) ).
% abs_triangle_ineq2_sym
thf(fact_2464_nonzero__abs__divide,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B3: A,A3: A] :
( ( B3
!= ( zero_zero @ A ) )
=> ( ( abs_abs @ A @ ( divide_divide @ A @ A3 @ B3 ) )
= ( divide_divide @ A @ ( abs_abs @ A @ A3 ) @ ( abs_abs @ A @ B3 ) ) ) ) ) ).
% nonzero_abs_divide
thf(fact_2465_abs__leI,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A3 ) @ B3 )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ A3 ) @ B3 ) ) ) ) ).
% abs_leI
thf(fact_2466_abs__le__D2,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ ( abs_abs @ A @ A3 ) @ B3 )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ A3 ) @ B3 ) ) ) ).
% abs_le_D2
thf(fact_2467_abs__le__iff,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ ( abs_abs @ A @ A3 ) @ B3 )
= ( ( ord_less_eq @ A @ A3 @ B3 )
& ( ord_less_eq @ A @ ( uminus_uminus @ A @ A3 ) @ B3 ) ) ) ) ).
% abs_le_iff
thf(fact_2468_abs__ge__minus__self,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A] : ( ord_less_eq @ A @ ( uminus_uminus @ A @ A3 ) @ ( abs_abs @ A @ A3 ) ) ) ).
% abs_ge_minus_self
thf(fact_2469_abs__less__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ ( abs_abs @ A @ A3 ) @ B3 )
= ( ( ord_less @ A @ A3 @ B3 )
& ( ord_less @ A @ ( uminus_uminus @ A @ A3 ) @ B3 ) ) ) ) ).
% abs_less_iff
thf(fact_2470_dense__eq0__I,axiom,
! [A: $tType] :
( ( ( ordere166539214618696060dd_abs @ A )
& ( dense_linorder @ A ) )
=> ! [X: A] :
( ! [E: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ E )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ X ) @ E ) )
=> ( X
= ( zero_zero @ A ) ) ) ) ).
% dense_eq0_I
thf(fact_2471_abs__eq__mult,axiom,
! [A: $tType] :
( ( ordered_ring_abs @ A )
=> ! [A3: A,B3: A] :
( ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
| ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) ) )
& ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B3 )
| ( ord_less_eq @ A @ B3 @ ( zero_zero @ A ) ) ) )
=> ( ( abs_abs @ A @ ( times_times @ A @ A3 @ B3 ) )
= ( times_times @ A @ ( abs_abs @ A @ A3 ) @ ( abs_abs @ A @ B3 ) ) ) ) ) ).
% abs_eq_mult
thf(fact_2472_abs__mult__pos,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( times_times @ A @ ( abs_abs @ A @ Y ) @ X )
= ( abs_abs @ A @ ( times_times @ A @ Y @ X ) ) ) ) ) ).
% abs_mult_pos
thf(fact_2473_abs__div__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [Y: A,X: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Y )
=> ( ( divide_divide @ A @ ( abs_abs @ A @ X ) @ Y )
= ( abs_abs @ A @ ( divide_divide @ A @ X @ Y ) ) ) ) ) ).
% abs_div_pos
thf(fact_2474_zero__le__power__abs,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,N: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ ( abs_abs @ A @ A3 ) @ N ) ) ) ).
% zero_le_power_abs
thf(fact_2475_abs__minus__le__zero,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A] : ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( abs_abs @ A @ A3 ) ) @ ( zero_zero @ A ) ) ) ).
% abs_minus_le_zero
thf(fact_2476_eq__abs__iff_H,axiom,
! [A: $tType] :
( ( linordered_ring @ A )
=> ! [A3: A,B3: A] :
( ( A3
= ( abs_abs @ A @ B3 ) )
= ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
& ( ( B3 = A3 )
| ( B3
= ( uminus_uminus @ A @ A3 ) ) ) ) ) ) ).
% eq_abs_iff'
thf(fact_2477_abs__eq__iff_H,axiom,
! [A: $tType] :
( ( linordered_ring @ A )
=> ! [A3: A,B3: A] :
( ( ( abs_abs @ A @ A3 )
= B3 )
= ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B3 )
& ( ( A3 = B3 )
| ( A3
= ( uminus_uminus @ A @ B3 ) ) ) ) ) ) ).
% abs_eq_iff'
thf(fact_2478_abs__if,axiom,
! [A: $tType] :
( ( abs_if @ A )
=> ( ( abs_abs @ A )
= ( ^ [A5: A] : ( if @ A @ ( ord_less @ A @ A5 @ ( zero_zero @ A ) ) @ ( uminus_uminus @ A @ A5 ) @ A5 ) ) ) ) ).
% abs_if
thf(fact_2479_abs__if__raw,axiom,
! [A: $tType] :
( ( abs_if @ A )
=> ( ( abs_abs @ A )
= ( ^ [A5: A] : ( if @ A @ ( ord_less @ A @ A5 @ ( zero_zero @ A ) ) @ ( uminus_uminus @ A @ A5 ) @ A5 ) ) ) ) ).
% abs_if_raw
thf(fact_2480_abs__of__neg,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A] :
( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( abs_abs @ A @ A3 )
= ( uminus_uminus @ A @ A3 ) ) ) ) ).
% abs_of_neg
thf(fact_2481_abs__triangle__ineq4,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A,B3: A] : ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ A3 @ B3 ) ) @ ( plus_plus @ A @ ( abs_abs @ A @ A3 ) @ ( abs_abs @ A @ B3 ) ) ) ) ).
% abs_triangle_ineq4
thf(fact_2482_abs__diff__triangle__ineq,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A,B3: A,C2: A,D2: A] : ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ ( plus_plus @ A @ A3 @ B3 ) @ ( plus_plus @ A @ C2 @ D2 ) ) ) @ ( plus_plus @ A @ ( abs_abs @ A @ ( minus_minus @ A @ A3 @ C2 ) ) @ ( abs_abs @ A @ ( minus_minus @ A @ B3 @ D2 ) ) ) ) ) ).
% abs_diff_triangle_ineq
thf(fact_2483_abs__diff__le__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A,A3: A,R3: A] :
( ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ X @ A3 ) ) @ R3 )
= ( ( ord_less_eq @ A @ ( minus_minus @ A @ A3 @ R3 ) @ X )
& ( ord_less_eq @ A @ X @ ( plus_plus @ A @ A3 @ R3 ) ) ) ) ) ).
% abs_diff_le_iff
thf(fact_2484_abs__diff__less__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A,A3: A,R3: A] :
( ( ord_less @ A @ ( abs_abs @ A @ ( minus_minus @ A @ X @ A3 ) ) @ R3 )
= ( ( ord_less @ A @ ( minus_minus @ A @ A3 @ R3 ) @ X )
& ( ord_less @ A @ X @ ( plus_plus @ A @ A3 @ R3 ) ) ) ) ) ).
% abs_diff_less_iff
thf(fact_2485_abs__real__def,axiom,
( ( abs_abs @ real )
= ( ^ [A5: real] : ( if @ real @ ( ord_less @ real @ A5 @ ( zero_zero @ real ) ) @ ( uminus_uminus @ real @ A5 ) @ A5 ) ) ) ).
% abs_real_def
thf(fact_2486_lemma__interval__lt,axiom,
! [A3: real,X: real,B3: real] :
( ( ord_less @ real @ A3 @ X )
=> ( ( ord_less @ real @ X @ B3 )
=> ? [D6: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D6 )
& ! [Y5: real] :
( ( ord_less @ real @ ( abs_abs @ real @ ( minus_minus @ real @ X @ Y5 ) ) @ D6 )
=> ( ( ord_less @ real @ A3 @ Y5 )
& ( ord_less @ real @ Y5 @ B3 ) ) ) ) ) ) ).
% lemma_interval_lt
thf(fact_2487_abs__add__one__gt__zero,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A] : ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ ( one_one @ A ) @ ( abs_abs @ A @ X ) ) ) ) ).
% abs_add_one_gt_zero
thf(fact_2488_of__int__leD,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: int,X: A] :
( ( ord_less_eq @ A @ ( abs_abs @ A @ ( ring_1_of_int @ A @ N ) ) @ X )
=> ( ( N
= ( zero_zero @ int ) )
| ( ord_less_eq @ A @ ( one_one @ A ) @ X ) ) ) ) ).
% of_int_leD
thf(fact_2489_of__int__lessD,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: int,X: A] :
( ( ord_less @ A @ ( abs_abs @ A @ ( ring_1_of_int @ A @ N ) ) @ X )
=> ( ( N
= ( zero_zero @ int ) )
| ( ord_less @ A @ ( one_one @ A ) @ X ) ) ) ) ).
% of_int_lessD
thf(fact_2490_lemma__interval,axiom,
! [A3: real,X: real,B3: real] :
( ( ord_less @ real @ A3 @ X )
=> ( ( ord_less @ real @ X @ B3 )
=> ? [D6: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D6 )
& ! [Y5: real] :
( ( ord_less @ real @ ( abs_abs @ real @ ( minus_minus @ real @ X @ Y5 ) ) @ D6 )
=> ( ( ord_less_eq @ real @ A3 @ Y5 )
& ( ord_less_eq @ real @ Y5 @ B3 ) ) ) ) ) ) ).
% lemma_interval
thf(fact_2491_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_2492_abs__le__square__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ ( abs_abs @ A @ X ) @ ( abs_abs @ A @ Y ) )
= ( ord_less_eq @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% abs_le_square_iff
thf(fact_2493_abs__square__eq__1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A] :
( ( ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( one_one @ A ) )
= ( ( abs_abs @ A @ X )
= ( one_one @ A ) ) ) ) ).
% abs_square_eq_1
thf(fact_2494_power__even__abs,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: nat,A3: A] :
( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( power_power @ A @ ( abs_abs @ A @ A3 ) @ N )
= ( power_power @ A @ A3 @ N ) ) ) ) ).
% power_even_abs
thf(fact_2495_power2__le__iff__abs__le,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
=> ( ( ord_less_eq @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( ord_less_eq @ A @ ( abs_abs @ A @ X ) @ Y ) ) ) ) ).
% power2_le_iff_abs_le
thf(fact_2496_abs__square__le__1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A] :
( ( ord_less_eq @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) )
= ( ord_less_eq @ A @ ( abs_abs @ A @ X ) @ ( one_one @ A ) ) ) ) ).
% abs_square_le_1
thf(fact_2497_abs__square__less__1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A] :
( ( ord_less @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) )
= ( ord_less @ A @ ( abs_abs @ A @ X ) @ ( one_one @ A ) ) ) ) ).
% abs_square_less_1
thf(fact_2498_power__mono__even,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: nat,A3: A,B3: A] :
( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( ord_less_eq @ A @ ( abs_abs @ A @ A3 ) @ ( abs_abs @ A @ B3 ) )
=> ( ord_less_eq @ A @ ( power_power @ A @ A3 @ N ) @ ( power_power @ A @ B3 @ N ) ) ) ) ) ).
% power_mono_even
thf(fact_2499_even__set__bit__iff,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M: nat,A3: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5668285175392031749et_bit @ A @ M @ A3 ) )
= ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
& ( M
!= ( zero_zero @ nat ) ) ) ) ) ).
% even_set_bit_iff
thf(fact_2500_even__unset__bit__iff,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M: nat,A3: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se2638667681897837118et_bit @ A @ M @ A3 ) )
= ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
| ( M
= ( zero_zero @ nat ) ) ) ) ) ).
% even_unset_bit_iff
thf(fact_2501_of__int__round__abs__le,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] : ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ ( archimedean_round @ A @ X ) ) @ X ) ) @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% of_int_round_abs_le
thf(fact_2502_round__unique_H,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,N: int] :
( ( ord_less @ A @ ( abs_abs @ A @ ( minus_minus @ A @ X @ ( ring_1_of_int @ A @ N ) ) ) @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
=> ( ( archimedean_round @ A @ X )
= N ) ) ) ).
% round_unique'
thf(fact_2503_abs__ln__one__plus__x__minus__x__bound__nonneg,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ X @ ( one_one @ real ) )
=> ( ord_less_eq @ real @ ( abs_abs @ real @ ( minus_minus @ real @ ( ln_ln @ real @ ( plus_plus @ real @ ( one_one @ real ) @ X ) ) @ X ) ) @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% abs_ln_one_plus_x_minus_x_bound_nonneg
thf(fact_2504_mult__less__iff1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z: A,X: A,Y: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Z )
=> ( ( ord_less @ A @ ( times_times @ A @ X @ Z ) @ ( times_times @ A @ Y @ Z ) )
= ( ord_less @ A @ X @ Y ) ) ) ) ).
% mult_less_iff1
thf(fact_2505_abs__ln__one__plus__x__minus__x__bound,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ X ) @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less_eq @ real @ ( abs_abs @ real @ ( minus_minus @ real @ ( ln_ln @ real @ ( plus_plus @ real @ ( one_one @ real ) @ X ) ) @ X ) ) @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% abs_ln_one_plus_x_minus_x_bound
thf(fact_2506_abs__sqrt__wlog,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [P: A > A > $o,X: A] :
( ! [X3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X3 )
=> ( P @ X3 @ ( power_power @ A @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
=> ( P @ ( abs_abs @ A @ X ) @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% abs_sqrt_wlog
thf(fact_2507_monoseq__arctan__series,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ X ) @ ( one_one @ real ) )
=> ( topological_monoseq @ real
@ ^ [N4: nat] : ( times_times @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( semiring_1_of_nat @ real @ ( plus_plus @ nat @ ( times_times @ nat @ N4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) @ ( power_power @ real @ X @ ( plus_plus @ nat @ ( times_times @ nat @ N4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) ) ) ).
% monoseq_arctan_series
thf(fact_2508_summable__arctan__series,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ X ) @ ( one_one @ real ) )
=> ( summable @ real
@ ^ [K3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ K3 ) @ ( times_times @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( semiring_1_of_nat @ real @ ( plus_plus @ nat @ ( times_times @ nat @ K3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) @ ( power_power @ real @ X @ ( plus_plus @ nat @ ( times_times @ nat @ K3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) ) ) ) ).
% summable_arctan_series
thf(fact_2509_round__altdef,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ( ( archimedean_round @ A )
= ( ^ [X2: A] : ( if @ int @ ( ord_less_eq @ A @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( archimedean_frac @ A @ X2 ) ) @ ( archimedean_ceiling @ A @ X2 ) @ ( archim6421214686448440834_floor @ A @ X2 ) ) ) ) ) ).
% round_altdef
thf(fact_2510_arctan__double,axiom,
! [X: real] :
( ( ord_less @ real @ ( abs_abs @ real @ X ) @ ( one_one @ real ) )
=> ( ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( arctan @ X ) )
= ( arctan @ ( divide_divide @ real @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ X ) @ ( minus_minus @ real @ ( one_one @ real ) @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% arctan_double
thf(fact_2511_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_2512_frac__frac,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( archimedean_frac @ A @ ( archimedean_frac @ A @ X ) )
= ( archimedean_frac @ A @ X ) ) ) ).
% frac_frac
thf(fact_2513_pochhammer__1,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A3: A] :
( ( comm_s3205402744901411588hammer @ A @ A3 @ ( one_one @ nat ) )
= A3 ) ) ).
% pochhammer_1
thf(fact_2514_arctan__zero__zero,axiom,
( ( arctan @ ( zero_zero @ real ) )
= ( zero_zero @ real ) ) ).
% arctan_zero_zero
thf(fact_2515_arctan__eq__zero__iff,axiom,
! [X: real] :
( ( ( arctan @ X )
= ( zero_zero @ real ) )
= ( X
= ( zero_zero @ real ) ) ) ).
% arctan_eq_zero_iff
thf(fact_2516_pochhammer__0,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A3: A] :
( ( comm_s3205402744901411588hammer @ A @ A3 @ ( zero_zero @ nat ) )
= ( one_one @ A ) ) ) ).
% pochhammer_0
thf(fact_2517_arctan__less__zero__iff,axiom,
! [X: real] :
( ( ord_less @ real @ ( arctan @ X ) @ ( zero_zero @ real ) )
= ( ord_less @ real @ X @ ( zero_zero @ real ) ) ) ).
% arctan_less_zero_iff
thf(fact_2518_zero__less__arctan__iff,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( arctan @ X ) )
= ( ord_less @ real @ ( zero_zero @ real ) @ X ) ) ).
% zero_less_arctan_iff
thf(fact_2519_arctan__le__zero__iff,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( arctan @ X ) @ ( zero_zero @ real ) )
= ( ord_less_eq @ real @ X @ ( zero_zero @ real ) ) ) ).
% arctan_le_zero_iff
thf(fact_2520_zero__le__arctan__iff,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( arctan @ X ) )
= ( ord_less_eq @ real @ ( zero_zero @ real ) @ X ) ) ).
% zero_le_arctan_iff
thf(fact_2521_zdvd1__eq,axiom,
! [X: int] :
( ( dvd_dvd @ int @ X @ ( one_one @ int ) )
= ( ( abs_abs @ int @ X )
= ( one_one @ int ) ) ) ).
% zdvd1_eq
thf(fact_2522_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_2523_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_2524_pochhammer__of__nat,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [X: nat,N: nat] :
( ( comm_s3205402744901411588hammer @ A @ ( semiring_1_of_nat @ A @ X ) @ N )
= ( semiring_1_of_nat @ A @ ( comm_s3205402744901411588hammer @ nat @ X @ N ) ) ) ) ).
% pochhammer_of_nat
thf(fact_2525_zdvd__antisym__abs,axiom,
! [A3: int,B3: int] :
( ( dvd_dvd @ int @ A3 @ B3 )
=> ( ( dvd_dvd @ int @ B3 @ A3 )
=> ( ( abs_abs @ int @ A3 )
= ( abs_abs @ int @ B3 ) ) ) ) ).
% zdvd_antisym_abs
thf(fact_2526_arctan__less__iff,axiom,
! [X: real,Y: real] :
( ( ord_less @ real @ ( arctan @ X ) @ ( arctan @ Y ) )
= ( ord_less @ real @ X @ Y ) ) ).
% arctan_less_iff
thf(fact_2527_arctan__monotone,axiom,
! [X: real,Y: real] :
( ( ord_less @ real @ X @ Y )
=> ( ord_less @ real @ ( arctan @ X ) @ ( arctan @ Y ) ) ) ).
% arctan_monotone
thf(fact_2528_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_2529_pochhammer__pos,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X: A,N: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ X )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( comm_s3205402744901411588hammer @ A @ X @ N ) ) ) ) ).
% pochhammer_pos
thf(fact_2530_pochhammer__eq__0__mono,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: A,N: nat,M: nat] :
( ( ( comm_s3205402744901411588hammer @ A @ A3 @ N )
= ( zero_zero @ A ) )
=> ( ( ord_less_eq @ nat @ N @ M )
=> ( ( comm_s3205402744901411588hammer @ A @ A3 @ M )
= ( zero_zero @ A ) ) ) ) ) ).
% pochhammer_eq_0_mono
thf(fact_2531_pochhammer__neq__0__mono,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: A,M: nat,N: nat] :
( ( ( comm_s3205402744901411588hammer @ A @ A3 @ M )
!= ( zero_zero @ A ) )
=> ( ( ord_less_eq @ nat @ N @ M )
=> ( ( comm_s3205402744901411588hammer @ A @ A3 @ N )
!= ( zero_zero @ A ) ) ) ) ) ).
% pochhammer_neq_0_mono
thf(fact_2532_frac__ge__0,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( archimedean_frac @ A @ X ) ) ) ).
% frac_ge_0
thf(fact_2533_frac__lt__1,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] : ( ord_less @ A @ ( archimedean_frac @ A @ X ) @ ( one_one @ A ) ) ) ).
% frac_lt_1
thf(fact_2534_frac__1__eq,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( archimedean_frac @ A @ ( plus_plus @ A @ X @ ( one_one @ A ) ) )
= ( archimedean_frac @ A @ X ) ) ) ).
% frac_1_eq
thf(fact_2535_zabs__def,axiom,
( ( abs_abs @ int )
= ( ^ [I4: int] : ( if @ int @ ( ord_less @ int @ I4 @ ( zero_zero @ int ) ) @ ( uminus_uminus @ int @ I4 ) @ I4 ) ) ) ).
% zabs_def
thf(fact_2536_pochhammer__nonneg,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X: A,N: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ X )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( comm_s3205402744901411588hammer @ A @ X @ N ) ) ) ) ).
% pochhammer_nonneg
thf(fact_2537_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_2538_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_2539_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_2540_frac__def,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ( ( archimedean_frac @ A )
= ( ^ [X2: A] : ( minus_minus @ A @ X2 @ ( ring_1_of_int @ A @ ( archim6421214686448440834_floor @ A @ X2 ) ) ) ) ) ) ).
% frac_def
thf(fact_2541_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_2542_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_2543_pochhammer__eq__0__iff,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: A,N: nat] :
( ( ( comm_s3205402744901411588hammer @ A @ A3 @ N )
= ( zero_zero @ A ) )
= ( ? [K3: nat] :
( ( ord_less @ nat @ K3 @ N )
& ( A3
= ( uminus_uminus @ A @ ( semiring_1_of_nat @ A @ K3 ) ) ) ) ) ) ) ).
% pochhammer_eq_0_iff
thf(fact_2544_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_2545_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_2546_even__add__abs__iff,axiom,
! [K: int,L: int] :
( ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( plus_plus @ int @ K @ ( abs_abs @ int @ L ) ) )
= ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( plus_plus @ int @ K @ L ) ) ) ).
% even_add_abs_iff
thf(fact_2547_even__abs__add__iff,axiom,
! [K: int,L: int] :
( ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( plus_plus @ int @ ( abs_abs @ int @ K ) @ L ) )
= ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( plus_plus @ int @ K @ L ) ) ) ).
% even_abs_add_iff
thf(fact_2548_frac__eq,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ( archimedean_frac @ A @ X )
= X )
= ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
& ( ord_less @ A @ X @ ( one_one @ A ) ) ) ) ) ).
% frac_eq
thf(fact_2549_frac__add,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Y: A] :
( ( ( ord_less @ A @ ( plus_plus @ A @ ( archimedean_frac @ A @ X ) @ ( archimedean_frac @ A @ Y ) ) @ ( one_one @ A ) )
=> ( ( archimedean_frac @ A @ ( plus_plus @ A @ X @ Y ) )
= ( plus_plus @ A @ ( archimedean_frac @ A @ X ) @ ( archimedean_frac @ A @ Y ) ) ) )
& ( ~ ( ord_less @ A @ ( plus_plus @ A @ ( archimedean_frac @ A @ X ) @ ( archimedean_frac @ A @ Y ) ) @ ( one_one @ A ) )
=> ( ( archimedean_frac @ A @ ( plus_plus @ A @ X @ Y ) )
= ( minus_minus @ A @ ( plus_plus @ A @ ( archimedean_frac @ A @ X ) @ ( archimedean_frac @ A @ Y ) ) @ ( one_one @ A ) ) ) ) ) ) ).
% frac_add
thf(fact_2550_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_2551_monoseq__realpow,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ X @ ( one_one @ real ) )
=> ( topological_monoseq @ real @ ( power_power @ real @ X ) ) ) ) ).
% monoseq_realpow
thf(fact_2552_incr__lemma,axiom,
! [D2: int,Z: int,X: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D2 )
=> ( ord_less @ int @ Z @ ( plus_plus @ int @ X @ ( times_times @ int @ ( plus_plus @ int @ ( abs_abs @ int @ ( minus_minus @ int @ X @ Z ) ) @ ( one_one @ int ) ) @ D2 ) ) ) ) ).
% incr_lemma
thf(fact_2553_decr__lemma,axiom,
! [D2: int,X: int,Z: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D2 )
=> ( ord_less @ int @ ( minus_minus @ int @ X @ ( times_times @ int @ ( plus_plus @ int @ ( abs_abs @ int @ ( minus_minus @ int @ X @ Z ) ) @ ( one_one @ int ) ) @ D2 ) ) @ Z ) ) ).
% decr_lemma
thf(fact_2554_pochhammer__absorb__comp,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [R3: A,K: nat] :
( ( times_times @ A @ ( minus_minus @ A @ R3 @ ( semiring_1_of_nat @ A @ K ) ) @ ( comm_s3205402744901411588hammer @ A @ ( uminus_uminus @ A @ R3 ) @ K ) )
= ( times_times @ A @ R3 @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ ( uminus_uminus @ A @ R3 ) @ ( one_one @ A ) ) @ K ) ) ) ) ).
% pochhammer_absorb_comp
thf(fact_2555_pochhammer__minus,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [B3: A,K: nat] :
( ( comm_s3205402744901411588hammer @ A @ ( uminus_uminus @ A @ B3 ) @ K )
= ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ K ) @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ ( minus_minus @ A @ B3 @ ( semiring_1_of_nat @ A @ K ) ) @ ( one_one @ A ) ) @ K ) ) ) ) ).
% pochhammer_minus
thf(fact_2556_pochhammer__minus_H,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [B3: A,K: nat] :
( ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ ( minus_minus @ A @ B3 @ ( semiring_1_of_nat @ A @ K ) ) @ ( one_one @ A ) ) @ K )
= ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ K ) @ ( comm_s3205402744901411588hammer @ A @ ( uminus_uminus @ A @ B3 ) @ K ) ) ) ) ).
% pochhammer_minus'
thf(fact_2557_floor__add,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Y: A] :
( ( ( ord_less @ A @ ( plus_plus @ A @ ( archimedean_frac @ A @ X ) @ ( archimedean_frac @ A @ Y ) ) @ ( one_one @ A ) )
=> ( ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ X @ Y ) )
= ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( archim6421214686448440834_floor @ A @ Y ) ) ) )
& ( ~ ( ord_less @ A @ ( plus_plus @ A @ ( archimedean_frac @ A @ X ) @ ( archimedean_frac @ A @ Y ) ) @ ( one_one @ A ) )
=> ( ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ X @ Y ) )
= ( plus_plus @ int @ ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( archim6421214686448440834_floor @ A @ Y ) ) @ ( one_one @ int ) ) ) ) ) ) ).
% floor_add
thf(fact_2558_nat0__intermed__int__val,axiom,
! [N: nat,F3: nat > int,K: int] :
( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ N )
=> ( ord_less_eq @ int @ ( abs_abs @ int @ ( minus_minus @ int @ ( F3 @ ( plus_plus @ nat @ I3 @ ( one_one @ nat ) ) ) @ ( F3 @ I3 ) ) ) @ ( one_one @ int ) ) )
=> ( ( ord_less_eq @ int @ ( F3 @ ( zero_zero @ nat ) ) @ K )
=> ( ( ord_less_eq @ int @ K @ ( F3 @ N ) )
=> ? [I3: nat] :
( ( ord_less_eq @ nat @ I3 @ N )
& ( ( F3 @ I3 )
= K ) ) ) ) ) ).
% nat0_intermed_int_val
thf(fact_2559_arctan__add,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ X ) @ ( one_one @ real ) )
=> ( ( ord_less @ real @ ( abs_abs @ real @ Y ) @ ( one_one @ real ) )
=> ( ( plus_plus @ real @ ( arctan @ X ) @ ( arctan @ Y ) )
= ( arctan @ ( divide_divide @ real @ ( plus_plus @ real @ X @ Y ) @ ( minus_minus @ real @ ( one_one @ real ) @ ( times_times @ real @ X @ Y ) ) ) ) ) ) ) ).
% arctan_add
thf(fact_2560_summable__divide__iff,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: nat > A,C2: A] :
( ( summable @ A
@ ^ [N4: nat] : ( divide_divide @ A @ ( F3 @ N4 ) @ C2 ) )
= ( ( C2
= ( zero_zero @ A ) )
| ( summable @ A @ F3 ) ) ) ) ).
% summable_divide_iff
thf(fact_2561_summable__cmult__iff,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [C2: A,F3: nat > A] :
( ( summable @ A
@ ^ [N4: nat] : ( times_times @ A @ C2 @ ( F3 @ N4 ) ) )
= ( ( C2
= ( zero_zero @ A ) )
| ( summable @ A @ F3 ) ) ) ) ).
% summable_cmult_iff
thf(fact_2562_summable__power__series,axiom,
! [F3: nat > real,Z: real] :
( ! [I3: nat] : ( ord_less_eq @ real @ ( F3 @ I3 ) @ ( one_one @ real ) )
=> ( ! [I3: nat] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( F3 @ I3 ) )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Z )
=> ( ( ord_less @ real @ Z @ ( one_one @ real ) )
=> ( summable @ real
@ ^ [I4: nat] : ( times_times @ real @ ( F3 @ I4 ) @ ( power_power @ real @ Z @ I4 ) ) ) ) ) ) ) ).
% summable_power_series
thf(fact_2563_summable__iff__shift,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: nat > A,K: nat] :
( ( summable @ A
@ ^ [N4: nat] : ( F3 @ ( plus_plus @ nat @ N4 @ K ) ) )
= ( summable @ A @ F3 ) ) ) ).
% summable_iff_shift
thf(fact_2564_summable__single,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [I: nat,F3: nat > A] :
( summable @ A
@ ^ [R5: nat] : ( if @ A @ ( R5 = I ) @ ( F3 @ R5 ) @ ( zero_zero @ A ) ) ) ) ).
% summable_single
thf(fact_2565_summable__zero,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ( summable @ A
@ ^ [N4: nat] : ( zero_zero @ A ) ) ) ).
% summable_zero
thf(fact_2566_summable__complex__of__real,axiom,
! [F3: nat > real] :
( ( summable @ complex
@ ^ [N4: nat] : ( real_Vector_of_real @ complex @ ( F3 @ N4 ) ) )
= ( summable @ real @ F3 ) ) ).
% summable_complex_of_real
thf(fact_2567_summable__const__iff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [C2: A] :
( ( summable @ A
@ ^ [Uu: nat] : C2 )
= ( C2
= ( zero_zero @ A ) ) ) ) ).
% summable_const_iff
thf(fact_2568_summable__add,axiom,
! [A: $tType] :
( ( ( topolo5987344860129210374id_add @ A )
& ( topological_t2_space @ A ) )
=> ! [F3: nat > A,G2: nat > A] :
( ( summable @ A @ F3 )
=> ( ( summable @ A @ G2 )
=> ( summable @ A
@ ^ [N4: nat] : ( plus_plus @ A @ ( F3 @ N4 ) @ ( G2 @ N4 ) ) ) ) ) ) ).
% summable_add
thf(fact_2569_summable__mult,axiom,
! [A: $tType] :
( ( real_V4412858255891104859lgebra @ A )
=> ! [F3: nat > A,C2: A] :
( ( summable @ A @ F3 )
=> ( summable @ A
@ ^ [N4: nat] : ( times_times @ A @ C2 @ ( F3 @ N4 ) ) ) ) ) ).
% summable_mult
thf(fact_2570_summable__mult2,axiom,
! [A: $tType] :
( ( real_V4412858255891104859lgebra @ A )
=> ! [F3: nat > A,C2: A] :
( ( summable @ A @ F3 )
=> ( summable @ A
@ ^ [N4: nat] : ( times_times @ A @ ( F3 @ N4 ) @ C2 ) ) ) ) ).
% summable_mult2
thf(fact_2571_summable__diff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: nat > A,G2: nat > A] :
( ( summable @ A @ F3 )
=> ( ( summable @ A @ G2 )
=> ( summable @ A
@ ^ [N4: nat] : ( minus_minus @ A @ ( F3 @ N4 ) @ ( G2 @ N4 ) ) ) ) ) ) ).
% summable_diff
thf(fact_2572_summable__divide,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: nat > A,C2: A] :
( ( summable @ A @ F3 )
=> ( summable @ A
@ ^ [N4: nat] : ( divide_divide @ A @ ( F3 @ N4 ) @ C2 ) ) ) ) ).
% summable_divide
thf(fact_2573_summable__minus,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: nat > A] :
( ( summable @ A @ F3 )
=> ( summable @ A
@ ^ [N4: nat] : ( uminus_uminus @ A @ ( F3 @ N4 ) ) ) ) ) ).
% summable_minus
thf(fact_2574_summable__minus__iff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: nat > A] :
( ( summable @ A
@ ^ [N4: nat] : ( uminus_uminus @ A @ ( F3 @ N4 ) ) )
= ( summable @ A @ F3 ) ) ) ).
% summable_minus_iff
thf(fact_2575_summable__ignore__initial__segment,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: nat > A,K: nat] :
( ( summable @ A @ F3 )
=> ( summable @ A
@ ^ [N4: nat] : ( F3 @ ( plus_plus @ nat @ N4 @ K ) ) ) ) ) ).
% summable_ignore_initial_segment
thf(fact_2576_summable__rabs__cancel,axiom,
! [F3: nat > real] :
( ( summable @ real
@ ^ [N4: nat] : ( abs_abs @ real @ ( F3 @ N4 ) ) )
=> ( summable @ real @ F3 ) ) ).
% summable_rabs_cancel
thf(fact_2577_summable__mult__D,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [C2: A,F3: nat > A] :
( ( summable @ A
@ ^ [N4: nat] : ( times_times @ A @ C2 @ ( F3 @ N4 ) ) )
=> ( ( C2
!= ( zero_zero @ A ) )
=> ( summable @ A @ F3 ) ) ) ) ).
% summable_mult_D
thf(fact_2578_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_2579_summable__of__real,axiom,
! [A: $tType] :
( ( ( real_V2191834092415804123ebra_1 @ A )
& ( real_V822414075346904944vector @ A ) )
=> ! [X5: nat > real] :
( ( summable @ real @ X5 )
=> ( summable @ A
@ ^ [N4: nat] : ( real_Vector_of_real @ A @ ( X5 @ N4 ) ) ) ) ) ).
% summable_of_real
thf(fact_2580_summable__0__powser,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [F3: nat > A] :
( summable @ A
@ ^ [N4: nat] : ( times_times @ A @ ( F3 @ N4 ) @ ( power_power @ A @ ( zero_zero @ A ) @ N4 ) ) ) ) ).
% summable_0_powser
thf(fact_2581_summable__zero__power_H,axiom,
! [A: $tType] :
( ( ( ring_1 @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [F3: nat > A] :
( summable @ A
@ ^ [N4: nat] : ( times_times @ A @ ( F3 @ N4 ) @ ( power_power @ A @ ( zero_zero @ A ) @ N4 ) ) ) ) ).
% summable_zero_power'
thf(fact_2582_summable__powser__ignore__initial__segment,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [F3: nat > A,M: nat,Z: A] :
( ( summable @ A
@ ^ [N4: nat] : ( times_times @ A @ ( F3 @ ( plus_plus @ nat @ N4 @ M ) ) @ ( power_power @ A @ Z @ N4 ) ) )
= ( summable @ A
@ ^ [N4: nat] : ( times_times @ A @ ( F3 @ N4 ) @ ( power_power @ A @ Z @ N4 ) ) ) ) ) ).
% summable_powser_ignore_initial_segment
thf(fact_2583_summable__rabs__comparison__test,axiom,
! [F3: nat > real,G2: nat > real] :
( ? [N8: nat] :
! [N2: nat] :
( ( ord_less_eq @ nat @ N8 @ N2 )
=> ( ord_less_eq @ real @ ( abs_abs @ real @ ( F3 @ N2 ) ) @ ( G2 @ N2 ) ) )
=> ( ( summable @ real @ G2 )
=> ( summable @ real
@ ^ [N4: nat] : ( abs_abs @ real @ ( F3 @ N4 ) ) ) ) ) ).
% summable_rabs_comparison_test
thf(fact_2584_arctan__series,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ X ) @ ( one_one @ real ) )
=> ( ( arctan @ X )
= ( suminf @ real
@ ^ [K3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ K3 ) @ ( times_times @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( semiring_1_of_nat @ real @ ( plus_plus @ nat @ ( times_times @ nat @ K3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) @ ( power_power @ real @ X @ ( plus_plus @ nat @ ( times_times @ nat @ K3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) ) ) ) ) ).
% arctan_series
thf(fact_2585_pochhammer__code,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ( ( comm_s3205402744901411588hammer @ A )
= ( ^ [A5: A,N4: nat] :
( if @ A
@ ( N4
= ( zero_zero @ nat ) )
@ ( one_one @ A )
@ ( set_fo6178422350223883121st_nat @ A
@ ^ [O: nat] : ( times_times @ A @ ( plus_plus @ A @ A5 @ ( semiring_1_of_nat @ A @ O ) ) )
@ ( zero_zero @ nat )
@ ( minus_minus @ nat @ N4 @ ( one_one @ nat ) )
@ ( one_one @ A ) ) ) ) ) ) ).
% pochhammer_code
thf(fact_2586_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_2587_of__int__code__if,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ( ( ring_1_of_int @ A )
= ( ^ [K3: int] :
( if @ A
@ ( K3
= ( zero_zero @ int ) )
@ ( zero_zero @ A )
@ ( if @ A @ ( ord_less @ int @ K3 @ ( zero_zero @ int ) ) @ ( uminus_uminus @ A @ ( ring_1_of_int @ A @ ( uminus_uminus @ int @ K3 ) ) )
@ ( if @ A
@ ( ( modulo_modulo @ int @ K3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) )
= ( zero_zero @ int ) )
@ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( ring_1_of_int @ A @ ( divide_divide @ int @ K3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) )
@ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( ring_1_of_int @ A @ ( divide_divide @ int @ K3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) @ ( one_one @ A ) ) ) ) ) ) ) ) ).
% of_int_code_if
thf(fact_2588_VEBT__internal_Obit__concat__def,axiom,
( vEBT_VEBT_bit_concat
= ( ^ [H: nat,L2: nat,D4: nat] : ( plus_plus @ nat @ ( times_times @ nat @ H @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ D4 ) ) @ L2 ) ) ) ).
% VEBT_internal.bit_concat_def
thf(fact_2589_set__decode__0,axiom,
! [X: nat] :
( ( member @ nat @ ( zero_zero @ nat ) @ ( nat_set_decode @ X ) )
= ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ X ) ) ) ).
% set_decode_0
thf(fact_2590_bits__mod__0,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A3: A] :
( ( modulo_modulo @ A @ ( zero_zero @ A ) @ A3 )
= ( zero_zero @ A ) ) ) ).
% bits_mod_0
thf(fact_2591_mod__0,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ! [A3: A] :
( ( modulo_modulo @ A @ ( zero_zero @ A ) @ A3 )
= ( zero_zero @ A ) ) ) ).
% mod_0
thf(fact_2592_mod__by__0,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ! [A3: A] :
( ( modulo_modulo @ A @ A3 @ ( zero_zero @ A ) )
= A3 ) ) ).
% mod_by_0
thf(fact_2593_mod__self,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ! [A3: A] :
( ( modulo_modulo @ A @ A3 @ A3 )
= ( zero_zero @ A ) ) ) ).
% mod_self
thf(fact_2594_suminf__zero,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topological_t2_space @ A ) )
=> ( ( suminf @ A
@ ^ [N4: nat] : ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% suminf_zero
thf(fact_2595_bits__mod__by__1,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A3: A] :
( ( modulo_modulo @ A @ A3 @ ( one_one @ A ) )
= ( zero_zero @ A ) ) ) ).
% bits_mod_by_1
thf(fact_2596_mod__by__1,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ! [A3: A] :
( ( modulo_modulo @ A @ A3 @ ( one_one @ A ) )
= ( zero_zero @ A ) ) ) ).
% mod_by_1
thf(fact_2597_mod__mult__self2__is__0,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A3: A,B3: A] :
( ( modulo_modulo @ A @ ( times_times @ A @ A3 @ B3 ) @ B3 )
= ( zero_zero @ A ) ) ) ).
% mod_mult_self2_is_0
thf(fact_2598_mod__mult__self1__is__0,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [B3: A,A3: A] :
( ( modulo_modulo @ A @ ( times_times @ A @ B3 @ A3 ) @ B3 )
= ( zero_zero @ A ) ) ) ).
% mod_mult_self1_is_0
thf(fact_2599_bits__mod__div__trivial,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A3: A,B3: A] :
( ( divide_divide @ A @ ( modulo_modulo @ A @ A3 @ B3 ) @ B3 )
= ( zero_zero @ A ) ) ) ).
% bits_mod_div_trivial
thf(fact_2600_mod__div__trivial,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A3: A,B3: A] :
( ( divide_divide @ A @ ( modulo_modulo @ A @ A3 @ B3 ) @ B3 )
= ( zero_zero @ A ) ) ) ).
% mod_div_trivial
thf(fact_2601_dvd__imp__mod__0,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ! [A3: A,B3: A] :
( ( dvd_dvd @ A @ A3 @ B3 )
=> ( ( modulo_modulo @ A @ B3 @ A3 )
= ( zero_zero @ A ) ) ) ) ).
% dvd_imp_mod_0
thf(fact_2602_set__decode__zero,axiom,
( ( nat_set_decode @ ( zero_zero @ nat ) )
= ( bot_bot @ ( set @ nat ) ) ) ).
% set_decode_zero
thf(fact_2603_mod__minus1__right,axiom,
! [A: $tType] :
( ( euclid8851590272496341667cancel @ A )
=> ! [A3: A] :
( ( modulo_modulo @ A @ A3 @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( zero_zero @ A ) ) ) ).
% mod_minus1_right
thf(fact_2604_mod__pos__pos__trivial,axiom,
! [K: int,L: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
=> ( ( ord_less @ int @ K @ L )
=> ( ( modulo_modulo @ int @ K @ L )
= K ) ) ) ).
% mod_pos_pos_trivial
thf(fact_2605_mod__neg__neg__trivial,axiom,
! [K: int,L: int] :
( ( ord_less_eq @ int @ K @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ L @ K )
=> ( ( modulo_modulo @ int @ K @ L )
= K ) ) ) ).
% mod_neg_neg_trivial
thf(fact_2606_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_2607_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_2608_even__mod__2__iff,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A3: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
= ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) ) ) ).
% even_mod_2_iff
thf(fact_2609_zmod__numeral__Bit0,axiom,
! [V2: num,W: num] :
( ( modulo_modulo @ int @ ( numeral_numeral @ int @ ( bit0 @ V2 ) ) @ ( numeral_numeral @ int @ ( bit0 @ W ) ) )
= ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( modulo_modulo @ int @ ( numeral_numeral @ int @ V2 ) @ ( numeral_numeral @ int @ W ) ) ) ) ).
% zmod_numeral_Bit0
thf(fact_2610_powser__zero,axiom,
! [A: $tType] :
( ( real_V2822296259951069270ebra_1 @ A )
=> ! [F3: nat > A] :
( ( suminf @ A
@ ^ [N4: nat] : ( times_times @ A @ ( F3 @ N4 ) @ ( power_power @ A @ ( zero_zero @ A ) @ N4 ) ) )
= ( F3 @ ( zero_zero @ nat ) ) ) ) ).
% powser_zero
thf(fact_2611_not__mod__2__eq__1__eq__0,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A3: A] :
( ( ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
!= ( one_one @ A ) )
= ( ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( zero_zero @ A ) ) ) ) ).
% not_mod_2_eq_1_eq_0
thf(fact_2612_not__mod__2__eq__0__eq__1,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A3: A] :
( ( ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
!= ( zero_zero @ A ) )
= ( ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( one_one @ A ) ) ) ) ).
% not_mod_2_eq_0_eq_1
thf(fact_2613_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_2614_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_2615_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_2616_even__succ__mod__exp,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A3: A,N: nat] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( modulo_modulo @ A @ ( plus_plus @ A @ ( one_one @ A ) @ A3 ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) )
= ( plus_plus @ A @ ( one_one @ A ) @ ( modulo_modulo @ A @ A3 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) ) ) ) ) ) ).
% even_succ_mod_exp
thf(fact_2617_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_2618_power__mod,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A3: A,B3: A,N: nat] :
( ( modulo_modulo @ A @ ( power_power @ A @ ( modulo_modulo @ A @ A3 @ B3 ) @ N ) @ B3 )
= ( modulo_modulo @ A @ ( power_power @ A @ A3 @ N ) @ B3 ) ) ) ).
% power_mod
thf(fact_2619_dvd__mod__imp__dvd,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ! [C2: A,A3: A,B3: A] :
( ( dvd_dvd @ A @ C2 @ ( modulo_modulo @ A @ A3 @ B3 ) )
=> ( ( dvd_dvd @ A @ C2 @ B3 )
=> ( dvd_dvd @ A @ C2 @ A3 ) ) ) ) ).
% dvd_mod_imp_dvd
thf(fact_2620_dvd__mod__iff,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ! [C2: A,B3: A,A3: A] :
( ( dvd_dvd @ A @ C2 @ B3 )
=> ( ( dvd_dvd @ A @ C2 @ ( modulo_modulo @ A @ A3 @ B3 ) )
= ( dvd_dvd @ A @ C2 @ A3 ) ) ) ) ).
% dvd_mod_iff
thf(fact_2621_suminf__of__real,axiom,
! [A: $tType] :
( ( ( real_V2191834092415804123ebra_1 @ A )
& ( real_V822414075346904944vector @ A ) )
=> ! [X5: nat > real] :
( ( summable @ real @ X5 )
=> ( ( real_Vector_of_real @ A @ ( suminf @ real @ X5 ) )
= ( suminf @ A
@ ^ [N4: nat] : ( real_Vector_of_real @ A @ ( X5 @ N4 ) ) ) ) ) ) ).
% suminf_of_real
thf(fact_2622_mod__plus__cong,axiom,
! [B3: int,B8: int,X: int,X7: int,Y: int,Y6: int,Z7: int] :
( ( B3 = B8 )
=> ( ( ( modulo_modulo @ int @ X @ B8 )
= ( modulo_modulo @ int @ X7 @ B8 ) )
=> ( ( ( modulo_modulo @ int @ Y @ B8 )
= ( modulo_modulo @ int @ Y6 @ B8 ) )
=> ( ( ( plus_plus @ int @ X7 @ Y6 )
= Z7 )
=> ( ( modulo_modulo @ int @ ( plus_plus @ int @ X @ Y ) @ B3 )
= ( modulo_modulo @ int @ Z7 @ B8 ) ) ) ) ) ) ).
% mod_plus_cong
thf(fact_2623_Word_Omod__minus__cong,axiom,
! [B3: int,B8: int,X: int,X7: int,Y: int,Y6: int,Z7: int] :
( ( B3 = B8 )
=> ( ( ( modulo_modulo @ int @ X @ B8 )
= ( modulo_modulo @ int @ X7 @ B8 ) )
=> ( ( ( modulo_modulo @ int @ Y @ B8 )
= ( modulo_modulo @ int @ Y6 @ B8 ) )
=> ( ( ( minus_minus @ int @ X7 @ Y6 )
= Z7 )
=> ( ( modulo_modulo @ int @ ( minus_minus @ int @ X @ Y ) @ B3 )
= ( modulo_modulo @ int @ Z7 @ B8 ) ) ) ) ) ) ).
% Word.mod_minus_cong
thf(fact_2624_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ord_less_eq @ A @ ( modulo_modulo @ A @ A3 @ B3 ) @ A3 ) ) ) ).
% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
thf(fact_2625_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [B3: A,A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ B3 )
=> ( ord_less @ A @ ( modulo_modulo @ A @ A3 @ B3 ) @ B3 ) ) ) ).
% unique_euclidean_semiring_numeral_class.pos_mod_bound
thf(fact_2626_cong__exp__iff__simps_I9_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M: num,Q3: num,N: num] :
( ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit0 @ M ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q3 ) ) )
= ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit0 @ N ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q3 ) ) ) )
= ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ Q3 ) )
= ( modulo_modulo @ A @ ( numeral_numeral @ A @ N ) @ ( numeral_numeral @ A @ Q3 ) ) ) ) ) ).
% cong_exp_iff_simps(9)
thf(fact_2627_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_2628_mod__eq__self__iff__div__eq__0,axiom,
! [A: $tType] :
( ( euclid3725896446679973847miring @ A )
=> ! [A3: A,B3: A] :
( ( ( modulo_modulo @ A @ A3 @ B3 )
= A3 )
= ( ( divide_divide @ A @ A3 @ B3 )
= ( zero_zero @ A ) ) ) ) ).
% mod_eq_self_iff_div_eq_0
thf(fact_2629_mod__eq__0__iff__dvd,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ! [A3: A,B3: A] :
( ( ( modulo_modulo @ A @ A3 @ B3 )
= ( zero_zero @ A ) )
= ( dvd_dvd @ A @ B3 @ A3 ) ) ) ).
% mod_eq_0_iff_dvd
thf(fact_2630_dvd__eq__mod__eq__0,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ( ( dvd_dvd @ A )
= ( ^ [A5: A,B5: A] :
( ( modulo_modulo @ A @ B5 @ A5 )
= ( zero_zero @ A ) ) ) ) ) ).
% dvd_eq_mod_eq_0
thf(fact_2631_mod__0__imp__dvd,axiom,
! [A: $tType] :
( ( semiring_modulo @ A )
=> ! [A3: A,B3: A] :
( ( ( modulo_modulo @ A @ A3 @ B3 )
= ( zero_zero @ A ) )
=> ( dvd_dvd @ A @ B3 @ A3 ) ) ) ).
% mod_0_imp_dvd
thf(fact_2632_dvd__minus__mod,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ! [B3: A,A3: A] : ( dvd_dvd @ A @ B3 @ ( minus_minus @ A @ A3 @ ( modulo_modulo @ A @ A3 @ B3 ) ) ) ) ).
% dvd_minus_mod
thf(fact_2633_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_2634_neg__mod__bound,axiom,
! [L: int,K: int] :
( ( ord_less @ int @ L @ ( zero_zero @ int ) )
=> ( ord_less @ int @ L @ ( modulo_modulo @ int @ K @ L ) ) ) ).
% neg_mod_bound
thf(fact_2635_Euclidean__Division_Opos__mod__bound,axiom,
! [L: int,K: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ L )
=> ( ord_less @ int @ ( modulo_modulo @ int @ K @ L ) @ L ) ) ).
% Euclidean_Division.pos_mod_bound
thf(fact_2636_zmod__zminus1__not__zero,axiom,
! [K: int,L: int] :
( ( ( modulo_modulo @ int @ ( uminus_uminus @ int @ K ) @ L )
!= ( zero_zero @ int ) )
=> ( ( modulo_modulo @ int @ K @ L )
!= ( zero_zero @ int ) ) ) ).
% zmod_zminus1_not_zero
thf(fact_2637_zmod__zminus2__not__zero,axiom,
! [K: int,L: int] :
( ( ( modulo_modulo @ int @ K @ ( uminus_uminus @ int @ L ) )
!= ( zero_zero @ int ) )
=> ( ( modulo_modulo @ int @ K @ L )
!= ( zero_zero @ int ) ) ) ).
% zmod_zminus2_not_zero
thf(fact_2638_zmod__eq__0D,axiom,
! [M: int,D2: int] :
( ( ( modulo_modulo @ int @ M @ D2 )
= ( zero_zero @ int ) )
=> ? [Q6: int] :
( M
= ( times_times @ int @ D2 @ Q6 ) ) ) ).
% zmod_eq_0D
thf(fact_2639_zmod__eq__0__iff,axiom,
! [M: int,D2: int] :
( ( ( modulo_modulo @ int @ M @ D2 )
= ( zero_zero @ int ) )
= ( ? [Q5: int] :
( M
= ( times_times @ int @ D2 @ Q5 ) ) ) ) ).
% zmod_eq_0_iff
thf(fact_2640_suminf__add,axiom,
! [A: $tType] :
( ( ( topolo5987344860129210374id_add @ A )
& ( topological_t2_space @ A ) )
=> ! [F3: nat > A,G2: nat > A] :
( ( summable @ A @ F3 )
=> ( ( summable @ A @ G2 )
=> ( ( plus_plus @ A @ ( suminf @ A @ F3 ) @ ( suminf @ A @ G2 ) )
= ( suminf @ A
@ ^ [N4: nat] : ( plus_plus @ A @ ( F3 @ N4 ) @ ( G2 @ N4 ) ) ) ) ) ) ) ).
% suminf_add
thf(fact_2641_suminf__mult2,axiom,
! [A: $tType] :
( ( real_V4412858255891104859lgebra @ A )
=> ! [F3: nat > A,C2: A] :
( ( summable @ A @ F3 )
=> ( ( times_times @ A @ ( suminf @ A @ F3 ) @ C2 )
= ( suminf @ A
@ ^ [N4: nat] : ( times_times @ A @ ( F3 @ N4 ) @ C2 ) ) ) ) ) ).
% suminf_mult2
thf(fact_2642_suminf__mult,axiom,
! [A: $tType] :
( ( real_V4412858255891104859lgebra @ A )
=> ! [F3: nat > A,C2: A] :
( ( summable @ A @ F3 )
=> ( ( suminf @ A
@ ^ [N4: nat] : ( times_times @ A @ C2 @ ( F3 @ N4 ) ) )
= ( times_times @ A @ C2 @ ( suminf @ A @ F3 ) ) ) ) ) ).
% suminf_mult
thf(fact_2643_suminf__diff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: nat > A,G2: nat > A] :
( ( summable @ A @ F3 )
=> ( ( summable @ A @ G2 )
=> ( ( minus_minus @ A @ ( suminf @ A @ F3 ) @ ( suminf @ A @ G2 ) )
= ( suminf @ A
@ ^ [N4: nat] : ( minus_minus @ A @ ( F3 @ N4 ) @ ( G2 @ N4 ) ) ) ) ) ) ) ).
% suminf_diff
thf(fact_2644_suminf__divide,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: nat > A,C2: A] :
( ( summable @ A @ F3 )
=> ( ( suminf @ A
@ ^ [N4: nat] : ( divide_divide @ A @ ( F3 @ N4 ) @ C2 ) )
= ( divide_divide @ A @ ( suminf @ A @ F3 ) @ C2 ) ) ) ) ).
% suminf_divide
thf(fact_2645_suminf__minus,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: nat > A] :
( ( summable @ A @ F3 )
=> ( ( suminf @ A
@ ^ [N4: nat] : ( uminus_uminus @ A @ ( F3 @ N4 ) ) )
= ( uminus_uminus @ A @ ( suminf @ A @ F3 ) ) ) ) ) ).
% suminf_minus
thf(fact_2646_suminf__nonneg,axiom,
! [A: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F3: nat > A] :
( ( summable @ A @ F3 )
=> ( ! [N2: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F3 @ N2 ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( suminf @ A @ F3 ) ) ) ) ) ).
% suminf_nonneg
thf(fact_2647_suminf__eq__zero__iff,axiom,
! [A: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F3: nat > A] :
( ( summable @ A @ F3 )
=> ( ! [N2: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F3 @ N2 ) )
=> ( ( ( suminf @ A @ F3 )
= ( zero_zero @ A ) )
= ( ! [N4: nat] :
( ( F3 @ N4 )
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% suminf_eq_zero_iff
thf(fact_2648_suminf__pos,axiom,
! [A: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F3: nat > A] :
( ( summable @ A @ F3 )
=> ( ! [N2: nat] : ( ord_less @ A @ ( zero_zero @ A ) @ ( F3 @ N2 ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( suminf @ A @ F3 ) ) ) ) ) ).
% suminf_pos
thf(fact_2649_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [B3: A,A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ B3 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( modulo_modulo @ A @ A3 @ B3 ) ) ) ) ).
% unique_euclidean_semiring_numeral_class.pos_mod_sign
thf(fact_2650_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ A3 @ B3 )
=> ( ( modulo_modulo @ A @ A3 @ B3 )
= A3 ) ) ) ) ).
% unique_euclidean_semiring_numeral_class.mod_less
thf(fact_2651_cong__exp__iff__simps_I2_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [N: num,Q3: num] :
( ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit0 @ N ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q3 ) ) )
= ( zero_zero @ A ) )
= ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ N ) @ ( numeral_numeral @ A @ Q3 ) )
= ( zero_zero @ A ) ) ) ) ).
% cong_exp_iff_simps(2)
thf(fact_2652_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_2653_cong__exp__iff__simps_I6_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [Q3: num,N: num] :
( ( modulo_modulo @ A @ ( numeral_numeral @ A @ one2 ) @ ( numeral_numeral @ A @ ( bit0 @ Q3 ) ) )
!= ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit0 @ N ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q3 ) ) ) ) ) ).
% cong_exp_iff_simps(6)
thf(fact_2654_cong__exp__iff__simps_I8_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M: num,Q3: num] :
( ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit0 @ M ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q3 ) ) )
!= ( modulo_modulo @ A @ ( numeral_numeral @ A @ one2 ) @ ( numeral_numeral @ A @ ( bit0 @ Q3 ) ) ) ) ) ).
% cong_exp_iff_simps(8)
thf(fact_2655_mult__div__mod__eq,axiom,
! [A: $tType] :
( ( semiring_modulo @ A )
=> ! [B3: A,A3: A] :
( ( plus_plus @ A @ ( times_times @ A @ B3 @ ( divide_divide @ A @ A3 @ B3 ) ) @ ( modulo_modulo @ A @ A3 @ B3 ) )
= A3 ) ) ).
% mult_div_mod_eq
thf(fact_2656_mod__mult__div__eq,axiom,
! [A: $tType] :
( ( semiring_modulo @ A )
=> ! [A3: A,B3: A] :
( ( plus_plus @ A @ ( modulo_modulo @ A @ A3 @ B3 ) @ ( times_times @ A @ B3 @ ( divide_divide @ A @ A3 @ B3 ) ) )
= A3 ) ) ).
% mod_mult_div_eq
thf(fact_2657_mod__div__mult__eq,axiom,
! [A: $tType] :
( ( semiring_modulo @ A )
=> ! [A3: A,B3: A] :
( ( plus_plus @ A @ ( modulo_modulo @ A @ A3 @ B3 ) @ ( times_times @ A @ ( divide_divide @ A @ A3 @ B3 ) @ B3 ) )
= A3 ) ) ).
% mod_div_mult_eq
thf(fact_2658_div__mult__mod__eq,axiom,
! [A: $tType] :
( ( semiring_modulo @ A )
=> ! [A3: A,B3: A] :
( ( plus_plus @ A @ ( times_times @ A @ ( divide_divide @ A @ A3 @ B3 ) @ B3 ) @ ( modulo_modulo @ A @ A3 @ B3 ) )
= A3 ) ) ).
% div_mult_mod_eq
thf(fact_2659_mod__div__decomp,axiom,
! [A: $tType] :
( ( semiring_modulo @ A )
=> ! [A3: A,B3: A] :
( A3
= ( plus_plus @ A @ ( times_times @ A @ ( divide_divide @ A @ A3 @ B3 ) @ B3 ) @ ( modulo_modulo @ A @ A3 @ B3 ) ) ) ) ).
% mod_div_decomp
thf(fact_2660_cancel__div__mod__rules_I1_J,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ! [A3: A,B3: A,C2: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ ( times_times @ A @ ( divide_divide @ A @ A3 @ B3 ) @ B3 ) @ ( modulo_modulo @ A @ A3 @ B3 ) ) @ C2 )
= ( plus_plus @ A @ A3 @ C2 ) ) ) ).
% cancel_div_mod_rules(1)
thf(fact_2661_cancel__div__mod__rules_I2_J,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ! [B3: A,A3: A,C2: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ ( times_times @ A @ B3 @ ( divide_divide @ A @ A3 @ B3 ) ) @ ( modulo_modulo @ A @ A3 @ B3 ) ) @ C2 )
= ( plus_plus @ A @ A3 @ C2 ) ) ) ).
% cancel_div_mod_rules(2)
thf(fact_2662_minus__mult__div__eq__mod,axiom,
! [A: $tType] :
( ( semiring_modulo @ A )
=> ! [A3: A,B3: A] :
( ( minus_minus @ A @ A3 @ ( times_times @ A @ B3 @ ( divide_divide @ A @ A3 @ B3 ) ) )
= ( modulo_modulo @ A @ A3 @ B3 ) ) ) ).
% minus_mult_div_eq_mod
thf(fact_2663_minus__mod__eq__mult__div,axiom,
! [A: $tType] :
( ( semiring_modulo @ A )
=> ! [A3: A,B3: A] :
( ( minus_minus @ A @ A3 @ ( modulo_modulo @ A @ A3 @ B3 ) )
= ( times_times @ A @ B3 @ ( divide_divide @ A @ A3 @ B3 ) ) ) ) ).
% minus_mod_eq_mult_div
thf(fact_2664_minus__mod__eq__div__mult,axiom,
! [A: $tType] :
( ( semiring_modulo @ A )
=> ! [A3: A,B3: A] :
( ( minus_minus @ A @ A3 @ ( modulo_modulo @ A @ A3 @ B3 ) )
= ( times_times @ A @ ( divide_divide @ A @ A3 @ B3 ) @ B3 ) ) ) ).
% minus_mod_eq_div_mult
thf(fact_2665_minus__div__mult__eq__mod,axiom,
! [A: $tType] :
( ( semiring_modulo @ A )
=> ! [A3: A,B3: A] :
( ( minus_minus @ A @ A3 @ ( times_times @ A @ ( divide_divide @ A @ A3 @ B3 ) @ B3 ) )
= ( modulo_modulo @ A @ A3 @ B3 ) ) ) ).
% minus_div_mult_eq_mod
thf(fact_2666_unit__imp__mod__eq__0,axiom,
! [A: $tType] :
( ( euclid3725896446679973847miring @ A )
=> ! [B3: A,A3: A] :
( ( dvd_dvd @ A @ B3 @ ( one_one @ A ) )
=> ( ( modulo_modulo @ A @ A3 @ B3 )
= ( zero_zero @ A ) ) ) ) ).
% unit_imp_mod_eq_0
thf(fact_2667_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_2668_pos__mod__conj,axiom,
! [B3: int,A3: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ B3 )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( modulo_modulo @ int @ A3 @ B3 ) )
& ( ord_less @ int @ ( modulo_modulo @ int @ A3 @ B3 ) @ B3 ) ) ) ).
% pos_mod_conj
thf(fact_2669_neg__mod__conj,axiom,
! [B3: int,A3: int] :
( ( ord_less @ int @ B3 @ ( zero_zero @ int ) )
=> ( ( ord_less_eq @ int @ ( modulo_modulo @ int @ A3 @ B3 ) @ ( zero_zero @ int ) )
& ( ord_less @ int @ B3 @ ( modulo_modulo @ int @ A3 @ B3 ) ) ) ) ).
% neg_mod_conj
thf(fact_2670_int__mod__ge,axiom,
! [A3: int,N: int] :
( ( ord_less @ int @ A3 @ N )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ N )
=> ( ord_less_eq @ int @ A3 @ ( modulo_modulo @ int @ A3 @ N ) ) ) ) ).
% int_mod_ge
thf(fact_2671_Euclidean__Division_Opos__mod__sign,axiom,
! [L: int,K: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ L )
=> ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( modulo_modulo @ int @ K @ L ) ) ) ).
% Euclidean_Division.pos_mod_sign
thf(fact_2672_neg__mod__sign,axiom,
! [L: int,K: int] :
( ( ord_less @ int @ L @ ( zero_zero @ int ) )
=> ( ord_less_eq @ int @ ( modulo_modulo @ int @ K @ L ) @ ( zero_zero @ int ) ) ) ).
% neg_mod_sign
thf(fact_2673_int__mod__lem,axiom,
! [N: int,B3: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ N )
=> ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ B3 )
& ( ord_less @ int @ B3 @ N ) )
= ( ( modulo_modulo @ int @ B3 @ N )
= B3 ) ) ) ).
% int_mod_lem
thf(fact_2674_int__mod__eq,axiom,
! [B3: int,N: int,A3: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ B3 )
=> ( ( ord_less @ int @ B3 @ N )
=> ( ( ( modulo_modulo @ int @ A3 @ N )
= ( modulo_modulo @ int @ B3 @ N ) )
=> ( ( modulo_modulo @ int @ A3 @ N )
= B3 ) ) ) ) ).
% int_mod_eq
thf(fact_2675_int__mod__le_H,axiom,
! [B3: int,N: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( minus_minus @ int @ B3 @ N ) )
=> ( ord_less_eq @ int @ ( modulo_modulo @ int @ B3 @ N ) @ ( minus_minus @ int @ B3 @ N ) ) ) ).
% int_mod_le'
thf(fact_2676_zmod__zminus2__eq__if,axiom,
! [A3: int,B3: int] :
( ( ( ( modulo_modulo @ int @ A3 @ B3 )
= ( zero_zero @ int ) )
=> ( ( modulo_modulo @ int @ A3 @ ( uminus_uminus @ int @ B3 ) )
= ( zero_zero @ int ) ) )
& ( ( ( modulo_modulo @ int @ A3 @ B3 )
!= ( zero_zero @ int ) )
=> ( ( modulo_modulo @ int @ A3 @ ( uminus_uminus @ int @ B3 ) )
= ( minus_minus @ int @ ( modulo_modulo @ int @ A3 @ B3 ) @ B3 ) ) ) ) ).
% zmod_zminus2_eq_if
thf(fact_2677_zmod__zminus1__eq__if,axiom,
! [A3: int,B3: int] :
( ( ( ( modulo_modulo @ int @ A3 @ B3 )
= ( zero_zero @ int ) )
=> ( ( modulo_modulo @ int @ ( uminus_uminus @ int @ A3 ) @ B3 )
= ( zero_zero @ int ) ) )
& ( ( ( modulo_modulo @ int @ A3 @ B3 )
!= ( zero_zero @ int ) )
=> ( ( modulo_modulo @ int @ ( uminus_uminus @ int @ A3 ) @ B3 )
= ( minus_minus @ int @ B3 @ ( modulo_modulo @ int @ A3 @ B3 ) ) ) ) ) ).
% zmod_zminus1_eq_if
thf(fact_2678_nonneg__mod__div,axiom,
! [A3: int,B3: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ A3 )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ B3 )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( modulo_modulo @ int @ A3 @ B3 ) )
& ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( divide_divide @ int @ A3 @ B3 ) ) ) ) ) ).
% nonneg_mod_div
thf(fact_2679_zdiv__mono__strict,axiom,
! [A2: int,B2: int,N: int] :
( ( ord_less @ int @ A2 @ B2 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ N )
=> ( ( ( modulo_modulo @ int @ A2 @ N )
= ( zero_zero @ int ) )
=> ( ( ( modulo_modulo @ int @ B2 @ N )
= ( zero_zero @ int ) )
=> ( ord_less @ int @ ( divide_divide @ int @ A2 @ N ) @ ( divide_divide @ int @ B2 @ N ) ) ) ) ) ) ).
% zdiv_mono_strict
thf(fact_2680_abs__mod__less,axiom,
! [L: int,K: int] :
( ( L
!= ( zero_zero @ int ) )
=> ( ord_less @ int @ ( abs_abs @ int @ ( modulo_modulo @ int @ K @ L ) ) @ ( abs_abs @ int @ L ) ) ) ).
% abs_mod_less
thf(fact_2681_div__mod__decomp__int,axiom,
! [A2: int,N: int] :
( A2
= ( plus_plus @ int @ ( times_times @ int @ ( divide_divide @ int @ A2 @ N ) @ N ) @ ( modulo_modulo @ int @ A2 @ N ) ) ) ).
% div_mod_decomp_int
thf(fact_2682_summable__rabs,axiom,
! [F3: nat > real] :
( ( summable @ real
@ ^ [N4: nat] : ( abs_abs @ real @ ( F3 @ N4 ) ) )
=> ( ord_less_eq @ real @ ( abs_abs @ real @ ( suminf @ real @ F3 ) )
@ ( suminf @ real
@ ^ [N4: nat] : ( abs_abs @ real @ ( F3 @ N4 ) ) ) ) ) ).
% summable_rabs
thf(fact_2683_suminf__pos2,axiom,
! [A: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F3: nat > A,I: nat] :
( ( summable @ A @ F3 )
=> ( ! [N2: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F3 @ N2 ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ ( F3 @ I ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( suminf @ A @ F3 ) ) ) ) ) ) ).
% suminf_pos2
thf(fact_2684_suminf__pos__iff,axiom,
! [A: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F3: nat > A] :
( ( summable @ A @ F3 )
=> ( ! [N2: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F3 @ N2 ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ ( suminf @ A @ F3 ) )
= ( ? [I4: nat] : ( ord_less @ A @ ( zero_zero @ A ) @ ( F3 @ I4 ) ) ) ) ) ) ) ).
% suminf_pos_iff
thf(fact_2685_mod__mult2__eq_H,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [A3: A,M: nat,N: nat] :
( ( modulo_modulo @ A @ A3 @ ( times_times @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) ) )
= ( plus_plus @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ M ) @ ( modulo_modulo @ A @ ( divide_divide @ A @ A3 @ ( semiring_1_of_nat @ A @ M ) ) @ ( semiring_1_of_nat @ A @ N ) ) ) @ ( modulo_modulo @ A @ A3 @ ( semiring_1_of_nat @ A @ M ) ) ) ) ) ).
% mod_mult2_eq'
thf(fact_2686_pos__mod__bound2,axiom,
! [A3: int] : ( ord_less @ int @ ( modulo_modulo @ int @ A3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ).
% pos_mod_bound2
thf(fact_2687_int__mod__ge_H,axiom,
! [B3: int,N: int] :
( ( ord_less @ int @ B3 @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ N )
=> ( ord_less_eq @ int @ ( plus_plus @ int @ B3 @ N ) @ ( modulo_modulo @ int @ B3 @ N ) ) ) ) ).
% int_mod_ge'
thf(fact_2688_mod__pos__neg__trivial,axiom,
! [K: int,L: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ K )
=> ( ( ord_less_eq @ int @ ( plus_plus @ int @ K @ L ) @ ( zero_zero @ int ) )
=> ( ( modulo_modulo @ int @ K @ L )
= ( plus_plus @ int @ K @ L ) ) ) ) ).
% mod_pos_neg_trivial
thf(fact_2689_mod__pos__geq,axiom,
! [L: int,K: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ L )
=> ( ( ord_less_eq @ int @ L @ K )
=> ( ( modulo_modulo @ int @ K @ L )
= ( modulo_modulo @ int @ ( minus_minus @ int @ K @ L ) @ L ) ) ) ) ).
% mod_pos_geq
thf(fact_2690_mod__int__pos__iff,axiom,
! [K: int,L: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( modulo_modulo @ int @ K @ L ) )
= ( ( dvd_dvd @ int @ L @ K )
| ( ( L
= ( zero_zero @ int ) )
& ( ord_less_eq @ int @ ( zero_zero @ int ) @ K ) )
| ( ord_less @ int @ ( zero_zero @ int ) @ L ) ) ) ).
% mod_int_pos_iff
thf(fact_2691_real__of__int__div__aux,axiom,
! [X: int,D2: int] :
( ( divide_divide @ real @ ( ring_1_of_int @ real @ X ) @ ( ring_1_of_int @ real @ D2 ) )
= ( plus_plus @ real @ ( ring_1_of_int @ real @ ( divide_divide @ int @ X @ D2 ) ) @ ( divide_divide @ real @ ( ring_1_of_int @ real @ ( modulo_modulo @ int @ X @ D2 ) ) @ ( ring_1_of_int @ real @ D2 ) ) ) ) ).
% real_of_int_div_aux
thf(fact_2692_binomial__antimono,axiom,
! [K: nat,K5: nat,N: nat] :
( ( ord_less_eq @ nat @ K @ K5 )
=> ( ( ord_less_eq @ nat @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ K )
=> ( ( ord_less_eq @ nat @ K5 @ N )
=> ( ord_less_eq @ nat @ ( binomial @ N @ K5 ) @ ( binomial @ N @ K ) ) ) ) ) ).
% binomial_antimono
thf(fact_2693_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_2694_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_2695_binomial__mono,axiom,
! [K: nat,K5: nat,N: nat] :
( ( ord_less_eq @ nat @ K @ K5 )
=> ( ( ord_less_eq @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K5 ) @ N )
=> ( ord_less_eq @ nat @ ( binomial @ N @ K ) @ ( binomial @ N @ K5 ) ) ) ) ).
% binomial_mono
thf(fact_2696_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [C2: A,A3: A,B3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( modulo_modulo @ A @ A3 @ ( times_times @ A @ B3 @ C2 ) )
= ( plus_plus @ A @ ( times_times @ A @ B3 @ ( modulo_modulo @ A @ ( divide_divide @ A @ A3 @ B3 ) @ C2 ) ) @ ( modulo_modulo @ A @ A3 @ B3 ) ) ) ) ) ).
% unique_euclidean_semiring_numeral_class.mod_mult2_eq
thf(fact_2697_even__iff__mod__2__eq__zero,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A3: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
= ( ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( zero_zero @ A ) ) ) ) ).
% even_iff_mod_2_eq_zero
thf(fact_2698_odd__iff__mod__2__eq__one,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A3: A] :
( ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) )
= ( ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( one_one @ A ) ) ) ) ).
% odd_iff_mod_2_eq_one
thf(fact_2699_fold__atLeastAtMost__nat_Oelims,axiom,
! [A: $tType,X: nat > A > A,Xa: nat,Xb: nat,Xc: A,Y: A] :
( ( ( set_fo6178422350223883121st_nat @ A @ X @ Xa @ Xb @ Xc )
= Y )
=> ( ( ( ord_less @ nat @ Xb @ Xa )
=> ( Y = Xc ) )
& ( ~ ( ord_less @ nat @ Xb @ Xa )
=> ( Y
= ( set_fo6178422350223883121st_nat @ A @ X @ ( plus_plus @ nat @ Xa @ ( one_one @ nat ) ) @ Xb @ ( X @ Xa @ Xc ) ) ) ) ) ) ).
% fold_atLeastAtMost_nat.elims
thf(fact_2700_fold__atLeastAtMost__nat_Osimps,axiom,
! [A: $tType] :
( ( set_fo6178422350223883121st_nat @ A )
= ( ^ [F2: nat > A > A,A5: nat,B5: nat,Acc: A] : ( if @ A @ ( ord_less @ nat @ B5 @ A5 ) @ Acc @ ( set_fo6178422350223883121st_nat @ A @ F2 @ ( plus_plus @ nat @ A5 @ ( one_one @ nat ) ) @ B5 @ ( F2 @ A5 @ Acc ) ) ) ) ) ).
% fold_atLeastAtMost_nat.simps
thf(fact_2701_pos__mod__sign2,axiom,
! [A3: int] : ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( modulo_modulo @ int @ A3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ).
% pos_mod_sign2
thf(fact_2702_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_2703_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_2704_mod__exp__less__eq__exp,axiom,
! [A3: int,N: nat] : ( ord_less @ int @ ( modulo_modulo @ int @ A3 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ).
% mod_exp_less_eq_exp
thf(fact_2705_mod__power__lem,axiom,
! [A3: int,M: nat,N: nat] :
( ( ord_less @ int @ ( one_one @ int ) @ A3 )
=> ( ( ( ord_less_eq @ nat @ M @ N )
=> ( ( modulo_modulo @ int @ ( power_power @ int @ A3 @ N ) @ ( power_power @ int @ A3 @ M ) )
= ( zero_zero @ int ) ) )
& ( ~ ( ord_less_eq @ nat @ M @ N )
=> ( ( modulo_modulo @ int @ ( power_power @ int @ A3 @ N ) @ ( power_power @ int @ A3 @ M ) )
= ( power_power @ int @ A3 @ N ) ) ) ) ) ).
% mod_power_lem
thf(fact_2706_int__mod__pos__eq,axiom,
! [A3: int,B3: int,Q3: int,R3: int] :
( ( A3
= ( plus_plus @ int @ ( times_times @ int @ B3 @ Q3 ) @ R3 ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ R3 )
=> ( ( ord_less @ int @ R3 @ B3 )
=> ( ( modulo_modulo @ int @ A3 @ B3 )
= R3 ) ) ) ) ).
% int_mod_pos_eq
thf(fact_2707_int__mod__neg__eq,axiom,
! [A3: int,B3: int,Q3: int,R3: int] :
( ( A3
= ( plus_plus @ int @ ( times_times @ int @ B3 @ Q3 ) @ R3 ) )
=> ( ( ord_less_eq @ int @ R3 @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ B3 @ R3 )
=> ( ( modulo_modulo @ int @ A3 @ B3 )
= R3 ) ) ) ) ).
% int_mod_neg_eq
thf(fact_2708_split__zmod,axiom,
! [P: int > $o,N: int,K: int] :
( ( P @ ( modulo_modulo @ int @ N @ K ) )
= ( ( ( K
= ( zero_zero @ int ) )
=> ( P @ N ) )
& ( ( ord_less @ int @ ( zero_zero @ int ) @ K )
=> ! [I4: int,J3: int] :
( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ J3 )
& ( ord_less @ int @ J3 @ K )
& ( N
= ( plus_plus @ int @ ( times_times @ int @ K @ I4 ) @ J3 ) ) )
=> ( P @ J3 ) ) )
& ( ( ord_less @ int @ K @ ( zero_zero @ int ) )
=> ! [I4: int,J3: int] :
( ( ( ord_less @ int @ K @ J3 )
& ( ord_less_eq @ int @ J3 @ ( zero_zero @ int ) )
& ( N
= ( plus_plus @ int @ ( times_times @ int @ K @ I4 ) @ J3 ) ) )
=> ( P @ J3 ) ) ) ) ) ).
% split_zmod
thf(fact_2709_minus__mod__int__eq,axiom,
! [L: int,K: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ L )
=> ( ( modulo_modulo @ int @ ( uminus_uminus @ int @ K ) @ L )
= ( minus_minus @ int @ ( minus_minus @ int @ L @ ( one_one @ int ) ) @ ( modulo_modulo @ int @ ( minus_minus @ int @ K @ ( one_one @ int ) ) @ L ) ) ) ) ).
% minus_mod_int_eq
thf(fact_2710_zmod__minus1,axiom,
! [B3: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ B3 )
=> ( ( modulo_modulo @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ B3 )
= ( minus_minus @ int @ B3 @ ( one_one @ int ) ) ) ) ).
% zmod_minus1
thf(fact_2711_mod__sub__if__z,axiom,
! [X: int,Z: int,Y: int] :
( ( ord_less @ int @ X @ Z )
=> ( ( ord_less @ int @ Y @ Z )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z )
=> ( ( ( ord_less_eq @ int @ Y @ X )
=> ( ( modulo_modulo @ int @ ( minus_minus @ int @ X @ Y ) @ Z )
= ( minus_minus @ int @ X @ Y ) ) )
& ( ~ ( ord_less_eq @ int @ Y @ X )
=> ( ( modulo_modulo @ int @ ( minus_minus @ int @ X @ Y ) @ Z )
= ( plus_plus @ int @ ( minus_minus @ int @ X @ Y ) @ Z ) ) ) ) ) ) ) ) ) ).
% mod_sub_if_z
thf(fact_2712_mod__add__if__z,axiom,
! [X: int,Z: int,Y: int] :
( ( ord_less @ int @ X @ Z )
=> ( ( ord_less @ int @ Y @ Z )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z )
=> ( ( ( ord_less @ int @ ( plus_plus @ int @ X @ Y ) @ Z )
=> ( ( modulo_modulo @ int @ ( plus_plus @ int @ X @ Y ) @ Z )
= ( plus_plus @ int @ X @ Y ) ) )
& ( ~ ( ord_less @ int @ ( plus_plus @ int @ X @ Y ) @ Z )
=> ( ( modulo_modulo @ int @ ( plus_plus @ int @ X @ Y ) @ Z )
= ( minus_minus @ int @ ( plus_plus @ int @ X @ Y ) @ Z ) ) ) ) ) ) ) ) ) ).
% mod_add_if_z
thf(fact_2713_zmod__zmult2__eq,axiom,
! [C2: int,A3: int,B3: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ C2 )
=> ( ( modulo_modulo @ int @ A3 @ ( times_times @ int @ B3 @ C2 ) )
= ( plus_plus @ int @ ( times_times @ int @ B3 @ ( modulo_modulo @ int @ ( divide_divide @ int @ A3 @ B3 ) @ C2 ) ) @ ( modulo_modulo @ int @ A3 @ B3 ) ) ) ) ).
% zmod_zmult2_eq
thf(fact_2714_zdiv__zminus2__eq__if,axiom,
! [B3: int,A3: int] :
( ( B3
!= ( zero_zero @ int ) )
=> ( ( ( ( modulo_modulo @ int @ A3 @ B3 )
= ( zero_zero @ int ) )
=> ( ( divide_divide @ int @ A3 @ ( uminus_uminus @ int @ B3 ) )
= ( uminus_uminus @ int @ ( divide_divide @ int @ A3 @ B3 ) ) ) )
& ( ( ( modulo_modulo @ int @ A3 @ B3 )
!= ( zero_zero @ int ) )
=> ( ( divide_divide @ int @ A3 @ ( uminus_uminus @ int @ B3 ) )
= ( minus_minus @ int @ ( uminus_uminus @ int @ ( divide_divide @ int @ A3 @ B3 ) ) @ ( one_one @ int ) ) ) ) ) ) ).
% zdiv_zminus2_eq_if
thf(fact_2715_zdiv__zminus1__eq__if,axiom,
! [B3: int,A3: int] :
( ( B3
!= ( zero_zero @ int ) )
=> ( ( ( ( modulo_modulo @ int @ A3 @ B3 )
= ( zero_zero @ int ) )
=> ( ( divide_divide @ int @ ( uminus_uminus @ int @ A3 ) @ B3 )
= ( uminus_uminus @ int @ ( divide_divide @ int @ A3 @ B3 ) ) ) )
& ( ( ( modulo_modulo @ int @ A3 @ B3 )
!= ( zero_zero @ int ) )
=> ( ( divide_divide @ int @ ( uminus_uminus @ int @ A3 ) @ B3 )
= ( minus_minus @ int @ ( uminus_uminus @ int @ ( divide_divide @ int @ A3 @ B3 ) ) @ ( one_one @ int ) ) ) ) ) ) ).
% zdiv_zminus1_eq_if
thf(fact_2716_binomial__strict__antimono,axiom,
! [K: nat,K5: nat,N: nat] :
( ( ord_less @ nat @ K @ K5 )
=> ( ( ord_less_eq @ nat @ N @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K ) )
=> ( ( ord_less_eq @ nat @ K5 @ N )
=> ( ord_less @ nat @ ( binomial @ N @ K5 ) @ ( binomial @ N @ K ) ) ) ) ) ).
% binomial_strict_antimono
thf(fact_2717_binomial__strict__mono,axiom,
! [K: nat,K5: nat,N: nat] :
( ( ord_less @ nat @ K @ K5 )
=> ( ( ord_less_eq @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K5 ) @ N )
=> ( ord_less @ nat @ ( binomial @ N @ K ) @ ( binomial @ N @ K5 ) ) ) ) ).
% binomial_strict_mono
thf(fact_2718_divmod__digit__0_I2_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [B3: A,A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ B3 )
=> ( ( ord_less @ A @ ( modulo_modulo @ A @ A3 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B3 ) ) @ B3 )
=> ( ( modulo_modulo @ A @ A3 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B3 ) )
= ( modulo_modulo @ A @ A3 @ B3 ) ) ) ) ) ).
% divmod_digit_0(2)
thf(fact_2719_bits__stable__imp__add__self,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A3: A] :
( ( ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= A3 )
=> ( ( plus_plus @ A @ A3 @ ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
= ( zero_zero @ A ) ) ) ) ).
% bits_stable_imp_add_self
thf(fact_2720_parity__cases,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A3: A] :
( ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
=> ( ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
!= ( zero_zero @ A ) ) )
=> ~ ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
=> ( ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
!= ( one_one @ A ) ) ) ) ) ).
% parity_cases
thf(fact_2721_mod2__eq__if,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A3: A] :
( ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
=> ( ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( zero_zero @ A ) ) )
& ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
=> ( ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( one_one @ A ) ) ) ) ) ).
% mod2_eq_if
thf(fact_2722_div__exp__mod__exp__eq,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A3: A,N: nat,M: nat] :
( ( modulo_modulo @ A @ ( divide_divide @ A @ A3 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) )
= ( divide_divide @ A @ ( modulo_modulo @ A @ A3 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( plus_plus @ nat @ N @ M ) ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% div_exp_mod_exp_eq
thf(fact_2723_axxmod2,axiom,
! [X: int] :
( ( ( modulo_modulo @ int @ ( plus_plus @ int @ ( plus_plus @ int @ ( one_one @ int ) @ X ) @ X ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) )
= ( one_one @ int ) )
& ( ( modulo_modulo @ int @ ( plus_plus @ int @ ( plus_plus @ int @ ( zero_zero @ int ) @ X ) @ X ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) )
= ( zero_zero @ int ) ) ) ).
% axxmod2
thf(fact_2724_z1pmod2,axiom,
! [B3: int] :
( ( modulo_modulo @ int @ ( plus_plus @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ B3 ) @ ( one_one @ int ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) )
= ( one_one @ int ) ) ).
% z1pmod2
thf(fact_2725_verit__le__mono__div__int,axiom,
! [A2: int,B2: int,N: int] :
( ( ord_less @ int @ A2 @ B2 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ N )
=> ( ord_less_eq @ int
@ ( plus_plus @ int @ ( divide_divide @ int @ A2 @ N )
@ ( if @ int
@ ( ( modulo_modulo @ int @ B2 @ N )
= ( zero_zero @ int ) )
@ ( one_one @ int )
@ ( zero_zero @ int ) ) )
@ ( divide_divide @ int @ B2 @ N ) ) ) ) ).
% verit_le_mono_div_int
thf(fact_2726_split__neg__lemma,axiom,
! [K: int,P: int > int > $o,N: int] :
( ( ord_less @ int @ K @ ( zero_zero @ int ) )
=> ( ( P @ ( divide_divide @ int @ N @ K ) @ ( modulo_modulo @ int @ N @ K ) )
= ( ! [I4: int,J3: int] :
( ( ( ord_less @ int @ K @ J3 )
& ( ord_less_eq @ int @ J3 @ ( zero_zero @ int ) )
& ( N
= ( plus_plus @ int @ ( times_times @ int @ K @ I4 ) @ J3 ) ) )
=> ( P @ I4 @ J3 ) ) ) ) ) ).
% split_neg_lemma
thf(fact_2727_split__pos__lemma,axiom,
! [K: int,P: int > int > $o,N: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ K )
=> ( ( P @ ( divide_divide @ int @ N @ K ) @ ( modulo_modulo @ int @ N @ K ) )
= ( ! [I4: int,J3: int] :
( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ J3 )
& ( ord_less @ int @ J3 @ K )
& ( N
= ( plus_plus @ int @ ( times_times @ int @ K @ I4 ) @ J3 ) ) )
=> ( P @ I4 @ J3 ) ) ) ) ) ).
% split_pos_lemma
thf(fact_2728_divmod__digit__0_I1_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [B3: A,A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ B3 )
=> ( ( ord_less @ A @ ( modulo_modulo @ A @ A3 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B3 ) ) @ B3 )
=> ( ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A3 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B3 ) ) )
= ( divide_divide @ A @ A3 @ B3 ) ) ) ) ) ).
% divmod_digit_0(1)
thf(fact_2729_mult__exp__mod__exp__eq,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [M: nat,N: nat,A3: A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( modulo_modulo @ A @ ( times_times @ A @ A3 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) )
= ( times_times @ A @ ( modulo_modulo @ A @ A3 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ N @ M ) ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) ) ) ) ) ).
% mult_exp_mod_exp_eq
thf(fact_2730_p1mod22k_H,axiom,
! [B3: int,N: nat] :
( ( modulo_modulo @ int @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ B3 ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) )
= ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( modulo_modulo @ int @ B3 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ) ) ) ).
% p1mod22k'
thf(fact_2731_p1mod22k,axiom,
! [B3: int,N: nat] :
( ( modulo_modulo @ int @ ( plus_plus @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ B3 ) @ ( one_one @ int ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) )
= ( plus_plus @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( modulo_modulo @ int @ B3 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ) @ ( one_one @ int ) ) ) ).
% p1mod22k
thf(fact_2732_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_2733_mod__double__modulus,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M: A,X: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ M )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( ( modulo_modulo @ A @ X @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) )
= ( modulo_modulo @ A @ X @ M ) )
| ( ( modulo_modulo @ A @ X @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) )
= ( plus_plus @ A @ ( modulo_modulo @ A @ X @ M ) @ M ) ) ) ) ) ) ).
% mod_double_modulus
thf(fact_2734_divmod__digit__1_I2_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B3 )
=> ( ( ord_less_eq @ A @ B3 @ ( modulo_modulo @ A @ A3 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B3 ) ) )
=> ( ( minus_minus @ A @ ( modulo_modulo @ A @ A3 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B3 ) ) @ B3 )
= ( modulo_modulo @ A @ A3 @ B3 ) ) ) ) ) ) ).
% divmod_digit_1(2)
thf(fact_2735_pos__zmod__mult__2,axiom,
! [A3: int,B3: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ A3 )
=> ( ( modulo_modulo @ int @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ B3 ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ A3 ) )
= ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( modulo_modulo @ int @ B3 @ A3 ) ) ) ) ) ).
% pos_zmod_mult_2
thf(fact_2736_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_2737_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_2738_sb__dec__lem_H,axiom,
! [K: nat,A3: int] :
( ( ord_less_eq @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K ) @ A3 )
=> ( ord_less_eq @ int @ ( modulo_modulo @ int @ ( plus_plus @ int @ A3 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K ) ) ) @ ( plus_plus @ int @ ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K ) ) @ A3 ) ) ) ).
% sb_dec_lem'
thf(fact_2739_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_2740_neg__zmod__mult__2,axiom,
! [A3: int,B3: int] :
( ( ord_less_eq @ int @ A3 @ ( zero_zero @ int ) )
=> ( ( modulo_modulo @ int @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ B3 ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ A3 ) )
= ( minus_minus @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( modulo_modulo @ int @ ( plus_plus @ int @ B3 @ ( one_one @ int ) ) @ A3 ) ) @ ( one_one @ int ) ) ) ) ).
% neg_zmod_mult_2
thf(fact_2741_sb__dec__lem,axiom,
! [K: nat,A3: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( plus_plus @ int @ ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K ) ) @ A3 ) )
=> ( ord_less_eq @ int @ ( modulo_modulo @ int @ ( plus_plus @ int @ A3 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K ) ) ) @ ( plus_plus @ int @ ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K ) ) @ A3 ) ) ) ).
% sb_dec_lem
thf(fact_2742_divmod__digit__1_I1_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B3 )
=> ( ( ord_less_eq @ A @ B3 @ ( modulo_modulo @ A @ A3 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B3 ) ) )
=> ( ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A3 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B3 ) ) ) @ ( one_one @ A ) )
= ( divide_divide @ A @ A3 @ B3 ) ) ) ) ) ) ).
% divmod_digit_1(1)
thf(fact_2743_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_2744_set__decode__def,axiom,
( nat_set_decode
= ( ^ [X2: nat] :
( collect @ nat
@ ^ [N4: nat] :
~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ X2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N4 ) ) ) ) ) ) ).
% set_decode_def
thf(fact_2745_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_2746_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_2747_binomial__n__0,axiom,
! [N: nat] :
( ( binomial @ N @ ( zero_zero @ nat ) )
= ( one_one @ nat ) ) ).
% binomial_n_0
thf(fact_2748_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_2749_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_2750_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_2751_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_2752_nat__mod__eq_H,axiom,
! [A3: nat,N: nat] :
( ( ord_less @ nat @ A3 @ N )
=> ( ( modulo_modulo @ nat @ A3 @ N )
= A3 ) ) ).
% nat_mod_eq'
thf(fact_2753_mod__less,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ N )
=> ( ( modulo_modulo @ nat @ M @ N )
= M ) ) ).
% mod_less
thf(fact_2754_binomial__n__n,axiom,
! [N: nat] :
( ( binomial @ N @ N )
= ( one_one @ nat ) ) ).
% binomial_n_n
thf(fact_2755_word__mod__no,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: num,B3: num] :
( ( modulo_modulo @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ A3 ) @ ( numeral_numeral @ ( word @ A ) @ B3 ) )
= ( ring_1_of_int @ ( word @ A ) @ ( modulo_modulo @ int @ ( semiring_1_unsigned @ A @ int @ ( numeral_numeral @ ( word @ A ) @ A3 ) ) @ ( semiring_1_unsigned @ A @ int @ ( numeral_numeral @ ( word @ A ) @ B3 ) ) ) ) ) ) ).
% word_mod_no
thf(fact_2756_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_2757_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_2758_unat__mod,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] :
( ( semiring_1_unsigned @ A @ nat @ ( modulo_modulo @ ( word @ A ) @ X @ Y ) )
= ( modulo_modulo @ nat @ ( semiring_1_unsigned @ A @ nat @ X ) @ ( semiring_1_unsigned @ A @ nat @ Y ) ) ) ) ).
% unat_mod
thf(fact_2759_unat__mod__distrib,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [V2: word @ A,W: word @ A] :
( ( semiring_1_unsigned @ A @ nat @ ( modulo_modulo @ ( word @ A ) @ V2 @ W ) )
= ( modulo_modulo @ nat @ ( semiring_1_unsigned @ A @ nat @ V2 ) @ ( semiring_1_unsigned @ A @ nat @ W ) ) ) ) ).
% unat_mod_distrib
thf(fact_2760_word__arith__nat__mod,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( modulo_modulo @ ( word @ A ) )
= ( ^ [A5: word @ A,B5: word @ A] : ( semiring_1_of_nat @ ( word @ A ) @ ( modulo_modulo @ nat @ ( semiring_1_unsigned @ A @ nat @ A5 ) @ ( semiring_1_unsigned @ A @ nat @ B5 ) ) ) ) ) ) ).
% word_arith_nat_mod
thf(fact_2761_nat__mod__eq,axiom,
! [B3: nat,N: nat,A3: nat] :
( ( ord_less @ nat @ B3 @ N )
=> ( ( ( modulo_modulo @ nat @ A3 @ N )
= ( modulo_modulo @ nat @ B3 @ N ) )
=> ( ( modulo_modulo @ nat @ A3 @ N )
= B3 ) ) ) ).
% nat_mod_eq
thf(fact_2762_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_2763_mod__word__less,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A,V2: word @ A] :
( ( ord_less @ ( word @ A ) @ W @ V2 )
=> ( ( modulo_modulo @ ( word @ A ) @ W @ V2 )
= W ) ) ) ).
% mod_word_less
thf(fact_2764_gcd__nat__induct,axiom,
! [P: nat > nat > $o,M: nat,N: nat] :
( ! [M4: nat] : ( P @ M4 @ ( zero_zero @ nat ) )
=> ( ! [M4: nat,N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( P @ N2 @ ( modulo_modulo @ nat @ M4 @ N2 ) )
=> ( P @ M4 @ N2 ) ) )
=> ( P @ M @ N ) ) ) ).
% gcd_nat_induct
thf(fact_2765_nat__mod__lem,axiom,
! [N: nat,B3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ nat @ B3 @ N )
= ( ( modulo_modulo @ nat @ B3 @ N )
= B3 ) ) ) ).
% nat_mod_lem
thf(fact_2766_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_2767_word__rot__lem,axiom,
! [L: nat,K: nat,D2: nat,N: nat] :
( ( ( plus_plus @ nat @ L @ K )
= ( plus_plus @ nat @ D2 @ ( modulo_modulo @ nat @ K @ L ) ) )
=> ( ( ord_less @ nat @ N @ L )
=> ( ( modulo_modulo @ nat @ ( plus_plus @ nat @ D2 @ N ) @ L )
= N ) ) ) ).
% word_rot_lem
thf(fact_2768_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_2769_mod__nat__sub,axiom,
! [X: nat,Z: nat,Y: nat] :
( ( ord_less @ nat @ X @ Z )
=> ( ( modulo_modulo @ nat @ ( minus_minus @ nat @ X @ Y ) @ Z )
= ( minus_minus @ nat @ X @ Y ) ) ) ).
% mod_nat_sub
thf(fact_2770_mod__if,axiom,
( ( modulo_modulo @ nat )
= ( ^ [M3: nat,N4: nat] : ( if @ nat @ ( ord_less @ nat @ M3 @ N4 ) @ M3 @ ( modulo_modulo @ nat @ ( minus_minus @ nat @ M3 @ N4 ) @ N4 ) ) ) ) ).
% mod_if
thf(fact_2771_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_2772_mod__eq__0D,axiom,
! [M: nat,D2: nat] :
( ( ( modulo_modulo @ nat @ M @ D2 )
= ( zero_zero @ nat ) )
=> ? [Q6: nat] :
( M
= ( times_times @ nat @ D2 @ Q6 ) ) ) ).
% mod_eq_0D
thf(fact_2773_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_2774_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_2775_uint__mod,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] :
( ( semiring_1_unsigned @ A @ int @ ( modulo_modulo @ ( word @ A ) @ X @ Y ) )
= ( modulo_modulo @ int @ ( semiring_1_unsigned @ A @ int @ X ) @ ( semiring_1_unsigned @ A @ int @ Y ) ) ) ) ).
% uint_mod
thf(fact_2776_uint__mod__distrib,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [V2: word @ A,W: word @ A] :
( ( semiring_1_unsigned @ A @ int @ ( modulo_modulo @ ( word @ A ) @ V2 @ W ) )
= ( modulo_modulo @ int @ ( semiring_1_unsigned @ A @ int @ V2 ) @ ( semiring_1_unsigned @ A @ int @ W ) ) ) ) ).
% uint_mod_distrib
thf(fact_2777_zmod__int,axiom,
! [A3: nat,B3: nat] :
( ( semiring_1_of_nat @ int @ ( modulo_modulo @ nat @ A3 @ B3 ) )
= ( modulo_modulo @ int @ ( semiring_1_of_nat @ int @ A3 ) @ ( semiring_1_of_nat @ int @ B3 ) ) ) ).
% zmod_int
thf(fact_2778_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_2779_div__less__mono,axiom,
! [A2: nat,B2: nat,N: nat] :
( ( ord_less @ nat @ A2 @ B2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ( modulo_modulo @ nat @ A2 @ N )
= ( zero_zero @ nat ) )
=> ( ( ( modulo_modulo @ nat @ B2 @ N )
= ( zero_zero @ nat ) )
=> ( ord_less @ nat @ ( divide_divide @ nat @ A2 @ N ) @ ( divide_divide @ nat @ B2 @ N ) ) ) ) ) ) ).
% div_less_mono
thf(fact_2780_mod__nat__add,axiom,
! [X: nat,Z: nat,Y: nat] :
( ( ord_less @ nat @ X @ Z )
=> ( ( ord_less @ nat @ Y @ Z )
=> ( ( ( ord_less @ nat @ ( plus_plus @ nat @ X @ Y ) @ Z )
=> ( ( modulo_modulo @ nat @ ( plus_plus @ nat @ X @ Y ) @ Z )
= ( plus_plus @ nat @ X @ Y ) ) )
& ( ~ ( ord_less @ nat @ ( plus_plus @ nat @ X @ Y ) @ Z )
=> ( ( modulo_modulo @ nat @ ( plus_plus @ nat @ X @ Y ) @ Z )
= ( minus_minus @ nat @ ( plus_plus @ nat @ X @ Y ) @ Z ) ) ) ) ) ) ).
% mod_nat_add
thf(fact_2781_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_2782_div__mod__decomp,axiom,
! [A2: nat,N: nat] :
( A2
= ( plus_plus @ nat @ ( times_times @ nat @ ( divide_divide @ nat @ A2 @ N ) @ N ) @ ( modulo_modulo @ nat @ A2 @ N ) ) ) ).
% div_mod_decomp
thf(fact_2783_word__mod__def,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( modulo_modulo @ ( word @ A ) )
= ( ^ [A5: word @ A,B5: word @ A] : ( ring_1_of_int @ ( word @ A ) @ ( modulo_modulo @ int @ ( semiring_1_unsigned @ A @ int @ A5 ) @ ( semiring_1_unsigned @ A @ int @ B5 ) ) ) ) ) ) ).
% word_mod_def
thf(fact_2784_word__mod__div__equality,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: word @ A,B3: word @ A] :
( ( plus_plus @ ( word @ A ) @ ( times_times @ ( word @ A ) @ ( divide_divide @ ( word @ A ) @ N @ B3 ) @ B3 ) @ ( modulo_modulo @ ( word @ A ) @ N @ B3 ) )
= N ) ) ).
% word_mod_div_equality
thf(fact_2785_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_2786_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_2787_mod__lemma,axiom,
! [C2: nat,R3: nat,B3: nat,Q3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ C2 )
=> ( ( ord_less @ nat @ R3 @ B3 )
=> ( ord_less @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ B3 @ ( modulo_modulo @ nat @ Q3 @ C2 ) ) @ R3 ) @ ( times_times @ nat @ B3 @ C2 ) ) ) ) ).
% mod_lemma
thf(fact_2788_split__mod,axiom,
! [P: nat > $o,M: nat,N: nat] :
( ( P @ ( modulo_modulo @ nat @ M @ N ) )
= ( ( ( N
= ( zero_zero @ nat ) )
=> ( P @ M ) )
& ( ( N
!= ( zero_zero @ nat ) )
=> ! [I4: nat,J3: nat] :
( ( ord_less @ nat @ J3 @ N )
=> ( ( M
= ( plus_plus @ nat @ ( times_times @ nat @ N @ I4 ) @ J3 ) )
=> ( P @ J3 ) ) ) ) ) ) ).
% split_mod
thf(fact_2789_diff__mod__le,axiom,
! [A3: nat,D2: nat,B3: nat] :
( ( ord_less @ nat @ A3 @ D2 )
=> ( ( dvd_dvd @ nat @ B3 @ D2 )
=> ( ord_less_eq @ nat @ ( minus_minus @ nat @ A3 @ ( modulo_modulo @ nat @ A3 @ B3 ) ) @ ( minus_minus @ nat @ D2 @ B3 ) ) ) ) ).
% diff_mod_le
thf(fact_2790_mod__nat__eqI,axiom,
! [R3: nat,N: nat,M: nat] :
( ( ord_less @ nat @ R3 @ N )
=> ( ( ord_less_eq @ nat @ R3 @ M )
=> ( ( dvd_dvd @ nat @ N @ ( minus_minus @ nat @ M @ R3 ) )
=> ( ( modulo_modulo @ nat @ M @ N )
= R3 ) ) ) ) ).
% mod_nat_eqI
thf(fact_2791_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_2792_real__of__nat__div__aux,axiom,
! [X: nat,D2: nat] :
( ( divide_divide @ real @ ( semiring_1_of_nat @ real @ X ) @ ( semiring_1_of_nat @ real @ D2 ) )
= ( plus_plus @ real @ ( semiring_1_of_nat @ real @ ( divide_divide @ nat @ X @ D2 ) ) @ ( divide_divide @ real @ ( semiring_1_of_nat @ real @ ( modulo_modulo @ nat @ X @ D2 ) ) @ ( semiring_1_of_nat @ real @ D2 ) ) ) ) ).
% real_of_nat_div_aux
thf(fact_2793_power__mod__div,axiom,
! [X: nat,N: nat,M: nat] :
( ( divide_divide @ nat @ ( modulo_modulo @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) )
= ( modulo_modulo @ nat @ ( divide_divide @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ N @ M ) ) ) ) ).
% power_mod_div
thf(fact_2794_verit__le__mono__div,axiom,
! [A2: nat,B2: nat,N: nat] :
( ( ord_less @ nat @ A2 @ B2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ord_less_eq @ nat
@ ( plus_plus @ nat @ ( divide_divide @ nat @ A2 @ N )
@ ( if @ nat
@ ( ( modulo_modulo @ nat @ B2 @ N )
= ( zero_zero @ nat ) )
@ ( one_one @ nat )
@ ( zero_zero @ nat ) ) )
@ ( divide_divide @ nat @ B2 @ N ) ) ) ) ).
% verit_le_mono_div
thf(fact_2795_choose__one,axiom,
! [N: nat] :
( ( binomial @ N @ ( one_one @ nat ) )
= N ) ).
% choose_one
thf(fact_2796_binomial__eq__0,axiom,
! [N: nat,K: nat] :
( ( ord_less @ nat @ N @ K )
=> ( ( binomial @ N @ K )
= ( zero_zero @ nat ) ) ) ).
% binomial_eq_0
thf(fact_2797_binomial__le__pow,axiom,
! [R3: nat,N: nat] :
( ( ord_less_eq @ nat @ R3 @ N )
=> ( ord_less_eq @ nat @ ( binomial @ N @ R3 ) @ ( power_power @ nat @ N @ R3 ) ) ) ).
% binomial_le_pow
thf(fact_2798_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_2799_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_2800_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_2801_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_2802_ln__series,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ X @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) )
=> ( ( ln_ln @ real @ X )
= ( suminf @ real
@ ^ [N4: nat] : ( times_times @ real @ ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ N4 ) @ ( divide_divide @ real @ ( one_one @ real ) @ ( semiring_1_of_nat @ real @ ( plus_plus @ nat @ N4 @ ( one_one @ nat ) ) ) ) ) @ ( power_power @ real @ ( minus_minus @ real @ X @ ( one_one @ real ) ) @ ( suc @ N4 ) ) ) ) ) ) ) ).
% ln_series
thf(fact_2803_pi__series,axiom,
( ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ ( bit0 @ one2 ) ) ) )
= ( suminf @ real
@ ^ [K3: nat] : ( divide_divide @ real @ ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ K3 ) @ ( one_one @ real ) ) @ ( semiring_1_of_nat @ real @ ( plus_plus @ nat @ ( times_times @ nat @ K3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) ) ) ).
% pi_series
thf(fact_2804_signed__take__bit__rec,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( ( bit_ri4674362597316999326ke_bit @ A )
= ( ^ [N4: nat,A5: A] :
( if @ A
@ ( N4
= ( zero_zero @ nat ) )
@ ( uminus_uminus @ A @ ( modulo_modulo @ A @ A5 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
@ ( plus_plus @ A @ ( modulo_modulo @ A @ A5 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_ri4674362597316999326ke_bit @ A @ ( minus_minus @ nat @ N4 @ ( one_one @ nat ) ) @ ( divide_divide @ A @ A5 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% signed_take_bit_rec
thf(fact_2805_binomial__code,axiom,
( binomial
= ( ^ [N4: nat,K3: nat] : ( if @ nat @ ( ord_less @ nat @ N4 @ K3 ) @ ( zero_zero @ nat ) @ ( if @ nat @ ( ord_less @ nat @ N4 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K3 ) ) @ ( binomial @ N4 @ ( minus_minus @ nat @ N4 @ K3 ) ) @ ( divide_divide @ nat @ ( set_fo6178422350223883121st_nat @ nat @ ( times_times @ nat ) @ ( plus_plus @ nat @ ( minus_minus @ nat @ N4 @ K3 ) @ ( one_one @ nat ) ) @ N4 @ ( one_one @ nat ) ) @ ( semiring_char_0_fact @ nat @ K3 ) ) ) ) ) ) ).
% binomial_code
thf(fact_2806_uint32_Osize__eq,axiom,
( ( size_size @ uint32 )
= ( ^ [P6: uint32] : ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% uint32.size_eq
thf(fact_2807_add__scale__eq__noteq,axiom,
! [A: $tType] :
( ( semiri1453513574482234551roduct @ A )
=> ! [R3: A,A3: A,B3: A,C2: A,D2: A] :
( ( R3
!= ( zero_zero @ A ) )
=> ( ( ( A3 = B3 )
& ( C2 != D2 ) )
=> ( ( plus_plus @ A @ A3 @ ( times_times @ A @ R3 @ C2 ) )
!= ( plus_plus @ A @ B3 @ ( times_times @ A @ R3 @ D2 ) ) ) ) ) ) ).
% add_scale_eq_noteq
thf(fact_2808_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_2809_nat_Oinject,axiom,
! [X22: nat,Y22: nat] :
( ( ( suc @ X22 )
= ( suc @ Y22 ) )
= ( X22 = Y22 ) ) ).
% nat.inject
thf(fact_2810_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_2811_Suc__mono,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ N )
=> ( ord_less @ nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_2812_lessI,axiom,
! [N: nat] : ( ord_less @ nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_2813_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_2814_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_2815_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_2816_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_2817_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_2818_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_2819_power__Suc0__right,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A3: A] :
( ( power_power @ A @ A3 @ ( suc @ ( zero_zero @ nat ) ) )
= A3 ) ) ).
% power_Suc0_right
thf(fact_2820_zero__less__Suc,axiom,
! [N: nat] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_2821_less__Suc0,axiom,
! [N: nat] :
( ( ord_less @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) )
= ( N
= ( zero_zero @ nat ) ) ) ).
% less_Suc0
thf(fact_2822_div__by__Suc__0,axiom,
! [M: nat] :
( ( divide_divide @ nat @ M @ ( suc @ ( zero_zero @ nat ) ) )
= M ) ).
% div_by_Suc_0
thf(fact_2823_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_2824_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_2825_power__Suc__0,axiom,
! [N: nat] :
( ( power_power @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% power_Suc_0
thf(fact_2826_nat__power__eq__Suc__0__iff,axiom,
! [X: nat,M: nat] :
( ( ( power_power @ nat @ X @ M )
= ( suc @ ( zero_zero @ nat ) ) )
= ( ( M
= ( zero_zero @ nat ) )
| ( X
= ( suc @ ( zero_zero @ nat ) ) ) ) ) ).
% nat_power_eq_Suc_0_iff
thf(fact_2827_diff__Suc__1,axiom,
! [N: nat] :
( ( minus_minus @ nat @ ( suc @ N ) @ ( one_one @ nat ) )
= N ) ).
% diff_Suc_1
thf(fact_2828_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_2829_dvd__1__left,axiom,
! [K: nat] : ( dvd_dvd @ nat @ ( suc @ ( zero_zero @ nat ) ) @ K ) ).
% dvd_1_left
thf(fact_2830_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_2831_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_2832_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_2833_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_2834_mod__by__Suc__0,axiom,
! [M: nat] :
( ( modulo_modulo @ nat @ M @ ( suc @ ( zero_zero @ nat ) ) )
= ( zero_zero @ nat ) ) ).
% mod_by_Suc_0
thf(fact_2835_fact__0,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ( ( semiring_char_0_fact @ A @ ( zero_zero @ nat ) )
= ( one_one @ A ) ) ) ).
% fact_0
thf(fact_2836_fact__1,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ( ( semiring_char_0_fact @ A @ ( one_one @ nat ) )
= ( one_one @ A ) ) ) ).
% fact_1
thf(fact_2837_binomial__1,axiom,
! [N: nat] :
( ( binomial @ N @ ( suc @ ( zero_zero @ nat ) ) )
= N ) ).
% binomial_1
thf(fact_2838_binomial__0__Suc,axiom,
! [K: nat] :
( ( binomial @ ( zero_zero @ nat ) @ ( suc @ K ) )
= ( zero_zero @ nat ) ) ).
% binomial_0_Suc
thf(fact_2839_pochhammer__Suc0,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A3: A] :
( ( comm_s3205402744901411588hammer @ A @ A3 @ ( suc @ ( zero_zero @ nat ) ) )
= A3 ) ) ).
% pochhammer_Suc0
thf(fact_2840_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_2841_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_2842_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_2843_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_2844_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_2845_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_2846_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_2847_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_2848_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_2849_Suc__numeral,axiom,
! [N: num] :
( ( suc @ ( numeral_numeral @ nat @ N ) )
= ( numeral_numeral @ nat @ ( plus_plus @ num @ N @ one2 ) ) ) ).
% Suc_numeral
thf(fact_2850_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_2851_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_2852_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_2853_Suc__1,axiom,
( ( suc @ ( one_one @ nat ) )
= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ).
% Suc_1
thf(fact_2854_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_2855_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_2856_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_2857_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_2858_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_2859_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_2860_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_2861_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_2862_Suc__unat__minus__one,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] :
( ( X
!= ( zero_zero @ ( word @ A ) ) )
=> ( ( suc @ ( semiring_1_unsigned @ A @ nat @ ( minus_minus @ ( word @ A ) @ X @ ( one_one @ ( word @ A ) ) ) ) )
= ( semiring_1_unsigned @ A @ nat @ X ) ) ) ) ).
% Suc_unat_minus_one
thf(fact_2863_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_2864_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_2865_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_2866_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_2867_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_2868_shiftl__of__Suc,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A,N: nat] :
( ( bit_Sh4282982442137083160shiftl @ A @ A3 @ ( suc @ N ) )
= ( bit_Sh4282982442137083160shiftl @ A @ ( times_times @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ N ) ) ) ).
% shiftl_of_Suc
thf(fact_2869_set__decode__Suc,axiom,
! [N: nat,X: nat] :
( ( member @ nat @ ( suc @ N ) @ ( nat_set_decode @ X ) )
= ( member @ nat @ N @ ( nat_set_decode @ ( divide_divide @ nat @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% set_decode_Suc
thf(fact_2870_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_2871_signed__take__bit__0,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [A3: A] :
( ( bit_ri4674362597316999326ke_bit @ A @ ( zero_zero @ nat ) @ A3 )
= ( uminus_uminus @ A @ ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% signed_take_bit_0
thf(fact_2872_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_2873_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_2874_Suc__inject,axiom,
! [X: nat,Y: nat] :
( ( ( suc @ X )
= ( suc @ Y ) )
=> ( X = Y ) ) ).
% Suc_inject
thf(fact_2875_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_2876_pi__neq__zero,axiom,
( pi
!= ( zero_zero @ real ) ) ).
% pi_neq_zero
thf(fact_2877_list__decode_Ocases,axiom,
! [X: nat] :
( ( X
!= ( zero_zero @ nat ) )
=> ~ ! [N2: nat] :
( X
!= ( suc @ N2 ) ) ) ).
% list_decode.cases
thf(fact_2878_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p_Ocases,axiom,
! [X: nat] :
( ( X
!= ( zero_zero @ nat ) )
=> ( ( X
!= ( suc @ ( zero_zero @ nat ) ) )
=> ~ ! [Va2: nat] :
( X
!= ( suc @ ( suc @ Va2 ) ) ) ) ) ).
% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d\<^sub>u\<^sub>p.cases
thf(fact_2879_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_2880_not0__implies__Suc,axiom,
! [N: nat] :
( ( N
!= ( zero_zero @ nat ) )
=> ? [M4: nat] :
( N
= ( suc @ M4 ) ) ) ).
% not0_implies_Suc
thf(fact_2881_Zero__not__Suc,axiom,
! [M: nat] :
( ( zero_zero @ nat )
!= ( suc @ M ) ) ).
% Zero_not_Suc
thf(fact_2882_Zero__neq__Suc,axiom,
! [M: nat] :
( ( zero_zero @ nat )
!= ( suc @ M ) ) ).
% Zero_neq_Suc
thf(fact_2883_Suc__neq__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= ( zero_zero @ nat ) ) ).
% Suc_neq_Zero
thf(fact_2884_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_2885_diff__induct,axiom,
! [P: nat > nat > $o,M: nat,N: nat] :
( ! [X3: nat] : ( P @ X3 @ ( zero_zero @ nat ) )
=> ( ! [Y4: nat] : ( P @ ( zero_zero @ nat ) @ ( suc @ Y4 ) )
=> ( ! [X3: nat,Y4: nat] :
( ( P @ X3 @ Y4 )
=> ( P @ ( suc @ X3 ) @ ( suc @ Y4 ) ) )
=> ( P @ M @ N ) ) ) ) ).
% diff_induct
thf(fact_2886_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_2887_old_Onat_Oexhaust,axiom,
! [Y: nat] :
( ( Y
!= ( zero_zero @ nat ) )
=> ~ ! [Nat3: nat] :
( Y
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_2888_nat_OdiscI,axiom,
! [Nat: nat,X22: nat] :
( ( Nat
= ( suc @ X22 ) )
=> ( Nat
!= ( zero_zero @ nat ) ) ) ).
% nat.discI
thf(fact_2889_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( ( zero_zero @ nat )
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_2890_old_Onat_Odistinct_I2_J,axiom,
! [Nat4: nat] :
( ( suc @ Nat4 )
!= ( zero_zero @ nat ) ) ).
% old.nat.distinct(2)
thf(fact_2891_nat_Odistinct_I1_J,axiom,
! [X22: nat] :
( ( zero_zero @ nat )
!= ( suc @ X22 ) ) ).
% nat.distinct(1)
thf(fact_2892_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_2893_strict__inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less @ nat @ I @ J )
=> ( ! [I3: nat] :
( ( J
= ( suc @ I3 ) )
=> ( P @ I3 ) )
=> ( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ J )
=> ( ( P @ ( suc @ I3 ) )
=> ( P @ I3 ) ) )
=> ( P @ I ) ) ) ) ).
% strict_inc_induct
thf(fact_2894_less__Suc__induct,axiom,
! [I: nat,J: nat,P: nat > nat > $o] :
( ( ord_less @ nat @ I @ J )
=> ( ! [I3: nat] : ( P @ I3 @ ( suc @ I3 ) )
=> ( ! [I3: nat,J2: nat,K2: nat] :
( ( ord_less @ nat @ I3 @ J2 )
=> ( ( ord_less @ nat @ J2 @ K2 )
=> ( ( P @ I3 @ J2 )
=> ( ( P @ J2 @ K2 )
=> ( P @ I3 @ K2 ) ) ) ) )
=> ( P @ I @ J ) ) ) ) ).
% less_Suc_induct
thf(fact_2895_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_2896_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_2897_less__antisym,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less @ nat @ N @ M )
=> ( ( ord_less @ nat @ N @ ( suc @ M ) )
=> ( M = N ) ) ) ).
% less_antisym
thf(fact_2898_Suc__less__eq2,axiom,
! [N: nat,M: nat] :
( ( ord_less @ nat @ ( suc @ N ) @ M )
= ( ? [M5: nat] :
( ( M
= ( suc @ M5 ) )
& ( ord_less @ nat @ N @ M5 ) ) ) ) ).
% Suc_less_eq2
thf(fact_2899_Nat_OAll__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( suc @ N ) )
=> ( P @ I4 ) ) )
= ( ( P @ N )
& ! [I4: nat] :
( ( ord_less @ nat @ I4 @ N )
=> ( P @ I4 ) ) ) ) ).
% Nat.All_less_Suc
thf(fact_2900_not__less__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less @ nat @ M @ N ) )
= ( ord_less @ nat @ N @ ( suc @ M ) ) ) ).
% not_less_eq
thf(fact_2901_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_2902_Ex__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I4: nat] :
( ( ord_less @ nat @ I4 @ ( suc @ N ) )
& ( P @ I4 ) ) )
= ( ( P @ N )
| ? [I4: nat] :
( ( ord_less @ nat @ I4 @ N )
& ( P @ I4 ) ) ) ) ).
% Ex_less_Suc
thf(fact_2903_less__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ N )
=> ( ord_less @ nat @ M @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_2904_less__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less @ nat @ M @ N )
=> ( M = N ) ) ) ).
% less_SucE
thf(fact_2905_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_2906_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_2907_Suc__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ ( suc @ M ) @ N )
=> ( ord_less @ nat @ M @ N ) ) ).
% Suc_lessD
thf(fact_2908_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_2909_nat__arith_Osuc1,axiom,
! [A2: nat,K: nat,A3: nat] :
( ( A2
= ( plus_plus @ nat @ K @ A3 ) )
=> ( ( suc @ A2 )
= ( plus_plus @ nat @ K @ ( suc @ A3 ) ) ) ) ).
% nat_arith.suc1
thf(fact_2910_add__Suc,axiom,
! [M: nat,N: nat] :
( ( plus_plus @ nat @ ( suc @ M ) @ N )
= ( suc @ ( plus_plus @ nat @ M @ N ) ) ) ).
% add_Suc
thf(fact_2911_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_2912_Suc__leD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ ( suc @ M ) @ N )
=> ( ord_less_eq @ nat @ M @ N ) ) ).
% Suc_leD
thf(fact_2913_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_2914_le__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ord_less_eq @ nat @ M @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_2915_Suc__le__D,axiom,
! [N: nat,M6: nat] :
( ( ord_less_eq @ nat @ ( suc @ N ) @ M6 )
=> ? [M4: nat] :
( M6
= ( suc @ M4 ) ) ) ).
% Suc_le_D
thf(fact_2916_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_2917_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq @ nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_2918_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_2919_full__nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M2: nat] :
( ( ord_less_eq @ nat @ ( suc @ M2 ) @ N2 )
=> ( P @ M2 ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% full_nat_induct
thf(fact_2920_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_2921_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,Y4: nat,Z3: nat] :
( ( R @ X3 @ Y4 )
=> ( ( R @ Y4 @ Z3 )
=> ( R @ X3 @ Z3 ) ) )
=> ( ! [N2: nat] : ( R @ N2 @ ( suc @ N2 ) )
=> ( R @ M @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_2922_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_2923_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_2924_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_2925_summable__Suc__iff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: nat > A] :
( ( summable @ A
@ ^ [N4: nat] : ( F3 @ ( suc @ N4 ) ) )
= ( summable @ A @ F3 ) ) ) ).
% summable_Suc_iff
thf(fact_2926_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_2927_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_2928_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_2929_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_2930_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_2931_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_2932_pi__gt__zero,axiom,
ord_less @ real @ ( zero_zero @ real ) @ pi ).
% pi_gt_zero
thf(fact_2933_pi__not__less__zero,axiom,
~ ( ord_less @ real @ pi @ ( zero_zero @ real ) ) ).
% pi_not_less_zero
thf(fact_2934_pi__ge__zero,axiom,
ord_less_eq @ real @ ( zero_zero @ real ) @ pi ).
% pi_ge_zero
thf(fact_2935_lift__Suc__mono__less__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [F3: nat > A,N: nat,M: nat] :
( ! [N2: nat] : ( ord_less @ A @ ( F3 @ N2 ) @ ( F3 @ ( suc @ N2 ) ) )
=> ( ( ord_less @ A @ ( F3 @ N ) @ ( F3 @ M ) )
= ( ord_less @ nat @ N @ M ) ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_2936_lift__Suc__mono__less,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [F3: nat > A,N: nat,N7: nat] :
( ! [N2: nat] : ( ord_less @ A @ ( F3 @ N2 ) @ ( F3 @ ( suc @ N2 ) ) )
=> ( ( ord_less @ nat @ N @ N7 )
=> ( ord_less @ A @ ( F3 @ N ) @ ( F3 @ N7 ) ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_2937_power__Suc2,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A3: A,N: nat] :
( ( power_power @ A @ A3 @ ( suc @ N ) )
= ( times_times @ A @ ( power_power @ A @ A3 @ N ) @ A3 ) ) ) ).
% power_Suc2
thf(fact_2938_power__Suc,axiom,
! [A: $tType] :
( ( power @ A )
=> ! [A3: A,N: nat] :
( ( power_power @ A @ A3 @ ( suc @ N ) )
= ( times_times @ A @ A3 @ ( power_power @ A @ A3 @ N ) ) ) ) ).
% power_Suc
thf(fact_2939_lift__Suc__antimono__le,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [F3: nat > A,N: nat,N7: nat] :
( ! [N2: nat] : ( ord_less_eq @ A @ ( F3 @ ( suc @ N2 ) ) @ ( F3 @ N2 ) )
=> ( ( ord_less_eq @ nat @ N @ N7 )
=> ( ord_less_eq @ A @ ( F3 @ N7 ) @ ( F3 @ N ) ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_2940_lift__Suc__mono__le,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [F3: nat > A,N: nat,N7: nat] :
( ! [N2: nat] : ( ord_less_eq @ A @ ( F3 @ N2 ) @ ( F3 @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq @ nat @ N @ N7 )
=> ( ord_less_eq @ A @ ( F3 @ N ) @ ( F3 @ N7 ) ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_2941_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_2942_Comparator__Generator_OAll__less__Suc,axiom,
! [X: nat,P: nat > $o] :
( ( ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( suc @ X ) )
=> ( P @ I4 ) ) )
= ( ( P @ ( zero_zero @ nat ) )
& ! [I4: nat] :
( ( ord_less @ nat @ I4 @ X )
=> ( P @ ( suc @ I4 ) ) ) ) ) ).
% Comparator_Generator.All_less_Suc
thf(fact_2943_forall__finite_I2_J,axiom,
! [P: nat > $o] :
( ( ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( suc @ ( zero_zero @ nat ) ) )
=> ( P @ I4 ) ) )
= ( P @ ( zero_zero @ nat ) ) ) ).
% forall_finite(2)
thf(fact_2944_forall__finite_I3_J,axiom,
! [X: nat,P: nat > $o] :
( ( ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( suc @ ( suc @ X ) ) )
=> ( P @ I4 ) ) )
= ( ( P @ ( zero_zero @ nat ) )
& ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( suc @ X ) )
=> ( P @ ( suc @ I4 ) ) ) ) ) ).
% forall_finite(3)
thf(fact_2945_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_2946_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ? [M4: nat] :
( N
= ( suc @ M4 ) ) ) ).
% gr0_implies_Suc
thf(fact_2947_All__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( suc @ N ) )
=> ( P @ I4 ) ) )
= ( ( P @ ( zero_zero @ nat ) )
& ! [I4: nat] :
( ( ord_less @ nat @ I4 @ N )
=> ( P @ ( suc @ I4 ) ) ) ) ) ).
% All_less_Suc2
thf(fact_2948_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
= ( ? [M3: nat] :
( N
= ( suc @ M3 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_2949_Ex__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I4: nat] :
( ( ord_less @ nat @ I4 @ ( suc @ N ) )
& ( P @ I4 ) ) )
= ( ( P @ ( zero_zero @ nat ) )
| ? [I4: nat] :
( ( ord_less @ nat @ I4 @ N )
& ( P @ ( suc @ I4 ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_2950_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_2951_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_2952_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_2953_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_2954_One__nat__def,axiom,
( ( one_one @ nat )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% One_nat_def
thf(fact_2955_less__natE,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ N )
=> ~ ! [Q6: nat] :
( N
!= ( suc @ ( plus_plus @ nat @ M @ Q6 ) ) ) ) ).
% less_natE
thf(fact_2956_less__add__Suc1,axiom,
! [I: nat,M: nat] : ( ord_less @ nat @ I @ ( suc @ ( plus_plus @ nat @ I @ M ) ) ) ).
% less_add_Suc1
thf(fact_2957_less__add__Suc2,axiom,
! [I: nat,M: nat] : ( ord_less @ nat @ I @ ( suc @ ( plus_plus @ nat @ M @ I ) ) ) ).
% less_add_Suc2
thf(fact_2958_less__iff__Suc__add,axiom,
( ( ord_less @ nat )
= ( ^ [M3: nat,N4: nat] :
? [K3: nat] :
( N4
= ( suc @ ( plus_plus @ nat @ M3 @ K3 ) ) ) ) ) ).
% less_iff_Suc_add
thf(fact_2959_less__imp__Suc__add,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ N )
=> ? [K2: nat] :
( N
= ( suc @ ( plus_plus @ nat @ M @ K2 ) ) ) ) ).
% less_imp_Suc_add
thf(fact_2960_Suc__leI,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ N )
=> ( ord_less_eq @ nat @ ( suc @ M ) @ N ) ) ).
% Suc_leI
thf(fact_2961_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_2962_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_2963_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_2964_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_2965_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_2966_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_2967_less__eq__Suc__le,axiom,
( ( ord_less @ nat )
= ( ^ [N4: nat] : ( ord_less_eq @ nat @ ( suc @ N4 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_2968_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_2969_nat__in__between__eq_I1_J,axiom,
! [A3: nat,B3: nat] :
( ( ( ord_less @ nat @ A3 @ B3 )
& ( ord_less_eq @ nat @ B3 @ ( suc @ A3 ) ) )
= ( B3
= ( suc @ A3 ) ) ) ).
% nat_in_between_eq(1)
thf(fact_2970_nat__in__between__eq_I2_J,axiom,
! [A3: nat,B3: nat] :
( ( ( ord_less_eq @ nat @ A3 @ B3 )
& ( ord_less @ nat @ B3 @ ( suc @ A3 ) ) )
= ( B3 = A3 ) ) ).
% nat_in_between_eq(2)
thf(fact_2971_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_2972_diff__less__Suc,axiom,
! [M: nat,N: nat] : ( ord_less @ nat @ ( minus_minus @ nat @ M @ N ) @ ( suc @ M ) ) ).
% diff_less_Suc
thf(fact_2973_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_2974_Suc__eq__plus1__left,axiom,
( suc
= ( plus_plus @ nat @ ( one_one @ nat ) ) ) ).
% Suc_eq_plus1_left
thf(fact_2975_plus__1__eq__Suc,axiom,
( ( plus_plus @ nat @ ( one_one @ nat ) )
= suc ) ).
% plus_1_eq_Suc
thf(fact_2976_Suc__eq__plus1,axiom,
( suc
= ( ^ [N4: nat] : ( plus_plus @ nat @ N4 @ ( one_one @ nat ) ) ) ) ).
% Suc_eq_plus1
thf(fact_2977_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_2978_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_2979_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_2980_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_2981_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_2982_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_2983_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_2984_mod__induct,axiom,
! [P: nat > $o,N: nat,P4: nat,M: nat] :
( ( P @ N )
=> ( ( ord_less @ nat @ N @ P4 )
=> ( ( ord_less @ nat @ M @ P4 )
=> ( ! [N2: nat] :
( ( ord_less @ nat @ N2 @ P4 )
=> ( ( P @ N2 )
=> ( P @ ( modulo_modulo @ nat @ ( suc @ N2 ) @ P4 ) ) ) )
=> ( P @ M ) ) ) ) ) ).
% mod_induct
thf(fact_2985_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_2986_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_2987_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_2988_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_2989_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_2990_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_2991_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_2992_power__inject__base,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,N: nat,B3: A] :
( ( ( power_power @ A @ A3 @ ( suc @ N ) )
= ( power_power @ A @ B3 @ ( suc @ N ) ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B3 )
=> ( A3 = B3 ) ) ) ) ) ).
% power_inject_base
thf(fact_2993_power__le__imp__le__base,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,N: nat,B3: A] :
( ( ord_less_eq @ A @ ( power_power @ A @ A3 @ ( suc @ N ) ) @ ( power_power @ A @ B3 @ ( suc @ N ) ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B3 )
=> ( ord_less_eq @ A @ A3 @ B3 ) ) ) ) ).
% power_le_imp_le_base
thf(fact_2994_power__gt1,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,N: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ A3 )
=> ( ord_less @ A @ ( one_one @ A ) @ ( power_power @ A @ A3 @ ( suc @ N ) ) ) ) ) ).
% power_gt1
thf(fact_2995_numeral__1__eq__Suc__0,axiom,
( ( numeral_numeral @ nat @ one2 )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% numeral_1_eq_Suc_0
thf(fact_2996_ex__least__nat__less,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ ( zero_zero @ nat ) )
=> ? [K2: nat] :
( ( ord_less @ nat @ K2 @ N )
& ! [I2: nat] :
( ( ord_less_eq @ nat @ I2 @ K2 )
=> ~ ( P @ I2 ) )
& ( P @ ( suc @ K2 ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_2997_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_2998_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_2999_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_3000_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_3001_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_3002_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_3003_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_3004_realpow__pos__nth2,axiom,
! [A3: real,N: nat] :
( ( ord_less @ real @ ( zero_zero @ real ) @ A3 )
=> ? [R2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R2 )
& ( ( power_power @ real @ R2 @ ( suc @ N ) )
= A3 ) ) ) ).
% realpow_pos_nth2
thf(fact_3005_unat__eq__1,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] :
( ( ( semiring_1_unsigned @ A @ nat @ X )
= ( suc @ ( zero_zero @ nat ) ) )
= ( X
= ( one_one @ ( word @ A ) ) ) ) ) ).
% unat_eq_1
thf(fact_3006_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_3007_int__ops_I4_J,axiom,
! [A3: nat] :
( ( semiring_1_of_nat @ int @ ( suc @ A3 ) )
= ( plus_plus @ int @ ( semiring_1_of_nat @ int @ A3 ) @ ( one_one @ int ) ) ) ).
% int_ops(4)
thf(fact_3008_zless__iff__Suc__zadd,axiom,
( ( ord_less @ int )
= ( ^ [W2: int,Z4: int] :
? [N4: nat] :
( Z4
= ( plus_plus @ int @ W2 @ ( semiring_1_of_nat @ int @ ( suc @ N4 ) ) ) ) ) ) ).
% zless_iff_Suc_zadd
thf(fact_3009_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_3010_powser__split__head_I3_J,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V8999393235501362500lgebra @ A ) )
=> ! [F3: nat > A,Z: A] :
( ( summable @ A
@ ^ [N4: nat] : ( times_times @ A @ ( F3 @ N4 ) @ ( power_power @ A @ Z @ N4 ) ) )
=> ( summable @ A
@ ^ [N4: nat] : ( times_times @ A @ ( F3 @ ( suc @ N4 ) ) @ ( power_power @ A @ Z @ N4 ) ) ) ) ) ).
% powser_split_head(3)
thf(fact_3011_summable__powser__split__head,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [F3: nat > A,Z: A] :
( ( summable @ A
@ ^ [N4: nat] : ( times_times @ A @ ( F3 @ ( suc @ N4 ) ) @ ( power_power @ A @ Z @ N4 ) ) )
= ( summable @ A
@ ^ [N4: nat] : ( times_times @ A @ ( F3 @ N4 ) @ ( power_power @ A @ Z @ N4 ) ) ) ) ) ).
% summable_powser_split_head
thf(fact_3012_signed__take__bit__Suc,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N: nat,A3: A] :
( ( bit_ri4674362597316999326ke_bit @ A @ ( suc @ N ) @ A3 )
= ( plus_plus @ A @ ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_ri4674362597316999326ke_bit @ A @ N @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% signed_take_bit_Suc
thf(fact_3013_fact__div__fact__le__pow,axiom,
! [R3: nat,N: nat] :
( ( ord_less_eq @ nat @ R3 @ N )
=> ( ord_less_eq @ nat @ ( divide_divide @ nat @ ( semiring_char_0_fact @ nat @ N ) @ ( semiring_char_0_fact @ nat @ ( minus_minus @ nat @ N @ R3 ) ) ) @ ( power_power @ nat @ N @ R3 ) ) ) ).
% fact_div_fact_le_pow
thf(fact_3014_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_3015_pi__less__4,axiom,
ord_less @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ ( bit0 @ one2 ) ) ) ).
% pi_less_4
thf(fact_3016_pi__ge__two,axiom,
ord_less_eq @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ).
% pi_ge_two
thf(fact_3017_power__Suc__le__self,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,N: nat] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ A3 @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( power_power @ A @ A3 @ ( suc @ N ) ) @ A3 ) ) ) ) ).
% power_Suc_le_self
thf(fact_3018_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_3019_power__Suc__less__one,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,N: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ A3 @ ( one_one @ A ) )
=> ( ord_less @ A @ ( power_power @ A @ A3 @ ( suc @ N ) ) @ ( one_one @ A ) ) ) ) ) ).
% power_Suc_less_one
thf(fact_3020_numeral__2__eq__2,axiom,
( ( numeral_numeral @ nat @ ( bit0 @ one2 ) )
= ( suc @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% numeral_2_eq_2
thf(fact_3021_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_3022_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_3023_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_3024_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_3025_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_3026_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_3027_div__if,axiom,
( ( divide_divide @ nat )
= ( ^ [M3: nat,N4: nat] :
( if @ nat
@ ( ( ord_less @ nat @ M3 @ N4 )
| ( N4
= ( zero_zero @ nat ) ) )
@ ( zero_zero @ nat )
@ ( suc @ ( divide_divide @ nat @ ( minus_minus @ nat @ M3 @ N4 ) @ N4 ) ) ) ) ) ).
% div_if
thf(fact_3028_add__eq__if,axiom,
( ( plus_plus @ nat )
= ( ^ [M3: nat,N4: nat] :
( if @ nat
@ ( M3
= ( zero_zero @ nat ) )
@ N4
@ ( suc @ ( plus_plus @ nat @ ( minus_minus @ nat @ M3 @ ( one_one @ nat ) ) @ N4 ) ) ) ) ) ).
% add_eq_if
thf(fact_3029_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_3030_div__nat__eqI,axiom,
! [N: nat,Q3: nat,M: nat] :
( ( ord_less_eq @ nat @ ( times_times @ nat @ N @ Q3 ) @ M )
=> ( ( ord_less @ nat @ M @ ( times_times @ nat @ N @ ( suc @ Q3 ) ) )
=> ( ( divide_divide @ nat @ M @ N )
= Q3 ) ) ) ).
% div_nat_eqI
thf(fact_3031_pochhammer__rec,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A3: A,N: nat] :
( ( comm_s3205402744901411588hammer @ A @ A3 @ ( suc @ N ) )
= ( times_times @ A @ A3 @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) @ N ) ) ) ) ).
% pochhammer_rec
thf(fact_3032_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_3033_pochhammer__Suc,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A3: A,N: nat] :
( ( comm_s3205402744901411588hammer @ A @ A3 @ ( suc @ N ) )
= ( times_times @ A @ ( comm_s3205402744901411588hammer @ A @ A3 @ N ) @ ( plus_plus @ A @ A3 @ ( semiring_1_of_nat @ A @ N ) ) ) ) ) ).
% pochhammer_Suc
thf(fact_3034_Suc__nat__number__of__add,axiom,
! [V2: num,N: nat] :
( ( suc @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ V2 ) @ N ) )
= ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( plus_plus @ num @ V2 @ one2 ) ) @ N ) ) ).
% Suc_nat_number_of_add
thf(fact_3035_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_3036_negD,axiom,
! [X: int] :
( ( ord_less @ int @ X @ ( zero_zero @ int ) )
=> ? [N2: nat] :
( X
= ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ ( suc @ N2 ) ) ) ) ) ).
% negD
thf(fact_3037_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_3038_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_3039_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_3040_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_3041_Suc__unat__diff__1,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ ( one_one @ ( word @ A ) ) @ X )
=> ( ( suc @ ( semiring_1_unsigned @ A @ nat @ ( minus_minus @ ( word @ A ) @ X @ ( one_one @ ( word @ A ) ) ) ) )
= ( semiring_1_unsigned @ A @ nat @ X ) ) ) ) ).
% Suc_unat_diff_1
thf(fact_3042_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_3043_suminf__split__head,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: nat > A] :
( ( summable @ A @ F3 )
=> ( ( suminf @ A
@ ^ [N4: nat] : ( F3 @ ( suc @ N4 ) ) )
= ( minus_minus @ A @ ( suminf @ A @ F3 ) @ ( F3 @ ( zero_zero @ nat ) ) ) ) ) ) ).
% suminf_split_head
thf(fact_3044_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_3045_num_Osize_I5_J,axiom,
! [X22: num] :
( ( size_size @ num @ ( bit0 @ X22 ) )
= ( plus_plus @ nat @ ( size_size @ num @ X22 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% num.size(5)
thf(fact_3046_even__signed__take__bit__iff,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [M: nat,A3: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_ri4674362597316999326ke_bit @ A @ M @ A3 ) )
= ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) ) ) ).
% even_signed_take_bit_iff
thf(fact_3047_pi__half__neq__zero,axiom,
( ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) )
!= ( zero_zero @ real ) ) ).
% pi_half_neq_zero
thf(fact_3048_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_3049_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_3050_nat__approx__posE,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [E2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ E2 )
=> ~ ! [N2: nat] :
~ ( ord_less @ A @ ( divide_divide @ A @ ( one_one @ A ) @ ( semiring_1_of_nat @ A @ ( suc @ N2 ) ) ) @ E2 ) ) ) ).
% nat_approx_posE
thf(fact_3051_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_3052_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_3053_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_3054_split__div_H,axiom,
! [P: nat > $o,M: nat,N: nat] :
( ( P @ ( divide_divide @ nat @ M @ N ) )
= ( ( ( N
= ( zero_zero @ nat ) )
& ( P @ ( zero_zero @ nat ) ) )
| ? [Q5: nat] :
( ( ord_less_eq @ nat @ ( times_times @ nat @ N @ Q5 ) @ M )
& ( ord_less @ nat @ M @ ( times_times @ nat @ N @ ( suc @ Q5 ) ) )
& ( P @ Q5 ) ) ) ) ).
% split_div'
thf(fact_3055_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_3056_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_3057_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_3058_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_3059_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_3060_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_3061_fact__code,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ( ( semiring_char_0_fact @ A )
= ( ^ [N4: nat] : ( semiring_1_of_nat @ A @ ( set_fo6178422350223883121st_nat @ nat @ ( times_times @ nat ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N4 @ ( one_one @ nat ) ) ) ) ) ) ).
% fact_code
thf(fact_3062_fact__num__eq__if,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ( ( semiring_char_0_fact @ A )
= ( ^ [M3: nat] :
( if @ A
@ ( M3
= ( zero_zero @ nat ) )
@ ( one_one @ A )
@ ( times_times @ A @ ( semiring_1_of_nat @ A @ M3 ) @ ( semiring_char_0_fact @ A @ ( minus_minus @ nat @ M3 @ ( one_one @ nat ) ) ) ) ) ) ) ) ).
% fact_num_eq_if
thf(fact_3063_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_3064_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_3065_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_3066_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_3067_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_3068_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_3069_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_3070_power__odd__eq,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A3: A,N: nat] :
( ( power_power @ A @ A3 @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) )
= ( times_times @ A @ A3 @ ( power_power @ A @ ( power_power @ A @ A3 @ N ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% power_odd_eq
thf(fact_3071_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_3072_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_3073_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_3074_arctan__ubound,axiom,
! [Y: real] : ( ord_less @ real @ ( arctan @ Y ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ).
% arctan_ubound
thf(fact_3075_arctan__one,axiom,
( ( arctan @ ( one_one @ real ) )
= ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ).
% arctan_one
thf(fact_3076_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_3077_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_3078_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_3079_nat__intermed__int__val,axiom,
! [M: nat,N: nat,F3: nat > int,K: int] :
( ! [I3: nat] :
( ( ( ord_less_eq @ nat @ M @ I3 )
& ( ord_less @ nat @ I3 @ N ) )
=> ( ord_less_eq @ int @ ( abs_abs @ int @ ( minus_minus @ int @ ( F3 @ ( suc @ I3 ) ) @ ( F3 @ I3 ) ) ) @ ( one_one @ int ) ) )
=> ( ( ord_less_eq @ nat @ M @ N )
=> ( ( ord_less_eq @ int @ ( F3 @ M ) @ K )
=> ( ( ord_less_eq @ int @ K @ ( F3 @ N ) )
=> ? [I3: nat] :
( ( ord_less_eq @ nat @ M @ I3 )
& ( ord_less_eq @ nat @ I3 @ N )
& ( ( F3 @ I3 )
= K ) ) ) ) ) ) ).
% nat_intermed_int_val
thf(fact_3080_add__0__iff,axiom,
! [A: $tType] :
( ( semiri1453513574482234551roduct @ A )
=> ! [B3: A,A3: A] :
( ( B3
= ( plus_plus @ A @ B3 @ A3 ) )
= ( A3
= ( zero_zero @ A ) ) ) ) ).
% add_0_iff
thf(fact_3081_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_3082_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_3083_powser__split__head_I1_J,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V8999393235501362500lgebra @ A ) )
=> ! [F3: nat > A,Z: A] :
( ( summable @ A
@ ^ [N4: nat] : ( times_times @ A @ ( F3 @ N4 ) @ ( power_power @ A @ Z @ N4 ) ) )
=> ( ( suminf @ A
@ ^ [N4: nat] : ( times_times @ A @ ( F3 @ N4 ) @ ( power_power @ A @ Z @ N4 ) ) )
= ( plus_plus @ A @ ( F3 @ ( zero_zero @ nat ) )
@ ( times_times @ A
@ ( suminf @ A
@ ^ [N4: nat] : ( times_times @ A @ ( F3 @ ( suc @ N4 ) ) @ ( power_power @ A @ Z @ N4 ) ) )
@ Z ) ) ) ) ) ).
% powser_split_head(1)
thf(fact_3084_powser__split__head_I2_J,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V8999393235501362500lgebra @ A ) )
=> ! [F3: nat > A,Z: A] :
( ( summable @ A
@ ^ [N4: nat] : ( times_times @ A @ ( F3 @ N4 ) @ ( power_power @ A @ Z @ N4 ) ) )
=> ( ( times_times @ A
@ ( suminf @ A
@ ^ [N4: nat] : ( times_times @ A @ ( F3 @ ( suc @ N4 ) ) @ ( power_power @ A @ Z @ N4 ) ) )
@ Z )
= ( minus_minus @ A
@ ( suminf @ A
@ ^ [N4: nat] : ( times_times @ A @ ( F3 @ N4 ) @ ( power_power @ A @ Z @ N4 ) ) )
@ ( F3 @ ( zero_zero @ nat ) ) ) ) ) ) ).
% powser_split_head(2)
thf(fact_3085_odd__0__le__power__imp__0__le,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,N: nat] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A3 @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% odd_0_le_power_imp_0_le
thf(fact_3086_odd__power__less__zero,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,N: nat] :
( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( power_power @ A @ A3 @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) @ ( zero_zero @ A ) ) ) ) ).
% odd_power_less_zero
thf(fact_3087_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_3088_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_3089_arctan__lbound,axiom,
! [Y: real] : ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ ( arctan @ Y ) ) ).
% arctan_lbound
thf(fact_3090_arctan__bounded,axiom,
! [Y: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ ( arctan @ Y ) )
& ( ord_less @ real @ ( arctan @ Y ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ).
% arctan_bounded
thf(fact_3091_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_3092_nat__ivt__aux,axiom,
! [N: nat,F3: nat > int,K: int] :
( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ N )
=> ( ord_less_eq @ int @ ( abs_abs @ int @ ( minus_minus @ int @ ( F3 @ ( suc @ I3 ) ) @ ( F3 @ I3 ) ) ) @ ( one_one @ int ) ) )
=> ( ( ord_less_eq @ int @ ( F3 @ ( zero_zero @ nat ) ) @ K )
=> ( ( ord_less_eq @ int @ K @ ( F3 @ N ) )
=> ? [I3: nat] :
( ( ord_less_eq @ nat @ I3 @ N )
& ( ( F3 @ I3 )
= K ) ) ) ) ) ).
% nat_ivt_aux
thf(fact_3093_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_3094_unset__bit__Suc,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A3: A] :
( ( bit_se2638667681897837118et_bit @ A @ ( suc @ N ) @ A3 )
= ( plus_plus @ A @ ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se2638667681897837118et_bit @ A @ N @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% unset_bit_Suc
thf(fact_3095_set__bit__Suc,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A3: A] :
( ( bit_se5668285175392031749et_bit @ A @ ( suc @ N ) @ A3 )
= ( plus_plus @ A @ ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5668285175392031749et_bit @ A @ N @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% set_bit_Suc
thf(fact_3096_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_3097_sb__inc__lem,axiom,
! [A3: int,K: nat] :
( ( ord_less @ int @ ( plus_plus @ int @ A3 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K ) ) @ ( zero_zero @ int ) )
=> ( ord_less_eq @ int @ ( plus_plus @ int @ ( plus_plus @ int @ A3 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( suc @ K ) ) ) @ ( modulo_modulo @ int @ ( plus_plus @ int @ A3 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( suc @ K ) ) ) ) ) ).
% sb_inc_lem
thf(fact_3098_sb__inc__lem_H,axiom,
! [A3: int,K: nat] :
( ( ord_less @ int @ A3 @ ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K ) ) )
=> ( ord_less_eq @ int @ ( plus_plus @ int @ ( plus_plus @ int @ A3 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( suc @ K ) ) ) @ ( modulo_modulo @ int @ ( plus_plus @ int @ A3 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( suc @ K ) ) ) ) ) ).
% sb_inc_lem'
thf(fact_3099_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_3100_lemma__termdiff3,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [H2: A,Z: A,K4: real,N: nat] :
( ( H2
!= ( zero_zero @ A ) )
=> ( ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ Z ) @ K4 )
=> ( ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ Z @ H2 ) ) @ K4 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( divide_divide @ A @ ( minus_minus @ A @ ( power_power @ A @ ( plus_plus @ A @ Z @ H2 ) @ N ) @ ( power_power @ A @ Z @ N ) ) @ H2 ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ ( power_power @ A @ Z @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) @ ( times_times @ real @ ( times_times @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ ( semiring_1_of_nat @ real @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) ) ) @ ( power_power @ real @ K4 @ ( minus_minus @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( real_V7770717601297561774m_norm @ A @ H2 ) ) ) ) ) ) ) ).
% lemma_termdiff3
thf(fact_3101_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_3102_Suc__if__eq,axiom,
! [A: $tType,F3: nat > A,H2: nat > A,G2: A,N: nat] :
( ! [N2: nat] :
( ( F3 @ ( suc @ N2 ) )
= ( H2 @ N2 ) )
=> ( ( ( F3 @ ( zero_zero @ nat ) )
= G2 )
=> ( ( ( N
= ( zero_zero @ nat ) )
=> ( ( F3 @ N )
= G2 ) )
& ( ( N
!= ( zero_zero @ nat ) )
=> ( ( F3 @ N )
= ( H2 @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ) ) ) ).
% Suc_if_eq
thf(fact_3103_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_3104_norm__eq__zero,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X: A] :
( ( ( real_V7770717601297561774m_norm @ A @ X )
= ( zero_zero @ real ) )
= ( X
= ( zero_zero @ A ) ) ) ) ).
% norm_eq_zero
thf(fact_3105_norm__zero,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ( ( real_V7770717601297561774m_norm @ A @ ( zero_zero @ A ) )
= ( zero_zero @ real ) ) ) ).
% norm_zero
thf(fact_3106_norm__one,axiom,
! [A: $tType] :
( ( real_V2822296259951069270ebra_1 @ A )
=> ( ( real_V7770717601297561774m_norm @ A @ ( one_one @ A ) )
= ( one_one @ real ) ) ) ).
% norm_one
thf(fact_3107_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_3108_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_3109_sin__pi,axiom,
( ( sin @ real @ pi )
= ( zero_zero @ real ) ) ).
% sin_pi
thf(fact_3110_zero__less__norm__iff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X: A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( real_V7770717601297561774m_norm @ A @ X ) )
= ( X
!= ( zero_zero @ A ) ) ) ) ).
% zero_less_norm_iff
thf(fact_3111_norm__le__zero__iff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X: A] :
( ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ ( zero_zero @ real ) )
= ( X
= ( zero_zero @ A ) ) ) ) ).
% norm_le_zero_iff
thf(fact_3112_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_3113_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_3114_norm__mult__numeral1,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [W: num,A3: A] :
( ( real_V7770717601297561774m_norm @ A @ ( times_times @ A @ ( numeral_numeral @ A @ W ) @ A3 ) )
= ( times_times @ real @ ( numeral_numeral @ real @ W ) @ ( real_V7770717601297561774m_norm @ A @ A3 ) ) ) ) ).
% norm_mult_numeral1
thf(fact_3115_norm__mult__numeral2,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [A3: A,W: num] :
( ( real_V7770717601297561774m_norm @ A @ ( times_times @ A @ A3 @ ( numeral_numeral @ A @ W ) ) )
= ( times_times @ real @ ( real_V7770717601297561774m_norm @ A @ A3 ) @ ( numeral_numeral @ real @ W ) ) ) ) ).
% norm_mult_numeral2
thf(fact_3116_norm__divide__numeral,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [A3: A,W: num] :
( ( real_V7770717601297561774m_norm @ A @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ W ) ) )
= ( divide_divide @ real @ ( real_V7770717601297561774m_norm @ A @ A3 ) @ ( numeral_numeral @ real @ W ) ) ) ) ).
% norm_divide_numeral
thf(fact_3117_sin__npi,axiom,
! [N: nat] :
( ( sin @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ pi ) )
= ( zero_zero @ real ) ) ).
% sin_npi
thf(fact_3118_sin__npi2,axiom,
! [N: nat] :
( ( sin @ real @ ( times_times @ real @ pi @ ( semiring_1_of_nat @ real @ N ) ) )
= ( zero_zero @ real ) ) ).
% sin_npi2
thf(fact_3119_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_3120_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_3121_sin__two__pi,axiom,
( ( sin @ real @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) )
= ( zero_zero @ real ) ) ).
% sin_two_pi
thf(fact_3122_sin__pi__half,axiom,
( ( sin @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
= ( one_one @ real ) ) ).
% sin_pi_half
thf(fact_3123_norm__of__real__add1,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [X: real] :
( ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ ( real_Vector_of_real @ A @ X ) @ ( one_one @ A ) ) )
= ( abs_abs @ real @ ( plus_plus @ real @ X @ ( one_one @ real ) ) ) ) ) ).
% norm_of_real_add1
thf(fact_3124_norm__of__real__addn,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [X: real,B3: num] :
( ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ ( real_Vector_of_real @ A @ X ) @ ( numeral_numeral @ A @ B3 ) ) )
= ( abs_abs @ real @ ( plus_plus @ real @ X @ ( numeral_numeral @ real @ B3 ) ) ) ) ) ).
% norm_of_real_addn
thf(fact_3125_sin__periodic,axiom,
! [X: real] :
( ( sin @ real @ ( plus_plus @ real @ X @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) ) )
= ( sin @ real @ X ) ) ).
% sin_periodic
thf(fact_3126_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_3127_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_3128_sin__2pi__minus,axiom,
! [X: real] :
( ( sin @ real @ ( minus_minus @ real @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) @ X ) )
= ( uminus_uminus @ real @ ( sin @ real @ X ) ) ) ).
% sin_2pi_minus
thf(fact_3129_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_3130_sin__x__le__x,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ord_less_eq @ real @ ( sin @ real @ X ) @ X ) ) ).
% sin_x_le_x
thf(fact_3131_sin__le__one,axiom,
! [X: real] : ( ord_less_eq @ real @ ( sin @ real @ X ) @ ( one_one @ real ) ) ).
% sin_le_one
thf(fact_3132_norm__not__less__zero,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X: A] :
~ ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ ( zero_zero @ real ) ) ) ).
% norm_not_less_zero
thf(fact_3133_norm__ge__zero,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X: A] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( real_V7770717601297561774m_norm @ A @ X ) ) ) ).
% norm_ge_zero
thf(fact_3134_norm__power,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [X: A,N: nat] :
( ( real_V7770717601297561774m_norm @ A @ ( power_power @ A @ X @ N ) )
= ( power_power @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ N ) ) ) ).
% norm_power
thf(fact_3135_summable__norm__cancel,axiom,
! [A: $tType] :
( ( real_Vector_banach @ A )
=> ! [F3: nat > A] :
( ( summable @ real
@ ^ [N4: nat] : ( real_V7770717601297561774m_norm @ A @ ( F3 @ N4 ) ) )
=> ( summable @ A @ F3 ) ) ) ).
% summable_norm_cancel
thf(fact_3136_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_3137_nonzero__norm__divide,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [B3: A,A3: A] :
( ( B3
!= ( zero_zero @ A ) )
=> ( ( real_V7770717601297561774m_norm @ A @ ( divide_divide @ A @ A3 @ B3 ) )
= ( divide_divide @ real @ ( real_V7770717601297561774m_norm @ A @ A3 ) @ ( real_V7770717601297561774m_norm @ A @ B3 ) ) ) ) ) ).
% nonzero_norm_divide
thf(fact_3138_sin__gt__zero,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ X @ pi )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( sin @ real @ X ) ) ) ) ).
% sin_gt_zero
thf(fact_3139_sin__x__ge__neg__x,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ord_less_eq @ real @ ( uminus_uminus @ real @ X ) @ ( sin @ real @ X ) ) ) ).
% sin_x_ge_neg_x
thf(fact_3140_sin__ge__zero,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ X @ pi )
=> ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( sin @ real @ X ) ) ) ) ).
% sin_ge_zero
thf(fact_3141_sin__ge__minus__one,axiom,
! [X: real] : ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ ( sin @ real @ X ) ) ).
% sin_ge_minus_one
thf(fact_3142_norm__mult__less,axiom,
! [A: $tType] :
( ( real_V4412858255891104859lgebra @ A )
=> ! [X: A,R3: real,Y: A,S2: real] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ R3 )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ Y ) @ S2 )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( times_times @ A @ X @ Y ) ) @ ( times_times @ real @ R3 @ S2 ) ) ) ) ) ).
% norm_mult_less
thf(fact_3143_abs__sin__le__one,axiom,
! [X: real] : ( ord_less_eq @ real @ ( abs_abs @ real @ ( sin @ real @ X ) ) @ ( one_one @ real ) ) ).
% abs_sin_le_one
thf(fact_3144_norm__triangle__lt,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X: A,Y: A,E2: real] :
( ( ord_less @ real @ ( plus_plus @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ ( real_V7770717601297561774m_norm @ A @ Y ) ) @ E2 )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ X @ Y ) ) @ E2 ) ) ) ).
% norm_triangle_lt
thf(fact_3145_norm__add__less,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X: A,R3: real,Y: A,S2: real] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ R3 )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ Y ) @ S2 )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ X @ Y ) ) @ ( plus_plus @ real @ R3 @ S2 ) ) ) ) ) ).
% norm_add_less
thf(fact_3146_norm__diff__triangle__less,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X: A,Y: A,E1: real,Z: A,E22: real] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ X @ Y ) ) @ E1 )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ Y @ Z ) ) @ E22 )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ X @ Z ) ) @ ( plus_plus @ real @ E1 @ E22 ) ) ) ) ) ).
% norm_diff_triangle_less
thf(fact_3147_norm__power__ineq,axiom,
! [A: $tType] :
( ( real_V2822296259951069270ebra_1 @ A )
=> ! [X: A,N: nat] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( power_power @ A @ X @ N ) ) @ ( power_power @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ N ) ) ) ).
% norm_power_ineq
thf(fact_3148_summable__norm__comparison__test,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: nat > A,G2: nat > real] :
( ? [N8: nat] :
! [N2: nat] :
( ( ord_less_eq @ nat @ N8 @ N2 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( F3 @ N2 ) ) @ ( G2 @ N2 ) ) )
=> ( ( summable @ real @ G2 )
=> ( summable @ real
@ ^ [N4: nat] : ( real_V7770717601297561774m_norm @ A @ ( F3 @ N4 ) ) ) ) ) ) ).
% summable_norm_comparison_test
thf(fact_3149_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_3150_sin__eq__0__pi,axiom,
! [X: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ pi ) @ X )
=> ( ( ord_less @ real @ X @ pi )
=> ( ( ( sin @ real @ X )
= ( zero_zero @ real ) )
=> ( X
= ( zero_zero @ real ) ) ) ) ) ).
% sin_eq_0_pi
thf(fact_3151_norm__less__p1,axiom,
! [A: $tType] :
( ( real_V2822296259951069270ebra_1 @ A )
=> ! [X: A] : ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ ( real_Vector_of_real @ A @ ( real_V7770717601297561774m_norm @ A @ X ) ) @ ( one_one @ A ) ) ) ) ) ).
% norm_less_p1
thf(fact_3152_sin__zero__pi__iff,axiom,
! [X: real] :
( ( ord_less @ real @ ( abs_abs @ real @ X ) @ pi )
=> ( ( ( sin @ real @ X )
= ( zero_zero @ real ) )
= ( X
= ( zero_zero @ real ) ) ) ) ).
% sin_zero_pi_iff
thf(fact_3153_sin__zero__iff__int2,axiom,
! [X: real] :
( ( ( sin @ real @ X )
= ( zero_zero @ real ) )
= ( ? [I4: int] :
( X
= ( times_times @ real @ ( ring_1_of_int @ real @ I4 ) @ pi ) ) ) ) ).
% sin_zero_iff_int2
thf(fact_3154_powser__inside,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V8999393235501362500lgebra @ A ) )
=> ! [F3: nat > A,X: A,Z: A] :
( ( summable @ A
@ ^ [N4: nat] : ( times_times @ A @ ( F3 @ N4 ) @ ( power_power @ A @ X @ N4 ) ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ Z ) @ ( real_V7770717601297561774m_norm @ A @ X ) )
=> ( summable @ A
@ ^ [N4: nat] : ( times_times @ A @ ( F3 @ N4 ) @ ( power_power @ A @ Z @ N4 ) ) ) ) ) ) ).
% powser_inside
thf(fact_3155_complete__algebra__summable__geometric,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X: A] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ ( one_one @ real ) )
=> ( summable @ A @ ( power_power @ A @ X ) ) ) ) ).
% complete_algebra_summable_geometric
thf(fact_3156_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_3157_sin__gt__zero__02,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ X @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( sin @ real @ X ) ) ) ) ).
% sin_gt_zero_02
thf(fact_3158_summable__norm,axiom,
! [A: $tType] :
( ( real_Vector_banach @ A )
=> ! [F3: nat > A] :
( ( summable @ real
@ ^ [N4: nat] : ( real_V7770717601297561774m_norm @ A @ ( F3 @ N4 ) ) )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( suminf @ A @ F3 ) )
@ ( suminf @ real
@ ^ [N4: nat] : ( real_V7770717601297561774m_norm @ A @ ( F3 @ N4 ) ) ) ) ) ) ).
% summable_norm
thf(fact_3159_lemma__NBseq__def2,axiom,
! [A: $tType,B: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [X5: A > B] :
( ( ? [K6: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K6 )
& ! [N4: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( X5 @ N4 ) ) @ K6 ) ) )
= ( ? [N9: nat] :
! [N4: A] : ( ord_less @ real @ ( real_V7770717601297561774m_norm @ B @ ( X5 @ N4 ) ) @ ( semiring_1_of_nat @ real @ ( suc @ N9 ) ) ) ) ) ) ).
% lemma_NBseq_def2
thf(fact_3160_lemma__NBseq__def,axiom,
! [A: $tType,B: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [X5: A > B] :
( ( ? [K6: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K6 )
& ! [N4: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( X5 @ N4 ) ) @ K6 ) ) )
= ( ? [N9: nat] :
! [N4: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( X5 @ N4 ) ) @ ( semiring_1_of_nat @ real @ ( suc @ N9 ) ) ) ) ) ) ).
% lemma_NBseq_def
thf(fact_3161_powser__insidea,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [F3: nat > A,X: A,Z: A] :
( ( summable @ A
@ ^ [N4: nat] : ( times_times @ A @ ( F3 @ N4 ) @ ( power_power @ A @ X @ N4 ) ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ Z ) @ ( real_V7770717601297561774m_norm @ A @ X ) )
=> ( summable @ real
@ ^ [N4: nat] : ( real_V7770717601297561774m_norm @ A @ ( times_times @ A @ ( F3 @ N4 ) @ ( power_power @ A @ Z @ N4 ) ) ) ) ) ) ) ).
% powser_insidea
thf(fact_3162_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_3163_square__norm__one,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [X: A] :
( ( ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( one_one @ A ) )
=> ( ( real_V7770717601297561774m_norm @ A @ X )
= ( one_one @ real ) ) ) ) ).
% square_norm_one
thf(fact_3164_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_3165_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_3166_suminf__exist__split,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [R3: real,F3: nat > A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R3 )
=> ( ( summable @ A @ F3 )
=> ? [N10: nat] :
! [N11: nat] :
( ( ord_less_eq @ nat @ N10 @ N11 )
=> ( ord_less @ real
@ ( real_V7770717601297561774m_norm @ A
@ ( suminf @ A
@ ^ [I4: nat] : ( F3 @ ( plus_plus @ nat @ I4 @ N11 ) ) ) )
@ R3 ) ) ) ) ) ).
% suminf_exist_split
thf(fact_3167_Abel__lemma,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [R3: real,R0: real,A3: nat > A,M7: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ R3 )
=> ( ( ord_less @ real @ R3 @ R0 )
=> ( ! [N2: nat] : ( ord_less_eq @ real @ ( times_times @ real @ ( real_V7770717601297561774m_norm @ A @ ( A3 @ N2 ) ) @ ( power_power @ real @ R0 @ N2 ) ) @ M7 )
=> ( summable @ real
@ ^ [N4: nat] : ( times_times @ real @ ( real_V7770717601297561774m_norm @ A @ ( A3 @ N4 ) ) @ ( power_power @ real @ R3 @ N4 ) ) ) ) ) ) ) ).
% Abel_lemma
thf(fact_3168_sin__gt__zero2,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ X @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( sin @ real @ X ) ) ) ) ).
% sin_gt_zero2
thf(fact_3169_sin__lt__zero,axiom,
! [X: real] :
( ( ord_less @ real @ pi @ X )
=> ( ( ord_less @ real @ X @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) )
=> ( ord_less @ real @ ( sin @ real @ X ) @ ( zero_zero @ real ) ) ) ) ).
% sin_lt_zero
thf(fact_3170_sin__monotone__2pi__le,axiom,
! [Y: real,X: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ Y )
=> ( ( ord_less_eq @ real @ Y @ X )
=> ( ( ord_less_eq @ real @ X @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less_eq @ real @ ( sin @ real @ Y ) @ ( sin @ real @ X ) ) ) ) ) ).
% sin_monotone_2pi_le
thf(fact_3171_sin__mono__le__eq,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X )
=> ( ( ord_less_eq @ real @ X @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ Y )
=> ( ( ord_less_eq @ real @ Y @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less_eq @ real @ ( sin @ real @ X ) @ ( sin @ real @ Y ) )
= ( ord_less_eq @ real @ X @ Y ) ) ) ) ) ) ).
% sin_mono_le_eq
thf(fact_3172_sin__inj__pi,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X )
=> ( ( ord_less_eq @ real @ X @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ Y )
=> ( ( ord_less_eq @ real @ Y @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ( sin @ real @ X )
= ( sin @ real @ Y ) )
=> ( X = Y ) ) ) ) ) ) ).
% sin_inj_pi
thf(fact_3173_summable__ratio__test,axiom,
! [A: $tType] :
( ( real_Vector_banach @ A )
=> ! [C2: real,N3: nat,F3: nat > A] :
( ( ord_less @ real @ C2 @ ( one_one @ real ) )
=> ( ! [N2: nat] :
( ( ord_less_eq @ nat @ N3 @ N2 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( F3 @ ( suc @ N2 ) ) ) @ ( times_times @ real @ C2 @ ( real_V7770717601297561774m_norm @ A @ ( F3 @ N2 ) ) ) ) )
=> ( summable @ A @ F3 ) ) ) ) ).
% summable_ratio_test
thf(fact_3174_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_3175_sin__le__zero,axiom,
! [X: real] :
( ( ord_less_eq @ real @ pi @ X )
=> ( ( ord_less @ real @ X @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) )
=> ( ord_less_eq @ real @ ( sin @ real @ X ) @ ( zero_zero @ real ) ) ) ) ).
% sin_le_zero
thf(fact_3176_sin__less__zero,axiom,
! [X: real] :
( ( ord_less @ real @ ( divide_divide @ real @ ( uminus_uminus @ real @ pi ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ X )
=> ( ( ord_less @ real @ X @ ( zero_zero @ real ) )
=> ( ord_less @ real @ ( sin @ real @ X ) @ ( zero_zero @ real ) ) ) ) ).
% sin_less_zero
thf(fact_3177_sin__monotone__2pi,axiom,
! [Y: real,X: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ Y )
=> ( ( ord_less @ real @ Y @ X )
=> ( ( ord_less_eq @ real @ X @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less @ real @ ( sin @ real @ Y ) @ ( sin @ real @ X ) ) ) ) ) ).
% sin_monotone_2pi
thf(fact_3178_sin__mono__less__eq,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X )
=> ( ( ord_less_eq @ real @ X @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ Y )
=> ( ( ord_less_eq @ real @ Y @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less @ real @ ( sin @ real @ X ) @ ( sin @ real @ Y ) )
= ( ord_less @ real @ X @ Y ) ) ) ) ) ) ).
% sin_mono_less_eq
thf(fact_3179_sin__total,axiom,
! [Y: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ Y )
=> ( ( ord_less_eq @ real @ Y @ ( one_one @ real ) )
=> ? [X3: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X3 )
& ( ord_less_eq @ real @ X3 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
& ( ( sin @ real @ X3 )
= Y )
& ! [Y5: real] :
( ( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ Y5 )
& ( ord_less_eq @ real @ Y5 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
& ( ( sin @ real @ Y5 )
= Y ) )
=> ( Y5 = X3 ) ) ) ) ) ).
% sin_total
thf(fact_3180_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_3181_sin__zero__iff__int,axiom,
! [X: real] :
( ( ( sin @ real @ X )
= ( zero_zero @ real ) )
= ( ? [I4: int] :
( ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ I4 )
& ( X
= ( times_times @ real @ ( ring_1_of_int @ real @ I4 ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% sin_zero_iff_int
thf(fact_3182_sin__zero__lemma,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ( sin @ real @ X )
= ( zero_zero @ real ) )
=> ? [N2: nat] :
( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
& ( X
= ( times_times @ real @ ( semiring_1_of_nat @ real @ N2 ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% sin_zero_lemma
thf(fact_3183_sin__zero__iff,axiom,
! [X: real] :
( ( ( sin @ real @ X )
= ( zero_zero @ real ) )
= ( ? [N4: nat] :
( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N4 )
& ( X
= ( times_times @ real @ ( semiring_1_of_nat @ real @ N4 ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) )
| ? [N4: nat] :
( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N4 )
& ( X
= ( uminus_uminus @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N4 ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ).
% sin_zero_iff
thf(fact_3184_sin__coeff__def,axiom,
( sin_coeff
= ( ^ [N4: nat] : ( if @ real @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N4 ) @ ( zero_zero @ real ) @ ( divide_divide @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ ( divide_divide @ nat @ ( minus_minus @ nat @ N4 @ ( suc @ ( zero_zero @ nat ) ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( semiring_char_0_fact @ real @ N4 ) ) ) ) ) ).
% sin_coeff_def
thf(fact_3185_cos__coeff__def,axiom,
( cos_coeff
= ( ^ [N4: nat] : ( if @ real @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N4 ) @ ( divide_divide @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ ( divide_divide @ nat @ N4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( semiring_char_0_fact @ real @ N4 ) ) @ ( zero_zero @ real ) ) ) ) ).
% cos_coeff_def
thf(fact_3186_flip__bit__Suc,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A3: A] :
( ( bit_se8732182000553998342ip_bit @ A @ ( suc @ N ) @ A3 )
= ( plus_plus @ A @ ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se8732182000553998342ip_bit @ A @ N @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% flip_bit_Suc
thf(fact_3187_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_3188_VEBT__internal_OT__vebt__buildupi_Oelims,axiom,
! [X: nat,Y: nat] :
( ( ( vEBT_V441764108873111860ildupi @ X )
= Y )
=> ( ( ( X
= ( zero_zero @ nat ) )
=> ( Y
!= ( suc @ ( zero_zero @ nat ) ) ) )
=> ( ( ( X
= ( suc @ ( zero_zero @ nat ) ) )
=> ( Y
!= ( suc @ ( zero_zero @ nat ) ) ) )
=> ~ ! [N2: nat] :
( ( X
= ( suc @ ( suc @ N2 ) ) )
=> ~ ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
=> ( Y
= ( suc @ ( suc @ ( suc @ ( plus_plus @ nat @ ( vEBT_V441764108873111860ildupi @ ( suc @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( times_times @ nat @ ( vEBT_V441764108873111860ildupi @ ( suc @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) )
& ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
=> ( Y
= ( suc @ ( suc @ ( suc @ ( plus_plus @ nat @ ( vEBT_V441764108873111860ildupi @ ( suc @ ( suc @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) @ ( times_times @ nat @ ( vEBT_V441764108873111860ildupi @ ( suc @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.T_vebt_buildupi.elims
thf(fact_3189_verit__eq__simplify_I9_J,axiom,
! [X32: num,Y32: num] :
( ( ( bit1 @ X32 )
= ( bit1 @ Y32 ) )
= ( X32 = Y32 ) ) ).
% verit_eq_simplify(9)
thf(fact_3190_semiring__norm_I90_J,axiom,
! [M: num,N: num] :
( ( ( bit1 @ M )
= ( bit1 @ N ) )
= ( M = N ) ) ).
% semiring_norm(90)
thf(fact_3191_semiring__norm_I88_J,axiom,
! [M: num,N: num] :
( ( bit0 @ M )
!= ( bit1 @ N ) ) ).
% semiring_norm(88)
thf(fact_3192_semiring__norm_I89_J,axiom,
! [M: num,N: num] :
( ( bit1 @ M )
!= ( bit0 @ N ) ) ).
% semiring_norm(89)
thf(fact_3193_semiring__norm_I84_J,axiom,
! [N: num] :
( one2
!= ( bit1 @ N ) ) ).
% semiring_norm(84)
thf(fact_3194_semiring__norm_I86_J,axiom,
! [M: num] :
( ( bit1 @ M )
!= one2 ) ).
% semiring_norm(86)
thf(fact_3195_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_3196_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_3197_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_3198_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_3199_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_3200_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_3201_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_3202_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_3203_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_3204_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_3205_semiring__norm_I70_J,axiom,
! [M: num] :
~ ( ord_less_eq @ num @ ( bit1 @ M ) @ one2 ) ).
% semiring_norm(70)
thf(fact_3206_semiring__norm_I77_J,axiom,
! [N: num] : ( ord_less @ num @ one2 @ ( bit1 @ N ) ) ).
% semiring_norm(77)
thf(fact_3207_sin__coeff__0,axiom,
( ( sin_coeff @ ( zero_zero @ nat ) )
= ( zero_zero @ real ) ) ).
% sin_coeff_0
thf(fact_3208_cos__coeff__0,axiom,
( ( cos_coeff @ ( zero_zero @ nat ) )
= ( one_one @ real ) ) ).
% cos_coeff_0
thf(fact_3209_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_3210_zdiv__numeral__Bit1,axiom,
! [V2: num,W: num] :
( ( divide_divide @ int @ ( numeral_numeral @ int @ ( bit1 @ V2 ) ) @ ( numeral_numeral @ int @ ( bit0 @ W ) ) )
= ( divide_divide @ int @ ( numeral_numeral @ int @ V2 ) @ ( numeral_numeral @ int @ W ) ) ) ).
% zdiv_numeral_Bit1
thf(fact_3211_semiring__norm_I3_J,axiom,
! [N: num] :
( ( plus_plus @ num @ one2 @ ( bit0 @ N ) )
= ( bit1 @ N ) ) ).
% semiring_norm(3)
thf(fact_3212_semiring__norm_I4_J,axiom,
! [N: num] :
( ( plus_plus @ num @ one2 @ ( bit1 @ N ) )
= ( bit0 @ ( plus_plus @ num @ N @ one2 ) ) ) ).
% semiring_norm(4)
thf(fact_3213_semiring__norm_I5_J,axiom,
! [M: num] :
( ( plus_plus @ num @ ( bit0 @ M ) @ one2 )
= ( bit1 @ M ) ) ).
% semiring_norm(5)
thf(fact_3214_semiring__norm_I8_J,axiom,
! [M: num] :
( ( plus_plus @ num @ ( bit1 @ M ) @ one2 )
= ( bit0 @ ( plus_plus @ num @ M @ one2 ) ) ) ).
% semiring_norm(8)
thf(fact_3215_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_3216_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_3217_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_3218_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_3219_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_3220_Suc__div__eq__add3__div__numeral,axiom,
! [M: nat,V2: num] :
( ( divide_divide @ nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ ( numeral_numeral @ nat @ V2 ) )
= ( divide_divide @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) @ M ) @ ( numeral_numeral @ nat @ V2 ) ) ) ).
% Suc_div_eq_add3_div_numeral
thf(fact_3221_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_3222_Suc__mod__eq__add3__mod__numeral,axiom,
! [M: nat,V2: num] :
( ( modulo_modulo @ nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ ( numeral_numeral @ nat @ V2 ) )
= ( modulo_modulo @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) @ M ) @ ( numeral_numeral @ nat @ V2 ) ) ) ).
% Suc_mod_eq_add3_mod_numeral
thf(fact_3223_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_3224_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_3225_zmod__numeral__Bit1,axiom,
! [V2: num,W: num] :
( ( modulo_modulo @ int @ ( numeral_numeral @ int @ ( bit1 @ V2 ) ) @ ( numeral_numeral @ int @ ( bit0 @ W ) ) )
= ( plus_plus @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( modulo_modulo @ int @ ( numeral_numeral @ int @ V2 ) @ ( numeral_numeral @ int @ W ) ) ) @ ( one_one @ int ) ) ) ).
% zmod_numeral_Bit1
thf(fact_3226_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_3227_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_3228_verit__eq__simplify_I14_J,axiom,
! [X22: num,X32: num] :
( ( bit0 @ X22 )
!= ( bit1 @ X32 ) ) ).
% verit_eq_simplify(14)
thf(fact_3229_verit__eq__simplify_I12_J,axiom,
! [X32: num] :
( one2
!= ( bit1 @ X32 ) ) ).
% verit_eq_simplify(12)
thf(fact_3230_num_Oexhaust,axiom,
! [Y: num] :
( ( Y != one2 )
=> ( ! [X23: num] :
( Y
!= ( bit0 @ X23 ) )
=> ~ ! [X33: num] :
( Y
!= ( bit1 @ X33 ) ) ) ) ).
% num.exhaust
thf(fact_3231_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_3232_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_3233_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_3234_VEBT__internal_OT__vebt__buildupi__gq__0,axiom,
! [N: nat] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( vEBT_V441764108873111860ildupi @ N ) ) ).
% VEBT_internal.T_vebt_buildupi_gq_0
thf(fact_3235_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_3236_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_3237_cong__exp__iff__simps_I13_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M: num,Q3: num,N: num] :
( ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit1 @ M ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q3 ) ) )
= ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit1 @ N ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q3 ) ) ) )
= ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ Q3 ) )
= ( modulo_modulo @ A @ ( numeral_numeral @ A @ N ) @ ( numeral_numeral @ A @ Q3 ) ) ) ) ) ).
% cong_exp_iff_simps(13)
thf(fact_3238_cong__exp__iff__simps_I12_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M: num,Q3: num,N: num] :
( ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit1 @ M ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q3 ) ) )
!= ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit0 @ N ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q3 ) ) ) ) ) ).
% cong_exp_iff_simps(12)
thf(fact_3239_cong__exp__iff__simps_I10_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M: num,Q3: num,N: num] :
( ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit0 @ M ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q3 ) ) )
!= ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit1 @ N ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q3 ) ) ) ) ) ).
% cong_exp_iff_simps(10)
thf(fact_3240_power__minus__Bit1,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [X: A,K: num] :
( ( power_power @ A @ ( uminus_uminus @ A @ X ) @ ( numeral_numeral @ nat @ ( bit1 @ K ) ) )
= ( uminus_uminus @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit1 @ K ) ) ) ) ) ) ).
% power_minus_Bit1
thf(fact_3241_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_3242_VEBT__internal_OT__vebt__buildupi__univ,axiom,
! [U2: nat,N: nat] :
( ( U2
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
=> ( ord_less_eq @ nat @ ( vEBT_V441764108873111860ildupi @ N ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) @ U2 ) ) ) ).
% VEBT_internal.T_vebt_buildupi_univ
thf(fact_3243_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_3244_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_3245_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_3246_cong__exp__iff__simps_I3_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [N: num,Q3: num] :
( ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit1 @ N ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q3 ) ) )
!= ( zero_zero @ A ) ) ) ).
% cong_exp_iff_simps(3)
thf(fact_3247_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_3248_power3__eq__cube,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A3: A] :
( ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) )
= ( times_times @ A @ ( times_times @ A @ A3 @ A3 ) @ A3 ) ) ) ).
% power3_eq_cube
thf(fact_3249_numeral__3__eq__3,axiom,
( ( numeral_numeral @ nat @ ( bit1 @ one2 ) )
= ( suc @ ( suc @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ).
% numeral_3_eq_3
thf(fact_3250_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_3251_num_Osize_I6_J,axiom,
! [X32: num] :
( ( size_size @ num @ ( bit1 @ X32 ) )
= ( plus_plus @ nat @ ( size_size @ num @ X32 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% num.size(6)
thf(fact_3252_cong__exp__iff__simps_I7_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [Q3: num,N: num] :
( ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ one2 ) @ ( numeral_numeral @ A @ ( bit0 @ Q3 ) ) )
= ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit1 @ N ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q3 ) ) ) )
= ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ N ) @ ( numeral_numeral @ A @ Q3 ) )
= ( zero_zero @ A ) ) ) ) ).
% cong_exp_iff_simps(7)
thf(fact_3253_cong__exp__iff__simps_I11_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M: num,Q3: num] :
( ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit1 @ M ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q3 ) ) )
= ( modulo_modulo @ A @ ( numeral_numeral @ A @ one2 ) @ ( numeral_numeral @ A @ ( bit0 @ Q3 ) ) ) )
= ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ Q3 ) )
= ( zero_zero @ A ) ) ) ) ).
% cong_exp_iff_simps(11)
thf(fact_3254_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_3255_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_3256_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_3257_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_3258_small__powers__of__2,axiom,
! [X: nat] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) @ X )
=> ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ X @ ( one_one @ nat ) ) ) ) ) ).
% small_powers_of_2
thf(fact_3259_even__flip__bit__iff,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M: nat,A3: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se8732182000553998342ip_bit @ A @ M @ A3 ) )
= ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
!= ( M
= ( zero_zero @ nat ) ) ) ) ) ).
% even_flip_bit_iff
thf(fact_3260_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_3261_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_3262_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_3263_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_3264_VEBT__internal_OT__vebt__buildupi_H_Oelims,axiom,
! [X: nat,Y: int] :
( ( ( vEBT_V9176841429113362141ildupi @ X )
= Y )
=> ( ( ( X
= ( zero_zero @ nat ) )
=> ( Y
!= ( one_one @ int ) ) )
=> ( ( ( X
= ( suc @ ( zero_zero @ nat ) ) )
=> ( Y
!= ( one_one @ int ) ) )
=> ~ ! [N2: nat] :
( ( X
= ( suc @ ( suc @ N2 ) ) )
=> ~ ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
=> ( Y
= ( plus_plus @ int @ ( numeral_numeral @ int @ ( bit1 @ one2 ) ) @ ( plus_plus @ int @ ( vEBT_V9176841429113362141ildupi @ ( suc @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( plus_plus @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ ( bit0 @ one2 ) ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( times_times @ int @ ( vEBT_V9176841429113362141ildupi @ ( suc @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) )
& ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
=> ( Y
= ( plus_plus @ int @ ( numeral_numeral @ int @ ( bit1 @ one2 ) ) @ ( plus_plus @ int @ ( vEBT_V9176841429113362141ildupi @ ( suc @ ( suc @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( plus_plus @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ ( bit0 @ one2 ) ) ) @ ( times_times @ int @ ( vEBT_V9176841429113362141ildupi @ ( suc @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.T_vebt_buildupi'.elims
thf(fact_3265_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_Oelims,axiom,
! [X: nat,Y: nat] :
( ( ( vEBT_V8646137997579335489_i_l_d @ X )
= Y )
=> ( ( ( X
= ( zero_zero @ nat ) )
=> ( Y
!= ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) ) )
=> ( ( ( X
= ( suc @ ( zero_zero @ nat ) ) )
=> ( Y
!= ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) ) )
=> ~ ! [Va2: nat] :
( ( X
= ( suc @ ( suc @ Va2 ) ) )
=> ~ ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ Va2 ) ) )
=> ( Y
= ( plus_plus @ nat @ ( one_one @ nat ) @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_V8646137997579335489_i_l_d @ ( divide_divide @ nat @ ( suc @ ( suc @ 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 ) ) )
=> ( Y
= ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) @ ( vEBT_V8646137997579335489_i_l_d @ ( suc @ ( divide_divide @ nat @ ( suc @ ( suc @ 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_3266_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p_Oelims,axiom,
! [X: nat,Y: nat] :
( ( ( vEBT_V8346862874174094_d_u_p @ X )
= Y )
=> ( ( ( X
= ( zero_zero @ nat ) )
=> ( Y
!= ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) ) )
=> ( ( ( X
= ( suc @ ( zero_zero @ nat ) ) )
=> ( Y
!= ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) ) )
=> ~ ! [Va2: nat] :
( ( X
= ( suc @ ( suc @ Va2 ) ) )
=> ~ ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ Va2 ) ) )
=> ( Y
= ( plus_plus @ nat @ ( one_one @ nat ) @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_V8346862874174094_d_u_p @ ( divide_divide @ nat @ ( suc @ ( suc @ 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 ) ) )
=> ( Y
= ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_V8346862874174094_d_u_p @ ( suc @ ( divide_divide @ nat @ ( suc @ ( suc @ 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_3267_VEBT__internal_OTb_Oelims,axiom,
! [X: nat,Y: int] :
( ( ( vEBT_VEBT_Tb @ X )
= Y )
=> ( ( ( X
= ( zero_zero @ nat ) )
=> ( Y
!= ( numeral_numeral @ int @ ( bit1 @ one2 ) ) ) )
=> ( ( ( X
= ( suc @ ( zero_zero @ nat ) ) )
=> ( Y
!= ( numeral_numeral @ int @ ( bit1 @ one2 ) ) ) )
=> ~ ! [N2: nat] :
( ( X
= ( suc @ ( suc @ N2 ) ) )
=> ~ ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
=> ( Y
= ( plus_plus @ int @ ( plus_plus @ int @ ( numeral_numeral @ int @ ( bit1 @ ( bit0 @ one2 ) ) ) @ ( vEBT_VEBT_Tb @ ( suc @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( times_times @ int @ ( vEBT_VEBT_Tb @ ( suc @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( suc @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) )
& ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
=> ( Y
= ( plus_plus @ int @ ( plus_plus @ int @ ( numeral_numeral @ int @ ( bit1 @ ( bit0 @ one2 ) ) ) @ ( vEBT_VEBT_Tb @ ( suc @ ( suc @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) @ ( times_times @ int @ ( vEBT_VEBT_Tb @ ( suc @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.Tb.elims
thf(fact_3268_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_3269_VEBT__internal_Obuildup__build__time,axiom,
! [N: nat] : ( ord_less @ nat @ ( vEBT_V8346862874174094_d_u_p @ N ) @ ( vEBT_V8646137997579335489_i_l_d @ N ) ) ).
% VEBT_internal.buildup_build_time
thf(fact_3270_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_3271_VEBT__internal_OTbuildupi__buildupi_H,axiom,
! [N: nat] :
( ( semiring_1_of_nat @ int @ ( vEBT_V441764108873111860ildupi @ N ) )
= ( vEBT_V9176841429113362141ildupi @ N ) ) ).
% VEBT_internal.Tbuildupi_buildupi'
thf(fact_3272_VEBT__internal_OTb__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 ) ) ) ) ).
% VEBT_internal.Tb_T_vebt_buildupi'
thf(fact_3273_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_3274_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_3275_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_3276_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_3277_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_3278_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_3279_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_3280_VEBT__internal_OTb__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 ) ) ) ) ).
% VEBT_internal.Tb_T_vebt_buildupi
thf(fact_3281_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_3282_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_3283_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_3284_vebt__inst_Otime__vebt__buildup,axiom,
! [U2: nat,N: nat] :
( ( U2
= ( 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 ) ) ) ) ) @ U2 ) ) ) ).
% vebt_inst.time_vebt_buildup
thf(fact_3285_vebt__buildupi__rule,axiom,
! [N: nat] : ( time_htt @ vEBT_VEBTi @ ( pure_assn @ ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) @ ( vEBT_vebt_buildupi @ N ) @ ( vEBT_Intf_vebt_assn @ N @ ( bot_bot @ ( set @ nat ) ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ).
% vebt_buildupi_rule
thf(fact_3286_VEBT__internal_OTb_H_Oelims,axiom,
! [X: nat,Y: nat] :
( ( ( vEBT_VEBT_Tb2 @ X )
= Y )
=> ( ( ( X
= ( zero_zero @ nat ) )
=> ( Y
!= ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) ) )
=> ( ( ( X
= ( suc @ ( zero_zero @ nat ) ) )
=> ( Y
!= ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) ) )
=> ~ ! [N2: nat] :
( ( X
= ( suc @ ( suc @ N2 ) ) )
=> ~ ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
=> ( Y
= ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ one2 ) ) ) @ ( vEBT_VEBT_Tb2 @ ( suc @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( times_times @ nat @ ( vEBT_VEBT_Tb2 @ ( suc @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) )
& ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
=> ( Y
= ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ one2 ) ) ) @ ( vEBT_VEBT_Tb2 @ ( suc @ ( suc @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) @ ( times_times @ nat @ ( vEBT_VEBT_Tb2 @ ( suc @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.Tb'.elims
thf(fact_3287_signed__take__bit__numeral__minus__bit1,axiom,
! [L: num,K: num] :
( ( bit_ri4674362597316999326ke_bit @ int @ ( numeral_numeral @ nat @ L ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ K ) ) ) )
= ( plus_plus @ int @ ( times_times @ int @ ( bit_ri4674362597316999326ke_bit @ int @ ( pred_numeral @ L ) @ ( minus_minus @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) @ ( one_one @ int ) ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( one_one @ int ) ) ) ).
% signed_take_bit_numeral_minus_bit1
thf(fact_3288_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_3289_pred__numeral__simps_I1_J,axiom,
( ( pred_numeral @ one2 )
= ( zero_zero @ nat ) ) ).
% pred_numeral_simps(1)
thf(fact_3290_eq__numeral__Suc,axiom,
! [K: num,N: nat] :
( ( ( numeral_numeral @ nat @ K )
= ( suc @ N ) )
= ( ( pred_numeral @ K )
= N ) ) ).
% eq_numeral_Suc
thf(fact_3291_Suc__eq__numeral,axiom,
! [N: nat,K: num] :
( ( ( suc @ N )
= ( numeral_numeral @ nat @ K ) )
= ( N
= ( pred_numeral @ K ) ) ) ).
% Suc_eq_numeral
thf(fact_3292_pred__numeral__simps_I3_J,axiom,
! [K: num] :
( ( pred_numeral @ ( bit1 @ K ) )
= ( numeral_numeral @ nat @ ( bit0 @ K ) ) ) ).
% pred_numeral_simps(3)
thf(fact_3293_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_3294_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_3295_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_3296_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_3297_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_3298_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_3299_signed__take__bit__numeral__bit0,axiom,
! [L: num,K: num] :
( ( bit_ri4674362597316999326ke_bit @ int @ ( numeral_numeral @ nat @ L ) @ ( numeral_numeral @ int @ ( bit0 @ K ) ) )
= ( times_times @ int @ ( bit_ri4674362597316999326ke_bit @ int @ ( pred_numeral @ L ) @ ( numeral_numeral @ int @ K ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ).
% signed_take_bit_numeral_bit0
thf(fact_3300_signed__take__bit__numeral__minus__bit0,axiom,
! [L: num,K: num] :
( ( bit_ri4674362597316999326ke_bit @ int @ ( numeral_numeral @ nat @ L ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ K ) ) ) )
= ( times_times @ int @ ( bit_ri4674362597316999326ke_bit @ int @ ( pred_numeral @ L ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ).
% signed_take_bit_numeral_minus_bit0
thf(fact_3301_signed__take__bit__numeral__bit1,axiom,
! [L: num,K: num] :
( ( bit_ri4674362597316999326ke_bit @ int @ ( numeral_numeral @ nat @ L ) @ ( numeral_numeral @ int @ ( bit1 @ K ) ) )
= ( plus_plus @ int @ ( times_times @ int @ ( bit_ri4674362597316999326ke_bit @ int @ ( pred_numeral @ L ) @ ( numeral_numeral @ int @ K ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( one_one @ int ) ) ) ).
% signed_take_bit_numeral_bit1
thf(fact_3302_numeral__eq__Suc,axiom,
( ( numeral_numeral @ nat )
= ( ^ [K3: num] : ( suc @ ( pred_numeral @ K3 ) ) ) ) ).
% numeral_eq_Suc
thf(fact_3303_pred__numeral__def,axiom,
( pred_numeral
= ( ^ [K3: num] : ( minus_minus @ nat @ ( numeral_numeral @ nat @ K3 ) @ ( one_one @ nat ) ) ) ) ).
% pred_numeral_def
thf(fact_3304_VEBT__internal_OTb__Tb_H,axiom,
( vEBT_VEBT_Tb
= ( ^ [T2: nat] : ( semiring_1_of_nat @ int @ ( vEBT_VEBT_Tb2 @ T2 ) ) ) ) ).
% VEBT_internal.Tb_Tb'
thf(fact_3305_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_3306_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_3307_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_3308_VEBT__internal_OTb__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 ) ) ) ) ).
% VEBT_internal.Tb_T_vebt_buildupi''
thf(fact_3309_vebt__inserti__rule,axiom,
! [X: nat,N: nat,S2: set @ nat,Ti: vEBT_VEBTi] :
( ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
=> ( time_htt @ vEBT_VEBTi @ ( vEBT_Intf_vebt_assn @ N @ S2 @ Ti ) @ ( vEBT_vebt_inserti @ Ti @ X ) @ ( vEBT_Intf_vebt_assn @ N @ ( sup_sup @ ( set @ nat ) @ S2 @ ( insert @ nat @ X @ ( bot_bot @ ( set @ nat ) ) ) ) ) @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) @ ( nat2 @ ( archimedean_ceiling @ real @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ N ) ) ) ) ) ) ) ) ).
% vebt_inserti_rule
thf(fact_3310_sincos__total__2pi,axiom,
! [X: real,Y: real] :
( ( ( plus_plus @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( one_one @ real ) )
=> ~ ! [T3: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ T3 )
=> ( ( ord_less @ real @ T3 @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) )
=> ( ( X
= ( cos @ real @ T3 ) )
=> ( Y
!= ( sin @ real @ T3 ) ) ) ) ) ) ).
% sincos_total_2pi
thf(fact_3311_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_3312_triangle__def,axiom,
( nat_triangle
= ( ^ [N4: nat] : ( divide_divide @ nat @ ( times_times @ nat @ N4 @ ( suc @ N4 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% triangle_def
thf(fact_3313_sin__paired,axiom,
! [X: real] :
( sums @ real
@ ^ [N4: nat] : ( times_times @ real @ ( divide_divide @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ N4 ) @ ( semiring_char_0_fact @ real @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N4 ) @ ( one_one @ nat ) ) ) ) @ ( power_power @ real @ X @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N4 ) @ ( one_one @ nat ) ) ) )
@ ( sin @ real @ X ) ) ).
% sin_paired
thf(fact_3314_nat__int,axiom,
! [N: nat] :
( ( nat2 @ ( semiring_1_of_nat @ int @ N ) )
= N ) ).
% nat_int
thf(fact_3315_sums__zero,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ( sums @ A
@ ^ [N4: nat] : ( zero_zero @ A )
@ ( zero_zero @ A ) ) ) ).
% sums_zero
thf(fact_3316_triangle__0,axiom,
( ( nat_triangle @ ( zero_zero @ nat ) )
= ( zero_zero @ nat ) ) ).
% triangle_0
thf(fact_3317_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_3318_nat__numeral,axiom,
! [K: num] :
( ( nat2 @ ( numeral_numeral @ int @ K ) )
= ( numeral_numeral @ nat @ K ) ) ).
% nat_numeral
thf(fact_3319_nat__1,axiom,
( ( nat2 @ ( one_one @ int ) )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% nat_1
thf(fact_3320_nat__0__iff,axiom,
! [I: int] :
( ( ( nat2 @ I )
= ( zero_zero @ nat ) )
= ( ord_less_eq @ int @ I @ ( zero_zero @ int ) ) ) ).
% nat_0_iff
thf(fact_3321_nat__le__0,axiom,
! [Z: int] :
( ( ord_less_eq @ int @ Z @ ( zero_zero @ int ) )
=> ( ( nat2 @ Z )
= ( zero_zero @ nat ) ) ) ).
% nat_le_0
thf(fact_3322_nat__neg__numeral,axiom,
! [K: num] :
( ( nat2 @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) )
= ( zero_zero @ nat ) ) ).
% nat_neg_numeral
thf(fact_3323_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_3324_nat__zminus__int,axiom,
! [N: nat] :
( ( nat2 @ ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N ) ) )
= ( zero_zero @ nat ) ) ).
% nat_zminus_int
thf(fact_3325_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_3326_cos__pi,axiom,
( ( cos @ real @ pi )
= ( uminus_uminus @ real @ ( one_one @ real ) ) ) ).
% cos_pi
thf(fact_3327_sin__cos__squared__add3,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A] :
( ( plus_plus @ A @ ( times_times @ A @ ( cos @ A @ X ) @ ( cos @ A @ X ) ) @ ( times_times @ A @ ( sin @ A @ X ) @ ( sin @ A @ X ) ) )
= ( one_one @ A ) ) ) ).
% sin_cos_squared_add3
thf(fact_3328_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_3329_diff__nat__numeral,axiom,
! [V2: num,V4: num] :
( ( minus_minus @ nat @ ( numeral_numeral @ nat @ V2 ) @ ( numeral_numeral @ nat @ V4 ) )
= ( nat2 @ ( minus_minus @ int @ ( numeral_numeral @ int @ V2 ) @ ( numeral_numeral @ int @ V4 ) ) ) ) ).
% diff_nat_numeral
thf(fact_3330_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_3331_nat__eq__numeral__power__cancel__iff,axiom,
! [Y: int,X: num,N: nat] :
( ( ( nat2 @ Y )
= ( power_power @ nat @ ( numeral_numeral @ nat @ X ) @ N ) )
= ( Y
= ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N ) ) ) ).
% nat_eq_numeral_power_cancel_iff
thf(fact_3332_numeral__power__eq__nat__cancel__iff,axiom,
! [X: num,N: nat,Y: int] :
( ( ( power_power @ nat @ ( numeral_numeral @ nat @ X ) @ N )
= ( nat2 @ Y ) )
= ( ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N )
= Y ) ) ).
% numeral_power_eq_nat_cancel_iff
thf(fact_3333_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_3334_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_3335_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_3336_powser__sums__zero__iff,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [A3: nat > A,X: A] :
( ( sums @ A
@ ^ [N4: nat] : ( times_times @ A @ ( A3 @ N4 ) @ ( power_power @ A @ ( zero_zero @ A ) @ N4 ) )
@ X )
= ( ( A3 @ ( zero_zero @ nat ) )
= X ) ) ) ).
% powser_sums_zero_iff
thf(fact_3337_nat__ceiling__le__eq,axiom,
! [X: real,A3: nat] :
( ( ord_less_eq @ nat @ ( nat2 @ ( archimedean_ceiling @ real @ X ) ) @ A3 )
= ( ord_less_eq @ real @ X @ ( semiring_1_of_nat @ real @ A3 ) ) ) ).
% nat_ceiling_le_eq
thf(fact_3338_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_3339_nat__numeral__diff__1,axiom,
! [V2: num] :
( ( minus_minus @ nat @ ( numeral_numeral @ nat @ V2 ) @ ( one_one @ nat ) )
= ( nat2 @ ( minus_minus @ int @ ( numeral_numeral @ int @ V2 ) @ ( one_one @ int ) ) ) ) ).
% nat_numeral_diff_1
thf(fact_3340_nat__less__numeral__power__cancel__iff,axiom,
! [A3: int,X: num,N: nat] :
( ( ord_less @ nat @ ( nat2 @ A3 ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ X ) @ N ) )
= ( ord_less @ int @ A3 @ ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N ) ) ) ).
% nat_less_numeral_power_cancel_iff
thf(fact_3341_numeral__power__less__nat__cancel__iff,axiom,
! [X: num,N: nat,A3: int] :
( ( ord_less @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ X ) @ N ) @ ( nat2 @ A3 ) )
= ( ord_less @ int @ ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N ) @ A3 ) ) ).
% numeral_power_less_nat_cancel_iff
thf(fact_3342_nat__le__numeral__power__cancel__iff,axiom,
! [A3: int,X: num,N: nat] :
( ( ord_less_eq @ nat @ ( nat2 @ A3 ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ X ) @ N ) )
= ( ord_less_eq @ int @ A3 @ ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N ) ) ) ).
% nat_le_numeral_power_cancel_iff
thf(fact_3343_numeral__power__le__nat__cancel__iff,axiom,
! [X: num,N: nat,A3: int] :
( ( ord_less_eq @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ X ) @ N ) @ ( nat2 @ A3 ) )
= ( ord_less_eq @ int @ ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N ) @ A3 ) ) ).
% numeral_power_le_nat_cancel_iff
thf(fact_3344_cos__pi__half,axiom,
( ( cos @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
= ( zero_zero @ real ) ) ).
% cos_pi_half
thf(fact_3345_cos__two__pi,axiom,
( ( cos @ real @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) )
= ( one_one @ real ) ) ).
% cos_two_pi
thf(fact_3346_cos__periodic,axiom,
! [X: real] :
( ( cos @ real @ ( plus_plus @ real @ X @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) ) )
= ( cos @ real @ X ) ) ).
% cos_periodic
thf(fact_3347_cos__2pi__minus,axiom,
! [X: real] :
( ( cos @ real @ ( minus_minus @ real @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) @ X ) )
= ( cos @ real @ X ) ) ).
% cos_2pi_minus
thf(fact_3348_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_3349_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_3350_sin__cos__squared__add,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A] :
( ( plus_plus @ A @ ( power_power @ A @ ( sin @ A @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ ( cos @ A @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( one_one @ A ) ) ) ).
% sin_cos_squared_add
thf(fact_3351_sin__cos__squared__add2,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A] :
( ( plus_plus @ A @ ( power_power @ A @ ( cos @ A @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ ( sin @ A @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( one_one @ A ) ) ) ).
% sin_cos_squared_add2
thf(fact_3352_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_3353_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_3354_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_3355_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_3356_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_3357_sums__of__real,axiom,
! [A: $tType] :
( ( ( real_V2191834092415804123ebra_1 @ A )
& ( real_V822414075346904944vector @ A ) )
=> ! [X5: nat > real,A3: real] :
( ( sums @ real @ X5 @ A3 )
=> ( sums @ A
@ ^ [N4: nat] : ( real_Vector_of_real @ A @ ( X5 @ N4 ) )
@ ( real_Vector_of_real @ A @ A3 ) ) ) ) ).
% sums_of_real
thf(fact_3358_sums__of__real__iff,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [F3: nat > real,C2: real] :
( ( sums @ A
@ ^ [N4: nat] : ( real_Vector_of_real @ A @ ( F3 @ N4 ) )
@ ( real_Vector_of_real @ A @ C2 ) )
= ( sums @ real @ F3 @ C2 ) ) ) ).
% sums_of_real_iff
thf(fact_3359_sums__0,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [F3: nat > A] :
( ! [N2: nat] :
( ( F3 @ N2 )
= ( zero_zero @ A ) )
=> ( sums @ A @ F3 @ ( zero_zero @ A ) ) ) ) ).
% sums_0
thf(fact_3360_sums__single,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [I: nat,F3: nat > A] :
( sums @ A
@ ^ [R5: nat] : ( if @ A @ ( R5 = I ) @ ( F3 @ R5 ) @ ( zero_zero @ A ) )
@ ( F3 @ I ) ) ) ).
% sums_single
thf(fact_3361_sums__add,axiom,
! [A: $tType] :
( ( ( topolo5987344860129210374id_add @ A )
& ( topological_t2_space @ A ) )
=> ! [F3: nat > A,A3: A,G2: nat > A,B3: A] :
( ( sums @ A @ F3 @ A3 )
=> ( ( sums @ A @ G2 @ B3 )
=> ( sums @ A
@ ^ [N4: nat] : ( plus_plus @ A @ ( F3 @ N4 ) @ ( G2 @ N4 ) )
@ ( plus_plus @ A @ A3 @ B3 ) ) ) ) ) ).
% sums_add
thf(fact_3362_sums__mult,axiom,
! [A: $tType] :
( ( real_V4412858255891104859lgebra @ A )
=> ! [F3: nat > A,A3: A,C2: A] :
( ( sums @ A @ F3 @ A3 )
=> ( sums @ A
@ ^ [N4: nat] : ( times_times @ A @ C2 @ ( F3 @ N4 ) )
@ ( times_times @ A @ C2 @ A3 ) ) ) ) ).
% sums_mult
thf(fact_3363_sums__mult2,axiom,
! [A: $tType] :
( ( real_V4412858255891104859lgebra @ A )
=> ! [F3: nat > A,A3: A,C2: A] :
( ( sums @ A @ F3 @ A3 )
=> ( sums @ A
@ ^ [N4: nat] : ( times_times @ A @ ( F3 @ N4 ) @ C2 )
@ ( times_times @ A @ A3 @ C2 ) ) ) ) ).
% sums_mult2
thf(fact_3364_sums__diff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: nat > A,A3: A,G2: nat > A,B3: A] :
( ( sums @ A @ F3 @ A3 )
=> ( ( sums @ A @ G2 @ B3 )
=> ( sums @ A
@ ^ [N4: nat] : ( minus_minus @ A @ ( F3 @ N4 ) @ ( G2 @ N4 ) )
@ ( minus_minus @ A @ A3 @ B3 ) ) ) ) ) ).
% sums_diff
thf(fact_3365_sums__divide,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: nat > A,A3: A,C2: A] :
( ( sums @ A @ F3 @ A3 )
=> ( sums @ A
@ ^ [N4: nat] : ( divide_divide @ A @ ( F3 @ N4 ) @ C2 )
@ ( divide_divide @ A @ A3 @ C2 ) ) ) ) ).
% sums_divide
thf(fact_3366_sums__minus,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: nat > A,A3: A] :
( ( sums @ A @ F3 @ A3 )
=> ( sums @ A
@ ^ [N4: nat] : ( uminus_uminus @ A @ ( F3 @ N4 ) )
@ ( uminus_uminus @ A @ A3 ) ) ) ) ).
% sums_minus
thf(fact_3367_nat__numeral__as__int,axiom,
( ( numeral_numeral @ nat )
= ( ^ [I4: num] : ( nat2 @ ( numeral_numeral @ int @ I4 ) ) ) ) ).
% nat_numeral_as_int
thf(fact_3368_nat__zero__as__int,axiom,
( ( zero_zero @ nat )
= ( nat2 @ ( zero_zero @ int ) ) ) ).
% nat_zero_as_int
thf(fact_3369_cos__le__one,axiom,
! [X: real] : ( ord_less_eq @ real @ ( cos @ real @ X ) @ ( one_one @ real ) ) ).
% cos_le_one
thf(fact_3370_nat__mono,axiom,
! [X: int,Y: int] :
( ( ord_less_eq @ int @ X @ Y )
=> ( ord_less_eq @ nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ).
% nat_mono
thf(fact_3371_nat__one__as__int,axiom,
( ( one_one @ nat )
= ( nat2 @ ( one_one @ int ) ) ) ).
% nat_one_as_int
thf(fact_3372_eq__nat__nat__iff,axiom,
! [Z: int,Z7: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z7 )
=> ( ( ( nat2 @ Z )
= ( nat2 @ Z7 ) )
= ( Z = Z7 ) ) ) ) ).
% eq_nat_nat_iff
thf(fact_3373_all__nat,axiom,
( ( ^ [P2: nat > $o] :
! [X4: nat] : ( P2 @ X4 ) )
= ( ^ [P3: nat > $o] :
! [X2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X2 )
=> ( P3 @ ( nat2 @ X2 ) ) ) ) ) ).
% all_nat
thf(fact_3374_ex__nat,axiom,
( ( ^ [P2: nat > $o] :
? [X4: nat] : ( P2 @ X4 ) )
= ( ^ [P3: nat > $o] :
? [X2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X2 )
& ( P3 @ ( nat2 @ X2 ) ) ) ) ) ).
% ex_nat
thf(fact_3375_sums__mult2__iff,axiom,
! [A: $tType] :
( ( ( field @ A )
& ( real_V4412858255891104859lgebra @ A ) )
=> ! [C2: A,F3: nat > A,D2: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( sums @ A
@ ^ [N4: nat] : ( times_times @ A @ ( F3 @ N4 ) @ C2 )
@ ( times_times @ A @ D2 @ C2 ) )
= ( sums @ A @ F3 @ D2 ) ) ) ) ).
% sums_mult2_iff
thf(fact_3376_sums__mult__iff,axiom,
! [A: $tType] :
( ( ( field @ A )
& ( real_V4412858255891104859lgebra @ A ) )
=> ! [C2: A,F3: nat > A,D2: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( sums @ A
@ ^ [N4: nat] : ( times_times @ A @ C2 @ ( F3 @ N4 ) )
@ ( times_times @ A @ C2 @ D2 ) )
= ( sums @ A @ F3 @ D2 ) ) ) ) ).
% sums_mult_iff
thf(fact_3377_unat__eq__nat__uint,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( semiring_1_unsigned @ A @ nat )
= ( ^ [W2: word @ A] : ( nat2 @ ( semiring_1_unsigned @ A @ int @ W2 ) ) ) ) ) ).
% unat_eq_nat_uint
thf(fact_3378_cos__arctan__not__zero,axiom,
! [X: real] :
( ( cos @ real @ ( arctan @ X ) )
!= ( zero_zero @ real ) ) ).
% cos_arctan_not_zero
thf(fact_3379_unset__bit__nat__def,axiom,
( ( bit_se2638667681897837118et_bit @ nat )
= ( ^ [M3: nat,N4: nat] : ( nat2 @ ( bit_se2638667681897837118et_bit @ int @ M3 @ ( semiring_1_of_nat @ int @ N4 ) ) ) ) ) ).
% unset_bit_nat_def
thf(fact_3380_cos__one__sin__zero,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A] :
( ( ( cos @ A @ X )
= ( one_one @ A ) )
=> ( ( sin @ A @ X )
= ( zero_zero @ A ) ) ) ) ).
% cos_one_sin_zero
thf(fact_3381_sums__mult__D,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [C2: A,F3: nat > A,A3: A] :
( ( sums @ A
@ ^ [N4: nat] : ( times_times @ A @ C2 @ ( F3 @ N4 ) )
@ A3 )
=> ( ( C2
!= ( zero_zero @ A ) )
=> ( sums @ A @ F3 @ ( divide_divide @ A @ A3 @ C2 ) ) ) ) ) ).
% sums_mult_D
thf(fact_3382_sums__Suc__imp,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: nat > A,S2: A] :
( ( ( F3 @ ( zero_zero @ nat ) )
= ( zero_zero @ A ) )
=> ( ( sums @ A
@ ^ [N4: nat] : ( F3 @ ( suc @ N4 ) )
@ S2 )
=> ( sums @ A @ F3 @ S2 ) ) ) ) ).
% sums_Suc_imp
thf(fact_3383_sums__Suc,axiom,
! [A: $tType] :
( ( ( topolo5987344860129210374id_add @ A )
& ( topological_t2_space @ A ) )
=> ! [F3: nat > A,L: A] :
( ( sums @ A
@ ^ [N4: nat] : ( F3 @ ( suc @ N4 ) )
@ L )
=> ( sums @ A @ F3 @ ( plus_plus @ A @ L @ ( F3 @ ( zero_zero @ nat ) ) ) ) ) ) ).
% sums_Suc
thf(fact_3384_sums__Suc__iff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: nat > A,S2: A] :
( ( sums @ A
@ ^ [N4: nat] : ( F3 @ ( suc @ N4 ) )
@ S2 )
= ( sums @ A @ F3 @ ( plus_plus @ A @ S2 @ ( F3 @ ( zero_zero @ nat ) ) ) ) ) ) ).
% sums_Suc_iff
thf(fact_3385_sums__zero__iff__shift,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [N: nat,F3: nat > A,S2: A] :
( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ N )
=> ( ( F3 @ I3 )
= ( zero_zero @ A ) ) )
=> ( ( sums @ A
@ ^ [I4: nat] : ( F3 @ ( plus_plus @ nat @ I4 @ N ) )
@ S2 )
= ( sums @ A @ F3 @ S2 ) ) ) ) ).
% sums_zero_iff_shift
thf(fact_3386_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_3387_cos__inj__pi,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ X @ pi )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y )
=> ( ( ord_less_eq @ real @ Y @ pi )
=> ( ( ( cos @ real @ X )
= ( cos @ real @ Y ) )
=> ( X = Y ) ) ) ) ) ) ).
% cos_inj_pi
thf(fact_3388_cos__mono__le__eq,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ X @ pi )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y )
=> ( ( ord_less_eq @ real @ Y @ pi )
=> ( ( ord_less_eq @ real @ ( cos @ real @ X ) @ ( cos @ real @ Y ) )
= ( ord_less_eq @ real @ Y @ X ) ) ) ) ) ) ).
% cos_mono_le_eq
thf(fact_3389_cos__monotone__0__pi__le,axiom,
! [Y: real,X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y )
=> ( ( ord_less_eq @ real @ Y @ X )
=> ( ( ord_less_eq @ real @ X @ pi )
=> ( ord_less_eq @ real @ ( cos @ real @ X ) @ ( cos @ real @ Y ) ) ) ) ) ).
% cos_monotone_0_pi_le
thf(fact_3390_cos__ge__minus__one,axiom,
! [X: real] : ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ ( cos @ real @ X ) ) ).
% cos_ge_minus_one
thf(fact_3391_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_3392_of__nat__ceiling,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [R3: A] : ( ord_less_eq @ A @ R3 @ ( semiring_1_of_nat @ A @ ( nat2 @ ( archimedean_ceiling @ A @ R3 ) ) ) ) ) ).
% of_nat_ceiling
thf(fact_3393_nat__le__iff,axiom,
! [X: int,N: nat] :
( ( ord_less_eq @ nat @ ( nat2 @ X ) @ N )
= ( ord_less_eq @ int @ X @ ( semiring_1_of_nat @ int @ N ) ) ) ).
% nat_le_iff
thf(fact_3394_nat__int__add,axiom,
! [A3: nat,B3: nat] :
( ( nat2 @ ( plus_plus @ int @ ( semiring_1_of_nat @ int @ A3 ) @ ( semiring_1_of_nat @ int @ B3 ) ) )
= ( plus_plus @ nat @ A3 @ B3 ) ) ).
% nat_int_add
thf(fact_3395_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_3396_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_3397_abs__cos__le__one,axiom,
! [X: real] : ( ord_less_eq @ real @ ( abs_abs @ real @ ( cos @ real @ X ) ) @ ( one_one @ real ) ) ).
% abs_cos_le_one
thf(fact_3398_nat__uint__less__helper,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y: word @ A,Z: nat,X: word @ A] :
( ( ( nat2 @ ( semiring_1_unsigned @ A @ int @ Y ) )
= Z )
=> ( ( ord_less @ ( word @ A ) @ X @ Y )
=> ( ord_less @ nat @ ( nat2 @ ( semiring_1_unsigned @ A @ int @ X ) ) @ Z ) ) ) ) ).
% nat_uint_less_helper
thf(fact_3399_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_3400_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_3401_real__nat__ceiling__ge,axiom,
! [X: real] : ( ord_less_eq @ real @ X @ ( semiring_1_of_nat @ real @ ( nat2 @ ( archimedean_ceiling @ real @ X ) ) ) ) ).
% real_nat_ceiling_ge
thf(fact_3402_nat__plus__as__int,axiom,
( ( plus_plus @ nat )
= ( ^ [A5: nat,B5: nat] : ( nat2 @ ( plus_plus @ int @ ( semiring_1_of_nat @ int @ A5 ) @ ( semiring_1_of_nat @ int @ B5 ) ) ) ) ) ).
% nat_plus_as_int
thf(fact_3403_nat__times__as__int,axiom,
( ( times_times @ nat )
= ( ^ [A5: nat,B5: nat] : ( nat2 @ ( times_times @ int @ ( semiring_1_of_nat @ int @ A5 ) @ ( semiring_1_of_nat @ int @ B5 ) ) ) ) ) ).
% nat_times_as_int
thf(fact_3404_nat__minus__as__int,axiom,
( ( minus_minus @ nat )
= ( ^ [A5: nat,B5: nat] : ( nat2 @ ( minus_minus @ int @ ( semiring_1_of_nat @ int @ A5 ) @ ( semiring_1_of_nat @ int @ B5 ) ) ) ) ) ).
% nat_minus_as_int
thf(fact_3405_nat__div__as__int,axiom,
( ( divide_divide @ nat )
= ( ^ [A5: nat,B5: nat] : ( nat2 @ ( divide_divide @ int @ ( semiring_1_of_nat @ int @ A5 ) @ ( semiring_1_of_nat @ int @ B5 ) ) ) ) ) ).
% nat_div_as_int
thf(fact_3406_nat__mod__as__int,axiom,
( ( modulo_modulo @ nat )
= ( ^ [A5: nat,B5: nat] : ( nat2 @ ( modulo_modulo @ int @ ( semiring_1_of_nat @ int @ A5 ) @ ( semiring_1_of_nat @ int @ B5 ) ) ) ) ) ).
% nat_mod_as_int
thf(fact_3407_powser__sums__if,axiom,
! [A: $tType] :
( ( ( ring_1 @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [M: nat,Z: A] :
( sums @ A
@ ^ [N4: nat] : ( times_times @ A @ ( if @ A @ ( N4 = M ) @ ( one_one @ A ) @ ( zero_zero @ A ) ) @ ( power_power @ A @ Z @ N4 ) )
@ ( power_power @ A @ Z @ M ) ) ) ).
% powser_sums_if
thf(fact_3408_powser__sums__zero,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [A3: nat > A] :
( sums @ A
@ ^ [N4: nat] : ( times_times @ A @ ( A3 @ N4 ) @ ( power_power @ A @ ( zero_zero @ A ) @ N4 ) )
@ ( A3 @ ( zero_zero @ nat ) ) ) ) ).
% powser_sums_zero
thf(fact_3409_sin__zero__norm__cos__one,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A] :
( ( ( sin @ A @ X )
= ( zero_zero @ A ) )
=> ( ( real_V7770717601297561774m_norm @ A @ ( cos @ A @ X ) )
= ( one_one @ real ) ) ) ) ).
% sin_zero_norm_cos_one
thf(fact_3410_cos__two__neq__zero,axiom,
( ( cos @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) )
!= ( zero_zero @ real ) ) ).
% cos_two_neq_zero
thf(fact_3411_of__nat__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [R3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ R3 )
=> ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ ( nat2 @ ( archim6421214686448440834_floor @ A @ R3 ) ) ) @ R3 ) ) ) ).
% of_nat_floor
thf(fact_3412_cos__mono__less__eq,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ X @ pi )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y )
=> ( ( ord_less_eq @ real @ Y @ pi )
=> ( ( ord_less @ real @ ( cos @ real @ X ) @ ( cos @ real @ Y ) )
= ( ord_less @ real @ Y @ X ) ) ) ) ) ) ).
% cos_mono_less_eq
thf(fact_3413_cos__monotone__0__pi,axiom,
! [Y: real,X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y )
=> ( ( ord_less @ real @ Y @ X )
=> ( ( ord_less_eq @ real @ X @ pi )
=> ( ord_less @ real @ ( cos @ real @ X ) @ ( cos @ real @ Y ) ) ) ) ) ).
% cos_monotone_0_pi
thf(fact_3414_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_3415_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_3416_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_3417_split__nat,axiom,
! [P: nat > $o,I: int] :
( ( P @ ( nat2 @ I ) )
= ( ! [N4: nat] :
( ( I
= ( semiring_1_of_nat @ int @ N4 ) )
=> ( P @ N4 ) )
& ( ( ord_less @ int @ I @ ( zero_zero @ int ) )
=> ( P @ ( zero_zero @ nat ) ) ) ) ) ).
% split_nat
thf(fact_3418_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_3419_nat__add__distrib,axiom,
! [Z: int,Z7: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z7 )
=> ( ( nat2 @ ( plus_plus @ int @ Z @ Z7 ) )
= ( plus_plus @ nat @ ( nat2 @ Z ) @ ( nat2 @ Z7 ) ) ) ) ) ).
% nat_add_distrib
thf(fact_3420_cos__monotone__minus__pi__0_H,axiom,
! [Y: real,X: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ pi ) @ Y )
=> ( ( ord_less_eq @ real @ Y @ X )
=> ( ( ord_less_eq @ real @ X @ ( zero_zero @ real ) )
=> ( ord_less_eq @ real @ ( cos @ real @ Y ) @ ( cos @ real @ X ) ) ) ) ) ).
% cos_monotone_minus_pi_0'
thf(fact_3421_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_3422_Suc__as__int,axiom,
( suc
= ( ^ [A5: nat] : ( nat2 @ ( plus_plus @ int @ ( semiring_1_of_nat @ int @ A5 ) @ ( one_one @ int ) ) ) ) ) ).
% Suc_as_int
thf(fact_3423_le__mult__nat__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [A3: A,B3: A] : ( ord_less_eq @ nat @ ( times_times @ nat @ ( nat2 @ ( archim6421214686448440834_floor @ A @ A3 ) ) @ ( nat2 @ ( archim6421214686448440834_floor @ A @ B3 ) ) ) @ ( nat2 @ ( archim6421214686448440834_floor @ A @ ( times_times @ A @ A3 @ B3 ) ) ) ) ) ).
% le_mult_nat_floor
thf(fact_3424_nat__mult__distrib,axiom,
! [Z: int,Z7: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z )
=> ( ( nat2 @ ( times_times @ int @ Z @ Z7 ) )
= ( times_times @ nat @ ( nat2 @ Z ) @ ( nat2 @ Z7 ) ) ) ) ).
% nat_mult_distrib
thf(fact_3425_nat__diff__distrib,axiom,
! [Z7: int,Z: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z7 )
=> ( ( ord_less_eq @ int @ Z7 @ Z )
=> ( ( nat2 @ ( minus_minus @ int @ Z @ Z7 ) )
= ( minus_minus @ nat @ ( nat2 @ Z ) @ ( nat2 @ Z7 ) ) ) ) ) ).
% nat_diff_distrib
thf(fact_3426_nat__diff__distrib_H,axiom,
! [X: int,Y: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y )
=> ( ( nat2 @ ( minus_minus @ int @ X @ Y ) )
= ( minus_minus @ nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ) ) ).
% nat_diff_distrib'
thf(fact_3427_nat__abs__triangle__ineq,axiom,
! [K: int,L: int] : ( ord_less_eq @ nat @ ( nat2 @ ( abs_abs @ int @ ( plus_plus @ int @ K @ L ) ) ) @ ( plus_plus @ nat @ ( nat2 @ ( abs_abs @ int @ K ) ) @ ( nat2 @ ( abs_abs @ int @ L ) ) ) ) ).
% nat_abs_triangle_ineq
thf(fact_3428_nat__div__distrib,axiom,
! [X: int,Y: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
=> ( ( nat2 @ ( divide_divide @ int @ X @ Y ) )
= ( divide_divide @ nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ) ).
% nat_div_distrib
thf(fact_3429_nat__div__distrib_H,axiom,
! [Y: int,X: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y )
=> ( ( nat2 @ ( divide_divide @ int @ X @ Y ) )
= ( divide_divide @ nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ) ).
% nat_div_distrib'
thf(fact_3430_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_3431_nat__mod__distrib,axiom,
! [X: int,Y: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y )
=> ( ( nat2 @ ( modulo_modulo @ int @ X @ Y ) )
= ( modulo_modulo @ nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ) ) ).
% nat_mod_distrib
thf(fact_3432_sin__zero__abs__cos__one,axiom,
! [X: real] :
( ( ( sin @ real @ X )
= ( zero_zero @ real ) )
=> ( ( abs_abs @ real @ ( cos @ real @ X ) )
= ( one_one @ real ) ) ) ).
% sin_zero_abs_cos_one
thf(fact_3433_div__abs__eq__div__nat,axiom,
! [K: int,L: int] :
( ( divide_divide @ int @ ( abs_abs @ int @ K ) @ ( abs_abs @ int @ L ) )
= ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ ( nat2 @ ( abs_abs @ int @ K ) ) @ ( nat2 @ ( abs_abs @ int @ L ) ) ) ) ) ).
% div_abs_eq_div_nat
thf(fact_3434_word__of__int__nat,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
=> ( ( ring_1_of_int @ ( word @ A ) @ X )
= ( semiring_1_of_nat @ ( word @ A ) @ ( nat2 @ X ) ) ) ) ) ).
% word_of_int_nat
thf(fact_3435_nat__floor__neg,axiom,
! [X: real] :
( ( ord_less_eq @ real @ X @ ( zero_zero @ real ) )
=> ( ( nat2 @ ( archim6421214686448440834_floor @ real @ X ) )
= ( zero_zero @ nat ) ) ) ).
% nat_floor_neg
thf(fact_3436_mod__abs__eq__div__nat,axiom,
! [K: int,L: int] :
( ( modulo_modulo @ int @ ( abs_abs @ int @ K ) @ ( abs_abs @ int @ L ) )
= ( semiring_1_of_nat @ int @ ( modulo_modulo @ nat @ ( nat2 @ ( abs_abs @ int @ K ) ) @ ( nat2 @ ( abs_abs @ int @ L ) ) ) ) ) ).
% mod_abs_eq_div_nat
thf(fact_3437_floor__eq3,axiom,
! [N: nat,X: real] :
( ( ord_less @ real @ ( semiring_1_of_nat @ real @ N ) @ X )
=> ( ( ord_less @ real @ X @ ( semiring_1_of_nat @ real @ ( suc @ N ) ) )
=> ( ( nat2 @ ( archim6421214686448440834_floor @ real @ X ) )
= N ) ) ) ).
% floor_eq3
thf(fact_3438_le__nat__floor,axiom,
! [X: nat,A3: real] :
( ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ X ) @ A3 )
=> ( ord_less_eq @ nat @ X @ ( nat2 @ ( archim6421214686448440834_floor @ real @ A3 ) ) ) ) ).
% le_nat_floor
thf(fact_3439_sin__double,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A] :
( ( sin @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X ) )
= ( times_times @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( sin @ A @ X ) ) @ ( cos @ A @ X ) ) ) ) ).
% sin_double
thf(fact_3440_cos__paired,axiom,
! [X: real] :
( sums @ real
@ ^ [N4: nat] : ( times_times @ real @ ( divide_divide @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ N4 ) @ ( semiring_char_0_fact @ real @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N4 ) ) ) @ ( power_power @ real @ X @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N4 ) ) )
@ ( cos @ real @ X ) ) ).
% cos_paired
thf(fact_3441_nat__2,axiom,
( ( nat2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) )
= ( suc @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% nat_2
thf(fact_3442_cos__two__less__zero,axiom,
ord_less @ real @ ( cos @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( zero_zero @ real ) ).
% cos_two_less_zero
thf(fact_3443_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_3444_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 ) )
& ! [Y5: real] :
( ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y5 )
& ( ord_less_eq @ real @ Y5 @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) )
& ( ( cos @ real @ Y5 )
= ( zero_zero @ real ) ) )
=> ( Y5 = X3 ) ) ) ).
% cos_is_zero
thf(fact_3445_cos__monotone__minus__pi__0,axiom,
! [Y: real,X: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ pi ) @ Y )
=> ( ( ord_less @ real @ Y @ X )
=> ( ( ord_less_eq @ real @ X @ ( zero_zero @ real ) )
=> ( ord_less @ real @ ( cos @ real @ Y ) @ ( cos @ real @ X ) ) ) ) ) ).
% cos_monotone_minus_pi_0
thf(fact_3446_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_3447_cos__total,axiom,
! [Y: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ Y )
=> ( ( ord_less_eq @ real @ Y @ ( one_one @ real ) )
=> ? [X3: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 )
& ( ord_less_eq @ real @ X3 @ pi )
& ( ( cos @ real @ X3 )
= Y )
& ! [Y5: real] :
( ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y5 )
& ( ord_less_eq @ real @ Y5 @ pi )
& ( ( cos @ real @ Y5 )
= Y ) )
=> ( Y5 = X3 ) ) ) ) ) ).
% cos_total
thf(fact_3448_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_3449_nat__mult__distrib__neg,axiom,
! [Z: int,Z7: int] :
( ( ord_less_eq @ int @ Z @ ( zero_zero @ int ) )
=> ( ( nat2 @ ( times_times @ int @ Z @ Z7 ) )
= ( times_times @ nat @ ( nat2 @ ( uminus_uminus @ int @ Z ) ) @ ( nat2 @ ( uminus_uminus @ int @ Z7 ) ) ) ) ) ).
% nat_mult_distrib_neg
thf(fact_3450_sincos__principal__value,axiom,
! [X: real] :
? [Y4: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ pi ) @ Y4 )
& ( ord_less_eq @ real @ Y4 @ pi )
& ( ( sin @ real @ Y4 )
= ( sin @ real @ X ) )
& ( ( cos @ real @ Y4 )
= ( cos @ real @ X ) ) ) ).
% sincos_principal_value
thf(fact_3451_nat__abs__int__diff,axiom,
! [A3: nat,B3: nat] :
( ( ( ord_less_eq @ nat @ A3 @ B3 )
=> ( ( nat2 @ ( abs_abs @ int @ ( minus_minus @ int @ ( semiring_1_of_nat @ int @ A3 ) @ ( semiring_1_of_nat @ int @ B3 ) ) ) )
= ( minus_minus @ nat @ B3 @ A3 ) ) )
& ( ~ ( ord_less_eq @ nat @ A3 @ B3 )
=> ( ( nat2 @ ( abs_abs @ int @ ( minus_minus @ int @ ( semiring_1_of_nat @ int @ A3 ) @ ( semiring_1_of_nat @ int @ B3 ) ) ) )
= ( minus_minus @ nat @ A3 @ B3 ) ) ) ) ).
% nat_abs_int_diff
thf(fact_3452_floor__eq4,axiom,
! [N: nat,X: real] :
( ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ N ) @ X )
=> ( ( ord_less @ real @ X @ ( semiring_1_of_nat @ real @ ( suc @ N ) ) )
=> ( ( nat2 @ ( archim6421214686448440834_floor @ real @ X ) )
= N ) ) ) ).
% floor_eq4
thf(fact_3453_diff__nat__eq__if,axiom,
! [Z7: int,Z: int] :
( ( ( ord_less @ int @ Z7 @ ( zero_zero @ int ) )
=> ( ( minus_minus @ nat @ ( nat2 @ Z ) @ ( nat2 @ Z7 ) )
= ( nat2 @ Z ) ) )
& ( ~ ( ord_less @ int @ Z7 @ ( zero_zero @ int ) )
=> ( ( minus_minus @ nat @ ( nat2 @ Z ) @ ( nat2 @ Z7 ) )
= ( if @ nat @ ( ord_less @ int @ ( minus_minus @ int @ Z @ Z7 ) @ ( zero_zero @ int ) ) @ ( zero_zero @ nat ) @ ( nat2 @ ( minus_minus @ int @ Z @ Z7 ) ) ) ) ) ) ).
% diff_nat_eq_if
thf(fact_3454_of__int__of__nat,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ( ( ring_1_of_int @ A )
= ( ^ [K3: int] : ( if @ A @ ( ord_less @ int @ K3 @ ( zero_zero @ int ) ) @ ( uminus_uminus @ A @ ( semiring_1_of_nat @ A @ ( nat2 @ ( uminus_uminus @ int @ K3 ) ) ) ) @ ( semiring_1_of_nat @ A @ ( nat2 @ K3 ) ) ) ) ) ) ).
% of_int_of_nat
thf(fact_3455_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_3456_sin__cos__le1,axiom,
! [X: real,Y: real] : ( ord_less_eq @ real @ ( abs_abs @ real @ ( plus_plus @ real @ ( times_times @ real @ ( sin @ real @ X ) @ ( sin @ real @ Y ) ) @ ( times_times @ real @ ( cos @ real @ X ) @ ( cos @ real @ Y ) ) ) ) @ ( one_one @ real ) ) ).
% sin_cos_le1
thf(fact_3457_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_3458_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_3459_cos__squared__eq,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A] :
( ( power_power @ A @ ( cos @ A @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ ( sin @ A @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% cos_squared_eq
thf(fact_3460_sin__squared__eq,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A] :
( ( power_power @ A @ ( sin @ A @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ ( cos @ A @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% sin_squared_eq
thf(fact_3461_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_3462_power__half__series,axiom,
( sums @ real
@ ^ [N4: nat] : ( power_power @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( suc @ N4 ) )
@ ( one_one @ real ) ) ).
% power_half_series
thf(fact_3463_cos__double__less__one,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ X @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) )
=> ( ord_less @ real @ ( cos @ real @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ X ) ) @ ( one_one @ real ) ) ) ) ).
% cos_double_less_one
thf(fact_3464_cos__gt__zero,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ X @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( cos @ real @ X ) ) ) ) ).
% cos_gt_zero
thf(fact_3465_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_3466_cos__one__2pi__int,axiom,
! [X: real] :
( ( ( cos @ real @ X )
= ( one_one @ real ) )
= ( ? [X2: int] :
( X
= ( times_times @ real @ ( times_times @ real @ ( ring_1_of_int @ real @ X2 ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ pi ) ) ) ) ).
% cos_one_2pi_int
thf(fact_3467_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_3468_cos__treble__cos,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A] :
( ( cos @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit1 @ one2 ) ) @ X ) )
= ( minus_minus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ ( cos @ A @ X ) @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit1 @ one2 ) ) @ ( cos @ A @ X ) ) ) ) ) ).
% cos_treble_cos
thf(fact_3469_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_3470_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_3471_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_3472_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_3473_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_3474_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_3475_cos__double,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A] :
( ( cos @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X ) )
= ( minus_minus @ A @ ( power_power @ A @ ( cos @ A @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ ( sin @ A @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% cos_double
thf(fact_3476_sums__if_H,axiom,
! [G2: nat > real,X: real] :
( ( sums @ real @ G2 @ X )
=> ( sums @ real
@ ^ [N4: nat] : ( if @ real @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N4 ) @ ( zero_zero @ real ) @ ( G2 @ ( divide_divide @ nat @ ( minus_minus @ nat @ N4 @ ( one_one @ nat ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
@ X ) ) ).
% sums_if'
thf(fact_3477_cos__sin__eq,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ( ( cos @ A )
= ( ^ [X2: A] : ( sin @ A @ ( minus_minus @ A @ ( divide_divide @ A @ ( real_Vector_of_real @ A @ pi ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ X2 ) ) ) ) ) ).
% cos_sin_eq
thf(fact_3478_sin__cos__eq,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ( ( sin @ A )
= ( ^ [X2: A] : ( cos @ A @ ( minus_minus @ A @ ( divide_divide @ A @ ( real_Vector_of_real @ A @ pi ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ X2 ) ) ) ) ) ).
% sin_cos_eq
thf(fact_3479_sums__if,axiom,
! [G2: nat > real,X: real,F3: nat > real,Y: real] :
( ( sums @ real @ G2 @ X )
=> ( ( sums @ real @ F3 @ Y )
=> ( sums @ real
@ ^ [N4: nat] : ( if @ real @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N4 ) @ ( F3 @ ( divide_divide @ nat @ N4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( G2 @ ( divide_divide @ nat @ ( minus_minus @ nat @ N4 @ ( one_one @ nat ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
@ ( plus_plus @ real @ X @ Y ) ) ) ) ).
% sums_if
thf(fact_3480_cos__gt__zero__pi,axiom,
! [X: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X )
=> ( ( ord_less @ real @ X @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( cos @ real @ X ) ) ) ) ).
% cos_gt_zero_pi
thf(fact_3481_cos__ge__zero,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X )
=> ( ( ord_less_eq @ real @ X @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( cos @ real @ X ) ) ) ) ).
% cos_ge_zero
thf(fact_3482_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_3483_cos__one__2pi,axiom,
! [X: real] :
( ( ( cos @ real @ X )
= ( one_one @ real ) )
= ( ? [X2: nat] :
( X
= ( times_times @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ X2 ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ pi ) )
| ? [X2: nat] :
( X
= ( uminus_uminus @ real @ ( times_times @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ X2 ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ pi ) ) ) ) ) ).
% cos_one_2pi
thf(fact_3484_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_3485_minus__sin__cos__eq,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A] :
( ( uminus_uminus @ A @ ( sin @ A @ X ) )
= ( cos @ A @ ( plus_plus @ A @ X @ ( divide_divide @ A @ ( real_Vector_of_real @ A @ pi ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% minus_sin_cos_eq
thf(fact_3486_sincos__total__pi,axiom,
! [Y: real,X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y )
=> ( ( ( plus_plus @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( one_one @ real ) )
=> ? [T3: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ T3 )
& ( ord_less_eq @ real @ T3 @ pi )
& ( X
= ( cos @ real @ T3 ) )
& ( Y
= ( sin @ real @ T3 ) ) ) ) ) ).
% sincos_total_pi
thf(fact_3487_sin__expansion__lemma,axiom,
! [X: real,M: nat] :
( ( sin @ real @ ( plus_plus @ real @ X @ ( divide_divide @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ ( suc @ M ) ) @ pi ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) )
= ( cos @ real @ ( plus_plus @ real @ X @ ( divide_divide @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ M ) @ pi ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ).
% sin_expansion_lemma
thf(fact_3488_cos__zero__iff__int,axiom,
! [X: real] :
( ( ( cos @ real @ X )
= ( zero_zero @ real ) )
= ( ? [I4: int] :
( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ I4 )
& ( X
= ( times_times @ real @ ( ring_1_of_int @ real @ I4 ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% cos_zero_iff_int
thf(fact_3489_powr__int,axiom,
! [X: real,I: int] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ I )
=> ( ( powr @ real @ X @ ( ring_1_of_int @ real @ I ) )
= ( power_power @ real @ X @ ( nat2 @ I ) ) ) )
& ( ~ ( ord_less_eq @ int @ ( zero_zero @ int ) @ I )
=> ( ( powr @ real @ X @ ( ring_1_of_int @ real @ I ) )
= ( divide_divide @ real @ ( one_one @ real ) @ ( power_power @ real @ X @ ( nat2 @ ( uminus_uminus @ int @ I ) ) ) ) ) ) ) ) ).
% powr_int
thf(fact_3490_cos__zero__lemma,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ( cos @ real @ X )
= ( zero_zero @ real ) )
=> ? [N2: nat] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
& ( X
= ( times_times @ real @ ( semiring_1_of_nat @ real @ N2 ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% cos_zero_lemma
thf(fact_3491_cos__zero__iff,axiom,
! [X: real] :
( ( ( cos @ real @ X )
= ( zero_zero @ real ) )
= ( ? [N4: nat] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N4 )
& ( X
= ( times_times @ real @ ( semiring_1_of_nat @ real @ N4 ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) )
| ? [N4: nat] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N4 )
& ( X
= ( uminus_uminus @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N4 ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ).
% cos_zero_iff
thf(fact_3492_cos__expansion__lemma,axiom,
! [X: real,M: nat] :
( ( cos @ real @ ( plus_plus @ real @ X @ ( divide_divide @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ ( suc @ M ) ) @ pi ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) )
= ( uminus_uminus @ real @ ( sin @ real @ ( plus_plus @ real @ X @ ( divide_divide @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ M ) @ pi ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% cos_expansion_lemma
thf(fact_3493_geometric__deriv__sums,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [Z: A] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ Z ) @ ( one_one @ real ) )
=> ( sums @ A
@ ^ [N4: nat] : ( times_times @ A @ ( semiring_1_of_nat @ A @ ( suc @ N4 ) ) @ ( power_power @ A @ Z @ N4 ) )
@ ( divide_divide @ A @ ( one_one @ A ) @ ( power_power @ A @ ( minus_minus @ A @ ( one_one @ A ) @ Z ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% geometric_deriv_sums
thf(fact_3494_sincos__total__pi__half,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y )
=> ( ( ( plus_plus @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( one_one @ real ) )
=> ? [T3: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ T3 )
& ( ord_less_eq @ real @ T3 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
& ( X
= ( cos @ real @ T3 ) )
& ( Y
= ( sin @ real @ T3 ) ) ) ) ) ) ).
% sincos_total_pi_half
thf(fact_3495_sincos__total__2pi__le,axiom,
! [X: real,Y: real] :
( ( ( plus_plus @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( one_one @ real ) )
=> ? [T3: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ T3 )
& ( ord_less_eq @ real @ T3 @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) )
& ( X
= ( cos @ real @ T3 ) )
& ( Y
= ( sin @ real @ T3 ) ) ) ) ).
% sincos_total_2pi_le
thf(fact_3496_vebt__deletei__rule,axiom,
! [N: nat,S2: set @ nat,Ti: vEBT_VEBTi,X: nat] : ( time_htt @ vEBT_VEBTi @ ( vEBT_Intf_vebt_assn @ N @ S2 @ Ti ) @ ( vEBT_vebt_deletei @ Ti @ X ) @ ( vEBT_Intf_vebt_assn @ N @ ( minus_minus @ ( set @ nat ) @ S2 @ ( insert @ nat @ X @ ( bot_bot @ ( set @ nat ) ) ) ) ) @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) ) @ ( nat2 @ ( archimedean_ceiling @ real @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ N ) ) ) ) ) ) ) ).
% vebt_deletei_rule
thf(fact_3497_diffs__equiv,axiom,
! [A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( ring_1 @ A ) )
=> ! [C2: nat > A,X: A] :
( ( summable @ A
@ ^ [N4: nat] : ( times_times @ A @ ( diffs @ A @ C2 @ N4 ) @ ( power_power @ A @ X @ N4 ) ) )
=> ( sums @ A
@ ^ [N4: nat] : ( times_times @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N4 ) @ ( C2 @ N4 ) ) @ ( power_power @ A @ X @ ( minus_minus @ nat @ N4 @ ( suc @ ( zero_zero @ nat ) ) ) ) )
@ ( suminf @ A
@ ^ [N4: nat] : ( times_times @ A @ ( diffs @ A @ C2 @ N4 ) @ ( power_power @ A @ X @ N4 ) ) ) ) ) ) ).
% diffs_equiv
thf(fact_3498_tan__double,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A] :
( ( ( cos @ A @ X )
!= ( zero_zero @ A ) )
=> ( ( ( cos @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X ) )
!= ( zero_zero @ A ) )
=> ( ( tan @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X ) )
= ( divide_divide @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( tan @ A @ X ) ) @ ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ ( tan @ A @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ).
% tan_double
thf(fact_3499_vebt__memberi__rule,axiom,
! [N: nat,S2: set @ nat,Ti: vEBT_VEBTi,X: nat] :
( time_htt @ $o @ ( vEBT_Intf_vebt_assn @ N @ S2 @ Ti ) @ ( vEBT_vebt_memberi @ Ti @ X )
@ ^ [R5: $o] :
( times_times @ assn @ ( vEBT_Intf_vebt_assn @ N @ S2 @ Ti )
@ ( pure_assn
@ ( R5
= ( member @ nat @ X @ S2 ) ) ) )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ one2 ) ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ one2 ) ) ) @ ( nat2 @ ( archimedean_ceiling @ real @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ N ) ) ) ) ) ) ) ).
% vebt_memberi_rule
thf(fact_3500_complex__unimodular__polar,axiom,
! [Z: complex] :
( ( ( real_V7770717601297561774m_norm @ complex @ Z )
= ( one_one @ real ) )
=> ~ ! [T3: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ T3 )
=> ( ( ord_less @ real @ T3 @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) )
=> ( Z
!= ( complex2 @ ( cos @ real @ T3 ) @ ( sin @ real @ T3 ) ) ) ) ) ) ).
% complex_unimodular_polar
thf(fact_3501_tan__pi,axiom,
( ( tan @ real @ pi )
= ( zero_zero @ real ) ) ).
% tan_pi
thf(fact_3502_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_3503_tan__npi,axiom,
! [N: nat] :
( ( tan @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ pi ) )
= ( zero_zero @ real ) ) ).
% tan_npi
thf(fact_3504_tan__periodic__n,axiom,
! [X: real,N: num] :
( ( tan @ real @ ( plus_plus @ real @ X @ ( times_times @ real @ ( numeral_numeral @ real @ N ) @ pi ) ) )
= ( tan @ real @ X ) ) ).
% tan_periodic_n
thf(fact_3505_tan__periodic__nat,axiom,
! [X: real,N: nat] :
( ( tan @ real @ ( plus_plus @ real @ X @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ pi ) ) )
= ( tan @ real @ X ) ) ).
% tan_periodic_nat
thf(fact_3506_norm__cos__sin,axiom,
! [T: real] :
( ( real_V7770717601297561774m_norm @ complex @ ( complex2 @ ( cos @ real @ T ) @ ( sin @ real @ T ) ) )
= ( one_one @ real ) ) ).
% norm_cos_sin
thf(fact_3507_tan__periodic,axiom,
! [X: real] :
( ( tan @ real @ ( plus_plus @ real @ X @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) ) )
= ( tan @ real @ X ) ) ).
% tan_periodic
thf(fact_3508_zero__complex_Ocode,axiom,
( ( zero_zero @ complex )
= ( complex2 @ ( zero_zero @ real ) @ ( zero_zero @ real ) ) ) ).
% zero_complex.code
thf(fact_3509_Complex__eq__0,axiom,
! [A3: real,B3: real] :
( ( ( complex2 @ A3 @ B3 )
= ( zero_zero @ complex ) )
= ( ( A3
= ( zero_zero @ real ) )
& ( B3
= ( zero_zero @ real ) ) ) ) ).
% Complex_eq_0
thf(fact_3510_diffs__of__real,axiom,
! [A: $tType] :
( ( real_V2191834092415804123ebra_1 @ A )
=> ! [F3: nat > real] :
( ( diffs @ A
@ ^ [N4: nat] : ( real_Vector_of_real @ A @ ( F3 @ N4 ) ) )
= ( ^ [N4: nat] : ( real_Vector_of_real @ A @ ( diffs @ real @ F3 @ N4 ) ) ) ) ) ).
% diffs_of_real
thf(fact_3511_one__complex_Ocode,axiom,
( ( one_one @ complex )
= ( complex2 @ ( one_one @ real ) @ ( zero_zero @ real ) ) ) ).
% one_complex.code
thf(fact_3512_Complex__eq__1,axiom,
! [A3: real,B3: real] :
( ( ( complex2 @ A3 @ B3 )
= ( one_one @ complex ) )
= ( ( A3
= ( one_one @ real ) )
& ( B3
= ( zero_zero @ real ) ) ) ) ).
% Complex_eq_1
thf(fact_3513_diffs__minus,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [C2: nat > A] :
( ( diffs @ A
@ ^ [N4: nat] : ( uminus_uminus @ A @ ( C2 @ N4 ) ) )
= ( ^ [N4: nat] : ( uminus_uminus @ A @ ( diffs @ A @ C2 @ N4 ) ) ) ) ) ).
% diffs_minus
thf(fact_3514_Complex__eq__numeral,axiom,
! [A3: real,B3: real,W: num] :
( ( ( complex2 @ A3 @ B3 )
= ( numeral_numeral @ complex @ W ) )
= ( ( A3
= ( numeral_numeral @ real @ W ) )
& ( B3
= ( zero_zero @ real ) ) ) ) ).
% Complex_eq_numeral
thf(fact_3515_complex__eq__cancel__iff2,axiom,
! [X: real,Y: real,Xa: real] :
( ( ( complex2 @ X @ Y )
= ( real_Vector_of_real @ complex @ Xa ) )
= ( ( X = Xa )
& ( Y
= ( zero_zero @ real ) ) ) ) ).
% complex_eq_cancel_iff2
thf(fact_3516_complex__of__real__code,axiom,
( ( real_Vector_of_real @ complex )
= ( ^ [X2: real] : ( complex2 @ X2 @ ( zero_zero @ real ) ) ) ) ).
% complex_of_real_code
thf(fact_3517_complex__of__real__def,axiom,
( ( real_Vector_of_real @ complex )
= ( ^ [R5: real] : ( complex2 @ R5 @ ( zero_zero @ real ) ) ) ) ).
% complex_of_real_def
thf(fact_3518_Complex__eq__neg__1,axiom,
! [A3: real,B3: real] :
( ( ( complex2 @ A3 @ B3 )
= ( uminus_uminus @ complex @ ( one_one @ complex ) ) )
= ( ( A3
= ( uminus_uminus @ real @ ( one_one @ real ) ) )
& ( B3
= ( zero_zero @ real ) ) ) ) ).
% Complex_eq_neg_1
thf(fact_3519_Complex__eq__neg__numeral,axiom,
! [A3: real,B3: real,W: num] :
( ( ( complex2 @ A3 @ B3 )
= ( uminus_uminus @ complex @ ( numeral_numeral @ complex @ W ) ) )
= ( ( A3
= ( uminus_uminus @ real @ ( numeral_numeral @ real @ W ) ) )
& ( B3
= ( zero_zero @ real ) ) ) ) ).
% Complex_eq_neg_numeral
thf(fact_3520_diffs__def,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ( ( diffs @ A )
= ( ^ [C6: nat > A,N4: nat] : ( times_times @ A @ ( semiring_1_of_nat @ A @ ( suc @ N4 ) ) @ ( C6 @ ( suc @ N4 ) ) ) ) ) ) ).
% diffs_def
thf(fact_3521_termdiff__converges__all,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [C2: nat > A,X: A] :
( ! [X3: A] :
( summable @ A
@ ^ [N4: nat] : ( times_times @ A @ ( C2 @ N4 ) @ ( power_power @ A @ X3 @ N4 ) ) )
=> ( summable @ A
@ ^ [N4: nat] : ( times_times @ A @ ( diffs @ A @ C2 @ N4 ) @ ( power_power @ A @ X @ N4 ) ) ) ) ) ).
% termdiff_converges_all
thf(fact_3522_tan__45,axiom,
( ( tan @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ ( bit0 @ one2 ) ) ) ) )
= ( one_one @ real ) ) ).
% tan_45
thf(fact_3523_tan__gt__zero,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ X @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( tan @ real @ X ) ) ) ) ).
% tan_gt_zero
thf(fact_3524_lemma__tan__total,axiom,
! [Y: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ Y )
=> ? [X3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
& ( ord_less @ real @ X3 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
& ( ord_less @ real @ Y @ ( tan @ real @ X3 ) ) ) ) ).
% lemma_tan_total
thf(fact_3525_tan__total,axiom,
! [Y: real] :
? [X3: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X3 )
& ( ord_less @ real @ X3 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
& ( ( tan @ real @ X3 )
= Y )
& ! [Y5: real] :
( ( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ Y5 )
& ( ord_less @ real @ Y5 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
& ( ( tan @ real @ Y5 )
= Y ) )
=> ( Y5 = X3 ) ) ) ).
% tan_total
thf(fact_3526_tan__monotone,axiom,
! [Y: real,X: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ Y )
=> ( ( ord_less @ real @ Y @ X )
=> ( ( ord_less @ real @ X @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less @ real @ ( tan @ real @ Y ) @ ( tan @ real @ X ) ) ) ) ) ).
% tan_monotone
thf(fact_3527_tan__monotone_H,axiom,
! [Y: real,X: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ Y )
=> ( ( ord_less @ real @ Y @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X )
=> ( ( ord_less @ real @ X @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less @ real @ Y @ X )
= ( ord_less @ real @ ( tan @ real @ Y ) @ ( tan @ real @ X ) ) ) ) ) ) ) ).
% tan_monotone'
thf(fact_3528_tan__mono__lt__eq,axiom,
! [X: real,Y: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X )
=> ( ( ord_less @ real @ X @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ Y )
=> ( ( ord_less @ real @ Y @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less @ real @ ( tan @ real @ X ) @ ( tan @ real @ Y ) )
= ( ord_less @ real @ X @ Y ) ) ) ) ) ) ).
% tan_mono_lt_eq
thf(fact_3529_lemma__tan__total1,axiom,
! [Y: real] :
? [X3: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X3 )
& ( ord_less @ real @ X3 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
& ( ( tan @ real @ X3 )
= Y ) ) ).
% lemma_tan_total1
thf(fact_3530_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_3531_tan__inverse,axiom,
! [Y: real] :
( ( divide_divide @ real @ ( one_one @ real ) @ ( tan @ real @ Y ) )
= ( tan @ real @ ( minus_minus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ Y ) ) ) ).
% tan_inverse
thf(fact_3532_vebt__heap__rules_I2_J,axiom,
! [N: nat,S2: set @ nat,Ti: vEBT_VEBTi,X: nat] :
( hoare_hoare_triple @ $o @ ( vEBT_Intf_vebt_assn @ N @ S2 @ Ti ) @ ( vEBT_vebt_memberi @ Ti @ X )
@ ^ [R5: $o] :
( times_times @ assn @ ( vEBT_Intf_vebt_assn @ N @ S2 @ Ti )
@ ( pure_assn
@ ( R5
= ( member @ nat @ X @ S2 ) ) ) ) ) ).
% vebt_heap_rules(2)
thf(fact_3533_add__tan__eq,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A,Y: A] :
( ( ( cos @ A @ X )
!= ( zero_zero @ A ) )
=> ( ( ( cos @ A @ Y )
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( tan @ A @ X ) @ ( tan @ A @ Y ) )
= ( divide_divide @ A @ ( sin @ A @ ( plus_plus @ A @ X @ Y ) ) @ ( times_times @ A @ ( cos @ A @ X ) @ ( cos @ A @ Y ) ) ) ) ) ) ) ).
% add_tan_eq
thf(fact_3534_termdiff__converges,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A,K4: real,C2: nat > A] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ K4 )
=> ( ! [X3: A] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X3 ) @ K4 )
=> ( summable @ A
@ ^ [N4: nat] : ( times_times @ A @ ( C2 @ N4 ) @ ( power_power @ A @ X3 @ N4 ) ) ) )
=> ( summable @ A
@ ^ [N4: nat] : ( times_times @ A @ ( diffs @ A @ C2 @ N4 ) @ ( power_power @ A @ X @ N4 ) ) ) ) ) ) ).
% termdiff_converges
thf(fact_3535_tan__total__pos,axiom,
! [Y: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y )
=> ? [X3: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 )
& ( ord_less @ real @ X3 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
& ( ( tan @ real @ X3 )
= Y ) ) ) ).
% tan_total_pos
thf(fact_3536_tan__pos__pi2__le,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ X @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( tan @ real @ X ) ) ) ) ).
% tan_pos_pi2_le
thf(fact_3537_tan__less__zero,axiom,
! [X: real] :
( ( ord_less @ real @ ( divide_divide @ real @ ( uminus_uminus @ real @ pi ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ X )
=> ( ( ord_less @ real @ X @ ( zero_zero @ real ) )
=> ( ord_less @ real @ ( tan @ real @ X ) @ ( zero_zero @ real ) ) ) ) ).
% tan_less_zero
thf(fact_3538_tan__mono__le,axiom,
! [X: real,Y: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X )
=> ( ( ord_less_eq @ real @ X @ Y )
=> ( ( ord_less @ real @ Y @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less_eq @ real @ ( tan @ real @ X ) @ ( tan @ real @ Y ) ) ) ) ) ).
% tan_mono_le
thf(fact_3539_tan__mono__le__eq,axiom,
! [X: real,Y: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X )
=> ( ( ord_less @ real @ X @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ Y )
=> ( ( ord_less @ real @ Y @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less_eq @ real @ ( tan @ real @ X ) @ ( tan @ real @ Y ) )
= ( ord_less_eq @ real @ X @ Y ) ) ) ) ) ) ).
% tan_mono_le_eq
thf(fact_3540_tan__bound__pi2,axiom,
! [X: real] :
( ( ord_less @ real @ ( abs_abs @ real @ X ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ ( bit0 @ one2 ) ) ) ) )
=> ( ord_less @ real @ ( abs_abs @ real @ ( tan @ real @ X ) ) @ ( one_one @ real ) ) ) ).
% tan_bound_pi2
thf(fact_3541_arctan,axiom,
! [Y: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ ( arctan @ Y ) )
& ( ord_less @ real @ ( arctan @ Y ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
& ( ( tan @ real @ ( arctan @ Y ) )
= Y ) ) ).
% arctan
thf(fact_3542_arctan__tan,axiom,
! [X: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X )
=> ( ( ord_less @ real @ X @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( arctan @ ( tan @ real @ X ) )
= X ) ) ) ).
% arctan_tan
thf(fact_3543_arctan__unique,axiom,
! [X: real,Y: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X )
=> ( ( ord_less @ real @ X @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ( tan @ real @ X )
= Y )
=> ( ( arctan @ Y )
= X ) ) ) ) ).
% arctan_unique
thf(fact_3544_lemma__tan__add1,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A,Y: A] :
( ( ( cos @ A @ X )
!= ( zero_zero @ A ) )
=> ( ( ( cos @ A @ Y )
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( tan @ A @ X ) @ ( tan @ A @ Y ) ) )
= ( divide_divide @ A @ ( cos @ A @ ( plus_plus @ A @ X @ Y ) ) @ ( times_times @ A @ ( cos @ A @ X ) @ ( cos @ A @ Y ) ) ) ) ) ) ) ).
% lemma_tan_add1
thf(fact_3545_tan__diff,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A,Y: A] :
( ( ( cos @ A @ X )
!= ( zero_zero @ A ) )
=> ( ( ( cos @ A @ Y )
!= ( zero_zero @ A ) )
=> ( ( ( cos @ A @ ( minus_minus @ A @ X @ Y ) )
!= ( zero_zero @ A ) )
=> ( ( tan @ A @ ( minus_minus @ A @ X @ Y ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( tan @ A @ X ) @ ( tan @ A @ Y ) ) @ ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( tan @ A @ X ) @ ( tan @ A @ Y ) ) ) ) ) ) ) ) ) ).
% tan_diff
thf(fact_3546_tan__add,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A,Y: A] :
( ( ( cos @ A @ X )
!= ( zero_zero @ A ) )
=> ( ( ( cos @ A @ Y )
!= ( zero_zero @ A ) )
=> ( ( ( cos @ A @ ( plus_plus @ A @ X @ Y ) )
!= ( zero_zero @ A ) )
=> ( ( tan @ A @ ( plus_plus @ A @ X @ Y ) )
= ( divide_divide @ A @ ( plus_plus @ A @ ( tan @ A @ X ) @ ( tan @ A @ Y ) ) @ ( minus_minus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( tan @ A @ X ) @ ( tan @ A @ Y ) ) ) ) ) ) ) ) ) ).
% tan_add
thf(fact_3547_tan__total__pi4,axiom,
! [X: real] :
( ( ord_less @ real @ ( abs_abs @ real @ X ) @ ( one_one @ real ) )
=> ? [Z3: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) @ Z3 )
& ( ord_less @ real @ Z3 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ ( bit0 @ one2 ) ) ) ) )
& ( ( tan @ real @ Z3 )
= X ) ) ) ).
% tan_total_pi4
thf(fact_3548_vebt__heap__rules_I8_J,axiom,
! [N: nat,S2: set @ nat,Ti: vEBT_VEBTi,X: nat] : ( hoare_hoare_triple @ vEBT_VEBTi @ ( vEBT_Intf_vebt_assn @ N @ S2 @ Ti ) @ ( vEBT_vebt_deletei @ Ti @ X ) @ ( vEBT_Intf_vebt_assn @ N @ ( minus_minus @ ( set @ nat ) @ S2 @ ( insert @ nat @ X @ ( bot_bot @ ( set @ nat ) ) ) ) ) ) ).
% vebt_heap_rules(8)
thf(fact_3549_tan__half,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ( ( tan @ A )
= ( ^ [X2: A] : ( divide_divide @ A @ ( sin @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X2 ) ) @ ( plus_plus @ A @ ( cos @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X2 ) ) @ ( one_one @ A ) ) ) ) ) ) ).
% tan_half
thf(fact_3550_sin__tan,axiom,
! [X: real] :
( ( ord_less @ real @ ( abs_abs @ real @ X ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( sin @ real @ X )
= ( divide_divide @ real @ ( tan @ real @ X ) @ ( sqrt @ ( plus_plus @ real @ ( one_one @ real ) @ ( power_power @ real @ ( tan @ real @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% sin_tan
thf(fact_3551_cos__tan,axiom,
! [X: real] :
( ( ord_less @ real @ ( abs_abs @ real @ X ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( cos @ real @ X )
= ( divide_divide @ real @ ( one_one @ real ) @ ( sqrt @ ( plus_plus @ real @ ( one_one @ real ) @ ( power_power @ real @ ( tan @ real @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% cos_tan
thf(fact_3552_cot__less__zero,axiom,
! [X: real] :
( ( ord_less @ real @ ( divide_divide @ real @ ( uminus_uminus @ real @ pi ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ X )
=> ( ( ord_less @ real @ X @ ( zero_zero @ real ) )
=> ( ord_less @ real @ ( cot @ real @ X ) @ ( zero_zero @ real ) ) ) ) ).
% cot_less_zero
thf(fact_3553_arccos__cos__eq__abs__2pi,axiom,
! [Theta: real] :
~ ! [K2: int] :
( ( arccos @ ( cos @ real @ Theta ) )
!= ( abs_abs @ real @ ( minus_minus @ real @ Theta @ ( times_times @ real @ ( ring_1_of_int @ real @ K2 ) @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) ) ) ) ) ).
% arccos_cos_eq_abs_2pi
thf(fact_3554_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_3555_real__sqrt__eq__iff,axiom,
! [X: real,Y: real] :
( ( ( sqrt @ X )
= ( sqrt @ Y ) )
= ( X = Y ) ) ).
% real_sqrt_eq_iff
thf(fact_3556_real__sqrt__eq__zero__cancel__iff,axiom,
! [X: real] :
( ( ( sqrt @ X )
= ( zero_zero @ real ) )
= ( X
= ( zero_zero @ real ) ) ) ).
% real_sqrt_eq_zero_cancel_iff
thf(fact_3557_real__sqrt__zero,axiom,
( ( sqrt @ ( zero_zero @ real ) )
= ( zero_zero @ real ) ) ).
% real_sqrt_zero
thf(fact_3558_real__sqrt__less__iff,axiom,
! [X: real,Y: real] :
( ( ord_less @ real @ ( sqrt @ X ) @ ( sqrt @ Y ) )
= ( ord_less @ real @ X @ Y ) ) ).
% real_sqrt_less_iff
thf(fact_3559_real__sqrt__le__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( sqrt @ X ) @ ( sqrt @ Y ) )
= ( ord_less_eq @ real @ X @ Y ) ) ).
% real_sqrt_le_iff
thf(fact_3560_real__sqrt__eq__1__iff,axiom,
! [X: real] :
( ( ( sqrt @ X )
= ( one_one @ real ) )
= ( X
= ( one_one @ real ) ) ) ).
% real_sqrt_eq_1_iff
thf(fact_3561_real__sqrt__one,axiom,
( ( sqrt @ ( one_one @ real ) )
= ( one_one @ real ) ) ).
% real_sqrt_one
thf(fact_3562_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_3563_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_3564_real__sqrt__lt__0__iff,axiom,
! [X: real] :
( ( ord_less @ real @ ( sqrt @ X ) @ ( zero_zero @ real ) )
= ( ord_less @ real @ X @ ( zero_zero @ real ) ) ) ).
% real_sqrt_lt_0_iff
thf(fact_3565_real__sqrt__gt__0__iff,axiom,
! [Y: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( sqrt @ Y ) )
= ( ord_less @ real @ ( zero_zero @ real ) @ Y ) ) ).
% real_sqrt_gt_0_iff
thf(fact_3566_real__sqrt__le__0__iff,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( sqrt @ X ) @ ( zero_zero @ real ) )
= ( ord_less_eq @ real @ X @ ( zero_zero @ real ) ) ) ).
% real_sqrt_le_0_iff
thf(fact_3567_real__sqrt__ge__0__iff,axiom,
! [Y: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( sqrt @ Y ) )
= ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y ) ) ).
% real_sqrt_ge_0_iff
thf(fact_3568_real__sqrt__lt__1__iff,axiom,
! [X: real] :
( ( ord_less @ real @ ( sqrt @ X ) @ ( one_one @ real ) )
= ( ord_less @ real @ X @ ( one_one @ real ) ) ) ).
% real_sqrt_lt_1_iff
thf(fact_3569_real__sqrt__gt__1__iff,axiom,
! [Y: real] :
( ( ord_less @ real @ ( one_one @ real ) @ ( sqrt @ Y ) )
= ( ord_less @ real @ ( one_one @ real ) @ Y ) ) ).
% real_sqrt_gt_1_iff
thf(fact_3570_real__sqrt__le__1__iff,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( sqrt @ X ) @ ( one_one @ real ) )
= ( ord_less_eq @ real @ X @ ( one_one @ real ) ) ) ).
% real_sqrt_le_1_iff
thf(fact_3571_real__sqrt__ge__1__iff,axiom,
! [Y: real] :
( ( ord_less_eq @ real @ ( one_one @ real ) @ ( sqrt @ Y ) )
= ( ord_less_eq @ real @ ( one_one @ real ) @ Y ) ) ).
% real_sqrt_ge_1_iff
thf(fact_3572_More__Word_Osint__0,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] :
( ( ( ring_1_signed @ A @ int @ X )
= ( zero_zero @ int ) )
= ( X
= ( zero_zero @ ( word @ A ) ) ) ) ) ).
% More_Word.sint_0
thf(fact_3573_Word_Oof__int__sint,axiom,
! [A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( ring_1 @ A ) )
=> ! [A3: word @ B] :
( ( ring_1_of_int @ A @ ( ring_1_signed @ B @ int @ A3 ) )
= ( ring_1_signed @ B @ A @ A3 ) ) ) ).
% Word.of_int_sint
thf(fact_3574_real__sqrt__abs2,axiom,
! [X: real] :
( ( sqrt @ ( times_times @ real @ X @ X ) )
= ( abs_abs @ real @ X ) ) ).
% real_sqrt_abs2
thf(fact_3575_real__sqrt__mult__self,axiom,
! [A3: real] :
( ( times_times @ real @ ( sqrt @ A3 ) @ ( sqrt @ A3 ) )
= ( abs_abs @ real @ A3 ) ) ).
% real_sqrt_mult_self
thf(fact_3576_arccos__1,axiom,
( ( arccos @ ( one_one @ real ) )
= ( zero_zero @ real ) ) ).
% arccos_1
thf(fact_3577_cot__pi,axiom,
( ( cot @ real @ pi )
= ( zero_zero @ real ) ) ).
% cot_pi
thf(fact_3578_real__sqrt__four,axiom,
( ( sqrt @ ( numeral_numeral @ real @ ( bit0 @ ( bit0 @ one2 ) ) ) )
= ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ).
% real_sqrt_four
thf(fact_3579_arccos__minus__1,axiom,
( ( arccos @ ( uminus_uminus @ real @ ( one_one @ real ) ) )
= pi ) ).
% arccos_minus_1
thf(fact_3580_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_3581_sint__minus1,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] :
( ( ( ring_1_signed @ A @ int @ X )
= ( uminus_uminus @ int @ ( one_one @ int ) ) )
= ( X
= ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) ) ) ) ).
% sint_minus1
thf(fact_3582_cos__arccos,axiom,
! [Y: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ Y )
=> ( ( ord_less_eq @ real @ Y @ ( one_one @ real ) )
=> ( ( cos @ real @ ( arccos @ Y ) )
= Y ) ) ) ).
% cos_arccos
thf(fact_3583_cot__npi,axiom,
! [N: nat] :
( ( cot @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ pi ) )
= ( zero_zero @ real ) ) ).
% cot_npi
thf(fact_3584_real__sqrt__abs,axiom,
! [X: real] :
( ( sqrt @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( abs_abs @ real @ X ) ) ).
% real_sqrt_abs
thf(fact_3585_real__sqrt__pow2,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( power_power @ real @ ( sqrt @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= X ) ) ).
% real_sqrt_pow2
thf(fact_3586_real__sqrt__pow2__iff,axiom,
! [X: real] :
( ( ( power_power @ real @ ( sqrt @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= X )
= ( ord_less_eq @ real @ ( zero_zero @ real ) @ X ) ) ).
% real_sqrt_pow2_iff
thf(fact_3587_real__sqrt__sum__squares__mult__squared__eq,axiom,
! [X: real,Y: real,Xa: real,Ya: real] :
( ( power_power @ real @ ( sqrt @ ( times_times @ real @ ( plus_plus @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( plus_plus @ real @ ( power_power @ real @ Xa @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Ya @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( times_times @ real @ ( plus_plus @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( plus_plus @ real @ ( power_power @ real @ Xa @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Ya @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% real_sqrt_sum_squares_mult_squared_eq
thf(fact_3588_arccos__0,axiom,
( ( arccos @ ( zero_zero @ real ) )
= ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ).
% arccos_0
thf(fact_3589_cot__periodic,axiom,
! [X: real] :
( ( cot @ real @ ( plus_plus @ real @ X @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) ) )
= ( cot @ real @ X ) ) ).
% cot_periodic
thf(fact_3590_signed__word__eqI,axiom,
! [A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( ring_char_0 @ A ) )
=> ! [V2: word @ B,W: word @ B] :
( ( ( ring_1_signed @ B @ A @ V2 )
= ( ring_1_signed @ B @ A @ W ) )
=> ( V2 = W ) ) ) ).
% signed_word_eqI
thf(fact_3591_word__eq__iff__signed,axiom,
! [A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( ring_char_0 @ A ) )
=> ( ( ^ [Y3: word @ B,Z2: word @ B] : Y3 = Z2 )
= ( ^ [V3: word @ B,W2: word @ B] :
( ( ring_1_signed @ B @ A @ V3 )
= ( ring_1_signed @ B @ A @ W2 ) ) ) ) ) ).
% word_eq_iff_signed
thf(fact_3592_More__Word_Oof__int__sint,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A] :
( ( ring_1_of_int @ ( word @ A ) @ ( ring_1_signed @ A @ int @ A3 ) )
= A3 ) ) ).
% More_Word.of_int_sint
thf(fact_3593_real__sqrt__minus,axiom,
! [X: real] :
( ( sqrt @ ( uminus_uminus @ real @ X ) )
= ( uminus_uminus @ real @ ( sqrt @ X ) ) ) ).
% real_sqrt_minus
thf(fact_3594_real__sqrt__power,axiom,
! [X: real,K: nat] :
( ( sqrt @ ( power_power @ real @ X @ K ) )
= ( power_power @ real @ ( sqrt @ X ) @ K ) ) ).
% real_sqrt_power
thf(fact_3595_real__sqrt__mult,axiom,
! [X: real,Y: real] :
( ( sqrt @ ( times_times @ real @ X @ Y ) )
= ( times_times @ real @ ( sqrt @ X ) @ ( sqrt @ Y ) ) ) ).
% real_sqrt_mult
thf(fact_3596_real__sqrt__divide,axiom,
! [X: real,Y: real] :
( ( sqrt @ ( divide_divide @ real @ X @ Y ) )
= ( divide_divide @ real @ ( sqrt @ X ) @ ( sqrt @ Y ) ) ) ).
% real_sqrt_divide
thf(fact_3597_real__sqrt__le__mono,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ X @ Y )
=> ( ord_less_eq @ real @ ( sqrt @ X ) @ ( sqrt @ Y ) ) ) ).
% real_sqrt_le_mono
thf(fact_3598_real__sqrt__less__mono,axiom,
! [X: real,Y: real] :
( ( ord_less @ real @ X @ Y )
=> ( ord_less @ real @ ( sqrt @ X ) @ ( sqrt @ Y ) ) ) ).
% real_sqrt_less_mono
thf(fact_3599_real__sqrt__gt__zero,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( sqrt @ X ) ) ) ).
% real_sqrt_gt_zero
thf(fact_3600_real__sqrt__eq__zero__cancel,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ( sqrt @ X )
= ( zero_zero @ real ) )
=> ( X
= ( zero_zero @ real ) ) ) ) ).
% real_sqrt_eq_zero_cancel
thf(fact_3601_real__sqrt__ge__zero,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( sqrt @ X ) ) ) ).
% real_sqrt_ge_zero
thf(fact_3602_real__sqrt__ge__one,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( one_one @ real ) @ X )
=> ( ord_less_eq @ real @ ( one_one @ real ) @ ( sqrt @ X ) ) ) ).
% real_sqrt_ge_one
thf(fact_3603_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_3604_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_3605_real__div__sqrt,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( divide_divide @ real @ X @ ( sqrt @ X ) )
= ( sqrt @ X ) ) ) ).
% real_div_sqrt
thf(fact_3606_sqrt__add__le__add__sqrt,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y )
=> ( ord_less_eq @ real @ ( sqrt @ ( plus_plus @ real @ X @ Y ) ) @ ( plus_plus @ real @ ( sqrt @ X ) @ ( sqrt @ Y ) ) ) ) ) ).
% sqrt_add_le_add_sqrt
thf(fact_3607_le__real__sqrt__sumsq,axiom,
! [X: real,Y: real] : ( ord_less_eq @ real @ X @ ( sqrt @ ( plus_plus @ real @ ( times_times @ real @ X @ X ) @ ( times_times @ real @ Y @ Y ) ) ) ) ).
% le_real_sqrt_sumsq
thf(fact_3608_arccos__le__arccos,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X )
=> ( ( ord_less_eq @ real @ X @ Y )
=> ( ( ord_less_eq @ real @ Y @ ( one_one @ real ) )
=> ( ord_less_eq @ real @ ( arccos @ Y ) @ ( arccos @ X ) ) ) ) ) ).
% arccos_le_arccos
thf(fact_3609_arccos__le__mono,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ X ) @ ( one_one @ real ) )
=> ( ( ord_less_eq @ real @ ( abs_abs @ real @ Y ) @ ( one_one @ real ) )
=> ( ( ord_less_eq @ real @ ( arccos @ X ) @ ( arccos @ Y ) )
= ( ord_less_eq @ real @ Y @ X ) ) ) ) ).
% arccos_le_mono
thf(fact_3610_arccos__eq__iff,axiom,
! [X: real,Y: real] :
( ( ( ord_less_eq @ real @ ( abs_abs @ real @ X ) @ ( one_one @ real ) )
& ( ord_less_eq @ real @ ( abs_abs @ real @ Y ) @ ( one_one @ real ) ) )
=> ( ( ( arccos @ X )
= ( arccos @ Y ) )
= ( X = Y ) ) ) ).
% arccos_eq_iff
thf(fact_3611_sqrt2__less__2,axiom,
ord_less @ real @ ( sqrt @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ).
% sqrt2_less_2
thf(fact_3612_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_3613_sin__arccos,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X )
=> ( ( ord_less_eq @ real @ X @ ( one_one @ real ) )
=> ( ( sin @ real @ ( arccos @ X ) )
= ( sqrt @ ( minus_minus @ real @ ( one_one @ real ) @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% sin_arccos
thf(fact_3614_sin__arccos__abs,axiom,
! [Y: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ Y ) @ ( one_one @ real ) )
=> ( ( sin @ real @ ( arccos @ Y ) )
= ( sqrt @ ( minus_minus @ real @ ( one_one @ real ) @ ( power_power @ real @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% sin_arccos_abs
thf(fact_3615_real__less__rsqrt,axiom,
! [X: real,Y: real] :
( ( ord_less @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ Y )
=> ( ord_less @ real @ X @ ( sqrt @ Y ) ) ) ).
% real_less_rsqrt
thf(fact_3616_sqrt__le__D,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( sqrt @ X ) @ Y )
=> ( ord_less_eq @ real @ X @ ( power_power @ real @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% sqrt_le_D
thf(fact_3617_real__le__rsqrt,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ Y )
=> ( ord_less_eq @ real @ X @ ( sqrt @ Y ) ) ) ).
% real_le_rsqrt
thf(fact_3618_arccos__lbound,axiom,
! [Y: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ Y )
=> ( ( ord_less_eq @ real @ Y @ ( one_one @ real ) )
=> ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( arccos @ Y ) ) ) ) ).
% arccos_lbound
thf(fact_3619_arccos__less__arccos,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X )
=> ( ( ord_less @ real @ X @ Y )
=> ( ( ord_less_eq @ real @ Y @ ( one_one @ real ) )
=> ( ord_less @ real @ ( arccos @ Y ) @ ( arccos @ X ) ) ) ) ) ).
% arccos_less_arccos
thf(fact_3620_arccos__less__mono,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ X ) @ ( one_one @ real ) )
=> ( ( ord_less_eq @ real @ ( abs_abs @ real @ Y ) @ ( one_one @ real ) )
=> ( ( ord_less @ real @ ( arccos @ X ) @ ( arccos @ Y ) )
= ( ord_less @ real @ Y @ X ) ) ) ) ).
% arccos_less_mono
thf(fact_3621_arccos__ubound,axiom,
! [Y: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ Y )
=> ( ( ord_less_eq @ real @ Y @ ( one_one @ real ) )
=> ( ord_less_eq @ real @ ( arccos @ Y ) @ pi ) ) ) ).
% arccos_ubound
thf(fact_3622_tan__60,axiom,
( ( tan @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit1 @ one2 ) ) ) )
= ( sqrt @ ( numeral_numeral @ real @ ( bit1 @ one2 ) ) ) ) ).
% tan_60
thf(fact_3623_arccos__cos,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ X @ pi )
=> ( ( arccos @ ( cos @ real @ X ) )
= X ) ) ) ).
% arccos_cos
thf(fact_3624_cos__arccos__abs,axiom,
! [Y: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ Y ) @ ( one_one @ real ) )
=> ( ( cos @ real @ ( arccos @ Y ) )
= Y ) ) ).
% cos_arccos_abs
thf(fact_3625_real__le__lsqrt,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y )
=> ( ( ord_less_eq @ real @ X @ ( power_power @ real @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
=> ( ord_less_eq @ real @ ( sqrt @ X ) @ Y ) ) ) ) ).
% real_le_lsqrt
thf(fact_3626_real__sqrt__unique,axiom,
! [Y: real,X: real] :
( ( ( power_power @ real @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= X )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y )
=> ( ( sqrt @ X )
= Y ) ) ) ).
% real_sqrt_unique
thf(fact_3627_lemma__real__divide__sqrt__less,axiom,
! [U2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ U2 )
=> ( ord_less @ real @ ( divide_divide @ real @ U2 @ ( sqrt @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ U2 ) ) ).
% lemma_real_divide_sqrt_less
thf(fact_3628_real__sqrt__sum__squares__eq__cancel,axiom,
! [X: real,Y: real] :
( ( ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
= X )
=> ( Y
= ( zero_zero @ real ) ) ) ).
% real_sqrt_sum_squares_eq_cancel
thf(fact_3629_real__sqrt__sum__squares__eq__cancel2,axiom,
! [X: real,Y: real] :
( ( ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
= Y )
=> ( X
= ( zero_zero @ real ) ) ) ).
% real_sqrt_sum_squares_eq_cancel2
thf(fact_3630_real__sqrt__sum__squares__ge1,axiom,
! [X: real,Y: real] : ( ord_less_eq @ real @ X @ ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% real_sqrt_sum_squares_ge1
thf(fact_3631_real__sqrt__sum__squares__ge2,axiom,
! [Y: real,X: real] : ( ord_less_eq @ real @ Y @ ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% real_sqrt_sum_squares_ge2
thf(fact_3632_real__sqrt__sum__squares__triangle__ineq,axiom,
! [A3: real,C2: real,B3: real,D2: real] : ( ord_less_eq @ real @ ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ ( plus_plus @ real @ A3 @ C2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ ( plus_plus @ real @ B3 @ D2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( plus_plus @ real @ ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ A3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ B3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ C2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ D2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% real_sqrt_sum_squares_triangle_ineq
thf(fact_3633_sqrt__ge__absD,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ X ) @ ( sqrt @ Y ) )
=> ( ord_less_eq @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ Y ) ) ).
% sqrt_ge_absD
thf(fact_3634_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_3635_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_3636_arccos__lt__bounded,axiom,
! [Y: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ Y )
=> ( ( ord_less @ real @ Y @ ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( arccos @ Y ) )
& ( ord_less @ real @ ( arccos @ Y ) @ pi ) ) ) ) ).
% arccos_lt_bounded
thf(fact_3637_arccos__bounded,axiom,
! [Y: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ Y )
=> ( ( ord_less_eq @ real @ Y @ ( one_one @ real ) )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( arccos @ Y ) )
& ( ord_less_eq @ real @ ( arccos @ Y ) @ pi ) ) ) ) ).
% arccos_bounded
thf(fact_3638_sin__arccos__nonzero,axiom,
! [X: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X )
=> ( ( ord_less @ real @ X @ ( one_one @ real ) )
=> ( ( sin @ real @ ( arccos @ X ) )
!= ( zero_zero @ real ) ) ) ) ).
% sin_arccos_nonzero
thf(fact_3639_arccos__cos2,axiom,
! [X: real] :
( ( ord_less_eq @ real @ X @ ( zero_zero @ real ) )
=> ( ( ord_less_eq @ real @ ( uminus_uminus @ real @ pi ) @ X )
=> ( ( arccos @ ( cos @ real @ X ) )
= ( uminus_uminus @ real @ X ) ) ) ) ).
% arccos_cos2
thf(fact_3640_arccos__minus,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X )
=> ( ( ord_less_eq @ real @ X @ ( one_one @ real ) )
=> ( ( arccos @ ( uminus_uminus @ real @ X ) )
= ( minus_minus @ real @ pi @ ( arccos @ X ) ) ) ) ) ).
% arccos_minus
thf(fact_3641_real__less__lsqrt,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y )
=> ( ( ord_less @ real @ X @ ( power_power @ real @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
=> ( ord_less @ real @ ( sqrt @ X ) @ Y ) ) ) ) ).
% real_less_lsqrt
thf(fact_3642_sqrt__sum__squares__le__sum,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y )
=> ( ord_less_eq @ real @ ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( plus_plus @ real @ X @ Y ) ) ) ) ).
% sqrt_sum_squares_le_sum
thf(fact_3643_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_3644_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_3645_real__sqrt__ge__abs1,axiom,
! [X: real,Y: real] : ( ord_less_eq @ real @ ( abs_abs @ real @ X ) @ ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% real_sqrt_ge_abs1
thf(fact_3646_real__sqrt__ge__abs2,axiom,
! [Y: real,X: real] : ( ord_less_eq @ real @ ( abs_abs @ real @ Y ) @ ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% real_sqrt_ge_abs2
thf(fact_3647_sqrt__sum__squares__le__sum__abs,axiom,
! [X: real,Y: real] : ( ord_less_eq @ real @ ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( plus_plus @ real @ ( abs_abs @ real @ X ) @ ( abs_abs @ real @ Y ) ) ) ).
% sqrt_sum_squares_le_sum_abs
thf(fact_3648_ln__sqrt,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ln_ln @ real @ ( sqrt @ X ) )
= ( divide_divide @ real @ ( ln_ln @ real @ X ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ).
% ln_sqrt
thf(fact_3649_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_3650_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_3651_arsinh__real__def,axiom,
( ( arsinh @ real )
= ( ^ [X2: real] : ( ln_ln @ real @ ( plus_plus @ real @ X2 @ ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ real ) ) ) ) ) ) ) ).
% arsinh_real_def
thf(fact_3652_complex__norm,axiom,
! [X: real,Y: real] :
( ( real_V7770717601297561774m_norm @ complex @ ( complex2 @ X @ Y ) )
= ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% complex_norm
thf(fact_3653_arccos,axiom,
! [Y: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ Y )
=> ( ( ord_less_eq @ real @ Y @ ( one_one @ real ) )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( arccos @ Y ) )
& ( ord_less_eq @ real @ ( arccos @ Y ) @ pi )
& ( ( cos @ real @ ( arccos @ Y ) )
= Y ) ) ) ) ).
% arccos
thf(fact_3654_arccos__minus__abs,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ X ) @ ( one_one @ real ) )
=> ( ( arccos @ ( uminus_uminus @ real @ X ) )
= ( minus_minus @ real @ pi @ ( arccos @ X ) ) ) ) ).
% arccos_minus_abs
thf(fact_3655_real__sqrt__power__even,axiom,
! [N: nat,X: real] :
( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( power_power @ real @ ( sqrt @ X ) @ N )
= ( power_power @ real @ X @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% real_sqrt_power_even
thf(fact_3656_arsinh__real__aux,axiom,
! [X: real] : ( ord_less @ real @ ( zero_zero @ real ) @ ( plus_plus @ real @ X @ ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ real ) ) ) ) ) ).
% arsinh_real_aux
thf(fact_3657_real__sqrt__sum__squares__mult__ge__zero,axiom,
! [X: real,Y: real,Xa: real,Ya: real] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( sqrt @ ( times_times @ real @ ( plus_plus @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( plus_plus @ real @ ( power_power @ real @ Xa @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Ya @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% real_sqrt_sum_squares_mult_ge_zero
thf(fact_3658_arith__geo__mean__sqrt,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y )
=> ( ord_less_eq @ real @ ( sqrt @ ( times_times @ real @ X @ Y ) ) @ ( divide_divide @ real @ ( plus_plus @ real @ X @ Y ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ).
% arith_geo_mean_sqrt
thf(fact_3659_powr__half__sqrt,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( powr @ real @ X @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
= ( sqrt @ X ) ) ) ).
% powr_half_sqrt
thf(fact_3660_tan__cot_H,axiom,
! [X: real] :
( ( tan @ real @ ( minus_minus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ X ) )
= ( cot @ real @ X ) ) ).
% tan_cot'
thf(fact_3661_cos__x__y__le__one,axiom,
! [X: real,Y: real] : ( ord_less_eq @ real @ ( abs_abs @ real @ ( divide_divide @ real @ X @ ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) @ ( one_one @ real ) ) ).
% cos_x_y_le_one
thf(fact_3662_real__sqrt__sum__squares__less,axiom,
! [X: real,U2: real,Y: real] :
( ( ord_less @ real @ ( abs_abs @ real @ X ) @ ( divide_divide @ real @ U2 @ ( sqrt @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) )
=> ( ( ord_less @ real @ ( abs_abs @ real @ Y ) @ ( divide_divide @ real @ U2 @ ( sqrt @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) )
=> ( ord_less @ real @ ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ U2 ) ) ) ).
% real_sqrt_sum_squares_less
thf(fact_3663_arcosh__real__def,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( one_one @ real ) @ X )
=> ( ( arcosh @ real @ X )
= ( ln_ln @ real @ ( plus_plus @ real @ X @ ( sqrt @ ( minus_minus @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ real ) ) ) ) ) ) ) ).
% arcosh_real_def
thf(fact_3664_cos__arctan,axiom,
! [X: real] :
( ( cos @ real @ ( arctan @ X ) )
= ( divide_divide @ real @ ( one_one @ real ) @ ( sqrt @ ( plus_plus @ real @ ( one_one @ real ) @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% cos_arctan
thf(fact_3665_sin__arctan,axiom,
! [X: real] :
( ( sin @ real @ ( arctan @ X ) )
= ( divide_divide @ real @ X @ ( sqrt @ ( plus_plus @ real @ ( one_one @ real ) @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% sin_arctan
thf(fact_3666_sint__above__size,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A,X: int] :
( ( ord_less_eq @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ ( size_size @ ( word @ A ) @ W ) @ ( one_one @ nat ) ) ) @ X )
=> ( ord_less @ int @ ( ring_1_signed @ A @ int @ W ) @ X ) ) ) ).
% sint_above_size
thf(fact_3667_sint__below__size,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: int,W: word @ A] :
( ( ord_less_eq @ int @ X @ ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ ( size_size @ ( word @ A ) @ W ) @ ( one_one @ nat ) ) ) ) )
=> ( ord_less_eq @ int @ X @ ( ring_1_signed @ A @ int @ W ) ) ) ) ).
% sint_below_size
thf(fact_3668_arccos__le__pi2,axiom,
! [Y: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y )
=> ( ( ord_less_eq @ real @ Y @ ( one_one @ real ) )
=> ( ord_less_eq @ real @ ( arccos @ Y ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ).
% arccos_le_pi2
thf(fact_3669_cot__gt__zero,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ X @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( cot @ real @ X ) ) ) ) ).
% cot_gt_zero
thf(fact_3670_sqrt__sum__squares__half__less,axiom,
! [X: real,U2: real,Y: real] :
( ( ord_less @ real @ X @ ( divide_divide @ real @ U2 @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less @ real @ Y @ ( divide_divide @ real @ U2 @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y )
=> ( ord_less @ real @ ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ U2 ) ) ) ) ) ).
% sqrt_sum_squares_half_less
thf(fact_3671_sin__cos__sqrt,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( sin @ real @ X ) )
=> ( ( sin @ real @ X )
= ( sqrt @ ( minus_minus @ real @ ( one_one @ real ) @ ( power_power @ real @ ( cos @ real @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% sin_cos_sqrt
thf(fact_3672_arctan__half,axiom,
( arctan
= ( ^ [X2: real] : ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( arctan @ ( divide_divide @ real @ X2 @ ( plus_plus @ real @ ( one_one @ real ) @ ( sqrt @ ( plus_plus @ real @ ( one_one @ real ) @ ( power_power @ real @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ).
% arctan_half
thf(fact_3673_cos__arcsin,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X )
=> ( ( ord_less_eq @ real @ X @ ( one_one @ real ) )
=> ( ( cos @ real @ ( arcsin @ X ) )
= ( sqrt @ ( minus_minus @ real @ ( one_one @ real ) @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% cos_arcsin
thf(fact_3674_arcsin,axiom,
! [Y: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ Y )
=> ( ( ord_less_eq @ real @ Y @ ( one_one @ real ) )
=> ( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ ( arcsin @ Y ) )
& ( ord_less_eq @ real @ ( arcsin @ Y ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
& ( ( sin @ real @ ( arcsin @ Y ) )
= Y ) ) ) ) ).
% arcsin
thf(fact_3675_arcsin__pi,axiom,
! [Y: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ Y )
=> ( ( ord_less_eq @ real @ Y @ ( one_one @ real ) )
=> ( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ ( arcsin @ Y ) )
& ( ord_less_eq @ real @ ( arcsin @ Y ) @ pi )
& ( ( sin @ real @ ( arcsin @ Y ) )
= Y ) ) ) ) ).
% arcsin_pi
thf(fact_3676_arcsin__le__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X )
=> ( ( ord_less_eq @ real @ X @ ( one_one @ real ) )
=> ( ( ord_less_eq @ real @ ( divide_divide @ real @ ( uminus_uminus @ real @ pi ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ Y )
=> ( ( ord_less_eq @ real @ Y @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less_eq @ real @ ( arcsin @ X ) @ Y )
= ( ord_less_eq @ real @ X @ ( sin @ real @ Y ) ) ) ) ) ) ) ).
% arcsin_le_iff
thf(fact_3677_le__arcsin__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X )
=> ( ( ord_less_eq @ real @ X @ ( one_one @ real ) )
=> ( ( ord_less_eq @ real @ ( divide_divide @ real @ ( uminus_uminus @ real @ pi ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ Y )
=> ( ( ord_less_eq @ real @ Y @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less_eq @ real @ Y @ ( arcsin @ X ) )
= ( ord_less_eq @ real @ ( sin @ real @ Y ) @ X ) ) ) ) ) ) ).
% le_arcsin_iff
thf(fact_3678_arcsin__0,axiom,
( ( arcsin @ ( zero_zero @ real ) )
= ( zero_zero @ real ) ) ).
% arcsin_0
thf(fact_3679_sin__arcsin,axiom,
! [Y: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ Y )
=> ( ( ord_less_eq @ real @ Y @ ( one_one @ real ) )
=> ( ( sin @ real @ ( arcsin @ Y ) )
= Y ) ) ) ).
% sin_arcsin
thf(fact_3680_arcsin__1,axiom,
( ( arcsin @ ( one_one @ real ) )
= ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ).
% arcsin_1
thf(fact_3681_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_3682_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_3683_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_3684_scast__eq,axiom,
! [A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( type_len @ A ) )
=> ( ( ring_1_signed @ B @ ( word @ A ) )
= ( ^ [W2: word @ B] : ( ring_1_of_int @ ( word @ A ) @ ( ring_1_signed @ B @ int @ W2 ) ) ) ) ) ).
% scast_eq
thf(fact_3685_arcsin__minus,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X )
=> ( ( ord_less_eq @ real @ X @ ( one_one @ real ) )
=> ( ( arcsin @ ( uminus_uminus @ real @ X ) )
= ( uminus_uminus @ real @ ( arcsin @ X ) ) ) ) ) ).
% arcsin_minus
thf(fact_3686_arcsin__le__arcsin,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X )
=> ( ( ord_less_eq @ real @ X @ Y )
=> ( ( ord_less_eq @ real @ Y @ ( one_one @ real ) )
=> ( ord_less_eq @ real @ ( arcsin @ X ) @ ( arcsin @ Y ) ) ) ) ) ).
% arcsin_le_arcsin
thf(fact_3687_arcsin__eq__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ X ) @ ( one_one @ real ) )
=> ( ( ord_less_eq @ real @ ( abs_abs @ real @ Y ) @ ( one_one @ real ) )
=> ( ( ( arcsin @ X )
= ( arcsin @ Y ) )
= ( X = Y ) ) ) ) ).
% arcsin_eq_iff
thf(fact_3688_arcsin__le__mono,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ X ) @ ( one_one @ real ) )
=> ( ( ord_less_eq @ real @ ( abs_abs @ real @ Y ) @ ( one_one @ real ) )
=> ( ( ord_less_eq @ real @ ( arcsin @ X ) @ ( arcsin @ Y ) )
= ( ord_less_eq @ real @ X @ Y ) ) ) ) ).
% arcsin_le_mono
thf(fact_3689_arcsin__less__arcsin,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X )
=> ( ( ord_less @ real @ X @ Y )
=> ( ( ord_less_eq @ real @ Y @ ( one_one @ real ) )
=> ( ord_less @ real @ ( arcsin @ X ) @ ( arcsin @ Y ) ) ) ) ) ).
% arcsin_less_arcsin
thf(fact_3690_arcsin__less__mono,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ X ) @ ( one_one @ real ) )
=> ( ( ord_less_eq @ real @ ( abs_abs @ real @ Y ) @ ( one_one @ real ) )
=> ( ( ord_less @ real @ ( arcsin @ X ) @ ( arcsin @ Y ) )
= ( ord_less @ real @ X @ Y ) ) ) ) ).
% arcsin_less_mono
thf(fact_3691_cos__arcsin__nonzero,axiom,
! [X: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X )
=> ( ( ord_less @ real @ X @ ( one_one @ real ) )
=> ( ( cos @ real @ ( arcsin @ X ) )
!= ( zero_zero @ real ) ) ) ) ).
% cos_arcsin_nonzero
thf(fact_3692_arcsin__lt__bounded,axiom,
! [Y: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ Y )
=> ( ( ord_less @ real @ Y @ ( one_one @ real ) )
=> ( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ ( arcsin @ Y ) )
& ( ord_less @ real @ ( arcsin @ Y ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% arcsin_lt_bounded
thf(fact_3693_arcsin__bounded,axiom,
! [Y: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ Y )
=> ( ( ord_less_eq @ real @ Y @ ( one_one @ real ) )
=> ( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ ( arcsin @ Y ) )
& ( ord_less_eq @ real @ ( arcsin @ Y ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% arcsin_bounded
thf(fact_3694_arcsin__ubound,axiom,
! [Y: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ Y )
=> ( ( ord_less_eq @ real @ Y @ ( one_one @ real ) )
=> ( ord_less_eq @ real @ ( arcsin @ Y ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ).
% arcsin_ubound
thf(fact_3695_arcsin__lbound,axiom,
! [Y: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ Y )
=> ( ( ord_less_eq @ real @ Y @ ( one_one @ real ) )
=> ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ ( arcsin @ Y ) ) ) ) ).
% arcsin_lbound
thf(fact_3696_arcsin__sin,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X )
=> ( ( ord_less_eq @ real @ X @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( arcsin @ ( sin @ real @ X ) )
= X ) ) ) ).
% arcsin_sin
thf(fact_3697_gbinomial__code,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( ( gbinomial @ A )
= ( ^ [A5: A,K3: nat] :
( if @ A
@ ( K3
= ( zero_zero @ nat ) )
@ ( one_one @ A )
@ ( divide_divide @ A
@ ( set_fo6178422350223883121st_nat @ A
@ ^ [L2: nat] : ( times_times @ A @ ( minus_minus @ A @ A5 @ ( semiring_1_of_nat @ A @ L2 ) ) )
@ ( zero_zero @ nat )
@ ( minus_minus @ nat @ K3 @ ( one_one @ nat ) )
@ ( one_one @ A ) )
@ ( semiring_char_0_fact @ A @ K3 ) ) ) ) ) ) ).
% gbinomial_code
thf(fact_3698_log__base__10__eq1,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( log @ ( numeral_numeral @ real @ ( bit0 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) @ X )
= ( times_times @ real @ ( divide_divide @ real @ ( ln_ln @ real @ ( exp @ real @ ( one_one @ real ) ) ) @ ( ln_ln @ real @ ( numeral_numeral @ real @ ( bit0 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) ) ) @ ( ln_ln @ real @ X ) ) ) ) ).
% log_base_10_eq1
thf(fact_3699_flip__bit__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A] :
( ( bit_se8732182000553998342ip_bit @ A @ ( zero_zero @ nat ) @ A3 )
= ( plus_plus @ A @ ( zero_neq_one_of_bool @ A @ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% flip_bit_0
thf(fact_3700_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_3701_nat__of__bool,axiom,
! [P: $o] :
( ( nat2 @ ( zero_neq_one_of_bool @ int @ P ) )
= ( zero_neq_one_of_bool @ nat @ P ) ) ).
% nat_of_bool
thf(fact_3702_scast__id,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( ring_1_signed @ A @ ( word @ A ) )
= ( ^ [W2: word @ A] : W2 ) ) ) ).
% scast_id
thf(fact_3703_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_3704_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_3705_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_3706_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_3707_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_3708_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_3709_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_3710_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_3711_of__int__of__bool,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [P: $o] :
( ( ring_1_of_int @ A @ ( zero_neq_one_of_bool @ int @ P ) )
= ( zero_neq_one_of_bool @ A @ P ) ) ) ).
% of_int_of_bool
thf(fact_3712_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_3713_gbinomial__1,axiom,
! [A: $tType] :
( ( ( semiring_char_0 @ A )
& ( semidom_divide @ A ) )
=> ! [A3: A] :
( ( gbinomial @ A @ A3 @ ( one_one @ nat ) )
= A3 ) ) ).
% gbinomial_1
thf(fact_3714_exp__less__mono,axiom,
! [X: real,Y: real] :
( ( ord_less @ real @ X @ Y )
=> ( ord_less @ real @ ( exp @ real @ X ) @ ( exp @ real @ Y ) ) ) ).
% exp_less_mono
thf(fact_3715_exp__less__cancel__iff,axiom,
! [X: real,Y: real] :
( ( ord_less @ real @ ( exp @ real @ X ) @ ( exp @ real @ Y ) )
= ( ord_less @ real @ X @ Y ) ) ).
% exp_less_cancel_iff
thf(fact_3716_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_3717_take__bit__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A] :
( ( bit_se2584673776208193580ke_bit @ A @ ( zero_zero @ nat ) @ A3 )
= ( zero_zero @ A ) ) ) ).
% take_bit_0
thf(fact_3718_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_3719_take__bit__numeral__1,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [L: num] :
( ( bit_se2584673776208193580ke_bit @ A @ ( numeral_numeral @ nat @ L ) @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% take_bit_numeral_1
thf(fact_3720_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_3721_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_3722_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_3723_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_3724_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_3725_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_3726_gbinomial__0_I1_J,axiom,
! [A: $tType] :
( ( ( semiring_char_0 @ A )
& ( semidom_divide @ A ) )
=> ! [A3: A] :
( ( gbinomial @ A @ A3 @ ( zero_zero @ nat ) )
= ( one_one @ A ) ) ) ).
% gbinomial_0(1)
thf(fact_3727_gbinomial__Suc0,axiom,
! [A: $tType] :
( ( ( semiring_char_0 @ A )
& ( semidom_divide @ A ) )
=> ! [A3: A] :
( ( gbinomial @ A @ A3 @ ( suc @ ( zero_zero @ nat ) ) )
= A3 ) ) ).
% gbinomial_Suc0
thf(fact_3728_exp__eq__one__iff,axiom,
! [X: real] :
( ( ( exp @ real @ X )
= ( one_one @ real ) )
= ( X
= ( zero_zero @ real ) ) ) ).
% exp_eq_one_iff
thf(fact_3729_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_3730_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_3731_one__less__exp__iff,axiom,
! [X: real] :
( ( ord_less @ real @ ( one_one @ real ) @ ( exp @ real @ X ) )
= ( ord_less @ real @ ( zero_zero @ real ) @ X ) ) ).
% one_less_exp_iff
thf(fact_3732_exp__less__one__iff,axiom,
! [X: real] :
( ( ord_less @ real @ ( exp @ real @ X ) @ ( one_one @ real ) )
= ( ord_less @ real @ X @ ( zero_zero @ real ) ) ) ).
% exp_less_one_iff
thf(fact_3733_one__le__exp__iff,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( one_one @ real ) @ ( exp @ real @ X ) )
= ( ord_less_eq @ real @ ( zero_zero @ real ) @ X ) ) ).
% one_le_exp_iff
thf(fact_3734_exp__le__one__iff,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( exp @ real @ X ) @ ( one_one @ real ) )
= ( ord_less_eq @ real @ X @ ( zero_zero @ real ) ) ) ).
% exp_le_one_iff
thf(fact_3735_exp__ln,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( exp @ real @ ( ln_ln @ real @ X ) )
= X ) ) ).
% exp_ln
thf(fact_3736_exp__ln__iff,axiom,
! [X: real] :
( ( ( exp @ real @ ( ln_ln @ real @ X ) )
= X )
= ( ord_less @ real @ ( zero_zero @ real ) @ X ) ) ).
% exp_ln_iff
thf(fact_3737_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_3738_odd__of__bool__self,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [P4: $o] :
( ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( zero_neq_one_of_bool @ A @ P4 ) ) )
= P4 ) ) ).
% odd_of_bool_self
thf(fact_3739_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_3740_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_3741_even__take__bit__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A3: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se2584673776208193580ke_bit @ A @ N @ A3 ) )
= ( ( N
= ( zero_zero @ nat ) )
| ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) ) ) ) ).
% even_take_bit_eq
thf(fact_3742_of__bool__half__eq__0,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [B3: $o] :
( ( divide_divide @ A @ ( zero_neq_one_of_bool @ A @ B3 ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( zero_zero @ A ) ) ) ).
% of_bool_half_eq_0
thf(fact_3743_take__bit__Suc__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A] :
( ( bit_se2584673776208193580ke_bit @ A @ ( suc @ ( zero_zero @ nat ) ) @ A3 )
= ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% take_bit_Suc_0
thf(fact_3744_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_3745_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_3746_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_3747_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_3748_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_3749_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_3750_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_3751_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_3752_of__bool__eq__iff,axiom,
! [A: $tType] :
( ( zero_neq_one @ A )
=> ! [P4: $o,Q3: $o] :
( ( ( zero_neq_one_of_bool @ A @ P4 )
= ( zero_neq_one_of_bool @ A @ Q3 ) )
= ( P4 = Q3 ) ) ) ).
% of_bool_eq_iff
thf(fact_3753_unsigned__take__bit__eq,axiom,
! [A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( bit_se359711467146920520ations @ A ) )
=> ! [N: nat,W: word @ B] :
( ( semiring_1_unsigned @ B @ A @ ( bit_se2584673776208193580ke_bit @ ( word @ B ) @ N @ W ) )
= ( bit_se2584673776208193580ke_bit @ A @ N @ ( semiring_1_unsigned @ B @ A @ W ) ) ) ) ).
% unsigned_take_bit_eq
thf(fact_3754_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_3755_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_3756_exp__less__cancel,axiom,
! [X: real,Y: real] :
( ( ord_less @ real @ ( exp @ real @ X ) @ ( exp @ real @ Y ) )
=> ( ord_less @ real @ X @ Y ) ) ).
% exp_less_cancel
thf(fact_3757_exp__not__eq__zero,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X: A] :
( ( exp @ A @ X )
!= ( zero_zero @ A ) ) ) ).
% exp_not_eq_zero
thf(fact_3758_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_3759_not__exp__less__zero,axiom,
! [X: real] :
~ ( ord_less @ real @ ( exp @ real @ X ) @ ( zero_zero @ real ) ) ).
% not_exp_less_zero
thf(fact_3760_exp__gt__zero,axiom,
! [X: real] : ( ord_less @ real @ ( zero_zero @ real ) @ ( exp @ real @ X ) ) ).
% exp_gt_zero
thf(fact_3761_exp__total,axiom,
! [Y: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ Y )
=> ? [X3: real] :
( ( exp @ real @ X3 )
= Y ) ) ).
% exp_total
thf(fact_3762_not__exp__le__zero,axiom,
! [X: real] :
~ ( ord_less_eq @ real @ ( exp @ real @ X ) @ ( zero_zero @ real ) ) ).
% not_exp_le_zero
thf(fact_3763_exp__ge__zero,axiom,
! [X: real] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( exp @ real @ X ) ) ).
% exp_ge_zero
thf(fact_3764_split__of__bool__asm,axiom,
! [A: $tType] :
( ( zero_neq_one @ A )
=> ! [P: A > $o,P4: $o] :
( ( P @ ( zero_neq_one_of_bool @ A @ P4 ) )
= ( ~ ( ( P4
& ~ ( P @ ( one_one @ A ) ) )
| ( ~ P4
& ~ ( P @ ( zero_zero @ A ) ) ) ) ) ) ) ).
% split_of_bool_asm
thf(fact_3765_split__of__bool,axiom,
! [A: $tType] :
( ( zero_neq_one @ A )
=> ! [P: A > $o,P4: $o] :
( ( P @ ( zero_neq_one_of_bool @ A @ P4 ) )
= ( ( P4
=> ( P @ ( one_one @ A ) ) )
& ( ~ P4
=> ( P @ ( zero_zero @ A ) ) ) ) ) ) ).
% split_of_bool
thf(fact_3766_of__bool__def,axiom,
! [A: $tType] :
( ( zero_neq_one @ A )
=> ( ( zero_neq_one_of_bool @ A )
= ( ^ [P6: $o] : ( if @ A @ P6 @ ( one_one @ A ) @ ( zero_zero @ A ) ) ) ) ) ).
% of_bool_def
thf(fact_3767_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_3768_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_3769_exp__gt__one,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ord_less @ real @ ( one_one @ real ) @ ( exp @ real @ X ) ) ) ).
% exp_gt_one
thf(fact_3770_exp__ge__add__one__self,axiom,
! [X: real] : ( ord_less_eq @ real @ ( plus_plus @ real @ ( one_one @ real ) @ X ) @ ( exp @ real @ X ) ) ).
% exp_ge_add_one_self
thf(fact_3771_gbinomial__Suc__Suc,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: A,K: nat] :
( ( gbinomial @ A @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) @ ( suc @ K ) )
= ( plus_plus @ A @ ( gbinomial @ A @ A3 @ K ) @ ( gbinomial @ A @ A3 @ ( suc @ K ) ) ) ) ) ).
% gbinomial_Suc_Suc
thf(fact_3772_exp__minus__inverse,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X: A] :
( ( times_times @ A @ ( exp @ A @ X ) @ ( exp @ A @ ( uminus_uminus @ A @ X ) ) )
= ( one_one @ A ) ) ) ).
% exp_minus_inverse
thf(fact_3773_exp__of__nat2__mult,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A,N: nat] :
( ( exp @ A @ ( times_times @ A @ X @ ( semiring_1_of_nat @ A @ N ) ) )
= ( power_power @ A @ ( exp @ A @ X ) @ N ) ) ) ).
% exp_of_nat2_mult
thf(fact_3774_exp__of__nat__mult,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [N: nat,X: A] :
( ( exp @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ X ) )
= ( power_power @ A @ ( exp @ A @ X ) @ N ) ) ) ).
% exp_of_nat_mult
thf(fact_3775_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_3776_log__ln,axiom,
( ( ln_ln @ real )
= ( log @ ( exp @ real @ ( one_one @ real ) ) ) ) ).
% log_ln
thf(fact_3777_exp__ge__add__one__self__aux,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ord_less_eq @ real @ ( plus_plus @ real @ ( one_one @ real ) @ X ) @ ( exp @ real @ X ) ) ) ).
% exp_ge_add_one_self_aux
thf(fact_3778_lemma__exp__total,axiom,
! [Y: real] :
( ( ord_less_eq @ real @ ( one_one @ real ) @ Y )
=> ? [X3: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 )
& ( ord_less_eq @ real @ X3 @ ( minus_minus @ real @ Y @ ( one_one @ real ) ) )
& ( ( exp @ real @ X3 )
= Y ) ) ) ).
% lemma_exp_total
thf(fact_3779_ln__ge__iff,axiom,
! [X: real,Y: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ Y @ ( ln_ln @ real @ X ) )
= ( ord_less_eq @ real @ ( exp @ real @ Y ) @ X ) ) ) ).
% ln_ge_iff
thf(fact_3780_gbinomial__addition__formula,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: A,K: nat] :
( ( gbinomial @ A @ A3 @ ( suc @ K ) )
= ( plus_plus @ A @ ( gbinomial @ A @ ( minus_minus @ A @ A3 @ ( one_one @ A ) ) @ ( suc @ K ) ) @ ( gbinomial @ A @ ( minus_minus @ A @ A3 @ ( one_one @ A ) ) @ K ) ) ) ) ).
% gbinomial_addition_formula
thf(fact_3781_gbinomial__absorb__comp,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: A,K: nat] :
( ( times_times @ A @ ( minus_minus @ A @ A3 @ ( semiring_1_of_nat @ A @ K ) ) @ ( gbinomial @ A @ A3 @ K ) )
= ( times_times @ A @ A3 @ ( gbinomial @ A @ ( minus_minus @ A @ A3 @ ( one_one @ A ) ) @ K ) ) ) ) ).
% gbinomial_absorb_comp
thf(fact_3782_gbinomial__mult__1,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: A,K: nat] :
( ( times_times @ A @ A3 @ ( gbinomial @ A @ A3 @ K ) )
= ( plus_plus @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ K ) @ ( gbinomial @ A @ A3 @ K ) ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( suc @ K ) ) @ ( gbinomial @ A @ A3 @ ( suc @ K ) ) ) ) ) ) ).
% gbinomial_mult_1
thf(fact_3783_gbinomial__mult__1_H,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: A,K: nat] :
( ( times_times @ A @ ( gbinomial @ A @ A3 @ K ) @ A3 )
= ( plus_plus @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ K ) @ ( gbinomial @ A @ A3 @ K ) ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( suc @ K ) ) @ ( gbinomial @ A @ A3 @ ( suc @ K ) ) ) ) ) ) ).
% gbinomial_mult_1'
thf(fact_3784_gbinomial__ge__n__over__k__pow__k,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [K: nat,A3: A] :
( ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ K ) @ A3 )
=> ( ord_less_eq @ A @ ( power_power @ A @ ( divide_divide @ A @ A3 @ ( semiring_1_of_nat @ A @ K ) ) @ K ) @ ( gbinomial @ A @ A3 @ K ) ) ) ) ).
% gbinomial_ge_n_over_k_pow_k
thf(fact_3785_ln__x__over__x__mono,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( exp @ real @ ( one_one @ real ) ) @ X )
=> ( ( ord_less_eq @ real @ X @ Y )
=> ( ord_less_eq @ real @ ( divide_divide @ real @ ( ln_ln @ real @ Y ) @ Y ) @ ( divide_divide @ real @ ( ln_ln @ real @ X ) @ X ) ) ) ) ).
% ln_x_over_x_mono
thf(fact_3786_powr__def,axiom,
! [A: $tType] :
( ( ln @ A )
=> ( ( powr @ A )
= ( ^ [X2: A,A5: A] :
( if @ A
@ ( X2
= ( zero_zero @ A ) )
@ ( zero_zero @ A )
@ ( exp @ A @ ( times_times @ A @ A5 @ ( ln_ln @ A @ X2 ) ) ) ) ) ) ) ).
% powr_def
thf(fact_3787_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_3788_take__bit__eq__mod,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se2584673776208193580ke_bit @ A )
= ( ^ [N4: nat,A5: A] : ( modulo_modulo @ A @ A5 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N4 ) ) ) ) ) ).
% take_bit_eq_mod
thf(fact_3789_exp__le,axiom,
ord_less_eq @ real @ ( exp @ real @ ( one_one @ real ) ) @ ( numeral_numeral @ real @ ( bit1 @ one2 ) ) ).
% exp_le
thf(fact_3790_of__bool__odd__eq__mod__2,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A3: A] :
( ( zero_neq_one_of_bool @ A
@ ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) )
= ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% of_bool_odd_eq_mod_2
thf(fact_3791_Suc__times__gbinomial,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K: nat,A3: A] :
( ( times_times @ A @ ( semiring_1_of_nat @ A @ ( suc @ K ) ) @ ( gbinomial @ A @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) @ ( suc @ K ) ) )
= ( times_times @ A @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) @ ( gbinomial @ A @ A3 @ K ) ) ) ) ).
% Suc_times_gbinomial
thf(fact_3792_gbinomial__absorption,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K: nat,A3: A] :
( ( times_times @ A @ ( semiring_1_of_nat @ A @ ( suc @ K ) ) @ ( gbinomial @ A @ A3 @ ( suc @ K ) ) )
= ( times_times @ A @ A3 @ ( gbinomial @ A @ ( minus_minus @ A @ A3 @ ( one_one @ A ) ) @ K ) ) ) ) ).
% gbinomial_absorption
thf(fact_3793_gbinomial__trinomial__revision,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K: nat,M: nat,A3: A] :
( ( ord_less_eq @ nat @ K @ M )
=> ( ( times_times @ A @ ( gbinomial @ A @ A3 @ M ) @ ( gbinomial @ A @ ( semiring_1_of_nat @ A @ M ) @ K ) )
= ( times_times @ A @ ( gbinomial @ A @ A3 @ K ) @ ( gbinomial @ A @ ( minus_minus @ A @ A3 @ ( semiring_1_of_nat @ A @ K ) ) @ ( minus_minus @ nat @ M @ K ) ) ) ) ) ) ).
% gbinomial_trinomial_revision
thf(fact_3794_exp__divide__power__eq,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [N: nat,X: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( power_power @ A @ ( exp @ A @ ( divide_divide @ A @ X @ ( semiring_1_of_nat @ A @ N ) ) ) @ N )
= ( exp @ A @ X ) ) ) ) ).
% exp_divide_power_eq
thf(fact_3795_take__bit__eq__0__iff,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A3: A] :
( ( ( bit_se2584673776208193580ke_bit @ A @ N @ A3 )
= ( zero_zero @ A ) )
= ( dvd_dvd @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) @ A3 ) ) ) ).
% take_bit_eq_0_iff
thf(fact_3796_bin__last__bintrunc,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [L: nat,N: A] :
( ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se2584673776208193580ke_bit @ A @ L @ N ) ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ L )
& ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% bin_last_bintrunc
thf(fact_3797_take__bit__numeral__bit0,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [L: num,K: num] :
( ( bit_se2584673776208193580ke_bit @ A @ ( numeral_numeral @ nat @ L ) @ ( numeral_numeral @ A @ ( bit0 @ K ) ) )
= ( times_times @ A @ ( bit_se2584673776208193580ke_bit @ A @ ( pred_numeral @ L ) @ ( numeral_numeral @ A @ K ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% take_bit_numeral_bit0
thf(fact_3798_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_3799_bits__induct,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [P: A > $o,A3: A] :
( ! [A6: A] :
( ( ( divide_divide @ A @ A6 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= A6 )
=> ( P @ A6 ) )
=> ( ! [A6: A,B7: $o] :
( ( P @ A6 )
=> ( ( ( divide_divide @ A @ ( plus_plus @ A @ ( zero_neq_one_of_bool @ A @ B7 ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A6 ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= A6 )
=> ( P @ ( plus_plus @ A @ ( zero_neq_one_of_bool @ A @ B7 ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A6 ) ) ) ) )
=> ( P @ A3 ) ) ) ) ).
% bits_induct
thf(fact_3800_gbinomial__rec,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: A,K: nat] :
( ( gbinomial @ A @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) @ ( suc @ K ) )
= ( times_times @ A @ ( gbinomial @ A @ A3 @ K ) @ ( divide_divide @ A @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) @ ( semiring_1_of_nat @ A @ ( suc @ K ) ) ) ) ) ) ).
% gbinomial_rec
thf(fact_3801_gbinomial__factors,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: A,K: nat] :
( ( gbinomial @ A @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) @ ( suc @ K ) )
= ( times_times @ A @ ( divide_divide @ A @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) @ ( semiring_1_of_nat @ A @ ( suc @ K ) ) ) @ ( gbinomial @ A @ A3 @ K ) ) ) ) ).
% gbinomial_factors
thf(fact_3802_gbinomial__negated__upper,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( ( gbinomial @ A )
= ( ^ [A5: A,K3: nat] : ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ K3 ) @ ( gbinomial @ A @ ( minus_minus @ A @ ( minus_minus @ A @ ( semiring_1_of_nat @ A @ K3 ) @ A5 ) @ ( one_one @ A ) ) @ K3 ) ) ) ) ) ).
% gbinomial_negated_upper
thf(fact_3803_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_3804_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_3805_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_3806_gbinomial__minus,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: A,K: nat] :
( ( gbinomial @ A @ ( uminus_uminus @ A @ A3 ) @ K )
= ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ K ) @ ( gbinomial @ A @ ( minus_minus @ A @ ( plus_plus @ A @ A3 @ ( semiring_1_of_nat @ A @ K ) ) @ ( one_one @ A ) ) @ K ) ) ) ) ).
% gbinomial_minus
thf(fact_3807_gbinomial__reduce__nat,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K: nat,A3: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ( gbinomial @ A @ A3 @ K )
= ( plus_plus @ A @ ( gbinomial @ A @ ( minus_minus @ A @ A3 @ ( one_one @ A ) ) @ ( minus_minus @ nat @ K @ ( one_one @ nat ) ) ) @ ( gbinomial @ A @ ( minus_minus @ A @ A3 @ ( one_one @ A ) ) @ K ) ) ) ) ) ).
% gbinomial_reduce_nat
thf(fact_3808_gbinomial__pochhammer,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( ( gbinomial @ A )
= ( ^ [A5: A,K3: nat] : ( divide_divide @ A @ ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ K3 ) @ ( comm_s3205402744901411588hammer @ A @ ( uminus_uminus @ A @ A5 ) @ K3 ) ) @ ( semiring_char_0_fact @ A @ K3 ) ) ) ) ) ).
% gbinomial_pochhammer
thf(fact_3809_gbinomial__pochhammer_H,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( ( gbinomial @ A )
= ( ^ [A5: A,K3: nat] : ( divide_divide @ A @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ ( minus_minus @ A @ A5 @ ( semiring_1_of_nat @ A @ K3 ) ) @ ( one_one @ A ) ) @ K3 ) @ ( semiring_char_0_fact @ A @ K3 ) ) ) ) ) ).
% gbinomial_pochhammer'
thf(fact_3810_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_3811_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_3812_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_3813_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_3814_take__bit__Suc,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A3: A] :
( ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N ) @ A3 )
= ( plus_plus @ A @ ( times_times @ A @ ( bit_se2584673776208193580ke_bit @ A @ N @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% take_bit_Suc
thf(fact_3815_exp__bound,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ X @ ( one_one @ real ) )
=> ( ord_less_eq @ real @ ( exp @ real @ X ) @ ( plus_plus @ real @ ( plus_plus @ real @ ( one_one @ real ) @ X ) @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% exp_bound
thf(fact_3816_stable__imp__take__bit__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A,N: nat] :
( ( ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= A3 )
=> ( ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
=> ( ( bit_se2584673776208193580ke_bit @ A @ N @ A3 )
= ( zero_zero @ A ) ) )
& ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
=> ( ( bit_se2584673776208193580ke_bit @ A @ N @ A3 )
= ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) @ ( one_one @ A ) ) ) ) ) ) ) ).
% stable_imp_take_bit_eq
thf(fact_3817_take__bit__numeral__bit1,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [L: num,K: num] :
( ( bit_se2584673776208193580ke_bit @ A @ ( numeral_numeral @ nat @ L ) @ ( numeral_numeral @ A @ ( bit1 @ K ) ) )
= ( plus_plus @ A @ ( times_times @ A @ ( bit_se2584673776208193580ke_bit @ A @ ( pred_numeral @ L ) @ ( numeral_numeral @ A @ K ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) ) ) ) ).
% take_bit_numeral_bit1
thf(fact_3818_real__exp__bound__lemma,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ X @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less_eq @ real @ ( exp @ real @ X ) @ ( plus_plus @ real @ ( one_one @ real ) @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ X ) ) ) ) ) ).
% real_exp_bound_lemma
thf(fact_3819_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_3820_exp__ge__one__plus__x__over__n__power__n,axiom,
! [N: nat,X: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( semiring_1_of_nat @ real @ N ) ) @ X )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ord_less_eq @ real @ ( power_power @ real @ ( plus_plus @ real @ ( one_one @ real ) @ ( divide_divide @ real @ X @ ( semiring_1_of_nat @ real @ N ) ) ) @ N ) @ ( exp @ real @ X ) ) ) ) ).
% exp_ge_one_plus_x_over_n_power_n
thf(fact_3821_exp__ge__one__minus__x__over__n__power__n,axiom,
! [X: real,N: nat] :
( ( ord_less_eq @ real @ X @ ( semiring_1_of_nat @ real @ N ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ord_less_eq @ real @ ( power_power @ real @ ( minus_minus @ real @ ( one_one @ real ) @ ( divide_divide @ real @ X @ ( semiring_1_of_nat @ real @ N ) ) ) @ N ) @ ( exp @ real @ ( uminus_uminus @ real @ X ) ) ) ) ) ).
% exp_ge_one_minus_x_over_n_power_n
thf(fact_3822_gbinomial__absorption_H,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K: nat,A3: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ( gbinomial @ A @ A3 @ K )
= ( times_times @ A @ ( divide_divide @ A @ A3 @ ( semiring_1_of_nat @ A @ K ) ) @ ( gbinomial @ A @ ( minus_minus @ A @ A3 @ ( one_one @ A ) ) @ ( minus_minus @ nat @ K @ ( one_one @ nat ) ) ) ) ) ) ) ).
% gbinomial_absorption'
thf(fact_3823_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_3824_exp__lower__Taylor__quadratic,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ord_less_eq @ real @ ( plus_plus @ real @ ( plus_plus @ real @ ( one_one @ real ) @ X ) @ ( divide_divide @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ ( exp @ real @ X ) ) ) ).
% exp_lower_Taylor_quadratic
thf(fact_3825_log__base__10__eq2,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( log @ ( numeral_numeral @ real @ ( bit0 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) @ X )
= ( times_times @ real @ ( log @ ( numeral_numeral @ real @ ( bit0 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) @ ( exp @ real @ ( one_one @ real ) ) ) @ ( ln_ln @ real @ X ) ) ) ) ).
% log_base_10_eq2
thf(fact_3826_tanh__real__altdef,axiom,
( ( tanh @ real )
= ( ^ [X2: real] : ( divide_divide @ real @ ( minus_minus @ real @ ( one_one @ real ) @ ( exp @ real @ ( times_times @ real @ ( uminus_uminus @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ X2 ) ) ) @ ( plus_plus @ real @ ( one_one @ real ) @ ( exp @ real @ ( times_times @ real @ ( uminus_uminus @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ X2 ) ) ) ) ) ) ).
% tanh_real_altdef
thf(fact_3827_take__bit__rec,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se2584673776208193580ke_bit @ A )
= ( ^ [N4: nat,A5: A] :
( if @ A
@ ( N4
= ( zero_zero @ nat ) )
@ ( zero_zero @ A )
@ ( plus_plus @ A @ ( times_times @ A @ ( bit_se2584673776208193580ke_bit @ A @ ( minus_minus @ nat @ N4 @ ( one_one @ nat ) ) @ ( divide_divide @ A @ A5 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ A @ A5 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% take_bit_rec
thf(fact_3828_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_3829_and__int_Oelims,axiom,
! [X: int,Xa: int,Y: int] :
( ( ( bit_se5824344872417868541ns_and @ int @ X @ Xa )
= Y )
=> ( ( ( ( member @ int @ X @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) )
& ( member @ int @ Xa @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) )
=> ( Y
= ( uminus_uminus @ int
@ ( zero_neq_one_of_bool @ int
@ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ X )
& ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ Xa ) ) ) ) ) )
& ( ~ ( ( member @ int @ X @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) )
& ( member @ int @ Xa @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) )
=> ( Y
= ( plus_plus @ int
@ ( zero_neq_one_of_bool @ int
@ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ X )
& ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ Xa ) ) )
@ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( divide_divide @ int @ X @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ Xa @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% and_int.elims
thf(fact_3830_and__int_Osimps,axiom,
( ( bit_se5824344872417868541ns_and @ int )
= ( ^ [K3: int,L2: int] :
( if @ int
@ ( ( member @ int @ K3 @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) )
& ( member @ int @ L2 @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) )
@ ( uminus_uminus @ int
@ ( zero_neq_one_of_bool @ int
@ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K3 )
& ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ L2 ) ) ) )
@ ( plus_plus @ int
@ ( zero_neq_one_of_bool @ int
@ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K3 )
& ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ L2 ) ) )
@ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( divide_divide @ int @ K3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ).
% and_int.simps
thf(fact_3831_arctan__def,axiom,
( arctan
= ( ^ [Y2: real] :
( the @ real
@ ^ [X2: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X2 )
& ( ord_less @ real @ X2 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
& ( ( tan @ real @ X2 )
= Y2 ) ) ) ) ) ).
% arctan_def
thf(fact_3832_AND__twice,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [W: A,M: A] :
( ( bit_se5824344872417868541ns_and @ A @ ( bit_se5824344872417868541ns_and @ A @ W @ M ) @ M )
= ( bit_se5824344872417868541ns_and @ A @ W @ M ) ) ) ).
% AND_twice
thf(fact_3833_and__zero__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A] :
( ( bit_se5824344872417868541ns_and @ A @ A3 @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% and_zero_eq
thf(fact_3834_zero__and__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A] :
( ( bit_se5824344872417868541ns_and @ A @ ( zero_zero @ A ) @ A3 )
= ( zero_zero @ A ) ) ) ).
% zero_and_eq
thf(fact_3835_bit_Oconj__zero__left,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X: A] :
( ( bit_se5824344872417868541ns_and @ A @ ( zero_zero @ A ) @ X )
= ( zero_zero @ A ) ) ) ).
% bit.conj_zero_left
thf(fact_3836_bit_Oconj__zero__right,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X: A] :
( ( bit_se5824344872417868541ns_and @ A @ X @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% bit.conj_zero_right
thf(fact_3837_and_Oleft__neutral,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [A3: A] :
( ( bit_se5824344872417868541ns_and @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ A3 )
= A3 ) ) ).
% and.left_neutral
thf(fact_3838_and_Oright__neutral,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [A3: A] :
( ( bit_se5824344872417868541ns_and @ A @ A3 @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= A3 ) ) ).
% and.right_neutral
thf(fact_3839_bit_Oconj__one__right,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X: A] :
( ( bit_se5824344872417868541ns_and @ A @ X @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= X ) ) ).
% bit.conj_one_right
thf(fact_3840_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_3841_and__nonnegative__int__iff,axiom,
! [K: int,L: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se5824344872417868541ns_and @ int @ K @ L ) )
= ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
| ( ord_less_eq @ int @ ( zero_zero @ int ) @ L ) ) ) ).
% and_nonnegative_int_iff
thf(fact_3842_and__negative__int__iff,axiom,
! [K: int,L: int] :
( ( ord_less @ int @ ( bit_se5824344872417868541ns_and @ int @ K @ L ) @ ( zero_zero @ int ) )
= ( ( ord_less @ int @ K @ ( zero_zero @ int ) )
& ( ord_less @ int @ L @ ( zero_zero @ int ) ) ) ) ).
% and_negative_int_iff
thf(fact_3843_pred__numeral__inc,axiom,
! [K: num] :
( ( pred_numeral @ ( inc @ K ) )
= ( numeral_numeral @ nat @ K ) ) ).
% pred_numeral_inc
thf(fact_3844_and__numerals_I8_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X: num] :
( ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ ( bit1 @ X ) ) @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% and_numerals(8)
thf(fact_3845_and__numerals_I2_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [Y: num] :
( ( bit_se5824344872417868541ns_and @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit1 @ Y ) ) )
= ( one_one @ A ) ) ) ).
% and_numerals(2)
thf(fact_3846_and__numerals_I1_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [Y: num] :
( ( bit_se5824344872417868541ns_and @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ Y ) ) )
= ( zero_zero @ A ) ) ) ).
% and_numerals(1)
thf(fact_3847_and__numerals_I5_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X: num] :
( ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ ( bit0 @ X ) ) @ ( one_one @ A ) )
= ( zero_zero @ A ) ) ) ).
% and_numerals(5)
thf(fact_3848_and__numerals_I3_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X: num,Y: num] :
( ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ ( bit0 @ X ) ) @ ( numeral_numeral @ A @ ( bit0 @ Y ) ) )
= ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ X ) @ ( numeral_numeral @ A @ Y ) ) ) ) ) ).
% and_numerals(3)
thf(fact_3849_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_3850_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_3851_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_3852_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_3853_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_3854_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_3855_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_3856_and__numerals_I4_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X: num,Y: num] :
( ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ ( bit0 @ X ) ) @ ( numeral_numeral @ A @ ( bit1 @ Y ) ) )
= ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ X ) @ ( numeral_numeral @ A @ Y ) ) ) ) ) ).
% and_numerals(4)
thf(fact_3857_and__numerals_I6_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X: num,Y: num] :
( ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ ( bit1 @ X ) ) @ ( numeral_numeral @ A @ ( bit0 @ Y ) ) )
= ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ X ) @ ( numeral_numeral @ A @ Y ) ) ) ) ) ).
% and_numerals(6)
thf(fact_3858_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_3859_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_3860_and__numerals_I7_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X: num,Y: num] :
( ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ ( bit1 @ X ) ) @ ( numeral_numeral @ A @ ( bit1 @ Y ) ) )
= ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ X ) @ ( numeral_numeral @ A @ Y ) ) ) ) ) ) ).
% and_numerals(7)
thf(fact_3861_signed__and__eq,axiom,
! [A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( bit_ri3973907225187159222ations @ A ) )
=> ! [V2: word @ B,W: word @ B] :
( ( ring_1_signed @ B @ A @ ( bit_se5824344872417868541ns_and @ ( word @ B ) @ V2 @ W ) )
= ( bit_se5824344872417868541ns_and @ A @ ( ring_1_signed @ B @ A @ V2 ) @ ( ring_1_signed @ B @ A @ W ) ) ) ) ).
% signed_and_eq
thf(fact_3862_num__induct,axiom,
! [P: num > $o,X: num] :
( ( P @ one2 )
=> ( ! [X3: num] :
( ( P @ X3 )
=> ( P @ ( inc @ X3 ) ) )
=> ( P @ X ) ) ) ).
% num_induct
thf(fact_3863_unsigned__and__eq,axiom,
! [A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( bit_se359711467146920520ations @ A ) )
=> ! [V2: word @ B,W: word @ B] :
( ( semiring_1_unsigned @ B @ A @ ( bit_se5824344872417868541ns_and @ ( word @ B ) @ V2 @ W ) )
= ( bit_se5824344872417868541ns_and @ A @ ( semiring_1_unsigned @ B @ A @ V2 ) @ ( semiring_1_unsigned @ B @ A @ W ) ) ) ) ).
% unsigned_and_eq
thf(fact_3864_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_3865_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_3866_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_3867_add__inc,axiom,
! [X: num,Y: num] :
( ( plus_plus @ num @ X @ ( inc @ Y ) )
= ( inc @ ( plus_plus @ num @ X @ Y ) ) ) ).
% add_inc
thf(fact_3868_and__eq__minus__1__iff,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [A3: A,B3: A] :
( ( ( bit_se5824344872417868541ns_and @ A @ A3 @ B3 )
= ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( ( A3
= ( uminus_uminus @ A @ ( one_one @ A ) ) )
& ( B3
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ) ).
% and_eq_minus_1_iff
thf(fact_3869_AND__lower,axiom,
! [X: int,Y: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
=> ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se5824344872417868541ns_and @ int @ X @ Y ) ) ) ).
% AND_lower
thf(fact_3870_AND__upper1,axiom,
! [X: int,Y: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
=> ( ord_less_eq @ int @ ( bit_se5824344872417868541ns_and @ int @ X @ Y ) @ X ) ) ).
% AND_upper1
thf(fact_3871_AND__upper2,axiom,
! [Y: int,X: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y )
=> ( ord_less_eq @ int @ ( bit_se5824344872417868541ns_and @ int @ X @ Y ) @ Y ) ) ).
% AND_upper2
thf(fact_3872_AND__upper1_H,axiom,
! [Y: int,Z: int,Ya: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y )
=> ( ( ord_less_eq @ int @ Y @ Z )
=> ( ord_less_eq @ int @ ( bit_se5824344872417868541ns_and @ int @ Y @ Ya ) @ Z ) ) ) ).
% AND_upper1'
thf(fact_3873_AND__upper2_H,axiom,
! [Y: int,Z: int,X: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y )
=> ( ( ord_less_eq @ int @ Y @ Z )
=> ( ord_less_eq @ int @ ( bit_se5824344872417868541ns_and @ int @ X @ Y ) @ Z ) ) ) ).
% AND_upper2'
thf(fact_3874_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_3875_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_3876_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_3877_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_3878_uint__take__bit__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,W: word @ A] :
( ( semiring_1_unsigned @ A @ int @ ( bit_se2584673776208193580ke_bit @ ( word @ A ) @ N @ W ) )
= ( bit_se2584673776208193580ke_bit @ int @ N @ ( semiring_1_unsigned @ A @ int @ W ) ) ) ) ).
% uint_take_bit_eq
thf(fact_3879_inc_Osimps_I1_J,axiom,
( ( inc @ one2 )
= ( bit0 @ one2 ) ) ).
% inc.simps(1)
thf(fact_3880_inc_Osimps_I2_J,axiom,
! [X: num] :
( ( inc @ ( bit0 @ X ) )
= ( bit1 @ X ) ) ).
% inc.simps(2)
thf(fact_3881_inc_Osimps_I3_J,axiom,
! [X: num] :
( ( inc @ ( bit1 @ X ) )
= ( bit0 @ ( inc @ X ) ) ) ).
% inc.simps(3)
thf(fact_3882_add__One,axiom,
! [X: num] :
( ( plus_plus @ num @ X @ one2 )
= ( inc @ X ) ) ).
% add_One
thf(fact_3883_ln__real__def,axiom,
( ( ln_ln @ real )
= ( ^ [X2: real] :
( the @ real
@ ^ [U: real] :
( ( exp @ real @ U )
= X2 ) ) ) ) ).
% ln_real_def
thf(fact_3884_suminf__def,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ( ( suminf @ A )
= ( ^ [F2: nat > A] : ( the @ A @ ( sums @ A @ F2 ) ) ) ) ) ).
% suminf_def
thf(fact_3885_mult__inc,axiom,
! [X: num,Y: num] :
( ( times_times @ num @ X @ ( inc @ Y ) )
= ( plus_plus @ num @ ( times_times @ num @ X @ Y ) @ X ) ) ).
% mult_inc
thf(fact_3886_AND__upper2_H_H,axiom,
! [Y: int,Z: int,X: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y )
=> ( ( ord_less @ int @ Y @ Z )
=> ( ord_less @ int @ ( bit_se5824344872417868541ns_and @ int @ X @ Y ) @ Z ) ) ) ).
% AND_upper2''
thf(fact_3887_AND__upper1_H_H,axiom,
! [Y: int,Z: int,Ya: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y )
=> ( ( ord_less @ int @ Y @ Z )
=> ( ord_less @ int @ ( bit_se5824344872417868541ns_and @ int @ Y @ Ya ) @ Z ) ) ) ).
% AND_upper1''
thf(fact_3888_and__less__eq,axiom,
! [L: int,K: int] :
( ( ord_less @ int @ L @ ( zero_zero @ int ) )
=> ( ord_less_eq @ int @ ( bit_se5824344872417868541ns_and @ int @ K @ L ) @ K ) ) ).
% and_less_eq
thf(fact_3889_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_3890_numeral__inc,axiom,
! [A: $tType] :
( ( numeral @ A )
=> ! [X: num] :
( ( numeral_numeral @ A @ ( inc @ X ) )
= ( plus_plus @ A @ ( numeral_numeral @ A @ X ) @ ( one_one @ A ) ) ) ) ).
% numeral_inc
thf(fact_3891_even__and__iff,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A,B3: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ A @ A3 @ B3 ) )
= ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
| ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B3 ) ) ) ) ).
% even_and_iff
thf(fact_3892_ln__neg__is__const,axiom,
! [X: real] :
( ( ord_less_eq @ real @ X @ ( zero_zero @ real ) )
=> ( ( ln_ln @ real @ X )
= ( the @ real
@ ^ [X2: real] : $false ) ) ) ).
% ln_neg_is_const
thf(fact_3893_even__and__iff__int,axiom,
! [K: int,L: int] :
( ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ K @ L ) )
= ( ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K )
| ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ L ) ) ) ).
% even_and_iff_int
thf(fact_3894_the__elem__def,axiom,
! [A: $tType] :
( ( the_elem @ A )
= ( ^ [X8: set @ A] :
( the @ A
@ ^ [X2: A] :
( X8
= ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% the_elem_def
thf(fact_3895_one__and__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A] :
( ( bit_se5824344872417868541ns_and @ A @ ( one_one @ A ) @ A3 )
= ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% one_and_eq
thf(fact_3896_and__one__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A] :
( ( bit_se5824344872417868541ns_and @ A @ A3 @ ( one_one @ A ) )
= ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% and_one_eq
thf(fact_3897_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_3898_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_3899_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_3900_take__bit__nat__def,axiom,
( ( bit_se2584673776208193580ke_bit @ nat )
= ( ^ [N4: nat,M3: nat] : ( modulo_modulo @ nat @ M3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N4 ) ) ) ) ).
% take_bit_nat_def
thf(fact_3901_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_3902_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_3903_take__bit__int__def,axiom,
( ( bit_se2584673776208193580ke_bit @ int )
= ( ^ [N4: nat,K3: int] : ( modulo_modulo @ int @ K3 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N4 ) ) ) ) ).
% take_bit_int_def
thf(fact_3904_take__bit__numeral__minus__bit1,axiom,
! [L: num,K: num] :
( ( bit_se2584673776208193580ke_bit @ int @ ( numeral_numeral @ nat @ L ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ K ) ) ) )
= ( plus_plus @ int @ ( times_times @ int @ ( bit_se2584673776208193580ke_bit @ int @ ( pred_numeral @ L ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( inc @ K ) ) ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( one_one @ int ) ) ) ).
% take_bit_numeral_minus_bit1
thf(fact_3905_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_3906_arccos__def,axiom,
( arccos
= ( ^ [Y2: real] :
( the @ real
@ ^ [X2: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
& ( ord_less_eq @ real @ X2 @ pi )
& ( ( cos @ real @ X2 )
= Y2 ) ) ) ) ) ).
% arccos_def
thf(fact_3907_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_3908_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_3909_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_3910_and__int__rec,axiom,
( ( bit_se5824344872417868541ns_and @ int )
= ( ^ [K3: int,L2: int] :
( plus_plus @ int
@ ( zero_neq_one_of_bool @ int
@ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K3 )
& ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ L2 ) ) )
@ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( divide_divide @ int @ K3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% and_int_rec
thf(fact_3911_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_3912_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_3913_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_3914_take__bit__numeral__minus__bit0,axiom,
! [L: num,K: num] :
( ( bit_se2584673776208193580ke_bit @ int @ ( numeral_numeral @ nat @ L ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ K ) ) ) )
= ( times_times @ int @ ( bit_se2584673776208193580ke_bit @ int @ ( pred_numeral @ L ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ).
% take_bit_numeral_minus_bit0
thf(fact_3915_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_3916_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_3917_signed__take__bit__eq__take__bit__shift,axiom,
( ( bit_ri4674362597316999326ke_bit @ int )
= ( ^ [N4: nat,K3: int] : ( minus_minus @ int @ ( bit_se2584673776208193580ke_bit @ int @ ( suc @ N4 ) @ ( plus_plus @ int @ K3 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N4 ) ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N4 ) ) ) ) ).
% signed_take_bit_eq_take_bit_shift
thf(fact_3918_pi__half,axiom,
( ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) )
= ( the @ real
@ ^ [X2: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
& ( ord_less_eq @ real @ X2 @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) )
& ( ( cos @ real @ X2 )
= ( zero_zero @ real ) ) ) ) ) ).
% pi_half
thf(fact_3919_pi__def,axiom,
( pi
= ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) )
@ ( the @ real
@ ^ [X2: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
& ( ord_less_eq @ real @ X2 @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) )
& ( ( cos @ real @ X2 )
= ( zero_zero @ real ) ) ) ) ) ) ).
% pi_def
thf(fact_3920_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_3921_and__int__unfold,axiom,
( ( bit_se5824344872417868541ns_and @ int )
= ( ^ [K3: int,L2: int] :
( if @ int
@ ( ( K3
= ( zero_zero @ int ) )
| ( L2
= ( zero_zero @ int ) ) )
@ ( zero_zero @ int )
@ ( if @ int
@ ( K3
= ( uminus_uminus @ int @ ( one_one @ int ) ) )
@ L2
@ ( if @ int
@ ( L2
= ( uminus_uminus @ int @ ( one_one @ int ) ) )
@ K3
@ ( plus_plus @ int @ ( times_times @ int @ ( modulo_modulo @ int @ K3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ int @ L2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( divide_divide @ int @ K3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ).
% and_int_unfold
thf(fact_3922_arcsin__def,axiom,
( arcsin
= ( ^ [Y2: real] :
( the @ real
@ ^ [X2: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X2 )
& ( ord_less_eq @ real @ X2 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
& ( ( sin @ real @ X2 )
= Y2 ) ) ) ) ) ).
% arcsin_def
thf(fact_3923_the__equality,axiom,
! [A: $tType,P: A > $o,A3: A] :
( ( P @ A3 )
=> ( ! [X3: A] :
( ( P @ X3 )
=> ( X3 = A3 ) )
=> ( ( the @ A @ P )
= A3 ) ) ) ).
% the_equality
thf(fact_3924_the__eq__trivial,axiom,
! [A: $tType,A3: A] :
( ( the @ A
@ ^ [X2: A] : X2 = A3 )
= A3 ) ).
% the_eq_trivial
thf(fact_3925_the__sym__eq__trivial,axiom,
! [A: $tType,X: A] :
( ( the @ A
@ ( ^ [Y3: A,Z2: A] : Y3 = Z2
@ X ) )
= X ) ).
% the_sym_eq_trivial
thf(fact_3926_modulo__int__def,axiom,
( ( modulo_modulo @ int )
= ( ^ [K3: int,L2: int] :
( if @ int
@ ( L2
= ( zero_zero @ int ) )
@ K3
@ ( if @ int
@ ( ( sgn_sgn @ int @ K3 )
= ( sgn_sgn @ int @ L2 ) )
@ ( times_times @ int @ ( sgn_sgn @ int @ L2 ) @ ( semiring_1_of_nat @ int @ ( modulo_modulo @ nat @ ( nat2 @ ( abs_abs @ int @ K3 ) ) @ ( nat2 @ ( abs_abs @ int @ L2 ) ) ) ) )
@ ( times_times @ int @ ( sgn_sgn @ int @ L2 )
@ ( minus_minus @ int
@ ( times_times @ int @ ( abs_abs @ int @ L2 )
@ ( zero_neq_one_of_bool @ int
@ ~ ( dvd_dvd @ int @ L2 @ K3 ) ) )
@ ( semiring_1_of_nat @ int @ ( modulo_modulo @ nat @ ( nat2 @ ( abs_abs @ int @ K3 ) ) @ ( nat2 @ ( abs_abs @ int @ L2 ) ) ) ) ) ) ) ) ) ) ).
% modulo_int_def
thf(fact_3927_sgn__sgn,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ! [A3: A] :
( ( sgn_sgn @ A @ ( sgn_sgn @ A @ A3 ) )
= ( sgn_sgn @ A @ A3 ) ) ) ).
% sgn_sgn
thf(fact_3928_sgn__zero,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ( ( sgn_sgn @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% sgn_zero
thf(fact_3929_sgn__0,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ( ( sgn_sgn @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% sgn_0
thf(fact_3930_sgn__one,axiom,
! [A: $tType] :
( ( real_V2822296259951069270ebra_1 @ A )
=> ( ( sgn_sgn @ A @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% sgn_one
thf(fact_3931_sgn__1,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ( ( sgn_sgn @ A @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% sgn_1
thf(fact_3932_power__sgn,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,N: nat] :
( ( sgn_sgn @ A @ ( power_power @ A @ A3 @ N ) )
= ( power_power @ A @ ( sgn_sgn @ A @ A3 ) @ N ) ) ) ).
% power_sgn
thf(fact_3933_idom__abs__sgn__class_Osgn__minus,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ! [A3: A] :
( ( sgn_sgn @ A @ ( uminus_uminus @ A @ A3 ) )
= ( uminus_uminus @ A @ ( sgn_sgn @ A @ A3 ) ) ) ) ).
% idom_abs_sgn_class.sgn_minus
thf(fact_3934_sgn__less,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( sgn_sgn @ A @ A3 ) @ ( zero_zero @ A ) )
= ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% sgn_less
thf(fact_3935_sgn__greater,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( sgn_sgn @ A @ A3 ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% sgn_greater
thf(fact_3936_word__and__max,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] :
( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) )
= X ) ) ).
% word_and_max
thf(fact_3937_word__bitwise__m1__simps_I2_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] :
( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) @ X )
= X ) ) ).
% word_bitwise_m1_simps(2)
thf(fact_3938_sgn__pos,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( sgn_sgn @ A @ A3 )
= ( one_one @ A ) ) ) ) ).
% sgn_pos
thf(fact_3939_abs__sgn__eq__1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( abs_abs @ A @ ( sgn_sgn @ A @ A3 ) )
= ( one_one @ A ) ) ) ) ).
% abs_sgn_eq_1
thf(fact_3940_sgn__mult__self__eq,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A] :
( ( times_times @ A @ ( sgn_sgn @ A @ A3 ) @ ( sgn_sgn @ A @ A3 ) )
= ( zero_neq_one_of_bool @ A
@ ( A3
!= ( zero_zero @ A ) ) ) ) ) ).
% sgn_mult_self_eq
thf(fact_3941_sgn__abs,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ! [A3: A] :
( ( abs_abs @ A @ ( sgn_sgn @ A @ A3 ) )
= ( zero_neq_one_of_bool @ A
@ ( A3
!= ( zero_zero @ A ) ) ) ) ) ).
% sgn_abs
thf(fact_3942_idom__abs__sgn__class_Oabs__sgn,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ! [A3: A] :
( ( sgn_sgn @ A @ ( abs_abs @ A @ A3 ) )
= ( zero_neq_one_of_bool @ A
@ ( A3
!= ( zero_zero @ A ) ) ) ) ) ).
% idom_abs_sgn_class.abs_sgn
thf(fact_3943_dvd__mult__sgn__iff,axiom,
! [L: int,K: int,R3: int] :
( ( dvd_dvd @ int @ L @ ( times_times @ int @ K @ ( sgn_sgn @ int @ R3 ) ) )
= ( ( dvd_dvd @ int @ L @ K )
| ( R3
= ( zero_zero @ int ) ) ) ) ).
% dvd_mult_sgn_iff
thf(fact_3944_dvd__sgn__mult__iff,axiom,
! [L: int,R3: int,K: int] :
( ( dvd_dvd @ int @ L @ ( times_times @ int @ ( sgn_sgn @ int @ R3 ) @ K ) )
= ( ( dvd_dvd @ int @ L @ K )
| ( R3
= ( zero_zero @ int ) ) ) ) ).
% dvd_sgn_mult_iff
thf(fact_3945_mult__sgn__dvd__iff,axiom,
! [L: int,R3: int,K: int] :
( ( dvd_dvd @ int @ ( times_times @ int @ L @ ( sgn_sgn @ int @ R3 ) ) @ K )
= ( ( dvd_dvd @ int @ L @ K )
& ( ( R3
= ( zero_zero @ int ) )
=> ( K
= ( zero_zero @ int ) ) ) ) ) ).
% mult_sgn_dvd_iff
thf(fact_3946_sgn__mult__dvd__iff,axiom,
! [R3: int,L: int,K: int] :
( ( dvd_dvd @ int @ ( times_times @ int @ ( sgn_sgn @ int @ R3 ) @ L ) @ K )
= ( ( dvd_dvd @ int @ L @ K )
& ( ( R3
= ( zero_zero @ int ) )
=> ( K
= ( zero_zero @ int ) ) ) ) ) ).
% sgn_mult_dvd_iff
thf(fact_3947_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_3948_sgn__neg,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A] :
( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( sgn_sgn @ A @ A3 )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ).
% sgn_neg
thf(fact_3949_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_3950_and__nat__numerals_I3_J,axiom,
! [X: num] :
( ( bit_se5824344872417868541ns_and @ nat @ ( numeral_numeral @ nat @ ( bit0 @ X ) ) @ ( suc @ ( zero_zero @ nat ) ) )
= ( zero_zero @ nat ) ) ).
% and_nat_numerals(3)
thf(fact_3951_and__nat__numerals_I1_J,axiom,
! [Y: num] :
( ( bit_se5824344872417868541ns_and @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( numeral_numeral @ nat @ ( bit0 @ Y ) ) )
= ( zero_zero @ nat ) ) ).
% and_nat_numerals(1)
thf(fact_3952_word__no__log__defs_I3_J,axiom,
! [C: $tType] :
( ( type_len @ C )
=> ! [A3: num,B3: num] :
( ( bit_se5824344872417868541ns_and @ ( word @ C ) @ ( numeral_numeral @ ( word @ C ) @ A3 ) @ ( numeral_numeral @ ( word @ C ) @ B3 ) )
= ( ring_1_of_int @ ( word @ C ) @ ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ A3 ) @ ( numeral_numeral @ int @ B3 ) ) ) ) ) ).
% word_no_log_defs(3)
thf(fact_3953_and__nat__numerals_I4_J,axiom,
! [X: num] :
( ( bit_se5824344872417868541ns_and @ nat @ ( numeral_numeral @ nat @ ( bit1 @ X ) ) @ ( suc @ ( zero_zero @ nat ) ) )
= ( one_one @ nat ) ) ).
% and_nat_numerals(4)
thf(fact_3954_and__nat__numerals_I2_J,axiom,
! [Y: num] :
( ( bit_se5824344872417868541ns_and @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( numeral_numeral @ nat @ ( bit1 @ Y ) ) )
= ( one_one @ nat ) ) ).
% and_nat_numerals(2)
thf(fact_3955_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_3956_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_3957_word__bitwise__1__simps_I4_J,axiom,
! [D: $tType] :
( ( type_len @ D )
=> ! [A3: num] :
( ( bit_se5824344872417868541ns_and @ ( word @ D ) @ ( numeral_numeral @ ( word @ D ) @ A3 ) @ ( one_one @ ( word @ D ) ) )
= ( ring_1_of_int @ ( word @ D ) @ ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ A3 ) @ ( one_one @ int ) ) ) ) ) ).
% word_bitwise_1_simps(4)
thf(fact_3958_word__bitwise__1__simps_I2_J,axiom,
! [B: $tType] :
( ( type_len @ B )
=> ! [B3: num] :
( ( bit_se5824344872417868541ns_and @ ( word @ B ) @ ( one_one @ ( word @ B ) ) @ ( numeral_numeral @ ( word @ B ) @ B3 ) )
= ( ring_1_of_int @ ( word @ B ) @ ( bit_se5824344872417868541ns_and @ int @ ( one_one @ int ) @ ( numeral_numeral @ int @ B3 ) ) ) ) ) ).
% word_bitwise_1_simps(2)
thf(fact_3959_word__no__log__defs_I4_J,axiom,
! [D: $tType] :
( ( type_len @ D )
=> ! [A3: num,B3: num] :
( ( bit_se5824344872417868541ns_and @ ( word @ D ) @ ( numeral_numeral @ ( word @ D ) @ A3 ) @ ( uminus_uminus @ ( word @ D ) @ ( numeral_numeral @ ( word @ D ) @ B3 ) ) )
= ( ring_1_of_int @ ( word @ D ) @ ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ A3 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ B3 ) ) ) ) ) ) ).
% word_no_log_defs(4)
thf(fact_3960_word__no__log__defs_I5_J,axiom,
! [E3: $tType] :
( ( type_len @ E3 )
=> ! [A3: num,B3: num] :
( ( bit_se5824344872417868541ns_and @ ( word @ E3 ) @ ( uminus_uminus @ ( word @ E3 ) @ ( numeral_numeral @ ( word @ E3 ) @ A3 ) ) @ ( numeral_numeral @ ( word @ E3 ) @ B3 ) )
= ( ring_1_of_int @ ( word @ E3 ) @ ( bit_se5824344872417868541ns_and @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ A3 ) ) @ ( numeral_numeral @ int @ B3 ) ) ) ) ) ).
% word_no_log_defs(5)
thf(fact_3961_word__no__log__defs_I6_J,axiom,
! [F: $tType] :
( ( type_len @ F )
=> ! [A3: num,B3: num] :
( ( bit_se5824344872417868541ns_and @ ( word @ F ) @ ( uminus_uminus @ ( word @ F ) @ ( numeral_numeral @ ( word @ F ) @ A3 ) ) @ ( uminus_uminus @ ( word @ F ) @ ( numeral_numeral @ ( word @ F ) @ B3 ) ) )
= ( ring_1_of_int @ ( word @ F ) @ ( bit_se5824344872417868541ns_and @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ A3 ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ B3 ) ) ) ) ) ) ).
% word_no_log_defs(6)
thf(fact_3962_word__bitwise__1__simps_I5_J,axiom,
! [E3: $tType] :
( ( type_len @ E3 )
=> ! [A3: num] :
( ( bit_se5824344872417868541ns_and @ ( word @ E3 ) @ ( uminus_uminus @ ( word @ E3 ) @ ( numeral_numeral @ ( word @ E3 ) @ A3 ) ) @ ( one_one @ ( word @ E3 ) ) )
= ( ring_1_of_int @ ( word @ E3 ) @ ( bit_se5824344872417868541ns_and @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ A3 ) ) @ ( one_one @ int ) ) ) ) ) ).
% word_bitwise_1_simps(5)
thf(fact_3963_word__bitwise__1__simps_I3_J,axiom,
! [C: $tType] :
( ( type_len @ C )
=> ! [B3: num] :
( ( bit_se5824344872417868541ns_and @ ( word @ C ) @ ( one_one @ ( word @ C ) ) @ ( uminus_uminus @ ( word @ C ) @ ( numeral_numeral @ ( word @ C ) @ B3 ) ) )
= ( ring_1_of_int @ ( word @ C ) @ ( bit_se5824344872417868541ns_and @ int @ ( one_one @ int ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ B3 ) ) ) ) ) ) ).
% word_bitwise_1_simps(3)
thf(fact_3964_word__log__esimps_I1_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] :
( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X @ ( zero_zero @ ( word @ A ) ) )
= ( zero_zero @ ( word @ A ) ) ) ) ).
% word_log_esimps(1)
thf(fact_3965_word__log__esimps_I7_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] :
( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ X )
= ( zero_zero @ ( word @ A ) ) ) ) ).
% word_log_esimps(7)
thf(fact_3966_word__bw__assocs_I1_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A,Z: word @ A] :
( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X @ Y ) @ Z )
= ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ Y @ Z ) ) ) ) ).
% word_bw_assocs(1)
thf(fact_3967_word__bw__comms_I1_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( bit_se5824344872417868541ns_and @ ( word @ A ) )
= ( ^ [X2: word @ A,Y2: word @ A] : ( bit_se5824344872417868541ns_and @ ( word @ A ) @ Y2 @ X2 ) ) ) ) ).
% word_bw_comms(1)
thf(fact_3968_word__bw__same_I1_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] :
( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X @ X )
= X ) ) ).
% word_bw_same(1)
thf(fact_3969_word__bw__lcs_I1_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y: word @ A,X: word @ A,Z: word @ A] :
( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ Y @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X @ Z ) )
= ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ Y @ Z ) ) ) ) ).
% word_bw_lcs(1)
thf(fact_3970_word__and__le2,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A,Y: word @ A] : ( ord_less_eq @ ( word @ A ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ A3 @ Y ) @ A3 ) ) ).
% word_and_le2
thf(fact_3971_word__and__le1,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y: word @ A,A3: word @ A] : ( ord_less_eq @ ( word @ A ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ Y @ A3 ) @ A3 ) ) ).
% word_and_le1
thf(fact_3972_sgn__mult,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ! [A3: A,B3: A] :
( ( sgn_sgn @ A @ ( times_times @ A @ A3 @ B3 ) )
= ( times_times @ A @ ( sgn_sgn @ A @ A3 ) @ ( sgn_sgn @ A @ B3 ) ) ) ) ).
% sgn_mult
thf(fact_3973_same__sgn__sgn__add,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [B3: A,A3: A] :
( ( ( sgn_sgn @ A @ B3 )
= ( sgn_sgn @ A @ A3 ) )
=> ( ( sgn_sgn @ A @ ( plus_plus @ A @ A3 @ B3 ) )
= ( sgn_sgn @ A @ A3 ) ) ) ) ).
% same_sgn_sgn_add
thf(fact_3974_sgn__eq__0__iff,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ! [A3: A] :
( ( ( sgn_sgn @ A @ A3 )
= ( zero_zero @ A ) )
= ( A3
= ( zero_zero @ A ) ) ) ) ).
% sgn_eq_0_iff
thf(fact_3975_sgn__0__0,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A] :
( ( ( sgn_sgn @ A @ A3 )
= ( zero_zero @ A ) )
= ( A3
= ( zero_zero @ A ) ) ) ) ).
% sgn_0_0
thf(fact_3976_sgn__zero__iff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X: A] :
( ( ( sgn_sgn @ A @ X )
= ( zero_zero @ A ) )
= ( X
= ( zero_zero @ A ) ) ) ) ).
% sgn_zero_iff
thf(fact_3977_sgn__not__eq__imp,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [B3: A,A3: A] :
( ( ( sgn_sgn @ A @ B3 )
!= ( sgn_sgn @ A @ A3 ) )
=> ( ( ( sgn_sgn @ A @ A3 )
!= ( zero_zero @ A ) )
=> ( ( ( sgn_sgn @ A @ B3 )
!= ( zero_zero @ A ) )
=> ( ( sgn_sgn @ A @ A3 )
= ( uminus_uminus @ A @ ( sgn_sgn @ A @ B3 ) ) ) ) ) ) ) ).
% sgn_not_eq_imp
thf(fact_3978_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_3979_int__sgnE,axiom,
! [K: int] :
~ ! [N2: nat,L3: int] :
( K
!= ( times_times @ int @ ( sgn_sgn @ int @ L3 ) @ ( semiring_1_of_nat @ int @ N2 ) ) ) ).
% int_sgnE
thf(fact_3980_same__sgn__abs__add,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [B3: A,A3: A] :
( ( ( sgn_sgn @ A @ B3 )
= ( sgn_sgn @ A @ A3 ) )
=> ( ( abs_abs @ A @ ( plus_plus @ A @ A3 @ B3 ) )
= ( plus_plus @ A @ ( abs_abs @ A @ A3 ) @ ( abs_abs @ A @ B3 ) ) ) ) ) ).
% same_sgn_abs_add
thf(fact_3981_linordered__idom__class_Oabs__sgn,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( ( abs_abs @ A )
= ( ^ [K3: A] : ( times_times @ A @ K3 @ ( sgn_sgn @ A @ K3 ) ) ) ) ) ).
% linordered_idom_class.abs_sgn
thf(fact_3982_abs__mult__sgn,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ! [A3: A] :
( ( times_times @ A @ ( abs_abs @ A @ A3 ) @ ( sgn_sgn @ A @ A3 ) )
= A3 ) ) ).
% abs_mult_sgn
thf(fact_3983_sgn__mult__abs,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ! [A3: A] :
( ( times_times @ A @ ( sgn_sgn @ A @ A3 ) @ ( abs_abs @ A @ A3 ) )
= A3 ) ) ).
% sgn_mult_abs
thf(fact_3984_mult__sgn__abs,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A] :
( ( times_times @ A @ ( sgn_sgn @ A @ X ) @ ( abs_abs @ A @ X ) )
= X ) ) ).
% mult_sgn_abs
thf(fact_3985_word__unat__and__lt,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,N: nat,Y: word @ A] :
( ( ( ord_less @ nat @ ( semiring_1_unsigned @ A @ nat @ X ) @ N )
| ( ord_less @ nat @ ( semiring_1_unsigned @ A @ nat @ Y ) @ N ) )
=> ( ord_less @ nat @ ( semiring_1_unsigned @ A @ nat @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X @ Y ) ) @ N ) ) ) ).
% word_unat_and_lt
thf(fact_3986_word__and__max__word,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,A3: word @ A] :
( ( X
= ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) )
=> ( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ A3 @ X )
= A3 ) ) ) ).
% word_and_max_word
thf(fact_3987_uint__and,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] :
( ( semiring_1_unsigned @ A @ int @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X @ Y ) )
= ( bit_se5824344872417868541ns_and @ int @ ( semiring_1_unsigned @ A @ int @ X ) @ ( semiring_1_unsigned @ A @ int @ Y ) ) ) ) ).
% uint_and
thf(fact_3988_word__wi__log__defs_I2_J,axiom,
! [B: $tType] :
( ( type_len @ B )
=> ! [A3: int,B3: int] :
( ( bit_se5824344872417868541ns_and @ ( word @ B ) @ ( ring_1_of_int @ ( word @ B ) @ A3 ) @ ( ring_1_of_int @ ( word @ B ) @ B3 ) )
= ( ring_1_of_int @ ( word @ B ) @ ( bit_se5824344872417868541ns_and @ int @ A3 @ B3 ) ) ) ) ).
% word_wi_log_defs(2)
thf(fact_3989_sgn__1__pos,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A] :
( ( ( sgn_sgn @ A @ A3 )
= ( one_one @ A ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% sgn_1_pos
thf(fact_3990_abs__sgn__eq,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A] :
( ( ( A3
= ( zero_zero @ A ) )
=> ( ( abs_abs @ A @ ( sgn_sgn @ A @ A3 ) )
= ( zero_zero @ A ) ) )
& ( ( A3
!= ( zero_zero @ A ) )
=> ( ( abs_abs @ A @ ( sgn_sgn @ A @ A3 ) )
= ( one_one @ A ) ) ) ) ) ).
% abs_sgn_eq
thf(fact_3991_sgn__mod,axiom,
! [L: int,K: int] :
( ( L
!= ( zero_zero @ int ) )
=> ( ~ ( dvd_dvd @ int @ L @ K )
=> ( ( sgn_sgn @ int @ ( modulo_modulo @ int @ K @ L ) )
= ( sgn_sgn @ int @ L ) ) ) ) ).
% sgn_mod
thf(fact_3992_and__nat__def,axiom,
( ( bit_se5824344872417868541ns_and @ nat )
= ( ^ [M3: nat,N4: nat] : ( nat2 @ ( bit_se5824344872417868541ns_and @ int @ ( semiring_1_of_nat @ int @ M3 ) @ ( semiring_1_of_nat @ int @ N4 ) ) ) ) ) ).
% and_nat_def
thf(fact_3993_word__and__def,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( bit_se5824344872417868541ns_and @ ( word @ A ) )
= ( ^ [A5: word @ A,B5: word @ A] : ( ring_1_of_int @ ( word @ A ) @ ( bit_se5824344872417868541ns_and @ int @ ( semiring_1_unsigned @ A @ int @ A5 ) @ ( semiring_1_unsigned @ A @ int @ B5 ) ) ) ) ) ) ).
% word_and_def
thf(fact_3994_sgn__if,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( ( sgn_sgn @ A )
= ( ^ [X2: A] :
( if @ A
@ ( X2
= ( zero_zero @ A ) )
@ ( zero_zero @ A )
@ ( if @ A @ ( ord_less @ A @ ( zero_zero @ A ) @ X2 ) @ ( one_one @ A ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ) ) ).
% sgn_if
thf(fact_3995_sgn__1__neg,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A] :
( ( ( sgn_sgn @ A @ A3 )
= ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% sgn_1_neg
thf(fact_3996_zsgn__def,axiom,
( ( sgn_sgn @ int )
= ( ^ [I4: int] :
( if @ int
@ ( I4
= ( zero_zero @ int ) )
@ ( zero_zero @ int )
@ ( if @ int @ ( ord_less @ int @ ( zero_zero @ int ) @ I4 ) @ ( one_one @ int ) @ ( uminus_uminus @ int @ ( one_one @ int ) ) ) ) ) ) ).
% zsgn_def
thf(fact_3997_norm__sgn,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X: A] :
( ( ( X
= ( zero_zero @ A ) )
=> ( ( real_V7770717601297561774m_norm @ A @ ( sgn_sgn @ A @ X ) )
= ( zero_zero @ real ) ) )
& ( ( X
!= ( zero_zero @ A ) )
=> ( ( real_V7770717601297561774m_norm @ A @ ( sgn_sgn @ A @ X ) )
= ( one_one @ real ) ) ) ) ) ).
% norm_sgn
thf(fact_3998_div__sgn__abs__cancel,axiom,
! [V2: int,K: int,L: int] :
( ( V2
!= ( zero_zero @ int ) )
=> ( ( divide_divide @ int @ ( times_times @ int @ ( sgn_sgn @ int @ V2 ) @ ( abs_abs @ int @ K ) ) @ ( times_times @ int @ ( sgn_sgn @ int @ V2 ) @ ( abs_abs @ int @ L ) ) )
= ( divide_divide @ int @ ( abs_abs @ int @ K ) @ ( abs_abs @ int @ L ) ) ) ) ).
% div_sgn_abs_cancel
thf(fact_3999_even__word__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( even_word @ A )
= ( ^ [A5: word @ A] :
( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ A5 @ ( one_one @ ( word @ A ) ) )
= ( zero_zero @ ( word @ A ) ) ) ) ) ) ).
% even_word_iff
thf(fact_4000_the1__equality,axiom,
! [A: $tType,P: A > $o,A3: A] :
( ? [X6: A] :
( ( P @ X6 )
& ! [Y4: A] :
( ( P @ Y4 )
=> ( Y4 = X6 ) ) )
=> ( ( P @ A3 )
=> ( ( the @ A @ P )
= A3 ) ) ) ).
% the1_equality
thf(fact_4001_the1I2,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ? [X6: A] :
( ( P @ X6 )
& ! [Y4: A] :
( ( P @ Y4 )
=> ( Y4 = X6 ) ) )
=> ( ! [X3: A] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( Q @ ( the @ A @ P ) ) ) ) ).
% the1I2
thf(fact_4002_If__def,axiom,
! [A: $tType] :
( ( if @ A )
= ( ^ [P3: $o,X2: A,Y2: A] :
( the @ A
@ ^ [Z4: A] :
( ( P3
=> ( Z4 = X2 ) )
& ( ~ P3
=> ( Z4 = Y2 ) ) ) ) ) ) ).
% If_def
thf(fact_4003_theI2,axiom,
! [A: $tType,P: A > $o,A3: A,Q: A > $o] :
( ( P @ A3 )
=> ( ! [X3: A] :
( ( P @ X3 )
=> ( X3 = A3 ) )
=> ( ! [X3: A] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( Q @ ( the @ A @ P ) ) ) ) ) ).
% theI2
thf(fact_4004_theI_H,axiom,
! [A: $tType,P: A > $o] :
( ? [X6: A] :
( ( P @ X6 )
& ! [Y4: A] :
( ( P @ Y4 )
=> ( Y4 = X6 ) ) )
=> ( P @ ( the @ A @ P ) ) ) ).
% theI'
thf(fact_4005_theI,axiom,
! [A: $tType,P: A > $o,A3: A] :
( ( P @ A3 )
=> ( ! [X3: A] :
( ( P @ X3 )
=> ( X3 = A3 ) )
=> ( P @ ( the @ A @ P ) ) ) ) ).
% theI
thf(fact_4006_div__noneq__sgn__abs,axiom,
! [L: int,K: int] :
( ( L
!= ( zero_zero @ int ) )
=> ( ( ( sgn_sgn @ int @ K )
!= ( sgn_sgn @ int @ L ) )
=> ( ( divide_divide @ int @ K @ L )
= ( minus_minus @ int @ ( uminus_uminus @ int @ ( divide_divide @ int @ ( abs_abs @ int @ K ) @ ( abs_abs @ int @ L ) ) )
@ ( zero_neq_one_of_bool @ int
@ ~ ( dvd_dvd @ int @ L @ K ) ) ) ) ) ) ).
% div_noneq_sgn_abs
thf(fact_4007_floor__real__def,axiom,
( ( archim6421214686448440834_floor @ real )
= ( ^ [X2: real] :
( the @ int
@ ^ [Z4: int] :
( ( ord_less_eq @ real @ ( ring_1_of_int @ real @ Z4 ) @ X2 )
& ( ord_less @ real @ X2 @ ( ring_1_of_int @ real @ ( plus_plus @ int @ Z4 @ ( one_one @ int ) ) ) ) ) ) ) ) ).
% floor_real_def
thf(fact_4008_and__nat__unfold,axiom,
( ( bit_se5824344872417868541ns_and @ nat )
= ( ^ [M3: nat,N4: nat] :
( if @ nat
@ ( ( M3
= ( zero_zero @ nat ) )
| ( N4
= ( zero_zero @ nat ) ) )
@ ( zero_zero @ nat )
@ ( plus_plus @ nat @ ( times_times @ nat @ ( modulo_modulo @ nat @ M3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ nat @ N4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ nat @ ( divide_divide @ nat @ M3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ nat @ N4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ).
% and_nat_unfold
thf(fact_4009_and__nat__rec,axiom,
( ( bit_se5824344872417868541ns_and @ nat )
= ( ^ [M3: nat,N4: nat] :
( plus_plus @ nat
@ ( zero_neq_one_of_bool @ nat
@ ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M3 )
& ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N4 ) ) )
@ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ nat @ ( divide_divide @ nat @ M3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ nat @ N4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% and_nat_rec
thf(fact_4010_divide__int__unfold,axiom,
! [L: int,K: int,N: nat,M: nat] :
( ( ( ( ( sgn_sgn @ int @ L )
= ( zero_zero @ int ) )
| ( ( sgn_sgn @ int @ K )
= ( zero_zero @ int ) )
| ( N
= ( zero_zero @ nat ) ) )
=> ( ( divide_divide @ int @ ( times_times @ int @ ( sgn_sgn @ int @ K ) @ ( semiring_1_of_nat @ int @ M ) ) @ ( times_times @ int @ ( sgn_sgn @ int @ L ) @ ( semiring_1_of_nat @ int @ N ) ) )
= ( zero_zero @ int ) ) )
& ( ~ ( ( ( sgn_sgn @ int @ L )
= ( zero_zero @ int ) )
| ( ( sgn_sgn @ int @ K )
= ( zero_zero @ int ) )
| ( N
= ( zero_zero @ nat ) ) )
=> ( ( ( ( sgn_sgn @ int @ K )
= ( sgn_sgn @ int @ L ) )
=> ( ( divide_divide @ int @ ( times_times @ int @ ( sgn_sgn @ int @ K ) @ ( semiring_1_of_nat @ int @ M ) ) @ ( times_times @ int @ ( sgn_sgn @ int @ L ) @ ( semiring_1_of_nat @ int @ N ) ) )
= ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ M @ N ) ) ) )
& ( ( ( sgn_sgn @ int @ K )
!= ( sgn_sgn @ int @ L ) )
=> ( ( divide_divide @ int @ ( times_times @ int @ ( sgn_sgn @ int @ K ) @ ( semiring_1_of_nat @ int @ M ) ) @ ( times_times @ int @ ( sgn_sgn @ int @ L ) @ ( semiring_1_of_nat @ int @ N ) ) )
= ( uminus_uminus @ int
@ ( semiring_1_of_nat @ int
@ ( plus_plus @ nat @ ( divide_divide @ nat @ M @ N )
@ ( zero_neq_one_of_bool @ nat
@ ~ ( dvd_dvd @ nat @ N @ M ) ) ) ) ) ) ) ) ) ) ).
% divide_int_unfold
thf(fact_4011_modulo__int__unfold,axiom,
! [L: int,K: int,N: nat,M: nat] :
( ( ( ( ( sgn_sgn @ int @ L )
= ( zero_zero @ int ) )
| ( ( sgn_sgn @ int @ K )
= ( zero_zero @ int ) )
| ( N
= ( zero_zero @ nat ) ) )
=> ( ( modulo_modulo @ int @ ( times_times @ int @ ( sgn_sgn @ int @ K ) @ ( semiring_1_of_nat @ int @ M ) ) @ ( times_times @ int @ ( sgn_sgn @ int @ L ) @ ( semiring_1_of_nat @ int @ N ) ) )
= ( times_times @ int @ ( sgn_sgn @ int @ K ) @ ( semiring_1_of_nat @ int @ M ) ) ) )
& ( ~ ( ( ( sgn_sgn @ int @ L )
= ( zero_zero @ int ) )
| ( ( sgn_sgn @ int @ K )
= ( zero_zero @ int ) )
| ( N
= ( zero_zero @ nat ) ) )
=> ( ( ( ( sgn_sgn @ int @ K )
= ( sgn_sgn @ int @ L ) )
=> ( ( modulo_modulo @ int @ ( times_times @ int @ ( sgn_sgn @ int @ K ) @ ( semiring_1_of_nat @ int @ M ) ) @ ( times_times @ int @ ( sgn_sgn @ int @ L ) @ ( semiring_1_of_nat @ int @ N ) ) )
= ( times_times @ int @ ( sgn_sgn @ int @ L ) @ ( semiring_1_of_nat @ int @ ( modulo_modulo @ nat @ M @ N ) ) ) ) )
& ( ( ( sgn_sgn @ int @ K )
!= ( sgn_sgn @ int @ L ) )
=> ( ( modulo_modulo @ int @ ( times_times @ int @ ( sgn_sgn @ int @ K ) @ ( semiring_1_of_nat @ int @ M ) ) @ ( times_times @ int @ ( sgn_sgn @ int @ L ) @ ( semiring_1_of_nat @ int @ N ) ) )
= ( times_times @ int @ ( sgn_sgn @ int @ L )
@ ( minus_minus @ int
@ ( semiring_1_of_nat @ int
@ ( times_times @ nat @ N
@ ( zero_neq_one_of_bool @ nat
@ ~ ( dvd_dvd @ nat @ N @ M ) ) ) )
@ ( semiring_1_of_nat @ int @ ( modulo_modulo @ nat @ M @ N ) ) ) ) ) ) ) ) ) ).
% modulo_int_unfold
thf(fact_4012_divide__int__def,axiom,
( ( divide_divide @ int )
= ( ^ [K3: int,L2: int] :
( if @ int
@ ( L2
= ( zero_zero @ int ) )
@ ( zero_zero @ int )
@ ( if @ int
@ ( ( sgn_sgn @ int @ K3 )
= ( sgn_sgn @ int @ L2 ) )
@ ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ ( nat2 @ ( abs_abs @ int @ K3 ) ) @ ( nat2 @ ( abs_abs @ int @ L2 ) ) ) )
@ ( uminus_uminus @ int
@ ( semiring_1_of_nat @ int
@ ( plus_plus @ nat @ ( divide_divide @ nat @ ( nat2 @ ( abs_abs @ int @ K3 ) ) @ ( nat2 @ ( abs_abs @ int @ L2 ) ) )
@ ( zero_neq_one_of_bool @ nat
@ ~ ( dvd_dvd @ int @ L2 @ K3 ) ) ) ) ) ) ) ) ) ).
% divide_int_def
thf(fact_4013_sgn__div__eq__sgn__mult,axiom,
! [A3: int,B3: int] :
( ( ( divide_divide @ int @ A3 @ B3 )
!= ( zero_zero @ int ) )
=> ( ( sgn_sgn @ int @ ( divide_divide @ int @ A3 @ B3 ) )
= ( sgn_sgn @ int @ ( times_times @ int @ A3 @ B3 ) ) ) ) ).
% sgn_div_eq_sgn_mult
thf(fact_4014_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_4015_signed__take__bit__eq__take__bit__minus,axiom,
( ( bit_ri4674362597316999326ke_bit @ int )
= ( ^ [N4: nat,K3: int] : ( minus_minus @ int @ ( bit_se2584673776208193580ke_bit @ int @ ( suc @ N4 ) @ K3 ) @ ( times_times @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( suc @ N4 ) ) @ ( zero_neq_one_of_bool @ int @ ( bit_se5641148757651400278ts_bit @ int @ K3 @ N4 ) ) ) ) ) ) ).
% signed_take_bit_eq_take_bit_minus
thf(fact_4016_num_Osize__gen_I3_J,axiom,
! [X32: num] :
( ( size_num @ ( bit1 @ X32 ) )
= ( plus_plus @ nat @ ( size_num @ X32 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% num.size_gen(3)
thf(fact_4017_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_4018_sgn__le__0__iff,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( sgn_sgn @ real @ X ) @ ( zero_zero @ real ) )
= ( ord_less_eq @ real @ X @ ( zero_zero @ real ) ) ) ).
% sgn_le_0_iff
thf(fact_4019_zero__le__sgn__iff,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( sgn_sgn @ real @ X ) )
= ( ord_less_eq @ real @ ( zero_zero @ real ) @ X ) ) ).
% zero_le_sgn_iff
thf(fact_4020_mask__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se2239418461657761734s_mask @ A @ ( zero_zero @ nat ) )
= ( zero_zero @ A ) ) ) ).
% mask_0
thf(fact_4021_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_4022_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_4023_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_4024_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_4025_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_4026_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_4027_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_4028_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_4029_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_4030_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_4031_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_4032_bit__0,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A3: A] :
( ( bit_se5641148757651400278ts_bit @ A @ A3 @ ( zero_zero @ nat ) )
= ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) ) ) ) ).
% bit_0
thf(fact_4033_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_4034_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_4035_bit__mod__2__iff,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A3: A,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ N )
= ( ( N
= ( zero_zero @ nat ) )
& ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) ) ) ) ).
% bit_mod_2_iff
thf(fact_4036_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_4037_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_4038_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_4039_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_4040_test__bit__cong,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [Xa: A,Y: A,X: nat] :
( ( Xa = Y )
=> ( ( bit_se5641148757651400278ts_bit @ A @ Xa @ X )
= ( bit_se5641148757651400278ts_bit @ A @ Y @ X ) ) ) ) ).
% test_bit_cong
thf(fact_4041_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_4042_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_4043_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_4044_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_4045_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_4046_bit__take__bit__iff,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M: nat,A3: A,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( bit_se2584673776208193580ke_bit @ A @ M @ A3 ) @ N )
= ( ( ord_less @ nat @ N @ M )
& ( bit_se5641148757651400278ts_bit @ A @ A3 @ N ) ) ) ) ).
% bit_take_bit_iff
thf(fact_4047_bit__of__bool__iff,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [B3: $o,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( zero_neq_one_of_bool @ A @ B3 ) @ N )
= ( B3
& ( N
= ( zero_zero @ nat ) ) ) ) ) ).
% bit_of_bool_iff
thf(fact_4048_mask__twice2,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,M: nat,X: word @ A] :
( ( ord_less_eq @ nat @ N @ M )
=> ( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ M ) ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) )
= ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) ) ) ) ).
% mask_twice2
thf(fact_4049_and__mask__eq__iff__le__mask,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A,N: nat] :
( ( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ W @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) )
= W )
= ( ord_less_eq @ ( word @ A ) @ W @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) ) ) ).
% and_mask_eq_iff_le_mask
thf(fact_4050_le__mask__imp__and__mask,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,N: nat] :
( ( ord_less_eq @ ( word @ A ) @ X @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) )
=> ( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) )
= X ) ) ) ).
% le_mask_imp_and_mask
thf(fact_4051_mask__eqs_I7_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A,N: nat,B3: word @ A] :
( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ A3 @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ B3 @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) )
= ( bit_se5824344872417868541ns_and @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ A3 @ B3 ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) ) ) ).
% mask_eqs(7)
thf(fact_4052_mask__eqs_I2_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A,B3: word @ A,N: nat] :
( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ A3 @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ B3 @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) )
= ( bit_se5824344872417868541ns_and @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ A3 @ B3 ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) ) ) ).
% mask_eqs(2)
thf(fact_4053_mask__eqs_I1_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A,N: nat,B3: word @ A] :
( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ A3 @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) @ B3 ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) )
= ( bit_se5824344872417868541ns_and @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ A3 @ B3 ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) ) ) ).
% mask_eqs(1)
thf(fact_4054_mask__eqs_I10_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A,N: nat] :
( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ ( uminus_uminus @ ( word @ A ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ A3 @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) )
= ( bit_se5824344872417868541ns_and @ ( word @ A ) @ ( uminus_uminus @ ( word @ A ) @ A3 ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) ) ) ).
% mask_eqs(10)
thf(fact_4055_mask__eqs_I8_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A,N: nat,B3: word @ A] :
( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ A3 @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ B3 @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) )
= ( bit_se5824344872417868541ns_and @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ A3 @ B3 ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) ) ) ).
% mask_eqs(8)
thf(fact_4056_mask__eqs_I4_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A,B3: word @ A,N: nat] :
( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ A3 @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ B3 @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) )
= ( bit_se5824344872417868541ns_and @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ A3 @ B3 ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) ) ) ).
% mask_eqs(4)
thf(fact_4057_mask__eqs_I3_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A,N: nat,B3: word @ A] :
( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ A3 @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) @ B3 ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) )
= ( bit_se5824344872417868541ns_and @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ A3 @ B3 ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) ) ) ).
% mask_eqs(3)
thf(fact_4058_mask__power__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,N: nat,K: nat] :
( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ ( power_power @ ( word @ A ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) @ K ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) )
= ( bit_se5824344872417868541ns_and @ ( word @ A ) @ ( power_power @ ( word @ A ) @ X @ K ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) ) ) ).
% mask_power_eq
thf(fact_4059_ucast__and__mask,axiom,
! [B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B ) )
=> ! [X: word @ B,N: nat] :
( ( semiring_1_unsigned @ B @ ( word @ A ) @ ( bit_se5824344872417868541ns_and @ ( word @ B ) @ X @ ( bit_se2239418461657761734s_mask @ ( word @ B ) @ N ) ) )
= ( bit_se5824344872417868541ns_and @ ( word @ A ) @ ( semiring_1_unsigned @ B @ ( word @ A ) @ X ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) ) ) ).
% ucast_and_mask
thf(fact_4060_mask__eqs_I5_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A,B3: word @ A,N: nat] :
( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ ( times_times @ ( word @ A ) @ A3 @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ B3 @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) )
= ( bit_se5824344872417868541ns_and @ ( word @ A ) @ ( times_times @ ( word @ A ) @ A3 @ B3 ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) ) ) ).
% mask_eqs(5)
thf(fact_4061_mask__eqs_I6_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [B3: word @ A,N: nat,A3: word @ A] :
( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ ( times_times @ ( word @ A ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ B3 @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) @ A3 ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) )
= ( bit_se5824344872417868541ns_and @ ( word @ A ) @ ( times_times @ ( word @ A ) @ B3 @ A3 ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) ) ) ).
% mask_eqs(6)
thf(fact_4062_mask__eqs_I9_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A,N: nat,B3: word @ A] :
( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ ( times_times @ ( word @ A ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ A3 @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ B3 @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) )
= ( bit_se5824344872417868541ns_and @ ( word @ A ) @ ( times_times @ ( word @ A ) @ A3 @ B3 ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) ) ) ).
% mask_eqs(9)
thf(fact_4063_less__eq__mask__iff__take__bit__eq__self,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A,N: nat] :
( ( ord_less_eq @ ( word @ A ) @ W @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) )
= ( ( bit_se2584673776208193580ke_bit @ ( word @ A ) @ N @ W )
= W ) ) ) ).
% less_eq_mask_iff_take_bit_eq_self
thf(fact_4064_mask__1,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se2239418461657761734s_mask @ A @ ( one_one @ nat ) )
= ( one_one @ A ) ) ) ).
% mask_1
thf(fact_4065_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_4066_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_4067_and__mask__wi,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [I: int,N: nat] :
( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ ( ring_1_of_int @ ( word @ A ) @ I ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) )
= ( ring_1_of_int @ ( word @ A ) @ ( bit_se2584673776208193580ke_bit @ int @ N @ I ) ) ) ) ).
% and_mask_wi
thf(fact_4068_sgn__real__def,axiom,
( ( sgn_sgn @ real )
= ( ^ [A5: real] :
( if @ real
@ ( A5
= ( zero_zero @ real ) )
@ ( zero_zero @ real )
@ ( if @ real @ ( ord_less @ real @ ( zero_zero @ real ) @ A5 ) @ ( one_one @ real ) @ ( uminus_uminus @ real @ ( one_one @ real ) ) ) ) ) ) ).
% sgn_real_def
thf(fact_4069_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_4070_mask__bin,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( bit_se2239418461657761734s_mask @ ( word @ A ) )
= ( ^ [N4: nat] : ( ring_1_of_int @ ( word @ A ) @ ( bit_se2584673776208193580ke_bit @ int @ N4 @ ( uminus_uminus @ int @ ( one_one @ int ) ) ) ) ) ) ) ).
% mask_bin
thf(fact_4071_and__mask__bintr,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 ) @ ( bit_se2584673776208193580ke_bit @ int @ N @ ( semiring_1_unsigned @ A @ int @ W ) ) ) ) ) ).
% and_mask_bintr
thf(fact_4072_exp__eq__0__imp__not__bit,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [N: nat,A3: A] :
( ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N )
= ( zero_zero @ A ) )
=> ~ ( bit_se5641148757651400278ts_bit @ A @ A3 @ N ) ) ) ).
% exp_eq_0_imp_not_bit
thf(fact_4073_bit__Suc,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A3: A,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ A3 @ ( suc @ N ) )
= ( bit_se5641148757651400278ts_bit @ A @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ N ) ) ) ).
% bit_Suc
thf(fact_4074_stable__imp__bit__iff__odd,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A3: A,N: nat] :
( ( ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= A3 )
=> ( ( bit_se5641148757651400278ts_bit @ A @ A3 @ N )
= ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) ) ) ) ) ).
% stable_imp_bit_iff_odd
thf(fact_4075_bit__iff__idd__imp__stable,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A3: A] :
( ! [N2: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ A3 @ N2 )
= ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) ) )
=> ( ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= A3 ) ) ) ).
% bit_iff_idd_imp_stable
thf(fact_4076_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_4077_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_4078_sgn__power__injE,axiom,
! [A3: real,N: nat,X: real,B3: real] :
( ( ( times_times @ real @ ( sgn_sgn @ real @ A3 ) @ ( power_power @ real @ ( abs_abs @ real @ A3 ) @ N ) )
= X )
=> ( ( X
= ( times_times @ real @ ( sgn_sgn @ real @ B3 ) @ ( power_power @ real @ ( abs_abs @ real @ B3 ) @ N ) ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( A3 = B3 ) ) ) ) ).
% sgn_power_injE
thf(fact_4079_int__bit__bound,axiom,
! [K: int] :
~ ! [N2: nat] :
( ! [M2: nat] :
( ( ord_less_eq @ nat @ N2 @ M2 )
=> ( ( bit_se5641148757651400278ts_bit @ int @ K @ M2 )
= ( bit_se5641148757651400278ts_bit @ int @ K @ N2 ) ) )
=> ~ ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( bit_se5641148757651400278ts_bit @ int @ K @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) )
= ( ~ ( bit_se5641148757651400278ts_bit @ int @ K @ N2 ) ) ) ) ) ).
% int_bit_bound
thf(fact_4080_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_4081_num_Osize__gen_I1_J,axiom,
( ( size_num @ one2 )
= ( zero_zero @ nat ) ) ).
% num.size_gen(1)
thf(fact_4082_bit__iff__odd,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ( ( bit_se5641148757651400278ts_bit @ A )
= ( ^ [A5: A,N4: nat] :
~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A5 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N4 ) ) ) ) ) ) ).
% bit_iff_odd
thf(fact_4083_and__exp__eq__0__iff__not__bit,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A,N: nat] :
( ( ( bit_se5824344872417868541ns_and @ A @ A3 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) )
= ( zero_zero @ A ) )
= ( ~ ( bit_se5641148757651400278ts_bit @ A @ A3 @ N ) ) ) ) ).
% and_exp_eq_0_iff_not_bit
thf(fact_4084_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_4085_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_4086_less__mask__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,N: nat] :
( ( ord_less @ ( word @ A ) @ X @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) )
=> ( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) )
= X ) ) ) ).
% less_mask_eq
thf(fact_4087_mask__eq__decr__exp,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( bit_se2239418461657761734s_mask @ ( word @ A ) )
= ( ^ [N4: nat] : ( minus_minus @ ( word @ A ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N4 ) @ ( one_one @ ( word @ A ) ) ) ) ) ) ).
% mask_eq_decr_exp
thf(fact_4088_bit__int__def,axiom,
( ( bit_se5641148757651400278ts_bit @ int )
= ( ^ [K3: int,N4: nat] :
~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( divide_divide @ int @ K3 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N4 ) ) ) ) ) ).
% bit_int_def
thf(fact_4089_even__bit__succ__iff,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A3: A,N: nat] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
=> ( ( bit_se5641148757651400278ts_bit @ A @ ( plus_plus @ A @ ( one_one @ A ) @ A3 ) @ N )
= ( ( bit_se5641148757651400278ts_bit @ A @ A3 @ N )
| ( N
= ( zero_zero @ nat ) ) ) ) ) ) ).
% even_bit_succ_iff
thf(fact_4090_odd__bit__iff__bit__pred,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A3: A,N: nat] :
( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
=> ( ( bit_se5641148757651400278ts_bit @ A @ A3 @ N )
= ( ( bit_se5641148757651400278ts_bit @ A @ ( minus_minus @ A @ A3 @ ( one_one @ A ) ) @ N )
| ( N
= ( zero_zero @ nat ) ) ) ) ) ) ).
% odd_bit_iff_bit_pred
thf(fact_4091_mask__eq__exp__minus__1,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se2239418461657761734s_mask @ A )
= ( ^ [N4: nat] : ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N4 ) @ ( one_one @ A ) ) ) ) ) ).
% mask_eq_exp_minus_1
thf(fact_4092_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_4093_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_4094_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_4095_is__aligned__AND__less__0,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [U2: word @ A,N: nat,V2: word @ A] :
( ( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ U2 @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) )
= ( zero_zero @ ( word @ A ) ) )
=> ( ( ord_less @ ( word @ A ) @ V2 @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) )
=> ( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ U2 @ V2 )
= ( zero_zero @ ( word @ A ) ) ) ) ) ) ).
% is_aligned_AND_less_0
thf(fact_4096_add__mask__fold,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,N: nat] :
( ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ X @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) ) @ ( one_one @ ( word @ A ) ) )
= ( plus_plus @ ( word @ A ) @ X @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) ) ) ).
% add_mask_fold
thf(fact_4097_bit__sum__mult__2__cases,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A,B3: A,N: nat] :
( ! [J2: nat] :
~ ( bit_se5641148757651400278ts_bit @ A @ A3 @ ( suc @ J2 ) )
=> ( ( bit_se5641148757651400278ts_bit @ A @ ( plus_plus @ A @ A3 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B3 ) ) @ N )
= ( ( ( N
= ( zero_zero @ nat ) )
=> ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) )
& ( ( N
!= ( zero_zero @ nat ) )
=> ( bit_se5641148757651400278ts_bit @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B3 ) @ N ) ) ) ) ) ) ).
% bit_sum_mult_2_cases
thf(fact_4098_bit__rec,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ( ( bit_se5641148757651400278ts_bit @ A )
= ( ^ [A5: A,N4: nat] :
( ( ( N4
= ( zero_zero @ nat ) )
=> ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A5 ) )
& ( ( N4
!= ( zero_zero @ nat ) )
=> ( bit_se5641148757651400278ts_bit @ A @ ( divide_divide @ A @ A5 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( minus_minus @ nat @ N4 @ ( one_one @ nat ) ) ) ) ) ) ) ) ).
% bit_rec
thf(fact_4099_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_4100_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_4101_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_4102_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_4103_and__mask__less__size,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,X: word @ A] :
( ( ord_less @ nat @ N @ ( size_size @ ( word @ A ) @ X ) )
=> ( ord_less @ ( word @ A ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% and_mask_less_size
thf(fact_4104_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_4105_word__mod__2p__is__mask,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,X: word @ A] :
( ( ord_less @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) )
=> ( ( modulo_modulo @ ( word @ A ) @ X @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) )
= ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) ) ) ) ).
% word_mod_2p_is_mask
thf(fact_4106_Bit__Operations_Oset__bit__eq,axiom,
( ( bit_se5668285175392031749et_bit @ int )
= ( ^ [N4: nat,K3: int] :
( plus_plus @ int @ K3
@ ( times_times @ int
@ ( zero_neq_one_of_bool @ int
@ ~ ( bit_se5641148757651400278ts_bit @ int @ K3 @ N4 ) )
@ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N4 ) ) ) ) ) ).
% Bit_Operations.set_bit_eq
thf(fact_4107_unset__bit__eq,axiom,
( ( bit_se2638667681897837118et_bit @ int )
= ( ^ [N4: nat,K3: int] : ( minus_minus @ int @ K3 @ ( times_times @ int @ ( zero_neq_one_of_bool @ int @ ( bit_se5641148757651400278ts_bit @ int @ K3 @ N4 ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N4 ) ) ) ) ) ).
% unset_bit_eq
thf(fact_4108_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_4109_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_4110_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_4111_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_4112_arctan__inverse,axiom,
! [X: real] :
( ( X
!= ( zero_zero @ real ) )
=> ( ( arctan @ ( divide_divide @ real @ ( one_one @ real ) @ X ) )
= ( minus_minus @ real @ ( divide_divide @ real @ ( times_times @ real @ ( sgn_sgn @ real @ X ) @ pi ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( arctan @ X ) ) ) ) ).
% arctan_inverse
thf(fact_4113_num_Osize__gen_I2_J,axiom,
! [X22: num] :
( ( size_num @ ( bit0 @ X22 ) )
= ( plus_plus @ nat @ ( size_num @ X22 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% num.size_gen(2)
thf(fact_4114_floor__rat__def,axiom,
( ( archim6421214686448440834_floor @ rat )
= ( ^ [X2: rat] :
( the @ int
@ ^ [Z4: int] :
( ( ord_less_eq @ rat @ ( ring_1_of_int @ rat @ Z4 ) @ X2 )
& ( ord_less @ rat @ X2 @ ( ring_1_of_int @ rat @ ( plus_plus @ int @ Z4 @ ( one_one @ int ) ) ) ) ) ) ) ) ).
% floor_rat_def
thf(fact_4115_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_4116_VEBT__internal_OT__vebt__buildupi_H_Opelims,axiom,
! [X: nat,Y: int] :
( ( ( vEBT_V9176841429113362141ildupi @ X )
= Y )
=> ( ( accp @ nat @ vEBT_V3352910403632780892pi_rel @ X )
=> ( ( ( X
= ( zero_zero @ nat ) )
=> ( ( Y
= ( one_one @ int ) )
=> ~ ( accp @ nat @ vEBT_V3352910403632780892pi_rel @ ( zero_zero @ nat ) ) ) )
=> ( ( ( X
= ( suc @ ( zero_zero @ nat ) ) )
=> ( ( Y
= ( one_one @ int ) )
=> ~ ( accp @ nat @ vEBT_V3352910403632780892pi_rel @ ( suc @ ( zero_zero @ nat ) ) ) ) )
=> ~ ! [N2: nat] :
( ( X
= ( suc @ ( suc @ N2 ) ) )
=> ( ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
=> ( Y
= ( plus_plus @ int @ ( numeral_numeral @ int @ ( bit1 @ one2 ) ) @ ( plus_plus @ int @ ( vEBT_V9176841429113362141ildupi @ ( suc @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( plus_plus @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ ( bit0 @ one2 ) ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( times_times @ int @ ( vEBT_V9176841429113362141ildupi @ ( suc @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) )
& ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
=> ( Y
= ( plus_plus @ int @ ( numeral_numeral @ int @ ( bit1 @ one2 ) ) @ ( plus_plus @ int @ ( vEBT_V9176841429113362141ildupi @ ( suc @ ( suc @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( plus_plus @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ ( bit0 @ one2 ) ) ) @ ( times_times @ int @ ( vEBT_V9176841429113362141ildupi @ ( suc @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) )
=> ~ ( accp @ nat @ vEBT_V3352910403632780892pi_rel @ ( suc @ ( suc @ N2 ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.T_vebt_buildupi'.pelims
thf(fact_4117_VEBT__internal_OTBOUND__buildupi,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( time_TBOUND @ vEBT_VEBTi @ ( vEBT_vebt_buildupi @ N ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% VEBT_internal.TBOUND_buildupi
thf(fact_4118_word__not__not,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] :
( ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ X ) )
= X ) ) ).
% word_not_not
thf(fact_4119_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_4120_bit_Oconj__cancel__right,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X: A] :
( ( bit_se5824344872417868541ns_and @ A @ X @ ( bit_ri4277139882892585799ns_not @ A @ X ) )
= ( zero_zero @ A ) ) ) ).
% bit.conj_cancel_right
thf(fact_4121_bit_Oconj__cancel__left,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X: A] :
( ( bit_se5824344872417868541ns_and @ A @ ( bit_ri4277139882892585799ns_not @ A @ X ) @ X )
= ( zero_zero @ A ) ) ) ).
% bit.conj_cancel_left
thf(fact_4122_and__and__not,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A,B3: word @ A] :
( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ A3 @ B3 ) @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ B3 ) )
= ( zero_zero @ ( word @ A ) ) ) ) ).
% and_and_not
thf(fact_4123_word__and__not,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] :
( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ X ) )
= ( zero_zero @ ( word @ A ) ) ) ) ).
% word_and_not
thf(fact_4124_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_4125_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_4126_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_4127_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_4128_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_4129_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_4130_word__add__not,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] :
( ( plus_plus @ ( word @ A ) @ X @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ X ) )
= ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) ) ) ).
% word_add_not
thf(fact_4131_even__not__iff,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [A3: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_ri4277139882892585799ns_not @ A @ A3 ) )
= ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) ) ) ) ).
% even_not_iff
thf(fact_4132_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_4133_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_4134_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_4135_word__and__nth,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A,N: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X @ Y ) @ N )
= ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ X @ N )
& ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ Y @ N ) ) ) ) ).
% word_and_nth
thf(fact_4136_sgn__rat__def,axiom,
( ( sgn_sgn @ rat )
= ( ^ [A5: rat] :
( if @ rat
@ ( A5
= ( zero_zero @ rat ) )
@ ( zero_zero @ rat )
@ ( if @ rat @ ( ord_less @ rat @ ( zero_zero @ rat ) @ A5 ) @ ( one_one @ rat ) @ ( uminus_uminus @ rat @ ( one_one @ rat ) ) ) ) ) ) ).
% sgn_rat_def
thf(fact_4137_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_4138_nth__0,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat] :
~ ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ N ) ) ).
% nth_0
thf(fact_4139_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_4140_not__switch,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [A3: A,X: A] :
( ( ( bit_ri4277139882892585799ns_not @ A @ A3 )
= X )
=> ( A3
= ( bit_ri4277139882892585799ns_not @ A @ X ) ) ) ) ).
% not_switch
thf(fact_4141_less__eq__rat__def,axiom,
( ( ord_less_eq @ rat )
= ( ^ [X2: rat,Y2: rat] :
( ( ord_less @ rat @ X2 @ Y2 )
| ( X2 = Y2 ) ) ) ) ).
% less_eq_rat_def
thf(fact_4142_obtain__pos__sum,axiom,
! [R3: rat] :
( ( ord_less @ rat @ ( zero_zero @ rat ) @ R3 )
=> ~ ! [S3: rat] :
( ( ord_less @ rat @ ( zero_zero @ rat ) @ S3 )
=> ! [T3: rat] :
( ( ord_less @ rat @ ( zero_zero @ rat ) @ T3 )
=> ( R3
!= ( plus_plus @ rat @ S3 @ T3 ) ) ) ) ) ).
% obtain_pos_sum
thf(fact_4143_abs__rat__def,axiom,
( ( abs_abs @ rat )
= ( ^ [A5: rat] : ( if @ rat @ ( ord_less @ rat @ A5 @ ( zero_zero @ rat ) ) @ ( uminus_uminus @ rat @ A5 ) @ A5 ) ) ) ).
% abs_rat_def
thf(fact_4144_split__word__eq__on__mask,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [M: word @ A] :
( ( ^ [Y3: word @ A,Z2: word @ A] : Y3 = Z2 )
= ( ^ [X2: word @ A,Y2: word @ A] :
( ( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X2 @ M )
= ( bit_se5824344872417868541ns_and @ ( word @ A ) @ Y2 @ M ) )
& ( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X2 @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ M ) )
= ( bit_se5824344872417868541ns_and @ ( word @ A ) @ Y2 @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ M ) ) ) ) ) ) ) ).
% split_word_eq_on_mask
thf(fact_4145_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_4146_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_4147_mask__eq__x__eq__0,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,W: word @ A] :
( ( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X @ W )
= X )
= ( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ W ) )
= ( zero_zero @ ( word @ A ) ) ) ) ) ).
% mask_eq_x_eq_0
thf(fact_4148_mask__eq__0__eq__x,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,W: word @ A] :
( ( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X @ W )
= ( zero_zero @ ( word @ A ) ) )
= ( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ W ) )
= X ) ) ) ).
% mask_eq_0_eq_x
thf(fact_4149_word__plus__and__or__coroll2,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,W: word @ A] :
( ( plus_plus @ ( word @ A ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X @ W ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ W ) ) )
= X ) ) ).
% word_plus_and_or_coroll2
thf(fact_4150_mask__eqI,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,N: nat,Y: word @ A] :
( ( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) )
= ( bit_se5824344872417868541ns_and @ ( word @ A ) @ Y @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) )
=> ( ( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) )
= ( bit_se5824344872417868541ns_and @ ( word @ A ) @ Y @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) ) )
=> ( X = Y ) ) ) ) ).
% mask_eqI
thf(fact_4151_test__bit__size,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A,N: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ W @ N )
=> ( ord_less @ nat @ N @ ( size_size @ ( word @ A ) @ W ) ) ) ) ).
% test_bit_size
thf(fact_4152_word__eqI,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [U2: word @ A,V2: word @ A] :
( ! [N2: nat] :
( ( ord_less @ nat @ N2 @ ( size_size @ ( word @ A ) @ U2 ) )
=> ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ U2 @ N2 )
= ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ V2 @ N2 ) ) )
=> ( U2 = V2 ) ) ) ).
% word_eqI
thf(fact_4153_test__bit__over,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,N: nat] :
( ( ord_less_eq @ nat @ ( size_size @ ( word @ A ) @ X ) @ N )
=> ~ ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ X @ N ) ) ) ).
% test_bit_over
thf(fact_4154_mask__nonnegative__int,axiom,
! [N: nat] : ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se2239418461657761734s_mask @ int @ N ) ) ).
% mask_nonnegative_int
thf(fact_4155_not__mask__negative__int,axiom,
! [N: nat] :
~ ( ord_less @ int @ ( bit_se2239418461657761734s_mask @ int @ N ) @ ( zero_zero @ int ) ) ).
% not_mask_negative_int
thf(fact_4156_test__bit__def_H,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) )
= ( ^ [A5: word @ A] : ( bit_se5641148757651400278ts_bit @ int @ ( semiring_1_unsigned @ A @ int @ A5 ) ) ) ) ) ).
% test_bit_def'
thf(fact_4157_word__test__bit__def,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) )
= ( ^ [A5: word @ A] : ( bit_se5641148757651400278ts_bit @ int @ ( semiring_1_unsigned @ A @ int @ A5 ) ) ) ) ) ).
% word_test_bit_def
thf(fact_4158_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_4159_minus__eq__not__plus__1,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( ( uminus_uminus @ A )
= ( ^ [A5: A] : ( plus_plus @ A @ ( bit_ri4277139882892585799ns_not @ A @ A5 ) @ ( one_one @ A ) ) ) ) ) ).
% minus_eq_not_plus_1
thf(fact_4160_minus__eq__not__minus__1,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( ( uminus_uminus @ A )
= ( ^ [A5: A] : ( bit_ri4277139882892585799ns_not @ A @ ( minus_minus @ A @ A5 @ ( one_one @ A ) ) ) ) ) ) ).
% minus_eq_not_minus_1
thf(fact_4161_not__eq__complement,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( ( bit_ri4277139882892585799ns_not @ A )
= ( ^ [A5: A] : ( minus_minus @ A @ ( uminus_uminus @ A @ A5 ) @ ( one_one @ A ) ) ) ) ) ).
% not_eq_complement
thf(fact_4162_mask__out__first__mask__some,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,N: nat,Y: word @ A,M: nat] :
( ( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) )
= Y )
=> ( ( ord_less_eq @ nat @ N @ M )
=> ( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ M ) ) )
= ( bit_se5824344872417868541ns_and @ ( word @ A ) @ Y @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ M ) ) ) ) ) ) ) ).
% mask_out_first_mask_some
thf(fact_4163_mask__lower__twice,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,M: nat,X: word @ A] :
( ( ord_less_eq @ nat @ N @ M )
=> ( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) ) @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ M ) ) )
= ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ M ) ) ) ) ) ) ).
% mask_lower_twice
thf(fact_4164_NOT__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( bit_ri4277139882892585799ns_not @ ( word @ A ) )
= ( ^ [X2: word @ A] : ( minus_minus @ ( word @ A ) @ ( uminus_uminus @ ( word @ A ) @ X2 ) @ ( one_one @ ( word @ A ) ) ) ) ) ) ).
% NOT_eq
thf(fact_4165_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_4166_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_4167_lsb__this__or__next,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] :
( ~ ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ X @ ( one_one @ ( word @ A ) ) ) @ ( zero_zero @ nat ) )
=> ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ X @ ( zero_zero @ nat ) ) ) ) ).
% lsb_this_or_next
thf(fact_4168_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_4169_subtract__mask_I2_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [P4: word @ A,N: nat] :
( ( minus_minus @ ( word @ A ) @ P4 @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ P4 @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) ) )
= ( bit_se5824344872417868541ns_and @ ( word @ A ) @ P4 @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) ) ) ).
% subtract_mask(2)
thf(fact_4170_subtract__mask_I1_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [P4: word @ A,N: nat] :
( ( minus_minus @ ( word @ A ) @ P4 @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ P4 @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) )
= ( bit_se5824344872417868541ns_and @ ( word @ A ) @ P4 @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) ) ) ) ).
% subtract_mask(1)
thf(fact_4171_mask__out__sub__mask,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,N: nat] :
( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) )
= ( minus_minus @ ( word @ A ) @ X @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) ) ) ) ).
% mask_out_sub_mask
thf(fact_4172_ucast__and__neg__mask,axiom,
! [B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B ) )
=> ! [X: word @ B,N: nat] :
( ( semiring_1_unsigned @ B @ ( word @ A ) @ ( bit_se5824344872417868541ns_and @ ( word @ B ) @ X @ ( bit_ri4277139882892585799ns_not @ ( word @ B ) @ ( bit_se2239418461657761734s_mask @ ( word @ B ) @ N ) ) ) )
= ( bit_se5824344872417868541ns_and @ ( word @ A ) @ ( semiring_1_unsigned @ B @ ( word @ A ) @ X ) @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) ) ) ) ).
% ucast_and_neg_mask
thf(fact_4173_word__leI,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [U2: word @ A,V2: word @ A] :
( ! [N2: nat] :
( ( ord_less @ nat @ N2 @ ( size_size @ ( word @ A ) @ U2 ) )
=> ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ U2 @ N2 )
=> ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ V2 @ N2 ) ) )
=> ( ord_less_eq @ ( word @ A ) @ U2 @ V2 ) ) ) ).
% word_leI
thf(fact_4174_nth__mask,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,I: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) @ I )
= ( ( ord_less @ nat @ I @ N )
& ( ord_less @ nat @ I @ ( size_size @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) ) ) ) ) ).
% nth_mask
thf(fact_4175_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_4176_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_4177_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_4178_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_4179_multiple__mask__trivia,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [M: nat,N: nat,X: word @ A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( plus_plus @ ( word @ A ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ M ) ) ) ) )
= ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ M ) ) ) ) ) ) ).
% multiple_mask_trivia
thf(fact_4180_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_4181_overflow__imp__lsb,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] :
( ( ( plus_plus @ ( word @ A ) @ X @ ( one_one @ ( word @ A ) ) )
= ( zero_zero @ ( word @ A ) ) )
=> ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ X @ ( zero_zero @ nat ) ) ) ) ).
% overflow_imp_lsb
thf(fact_4182_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_4183_test__bit__bin_H,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) )
= ( ^ [W2: word @ A,N4: nat] :
( ( ord_less @ nat @ N4 @ ( size_size @ ( word @ A ) @ W2 ) )
& ( bit_se5641148757651400278ts_bit @ int @ ( semiring_1_unsigned @ A @ int @ W2 ) @ N4 ) ) ) ) ) ).
% test_bit_bin'
thf(fact_4184_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_4185_bang__is__le,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,M: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ X @ M )
=> ( ord_less_eq @ ( word @ A ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ M ) @ X ) ) ) ).
% bang_is_le
thf(fact_4186_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_4187_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_4188_bit__nat__def,axiom,
( ( bit_se5641148757651400278ts_bit @ nat )
= ( ^ [M3: nat,N4: nat] :
~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ M3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N4 ) ) ) ) ) ).
% bit_nat_def
thf(fact_4189_bit__not__iff__eq,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [A3: A,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( bit_ri4277139882892585799ns_not @ A @ A3 ) @ N )
= ( ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N )
!= ( zero_zero @ A ) )
& ~ ( bit_se5641148757651400278ts_bit @ A @ A3 @ N ) ) ) ) ).
% bit_not_iff_eq
thf(fact_4190_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_4191_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_4192_odd__iff__lsb,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] :
( ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( semiring_1_unsigned @ A @ nat @ X ) ) )
= ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ X @ ( zero_zero @ nat ) ) ) ) ).
% odd_iff_lsb
thf(fact_4193_and__neq__0__is__nth,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y: word @ A,N: nat,X: word @ A] :
( ( Y
= ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) )
=> ( ( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X @ Y )
!= ( zero_zero @ ( word @ A ) ) )
= ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ X @ N ) ) ) ) ).
% and_neq_0_is_nth
thf(fact_4194_nth__is__and__neq__0,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) )
= ( ^ [X2: word @ A,N4: nat] :
( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X2 @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N4 ) )
!= ( zero_zero @ ( word @ A ) ) ) ) ) ) ).
% nth_is_and_neq_0
thf(fact_4195_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_4196_mask__int__def,axiom,
( ( bit_se2239418461657761734s_mask @ int )
= ( ^ [N4: nat] : ( minus_minus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N4 ) @ ( one_one @ int ) ) ) ) ).
% mask_int_def
thf(fact_4197_mask__nat__def,axiom,
( ( bit_se2239418461657761734s_mask @ nat )
= ( ^ [N4: nat] : ( minus_minus @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N4 ) @ ( one_one @ nat ) ) ) ) ).
% mask_nat_def
thf(fact_4198_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_4199_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_4200_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_Opelims,axiom,
! [X: nat,Y: nat] :
( ( ( vEBT_V8646137997579335489_i_l_d @ X )
= Y )
=> ( ( accp @ nat @ vEBT_V5144397997797733112_d_rel @ X )
=> ( ( ( X
= ( zero_zero @ nat ) )
=> ( ( Y
= ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) )
=> ~ ( accp @ nat @ vEBT_V5144397997797733112_d_rel @ ( zero_zero @ nat ) ) ) )
=> ( ( ( X
= ( suc @ ( zero_zero @ nat ) ) )
=> ( ( Y
= ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) )
=> ~ ( accp @ nat @ vEBT_V5144397997797733112_d_rel @ ( suc @ ( zero_zero @ nat ) ) ) ) )
=> ~ ! [Va2: nat] :
( ( X
= ( suc @ ( suc @ Va2 ) ) )
=> ( ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ Va2 ) ) )
=> ( Y
= ( plus_plus @ nat @ ( one_one @ nat ) @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_V8646137997579335489_i_l_d @ ( divide_divide @ nat @ ( suc @ ( suc @ 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 ) ) )
=> ( Y
= ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) @ ( vEBT_V8646137997579335489_i_l_d @ ( suc @ ( divide_divide @ nat @ ( suc @ ( suc @ 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_4201_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p_Opelims,axiom,
! [X: nat,Y: nat] :
( ( ( vEBT_V8346862874174094_d_u_p @ X )
= Y )
=> ( ( accp @ nat @ vEBT_V1247956027447740395_p_rel @ X )
=> ( ( ( X
= ( zero_zero @ nat ) )
=> ( ( Y
= ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) )
=> ~ ( accp @ nat @ vEBT_V1247956027447740395_p_rel @ ( zero_zero @ nat ) ) ) )
=> ( ( ( X
= ( suc @ ( zero_zero @ nat ) ) )
=> ( ( Y
= ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) )
=> ~ ( accp @ nat @ vEBT_V1247956027447740395_p_rel @ ( suc @ ( zero_zero @ nat ) ) ) ) )
=> ~ ! [Va2: nat] :
( ( X
= ( suc @ ( suc @ Va2 ) ) )
=> ( ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ Va2 ) ) )
=> ( Y
= ( plus_plus @ nat @ ( one_one @ nat ) @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_V8346862874174094_d_u_p @ ( divide_divide @ nat @ ( suc @ ( suc @ 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 ) ) )
=> ( Y
= ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_V8346862874174094_d_u_p @ ( suc @ ( divide_divide @ nat @ ( suc @ ( suc @ 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_4202_VEBT__internal_OTb_Opelims,axiom,
! [X: nat,Y: int] :
( ( ( vEBT_VEBT_Tb @ X )
= Y )
=> ( ( accp @ nat @ vEBT_VEBT_Tb_rel2 @ X )
=> ( ( ( X
= ( zero_zero @ nat ) )
=> ( ( Y
= ( numeral_numeral @ int @ ( bit1 @ one2 ) ) )
=> ~ ( accp @ nat @ vEBT_VEBT_Tb_rel2 @ ( zero_zero @ nat ) ) ) )
=> ( ( ( X
= ( suc @ ( zero_zero @ nat ) ) )
=> ( ( Y
= ( numeral_numeral @ int @ ( bit1 @ one2 ) ) )
=> ~ ( accp @ nat @ vEBT_VEBT_Tb_rel2 @ ( suc @ ( zero_zero @ nat ) ) ) ) )
=> ~ ! [N2: nat] :
( ( X
= ( suc @ ( suc @ N2 ) ) )
=> ( ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
=> ( Y
= ( plus_plus @ int @ ( plus_plus @ int @ ( numeral_numeral @ int @ ( bit1 @ ( bit0 @ one2 ) ) ) @ ( vEBT_VEBT_Tb @ ( suc @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( times_times @ int @ ( vEBT_VEBT_Tb @ ( suc @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( suc @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) )
& ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
=> ( Y
= ( plus_plus @ int @ ( plus_plus @ int @ ( numeral_numeral @ int @ ( bit1 @ ( bit0 @ one2 ) ) ) @ ( vEBT_VEBT_Tb @ ( suc @ ( suc @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) @ ( times_times @ int @ ( vEBT_VEBT_Tb @ ( suc @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) )
=> ~ ( accp @ nat @ vEBT_VEBT_Tb_rel2 @ ( suc @ ( suc @ N2 ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.Tb.pelims
thf(fact_4203_VEBT__internal_OTb_H_Opelims,axiom,
! [X: nat,Y: nat] :
( ( ( vEBT_VEBT_Tb2 @ X )
= Y )
=> ( ( accp @ nat @ vEBT_VEBT_Tb_rel @ X )
=> ( ( ( X
= ( zero_zero @ nat ) )
=> ( ( Y
= ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) )
=> ~ ( accp @ nat @ vEBT_VEBT_Tb_rel @ ( zero_zero @ nat ) ) ) )
=> ( ( ( X
= ( suc @ ( zero_zero @ nat ) ) )
=> ( ( Y
= ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) )
=> ~ ( accp @ nat @ vEBT_VEBT_Tb_rel @ ( suc @ ( zero_zero @ nat ) ) ) ) )
=> ~ ! [N2: nat] :
( ( X
= ( suc @ ( suc @ N2 ) ) )
=> ( ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
=> ( Y
= ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ one2 ) ) ) @ ( vEBT_VEBT_Tb2 @ ( suc @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( times_times @ nat @ ( vEBT_VEBT_Tb2 @ ( suc @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) )
& ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
=> ( Y
= ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ one2 ) ) ) @ ( vEBT_VEBT_Tb2 @ ( suc @ ( suc @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) @ ( times_times @ nat @ ( vEBT_VEBT_Tb2 @ ( suc @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) )
=> ~ ( accp @ nat @ vEBT_VEBT_Tb_rel @ ( suc @ ( suc @ N2 ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.Tb'.pelims
thf(fact_4204_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_4205_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_4206_word__no__log__defs_I1_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: num] :
( ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ A3 ) )
= ( ring_1_of_int @ ( word @ A ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ A3 ) ) ) ) ) ).
% word_no_log_defs(1)
thf(fact_4207_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_4208_word__no__log__defs_I2_J,axiom,
! [B: $tType] :
( ( type_len @ B )
=> ! [A3: num] :
( ( bit_ri4277139882892585799ns_not @ ( word @ B ) @ ( uminus_uminus @ ( word @ B ) @ ( numeral_numeral @ ( word @ B ) @ A3 ) ) )
= ( ring_1_of_int @ ( word @ B ) @ ( bit_ri4277139882892585799ns_not @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ A3 ) ) ) ) ) ) ).
% word_no_log_defs(2)
thf(fact_4209_test__bit__eq__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [U2: word @ A,V2: word @ A] :
( ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ U2 )
= ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ V2 ) )
= ( U2 = V2 ) ) ) ).
% test_bit_eq_iff
thf(fact_4210_word__eqD,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [U2: word @ A,V2: word @ A,X: nat] :
( ( U2 = V2 )
=> ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ U2 @ X )
= ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ V2 @ X ) ) ) ) ).
% word_eqD
thf(fact_4211_bang__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( ^ [Y3: word @ A,Z2: word @ A] : Y3 = Z2 )
= ( ^ [X2: word @ A,Y2: word @ A] :
! [N4: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ X2 @ N4 )
= ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ Y2 @ N4 ) ) ) ) ) ).
% bang_eq
thf(fact_4212_word__wi__log__defs_I1_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: int] :
( ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ ( ring_1_of_int @ ( word @ A ) @ A3 ) )
= ( ring_1_of_int @ ( word @ A ) @ ( bit_ri4277139882892585799ns_not @ int @ A3 ) ) ) ) ).
% word_wi_log_defs(1)
thf(fact_4213_not__int__def,axiom,
( ( bit_ri4277139882892585799ns_not @ int )
= ( ^ [K3: int] : ( minus_minus @ int @ ( uminus_uminus @ int @ K3 ) @ ( one_one @ int ) ) ) ) ).
% not_int_def
thf(fact_4214_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_4215_word__not__def,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( bit_ri4277139882892585799ns_not @ ( word @ A ) )
= ( ^ [A5: word @ A] : ( ring_1_of_int @ ( word @ A ) @ ( bit_ri4277139882892585799ns_not @ int @ ( semiring_1_unsigned @ A @ int @ A5 ) ) ) ) ) ) ).
% word_not_def
thf(fact_4216_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_4217_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_4218_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_4219_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_4220_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_4221_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_4222_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_4223_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_4224_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_4225_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_4226_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_4227_not__int__rec,axiom,
( ( bit_ri4277139882892585799ns_not @ int )
= ( ^ [K3: int] : ( plus_plus @ int @ ( zero_neq_one_of_bool @ int @ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K3 ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_ri4277139882892585799ns_not @ int @ ( divide_divide @ int @ K3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% not_int_rec
thf(fact_4228_VEBT__internal_OT__vebt__buildupi_Opelims,axiom,
! [X: nat,Y: nat] :
( ( ( vEBT_V441764108873111860ildupi @ X )
= Y )
=> ( ( accp @ nat @ vEBT_V2957053500504383685pi_rel @ X )
=> ( ( ( X
= ( zero_zero @ nat ) )
=> ( ( Y
= ( suc @ ( zero_zero @ nat ) ) )
=> ~ ( accp @ nat @ vEBT_V2957053500504383685pi_rel @ ( zero_zero @ nat ) ) ) )
=> ( ( ( X
= ( suc @ ( zero_zero @ nat ) ) )
=> ( ( Y
= ( suc @ ( zero_zero @ nat ) ) )
=> ~ ( accp @ nat @ vEBT_V2957053500504383685pi_rel @ ( suc @ ( zero_zero @ nat ) ) ) ) )
=> ~ ! [N2: nat] :
( ( X
= ( suc @ ( suc @ N2 ) ) )
=> ( ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
=> ( Y
= ( suc @ ( suc @ ( suc @ ( plus_plus @ nat @ ( vEBT_V441764108873111860ildupi @ ( suc @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( times_times @ nat @ ( vEBT_V441764108873111860ildupi @ ( suc @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) )
& ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
=> ( Y
= ( suc @ ( suc @ ( suc @ ( plus_plus @ nat @ ( vEBT_V441764108873111860ildupi @ ( suc @ ( suc @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) @ ( times_times @ nat @ ( vEBT_V441764108873111860ildupi @ ( suc @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) )
=> ~ ( accp @ nat @ vEBT_V2957053500504383685pi_rel @ ( suc @ ( suc @ N2 ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.T_vebt_buildupi.pelims
thf(fact_4229_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_4230_xor__int__unfold,axiom,
( ( bit_se5824344971392196577ns_xor @ int )
= ( ^ [K3: int,L2: int] :
( if @ int
@ ( K3
= ( uminus_uminus @ int @ ( one_one @ int ) ) )
@ ( bit_ri4277139882892585799ns_not @ int @ L2 )
@ ( if @ int
@ ( L2
= ( uminus_uminus @ int @ ( one_one @ int ) ) )
@ ( bit_ri4277139882892585799ns_not @ int @ K3 )
@ ( if @ int
@ ( K3
= ( zero_zero @ int ) )
@ L2
@ ( if @ int
@ ( L2
= ( zero_zero @ int ) )
@ K3
@ ( plus_plus @ int @ ( abs_abs @ int @ ( minus_minus @ int @ ( modulo_modulo @ int @ K3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ int @ L2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344971392196577ns_xor @ int @ ( divide_divide @ int @ K3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ).
% xor_int_unfold
thf(fact_4231_httI__TBOUND,axiom,
! [A: $tType,P: assn,C2: heap_Time_Heap @ A,Q: A > assn,T: nat] :
( ( hoare_hoare_triple @ A @ P @ C2 @ Q )
=> ( ( time_TBOUND @ A @ C2 @ T )
=> ( time_htt @ A @ P @ C2 @ Q @ T ) ) ) ).
% httI_TBOUND
thf(fact_4232_xor_Oright__neutral,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A] :
( ( bit_se5824344971392196577ns_xor @ A @ A3 @ ( zero_zero @ A ) )
= A3 ) ) ).
% xor.right_neutral
thf(fact_4233_xor_Oleft__neutral,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A] :
( ( bit_se5824344971392196577ns_xor @ A @ ( zero_zero @ A ) @ A3 )
= A3 ) ) ).
% xor.left_neutral
thf(fact_4234_xor__self__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A] :
( ( bit_se5824344971392196577ns_xor @ A @ A3 @ A3 )
= ( zero_zero @ A ) ) ) ).
% xor_self_eq
thf(fact_4235_bit_Oxor__self,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X: A] :
( ( bit_se5824344971392196577ns_xor @ A @ X @ X )
= ( zero_zero @ A ) ) ) ).
% bit.xor_self
thf(fact_4236_xor__nonnegative__int__iff,axiom,
! [K: int,L: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se5824344971392196577ns_xor @ int @ K @ L ) )
= ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
= ( ord_less_eq @ int @ ( zero_zero @ int ) @ L ) ) ) ).
% xor_nonnegative_int_iff
thf(fact_4237_xor__negative__int__iff,axiom,
! [K: int,L: int] :
( ( ord_less @ int @ ( bit_se5824344971392196577ns_xor @ int @ K @ L ) @ ( zero_zero @ int ) )
= ( ( ord_less @ int @ K @ ( zero_zero @ int ) )
!= ( ord_less @ int @ L @ ( zero_zero @ int ) ) ) ) ).
% xor_negative_int_iff
thf(fact_4238_bit_Oxor__cancel__right,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X: A] :
( ( bit_se5824344971392196577ns_xor @ A @ X @ ( bit_ri4277139882892585799ns_not @ A @ X ) )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% bit.xor_cancel_right
thf(fact_4239_bit_Oxor__cancel__left,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X: A] :
( ( bit_se5824344971392196577ns_xor @ A @ ( bit_ri4277139882892585799ns_not @ A @ X ) @ X )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% bit.xor_cancel_left
thf(fact_4240_bit_Oxor__one__right,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X: A] :
( ( bit_se5824344971392196577ns_xor @ A @ X @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( bit_ri4277139882892585799ns_not @ A @ X ) ) ) ).
% bit.xor_one_right
thf(fact_4241_bit_Oxor__one__left,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X: A] :
( ( bit_se5824344971392196577ns_xor @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ X )
= ( bit_ri4277139882892585799ns_not @ A @ X ) ) ) ).
% bit.xor_one_left
thf(fact_4242_xor__numerals_I3_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X: num,Y: num] :
( ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ ( bit0 @ X ) ) @ ( numeral_numeral @ A @ ( bit0 @ Y ) ) )
= ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ X ) @ ( numeral_numeral @ A @ Y ) ) ) ) ) ).
% xor_numerals(3)
thf(fact_4243_xor__numerals_I8_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X: num] :
( ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ ( bit1 @ X ) ) @ ( one_one @ A ) )
= ( numeral_numeral @ A @ ( bit0 @ X ) ) ) ) ).
% xor_numerals(8)
thf(fact_4244_xor__numerals_I5_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X: num] :
( ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ ( bit0 @ X ) ) @ ( one_one @ A ) )
= ( numeral_numeral @ A @ ( bit1 @ X ) ) ) ) ).
% xor_numerals(5)
thf(fact_4245_xor__numerals_I2_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [Y: num] :
( ( bit_se5824344971392196577ns_xor @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit1 @ Y ) ) )
= ( numeral_numeral @ A @ ( bit0 @ Y ) ) ) ) ).
% xor_numerals(2)
thf(fact_4246_xor__numerals_I1_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [Y: num] :
( ( bit_se5824344971392196577ns_xor @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ Y ) ) )
= ( numeral_numeral @ A @ ( bit1 @ Y ) ) ) ) ).
% xor_numerals(1)
thf(fact_4247_xor__numerals_I7_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X: num,Y: num] :
( ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ ( bit1 @ X ) ) @ ( numeral_numeral @ A @ ( bit1 @ Y ) ) )
= ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ X ) @ ( numeral_numeral @ A @ Y ) ) ) ) ) ).
% xor_numerals(7)
thf(fact_4248_xor__numerals_I6_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X: num,Y: num] :
( ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ ( bit1 @ X ) ) @ ( numeral_numeral @ A @ ( bit0 @ Y ) ) )
= ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ X ) @ ( numeral_numeral @ A @ Y ) ) ) ) ) ) ).
% xor_numerals(6)
thf(fact_4249_xor__numerals_I4_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X: num,Y: num] :
( ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ ( bit0 @ X ) ) @ ( numeral_numeral @ A @ ( bit1 @ Y ) ) )
= ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ X ) @ ( numeral_numeral @ A @ Y ) ) ) ) ) ) ).
% xor_numerals(4)
thf(fact_4250_int__xor__code_I2_J,axiom,
! [I: int] :
( ( bit_se5824344971392196577ns_xor @ int @ I @ ( zero_zero @ int ) )
= I ) ).
% int_xor_code(2)
thf(fact_4251_int__xor__code_I1_J,axiom,
! [J: int] :
( ( bit_se5824344971392196577ns_xor @ int @ ( zero_zero @ int ) @ J )
= J ) ).
% int_xor_code(1)
thf(fact_4252_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_4253_unsigned__xor__eq,axiom,
! [A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( bit_se359711467146920520ations @ A ) )
=> ! [V2: word @ B,W: word @ B] :
( ( semiring_1_unsigned @ B @ A @ ( bit_se5824344971392196577ns_xor @ ( word @ B ) @ V2 @ W ) )
= ( bit_se5824344971392196577ns_xor @ A @ ( semiring_1_unsigned @ B @ A @ V2 ) @ ( semiring_1_unsigned @ B @ A @ W ) ) ) ) ).
% unsigned_xor_eq
thf(fact_4254_signed__xor__eq,axiom,
! [A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( bit_ri3973907225187159222ations @ A ) )
=> ! [V2: word @ B,W: word @ B] :
( ( ring_1_signed @ B @ A @ ( bit_se5824344971392196577ns_xor @ ( word @ B ) @ V2 @ W ) )
= ( bit_se5824344971392196577ns_xor @ A @ ( ring_1_signed @ B @ A @ V2 ) @ ( ring_1_signed @ B @ A @ W ) ) ) ) ).
% signed_xor_eq
thf(fact_4255_XOR__lower,axiom,
! [X: int,Y: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y )
=> ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se5824344971392196577ns_xor @ int @ X @ Y ) ) ) ) ).
% XOR_lower
thf(fact_4256_even__xor__iff,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A,B3: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5824344971392196577ns_xor @ A @ A3 @ B3 ) )
= ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
= ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B3 ) ) ) ) ).
% even_xor_iff
thf(fact_4257_XOR__upper,axiom,
! [X: int,N: nat,Y: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
=> ( ( ord_less @ int @ X @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) )
=> ( ( ord_less @ int @ Y @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) )
=> ( ord_less @ int @ ( bit_se5824344971392196577ns_xor @ int @ X @ Y ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ) ) ) ).
% XOR_upper
thf(fact_4258_test__bit__int__code_I1_J,axiom,
! [N: nat] :
~ ( bit_se5641148757651400278ts_bit @ int @ ( zero_zero @ int ) @ N ) ).
% test_bit_int_code(1)
thf(fact_4259_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_4260_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_4261_xor__int__rec,axiom,
( ( bit_se5824344971392196577ns_xor @ int )
= ( ^ [K3: int,L2: int] :
( plus_plus @ int
@ ( zero_neq_one_of_bool @ int
@ ( ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K3 ) )
!= ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ L2 ) ) ) )
@ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344971392196577ns_xor @ int @ ( divide_divide @ int @ K3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% xor_int_rec
thf(fact_4262_one__xor__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A] :
( ( bit_se5824344971392196577ns_xor @ A @ ( one_one @ A ) @ A3 )
= ( minus_minus @ A @ ( plus_plus @ A @ A3 @ ( zero_neq_one_of_bool @ A @ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) ) )
@ ( zero_neq_one_of_bool @ A
@ ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) ) ) ) ) ).
% one_xor_eq
thf(fact_4263_xor__one__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A] :
( ( bit_se5824344971392196577ns_xor @ A @ A3 @ ( one_one @ A ) )
= ( minus_minus @ A @ ( plus_plus @ A @ A3 @ ( zero_neq_one_of_bool @ A @ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) ) )
@ ( zero_neq_one_of_bool @ A
@ ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) ) ) ) ) ).
% xor_one_eq
thf(fact_4264_htt__htD,axiom,
! [A: $tType,P: assn,C2: heap_Time_Heap @ A,Q: A > assn,T: nat] :
( ( time_htt @ A @ P @ C2 @ Q @ T )
=> ( hoare_hoare_triple @ A @ P @ C2 @ Q ) ) ).
% htt_htD
thf(fact_4265_neg__mask__add__mask,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,N: nat] :
( ( plus_plus @ ( word @ A ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) ) @ ( minus_minus @ ( word @ A ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) @ ( one_one @ ( word @ A ) ) ) )
= ( bit_se1065995026697491101ons_or @ ( word @ A ) @ X @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) ) ) ).
% neg_mask_add_mask
thf(fact_4266_monoseq__minus,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: nat > A] :
( ( topological_monoseq @ A @ A3 )
=> ( topological_monoseq @ A
@ ^ [N4: nat] : ( uminus_uminus @ A @ ( A3 @ N4 ) ) ) ) ) ).
% monoseq_minus
thf(fact_4267_word__bitwise__1__simps_I13_J,axiom,
! [M8: $tType] :
( ( type_len @ M8 )
=> ! [A3: num] :
( ( bit_se5824344971392196577ns_xor @ ( word @ M8 ) @ ( uminus_uminus @ ( word @ M8 ) @ ( numeral_numeral @ ( word @ M8 ) @ A3 ) ) @ ( one_one @ ( word @ M8 ) ) )
= ( ring_1_of_int @ ( word @ M8 ) @ ( bit_se5824344971392196577ns_xor @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ A3 ) ) @ ( one_one @ int ) ) ) ) ) ).
% word_bitwise_1_simps(13)
thf(fact_4268_word__bitwise__1__simps_I11_J,axiom,
! [K7: $tType] :
( ( type_len @ K7 )
=> ! [B3: num] :
( ( bit_se5824344971392196577ns_xor @ ( word @ K7 ) @ ( one_one @ ( word @ K7 ) ) @ ( uminus_uminus @ ( word @ K7 ) @ ( numeral_numeral @ ( word @ K7 ) @ B3 ) ) )
= ( ring_1_of_int @ ( word @ K7 ) @ ( bit_se5824344971392196577ns_xor @ int @ ( one_one @ int ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ B3 ) ) ) ) ) ) ).
% word_bitwise_1_simps(11)
thf(fact_4269_or_Oright__neutral,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A] :
( ( bit_se1065995026697491101ons_or @ A @ A3 @ ( zero_zero @ A ) )
= A3 ) ) ).
% or.right_neutral
thf(fact_4270_or_Oleft__neutral,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A] :
( ( bit_se1065995026697491101ons_or @ A @ ( zero_zero @ A ) @ A3 )
= A3 ) ) ).
% or.left_neutral
thf(fact_4271_word__ao__absorbs_I1_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] :
( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X @ ( bit_se1065995026697491101ons_or @ ( word @ A ) @ Y @ X ) )
= X ) ) ).
% word_ao_absorbs(1)
thf(fact_4272_word__ao__absorbs_I2_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] :
( ( bit_se1065995026697491101ons_or @ ( word @ A ) @ X @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ Y @ X ) )
= X ) ) ).
% word_ao_absorbs(2)
thf(fact_4273_word__ao__absorbs_I3_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] :
( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X @ ( bit_se1065995026697491101ons_or @ ( word @ A ) @ X @ Y ) )
= X ) ) ).
% word_ao_absorbs(3)
thf(fact_4274_word__ao__absorbs_I4_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y: word @ A,X: word @ A] :
( ( bit_se1065995026697491101ons_or @ ( word @ A ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ Y @ X ) @ X )
= X ) ) ).
% word_ao_absorbs(4)
thf(fact_4275_word__ao__absorbs_I5_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y: word @ A,X: word @ A] :
( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ ( bit_se1065995026697491101ons_or @ ( word @ A ) @ Y @ X ) @ X )
= X ) ) ).
% word_ao_absorbs(5)
thf(fact_4276_word__ao__absorbs_I6_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] :
( ( bit_se1065995026697491101ons_or @ ( word @ A ) @ X @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X @ Y ) )
= X ) ) ).
% word_ao_absorbs(6)
thf(fact_4277_word__ao__absorbs_I7_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] :
( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ ( bit_se1065995026697491101ons_or @ ( word @ A ) @ X @ Y ) @ X )
= X ) ) ).
% word_ao_absorbs(7)
thf(fact_4278_word__ao__absorbs_I8_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] :
( ( bit_se1065995026697491101ons_or @ ( word @ A ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X @ Y ) @ X )
= X ) ) ).
% word_ao_absorbs(8)
thf(fact_4279_bit_Odisj__one__left,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X: A] :
( ( bit_se1065995026697491101ons_or @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ X )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% bit.disj_one_left
thf(fact_4280_bit_Odisj__one__right,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X: A] :
( ( bit_se1065995026697491101ons_or @ A @ X @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% bit.disj_one_right
thf(fact_4281_word__plus__and__or,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] :
( ( plus_plus @ ( word @ A ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X @ Y ) @ ( bit_se1065995026697491101ons_or @ ( word @ A ) @ X @ Y ) )
= ( plus_plus @ ( word @ A ) @ X @ Y ) ) ) ).
% word_plus_and_or
thf(fact_4282_word__bitwise__m1__simps_I4_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] :
( ( bit_se1065995026697491101ons_or @ ( word @ A ) @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) @ X )
= ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) ) ) ).
% word_bitwise_m1_simps(4)
thf(fact_4283_word__or__max,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] :
( ( bit_se1065995026697491101ons_or @ ( word @ A ) @ X @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) )
= ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) ) ) ).
% word_or_max
thf(fact_4284_or__numerals_I8_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X: num] :
( ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ ( bit1 @ X ) ) @ ( one_one @ A ) )
= ( numeral_numeral @ A @ ( bit1 @ X ) ) ) ) ).
% or_numerals(8)
thf(fact_4285_or__numerals_I2_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [Y: num] :
( ( bit_se1065995026697491101ons_or @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit1 @ Y ) ) )
= ( numeral_numeral @ A @ ( bit1 @ Y ) ) ) ) ).
% or_numerals(2)
thf(fact_4286_bit_Odisj__cancel__left,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X: A] :
( ( bit_se1065995026697491101ons_or @ A @ ( bit_ri4277139882892585799ns_not @ A @ X ) @ X )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% bit.disj_cancel_left
thf(fact_4287_bit_Odisj__cancel__right,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X: A] :
( ( bit_se1065995026697491101ons_or @ A @ X @ ( bit_ri4277139882892585799ns_not @ A @ X ) )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% bit.disj_cancel_right
thf(fact_4288_word__or__not,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] :
( ( bit_se1065995026697491101ons_or @ ( word @ A ) @ X @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ X ) )
= ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) ) ) ).
% word_or_not
thf(fact_4289_word__bitwise__m1__simps_I7_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] :
( ( bit_se5824344971392196577ns_xor @ ( word @ A ) @ X @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) )
= ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ X ) ) ) ).
% word_bitwise_m1_simps(7)
thf(fact_4290_word__bitwise__m1__simps_I6_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] :
( ( bit_se5824344971392196577ns_xor @ ( word @ A ) @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) @ X )
= ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ X ) ) ) ).
% word_bitwise_m1_simps(6)
thf(fact_4291_or__numerals_I3_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X: num,Y: num] :
( ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ ( bit0 @ X ) ) @ ( numeral_numeral @ A @ ( bit0 @ Y ) ) )
= ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ X ) @ ( numeral_numeral @ A @ Y ) ) ) ) ) ).
% or_numerals(3)
thf(fact_4292_or__numerals_I1_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [Y: num] :
( ( bit_se1065995026697491101ons_or @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ Y ) ) )
= ( numeral_numeral @ A @ ( bit1 @ Y ) ) ) ) ).
% or_numerals(1)
thf(fact_4293_or__numerals_I5_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X: num] :
( ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ ( bit0 @ X ) ) @ ( one_one @ A ) )
= ( numeral_numeral @ A @ ( bit1 @ X ) ) ) ) ).
% or_numerals(5)
thf(fact_4294_word__no__log__defs_I11_J,axiom,
! [K7: $tType] :
( ( type_len @ K7 )
=> ! [A3: num,B3: num] :
( ( bit_se5824344971392196577ns_xor @ ( word @ K7 ) @ ( numeral_numeral @ ( word @ K7 ) @ A3 ) @ ( numeral_numeral @ ( word @ K7 ) @ B3 ) )
= ( ring_1_of_int @ ( word @ K7 ) @ ( bit_se5824344971392196577ns_xor @ int @ ( numeral_numeral @ int @ A3 ) @ ( numeral_numeral @ int @ B3 ) ) ) ) ) ).
% word_no_log_defs(11)
thf(fact_4295_xor__nat__numerals_I4_J,axiom,
! [X: num] :
( ( bit_se5824344971392196577ns_xor @ nat @ ( numeral_numeral @ nat @ ( bit1 @ X ) ) @ ( suc @ ( zero_zero @ nat ) ) )
= ( numeral_numeral @ nat @ ( bit0 @ X ) ) ) ).
% xor_nat_numerals(4)
thf(fact_4296_xor__nat__numerals_I3_J,axiom,
! [X: num] :
( ( bit_se5824344971392196577ns_xor @ nat @ ( numeral_numeral @ nat @ ( bit0 @ X ) ) @ ( suc @ ( zero_zero @ nat ) ) )
= ( numeral_numeral @ nat @ ( bit1 @ X ) ) ) ).
% xor_nat_numerals(3)
thf(fact_4297_xor__nat__numerals_I2_J,axiom,
! [Y: num] :
( ( bit_se5824344971392196577ns_xor @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( numeral_numeral @ nat @ ( bit1 @ Y ) ) )
= ( numeral_numeral @ nat @ ( bit0 @ Y ) ) ) ).
% xor_nat_numerals(2)
thf(fact_4298_xor__nat__numerals_I1_J,axiom,
! [Y: num] :
( ( bit_se5824344971392196577ns_xor @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( numeral_numeral @ nat @ ( bit0 @ Y ) ) )
= ( numeral_numeral @ nat @ ( bit1 @ Y ) ) ) ).
% xor_nat_numerals(1)
thf(fact_4299_word__bitwise__1__simps_I10_J,axiom,
! [J4: $tType] :
( ( type_len @ J4 )
=> ! [B3: num] :
( ( bit_se5824344971392196577ns_xor @ ( word @ J4 ) @ ( one_one @ ( word @ J4 ) ) @ ( numeral_numeral @ ( word @ J4 ) @ B3 ) )
= ( ring_1_of_int @ ( word @ J4 ) @ ( bit_se5824344971392196577ns_xor @ int @ ( one_one @ int ) @ ( numeral_numeral @ int @ B3 ) ) ) ) ) ).
% word_bitwise_1_simps(10)
thf(fact_4300_word__bitwise__1__simps_I12_J,axiom,
! [L4: $tType] :
( ( type_len @ L4 )
=> ! [A3: num] :
( ( bit_se5824344971392196577ns_xor @ ( word @ L4 ) @ ( numeral_numeral @ ( word @ L4 ) @ A3 ) @ ( one_one @ ( word @ L4 ) ) )
= ( ring_1_of_int @ ( word @ L4 ) @ ( bit_se5824344971392196577ns_xor @ int @ ( numeral_numeral @ int @ A3 ) @ ( one_one @ int ) ) ) ) ) ).
% word_bitwise_1_simps(12)
thf(fact_4301_word__no__log__defs_I14_J,axiom,
! [N12: $tType] :
( ( type_len @ N12 )
=> ! [A3: num,B3: num] :
( ( bit_se5824344971392196577ns_xor @ ( word @ N12 ) @ ( uminus_uminus @ ( word @ N12 ) @ ( numeral_numeral @ ( word @ N12 ) @ A3 ) ) @ ( uminus_uminus @ ( word @ N12 ) @ ( numeral_numeral @ ( word @ N12 ) @ B3 ) ) )
= ( ring_1_of_int @ ( word @ N12 ) @ ( bit_se5824344971392196577ns_xor @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ A3 ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ B3 ) ) ) ) ) ) ).
% word_no_log_defs(14)
thf(fact_4302_word__no__log__defs_I13_J,axiom,
! [M8: $tType] :
( ( type_len @ M8 )
=> ! [A3: num,B3: num] :
( ( bit_se5824344971392196577ns_xor @ ( word @ M8 ) @ ( uminus_uminus @ ( word @ M8 ) @ ( numeral_numeral @ ( word @ M8 ) @ A3 ) ) @ ( numeral_numeral @ ( word @ M8 ) @ B3 ) )
= ( ring_1_of_int @ ( word @ M8 ) @ ( bit_se5824344971392196577ns_xor @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ A3 ) ) @ ( numeral_numeral @ int @ B3 ) ) ) ) ) ).
% word_no_log_defs(13)
thf(fact_4303_word__no__log__defs_I12_J,axiom,
! [L4: $tType] :
( ( type_len @ L4 )
=> ! [A3: num,B3: num] :
( ( bit_se5824344971392196577ns_xor @ ( word @ L4 ) @ ( numeral_numeral @ ( word @ L4 ) @ A3 ) @ ( uminus_uminus @ ( word @ L4 ) @ ( numeral_numeral @ ( word @ L4 ) @ B3 ) ) )
= ( ring_1_of_int @ ( word @ L4 ) @ ( bit_se5824344971392196577ns_xor @ int @ ( numeral_numeral @ int @ A3 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ B3 ) ) ) ) ) ) ).
% word_no_log_defs(12)
thf(fact_4304_or__numerals_I4_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X: num,Y: num] :
( ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ ( bit0 @ X ) ) @ ( numeral_numeral @ A @ ( bit1 @ Y ) ) )
= ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ X ) @ ( numeral_numeral @ A @ Y ) ) ) ) ) ) ).
% or_numerals(4)
thf(fact_4305_or__numerals_I6_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X: num,Y: num] :
( ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ ( bit1 @ X ) ) @ ( numeral_numeral @ A @ ( bit0 @ Y ) ) )
= ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ X ) @ ( numeral_numeral @ A @ Y ) ) ) ) ) ) ).
% or_numerals(6)
thf(fact_4306_or__numerals_I7_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X: num,Y: num] :
( ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ ( bit1 @ X ) ) @ ( numeral_numeral @ A @ ( bit1 @ Y ) ) )
= ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ X ) @ ( numeral_numeral @ A @ Y ) ) ) ) ) ) ).
% or_numerals(7)
thf(fact_4307_word__xor__nth,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A,N: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( bit_se5824344971392196577ns_xor @ ( word @ A ) @ X @ Y ) @ N )
= ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ X @ N )
!= ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ Y @ N ) ) ) ) ).
% word_xor_nth
thf(fact_4308_word__xor__and__or,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( bit_se5824344971392196577ns_xor @ ( word @ A ) )
= ( ^ [X2: word @ A,Y2: word @ A] : ( bit_se1065995026697491101ons_or @ ( word @ A ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X2 @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ Y2 ) ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ X2 ) @ Y2 ) ) ) ) ) ).
% word_xor_and_or
thf(fact_4309_word__log__esimps_I5_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] :
( ( bit_se5824344971392196577ns_xor @ ( word @ A ) @ X @ ( zero_zero @ ( word @ A ) ) )
= X ) ) ).
% word_log_esimps(5)
thf(fact_4310_word__log__esimps_I11_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] :
( ( bit_se5824344971392196577ns_xor @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ X )
= X ) ) ).
% word_log_esimps(11)
thf(fact_4311_word__bw__same_I3_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] :
( ( bit_se5824344971392196577ns_xor @ ( word @ A ) @ X @ X )
= ( zero_zero @ ( word @ A ) ) ) ) ).
% word_bw_same(3)
thf(fact_4312_word__bw__lcs_I3_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y: word @ A,X: word @ A,Z: word @ A] :
( ( bit_se5824344971392196577ns_xor @ ( word @ A ) @ Y @ ( bit_se5824344971392196577ns_xor @ ( word @ A ) @ X @ Z ) )
= ( bit_se5824344971392196577ns_xor @ ( word @ A ) @ X @ ( bit_se5824344971392196577ns_xor @ ( word @ A ) @ Y @ Z ) ) ) ) ).
% word_bw_lcs(3)
thf(fact_4313_word__bw__comms_I3_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( bit_se5824344971392196577ns_xor @ ( word @ A ) )
= ( ^ [X2: word @ A,Y2: word @ A] : ( bit_se5824344971392196577ns_xor @ ( word @ A ) @ Y2 @ X2 ) ) ) ) ).
% word_bw_comms(3)
thf(fact_4314_word__bw__assocs_I3_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A,Z: word @ A] :
( ( bit_se5824344971392196577ns_xor @ ( word @ A ) @ ( bit_se5824344971392196577ns_xor @ ( word @ A ) @ X @ Y ) @ Z )
= ( bit_se5824344971392196577ns_xor @ ( word @ A ) @ X @ ( bit_se5824344971392196577ns_xor @ ( word @ A ) @ Y @ Z ) ) ) ) ).
% word_bw_assocs(3)
thf(fact_4315_swap__with__xor,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,A3: word @ A,B3: word @ A,Y: word @ A,Z: word @ A] :
( ( X
= ( bit_se5824344971392196577ns_xor @ ( word @ A ) @ A3 @ B3 ) )
=> ( ( Y
= ( bit_se5824344971392196577ns_xor @ ( word @ A ) @ B3 @ X ) )
=> ( ( Z
= ( bit_se5824344971392196577ns_xor @ ( word @ A ) @ X @ Y ) )
=> ( ( Z = B3 )
& ( Y = A3 ) ) ) ) ) ) ).
% swap_with_xor
thf(fact_4316_le__word__or1,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [C2: word @ A,Y: word @ A] : ( ord_less_eq @ ( word @ A ) @ C2 @ ( bit_se1065995026697491101ons_or @ ( word @ A ) @ Y @ C2 ) ) ) ).
% le_word_or1
thf(fact_4317_le__word__or2,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] : ( ord_less_eq @ ( word @ A ) @ X @ ( bit_se1065995026697491101ons_or @ ( word @ A ) @ X @ Y ) ) ) ).
% le_word_or2
thf(fact_4318_word__combine__masks,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A,M: word @ A,Z: word @ A,M6: word @ A,Z7: word @ A] :
( ( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ W @ M )
= Z )
=> ( ( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ W @ M6 )
= Z7 )
=> ( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ W @ ( bit_se1065995026697491101ons_or @ ( word @ A ) @ M @ M6 ) )
= ( bit_se1065995026697491101ons_or @ ( word @ A ) @ Z @ Z7 ) ) ) ) ) ).
% word_combine_masks
thf(fact_4319_word__oa__dist2,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A,Z: word @ A] :
( ( bit_se1065995026697491101ons_or @ ( word @ A ) @ X @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ Y @ Z ) )
= ( bit_se5824344872417868541ns_and @ ( word @ A ) @ ( bit_se1065995026697491101ons_or @ ( word @ A ) @ X @ Y ) @ ( bit_se1065995026697491101ons_or @ ( word @ A ) @ X @ Z ) ) ) ) ).
% word_oa_dist2
thf(fact_4320_word__ao__equiv,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A,W4: word @ A] :
( ( W
= ( bit_se1065995026697491101ons_or @ ( word @ A ) @ W @ W4 ) )
= ( W4
= ( bit_se5824344872417868541ns_and @ ( word @ A ) @ W @ W4 ) ) ) ) ).
% word_ao_equiv
thf(fact_4321_word__ao__dist2,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A,Z: word @ A] :
( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X @ ( bit_se1065995026697491101ons_or @ ( word @ A ) @ Y @ Z ) )
= ( bit_se1065995026697491101ons_or @ ( word @ A ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X @ Y ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X @ Z ) ) ) ) ).
% word_ao_dist2
thf(fact_4322_word__oa__dist,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A,Z: word @ A] :
( ( bit_se1065995026697491101ons_or @ ( word @ A ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X @ Y ) @ Z )
= ( bit_se5824344872417868541ns_and @ ( word @ A ) @ ( bit_se1065995026697491101ons_or @ ( word @ A ) @ X @ Z ) @ ( bit_se1065995026697491101ons_or @ ( word @ A ) @ Y @ Z ) ) ) ) ).
% word_oa_dist
thf(fact_4323_word__ao__dist,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A,Z: word @ A] :
( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ ( bit_se1065995026697491101ons_or @ ( word @ A ) @ X @ Y ) @ Z )
= ( bit_se1065995026697491101ons_or @ ( word @ A ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X @ Z ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ Y @ Z ) ) ) ) ).
% word_ao_dist
thf(fact_4324_leoa,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A,X: word @ A,Y: word @ A] :
( ( W
= ( bit_se1065995026697491101ons_or @ ( word @ A ) @ X @ Y ) )
=> ( Y
= ( bit_se5824344872417868541ns_and @ ( word @ A ) @ W @ Y ) ) ) ) ).
% leoa
thf(fact_4325_leao,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W4: word @ A,X7: word @ A,Y6: word @ A] :
( ( W4
= ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X7 @ Y6 ) )
=> ( X7
= ( bit_se1065995026697491101ons_or @ ( word @ A ) @ X7 @ W4 ) ) ) ) ).
% leao
thf(fact_4326_signed__or__eq,axiom,
! [A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( bit_ri3973907225187159222ations @ A ) )
=> ! [V2: word @ B,W: word @ B] :
( ( ring_1_signed @ B @ A @ ( bit_se1065995026697491101ons_or @ ( word @ B ) @ V2 @ W ) )
= ( bit_se1065995026697491101ons_or @ A @ ( ring_1_signed @ B @ A @ V2 ) @ ( ring_1_signed @ B @ A @ W ) ) ) ) ).
% signed_or_eq
thf(fact_4327_unsigned__or__eq,axiom,
! [A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( bit_se359711467146920520ations @ A ) )
=> ! [V2: word @ B,W: word @ B] :
( ( semiring_1_unsigned @ B @ A @ ( bit_se1065995026697491101ons_or @ ( word @ B ) @ V2 @ W ) )
= ( bit_se1065995026697491101ons_or @ A @ ( semiring_1_unsigned @ B @ A @ V2 ) @ ( semiring_1_unsigned @ B @ A @ W ) ) ) ) ).
% unsigned_or_eq
thf(fact_4328_word__bw__assocs_I2_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A,Z: word @ A] :
( ( bit_se1065995026697491101ons_or @ ( word @ A ) @ ( bit_se1065995026697491101ons_or @ ( word @ A ) @ X @ Y ) @ Z )
= ( bit_se1065995026697491101ons_or @ ( word @ A ) @ X @ ( bit_se1065995026697491101ons_or @ ( word @ A ) @ Y @ Z ) ) ) ) ).
% word_bw_assocs(2)
thf(fact_4329_word__bw__comms_I2_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( bit_se1065995026697491101ons_or @ ( word @ A ) )
= ( ^ [X2: word @ A,Y2: word @ A] : ( bit_se1065995026697491101ons_or @ ( word @ A ) @ Y2 @ X2 ) ) ) ) ).
% word_bw_comms(2)
thf(fact_4330_word__bw__same_I2_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] :
( ( bit_se1065995026697491101ons_or @ ( word @ A ) @ X @ X )
= X ) ) ).
% word_bw_same(2)
thf(fact_4331_word__bw__lcs_I2_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y: word @ A,X: word @ A,Z: word @ A] :
( ( bit_se1065995026697491101ons_or @ ( word @ A ) @ Y @ ( bit_se1065995026697491101ons_or @ ( word @ A ) @ X @ Z ) )
= ( bit_se1065995026697491101ons_or @ ( word @ A ) @ X @ ( bit_se1065995026697491101ons_or @ ( word @ A ) @ Y @ Z ) ) ) ) ).
% word_bw_lcs(2)
thf(fact_4332_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_4333_ucast__or__distrib,axiom,
! [B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B ) )
=> ! [X: word @ A,Y: word @ A] :
( ( semiring_1_unsigned @ A @ ( word @ B ) @ ( bit_se1065995026697491101ons_or @ ( word @ A ) @ X @ Y ) )
= ( bit_se1065995026697491101ons_or @ ( word @ B ) @ ( semiring_1_unsigned @ A @ ( word @ B ) @ X ) @ ( semiring_1_unsigned @ A @ ( word @ B ) @ Y ) ) ) ) ).
% ucast_or_distrib
thf(fact_4334_word__log__esimps_I9_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] :
( ( bit_se1065995026697491101ons_or @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ X )
= X ) ) ).
% word_log_esimps(9)
thf(fact_4335_word__log__esimps_I3_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] :
( ( bit_se1065995026697491101ons_or @ ( word @ A ) @ X @ ( zero_zero @ ( word @ A ) ) )
= X ) ) ).
% word_log_esimps(3)
thf(fact_4336_word__or__zero,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A,B3: word @ A] :
( ( ( bit_se1065995026697491101ons_or @ ( word @ A ) @ A3 @ B3 )
= ( zero_zero @ ( word @ A ) ) )
= ( ( A3
= ( zero_zero @ ( word @ A ) ) )
& ( B3
= ( zero_zero @ ( word @ A ) ) ) ) ) ) ).
% word_or_zero
thf(fact_4337_bit_Odisj__zero__right,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X: A] :
( ( bit_se1065995026697491101ons_or @ A @ X @ ( zero_zero @ A ) )
= X ) ) ).
% bit.disj_zero_right
thf(fact_4338_or__eq__0__iff,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A,B3: A] :
( ( ( bit_se1065995026697491101ons_or @ A @ A3 @ B3 )
= ( zero_zero @ A ) )
= ( ( A3
= ( zero_zero @ A ) )
& ( B3
= ( zero_zero @ A ) ) ) ) ) ).
% or_eq_0_iff
thf(fact_4339_word__or__nth,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A,N: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( bit_se1065995026697491101ons_or @ ( word @ A ) @ X @ Y ) @ N )
= ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ X @ N )
| ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ Y @ N ) ) ) ) ).
% word_or_nth
thf(fact_4340_word__ao__nth,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A,N: nat] :
( ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( bit_se1065995026697491101ons_or @ ( word @ A ) @ X @ Y ) @ N )
= ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ X @ N )
| ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ Y @ N ) ) )
& ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X @ Y ) @ N )
= ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ X @ N )
& ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ Y @ N ) ) ) ) ) ).
% word_ao_nth
thf(fact_4341_uint__xor,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] :
( ( semiring_1_unsigned @ A @ int @ ( bit_se5824344971392196577ns_xor @ ( word @ A ) @ X @ Y ) )
= ( bit_se5824344971392196577ns_xor @ int @ ( semiring_1_unsigned @ A @ int @ X ) @ ( semiring_1_unsigned @ A @ int @ Y ) ) ) ) ).
% uint_xor
thf(fact_4342_word__not__dist_I1_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] :
( ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ ( bit_se1065995026697491101ons_or @ ( word @ A ) @ X @ Y ) )
= ( bit_se5824344872417868541ns_and @ ( word @ A ) @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ X ) @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ Y ) ) ) ) ).
% word_not_dist(1)
thf(fact_4343_word__not__dist_I2_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] :
( ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X @ Y ) )
= ( bit_se1065995026697491101ons_or @ ( word @ A ) @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ X ) @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ Y ) ) ) ) ).
% word_not_dist(2)
thf(fact_4344_word__wi__log__defs_I4_J,axiom,
! [D: $tType] :
( ( type_len @ D )
=> ! [A3: int,B3: int] :
( ( bit_se5824344971392196577ns_xor @ ( word @ D ) @ ( ring_1_of_int @ ( word @ D ) @ A3 ) @ ( ring_1_of_int @ ( word @ D ) @ B3 ) )
= ( ring_1_of_int @ ( word @ D ) @ ( bit_se5824344971392196577ns_xor @ int @ A3 @ B3 ) ) ) ) ).
% word_wi_log_defs(4)
thf(fact_4345_word__ops__nth__size,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,X: word @ A,Y: word @ A] :
( ( ord_less @ nat @ N @ ( size_size @ ( word @ A ) @ X ) )
=> ( ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( bit_se1065995026697491101ons_or @ ( word @ A ) @ X @ Y ) @ N )
= ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ X @ N )
| ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ Y @ N ) ) )
& ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X @ Y ) @ N )
= ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ X @ N )
& ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ Y @ N ) ) )
& ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( bit_se5824344971392196577ns_xor @ ( word @ A ) @ X @ Y ) @ N )
= ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ X @ N )
!= ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ Y @ N ) ) )
& ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ X ) @ N )
= ( ~ ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ X @ N ) ) ) ) ) ) ).
% word_ops_nth_size
thf(fact_4346_word__plus__and__or__coroll,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] :
( ( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X @ Y )
= ( zero_zero @ ( word @ A ) ) )
=> ( ( plus_plus @ ( word @ A ) @ X @ Y )
= ( bit_se1065995026697491101ons_or @ ( word @ A ) @ X @ Y ) ) ) ) ).
% word_plus_and_or_coroll
thf(fact_4347_word__xor__def,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( bit_se5824344971392196577ns_xor @ ( word @ A ) )
= ( ^ [A5: word @ A,B5: word @ A] : ( ring_1_of_int @ ( word @ A ) @ ( bit_se5824344971392196577ns_xor @ int @ ( semiring_1_unsigned @ A @ int @ A5 ) @ ( semiring_1_unsigned @ A @ int @ B5 ) ) ) ) ) ) ).
% word_xor_def
thf(fact_4348_xor__nat__def,axiom,
( ( bit_se5824344971392196577ns_xor @ nat )
= ( ^ [M3: nat,N4: nat] : ( nat2 @ ( bit_se5824344971392196577ns_xor @ int @ ( semiring_1_of_nat @ int @ M3 ) @ ( semiring_1_of_nat @ int @ N4 ) ) ) ) ) ).
% xor_nat_def
thf(fact_4349_or__not__mask__nop,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,N: nat] :
( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ ( bit_se1065995026697491101ons_or @ ( word @ A ) @ X @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) )
= ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) ) ) ).
% or_not_mask_nop
thf(fact_4350_mask__or__not__mask,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,N: nat] :
( ( bit_se1065995026697491101ons_or @ ( word @ A ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) ) )
= X ) ) ).
% mask_or_not_mask
thf(fact_4351_even__or__iff,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A,B3: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ A @ A3 @ B3 ) )
= ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
& ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B3 ) ) ) ) ).
% even_or_iff
thf(fact_4352_bit_Ocomplement__unique,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [A3: A,X: A,Y: A] :
( ( ( bit_se5824344872417868541ns_and @ A @ A3 @ X )
= ( zero_zero @ A ) )
=> ( ( ( bit_se1065995026697491101ons_or @ A @ A3 @ X )
= ( uminus_uminus @ A @ ( one_one @ A ) ) )
=> ( ( ( bit_se5824344872417868541ns_and @ A @ A3 @ Y )
= ( zero_zero @ A ) )
=> ( ( ( bit_se1065995026697491101ons_or @ A @ A3 @ Y )
= ( uminus_uminus @ A @ ( one_one @ A ) ) )
=> ( X = Y ) ) ) ) ) ) ).
% bit.complement_unique
thf(fact_4353_mask__subsume,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,M: nat,X: word @ A,Y: word @ A] :
( ( ord_less_eq @ nat @ N @ M )
=> ( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ ( bit_se1065995026697491101ons_or @ ( word @ A ) @ X @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ Y @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) ) @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ M ) ) )
= ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ M ) ) ) ) ) ) ).
% mask_subsume
thf(fact_4354_bit_Ocompl__unique,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X: A,Y: A] :
( ( ( bit_se5824344872417868541ns_and @ A @ X @ Y )
= ( zero_zero @ A ) )
=> ( ( ( bit_se1065995026697491101ons_or @ A @ X @ Y )
= ( uminus_uminus @ A @ ( one_one @ A ) ) )
=> ( ( bit_ri4277139882892585799ns_not @ A @ X )
= Y ) ) ) ) ).
% bit.compl_unique
thf(fact_4355_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_4356_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_4357_or__one__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A] :
( ( bit_se1065995026697491101ons_or @ A @ A3 @ ( one_one @ A ) )
= ( plus_plus @ A @ A3 @ ( zero_neq_one_of_bool @ A @ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) ) ) ) ) ).
% or_one_eq
thf(fact_4358_one__or__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A] :
( ( bit_se1065995026697491101ons_or @ A @ ( one_one @ A ) @ A3 )
= ( plus_plus @ A @ A3 @ ( zero_neq_one_of_bool @ A @ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) ) ) ) ) ).
% one_or_eq
thf(fact_4359_xor__nat__unfold,axiom,
( ( bit_se5824344971392196577ns_xor @ nat )
= ( ^ [M3: nat,N4: nat] :
( if @ nat
@ ( M3
= ( zero_zero @ nat ) )
@ N4
@ ( if @ nat
@ ( N4
= ( zero_zero @ nat ) )
@ M3
@ ( plus_plus @ nat @ ( modulo_modulo @ nat @ ( plus_plus @ nat @ ( modulo_modulo @ nat @ M3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ nat @ N4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( bit_se5824344971392196577ns_xor @ nat @ ( divide_divide @ nat @ M3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ nat @ N4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% xor_nat_unfold
thf(fact_4360_xor__nat__rec,axiom,
( ( bit_se5824344971392196577ns_xor @ nat )
= ( ^ [M3: nat,N4: nat] :
( plus_plus @ nat
@ ( zero_neq_one_of_bool @ nat
@ ( ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M3 ) )
!= ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N4 ) ) ) )
@ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( bit_se5824344971392196577ns_xor @ nat @ ( divide_divide @ nat @ M3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ nat @ N4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% xor_nat_rec
thf(fact_4361_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_4362_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_4363_word__ops__lsb,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] :
( ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( bit_se1065995026697491101ons_or @ ( word @ A ) @ X @ Y ) @ ( zero_zero @ nat ) )
= ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ X @ ( zero_zero @ nat ) )
| ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ Y @ ( zero_zero @ nat ) ) ) )
& ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X @ Y ) @ ( zero_zero @ nat ) )
= ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ X @ ( zero_zero @ nat ) )
& ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ Y @ ( zero_zero @ nat ) ) ) )
& ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( bit_se5824344971392196577ns_xor @ ( word @ A ) @ X @ Y ) @ ( zero_zero @ nat ) )
= ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ X @ ( zero_zero @ nat ) )
!= ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ Y @ ( zero_zero @ nat ) ) ) )
& ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ X ) @ ( zero_zero @ nat ) )
= ( ~ ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ X @ ( zero_zero @ nat ) ) ) ) ) ) ).
% word_ops_lsb
thf(fact_4364_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_4365_word__bitwise__1__simps_I7_J,axiom,
! [G3: $tType] :
( ( type_len @ G3 )
=> ! [B3: num] :
( ( bit_se1065995026697491101ons_or @ ( word @ G3 ) @ ( one_one @ ( word @ G3 ) ) @ ( uminus_uminus @ ( word @ G3 ) @ ( numeral_numeral @ ( word @ G3 ) @ B3 ) ) )
= ( ring_1_of_int @ ( word @ G3 ) @ ( bit_se1065995026697491101ons_or @ int @ ( one_one @ int ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ B3 ) ) ) ) ) ) ).
% word_bitwise_1_simps(7)
thf(fact_4366_word__bitwise__1__simps_I9_J,axiom,
! [I5: $tType] :
( ( type_len @ I5 )
=> ! [A3: num] :
( ( bit_se1065995026697491101ons_or @ ( word @ I5 ) @ ( uminus_uminus @ ( word @ I5 ) @ ( numeral_numeral @ ( word @ I5 ) @ A3 ) ) @ ( one_one @ ( word @ I5 ) ) )
= ( ring_1_of_int @ ( word @ I5 ) @ ( bit_se1065995026697491101ons_or @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ A3 ) ) @ ( one_one @ int ) ) ) ) ) ).
% word_bitwise_1_simps(9)
thf(fact_4367_or__nonnegative__int__iff,axiom,
! [K: int,L: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se1065995026697491101ons_or @ int @ K @ L ) )
= ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
& ( ord_less_eq @ int @ ( zero_zero @ int ) @ L ) ) ) ).
% or_nonnegative_int_iff
thf(fact_4368_or__negative__int__iff,axiom,
! [K: int,L: int] :
( ( ord_less @ int @ ( bit_se1065995026697491101ons_or @ int @ K @ L ) @ ( zero_zero @ int ) )
= ( ( ord_less @ int @ K @ ( zero_zero @ int ) )
| ( ord_less @ int @ L @ ( zero_zero @ int ) ) ) ) ).
% or_negative_int_iff
thf(fact_4369_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_4370_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_4371_or__nat__numerals_I4_J,axiom,
! [X: num] :
( ( bit_se1065995026697491101ons_or @ nat @ ( numeral_numeral @ nat @ ( bit1 @ X ) ) @ ( suc @ ( zero_zero @ nat ) ) )
= ( numeral_numeral @ nat @ ( bit1 @ X ) ) ) ).
% or_nat_numerals(4)
thf(fact_4372_or__nat__numerals_I2_J,axiom,
! [Y: num] :
( ( bit_se1065995026697491101ons_or @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( numeral_numeral @ nat @ ( bit1 @ Y ) ) )
= ( numeral_numeral @ nat @ ( bit1 @ Y ) ) ) ).
% or_nat_numerals(2)
thf(fact_4373_word__no__log__defs_I7_J,axiom,
! [G3: $tType] :
( ( type_len @ G3 )
=> ! [A3: num,B3: num] :
( ( bit_se1065995026697491101ons_or @ ( word @ G3 ) @ ( numeral_numeral @ ( word @ G3 ) @ A3 ) @ ( numeral_numeral @ ( word @ G3 ) @ B3 ) )
= ( ring_1_of_int @ ( word @ G3 ) @ ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ A3 ) @ ( numeral_numeral @ int @ B3 ) ) ) ) ) ).
% word_no_log_defs(7)
thf(fact_4374_or__nat__numerals_I1_J,axiom,
! [Y: num] :
( ( bit_se1065995026697491101ons_or @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( numeral_numeral @ nat @ ( bit0 @ Y ) ) )
= ( numeral_numeral @ nat @ ( bit1 @ Y ) ) ) ).
% or_nat_numerals(1)
thf(fact_4375_or__nat__numerals_I3_J,axiom,
! [X: num] :
( ( bit_se1065995026697491101ons_or @ nat @ ( numeral_numeral @ nat @ ( bit0 @ X ) ) @ ( suc @ ( zero_zero @ nat ) ) )
= ( numeral_numeral @ nat @ ( bit1 @ X ) ) ) ).
% or_nat_numerals(3)
thf(fact_4376_word__bitwise__1__simps_I8_J,axiom,
! [H3: $tType] :
( ( type_len @ H3 )
=> ! [A3: num] :
( ( bit_se1065995026697491101ons_or @ ( word @ H3 ) @ ( numeral_numeral @ ( word @ H3 ) @ A3 ) @ ( one_one @ ( word @ H3 ) ) )
= ( ring_1_of_int @ ( word @ H3 ) @ ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ A3 ) @ ( one_one @ int ) ) ) ) ) ).
% word_bitwise_1_simps(8)
thf(fact_4377_word__bitwise__1__simps_I6_J,axiom,
! [F: $tType] :
( ( type_len @ F )
=> ! [B3: num] :
( ( bit_se1065995026697491101ons_or @ ( word @ F ) @ ( one_one @ ( word @ F ) ) @ ( numeral_numeral @ ( word @ F ) @ B3 ) )
= ( ring_1_of_int @ ( word @ F ) @ ( bit_se1065995026697491101ons_or @ int @ ( one_one @ int ) @ ( numeral_numeral @ int @ B3 ) ) ) ) ) ).
% word_bitwise_1_simps(6)
thf(fact_4378_word__no__log__defs_I8_J,axiom,
! [H3: $tType] :
( ( type_len @ H3 )
=> ! [A3: num,B3: num] :
( ( bit_se1065995026697491101ons_or @ ( word @ H3 ) @ ( numeral_numeral @ ( word @ H3 ) @ A3 ) @ ( uminus_uminus @ ( word @ H3 ) @ ( numeral_numeral @ ( word @ H3 ) @ B3 ) ) )
= ( ring_1_of_int @ ( word @ H3 ) @ ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ A3 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ B3 ) ) ) ) ) ) ).
% word_no_log_defs(8)
thf(fact_4379_word__no__log__defs_I9_J,axiom,
! [I5: $tType] :
( ( type_len @ I5 )
=> ! [A3: num,B3: num] :
( ( bit_se1065995026697491101ons_or @ ( word @ I5 ) @ ( uminus_uminus @ ( word @ I5 ) @ ( numeral_numeral @ ( word @ I5 ) @ A3 ) ) @ ( numeral_numeral @ ( word @ I5 ) @ B3 ) )
= ( ring_1_of_int @ ( word @ I5 ) @ ( bit_se1065995026697491101ons_or @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ A3 ) ) @ ( numeral_numeral @ int @ B3 ) ) ) ) ) ).
% word_no_log_defs(9)
thf(fact_4380_word__no__log__defs_I10_J,axiom,
! [J4: $tType] :
( ( type_len @ J4 )
=> ! [A3: num,B3: num] :
( ( bit_se1065995026697491101ons_or @ ( word @ J4 ) @ ( uminus_uminus @ ( word @ J4 ) @ ( numeral_numeral @ ( word @ J4 ) @ A3 ) ) @ ( uminus_uminus @ ( word @ J4 ) @ ( numeral_numeral @ ( word @ J4 ) @ B3 ) ) )
= ( ring_1_of_int @ ( word @ J4 ) @ ( bit_se1065995026697491101ons_or @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ A3 ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ B3 ) ) ) ) ) ) ).
% word_no_log_defs(10)
thf(fact_4381_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_4382_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_4383_int__or__code_I2_J,axiom,
! [I: int] :
( ( bit_se1065995026697491101ons_or @ int @ I @ ( zero_zero @ int ) )
= I ) ).
% int_or_code(2)
thf(fact_4384_int__or__code_I1_J,axiom,
! [J: int] :
( ( bit_se1065995026697491101ons_or @ int @ ( zero_zero @ int ) @ J )
= J ) ).
% int_or_code(1)
thf(fact_4385_or__nat__def,axiom,
( ( bit_se1065995026697491101ons_or @ nat )
= ( ^ [M3: nat,N4: nat] : ( nat2 @ ( bit_se1065995026697491101ons_or @ int @ ( semiring_1_of_nat @ int @ M3 ) @ ( semiring_1_of_nat @ int @ N4 ) ) ) ) ) ).
% or_nat_def
thf(fact_4386_abstract__boolean__algebra__sym__diff_Oxor__def,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One: A,Xor: A > A > A,X: A,Y: A] :
( ( boolea3799213064322606851m_diff @ A @ Conj @ Disj @ Compl @ Zero @ One @ Xor )
=> ( ( Xor @ X @ Y )
= ( Disj @ ( Conj @ X @ ( Compl @ Y ) ) @ ( Conj @ ( Compl @ X ) @ Y ) ) ) ) ).
% abstract_boolean_algebra_sym_diff.xor_def
thf(fact_4387_abstract__boolean__algebra__sym__diff_Oxor__def2,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One: A,Xor: A > A > A,X: A,Y: A] :
( ( boolea3799213064322606851m_diff @ A @ Conj @ Disj @ Compl @ Zero @ One @ Xor )
=> ( ( Xor @ X @ Y )
= ( Conj @ ( Disj @ X @ Y ) @ ( Disj @ ( Compl @ X ) @ ( Compl @ Y ) ) ) ) ) ).
% abstract_boolean_algebra_sym_diff.xor_def2
thf(fact_4388_abstract__boolean__algebra__sym__diff_Oxor__self,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One: A,Xor: A > A > A,X: A] :
( ( boolea3799213064322606851m_diff @ A @ Conj @ Disj @ Compl @ Zero @ One @ Xor )
=> ( ( Xor @ X @ X )
= Zero ) ) ).
% abstract_boolean_algebra_sym_diff.xor_self
thf(fact_4389_abstract__boolean__algebra__sym__diff_Oxor__one__left,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One: A,Xor: A > A > A,X: A] :
( ( boolea3799213064322606851m_diff @ A @ Conj @ Disj @ Compl @ Zero @ One @ Xor )
=> ( ( Xor @ One @ X )
= ( Compl @ X ) ) ) ).
% abstract_boolean_algebra_sym_diff.xor_one_left
thf(fact_4390_abstract__boolean__algebra__sym__diff_Oxor__left__self,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One: A,Xor: A > A > A,X: A,Y: A] :
( ( boolea3799213064322606851m_diff @ A @ Conj @ Disj @ Compl @ Zero @ One @ Xor )
=> ( ( Xor @ X @ ( Xor @ X @ Y ) )
= Y ) ) ).
% abstract_boolean_algebra_sym_diff.xor_left_self
thf(fact_4391_abstract__boolean__algebra__sym__diff_Oxor__one__right,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One: A,Xor: A > A > A,X: A] :
( ( boolea3799213064322606851m_diff @ A @ Conj @ Disj @ Compl @ Zero @ One @ Xor )
=> ( ( Xor @ X @ One )
= ( Compl @ X ) ) ) ).
% abstract_boolean_algebra_sym_diff.xor_one_right
thf(fact_4392_abstract__boolean__algebra__sym__diff_Oxor__compl__left,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One: A,Xor: A > A > A,X: A,Y: A] :
( ( boolea3799213064322606851m_diff @ A @ Conj @ Disj @ Compl @ Zero @ One @ Xor )
=> ( ( Xor @ ( Compl @ X ) @ Y )
= ( Compl @ ( Xor @ X @ Y ) ) ) ) ).
% abstract_boolean_algebra_sym_diff.xor_compl_left
thf(fact_4393_abstract__boolean__algebra__sym__diff_Oxor__cancel__left,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One: A,Xor: A > A > A,X: A] :
( ( boolea3799213064322606851m_diff @ A @ Conj @ Disj @ Compl @ Zero @ One @ Xor )
=> ( ( Xor @ ( Compl @ X ) @ X )
= One ) ) ).
% abstract_boolean_algebra_sym_diff.xor_cancel_left
thf(fact_4394_abstract__boolean__algebra__sym__diff_Oxor__compl__right,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One: A,Xor: A > A > A,X: A,Y: A] :
( ( boolea3799213064322606851m_diff @ A @ Conj @ Disj @ Compl @ Zero @ One @ Xor )
=> ( ( Xor @ X @ ( Compl @ Y ) )
= ( Compl @ ( Xor @ X @ Y ) ) ) ) ).
% abstract_boolean_algebra_sym_diff.xor_compl_right
thf(fact_4395_abstract__boolean__algebra__sym__diff_Oconj__xor__distrib,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One: A,Xor: A > A > A,X: A,Y: A,Z: A] :
( ( boolea3799213064322606851m_diff @ A @ Conj @ Disj @ Compl @ Zero @ One @ Xor )
=> ( ( Conj @ X @ ( Xor @ Y @ Z ) )
= ( Xor @ ( Conj @ X @ Y ) @ ( Conj @ X @ Z ) ) ) ) ).
% abstract_boolean_algebra_sym_diff.conj_xor_distrib
thf(fact_4396_abstract__boolean__algebra__sym__diff_Oxor__cancel__right,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One: A,Xor: A > A > A,X: A] :
( ( boolea3799213064322606851m_diff @ A @ Conj @ Disj @ Compl @ Zero @ One @ Xor )
=> ( ( Xor @ X @ ( Compl @ X ) )
= One ) ) ).
% abstract_boolean_algebra_sym_diff.xor_cancel_right
thf(fact_4397_abstract__boolean__algebra__sym__diff_Oconj__xor__distrib2,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One: A,Xor: A > A > A,Y: A,Z: A,X: A] :
( ( boolea3799213064322606851m_diff @ A @ Conj @ Disj @ Compl @ Zero @ One @ Xor )
=> ( ( Conj @ ( Xor @ Y @ Z ) @ X )
= ( Xor @ ( Conj @ Y @ X ) @ ( Conj @ Z @ X ) ) ) ) ).
% abstract_boolean_algebra_sym_diff.conj_xor_distrib2
thf(fact_4398_or__greater__eq,axiom,
! [L: int,K: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ L )
=> ( ord_less_eq @ int @ K @ ( bit_se1065995026697491101ons_or @ int @ K @ L ) ) ) ).
% or_greater_eq
thf(fact_4399_OR__lower,axiom,
! [X: int,Y: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y )
=> ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se1065995026697491101ons_or @ int @ X @ Y ) ) ) ) ).
% OR_lower
thf(fact_4400_uint__or,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] :
( ( semiring_1_unsigned @ A @ int @ ( bit_se1065995026697491101ons_or @ ( word @ A ) @ X @ Y ) )
= ( bit_se1065995026697491101ons_or @ int @ ( semiring_1_unsigned @ A @ int @ X ) @ ( semiring_1_unsigned @ A @ int @ Y ) ) ) ) ).
% uint_or
thf(fact_4401_word__wi__log__defs_I3_J,axiom,
! [C: $tType] :
( ( type_len @ C )
=> ! [A3: int,B3: int] :
( ( bit_se1065995026697491101ons_or @ ( word @ C ) @ ( ring_1_of_int @ ( word @ C ) @ A3 ) @ ( ring_1_of_int @ ( word @ C ) @ B3 ) )
= ( ring_1_of_int @ ( word @ C ) @ ( bit_se1065995026697491101ons_or @ int @ A3 @ B3 ) ) ) ) ).
% word_wi_log_defs(3)
thf(fact_4402_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_4403_word__or__def,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( bit_se1065995026697491101ons_or @ ( word @ A ) )
= ( ^ [A5: word @ A,B5: word @ A] : ( ring_1_of_int @ ( word @ A ) @ ( bit_se1065995026697491101ons_or @ int @ ( semiring_1_unsigned @ A @ int @ A5 ) @ ( semiring_1_unsigned @ A @ int @ B5 ) ) ) ) ) ) ).
% word_or_def
thf(fact_4404_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_4405_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_4406_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_4407_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_4408_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_4409_OR__upper,axiom,
! [X: int,N: nat,Y: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
=> ( ( ord_less @ int @ X @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) )
=> ( ( ord_less @ int @ Y @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) )
=> ( ord_less @ int @ ( bit_se1065995026697491101ons_or @ int @ X @ Y ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ) ) ) ).
% OR_upper
thf(fact_4410_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_4411_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_4412_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_4413_or__nat__rec,axiom,
( ( bit_se1065995026697491101ons_or @ nat )
= ( ^ [M3: nat,N4: nat] :
( plus_plus @ nat
@ ( zero_neq_one_of_bool @ nat
@ ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M3 )
| ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N4 ) ) )
@ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ nat @ ( divide_divide @ nat @ M3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ nat @ N4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% or_nat_rec
thf(fact_4414_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_4415_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_4416_or__int__rec,axiom,
( ( bit_se1065995026697491101ons_or @ int )
= ( ^ [K3: int,L2: int] :
( plus_plus @ int
@ ( zero_neq_one_of_bool @ int
@ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K3 )
| ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ L2 ) ) )
@ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ int @ ( divide_divide @ int @ K3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% or_int_rec
thf(fact_4417_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_4418_lsb__odd,axiom,
! [A: $tType] :
( ( least_6119777620449941438nt_lsb @ A )
=> ( ( least_8051144512741203767sb_lsb @ A )
= ( ^ [A5: A] :
~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A5 ) ) ) ) ).
% lsb_odd
thf(fact_4419_or__int__unfold,axiom,
( ( bit_se1065995026697491101ons_or @ int )
= ( ^ [K3: int,L2: int] :
( if @ int
@ ( ( K3
= ( uminus_uminus @ int @ ( one_one @ int ) ) )
| ( L2
= ( uminus_uminus @ int @ ( one_one @ int ) ) ) )
@ ( uminus_uminus @ int @ ( one_one @ int ) )
@ ( if @ int
@ ( K3
= ( zero_zero @ int ) )
@ L2
@ ( if @ int
@ ( L2
= ( zero_zero @ int ) )
@ K3
@ ( plus_plus @ int @ ( ord_max @ int @ ( modulo_modulo @ int @ K3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ int @ L2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ int @ ( divide_divide @ int @ K3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ).
% or_int_unfold
thf(fact_4420_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_4421_int__lsb__numeral_I1_J,axiom,
~ ( least_8051144512741203767sb_lsb @ int @ ( zero_zero @ int ) ) ).
% int_lsb_numeral(1)
thf(fact_4422_int__lsb__numeral_I2_J,axiom,
least_8051144512741203767sb_lsb @ int @ ( one_one @ int ) ).
% int_lsb_numeral(2)
thf(fact_4423_max_Oright__idem,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B3: A] :
( ( ord_max @ A @ ( ord_max @ A @ A3 @ B3 ) @ B3 )
= ( ord_max @ A @ A3 @ B3 ) ) ) ).
% max.right_idem
thf(fact_4424_max_Oleft__idem,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B3: A] :
( ( ord_max @ A @ A3 @ ( ord_max @ A @ A3 @ B3 ) )
= ( ord_max @ A @ A3 @ B3 ) ) ) ).
% max.left_idem
thf(fact_4425_max_Oidem,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A] :
( ( ord_max @ A @ A3 @ A3 )
= A3 ) ) ).
% max.idem
thf(fact_4426_int__lsb__numeral_I6_J,axiom,
! [W: num] :
~ ( least_8051144512741203767sb_lsb @ int @ ( numeral_numeral @ int @ ( bit0 @ W ) ) ) ).
% int_lsb_numeral(6)
thf(fact_4427_int__lsb__numeral_I3_J,axiom,
least_8051144512741203767sb_lsb @ int @ ( numeral_numeral @ int @ one2 ) ).
% int_lsb_numeral(3)
thf(fact_4428_int__lsb__numeral_I7_J,axiom,
! [W: num] : ( least_8051144512741203767sb_lsb @ int @ ( numeral_numeral @ int @ ( bit1 @ W ) ) ) ).
% int_lsb_numeral(7)
thf(fact_4429_max_Obounded__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B3: A,C2: A,A3: A] :
( ( ord_less_eq @ A @ ( ord_max @ A @ B3 @ C2 ) @ A3 )
= ( ( ord_less_eq @ A @ B3 @ A3 )
& ( ord_less_eq @ A @ C2 @ A3 ) ) ) ) ).
% max.bounded_iff
thf(fact_4430_max_Oabsorb2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ord_max @ A @ A3 @ B3 )
= B3 ) ) ) ).
% max.absorb2
thf(fact_4431_max_Oabsorb1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B3: A,A3: A] :
( ( ord_less_eq @ A @ B3 @ A3 )
=> ( ( ord_max @ A @ A3 @ B3 )
= A3 ) ) ) ).
% max.absorb1
thf(fact_4432_max__less__iff__conj,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A,Z: A] :
( ( ord_less @ A @ ( ord_max @ A @ X @ Y ) @ Z )
= ( ( ord_less @ A @ X @ Z )
& ( ord_less @ A @ Y @ Z ) ) ) ) ).
% max_less_iff_conj
thf(fact_4433_max_Oabsorb4,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ A3 @ B3 )
=> ( ( ord_max @ A @ A3 @ B3 )
= B3 ) ) ) ).
% max.absorb4
thf(fact_4434_max_Oabsorb3,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B3: A,A3: A] :
( ( ord_less @ A @ B3 @ A3 )
=> ( ( ord_max @ A @ A3 @ B3 )
= A3 ) ) ) ).
% max.absorb3
thf(fact_4435_max__bot,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [X: A] :
( ( ord_max @ A @ ( bot_bot @ A ) @ X )
= X ) ) ).
% max_bot
thf(fact_4436_max__bot2,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [X: A] :
( ( ord_max @ A @ X @ ( bot_bot @ A ) )
= X ) ) ).
% max_bot2
thf(fact_4437_int__lsb__numeral_I4_J,axiom,
least_8051144512741203767sb_lsb @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) ).
% int_lsb_numeral(4)
thf(fact_4438_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_4439_max__0__1_I4_J,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [X: num] :
( ( ord_max @ A @ ( numeral_numeral @ A @ X ) @ ( zero_zero @ A ) )
= ( numeral_numeral @ A @ X ) ) ) ).
% max_0_1(4)
thf(fact_4440_max__0__1_I3_J,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [X: num] :
( ( ord_max @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ X ) )
= ( numeral_numeral @ A @ X ) ) ) ).
% max_0_1(3)
thf(fact_4441_max__number__of_I1_J,axiom,
! [A: $tType] :
( ( ( numeral @ A )
& ( ord @ A ) )
=> ! [U2: num,V2: num] :
( ( ( ord_less_eq @ A @ ( numeral_numeral @ A @ U2 ) @ ( numeral_numeral @ A @ V2 ) )
=> ( ( ord_max @ A @ ( numeral_numeral @ A @ U2 ) @ ( numeral_numeral @ A @ V2 ) )
= ( numeral_numeral @ A @ V2 ) ) )
& ( ~ ( ord_less_eq @ A @ ( numeral_numeral @ A @ U2 ) @ ( numeral_numeral @ A @ V2 ) )
=> ( ( ord_max @ A @ ( numeral_numeral @ A @ U2 ) @ ( numeral_numeral @ A @ V2 ) )
= ( numeral_numeral @ A @ U2 ) ) ) ) ) ).
% max_number_of(1)
thf(fact_4442_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_4443_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_4444_max__0__1_I6_J,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [X: num] :
( ( ord_max @ A @ ( numeral_numeral @ A @ X ) @ ( one_one @ A ) )
= ( numeral_numeral @ A @ X ) ) ) ).
% max_0_1(6)
thf(fact_4445_max__0__1_I5_J,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [X: num] :
( ( ord_max @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ X ) )
= ( numeral_numeral @ A @ X ) ) ) ).
% max_0_1(5)
thf(fact_4446_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_4447_int__lsb__numeral_I5_J,axiom,
least_8051144512741203767sb_lsb @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ one2 ) ) ).
% int_lsb_numeral(5)
thf(fact_4448_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_4449_max__number__of_I4_J,axiom,
! [A: $tType] :
( ( ( uminus @ A )
& ( numeral @ A )
& ( ord @ A ) )
=> ! [U2: num,V2: num] :
( ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U2 ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) )
=> ( ( ord_max @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U2 ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U2 ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) )
=> ( ( ord_max @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U2 ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ U2 ) ) ) ) ) ) ).
% max_number_of(4)
thf(fact_4450_max__number__of_I3_J,axiom,
! [A: $tType] :
( ( ( uminus @ A )
& ( numeral @ A )
& ( ord @ A ) )
=> ! [U2: num,V2: num] :
( ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U2 ) ) @ ( numeral_numeral @ A @ V2 ) )
=> ( ( ord_max @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U2 ) ) @ ( numeral_numeral @ A @ V2 ) )
= ( numeral_numeral @ A @ V2 ) ) )
& ( ~ ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U2 ) ) @ ( numeral_numeral @ A @ V2 ) )
=> ( ( ord_max @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U2 ) ) @ ( numeral_numeral @ A @ V2 ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ U2 ) ) ) ) ) ) ).
% max_number_of(3)
thf(fact_4451_max__number__of_I2_J,axiom,
! [A: $tType] :
( ( ( uminus @ A )
& ( numeral @ A )
& ( ord @ A ) )
=> ! [U2: num,V2: num] :
( ( ( ord_less_eq @ A @ ( numeral_numeral @ A @ U2 ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) )
=> ( ( ord_max @ A @ ( numeral_numeral @ A @ U2 ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( numeral_numeral @ A @ U2 ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) )
=> ( ( ord_max @ A @ ( numeral_numeral @ A @ U2 ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) )
= ( numeral_numeral @ A @ U2 ) ) ) ) ) ).
% max_number_of(2)
thf(fact_4452_max__def__raw,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( ord_max @ A )
= ( ^ [A5: A,B5: A] : ( if @ A @ ( ord_less_eq @ A @ A5 @ B5 ) @ B5 @ A5 ) ) ) ) ).
% max_def_raw
thf(fact_4453_of__int__max,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: int,Y: int] :
( ( ring_1_of_int @ A @ ( ord_max @ int @ X @ Y ) )
= ( ord_max @ A @ ( ring_1_of_int @ A @ X ) @ ( ring_1_of_int @ A @ Y ) ) ) ) ).
% of_int_max
thf(fact_4454_max_Ostrict__coboundedI2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [C2: A,B3: A,A3: A] :
( ( ord_less @ A @ C2 @ B3 )
=> ( ord_less @ A @ C2 @ ( ord_max @ A @ A3 @ B3 ) ) ) ) ).
% max.strict_coboundedI2
thf(fact_4455_max_Ostrict__coboundedI1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [C2: A,A3: A,B3: A] :
( ( ord_less @ A @ C2 @ A3 )
=> ( ord_less @ A @ C2 @ ( ord_max @ A @ A3 @ B3 ) ) ) ) ).
% max.strict_coboundedI1
thf(fact_4456_max_Ostrict__order__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less @ A )
= ( ^ [B5: A,A5: A] :
( ( A5
= ( ord_max @ A @ A5 @ B5 ) )
& ( A5 != B5 ) ) ) ) ) ).
% max.strict_order_iff
thf(fact_4457_max_Ostrict__boundedE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B3: A,C2: A,A3: A] :
( ( ord_less @ A @ ( ord_max @ A @ B3 @ C2 ) @ A3 )
=> ~ ( ( ord_less @ A @ B3 @ A3 )
=> ~ ( ord_less @ A @ C2 @ A3 ) ) ) ) ).
% max.strict_boundedE
thf(fact_4458_less__max__iff__disj,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Z: A,X: A,Y: A] :
( ( ord_less @ A @ Z @ ( ord_max @ A @ X @ Y ) )
= ( ( ord_less @ A @ Z @ X )
| ( ord_less @ A @ Z @ Y ) ) ) ) ).
% less_max_iff_disj
thf(fact_4459_max_OcoboundedI2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [C2: A,B3: A,A3: A] :
( ( ord_less_eq @ A @ C2 @ B3 )
=> ( ord_less_eq @ A @ C2 @ ( ord_max @ A @ A3 @ B3 ) ) ) ) ).
% max.coboundedI2
thf(fact_4460_max_OcoboundedI1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [C2: A,A3: A,B3: A] :
( ( ord_less_eq @ A @ C2 @ A3 )
=> ( ord_less_eq @ A @ C2 @ ( ord_max @ A @ A3 @ B3 ) ) ) ) ).
% max.coboundedI1
thf(fact_4461_max_Oabsorb__iff2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A5: A,B5: A] :
( ( ord_max @ A @ A5 @ B5 )
= B5 ) ) ) ) ).
% max.absorb_iff2
thf(fact_4462_max_Oabsorb__iff1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B5: A,A5: A] :
( ( ord_max @ A @ A5 @ B5 )
= A5 ) ) ) ) ).
% max.absorb_iff1
thf(fact_4463_le__max__iff__disj,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Z: A,X: A,Y: A] :
( ( ord_less_eq @ A @ Z @ ( ord_max @ A @ X @ Y ) )
= ( ( ord_less_eq @ A @ Z @ X )
| ( ord_less_eq @ A @ Z @ Y ) ) ) ) ).
% le_max_iff_disj
thf(fact_4464_max_Ocobounded2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B3: A,A3: A] : ( ord_less_eq @ A @ B3 @ ( ord_max @ A @ A3 @ B3 ) ) ) ).
% max.cobounded2
thf(fact_4465_max_Ocobounded1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B3: A] : ( ord_less_eq @ A @ A3 @ ( ord_max @ A @ A3 @ B3 ) ) ) ).
% max.cobounded1
thf(fact_4466_max_Oorder__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B5: A,A5: A] :
( A5
= ( ord_max @ A @ A5 @ B5 ) ) ) ) ) ).
% max.order_iff
thf(fact_4467_max_OboundedI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B3: A,A3: A,C2: A] :
( ( ord_less_eq @ A @ B3 @ A3 )
=> ( ( ord_less_eq @ A @ C2 @ A3 )
=> ( ord_less_eq @ A @ ( ord_max @ A @ B3 @ C2 ) @ A3 ) ) ) ) ).
% max.boundedI
thf(fact_4468_max_OboundedE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B3: A,C2: A,A3: A] :
( ( ord_less_eq @ A @ ( ord_max @ A @ B3 @ C2 ) @ A3 )
=> ~ ( ( ord_less_eq @ A @ B3 @ A3 )
=> ~ ( ord_less_eq @ A @ C2 @ A3 ) ) ) ) ).
% max.boundedE
thf(fact_4469_max_OorderI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B3: A] :
( ( A3
= ( ord_max @ A @ A3 @ B3 ) )
=> ( ord_less_eq @ A @ B3 @ A3 ) ) ) ).
% max.orderI
thf(fact_4470_max_OorderE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B3: A,A3: A] :
( ( ord_less_eq @ A @ B3 @ A3 )
=> ( A3
= ( ord_max @ A @ A3 @ B3 ) ) ) ) ).
% max.orderE
thf(fact_4471_max_Omono,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [C2: A,A3: A,D2: A,B3: A] :
( ( ord_less_eq @ A @ C2 @ A3 )
=> ( ( ord_less_eq @ A @ D2 @ B3 )
=> ( ord_less_eq @ A @ ( ord_max @ A @ C2 @ D2 ) @ ( ord_max @ A @ A3 @ B3 ) ) ) ) ) ).
% max.mono
thf(fact_4472_max__absorb2,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_max @ A @ X @ Y )
= Y ) ) ) ).
% max_absorb2
thf(fact_4473_max__absorb1,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ Y @ X )
=> ( ( ord_max @ A @ X @ Y )
= X ) ) ) ).
% max_absorb1
thf(fact_4474_max__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( ord_max @ A )
= ( ^ [A5: A,B5: A] : ( if @ A @ ( ord_less_eq @ A @ A5 @ B5 ) @ B5 @ A5 ) ) ) ) ).
% max_def
thf(fact_4475_sup__max,axiom,
! [A: $tType] :
( ( ( semilattice_sup @ A )
& ( linorder @ A ) )
=> ( ( sup_sup @ A )
= ( ord_max @ A ) ) ) ).
% sup_max
thf(fact_4476_sup__int__def,axiom,
( ( sup_sup @ int )
= ( ord_max @ int ) ) ).
% sup_int_def
thf(fact_4477_max_Oleft__commute,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B3: A,A3: A,C2: A] :
( ( ord_max @ A @ B3 @ ( ord_max @ A @ A3 @ C2 ) )
= ( ord_max @ A @ A3 @ ( ord_max @ A @ B3 @ C2 ) ) ) ) ).
% max.left_commute
thf(fact_4478_max_Ocommute,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_max @ A )
= ( ^ [A5: A,B5: A] : ( ord_max @ A @ B5 @ A5 ) ) ) ) ).
% max.commute
thf(fact_4479_max_Oassoc,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B3: A,C2: A] :
( ( ord_max @ A @ ( ord_max @ A @ A3 @ B3 ) @ C2 )
= ( ord_max @ A @ A3 @ ( ord_max @ A @ B3 @ C2 ) ) ) ) ).
% max.assoc
thf(fact_4480_of__nat__max,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [X: nat,Y: nat] :
( ( semiring_1_of_nat @ A @ ( ord_max @ nat @ X @ Y ) )
= ( ord_max @ A @ ( semiring_1_of_nat @ A @ X ) @ ( semiring_1_of_nat @ A @ Y ) ) ) ) ).
% of_nat_max
thf(fact_4481_max__diff__distrib__left,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [X: A,Y: A,Z: A] :
( ( minus_minus @ A @ ( ord_max @ A @ X @ Y ) @ Z )
= ( ord_max @ A @ ( minus_minus @ A @ X @ Z ) @ ( minus_minus @ A @ Y @ Z ) ) ) ) ).
% max_diff_distrib_left
thf(fact_4482_max__add__distrib__right,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [X: A,Y: A,Z: A] :
( ( plus_plus @ A @ X @ ( ord_max @ A @ Y @ Z ) )
= ( ord_max @ A @ ( plus_plus @ A @ X @ Y ) @ ( plus_plus @ A @ X @ Z ) ) ) ) ).
% max_add_distrib_right
thf(fact_4483_max__add__distrib__left,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [X: A,Y: A,Z: A] :
( ( plus_plus @ A @ ( ord_max @ A @ X @ Y ) @ Z )
= ( ord_max @ A @ ( plus_plus @ A @ X @ Z ) @ ( plus_plus @ A @ Y @ Z ) ) ) ) ).
% max_add_distrib_left
thf(fact_4484_lsb__int__def,axiom,
( ( least_8051144512741203767sb_lsb @ int )
= ( ^ [I4: int] : ( bit_se5641148757651400278ts_bit @ int @ I4 @ ( zero_zero @ nat ) ) ) ) ).
% lsb_int_def
thf(fact_4485_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_4486_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_4487_bin__last__conv__lsb,axiom,
( ( ^ [A5: int] :
~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ A5 ) )
= ( least_8051144512741203767sb_lsb @ int ) ) ).
% bin_last_conv_lsb
thf(fact_4488_lsb__word__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( least_8051144512741203767sb_lsb @ ( word @ A ) )
= ( ^ [A5: word @ A] :
~ ( dvd_dvd @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ A5 ) ) ) ) ).
% lsb_word_eq
thf(fact_4489_word__lsb__def,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( least_8051144512741203767sb_lsb @ ( word @ A ) )
= ( ^ [A5: word @ A] :
~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( semiring_1_unsigned @ A @ int @ A5 ) ) ) ) ) ).
% word_lsb_def
thf(fact_4490_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_4491_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_4492_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_4493_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_4494_powr__real__of__int,axiom,
! [X: real,N: int] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ N )
=> ( ( powr @ real @ X @ ( ring_1_of_int @ real @ N ) )
= ( power_power @ real @ X @ ( nat2 @ N ) ) ) )
& ( ~ ( ord_less_eq @ int @ ( zero_zero @ int ) @ N )
=> ( ( powr @ real @ X @ ( ring_1_of_int @ real @ N ) )
= ( inverse_inverse @ real @ ( power_power @ real @ X @ ( nat2 @ ( uminus_uminus @ int @ N ) ) ) ) ) ) ) ) ).
% powr_real_of_int
thf(fact_4495_sofl__test,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] :
( ( ( plus_plus @ int @ ( ring_1_signed @ A @ int @ X ) @ ( ring_1_signed @ A @ int @ Y ) )
= ( ring_1_signed @ A @ int @ ( plus_plus @ ( word @ A ) @ X @ Y ) ) )
= ( ( bit_se4197421643247451524op_bit @ ( word @ A ) @ ( minus_minus @ nat @ ( size_size @ ( word @ A ) @ X ) @ ( one_one @ nat ) ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ ( bit_se5824344971392196577ns_xor @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ X @ Y ) @ X ) @ ( bit_se5824344971392196577ns_xor @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ X @ Y ) @ Y ) ) )
= ( zero_zero @ ( word @ A ) ) ) ) ) ).
% sofl_test
thf(fact_4496_max__enat__simps_I2_J,axiom,
! [Q3: extended_enat] :
( ( ord_max @ extended_enat @ Q3 @ ( zero_zero @ extended_enat ) )
= Q3 ) ).
% max_enat_simps(2)
thf(fact_4497_max__enat__simps_I3_J,axiom,
! [Q3: extended_enat] :
( ( ord_max @ extended_enat @ ( zero_zero @ extended_enat ) @ Q3 )
= Q3 ) ).
% max_enat_simps(3)
thf(fact_4498_inverse__zero,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ( ( inverse_inverse @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% inverse_zero
thf(fact_4499_inverse__nonzero__iff__nonzero,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A] :
( ( ( inverse_inverse @ A @ A3 )
= ( zero_zero @ A ) )
= ( A3
= ( zero_zero @ A ) ) ) ) ).
% inverse_nonzero_iff_nonzero
thf(fact_4500_inverse__eq__1__iff,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [X: A] :
( ( ( inverse_inverse @ A @ X )
= ( one_one @ A ) )
= ( X
= ( one_one @ A ) ) ) ) ).
% inverse_eq_1_iff
thf(fact_4501_inverse__1,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ( ( inverse_inverse @ A @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% inverse_1
thf(fact_4502_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_4503_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_4504_max__0R,axiom,
! [N: nat] :
( ( ord_max @ nat @ N @ ( zero_zero @ nat ) )
= N ) ).
% max_0R
thf(fact_4505_max__0L,axiom,
! [N: nat] :
( ( ord_max @ nat @ ( zero_zero @ nat ) @ N )
= N ) ).
% max_0L
thf(fact_4506_max__nat_Oright__neutral,axiom,
! [A3: nat] :
( ( ord_max @ nat @ A3 @ ( zero_zero @ nat ) )
= A3 ) ).
% max_nat.right_neutral
thf(fact_4507_max__nat_Oneutr__eq__iff,axiom,
! [A3: nat,B3: nat] :
( ( ( zero_zero @ nat )
= ( ord_max @ nat @ A3 @ B3 ) )
= ( ( A3
= ( zero_zero @ nat ) )
& ( B3
= ( zero_zero @ nat ) ) ) ) ).
% max_nat.neutr_eq_iff
thf(fact_4508_max__nat_Oleft__neutral,axiom,
! [A3: nat] :
( ( ord_max @ nat @ ( zero_zero @ nat ) @ A3 )
= A3 ) ).
% max_nat.left_neutral
thf(fact_4509_max__nat_Oeq__neutr__iff,axiom,
! [A3: nat,B3: nat] :
( ( ( ord_max @ nat @ A3 @ B3 )
= ( zero_zero @ nat ) )
= ( ( A3
= ( zero_zero @ nat ) )
& ( B3
= ( zero_zero @ nat ) ) ) ) ).
% max_nat.eq_neutr_iff
thf(fact_4510_inverse__nonpositive__iff__nonpositive,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ ( inverse_inverse @ A @ A3 ) @ ( zero_zero @ A ) )
= ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% inverse_nonpositive_iff_nonpositive
thf(fact_4511_inverse__nonnegative__iff__nonnegative,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( inverse_inverse @ A @ A3 ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% inverse_nonnegative_iff_nonnegative
thf(fact_4512_inverse__less__iff__less,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B3 )
=> ( ( ord_less @ A @ ( inverse_inverse @ A @ A3 ) @ ( inverse_inverse @ A @ B3 ) )
= ( ord_less @ A @ B3 @ A3 ) ) ) ) ) ).
% inverse_less_iff_less
thf(fact_4513_inverse__less__iff__less__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ B3 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( inverse_inverse @ A @ A3 ) @ ( inverse_inverse @ A @ B3 ) )
= ( ord_less @ A @ B3 @ A3 ) ) ) ) ) ).
% inverse_less_iff_less_neg
thf(fact_4514_inverse__negative__iff__negative,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( inverse_inverse @ A @ A3 ) @ ( zero_zero @ A ) )
= ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% inverse_negative_iff_negative
thf(fact_4515_inverse__positive__iff__positive,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( inverse_inverse @ A @ A3 ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% inverse_positive_iff_positive
thf(fact_4516_drop__bit__of__bool,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,B3: $o] :
( ( bit_se4197421643247451524op_bit @ A @ N @ ( zero_neq_one_of_bool @ A @ B3 ) )
= ( zero_neq_one_of_bool @ A
@ ( ( N
= ( zero_zero @ nat ) )
& B3 ) ) ) ) ).
% drop_bit_of_bool
thf(fact_4517_norm__ii,axiom,
( ( real_V7770717601297561774m_norm @ complex @ imaginary_unit )
= ( one_one @ real ) ) ).
% norm_ii
thf(fact_4518_inverse__le__iff__le,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B3 )
=> ( ( ord_less_eq @ A @ ( inverse_inverse @ A @ A3 ) @ ( inverse_inverse @ A @ B3 ) )
= ( ord_less_eq @ A @ B3 @ A3 ) ) ) ) ) ).
% inverse_le_iff_le
thf(fact_4519_inverse__le__iff__le__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ B3 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( inverse_inverse @ A @ A3 ) @ ( inverse_inverse @ A @ B3 ) )
= ( ord_less_eq @ A @ B3 @ A3 ) ) ) ) ) ).
% inverse_le_iff_le_neg
thf(fact_4520_right__inverse,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ A3 @ ( inverse_inverse @ A @ A3 ) )
= ( one_one @ A ) ) ) ) ).
% right_inverse
thf(fact_4521_left__inverse,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ ( inverse_inverse @ A @ A3 ) @ A3 )
= ( one_one @ A ) ) ) ) ).
% left_inverse
thf(fact_4522_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_4523_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_4524_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_4525_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_4526_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_4527_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_4528_i__squared,axiom,
( ( times_times @ complex @ imaginary_unit @ imaginary_unit )
= ( uminus_uminus @ complex @ ( one_one @ complex ) ) ) ).
% i_squared
thf(fact_4529_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_4530_drop__bit__numeral__bit0,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [L: num,K: num] :
( ( bit_se4197421643247451524op_bit @ A @ ( numeral_numeral @ nat @ L ) @ ( numeral_numeral @ A @ ( bit0 @ K ) ) )
= ( bit_se4197421643247451524op_bit @ A @ ( pred_numeral @ L ) @ ( numeral_numeral @ A @ K ) ) ) ) ).
% drop_bit_numeral_bit0
thf(fact_4531_drop__bit__numeral__bit1,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [L: num,K: num] :
( ( bit_se4197421643247451524op_bit @ A @ ( numeral_numeral @ nat @ L ) @ ( numeral_numeral @ A @ ( bit1 @ K ) ) )
= ( bit_se4197421643247451524op_bit @ A @ ( pred_numeral @ L ) @ ( numeral_numeral @ A @ K ) ) ) ) ).
% drop_bit_numeral_bit1
thf(fact_4532_power2__i,axiom,
( ( power_power @ complex @ imaginary_unit @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( uminus_uminus @ complex @ ( one_one @ complex ) ) ) ).
% power2_i
thf(fact_4533_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_4534_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_4535_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_4536_nat__mult__max__right,axiom,
! [M: nat,N: nat,Q3: nat] :
( ( times_times @ nat @ M @ ( ord_max @ nat @ N @ Q3 ) )
= ( ord_max @ nat @ ( times_times @ nat @ M @ N ) @ ( times_times @ nat @ M @ Q3 ) ) ) ).
% nat_mult_max_right
thf(fact_4537_nat__mult__max__left,axiom,
! [M: nat,N: nat,Q3: nat] :
( ( times_times @ nat @ ( ord_max @ nat @ M @ N ) @ Q3 )
= ( ord_max @ nat @ ( times_times @ nat @ M @ Q3 ) @ ( times_times @ nat @ N @ Q3 ) ) ) ).
% nat_mult_max_left
thf(fact_4538_sup__nat__def,axiom,
( ( sup_sup @ nat )
= ( ord_max @ nat ) ) ).
% sup_nat_def
thf(fact_4539_nat__add__max__left,axiom,
! [M: nat,N: nat,Q3: nat] :
( ( plus_plus @ nat @ ( ord_max @ nat @ M @ N ) @ Q3 )
= ( ord_max @ nat @ ( plus_plus @ nat @ M @ Q3 ) @ ( plus_plus @ nat @ N @ Q3 ) ) ) ).
% nat_add_max_left
thf(fact_4540_nat__add__max__right,axiom,
! [M: nat,N: nat,Q3: nat] :
( ( plus_plus @ nat @ M @ ( ord_max @ nat @ N @ Q3 ) )
= ( ord_max @ nat @ ( plus_plus @ nat @ M @ N ) @ ( plus_plus @ nat @ M @ Q3 ) ) ) ).
% nat_add_max_right
thf(fact_4541_complex__i__not__one,axiom,
( imaginary_unit
!= ( one_one @ complex ) ) ).
% complex_i_not_one
thf(fact_4542_nonzero__norm__inverse,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [A3: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( real_V7770717601297561774m_norm @ A @ ( inverse_inverse @ A @ A3 ) )
= ( inverse_inverse @ real @ ( real_V7770717601297561774m_norm @ A @ A3 ) ) ) ) ) ).
% nonzero_norm_inverse
thf(fact_4543_nonzero__of__real__inverse,axiom,
! [A: $tType] :
( ( real_V5047593784448816457lgebra @ A )
=> ! [X: real] :
( ( X
!= ( zero_zero @ real ) )
=> ( ( real_Vector_of_real @ A @ ( inverse_inverse @ real @ X ) )
= ( inverse_inverse @ A @ ( real_Vector_of_real @ A @ X ) ) ) ) ) ).
% nonzero_of_real_inverse
thf(fact_4544_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_4545_inverse__zero__imp__zero,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A] :
( ( ( inverse_inverse @ A @ A3 )
= ( zero_zero @ A ) )
=> ( A3
= ( zero_zero @ A ) ) ) ) ).
% inverse_zero_imp_zero
thf(fact_4546_nonzero__inverse__eq__imp__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A,B3: A] :
( ( ( inverse_inverse @ A @ A3 )
= ( inverse_inverse @ A @ B3 ) )
=> ( ( A3
!= ( zero_zero @ A ) )
=> ( ( B3
!= ( zero_zero @ A ) )
=> ( A3 = B3 ) ) ) ) ) ).
% nonzero_inverse_eq_imp_eq
thf(fact_4547_nonzero__inverse__inverse__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( inverse_inverse @ A @ ( inverse_inverse @ A @ A3 ) )
= A3 ) ) ) ).
% nonzero_inverse_inverse_eq
thf(fact_4548_nonzero__imp__inverse__nonzero,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( inverse_inverse @ A @ A3 )
!= ( zero_zero @ A ) ) ) ) ).
% nonzero_imp_inverse_nonzero
thf(fact_4549_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_4550_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_4551_power__inverse,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A,N: nat] :
( ( power_power @ A @ ( inverse_inverse @ A @ A3 ) @ N )
= ( inverse_inverse @ A @ ( power_power @ A @ A3 @ N ) ) ) ) ).
% power_inverse
thf(fact_4552_real__sqrt__inverse,axiom,
! [X: real] :
( ( sqrt @ ( inverse_inverse @ real @ X ) )
= ( inverse_inverse @ real @ ( sqrt @ X ) ) ) ).
% real_sqrt_inverse
thf(fact_4553_complex__i__not__zero,axiom,
( imaginary_unit
!= ( zero_zero @ complex ) ) ).
% complex_i_not_zero
thf(fact_4554_norm__inverse__le__norm,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [R3: real,X: A] :
( ( ord_less_eq @ real @ R3 @ ( real_V7770717601297561774m_norm @ A @ X ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ R3 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( inverse_inverse @ A @ X ) ) @ ( inverse_inverse @ real @ R3 ) ) ) ) ) ).
% norm_inverse_le_norm
thf(fact_4555_inverse__less__imp__less,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ ( inverse_inverse @ A @ A3 ) @ ( inverse_inverse @ A @ B3 ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ord_less @ A @ B3 @ A3 ) ) ) ) ).
% inverse_less_imp_less
thf(fact_4556_less__imp__inverse__less,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ A3 @ B3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ord_less @ A @ ( inverse_inverse @ A @ B3 ) @ ( inverse_inverse @ A @ A3 ) ) ) ) ) ).
% less_imp_inverse_less
thf(fact_4557_inverse__less__imp__less__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ ( inverse_inverse @ A @ A3 ) @ ( inverse_inverse @ A @ B3 ) )
=> ( ( ord_less @ A @ B3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ B3 @ A3 ) ) ) ) ).
% inverse_less_imp_less_neg
thf(fact_4558_less__imp__inverse__less__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ A3 @ B3 )
=> ( ( ord_less @ A @ B3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( inverse_inverse @ A @ B3 ) @ ( inverse_inverse @ A @ A3 ) ) ) ) ) ).
% less_imp_inverse_less_neg
thf(fact_4559_inverse__negative__imp__negative,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( inverse_inverse @ A @ A3 ) @ ( zero_zero @ A ) )
=> ( ( A3
!= ( zero_zero @ A ) )
=> ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ) ).
% inverse_negative_imp_negative
thf(fact_4560_inverse__positive__imp__positive,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( inverse_inverse @ A @ A3 ) )
=> ( ( A3
!= ( zero_zero @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ A3 ) ) ) ) ).
% inverse_positive_imp_positive
thf(fact_4561_negative__imp__inverse__negative,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A] :
( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( inverse_inverse @ A @ A3 ) @ ( zero_zero @ A ) ) ) ) ).
% negative_imp_inverse_negative
thf(fact_4562_positive__imp__inverse__positive,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( inverse_inverse @ A @ A3 ) ) ) ) ).
% positive_imp_inverse_positive
thf(fact_4563_nonzero__inverse__mult__distrib,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A,B3: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( B3
!= ( zero_zero @ A ) )
=> ( ( inverse_inverse @ A @ ( times_times @ A @ A3 @ B3 ) )
= ( times_times @ A @ ( inverse_inverse @ A @ B3 ) @ ( inverse_inverse @ A @ A3 ) ) ) ) ) ) ).
% nonzero_inverse_mult_distrib
thf(fact_4564_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_4565_inverse__unique,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A,B3: A] :
( ( ( times_times @ A @ A3 @ B3 )
= ( one_one @ A ) )
=> ( ( inverse_inverse @ A @ A3 )
= B3 ) ) ) ).
% inverse_unique
thf(fact_4566_nonzero__inverse__minus__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( inverse_inverse @ A @ ( uminus_uminus @ A @ A3 ) )
= ( uminus_uminus @ A @ ( inverse_inverse @ A @ A3 ) ) ) ) ) ).
% nonzero_inverse_minus_eq
thf(fact_4567_inverse__eq__divide,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ( ( inverse_inverse @ A )
= ( divide_divide @ A @ ( one_one @ A ) ) ) ) ).
% inverse_eq_divide
thf(fact_4568_power__mult__power__inverse__commute,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X: A,M: nat,N: nat] :
( ( times_times @ A @ ( power_power @ A @ X @ M ) @ ( power_power @ A @ ( inverse_inverse @ A @ X ) @ N ) )
= ( times_times @ A @ ( power_power @ A @ ( inverse_inverse @ A @ X ) @ N ) @ ( power_power @ A @ X @ M ) ) ) ) ).
% power_mult_power_inverse_commute
thf(fact_4569_power__mult__inverse__distrib,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X: A,M: nat] :
( ( times_times @ A @ ( power_power @ A @ X @ M ) @ ( inverse_inverse @ A @ X ) )
= ( times_times @ A @ ( inverse_inverse @ A @ X ) @ ( power_power @ A @ X @ M ) ) ) ) ).
% power_mult_inverse_distrib
thf(fact_4570_mult__inverse__of__nat__commute,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Xa: nat,X: A] :
( ( times_times @ A @ ( inverse_inverse @ A @ ( semiring_1_of_nat @ A @ Xa ) ) @ X )
= ( times_times @ A @ X @ ( inverse_inverse @ A @ ( semiring_1_of_nat @ A @ Xa ) ) ) ) ) ).
% mult_inverse_of_nat_commute
thf(fact_4571_nonzero__abs__inverse,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( abs_abs @ A @ ( inverse_inverse @ A @ A3 ) )
= ( inverse_inverse @ A @ ( abs_abs @ A @ A3 ) ) ) ) ) ).
% nonzero_abs_inverse
thf(fact_4572_mult__inverse__of__int__commute,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Xa: int,X: A] :
( ( times_times @ A @ ( inverse_inverse @ A @ ( ring_1_of_int @ A @ Xa ) ) @ X )
= ( times_times @ A @ X @ ( inverse_inverse @ A @ ( ring_1_of_int @ A @ Xa ) ) ) ) ) ).
% mult_inverse_of_int_commute
thf(fact_4573_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_4574_divide__real__def,axiom,
( ( divide_divide @ real )
= ( ^ [X2: real,Y2: real] : ( times_times @ real @ X2 @ ( inverse_inverse @ real @ Y2 ) ) ) ) ).
% divide_real_def
thf(fact_4575_take__bit__eq__self__iff__drop__bit__eq__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A3: A] :
( ( ( bit_se2584673776208193580ke_bit @ A @ N @ A3 )
= A3 )
= ( ( bit_se4197421643247451524op_bit @ A @ N @ A3 )
= ( zero_zero @ A ) ) ) ) ).
% take_bit_eq_self_iff_drop_bit_eq_0
thf(fact_4576_drop__bit__mask__eq,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [M: nat,N: nat] :
( ( bit_se4197421643247451524op_bit @ A @ M @ ( bit_se2239418461657761734s_mask @ A @ N ) )
= ( bit_se2239418461657761734s_mask @ A @ ( minus_minus @ nat @ N @ M ) ) ) ) ).
% drop_bit_mask_eq
thf(fact_4577_exp__fdiffs,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ( ( diffs @ A
@ ^ [N4: nat] : ( inverse_inverse @ A @ ( semiring_char_0_fact @ A @ N4 ) ) )
= ( ^ [N4: nat] : ( inverse_inverse @ A @ ( semiring_char_0_fact @ A @ N4 ) ) ) ) ) ).
% exp_fdiffs
thf(fact_4578_inverse__le__imp__le,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ ( inverse_inverse @ A @ A3 ) @ ( inverse_inverse @ A @ B3 ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ord_less_eq @ A @ B3 @ A3 ) ) ) ) ).
% inverse_le_imp_le
thf(fact_4579_le__imp__inverse__le,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ord_less_eq @ A @ ( inverse_inverse @ A @ B3 ) @ ( inverse_inverse @ A @ A3 ) ) ) ) ) ).
% le_imp_inverse_le
thf(fact_4580_inverse__le__imp__le__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ ( inverse_inverse @ A @ A3 ) @ ( inverse_inverse @ A @ B3 ) )
=> ( ( ord_less @ A @ B3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ B3 @ A3 ) ) ) ) ).
% inverse_le_imp_le_neg
thf(fact_4581_le__imp__inverse__le__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ord_less @ A @ B3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( inverse_inverse @ A @ B3 ) @ ( inverse_inverse @ A @ A3 ) ) ) ) ) ).
% le_imp_inverse_le_neg
thf(fact_4582_inverse__le__1__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A] :
( ( ord_less_eq @ A @ ( inverse_inverse @ A @ X ) @ ( one_one @ A ) )
= ( ( ord_less_eq @ A @ X @ ( zero_zero @ A ) )
| ( ord_less_eq @ A @ ( one_one @ A ) @ X ) ) ) ) ).
% inverse_le_1_iff
thf(fact_4583_one__less__inverse,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ A3 @ ( one_one @ A ) )
=> ( ord_less @ A @ ( one_one @ A ) @ ( inverse_inverse @ A @ A3 ) ) ) ) ) ).
% one_less_inverse
thf(fact_4584_one__less__inverse__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A] :
( ( ord_less @ A @ ( one_one @ A ) @ ( inverse_inverse @ A @ X ) )
= ( ( ord_less @ A @ ( zero_zero @ A ) @ X )
& ( ord_less @ A @ X @ ( one_one @ A ) ) ) ) ) ).
% one_less_inverse_iff
thf(fact_4585_division__ring__inverse__add,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A,B3: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( B3
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( inverse_inverse @ A @ A3 ) @ ( inverse_inverse @ A @ B3 ) )
= ( times_times @ A @ ( times_times @ A @ ( inverse_inverse @ A @ A3 ) @ ( plus_plus @ A @ A3 @ B3 ) ) @ ( inverse_inverse @ A @ B3 ) ) ) ) ) ) ).
% division_ring_inverse_add
thf(fact_4586_inverse__add,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A3: A,B3: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( B3
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( inverse_inverse @ A @ A3 ) @ ( inverse_inverse @ A @ B3 ) )
= ( times_times @ A @ ( times_times @ A @ ( plus_plus @ A @ A3 @ B3 ) @ ( inverse_inverse @ A @ A3 ) ) @ ( inverse_inverse @ A @ B3 ) ) ) ) ) ) ).
% inverse_add
thf(fact_4587_field__class_Ofield__inverse,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A3: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ ( inverse_inverse @ A @ A3 ) @ A3 )
= ( one_one @ A ) ) ) ) ).
% field_class.field_inverse
thf(fact_4588_division__ring__inverse__diff,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A,B3: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( B3
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( inverse_inverse @ A @ A3 ) @ ( inverse_inverse @ A @ B3 ) )
= ( times_times @ A @ ( times_times @ A @ ( inverse_inverse @ A @ A3 ) @ ( minus_minus @ A @ B3 @ A3 ) ) @ ( inverse_inverse @ A @ B3 ) ) ) ) ) ) ).
% division_ring_inverse_diff
thf(fact_4589_nonzero__inverse__eq__divide,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( inverse_inverse @ A @ A3 )
= ( divide_divide @ A @ ( one_one @ A ) @ A3 ) ) ) ) ).
% nonzero_inverse_eq_divide
thf(fact_4590_max__lt,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A,B3: word @ A,C2: word @ A] :
( ( semiring_1_unsigned @ A @ nat @ ( divide_divide @ ( word @ A ) @ ( ord_max @ ( word @ A ) @ A3 @ B3 ) @ C2 ) )
= ( divide_divide @ nat @ ( semiring_1_unsigned @ A @ nat @ ( ord_max @ ( word @ A ) @ A3 @ B3 ) ) @ ( semiring_1_unsigned @ A @ nat @ C2 ) ) ) ) ).
% max_lt
thf(fact_4591_bit__word__iff__drop__bit__and,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) )
= ( ^ [A5: word @ A,N4: nat] :
( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ ( bit_se4197421643247451524op_bit @ ( word @ A ) @ N4 @ A5 ) @ ( one_one @ ( word @ A ) ) )
= ( one_one @ ( word @ A ) ) ) ) ) ) ).
% bit_word_iff_drop_bit_and
thf(fact_4592_inverse__powr,axiom,
! [Y: real,A3: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y )
=> ( ( powr @ real @ ( inverse_inverse @ real @ Y ) @ A3 )
= ( inverse_inverse @ real @ ( powr @ real @ Y @ A3 ) ) ) ) ).
% inverse_powr
thf(fact_4593_and__not__mask__twice,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A,N: nat,M: nat] :
( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ W @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) ) @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ M ) ) )
= ( bit_se5824344872417868541ns_and @ ( word @ A ) @ W @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ ( ord_max @ nat @ M @ N ) ) ) ) ) ) ).
% and_not_mask_twice
thf(fact_4594_mask__lower__twice2,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A,N: nat,M: nat] :
( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ A3 @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) ) @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ M ) ) )
= ( bit_se5824344872417868541ns_and @ ( word @ A ) @ A3 @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ ( ord_max @ nat @ N @ M ) ) ) ) ) ) ).
% mask_lower_twice2
thf(fact_4595_neg__mask__combine,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: nat,B3: nat] :
( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ A3 ) ) @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ B3 ) ) )
= ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ ( ord_max @ nat @ A3 @ B3 ) ) ) ) ) ).
% neg_mask_combine
thf(fact_4596_neg__mask__twice,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,N: nat,M: nat] :
( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ M ) ) ) )
= ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ ( ord_max @ nat @ N @ M ) ) ) ) ) ) ).
% neg_mask_twice
thf(fact_4597_bit__iff__and__drop__bit__eq__1,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se5641148757651400278ts_bit @ A )
= ( ^ [A5: A,N4: nat] :
( ( bit_se5824344872417868541ns_and @ A @ ( bit_se4197421643247451524op_bit @ A @ N4 @ A5 ) @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ) ) ).
% bit_iff_and_drop_bit_eq_1
thf(fact_4598_Complex__eq__i,axiom,
! [X: real,Y: real] :
( ( ( complex2 @ X @ Y )
= imaginary_unit )
= ( ( X
= ( zero_zero @ real ) )
& ( Y
= ( one_one @ real ) ) ) ) ).
% Complex_eq_i
thf(fact_4599_imaginary__unit_Ocode,axiom,
( imaginary_unit
= ( complex2 @ ( zero_zero @ real ) @ ( one_one @ real ) ) ) ).
% imaginary_unit.code
thf(fact_4600_one__le__inverse,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ A3 @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ ( inverse_inverse @ A @ A3 ) ) ) ) ) ).
% one_le_inverse
thf(fact_4601_inverse__less__1__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A] :
( ( ord_less @ A @ ( inverse_inverse @ A @ X ) @ ( one_one @ A ) )
= ( ( ord_less_eq @ A @ X @ ( zero_zero @ A ) )
| ( ord_less @ A @ ( one_one @ A ) @ X ) ) ) ) ).
% inverse_less_1_iff
thf(fact_4602_one__le__inverse__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A] :
( ( ord_less_eq @ A @ ( one_one @ A ) @ ( inverse_inverse @ A @ X ) )
= ( ( ord_less @ A @ ( zero_zero @ A ) @ X )
& ( ord_less_eq @ A @ X @ ( one_one @ A ) ) ) ) ) ).
% one_le_inverse_iff
thf(fact_4603_inverse__le__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ ( inverse_inverse @ A @ A3 ) @ ( inverse_inverse @ A @ B3 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ B3 ) )
=> ( ord_less_eq @ A @ B3 @ A3 ) )
& ( ( ord_less_eq @ A @ ( times_times @ A @ A3 @ B3 ) @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ A3 @ B3 ) ) ) ) ) ).
% inverse_le_iff
thf(fact_4604_inverse__less__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ ( inverse_inverse @ A @ A3 ) @ ( inverse_inverse @ A @ B3 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ B3 ) )
=> ( ord_less @ A @ B3 @ A3 ) )
& ( ( ord_less_eq @ A @ ( times_times @ A @ A3 @ B3 ) @ ( zero_zero @ A ) )
=> ( ord_less @ A @ A3 @ B3 ) ) ) ) ) ).
% inverse_less_iff
thf(fact_4605_inverse__diff__inverse,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A,B3: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( B3
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( inverse_inverse @ A @ A3 ) @ ( inverse_inverse @ A @ B3 ) )
= ( uminus_uminus @ A @ ( times_times @ A @ ( times_times @ A @ ( inverse_inverse @ A @ A3 ) @ ( minus_minus @ A @ A3 @ B3 ) ) @ ( inverse_inverse @ A @ B3 ) ) ) ) ) ) ) ).
% inverse_diff_inverse
thf(fact_4606_reals__Archimedean,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ X )
=> ? [N2: nat] : ( ord_less @ A @ ( inverse_inverse @ A @ ( semiring_1_of_nat @ A @ ( suc @ N2 ) ) ) @ X ) ) ) ).
% reals_Archimedean
thf(fact_4607_complex__of__real__i,axiom,
! [R3: real] :
( ( times_times @ complex @ ( real_Vector_of_real @ complex @ R3 ) @ imaginary_unit )
= ( complex2 @ ( zero_zero @ real ) @ R3 ) ) ).
% complex_of_real_i
thf(fact_4608_i__complex__of__real,axiom,
! [R3: real] :
( ( times_times @ complex @ imaginary_unit @ ( real_Vector_of_real @ complex @ R3 ) )
= ( complex2 @ ( zero_zero @ real ) @ R3 ) ) ).
% i_complex_of_real
thf(fact_4609_forall__pos__mono__1,axiom,
! [P: real > $o,E2: real] :
( ! [D6: real,E: real] :
( ( ord_less @ real @ D6 @ E )
=> ( ( P @ D6 )
=> ( P @ E ) ) )
=> ( ! [N2: nat] : ( P @ ( inverse_inverse @ real @ ( semiring_1_of_nat @ real @ ( suc @ N2 ) ) ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ E2 )
=> ( P @ E2 ) ) ) ) ).
% forall_pos_mono_1
thf(fact_4610_real__arch__inverse,axiom,
! [E2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E2 )
= ( ? [N4: nat] :
( ( N4
!= ( zero_zero @ nat ) )
& ( ord_less @ real @ ( zero_zero @ real ) @ ( inverse_inverse @ real @ ( semiring_1_of_nat @ real @ N4 ) ) )
& ( ord_less @ real @ ( inverse_inverse @ real @ ( semiring_1_of_nat @ real @ N4 ) ) @ E2 ) ) ) ) ).
% real_arch_inverse
thf(fact_4611_forall__pos__mono,axiom,
! [P: real > $o,E2: real] :
( ! [D6: real,E: real] :
( ( ord_less @ real @ D6 @ E )
=> ( ( P @ D6 )
=> ( P @ E ) ) )
=> ( ! [N2: nat] :
( ( N2
!= ( zero_zero @ nat ) )
=> ( P @ ( inverse_inverse @ real @ ( semiring_1_of_nat @ real @ N2 ) ) ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ E2 )
=> ( P @ E2 ) ) ) ) ).
% forall_pos_mono
thf(fact_4612_sqrt__divide__self__eq,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( divide_divide @ real @ ( sqrt @ X ) @ X )
= ( inverse_inverse @ real @ ( sqrt @ X ) ) ) ) ).
% sqrt_divide_self_eq
thf(fact_4613_drop__bit__half,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A3: A] :
( ( bit_se4197421643247451524op_bit @ A @ N @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
= ( divide_divide @ A @ ( bit_se4197421643247451524op_bit @ A @ N @ A3 ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% drop_bit_half
thf(fact_4614_stable__imp__drop__bit__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A,N: nat] :
( ( ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= A3 )
=> ( ( bit_se4197421643247451524op_bit @ A @ N @ A3 )
= A3 ) ) ) ).
% stable_imp_drop_bit_eq
thf(fact_4615_ln__inverse,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ln_ln @ real @ ( inverse_inverse @ real @ X ) )
= ( uminus_uminus @ real @ ( ln_ln @ real @ X ) ) ) ) ).
% ln_inverse
thf(fact_4616_summable__exp,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A] :
( summable @ A
@ ^ [N4: nat] : ( times_times @ A @ ( inverse_inverse @ A @ ( semiring_char_0_fact @ A @ N4 ) ) @ ( power_power @ A @ X @ N4 ) ) ) ) ).
% summable_exp
thf(fact_4617_ex__inverse__of__nat__less,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ X )
=> ? [N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
& ( ord_less @ A @ ( inverse_inverse @ A @ ( semiring_1_of_nat @ A @ N2 ) ) @ X ) ) ) ) ).
% ex_inverse_of_nat_less
thf(fact_4618_power__diff__conv__inverse,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X: A,M: nat,N: nat] :
( ( X
!= ( zero_zero @ A ) )
=> ( ( ord_less_eq @ nat @ M @ N )
=> ( ( power_power @ A @ X @ ( minus_minus @ nat @ N @ M ) )
= ( times_times @ A @ ( power_power @ A @ X @ N ) @ ( power_power @ A @ ( inverse_inverse @ A @ X ) @ M ) ) ) ) ) ) ).
% power_diff_conv_inverse
thf(fact_4619_log__inverse,axiom,
! [A3: real,X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ A3 )
=> ( ( A3
!= ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( log @ A3 @ ( inverse_inverse @ real @ X ) )
= ( uminus_uminus @ real @ ( log @ A3 @ X ) ) ) ) ) ) ).
% log_inverse
thf(fact_4620_drop__bit__Suc,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A3: A] :
( ( bit_se4197421643247451524op_bit @ A @ ( suc @ N ) @ A3 )
= ( bit_se4197421643247451524op_bit @ A @ N @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% drop_bit_Suc
thf(fact_4621_drop__bit__eq__div,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se4197421643247451524op_bit @ A )
= ( ^ [N4: nat,A5: A] : ( divide_divide @ A @ A5 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N4 ) ) ) ) ) ).
% drop_bit_eq_div
thf(fact_4622_bit__iff__odd__drop__bit,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se5641148757651400278ts_bit @ A )
= ( ^ [A5: A,N4: nat] :
~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se4197421643247451524op_bit @ A @ N4 @ A5 ) ) ) ) ) ).
% bit_iff_odd_drop_bit
thf(fact_4623_even__drop__bit__iff__not__bit,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A3: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se4197421643247451524op_bit @ A @ N @ A3 ) )
= ( ~ ( bit_se5641148757651400278ts_bit @ A @ A3 @ N ) ) ) ) ).
% even_drop_bit_iff_not_bit
thf(fact_4624_exp__plus__inverse__exp,axiom,
! [X: real] : ( ord_less_eq @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( plus_plus @ real @ ( exp @ real @ X ) @ ( inverse_inverse @ real @ ( exp @ real @ X ) ) ) ) ).
% exp_plus_inverse_exp
thf(fact_4625_bit__twiddle__max,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] :
( ( bit_se5824344971392196577ns_xor @ ( word @ A ) @ X @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ ( bit_se5824344971392196577ns_xor @ ( word @ A ) @ X @ Y ) @ ( if @ ( word @ A ) @ ( ord_less @ ( word @ A ) @ X @ Y ) @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) @ ( zero_zero @ ( word @ A ) ) ) ) )
= ( ord_max @ ( word @ A ) @ X @ Y ) ) ) ).
% bit_twiddle_max
thf(fact_4626_cmod__unit__one,axiom,
! [A3: real] :
( ( real_V7770717601297561774m_norm @ complex @ ( plus_plus @ complex @ ( real_Vector_of_real @ complex @ ( cos @ real @ A3 ) ) @ ( times_times @ complex @ imaginary_unit @ ( real_Vector_of_real @ complex @ ( sin @ real @ A3 ) ) ) ) )
= ( one_one @ real ) ) ).
% cmod_unit_one
thf(fact_4627_plus__inverse__ge__2,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ord_less_eq @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( plus_plus @ real @ X @ ( inverse_inverse @ real @ X ) ) ) ) ).
% plus_inverse_ge_2
thf(fact_4628_real__inv__sqrt__pow2,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( power_power @ real @ ( inverse_inverse @ real @ ( sqrt @ X ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( inverse_inverse @ real @ X ) ) ) ).
% real_inv_sqrt_pow2
thf(fact_4629_tan__cot,axiom,
! [X: real] :
( ( tan @ real @ ( minus_minus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ X ) )
= ( inverse_inverse @ real @ ( tan @ real @ X ) ) ) ).
% tan_cot
thf(fact_4630_drop__bit__rec,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se4197421643247451524op_bit @ A )
= ( ^ [N4: nat,A5: A] :
( if @ A
@ ( N4
= ( zero_zero @ nat ) )
@ A5
@ ( bit_se4197421643247451524op_bit @ A @ ( minus_minus @ nat @ N4 @ ( one_one @ nat ) ) @ ( divide_divide @ A @ A5 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% drop_bit_rec
thf(fact_4631_real__le__x__sinh,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ord_less_eq @ real @ X @ ( divide_divide @ real @ ( minus_minus @ real @ ( exp @ real @ X ) @ ( inverse_inverse @ real @ ( exp @ real @ X ) ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ).
% real_le_x_sinh
thf(fact_4632_real__le__abs__sinh,axiom,
! [X: real] : ( ord_less_eq @ real @ ( abs_abs @ real @ X ) @ ( abs_abs @ real @ ( divide_divide @ real @ ( minus_minus @ real @ ( exp @ real @ X ) @ ( inverse_inverse @ real @ ( exp @ real @ X ) ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ).
% real_le_abs_sinh
thf(fact_4633_tan__sec,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A] :
( ( ( cos @ A @ X )
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( one_one @ A ) @ ( power_power @ A @ ( tan @ A @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( power_power @ A @ ( inverse_inverse @ A @ ( cos @ A @ X ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% tan_sec
thf(fact_4634_or__nat__unfold,axiom,
( ( bit_se1065995026697491101ons_or @ nat )
= ( ^ [M3: nat,N4: nat] :
( if @ nat
@ ( M3
= ( zero_zero @ nat ) )
@ N4
@ ( if @ nat
@ ( N4
= ( zero_zero @ nat ) )
@ M3
@ ( plus_plus @ nat @ ( ord_max @ nat @ ( modulo_modulo @ nat @ M3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ nat @ N4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ nat @ ( divide_divide @ nat @ M3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ nat @ N4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% or_nat_unfold
thf(fact_4635_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_4636_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_4637_Arg__ii,axiom,
( ( arg @ imaginary_unit )
= ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ).
% Arg_ii
thf(fact_4638_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_4639_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_4640_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_4641_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_4642_csqrt__eq__0,axiom,
! [Z: complex] :
( ( ( csqrt @ Z )
= ( zero_zero @ complex ) )
= ( Z
= ( zero_zero @ complex ) ) ) ).
% csqrt_eq_0
thf(fact_4643_csqrt__0,axiom,
( ( csqrt @ ( zero_zero @ complex ) )
= ( zero_zero @ complex ) ) ).
% csqrt_0
thf(fact_4644_csqrt__eq__1,axiom,
! [Z: complex] :
( ( ( csqrt @ Z )
= ( one_one @ complex ) )
= ( Z
= ( one_one @ complex ) ) ) ).
% csqrt_eq_1
thf(fact_4645_csqrt__1,axiom,
( ( csqrt @ ( one_one @ complex ) )
= ( one_one @ complex ) ) ).
% csqrt_1
thf(fact_4646_norm__cis,axiom,
! [A3: real] :
( ( real_V7770717601297561774m_norm @ complex @ ( cis @ A3 ) )
= ( one_one @ real ) ) ).
% norm_cis
thf(fact_4647_cis__zero,axiom,
( ( cis @ ( zero_zero @ real ) )
= ( one_one @ complex ) ) ).
% cis_zero
thf(fact_4648_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_4649_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_4650_cis__pi,axiom,
( ( cis @ pi )
= ( uminus_uminus @ complex @ ( one_one @ complex ) ) ) ).
% cis_pi
thf(fact_4651_drop__bit__numeral__minus__bit0,axiom,
! [L: num,K: num] :
( ( bit_se4197421643247451524op_bit @ int @ ( numeral_numeral @ nat @ L ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ K ) ) ) )
= ( bit_se4197421643247451524op_bit @ int @ ( pred_numeral @ L ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) ) ) ).
% drop_bit_numeral_minus_bit0
thf(fact_4652_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_4653_power2__csqrt,axiom,
! [Z: complex] :
( ( power_power @ complex @ ( csqrt @ Z ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= Z ) ).
% power2_csqrt
thf(fact_4654_drop__bit__numeral__minus__bit1,axiom,
! [L: num,K: num] :
( ( bit_se4197421643247451524op_bit @ int @ ( numeral_numeral @ nat @ L ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ K ) ) ) )
= ( bit_se4197421643247451524op_bit @ int @ ( pred_numeral @ L ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( inc @ K ) ) ) ) ) ).
% drop_bit_numeral_minus_bit1
thf(fact_4655_cis__pi__half,axiom,
( ( cis @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
= imaginary_unit ) ).
% cis_pi_half
thf(fact_4656_cis__2pi,axiom,
( ( cis @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) )
= ( one_one @ complex ) ) ).
% cis_2pi
thf(fact_4657_sup__enat__def,axiom,
( ( sup_sup @ extended_enat )
= ( ord_max @ extended_enat ) ) ).
% sup_enat_def
thf(fact_4658_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_4659_cis__Arg,axiom,
! [Z: complex] :
( ( Z
!= ( zero_zero @ complex ) )
=> ( ( cis @ ( arg @ Z ) )
= ( sgn_sgn @ complex @ Z ) ) ) ).
% cis_Arg
thf(fact_4660_cis__neq__zero,axiom,
! [A3: real] :
( ( cis @ A3 )
!= ( zero_zero @ complex ) ) ).
% cis_neq_zero
thf(fact_4661_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_4662_unat__drop__bit__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,W: word @ A] :
( ( semiring_1_unsigned @ A @ nat @ ( bit_se4197421643247451524op_bit @ ( word @ A ) @ N @ W ) )
= ( bit_se4197421643247451524op_bit @ nat @ N @ ( semiring_1_unsigned @ A @ nat @ W ) ) ) ) ).
% unat_drop_bit_eq
thf(fact_4663_DeMoivre,axiom,
! [A3: real,N: nat] :
( ( power_power @ complex @ ( cis @ A3 ) @ N )
= ( cis @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ A3 ) ) ) ).
% DeMoivre
thf(fact_4664_cis__Arg__unique,axiom,
! [Z: complex,X: real] :
( ( ( sgn_sgn @ complex @ Z )
= ( cis @ X ) )
=> ( ( ord_less @ real @ ( uminus_uminus @ real @ pi ) @ X )
=> ( ( ord_less_eq @ real @ X @ pi )
=> ( ( arg @ Z )
= X ) ) ) ) ).
% cis_Arg_unique
thf(fact_4665_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_4666_Arg__zero,axiom,
( ( arg @ ( zero_zero @ complex ) )
= ( zero_zero @ real ) ) ).
% Arg_zero
thf(fact_4667_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_4668_drop__bit__int__def,axiom,
( ( bit_se4197421643247451524op_bit @ int )
= ( ^ [N4: nat,K3: int] : ( divide_divide @ int @ K3 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N4 ) ) ) ) ).
% drop_bit_int_def
thf(fact_4669_drop__bit__nat__def,axiom,
( ( bit_se4197421643247451524op_bit @ nat )
= ( ^ [N4: nat,M3: nat] : ( divide_divide @ nat @ M3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N4 ) ) ) ) ).
% drop_bit_nat_def
thf(fact_4670_of__real__sqrt,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( real_Vector_of_real @ complex @ ( sqrt @ X ) )
= ( csqrt @ ( real_Vector_of_real @ complex @ X ) ) ) ) ).
% of_real_sqrt
thf(fact_4671_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_4672_complex__inverse,axiom,
! [A3: real,B3: real] :
( ( inverse_inverse @ complex @ ( complex2 @ A3 @ B3 ) )
= ( complex2 @ ( divide_divide @ real @ A3 @ ( plus_plus @ real @ ( power_power @ real @ A3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ B3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( divide_divide @ real @ ( uminus_uminus @ real @ B3 ) @ ( plus_plus @ real @ ( power_power @ real @ A3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ B3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% complex_inverse
thf(fact_4673_exp__first__two__terms,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ( ( exp @ A )
= ( ^ [X2: A] :
( plus_plus @ A @ ( plus_plus @ A @ ( one_one @ A ) @ X2 )
@ ( suminf @ A
@ ^ [N4: nat] : ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ ( plus_plus @ nat @ N4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( power_power @ A @ X2 @ ( plus_plus @ nat @ N4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% exp_first_two_terms
thf(fact_4674_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_4675_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_4676_scaleR__zero__right,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [A3: real] :
( ( real_V8093663219630862766scaleR @ A @ A3 @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% scaleR_zero_right
thf(fact_4677_scaleR__cancel__right,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [A3: real,X: A,B3: real] :
( ( ( real_V8093663219630862766scaleR @ A @ A3 @ X )
= ( real_V8093663219630862766scaleR @ A @ B3 @ X ) )
= ( ( A3 = B3 )
| ( X
= ( zero_zero @ A ) ) ) ) ) ).
% scaleR_cancel_right
thf(fact_4678_scaleR__cancel__left,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [A3: real,X: A,Y: A] :
( ( ( real_V8093663219630862766scaleR @ A @ A3 @ X )
= ( real_V8093663219630862766scaleR @ A @ A3 @ Y ) )
= ( ( X = Y )
| ( A3
= ( zero_zero @ real ) ) ) ) ) ).
% scaleR_cancel_left
thf(fact_4679_scaleR__one,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [X: A] :
( ( real_V8093663219630862766scaleR @ A @ ( one_one @ real ) @ X )
= X ) ) ).
% scaleR_one
thf(fact_4680_shiftr__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat] :
( ( bit_Sh4282982442137083166shiftr @ A @ ( zero_zero @ A ) @ N )
= ( zero_zero @ A ) ) ) ).
% shiftr_0
thf(fact_4681_shiftr__of__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A] :
( ( bit_Sh4282982442137083166shiftr @ A @ A3 @ ( zero_zero @ nat ) )
= A3 ) ) ).
% shiftr_of_0
thf(fact_4682_scaleR__zero__left,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [X: A] :
( ( real_V8093663219630862766scaleR @ A @ ( zero_zero @ real ) @ X )
= ( zero_zero @ A ) ) ) ).
% scaleR_zero_left
thf(fact_4683_scaleR__eq__0__iff,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [A3: real,X: A] :
( ( ( real_V8093663219630862766scaleR @ A @ A3 @ X )
= ( zero_zero @ A ) )
= ( ( A3
= ( zero_zero @ real ) )
| ( X
= ( zero_zero @ A ) ) ) ) ) ).
% scaleR_eq_0_iff
thf(fact_4684_scaleR__eq__iff,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [B3: A,U2: real,A3: A] :
( ( ( plus_plus @ A @ B3 @ ( real_V8093663219630862766scaleR @ A @ U2 @ A3 ) )
= ( plus_plus @ A @ A3 @ ( real_V8093663219630862766scaleR @ A @ U2 @ B3 ) ) )
= ( ( A3 = B3 )
| ( U2
= ( one_one @ real ) ) ) ) ) ).
% scaleR_eq_iff
thf(fact_4685_scaleR__power,axiom,
! [A: $tType] :
( ( real_V2822296259951069270ebra_1 @ A )
=> ! [X: real,Y: A,N: nat] :
( ( power_power @ A @ ( real_V8093663219630862766scaleR @ A @ X @ Y ) @ N )
= ( real_V8093663219630862766scaleR @ A @ ( power_power @ real @ X @ N ) @ ( power_power @ A @ Y @ N ) ) ) ) ).
% scaleR_power
thf(fact_4686_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_4687_scaleR__minus1__left,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [X: A] :
( ( real_V8093663219630862766scaleR @ A @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X )
= ( uminus_uminus @ A @ X ) ) ) ).
% scaleR_minus1_left
thf(fact_4688_scaleR__collapse,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [U2: real,A3: A] :
( ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ ( minus_minus @ real @ ( one_one @ real ) @ U2 ) @ A3 ) @ ( real_V8093663219630862766scaleR @ A @ U2 @ A3 ) )
= A3 ) ) ).
% scaleR_collapse
thf(fact_4689_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_4690_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_4691_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_4692_scaleR__times,axiom,
! [A: $tType] :
( ( real_V2191834092415804123ebra_1 @ A )
=> ! [U2: num,W: num,A3: A] :
( ( real_V8093663219630862766scaleR @ A @ ( numeral_numeral @ real @ U2 ) @ ( times_times @ A @ ( numeral_numeral @ A @ W ) @ A3 ) )
= ( real_V8093663219630862766scaleR @ A @ ( times_times @ real @ ( numeral_numeral @ real @ U2 ) @ ( numeral_numeral @ real @ W ) ) @ A3 ) ) ) ).
% scaleR_times
thf(fact_4693_inverse__scaleR__times,axiom,
! [A: $tType] :
( ( real_V2191834092415804123ebra_1 @ A )
=> ! [V2: num,W: num,A3: A] :
( ( real_V8093663219630862766scaleR @ A @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ V2 ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ W ) @ A3 ) )
= ( real_V8093663219630862766scaleR @ A @ ( divide_divide @ real @ ( numeral_numeral @ real @ W ) @ ( numeral_numeral @ real @ V2 ) ) @ A3 ) ) ) ).
% inverse_scaleR_times
thf(fact_4694_fraction__scaleR__times,axiom,
! [A: $tType] :
( ( real_V2191834092415804123ebra_1 @ A )
=> ! [U2: num,V2: num,W: num,A3: A] :
( ( real_V8093663219630862766scaleR @ A @ ( divide_divide @ real @ ( numeral_numeral @ real @ U2 ) @ ( numeral_numeral @ real @ V2 ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ W ) @ A3 ) )
= ( real_V8093663219630862766scaleR @ A @ ( divide_divide @ real @ ( times_times @ real @ ( numeral_numeral @ real @ U2 ) @ ( numeral_numeral @ real @ W ) ) @ ( numeral_numeral @ real @ V2 ) ) @ A3 ) ) ) ).
% fraction_scaleR_times
thf(fact_4695_scaleR__half__double,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [A3: A] :
( ( real_V8093663219630862766scaleR @ A @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( plus_plus @ A @ A3 @ A3 ) )
= A3 ) ) ).
% scaleR_half_double
thf(fact_4696_summable__scaleR__right,axiom,
! [B: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [X5: nat > B,R3: real] :
( ( summable @ B @ X5 )
=> ( summable @ B
@ ^ [N4: nat] : ( real_V8093663219630862766scaleR @ B @ R3 @ ( X5 @ N4 ) ) ) ) ) ).
% summable_scaleR_right
thf(fact_4697_scaleR__left__imp__eq,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [A3: real,X: A,Y: A] :
( ( A3
!= ( zero_zero @ real ) )
=> ( ( ( real_V8093663219630862766scaleR @ A @ A3 @ X )
= ( real_V8093663219630862766scaleR @ A @ A3 @ Y ) )
=> ( X = Y ) ) ) ) ).
% scaleR_left_imp_eq
thf(fact_4698_scaleR__right__imp__eq,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [X: A,A3: real,B3: real] :
( ( X
!= ( zero_zero @ A ) )
=> ( ( ( real_V8093663219630862766scaleR @ A @ A3 @ X )
= ( real_V8093663219630862766scaleR @ A @ B3 @ X ) )
=> ( A3 = B3 ) ) ) ) ).
% scaleR_right_imp_eq
thf(fact_4699_sums__scaleR__right,axiom,
! [B: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [X5: nat > B,A3: B,R3: real] :
( ( sums @ B @ X5 @ A3 )
=> ( sums @ B
@ ^ [N4: nat] : ( real_V8093663219630862766scaleR @ B @ R3 @ ( X5 @ N4 ) )
@ ( real_V8093663219630862766scaleR @ B @ R3 @ A3 ) ) ) ) ).
% sums_scaleR_right
thf(fact_4700_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_4701_suminf__scaleR__right,axiom,
! [B: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [X5: nat > B,R3: real] :
( ( summable @ B @ X5 )
=> ( ( real_V8093663219630862766scaleR @ B @ R3 @ ( suminf @ B @ X5 ) )
= ( suminf @ B
@ ^ [N4: nat] : ( real_V8093663219630862766scaleR @ B @ R3 @ ( X5 @ N4 ) ) ) ) ) ) ).
% suminf_scaleR_right
thf(fact_4702_summable__scaleR__left,axiom,
! [B: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [X5: nat > real,X: B] :
( ( summable @ real @ X5 )
=> ( summable @ B
@ ^ [N4: nat] : ( real_V8093663219630862766scaleR @ B @ ( X5 @ N4 ) @ X ) ) ) ) ).
% summable_scaleR_left
thf(fact_4703_sums__scaleR__left,axiom,
! [B: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [X5: nat > real,A3: real,X: B] :
( ( sums @ real @ X5 @ A3 )
=> ( sums @ B
@ ^ [N4: nat] : ( real_V8093663219630862766scaleR @ B @ ( X5 @ N4 ) @ X )
@ ( real_V8093663219630862766scaleR @ B @ A3 @ X ) ) ) ) ).
% sums_scaleR_left
thf(fact_4704_scaleR__right__mono__neg,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [B3: real,A3: real,C2: A] :
( ( ord_less_eq @ real @ B3 @ A3 )
=> ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ A3 @ C2 ) @ ( real_V8093663219630862766scaleR @ A @ B3 @ C2 ) ) ) ) ) ).
% scaleR_right_mono_neg
thf(fact_4705_scaleR__right__mono,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [A3: real,B3: real,X: A] :
( ( ord_less_eq @ real @ A3 @ B3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ A3 @ X ) @ ( real_V8093663219630862766scaleR @ A @ B3 @ X ) ) ) ) ) ).
% scaleR_right_mono
thf(fact_4706_scaleR__le__cancel__left,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C2: real,A3: A,B3: A] :
( ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ C2 @ A3 ) @ ( real_V8093663219630862766scaleR @ A @ C2 @ B3 ) )
= ( ( ( ord_less @ real @ ( zero_zero @ real ) @ C2 )
=> ( ord_less_eq @ A @ A3 @ B3 ) )
& ( ( ord_less @ real @ C2 @ ( zero_zero @ real ) )
=> ( ord_less_eq @ A @ B3 @ A3 ) ) ) ) ) ).
% scaleR_le_cancel_left
thf(fact_4707_scaleR__le__cancel__left__neg,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C2: real,A3: A,B3: A] :
( ( ord_less @ real @ C2 @ ( zero_zero @ real ) )
=> ( ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ C2 @ A3 ) @ ( real_V8093663219630862766scaleR @ A @ C2 @ B3 ) )
= ( ord_less_eq @ A @ B3 @ A3 ) ) ) ) ).
% scaleR_le_cancel_left_neg
thf(fact_4708_scaleR__le__cancel__left__pos,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C2: real,A3: A,B3: A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ C2 )
=> ( ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ C2 @ A3 ) @ ( real_V8093663219630862766scaleR @ A @ C2 @ B3 ) )
= ( ord_less_eq @ A @ A3 @ B3 ) ) ) ) ).
% scaleR_le_cancel_left_pos
thf(fact_4709_scaleR__left__mono__neg,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [B3: A,A3: A,C2: real] :
( ( ord_less_eq @ A @ B3 @ A3 )
=> ( ( ord_less_eq @ real @ C2 @ ( zero_zero @ real ) )
=> ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ C2 @ A3 ) @ ( real_V8093663219630862766scaleR @ A @ C2 @ B3 ) ) ) ) ) ).
% scaleR_left_mono_neg
thf(fact_4710_scaleR__left__mono,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [X: A,Y: A,A3: real] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ A3 )
=> ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ A3 @ X ) @ ( real_V8093663219630862766scaleR @ A @ A3 @ Y ) ) ) ) ) ).
% scaleR_left_mono
thf(fact_4711_vector__fraction__eq__iff,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [U2: real,V2: real,A3: A,X: A] :
( ( ( real_V8093663219630862766scaleR @ A @ ( divide_divide @ real @ U2 @ V2 ) @ A3 )
= X )
= ( ( ( V2
= ( zero_zero @ real ) )
=> ( X
= ( zero_zero @ A ) ) )
& ( ( V2
!= ( zero_zero @ real ) )
=> ( ( real_V8093663219630862766scaleR @ A @ U2 @ A3 )
= ( real_V8093663219630862766scaleR @ A @ V2 @ X ) ) ) ) ) ) ).
% vector_fraction_eq_iff
thf(fact_4712_eq__vector__fraction__iff,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [X: A,U2: real,V2: real,A3: A] :
( ( X
= ( real_V8093663219630862766scaleR @ A @ ( divide_divide @ real @ U2 @ V2 ) @ A3 ) )
= ( ( ( V2
= ( zero_zero @ real ) )
=> ( X
= ( zero_zero @ A ) ) )
& ( ( V2
!= ( zero_zero @ real ) )
=> ( ( real_V8093663219630862766scaleR @ A @ V2 @ X )
= ( real_V8093663219630862766scaleR @ A @ U2 @ A3 ) ) ) ) ) ) ).
% eq_vector_fraction_iff
thf(fact_4713_zero__le__scaleR__iff,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [A3: real,B3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( real_V8093663219630862766scaleR @ A @ A3 @ B3 ) )
= ( ( ( ord_less @ real @ ( zero_zero @ real ) @ A3 )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ B3 ) )
| ( ( ord_less @ real @ A3 @ ( zero_zero @ real ) )
& ( ord_less_eq @ A @ B3 @ ( zero_zero @ A ) ) )
| ( A3
= ( zero_zero @ real ) ) ) ) ) ).
% zero_le_scaleR_iff
thf(fact_4714_scaleR__le__0__iff,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [A3: real,B3: A] :
( ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ A3 @ B3 ) @ ( zero_zero @ A ) )
= ( ( ( ord_less @ real @ ( zero_zero @ real ) @ A3 )
& ( ord_less_eq @ A @ B3 @ ( zero_zero @ A ) ) )
| ( ( ord_less @ real @ A3 @ ( zero_zero @ real ) )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ B3 ) )
| ( A3
= ( zero_zero @ real ) ) ) ) ) ).
% scaleR_le_0_iff
thf(fact_4715_scaleR__nonpos__nonpos,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [A3: real,B3: A] :
( ( ord_less_eq @ real @ A3 @ ( zero_zero @ real ) )
=> ( ( ord_less_eq @ A @ B3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( real_V8093663219630862766scaleR @ A @ A3 @ B3 ) ) ) ) ) ).
% scaleR_nonpos_nonpos
thf(fact_4716_scaleR__nonpos__nonneg,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [A3: real,X: A] :
( ( ord_less_eq @ real @ A3 @ ( zero_zero @ real ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ A3 @ X ) @ ( zero_zero @ A ) ) ) ) ) ).
% scaleR_nonpos_nonneg
thf(fact_4717_scaleR__nonneg__nonpos,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [A3: real,X: A] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ A3 )
=> ( ( ord_less_eq @ A @ X @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ A3 @ X ) @ ( zero_zero @ A ) ) ) ) ) ).
% scaleR_nonneg_nonpos
thf(fact_4718_scaleR__nonneg__nonneg,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [A3: real,X: A] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( real_V8093663219630862766scaleR @ A @ A3 @ X ) ) ) ) ) ).
% scaleR_nonneg_nonneg
thf(fact_4719_split__scaleR__pos__le,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [A3: real,B3: A] :
( ( ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ A3 )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ B3 ) )
| ( ( ord_less_eq @ real @ A3 @ ( zero_zero @ real ) )
& ( ord_less_eq @ A @ B3 @ ( zero_zero @ A ) ) ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( real_V8093663219630862766scaleR @ A @ A3 @ B3 ) ) ) ) ).
% split_scaleR_pos_le
thf(fact_4720_split__scaleR__neg__le,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [A3: real,X: A] :
( ( ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ A3 )
& ( ord_less_eq @ A @ X @ ( zero_zero @ A ) ) )
| ( ( ord_less_eq @ real @ A3 @ ( zero_zero @ real ) )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ X ) ) )
=> ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ A3 @ X ) @ ( zero_zero @ A ) ) ) ) ).
% split_scaleR_neg_le
thf(fact_4721_scaleR__mono_H,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [A3: real,B3: real,C2: A,D2: A] :
( ( ord_less_eq @ real @ A3 @ B3 )
=> ( ( ord_less_eq @ A @ C2 @ D2 )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ A3 @ C2 ) @ ( real_V8093663219630862766scaleR @ A @ B3 @ D2 ) ) ) ) ) ) ) ).
% scaleR_mono'
thf(fact_4722_scaleR__mono,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [A3: real,B3: real,X: A,Y: A] :
( ( ord_less_eq @ real @ A3 @ B3 )
=> ( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ B3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ A3 @ X ) @ ( real_V8093663219630862766scaleR @ A @ B3 @ Y ) ) ) ) ) ) ) ).
% scaleR_mono
thf(fact_4723_suminf__scaleR__left,axiom,
! [B: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [X5: nat > real,X: B] :
( ( summable @ real @ X5 )
=> ( ( real_V8093663219630862766scaleR @ B @ ( suminf @ real @ X5 ) @ X )
= ( suminf @ B
@ ^ [N4: nat] : ( real_V8093663219630862766scaleR @ B @ ( X5 @ N4 ) @ X ) ) ) ) ) ).
% suminf_scaleR_left
thf(fact_4724_scaleR__2,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [X: A] :
( ( real_V8093663219630862766scaleR @ A @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ X )
= ( plus_plus @ A @ X @ X ) ) ) ).
% scaleR_2
thf(fact_4725_scaleR__left__le__one__le,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [X: A,A3: real] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less_eq @ real @ A3 @ ( one_one @ real ) )
=> ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ A3 @ X ) @ X ) ) ) ) ).
% scaleR_left_le_one_le
thf(fact_4726_real__vector__eq__affinity,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [M: real,Y: A,X: A,C2: A] :
( ( M
!= ( zero_zero @ real ) )
=> ( ( Y
= ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ M @ X ) @ C2 ) )
= ( ( minus_minus @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ M ) @ Y ) @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ M ) @ C2 ) )
= X ) ) ) ) ).
% real_vector_eq_affinity
thf(fact_4727_real__vector__affinity__eq,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [M: real,X: A,C2: A,Y: A] :
( ( M
!= ( zero_zero @ real ) )
=> ( ( ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ M @ X ) @ C2 )
= Y )
= ( X
= ( minus_minus @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ M ) @ Y ) @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ M ) @ C2 ) ) ) ) ) ) ).
% real_vector_affinity_eq
thf(fact_4728_neg__le__divideR__eq,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C2: real,A3: A,B3: A] :
( ( ord_less @ real @ C2 @ ( zero_zero @ real ) )
=> ( ( ord_less_eq @ A @ A3 @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C2 ) @ B3 ) )
= ( ord_less_eq @ A @ B3 @ ( real_V8093663219630862766scaleR @ A @ C2 @ A3 ) ) ) ) ) ).
% neg_le_divideR_eq
thf(fact_4729_neg__divideR__le__eq,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C2: real,B3: A,A3: A] :
( ( ord_less @ real @ C2 @ ( zero_zero @ real ) )
=> ( ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C2 ) @ B3 ) @ A3 )
= ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ C2 @ A3 ) @ B3 ) ) ) ) ).
% neg_divideR_le_eq
thf(fact_4730_pos__le__divideR__eq,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C2: real,A3: A,B3: A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ C2 )
=> ( ( ord_less_eq @ A @ A3 @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C2 ) @ B3 ) )
= ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ C2 @ A3 ) @ B3 ) ) ) ) ).
% pos_le_divideR_eq
thf(fact_4731_pos__divideR__le__eq,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C2: real,B3: A,A3: A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ C2 )
=> ( ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C2 ) @ B3 ) @ A3 )
= ( ord_less_eq @ A @ B3 @ ( real_V8093663219630862766scaleR @ A @ C2 @ A3 ) ) ) ) ) ).
% pos_divideR_le_eq
thf(fact_4732_pos__divideR__less__eq,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C2: real,B3: A,A3: A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ C2 )
=> ( ( ord_less @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C2 ) @ B3 ) @ A3 )
= ( ord_less @ A @ B3 @ ( real_V8093663219630862766scaleR @ A @ C2 @ A3 ) ) ) ) ) ).
% pos_divideR_less_eq
thf(fact_4733_pos__less__divideR__eq,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C2: real,A3: A,B3: A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ C2 )
=> ( ( ord_less @ A @ A3 @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C2 ) @ B3 ) )
= ( ord_less @ A @ ( real_V8093663219630862766scaleR @ A @ C2 @ A3 ) @ B3 ) ) ) ) ).
% pos_less_divideR_eq
thf(fact_4734_neg__divideR__less__eq,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C2: real,B3: A,A3: A] :
( ( ord_less @ real @ C2 @ ( zero_zero @ real ) )
=> ( ( ord_less @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C2 ) @ B3 ) @ A3 )
= ( ord_less @ A @ ( real_V8093663219630862766scaleR @ A @ C2 @ A3 ) @ B3 ) ) ) ) ).
% neg_divideR_less_eq
thf(fact_4735_neg__less__divideR__eq,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C2: real,A3: A,B3: A] :
( ( ord_less @ real @ C2 @ ( zero_zero @ real ) )
=> ( ( ord_less @ A @ A3 @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C2 ) @ B3 ) )
= ( ord_less @ A @ B3 @ ( real_V8093663219630862766scaleR @ A @ C2 @ A3 ) ) ) ) ) ).
% neg_less_divideR_eq
thf(fact_4736_nonzero__inverse__scaleR__distrib,axiom,
! [A: $tType] :
( ( real_V5047593784448816457lgebra @ A )
=> ! [A3: real,X: A] :
( ( A3
!= ( zero_zero @ real ) )
=> ( ( X
!= ( zero_zero @ A ) )
=> ( ( inverse_inverse @ A @ ( real_V8093663219630862766scaleR @ A @ A3 @ X ) )
= ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ A3 ) @ ( inverse_inverse @ A @ X ) ) ) ) ) ) ).
% nonzero_inverse_scaleR_distrib
thf(fact_4737_summable__exp__generic,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X: A] :
( summable @ A
@ ^ [N4: nat] : ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ N4 ) ) @ ( power_power @ A @ X @ N4 ) ) ) ) ).
% summable_exp_generic
thf(fact_4738_sin__converges,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X: A] :
( sums @ A
@ ^ [N4: nat] : ( real_V8093663219630862766scaleR @ A @ ( sin_coeff @ N4 ) @ ( power_power @ A @ X @ N4 ) )
@ ( sin @ A @ X ) ) ) ).
% sin_converges
thf(fact_4739_sin__def,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ( ( sin @ A )
= ( ^ [X2: A] :
( suminf @ A
@ ^ [N4: nat] : ( real_V8093663219630862766scaleR @ A @ ( sin_coeff @ N4 ) @ ( power_power @ A @ X2 @ N4 ) ) ) ) ) ) ).
% sin_def
thf(fact_4740_cos__converges,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X: A] :
( sums @ A
@ ^ [N4: nat] : ( real_V8093663219630862766scaleR @ A @ ( cos_coeff @ N4 ) @ ( power_power @ A @ X @ N4 ) )
@ ( cos @ A @ X ) ) ) ).
% cos_converges
thf(fact_4741_cos__def,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ( ( cos @ A )
= ( ^ [X2: A] :
( suminf @ A
@ ^ [N4: nat] : ( real_V8093663219630862766scaleR @ A @ ( cos_coeff @ N4 ) @ ( power_power @ A @ X2 @ N4 ) ) ) ) ) ) ).
% cos_def
thf(fact_4742_summable__norm__sin,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X: A] :
( summable @ real
@ ^ [N4: nat] : ( real_V7770717601297561774m_norm @ A @ ( real_V8093663219630862766scaleR @ A @ ( sin_coeff @ N4 ) @ ( power_power @ A @ X @ N4 ) ) ) ) ) ).
% summable_norm_sin
thf(fact_4743_summable__norm__cos,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X: A] :
( summable @ real
@ ^ [N4: nat] : ( real_V7770717601297561774m_norm @ A @ ( real_V8093663219630862766scaleR @ A @ ( cos_coeff @ N4 ) @ ( power_power @ A @ X @ N4 ) ) ) ) ) ).
% summable_norm_cos
thf(fact_4744_pos__le__minus__divideR__eq,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C2: real,A3: A,B3: A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ C2 )
=> ( ( ord_less_eq @ A @ A3 @ ( uminus_uminus @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C2 ) @ B3 ) ) )
= ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ C2 @ A3 ) @ ( uminus_uminus @ A @ B3 ) ) ) ) ) ).
% pos_le_minus_divideR_eq
thf(fact_4745_pos__minus__divideR__le__eq,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C2: real,B3: A,A3: A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ C2 )
=> ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C2 ) @ B3 ) ) @ A3 )
= ( ord_less_eq @ A @ ( uminus_uminus @ A @ B3 ) @ ( real_V8093663219630862766scaleR @ A @ C2 @ A3 ) ) ) ) ) ).
% pos_minus_divideR_le_eq
thf(fact_4746_neg__le__minus__divideR__eq,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C2: real,A3: A,B3: A] :
( ( ord_less @ real @ C2 @ ( zero_zero @ real ) )
=> ( ( ord_less_eq @ A @ A3 @ ( uminus_uminus @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C2 ) @ B3 ) ) )
= ( ord_less_eq @ A @ ( uminus_uminus @ A @ B3 ) @ ( real_V8093663219630862766scaleR @ A @ C2 @ A3 ) ) ) ) ) ).
% neg_le_minus_divideR_eq
thf(fact_4747_neg__minus__divideR__le__eq,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C2: real,B3: A,A3: A] :
( ( ord_less @ real @ C2 @ ( zero_zero @ real ) )
=> ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C2 ) @ B3 ) ) @ A3 )
= ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ C2 @ A3 ) @ ( uminus_uminus @ A @ B3 ) ) ) ) ) ).
% neg_minus_divideR_le_eq
thf(fact_4748_pos__less__minus__divideR__eq,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C2: real,A3: A,B3: A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ C2 )
=> ( ( ord_less @ A @ A3 @ ( uminus_uminus @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C2 ) @ B3 ) ) )
= ( ord_less @ A @ ( real_V8093663219630862766scaleR @ A @ C2 @ A3 ) @ ( uminus_uminus @ A @ B3 ) ) ) ) ) ).
% pos_less_minus_divideR_eq
thf(fact_4749_pos__minus__divideR__less__eq,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C2: real,B3: A,A3: A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ C2 )
=> ( ( ord_less @ A @ ( uminus_uminus @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C2 ) @ B3 ) ) @ A3 )
= ( ord_less @ A @ ( uminus_uminus @ A @ B3 ) @ ( real_V8093663219630862766scaleR @ A @ C2 @ A3 ) ) ) ) ) ).
% pos_minus_divideR_less_eq
thf(fact_4750_neg__less__minus__divideR__eq,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C2: real,A3: A,B3: A] :
( ( ord_less @ real @ C2 @ ( zero_zero @ real ) )
=> ( ( ord_less @ A @ A3 @ ( uminus_uminus @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C2 ) @ B3 ) ) )
= ( ord_less @ A @ ( uminus_uminus @ A @ B3 ) @ ( real_V8093663219630862766scaleR @ A @ C2 @ A3 ) ) ) ) ) ).
% neg_less_minus_divideR_eq
thf(fact_4751_neg__minus__divideR__less__eq,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C2: real,B3: A,A3: A] :
( ( ord_less @ real @ C2 @ ( zero_zero @ real ) )
=> ( ( ord_less @ A @ ( uminus_uminus @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C2 ) @ B3 ) ) @ A3 )
= ( ord_less @ A @ ( real_V8093663219630862766scaleR @ A @ C2 @ A3 ) @ ( uminus_uminus @ A @ B3 ) ) ) ) ) ).
% neg_minus_divideR_less_eq
thf(fact_4752_exp__converges,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X: A] :
( sums @ A
@ ^ [N4: nat] : ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ N4 ) ) @ ( power_power @ A @ X @ N4 ) )
@ ( exp @ A @ X ) ) ) ).
% exp_converges
thf(fact_4753_exp__def,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ( ( exp @ A )
= ( ^ [X2: A] :
( suminf @ A
@ ^ [N4: nat] : ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ N4 ) ) @ ( power_power @ A @ X2 @ N4 ) ) ) ) ) ) ).
% exp_def
thf(fact_4754_summable__norm__exp,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X: A] :
( summable @ real
@ ^ [N4: nat] : ( real_V7770717601297561774m_norm @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ N4 ) ) @ ( power_power @ A @ X @ N4 ) ) ) ) ) ).
% summable_norm_exp
thf(fact_4755_sin__minus__converges,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X: A] :
( sums @ A
@ ^ [N4: nat] : ( uminus_uminus @ A @ ( real_V8093663219630862766scaleR @ A @ ( sin_coeff @ N4 ) @ ( power_power @ A @ ( uminus_uminus @ A @ X ) @ N4 ) ) )
@ ( sin @ A @ X ) ) ) ).
% sin_minus_converges
thf(fact_4756_cos__minus__converges,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X: A] :
( sums @ A
@ ^ [N4: nat] : ( real_V8093663219630862766scaleR @ A @ ( cos_coeff @ N4 ) @ ( power_power @ A @ ( uminus_uminus @ A @ X ) @ N4 ) )
@ ( cos @ A @ X ) ) ) ).
% cos_minus_converges
thf(fact_4757_exp__first__term,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ( ( exp @ A )
= ( ^ [X2: A] :
( plus_plus @ A @ ( one_one @ A )
@ ( suminf @ A
@ ^ [N4: nat] : ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ ( suc @ N4 ) ) ) @ ( power_power @ A @ X2 @ ( suc @ N4 ) ) ) ) ) ) ) ) ).
% exp_first_term
thf(fact_4758_shiftr__eq__div,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_Sh4282982442137083166shiftr @ A )
= ( ^ [X2: A,N4: nat] : ( divide_divide @ A @ X2 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N4 ) ) ) ) ) ).
% shiftr_eq_div
thf(fact_4759_sinh__converges,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X: A] :
( sums @ A
@ ^ [N4: nat] : ( if @ A @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N4 ) @ ( zero_zero @ A ) @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ N4 ) ) @ ( power_power @ A @ X @ N4 ) ) )
@ ( sinh @ A @ X ) ) ) ).
% sinh_converges
thf(fact_4760_cosh__converges,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X: A] :
( sums @ A
@ ^ [N4: nat] : ( if @ A @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N4 ) @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ N4 ) ) @ ( power_power @ A @ X @ N4 ) ) @ ( zero_zero @ A ) )
@ ( cosh @ A @ X ) ) ) ).
% cosh_converges
thf(fact_4761_sin__x__sin__y,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A,Y: A] :
( sums @ A
@ ^ [P6: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [N4: nat] :
( if @ A
@ ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ P6 )
& ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N4 ) )
@ ( times_times @ A @ ( real_V8093663219630862766scaleR @ A @ ( uminus_uminus @ real @ ( divide_divide @ real @ ( ring_1_of_int @ real @ ( times_times @ int @ ( power_power @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( divide_divide @ nat @ P6 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( semiring_1_of_nat @ int @ ( binomial @ P6 @ N4 ) ) ) ) @ ( semiring_char_0_fact @ real @ P6 ) ) ) @ ( power_power @ A @ X @ N4 ) ) @ ( power_power @ A @ Y @ ( minus_minus @ nat @ P6 @ N4 ) ) )
@ ( zero_zero @ A ) )
@ ( set_ord_atMost @ nat @ P6 ) )
@ ( times_times @ A @ ( sin @ A @ X ) @ ( sin @ A @ Y ) ) ) ) ).
% sin_x_sin_y
thf(fact_4762_sinh__real__zero__iff,axiom,
! [X: real] :
( ( ( sinh @ real @ X )
= ( zero_zero @ real ) )
= ( X
= ( zero_zero @ real ) ) ) ).
% sinh_real_zero_iff
thf(fact_4763_sinh__real__less__iff,axiom,
! [X: real,Y: real] :
( ( ord_less @ real @ ( sinh @ real @ X ) @ ( sinh @ real @ Y ) )
= ( ord_less @ real @ X @ Y ) ) ).
% sinh_real_less_iff
thf(fact_4764_sinh__real__neg__iff,axiom,
! [X: real] :
( ( ord_less @ real @ ( sinh @ real @ X ) @ ( zero_zero @ real ) )
= ( ord_less @ real @ X @ ( zero_zero @ real ) ) ) ).
% sinh_real_neg_iff
thf(fact_4765_sinh__real__pos__iff,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( sinh @ real @ X ) )
= ( ord_less @ real @ ( zero_zero @ real ) @ X ) ) ).
% sinh_real_pos_iff
thf(fact_4766_sinh__real__nonneg__iff,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( sinh @ real @ X ) )
= ( ord_less_eq @ real @ ( zero_zero @ real ) @ X ) ) ).
% sinh_real_nonneg_iff
thf(fact_4767_sinh__real__nonpos__iff,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( sinh @ real @ X ) @ ( zero_zero @ real ) )
= ( ord_less_eq @ real @ X @ ( zero_zero @ real ) ) ) ).
% sinh_real_nonpos_iff
thf(fact_4768_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_4769_of__nat__sum,axiom,
! [A: $tType,B: $tType] :
( ( semiring_1 @ A )
=> ! [F3: B > nat,A2: set @ B] :
( ( semiring_1_of_nat @ A @ ( groups7311177749621191930dd_sum @ B @ nat @ F3 @ A2 ) )
= ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X2: B] : ( semiring_1_of_nat @ A @ ( F3 @ X2 ) )
@ A2 ) ) ) ).
% of_nat_sum
thf(fact_4770_of__int__sum,axiom,
! [A: $tType,B: $tType] :
( ( ring_1 @ A )
=> ! [F3: B > int,A2: set @ B] :
( ( ring_1_of_int @ A @ ( groups7311177749621191930dd_sum @ B @ int @ F3 @ A2 ) )
= ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X2: B] : ( ring_1_of_int @ A @ ( F3 @ X2 ) )
@ A2 ) ) ) ).
% of_int_sum
thf(fact_4771_of__real__sum,axiom,
! [A: $tType,B: $tType] :
( ( real_V2191834092415804123ebra_1 @ A )
=> ! [F3: B > real,S2: set @ B] :
( ( real_Vector_of_real @ A @ ( groups7311177749621191930dd_sum @ B @ real @ F3 @ S2 ) )
= ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X2: B] : ( real_Vector_of_real @ A @ ( F3 @ X2 ) )
@ S2 ) ) ) ).
% of_real_sum
thf(fact_4772_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_4773_atMost__0,axiom,
( ( set_ord_atMost @ nat @ ( zero_zero @ nat ) )
= ( insert @ nat @ ( zero_zero @ nat ) @ ( bot_bot @ ( set @ nat ) ) ) ) ).
% atMost_0
thf(fact_4774_scaleR__left_Osum,axiom,
! [A: $tType,C: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [G2: C > real,A2: set @ C,X: A] :
( ( real_V8093663219630862766scaleR @ A @ ( groups7311177749621191930dd_sum @ C @ real @ G2 @ A2 ) @ X )
= ( groups7311177749621191930dd_sum @ C @ A
@ ^ [X2: C] : ( real_V8093663219630862766scaleR @ A @ ( G2 @ X2 ) @ X )
@ A2 ) ) ) ).
% scaleR_left.sum
thf(fact_4775_scaleR__right_Osum,axiom,
! [A: $tType,C: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [A3: real,G2: C > A,A2: set @ C] :
( ( real_V8093663219630862766scaleR @ A @ A3 @ ( groups7311177749621191930dd_sum @ C @ A @ G2 @ A2 ) )
= ( groups7311177749621191930dd_sum @ C @ A
@ ^ [X2: C] : ( real_V8093663219630862766scaleR @ A @ A3 @ ( G2 @ X2 ) )
@ A2 ) ) ) ).
% scaleR_right.sum
thf(fact_4776_scaleR__sum__left,axiom,
! [A: $tType,C: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [F3: C > real,A2: set @ C,X: A] :
( ( real_V8093663219630862766scaleR @ A @ ( groups7311177749621191930dd_sum @ C @ real @ F3 @ A2 ) @ X )
= ( groups7311177749621191930dd_sum @ C @ A
@ ^ [A5: C] : ( real_V8093663219630862766scaleR @ A @ ( F3 @ A5 ) @ X )
@ A2 ) ) ) ).
% scaleR_sum_left
thf(fact_4777_scaleR__sum__right,axiom,
! [A: $tType,C: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [A3: real,F3: C > A,A2: set @ C] :
( ( real_V8093663219630862766scaleR @ A @ A3 @ ( groups7311177749621191930dd_sum @ C @ A @ F3 @ A2 ) )
= ( groups7311177749621191930dd_sum @ C @ A
@ ^ [X2: C] : ( real_V8093663219630862766scaleR @ A @ A3 @ ( F3 @ X2 ) )
@ A2 ) ) ) ).
% scaleR_sum_right
thf(fact_4778_atMost__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( set_ord_atMost @ A )
= ( ^ [U: A] :
( collect @ A
@ ^ [X2: A] : ( ord_less_eq @ A @ X2 @ U ) ) ) ) ) ).
% atMost_def
thf(fact_4779_bounded__imp__summable,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( linord2810124833399127020strict @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [A3: nat > A,B2: A] :
( ! [N2: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( A3 @ N2 ) )
=> ( ! [N2: nat] : ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ A3 @ ( set_ord_atMost @ nat @ N2 ) ) @ B2 )
=> ( summable @ A @ A3 ) ) ) ) ).
% bounded_imp_summable
thf(fact_4780_norm__sum,axiom,
! [A: $tType,B: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: B > A,A2: set @ B] :
( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ A2 ) )
@ ( groups7311177749621191930dd_sum @ B @ real
@ ^ [I4: B] : ( real_V7770717601297561774m_norm @ A @ ( F3 @ I4 ) )
@ A2 ) ) ) ).
% norm_sum
thf(fact_4781_sinh__less__cosh__real,axiom,
! [X: real] : ( ord_less @ real @ ( sinh @ real @ X ) @ ( cosh @ real @ X ) ) ).
% sinh_less_cosh_real
thf(fact_4782_not__empty__eq__Iic__eq__empty,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [H2: A] :
( ( bot_bot @ ( set @ A ) )
!= ( set_ord_atMost @ A @ H2 ) ) ) ).
% not_empty_eq_Iic_eq_empty
thf(fact_4783_summable__sum,axiom,
! [I5: $tType,A: $tType] :
( ( ( topolo5987344860129210374id_add @ A )
& ( topological_t2_space @ A ) )
=> ! [I6: set @ I5,F3: I5 > nat > A] :
( ! [I3: I5] :
( ( member @ I5 @ I3 @ I6 )
=> ( summable @ A @ ( F3 @ I3 ) ) )
=> ( summable @ A
@ ^ [N4: nat] :
( groups7311177749621191930dd_sum @ I5 @ A
@ ^ [I4: I5] : ( F3 @ I4 @ N4 )
@ I6 ) ) ) ) ).
% summable_sum
thf(fact_4784_sums__sum,axiom,
! [A: $tType,I5: $tType] :
( ( ( topolo5987344860129210374id_add @ A )
& ( topological_t2_space @ A ) )
=> ! [I6: set @ I5,F3: I5 > nat > A,X: I5 > A] :
( ! [I3: I5] :
( ( member @ I5 @ I3 @ I6 )
=> ( sums @ A @ ( F3 @ I3 ) @ ( X @ I3 ) ) )
=> ( sums @ A
@ ^ [N4: nat] :
( groups7311177749621191930dd_sum @ I5 @ A
@ ^ [I4: I5] : ( F3 @ I4 @ N4 )
@ I6 )
@ ( groups7311177749621191930dd_sum @ I5 @ A @ X @ I6 ) ) ) ) ).
% sums_sum
thf(fact_4785_mod__sum__eq,axiom,
! [B: $tType,A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [F3: B > A,A3: A,A2: set @ B] :
( ( modulo_modulo @ A
@ ( groups7311177749621191930dd_sum @ B @ A
@ ^ [I4: B] : ( modulo_modulo @ A @ ( F3 @ I4 ) @ A3 )
@ A2 )
@ A3 )
= ( modulo_modulo @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ A2 ) @ A3 ) ) ) ).
% mod_sum_eq
thf(fact_4786_sum__choose__upper,axiom,
! [M: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [K3: nat] : ( binomial @ K3 @ M )
@ ( set_ord_atMost @ nat @ N ) )
= ( binomial @ ( suc @ N ) @ ( suc @ M ) ) ) ).
% sum_choose_upper
thf(fact_4787_sum__telescope,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [F3: nat > A,I: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( minus_minus @ A @ ( F3 @ I4 ) @ ( F3 @ ( suc @ I4 ) ) )
@ ( set_ord_atMost @ nat @ I ) )
= ( minus_minus @ A @ ( F3 @ ( zero_zero @ nat ) ) @ ( F3 @ ( suc @ I ) ) ) ) ) ).
% sum_telescope
thf(fact_4788_cosh__real__nonzero,axiom,
! [X: real] :
( ( cosh @ real @ X )
!= ( zero_zero @ real ) ) ).
% cosh_real_nonzero
thf(fact_4789_sum_OatMost__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G2: nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G2 @ ( set_ord_atMost @ nat @ ( suc @ N ) ) )
= ( plus_plus @ A @ ( G2 @ ( zero_zero @ nat ) )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( G2 @ ( suc @ I4 ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ) ).
% sum.atMost_Suc_shift
thf(fact_4790_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_4791_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_4792_sum__choose__lower,axiom,
! [R3: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [K3: nat] : ( binomial @ ( plus_plus @ nat @ R3 @ K3 ) @ K3 )
@ ( set_ord_atMost @ nat @ N ) )
= ( binomial @ ( suc @ ( plus_plus @ nat @ R3 @ N ) ) @ N ) ) ).
% sum_choose_lower
thf(fact_4793_polyfun__eq__coeffs,axiom,
! [A: $tType] :
( ( ( real_V8999393235501362500lgebra @ A )
& ( idom @ A ) )
=> ! [C2: nat > A,N: nat,D2: nat > A] :
( ( ! [X2: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( C2 @ I4 ) @ ( power_power @ A @ X2 @ I4 ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( D2 @ I4 ) @ ( power_power @ A @ X2 @ I4 ) )
@ ( set_ord_atMost @ nat @ N ) ) ) )
= ( ! [I4: nat] :
( ( ord_less_eq @ nat @ I4 @ N )
=> ( ( C2 @ I4 )
= ( D2 @ I4 ) ) ) ) ) ) ).
% polyfun_eq_coeffs
thf(fact_4794_polyfun__eq__0,axiom,
! [A: $tType] :
( ( ( real_V8999393235501362500lgebra @ A )
& ( idom @ A ) )
=> ! [C2: nat > A,N: nat] :
( ( ! [X2: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( C2 @ I4 ) @ ( power_power @ A @ X2 @ I4 ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( zero_zero @ A ) ) )
= ( ! [I4: nat] :
( ( ord_less_eq @ nat @ I4 @ N )
=> ( ( C2 @ I4 )
= ( zero_zero @ A ) ) ) ) ) ) ).
% polyfun_eq_0
thf(fact_4795_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
@ ^ [I4: nat] : ( times_times @ A @ ( C2 @ I4 ) @ ( power_power @ A @ W3 @ I4 ) )
@ ( 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_4796_gbinomial__parallel__sum,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( gbinomial @ A @ ( plus_plus @ A @ A3 @ ( semiring_1_of_nat @ A @ K3 ) ) @ K3 )
@ ( set_ord_atMost @ nat @ N ) )
= ( gbinomial @ A @ ( plus_plus @ A @ ( plus_plus @ A @ A3 @ ( semiring_1_of_nat @ A @ N ) ) @ ( one_one @ A ) ) @ N ) ) ) ).
% gbinomial_parallel_sum
thf(fact_4797_sum__choose__diagonal,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [K3: nat] : ( binomial @ ( minus_minus @ nat @ N @ K3 ) @ ( minus_minus @ nat @ M @ K3 ) )
@ ( set_ord_atMost @ nat @ M ) )
= ( binomial @ ( suc @ N ) @ M ) ) ) ).
% sum_choose_diagonal
thf(fact_4798_vandermonde,axiom,
! [M: nat,N: nat,R3: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [K3: nat] : ( times_times @ nat @ ( binomial @ M @ K3 ) @ ( binomial @ N @ ( minus_minus @ nat @ R3 @ K3 ) ) )
@ ( set_ord_atMost @ nat @ R3 ) )
= ( binomial @ ( plus_plus @ nat @ M @ N ) @ R3 ) ) ).
% vandermonde
thf(fact_4799_cosh__real__pos,axiom,
! [X: real] : ( ord_less @ real @ ( zero_zero @ real ) @ ( cosh @ real @ X ) ) ).
% cosh_real_pos
thf(fact_4800_cosh__real__nonneg,axiom,
! [X: real] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( cosh @ real @ X ) ) ).
% cosh_real_nonneg
thf(fact_4801_cosh__real__nonneg__le__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y )
=> ( ( ord_less_eq @ real @ ( cosh @ real @ X ) @ ( cosh @ real @ Y ) )
= ( ord_less_eq @ real @ X @ Y ) ) ) ) ).
% cosh_real_nonneg_le_iff
thf(fact_4802_cosh__real__nonpos__le__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ X @ ( zero_zero @ real ) )
=> ( ( ord_less_eq @ real @ Y @ ( zero_zero @ real ) )
=> ( ( ord_less_eq @ real @ ( cosh @ real @ X ) @ ( cosh @ real @ Y ) )
= ( ord_less_eq @ real @ Y @ X ) ) ) ) ).
% cosh_real_nonpos_le_iff
thf(fact_4803_atMost__Suc,axiom,
! [K: nat] :
( ( set_ord_atMost @ nat @ ( suc @ K ) )
= ( insert @ nat @ ( suc @ K ) @ ( set_ord_atMost @ nat @ K ) ) ) ).
% atMost_Suc
thf(fact_4804_sinh__double,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A] :
( ( sinh @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X ) )
= ( times_times @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( sinh @ A @ X ) ) @ ( cosh @ A @ X ) ) ) ) ).
% sinh_double
thf(fact_4805_cosh__real__ge__1,axiom,
! [X: real] : ( ord_less_eq @ real @ ( one_one @ real ) @ ( cosh @ real @ X ) ) ).
% cosh_real_ge_1
thf(fact_4806_sum__gp__basic,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [X: A,N: nat] :
( ( times_times @ A @ ( minus_minus @ A @ ( one_one @ A ) @ X ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_ord_atMost @ nat @ N ) ) )
= ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ X @ ( suc @ N ) ) ) ) ) ).
% sum_gp_basic
thf(fact_4807_suminf__sum,axiom,
! [A: $tType,I5: $tType] :
( ( ( topolo5987344860129210374id_add @ A )
& ( topological_t2_space @ A ) )
=> ! [I6: set @ I5,F3: I5 > nat > A] :
( ! [I3: I5] :
( ( member @ I5 @ I3 @ I6 )
=> ( summable @ A @ ( F3 @ I3 ) ) )
=> ( ( suminf @ A
@ ^ [N4: nat] :
( groups7311177749621191930dd_sum @ I5 @ A
@ ^ [I4: I5] : ( F3 @ I4 @ N4 )
@ I6 ) )
= ( groups7311177749621191930dd_sum @ I5 @ A
@ ^ [I4: I5] : ( suminf @ A @ ( F3 @ I4 ) )
@ I6 ) ) ) ) ).
% suminf_sum
thf(fact_4808_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_4809_binomial,axiom,
! [A3: nat,B3: nat,N: nat] :
( ( power_power @ nat @ ( plus_plus @ nat @ A3 @ B3 ) @ N )
= ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [K3: nat] : ( times_times @ nat @ ( times_times @ nat @ ( semiring_1_of_nat @ nat @ ( binomial @ N @ K3 ) ) @ ( power_power @ nat @ A3 @ K3 ) ) @ ( power_power @ nat @ B3 @ ( minus_minus @ nat @ N @ K3 ) ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ).
% binomial
thf(fact_4810_summable__Cauchy__product,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V4412858255891104859lgebra @ A ) )
=> ! [A3: nat > A,B3: nat > A] :
( ( summable @ real
@ ^ [K3: nat] : ( real_V7770717601297561774m_norm @ A @ ( A3 @ K3 ) ) )
=> ( ( summable @ real
@ ^ [K3: nat] : ( real_V7770717601297561774m_norm @ A @ ( B3 @ K3 ) ) )
=> ( summable @ A
@ ^ [K3: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( A3 @ I4 ) @ ( B3 @ ( minus_minus @ nat @ K3 @ I4 ) ) )
@ ( set_ord_atMost @ nat @ K3 ) ) ) ) ) ) ).
% summable_Cauchy_product
thf(fact_4811_Cauchy__product,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V4412858255891104859lgebra @ A ) )
=> ! [A3: nat > A,B3: nat > A] :
( ( summable @ real
@ ^ [K3: nat] : ( real_V7770717601297561774m_norm @ A @ ( A3 @ K3 ) ) )
=> ( ( summable @ real
@ ^ [K3: nat] : ( real_V7770717601297561774m_norm @ A @ ( B3 @ K3 ) ) )
=> ( ( times_times @ A @ ( suminf @ A @ A3 ) @ ( suminf @ A @ B3 ) )
= ( suminf @ A
@ ^ [K3: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( A3 @ I4 ) @ ( B3 @ ( minus_minus @ nat @ K3 @ I4 ) ) )
@ ( set_ord_atMost @ nat @ K3 ) ) ) ) ) ) ) ).
% Cauchy_product
thf(fact_4812_sum_Oin__pairs__0,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G2: nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G2 @ ( set_ord_atMost @ nat @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( plus_plus @ A @ ( G2 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I4 ) ) @ ( G2 @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I4 ) ) ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% sum.in_pairs_0
thf(fact_4813_polynomial__product,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [M: nat,A3: nat > A,N: nat,B3: nat > A,X: A] :
( ! [I3: nat] :
( ( ord_less @ nat @ M @ I3 )
=> ( ( A3 @ I3 )
= ( zero_zero @ A ) ) )
=> ( ! [J2: nat] :
( ( ord_less @ nat @ N @ J2 )
=> ( ( B3 @ J2 )
= ( zero_zero @ A ) ) )
=> ( ( times_times @ A
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( A3 @ I4 ) @ ( power_power @ A @ X @ I4 ) )
@ ( set_ord_atMost @ nat @ M ) )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [J3: nat] : ( times_times @ A @ ( B3 @ J3 ) @ ( power_power @ A @ X @ J3 ) )
@ ( set_ord_atMost @ nat @ N ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [R5: nat] :
( times_times @ A
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( times_times @ A @ ( A3 @ K3 ) @ ( B3 @ ( minus_minus @ nat @ R5 @ K3 ) ) )
@ ( set_ord_atMost @ nat @ R5 ) )
@ ( power_power @ A @ X @ R5 ) )
@ ( set_ord_atMost @ nat @ ( plus_plus @ nat @ M @ N ) ) ) ) ) ) ) ).
% polynomial_product
thf(fact_4814_gbinomial__sum__lower__neg,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: A,M: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( times_times @ A @ ( gbinomial @ A @ A3 @ K3 ) @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ K3 ) )
@ ( set_ord_atMost @ nat @ M ) )
= ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ M ) @ ( gbinomial @ A @ ( minus_minus @ A @ A3 @ ( one_one @ A ) ) @ M ) ) ) ) ).
% gbinomial_sum_lower_neg
thf(fact_4815_polynomial__product__nat,axiom,
! [M: nat,A3: nat > nat,N: nat,B3: nat > nat,X: nat] :
( ! [I3: nat] :
( ( ord_less @ nat @ M @ I3 )
=> ( ( A3 @ I3 )
= ( zero_zero @ nat ) ) )
=> ( ! [J2: nat] :
( ( ord_less @ nat @ N @ J2 )
=> ( ( B3 @ J2 )
= ( zero_zero @ nat ) ) )
=> ( ( times_times @ nat
@ ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [I4: nat] : ( times_times @ nat @ ( A3 @ I4 ) @ ( power_power @ nat @ X @ I4 ) )
@ ( set_ord_atMost @ nat @ M ) )
@ ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [J3: nat] : ( times_times @ nat @ ( B3 @ J3 ) @ ( power_power @ nat @ X @ J3 ) )
@ ( set_ord_atMost @ nat @ N ) ) )
= ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [R5: nat] :
( times_times @ nat
@ ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [K3: nat] : ( times_times @ nat @ ( A3 @ K3 ) @ ( B3 @ ( minus_minus @ nat @ R5 @ K3 ) ) )
@ ( set_ord_atMost @ nat @ R5 ) )
@ ( power_power @ nat @ X @ R5 ) )
@ ( set_ord_atMost @ nat @ ( plus_plus @ nat @ M @ N ) ) ) ) ) ) ).
% polynomial_product_nat
thf(fact_4816_binomial__ring,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A3: A,B3: A,N: nat] :
( ( power_power @ A @ ( plus_plus @ A @ A3 @ B3 ) @ N )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( times_times @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( binomial @ N @ K3 ) ) @ ( power_power @ A @ A3 @ K3 ) ) @ ( power_power @ A @ B3 @ ( minus_minus @ nat @ N @ K3 ) ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% binomial_ring
thf(fact_4817_choose__square__sum,axiom,
! [N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [K3: nat] : ( power_power @ nat @ ( binomial @ N @ K3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( binomial @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ N ) ) ).
% choose_square_sum
thf(fact_4818_pochhammer__binomial__sum,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [A3: A,B3: A,N: nat] :
( ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ A3 @ B3 ) @ N )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( times_times @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( binomial @ N @ K3 ) ) @ ( comm_s3205402744901411588hammer @ A @ A3 @ K3 ) ) @ ( comm_s3205402744901411588hammer @ A @ B3 @ ( minus_minus @ nat @ N @ K3 ) ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% pochhammer_binomial_sum
thf(fact_4819_cosh__real__nonpos__less__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ X @ ( zero_zero @ real ) )
=> ( ( ord_less_eq @ real @ Y @ ( zero_zero @ real ) )
=> ( ( ord_less @ real @ ( cosh @ real @ X ) @ ( cosh @ real @ Y ) )
= ( ord_less @ real @ Y @ X ) ) ) ) ).
% cosh_real_nonpos_less_iff
thf(fact_4820_cosh__real__nonneg__less__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y )
=> ( ( ord_less @ real @ ( cosh @ real @ X ) @ ( cosh @ real @ Y ) )
= ( ord_less @ real @ X @ Y ) ) ) ) ).
% cosh_real_nonneg_less_iff
thf(fact_4821_cosh__real__strict__mono,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ X @ Y )
=> ( ord_less @ real @ ( cosh @ real @ X ) @ ( cosh @ real @ Y ) ) ) ) ).
% cosh_real_strict_mono
thf(fact_4822_Cauchy__product__sums,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V4412858255891104859lgebra @ A ) )
=> ! [A3: nat > A,B3: nat > A] :
( ( summable @ real
@ ^ [K3: nat] : ( real_V7770717601297561774m_norm @ A @ ( A3 @ K3 ) ) )
=> ( ( summable @ real
@ ^ [K3: nat] : ( real_V7770717601297561774m_norm @ A @ ( B3 @ K3 ) ) )
=> ( sums @ A
@ ^ [K3: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( A3 @ I4 ) @ ( B3 @ ( minus_minus @ nat @ K3 @ I4 ) ) )
@ ( set_ord_atMost @ nat @ K3 ) )
@ ( times_times @ A @ ( suminf @ A @ A3 ) @ ( suminf @ A @ B3 ) ) ) ) ) ) ).
% Cauchy_product_sums
thf(fact_4823_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_4824_cosh__square__eq,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A] :
( ( power_power @ A @ ( cosh @ A @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( plus_plus @ A @ ( power_power @ A @ ( sinh @ A @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) ) ) ) ).
% cosh_square_eq
thf(fact_4825_hyperbolic__pythagoras,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A] :
( ( minus_minus @ A @ ( power_power @ A @ ( cosh @ A @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ ( sinh @ A @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( one_one @ A ) ) ) ).
% hyperbolic_pythagoras
thf(fact_4826_sinh__square__eq,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A] :
( ( power_power @ A @ ( sinh @ A @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( minus_minus @ A @ ( power_power @ A @ ( cosh @ A @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) ) ) ) ).
% sinh_square_eq
thf(fact_4827_sum__power__add,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [X: A,M: nat,I6: set @ nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( power_power @ A @ X @ ( plus_plus @ nat @ M @ I4 ) )
@ I6 )
= ( times_times @ A @ ( power_power @ A @ X @ M ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ I6 ) ) ) ) ).
% sum_power_add
thf(fact_4828_sum_Ozero__middle,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [P4: nat,K: nat,G2: nat > A,H2: nat > A] :
( ( ord_less_eq @ nat @ ( one_one @ nat ) @ P4 )
=> ( ( ord_less_eq @ nat @ K @ P4 )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [J3: nat] : ( if @ A @ ( ord_less @ nat @ J3 @ K ) @ ( G2 @ J3 ) @ ( if @ A @ ( J3 = K ) @ ( zero_zero @ A ) @ ( H2 @ ( minus_minus @ nat @ J3 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) )
@ ( set_ord_atMost @ nat @ P4 ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [J3: nat] : ( if @ A @ ( ord_less @ nat @ J3 @ K ) @ ( G2 @ J3 ) @ ( H2 @ J3 ) )
@ ( set_ord_atMost @ nat @ ( minus_minus @ nat @ P4 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ) ) ).
% sum.zero_middle
thf(fact_4829_arcosh__cosh__real,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( arcosh @ real @ ( cosh @ real @ X ) )
= X ) ) ).
% arcosh_cosh_real
thf(fact_4830_gbinomial__partial__sum__poly,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [M: nat,A3: A,X: A,Y: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( times_times @ A @ ( times_times @ A @ ( gbinomial @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ M ) @ A3 ) @ K3 ) @ ( power_power @ A @ X @ K3 ) ) @ ( power_power @ A @ Y @ ( minus_minus @ nat @ M @ K3 ) ) )
@ ( set_ord_atMost @ nat @ M ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( times_times @ A @ ( times_times @ A @ ( gbinomial @ A @ ( uminus_uminus @ A @ A3 ) @ K3 ) @ ( power_power @ A @ ( uminus_uminus @ A @ X ) @ K3 ) ) @ ( power_power @ A @ ( plus_plus @ A @ X @ Y ) @ ( minus_minus @ nat @ M @ K3 ) ) )
@ ( set_ord_atMost @ nat @ M ) ) ) ) ).
% gbinomial_partial_sum_poly
thf(fact_4831_exp__series__add__commuting,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X: A,Y: A,N: nat] :
( ( ( times_times @ A @ X @ Y )
= ( times_times @ A @ Y @ X ) )
=> ( ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ A @ ( plus_plus @ A @ X @ Y ) @ N ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ I4 ) ) @ ( power_power @ A @ X @ I4 ) ) @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ ( minus_minus @ nat @ N @ I4 ) ) ) @ ( power_power @ A @ Y @ ( minus_minus @ nat @ N @ I4 ) ) ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ) ).
% exp_series_add_commuting
thf(fact_4832_cosh__double,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A] :
( ( cosh @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X ) )
= ( plus_plus @ A @ ( power_power @ A @ ( cosh @ A @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ ( sinh @ A @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% cosh_double
thf(fact_4833_root__polyfun,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [N: nat,Z: A,A3: A] :
( ( ord_less_eq @ nat @ ( one_one @ nat ) @ N )
=> ( ( ( power_power @ A @ Z @ N )
= A3 )
= ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] :
( times_times @ A
@ ( if @ A
@ ( I4
= ( zero_zero @ nat ) )
@ ( uminus_uminus @ A @ A3 )
@ ( if @ A @ ( I4 = N ) @ ( one_one @ A ) @ ( zero_zero @ A ) ) )
@ ( power_power @ A @ Z @ I4 ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( zero_zero @ A ) ) ) ) ) ).
% root_polyfun
thf(fact_4834_sum__gp0,axiom,
! [A: $tType] :
( ( ( division_ring @ A )
& ( comm_ring @ A ) )
=> ! [X: A,N: nat] :
( ( ( X
= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_ord_atMost @ nat @ N ) )
= ( semiring_1_of_nat @ A @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) ) ) )
& ( ( X
!= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_ord_atMost @ nat @ N ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ X @ ( suc @ N ) ) ) @ ( minus_minus @ A @ ( one_one @ A ) @ X ) ) ) ) ) ) ).
% sum_gp0
thf(fact_4835_choose__alternating__linear__sum,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [N: nat] :
( ( N
!= ( one_one @ nat ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ I4 ) @ ( semiring_1_of_nat @ A @ I4 ) ) @ ( semiring_1_of_nat @ A @ ( binomial @ N @ I4 ) ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( zero_zero @ A ) ) ) ) ).
% choose_alternating_linear_sum
thf(fact_4836_gbinomial__sum__nat__pow2,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [M: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( divide_divide @ A @ ( gbinomial @ A @ ( semiring_1_of_nat @ A @ ( plus_plus @ nat @ M @ K3 ) ) @ K3 ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ K3 ) )
@ ( set_ord_atMost @ nat @ M ) )
= ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) ) ) ).
% gbinomial_sum_nat_pow2
thf(fact_4837_gbinomial__partial__sum__poly__xpos,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [M: nat,A3: A,X: A,Y: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( times_times @ A @ ( times_times @ A @ ( gbinomial @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ M ) @ A3 ) @ K3 ) @ ( power_power @ A @ X @ K3 ) ) @ ( power_power @ A @ Y @ ( minus_minus @ nat @ M @ K3 ) ) )
@ ( set_ord_atMost @ nat @ M ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( times_times @ A @ ( times_times @ A @ ( gbinomial @ A @ ( minus_minus @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ K3 ) @ A3 ) @ ( one_one @ A ) ) @ K3 ) @ ( power_power @ A @ X @ K3 ) ) @ ( power_power @ A @ ( plus_plus @ A @ X @ Y ) @ ( minus_minus @ nat @ M @ K3 ) ) )
@ ( set_ord_atMost @ nat @ M ) ) ) ) ).
% gbinomial_partial_sum_poly_xpos
thf(fact_4838_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_4839_choose__linear__sum,axiom,
! [N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [I4: nat] : ( times_times @ nat @ I4 @ ( binomial @ N @ I4 ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( times_times @ nat @ N @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ).
% choose_linear_sum
thf(fact_4840_choose__alternating__sum,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ I4 ) @ ( semiring_1_of_nat @ A @ ( binomial @ N @ I4 ) ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( zero_zero @ A ) ) ) ) ).
% choose_alternating_sum
thf(fact_4841_polyfun__extremal__lemma,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [E2: real,C2: nat > A,N: nat] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E2 )
=> ? [M9: real] :
! [Z6: A] :
( ( ord_less_eq @ real @ M9 @ ( real_V7770717601297561774m_norm @ A @ Z6 ) )
=> ( ord_less_eq @ real
@ ( real_V7770717601297561774m_norm @ A
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( C2 @ I4 ) @ ( power_power @ A @ Z6 @ I4 ) )
@ ( set_ord_atMost @ nat @ N ) ) )
@ ( times_times @ real @ E2 @ ( power_power @ real @ ( real_V7770717601297561774m_norm @ A @ Z6 ) @ ( suc @ N ) ) ) ) ) ) ) ).
% polyfun_extremal_lemma
thf(fact_4842_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_4843_gbinomial__partial__row__sum,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: A,M: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( times_times @ A @ ( gbinomial @ A @ A3 @ K3 ) @ ( minus_minus @ A @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( semiring_1_of_nat @ A @ K3 ) ) )
@ ( set_ord_atMost @ nat @ M ) )
= ( times_times @ A @ ( divide_divide @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ M ) @ ( one_one @ A ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( gbinomial @ A @ A3 @ ( plus_plus @ nat @ M @ ( one_one @ nat ) ) ) ) ) ) ).
% gbinomial_partial_row_sum
thf(fact_4844_choose__even__sum,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( if @ A @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I4 ) @ ( semiring_1_of_nat @ A @ ( binomial @ N @ I4 ) ) @ ( zero_zero @ A ) )
@ ( set_ord_atMost @ nat @ N ) ) )
= ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% choose_even_sum
thf(fact_4845_choose__odd__sum,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] :
( if @ A
@ ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I4 )
@ ( semiring_1_of_nat @ A @ ( binomial @ N @ I4 ) )
@ ( zero_zero @ A ) )
@ ( set_ord_atMost @ nat @ N ) ) )
= ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% choose_odd_sum
thf(fact_4846_tanh__add,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A,Y: A] :
( ( ( cosh @ A @ X )
!= ( zero_zero @ A ) )
=> ( ( ( cosh @ A @ Y )
!= ( zero_zero @ A ) )
=> ( ( tanh @ A @ ( plus_plus @ A @ X @ Y ) )
= ( divide_divide @ A @ ( plus_plus @ A @ ( tanh @ A @ X ) @ ( tanh @ A @ Y ) ) @ ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( tanh @ A @ X ) @ ( tanh @ A @ Y ) ) ) ) ) ) ) ) ).
% tanh_add
thf(fact_4847_mask__eq__sum__exp,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [N: nat] :
( ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) @ ( one_one @ A ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
@ ( collect @ nat
@ ^ [Q5: nat] : ( ord_less @ nat @ Q5 @ N ) ) ) ) ) ).
% mask_eq_sum_exp
thf(fact_4848_sinh__zero__iff,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A] :
( ( ( sinh @ A @ X )
= ( zero_zero @ A ) )
= ( member @ A @ ( exp @ A @ X ) @ ( insert @ A @ ( one_one @ A ) @ ( insert @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% sinh_zero_iff
thf(fact_4849_cosh__field__def,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ( ( cosh @ A )
= ( ^ [Z4: A] : ( divide_divide @ A @ ( plus_plus @ A @ ( exp @ A @ Z4 ) @ ( exp @ A @ ( uminus_uminus @ A @ Z4 ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% cosh_field_def
thf(fact_4850_mask__eq__sum__exp__nat,axiom,
! [N: nat] :
( ( minus_minus @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ ( suc @ ( zero_zero @ nat ) ) )
= ( groups7311177749621191930dd_sum @ nat @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
@ ( collect @ nat
@ ^ [Q5: nat] : ( ord_less @ nat @ Q5 @ N ) ) ) ) ).
% mask_eq_sum_exp_nat
thf(fact_4851_sinh__field__def,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ( ( sinh @ A )
= ( ^ [Z4: A] : ( divide_divide @ A @ ( minus_minus @ A @ ( exp @ A @ Z4 ) @ ( exp @ A @ ( uminus_uminus @ A @ Z4 ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% sinh_field_def
thf(fact_4852_cosh__zero__iff,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A] :
( ( ( cosh @ A @ X )
= ( zero_zero @ A ) )
= ( ( power_power @ A @ ( exp @ A @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ).
% cosh_zero_iff
thf(fact_4853_cosh__def,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ( ( cosh @ A )
= ( ^ [X2: A] : ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( plus_plus @ A @ ( exp @ A @ X2 ) @ ( exp @ A @ ( uminus_uminus @ A @ X2 ) ) ) ) ) ) ) ).
% cosh_def
thf(fact_4854_cosh__ln__real,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( cosh @ real @ ( ln_ln @ real @ X ) )
= ( divide_divide @ real @ ( plus_plus @ real @ X @ ( inverse_inverse @ real @ X ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ).
% cosh_ln_real
thf(fact_4855_sinh__def,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ( ( sinh @ A )
= ( ^ [X2: A] : ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( minus_minus @ A @ ( exp @ A @ X2 ) @ ( exp @ A @ ( uminus_uminus @ A @ X2 ) ) ) ) ) ) ) ).
% sinh_def
thf(fact_4856_sinh__ln__real,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( sinh @ real @ ( ln_ln @ real @ X ) )
= ( divide_divide @ real @ ( minus_minus @ real @ X @ ( inverse_inverse @ real @ X ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ).
% sinh_ln_real
thf(fact_4857_cos__x__cos__y,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A,Y: A] :
( sums @ A
@ ^ [P6: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [N4: nat] :
( if @ A
@ ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ P6 )
& ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N4 ) )
@ ( times_times @ A @ ( real_V8093663219630862766scaleR @ A @ ( divide_divide @ real @ ( ring_1_of_int @ real @ ( times_times @ int @ ( power_power @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( divide_divide @ nat @ P6 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( semiring_1_of_nat @ int @ ( binomial @ P6 @ N4 ) ) ) ) @ ( semiring_char_0_fact @ real @ P6 ) ) @ ( power_power @ A @ X @ N4 ) ) @ ( power_power @ A @ Y @ ( minus_minus @ nat @ P6 @ N4 ) ) )
@ ( zero_zero @ A ) )
@ ( set_ord_atMost @ nat @ P6 ) )
@ ( times_times @ A @ ( cos @ A @ X ) @ ( cos @ A @ Y ) ) ) ) ).
% cos_x_cos_y
thf(fact_4858_sums__cos__x__plus__y,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A,Y: A] :
( sums @ A
@ ^ [P6: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [N4: nat] : ( if @ A @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ P6 ) @ ( 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 @ P6 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( semiring_1_of_nat @ int @ ( binomial @ P6 @ N4 ) ) ) ) @ ( semiring_char_0_fact @ real @ P6 ) ) @ ( power_power @ A @ X @ N4 ) ) @ ( power_power @ A @ Y @ ( minus_minus @ nat @ P6 @ N4 ) ) ) @ ( zero_zero @ A ) )
@ ( set_ord_atMost @ nat @ P6 ) )
@ ( cos @ A @ ( plus_plus @ A @ X @ Y ) ) ) ) ).
% sums_cos_x_plus_y
thf(fact_4859_sum__abs__ge__zero,axiom,
! [B: $tType,A: $tType] :
( ( ordere166539214618696060dd_abs @ B )
=> ! [F3: A > B,A2: set @ A] :
( ord_less_eq @ B @ ( zero_zero @ B )
@ ( groups7311177749621191930dd_sum @ A @ B
@ ^ [I4: A] : ( abs_abs @ B @ ( F3 @ I4 ) )
@ A2 ) ) ) ).
% sum_abs_ge_zero
thf(fact_4860_sum__abs,axiom,
! [B: $tType,A: $tType] :
( ( ordere166539214618696060dd_abs @ B )
=> ! [F3: A > B,A2: set @ A] :
( ord_less_eq @ B @ ( abs_abs @ B @ ( groups7311177749621191930dd_sum @ A @ B @ F3 @ A2 ) )
@ ( groups7311177749621191930dd_sum @ A @ B
@ ^ [I4: A] : ( abs_abs @ B @ ( F3 @ I4 ) )
@ A2 ) ) ) ).
% sum_abs
thf(fact_4861_sum_Oempty,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G2: B > A] :
( ( groups7311177749621191930dd_sum @ B @ A @ G2 @ ( bot_bot @ ( set @ B ) ) )
= ( zero_zero @ A ) ) ) ).
% sum.empty
thf(fact_4862_sum_Oneutral__const,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A2: set @ B] :
( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [Uu: B] : ( zero_zero @ A )
@ A2 )
= ( zero_zero @ A ) ) ) ).
% sum.neutral_const
thf(fact_4863_abs__sum__abs,axiom,
! [B: $tType,A: $tType] :
( ( ordere166539214618696060dd_abs @ B )
=> ! [F3: A > B,A2: set @ A] :
( ( abs_abs @ B
@ ( groups7311177749621191930dd_sum @ A @ B
@ ^ [A5: A] : ( abs_abs @ B @ ( F3 @ A5 ) )
@ A2 ) )
= ( groups7311177749621191930dd_sum @ A @ B
@ ^ [A5: A] : ( abs_abs @ B @ ( F3 @ A5 ) )
@ A2 ) ) ) ).
% abs_sum_abs
thf(fact_4864_int__sum,axiom,
! [B: $tType,F3: B > nat,A2: set @ B] :
( ( semiring_1_of_nat @ int @ ( groups7311177749621191930dd_sum @ B @ nat @ F3 @ A2 ) )
= ( groups7311177749621191930dd_sum @ B @ int
@ ^ [X2: B] : ( semiring_1_of_nat @ int @ ( F3 @ X2 ) )
@ A2 ) ) ).
% int_sum
thf(fact_4865_sum__subtractf__nat,axiom,
! [A: $tType,A2: set @ A,G2: A > nat,F3: A > nat] :
( ! [X3: A] :
( ( member @ A @ X3 @ A2 )
=> ( ord_less_eq @ nat @ ( G2 @ X3 ) @ ( F3 @ X3 ) ) )
=> ( ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [X2: A] : ( minus_minus @ nat @ ( F3 @ X2 ) @ ( G2 @ X2 ) )
@ A2 )
= ( minus_minus @ nat @ ( groups7311177749621191930dd_sum @ A @ nat @ F3 @ A2 ) @ ( groups7311177749621191930dd_sum @ A @ nat @ G2 @ A2 ) ) ) ) ).
% sum_subtractf_nat
thf(fact_4866_Complex__sum_H,axiom,
! [A: $tType,F3: A > real,S2: set @ A] :
( ( groups7311177749621191930dd_sum @ A @ complex
@ ^ [X2: A] : ( complex2 @ ( F3 @ X2 ) @ ( zero_zero @ real ) )
@ S2 )
= ( complex2 @ ( groups7311177749621191930dd_sum @ A @ real @ F3 @ S2 ) @ ( zero_zero @ real ) ) ) ).
% Complex_sum'
thf(fact_4867_sum__SucD,axiom,
! [A: $tType,F3: A > nat,A2: set @ A,N: nat] :
( ( ( groups7311177749621191930dd_sum @ A @ nat @ F3 @ A2 )
= ( suc @ N ) )
=> ? [X3: A] :
( ( member @ A @ X3 @ A2 )
& ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F3 @ X3 ) ) ) ) ).
% sum_SucD
thf(fact_4868_sum__diff1__nat,axiom,
! [A: $tType,A3: A,A2: set @ A,F3: A > nat] :
( ( ( member @ A @ A3 @ A2 )
=> ( ( groups7311177749621191930dd_sum @ A @ nat @ F3 @ ( minus_minus @ ( set @ A ) @ A2 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( minus_minus @ nat @ ( groups7311177749621191930dd_sum @ A @ nat @ F3 @ A2 ) @ ( F3 @ A3 ) ) ) )
& ( ~ ( member @ A @ A3 @ A2 )
=> ( ( groups7311177749621191930dd_sum @ A @ nat @ F3 @ ( minus_minus @ ( set @ A ) @ A2 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( groups7311177749621191930dd_sum @ A @ nat @ F3 @ A2 ) ) ) ) ).
% sum_diff1_nat
thf(fact_4869_sum__nth__roots,axiom,
! [N: nat,C2: complex] :
( ( ord_less @ nat @ ( one_one @ nat ) @ N )
=> ( ( groups7311177749621191930dd_sum @ complex @ complex
@ ^ [X2: complex] : X2
@ ( collect @ complex
@ ^ [Z4: complex] :
( ( power_power @ complex @ Z4 @ N )
= C2 ) ) )
= ( zero_zero @ complex ) ) ) ).
% sum_nth_roots
thf(fact_4870_sum_Oswap,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( comm_monoid_add @ A )
=> ! [G2: B > C > A,B2: set @ C,A2: set @ B] :
( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [I4: B] : ( groups7311177749621191930dd_sum @ C @ A @ ( G2 @ I4 ) @ B2 )
@ A2 )
= ( groups7311177749621191930dd_sum @ C @ A
@ ^ [J3: C] :
( groups7311177749621191930dd_sum @ B @ A
@ ^ [I4: B] : ( G2 @ I4 @ J3 )
@ A2 )
@ B2 ) ) ) ).
% sum.swap
thf(fact_4871_sum__roots__unity,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( one_one @ nat ) @ N )
=> ( ( groups7311177749621191930dd_sum @ complex @ complex
@ ^ [X2: complex] : X2
@ ( collect @ complex
@ ^ [Z4: complex] :
( ( power_power @ complex @ Z4 @ N )
= ( one_one @ complex ) ) ) )
= ( zero_zero @ complex ) ) ) ).
% sum_roots_unity
thf(fact_4872_sum_Oneutral,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A2: set @ B,G2: B > A] :
( ! [X3: B] :
( ( member @ B @ X3 @ A2 )
=> ( ( G2 @ X3 )
= ( zero_zero @ A ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G2 @ A2 )
= ( zero_zero @ A ) ) ) ) ).
% sum.neutral
thf(fact_4873_sum_Onot__neutral__contains__not__neutral,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G2: B > A,A2: set @ B] :
( ( ( groups7311177749621191930dd_sum @ B @ A @ G2 @ A2 )
!= ( zero_zero @ A ) )
=> ~ ! [A6: B] :
( ( member @ B @ A6 @ A2 )
=> ( ( G2 @ A6 )
= ( zero_zero @ A ) ) ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_4874_sum__mono,axiom,
! [A: $tType,B: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [K4: set @ B,F3: B > A,G2: B > A] :
( ! [I3: B] :
( ( member @ B @ I3 @ K4 )
=> ( ord_less_eq @ A @ ( F3 @ I3 ) @ ( G2 @ I3 ) ) )
=> ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ K4 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G2 @ K4 ) ) ) ) ).
% sum_mono
thf(fact_4875_sum_Odistrib,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [G2: B > A,H2: B > A,A2: set @ B] :
( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X2: B] : ( plus_plus @ A @ ( G2 @ X2 ) @ ( H2 @ X2 ) )
@ A2 )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ G2 @ A2 ) @ ( groups7311177749621191930dd_sum @ B @ A @ H2 @ A2 ) ) ) ) ).
% sum.distrib
thf(fact_4876_sum__product,axiom,
! [B: $tType,C: $tType,A: $tType] :
( ( semiring_0 @ B )
=> ! [F3: A > B,A2: set @ A,G2: C > B,B2: set @ C] :
( ( times_times @ B @ ( groups7311177749621191930dd_sum @ A @ B @ F3 @ A2 ) @ ( groups7311177749621191930dd_sum @ C @ B @ G2 @ B2 ) )
= ( groups7311177749621191930dd_sum @ A @ B
@ ^ [I4: A] :
( groups7311177749621191930dd_sum @ C @ B
@ ^ [J3: C] : ( times_times @ B @ ( F3 @ I4 ) @ ( G2 @ J3 ) )
@ B2 )
@ A2 ) ) ) ).
% sum_product
thf(fact_4877_sum__distrib__right,axiom,
! [A: $tType,B: $tType] :
( ( semiring_0 @ A )
=> ! [F3: B > A,A2: set @ B,R3: A] :
( ( times_times @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ A2 ) @ R3 )
= ( groups7311177749621191930dd_sum @ B @ A
@ ^ [N4: B] : ( times_times @ A @ ( F3 @ N4 ) @ R3 )
@ A2 ) ) ) ).
% sum_distrib_right
thf(fact_4878_sum__distrib__left,axiom,
! [A: $tType,B: $tType] :
( ( semiring_0 @ A )
=> ! [R3: A,F3: B > A,A2: set @ B] :
( ( times_times @ A @ R3 @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ A2 ) )
= ( groups7311177749621191930dd_sum @ B @ A
@ ^ [N4: B] : ( times_times @ A @ R3 @ ( F3 @ N4 ) )
@ A2 ) ) ) ).
% sum_distrib_left
thf(fact_4879_sum__subtractf,axiom,
! [A: $tType,B: $tType] :
( ( ab_group_add @ A )
=> ! [F3: B > A,G2: B > A,A2: set @ B] :
( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X2: B] : ( minus_minus @ A @ ( F3 @ X2 ) @ ( G2 @ X2 ) )
@ A2 )
= ( minus_minus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ A2 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G2 @ A2 ) ) ) ) ).
% sum_subtractf
thf(fact_4880_sum__divide__distrib,axiom,
! [A: $tType,B: $tType] :
( ( field @ A )
=> ! [F3: B > A,A2: set @ B,R3: A] :
( ( divide_divide @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ A2 ) @ R3 )
= ( groups7311177749621191930dd_sum @ B @ A
@ ^ [N4: B] : ( divide_divide @ A @ ( F3 @ N4 ) @ R3 )
@ A2 ) ) ) ).
% sum_divide_distrib
thf(fact_4881_sum__negf,axiom,
! [A: $tType,B: $tType] :
( ( ab_group_add @ A )
=> ! [F3: B > A,A2: set @ B] :
( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X2: B] : ( uminus_uminus @ A @ ( F3 @ X2 ) )
@ A2 )
= ( uminus_uminus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ A2 ) ) ) ) ).
% sum_negf
thf(fact_4882_sum__nonpos,axiom,
! [B: $tType,A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A2: set @ B,F3: B > A] :
( ! [X3: B] :
( ( member @ B @ X3 @ A2 )
=> ( ord_less_eq @ A @ ( F3 @ X3 ) @ ( zero_zero @ A ) ) )
=> ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ A2 ) @ ( zero_zero @ A ) ) ) ) ).
% sum_nonpos
thf(fact_4883_sum__nonneg,axiom,
! [A: $tType,B: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A2: set @ B,F3: B > A] :
( ! [X3: B] :
( ( member @ B @ X3 @ A2 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F3 @ X3 ) ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ A2 ) ) ) ) ).
% sum_nonneg
thf(fact_4884_sum__cong__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A2: set @ nat,F3: nat > A,G2: nat > A] :
( ~ ( member @ nat @ ( zero_zero @ nat ) @ A2 )
=> ( ! [X3: nat] :
( ( member @ nat @ ( suc @ X3 ) @ A2 )
=> ( ( F3 @ ( suc @ X3 ) )
= ( G2 @ ( suc @ X3 ) ) ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ A2 )
= ( groups7311177749621191930dd_sum @ nat @ A @ G2 @ A2 ) ) ) ) ) ).
% sum_cong_Suc
thf(fact_4885_convex__sum__bound__le,axiom,
! [A: $tType,B: $tType] :
( ( linordered_idom @ B )
=> ! [I6: set @ A,X: A > B,A3: A > B,B3: B,Delta: B] :
( ! [I3: A] :
( ( member @ A @ I3 @ I6 )
=> ( ord_less_eq @ B @ ( zero_zero @ B ) @ ( X @ I3 ) ) )
=> ( ( ( groups7311177749621191930dd_sum @ A @ B @ X @ I6 )
= ( one_one @ B ) )
=> ( ! [I3: A] :
( ( member @ A @ I3 @ I6 )
=> ( ord_less_eq @ B @ ( abs_abs @ B @ ( minus_minus @ B @ ( A3 @ I3 ) @ B3 ) ) @ Delta ) )
=> ( ord_less_eq @ B
@ ( abs_abs @ B
@ ( minus_minus @ B
@ ( groups7311177749621191930dd_sum @ A @ B
@ ^ [I4: A] : ( times_times @ B @ ( A3 @ I4 ) @ ( X @ I4 ) )
@ I6 )
@ B3 ) )
@ Delta ) ) ) ) ) ).
% convex_sum_bound_le
thf(fact_4886_Maclaurin__cos__expansion2,axiom,
! [X: real,N: nat] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ? [T3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ T3 )
& ( ord_less @ real @ T3 @ X )
& ( ( cos @ real @ X )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M3: nat] : ( times_times @ real @ ( cos_coeff @ M3 ) @ ( power_power @ real @ X @ M3 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( cos @ real @ ( plus_plus @ real @ T3 @ ( times_times @ real @ ( times_times @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( semiring_1_of_nat @ real @ N ) ) @ pi ) ) ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ X @ N ) ) ) ) ) ) ) ).
% Maclaurin_cos_expansion2
thf(fact_4887_Maclaurin__minus__cos__expansion,axiom,
! [N: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ X @ ( zero_zero @ real ) )
=> ? [T3: real] :
( ( ord_less @ real @ X @ T3 )
& ( ord_less @ real @ T3 @ ( zero_zero @ real ) )
& ( ( cos @ real @ X )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M3: nat] : ( times_times @ real @ ( cos_coeff @ M3 ) @ ( power_power @ real @ X @ M3 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( cos @ real @ ( plus_plus @ real @ T3 @ ( times_times @ real @ ( times_times @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( semiring_1_of_nat @ real @ N ) ) @ pi ) ) ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ X @ N ) ) ) ) ) ) ) ).
% Maclaurin_minus_cos_expansion
thf(fact_4888_Maclaurin__sin__expansion3,axiom,
! [N: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ? [T3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ T3 )
& ( ord_less @ real @ T3 @ X )
& ( ( sin @ real @ X )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M3: nat] : ( times_times @ real @ ( sin_coeff @ M3 ) @ ( power_power @ real @ X @ M3 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( sin @ real @ ( plus_plus @ real @ T3 @ ( times_times @ real @ ( times_times @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( semiring_1_of_nat @ real @ N ) ) @ pi ) ) ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ X @ N ) ) ) ) ) ) ) ).
% Maclaurin_sin_expansion3
thf(fact_4889_lessThan__iff,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [I: A,K: A] :
( ( member @ A @ I @ ( set_ord_lessThan @ A @ K ) )
= ( ord_less @ A @ I @ K ) ) ) ).
% lessThan_iff
thf(fact_4890_lessThan__0,axiom,
( ( set_ord_lessThan @ nat @ ( zero_zero @ nat ) )
= ( bot_bot @ ( set @ nat ) ) ) ).
% lessThan_0
thf(fact_4891_single__Diff__lessThan,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [K: A] :
( ( minus_minus @ ( set @ A ) @ ( insert @ A @ K @ ( bot_bot @ ( set @ A ) ) ) @ ( set_ord_lessThan @ A @ K ) )
= ( insert @ A @ K @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% single_Diff_lessThan
thf(fact_4892_sumr__cos__zero__one,axiom,
! [N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M3: nat] : ( times_times @ real @ ( cos_coeff @ M3 ) @ ( power_power @ real @ ( zero_zero @ real ) @ M3 ) )
@ ( set_ord_lessThan @ nat @ ( suc @ N ) ) )
= ( one_one @ real ) ) ).
% sumr_cos_zero_one
thf(fact_4893_lessThan__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( set_ord_lessThan @ A )
= ( ^ [U: A] :
( collect @ A
@ ^ [X2: A] : ( ord_less @ A @ X2 @ U ) ) ) ) ) ).
% lessThan_def
thf(fact_4894_lessThan__non__empty,axiom,
! [A: $tType] :
( ( no_bot @ A )
=> ! [X: A] :
( ( set_ord_lessThan @ A @ X )
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% lessThan_non_empty
thf(fact_4895_Iio__eq__empty__iff,axiom,
! [A: $tType] :
( ( ( linorder @ A )
& ( order_bot @ A ) )
=> ! [N: A] :
( ( ( set_ord_lessThan @ A @ N )
= ( bot_bot @ ( set @ A ) ) )
= ( N
= ( bot_bot @ A ) ) ) ) ).
% Iio_eq_empty_iff
thf(fact_4896_lessThan__strict__subset__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [M: A,N: A] :
( ( ord_less @ ( set @ A ) @ ( set_ord_lessThan @ A @ M ) @ ( set_ord_lessThan @ A @ N ) )
= ( ord_less @ A @ M @ N ) ) ) ).
% lessThan_strict_subset_iff
thf(fact_4897_lessThan__empty__iff,axiom,
! [N: nat] :
( ( ( set_ord_lessThan @ nat @ N )
= ( bot_bot @ ( set @ nat ) ) )
= ( N
= ( zero_zero @ nat ) ) ) ).
% lessThan_empty_iff
thf(fact_4898_lessThan__Suc,axiom,
! [K: nat] :
( ( set_ord_lessThan @ nat @ ( suc @ K ) )
= ( insert @ nat @ K @ ( set_ord_lessThan @ nat @ K ) ) ) ).
% lessThan_Suc
thf(fact_4899_Iic__subset__Iio__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_ord_atMost @ A @ A3 ) @ ( set_ord_lessThan @ A @ B3 ) )
= ( ord_less @ A @ A3 @ B3 ) ) ) ).
% Iic_subset_Iio_iff
thf(fact_4900_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_4901_sum_Onat__diff__reindex,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G2: nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( G2 @ ( minus_minus @ nat @ N @ ( suc @ I4 ) ) )
@ ( set_ord_lessThan @ nat @ N ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ G2 @ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% sum.nat_diff_reindex
thf(fact_4902_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
@ ^ [X2: A] : ( minus_minus @ nat @ ( P @ X2 ) @ ( Q @ X2 ) )
@ ( set_ord_lessThan @ A @ N ) ) ) ) ) ).
% sum_diff_distrib
thf(fact_4903_ivl__disj__un__singleton_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [U2: A] :
( ( sup_sup @ ( set @ A ) @ ( set_ord_lessThan @ A @ U2 ) @ ( insert @ A @ U2 @ ( bot_bot @ ( set @ A ) ) ) )
= ( set_ord_atMost @ A @ U2 ) ) ) ).
% ivl_disj_un_singleton(2)
thf(fact_4904_sum_OlessThan__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G2: nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G2 @ ( set_ord_lessThan @ nat @ ( suc @ N ) ) )
= ( plus_plus @ A @ ( G2 @ ( zero_zero @ nat ) )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( G2 @ ( suc @ I4 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% sum.lessThan_Suc_shift
thf(fact_4905_sum__lessThan__telescope,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [F3: nat > A,M: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [N4: nat] : ( minus_minus @ A @ ( F3 @ ( suc @ N4 ) ) @ ( F3 @ N4 ) )
@ ( set_ord_lessThan @ nat @ M ) )
= ( minus_minus @ A @ ( F3 @ M ) @ ( F3 @ ( zero_zero @ nat ) ) ) ) ) ).
% sum_lessThan_telescope
thf(fact_4906_sum__lessThan__telescope_H,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [F3: nat > A,M: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [N4: nat] : ( minus_minus @ A @ ( F3 @ N4 ) @ ( F3 @ ( suc @ N4 ) ) )
@ ( set_ord_lessThan @ nat @ M ) )
= ( minus_minus @ A @ ( F3 @ ( zero_zero @ nat ) ) @ ( F3 @ M ) ) ) ) ).
% sum_lessThan_telescope'
thf(fact_4907_sumr__diff__mult__const2,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [F3: nat > A,N: nat,R3: A] :
( ( minus_minus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_ord_lessThan @ nat @ N ) ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ R3 ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( minus_minus @ A @ ( F3 @ I4 ) @ R3 )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% sumr_diff_mult_const2
thf(fact_4908_summableI__nonneg__bounded,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( ordere6911136660526730532id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F3: nat > A,X: A] :
( ! [N2: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F3 @ N2 ) )
=> ( ! [N2: nat] : ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_ord_lessThan @ nat @ N2 ) ) @ X )
=> ( summable @ A @ F3 ) ) ) ) ).
% summableI_nonneg_bounded
thf(fact_4909_sums__iff__shift,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: nat > A,N: nat,S2: A] :
( ( sums @ A
@ ^ [I4: nat] : ( F3 @ ( plus_plus @ nat @ I4 @ N ) )
@ S2 )
= ( sums @ A @ F3 @ ( plus_plus @ A @ S2 @ ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ) ).
% sums_iff_shift
thf(fact_4910_sums__iff__shift_H,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: nat > A,N: nat,S2: A] :
( ( sums @ A
@ ^ [I4: nat] : ( F3 @ ( plus_plus @ nat @ I4 @ N ) )
@ ( minus_minus @ A @ S2 @ ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_ord_lessThan @ nat @ N ) ) ) )
= ( sums @ A @ F3 @ S2 ) ) ) ).
% sums_iff_shift'
thf(fact_4911_sums__split__initial__segment,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: nat > A,S2: A,N: nat] :
( ( sums @ A @ F3 @ S2 )
=> ( sums @ A
@ ^ [I4: nat] : ( F3 @ ( plus_plus @ nat @ I4 @ N ) )
@ ( minus_minus @ A @ S2 @ ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ) ).
% sums_split_initial_segment
thf(fact_4912_power__diff__1__eq,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [X: A,N: nat] :
( ( minus_minus @ A @ ( power_power @ A @ X @ N ) @ ( one_one @ A ) )
= ( times_times @ A @ ( minus_minus @ A @ X @ ( one_one @ A ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% power_diff_1_eq
thf(fact_4913_one__diff__power__eq,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [X: A,N: nat] :
( ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ X @ N ) )
= ( times_times @ A @ ( minus_minus @ A @ ( one_one @ A ) @ X ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% one_diff_power_eq
thf(fact_4914_geometric__sum,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [X: A,N: nat] :
( ( X
!= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_ord_lessThan @ nat @ N ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( power_power @ A @ X @ N ) @ ( one_one @ A ) ) @ ( minus_minus @ A @ X @ ( one_one @ A ) ) ) ) ) ) ).
% geometric_sum
thf(fact_4915_sum_OatMost__shift,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G2: nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G2 @ ( set_ord_atMost @ nat @ N ) )
= ( plus_plus @ A @ ( G2 @ ( zero_zero @ nat ) )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( G2 @ ( suc @ I4 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% sum.atMost_shift
thf(fact_4916_suminf__split__initial__segment,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: nat > A,K: nat] :
( ( summable @ A @ F3 )
=> ( ( suminf @ A @ F3 )
= ( plus_plus @ A
@ ( suminf @ A
@ ^ [N4: nat] : ( F3 @ ( plus_plus @ nat @ N4 @ K ) ) )
@ ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_ord_lessThan @ nat @ K ) ) ) ) ) ) ).
% suminf_split_initial_segment
thf(fact_4917_suminf__minus__initial__segment,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: nat > A,K: nat] :
( ( summable @ A @ F3 )
=> ( ( suminf @ A
@ ^ [N4: nat] : ( F3 @ ( plus_plus @ nat @ N4 @ K ) ) )
= ( minus_minus @ A @ ( suminf @ A @ F3 ) @ ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_ord_lessThan @ nat @ K ) ) ) ) ) ) ).
% suminf_minus_initial_segment
thf(fact_4918_sum__less__suminf,axiom,
! [A: $tType] :
( ( ( ordere8940638589300402666id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F3: nat > A,N: nat] :
( ( summable @ A @ F3 )
=> ( ! [M4: nat] :
( ( ord_less_eq @ nat @ N @ M4 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( F3 @ M4 ) ) )
=> ( ord_less @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_ord_lessThan @ nat @ N ) ) @ ( suminf @ A @ F3 ) ) ) ) ) ).
% sum_less_suminf
thf(fact_4919_lemma__termdiff1,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [Z: A,H2: A,M: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [P6: nat] : ( minus_minus @ A @ ( times_times @ A @ ( power_power @ A @ ( plus_plus @ A @ Z @ H2 ) @ ( minus_minus @ nat @ M @ P6 ) ) @ ( power_power @ A @ Z @ P6 ) ) @ ( power_power @ A @ Z @ M ) )
@ ( set_ord_lessThan @ nat @ M ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [P6: nat] : ( times_times @ A @ ( power_power @ A @ Z @ P6 ) @ ( minus_minus @ A @ ( power_power @ A @ ( plus_plus @ A @ Z @ H2 ) @ ( minus_minus @ nat @ M @ P6 ) ) @ ( power_power @ A @ Z @ ( minus_minus @ nat @ M @ P6 ) ) ) )
@ ( set_ord_lessThan @ nat @ M ) ) ) ) ).
% lemma_termdiff1
thf(fact_4920_sum__gp__strict,axiom,
! [A: $tType] :
( ( ( division_ring @ A )
& ( comm_ring @ A ) )
=> ! [X: A,N: nat] :
( ( ( X
= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_ord_lessThan @ nat @ N ) )
= ( semiring_1_of_nat @ A @ N ) ) )
& ( ( X
!= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_ord_lessThan @ nat @ N ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ X @ N ) ) @ ( minus_minus @ A @ ( one_one @ A ) @ X ) ) ) ) ) ) ).
% sum_gp_strict
thf(fact_4921_diff__power__eq__sum,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [X: A,N: nat,Y: A] :
( ( minus_minus @ A @ ( power_power @ A @ X @ ( suc @ N ) ) @ ( power_power @ A @ Y @ ( suc @ N ) ) )
= ( times_times @ A @ ( minus_minus @ A @ X @ Y )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [P6: nat] : ( times_times @ A @ ( power_power @ A @ X @ P6 ) @ ( power_power @ A @ Y @ ( minus_minus @ nat @ N @ P6 ) ) )
@ ( set_ord_lessThan @ nat @ ( suc @ N ) ) ) ) ) ) ).
% diff_power_eq_sum
thf(fact_4922_power__diff__sumr2,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [X: A,N: nat,Y: A] :
( ( minus_minus @ A @ ( power_power @ A @ X @ N ) @ ( power_power @ A @ Y @ N ) )
= ( times_times @ A @ ( minus_minus @ A @ X @ Y )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( power_power @ A @ Y @ ( minus_minus @ nat @ N @ ( suc @ I4 ) ) ) @ ( power_power @ A @ X @ I4 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% power_diff_sumr2
thf(fact_4923_polyfun__linear__factor__root,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [C2: nat > A,A3: A,N: nat] :
( ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( C2 @ I4 ) @ ( power_power @ A @ A3 @ I4 ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( zero_zero @ A ) )
=> ~ ! [B7: nat > A] :
~ ! [Z6: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( C2 @ I4 ) @ ( power_power @ A @ Z6 @ I4 ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( times_times @ A @ ( minus_minus @ A @ Z6 @ A3 )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( B7 @ I4 ) @ ( power_power @ A @ Z6 @ I4 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ) ).
% polyfun_linear_factor_root
thf(fact_4924_polyfun__linear__factor,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [C2: nat > A,N: nat,A3: A] :
? [B7: nat > A] :
! [Z6: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( C2 @ I4 ) @ ( power_power @ A @ Z6 @ I4 ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( plus_plus @ A
@ ( times_times @ A @ ( minus_minus @ A @ Z6 @ A3 )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( B7 @ I4 ) @ ( power_power @ A @ Z6 @ I4 ) )
@ ( set_ord_lessThan @ nat @ N ) ) )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( C2 @ I4 ) @ ( power_power @ A @ A3 @ I4 ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ) ).
% polyfun_linear_factor
thf(fact_4925_real__sum__nat__ivl__bounded2,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N: nat,F3: nat > A,K4: A,K: nat] :
( ! [P7: nat] :
( ( ord_less @ nat @ P7 @ N )
=> ( ord_less_eq @ A @ ( F3 @ P7 ) @ K4 ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ K4 )
=> ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_ord_lessThan @ nat @ ( minus_minus @ nat @ N @ K ) ) ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ K4 ) ) ) ) ) ).
% real_sum_nat_ivl_bounded2
thf(fact_4926_sum__less__suminf2,axiom,
! [A: $tType] :
( ( ( ordere8940638589300402666id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F3: nat > A,N: nat,I: nat] :
( ( summable @ A @ F3 )
=> ( ! [M4: nat] :
( ( ord_less_eq @ nat @ N @ M4 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F3 @ M4 ) ) )
=> ( ( ord_less_eq @ nat @ N @ I )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ ( F3 @ I ) )
=> ( ord_less @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_ord_lessThan @ nat @ N ) ) @ ( suminf @ A @ F3 ) ) ) ) ) ) ) ).
% sum_less_suminf2
thf(fact_4927_one__diff__power__eq_H,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [X: A,N: nat] :
( ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ X @ N ) )
= ( times_times @ A @ ( minus_minus @ A @ ( one_one @ A ) @ X )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( power_power @ A @ X @ ( minus_minus @ nat @ N @ ( suc @ I4 ) ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% one_diff_power_eq'
thf(fact_4928_Maclaurin__zero,axiom,
! [A: $tType] :
( ( zero @ A )
=> ! [X: real,N: nat,Diff: nat > A > real] :
( ( X
= ( zero_zero @ real ) )
=> ( ( N
!= ( zero_zero @ nat ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M3: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M3 @ ( zero_zero @ A ) ) @ ( semiring_char_0_fact @ real @ M3 ) ) @ ( power_power @ real @ X @ M3 ) )
@ ( set_ord_lessThan @ nat @ N ) )
= ( Diff @ ( zero_zero @ nat ) @ ( zero_zero @ A ) ) ) ) ) ) ).
% Maclaurin_zero
thf(fact_4929_Maclaurin__lemma,axiom,
! [H2: real,F3: real > real,J: nat > real,N: nat] :
( ( ord_less @ real @ ( zero_zero @ real ) @ H2 )
=> ? [B6: real] :
( ( F3 @ H2 )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M3: nat] : ( times_times @ real @ ( divide_divide @ real @ ( J @ M3 ) @ ( semiring_char_0_fact @ real @ M3 ) ) @ ( power_power @ real @ H2 @ M3 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ B6 @ ( divide_divide @ real @ ( power_power @ real @ H2 @ N ) @ ( semiring_char_0_fact @ real @ N ) ) ) ) ) ) ).
% Maclaurin_lemma
thf(fact_4930_sum__split__even__odd,axiom,
! [F3: nat > real,G2: nat > real,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I4: nat] : ( if @ real @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I4 ) @ ( F3 @ I4 ) @ ( G2 @ I4 ) )
@ ( set_ord_lessThan @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I4: nat] : ( F3 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I4 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I4: nat] : ( G2 @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I4 ) @ ( one_one @ nat ) ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% sum_split_even_odd
thf(fact_4931_Maclaurin__exp__le,axiom,
! [X: real,N: nat] :
? [T3: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ T3 ) @ ( abs_abs @ real @ X ) )
& ( ( exp @ real @ X )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M3: nat] : ( divide_divide @ real @ ( power_power @ real @ X @ M3 ) @ ( semiring_char_0_fact @ real @ M3 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( exp @ real @ T3 ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ X @ N ) ) ) ) ) ).
% Maclaurin_exp_le
thf(fact_4932_polyfun__diff__alt,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [N: nat,A3: nat > A,X: A,Y: A] :
( ( ord_less_eq @ nat @ ( one_one @ nat ) @ N )
=> ( ( minus_minus @ A
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( A3 @ I4 ) @ ( power_power @ A @ X @ I4 ) )
@ ( set_ord_atMost @ nat @ N ) )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( A3 @ I4 ) @ ( power_power @ A @ Y @ I4 ) )
@ ( set_ord_atMost @ nat @ N ) ) )
= ( times_times @ A @ ( minus_minus @ A @ X @ Y )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [J3: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( times_times @ A @ ( times_times @ A @ ( A3 @ ( plus_plus @ nat @ ( plus_plus @ nat @ J3 @ K3 ) @ ( one_one @ nat ) ) ) @ ( power_power @ A @ Y @ K3 ) ) @ ( power_power @ A @ X @ J3 ) )
@ ( set_ord_lessThan @ nat @ ( minus_minus @ nat @ N @ J3 ) ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ) ).
% polyfun_diff_alt
thf(fact_4933_exp__first__terms,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [K: nat] :
( ( exp @ A )
= ( ^ [X2: A] :
( plus_plus @ A
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [N4: nat] : ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ N4 ) ) @ ( power_power @ A @ X2 @ N4 ) )
@ ( set_ord_lessThan @ nat @ K ) )
@ ( suminf @ A
@ ^ [N4: nat] : ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ ( plus_plus @ nat @ N4 @ K ) ) ) @ ( power_power @ A @ X2 @ ( plus_plus @ nat @ N4 @ K ) ) ) ) ) ) ) ) ).
% exp_first_terms
thf(fact_4934_Maclaurin__sin__bound,axiom,
! [X: real,N: nat] :
( ord_less_eq @ real
@ ( abs_abs @ real
@ ( minus_minus @ real @ ( sin @ real @ X )
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M3: nat] : ( times_times @ real @ ( sin_coeff @ M3 ) @ ( power_power @ real @ X @ M3 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) )
@ ( times_times @ real @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ ( abs_abs @ real @ X ) @ N ) ) ) ).
% Maclaurin_sin_bound
thf(fact_4935_sum__pos__lt__pair,axiom,
! [F3: nat > real,K: nat] :
( ( summable @ real @ F3 )
=> ( ! [D6: nat] : ( ord_less @ real @ ( zero_zero @ real ) @ ( plus_plus @ real @ ( F3 @ ( plus_plus @ nat @ K @ ( times_times @ nat @ ( suc @ ( suc @ ( zero_zero @ nat ) ) ) @ D6 ) ) ) @ ( F3 @ ( plus_plus @ nat @ K @ ( plus_plus @ nat @ ( times_times @ nat @ ( suc @ ( suc @ ( zero_zero @ nat ) ) ) @ D6 ) @ ( one_one @ nat ) ) ) ) ) )
=> ( ord_less @ real @ ( groups7311177749621191930dd_sum @ nat @ real @ F3 @ ( set_ord_lessThan @ nat @ K ) ) @ ( suminf @ real @ F3 ) ) ) ) ).
% sum_pos_lt_pair
thf(fact_4936_Maclaurin__exp__lt,axiom,
! [X: real,N: nat] :
( ( X
!= ( zero_zero @ real ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ? [T3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( abs_abs @ real @ T3 ) )
& ( ord_less @ real @ ( abs_abs @ real @ T3 ) @ ( abs_abs @ real @ X ) )
& ( ( exp @ real @ X )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M3: nat] : ( divide_divide @ real @ ( power_power @ real @ X @ M3 ) @ ( semiring_char_0_fact @ real @ M3 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( exp @ real @ T3 ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ X @ N ) ) ) ) ) ) ) ).
% Maclaurin_exp_lt
thf(fact_4937_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
@ ^ [P6: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [Q5: nat] : ( times_times @ A @ ( power_power @ A @ ( plus_plus @ A @ Z @ H2 ) @ Q5 ) @ ( power_power @ A @ Z @ ( minus_minus @ nat @ ( minus_minus @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ Q5 ) ) )
@ ( set_ord_lessThan @ nat @ ( minus_minus @ nat @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) @ P6 ) ) )
@ ( set_ord_lessThan @ nat @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ) ) ).
% lemma_termdiff2
thf(fact_4938_Maclaurin__sin__expansion,axiom,
! [X: real,N: nat] :
? [T3: real] :
( ( sin @ real @ X )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M3: nat] : ( times_times @ real @ ( sin_coeff @ M3 ) @ ( power_power @ real @ X @ M3 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( sin @ real @ ( plus_plus @ real @ T3 @ ( times_times @ real @ ( times_times @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( semiring_1_of_nat @ real @ N ) ) @ pi ) ) ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ X @ N ) ) ) ) ).
% Maclaurin_sin_expansion
thf(fact_4939_Maclaurin__sin__expansion2,axiom,
! [X: real,N: nat] :
? [T3: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ T3 ) @ ( abs_abs @ real @ X ) )
& ( ( sin @ real @ X )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M3: nat] : ( times_times @ real @ ( sin_coeff @ M3 ) @ ( power_power @ real @ X @ M3 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( sin @ real @ ( plus_plus @ real @ T3 @ ( times_times @ real @ ( times_times @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( semiring_1_of_nat @ real @ N ) ) @ pi ) ) ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ X @ N ) ) ) ) ) ).
% Maclaurin_sin_expansion2
thf(fact_4940_Maclaurin__cos__expansion,axiom,
! [X: real,N: nat] :
? [T3: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ T3 ) @ ( abs_abs @ real @ X ) )
& ( ( cos @ real @ X )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M3: nat] : ( times_times @ real @ ( cos_coeff @ M3 ) @ ( power_power @ real @ X @ M3 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( cos @ real @ ( plus_plus @ real @ T3 @ ( times_times @ real @ ( times_times @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( semiring_1_of_nat @ real @ N ) ) @ pi ) ) ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ X @ N ) ) ) ) ) ).
% Maclaurin_cos_expansion
thf(fact_4941_Maclaurin__sin__expansion4,axiom,
! [X: real,N: nat] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ? [T3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ T3 )
& ( ord_less_eq @ real @ T3 @ X )
& ( ( sin @ real @ X )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M3: nat] : ( times_times @ real @ ( sin_coeff @ M3 ) @ ( power_power @ real @ X @ M3 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( sin @ real @ ( plus_plus @ real @ T3 @ ( times_times @ real @ ( times_times @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( semiring_1_of_nat @ real @ N ) ) @ pi ) ) ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ X @ N ) ) ) ) ) ) ).
% Maclaurin_sin_expansion4
thf(fact_4942_bij__betw__roots__unity,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( bij_betw @ nat @ complex
@ ^ [K3: nat] : ( cis @ ( divide_divide @ real @ ( times_times @ real @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) @ ( semiring_1_of_nat @ real @ K3 ) ) @ ( semiring_1_of_nat @ real @ N ) ) )
@ ( set_ord_lessThan @ nat @ N )
@ ( collect @ complex
@ ^ [Z4: complex] :
( ( power_power @ complex @ Z4 @ N )
= ( one_one @ complex ) ) ) ) ) ).
% bij_betw_roots_unity
thf(fact_4943_sum__gp,axiom,
! [A: $tType] :
( ( ( division_ring @ A )
& ( comm_ring @ A ) )
=> ! [N: nat,M: nat,X: A] :
( ( ( ord_less @ nat @ N @ M )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( zero_zero @ A ) ) )
& ( ~ ( ord_less @ nat @ N @ M )
=> ( ( ( X
= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) @ M ) ) ) )
& ( ( X
!= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( power_power @ A @ X @ M ) @ ( power_power @ A @ X @ ( suc @ N ) ) ) @ ( minus_minus @ A @ ( one_one @ A ) @ X ) ) ) ) ) ) ) ) ).
% sum_gp
thf(fact_4944_gchoose__row__sum__weighted,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [R3: A,M: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( times_times @ A @ ( gbinomial @ A @ R3 @ K3 ) @ ( minus_minus @ A @ ( divide_divide @ A @ R3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( semiring_1_of_nat @ A @ K3 ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ M ) )
= ( times_times @ A @ ( divide_divide @ A @ ( semiring_1_of_nat @ A @ ( suc @ M ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( gbinomial @ A @ R3 @ ( suc @ M ) ) ) ) ) ).
% gchoose_row_sum_weighted
thf(fact_4945_atLeastatMost__empty__iff2,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: A,B3: A] :
( ( ( bot_bot @ ( set @ A ) )
= ( set_or1337092689740270186AtMost @ A @ A3 @ B3 ) )
= ( ~ ( ord_less_eq @ A @ A3 @ B3 ) ) ) ) ).
% atLeastatMost_empty_iff2
thf(fact_4946_atLeastatMost__empty__iff,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: A,B3: A] :
( ( ( set_or1337092689740270186AtMost @ A @ A3 @ B3 )
= ( bot_bot @ ( set @ A ) ) )
= ( ~ ( ord_less_eq @ A @ A3 @ B3 ) ) ) ) ).
% atLeastatMost_empty_iff
thf(fact_4947_atLeastatMost__empty,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B3: A,A3: A] :
( ( ord_less @ A @ B3 @ A3 )
=> ( ( set_or1337092689740270186AtMost @ A @ A3 @ B3 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% atLeastatMost_empty
thf(fact_4948_atLeastAtMost__singleton__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A,B3: A,C2: A] :
( ( ( set_or1337092689740270186AtMost @ A @ A3 @ B3 )
= ( insert @ A @ C2 @ ( bot_bot @ ( set @ A ) ) ) )
= ( ( A3 = B3 )
& ( B3 = C2 ) ) ) ) ).
% atLeastAtMost_singleton_iff
thf(fact_4949_atLeastAtMost__singleton,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A] :
( ( set_or1337092689740270186AtMost @ A @ A3 @ A3 )
= ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% atLeastAtMost_singleton
thf(fact_4950_sum_Ocl__ivl__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [N: nat,M: nat,G2: nat > A] :
( ( ( ord_less @ nat @ ( suc @ N ) @ M )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G2 @ ( set_or1337092689740270186AtMost @ nat @ M @ ( suc @ N ) ) )
= ( zero_zero @ A ) ) )
& ( ~ ( ord_less @ nat @ ( suc @ N ) @ M )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G2 @ ( set_or1337092689740270186AtMost @ nat @ M @ ( suc @ N ) ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G2 @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) @ ( G2 @ ( suc @ N ) ) ) ) ) ) ) ).
% sum.cl_ivl_Suc
thf(fact_4951_sum_Oreindex__bij__betw,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comm_monoid_add @ A )
=> ! [H2: B > C,S: set @ B,T4: set @ C,G2: C > A] :
( ( bij_betw @ B @ C @ H2 @ S @ T4 )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X2: B] : ( G2 @ ( H2 @ X2 ) )
@ S )
= ( groups7311177749621191930dd_sum @ C @ A @ G2 @ T4 ) ) ) ) ).
% sum.reindex_bij_betw
thf(fact_4952_ivl__disj__un__two__touch_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,M: A,U2: A] :
( ( ord_less_eq @ A @ L @ M )
=> ( ( ord_less_eq @ A @ M @ U2 )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ L @ M ) @ ( set_or1337092689740270186AtMost @ A @ M @ U2 ) )
= ( set_or1337092689740270186AtMost @ A @ L @ U2 ) ) ) ) ) ).
% ivl_disj_un_two_touch(4)
thf(fact_4953_atLeastAtMost__singleton_H,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A,B3: A] :
( ( A3 = B3 )
=> ( ( set_or1337092689740270186AtMost @ A @ A3 @ B3 )
= ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% atLeastAtMost_singleton'
thf(fact_4954_ex__nat__less,axiom,
! [N: nat,P: nat > $o] :
( ( ? [M3: nat] :
( ( ord_less_eq @ nat @ M3 @ N )
& ( P @ M3 ) ) )
= ( ? [X2: nat] :
( ( member @ nat @ X2 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
& ( P @ X2 ) ) ) ) ).
% ex_nat_less
thf(fact_4955_all__nat__less,axiom,
! [N: nat,P: nat > $o] :
( ( ! [M3: nat] :
( ( ord_less_eq @ nat @ M3 @ N )
=> ( P @ M3 ) ) )
= ( ! [X2: nat] :
( ( member @ nat @ X2 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
=> ( P @ X2 ) ) ) ) ).
% all_nat_less
thf(fact_4956_atMost__atLeast0,axiom,
( ( set_ord_atMost @ nat )
= ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) ) ) ).
% atMost_atLeast0
thf(fact_4957_sum_Oshift__bounds__cl__Suc__ivl,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G2: nat > A,M: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G2 @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ ( suc @ N ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( G2 @ ( suc @ I4 ) )
@ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ).
% sum.shift_bounds_cl_Suc_ivl
thf(fact_4958_sum_Oshift__bounds__cl__nat__ivl,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G2: nat > A,M: nat,K: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G2 @ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ M @ K ) @ ( plus_plus @ nat @ N @ K ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( G2 @ ( plus_plus @ nat @ I4 @ K ) )
@ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ).
% sum.shift_bounds_cl_nat_ivl
thf(fact_4959_atLeastatMost__psubset__iff,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: A,B3: A,C2: A,D2: A] :
( ( ord_less @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A3 @ B3 ) @ ( set_or1337092689740270186AtMost @ A @ C2 @ D2 ) )
= ( ( ~ ( ord_less_eq @ A @ A3 @ B3 )
| ( ( ord_less_eq @ A @ C2 @ A3 )
& ( ord_less_eq @ A @ B3 @ D2 )
& ( ( ord_less @ A @ C2 @ A3 )
| ( ord_less @ A @ B3 @ D2 ) ) ) )
& ( ord_less_eq @ A @ C2 @ D2 ) ) ) ) ).
% atLeastatMost_psubset_iff
thf(fact_4960_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_4961_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_4962_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_4963_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_4964_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_4965_sum_OatLeastAtMost__rev,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G2: nat > A,N: nat,M: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G2 @ ( set_or1337092689740270186AtMost @ nat @ N @ M ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( G2 @ ( minus_minus @ nat @ ( plus_plus @ nat @ M @ N ) @ I4 ) )
@ ( set_or1337092689740270186AtMost @ nat @ N @ M ) ) ) ) ).
% sum.atLeastAtMost_rev
thf(fact_4966_ivl__disj__un__one_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,U2: A] :
( ( ord_less_eq @ A @ L @ U2 )
=> ( ( sup_sup @ ( set @ A ) @ ( set_ord_lessThan @ A @ L ) @ ( set_or1337092689740270186AtMost @ A @ L @ U2 ) )
= ( set_ord_atMost @ A @ U2 ) ) ) ) ).
% ivl_disj_un_one(4)
thf(fact_4967_sum__shift__lb__Suc0__0,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [F3: nat > A,K: nat] :
( ( ( F3 @ ( zero_zero @ nat ) )
= ( zero_zero @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ ( zero_zero @ nat ) ) @ K ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ K ) ) ) ) ) ).
% sum_shift_lb_Suc0_0
thf(fact_4968_sum_OatLeast0__atMost__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G2: nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G2 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G2 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) @ ( G2 @ ( suc @ N ) ) ) ) ) ).
% sum.atLeast0_atMost_Suc
thf(fact_4969_sum_OSuc__reindex__ivl,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [M: nat,N: nat,G2: nat > A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G2 @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) @ ( G2 @ ( suc @ N ) ) )
= ( plus_plus @ A @ ( G2 @ M )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( G2 @ ( suc @ I4 ) )
@ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ) ) ).
% sum.Suc_reindex_ivl
thf(fact_4970_sum__Suc__diff,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [M: nat,N: nat,F3: nat > A] :
( ( ord_less_eq @ nat @ M @ ( suc @ N ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( minus_minus @ A @ ( F3 @ ( suc @ I4 ) ) @ ( F3 @ I4 ) )
@ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( minus_minus @ A @ ( F3 @ ( suc @ N ) ) @ ( F3 @ M ) ) ) ) ) ).
% sum_Suc_diff
thf(fact_4971_sum_OatLeast1__atMost__eq,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G2: nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G2 @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( G2 @ ( suc @ K3 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% sum.atLeast1_atMost_eq
thf(fact_4972_sum__bounds__lt__plus1,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [F3: nat > A,Mm: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( F3 @ ( suc @ K3 ) )
@ ( set_ord_lessThan @ nat @ Mm ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_or1337092689740270186AtMost @ nat @ ( one_one @ nat ) @ Mm ) ) ) ) ).
% sum_bounds_lt_plus1
thf(fact_4973_sum_Onested__swap_H,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A3: nat > nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( groups7311177749621191930dd_sum @ nat @ A @ ( A3 @ I4 ) @ ( set_ord_lessThan @ nat @ I4 ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [J3: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( A3 @ I4 @ J3 )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ J3 ) @ N ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% sum.nested_swap'
thf(fact_4974_sum__atLeastAtMost__code,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [F3: nat > A,A3: nat,B3: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_or1337092689740270186AtMost @ nat @ A3 @ B3 ) )
= ( set_fo6178422350223883121st_nat @ A
@ ^ [A5: nat] : ( plus_plus @ A @ ( F3 @ A5 ) )
@ A3
@ B3
@ ( zero_zero @ A ) ) ) ) ).
% sum_atLeastAtMost_code
thf(fact_4975_sum_Oub__add__nat,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [M: nat,N: nat,G2: nat > A,P4: nat] :
( ( ord_less_eq @ nat @ M @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G2 @ ( set_or1337092689740270186AtMost @ nat @ M @ ( plus_plus @ nat @ N @ P4 ) ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G2 @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ G2 @ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) @ ( plus_plus @ nat @ N @ P4 ) ) ) ) ) ) ) ).
% sum.ub_add_nat
thf(fact_4976_sum__up__index__split,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [F3: nat > A,M: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_ord_atMost @ nat @ ( plus_plus @ nat @ M @ N ) ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_ord_atMost @ nat @ M ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ ( plus_plus @ nat @ M @ N ) ) ) ) ) ) ).
% sum_up_index_split
thf(fact_4977_sum__natinterval__diff,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [M: nat,N: nat,F3: nat > A] :
( ( ( ord_less_eq @ nat @ M @ N )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( minus_minus @ A @ ( F3 @ K3 ) @ ( F3 @ ( plus_plus @ nat @ K3 @ ( one_one @ nat ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( minus_minus @ A @ ( F3 @ M ) @ ( F3 @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) ) ) ) )
& ( ~ ( ord_less_eq @ nat @ M @ N )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( minus_minus @ A @ ( F3 @ K3 ) @ ( F3 @ ( plus_plus @ nat @ K3 @ ( one_one @ nat ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( zero_zero @ A ) ) ) ) ) ).
% sum_natinterval_diff
thf(fact_4978_sum__telescope_H_H,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [M: nat,N: nat,F3: nat > A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( minus_minus @ A @ ( F3 @ K3 ) @ ( F3 @ ( minus_minus @ nat @ K3 @ ( one_one @ nat ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ N ) )
= ( minus_minus @ A @ ( F3 @ N ) @ ( F3 @ M ) ) ) ) ) ).
% sum_telescope''
thf(fact_4979_sum__power__shift,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [M: nat,N: nat,X: A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( times_times @ A @ ( power_power @ A @ X @ M ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_ord_atMost @ nat @ ( minus_minus @ nat @ N @ M ) ) ) ) ) ) ) ).
% sum_power_shift
thf(fact_4980_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_4981_summable__partial__sum__bound,axiom,
! [A: $tType] :
( ( real_Vector_banach @ A )
=> ! [F3: nat > A,E2: real] :
( ( summable @ A @ F3 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ E2 )
=> ~ ! [N10: nat] :
~ ! [M2: nat] :
( ( ord_less_eq @ nat @ N10 @ M2 )
=> ! [N11: nat] : ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N11 ) ) ) @ E2 ) ) ) ) ) ).
% summable_partial_sum_bound
thf(fact_4982_sum__gp__multiplied,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [M: nat,N: nat,X: A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( times_times @ A @ ( minus_minus @ A @ ( one_one @ A ) @ X ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) )
= ( minus_minus @ A @ ( power_power @ A @ X @ M ) @ ( power_power @ A @ X @ ( suc @ N ) ) ) ) ) ) ).
% sum_gp_multiplied
thf(fact_4983_sum_Oin__pairs,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G2: nat > A,M: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G2 @ ( set_or1337092689740270186AtMost @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( plus_plus @ A @ ( G2 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I4 ) ) @ ( G2 @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I4 ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ).
% sum.in_pairs
thf(fact_4984_polyfun__eq__const,axiom,
! [A: $tType] :
( ( ( real_V8999393235501362500lgebra @ A )
& ( idom @ A ) )
=> ! [C2: nat > A,N: nat,K: A] :
( ( ! [X2: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( C2 @ I4 ) @ ( power_power @ A @ X2 @ I4 ) )
@ ( set_ord_atMost @ nat @ N ) )
= K ) )
= ( ( ( C2 @ ( zero_zero @ nat ) )
= K )
& ! [X2: nat] :
( ( member @ nat @ X2 @ ( set_or1337092689740270186AtMost @ nat @ ( one_one @ nat ) @ N ) )
=> ( ( C2 @ X2 )
= ( zero_zero @ A ) ) ) ) ) ) ).
% polyfun_eq_const
thf(fact_4985_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_4986_gauss__sum__nat,axiom,
! [N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [X2: nat] : X2
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
= ( divide_divide @ nat @ ( times_times @ nat @ N @ ( suc @ N ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% gauss_sum_nat
thf(fact_4987_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_4988_double__arith__series,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A3: A,D2: A,N: nat] :
( ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( plus_plus @ A @ A3 @ ( times_times @ A @ ( semiring_1_of_nat @ A @ I4 ) @ D2 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) )
= ( times_times @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N ) @ ( one_one @ A ) ) @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ D2 ) ) ) ) ) ).
% double_arith_series
thf(fact_4989_arith__series__nat,axiom,
! [A3: nat,D2: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [I4: nat] : ( plus_plus @ nat @ A3 @ ( times_times @ nat @ I4 @ D2 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
= ( divide_divide @ nat @ ( times_times @ nat @ ( suc @ N ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ A3 ) @ ( times_times @ nat @ N @ D2 ) ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% arith_series_nat
thf(fact_4990_Sum__Icc__nat,axiom,
! [M: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [X2: nat] : X2
@ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( divide_divide @ nat @ ( minus_minus @ nat @ ( times_times @ nat @ N @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) ) @ ( times_times @ nat @ M @ ( minus_minus @ nat @ M @ ( one_one @ nat ) ) ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% Sum_Icc_nat
thf(fact_4991_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_4992_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_4993_arith__series,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [A3: A,D2: A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( plus_plus @ A @ A3 @ ( times_times @ A @ ( semiring_1_of_nat @ A @ I4 ) @ D2 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
= ( divide_divide @ A @ ( times_times @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N ) @ ( one_one @ A ) ) @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ D2 ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% arith_series
thf(fact_4994_sum__gp__offset,axiom,
! [A: $tType] :
( ( ( division_ring @ A )
& ( comm_ring @ A ) )
=> ! [X: A,M: nat,N: nat] :
( ( ( X
= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_or1337092689740270186AtMost @ nat @ M @ ( plus_plus @ nat @ M @ N ) ) )
= ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N ) @ ( one_one @ A ) ) ) )
& ( ( X
!= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_or1337092689740270186AtMost @ nat @ M @ ( plus_plus @ nat @ M @ N ) ) )
= ( divide_divide @ A @ ( times_times @ A @ ( power_power @ A @ X @ M ) @ ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ X @ ( suc @ N ) ) ) ) @ ( minus_minus @ A @ ( one_one @ A ) @ X ) ) ) ) ) ) ).
% sum_gp_offset
thf(fact_4995_polyfun__diff,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [N: nat,A3: nat > A,X: A,Y: A] :
( ( ord_less_eq @ nat @ ( one_one @ nat ) @ N )
=> ( ( minus_minus @ A
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( A3 @ I4 ) @ ( power_power @ A @ X @ I4 ) )
@ ( set_ord_atMost @ nat @ N ) )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( A3 @ I4 ) @ ( power_power @ A @ Y @ I4 ) )
@ ( set_ord_atMost @ nat @ N ) ) )
= ( times_times @ A @ ( minus_minus @ A @ X @ Y )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [J3: nat] :
( times_times @ A
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( A3 @ I4 ) @ ( power_power @ A @ Y @ ( minus_minus @ nat @ ( minus_minus @ nat @ I4 @ J3 ) @ ( one_one @ nat ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ J3 ) @ N ) )
@ ( power_power @ A @ X @ J3 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ) ).
% polyfun_diff
thf(fact_4996_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_4997_pochhammer__times__pochhammer__half,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [Z: A,N: nat] :
( ( times_times @ A @ ( comm_s3205402744901411588hammer @ A @ Z @ ( suc @ N ) ) @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ Z @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ ( suc @ N ) ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [K3: nat] : ( plus_plus @ A @ Z @ ( divide_divide @ A @ ( semiring_1_of_nat @ A @ K3 ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ ( one_one @ nat ) ) ) ) ) ) ).
% pochhammer_times_pochhammer_half
thf(fact_4998_notIn__Un__bij__betw,axiom,
! [A: $tType,B: $tType,B3: A,A2: set @ A,F3: A > B,A9: set @ B] :
( ~ ( member @ A @ B3 @ A2 )
=> ( ~ ( member @ B @ ( F3 @ B3 ) @ A9 )
=> ( ( bij_betw @ A @ B @ F3 @ A2 @ A9 )
=> ( bij_betw @ A @ B @ F3 @ ( sup_sup @ ( set @ A ) @ A2 @ ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( sup_sup @ ( set @ B ) @ A9 @ ( insert @ B @ ( F3 @ B3 ) @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ).
% notIn_Un_bij_betw
thf(fact_4999_notIn__Un__bij__betw3,axiom,
! [A: $tType,B: $tType,B3: A,A2: set @ A,F3: A > B,A9: set @ B] :
( ~ ( member @ A @ B3 @ A2 )
=> ( ~ ( member @ B @ ( F3 @ B3 ) @ A9 )
=> ( ( bij_betw @ A @ B @ F3 @ A2 @ A9 )
= ( bij_betw @ A @ B @ F3 @ ( sup_sup @ ( set @ A ) @ A2 @ ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( sup_sup @ ( set @ B ) @ A9 @ ( insert @ B @ ( F3 @ B3 ) @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ).
% notIn_Un_bij_betw3
thf(fact_5000_prod_Oneutral__const,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A2: set @ B] :
( ( groups7121269368397514597t_prod @ B @ A
@ ^ [Uu: B] : ( one_one @ A )
@ A2 )
= ( one_one @ A ) ) ) ).
% prod.neutral_const
thf(fact_5001_of__nat__prod,axiom,
! [A: $tType,B: $tType] :
( ( comm_semiring_1 @ A )
=> ! [F3: B > nat,A2: set @ B] :
( ( semiring_1_of_nat @ A @ ( groups7121269368397514597t_prod @ B @ nat @ F3 @ A2 ) )
= ( groups7121269368397514597t_prod @ B @ A
@ ^ [X2: B] : ( semiring_1_of_nat @ A @ ( F3 @ X2 ) )
@ A2 ) ) ) ).
% of_nat_prod
thf(fact_5002_of__int__prod,axiom,
! [A: $tType,B: $tType] :
( ( comm_ring_1 @ A )
=> ! [F3: B > int,A2: set @ B] :
( ( ring_1_of_int @ A @ ( groups7121269368397514597t_prod @ B @ int @ F3 @ A2 ) )
= ( groups7121269368397514597t_prod @ B @ A
@ ^ [X2: B] : ( ring_1_of_int @ A @ ( F3 @ X2 ) )
@ A2 ) ) ) ).
% of_int_prod
thf(fact_5003_of__real__prod,axiom,
! [A: $tType,B: $tType] :
( ( ( comm_monoid_mult @ A )
& ( real_V2191834092415804123ebra_1 @ A ) )
=> ! [F3: B > real,S2: set @ B] :
( ( real_Vector_of_real @ A @ ( groups7121269368397514597t_prod @ B @ real @ F3 @ S2 ) )
= ( groups7121269368397514597t_prod @ B @ A
@ ^ [X2: B] : ( real_Vector_of_real @ A @ ( F3 @ X2 ) )
@ S2 ) ) ) ).
% of_real_prod
thf(fact_5004_prod_Oempty,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G2: B > A] :
( ( groups7121269368397514597t_prod @ B @ A @ G2 @ ( bot_bot @ ( set @ B ) ) )
= ( one_one @ A ) ) ) ).
% prod.empty
thf(fact_5005_prod_Ocl__ivl__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [N: nat,M: nat,G2: nat > A] :
( ( ( ord_less @ nat @ ( suc @ N ) @ M )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_or1337092689740270186AtMost @ nat @ M @ ( suc @ N ) ) )
= ( one_one @ A ) ) )
& ( ~ ( ord_less @ nat @ ( suc @ N ) @ M )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_or1337092689740270186AtMost @ nat @ M @ ( suc @ N ) ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) @ ( G2 @ ( suc @ N ) ) ) ) ) ) ) ).
% prod.cl_ivl_Suc
thf(fact_5006_prod_Oreindex__bij__betw,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comm_monoid_mult @ A )
=> ! [H2: B > C,S: set @ B,T4: set @ C,G2: C > A] :
( ( bij_betw @ B @ C @ H2 @ S @ T4 )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [X2: B] : ( G2 @ ( H2 @ X2 ) )
@ S )
= ( groups7121269368397514597t_prod @ C @ A @ G2 @ T4 ) ) ) ) ).
% prod.reindex_bij_betw
thf(fact_5007_prod__nonneg,axiom,
! [A: $tType,B: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: set @ B,F3: B > A] :
( ! [X3: B] :
( ( member @ B @ X3 @ A2 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F3 @ X3 ) ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( groups7121269368397514597t_prod @ B @ A @ F3 @ A2 ) ) ) ) ).
% prod_nonneg
thf(fact_5008_prod__mono,axiom,
! [A: $tType,B: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: set @ B,F3: B > A,G2: B > A] :
( ! [I3: B] :
( ( member @ B @ I3 @ A2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F3 @ I3 ) )
& ( ord_less_eq @ A @ ( F3 @ I3 ) @ ( G2 @ I3 ) ) ) )
=> ( ord_less_eq @ A @ ( groups7121269368397514597t_prod @ B @ A @ F3 @ A2 ) @ ( groups7121269368397514597t_prod @ B @ A @ G2 @ A2 ) ) ) ) ).
% prod_mono
thf(fact_5009_prod__pos,axiom,
! [A: $tType,B: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: set @ B,F3: B > A] :
( ! [X3: B] :
( ( member @ B @ X3 @ A2 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( F3 @ X3 ) ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( groups7121269368397514597t_prod @ B @ A @ F3 @ A2 ) ) ) ) ).
% prod_pos
thf(fact_5010_prod__power__distrib,axiom,
! [B: $tType,A: $tType] :
( ( comm_semiring_1 @ B )
=> ! [F3: A > B,A2: set @ A,N: nat] :
( ( power_power @ B @ ( groups7121269368397514597t_prod @ A @ B @ F3 @ A2 ) @ N )
= ( groups7121269368397514597t_prod @ A @ B
@ ^ [X2: A] : ( power_power @ B @ ( F3 @ X2 ) @ N )
@ A2 ) ) ) ).
% prod_power_distrib
thf(fact_5011_prod__dividef,axiom,
! [A: $tType,B: $tType] :
( ( field @ A )
=> ! [F3: B > A,G2: B > A,A2: set @ B] :
( ( groups7121269368397514597t_prod @ B @ A
@ ^ [X2: B] : ( divide_divide @ A @ ( F3 @ X2 ) @ ( G2 @ X2 ) )
@ A2 )
= ( divide_divide @ A @ ( groups7121269368397514597t_prod @ B @ A @ F3 @ A2 ) @ ( groups7121269368397514597t_prod @ B @ A @ G2 @ A2 ) ) ) ) ).
% prod_dividef
thf(fact_5012_prod_Oneutral,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A2: set @ B,G2: B > A] :
( ! [X3: B] :
( ( member @ B @ X3 @ A2 )
=> ( ( G2 @ X3 )
= ( one_one @ A ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G2 @ A2 )
= ( one_one @ A ) ) ) ) ).
% prod.neutral
thf(fact_5013_prod_Onot__neutral__contains__not__neutral,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G2: B > A,A2: set @ B] :
( ( ( groups7121269368397514597t_prod @ B @ A @ G2 @ A2 )
!= ( one_one @ A ) )
=> ~ ! [A6: B] :
( ( member @ B @ A6 @ A2 )
=> ( ( G2 @ A6 )
= ( one_one @ A ) ) ) ) ) ).
% prod.not_neutral_contains_not_neutral
thf(fact_5014_abs__prod,axiom,
! [A: $tType,B: $tType] :
( ( linordered_field @ A )
=> ! [F3: B > A,A2: set @ B] :
( ( abs_abs @ A @ ( groups7121269368397514597t_prod @ B @ A @ F3 @ A2 ) )
= ( groups7121269368397514597t_prod @ B @ A
@ ^ [X2: B] : ( abs_abs @ A @ ( F3 @ X2 ) )
@ A2 ) ) ) ).
% abs_prod
thf(fact_5015_prod_Oswap,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G2: B > C > A,B2: set @ C,A2: set @ B] :
( ( groups7121269368397514597t_prod @ B @ A
@ ^ [I4: B] : ( groups7121269368397514597t_prod @ C @ A @ ( G2 @ I4 ) @ B2 )
@ A2 )
= ( groups7121269368397514597t_prod @ C @ A
@ ^ [J3: C] :
( groups7121269368397514597t_prod @ B @ A
@ ^ [I4: B] : ( G2 @ I4 @ J3 )
@ A2 )
@ B2 ) ) ) ).
% prod.swap
thf(fact_5016_prod_Odistrib,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G2: B > A,H2: B > A,A2: set @ B] :
( ( groups7121269368397514597t_prod @ B @ A
@ ^ [X2: B] : ( times_times @ A @ ( G2 @ X2 ) @ ( H2 @ X2 ) )
@ A2 )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ G2 @ A2 ) @ ( groups7121269368397514597t_prod @ B @ A @ H2 @ A2 ) ) ) ) ).
% prod.distrib
thf(fact_5017_norm__prod__le,axiom,
! [A: $tType,B: $tType] :
( ( ( comm_monoid_mult @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [F3: B > A,A2: set @ B] :
( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( groups7121269368397514597t_prod @ B @ A @ F3 @ A2 ) )
@ ( groups7121269368397514597t_prod @ B @ real
@ ^ [A5: B] : ( real_V7770717601297561774m_norm @ A @ ( F3 @ A5 ) )
@ A2 ) ) ) ).
% norm_prod_le
thf(fact_5018_prod__norm,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V8999393235501362500lgebra @ B )
& ( comm_semiring_1 @ B ) )
=> ! [F3: A > B,A2: set @ A] :
( ( groups7121269368397514597t_prod @ A @ real
@ ^ [X2: A] : ( real_V7770717601297561774m_norm @ B @ ( F3 @ X2 ) )
@ A2 )
= ( real_V7770717601297561774m_norm @ B @ ( groups7121269368397514597t_prod @ A @ B @ F3 @ A2 ) ) ) ) ).
% prod_norm
thf(fact_5019_mod__prod__eq,axiom,
! [B: $tType,A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [F3: B > A,A3: A,A2: set @ B] :
( ( modulo_modulo @ A
@ ( groups7121269368397514597t_prod @ B @ A
@ ^ [I4: B] : ( modulo_modulo @ A @ ( F3 @ I4 ) @ A3 )
@ A2 )
@ A3 )
= ( modulo_modulo @ A @ ( groups7121269368397514597t_prod @ B @ A @ F3 @ A2 ) @ A3 ) ) ) ).
% mod_prod_eq
thf(fact_5020_prod__ge__1,axiom,
! [A: $tType,B: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [A2: set @ B,F3: B > A] :
( ! [X3: B] :
( ( member @ B @ X3 @ A2 )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ ( F3 @ X3 ) ) )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ ( groups7121269368397514597t_prod @ B @ A @ F3 @ A2 ) ) ) ) ).
% prod_ge_1
thf(fact_5021_range__subset__card,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A,B3: word @ A,C2: word @ A,D2: word @ A] :
( ( ord_less_eq @ ( set @ ( word @ A ) ) @ ( set_or1337092689740270186AtMost @ ( word @ A ) @ A3 @ B3 ) @ ( set_or1337092689740270186AtMost @ ( word @ A ) @ C2 @ D2 ) )
=> ( ( ord_less_eq @ ( word @ A ) @ A3 @ B3 )
=> ( ( ord_less_eq @ ( word @ A ) @ C2 @ D2 )
& ( ord_less_eq @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ B3 @ A3 ) @ ( minus_minus @ ( word @ A ) @ D2 @ C2 ) ) ) ) ) ) ).
% range_subset_card
thf(fact_5022_prod_Oshift__bounds__cl__Suc__ivl,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G2: nat > A,M: nat,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ ( suc @ N ) ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( G2 @ ( suc @ I4 ) )
@ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ).
% prod.shift_bounds_cl_Suc_ivl
thf(fact_5023_power__sum,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [C2: A,F3: B > nat,A2: set @ B] :
( ( power_power @ A @ C2 @ ( groups7311177749621191930dd_sum @ B @ nat @ F3 @ A2 ) )
= ( groups7121269368397514597t_prod @ B @ A
@ ^ [A5: B] : ( power_power @ A @ C2 @ ( F3 @ A5 ) )
@ A2 ) ) ) ).
% power_sum
thf(fact_5024_prod_Oshift__bounds__cl__nat__ivl,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G2: nat > A,M: nat,K: nat,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ M @ K ) @ ( plus_plus @ nat @ N @ K ) ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( G2 @ ( plus_plus @ nat @ I4 @ K ) )
@ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ).
% prod.shift_bounds_cl_nat_ivl
thf(fact_5025_prod__le__1,axiom,
! [B: $tType,A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [A2: set @ B,F3: B > A] :
( ! [X3: B] :
( ( member @ B @ X3 @ A2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F3 @ X3 ) )
& ( ord_less_eq @ A @ ( F3 @ X3 ) @ ( one_one @ A ) ) ) )
=> ( ord_less_eq @ A @ ( groups7121269368397514597t_prod @ B @ A @ F3 @ A2 ) @ ( one_one @ A ) ) ) ) ).
% prod_le_1
thf(fact_5026_bset_I1_J,axiom,
! [D3: int,B2: set @ int,P: int > $o,Q: int > $o] :
( ! [X3: int] :
( ! [Xa2: int] :
( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D3 ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ B2 )
=> ( X3
!= ( plus_plus @ int @ Xb2 @ Xa2 ) ) ) )
=> ( ( P @ X3 )
=> ( P @ ( minus_minus @ int @ X3 @ D3 ) ) ) )
=> ( ! [X3: int] :
( ! [Xa2: int] :
( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D3 ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ B2 )
=> ( X3
!= ( plus_plus @ int @ Xb2 @ Xa2 ) ) ) )
=> ( ( Q @ X3 )
=> ( Q @ ( minus_minus @ int @ X3 @ D3 ) ) ) )
=> ! [X6: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D3 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ B2 )
=> ( X6
!= ( plus_plus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ( P @ X6 )
& ( Q @ X6 ) )
=> ( ( P @ ( minus_minus @ int @ X6 @ D3 ) )
& ( Q @ ( minus_minus @ int @ X6 @ D3 ) ) ) ) ) ) ) ).
% bset(1)
thf(fact_5027_bset_I2_J,axiom,
! [D3: int,B2: set @ int,P: int > $o,Q: int > $o] :
( ! [X3: int] :
( ! [Xa2: int] :
( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D3 ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ B2 )
=> ( X3
!= ( plus_plus @ int @ Xb2 @ Xa2 ) ) ) )
=> ( ( P @ X3 )
=> ( P @ ( minus_minus @ int @ X3 @ D3 ) ) ) )
=> ( ! [X3: int] :
( ! [Xa2: int] :
( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D3 ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ B2 )
=> ( X3
!= ( plus_plus @ int @ Xb2 @ Xa2 ) ) ) )
=> ( ( Q @ X3 )
=> ( Q @ ( minus_minus @ int @ X3 @ D3 ) ) ) )
=> ! [X6: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D3 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ B2 )
=> ( X6
!= ( plus_plus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ( P @ X6 )
| ( Q @ X6 ) )
=> ( ( P @ ( minus_minus @ int @ X6 @ D3 ) )
| ( Q @ ( minus_minus @ int @ X6 @ D3 ) ) ) ) ) ) ) ).
% bset(2)
thf(fact_5028_aset_I1_J,axiom,
! [D3: int,A2: set @ int,P: int > $o,Q: int > $o] :
( ! [X3: int] :
( ! [Xa2: int] :
( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D3 ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ A2 )
=> ( X3
!= ( minus_minus @ int @ Xb2 @ Xa2 ) ) ) )
=> ( ( P @ X3 )
=> ( P @ ( plus_plus @ int @ X3 @ D3 ) ) ) )
=> ( ! [X3: int] :
( ! [Xa2: int] :
( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D3 ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ A2 )
=> ( X3
!= ( minus_minus @ int @ Xb2 @ Xa2 ) ) ) )
=> ( ( Q @ X3 )
=> ( Q @ ( plus_plus @ int @ X3 @ D3 ) ) ) )
=> ! [X6: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D3 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ A2 )
=> ( X6
!= ( minus_minus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ( P @ X6 )
& ( Q @ X6 ) )
=> ( ( P @ ( plus_plus @ int @ X6 @ D3 ) )
& ( Q @ ( plus_plus @ int @ X6 @ D3 ) ) ) ) ) ) ) ).
% aset(1)
thf(fact_5029_aset_I2_J,axiom,
! [D3: int,A2: set @ int,P: int > $o,Q: int > $o] :
( ! [X3: int] :
( ! [Xa2: int] :
( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D3 ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ A2 )
=> ( X3
!= ( minus_minus @ int @ Xb2 @ Xa2 ) ) ) )
=> ( ( P @ X3 )
=> ( P @ ( plus_plus @ int @ X3 @ D3 ) ) ) )
=> ( ! [X3: int] :
( ! [Xa2: int] :
( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D3 ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ A2 )
=> ( X3
!= ( minus_minus @ int @ Xb2 @ Xa2 ) ) ) )
=> ( ( Q @ X3 )
=> ( Q @ ( plus_plus @ int @ X3 @ D3 ) ) ) )
=> ! [X6: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D3 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ A2 )
=> ( X6
!= ( minus_minus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ( P @ X6 )
| ( Q @ X6 ) )
=> ( ( P @ ( plus_plus @ int @ X6 @ D3 ) )
| ( Q @ ( plus_plus @ int @ X6 @ D3 ) ) ) ) ) ) ) ).
% aset(2)
thf(fact_5030_prod_Onat__diff__reindex,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G2: nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( G2 @ ( minus_minus @ nat @ N @ ( suc @ I4 ) ) )
@ ( set_ord_lessThan @ nat @ N ) )
= ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% prod.nat_diff_reindex
thf(fact_5031_prod_OatLeastAtMost__rev,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G2: nat > A,N: nat,M: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_or1337092689740270186AtMost @ nat @ N @ M ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( G2 @ ( minus_minus @ nat @ ( plus_plus @ nat @ M @ N ) @ I4 ) )
@ ( set_or1337092689740270186AtMost @ nat @ N @ M ) ) ) ) ).
% prod.atLeastAtMost_rev
thf(fact_5032_prod_OatLeast0__atMost__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G2: nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) @ ( G2 @ ( suc @ N ) ) ) ) ) ).
% prod.atLeast0_atMost_Suc
thf(fact_5033_word__atLeastLessThan__Suc__atLeastAtMost__union,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [M: word @ A,L: word @ A,U2: word @ A] :
( ( M
!= ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) )
=> ( ( ord_less_eq @ ( word @ A ) @ L @ M )
=> ( ( ord_less_eq @ ( word @ A ) @ M @ U2 )
=> ( ( sup_sup @ ( set @ ( word @ A ) ) @ ( set_or1337092689740270186AtMost @ ( word @ A ) @ L @ M ) @ ( set_or1337092689740270186AtMost @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ M @ ( one_one @ ( word @ A ) ) ) @ U2 ) )
= ( set_or1337092689740270186AtMost @ ( word @ A ) @ L @ U2 ) ) ) ) ) ) ).
% word_atLeastLessThan_Suc_atLeastAtMost_union
thf(fact_5034_aset_I10_J,axiom,
! [D2: int,D3: int,A2: set @ int,T: int] :
( ( dvd_dvd @ int @ D2 @ D3 )
=> ! [X6: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D3 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ A2 )
=> ( X6
!= ( minus_minus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ~ ( dvd_dvd @ int @ D2 @ ( plus_plus @ int @ X6 @ T ) )
=> ~ ( dvd_dvd @ int @ D2 @ ( plus_plus @ int @ ( plus_plus @ int @ X6 @ D3 ) @ T ) ) ) ) ) ).
% aset(10)
thf(fact_5035_aset_I9_J,axiom,
! [D2: int,D3: int,A2: set @ int,T: int] :
( ( dvd_dvd @ int @ D2 @ D3 )
=> ! [X6: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D3 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ A2 )
=> ( X6
!= ( minus_minus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( dvd_dvd @ int @ D2 @ ( plus_plus @ int @ X6 @ T ) )
=> ( dvd_dvd @ int @ D2 @ ( plus_plus @ int @ ( plus_plus @ int @ X6 @ D3 ) @ T ) ) ) ) ) ).
% aset(9)
thf(fact_5036_bset_I10_J,axiom,
! [D2: int,D3: int,B2: set @ int,T: int] :
( ( dvd_dvd @ int @ D2 @ D3 )
=> ! [X6: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D3 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ B2 )
=> ( X6
!= ( plus_plus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ~ ( dvd_dvd @ int @ D2 @ ( plus_plus @ int @ X6 @ T ) )
=> ~ ( dvd_dvd @ int @ D2 @ ( plus_plus @ int @ ( minus_minus @ int @ X6 @ D3 ) @ T ) ) ) ) ) ).
% bset(10)
thf(fact_5037_bset_I9_J,axiom,
! [D2: int,D3: int,B2: set @ int,T: int] :
( ( dvd_dvd @ int @ D2 @ D3 )
=> ! [X6: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D3 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ B2 )
=> ( X6
!= ( plus_plus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( dvd_dvd @ int @ D2 @ ( plus_plus @ int @ X6 @ T ) )
=> ( dvd_dvd @ int @ D2 @ ( plus_plus @ int @ ( minus_minus @ int @ X6 @ D3 ) @ T ) ) ) ) ) ).
% bset(9)
thf(fact_5038_prod_OlessThan__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G2: nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_ord_lessThan @ nat @ ( suc @ N ) ) )
= ( times_times @ A @ ( G2 @ ( zero_zero @ nat ) )
@ ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( G2 @ ( suc @ I4 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% prod.lessThan_Suc_shift
thf(fact_5039_prod_OSuc__reindex__ivl,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M: nat,N: nat,G2: nat > A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) @ ( G2 @ ( suc @ N ) ) )
= ( times_times @ A @ ( G2 @ M )
@ ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( G2 @ ( suc @ I4 ) )
@ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ) ) ).
% prod.Suc_reindex_ivl
thf(fact_5040_prod_OatMost__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G2: nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_ord_atMost @ nat @ ( suc @ N ) ) )
= ( times_times @ A @ ( G2 @ ( zero_zero @ nat ) )
@ ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( G2 @ ( suc @ I4 ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ) ).
% prod.atMost_Suc_shift
thf(fact_5041_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_5042_prod_OatLeast1__atMost__eq,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G2: nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [K3: nat] : ( G2 @ ( suc @ K3 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% prod.atLeast1_atMost_eq
thf(fact_5043_fact__prod,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ( ( semiring_char_0_fact @ A )
= ( ^ [N4: nat] :
( semiring_1_of_nat @ A
@ ( groups7121269368397514597t_prod @ nat @ nat
@ ^ [X2: nat] : X2
@ ( set_or1337092689740270186AtMost @ nat @ ( one_one @ nat ) @ N4 ) ) ) ) ) ) ).
% fact_prod
thf(fact_5044_prod_Onested__swap_H,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: nat > nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( groups7121269368397514597t_prod @ nat @ A @ ( A3 @ I4 ) @ ( set_ord_lessThan @ nat @ I4 ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [J3: nat] :
( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( A3 @ I4 @ J3 )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ J3 ) @ N ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% prod.nested_swap'
thf(fact_5045_prod__atLeastAtMost__code,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [F3: nat > A,A3: nat,B3: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ F3 @ ( set_or1337092689740270186AtMost @ nat @ A3 @ B3 ) )
= ( set_fo6178422350223883121st_nat @ A
@ ^ [A5: nat] : ( times_times @ A @ ( F3 @ A5 ) )
@ A3
@ B3
@ ( one_one @ A ) ) ) ) ).
% prod_atLeastAtMost_code
thf(fact_5046_prod_Oub__add__nat,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M: nat,N: nat,G2: nat > A,P4: nat] :
( ( ord_less_eq @ nat @ M @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_or1337092689740270186AtMost @ nat @ M @ ( plus_plus @ nat @ N @ P4 ) ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) @ ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) @ ( plus_plus @ nat @ N @ P4 ) ) ) ) ) ) ) ).
% prod.ub_add_nat
thf(fact_5047_bset_I3_J,axiom,
! [D3: int,T: int,B2: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D3 )
=> ( ( member @ int @ ( minus_minus @ int @ T @ ( one_one @ int ) ) @ B2 )
=> ! [X6: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D3 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ B2 )
=> ( X6
!= ( plus_plus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( X6 = T )
=> ( ( minus_minus @ int @ X6 @ D3 )
= T ) ) ) ) ) ).
% bset(3)
thf(fact_5048_bset_I4_J,axiom,
! [D3: int,T: int,B2: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D3 )
=> ( ( member @ int @ T @ B2 )
=> ! [X6: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D3 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ B2 )
=> ( X6
!= ( plus_plus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( X6 != T )
=> ( ( minus_minus @ int @ X6 @ D3 )
!= T ) ) ) ) ) ).
% bset(4)
thf(fact_5049_bset_I5_J,axiom,
! [D3: int,B2: set @ int,T: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D3 )
=> ! [X6: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D3 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ B2 )
=> ( X6
!= ( plus_plus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ord_less @ int @ X6 @ T )
=> ( ord_less @ int @ ( minus_minus @ int @ X6 @ D3 ) @ T ) ) ) ) ).
% bset(5)
thf(fact_5050_bset_I7_J,axiom,
! [D3: int,T: int,B2: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D3 )
=> ( ( member @ int @ T @ B2 )
=> ! [X6: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D3 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ B2 )
=> ( X6
!= ( plus_plus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ord_less @ int @ T @ X6 )
=> ( ord_less @ int @ T @ ( minus_minus @ int @ X6 @ D3 ) ) ) ) ) ) ).
% bset(7)
thf(fact_5051_aset_I3_J,axiom,
! [D3: int,T: int,A2: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D3 )
=> ( ( member @ int @ ( plus_plus @ int @ T @ ( one_one @ int ) ) @ A2 )
=> ! [X6: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D3 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ A2 )
=> ( X6
!= ( minus_minus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( X6 = T )
=> ( ( plus_plus @ int @ X6 @ D3 )
= T ) ) ) ) ) ).
% aset(3)
thf(fact_5052_aset_I4_J,axiom,
! [D3: int,T: int,A2: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D3 )
=> ( ( member @ int @ T @ A2 )
=> ! [X6: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D3 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ A2 )
=> ( X6
!= ( minus_minus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( X6 != T )
=> ( ( plus_plus @ int @ X6 @ D3 )
!= T ) ) ) ) ) ).
% aset(4)
thf(fact_5053_aset_I5_J,axiom,
! [D3: int,T: int,A2: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D3 )
=> ( ( member @ int @ T @ A2 )
=> ! [X6: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D3 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ A2 )
=> ( X6
!= ( minus_minus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ord_less @ int @ X6 @ T )
=> ( ord_less @ int @ ( plus_plus @ int @ X6 @ D3 ) @ T ) ) ) ) ) ).
% aset(5)
thf(fact_5054_aset_I7_J,axiom,
! [D3: int,A2: set @ int,T: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D3 )
=> ! [X6: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D3 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ A2 )
=> ( X6
!= ( minus_minus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ord_less @ int @ T @ X6 )
=> ( ord_less @ int @ T @ ( plus_plus @ int @ X6 @ D3 ) ) ) ) ) ).
% aset(7)
thf(fact_5055_periodic__finite__ex,axiom,
! [D2: int,P: int > $o] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D2 )
=> ( ! [X3: int,K2: int] :
( ( P @ X3 )
= ( P @ ( minus_minus @ int @ X3 @ ( times_times @ int @ K2 @ D2 ) ) ) )
=> ( ( ? [X8: int] : ( P @ X8 ) )
= ( ? [X2: int] :
( ( member @ int @ X2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D2 ) )
& ( P @ X2 ) ) ) ) ) ) ).
% periodic_finite_ex
thf(fact_5056_word__subset__less,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,R3: word @ A,Y: word @ A,S2: word @ A] :
( ( ord_less_eq @ ( set @ ( word @ A ) ) @ ( set_or1337092689740270186AtMost @ ( word @ A ) @ X @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ X @ R3 ) @ ( one_one @ ( word @ A ) ) ) ) @ ( set_or1337092689740270186AtMost @ ( word @ A ) @ Y @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ Y @ S2 ) @ ( one_one @ ( word @ A ) ) ) ) )
=> ( ( ord_less_eq @ ( word @ A ) @ X @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ X @ R3 ) @ ( one_one @ ( word @ A ) ) ) )
=> ( ( ord_less_eq @ ( word @ A ) @ Y @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ Y @ S2 ) @ ( one_one @ ( word @ A ) ) ) )
=> ( ( S2
!= ( zero_zero @ ( word @ A ) ) )
=> ( ord_less_eq @ ( word @ A ) @ R3 @ S2 ) ) ) ) ) ) ).
% word_subset_less
thf(fact_5057_norm__prod__diff,axiom,
! [A: $tType,I5: $tType] :
( ( ( comm_monoid_mult @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [I6: set @ I5,Z: I5 > A,W: I5 > A] :
( ! [I3: I5] :
( ( member @ I5 @ I3 @ I6 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( Z @ I3 ) ) @ ( one_one @ real ) ) )
=> ( ! [I3: I5] :
( ( member @ I5 @ I3 @ I6 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( W @ I3 ) ) @ ( one_one @ real ) ) )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( groups7121269368397514597t_prod @ I5 @ A @ Z @ I6 ) @ ( groups7121269368397514597t_prod @ I5 @ A @ W @ I6 ) ) )
@ ( groups7311177749621191930dd_sum @ I5 @ real
@ ^ [I4: I5] : ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( Z @ I4 ) @ ( W @ I4 ) ) )
@ I6 ) ) ) ) ) ).
% norm_prod_diff
thf(fact_5058_prod_OatMost__shift,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G2: nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_ord_atMost @ nat @ N ) )
= ( times_times @ A @ ( G2 @ ( zero_zero @ nat ) )
@ ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( G2 @ ( suc @ I4 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% prod.atMost_shift
thf(fact_5059_fact__eq__fact__times,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq @ nat @ N @ M )
=> ( ( semiring_char_0_fact @ nat @ M )
= ( times_times @ nat @ ( semiring_char_0_fact @ nat @ N )
@ ( groups7121269368397514597t_prod @ nat @ nat
@ ^ [X2: nat] : X2
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ N ) @ M ) ) ) ) ) ).
% fact_eq_fact_times
thf(fact_5060_simp__from__to,axiom,
( ( set_or1337092689740270186AtMost @ int )
= ( ^ [I4: int,J3: int] : ( if @ ( set @ int ) @ ( ord_less @ int @ J3 @ I4 ) @ ( bot_bot @ ( set @ int ) ) @ ( insert @ int @ I4 @ ( set_or1337092689740270186AtMost @ int @ ( plus_plus @ int @ I4 @ ( one_one @ int ) ) @ J3 ) ) ) ) ) ).
% simp_from_to
thf(fact_5061_bset_I6_J,axiom,
! [D3: int,B2: set @ int,T: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D3 )
=> ! [X6: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D3 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ B2 )
=> ( X6
!= ( plus_plus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ord_less_eq @ int @ X6 @ T )
=> ( ord_less_eq @ int @ ( minus_minus @ int @ X6 @ D3 ) @ T ) ) ) ) ).
% bset(6)
thf(fact_5062_bset_I8_J,axiom,
! [D3: int,T: int,B2: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D3 )
=> ( ( member @ int @ ( minus_minus @ int @ T @ ( one_one @ int ) ) @ B2 )
=> ! [X6: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D3 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ B2 )
=> ( X6
!= ( plus_plus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ord_less_eq @ int @ T @ X6 )
=> ( ord_less_eq @ int @ T @ ( minus_minus @ int @ X6 @ D3 ) ) ) ) ) ) ).
% bset(8)
thf(fact_5063_aset_I6_J,axiom,
! [D3: int,T: int,A2: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D3 )
=> ( ( member @ int @ ( plus_plus @ int @ T @ ( one_one @ int ) ) @ A2 )
=> ! [X6: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D3 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ A2 )
=> ( X6
!= ( minus_minus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ord_less_eq @ int @ X6 @ T )
=> ( ord_less_eq @ int @ ( plus_plus @ int @ X6 @ D3 ) @ T ) ) ) ) ) ).
% aset(6)
thf(fact_5064_aset_I8_J,axiom,
! [D3: int,A2: set @ int,T: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D3 )
=> ! [X6: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D3 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ A2 )
=> ( X6
!= ( minus_minus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ord_less_eq @ int @ T @ X6 )
=> ( ord_less_eq @ int @ T @ ( plus_plus @ int @ X6 @ D3 ) ) ) ) ) ).
% aset(8)
thf(fact_5065_cppi,axiom,
! [D3: int,P: int > $o,P5: int > $o,A2: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D3 )
=> ( ? [Z6: int] :
! [X3: int] :
( ( ord_less @ int @ Z6 @ X3 )
=> ( ( P @ X3 )
= ( P5 @ X3 ) ) )
=> ( ! [X3: int] :
( ! [Xa2: int] :
( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D3 ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ A2 )
=> ( X3
!= ( minus_minus @ int @ Xb2 @ Xa2 ) ) ) )
=> ( ( P @ X3 )
=> ( P @ ( plus_plus @ int @ X3 @ D3 ) ) ) )
=> ( ! [X3: int,K2: int] :
( ( P5 @ X3 )
= ( P5 @ ( minus_minus @ int @ X3 @ ( times_times @ int @ K2 @ D3 ) ) ) )
=> ( ( ? [X8: int] : ( P @ X8 ) )
= ( ? [X2: int] :
( ( member @ int @ X2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D3 ) )
& ( P5 @ X2 ) )
| ? [X2: int] :
( ( member @ int @ X2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D3 ) )
& ? [Y2: int] :
( ( member @ int @ Y2 @ A2 )
& ( P @ ( minus_minus @ int @ Y2 @ X2 ) ) ) ) ) ) ) ) ) ) ).
% cppi
thf(fact_5066_cpmi,axiom,
! [D3: int,P: int > $o,P5: int > $o,B2: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D3 )
=> ( ? [Z6: int] :
! [X3: int] :
( ( ord_less @ int @ X3 @ Z6 )
=> ( ( P @ X3 )
= ( P5 @ X3 ) ) )
=> ( ! [X3: int] :
( ! [Xa2: int] :
( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D3 ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ B2 )
=> ( X3
!= ( plus_plus @ int @ Xb2 @ Xa2 ) ) ) )
=> ( ( P @ X3 )
=> ( P @ ( minus_minus @ int @ X3 @ D3 ) ) ) )
=> ( ! [X3: int,K2: int] :
( ( P5 @ X3 )
= ( P5 @ ( minus_minus @ int @ X3 @ ( times_times @ int @ K2 @ D3 ) ) ) )
=> ( ( ? [X8: int] : ( P @ X8 ) )
= ( ? [X2: int] :
( ( member @ int @ X2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D3 ) )
& ( P5 @ X2 ) )
| ? [X2: int] :
( ( member @ int @ X2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D3 ) )
& ? [Y2: int] :
( ( member @ int @ Y2 @ B2 )
& ( P @ ( plus_plus @ int @ Y2 @ X2 ) ) ) ) ) ) ) ) ) ) ).
% cpmi
thf(fact_5067_pochhammer__Suc__prod,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A3: A,N: nat] :
( ( comm_s3205402744901411588hammer @ A @ A3 @ ( suc @ N ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( plus_plus @ A @ A3 @ ( semiring_1_of_nat @ A @ I4 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% pochhammer_Suc_prod
thf(fact_5068_pochhammer__prod__rev,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ( ( comm_s3205402744901411588hammer @ A )
= ( ^ [A5: A,N4: nat] :
( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( plus_plus @ A @ A5 @ ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ N4 @ I4 ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( one_one @ nat ) @ N4 ) ) ) ) ) ).
% pochhammer_prod_rev
thf(fact_5069_fact__div__fact,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq @ nat @ N @ M )
=> ( ( divide_divide @ nat @ ( semiring_char_0_fact @ nat @ M ) @ ( semiring_char_0_fact @ nat @ N ) )
= ( groups7121269368397514597t_prod @ nat @ nat
@ ^ [X2: nat] : X2
@ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) @ M ) ) ) ) ).
% fact_div_fact
thf(fact_5070_prod_Oin__pairs,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G2: nat > A,M: nat,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_or1337092689740270186AtMost @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( G2 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I4 ) ) @ ( G2 @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I4 ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ).
% prod.in_pairs
thf(fact_5071_prod_Oin__pairs__0,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G2: nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_ord_atMost @ nat @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( G2 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I4 ) ) @ ( G2 @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I4 ) ) ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% prod.in_pairs_0
thf(fact_5072_pochhammer__Suc__prod__rev,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A3: A,N: nat] :
( ( comm_s3205402744901411588hammer @ A @ A3 @ ( suc @ N ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( plus_plus @ A @ A3 @ ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ N @ I4 ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% pochhammer_Suc_prod_rev
thf(fact_5073_bij__betw__empty2,axiom,
! [B: $tType,A: $tType,F3: A > B,A2: set @ A] :
( ( bij_betw @ A @ B @ F3 @ A2 @ ( bot_bot @ ( set @ B ) ) )
=> ( A2
= ( bot_bot @ ( set @ A ) ) ) ) ).
% bij_betw_empty2
thf(fact_5074_bij__betw__empty1,axiom,
! [A: $tType,B: $tType,F3: A > B,A2: set @ B] :
( ( bij_betw @ A @ B @ F3 @ ( bot_bot @ ( set @ A ) ) @ A2 )
=> ( A2
= ( bot_bot @ ( set @ B ) ) ) ) ).
% bij_betw_empty1
thf(fact_5075_prod_Ozero__middle,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [P4: nat,K: nat,G2: nat > A,H2: nat > A] :
( ( ord_less_eq @ nat @ ( one_one @ nat ) @ P4 )
=> ( ( ord_less_eq @ nat @ K @ P4 )
=> ( ( groups7121269368397514597t_prod @ nat @ A
@ ^ [J3: nat] : ( if @ A @ ( ord_less @ nat @ J3 @ K ) @ ( G2 @ J3 ) @ ( if @ A @ ( J3 = K ) @ ( one_one @ A ) @ ( H2 @ ( minus_minus @ nat @ J3 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) )
@ ( set_ord_atMost @ nat @ P4 ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [J3: nat] : ( if @ A @ ( ord_less @ nat @ J3 @ K ) @ ( G2 @ J3 ) @ ( H2 @ J3 ) )
@ ( set_ord_atMost @ nat @ ( minus_minus @ nat @ P4 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ) ) ).
% prod.zero_middle
thf(fact_5076_gbinomial__Suc,axiom,
! [A: $tType] :
( ( ( semiring_char_0 @ A )
& ( semidom_divide @ A ) )
=> ! [A3: A,K: nat] :
( ( gbinomial @ A @ A3 @ ( suc @ K ) )
= ( divide_divide @ A
@ ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( minus_minus @ A @ A3 @ ( semiring_1_of_nat @ A @ I4 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ K ) )
@ ( semiring_char_0_fact @ A @ ( suc @ K ) ) ) ) ) ).
% gbinomial_Suc
thf(fact_5077_Sum__Icc__int,axiom,
! [M: int,N: int] :
( ( ord_less_eq @ int @ M @ N )
=> ( ( groups7311177749621191930dd_sum @ int @ int
@ ^ [X2: int] : X2
@ ( set_or1337092689740270186AtMost @ int @ M @ N ) )
= ( divide_divide @ int @ ( minus_minus @ int @ ( times_times @ int @ N @ ( plus_plus @ int @ N @ ( one_one @ int ) ) ) @ ( times_times @ int @ M @ ( minus_minus @ int @ M @ ( one_one @ int ) ) ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ).
% Sum_Icc_int
thf(fact_5078_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
@ ^ [Z4: complex] :
( ( power_power @ complex @ Z4 @ N )
= ( one_one @ complex ) ) )
@ ( collect @ complex
@ ^ [Z4: complex] :
( ( power_power @ complex @ Z4 @ N )
= C2 ) ) ) ) ) ).
% bij_betw_nth_root_unity
thf(fact_5079_Arg__def,axiom,
( arg
= ( ^ [Z4: complex] :
( if @ real
@ ( Z4
= ( zero_zero @ complex ) )
@ ( zero_zero @ real )
@ ( fChoice @ real
@ ^ [A5: real] :
( ( ( sgn_sgn @ complex @ Z4 )
= ( cis @ A5 ) )
& ( ord_less @ real @ ( uminus_uminus @ real @ pi ) @ A5 )
& ( ord_less_eq @ real @ A5 @ pi ) ) ) ) ) ) ).
% Arg_def
thf(fact_5080_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_5081_real__root__zero,axiom,
! [N: nat] :
( ( root @ N @ ( zero_zero @ real ) )
= ( zero_zero @ real ) ) ).
% real_root_zero
thf(fact_5082_Ints__sum,axiom,
! [A: $tType,B: $tType] :
( ( ring_1 @ B )
=> ! [A2: set @ A,F3: A > B] :
( ! [X3: A] :
( ( member @ A @ X3 @ A2 )
=> ( member @ B @ ( F3 @ X3 ) @ ( ring_1_Ints @ B ) ) )
=> ( member @ B @ ( groups7311177749621191930dd_sum @ A @ B @ F3 @ A2 ) @ ( ring_1_Ints @ B ) ) ) ) ).
% Ints_sum
thf(fact_5083_Ints__prod,axiom,
! [A: $tType,B: $tType] :
( ( ( comm_monoid_mult @ B )
& ( ring_1 @ B ) )
=> ! [A2: set @ A,F3: A > B] :
( ! [X3: A] :
( ( member @ A @ X3 @ A2 )
=> ( member @ B @ ( F3 @ X3 ) @ ( ring_1_Ints @ B ) ) )
=> ( member @ B @ ( groups7121269368397514597t_prod @ A @ B @ F3 @ A2 ) @ ( ring_1_Ints @ B ) ) ) ) ).
% Ints_prod
thf(fact_5084_frac__in__Ints__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( member @ A @ ( archimedean_frac @ A @ X ) @ ( ring_1_Ints @ A ) )
= ( member @ A @ X @ ( ring_1_Ints @ A ) ) ) ) ).
% frac_in_Ints_iff
thf(fact_5085_real__root__Suc__0,axiom,
! [X: real] :
( ( root @ ( suc @ ( zero_zero @ nat ) ) @ X )
= X ) ).
% real_root_Suc_0
thf(fact_5086_real__root__eq__iff,axiom,
! [N: nat,X: real,Y: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ( root @ N @ X )
= ( root @ N @ Y ) )
= ( X = Y ) ) ) ).
% real_root_eq_iff
thf(fact_5087_root__0,axiom,
! [X: real] :
( ( root @ ( zero_zero @ nat ) @ X )
= ( zero_zero @ real ) ) ).
% root_0
thf(fact_5088_frac__eq__0__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ( archimedean_frac @ A @ X )
= ( zero_zero @ A ) )
= ( member @ A @ X @ ( ring_1_Ints @ A ) ) ) ) ).
% frac_eq_0_iff
thf(fact_5089_some__insert__self,axiom,
! [A: $tType,S: set @ A] :
( ( S
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( insert @ A
@ ( fChoice @ A
@ ^ [X2: A] : ( member @ A @ X2 @ S ) )
@ S )
= S ) ) ).
% some_insert_self
thf(fact_5090_real__root__eq__0__iff,axiom,
! [N: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ( root @ N @ X )
= ( zero_zero @ real ) )
= ( X
= ( zero_zero @ real ) ) ) ) ).
% real_root_eq_0_iff
thf(fact_5091_real__root__less__iff,axiom,
! [N: nat,X: real,Y: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ ( root @ N @ X ) @ ( root @ N @ Y ) )
= ( ord_less @ real @ X @ Y ) ) ) ).
% real_root_less_iff
thf(fact_5092_real__root__le__iff,axiom,
! [N: nat,X: real,Y: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less_eq @ real @ ( root @ N @ X ) @ ( root @ N @ Y ) )
= ( ord_less_eq @ real @ X @ Y ) ) ) ).
% real_root_le_iff
thf(fact_5093_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_5094_real__root__eq__1__iff,axiom,
! [N: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ( root @ N @ X )
= ( one_one @ real ) )
= ( X
= ( one_one @ real ) ) ) ) ).
% real_root_eq_1_iff
thf(fact_5095_floor__add2,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Y: A] :
( ( ( member @ A @ X @ ( ring_1_Ints @ A ) )
| ( member @ A @ Y @ ( ring_1_Ints @ A ) ) )
=> ( ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ X @ Y ) )
= ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( archim6421214686448440834_floor @ A @ Y ) ) ) ) ) ).
% floor_add2
thf(fact_5096_frac__gt__0__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( archimedean_frac @ A @ X ) )
= ( ~ ( member @ A @ X @ ( ring_1_Ints @ A ) ) ) ) ) ).
% frac_gt_0_iff
thf(fact_5097_real__root__gt__0__iff,axiom,
! [N: nat,Y: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( root @ N @ Y ) )
= ( ord_less @ real @ ( zero_zero @ real ) @ Y ) ) ) ).
% real_root_gt_0_iff
thf(fact_5098_real__root__lt__0__iff,axiom,
! [N: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ ( root @ N @ X ) @ ( zero_zero @ real ) )
= ( ord_less @ real @ X @ ( zero_zero @ real ) ) ) ) ).
% real_root_lt_0_iff
thf(fact_5099_real__root__le__0__iff,axiom,
! [N: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less_eq @ real @ ( root @ N @ X ) @ ( zero_zero @ real ) )
= ( ord_less_eq @ real @ X @ ( zero_zero @ real ) ) ) ) ).
% real_root_le_0_iff
thf(fact_5100_real__root__ge__0__iff,axiom,
! [N: nat,Y: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( root @ N @ Y ) )
= ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y ) ) ) ).
% real_root_ge_0_iff
thf(fact_5101_real__root__gt__1__iff,axiom,
! [N: nat,Y: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ ( one_one @ real ) @ ( root @ N @ Y ) )
= ( ord_less @ real @ ( one_one @ real ) @ Y ) ) ) ).
% real_root_gt_1_iff
thf(fact_5102_real__root__lt__1__iff,axiom,
! [N: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ ( root @ N @ X ) @ ( one_one @ real ) )
= ( ord_less @ real @ X @ ( one_one @ real ) ) ) ) ).
% real_root_lt_1_iff
thf(fact_5103_real__root__le__1__iff,axiom,
! [N: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less_eq @ real @ ( root @ N @ X ) @ ( one_one @ real ) )
= ( ord_less_eq @ real @ X @ ( one_one @ real ) ) ) ) ).
% real_root_le_1_iff
thf(fact_5104_real__root__ge__1__iff,axiom,
! [N: nat,Y: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less_eq @ real @ ( one_one @ real ) @ ( root @ N @ Y ) )
= ( ord_less_eq @ real @ ( one_one @ real ) @ Y ) ) ) ).
% real_root_ge_1_iff
thf(fact_5105_real__root__pow__pos2,axiom,
! [N: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( power_power @ real @ ( root @ N @ X ) @ N )
= X ) ) ) ).
% real_root_pow_pos2
thf(fact_5106_int__prod,axiom,
! [B: $tType,F3: B > nat,A2: set @ B] :
( ( semiring_1_of_nat @ int @ ( groups7121269368397514597t_prod @ B @ nat @ F3 @ A2 ) )
= ( groups7121269368397514597t_prod @ B @ int
@ ^ [X2: B] : ( semiring_1_of_nat @ int @ ( F3 @ X2 ) )
@ A2 ) ) ).
% int_prod
thf(fact_5107_Ints__power,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [A3: A,N: nat] :
( ( member @ A @ A3 @ ( ring_1_Ints @ A ) )
=> ( member @ A @ ( power_power @ A @ A3 @ N ) @ ( ring_1_Ints @ A ) ) ) ) ).
% Ints_power
thf(fact_5108_real__root__divide,axiom,
! [N: nat,X: real,Y: real] :
( ( root @ N @ ( divide_divide @ real @ X @ Y ) )
= ( divide_divide @ real @ ( root @ N @ X ) @ ( root @ N @ Y ) ) ) ).
% real_root_divide
thf(fact_5109_Ints__add,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [A3: A,B3: A] :
( ( member @ A @ A3 @ ( ring_1_Ints @ A ) )
=> ( ( member @ A @ B3 @ ( ring_1_Ints @ A ) )
=> ( member @ A @ ( plus_plus @ A @ A3 @ B3 ) @ ( ring_1_Ints @ A ) ) ) ) ) ).
% Ints_add
thf(fact_5110_Ints__0,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ( member @ A @ ( zero_zero @ A ) @ ( ring_1_Ints @ A ) ) ) ).
% Ints_0
thf(fact_5111_real__root__minus,axiom,
! [N: nat,X: real] :
( ( root @ N @ ( uminus_uminus @ real @ X ) )
= ( uminus_uminus @ real @ ( root @ N @ X ) ) ) ).
% real_root_minus
thf(fact_5112_minus__in__Ints__iff,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [X: A] :
( ( member @ A @ ( uminus_uminus @ A @ X ) @ ( ring_1_Ints @ A ) )
= ( member @ A @ X @ ( ring_1_Ints @ A ) ) ) ) ).
% minus_in_Ints_iff
thf(fact_5113_Ints__minus,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [A3: A] :
( ( member @ A @ A3 @ ( ring_1_Ints @ A ) )
=> ( member @ A @ ( uminus_uminus @ A @ A3 ) @ ( ring_1_Ints @ A ) ) ) ) ).
% Ints_minus
thf(fact_5114_Ints__1,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ( member @ A @ ( one_one @ A ) @ ( ring_1_Ints @ A ) ) ) ).
% Ints_1
thf(fact_5115_Ints__abs,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A] :
( ( member @ A @ A3 @ ( ring_1_Ints @ A ) )
=> ( member @ A @ ( abs_abs @ A @ A3 ) @ ( ring_1_Ints @ A ) ) ) ) ).
% Ints_abs
thf(fact_5116_some__elem,axiom,
! [A: $tType,S: set @ A] :
( ( S
!= ( bot_bot @ ( set @ A ) ) )
=> ( member @ A
@ ( fChoice @ A
@ ^ [X2: A] : ( member @ A @ X2 @ S ) )
@ S ) ) ).
% some_elem
thf(fact_5117_Ints__diff,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [A3: A,B3: A] :
( ( member @ A @ A3 @ ( ring_1_Ints @ A ) )
=> ( ( member @ A @ B3 @ ( ring_1_Ints @ A ) )
=> ( member @ A @ ( minus_minus @ A @ A3 @ B3 ) @ ( ring_1_Ints @ A ) ) ) ) ) ).
% Ints_diff
thf(fact_5118_verit__sko__ex_H,axiom,
! [A: $tType,P: A > $o,A2: $o] :
( ( ( P @ ( fChoice @ A @ P ) )
= A2 )
=> ( ( ? [X8: A] : ( P @ X8 ) )
= A2 ) ) ).
% verit_sko_ex'
thf(fact_5119_verit__sko__forall,axiom,
! [A: $tType] :
( ( ^ [P2: A > $o] :
! [X4: A] : ( P2 @ X4 ) )
= ( ^ [P3: A > $o] :
( P3
@ ( fChoice @ A
@ ^ [X2: A] :
~ ( P3 @ X2 ) ) ) ) ) ).
% verit_sko_forall
thf(fact_5120_verit__sko__forall_H,axiom,
! [A: $tType,P: A > $o,A2: $o] :
( ( ( P
@ ( fChoice @ A
@ ^ [X2: A] :
~ ( P @ X2 ) ) )
= A2 )
=> ( ( ! [X8: A] : ( P @ X8 ) )
= A2 ) ) ).
% verit_sko_forall'
thf(fact_5121_verit__sko__forall_H_H,axiom,
! [A: $tType,B2: A,A2: A,P: A > $o] :
( ( B2 = A2 )
=> ( ( ( fChoice @ A @ P )
= A2 )
= ( ( fChoice @ A @ P )
= B2 ) ) ) ).
% verit_sko_forall''
thf(fact_5122_verit__sko__ex__indirect,axiom,
! [A: $tType,X: A,P: A > $o] :
( ( X
= ( fChoice @ A @ P ) )
=> ( ( ? [X8: A] : ( P @ X8 ) )
= ( P @ X ) ) ) ).
% verit_sko_ex_indirect
thf(fact_5123_verit__sko__ex__indirect2,axiom,
! [A: $tType,X: A,P: A > $o,P5: A > $o] :
( ( X
= ( fChoice @ A @ P ) )
=> ( ! [X3: A] :
( ( P @ X3 )
= ( P5 @ X3 ) )
=> ( ( ? [X8: A] : ( P5 @ X8 ) )
= ( P @ X ) ) ) ) ).
% verit_sko_ex_indirect2
thf(fact_5124_verit__sko__forall__indirect,axiom,
! [A: $tType,X: A,P: A > $o] :
( ( X
= ( fChoice @ A
@ ^ [X2: A] :
~ ( P @ X2 ) ) )
=> ( ( ! [X8: A] : ( P @ X8 ) )
= ( P @ X ) ) ) ).
% verit_sko_forall_indirect
thf(fact_5125_verit__sko__forall__indirect2,axiom,
! [A: $tType,X: A,P: A > $o,P5: A > $o] :
( ( X
= ( fChoice @ A
@ ^ [X2: A] :
~ ( P @ X2 ) ) )
=> ( ! [X3: A] :
( ( P @ X3 )
= ( P5 @ X3 ) )
=> ( ( ! [X8: A] : ( P5 @ X8 ) )
= ( P @ X ) ) ) ) ).
% verit_sko_forall_indirect2
thf(fact_5126_real__root__commute,axiom,
! [M: nat,N: nat,X: real] :
( ( root @ M @ ( root @ N @ X ) )
= ( root @ N @ ( root @ M @ X ) ) ) ).
% real_root_commute
thf(fact_5127_Ints__numeral,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [N: num] : ( member @ A @ ( numeral_numeral @ A @ N ) @ ( ring_1_Ints @ A ) ) ) ).
% Ints_numeral
thf(fact_5128_Ints__cases,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [Q3: A] :
( ( member @ A @ Q3 @ ( ring_1_Ints @ A ) )
=> ~ ! [Z3: int] :
( Q3
!= ( ring_1_of_int @ A @ Z3 ) ) ) ) ).
% Ints_cases
thf(fact_5129_Ints__induct,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [Q3: A,P: A > $o] :
( ( member @ A @ Q3 @ ( ring_1_Ints @ A ) )
=> ( ! [Z3: int] : ( P @ ( ring_1_of_int @ A @ Z3 ) )
=> ( P @ Q3 ) ) ) ) ).
% Ints_induct
thf(fact_5130_Ints__of__int,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [Z: int] : ( member @ A @ ( ring_1_of_int @ A @ Z ) @ ( ring_1_Ints @ A ) ) ) ).
% Ints_of_int
thf(fact_5131_Ints__mult,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [A3: A,B3: A] :
( ( member @ A @ A3 @ ( ring_1_Ints @ A ) )
=> ( ( member @ A @ B3 @ ( ring_1_Ints @ A ) )
=> ( member @ A @ ( times_times @ A @ A3 @ B3 ) @ ( ring_1_Ints @ A ) ) ) ) ) ).
% Ints_mult
thf(fact_5132_real__root__mult__exp,axiom,
! [M: nat,N: nat,X: real] :
( ( root @ ( times_times @ nat @ M @ N ) @ X )
= ( root @ M @ ( root @ N @ X ) ) ) ).
% real_root_mult_exp
thf(fact_5133_real__root__mult,axiom,
! [N: nat,X: real,Y: real] :
( ( root @ N @ ( times_times @ real @ X @ Y ) )
= ( times_times @ real @ ( root @ N @ X ) @ ( root @ N @ Y ) ) ) ).
% real_root_mult
thf(fact_5134_real__root__inverse,axiom,
! [N: nat,X: real] :
( ( root @ N @ ( inverse_inverse @ real @ X ) )
= ( inverse_inverse @ real @ ( root @ N @ X ) ) ) ).
% real_root_inverse
thf(fact_5135_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_5136_real__root__pos__pos__le,axiom,
! [X: real,N: nat] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( root @ N @ X ) ) ) ).
% real_root_pos_pos_le
thf(fact_5137_Ints__double__eq__0__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [A3: A] :
( ( member @ A @ A3 @ ( ring_1_Ints @ A ) )
=> ( ( ( plus_plus @ A @ A3 @ A3 )
= ( zero_zero @ A ) )
= ( A3
= ( zero_zero @ A ) ) ) ) ) ).
% Ints_double_eq_0_iff
thf(fact_5138_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
@ ^ [X2: int] : X2
@ ( set_or1337092689740270186AtMost @ int @ ( semiring_1_of_nat @ int @ I ) @ ( semiring_1_of_nat @ int @ J ) ) ) ) ).
% prod_int_eq
thf(fact_5139_real__root__less__mono,axiom,
! [N: nat,X: real,Y: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ X @ Y )
=> ( ord_less @ real @ ( root @ N @ X ) @ ( root @ N @ Y ) ) ) ) ).
% real_root_less_mono
thf(fact_5140_real__root__le__mono,axiom,
! [N: nat,X: real,Y: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less_eq @ real @ X @ Y )
=> ( ord_less_eq @ real @ ( root @ N @ X ) @ ( root @ N @ Y ) ) ) ) ).
% real_root_le_mono
thf(fact_5141_real__root__power,axiom,
! [N: nat,X: real,K: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( root @ N @ ( power_power @ real @ X @ K ) )
= ( power_power @ real @ ( root @ N @ X ) @ K ) ) ) ).
% real_root_power
thf(fact_5142_real__root__abs,axiom,
! [N: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( root @ N @ ( abs_abs @ real @ X ) )
= ( abs_abs @ real @ ( root @ N @ X ) ) ) ) ).
% real_root_abs
thf(fact_5143_sgn__root,axiom,
! [N: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( sgn_sgn @ real @ ( root @ N @ X ) )
= ( sgn_sgn @ real @ X ) ) ) ).
% sgn_root
thf(fact_5144_Ints__odd__nonzero,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [A3: A] :
( ( member @ A @ A3 @ ( ring_1_Ints @ A ) )
=> ( ( plus_plus @ A @ ( plus_plus @ A @ ( one_one @ A ) @ A3 ) @ A3 )
!= ( zero_zero @ A ) ) ) ) ).
% Ints_odd_nonzero
thf(fact_5145_prod__int__plus__eq,axiom,
! [I: nat,J: nat] :
( ( groups7121269368397514597t_prod @ nat @ int @ ( semiring_1_of_nat @ int ) @ ( set_or1337092689740270186AtMost @ nat @ I @ ( plus_plus @ nat @ I @ J ) ) )
= ( groups7121269368397514597t_prod @ int @ int
@ ^ [X2: int] : X2
@ ( set_or1337092689740270186AtMost @ int @ ( semiring_1_of_nat @ int @ I ) @ ( semiring_1_of_nat @ int @ ( plus_plus @ nat @ I @ J ) ) ) ) ) ).
% prod_int_plus_eq
thf(fact_5146_of__int__divide__in__Ints,axiom,
! [A: $tType] :
( ( idom_divide @ A )
=> ! [B3: int,A3: int] :
( ( dvd_dvd @ int @ B3 @ A3 )
=> ( member @ A @ ( divide_divide @ A @ ( ring_1_of_int @ A @ A3 ) @ ( ring_1_of_int @ A @ B3 ) ) @ ( ring_1_Ints @ A ) ) ) ) ).
% of_int_divide_in_Ints
thf(fact_5147_real__root__gt__zero,axiom,
! [N: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( root @ N @ X ) ) ) ) ).
% real_root_gt_zero
thf(fact_5148_real__root__strict__decreasing,axiom,
! [N: nat,N3: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ nat @ N @ N3 )
=> ( ( ord_less @ real @ ( one_one @ real ) @ X )
=> ( ord_less @ real @ ( root @ N3 @ X ) @ ( root @ N @ X ) ) ) ) ) ).
% real_root_strict_decreasing
thf(fact_5149_sqrt__def,axiom,
( sqrt
= ( root @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% sqrt_def
thf(fact_5150_root__abs__power,axiom,
! [N: nat,Y: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( abs_abs @ real @ ( root @ N @ ( power_power @ real @ Y @ N ) ) )
= ( abs_abs @ real @ Y ) ) ) ).
% root_abs_power
thf(fact_5151_Ints__odd__less__0,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A] :
( ( member @ A @ A3 @ ( ring_1_Ints @ A ) )
=> ( ( ord_less @ A @ ( plus_plus @ A @ ( plus_plus @ A @ ( one_one @ A ) @ A3 ) @ A3 ) @ ( zero_zero @ A ) )
= ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ) ).
% Ints_odd_less_0
thf(fact_5152_Ints__nonzero__abs__ge1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A] :
( ( member @ A @ X @ ( ring_1_Ints @ A ) )
=> ( ( X
!= ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ ( abs_abs @ A @ X ) ) ) ) ) ).
% Ints_nonzero_abs_ge1
thf(fact_5153_Ints__nonzero__abs__less1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A] :
( ( member @ A @ X @ ( ring_1_Ints @ A ) )
=> ( ( ord_less @ A @ ( abs_abs @ A @ X ) @ ( one_one @ A ) )
=> ( X
= ( zero_zero @ A ) ) ) ) ) ).
% Ints_nonzero_abs_less1
thf(fact_5154_Ints__eq__abs__less1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A,Y: A] :
( ( member @ A @ X @ ( ring_1_Ints @ A ) )
=> ( ( member @ A @ Y @ ( ring_1_Ints @ A ) )
=> ( ( X = Y )
= ( ord_less @ A @ ( abs_abs @ A @ ( minus_minus @ A @ X @ Y ) ) @ ( one_one @ A ) ) ) ) ) ) ).
% Ints_eq_abs_less1
thf(fact_5155_sin__times__pi__eq__0,axiom,
! [X: real] :
( ( ( sin @ real @ ( times_times @ real @ X @ pi ) )
= ( zero_zero @ real ) )
= ( member @ real @ X @ ( ring_1_Ints @ real ) ) ) ).
% sin_times_pi_eq_0
thf(fact_5156_real__root__pos__pos,axiom,
! [N: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( root @ N @ X ) ) ) ) ).
% real_root_pos_pos
thf(fact_5157_real__root__strict__increasing,axiom,
! [N: nat,N3: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ nat @ N @ N3 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ X @ ( one_one @ real ) )
=> ( ord_less @ real @ ( root @ N @ X ) @ ( root @ N3 @ X ) ) ) ) ) ) ).
% real_root_strict_increasing
thf(fact_5158_real__root__decreasing,axiom,
! [N: nat,N3: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less_eq @ nat @ N @ N3 )
=> ( ( ord_less_eq @ real @ ( one_one @ real ) @ X )
=> ( ord_less_eq @ real @ ( root @ N3 @ X ) @ ( root @ N @ X ) ) ) ) ) ).
% real_root_decreasing
thf(fact_5159_odd__real__root__pow,axiom,
! [N: nat,X: real] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( power_power @ real @ ( root @ N @ X ) @ N )
= X ) ) ).
% odd_real_root_pow
thf(fact_5160_odd__real__root__unique,axiom,
! [N: nat,Y: real,X: real] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( ( power_power @ real @ Y @ N )
= X )
=> ( ( root @ N @ X )
= Y ) ) ) ).
% odd_real_root_unique
thf(fact_5161_odd__real__root__power__cancel,axiom,
! [N: nat,X: real] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( root @ N @ ( power_power @ real @ X @ N ) )
= X ) ) ).
% odd_real_root_power_cancel
thf(fact_5162_real__root__pow__pos,axiom,
! [N: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( power_power @ real @ ( root @ N @ X ) @ N )
= X ) ) ) ).
% real_root_pow_pos
thf(fact_5163_real__root__power__cancel,axiom,
! [N: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( root @ N @ ( power_power @ real @ X @ N ) )
= X ) ) ) ).
% real_root_power_cancel
thf(fact_5164_real__root__pos__unique,axiom,
! [N: nat,Y: real,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y )
=> ( ( ( power_power @ real @ Y @ N )
= X )
=> ( ( root @ N @ X )
= Y ) ) ) ) ).
% real_root_pos_unique
thf(fact_5165_frac__neg,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ( member @ A @ X @ ( ring_1_Ints @ A ) )
=> ( ( archimedean_frac @ A @ ( uminus_uminus @ A @ X ) )
= ( zero_zero @ A ) ) )
& ( ~ ( member @ A @ X @ ( ring_1_Ints @ A ) )
=> ( ( archimedean_frac @ A @ ( uminus_uminus @ A @ X ) )
= ( minus_minus @ A @ ( one_one @ A ) @ ( archimedean_frac @ A @ X ) ) ) ) ) ) ).
% frac_neg
thf(fact_5166_real__root__increasing,axiom,
! [N: nat,N3: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less_eq @ nat @ N @ N3 )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ X @ ( one_one @ real ) )
=> ( ord_less_eq @ real @ ( root @ N @ X ) @ ( root @ N3 @ X ) ) ) ) ) ) ).
% real_root_increasing
thf(fact_5167_sgn__power__root,axiom,
! [N: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( times_times @ real @ ( sgn_sgn @ real @ ( root @ N @ X ) ) @ ( power_power @ real @ ( abs_abs @ real @ ( root @ N @ X ) ) @ N ) )
= X ) ) ).
% sgn_power_root
thf(fact_5168_root__sgn__power,axiom,
! [N: nat,Y: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( root @ N @ ( times_times @ real @ ( sgn_sgn @ real @ Y ) @ ( power_power @ real @ ( abs_abs @ real @ Y ) @ N ) ) )
= Y ) ) ).
% root_sgn_power
thf(fact_5169_le__mult__floor__Ints,axiom,
! [A: $tType,B: $tType] :
( ( ( archim2362893244070406136eiling @ B )
& ( linordered_idom @ A ) )
=> ! [A3: B,B3: B] :
( ( ord_less_eq @ B @ ( zero_zero @ B ) @ A3 )
=> ( ( member @ B @ A3 @ ( ring_1_Ints @ B ) )
=> ( ord_less_eq @ A @ ( ring_1_of_int @ A @ ( times_times @ int @ ( archim6421214686448440834_floor @ B @ A3 ) @ ( archim6421214686448440834_floor @ B @ B3 ) ) ) @ ( ring_1_of_int @ A @ ( archim6421214686448440834_floor @ B @ ( times_times @ B @ A3 @ B3 ) ) ) ) ) ) ) ).
% le_mult_floor_Ints
thf(fact_5170_mult__ceiling__le__Ints,axiom,
! [A: $tType,B: $tType] :
( ( ( archim2362893244070406136eiling @ B )
& ( linordered_idom @ A ) )
=> ! [A3: B,B3: B] :
( ( ord_less_eq @ B @ ( zero_zero @ B ) @ A3 )
=> ( ( member @ B @ A3 @ ( ring_1_Ints @ B ) )
=> ( ord_less_eq @ A @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ B @ ( times_times @ B @ A3 @ B3 ) ) ) @ ( ring_1_of_int @ A @ ( times_times @ int @ ( archimedean_ceiling @ B @ A3 ) @ ( archimedean_ceiling @ B @ B3 ) ) ) ) ) ) ) ).
% mult_ceiling_le_Ints
thf(fact_5171_frac__unique__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,A3: A] :
( ( ( archimedean_frac @ A @ X )
= A3 )
= ( ( member @ A @ ( minus_minus @ A @ X @ A3 ) @ ( ring_1_Ints @ A ) )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
& ( ord_less @ A @ A3 @ ( one_one @ A ) ) ) ) ) ).
% frac_unique_iff
thf(fact_5172_log__root,axiom,
! [N: nat,A3: real,B3: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ A3 )
=> ( ( log @ B3 @ ( root @ N @ A3 ) )
= ( divide_divide @ real @ ( log @ B3 @ A3 ) @ ( semiring_1_of_nat @ real @ N ) ) ) ) ) ).
% log_root
thf(fact_5173_log__base__root,axiom,
! [N: nat,B3: real,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ B3 )
=> ( ( log @ ( root @ N @ B3 ) @ X )
= ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ ( log @ B3 @ X ) ) ) ) ) ).
% log_base_root
thf(fact_5174_ln__root,axiom,
! [N: nat,B3: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ B3 )
=> ( ( ln_ln @ real @ ( root @ N @ B3 ) )
= ( divide_divide @ real @ ( ln_ln @ real @ B3 ) @ ( semiring_1_of_nat @ real @ N ) ) ) ) ) ).
% ln_root
thf(fact_5175_split__root,axiom,
! [P: real > $o,N: nat,X: real] :
( ( P @ ( root @ N @ X ) )
= ( ( ( N
= ( zero_zero @ nat ) )
=> ( P @ ( zero_zero @ real ) ) )
& ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ! [Y2: real] :
( ( ( times_times @ real @ ( sgn_sgn @ real @ Y2 ) @ ( power_power @ real @ ( abs_abs @ real @ Y2 ) @ N ) )
= X )
=> ( P @ Y2 ) ) ) ) ) ).
% split_root
thf(fact_5176_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_5177_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_5178_root__powr__inverse,axiom,
! [N: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( root @ N @ X )
= ( powr @ real @ X @ ( divide_divide @ real @ ( one_one @ real ) @ ( semiring_1_of_nat @ real @ N ) ) ) ) ) ) ).
% root_powr_inverse
thf(fact_5179_some__sym__eq__trivial,axiom,
! [A: $tType,X: A] :
( ( fChoice @ A
@ ( ^ [Y3: A,Z2: A] : Y3 = Z2
@ X ) )
= X ) ).
% some_sym_eq_trivial
thf(fact_5180_some__eq__trivial,axiom,
! [A: $tType,X: A] :
( ( fChoice @ A
@ ^ [Y2: A] : Y2 = X )
= X ) ).
% some_eq_trivial
thf(fact_5181_some__equality,axiom,
! [A: $tType,P: A > $o,A3: A] :
( ( P @ A3 )
=> ( ! [X3: A] :
( ( P @ X3 )
=> ( X3 = A3 ) )
=> ( ( fChoice @ A @ P )
= A3 ) ) ) ).
% some_equality
thf(fact_5182_someI2,axiom,
! [A: $tType,P: A > $o,A3: A,Q: A > $o] :
( ( P @ A3 )
=> ( ! [X3: A] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( Q @ ( fChoice @ A @ P ) ) ) ) ).
% someI2
thf(fact_5183_someI__ex,axiom,
! [A: $tType,P: A > $o] :
( ? [X_12: A] : ( P @ X_12 )
=> ( P @ ( fChoice @ A @ P ) ) ) ).
% someI_ex
thf(fact_5184_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_5185_someI2__bex,axiom,
! [A: $tType,A2: set @ A,P: A > $o,Q: A > $o] :
( ? [X6: A] :
( ( member @ A @ X6 @ A2 )
& ( P @ X6 ) )
=> ( ! [X3: A] :
( ( ( member @ A @ X3 @ A2 )
& ( P @ X3 ) )
=> ( Q @ X3 ) )
=> ( Q
@ ( fChoice @ A
@ ^ [X2: A] :
( ( member @ A @ X2 @ A2 )
& ( P @ X2 ) ) ) ) ) ) ).
% someI2_bex
thf(fact_5186_some__eq__ex,axiom,
! [A: $tType,P: A > $o] :
( ( P @ ( fChoice @ A @ P ) )
= ( ? [X8: A] : ( P @ X8 ) ) ) ).
% some_eq_ex
thf(fact_5187_some1__equality,axiom,
! [A: $tType,P: A > $o,A3: A] :
( ? [X6: A] :
( ( P @ X6 )
& ! [Y4: A] :
( ( P @ Y4 )
=> ( Y4 = X6 ) ) )
=> ( ( P @ A3 )
=> ( ( fChoice @ A @ P )
= A3 ) ) ) ).
% some1_equality
thf(fact_5188_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 )
=> ? [Y5: A] :
( ( P @ ( suc @ N2 ) @ Y5 )
& ( Q @ N2 @ X3 @ Y5 ) ) )
=> ? [F5: nat > A] :
! [N11: nat] :
( ( P @ N11 @ ( F5 @ N11 ) )
& ( Q @ N11 @ ( F5 @ N11 ) @ ( F5 @ ( suc @ N11 ) ) ) ) ) ) ).
% dependent_nat_choice
thf(fact_5189_some__in__eq,axiom,
! [A: $tType,A2: set @ A] :
( ( member @ A
@ ( fChoice @ A
@ ^ [X2: A] : ( member @ A @ X2 @ A2 ) )
@ A2 )
= ( A2
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% some_in_eq
thf(fact_5190_frame__rule__left,axiom,
! [A: $tType,P: assn,C2: heap_Time_Heap @ A,Q: A > assn,R: assn] :
( ( hoare_hoare_triple @ A @ P @ C2 @ Q )
=> ( hoare_hoare_triple @ A @ ( times_times @ assn @ R @ P ) @ C2
@ ^ [X2: A] : ( times_times @ assn @ R @ ( Q @ X2 ) ) ) ) ).
% frame_rule_left
thf(fact_5191_div__half__nat,axiom,
! [Y: nat,X: nat] :
( ( Y
!= ( zero_zero @ nat ) )
=> ( ( product_Pair @ nat @ nat @ ( divide_divide @ nat @ X @ Y ) @ ( modulo_modulo @ nat @ X @ Y ) )
= ( if @ ( product_prod @ nat @ nat ) @ ( ord_less_eq @ nat @ Y @ ( minus_minus @ nat @ X @ ( times_times @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( divide_divide @ nat @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ Y ) ) @ Y ) ) ) @ ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( divide_divide @ nat @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ Y ) ) @ ( one_one @ nat ) ) @ ( minus_minus @ nat @ ( minus_minus @ nat @ X @ ( times_times @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( divide_divide @ nat @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ Y ) ) @ Y ) ) @ Y ) ) @ ( product_Pair @ nat @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( divide_divide @ nat @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ Y ) ) @ ( minus_minus @ nat @ X @ ( times_times @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( divide_divide @ nat @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ Y ) ) @ Y ) ) ) ) ) ) ).
% div_half_nat
thf(fact_5192_concat__bit__Suc,axiom,
! [N: nat,K: int,L: int] :
( ( bit_concat_bit @ ( suc @ N ) @ K @ L )
= ( plus_plus @ int @ ( modulo_modulo @ int @ K @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_concat_bit @ N @ ( divide_divide @ int @ K @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ L ) ) ) ) ).
% concat_bit_Suc
thf(fact_5193_concat__bit__0,axiom,
! [K: int,L: int] :
( ( bit_concat_bit @ ( zero_zero @ nat ) @ K @ L )
= L ) ).
% concat_bit_0
thf(fact_5194_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_5195_concat__bit__nonnegative__iff,axiom,
! [N: nat,K: int,L: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_concat_bit @ N @ K @ L ) )
= ( ord_less_eq @ int @ ( zero_zero @ int ) @ L ) ) ).
% concat_bit_nonnegative_iff
thf(fact_5196_concat__bit__negative__iff,axiom,
! [N: nat,K: int,L: int] :
( ( ord_less @ int @ ( bit_concat_bit @ N @ K @ L ) @ ( zero_zero @ int ) )
= ( ord_less @ int @ L @ ( zero_zero @ int ) ) ) ).
% concat_bit_negative_iff
thf(fact_5197_pred__equals__eq2,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ A @ B )] :
( ( ( ^ [X2: A,Y2: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y2 ) @ R ) )
= ( ^ [X2: A,Y2: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y2 ) @ S ) ) )
= ( R = S ) ) ).
% pred_equals_eq2
thf(fact_5198_bot__empty__eq2,axiom,
! [B: $tType,A: $tType] :
( ( bot_bot @ ( A > B > $o ) )
= ( ^ [X2: A,Y2: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y2 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ) ) ).
% bot_empty_eq2
thf(fact_5199_sup__Un__eq2,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ A @ B )] :
( ( sup_sup @ ( A > B > $o )
@ ^ [X2: A,Y2: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y2 ) @ R )
@ ^ [X2: A,Y2: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y2 ) @ S ) )
= ( ^ [X2: A,Y2: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y2 ) @ ( sup_sup @ ( set @ ( product_prod @ A @ B ) ) @ R @ S ) ) ) ) ).
% sup_Un_eq2
thf(fact_5200_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 )
@ ^ [X2: A,Y2: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y2 ) @ R )
@ ^ [X2: A,Y2: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y2 ) @ S ) )
= ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R @ S ) ) ).
% pred_subset_eq2
thf(fact_5201_subrelI,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ B ),S2: set @ ( product_prod @ A @ B )] :
( ! [X3: A,Y4: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y4 ) @ R3 )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y4 ) @ S2 ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R3 @ S2 ) ) ).
% subrelI
thf(fact_5202_is__hoare__triple,axiom,
! [A: $tType,P: assn,C2: heap_Time_Heap @ A,Q: A > assn] :
( ( hoare_hoare_triple @ A @ P @ C2 @ Q )
=> ( hoare_hoare_triple @ A @ P @ C2 @ Q ) ) ).
% is_hoare_triple
thf(fact_5203_bit__concat__bit__iff,axiom,
! [M: nat,K: int,L: int,N: nat] :
( ( bit_se5641148757651400278ts_bit @ int @ ( bit_concat_bit @ M @ K @ L ) @ N )
= ( ( ( ord_less @ nat @ N @ M )
& ( bit_se5641148757651400278ts_bit @ int @ K @ N ) )
| ( ( ord_less_eq @ nat @ M @ N )
& ( bit_se5641148757651400278ts_bit @ int @ L @ ( minus_minus @ nat @ N @ M ) ) ) ) ) ).
% bit_concat_bit_iff
thf(fact_5204_norm__assertion__simps_I2_J,axiom,
! [A3: assn] :
( ( times_times @ assn @ A3 @ ( one_one @ assn ) )
= A3 ) ).
% norm_assertion_simps(2)
thf(fact_5205_norm__assertion__simps_I1_J,axiom,
! [A3: assn] :
( ( times_times @ assn @ ( one_one @ assn ) @ A3 )
= A3 ) ).
% norm_assertion_simps(1)
thf(fact_5206_norm__assertion__simps_I5_J,axiom,
! [X: assn] :
( ( sup_sup @ assn @ ( bot_bot @ assn ) @ X )
= X ) ).
% norm_assertion_simps(5)
thf(fact_5207_norm__assertion__simps_I6_J,axiom,
! [X: assn] :
( ( sup_sup @ assn @ X @ ( bot_bot @ assn ) )
= X ) ).
% norm_assertion_simps(6)
thf(fact_5208_divmod__step__eq,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [L: num,R3: A,Q3: A] :
( ( ( ord_less_eq @ A @ ( numeral_numeral @ A @ L ) @ R3 )
=> ( ( unique1321980374590559556d_step @ A @ L @ ( product_Pair @ A @ A @ Q3 @ R3 ) )
= ( product_Pair @ A @ A @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Q3 ) @ ( one_one @ A ) ) @ ( minus_minus @ A @ R3 @ ( numeral_numeral @ A @ L ) ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( numeral_numeral @ A @ L ) @ R3 )
=> ( ( unique1321980374590559556d_step @ A @ L @ ( product_Pair @ A @ A @ Q3 @ R3 ) )
= ( product_Pair @ A @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Q3 ) @ R3 ) ) ) ) ) ).
% divmod_step_eq
thf(fact_5209_divides__aux__eq,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [Q3: A,R3: A] :
( ( unique5940410009612947441es_aux @ A @ ( product_Pair @ A @ A @ Q3 @ R3 ) )
= ( R3
= ( zero_zero @ A ) ) ) ) ).
% divides_aux_eq
thf(fact_5210_fun__of__rel__def,axiom,
! [A: $tType,B: $tType] :
( ( fun_of_rel @ B @ A )
= ( ^ [R6: set @ ( product_prod @ B @ A ),X2: B] :
( fChoice @ A
@ ^ [Y2: A] : ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X2 @ Y2 ) @ R6 ) ) ) ) ).
% fun_of_rel_def
thf(fact_5211_predicate2I,axiom,
! [B: $tType,A: $tType,P: A > B > $o,Q: A > B > $o] :
( ! [X3: A,Y4: B] :
( ( P @ X3 @ Y4 )
=> ( Q @ X3 @ Y4 ) )
=> ( ord_less_eq @ ( A > B > $o ) @ P @ Q ) ) ).
% predicate2I
thf(fact_5212_predicate2D,axiom,
! [A: $tType,B: $tType,P: A > B > $o,Q: A > B > $o,X: A,Y: B] :
( ( ord_less_eq @ ( A > B > $o ) @ P @ Q )
=> ( ( P @ X @ Y )
=> ( Q @ X @ Y ) ) ) ).
% predicate2D
thf(fact_5213_rev__predicate2D,axiom,
! [A: $tType,B: $tType,P: A > B > $o,X: A,Y: B,Q: A > B > $o] :
( ( P @ X @ Y )
=> ( ( ord_less_eq @ ( A > B > $o ) @ P @ Q )
=> ( Q @ X @ Y ) ) ) ).
% rev_predicate2D
thf(fact_5214_bot2E,axiom,
! [A: $tType,B: $tType,X: A,Y: B] :
~ ( bot_bot @ ( A > B > $o ) @ X @ Y ) ).
% bot2E
thf(fact_5215_VEBT__internal_Oreplicatei_Ocases,axiom,
! [A: $tType,X: product_prod @ nat @ ( heap_Time_Heap @ A )] :
( ! [X3: heap_Time_Heap @ A] :
( X
!= ( product_Pair @ nat @ ( heap_Time_Heap @ A ) @ ( zero_zero @ nat ) @ X3 ) )
=> ~ ! [N2: nat,X3: heap_Time_Heap @ A] :
( X
!= ( product_Pair @ nat @ ( heap_Time_Heap @ A ) @ ( suc @ N2 ) @ X3 ) ) ) ).
% VEBT_internal.replicatei.cases
thf(fact_5216_xor__num_Ocases,axiom,
! [X: product_prod @ num @ num] :
( ( X
!= ( product_Pair @ num @ num @ one2 @ one2 ) )
=> ( ! [N2: num] :
( X
!= ( product_Pair @ num @ num @ one2 @ ( bit0 @ N2 ) ) )
=> ( ! [N2: num] :
( X
!= ( product_Pair @ num @ num @ one2 @ ( bit1 @ N2 ) ) )
=> ( ! [M4: num] :
( X
!= ( product_Pair @ num @ num @ ( bit0 @ M4 ) @ one2 ) )
=> ( ! [M4: num,N2: num] :
( X
!= ( product_Pair @ num @ num @ ( bit0 @ M4 ) @ ( bit0 @ N2 ) ) )
=> ( ! [M4: num,N2: num] :
( X
!= ( product_Pair @ num @ num @ ( bit0 @ M4 ) @ ( bit1 @ N2 ) ) )
=> ( ! [M4: num] :
( X
!= ( product_Pair @ num @ num @ ( bit1 @ M4 ) @ one2 ) )
=> ( ! [M4: num,N2: num] :
( X
!= ( product_Pair @ num @ num @ ( bit1 @ M4 ) @ ( bit0 @ N2 ) ) )
=> ~ ! [M4: num,N2: num] :
( X
!= ( product_Pair @ num @ num @ ( bit1 @ M4 ) @ ( bit1 @ N2 ) ) ) ) ) ) ) ) ) ) ) ).
% xor_num.cases
thf(fact_5217_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_5218_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_5219_neg__eucl__rel__int__mult__2,axiom,
! [B3: int,A3: int,Q3: int,R3: int] :
( ( ord_less_eq @ int @ B3 @ ( zero_zero @ int ) )
=> ( ( eucl_rel_int @ ( plus_plus @ int @ A3 @ ( one_one @ int ) ) @ B3 @ ( product_Pair @ int @ int @ Q3 @ R3 ) )
=> ( eucl_rel_int @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ A3 ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ B3 ) @ ( product_Pair @ int @ int @ Q3 @ ( minus_minus @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ R3 ) @ ( one_one @ int ) ) ) ) ) ) ).
% neg_eucl_rel_int_mult_2
thf(fact_5220_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_5221_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_5222_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_5223_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_5224_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_5225_eucl__rel__int__dividesI,axiom,
! [L: int,K: int,Q3: int] :
( ( L
!= ( zero_zero @ int ) )
=> ( ( K
= ( times_times @ int @ Q3 @ L ) )
=> ( eucl_rel_int @ K @ L @ ( product_Pair @ int @ int @ Q3 @ ( zero_zero @ int ) ) ) ) ) ).
% eucl_rel_int_dividesI
thf(fact_5226_divmod__int__def,axiom,
( ( unique8689654367752047608divmod @ int )
= ( ^ [M3: num,N4: num] : ( product_Pair @ int @ int @ ( divide_divide @ int @ ( numeral_numeral @ int @ M3 ) @ ( numeral_numeral @ int @ N4 ) ) @ ( modulo_modulo @ int @ ( numeral_numeral @ int @ M3 ) @ ( numeral_numeral @ int @ N4 ) ) ) ) ) ).
% divmod_int_def
thf(fact_5227_divmod_H__nat__def,axiom,
( ( unique8689654367752047608divmod @ nat )
= ( ^ [M3: num,N4: num] : ( product_Pair @ nat @ nat @ ( divide_divide @ nat @ ( numeral_numeral @ nat @ M3 ) @ ( numeral_numeral @ nat @ N4 ) ) @ ( modulo_modulo @ nat @ ( numeral_numeral @ nat @ M3 ) @ ( numeral_numeral @ nat @ N4 ) ) ) ) ) ).
% divmod'_nat_def
thf(fact_5228_divmod__def,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ( ( unique8689654367752047608divmod @ A )
= ( ^ [M3: num,N4: num] : ( product_Pair @ A @ A @ ( divide_divide @ A @ ( numeral_numeral @ A @ M3 ) @ ( numeral_numeral @ A @ N4 ) ) @ ( modulo_modulo @ A @ ( numeral_numeral @ A @ M3 ) @ ( numeral_numeral @ A @ N4 ) ) ) ) ) ) ).
% divmod_def
thf(fact_5229_zminus1__lemma,axiom,
! [A3: int,B3: int,Q3: int,R3: int] :
( ( eucl_rel_int @ A3 @ B3 @ ( product_Pair @ int @ int @ Q3 @ R3 ) )
=> ( ( B3
!= ( zero_zero @ int ) )
=> ( eucl_rel_int @ ( uminus_uminus @ int @ A3 ) @ B3
@ ( product_Pair @ int @ int
@ ( if @ int
@ ( R3
= ( zero_zero @ int ) )
@ ( uminus_uminus @ int @ Q3 )
@ ( minus_minus @ int @ ( uminus_uminus @ int @ Q3 ) @ ( one_one @ int ) ) )
@ ( if @ int
@ ( R3
= ( zero_zero @ int ) )
@ ( zero_zero @ int )
@ ( minus_minus @ int @ B3 @ R3 ) ) ) ) ) ) ).
% zminus1_lemma
thf(fact_5230_eucl__rel__int__iff,axiom,
! [K: int,L: int,Q3: int,R3: int] :
( ( eucl_rel_int @ K @ L @ ( product_Pair @ int @ int @ Q3 @ R3 ) )
= ( ( K
= ( plus_plus @ int @ ( times_times @ int @ L @ Q3 ) @ R3 ) )
& ( ( ord_less @ int @ ( zero_zero @ int ) @ L )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ R3 )
& ( ord_less @ int @ R3 @ L ) ) )
& ( ~ ( ord_less @ int @ ( zero_zero @ int ) @ L )
=> ( ( ( ord_less @ int @ L @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ L @ R3 )
& ( ord_less_eq @ int @ R3 @ ( zero_zero @ int ) ) ) )
& ( ~ ( ord_less @ int @ L @ ( zero_zero @ int ) )
=> ( Q3
= ( zero_zero @ int ) ) ) ) ) ) ) ).
% eucl_rel_int_iff
thf(fact_5231_eucl__rel__int__remainderI,axiom,
! [R3: int,L: int,K: int,Q3: int] :
( ( ( sgn_sgn @ int @ R3 )
= ( sgn_sgn @ int @ L ) )
=> ( ( ord_less @ int @ ( abs_abs @ int @ R3 ) @ ( abs_abs @ int @ L ) )
=> ( ( K
= ( plus_plus @ int @ ( times_times @ int @ Q3 @ L ) @ R3 ) )
=> ( eucl_rel_int @ K @ L @ ( product_Pair @ int @ int @ Q3 @ R3 ) ) ) ) ) ).
% eucl_rel_int_remainderI
thf(fact_5232_eucl__rel__int_Osimps,axiom,
( eucl_rel_int
= ( ^ [A1: int,A22: int,A32: product_prod @ int @ int] :
( ? [K3: int] :
( ( A1 = K3 )
& ( A22
= ( zero_zero @ int ) )
& ( A32
= ( product_Pair @ int @ int @ ( zero_zero @ int ) @ K3 ) ) )
| ? [L2: int,K3: int,Q5: int] :
( ( A1 = K3 )
& ( A22 = L2 )
& ( A32
= ( product_Pair @ int @ int @ Q5 @ ( zero_zero @ int ) ) )
& ( L2
!= ( zero_zero @ int ) )
& ( K3
= ( times_times @ int @ Q5 @ L2 ) ) )
| ? [R5: int,L2: int,K3: int,Q5: int] :
( ( A1 = K3 )
& ( A22 = L2 )
& ( A32
= ( product_Pair @ int @ int @ Q5 @ R5 ) )
& ( ( sgn_sgn @ int @ R5 )
= ( sgn_sgn @ int @ L2 ) )
& ( ord_less @ int @ ( abs_abs @ int @ R5 ) @ ( abs_abs @ int @ L2 ) )
& ( K3
= ( plus_plus @ int @ ( times_times @ int @ Q5 @ L2 ) @ R5 ) ) ) ) ) ) ).
% eucl_rel_int.simps
thf(fact_5233_eucl__rel__int_Ocases,axiom,
! [A12: int,A23: int,A33: product_prod @ int @ int] :
( ( eucl_rel_int @ A12 @ A23 @ A33 )
=> ( ( ( A23
= ( zero_zero @ int ) )
=> ( A33
!= ( product_Pair @ int @ int @ ( zero_zero @ int ) @ A12 ) ) )
=> ( ! [Q6: int] :
( ( A33
= ( product_Pair @ int @ int @ Q6 @ ( zero_zero @ int ) ) )
=> ( ( A23
!= ( zero_zero @ int ) )
=> ( A12
!= ( times_times @ int @ Q6 @ A23 ) ) ) )
=> ~ ! [R2: int,Q6: int] :
( ( A33
= ( product_Pair @ int @ int @ Q6 @ R2 ) )
=> ( ( ( sgn_sgn @ int @ R2 )
= ( sgn_sgn @ int @ A23 ) )
=> ( ( ord_less @ int @ ( abs_abs @ int @ R2 ) @ ( abs_abs @ int @ A23 ) )
=> ( A12
!= ( plus_plus @ int @ ( times_times @ int @ Q6 @ A23 ) @ R2 ) ) ) ) ) ) ) ) ).
% eucl_rel_int.cases
thf(fact_5234_divmod__divmod__step,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ( ( unique8689654367752047608divmod @ A )
= ( ^ [M3: num,N4: num] : ( if @ ( product_prod @ A @ A ) @ ( ord_less @ num @ M3 @ N4 ) @ ( product_Pair @ A @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ M3 ) ) @ ( unique1321980374590559556d_step @ A @ N4 @ ( unique8689654367752047608divmod @ A @ M3 @ ( bit0 @ N4 ) ) ) ) ) ) ) ).
% divmod_divmod_step
thf(fact_5235_pos__eucl__rel__int__mult__2,axiom,
! [B3: int,A3: int,Q3: int,R3: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ B3 )
=> ( ( eucl_rel_int @ A3 @ B3 @ ( product_Pair @ int @ int @ Q3 @ R3 ) )
=> ( eucl_rel_int @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ A3 ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ B3 ) @ ( product_Pair @ int @ int @ Q3 @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ R3 ) ) ) ) ) ) ).
% pos_eucl_rel_int_mult_2
thf(fact_5236_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_5237_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_5238_Divides_Oadjust__div__eq,axiom,
! [Q3: int,R3: int] :
( ( adjust_div @ ( product_Pair @ int @ int @ Q3 @ R3 ) )
= ( plus_plus @ int @ Q3
@ ( zero_neq_one_of_bool @ int
@ ( R3
!= ( zero_zero @ int ) ) ) ) ) ).
% Divides.adjust_div_eq
thf(fact_5239_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_5240_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_5241_sgn__integer__code,axiom,
( ( sgn_sgn @ code_integer )
= ( ^ [K3: code_integer] :
( if @ code_integer
@ ( K3
= ( zero_zero @ code_integer ) )
@ ( zero_zero @ code_integer )
@ ( if @ code_integer @ ( ord_less @ code_integer @ K3 @ ( zero_zero @ code_integer ) ) @ ( uminus_uminus @ code_integer @ ( one_one @ code_integer ) ) @ ( one_one @ code_integer ) ) ) ) ) ).
% sgn_integer_code
thf(fact_5242_and__int_Opelims,axiom,
! [X: int,Xa: int,Y: int] :
( ( ( bit_se5824344872417868541ns_and @ int @ X @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ int @ int ) @ bit_and_int_rel @ ( product_Pair @ int @ int @ X @ Xa ) )
=> ~ ( ( ( ( ( member @ int @ X @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) )
& ( member @ int @ Xa @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) )
=> ( Y
= ( uminus_uminus @ int
@ ( zero_neq_one_of_bool @ int
@ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ X )
& ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ Xa ) ) ) ) ) )
& ( ~ ( ( member @ int @ X @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) )
& ( member @ int @ Xa @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) )
=> ( Y
= ( plus_plus @ int
@ ( zero_neq_one_of_bool @ int
@ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ X )
& ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ Xa ) ) )
@ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( divide_divide @ int @ X @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ Xa @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) )
=> ~ ( accp @ ( product_prod @ int @ int ) @ bit_and_int_rel @ ( product_Pair @ int @ int @ X @ Xa ) ) ) ) ) ).
% and_int.pelims
thf(fact_5243_and__int_Opsimps,axiom,
! [K: int,L: int] :
( ( accp @ ( product_prod @ int @ int ) @ bit_and_int_rel @ ( product_Pair @ int @ int @ K @ L ) )
=> ( ( ( ( member @ int @ K @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) )
& ( member @ int @ L @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) )
=> ( ( bit_se5824344872417868541ns_and @ int @ K @ L )
= ( uminus_uminus @ int
@ ( zero_neq_one_of_bool @ int
@ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K )
& ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ L ) ) ) ) ) )
& ( ~ ( ( member @ int @ K @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) )
& ( member @ int @ L @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) )
=> ( ( bit_se5824344872417868541ns_and @ int @ K @ L )
= ( plus_plus @ int
@ ( zero_neq_one_of_bool @ int
@ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K )
& ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ L ) ) )
@ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( divide_divide @ int @ K @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% and_int.psimps
thf(fact_5244_zero__natural_Orsp,axiom,
( ( zero_zero @ nat )
= ( zero_zero @ nat ) ) ).
% zero_natural.rsp
thf(fact_5245_one__natural_Orsp,axiom,
( ( one_one @ nat )
= ( one_one @ nat ) ) ).
% one_natural.rsp
thf(fact_5246_zero__integer_Orsp,axiom,
( ( zero_zero @ int )
= ( zero_zero @ int ) ) ).
% zero_integer.rsp
thf(fact_5247_one__integer_Orsp,axiom,
( ( one_one @ int )
= ( one_one @ int ) ) ).
% one_integer.rsp
thf(fact_5248_uminus__integer__code_I1_J,axiom,
( ( uminus_uminus @ code_integer @ ( zero_zero @ code_integer ) )
= ( zero_zero @ code_integer ) ) ).
% uminus_integer_code(1)
thf(fact_5249_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_5250_times__integer__code_I2_J,axiom,
! [L: code_integer] :
( ( times_times @ code_integer @ ( zero_zero @ code_integer ) @ L )
= ( zero_zero @ code_integer ) ) ).
% times_integer_code(2)
thf(fact_5251_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_5252_less__integer__code_I1_J,axiom,
~ ( ord_less @ code_integer @ ( zero_zero @ code_integer ) @ ( zero_zero @ code_integer ) ) ).
% less_integer_code(1)
thf(fact_5253_abs__integer__code,axiom,
( ( abs_abs @ code_integer )
= ( ^ [K3: code_integer] : ( if @ code_integer @ ( ord_less @ code_integer @ K3 @ ( zero_zero @ code_integer ) ) @ ( uminus_uminus @ code_integer @ K3 ) @ K3 ) ) ) ).
% abs_integer_code
thf(fact_5254_and__int_Opinduct,axiom,
! [A0: int,A12: int,P: int > int > $o] :
( ( accp @ ( product_prod @ int @ int ) @ bit_and_int_rel @ ( product_Pair @ int @ int @ A0 @ A12 ) )
=> ( ! [K2: int,L3: int] :
( ( accp @ ( product_prod @ int @ int ) @ bit_and_int_rel @ ( product_Pair @ int @ int @ K2 @ L3 ) )
=> ( ( ~ ( ( member @ int @ K2 @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) )
& ( member @ int @ L3 @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) )
=> ( P @ ( divide_divide @ int @ K2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) )
=> ( P @ K2 @ L3 ) ) )
=> ( P @ A0 @ A12 ) ) ) ).
% and_int.pinduct
thf(fact_5255_plus__integer__code_I2_J,axiom,
! [L: code_integer] :
( ( plus_plus @ code_integer @ ( zero_zero @ code_integer ) @ L )
= L ) ).
% plus_integer_code(2)
thf(fact_5256_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_5257_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_5258_minus__integer__code_I2_J,axiom,
! [L: code_integer] :
( ( minus_minus @ code_integer @ ( zero_zero @ code_integer ) @ L )
= ( uminus_uminus @ code_integer @ L ) ) ).
% minus_integer_code(2)
thf(fact_5259_upto_Opinduct,axiom,
! [A0: int,A12: int,P: int > int > $o] :
( ( accp @ ( product_prod @ int @ int ) @ upto_rel @ ( product_Pair @ int @ int @ A0 @ A12 ) )
=> ( ! [I3: int,J2: int] :
( ( accp @ ( product_prod @ int @ int ) @ upto_rel @ ( product_Pair @ int @ int @ I3 @ J2 ) )
=> ( ( ( ord_less_eq @ int @ I3 @ J2 )
=> ( P @ ( plus_plus @ int @ I3 @ ( one_one @ int ) ) @ J2 ) )
=> ( P @ I3 @ J2 ) ) )
=> ( P @ A0 @ A12 ) ) ) ).
% upto.pinduct
thf(fact_5260_bitNOT__integer__code,axiom,
( ( bit_ri4277139882892585799ns_not @ code_integer )
= ( ^ [I4: code_integer] : ( minus_minus @ code_integer @ ( uminus_uminus @ code_integer @ I4 ) @ ( one_one @ code_integer ) ) ) ) ).
% bitNOT_integer_code
thf(fact_5261_integer__of__int__code,axiom,
( code_integer_of_int
= ( ^ [K3: int] :
( if @ code_integer @ ( ord_less @ int @ K3 @ ( zero_zero @ int ) ) @ ( uminus_uminus @ code_integer @ ( code_integer_of_int @ ( uminus_uminus @ int @ K3 ) ) )
@ ( if @ code_integer
@ ( K3
= ( zero_zero @ int ) )
@ ( zero_zero @ code_integer )
@ ( if @ code_integer
@ ( ( modulo_modulo @ int @ K3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) )
= ( zero_zero @ int ) )
@ ( times_times @ code_integer @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) @ ( code_integer_of_int @ ( divide_divide @ int @ K3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) )
@ ( plus_plus @ code_integer @ ( times_times @ code_integer @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) @ ( code_integer_of_int @ ( divide_divide @ int @ K3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) @ ( one_one @ code_integer ) ) ) ) ) ) ) ).
% integer_of_int_code
thf(fact_5262_zero__integer__def,axiom,
( ( zero_zero @ code_integer )
= ( code_integer_of_int @ ( zero_zero @ int ) ) ) ).
% zero_integer_def
thf(fact_5263_less__integer_Oabs__eq,axiom,
! [Xa: int,X: int] :
( ( ord_less @ code_integer @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X ) )
= ( ord_less @ int @ Xa @ X ) ) ).
% less_integer.abs_eq
thf(fact_5264_one__integer__def,axiom,
( ( one_one @ code_integer )
= ( code_integer_of_int @ ( one_one @ int ) ) ) ).
% one_integer_def
thf(fact_5265_lsb__integer__code,axiom,
( ( least_8051144512741203767sb_lsb @ code_integer )
= ( ^ [X2: code_integer] : ( bit_se5641148757651400278ts_bit @ code_integer @ X2 @ ( zero_zero @ nat ) ) ) ) ).
% lsb_integer_code
thf(fact_5266_shiftr__integer__conv__div__pow2,axiom,
( ( bit_se4197421643247451524op_bit @ code_integer )
= ( ^ [N4: nat,X2: code_integer] : ( divide_divide @ code_integer @ X2 @ ( power_power @ code_integer @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) @ N4 ) ) ) ) ).
% shiftr_integer_conv_div_pow2
thf(fact_5267_Bit__integer_Oabs__eq,axiom,
! [Xa: int,X: $o] :
( ( bits_Bit_integer @ ( code_integer_of_int @ Xa ) @ X )
= ( code_integer_of_int @ ( plus_plus @ int @ ( zero_neq_one_of_bool @ int @ X ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ Xa ) ) ) ) ).
% Bit_integer.abs_eq
thf(fact_5268_div__half__word,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y: word @ A,X: word @ A] :
( ( Y
!= ( zero_zero @ ( word @ A ) ) )
=> ( ( product_Pair @ ( word @ A ) @ ( word @ A ) @ ( divide_divide @ ( word @ A ) @ X @ Y ) @ ( modulo_modulo @ ( word @ A ) @ X @ Y ) )
= ( if @ ( product_prod @ ( word @ A ) @ ( word @ A ) ) @ ( ord_less_eq @ ( word @ A ) @ Y @ ( minus_minus @ ( word @ A ) @ X @ ( times_times @ ( word @ A ) @ ( bit_se4730199178511100633sh_bit @ ( word @ A ) @ ( one_one @ nat ) @ ( divide_divide @ ( word @ A ) @ ( bit_se4197421643247451524op_bit @ ( word @ A ) @ ( one_one @ nat ) @ X ) @ Y ) ) @ Y ) ) ) @ ( product_Pair @ ( word @ A ) @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ ( bit_se4730199178511100633sh_bit @ ( word @ A ) @ ( one_one @ nat ) @ ( divide_divide @ ( word @ A ) @ ( bit_se4197421643247451524op_bit @ ( word @ A ) @ ( one_one @ nat ) @ X ) @ Y ) ) @ ( one_one @ ( word @ A ) ) ) @ ( minus_minus @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ X @ ( times_times @ ( word @ A ) @ ( bit_se4730199178511100633sh_bit @ ( word @ A ) @ ( one_one @ nat ) @ ( divide_divide @ ( word @ A ) @ ( bit_se4197421643247451524op_bit @ ( word @ A ) @ ( one_one @ nat ) @ X ) @ Y ) ) @ Y ) ) @ Y ) ) @ ( product_Pair @ ( word @ A ) @ ( word @ A ) @ ( bit_se4730199178511100633sh_bit @ ( word @ A ) @ ( one_one @ nat ) @ ( divide_divide @ ( word @ A ) @ ( bit_se4197421643247451524op_bit @ ( word @ A ) @ ( one_one @ nat ) @ X ) @ Y ) ) @ ( minus_minus @ ( word @ A ) @ X @ ( times_times @ ( word @ A ) @ ( bit_se4730199178511100633sh_bit @ ( word @ A ) @ ( one_one @ nat ) @ ( divide_divide @ ( word @ A ) @ ( bit_se4197421643247451524op_bit @ ( word @ A ) @ ( one_one @ nat ) @ X ) @ Y ) ) @ Y ) ) ) ) ) ) ) ).
% div_half_word
thf(fact_5269_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_5270_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_5271_push__bit__eq__0__iff,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [N: nat,A3: A] :
( ( ( bit_se4730199178511100633sh_bit @ A @ N @ A3 )
= ( zero_zero @ A ) )
= ( A3
= ( zero_zero @ A ) ) ) ) ).
% push_bit_eq_0_iff
thf(fact_5272_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_5273_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_5274_pred__numeral__simps_I2_J,axiom,
! [K: num] :
( ( pred_numeral @ ( bit0 @ K ) )
= ( numeral_numeral @ nat @ ( bitM @ K ) ) ) ).
% pred_numeral_simps(2)
thf(fact_5275_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_5276_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_5277_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_5278_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_5279_push__bit__numeral,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [L: num,K: num] :
( ( bit_se4730199178511100633sh_bit @ A @ ( numeral_numeral @ nat @ L ) @ ( numeral_numeral @ A @ K ) )
= ( bit_se4730199178511100633sh_bit @ A @ ( pred_numeral @ L ) @ ( numeral_numeral @ A @ ( bit0 @ K ) ) ) ) ) ).
% push_bit_numeral
thf(fact_5280_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_5281_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_5282_push__bit__Suc,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A3: A] :
( ( bit_se4730199178511100633sh_bit @ A @ ( suc @ N ) @ A3 )
= ( bit_se4730199178511100633sh_bit @ A @ N @ ( times_times @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% push_bit_Suc
thf(fact_5283_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_5284_even__push__bit__iff,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A3: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se4730199178511100633sh_bit @ A @ N @ A3 ) )
= ( ( N
!= ( zero_zero @ nat ) )
| ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) ) ) ) ).
% even_push_bit_iff
thf(fact_5285_push__bit__minus__numeral,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [L: num,K: num] :
( ( bit_se4730199178511100633sh_bit @ A @ ( numeral_numeral @ nat @ L ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ K ) ) )
= ( bit_se4730199178511100633sh_bit @ A @ ( pred_numeral @ L ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( bit0 @ K ) ) ) ) ) ) ).
% push_bit_minus_numeral
thf(fact_5286_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_5287_semiring__norm_I26_J,axiom,
( ( bitM @ one2 )
= one2 ) ).
% semiring_norm(26)
thf(fact_5288_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_5289_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_5290_semiring__norm_I28_J,axiom,
! [N: num] :
( ( bitM @ ( bit1 @ N ) )
= ( bit1 @ ( bit0 @ N ) ) ) ).
% semiring_norm(28)
thf(fact_5291_semiring__norm_I27_J,axiom,
! [N: num] :
( ( bitM @ ( bit0 @ N ) )
= ( bit1 @ ( bitM @ N ) ) ) ).
% semiring_norm(27)
thf(fact_5292_inc__BitM__eq,axiom,
! [N: num] :
( ( inc @ ( bitM @ N ) )
= ( bit0 @ N ) ) ).
% inc_BitM_eq
thf(fact_5293_BitM__inc__eq,axiom,
! [N: num] :
( ( bitM @ ( inc @ N ) )
= ( bit1 @ N ) ) ).
% BitM_inc_eq
thf(fact_5294_div__push__bit__of__1__eq__drop__bit,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A,N: nat] :
( ( divide_divide @ A @ A3 @ ( bit_se4730199178511100633sh_bit @ A @ N @ ( one_one @ A ) ) )
= ( bit_se4197421643247451524op_bit @ A @ N @ A3 ) ) ) ).
% div_push_bit_of_1_eq_drop_bit
thf(fact_5295_set__bit__eq__or,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se5668285175392031749et_bit @ A )
= ( ^ [N4: nat,A5: A] : ( bit_se1065995026697491101ons_or @ A @ A5 @ ( bit_se4730199178511100633sh_bit @ A @ N4 @ ( one_one @ A ) ) ) ) ) ) ).
% set_bit_eq_or
thf(fact_5296_flip__bit__eq__xor,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se8732182000553998342ip_bit @ A )
= ( ^ [N4: nat,A5: A] : ( bit_se5824344971392196577ns_xor @ A @ A5 @ ( bit_se4730199178511100633sh_bit @ A @ N4 @ ( one_one @ A ) ) ) ) ) ) ).
% flip_bit_eq_xor
thf(fact_5297_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_5298_BitM__plus__one,axiom,
! [N: num] :
( ( plus_plus @ num @ ( bitM @ N ) @ one2 )
= ( bit0 @ N ) ) ).
% BitM_plus_one
thf(fact_5299_one__plus__BitM,axiom,
! [N: num] :
( ( plus_plus @ num @ one2 @ ( bitM @ N ) )
= ( bit0 @ N ) ) ).
% one_plus_BitM
thf(fact_5300_push__bit__double,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A3: A] :
( ( bit_se4730199178511100633sh_bit @ A @ N @ ( times_times @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
= ( times_times @ A @ ( bit_se4730199178511100633sh_bit @ A @ N @ A3 ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% push_bit_double
thf(fact_5301_bit__iff__and__push__bit__not__eq__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se5641148757651400278ts_bit @ A )
= ( ^ [A5: A,N4: nat] :
( ( bit_se5824344872417868541ns_and @ A @ A5 @ ( bit_se4730199178511100633sh_bit @ A @ N4 @ ( one_one @ A ) ) )
!= ( zero_zero @ A ) ) ) ) ) ).
% bit_iff_and_push_bit_not_eq_0
thf(fact_5302_unset__bit__eq__and__not,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( ( bit_se2638667681897837118et_bit @ A )
= ( ^ [N4: nat,A5: A] : ( bit_se5824344872417868541ns_and @ A @ A5 @ ( bit_ri4277139882892585799ns_not @ A @ ( bit_se4730199178511100633sh_bit @ A @ N4 @ ( one_one @ A ) ) ) ) ) ) ) ).
% unset_bit_eq_and_not
thf(fact_5303_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_5304_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_5305_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_5306_push__bit__eq__mult,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se4730199178511100633sh_bit @ A )
= ( ^ [N4: nat,A5: A] : ( times_times @ A @ A5 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N4 ) ) ) ) ) ).
% push_bit_eq_mult
thf(fact_5307_exp__dvdE,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A3: A] :
( ( dvd_dvd @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) @ A3 )
=> ~ ! [B7: A] :
( A3
!= ( bit_se4730199178511100633sh_bit @ A @ N @ B7 ) ) ) ) ).
% exp_dvdE
thf(fact_5308_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_5309_signed__take__bit__code,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( ( bit_ri4674362597316999326ke_bit @ A )
= ( ^ [N4: nat,A5: A] : ( if @ A @ ( bit_se5641148757651400278ts_bit @ A @ ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N4 ) @ A5 ) @ N4 ) @ ( plus_plus @ A @ ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N4 ) @ A5 ) @ ( bit_se4730199178511100633sh_bit @ A @ ( suc @ N4 ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) @ ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N4 ) @ A5 ) ) ) ) ) ).
% signed_take_bit_code
thf(fact_5310_set__bits__aux__Suc,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [F3: nat > $o,N: nat,W: word @ A] :
( ( code_T2661198915054445665ts_aux @ A @ F3 @ ( suc @ N ) @ W )
= ( code_T2661198915054445665ts_aux @ A @ F3 @ N @ ( bit_se1065995026697491101ons_or @ ( word @ A ) @ ( bit_se4730199178511100633sh_bit @ ( word @ A ) @ ( one_one @ nat ) @ W ) @ ( if @ ( word @ A ) @ ( F3 @ N ) @ ( one_one @ ( word @ A ) ) @ ( zero_zero @ ( word @ A ) ) ) ) ) ) ) ).
% set_bits_aux_Suc
thf(fact_5311_set__bits__aux__rec,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( code_T2661198915054445665ts_aux @ A )
= ( ^ [F2: nat > $o,N4: nat,W2: word @ A] :
( if @ ( word @ A )
@ ( N4
= ( zero_zero @ nat ) )
@ W2
@ ( code_T2661198915054445665ts_aux @ A @ F2 @ ( minus_minus @ nat @ N4 @ ( one_one @ nat ) ) @ ( bit_se1065995026697491101ons_or @ ( word @ A ) @ ( bit_se4730199178511100633sh_bit @ ( word @ A ) @ ( one_one @ nat ) @ W2 ) @ ( if @ ( word @ A ) @ ( F2 @ ( minus_minus @ nat @ N4 @ ( one_one @ nat ) ) ) @ ( one_one @ ( word @ A ) ) @ ( zero_zero @ ( word @ A ) ) ) ) ) ) ) ) ) ).
% set_bits_aux_rec
thf(fact_5312_test__bit__split,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( ( type_len @ B )
& ( type_len @ C )
& ( type_len @ A ) )
=> ! [C2: word @ C,A3: word @ A,B3: word @ B] :
( ( ( word_split @ C @ A @ B @ C2 )
= ( product_Pair @ ( word @ A ) @ ( word @ B ) @ A3 @ B3 ) )
=> ( ! [N11: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ B ) @ B3 @ N11 )
= ( ( ord_less @ nat @ N11 @ ( size_size @ ( word @ B ) @ B3 ) )
& ( bit_se5641148757651400278ts_bit @ ( word @ C ) @ C2 @ N11 ) ) )
& ! [M2: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ A3 @ M2 )
= ( ( ord_less @ nat @ M2 @ ( size_size @ ( word @ A ) @ A3 ) )
& ( bit_se5641148757651400278ts_bit @ ( word @ C ) @ C2 @ ( plus_plus @ nat @ M2 @ ( size_size @ ( word @ B ) @ B3 ) ) ) ) ) ) ) ) ).
% test_bit_split
thf(fact_5313_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_5314_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_5315_concat__bit__of__zero__1,axiom,
! [N: nat,L: int] :
( ( bit_concat_bit @ N @ ( zero_zero @ int ) @ L )
= ( bit_se4730199178511100633sh_bit @ int @ N @ L ) ) ).
% concat_bit_of_zero_1
thf(fact_5316_set__bits__aux__0,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [F3: nat > $o,W: word @ A] :
( ( code_T2661198915054445665ts_aux @ A @ F3 @ ( zero_zero @ nat ) @ W )
= W ) ) ).
% set_bits_aux_0
thf(fact_5317_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_5318_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_5319_flip__bit__nat__def,axiom,
( ( bit_se8732182000553998342ip_bit @ nat )
= ( ^ [M3: nat,N4: nat] : ( bit_se5824344971392196577ns_xor @ nat @ N4 @ ( bit_se4730199178511100633sh_bit @ nat @ M3 @ ( one_one @ nat ) ) ) ) ) ).
% flip_bit_nat_def
thf(fact_5320_set__bit__nat__def,axiom,
( ( bit_se5668285175392031749et_bit @ nat )
= ( ^ [M3: nat,N4: nat] : ( bit_se1065995026697491101ons_or @ nat @ N4 @ ( bit_se4730199178511100633sh_bit @ nat @ M3 @ ( one_one @ nat ) ) ) ) ) ).
% set_bit_nat_def
thf(fact_5321_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_5322_Bit__Operations_Oset__bit__int__def,axiom,
( ( bit_se5668285175392031749et_bit @ int )
= ( ^ [N4: nat,K3: int] : ( bit_se1065995026697491101ons_or @ int @ K3 @ ( bit_se4730199178511100633sh_bit @ int @ N4 @ ( one_one @ int ) ) ) ) ) ).
% Bit_Operations.set_bit_int_def
thf(fact_5323_flip__bit__int__def,axiom,
( ( bit_se8732182000553998342ip_bit @ int )
= ( ^ [N4: nat,K3: int] : ( bit_se5824344971392196577ns_xor @ int @ K3 @ ( bit_se4730199178511100633sh_bit @ int @ N4 @ ( one_one @ int ) ) ) ) ) ).
% flip_bit_int_def
thf(fact_5324_unset__bit__int__def,axiom,
( ( bit_se2638667681897837118et_bit @ int )
= ( ^ [N4: nat,K3: int] : ( bit_se5824344872417868541ns_and @ int @ K3 @ ( bit_ri4277139882892585799ns_not @ int @ ( bit_se4730199178511100633sh_bit @ int @ N4 @ ( one_one @ int ) ) ) ) ) ) ).
% unset_bit_int_def
thf(fact_5325_push__bit__int__def,axiom,
( ( bit_se4730199178511100633sh_bit @ int )
= ( ^ [N4: nat,K3: int] : ( times_times @ int @ K3 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N4 ) ) ) ) ).
% push_bit_int_def
thf(fact_5326_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_5327_push__bit__nat__def,axiom,
( ( bit_se4730199178511100633sh_bit @ nat )
= ( ^ [N4: nat,M3: nat] : ( times_times @ nat @ M3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N4 ) ) ) ) ).
% push_bit_nat_def
thf(fact_5328_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_5329_shiftl__integer__conv__mult__pow2,axiom,
( ( bit_se4730199178511100633sh_bit @ code_integer )
= ( ^ [N4: nat,X2: code_integer] : ( times_times @ code_integer @ X2 @ ( power_power @ code_integer @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) @ N4 ) ) ) ) ).
% shiftl_integer_conv_mult_pow2
thf(fact_5330_test__bit__split__eq,axiom,
! [B: $tType,C: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ C )
& ( type_len @ B ) )
=> ! [C2: word @ C,A3: word @ A,B3: word @ B] :
( ( ( word_split @ C @ A @ B @ C2 )
= ( product_Pair @ ( word @ A ) @ ( word @ B ) @ A3 @ B3 ) )
= ( ! [N4: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ B ) @ B3 @ N4 )
= ( ( ord_less @ nat @ N4 @ ( size_size @ ( word @ B ) @ B3 ) )
& ( bit_se5641148757651400278ts_bit @ ( word @ C ) @ C2 @ N4 ) ) )
& ! [M3: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ A3 @ M3 )
= ( ( ord_less @ nat @ M3 @ ( size_size @ ( word @ A ) @ A3 ) )
& ( bit_se5641148757651400278ts_bit @ ( word @ C ) @ C2 @ ( plus_plus @ nat @ M3 @ ( size_size @ ( word @ B ) @ B3 ) ) ) ) ) ) ) ) ).
% test_bit_split_eq
thf(fact_5331_test__bit__split_H,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( ( type_len @ B )
& ( type_len @ C )
& ( type_len @ A ) )
=> ! [C2: word @ C,A3: word @ A,B3: word @ B] :
( ( ( word_split @ C @ A @ B @ C2 )
= ( product_Pair @ ( word @ A ) @ ( word @ B ) @ A3 @ B3 ) )
=> ! [N11: nat,M2: nat] :
( ( ( bit_se5641148757651400278ts_bit @ ( word @ B ) @ B3 @ N11 )
= ( ( ord_less @ nat @ N11 @ ( size_size @ ( word @ B ) @ B3 ) )
& ( bit_se5641148757651400278ts_bit @ ( word @ C ) @ C2 @ N11 ) ) )
& ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ A3 @ M2 )
= ( ( ord_less @ nat @ M2 @ ( size_size @ ( word @ A ) @ A3 ) )
& ( bit_se5641148757651400278ts_bit @ ( word @ C ) @ C2 @ ( plus_plus @ nat @ M2 @ ( size_size @ ( word @ B ) @ B3 ) ) ) ) ) ) ) ) ).
% test_bit_split'
thf(fact_5332_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_5333_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_5334_divmod__algorithm__code_I6_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M: num,N: num] :
( ( unique8689654367752047608divmod @ A @ ( bit1 @ M ) @ ( bit0 @ N ) )
= ( product_case_prod @ A @ A @ ( product_prod @ A @ A )
@ ^ [Q5: A,R5: A] : ( product_Pair @ A @ A @ Q5 @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ R5 ) @ ( one_one @ A ) ) )
@ ( unique8689654367752047608divmod @ A @ M @ N ) ) ) ) ).
% divmod_algorithm_code(6)
thf(fact_5335_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_5336_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_5337_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_5338_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_5339_divmod__algorithm__code_I5_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M: num,N: num] :
( ( unique8689654367752047608divmod @ A @ ( bit0 @ M ) @ ( bit0 @ N ) )
= ( product_case_prod @ A @ A @ ( product_prod @ A @ A )
@ ^ [Q5: A,R5: A] : ( product_Pair @ A @ A @ Q5 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ R5 ) )
@ ( unique8689654367752047608divmod @ A @ M @ N ) ) ) ) ).
% divmod_algorithm_code(5)
thf(fact_5340_or__not__num__neg_Osimps_I1_J,axiom,
( ( bit_or_not_num_neg @ one2 @ one2 )
= one2 ) ).
% or_not_num_neg.simps(1)
thf(fact_5341_nested__case__prod__simp,axiom,
! [A: $tType,D: $tType,C: $tType,B: $tType] :
( ( product_case_prod @ B @ C @ ( D > A ) )
= ( ^ [F2: B > C > D > A,X2: product_prod @ B @ C,Y2: D] :
( product_case_prod @ B @ C @ A
@ ^ [A5: B,B5: C] : ( F2 @ A5 @ B5 @ Y2 )
@ X2 ) ) ) ).
% nested_case_prod_simp
thf(fact_5342_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_5343_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_5344_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_5345_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_5346_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_5347_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_5348_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_5349_or__not__num__neg_Oelims,axiom,
! [X: num,Xa: num,Y: num] :
( ( ( bit_or_not_num_neg @ X @ Xa )
= Y )
=> ( ( ( X = one2 )
=> ( ( Xa = one2 )
=> ( Y != one2 ) ) )
=> ( ( ( X = one2 )
=> ! [M4: num] :
( ( Xa
= ( bit0 @ M4 ) )
=> ( Y
!= ( bit1 @ M4 ) ) ) )
=> ( ( ( X = one2 )
=> ! [M4: num] :
( ( Xa
= ( bit1 @ M4 ) )
=> ( Y
!= ( bit1 @ M4 ) ) ) )
=> ( ( ? [N2: num] :
( X
= ( bit0 @ N2 ) )
=> ( ( Xa = one2 )
=> ( Y
!= ( bit0 @ one2 ) ) ) )
=> ( ! [N2: num] :
( ( X
= ( bit0 @ N2 ) )
=> ! [M4: num] :
( ( Xa
= ( bit0 @ M4 ) )
=> ( Y
!= ( bitM @ ( bit_or_not_num_neg @ N2 @ M4 ) ) ) ) )
=> ( ! [N2: num] :
( ( X
= ( bit0 @ N2 ) )
=> ! [M4: num] :
( ( Xa
= ( bit1 @ M4 ) )
=> ( Y
!= ( bit0 @ ( bit_or_not_num_neg @ N2 @ M4 ) ) ) ) )
=> ( ( ? [N2: num] :
( X
= ( bit1 @ N2 ) )
=> ( ( Xa = one2 )
=> ( Y != one2 ) ) )
=> ( ! [N2: num] :
( ( X
= ( bit1 @ N2 ) )
=> ! [M4: num] :
( ( Xa
= ( bit0 @ M4 ) )
=> ( Y
!= ( bitM @ ( bit_or_not_num_neg @ N2 @ M4 ) ) ) ) )
=> ~ ! [N2: num] :
( ( X
= ( bit1 @ N2 ) )
=> ! [M4: num] :
( ( Xa
= ( bit1 @ M4 ) )
=> ( Y
!= ( bitM @ ( bit_or_not_num_neg @ N2 @ M4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% or_not_num_neg.elims
thf(fact_5350_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_5351_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_5352_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_5353_divmod__step__nat__def,axiom,
( ( unique1321980374590559556d_step @ nat )
= ( ^ [L2: num] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [Q5: nat,R5: nat] : ( if @ ( product_prod @ nat @ nat ) @ ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ L2 ) @ R5 ) @ ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Q5 ) @ ( one_one @ nat ) ) @ ( minus_minus @ nat @ R5 @ ( numeral_numeral @ nat @ L2 ) ) ) @ ( product_Pair @ nat @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Q5 ) @ R5 ) ) ) ) ) ).
% divmod_step_nat_def
thf(fact_5354_divmod__step__int__def,axiom,
( ( unique1321980374590559556d_step @ int )
= ( ^ [L2: num] :
( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [Q5: int,R5: int] : ( if @ ( product_prod @ int @ int ) @ ( ord_less_eq @ int @ ( numeral_numeral @ int @ L2 ) @ R5 ) @ ( product_Pair @ int @ int @ ( plus_plus @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ Q5 ) @ ( one_one @ int ) ) @ ( minus_minus @ int @ R5 @ ( numeral_numeral @ int @ L2 ) ) ) @ ( product_Pair @ int @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ Q5 ) @ R5 ) ) ) ) ) ).
% divmod_step_int_def
thf(fact_5355_divmod__step__integer__def,axiom,
( ( unique1321980374590559556d_step @ code_integer )
= ( ^ [L2: num] :
( product_case_prod @ code_integer @ code_integer @ ( product_prod @ code_integer @ code_integer )
@ ^ [Q5: code_integer,R5: code_integer] : ( if @ ( product_prod @ code_integer @ code_integer ) @ ( ord_less_eq @ code_integer @ ( numeral_numeral @ code_integer @ L2 ) @ R5 ) @ ( product_Pair @ code_integer @ code_integer @ ( plus_plus @ code_integer @ ( times_times @ code_integer @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) @ Q5 ) @ ( one_one @ code_integer ) ) @ ( minus_minus @ code_integer @ R5 @ ( numeral_numeral @ code_integer @ L2 ) ) ) @ ( product_Pair @ code_integer @ code_integer @ ( times_times @ code_integer @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) @ Q5 ) @ R5 ) ) ) ) ) ).
% divmod_step_integer_def
thf(fact_5356_divmod__step__def,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ( ( unique1321980374590559556d_step @ A )
= ( ^ [L2: num] :
( product_case_prod @ A @ A @ ( product_prod @ A @ A )
@ ^ [Q5: A,R5: A] : ( if @ ( product_prod @ A @ A ) @ ( ord_less_eq @ A @ ( numeral_numeral @ A @ L2 ) @ R5 ) @ ( product_Pair @ A @ A @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Q5 ) @ ( one_one @ A ) ) @ ( minus_minus @ A @ R5 @ ( numeral_numeral @ A @ L2 ) ) ) @ ( product_Pair @ A @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Q5 ) @ R5 ) ) ) ) ) ) ).
% divmod_step_def
thf(fact_5357_case__prod__conv,axiom,
! [B: $tType,A: $tType,C: $tType,F3: B > C > A,A3: B,B3: C] :
( ( product_case_prod @ B @ C @ A @ F3 @ ( product_Pair @ B @ C @ A3 @ B3 ) )
= ( F3 @ A3 @ B3 ) ) ).
% case_prod_conv
thf(fact_5358_divmod__nat__if,axiom,
( divmod_nat
= ( ^ [M3: nat,N4: nat] :
( if @ ( product_prod @ nat @ nat )
@ ( ( N4
= ( zero_zero @ nat ) )
| ( ord_less @ nat @ M3 @ N4 ) )
@ ( product_Pair @ nat @ nat @ ( zero_zero @ nat ) @ M3 )
@ ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [Q5: nat] : ( product_Pair @ nat @ nat @ ( suc @ Q5 ) )
@ ( divmod_nat @ ( minus_minus @ nat @ M3 @ N4 ) @ N4 ) ) ) ) ) ).
% divmod_nat_if
thf(fact_5359_hash__code__prod__simps,axiom,
! [A: $tType,B: $tType,H_a: A > uint32,H_b: B > uint32,X: A,Xa: B] :
( ( hash_hash_code_prod @ A @ B @ H_a @ H_b @ ( product_Pair @ A @ B @ X @ Xa ) )
= ( plus_plus @ uint32 @ ( times_times @ uint32 @ ( H_a @ X ) @ ( numeral_numeral @ uint32 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( plus_plus @ uint32 @ ( times_times @ uint32 @ ( H_b @ Xa ) @ ( numeral_numeral @ uint32 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numeral_numeral @ uint32 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% hash_code_prod_simps
thf(fact_5360_Eps__case__prod__eq,axiom,
! [A: $tType,B: $tType,X: A,Y: B] :
( ( fChoice @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [X9: A,Y7: B] :
( ( X = X9 )
& ( Y = Y7 ) ) ) )
= ( product_Pair @ A @ B @ X @ Y ) ) ).
% Eps_case_prod_eq
thf(fact_5361_The__split__eq,axiom,
! [A: $tType,B: $tType,X: A,Y: B] :
( ( the @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [X9: A,Y7: B] :
( ( X = X9 )
& ( Y = Y7 ) ) ) )
= ( product_Pair @ A @ B @ X @ Y ) ) ).
% The_split_eq
thf(fact_5362_mem__case__prodI2,axiom,
! [C: $tType,B: $tType,A: $tType,P4: product_prod @ A @ B,Z: C,C2: A > B > ( set @ C )] :
( ! [A6: A,B7: B] :
( ( P4
= ( product_Pair @ A @ B @ A6 @ B7 ) )
=> ( member @ C @ Z @ ( C2 @ A6 @ B7 ) ) )
=> ( member @ C @ Z @ ( product_case_prod @ A @ B @ ( set @ C ) @ C2 @ P4 ) ) ) ).
% mem_case_prodI2
thf(fact_5363_mem__case__prodI,axiom,
! [A: $tType,B: $tType,C: $tType,Z: A,C2: B > C > ( set @ A ),A3: B,B3: C] :
( ( member @ A @ Z @ ( C2 @ A3 @ B3 ) )
=> ( member @ A @ Z @ ( product_case_prod @ B @ C @ ( set @ A ) @ C2 @ ( product_Pair @ B @ C @ A3 @ B3 ) ) ) ) ).
% mem_case_prodI
thf(fact_5364_case__prodI2_H,axiom,
! [A: $tType,B: $tType,C: $tType,P4: product_prod @ A @ B,C2: A > B > C > $o,X: C] :
( ! [A6: A,B7: B] :
( ( ( product_Pair @ A @ B @ A6 @ B7 )
= P4 )
=> ( C2 @ A6 @ B7 @ X ) )
=> ( product_case_prod @ A @ B @ ( C > $o ) @ C2 @ P4 @ X ) ) ).
% case_prodI2'
thf(fact_5365_case__prodI2,axiom,
! [B: $tType,A: $tType,P4: product_prod @ A @ B,C2: A > B > $o] :
( ! [A6: A,B7: B] :
( ( P4
= ( product_Pair @ A @ B @ A6 @ B7 ) )
=> ( C2 @ A6 @ B7 ) )
=> ( product_case_prod @ A @ B @ $o @ C2 @ P4 ) ) ).
% case_prodI2
thf(fact_5366_case__prodI,axiom,
! [A: $tType,B: $tType,F3: A > B > $o,A3: A,B3: B] :
( ( F3 @ A3 @ B3 )
=> ( product_case_prod @ A @ B @ $o @ F3 @ ( product_Pair @ A @ B @ A3 @ B3 ) ) ) ).
% case_prodI
thf(fact_5367_split__paired__Eps,axiom,
! [B: $tType,A: $tType] :
( ( fChoice @ ( product_prod @ A @ B ) )
= ( ^ [P3: ( product_prod @ A @ B ) > $o] :
( fChoice @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [A5: A,B5: B] : ( P3 @ ( product_Pair @ A @ B @ A5 @ B5 ) ) ) ) ) ) ).
% split_paired_Eps
thf(fact_5368_case__prodE_H,axiom,
! [B: $tType,A: $tType,C: $tType,C2: A > B > C > $o,P4: product_prod @ A @ B,Z: C] :
( ( product_case_prod @ A @ B @ ( C > $o ) @ C2 @ P4 @ Z )
=> ~ ! [X3: A,Y4: B] :
( ( P4
= ( product_Pair @ A @ B @ X3 @ Y4 ) )
=> ~ ( C2 @ X3 @ Y4 @ Z ) ) ) ).
% case_prodE'
thf(fact_5369_case__prodD_H,axiom,
! [B: $tType,A: $tType,C: $tType,R: A > B > C > $o,A3: A,B3: B,C2: C] :
( ( product_case_prod @ A @ B @ ( C > $o ) @ R @ ( product_Pair @ A @ B @ A3 @ B3 ) @ C2 )
=> ( R @ A3 @ B3 @ C2 ) ) ).
% case_prodD'
thf(fact_5370_case__prodE,axiom,
! [A: $tType,B: $tType,C2: A > B > $o,P4: product_prod @ A @ B] :
( ( product_case_prod @ A @ B @ $o @ C2 @ P4 )
=> ~ ! [X3: A,Y4: B] :
( ( P4
= ( product_Pair @ A @ B @ X3 @ Y4 ) )
=> ~ ( C2 @ X3 @ Y4 ) ) ) ).
% case_prodE
thf(fact_5371_case__prodD,axiom,
! [A: $tType,B: $tType,F3: A > B > $o,A3: A,B3: B] :
( ( product_case_prod @ A @ B @ $o @ F3 @ ( product_Pair @ A @ B @ A3 @ B3 ) )
=> ( F3 @ A3 @ B3 ) ) ).
% case_prodD
thf(fact_5372_TBOUND__prod__case,axiom,
! [C: $tType,B: $tType,A: $tType,T: product_prod @ A @ B,F3: A > B > ( heap_Time_Heap @ C ),Bnd: A > B > nat] :
( ! [A6: A,B7: B] :
( ( T
= ( product_Pair @ A @ B @ A6 @ B7 ) )
=> ( time_TBOUND @ C @ ( F3 @ A6 @ B7 ) @ ( Bnd @ A6 @ B7 ) ) )
=> ( time_TBOUND @ C @ ( product_case_prod @ A @ B @ ( heap_Time_Heap @ C ) @ F3 @ T ) @ ( product_case_prod @ A @ B @ nat @ Bnd @ T ) ) ) ).
% TBOUND_prod_case
thf(fact_5373_sum_Otriangle__reindex__eq,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G2: nat > nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ ( product_prod @ nat @ nat ) @ A @ ( product_case_prod @ nat @ nat @ A @ G2 )
@ ( collect @ ( product_prod @ nat @ nat )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [I4: nat,J3: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ I4 @ J3 ) @ N ) ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( G2 @ I4 @ ( minus_minus @ nat @ K3 @ I4 ) )
@ ( set_ord_atMost @ nat @ K3 ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% sum.triangle_reindex_eq
thf(fact_5374_case__prod__rule,axiom,
! [A: $tType,B: $tType,C: $tType,X: product_prod @ A @ B,P: assn,F3: A > B > ( heap_Time_Heap @ C ),Q: C > assn] :
( ! [A6: A,B7: B] :
( ( X
= ( product_Pair @ A @ B @ A6 @ B7 ) )
=> ( hoare_hoare_triple @ C @ P @ ( F3 @ A6 @ B7 ) @ Q ) )
=> ( hoare_hoare_triple @ C @ P @ ( product_case_prod @ A @ B @ ( heap_Time_Heap @ C ) @ F3 @ X ) @ Q ) ) ).
% case_prod_rule
thf(fact_5375_prod_Otriangle__reindex__eq,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G2: nat > nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ ( product_prod @ nat @ nat ) @ A @ ( product_case_prod @ nat @ nat @ A @ G2 )
@ ( collect @ ( product_prod @ nat @ nat )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [I4: nat,J3: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ I4 @ J3 ) @ N ) ) ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [K3: nat] :
( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( G2 @ I4 @ ( minus_minus @ nat @ K3 @ I4 ) )
@ ( set_ord_atMost @ nat @ K3 ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% prod.triangle_reindex_eq
thf(fact_5376_sum_Otriangle__reindex,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G2: nat > nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ ( product_prod @ nat @ nat ) @ A @ ( product_case_prod @ nat @ nat @ A @ G2 )
@ ( collect @ ( product_prod @ nat @ nat )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [I4: nat,J3: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ I4 @ J3 ) @ N ) ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( G2 @ I4 @ ( minus_minus @ nat @ K3 @ I4 ) )
@ ( set_ord_atMost @ nat @ K3 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% sum.triangle_reindex
thf(fact_5377_prod_Otriangle__reindex,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G2: nat > nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ ( product_prod @ nat @ nat ) @ A @ ( product_case_prod @ nat @ nat @ A @ G2 )
@ ( collect @ ( product_prod @ nat @ nat )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [I4: nat,J3: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ I4 @ J3 ) @ N ) ) ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [K3: nat] :
( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( G2 @ I4 @ ( minus_minus @ nat @ K3 @ I4 ) )
@ ( set_ord_atMost @ nat @ K3 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% prod.triangle_reindex
thf(fact_5378_prod_Ocase__distrib,axiom,
! [C: $tType,D: $tType,B: $tType,A: $tType,H2: C > D,F3: A > B > C,Prod: product_prod @ A @ B] :
( ( H2 @ ( product_case_prod @ A @ B @ C @ F3 @ Prod ) )
= ( product_case_prod @ A @ B @ D
@ ^ [X1: A,X24: B] : ( H2 @ ( F3 @ X1 @ X24 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_5379_Divides_Oadjust__div__def,axiom,
( adjust_div
= ( product_case_prod @ int @ int @ int
@ ^ [Q5: int,R5: int] :
( plus_plus @ int @ Q5
@ ( zero_neq_one_of_bool @ int
@ ( R5
!= ( zero_zero @ int ) ) ) ) ) ) ).
% Divides.adjust_div_def
thf(fact_5380_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,Y4: C] :
( ( Z
= ( product_Pair @ B @ C @ X3 @ Y4 ) )
=> ~ ( Q @ ( P @ X3 @ Y4 ) ) ) ) ).
% case_prodE2
thf(fact_5381_case__prod__eta,axiom,
! [C: $tType,B: $tType,A: $tType,F3: ( product_prod @ A @ B ) > C] :
( ( product_case_prod @ A @ B @ C
@ ^ [X2: A,Y2: B] : ( F3 @ ( product_Pair @ A @ B @ X2 @ Y2 ) ) )
= F3 ) ).
% case_prod_eta
thf(fact_5382_cond__case__prod__eta,axiom,
! [C: $tType,B: $tType,A: $tType,F3: A > B > C,G2: ( product_prod @ A @ B ) > C] :
( ! [X3: A,Y4: B] :
( ( F3 @ X3 @ Y4 )
= ( G2 @ ( product_Pair @ A @ B @ X3 @ Y4 ) ) )
=> ( ( product_case_prod @ A @ B @ C @ F3 )
= G2 ) ) ).
% cond_case_prod_eta
thf(fact_5383_of__nat__code__if,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ( ( semiring_1_of_nat @ A )
= ( ^ [N4: nat] :
( if @ A
@ ( N4
= ( zero_zero @ nat ) )
@ ( zero_zero @ A )
@ ( product_case_prod @ nat @ nat @ A
@ ^ [M3: nat,Q5: nat] :
( if @ A
@ ( Q5
= ( zero_zero @ nat ) )
@ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ A @ M3 ) )
@ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ A @ M3 ) ) @ ( one_one @ A ) ) )
@ ( divmod_nat @ N4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% of_nat_code_if
thf(fact_5384_int__ge__less__than__def,axiom,
( int_ge_less_than
= ( ^ [D4: int] :
( collect @ ( product_prod @ int @ int )
@ ( product_case_prod @ int @ int @ $o
@ ^ [Z8: int,Z4: int] :
( ( ord_less_eq @ int @ D4 @ Z8 )
& ( ord_less @ int @ Z8 @ Z4 ) ) ) ) ) ) ).
% int_ge_less_than_def
thf(fact_5385_int__ge__less__than2__def,axiom,
( int_ge_less_than2
= ( ^ [D4: int] :
( collect @ ( product_prod @ int @ int )
@ ( product_case_prod @ int @ int @ $o
@ ^ [Z8: int,Z4: int] :
( ( ord_less_eq @ int @ D4 @ Z4 )
& ( ord_less @ int @ Z8 @ Z4 ) ) ) ) ) ) ).
% int_ge_less_than2_def
thf(fact_5386_split__part,axiom,
! [B: $tType,A: $tType,P: $o,Q: A > B > $o] :
( ( product_case_prod @ A @ B @ $o
@ ^ [A5: A,B5: B] :
( P
& ( Q @ A5 @ B5 ) ) )
= ( ^ [Ab2: product_prod @ A @ B] :
( P
& ( product_case_prod @ A @ B @ $o @ Q @ Ab2 ) ) ) ) ).
% split_part
thf(fact_5387_prod_Odisc__eq__case,axiom,
! [B: $tType,A: $tType,Prod: product_prod @ A @ B] :
( product_case_prod @ A @ B @ $o
@ ^ [Uu: A,Uv: B] : $true
@ Prod ) ).
% prod.disc_eq_case
thf(fact_5388_dup__1,axiom,
( ( code_dup @ ( one_one @ code_integer ) )
= ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) ).
% dup_1
thf(fact_5389_rat__inverse__code,axiom,
! [P4: rat] :
( ( quotient_of @ ( inverse_inverse @ rat @ P4 ) )
= ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [A5: int,B5: int] :
( if @ ( product_prod @ int @ int )
@ ( A5
= ( zero_zero @ int ) )
@ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) )
@ ( product_Pair @ int @ int @ ( times_times @ int @ ( sgn_sgn @ int @ A5 ) @ B5 ) @ ( abs_abs @ int @ A5 ) ) )
@ ( quotient_of @ P4 ) ) ) ).
% rat_inverse_code
thf(fact_5390_bin__rest__integer_Oabs__eq,axiom,
! [X: int] :
( ( bits_b2549910563261871055nteger @ ( code_integer_of_int @ X ) )
= ( code_integer_of_int @ ( divide_divide @ int @ X @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ).
% bin_rest_integer.abs_eq
thf(fact_5391_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_5392_rat__one__code,axiom,
( ( quotient_of @ ( one_one @ rat ) )
= ( product_Pair @ int @ int @ ( one_one @ int ) @ ( one_one @ int ) ) ) ).
% rat_one_code
thf(fact_5393_rat__zero__code,axiom,
( ( quotient_of @ ( zero_zero @ rat ) )
= ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) ) ) ).
% rat_zero_code
thf(fact_5394_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_5395_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_5396_Code__Numeral_Odup__code_I1_J,axiom,
( ( code_dup @ ( zero_zero @ code_integer ) )
= ( zero_zero @ code_integer ) ) ).
% Code_Numeral.dup_code(1)
thf(fact_5397_quotient__of__denom__pos,axiom,
! [R3: rat,P4: int,Q3: int] :
( ( ( quotient_of @ R3 )
= ( product_Pair @ int @ int @ P4 @ Q3 ) )
=> ( ord_less @ int @ ( zero_zero @ int ) @ Q3 ) ) ).
% quotient_of_denom_pos
thf(fact_5398_rat__uminus__code,axiom,
! [P4: rat] :
( ( quotient_of @ ( uminus_uminus @ rat @ P4 ) )
= ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [A5: int] : ( product_Pair @ int @ int @ ( uminus_uminus @ int @ A5 ) )
@ ( quotient_of @ P4 ) ) ) ).
% rat_uminus_code
thf(fact_5399_rat__less__code,axiom,
( ( ord_less @ rat )
= ( ^ [P6: rat,Q5: rat] :
( product_case_prod @ int @ int @ $o
@ ^ [A5: int,C6: int] :
( product_case_prod @ int @ int @ $o
@ ^ [B5: int,D4: int] : ( ord_less @ int @ ( times_times @ int @ A5 @ D4 ) @ ( times_times @ int @ C6 @ B5 ) )
@ ( quotient_of @ Q5 ) )
@ ( quotient_of @ P6 ) ) ) ) ).
% rat_less_code
thf(fact_5400_rat__floor__code,axiom,
( ( archim6421214686448440834_floor @ rat )
= ( ^ [P6: rat] : ( product_case_prod @ int @ int @ int @ ( divide_divide @ int ) @ ( quotient_of @ P6 ) ) ) ) ).
% rat_floor_code
thf(fact_5401_rat__abs__code,axiom,
! [P4: rat] :
( ( quotient_of @ ( abs_abs @ rat @ P4 ) )
= ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [A5: int] : ( product_Pair @ int @ int @ ( abs_abs @ int @ A5 ) )
@ ( quotient_of @ P4 ) ) ) ).
% rat_abs_code
thf(fact_5402_rat__less__eq__code,axiom,
( ( ord_less_eq @ rat )
= ( ^ [P6: rat,Q5: rat] :
( product_case_prod @ int @ int @ $o
@ ^ [A5: int,C6: int] :
( product_case_prod @ int @ int @ $o
@ ^ [B5: int,D4: int] : ( ord_less_eq @ int @ ( times_times @ int @ A5 @ D4 ) @ ( times_times @ int @ C6 @ B5 ) )
@ ( quotient_of @ Q5 ) )
@ ( quotient_of @ P6 ) ) ) ) ).
% rat_less_eq_code
thf(fact_5403_bin__rest__integer__code,axiom,
( bits_b2549910563261871055nteger
= ( ^ [I4: code_integer] : ( divide_divide @ code_integer @ I4 @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) ) ) ).
% bin_rest_integer_code
thf(fact_5404_quotient__of__int,axiom,
! [A3: int] :
( ( quotient_of @ ( of_int @ A3 ) )
= ( product_Pair @ int @ int @ A3 @ ( one_one @ int ) ) ) ).
% quotient_of_int
thf(fact_5405_bitXOR__integer__unfold,axiom,
( ( bit_se5824344971392196577ns_xor @ code_integer )
= ( ^ [X2: code_integer,Y2: code_integer] :
( if @ code_integer
@ ( X2
= ( zero_zero @ code_integer ) )
@ Y2
@ ( if @ code_integer
@ ( X2
= ( uminus_uminus @ code_integer @ ( one_one @ code_integer ) ) )
@ ( bit_ri4277139882892585799ns_not @ code_integer @ Y2 )
@ ( bits_Bit_integer @ ( bit_se5824344971392196577ns_xor @ code_integer @ ( bits_b2549910563261871055nteger @ X2 ) @ ( bits_b2549910563261871055nteger @ Y2 ) )
@ ( ( ~ ( bits_b8758750999018896077nteger @ X2 ) )
= ( bits_b8758750999018896077nteger @ Y2 ) ) ) ) ) ) ) ).
% bitXOR_integer_unfold
thf(fact_5406_rat__plus__code,axiom,
! [P4: rat,Q3: rat] :
( ( quotient_of @ ( plus_plus @ rat @ P4 @ Q3 ) )
= ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [A5: int,C6: int] :
( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [B5: int,D4: int] : ( normalize @ ( product_Pair @ int @ int @ ( plus_plus @ int @ ( times_times @ int @ A5 @ D4 ) @ ( times_times @ int @ B5 @ C6 ) ) @ ( times_times @ int @ C6 @ D4 ) ) )
@ ( quotient_of @ Q3 ) )
@ ( quotient_of @ P4 ) ) ) ).
% rat_plus_code
thf(fact_5407_normalize__denom__zero,axiom,
! [P4: int] :
( ( normalize @ ( product_Pair @ int @ int @ P4 @ ( zero_zero @ int ) ) )
= ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) ) ) ).
% normalize_denom_zero
thf(fact_5408_normalize__negative,axiom,
! [Q3: int,P4: int] :
( ( ord_less @ int @ Q3 @ ( zero_zero @ int ) )
=> ( ( normalize @ ( product_Pair @ int @ int @ P4 @ Q3 ) )
= ( normalize @ ( product_Pair @ int @ int @ ( uminus_uminus @ int @ P4 ) @ ( uminus_uminus @ int @ Q3 ) ) ) ) ) ).
% normalize_negative
thf(fact_5409_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_5410_bin__last__integer__code,axiom,
( bits_b8758750999018896077nteger
= ( ^ [I4: code_integer] :
( ( bit_se5824344872417868541ns_and @ code_integer @ I4 @ ( one_one @ code_integer ) )
!= ( zero_zero @ code_integer ) ) ) ) ).
% bin_last_integer_code
thf(fact_5411_normalize__denom__pos,axiom,
! [R3: product_prod @ int @ int,P4: int,Q3: int] :
( ( ( normalize @ R3 )
= ( product_Pair @ int @ int @ P4 @ Q3 ) )
=> ( ord_less @ int @ ( zero_zero @ int ) @ Q3 ) ) ).
% normalize_denom_pos
thf(fact_5412_normalize__crossproduct,axiom,
! [Q3: int,S2: int,P4: int,R3: int] :
( ( Q3
!= ( zero_zero @ int ) )
=> ( ( S2
!= ( zero_zero @ int ) )
=> ( ( ( normalize @ ( product_Pair @ int @ int @ P4 @ Q3 ) )
= ( normalize @ ( product_Pair @ int @ int @ R3 @ S2 ) ) )
=> ( ( times_times @ int @ P4 @ S2 )
= ( times_times @ int @ R3 @ Q3 ) ) ) ) ) ).
% normalize_crossproduct
thf(fact_5413_bin__last__integer__nbe,axiom,
( bits_b8758750999018896077nteger
= ( ^ [I4: code_integer] :
( ( modulo_modulo @ code_integer @ I4 @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) )
!= ( zero_zero @ code_integer ) ) ) ) ).
% bin_last_integer_nbe
thf(fact_5414_rat__divide__code,axiom,
! [P4: rat,Q3: rat] :
( ( quotient_of @ ( divide_divide @ rat @ P4 @ Q3 ) )
= ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [A5: int,C6: int] :
( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [B5: int,D4: int] : ( normalize @ ( product_Pair @ int @ int @ ( times_times @ int @ A5 @ D4 ) @ ( times_times @ int @ C6 @ B5 ) ) )
@ ( quotient_of @ Q3 ) )
@ ( quotient_of @ P4 ) ) ) ).
% rat_divide_code
thf(fact_5415_rat__times__code,axiom,
! [P4: rat,Q3: rat] :
( ( quotient_of @ ( times_times @ rat @ P4 @ Q3 ) )
= ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [A5: int,C6: int] :
( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [B5: int,D4: int] : ( normalize @ ( product_Pair @ int @ int @ ( times_times @ int @ A5 @ B5 ) @ ( times_times @ int @ C6 @ D4 ) ) )
@ ( quotient_of @ Q3 ) )
@ ( quotient_of @ P4 ) ) ) ).
% rat_times_code
thf(fact_5416_bin__last__integer_Oabs__eq,axiom,
! [X: int] :
( ( bits_b8758750999018896077nteger @ ( code_integer_of_int @ X ) )
= ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ X ) ) ) ).
% bin_last_integer.abs_eq
thf(fact_5417_rat__minus__code,axiom,
! [P4: rat,Q3: rat] :
( ( quotient_of @ ( minus_minus @ rat @ P4 @ Q3 ) )
= ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [A5: int,C6: int] :
( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [B5: int,D4: int] : ( normalize @ ( product_Pair @ int @ int @ ( minus_minus @ int @ ( times_times @ int @ A5 @ D4 ) @ ( times_times @ int @ B5 @ C6 ) ) @ ( times_times @ int @ C6 @ D4 ) ) )
@ ( quotient_of @ Q3 ) )
@ ( quotient_of @ P4 ) ) ) ).
% rat_minus_code
thf(fact_5418_bitOR__integer__unfold,axiom,
( ( bit_se1065995026697491101ons_or @ code_integer )
= ( ^ [X2: code_integer,Y2: code_integer] :
( if @ code_integer
@ ( X2
= ( zero_zero @ code_integer ) )
@ Y2
@ ( if @ code_integer
@ ( X2
= ( uminus_uminus @ code_integer @ ( one_one @ code_integer ) ) )
@ ( uminus_uminus @ code_integer @ ( one_one @ code_integer ) )
@ ( bits_Bit_integer @ ( bit_se1065995026697491101ons_or @ code_integer @ ( bits_b2549910563261871055nteger @ X2 ) @ ( bits_b2549910563261871055nteger @ Y2 ) )
@ ( ( bits_b8758750999018896077nteger @ X2 )
| ( bits_b8758750999018896077nteger @ Y2 ) ) ) ) ) ) ) ).
% bitOR_integer_unfold
thf(fact_5419_bitAND__integer__unfold,axiom,
( ( bit_se5824344872417868541ns_and @ code_integer )
= ( ^ [X2: code_integer,Y2: code_integer] :
( if @ code_integer
@ ( X2
= ( zero_zero @ code_integer ) )
@ ( zero_zero @ code_integer )
@ ( if @ code_integer
@ ( X2
= ( uminus_uminus @ code_integer @ ( one_one @ code_integer ) ) )
@ Y2
@ ( bits_Bit_integer @ ( bit_se5824344872417868541ns_and @ code_integer @ ( bits_b2549910563261871055nteger @ X2 ) @ ( bits_b2549910563261871055nteger @ Y2 ) )
@ ( ( bits_b8758750999018896077nteger @ X2 )
& ( bits_b8758750999018896077nteger @ Y2 ) ) ) ) ) ) ) ).
% bitAND_integer_unfold
thf(fact_5420_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_5421_case__prod__app,axiom,
! [A: $tType,D: $tType,C: $tType,B: $tType] :
( ( product_case_prod @ B @ C @ ( D > A ) )
= ( ^ [F2: B > C > D > A,X2: product_prod @ B @ C,Y2: D] :
( product_case_prod @ B @ C @ A
@ ^ [L2: B,R5: C] : ( F2 @ L2 @ R5 @ Y2 )
@ X2 ) ) ) ).
% case_prod_app
thf(fact_5422_case__prod__Pair__iden,axiom,
! [B: $tType,A: $tType,P4: product_prod @ A @ B] :
( ( product_case_prod @ A @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B ) @ P4 )
= P4 ) ).
% case_prod_Pair_iden
thf(fact_5423_fun__cong__unused__0,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( zero @ B )
=> ! [F3: ( A > B ) > C,G2: C] :
( ( F3
= ( ^ [X2: A > B] : G2 ) )
=> ( ( F3
@ ^ [X2: A] : ( zero_zero @ B ) )
= G2 ) ) ) ).
% fun_cong_unused_0
thf(fact_5424_eq__subset,axiom,
! [A: $tType,P: A > A > $o] :
( ord_less_eq @ ( A > A > $o )
@ ^ [Y3: A,Z2: A] : Y3 = Z2
@ ^ [A5: A,B5: A] :
( ( P @ A5 @ B5 )
| ( A5 = B5 ) ) ) ).
% eq_subset
thf(fact_5425_Frct__code__post_I2_J,axiom,
! [A3: int] :
( ( frct @ ( product_Pair @ int @ int @ A3 @ ( zero_zero @ int ) ) )
= ( zero_zero @ rat ) ) ).
% Frct_code_post(2)
thf(fact_5426_Frct__code__post_I1_J,axiom,
! [A3: int] :
( ( frct @ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ A3 ) )
= ( zero_zero @ rat ) ) ).
% Frct_code_post(1)
thf(fact_5427_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_5428_Frct__code__post_I6_J,axiom,
! [K: num,L: num] :
( ( frct @ ( product_Pair @ int @ int @ ( numeral_numeral @ int @ K ) @ ( numeral_numeral @ int @ L ) ) )
= ( divide_divide @ rat @ ( numeral_numeral @ rat @ K ) @ ( numeral_numeral @ rat @ L ) ) ) ).
% Frct_code_post(6)
thf(fact_5429_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_5430_set__bit__integer__conv__masks,axiom,
( ( generi7602027413899671122et_bit @ code_integer )
= ( ^ [X2: code_integer,I4: nat,B5: $o] : ( if @ code_integer @ B5 @ ( bit_se1065995026697491101ons_or @ code_integer @ X2 @ ( bit_se4730199178511100633sh_bit @ code_integer @ I4 @ ( one_one @ code_integer ) ) ) @ ( bit_se5824344872417868541ns_and @ code_integer @ X2 @ ( bit_ri4277139882892585799ns_not @ code_integer @ ( bit_se4730199178511100633sh_bit @ code_integer @ I4 @ ( one_one @ code_integer ) ) ) ) ) ) ) ).
% set_bit_integer_conv_masks
thf(fact_5431_Uint32__code,axiom,
( uint322
= ( ^ [I4: code_integer] : ( if @ uint32 @ ( bit_se5641148757651400278ts_bit @ code_integer @ ( bit_se5824344872417868541ns_and @ code_integer @ I4 @ ( 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 @ I4 @ ( 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 @ I4 @ ( 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_5432_word__cat__split__alt,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B )
& ( type_len @ C ) )
=> ! [W: word @ A,U2: word @ B,V2: word @ C] :
( ( ord_less_eq @ nat @ ( size_size @ ( word @ A ) @ W ) @ ( plus_plus @ nat @ ( size_size @ ( word @ B ) @ U2 ) @ ( size_size @ ( word @ C ) @ V2 ) ) )
=> ( ( ( word_split @ A @ B @ C @ W )
= ( product_Pair @ ( word @ B ) @ ( word @ C ) @ U2 @ V2 ) )
=> ( ( word_cat @ B @ C @ A @ U2 @ V2 )
= W ) ) ) ) ).
% word_cat_split_alt
thf(fact_5433_word__cat__id,axiom,
! [A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( type_len @ A ) )
=> ( ( word_cat @ B @ A @ A )
= ( ^ [A5: word @ B,B5: word @ A] : B5 ) ) ) ).
% word_cat_id
thf(fact_5434_test__bit__cat,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( ( type_len @ C )
& ( type_len @ B )
& ( type_len @ A ) )
=> ! [A3: word @ B,B3: word @ C,N: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( word_cat @ B @ C @ A @ A3 @ B3 ) @ N )
= ( ( ord_less @ nat @ N @ ( size_size @ ( word @ A ) @ ( word_cat @ B @ C @ A @ A3 @ B3 ) ) )
& ( ( ord_less @ nat @ N @ ( size_size @ ( word @ C ) @ B3 ) )
=> ( bit_se5641148757651400278ts_bit @ ( word @ C ) @ B3 @ N ) )
& ( ~ ( ord_less @ nat @ N @ ( size_size @ ( word @ C ) @ B3 ) )
=> ( bit_se5641148757651400278ts_bit @ ( word @ B ) @ A3 @ ( minus_minus @ nat @ N @ ( size_size @ ( word @ C ) @ B3 ) ) ) ) ) ) ) ).
% test_bit_cat
thf(fact_5435_word__cat__split__size,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( ( type_len @ C )
& ( type_len @ A )
& ( type_len @ B ) )
=> ! [T: word @ A,U2: word @ B,V2: word @ C] :
( ( ord_less_eq @ nat @ ( size_size @ ( word @ A ) @ T ) @ ( plus_plus @ nat @ ( size_size @ ( word @ B ) @ U2 ) @ ( size_size @ ( word @ C ) @ V2 ) ) )
=> ( ( ( product_Pair @ ( word @ B ) @ ( word @ C ) @ U2 @ V2 )
= ( word_split @ A @ B @ C @ T ) )
=> ( T
= ( word_cat @ B @ C @ A @ U2 @ V2 ) ) ) ) ) ).
% word_cat_split_size
thf(fact_5436_word__split__cat__alt,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( ( type_len @ C )
& ( type_len @ B )
& ( type_len @ A ) )
=> ! [W: word @ A,U2: word @ B,V2: word @ C] :
( ( W
= ( word_cat @ B @ C @ A @ U2 @ V2 ) )
=> ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( size_size @ ( word @ B ) @ U2 ) @ ( size_size @ ( word @ C ) @ V2 ) ) @ ( size_size @ ( word @ A ) @ W ) )
=> ( ( word_split @ A @ B @ C @ W )
= ( product_Pair @ ( word @ B ) @ ( word @ C ) @ U2 @ V2 ) ) ) ) ) ).
% word_split_cat_alt
thf(fact_5437_bit__set__bit__iff__2n,axiom,
! [A: $tType] :
( ( generic_set_set_bit @ A )
=> ! [A3: A,M: nat,B3: $o,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( generi7602027413899671122et_bit @ A @ A3 @ M @ B3 ) @ N )
= ( ( ( M = N )
=> B3 )
& ( ( M != N )
=> ( bit_se5641148757651400278ts_bit @ A @ A3 @ N ) )
& ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N )
!= ( zero_zero @ A ) ) ) ) ) ).
% bit_set_bit_iff_2n
thf(fact_5438_Uint32__signed__def,axiom,
( uint32_signed
= ( ^ [I4: code_integer] :
( if @ uint32
@ ( ( ord_less @ code_integer @ I4 @ ( 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 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ I4 ) )
@ ( undefined @ ( ( code_integer > uint32 ) > code_integer > uint32 ) @ uint322 @ I4 )
@ ( uint322 @ I4 ) ) ) ) ).
% Uint32_signed_def
thf(fact_5439_cat__slices,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( type_len @ A )
& ( type_len @ C ) )
=> ! [A3: word @ A,N: nat,C2: word @ B,B3: word @ C] :
( ( A3
= ( slice2 @ B @ A @ N @ C2 ) )
=> ( ( B3
= ( slice2 @ B @ C @ ( zero_zero @ nat ) @ C2 ) )
=> ( ( N
= ( size_size @ ( word @ C ) @ B3 ) )
=> ( ( ord_less_eq @ nat @ ( size_size @ ( word @ B ) @ C2 ) @ ( plus_plus @ nat @ ( size_size @ ( word @ A ) @ A3 ) @ ( size_size @ ( word @ C ) @ B3 ) ) )
=> ( ( word_cat @ A @ C @ B @ A3 @ B3 )
= C2 ) ) ) ) ) ) ).
% cat_slices
thf(fact_5440_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_5441_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_5442_slice__id,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [T: word @ A] :
( ( slice2 @ A @ A @ ( zero_zero @ nat ) @ T )
= T ) ) ).
% slice_id
thf(fact_5443_test__bit__set__gen,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A,N: nat,X: $o,M: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( generi7602027413899671122et_bit @ ( word @ A ) @ W @ N @ X ) @ M )
= ( ( ( M = N )
=> ( ( ord_less @ nat @ N @ ( size_size @ ( word @ A ) @ W ) )
& X ) )
& ( ( M != N )
=> ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ W @ M ) ) ) ) ) ).
% test_bit_set_gen
thf(fact_5444_test__bit__set,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A,N: nat,X: $o] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( generi7602027413899671122et_bit @ ( word @ A ) @ W @ N @ X ) @ N )
= ( ( ord_less @ nat @ N @ ( size_size @ ( word @ A ) @ W ) )
& X ) ) ) ).
% test_bit_set
thf(fact_5445_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_5446_slice__cat2,axiom,
! [B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B ) )
=> ! [A3: word @ B,T: word @ A] :
( ( slice2 @ A @ A @ ( zero_zero @ nat ) @ ( word_cat @ B @ A @ A @ A3 @ T ) )
= T ) ) ).
% slice_cat2
thf(fact_5447_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_5448_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_5449_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_5450_slice__cat1,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( type_len @ A )
& ( type_len @ C ) )
=> ! [A3: word @ B,B3: word @ C] :
( ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( size_size @ ( word @ B ) @ A3 ) @ ( size_size @ ( word @ C ) @ B3 ) ) @ ( size_size @ ( word @ A ) @ ( word_cat @ B @ C @ A @ A3 @ B3 ) ) )
=> ( ( slice2 @ A @ B @ ( size_size @ ( word @ C ) @ B3 ) @ ( word_cat @ B @ C @ A @ A3 @ B3 ) )
= A3 ) ) ) ).
% slice_cat1
thf(fact_5451_split__slices,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( ( type_len @ C )
& ( type_len @ B )
& ( type_len @ A ) )
=> ! [W: word @ C,U2: word @ A,V2: word @ B] :
( ( ( word_split @ C @ A @ B @ W )
= ( product_Pair @ ( word @ A ) @ ( word @ B ) @ U2 @ V2 ) )
=> ( ( U2
= ( slice2 @ C @ A @ ( size_size @ ( word @ B ) @ V2 ) @ W ) )
& ( V2
= ( slice2 @ C @ B @ ( zero_zero @ nat ) @ W ) ) ) ) ) ).
% split_slices
thf(fact_5452_int__set__bit__conv__ops,axiom,
( ( generi7602027413899671122et_bit @ int )
= ( ^ [I4: int,N4: nat,B5: $o] : ( if @ int @ B5 @ ( bit_se1065995026697491101ons_or @ int @ I4 @ ( bit_se4730199178511100633sh_bit @ int @ N4 @ ( one_one @ int ) ) ) @ ( bit_se5824344872417868541ns_and @ int @ I4 @ ( bit_ri4277139882892585799ns_not @ int @ ( bit_se4730199178511100633sh_bit @ int @ N4 @ ( one_one @ int ) ) ) ) ) ) ) ).
% int_set_bit_conv_ops
thf(fact_5453_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_5454_size__empty,axiom,
! [A: $tType] :
( ( size_size @ ( multiset @ A ) @ ( zero_zero @ ( multiset @ A ) ) )
= ( zero_zero @ nat ) ) ).
% size_empty
thf(fact_5455_set__encode__def,axiom,
( nat_set_encode
= ( groups7311177749621191930dd_sum @ nat @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% set_encode_def
thf(fact_5456_union__eq__empty,axiom,
! [A: $tType,M7: multiset @ A,N3: multiset @ A] :
( ( ( plus_plus @ ( multiset @ A ) @ M7 @ N3 )
= ( zero_zero @ ( multiset @ A ) ) )
= ( ( M7
= ( zero_zero @ ( multiset @ A ) ) )
& ( N3
= ( zero_zero @ ( multiset @ A ) ) ) ) ) ).
% union_eq_empty
thf(fact_5457_empty__eq__union,axiom,
! [A: $tType,M7: multiset @ A,N3: multiset @ A] :
( ( ( zero_zero @ ( multiset @ A ) )
= ( plus_plus @ ( multiset @ A ) @ M7 @ N3 ) )
= ( ( M7
= ( zero_zero @ ( multiset @ A ) ) )
& ( N3
= ( zero_zero @ ( multiset @ A ) ) ) ) ) ).
% empty_eq_union
thf(fact_5458_subset__mset_Ozero__eq__add__iff__both__eq__0,axiom,
! [A: $tType,X: multiset @ A,Y: multiset @ A] :
( ( ( zero_zero @ ( multiset @ A ) )
= ( plus_plus @ ( multiset @ A ) @ X @ Y ) )
= ( ( X
= ( zero_zero @ ( multiset @ A ) ) )
& ( Y
= ( zero_zero @ ( multiset @ A ) ) ) ) ) ).
% subset_mset.zero_eq_add_iff_both_eq_0
thf(fact_5459_subset__mset_Oadd__eq__0__iff__both__eq__0,axiom,
! [A: $tType,X: multiset @ A,Y: multiset @ A] :
( ( ( plus_plus @ ( multiset @ A ) @ X @ Y )
= ( zero_zero @ ( multiset @ A ) ) )
= ( ( X
= ( zero_zero @ ( multiset @ A ) ) )
& ( Y
= ( zero_zero @ ( multiset @ A ) ) ) ) ) ).
% subset_mset.add_eq_0_iff_both_eq_0
thf(fact_5460_set__encode__empty,axiom,
( ( nat_set_encode @ ( bot_bot @ ( set @ nat ) ) )
= ( zero_zero @ nat ) ) ).
% set_encode_empty
thf(fact_5461_Multiset_Odiff__cancel,axiom,
! [A: $tType,A2: multiset @ A] :
( ( minus_minus @ ( multiset @ A ) @ A2 @ A2 )
= ( zero_zero @ ( multiset @ A ) ) ) ).
% Multiset.diff_cancel
thf(fact_5462_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_5463_union__diff__assoc,axiom,
! [A: $tType,C3: multiset @ A,B2: multiset @ A,A2: multiset @ A] :
( ( ( minus_minus @ ( multiset @ A ) @ C3 @ B2 )
= ( zero_zero @ ( multiset @ A ) ) )
=> ( ( minus_minus @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ A2 @ B2 ) @ C3 )
= ( plus_plus @ ( multiset @ A ) @ A2 @ ( minus_minus @ ( multiset @ A ) @ B2 @ C3 ) ) ) ) ).
% union_diff_assoc
thf(fact_5464_empty__neutral_I1_J,axiom,
! [A: $tType,X: multiset @ A] :
( ( plus_plus @ ( multiset @ A ) @ ( zero_zero @ ( multiset @ A ) ) @ X )
= X ) ).
% empty_neutral(1)
thf(fact_5465_empty__neutral_I2_J,axiom,
! [A: $tType,X: multiset @ A] :
( ( plus_plus @ ( multiset @ A ) @ X @ ( zero_zero @ ( multiset @ A ) ) )
= X ) ).
% empty_neutral(2)
thf(fact_5466_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_5467_set__encode__insert,axiom,
! [A2: set @ nat,N: nat] :
( ( finite_finite2 @ nat @ A2 )
=> ( ~ ( member @ nat @ N @ A2 )
=> ( ( nat_set_encode @ ( insert @ nat @ N @ A2 ) )
= ( plus_plus @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ ( nat_set_encode @ A2 ) ) ) ) ) ).
% set_encode_insert
thf(fact_5468_sdiv__word__min,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A,B3: word @ A] : ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ ( size_size @ ( word @ A ) @ A3 ) @ ( one_one @ nat ) ) ) ) @ ( signed7115095781618012415divide @ int @ ( ring_1_signed @ A @ int @ A3 ) @ ( ring_1_signed @ A @ int @ B3 ) ) ) ) ).
% sdiv_word_min
thf(fact_5469_Cauchy__iff2,axiom,
( ( topolo3814608138187158403Cauchy @ real )
= ( ^ [X8: nat > real] :
! [J3: nat] :
? [M10: nat] :
! [M3: nat] :
( ( ord_less_eq @ nat @ M10 @ M3 )
=> ! [N4: nat] :
( ( ord_less_eq @ nat @ M10 @ N4 )
=> ( ord_less @ real @ ( abs_abs @ real @ ( minus_minus @ real @ ( X8 @ M3 ) @ ( X8 @ N4 ) ) ) @ ( inverse_inverse @ real @ ( semiring_1_of_nat @ real @ ( suc @ J3 ) ) ) ) ) ) ) ) ).
% Cauchy_iff2
thf(fact_5470_List_Ofinite__set,axiom,
! [A: $tType,Xs: list @ A] : ( finite_finite2 @ A @ ( set2 @ A @ Xs ) ) ).
% List.finite_set
thf(fact_5471_sdiv__int__div__0,axiom,
! [X: int] :
( ( signed7115095781618012415divide @ int @ X @ ( zero_zero @ int ) )
= ( zero_zero @ int ) ) ).
% sdiv_int_div_0
thf(fact_5472_sdiv__int__0__div,axiom,
! [X: int] :
( ( signed7115095781618012415divide @ int @ ( zero_zero @ int ) @ X )
= ( zero_zero @ int ) ) ).
% sdiv_int_0_div
thf(fact_5473_int__sdiv__simps_I2_J,axiom,
! [A3: int] :
( ( signed7115095781618012415divide @ int @ A3 @ ( zero_zero @ int ) )
= ( zero_zero @ int ) ) ).
% int_sdiv_simps(2)
thf(fact_5474_int__sdiv__simps_I1_J,axiom,
! [A3: int] :
( ( signed7115095781618012415divide @ int @ A3 @ ( one_one @ int ) )
= A3 ) ).
% int_sdiv_simps(1)
thf(fact_5475_sum__eq__0__iff,axiom,
! [A: $tType,B: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [F4: set @ B,F3: B > A] :
( ( finite_finite2 @ B @ F4 )
=> ( ( ( groups7311177749621191930dd_sum @ B @ A @ F3 @ F4 )
= ( zero_zero @ A ) )
= ( ! [X2: B] :
( ( member @ B @ X2 @ F4 )
=> ( ( F3 @ X2 )
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% sum_eq_0_iff
thf(fact_5476_sum_Oinfinite,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A2: set @ B,G2: B > A] :
( ~ ( finite_finite2 @ B @ A2 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G2 @ A2 )
= ( zero_zero @ A ) ) ) ) ).
% sum.infinite
thf(fact_5477_infinite__Icc__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A3: A,B3: A] :
( ( ~ ( finite_finite2 @ A @ ( set_or1337092689740270186AtMost @ A @ A3 @ B3 ) ) )
= ( ord_less @ A @ A3 @ B3 ) ) ) ).
% infinite_Icc_iff
thf(fact_5478_prod__zero__iff,axiom,
! [A: $tType,B: $tType] :
( ( semidom @ A )
=> ! [A2: set @ B,F3: B > A] :
( ( finite_finite2 @ B @ A2 )
=> ( ( ( groups7121269368397514597t_prod @ B @ A @ F3 @ A2 )
= ( zero_zero @ A ) )
= ( ? [X2: B] :
( ( member @ B @ X2 @ A2 )
& ( ( F3 @ X2 )
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% prod_zero_iff
thf(fact_5479_prod_Oinfinite,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A2: set @ B,G2: B > A] :
( ~ ( finite_finite2 @ B @ A2 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G2 @ A2 )
= ( one_one @ A ) ) ) ) ).
% prod.infinite
thf(fact_5480_int__sdiv__same__is__1,axiom,
! [A3: int,B3: int] :
( ( A3
!= ( zero_zero @ int ) )
=> ( ( ( signed7115095781618012415divide @ int @ A3 @ B3 )
= A3 )
= ( B3
= ( one_one @ int ) ) ) ) ).
% int_sdiv_same_is_1
thf(fact_5481_int__sdiv__simps_I3_J,axiom,
! [A3: int] :
( ( signed7115095781618012415divide @ int @ A3 @ ( uminus_uminus @ int @ ( one_one @ int ) ) )
= ( uminus_uminus @ int @ A3 ) ) ).
% int_sdiv_simps(3)
thf(fact_5482_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_5483_prod__eq__1__iff,axiom,
! [A: $tType,A2: set @ A,F3: A > nat] :
( ( finite_finite2 @ A @ A2 )
=> ( ( ( groups7121269368397514597t_prod @ A @ nat @ F3 @ A2 )
= ( one_one @ nat ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ A2 )
=> ( ( F3 @ X2 )
= ( one_one @ nat ) ) ) ) ) ) ).
% prod_eq_1_iff
thf(fact_5484_sum_Odelta,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add @ A )
=> ! [S: set @ B,A3: B,B3: B > A] :
( ( finite_finite2 @ B @ S )
=> ( ( ( member @ B @ A3 @ S )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [K3: B] : ( if @ A @ ( K3 = A3 ) @ ( B3 @ K3 ) @ ( zero_zero @ A ) )
@ S )
= ( B3 @ A3 ) ) )
& ( ~ ( member @ B @ A3 @ S )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [K3: B] : ( if @ A @ ( K3 = A3 ) @ ( B3 @ K3 ) @ ( zero_zero @ A ) )
@ S )
= ( zero_zero @ A ) ) ) ) ) ) ).
% sum.delta
thf(fact_5485_sum_Odelta_H,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add @ A )
=> ! [S: set @ B,A3: B,B3: B > A] :
( ( finite_finite2 @ B @ S )
=> ( ( ( member @ B @ A3 @ S )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [K3: B] : ( if @ A @ ( A3 = K3 ) @ ( B3 @ K3 ) @ ( zero_zero @ A ) )
@ S )
= ( B3 @ A3 ) ) )
& ( ~ ( member @ B @ A3 @ S )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [K3: B] : ( if @ A @ ( A3 = K3 ) @ ( B3 @ K3 ) @ ( zero_zero @ A ) )
@ S )
= ( zero_zero @ A ) ) ) ) ) ) ).
% sum.delta'
thf(fact_5486_prod_Odelta_H,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S: set @ B,A3: B,B3: B > A] :
( ( finite_finite2 @ B @ S )
=> ( ( ( member @ B @ A3 @ S )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K3: B] : ( if @ A @ ( A3 = K3 ) @ ( B3 @ K3 ) @ ( one_one @ A ) )
@ S )
= ( B3 @ A3 ) ) )
& ( ~ ( member @ B @ A3 @ S )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K3: B] : ( if @ A @ ( A3 = K3 ) @ ( B3 @ K3 ) @ ( one_one @ A ) )
@ S )
= ( one_one @ A ) ) ) ) ) ) ).
% prod.delta'
thf(fact_5487_prod_Odelta,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S: set @ B,A3: B,B3: B > A] :
( ( finite_finite2 @ B @ S )
=> ( ( ( member @ B @ A3 @ S )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K3: B] : ( if @ A @ ( K3 = A3 ) @ ( B3 @ K3 ) @ ( one_one @ A ) )
@ S )
= ( B3 @ A3 ) ) )
& ( ~ ( member @ B @ A3 @ S )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K3: B] : ( if @ A @ ( K3 = A3 ) @ ( B3 @ K3 ) @ ( one_one @ A ) )
@ S )
= ( one_one @ A ) ) ) ) ) ) ).
% prod.delta
thf(fact_5488_summable__If__finite__set,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [A2: set @ nat,F3: nat > A] :
( ( finite_finite2 @ nat @ A2 )
=> ( summable @ A
@ ^ [R5: nat] : ( if @ A @ ( member @ nat @ R5 @ A2 ) @ ( F3 @ R5 ) @ ( zero_zero @ A ) ) ) ) ) ).
% summable_If_finite_set
thf(fact_5489_summable__If__finite,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [P: nat > $o,F3: nat > A] :
( ( finite_finite2 @ nat @ ( collect @ nat @ P ) )
=> ( summable @ A
@ ^ [R5: nat] : ( if @ A @ ( P @ R5 ) @ ( F3 @ R5 ) @ ( zero_zero @ A ) ) ) ) ) ).
% summable_If_finite
thf(fact_5490_sum_Oinsert,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A2: set @ B,X: B,G2: B > A] :
( ( finite_finite2 @ B @ A2 )
=> ( ~ ( member @ B @ X @ A2 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G2 @ ( insert @ B @ X @ A2 ) )
= ( plus_plus @ A @ ( G2 @ X ) @ ( groups7311177749621191930dd_sum @ B @ A @ G2 @ A2 ) ) ) ) ) ) ).
% sum.insert
thf(fact_5491_prod_Oinsert,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A2: set @ B,X: B,G2: B > A] :
( ( finite_finite2 @ B @ A2 )
=> ( ~ ( member @ B @ X @ A2 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G2 @ ( insert @ B @ X @ A2 ) )
= ( times_times @ A @ ( G2 @ X ) @ ( groups7121269368397514597t_prod @ B @ A @ G2 @ A2 ) ) ) ) ) ) ).
% prod.insert
thf(fact_5492_int__sdiv__negated__is__minus1,axiom,
! [A3: int,B3: int] :
( ( A3
!= ( zero_zero @ int ) )
=> ( ( ( signed7115095781618012415divide @ int @ A3 @ B3 )
= ( uminus_uminus @ int @ A3 ) )
= ( B3
= ( uminus_uminus @ int @ ( one_one @ int ) ) ) ) ) ).
% int_sdiv_negated_is_minus1
thf(fact_5493_prod__pos__nat__iff,axiom,
! [A: $tType,A2: set @ A,F3: A > nat] :
( ( finite_finite2 @ A @ A2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( groups7121269368397514597t_prod @ A @ nat @ F3 @ A2 ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ A2 )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F3 @ X2 ) ) ) ) ) ) ).
% prod_pos_nat_iff
thf(fact_5494_sum__zero__power,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: set @ nat,C2: nat > A] :
( ( ( ( finite_finite2 @ nat @ A2 )
& ( member @ nat @ ( zero_zero @ nat ) @ A2 ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( C2 @ I4 ) @ ( power_power @ A @ ( zero_zero @ A ) @ I4 ) )
@ A2 )
= ( C2 @ ( zero_zero @ nat ) ) ) )
& ( ~ ( ( finite_finite2 @ nat @ A2 )
& ( member @ nat @ ( zero_zero @ nat ) @ A2 ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( C2 @ I4 ) @ ( power_power @ A @ ( zero_zero @ A ) @ I4 ) )
@ A2 )
= ( zero_zero @ A ) ) ) ) ) ).
% sum_zero_power
thf(fact_5495_sum__zero__power_H,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A2: set @ nat,C2: nat > A,D2: nat > A] :
( ( ( ( finite_finite2 @ nat @ A2 )
& ( member @ nat @ ( zero_zero @ nat ) @ A2 ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( divide_divide @ A @ ( times_times @ A @ ( C2 @ I4 ) @ ( power_power @ A @ ( zero_zero @ A ) @ I4 ) ) @ ( D2 @ I4 ) )
@ A2 )
= ( divide_divide @ A @ ( C2 @ ( zero_zero @ nat ) ) @ ( D2 @ ( zero_zero @ nat ) ) ) ) )
& ( ~ ( ( finite_finite2 @ nat @ A2 )
& ( member @ nat @ ( zero_zero @ nat ) @ A2 ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( divide_divide @ A @ ( times_times @ A @ ( C2 @ I4 ) @ ( power_power @ A @ ( zero_zero @ A ) @ I4 ) ) @ ( D2 @ I4 ) )
@ A2 )
= ( zero_zero @ A ) ) ) ) ) ).
% sum_zero_power'
thf(fact_5496_union__le__mono1,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [B2: multiset @ A,D3: multiset @ A,C3: multiset @ A] :
( ( ord_less @ ( multiset @ A ) @ B2 @ D3 )
=> ( ord_less @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ B2 @ C3 ) @ ( plus_plus @ ( multiset @ A ) @ D3 @ C3 ) ) ) ) ).
% union_le_mono1
thf(fact_5497_union__le__mono2,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [B2: multiset @ A,D3: multiset @ A,C3: multiset @ A] :
( ( ord_less @ ( multiset @ A ) @ B2 @ D3 )
=> ( ord_less @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ C3 @ B2 ) @ ( plus_plus @ ( multiset @ A ) @ C3 @ D3 ) ) ) ) ).
% union_le_mono2
thf(fact_5498_union__less__mono,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A2: multiset @ A,C3: multiset @ A,B2: multiset @ A,D3: multiset @ A] :
( ( ord_less @ ( multiset @ A ) @ A2 @ C3 )
=> ( ( ord_less @ ( multiset @ A ) @ B2 @ D3 )
=> ( ord_less @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ A2 @ B2 ) @ ( plus_plus @ ( multiset @ A ) @ C3 @ D3 ) ) ) ) ) ).
% union_less_mono
thf(fact_5499_finite__lists__length__eq,axiom,
! [A: $tType,A2: set @ A,N: nat] :
( ( finite_finite2 @ A @ A2 )
=> ( finite_finite2 @ ( list @ A )
@ ( collect @ ( list @ A )
@ ^ [Xs3: list @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs3 ) @ A2 )
& ( ( size_size @ ( list @ A ) @ Xs3 )
= N ) ) ) ) ) ).
% finite_lists_length_eq
thf(fact_5500_sum__eq__empty__iff,axiom,
! [B: $tType,A: $tType,A2: set @ A,F3: A > ( multiset @ B )] :
( ( finite_finite2 @ A @ A2 )
=> ( ( ( groups7311177749621191930dd_sum @ A @ ( multiset @ B ) @ F3 @ A2 )
= ( zero_zero @ ( multiset @ B ) ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ A2 )
=> ( ( F3 @ X2 )
= ( zero_zero @ ( multiset @ B ) ) ) ) ) ) ) ).
% sum_eq_empty_iff
thf(fact_5501_prod_Oswap__restrict,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A2: set @ B,B2: set @ C,G2: B > C > A,R: B > C > $o] :
( ( finite_finite2 @ B @ A2 )
=> ( ( finite_finite2 @ C @ B2 )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [X2: B] :
( groups7121269368397514597t_prod @ C @ A @ ( G2 @ X2 )
@ ( collect @ C
@ ^ [Y2: C] :
( ( member @ C @ Y2 @ B2 )
& ( R @ X2 @ Y2 ) ) ) )
@ A2 )
= ( groups7121269368397514597t_prod @ C @ A
@ ^ [Y2: C] :
( groups7121269368397514597t_prod @ B @ A
@ ^ [X2: B] : ( G2 @ X2 @ Y2 )
@ ( collect @ B
@ ^ [X2: B] :
( ( member @ B @ X2 @ A2 )
& ( R @ X2 @ Y2 ) ) ) )
@ B2 ) ) ) ) ) ).
% prod.swap_restrict
thf(fact_5502_sum_Oswap__restrict,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( comm_monoid_add @ A )
=> ! [A2: set @ B,B2: set @ C,G2: B > C > A,R: B > C > $o] :
( ( finite_finite2 @ B @ A2 )
=> ( ( finite_finite2 @ C @ B2 )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X2: B] :
( groups7311177749621191930dd_sum @ C @ A @ ( G2 @ X2 )
@ ( collect @ C
@ ^ [Y2: C] :
( ( member @ C @ Y2 @ B2 )
& ( R @ X2 @ Y2 ) ) ) )
@ A2 )
= ( groups7311177749621191930dd_sum @ C @ A
@ ^ [Y2: C] :
( groups7311177749621191930dd_sum @ B @ A
@ ^ [X2: B] : ( G2 @ X2 @ Y2 )
@ ( collect @ B
@ ^ [X2: B] :
( ( member @ B @ X2 @ A2 )
& ( R @ X2 @ Y2 ) ) ) )
@ B2 ) ) ) ) ) ).
% sum.swap_restrict
thf(fact_5503_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_5504_finite__list,axiom,
! [A: $tType,A2: set @ A] :
( ( finite_finite2 @ A @ A2 )
=> ? [Xs2: list @ A] :
( ( set2 @ A @ Xs2 )
= A2 ) ) ).
% finite_list
thf(fact_5505_finite__M__bounded__by__nat,axiom,
! [P: nat > $o,I: nat] :
( finite_finite2 @ nat
@ ( collect @ nat
@ ^ [K3: nat] :
( ( P @ K3 )
& ( ord_less @ nat @ K3 @ I ) ) ) ) ).
% finite_M_bounded_by_nat
thf(fact_5506_bounded__nat__set__is__finite,axiom,
! [N3: set @ nat,N: nat] :
( ! [X3: nat] :
( ( member @ nat @ X3 @ N3 )
=> ( ord_less @ nat @ X3 @ N ) )
=> ( finite_finite2 @ nat @ N3 ) ) ).
% bounded_nat_set_is_finite
thf(fact_5507_finite__nat__set__iff__bounded,axiom,
( ( finite_finite2 @ nat )
= ( ^ [N9: set @ nat] :
? [M3: nat] :
! [X2: nat] :
( ( member @ nat @ X2 @ N9 )
=> ( ord_less @ nat @ X2 @ M3 ) ) ) ) ).
% finite_nat_set_iff_bounded
thf(fact_5508_finite__less__ub,axiom,
! [F3: nat > nat,U2: nat] :
( ! [N2: nat] : ( ord_less_eq @ nat @ N2 @ ( F3 @ N2 ) )
=> ( finite_finite2 @ nat
@ ( collect @ nat
@ ^ [N4: nat] : ( ord_less_eq @ nat @ ( F3 @ N4 ) @ U2 ) ) ) ) ).
% finite_less_ub
thf(fact_5509_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_5510_ex__min__if__finite,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [S: set @ A] :
( ( finite_finite2 @ A @ S )
=> ( ( S
!= ( bot_bot @ ( set @ A ) ) )
=> ? [X3: A] :
( ( member @ A @ X3 @ S )
& ~ ? [Xa2: A] :
( ( member @ A @ Xa2 @ S )
& ( ord_less @ A @ Xa2 @ X3 ) ) ) ) ) ) ).
% ex_min_if_finite
thf(fact_5511_infinite__growing,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X5: set @ A] :
( ( X5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ X5 )
=> ? [Xa2: A] :
( ( member @ A @ Xa2 @ X5 )
& ( ord_less @ A @ X3 @ Xa2 ) ) )
=> ~ ( finite_finite2 @ A @ X5 ) ) ) ) ).
% infinite_growing
thf(fact_5512_infinite__Icc,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ A3 @ B3 )
=> ~ ( finite_finite2 @ A @ ( set_or1337092689740270186AtMost @ A @ A3 @ B3 ) ) ) ) ).
% infinite_Icc
thf(fact_5513_prod__zero,axiom,
! [B: $tType,A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A2: set @ B,F3: B > A] :
( ( finite_finite2 @ B @ A2 )
=> ( ? [X6: B] :
( ( member @ B @ X6 @ A2 )
& ( ( F3 @ X6 )
= ( zero_zero @ A ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ F3 @ A2 )
= ( zero_zero @ A ) ) ) ) ) ).
% prod_zero
thf(fact_5514_finite__lists__length__le,axiom,
! [A: $tType,A2: set @ A,N: nat] :
( ( finite_finite2 @ A @ A2 )
=> ( finite_finite2 @ ( list @ A )
@ ( collect @ ( list @ A )
@ ^ [Xs3: list @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs3 ) @ A2 )
& ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs3 ) @ N ) ) ) ) ) ).
% finite_lists_length_le
thf(fact_5515_summable__finite,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [N3: set @ nat,F3: nat > A] :
( ( finite_finite2 @ nat @ N3 )
=> ( ! [N2: nat] :
( ~ ( member @ nat @ N2 @ N3 )
=> ( ( F3 @ N2 )
= ( zero_zero @ A ) ) )
=> ( summable @ A @ F3 ) ) ) ) ).
% summable_finite
thf(fact_5516_subset__eq__atLeast0__atMost__finite,axiom,
! [N3: set @ nat,N: nat] :
( ( ord_less_eq @ ( set @ nat ) @ N3 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
=> ( finite_finite2 @ nat @ N3 ) ) ).
% subset_eq_atLeast0_atMost_finite
thf(fact_5517_sum_Ofinite__Collect__op,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [I6: set @ B,X: B > A,Y: B > A] :
( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [I4: B] :
( ( member @ B @ I4 @ I6 )
& ( ( X @ I4 )
!= ( zero_zero @ A ) ) ) ) )
=> ( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [I4: B] :
( ( member @ B @ I4 @ I6 )
& ( ( Y @ I4 )
!= ( zero_zero @ A ) ) ) ) )
=> ( finite_finite2 @ B
@ ( collect @ B
@ ^ [I4: B] :
( ( member @ B @ I4 @ I6 )
& ( ( plus_plus @ A @ ( X @ I4 ) @ ( Y @ I4 ) )
!= ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% sum.finite_Collect_op
thf(fact_5518_prod_Ofinite__Collect__op,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [I6: set @ B,X: B > A,Y: B > A] :
( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [I4: B] :
( ( member @ B @ I4 @ I6 )
& ( ( X @ I4 )
!= ( one_one @ A ) ) ) ) )
=> ( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [I4: B] :
( ( member @ B @ I4 @ I6 )
& ( ( Y @ I4 )
!= ( one_one @ A ) ) ) ) )
=> ( finite_finite2 @ B
@ ( collect @ B
@ ^ [I4: B] :
( ( member @ B @ I4 @ I6 )
& ( ( times_times @ A @ ( X @ I4 ) @ ( Y @ I4 ) )
!= ( one_one @ A ) ) ) ) ) ) ) ) ).
% prod.finite_Collect_op
thf(fact_5519_set__encode__inf,axiom,
! [A2: set @ nat] :
( ~ ( finite_finite2 @ nat @ A2 )
=> ( ( nat_set_encode @ A2 )
= ( zero_zero @ nat ) ) ) ).
% set_encode_inf
thf(fact_5520_filter__preserves__multiset,axiom,
! [A: $tType,M7: A > nat,P: A > $o] :
( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X2: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( M7 @ X2 ) ) ) )
=> ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X2: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( if @ nat @ ( P @ X2 ) @ ( M7 @ X2 ) @ ( zero_zero @ nat ) ) ) ) ) ) ).
% filter_preserves_multiset
thf(fact_5521_sum_Ointer__filter,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A2: set @ B,G2: B > A,P: B > $o] :
( ( finite_finite2 @ B @ A2 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G2
@ ( collect @ B
@ ^ [X2: B] :
( ( member @ B @ X2 @ A2 )
& ( P @ X2 ) ) ) )
= ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X2: B] : ( if @ A @ ( P @ X2 ) @ ( G2 @ X2 ) @ ( zero_zero @ A ) )
@ A2 ) ) ) ) ).
% sum.inter_filter
thf(fact_5522_prod_Ointer__filter,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A2: set @ B,G2: B > A,P: B > $o] :
( ( finite_finite2 @ B @ A2 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G2
@ ( collect @ B
@ ^ [X2: B] :
( ( member @ B @ X2 @ A2 )
& ( P @ X2 ) ) ) )
= ( groups7121269368397514597t_prod @ B @ A
@ ^ [X2: B] : ( if @ A @ ( P @ X2 ) @ ( G2 @ X2 ) @ ( one_one @ A ) )
@ A2 ) ) ) ) ).
% prod.inter_filter
thf(fact_5523_finite__int__segment,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [A3: A,B3: A] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X2: A] :
( ( member @ A @ X2 @ ( ring_1_Ints @ A ) )
& ( ord_less_eq @ A @ A3 @ X2 )
& ( ord_less_eq @ A @ X2 @ B3 ) ) ) ) ) ).
% finite_int_segment
thf(fact_5524_sum__le__included,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [S2: set @ B,T: set @ C,G2: C > A,I: C > B,F3: B > A] :
( ( finite_finite2 @ B @ S2 )
=> ( ( finite_finite2 @ C @ T )
=> ( ! [X3: C] :
( ( member @ C @ X3 @ T )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( G2 @ X3 ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ S2 )
=> ? [Xa2: C] :
( ( member @ C @ Xa2 @ T )
& ( ( I @ Xa2 )
= X3 )
& ( ord_less_eq @ A @ ( F3 @ X3 ) @ ( G2 @ Xa2 ) ) ) )
=> ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ S2 ) @ ( groups7311177749621191930dd_sum @ C @ A @ G2 @ T ) ) ) ) ) ) ) ).
% sum_le_included
thf(fact_5525_sum__nonneg__eq__0__iff,axiom,
! [A: $tType,B: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A2: set @ B,F3: B > A] :
( ( finite_finite2 @ B @ A2 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ A2 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F3 @ X3 ) ) )
=> ( ( ( groups7311177749621191930dd_sum @ B @ A @ F3 @ A2 )
= ( zero_zero @ A ) )
= ( ! [X2: B] :
( ( member @ B @ X2 @ A2 )
=> ( ( F3 @ X2 )
= ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
thf(fact_5526_sum__strict__mono__ex1,axiom,
! [A: $tType,I5: $tType] :
( ( ordere8940638589300402666id_add @ A )
=> ! [A2: set @ I5,F3: I5 > A,G2: I5 > A] :
( ( finite_finite2 @ I5 @ A2 )
=> ( ! [X3: I5] :
( ( member @ I5 @ X3 @ A2 )
=> ( ord_less_eq @ A @ ( F3 @ X3 ) @ ( G2 @ X3 ) ) )
=> ( ? [X6: I5] :
( ( member @ I5 @ X6 @ A2 )
& ( ord_less @ A @ ( F3 @ X6 ) @ ( G2 @ X6 ) ) )
=> ( ord_less @ A @ ( groups7311177749621191930dd_sum @ I5 @ A @ F3 @ A2 ) @ ( groups7311177749621191930dd_sum @ I5 @ A @ G2 @ A2 ) ) ) ) ) ) ).
% sum_strict_mono_ex1
thf(fact_5527_sum_Orelated,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [R: A > A > $o,S: set @ B,H2: B > A,G2: B > A] :
( ( R @ ( zero_zero @ A ) @ ( zero_zero @ A ) )
=> ( ! [X12: A,Y1: A,X23: A,Y23: A] :
( ( ( R @ X12 @ X23 )
& ( R @ Y1 @ Y23 ) )
=> ( R @ ( plus_plus @ A @ X12 @ Y1 ) @ ( plus_plus @ A @ X23 @ Y23 ) ) )
=> ( ( finite_finite2 @ B @ S )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ S )
=> ( R @ ( H2 @ X3 ) @ ( G2 @ X3 ) ) )
=> ( R @ ( groups7311177749621191930dd_sum @ B @ A @ H2 @ S ) @ ( groups7311177749621191930dd_sum @ B @ A @ G2 @ S ) ) ) ) ) ) ) ).
% sum.related
thf(fact_5528_finite__ranking__induct,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [S: set @ B,P: ( set @ B ) > $o,F3: B > A] :
( ( finite_finite2 @ B @ S )
=> ( ( P @ ( bot_bot @ ( set @ B ) ) )
=> ( ! [X3: B,S6: set @ B] :
( ( finite_finite2 @ B @ S6 )
=> ( ! [Y5: B] :
( ( member @ B @ Y5 @ S6 )
=> ( ord_less_eq @ A @ ( F3 @ Y5 ) @ ( F3 @ X3 ) ) )
=> ( ( P @ S6 )
=> ( P @ ( insert @ B @ X3 @ S6 ) ) ) ) )
=> ( P @ S ) ) ) ) ) ).
% finite_ranking_induct
thf(fact_5529_finite__linorder__min__induct,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ A2 )
=> ( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [B7: A,A10: set @ A] :
( ( finite_finite2 @ A @ A10 )
=> ( ! [X6: A] :
( ( member @ A @ X6 @ A10 )
=> ( ord_less @ A @ B7 @ X6 ) )
=> ( ( P @ A10 )
=> ( P @ ( insert @ A @ B7 @ A10 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ) ).
% finite_linorder_min_induct
thf(fact_5530_finite__linorder__max__induct,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ A2 )
=> ( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [B7: A,A10: set @ A] :
( ( finite_finite2 @ A @ A10 )
=> ( ! [X6: A] :
( ( member @ A @ X6 @ A10 )
=> ( ord_less @ A @ X6 @ B7 ) )
=> ( ( P @ A10 )
=> ( P @ ( insert @ A @ B7 @ A10 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ) ).
% finite_linorder_max_induct
thf(fact_5531_sum__strict__mono,axiom,
! [A: $tType,B: $tType] :
( ( strict7427464778891057005id_add @ A )
=> ! [A2: set @ B,F3: B > A,G2: B > A] :
( ( finite_finite2 @ B @ A2 )
=> ( ( A2
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ A2 )
=> ( ord_less @ A @ ( F3 @ X3 ) @ ( G2 @ X3 ) ) )
=> ( ord_less @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ A2 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G2 @ A2 ) ) ) ) ) ) ).
% sum_strict_mono
thf(fact_5532_prod_Orelated,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [R: A > A > $o,S: set @ B,H2: B > A,G2: B > A] :
( ( R @ ( one_one @ A ) @ ( one_one @ A ) )
=> ( ! [X12: A,Y1: A,X23: A,Y23: A] :
( ( ( R @ X12 @ X23 )
& ( R @ Y1 @ Y23 ) )
=> ( R @ ( times_times @ A @ X12 @ Y1 ) @ ( times_times @ A @ X23 @ Y23 ) ) )
=> ( ( finite_finite2 @ B @ S )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ S )
=> ( R @ ( H2 @ X3 ) @ ( G2 @ X3 ) ) )
=> ( R @ ( groups7121269368397514597t_prod @ B @ A @ H2 @ S ) @ ( groups7121269368397514597t_prod @ B @ A @ G2 @ S ) ) ) ) ) ) ) ).
% prod.related
thf(fact_5533_sum_Oinsert__if,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A2: set @ B,X: B,G2: B > A] :
( ( finite_finite2 @ B @ A2 )
=> ( ( ( member @ B @ X @ A2 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G2 @ ( insert @ B @ X @ A2 ) )
= ( groups7311177749621191930dd_sum @ B @ A @ G2 @ A2 ) ) )
& ( ~ ( member @ B @ X @ A2 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G2 @ ( insert @ B @ X @ A2 ) )
= ( plus_plus @ A @ ( G2 @ X ) @ ( groups7311177749621191930dd_sum @ B @ A @ G2 @ A2 ) ) ) ) ) ) ) ).
% sum.insert_if
thf(fact_5534_sum_Oreindex__bij__witness__not__neutral,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comm_monoid_add @ A )
=> ! [S7: set @ B,T5: set @ C,S: set @ B,I: C > B,J: B > C,T4: set @ C,G2: B > A,H2: C > A] :
( ( finite_finite2 @ B @ S7 )
=> ( ( finite_finite2 @ C @ T5 )
=> ( ! [A6: B] :
( ( member @ B @ A6 @ ( minus_minus @ ( set @ B ) @ S @ S7 ) )
=> ( ( I @ ( J @ A6 ) )
= A6 ) )
=> ( ! [A6: B] :
( ( member @ B @ A6 @ ( minus_minus @ ( set @ B ) @ S @ S7 ) )
=> ( member @ C @ ( J @ A6 ) @ ( minus_minus @ ( set @ C ) @ T4 @ T5 ) ) )
=> ( ! [B7: C] :
( ( member @ C @ B7 @ ( minus_minus @ ( set @ C ) @ T4 @ T5 ) )
=> ( ( J @ ( I @ B7 ) )
= B7 ) )
=> ( ! [B7: C] :
( ( member @ C @ B7 @ ( minus_minus @ ( set @ C ) @ T4 @ T5 ) )
=> ( member @ B @ ( I @ B7 ) @ ( minus_minus @ ( set @ B ) @ S @ S7 ) ) )
=> ( ! [A6: B] :
( ( member @ B @ A6 @ S7 )
=> ( ( G2 @ A6 )
= ( zero_zero @ A ) ) )
=> ( ! [B7: C] :
( ( member @ C @ B7 @ T5 )
=> ( ( H2 @ B7 )
= ( zero_zero @ A ) ) )
=> ( ! [A6: B] :
( ( member @ B @ A6 @ S )
=> ( ( H2 @ ( J @ A6 ) )
= ( G2 @ A6 ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G2 @ S )
= ( groups7311177749621191930dd_sum @ C @ A @ H2 @ T4 ) ) ) ) ) ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness_not_neutral
thf(fact_5535_prod_Oinsert__if,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A2: set @ B,X: B,G2: B > A] :
( ( finite_finite2 @ B @ A2 )
=> ( ( ( member @ B @ X @ A2 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G2 @ ( insert @ B @ X @ A2 ) )
= ( groups7121269368397514597t_prod @ B @ A @ G2 @ A2 ) ) )
& ( ~ ( member @ B @ X @ A2 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G2 @ ( insert @ B @ X @ A2 ) )
= ( times_times @ A @ ( G2 @ X ) @ ( groups7121269368397514597t_prod @ B @ A @ G2 @ A2 ) ) ) ) ) ) ) ).
% prod.insert_if
thf(fact_5536_prod_Oreindex__bij__witness__not__neutral,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S7: set @ B,T5: set @ C,S: set @ B,I: C > B,J: B > C,T4: set @ C,G2: B > A,H2: C > A] :
( ( finite_finite2 @ B @ S7 )
=> ( ( finite_finite2 @ C @ T5 )
=> ( ! [A6: B] :
( ( member @ B @ A6 @ ( minus_minus @ ( set @ B ) @ S @ S7 ) )
=> ( ( I @ ( J @ A6 ) )
= A6 ) )
=> ( ! [A6: B] :
( ( member @ B @ A6 @ ( minus_minus @ ( set @ B ) @ S @ S7 ) )
=> ( member @ C @ ( J @ A6 ) @ ( minus_minus @ ( set @ C ) @ T4 @ T5 ) ) )
=> ( ! [B7: C] :
( ( member @ C @ B7 @ ( minus_minus @ ( set @ C ) @ T4 @ T5 ) )
=> ( ( J @ ( I @ B7 ) )
= B7 ) )
=> ( ! [B7: C] :
( ( member @ C @ B7 @ ( minus_minus @ ( set @ C ) @ T4 @ T5 ) )
=> ( member @ B @ ( I @ B7 ) @ ( minus_minus @ ( set @ B ) @ S @ S7 ) ) )
=> ( ! [A6: B] :
( ( member @ B @ A6 @ S7 )
=> ( ( G2 @ A6 )
= ( one_one @ A ) ) )
=> ( ! [B7: C] :
( ( member @ C @ B7 @ T5 )
=> ( ( H2 @ B7 )
= ( one_one @ A ) ) )
=> ( ! [A6: B] :
( ( member @ B @ A6 @ S )
=> ( ( H2 @ ( J @ A6 ) )
= ( G2 @ A6 ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G2 @ S )
= ( groups7121269368397514597t_prod @ C @ A @ H2 @ T4 ) ) ) ) ) ) ) ) ) ) ) ) ).
% prod.reindex_bij_witness_not_neutral
thf(fact_5537_sum__eq__Suc0__iff,axiom,
! [A: $tType,A2: set @ A,F3: A > nat] :
( ( finite_finite2 @ A @ A2 )
=> ( ( ( groups7311177749621191930dd_sum @ A @ nat @ F3 @ A2 )
= ( suc @ ( zero_zero @ nat ) ) )
= ( ? [X2: A] :
( ( member @ A @ X2 @ A2 )
& ( ( F3 @ X2 )
= ( suc @ ( zero_zero @ nat ) ) )
& ! [Y2: A] :
( ( member @ A @ Y2 @ A2 )
=> ( ( X2 != Y2 )
=> ( ( F3 @ Y2 )
= ( zero_zero @ nat ) ) ) ) ) ) ) ) ).
% sum_eq_Suc0_iff
thf(fact_5538_sum__eq__1__iff,axiom,
! [A: $tType,A2: set @ A,F3: A > nat] :
( ( finite_finite2 @ A @ A2 )
=> ( ( ( groups7311177749621191930dd_sum @ A @ nat @ F3 @ A2 )
= ( one_one @ nat ) )
= ( ? [X2: A] :
( ( member @ A @ X2 @ A2 )
& ( ( F3 @ X2 )
= ( one_one @ nat ) )
& ! [Y2: A] :
( ( member @ A @ Y2 @ A2 )
=> ( ( X2 != Y2 )
=> ( ( F3 @ Y2 )
= ( zero_zero @ nat ) ) ) ) ) ) ) ) ).
% sum_eq_1_iff
thf(fact_5539_sum__nonneg__0,axiom,
! [B: $tType,A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [S2: set @ B,F3: B > A,I: B] :
( ( finite_finite2 @ B @ S2 )
=> ( ! [I3: B] :
( ( member @ B @ I3 @ S2 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F3 @ I3 ) ) )
=> ( ( ( groups7311177749621191930dd_sum @ B @ A @ F3 @ S2 )
= ( zero_zero @ A ) )
=> ( ( member @ B @ I @ S2 )
=> ( ( F3 @ I )
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% sum_nonneg_0
thf(fact_5540_sum__nonneg__leq__bound,axiom,
! [B: $tType,A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [S2: set @ B,F3: B > A,B2: A,I: B] :
( ( finite_finite2 @ B @ S2 )
=> ( ! [I3: B] :
( ( member @ B @ I3 @ S2 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F3 @ I3 ) ) )
=> ( ( ( groups7311177749621191930dd_sum @ B @ A @ F3 @ S2 )
= B2 )
=> ( ( member @ B @ I @ S2 )
=> ( ord_less_eq @ A @ ( F3 @ I ) @ B2 ) ) ) ) ) ) ).
% sum_nonneg_leq_bound
thf(fact_5541_add__mset__in__multiset,axiom,
! [A: $tType,M7: A > nat,A3: A] :
( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X2: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( M7 @ X2 ) ) ) )
=> ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X2: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( if @ nat @ ( X2 = A3 ) @ ( suc @ ( M7 @ X2 ) ) @ ( M7 @ X2 ) ) ) ) ) ) ).
% add_mset_in_multiset
thf(fact_5542_diff__preserves__multiset,axiom,
! [A: $tType,M7: A > nat,N3: A > nat] :
( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X2: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( M7 @ X2 ) ) ) )
=> ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X2: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ ( M7 @ X2 ) @ ( N3 @ X2 ) ) ) ) ) ) ).
% diff_preserves_multiset
thf(fact_5543_sum_Osetdiff__irrelevant,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A2: set @ B,G2: B > A] :
( ( finite_finite2 @ B @ A2 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G2
@ ( minus_minus @ ( set @ B ) @ A2
@ ( collect @ B
@ ^ [X2: B] :
( ( G2 @ X2 )
= ( zero_zero @ A ) ) ) ) )
= ( groups7311177749621191930dd_sum @ B @ A @ G2 @ A2 ) ) ) ) ).
% sum.setdiff_irrelevant
thf(fact_5544_finite__divisors__nat,axiom,
! [M: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
=> ( finite_finite2 @ nat
@ ( collect @ nat
@ ^ [D4: nat] : ( dvd_dvd @ nat @ D4 @ M ) ) ) ) ).
% finite_divisors_nat
thf(fact_5545_prod_Osetdiff__irrelevant,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A2: set @ B,G2: B > A] :
( ( finite_finite2 @ B @ A2 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G2
@ ( minus_minus @ ( set @ B ) @ A2
@ ( collect @ B
@ ^ [X2: B] :
( ( G2 @ X2 )
= ( one_one @ A ) ) ) ) )
= ( groups7121269368397514597t_prod @ B @ A @ G2 @ A2 ) ) ) ) ).
% prod.setdiff_irrelevant
thf(fact_5546_sums__If__finite__set,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [A2: set @ nat,F3: nat > A] :
( ( finite_finite2 @ nat @ A2 )
=> ( sums @ A
@ ^ [R5: nat] : ( if @ A @ ( member @ nat @ R5 @ A2 ) @ ( F3 @ R5 ) @ ( zero_zero @ A ) )
@ ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ A2 ) ) ) ) ).
% sums_If_finite_set
thf(fact_5547_sums__If__finite,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [P: nat > $o,F3: nat > A] :
( ( finite_finite2 @ nat @ ( collect @ nat @ P ) )
=> ( sums @ A
@ ^ [R5: nat] : ( if @ A @ ( P @ R5 ) @ ( F3 @ R5 ) @ ( zero_zero @ A ) )
@ ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( collect @ nat @ P ) ) ) ) ) ).
% sums_If_finite
thf(fact_5548_sums__finite,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [N3: set @ nat,F3: nat > A] :
( ( finite_finite2 @ nat @ N3 )
=> ( ! [N2: nat] :
( ~ ( member @ nat @ N2 @ N3 )
=> ( ( F3 @ N2 )
= ( zero_zero @ A ) ) )
=> ( sums @ A @ F3 @ ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ N3 ) ) ) ) ) ).
% sums_finite
thf(fact_5549_suminf__finite,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topological_t2_space @ A ) )
=> ! [N3: set @ nat,F3: nat > A] :
( ( finite_finite2 @ nat @ N3 )
=> ( ! [N2: nat] :
( ~ ( member @ nat @ N2 @ N3 )
=> ( ( F3 @ N2 )
= ( zero_zero @ A ) ) )
=> ( ( suminf @ A @ F3 )
= ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ N3 ) ) ) ) ) ).
% suminf_finite
thf(fact_5550_finite__abs__int__segment,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [A3: A] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [K3: A] :
( ( member @ A @ K3 @ ( ring_1_Ints @ A ) )
& ( ord_less_eq @ A @ ( abs_abs @ A @ K3 ) @ A3 ) ) ) ) ) ).
% finite_abs_int_segment
thf(fact_5551_exp__sum,axiom,
! [B: $tType,A: $tType] :
( ( ( comm_monoid_mult @ B )
& ( real_Vector_banach @ B )
& ( real_V2822296259951069270ebra_1 @ B ) )
=> ! [I6: set @ A,F3: A > B] :
( ( finite_finite2 @ A @ I6 )
=> ( ( exp @ B @ ( groups7311177749621191930dd_sum @ A @ B @ F3 @ I6 ) )
= ( groups7121269368397514597t_prod @ A @ B
@ ^ [X2: A] : ( exp @ B @ ( F3 @ X2 ) )
@ I6 ) ) ) ) ).
% exp_sum
thf(fact_5552_sgn__sdiv__eq__sgn__mult,axiom,
! [A3: int,B3: int] :
( ( ( signed7115095781618012415divide @ int @ A3 @ B3 )
!= ( zero_zero @ int ) )
=> ( ( sgn_sgn @ int @ ( signed7115095781618012415divide @ int @ A3 @ B3 ) )
= ( sgn_sgn @ int @ ( times_times @ int @ A3 @ B3 ) ) ) ) ).
% sgn_sdiv_eq_sgn_mult
thf(fact_5553_sum__pos2,axiom,
! [A: $tType,B: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [I6: set @ B,I: B,F3: B > A] :
( ( finite_finite2 @ B @ I6 )
=> ( ( member @ B @ I @ I6 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ ( F3 @ I ) )
=> ( ! [I3: B] :
( ( member @ B @ I3 @ I6 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F3 @ I3 ) ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ I6 ) ) ) ) ) ) ) ).
% sum_pos2
thf(fact_5554_sum__pos,axiom,
! [A: $tType,B: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [I6: set @ B,F3: B > A] :
( ( finite_finite2 @ B @ I6 )
=> ( ( I6
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [I3: B] :
( ( member @ B @ I3 @ I6 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( F3 @ I3 ) ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ I6 ) ) ) ) ) ) ).
% sum_pos
thf(fact_5555_less__1__prod2,axiom,
! [B: $tType,A: $tType] :
( ( linordered_idom @ B )
=> ! [I6: set @ A,I: A,F3: A > B] :
( ( finite_finite2 @ A @ I6 )
=> ( ( member @ A @ I @ I6 )
=> ( ( ord_less @ B @ ( one_one @ B ) @ ( F3 @ I ) )
=> ( ! [I3: A] :
( ( member @ A @ I3 @ I6 )
=> ( ord_less_eq @ B @ ( one_one @ B ) @ ( F3 @ I3 ) ) )
=> ( ord_less @ B @ ( one_one @ B ) @ ( groups7121269368397514597t_prod @ A @ B @ F3 @ I6 ) ) ) ) ) ) ) ).
% less_1_prod2
thf(fact_5556_less__1__prod,axiom,
! [B: $tType,A: $tType] :
( ( linordered_idom @ B )
=> ! [I6: set @ A,F3: A > B] :
( ( finite_finite2 @ A @ I6 )
=> ( ( I6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [I3: A] :
( ( member @ A @ I3 @ I6 )
=> ( ord_less @ B @ ( one_one @ B ) @ ( F3 @ I3 ) ) )
=> ( ord_less @ B @ ( one_one @ B ) @ ( groups7121269368397514597t_prod @ A @ B @ F3 @ I6 ) ) ) ) ) ) ).
% less_1_prod
thf(fact_5557_sum_Osame__carrier,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [C3: set @ B,A2: set @ B,B2: set @ B,G2: B > A,H2: B > A] :
( ( finite_finite2 @ B @ C3 )
=> ( ( ord_less_eq @ ( set @ B ) @ A2 @ C3 )
=> ( ( ord_less_eq @ ( set @ B ) @ B2 @ C3 )
=> ( ! [A6: B] :
( ( member @ B @ A6 @ ( minus_minus @ ( set @ B ) @ C3 @ A2 ) )
=> ( ( G2 @ A6 )
= ( zero_zero @ A ) ) )
=> ( ! [B7: B] :
( ( member @ B @ B7 @ ( minus_minus @ ( set @ B ) @ C3 @ B2 ) )
=> ( ( H2 @ B7 )
= ( zero_zero @ A ) ) )
=> ( ( ( groups7311177749621191930dd_sum @ B @ A @ G2 @ A2 )
= ( groups7311177749621191930dd_sum @ B @ A @ H2 @ B2 ) )
= ( ( groups7311177749621191930dd_sum @ B @ A @ G2 @ C3 )
= ( groups7311177749621191930dd_sum @ B @ A @ H2 @ C3 ) ) ) ) ) ) ) ) ) ).
% sum.same_carrier
thf(fact_5558_sum_Osame__carrierI,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [C3: set @ B,A2: set @ B,B2: set @ B,G2: B > A,H2: B > A] :
( ( finite_finite2 @ B @ C3 )
=> ( ( ord_less_eq @ ( set @ B ) @ A2 @ C3 )
=> ( ( ord_less_eq @ ( set @ B ) @ B2 @ C3 )
=> ( ! [A6: B] :
( ( member @ B @ A6 @ ( minus_minus @ ( set @ B ) @ C3 @ A2 ) )
=> ( ( G2 @ A6 )
= ( zero_zero @ A ) ) )
=> ( ! [B7: B] :
( ( member @ B @ B7 @ ( minus_minus @ ( set @ B ) @ C3 @ B2 ) )
=> ( ( H2 @ B7 )
= ( zero_zero @ A ) ) )
=> ( ( ( groups7311177749621191930dd_sum @ B @ A @ G2 @ C3 )
= ( groups7311177749621191930dd_sum @ B @ A @ H2 @ C3 ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G2 @ A2 )
= ( groups7311177749621191930dd_sum @ B @ A @ H2 @ B2 ) ) ) ) ) ) ) ) ) ).
% sum.same_carrierI
thf(fact_5559_sum_Omono__neutral__left,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [T4: set @ B,S: set @ B,G2: B > A] :
( ( finite_finite2 @ B @ T4 )
=> ( ( ord_less_eq @ ( set @ B ) @ S @ T4 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T4 @ S ) )
=> ( ( G2 @ X3 )
= ( zero_zero @ A ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G2 @ S )
= ( groups7311177749621191930dd_sum @ B @ A @ G2 @ T4 ) ) ) ) ) ) ).
% sum.mono_neutral_left
thf(fact_5560_sum_Omono__neutral__right,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [T4: set @ B,S: set @ B,G2: B > A] :
( ( finite_finite2 @ B @ T4 )
=> ( ( ord_less_eq @ ( set @ B ) @ S @ T4 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T4 @ S ) )
=> ( ( G2 @ X3 )
= ( zero_zero @ A ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G2 @ T4 )
= ( groups7311177749621191930dd_sum @ B @ A @ G2 @ S ) ) ) ) ) ) ).
% sum.mono_neutral_right
thf(fact_5561_sum_Omono__neutral__cong__left,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [T4: set @ B,S: set @ B,H2: B > A,G2: B > A] :
( ( finite_finite2 @ B @ T4 )
=> ( ( ord_less_eq @ ( set @ B ) @ S @ T4 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T4 @ S ) )
=> ( ( H2 @ X3 )
= ( zero_zero @ A ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ S )
=> ( ( G2 @ X3 )
= ( H2 @ X3 ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G2 @ S )
= ( groups7311177749621191930dd_sum @ B @ A @ H2 @ T4 ) ) ) ) ) ) ) ).
% sum.mono_neutral_cong_left
thf(fact_5562_sum_Omono__neutral__cong__right,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [T4: set @ B,S: set @ B,G2: B > A,H2: B > A] :
( ( finite_finite2 @ B @ T4 )
=> ( ( ord_less_eq @ ( set @ B ) @ S @ T4 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T4 @ S ) )
=> ( ( G2 @ X3 )
= ( zero_zero @ A ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ S )
=> ( ( G2 @ X3 )
= ( H2 @ X3 ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G2 @ T4 )
= ( groups7311177749621191930dd_sum @ B @ A @ H2 @ S ) ) ) ) ) ) ) ).
% sum.mono_neutral_cong_right
thf(fact_5563_prod_Omono__neutral__cong__right,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [T4: set @ B,S: set @ B,G2: B > A,H2: B > A] :
( ( finite_finite2 @ B @ T4 )
=> ( ( ord_less_eq @ ( set @ B ) @ S @ T4 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T4 @ S ) )
=> ( ( G2 @ X3 )
= ( one_one @ A ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ S )
=> ( ( G2 @ X3 )
= ( H2 @ X3 ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G2 @ T4 )
= ( groups7121269368397514597t_prod @ B @ A @ H2 @ S ) ) ) ) ) ) ) ).
% prod.mono_neutral_cong_right
thf(fact_5564_prod_Omono__neutral__cong__left,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [T4: set @ B,S: set @ B,H2: B > A,G2: B > A] :
( ( finite_finite2 @ B @ T4 )
=> ( ( ord_less_eq @ ( set @ B ) @ S @ T4 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T4 @ S ) )
=> ( ( H2 @ X3 )
= ( one_one @ A ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ S )
=> ( ( G2 @ X3 )
= ( H2 @ X3 ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G2 @ S )
= ( groups7121269368397514597t_prod @ B @ A @ H2 @ T4 ) ) ) ) ) ) ) ).
% prod.mono_neutral_cong_left
thf(fact_5565_prod_Omono__neutral__right,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [T4: set @ B,S: set @ B,G2: B > A] :
( ( finite_finite2 @ B @ T4 )
=> ( ( ord_less_eq @ ( set @ B ) @ S @ T4 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T4 @ S ) )
=> ( ( G2 @ X3 )
= ( one_one @ A ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G2 @ T4 )
= ( groups7121269368397514597t_prod @ B @ A @ G2 @ S ) ) ) ) ) ) ).
% prod.mono_neutral_right
thf(fact_5566_prod_Omono__neutral__left,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [T4: set @ B,S: set @ B,G2: B > A] :
( ( finite_finite2 @ B @ T4 )
=> ( ( ord_less_eq @ ( set @ B ) @ S @ T4 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T4 @ S ) )
=> ( ( G2 @ X3 )
= ( one_one @ A ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G2 @ S )
= ( groups7121269368397514597t_prod @ B @ A @ G2 @ T4 ) ) ) ) ) ) ).
% prod.mono_neutral_left
thf(fact_5567_prod_Osame__carrierI,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [C3: set @ B,A2: set @ B,B2: set @ B,G2: B > A,H2: B > A] :
( ( finite_finite2 @ B @ C3 )
=> ( ( ord_less_eq @ ( set @ B ) @ A2 @ C3 )
=> ( ( ord_less_eq @ ( set @ B ) @ B2 @ C3 )
=> ( ! [A6: B] :
( ( member @ B @ A6 @ ( minus_minus @ ( set @ B ) @ C3 @ A2 ) )
=> ( ( G2 @ A6 )
= ( one_one @ A ) ) )
=> ( ! [B7: B] :
( ( member @ B @ B7 @ ( minus_minus @ ( set @ B ) @ C3 @ B2 ) )
=> ( ( H2 @ B7 )
= ( one_one @ A ) ) )
=> ( ( ( groups7121269368397514597t_prod @ B @ A @ G2 @ C3 )
= ( groups7121269368397514597t_prod @ B @ A @ H2 @ C3 ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G2 @ A2 )
= ( groups7121269368397514597t_prod @ B @ A @ H2 @ B2 ) ) ) ) ) ) ) ) ) ).
% prod.same_carrierI
thf(fact_5568_prod_Osame__carrier,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [C3: set @ B,A2: set @ B,B2: set @ B,G2: B > A,H2: B > A] :
( ( finite_finite2 @ B @ C3 )
=> ( ( ord_less_eq @ ( set @ B ) @ A2 @ C3 )
=> ( ( ord_less_eq @ ( set @ B ) @ B2 @ C3 )
=> ( ! [A6: B] :
( ( member @ B @ A6 @ ( minus_minus @ ( set @ B ) @ C3 @ A2 ) )
=> ( ( G2 @ A6 )
= ( one_one @ A ) ) )
=> ( ! [B7: B] :
( ( member @ B @ B7 @ ( minus_minus @ ( set @ B ) @ C3 @ B2 ) )
=> ( ( H2 @ B7 )
= ( one_one @ A ) ) )
=> ( ( ( groups7121269368397514597t_prod @ B @ A @ G2 @ A2 )
= ( groups7121269368397514597t_prod @ B @ A @ H2 @ B2 ) )
= ( ( groups7121269368397514597t_prod @ B @ A @ G2 @ C3 )
= ( groups7121269368397514597t_prod @ B @ A @ H2 @ C3 ) ) ) ) ) ) ) ) ) ).
% prod.same_carrier
thf(fact_5569_infinite__imp__bij__betw,axiom,
! [A: $tType,A2: set @ A,A3: A] :
( ~ ( finite_finite2 @ A @ A2 )
=> ? [H4: A > A] : ( bij_betw @ A @ A @ H4 @ A2 @ ( minus_minus @ ( set @ A ) @ A2 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% infinite_imp_bij_betw
thf(fact_5570_infinite__imp__bij__betw2,axiom,
! [A: $tType,A2: set @ A,A3: A] :
( ~ ( finite_finite2 @ A @ A2 )
=> ? [H4: A > A] : ( bij_betw @ A @ A @ H4 @ A2 @ ( sup_sup @ ( set @ A ) @ A2 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% infinite_imp_bij_betw2
thf(fact_5571_finite__roots__unity,axiom,
! [A: $tType] :
( ( ( real_V8999393235501362500lgebra @ A )
& ( idom @ A ) )
=> ! [N: nat] :
( ( ord_less_eq @ nat @ ( one_one @ nat ) @ N )
=> ( finite_finite2 @ A
@ ( collect @ A
@ ^ [Z4: A] :
( ( power_power @ A @ Z4 @ N )
= ( one_one @ A ) ) ) ) ) ) ).
% finite_roots_unity
thf(fact_5572_sum_Oreindex__bij__betw__not__neutral,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comm_monoid_add @ A )
=> ! [S7: set @ B,T5: set @ C,H2: B > C,S: set @ B,T4: set @ C,G2: C > A] :
( ( finite_finite2 @ B @ S7 )
=> ( ( finite_finite2 @ C @ T5 )
=> ( ( bij_betw @ B @ C @ H2 @ ( minus_minus @ ( set @ B ) @ S @ S7 ) @ ( minus_minus @ ( set @ C ) @ T4 @ T5 ) )
=> ( ! [A6: B] :
( ( member @ B @ A6 @ S7 )
=> ( ( G2 @ ( H2 @ A6 ) )
= ( zero_zero @ A ) ) )
=> ( ! [B7: C] :
( ( member @ C @ B7 @ T5 )
=> ( ( G2 @ B7 )
= ( zero_zero @ A ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X2: B] : ( G2 @ ( H2 @ X2 ) )
@ S )
= ( groups7311177749621191930dd_sum @ C @ A @ G2 @ T4 ) ) ) ) ) ) ) ) ).
% sum.reindex_bij_betw_not_neutral
thf(fact_5573_sums__If__finite__set_H,axiom,
! [A: $tType] :
( ( ( topolo1287966508704411220up_add @ A )
& ( topological_t2_space @ A ) )
=> ! [G2: nat > A,S: A,A2: set @ nat,S7: A,F3: nat > A] :
( ( sums @ A @ G2 @ S )
=> ( ( finite_finite2 @ nat @ A2 )
=> ( ( S7
= ( plus_plus @ A @ S
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [N4: nat] : ( minus_minus @ A @ ( F3 @ N4 ) @ ( G2 @ N4 ) )
@ A2 ) ) )
=> ( sums @ A
@ ^ [N4: nat] : ( if @ A @ ( member @ nat @ N4 @ A2 ) @ ( F3 @ N4 ) @ ( G2 @ N4 ) )
@ S7 ) ) ) ) ) ).
% sums_If_finite_set'
thf(fact_5574_prod_Oreindex__bij__betw__not__neutral,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S7: set @ B,T5: set @ C,H2: B > C,S: set @ B,T4: set @ C,G2: C > A] :
( ( finite_finite2 @ B @ S7 )
=> ( ( finite_finite2 @ C @ T5 )
=> ( ( bij_betw @ B @ C @ H2 @ ( minus_minus @ ( set @ B ) @ S @ S7 ) @ ( minus_minus @ ( set @ C ) @ T4 @ T5 ) )
=> ( ! [A6: B] :
( ( member @ B @ A6 @ S7 )
=> ( ( G2 @ ( H2 @ A6 ) )
= ( one_one @ A ) ) )
=> ( ! [B7: C] :
( ( member @ C @ B7 @ T5 )
=> ( ( G2 @ B7 )
= ( one_one @ A ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [X2: B] : ( G2 @ ( H2 @ X2 ) )
@ S )
= ( groups7121269368397514597t_prod @ C @ A @ G2 @ T4 ) ) ) ) ) ) ) ) ).
% prod.reindex_bij_betw_not_neutral
thf(fact_5575_prod__mono__strict,axiom,
! [A: $tType,B: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: set @ B,F3: B > A,G2: B > A] :
( ( finite_finite2 @ B @ A2 )
=> ( ! [I3: B] :
( ( member @ B @ I3 @ A2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F3 @ I3 ) )
& ( ord_less @ A @ ( F3 @ I3 ) @ ( G2 @ I3 ) ) ) )
=> ( ( A2
!= ( bot_bot @ ( set @ B ) ) )
=> ( ord_less @ A @ ( groups7121269368397514597t_prod @ B @ A @ F3 @ A2 ) @ ( groups7121269368397514597t_prod @ B @ A @ G2 @ A2 ) ) ) ) ) ) ).
% prod_mono_strict
thf(fact_5576_even__prod__iff,axiom,
! [A: $tType,B: $tType] :
( ( semiring_parity @ A )
=> ! [A2: set @ B,F3: B > A] :
( ( finite_finite2 @ B @ A2 )
=> ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( groups7121269368397514597t_prod @ B @ A @ F3 @ A2 ) )
= ( ? [X2: B] :
( ( member @ B @ X2 @ A2 )
& ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( F3 @ X2 ) ) ) ) ) ) ) ).
% even_prod_iff
thf(fact_5577_sum__mono2,axiom,
! [A: $tType,B: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [B2: set @ B,A2: set @ B,F3: B > A] :
( ( finite_finite2 @ B @ B2 )
=> ( ( ord_less_eq @ ( set @ B ) @ A2 @ B2 )
=> ( ! [B7: B] :
( ( member @ B @ B7 @ ( minus_minus @ ( set @ B ) @ B2 @ A2 ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F3 @ B7 ) ) )
=> ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ A2 ) @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ B2 ) ) ) ) ) ) ).
% sum_mono2
thf(fact_5578_sum_Oremove,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A2: set @ B,X: B,G2: B > A] :
( ( finite_finite2 @ B @ A2 )
=> ( ( member @ B @ X @ A2 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G2 @ A2 )
= ( plus_plus @ A @ ( G2 @ X ) @ ( groups7311177749621191930dd_sum @ B @ A @ G2 @ ( minus_minus @ ( set @ B ) @ A2 @ ( insert @ B @ X @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ) ).
% sum.remove
thf(fact_5579_sum_Oinsert__remove,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A2: set @ B,G2: B > A,X: B] :
( ( finite_finite2 @ B @ A2 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G2 @ ( insert @ B @ X @ A2 ) )
= ( plus_plus @ A @ ( G2 @ X ) @ ( groups7311177749621191930dd_sum @ B @ A @ G2 @ ( minus_minus @ ( set @ B ) @ A2 @ ( insert @ B @ X @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ).
% sum.insert_remove
thf(fact_5580_sum__diff1,axiom,
! [A: $tType,B: $tType] :
( ( ab_group_add @ A )
=> ! [A2: set @ B,A3: B,F3: B > A] :
( ( finite_finite2 @ B @ A2 )
=> ( ( ( member @ B @ A3 @ A2 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ F3 @ ( minus_minus @ ( set @ B ) @ A2 @ ( insert @ B @ A3 @ ( bot_bot @ ( set @ B ) ) ) ) )
= ( minus_minus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ A2 ) @ ( F3 @ A3 ) ) ) )
& ( ~ ( member @ B @ A3 @ A2 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ F3 @ ( minus_minus @ ( set @ B ) @ A2 @ ( insert @ B @ A3 @ ( bot_bot @ ( set @ B ) ) ) ) )
= ( groups7311177749621191930dd_sum @ B @ A @ F3 @ A2 ) ) ) ) ) ) ).
% sum_diff1
thf(fact_5581_prod_Oremove,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A2: set @ B,X: B,G2: B > A] :
( ( finite_finite2 @ B @ A2 )
=> ( ( member @ B @ X @ A2 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G2 @ A2 )
= ( times_times @ A @ ( G2 @ X ) @ ( groups7121269368397514597t_prod @ B @ A @ G2 @ ( minus_minus @ ( set @ B ) @ A2 @ ( insert @ B @ X @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ) ).
% prod.remove
thf(fact_5582_prod_Oinsert__remove,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A2: set @ B,G2: B > A,X: B] :
( ( finite_finite2 @ B @ A2 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G2 @ ( insert @ B @ X @ A2 ) )
= ( times_times @ A @ ( G2 @ X ) @ ( groups7121269368397514597t_prod @ B @ A @ G2 @ ( minus_minus @ ( set @ B ) @ A2 @ ( insert @ B @ X @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ).
% prod.insert_remove
thf(fact_5583_sum__le__suminf,axiom,
! [A: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F3: nat > A,I6: set @ nat] :
( ( summable @ A @ F3 )
=> ( ( finite_finite2 @ nat @ I6 )
=> ( ! [N2: nat] :
( ( member @ nat @ N2 @ ( uminus_uminus @ ( set @ nat ) @ I6 ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F3 @ N2 ) ) )
=> ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ I6 ) @ ( suminf @ A @ F3 ) ) ) ) ) ) ).
% sum_le_suminf
thf(fact_5584_sum_Odelta__remove,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [S: set @ B,A3: B,B3: B > A,C2: B > A] :
( ( finite_finite2 @ B @ S )
=> ( ( ( member @ B @ A3 @ S )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [K3: B] : ( if @ A @ ( K3 = A3 ) @ ( B3 @ K3 ) @ ( C2 @ K3 ) )
@ S )
= ( plus_plus @ A @ ( B3 @ A3 ) @ ( groups7311177749621191930dd_sum @ B @ A @ C2 @ ( minus_minus @ ( set @ B ) @ S @ ( insert @ B @ A3 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) )
& ( ~ ( member @ B @ A3 @ S )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [K3: B] : ( if @ A @ ( K3 = A3 ) @ ( B3 @ K3 ) @ ( C2 @ K3 ) )
@ S )
= ( groups7311177749621191930dd_sum @ B @ A @ C2 @ ( minus_minus @ ( set @ B ) @ S @ ( insert @ B @ A3 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ) ).
% sum.delta_remove
thf(fact_5585_prod_Odelta__remove,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S: set @ B,A3: B,B3: B > A,C2: B > A] :
( ( finite_finite2 @ B @ S )
=> ( ( ( member @ B @ A3 @ S )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K3: B] : ( if @ A @ ( K3 = A3 ) @ ( B3 @ K3 ) @ ( C2 @ K3 ) )
@ S )
= ( times_times @ A @ ( B3 @ A3 ) @ ( groups7121269368397514597t_prod @ B @ A @ C2 @ ( minus_minus @ ( set @ B ) @ S @ ( insert @ B @ A3 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) )
& ( ~ ( member @ B @ A3 @ S )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K3: B] : ( if @ A @ ( K3 = A3 ) @ ( B3 @ K3 ) @ ( C2 @ K3 ) )
@ S )
= ( groups7121269368397514597t_prod @ B @ A @ C2 @ ( minus_minus @ ( set @ B ) @ S @ ( insert @ B @ A3 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ) ).
% prod.delta_remove
thf(fact_5586_CauchyD,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X5: nat > A,E2: real] :
( ( topolo3814608138187158403Cauchy @ A @ X5 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ E2 )
=> ? [M9: nat] :
! [M2: nat] :
( ( ord_less_eq @ nat @ M9 @ M2 )
=> ! [N11: nat] :
( ( ord_less_eq @ nat @ M9 @ N11 )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( X5 @ M2 ) @ ( X5 @ N11 ) ) ) @ E2 ) ) ) ) ) ) ).
% CauchyD
thf(fact_5587_CauchyI,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X5: nat > A] :
( ! [E: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E )
=> ? [M11: nat] :
! [M4: nat] :
( ( ord_less_eq @ nat @ M11 @ M4 )
=> ! [N2: nat] :
( ( ord_less_eq @ nat @ M11 @ N2 )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( X5 @ M4 ) @ ( X5 @ N2 ) ) ) @ E ) ) ) )
=> ( topolo3814608138187158403Cauchy @ A @ X5 ) ) ) ).
% CauchyI
thf(fact_5588_Cauchy__iff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ( ( topolo3814608138187158403Cauchy @ A )
= ( ^ [X8: nat > A] :
! [E4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E4 )
=> ? [M10: nat] :
! [M3: nat] :
( ( ord_less_eq @ nat @ M10 @ M3 )
=> ! [N4: nat] :
( ( ord_less_eq @ nat @ M10 @ N4 )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( X8 @ M3 ) @ ( X8 @ N4 ) ) ) @ E4 ) ) ) ) ) ) ) ).
% Cauchy_iff
thf(fact_5589_sum__strict__mono2,axiom,
! [B: $tType,A: $tType] :
( ( ordere8940638589300402666id_add @ B )
=> ! [B2: set @ A,A2: set @ A,B3: A,F3: A > B] :
( ( finite_finite2 @ A @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ( member @ A @ B3 @ ( minus_minus @ ( set @ A ) @ B2 @ A2 ) )
=> ( ( ord_less @ B @ ( zero_zero @ B ) @ ( F3 @ B3 ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ B2 )
=> ( ord_less_eq @ B @ ( zero_zero @ B ) @ ( F3 @ X3 ) ) )
=> ( ord_less @ B @ ( groups7311177749621191930dd_sum @ A @ B @ F3 @ A2 ) @ ( groups7311177749621191930dd_sum @ A @ B @ F3 @ B2 ) ) ) ) ) ) ) ) ).
% sum_strict_mono2
thf(fact_5590_member__le__sum,axiom,
! [B: $tType,C: $tType] :
( ( ( ordere6911136660526730532id_add @ B )
& ( semiring_1 @ B ) )
=> ! [I: C,A2: set @ C,F3: C > B] :
( ( member @ C @ I @ A2 )
=> ( ! [X3: C] :
( ( member @ C @ X3 @ ( minus_minus @ ( set @ C ) @ A2 @ ( insert @ C @ I @ ( bot_bot @ ( set @ C ) ) ) ) )
=> ( ord_less_eq @ B @ ( zero_zero @ B ) @ ( F3 @ X3 ) ) )
=> ( ( finite_finite2 @ C @ A2 )
=> ( ord_less_eq @ B @ ( F3 @ I ) @ ( groups7311177749621191930dd_sum @ C @ B @ F3 @ A2 ) ) ) ) ) ) ).
% member_le_sum
thf(fact_5591_prod__mono2,axiom,
! [B: $tType,A: $tType] :
( ( linordered_idom @ B )
=> ! [B2: set @ A,A2: set @ A,F3: A > B] :
( ( finite_finite2 @ A @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ! [B7: A] :
( ( member @ A @ B7 @ ( minus_minus @ ( set @ A ) @ B2 @ A2 ) )
=> ( ord_less_eq @ B @ ( one_one @ B ) @ ( F3 @ B7 ) ) )
=> ( ! [A6: A] :
( ( member @ A @ A6 @ A2 )
=> ( ord_less_eq @ B @ ( zero_zero @ B ) @ ( F3 @ A6 ) ) )
=> ( ord_less_eq @ B @ ( groups7121269368397514597t_prod @ A @ B @ F3 @ A2 ) @ ( groups7121269368397514597t_prod @ A @ B @ F3 @ B2 ) ) ) ) ) ) ) ).
% prod_mono2
thf(fact_5592_prod__diff1,axiom,
! [A: $tType,B: $tType] :
( ( semidom_divide @ A )
=> ! [A2: set @ B,F3: B > A,A3: B] :
( ( finite_finite2 @ B @ A2 )
=> ( ( ( F3 @ A3 )
!= ( zero_zero @ A ) )
=> ( ( ( member @ B @ A3 @ A2 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ F3 @ ( minus_minus @ ( set @ B ) @ A2 @ ( insert @ B @ A3 @ ( bot_bot @ ( set @ B ) ) ) ) )
= ( divide_divide @ A @ ( groups7121269368397514597t_prod @ B @ A @ F3 @ A2 ) @ ( F3 @ A3 ) ) ) )
& ( ~ ( member @ B @ A3 @ A2 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ F3 @ ( minus_minus @ ( set @ B ) @ A2 @ ( insert @ B @ A3 @ ( bot_bot @ ( set @ B ) ) ) ) )
= ( groups7121269368397514597t_prod @ B @ A @ F3 @ A2 ) ) ) ) ) ) ) ).
% prod_diff1
thf(fact_5593_even__set__encode__iff,axiom,
! [A2: set @ nat] :
( ( finite_finite2 @ nat @ A2 )
=> ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( nat_set_encode @ A2 ) )
= ( ~ ( member @ nat @ ( zero_zero @ nat ) @ A2 ) ) ) ) ).
% even_set_encode_iff
thf(fact_5594_ln__prod,axiom,
! [A: $tType,I6: set @ A,F3: A > real] :
( ( finite_finite2 @ A @ I6 )
=> ( ! [I3: A] :
( ( member @ A @ I3 @ I6 )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( F3 @ I3 ) ) )
=> ( ( ln_ln @ real @ ( groups7121269368397514597t_prod @ A @ real @ F3 @ I6 ) )
= ( groups7311177749621191930dd_sum @ A @ real
@ ^ [X2: A] : ( ln_ln @ real @ ( F3 @ X2 ) )
@ I6 ) ) ) ) ).
% ln_prod
thf(fact_5595_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
@ ^ [Z4: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( C2 @ I4 ) @ ( power_power @ A @ Z4 @ I4 ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% polyfun_roots_finite
thf(fact_5596_polyfun__finite__roots,axiom,
! [A: $tType] :
( ( ( real_V8999393235501362500lgebra @ A )
& ( idom @ A ) )
=> ! [C2: nat > A,N: nat] :
( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X2: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( C2 @ I4 ) @ ( power_power @ A @ X2 @ I4 ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( zero_zero @ A ) ) ) )
= ( ? [I4: nat] :
( ( ord_less_eq @ nat @ I4 @ N )
& ( ( C2 @ I4 )
!= ( zero_zero @ A ) ) ) ) ) ) ).
% polyfun_finite_roots
thf(fact_5597_sdiv__word__max,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A,B3: word @ A] : ( ord_less_eq @ int @ ( signed7115095781618012415divide @ int @ ( ring_1_signed @ A @ int @ A3 ) @ ( ring_1_signed @ A @ int @ B3 ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ ( size_size @ ( word @ A ) @ A3 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ).
% sdiv_word_max
thf(fact_5598_finite__Diff__insert,axiom,
! [A: $tType,A2: set @ A,A3: A,B2: set @ A] :
( ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ A2 @ ( insert @ A @ A3 @ B2 ) ) )
= ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ A2 @ B2 ) ) ) ).
% finite_Diff_insert
thf(fact_5599_finite__Collect__le__nat,axiom,
! [K: nat] :
( finite_finite2 @ nat
@ ( collect @ nat
@ ^ [N4: nat] : ( ord_less_eq @ nat @ N4 @ K ) ) ) ).
% finite_Collect_le_nat
thf(fact_5600_finite__Collect__less__nat,axiom,
! [K: nat] :
( finite_finite2 @ nat
@ ( collect @ nat
@ ^ [N4: nat] : ( ord_less @ nat @ N4 @ K ) ) ) ).
% finite_Collect_less_nat
thf(fact_5601_finite__Collect__disjI,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X2: A] :
( ( P @ X2 )
| ( Q @ X2 ) ) ) )
= ( ( finite_finite2 @ A @ ( collect @ A @ P ) )
& ( finite_finite2 @ A @ ( collect @ A @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_5602_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
@ ^ [X2: A] :
( ( P @ X2 )
& ( Q @ X2 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_5603_finite__interval__int1,axiom,
! [A3: int,B3: int] :
( finite_finite2 @ int
@ ( collect @ int
@ ^ [I4: int] :
( ( ord_less_eq @ int @ A3 @ I4 )
& ( ord_less_eq @ int @ I4 @ B3 ) ) ) ) ).
% finite_interval_int1
thf(fact_5604_finite__interval__int4,axiom,
! [A3: int,B3: int] :
( finite_finite2 @ int
@ ( collect @ int
@ ^ [I4: int] :
( ( ord_less @ int @ A3 @ I4 )
& ( ord_less @ int @ I4 @ B3 ) ) ) ) ).
% finite_interval_int4
thf(fact_5605_finite__insert,axiom,
! [A: $tType,A3: A,A2: set @ A] :
( ( finite_finite2 @ A @ ( insert @ A @ A3 @ A2 ) )
= ( finite_finite2 @ A @ A2 ) ) ).
% finite_insert
thf(fact_5606_finite__Un,axiom,
! [A: $tType,F4: set @ A,G4: set @ A] :
( ( finite_finite2 @ A @ ( sup_sup @ ( set @ A ) @ F4 @ G4 ) )
= ( ( finite_finite2 @ A @ F4 )
& ( finite_finite2 @ A @ G4 ) ) ) ).
% finite_Un
thf(fact_5607_finite__interval__int3,axiom,
! [A3: int,B3: int] :
( finite_finite2 @ int
@ ( collect @ int
@ ^ [I4: int] :
( ( ord_less @ int @ A3 @ I4 )
& ( ord_less_eq @ int @ I4 @ B3 ) ) ) ) ).
% finite_interval_int3
thf(fact_5608_finite__interval__int2,axiom,
! [A3: int,B3: int] :
( finite_finite2 @ int
@ ( collect @ int
@ ^ [I4: int] :
( ( ord_less_eq @ int @ A3 @ I4 )
& ( ord_less @ int @ I4 @ B3 ) ) ) ) ).
% finite_interval_int2
thf(fact_5609_finite__nth__roots,axiom,
! [N: nat,C2: complex] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( finite_finite2 @ complex
@ ( collect @ complex
@ ^ [Z4: complex] :
( ( power_power @ complex @ Z4 @ N )
= C2 ) ) ) ) ).
% finite_nth_roots
thf(fact_5610_finite__Collect__subsets,axiom,
! [A: $tType,A2: set @ A] :
( ( finite_finite2 @ A @ A2 )
=> ( finite_finite2 @ ( set @ A )
@ ( collect @ ( set @ A )
@ ^ [B4: set @ A] : ( ord_less_eq @ ( set @ A ) @ B4 @ A2 ) ) ) ) ).
% finite_Collect_subsets
thf(fact_5611_mset__le__asym,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [M7: multiset @ A,N3: multiset @ A] :
( ( ord_less @ ( multiset @ A ) @ M7 @ N3 )
=> ~ ( ord_less @ ( multiset @ A ) @ N3 @ M7 ) ) ) ).
% mset_le_asym
thf(fact_5612_mset__le__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [K4: multiset @ A,M7: multiset @ A,N3: multiset @ A] :
( ( ord_less @ ( multiset @ A ) @ K4 @ M7 )
=> ( ( ord_less @ ( multiset @ A ) @ M7 @ N3 )
=> ( ord_less @ ( multiset @ A ) @ K4 @ N3 ) ) ) ) ).
% mset_le_trans
thf(fact_5613_mset__le__irrefl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [M7: multiset @ A] :
~ ( ord_less @ ( multiset @ A ) @ M7 @ M7 ) ) ).
% mset_le_irrefl
thf(fact_5614_mset__le__not__sym,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [M7: multiset @ A,N3: multiset @ A] :
( ( ord_less @ ( multiset @ A ) @ M7 @ N3 )
=> ~ ( ord_less @ ( multiset @ A ) @ N3 @ M7 ) ) ) ).
% mset_le_not_sym
thf(fact_5615_mset__le__not__refl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [M7: multiset @ A] :
~ ( ord_less @ ( multiset @ A ) @ M7 @ M7 ) ) ).
% mset_le_not_refl
thf(fact_5616_less__eq__multiset__def,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( ( ord_less_eq @ ( multiset @ A ) )
= ( ^ [M10: multiset @ A,N9: multiset @ A] :
( ( ord_less @ ( multiset @ A ) @ M10 @ N9 )
| ( M10 = N9 ) ) ) ) ) ).
% less_eq_multiset_def
thf(fact_5617_finite__maxlen,axiom,
! [A: $tType,M7: set @ ( list @ A )] :
( ( finite_finite2 @ ( list @ A ) @ M7 )
=> ? [N2: nat] :
! [X6: list @ A] :
( ( member @ ( list @ A ) @ X6 @ M7 )
=> ( ord_less @ nat @ ( size_size @ ( list @ A ) @ X6 ) @ N2 ) ) ) ).
% finite_maxlen
thf(fact_5618_finite__divisors__int,axiom,
! [I: int] :
( ( I
!= ( zero_zero @ int ) )
=> ( finite_finite2 @ int
@ ( collect @ int
@ ^ [D4: int] : ( dvd_dvd @ int @ D4 @ I ) ) ) ) ).
% finite_divisors_int
thf(fact_5619_pigeonhole__infinite__rel,axiom,
! [B: $tType,A: $tType,A2: set @ A,B2: set @ B,R: A > B > $o] :
( ~ ( finite_finite2 @ A @ A2 )
=> ( ( finite_finite2 @ B @ B2 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ A2 )
=> ? [Xa2: B] :
( ( member @ B @ Xa2 @ B2 )
& ( R @ X3 @ Xa2 ) ) )
=> ? [X3: B] :
( ( member @ B @ X3 @ B2 )
& ~ ( finite_finite2 @ A
@ ( collect @ A
@ ^ [A5: A] :
( ( member @ A @ A5 @ A2 )
& ( R @ A5 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_5620_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_5621_infinite__imp__nonempty,axiom,
! [A: $tType,S: set @ A] :
( ~ ( finite_finite2 @ A @ S )
=> ( S
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% infinite_imp_nonempty
thf(fact_5622_finite_OemptyI,axiom,
! [A: $tType] : ( finite_finite2 @ A @ ( bot_bot @ ( set @ A ) ) ) ).
% finite.emptyI
thf(fact_5623_finite_OinsertI,axiom,
! [A: $tType,A2: set @ A,A3: A] :
( ( finite_finite2 @ A @ A2 )
=> ( finite_finite2 @ A @ ( insert @ A @ A3 @ A2 ) ) ) ).
% finite.insertI
thf(fact_5624_finite__UnI,axiom,
! [A: $tType,F4: set @ A,G4: set @ A] :
( ( finite_finite2 @ A @ F4 )
=> ( ( finite_finite2 @ A @ G4 )
=> ( finite_finite2 @ A @ ( sup_sup @ ( set @ A ) @ F4 @ G4 ) ) ) ) ).
% finite_UnI
thf(fact_5625_Un__infinite,axiom,
! [A: $tType,S: set @ A,T4: set @ A] :
( ~ ( finite_finite2 @ A @ S )
=> ~ ( finite_finite2 @ A @ ( sup_sup @ ( set @ A ) @ S @ T4 ) ) ) ).
% Un_infinite
thf(fact_5626_infinite__Un,axiom,
! [A: $tType,S: set @ A,T4: set @ A] :
( ( ~ ( finite_finite2 @ A @ ( sup_sup @ ( set @ A ) @ S @ T4 ) ) )
= ( ~ ( finite_finite2 @ A @ S )
| ~ ( finite_finite2 @ A @ T4 ) ) ) ).
% infinite_Un
thf(fact_5627_finite__psubset__induct,axiom,
! [A: $tType,A2: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ A2 )
=> ( ! [A10: set @ A] :
( ( finite_finite2 @ A @ A10 )
=> ( ! [B10: set @ A] :
( ( ord_less @ ( set @ A ) @ B10 @ A10 )
=> ( P @ B10 ) )
=> ( P @ A10 ) ) )
=> ( P @ A2 ) ) ) ).
% finite_psubset_induct
thf(fact_5628_finite__has__maximal,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: set @ A] :
( ( finite_finite2 @ A @ A2 )
=> ( ( A2
!= ( bot_bot @ ( set @ A ) ) )
=> ? [X3: A] :
( ( member @ A @ X3 @ A2 )
& ! [Xa2: A] :
( ( member @ A @ Xa2 @ A2 )
=> ( ( ord_less_eq @ A @ X3 @ Xa2 )
=> ( X3 = Xa2 ) ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_5629_finite__has__minimal,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: set @ A] :
( ( finite_finite2 @ A @ A2 )
=> ( ( A2
!= ( bot_bot @ ( set @ A ) ) )
=> ? [X3: A] :
( ( member @ A @ X3 @ A2 )
& ! [Xa2: A] :
( ( member @ A @ Xa2 @ A2 )
=> ( ( ord_less_eq @ A @ Xa2 @ X3 )
=> ( X3 = Xa2 ) ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_5630_finite_Ocases,axiom,
! [A: $tType,A3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ~ ! [A10: set @ A] :
( ? [A6: A] :
( A3
= ( insert @ A @ A6 @ A10 ) )
=> ~ ( finite_finite2 @ A @ A10 ) ) ) ) ).
% finite.cases
thf(fact_5631_finite_Osimps,axiom,
! [A: $tType] :
( ( finite_finite2 @ A )
= ( ^ [A5: set @ A] :
( ( A5
= ( bot_bot @ ( set @ A ) ) )
| ? [A4: set @ A,B5: A] :
( ( A5
= ( insert @ A @ B5 @ A4 ) )
& ( finite_finite2 @ A @ A4 ) ) ) ) ) ).
% finite.simps
thf(fact_5632_finite__induct,axiom,
! [A: $tType,F4: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ F4 )
=> ( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A,F6: set @ A] :
( ( finite_finite2 @ A @ F6 )
=> ( ~ ( member @ A @ X3 @ F6 )
=> ( ( P @ F6 )
=> ( P @ ( insert @ A @ X3 @ F6 ) ) ) ) )
=> ( P @ F4 ) ) ) ) ).
% finite_induct
thf(fact_5633_finite__ne__induct,axiom,
! [A: $tType,F4: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ F4 )
=> ( ( F4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A] : ( P @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
=> ( ! [X3: A,F6: set @ A] :
( ( finite_finite2 @ A @ F6 )
=> ( ( F6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ~ ( member @ A @ X3 @ F6 )
=> ( ( P @ F6 )
=> ( P @ ( insert @ A @ X3 @ F6 ) ) ) ) ) )
=> ( P @ F4 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_5634_infinite__finite__induct,axiom,
! [A: $tType,P: ( set @ A ) > $o,A2: set @ A] :
( ! [A10: set @ A] :
( ~ ( finite_finite2 @ A @ A10 )
=> ( P @ A10 ) )
=> ( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A,F6: set @ A] :
( ( finite_finite2 @ A @ F6 )
=> ( ~ ( member @ A @ X3 @ F6 )
=> ( ( P @ F6 )
=> ( P @ ( insert @ A @ X3 @ F6 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% infinite_finite_induct
thf(fact_5635_finite__subset__induct_H,axiom,
! [A: $tType,F4: set @ A,A2: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ F4 )
=> ( ( ord_less_eq @ ( set @ A ) @ F4 @ A2 )
=> ( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [A6: A,F6: set @ A] :
( ( finite_finite2 @ A @ F6 )
=> ( ( member @ A @ A6 @ A2 )
=> ( ( ord_less_eq @ ( set @ A ) @ F6 @ A2 )
=> ( ~ ( member @ A @ A6 @ F6 )
=> ( ( P @ F6 )
=> ( P @ ( insert @ A @ A6 @ F6 ) ) ) ) ) ) )
=> ( P @ F4 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_5636_finite__subset__induct,axiom,
! [A: $tType,F4: set @ A,A2: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ F4 )
=> ( ( ord_less_eq @ ( set @ A ) @ F4 @ A2 )
=> ( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [A6: A,F6: set @ A] :
( ( finite_finite2 @ A @ F6 )
=> ( ( member @ A @ A6 @ A2 )
=> ( ~ ( member @ A @ A6 @ F6 )
=> ( ( P @ F6 )
=> ( P @ ( insert @ A @ A6 @ F6 ) ) ) ) ) )
=> ( P @ F4 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_5637_finite__empty__induct,axiom,
! [A: $tType,A2: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ A2 )
=> ( ( P @ A2 )
=> ( ! [A6: A,A10: set @ A] :
( ( finite_finite2 @ A @ A10 )
=> ( ( member @ A @ A6 @ A10 )
=> ( ( P @ A10 )
=> ( P @ ( minus_minus @ ( set @ A ) @ A10 @ ( insert @ A @ A6 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) )
=> ( P @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% finite_empty_induct
thf(fact_5638_infinite__coinduct,axiom,
! [A: $tType,X5: ( set @ A ) > $o,A2: set @ A] :
( ( X5 @ A2 )
=> ( ! [A10: set @ A] :
( ( X5 @ A10 )
=> ? [X6: A] :
( ( member @ A @ X6 @ A10 )
& ( ( X5 @ ( minus_minus @ ( set @ A ) @ A10 @ ( insert @ A @ X6 @ ( bot_bot @ ( set @ A ) ) ) ) )
| ~ ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ A10 @ ( insert @ A @ X6 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) )
=> ~ ( finite_finite2 @ A @ A2 ) ) ) ).
% infinite_coinduct
thf(fact_5639_infinite__remove,axiom,
! [A: $tType,S: set @ A,A3: A] :
( ~ ( finite_finite2 @ A @ S )
=> ~ ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ S @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% infinite_remove
thf(fact_5640_finite__remove__induct,axiom,
! [A: $tType,B2: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ B2 )
=> ( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [A10: set @ A] :
( ( finite_finite2 @ A @ A10 )
=> ( ( A10
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ A ) @ A10 @ B2 )
=> ( ! [X6: A] :
( ( member @ A @ X6 @ A10 )
=> ( P @ ( minus_minus @ ( set @ A ) @ A10 @ ( insert @ A @ X6 @ ( bot_bot @ ( set @ A ) ) ) ) ) )
=> ( P @ A10 ) ) ) ) )
=> ( P @ B2 ) ) ) ) ).
% finite_remove_induct
thf(fact_5641_remove__induct,axiom,
! [A: $tType,P: ( set @ A ) > $o,B2: set @ A] :
( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ( ~ ( finite_finite2 @ A @ B2 )
=> ( P @ B2 ) )
=> ( ! [A10: set @ A] :
( ( finite_finite2 @ A @ A10 )
=> ( ( A10
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ A ) @ A10 @ B2 )
=> ( ! [X6: A] :
( ( member @ A @ X6 @ A10 )
=> ( P @ ( minus_minus @ ( set @ A ) @ A10 @ ( insert @ A @ X6 @ ( bot_bot @ ( set @ A ) ) ) ) ) )
=> ( P @ A10 ) ) ) ) )
=> ( P @ B2 ) ) ) ) ).
% remove_induct
thf(fact_5642_finite__induct__select,axiom,
! [A: $tType,S: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ S )
=> ( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [T6: set @ A] :
( ( ord_less @ ( set @ A ) @ T6 @ S )
=> ( ( P @ T6 )
=> ? [X6: A] :
( ( member @ A @ X6 @ ( minus_minus @ ( set @ A ) @ S @ T6 ) )
& ( P @ ( insert @ A @ X6 @ T6 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_induct_select
thf(fact_5643_sum__count__set,axiom,
! [A: $tType,Xs: list @ A,X5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ X5 )
=> ( ( finite_finite2 @ A @ X5 )
=> ( ( groups7311177749621191930dd_sum @ A @ nat @ ( count_list @ A @ Xs ) @ X5 )
= ( size_size @ ( list @ A ) @ Xs ) ) ) ) ).
% sum_count_set
thf(fact_5644_sum__diff1_H__aux,axiom,
! [B: $tType,A: $tType] :
( ( ab_group_add @ B )
=> ! [F4: set @ A,I6: set @ A,F3: A > B,I: A] :
( ( finite_finite2 @ A @ F4 )
=> ( ( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [I4: A] :
( ( member @ A @ I4 @ I6 )
& ( ( F3 @ I4 )
!= ( zero_zero @ B ) ) ) )
@ F4 )
=> ( ( ( member @ A @ I @ I6 )
=> ( ( groups1027152243600224163dd_sum @ A @ B @ F3 @ ( minus_minus @ ( set @ A ) @ I6 @ ( insert @ A @ I @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( minus_minus @ B @ ( groups1027152243600224163dd_sum @ A @ B @ F3 @ I6 ) @ ( F3 @ I ) ) ) )
& ( ~ ( member @ A @ I @ I6 )
=> ( ( groups1027152243600224163dd_sum @ A @ B @ F3 @ ( minus_minus @ ( set @ A ) @ I6 @ ( insert @ A @ I @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( groups1027152243600224163dd_sum @ A @ B @ F3 @ I6 ) ) ) ) ) ) ) ).
% sum_diff1'_aux
thf(fact_5645_int__of__nat__def,axiom,
( code_T6385005292777649522of_nat
= ( semiring_1_of_nat @ int ) ) ).
% int_of_nat_def
thf(fact_5646_sum_Oempty_H,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add @ A )
=> ! [P4: B > A] :
( ( groups1027152243600224163dd_sum @ B @ A @ P4 @ ( bot_bot @ ( set @ B ) ) )
= ( zero_zero @ A ) ) ) ).
% sum.empty'
thf(fact_5647_count__notin,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ~ ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ( count_list @ A @ Xs @ X )
= ( zero_zero @ nat ) ) ) ).
% count_notin
thf(fact_5648_sum_Oinsert_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [I6: set @ B,P4: B > A,I: B] :
( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X2: B] :
( ( member @ B @ X2 @ I6 )
& ( ( P4 @ X2 )
!= ( zero_zero @ A ) ) ) ) )
=> ( ( ( member @ B @ I @ I6 )
=> ( ( groups1027152243600224163dd_sum @ B @ A @ P4 @ ( insert @ B @ I @ I6 ) )
= ( groups1027152243600224163dd_sum @ B @ A @ P4 @ I6 ) ) )
& ( ~ ( member @ B @ I @ I6 )
=> ( ( groups1027152243600224163dd_sum @ B @ A @ P4 @ ( insert @ B @ I @ I6 ) )
= ( plus_plus @ A @ ( P4 @ I ) @ ( groups1027152243600224163dd_sum @ B @ A @ P4 @ I6 ) ) ) ) ) ) ) ).
% sum.insert'
thf(fact_5649_sum_Onon__neutral_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [G2: B > A,I6: set @ B] :
( ( groups1027152243600224163dd_sum @ B @ A @ G2
@ ( collect @ B
@ ^ [X2: B] :
( ( member @ B @ X2 @ I6 )
& ( ( G2 @ X2 )
!= ( zero_zero @ A ) ) ) ) )
= ( groups1027152243600224163dd_sum @ B @ A @ G2 @ I6 ) ) ) ).
% sum.non_neutral'
thf(fact_5650_sum_Odistrib__triv_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [I6: set @ B,G2: B > A,H2: B > A] :
( ( finite_finite2 @ B @ I6 )
=> ( ( groups1027152243600224163dd_sum @ B @ A
@ ^ [I4: B] : ( plus_plus @ A @ ( G2 @ I4 ) @ ( H2 @ I4 ) )
@ I6 )
= ( plus_plus @ A @ ( groups1027152243600224163dd_sum @ B @ A @ G2 @ I6 ) @ ( groups1027152243600224163dd_sum @ B @ A @ H2 @ I6 ) ) ) ) ) ).
% sum.distrib_triv'
thf(fact_5651_sum_Omono__neutral__cong__right_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [S: set @ B,T4: set @ B,G2: B > A,H2: B > A] :
( ( ord_less_eq @ ( set @ B ) @ S @ T4 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T4 @ S ) )
=> ( ( G2 @ X3 )
= ( zero_zero @ A ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ S )
=> ( ( G2 @ X3 )
= ( H2 @ X3 ) ) )
=> ( ( groups1027152243600224163dd_sum @ B @ A @ G2 @ T4 )
= ( groups1027152243600224163dd_sum @ B @ A @ H2 @ S ) ) ) ) ) ) ).
% sum.mono_neutral_cong_right'
thf(fact_5652_sum_Omono__neutral__cong__left_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [S: set @ B,T4: set @ B,H2: B > A,G2: B > A] :
( ( ord_less_eq @ ( set @ B ) @ S @ T4 )
=> ( ! [I3: B] :
( ( member @ B @ I3 @ ( minus_minus @ ( set @ B ) @ T4 @ S ) )
=> ( ( H2 @ I3 )
= ( zero_zero @ A ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ S )
=> ( ( G2 @ X3 )
= ( H2 @ X3 ) ) )
=> ( ( groups1027152243600224163dd_sum @ B @ A @ G2 @ S )
= ( groups1027152243600224163dd_sum @ B @ A @ H2 @ T4 ) ) ) ) ) ) ).
% sum.mono_neutral_cong_left'
thf(fact_5653_sum_Omono__neutral__right_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [S: set @ B,T4: set @ B,G2: B > A] :
( ( ord_less_eq @ ( set @ B ) @ S @ T4 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T4 @ S ) )
=> ( ( G2 @ X3 )
= ( zero_zero @ A ) ) )
=> ( ( groups1027152243600224163dd_sum @ B @ A @ G2 @ T4 )
= ( groups1027152243600224163dd_sum @ B @ A @ G2 @ S ) ) ) ) ) ).
% sum.mono_neutral_right'
thf(fact_5654_sum_Omono__neutral__left_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [S: set @ B,T4: set @ B,G2: B > A] :
( ( ord_less_eq @ ( set @ B ) @ S @ T4 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T4 @ S ) )
=> ( ( G2 @ X3 )
= ( zero_zero @ A ) ) )
=> ( ( groups1027152243600224163dd_sum @ B @ A @ G2 @ S )
= ( groups1027152243600224163dd_sum @ B @ A @ G2 @ T4 ) ) ) ) ) ).
% sum.mono_neutral_left'
thf(fact_5655_sum_Odistrib_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [I6: set @ B,G2: B > A,H2: B > A] :
( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X2: B] :
( ( member @ B @ X2 @ I6 )
& ( ( G2 @ X2 )
!= ( zero_zero @ A ) ) ) ) )
=> ( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X2: B] :
( ( member @ B @ X2 @ I6 )
& ( ( H2 @ X2 )
!= ( zero_zero @ A ) ) ) ) )
=> ( ( groups1027152243600224163dd_sum @ B @ A
@ ^ [I4: B] : ( plus_plus @ A @ ( G2 @ I4 ) @ ( H2 @ I4 ) )
@ I6 )
= ( plus_plus @ A @ ( groups1027152243600224163dd_sum @ B @ A @ G2 @ I6 ) @ ( groups1027152243600224163dd_sum @ B @ A @ H2 @ I6 ) ) ) ) ) ) ).
% sum.distrib'
thf(fact_5656_sum_OG__def,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ( ( groups1027152243600224163dd_sum @ B @ A )
= ( ^ [P6: B > A,I7: set @ B] :
( if @ A
@ ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X2: B] :
( ( member @ B @ X2 @ I7 )
& ( ( P6 @ X2 )
!= ( zero_zero @ A ) ) ) ) )
@ ( groups7311177749621191930dd_sum @ B @ A @ P6
@ ( collect @ B
@ ^ [X2: B] :
( ( member @ B @ X2 @ I7 )
& ( ( P6 @ X2 )
!= ( zero_zero @ A ) ) ) ) )
@ ( zero_zero @ A ) ) ) ) ) ).
% sum.G_def
thf(fact_5657_count__le__length,axiom,
! [A: $tType,Xs: list @ A,X: A] : ( ord_less_eq @ nat @ ( count_list @ A @ Xs @ X ) @ ( size_size @ ( list @ A ) @ Xs ) ) ).
% count_le_length
thf(fact_5658_sum__diff1_H,axiom,
! [B: $tType,A: $tType] :
( ( ab_group_add @ B )
=> ! [I6: set @ A,F3: A > B,I: A] :
( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [I4: A] :
( ( member @ A @ I4 @ I6 )
& ( ( F3 @ I4 )
!= ( zero_zero @ B ) ) ) ) )
=> ( ( ( member @ A @ I @ I6 )
=> ( ( groups1027152243600224163dd_sum @ A @ B @ F3 @ ( minus_minus @ ( set @ A ) @ I6 @ ( insert @ A @ I @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( minus_minus @ B @ ( groups1027152243600224163dd_sum @ A @ B @ F3 @ I6 ) @ ( F3 @ I ) ) ) )
& ( ~ ( member @ A @ I @ I6 )
=> ( ( groups1027152243600224163dd_sum @ A @ B @ F3 @ ( minus_minus @ ( set @ A ) @ I6 @ ( insert @ A @ I @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( groups1027152243600224163dd_sum @ A @ B @ F3 @ I6 ) ) ) ) ) ) ).
% sum_diff1'
thf(fact_5659_infinite__int__iff__unbounded,axiom,
! [S: set @ int] :
( ( ~ ( finite_finite2 @ int @ S ) )
= ( ! [M3: int] :
? [N4: int] :
( ( ord_less @ int @ M3 @ ( abs_abs @ int @ N4 ) )
& ( member @ int @ N4 @ S ) ) ) ) ).
% infinite_int_iff_unbounded
thf(fact_5660_Sum__Ico__nat,axiom,
! [M: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [X2: nat] : X2
@ ( set_or7035219750837199246ssThan @ nat @ M @ N ) )
= ( divide_divide @ nat @ ( minus_minus @ nat @ ( times_times @ nat @ N @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) @ ( times_times @ nat @ M @ ( minus_minus @ nat @ M @ ( one_one @ nat ) ) ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% Sum_Ico_nat
thf(fact_5661_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_5662_atLeastLessThan__iff,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [I: A,L: A,U2: A] :
( ( member @ A @ I @ ( set_or7035219750837199246ssThan @ A @ L @ U2 ) )
= ( ( ord_less_eq @ A @ L @ I )
& ( ord_less @ A @ I @ U2 ) ) ) ) ).
% atLeastLessThan_iff
thf(fact_5663_atLeastLessThan__empty,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B3: A,A3: A] :
( ( ord_less_eq @ A @ B3 @ A3 )
=> ( ( set_or7035219750837199246ssThan @ A @ A3 @ B3 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% atLeastLessThan_empty
thf(fact_5664_atLeastLessThan__empty__iff,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: A,B3: A] :
( ( ( set_or7035219750837199246ssThan @ A @ A3 @ B3 )
= ( bot_bot @ ( set @ A ) ) )
= ( ~ ( ord_less @ A @ A3 @ B3 ) ) ) ) ).
% atLeastLessThan_empty_iff
thf(fact_5665_atLeastLessThan__empty__iff2,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: A,B3: A] :
( ( ( bot_bot @ ( set @ A ) )
= ( set_or7035219750837199246ssThan @ A @ A3 @ B3 ) )
= ( ~ ( ord_less @ A @ A3 @ B3 ) ) ) ) ).
% atLeastLessThan_empty_iff2
thf(fact_5666_infinite__Ico__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A3: A,B3: A] :
( ( ~ ( finite_finite2 @ A @ ( set_or7035219750837199246ssThan @ A @ A3 @ B3 ) ) )
= ( ord_less @ A @ A3 @ B3 ) ) ) ).
% infinite_Ico_iff
thf(fact_5667_atLeastLessThan__singleton,axiom,
! [M: nat] :
( ( set_or7035219750837199246ssThan @ nat @ M @ ( suc @ M ) )
= ( insert @ nat @ M @ ( bot_bot @ ( set @ nat ) ) ) ) ).
% atLeastLessThan_singleton
thf(fact_5668_sum_Oop__ivl__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [N: nat,M: nat,G2: nat > A] :
( ( ( ord_less @ nat @ N @ M )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G2 @ ( set_or7035219750837199246ssThan @ nat @ M @ ( suc @ N ) ) )
= ( zero_zero @ A ) ) )
& ( ~ ( ord_less @ nat @ N @ M )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G2 @ ( set_or7035219750837199246ssThan @ nat @ M @ ( suc @ N ) ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G2 @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) @ ( G2 @ N ) ) ) ) ) ) ).
% sum.op_ivl_Suc
thf(fact_5669_prod_Oop__ivl__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [N: nat,M: nat,G2: nat > A] :
( ( ( ord_less @ nat @ N @ M )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_or7035219750837199246ssThan @ nat @ M @ ( suc @ N ) ) )
= ( one_one @ A ) ) )
& ( ~ ( ord_less @ nat @ N @ M )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_or7035219750837199246ssThan @ nat @ M @ ( suc @ N ) ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) @ ( G2 @ N ) ) ) ) ) ) ).
% prod.op_ivl_Suc
thf(fact_5670_infinite__Ico,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ A3 @ B3 )
=> ~ ( finite_finite2 @ A @ ( set_or7035219750837199246ssThan @ A @ A3 @ B3 ) ) ) ) ).
% infinite_Ico
thf(fact_5671_ivl__disj__un__two_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,M: A,U2: A] :
( ( ord_less_eq @ A @ L @ M )
=> ( ( ord_less_eq @ A @ M @ U2 )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ L @ M ) @ ( set_or7035219750837199246ssThan @ A @ M @ U2 ) )
= ( set_or7035219750837199246ssThan @ A @ L @ U2 ) ) ) ) ) ).
% ivl_disj_un_two(3)
thf(fact_5672_atLeastLessThan0,axiom,
! [M: nat] :
( ( set_or7035219750837199246ssThan @ nat @ M @ ( zero_zero @ nat ) )
= ( bot_bot @ ( set @ nat ) ) ) ).
% atLeastLessThan0
thf(fact_5673_lessThan__atLeast0,axiom,
( ( set_ord_lessThan @ nat )
= ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) ) ) ).
% lessThan_atLeast0
thf(fact_5674_all__nat__less__eq,axiom,
! [N: nat,P: nat > $o] :
( ( ! [M3: nat] :
( ( ord_less @ nat @ M3 @ N )
=> ( P @ M3 ) ) )
= ( ! [X2: nat] :
( ( member @ nat @ X2 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) )
=> ( P @ X2 ) ) ) ) ).
% all_nat_less_eq
thf(fact_5675_ex__nat__less__eq,axiom,
! [N: nat,P: nat > $o] :
( ( ? [M3: nat] :
( ( ord_less @ nat @ M3 @ N )
& ( P @ M3 ) ) )
= ( ? [X2: nat] :
( ( member @ nat @ X2 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) )
& ( P @ X2 ) ) ) ) ).
% ex_nat_less_eq
thf(fact_5676_atLeastLessThan__eq__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B3: A,C2: A,D2: A] :
( ( ord_less @ A @ A3 @ B3 )
=> ( ( ord_less @ A @ C2 @ D2 )
=> ( ( ( set_or7035219750837199246ssThan @ A @ A3 @ B3 )
= ( set_or7035219750837199246ssThan @ A @ C2 @ D2 ) )
= ( ( A3 = C2 )
& ( B3 = D2 ) ) ) ) ) ) ).
% atLeastLessThan_eq_iff
thf(fact_5677_atLeastLessThan__inj_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B3: A,C2: A,D2: A] :
( ( ( set_or7035219750837199246ssThan @ A @ A3 @ B3 )
= ( set_or7035219750837199246ssThan @ A @ C2 @ D2 ) )
=> ( ( ord_less @ A @ A3 @ B3 )
=> ( ( ord_less @ A @ C2 @ D2 )
=> ( A3 = C2 ) ) ) ) ) ).
% atLeastLessThan_inj(1)
thf(fact_5678_atLeastLessThan__inj_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B3: A,C2: A,D2: A] :
( ( ( set_or7035219750837199246ssThan @ A @ A3 @ B3 )
= ( set_or7035219750837199246ssThan @ A @ C2 @ D2 ) )
=> ( ( ord_less @ A @ A3 @ B3 )
=> ( ( ord_less @ A @ C2 @ D2 )
=> ( B3 = D2 ) ) ) ) ) ).
% atLeastLessThan_inj(2)
thf(fact_5679_sum_Oshift__bounds__Suc__ivl,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G2: nat > A,M: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G2 @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M ) @ ( suc @ N ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( G2 @ ( suc @ I4 ) )
@ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) ) ) ).
% sum.shift_bounds_Suc_ivl
thf(fact_5680_sum_Oshift__bounds__nat__ivl,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G2: nat > A,M: nat,K: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G2 @ ( set_or7035219750837199246ssThan @ nat @ ( plus_plus @ nat @ M @ K ) @ ( plus_plus @ nat @ N @ K ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( G2 @ ( plus_plus @ nat @ I4 @ K ) )
@ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) ) ) ).
% sum.shift_bounds_nat_ivl
thf(fact_5681_prod_Oshift__bounds__Suc__ivl,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G2: nat > A,M: nat,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M ) @ ( suc @ N ) ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( G2 @ ( suc @ I4 ) )
@ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) ) ) ).
% prod.shift_bounds_Suc_ivl
thf(fact_5682_prod_Oshift__bounds__nat__ivl,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G2: nat > A,M: nat,K: nat,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_or7035219750837199246ssThan @ nat @ ( plus_plus @ nat @ M @ K ) @ ( plus_plus @ nat @ N @ K ) ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( G2 @ ( plus_plus @ nat @ I4 @ K ) )
@ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) ) ) ).
% prod.shift_bounds_nat_ivl
thf(fact_5683_sum_Oivl__cong,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( comm_monoid_add @ A ) )
=> ! [A3: B,C2: B,B3: B,D2: B,G2: B > A,H2: B > A] :
( ( A3 = C2 )
=> ( ( B3 = D2 )
=> ( ! [X3: B] :
( ( ord_less_eq @ B @ C2 @ X3 )
=> ( ( ord_less @ B @ X3 @ D2 )
=> ( ( G2 @ X3 )
= ( H2 @ X3 ) ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G2 @ ( set_or7035219750837199246ssThan @ B @ A3 @ B3 ) )
= ( groups7311177749621191930dd_sum @ B @ A @ H2 @ ( set_or7035219750837199246ssThan @ B @ C2 @ D2 ) ) ) ) ) ) ) ).
% sum.ivl_cong
thf(fact_5684_prod_Oivl__cong,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( comm_monoid_mult @ A ) )
=> ! [A3: B,C2: B,B3: B,D2: B,G2: B > A,H2: B > A] :
( ( A3 = C2 )
=> ( ( B3 = D2 )
=> ( ! [X3: B] :
( ( ord_less_eq @ B @ C2 @ X3 )
=> ( ( ord_less @ B @ X3 @ D2 )
=> ( ( G2 @ X3 )
= ( H2 @ X3 ) ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G2 @ ( set_or7035219750837199246ssThan @ B @ A3 @ B3 ) )
= ( groups7121269368397514597t_prod @ B @ A @ H2 @ ( set_or7035219750837199246ssThan @ B @ C2 @ D2 ) ) ) ) ) ) ) ).
% prod.ivl_cong
thf(fact_5685_ivl__disj__un__two_I7_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,M: A,U2: A] :
( ( ord_less_eq @ A @ L @ M )
=> ( ( ord_less_eq @ A @ M @ U2 )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ L @ M ) @ ( set_or1337092689740270186AtMost @ A @ M @ U2 ) )
= ( set_or1337092689740270186AtMost @ A @ L @ U2 ) ) ) ) ) ).
% ivl_disj_un_two(7)
thf(fact_5686_ivl__disj__un__one_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,U2: A] :
( ( ord_less_eq @ A @ L @ U2 )
=> ( ( sup_sup @ ( set @ A ) @ ( set_ord_lessThan @ A @ L ) @ ( set_or7035219750837199246ssThan @ A @ L @ U2 ) )
= ( set_ord_lessThan @ A @ U2 ) ) ) ) ).
% ivl_disj_un_one(2)
thf(fact_5687_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_5688_atLeastLessThan__add__Un,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( set_or7035219750837199246ssThan @ nat @ I @ ( plus_plus @ nat @ J @ K ) )
= ( sup_sup @ ( set @ nat ) @ ( set_or7035219750837199246ssThan @ nat @ I @ J ) @ ( set_or7035219750837199246ssThan @ nat @ J @ ( plus_plus @ nat @ J @ K ) ) ) ) ) ).
% atLeastLessThan_add_Un
thf(fact_5689_subset__eq__atLeast0__lessThan__finite,axiom,
! [N3: set @ nat,N: nat] :
( ( ord_less_eq @ ( set @ nat ) @ N3 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) )
=> ( finite_finite2 @ nat @ N3 ) ) ).
% subset_eq_atLeast0_lessThan_finite
thf(fact_5690_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: A,B3: A,C2: A,D2: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A3 @ B3 ) @ ( set_or7035219750837199246ssThan @ A @ C2 @ D2 ) )
= ( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ A @ C2 @ A3 )
& ( ord_less @ A @ B3 @ D2 ) ) ) ) ) ).
% atLeastAtMost_subseteq_atLeastLessThan_iff
thf(fact_5691_atLeastLessThan__subseteq__atLeastAtMost__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A3: A,B3: A,C2: A,D2: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ A3 @ B3 ) @ ( set_or1337092689740270186AtMost @ A @ C2 @ D2 ) )
= ( ( ord_less @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ A @ C2 @ A3 )
& ( ord_less_eq @ A @ B3 @ D2 ) ) ) ) ) ).
% atLeastLessThan_subseteq_atLeastAtMost_iff
thf(fact_5692_ivl__disj__un__two__touch_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,M: A,U2: A] :
( ( ord_less_eq @ A @ L @ M )
=> ( ( ord_less @ A @ M @ U2 )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ L @ M ) @ ( set_or7035219750837199246ssThan @ A @ M @ U2 ) )
= ( set_or7035219750837199246ssThan @ A @ L @ U2 ) ) ) ) ) ).
% ivl_disj_un_two_touch(2)
thf(fact_5693_sum__shift__lb__Suc0__0__upt,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [F3: nat > A,K: nat] :
( ( ( F3 @ ( zero_zero @ nat ) )
= ( zero_zero @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ ( zero_zero @ nat ) ) @ K ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ K ) ) ) ) ) ).
% sum_shift_lb_Suc0_0_upt
thf(fact_5694_sum_OatLeast0__lessThan__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G2: nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G2 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G2 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) @ ( G2 @ N ) ) ) ) ).
% sum.atLeast0_lessThan_Suc
thf(fact_5695_sum_OatLeast__Suc__lessThan,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [M: nat,N: nat,G2: nat > A] :
( ( ord_less @ nat @ M @ N )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G2 @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) )
= ( plus_plus @ A @ ( G2 @ M ) @ ( groups7311177749621191930dd_sum @ nat @ A @ G2 @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M ) @ N ) ) ) ) ) ) ).
% sum.atLeast_Suc_lessThan
thf(fact_5696_atLeastLessThan__eq__atLeastAtMost__diff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( set_or7035219750837199246ssThan @ A )
= ( ^ [A5: A,B5: A] : ( minus_minus @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A5 @ B5 ) @ ( insert @ A @ B5 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% atLeastLessThan_eq_atLeastAtMost_diff
thf(fact_5697_prod_OatLeast0__lessThan__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G2: nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) @ ( G2 @ N ) ) ) ) ).
% prod.atLeast0_lessThan_Suc
thf(fact_5698_prod_OatLeast__Suc__lessThan,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M: nat,N: nat,G2: nat > A] :
( ( ord_less @ nat @ M @ N )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) )
= ( times_times @ A @ ( G2 @ M ) @ ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M ) @ N ) ) ) ) ) ) ).
% prod.atLeast_Suc_lessThan
thf(fact_5699_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_5700_le__mask__high__bits,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A,N: nat] :
( ( ord_less_eq @ ( word @ A ) @ W @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) )
= ( ! [X2: nat] :
( ( member @ nat @ X2 @ ( set_or7035219750837199246ssThan @ nat @ N @ ( size_size @ ( word @ A ) @ W ) ) )
=> ~ ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ W @ X2 ) ) ) ) ) ).
% le_mask_high_bits
thf(fact_5701_sum__Suc__diff_H,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [M: nat,N: nat,F3: nat > A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( minus_minus @ A @ ( F3 @ ( suc @ I4 ) ) @ ( F3 @ I4 ) )
@ ( set_or7035219750837199246ssThan @ nat @ M @ N ) )
= ( minus_minus @ A @ ( F3 @ N ) @ ( F3 @ M ) ) ) ) ) ).
% sum_Suc_diff'
thf(fact_5702_sum_OatLeastLessThan__rev,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G2: nat > A,N: nat,M: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G2 @ ( set_or7035219750837199246ssThan @ nat @ N @ M ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( G2 @ ( minus_minus @ nat @ ( plus_plus @ nat @ M @ N ) @ ( suc @ I4 ) ) )
@ ( set_or7035219750837199246ssThan @ nat @ N @ M ) ) ) ) ).
% sum.atLeastLessThan_rev
thf(fact_5703_sum_Onested__swap,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A3: nat > nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( groups7311177749621191930dd_sum @ nat @ A @ ( A3 @ I4 ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ I4 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [J3: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( A3 @ I4 @ J3 )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ J3 ) @ N ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% sum.nested_swap
thf(fact_5704_prod_OatLeastLessThan__rev,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G2: nat > A,N: nat,M: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_or7035219750837199246ssThan @ nat @ N @ M ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( G2 @ ( minus_minus @ nat @ ( plus_plus @ nat @ M @ N ) @ ( suc @ I4 ) ) )
@ ( set_or7035219750837199246ssThan @ nat @ N @ M ) ) ) ) ).
% prod.atLeastLessThan_rev
thf(fact_5705_prod_Onested__swap,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: nat > nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( groups7121269368397514597t_prod @ nat @ A @ ( A3 @ I4 ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ I4 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [J3: nat] :
( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( A3 @ I4 @ J3 )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ J3 ) @ N ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% prod.nested_swap
thf(fact_5706_sum_Onat__group,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G2: nat > A,K: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [M3: nat] : ( groups7311177749621191930dd_sum @ nat @ A @ G2 @ ( set_or7035219750837199246ssThan @ nat @ ( times_times @ nat @ M3 @ K ) @ ( plus_plus @ nat @ ( times_times @ nat @ M3 @ K ) @ K ) ) )
@ ( set_ord_lessThan @ nat @ N ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ G2 @ ( set_ord_lessThan @ nat @ ( times_times @ nat @ N @ K ) ) ) ) ) ).
% sum.nat_group
thf(fact_5707_prod_Onat__group,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G2: nat > A,K: nat,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A
@ ^ [M3: nat] : ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_or7035219750837199246ssThan @ nat @ ( times_times @ nat @ M3 @ K ) @ ( plus_plus @ nat @ ( times_times @ nat @ M3 @ K ) @ K ) ) )
@ ( set_ord_lessThan @ nat @ N ) )
= ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_ord_lessThan @ nat @ ( times_times @ nat @ N @ K ) ) ) ) ) ).
% prod.nat_group
thf(fact_5708_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_5709_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_5710_ivl__disj__un__singleton_I6_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,U2: A] :
( ( ord_less_eq @ A @ L @ U2 )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ L @ U2 ) @ ( insert @ A @ U2 @ ( bot_bot @ ( set @ A ) ) ) )
= ( set_or1337092689740270186AtMost @ A @ L @ U2 ) ) ) ) ).
% ivl_disj_un_singleton(6)
thf(fact_5711_sum_Ohead__if,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [N: nat,M: nat,G2: nat > A] :
( ( ( ord_less @ nat @ N @ M )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G2 @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( zero_zero @ A ) ) )
& ( ~ ( ord_less @ nat @ N @ M )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G2 @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G2 @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) @ ( G2 @ N ) ) ) ) ) ) ).
% sum.head_if
thf(fact_5712_prod_Ohead__if,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [N: nat,M: nat,G2: nat > A] :
( ( ( ord_less @ nat @ N @ M )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( one_one @ A ) ) )
& ( ~ ( ord_less @ nat @ N @ M )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) @ ( G2 @ N ) ) ) ) ) ) ).
% prod.head_if
thf(fact_5713_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_5714_fact__prod__Suc,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ( ( semiring_char_0_fact @ A )
= ( ^ [N4: nat] : ( semiring_1_of_nat @ A @ ( groups7121269368397514597t_prod @ nat @ nat @ suc @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N4 ) ) ) ) ) ) ).
% fact_prod_Suc
thf(fact_5715_sum_OatLeastLessThan__rev__at__least__Suc__atMost,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G2: nat > A,N: nat,M: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G2 @ ( set_or7035219750837199246ssThan @ nat @ N @ M ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( G2 @ ( minus_minus @ nat @ ( plus_plus @ nat @ M @ N ) @ I4 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ N ) @ M ) ) ) ) ).
% sum.atLeastLessThan_rev_at_least_Suc_atMost
thf(fact_5716_prod_OatLeastLessThan__rev__at__least__Suc__atMost,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G2: nat > A,N: nat,M: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_or7035219750837199246ssThan @ nat @ N @ M ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( G2 @ ( minus_minus @ nat @ ( plus_plus @ nat @ M @ N ) @ I4 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ N ) @ M ) ) ) ) ).
% prod.atLeastLessThan_rev_at_least_Suc_atMost
thf(fact_5717_pochhammer__prod,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ( ( comm_s3205402744901411588hammer @ A )
= ( ^ [A5: A,N4: nat] :
( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( plus_plus @ A @ A5 @ ( semiring_1_of_nat @ A @ I4 ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N4 ) ) ) ) ) ).
% pochhammer_prod
thf(fact_5718_fact__prod__rev,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ( ( semiring_char_0_fact @ A )
= ( ^ [N4: nat] : ( semiring_1_of_nat @ A @ ( groups7121269368397514597t_prod @ nat @ nat @ ( minus_minus @ nat @ N4 ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N4 ) ) ) ) ) ) ).
% fact_prod_rev
thf(fact_5719_summable__Cauchy,axiom,
! [A: $tType] :
( ( real_Vector_banach @ A )
=> ( ( summable @ A )
= ( ^ [F2: nat > A] :
! [E4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E4 )
=> ? [N9: nat] :
! [M3: nat] :
( ( ord_less_eq @ nat @ N9 @ M3 )
=> ! [N4: nat] : ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_or7035219750837199246ssThan @ nat @ M3 @ N4 ) ) ) @ E4 ) ) ) ) ) ) ).
% summable_Cauchy
thf(fact_5720_sums__group,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [F3: nat > A,S2: A,K: nat] :
( ( sums @ A @ F3 @ S2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( sums @ A
@ ^ [N4: nat] : ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_or7035219750837199246ssThan @ nat @ ( times_times @ nat @ N4 @ K ) @ ( plus_plus @ nat @ ( times_times @ nat @ N4 @ K ) @ K ) ) )
@ S2 ) ) ) ) ).
% sums_group
thf(fact_5721_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_5722_take__bit__sum,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ( ( bit_se2584673776208193580ke_bit @ A )
= ( ^ [N4: nat,A5: A] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( bit_se4730199178511100633sh_bit @ A @ K3 @ ( zero_neq_one_of_bool @ A @ ( bit_se5641148757651400278ts_bit @ A @ A5 @ K3 ) ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N4 ) ) ) ) ) ).
% take_bit_sum
thf(fact_5723_finite__transitivity__chain,axiom,
! [A: $tType,A2: set @ A,R: A > A > $o] :
( ( finite_finite2 @ A @ A2 )
=> ( ! [X3: A] :
~ ( R @ X3 @ X3 )
=> ( ! [X3: A,Y4: A,Z3: A] :
( ( R @ X3 @ Y4 )
=> ( ( R @ Y4 @ Z3 )
=> ( R @ X3 @ Z3 ) ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ A2 )
=> ? [Y5: A] :
( ( member @ A @ Y5 @ A2 )
& ( R @ X3 @ Y5 ) ) )
=> ( A2
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% finite_transitivity_chain
thf(fact_5724_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_5725_binomial__altdef__of__nat,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K: nat,N: nat] :
( ( ord_less_eq @ nat @ K @ N )
=> ( ( semiring_1_of_nat @ A @ ( binomial @ N @ K ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( divide_divide @ A @ ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ N @ I4 ) ) @ ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ K @ I4 ) ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ K ) ) ) ) ) ).
% binomial_altdef_of_nat
thf(fact_5726_gbinomial__altdef__of__nat,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( ( gbinomial @ A )
= ( ^ [A5: A,K3: nat] :
( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( divide_divide @ A @ ( minus_minus @ A @ A5 @ ( semiring_1_of_nat @ A @ I4 ) ) @ ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ K3 @ I4 ) ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ K3 ) ) ) ) ) ).
% gbinomial_altdef_of_nat
thf(fact_5727_infinite__nat__iff__unbounded,axiom,
! [S: set @ nat] :
( ( ~ ( finite_finite2 @ nat @ S ) )
= ( ! [M3: nat] :
? [N4: nat] :
( ( ord_less @ nat @ M3 @ N4 )
& ( member @ nat @ N4 @ S ) ) ) ) ).
% infinite_nat_iff_unbounded
thf(fact_5728_unbounded__k__infinite,axiom,
! [K: nat,S: set @ nat] :
( ! [M4: nat] :
( ( ord_less @ nat @ K @ M4 )
=> ? [N11: nat] :
( ( ord_less @ nat @ M4 @ N11 )
& ( member @ nat @ N11 @ S ) ) )
=> ~ ( finite_finite2 @ nat @ S ) ) ).
% unbounded_k_infinite
thf(fact_5729_gbinomial__mult__fact,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K: nat,A3: A] :
( ( times_times @ A @ ( semiring_char_0_fact @ A @ K ) @ ( gbinomial @ A @ A3 @ K ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( minus_minus @ A @ A3 @ ( semiring_1_of_nat @ A @ I4 ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ K ) ) ) ) ).
% gbinomial_mult_fact
thf(fact_5730_gbinomial__mult__fact_H,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: A,K: nat] :
( ( times_times @ A @ ( gbinomial @ A @ A3 @ K ) @ ( semiring_char_0_fact @ A @ K ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( minus_minus @ A @ A3 @ ( semiring_1_of_nat @ A @ I4 ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ K ) ) ) ) ).
% gbinomial_mult_fact'
thf(fact_5731_gbinomial__prod__rev,axiom,
! [A: $tType] :
( ( ( semiring_char_0 @ A )
& ( semidom_divide @ A ) )
=> ( ( gbinomial @ A )
= ( ^ [A5: A,K3: nat] :
( divide_divide @ A
@ ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( minus_minus @ A @ A5 @ ( semiring_1_of_nat @ A @ I4 ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ K3 ) )
@ ( semiring_char_0_fact @ A @ K3 ) ) ) ) ) ).
% gbinomial_prod_rev
thf(fact_5732_Chebyshev__sum__upper,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: nat,A3: nat > A,B3: nat > A] :
( ! [I3: nat,J2: nat] :
( ( ord_less_eq @ nat @ I3 @ J2 )
=> ( ( ord_less @ nat @ J2 @ N )
=> ( ord_less_eq @ A @ ( A3 @ I3 ) @ ( A3 @ J2 ) ) ) )
=> ( ! [I3: nat,J2: nat] :
( ( ord_less_eq @ nat @ I3 @ J2 )
=> ( ( ord_less @ nat @ J2 @ N )
=> ( ord_less_eq @ A @ ( B3 @ J2 ) @ ( B3 @ I3 ) ) ) )
=> ( ord_less_eq @ A
@ ( times_times @ A @ ( semiring_1_of_nat @ A @ N )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( times_times @ A @ ( A3 @ K3 ) @ ( B3 @ K3 ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) )
@ ( times_times @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ A3 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ B3 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ) ) ) ).
% Chebyshev_sum_upper
thf(fact_5733_Chebyshev__sum__upper__nat,axiom,
! [N: nat,A3: nat > nat,B3: nat > nat] :
( ! [I3: nat,J2: nat] :
( ( ord_less_eq @ nat @ I3 @ J2 )
=> ( ( ord_less @ nat @ J2 @ N )
=> ( ord_less_eq @ nat @ ( A3 @ I3 ) @ ( A3 @ J2 ) ) ) )
=> ( ! [I3: nat,J2: nat] :
( ( ord_less_eq @ nat @ I3 @ J2 )
=> ( ( ord_less @ nat @ J2 @ N )
=> ( ord_less_eq @ nat @ ( B3 @ J2 ) @ ( B3 @ I3 ) ) ) )
=> ( ord_less_eq @ nat
@ ( times_times @ nat @ N
@ ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [I4: nat] : ( times_times @ nat @ ( A3 @ I4 ) @ ( B3 @ I4 ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) )
@ ( times_times @ nat @ ( groups7311177749621191930dd_sum @ nat @ nat @ A3 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) @ ( groups7311177749621191930dd_sum @ nat @ nat @ B3 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ) ) ).
% Chebyshev_sum_upper_nat
thf(fact_5734_smod__int__range,axiom,
! [B3: int,A3: int] :
( ( B3
!= ( zero_zero @ int ) )
=> ( member @ int @ ( signed6721504322012087516modulo @ int @ A3 @ B3 ) @ ( set_or1337092689740270186AtMost @ int @ ( plus_plus @ int @ ( uminus_uminus @ int @ ( abs_abs @ int @ B3 ) ) @ ( one_one @ int ) ) @ ( minus_minus @ int @ ( abs_abs @ int @ B3 ) @ ( one_one @ int ) ) ) ) ) ).
% smod_int_range
thf(fact_5735_smod__int__mod__0,axiom,
! [X: int] :
( ( signed6721504322012087516modulo @ int @ X @ ( zero_zero @ int ) )
= X ) ).
% smod_int_mod_0
thf(fact_5736_smod__int__0__mod,axiom,
! [X: int] :
( ( signed6721504322012087516modulo @ int @ ( zero_zero @ int ) @ X )
= ( zero_zero @ int ) ) ).
% smod_int_0_mod
thf(fact_5737_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_5738_finite__atLeastZeroLessThan__integer,axiom,
! [U2: code_integer] : ( finite_finite2 @ code_integer @ ( set_or7035219750837199246ssThan @ code_integer @ ( zero_zero @ code_integer ) @ U2 ) ) ).
% finite_atLeastZeroLessThan_integer
thf(fact_5739_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_5740_finite__atLeastZeroLessThan__int,axiom,
! [U2: int] : ( finite_finite2 @ int @ ( set_or7035219750837199246ssThan @ int @ ( zero_zero @ int ) @ U2 ) ) ).
% finite_atLeastZeroLessThan_int
thf(fact_5741_atLeastLessThanPlusOne__atLeastAtMost__integer,axiom,
! [L: code_integer,U2: code_integer] :
( ( set_or7035219750837199246ssThan @ code_integer @ L @ ( plus_plus @ code_integer @ U2 @ ( one_one @ code_integer ) ) )
= ( set_or1337092689740270186AtMost @ code_integer @ L @ U2 ) ) ).
% atLeastLessThanPlusOne_atLeastAtMost_integer
thf(fact_5742_atLeastLessThanPlusOne__atLeastAtMost__int,axiom,
! [L: int,U2: int] :
( ( set_or7035219750837199246ssThan @ int @ L @ ( plus_plus @ int @ U2 @ ( one_one @ int ) ) )
= ( set_or1337092689740270186AtMost @ int @ L @ U2 ) ) ).
% atLeastLessThanPlusOne_atLeastAtMost_int
thf(fact_5743_smod__int__compares_I1_J,axiom,
! [A3: int,B3: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ A3 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ B3 )
=> ( ord_less @ int @ ( signed6721504322012087516modulo @ int @ A3 @ B3 ) @ B3 ) ) ) ).
% smod_int_compares(1)
thf(fact_5744_smod__int__compares_I2_J,axiom,
! [A3: int,B3: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ A3 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ B3 )
=> ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( signed6721504322012087516modulo @ int @ A3 @ B3 ) ) ) ) ).
% smod_int_compares(2)
thf(fact_5745_smod__int__compares_I4_J,axiom,
! [A3: int,B3: int] :
( ( ord_less_eq @ int @ A3 @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ B3 )
=> ( ord_less_eq @ int @ ( signed6721504322012087516modulo @ int @ A3 @ B3 ) @ ( zero_zero @ int ) ) ) ) ).
% smod_int_compares(4)
thf(fact_5746_smod__int__compares_I6_J,axiom,
! [A3: int,B3: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ A3 )
=> ( ( ord_less @ int @ B3 @ ( zero_zero @ int ) )
=> ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( signed6721504322012087516modulo @ int @ A3 @ B3 ) ) ) ) ).
% smod_int_compares(6)
thf(fact_5747_smod__int__compares_I7_J,axiom,
! [A3: int,B3: int] :
( ( ord_less_eq @ int @ A3 @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ B3 @ ( zero_zero @ int ) )
=> ( ord_less_eq @ int @ ( signed6721504322012087516modulo @ int @ A3 @ B3 ) @ ( zero_zero @ int ) ) ) ) ).
% smod_int_compares(7)
thf(fact_5748_smod__int__compares_I8_J,axiom,
! [A3: int,B3: int] :
( ( ord_less_eq @ int @ A3 @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ B3 @ ( zero_zero @ int ) )
=> ( ord_less_eq @ int @ B3 @ ( signed6721504322012087516modulo @ int @ A3 @ B3 ) ) ) ) ).
% smod_int_compares(8)
thf(fact_5749_smod__mod__positive,axiom,
! [A3: int,B3: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ A3 )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ B3 )
=> ( ( signed6721504322012087516modulo @ int @ A3 @ B3 )
= ( modulo_modulo @ int @ A3 @ B3 ) ) ) ) ).
% smod_mod_positive
thf(fact_5750_word__atLeastLessThan__Suc__atLeastAtMost,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [U2: word @ A,L: word @ A] :
( ( U2
!= ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) )
=> ( ( set_or7035219750837199246ssThan @ ( word @ A ) @ L @ ( plus_plus @ ( word @ A ) @ U2 @ ( one_one @ ( word @ A ) ) ) )
= ( set_or1337092689740270186AtMost @ ( word @ A ) @ L @ U2 ) ) ) ) ).
% word_atLeastLessThan_Suc_atLeastAtMost
thf(fact_5751_word__range__minus__1,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [B3: word @ A,A3: word @ A] :
( ( B3
!= ( zero_zero @ ( word @ A ) ) )
=> ( ( set_or1337092689740270186AtMost @ ( word @ A ) @ A3 @ ( minus_minus @ ( word @ A ) @ B3 @ ( one_one @ ( word @ A ) ) ) )
= ( set_or7035219750837199246ssThan @ ( word @ A ) @ A3 @ B3 ) ) ) ) ).
% word_range_minus_1
thf(fact_5752_smod__int__compares_I3_J,axiom,
! [A3: int,B3: int] :
( ( ord_less_eq @ int @ A3 @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ B3 )
=> ( ord_less @ int @ ( uminus_uminus @ int @ B3 ) @ ( signed6721504322012087516modulo @ int @ A3 @ B3 ) ) ) ) ).
% smod_int_compares(3)
thf(fact_5753_smod__int__compares_I5_J,axiom,
! [A3: int,B3: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ A3 )
=> ( ( ord_less @ int @ B3 @ ( zero_zero @ int ) )
=> ( ord_less @ int @ ( signed6721504322012087516modulo @ int @ A3 @ B3 ) @ ( uminus_uminus @ int @ B3 ) ) ) ) ).
% smod_int_compares(5)
thf(fact_5754_word__count__from__top,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: word @ A] :
( ( N
!= ( zero_zero @ ( word @ A ) ) )
=> ( ( set_or7035219750837199246ssThan @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ N )
= ( sup_sup @ ( set @ ( word @ A ) ) @ ( set_or7035219750837199246ssThan @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ ( minus_minus @ ( word @ A ) @ N @ ( one_one @ ( word @ A ) ) ) ) @ ( insert @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ N @ ( one_one @ ( word @ A ) ) ) @ ( bot_bot @ ( set @ ( word @ A ) ) ) ) ) ) ) ) ).
% word_count_from_top
thf(fact_5755_Multiset_Ois__empty__def,axiom,
! [A: $tType] :
( ( is_empty @ A )
= ( ^ [A4: multiset @ A] :
( A4
= ( zero_zero @ ( multiset @ A ) ) ) ) ) ).
% Multiset.is_empty_def
thf(fact_5756_mod__word__minus__1__minus__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [B3: num] :
( ( modulo_modulo @ ( word @ A ) @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ B3 ) ) )
= ( ring_1_of_int @ ( word @ A ) @ ( modulo_modulo @ int @ ( minus_minus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) @ ( one_one @ int ) ) @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ B3 ) ) ) ) ) ) ) ).
% mod_word_minus_1_minus_numeral
thf(fact_5757_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_5758_take__bit__length__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A] :
( ( bit_se2584673776208193580ke_bit @ ( word @ A ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ W )
= W ) ) ).
% take_bit_length_eq
thf(fact_5759_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_5760_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_5761_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_5762_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_5763_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_5764_less__eq__word__numeral__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: num,B3: num] :
( ( ord_less_eq @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ A3 ) @ ( numeral_numeral @ ( word @ A ) @ B3 ) )
= ( ord_less_eq @ int @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( numeral_numeral @ int @ A3 ) ) @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( numeral_numeral @ int @ B3 ) ) ) ) ) ).
% less_eq_word_numeral_numeral
thf(fact_5765_less__word__numeral__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: num,B3: num] :
( ( ord_less @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ A3 ) @ ( numeral_numeral @ ( word @ A ) @ B3 ) )
= ( ord_less @ int @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( numeral_numeral @ int @ A3 ) ) @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( numeral_numeral @ int @ B3 ) ) ) ) ) ).
% less_word_numeral_numeral
thf(fact_5766_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_5767_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_5768_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_5769_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_5770_unat__lt2p,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] : ( ord_less @ nat @ ( semiring_1_unsigned @ A @ nat @ X ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ).
% unat_lt2p
thf(fact_5771_uint__lt2p,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] : ( ord_less @ int @ ( semiring_1_unsigned @ A @ int @ X ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ).
% uint_lt2p
thf(fact_5772_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_5773_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_5774_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_5775_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_5776_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_5777_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_5778_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_5779_div__word__numeral__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: num,B3: num] :
( ( divide_divide @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ A3 ) @ ( numeral_numeral @ ( word @ A ) @ B3 ) )
= ( ring_1_of_int @ ( word @ A ) @ ( divide_divide @ int @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( numeral_numeral @ int @ A3 ) ) @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( numeral_numeral @ int @ B3 ) ) ) ) ) ) ).
% div_word_numeral_numeral
thf(fact_5780_mod__word__numeral__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: num,B3: num] :
( ( modulo_modulo @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ A3 ) @ ( numeral_numeral @ ( word @ A ) @ B3 ) )
= ( ring_1_of_int @ ( word @ A ) @ ( modulo_modulo @ int @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( numeral_numeral @ int @ A3 ) ) @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( numeral_numeral @ int @ B3 ) ) ) ) ) ) ).
% mod_word_numeral_numeral
thf(fact_5781_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_5782_less__eq__word__minus__numeral__minus__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: num,B3: num] :
( ( ord_less_eq @ ( word @ A ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ A3 ) ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ B3 ) ) )
= ( ord_less_eq @ int @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ A3 ) ) ) @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ B3 ) ) ) ) ) ) ).
% less_eq_word_minus_numeral_minus_numeral
thf(fact_5783_less__eq__word__numeral__minus__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: num,B3: num] :
( ( ord_less_eq @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ A3 ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ B3 ) ) )
= ( ord_less_eq @ int @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( numeral_numeral @ int @ A3 ) ) @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ B3 ) ) ) ) ) ) ).
% less_eq_word_numeral_minus_numeral
thf(fact_5784_less__eq__word__minus__numeral__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: num,B3: num] :
( ( ord_less_eq @ ( word @ A ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ A3 ) ) @ ( numeral_numeral @ ( word @ A ) @ B3 ) )
= ( ord_less_eq @ int @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ A3 ) ) ) @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( numeral_numeral @ int @ B3 ) ) ) ) ) ).
% less_eq_word_minus_numeral_numeral
thf(fact_5785_less__word__minus__numeral__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: num,B3: num] :
( ( ord_less @ ( word @ A ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ A3 ) ) @ ( numeral_numeral @ ( word @ A ) @ B3 ) )
= ( ord_less @ int @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ A3 ) ) ) @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( numeral_numeral @ int @ B3 ) ) ) ) ) ).
% less_word_minus_numeral_numeral
thf(fact_5786_less__word__numeral__minus__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: num,B3: num] :
( ( ord_less @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ A3 ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ B3 ) ) )
= ( ord_less @ int @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( numeral_numeral @ int @ A3 ) ) @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ B3 ) ) ) ) ) ) ).
% less_word_numeral_minus_numeral
thf(fact_5787_less__word__minus__numeral__minus__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: num,B3: num] :
( ( ord_less @ ( word @ A ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ A3 ) ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ B3 ) ) )
= ( ord_less @ int @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ A3 ) ) ) @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ B3 ) ) ) ) ) ) ).
% less_word_minus_numeral_minus_numeral
thf(fact_5788_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_5789_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_5790_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_5791_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_5792_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_5793_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_5794_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_5795_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_5796_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_5797_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_5798_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_5799_div__word__minus__numeral__minus__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: num,B3: num] :
( ( divide_divide @ ( word @ A ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ A3 ) ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ B3 ) ) )
= ( ring_1_of_int @ ( word @ A ) @ ( divide_divide @ int @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ A3 ) ) ) @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ B3 ) ) ) ) ) ) ) ).
% div_word_minus_numeral_minus_numeral
thf(fact_5800_div__word__numeral__minus__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: num,B3: num] :
( ( divide_divide @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ A3 ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ B3 ) ) )
= ( ring_1_of_int @ ( word @ A ) @ ( divide_divide @ int @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( numeral_numeral @ int @ A3 ) ) @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ B3 ) ) ) ) ) ) ) ).
% div_word_numeral_minus_numeral
thf(fact_5801_div__word__minus__numeral__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: num,B3: num] :
( ( divide_divide @ ( word @ A ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ A3 ) ) @ ( numeral_numeral @ ( word @ A ) @ B3 ) )
= ( 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 @ A3 ) ) ) @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( numeral_numeral @ int @ B3 ) ) ) ) ) ) ).
% div_word_minus_numeral_numeral
thf(fact_5802_mod__word__minus__numeral__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: num,B3: num] :
( ( modulo_modulo @ ( word @ A ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ A3 ) ) @ ( numeral_numeral @ ( word @ A ) @ B3 ) )
= ( 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 @ A3 ) ) ) @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( numeral_numeral @ int @ B3 ) ) ) ) ) ) ).
% mod_word_minus_numeral_numeral
thf(fact_5803_mod__word__numeral__minus__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: num,B3: num] :
( ( modulo_modulo @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ A3 ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ B3 ) ) )
= ( ring_1_of_int @ ( word @ A ) @ ( modulo_modulo @ int @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( numeral_numeral @ int @ A3 ) ) @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ B3 ) ) ) ) ) ) ) ).
% mod_word_numeral_minus_numeral
thf(fact_5804_mod__word__minus__numeral__minus__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: num,B3: num] :
( ( modulo_modulo @ ( word @ A ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ A3 ) ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ B3 ) ) )
= ( ring_1_of_int @ ( word @ A ) @ ( modulo_modulo @ int @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ A3 ) ) ) @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ B3 ) ) ) ) ) ) ) ).
% mod_word_minus_numeral_minus_numeral
thf(fact_5805_word__less__sub__le,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,X: word @ A] :
( ( ord_less @ nat @ N @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ( ord_less_eq @ ( word @ A ) @ X @ ( minus_minus @ ( word @ A ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) @ ( one_one @ ( word @ A ) ) ) )
= ( ord_less @ ( word @ A ) @ X @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) ) ) ) ) ).
% word_less_sub_le
thf(fact_5806_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_5807_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_5808_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_5809_less__word__numeral__minus__1,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: num] :
( ( ord_less @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ A3 ) @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) )
= ( ord_less @ int @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( numeral_numeral @ int @ A3 ) ) @ ( minus_minus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) @ ( one_one @ int ) ) ) ) ) ).
% less_word_numeral_minus_1
thf(fact_5810_less__word__minus__numeral__minus__1,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: num] :
( ( ord_less @ ( word @ A ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ A3 ) ) @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) )
= ( ord_less @ int @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ A3 ) ) ) @ ( minus_minus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) @ ( one_one @ int ) ) ) ) ) ).
% less_word_minus_numeral_minus_1
thf(fact_5811_div__word__minus__1__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [B3: num] :
( ( divide_divide @ ( word @ A ) @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) @ ( numeral_numeral @ ( word @ A ) @ B3 ) )
= ( ring_1_of_int @ ( word @ A ) @ ( divide_divide @ int @ ( minus_minus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) @ ( one_one @ int ) ) @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( numeral_numeral @ int @ B3 ) ) ) ) ) ) ).
% div_word_minus_1_numeral
thf(fact_5812_mod__word__minus__1__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [B3: num] :
( ( modulo_modulo @ ( word @ A ) @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) @ ( numeral_numeral @ ( word @ A ) @ B3 ) )
= ( ring_1_of_int @ ( word @ A ) @ ( modulo_modulo @ int @ ( minus_minus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) @ ( one_one @ int ) ) @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( numeral_numeral @ int @ B3 ) ) ) ) ) ) ).
% mod_word_minus_1_numeral
thf(fact_5813_div__word__minus__1__minus__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [B3: num] :
( ( divide_divide @ ( word @ A ) @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ B3 ) ) )
= ( ring_1_of_int @ ( word @ A ) @ ( divide_divide @ int @ ( minus_minus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) @ ( one_one @ int ) ) @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ B3 ) ) ) ) ) ) ) ).
% div_word_minus_1_minus_numeral
thf(fact_5814_ucast__mask__drop,axiom,
! [A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( type_len @ A ) )
=> ! [N: nat,X: word @ B] :
( ( ord_less_eq @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ N )
=> ( ( semiring_1_unsigned @ B @ ( word @ A ) @ ( bit_se5824344872417868541ns_and @ ( word @ B ) @ X @ ( bit_se2239418461657761734s_mask @ ( word @ B ) @ N ) ) )
= ( semiring_1_unsigned @ B @ ( word @ A ) @ X ) ) ) ) ).
% ucast_mask_drop
thf(fact_5815_uint__word__arith__bintrs_I3_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A,B3: word @ A] :
( ( semiring_1_unsigned @ A @ int @ ( times_times @ ( word @ A ) @ A3 @ B3 ) )
= ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( times_times @ int @ ( semiring_1_unsigned @ A @ int @ A3 ) @ ( semiring_1_unsigned @ A @ int @ B3 ) ) ) ) ) ).
% uint_word_arith_bintrs(3)
thf(fact_5816_signed__take__bit__eq,axiom,
! [A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( bit_ri3973907225187159222ations @ A ) )
=> ! [N: nat,W: word @ B] :
( ( ( ord_less @ nat @ N @ ( type_len0_len_of @ B @ ( type2 @ B ) ) )
=> ( ( ring_1_signed @ B @ A @ ( bit_se2584673776208193580ke_bit @ ( word @ B ) @ N @ W ) )
= ( bit_se2584673776208193580ke_bit @ A @ N @ ( ring_1_signed @ B @ A @ W ) ) ) )
& ( ~ ( ord_less @ nat @ N @ ( type_len0_len_of @ B @ ( type2 @ B ) ) )
=> ( ( ring_1_signed @ B @ A @ ( bit_se2584673776208193580ke_bit @ ( word @ B ) @ N @ W ) )
= ( ring_1_signed @ B @ A @ W ) ) ) ) ) ).
% signed_take_bit_eq
thf(fact_5817_word__cat__inj,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( type_len @ A )
& ( type_len @ C ) )
=> ! [A3: word @ A,B3: word @ B,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 @ A3 @ B3 )
= ( word_cat @ A @ B @ C @ C2 @ D2 ) )
= ( ( A3 = C2 )
& ( B3 = D2 ) ) ) ) ) ).
% word_cat_inj
thf(fact_5818_ucast__ucast__add,axiom,
! [A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( type_len @ A ) )
=> ! [X: word @ A,Y: word @ B] :
( ( ord_less_eq @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( type_len0_len_of @ B @ ( type2 @ B ) ) )
=> ( ( semiring_1_unsigned @ B @ ( word @ A ) @ ( plus_plus @ ( word @ B ) @ ( semiring_1_unsigned @ A @ ( word @ B ) @ X ) @ Y ) )
= ( plus_plus @ ( word @ A ) @ X @ ( semiring_1_unsigned @ B @ ( word @ A ) @ Y ) ) ) ) ) ).
% ucast_ucast_add
thf(fact_5819_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_5820_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_5821_eq__ucast__ucast__eq,axiom,
! [A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( type_len @ A ) )
=> ! [X: word @ A,Y: word @ B] :
( ( ord_less_eq @ nat @ ( type_len0_len_of @ B @ ( type2 @ B ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ( X
= ( semiring_1_unsigned @ B @ ( word @ A ) @ Y ) )
=> ( ( semiring_1_unsigned @ A @ ( word @ B ) @ X )
= Y ) ) ) ) ).
% eq_ucast_ucast_eq
thf(fact_5822_up__ucast__inj__eq,axiom,
! [B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B ) )
=> ! [X: word @ A,Y: word @ A] :
( ( ord_less_eq @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( type_len0_len_of @ B @ ( type2 @ B ) ) )
=> ( ( ( semiring_1_unsigned @ A @ ( word @ B ) @ X )
= ( semiring_1_unsigned @ A @ ( word @ B ) @ Y ) )
= ( X = Y ) ) ) ) ).
% up_ucast_inj_eq
thf(fact_5823_ucast__ucast__eq,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( type_len @ A )
& ( type_len @ C ) )
=> ! [X: word @ A,Y: word @ B] :
( ( ( semiring_1_unsigned @ A @ ( word @ C ) @ X )
= ( semiring_1_unsigned @ A @ ( word @ C ) @ ( semiring_1_unsigned @ B @ ( word @ A ) @ Y ) ) )
=> ( ( ord_less_eq @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( type_len0_len_of @ B @ ( type2 @ B ) ) )
=> ( ( ord_less_eq @ nat @ ( type_len0_len_of @ B @ ( type2 @ B ) ) @ ( type_len0_len_of @ C @ ( type2 @ C ) ) )
=> ( X
= ( semiring_1_unsigned @ B @ ( word @ A ) @ Y ) ) ) ) ) ) ).
% ucast_ucast_eq
thf(fact_5824_up__ucast__inj,axiom,
! [B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B ) )
=> ! [X: word @ A,Y: word @ A] :
( ( ( semiring_1_unsigned @ A @ ( word @ B ) @ X )
= ( semiring_1_unsigned @ A @ ( word @ B ) @ Y ) )
=> ( ( ord_less_eq @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( type_len0_len_of @ B @ ( type2 @ B ) ) )
=> ( X = Y ) ) ) ) ).
% up_ucast_inj
thf(fact_5825_up__scast__inj__eq,axiom,
! [B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B ) )
=> ! [X: word @ A,Y: word @ A] :
( ( ord_less_eq @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( type_len0_len_of @ B @ ( type2 @ B ) ) )
=> ( ( ( ring_1_signed @ A @ ( word @ B ) @ X )
= ( ring_1_signed @ A @ ( word @ B ) @ Y ) )
= ( X = Y ) ) ) ) ).
% up_scast_inj_eq
thf(fact_5826_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_5827_unat__ucast__up__simp,axiom,
! [B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B ) )
=> ! [X: word @ A] :
( ( ord_less_eq @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( type_len0_len_of @ B @ ( type2 @ B ) ) )
=> ( ( semiring_1_unsigned @ B @ nat @ ( semiring_1_unsigned @ A @ ( word @ B ) @ X ) )
= ( semiring_1_unsigned @ A @ nat @ X ) ) ) ) ).
% unat_ucast_up_simp
thf(fact_5828_less__ucast__ucast__less,axiom,
! [A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( type_len @ A ) )
=> ! [X: word @ A,Y: word @ B] :
( ( ord_less_eq @ nat @ ( type_len0_len_of @ B @ ( type2 @ B ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ( ord_less @ ( word @ A ) @ X @ ( semiring_1_unsigned @ B @ ( word @ A ) @ Y ) )
=> ( ord_less @ ( word @ B ) @ ( semiring_1_unsigned @ A @ ( word @ B ) @ X ) @ Y ) ) ) ) ).
% less_ucast_ucast_less
thf(fact_5829_ucast__less__ucast,axiom,
! [B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B ) )
=> ! [X: word @ A,Y: word @ A] :
( ( ord_less_eq @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( type_len0_len_of @ B @ ( type2 @ B ) ) )
=> ( ( ord_less @ ( word @ B ) @ ( semiring_1_unsigned @ A @ ( word @ B ) @ X ) @ ( semiring_1_unsigned @ A @ ( word @ B ) @ Y ) )
= ( ord_less @ ( word @ A ) @ X @ Y ) ) ) ) ).
% ucast_less_ucast
thf(fact_5830_ucast__up__mono,axiom,
! [B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B ) )
=> ! [X: word @ A,Y: word @ A] :
( ( ord_less_eq @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( type_len0_len_of @ B @ ( type2 @ B ) ) )
=> ( ( ord_less @ ( word @ A ) @ X @ Y )
=> ( ord_less @ ( word @ B ) @ ( semiring_1_unsigned @ A @ ( word @ B ) @ X ) @ ( semiring_1_unsigned @ A @ ( word @ B ) @ Y ) ) ) ) ) ).
% ucast_up_mono
thf(fact_5831_ucast__up__mono__le,axiom,
! [B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B ) )
=> ! [X: word @ A,Y: word @ A] :
( ( ord_less_eq @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( type_len0_len_of @ B @ ( type2 @ B ) ) )
=> ( ( ord_less_eq @ ( word @ A ) @ X @ Y )
=> ( ord_less_eq @ ( word @ B ) @ ( semiring_1_unsigned @ A @ ( word @ B ) @ X ) @ ( semiring_1_unsigned @ A @ ( word @ B ) @ Y ) ) ) ) ) ).
% ucast_up_mono_le
thf(fact_5832_ucast__le__ucast,axiom,
! [B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B ) )
=> ! [X: word @ A,Y: word @ A] :
( ( ord_less_eq @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( type_len0_len_of @ B @ ( type2 @ B ) ) )
=> ( ( ord_less_eq @ ( word @ B ) @ ( semiring_1_unsigned @ A @ ( word @ B ) @ X ) @ ( semiring_1_unsigned @ A @ ( word @ B ) @ Y ) )
= ( ord_less_eq @ ( word @ A ) @ X @ Y ) ) ) ) ).
% ucast_le_ucast
thf(fact_5833_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_5834_up__scast__inj,axiom,
! [B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B ) )
=> ! [X: word @ A,Y: word @ A] :
( ( ( ring_1_signed @ A @ ( word @ B ) @ X )
= ( ring_1_signed @ A @ ( word @ B ) @ Y ) )
=> ( ( ord_less_eq @ nat @ ( size_size @ ( word @ A ) @ X ) @ ( type_len0_len_of @ B @ ( type2 @ B ) ) )
=> ( X = Y ) ) ) ) ).
% up_scast_inj
thf(fact_5835_word__of__int__less__eq__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [K: int,L: int] :
( ( ord_less_eq @ ( word @ A ) @ ( ring_1_of_int @ ( word @ A ) @ K ) @ ( ring_1_of_int @ ( word @ A ) @ L ) )
= ( ord_less_eq @ int @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ K ) @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ L ) ) ) ) ).
% word_of_int_less_eq_iff
thf(fact_5836_uint__word__arith__bintrs_I4_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A] :
( ( semiring_1_unsigned @ A @ int @ ( uminus_uminus @ ( word @ A ) @ A3 ) )
= ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( uminus_uminus @ int @ ( semiring_1_unsigned @ A @ int @ A3 ) ) ) ) ) ).
% uint_word_arith_bintrs(4)
thf(fact_5837_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_5838_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_5839_unsigned__not__eq,axiom,
! [A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( bit_ri3973907225187159222ations @ A ) )
=> ! [W: word @ B] :
( ( semiring_1_unsigned @ B @ A @ ( bit_ri4277139882892585799ns_not @ ( word @ B ) @ W ) )
= ( bit_se2584673776208193580ke_bit @ A @ ( type_len0_len_of @ B @ ( type2 @ B ) ) @ ( bit_ri4277139882892585799ns_not @ A @ ( semiring_1_unsigned @ B @ A @ W ) ) ) ) ) ).
% unsigned_not_eq
thf(fact_5840_uint__word__arith__bintrs_I2_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A,B3: word @ A] :
( ( semiring_1_unsigned @ A @ int @ ( minus_minus @ ( word @ A ) @ A3 @ B3 ) )
= ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( minus_minus @ int @ ( semiring_1_unsigned @ A @ int @ A3 ) @ ( semiring_1_unsigned @ A @ int @ B3 ) ) ) ) ) ).
% uint_word_arith_bintrs(2)
thf(fact_5841_test__bit__bin,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) )
= ( ^ [W2: word @ A,N4: nat] :
( ( ord_less @ nat @ N4 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
& ( bit_se5641148757651400278ts_bit @ int @ ( semiring_1_unsigned @ A @ int @ W2 ) @ N4 ) ) ) ) ) ).
% test_bit_bin
thf(fact_5842_bit__uint__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A,N: nat] :
( ( bit_se5641148757651400278ts_bit @ int @ ( semiring_1_unsigned @ A @ int @ W ) @ N )
= ( ( ord_less @ nat @ N @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
& ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ W @ N ) ) ) ) ).
% bit_uint_iff
thf(fact_5843_ucast__sub__ucast,axiom,
! [B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B ) )
=> ! [Y: word @ A,X: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ Y @ X )
=> ( ( ord_less_eq @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( type_len0_len_of @ B @ ( type2 @ B ) ) )
=> ( ( semiring_1_unsigned @ A @ ( word @ B ) @ ( minus_minus @ ( word @ A ) @ X @ Y ) )
= ( minus_minus @ ( word @ B ) @ ( semiring_1_unsigned @ A @ ( word @ B ) @ X ) @ ( semiring_1_unsigned @ A @ ( word @ B ) @ Y ) ) ) ) ) ) ).
% ucast_sub_ucast
thf(fact_5844_word__and__full__mask__simp,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] :
( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
= X ) ) ).
% word_and_full_mask_simp
thf(fact_5845_ucast__ucast__mask,axiom,
! [B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B ) )
=> ! [X: word @ B] :
( ( semiring_1_unsigned @ A @ ( word @ B ) @ ( semiring_1_unsigned @ B @ ( word @ A ) @ X ) )
= ( bit_se5824344872417868541ns_and @ ( word @ B ) @ X @ ( bit_se2239418461657761734s_mask @ ( word @ B ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).
% ucast_ucast_mask
thf(fact_5846_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_5847_uint__sint,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( semiring_1_unsigned @ A @ int )
= ( ^ [W2: word @ A] : ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( ring_1_signed @ A @ int @ W2 ) ) ) ) ) ).
% uint_sint
thf(fact_5848_word__of__int__eq__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [K: int,L: int] :
( ( ( ring_1_of_int @ ( word @ A ) @ K )
= ( ring_1_of_int @ ( word @ A ) @ L ) )
= ( ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ K )
= ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ L ) ) ) ) ).
% word_of_int_eq_iff
thf(fact_5849_uint__word__of__int__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [K: int] :
( ( semiring_1_unsigned @ A @ int @ ( ring_1_of_int @ ( word @ A ) @ K ) )
= ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ K ) ) ) ).
% uint_word_of_int_eq
thf(fact_5850_unsigned__ucast__eq,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( bit_se359711467146920520ations @ A )
& ( type_len @ C ) )
=> ! [W: word @ B] :
( ( semiring_1_unsigned @ C @ A @ ( semiring_1_unsigned @ B @ ( word @ C ) @ W ) )
= ( bit_se2584673776208193580ke_bit @ A @ ( type_len0_len_of @ C @ ( type2 @ C ) ) @ ( semiring_1_unsigned @ B @ A @ W ) ) ) ) ).
% unsigned_ucast_eq
thf(fact_5851_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_5852_not__bit__length,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A] :
~ ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ W @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ).
% not_bit_length
thf(fact_5853_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_5854_num__of__bintr_H,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: num,B3: num] :
( ( ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( numeral_numeral @ int @ A3 ) )
= ( numeral_numeral @ int @ B3 ) )
=> ( ( numeral_numeral @ ( word @ A ) @ A3 )
= ( numeral_numeral @ ( word @ A ) @ B3 ) ) ) ) ).
% num_of_bintr'
thf(fact_5855_word__size,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( size_size @ ( word @ A ) )
= ( ^ [W2: word @ A] : ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ).
% word_size
thf(fact_5856_size__word_Orep__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( size_size @ ( word @ A ) )
= ( ^ [X2: word @ A] : ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ).
% size_word.rep_eq
thf(fact_5857_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_5858_size__0__same,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A,V2: word @ A] :
( ( ( type_len0_len_of @ A @ ( type2 @ A ) )
= ( zero_zero @ nat ) )
=> ( W = V2 ) ) ) ).
% size_0_same
thf(fact_5859_degenerate__word,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] :
( ( ( type_len0_len_of @ A @ ( type2 @ A ) )
= ( one_one @ nat ) )
=> ( ( X
= ( zero_zero @ ( word @ A ) ) )
| ( X
= ( one_one @ ( word @ A ) ) ) ) ) ) ).
% degenerate_word
thf(fact_5860_word__of__int__less__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [K: int,L: int] :
( ( ord_less @ ( word @ A ) @ ( ring_1_of_int @ ( word @ A ) @ K ) @ ( ring_1_of_int @ ( word @ A ) @ L ) )
= ( ord_less @ int @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ K ) @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ L ) ) ) ) ).
% word_of_int_less_iff
thf(fact_5861_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_5862_uint__word__arith__bintrs_I1_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A,B3: word @ A] :
( ( semiring_1_unsigned @ A @ int @ ( plus_plus @ ( word @ A ) @ A3 @ B3 ) )
= ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( plus_plus @ int @ ( semiring_1_unsigned @ A @ int @ A3 ) @ ( semiring_1_unsigned @ A @ int @ B3 ) ) ) ) ) ).
% uint_word_arith_bintrs(1)
thf(fact_5863_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_5864_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_5865_num__abs__bintr,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( numeral_numeral @ ( word @ A ) )
= ( ^ [X2: num] : ( ring_1_of_int @ ( word @ A ) @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( numeral_numeral @ int @ X2 ) ) ) ) ) ) ).
% num_abs_bintr
thf(fact_5866_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_5867_bit__set__bit__word__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A,M: nat,B3: $o,N: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( generi7602027413899671122et_bit @ ( word @ A ) @ W @ M @ B3 ) @ N )
= ( ( ( M = N )
=> ( ( ord_less @ nat @ N @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
& B3 ) )
& ( ( M != N )
=> ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ W @ N ) ) ) ) ) ).
% bit_set_bit_word_iff
thf(fact_5868_neg__test__bit,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,N: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ X ) @ N )
= ( ~ ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ X @ N )
& ( ord_less @ nat @ N @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).
% neg_test_bit
thf(fact_5869_test__bit__conj__lt,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,M: nat] :
( ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ X @ M )
& ( ord_less @ nat @ M @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
= ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ X @ M ) ) ) ).
% test_bit_conj_lt
thf(fact_5870_bit__imp__le__length,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A,N: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ W @ N )
=> ( ord_less @ nat @ N @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ).
% bit_imp_le_length
thf(fact_5871_word__eq__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( ^ [Y3: word @ A,Z2: word @ A] : Y3 = Z2 )
= ( ^ [X2: word @ A,Y2: word @ A] :
! [N4: nat] :
( ( ord_less @ nat @ N4 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ X2 @ N4 )
= ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ Y2 @ N4 ) ) ) ) ) ) ).
% word_eq_iff
thf(fact_5872_bit__word__eqI,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A,B3: word @ A] :
( ! [N2: nat] :
( ( ord_less @ nat @ N2 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ A3 @ N2 )
= ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ B3 @ N2 ) ) )
=> ( A3 = B3 ) ) ) ).
% bit_word_eqI
thf(fact_5873_bit__word__ucast__iff,axiom,
! [B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B ) )
=> ! [W: word @ A,N: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ B ) @ ( semiring_1_unsigned @ A @ ( word @ B ) @ W ) @ N )
= ( ( ord_less @ nat @ N @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
& ( ord_less @ nat @ N @ ( type_len0_len_of @ B @ ( type2 @ B ) ) )
& ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ W @ N ) ) ) ) ).
% bit_word_ucast_iff
thf(fact_5874_nth__ucast,axiom,
! [B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B ) )
=> ! [W: word @ B,N: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( semiring_1_unsigned @ B @ ( word @ A ) @ W ) @ N )
= ( ( bit_se5641148757651400278ts_bit @ ( word @ B ) @ W @ N )
& ( ord_less @ nat @ N @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).
% nth_ucast
thf(fact_5875_bit__ucast__iff,axiom,
! [A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( type_len @ A ) )
=> ! [A3: word @ B,N: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( semiring_1_unsigned @ B @ ( word @ A ) @ A3 ) @ N )
= ( ( ord_less @ nat @ N @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
& ( bit_se5641148757651400278ts_bit @ ( word @ B ) @ A3 @ N ) ) ) ) ).
% bit_ucast_iff
thf(fact_5876_bin__nth__uint__imp,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A,N: nat] :
( ( bit_se5641148757651400278ts_bit @ int @ ( semiring_1_unsigned @ A @ int @ W ) @ N )
=> ( ord_less @ nat @ N @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ).
% bin_nth_uint_imp
thf(fact_5877_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_5878_ucast__less__ucast__weak,axiom,
! [B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B ) )
=> ! [X: word @ A,Y: word @ A] :
( ( ord_less @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( type_len0_len_of @ B @ ( type2 @ B ) ) )
=> ( ( ord_less @ ( word @ B ) @ ( semiring_1_unsigned @ A @ ( word @ B ) @ X ) @ ( semiring_1_unsigned @ A @ ( word @ B ) @ Y ) )
= ( ord_less @ ( word @ A ) @ X @ Y ) ) ) ) ).
% ucast_less_ucast_weak
thf(fact_5879_ucast__ucast__id,axiom,
! [B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B ) )
=> ! [X: word @ A] :
( ( ord_less @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( type_len0_len_of @ B @ ( type2 @ B ) ) )
=> ( ( semiring_1_unsigned @ B @ ( word @ A ) @ ( semiring_1_unsigned @ A @ ( word @ B ) @ X ) )
= X ) ) ) ).
% ucast_ucast_id
thf(fact_5880_bit__word__of__int__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [K: int,N: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( ring_1_of_int @ ( word @ A ) @ K ) @ N )
= ( ( ord_less @ nat @ N @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
& ( bit_se5641148757651400278ts_bit @ int @ K @ N ) ) ) ) ).
% bit_word_of_int_iff
thf(fact_5881_test__bit__wi,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: int,N: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( ring_1_of_int @ ( word @ A ) @ X ) @ N )
= ( ( ord_less @ nat @ N @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
& ( bit_se5641148757651400278ts_bit @ int @ X @ N ) ) ) ) ).
% test_bit_wi
thf(fact_5882_signed__not__eq,axiom,
! [A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( bit_ri3973907225187159222ations @ A ) )
=> ! [W: word @ B] :
( ( ring_1_signed @ B @ A @ ( bit_ri4277139882892585799ns_not @ ( word @ B ) @ W ) )
= ( bit_ri4674362597316999326ke_bit @ A @ ( type_len0_len_of @ B @ ( type2 @ B ) ) @ ( bit_ri4277139882892585799ns_not @ A @ ( ring_1_signed @ B @ A @ W ) ) ) ) ) ).
% signed_not_eq
thf(fact_5883_unsigned__drop__bit__eq,axiom,
! [A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( bit_un5681908812861735899ations @ A ) )
=> ! [N: nat,W: word @ B] :
( ( semiring_1_unsigned @ B @ A @ ( bit_se4197421643247451524op_bit @ ( word @ B ) @ N @ W ) )
= ( bit_se4197421643247451524op_bit @ A @ N @ ( bit_se2584673776208193580ke_bit @ A @ ( type_len0_len_of @ B @ ( type2 @ B ) ) @ ( semiring_1_unsigned @ B @ A @ W ) ) ) ) ) ).
% unsigned_drop_bit_eq
thf(fact_5884_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_5885_unsigned__push__bit__eq,axiom,
! [A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( bit_se359711467146920520ations @ A ) )
=> ! [N: nat,W: word @ B] :
( ( semiring_1_unsigned @ B @ A @ ( bit_se4730199178511100633sh_bit @ ( word @ B ) @ N @ W ) )
= ( bit_se2584673776208193580ke_bit @ A @ ( type_len0_len_of @ B @ ( type2 @ B ) ) @ ( bit_se4730199178511100633sh_bit @ A @ N @ ( semiring_1_unsigned @ B @ A @ W ) ) ) ) ) ).
% unsigned_push_bit_eq
thf(fact_5886_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_5887_bit__word__cat__iff,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( type_len @ A )
& ( type_len @ C ) )
=> ! [V2: word @ A,W: word @ B,N: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ C ) @ ( word_cat @ A @ B @ C @ V2 @ W ) @ N )
= ( ( ord_less @ nat @ N @ ( type_len0_len_of @ C @ ( type2 @ C ) ) )
& ( ( ord_less @ nat @ N @ ( type_len0_len_of @ B @ ( type2 @ B ) ) )
=> ( bit_se5641148757651400278ts_bit @ ( word @ B ) @ W @ N ) )
& ( ~ ( ord_less @ nat @ N @ ( type_len0_len_of @ B @ ( type2 @ B ) ) )
=> ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ V2 @ ( minus_minus @ nat @ N @ ( type_len0_len_of @ B @ ( type2 @ B ) ) ) ) ) ) ) ) ).
% bit_word_cat_iff
thf(fact_5888_bit__set__bit__aux,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [F3: nat > $o,N: nat,W: word @ A,M: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( code_T2661198915054445665ts_aux @ A @ F3 @ N @ W ) @ M )
= ( ( ord_less @ nat @ M @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
& ( ( ord_less @ nat @ M @ N )
=> ( F3 @ M ) )
& ( ~ ( ord_less @ nat @ M @ N )
=> ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ W @ ( minus_minus @ nat @ M @ N ) ) ) ) ) ) ).
% bit_set_bit_aux
thf(fact_5889_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_5890_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_5891_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_5892_word__cat__eq_H,axiom,
! [B: $tType,C: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ C )
& ( type_len @ B ) )
=> ( ( word_cat @ A @ B @ C )
= ( ^ [A5: word @ A,B5: word @ B] : ( ring_1_of_int @ ( word @ C ) @ ( bit_concat_bit @ ( type_len0_len_of @ B @ ( type2 @ B ) ) @ ( semiring_1_unsigned @ B @ int @ B5 ) @ ( semiring_1_unsigned @ A @ int @ A5 ) ) ) ) ) ) ).
% word_cat_eq'
thf(fact_5893_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 )
= ( ^ [V3: 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 ) @ V3 ) ) @ ( semiring_1_unsigned @ B @ ( word @ C ) @ W2 ) ) ) ) ) ).
% word_cat_bin'
thf(fact_5894_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_5895_signed__take__bit__decr__length__iff,axiom,
! [A: $tType,B: $tType] :
( ( ( bit_ri3973907225187159222ations @ B )
& ( type_len @ A ) )
=> ! [K: B,L: B] :
( ( ( bit_ri4674362597316999326ke_bit @ B @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) @ K )
= ( bit_ri4674362597316999326ke_bit @ B @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) @ L ) )
= ( ( bit_se2584673776208193580ke_bit @ B @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ K )
= ( bit_se2584673776208193580ke_bit @ B @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ L ) ) ) ) ).
% signed_take_bit_decr_length_iff
thf(fact_5896_num__of__sbintr_H,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: num,B3: num] :
( ( ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( numeral_numeral @ int @ A3 ) )
= ( numeral_numeral @ int @ B3 ) )
=> ( ( numeral_numeral @ ( word @ A ) @ A3 )
= ( numeral_numeral @ ( word @ A ) @ B3 ) ) ) ) ).
% num_of_sbintr'
thf(fact_5897_word__split__bin_H,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B )
& ( type_len @ C ) )
=> ( ( word_split @ A @ B @ C )
= ( ^ [W2: word @ A] : ( product_Pair @ ( word @ B ) @ ( word @ C ) @ ( semiring_1_unsigned @ A @ ( word @ B ) @ ( bit_se4197421643247451524op_bit @ ( word @ A ) @ ( type_len0_len_of @ C @ ( type2 @ C ) ) @ W2 ) ) @ ( semiring_1_unsigned @ A @ ( word @ C ) @ W2 ) ) ) ) ) ).
% word_split_bin'
thf(fact_5898_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_5899_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_5900_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_5901_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_5902_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_5903_word__of__int__bin__cat__eq__iff,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B )
& ( type_len @ C ) )
=> ! [B3: word @ B,A3: word @ A,D2: word @ B,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 @ B3 ) @ ( semiring_1_unsigned @ A @ int @ A3 ) ) )
= ( ring_1_of_int @ ( word @ C ) @ ( bit_concat_bit @ ( type_len0_len_of @ B @ ( type2 @ B ) ) @ ( semiring_1_unsigned @ B @ int @ D2 ) @ ( semiring_1_unsigned @ A @ int @ C2 ) ) ) )
= ( ( B3 = D2 )
& ( A3 = C2 ) ) ) ) ) ).
% word_of_int_bin_cat_eq_iff
thf(fact_5904_mask__exceed,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,X: word @ A] :
( ( ord_less_eq @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ N )
=> ( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) )
= ( zero_zero @ ( word @ A ) ) ) ) ) ).
% mask_exceed
thf(fact_5905_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_5906_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_5907_word__nchotomy,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W5: word @ A] :
? [N2: nat] :
( ( W5
= ( 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_5908_word__nat__cases,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] :
~ ! [N2: nat] :
( ( X
= ( semiring_1_of_nat @ ( word @ A ) @ N2 ) )
=> ~ ( ord_less @ nat @ N2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).
% word_nat_cases
thf(fact_5909_of__nat__inj,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: nat,Y: nat] :
( ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
=> ( ( ord_less @ nat @ Y @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
=> ( ( ( semiring_1_of_nat @ ( word @ A ) @ X )
= ( semiring_1_of_nat @ ( word @ A ) @ Y ) )
= ( X = Y ) ) ) ) ) ).
% of_nat_inj
thf(fact_5910_word__of__nat__inj,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: nat,Y: nat] :
( ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
=> ( ( ord_less @ nat @ Y @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
=> ( ( ( semiring_1_of_nat @ ( word @ A ) @ X )
= ( semiring_1_of_nat @ ( word @ A ) @ Y ) )
=> ( X = Y ) ) ) ) ) ).
% word_of_nat_inj
thf(fact_5911_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_5912_word__power__increasing,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: nat,Y: nat] :
( ( ord_less @ ( word @ A ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ X ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ Y ) )
=> ( ( ord_less @ nat @ X @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ( ord_less @ nat @ Y @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ord_less @ nat @ X @ Y ) ) ) ) ) ).
% word_power_increasing
thf(fact_5913_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_5914_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_5915_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_5916_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_5917_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_5918_of__nat__neq__iff__word,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: nat,Y: nat] :
( ( ( modulo_modulo @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
!= ( modulo_modulo @ nat @ Y @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) )
=> ( ( ( semiring_1_of_nat @ ( word @ A ) @ X )
!= ( semiring_1_of_nat @ ( word @ A ) @ Y ) )
= ( X != Y ) ) ) ) ).
% of_nat_neq_iff_word
thf(fact_5919_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_5920_num__abs__sbintr,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( numeral_numeral @ ( word @ A ) )
= ( ^ [X2: num] : ( ring_1_of_int @ ( word @ A ) @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( numeral_numeral @ int @ X2 ) ) ) ) ) ) ).
% num_abs_sbintr
thf(fact_5921_ucast__mono,axiom,
! [A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( type_len @ A ) )
=> ! [X: word @ B,Y: word @ B] :
( ( ord_less @ ( word @ B ) @ X @ Y )
=> ( ( ord_less @ ( word @ B ) @ Y @ ( power_power @ ( word @ B ) @ ( numeral_numeral @ ( word @ B ) @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
=> ( ord_less @ ( word @ A ) @ ( semiring_1_unsigned @ B @ ( word @ A ) @ X ) @ ( semiring_1_unsigned @ B @ ( word @ A ) @ Y ) ) ) ) ) ).
% ucast_mono
thf(fact_5922_ucast__ucast__len,axiom,
! [B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B ) )
=> ! [X: word @ A] :
( ( ord_less @ ( word @ A ) @ X @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ B @ ( type2 @ B ) ) ) )
=> ( ( semiring_1_unsigned @ B @ ( word @ A ) @ ( semiring_1_unsigned @ A @ ( word @ B ) @ X ) )
= X ) ) ) ).
% ucast_ucast_len
thf(fact_5923_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_5924_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_5925_sint__word__ariths_I1_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A,B3: word @ A] :
( ( ring_1_signed @ A @ int @ ( plus_plus @ ( word @ A ) @ A3 @ B3 ) )
= ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( plus_plus @ int @ ( ring_1_signed @ A @ int @ A3 ) @ ( ring_1_signed @ A @ int @ B3 ) ) ) ) ) ).
% sint_word_ariths(1)
thf(fact_5926_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_5927_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_5928_sint__word__ariths_I4_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A] :
( ( ring_1_signed @ A @ int @ ( uminus_uminus @ ( word @ A ) @ A3 ) )
= ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( uminus_uminus @ int @ ( ring_1_signed @ A @ int @ A3 ) ) ) ) ) ).
% sint_word_ariths(4)
thf(fact_5929_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_5930_sint__word__ariths_I2_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A,B3: word @ A] :
( ( ring_1_signed @ A @ int @ ( minus_minus @ ( word @ A ) @ A3 @ B3 ) )
= ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( minus_minus @ int @ ( ring_1_signed @ A @ int @ A3 ) @ ( ring_1_signed @ A @ int @ B3 ) ) ) ) ) ).
% sint_word_ariths(2)
thf(fact_5931_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_5932_sint__word__ariths_I3_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A,B3: word @ A] :
( ( ring_1_signed @ A @ int @ ( times_times @ ( word @ A ) @ A3 @ B3 ) )
= ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( times_times @ int @ ( ring_1_signed @ A @ int @ A3 ) @ ( ring_1_signed @ A @ int @ B3 ) ) ) ) ) ).
% sint_word_ariths(3)
thf(fact_5933_less__Suc__unat__less__bound,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,X: word @ A] :
( ( ord_less @ nat @ N @ ( suc @ ( semiring_1_unsigned @ A @ nat @ X ) ) )
=> ( ord_less @ nat @ N @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).
% less_Suc_unat_less_bound
thf(fact_5934_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_5935_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_5936_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_5937_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_5938_unat__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [B3: num] :
( ( semiring_1_unsigned @ A @ nat @ ( numeral_numeral @ ( word @ A ) @ B3 ) )
= ( modulo_modulo @ nat @ ( numeral_numeral @ nat @ B3 ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).
% unat_numeral
thf(fact_5939_of__nat__mono__maybe,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: nat,Y: nat] :
( ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
=> ( ( ord_less @ nat @ Y @ X )
=> ( ord_less @ ( word @ A ) @ ( semiring_1_of_nat @ ( word @ A ) @ Y ) @ ( semiring_1_of_nat @ ( word @ A ) @ X ) ) ) ) ) ).
% of_nat_mono_maybe
thf(fact_5940_of__nat__mono__maybe_H,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: nat,Y: nat] :
( ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
=> ( ( ord_less @ nat @ Y @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
=> ( ( ord_less @ nat @ Y @ X )
= ( ord_less @ ( word @ A ) @ ( semiring_1_of_nat @ ( word @ A ) @ Y ) @ ( semiring_1_of_nat @ ( word @ A ) @ X ) ) ) ) ) ) ).
% of_nat_mono_maybe'
thf(fact_5941_unat__split,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [P: nat > $o,X: word @ A] :
( ( P @ ( semiring_1_unsigned @ A @ nat @ X ) )
= ( ! [N4: nat] :
( ( ( ( semiring_1_of_nat @ ( word @ A ) @ N4 )
= X )
& ( ord_less @ nat @ N4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) )
=> ( P @ N4 ) ) ) ) ) ).
% unat_split
thf(fact_5942_of__nat__inverse,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [R3: nat,A3: word @ A] :
( ( ( semiring_1_of_nat @ ( word @ A ) @ R3 )
= A3 )
=> ( ( ord_less @ nat @ R3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
=> ( ( semiring_1_unsigned @ A @ nat @ A3 )
= R3 ) ) ) ) ).
% of_nat_inverse
thf(fact_5943_unat__split__asm,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [P: nat > $o,X: word @ A] :
( ( P @ ( semiring_1_unsigned @ A @ nat @ X ) )
= ( ~ ? [N4: nat] :
( ( ( semiring_1_of_nat @ ( word @ A ) @ N4 )
= X )
& ( ord_less @ nat @ N4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
& ~ ( P @ N4 ) ) ) ) ) ).
% unat_split_asm
thf(fact_5944_unat__eq__of__nat,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,X: word @ A] :
( ( ord_less @ nat @ N @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
=> ( ( ( semiring_1_unsigned @ A @ nat @ X )
= N )
= ( X
= ( semiring_1_of_nat @ ( word @ A ) @ N ) ) ) ) ) ).
% unat_eq_of_nat
thf(fact_5945_unat__of__nat__len,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: nat] :
( ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
=> ( ( semiring_1_unsigned @ A @ nat @ ( semiring_1_of_nat @ ( word @ A ) @ X ) )
= X ) ) ) ).
% unat_of_nat_len
thf(fact_5946_x__less__2__0__1_H,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] :
( ( ( type_len0_len_of @ A @ ( type2 @ A ) )
!= ( one_one @ nat ) )
=> ( ( ord_less @ ( word @ A ) @ X @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) )
=> ( ( X
= ( zero_zero @ ( word @ A ) ) )
| ( X
= ( one_one @ ( word @ A ) ) ) ) ) ) ) ).
% x_less_2_0_1'
thf(fact_5947_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_5948_Word_Oof__nat__0,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [M: nat] :
( ( ( semiring_1_of_nat @ ( word @ A ) @ M )
= ( zero_zero @ ( word @ A ) ) )
= ( ? [Q5: nat] :
( M
= ( times_times @ nat @ Q5 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ) ) ).
% Word.of_nat_0
thf(fact_5949_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_5950_ucast__of__nat__small,axiom,
! [A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( type_len @ A ) )
=> ! [X: nat] :
( ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
=> ( ( semiring_1_unsigned @ A @ ( word @ B ) @ ( semiring_1_of_nat @ ( word @ A ) @ X ) )
= ( semiring_1_of_nat @ ( word @ B ) @ X ) ) ) ) ).
% ucast_of_nat_small
thf(fact_5951_uint__sub__lt2p,axiom,
! [B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B ) )
=> ! [X: word @ A,Y: word @ B] : ( ord_less @ int @ ( minus_minus @ int @ ( semiring_1_unsigned @ A @ int @ X ) @ ( semiring_1_unsigned @ B @ int @ Y ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ).
% uint_sub_lt2p
thf(fact_5952_uint__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [B3: num] :
( ( semiring_1_unsigned @ A @ int @ ( numeral_numeral @ ( word @ A ) @ B3 ) )
= ( modulo_modulo @ int @ ( numeral_numeral @ int @ B3 ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).
% uint_numeral
thf(fact_5953_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_5954_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_5955_unat__ucast,axiom,
! [B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B ) )
=> ! [X: word @ B] :
( ( semiring_1_unsigned @ A @ nat @ ( semiring_1_unsigned @ B @ ( word @ A ) @ X ) )
= ( modulo_modulo @ nat @ ( semiring_1_unsigned @ B @ nat @ X ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).
% unat_ucast
thf(fact_5956_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_5957_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_5958_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_5959_unat__of__nat,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: nat] :
( ( semiring_1_unsigned @ A @ nat @ ( semiring_1_of_nat @ ( word @ A ) @ X ) )
= ( modulo_modulo @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).
% unat_of_nat
thf(fact_5960_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_5961_ucast__less,axiom,
! [A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( type_len @ A ) )
=> ! [X: word @ B] :
( ( ord_less @ nat @ ( type_len0_len_of @ B @ ( type2 @ B ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ord_less @ ( word @ A ) @ ( semiring_1_unsigned @ B @ ( word @ A ) @ X ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ B @ ( type2 @ B ) ) ) ) ) ) ).
% ucast_less
thf(fact_5962_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_5963_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_5964_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_5965_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_5966_complement__nth__w2p,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N7: nat,N: nat] :
( ( ord_less @ nat @ N7 @ ( 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 ) ) @ N7 )
= ( N7 != N ) ) ) ) ).
% complement_nth_w2p
thf(fact_5967_upper__trivial,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] :
( ( X
!= ( minus_minus @ ( word @ A ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) @ ( one_one @ ( word @ A ) ) ) )
=> ( ord_less @ ( word @ A ) @ X @ ( minus_minus @ ( word @ A ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) @ ( one_one @ ( word @ A ) ) ) ) ) ) ).
% upper_trivial
thf(fact_5968_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_5969_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_5970_word__power__less__diff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,Q3: word @ A,M: nat] :
( ( ord_less @ ( word @ A ) @ ( times_times @ ( word @ A ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) @ Q3 ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ M ) )
=> ( ( ord_less @ ( word @ A ) @ Q3 @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ N ) ) )
=> ( ord_less @ ( word @ A ) @ Q3 @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ M @ N ) ) ) ) ) ) ).
% word_power_less_diff
thf(fact_5971_unat__word__ariths_I7_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A,B3: word @ A] :
( ( semiring_1_unsigned @ A @ nat @ ( modulo_modulo @ ( word @ A ) @ A3 @ B3 ) )
= ( modulo_modulo @ nat @ ( modulo_modulo @ nat @ ( semiring_1_unsigned @ A @ nat @ A3 ) @ ( semiring_1_unsigned @ A @ nat @ B3 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).
% unat_word_ariths(7)
thf(fact_5972_ucast__mono__le,axiom,
! [B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B ) )
=> ! [X: word @ A,Y: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ X @ Y )
=> ( ( ord_less @ ( word @ A ) @ Y @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ B @ ( type2 @ B ) ) ) )
=> ( ord_less_eq @ ( word @ B ) @ ( semiring_1_unsigned @ A @ ( word @ B ) @ X ) @ ( semiring_1_unsigned @ A @ ( word @ B ) @ Y ) ) ) ) ) ).
% ucast_mono_le
thf(fact_5973_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_5974_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_5975_msb0,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A,I: nat] :
( ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( bit_se1065995026697491101ons_or @ ( word @ A ) @ X @ Y ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ I ) ) )
= ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ X @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ I ) ) )
| ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ Y @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ I ) ) ) ) )
& ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X @ Y ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ I ) ) )
= ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ X @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ I ) ) )
& ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ Y @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ I ) ) ) ) )
& ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( bit_se5824344971392196577ns_xor @ ( word @ A ) @ X @ Y ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ I ) ) )
= ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ X @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ I ) ) )
!= ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ Y @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ I ) ) ) ) )
& ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ X ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ I ) ) )
= ( ~ ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ X @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ I ) ) ) ) ) ) ) ).
% msb0
thf(fact_5976_unat__add__lem,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] :
( ( ord_less @ nat @ ( plus_plus @ nat @ ( semiring_1_unsigned @ A @ nat @ X ) @ ( semiring_1_unsigned @ A @ nat @ Y ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
= ( ( semiring_1_unsigned @ A @ nat @ ( plus_plus @ ( word @ A ) @ X @ Y ) )
= ( plus_plus @ nat @ ( semiring_1_unsigned @ A @ nat @ X ) @ ( semiring_1_unsigned @ A @ nat @ Y ) ) ) ) ) ).
% unat_add_lem
thf(fact_5977_unat__add__lem_H,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] :
( ( ord_less @ nat @ ( plus_plus @ nat @ ( semiring_1_unsigned @ A @ nat @ X ) @ ( semiring_1_unsigned @ A @ nat @ Y ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
=> ( ( semiring_1_unsigned @ A @ nat @ ( plus_plus @ ( word @ A ) @ X @ Y ) )
= ( plus_plus @ nat @ ( semiring_1_unsigned @ A @ nat @ X ) @ ( semiring_1_unsigned @ A @ nat @ Y ) ) ) ) ) ).
% unat_add_lem'
thf(fact_5978_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_5979_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_5980_of__nat__mono__maybe__le,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: nat,Y: nat] :
( ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
=> ( ( ord_less @ nat @ Y @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
=> ( ( ord_less_eq @ nat @ Y @ X )
= ( ord_less_eq @ ( word @ A ) @ ( semiring_1_of_nat @ ( word @ A ) @ Y ) @ ( semiring_1_of_nat @ ( word @ A ) @ X ) ) ) ) ) ) ).
% of_nat_mono_maybe_le
thf(fact_5981_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_5982_unat__word__ariths_I1_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A,B3: word @ A] :
( ( semiring_1_unsigned @ A @ nat @ ( plus_plus @ ( word @ A ) @ A3 @ B3 ) )
= ( modulo_modulo @ nat @ ( plus_plus @ nat @ ( semiring_1_unsigned @ A @ nat @ A3 ) @ ( semiring_1_unsigned @ A @ nat @ B3 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).
% unat_word_ariths(1)
thf(fact_5983_bool__mask_H,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] :
( ( ord_less @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ( ord_less @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X @ ( one_one @ ( word @ A ) ) ) )
= ( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X @ ( one_one @ ( word @ A ) ) )
= ( one_one @ ( word @ A ) ) ) ) ) ) ).
% bool_mask'
thf(fact_5984_uint__range_H,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( semiring_1_unsigned @ A @ int @ X ) )
& ( ord_less @ int @ ( semiring_1_unsigned @ A @ int @ X ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).
% uint_range'
thf(fact_5985_sint__lt,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] : ( ord_less @ int @ ( ring_1_signed @ A @ int @ X ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) ) ) ) ).
% sint_lt
thf(fact_5986_ucast__mono__le_H,axiom,
! [B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B ) )
=> ! [Y: word @ A,X: word @ A] :
( ( ord_less @ nat @ ( semiring_1_unsigned @ A @ nat @ Y ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ B @ ( type2 @ B ) ) ) )
=> ( ( ord_less @ nat @ ( type_len0_len_of @ B @ ( type2 @ B ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ( ord_less_eq @ ( word @ A ) @ X @ Y )
=> ( ord_less_eq @ ( word @ B ) @ ( semiring_1_unsigned @ A @ ( word @ B ) @ X ) @ ( semiring_1_unsigned @ A @ ( word @ B ) @ Y ) ) ) ) ) ) ).
% ucast_mono_le'
thf(fact_5987_of__nat__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,W: word @ A] :
( ( ( semiring_1_of_nat @ ( word @ A ) @ N )
= W )
= ( ? [Q5: nat] :
( N
= ( plus_plus @ nat @ ( semiring_1_unsigned @ A @ nat @ W ) @ ( times_times @ nat @ Q5 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ) ) ) ).
% of_nat_eq
thf(fact_5988_unat__mult__lem,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] :
( ( ord_less @ nat @ ( times_times @ nat @ ( semiring_1_unsigned @ A @ nat @ X ) @ ( semiring_1_unsigned @ A @ nat @ Y ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
= ( ( semiring_1_unsigned @ A @ nat @ ( times_times @ ( word @ A ) @ X @ Y ) )
= ( times_times @ nat @ ( semiring_1_unsigned @ A @ nat @ X ) @ ( semiring_1_unsigned @ A @ nat @ Y ) ) ) ) ) ).
% unat_mult_lem
thf(fact_5989_word__of__int__inj,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: int,Y: int] :
( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
& ( ord_less @ int @ X @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) )
=> ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y )
& ( ord_less @ int @ Y @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) )
=> ( ( ( ring_1_of_int @ ( word @ A ) @ X )
= ( ring_1_of_int @ ( word @ A ) @ Y ) )
= ( X = Y ) ) ) ) ) ).
% word_of_int_inj
thf(fact_5990_word__int__cases,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] :
~ ! [N2: int] :
( ( X
= ( ring_1_of_int @ ( word @ A ) @ N2 ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ N2 )
=> ~ ( ord_less @ int @ N2 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ) ).
% word_int_cases
thf(fact_5991_unat__ucast__no__overflow__le,axiom,
! [A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( type_len @ A ) )
=> ! [B3: word @ B,F3: word @ A] :
( ( ord_less @ nat @ ( semiring_1_unsigned @ B @ nat @ B3 ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
=> ( ( ord_less @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( type_len0_len_of @ B @ ( type2 @ B ) ) )
=> ( ( ord_less @ ( word @ B ) @ ( semiring_1_unsigned @ A @ ( word @ B ) @ F3 ) @ B3 )
= ( ord_less @ nat @ ( semiring_1_unsigned @ A @ nat @ F3 ) @ ( semiring_1_unsigned @ B @ nat @ B3 ) ) ) ) ) ) ).
% unat_ucast_no_overflow_le
thf(fact_5992_uint__m2p__not__non__neg,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] :
~ ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( minus_minus @ int @ ( semiring_1_unsigned @ A @ int @ X ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).
% uint_m2p_not_non_neg
thf(fact_5993_uint__m2p__neg,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] : ( ord_less @ int @ ( minus_minus @ int @ ( semiring_1_unsigned @ A @ int @ X ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) @ ( zero_zero @ int ) ) ) ).
% uint_m2p_neg
thf(fact_5994_unat__ucast__less__no__overflow,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,F3: word @ A] :
( ( ord_less @ nat @ N @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
=> ( ( ord_less @ nat @ ( semiring_1_unsigned @ A @ nat @ F3 ) @ N )
=> ( ord_less @ ( word @ A ) @ F3 @ ( semiring_1_of_nat @ ( word @ A ) @ N ) ) ) ) ) ).
% unat_ucast_less_no_overflow
thf(fact_5995_unat__ucast__less__no__overflow__simp,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,F3: word @ A] :
( ( ord_less @ nat @ N @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
=> ( ( ord_less @ nat @ ( semiring_1_unsigned @ A @ nat @ F3 ) @ N )
= ( ord_less @ ( word @ A ) @ F3 @ ( semiring_1_of_nat @ ( word @ A ) @ N ) ) ) ) ) ).
% unat_ucast_less_no_overflow_simp
thf(fact_5996_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_5997_nth__bounded,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,N: nat,M: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ X @ N )
=> ( ( ord_less @ ( word @ A ) @ X @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ M ) )
=> ( ( ord_less_eq @ nat @ M @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ord_less @ nat @ N @ M ) ) ) ) ) ).
% nth_bounded
thf(fact_5998_upper__bits__unset__is__l2p,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,P4: word @ A] :
( ( ord_less @ nat @ N @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ( ! [N13: nat] :
( ( ord_less_eq @ nat @ N @ N13 )
=> ( ( ord_less @ nat @ N13 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ~ ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ P4 @ N13 ) ) ) )
= ( ord_less @ ( word @ A ) @ P4 @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) ) ) ) ) ).
% upper_bits_unset_is_l2p
thf(fact_5999_less__2p__is__upper__bits__unset,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [P4: word @ A,N: nat] :
( ( ord_less @ ( word @ A ) @ P4 @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) )
= ( ( ord_less @ nat @ N @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
& ! [N13: nat] :
( ( ord_less_eq @ nat @ N @ N13 )
=> ( ( ord_less @ nat @ N13 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ~ ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ P4 @ N13 ) ) ) ) ) ) ).
% less_2p_is_upper_bits_unset
thf(fact_6000_uint__add__lem,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] :
( ( ord_less @ int @ ( plus_plus @ int @ ( semiring_1_unsigned @ A @ int @ X ) @ ( semiring_1_unsigned @ A @ int @ Y ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
= ( ( semiring_1_unsigned @ A @ int @ ( plus_plus @ ( word @ A ) @ X @ Y ) )
= ( plus_plus @ int @ ( semiring_1_unsigned @ A @ int @ X ) @ ( semiring_1_unsigned @ A @ int @ Y ) ) ) ) ) ).
% uint_add_lem
thf(fact_6001_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_6002_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_6003_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_6004_uint__word__ariths_I1_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A,B3: word @ A] :
( ( semiring_1_unsigned @ A @ int @ ( plus_plus @ ( word @ A ) @ A3 @ B3 ) )
= ( modulo_modulo @ int @ ( plus_plus @ int @ ( semiring_1_unsigned @ A @ int @ A3 ) @ ( semiring_1_unsigned @ A @ int @ B3 ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).
% uint_word_ariths(1)
thf(fact_6005_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_6006_unat__word__ariths_I2_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A,B3: word @ A] :
( ( semiring_1_unsigned @ A @ nat @ ( times_times @ ( word @ A ) @ A3 @ B3 ) )
= ( modulo_modulo @ nat @ ( times_times @ nat @ ( semiring_1_unsigned @ A @ nat @ A3 ) @ ( semiring_1_unsigned @ A @ nat @ B3 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).
% unat_word_ariths(2)
thf(fact_6007_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_6008_unat__word__ariths_I6_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A,B3: word @ A] :
( ( semiring_1_unsigned @ A @ nat @ ( divide_divide @ ( word @ A ) @ A3 @ B3 ) )
= ( modulo_modulo @ nat @ ( divide_divide @ nat @ ( semiring_1_unsigned @ A @ nat @ A3 ) @ ( semiring_1_unsigned @ A @ nat @ B3 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).
% unat_word_ariths(6)
thf(fact_6009_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_6010_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_6011_uint__word__ariths_I4_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A] :
( ( semiring_1_unsigned @ A @ int @ ( uminus_uminus @ ( word @ A ) @ A3 ) )
= ( modulo_modulo @ int @ ( uminus_uminus @ int @ ( semiring_1_unsigned @ A @ int @ A3 ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).
% uint_word_ariths(4)
thf(fact_6012_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_6013_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_6014_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_6015_sint__1__cases,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A] :
( ( ( ( type_len0_len_of @ A @ ( type2 @ A ) )
= ( one_one @ nat ) )
=> ( ( A3
= ( zero_zero @ ( word @ A ) ) )
=> ( ( ring_1_signed @ A @ int @ A3 )
!= ( zero_zero @ int ) ) ) )
=> ( ( ( ( type_len0_len_of @ A @ ( type2 @ A ) )
= ( one_one @ nat ) )
=> ( ( A3
= ( one_one @ ( word @ A ) ) )
=> ( ( ring_1_signed @ A @ int @ ( one_one @ ( word @ A ) ) )
!= ( uminus_uminus @ int @ ( one_one @ int ) ) ) ) )
=> ~ ( ( ord_less @ nat @ ( one_one @ nat ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ( ring_1_signed @ A @ int @ ( one_one @ ( word @ A ) ) )
!= ( one_one @ int ) ) ) ) ) ) ).
% sint_1_cases
thf(fact_6016_uint__word__ariths_I2_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A,B3: word @ A] :
( ( semiring_1_unsigned @ A @ int @ ( minus_minus @ ( word @ A ) @ A3 @ B3 ) )
= ( modulo_modulo @ int @ ( minus_minus @ int @ ( semiring_1_unsigned @ A @ int @ A3 ) @ ( semiring_1_unsigned @ A @ int @ B3 ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).
% uint_word_ariths(2)
thf(fact_6017_uint__mult__lem,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] :
( ( ord_less @ int @ ( times_times @ int @ ( semiring_1_unsigned @ A @ int @ X ) @ ( semiring_1_unsigned @ A @ int @ Y ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
= ( ( semiring_1_unsigned @ A @ int @ ( times_times @ ( word @ A ) @ X @ Y ) )
= ( times_times @ int @ ( semiring_1_unsigned @ A @ int @ X ) @ ( semiring_1_unsigned @ A @ int @ Y ) ) ) ) ) ).
% uint_mult_lem
thf(fact_6018_uint__word__ariths_I3_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A,B3: word @ A] :
( ( semiring_1_unsigned @ A @ int @ ( times_times @ ( word @ A ) @ A3 @ B3 ) )
= ( modulo_modulo @ int @ ( times_times @ int @ ( semiring_1_unsigned @ A @ int @ A3 ) @ ( semiring_1_unsigned @ A @ int @ B3 ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).
% uint_word_ariths(3)
thf(fact_6019_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_6020_word__power__mod__div,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,M: nat,X: word @ A] :
( ( ord_less @ nat @ N @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ( ord_less @ nat @ M @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ( divide_divide @ ( word @ A ) @ ( modulo_modulo @ ( word @ A ) @ X @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ M ) )
= ( modulo_modulo @ ( word @ A ) @ ( divide_divide @ ( word @ A ) @ X @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ M ) ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ N @ M ) ) ) ) ) ) ) ).
% word_power_mod_div
thf(fact_6021_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_6022_msb1,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] :
( ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( bit_se1065995026697491101ons_or @ ( word @ A ) @ X @ Y ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) )
= ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ X @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) )
| ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ Y @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ) )
& ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X @ Y ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) )
= ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ X @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) )
& ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ Y @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ) )
& ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( bit_se5824344971392196577ns_xor @ ( word @ A ) @ X @ Y ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) )
= ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ X @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) )
!= ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ Y @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ) )
& ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ X ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) )
= ( ~ ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ X @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ) ).
% msb1
thf(fact_6023_unat__plus__if_H,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A,B3: word @ A] :
( ( ( ord_less @ nat @ ( plus_plus @ nat @ ( semiring_1_unsigned @ A @ nat @ A3 ) @ ( semiring_1_unsigned @ A @ nat @ B3 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
=> ( ( semiring_1_unsigned @ A @ nat @ ( plus_plus @ ( word @ A ) @ A3 @ B3 ) )
= ( plus_plus @ nat @ ( semiring_1_unsigned @ A @ nat @ A3 ) @ ( semiring_1_unsigned @ A @ nat @ B3 ) ) ) )
& ( ~ ( ord_less @ nat @ ( plus_plus @ nat @ ( semiring_1_unsigned @ A @ nat @ A3 ) @ ( semiring_1_unsigned @ A @ nat @ B3 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
=> ( ( semiring_1_unsigned @ A @ nat @ ( plus_plus @ ( word @ A ) @ A3 @ B3 ) )
= ( minus_minus @ nat @ ( plus_plus @ nat @ ( semiring_1_unsigned @ A @ nat @ A3 ) @ ( semiring_1_unsigned @ A @ nat @ B3 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ) ) ).
% unat_plus_if'
thf(fact_6024_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_6025_unat__sub__if_H,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y: word @ A,X: word @ A] :
( ( ( ord_less_eq @ nat @ ( semiring_1_unsigned @ A @ nat @ Y ) @ ( semiring_1_unsigned @ A @ nat @ X ) )
=> ( ( semiring_1_unsigned @ A @ nat @ ( minus_minus @ ( word @ A ) @ X @ Y ) )
= ( minus_minus @ nat @ ( semiring_1_unsigned @ A @ nat @ X ) @ ( semiring_1_unsigned @ A @ nat @ Y ) ) ) )
& ( ~ ( ord_less_eq @ nat @ ( semiring_1_unsigned @ A @ nat @ Y ) @ ( semiring_1_unsigned @ A @ nat @ X ) )
=> ( ( semiring_1_unsigned @ A @ nat @ ( minus_minus @ ( word @ A ) @ X @ Y ) )
= ( minus_minus @ nat @ ( plus_plus @ nat @ ( semiring_1_unsigned @ A @ nat @ X ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) @ ( semiring_1_unsigned @ A @ nat @ Y ) ) ) ) ) ) ).
% unat_sub_if'
thf(fact_6026_no__olen__add__nat,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ X @ ( plus_plus @ ( word @ A ) @ X @ Y ) )
= ( ord_less @ nat @ ( plus_plus @ nat @ ( semiring_1_unsigned @ A @ nat @ X ) @ ( semiring_1_unsigned @ A @ nat @ Y ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).
% no_olen_add_nat
thf(fact_6027_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_6028_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_6029_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_6030_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_6031_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_6032_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_6033_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_6034_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_6035_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_6036_sint__ge,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] : ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) ) ) @ ( ring_1_signed @ A @ int @ X ) ) ) ).
% sint_ge
thf(fact_6037_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_6038_word__of__int__inverse,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [R3: int,A3: word @ A] :
( ( ( ring_1_of_int @ ( word @ A ) @ R3 )
= A3 )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ R3 )
=> ( ( ord_less @ int @ R3 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
=> ( ( semiring_1_unsigned @ A @ int @ A3 )
= R3 ) ) ) ) ) ).
% word_of_int_inverse
thf(fact_6039_uint__split__asm,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [P: int > $o,X: word @ A] :
( ( P @ ( semiring_1_unsigned @ A @ int @ X ) )
= ( ~ ? [I4: int] :
( ( ( ring_1_of_int @ ( word @ A ) @ I4 )
= X )
& ( ord_less_eq @ int @ ( zero_zero @ int ) @ I4 )
& ( ord_less @ int @ I4 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
& ~ ( P @ I4 ) ) ) ) ) ).
% uint_split_asm
thf(fact_6040_uint__split,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [P: int > $o,X: word @ A] :
( ( P @ ( semiring_1_unsigned @ A @ int @ X ) )
= ( ! [I4: int] :
( ( ( ( ring_1_of_int @ ( word @ A ) @ I4 )
= X )
& ( ord_less_eq @ int @ ( zero_zero @ int ) @ I4 )
& ( ord_less @ int @ I4 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) )
=> ( P @ I4 ) ) ) ) ) ).
% uint_split
thf(fact_6041_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_6042_div__lt_H,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [I: word @ A,K: word @ A,X: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ I @ ( divide_divide @ ( word @ A ) @ K @ X ) )
=> ( ord_less @ nat @ ( times_times @ nat @ ( semiring_1_unsigned @ A @ nat @ I ) @ ( semiring_1_unsigned @ A @ nat @ X ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).
% div_lt'
thf(fact_6043_div__lt_H_H,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [I: word @ A,K: word @ A,X: word @ A] :
( ( ord_less @ ( word @ A ) @ I @ ( divide_divide @ ( word @ A ) @ K @ X ) )
=> ( ord_less @ nat @ ( times_times @ nat @ ( semiring_1_unsigned @ A @ nat @ I ) @ ( semiring_1_unsigned @ A @ nat @ X ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).
% div_lt''
thf(fact_6044_double__eq__zero__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A] :
( ( ( times_times @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ A3 )
= ( zero_zero @ ( word @ A ) ) )
= ( ( A3
= ( zero_zero @ ( word @ A ) ) )
| ( A3
= ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ) ).
% double_eq_zero_iff
thf(fact_6045_word__le__exists_H,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ X @ Y )
=> ? [Z3: word @ A] :
( ( Y
= ( plus_plus @ ( word @ A ) @ X @ Z3 ) )
& ( ord_less @ int @ ( plus_plus @ int @ ( semiring_1_unsigned @ A @ int @ X ) @ ( semiring_1_unsigned @ A @ int @ Z3 ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ) ).
% word_le_exists'
thf(fact_6046_no__olen__add_H,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ X @ ( plus_plus @ ( word @ A ) @ Y @ X ) )
= ( ord_less @ int @ ( plus_plus @ int @ ( semiring_1_unsigned @ A @ int @ Y ) @ ( semiring_1_unsigned @ A @ int @ X ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).
% no_olen_add'
thf(fact_6047_no__olen__add,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ X @ ( plus_plus @ ( word @ A ) @ X @ Y ) )
= ( ord_less @ int @ ( plus_plus @ int @ ( semiring_1_unsigned @ A @ int @ X ) @ ( semiring_1_unsigned @ A @ int @ Y ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).
% no_olen_add
thf(fact_6048_More__Word_Oof__nat__power,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [P4: nat,X: nat] :
( ( ord_less @ nat @ P4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ X ) )
=> ( ( ord_less @ nat @ X @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ord_less @ ( word @ A ) @ ( semiring_1_of_nat @ ( word @ A ) @ P4 ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ X ) ) ) ) ) ).
% More_Word.of_nat_power
thf(fact_6049_uint__plus__if_H,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A,B3: word @ A] :
( ( ( ord_less @ int @ ( plus_plus @ int @ ( semiring_1_unsigned @ A @ int @ A3 ) @ ( semiring_1_unsigned @ A @ int @ B3 ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
=> ( ( semiring_1_unsigned @ A @ int @ ( plus_plus @ ( word @ A ) @ A3 @ B3 ) )
= ( plus_plus @ int @ ( semiring_1_unsigned @ A @ int @ A3 ) @ ( semiring_1_unsigned @ A @ int @ B3 ) ) ) )
& ( ~ ( ord_less @ int @ ( plus_plus @ int @ ( semiring_1_unsigned @ A @ int @ A3 ) @ ( semiring_1_unsigned @ A @ int @ B3 ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
=> ( ( semiring_1_unsigned @ A @ int @ ( plus_plus @ ( word @ A ) @ A3 @ B3 ) )
= ( minus_minus @ int @ ( plus_plus @ int @ ( semiring_1_unsigned @ A @ int @ A3 ) @ ( semiring_1_unsigned @ A @ int @ B3 ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ) ) ).
% uint_plus_if'
thf(fact_6050_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_6051_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_6052_uint__sub__if_H,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [B3: word @ A,A3: word @ A] :
( ( ( ord_less_eq @ int @ ( semiring_1_unsigned @ A @ int @ B3 ) @ ( semiring_1_unsigned @ A @ int @ A3 ) )
=> ( ( semiring_1_unsigned @ A @ int @ ( minus_minus @ ( word @ A ) @ A3 @ B3 ) )
= ( minus_minus @ int @ ( semiring_1_unsigned @ A @ int @ A3 ) @ ( semiring_1_unsigned @ A @ int @ B3 ) ) ) )
& ( ~ ( ord_less_eq @ int @ ( semiring_1_unsigned @ A @ int @ B3 ) @ ( semiring_1_unsigned @ A @ int @ A3 ) )
=> ( ( semiring_1_unsigned @ A @ int @ ( minus_minus @ ( word @ A ) @ A3 @ B3 ) )
= ( plus_plus @ int @ ( minus_minus @ int @ ( semiring_1_unsigned @ A @ int @ A3 ) @ ( semiring_1_unsigned @ A @ int @ B3 ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ) ) ).
% uint_sub_if'
thf(fact_6053_word__less__two__pow__divI,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,N: nat,M: nat] :
( ( ord_less @ ( word @ A ) @ X @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ N @ M ) ) )
=> ( ( ord_less_eq @ nat @ M @ N )
=> ( ( ord_less @ nat @ N @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ord_less @ ( word @ A ) @ X @ ( divide_divide @ ( word @ A ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ M ) ) ) ) ) ) ) ).
% word_less_two_pow_divI
thf(fact_6054_uint__neg__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [B3: num] :
( ( semiring_1_unsigned @ A @ int @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ B3 ) ) )
= ( modulo_modulo @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ B3 ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).
% uint_neg_numeral
thf(fact_6055_word__power__nonzero,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,N: nat] :
( ( ord_less @ ( word @ A ) @ X @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ N ) ) )
=> ( ( ord_less @ nat @ N @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ( X
!= ( zero_zero @ ( word @ A ) ) )
=> ( ( times_times @ ( word @ A ) @ X @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) )
!= ( zero_zero @ ( word @ A ) ) ) ) ) ) ) ).
% word_power_nonzero
thf(fact_6056_mult__pow2__inj,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [M: nat,N: nat,X: word @ A,Y: word @ A] :
( ( ord_less_eq @ nat @ ( plus_plus @ nat @ M @ N ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ( ord_less_eq @ ( word @ A ) @ X @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ M ) )
=> ( ( ord_less_eq @ ( word @ A ) @ Y @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ M ) )
=> ( ( ( times_times @ ( word @ A ) @ X @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) )
= ( times_times @ ( word @ A ) @ Y @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) ) )
=> ( X = Y ) ) ) ) ) ) ).
% mult_pow2_inj
thf(fact_6057_div__lt__uint_H,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [I: word @ A,K: word @ A,X: word @ A] :
( ( ord_less_eq @ ( word @ A ) @ I @ ( divide_divide @ ( word @ A ) @ K @ X ) )
=> ( ord_less @ int @ ( times_times @ int @ ( semiring_1_unsigned @ A @ int @ I ) @ ( semiring_1_unsigned @ A @ int @ X ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).
% div_lt_uint'
thf(fact_6058_div__lt__uint_H_H,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [I: word @ A,K: word @ A,X: word @ A] :
( ( ord_less @ ( word @ A ) @ I @ ( divide_divide @ ( word @ A ) @ K @ X ) )
=> ( ord_less @ int @ ( times_times @ int @ ( semiring_1_unsigned @ A @ int @ I ) @ ( semiring_1_unsigned @ A @ int @ X ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).
% div_lt_uint''
thf(fact_6059_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_6060_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_6061_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_6062_int__eq__sint,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: nat] :
( ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) ) )
=> ( ( ring_1_signed @ A @ int @ ( semiring_1_of_nat @ ( word @ A ) @ X ) )
= ( semiring_1_of_nat @ int @ X ) ) ) ) ).
% int_eq_sint
thf(fact_6063_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_6064_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_6065_smod__word__max,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A,B3: word @ A] : ( ord_less @ int @ ( signed6721504322012087516modulo @ int @ ( ring_1_signed @ A @ int @ A3 ) @ ( ring_1_signed @ A @ int @ B3 ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ).
% smod_word_max
thf(fact_6066_le2p__bits__unset,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [P4: word @ A,N: nat] :
( ( ord_less_eq @ ( word @ A ) @ P4 @ ( minus_minus @ ( word @ A ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) @ ( one_one @ ( word @ A ) ) ) )
=> ! [N5: nat] :
( ( ord_less_eq @ nat @ N @ N5 )
=> ( ( ord_less @ nat @ N5 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ~ ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ P4 @ N5 ) ) ) ) ) ).
% le2p_bits_unset
thf(fact_6067_le__2p__upper__bits,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [P4: word @ A,N: nat] :
( ( ord_less_eq @ ( word @ A ) @ P4 @ ( minus_minus @ ( word @ A ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) @ ( one_one @ ( word @ A ) ) ) )
=> ( ( ord_less @ nat @ N @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ! [N5: nat] :
( ( ord_less_eq @ nat @ N @ N5 )
=> ( ( ord_less @ nat @ N5 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ~ ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ P4 @ N5 ) ) ) ) ) ) ).
% le_2p_upper_bits
thf(fact_6068_word__add__offset__less,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y: word @ A,N: nat,X: word @ A,M: nat,Sz: nat] :
( ( ord_less @ ( word @ A ) @ Y @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) )
=> ( ( ord_less @ ( word @ A ) @ X @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ M ) )
=> ( ( ord_less @ nat @ Sz @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ( ord_less @ ( word @ A ) @ X @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ N ) ) )
=> ( ( Sz
= ( plus_plus @ nat @ M @ N ) )
=> ( ord_less @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ ( times_times @ ( word @ A ) @ X @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) ) @ Y ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ Sz ) ) ) ) ) ) ) ) ).
% word_add_offset_less
thf(fact_6069_div__power__helper,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: nat,Y: nat] :
( ( ord_less_eq @ nat @ X @ Y )
=> ( ( ord_less @ nat @ Y @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ( divide_divide @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ Y ) @ ( one_one @ ( word @ A ) ) ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ X ) )
= ( minus_minus @ ( word @ A ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Y @ X ) ) @ ( one_one @ ( word @ A ) ) ) ) ) ) ) ).
% div_power_helper
thf(fact_6070_even__mult__exp__div__word__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A,M: nat,N: nat] :
( ( dvd_dvd @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( divide_divide @ ( word @ A ) @ ( times_times @ ( word @ A ) @ A3 @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ M ) ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) ) )
= ( ~ ( ( ord_less_eq @ nat @ M @ N )
& ( ord_less @ nat @ N @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
& ~ ( dvd_dvd @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( divide_divide @ ( word @ A ) @ A3 @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ N @ M ) ) ) ) ) ) ) ) ).
% even_mult_exp_div_word_iff
thf(fact_6071_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_6072_sint__of__nat__le,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [B3: nat,A3: nat] :
( ( ord_less @ nat @ B3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) ) )
=> ( ( ord_less_eq @ nat @ A3 @ B3 )
=> ( ord_less_eq @ int @ ( ring_1_signed @ A @ int @ ( semiring_1_of_nat @ ( word @ A ) @ A3 ) ) @ ( ring_1_signed @ A @ int @ ( semiring_1_of_nat @ ( word @ A ) @ B3 ) ) ) ) ) ) ).
% sint_of_nat_le
thf(fact_6073_sint__of__nat__ge__zero,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: nat] :
( ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) ) )
=> ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( ring_1_signed @ A @ int @ ( semiring_1_of_nat @ ( word @ A ) @ X ) ) ) ) ) ).
% sint_of_nat_ge_zero
thf(fact_6074_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_6075_sint__of__int__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: int] :
( ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) ) ) @ X )
=> ( ( ord_less @ int @ X @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) ) )
=> ( ( ring_1_signed @ A @ int @ ( ring_1_of_int @ ( word @ A ) @ X ) )
= X ) ) ) ) ).
% sint_of_int_eq
thf(fact_6076_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_6077_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_6078_sint__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [B3: num] :
( ( ring_1_signed @ A @ int @ ( numeral_numeral @ ( word @ A ) @ B3 ) )
= ( minus_minus @ int @ ( modulo_modulo @ int @ ( plus_plus @ int @ ( numeral_numeral @ int @ B3 ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) ) ) ) ) ).
% sint_numeral
thf(fact_6079_smod__word__min,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A,B3: word @ A] : ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ) @ ( signed6721504322012087516modulo @ int @ ( ring_1_signed @ A @ int @ A3 ) @ ( ring_1_signed @ A @ int @ B3 ) ) ) ) ).
% smod_word_min
thf(fact_6080_int__word__sint,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: int] :
( ( ring_1_signed @ A @ int @ ( ring_1_of_int @ ( word @ A ) @ X ) )
= ( minus_minus @ int @ ( modulo_modulo @ int @ ( plus_plus @ int @ X @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) ) ) ) ) ).
% int_word_sint
thf(fact_6081_Word_Oword__div__mult,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y: word @ A,X: word @ A] :
( ( ord_less @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ Y )
=> ( ( ord_less @ nat @ ( times_times @ nat @ ( semiring_1_unsigned @ A @ nat @ X ) @ ( semiring_1_unsigned @ A @ nat @ Y ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
=> ( ( divide_divide @ ( word @ A ) @ ( times_times @ ( word @ A ) @ X @ Y ) @ Y )
= X ) ) ) ) ).
% Word.word_div_mult
thf(fact_6082_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_6083_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_6084_word__bit__induct,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [P: ( word @ A ) > $o,A3: word @ A] :
( ( P @ ( zero_zero @ ( word @ A ) ) )
=> ( ! [A6: word @ A] :
( ( P @ A6 )
=> ( ( ord_less @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ A6 )
=> ( ( ord_less @ ( word @ A ) @ A6 @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) ) )
=> ( P @ ( times_times @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ A6 ) ) ) ) )
=> ( ! [A6: word @ A] :
( ( P @ A6 )
=> ( ( ord_less @ ( word @ A ) @ A6 @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) ) )
=> ( P @ ( plus_plus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) @ ( times_times @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ A6 ) ) ) ) )
=> ( P @ A3 ) ) ) ) ) ).
% word_bit_induct
thf(fact_6085_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_6086_bit__word__half__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A,B3: $o] :
( ( ord_less @ ( word @ A ) @ A3 @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) ) )
=> ( ( divide_divide @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ ( zero_neq_one_of_bool @ ( word @ A ) @ B3 ) @ ( times_times @ ( word @ A ) @ A3 @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) ) ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) )
= A3 ) ) ) ).
% bit_word_half_eq
thf(fact_6087_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_6088_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_6089_alignUp__div__helper,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [K: nat,N: nat,X: word @ A,A3: word @ A] :
( ( ord_less @ nat @ K @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ N ) ) )
=> ( ( X
= ( times_times @ ( word @ A ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) @ ( semiring_1_of_nat @ ( word @ A ) @ K ) ) )
=> ( ( ord_less_eq @ ( word @ A ) @ A3 @ X )
=> ( ( ord_less @ nat @ N @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ( ( modulo_modulo @ ( word @ A ) @ A3 @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) )
!= ( zero_zero @ ( word @ A ) ) )
=> ( ord_less @ ( word @ A ) @ ( divide_divide @ ( word @ A ) @ A3 @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) ) @ ( semiring_1_of_nat @ ( word @ A ) @ K ) ) ) ) ) ) ) ) ).
% alignUp_div_helper
thf(fact_6090_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_6091_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_6092_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_6093_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_6094_len__num0,axiom,
( ( type_len0_len_of @ numeral_num0 )
= ( ^ [Uu2: itself @ numeral_num0] : ( zero_zero @ nat ) ) ) ).
% len_num0
thf(fact_6095_len__of__finite__1__def,axiom,
( ( type_len0_len_of @ finite_1 )
= ( ^ [X2: itself @ finite_1] : ( one_one @ nat ) ) ) ).
% len_of_finite_1_def
thf(fact_6096_len__num1,axiom,
( ( type_len0_len_of @ numeral_num1 )
= ( ^ [Uu2: itself @ numeral_num1] : ( one_one @ nat ) ) ) ).
% len_num1
thf(fact_6097_len__of__finite__2__def,axiom,
( ( type_len0_len_of @ finite_2 )
= ( ^ [X2: itself @ finite_2] : ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% len_of_finite_2_def
thf(fact_6098_len__of__finite__3__def,axiom,
( ( type_len0_len_of @ finite_3 )
= ( ^ [X2: itself @ finite_3] : ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ).
% len_of_finite_3_def
thf(fact_6099_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_6100_len__bit0,axiom,
! [A: $tType] :
( ( type_len0 @ A )
=> ( ( type_len0_len_of @ ( numeral_bit0 @ A ) )
= ( ^ [Uu2: itself @ ( numeral_bit0 @ A )] : ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).
% len_bit0
thf(fact_6101_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_6102_len__bit1,axiom,
! [A: $tType] :
( ( type_len0 @ A )
=> ( ( type_len0_len_of @ ( numeral_bit1 @ A ) )
= ( ^ [Uu2: 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_6103_divmod__via__sdivmod,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y: word @ A,X: word @ A] :
( ( Y
!= ( zero_zero @ ( word @ A ) ) )
=> ( ( ( ord_less_eq @ ( word @ A ) @ ( bit_se4730199178511100633sh_bit @ ( word @ A ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( one_one @ ( word @ A ) ) ) @ Y )
=> ( ( ( ord_less @ ( word @ A ) @ X @ Y )
=> ( ( product_Pair @ ( word @ A ) @ ( word @ A ) @ ( divide_divide @ ( word @ A ) @ X @ Y ) @ ( modulo_modulo @ ( word @ A ) @ X @ Y ) )
= ( product_Pair @ ( word @ A ) @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ X ) ) )
& ( ~ ( ord_less @ ( word @ A ) @ X @ Y )
=> ( ( product_Pair @ ( word @ A ) @ ( word @ A ) @ ( divide_divide @ ( word @ A ) @ X @ Y ) @ ( modulo_modulo @ ( word @ A ) @ X @ Y ) )
= ( product_Pair @ ( word @ A ) @ ( word @ A ) @ ( one_one @ ( word @ A ) ) @ ( minus_minus @ ( word @ A ) @ X @ Y ) ) ) ) ) )
& ( ~ ( ord_less_eq @ ( word @ A ) @ ( bit_se4730199178511100633sh_bit @ ( word @ A ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( one_one @ ( word @ A ) ) ) @ Y )
=> ( ( product_Pair @ ( word @ A ) @ ( word @ A ) @ ( divide_divide @ ( word @ A ) @ X @ Y ) @ ( modulo_modulo @ ( word @ A ) @ X @ Y ) )
= ( if @ ( product_prod @ ( word @ A ) @ ( word @ A ) ) @ ( ord_less_eq @ ( word @ A ) @ Y @ ( minus_minus @ ( word @ A ) @ X @ ( times_times @ ( word @ A ) @ ( bit_se4730199178511100633sh_bit @ ( word @ A ) @ ( one_one @ nat ) @ ( signed7115095781618012415divide @ ( word @ A ) @ ( bit_se4197421643247451524op_bit @ ( word @ A ) @ ( one_one @ nat ) @ X ) @ Y ) ) @ Y ) ) ) @ ( product_Pair @ ( word @ A ) @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ ( bit_se4730199178511100633sh_bit @ ( word @ A ) @ ( one_one @ nat ) @ ( signed7115095781618012415divide @ ( word @ A ) @ ( bit_se4197421643247451524op_bit @ ( word @ A ) @ ( one_one @ nat ) @ X ) @ Y ) ) @ ( one_one @ ( word @ A ) ) ) @ ( minus_minus @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ X @ ( times_times @ ( word @ A ) @ ( bit_se4730199178511100633sh_bit @ ( word @ A ) @ ( one_one @ nat ) @ ( signed7115095781618012415divide @ ( word @ A ) @ ( bit_se4197421643247451524op_bit @ ( word @ A ) @ ( one_one @ nat ) @ X ) @ Y ) ) @ Y ) ) @ Y ) ) @ ( product_Pair @ ( word @ A ) @ ( word @ A ) @ ( bit_se4730199178511100633sh_bit @ ( word @ A ) @ ( one_one @ nat ) @ ( signed7115095781618012415divide @ ( word @ A ) @ ( bit_se4197421643247451524op_bit @ ( word @ A ) @ ( one_one @ nat ) @ X ) @ Y ) ) @ ( minus_minus @ ( word @ A ) @ X @ ( times_times @ ( word @ A ) @ ( bit_se4730199178511100633sh_bit @ ( word @ A ) @ ( one_one @ nat ) @ ( signed7115095781618012415divide @ ( word @ A ) @ ( bit_se4197421643247451524op_bit @ ( word @ A ) @ ( one_one @ nat ) @ X ) @ Y ) ) @ Y ) ) ) ) ) ) ) ) ) ).
% divmod_via_sdivmod
thf(fact_6104_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_6105_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_6106_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_6107_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_6108_word__sdiv__div1,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A] :
( ( signed7115095781618012415divide @ ( word @ A ) @ A3 @ ( one_one @ ( word @ A ) ) )
= A3 ) ) ).
% word_sdiv_div1
thf(fact_6109_word__sdiv__div0,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A] :
( ( signed7115095781618012415divide @ ( word @ A ) @ A3 @ ( zero_zero @ ( word @ A ) ) )
= ( zero_zero @ ( word @ A ) ) ) ) ).
% word_sdiv_div0
thf(fact_6110_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_6111_word__sdiv__div__minus1,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A] :
( ( signed7115095781618012415divide @ ( word @ A ) @ A3 @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) )
= ( uminus_uminus @ ( word @ A ) @ A3 ) ) ) ).
% word_sdiv_div_minus1
thf(fact_6112_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_6113_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_6114_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_6115_sdiv__word__numeral__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: num,B3: num] :
( ( signed7115095781618012415divide @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ A3 ) @ ( numeral_numeral @ ( word @ A ) @ B3 ) )
= ( ring_1_of_int @ ( word @ A ) @ ( signed7115095781618012415divide @ int @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( numeral_numeral @ int @ A3 ) ) @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( numeral_numeral @ int @ B3 ) ) ) ) ) ) ).
% sdiv_word_numeral_numeral
thf(fact_6116_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_6117_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_6118_sdiv__word__minus__numeral__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: num,B3: num] :
( ( signed7115095781618012415divide @ ( word @ A ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ A3 ) ) @ ( numeral_numeral @ ( word @ A ) @ B3 ) )
= ( ring_1_of_int @ ( word @ A ) @ ( signed7115095781618012415divide @ int @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ A3 ) ) ) @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( numeral_numeral @ int @ B3 ) ) ) ) ) ) ).
% sdiv_word_minus_numeral_numeral
thf(fact_6119_sdiv__word__numeral__minus__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: num,B3: num] :
( ( signed7115095781618012415divide @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ A3 ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ B3 ) ) )
= ( ring_1_of_int @ ( word @ A ) @ ( signed7115095781618012415divide @ int @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( numeral_numeral @ int @ A3 ) ) @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ B3 ) ) ) ) ) ) ) ).
% sdiv_word_numeral_minus_numeral
thf(fact_6120_sdiv__word__minus__numeral__minus__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: num,B3: num] :
( ( signed7115095781618012415divide @ ( word @ A ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ A3 ) ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ B3 ) ) )
= ( ring_1_of_int @ ( word @ A ) @ ( signed7115095781618012415divide @ int @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ A3 ) ) ) @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ B3 ) ) ) ) ) ) ) ).
% sdiv_word_minus_numeral_minus_numeral
thf(fact_6121_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_6122_sint__signed__drop__bit__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,W: word @ A] :
( ( ring_1_signed @ A @ int @ ( signed_drop_bit @ A @ N @ W ) )
= ( bit_se4197421643247451524op_bit @ int @ N @ ( ring_1_signed @ A @ int @ W ) ) ) ) ).
% sint_signed_drop_bit_eq
thf(fact_6123_word__sdiv__numerals_I1_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: num,Y: num] :
( ( signed7115095781618012415divide @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ X ) @ ( numeral_numeral @ ( word @ A ) @ Y ) )
= ( ring_1_of_int @ ( word @ A ) @ ( signed7115095781618012415divide @ int @ ( ring_1_signed @ A @ int @ ( numeral_numeral @ ( word @ A ) @ X ) ) @ ( ring_1_signed @ A @ int @ ( numeral_numeral @ ( word @ A ) @ Y ) ) ) ) ) ) ).
% word_sdiv_numerals(1)
thf(fact_6124_word__sdiv__numerals__lhs_I1_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: num,W: word @ A] :
( ( signed7115095781618012415divide @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ X ) @ W )
= ( ring_1_of_int @ ( word @ A ) @ ( signed7115095781618012415divide @ int @ ( ring_1_signed @ A @ int @ ( numeral_numeral @ ( word @ A ) @ X ) ) @ ( ring_1_signed @ A @ int @ W ) ) ) ) ) ).
% word_sdiv_numerals_lhs(1)
thf(fact_6125_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_6126_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_6127_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_6128_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_6129_word__sdiv__numerals_I3_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y: num] :
( ( signed7115095781618012415divide @ ( word @ A ) @ ( one_one @ ( word @ A ) ) @ ( numeral_numeral @ ( word @ A ) @ Y ) )
= ( ring_1_of_int @ ( word @ A ) @ ( signed7115095781618012415divide @ int @ ( ring_1_signed @ A @ int @ ( one_one @ ( word @ A ) ) ) @ ( ring_1_signed @ A @ int @ ( numeral_numeral @ ( word @ A ) @ Y ) ) ) ) ) ) ).
% word_sdiv_numerals(3)
thf(fact_6130_word__sdiv__numerals_I7_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: num] :
( ( signed7115095781618012415divide @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ X ) @ ( one_one @ ( word @ A ) ) )
= ( ring_1_of_int @ ( word @ A ) @ ( signed7115095781618012415divide @ int @ ( ring_1_signed @ A @ int @ ( numeral_numeral @ ( word @ A ) @ X ) ) @ ( ring_1_signed @ A @ int @ ( one_one @ ( word @ A ) ) ) ) ) ) ) ).
% word_sdiv_numerals(7)
thf(fact_6131_word__sdiv__numerals_I4_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: num] :
( ( signed7115095781618012415divide @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ X ) @ ( zero_zero @ ( word @ A ) ) )
= ( ring_1_of_int @ ( word @ A ) @ ( signed7115095781618012415divide @ int @ ( ring_1_signed @ A @ int @ ( numeral_numeral @ ( word @ A ) @ X ) ) @ ( ring_1_signed @ A @ int @ ( zero_zero @ ( word @ A ) ) ) ) ) ) ) ).
% word_sdiv_numerals(4)
thf(fact_6132_word__sdiv__numerals_I2_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y: num] :
( ( signed7115095781618012415divide @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ ( numeral_numeral @ ( word @ A ) @ Y ) )
= ( ring_1_of_int @ ( word @ A ) @ ( signed7115095781618012415divide @ int @ ( ring_1_signed @ A @ int @ ( zero_zero @ ( word @ A ) ) ) @ ( ring_1_signed @ A @ int @ ( numeral_numeral @ ( word @ A ) @ Y ) ) ) ) ) ) ).
% word_sdiv_numerals(2)
thf(fact_6133_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_6134_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_6135_signed__div__arith,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A,B3: word @ A] :
( ( ring_1_signed @ A @ int @ ( signed7115095781618012415divide @ ( word @ A ) @ A3 @ B3 ) )
= ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( signed7115095781618012415divide @ int @ ( ring_1_signed @ A @ int @ A3 ) @ ( ring_1_signed @ A @ int @ B3 ) ) ) ) ) ).
% signed_div_arith
thf(fact_6136_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_6137_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_6138_word__int__split__asm,axiom,
! [A: $tType,B: $tType] :
( ( type_len @ B )
=> ! [P: A > $o,F3: int > A,X: word @ B] :
( ( P @ ( word_int_case @ A @ B @ F3 @ X ) )
= ( ~ ? [N4: int] :
( ( X
= ( ring_1_of_int @ ( word @ B ) @ N4 ) )
& ( ord_less_eq @ int @ ( zero_zero @ int ) @ N4 )
& ( ord_less @ int @ N4 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ B @ ( type2 @ B ) ) ) )
& ~ ( P @ ( F3 @ N4 ) ) ) ) ) ) ).
% word_int_split_asm
thf(fact_6139_word__int__split,axiom,
! [A: $tType,B: $tType] :
( ( type_len @ B )
=> ! [P: A > $o,F3: int > A,X: word @ B] :
( ( P @ ( word_int_case @ A @ B @ F3 @ X ) )
= ( ! [I4: int] :
( ( ( X
= ( ring_1_of_int @ ( word @ B ) @ I4 ) )
& ( ord_less_eq @ int @ ( zero_zero @ int ) @ I4 )
& ( ord_less @ int @ I4 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ B @ ( type2 @ B ) ) ) ) )
=> ( P @ ( F3 @ I4 ) ) ) ) ) ) ).
% word_int_split
thf(fact_6140_word__int__case__eq__uint,axiom,
! [B: $tType,A: $tType] :
( ( type_len @ B )
=> ( ( word_int_case @ A @ B )
= ( ^ [F2: int > A,W2: word @ B] : ( F2 @ ( semiring_1_unsigned @ B @ int @ W2 ) ) ) ) ) ).
% word_int_case_eq_uint
thf(fact_6141_word__int__case__wi,axiom,
! [A: $tType,B: $tType] :
( ( type_len @ B )
=> ! [F3: int > A,I: int] :
( ( word_int_case @ A @ B @ F3 @ ( ring_1_of_int @ ( word @ B ) @ I ) )
= ( F3 @ ( 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_6142_smod__word__minus__numeral__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: num,B3: num] :
( ( signed6721504322012087516modulo @ ( word @ A ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ A3 ) ) @ ( numeral_numeral @ ( word @ A ) @ B3 ) )
= ( ring_1_of_int @ ( word @ A ) @ ( signed6721504322012087516modulo @ int @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ A3 ) ) ) @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( numeral_numeral @ int @ B3 ) ) ) ) ) ) ).
% smod_word_minus_numeral_numeral
thf(fact_6143_smod__word__numeral__minus__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: num,B3: num] :
( ( signed6721504322012087516modulo @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ A3 ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ B3 ) ) )
= ( ring_1_of_int @ ( word @ A ) @ ( signed6721504322012087516modulo @ int @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( numeral_numeral @ int @ A3 ) ) @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ B3 ) ) ) ) ) ) ) ).
% smod_word_numeral_minus_numeral
thf(fact_6144_smod__word__mod__0,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] :
( ( signed6721504322012087516modulo @ ( word @ A ) @ X @ ( zero_zero @ ( word @ A ) ) )
= X ) ) ).
% smod_word_mod_0
thf(fact_6145_smod__word__0__mod,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] :
( ( signed6721504322012087516modulo @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ X )
= ( zero_zero @ ( word @ A ) ) ) ) ).
% smod_word_0_mod
thf(fact_6146_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_6147_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_6148_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_6149_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_6150_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_6151_smod__word__numeral__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: num,B3: num] :
( ( signed6721504322012087516modulo @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ A3 ) @ ( numeral_numeral @ ( word @ A ) @ B3 ) )
= ( ring_1_of_int @ ( word @ A ) @ ( signed6721504322012087516modulo @ int @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( numeral_numeral @ int @ A3 ) ) @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( numeral_numeral @ int @ B3 ) ) ) ) ) ) ).
% smod_word_numeral_numeral
thf(fact_6152_smod__word__minus__numeral__minus__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: num,B3: num] :
( ( signed6721504322012087516modulo @ ( word @ A ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ A3 ) ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ B3 ) ) )
= ( ring_1_of_int @ ( word @ A ) @ ( signed6721504322012087516modulo @ int @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ A3 ) ) ) @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ B3 ) ) ) ) ) ) ) ).
% smod_word_minus_numeral_minus_numeral
thf(fact_6153_word__smod__numerals_I1_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: num,Y: num] :
( ( signed6721504322012087516modulo @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ X ) @ ( numeral_numeral @ ( word @ A ) @ Y ) )
= ( ring_1_of_int @ ( word @ A ) @ ( signed6721504322012087516modulo @ int @ ( ring_1_signed @ A @ int @ ( numeral_numeral @ ( word @ A ) @ X ) ) @ ( ring_1_signed @ A @ int @ ( numeral_numeral @ ( word @ A ) @ Y ) ) ) ) ) ) ).
% word_smod_numerals(1)
thf(fact_6154_word__smod__numerals__lhs_I1_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: num,W: word @ A] :
( ( signed6721504322012087516modulo @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ X ) @ W )
= ( ring_1_of_int @ ( word @ A ) @ ( signed6721504322012087516modulo @ int @ ( ring_1_signed @ A @ int @ ( numeral_numeral @ ( word @ A ) @ X ) ) @ ( ring_1_signed @ A @ int @ W ) ) ) ) ) ).
% word_smod_numerals_lhs(1)
thf(fact_6155_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_6156_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_6157_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_6158_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_6159_word__smod__numerals_I3_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y: num] :
( ( signed6721504322012087516modulo @ ( word @ A ) @ ( one_one @ ( word @ A ) ) @ ( numeral_numeral @ ( word @ A ) @ Y ) )
= ( ring_1_of_int @ ( word @ A ) @ ( signed6721504322012087516modulo @ int @ ( ring_1_signed @ A @ int @ ( one_one @ ( word @ A ) ) ) @ ( ring_1_signed @ A @ int @ ( numeral_numeral @ ( word @ A ) @ Y ) ) ) ) ) ) ).
% word_smod_numerals(3)
thf(fact_6160_word__smod__numerals_I7_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: num] :
( ( signed6721504322012087516modulo @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ X ) @ ( one_one @ ( word @ A ) ) )
= ( ring_1_of_int @ ( word @ A ) @ ( signed6721504322012087516modulo @ int @ ( ring_1_signed @ A @ int @ ( numeral_numeral @ ( word @ A ) @ X ) ) @ ( ring_1_signed @ A @ int @ ( one_one @ ( word @ A ) ) ) ) ) ) ) ).
% word_smod_numerals(7)
thf(fact_6161_word__smod__numerals_I4_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: num] :
( ( signed6721504322012087516modulo @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ X ) @ ( zero_zero @ ( word @ A ) ) )
= ( ring_1_of_int @ ( word @ A ) @ ( signed6721504322012087516modulo @ int @ ( ring_1_signed @ A @ int @ ( numeral_numeral @ ( word @ A ) @ X ) ) @ ( ring_1_signed @ A @ int @ ( zero_zero @ ( word @ A ) ) ) ) ) ) ) ).
% word_smod_numerals(4)
thf(fact_6162_word__smod__numerals_I2_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y: num] :
( ( signed6721504322012087516modulo @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ ( numeral_numeral @ ( word @ A ) @ Y ) )
= ( ring_1_of_int @ ( word @ A ) @ ( signed6721504322012087516modulo @ int @ ( ring_1_signed @ A @ int @ ( zero_zero @ ( word @ A ) ) ) @ ( ring_1_signed @ A @ int @ ( numeral_numeral @ ( word @ A ) @ Y ) ) ) ) ) ) ).
% word_smod_numerals(2)
thf(fact_6163_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_6164_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_6165_signed__mod__arith,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A,B3: word @ A] :
( ( ring_1_signed @ A @ int @ ( signed6721504322012087516modulo @ ( word @ A ) @ A3 @ B3 ) )
= ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( signed6721504322012087516modulo @ int @ ( ring_1_signed @ A @ int @ A3 ) @ ( ring_1_signed @ A @ int @ B3 ) ) ) ) ) ).
% signed_mod_arith
thf(fact_6166_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_6167_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_6168_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_6169_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_6170_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_6171_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_6172_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_6173_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_6174_uint__word__ariths_I6_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A] :
( ( semiring_1_unsigned @ A @ int @ ( word_pred @ A @ A3 ) )
= ( modulo_modulo @ int @ ( minus_minus @ int @ ( semiring_1_unsigned @ A @ int @ A3 ) @ ( one_one @ int ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).
% uint_word_ariths(6)
thf(fact_6175_slice1__def,axiom,
! [A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( type_len @ A ) )
=> ( ( slice1 @ A @ B )
= ( ^ [N4: nat,W2: word @ A] : ( if @ ( word @ B ) @ ( ord_less @ nat @ N4 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) @ ( semiring_1_unsigned @ A @ ( word @ B ) @ ( bit_se4197421643247451524op_bit @ ( word @ A ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ N4 ) @ W2 ) ) @ ( bit_se4730199178511100633sh_bit @ ( word @ B ) @ ( minus_minus @ nat @ N4 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) @ ( semiring_1_unsigned @ A @ ( word @ B ) @ W2 ) ) ) ) ) ) ).
% slice1_def
thf(fact_6176_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_6177_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_6178_succ__pred__no_I4_J,axiom,
! [D: $tType] :
( ( type_len @ D )
=> ! [W: num] :
( ( word_pred @ D @ ( uminus_uminus @ ( word @ D ) @ ( numeral_numeral @ ( word @ D ) @ W ) ) )
= ( minus_minus @ ( word @ D ) @ ( uminus_uminus @ ( word @ D ) @ ( numeral_numeral @ ( word @ D ) @ W ) ) @ ( one_one @ ( word @ D ) ) ) ) ) ).
% succ_pred_no(4)
thf(fact_6179_word__m1__ge,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y: word @ A] : ( ord_less_eq @ ( word @ A ) @ Y @ ( word_pred @ A @ ( zero_zero @ ( word @ A ) ) ) ) ) ).
% word_m1_ge
thf(fact_6180_word__not__simps_I2_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y: word @ A] :
~ ( ord_less @ ( word @ A ) @ ( word_pred @ A @ ( zero_zero @ ( word @ A ) ) ) @ Y ) ) ).
% word_not_simps(2)
thf(fact_6181_word__pred__m1,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( word_pred @ A )
= ( ^ [A5: word @ A] : ( minus_minus @ ( word @ A ) @ A5 @ ( one_one @ ( word @ A ) ) ) ) ) ) ).
% word_pred_m1
thf(fact_6182_mask__eqs_I12_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A,N: nat] :
( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ ( word_pred @ A @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ A3 @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) )
= ( bit_se5824344872417868541ns_and @ ( word @ A ) @ ( word_pred @ A @ A3 ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) ) ) ).
% mask_eqs(12)
thf(fact_6183_ucast__slice1,axiom,
! [A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( type_len @ A ) )
=> ( ( semiring_1_unsigned @ B @ ( word @ A ) )
= ( ^ [W2: word @ B] : ( slice1 @ B @ A @ ( size_size @ ( word @ B ) @ W2 ) @ W2 ) ) ) ) ).
% ucast_slice1
thf(fact_6184_wi__hom__pred,axiom,
! [F: $tType] :
( ( type_len @ F )
=> ! [A3: int] :
( ( word_pred @ F @ ( ring_1_of_int @ ( word @ F ) @ A3 ) )
= ( ring_1_of_int @ ( word @ F ) @ ( minus_minus @ int @ A3 @ ( one_one @ int ) ) ) ) ) ).
% wi_hom_pred
thf(fact_6185_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_6186_word__pred__alt,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( word_pred @ A )
= ( ^ [A5: word @ A] : ( ring_1_of_int @ ( word @ A ) @ ( minus_minus @ int @ ( semiring_1_unsigned @ A @ int @ A5 ) @ ( one_one @ int ) ) ) ) ) ) ).
% word_pred_alt
thf(fact_6187_Word_Oslice__def,axiom,
! [A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( type_len @ A ) )
=> ( ( slice2 @ A @ B )
= ( ^ [N4: nat] : ( slice1 @ A @ B @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ N4 ) ) ) ) ) ).
% Word.slice_def
thf(fact_6188_uint__word__arith__bintrs_I6_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A] :
( ( semiring_1_unsigned @ A @ int @ ( word_pred @ A @ A3 ) )
= ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( minus_minus @ int @ ( semiring_1_unsigned @ A @ int @ A3 ) @ ( one_one @ int ) ) ) ) ) ).
% uint_word_arith_bintrs(6)
thf(fact_6189_sint__word__ariths_I6_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A] :
( ( ring_1_signed @ A @ int @ ( word_pred @ A @ A3 ) )
= ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( minus_minus @ int @ ( ring_1_signed @ A @ int @ A3 ) @ ( one_one @ int ) ) ) ) ) ).
% sint_word_ariths(6)
thf(fact_6190_uint__word__ariths_I5_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A] :
( ( semiring_1_unsigned @ A @ int @ ( word_succ @ A @ A3 ) )
= ( modulo_modulo @ int @ ( plus_plus @ int @ ( semiring_1_unsigned @ A @ int @ A3 ) @ ( one_one @ int ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).
% uint_word_ariths(5)
thf(fact_6191_of__nat__less__two__pow__div__set,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,M: nat] :
( ( ord_less @ nat @ N @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
=> ( ( collect @ ( word @ A )
@ ^ [X2: word @ A] : ( ord_less @ ( word @ A ) @ X2 @ ( divide_divide @ ( word @ A ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ M ) ) ) )
= ( image @ nat @ ( word @ A ) @ ( semiring_1_of_nat @ ( word @ A ) )
@ ( collect @ nat
@ ^ [K3: nat] : ( ord_less @ nat @ K3 @ ( divide_divide @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) ) ) ) ) ) ) ) ).
% of_nat_less_two_pow_div_set
thf(fact_6192_image__eqI,axiom,
! [A: $tType,B: $tType,B3: A,F3: B > A,X: B,A2: set @ B] :
( ( B3
= ( F3 @ X ) )
=> ( ( member @ B @ X @ A2 )
=> ( member @ A @ B3 @ ( image @ B @ A @ F3 @ A2 ) ) ) ) ).
% image_eqI
thf(fact_6193_image__ident,axiom,
! [A: $tType,Y8: set @ A] :
( ( image @ A @ A
@ ^ [X2: A] : X2
@ Y8 )
= Y8 ) ).
% image_ident
thf(fact_6194_image__is__empty,axiom,
! [A: $tType,B: $tType,F3: B > A,A2: set @ B] :
( ( ( image @ B @ A @ F3 @ A2 )
= ( bot_bot @ ( set @ A ) ) )
= ( A2
= ( bot_bot @ ( set @ B ) ) ) ) ).
% image_is_empty
thf(fact_6195_empty__is__image,axiom,
! [A: $tType,B: $tType,F3: B > A,A2: set @ B] :
( ( ( bot_bot @ ( set @ A ) )
= ( image @ B @ A @ F3 @ A2 ) )
= ( A2
= ( bot_bot @ ( set @ B ) ) ) ) ).
% empty_is_image
thf(fact_6196_image__empty,axiom,
! [B: $tType,A: $tType,F3: B > A] :
( ( image @ B @ A @ F3 @ ( bot_bot @ ( set @ B ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% image_empty
thf(fact_6197_insert__image,axiom,
! [B: $tType,A: $tType,X: A,A2: set @ A,F3: A > B] :
( ( member @ A @ X @ A2 )
=> ( ( insert @ B @ ( F3 @ X ) @ ( image @ A @ B @ F3 @ A2 ) )
= ( image @ A @ B @ F3 @ A2 ) ) ) ).
% insert_image
thf(fact_6198_image__insert,axiom,
! [A: $tType,B: $tType,F3: B > A,A3: B,B2: set @ B] :
( ( image @ B @ A @ F3 @ ( insert @ B @ A3 @ B2 ) )
= ( insert @ A @ ( F3 @ A3 ) @ ( image @ B @ A @ F3 @ B2 ) ) ) ).
% image_insert
thf(fact_6199_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_6200_bij__betw__of__nat,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [N3: set @ nat,A2: set @ A] :
( ( bij_betw @ nat @ A @ ( semiring_1_of_nat @ A ) @ N3 @ A2 )
= ( ( image @ nat @ A @ ( semiring_1_of_nat @ A ) @ N3 )
= A2 ) ) ) ).
% bij_betw_of_nat
thf(fact_6201_image__add__atLeastAtMost_H,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [K: A,I: A,J: A] :
( ( image @ A @ A
@ ^ [N4: A] : ( plus_plus @ A @ N4 @ K )
@ ( set_or1337092689740270186AtMost @ A @ I @ J ) )
= ( set_or1337092689740270186AtMost @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ K ) ) ) ) ).
% image_add_atLeastAtMost'
thf(fact_6202_image__add__atLeastLessThan_H,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [K: A,I: A,J: A] :
( ( image @ A @ A
@ ^ [N4: A] : ( plus_plus @ A @ N4 @ K )
@ ( set_or7035219750837199246ssThan @ A @ I @ J ) )
= ( set_or7035219750837199246ssThan @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ K ) ) ) ) ).
% image_add_atLeastLessThan'
thf(fact_6203_image__minus__const__atLeastAtMost_H,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [D2: A,A3: A,B3: A] :
( ( image @ A @ A
@ ^ [T2: A] : ( minus_minus @ A @ T2 @ D2 )
@ ( set_or1337092689740270186AtMost @ A @ A3 @ B3 ) )
= ( set_or1337092689740270186AtMost @ A @ ( minus_minus @ A @ A3 @ D2 ) @ ( minus_minus @ A @ B3 @ D2 ) ) ) ) ).
% image_minus_const_atLeastAtMost'
thf(fact_6204_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_6205_image__mult__atLeastAtMost,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [D2: A,A3: A,B3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ D2 )
=> ( ( image @ A @ A @ ( times_times @ A @ D2 ) @ ( set_or1337092689740270186AtMost @ A @ A3 @ B3 ) )
= ( set_or1337092689740270186AtMost @ A @ ( times_times @ A @ D2 @ A3 ) @ ( times_times @ A @ D2 @ B3 ) ) ) ) ) ).
% image_mult_atLeastAtMost
thf(fact_6206_image__divide__atLeastAtMost,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [D2: A,A3: A,B3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ D2 )
=> ( ( image @ A @ A
@ ^ [C6: A] : ( divide_divide @ A @ C6 @ D2 )
@ ( set_or1337092689740270186AtMost @ A @ A3 @ B3 ) )
= ( set_or1337092689740270186AtMost @ A @ ( divide_divide @ A @ A3 @ D2 ) @ ( divide_divide @ A @ B3 @ D2 ) ) ) ) ) ).
% image_divide_atLeastAtMost
thf(fact_6207_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_6208_word__pred__succ,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A] :
( ( word_pred @ A @ ( word_succ @ A @ A3 ) )
= A3 ) ) ).
% word_pred_succ
thf(fact_6209_word__succ__pred,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A] :
( ( word_succ @ A @ ( word_pred @ A @ A3 ) )
= A3 ) ) ).
% word_succ_pred
thf(fact_6210_finite__conv__nat__seg__image,axiom,
! [A: $tType] :
( ( finite_finite2 @ A )
= ( ^ [A4: set @ A] :
? [N4: nat,F2: nat > A] :
( A4
= ( image @ nat @ A @ F2
@ ( collect @ nat
@ ^ [I4: nat] : ( ord_less @ nat @ I4 @ N4 ) ) ) ) ) ) ).
% finite_conv_nat_seg_image
thf(fact_6211_nat__seg__image__imp__finite,axiom,
! [A: $tType,A2: set @ A,F3: nat > A,N: nat] :
( ( A2
= ( image @ nat @ A @ F3
@ ( collect @ nat
@ ^ [I4: nat] : ( ord_less @ nat @ I4 @ N ) ) ) )
=> ( finite_finite2 @ A @ A2 ) ) ).
% nat_seg_image_imp_finite
thf(fact_6212_pigeonhole__infinite,axiom,
! [B: $tType,A: $tType,A2: set @ A,F3: A > B] :
( ~ ( finite_finite2 @ A @ A2 )
=> ( ( finite_finite2 @ B @ ( image @ A @ B @ F3 @ A2 ) )
=> ? [X3: A] :
( ( member @ A @ X3 @ A2 )
& ~ ( finite_finite2 @ A
@ ( collect @ A
@ ^ [A5: A] :
( ( member @ A @ A5 @ A2 )
& ( ( F3 @ A5 )
= ( F3 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_6213_rev__image__eqI,axiom,
! [B: $tType,A: $tType,X: A,A2: set @ A,B3: B,F3: A > B] :
( ( member @ A @ X @ A2 )
=> ( ( B3
= ( F3 @ X ) )
=> ( member @ B @ B3 @ ( image @ A @ B @ F3 @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_6214_ball__imageD,axiom,
! [A: $tType,B: $tType,F3: B > A,A2: set @ B,P: A > $o] :
( ! [X3: A] :
( ( member @ A @ X3 @ ( image @ B @ A @ F3 @ A2 ) )
=> ( P @ X3 ) )
=> ! [X6: B] :
( ( member @ B @ X6 @ A2 )
=> ( P @ ( F3 @ X6 ) ) ) ) ).
% ball_imageD
thf(fact_6215_image__cong,axiom,
! [B: $tType,A: $tType,M7: set @ A,N3: set @ A,F3: A > B,G2: A > B] :
( ( M7 = N3 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ N3 )
=> ( ( F3 @ X3 )
= ( G2 @ X3 ) ) )
=> ( ( image @ A @ B @ F3 @ M7 )
= ( image @ A @ B @ G2 @ N3 ) ) ) ) ).
% image_cong
thf(fact_6216_bex__imageD,axiom,
! [A: $tType,B: $tType,F3: B > A,A2: set @ B,P: A > $o] :
( ? [X6: A] :
( ( member @ A @ X6 @ ( image @ B @ A @ F3 @ A2 ) )
& ( P @ X6 ) )
=> ? [X3: B] :
( ( member @ B @ X3 @ A2 )
& ( P @ ( F3 @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_6217_image__iff,axiom,
! [A: $tType,B: $tType,Z: A,F3: B > A,A2: set @ B] :
( ( member @ A @ Z @ ( image @ B @ A @ F3 @ A2 ) )
= ( ? [X2: B] :
( ( member @ B @ X2 @ A2 )
& ( Z
= ( F3 @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_6218_imageI,axiom,
! [B: $tType,A: $tType,X: A,A2: set @ A,F3: A > B] :
( ( member @ A @ X @ A2 )
=> ( member @ B @ ( F3 @ X ) @ ( image @ A @ B @ F3 @ A2 ) ) ) ).
% imageI
thf(fact_6219_Compr__image__eq,axiom,
! [A: $tType,B: $tType,F3: B > A,A2: set @ B,P: A > $o] :
( ( collect @ A
@ ^ [X2: A] :
( ( member @ A @ X2 @ ( image @ B @ A @ F3 @ A2 ) )
& ( P @ X2 ) ) )
= ( image @ B @ A @ F3
@ ( collect @ B
@ ^ [X2: B] :
( ( member @ B @ X2 @ A2 )
& ( P @ ( F3 @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_6220_image__image,axiom,
! [A: $tType,B: $tType,C: $tType,F3: B > A,G2: C > B,A2: set @ C] :
( ( image @ B @ A @ F3 @ ( image @ C @ B @ G2 @ A2 ) )
= ( image @ C @ A
@ ^ [X2: C] : ( F3 @ ( G2 @ X2 ) )
@ A2 ) ) ).
% image_image
thf(fact_6221_imageE,axiom,
! [A: $tType,B: $tType,B3: A,F3: B > A,A2: set @ B] :
( ( member @ A @ B3 @ ( image @ B @ A @ F3 @ A2 ) )
=> ~ ! [X3: B] :
( ( B3
= ( F3 @ X3 ) )
=> ~ ( member @ B @ X3 @ A2 ) ) ) ).
% imageE
thf(fact_6222_image__Un,axiom,
! [A: $tType,B: $tType,F3: B > A,A2: set @ B,B2: set @ B] :
( ( image @ B @ A @ F3 @ ( sup_sup @ ( set @ B ) @ A2 @ B2 ) )
= ( sup_sup @ ( set @ A ) @ ( image @ B @ A @ F3 @ A2 ) @ ( image @ B @ A @ F3 @ B2 ) ) ) ).
% image_Un
thf(fact_6223_image__diff__subset,axiom,
! [A: $tType,B: $tType,F3: B > A,A2: set @ B,B2: set @ B] : ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ ( image @ B @ A @ F3 @ A2 ) @ ( image @ B @ A @ F3 @ B2 ) ) @ ( image @ B @ A @ F3 @ ( minus_minus @ ( set @ B ) @ A2 @ B2 ) ) ) ).
% image_diff_subset
thf(fact_6224_subset__image__iff,axiom,
! [A: $tType,B: $tType,B2: set @ A,F3: B > A,A2: set @ B] :
( ( ord_less_eq @ ( set @ A ) @ B2 @ ( image @ B @ A @ F3 @ A2 ) )
= ( ? [AA: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ AA @ A2 )
& ( B2
= ( image @ B @ A @ F3 @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_6225_image__subset__iff,axiom,
! [A: $tType,B: $tType,F3: B > A,A2: set @ B,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ F3 @ A2 ) @ B2 )
= ( ! [X2: B] :
( ( member @ B @ X2 @ A2 )
=> ( member @ A @ ( F3 @ X2 ) @ B2 ) ) ) ) ).
% image_subset_iff
thf(fact_6226_subset__imageE,axiom,
! [A: $tType,B: $tType,B2: set @ A,F3: B > A,A2: set @ B] :
( ( ord_less_eq @ ( set @ A ) @ B2 @ ( image @ B @ A @ F3 @ A2 ) )
=> ~ ! [C7: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ C7 @ A2 )
=> ( B2
!= ( image @ B @ A @ F3 @ C7 ) ) ) ) ).
% subset_imageE
thf(fact_6227_image__subsetI,axiom,
! [A: $tType,B: $tType,A2: set @ A,F3: A > B,B2: set @ B] :
( ! [X3: A] :
( ( member @ A @ X3 @ A2 )
=> ( member @ B @ ( F3 @ X3 ) @ B2 ) )
=> ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F3 @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_6228_image__mono,axiom,
! [B: $tType,A: $tType,A2: set @ A,B2: set @ A,F3: A > B] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F3 @ A2 ) @ ( image @ A @ B @ F3 @ B2 ) ) ) ).
% image_mono
thf(fact_6229_image__constant,axiom,
! [A: $tType,B: $tType,X: A,A2: set @ A,C2: B] :
( ( member @ A @ X @ A2 )
=> ( ( image @ A @ B
@ ^ [X2: A] : C2
@ A2 )
= ( insert @ B @ C2 @ ( bot_bot @ ( set @ B ) ) ) ) ) ).
% image_constant
thf(fact_6230_image__constant__conv,axiom,
! [B: $tType,A: $tType,A2: set @ B,C2: A] :
( ( ( A2
= ( bot_bot @ ( set @ B ) ) )
=> ( ( image @ B @ A
@ ^ [X2: B] : C2
@ A2 )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( A2
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( image @ B @ A
@ ^ [X2: B] : C2
@ A2 )
= ( insert @ A @ C2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% image_constant_conv
thf(fact_6231_sum_Oimage__gen,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [S: set @ B,H2: B > A,G2: B > C] :
( ( finite_finite2 @ B @ S )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ H2 @ S )
= ( groups7311177749621191930dd_sum @ C @ A
@ ^ [Y2: C] :
( groups7311177749621191930dd_sum @ B @ A @ H2
@ ( collect @ B
@ ^ [X2: B] :
( ( member @ B @ X2 @ S )
& ( ( G2 @ X2 )
= Y2 ) ) ) )
@ ( image @ B @ C @ G2 @ S ) ) ) ) ) ).
% sum.image_gen
thf(fact_6232_translation__subtract__diff,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A3: A,S2: set @ A,T: set @ A] :
( ( image @ A @ A
@ ^ [X2: A] : ( minus_minus @ A @ X2 @ A3 )
@ ( minus_minus @ ( set @ A ) @ S2 @ T ) )
= ( minus_minus @ ( set @ A )
@ ( image @ A @ A
@ ^ [X2: A] : ( minus_minus @ A @ X2 @ A3 )
@ S2 )
@ ( image @ A @ A
@ ^ [X2: A] : ( minus_minus @ A @ X2 @ A3 )
@ T ) ) ) ) ).
% translation_subtract_diff
thf(fact_6233_prod_Oimage__gen,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S: set @ B,H2: B > A,G2: B > C] :
( ( finite_finite2 @ B @ S )
=> ( ( groups7121269368397514597t_prod @ B @ A @ H2 @ S )
= ( groups7121269368397514597t_prod @ C @ A
@ ^ [Y2: C] :
( groups7121269368397514597t_prod @ B @ A @ H2
@ ( collect @ B
@ ^ [X2: B] :
( ( member @ B @ X2 @ S )
& ( ( G2 @ X2 )
= Y2 ) ) ) )
@ ( image @ B @ C @ G2 @ S ) ) ) ) ) ).
% prod.image_gen
thf(fact_6234_translation__subtract__Compl,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A3: A,T: set @ A] :
( ( image @ A @ A
@ ^ [X2: A] : ( minus_minus @ A @ X2 @ A3 )
@ ( uminus_uminus @ ( set @ A ) @ T ) )
= ( uminus_uminus @ ( set @ A )
@ ( image @ A @ A
@ ^ [X2: A] : ( minus_minus @ A @ X2 @ A3 )
@ T ) ) ) ) ).
% translation_subtract_Compl
thf(fact_6235_the__elem__image__unique,axiom,
! [B: $tType,A: $tType,A2: set @ A,F3: A > B,X: A] :
( ( A2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [Y4: A] :
( ( member @ A @ Y4 @ A2 )
=> ( ( F3 @ Y4 )
= ( F3 @ X ) ) )
=> ( ( the_elem @ B @ ( image @ A @ B @ F3 @ A2 ) )
= ( F3 @ X ) ) ) ) ).
% the_elem_image_unique
thf(fact_6236_Abs__fnat__hom__Suc,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: nat] :
( ( word_succ @ A @ ( semiring_1_of_nat @ ( word @ A ) @ A3 ) )
= ( semiring_1_of_nat @ ( word @ A ) @ ( suc @ A3 ) ) ) ) ).
% Abs_fnat_hom_Suc
thf(fact_6237_word__succ__p1,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( word_succ @ A )
= ( ^ [A5: word @ A] : ( plus_plus @ ( word @ A ) @ A5 @ ( one_one @ ( word @ A ) ) ) ) ) ) ).
% word_succ_p1
thf(fact_6238_mask__eqs_I11_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A,N: nat] :
( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ ( word_succ @ A @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ A3 @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) )
= ( bit_se5824344872417868541ns_and @ ( word @ A ) @ ( word_succ @ A @ A3 ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) ) ) ).
% mask_eqs(11)
thf(fact_6239_word__mult__succ,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A,B3: word @ A] :
( ( times_times @ ( word @ A ) @ ( word_succ @ A @ A3 ) @ B3 )
= ( plus_plus @ ( word @ A ) @ B3 @ ( times_times @ ( word @ A ) @ A3 @ B3 ) ) ) ) ).
% word_mult_succ
thf(fact_6240_sum_Ogroup,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [S: set @ B,T4: set @ C,G2: B > C,H2: B > A] :
( ( finite_finite2 @ B @ S )
=> ( ( finite_finite2 @ C @ T4 )
=> ( ( ord_less_eq @ ( set @ C ) @ ( image @ B @ C @ G2 @ S ) @ T4 )
=> ( ( groups7311177749621191930dd_sum @ C @ A
@ ^ [Y2: C] :
( groups7311177749621191930dd_sum @ B @ A @ H2
@ ( collect @ B
@ ^ [X2: B] :
( ( member @ B @ X2 @ S )
& ( ( G2 @ X2 )
= Y2 ) ) ) )
@ T4 )
= ( groups7311177749621191930dd_sum @ B @ A @ H2 @ S ) ) ) ) ) ) ).
% sum.group
thf(fact_6241_prod_Ogroup,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S: set @ B,T4: set @ C,G2: B > C,H2: B > A] :
( ( finite_finite2 @ B @ S )
=> ( ( finite_finite2 @ C @ T4 )
=> ( ( ord_less_eq @ ( set @ C ) @ ( image @ B @ C @ G2 @ S ) @ T4 )
=> ( ( groups7121269368397514597t_prod @ C @ A
@ ^ [Y2: C] :
( groups7121269368397514597t_prod @ B @ A @ H2
@ ( collect @ B
@ ^ [X2: B] :
( ( member @ B @ X2 @ S )
& ( ( G2 @ X2 )
= Y2 ) ) ) )
@ T4 )
= ( groups7121269368397514597t_prod @ B @ A @ H2 @ S ) ) ) ) ) ) ).
% prod.group
thf(fact_6242_scaleR__image__atLeastAtMost,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C2: real,X: A,Y: A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ C2 )
=> ( ( image @ A @ A @ ( real_V8093663219630862766scaleR @ A @ C2 ) @ ( set_or1337092689740270186AtMost @ A @ X @ Y ) )
= ( set_or1337092689740270186AtMost @ A @ ( real_V8093663219630862766scaleR @ A @ C2 @ X ) @ ( real_V8093663219630862766scaleR @ A @ C2 @ Y ) ) ) ) ) ).
% scaleR_image_atLeastAtMost
thf(fact_6243_wi__hom__succ,axiom,
! [E3: $tType] :
( ( type_len @ E3 )
=> ! [A3: int] :
( ( word_succ @ E3 @ ( ring_1_of_int @ ( word @ E3 ) @ A3 ) )
= ( ring_1_of_int @ ( word @ E3 ) @ ( plus_plus @ int @ A3 @ ( one_one @ int ) ) ) ) ) ).
% wi_hom_succ
thf(fact_6244_word__arith__nat__Suc,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( word_succ @ A )
= ( ^ [A5: word @ A] : ( semiring_1_of_nat @ ( word @ A ) @ ( suc @ ( semiring_1_unsigned @ A @ nat @ A5 ) ) ) ) ) ) ).
% word_arith_nat_Suc
thf(fact_6245_image__mult__atLeastAtMost__if,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,X: A,Y: A] :
( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( image @ A @ A @ ( times_times @ A @ C2 ) @ ( set_or1337092689740270186AtMost @ A @ X @ Y ) )
= ( set_or1337092689740270186AtMost @ A @ ( times_times @ A @ C2 @ X ) @ ( times_times @ A @ C2 @ Y ) ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less_eq @ A @ X @ Y )
=> ( ( image @ A @ A @ ( times_times @ A @ C2 ) @ ( set_or1337092689740270186AtMost @ A @ X @ Y ) )
= ( set_or1337092689740270186AtMost @ A @ ( times_times @ A @ C2 @ Y ) @ ( times_times @ A @ C2 @ X ) ) ) )
& ( ~ ( ord_less_eq @ A @ X @ Y )
=> ( ( image @ A @ A @ ( times_times @ A @ C2 ) @ ( set_or1337092689740270186AtMost @ A @ X @ Y ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% image_mult_atLeastAtMost_if
thf(fact_6246_word__succ__alt,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( word_succ @ A )
= ( ^ [A5: word @ A] : ( ring_1_of_int @ ( word @ A ) @ ( plus_plus @ int @ ( semiring_1_unsigned @ A @ int @ A5 ) @ ( one_one @ int ) ) ) ) ) ) ).
% word_succ_alt
thf(fact_6247_image__mult__atLeastAtMost__if_H,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y: A,C2: A] :
( ( ( ord_less_eq @ A @ X @ Y )
=> ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( image @ A @ A
@ ^ [X2: A] : ( times_times @ A @ X2 @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ X @ Y ) )
= ( set_or1337092689740270186AtMost @ A @ ( times_times @ A @ X @ C2 ) @ ( times_times @ A @ Y @ C2 ) ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( image @ A @ A
@ ^ [X2: A] : ( times_times @ A @ X2 @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ X @ Y ) )
= ( set_or1337092689740270186AtMost @ A @ ( times_times @ A @ Y @ C2 ) @ ( times_times @ A @ X @ C2 ) ) ) ) ) )
& ( ~ ( ord_less_eq @ A @ X @ Y )
=> ( ( image @ A @ A
@ ^ [X2: A] : ( times_times @ A @ X2 @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ X @ Y ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% image_mult_atLeastAtMost_if'
thf(fact_6248_image__affinity__atLeastAtMost,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B3: A,M: A,C2: A] :
( ( ( ( set_or1337092689740270186AtMost @ A @ A3 @ B3 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( image @ A @ A
@ ^ [X2: A] : ( plus_plus @ A @ ( times_times @ A @ M @ X2 ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A3 @ B3 ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( ( set_or1337092689740270186AtMost @ A @ A3 @ B3 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ M )
=> ( ( image @ A @ A
@ ^ [X2: A] : ( plus_plus @ A @ ( times_times @ A @ M @ X2 ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A3 @ B3 ) )
= ( set_or1337092689740270186AtMost @ A @ ( plus_plus @ A @ ( times_times @ A @ M @ A3 ) @ C2 ) @ ( plus_plus @ A @ ( times_times @ A @ M @ B3 ) @ C2 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ M )
=> ( ( image @ A @ A
@ ^ [X2: A] : ( plus_plus @ A @ ( times_times @ A @ M @ X2 ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A3 @ B3 ) )
= ( set_or1337092689740270186AtMost @ A @ ( plus_plus @ A @ ( times_times @ A @ M @ B3 ) @ C2 ) @ ( plus_plus @ A @ ( times_times @ A @ M @ A3 ) @ C2 ) ) ) ) ) ) ) ) ).
% image_affinity_atLeastAtMost
thf(fact_6249_image__affinity__atLeastAtMost__diff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B3: A,M: A,C2: A] :
( ( ( ( set_or1337092689740270186AtMost @ A @ A3 @ B3 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( image @ A @ A
@ ^ [X2: A] : ( minus_minus @ A @ ( times_times @ A @ M @ X2 ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A3 @ B3 ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( ( set_or1337092689740270186AtMost @ A @ A3 @ B3 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ M )
=> ( ( image @ A @ A
@ ^ [X2: A] : ( minus_minus @ A @ ( times_times @ A @ M @ X2 ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A3 @ B3 ) )
= ( set_or1337092689740270186AtMost @ A @ ( minus_minus @ A @ ( times_times @ A @ M @ A3 ) @ C2 ) @ ( minus_minus @ A @ ( times_times @ A @ M @ B3 ) @ C2 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ M )
=> ( ( image @ A @ A
@ ^ [X2: A] : ( minus_minus @ A @ ( times_times @ A @ M @ X2 ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A3 @ B3 ) )
= ( set_or1337092689740270186AtMost @ A @ ( minus_minus @ A @ ( times_times @ A @ M @ B3 ) @ C2 ) @ ( minus_minus @ A @ ( times_times @ A @ M @ A3 ) @ C2 ) ) ) ) ) ) ) ) ).
% image_affinity_atLeastAtMost_diff
thf(fact_6250_image__affinity__atLeastAtMost__div,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B3: A,M: A,C2: A] :
( ( ( ( set_or1337092689740270186AtMost @ A @ A3 @ B3 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( image @ A @ A
@ ^ [X2: A] : ( plus_plus @ A @ ( divide_divide @ A @ X2 @ M ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A3 @ B3 ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( ( set_or1337092689740270186AtMost @ A @ A3 @ B3 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ M )
=> ( ( image @ A @ A
@ ^ [X2: A] : ( plus_plus @ A @ ( divide_divide @ A @ X2 @ M ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A3 @ B3 ) )
= ( set_or1337092689740270186AtMost @ A @ ( plus_plus @ A @ ( divide_divide @ A @ A3 @ M ) @ C2 ) @ ( plus_plus @ A @ ( divide_divide @ A @ B3 @ M ) @ C2 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ M )
=> ( ( image @ A @ A
@ ^ [X2: A] : ( plus_plus @ A @ ( divide_divide @ A @ X2 @ M ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A3 @ B3 ) )
= ( set_or1337092689740270186AtMost @ A @ ( plus_plus @ A @ ( divide_divide @ A @ B3 @ M ) @ C2 ) @ ( plus_plus @ A @ ( divide_divide @ A @ A3 @ M ) @ C2 ) ) ) ) ) ) ) ) ).
% image_affinity_atLeastAtMost_div
thf(fact_6251_image__affinity__atLeastAtMost__div__diff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B3: A,M: A,C2: A] :
( ( ( ( set_or1337092689740270186AtMost @ A @ A3 @ B3 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( image @ A @ A
@ ^ [X2: A] : ( minus_minus @ A @ ( divide_divide @ A @ X2 @ M ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A3 @ B3 ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( ( set_or1337092689740270186AtMost @ A @ A3 @ B3 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ M )
=> ( ( image @ A @ A
@ ^ [X2: A] : ( minus_minus @ A @ ( divide_divide @ A @ X2 @ M ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A3 @ B3 ) )
= ( set_or1337092689740270186AtMost @ A @ ( minus_minus @ A @ ( divide_divide @ A @ A3 @ M ) @ C2 ) @ ( minus_minus @ A @ ( divide_divide @ A @ B3 @ M ) @ C2 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ M )
=> ( ( image @ A @ A
@ ^ [X2: A] : ( minus_minus @ A @ ( divide_divide @ A @ X2 @ M ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A3 @ B3 ) )
= ( set_or1337092689740270186AtMost @ A @ ( minus_minus @ A @ ( divide_divide @ A @ B3 @ M ) @ C2 ) @ ( minus_minus @ A @ ( divide_divide @ A @ A3 @ M ) @ C2 ) ) ) ) ) ) ) ) ).
% image_affinity_atLeastAtMost_div_diff
thf(fact_6252_word__sp__01,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType] :
( ( ( type_len @ D )
& ( 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 @ D @ ( one_one @ ( word @ D ) ) )
= ( zero_zero @ ( word @ D ) ) ) ) ) ).
% word_sp_01
thf(fact_6253_uint__word__arith__bintrs_I5_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A] :
( ( semiring_1_unsigned @ A @ int @ ( word_succ @ A @ A3 ) )
= ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( plus_plus @ int @ ( semiring_1_unsigned @ A @ int @ A3 ) @ ( one_one @ int ) ) ) ) ) ).
% uint_word_arith_bintrs(5)
thf(fact_6254_sint__word__ariths_I5_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A] :
( ( ring_1_signed @ A @ int @ ( word_succ @ A @ A3 ) )
= ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( plus_plus @ int @ ( ring_1_signed @ A @ int @ A3 ) @ ( one_one @ int ) ) ) ) ) ).
% sint_word_ariths(5)
thf(fact_6255_unat__word__ariths_I3_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A] :
( ( semiring_1_unsigned @ A @ nat @ ( word_succ @ A @ A3 ) )
= ( modulo_modulo @ nat @ ( suc @ ( semiring_1_unsigned @ A @ nat @ A3 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).
% unat_word_ariths(3)
thf(fact_6256_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_6257_sless__eq__word__minus__numeral__minus__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: num,B3: num] :
( ( word_sle @ A @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ A3 ) ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ B3 ) ) )
= ( ord_less_eq @ int @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ A3 ) ) ) @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ B3 ) ) ) ) ) ) ).
% sless_eq_word_minus_numeral_minus_numeral
thf(fact_6258_bij__betw__Suc,axiom,
! [M7: set @ nat,N3: set @ nat] :
( ( bij_betw @ nat @ nat @ suc @ M7 @ N3 )
= ( ( image @ nat @ nat @ suc @ M7 )
= N3 ) ) ).
% bij_betw_Suc
thf(fact_6259_signed_Oorder__refl,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] : ( word_sle @ A @ X @ X ) ) ).
% signed.order_refl
thf(fact_6260_signed_Odual__order_Orefl,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A] : ( word_sle @ A @ A3 @ A3 ) ) ).
% signed.dual_order.refl
thf(fact_6261_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_6262_pair__imageI,axiom,
! [C: $tType,B: $tType,A: $tType,A3: A,B3: B,A2: set @ ( product_prod @ A @ B ),F3: A > B > C] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B3 ) @ A2 )
=> ( member @ C @ ( F3 @ A3 @ B3 ) @ ( image @ ( product_prod @ A @ B ) @ C @ ( product_case_prod @ A @ B @ C @ F3 ) @ A2 ) ) ) ).
% pair_imageI
thf(fact_6263_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_6264_word__sle__no,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: num,B3: num] :
( ( word_sle @ A @ ( numeral_numeral @ ( word @ A ) @ A3 ) @ ( numeral_numeral @ ( word @ A ) @ B3 ) )
= ( ord_less_eq @ int @ ( ring_1_signed @ A @ int @ ( numeral_numeral @ ( word @ A ) @ A3 ) ) @ ( ring_1_signed @ A @ int @ ( numeral_numeral @ ( word @ A ) @ B3 ) ) ) ) ) ).
% word_sle_no
thf(fact_6265_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_6266_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_6267_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_6268_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_6269_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_6270_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_6271_sless__eq__word__numeral__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: num,B3: num] :
( ( word_sle @ A @ ( numeral_numeral @ ( word @ A ) @ A3 ) @ ( numeral_numeral @ ( word @ A ) @ B3 ) )
= ( ord_less_eq @ int @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( numeral_numeral @ int @ A3 ) ) @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( numeral_numeral @ int @ B3 ) ) ) ) ) ).
% sless_eq_word_numeral_numeral
thf(fact_6272_sless__eq__word__minus__numeral__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: num,B3: num] :
( ( word_sle @ A @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ A3 ) ) @ ( numeral_numeral @ ( word @ A ) @ B3 ) )
= ( ord_less_eq @ int @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ A3 ) ) ) @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( numeral_numeral @ int @ B3 ) ) ) ) ) ).
% sless_eq_word_minus_numeral_numeral
thf(fact_6273_sless__eq__word__numeral__minus__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: num,B3: num] :
( ( word_sle @ A @ ( numeral_numeral @ ( word @ A ) @ A3 ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ B3 ) ) )
= ( ord_less_eq @ int @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( numeral_numeral @ int @ A3 ) ) @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ B3 ) ) ) ) ) ) ).
% sless_eq_word_numeral_minus_numeral
thf(fact_6274_zero__notin__Suc__image,axiom,
! [A2: set @ nat] :
~ ( member @ nat @ ( zero_zero @ nat ) @ ( image @ nat @ nat @ suc @ A2 ) ) ).
% zero_notin_Suc_image
thf(fact_6275_word__sle__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( word_sle @ A )
= ( ^ [A5: word @ A,B5: word @ A] : ( ord_less_eq @ int @ ( ring_1_signed @ A @ int @ A5 ) @ ( ring_1_signed @ A @ int @ B5 ) ) ) ) ) ).
% word_sle_eq
thf(fact_6276_signed_Olift__Suc__antimono__le,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [F3: nat > ( word @ A ),N: nat,N7: nat] :
( ! [N2: nat] : ( word_sle @ A @ ( F3 @ ( suc @ N2 ) ) @ ( F3 @ N2 ) )
=> ( ( ord_less_eq @ nat @ N @ N7 )
=> ( word_sle @ A @ ( F3 @ N7 ) @ ( F3 @ N ) ) ) ) ) ).
% signed.lift_Suc_antimono_le
thf(fact_6277_signed_Olift__Suc__mono__le,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [F3: nat > ( word @ A ),N: nat,N7: nat] :
( ! [N2: nat] : ( word_sle @ A @ ( F3 @ N2 ) @ ( F3 @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq @ nat @ N @ N7 )
=> ( word_sle @ A @ ( F3 @ N ) @ ( F3 @ N7 ) ) ) ) ) ).
% signed.lift_Suc_mono_le
thf(fact_6278_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_6279_signed_Olinear,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] :
( ( word_sle @ A @ X @ Y )
| ( word_sle @ A @ Y @ X ) ) ) ).
% signed.linear
thf(fact_6280_signed_Onle__le,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A,B3: word @ A] :
( ( ~ ( word_sle @ A @ A3 @ B3 ) )
= ( ( word_sle @ A @ B3 @ A3 )
& ( B3 != A3 ) ) ) ) ).
% signed.nle_le
thf(fact_6281_signed_Oeq__refl,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] :
( ( X = Y )
=> ( word_sle @ A @ X @ Y ) ) ) ).
% signed.eq_refl
thf(fact_6282_signed_Ole__cases,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] :
( ~ ( word_sle @ A @ X @ Y )
=> ( word_sle @ A @ Y @ X ) ) ) ).
% signed.le_cases
thf(fact_6283_signed_Ole__cases3,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A,Z: word @ A] :
( ( ( word_sle @ A @ X @ Y )
=> ~ ( word_sle @ A @ Y @ Z ) )
=> ( ( ( word_sle @ A @ Y @ X )
=> ~ ( word_sle @ A @ X @ Z ) )
=> ( ( ( word_sle @ A @ X @ Z )
=> ~ ( word_sle @ A @ Z @ Y ) )
=> ( ( ( word_sle @ A @ Z @ Y )
=> ~ ( word_sle @ A @ Y @ X ) )
=> ( ( ( word_sle @ A @ Y @ Z )
=> ~ ( word_sle @ A @ Z @ X ) )
=> ~ ( ( word_sle @ A @ Z @ X )
=> ~ ( word_sle @ A @ X @ Y ) ) ) ) ) ) ) ) ).
% signed.le_cases3
thf(fact_6284_signed_Oorder_Otrans,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A,B3: word @ A,C2: word @ A] :
( ( word_sle @ A @ A3 @ B3 )
=> ( ( word_sle @ A @ B3 @ C2 )
=> ( word_sle @ A @ A3 @ C2 ) ) ) ) ).
% signed.order.trans
thf(fact_6285_signed_Oorder__trans,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A,Z: word @ A] :
( ( word_sle @ A @ X @ Y )
=> ( ( word_sle @ A @ Y @ Z )
=> ( word_sle @ A @ X @ Z ) ) ) ) ).
% signed.order_trans
thf(fact_6286_signed_Oantisym__conv,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y: word @ A,X: word @ A] :
( ( word_sle @ A @ Y @ X )
=> ( ( word_sle @ A @ X @ Y )
= ( X = Y ) ) ) ) ).
% signed.antisym_conv
thf(fact_6287_signed_Oorder_Oeq__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( ^ [Y3: word @ A,Z2: word @ A] : Y3 = Z2 )
= ( ^ [A5: word @ A,B5: word @ A] :
( ( word_sle @ A @ A5 @ B5 )
& ( word_sle @ A @ B5 @ A5 ) ) ) ) ) ).
% signed.order.eq_iff
thf(fact_6288_signed_Oorder__eq__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( ^ [Y3: word @ A,Z2: word @ A] : Y3 = Z2 )
= ( ^ [X2: word @ A,Y2: word @ A] :
( ( word_sle @ A @ X2 @ Y2 )
& ( word_sle @ A @ Y2 @ X2 ) ) ) ) ) ).
% signed.order_eq_iff
thf(fact_6289_signed_Olinorder__wlog,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [P: ( word @ A ) > ( word @ A ) > $o,A3: word @ A,B3: word @ A] :
( ! [A6: word @ A,B7: word @ A] :
( ( word_sle @ A @ A6 @ B7 )
=> ( P @ A6 @ B7 ) )
=> ( ! [A6: word @ A,B7: word @ A] :
( ( P @ B7 @ A6 )
=> ( P @ A6 @ B7 ) )
=> ( P @ A3 @ B3 ) ) ) ) ).
% signed.linorder_wlog
thf(fact_6290_signed_Oorder_Oantisym,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A,B3: word @ A] :
( ( word_sle @ A @ A3 @ B3 )
=> ( ( word_sle @ A @ B3 @ A3 )
=> ( A3 = B3 ) ) ) ) ).
% signed.order.antisym
thf(fact_6291_signed_Oorder__antisym,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] :
( ( word_sle @ A @ X @ Y )
=> ( ( word_sle @ A @ Y @ X )
=> ( X = Y ) ) ) ) ).
% signed.order_antisym
thf(fact_6292_signed_Oord__eq__le__trans,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A,B3: word @ A,C2: word @ A] :
( ( A3 = B3 )
=> ( ( word_sle @ A @ B3 @ C2 )
=> ( word_sle @ A @ A3 @ C2 ) ) ) ) ).
% signed.ord_eq_le_trans
thf(fact_6293_signed_Oord__le__eq__trans,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A,B3: word @ A,C2: word @ A] :
( ( word_sle @ A @ A3 @ B3 )
=> ( ( B3 = C2 )
=> ( word_sle @ A @ A3 @ C2 ) ) ) ) ).
% signed.ord_le_eq_trans
thf(fact_6294_signed_Odual__order_Otrans,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [B3: word @ A,A3: word @ A,C2: word @ A] :
( ( word_sle @ A @ B3 @ A3 )
=> ( ( word_sle @ A @ C2 @ B3 )
=> ( word_sle @ A @ C2 @ A3 ) ) ) ) ).
% signed.dual_order.trans
thf(fact_6295_signed_Odual__order_Oeq__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( ^ [Y3: word @ A,Z2: word @ A] : Y3 = Z2 )
= ( ^ [A5: word @ A,B5: word @ A] :
( ( word_sle @ A @ B5 @ A5 )
& ( word_sle @ A @ A5 @ B5 ) ) ) ) ) ).
% signed.dual_order.eq_iff
thf(fact_6296_signed_Odual__order_Oantisym,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [B3: word @ A,A3: word @ A] :
( ( word_sle @ A @ B3 @ A3 )
=> ( ( word_sle @ A @ A3 @ B3 )
=> ( A3 = B3 ) ) ) ) ).
% signed.dual_order.antisym
thf(fact_6297_signed_Ofinite__has__maximal,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: set @ ( word @ A )] :
( ( finite_finite2 @ ( word @ A ) @ A2 )
=> ( ( A2
!= ( bot_bot @ ( set @ ( word @ A ) ) ) )
=> ? [X3: word @ A] :
( ( member @ ( word @ A ) @ X3 @ A2 )
& ! [Xa2: word @ A] :
( ( member @ ( word @ A ) @ Xa2 @ A2 )
=> ( ( word_sle @ A @ X3 @ Xa2 )
=> ( X3 = Xa2 ) ) ) ) ) ) ) ).
% signed.finite_has_maximal
thf(fact_6298_signed_Ofinite__has__minimal,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: set @ ( word @ A )] :
( ( finite_finite2 @ ( word @ A ) @ A2 )
=> ( ( A2
!= ( bot_bot @ ( set @ ( word @ A ) ) ) )
=> ? [X3: word @ A] :
( ( member @ ( word @ A ) @ X3 @ A2 )
& ! [Xa2: word @ A] :
( ( member @ ( word @ A ) @ Xa2 @ A2 )
=> ( ( word_sle @ A @ Xa2 @ X3 )
=> ( X3 = Xa2 ) ) ) ) ) ) ) ).
% signed.finite_has_minimal
thf(fact_6299_signed_Ofinite__has__maximal2,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: set @ ( word @ A ),A3: word @ A] :
( ( finite_finite2 @ ( word @ A ) @ A2 )
=> ( ( member @ ( word @ A ) @ A3 @ A2 )
=> ? [X3: word @ A] :
( ( member @ ( word @ A ) @ X3 @ A2 )
& ( word_sle @ A @ A3 @ X3 )
& ! [Xa2: word @ A] :
( ( member @ ( word @ A ) @ Xa2 @ A2 )
=> ( ( word_sle @ A @ X3 @ Xa2 )
=> ( X3 = Xa2 ) ) ) ) ) ) ) ).
% signed.finite_has_maximal2
thf(fact_6300_signed_Ofinite__has__minimal2,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: set @ ( word @ A ),A3: word @ A] :
( ( finite_finite2 @ ( word @ A ) @ A2 )
=> ( ( member @ ( word @ A ) @ A3 @ A2 )
=> ? [X3: word @ A] :
( ( member @ ( word @ A ) @ X3 @ A2 )
& ( word_sle @ A @ X3 @ A3 )
& ! [Xa2: word @ A] :
( ( member @ ( word @ A ) @ Xa2 @ A2 )
=> ( ( word_sle @ A @ Xa2 @ X3 )
=> ( X3 = Xa2 ) ) ) ) ) ) ) ).
% signed.finite_has_minimal2
thf(fact_6301_in__image__insert__iff,axiom,
! [A: $tType,B2: set @ ( set @ A ),X: A,A2: set @ A] :
( ! [C7: set @ A] :
( ( member @ ( set @ A ) @ C7 @ B2 )
=> ~ ( member @ A @ X @ C7 ) )
=> ( ( member @ ( set @ A ) @ A2 @ ( image @ ( set @ A ) @ ( set @ A ) @ ( insert @ A @ X ) @ B2 ) )
= ( ( member @ A @ X @ A2 )
& ( member @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A2 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) @ B2 ) ) ) ) ).
% in_image_insert_iff
thf(fact_6302_signed_Ofinite__ranking__induct,axiom,
! [A: $tType,B: $tType] :
( ( type_len @ A )
=> ! [S: set @ B,P: ( set @ B ) > $o,F3: B > ( word @ A )] :
( ( finite_finite2 @ B @ S )
=> ( ( P @ ( bot_bot @ ( set @ B ) ) )
=> ( ! [X3: B,S6: set @ B] :
( ( finite_finite2 @ B @ S6 )
=> ( ! [Y5: B] :
( ( member @ B @ Y5 @ S6 )
=> ( word_sle @ A @ ( F3 @ Y5 ) @ ( F3 @ X3 ) ) )
=> ( ( P @ S6 )
=> ( P @ ( insert @ B @ X3 @ S6 ) ) ) ) )
=> ( P @ S ) ) ) ) ) ).
% signed.finite_ranking_induct
thf(fact_6303_image__int__atLeastAtMost,axiom,
! [A3: nat,B3: nat] :
( ( image @ nat @ int @ ( semiring_1_of_nat @ int ) @ ( set_or1337092689740270186AtMost @ nat @ A3 @ B3 ) )
= ( set_or1337092689740270186AtMost @ int @ ( semiring_1_of_nat @ int @ A3 ) @ ( semiring_1_of_nat @ int @ B3 ) ) ) ).
% image_int_atLeastAtMost
thf(fact_6304_image__int__atLeastLessThan,axiom,
! [A3: nat,B3: nat] :
( ( image @ nat @ int @ ( semiring_1_of_nat @ int ) @ ( set_or7035219750837199246ssThan @ nat @ A3 @ B3 ) )
= ( set_or7035219750837199246ssThan @ int @ ( semiring_1_of_nat @ int @ A3 ) @ ( semiring_1_of_nat @ int @ B3 ) ) ) ).
% image_int_atLeastLessThan
thf(fact_6305_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_6306_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_6307_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_6308_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_6309_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_6310_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_6311_image__add__int__atLeastLessThan,axiom,
! [L: int,U2: int] :
( ( image @ int @ int
@ ^ [X2: int] : ( plus_plus @ int @ X2 @ L )
@ ( set_or7035219750837199246ssThan @ int @ ( zero_zero @ int ) @ ( minus_minus @ int @ U2 @ L ) ) )
= ( set_or7035219750837199246ssThan @ int @ L @ U2 ) ) ).
% image_add_int_atLeastLessThan
thf(fact_6312_word__roti__conv__mod_H,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( word_roti @ A )
= ( ^ [N4: int,W2: word @ A] : ( word_roti @ A @ ( modulo_modulo @ int @ N4 @ ( semiring_1_of_nat @ int @ ( size_size @ ( word @ A ) @ W2 ) ) ) @ W2 ) ) ) ) ).
% word_roti_conv_mod'
thf(fact_6313_image__add__integer__atLeastLessThan,axiom,
! [L: code_integer,U2: code_integer] :
( ( image @ code_integer @ code_integer
@ ^ [X2: code_integer] : ( plus_plus @ code_integer @ X2 @ L )
@ ( set_or7035219750837199246ssThan @ code_integer @ ( zero_zero @ code_integer ) @ ( minus_minus @ code_integer @ U2 @ L ) ) )
= ( set_or7035219750837199246ssThan @ code_integer @ L @ U2 ) ) ).
% image_add_integer_atLeastLessThan
thf(fact_6314_image__atLeastZeroLessThan__int,axiom,
! [U2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ U2 )
=> ( ( set_or7035219750837199246ssThan @ int @ ( zero_zero @ int ) @ U2 )
= ( image @ nat @ int @ ( semiring_1_of_nat @ int ) @ ( set_ord_lessThan @ nat @ ( nat2 @ U2 ) ) ) ) ) ).
% image_atLeastZeroLessThan_int
thf(fact_6315_image__minus__const__atLeastLessThan__nat,axiom,
! [C2: nat,Y: nat,X: nat] :
( ( ( ord_less @ nat @ C2 @ Y )
=> ( ( image @ nat @ nat
@ ^ [I4: nat] : ( minus_minus @ nat @ I4 @ C2 )
@ ( set_or7035219750837199246ssThan @ nat @ X @ Y ) )
= ( set_or7035219750837199246ssThan @ nat @ ( minus_minus @ nat @ X @ C2 ) @ ( minus_minus @ nat @ Y @ C2 ) ) ) )
& ( ~ ( ord_less @ nat @ C2 @ Y )
=> ( ( ( ord_less @ nat @ X @ Y )
=> ( ( image @ nat @ nat
@ ^ [I4: nat] : ( minus_minus @ nat @ I4 @ C2 )
@ ( set_or7035219750837199246ssThan @ nat @ X @ Y ) )
= ( insert @ nat @ ( zero_zero @ nat ) @ ( bot_bot @ ( set @ nat ) ) ) ) )
& ( ~ ( ord_less @ nat @ X @ Y )
=> ( ( image @ nat @ nat
@ ^ [I4: nat] : ( minus_minus @ nat @ I4 @ C2 )
@ ( set_or7035219750837199246ssThan @ nat @ X @ Y ) )
= ( bot_bot @ ( set @ nat ) ) ) ) ) ) ) ).
% image_minus_const_atLeastLessThan_nat
thf(fact_6316_word__roti__conv__mod,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( word_roti @ A )
= ( ^ [N4: int] : ( word_roti @ A @ ( modulo_modulo @ int @ N4 @ ( semiring_1_of_nat @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ) ) ).
% word_roti_conv_mod
thf(fact_6317_word__0__sle__from__less,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] :
( ( ord_less @ ( word @ A ) @ X @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) ) )
=> ( word_sle @ A @ ( zero_zero @ ( word @ A ) ) @ X ) ) ) ).
% word_0_sle_from_less
thf(fact_6318_sless__word__minus__numeral__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: num,B3: num] :
( ( word_sless @ A @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ A3 ) ) @ ( numeral_numeral @ ( word @ A ) @ B3 ) )
= ( ord_less @ int @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ A3 ) ) ) @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( numeral_numeral @ int @ B3 ) ) ) ) ) ).
% sless_word_minus_numeral_numeral
thf(fact_6319_sless__word__numeral__minus__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: num,B3: num] :
( ( word_sless @ A @ ( numeral_numeral @ ( word @ A ) @ A3 ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ B3 ) ) )
= ( ord_less @ int @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( numeral_numeral @ int @ A3 ) ) @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ B3 ) ) ) ) ) ) ).
% sless_word_numeral_minus_numeral
thf(fact_6320_word__sless__no,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: num,B3: num] :
( ( word_sless @ A @ ( numeral_numeral @ ( word @ A ) @ A3 ) @ ( numeral_numeral @ ( word @ A ) @ B3 ) )
= ( ( word_sle @ A @ ( numeral_numeral @ ( word @ A ) @ A3 ) @ ( numeral_numeral @ ( word @ A ) @ B3 ) )
& ( ( numeral_numeral @ ( word @ A ) @ A3 )
!= ( numeral_numeral @ ( word @ A ) @ B3 ) ) ) ) ) ).
% word_sless_no
thf(fact_6321_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_6322_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_6323_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_6324_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_6325_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_6326_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_6327_sless__word__numeral__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: num,B3: num] :
( ( word_sless @ A @ ( numeral_numeral @ ( word @ A ) @ A3 ) @ ( numeral_numeral @ ( word @ A ) @ B3 ) )
= ( ord_less @ int @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( numeral_numeral @ int @ A3 ) ) @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( numeral_numeral @ int @ B3 ) ) ) ) ) ).
% sless_word_numeral_numeral
thf(fact_6328_sless__word__minus__numeral__minus__numeral,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: num,B3: num] :
( ( word_sless @ A @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ A3 ) ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ B3 ) ) )
= ( ord_less @ int @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ A3 ) ) ) @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ B3 ) ) ) ) ) ) ).
% sless_word_minus_numeral_minus_numeral
thf(fact_6329_signed_OleD,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y: word @ A,X: word @ A] :
( ( word_sle @ A @ Y @ X )
=> ~ ( word_sless @ A @ X @ Y ) ) ) ).
% signed.leD
thf(fact_6330_signed_OleI,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] :
( ~ ( word_sless @ A @ X @ Y )
=> ( word_sle @ A @ Y @ X ) ) ) ).
% signed.leI
thf(fact_6331_signed_Onot__le,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] :
( ( ~ ( word_sle @ A @ X @ Y ) )
= ( word_sless @ A @ Y @ X ) ) ) ).
% signed.not_le
thf(fact_6332_signed_Ole__less,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( word_sle @ A )
= ( ^ [X2: word @ A,Y2: word @ A] :
( ( word_sless @ A @ X2 @ Y2 )
| ( X2 = Y2 ) ) ) ) ) ).
% signed.le_less
thf(fact_6333_signed_Onless__le,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A,B3: word @ A] :
( ( ~ ( word_sless @ A @ A3 @ B3 ) )
= ( ~ ( word_sle @ A @ A3 @ B3 )
| ( A3 = B3 ) ) ) ) ).
% signed.nless_le
thf(fact_6334_signed_Onot__less,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] :
( ( ~ ( word_sless @ A @ X @ Y ) )
= ( word_sle @ A @ Y @ X ) ) ) ).
% signed.not_less
thf(fact_6335_signed_Oless__imp__le,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] :
( ( word_sless @ A @ X @ Y )
=> ( word_sle @ A @ X @ Y ) ) ) ).
% signed.less_imp_le
thf(fact_6336_signed_Ole__neq__trans,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A,B3: word @ A] :
( ( word_sle @ A @ A3 @ B3 )
=> ( ( A3 != B3 )
=> ( word_sless @ A @ A3 @ B3 ) ) ) ) ).
% signed.le_neq_trans
thf(fact_6337_signed_Oantisym__conv1,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] :
( ~ ( word_sless @ A @ X @ Y )
=> ( ( word_sle @ A @ X @ Y )
= ( X = Y ) ) ) ) ).
% signed.antisym_conv1
thf(fact_6338_signed_Oantisym__conv2,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] :
( ( word_sle @ A @ X @ Y )
=> ( ( ~ ( word_sless @ A @ X @ Y ) )
= ( X = Y ) ) ) ) ).
% signed.antisym_conv2
thf(fact_6339_signed_Ole__less__trans,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A,Z: word @ A] :
( ( word_sle @ A @ X @ Y )
=> ( ( word_sless @ A @ Y @ Z )
=> ( word_sless @ A @ X @ Z ) ) ) ) ).
% signed.le_less_trans
thf(fact_6340_signed_Oless__le__trans,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A,Z: word @ A] :
( ( word_sless @ A @ X @ Y )
=> ( ( word_sle @ A @ Y @ Z )
=> ( word_sless @ A @ X @ Z ) ) ) ) ).
% signed.less_le_trans
thf(fact_6341_signed_Ole__less__linear,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] :
( ( word_sle @ A @ X @ Y )
| ( word_sless @ A @ Y @ X ) ) ) ).
% signed.le_less_linear
thf(fact_6342_signed_Oless__le__not__le,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( word_sless @ A )
= ( ^ [X2: word @ A,Y2: word @ A] :
( ( word_sle @ A @ X2 @ Y2 )
& ~ ( word_sle @ A @ Y2 @ X2 ) ) ) ) ) ).
% signed.less_le_not_le
thf(fact_6343_signed_Onot__le__imp__less,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y: word @ A,X: word @ A] :
( ~ ( word_sle @ A @ Y @ X )
=> ( word_sless @ A @ X @ Y ) ) ) ).
% signed.not_le_imp_less
thf(fact_6344_signed_Ole__imp__less__or__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] :
( ( word_sle @ A @ X @ Y )
=> ( ( word_sless @ A @ X @ Y )
| ( X = Y ) ) ) ) ).
% signed.le_imp_less_or_eq
thf(fact_6345_signed_Oorder_Ostrict__trans1,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A,B3: word @ A,C2: word @ A] :
( ( word_sle @ A @ A3 @ B3 )
=> ( ( word_sless @ A @ B3 @ C2 )
=> ( word_sless @ A @ A3 @ C2 ) ) ) ) ).
% signed.order.strict_trans1
thf(fact_6346_signed_Oorder_Ostrict__trans2,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A,B3: word @ A,C2: word @ A] :
( ( word_sless @ A @ A3 @ B3 )
=> ( ( word_sle @ A @ B3 @ C2 )
=> ( word_sless @ A @ A3 @ C2 ) ) ) ) ).
% signed.order.strict_trans2
thf(fact_6347_signed_Oorder_Ostrict__iff__not,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( word_sless @ A )
= ( ^ [A5: word @ A,B5: word @ A] :
( ( word_sle @ A @ A5 @ B5 )
& ~ ( word_sle @ A @ B5 @ A5 ) ) ) ) ) ).
% signed.order.strict_iff_not
thf(fact_6348_signed_Oorder_Oorder__iff__strict,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( word_sle @ A )
= ( ^ [A5: word @ A,B5: word @ A] :
( ( word_sless @ A @ A5 @ B5 )
| ( A5 = B5 ) ) ) ) ) ).
% signed.order.order_iff_strict
thf(fact_6349_signed_Oorder_Ostrict__iff__order,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( word_sless @ A )
= ( ^ [A5: word @ A,B5: word @ A] :
( ( word_sle @ A @ A5 @ B5 )
& ( A5 != B5 ) ) ) ) ) ).
% signed.order.strict_iff_order
thf(fact_6350_signed_Odual__order_Ostrict__trans1,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [B3: word @ A,A3: word @ A,C2: word @ A] :
( ( word_sle @ A @ B3 @ A3 )
=> ( ( word_sless @ A @ C2 @ B3 )
=> ( word_sless @ A @ C2 @ A3 ) ) ) ) ).
% signed.dual_order.strict_trans1
thf(fact_6351_signed_Odual__order_Ostrict__trans2,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [B3: word @ A,A3: word @ A,C2: word @ A] :
( ( word_sless @ A @ B3 @ A3 )
=> ( ( word_sle @ A @ C2 @ B3 )
=> ( word_sless @ A @ C2 @ A3 ) ) ) ) ).
% signed.dual_order.strict_trans2
thf(fact_6352_signed_Odual__order_Ostrict__iff__not,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( word_sless @ A )
= ( ^ [B5: word @ A,A5: word @ A] :
( ( word_sle @ A @ B5 @ A5 )
& ~ ( word_sle @ A @ A5 @ B5 ) ) ) ) ) ).
% signed.dual_order.strict_iff_not
thf(fact_6353_signed_Oorder_Ostrict__implies__order,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A,B3: word @ A] :
( ( word_sless @ A @ A3 @ B3 )
=> ( word_sle @ A @ A3 @ B3 ) ) ) ).
% signed.order.strict_implies_order
thf(fact_6354_signed_Odual__order_Oorder__iff__strict,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( word_sle @ A )
= ( ^ [B5: word @ A,A5: word @ A] :
( ( word_sless @ A @ B5 @ A5 )
| ( A5 = B5 ) ) ) ) ) ).
% signed.dual_order.order_iff_strict
thf(fact_6355_signed_Odual__order_Ostrict__iff__order,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( word_sless @ A )
= ( ^ [B5: word @ A,A5: word @ A] :
( ( word_sle @ A @ B5 @ A5 )
& ( A5 != B5 ) ) ) ) ) ).
% signed.dual_order.strict_iff_order
thf(fact_6356_signed_Odual__order_Ostrict__implies__order,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [B3: word @ A,A3: word @ A] :
( ( word_sless @ A @ B3 @ A3 )
=> ( word_sle @ A @ B3 @ A3 ) ) ) ).
% signed.dual_order.strict_implies_order
thf(fact_6357_signed_Oorder_Onot__eq__order__implies__strict,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A,B3: word @ A] :
( ( A3 != B3 )
=> ( ( word_sle @ A @ A3 @ B3 )
=> ( word_sless @ A @ A3 @ B3 ) ) ) ) ).
% signed.order.not_eq_order_implies_strict
thf(fact_6358_signed_Odual__order_Onot__eq__order__implies__strict,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A,B3: word @ A] :
( ( A3 != B3 )
=> ( ( word_sle @ A @ B3 @ A3 )
=> ( word_sless @ A @ B3 @ A3 ) ) ) ) ).
% signed.dual_order.not_eq_order_implies_strict
thf(fact_6359_word__sless__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( word_sless @ A )
= ( ^ [X2: word @ A,Y2: word @ A] :
( ( word_sle @ A @ X2 @ Y2 )
& ( X2 != Y2 ) ) ) ) ) ).
% word_sless_eq
thf(fact_6360_finite__word,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: set @ ( word @ A )] : ( finite_finite2 @ ( word @ A ) @ A2 ) ) ).
% finite_word
thf(fact_6361_signed_Oinfinite__growing,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X5: set @ ( word @ A )] :
( ( X5
!= ( bot_bot @ ( set @ ( word @ A ) ) ) )
=> ( ! [X3: word @ A] :
( ( member @ ( word @ A ) @ X3 @ X5 )
=> ? [Xa2: word @ A] :
( ( member @ ( word @ A ) @ Xa2 @ X5 )
& ( word_sless @ A @ X3 @ Xa2 ) ) )
=> ~ ( finite_finite2 @ ( word @ A ) @ X5 ) ) ) ) ).
% signed.infinite_growing
thf(fact_6362_signed_Olift__Suc__mono__less,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [F3: nat > ( word @ A ),N: nat,N7: nat] :
( ! [N2: nat] : ( word_sless @ A @ ( F3 @ N2 ) @ ( F3 @ ( suc @ N2 ) ) )
=> ( ( ord_less @ nat @ N @ N7 )
=> ( word_sless @ A @ ( F3 @ N ) @ ( F3 @ N7 ) ) ) ) ) ).
% signed.lift_Suc_mono_less
thf(fact_6363_signed_Olift__Suc__mono__less__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [F3: nat > ( word @ A ),N: nat,M: nat] :
( ! [N2: nat] : ( word_sless @ A @ ( F3 @ N2 ) @ ( F3 @ ( suc @ N2 ) ) )
=> ( ( word_sless @ A @ ( F3 @ N ) @ ( F3 @ M ) )
= ( ord_less @ nat @ N @ M ) ) ) ) ).
% signed.lift_Suc_mono_less_iff
thf(fact_6364_signed_Odual__order_Ostrict__implies__not__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [B3: word @ A,A3: word @ A] :
( ( word_sless @ A @ B3 @ A3 )
=> ( A3 != B3 ) ) ) ).
% signed.dual_order.strict_implies_not_eq
thf(fact_6365_signed_Oorder_Ostrict__implies__not__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A,B3: word @ A] :
( ( word_sless @ A @ A3 @ B3 )
=> ( A3 != B3 ) ) ) ).
% signed.order.strict_implies_not_eq
thf(fact_6366_signed_Odual__order_Ostrict__trans,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [B3: word @ A,A3: word @ A,C2: word @ A] :
( ( word_sless @ A @ B3 @ A3 )
=> ( ( word_sless @ A @ C2 @ B3 )
=> ( word_sless @ A @ C2 @ A3 ) ) ) ) ).
% signed.dual_order.strict_trans
thf(fact_6367_signed_Onot__less__iff__gr__or__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] :
( ( ~ ( word_sless @ A @ X @ Y ) )
= ( ( word_sless @ A @ Y @ X )
| ( X = Y ) ) ) ) ).
% signed.not_less_iff_gr_or_eq
thf(fact_6368_signed_Oorder_Ostrict__trans,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A,B3: word @ A,C2: word @ A] :
( ( word_sless @ A @ A3 @ B3 )
=> ( ( word_sless @ A @ B3 @ C2 )
=> ( word_sless @ A @ A3 @ C2 ) ) ) ) ).
% signed.order.strict_trans
thf(fact_6369_signed_Olinorder__less__wlog,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [P: ( word @ A ) > ( word @ A ) > $o,A3: word @ A,B3: word @ A] :
( ! [A6: word @ A,B7: word @ A] :
( ( word_sless @ A @ A6 @ B7 )
=> ( P @ A6 @ B7 ) )
=> ( ! [A6: word @ A] : ( P @ A6 @ A6 )
=> ( ! [A6: word @ A,B7: word @ A] :
( ( P @ B7 @ A6 )
=> ( P @ A6 @ B7 ) )
=> ( P @ A3 @ B3 ) ) ) ) ) ).
% signed.linorder_less_wlog
thf(fact_6370_signed_Oord__less__eq__trans,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A,B3: word @ A,C2: word @ A] :
( ( word_sless @ A @ A3 @ B3 )
=> ( ( B3 = C2 )
=> ( word_sless @ A @ A3 @ C2 ) ) ) ) ).
% signed.ord_less_eq_trans
thf(fact_6371_signed_Oord__eq__less__trans,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A,B3: word @ A,C2: word @ A] :
( ( A3 = B3 )
=> ( ( word_sless @ A @ B3 @ C2 )
=> ( word_sless @ A @ A3 @ C2 ) ) ) ) ).
% signed.ord_eq_less_trans
thf(fact_6372_signed_Oless__imp__not__less,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] :
( ( word_sless @ A @ X @ Y )
=> ~ ( word_sless @ A @ Y @ X ) ) ) ).
% signed.less_imp_not_less
thf(fact_6373_signed_Odual__order_Oirrefl,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A] :
~ ( word_sless @ A @ A3 @ A3 ) ) ).
% signed.dual_order.irrefl
thf(fact_6374_signed_Oless__imp__not__eq2,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] :
( ( word_sless @ A @ X @ Y )
=> ( Y != X ) ) ) ).
% signed.less_imp_not_eq2
thf(fact_6375_signed_Oless__imp__not__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] :
( ( word_sless @ A @ X @ Y )
=> ( X != Y ) ) ) ).
% signed.less_imp_not_eq
thf(fact_6376_signed_Odual__order_Oasym,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [B3: word @ A,A3: word @ A] :
( ( word_sless @ A @ B3 @ A3 )
=> ~ ( word_sless @ A @ A3 @ B3 ) ) ) ).
% signed.dual_order.asym
thf(fact_6377_signed_Olinorder__cases,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] :
( ~ ( word_sless @ A @ X @ Y )
=> ( ( X != Y )
=> ( word_sless @ A @ Y @ X ) ) ) ) ).
% signed.linorder_cases
thf(fact_6378_signed_Oless__imp__triv,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A,P: $o] :
( ( word_sless @ A @ X @ Y )
=> ( ( word_sless @ A @ Y @ X )
=> P ) ) ) ).
% signed.less_imp_triv
thf(fact_6379_signed_Oantisym__conv3,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y: word @ A,X: word @ A] :
( ~ ( word_sless @ A @ Y @ X )
=> ( ( ~ ( word_sless @ A @ X @ Y ) )
= ( X = Y ) ) ) ) ).
% signed.antisym_conv3
thf(fact_6380_signed_Oless__not__sym,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] :
( ( word_sless @ A @ X @ Y )
=> ~ ( word_sless @ A @ Y @ X ) ) ) ).
% signed.less_not_sym
thf(fact_6381_signed_Oless__imp__neq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] :
( ( word_sless @ A @ X @ Y )
=> ( X != Y ) ) ) ).
% signed.less_imp_neq
thf(fact_6382_signed_Oless__linear,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] :
( ( word_sless @ A @ X @ Y )
| ( X = Y )
| ( word_sless @ A @ Y @ X ) ) ) ).
% signed.less_linear
thf(fact_6383_signed_Oless__irrefl,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] :
~ ( word_sless @ A @ X @ X ) ) ).
% signed.less_irrefl
thf(fact_6384_signed_Oorder_Oasym,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A,B3: word @ A] :
( ( word_sless @ A @ A3 @ B3 )
=> ~ ( word_sless @ A @ B3 @ A3 ) ) ) ).
% signed.order.asym
thf(fact_6385_signed_Oless__trans,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A,Z: word @ A] :
( ( word_sless @ A @ X @ Y )
=> ( ( word_sless @ A @ Y @ Z )
=> ( word_sless @ A @ X @ Z ) ) ) ) ).
% signed.less_trans
thf(fact_6386_signed_Oless__asym_H,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A,B3: word @ A] :
( ( word_sless @ A @ A3 @ B3 )
=> ~ ( word_sless @ A @ B3 @ A3 ) ) ) ).
% signed.less_asym'
thf(fact_6387_signed_Oless__asym,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] :
( ( word_sless @ A @ X @ Y )
=> ~ ( word_sless @ A @ Y @ X ) ) ) ).
% signed.less_asym
thf(fact_6388_signed_Oneq__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] :
( ( X != Y )
= ( ( word_sless @ A @ X @ Y )
| ( word_sless @ A @ Y @ X ) ) ) ) ).
% signed.neq_iff
thf(fact_6389_signed_OneqE,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] :
( ( X != Y )
=> ( ~ ( word_sless @ A @ X @ Y )
=> ( word_sless @ A @ Y @ X ) ) ) ) ).
% signed.neqE
thf(fact_6390_word__sless__alt,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( word_sless @ A )
= ( ^ [A5: word @ A,B5: word @ A] : ( ord_less @ int @ ( ring_1_signed @ A @ int @ A5 ) @ ( ring_1_signed @ A @ int @ B5 ) ) ) ) ) ).
% word_sless_alt
thf(fact_6391_signed_Ofinite__linorder__min__induct,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: set @ ( word @ A ),P: ( set @ ( word @ A ) ) > $o] :
( ( finite_finite2 @ ( word @ A ) @ A2 )
=> ( ( P @ ( bot_bot @ ( set @ ( word @ A ) ) ) )
=> ( ! [B7: word @ A,A10: set @ ( word @ A )] :
( ( finite_finite2 @ ( word @ A ) @ A10 )
=> ( ! [X6: word @ A] :
( ( member @ ( word @ A ) @ X6 @ A10 )
=> ( word_sless @ A @ B7 @ X6 ) )
=> ( ( P @ A10 )
=> ( P @ ( insert @ ( word @ A ) @ B7 @ A10 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ) ).
% signed.finite_linorder_min_induct
thf(fact_6392_signed_Ofinite__linorder__max__induct,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A2: set @ ( word @ A ),P: ( set @ ( word @ A ) ) > $o] :
( ( finite_finite2 @ ( word @ A ) @ A2 )
=> ( ( P @ ( bot_bot @ ( set @ ( word @ A ) ) ) )
=> ( ! [B7: word @ A,A10: set @ ( word @ A )] :
( ( finite_finite2 @ ( word @ A ) @ A10 )
=> ( ! [X6: word @ A] :
( ( member @ ( word @ A ) @ X6 @ A10 )
=> ( word_sless @ A @ X6 @ B7 ) )
=> ( ( P @ A10 )
=> ( P @ ( insert @ ( word @ A ) @ B7 @ A10 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ) ).
% signed.finite_linorder_max_induct
thf(fact_6393_word__sless__sint__le,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,Y: word @ A] :
( ( word_sless @ A @ X @ Y )
=> ( ord_less_eq @ int @ ( ring_1_signed @ A @ int @ X ) @ ( minus_minus @ int @ ( ring_1_signed @ A @ int @ Y ) @ ( one_one @ int ) ) ) ) ) ).
% word_sless_sint_le
thf(fact_6394_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_6395_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_6396_min_Oidem,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A] :
( ( ord_min @ A @ A3 @ A3 )
= A3 ) ) ).
% min.idem
thf(fact_6397_min_Oleft__idem,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B3: A] :
( ( ord_min @ A @ A3 @ ( ord_min @ A @ A3 @ B3 ) )
= ( ord_min @ A @ A3 @ B3 ) ) ) ).
% min.left_idem
thf(fact_6398_min_Oright__idem,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B3: A] :
( ( ord_min @ A @ ( ord_min @ A @ A3 @ B3 ) @ B3 )
= ( ord_min @ A @ A3 @ B3 ) ) ) ).
% min.right_idem
thf(fact_6399_min_Obounded__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B3: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ ( ord_min @ A @ B3 @ C2 ) )
= ( ( ord_less_eq @ A @ A3 @ B3 )
& ( ord_less_eq @ A @ A3 @ C2 ) ) ) ) ).
% min.bounded_iff
thf(fact_6400_min_Oabsorb2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B3: A,A3: A] :
( ( ord_less_eq @ A @ B3 @ A3 )
=> ( ( ord_min @ A @ A3 @ B3 )
= B3 ) ) ) ).
% min.absorb2
thf(fact_6401_min_Oabsorb1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ord_min @ A @ A3 @ B3 )
= A3 ) ) ) ).
% min.absorb1
thf(fact_6402_min__simps_I2_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B3: A,A3: A] :
( ( ord_less @ A @ B3 @ A3 )
=> ( ( ord_min @ A @ A3 @ B3 )
= B3 ) ) ) ).
% min_simps(2)
thf(fact_6403_min__simps_I1_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ A3 @ B3 )
=> ( ( ord_min @ A @ A3 @ B3 )
= A3 ) ) ) ).
% min_simps(1)
thf(fact_6404_min__less__self__conv_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ ( ord_min @ A @ A3 @ B3 ) @ B3 )
= ( ord_less @ A @ A3 @ B3 ) ) ) ).
% min_less_self_conv(2)
thf(fact_6405_min__less__self__conv_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ ( ord_min @ A @ A3 @ B3 ) @ A3 )
= ( ord_less @ A @ B3 @ A3 ) ) ) ).
% min_less_self_conv(1)
thf(fact_6406_min__arg__not__ge_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [M: A,N: A] :
( ( ~ ( ord_less @ A @ ( ord_min @ A @ M @ N ) @ N ) )
= ( ( ord_min @ A @ M @ N )
= N ) ) ) ).
% min_arg_not_ge(2)
thf(fact_6407_min__arg__not__ge_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [M: A,N: A] :
( ( ~ ( ord_less @ A @ ( ord_min @ A @ M @ N ) @ M ) )
= ( ( ord_min @ A @ M @ N )
= M ) ) ) ).
% min_arg_not_ge(1)
thf(fact_6408_min_Oabsorb3,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ A3 @ B3 )
=> ( ( ord_min @ A @ A3 @ B3 )
= A3 ) ) ) ).
% min.absorb3
thf(fact_6409_min_Oabsorb4,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B3: A,A3: A] :
( ( ord_less @ A @ B3 @ A3 )
=> ( ( ord_min @ A @ A3 @ B3 )
= B3 ) ) ) ).
% min.absorb4
thf(fact_6410_min__less__iff__conj,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Z: A,X: A,Y: A] :
( ( ord_less @ A @ Z @ ( ord_min @ A @ X @ Y ) )
= ( ( ord_less @ A @ Z @ X )
& ( ord_less @ A @ Z @ Y ) ) ) ) ).
% min_less_iff_conj
thf(fact_6411_min__bot2,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [X: A] :
( ( ord_min @ A @ X @ ( bot_bot @ A ) )
= ( bot_bot @ A ) ) ) ).
% min_bot2
thf(fact_6412_min__bot,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [X: A] :
( ( ord_min @ A @ ( bot_bot @ A ) @ X )
= ( bot_bot @ A ) ) ) ).
% min_bot
thf(fact_6413_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_6414_min__0L,axiom,
! [N: nat] :
( ( ord_min @ nat @ ( zero_zero @ nat ) @ N )
= ( zero_zero @ nat ) ) ).
% min_0L
thf(fact_6415_min__0R,axiom,
! [N: nat] :
( ( ord_min @ nat @ N @ ( zero_zero @ nat ) )
= ( zero_zero @ nat ) ) ).
% min_0R
thf(fact_6416_max__min__same_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( ord_max @ A @ X @ ( ord_min @ A @ X @ Y ) )
= X ) ) ).
% max_min_same(1)
thf(fact_6417_max__min__same_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( ord_max @ A @ ( ord_min @ A @ X @ Y ) @ X )
= X ) ) ).
% max_min_same(2)
thf(fact_6418_max__min__same_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( ord_max @ A @ ( ord_min @ A @ X @ Y ) @ Y )
= Y ) ) ).
% max_min_same(3)
thf(fact_6419_max__min__same_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Y: A,X: A] :
( ( ord_max @ A @ Y @ ( ord_min @ A @ X @ Y ) )
= Y ) ) ).
% max_min_same(4)
thf(fact_6420_min__minus_H,axiom,
! [M: nat,K: nat] :
( ( ord_min @ nat @ ( minus_minus @ nat @ M @ K ) @ M )
= ( minus_minus @ nat @ M @ K ) ) ).
% min_minus'
thf(fact_6421_min__minus,axiom,
! [M: nat,K: nat] :
( ( ord_min @ nat @ M @ ( minus_minus @ nat @ M @ K ) )
= ( minus_minus @ nat @ M @ K ) ) ).
% min_minus
thf(fact_6422_min__0__1_I4_J,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [X: num] :
( ( ord_min @ A @ ( numeral_numeral @ A @ X ) @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% min_0_1(4)
thf(fact_6423_min__0__1_I3_J,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [X: num] :
( ( ord_min @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ X ) )
= ( zero_zero @ A ) ) ) ).
% min_0_1(3)
thf(fact_6424_min__number__of_I1_J,axiom,
! [A: $tType] :
( ( ( numeral @ A )
& ( ord @ A ) )
=> ! [U2: num,V2: num] :
( ( ( ord_less_eq @ A @ ( numeral_numeral @ A @ U2 ) @ ( numeral_numeral @ A @ V2 ) )
=> ( ( ord_min @ A @ ( numeral_numeral @ A @ U2 ) @ ( numeral_numeral @ A @ V2 ) )
= ( numeral_numeral @ A @ U2 ) ) )
& ( ~ ( ord_less_eq @ A @ ( numeral_numeral @ A @ U2 ) @ ( numeral_numeral @ A @ V2 ) )
=> ( ( ord_min @ A @ ( numeral_numeral @ A @ U2 ) @ ( numeral_numeral @ A @ V2 ) )
= ( numeral_numeral @ A @ V2 ) ) ) ) ) ).
% min_number_of(1)
thf(fact_6425_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_6426_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_6427_min__0__1_I6_J,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [X: num] :
( ( ord_min @ A @ ( numeral_numeral @ A @ X ) @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% min_0_1(6)
thf(fact_6428_min__0__1_I5_J,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [X: num] :
( ( ord_min @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ X ) )
= ( one_one @ A ) ) ) ).
% min_0_1(5)
thf(fact_6429_min__Suc__gt_I1_J,axiom,
! [A3: nat,B3: nat] :
( ( ord_less @ nat @ A3 @ B3 )
=> ( ( ord_min @ nat @ ( suc @ A3 ) @ B3 )
= ( suc @ A3 ) ) ) ).
% min_Suc_gt(1)
thf(fact_6430_min__Suc__gt_I2_J,axiom,
! [A3: nat,B3: nat] :
( ( ord_less @ nat @ A3 @ B3 )
=> ( ( ord_min @ nat @ B3 @ ( suc @ A3 ) )
= ( suc @ A3 ) ) ) ).
% min_Suc_gt(2)
thf(fact_6431_rev__min__pm1,axiom,
! [A3: nat,B3: nat] :
( ( plus_plus @ nat @ ( minus_minus @ nat @ A3 @ B3 ) @ ( ord_min @ nat @ B3 @ A3 ) )
= A3 ) ).
% rev_min_pm1
thf(fact_6432_rev__min__pm,axiom,
! [B3: nat,A3: nat] :
( ( plus_plus @ nat @ ( ord_min @ nat @ B3 @ A3 ) @ ( minus_minus @ nat @ A3 @ B3 ) )
= A3 ) ).
% rev_min_pm
thf(fact_6433_min__pm1,axiom,
! [A3: nat,B3: nat] :
( ( plus_plus @ nat @ ( minus_minus @ nat @ A3 @ B3 ) @ ( ord_min @ nat @ A3 @ B3 ) )
= A3 ) ).
% min_pm1
thf(fact_6434_min__pm,axiom,
! [A3: nat,B3: nat] :
( ( plus_plus @ nat @ ( ord_min @ nat @ A3 @ B3 ) @ ( minus_minus @ nat @ A3 @ B3 ) )
= A3 ) ).
% min_pm
thf(fact_6435_min__number__of_I2_J,axiom,
! [A: $tType] :
( ( ( uminus @ A )
& ( numeral @ A )
& ( ord @ A ) )
=> ! [U2: num,V2: num] :
( ( ( ord_less_eq @ A @ ( numeral_numeral @ A @ U2 ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) )
=> ( ( ord_min @ A @ ( numeral_numeral @ A @ U2 ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) )
= ( numeral_numeral @ A @ U2 ) ) )
& ( ~ ( ord_less_eq @ A @ ( numeral_numeral @ A @ U2 ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) )
=> ( ( ord_min @ A @ ( numeral_numeral @ A @ U2 ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) ) ) ) ) ).
% min_number_of(2)
thf(fact_6436_min__number__of_I3_J,axiom,
! [A: $tType] :
( ( ( uminus @ A )
& ( numeral @ A )
& ( ord @ A ) )
=> ! [U2: num,V2: num] :
( ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U2 ) ) @ ( numeral_numeral @ A @ V2 ) )
=> ( ( ord_min @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U2 ) ) @ ( numeral_numeral @ A @ V2 ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ U2 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U2 ) ) @ ( numeral_numeral @ A @ V2 ) )
=> ( ( ord_min @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U2 ) ) @ ( numeral_numeral @ A @ V2 ) )
= ( numeral_numeral @ A @ V2 ) ) ) ) ) ).
% min_number_of(3)
thf(fact_6437_min__number__of_I4_J,axiom,
! [A: $tType] :
( ( ( uminus @ A )
& ( numeral @ A )
& ( ord @ A ) )
=> ! [U2: num,V2: num] :
( ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U2 ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) )
=> ( ( ord_min @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U2 ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ U2 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U2 ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) )
=> ( ( ord_min @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U2 ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) ) ) ) ) ).
% min_number_of(4)
thf(fact_6438_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_6439_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_6440_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_6441_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_6442_min__def__raw,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( ord_min @ A )
= ( ^ [A5: A,B5: A] : ( if @ A @ ( ord_less_eq @ A @ A5 @ B5 ) @ A5 @ B5 ) ) ) ) ).
% min_def_raw
thf(fact_6443_min__le__iff__disj,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A,Z: A] :
( ( ord_less_eq @ A @ ( ord_min @ A @ X @ Y ) @ Z )
= ( ( ord_less_eq @ A @ X @ Z )
| ( ord_less_eq @ A @ Y @ Z ) ) ) ) ).
% min_le_iff_disj
thf(fact_6444_min_OcoboundedI2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B3: A,C2: A,A3: A] :
( ( ord_less_eq @ A @ B3 @ C2 )
=> ( ord_less_eq @ A @ ( ord_min @ A @ A3 @ B3 ) @ C2 ) ) ) ).
% min.coboundedI2
thf(fact_6445_min_OcoboundedI1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,C2: A,B3: A] :
( ( ord_less_eq @ A @ A3 @ C2 )
=> ( ord_less_eq @ A @ ( ord_min @ A @ A3 @ B3 ) @ C2 ) ) ) ).
% min.coboundedI1
thf(fact_6446_min_Oabsorb__iff2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B5: A,A5: A] :
( ( ord_min @ A @ A5 @ B5 )
= B5 ) ) ) ) ).
% min.absorb_iff2
thf(fact_6447_min_Oabsorb__iff1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A5: A,B5: A] :
( ( ord_min @ A @ A5 @ B5 )
= A5 ) ) ) ) ).
% min.absorb_iff1
thf(fact_6448_min_Ocobounded2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B3: A] : ( ord_less_eq @ A @ ( ord_min @ A @ A3 @ B3 ) @ B3 ) ) ).
% min.cobounded2
thf(fact_6449_min_Ocobounded1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B3: A] : ( ord_less_eq @ A @ ( ord_min @ A @ A3 @ B3 ) @ A3 ) ) ).
% min.cobounded1
thf(fact_6450_min_Oorder__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A5: A,B5: A] :
( A5
= ( ord_min @ A @ A5 @ B5 ) ) ) ) ) ).
% min.order_iff
thf(fact_6451_min_OboundedI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B3: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ A @ A3 @ C2 )
=> ( ord_less_eq @ A @ A3 @ ( ord_min @ A @ B3 @ C2 ) ) ) ) ) ).
% min.boundedI
thf(fact_6452_min_OboundedE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B3: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ ( ord_min @ A @ B3 @ C2 ) )
=> ~ ( ( ord_less_eq @ A @ A3 @ B3 )
=> ~ ( ord_less_eq @ A @ A3 @ C2 ) ) ) ) ).
% min.boundedE
thf(fact_6453_min_OorderI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B3: A] :
( ( A3
= ( ord_min @ A @ A3 @ B3 ) )
=> ( ord_less_eq @ A @ A3 @ B3 ) ) ) ).
% min.orderI
thf(fact_6454_min_OorderE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( A3
= ( ord_min @ A @ A3 @ B3 ) ) ) ) ).
% min.orderE
thf(fact_6455_min_Omono,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,C2: A,B3: A,D2: A] :
( ( ord_less_eq @ A @ A3 @ C2 )
=> ( ( ord_less_eq @ A @ B3 @ D2 )
=> ( ord_less_eq @ A @ ( ord_min @ A @ A3 @ B3 ) @ ( ord_min @ A @ C2 @ D2 ) ) ) ) ) ).
% min.mono
thf(fact_6456_min__absorb2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ Y @ X )
=> ( ( ord_min @ A @ X @ Y )
= Y ) ) ) ).
% min_absorb2
thf(fact_6457_min__absorb1,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_min @ A @ X @ Y )
= X ) ) ) ).
% min_absorb1
thf(fact_6458_min__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( ord_min @ A )
= ( ^ [A5: A,B5: A] : ( if @ A @ ( ord_less_eq @ A @ A5 @ B5 ) @ A5 @ B5 ) ) ) ) ).
% min_def
thf(fact_6459_min__add__distrib__left,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [X: A,Y: A,Z: A] :
( ( plus_plus @ A @ ( ord_min @ A @ X @ Y ) @ Z )
= ( ord_min @ A @ ( plus_plus @ A @ X @ Z ) @ ( plus_plus @ A @ Y @ Z ) ) ) ) ).
% min_add_distrib_left
thf(fact_6460_min__add__distrib__right,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [X: A,Y: A,Z: A] :
( ( plus_plus @ A @ X @ ( ord_min @ A @ Y @ Z ) )
= ( ord_min @ A @ ( plus_plus @ A @ X @ Y ) @ ( plus_plus @ A @ X @ Z ) ) ) ) ).
% min_add_distrib_right
thf(fact_6461_min__diff__distrib__left,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [X: A,Y: A,Z: A] :
( ( minus_minus @ A @ ( ord_min @ A @ X @ Y ) @ Z )
= ( ord_min @ A @ ( minus_minus @ A @ X @ Z ) @ ( minus_minus @ A @ Y @ Z ) ) ) ) ).
% min_diff_distrib_left
thf(fact_6462_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_6463_nat__mult__min__left,axiom,
! [M: nat,N: nat,Q3: nat] :
( ( times_times @ nat @ ( ord_min @ nat @ M @ N ) @ Q3 )
= ( ord_min @ nat @ ( times_times @ nat @ M @ Q3 ) @ ( times_times @ nat @ N @ Q3 ) ) ) ).
% nat_mult_min_left
thf(fact_6464_nat__mult__min__right,axiom,
! [M: nat,N: nat,Q3: nat] :
( ( times_times @ nat @ M @ ( ord_min @ nat @ N @ Q3 ) )
= ( ord_min @ nat @ ( times_times @ nat @ M @ N ) @ ( times_times @ nat @ M @ Q3 ) ) ) ).
% nat_mult_min_right
thf(fact_6465_min__max__distrib2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B3: A,C2: A] :
( ( ord_min @ A @ A3 @ ( ord_max @ A @ B3 @ C2 ) )
= ( ord_max @ A @ ( ord_min @ A @ A3 @ B3 ) @ ( ord_min @ A @ A3 @ C2 ) ) ) ) ).
% min_max_distrib2
thf(fact_6466_min__max__distrib1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B3: A,C2: A,A3: A] :
( ( ord_min @ A @ ( ord_max @ A @ B3 @ C2 ) @ A3 )
= ( ord_max @ A @ ( ord_min @ A @ B3 @ A3 ) @ ( ord_min @ A @ C2 @ A3 ) ) ) ) ).
% min_max_distrib1
thf(fact_6467_max__min__distrib2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B3: A,C2: A] :
( ( ord_max @ A @ A3 @ ( ord_min @ A @ B3 @ C2 ) )
= ( ord_min @ A @ ( ord_max @ A @ A3 @ B3 ) @ ( ord_max @ A @ A3 @ C2 ) ) ) ) ).
% max_min_distrib2
thf(fact_6468_max__min__distrib1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B3: A,C2: A,A3: A] :
( ( ord_max @ A @ ( ord_min @ A @ B3 @ C2 ) @ A3 )
= ( ord_min @ A @ ( ord_max @ A @ B3 @ A3 ) @ ( ord_max @ A @ C2 @ A3 ) ) ) ) ).
% max_min_distrib1
thf(fact_6469_min_Oassoc,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B3: A,C2: A] :
( ( ord_min @ A @ ( ord_min @ A @ A3 @ B3 ) @ C2 )
= ( ord_min @ A @ A3 @ ( ord_min @ A @ B3 @ C2 ) ) ) ) ).
% min.assoc
thf(fact_6470_min_Ocommute,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_min @ A )
= ( ^ [A5: A,B5: A] : ( ord_min @ A @ B5 @ A5 ) ) ) ) ).
% min.commute
thf(fact_6471_min_Oleft__commute,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B3: A,A3: A,C2: A] :
( ( ord_min @ A @ B3 @ ( ord_min @ A @ A3 @ C2 ) )
= ( ord_min @ A @ A3 @ ( ord_min @ A @ B3 @ C2 ) ) ) ) ).
% min.left_commute
thf(fact_6472_of__int__min,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: int,Y: int] :
( ( ring_1_of_int @ A @ ( ord_min @ int @ X @ Y ) )
= ( ord_min @ A @ ( ring_1_of_int @ A @ X ) @ ( ring_1_of_int @ A @ Y ) ) ) ) ).
% of_int_min
thf(fact_6473_of__nat__min,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [X: nat,Y: nat] :
( ( semiring_1_of_nat @ A @ ( ord_min @ nat @ X @ Y ) )
= ( ord_min @ A @ ( semiring_1_of_nat @ A @ X ) @ ( semiring_1_of_nat @ A @ Y ) ) ) ) ).
% of_nat_min
thf(fact_6474_minus__min__eq__max,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [X: A,Y: A] :
( ( uminus_uminus @ A @ ( ord_min @ A @ X @ Y ) )
= ( ord_max @ A @ ( uminus_uminus @ A @ X ) @ ( uminus_uminus @ A @ Y ) ) ) ) ).
% minus_min_eq_max
thf(fact_6475_minus__max__eq__min,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [X: A,Y: A] :
( ( uminus_uminus @ A @ ( ord_max @ A @ X @ Y ) )
= ( ord_min @ A @ ( uminus_uminus @ A @ X ) @ ( uminus_uminus @ A @ Y ) ) ) ) ).
% minus_max_eq_min
thf(fact_6476_min__less__iff__disj,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A,Z: A] :
( ( ord_less @ A @ ( ord_min @ A @ X @ Y ) @ Z )
= ( ( ord_less @ A @ X @ Z )
| ( ord_less @ A @ Y @ Z ) ) ) ) ).
% min_less_iff_disj
thf(fact_6477_min_Ostrict__boundedE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B3: A,C2: A] :
( ( ord_less @ A @ A3 @ ( ord_min @ A @ B3 @ C2 ) )
=> ~ ( ( ord_less @ A @ A3 @ B3 )
=> ~ ( ord_less @ A @ A3 @ C2 ) ) ) ) ).
% min.strict_boundedE
thf(fact_6478_min_Ostrict__order__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less @ A )
= ( ^ [A5: A,B5: A] :
( ( A5
= ( ord_min @ A @ A5 @ B5 ) )
& ( A5 != B5 ) ) ) ) ) ).
% min.strict_order_iff
thf(fact_6479_min_Ostrict__coboundedI1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,C2: A,B3: A] :
( ( ord_less @ A @ A3 @ C2 )
=> ( ord_less @ A @ ( ord_min @ A @ A3 @ B3 ) @ C2 ) ) ) ).
% min.strict_coboundedI1
thf(fact_6480_min_Ostrict__coboundedI2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B3: A,C2: A,A3: A] :
( ( ord_less @ A @ B3 @ C2 )
=> ( ord_less @ A @ ( ord_min @ A @ A3 @ B3 ) @ C2 ) ) ) ).
% min.strict_coboundedI2
thf(fact_6481_mask__and__mask,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: nat,B3: nat] :
( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ A3 ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ B3 ) )
= ( bit_se2239418461657761734s_mask @ ( word @ A ) @ ( ord_min @ nat @ A3 @ B3 ) ) ) ) ).
% mask_and_mask
thf(fact_6482_mask__twice,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A,N: nat,M: nat] :
( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ M ) )
= ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ ( ord_min @ nat @ M @ N ) ) ) ) ) ).
% mask_twice
thf(fact_6483_max__mult__distrib__left,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [P4: A,X: A,Y: A] :
( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ P4 )
=> ( ( times_times @ A @ P4 @ ( ord_max @ A @ X @ Y ) )
= ( ord_max @ A @ ( times_times @ A @ P4 @ X ) @ ( times_times @ A @ P4 @ Y ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ P4 )
=> ( ( times_times @ A @ P4 @ ( ord_max @ A @ X @ Y ) )
= ( ord_min @ A @ ( times_times @ A @ P4 @ X ) @ ( times_times @ A @ P4 @ Y ) ) ) ) ) ) ).
% max_mult_distrib_left
thf(fact_6484_min__mult__distrib__left,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [P4: A,X: A,Y: A] :
( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ P4 )
=> ( ( times_times @ A @ P4 @ ( ord_min @ A @ X @ Y ) )
= ( ord_min @ A @ ( times_times @ A @ P4 @ X ) @ ( times_times @ A @ P4 @ Y ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ P4 )
=> ( ( times_times @ A @ P4 @ ( ord_min @ A @ X @ Y ) )
= ( ord_max @ A @ ( times_times @ A @ P4 @ X ) @ ( times_times @ A @ P4 @ Y ) ) ) ) ) ) ).
% min_mult_distrib_left
thf(fact_6485_max__mult__distrib__right,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [P4: A,X: A,Y: A] :
( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ P4 )
=> ( ( times_times @ A @ ( ord_max @ A @ X @ Y ) @ P4 )
= ( ord_max @ A @ ( times_times @ A @ X @ P4 ) @ ( times_times @ A @ Y @ P4 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ P4 )
=> ( ( times_times @ A @ ( ord_max @ A @ X @ Y ) @ P4 )
= ( ord_min @ A @ ( times_times @ A @ X @ P4 ) @ ( times_times @ A @ Y @ P4 ) ) ) ) ) ) ).
% max_mult_distrib_right
thf(fact_6486_min__mult__distrib__right,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [P4: A,X: A,Y: A] :
( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ P4 )
=> ( ( times_times @ A @ ( ord_min @ A @ X @ Y ) @ P4 )
= ( ord_min @ A @ ( times_times @ A @ X @ P4 ) @ ( times_times @ A @ Y @ P4 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ P4 )
=> ( ( times_times @ A @ ( ord_min @ A @ X @ Y ) @ P4 )
= ( ord_max @ A @ ( times_times @ A @ X @ P4 ) @ ( times_times @ A @ Y @ P4 ) ) ) ) ) ) ).
% min_mult_distrib_right
thf(fact_6487_min__divide__distrib__right,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [P4: A,X: A,Y: A] :
( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ P4 )
=> ( ( divide_divide @ A @ ( ord_min @ A @ X @ Y ) @ P4 )
= ( ord_min @ A @ ( divide_divide @ A @ X @ P4 ) @ ( divide_divide @ A @ Y @ P4 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ P4 )
=> ( ( divide_divide @ A @ ( ord_min @ A @ X @ Y ) @ P4 )
= ( ord_max @ A @ ( divide_divide @ A @ X @ P4 ) @ ( divide_divide @ A @ Y @ P4 ) ) ) ) ) ) ).
% min_divide_distrib_right
thf(fact_6488_max__divide__distrib__right,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [P4: A,X: A,Y: A] :
( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ P4 )
=> ( ( divide_divide @ A @ ( ord_max @ A @ X @ Y ) @ P4 )
= ( ord_max @ A @ ( divide_divide @ A @ X @ P4 ) @ ( divide_divide @ A @ Y @ P4 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ P4 )
=> ( ( divide_divide @ A @ ( ord_max @ A @ X @ Y ) @ P4 )
= ( ord_min @ A @ ( divide_divide @ A @ X @ P4 ) @ ( divide_divide @ A @ Y @ P4 ) ) ) ) ) ) ).
% max_divide_distrib_right
thf(fact_6489_ucast__mask__eq,axiom,
! [A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( type_len @ A ) )
=> ! [N: nat] :
( ( semiring_1_unsigned @ B @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ B ) @ N ) )
= ( bit_se2239418461657761734s_mask @ ( word @ A ) @ ( ord_min @ nat @ ( type_len0_len_of @ B @ ( type2 @ B ) ) @ N ) ) ) ) ).
% ucast_mask_eq
thf(fact_6490_mod__exp__eq,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A3: A,M: nat,N: nat] :
( ( modulo_modulo @ A @ ( modulo_modulo @ A @ A3 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) )
= ( modulo_modulo @ A @ A3 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( ord_min @ nat @ M @ N ) ) ) ) ) ).
% mod_exp_eq
thf(fact_6491_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_6492_Word_Obit__mask__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [M: nat,N: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ M ) @ N )
= ( ord_less @ nat @ N @ ( ord_min @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ M ) ) ) ) ).
% Word.bit_mask_iff
thf(fact_6493_uint__mask__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat] :
( ( semiring_1_unsigned @ A @ int @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) )
= ( bit_se2239418461657761734s_mask @ int @ ( ord_min @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ N ) ) ) ) ).
% uint_mask_eq
thf(fact_6494_unat__mask__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat] :
( ( semiring_1_unsigned @ A @ nat @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) )
= ( bit_se2239418461657761734s_mask @ nat @ ( ord_min @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ N ) ) ) ) ).
% unat_mask_eq
thf(fact_6495_and__mask__wi_H,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [I: int,N: nat] :
( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ ( ring_1_of_int @ ( word @ A ) @ I ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N ) )
= ( ring_1_of_int @ ( word @ A ) @ ( bit_se2584673776208193580ke_bit @ int @ ( ord_min @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ N ) @ I ) ) ) ) ).
% and_mask_wi'
thf(fact_6496_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_6497_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_6498_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_6499_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_6500_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_6501_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_6502_top__apply,axiom,
! [C: $tType,D: $tType] :
( ( top @ C )
=> ( ( top_top @ ( D > C ) )
= ( ^ [X2: D] : ( top_top @ C ) ) ) ) ).
% top_apply
thf(fact_6503_UNIV__I,axiom,
! [A: $tType,X: A] : ( member @ A @ X @ ( top_top @ ( set @ A ) ) ) ).
% UNIV_I
thf(fact_6504_min__enat__simps_I3_J,axiom,
! [Q3: extended_enat] :
( ( ord_min @ extended_enat @ ( zero_zero @ extended_enat ) @ Q3 )
= ( zero_zero @ extended_enat ) ) ).
% min_enat_simps(3)
thf(fact_6505_min__enat__simps_I2_J,axiom,
! [Q3: extended_enat] :
( ( ord_min @ extended_enat @ Q3 @ ( zero_zero @ extended_enat ) )
= ( zero_zero @ extended_enat ) ) ).
% min_enat_simps(2)
thf(fact_6506_sup__top__right,axiom,
! [A: $tType] :
( ( bounded_lattice_top @ A )
=> ! [X: A] :
( ( sup_sup @ A @ X @ ( top_top @ A ) )
= ( top_top @ A ) ) ) ).
% sup_top_right
thf(fact_6507_sup__top__left,axiom,
! [A: $tType] :
( ( bounded_lattice_top @ A )
=> ! [X: A] :
( ( sup_sup @ A @ ( top_top @ A ) @ X )
= ( top_top @ A ) ) ) ).
% sup_top_left
thf(fact_6508_boolean__algebra_Odisj__one__right,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A] :
( ( sup_sup @ A @ X @ ( top_top @ A ) )
= ( top_top @ A ) ) ) ).
% boolean_algebra.disj_one_right
thf(fact_6509_boolean__algebra_Odisj__one__left,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A] :
( ( sup_sup @ A @ ( top_top @ A ) @ X )
= ( top_top @ A ) ) ) ).
% boolean_algebra.disj_one_left
thf(fact_6510_min__top,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [X: A] :
( ( ord_min @ A @ ( top_top @ A ) @ X )
= X ) ) ).
% min_top
thf(fact_6511_min__top2,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [X: A] :
( ( ord_min @ A @ X @ ( top_top @ A ) )
= X ) ) ).
% min_top2
thf(fact_6512_max__top,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [X: A] :
( ( ord_max @ A @ ( top_top @ A ) @ X )
= ( top_top @ A ) ) ) ).
% max_top
thf(fact_6513_max__top2,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [X: A] :
( ( ord_max @ A @ X @ ( top_top @ A ) )
= ( top_top @ A ) ) ) ).
% max_top2
thf(fact_6514_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_6515_finite__Collect__not,axiom,
! [A: $tType,P: A > $o] :
( ( finite_finite2 @ A @ ( collect @ A @ P ) )
=> ( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X2: A] :
~ ( P @ X2 ) ) )
= ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) ) ) ) ).
% finite_Collect_not
thf(fact_6516_boolean__algebra_Ocompl__zero,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ( ( uminus_uminus @ A @ ( bot_bot @ A ) )
= ( top_top @ A ) ) ) ).
% boolean_algebra.compl_zero
thf(fact_6517_boolean__algebra_Ocompl__one,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ( ( uminus_uminus @ A @ ( top_top @ A ) )
= ( bot_bot @ A ) ) ) ).
% boolean_algebra.compl_one
thf(fact_6518_possible__bit__word,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [M: nat] :
( ( bit_se6407376104438227557le_bit @ ( word @ A ) @ ( type2 @ ( word @ A ) ) @ M )
= ( ord_less @ nat @ M @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ).
% possible_bit_word
thf(fact_6519_boolean__algebra_Odisj__cancel__right,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A] :
( ( sup_sup @ A @ X @ ( uminus_uminus @ A @ X ) )
= ( top_top @ A ) ) ) ).
% boolean_algebra.disj_cancel_right
thf(fact_6520_boolean__algebra_Odisj__cancel__left,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A] :
( ( sup_sup @ A @ ( uminus_uminus @ A @ X ) @ X )
= ( top_top @ A ) ) ) ).
% boolean_algebra.disj_cancel_left
thf(fact_6521_sup__compl__top__left2,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,Y: A] :
( ( sup_sup @ A @ X @ ( sup_sup @ A @ ( uminus_uminus @ A @ X ) @ Y ) )
= ( top_top @ A ) ) ) ).
% sup_compl_top_left2
thf(fact_6522_sup__compl__top__left1,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,Y: A] :
( ( sup_sup @ A @ ( uminus_uminus @ A @ X ) @ ( sup_sup @ A @ X @ Y ) )
= ( top_top @ A ) ) ) ).
% sup_compl_top_left1
thf(fact_6523_Diff__UNIV,axiom,
! [A: $tType,A2: set @ A] :
( ( minus_minus @ ( set @ A ) @ A2 @ ( top_top @ ( set @ A ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Diff_UNIV
thf(fact_6524_surj__diff__right,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A3: A] :
( ( image @ A @ A
@ ^ [X2: A] : ( minus_minus @ A @ X2 @ A3 )
@ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ A ) ) ) ) ).
% surj_diff_right
thf(fact_6525_range__constant,axiom,
! [B: $tType,A: $tType,X: A] :
( ( image @ B @ A
@ ^ [Uu: B] : X
@ ( top_top @ ( set @ B ) ) )
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ).
% range_constant
thf(fact_6526_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_6527_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_6528_Compl__eq__Diff__UNIV,axiom,
! [A: $tType] :
( ( uminus_uminus @ ( set @ A ) )
= ( minus_minus @ ( set @ A ) @ ( top_top @ ( set @ A ) ) ) ) ).
% Compl_eq_Diff_UNIV
thf(fact_6529_Compl__partition2,axiom,
! [A: $tType,A2: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A2 ) @ A2 )
= ( top_top @ ( set @ A ) ) ) ).
% Compl_partition2
thf(fact_6530_Compl__partition,axiom,
! [A: $tType,A2: set @ A] :
( ( sup_sup @ ( set @ A ) @ A2 @ ( uminus_uminus @ ( set @ A ) @ A2 ) )
= ( top_top @ ( set @ A ) ) ) ).
% Compl_partition
thf(fact_6531_atLeastAtMost__eq__UNIV__iff,axiom,
! [A: $tType] :
( ( bounded_lattice @ A )
=> ! [X: A,Y: A] :
( ( ( set_or1337092689740270186AtMost @ A @ X @ Y )
= ( top_top @ ( set @ A ) ) )
= ( ( X
= ( bot_bot @ A ) )
& ( Y
= ( top_top @ A ) ) ) ) ) ).
% atLeastAtMost_eq_UNIV_iff
thf(fact_6532_empty__not__UNIV,axiom,
! [A: $tType] :
( ( bot_bot @ ( set @ A ) )
!= ( top_top @ ( set @ A ) ) ) ).
% empty_not_UNIV
thf(fact_6533_insert__UNIV,axiom,
! [A: $tType,X: A] :
( ( insert @ A @ X @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ A ) ) ) ).
% insert_UNIV
thf(fact_6534_top_Onot__eq__extremum,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [A3: A] :
( ( A3
!= ( top_top @ A ) )
= ( ord_less @ A @ A3 @ ( top_top @ A ) ) ) ) ).
% top.not_eq_extremum
thf(fact_6535_top_Oextremum__strict,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [A3: A] :
~ ( ord_less @ A @ ( top_top @ A ) @ A3 ) ) ).
% top.extremum_strict
thf(fact_6536_sup__cancel__left1,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,A3: A,B3: A] :
( ( sup_sup @ A @ ( sup_sup @ A @ X @ A3 ) @ ( sup_sup @ A @ ( uminus_uminus @ A @ X ) @ B3 ) )
= ( top_top @ A ) ) ) ).
% sup_cancel_left1
thf(fact_6537_sup__cancel__left2,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,A3: A,B3: A] :
( ( sup_sup @ A @ ( sup_sup @ A @ ( uminus_uminus @ A @ X ) @ A3 ) @ ( sup_sup @ A @ X @ B3 ) )
= ( top_top @ A ) ) ) ).
% sup_cancel_left2
thf(fact_6538_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_6539_UNIV__def,axiom,
! [A: $tType] :
( ( top_top @ ( set @ A ) )
= ( collect @ A
@ ^ [X2: A] : $true ) ) ).
% UNIV_def
thf(fact_6540_UNIV__witness,axiom,
! [A: $tType] :
? [X3: A] : ( member @ A @ X3 @ ( top_top @ ( set @ A ) ) ) ).
% UNIV_witness
thf(fact_6541_UNIV__eq__I,axiom,
! [A: $tType,A2: set @ A] :
( ! [X3: A] : ( member @ A @ X3 @ A2 )
=> ( ( top_top @ ( set @ A ) )
= A2 ) ) ).
% UNIV_eq_I
thf(fact_6542_Un__UNIV__right,axiom,
! [A: $tType,A2: set @ A] :
( ( sup_sup @ ( set @ A ) @ A2 @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ A ) ) ) ).
% Un_UNIV_right
thf(fact_6543_Un__UNIV__left,axiom,
! [A: $tType,B2: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( top_top @ ( set @ A ) ) @ B2 )
= ( top_top @ ( set @ A ) ) ) ).
% Un_UNIV_left
thf(fact_6544_top__greatest,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [A3: A] : ( ord_less_eq @ A @ A3 @ ( top_top @ A ) ) ) ).
% top_greatest
thf(fact_6545_top_Oextremum__unique,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ ( top_top @ A ) @ A3 )
= ( A3
= ( top_top @ A ) ) ) ) ).
% top.extremum_unique
thf(fact_6546_top_Oextremum__uniqueI,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ ( top_top @ A ) @ A3 )
=> ( A3
= ( top_top @ A ) ) ) ) ).
% top.extremum_uniqueI
thf(fact_6547_subset__UNIV,axiom,
! [A: $tType,A2: set @ A] : ( ord_less_eq @ ( set @ A ) @ A2 @ ( top_top @ ( set @ A ) ) ) ).
% subset_UNIV
thf(fact_6548_perfect__space__class_OUNIV__not__singleton,axiom,
! [A: $tType] :
( ( topolo8386298272705272623_space @ A )
=> ! [X: A] :
( ( top_top @ ( set @ A ) )
!= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% perfect_space_class.UNIV_not_singleton
thf(fact_6549_possible__bit,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [N: nat] : ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N ) ) ).
% possible_bit
thf(fact_6550_Compl__UNIV__eq,axiom,
! [A: $tType] :
( ( uminus_uminus @ ( set @ A ) @ ( top_top @ ( set @ A ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Compl_UNIV_eq
thf(fact_6551_Compl__empty__eq,axiom,
! [A: $tType] :
( ( uminus_uminus @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) )
= ( top_top @ ( set @ A ) ) ) ).
% Compl_empty_eq
thf(fact_6552_range__subsetD,axiom,
! [B: $tType,A: $tType,F3: B > A,B2: set @ A,I: B] :
( ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) ) @ B2 )
=> ( member @ A @ ( F3 @ I ) @ B2 ) ) ).
% range_subsetD
thf(fact_6553_rangeE,axiom,
! [A: $tType,B: $tType,B3: A,F3: B > A] :
( ( member @ A @ B3 @ ( image @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) ) )
=> ~ ! [X3: B] :
( B3
!= ( F3 @ X3 ) ) ) ).
% rangeE
thf(fact_6554_rangeI,axiom,
! [A: $tType,B: $tType,F3: B > A,X: B] : ( member @ A @ ( F3 @ X ) @ ( image @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) ) ) ).
% rangeI
thf(fact_6555_range__eqI,axiom,
! [A: $tType,B: $tType,B3: A,F3: B > A,X: B] :
( ( B3
= ( F3 @ X ) )
=> ( member @ A @ B3 @ ( image @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) ) ) ) ).
% range_eqI
thf(fact_6556_range__composition,axiom,
! [A: $tType,C: $tType,B: $tType,F3: C > A,G2: B > C] :
( ( image @ B @ A
@ ^ [X2: B] : ( F3 @ ( G2 @ X2 ) )
@ ( top_top @ ( set @ B ) ) )
= ( image @ C @ A @ F3 @ ( image @ B @ C @ G2 @ ( top_top @ ( set @ B ) ) ) ) ) ).
% range_composition
thf(fact_6557_finite__range__imageI,axiom,
! [C: $tType,A: $tType,B: $tType,G2: B > A,F3: A > C] :
( ( finite_finite2 @ A @ ( image @ B @ A @ G2 @ ( top_top @ ( set @ B ) ) ) )
=> ( finite_finite2 @ C
@ ( image @ B @ C
@ ^ [X2: B] : ( F3 @ ( G2 @ X2 ) )
@ ( top_top @ ( set @ B ) ) ) ) ) ).
% finite_range_imageI
thf(fact_6558_sup__shunt,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,Y: A] :
( ( ( sup_sup @ A @ X @ Y )
= ( top_top @ A ) )
= ( ord_less_eq @ A @ ( uminus_uminus @ A @ X ) @ Y ) ) ) ).
% sup_shunt
thf(fact_6559_range__eq__singletonD,axiom,
! [B: $tType,A: $tType,F3: B > A,A3: A,X: B] :
( ( ( image @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) )
= ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) )
=> ( ( F3 @ X )
= A3 ) ) ).
% range_eq_singletonD
thf(fact_6560_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_6561_bit__unsigned__iff,axiom,
! [A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( bit_semiring_bits @ A ) )
=> ! [W: word @ B,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( semiring_1_unsigned @ B @ A @ W ) @ N )
= ( ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N )
& ( bit_se5641148757651400278ts_bit @ ( word @ B ) @ W @ N ) ) ) ) ).
% bit_unsigned_iff
thf(fact_6562_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_6563_possible__bit__def,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ( ( bit_se6407376104438227557le_bit @ A )
= ( ^ [Tyrep: itself @ A,N4: nat] :
( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N4 )
!= ( zero_zero @ A ) ) ) ) ) ).
% possible_bit_def
thf(fact_6564_bit__minus__iff,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [A3: A,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( uminus_uminus @ A @ A3 ) @ N )
= ( ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N )
& ~ ( bit_se5641148757651400278ts_bit @ A @ ( minus_minus @ A @ A3 @ ( one_one @ A ) ) @ N ) ) ) ) ).
% bit_minus_iff
thf(fact_6565_bit__twiddle__min,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Y: word @ A,X: word @ A] :
( ( bit_se5824344971392196577ns_xor @ ( word @ A ) @ Y @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ ( bit_se5824344971392196577ns_xor @ ( word @ A ) @ X @ Y ) @ ( if @ ( word @ A ) @ ( ord_less @ ( word @ A ) @ X @ Y ) @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) @ ( zero_zero @ ( word @ A ) ) ) ) )
= ( ord_min @ ( word @ A ) @ X @ Y ) ) ) ).
% bit_twiddle_min
thf(fact_6566_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_6567_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_6568_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_6569_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_6570_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_6571_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_6572_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_6573_ucast__range__less,axiom,
! [A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( type_len @ A ) )
=> ( ( ord_less @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( type_len0_len_of @ B @ ( type2 @ B ) ) )
=> ( ( image @ ( word @ A ) @ ( word @ B ) @ ( semiring_1_unsigned @ A @ ( word @ B ) ) @ ( top_top @ ( set @ ( word @ A ) ) ) )
= ( collect @ ( word @ B )
@ ^ [X2: word @ B] : ( ord_less @ ( word @ B ) @ X2 @ ( power_power @ ( word @ B ) @ ( numeral_numeral @ ( word @ B ) @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ) ) ).
% ucast_range_less
thf(fact_6574_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_6575_bit__double__iff,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A3: A,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) @ N )
= ( ( bit_se5641148757651400278ts_bit @ A @ A3 @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) )
& ( N
!= ( zero_zero @ nat ) )
& ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N ) ) ) ) ).
% bit_double_iff
thf(fact_6576_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_6577_uint__word__rotr__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,W: word @ A] :
( ( semiring_1_unsigned @ A @ int @ ( word_rotr @ A @ N @ W ) )
= ( bit_concat_bit @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( modulo_modulo @ nat @ N @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) @ ( bit_se4197421643247451524op_bit @ int @ ( modulo_modulo @ nat @ N @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) @ ( semiring_1_unsigned @ A @ int @ W ) ) @ ( semiring_1_unsigned @ A @ int @ ( bit_se2584673776208193580ke_bit @ ( word @ A ) @ ( modulo_modulo @ nat @ N @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) @ W ) ) ) ) ) ).
% uint_word_rotr_eq
thf(fact_6578_word__roti_Oabs__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Xa: int,X: int] :
( ( word_roti @ A @ Xa @ ( word2 @ A @ X ) )
= ( word2 @ A @ ( bit_concat_bit @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( nat2 @ ( modulo_modulo @ int @ Xa @ ( semiring_1_of_nat @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) @ ( bit_se4197421643247451524op_bit @ int @ ( nat2 @ ( modulo_modulo @ int @ Xa @ ( semiring_1_of_nat @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ X ) ) @ ( bit_se2584673776208193580ke_bit @ int @ ( nat2 @ ( modulo_modulo @ int @ Xa @ ( semiring_1_of_nat @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) @ X ) ) ) ) ) ).
% word_roti.abs_eq
thf(fact_6579_assn__basic__inequalities_I1_J,axiom,
( ( top_top @ assn )
!= ( one_one @ assn ) ) ).
% assn_basic_inequalities(1)
thf(fact_6580_assn__basic__inequalities_I5_J,axiom,
( ( top_top @ assn )
!= ( bot_bot @ assn ) ) ).
% assn_basic_inequalities(5)
thf(fact_6581_range__mult,axiom,
! [A3: real] :
( ( ( A3
= ( zero_zero @ real ) )
=> ( ( image @ real @ real @ ( times_times @ real @ A3 ) @ ( top_top @ ( set @ real ) ) )
= ( insert @ real @ ( zero_zero @ real ) @ ( bot_bot @ ( set @ real ) ) ) ) )
& ( ( A3
!= ( zero_zero @ real ) )
=> ( ( image @ real @ real @ ( times_times @ real @ A3 ) @ ( top_top @ ( set @ real ) ) )
= ( top_top @ ( set @ real ) ) ) ) ) ).
% range_mult
thf(fact_6582_Word__eq__word__of__int,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( word2 @ A )
= ( ring_1_of_int @ ( word @ A ) ) ) ) ).
% Word_eq_word_of_int
thf(fact_6583_Collect__const__case__prod,axiom,
! [B: $tType,A: $tType,P: $o] :
( ( P
=> ( ( collect @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [A5: A,B5: B] : P ) )
= ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) )
& ( ~ P
=> ( ( collect @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [A5: A,B5: B] : P ) )
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ) ) ).
% Collect_const_case_prod
thf(fact_6584_top__empty__eq,axiom,
! [A: $tType] :
( ( top_top @ ( A > $o ) )
= ( ^ [X2: A] : ( member @ A @ X2 @ ( top_top @ ( set @ A ) ) ) ) ) ).
% top_empty_eq
thf(fact_6585_top__set__def,axiom,
! [A: $tType] :
( ( top_top @ ( set @ A ) )
= ( collect @ A @ ( top_top @ ( A > $o ) ) ) ) ).
% top_set_def
thf(fact_6586_top__empty__eq2,axiom,
! [B: $tType,A: $tType] :
( ( top_top @ ( A > B > $o ) )
= ( ^ [X2: A,Y2: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y2 ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) ) ) ).
% top_empty_eq2
thf(fact_6587_infinite__UNIV__listI,axiom,
! [A: $tType] :
~ ( finite_finite2 @ ( list @ A ) @ ( top_top @ ( set @ ( list @ A ) ) ) ) ).
% infinite_UNIV_listI
thf(fact_6588_infinite__UNIV__int,axiom,
~ ( finite_finite2 @ int @ ( top_top @ ( set @ int ) ) ) ).
% infinite_UNIV_int
thf(fact_6589_xor__word_Oabs__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Xa: int,X: int] :
( ( bit_se5824344971392196577ns_xor @ ( word @ A ) @ ( word2 @ A @ Xa ) @ ( word2 @ A @ X ) )
= ( word2 @ A @ ( bit_se5824344971392196577ns_xor @ int @ Xa @ X ) ) ) ) ).
% xor_word.abs_eq
thf(fact_6590_mask__word_Oabs__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( bit_se2239418461657761734s_mask @ ( word @ A ) )
= ( ^ [X2: nat] : ( word2 @ A @ ( bit_se2239418461657761734s_mask @ int @ X2 ) ) ) ) ) ).
% mask_word.abs_eq
thf(fact_6591_push__bit__word_Oabs__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Xa: nat,X: int] :
( ( bit_se4730199178511100633sh_bit @ ( word @ A ) @ Xa @ ( word2 @ A @ X ) )
= ( word2 @ A @ ( bit_se4730199178511100633sh_bit @ int @ Xa @ X ) ) ) ) ).
% push_bit_word.abs_eq
thf(fact_6592_not__word_Oabs__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: int] :
( ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ ( word2 @ A @ X ) )
= ( word2 @ A @ ( bit_ri4277139882892585799ns_not @ int @ X ) ) ) ) ).
% not_word.abs_eq
thf(fact_6593_or__word_Oabs__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Xa: int,X: int] :
( ( bit_se1065995026697491101ons_or @ ( word @ A ) @ ( word2 @ A @ Xa ) @ ( word2 @ A @ X ) )
= ( word2 @ A @ ( bit_se1065995026697491101ons_or @ int @ Xa @ X ) ) ) ) ).
% or_word.abs_eq
thf(fact_6594_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_6595_plus__word_Oabs__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Xa: int,X: int] :
( ( plus_plus @ ( word @ A ) @ ( word2 @ A @ Xa ) @ ( word2 @ A @ X ) )
= ( word2 @ A @ ( plus_plus @ int @ Xa @ X ) ) ) ) ).
% plus_word.abs_eq
thf(fact_6596_word__rotate_Oword__rot__logs_I4_J,axiom,
! [B: $tType] :
( ( type_len @ B )
=> ! [N: nat,X: word @ B,Y: word @ B] :
( ( word_rotr @ B @ N @ ( bit_se5824344872417868541ns_and @ ( word @ B ) @ X @ Y ) )
= ( bit_se5824344872417868541ns_and @ ( word @ B ) @ ( word_rotr @ B @ N @ X ) @ ( word_rotr @ B @ N @ Y ) ) ) ) ).
% word_rotate.word_rot_logs(4)
thf(fact_6597_word__rotate_Oword__rot__logs_I6_J,axiom,
! [B: $tType] :
( ( type_len @ B )
=> ! [N: nat,X: word @ B,Y: word @ B] :
( ( word_rotr @ B @ N @ ( bit_se1065995026697491101ons_or @ ( word @ B ) @ X @ Y ) )
= ( bit_se1065995026697491101ons_or @ ( word @ B ) @ ( word_rotr @ B @ N @ X ) @ ( word_rotr @ B @ N @ Y ) ) ) ) ).
% word_rotate.word_rot_logs(6)
thf(fact_6598_word__rotate_Oword__rot__logs_I2_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,V2: word @ A] :
( ( word_rotr @ A @ N @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ V2 ) )
= ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ ( word_rotr @ A @ N @ V2 ) ) ) ) ).
% word_rotate.word_rot_logs(2)
thf(fact_6599_unset__bit__word_Oabs__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Xa: nat,X: int] :
( ( bit_se2638667681897837118et_bit @ ( word @ A ) @ Xa @ ( word2 @ A @ X ) )
= ( word2 @ A @ ( bit_se2638667681897837118et_bit @ int @ Xa @ X ) ) ) ) ).
% unset_bit_word.abs_eq
thf(fact_6600_word_Oabs__induct,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [P: ( word @ A ) > $o,X: word @ A] :
( ! [Y4: int] : ( P @ ( word2 @ A @ Y4 ) )
=> ( P @ X ) ) ) ).
% word.abs_induct
thf(fact_6601_set__bit__word_Oabs__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Xa: nat,X: int] :
( ( bit_se5668285175392031749et_bit @ ( word @ A ) @ Xa @ ( word2 @ A @ X ) )
= ( word2 @ A @ ( bit_se5668285175392031749et_bit @ int @ Xa @ X ) ) ) ) ).
% set_bit_word.abs_eq
thf(fact_6602_flip__bit__word_Oabs__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Xa: nat,X: int] :
( ( bit_se8732182000553998342ip_bit @ ( word @ A ) @ Xa @ ( word2 @ A @ X ) )
= ( word2 @ A @ ( bit_se8732182000553998342ip_bit @ int @ Xa @ X ) ) ) ) ).
% flip_bit_word.abs_eq
thf(fact_6603_one__word__def,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( one_one @ ( word @ A ) )
= ( word2 @ A @ ( one_one @ int ) ) ) ) ).
% one_word_def
thf(fact_6604_word__rotate_Oword__rot__logs_I8_J,axiom,
! [B: $tType] :
( ( type_len @ B )
=> ! [N: nat,X: word @ B,Y: word @ B] :
( ( word_rotr @ B @ N @ ( bit_se5824344971392196577ns_xor @ ( word @ B ) @ X @ Y ) )
= ( bit_se5824344971392196577ns_xor @ ( word @ B ) @ ( word_rotr @ B @ N @ X ) @ ( word_rotr @ B @ N @ Y ) ) ) ) ).
% word_rotate.word_rot_logs(8)
thf(fact_6605_zero__word__def,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( zero_zero @ ( word @ A ) )
= ( word2 @ A @ ( zero_zero @ int ) ) ) ) ).
% zero_word_def
thf(fact_6606_and__word_Oabs__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Xa: int,X: int] :
( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ ( word2 @ A @ Xa ) @ ( word2 @ A @ X ) )
= ( word2 @ A @ ( bit_se5824344872417868541ns_and @ int @ Xa @ X ) ) ) ) ).
% and_word.abs_eq
thf(fact_6607_times__word_Oabs__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Xa: int,X: int] :
( ( times_times @ ( word @ A ) @ ( word2 @ A @ Xa ) @ ( word2 @ A @ X ) )
= ( word2 @ A @ ( times_times @ int @ Xa @ X ) ) ) ) ).
% times_word.abs_eq
thf(fact_6608_minus__word_Oabs__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Xa: int,X: int] :
( ( minus_minus @ ( word @ A ) @ ( word2 @ A @ Xa ) @ ( word2 @ A @ X ) )
= ( word2 @ A @ ( minus_minus @ int @ Xa @ X ) ) ) ) ).
% minus_word.abs_eq
thf(fact_6609_uminus__word_Oabs__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: int] :
( ( uminus_uminus @ ( word @ A ) @ ( word2 @ A @ X ) )
= ( word2 @ A @ ( uminus_uminus @ int @ X ) ) ) ) ).
% uminus_word.abs_eq
thf(fact_6610_word_Oabs__eq__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: int,Y: int] :
( ( ( word2 @ A @ X )
= ( word2 @ A @ Y ) )
= ( ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ X )
= ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ Y ) ) ) ) ).
% word.abs_eq_iff
thf(fact_6611_size__word_Oabs__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: int] :
( ( size_size @ ( word @ A ) @ ( word2 @ A @ X ) )
= ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ).
% size_word.abs_eq
thf(fact_6612_word__succ_Oabs__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: int] :
( ( word_succ @ A @ ( word2 @ A @ X ) )
= ( word2 @ A @ ( plus_plus @ int @ X @ ( one_one @ int ) ) ) ) ) ).
% word_succ.abs_eq
thf(fact_6613_word__pred_Oabs__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: int] :
( ( word_pred @ A @ ( word2 @ A @ X ) )
= ( word2 @ A @ ( minus_minus @ int @ X @ ( one_one @ int ) ) ) ) ) ).
% word_pred.abs_eq
thf(fact_6614_Ints__def,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ( ( ring_1_Ints @ A )
= ( image @ int @ A @ ( ring_1_of_int @ A ) @ ( top_top @ ( set @ int ) ) ) ) ) ).
% Ints_def
thf(fact_6615_word__rotr_Oabs__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Xa: nat,X: int] :
( ( word_rotr @ A @ Xa @ ( word2 @ A @ X ) )
= ( word2 @ A @ ( bit_concat_bit @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( modulo_modulo @ nat @ Xa @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) @ ( bit_se4197421643247451524op_bit @ int @ ( modulo_modulo @ nat @ Xa @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ X ) ) @ ( bit_se2584673776208193580ke_bit @ int @ ( modulo_modulo @ nat @ Xa @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) @ X ) ) ) ) ) ).
% word_rotr.abs_eq
thf(fact_6616_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_6617_word__int__case_Oabs__eq,axiom,
! [B: $tType,A: $tType] :
( ( type_len @ A )
=> ! [Xa: int > B,X: int] :
( ( word_int_case @ B @ A @ Xa @ ( word2 @ A @ X ) )
= ( Xa @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ X ) ) ) ) ).
% word_int_case.abs_eq
thf(fact_6618_less__eq__word_Oabs__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Xa: int,X: int] :
( ( ord_less_eq @ ( word @ A ) @ ( word2 @ A @ Xa ) @ ( word2 @ A @ X ) )
= ( ord_less_eq @ int @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ Xa ) @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ X ) ) ) ) ).
% less_eq_word.abs_eq
thf(fact_6619_less__word_Oabs__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Xa: int,X: int] :
( ( ord_less @ ( word @ A ) @ ( word2 @ A @ Xa ) @ ( word2 @ A @ X ) )
= ( ord_less @ int @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ Xa ) @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ X ) ) ) ) ).
% less_word.abs_eq
thf(fact_6620_bit__word_Oabs__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: int] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( word2 @ A @ X ) )
= ( ^ [N4: nat] :
( ( ord_less @ nat @ N4 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
& ( bit_se5641148757651400278ts_bit @ int @ X @ N4 ) ) ) ) ) ).
% bit_word.abs_eq
thf(fact_6621_divide__word_Oabs__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Xa: int,X: int] :
( ( divide_divide @ ( word @ A ) @ ( word2 @ A @ Xa ) @ ( word2 @ A @ X ) )
= ( word2 @ A @ ( divide_divide @ int @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ Xa ) @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ X ) ) ) ) ) ).
% divide_word.abs_eq
thf(fact_6622_modulo__word_Oabs__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Xa: int,X: int] :
( ( modulo_modulo @ ( word @ A ) @ ( word2 @ A @ Xa ) @ ( word2 @ A @ X ) )
= ( word2 @ A @ ( modulo_modulo @ int @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ Xa ) @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ X ) ) ) ) ) ).
% modulo_word.abs_eq
thf(fact_6623_take__bit__word_Oabs__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Xa: nat,X: int] :
( ( bit_se2584673776208193580ke_bit @ ( word @ A ) @ Xa @ ( word2 @ A @ X ) )
= ( word2 @ A @ ( bit_se2584673776208193580ke_bit @ int @ ( ord_min @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ Xa ) @ X ) ) ) ) ).
% take_bit_word.abs_eq
thf(fact_6624_drop__bit__word_Oabs__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Xa: nat,X: int] :
( ( bit_se4197421643247451524op_bit @ ( word @ A ) @ Xa @ ( word2 @ A @ X ) )
= ( word2 @ A @ ( bit_se4197421643247451524op_bit @ int @ Xa @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ X ) ) ) ) ) ).
% drop_bit_word.abs_eq
thf(fact_6625_word__cat_Oabs__eq,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B )
& ( type_len @ C ) )
=> ! [Xa: int,X: int] :
( ( word_cat @ A @ B @ C @ ( word2 @ A @ Xa ) @ ( word2 @ B @ X ) )
= ( word2 @ C @ ( bit_concat_bit @ ( type_len0_len_of @ B @ ( type2 @ B ) ) @ X @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ Xa ) ) ) ) ) ).
% word_cat.abs_eq
thf(fact_6626_unsigned_Oabs__eq,axiom,
! [A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( semiring_1 @ A ) )
=> ! [X: int] :
( ( semiring_1_unsigned @ B @ A @ ( word2 @ B @ X ) )
= ( semiring_1_of_nat @ A @ ( nat2 @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ B @ ( type2 @ B ) ) @ X ) ) ) ) ) ).
% unsigned.abs_eq
thf(fact_6627_range__mod,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( image @ nat @ nat
@ ^ [M3: nat] : ( modulo_modulo @ nat @ M3 @ N )
@ ( top_top @ ( set @ nat ) ) )
= ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ).
% range_mod
thf(fact_6628_word__sle_Oabs__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Xa: int,X: int] :
( ( word_sle @ A @ ( word2 @ A @ Xa ) @ ( word2 @ A @ X ) )
= ( ord_less_eq @ int @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) @ Xa ) @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) @ X ) ) ) ) ).
% word_sle.abs_eq
thf(fact_6629_word__sless_Oabs__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Xa: int,X: int] :
( ( word_sless @ A @ ( word2 @ A @ Xa ) @ ( word2 @ A @ X ) )
= ( ord_less @ int @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) @ Xa ) @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) @ X ) ) ) ) ).
% word_sless.abs_eq
thf(fact_6630_signed_Oabs__eq,axiom,
! [A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( ring_1 @ A ) )
=> ! [X: int] :
( ( ring_1_signed @ B @ A @ ( word2 @ B @ X ) )
= ( ring_1_of_int @ A @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ B @ ( type2 @ B ) ) @ ( suc @ ( zero_zero @ nat ) ) ) @ X ) ) ) ) ).
% signed.abs_eq
thf(fact_6631_signed__drop__bit_Oabs__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Xa: nat,X: int] :
( ( signed_drop_bit @ A @ Xa @ ( word2 @ A @ X ) )
= ( word2 @ A @ ( bit_se4197421643247451524op_bit @ int @ Xa @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) @ X ) ) ) ) ) ).
% signed_drop_bit.abs_eq
thf(fact_6632_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_6633_signed__modulo__word_Oabs__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Xa: int,X: int] :
( ( signed6721504322012087516modulo @ ( word @ A ) @ ( word2 @ A @ Xa ) @ ( word2 @ A @ X ) )
= ( word2 @ A @ ( signed6721504322012087516modulo @ int @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) @ Xa ) @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) @ X ) ) ) ) ) ).
% signed_modulo_word.abs_eq
thf(fact_6634_signed__divide__word_Oabs__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Xa: int,X: int] :
( ( signed7115095781618012415divide @ ( word @ A ) @ ( word2 @ A @ Xa ) @ ( word2 @ A @ X ) )
= ( word2 @ A @ ( signed7115095781618012415divide @ int @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) @ Xa ) @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) @ X ) ) ) ) ) ).
% signed_divide_word.abs_eq
thf(fact_6635_signed__cast_Oabs__eq,axiom,
! [B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B ) )
=> ! [X: int] :
( ( signed_cast @ A @ B @ ( word2 @ A @ X ) )
= ( word2 @ B @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) @ X ) ) ) ) ).
% signed_cast.abs_eq
thf(fact_6636_the__signed__int_Oabs__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: int] :
( ( the_signed_int @ A @ ( word2 @ A @ X ) )
= ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) @ X ) ) ) ).
% the_signed_int.abs_eq
thf(fact_6637_top1I,axiom,
! [A: $tType,X: A] : ( top_top @ ( A > $o ) @ X ) ).
% top1I
thf(fact_6638_top2I,axiom,
! [A: $tType,B: $tType,X: A,Y: B] : ( top_top @ ( A > B > $o ) @ X @ Y ) ).
% top2I
thf(fact_6639_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_6640_of__nat_Oabs__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( of_nat @ A )
= ( ^ [X2: nat] : ( word2 @ A @ ( semiring_1_of_nat @ int @ ( bit_se2584673776208193580ke_bit @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ X2 ) ) ) ) ) ) ).
% of_nat.abs_eq
thf(fact_6641_the__nat_Oabs__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: int] :
( ( the_nat @ A @ ( word2 @ A @ X ) )
= ( nat2 @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ X ) ) ) ) ).
% the_nat.abs_eq
thf(fact_6642_cast_Oabs__eq,axiom,
! [B: $tType,A: $tType] :
( ( ( type_len @ A )
& ( type_len @ B ) )
=> ! [X: int] :
( ( cast @ A @ B @ ( word2 @ A @ X ) )
= ( word2 @ B @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ X ) ) ) ) ).
% cast.abs_eq
thf(fact_6643_Word_Oof__int_Oabs__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( of_int2 @ A )
= ( ^ [X2: int] : ( word2 @ A @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ X2 ) ) ) ) ) ).
% Word.of_int.abs_eq
thf(fact_6644_root__def,axiom,
( root
= ( ^ [N4: nat,X2: real] :
( if @ real
@ ( N4
= ( zero_zero @ nat ) )
@ ( zero_zero @ real )
@ ( the_inv_into @ real @ real @ ( top_top @ ( set @ real ) )
@ ^ [Y2: real] : ( times_times @ real @ ( sgn_sgn @ real @ Y2 ) @ ( power_power @ real @ ( abs_abs @ real @ Y2 ) @ N4 ) )
@ X2 ) ) ) ) ).
% root_def
thf(fact_6645_the__inv__into__def,axiom,
! [B: $tType,A: $tType] :
( ( the_inv_into @ A @ B )
= ( ^ [A4: set @ A,F2: A > B,X2: B] :
( the @ A
@ ^ [Y2: A] :
( ( member @ A @ Y2 @ A4 )
& ( ( F2 @ Y2 )
= X2 ) ) ) ) ) ).
% the_inv_into_def
thf(fact_6646_word__roti__eq__word__rotr__word__rotl,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( word_roti @ A )
= ( ^ [I4: int,W2: word @ A] : ( if @ ( word @ A ) @ ( ord_less_eq @ int @ ( zero_zero @ int ) @ I4 ) @ ( word_rotr @ A @ ( nat2 @ I4 ) @ W2 ) @ ( word_rotl @ A @ ( nat2 @ ( uminus_uminus @ int @ I4 ) ) @ W2 ) ) ) ) ) ).
% word_roti_eq_word_rotr_word_rotl
thf(fact_6647_word__rotl_Oabs__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [Xa: nat,X: int] :
( ( word_rotl @ A @ Xa @ ( word2 @ A @ X ) )
= ( word2 @ A @ ( bit_concat_bit @ ( modulo_modulo @ nat @ Xa @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) @ ( bit_se4197421643247451524op_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( modulo_modulo @ nat @ Xa @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ X ) ) @ ( bit_se2584673776208193580ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( modulo_modulo @ nat @ Xa @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) @ X ) ) ) ) ) ).
% word_rotl.abs_eq
thf(fact_6648_word__rot__lr,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [K: nat,V2: word @ A] :
( ( word_rotr @ A @ K @ ( word_rotl @ A @ K @ V2 ) )
= V2 ) ) ).
% word_rot_lr
thf(fact_6649_word__rot__rl,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [K: nat,V2: word @ A] :
( ( word_rotl @ A @ K @ ( word_rotr @ A @ K @ V2 ) )
= V2 ) ) ).
% word_rot_rl
thf(fact_6650_word__rot__gal,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,V2: word @ A,W: word @ A] :
( ( ( word_rotr @ A @ N @ V2 )
= W )
= ( ( word_rotl @ A @ N @ W )
= V2 ) ) ) ).
% word_rot_gal
thf(fact_6651_word__rot__gal_H,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [W: word @ A,N: nat,V2: word @ A] :
( ( W
= ( word_rotr @ A @ N @ V2 ) )
= ( V2
= ( word_rotl @ A @ N @ W ) ) ) ) ).
% word_rot_gal'
thf(fact_6652_word__rotate_Oword__rot__logs_I7_J,axiom,
! [B: $tType] :
( ( type_len @ B )
=> ! [N: nat,X: word @ B,Y: word @ B] :
( ( word_rotl @ B @ N @ ( bit_se5824344971392196577ns_xor @ ( word @ B ) @ X @ Y ) )
= ( bit_se5824344971392196577ns_xor @ ( word @ B ) @ ( word_rotl @ B @ N @ X ) @ ( word_rotl @ B @ N @ Y ) ) ) ) ).
% word_rotate.word_rot_logs(7)
thf(fact_6653_word__rotate_Oword__rot__logs_I1_J,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat,V2: word @ A] :
( ( word_rotl @ A @ N @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ V2 ) )
= ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ ( word_rotl @ A @ N @ V2 ) ) ) ) ).
% word_rotate.word_rot_logs(1)
thf(fact_6654_word__rotate_Oword__rot__logs_I5_J,axiom,
! [B: $tType] :
( ( type_len @ B )
=> ! [N: nat,X: word @ B,Y: word @ B] :
( ( word_rotl @ B @ N @ ( bit_se1065995026697491101ons_or @ ( word @ B ) @ X @ Y ) )
= ( bit_se1065995026697491101ons_or @ ( word @ B ) @ ( word_rotl @ B @ N @ X ) @ ( word_rotl @ B @ N @ Y ) ) ) ) ).
% word_rotate.word_rot_logs(5)
thf(fact_6655_word__rotate_Oword__rot__logs_I3_J,axiom,
! [B: $tType] :
( ( type_len @ B )
=> ! [N: nat,X: word @ B,Y: word @ B] :
( ( word_rotl @ B @ N @ ( bit_se5824344872417868541ns_and @ ( word @ B ) @ X @ Y ) )
= ( bit_se5824344872417868541ns_and @ ( word @ B ) @ ( word_rotl @ B @ N @ X ) @ ( word_rotl @ B @ N @ Y ) ) ) ) ).
% word_rotate.word_rot_logs(3)
thf(fact_6656_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_6657_word__rotl__eq__word__rotr,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( word_rotl @ A )
= ( ^ [N4: nat] : ( word_rotr @ A @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( modulo_modulo @ nat @ N4 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ) ) ).
% word_rotl_eq_word_rotr
thf(fact_6658_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_6659_the__int_Oabs__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: int] :
( ( the_int @ A @ ( word2 @ A @ X ) )
= ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ X ) ) ) ).
% the_int.abs_eq
thf(fact_6660_CHAR__pos__iff,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( semiri4206861660011772517g_char @ A @ ( type2 @ A ) ) )
= ( ? [N4: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N4 )
& ( ( semiring_1_of_nat @ A @ N4 )
= ( zero_zero @ A ) ) ) ) ) ) ).
% CHAR_pos_iff
thf(fact_6661_CHAR__eq__0,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ( ( semiri4206861660011772517g_char @ A @ ( type2 @ A ) )
= ( zero_zero @ nat ) ) ) ).
% CHAR_eq_0
thf(fact_6662_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_6663_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_6664_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_6665_CHAR__eq0__iff,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ( ( ( semiri4206861660011772517g_char @ A @ ( type2 @ A ) )
= ( zero_zero @ nat ) )
= ( ! [N4: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N4 )
=> ( ( semiring_1_of_nat @ A @ N4 )
!= ( zero_zero @ A ) ) ) ) ) ) ).
% CHAR_eq0_iff
thf(fact_6666_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_6667_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_6668_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_6669_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_6670_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_6671_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_6672_word__msb__0,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ~ ( most_s684356279273892711sb_msb @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) ) ) ).
% word_msb_0
thf(fact_6673_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_6674_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_6675_word__sless__msb__less,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( word_sless @ A )
= ( ^ [X2: word @ A,Y2: word @ A] :
( ( ( most_s684356279273892711sb_msb @ ( word @ A ) @ Y2 )
=> ( most_s684356279273892711sb_msb @ ( word @ A ) @ X2 ) )
& ( ( ( most_s684356279273892711sb_msb @ ( word @ A ) @ X2 )
& ~ ( most_s684356279273892711sb_msb @ ( word @ A ) @ Y2 ) )
| ( ord_less @ ( word @ A ) @ X2 @ Y2 ) ) ) ) ) ) ).
% word_sless_msb_less
thf(fact_6676_revcast__slice1,axiom,
! [A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( type_len @ A ) )
=> ! [W: word @ B] :
( ( slice1 @ B @ A @ ( size_size @ ( word @ A ) @ ( revcast @ B @ A @ W ) ) @ W )
= ( revcast @ B @ A @ W ) ) ) ).
% revcast_slice1
thf(fact_6677_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_6678_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_6679_revcast__def,axiom,
! [A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( type_len @ A ) )
=> ( ( revcast @ A @ B )
= ( slice1 @ A @ B @ ( type_len0_len_of @ B @ ( type2 @ B ) ) ) ) ) ).
% revcast_def
thf(fact_6680_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_6681_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_6682_msb__word__of__int,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: int] :
( ( most_s684356279273892711sb_msb @ ( word @ A ) @ ( ring_1_of_int @ ( word @ A ) @ X ) )
= ( bit_se5641148757651400278ts_bit @ int @ X @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) ) ) ) ).
% msb_word_of_int
thf(fact_6683_not__msb__from__less,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [V2: word @ A] :
( ( ord_less @ ( word @ A ) @ V2 @ ( 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 ) @ V2 ) ) ) ).
% not_msb_from_less
thf(fact_6684_word__sint__msb__eq,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( ring_1_signed @ A @ int )
= ( ^ [X2: word @ A] : ( minus_minus @ int @ ( semiring_1_unsigned @ A @ int @ X2 ) @ ( if @ int @ ( most_s684356279273892711sb_msb @ ( word @ A ) @ X2 ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( size_size @ ( word @ A ) @ X2 ) ) @ ( zero_zero @ int ) ) ) ) ) ) ).
% word_sint_msb_eq
thf(fact_6685_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_6686_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_6687_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_6688_inj__on__empty,axiom,
! [B: $tType,A: $tType,F3: A > B] : ( inj_on @ A @ B @ F3 @ ( bot_bot @ ( set @ A ) ) ) ).
% inj_on_empty
thf(fact_6689_inj__mult__left,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [A3: A] :
( ( inj_on @ A @ A @ ( times_times @ A @ A3 ) @ ( top_top @ ( set @ A ) ) )
= ( A3
!= ( zero_zero @ A ) ) ) ) ).
% inj_mult_left
thf(fact_6690_inj__divide__right,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A3: A] :
( ( inj_on @ A @ A
@ ^ [B5: A] : ( divide_divide @ A @ B5 @ A3 )
@ ( top_top @ ( set @ A ) ) )
= ( A3
!= ( zero_zero @ A ) ) ) ) ).
% inj_divide_right
thf(fact_6691_inj__on__insert,axiom,
! [B: $tType,A: $tType,F3: A > B,A3: A,A2: set @ A] :
( ( inj_on @ A @ B @ F3 @ ( insert @ A @ A3 @ A2 ) )
= ( ( inj_on @ A @ B @ F3 @ A2 )
& ~ ( member @ B @ ( F3 @ A3 ) @ ( image @ A @ B @ F3 @ ( minus_minus @ ( set @ A ) @ A2 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% inj_on_insert
thf(fact_6692_nth__image__indices,axiom,
! [A: $tType,L: list @ A] :
( ( image @ nat @ A @ ( nth @ A @ L ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ L ) ) )
= ( set2 @ A @ L ) ) ).
% nth_image_indices
thf(fact_6693_linorder__injI,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ A )
=> ! [F3: A > B] :
( ! [X3: A,Y4: A] :
( ( ord_less @ A @ X3 @ Y4 )
=> ( ( F3 @ X3 )
!= ( F3 @ Y4 ) ) )
=> ( inj_on @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) ) ) ) ).
% linorder_injI
thf(fact_6694_sorted__list__of__set_Oinj__on,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( inj_on @ A @ A
@ ^ [X2: A] : X2
@ ( top_top @ ( set @ A ) ) ) ) ).
% sorted_list_of_set.inj_on
thf(fact_6695_inj__fun,axiom,
! [B: $tType,C: $tType,A: $tType,F3: A > B] :
( ( inj_on @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) )
=> ( inj_on @ A @ ( C > B )
@ ^ [X2: A,Y2: C] : ( F3 @ X2 )
@ ( top_top @ ( set @ A ) ) ) ) ).
% inj_fun
thf(fact_6696_inj__diff__right,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A3: A] :
( inj_on @ A @ A
@ ^ [B5: A] : ( minus_minus @ A @ B5 @ A3 )
@ ( top_top @ ( set @ A ) ) ) ) ).
% inj_diff_right
thf(fact_6697_finite__inverse__image,axiom,
! [A: $tType,B: $tType,A2: set @ A,F3: B > A] :
( ( finite_finite2 @ A @ A2 )
=> ( ( inj_on @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) )
=> ( finite_finite2 @ B
@ ( collect @ B
@ ^ [J3: B] : ( member @ A @ ( F3 @ J3 ) @ A2 ) ) ) ) ) ).
% finite_inverse_image
thf(fact_6698_finite__Collect,axiom,
! [A: $tType,B: $tType,S: set @ A,F3: B > A] :
( ( finite_finite2 @ A @ S )
=> ( ( inj_on @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) )
=> ( finite_finite2 @ B
@ ( collect @ B
@ ^ [A5: B] : ( member @ A @ ( F3 @ A5 ) @ S ) ) ) ) ) ).
% finite_Collect
thf(fact_6699_inj__img__insertE,axiom,
! [B: $tType,A: $tType,F3: A > B,A2: set @ A,X: B,B2: set @ B] :
( ( inj_on @ A @ B @ F3 @ A2 )
=> ( ~ ( member @ B @ X @ B2 )
=> ( ( ( insert @ B @ X @ B2 )
= ( image @ A @ B @ F3 @ A2 ) )
=> ~ ! [X10: A,A8: set @ A] :
( ~ ( member @ A @ X10 @ A8 )
=> ( ( A2
= ( insert @ A @ X10 @ A8 ) )
=> ( ( X
= ( F3 @ X10 ) )
=> ( B2
!= ( image @ A @ B @ F3 @ A8 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_6700_inj__on__strict__subset,axiom,
! [B: $tType,A: $tType,F3: A > B,B2: set @ A,A2: set @ A] :
( ( inj_on @ A @ B @ F3 @ B2 )
=> ( ( ord_less @ ( set @ A ) @ A2 @ B2 )
=> ( ord_less @ ( set @ B ) @ ( image @ A @ B @ F3 @ A2 ) @ ( image @ A @ B @ F3 @ B2 ) ) ) ) ).
% inj_on_strict_subset
thf(fact_6701_sum_Oimage__eq,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [G2: B > A,A2: set @ B] :
( ( inj_on @ B @ A @ G2 @ A2 )
=> ( ( groups7311177749621191930dd_sum @ A @ A
@ ^ [X2: A] : X2
@ ( image @ B @ A @ G2 @ A2 ) )
= ( groups7311177749621191930dd_sum @ B @ A @ G2 @ A2 ) ) ) ) ).
% sum.image_eq
thf(fact_6702_prod_Oimage__eq,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G2: B > A,A2: set @ B] :
( ( inj_on @ B @ A @ G2 @ A2 )
=> ( ( groups7121269368397514597t_prod @ A @ A
@ ^ [X2: A] : X2
@ ( image @ B @ A @ G2 @ A2 ) )
= ( groups7121269368397514597t_prod @ B @ A @ G2 @ A2 ) ) ) ) ).
% prod.image_eq
thf(fact_6703_inj__on__Un__image__eq__iff,axiom,
! [B: $tType,A: $tType,F3: A > B,A2: set @ A,B2: set @ A] :
( ( inj_on @ A @ B @ F3 @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) )
=> ( ( ( image @ A @ B @ F3 @ A2 )
= ( image @ A @ B @ F3 @ B2 ) )
= ( A2 = B2 ) ) ) ).
% inj_on_Un_image_eq_iff
thf(fact_6704_finite__inverse__image__gen,axiom,
! [A: $tType,B: $tType,A2: set @ A,F3: B > A,D3: set @ B] :
( ( finite_finite2 @ A @ A2 )
=> ( ( inj_on @ B @ A @ F3 @ D3 )
=> ( finite_finite2 @ B
@ ( collect @ B
@ ^ [J3: B] :
( ( member @ B @ J3 @ D3 )
& ( member @ A @ ( F3 @ J3 ) @ A2 ) ) ) ) ) ) ).
% finite_inverse_image_gen
thf(fact_6705_linorder__inj__onI,axiom,
! [B: $tType,A: $tType] :
( ( order @ A )
=> ! [A2: set @ A,F3: A > B] :
( ! [X3: A,Y4: A] :
( ( ord_less @ A @ X3 @ Y4 )
=> ( ( member @ A @ X3 @ A2 )
=> ( ( member @ A @ Y4 @ A2 )
=> ( ( F3 @ X3 )
!= ( F3 @ Y4 ) ) ) ) )
=> ( ! [X3: A,Y4: A] :
( ( member @ A @ X3 @ A2 )
=> ( ( member @ A @ Y4 @ A2 )
=> ( ( ord_less_eq @ A @ X3 @ Y4 )
| ( ord_less_eq @ A @ Y4 @ X3 ) ) ) )
=> ( inj_on @ A @ B @ F3 @ A2 ) ) ) ) ).
% linorder_inj_onI
thf(fact_6706_inj__on__add_H,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [A3: A,A2: set @ A] :
( inj_on @ A @ A
@ ^ [B5: A] : ( plus_plus @ A @ B5 @ A3 )
@ A2 ) ) ).
% inj_on_add'
thf(fact_6707_inj__on__id2,axiom,
! [A: $tType,A2: set @ A] :
( inj_on @ A @ A
@ ^ [X2: A] : X2
@ A2 ) ).
% inj_on_id2
thf(fact_6708_inj__on__mult,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [A3: A,A2: set @ A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( inj_on @ A @ A @ ( times_times @ A @ A3 ) @ A2 ) ) ) ).
% inj_on_mult
thf(fact_6709_obtain__list__from__elements,axiom,
! [A: $tType,N: nat,P: A > nat > $o] :
( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ N )
=> ? [Li: A] : ( P @ Li @ I3 ) )
=> ~ ! [L3: list @ A] :
( ( ( size_size @ ( list @ A ) @ L3 )
= N )
=> ~ ! [I2: nat] :
( ( ord_less @ nat @ I2 @ N )
=> ( P @ ( nth @ A @ L3 @ I2 ) @ I2 ) ) ) ) ).
% obtain_list_from_elements
thf(fact_6710_nth__equalityI,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ A ) @ Ys ) )
=> ( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( nth @ A @ Xs @ I3 )
= ( nth @ A @ Ys @ I3 ) ) )
=> ( Xs = Ys ) ) ) ).
% nth_equalityI
thf(fact_6711_Skolem__list__nth,axiom,
! [A: $tType,K: nat,P: nat > A > $o] :
( ( ! [I4: nat] :
( ( ord_less @ nat @ I4 @ K )
=> ? [X8: A] : ( P @ I4 @ X8 ) ) )
= ( ? [Xs3: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs3 )
= K )
& ! [I4: nat] :
( ( ord_less @ nat @ I4 @ K )
=> ( P @ I4 @ ( nth @ A @ Xs3 @ I4 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_6712_list__eq__iff__nth__eq,axiom,
! [A: $tType] :
( ( ^ [Y3: list @ A,Z2: list @ A] : Y3 = Z2 )
= ( ^ [Xs3: list @ A,Ys3: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs3 )
= ( size_size @ ( list @ A ) @ Ys3 ) )
& ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( size_size @ ( list @ A ) @ Xs3 ) )
=> ( ( nth @ A @ Xs3 @ I4 )
= ( nth @ A @ Ys3 @ I4 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_6713_inj__on__iff__surj,axiom,
! [A: $tType,B: $tType,A2: set @ A,A9: set @ B] :
( ( A2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ? [F2: A > B] :
( ( inj_on @ A @ B @ F2 @ A2 )
& ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F2 @ A2 ) @ A9 ) ) )
= ( ? [G: B > A] :
( ( image @ B @ A @ G @ A9 )
= A2 ) ) ) ) ).
% inj_on_iff_surj
thf(fact_6714_all__set__conv__nth,axiom,
! [A: $tType,L: list @ A,P: A > $o] :
( ( ! [X2: A] :
( ( member @ A @ X2 @ ( set2 @ A @ L ) )
=> ( P @ X2 ) ) )
= ( ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( size_size @ ( list @ A ) @ L ) )
=> ( P @ ( nth @ A @ L @ I4 ) ) ) ) ) ).
% all_set_conv_nth
thf(fact_6715_nth__mem,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( member @ A @ ( nth @ A @ Xs @ N ) @ ( set2 @ A @ Xs ) ) ) ).
% nth_mem
thf(fact_6716_list__ball__nth,axiom,
! [A: $tType,N: nat,Xs: list @ A,P: A > $o] :
( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( P @ X3 ) )
=> ( P @ ( nth @ A @ Xs @ N ) ) ) ) ).
% list_ball_nth
thf(fact_6717_in__set__conv__nth,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
= ( ? [I4: nat] :
( ( ord_less @ nat @ I4 @ ( size_size @ ( list @ A ) @ Xs ) )
& ( ( nth @ A @ Xs @ I4 )
= X ) ) ) ) ).
% in_set_conv_nth
thf(fact_6718_all__nth__imp__all__set,axiom,
! [A: $tType,Xs: list @ A,P: A > $o,X: A] :
( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( P @ ( nth @ A @ Xs @ I3 ) ) )
=> ( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( P @ X ) ) ) ).
% all_nth_imp_all_set
thf(fact_6719_all__set__conv__all__nth,axiom,
! [A: $tType,Xs: list @ A,P: A > $o] :
( ( ! [X2: A] :
( ( member @ A @ X2 @ ( set2 @ A @ Xs ) )
=> ( P @ X2 ) ) )
= ( ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( P @ ( nth @ A @ Xs @ I4 ) ) ) ) ) ).
% all_set_conv_all_nth
thf(fact_6720_VEBT__internal_Ointhall,axiom,
! [A: $tType,Xs: list @ A,P: A > $o,N: nat] :
( ! [X3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( P @ X3 ) )
=> ( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( P @ ( nth @ A @ Xs @ N ) ) ) ) ).
% VEBT_internal.inthall
thf(fact_6721_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_6722_in__set__image__conv__nth,axiom,
! [B: $tType,A: $tType,F3: B > A,X: B,L: list @ B] :
( ( member @ A @ ( F3 @ X ) @ ( image @ B @ A @ F3 @ ( set2 @ B @ L ) ) )
= ( ? [I4: nat] :
( ( ord_less @ nat @ I4 @ ( size_size @ ( list @ B ) @ L ) )
& ( ( F3 @ ( nth @ B @ L @ I4 ) )
= ( F3 @ X ) ) ) ) ) ).
% in_set_image_conv_nth
thf(fact_6723_set__image__eq__pointwiseI,axiom,
! [B: $tType,A: $tType,L: list @ A,L5: list @ A,F3: A > B] :
( ( ( size_size @ ( list @ A ) @ L )
= ( size_size @ ( list @ A ) @ L5 ) )
=> ( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ L ) )
=> ( ( F3 @ ( nth @ A @ L @ I3 ) )
= ( F3 @ ( nth @ A @ L5 @ I3 ) ) ) )
=> ( ( image @ A @ B @ F3 @ ( set2 @ A @ L ) )
= ( image @ A @ B @ F3 @ ( set2 @ A @ L5 ) ) ) ) ) ).
% set_image_eq_pointwiseI
thf(fact_6724_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_6725_horner__sum__eq__sum,axiom,
! [A: $tType,B: $tType] :
( ( comm_semiring_1 @ A )
=> ( ( groups4207007520872428315er_sum @ B @ A )
= ( ^ [F2: B > A,A5: A,Xs3: list @ B] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [N4: nat] : ( times_times @ A @ ( F2 @ ( nth @ B @ Xs3 @ N4 ) ) @ ( power_power @ A @ A5 @ N4 ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ B ) @ Xs3 ) ) ) ) ) ) ).
% horner_sum_eq_sum
thf(fact_6726_product__nth,axiom,
! [A: $tType,B: $tType,N: nat,Xs: list @ A,Ys: list @ B] :
( ( ord_less @ nat @ N @ ( times_times @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ B ) @ Ys ) ) )
=> ( ( nth @ ( product_prod @ A @ B ) @ ( product @ A @ B @ Xs @ Ys ) @ N )
= ( product_Pair @ A @ B @ ( nth @ A @ Xs @ ( divide_divide @ nat @ N @ ( size_size @ ( list @ B ) @ Ys ) ) ) @ ( nth @ B @ Ys @ ( modulo_modulo @ nat @ N @ ( size_size @ ( list @ B ) @ Ys ) ) ) ) ) ) ).
% product_nth
thf(fact_6727_length__product,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B] :
( ( size_size @ ( list @ ( product_prod @ A @ B ) ) @ ( product @ A @ B @ Xs @ Ys ) )
= ( times_times @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ B ) @ Ys ) ) ) ).
% length_product
thf(fact_6728_inj__graph,axiom,
! [B: $tType,A: $tType] :
( inj_on @ ( A > B ) @ ( set @ ( product_prod @ A @ B ) )
@ ^ [F2: A > B] :
( collect @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [X2: A,Y2: B] :
( Y2
= ( F2 @ X2 ) ) ) )
@ ( top_top @ ( set @ ( A > B ) ) ) ) ).
% inj_graph
thf(fact_6729_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_6730_inj__signed,axiom,
! [A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( ring_char_0 @ A ) )
=> ( inj_on @ ( word @ B ) @ A @ ( ring_1_signed @ B @ A ) @ ( top_top @ ( set @ ( word @ B ) ) ) ) ) ).
% inj_signed
thf(fact_6731_unat__inj,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( inj_on @ ( word @ A ) @ nat @ ( semiring_1_unsigned @ A @ nat ) @ ( top_top @ ( set @ ( word @ A ) ) ) ) ) ).
% unat_inj
thf(fact_6732_inj__unsigned,axiom,
! [A: $tType,B: $tType] :
( ( ( type_len @ B )
& ( semiring_char_0 @ A ) )
=> ( inj_on @ ( word @ B ) @ A @ ( semiring_1_unsigned @ B @ A ) @ ( top_top @ ( set @ ( word @ B ) ) ) ) ) ).
% inj_unsigned
thf(fact_6733_signed_Osorted__list__of__set_Oinj__on,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( inj_on @ ( word @ A ) @ ( word @ A )
@ ^ [X2: word @ A] : X2
@ ( top_top @ ( set @ ( word @ A ) ) ) ) ) ).
% signed.sorted_list_of_set.inj_on
thf(fact_6734_inj__singleton,axiom,
! [A: $tType,A2: set @ A] :
( inj_on @ A @ ( set @ A )
@ ^ [X2: A] : ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) )
@ A2 ) ).
% inj_singleton
thf(fact_6735_swap__inj__on,axiom,
! [B: $tType,A: $tType,A2: set @ ( product_prod @ A @ B )] :
( inj_on @ ( product_prod @ A @ B ) @ ( product_prod @ B @ A )
@ ( product_case_prod @ A @ B @ ( product_prod @ B @ A )
@ ^ [I4: A,J3: B] : ( product_Pair @ B @ A @ J3 @ I4 ) )
@ A2 ) ).
% swap_inj_on
thf(fact_6736_inj__on__diff__nat,axiom,
! [N3: set @ nat,K: nat] :
( ! [N2: nat] :
( ( member @ nat @ N2 @ N3 )
=> ( ord_less_eq @ nat @ K @ N2 ) )
=> ( inj_on @ nat @ nat
@ ^ [N4: nat] : ( minus_minus @ nat @ N4 @ K )
@ N3 ) ) ).
% inj_on_diff_nat
thf(fact_6737_inj__on__convol__ident,axiom,
! [B: $tType,A: $tType,F3: A > B,X5: set @ A] :
( inj_on @ A @ ( product_prod @ A @ B )
@ ^ [X2: A] : ( product_Pair @ A @ B @ X2 @ ( F3 @ X2 ) )
@ X5 ) ).
% inj_on_convol_ident
thf(fact_6738_inj__Pair_I1_J,axiom,
! [B: $tType,A: $tType,C2: A > B,S: set @ A] :
( inj_on @ A @ ( product_prod @ A @ B )
@ ^ [X2: A] : ( product_Pair @ A @ B @ X2 @ ( C2 @ X2 ) )
@ S ) ).
% inj_Pair(1)
thf(fact_6739_inj__Pair_I2_J,axiom,
! [B: $tType,A: $tType,C2: A > B,S: set @ A] :
( inj_on @ A @ ( product_prod @ B @ A )
@ ^ [X2: A] : ( product_Pair @ B @ A @ ( C2 @ X2 ) @ X2 )
@ S ) ).
% inj_Pair(2)
thf(fact_6740_inj__Suc,axiom,
! [N3: set @ nat] : ( inj_on @ nat @ nat @ suc @ N3 ) ).
% inj_Suc
thf(fact_6741_inj__on__of__nat,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [N3: set @ nat] : ( inj_on @ nat @ A @ ( semiring_1_of_nat @ A ) @ N3 ) ) ).
% inj_on_of_nat
thf(fact_6742_finite__imp__nat__seg__image__inj__on,axiom,
! [A: $tType,A2: set @ A] :
( ( finite_finite2 @ A @ A2 )
=> ? [N2: nat,F5: nat > A] :
( ( A2
= ( image @ nat @ A @ F5
@ ( collect @ nat
@ ^ [I4: nat] : ( ord_less @ nat @ I4 @ N2 ) ) ) )
& ( inj_on @ nat @ A @ F5
@ ( collect @ nat
@ ^ [I4: nat] : ( ord_less @ nat @ I4 @ N2 ) ) ) ) ) ).
% finite_imp_nat_seg_image_inj_on
thf(fact_6743_finite__imp__inj__to__nat__seg_H,axiom,
! [A: $tType,A2: set @ A] :
( ( finite_finite2 @ A @ A2 )
=> ~ ! [F5: A > nat] :
( ? [N2: nat] :
( ( image @ A @ nat @ F5 @ A2 )
= ( collect @ nat
@ ^ [I4: nat] : ( ord_less @ nat @ I4 @ N2 ) ) )
=> ~ ( inj_on @ A @ nat @ F5 @ A2 ) ) ) ).
% finite_imp_inj_to_nat_seg'
thf(fact_6744_finite__imp__inj__to__nat__seg,axiom,
! [A: $tType,A2: set @ A] :
( ( finite_finite2 @ A @ A2 )
=> ? [F5: A > nat,N2: nat] :
( ( ( image @ A @ nat @ F5 @ A2 )
= ( collect @ nat
@ ^ [I4: nat] : ( ord_less @ nat @ I4 @ N2 ) ) )
& ( inj_on @ A @ nat @ F5 @ A2 ) ) ) ).
% finite_imp_inj_to_nat_seg
thf(fact_6745_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_6746_inj__sgn__power,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( inj_on @ real @ real
@ ^ [Y2: real] : ( times_times @ real @ ( sgn_sgn @ real @ Y2 ) @ ( power_power @ real @ ( abs_abs @ real @ Y2 ) @ N ) )
@ ( top_top @ ( set @ real ) ) ) ) ).
% inj_sgn_power
thf(fact_6747_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_6748_slice__nth,axiom,
! [A: $tType,From: nat,To: nat,Xs: list @ A,I: nat] :
( ( ord_less @ nat @ From @ To )
=> ( ( ord_less_eq @ nat @ To @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ord_less @ nat @ I @ ( minus_minus @ nat @ To @ From ) )
=> ( ( nth @ A @ ( slice @ A @ From @ To @ Xs ) @ I )
= ( nth @ A @ Xs @ ( plus_plus @ nat @ From @ I ) ) ) ) ) ) ).
% slice_nth
thf(fact_6749_If__the__inv__into__in__Func,axiom,
! [B: $tType,A: $tType,G2: A > B,C3: set @ A,B2: set @ A,X: A] :
( ( inj_on @ A @ B @ G2 @ C3 )
=> ( ( ord_less_eq @ ( set @ A ) @ C3 @ ( sup_sup @ ( set @ A ) @ B2 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( member @ ( B > A )
@ ^ [I4: B] : ( if @ A @ ( member @ B @ I4 @ ( image @ A @ B @ G2 @ C3 ) ) @ ( the_inv_into @ A @ B @ C3 @ G2 @ I4 ) @ X )
@ ( bNF_Wellorder_Func @ B @ A @ ( top_top @ ( set @ B ) ) @ ( sup_sup @ ( set @ A ) @ B2 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% If_the_inv_into_in_Func
thf(fact_6750_slice__complete,axiom,
! [A: $tType,Xs: list @ A] :
( ( slice @ A @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Xs ) @ Xs )
= Xs ) ).
% slice_complete
thf(fact_6751_Func__empty,axiom,
! [B: $tType,A: $tType,B2: set @ B] :
( ( bNF_Wellorder_Func @ A @ B @ ( bot_bot @ ( set @ A ) ) @ B2 )
= ( insert @ ( A > B )
@ ^ [X2: A] : ( undefined @ B )
@ ( bot_bot @ ( set @ ( A > B ) ) ) ) ) ).
% Func_empty
thf(fact_6752_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_6753_list_Oinject,axiom,
! [A: $tType,X21: A,X222: list @ A,Y21: A,Y222: list @ A] :
( ( ( cons @ A @ X21 @ X222 )
= ( cons @ A @ Y21 @ Y222 ) )
= ( ( X21 = Y21 )
& ( X222 = Y222 ) ) ) ).
% list.inject
thf(fact_6754_nth__Cons__Suc,axiom,
! [A: $tType,X: A,Xs: list @ A,N: nat] :
( ( nth @ A @ ( cons @ A @ X @ Xs ) @ ( suc @ N ) )
= ( nth @ A @ Xs @ N ) ) ).
% nth_Cons_Suc
thf(fact_6755_length__nth__simps_I4_J,axiom,
! [B: $tType,X: B,Xs: list @ B,N: nat] :
( ( nth @ B @ ( cons @ B @ X @ Xs ) @ ( suc @ N ) )
= ( nth @ B @ Xs @ N ) ) ).
% length_nth_simps(4)
thf(fact_6756_length__nth__simps_I3_J,axiom,
! [B: $tType,X: B,Xs: list @ B] :
( ( nth @ B @ ( cons @ B @ X @ Xs ) @ ( zero_zero @ nat ) )
= X ) ).
% length_nth_simps(3)
thf(fact_6757_nth__Cons__0,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( nth @ A @ ( cons @ A @ X @ Xs ) @ ( zero_zero @ nat ) )
= X ) ).
% nth_Cons_0
thf(fact_6758_list_Osimps_I15_J,axiom,
! [A: $tType,X21: A,X222: list @ A] :
( ( set2 @ A @ ( cons @ A @ X21 @ X222 ) )
= ( insert @ A @ X21 @ ( set2 @ A @ X222 ) ) ) ).
% list.simps(15)
thf(fact_6759_nth__Cons__numeral,axiom,
! [A: $tType,X: A,Xs: list @ A,V2: num] :
( ( nth @ A @ ( cons @ A @ X @ Xs ) @ ( numeral_numeral @ nat @ V2 ) )
= ( nth @ A @ Xs @ ( minus_minus @ nat @ ( numeral_numeral @ nat @ V2 ) @ ( one_one @ nat ) ) ) ) ).
% nth_Cons_numeral
thf(fact_6760_nth__Cons__pos,axiom,
! [A: $tType,N: nat,X: A,Xs: list @ A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( nth @ A @ ( cons @ A @ X @ Xs ) @ N )
= ( nth @ A @ Xs @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ).
% nth_Cons_pos
thf(fact_6761_inj__on__Cons1,axiom,
! [A: $tType,X: A,A2: set @ ( list @ A )] : ( inj_on @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X ) @ A2 ) ).
% inj_on_Cons1
thf(fact_6762_inj__split__Cons,axiom,
! [A: $tType,X5: set @ ( product_prod @ ( list @ A ) @ A )] :
( inj_on @ ( product_prod @ ( list @ A ) @ A ) @ ( list @ A )
@ ( product_case_prod @ ( list @ A ) @ A @ ( list @ A )
@ ^ [Xs3: list @ A,N4: A] : ( cons @ A @ N4 @ Xs3 ) )
@ X5 ) ).
% inj_split_Cons
thf(fact_6763_length__Suc__conv,axiom,
! [A: $tType,Xs: list @ A,N: nat] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( suc @ N ) )
= ( ? [Y2: A,Ys3: list @ A] :
( ( Xs
= ( cons @ A @ Y2 @ Ys3 ) )
& ( ( size_size @ ( list @ A ) @ Ys3 )
= N ) ) ) ) ).
% length_Suc_conv
thf(fact_6764_Suc__length__conv,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( ( suc @ N )
= ( size_size @ ( list @ A ) @ Xs ) )
= ( ? [Y2: A,Ys3: list @ A] :
( ( Xs
= ( cons @ A @ Y2 @ Ys3 ) )
& ( ( size_size @ ( list @ A ) @ Ys3 )
= N ) ) ) ) ).
% Suc_length_conv
thf(fact_6765_length__nth__simps_I2_J,axiom,
! [B: $tType,X: B,Xs: list @ B] :
( ( size_size @ ( list @ B ) @ ( cons @ B @ X @ Xs ) )
= ( suc @ ( size_size @ ( list @ B ) @ Xs ) ) ) ).
% length_nth_simps(2)
thf(fact_6766_impossible__Cons,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,X: A] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ A ) @ Ys ) )
=> ( Xs
!= ( cons @ A @ X @ Ys ) ) ) ).
% impossible_Cons
thf(fact_6767_list_Oset__intros_I2_J,axiom,
! [A: $tType,Y: A,X222: list @ A,X21: A] :
( ( member @ A @ Y @ ( set2 @ A @ X222 ) )
=> ( member @ A @ Y @ ( set2 @ A @ ( cons @ A @ X21 @ X222 ) ) ) ) ).
% list.set_intros(2)
thf(fact_6768_list_Oset__intros_I1_J,axiom,
! [A: $tType,X21: A,X222: list @ A] : ( member @ A @ X21 @ ( set2 @ A @ ( cons @ A @ X21 @ X222 ) ) ) ).
% list.set_intros(1)
thf(fact_6769_list_Oset__cases,axiom,
! [A: $tType,E2: A,A3: list @ A] :
( ( member @ A @ E2 @ ( set2 @ A @ A3 ) )
=> ( ! [Z23: list @ A] :
( A3
!= ( cons @ A @ E2 @ Z23 ) )
=> ~ ! [Z12: A,Z23: list @ A] :
( ( A3
= ( cons @ A @ Z12 @ Z23 ) )
=> ~ ( member @ A @ E2 @ ( set2 @ A @ Z23 ) ) ) ) ) ).
% list.set_cases
thf(fact_6770_set__ConsD,axiom,
! [A: $tType,Y: A,X: A,Xs: list @ A] :
( ( member @ A @ Y @ ( set2 @ A @ ( cons @ A @ X @ Xs ) ) )
=> ( ( Y = X )
| ( member @ A @ Y @ ( set2 @ A @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_6771_not__Cons__self2,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( cons @ A @ X @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_6772_set__subset__Cons,axiom,
! [A: $tType,Xs: list @ A,X: A] : ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ ( cons @ A @ X @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_6773_Suc__le__length__iff,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( ord_less_eq @ nat @ ( suc @ N ) @ ( size_size @ ( list @ A ) @ Xs ) )
= ( ? [X2: A,Ys3: list @ A] :
( ( Xs
= ( cons @ A @ X2 @ Ys3 ) )
& ( ord_less_eq @ nat @ N @ ( size_size @ ( list @ A ) @ Ys3 ) ) ) ) ) ).
% Suc_le_length_iff
thf(fact_6774_count__list_Osimps_I2_J,axiom,
! [A: $tType,X: A,Y: A,Xs: list @ A] :
( ( ( X = Y )
=> ( ( count_list @ A @ ( cons @ A @ X @ Xs ) @ Y )
= ( plus_plus @ nat @ ( count_list @ A @ Xs @ Y ) @ ( one_one @ nat ) ) ) )
& ( ( X != Y )
=> ( ( count_list @ A @ ( cons @ A @ X @ Xs ) @ Y )
= ( count_list @ A @ Xs @ Y ) ) ) ) ).
% count_list.simps(2)
thf(fact_6775_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_6776_nth__Cons_H,axiom,
! [A: $tType,N: nat,X: A,Xs: list @ A] :
( ( ( N
= ( zero_zero @ nat ) )
=> ( ( nth @ A @ ( cons @ A @ X @ Xs ) @ N )
= X ) )
& ( ( N
!= ( zero_zero @ nat ) )
=> ( ( nth @ A @ ( cons @ A @ X @ Xs ) @ N )
= ( nth @ A @ Xs @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ) ).
% nth_Cons'
thf(fact_6777_nth__non__equal__first__eq,axiom,
! [A: $tType,X: A,Y: A,Xs: list @ A,N: nat] :
( ( X != Y )
=> ( ( ( nth @ A @ ( cons @ A @ X @ Xs ) @ N )
= Y )
= ( ( ( nth @ A @ Xs @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) )
= Y )
& ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% nth_non_equal_first_eq
thf(fact_6778_nth__equal__first__eq,axiom,
! [A: $tType,X: A,Xs: list @ A,N: nat] :
( ~ ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ( ord_less_eq @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ( nth @ A @ ( cons @ A @ X @ Xs ) @ N )
= X )
= ( N
= ( zero_zero @ nat ) ) ) ) ) ).
% nth_equal_first_eq
thf(fact_6779_slice__Cons,axiom,
! [A: $tType,Begin: nat,End: nat,X: A,Xs: list @ A] :
( ( ( ( Begin
= ( zero_zero @ nat ) )
& ( ord_less @ nat @ ( zero_zero @ nat ) @ End ) )
=> ( ( slice @ A @ Begin @ End @ ( cons @ A @ X @ Xs ) )
= ( cons @ A @ X @ ( slice @ A @ Begin @ ( minus_minus @ nat @ End @ ( one_one @ nat ) ) @ Xs ) ) ) )
& ( ~ ( ( Begin
= ( zero_zero @ nat ) )
& ( ord_less @ nat @ ( zero_zero @ nat ) @ End ) )
=> ( ( slice @ A @ Begin @ End @ ( cons @ A @ X @ Xs ) )
= ( slice @ A @ ( minus_minus @ nat @ Begin @ ( one_one @ nat ) ) @ ( minus_minus @ nat @ End @ ( one_one @ nat ) ) @ Xs ) ) ) ) ).
% slice_Cons
thf(fact_6780_Func__non__emp,axiom,
! [A: $tType,B: $tType,B2: set @ A,A2: set @ B] :
( ( B2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( bNF_Wellorder_Func @ B @ A @ A2 @ B2 )
!= ( bot_bot @ ( set @ ( B > A ) ) ) ) ) ).
% Func_non_emp
thf(fact_6781_Func__is__emp,axiom,
! [A: $tType,B: $tType,A2: set @ A,B2: set @ B] :
( ( ( bNF_Wellorder_Func @ A @ B @ A2 @ B2 )
= ( bot_bot @ ( set @ ( A > B ) ) ) )
= ( ( A2
!= ( bot_bot @ ( set @ A ) ) )
& ( B2
= ( bot_bot @ ( set @ B ) ) ) ) ) ).
% Func_is_emp
thf(fact_6782_hash__code__list__simps_I2_J,axiom,
! [A: $tType,H_a: A > uint32,X: A,Xa: list @ A] :
( ( hash_hash_code_list @ A @ H_a @ ( cons @ A @ X @ Xa ) )
= ( plus_plus @ uint32 @ ( times_times @ uint32 @ ( H_a @ X ) @ ( numeral_numeral @ uint32 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( plus_plus @ uint32 @ ( times_times @ uint32 @ ( hash_hash_code_list @ A @ H_a @ Xa ) @ ( numeral_numeral @ uint32 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numeral_numeral @ uint32 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% hash_code_list_simps(2)
thf(fact_6783_Func__map__surj,axiom,
! [C: $tType,A: $tType,D: $tType,B: $tType,F1: B > A,A13: set @ B,B1: set @ A,F22: C > D,B22: set @ C,A24: set @ D] :
( ( ( image @ B @ A @ F1 @ A13 )
= B1 )
=> ( ( inj_on @ C @ D @ F22 @ B22 )
=> ( ( ord_less_eq @ ( set @ D ) @ ( image @ C @ D @ F22 @ B22 ) @ A24 )
=> ( ( ( B22
= ( bot_bot @ ( set @ C ) ) )
=> ( A24
= ( bot_bot @ ( set @ D ) ) ) )
=> ( ( bNF_Wellorder_Func @ C @ A @ B22 @ B1 )
= ( image @ ( D > B ) @ ( C > A ) @ ( bNF_We4925052301507509544nc_map @ C @ B @ A @ D @ B22 @ F1 @ F22 ) @ ( bNF_Wellorder_Func @ D @ B @ A24 @ A13 ) ) ) ) ) ) ) ).
% Func_map_surj
thf(fact_6784_upto__aux__rec,axiom,
( upto_aux
= ( ^ [I4: int,J3: int,Js: list @ int] : ( if @ ( list @ int ) @ ( ord_less @ int @ J3 @ I4 ) @ Js @ ( upto_aux @ I4 @ ( minus_minus @ int @ J3 @ ( one_one @ int ) ) @ ( cons @ int @ J3 @ Js ) ) ) ) ) ).
% upto_aux_rec
thf(fact_6785_image__Collect__subsetI,axiom,
! [A: $tType,B: $tType,P: A > $o,F3: A > B,B2: set @ B] :
( ! [X3: A] :
( ( P @ X3 )
=> ( member @ B @ ( F3 @ X3 ) @ B2 ) )
=> ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F3 @ ( collect @ A @ P ) ) @ B2 ) ) ).
% image_Collect_subsetI
thf(fact_6786_Collect__restrict,axiom,
! [A: $tType,X5: set @ A,P: A > $o] :
( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [X2: A] :
( ( member @ A @ X2 @ X5 )
& ( P @ X2 ) ) )
@ X5 ) ).
% Collect_restrict
thf(fact_6787_prop__restrict,axiom,
! [A: $tType,X: A,Z9: set @ A,X5: set @ A,P: A > $o] :
( ( member @ A @ X @ Z9 )
=> ( ( ord_less_eq @ ( set @ A ) @ Z9
@ ( collect @ A
@ ^ [X2: A] :
( ( member @ A @ X2 @ X5 )
& ( P @ X2 ) ) ) )
=> ( P @ X ) ) ) ).
% prop_restrict
thf(fact_6788_subset__emptyI,axiom,
! [A: $tType,A2: set @ A] :
( ! [X3: A] :
~ ( member @ A @ X3 @ A2 )
=> ( ord_less_eq @ ( set @ A ) @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% subset_emptyI
thf(fact_6789_insert__subsetI,axiom,
! [A: $tType,X: A,A2: set @ A,X5: set @ A] :
( ( member @ A @ X @ A2 )
=> ( ( ord_less_eq @ ( set @ A ) @ X5 @ A2 )
=> ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X @ X5 ) @ A2 ) ) ) ).
% insert_subsetI
thf(fact_6790_DERIV__even__real__root,axiom,
! [N: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( ord_less @ real @ X @ ( zero_zero @ real ) )
=> ( has_field_derivative @ real @ ( root @ N ) @ ( inverse_inverse @ real @ ( times_times @ real @ ( uminus_uminus @ real @ ( semiring_1_of_nat @ real @ N ) ) @ ( power_power @ real @ ( root @ N @ X ) @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ).
% DERIV_even_real_root
thf(fact_6791_DERIV__real__root__generic,axiom,
! [N: nat,X: real,D3: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( X
!= ( zero_zero @ real ) )
=> ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( D3
= ( inverse_inverse @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ ( power_power @ real @ ( root @ N @ X ) @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ) )
=> ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( ord_less @ real @ X @ ( zero_zero @ real ) )
=> ( D3
= ( uminus_uminus @ real @ ( inverse_inverse @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ ( power_power @ real @ ( root @ N @ X ) @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ) ) )
=> ( ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( D3
= ( inverse_inverse @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ ( power_power @ real @ ( root @ N @ X ) @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) )
=> ( has_field_derivative @ real @ ( root @ N ) @ D3 @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ) ) ).
% DERIV_real_root_generic
thf(fact_6792_at__within__empty,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [A3: A] :
( ( topolo174197925503356063within @ A @ A3 @ ( bot_bot @ ( set @ A ) ) )
= ( bot_bot @ ( filter @ A ) ) ) ) ).
% at_within_empty
thf(fact_6793_at__neq__bot,axiom,
! [A: $tType] :
( ( topolo8386298272705272623_space @ A )
=> ! [A3: A] :
( ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) )
!= ( bot_bot @ ( filter @ A ) ) ) ) ).
% at_neq_bot
thf(fact_6794_trivial__limit__at__left__real,axiom,
! [A: $tType] :
( ( ( dense_order @ A )
& ( no_bot @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [X: A] :
( ( topolo174197925503356063within @ A @ X @ ( set_ord_lessThan @ A @ X ) )
!= ( bot_bot @ ( filter @ A ) ) ) ) ).
% trivial_limit_at_left_real
thf(fact_6795_at__discrete,axiom,
! [A: $tType] :
( ( topolo8865339358273720382pology @ A )
=> ( ( topolo174197925503356063within @ A )
= ( ^ [X2: A,S8: set @ A] : ( bot_bot @ ( filter @ A ) ) ) ) ) ).
% at_discrete
thf(fact_6796_trivial__limit__at__left__bot,axiom,
! [A: $tType] :
( ( ( order_bot @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ( ( topolo174197925503356063within @ A @ ( bot_bot @ A ) @ ( set_ord_lessThan @ A @ ( bot_bot @ A ) ) )
= ( bot_bot @ ( filter @ A ) ) ) ) ).
% trivial_limit_at_left_bot
thf(fact_6797_at__within__union,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [X: A,S: set @ A,T4: set @ A] :
( ( topolo174197925503356063within @ A @ X @ ( sup_sup @ ( set @ A ) @ S @ T4 ) )
= ( sup_sup @ ( filter @ A ) @ ( topolo174197925503356063within @ A @ X @ S ) @ ( topolo174197925503356063within @ A @ X @ T4 ) ) ) ) ).
% at_within_union
thf(fact_6798_has__real__derivative__neg__dec__right,axiom,
! [F3: real > real,L: real,X: real,S: set @ real] :
( ( has_field_derivative @ real @ F3 @ L @ ( topolo174197925503356063within @ real @ X @ S ) )
=> ( ( ord_less @ real @ L @ ( zero_zero @ real ) )
=> ? [D6: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D6 )
& ! [H5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ H5 )
=> ( ( member @ real @ ( plus_plus @ real @ X @ H5 ) @ S )
=> ( ( ord_less @ real @ H5 @ D6 )
=> ( ord_less @ real @ ( F3 @ ( plus_plus @ real @ X @ H5 ) ) @ ( F3 @ X ) ) ) ) ) ) ) ) ).
% has_real_derivative_neg_dec_right
thf(fact_6799_has__real__derivative__pos__inc__right,axiom,
! [F3: real > real,L: real,X: real,S: set @ real] :
( ( has_field_derivative @ real @ F3 @ L @ ( topolo174197925503356063within @ real @ X @ S ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ L )
=> ? [D6: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D6 )
& ! [H5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ H5 )
=> ( ( member @ real @ ( plus_plus @ real @ X @ H5 ) @ S )
=> ( ( ord_less @ real @ H5 @ D6 )
=> ( ord_less @ real @ ( F3 @ X ) @ ( F3 @ ( plus_plus @ real @ X @ H5 ) ) ) ) ) ) ) ) ) ).
% has_real_derivative_pos_inc_right
thf(fact_6800_has__real__derivative__pos__inc__left,axiom,
! [F3: real > real,L: real,X: real,S: set @ real] :
( ( has_field_derivative @ real @ F3 @ L @ ( topolo174197925503356063within @ real @ X @ S ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ L )
=> ? [D6: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D6 )
& ! [H5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ H5 )
=> ( ( member @ real @ ( minus_minus @ real @ X @ H5 ) @ S )
=> ( ( ord_less @ real @ H5 @ D6 )
=> ( ord_less @ real @ ( F3 @ ( minus_minus @ real @ X @ H5 ) ) @ ( F3 @ X ) ) ) ) ) ) ) ) ).
% has_real_derivative_pos_inc_left
thf(fact_6801_has__real__derivative__neg__dec__left,axiom,
! [F3: real > real,L: real,X: real,S: set @ real] :
( ( has_field_derivative @ real @ F3 @ L @ ( topolo174197925503356063within @ real @ X @ S ) )
=> ( ( ord_less @ real @ L @ ( zero_zero @ real ) )
=> ? [D6: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D6 )
& ! [H5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ H5 )
=> ( ( member @ real @ ( minus_minus @ real @ X @ H5 ) @ S )
=> ( ( ord_less @ real @ H5 @ D6 )
=> ( ord_less @ real @ ( F3 @ X ) @ ( F3 @ ( minus_minus @ real @ X @ H5 ) ) ) ) ) ) ) ) ) ).
% has_real_derivative_neg_dec_left
thf(fact_6802_field__differentiable__minus,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,F7: A,F4: filter @ A] :
( ( has_field_derivative @ A @ F3 @ F7 @ F4 )
=> ( has_field_derivative @ A
@ ^ [Z4: A] : ( uminus_uminus @ A @ ( F3 @ Z4 ) )
@ ( uminus_uminus @ A @ F7 )
@ F4 ) ) ) ).
% field_differentiable_minus
thf(fact_6803_DERIV__minus,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,D3: A,X: A,S2: set @ A] :
( ( has_field_derivative @ A @ F3 @ D3 @ ( topolo174197925503356063within @ A @ X @ S2 ) )
=> ( has_field_derivative @ A
@ ^ [X2: A] : ( uminus_uminus @ A @ ( F3 @ X2 ) )
@ ( uminus_uminus @ A @ D3 )
@ ( topolo174197925503356063within @ A @ X @ S2 ) ) ) ) ).
% DERIV_minus
thf(fact_6804_DERIV__sum,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( real_V3459762299906320749_field @ B )
=> ! [S: set @ A,F3: B > A > B,F7: C > A > B,X: C,F4: filter @ B] :
( ! [N2: A] :
( ( member @ A @ N2 @ S )
=> ( has_field_derivative @ B
@ ^ [X2: B] : ( F3 @ X2 @ N2 )
@ ( F7 @ X @ N2 )
@ F4 ) )
=> ( has_field_derivative @ B
@ ^ [X2: B] : ( groups7311177749621191930dd_sum @ A @ B @ ( F3 @ X2 ) @ S )
@ ( groups7311177749621191930dd_sum @ A @ B @ ( F7 @ X ) @ S )
@ F4 ) ) ) ).
% DERIV_sum
thf(fact_6805_DERIV__inverse_H,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,D3: A,X: A,S2: set @ A] :
( ( has_field_derivative @ A @ F3 @ D3 @ ( topolo174197925503356063within @ A @ X @ S2 ) )
=> ( ( ( F3 @ X )
!= ( zero_zero @ A ) )
=> ( has_field_derivative @ A
@ ^ [X2: A] : ( inverse_inverse @ A @ ( F3 @ X2 ) )
@ ( uminus_uminus @ A @ ( times_times @ A @ ( times_times @ A @ ( inverse_inverse @ A @ ( F3 @ X ) ) @ D3 ) @ ( inverse_inverse @ A @ ( F3 @ X ) ) ) )
@ ( topolo174197925503356063within @ A @ X @ S2 ) ) ) ) ) ).
% DERIV_inverse'
thf(fact_6806_DERIV__const,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [K: A,F4: filter @ A] :
( has_field_derivative @ A
@ ^ [X2: A] : K
@ ( zero_zero @ A )
@ F4 ) ) ).
% DERIV_const
thf(fact_6807_has__field__derivative__scaleR__right,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,D3: A,F4: filter @ A,C2: real] :
( ( has_field_derivative @ A @ F3 @ D3 @ F4 )
=> ( has_field_derivative @ A
@ ^ [X2: A] : ( real_V8093663219630862766scaleR @ A @ C2 @ ( F3 @ X2 ) )
@ ( real_V8093663219630862766scaleR @ A @ C2 @ D3 )
@ F4 ) ) ) ).
% has_field_derivative_scaleR_right
thf(fact_6808_DERIV__ident,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F4: filter @ A] :
( has_field_derivative @ A
@ ^ [X2: A] : X2
@ ( one_one @ A )
@ F4 ) ) ).
% DERIV_ident
thf(fact_6809_DERIV__cmult,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,D3: A,X: A,S2: set @ A,C2: A] :
( ( has_field_derivative @ A @ F3 @ D3 @ ( topolo174197925503356063within @ A @ X @ S2 ) )
=> ( has_field_derivative @ A
@ ^ [X2: A] : ( times_times @ A @ C2 @ ( F3 @ X2 ) )
@ ( times_times @ A @ C2 @ D3 )
@ ( topolo174197925503356063within @ A @ X @ S2 ) ) ) ) ).
% DERIV_cmult
thf(fact_6810_DERIV__cmult__right,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,D3: A,X: A,S2: set @ A,C2: A] :
( ( has_field_derivative @ A @ F3 @ D3 @ ( topolo174197925503356063within @ A @ X @ S2 ) )
=> ( has_field_derivative @ A
@ ^ [X2: A] : ( times_times @ A @ ( F3 @ X2 ) @ C2 )
@ ( times_times @ A @ D3 @ C2 )
@ ( topolo174197925503356063within @ A @ X @ S2 ) ) ) ) ).
% DERIV_cmult_right
thf(fact_6811_has__field__derivative__cosh,axiom,
! [A11: $tType] :
( ( ( real_Vector_banach @ A11 )
& ( real_V3459762299906320749_field @ A11 ) )
=> ! [G2: A11 > A11,Db: A11,X: A11,S2: set @ A11] :
( ( has_field_derivative @ A11 @ G2 @ Db @ ( topolo174197925503356063within @ A11 @ X @ S2 ) )
=> ( has_field_derivative @ A11
@ ^ [X2: A11] : ( cosh @ A11 @ ( G2 @ X2 ) )
@ ( times_times @ A11 @ ( sinh @ A11 @ ( G2 @ X ) ) @ Db )
@ ( topolo174197925503356063within @ A11 @ X @ S2 ) ) ) ) ).
% has_field_derivative_cosh
thf(fact_6812_has__field__derivative__sinh,axiom,
! [A11: $tType] :
( ( ( real_Vector_banach @ A11 )
& ( real_V3459762299906320749_field @ A11 ) )
=> ! [G2: A11 > A11,Db: A11,X: A11,S2: set @ A11] :
( ( has_field_derivative @ A11 @ G2 @ Db @ ( topolo174197925503356063within @ A11 @ X @ S2 ) )
=> ( has_field_derivative @ A11
@ ^ [X2: A11] : ( sinh @ A11 @ ( G2 @ X2 ) )
@ ( times_times @ A11 @ ( cosh @ A11 @ ( G2 @ X ) ) @ Db )
@ ( topolo174197925503356063within @ A11 @ X @ S2 ) ) ) ) ).
% has_field_derivative_sinh
thf(fact_6813_DERIV__cdivide,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,D3: A,X: A,S2: set @ A,C2: A] :
( ( has_field_derivative @ A @ F3 @ D3 @ ( topolo174197925503356063within @ A @ X @ S2 ) )
=> ( has_field_derivative @ A
@ ^ [X2: A] : ( divide_divide @ A @ ( F3 @ X2 ) @ C2 )
@ ( divide_divide @ A @ D3 @ C2 )
@ ( topolo174197925503356063within @ A @ X @ S2 ) ) ) ) ).
% DERIV_cdivide
thf(fact_6814_field__differentiable__diff,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,F7: A,F4: filter @ A,G2: A > A,G5: A] :
( ( has_field_derivative @ A @ F3 @ F7 @ F4 )
=> ( ( has_field_derivative @ A @ G2 @ G5 @ F4 )
=> ( has_field_derivative @ A
@ ^ [Z4: A] : ( minus_minus @ A @ ( F3 @ Z4 ) @ ( G2 @ Z4 ) )
@ ( minus_minus @ A @ F7 @ G5 )
@ F4 ) ) ) ) ).
% field_differentiable_diff
thf(fact_6815_DERIV__diff,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,D3: A,X: A,S2: set @ A,G2: A > A,E5: A] :
( ( has_field_derivative @ A @ F3 @ D3 @ ( topolo174197925503356063within @ A @ X @ S2 ) )
=> ( ( has_field_derivative @ A @ G2 @ E5 @ ( topolo174197925503356063within @ A @ X @ S2 ) )
=> ( has_field_derivative @ A
@ ^ [X2: A] : ( minus_minus @ A @ ( F3 @ X2 ) @ ( G2 @ X2 ) )
@ ( minus_minus @ A @ D3 @ E5 )
@ ( topolo174197925503356063within @ A @ X @ S2 ) ) ) ) ) ).
% DERIV_diff
thf(fact_6816_DERIV__mult,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,Da: A,X: A,S2: set @ A,G2: A > A,Db: A] :
( ( has_field_derivative @ A @ F3 @ Da @ ( topolo174197925503356063within @ A @ X @ S2 ) )
=> ( ( has_field_derivative @ A @ G2 @ Db @ ( topolo174197925503356063within @ A @ X @ S2 ) )
=> ( has_field_derivative @ A
@ ^ [X2: A] : ( times_times @ A @ ( F3 @ X2 ) @ ( G2 @ X2 ) )
@ ( plus_plus @ A @ ( times_times @ A @ Da @ ( G2 @ X ) ) @ ( times_times @ A @ Db @ ( F3 @ X ) ) )
@ ( topolo174197925503356063within @ A @ X @ S2 ) ) ) ) ) ).
% DERIV_mult
thf(fact_6817_DERIV__mult_H,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,D3: A,X: A,S2: set @ A,G2: A > A,E5: A] :
( ( has_field_derivative @ A @ F3 @ D3 @ ( topolo174197925503356063within @ A @ X @ S2 ) )
=> ( ( has_field_derivative @ A @ G2 @ E5 @ ( topolo174197925503356063within @ A @ X @ S2 ) )
=> ( has_field_derivative @ A
@ ^ [X2: A] : ( times_times @ A @ ( F3 @ X2 ) @ ( G2 @ X2 ) )
@ ( plus_plus @ A @ ( times_times @ A @ ( F3 @ X ) @ E5 ) @ ( times_times @ A @ D3 @ ( G2 @ X ) ) )
@ ( topolo174197925503356063within @ A @ X @ S2 ) ) ) ) ) ).
% DERIV_mult'
thf(fact_6818_field__differentiable__add,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,F7: A,F4: filter @ A,G2: A > A,G5: A] :
( ( has_field_derivative @ A @ F3 @ F7 @ F4 )
=> ( ( has_field_derivative @ A @ G2 @ G5 @ F4 )
=> ( has_field_derivative @ A
@ ^ [Z4: A] : ( plus_plus @ A @ ( F3 @ Z4 ) @ ( G2 @ Z4 ) )
@ ( plus_plus @ A @ F7 @ G5 )
@ F4 ) ) ) ) ).
% field_differentiable_add
thf(fact_6819_DERIV__add,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,D3: A,X: A,S2: set @ A,G2: A > A,E5: A] :
( ( has_field_derivative @ A @ F3 @ D3 @ ( topolo174197925503356063within @ A @ X @ S2 ) )
=> ( ( has_field_derivative @ A @ G2 @ E5 @ ( topolo174197925503356063within @ A @ X @ S2 ) )
=> ( has_field_derivative @ A
@ ^ [X2: A] : ( plus_plus @ A @ ( F3 @ X2 ) @ ( G2 @ X2 ) )
@ ( plus_plus @ A @ D3 @ E5 )
@ ( topolo174197925503356063within @ A @ X @ S2 ) ) ) ) ) ).
% DERIV_add
thf(fact_6820_DERIV__divide,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,D3: A,X: A,S2: set @ A,G2: A > A,E5: A] :
( ( has_field_derivative @ A @ F3 @ D3 @ ( topolo174197925503356063within @ A @ X @ S2 ) )
=> ( ( has_field_derivative @ A @ G2 @ E5 @ ( topolo174197925503356063within @ A @ X @ S2 ) )
=> ( ( ( G2 @ X )
!= ( zero_zero @ A ) )
=> ( has_field_derivative @ A
@ ^ [X2: A] : ( divide_divide @ A @ ( F3 @ X2 ) @ ( G2 @ X2 ) )
@ ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ D3 @ ( G2 @ X ) ) @ ( times_times @ A @ ( F3 @ X ) @ E5 ) ) @ ( times_times @ A @ ( G2 @ X ) @ ( G2 @ X ) ) )
@ ( topolo174197925503356063within @ A @ X @ S2 ) ) ) ) ) ) ).
% DERIV_divide
thf(fact_6821_MVT2,axiom,
! [A3: real,B3: real,F3: real > real,F7: real > real] :
( ( ord_less @ real @ A3 @ B3 )
=> ( ! [X3: real] :
( ( ord_less_eq @ real @ A3 @ X3 )
=> ( ( ord_less_eq @ real @ X3 @ B3 )
=> ( has_field_derivative @ real @ F3 @ ( F7 @ X3 ) @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) ) ) )
=> ? [Z3: real] :
( ( ord_less @ real @ A3 @ Z3 )
& ( ord_less @ real @ Z3 @ B3 )
& ( ( minus_minus @ real @ ( F3 @ B3 ) @ ( F3 @ A3 ) )
= ( times_times @ real @ ( minus_minus @ real @ B3 @ A3 ) @ ( F7 @ Z3 ) ) ) ) ) ) ).
% MVT2
thf(fact_6822_DERIV__neg__imp__decreasing,axiom,
! [A3: real,B3: real,F3: real > real] :
( ( ord_less @ real @ A3 @ B3 )
=> ( ! [X3: real] :
( ( ord_less_eq @ real @ A3 @ X3 )
=> ( ( ord_less_eq @ real @ X3 @ B3 )
=> ? [Y5: real] :
( ( has_field_derivative @ real @ F3 @ Y5 @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) )
& ( ord_less @ real @ Y5 @ ( zero_zero @ real ) ) ) ) )
=> ( ord_less @ real @ ( F3 @ B3 ) @ ( F3 @ A3 ) ) ) ) ).
% DERIV_neg_imp_decreasing
thf(fact_6823_DERIV__pos__imp__increasing,axiom,
! [A3: real,B3: real,F3: real > real] :
( ( ord_less @ real @ A3 @ B3 )
=> ( ! [X3: real] :
( ( ord_less_eq @ real @ A3 @ X3 )
=> ( ( ord_less_eq @ real @ X3 @ B3 )
=> ? [Y5: real] :
( ( has_field_derivative @ real @ F3 @ Y5 @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) )
& ( ord_less @ real @ ( zero_zero @ real ) @ Y5 ) ) ) )
=> ( ord_less @ real @ ( F3 @ A3 ) @ ( F3 @ B3 ) ) ) ) ).
% DERIV_pos_imp_increasing
thf(fact_6824_DERIV__nonneg__imp__nondecreasing,axiom,
! [A3: real,B3: real,F3: real > real] :
( ( ord_less_eq @ real @ A3 @ B3 )
=> ( ! [X3: real] :
( ( ord_less_eq @ real @ A3 @ X3 )
=> ( ( ord_less_eq @ real @ X3 @ B3 )
=> ? [Y5: real] :
( ( has_field_derivative @ real @ F3 @ Y5 @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) )
& ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y5 ) ) ) )
=> ( ord_less_eq @ real @ ( F3 @ A3 ) @ ( F3 @ B3 ) ) ) ) ).
% DERIV_nonneg_imp_nondecreasing
thf(fact_6825_DERIV__nonpos__imp__nonincreasing,axiom,
! [A3: real,B3: real,F3: real > real] :
( ( ord_less_eq @ real @ A3 @ B3 )
=> ( ! [X3: real] :
( ( ord_less_eq @ real @ A3 @ X3 )
=> ( ( ord_less_eq @ real @ X3 @ B3 )
=> ? [Y5: real] :
( ( has_field_derivative @ real @ F3 @ Y5 @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) )
& ( ord_less_eq @ real @ Y5 @ ( zero_zero @ real ) ) ) ) )
=> ( ord_less_eq @ real @ ( F3 @ B3 ) @ ( F3 @ A3 ) ) ) ) ).
% DERIV_nonpos_imp_nonincreasing
thf(fact_6826_DERIV__shift,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,Y: A,X: A,Z: A] :
( ( has_field_derivative @ A @ F3 @ Y @ ( topolo174197925503356063within @ A @ ( plus_plus @ A @ X @ Z ) @ ( top_top @ ( set @ A ) ) ) )
= ( has_field_derivative @ A
@ ^ [X2: A] : ( F3 @ ( plus_plus @ A @ X2 @ Z ) )
@ Y
@ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% DERIV_shift
thf(fact_6827_DERIV__fun__exp,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [G2: A > A,M: A,X: A] :
( ( has_field_derivative @ A @ G2 @ M @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) )
=> ( has_field_derivative @ A
@ ^ [X2: A] : ( exp @ A @ ( G2 @ X2 ) )
@ ( times_times @ A @ ( exp @ A @ ( G2 @ X ) ) @ M )
@ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% DERIV_fun_exp
thf(fact_6828_DERIV__fun__sin,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [G2: A > A,M: A,X: A] :
( ( has_field_derivative @ A @ G2 @ M @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) )
=> ( has_field_derivative @ A
@ ^ [X2: A] : ( sin @ A @ ( G2 @ X2 ) )
@ ( times_times @ A @ ( cos @ A @ ( G2 @ X ) ) @ M )
@ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% DERIV_fun_sin
thf(fact_6829_DERIV__chain__s,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [S2: set @ A,G2: A > A,G5: A > A,F3: A > A,F7: A,X: A] :
( ! [X3: A] :
( ( member @ A @ X3 @ S2 )
=> ( has_field_derivative @ A @ G2 @ ( G5 @ X3 ) @ ( topolo174197925503356063within @ A @ X3 @ ( top_top @ ( set @ A ) ) ) ) )
=> ( ( has_field_derivative @ A @ F3 @ F7 @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) )
=> ( ( member @ A @ ( F3 @ X ) @ S2 )
=> ( has_field_derivative @ A
@ ^ [X2: A] : ( G2 @ ( F3 @ X2 ) )
@ ( times_times @ A @ F7 @ ( G5 @ ( F3 @ X ) ) )
@ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ).
% DERIV_chain_s
thf(fact_6830_DERIV__chain3,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [G2: A > A,G5: A > A,F3: A > A,F7: A,X: A] :
( ! [X3: A] : ( has_field_derivative @ A @ G2 @ ( G5 @ X3 ) @ ( topolo174197925503356063within @ A @ X3 @ ( top_top @ ( set @ A ) ) ) )
=> ( ( has_field_derivative @ A @ F3 @ F7 @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) )
=> ( has_field_derivative @ A
@ ^ [X2: A] : ( G2 @ ( F3 @ X2 ) )
@ ( times_times @ A @ F7 @ ( G5 @ ( F3 @ X ) ) )
@ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% DERIV_chain3
thf(fact_6831_DERIV__chain2,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,Da: A,G2: A > A,X: A,Db: A,S2: set @ A] :
( ( has_field_derivative @ A @ F3 @ Da @ ( topolo174197925503356063within @ A @ ( G2 @ X ) @ ( top_top @ ( set @ A ) ) ) )
=> ( ( has_field_derivative @ A @ G2 @ Db @ ( topolo174197925503356063within @ A @ X @ S2 ) )
=> ( has_field_derivative @ A
@ ^ [X2: A] : ( F3 @ ( G2 @ X2 ) )
@ ( times_times @ A @ Da @ Db )
@ ( topolo174197925503356063within @ A @ X @ S2 ) ) ) ) ) ).
% DERIV_chain2
thf(fact_6832_DERIV__chain_H,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,D3: A,X: A,S2: set @ A,G2: A > A,E5: A] :
( ( has_field_derivative @ A @ F3 @ D3 @ ( topolo174197925503356063within @ A @ X @ S2 ) )
=> ( ( has_field_derivative @ A @ G2 @ E5 @ ( topolo174197925503356063within @ A @ ( F3 @ X ) @ ( top_top @ ( set @ A ) ) ) )
=> ( has_field_derivative @ A
@ ^ [X2: A] : ( G2 @ ( F3 @ X2 ) )
@ ( times_times @ A @ E5 @ D3 )
@ ( topolo174197925503356063within @ A @ X @ S2 ) ) ) ) ) ).
% DERIV_chain'
thf(fact_6833_DERIV__isconst__all,axiom,
! [F3: real > real,X: real,Y: real] :
( ! [X3: real] : ( has_field_derivative @ real @ F3 @ ( zero_zero @ real ) @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( F3 @ X )
= ( F3 @ Y ) ) ) ).
% DERIV_isconst_all
thf(fact_6834_DERIV__ln,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( has_field_derivative @ real @ ( ln_ln @ real ) @ ( inverse_inverse @ real @ X ) @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) ) ) ).
% DERIV_ln
thf(fact_6835_DERIV__mirror,axiom,
! [F3: real > real,Y: real,X: real] :
( ( has_field_derivative @ real @ F3 @ Y @ ( topolo174197925503356063within @ real @ ( uminus_uminus @ real @ X ) @ ( top_top @ ( set @ real ) ) ) )
= ( has_field_derivative @ real
@ ^ [X2: real] : ( F3 @ ( uminus_uminus @ real @ X2 ) )
@ ( uminus_uminus @ real @ Y )
@ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) ) ) ).
% DERIV_mirror
thf(fact_6836_DERIV__fun__cos,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [G2: A > A,M: A,X: A] :
( ( has_field_derivative @ A @ G2 @ M @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) )
=> ( has_field_derivative @ A
@ ^ [X2: A] : ( cos @ A @ ( G2 @ X2 ) )
@ ( times_times @ A @ ( uminus_uminus @ A @ ( sin @ A @ ( G2 @ X ) ) ) @ M )
@ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% DERIV_fun_cos
thf(fact_6837_DERIV__cos__add,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [K: A,Xa: A] :
( has_field_derivative @ A
@ ^ [X2: A] : ( cos @ A @ ( plus_plus @ A @ X2 @ K ) )
@ ( uminus_uminus @ A @ ( sin @ A @ ( plus_plus @ A @ Xa @ K ) ) )
@ ( topolo174197925503356063within @ A @ Xa @ ( top_top @ ( set @ A ) ) ) ) ) ).
% DERIV_cos_add
thf(fact_6838_DERIV__pos__inc__right,axiom,
! [F3: real > real,L: real,X: real] :
( ( has_field_derivative @ real @ F3 @ L @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ L )
=> ? [D6: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D6 )
& ! [H5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ H5 )
=> ( ( ord_less @ real @ H5 @ D6 )
=> ( ord_less @ real @ ( F3 @ X ) @ ( F3 @ ( plus_plus @ real @ X @ H5 ) ) ) ) ) ) ) ) ).
% DERIV_pos_inc_right
thf(fact_6839_DERIV__neg__dec__right,axiom,
! [F3: real > real,L: real,X: real] :
( ( has_field_derivative @ real @ F3 @ L @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
=> ( ( ord_less @ real @ L @ ( zero_zero @ real ) )
=> ? [D6: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D6 )
& ! [H5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ H5 )
=> ( ( ord_less @ real @ H5 @ D6 )
=> ( ord_less @ real @ ( F3 @ ( plus_plus @ real @ X @ H5 ) ) @ ( F3 @ X ) ) ) ) ) ) ) ).
% DERIV_neg_dec_right
thf(fact_6840_DERIV__local__const,axiom,
! [F3: real > real,L: real,X: real,D2: real] :
( ( has_field_derivative @ real @ F3 @ L @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ D2 )
=> ( ! [Y4: real] :
( ( ord_less @ real @ ( abs_abs @ real @ ( minus_minus @ real @ X @ Y4 ) ) @ D2 )
=> ( ( F3 @ X )
= ( F3 @ Y4 ) ) )
=> ( L
= ( zero_zero @ real ) ) ) ) ) ).
% DERIV_local_const
thf(fact_6841_DERIV__pos__inc__left,axiom,
! [F3: real > real,L: real,X: real] :
( ( has_field_derivative @ real @ F3 @ L @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ L )
=> ? [D6: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D6 )
& ! [H5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ H5 )
=> ( ( ord_less @ real @ H5 @ D6 )
=> ( ord_less @ real @ ( F3 @ ( minus_minus @ real @ X @ H5 ) ) @ ( F3 @ X ) ) ) ) ) ) ) ).
% DERIV_pos_inc_left
thf(fact_6842_DERIV__neg__dec__left,axiom,
! [F3: real > real,L: real,X: real] :
( ( has_field_derivative @ real @ F3 @ L @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
=> ( ( ord_less @ real @ L @ ( zero_zero @ real ) )
=> ? [D6: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D6 )
& ! [H5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ H5 )
=> ( ( ord_less @ real @ H5 @ D6 )
=> ( ord_less @ real @ ( F3 @ X ) @ ( F3 @ ( minus_minus @ real @ X @ H5 ) ) ) ) ) ) ) ) ).
% DERIV_neg_dec_left
thf(fact_6843_deriv__nonneg__imp__mono,axiom,
! [A3: real,B3: real,G2: real > real,G5: real > real] :
( ! [X3: real] :
( ( member @ real @ X3 @ ( set_or1337092689740270186AtMost @ real @ A3 @ B3 ) )
=> ( has_field_derivative @ real @ G2 @ ( G5 @ X3 ) @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) ) )
=> ( ! [X3: real] :
( ( member @ real @ X3 @ ( set_or1337092689740270186AtMost @ real @ A3 @ B3 ) )
=> ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( G5 @ X3 ) ) )
=> ( ( ord_less_eq @ real @ A3 @ B3 )
=> ( ord_less_eq @ real @ ( G2 @ A3 ) @ ( G2 @ B3 ) ) ) ) ) ).
% deriv_nonneg_imp_mono
thf(fact_6844_DERIV__at__within__shift,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,Y: A,Z: A,X: A,S: set @ A] :
( ( has_field_derivative @ A @ F3 @ Y @ ( topolo174197925503356063within @ A @ ( plus_plus @ A @ Z @ X ) @ ( image @ A @ A @ ( plus_plus @ A @ Z ) @ S ) ) )
= ( has_field_derivative @ A
@ ^ [X2: A] : ( F3 @ ( plus_plus @ A @ Z @ X2 ) )
@ Y
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ).
% DERIV_at_within_shift
thf(fact_6845_DERIV__power__Suc,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,D3: A,X: A,S2: set @ A,N: nat] :
( ( has_field_derivative @ A @ F3 @ D3 @ ( topolo174197925503356063within @ A @ X @ S2 ) )
=> ( has_field_derivative @ A
@ ^ [X2: A] : ( power_power @ A @ ( F3 @ X2 ) @ ( suc @ N ) )
@ ( times_times @ A @ ( plus_plus @ A @ ( one_one @ A ) @ ( semiring_1_of_nat @ A @ N ) ) @ ( times_times @ A @ D3 @ ( power_power @ A @ ( F3 @ X ) @ N ) ) )
@ ( topolo174197925503356063within @ A @ X @ S2 ) ) ) ) ).
% DERIV_power_Suc
thf(fact_6846_DERIV__const__average,axiom,
! [A3: real,B3: real,V2: real > real,K: real] :
( ( A3 != B3 )
=> ( ! [X3: real] : ( has_field_derivative @ real @ V2 @ K @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( V2 @ ( divide_divide @ real @ ( plus_plus @ real @ A3 @ B3 ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
= ( divide_divide @ real @ ( plus_plus @ real @ ( V2 @ A3 ) @ ( V2 @ B3 ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ).
% DERIV_const_average
thf(fact_6847_DERIV__power,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,D3: A,X: A,S2: set @ A,N: nat] :
( ( has_field_derivative @ A @ F3 @ D3 @ ( topolo174197925503356063within @ A @ X @ S2 ) )
=> ( has_field_derivative @ A
@ ^ [X2: A] : ( power_power @ A @ ( F3 @ X2 ) @ N )
@ ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ ( times_times @ A @ D3 @ ( power_power @ A @ ( F3 @ X ) @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) ) ) )
@ ( topolo174197925503356063within @ A @ X @ S2 ) ) ) ) ).
% DERIV_power
thf(fact_6848_DERIV__inverse,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [X: A,S2: set @ A] :
( ( X
!= ( zero_zero @ A ) )
=> ( has_field_derivative @ A @ ( inverse_inverse @ A ) @ ( uminus_uminus @ A @ ( power_power @ A @ ( inverse_inverse @ A @ X ) @ ( suc @ ( suc @ ( zero_zero @ nat ) ) ) ) ) @ ( topolo174197925503356063within @ A @ X @ S2 ) ) ) ) ).
% DERIV_inverse
thf(fact_6849_DERIV__local__min,axiom,
! [F3: real > real,L: real,X: real,D2: real] :
( ( has_field_derivative @ real @ F3 @ L @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ D2 )
=> ( ! [Y4: real] :
( ( ord_less @ real @ ( abs_abs @ real @ ( minus_minus @ real @ X @ Y4 ) ) @ D2 )
=> ( ord_less_eq @ real @ ( F3 @ X ) @ ( F3 @ Y4 ) ) )
=> ( L
= ( zero_zero @ real ) ) ) ) ) ).
% DERIV_local_min
thf(fact_6850_DERIV__local__max,axiom,
! [F3: real > real,L: real,X: real,D2: real] :
( ( has_field_derivative @ real @ F3 @ L @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ D2 )
=> ( ! [Y4: real] :
( ( ord_less @ real @ ( abs_abs @ real @ ( minus_minus @ real @ X @ Y4 ) ) @ D2 )
=> ( ord_less_eq @ real @ ( F3 @ Y4 ) @ ( F3 @ X ) ) )
=> ( L
= ( zero_zero @ real ) ) ) ) ) ).
% DERIV_local_max
thf(fact_6851_DERIV__ln__divide,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( has_field_derivative @ real @ ( ln_ln @ real ) @ ( divide_divide @ real @ ( one_one @ real ) @ X ) @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) ) ) ).
% DERIV_ln_divide
thf(fact_6852_DERIV__pow,axiom,
! [N: nat,X: real,S2: set @ real] :
( has_field_derivative @ real
@ ^ [X2: real] : ( power_power @ real @ X2 @ N )
@ ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ ( power_power @ real @ X @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) ) )
@ ( topolo174197925503356063within @ real @ X @ S2 ) ) ).
% DERIV_pow
thf(fact_6853_termdiffs__strong__converges__everywhere,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [C2: nat > A,X: A] :
( ! [Y4: A] :
( summable @ A
@ ^ [N4: nat] : ( times_times @ A @ ( C2 @ N4 ) @ ( power_power @ A @ Y4 @ N4 ) ) )
=> ( has_field_derivative @ A
@ ^ [X2: A] :
( suminf @ A
@ ^ [N4: nat] : ( times_times @ A @ ( C2 @ N4 ) @ ( power_power @ A @ X2 @ N4 ) ) )
@ ( suminf @ A
@ ^ [N4: nat] : ( times_times @ A @ ( diffs @ A @ C2 @ N4 ) @ ( power_power @ A @ X @ N4 ) ) )
@ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% termdiffs_strong_converges_everywhere
thf(fact_6854_at__within__Icc__at,axiom,
! [A: $tType] :
( ( topolo2564578578187576103pology @ A )
=> ! [A3: A,X: A,B3: A] :
( ( ord_less @ A @ A3 @ X )
=> ( ( ord_less @ A @ X @ B3 )
=> ( ( topolo174197925503356063within @ A @ X @ ( set_or1337092689740270186AtMost @ A @ A3 @ B3 ) )
= ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% at_within_Icc_at
thf(fact_6855_DERIV__fun__pow,axiom,
! [G2: real > real,M: real,X: real,N: nat] :
( ( has_field_derivative @ real @ G2 @ M @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
=> ( has_field_derivative @ real
@ ^ [X2: real] : ( power_power @ real @ ( G2 @ X2 ) @ N )
@ ( times_times @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ ( power_power @ real @ ( G2 @ X ) @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) @ M )
@ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) ) ) ).
% DERIV_fun_pow
thf(fact_6856_at__within__Icc__at__left,axiom,
! [A: $tType] :
( ( topolo2564578578187576103pology @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ A3 @ B3 )
=> ( ( topolo174197925503356063within @ A @ B3 @ ( set_or1337092689740270186AtMost @ A @ A3 @ B3 ) )
= ( topolo174197925503356063within @ A @ B3 @ ( set_ord_lessThan @ A @ B3 ) ) ) ) ) ).
% at_within_Icc_at_left
thf(fact_6857_DERIV__quotient,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,D2: A,X: A,S2: set @ A,G2: A > A,E2: A] :
( ( has_field_derivative @ A @ F3 @ D2 @ ( topolo174197925503356063within @ A @ X @ S2 ) )
=> ( ( has_field_derivative @ A @ G2 @ E2 @ ( topolo174197925503356063within @ A @ X @ S2 ) )
=> ( ( ( G2 @ X )
!= ( zero_zero @ A ) )
=> ( has_field_derivative @ A
@ ^ [Y2: A] : ( divide_divide @ A @ ( F3 @ Y2 ) @ ( G2 @ Y2 ) )
@ ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ D2 @ ( G2 @ X ) ) @ ( times_times @ A @ E2 @ ( F3 @ X ) ) ) @ ( power_power @ A @ ( G2 @ X ) @ ( suc @ ( suc @ ( zero_zero @ nat ) ) ) ) )
@ ( topolo174197925503356063within @ A @ X @ S2 ) ) ) ) ) ) ).
% DERIV_quotient
thf(fact_6858_DERIV__inverse__fun,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,D2: A,X: A,S2: set @ A] :
( ( has_field_derivative @ A @ F3 @ D2 @ ( topolo174197925503356063within @ A @ X @ S2 ) )
=> ( ( ( F3 @ X )
!= ( zero_zero @ A ) )
=> ( has_field_derivative @ A
@ ^ [X2: A] : ( inverse_inverse @ A @ ( F3 @ X2 ) )
@ ( uminus_uminus @ A @ ( times_times @ A @ D2 @ ( inverse_inverse @ A @ ( power_power @ A @ ( F3 @ X ) @ ( suc @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) )
@ ( topolo174197925503356063within @ A @ X @ S2 ) ) ) ) ) ).
% DERIV_inverse_fun
thf(fact_6859_termdiffs__sums__strong,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [K4: real,C2: nat > A,F3: A > A,F7: A,Z: A] :
( ! [Z3: A] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ Z3 ) @ K4 )
=> ( sums @ A
@ ^ [N4: nat] : ( times_times @ A @ ( C2 @ N4 ) @ ( power_power @ A @ Z3 @ N4 ) )
@ ( F3 @ Z3 ) ) )
=> ( ( has_field_derivative @ A @ F3 @ F7 @ ( topolo174197925503356063within @ A @ Z @ ( top_top @ ( set @ A ) ) ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ Z ) @ K4 )
=> ( sums @ A
@ ^ [N4: nat] : ( times_times @ A @ ( diffs @ A @ C2 @ N4 ) @ ( power_power @ A @ Z @ N4 ) )
@ F7 ) ) ) ) ) ).
% termdiffs_sums_strong
thf(fact_6860_has__real__derivative__powr,axiom,
! [Z: real,R3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ Z )
=> ( has_field_derivative @ real
@ ^ [Z4: real] : ( powr @ real @ Z4 @ R3 )
@ ( times_times @ real @ R3 @ ( powr @ real @ Z @ ( minus_minus @ real @ R3 @ ( one_one @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ Z @ ( top_top @ ( set @ real ) ) ) ) ) ).
% has_real_derivative_powr
thf(fact_6861_termdiffs,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [C2: nat > A,K4: A,X: A] :
( ( summable @ A
@ ^ [N4: nat] : ( times_times @ A @ ( C2 @ N4 ) @ ( power_power @ A @ K4 @ N4 ) ) )
=> ( ( summable @ A
@ ^ [N4: nat] : ( times_times @ A @ ( diffs @ A @ C2 @ N4 ) @ ( power_power @ A @ K4 @ N4 ) ) )
=> ( ( summable @ A
@ ^ [N4: nat] : ( times_times @ A @ ( diffs @ A @ ( diffs @ A @ C2 ) @ N4 ) @ ( power_power @ A @ K4 @ N4 ) ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ ( real_V7770717601297561774m_norm @ A @ K4 ) )
=> ( has_field_derivative @ A
@ ^ [X2: A] :
( suminf @ A
@ ^ [N4: nat] : ( times_times @ A @ ( C2 @ N4 ) @ ( power_power @ A @ X2 @ N4 ) ) )
@ ( suminf @ A
@ ^ [N4: nat] : ( times_times @ A @ ( diffs @ A @ C2 @ N4 ) @ ( power_power @ A @ X @ N4 ) ) )
@ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ) ).
% termdiffs
thf(fact_6862_termdiffs__strong,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [C2: nat > A,K4: A,X: A] :
( ( summable @ A
@ ^ [N4: nat] : ( times_times @ A @ ( C2 @ N4 ) @ ( power_power @ A @ K4 @ N4 ) ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ ( real_V7770717601297561774m_norm @ A @ K4 ) )
=> ( has_field_derivative @ A
@ ^ [X2: A] :
( suminf @ A
@ ^ [N4: nat] : ( times_times @ A @ ( C2 @ N4 ) @ ( power_power @ A @ X2 @ N4 ) ) )
@ ( suminf @ A
@ ^ [N4: nat] : ( times_times @ A @ ( diffs @ A @ C2 @ N4 ) @ ( power_power @ A @ X @ N4 ) ) )
@ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% termdiffs_strong
thf(fact_6863_termdiffs__strong_H,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [K4: real,C2: nat > A,Z: A] :
( ! [Z3: A] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ Z3 ) @ K4 )
=> ( summable @ A
@ ^ [N4: nat] : ( times_times @ A @ ( C2 @ N4 ) @ ( power_power @ A @ Z3 @ N4 ) ) ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ Z ) @ K4 )
=> ( has_field_derivative @ A
@ ^ [Z4: A] :
( suminf @ A
@ ^ [N4: nat] : ( times_times @ A @ ( C2 @ N4 ) @ ( power_power @ A @ Z4 @ N4 ) ) )
@ ( suminf @ A
@ ^ [N4: nat] : ( times_times @ A @ ( diffs @ A @ C2 @ N4 ) @ ( power_power @ A @ Z @ N4 ) ) )
@ ( topolo174197925503356063within @ A @ Z @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% termdiffs_strong'
thf(fact_6864_DERIV__fun__powr,axiom,
! [G2: real > real,M: real,X: real,R3: real] :
( ( has_field_derivative @ real @ G2 @ M @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( G2 @ X ) )
=> ( has_field_derivative @ real
@ ^ [X2: real] : ( powr @ real @ ( G2 @ X2 ) @ R3 )
@ ( times_times @ real @ ( times_times @ real @ R3 @ ( powr @ real @ ( G2 @ X ) @ ( minus_minus @ real @ R3 @ ( semiring_1_of_nat @ real @ ( one_one @ nat ) ) ) ) ) @ M )
@ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) ) ) ) ).
% DERIV_fun_powr
thf(fact_6865_DERIV__log,axiom,
! [X: real,B3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( has_field_derivative @ real @ ( log @ B3 ) @ ( divide_divide @ real @ ( one_one @ real ) @ ( times_times @ real @ ( ln_ln @ real @ B3 ) @ X ) ) @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) ) ) ).
% DERIV_log
thf(fact_6866_DERIV__powr,axiom,
! [G2: real > real,M: real,X: real,F3: real > real,R3: real] :
( ( has_field_derivative @ real @ G2 @ M @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( G2 @ X ) )
=> ( ( has_field_derivative @ real @ F3 @ R3 @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
=> ( has_field_derivative @ real
@ ^ [X2: real] : ( powr @ real @ ( G2 @ X2 ) @ ( F3 @ X2 ) )
@ ( times_times @ real @ ( powr @ real @ ( G2 @ X ) @ ( F3 @ X ) ) @ ( plus_plus @ real @ ( times_times @ real @ R3 @ ( ln_ln @ real @ ( G2 @ X ) ) ) @ ( divide_divide @ real @ ( times_times @ real @ M @ ( F3 @ X ) ) @ ( G2 @ X ) ) ) )
@ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ).
% DERIV_powr
thf(fact_6867_DERIV__tan,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A] :
( ( ( cos @ A @ X )
!= ( zero_zero @ A ) )
=> ( has_field_derivative @ A @ ( tan @ A ) @ ( inverse_inverse @ A @ ( power_power @ A @ ( cos @ A @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% DERIV_tan
thf(fact_6868_DERIV__real__sqrt,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( has_field_derivative @ real @ sqrt @ ( divide_divide @ real @ ( inverse_inverse @ real @ ( sqrt @ X ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) ) ) ).
% DERIV_real_sqrt
thf(fact_6869_DERIV__arctan,axiom,
! [X: real] : ( has_field_derivative @ real @ arctan @ ( inverse_inverse @ real @ ( plus_plus @ real @ ( one_one @ real ) @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) ) ).
% DERIV_arctan
thf(fact_6870_arsinh__real__has__field__derivative,axiom,
! [X: real,A2: set @ real] : ( has_field_derivative @ real @ ( arsinh @ real ) @ ( divide_divide @ real @ ( one_one @ real ) @ ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ real ) ) ) ) @ ( topolo174197925503356063within @ real @ X @ A2 ) ) ).
% arsinh_real_has_field_derivative
thf(fact_6871_DERIV__cot,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A] :
( ( ( sin @ A @ X )
!= ( zero_zero @ A ) )
=> ( has_field_derivative @ A @ ( cot @ A ) @ ( uminus_uminus @ A @ ( inverse_inverse @ A @ ( power_power @ A @ ( sin @ A @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% DERIV_cot
thf(fact_6872_has__field__derivative__tanh,axiom,
! [A11: $tType] :
( ( ( real_Vector_banach @ A11 )
& ( real_V3459762299906320749_field @ A11 ) )
=> ! [G2: A11 > A11,X: A11,Db: A11,S2: set @ A11] :
( ( ( cosh @ A11 @ ( G2 @ X ) )
!= ( zero_zero @ A11 ) )
=> ( ( has_field_derivative @ A11 @ G2 @ Db @ ( topolo174197925503356063within @ A11 @ X @ S2 ) )
=> ( has_field_derivative @ A11
@ ^ [X2: A11] : ( tanh @ A11 @ ( G2 @ X2 ) )
@ ( times_times @ A11 @ ( minus_minus @ A11 @ ( one_one @ A11 ) @ ( power_power @ A11 @ ( tanh @ A11 @ ( G2 @ X ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ Db )
@ ( topolo174197925503356063within @ A11 @ X @ S2 ) ) ) ) ) ).
% has_field_derivative_tanh
thf(fact_6873_DERIV__real__sqrt__generic,axiom,
! [X: real,D3: real] :
( ( X
!= ( zero_zero @ real ) )
=> ( ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( D3
= ( divide_divide @ real @ ( inverse_inverse @ real @ ( sqrt @ X ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) )
=> ( ( ( ord_less @ real @ X @ ( zero_zero @ real ) )
=> ( D3
= ( divide_divide @ real @ ( uminus_uminus @ real @ ( inverse_inverse @ real @ ( sqrt @ X ) ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) )
=> ( has_field_derivative @ real @ sqrt @ D3 @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ).
% DERIV_real_sqrt_generic
thf(fact_6874_arcosh__real__has__field__derivative,axiom,
! [X: real,A2: set @ real] :
( ( ord_less @ real @ ( one_one @ real ) @ X )
=> ( has_field_derivative @ real @ ( arcosh @ real ) @ ( divide_divide @ real @ ( one_one @ real ) @ ( sqrt @ ( minus_minus @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ real ) ) ) ) @ ( topolo174197925503356063within @ real @ X @ A2 ) ) ) ).
% arcosh_real_has_field_derivative
thf(fact_6875_artanh__real__has__field__derivative,axiom,
! [X: real,A2: set @ real] :
( ( ord_less @ real @ ( abs_abs @ real @ X ) @ ( one_one @ real ) )
=> ( has_field_derivative @ real @ ( artanh @ real ) @ ( divide_divide @ real @ ( one_one @ real ) @ ( minus_minus @ real @ ( one_one @ real ) @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( topolo174197925503356063within @ real @ X @ A2 ) ) ) ).
% artanh_real_has_field_derivative
thf(fact_6876_DERIV__real__root,axiom,
! [N: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( has_field_derivative @ real @ ( root @ N ) @ ( inverse_inverse @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ ( power_power @ real @ ( root @ N @ X ) @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) ) ) ) ).
% DERIV_real_root
thf(fact_6877_DERIV__arccos,axiom,
! [X: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X )
=> ( ( ord_less @ real @ X @ ( one_one @ real ) )
=> ( has_field_derivative @ real @ arccos @ ( inverse_inverse @ real @ ( uminus_uminus @ real @ ( sqrt @ ( minus_minus @ real @ ( one_one @ real ) @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) ) ) ) ).
% DERIV_arccos
thf(fact_6878_DERIV__arcsin,axiom,
! [X: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X )
=> ( ( ord_less @ real @ X @ ( one_one @ real ) )
=> ( has_field_derivative @ real @ arcsin @ ( inverse_inverse @ real @ ( sqrt @ ( minus_minus @ real @ ( one_one @ real ) @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) ) ) ) ).
% DERIV_arcsin
thf(fact_6879_Maclaurin__all__le,axiom,
! [Diff: nat > real > real,F3: real > real,X: real,N: nat] :
( ( ( Diff @ ( zero_zero @ nat ) )
= F3 )
=> ( ! [M4: nat,X3: real] : ( has_field_derivative @ real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ X3 ) @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) )
=> ? [T3: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ T3 ) @ ( abs_abs @ real @ X ) )
& ( ( F3 @ X )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M3: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M3 @ ( zero_zero @ real ) ) @ ( semiring_char_0_fact @ real @ M3 ) ) @ ( power_power @ real @ X @ M3 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N @ T3 ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ X @ N ) ) ) ) ) ) ) ).
% Maclaurin_all_le
thf(fact_6880_Maclaurin__all__le__objl,axiom,
! [Diff: nat > real > real,F3: real > real,X: real,N: nat] :
( ( ( ( Diff @ ( zero_zero @ nat ) )
= F3 )
& ! [M4: nat,X3: real] : ( has_field_derivative @ real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ X3 ) @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) ) )
=> ? [T3: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ T3 ) @ ( abs_abs @ real @ X ) )
& ( ( F3 @ X )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M3: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M3 @ ( zero_zero @ real ) ) @ ( semiring_char_0_fact @ real @ M3 ) ) @ ( power_power @ real @ X @ M3 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N @ T3 ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ X @ N ) ) ) ) ) ) ).
% Maclaurin_all_le_objl
thf(fact_6881_DERIV__odd__real__root,axiom,
! [N: nat,X: real] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( X
!= ( zero_zero @ real ) )
=> ( has_field_derivative @ real @ ( root @ N ) @ ( inverse_inverse @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ ( power_power @ real @ ( root @ N @ X ) @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) ) ) ) ).
% DERIV_odd_real_root
thf(fact_6882_Maclaurin,axiom,
! [H2: real,N: nat,Diff: nat > real > real,F3: real > real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ H2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ( Diff @ ( zero_zero @ nat ) )
= F3 )
=> ( ! [M4: nat,T3: real] :
( ( ( ord_less @ nat @ M4 @ N )
& ( ord_less_eq @ real @ ( zero_zero @ real ) @ T3 )
& ( ord_less_eq @ real @ T3 @ H2 ) )
=> ( has_field_derivative @ real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T3 ) @ ( topolo174197925503356063within @ real @ T3 @ ( top_top @ ( set @ real ) ) ) ) )
=> ? [T3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ T3 )
& ( ord_less @ real @ T3 @ H2 )
& ( ( F3 @ H2 )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M3: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M3 @ ( zero_zero @ real ) ) @ ( semiring_char_0_fact @ real @ M3 ) ) @ ( power_power @ real @ H2 @ M3 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N @ T3 ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ H2 @ N ) ) ) ) ) ) ) ) ) ).
% Maclaurin
thf(fact_6883_Maclaurin2,axiom,
! [H2: real,Diff: nat > real > real,F3: real > real,N: nat] :
( ( ord_less @ real @ ( zero_zero @ real ) @ H2 )
=> ( ( ( Diff @ ( zero_zero @ nat ) )
= F3 )
=> ( ! [M4: nat,T3: real] :
( ( ( ord_less @ nat @ M4 @ N )
& ( ord_less_eq @ real @ ( zero_zero @ real ) @ T3 )
& ( ord_less_eq @ real @ T3 @ H2 ) )
=> ( has_field_derivative @ real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T3 ) @ ( topolo174197925503356063within @ real @ T3 @ ( top_top @ ( set @ real ) ) ) ) )
=> ? [T3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ T3 )
& ( ord_less_eq @ real @ T3 @ H2 )
& ( ( F3 @ H2 )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M3: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M3 @ ( zero_zero @ real ) ) @ ( semiring_char_0_fact @ real @ M3 ) ) @ ( power_power @ real @ H2 @ M3 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N @ T3 ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ H2 @ N ) ) ) ) ) ) ) ) ).
% Maclaurin2
thf(fact_6884_Maclaurin__minus,axiom,
! [H2: real,N: nat,Diff: nat > real > real,F3: real > real] :
( ( ord_less @ real @ H2 @ ( zero_zero @ real ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ( Diff @ ( zero_zero @ nat ) )
= F3 )
=> ( ! [M4: nat,T3: real] :
( ( ( ord_less @ nat @ M4 @ N )
& ( ord_less_eq @ real @ H2 @ T3 )
& ( ord_less_eq @ real @ T3 @ ( zero_zero @ real ) ) )
=> ( has_field_derivative @ real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T3 ) @ ( topolo174197925503356063within @ real @ T3 @ ( top_top @ ( set @ real ) ) ) ) )
=> ? [T3: real] :
( ( ord_less @ real @ H2 @ T3 )
& ( ord_less @ real @ T3 @ ( zero_zero @ real ) )
& ( ( F3 @ H2 )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M3: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M3 @ ( zero_zero @ real ) ) @ ( semiring_char_0_fact @ real @ M3 ) ) @ ( power_power @ real @ H2 @ M3 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N @ T3 ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ H2 @ N ) ) ) ) ) ) ) ) ) ).
% Maclaurin_minus
thf(fact_6885_Maclaurin__all__lt,axiom,
! [Diff: nat > real > real,F3: real > real,N: nat,X: real] :
( ( ( Diff @ ( zero_zero @ nat ) )
= F3 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( X
!= ( zero_zero @ real ) )
=> ( ! [M4: nat,X3: real] : ( has_field_derivative @ real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ X3 ) @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) )
=> ? [T3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( abs_abs @ real @ T3 ) )
& ( ord_less @ real @ ( abs_abs @ real @ T3 ) @ ( abs_abs @ real @ X ) )
& ( ( F3 @ X )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M3: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M3 @ ( zero_zero @ real ) ) @ ( semiring_char_0_fact @ real @ M3 ) ) @ ( power_power @ real @ X @ M3 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N @ T3 ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ X @ N ) ) ) ) ) ) ) ) ) ).
% Maclaurin_all_lt
thf(fact_6886_Maclaurin__bi__le,axiom,
! [Diff: nat > real > real,F3: real > real,N: nat,X: real] :
( ( ( Diff @ ( zero_zero @ nat ) )
= F3 )
=> ( ! [M4: nat,T3: real] :
( ( ( ord_less @ nat @ M4 @ N )
& ( ord_less_eq @ real @ ( abs_abs @ real @ T3 ) @ ( abs_abs @ real @ X ) ) )
=> ( has_field_derivative @ real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T3 ) @ ( topolo174197925503356063within @ real @ T3 @ ( top_top @ ( set @ real ) ) ) ) )
=> ? [T3: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ T3 ) @ ( abs_abs @ real @ X ) )
& ( ( F3 @ X )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M3: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M3 @ ( zero_zero @ real ) ) @ ( semiring_char_0_fact @ real @ M3 ) ) @ ( power_power @ real @ X @ M3 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N @ T3 ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ X @ N ) ) ) ) ) ) ) ).
% Maclaurin_bi_le
thf(fact_6887_Taylor__down,axiom,
! [N: nat,Diff: nat > real > real,F3: real > real,A3: real,B3: real,C2: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ( Diff @ ( zero_zero @ nat ) )
= F3 )
=> ( ! [M4: nat,T3: real] :
( ( ( ord_less @ nat @ M4 @ N )
& ( ord_less_eq @ real @ A3 @ T3 )
& ( ord_less_eq @ real @ T3 @ B3 ) )
=> ( has_field_derivative @ real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T3 ) @ ( topolo174197925503356063within @ real @ T3 @ ( top_top @ ( set @ real ) ) ) ) )
=> ( ( ord_less @ real @ A3 @ C2 )
=> ( ( ord_less_eq @ real @ C2 @ B3 )
=> ? [T3: real] :
( ( ord_less @ real @ A3 @ T3 )
& ( ord_less @ real @ T3 @ C2 )
& ( ( F3 @ A3 )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M3: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M3 @ C2 ) @ ( semiring_char_0_fact @ real @ M3 ) ) @ ( power_power @ real @ ( minus_minus @ real @ A3 @ C2 ) @ M3 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N @ T3 ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ ( minus_minus @ real @ A3 @ C2 ) @ N ) ) ) ) ) ) ) ) ) ) ).
% Taylor_down
thf(fact_6888_Taylor__up,axiom,
! [N: nat,Diff: nat > real > real,F3: real > real,A3: real,B3: real,C2: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ( Diff @ ( zero_zero @ nat ) )
= F3 )
=> ( ! [M4: nat,T3: real] :
( ( ( ord_less @ nat @ M4 @ N )
& ( ord_less_eq @ real @ A3 @ T3 )
& ( ord_less_eq @ real @ T3 @ B3 ) )
=> ( has_field_derivative @ real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T3 ) @ ( topolo174197925503356063within @ real @ T3 @ ( top_top @ ( set @ real ) ) ) ) )
=> ( ( ord_less_eq @ real @ A3 @ C2 )
=> ( ( ord_less @ real @ C2 @ B3 )
=> ? [T3: real] :
( ( ord_less @ real @ C2 @ T3 )
& ( ord_less @ real @ T3 @ B3 )
& ( ( F3 @ B3 )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M3: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M3 @ C2 ) @ ( semiring_char_0_fact @ real @ M3 ) ) @ ( power_power @ real @ ( minus_minus @ real @ B3 @ C2 ) @ M3 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N @ T3 ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ ( minus_minus @ real @ B3 @ C2 ) @ N ) ) ) ) ) ) ) ) ) ) ).
% Taylor_up
thf(fact_6889_Taylor,axiom,
! [N: nat,Diff: nat > real > real,F3: real > real,A3: real,B3: real,C2: real,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ( Diff @ ( zero_zero @ nat ) )
= F3 )
=> ( ! [M4: nat,T3: real] :
( ( ( ord_less @ nat @ M4 @ N )
& ( ord_less_eq @ real @ A3 @ T3 )
& ( ord_less_eq @ real @ T3 @ B3 ) )
=> ( has_field_derivative @ real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T3 ) @ ( topolo174197925503356063within @ real @ T3 @ ( top_top @ ( set @ real ) ) ) ) )
=> ( ( ord_less_eq @ real @ A3 @ C2 )
=> ( ( ord_less_eq @ real @ C2 @ B3 )
=> ( ( ord_less_eq @ real @ A3 @ X )
=> ( ( ord_less_eq @ real @ X @ B3 )
=> ( ( X != C2 )
=> ? [T3: real] :
( ( ( ord_less @ real @ X @ C2 )
=> ( ( ord_less @ real @ X @ T3 )
& ( ord_less @ real @ T3 @ C2 ) ) )
& ( ~ ( ord_less @ real @ X @ C2 )
=> ( ( ord_less @ real @ C2 @ T3 )
& ( ord_less @ real @ T3 @ X ) ) )
& ( ( F3 @ X )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M3: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M3 @ C2 ) @ ( semiring_char_0_fact @ real @ M3 ) ) @ ( power_power @ real @ ( minus_minus @ real @ X @ C2 ) @ M3 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N @ T3 ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ ( minus_minus @ real @ X @ C2 ) @ N ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% Taylor
thf(fact_6890_Maclaurin__lemma2,axiom,
! [N: nat,H2: real,Diff: nat > real > real,K: nat,B2: real] :
( ! [M4: nat,T3: real] :
( ( ( ord_less @ nat @ M4 @ N )
& ( ord_less_eq @ real @ ( zero_zero @ real ) @ T3 )
& ( ord_less_eq @ real @ T3 @ H2 ) )
=> ( has_field_derivative @ real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T3 ) @ ( topolo174197925503356063within @ real @ T3 @ ( top_top @ ( set @ real ) ) ) ) )
=> ( ( N
= ( suc @ K ) )
=> ! [M2: nat,T7: real] :
( ( ( ord_less @ nat @ M2 @ N )
& ( ord_less_eq @ real @ ( zero_zero @ real ) @ T7 )
& ( ord_less_eq @ real @ T7 @ H2 ) )
=> ( has_field_derivative @ real
@ ^ [U: real] :
( minus_minus @ real @ ( Diff @ M2 @ U )
@ ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [P6: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ ( plus_plus @ nat @ M2 @ P6 ) @ ( zero_zero @ real ) ) @ ( semiring_char_0_fact @ real @ P6 ) ) @ ( power_power @ real @ U @ P6 ) )
@ ( set_ord_lessThan @ nat @ ( minus_minus @ nat @ N @ M2 ) ) )
@ ( times_times @ real @ B2 @ ( divide_divide @ real @ ( power_power @ real @ U @ ( minus_minus @ nat @ N @ M2 ) ) @ ( semiring_char_0_fact @ real @ ( minus_minus @ nat @ N @ M2 ) ) ) ) ) )
@ ( minus_minus @ real @ ( Diff @ ( suc @ M2 ) @ T7 )
@ ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [P6: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ ( plus_plus @ nat @ ( suc @ M2 ) @ P6 ) @ ( zero_zero @ real ) ) @ ( semiring_char_0_fact @ real @ P6 ) ) @ ( power_power @ real @ T7 @ P6 ) )
@ ( set_ord_lessThan @ nat @ ( minus_minus @ nat @ N @ ( suc @ M2 ) ) ) )
@ ( times_times @ real @ B2 @ ( divide_divide @ real @ ( power_power @ real @ T7 @ ( minus_minus @ nat @ N @ ( suc @ M2 ) ) ) @ ( semiring_char_0_fact @ real @ ( minus_minus @ nat @ N @ ( suc @ M2 ) ) ) ) ) ) )
@ ( topolo174197925503356063within @ real @ T7 @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ).
% Maclaurin_lemma2
thf(fact_6891_DERIV__arctan__series,axiom,
! [X: real] :
( ( ord_less @ real @ ( abs_abs @ real @ X ) @ ( one_one @ real ) )
=> ( has_field_derivative @ real
@ ^ [X9: real] :
( suminf @ real
@ ^ [K3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ K3 ) @ ( times_times @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( semiring_1_of_nat @ real @ ( plus_plus @ nat @ ( times_times @ nat @ K3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) @ ( power_power @ real @ X9 @ ( plus_plus @ nat @ ( times_times @ nat @ K3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) ) )
@ ( suminf @ real
@ ^ [K3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ K3 ) @ ( power_power @ real @ X @ ( times_times @ nat @ K3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) ) ) ).
% DERIV_arctan_series
thf(fact_6892_DERIV__power__series_H,axiom,
! [R: real,F3: nat > real,X0: real] :
( ! [X3: real] :
( ( member @ real @ X3 @ ( set_or5935395276787703475ssThan @ real @ ( uminus_uminus @ real @ R ) @ R ) )
=> ( summable @ real
@ ^ [N4: nat] : ( times_times @ real @ ( times_times @ real @ ( F3 @ N4 ) @ ( semiring_1_of_nat @ real @ ( suc @ N4 ) ) ) @ ( power_power @ real @ X3 @ N4 ) ) ) )
=> ( ( member @ real @ X0 @ ( set_or5935395276787703475ssThan @ real @ ( uminus_uminus @ real @ R ) @ R ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ R )
=> ( has_field_derivative @ real
@ ^ [X2: real] :
( suminf @ real
@ ^ [N4: nat] : ( times_times @ real @ ( F3 @ N4 ) @ ( power_power @ real @ X2 @ ( suc @ N4 ) ) ) )
@ ( suminf @ real
@ ^ [N4: nat] : ( times_times @ real @ ( times_times @ real @ ( F3 @ N4 ) @ ( semiring_1_of_nat @ real @ ( suc @ N4 ) ) ) @ ( power_power @ real @ X0 @ N4 ) ) )
@ ( topolo174197925503356063within @ real @ X0 @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ).
% DERIV_power_series'
thf(fact_6893_has__derivative__arcsin,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [G2: A > real,X: A,G5: A > real,S2: set @ A] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ ( G2 @ X ) )
=> ( ( ord_less @ real @ ( G2 @ X ) @ ( one_one @ real ) )
=> ( ( has_derivative @ A @ real @ G2 @ G5 @ ( topolo174197925503356063within @ A @ X @ S2 ) )
=> ( has_derivative @ A @ real
@ ^ [X2: A] : ( arcsin @ ( G2 @ X2 ) )
@ ^ [X2: A] : ( times_times @ real @ ( G5 @ X2 ) @ ( inverse_inverse @ real @ ( sqrt @ ( minus_minus @ real @ ( one_one @ real ) @ ( power_power @ real @ ( G2 @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) )
@ ( topolo174197925503356063within @ A @ X @ S2 ) ) ) ) ) ) ).
% has_derivative_arcsin
thf(fact_6894_greaterThanLessThan__iff,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [I: A,L: A,U2: A] :
( ( member @ A @ I @ ( set_or5935395276787703475ssThan @ A @ L @ U2 ) )
= ( ( ord_less @ A @ L @ I )
& ( ord_less @ A @ I @ U2 ) ) ) ) ).
% greaterThanLessThan_iff
thf(fact_6895_greaterThanLessThan__empty,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L: A,K: A] :
( ( ord_less_eq @ A @ L @ K )
=> ( ( set_or5935395276787703475ssThan @ A @ K @ L )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% greaterThanLessThan_empty
thf(fact_6896_greaterThanLessThan__empty__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A3: A,B3: A] :
( ( ( set_or5935395276787703475ssThan @ A @ A3 @ B3 )
= ( bot_bot @ ( set @ A ) ) )
= ( ord_less_eq @ A @ B3 @ A3 ) ) ) ).
% greaterThanLessThan_empty_iff
thf(fact_6897_greaterThanLessThan__empty__iff2,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A3: A,B3: A] :
( ( ( bot_bot @ ( set @ A ) )
= ( set_or5935395276787703475ssThan @ A @ A3 @ B3 ) )
= ( ord_less_eq @ A @ B3 @ A3 ) ) ) ).
% greaterThanLessThan_empty_iff2
thf(fact_6898_infinite__Ioo__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A3: A,B3: A] :
( ( ~ ( finite_finite2 @ A @ ( set_or5935395276787703475ssThan @ A @ A3 @ B3 ) ) )
= ( ord_less @ A @ A3 @ B3 ) ) ) ).
% infinite_Ioo_iff
thf(fact_6899_has__derivative__scaleR,axiom,
! [C: $tType,D: $tType] :
( ( ( real_V822414075346904944vector @ D )
& ( real_V822414075346904944vector @ C ) )
=> ! [F3: D > real,F7: D > real,X: D,S2: set @ D,G2: D > C,G5: D > C] :
( ( has_derivative @ D @ real @ F3 @ F7 @ ( topolo174197925503356063within @ D @ X @ S2 ) )
=> ( ( has_derivative @ D @ C @ G2 @ G5 @ ( topolo174197925503356063within @ D @ X @ S2 ) )
=> ( has_derivative @ D @ C
@ ^ [X2: D] : ( real_V8093663219630862766scaleR @ C @ ( F3 @ X2 ) @ ( G2 @ X2 ) )
@ ^ [H: D] : ( plus_plus @ C @ ( real_V8093663219630862766scaleR @ C @ ( F3 @ X ) @ ( G5 @ H ) ) @ ( real_V8093663219630862766scaleR @ C @ ( F7 @ H ) @ ( G2 @ X ) ) )
@ ( topolo174197925503356063within @ D @ X @ S2 ) ) ) ) ) ).
% has_derivative_scaleR
thf(fact_6900_has__derivative__compose,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B )
& ( real_V822414075346904944vector @ C ) )
=> ! [F3: A > B,F7: A > B,X: A,S2: set @ A,G2: B > C,G5: B > C] :
( ( has_derivative @ A @ B @ F3 @ F7 @ ( topolo174197925503356063within @ A @ X @ S2 ) )
=> ( ( has_derivative @ B @ C @ G2 @ G5 @ ( topolo174197925503356063within @ B @ ( F3 @ X ) @ ( top_top @ ( set @ B ) ) ) )
=> ( has_derivative @ A @ C
@ ^ [X2: A] : ( G2 @ ( F3 @ X2 ) )
@ ^ [X2: A] : ( G5 @ ( F7 @ X2 ) )
@ ( topolo174197925503356063within @ A @ X @ S2 ) ) ) ) ) ).
% has_derivative_compose
thf(fact_6901_has__derivative__in__compose,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B )
& ( real_V822414075346904944vector @ C ) )
=> ! [F3: A > B,F7: A > B,X: A,S2: set @ A,G2: B > C,G5: B > C] :
( ( has_derivative @ A @ B @ F3 @ F7 @ ( topolo174197925503356063within @ A @ X @ S2 ) )
=> ( ( has_derivative @ B @ C @ G2 @ G5 @ ( topolo174197925503356063within @ B @ ( F3 @ X ) @ ( image @ A @ B @ F3 @ S2 ) ) )
=> ( has_derivative @ A @ C
@ ^ [X2: A] : ( G2 @ ( F3 @ X2 ) )
@ ^ [X2: A] : ( G5 @ ( F7 @ X2 ) )
@ ( topolo174197925503356063within @ A @ X @ S2 ) ) ) ) ) ).
% has_derivative_in_compose
thf(fact_6902_has__derivative__add,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F3: A > B,F7: A > B,F4: filter @ A,G2: A > B,G5: A > B] :
( ( has_derivative @ A @ B @ F3 @ F7 @ F4 )
=> ( ( has_derivative @ A @ B @ G2 @ G5 @ F4 )
=> ( has_derivative @ A @ B
@ ^ [X2: A] : ( plus_plus @ B @ ( F3 @ X2 ) @ ( G2 @ X2 ) )
@ ^ [X2: A] : ( plus_plus @ B @ ( F7 @ X2 ) @ ( G5 @ X2 ) )
@ F4 ) ) ) ) ).
% has_derivative_add
thf(fact_6903_has__derivative__diff,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F3: A > B,F7: A > B,F4: filter @ A,G2: A > B,G5: A > B] :
( ( has_derivative @ A @ B @ F3 @ F7 @ F4 )
=> ( ( has_derivative @ A @ B @ G2 @ G5 @ F4 )
=> ( has_derivative @ A @ B
@ ^ [X2: A] : ( minus_minus @ B @ ( F3 @ X2 ) @ ( G2 @ X2 ) )
@ ^ [X2: A] : ( minus_minus @ B @ ( F7 @ X2 ) @ ( G5 @ X2 ) )
@ F4 ) ) ) ) ).
% has_derivative_diff
thf(fact_6904_has__derivative__mult__right,axiom,
! [A: $tType,C: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V4412858255891104859lgebra @ A ) )
=> ! [G2: C > A,G5: C > A,F4: filter @ C,X: A] :
( ( has_derivative @ C @ A @ G2 @ G5 @ F4 )
=> ( has_derivative @ C @ A
@ ^ [X2: C] : ( times_times @ A @ X @ ( G2 @ X2 ) )
@ ^ [X2: C] : ( times_times @ A @ X @ ( G5 @ X2 ) )
@ F4 ) ) ) ).
% has_derivative_mult_right
thf(fact_6905_has__derivative__mult__left,axiom,
! [A: $tType,C: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V4412858255891104859lgebra @ A ) )
=> ! [G2: C > A,G5: C > A,F4: filter @ C,Y: A] :
( ( has_derivative @ C @ A @ G2 @ G5 @ F4 )
=> ( has_derivative @ C @ A
@ ^ [X2: C] : ( times_times @ A @ ( G2 @ X2 ) @ Y )
@ ^ [X2: C] : ( times_times @ A @ ( G5 @ X2 ) @ Y )
@ F4 ) ) ) ).
% has_derivative_mult_left
thf(fact_6906_has__derivative__of__real,axiom,
! [A: $tType,C: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V2191834092415804123ebra_1 @ A )
& ( real_V822414075346904944vector @ A ) )
=> ! [G2: C > real,G5: C > real,F4: filter @ C] :
( ( has_derivative @ C @ real @ G2 @ G5 @ F4 )
=> ( has_derivative @ C @ A
@ ^ [X2: C] : ( real_Vector_of_real @ A @ ( G2 @ X2 ) )
@ ^ [X2: C] : ( real_Vector_of_real @ A @ ( G5 @ X2 ) )
@ F4 ) ) ) ).
% has_derivative_of_real
thf(fact_6907_has__derivative__ident,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F4: filter @ A] :
( has_derivative @ A @ A
@ ^ [X2: A] : X2
@ ^ [X2: A] : X2
@ F4 ) ) ).
% has_derivative_ident
thf(fact_6908_has__derivative__scaleR__right,axiom,
! [B: $tType,C: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V822414075346904944vector @ B ) )
=> ! [G2: C > B,G5: C > B,F4: filter @ C,R3: real] :
( ( has_derivative @ C @ B @ G2 @ G5 @ F4 )
=> ( has_derivative @ C @ B
@ ^ [X2: C] : ( real_V8093663219630862766scaleR @ B @ R3 @ ( G2 @ X2 ) )
@ ^ [X2: C] : ( real_V8093663219630862766scaleR @ B @ R3 @ ( G5 @ X2 ) )
@ F4 ) ) ) ).
% has_derivative_scaleR_right
thf(fact_6909_has__derivative__scaleR__left,axiom,
! [B: $tType,C: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V822414075346904944vector @ B ) )
=> ! [G2: C > real,G5: C > real,F4: filter @ C,X: B] :
( ( has_derivative @ C @ real @ G2 @ G5 @ F4 )
=> ( has_derivative @ C @ B
@ ^ [X2: C] : ( real_V8093663219630862766scaleR @ B @ ( G2 @ X2 ) @ X )
@ ^ [X2: C] : ( real_V8093663219630862766scaleR @ B @ ( G5 @ X2 ) @ X )
@ F4 ) ) ) ).
% has_derivative_scaleR_left
thf(fact_6910_has__derivative__const,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [C2: B,F4: filter @ A] :
( has_derivative @ A @ B
@ ^ [X2: A] : C2
@ ^ [X2: A] : ( zero_zero @ B )
@ F4 ) ) ).
% has_derivative_const
thf(fact_6911_has__derivative__sum,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( ( real_V822414075346904944vector @ B )
& ( real_V822414075346904944vector @ C ) )
=> ! [I6: set @ A,F3: A > B > C,F7: A > B > C,F4: filter @ B] :
( ! [I3: A] :
( ( member @ A @ I3 @ I6 )
=> ( has_derivative @ B @ C @ ( F3 @ I3 ) @ ( F7 @ I3 ) @ F4 ) )
=> ( has_derivative @ B @ C
@ ^ [X2: B] :
( groups7311177749621191930dd_sum @ A @ C
@ ^ [I4: A] : ( F3 @ I4 @ X2 )
@ I6 )
@ ^ [X2: B] :
( groups7311177749621191930dd_sum @ A @ C
@ ^ [I4: A] : ( F7 @ I4 @ X2 )
@ I6 )
@ F4 ) ) ) ).
% has_derivative_sum
thf(fact_6912_has__derivative__minus,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F3: A > B,F7: A > B,F4: filter @ A] :
( ( has_derivative @ A @ B @ F3 @ F7 @ F4 )
=> ( has_derivative @ A @ B
@ ^ [X2: A] : ( uminus_uminus @ B @ ( F3 @ X2 ) )
@ ^ [X2: A] : ( uminus_uminus @ B @ ( F7 @ X2 ) )
@ F4 ) ) ) ).
% has_derivative_minus
thf(fact_6913_infinite__Ioo,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ A3 @ B3 )
=> ~ ( finite_finite2 @ A @ ( set_or5935395276787703475ssThan @ A @ A3 @ B3 ) ) ) ) ).
% infinite_Ioo
thf(fact_6914_has__derivative__mult,axiom,
! [A: $tType,D: $tType] :
( ( ( real_V822414075346904944vector @ D )
& ( real_V4412858255891104859lgebra @ A ) )
=> ! [F3: D > A,F7: D > A,X: D,S2: set @ D,G2: D > A,G5: D > A] :
( ( has_derivative @ D @ A @ F3 @ F7 @ ( topolo174197925503356063within @ D @ X @ S2 ) )
=> ( ( has_derivative @ D @ A @ G2 @ G5 @ ( topolo174197925503356063within @ D @ X @ S2 ) )
=> ( has_derivative @ D @ A
@ ^ [X2: D] : ( times_times @ A @ ( F3 @ X2 ) @ ( G2 @ X2 ) )
@ ^ [H: D] : ( plus_plus @ A @ ( times_times @ A @ ( F3 @ X ) @ ( G5 @ H ) ) @ ( times_times @ A @ ( F7 @ H ) @ ( G2 @ X ) ) )
@ ( topolo174197925503356063within @ D @ X @ S2 ) ) ) ) ) ).
% has_derivative_mult
thf(fact_6915_has__derivative__zero__unique,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F4: A > B,X: A] :
( ( has_derivative @ A @ B
@ ^ [X2: A] : ( zero_zero @ B )
@ F4
@ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) )
=> ( F4
= ( ^ [H: A] : ( zero_zero @ B ) ) ) ) ) ).
% has_derivative_zero_unique
thf(fact_6916_has__derivative__in__compose2,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [T: set @ A,G2: A > B,G5: A > A > B,F3: C > A,S2: set @ C,X: C,F7: C > A] :
( ! [X3: A] :
( ( member @ A @ X3 @ T )
=> ( has_derivative @ A @ B @ G2 @ ( G5 @ X3 ) @ ( topolo174197925503356063within @ A @ X3 @ T ) ) )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image @ C @ A @ F3 @ S2 ) @ T )
=> ( ( member @ C @ X @ S2 )
=> ( ( has_derivative @ C @ A @ F3 @ F7 @ ( topolo174197925503356063within @ C @ X @ S2 ) )
=> ( has_derivative @ C @ B
@ ^ [X2: C] : ( G2 @ ( F3 @ X2 ) )
@ ^ [Y2: C] : ( G5 @ ( F3 @ X ) @ ( F7 @ Y2 ) )
@ ( topolo174197925503356063within @ C @ X @ S2 ) ) ) ) ) ) ) ).
% has_derivative_in_compose2
thf(fact_6917_has__derivative__exp,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [G2: A > real,G5: A > real,X: A,S2: set @ A] :
( ( has_derivative @ A @ real @ G2 @ G5 @ ( topolo174197925503356063within @ A @ X @ S2 ) )
=> ( has_derivative @ A @ real
@ ^ [X2: A] : ( exp @ real @ ( G2 @ X2 ) )
@ ^ [X2: A] : ( times_times @ real @ ( G5 @ X2 ) @ ( exp @ real @ ( G2 @ X ) ) )
@ ( topolo174197925503356063within @ A @ X @ S2 ) ) ) ) ).
% has_derivative_exp
thf(fact_6918_greaterThanLessThan__subseteq__greaterThanLessThan,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A3: A,B3: A,C2: A,D2: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ A3 @ B3 ) @ ( set_or5935395276787703475ssThan @ A @ C2 @ D2 ) )
= ( ( ord_less @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ A @ C2 @ A3 )
& ( ord_less_eq @ A @ B3 @ D2 ) ) ) ) ) ).
% greaterThanLessThan_subseteq_greaterThanLessThan
thf(fact_6919_tanh__real__bounds,axiom,
! [X: real] : ( member @ real @ ( tanh @ real @ X ) @ ( set_or5935395276787703475ssThan @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ ( one_one @ real ) ) ) ).
% tanh_real_bounds
thf(fact_6920_has__derivative__sinh,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [G2: A > A,Db: A,X: A,S2: set @ A] :
( ( has_derivative @ A @ A @ G2 @ ( times_times @ A @ Db ) @ ( topolo174197925503356063within @ A @ X @ S2 ) )
=> ( has_derivative @ A @ A
@ ^ [X2: A] : ( sinh @ A @ ( G2 @ X2 ) )
@ ( times_times @ A @ ( times_times @ A @ ( cosh @ A @ ( G2 @ X ) ) @ Db ) )
@ ( topolo174197925503356063within @ A @ X @ S2 ) ) ) ) ).
% has_derivative_sinh
thf(fact_6921_has__derivative__cosh,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [G2: A > A,Db: A,X: A,S2: set @ A] :
( ( has_derivative @ A @ A @ G2 @ ( times_times @ A @ Db ) @ ( topolo174197925503356063within @ A @ X @ S2 ) )
=> ( has_derivative @ A @ A
@ ^ [X2: A] : ( cosh @ A @ ( G2 @ X2 ) )
@ ( times_times @ A @ ( times_times @ A @ ( sinh @ A @ ( G2 @ X ) ) @ Db ) )
@ ( topolo174197925503356063within @ A @ X @ S2 ) ) ) ) ).
% has_derivative_cosh
thf(fact_6922_has__derivative__sin,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [G2: A > real,G5: A > real,X: A,S2: set @ A] :
( ( has_derivative @ A @ real @ G2 @ G5 @ ( topolo174197925503356063within @ A @ X @ S2 ) )
=> ( has_derivative @ A @ real
@ ^ [X2: A] : ( sin @ real @ ( G2 @ X2 ) )
@ ^ [X2: A] : ( times_times @ real @ ( G5 @ X2 ) @ ( cos @ real @ ( G2 @ X ) ) )
@ ( topolo174197925503356063within @ A @ X @ S2 ) ) ) ) ).
% has_derivative_sin
thf(fact_6923_greaterThanLessThan__subseteq__atLeastAtMost__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A3: A,B3: A,C2: A,D2: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ A3 @ B3 ) @ ( set_or1337092689740270186AtMost @ A @ C2 @ D2 ) )
= ( ( ord_less @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ A @ C2 @ A3 )
& ( ord_less_eq @ A @ B3 @ D2 ) ) ) ) ) ).
% greaterThanLessThan_subseteq_atLeastAtMost_iff
thf(fact_6924_greaterThanLessThan__subseteq__atLeastLessThan__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A3: A,B3: A,C2: A,D2: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ A3 @ B3 ) @ ( set_or7035219750837199246ssThan @ A @ C2 @ D2 ) )
= ( ( ord_less @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ A @ C2 @ A3 )
& ( ord_less_eq @ A @ B3 @ D2 ) ) ) ) ) ).
% greaterThanLessThan_subseteq_atLeastLessThan_iff
thf(fact_6925_ivl__disj__un__two_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,M: A,U2: A] :
( ( ord_less @ A @ L @ M )
=> ( ( ord_less_eq @ A @ M @ U2 )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ L @ M ) @ ( set_or7035219750837199246ssThan @ A @ M @ U2 ) )
= ( set_or5935395276787703475ssThan @ A @ L @ U2 ) ) ) ) ) ).
% ivl_disj_un_two(1)
thf(fact_6926_atLeastAtMost__diff__ends,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B3: A] :
( ( minus_minus @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A3 @ B3 ) @ ( insert @ A @ A3 @ ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( set_or5935395276787703475ssThan @ A @ A3 @ B3 ) ) ) ).
% atLeastAtMost_diff_ends
thf(fact_6927_ivl__disj__un__one_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,U2: A] :
( ( ord_less @ A @ L @ U2 )
=> ( ( sup_sup @ ( set @ A ) @ ( set_ord_atMost @ A @ L ) @ ( set_or5935395276787703475ssThan @ A @ L @ U2 ) )
= ( set_ord_lessThan @ A @ U2 ) ) ) ) ).
% ivl_disj_un_one(1)
thf(fact_6928_has__derivative__divide_H,axiom,
! [A: $tType,C: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V3459762299906320749_field @ A ) )
=> ! [F3: C > A,F7: C > A,X: C,S: set @ C,G2: C > A,G5: C > A] :
( ( has_derivative @ C @ A @ F3 @ F7 @ ( topolo174197925503356063within @ C @ X @ S ) )
=> ( ( has_derivative @ C @ A @ G2 @ G5 @ ( topolo174197925503356063within @ C @ X @ S ) )
=> ( ( ( G2 @ X )
!= ( zero_zero @ A ) )
=> ( has_derivative @ C @ A
@ ^ [X2: C] : ( divide_divide @ A @ ( F3 @ X2 ) @ ( G2 @ X2 ) )
@ ^ [H: C] : ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ ( F7 @ H ) @ ( G2 @ X ) ) @ ( times_times @ A @ ( F3 @ X ) @ ( G5 @ H ) ) ) @ ( times_times @ A @ ( G2 @ X ) @ ( G2 @ X ) ) )
@ ( topolo174197925503356063within @ C @ X @ S ) ) ) ) ) ) ).
% has_derivative_divide'
thf(fact_6929_has__derivative__inverse,axiom,
! [A: $tType,C: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V8999393235501362500lgebra @ A ) )
=> ! [F3: C > A,X: C,F7: C > A,S: set @ C] :
( ( ( F3 @ X )
!= ( zero_zero @ A ) )
=> ( ( has_derivative @ C @ A @ F3 @ F7 @ ( topolo174197925503356063within @ C @ X @ S ) )
=> ( has_derivative @ C @ A
@ ^ [X2: C] : ( inverse_inverse @ A @ ( F3 @ X2 ) )
@ ^ [H: C] : ( uminus_uminus @ A @ ( times_times @ A @ ( times_times @ A @ ( inverse_inverse @ A @ ( F3 @ X ) ) @ ( F7 @ H ) ) @ ( inverse_inverse @ A @ ( F3 @ X ) ) ) )
@ ( topolo174197925503356063within @ C @ X @ S ) ) ) ) ) ).
% has_derivative_inverse
thf(fact_6930_has__derivative__inverse_H,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [X: A,S: set @ A] :
( ( X
!= ( zero_zero @ A ) )
=> ( has_derivative @ A @ A @ ( inverse_inverse @ A )
@ ^ [H: A] : ( uminus_uminus @ A @ ( times_times @ A @ ( times_times @ A @ ( inverse_inverse @ A @ X ) @ H ) @ ( inverse_inverse @ A @ X ) ) )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ).
% has_derivative_inverse'
thf(fact_6931_DERIV__compose__FDERIV,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: real > real,F7: real,G2: A > real,X: A,G5: A > real,S2: set @ A] :
( ( has_field_derivative @ real @ F3 @ F7 @ ( topolo174197925503356063within @ real @ ( G2 @ X ) @ ( top_top @ ( set @ real ) ) ) )
=> ( ( has_derivative @ A @ real @ G2 @ G5 @ ( topolo174197925503356063within @ A @ X @ S2 ) )
=> ( has_derivative @ A @ real
@ ^ [X2: A] : ( F3 @ ( G2 @ X2 ) )
@ ^ [X2: A] : ( times_times @ real @ ( G5 @ X2 ) @ F7 )
@ ( topolo174197925503356063within @ A @ X @ S2 ) ) ) ) ) ).
% DERIV_compose_FDERIV
thf(fact_6932_has__derivative__cos,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [G2: A > real,G5: A > real,X: A,S2: set @ A] :
( ( has_derivative @ A @ real @ G2 @ G5 @ ( topolo174197925503356063within @ A @ X @ S2 ) )
=> ( has_derivative @ A @ real
@ ^ [X2: A] : ( cos @ real @ ( G2 @ X2 ) )
@ ^ [X2: A] : ( times_times @ real @ ( G5 @ X2 ) @ ( uminus_uminus @ real @ ( sin @ real @ ( G2 @ X ) ) ) )
@ ( topolo174197925503356063within @ A @ X @ S2 ) ) ) ) ).
% has_derivative_cos
thf(fact_6933_DERIV__isconst3,axiom,
! [A3: real,B3: real,X: real,Y: real,F3: real > real] :
( ( ord_less @ real @ A3 @ B3 )
=> ( ( member @ real @ X @ ( set_or5935395276787703475ssThan @ real @ A3 @ B3 ) )
=> ( ( member @ real @ Y @ ( set_or5935395276787703475ssThan @ real @ A3 @ B3 ) )
=> ( ! [X3: real] :
( ( member @ real @ X3 @ ( set_or5935395276787703475ssThan @ real @ A3 @ B3 ) )
=> ( has_field_derivative @ real @ F3 @ ( zero_zero @ real ) @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) ) )
=> ( ( F3 @ X )
= ( F3 @ Y ) ) ) ) ) ) ).
% DERIV_isconst3
thf(fact_6934_ivl__disj__un__singleton_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,U2: A] :
( ( ord_less @ A @ L @ U2 )
=> ( ( sup_sup @ ( set @ A ) @ ( insert @ A @ L @ ( bot_bot @ ( set @ A ) ) ) @ ( set_or5935395276787703475ssThan @ A @ L @ U2 ) )
= ( set_or7035219750837199246ssThan @ A @ L @ U2 ) ) ) ) ).
% ivl_disj_un_singleton(3)
thf(fact_6935_ivl__disj__un__two_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,M: A,U2: A] :
( ( ord_less_eq @ A @ L @ M )
=> ( ( ord_less @ A @ M @ U2 )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ L @ M ) @ ( set_or5935395276787703475ssThan @ A @ M @ U2 ) )
= ( set_or7035219750837199246ssThan @ A @ L @ U2 ) ) ) ) ) ).
% ivl_disj_un_two(4)
thf(fact_6936_has__derivative__power,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V3459762299906320749_field @ B ) )
=> ! [F3: A > B,F7: A > B,X: A,S: set @ A,N: nat] :
( ( has_derivative @ A @ B @ F3 @ F7 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( has_derivative @ A @ B
@ ^ [X2: A] : ( power_power @ B @ ( F3 @ X2 ) @ N )
@ ^ [Y2: A] : ( times_times @ B @ ( times_times @ B @ ( semiring_1_of_nat @ B @ N ) @ ( F7 @ Y2 ) ) @ ( power_power @ B @ ( F3 @ X ) @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ).
% has_derivative_power
thf(fact_6937_has__derivative__ln,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [G2: A > real,X: A,G5: A > real,S2: set @ A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( G2 @ X ) )
=> ( ( has_derivative @ A @ real @ G2 @ G5 @ ( topolo174197925503356063within @ A @ X @ S2 ) )
=> ( has_derivative @ A @ real
@ ^ [X2: A] : ( ln_ln @ real @ ( G2 @ X2 ) )
@ ^ [X2: A] : ( times_times @ real @ ( G5 @ X2 ) @ ( inverse_inverse @ real @ ( G2 @ X ) ) )
@ ( topolo174197925503356063within @ A @ X @ S2 ) ) ) ) ) ).
% has_derivative_ln
thf(fact_6938_has__derivative__divide,axiom,
! [A: $tType,C: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V8999393235501362500lgebra @ A ) )
=> ! [F3: C > A,F7: C > A,X: C,S: set @ C,G2: C > A,G5: C > A] :
( ( has_derivative @ C @ A @ F3 @ F7 @ ( topolo174197925503356063within @ C @ X @ S ) )
=> ( ( has_derivative @ C @ A @ G2 @ G5 @ ( topolo174197925503356063within @ C @ X @ S ) )
=> ( ( ( G2 @ X )
!= ( zero_zero @ A ) )
=> ( has_derivative @ C @ A
@ ^ [X2: C] : ( divide_divide @ A @ ( F3 @ X2 ) @ ( G2 @ X2 ) )
@ ^ [H: C] : ( plus_plus @ A @ ( times_times @ A @ ( uminus_uminus @ A @ ( F3 @ X ) ) @ ( times_times @ A @ ( times_times @ A @ ( inverse_inverse @ A @ ( G2 @ X ) ) @ ( G5 @ H ) ) @ ( inverse_inverse @ A @ ( G2 @ X ) ) ) ) @ ( divide_divide @ A @ ( F7 @ H ) @ ( G2 @ X ) ) )
@ ( topolo174197925503356063within @ C @ X @ S ) ) ) ) ) ) ).
% has_derivative_divide
thf(fact_6939_has__derivative__prod,axiom,
! [B: $tType,I5: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V3459762299906320749_field @ B ) )
=> ! [I6: set @ I5,F3: I5 > A > B,F7: I5 > A > B,X: A,S: set @ A] :
( ! [I3: I5] :
( ( member @ I5 @ I3 @ I6 )
=> ( has_derivative @ A @ B @ ( F3 @ I3 ) @ ( F7 @ I3 ) @ ( topolo174197925503356063within @ A @ X @ S ) ) )
=> ( has_derivative @ A @ B
@ ^ [X2: A] :
( groups7121269368397514597t_prod @ I5 @ B
@ ^ [I4: I5] : ( F3 @ I4 @ X2 )
@ I6 )
@ ^ [Y2: A] :
( groups7311177749621191930dd_sum @ I5 @ B
@ ^ [I4: I5] :
( times_times @ B @ ( F7 @ I4 @ Y2 )
@ ( groups7121269368397514597t_prod @ I5 @ B
@ ^ [J3: I5] : ( F3 @ J3 @ X )
@ ( minus_minus @ ( set @ I5 ) @ I6 @ ( insert @ I5 @ I4 @ ( bot_bot @ ( set @ I5 ) ) ) ) ) )
@ I6 )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ).
% has_derivative_prod
thf(fact_6940_has__derivative__powr,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [G2: A > real,G5: A > real,X: A,X5: set @ A,F3: A > real,F7: A > real] :
( ( has_derivative @ A @ real @ G2 @ G5 @ ( topolo174197925503356063within @ A @ X @ X5 ) )
=> ( ( has_derivative @ A @ real @ F3 @ F7 @ ( topolo174197925503356063within @ A @ X @ X5 ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( G2 @ X ) )
=> ( ( member @ A @ X @ X5 )
=> ( has_derivative @ A @ real
@ ^ [X2: A] : ( powr @ real @ ( G2 @ X2 ) @ ( F3 @ X2 ) )
@ ^ [H: A] : ( times_times @ real @ ( powr @ real @ ( G2 @ X ) @ ( F3 @ X ) ) @ ( plus_plus @ real @ ( times_times @ real @ ( F7 @ H ) @ ( ln_ln @ real @ ( G2 @ X ) ) ) @ ( divide_divide @ real @ ( times_times @ real @ ( G5 @ H ) @ ( F3 @ X ) ) @ ( G2 @ X ) ) ) )
@ ( topolo174197925503356063within @ A @ X @ X5 ) ) ) ) ) ) ) ).
% has_derivative_powr
thf(fact_6941_has__derivative__real__sqrt,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [G2: A > real,X: A,G5: A > real,S2: set @ A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( G2 @ X ) )
=> ( ( has_derivative @ A @ real @ G2 @ G5 @ ( topolo174197925503356063within @ A @ X @ S2 ) )
=> ( has_derivative @ A @ real
@ ^ [X2: A] : ( sqrt @ ( G2 @ X2 ) )
@ ^ [X2: A] : ( times_times @ real @ ( G5 @ X2 ) @ ( divide_divide @ real @ ( inverse_inverse @ real @ ( sqrt @ ( G2 @ X ) ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
@ ( topolo174197925503356063within @ A @ X @ S2 ) ) ) ) ) ).
% has_derivative_real_sqrt
thf(fact_6942_has__derivative__arctan,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [G2: A > real,G5: A > real,X: A,S2: set @ A] :
( ( has_derivative @ A @ real @ G2 @ G5 @ ( topolo174197925503356063within @ A @ X @ S2 ) )
=> ( has_derivative @ A @ real
@ ^ [X2: A] : ( arctan @ ( G2 @ X2 ) )
@ ^ [X2: A] : ( times_times @ real @ ( G5 @ X2 ) @ ( inverse_inverse @ real @ ( plus_plus @ real @ ( one_one @ real ) @ ( power_power @ real @ ( G2 @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( topolo174197925503356063within @ A @ X @ S2 ) ) ) ) ).
% has_derivative_arctan
thf(fact_6943_has__derivative__tan,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [G2: A > real,X: A,G5: A > real,S2: set @ A] :
( ( ( cos @ real @ ( G2 @ X ) )
!= ( zero_zero @ real ) )
=> ( ( has_derivative @ A @ real @ G2 @ G5 @ ( topolo174197925503356063within @ A @ X @ S2 ) )
=> ( has_derivative @ A @ real
@ ^ [X2: A] : ( tan @ real @ ( G2 @ X2 ) )
@ ^ [X2: A] : ( times_times @ real @ ( G5 @ X2 ) @ ( inverse_inverse @ real @ ( power_power @ real @ ( cos @ real @ ( G2 @ X ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
@ ( topolo174197925503356063within @ A @ X @ S2 ) ) ) ) ) ).
% has_derivative_tan
thf(fact_6944_DERIV__series_H,axiom,
! [F3: real > nat > real,F7: real > nat > real,X0: real,A3: real,B3: real,L6: nat > real] :
( ! [N2: nat] :
( has_field_derivative @ real
@ ^ [X2: real] : ( F3 @ X2 @ N2 )
@ ( F7 @ X0 @ N2 )
@ ( topolo174197925503356063within @ real @ X0 @ ( top_top @ ( set @ real ) ) ) )
=> ( ! [X3: real] :
( ( member @ real @ X3 @ ( set_or5935395276787703475ssThan @ real @ A3 @ B3 ) )
=> ( summable @ real @ ( F3 @ X3 ) ) )
=> ( ( member @ real @ X0 @ ( set_or5935395276787703475ssThan @ real @ A3 @ B3 ) )
=> ( ( summable @ real @ ( F7 @ X0 ) )
=> ( ( summable @ real @ L6 )
=> ( ! [N2: nat,X3: real,Y4: real] :
( ( member @ real @ X3 @ ( set_or5935395276787703475ssThan @ real @ A3 @ B3 ) )
=> ( ( member @ real @ Y4 @ ( set_or5935395276787703475ssThan @ real @ A3 @ B3 ) )
=> ( ord_less_eq @ real @ ( abs_abs @ real @ ( minus_minus @ real @ ( F3 @ X3 @ N2 ) @ ( F3 @ Y4 @ N2 ) ) ) @ ( times_times @ real @ ( L6 @ N2 ) @ ( abs_abs @ real @ ( minus_minus @ real @ X3 @ Y4 ) ) ) ) ) )
=> ( has_field_derivative @ real
@ ^ [X2: real] : ( suminf @ real @ ( F3 @ X2 ) )
@ ( suminf @ real @ ( F7 @ X0 ) )
@ ( topolo174197925503356063within @ real @ X0 @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ) ) ) ).
% DERIV_series'
thf(fact_6945_has__derivative__arccos,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [G2: A > real,X: A,G5: A > real,S2: set @ A] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ ( G2 @ X ) )
=> ( ( ord_less @ real @ ( G2 @ X ) @ ( one_one @ real ) )
=> ( ( has_derivative @ A @ real @ G2 @ G5 @ ( topolo174197925503356063within @ A @ X @ S2 ) )
=> ( has_derivative @ A @ real
@ ^ [X2: A] : ( arccos @ ( G2 @ X2 ) )
@ ^ [X2: A] : ( times_times @ real @ ( G5 @ X2 ) @ ( inverse_inverse @ real @ ( uminus_uminus @ real @ ( sqrt @ ( minus_minus @ real @ ( one_one @ real ) @ ( power_power @ real @ ( G2 @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
@ ( topolo174197925503356063within @ A @ X @ S2 ) ) ) ) ) ) ).
% has_derivative_arccos
thf(fact_6946_has__derivative__floor,axiom,
! [Aa: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( archim2362893244070406136eiling @ Aa )
& ( topolo2564578578187576103pology @ Aa ) )
=> ! [G2: A > real,X: A,F3: real > Aa,G5: A > real,S2: set @ A] :
( ( topolo3448309680560233919inuous @ real @ Aa @ ( topolo174197925503356063within @ real @ ( G2 @ X ) @ ( top_top @ ( set @ real ) ) ) @ F3 )
=> ( ~ ( member @ Aa @ ( F3 @ ( G2 @ X ) ) @ ( ring_1_Ints @ Aa ) )
=> ( ( has_derivative @ A @ real @ G2 @ G5 @ ( topolo174197925503356063within @ A @ X @ S2 ) )
=> ( has_derivative @ A @ real
@ ^ [X2: A] : ( ring_1_of_int @ real @ ( archim6421214686448440834_floor @ Aa @ ( F3 @ ( G2 @ X2 ) ) ) )
@ ^ [X2: A] : ( times_times @ real @ ( G5 @ X2 ) @ ( zero_zero @ real ) )
@ ( topolo174197925503356063within @ A @ X @ S2 ) ) ) ) ) ) ).
% has_derivative_floor
thf(fact_6947_termdiffs__aux,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [C2: nat > A,K4: A,X: A] :
( ( summable @ A
@ ^ [N4: nat] : ( times_times @ A @ ( diffs @ A @ ( diffs @ A @ C2 ) @ N4 ) @ ( power_power @ A @ K4 @ N4 ) ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ ( real_V7770717601297561774m_norm @ A @ K4 ) )
=> ( filterlim @ A @ A
@ ^ [H: A] :
( suminf @ A
@ ^ [N4: nat] : ( times_times @ A @ ( C2 @ N4 ) @ ( minus_minus @ A @ ( divide_divide @ A @ ( minus_minus @ A @ ( power_power @ A @ ( plus_plus @ A @ X @ H ) @ N4 ) @ ( power_power @ A @ X @ N4 ) ) @ H ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N4 ) @ ( power_power @ A @ X @ ( minus_minus @ nat @ N4 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% termdiffs_aux
thf(fact_6948_tendsto__const,axiom,
! [A: $tType,B: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [K: A,F4: filter @ B] :
( filterlim @ B @ A
@ ^ [X2: B] : K
@ ( topolo7230453075368039082e_nhds @ A @ K )
@ F4 ) ) ).
% tendsto_const
thf(fact_6949_tendsto__ident__at,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [A3: A,S2: set @ A] :
( filterlim @ A @ A
@ ^ [X2: A] : X2
@ ( topolo7230453075368039082e_nhds @ A @ A3 )
@ ( topolo174197925503356063within @ A @ A3 @ S2 ) ) ) ).
% tendsto_ident_at
thf(fact_6950_tendsto__mult__right__iff,axiom,
! [A: $tType,B: $tType] :
( ( ( field @ A )
& ( topolo4211221413907600880p_mult @ A ) )
=> ! [C2: A,F3: B > A,L: A,F4: filter @ B] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( filterlim @ B @ A
@ ^ [X2: B] : ( times_times @ A @ ( F3 @ X2 ) @ C2 )
@ ( topolo7230453075368039082e_nhds @ A @ ( times_times @ A @ L @ C2 ) )
@ F4 )
= ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ F4 ) ) ) ) ).
% tendsto_mult_right_iff
thf(fact_6951_tendsto__mult__left__iff,axiom,
! [A: $tType,B: $tType] :
( ( ( field @ A )
& ( topolo4211221413907600880p_mult @ A ) )
=> ! [C2: A,F3: B > A,L: A,F4: filter @ B] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( filterlim @ B @ A
@ ^ [X2: B] : ( times_times @ A @ C2 @ ( F3 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( times_times @ A @ C2 @ L ) )
@ F4 )
= ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ F4 ) ) ) ) ).
% tendsto_mult_left_iff
thf(fact_6952_power__tendsto__0__iff,axiom,
! [A: $tType,N: nat,F3: A > real,F4: filter @ A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( filterlim @ A @ real
@ ^ [X2: A] : ( power_power @ real @ ( F3 @ X2 ) @ N )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ F4 )
= ( filterlim @ A @ real @ F3 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ F4 ) ) ) ).
% power_tendsto_0_iff
thf(fact_6953_continuous__ident,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [X: A,S: set @ A] :
( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ X @ S )
@ ^ [X2: A] : X2 ) ) ).
% continuous_ident
thf(fact_6954_has__field__derivative__iff,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,D3: A,X: A,S: set @ A] :
( ( has_field_derivative @ A @ F3 @ D3 @ ( topolo174197925503356063within @ A @ X @ S ) )
= ( filterlim @ A @ A
@ ^ [Y2: A] : ( divide_divide @ A @ ( minus_minus @ A @ ( F3 @ Y2 ) @ ( F3 @ X ) ) @ ( minus_minus @ A @ Y2 @ X ) )
@ ( topolo7230453075368039082e_nhds @ A @ D3 )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ).
% has_field_derivative_iff
thf(fact_6955_has__field__derivativeD,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,D3: A,X: A,S: set @ A] :
( ( has_field_derivative @ A @ F3 @ D3 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( filterlim @ A @ A
@ ^ [Y2: A] : ( divide_divide @ A @ ( minus_minus @ A @ ( F3 @ Y2 ) @ ( F3 @ X ) ) @ ( minus_minus @ A @ Y2 @ X ) )
@ ( topolo7230453075368039082e_nhds @ A @ D3 )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ).
% has_field_derivativeD
thf(fact_6956_real__LIM__sandwich__zero,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [F3: A > real,A3: A,G2: A > real] :
( ( filterlim @ A @ real @ F3 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) )
=> ( ! [X3: A] :
( ( X3 != A3 )
=> ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( G2 @ X3 ) ) )
=> ( ! [X3: A] :
( ( X3 != A3 )
=> ( ord_less_eq @ real @ ( G2 @ X3 ) @ ( F3 @ X3 ) ) )
=> ( filterlim @ A @ real @ G2 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ).
% real_LIM_sandwich_zero
thf(fact_6957_isCont__iff,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [X: A,F3: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) @ F3 )
= ( filterlim @ A @ B
@ ^ [H: A] : ( F3 @ ( plus_plus @ A @ X @ H ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( F3 @ X ) )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% isCont_iff
thf(fact_6958_LIM__isCont__iff,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [F3: A > B,A3: A] :
( ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ ( F3 @ A3 ) ) @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) )
= ( filterlim @ A @ B
@ ^ [H: A] : ( F3 @ ( plus_plus @ A @ A3 @ H ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( F3 @ A3 ) )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% LIM_isCont_iff
thf(fact_6959_LIM__offset__zero,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [F3: A > B,L6: B,A3: A] :
( ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ L6 ) @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) )
=> ( filterlim @ A @ B
@ ^ [H: A] : ( F3 @ ( plus_plus @ A @ A3 @ H ) )
@ ( topolo7230453075368039082e_nhds @ B @ L6 )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% LIM_offset_zero
thf(fact_6960_LIM__offset__zero__cancel,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [F3: A > B,A3: A,L6: B] :
( ( filterlim @ A @ B
@ ^ [H: A] : ( F3 @ ( plus_plus @ A @ A3 @ H ) )
@ ( topolo7230453075368039082e_nhds @ B @ L6 )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) )
=> ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ L6 ) @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% LIM_offset_zero_cancel
thf(fact_6961_LIM__offset,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [F3: A > B,L6: B,A3: A,K: A] :
( ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ L6 ) @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) )
=> ( filterlim @ A @ B
@ ^ [X2: A] : ( F3 @ ( plus_plus @ A @ X2 @ K ) )
@ ( topolo7230453075368039082e_nhds @ B @ L6 )
@ ( topolo174197925503356063within @ A @ ( minus_minus @ A @ A3 @ K ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% LIM_offset
thf(fact_6962_isCont__Pair,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( topolo4958980785337419405_space @ B )
& ( topolo4958980785337419405_space @ C ) )
=> ! [A3: A,F3: A > B,G2: A > C] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) @ F3 )
=> ( ( topolo3448309680560233919inuous @ A @ C @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) @ G2 )
=> ( topolo3448309680560233919inuous @ A @ ( product_prod @ B @ C ) @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) )
@ ^ [X2: A] : ( product_Pair @ B @ C @ ( F3 @ X2 ) @ ( G2 @ X2 ) ) ) ) ) ) ).
% isCont_Pair
thf(fact_6963_filterlim__at__If,axiom,
! [B: $tType,A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [F3: A > B,G4: filter @ B,X: A,P: A > $o,G2: A > B] :
( ( filterlim @ A @ B @ F3 @ G4 @ ( topolo174197925503356063within @ A @ X @ ( collect @ A @ P ) ) )
=> ( ( filterlim @ A @ B @ G2 @ G4
@ ( topolo174197925503356063within @ A @ X
@ ( collect @ A
@ ^ [X2: A] :
~ ( P @ X2 ) ) ) )
=> ( filterlim @ A @ B
@ ^ [X2: A] : ( if @ B @ ( P @ X2 ) @ ( F3 @ X2 ) @ ( G2 @ X2 ) )
@ G4
@ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% filterlim_at_If
thf(fact_6964_isCont__o2,axiom,
! [B: $tType,C: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( topolo4958980785337419405_space @ C )
& ( topological_t2_space @ B ) )
=> ! [A3: A,F3: A > B,G2: B > C] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) @ F3 )
=> ( ( topolo3448309680560233919inuous @ B @ C @ ( topolo174197925503356063within @ B @ ( F3 @ A3 ) @ ( top_top @ ( set @ B ) ) ) @ G2 )
=> ( topolo3448309680560233919inuous @ A @ C @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) )
@ ^ [X2: A] : ( G2 @ ( F3 @ X2 ) ) ) ) ) ) ).
% isCont_o2
thf(fact_6965_LIM__const__eq,axiom,
! [A: $tType,B: $tType] :
( ( ( topological_t2_space @ B )
& ( topolo8386298272705272623_space @ A ) )
=> ! [K: B,L6: B,A3: A] :
( ( filterlim @ A @ B
@ ^ [X2: A] : K
@ ( topolo7230453075368039082e_nhds @ B @ L6 )
@ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) )
=> ( K = L6 ) ) ) ).
% LIM_const_eq
thf(fact_6966_tendsto__compose,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( ( topolo4958980785337419405_space @ B )
& ( topolo4958980785337419405_space @ A ) )
=> ! [G2: A > B,L: A,F3: C > A,F4: filter @ C] :
( ( filterlim @ A @ B @ G2 @ ( topolo7230453075368039082e_nhds @ B @ ( G2 @ L ) ) @ ( topolo174197925503356063within @ A @ L @ ( top_top @ ( set @ A ) ) ) )
=> ( ( filterlim @ C @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ F4 )
=> ( filterlim @ C @ B
@ ^ [X2: C] : ( G2 @ ( F3 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( G2 @ L ) )
@ F4 ) ) ) ) ).
% tendsto_compose
thf(fact_6967_LIM__const__not__eq,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo8386298272705272623_space @ A )
& ( topological_t2_space @ B ) )
=> ! [K: B,L6: B,A3: A] :
( ( K != L6 )
=> ~ ( filterlim @ A @ B
@ ^ [X2: A] : K
@ ( topolo7230453075368039082e_nhds @ B @ L6 )
@ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% LIM_const_not_eq
thf(fact_6968_isCont__tendsto__compose,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( ( topolo4958980785337419405_space @ B )
& ( topological_t2_space @ A ) )
=> ! [L: A,G2: A > B,F3: C > A,F4: filter @ C] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ L @ ( top_top @ ( set @ A ) ) ) @ G2 )
=> ( ( filterlim @ C @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ F4 )
=> ( filterlim @ C @ B
@ ^ [X2: C] : ( G2 @ ( F3 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( G2 @ L ) )
@ F4 ) ) ) ) ).
% isCont_tendsto_compose
thf(fact_6969_continuous__within__compose3,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( ( topological_t2_space @ C )
& ( topolo4958980785337419405_space @ B )
& ( topological_t2_space @ A ) )
=> ! [F3: C > A,X: C,G2: A > B,S2: set @ C] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ ( F3 @ X ) @ ( top_top @ ( set @ A ) ) ) @ G2 )
=> ( ( topolo3448309680560233919inuous @ C @ A @ ( topolo174197925503356063within @ C @ X @ S2 ) @ F3 )
=> ( topolo3448309680560233919inuous @ C @ B @ ( topolo174197925503356063within @ C @ X @ S2 )
@ ^ [X2: C] : ( G2 @ ( F3 @ X2 ) ) ) ) ) ) ).
% continuous_within_compose3
thf(fact_6970_isCont__norm,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [A3: A,F3: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) @ F3 )
=> ( topolo3448309680560233919inuous @ A @ real @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) )
@ ^ [X2: A] : ( real_V7770717601297561774m_norm @ B @ ( F3 @ X2 ) ) ) ) ) ).
% isCont_norm
thf(fact_6971_isCont__of__real,axiom,
! [A: $tType,C: $tType] :
( ( ( topological_t2_space @ C )
& ( real_V2191834092415804123ebra_1 @ A )
& ( real_V822414075346904944vector @ A ) )
=> ! [A3: C,G2: C > real] :
( ( topolo3448309680560233919inuous @ C @ real @ ( topolo174197925503356063within @ C @ A3 @ ( top_top @ ( set @ C ) ) ) @ G2 )
=> ( topolo3448309680560233919inuous @ C @ A @ ( topolo174197925503356063within @ C @ A3 @ ( top_top @ ( set @ C ) ) )
@ ^ [X2: C] : ( real_Vector_of_real @ A @ ( G2 @ X2 ) ) ) ) ) ).
% isCont_of_real
thf(fact_6972_isCont__scaleR,axiom,
! [C: $tType,D: $tType] :
( ( ( topological_t2_space @ D )
& ( real_V822414075346904944vector @ C ) )
=> ! [A3: D,F3: D > real,G2: D > C] :
( ( topolo3448309680560233919inuous @ D @ real @ ( topolo174197925503356063within @ D @ A3 @ ( top_top @ ( set @ D ) ) ) @ F3 )
=> ( ( topolo3448309680560233919inuous @ D @ C @ ( topolo174197925503356063within @ D @ A3 @ ( top_top @ ( set @ D ) ) ) @ G2 )
=> ( topolo3448309680560233919inuous @ D @ C @ ( topolo174197925503356063within @ D @ A3 @ ( top_top @ ( set @ D ) ) )
@ ^ [X2: D] : ( real_V8093663219630862766scaleR @ C @ ( F3 @ X2 ) @ ( G2 @ X2 ) ) ) ) ) ) ).
% isCont_scaleR
thf(fact_6973_LIM__not__zero,axiom,
! [Aa: $tType,A: $tType] :
( ( ( topolo8386298272705272623_space @ A )
& ( zero @ Aa )
& ( topological_t2_space @ Aa ) )
=> ! [K: Aa,A3: A] :
( ( K
!= ( zero_zero @ Aa ) )
=> ~ ( filterlim @ A @ Aa
@ ^ [X2: A] : K
@ ( topolo7230453075368039082e_nhds @ Aa @ ( zero_zero @ Aa ) )
@ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% LIM_not_zero
thf(fact_6974_isCont__LIM__compose2,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( topolo4958980785337419405_space @ B )
& ( topolo4958980785337419405_space @ C ) )
=> ! [A3: A,F3: A > B,G2: B > C,L: C] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) @ F3 )
=> ( ( filterlim @ B @ C @ G2 @ ( topolo7230453075368039082e_nhds @ C @ L ) @ ( topolo174197925503356063within @ B @ ( F3 @ A3 ) @ ( top_top @ ( set @ B ) ) ) )
=> ( ? [D5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D5 )
& ! [X3: A] :
( ( ( X3 != A3 )
& ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ X3 @ A3 ) ) @ D5 ) )
=> ( ( F3 @ X3 )
!= ( F3 @ A3 ) ) ) )
=> ( filterlim @ A @ C
@ ^ [X2: A] : ( G2 @ ( F3 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ C @ L )
@ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ).
% isCont_LIM_compose2
thf(fact_6975_continuous__within__compose2,axiom,
! [B: $tType,C: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( topolo4958980785337419405_space @ C )
& ( topological_t2_space @ B ) )
=> ! [X: A,S2: set @ A,F3: A > B,G2: B > C] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ X @ S2 ) @ F3 )
=> ( ( topolo3448309680560233919inuous @ B @ C @ ( topolo174197925503356063within @ B @ ( F3 @ X ) @ ( image @ A @ B @ F3 @ S2 ) ) @ G2 )
=> ( topolo3448309680560233919inuous @ A @ C @ ( topolo174197925503356063within @ A @ X @ S2 )
@ ^ [X2: A] : ( G2 @ ( F3 @ X2 ) ) ) ) ) ) ).
% continuous_within_compose2
thf(fact_6976_filterlim__sup,axiom,
! [B: $tType,A: $tType,F3: A > B,F4: filter @ B,F12: filter @ A,F23: filter @ A] :
( ( filterlim @ A @ B @ F3 @ F4 @ F12 )
=> ( ( filterlim @ A @ B @ F3 @ F4 @ F23 )
=> ( filterlim @ A @ B @ F3 @ F4 @ ( sup_sup @ ( filter @ A ) @ F12 @ F23 ) ) ) ) ).
% filterlim_sup
thf(fact_6977_continuous__trivial__limit,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [Net: filter @ A,F3: A > B] :
( ( Net
= ( bot_bot @ ( filter @ A ) ) )
=> ( topolo3448309680560233919inuous @ A @ B @ Net @ F3 ) ) ) ).
% continuous_trivial_limit
thf(fact_6978_continuous__bot,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [F3: A > B] : ( topolo3448309680560233919inuous @ A @ B @ ( bot_bot @ ( filter @ A ) ) @ F3 ) ) ).
% continuous_bot
thf(fact_6979_nhds__neq__bot,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [A3: A] :
( ( topolo7230453075368039082e_nhds @ A @ A3 )
!= ( bot_bot @ ( filter @ A ) ) ) ) ).
% nhds_neq_bot
thf(fact_6980_tendsto__bot,axiom,
! [A: $tType,B: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [F3: B > A,A3: A] : ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ A3 ) @ ( bot_bot @ ( filter @ B ) ) ) ) ).
% tendsto_bot
thf(fact_6981_tendsto__unique,axiom,
! [B: $tType,A: $tType] :
( ( topological_t2_space @ A )
=> ! [F4: filter @ B,F3: B > A,A3: A,B3: A] :
( ( F4
!= ( bot_bot @ ( filter @ B ) ) )
=> ( ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ A3 ) @ F4 )
=> ( ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ B3 ) @ F4 )
=> ( A3 = B3 ) ) ) ) ) ).
% tendsto_unique
thf(fact_6982_tendsto__const__iff,axiom,
! [B: $tType,A: $tType] :
( ( topological_t2_space @ A )
=> ! [F4: filter @ B,A3: A,B3: A] :
( ( F4
!= ( bot_bot @ ( filter @ B ) ) )
=> ( ( filterlim @ B @ A
@ ^ [X2: B] : A3
@ ( topolo7230453075368039082e_nhds @ A @ B3 )
@ F4 )
= ( A3 = B3 ) ) ) ) ).
% tendsto_const_iff
thf(fact_6983_LIM__D,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F3: A > B,L6: B,A3: A,R3: real] :
( ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ L6 ) @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ R3 )
=> ? [S3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ S3 )
& ! [X6: A] :
( ( ( X6 != A3 )
& ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ X6 @ A3 ) ) @ S3 ) )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ B @ ( minus_minus @ B @ ( F3 @ X6 ) @ L6 ) ) @ R3 ) ) ) ) ) ) ).
% LIM_D
thf(fact_6984_LIM__I,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [A3: A,F3: A > B,L6: B] :
( ! [R2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R2 )
=> ? [S9: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ S9 )
& ! [X3: A] :
( ( ( X3 != A3 )
& ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ X3 @ A3 ) ) @ S9 ) )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ B @ ( minus_minus @ B @ ( F3 @ X3 ) @ L6 ) ) @ R2 ) ) ) )
=> ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ L6 ) @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% LIM_I
thf(fact_6985_LIM__eq,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F3: A > B,L6: B,A3: A] :
( ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ L6 ) @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) )
= ( ! [R5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R5 )
=> ? [S5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ S5 )
& ! [X2: A] :
( ( ( X2 != A3 )
& ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ X2 @ A3 ) ) @ S5 ) )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ B @ ( minus_minus @ B @ ( F3 @ X2 ) @ L6 ) ) @ R5 ) ) ) ) ) ) ) ).
% LIM_eq
thf(fact_6986_LIM__equal2,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [R: real,A3: A,F3: A > B,G2: A > B,L: B] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R )
=> ( ! [X3: A] :
( ( X3 != A3 )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ X3 @ A3 ) ) @ R )
=> ( ( F3 @ X3 )
= ( G2 @ X3 ) ) ) )
=> ( ( filterlim @ A @ B @ G2 @ ( topolo7230453075368039082e_nhds @ B @ L ) @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) )
=> ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ L ) @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ).
% LIM_equal2
thf(fact_6987_filterlim__mono,axiom,
! [B: $tType,A: $tType,F3: A > B,F23: filter @ B,F12: filter @ A,F24: filter @ B,F13: filter @ A] :
( ( filterlim @ A @ B @ F3 @ F23 @ F12 )
=> ( ( ord_less_eq @ ( filter @ B ) @ F23 @ F24 )
=> ( ( ord_less_eq @ ( filter @ A ) @ F13 @ F12 )
=> ( filterlim @ A @ B @ F3 @ F24 @ F13 ) ) ) ) ).
% filterlim_mono
thf(fact_6988_tendsto__min,axiom,
! [A: $tType,B: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [X5: B > A,X: A,Net: filter @ B,Y8: B > A,Y: A] :
( ( filterlim @ B @ A @ X5 @ ( topolo7230453075368039082e_nhds @ A @ X ) @ Net )
=> ( ( filterlim @ B @ A @ Y8 @ ( topolo7230453075368039082e_nhds @ A @ Y ) @ Net )
=> ( filterlim @ B @ A
@ ^ [X2: B] : ( ord_min @ A @ ( X5 @ X2 ) @ ( Y8 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( ord_min @ A @ X @ Y ) )
@ Net ) ) ) ) ).
% tendsto_min
thf(fact_6989_continuous__min,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( topolo1944317154257567458pology @ B ) )
=> ! [F4: filter @ A,F3: A > B,G2: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ F4 @ F3 )
=> ( ( topolo3448309680560233919inuous @ A @ B @ F4 @ G2 )
=> ( topolo3448309680560233919inuous @ A @ B @ F4
@ ^ [X2: A] : ( ord_min @ B @ ( F3 @ X2 ) @ ( G2 @ X2 ) ) ) ) ) ) ).
% continuous_min
thf(fact_6990_tendsto__arcosh,axiom,
! [B: $tType,F3: B > real,A3: real,F4: filter @ B] :
( ( filterlim @ B @ real @ F3 @ ( topolo7230453075368039082e_nhds @ real @ A3 ) @ F4 )
=> ( ( ord_less @ real @ ( one_one @ real ) @ A3 )
=> ( filterlim @ B @ real
@ ^ [X2: B] : ( arcosh @ real @ ( F3 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( arcosh @ real @ A3 ) )
@ F4 ) ) ) ).
% tendsto_arcosh
thf(fact_6991_continuous__minus,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( topolo1633459387980952147up_add @ B ) )
=> ! [F4: filter @ A,F3: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ F4 @ F3 )
=> ( topolo3448309680560233919inuous @ A @ B @ F4
@ ^ [X2: A] : ( uminus_uminus @ B @ ( F3 @ X2 ) ) ) ) ) ).
% continuous_minus
thf(fact_6992_tendsto__minus,axiom,
! [A: $tType,B: $tType] :
( ( topolo1633459387980952147up_add @ A )
=> ! [F3: B > A,A3: A,F4: filter @ B] :
( ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ A3 ) @ F4 )
=> ( filterlim @ B @ A
@ ^ [X2: B] : ( uminus_uminus @ A @ ( F3 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( uminus_uminus @ A @ A3 ) )
@ F4 ) ) ) ).
% tendsto_minus
thf(fact_6993_tendsto__minus__cancel,axiom,
! [A: $tType,B: $tType] :
( ( topolo1633459387980952147up_add @ A )
=> ! [F3: B > A,A3: A,F4: filter @ B] :
( ( filterlim @ B @ A
@ ^ [X2: B] : ( uminus_uminus @ A @ ( F3 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( uminus_uminus @ A @ A3 ) )
@ F4 )
=> ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ A3 ) @ F4 ) ) ) ).
% tendsto_minus_cancel
thf(fact_6994_tendsto__minus__cancel__left,axiom,
! [B: $tType,A: $tType] :
( ( topolo1633459387980952147up_add @ B )
=> ! [F3: A > B,Y: B,F4: filter @ A] :
( ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ ( uminus_uminus @ B @ Y ) ) @ F4 )
= ( filterlim @ A @ B
@ ^ [X2: A] : ( uminus_uminus @ B @ ( F3 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ B @ Y )
@ F4 ) ) ) ).
% tendsto_minus_cancel_left
thf(fact_6995_continuous__sum,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( ( topological_t2_space @ B )
& ( topolo5987344860129210374id_add @ C ) )
=> ! [I6: set @ A,F4: filter @ B,F3: A > B > C] :
( ! [I3: A] :
( ( member @ A @ I3 @ I6 )
=> ( topolo3448309680560233919inuous @ B @ C @ F4 @ ( F3 @ I3 ) ) )
=> ( topolo3448309680560233919inuous @ B @ C @ F4
@ ^ [X2: B] :
( groups7311177749621191930dd_sum @ A @ C
@ ^ [I4: A] : ( F3 @ I4 @ X2 )
@ I6 ) ) ) ) ).
% continuous_sum
thf(fact_6996_tendsto__sum,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( topolo5987344860129210374id_add @ C )
=> ! [I6: set @ A,F3: A > B > C,A3: A > C,F4: filter @ B] :
( ! [I3: A] :
( ( member @ A @ I3 @ I6 )
=> ( filterlim @ B @ C @ ( F3 @ I3 ) @ ( topolo7230453075368039082e_nhds @ C @ ( A3 @ I3 ) ) @ F4 ) )
=> ( filterlim @ B @ C
@ ^ [X2: B] :
( groups7311177749621191930dd_sum @ A @ C
@ ^ [I4: A] : ( F3 @ I4 @ X2 )
@ I6 )
@ ( topolo7230453075368039082e_nhds @ C @ ( groups7311177749621191930dd_sum @ A @ C @ A3 @ I6 ) )
@ F4 ) ) ) ).
% tendsto_sum
thf(fact_6997_tendsto__null__sum,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( topolo5987344860129210374id_add @ C )
=> ! [I6: set @ B,F3: A > B > C,F4: filter @ A] :
( ! [I3: B] :
( ( member @ B @ I3 @ I6 )
=> ( filterlim @ A @ C
@ ^ [X2: A] : ( F3 @ X2 @ I3 )
@ ( topolo7230453075368039082e_nhds @ C @ ( zero_zero @ C ) )
@ F4 ) )
=> ( filterlim @ A @ C
@ ^ [I4: A] : ( groups7311177749621191930dd_sum @ B @ C @ ( F3 @ I4 ) @ I6 )
@ ( topolo7230453075368039082e_nhds @ C @ ( zero_zero @ C ) )
@ F4 ) ) ) ).
% tendsto_null_sum
thf(fact_6998_tendsto__null__power,axiom,
! [B: $tType,A: $tType] :
( ( real_V2822296259951069270ebra_1 @ B )
=> ! [F3: A > B,F4: filter @ A,N: nat] :
( ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) ) @ F4 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( filterlim @ A @ B
@ ^ [X2: A] : ( power_power @ B @ ( F3 @ X2 ) @ N )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ F4 ) ) ) ) ).
% tendsto_null_power
thf(fact_6999_tendsto__log,axiom,
! [A: $tType,F3: A > real,A3: real,F4: filter @ A,G2: A > real,B3: real] :
( ( filterlim @ A @ real @ F3 @ ( topolo7230453075368039082e_nhds @ real @ A3 ) @ F4 )
=> ( ( filterlim @ A @ real @ G2 @ ( topolo7230453075368039082e_nhds @ real @ B3 ) @ F4 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ A3 )
=> ( ( A3
!= ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ B3 )
=> ( filterlim @ A @ real
@ ^ [X2: A] : ( log @ ( F3 @ X2 ) @ ( G2 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( log @ A3 @ B3 ) )
@ F4 ) ) ) ) ) ) ).
% tendsto_log
thf(fact_7000_tendsto__divide__zero,axiom,
! [A: $tType,B: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: B > A,F4: filter @ B,C2: A] :
( ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ F4 )
=> ( filterlim @ B @ A
@ ^ [X2: B] : ( divide_divide @ A @ ( F3 @ X2 ) @ C2 )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ F4 ) ) ) ).
% tendsto_divide_zero
thf(fact_7001_tendsto__divide,axiom,
! [A: $tType,B: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: B > A,A3: A,F4: filter @ B,G2: B > A,B3: A] :
( ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ A3 ) @ F4 )
=> ( ( filterlim @ B @ A @ G2 @ ( topolo7230453075368039082e_nhds @ A @ B3 ) @ F4 )
=> ( ( B3
!= ( zero_zero @ A ) )
=> ( filterlim @ B @ A
@ ^ [X2: B] : ( divide_divide @ A @ ( F3 @ X2 ) @ ( G2 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( divide_divide @ A @ A3 @ B3 ) )
@ F4 ) ) ) ) ) ).
% tendsto_divide
thf(fact_7002_tendsto__mult__right__zero,axiom,
! [A: $tType,D: $tType] :
( ( real_V4412858255891104859lgebra @ A )
=> ! [F3: D > A,F4: filter @ D,C2: A] :
( ( filterlim @ D @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ F4 )
=> ( filterlim @ D @ A
@ ^ [X2: D] : ( times_times @ A @ C2 @ ( F3 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ F4 ) ) ) ).
% tendsto_mult_right_zero
thf(fact_7003_tendsto__mult__left__zero,axiom,
! [A: $tType,D: $tType] :
( ( real_V4412858255891104859lgebra @ A )
=> ! [F3: D > A,F4: filter @ D,C2: A] :
( ( filterlim @ D @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ F4 )
=> ( filterlim @ D @ A
@ ^ [X2: D] : ( times_times @ A @ ( F3 @ X2 ) @ C2 )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ F4 ) ) ) ).
% tendsto_mult_left_zero
thf(fact_7004_tendsto__mult__zero,axiom,
! [A: $tType,D: $tType] :
( ( real_V4412858255891104859lgebra @ A )
=> ! [F3: D > A,F4: filter @ D,G2: D > A] :
( ( filterlim @ D @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ F4 )
=> ( ( filterlim @ D @ A @ G2 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ F4 )
=> ( filterlim @ D @ A
@ ^ [X2: D] : ( times_times @ A @ ( F3 @ X2 ) @ ( G2 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ F4 ) ) ) ) ).
% tendsto_mult_zero
thf(fact_7005_tendsto__inverse,axiom,
! [A: $tType,B: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [F3: B > A,A3: A,F4: filter @ B] :
( ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ A3 ) @ F4 )
=> ( ( A3
!= ( zero_zero @ A ) )
=> ( filterlim @ B @ A
@ ^ [X2: B] : ( inverse_inverse @ A @ ( F3 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( inverse_inverse @ A @ A3 ) )
@ F4 ) ) ) ) ).
% tendsto_inverse
thf(fact_7006_tendsto__rabs__zero__cancel,axiom,
! [A: $tType,F3: A > real,F4: filter @ A] :
( ( filterlim @ A @ real
@ ^ [X2: A] : ( abs_abs @ real @ ( F3 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ F4 )
=> ( filterlim @ A @ real @ F3 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ F4 ) ) ).
% tendsto_rabs_zero_cancel
thf(fact_7007_tendsto__rabs__zero__iff,axiom,
! [A: $tType,F3: A > real,F4: filter @ A] :
( ( filterlim @ A @ real
@ ^ [X2: A] : ( abs_abs @ real @ ( F3 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ F4 )
= ( filterlim @ A @ real @ F3 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ F4 ) ) ).
% tendsto_rabs_zero_iff
thf(fact_7008_tendsto__rabs__zero,axiom,
! [A: $tType,F3: A > real,F4: filter @ A] :
( ( filterlim @ A @ real @ F3 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ F4 )
=> ( filterlim @ A @ real
@ ^ [X2: A] : ( abs_abs @ real @ ( F3 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ F4 ) ) ).
% tendsto_rabs_zero
thf(fact_7009_tendsto__tanh,axiom,
! [A: $tType,C: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [F3: C > A,A3: A,F4: filter @ C] :
( ( filterlim @ C @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ A3 ) @ F4 )
=> ( ( ( cosh @ A @ A3 )
!= ( zero_zero @ A ) )
=> ( filterlim @ C @ A
@ ^ [X2: C] : ( tanh @ A @ ( F3 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( tanh @ A @ A3 ) )
@ F4 ) ) ) ) ).
% tendsto_tanh
thf(fact_7010_tendsto__cot,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [F3: A > A,A3: A,F4: filter @ A] :
( ( filterlim @ A @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ A3 ) @ F4 )
=> ( ( ( sin @ A @ A3 )
!= ( zero_zero @ A ) )
=> ( filterlim @ A @ A
@ ^ [X2: A] : ( cot @ A @ ( F3 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( cot @ A @ A3 ) )
@ F4 ) ) ) ) ).
% tendsto_cot
thf(fact_7011_tendsto__tan,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [F3: A > A,A3: A,F4: filter @ A] :
( ( filterlim @ A @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ A3 ) @ F4 )
=> ( ( ( cos @ A @ A3 )
!= ( zero_zero @ A ) )
=> ( filterlim @ A @ A
@ ^ [X2: A] : ( tan @ A @ ( F3 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( tan @ A @ A3 ) )
@ F4 ) ) ) ) ).
% tendsto_tan
thf(fact_7012_tendsto__ln,axiom,
! [A: $tType,F3: A > real,A3: real,F4: filter @ A] :
( ( filterlim @ A @ real @ F3 @ ( topolo7230453075368039082e_nhds @ real @ A3 ) @ F4 )
=> ( ( A3
!= ( zero_zero @ real ) )
=> ( filterlim @ A @ real
@ ^ [X2: A] : ( ln_ln @ real @ ( F3 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( ln_ln @ real @ A3 ) )
@ F4 ) ) ) ).
% tendsto_ln
thf(fact_7013_tendsto__powr,axiom,
! [A: $tType,F3: A > real,A3: real,F4: filter @ A,G2: A > real,B3: real] :
( ( filterlim @ A @ real @ F3 @ ( topolo7230453075368039082e_nhds @ real @ A3 ) @ F4 )
=> ( ( filterlim @ A @ real @ G2 @ ( topolo7230453075368039082e_nhds @ real @ B3 ) @ F4 )
=> ( ( A3
!= ( zero_zero @ real ) )
=> ( filterlim @ A @ real
@ ^ [X2: A] : ( powr @ real @ ( F3 @ X2 ) @ ( G2 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( powr @ real @ A3 @ B3 ) )
@ F4 ) ) ) ) ).
% tendsto_powr
thf(fact_7014_tendsto__norm__zero__cancel,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [F3: A > B,F4: filter @ A] :
( ( filterlim @ A @ real
@ ^ [X2: A] : ( real_V7770717601297561774m_norm @ B @ ( F3 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ F4 )
=> ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) ) @ F4 ) ) ) ).
% tendsto_norm_zero_cancel
thf(fact_7015_tendsto__norm__zero__iff,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [F3: A > B,F4: filter @ A] :
( ( filterlim @ A @ real
@ ^ [X2: A] : ( real_V7770717601297561774m_norm @ B @ ( F3 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ F4 )
= ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) ) @ F4 ) ) ) ).
% tendsto_norm_zero_iff
thf(fact_7016_tendsto__norm__zero,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [F3: A > B,F4: filter @ A] :
( ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) ) @ F4 )
=> ( filterlim @ A @ real
@ ^ [X2: A] : ( real_V7770717601297561774m_norm @ B @ ( F3 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ F4 ) ) ) ).
% tendsto_norm_zero
thf(fact_7017_tendsto__sgn,axiom,
! [A: $tType,B: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: B > A,L: A,F4: filter @ B] :
( ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ F4 )
=> ( ( L
!= ( zero_zero @ A ) )
=> ( filterlim @ B @ A
@ ^ [X2: B] : ( sgn_sgn @ A @ ( F3 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( sgn_sgn @ A @ L ) )
@ F4 ) ) ) ) ).
% tendsto_sgn
thf(fact_7018_tendsto__one__prod_H,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( topolo4987421752381908075d_mult @ C )
=> ! [I6: set @ B,F3: A > B > C,F4: filter @ A] :
( ! [I3: B] :
( ( member @ B @ I3 @ I6 )
=> ( filterlim @ A @ C
@ ^ [X2: A] : ( F3 @ X2 @ I3 )
@ ( topolo7230453075368039082e_nhds @ C @ ( one_one @ C ) )
@ F4 ) )
=> ( filterlim @ A @ C
@ ^ [I4: A] : ( groups7121269368397514597t_prod @ B @ C @ ( F3 @ I4 ) @ I6 )
@ ( topolo7230453075368039082e_nhds @ C @ ( one_one @ C ) )
@ F4 ) ) ) ).
% tendsto_one_prod'
thf(fact_7019_continuous__prod_H,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( ( topological_t2_space @ B )
& ( topolo4987421752381908075d_mult @ C ) )
=> ! [I6: set @ A,F4: filter @ B,F3: A > B > C] :
( ! [I3: A] :
( ( member @ A @ I3 @ I6 )
=> ( topolo3448309680560233919inuous @ B @ C @ F4 @ ( F3 @ I3 ) ) )
=> ( topolo3448309680560233919inuous @ B @ C @ F4
@ ^ [X2: B] :
( groups7121269368397514597t_prod @ A @ C
@ ^ [I4: A] : ( F3 @ I4 @ X2 )
@ I6 ) ) ) ) ).
% continuous_prod'
thf(fact_7020_continuous__prod,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( ( topological_t2_space @ B )
& ( real_V4412858255891104859lgebra @ C )
& ( comm_ring_1 @ C ) )
=> ! [S: set @ A,F4: filter @ B,F3: A > B > C] :
( ! [I3: A] :
( ( member @ A @ I3 @ S )
=> ( topolo3448309680560233919inuous @ B @ C @ F4 @ ( F3 @ I3 ) ) )
=> ( topolo3448309680560233919inuous @ B @ C @ F4
@ ^ [X2: B] :
( groups7121269368397514597t_prod @ A @ C
@ ^ [I4: A] : ( F3 @ I4 @ X2 )
@ S ) ) ) ) ).
% continuous_prod
thf(fact_7021_tendsto__prod_H,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( topolo4987421752381908075d_mult @ C )
=> ! [I6: set @ A,F3: A > B > C,A3: A > C,F4: filter @ B] :
( ! [I3: A] :
( ( member @ A @ I3 @ I6 )
=> ( filterlim @ B @ C @ ( F3 @ I3 ) @ ( topolo7230453075368039082e_nhds @ C @ ( A3 @ I3 ) ) @ F4 ) )
=> ( filterlim @ B @ C
@ ^ [X2: B] :
( groups7121269368397514597t_prod @ A @ C
@ ^ [I4: A] : ( F3 @ I4 @ X2 )
@ I6 )
@ ( topolo7230453075368039082e_nhds @ C @ ( groups7121269368397514597t_prod @ A @ C @ A3 @ I6 ) )
@ F4 ) ) ) ).
% tendsto_prod'
thf(fact_7022_tendsto__prod,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( ( real_V4412858255891104859lgebra @ C )
& ( comm_ring_1 @ C ) )
=> ! [S: set @ A,F3: A > B > C,L6: A > C,F4: filter @ B] :
( ! [I3: A] :
( ( member @ A @ I3 @ S )
=> ( filterlim @ B @ C @ ( F3 @ I3 ) @ ( topolo7230453075368039082e_nhds @ C @ ( L6 @ I3 ) ) @ F4 ) )
=> ( filterlim @ B @ C
@ ^ [X2: B] :
( groups7121269368397514597t_prod @ A @ C
@ ^ [I4: A] : ( F3 @ I4 @ X2 )
@ S )
@ ( topolo7230453075368039082e_nhds @ C @ ( groups7121269368397514597t_prod @ A @ C @ L6 @ S ) )
@ F4 ) ) ) ).
% tendsto_prod
thf(fact_7023_continuous__power_H,axiom,
! [B: $tType,C: $tType] :
( ( ( topological_t2_space @ C )
& ( topolo1898628316856586783d_mult @ B ) )
=> ! [F4: filter @ C,F3: C > B,G2: C > nat] :
( ( topolo3448309680560233919inuous @ C @ B @ F4 @ F3 )
=> ( ( topolo3448309680560233919inuous @ C @ nat @ F4 @ G2 )
=> ( topolo3448309680560233919inuous @ C @ B @ F4
@ ^ [X2: C] : ( power_power @ B @ ( F3 @ X2 ) @ ( G2 @ X2 ) ) ) ) ) ) ).
% continuous_power'
thf(fact_7024_tendsto__power__strong,axiom,
! [B: $tType,C: $tType] :
( ( topolo1898628316856586783d_mult @ B )
=> ! [F3: C > B,A3: B,F4: filter @ C,G2: C > nat,B3: nat] :
( ( filterlim @ C @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ A3 ) @ F4 )
=> ( ( filterlim @ C @ nat @ G2 @ ( topolo7230453075368039082e_nhds @ nat @ B3 ) @ F4 )
=> ( filterlim @ C @ B
@ ^ [X2: C] : ( power_power @ B @ ( F3 @ X2 ) @ ( G2 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( power_power @ B @ A3 @ B3 ) )
@ F4 ) ) ) ) ).
% tendsto_power_strong
thf(fact_7025_tendsto__power,axiom,
! [B: $tType,A: $tType] :
( ( ( power @ B )
& ( real_V4412858255891104859lgebra @ B ) )
=> ! [F3: A > B,A3: B,F4: filter @ A,N: nat] :
( ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ A3 ) @ F4 )
=> ( filterlim @ A @ B
@ ^ [X2: A] : ( power_power @ B @ ( F3 @ X2 ) @ N )
@ ( topolo7230453075368039082e_nhds @ B @ ( power_power @ B @ A3 @ N ) )
@ F4 ) ) ) ).
% tendsto_power
thf(fact_7026_continuous__power,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( power @ B )
& ( real_V4412858255891104859lgebra @ B ) )
=> ! [F4: filter @ A,F3: A > B,N: nat] :
( ( topolo3448309680560233919inuous @ A @ B @ F4 @ F3 )
=> ( topolo3448309680560233919inuous @ A @ B @ F4
@ ^ [X2: A] : ( power_power @ B @ ( F3 @ X2 ) @ N ) ) ) ) ).
% continuous_power
thf(fact_7027_tendsto__real__root,axiom,
! [A: $tType,F3: A > real,X: real,F4: filter @ A,N: nat] :
( ( filterlim @ A @ real @ F3 @ ( topolo7230453075368039082e_nhds @ real @ X ) @ F4 )
=> ( filterlim @ A @ real
@ ^ [X2: A] : ( root @ N @ ( F3 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( root @ N @ X ) )
@ F4 ) ) ).
% tendsto_real_root
thf(fact_7028_continuous__sinh,axiom,
! [A: $tType,C: $tType] :
( ( ( topological_t2_space @ C )
& ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [F4: filter @ C,F3: C > A] :
( ( topolo3448309680560233919inuous @ C @ A @ F4 @ F3 )
=> ( topolo3448309680560233919inuous @ C @ A @ F4
@ ^ [X2: C] : ( sinh @ A @ ( F3 @ X2 ) ) ) ) ) ).
% continuous_sinh
thf(fact_7029_continuous__cosh,axiom,
! [A: $tType,C: $tType] :
( ( ( topological_t2_space @ C )
& ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [F4: filter @ C,F3: C > A] :
( ( topolo3448309680560233919inuous @ C @ A @ F4 @ F3 )
=> ( topolo3448309680560233919inuous @ C @ A @ F4
@ ^ [X2: C] : ( cosh @ A @ ( F3 @ X2 ) ) ) ) ) ).
% continuous_cosh
thf(fact_7030_tendsto__sinh,axiom,
! [A: $tType,C: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [F3: C > A,A3: A,F4: filter @ C] :
( ( filterlim @ C @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ A3 ) @ F4 )
=> ( filterlim @ C @ A
@ ^ [X2: C] : ( sinh @ A @ ( F3 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( sinh @ A @ A3 ) )
@ F4 ) ) ) ).
% tendsto_sinh
thf(fact_7031_tendsto__cosh,axiom,
! [A: $tType,C: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [F3: C > A,A3: A,F4: filter @ C] :
( ( filterlim @ C @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ A3 ) @ F4 )
=> ( filterlim @ C @ A
@ ^ [X2: C] : ( cosh @ A @ ( F3 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( cosh @ A @ A3 ) )
@ F4 ) ) ) ).
% tendsto_cosh
thf(fact_7032_tendsto__scaleR,axiom,
! [C: $tType,D: $tType] :
( ( real_V822414075346904944vector @ C )
=> ! [F3: D > real,A3: real,F4: filter @ D,G2: D > C,B3: C] :
( ( filterlim @ D @ real @ F3 @ ( topolo7230453075368039082e_nhds @ real @ A3 ) @ F4 )
=> ( ( filterlim @ D @ C @ G2 @ ( topolo7230453075368039082e_nhds @ C @ B3 ) @ F4 )
=> ( filterlim @ D @ C
@ ^ [X2: D] : ( real_V8093663219630862766scaleR @ C @ ( F3 @ X2 ) @ ( G2 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ C @ ( real_V8093663219630862766scaleR @ C @ A3 @ B3 ) )
@ F4 ) ) ) ) ).
% tendsto_scaleR
thf(fact_7033_continuous__scaleR,axiom,
! [C: $tType,D: $tType] :
( ( ( topological_t2_space @ D )
& ( real_V822414075346904944vector @ C ) )
=> ! [F4: filter @ D,F3: D > real,G2: D > C] :
( ( topolo3448309680560233919inuous @ D @ real @ F4 @ F3 )
=> ( ( topolo3448309680560233919inuous @ D @ C @ F4 @ G2 )
=> ( topolo3448309680560233919inuous @ D @ C @ F4
@ ^ [X2: D] : ( real_V8093663219630862766scaleR @ C @ ( F3 @ X2 ) @ ( G2 @ X2 ) ) ) ) ) ) ).
% continuous_scaleR
thf(fact_7034_continuous__exp,axiom,
! [A: $tType,C: $tType] :
( ( ( topological_t2_space @ C )
& ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [F4: filter @ C,F3: C > A] :
( ( topolo3448309680560233919inuous @ C @ A @ F4 @ F3 )
=> ( topolo3448309680560233919inuous @ C @ A @ F4
@ ^ [X2: C] : ( exp @ A @ ( F3 @ X2 ) ) ) ) ) ).
% continuous_exp
thf(fact_7035_tendsto__exp,axiom,
! [A: $tType,C: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [F3: C > A,A3: A,F4: filter @ C] :
( ( filterlim @ C @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ A3 ) @ F4 )
=> ( filterlim @ C @ A
@ ^ [X2: C] : ( exp @ A @ ( F3 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( exp @ A @ A3 ) )
@ F4 ) ) ) ).
% tendsto_exp
thf(fact_7036_continuous__cos,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_Vector_banach @ B )
& ( real_V3459762299906320749_field @ B ) )
=> ! [F4: filter @ A,F3: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ F4 @ F3 )
=> ( topolo3448309680560233919inuous @ A @ B @ F4
@ ^ [X2: A] : ( cos @ B @ ( F3 @ X2 ) ) ) ) ) ).
% continuous_cos
thf(fact_7037_tendsto__real__sqrt,axiom,
! [A: $tType,F3: A > real,X: real,F4: filter @ A] :
( ( filterlim @ A @ real @ F3 @ ( topolo7230453075368039082e_nhds @ real @ X ) @ F4 )
=> ( filterlim @ A @ real
@ ^ [X2: A] : ( sqrt @ ( F3 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( sqrt @ X ) )
@ F4 ) ) ).
% tendsto_real_sqrt
thf(fact_7038_filterlim__compose,axiom,
! [B: $tType,A: $tType,C: $tType,G2: A > B,F32: filter @ B,F23: filter @ A,F3: C > A,F12: filter @ C] :
( ( filterlim @ A @ B @ G2 @ F32 @ F23 )
=> ( ( filterlim @ C @ A @ F3 @ F23 @ F12 )
=> ( filterlim @ C @ B
@ ^ [X2: C] : ( G2 @ ( F3 @ X2 ) )
@ F32
@ F12 ) ) ) ).
% filterlim_compose
thf(fact_7039_filterlim__ident,axiom,
! [A: $tType,F4: filter @ A] :
( filterlim @ A @ A
@ ^ [X2: A] : X2
@ F4
@ F4 ) ).
% filterlim_ident
thf(fact_7040_tendsto__arctan,axiom,
! [A: $tType,F3: A > real,X: real,F4: filter @ A] :
( ( filterlim @ A @ real @ F3 @ ( topolo7230453075368039082e_nhds @ real @ X ) @ F4 )
=> ( filterlim @ A @ real
@ ^ [X2: A] : ( arctan @ ( F3 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( arctan @ X ) )
@ F4 ) ) ).
% tendsto_arctan
thf(fact_7041_tendsto__of__real,axiom,
! [A: $tType,C: $tType] :
( ( ( real_V2191834092415804123ebra_1 @ A )
& ( real_V822414075346904944vector @ A ) )
=> ! [G2: C > real,A3: real,F4: filter @ C] :
( ( filterlim @ C @ real @ G2 @ ( topolo7230453075368039082e_nhds @ real @ A3 ) @ F4 )
=> ( filterlim @ C @ A
@ ^ [X2: C] : ( real_Vector_of_real @ A @ ( G2 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( real_Vector_of_real @ A @ A3 ) )
@ F4 ) ) ) ).
% tendsto_of_real
thf(fact_7042_continuous__of__real,axiom,
! [A: $tType,C: $tType] :
( ( ( topological_t2_space @ C )
& ( real_V2191834092415804123ebra_1 @ A )
& ( real_V822414075346904944vector @ A ) )
=> ! [F4: filter @ C,G2: C > real] :
( ( topolo3448309680560233919inuous @ C @ real @ F4 @ G2 )
=> ( topolo3448309680560233919inuous @ C @ A @ F4
@ ^ [X2: C] : ( real_Vector_of_real @ A @ ( G2 @ X2 ) ) ) ) ) ).
% continuous_of_real
thf(fact_7043_tendsto__of__real__iff,axiom,
! [A: $tType,B: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [F3: B > real,C2: real,F4: filter @ B] :
( ( filterlim @ B @ A
@ ^ [X2: B] : ( real_Vector_of_real @ A @ ( F3 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( real_Vector_of_real @ A @ C2 ) )
@ F4 )
= ( filterlim @ B @ real @ F3 @ ( topolo7230453075368039082e_nhds @ real @ C2 ) @ F4 ) ) ) ).
% tendsto_of_real_iff
thf(fact_7044_tendsto__sin,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_Vector_banach @ B )
& ( real_V3459762299906320749_field @ B ) )
=> ! [F3: A > B,A3: B,F4: filter @ A] :
( ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ A3 ) @ F4 )
=> ( filterlim @ A @ B
@ ^ [X2: A] : ( sin @ B @ ( F3 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( sin @ B @ A3 ) )
@ F4 ) ) ) ).
% tendsto_sin
thf(fact_7045_tendsto__Complex,axiom,
! [A: $tType,F3: A > real,A3: real,F4: filter @ A,G2: A > real,B3: real] :
( ( filterlim @ A @ real @ F3 @ ( topolo7230453075368039082e_nhds @ real @ A3 ) @ F4 )
=> ( ( filterlim @ A @ real @ G2 @ ( topolo7230453075368039082e_nhds @ real @ B3 ) @ F4 )
=> ( filterlim @ A @ complex
@ ^ [X2: A] : ( complex2 @ ( F3 @ X2 ) @ ( G2 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ complex @ ( complex2 @ A3 @ B3 ) )
@ F4 ) ) ) ).
% tendsto_Complex
thf(fact_7046_continuous__norm,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F4: filter @ A,F3: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ F4 @ F3 )
=> ( topolo3448309680560233919inuous @ A @ real @ F4
@ ^ [X2: A] : ( real_V7770717601297561774m_norm @ B @ ( F3 @ X2 ) ) ) ) ) ).
% continuous_norm
thf(fact_7047_tendsto__norm,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [F3: A > B,A3: B,F4: filter @ A] :
( ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ A3 ) @ F4 )
=> ( filterlim @ A @ real
@ ^ [X2: A] : ( real_V7770717601297561774m_norm @ B @ ( F3 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( real_V7770717601297561774m_norm @ B @ A3 ) )
@ F4 ) ) ) ).
% tendsto_norm
thf(fact_7048_tendsto__cos,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_Vector_banach @ B )
& ( real_V3459762299906320749_field @ B ) )
=> ! [F3: A > B,A3: B,F4: filter @ A] :
( ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ A3 ) @ F4 )
=> ( filterlim @ A @ B
@ ^ [X2: A] : ( cos @ B @ ( F3 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( cos @ B @ A3 ) )
@ F4 ) ) ) ).
% tendsto_cos
thf(fact_7049_tendsto__arsinh,axiom,
! [B: $tType,F3: B > real,A3: real,F4: filter @ B] :
( ( filterlim @ B @ real @ F3 @ ( topolo7230453075368039082e_nhds @ real @ A3 ) @ F4 )
=> ( filterlim @ B @ real
@ ^ [X2: B] : ( arsinh @ real @ ( F3 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( arsinh @ real @ A3 ) )
@ F4 ) ) ).
% tendsto_arsinh
thf(fact_7050_continuous__sin,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_Vector_banach @ B )
& ( real_V3459762299906320749_field @ B ) )
=> ! [F4: filter @ A,F3: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ F4 @ F3 )
=> ( topolo3448309680560233919inuous @ A @ B @ F4
@ ^ [X2: A] : ( sin @ B @ ( F3 @ X2 ) ) ) ) ) ).
% continuous_sin
thf(fact_7051_continuous__const,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [F4: filter @ A,C2: B] :
( topolo3448309680560233919inuous @ A @ B @ F4
@ ^ [X2: A] : C2 ) ) ).
% continuous_const
thf(fact_7052_tendsto__max,axiom,
! [A: $tType,B: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [X5: B > A,X: A,Net: filter @ B,Y8: B > A,Y: A] :
( ( filterlim @ B @ A @ X5 @ ( topolo7230453075368039082e_nhds @ A @ X ) @ Net )
=> ( ( filterlim @ B @ A @ Y8 @ ( topolo7230453075368039082e_nhds @ A @ Y ) @ Net )
=> ( filterlim @ B @ A
@ ^ [X2: B] : ( ord_max @ A @ ( X5 @ X2 ) @ ( Y8 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( ord_max @ A @ X @ Y ) )
@ Net ) ) ) ) ).
% tendsto_max
thf(fact_7053_continuous__max,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( topolo1944317154257567458pology @ B ) )
=> ! [F4: filter @ A,F3: A > B,G2: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ F4 @ F3 )
=> ( ( topolo3448309680560233919inuous @ A @ B @ F4 @ G2 )
=> ( topolo3448309680560233919inuous @ A @ B @ F4
@ ^ [X2: A] : ( ord_max @ B @ ( F3 @ X2 ) @ ( G2 @ X2 ) ) ) ) ) ) ).
% continuous_max
thf(fact_7054_tendsto__of__int__ceiling,axiom,
! [B: $tType,C: $tType,A: $tType] :
( ( ( ring_1 @ C )
& ( topolo4958980785337419405_space @ C )
& ( archim2362893244070406136eiling @ B )
& ( topolo2564578578187576103pology @ B ) )
=> ! [F3: A > B,L: B,F4: filter @ A] :
( ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ L ) @ F4 )
=> ( ~ ( member @ B @ L @ ( ring_1_Ints @ B ) )
=> ( filterlim @ A @ C
@ ^ [X2: A] : ( ring_1_of_int @ C @ ( archimedean_ceiling @ B @ ( F3 @ X2 ) ) )
@ ( topolo7230453075368039082e_nhds @ C @ ( ring_1_of_int @ C @ ( archimedean_ceiling @ B @ L ) ) )
@ F4 ) ) ) ) ).
% tendsto_of_int_ceiling
thf(fact_7055_tendsto__of__int__floor,axiom,
! [B: $tType,C: $tType,A: $tType] :
( ( ( ring_1 @ C )
& ( topolo4958980785337419405_space @ C )
& ( archim2362893244070406136eiling @ B )
& ( topolo2564578578187576103pology @ B ) )
=> ! [F3: A > B,L: B,F4: filter @ A] :
( ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ L ) @ F4 )
=> ( ~ ( member @ B @ L @ ( ring_1_Ints @ B ) )
=> ( filterlim @ A @ C
@ ^ [X2: A] : ( ring_1_of_int @ C @ ( archim6421214686448440834_floor @ B @ ( F3 @ X2 ) ) )
@ ( topolo7230453075368039082e_nhds @ C @ ( ring_1_of_int @ C @ ( archim6421214686448440834_floor @ B @ L ) ) )
@ F4 ) ) ) ) ).
% tendsto_of_int_floor
thf(fact_7056_tendsto__rabs,axiom,
! [A: $tType,F3: A > real,L: real,F4: filter @ A] :
( ( filterlim @ A @ real @ F3 @ ( topolo7230453075368039082e_nhds @ real @ L ) @ F4 )
=> ( filterlim @ A @ real
@ ^ [X2: A] : ( abs_abs @ real @ ( F3 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( abs_abs @ real @ L ) )
@ F4 ) ) ).
% tendsto_rabs
thf(fact_7057_tendsto__Pair,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( topolo4958980785337419405_space @ B )
& ( topolo4958980785337419405_space @ C ) )
=> ! [F3: A > B,A3: B,F4: filter @ A,G2: A > C,B3: C] :
( ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ A3 ) @ F4 )
=> ( ( filterlim @ A @ C @ G2 @ ( topolo7230453075368039082e_nhds @ C @ B3 ) @ F4 )
=> ( filterlim @ A @ ( product_prod @ B @ C )
@ ^ [X2: A] : ( product_Pair @ B @ C @ ( F3 @ X2 ) @ ( G2 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ A3 @ B3 ) )
@ F4 ) ) ) ) ).
% tendsto_Pair
thf(fact_7058_continuous__Pair,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( topolo4958980785337419405_space @ B )
& ( topolo4958980785337419405_space @ C ) )
=> ! [F4: filter @ A,F3: A > B,G2: A > C] :
( ( topolo3448309680560233919inuous @ A @ B @ F4 @ F3 )
=> ( ( topolo3448309680560233919inuous @ A @ C @ F4 @ G2 )
=> ( topolo3448309680560233919inuous @ A @ ( product_prod @ B @ C ) @ F4
@ ^ [X2: A] : ( product_Pair @ B @ C @ ( F3 @ X2 ) @ ( G2 @ X2 ) ) ) ) ) ) ).
% continuous_Pair
thf(fact_7059_tendsto__mult__one,axiom,
! [B: $tType,D: $tType] :
( ( topolo1898628316856586783d_mult @ B )
=> ! [F3: D > B,F4: filter @ D,G2: D > B] :
( ( filterlim @ D @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ ( one_one @ B ) ) @ F4 )
=> ( ( filterlim @ D @ B @ G2 @ ( topolo7230453075368039082e_nhds @ B @ ( one_one @ B ) ) @ F4 )
=> ( filterlim @ D @ B
@ ^ [X2: D] : ( times_times @ B @ ( F3 @ X2 ) @ ( G2 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( one_one @ B ) )
@ F4 ) ) ) ) ).
% tendsto_mult_one
thf(fact_7060_tendsto__mult__right,axiom,
! [A: $tType,B: $tType] :
( ( topolo4211221413907600880p_mult @ A )
=> ! [F3: B > A,L: A,F4: filter @ B,C2: A] :
( ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ F4 )
=> ( filterlim @ B @ A
@ ^ [X2: B] : ( times_times @ A @ ( F3 @ X2 ) @ C2 )
@ ( topolo7230453075368039082e_nhds @ A @ ( times_times @ A @ L @ C2 ) )
@ F4 ) ) ) ).
% tendsto_mult_right
thf(fact_7061_tendsto__mult__left,axiom,
! [A: $tType,B: $tType] :
( ( topolo4211221413907600880p_mult @ A )
=> ! [F3: B > A,L: A,F4: filter @ B,C2: A] :
( ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ F4 )
=> ( filterlim @ B @ A
@ ^ [X2: B] : ( times_times @ A @ C2 @ ( F3 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( times_times @ A @ C2 @ L ) )
@ F4 ) ) ) ).
% tendsto_mult_left
thf(fact_7062_tendsto__mult,axiom,
! [A: $tType,B: $tType] :
( ( topolo4211221413907600880p_mult @ A )
=> ! [F3: B > A,A3: A,F4: filter @ B,G2: B > A,B3: A] :
( ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ A3 ) @ F4 )
=> ( ( filterlim @ B @ A @ G2 @ ( topolo7230453075368039082e_nhds @ A @ B3 ) @ F4 )
=> ( filterlim @ B @ A
@ ^ [X2: B] : ( times_times @ A @ ( F3 @ X2 ) @ ( G2 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( times_times @ A @ A3 @ B3 ) )
@ F4 ) ) ) ) ).
% tendsto_mult
thf(fact_7063_continuous__mult__right,axiom,
! [A: $tType,B: $tType] :
( ( ( topological_t2_space @ B )
& ( real_V4412858255891104859lgebra @ A ) )
=> ! [F4: filter @ B,F3: B > A,C2: A] :
( ( topolo3448309680560233919inuous @ B @ A @ F4 @ F3 )
=> ( topolo3448309680560233919inuous @ B @ A @ F4
@ ^ [X2: B] : ( times_times @ A @ ( F3 @ X2 ) @ C2 ) ) ) ) ).
% continuous_mult_right
thf(fact_7064_continuous__mult__left,axiom,
! [A: $tType,B: $tType] :
( ( ( topological_t2_space @ B )
& ( real_V4412858255891104859lgebra @ A ) )
=> ! [F4: filter @ B,F3: B > A,C2: A] :
( ( topolo3448309680560233919inuous @ B @ A @ F4 @ F3 )
=> ( topolo3448309680560233919inuous @ B @ A @ F4
@ ^ [X2: B] : ( times_times @ A @ C2 @ ( F3 @ X2 ) ) ) ) ) ).
% continuous_mult_left
thf(fact_7065_continuous__mult_H,axiom,
! [B: $tType,D: $tType] :
( ( ( topological_t2_space @ D )
& ( topolo4211221413907600880p_mult @ B ) )
=> ! [F4: filter @ D,F3: D > B,G2: D > B] :
( ( topolo3448309680560233919inuous @ D @ B @ F4 @ F3 )
=> ( ( topolo3448309680560233919inuous @ D @ B @ F4 @ G2 )
=> ( topolo3448309680560233919inuous @ D @ B @ F4
@ ^ [X2: D] : ( times_times @ B @ ( F3 @ X2 ) @ ( G2 @ X2 ) ) ) ) ) ) ).
% continuous_mult'
thf(fact_7066_continuous__mult,axiom,
! [A: $tType,D: $tType] :
( ( ( topological_t2_space @ D )
& ( real_V4412858255891104859lgebra @ A ) )
=> ! [F4: filter @ D,F3: D > A,G2: D > A] :
( ( topolo3448309680560233919inuous @ D @ A @ F4 @ F3 )
=> ( ( topolo3448309680560233919inuous @ D @ A @ F4 @ G2 )
=> ( topolo3448309680560233919inuous @ D @ A @ F4
@ ^ [X2: D] : ( times_times @ A @ ( F3 @ X2 ) @ ( G2 @ X2 ) ) ) ) ) ) ).
% continuous_mult
thf(fact_7067_tendsto__diff,axiom,
! [A: $tType,B: $tType] :
( ( topolo1633459387980952147up_add @ A )
=> ! [F3: B > A,A3: A,F4: filter @ B,G2: B > A,B3: A] :
( ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ A3 ) @ F4 )
=> ( ( filterlim @ B @ A @ G2 @ ( topolo7230453075368039082e_nhds @ A @ B3 ) @ F4 )
=> ( filterlim @ B @ A
@ ^ [X2: B] : ( minus_minus @ A @ ( F3 @ X2 ) @ ( G2 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( minus_minus @ A @ A3 @ B3 ) )
@ F4 ) ) ) ) ).
% tendsto_diff
thf(fact_7068_continuous__diff,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( topolo1633459387980952147up_add @ B ) )
=> ! [F4: filter @ A,F3: A > B,G2: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ F4 @ F3 )
=> ( ( topolo3448309680560233919inuous @ A @ B @ F4 @ G2 )
=> ( topolo3448309680560233919inuous @ A @ B @ F4
@ ^ [X2: A] : ( minus_minus @ B @ ( F3 @ X2 ) @ ( G2 @ X2 ) ) ) ) ) ) ).
% continuous_diff
thf(fact_7069_continuous__add,axiom,
! [B: $tType,D: $tType] :
( ( ( topological_t2_space @ D )
& ( topolo6943815403480290642id_add @ B ) )
=> ! [F4: filter @ D,F3: D > B,G2: D > B] :
( ( topolo3448309680560233919inuous @ D @ B @ F4 @ F3 )
=> ( ( topolo3448309680560233919inuous @ D @ B @ F4 @ G2 )
=> ( topolo3448309680560233919inuous @ D @ B @ F4
@ ^ [X2: D] : ( plus_plus @ B @ ( F3 @ X2 ) @ ( G2 @ X2 ) ) ) ) ) ) ).
% continuous_add
thf(fact_7070_tendsto__add,axiom,
! [A: $tType,B: $tType] :
( ( topolo6943815403480290642id_add @ A )
=> ! [F3: B > A,A3: A,F4: filter @ B,G2: B > A,B3: A] :
( ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ A3 ) @ F4 )
=> ( ( filterlim @ B @ A @ G2 @ ( topolo7230453075368039082e_nhds @ A @ B3 ) @ F4 )
=> ( filterlim @ B @ A
@ ^ [X2: B] : ( plus_plus @ A @ ( F3 @ X2 ) @ ( G2 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( plus_plus @ A @ A3 @ B3 ) )
@ F4 ) ) ) ) ).
% tendsto_add
thf(fact_7071_tendsto__add__const__iff,axiom,
! [A: $tType,B: $tType] :
( ( topolo1633459387980952147up_add @ A )
=> ! [C2: A,F3: B > A,D2: A,F4: filter @ B] :
( ( filterlim @ B @ A
@ ^ [X2: B] : ( plus_plus @ A @ C2 @ ( F3 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( plus_plus @ A @ C2 @ D2 ) )
@ F4 )
= ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ D2 ) @ F4 ) ) ) ).
% tendsto_add_const_iff
thf(fact_7072_tendsto__add__zero,axiom,
! [B: $tType,D: $tType] :
( ( topolo6943815403480290642id_add @ B )
=> ! [F3: D > B,F4: filter @ D,G2: D > B] :
( ( filterlim @ D @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) ) @ F4 )
=> ( ( filterlim @ D @ B @ G2 @ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) ) @ F4 )
=> ( filterlim @ D @ B
@ ^ [X2: D] : ( plus_plus @ B @ ( F3 @ X2 ) @ ( G2 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ F4 ) ) ) ) ).
% tendsto_add_zero
thf(fact_7073_Lim__transform__eq,axiom,
! [A: $tType,B: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: B > A,G2: B > A,F4: filter @ B,A3: A] :
( ( filterlim @ B @ A
@ ^ [X2: B] : ( minus_minus @ A @ ( F3 @ X2 ) @ ( G2 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ F4 )
=> ( ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ A3 ) @ F4 )
= ( filterlim @ B @ A @ G2 @ ( topolo7230453075368039082e_nhds @ A @ A3 ) @ F4 ) ) ) ) ).
% Lim_transform_eq
thf(fact_7074_LIM__zero__cancel,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [F3: A > B,L: B,F4: filter @ A] :
( ( filterlim @ A @ B
@ ^ [X2: A] : ( minus_minus @ B @ ( F3 @ X2 ) @ L )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ F4 )
=> ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ L ) @ F4 ) ) ) ).
% LIM_zero_cancel
thf(fact_7075_Lim__transform2,axiom,
! [A: $tType,B: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: B > A,A3: A,F4: filter @ B,G2: B > A] :
( ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ A3 ) @ F4 )
=> ( ( filterlim @ B @ A
@ ^ [X2: B] : ( minus_minus @ A @ ( F3 @ X2 ) @ ( G2 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ F4 )
=> ( filterlim @ B @ A @ G2 @ ( topolo7230453075368039082e_nhds @ A @ A3 ) @ F4 ) ) ) ) ).
% Lim_transform2
thf(fact_7076_Lim__transform,axiom,
! [A: $tType,B: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [G2: B > A,A3: A,F4: filter @ B,F3: B > A] :
( ( filterlim @ B @ A @ G2 @ ( topolo7230453075368039082e_nhds @ A @ A3 ) @ F4 )
=> ( ( filterlim @ B @ A
@ ^ [X2: B] : ( minus_minus @ A @ ( F3 @ X2 ) @ ( G2 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ F4 )
=> ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ A3 ) @ F4 ) ) ) ) ).
% Lim_transform
thf(fact_7077_LIM__zero__iff,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [F3: A > B,L: B,F4: filter @ A] :
( ( filterlim @ A @ B
@ ^ [X2: A] : ( minus_minus @ B @ ( F3 @ X2 ) @ L )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ F4 )
= ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ L ) @ F4 ) ) ) ).
% LIM_zero_iff
thf(fact_7078_LIM__zero,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [F3: A > B,L: B,F4: filter @ A] :
( ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ L ) @ F4 )
=> ( filterlim @ A @ B
@ ^ [X2: A] : ( minus_minus @ B @ ( F3 @ X2 ) @ L )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ F4 ) ) ) ).
% LIM_zero
thf(fact_7079_tendsto__artanh,axiom,
! [A: $tType,F3: A > real,A3: real,F4: filter @ A] :
( ( filterlim @ A @ real @ F3 @ ( topolo7230453075368039082e_nhds @ real @ A3 ) @ F4 )
=> ( ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ A3 )
=> ( ( ord_less @ real @ A3 @ ( one_one @ real ) )
=> ( filterlim @ A @ real
@ ^ [X2: A] : ( artanh @ real @ ( F3 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( artanh @ real @ A3 ) )
@ F4 ) ) ) ) ).
% tendsto_artanh
thf(fact_7080_filterlim__top,axiom,
! [B: $tType,A: $tType,F3: A > B,F4: filter @ A] : ( filterlim @ A @ B @ F3 @ ( top_top @ ( filter @ B ) ) @ F4 ) ).
% filterlim_top
thf(fact_7081_DERIV__LIM__iff,axiom,
! [A: $tType] :
( ( ( inverse @ A )
& ( real_V822414075346904944vector @ A ) )
=> ! [F3: A > A,A3: A,D3: A] :
( ( filterlim @ A @ A
@ ^ [H: A] : ( divide_divide @ A @ ( minus_minus @ A @ ( F3 @ ( plus_plus @ A @ A3 @ H ) ) @ ( F3 @ A3 ) ) @ H )
@ ( topolo7230453075368039082e_nhds @ A @ D3 )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) )
= ( filterlim @ A @ A
@ ^ [X2: A] : ( divide_divide @ A @ ( minus_minus @ A @ ( F3 @ X2 ) @ ( F3 @ A3 ) ) @ ( minus_minus @ A @ X2 @ A3 ) )
@ ( topolo7230453075368039082e_nhds @ A @ D3 )
@ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% DERIV_LIM_iff
thf(fact_7082_LIM__fun__less__zero,axiom,
! [F3: real > real,L: real,C2: real] :
( ( filterlim @ real @ real @ F3 @ ( topolo7230453075368039082e_nhds @ real @ L ) @ ( topolo174197925503356063within @ real @ C2 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( ord_less @ real @ L @ ( zero_zero @ real ) )
=> ? [R2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R2 )
& ! [X6: real] :
( ( ( X6 != C2 )
& ( ord_less @ real @ ( abs_abs @ real @ ( minus_minus @ real @ C2 @ X6 ) ) @ R2 ) )
=> ( ord_less @ real @ ( F3 @ X6 ) @ ( zero_zero @ real ) ) ) ) ) ) ).
% LIM_fun_less_zero
thf(fact_7083_LIM__fun__not__zero,axiom,
! [F3: real > real,L: real,C2: real] :
( ( filterlim @ real @ real @ F3 @ ( topolo7230453075368039082e_nhds @ real @ L ) @ ( topolo174197925503356063within @ real @ C2 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( L
!= ( zero_zero @ real ) )
=> ? [R2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R2 )
& ! [X6: real] :
( ( ( X6 != C2 )
& ( ord_less @ real @ ( abs_abs @ real @ ( minus_minus @ real @ C2 @ X6 ) ) @ R2 ) )
=> ( ( F3 @ X6 )
!= ( zero_zero @ real ) ) ) ) ) ) ).
% LIM_fun_not_zero
thf(fact_7084_LIM__fun__gt__zero,axiom,
! [F3: real > real,L: real,C2: real] :
( ( filterlim @ real @ real @ F3 @ ( topolo7230453075368039082e_nhds @ real @ L ) @ ( topolo174197925503356063within @ real @ C2 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ L )
=> ? [R2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R2 )
& ! [X6: real] :
( ( ( X6 != C2 )
& ( ord_less @ real @ ( abs_abs @ real @ ( minus_minus @ real @ C2 @ X6 ) ) @ R2 ) )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( F3 @ X6 ) ) ) ) ) ) ).
% LIM_fun_gt_zero
thf(fact_7085_LIM__compose2,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( topolo4958980785337419405_space @ B )
& ( topolo4958980785337419405_space @ C ) )
=> ! [F3: A > B,B3: B,A3: A,G2: B > C,C2: C] :
( ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ B3 ) @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) )
=> ( ( filterlim @ B @ C @ G2 @ ( topolo7230453075368039082e_nhds @ C @ C2 ) @ ( topolo174197925503356063within @ B @ B3 @ ( top_top @ ( set @ B ) ) ) )
=> ( ? [D5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D5 )
& ! [X3: A] :
( ( ( X3 != A3 )
& ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ X3 @ A3 ) ) @ D5 ) )
=> ( ( F3 @ X3 )
!= B3 ) ) )
=> ( filterlim @ A @ C
@ ^ [X2: A] : ( G2 @ ( F3 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ C @ C2 )
@ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ).
% LIM_compose2
thf(fact_7086_isCont__real__sqrt,axiom,
! [X: real] : ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) @ sqrt ) ).
% isCont_real_sqrt
thf(fact_7087_isCont__real__root,axiom,
! [X: real,N: nat] : ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) @ ( root @ N ) ) ).
% isCont_real_root
thf(fact_7088_continuous__at__within__divide,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_V3459762299906320749_field @ B ) )
=> ! [A3: A,S2: set @ A,F3: A > B,G2: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A3 @ S2 ) @ F3 )
=> ( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A3 @ S2 ) @ G2 )
=> ( ( ( G2 @ A3 )
!= ( zero_zero @ B ) )
=> ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A3 @ S2 )
@ ^ [X2: A] : ( divide_divide @ B @ ( F3 @ X2 ) @ ( G2 @ X2 ) ) ) ) ) ) ) ).
% continuous_at_within_divide
thf(fact_7089_isCont__add,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( topolo6943815403480290642id_add @ B ) )
=> ! [A3: A,F3: A > B,G2: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) @ F3 )
=> ( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) @ G2 )
=> ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) )
@ ^ [X2: A] : ( plus_plus @ B @ ( F3 @ X2 ) @ ( G2 @ X2 ) ) ) ) ) ) ).
% isCont_add
thf(fact_7090_isCont__mult,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_V4412858255891104859lgebra @ B ) )
=> ! [A3: A,F3: A > B,G2: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) @ F3 )
=> ( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) @ G2 )
=> ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) )
@ ^ [X2: A] : ( times_times @ B @ ( F3 @ X2 ) @ ( G2 @ X2 ) ) ) ) ) ) ).
% isCont_mult
thf(fact_7091_isCont__diff,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [A3: A,F3: A > B,G2: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) @ F3 )
=> ( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) @ G2 )
=> ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) )
@ ^ [X2: A] : ( minus_minus @ B @ ( F3 @ X2 ) @ ( G2 @ X2 ) ) ) ) ) ) ).
% isCont_diff
thf(fact_7092_isCont__power,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( power @ B )
& ( real_V4412858255891104859lgebra @ B ) )
=> ! [A3: A,F3: A > B,N: nat] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) @ F3 )
=> ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) )
@ ^ [X2: A] : ( power_power @ B @ ( F3 @ X2 ) @ N ) ) ) ) ).
% isCont_power
thf(fact_7093_isCont__minus,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [A3: A,F3: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) @ F3 )
=> ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) )
@ ^ [X2: A] : ( uminus_uminus @ B @ ( F3 @ X2 ) ) ) ) ) ).
% isCont_minus
thf(fact_7094_continuous__at__within__inverse,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_V8999393235501362500lgebra @ B ) )
=> ! [A3: A,S2: set @ A,F3: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A3 @ S2 ) @ F3 )
=> ( ( ( F3 @ A3 )
!= ( zero_zero @ B ) )
=> ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A3 @ S2 )
@ ^ [X2: A] : ( inverse_inverse @ B @ ( F3 @ X2 ) ) ) ) ) ) ).
% continuous_at_within_inverse
thf(fact_7095_isCont__sum,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( ( topological_t2_space @ B )
& ( topolo5987344860129210374id_add @ C ) )
=> ! [A2: set @ A,A3: B,F3: A > B > C] :
( ! [X3: A] :
( ( member @ A @ X3 @ A2 )
=> ( topolo3448309680560233919inuous @ B @ C @ ( topolo174197925503356063within @ B @ A3 @ ( top_top @ ( set @ B ) ) ) @ ( F3 @ X3 ) ) )
=> ( topolo3448309680560233919inuous @ B @ C @ ( topolo174197925503356063within @ B @ A3 @ ( top_top @ ( set @ B ) ) )
@ ^ [X2: B] :
( groups7311177749621191930dd_sum @ A @ C
@ ^ [I4: A] : ( F3 @ I4 @ X2 )
@ A2 ) ) ) ) ).
% isCont_sum
thf(fact_7096_continuous__at__within__sgn,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [A3: A,S2: set @ A,F3: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A3 @ S2 ) @ F3 )
=> ( ( ( F3 @ A3 )
!= ( zero_zero @ B ) )
=> ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A3 @ S2 )
@ ^ [X2: A] : ( sgn_sgn @ B @ ( F3 @ X2 ) ) ) ) ) ) ).
% continuous_at_within_sgn
thf(fact_7097_isCont__cos_H,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_Vector_banach @ B )
& ( real_V3459762299906320749_field @ B ) )
=> ! [A3: A,F3: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) @ F3 )
=> ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) )
@ ^ [X2: A] : ( cos @ B @ ( F3 @ X2 ) ) ) ) ) ).
% isCont_cos'
thf(fact_7098_isCont__sin_H,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_Vector_banach @ B )
& ( real_V3459762299906320749_field @ B ) )
=> ! [A3: A,F3: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) @ F3 )
=> ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) )
@ ^ [X2: A] : ( sin @ B @ ( F3 @ X2 ) ) ) ) ) ).
% isCont_sin'
thf(fact_7099_isCont__exp_H,axiom,
! [A: $tType,C: $tType] :
( ( ( topological_t2_space @ C )
& ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [A3: C,F3: C > A] :
( ( topolo3448309680560233919inuous @ C @ A @ ( topolo174197925503356063within @ C @ A3 @ ( top_top @ ( set @ C ) ) ) @ F3 )
=> ( topolo3448309680560233919inuous @ C @ A @ ( topolo174197925503356063within @ C @ A3 @ ( top_top @ ( set @ C ) ) )
@ ^ [X2: C] : ( exp @ A @ ( F3 @ X2 ) ) ) ) ) ).
% isCont_exp'
thf(fact_7100_isCont__pochhammer,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [Z: A,N: nat] :
( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ Z @ ( top_top @ ( set @ A ) ) )
@ ^ [Z4: A] : ( comm_s3205402744901411588hammer @ A @ Z4 @ N ) ) ) ).
% isCont_pochhammer
thf(fact_7101_DERIV__def,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,D3: A,X: A] :
( ( has_field_derivative @ A @ F3 @ D3 @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) )
= ( filterlim @ A @ A
@ ^ [H: A] : ( divide_divide @ A @ ( minus_minus @ A @ ( F3 @ ( plus_plus @ A @ X @ H ) ) @ ( F3 @ X ) ) @ H )
@ ( topolo7230453075368039082e_nhds @ A @ D3 )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% DERIV_def
thf(fact_7102_DERIV__D,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,D3: A,X: A] :
( ( has_field_derivative @ A @ F3 @ D3 @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) )
=> ( filterlim @ A @ A
@ ^ [H: A] : ( divide_divide @ A @ ( minus_minus @ A @ ( F3 @ ( plus_plus @ A @ X @ H ) ) @ ( F3 @ X ) ) @ H )
@ ( topolo7230453075368039082e_nhds @ A @ D3 )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% DERIV_D
thf(fact_7103_lim__exp__minus__1,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ( filterlim @ A @ A
@ ^ [Z4: A] : ( divide_divide @ A @ ( minus_minus @ A @ ( exp @ A @ Z4 ) @ ( one_one @ A ) ) @ Z4 )
@ ( topolo7230453075368039082e_nhds @ A @ ( one_one @ A ) )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ).
% lim_exp_minus_1
thf(fact_7104_atLeastPlusOneLessThan__greaterThanLessThan__int,axiom,
! [L: int,U2: int] :
( ( set_or7035219750837199246ssThan @ int @ ( plus_plus @ int @ L @ ( one_one @ int ) ) @ U2 )
= ( set_or5935395276787703475ssThan @ int @ L @ U2 ) ) ).
% atLeastPlusOneLessThan_greaterThanLessThan_int
thf(fact_7105_lemma__termdiff4,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [K: real,F3: A > B,K4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K )
=> ( ! [H4: A] :
( ( H4
!= ( zero_zero @ A ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ H4 ) @ K )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( F3 @ H4 ) ) @ ( times_times @ real @ K4 @ ( real_V7770717601297561774m_norm @ A @ H4 ) ) ) ) )
=> ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) ) @ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% lemma_termdiff4
thf(fact_7106_isCont__inverse__function2,axiom,
! [A3: real,X: real,B3: real,G2: real > real,F3: real > real] :
( ( ord_less @ real @ A3 @ X )
=> ( ( ord_less @ real @ X @ B3 )
=> ( ! [Z3: real] :
( ( ord_less_eq @ real @ A3 @ Z3 )
=> ( ( ord_less_eq @ real @ Z3 @ B3 )
=> ( ( G2 @ ( F3 @ Z3 ) )
= Z3 ) ) )
=> ( ! [Z3: real] :
( ( ord_less_eq @ real @ A3 @ Z3 )
=> ( ( ord_less_eq @ real @ Z3 @ B3 )
=> ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ Z3 @ ( top_top @ ( set @ real ) ) ) @ F3 ) ) )
=> ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ ( F3 @ X ) @ ( top_top @ ( set @ real ) ) ) @ G2 ) ) ) ) ) ).
% isCont_inverse_function2
thf(fact_7107_field__has__derivative__at,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,D3: A,X: A] :
( ( has_derivative @ A @ A @ F3 @ ( times_times @ A @ D3 ) @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) )
= ( filterlim @ A @ A
@ ^ [H: A] : ( divide_divide @ A @ ( minus_minus @ A @ ( F3 @ ( plus_plus @ A @ X @ H ) ) @ ( F3 @ X ) ) @ H )
@ ( topolo7230453075368039082e_nhds @ A @ D3 )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% field_has_derivative_at
thf(fact_7108_isCont__divide,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_V3459762299906320749_field @ B ) )
=> ! [A3: A,F3: A > B,G2: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) @ F3 )
=> ( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) @ G2 )
=> ( ( ( G2 @ A3 )
!= ( zero_zero @ B ) )
=> ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) )
@ ^ [X2: A] : ( divide_divide @ B @ ( F3 @ X2 ) @ ( G2 @ X2 ) ) ) ) ) ) ) ).
% isCont_divide
thf(fact_7109_isCont__ln,axiom,
! [X: real] :
( ( X
!= ( zero_zero @ real ) )
=> ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) @ ( ln_ln @ real ) ) ) ).
% isCont_ln
thf(fact_7110_isCont__sgn,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [A3: A,F3: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) @ F3 )
=> ( ( ( F3 @ A3 )
!= ( zero_zero @ B ) )
=> ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) )
@ ^ [X2: A] : ( sgn_sgn @ B @ ( F3 @ X2 ) ) ) ) ) ) ).
% isCont_sgn
thf(fact_7111_filterlim__at__to__0,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: A > B,F4: filter @ B,A3: A] :
( ( filterlim @ A @ B @ F3 @ F4 @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) )
= ( filterlim @ A @ B
@ ^ [X2: A] : ( F3 @ ( plus_plus @ A @ X2 @ A3 ) )
@ F4
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% filterlim_at_to_0
thf(fact_7112_continuous__within__tan,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A,S2: set @ A,F3: A > A] :
( ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ X @ S2 ) @ F3 )
=> ( ( ( cos @ A @ ( F3 @ X ) )
!= ( zero_zero @ A ) )
=> ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ X @ S2 )
@ ^ [X2: A] : ( tan @ A @ ( F3 @ X2 ) ) ) ) ) ) ).
% continuous_within_tan
thf(fact_7113_atLeastPlusOneLessThan__greaterThanLessThan__integer,axiom,
! [L: code_integer,U2: code_integer] :
( ( set_or7035219750837199246ssThan @ code_integer @ ( plus_plus @ code_integer @ L @ ( one_one @ code_integer ) ) @ U2 )
= ( set_or5935395276787703475ssThan @ code_integer @ L @ U2 ) ) ).
% atLeastPlusOneLessThan_greaterThanLessThan_integer
thf(fact_7114_continuous__within__cot,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A,S2: set @ A,F3: A > A] :
( ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ X @ S2 ) @ F3 )
=> ( ( ( sin @ A @ ( F3 @ X ) )
!= ( zero_zero @ A ) )
=> ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ X @ S2 )
@ ^ [X2: A] : ( cot @ A @ ( F3 @ X2 ) ) ) ) ) ) ).
% continuous_within_cot
thf(fact_7115_continuous__at__within__tanh,axiom,
! [A: $tType,C: $tType] :
( ( ( topological_t2_space @ C )
& ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: C,A2: set @ C,F3: C > A] :
( ( topolo3448309680560233919inuous @ C @ A @ ( topolo174197925503356063within @ C @ X @ A2 ) @ F3 )
=> ( ( ( cosh @ A @ ( F3 @ X ) )
!= ( zero_zero @ A ) )
=> ( topolo3448309680560233919inuous @ C @ A @ ( topolo174197925503356063within @ C @ X @ A2 )
@ ^ [X2: C] : ( tanh @ A @ ( F3 @ X2 ) ) ) ) ) ) ).
% continuous_at_within_tanh
thf(fact_7116_isCont__has__Ub,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [A3: real,B3: real,F3: real > A] :
( ( ord_less_eq @ real @ A3 @ B3 )
=> ( ! [X3: real] :
( ( ( ord_less_eq @ real @ A3 @ X3 )
& ( ord_less_eq @ real @ X3 @ B3 ) )
=> ( topolo3448309680560233919inuous @ real @ A @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) @ F3 ) )
=> ? [M9: A] :
( ! [X6: real] :
( ( ( ord_less_eq @ real @ A3 @ X6 )
& ( ord_less_eq @ real @ X6 @ B3 ) )
=> ( ord_less_eq @ A @ ( F3 @ X6 ) @ M9 ) )
& ! [N8: A] :
( ( ord_less @ A @ N8 @ M9 )
=> ? [X3: real] :
( ( ord_less_eq @ real @ A3 @ X3 )
& ( ord_less_eq @ real @ X3 @ B3 )
& ( ord_less @ A @ N8 @ ( F3 @ X3 ) ) ) ) ) ) ) ) ).
% isCont_has_Ub
thf(fact_7117_isCont__tan,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A] :
( ( ( cos @ A @ X )
!= ( zero_zero @ A ) )
=> ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) @ ( tan @ A ) ) ) ) ).
% isCont_tan
thf(fact_7118_isCont__cot,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A] :
( ( ( sin @ A @ X )
!= ( zero_zero @ A ) )
=> ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) @ ( cot @ A ) ) ) ) ).
% isCont_cot
thf(fact_7119_isCont__tanh,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A] :
( ( ( cosh @ A @ X )
!= ( zero_zero @ A ) )
=> ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) @ ( tanh @ A ) ) ) ) ).
% isCont_tanh
thf(fact_7120_powser__limit__0__strong,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [S2: real,A3: nat > A,F3: A > A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ S2 )
=> ( ! [X3: A] :
( ( X3
!= ( zero_zero @ A ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X3 ) @ S2 )
=> ( sums @ A
@ ^ [N4: nat] : ( times_times @ A @ ( A3 @ N4 ) @ ( power_power @ A @ X3 @ N4 ) )
@ ( F3 @ X3 ) ) ) )
=> ( filterlim @ A @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ ( A3 @ ( zero_zero @ nat ) ) ) @ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% powser_limit_0_strong
thf(fact_7121_powser__limit__0,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [S2: real,A3: nat > A,F3: A > A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ S2 )
=> ( ! [X3: A] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X3 ) @ S2 )
=> ( sums @ A
@ ^ [N4: nat] : ( times_times @ A @ ( A3 @ N4 ) @ ( power_power @ A @ X3 @ N4 ) )
@ ( F3 @ X3 ) ) )
=> ( filterlim @ A @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ ( A3 @ ( zero_zero @ nat ) ) ) @ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% powser_limit_0
thf(fact_7122_lemma__termdiff5,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_Vector_banach @ B ) )
=> ! [K: real,F3: nat > real,G2: A > nat > B] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K )
=> ( ( summable @ real @ F3 )
=> ( ! [H4: A,N2: nat] :
( ( H4
!= ( zero_zero @ A ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ H4 ) @ K )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( G2 @ H4 @ N2 ) ) @ ( times_times @ real @ ( F3 @ N2 ) @ ( real_V7770717601297561774m_norm @ A @ H4 ) ) ) ) )
=> ( filterlim @ A @ B
@ ^ [H: A] : ( suminf @ B @ ( G2 @ H ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ).
% lemma_termdiff5
thf(fact_7123_isCont__tan_H,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [A3: A,F3: A > A] :
( ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) @ F3 )
=> ( ( ( cos @ A @ ( F3 @ A3 ) )
!= ( zero_zero @ A ) )
=> ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) )
@ ^ [X2: A] : ( tan @ A @ ( F3 @ X2 ) ) ) ) ) ) ).
% isCont_tan'
thf(fact_7124_isCont__arcosh,axiom,
! [X: real] :
( ( ord_less @ real @ ( one_one @ real ) @ X )
=> ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) @ ( arcosh @ real ) ) ) ).
% isCont_arcosh
thf(fact_7125_LIM__cos__div__sin,axiom,
( filterlim @ real @ real
@ ^ [X2: real] : ( divide_divide @ real @ ( cos @ real @ X2 ) @ ( sin @ real @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ ( topolo174197925503356063within @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( top_top @ ( set @ real ) ) ) ) ).
% LIM_cos_div_sin
thf(fact_7126_isCont__cot_H,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [A3: A,F3: A > A] :
( ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) @ F3 )
=> ( ( ( sin @ A @ ( F3 @ A3 ) )
!= ( zero_zero @ A ) )
=> ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) )
@ ^ [X2: A] : ( cot @ A @ ( F3 @ X2 ) ) ) ) ) ) ).
% isCont_cot'
thf(fact_7127_DERIV__inverse__function,axiom,
! [F3: real > real,D3: real,G2: real > real,X: real,A3: real,B3: real] :
( ( has_field_derivative @ real @ F3 @ D3 @ ( topolo174197925503356063within @ real @ ( G2 @ X ) @ ( top_top @ ( set @ real ) ) ) )
=> ( ( D3
!= ( zero_zero @ real ) )
=> ( ( ord_less @ real @ A3 @ X )
=> ( ( ord_less @ real @ X @ B3 )
=> ( ! [Y4: real] :
( ( ord_less @ real @ A3 @ Y4 )
=> ( ( ord_less @ real @ Y4 @ B3 )
=> ( ( F3 @ ( G2 @ Y4 ) )
= Y4 ) ) )
=> ( ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) @ G2 )
=> ( has_field_derivative @ real @ G2 @ ( inverse_inverse @ real @ D3 ) @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ) ) ) ).
% DERIV_inverse_function
thf(fact_7128_isCont__polynom,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [A3: A,C2: nat > A,N: nat] :
( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) )
@ ^ [W2: A] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( C2 @ I4 ) @ ( power_power @ A @ W2 @ I4 ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% isCont_polynom
thf(fact_7129_isCont__powser__converges__everywhere,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [C2: nat > A,X: A] :
( ! [Y4: A] :
( summable @ A
@ ^ [N4: nat] : ( times_times @ A @ ( C2 @ N4 ) @ ( power_power @ A @ Y4 @ N4 ) ) )
=> ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) )
@ ^ [X2: A] :
( suminf @ A
@ ^ [N4: nat] : ( times_times @ A @ ( C2 @ N4 ) @ ( power_power @ A @ X2 @ N4 ) ) ) ) ) ) ).
% isCont_powser_converges_everywhere
thf(fact_7130_isCont__arccos,axiom,
! [X: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X )
=> ( ( ord_less @ real @ X @ ( one_one @ real ) )
=> ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) @ arccos ) ) ) ).
% isCont_arccos
thf(fact_7131_isCont__arcsin,axiom,
! [X: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X )
=> ( ( ord_less @ real @ X @ ( one_one @ real ) )
=> ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) @ arcsin ) ) ) ).
% isCont_arcsin
thf(fact_7132_LIM__less__bound,axiom,
! [B3: real,X: real,F3: real > real] :
( ( ord_less @ real @ B3 @ X )
=> ( ! [X3: real] :
( ( member @ real @ X3 @ ( set_or5935395276787703475ssThan @ real @ B3 @ X ) )
=> ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( F3 @ X3 ) ) )
=> ( ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) @ F3 )
=> ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( F3 @ X ) ) ) ) ) ).
% LIM_less_bound
thf(fact_7133_isCont__artanh,axiom,
! [X: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X )
=> ( ( ord_less @ real @ X @ ( one_one @ real ) )
=> ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) @ ( artanh @ real ) ) ) ) ).
% isCont_artanh
thf(fact_7134_isCont__inverse__function,axiom,
! [D2: real,X: real,G2: real > real,F3: real > real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D2 )
=> ( ! [Z3: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ ( minus_minus @ real @ Z3 @ X ) ) @ D2 )
=> ( ( G2 @ ( F3 @ Z3 ) )
= Z3 ) )
=> ( ! [Z3: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ ( minus_minus @ real @ Z3 @ X ) ) @ D2 )
=> ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ Z3 @ ( top_top @ ( set @ real ) ) ) @ F3 ) )
=> ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ ( F3 @ X ) @ ( top_top @ ( set @ real ) ) ) @ G2 ) ) ) ) ).
% isCont_inverse_function
thf(fact_7135_GMVT_H,axiom,
! [A3: real,B3: real,F3: real > real,G2: real > real,G5: real > real,F7: real > real] :
( ( ord_less @ real @ A3 @ B3 )
=> ( ! [Z3: real] :
( ( ord_less_eq @ real @ A3 @ Z3 )
=> ( ( ord_less_eq @ real @ Z3 @ B3 )
=> ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ Z3 @ ( top_top @ ( set @ real ) ) ) @ F3 ) ) )
=> ( ! [Z3: real] :
( ( ord_less_eq @ real @ A3 @ Z3 )
=> ( ( ord_less_eq @ real @ Z3 @ B3 )
=> ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ Z3 @ ( top_top @ ( set @ real ) ) ) @ G2 ) ) )
=> ( ! [Z3: real] :
( ( ord_less @ real @ A3 @ Z3 )
=> ( ( ord_less @ real @ Z3 @ B3 )
=> ( has_field_derivative @ real @ G2 @ ( G5 @ Z3 ) @ ( topolo174197925503356063within @ real @ Z3 @ ( top_top @ ( set @ real ) ) ) ) ) )
=> ( ! [Z3: real] :
( ( ord_less @ real @ A3 @ Z3 )
=> ( ( ord_less @ real @ Z3 @ B3 )
=> ( has_field_derivative @ real @ F3 @ ( F7 @ Z3 ) @ ( topolo174197925503356063within @ real @ Z3 @ ( top_top @ ( set @ real ) ) ) ) ) )
=> ? [C5: real] :
( ( ord_less @ real @ A3 @ C5 )
& ( ord_less @ real @ C5 @ B3 )
& ( ( times_times @ real @ ( minus_minus @ real @ ( F3 @ B3 ) @ ( F3 @ A3 ) ) @ ( G5 @ C5 ) )
= ( times_times @ real @ ( minus_minus @ real @ ( G2 @ B3 ) @ ( G2 @ A3 ) ) @ ( F7 @ C5 ) ) ) ) ) ) ) ) ) ).
% GMVT'
thf(fact_7136_floor__has__real__derivative,axiom,
! [A: $tType] :
( ( ( archim2362893244070406136eiling @ A )
& ( topolo2564578578187576103pology @ A ) )
=> ! [X: real,F3: real > A] :
( ( topolo3448309680560233919inuous @ real @ A @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) @ F3 )
=> ( ~ ( member @ A @ ( F3 @ X ) @ ( ring_1_Ints @ A ) )
=> ( has_field_derivative @ real
@ ^ [X2: real] : ( ring_1_of_int @ real @ ( archim6421214686448440834_floor @ A @ ( F3 @ X2 ) ) )
@ ( zero_zero @ real )
@ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ).
% floor_has_real_derivative
thf(fact_7137_isCont__powser,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [C2: nat > A,K4: A,X: A] :
( ( summable @ A
@ ^ [N4: nat] : ( times_times @ A @ ( C2 @ N4 ) @ ( power_power @ A @ K4 @ N4 ) ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ ( real_V7770717601297561774m_norm @ A @ K4 ) )
=> ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) )
@ ^ [X2: A] :
( suminf @ A
@ ^ [N4: nat] : ( times_times @ A @ ( C2 @ N4 ) @ ( power_power @ A @ X2 @ N4 ) ) ) ) ) ) ) ).
% isCont_powser
thf(fact_7138_isCont__powser_H,axiom,
! [Aa: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_Vector_banach @ Aa )
& ( real_V3459762299906320749_field @ Aa ) )
=> ! [A3: A,F3: A > Aa,C2: nat > Aa,K4: Aa] :
( ( topolo3448309680560233919inuous @ A @ Aa @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) @ F3 )
=> ( ( summable @ Aa
@ ^ [N4: nat] : ( times_times @ Aa @ ( C2 @ N4 ) @ ( power_power @ Aa @ K4 @ N4 ) ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ Aa @ ( F3 @ A3 ) ) @ ( real_V7770717601297561774m_norm @ Aa @ K4 ) )
=> ( topolo3448309680560233919inuous @ A @ Aa @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) )
@ ^ [X2: A] :
( suminf @ Aa
@ ^ [N4: nat] : ( times_times @ Aa @ ( C2 @ N4 ) @ ( power_power @ Aa @ ( F3 @ X2 ) @ N4 ) ) ) ) ) ) ) ) ).
% isCont_powser'
thf(fact_7139_summable__Leibniz_I3_J,axiom,
! [A3: nat > real] :
( ( filterlim @ nat @ real @ A3 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) )
=> ( ( topological_monoseq @ real @ A3 )
=> ( ( ord_less @ real @ ( A3 @ ( zero_zero @ nat ) ) @ ( zero_zero @ real ) )
=> ! [N11: nat] :
( member @ real
@ ( suminf @ real
@ ^ [I4: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I4 ) @ ( A3 @ I4 ) ) )
@ ( set_or1337092689740270186AtMost @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I4: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I4 ) @ ( A3 @ I4 ) )
@ ( set_ord_lessThan @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N11 ) @ ( one_one @ nat ) ) ) )
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I4: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I4 ) @ ( A3 @ I4 ) )
@ ( set_ord_lessThan @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N11 ) ) ) ) ) ) ) ) ).
% summable_Leibniz(3)
thf(fact_7140_summable__Leibniz_I2_J,axiom,
! [A3: nat > real] :
( ( filterlim @ nat @ real @ A3 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) )
=> ( ( topological_monoseq @ real @ A3 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( A3 @ ( zero_zero @ nat ) ) )
=> ! [N11: nat] :
( member @ real
@ ( suminf @ real
@ ^ [I4: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I4 ) @ ( A3 @ I4 ) ) )
@ ( set_or1337092689740270186AtMost @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I4: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I4 ) @ ( A3 @ I4 ) )
@ ( set_ord_lessThan @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N11 ) ) )
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I4: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I4 ) @ ( A3 @ I4 ) )
@ ( set_ord_lessThan @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N11 ) @ ( one_one @ nat ) ) ) ) ) ) ) ) ) ).
% summable_Leibniz(2)
thf(fact_7141_trivial__limit__sequentially,axiom,
( ( at_top @ nat )
!= ( bot_bot @ ( filter @ nat ) ) ) ).
% trivial_limit_sequentially
thf(fact_7142_tendsto__zero__mult__left__iff,axiom,
! [A: $tType] :
( ( ( field @ A )
& ( topolo4211221413907600880p_mult @ A ) )
=> ! [C2: A,A3: nat > A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( filterlim @ nat @ A
@ ^ [N4: nat] : ( times_times @ A @ C2 @ ( A3 @ N4 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ ( at_top @ nat ) )
= ( filterlim @ nat @ A @ A3 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ ( at_top @ nat ) ) ) ) ) ).
% tendsto_zero_mult_left_iff
thf(fact_7143_tendsto__zero__mult__right__iff,axiom,
! [A: $tType] :
( ( ( field @ A )
& ( topolo4211221413907600880p_mult @ A ) )
=> ! [C2: A,A3: nat > A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( filterlim @ nat @ A
@ ^ [N4: nat] : ( times_times @ A @ ( A3 @ N4 ) @ C2 )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ ( at_top @ nat ) )
= ( filterlim @ nat @ A @ A3 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ ( at_top @ nat ) ) ) ) ) ).
% tendsto_zero_mult_right_iff
thf(fact_7144_tendsto__zero__divide__iff,axiom,
! [A: $tType] :
( ( ( field @ A )
& ( topolo4211221413907600880p_mult @ A ) )
=> ! [C2: A,A3: nat > A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( filterlim @ nat @ A
@ ^ [N4: nat] : ( divide_divide @ A @ ( A3 @ N4 ) @ C2 )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ ( at_top @ nat ) )
= ( filterlim @ nat @ A @ A3 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ ( at_top @ nat ) ) ) ) ) ).
% tendsto_zero_divide_iff
thf(fact_7145_filterlim__Suc,axiom,
filterlim @ nat @ nat @ suc @ ( at_top @ nat ) @ ( at_top @ nat ) ).
% filterlim_Suc
thf(fact_7146_approx__from__below__dense__linorder,axiom,
! [A: $tType] :
( ( ( dense_linorder @ A )
& ( topolo3112930676232923870pology @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [Y: A,X: A] :
( ( ord_less @ A @ Y @ X )
=> ? [U3: nat > A] :
( ! [N11: nat] : ( ord_less @ A @ ( U3 @ N11 ) @ X )
& ( filterlim @ nat @ A @ U3 @ ( topolo7230453075368039082e_nhds @ A @ X ) @ ( at_top @ nat ) ) ) ) ) ).
% approx_from_below_dense_linorder
thf(fact_7147_approx__from__above__dense__linorder,axiom,
! [A: $tType] :
( ( ( dense_linorder @ A )
& ( topolo3112930676232923870pology @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ? [U3: nat > A] :
( ! [N11: nat] : ( ord_less @ A @ X @ ( U3 @ N11 ) )
& ( filterlim @ nat @ A @ U3 @ ( topolo7230453075368039082e_nhds @ A @ X ) @ ( at_top @ nat ) ) ) ) ) ).
% approx_from_above_dense_linorder
thf(fact_7148_continuous__real__root,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [F4: filter @ A,F3: A > real,N: nat] :
( ( topolo3448309680560233919inuous @ A @ real @ F4 @ F3 )
=> ( topolo3448309680560233919inuous @ A @ real @ F4
@ ^ [X2: A] : ( root @ N @ ( F3 @ X2 ) ) ) ) ) ).
% continuous_real_root
thf(fact_7149_continuous__arsinh,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [F4: filter @ A,F3: A > real] :
( ( topolo3448309680560233919inuous @ A @ real @ F4 @ F3 )
=> ( topolo3448309680560233919inuous @ A @ real @ F4
@ ^ [X2: A] : ( arsinh @ real @ ( F3 @ X2 ) ) ) ) ) ).
% continuous_arsinh
thf(fact_7150_continuous__real__sqrt,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [F4: filter @ A,F3: A > real] :
( ( topolo3448309680560233919inuous @ A @ real @ F4 @ F3 )
=> ( topolo3448309680560233919inuous @ A @ real @ F4
@ ^ [X2: A] : ( sqrt @ ( F3 @ X2 ) ) ) ) ) ).
% continuous_real_sqrt
thf(fact_7151_filterlim__sequentially__Suc,axiom,
! [A: $tType,F3: nat > A,F4: filter @ A] :
( ( filterlim @ nat @ A
@ ^ [X2: nat] : ( F3 @ ( suc @ X2 ) )
@ F4
@ ( at_top @ nat ) )
= ( filterlim @ nat @ A @ F3 @ F4 @ ( at_top @ nat ) ) ) ).
% filterlim_sequentially_Suc
thf(fact_7152_continuous__arctan,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [F4: filter @ A,F3: A > real] :
( ( topolo3448309680560233919inuous @ A @ real @ F4 @ F3 )
=> ( topolo3448309680560233919inuous @ A @ real @ F4
@ ^ [X2: A] : ( arctan @ ( F3 @ X2 ) ) ) ) ) ).
% continuous_arctan
thf(fact_7153_LIMSEQ__Suc,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [F3: nat > A,L: A] :
( ( filterlim @ nat @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ ( at_top @ nat ) )
=> ( filterlim @ nat @ A
@ ^ [N4: nat] : ( F3 @ ( suc @ N4 ) )
@ ( topolo7230453075368039082e_nhds @ A @ L )
@ ( at_top @ nat ) ) ) ) ).
% LIMSEQ_Suc
thf(fact_7154_LIMSEQ__imp__Suc,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [F3: nat > A,L: A] :
( ( filterlim @ nat @ A
@ ^ [N4: nat] : ( F3 @ ( suc @ N4 ) )
@ ( topolo7230453075368039082e_nhds @ A @ L )
@ ( at_top @ nat ) )
=> ( filterlim @ nat @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ ( at_top @ nat ) ) ) ) ).
% LIMSEQ_imp_Suc
thf(fact_7155_LIMSEQ__const__iff,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [K: A,L: A] :
( ( filterlim @ nat @ A
@ ^ [N4: nat] : K
@ ( topolo7230453075368039082e_nhds @ A @ L )
@ ( at_top @ nat ) )
= ( K = L ) ) ) ).
% LIMSEQ_const_iff
thf(fact_7156_continuous__rabs,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [F4: filter @ A,F3: A > real] :
( ( topolo3448309680560233919inuous @ A @ real @ F4 @ F3 )
=> ( topolo3448309680560233919inuous @ A @ real @ F4
@ ^ [X2: A] : ( abs_abs @ real @ ( F3 @ X2 ) ) ) ) ) ).
% continuous_rabs
thf(fact_7157_seq__offset__neg,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [F3: nat > A,L: A,K: nat] :
( ( filterlim @ nat @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ ( at_top @ nat ) )
=> ( filterlim @ nat @ A
@ ^ [I4: nat] : ( F3 @ ( minus_minus @ nat @ I4 @ K ) )
@ ( topolo7230453075368039082e_nhds @ A @ L )
@ ( at_top @ nat ) ) ) ) ).
% seq_offset_neg
thf(fact_7158_LIMSEQ__offset,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [F3: nat > A,K: nat,A3: A] :
( ( filterlim @ nat @ A
@ ^ [N4: nat] : ( F3 @ ( plus_plus @ nat @ N4 @ K ) )
@ ( topolo7230453075368039082e_nhds @ A @ A3 )
@ ( at_top @ nat ) )
=> ( filterlim @ nat @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ A3 ) @ ( at_top @ nat ) ) ) ) ).
% LIMSEQ_offset
thf(fact_7159_LIMSEQ__ignore__initial__segment,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [F3: nat > A,A3: A,K: nat] :
( ( filterlim @ nat @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ A3 ) @ ( at_top @ nat ) )
=> ( filterlim @ nat @ A
@ ^ [N4: nat] : ( F3 @ ( plus_plus @ nat @ N4 @ K ) )
@ ( topolo7230453075368039082e_nhds @ A @ A3 )
@ ( at_top @ nat ) ) ) ) ).
% LIMSEQ_ignore_initial_segment
thf(fact_7160_less__filter__def,axiom,
! [A: $tType] :
( ( ord_less @ ( filter @ A ) )
= ( ^ [F8: filter @ A,F9: filter @ A] :
( ( ord_less_eq @ ( filter @ A ) @ F8 @ F9 )
& ~ ( ord_less_eq @ ( filter @ A ) @ F9 @ F8 ) ) ) ) ).
% less_filter_def
thf(fact_7161_trivial__limit__at__top__linorder,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( at_top @ A )
!= ( bot_bot @ ( filter @ A ) ) ) ) ).
% trivial_limit_at_top_linorder
thf(fact_7162_summable__LIMSEQ__zero,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: nat > A] :
( ( summable @ A @ F3 )
=> ( filterlim @ nat @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ ( at_top @ nat ) ) ) ) ).
% summable_LIMSEQ_zero
thf(fact_7163_isCont__rabs,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [A3: A,F3: A > real] :
( ( topolo3448309680560233919inuous @ A @ real @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) @ F3 )
=> ( topolo3448309680560233919inuous @ A @ real @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) )
@ ^ [X2: A] : ( abs_abs @ real @ ( F3 @ X2 ) ) ) ) ) ).
% isCont_rabs
thf(fact_7164_continuous__at__within__powr,axiom,
! [C: $tType] :
( ( topological_t2_space @ C )
=> ! [A3: C,S2: set @ C,F3: C > real,G2: C > real] :
( ( topolo3448309680560233919inuous @ C @ real @ ( topolo174197925503356063within @ C @ A3 @ S2 ) @ F3 )
=> ( ( topolo3448309680560233919inuous @ C @ real @ ( topolo174197925503356063within @ C @ A3 @ S2 ) @ G2 )
=> ( ( ( F3 @ A3 )
!= ( zero_zero @ real ) )
=> ( topolo3448309680560233919inuous @ C @ real @ ( topolo174197925503356063within @ C @ A3 @ S2 )
@ ^ [X2: C] : ( powr @ real @ ( F3 @ X2 ) @ ( G2 @ X2 ) ) ) ) ) ) ) ).
% continuous_at_within_powr
thf(fact_7165_continuous__within__ln,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [X: A,S2: set @ A,F3: A > real] :
( ( topolo3448309680560233919inuous @ A @ real @ ( topolo174197925503356063within @ A @ X @ S2 ) @ F3 )
=> ( ( ( F3 @ X )
!= ( zero_zero @ real ) )
=> ( topolo3448309680560233919inuous @ A @ real @ ( topolo174197925503356063within @ A @ X @ S2 )
@ ^ [X2: A] : ( ln_ln @ real @ ( F3 @ X2 ) ) ) ) ) ) ).
% continuous_within_ln
thf(fact_7166_mult__nat__right__at__top,axiom,
! [C2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ C2 )
=> ( filterlim @ nat @ nat
@ ^ [X2: nat] : ( times_times @ nat @ X2 @ C2 )
@ ( at_top @ nat )
@ ( at_top @ nat ) ) ) ).
% mult_nat_right_at_top
thf(fact_7167_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_7168_LIMSEQ__lessThan__iff__atMost,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [F3: ( set @ nat ) > A,X: A] :
( ( filterlim @ nat @ A
@ ^ [N4: nat] : ( F3 @ ( set_ord_lessThan @ nat @ N4 ) )
@ ( topolo7230453075368039082e_nhds @ A @ X )
@ ( at_top @ nat ) )
= ( filterlim @ nat @ A
@ ^ [N4: nat] : ( F3 @ ( set_ord_atMost @ nat @ N4 ) )
@ ( topolo7230453075368039082e_nhds @ A @ X )
@ ( at_top @ nat ) ) ) ) ).
% LIMSEQ_lessThan_iff_atMost
thf(fact_7169_LIMSEQ__root,axiom,
( filterlim @ nat @ real
@ ^ [N4: nat] : ( root @ N4 @ ( semiring_1_of_nat @ real @ N4 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( one_one @ real ) )
@ ( at_top @ nat ) ) ).
% LIMSEQ_root
thf(fact_7170_isCont__powr,axiom,
! [C: $tType] :
( ( topological_t2_space @ C )
=> ! [A3: C,F3: C > real,G2: C > real] :
( ( topolo3448309680560233919inuous @ C @ real @ ( topolo174197925503356063within @ C @ A3 @ ( top_top @ ( set @ C ) ) ) @ F3 )
=> ( ( topolo3448309680560233919inuous @ C @ real @ ( topolo174197925503356063within @ C @ A3 @ ( top_top @ ( set @ C ) ) ) @ G2 )
=> ( ( ( F3 @ A3 )
!= ( zero_zero @ real ) )
=> ( topolo3448309680560233919inuous @ C @ real @ ( topolo174197925503356063within @ C @ A3 @ ( top_top @ ( set @ C ) ) )
@ ^ [X2: C] : ( powr @ real @ ( F3 @ X2 ) @ ( G2 @ X2 ) ) ) ) ) ) ) ).
% isCont_powr
thf(fact_7171_isCont__ln_H,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [X: A,F3: A > real] :
( ( topolo3448309680560233919inuous @ A @ real @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) @ F3 )
=> ( ( ( F3 @ X )
!= ( zero_zero @ real ) )
=> ( topolo3448309680560233919inuous @ A @ real @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) )
@ ^ [X2: A] : ( ln_ln @ real @ ( F3 @ X2 ) ) ) ) ) ) ).
% isCont_ln'
thf(fact_7172_lim__const__over__n,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [A3: A] :
( filterlim @ nat @ A
@ ^ [N4: nat] : ( divide_divide @ A @ A3 @ ( semiring_1_of_nat @ A @ N4 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ ( at_top @ nat ) ) ) ).
% lim_const_over_n
thf(fact_7173_LIMSEQ__linear,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [X5: nat > A,X: A,L: nat] :
( ( filterlim @ nat @ A @ X5 @ ( topolo7230453075368039082e_nhds @ A @ X ) @ ( at_top @ nat ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ L )
=> ( filterlim @ nat @ A
@ ^ [N4: nat] : ( X5 @ ( times_times @ nat @ N4 @ L ) )
@ ( topolo7230453075368039082e_nhds @ A @ X )
@ ( at_top @ nat ) ) ) ) ) ).
% LIMSEQ_linear
thf(fact_7174_lim__inverse__n,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ( filterlim @ nat @ A
@ ^ [N4: nat] : ( inverse_inverse @ A @ ( semiring_1_of_nat @ A @ N4 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ ( at_top @ nat ) ) ) ).
% lim_inverse_n
thf(fact_7175_LIMSEQ__SEQ__conv2,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo3112930676232923870pology @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [A3: A,F3: A > B,L: B] :
( ! [S6: nat > A] :
( ( ! [N11: nat] :
( ( S6 @ N11 )
!= A3 )
& ( filterlim @ nat @ A @ S6 @ ( topolo7230453075368039082e_nhds @ A @ A3 ) @ ( at_top @ nat ) ) )
=> ( filterlim @ nat @ B
@ ^ [N4: nat] : ( F3 @ ( S6 @ N4 ) )
@ ( topolo7230453075368039082e_nhds @ B @ L )
@ ( at_top @ nat ) ) )
=> ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ L ) @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% LIMSEQ_SEQ_conv2
thf(fact_7176_LIMSEQ__SEQ__conv1,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo4958980785337419405_space @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [F3: A > B,L: B,A3: A] :
( ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ L ) @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) )
=> ! [S10: nat > A] :
( ( ! [N2: nat] :
( ( S10 @ N2 )
!= A3 )
& ( filterlim @ nat @ A @ S10 @ ( topolo7230453075368039082e_nhds @ A @ A3 ) @ ( at_top @ nat ) ) )
=> ( filterlim @ nat @ B
@ ^ [N4: nat] : ( F3 @ ( S10 @ N4 ) )
@ ( topolo7230453075368039082e_nhds @ B @ L )
@ ( at_top @ nat ) ) ) ) ) ).
% LIMSEQ_SEQ_conv1
thf(fact_7177_LIMSEQ__SEQ__conv,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo3112930676232923870pology @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [A3: A,X5: A > B,L6: B] :
( ( ! [S8: nat > A] :
( ( ! [N4: nat] :
( ( S8 @ N4 )
!= A3 )
& ( filterlim @ nat @ A @ S8 @ ( topolo7230453075368039082e_nhds @ A @ A3 ) @ ( at_top @ nat ) ) )
=> ( filterlim @ nat @ B
@ ^ [N4: nat] : ( X5 @ ( S8 @ N4 ) )
@ ( topolo7230453075368039082e_nhds @ B @ L6 )
@ ( at_top @ nat ) ) ) )
= ( filterlim @ A @ B @ X5 @ ( topolo7230453075368039082e_nhds @ B @ L6 ) @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% LIMSEQ_SEQ_conv
thf(fact_7178_telescope__summable,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: nat > A,C2: A] :
( ( filterlim @ nat @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ C2 ) @ ( at_top @ nat ) )
=> ( summable @ A
@ ^ [N4: nat] : ( minus_minus @ A @ ( F3 @ ( suc @ N4 ) ) @ ( F3 @ N4 ) ) ) ) ) ).
% telescope_summable
thf(fact_7179_telescope__summable_H,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: nat > A,C2: A] :
( ( filterlim @ nat @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ C2 ) @ ( at_top @ nat ) )
=> ( summable @ A
@ ^ [N4: nat] : ( minus_minus @ A @ ( F3 @ N4 ) @ ( F3 @ ( suc @ N4 ) ) ) ) ) ) ).
% telescope_summable'
thf(fact_7180_nested__sequence__unique,axiom,
! [F3: nat > real,G2: nat > real] :
( ! [N2: nat] : ( ord_less_eq @ real @ ( F3 @ N2 ) @ ( F3 @ ( suc @ N2 ) ) )
=> ( ! [N2: nat] : ( ord_less_eq @ real @ ( G2 @ ( suc @ N2 ) ) @ ( G2 @ N2 ) )
=> ( ! [N2: nat] : ( ord_less_eq @ real @ ( F3 @ N2 ) @ ( G2 @ N2 ) )
=> ( ( filterlim @ nat @ real
@ ^ [N4: nat] : ( minus_minus @ real @ ( F3 @ N4 ) @ ( G2 @ N4 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ ( at_top @ nat ) )
=> ? [L3: real] :
( ! [N11: nat] : ( ord_less_eq @ real @ ( F3 @ N11 ) @ L3 )
& ( filterlim @ nat @ real @ F3 @ ( topolo7230453075368039082e_nhds @ real @ L3 ) @ ( at_top @ nat ) )
& ! [N11: nat] : ( ord_less_eq @ real @ L3 @ ( G2 @ N11 ) )
& ( filterlim @ nat @ real @ G2 @ ( topolo7230453075368039082e_nhds @ real @ L3 ) @ ( at_top @ nat ) ) ) ) ) ) ) ).
% nested_sequence_unique
thf(fact_7181_LIMSEQ__inverse__zero,axiom,
! [X5: nat > real] :
( ! [R2: real] :
? [N8: nat] :
! [N2: nat] :
( ( ord_less_eq @ nat @ N8 @ N2 )
=> ( ord_less @ real @ R2 @ ( X5 @ N2 ) ) )
=> ( filterlim @ nat @ real
@ ^ [N4: nat] : ( inverse_inverse @ real @ ( X5 @ N4 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ ( at_top @ nat ) ) ) ).
% LIMSEQ_inverse_zero
thf(fact_7182_lim__inverse__n_H,axiom,
( filterlim @ nat @ real
@ ^ [N4: nat] : ( divide_divide @ real @ ( one_one @ real ) @ ( semiring_1_of_nat @ real @ N4 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ ( at_top @ nat ) ) ).
% lim_inverse_n'
thf(fact_7183_LIMSEQ__inverse__real__of__nat,axiom,
( filterlim @ nat @ real
@ ^ [N4: nat] : ( inverse_inverse @ real @ ( semiring_1_of_nat @ real @ ( suc @ N4 ) ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ ( at_top @ nat ) ) ).
% LIMSEQ_inverse_real_of_nat
thf(fact_7184_LIMSEQ__root__const,axiom,
! [C2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ C2 )
=> ( filterlim @ nat @ real
@ ^ [N4: nat] : ( root @ N4 @ C2 )
@ ( topolo7230453075368039082e_nhds @ real @ ( one_one @ real ) )
@ ( at_top @ nat ) ) ) ).
% LIMSEQ_root_const
thf(fact_7185_LIMSEQ__inverse__real__of__nat__add,axiom,
! [R3: real] :
( filterlim @ nat @ real
@ ^ [N4: nat] : ( plus_plus @ real @ R3 @ ( inverse_inverse @ real @ ( semiring_1_of_nat @ real @ ( suc @ N4 ) ) ) )
@ ( topolo7230453075368039082e_nhds @ real @ R3 )
@ ( at_top @ nat ) ) ).
% LIMSEQ_inverse_real_of_nat_add
thf(fact_7186_sums__def,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ( ( sums @ A )
= ( ^ [F2: nat > A,S5: A] :
( filterlim @ nat @ A
@ ^ [N4: nat] : ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_ord_lessThan @ nat @ N4 ) )
@ ( topolo7230453075368039082e_nhds @ A @ S5 )
@ ( at_top @ nat ) ) ) ) ) ).
% sums_def
thf(fact_7187_sums__def__le,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ( ( sums @ A )
= ( ^ [F2: nat > A,S5: A] :
( filterlim @ nat @ A
@ ^ [N4: nat] : ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_ord_atMost @ nat @ N4 ) )
@ ( topolo7230453075368039082e_nhds @ A @ S5 )
@ ( at_top @ nat ) ) ) ) ) ).
% sums_def_le
thf(fact_7188_continuous__at__within__log,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [A3: A,S2: set @ A,F3: A > real,G2: A > real] :
( ( topolo3448309680560233919inuous @ A @ real @ ( topolo174197925503356063within @ A @ A3 @ S2 ) @ F3 )
=> ( ( topolo3448309680560233919inuous @ A @ real @ ( topolo174197925503356063within @ A @ A3 @ S2 ) @ G2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( F3 @ A3 ) )
=> ( ( ( F3 @ A3 )
!= ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( G2 @ A3 ) )
=> ( topolo3448309680560233919inuous @ A @ real @ ( topolo174197925503356063within @ A @ A3 @ S2 )
@ ^ [X2: A] : ( log @ ( F3 @ X2 ) @ ( G2 @ X2 ) ) ) ) ) ) ) ) ) ).
% continuous_at_within_log
thf(fact_7189_increasing__LIMSEQ,axiom,
! [F3: nat > real,L: real] :
( ! [N2: nat] : ( ord_less_eq @ real @ ( F3 @ N2 ) @ ( F3 @ ( suc @ N2 ) ) )
=> ( ! [N2: nat] : ( ord_less_eq @ real @ ( F3 @ N2 ) @ L )
=> ( ! [E: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E )
=> ? [N11: nat] : ( ord_less_eq @ real @ L @ ( plus_plus @ real @ ( F3 @ N11 ) @ E ) ) )
=> ( filterlim @ nat @ real @ F3 @ ( topolo7230453075368039082e_nhds @ real @ L ) @ ( at_top @ nat ) ) ) ) ) ).
% increasing_LIMSEQ
thf(fact_7190_lim__1__over__n,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ( filterlim @ nat @ A
@ ^ [N4: nat] : ( divide_divide @ A @ ( one_one @ A ) @ ( semiring_1_of_nat @ A @ N4 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ ( at_top @ nat ) ) ) ).
% lim_1_over_n
thf(fact_7191_LIMSEQ__Suc__n__over__n,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ( filterlim @ nat @ A
@ ^ [N4: nat] : ( divide_divide @ A @ ( semiring_1_of_nat @ A @ ( suc @ N4 ) ) @ ( semiring_1_of_nat @ A @ N4 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( one_one @ A ) )
@ ( at_top @ nat ) ) ) ).
% LIMSEQ_Suc_n_over_n
thf(fact_7192_LIMSEQ__n__over__Suc__n,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ( filterlim @ nat @ A
@ ^ [N4: nat] : ( divide_divide @ A @ ( semiring_1_of_nat @ A @ N4 ) @ ( semiring_1_of_nat @ A @ ( suc @ N4 ) ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( one_one @ A ) )
@ ( at_top @ nat ) ) ) ).
% LIMSEQ_n_over_Suc_n
thf(fact_7193_telescope__sums,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: nat > A,C2: A] :
( ( filterlim @ nat @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ C2 ) @ ( at_top @ nat ) )
=> ( sums @ A
@ ^ [N4: nat] : ( minus_minus @ A @ ( F3 @ ( suc @ N4 ) ) @ ( F3 @ N4 ) )
@ ( minus_minus @ A @ C2 @ ( F3 @ ( zero_zero @ nat ) ) ) ) ) ) ).
% telescope_sums
thf(fact_7194_telescope__sums_H,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: nat > A,C2: A] :
( ( filterlim @ nat @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ C2 ) @ ( at_top @ nat ) )
=> ( sums @ A
@ ^ [N4: nat] : ( minus_minus @ A @ ( F3 @ N4 ) @ ( F3 @ ( suc @ N4 ) ) )
@ ( minus_minus @ A @ ( F3 @ ( zero_zero @ nat ) ) @ C2 ) ) ) ) ).
% telescope_sums'
thf(fact_7195_LIMSEQ__realpow__zero,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ X @ ( one_one @ real ) )
=> ( filterlim @ nat @ real @ ( power_power @ real @ X ) @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) ) ) ) ).
% LIMSEQ_realpow_zero
thf(fact_7196_LIMSEQ__divide__realpow__zero,axiom,
! [X: real,A3: real] :
( ( ord_less @ real @ ( one_one @ real ) @ X )
=> ( filterlim @ nat @ real
@ ^ [N4: nat] : ( divide_divide @ real @ A3 @ ( power_power @ real @ X @ N4 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ ( at_top @ nat ) ) ) ).
% LIMSEQ_divide_realpow_zero
thf(fact_7197_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_7198_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_7199_LIMSEQ__inverse__realpow__zero,axiom,
! [X: real] :
( ( ord_less @ real @ ( one_one @ real ) @ X )
=> ( filterlim @ nat @ real
@ ^ [N4: nat] : ( inverse_inverse @ real @ ( power_power @ real @ X @ N4 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ ( at_top @ nat ) ) ) ).
% LIMSEQ_inverse_realpow_zero
thf(fact_7200_sums__def_H,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ( ( sums @ A )
= ( ^ [F2: nat > A,S5: A] :
( filterlim @ nat @ A
@ ^ [N4: nat] : ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N4 ) )
@ ( topolo7230453075368039082e_nhds @ A @ S5 )
@ ( at_top @ nat ) ) ) ) ) ).
% sums_def'
thf(fact_7201_LIMSEQ__inverse__real__of__nat__add__minus,axiom,
! [R3: real] :
( filterlim @ nat @ real
@ ^ [N4: nat] : ( plus_plus @ real @ R3 @ ( uminus_uminus @ real @ ( inverse_inverse @ real @ ( semiring_1_of_nat @ real @ ( suc @ N4 ) ) ) ) )
@ ( topolo7230453075368039082e_nhds @ real @ R3 )
@ ( at_top @ nat ) ) ).
% LIMSEQ_inverse_real_of_nat_add_minus
thf(fact_7202_root__test__convergence,axiom,
! [A: $tType] :
( ( real_Vector_banach @ A )
=> ! [F3: nat > A,X: real] :
( ( filterlim @ nat @ real
@ ^ [N4: nat] : ( root @ N4 @ ( real_V7770717601297561774m_norm @ A @ ( F3 @ N4 ) ) )
@ ( topolo7230453075368039082e_nhds @ real @ X )
@ ( at_top @ nat ) )
=> ( ( ord_less @ real @ X @ ( one_one @ real ) )
=> ( summable @ A @ F3 ) ) ) ) ).
% root_test_convergence
thf(fact_7203_summable__LIMSEQ,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topological_t2_space @ A ) )
=> ! [F3: nat > A] :
( ( summable @ A @ F3 )
=> ( filterlim @ nat @ A
@ ^ [N4: nat] : ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_ord_lessThan @ nat @ N4 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( suminf @ A @ F3 ) )
@ ( at_top @ nat ) ) ) ) ).
% summable_LIMSEQ
thf(fact_7204_summable__LIMSEQ_H,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topological_t2_space @ A ) )
=> ! [F3: nat > A] :
( ( summable @ A @ F3 )
=> ( filterlim @ nat @ A
@ ^ [N4: nat] : ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_ord_atMost @ nat @ N4 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( suminf @ A @ F3 ) )
@ ( at_top @ nat ) ) ) ) ).
% summable_LIMSEQ'
thf(fact_7205_isCont__log,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [A3: A,F3: A > real,G2: A > real] :
( ( topolo3448309680560233919inuous @ A @ real @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) @ F3 )
=> ( ( topolo3448309680560233919inuous @ A @ real @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) @ G2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( F3 @ A3 ) )
=> ( ( ( F3 @ A3 )
!= ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( G2 @ A3 ) )
=> ( topolo3448309680560233919inuous @ A @ real @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) )
@ ^ [X2: A] : ( log @ ( F3 @ X2 ) @ ( G2 @ X2 ) ) ) ) ) ) ) ) ) ).
% isCont_log
thf(fact_7206_LIMSEQ__D,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X5: nat > A,L6: A,R3: real] :
( ( filterlim @ nat @ A @ X5 @ ( topolo7230453075368039082e_nhds @ A @ L6 ) @ ( at_top @ nat ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ R3 )
=> ? [No: nat] :
! [N11: nat] :
( ( ord_less_eq @ nat @ No @ N11 )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( X5 @ N11 ) @ L6 ) ) @ R3 ) ) ) ) ) ).
% LIMSEQ_D
thf(fact_7207_LIMSEQ__I,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X5: nat > A,L6: A] :
( ! [R2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R2 )
=> ? [No2: nat] :
! [N2: nat] :
( ( ord_less_eq @ nat @ No2 @ N2 )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( X5 @ N2 ) @ L6 ) ) @ R2 ) ) )
=> ( filterlim @ nat @ A @ X5 @ ( topolo7230453075368039082e_nhds @ A @ L6 ) @ ( at_top @ nat ) ) ) ) ).
% LIMSEQ_I
thf(fact_7208_LIMSEQ__iff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X5: nat > A,L6: A] :
( ( filterlim @ nat @ A @ X5 @ ( topolo7230453075368039082e_nhds @ A @ L6 ) @ ( at_top @ nat ) )
= ( ! [R5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R5 )
=> ? [No3: nat] :
! [N4: nat] :
( ( ord_less_eq @ nat @ No3 @ N4 )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( X5 @ N4 ) @ L6 ) ) @ R5 ) ) ) ) ) ) ).
% LIMSEQ_iff
thf(fact_7209_LIMSEQ__power__zero,axiom,
! [A: $tType] :
( ( real_V2822296259951069270ebra_1 @ A )
=> ! [X: A] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ ( one_one @ real ) )
=> ( filterlim @ nat @ A @ ( power_power @ A @ X ) @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ ( at_top @ nat ) ) ) ) ).
% LIMSEQ_power_zero
thf(fact_7210_tendsto__exp__limit__sequentially,axiom,
! [X: real] :
( filterlim @ nat @ real
@ ^ [N4: nat] : ( power_power @ real @ ( plus_plus @ real @ ( one_one @ real ) @ ( divide_divide @ real @ X @ ( semiring_1_of_nat @ real @ N4 ) ) ) @ N4 )
@ ( topolo7230453075368039082e_nhds @ real @ ( exp @ real @ X ) )
@ ( at_top @ nat ) ) ).
% tendsto_exp_limit_sequentially
thf(fact_7211_tendsto__power__zero,axiom,
! [A: $tType,B: $tType] :
( ( real_V2822296259951069270ebra_1 @ A )
=> ! [F3: B > nat,F4: filter @ B,X: A] :
( ( filterlim @ B @ nat @ F3 @ ( at_top @ nat ) @ F4 )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ ( one_one @ real ) )
=> ( filterlim @ B @ A
@ ^ [Y2: B] : ( power_power @ A @ X @ ( F3 @ Y2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ F4 ) ) ) ) ).
% tendsto_power_zero
thf(fact_7212_LIMSEQ__inverse__real__of__nat__add__minus__mult,axiom,
! [R3: real] :
( filterlim @ nat @ real
@ ^ [N4: nat] : ( times_times @ real @ R3 @ ( plus_plus @ real @ ( one_one @ real ) @ ( uminus_uminus @ real @ ( inverse_inverse @ real @ ( semiring_1_of_nat @ real @ ( suc @ N4 ) ) ) ) ) )
@ ( topolo7230453075368039082e_nhds @ real @ R3 )
@ ( at_top @ nat ) ) ).
% LIMSEQ_inverse_real_of_nat_add_minus_mult
thf(fact_7213_LIMSEQ__norm__0,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: nat > A] :
( ! [N2: nat] : ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( F3 @ N2 ) ) @ ( divide_divide @ real @ ( one_one @ real ) @ ( semiring_1_of_nat @ real @ ( suc @ N2 ) ) ) )
=> ( filterlim @ nat @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ ( at_top @ nat ) ) ) ) ).
% LIMSEQ_norm_0
thf(fact_7214_summable__Leibniz_I1_J,axiom,
! [A3: nat > real] :
( ( filterlim @ nat @ real @ A3 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) )
=> ( ( topological_monoseq @ real @ A3 )
=> ( summable @ real
@ ^ [N4: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ N4 ) @ ( A3 @ N4 ) ) ) ) ) ).
% summable_Leibniz(1)
thf(fact_7215_field__derivative__lim__unique,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,Df: A,Z: A,S2: nat > A,A3: A] :
( ( has_field_derivative @ A @ F3 @ Df @ ( topolo174197925503356063within @ A @ Z @ ( top_top @ ( set @ A ) ) ) )
=> ( ( filterlim @ nat @ A @ S2 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ ( at_top @ nat ) )
=> ( ! [N2: nat] :
( ( S2 @ N2 )
!= ( zero_zero @ A ) )
=> ( ( filterlim @ nat @ A
@ ^ [N4: nat] : ( divide_divide @ A @ ( minus_minus @ A @ ( F3 @ ( plus_plus @ A @ Z @ ( S2 @ N4 ) ) ) @ ( F3 @ Z ) ) @ ( S2 @ N4 ) )
@ ( topolo7230453075368039082e_nhds @ A @ A3 )
@ ( at_top @ nat ) )
=> ( Df = A3 ) ) ) ) ) ) ).
% field_derivative_lim_unique
thf(fact_7216_powser__times__n__limit__0,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V8999393235501362500lgebra @ A ) )
=> ! [X: A] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ ( one_one @ real ) )
=> ( filterlim @ nat @ A
@ ^ [N4: nat] : ( times_times @ A @ ( semiring_1_of_nat @ A @ N4 ) @ ( power_power @ A @ X @ N4 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ ( at_top @ nat ) ) ) ) ).
% powser_times_n_limit_0
thf(fact_7217_lim__n__over__pown,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A] :
( ( ord_less @ real @ ( one_one @ real ) @ ( real_V7770717601297561774m_norm @ A @ X ) )
=> ( filterlim @ nat @ A
@ ^ [N4: nat] : ( divide_divide @ A @ ( semiring_1_of_nat @ A @ N4 ) @ ( power_power @ A @ X @ N4 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ ( at_top @ nat ) ) ) ) ).
% lim_n_over_pown
thf(fact_7218_summable,axiom,
! [A3: nat > real] :
( ( filterlim @ nat @ real @ A3 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) )
=> ( ! [N2: nat] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( A3 @ N2 ) )
=> ( ! [N2: nat] : ( ord_less_eq @ real @ ( A3 @ ( suc @ N2 ) ) @ ( A3 @ N2 ) )
=> ( summable @ real
@ ^ [N4: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ N4 ) @ ( A3 @ N4 ) ) ) ) ) ) ).
% summable
thf(fact_7219_cos__diff__limit__1,axiom,
! [Theta: nat > real,Theta2: real] :
( ( filterlim @ nat @ real
@ ^ [J3: nat] : ( cos @ real @ ( minus_minus @ real @ ( Theta @ J3 ) @ Theta2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( one_one @ real ) )
@ ( at_top @ nat ) )
=> ~ ! [K2: nat > int] :
~ ( filterlim @ nat @ real
@ ^ [J3: nat] : ( minus_minus @ real @ ( Theta @ J3 ) @ ( times_times @ real @ ( ring_1_of_int @ real @ ( K2 @ J3 ) ) @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) ) )
@ ( topolo7230453075368039082e_nhds @ real @ Theta2 )
@ ( at_top @ nat ) ) ) ).
% cos_diff_limit_1
thf(fact_7220_cos__limit__1,axiom,
! [Theta: nat > real] :
( ( filterlim @ nat @ real
@ ^ [J3: nat] : ( cos @ real @ ( Theta @ J3 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( one_one @ real ) )
@ ( at_top @ nat ) )
=> ? [K2: nat > int] :
( filterlim @ nat @ real
@ ^ [J3: nat] : ( minus_minus @ real @ ( Theta @ J3 ) @ ( times_times @ real @ ( ring_1_of_int @ real @ ( K2 @ J3 ) ) @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ ( at_top @ nat ) ) ) ).
% cos_limit_1
thf(fact_7221_summable__Leibniz_I4_J,axiom,
! [A3: nat > real] :
( ( filterlim @ nat @ real @ A3 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) )
=> ( ( topological_monoseq @ real @ A3 )
=> ( filterlim @ nat @ real
@ ^ [N4: nat] :
( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I4: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I4 ) @ ( A3 @ I4 ) )
@ ( set_ord_lessThan @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N4 ) ) )
@ ( topolo7230453075368039082e_nhds @ real
@ ( suminf @ real
@ ^ [I4: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I4 ) @ ( A3 @ I4 ) ) ) )
@ ( at_top @ nat ) ) ) ) ).
% summable_Leibniz(4)
thf(fact_7222_zeroseq__arctan__series,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ X ) @ ( one_one @ real ) )
=> ( filterlim @ nat @ real
@ ^ [N4: nat] : ( times_times @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( semiring_1_of_nat @ real @ ( plus_plus @ nat @ ( times_times @ nat @ N4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) @ ( power_power @ real @ X @ ( plus_plus @ nat @ ( times_times @ nat @ N4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ ( at_top @ nat ) ) ) ).
% zeroseq_arctan_series
thf(fact_7223_summable__Leibniz_H_I3_J,axiom,
! [A3: nat > real] :
( ( filterlim @ nat @ real @ A3 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) )
=> ( ! [N2: nat] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( A3 @ N2 ) )
=> ( ! [N2: nat] : ( ord_less_eq @ real @ ( A3 @ ( suc @ N2 ) ) @ ( A3 @ N2 ) )
=> ( filterlim @ nat @ real
@ ^ [N4: nat] :
( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I4: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I4 ) @ ( A3 @ I4 ) )
@ ( set_ord_lessThan @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N4 ) ) )
@ ( topolo7230453075368039082e_nhds @ real
@ ( suminf @ real
@ ^ [I4: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I4 ) @ ( A3 @ I4 ) ) ) )
@ ( at_top @ nat ) ) ) ) ) ).
% summable_Leibniz'(3)
thf(fact_7224_summable__Leibniz_H_I2_J,axiom,
! [A3: nat > real,N: nat] :
( ( filterlim @ nat @ real @ A3 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) )
=> ( ! [N2: nat] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( A3 @ N2 ) )
=> ( ! [N2: nat] : ( ord_less_eq @ real @ ( A3 @ ( suc @ N2 ) ) @ ( A3 @ N2 ) )
=> ( ord_less_eq @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I4: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I4 ) @ ( A3 @ I4 ) )
@ ( set_ord_lessThan @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) )
@ ( suminf @ real
@ ^ [I4: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I4 ) @ ( A3 @ I4 ) ) ) ) ) ) ) ).
% summable_Leibniz'(2)
thf(fact_7225_sums__alternating__upper__lower,axiom,
! [A3: nat > real] :
( ! [N2: nat] : ( ord_less_eq @ real @ ( A3 @ ( suc @ N2 ) ) @ ( A3 @ N2 ) )
=> ( ! [N2: nat] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( A3 @ N2 ) )
=> ( ( filterlim @ nat @ real @ A3 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) )
=> ? [L3: real] :
( ! [N11: nat] :
( ord_less_eq @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I4: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I4 ) @ ( A3 @ I4 ) )
@ ( set_ord_lessThan @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N11 ) ) )
@ L3 )
& ( filterlim @ nat @ real
@ ^ [N4: nat] :
( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I4: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I4 ) @ ( A3 @ I4 ) )
@ ( set_ord_lessThan @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N4 ) ) )
@ ( topolo7230453075368039082e_nhds @ real @ L3 )
@ ( at_top @ nat ) )
& ! [N11: nat] :
( ord_less_eq @ real @ L3
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I4: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I4 ) @ ( A3 @ I4 ) )
@ ( set_ord_lessThan @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N11 ) @ ( one_one @ nat ) ) ) ) )
& ( filterlim @ nat @ real
@ ^ [N4: nat] :
( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I4: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I4 ) @ ( A3 @ I4 ) )
@ ( set_ord_lessThan @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N4 ) @ ( one_one @ nat ) ) ) )
@ ( topolo7230453075368039082e_nhds @ real @ L3 )
@ ( at_top @ nat ) ) ) ) ) ) ).
% sums_alternating_upper_lower
thf(fact_7226_summable__Leibniz_I5_J,axiom,
! [A3: nat > real] :
( ( filterlim @ nat @ real @ A3 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) )
=> ( ( topological_monoseq @ real @ A3 )
=> ( filterlim @ nat @ real
@ ^ [N4: nat] :
( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I4: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I4 ) @ ( A3 @ I4 ) )
@ ( set_ord_lessThan @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N4 ) @ ( one_one @ nat ) ) ) )
@ ( topolo7230453075368039082e_nhds @ real
@ ( suminf @ real
@ ^ [I4: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I4 ) @ ( A3 @ I4 ) ) ) )
@ ( at_top @ nat ) ) ) ) ).
% summable_Leibniz(5)
thf(fact_7227_summable__Leibniz_H_I4_J,axiom,
! [A3: nat > real,N: nat] :
( ( filterlim @ nat @ real @ A3 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) )
=> ( ! [N2: nat] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( A3 @ N2 ) )
=> ( ! [N2: nat] : ( ord_less_eq @ real @ ( A3 @ ( suc @ N2 ) ) @ ( A3 @ N2 ) )
=> ( ord_less_eq @ real
@ ( suminf @ real
@ ^ [I4: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I4 ) @ ( A3 @ I4 ) ) )
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I4: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I4 ) @ ( A3 @ I4 ) )
@ ( set_ord_lessThan @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ ( one_one @ nat ) ) ) ) ) ) ) ) ).
% summable_Leibniz'(4)
thf(fact_7228_summable__Leibniz_H_I5_J,axiom,
! [A3: nat > real] :
( ( filterlim @ nat @ real @ A3 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) )
=> ( ! [N2: nat] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( A3 @ N2 ) )
=> ( ! [N2: nat] : ( ord_less_eq @ real @ ( A3 @ ( suc @ N2 ) ) @ ( A3 @ N2 ) )
=> ( filterlim @ nat @ real
@ ^ [N4: nat] :
( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I4: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I4 ) @ ( A3 @ I4 ) )
@ ( set_ord_lessThan @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N4 ) @ ( one_one @ nat ) ) ) )
@ ( topolo7230453075368039082e_nhds @ real
@ ( suminf @ real
@ ^ [I4: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I4 ) @ ( A3 @ I4 ) ) ) )
@ ( at_top @ nat ) ) ) ) ) ).
% summable_Leibniz'(5)
thf(fact_7229_has__derivative__at2,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F3: A > B,F7: A > B,X: A] :
( ( has_derivative @ A @ B @ F3 @ F7 @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) )
= ( ( real_V3181309239436604168linear @ A @ B @ F7 )
& ( filterlim @ A @ B
@ ^ [Y2: A] : ( real_V8093663219630862766scaleR @ B @ ( divide_divide @ real @ ( one_one @ real ) @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ Y2 @ X ) ) ) @ ( minus_minus @ B @ ( F3 @ Y2 ) @ ( plus_plus @ B @ ( F3 @ X ) @ ( F7 @ ( minus_minus @ A @ Y2 @ X ) ) ) ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% has_derivative_at2
thf(fact_7230_has__derivative__at,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F3: A > B,D3: A > B,X: A] :
( ( has_derivative @ A @ B @ F3 @ D3 @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) )
= ( ( real_V3181309239436604168linear @ A @ B @ D3 )
& ( filterlim @ A @ real
@ ^ [H: A] : ( divide_divide @ real @ ( real_V7770717601297561774m_norm @ B @ ( minus_minus @ B @ ( minus_minus @ B @ ( F3 @ ( plus_plus @ A @ X @ H ) ) @ ( F3 @ X ) ) @ ( D3 @ 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_7231_filterlim__int__sequentially,axiom,
filterlim @ nat @ int @ ( semiring_1_of_nat @ int ) @ ( at_top @ int ) @ ( at_top @ nat ) ).
% filterlim_int_sequentially
thf(fact_7232_exp__at__top,axiom,
filterlim @ real @ real @ ( exp @ real ) @ ( at_top @ real ) @ ( at_top @ real ) ).
% exp_at_top
thf(fact_7233_sqrt__at__top,axiom,
filterlim @ real @ real @ sqrt @ ( at_top @ real ) @ ( at_top @ real ) ).
% sqrt_at_top
thf(fact_7234_ln__at__top,axiom,
filterlim @ real @ real @ ( ln_ln @ real ) @ ( at_top @ real ) @ ( at_top @ real ) ).
% ln_at_top
thf(fact_7235_bounded__linear_Ocontinuous,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( ( topological_t2_space @ C )
& ( real_V822414075346904944vector @ B )
& ( real_V822414075346904944vector @ A ) )
=> ! [F3: A > B,F4: filter @ C,G2: C > A] :
( ( real_V3181309239436604168linear @ A @ B @ F3 )
=> ( ( topolo3448309680560233919inuous @ C @ A @ F4 @ G2 )
=> ( topolo3448309680560233919inuous @ C @ B @ F4
@ ^ [X2: C] : ( F3 @ ( G2 @ X2 ) ) ) ) ) ) ).
% bounded_linear.continuous
thf(fact_7236_bounded__linear_Otendsto,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( ( real_V822414075346904944vector @ B )
& ( real_V822414075346904944vector @ A ) )
=> ! [F3: A > B,G2: C > A,A3: A,F4: filter @ C] :
( ( real_V3181309239436604168linear @ A @ B @ F3 )
=> ( ( filterlim @ C @ A @ G2 @ ( topolo7230453075368039082e_nhds @ A @ A3 ) @ F4 )
=> ( filterlim @ C @ B
@ ^ [X2: C] : ( F3 @ ( G2 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( F3 @ A3 ) )
@ F4 ) ) ) ) ).
% bounded_linear.tendsto
thf(fact_7237_filterlim__at__top__mult__at__top,axiom,
! [A: $tType,F3: A > real,F4: filter @ A,G2: A > real] :
( ( filterlim @ A @ real @ F3 @ ( at_top @ real ) @ F4 )
=> ( ( filterlim @ A @ real @ G2 @ ( at_top @ real ) @ F4 )
=> ( filterlim @ A @ real
@ ^ [X2: A] : ( times_times @ real @ ( F3 @ X2 ) @ ( G2 @ X2 ) )
@ ( at_top @ real )
@ F4 ) ) ) ).
% filterlim_at_top_mult_at_top
thf(fact_7238_filterlim__at__top__add__at__top,axiom,
! [A: $tType,F3: A > real,F4: filter @ A,G2: A > real] :
( ( filterlim @ A @ real @ F3 @ ( at_top @ real ) @ F4 )
=> ( ( filterlim @ A @ real @ G2 @ ( at_top @ real ) @ F4 )
=> ( filterlim @ A @ real
@ ^ [X2: A] : ( plus_plus @ real @ ( F3 @ X2 ) @ ( G2 @ X2 ) )
@ ( at_top @ real )
@ F4 ) ) ) ).
% filterlim_at_top_add_at_top
thf(fact_7239_bounded__linear_OCauchy,axiom,
! [A: $tType,B: $tType] :
( ( ( real_V822414075346904944vector @ B )
& ( real_V822414075346904944vector @ A ) )
=> ! [F3: A > B,X5: nat > A] :
( ( real_V3181309239436604168linear @ A @ B @ F3 )
=> ( ( topolo3814608138187158403Cauchy @ A @ X5 )
=> ( topolo3814608138187158403Cauchy @ B
@ ^ [N4: nat] : ( F3 @ ( X5 @ N4 ) ) ) ) ) ) ).
% bounded_linear.Cauchy
thf(fact_7240_bounded__linear__minus,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F3: A > B] :
( ( real_V3181309239436604168linear @ A @ B @ F3 )
=> ( real_V3181309239436604168linear @ A @ B
@ ^ [X2: A] : ( uminus_uminus @ B @ ( F3 @ X2 ) ) ) ) ) ).
% bounded_linear_minus
thf(fact_7241_bounded__linear__sum,axiom,
! [I5: $tType,B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [I6: set @ I5,F3: I5 > A > B] :
( ! [I3: I5] :
( ( member @ I5 @ I3 @ I6 )
=> ( real_V3181309239436604168linear @ A @ B @ ( F3 @ I3 ) ) )
=> ( real_V3181309239436604168linear @ A @ B
@ ^ [X2: A] :
( groups7311177749621191930dd_sum @ I5 @ B
@ ^ [I4: I5] : ( F3 @ I4 @ X2 )
@ I6 ) ) ) ) ).
% bounded_linear_sum
thf(fact_7242_bounded__linear__zero,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ( real_V3181309239436604168linear @ A @ B
@ ^ [X2: A] : ( zero_zero @ B ) ) ) ).
% bounded_linear_zero
thf(fact_7243_bounded__linear__scaleR__left,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X: A] :
( real_V3181309239436604168linear @ real @ A
@ ^ [R5: real] : ( real_V8093663219630862766scaleR @ A @ R5 @ X ) ) ) ).
% bounded_linear_scaleR_left
thf(fact_7244_bounded__linear__const__scaleR,axiom,
! [B: $tType,C: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V822414075346904944vector @ B ) )
=> ! [G2: C > B,R3: real] :
( ( real_V3181309239436604168linear @ C @ B @ G2 )
=> ( real_V3181309239436604168linear @ C @ B
@ ^ [X2: C] : ( real_V8093663219630862766scaleR @ B @ R3 @ ( G2 @ X2 ) ) ) ) ) ).
% bounded_linear_const_scaleR
thf(fact_7245_bounded__linear__scaleR__const,axiom,
! [B: $tType,C: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V822414075346904944vector @ B ) )
=> ! [G2: C > real,X: B] :
( ( real_V3181309239436604168linear @ C @ real @ G2 )
=> ( real_V3181309239436604168linear @ C @ B
@ ^ [X2: C] : ( real_V8093663219630862766scaleR @ B @ ( G2 @ X2 ) @ X ) ) ) ) ).
% bounded_linear_scaleR_const
thf(fact_7246_bounded__linear__scaleR__right,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [R3: real] : ( real_V3181309239436604168linear @ A @ A @ ( real_V8093663219630862766scaleR @ A @ R3 ) ) ) ).
% bounded_linear_scaleR_right
thf(fact_7247_bounded__linear_Osums,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F3: A > B,X5: nat > A,A3: A] :
( ( real_V3181309239436604168linear @ A @ B @ F3 )
=> ( ( sums @ A @ X5 @ A3 )
=> ( sums @ B
@ ^ [N4: nat] : ( F3 @ ( X5 @ N4 ) )
@ ( F3 @ A3 ) ) ) ) ) ).
% bounded_linear.sums
thf(fact_7248_bounded__linear__compose,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V822414075346904944vector @ B )
& ( real_V822414075346904944vector @ A ) )
=> ! [F3: A > B,G2: C > A] :
( ( real_V3181309239436604168linear @ A @ B @ F3 )
=> ( ( real_V3181309239436604168linear @ C @ A @ G2 )
=> ( real_V3181309239436604168linear @ C @ B
@ ^ [X2: C] : ( F3 @ ( G2 @ X2 ) ) ) ) ) ) ).
% bounded_linear_compose
thf(fact_7249_bounded__linear__ident,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ( real_V3181309239436604168linear @ A @ A
@ ^ [X2: A] : X2 ) ) ).
% bounded_linear_ident
thf(fact_7250_bounded__linear_Osummable,axiom,
! [A: $tType,B: $tType] :
( ( ( real_V822414075346904944vector @ B )
& ( real_V822414075346904944vector @ A ) )
=> ! [F3: A > B,X5: nat > A] :
( ( real_V3181309239436604168linear @ A @ B @ F3 )
=> ( ( summable @ A @ X5 )
=> ( summable @ B
@ ^ [N4: nat] : ( F3 @ ( X5 @ N4 ) ) ) ) ) ) ).
% bounded_linear.summable
thf(fact_7251_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_7252_bounded__linear__mult__right,axiom,
! [A: $tType] :
( ( real_V4412858255891104859lgebra @ A )
=> ! [X: A] : ( real_V3181309239436604168linear @ A @ A @ ( times_times @ A @ X ) ) ) ).
% bounded_linear_mult_right
thf(fact_7253_bounded__linear__mult__const,axiom,
! [A: $tType,C: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V4412858255891104859lgebra @ A ) )
=> ! [G2: C > A,Y: A] :
( ( real_V3181309239436604168linear @ C @ A @ G2 )
=> ( real_V3181309239436604168linear @ C @ A
@ ^ [X2: C] : ( times_times @ A @ ( G2 @ X2 ) @ Y ) ) ) ) ).
% bounded_linear_mult_const
thf(fact_7254_bounded__linear__const__mult,axiom,
! [A: $tType,C: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V4412858255891104859lgebra @ A ) )
=> ! [G2: C > A,X: A] :
( ( real_V3181309239436604168linear @ C @ A @ G2 )
=> ( real_V3181309239436604168linear @ C @ A
@ ^ [X2: C] : ( times_times @ A @ X @ ( G2 @ X2 ) ) ) ) ) ).
% bounded_linear_const_mult
thf(fact_7255_bounded__linear__mult__left,axiom,
! [A: $tType] :
( ( real_V4412858255891104859lgebra @ A )
=> ! [Y: A] :
( real_V3181309239436604168linear @ A @ A
@ ^ [X2: A] : ( times_times @ A @ X2 @ Y ) ) ) ).
% bounded_linear_mult_left
thf(fact_7256_bounded__linear__divide,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [Y: A] :
( real_V3181309239436604168linear @ A @ A
@ ^ [X2: A] : ( divide_divide @ A @ X2 @ Y ) ) ) ).
% bounded_linear_divide
thf(fact_7257_bounded__linear__sub,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F3: A > B,G2: A > B] :
( ( real_V3181309239436604168linear @ A @ B @ F3 )
=> ( ( real_V3181309239436604168linear @ A @ B @ G2 )
=> ( real_V3181309239436604168linear @ A @ B
@ ^ [X2: A] : ( minus_minus @ B @ ( F3 @ X2 ) @ ( G2 @ X2 ) ) ) ) ) ) ).
% bounded_linear_sub
thf(fact_7258_bounded__linear__add,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F3: A > B,G2: A > B] :
( ( real_V3181309239436604168linear @ A @ B @ F3 )
=> ( ( real_V3181309239436604168linear @ A @ B @ G2 )
=> ( real_V3181309239436604168linear @ A @ B
@ ^ [X2: A] : ( plus_plus @ B @ ( F3 @ X2 ) @ ( G2 @ X2 ) ) ) ) ) ) ).
% bounded_linear_add
thf(fact_7259_bounded__linear_Osuminf,axiom,
! [A: $tType,B: $tType] :
( ( ( real_V822414075346904944vector @ B )
& ( real_V822414075346904944vector @ A ) )
=> ! [F3: A > B,X5: nat > A] :
( ( real_V3181309239436604168linear @ A @ B @ F3 )
=> ( ( summable @ A @ X5 )
=> ( ( F3 @ ( suminf @ A @ X5 ) )
= ( suminf @ B
@ ^ [N4: nat] : ( F3 @ ( X5 @ N4 ) ) ) ) ) ) ) ).
% bounded_linear.suminf
thf(fact_7260_bounded__linear_Ohas__derivative,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F3: A > B,G2: C > A,G5: C > A,F4: filter @ C] :
( ( real_V3181309239436604168linear @ A @ B @ F3 )
=> ( ( has_derivative @ C @ A @ G2 @ G5 @ F4 )
=> ( has_derivative @ C @ B
@ ^ [X2: C] : ( F3 @ ( G2 @ X2 ) )
@ ^ [X2: C] : ( F3 @ ( G5 @ X2 ) )
@ F4 ) ) ) ) ).
% bounded_linear.has_derivative
thf(fact_7261_bounded__linear_Otendsto__zero,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F3: A > B,G2: C > A,F4: filter @ C] :
( ( real_V3181309239436604168linear @ A @ B @ F3 )
=> ( ( filterlim @ C @ A @ G2 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ F4 )
=> ( filterlim @ C @ B
@ ^ [X2: C] : ( F3 @ ( G2 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ F4 ) ) ) ) ).
% bounded_linear.tendsto_zero
thf(fact_7262_bounded__linear_OisCont,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( ( topological_t2_space @ C )
& ( real_V822414075346904944vector @ B )
& ( real_V822414075346904944vector @ A ) )
=> ! [F3: A > B,A3: C,G2: C > A] :
( ( real_V3181309239436604168linear @ A @ B @ F3 )
=> ( ( topolo3448309680560233919inuous @ C @ A @ ( topolo174197925503356063within @ C @ A3 @ ( top_top @ ( set @ C ) ) ) @ G2 )
=> ( topolo3448309680560233919inuous @ C @ B @ ( topolo174197925503356063within @ C @ A3 @ ( top_top @ ( set @ C ) ) )
@ ^ [X2: C] : ( F3 @ ( G2 @ X2 ) ) ) ) ) ) ).
% bounded_linear.isCont
thf(fact_7263_filterlim__tendsto__add__at__top,axiom,
! [A: $tType,F3: A > real,C2: real,F4: filter @ A,G2: A > real] :
( ( filterlim @ A @ real @ F3 @ ( topolo7230453075368039082e_nhds @ real @ C2 ) @ F4 )
=> ( ( filterlim @ A @ real @ G2 @ ( at_top @ real ) @ F4 )
=> ( filterlim @ A @ real
@ ^ [X2: A] : ( plus_plus @ real @ ( F3 @ X2 ) @ ( G2 @ X2 ) )
@ ( at_top @ real )
@ F4 ) ) ) ).
% filterlim_tendsto_add_at_top
thf(fact_7264_filterlim__real__sequentially,axiom,
filterlim @ nat @ real @ ( semiring_1_of_nat @ real ) @ ( at_top @ real ) @ ( at_top @ nat ) ).
% filterlim_real_sequentially
thf(fact_7265_bounded__linear_Ononneg__bounded,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F3: A > B] :
( ( real_V3181309239436604168linear @ A @ B @ F3 )
=> ? [K8: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ K8 )
& ! [X6: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( F3 @ X6 ) ) @ ( times_times @ real @ ( real_V7770717601297561774m_norm @ A @ X6 ) @ K8 ) ) ) ) ) ).
% bounded_linear.nonneg_bounded
thf(fact_7266_has__derivative__within__singleton__iff,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F3: A > B,G2: A > B,X: A] :
( ( has_derivative @ A @ B @ F3 @ G2 @ ( topolo174197925503356063within @ A @ X @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( real_V3181309239436604168linear @ A @ B @ G2 ) ) ) ).
% has_derivative_within_singleton_iff
thf(fact_7267_filterlim__pow__at__top,axiom,
! [A: $tType,N: nat,F3: A > real,F4: filter @ A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( filterlim @ A @ real @ F3 @ ( at_top @ real ) @ F4 )
=> ( filterlim @ A @ real
@ ^ [X2: A] : ( power_power @ real @ ( F3 @ X2 ) @ N )
@ ( at_top @ real )
@ F4 ) ) ) ).
% filterlim_pow_at_top
thf(fact_7268_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_7269_real__tendsto__divide__at__top,axiom,
! [A: $tType,F3: A > real,C2: real,F4: filter @ A,G2: A > real] :
( ( filterlim @ A @ real @ F3 @ ( topolo7230453075368039082e_nhds @ real @ C2 ) @ F4 )
=> ( ( filterlim @ A @ real @ G2 @ ( at_top @ real ) @ F4 )
=> ( filterlim @ A @ real
@ ^ [X2: A] : ( divide_divide @ real @ ( F3 @ X2 ) @ ( G2 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ F4 ) ) ) ).
% real_tendsto_divide_at_top
thf(fact_7270_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_7271_tendsto__inverse__0__at__top,axiom,
! [A: $tType,F3: A > real,F4: filter @ A] :
( ( filterlim @ A @ real @ F3 @ ( at_top @ real ) @ F4 )
=> ( filterlim @ A @ real
@ ^ [X2: A] : ( inverse_inverse @ real @ ( F3 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ F4 ) ) ).
% tendsto_inverse_0_at_top
thf(fact_7272_filterlim__int__of__nat__at__topD,axiom,
! [A: $tType,F3: int > A,F4: filter @ A] :
( ( filterlim @ nat @ A
@ ^ [X2: nat] : ( F3 @ ( semiring_1_of_nat @ int @ X2 ) )
@ F4
@ ( at_top @ nat ) )
=> ( filterlim @ int @ A @ F3 @ F4 @ ( at_top @ int ) ) ) ).
% filterlim_int_of_nat_at_topD
thf(fact_7273_bounded__linear_Opos__bounded,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F3: A > B] :
( ( real_V3181309239436604168linear @ A @ B @ F3 )
=> ? [K8: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K8 )
& ! [X6: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( F3 @ X6 ) ) @ ( times_times @ real @ ( real_V7770717601297561774m_norm @ A @ X6 ) @ K8 ) ) ) ) ) ).
% bounded_linear.pos_bounded
thf(fact_7274_filterlim__sequentially__iff__filterlim__real,axiom,
! [A: $tType,F3: A > nat,F4: filter @ A] :
( ( filterlim @ A @ nat @ F3 @ ( at_top @ nat ) @ F4 )
= ( filterlim @ A @ real
@ ^ [X2: A] : ( semiring_1_of_nat @ real @ ( F3 @ X2 ) )
@ ( at_top @ real )
@ F4 ) ) ).
% filterlim_sequentially_iff_filterlim_real
thf(fact_7275_filterlim__tendsto__pos__mult__at__top,axiom,
! [A: $tType,F3: A > real,C2: real,F4: filter @ A,G2: A > real] :
( ( filterlim @ A @ real @ F3 @ ( topolo7230453075368039082e_nhds @ real @ C2 ) @ F4 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ C2 )
=> ( ( filterlim @ A @ real @ G2 @ ( at_top @ real ) @ F4 )
=> ( filterlim @ A @ real
@ ^ [X2: A] : ( times_times @ real @ ( F3 @ X2 ) @ ( G2 @ X2 ) )
@ ( at_top @ real )
@ F4 ) ) ) ) ).
% filterlim_tendsto_pos_mult_at_top
thf(fact_7276_filterlim__at__top__mult__tendsto__pos,axiom,
! [A: $tType,F3: A > real,C2: real,F4: filter @ A,G2: A > real] :
( ( filterlim @ A @ real @ F3 @ ( topolo7230453075368039082e_nhds @ real @ C2 ) @ F4 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ C2 )
=> ( ( filterlim @ A @ real @ G2 @ ( at_top @ real ) @ F4 )
=> ( filterlim @ A @ real
@ ^ [X2: A] : ( times_times @ real @ ( G2 @ X2 ) @ ( F3 @ X2 ) )
@ ( at_top @ real )
@ F4 ) ) ) ) ).
% filterlim_at_top_mult_tendsto_pos
thf(fact_7277_tendsto__neg__powr,axiom,
! [A: $tType,S2: real,F3: A > real,F4: filter @ A] :
( ( ord_less @ real @ S2 @ ( zero_zero @ real ) )
=> ( ( filterlim @ A @ real @ F3 @ ( at_top @ real ) @ F4 )
=> ( filterlim @ A @ real
@ ^ [X2: A] : ( powr @ real @ ( F3 @ X2 ) @ S2 )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ F4 ) ) ) ).
% tendsto_neg_powr
thf(fact_7278_ln__x__over__x__tendsto__0,axiom,
( filterlim @ real @ real
@ ^ [X2: real] : ( divide_divide @ real @ ( ln_ln @ real @ X2 ) @ X2 )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ ( at_top @ real ) ) ).
% ln_x_over_x_tendsto_0
thf(fact_7279_tendsto__at__topI__sequentially,axiom,
! [B: $tType] :
( ( topolo3112930676232923870pology @ B )
=> ! [F3: real > B,Y: B] :
( ! [X11: nat > real] :
( ( filterlim @ nat @ real @ X11 @ ( at_top @ real ) @ ( at_top @ nat ) )
=> ( filterlim @ nat @ B
@ ^ [N4: nat] : ( F3 @ ( X11 @ N4 ) )
@ ( topolo7230453075368039082e_nhds @ B @ Y )
@ ( at_top @ nat ) ) )
=> ( filterlim @ real @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ Y ) @ ( at_top @ real ) ) ) ) ).
% tendsto_at_topI_sequentially
thf(fact_7280_tendsto__power__div__exp__0,axiom,
! [K: nat] :
( filterlim @ real @ real
@ ^ [X2: real] : ( divide_divide @ real @ ( power_power @ real @ X2 @ K ) @ ( exp @ real @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ ( at_top @ real ) ) ).
% tendsto_power_div_exp_0
thf(fact_7281_tendsto__exp__limit__at__top,axiom,
! [X: real] :
( filterlim @ real @ real
@ ^ [Y2: real] : ( powr @ real @ ( plus_plus @ real @ ( one_one @ real ) @ ( divide_divide @ real @ X @ Y2 ) ) @ Y2 )
@ ( topolo7230453075368039082e_nhds @ real @ ( exp @ real @ X ) )
@ ( at_top @ real ) ) ).
% tendsto_exp_limit_at_top
thf(fact_7282_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_7283_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_7284_has__derivative__iff__norm,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F3: A > B,F7: A > B,X: A,S2: set @ A] :
( ( has_derivative @ A @ B @ F3 @ F7 @ ( topolo174197925503356063within @ A @ X @ S2 ) )
= ( ( real_V3181309239436604168linear @ A @ B @ F7 )
& ( filterlim @ A @ real
@ ^ [Y2: A] : ( divide_divide @ real @ ( real_V7770717601297561774m_norm @ B @ ( minus_minus @ B @ ( minus_minus @ B @ ( F3 @ Y2 ) @ ( F3 @ X ) ) @ ( F7 @ ( minus_minus @ A @ Y2 @ X ) ) ) ) @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ Y2 @ X ) ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ ( topolo174197925503356063within @ A @ X @ S2 ) ) ) ) ) ).
% has_derivative_iff_norm
thf(fact_7285_DERIV__neg__imp__decreasing__at__top,axiom,
! [B3: real,F3: real > real,Flim: real] :
( ! [X3: real] :
( ( ord_less_eq @ real @ B3 @ X3 )
=> ? [Y5: real] :
( ( has_field_derivative @ real @ F3 @ Y5 @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) )
& ( ord_less @ real @ Y5 @ ( zero_zero @ real ) ) ) )
=> ( ( filterlim @ real @ real @ F3 @ ( topolo7230453075368039082e_nhds @ real @ Flim ) @ ( at_top @ real ) )
=> ( ord_less @ real @ Flim @ ( F3 @ B3 ) ) ) ) ).
% DERIV_neg_imp_decreasing_at_top
thf(fact_7286_has__derivative__at__within,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F3: A > B,F7: A > B,X: A,S2: set @ A] :
( ( has_derivative @ A @ B @ F3 @ F7 @ ( topolo174197925503356063within @ A @ X @ S2 ) )
= ( ( real_V3181309239436604168linear @ A @ B @ F7 )
& ( filterlim @ A @ B
@ ^ [Y2: A] : ( real_V8093663219630862766scaleR @ B @ ( inverse_inverse @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ Y2 @ X ) ) ) @ ( minus_minus @ B @ ( minus_minus @ B @ ( F3 @ Y2 ) @ ( F3 @ X ) ) @ ( F7 @ ( minus_minus @ A @ Y2 @ X ) ) ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ ( topolo174197925503356063within @ A @ X @ S2 ) ) ) ) ) ).
% has_derivative_at_within
thf(fact_7287_has__derivativeI,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F7: A > B,X: A,F3: A > B,S2: set @ A] :
( ( real_V3181309239436604168linear @ A @ B @ F7 )
=> ( ( filterlim @ A @ B
@ ^ [Y2: A] : ( real_V8093663219630862766scaleR @ B @ ( inverse_inverse @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ Y2 @ X ) ) ) @ ( minus_minus @ B @ ( minus_minus @ B @ ( F3 @ Y2 ) @ ( F3 @ X ) ) @ ( F7 @ ( minus_minus @ A @ Y2 @ X ) ) ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ ( topolo174197925503356063within @ A @ X @ S2 ) )
=> ( has_derivative @ A @ B @ F3 @ F7 @ ( topolo174197925503356063within @ A @ X @ S2 ) ) ) ) ) ).
% has_derivativeI
thf(fact_7288_has__derivative__iff__Ex,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F3: A > B,F7: A > B,X: A] :
( ( has_derivative @ A @ B @ F3 @ F7 @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) )
= ( ( real_V3181309239436604168linear @ A @ B @ F7 )
& ? [E4: A > B] :
( ! [H: A] :
( ( F3 @ ( plus_plus @ A @ X @ H ) )
= ( plus_plus @ B @ ( plus_plus @ B @ ( F3 @ X ) @ ( 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_7289_has__derivative__within,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F3: A > B,F7: A > B,X: A,S2: set @ A] :
( ( has_derivative @ A @ B @ F3 @ F7 @ ( topolo174197925503356063within @ A @ X @ S2 ) )
= ( ( real_V3181309239436604168linear @ A @ B @ F7 )
& ( filterlim @ A @ B
@ ^ [Y2: A] : ( real_V8093663219630862766scaleR @ B @ ( divide_divide @ real @ ( one_one @ real ) @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ Y2 @ X ) ) ) @ ( minus_minus @ B @ ( F3 @ Y2 ) @ ( plus_plus @ B @ ( F3 @ X ) @ ( F7 @ ( minus_minus @ A @ Y2 @ X ) ) ) ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ ( topolo174197925503356063within @ A @ X @ S2 ) ) ) ) ) ).
% has_derivative_within
thf(fact_7290_has__derivative__def,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ( ( has_derivative @ A @ B )
= ( ^ [F2: A > B,F10: A > B,F8: filter @ A] :
( ( real_V3181309239436604168linear @ A @ B @ F10 )
& ( filterlim @ A @ B
@ ^ [Y2: A] :
( real_V8093663219630862766scaleR @ B
@ ( inverse_inverse @ real
@ ( real_V7770717601297561774m_norm @ A
@ ( minus_minus @ A @ Y2
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F8
@ ^ [X2: A] : X2 ) ) ) )
@ ( minus_minus @ B
@ ( minus_minus @ B @ ( F2 @ Y2 )
@ ( F2
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F8
@ ^ [X2: A] : X2 ) ) )
@ ( F10
@ ( minus_minus @ A @ Y2
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F8
@ ^ [X2: A] : X2 ) ) ) ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ F8 ) ) ) ) ) ).
% has_derivative_def
thf(fact_7291_has__derivative__at__within__iff__Ex,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [X: A,S: set @ A,F3: A > B,F7: A > B] :
( ( member @ A @ X @ S )
=> ( ( topolo1002775350975398744n_open @ A @ S )
=> ( ( has_derivative @ A @ B @ F3 @ F7 @ ( topolo174197925503356063within @ A @ X @ S ) )
= ( ( real_V3181309239436604168linear @ A @ B @ F7 )
& ? [E4: A > B] :
( ! [H: A] :
( ( member @ A @ ( plus_plus @ A @ X @ H ) @ S )
=> ( ( F3 @ ( plus_plus @ A @ X @ H ) )
= ( plus_plus @ B @ ( plus_plus @ B @ ( F3 @ X ) @ ( 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_7292_lim__const,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [A3: A] :
( ( topolo3827282254853284352ce_Lim @ nat @ A @ ( at_top @ nat )
@ ^ [M3: nat] : A3 )
= A3 ) ) ).
% lim_const
thf(fact_7293_open__empty,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ( topolo1002775350975398744n_open @ A @ ( bot_bot @ ( set @ A ) ) ) ) ).
% open_empty
thf(fact_7294_open__Un,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [S: set @ A,T4: set @ A] :
( ( topolo1002775350975398744n_open @ A @ S )
=> ( ( topolo1002775350975398744n_open @ A @ T4 )
=> ( topolo1002775350975398744n_open @ A @ ( sup_sup @ ( set @ A ) @ S @ T4 ) ) ) ) ) ).
% open_Un
thf(fact_7295_open__right,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [S: set @ A,X: A,Y: A] :
( ( topolo1002775350975398744n_open @ A @ S )
=> ( ( member @ A @ X @ S )
=> ( ( ord_less @ A @ X @ Y )
=> ? [B7: A] :
( ( ord_less @ A @ X @ B7 )
& ( ord_less_eq @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ X @ B7 ) @ S ) ) ) ) ) ) ).
% open_right
thf(fact_7296_not__open__singleton,axiom,
! [A: $tType] :
( ( topolo8386298272705272623_space @ A )
=> ! [X: A] :
~ ( topolo1002775350975398744n_open @ A @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% not_open_singleton
thf(fact_7297_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
@ ^ [X2: A] :
( ( P @ X2 )
& ( Q @ X2 ) ) ) ) ) ) ) ).
% open_Collect_conj
thf(fact_7298_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
@ ^ [X2: A] :
( ( P @ X2 )
| ( Q @ X2 ) ) ) ) ) ) ) ).
% open_Collect_disj
thf(fact_7299_open__Collect__const,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [P: $o] :
( topolo1002775350975398744n_open @ A
@ ( collect @ A
@ ^ [X2: A] : P ) ) ) ).
% open_Collect_const
thf(fact_7300_Lim__ident__at,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [X: A,S2: set @ A] :
( ( ( topolo174197925503356063within @ A @ X @ S2 )
!= ( bot_bot @ ( filter @ A ) ) )
=> ( ( topolo3827282254853284352ce_Lim @ A @ A @ ( topolo174197925503356063within @ A @ X @ S2 )
@ ^ [X2: A] : X2 )
= X ) ) ) ).
% Lim_ident_at
thf(fact_7301_tendsto__Lim,axiom,
! [A: $tType,B: $tType] :
( ( topological_t2_space @ B )
=> ! [Net: filter @ A,F3: A > B,L: B] :
( ( Net
!= ( bot_bot @ ( filter @ A ) ) )
=> ( ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ L ) @ Net )
=> ( ( topolo3827282254853284352ce_Lim @ A @ B @ Net @ F3 )
= L ) ) ) ) ).
% tendsto_Lim
thf(fact_7302_continuous__divide,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_V3459762299906320749_field @ B ) )
=> ! [F4: filter @ A,F3: A > B,G2: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ F4 @ F3 )
=> ( ( topolo3448309680560233919inuous @ A @ B @ F4 @ G2 )
=> ( ( ( G2
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F4
@ ^ [X2: A] : X2 ) )
!= ( zero_zero @ B ) )
=> ( topolo3448309680560233919inuous @ A @ B @ F4
@ ^ [X2: A] : ( divide_divide @ B @ ( F3 @ X2 ) @ ( G2 @ X2 ) ) ) ) ) ) ) ).
% continuous_divide
thf(fact_7303_continuous__inverse,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_V8999393235501362500lgebra @ B ) )
=> ! [F4: filter @ A,F3: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ F4 @ F3 )
=> ( ( ( F3
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F4
@ ^ [X2: A] : X2 ) )
!= ( zero_zero @ B ) )
=> ( topolo3448309680560233919inuous @ A @ B @ F4
@ ^ [X2: A] : ( inverse_inverse @ B @ ( F3 @ X2 ) ) ) ) ) ) ).
% continuous_inverse
thf(fact_7304_continuous__def,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ( ( topolo3448309680560233919inuous @ A @ B )
= ( ^ [F8: filter @ A,F2: A > B] :
( filterlim @ A @ B @ F2
@ ( topolo7230453075368039082e_nhds @ B
@ ( F2
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F8
@ ^ [X2: A] : X2 ) ) )
@ F8 ) ) ) ) ).
% continuous_def
thf(fact_7305_continuous__sgn,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F4: filter @ A,F3: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ F4 @ F3 )
=> ( ( ( F3
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F4
@ ^ [X2: A] : X2 ) )
!= ( zero_zero @ B ) )
=> ( topolo3448309680560233919inuous @ A @ B @ F4
@ ^ [X2: A] : ( sgn_sgn @ B @ ( F3 @ X2 ) ) ) ) ) ) ).
% continuous_sgn
thf(fact_7306_t2__space__class_OLim__def,axiom,
! [A: $tType,F: $tType] :
( ( topological_t2_space @ A )
=> ( ( topolo3827282254853284352ce_Lim @ F @ A )
= ( ^ [A4: filter @ F,F2: F > A] :
( the @ A
@ ^ [L2: A] : ( filterlim @ F @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L2 ) @ A4 ) ) ) ) ) ).
% t2_space_class.Lim_def
thf(fact_7307_continuous__powr,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [F4: filter @ A,F3: A > real,G2: A > real] :
( ( topolo3448309680560233919inuous @ A @ real @ F4 @ F3 )
=> ( ( topolo3448309680560233919inuous @ A @ real @ F4 @ G2 )
=> ( ( ( F3
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F4
@ ^ [X2: A] : X2 ) )
!= ( zero_zero @ real ) )
=> ( topolo3448309680560233919inuous @ A @ real @ F4
@ ^ [X2: A] : ( powr @ real @ ( F3 @ X2 ) @ ( G2 @ X2 ) ) ) ) ) ) ) ).
% continuous_powr
thf(fact_7308_continuous__ln,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [F4: filter @ A,F3: A > real] :
( ( topolo3448309680560233919inuous @ A @ real @ F4 @ F3 )
=> ( ( ( F3
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F4
@ ^ [X2: A] : X2 ) )
!= ( zero_zero @ real ) )
=> ( topolo3448309680560233919inuous @ A @ real @ F4
@ ^ [X2: A] : ( ln_ln @ real @ ( F3 @ X2 ) ) ) ) ) ) ).
% continuous_ln
thf(fact_7309_at__eq__bot__iff,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [A3: A] :
( ( ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) )
= ( bot_bot @ ( filter @ A ) ) )
= ( topolo1002775350975398744n_open @ A @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% at_eq_bot_iff
thf(fact_7310_suminf__eq__lim,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topological_t2_space @ A ) )
=> ( ( suminf @ A )
= ( ^ [F2: nat > A] :
( topolo3827282254853284352ce_Lim @ nat @ A @ ( at_top @ nat )
@ ^ [N4: nat] : ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_ord_lessThan @ nat @ N4 ) ) ) ) ) ) ).
% suminf_eq_lim
thf(fact_7311_Topological__Spaces_Olim__def,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [X5: nat > A] :
( ( topolo3827282254853284352ce_Lim @ nat @ A @ ( at_top @ nat ) @ X5 )
= ( the @ A
@ ^ [L7: A] : ( filterlim @ nat @ A @ X5 @ ( topolo7230453075368039082e_nhds @ A @ L7 ) @ ( at_top @ nat ) ) ) ) ) ).
% Topological_Spaces.lim_def
thf(fact_7312_continuous__tan,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [F4: filter @ A,F3: A > A] :
( ( topolo3448309680560233919inuous @ A @ A @ F4 @ F3 )
=> ( ( ( cos @ A
@ ( F3
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F4
@ ^ [X2: A] : X2 ) ) )
!= ( zero_zero @ A ) )
=> ( topolo3448309680560233919inuous @ A @ A @ F4
@ ^ [X2: A] : ( tan @ A @ ( F3 @ X2 ) ) ) ) ) ) ).
% continuous_tan
thf(fact_7313_continuous__cot,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [F4: filter @ A,F3: A > A] :
( ( topolo3448309680560233919inuous @ A @ A @ F4 @ F3 )
=> ( ( ( sin @ A
@ ( F3
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F4
@ ^ [X2: A] : X2 ) ) )
!= ( zero_zero @ A ) )
=> ( topolo3448309680560233919inuous @ A @ A @ F4
@ ^ [X2: A] : ( cot @ A @ ( F3 @ X2 ) ) ) ) ) ) ).
% continuous_cot
thf(fact_7314_continuous__tanh,axiom,
! [A: $tType,C: $tType] :
( ( ( topological_t2_space @ C )
& ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [F4: filter @ C,F3: C > A] :
( ( topolo3448309680560233919inuous @ C @ A @ F4 @ F3 )
=> ( ( ( cosh @ A
@ ( F3
@ ( topolo3827282254853284352ce_Lim @ C @ C @ F4
@ ^ [X2: C] : X2 ) ) )
!= ( zero_zero @ A ) )
=> ( topolo3448309680560233919inuous @ C @ A @ F4
@ ^ [X2: C] : ( tanh @ A @ ( F3 @ X2 ) ) ) ) ) ) ).
% continuous_tanh
thf(fact_7315_continuous__arcosh,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [F4: filter @ A,F3: A > real] :
( ( topolo3448309680560233919inuous @ A @ real @ F4 @ F3 )
=> ( ( ord_less @ real @ ( one_one @ real )
@ ( F3
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F4
@ ^ [X2: A] : X2 ) ) )
=> ( topolo3448309680560233919inuous @ A @ real @ F4
@ ^ [X2: A] : ( arcosh @ real @ ( F3 @ X2 ) ) ) ) ) ) ).
% continuous_arcosh
thf(fact_7316_continuous__log,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [F4: filter @ A,F3: A > real,G2: A > real] :
( ( topolo3448309680560233919inuous @ A @ real @ F4 @ F3 )
=> ( ( topolo3448309680560233919inuous @ A @ real @ F4 @ G2 )
=> ( ( ord_less @ real @ ( zero_zero @ real )
@ ( F3
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F4
@ ^ [X2: A] : X2 ) ) )
=> ( ( ( F3
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F4
@ ^ [X2: A] : X2 ) )
!= ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real )
@ ( G2
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F4
@ ^ [X2: A] : X2 ) ) )
=> ( topolo3448309680560233919inuous @ A @ real @ F4
@ ^ [X2: A] : ( log @ ( F3 @ X2 ) @ ( G2 @ X2 ) ) ) ) ) ) ) ) ) ).
% continuous_log
thf(fact_7317_continuous__artanh,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [F4: filter @ A,F3: A > real] :
( ( topolo3448309680560233919inuous @ A @ real @ F4 @ F3 )
=> ( ( member @ real
@ ( F3
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F4
@ ^ [X2: A] : X2 ) )
@ ( set_or5935395276787703475ssThan @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ ( one_one @ real ) ) )
=> ( topolo3448309680560233919inuous @ A @ real @ F4
@ ^ [X2: A] : ( artanh @ real @ ( F3 @ X2 ) ) ) ) ) ) ).
% continuous_artanh
thf(fact_7318_tendsto__offset__zero__iff,axiom,
! [C: $tType,D: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( topolo4958980785337419405_space @ D )
& ( zero @ C ) )
=> ! [A3: A,S: set @ A,F3: A > D,L6: D] :
( ( nO_MATCH @ C @ A @ ( zero_zero @ C ) @ A3 )
=> ( ( member @ A @ A3 @ S )
=> ( ( topolo1002775350975398744n_open @ A @ S )
=> ( ( filterlim @ A @ D @ F3 @ ( topolo7230453075368039082e_nhds @ D @ L6 ) @ ( topolo174197925503356063within @ A @ A3 @ S ) )
= ( filterlim @ A @ D
@ ^ [H: A] : ( F3 @ ( plus_plus @ A @ A3 @ H ) )
@ ( topolo7230453075368039082e_nhds @ D @ L6 )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ) ).
% tendsto_offset_zero_iff
thf(fact_7319_has__derivativeI__sandwich,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [E2: real,F7: A > B,S2: set @ A,X: A,F3: A > B,H6: A > real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E2 )
=> ( ( real_V3181309239436604168linear @ A @ B @ F7 )
=> ( ! [Y4: A] :
( ( member @ A @ Y4 @ S2 )
=> ( ( Y4 != X )
=> ( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ Y4 @ X ) @ E2 )
=> ( ord_less_eq @ real @ ( divide_divide @ real @ ( real_V7770717601297561774m_norm @ B @ ( minus_minus @ B @ ( minus_minus @ B @ ( F3 @ Y4 ) @ ( F3 @ X ) ) @ ( F7 @ ( minus_minus @ A @ Y4 @ X ) ) ) ) @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ Y4 @ X ) ) ) @ ( H6 @ Y4 ) ) ) ) )
=> ( ( filterlim @ A @ real @ H6 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( topolo174197925503356063within @ A @ X @ S2 ) )
=> ( has_derivative @ A @ B @ F3 @ F7 @ ( topolo174197925503356063within @ A @ X @ S2 ) ) ) ) ) ) ) ).
% has_derivativeI_sandwich
thf(fact_7320_dist__eq__0__iff,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X: A,Y: A] :
( ( ( real_V557655796197034286t_dist @ A @ X @ Y )
= ( zero_zero @ real ) )
= ( X = Y ) ) ) ).
% dist_eq_0_iff
thf(fact_7321_dist__self,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X: A] :
( ( real_V557655796197034286t_dist @ A @ X @ X )
= ( zero_zero @ real ) ) ) ).
% dist_self
thf(fact_7322_dist__0__norm,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X: A] :
( ( real_V557655796197034286t_dist @ A @ ( zero_zero @ A ) @ X )
= ( real_V7770717601297561774m_norm @ A @ X ) ) ) ).
% dist_0_norm
thf(fact_7323_zero__less__dist__iff,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( real_V557655796197034286t_dist @ A @ X @ Y ) )
= ( X != Y ) ) ) ).
% zero_less_dist_iff
thf(fact_7324_dist__le__zero__iff,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ real @ ( real_V557655796197034286t_dist @ A @ X @ Y ) @ ( zero_zero @ real ) )
= ( X = Y ) ) ) ).
% dist_le_zero_iff
thf(fact_7325_div__add__self2__no__field,axiom,
! [B: $tType,A: $tType] :
( ( ( euclid4440199948858584721cancel @ A )
& ( field @ B ) )
=> ! [X: B,B3: A,A3: A] :
( ( nO_MATCH @ B @ A @ X @ B3 )
=> ( ( B3
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ A3 @ B3 ) @ B3 )
= ( plus_plus @ A @ ( divide_divide @ A @ A3 @ B3 ) @ ( one_one @ A ) ) ) ) ) ) ).
% div_add_self2_no_field
thf(fact_7326_div__add__self1__no__field,axiom,
! [B: $tType,A: $tType] :
( ( ( euclid4440199948858584721cancel @ A )
& ( field @ B ) )
=> ! [X: B,B3: A,A3: A] :
( ( nO_MATCH @ B @ A @ X @ B3 )
=> ( ( B3
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ B3 @ A3 ) @ B3 )
= ( plus_plus @ A @ ( divide_divide @ A @ A3 @ B3 ) @ ( one_one @ A ) ) ) ) ) ) ).
% div_add_self1_no_field
thf(fact_7327_open__dist,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ( ( topolo1002775350975398744n_open @ A )
= ( ^ [S8: set @ A] :
! [X2: A] :
( ( member @ A @ X2 @ S8 )
=> ? [E4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E4 )
& ! [Y2: A] :
( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ Y2 @ X2 ) @ E4 )
=> ( member @ A @ Y2 @ S8 ) ) ) ) ) ) ) ).
% open_dist
thf(fact_7328_open__ball,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X: A,D2: real] :
( topolo1002775350975398744n_open @ A
@ ( collect @ A
@ ^ [Y2: A] : ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X @ Y2 ) @ D2 ) ) ) ) ).
% open_ball
thf(fact_7329_continuous__dist,axiom,
! [A: $tType,D: $tType] :
( ( ( topological_t2_space @ D )
& ( real_V7819770556892013058_space @ A ) )
=> ! [F4: filter @ D,F3: D > A,G2: D > A] :
( ( topolo3448309680560233919inuous @ D @ A @ F4 @ F3 )
=> ( ( topolo3448309680560233919inuous @ D @ A @ F4 @ G2 )
=> ( topolo3448309680560233919inuous @ D @ real @ F4
@ ^ [X2: D] : ( real_V557655796197034286t_dist @ A @ ( F3 @ X2 ) @ ( G2 @ X2 ) ) ) ) ) ) ).
% continuous_dist
thf(fact_7330_zero__le__dist,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X: A,Y: A] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( real_V557655796197034286t_dist @ A @ X @ Y ) ) ) ).
% zero_le_dist
thf(fact_7331_norm__conv__dist,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ( ( real_V7770717601297561774m_norm @ A )
= ( ^ [X2: A] : ( real_V557655796197034286t_dist @ A @ X2 @ ( zero_zero @ A ) ) ) ) ) ).
% norm_conv_dist
thf(fact_7332_dist__triangle__less__add,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X13: A,Y: A,E1: real,X22: A,E22: real] :
( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X13 @ Y ) @ E1 )
=> ( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X22 @ Y ) @ E22 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X13 @ X22 ) @ ( plus_plus @ real @ E1 @ E22 ) ) ) ) ) ).
% dist_triangle_less_add
thf(fact_7333_dist__triangle__lt,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X: A,Z: A,Y: A,E2: real] :
( ( ord_less @ real @ ( plus_plus @ real @ ( real_V557655796197034286t_dist @ A @ X @ Z ) @ ( real_V557655796197034286t_dist @ A @ Y @ Z ) ) @ E2 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X @ Y ) @ E2 ) ) ) ).
% dist_triangle_lt
thf(fact_7334_dist__commute__lessI,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [Y: A,X: A,E2: real] :
( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ Y @ X ) @ E2 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X @ Y ) @ E2 ) ) ) ).
% dist_commute_lessI
thf(fact_7335_dist__not__less__zero,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X: A,Y: A] :
~ ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X @ Y ) @ ( zero_zero @ real ) ) ) ).
% dist_not_less_zero
thf(fact_7336_dist__pos__lt,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X: A,Y: A] :
( ( X != Y )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( real_V557655796197034286t_dist @ A @ X @ Y ) ) ) ) ).
% dist_pos_lt
thf(fact_7337_has__field__derivative__transform__within,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,F7: A,A3: A,S: set @ A,D2: real,G2: A > A] :
( ( has_field_derivative @ A @ F3 @ F7 @ ( topolo174197925503356063within @ A @ A3 @ S ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ D2 )
=> ( ( member @ A @ A3 @ S )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ S )
=> ( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X3 @ A3 ) @ D2 )
=> ( ( F3 @ X3 )
= ( G2 @ X3 ) ) ) )
=> ( has_field_derivative @ A @ G2 @ F7 @ ( topolo174197925503356063within @ A @ A3 @ S ) ) ) ) ) ) ) ).
% has_field_derivative_transform_within
thf(fact_7338_has__derivative__transform__within,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F3: A > B,F7: A > B,X: A,S2: set @ A,D2: real,G2: A > B] :
( ( has_derivative @ A @ B @ F3 @ F7 @ ( topolo174197925503356063within @ A @ X @ S2 ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ D2 )
=> ( ( member @ A @ X @ S2 )
=> ( ! [X10: A] :
( ( member @ A @ X10 @ S2 )
=> ( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X10 @ X ) @ D2 )
=> ( ( F3 @ X10 )
= ( G2 @ X10 ) ) ) )
=> ( has_derivative @ A @ B @ G2 @ F7 @ ( topolo174197925503356063within @ A @ X @ S2 ) ) ) ) ) ) ) ).
% has_derivative_transform_within
thf(fact_7339_Cauchy__def,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ( ( topolo3814608138187158403Cauchy @ A )
= ( ^ [X8: nat > A] :
! [E4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E4 )
=> ? [M10: nat] :
! [M3: nat] :
( ( ord_less_eq @ nat @ M10 @ M3 )
=> ! [N4: nat] :
( ( ord_less_eq @ nat @ M10 @ N4 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( X8 @ M3 ) @ ( X8 @ N4 ) ) @ E4 ) ) ) ) ) ) ) ).
% Cauchy_def
thf(fact_7340_Cauchy__altdef2,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ( ( topolo3814608138187158403Cauchy @ A )
= ( ^ [S5: nat > A] :
! [E4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E4 )
=> ? [N9: nat] :
! [N4: nat] :
( ( ord_less_eq @ nat @ N9 @ N4 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( S5 @ N4 ) @ ( S5 @ N9 ) ) @ E4 ) ) ) ) ) ) ).
% Cauchy_altdef2
thf(fact_7341_metric__CauchyD,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X5: nat > A,E2: real] :
( ( topolo3814608138187158403Cauchy @ A @ X5 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ E2 )
=> ? [M9: nat] :
! [M2: nat] :
( ( ord_less_eq @ nat @ M9 @ M2 )
=> ! [N11: nat] :
( ( ord_less_eq @ nat @ M9 @ N11 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( X5 @ M2 ) @ ( X5 @ N11 ) ) @ E2 ) ) ) ) ) ) ).
% metric_CauchyD
thf(fact_7342_metric__CauchyI,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X5: nat > A] :
( ! [E: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E )
=> ? [M11: nat] :
! [M4: nat] :
( ( ord_less_eq @ nat @ M11 @ M4 )
=> ! [N2: nat] :
( ( ord_less_eq @ nat @ M11 @ N2 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( X5 @ M4 ) @ ( X5 @ N2 ) ) @ E ) ) ) )
=> ( topolo3814608138187158403Cauchy @ A @ X5 ) ) ) ).
% metric_CauchyI
thf(fact_7343_tendsto__dist,axiom,
! [A: $tType,B: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [F3: B > A,L: A,F4: filter @ B,G2: B > A,M: A] :
( ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ F4 )
=> ( ( filterlim @ B @ A @ G2 @ ( topolo7230453075368039082e_nhds @ A @ M ) @ F4 )
=> ( filterlim @ B @ real
@ ^ [X2: B] : ( real_V557655796197034286t_dist @ A @ ( F3 @ X2 ) @ ( G2 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( real_V557655796197034286t_dist @ A @ L @ M ) )
@ F4 ) ) ) ) ).
% tendsto_dist
thf(fact_7344_distrib__left__NO__MATCH,axiom,
! [B: $tType,A: $tType] :
( ( semiring @ A )
=> ! [X: B,Y: B,A3: A,B3: A,C2: A] :
( ( nO_MATCH @ B @ A @ ( divide_divide @ B @ X @ Y ) @ A3 )
=> ( ( times_times @ A @ A3 @ ( plus_plus @ A @ B3 @ C2 ) )
= ( plus_plus @ A @ ( times_times @ A @ A3 @ B3 ) @ ( times_times @ A @ A3 @ C2 ) ) ) ) ) ).
% distrib_left_NO_MATCH
thf(fact_7345_distrib__right__NO__MATCH,axiom,
! [B: $tType,A: $tType] :
( ( semiring @ A )
=> ! [X: B,Y: B,C2: A,A3: A,B3: A] :
( ( nO_MATCH @ B @ A @ ( divide_divide @ B @ X @ Y ) @ C2 )
=> ( ( times_times @ A @ ( plus_plus @ A @ A3 @ B3 ) @ C2 )
= ( plus_plus @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B3 @ C2 ) ) ) ) ) ).
% distrib_right_NO_MATCH
thf(fact_7346_left__diff__distrib__NO__MATCH,axiom,
! [B: $tType,A: $tType] :
( ( ring @ A )
=> ! [X: B,Y: B,C2: A,A3: A,B3: A] :
( ( nO_MATCH @ B @ A @ ( divide_divide @ B @ X @ Y ) @ C2 )
=> ( ( times_times @ A @ ( minus_minus @ A @ A3 @ B3 ) @ C2 )
= ( minus_minus @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B3 @ C2 ) ) ) ) ) ).
% left_diff_distrib_NO_MATCH
thf(fact_7347_right__diff__distrib__NO__MATCH,axiom,
! [B: $tType,A: $tType] :
( ( ring @ A )
=> ! [X: B,Y: B,A3: A,B3: A,C2: A] :
( ( nO_MATCH @ B @ A @ ( divide_divide @ B @ X @ Y ) @ A3 )
=> ( ( times_times @ A @ A3 @ ( minus_minus @ A @ B3 @ C2 ) )
= ( minus_minus @ A @ ( times_times @ A @ A3 @ B3 ) @ ( times_times @ A @ A3 @ C2 ) ) ) ) ) ).
% right_diff_distrib_NO_MATCH
thf(fact_7348_dist__triangle__half__r,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [Y: A,X13: A,E2: real,X22: A] :
( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ Y @ X13 ) @ ( divide_divide @ real @ E2 @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ Y @ X22 ) @ ( divide_divide @ real @ E2 @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X13 @ X22 ) @ E2 ) ) ) ) ).
% dist_triangle_half_r
thf(fact_7349_dist__triangle__half__l,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X13: A,Y: A,E2: real,X22: A] :
( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X13 @ Y ) @ ( divide_divide @ real @ E2 @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X22 @ Y ) @ ( divide_divide @ real @ E2 @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X13 @ X22 ) @ E2 ) ) ) ) ).
% dist_triangle_half_l
thf(fact_7350_dist__triangle__third,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X13: A,X22: A,E2: real,X32: A,X42: A] :
( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X13 @ X22 ) @ ( divide_divide @ real @ E2 @ ( numeral_numeral @ real @ ( bit1 @ one2 ) ) ) )
=> ( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X22 @ X32 ) @ ( divide_divide @ real @ E2 @ ( numeral_numeral @ real @ ( bit1 @ one2 ) ) ) )
=> ( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X32 @ X42 ) @ ( divide_divide @ real @ E2 @ ( numeral_numeral @ real @ ( bit1 @ one2 ) ) ) )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X13 @ X42 ) @ E2 ) ) ) ) ) ).
% dist_triangle_third
thf(fact_7351_Lim__transform__within,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V7819770556892013058_space @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [F3: A > B,L: B,X: A,S: set @ A,D2: real,G2: A > B] :
( ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ L ) @ ( topolo174197925503356063within @ A @ X @ 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 @ X ) )
=> ( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X10 @ X ) @ D2 )
=> ( ( F3 @ X10 )
= ( G2 @ X10 ) ) ) ) )
=> ( filterlim @ A @ B @ G2 @ ( topolo7230453075368039082e_nhds @ B @ L ) @ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ) ).
% Lim_transform_within
thf(fact_7352_filterlim__transform__within,axiom,
! [B: $tType,A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [G2: A > B,G4: filter @ B,X: A,S: set @ A,F4: filter @ B,D2: real,F3: A > B] :
( ( filterlim @ A @ B @ G2 @ G4 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( ( ord_less_eq @ ( filter @ B ) @ G4 @ F4 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ D2 )
=> ( ! [X10: A] :
( ( member @ A @ X10 @ S )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( real_V557655796197034286t_dist @ A @ X10 @ X ) )
=> ( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X10 @ X ) @ D2 )
=> ( ( F3 @ X10 )
= ( G2 @ X10 ) ) ) ) )
=> ( filterlim @ A @ B @ F3 @ F4 @ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ) ) ).
% filterlim_transform_within
thf(fact_7353_Cauchy__altdef,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ( ( topolo3814608138187158403Cauchy @ A )
= ( ^ [F2: nat > A] :
! [E4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E4 )
=> ? [M10: nat] :
! [M3: nat] :
( ( ord_less_eq @ nat @ M10 @ M3 )
=> ! [N4: nat] :
( ( ord_less @ nat @ M3 @ N4 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( F2 @ M3 ) @ ( F2 @ N4 ) ) @ E4 ) ) ) ) ) ) ) ).
% Cauchy_altdef
thf(fact_7354_CauchyI_H,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X5: nat > A] :
( ! [E: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E )
=> ? [M11: nat] :
! [M4: nat] :
( ( ord_less_eq @ nat @ M11 @ M4 )
=> ! [N2: nat] :
( ( ord_less @ nat @ M4 @ N2 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( X5 @ M4 ) @ ( X5 @ N2 ) ) @ E ) ) ) )
=> ( topolo3814608138187158403Cauchy @ A @ X5 ) ) ) ).
% CauchyI'
thf(fact_7355_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_7356_tendsto__dist__iff,axiom,
! [B: $tType,A: $tType] :
( ( real_V7819770556892013058_space @ B )
=> ! [F3: A > B,L: B,F4: filter @ A] :
( ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ L ) @ F4 )
= ( filterlim @ A @ real
@ ^ [X2: A] : ( real_V557655796197034286t_dist @ B @ ( F3 @ X2 ) @ L )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ F4 ) ) ) ).
% tendsto_dist_iff
thf(fact_7357_LIM__def,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V7819770556892013058_space @ A )
& ( real_V7819770556892013058_space @ B ) )
=> ! [F3: A > B,L6: B,A3: A] :
( ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ L6 ) @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) )
= ( ! [R5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R5 )
=> ? [S5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ S5 )
& ! [X2: A] :
( ( ( X2 != A3 )
& ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X2 @ A3 ) @ S5 ) )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ B @ ( F3 @ X2 ) @ L6 ) @ R5 ) ) ) ) ) ) ) ).
% LIM_def
thf(fact_7358_metric__LIM__D,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V7819770556892013058_space @ A )
& ( real_V7819770556892013058_space @ B ) )
=> ! [F3: A > B,L6: B,A3: A,R3: real] :
( ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ L6 ) @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ R3 )
=> ? [S3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ S3 )
& ! [X6: A] :
( ( ( X6 != A3 )
& ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X6 @ A3 ) @ S3 ) )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ B @ ( F3 @ X6 ) @ L6 ) @ R3 ) ) ) ) ) ) ).
% metric_LIM_D
thf(fact_7359_metric__LIM__I,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V7819770556892013058_space @ A )
& ( real_V7819770556892013058_space @ B ) )
=> ! [A3: A,F3: A > B,L6: B] :
( ! [R2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R2 )
=> ? [S9: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ S9 )
& ! [X3: A] :
( ( ( X3 != A3 )
& ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X3 @ A3 ) @ S9 ) )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ B @ ( F3 @ X3 ) @ L6 ) @ R2 ) ) ) )
=> ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ L6 ) @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% metric_LIM_I
thf(fact_7360_metric__LIM__equal2,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V7819770556892013058_space @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [G2: A > B,L: B,A3: A,R: real,F3: A > B] :
( ( filterlim @ A @ B @ G2 @ ( topolo7230453075368039082e_nhds @ B @ L ) @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ R )
=> ( ! [X3: A] :
( ( X3 != A3 )
=> ( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X3 @ A3 ) @ R )
=> ( ( F3 @ X3 )
= ( G2 @ X3 ) ) ) )
=> ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ L ) @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ).
% metric_LIM_equal2
thf(fact_7361_metric__LIMSEQ__D,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X5: nat > A,L6: A,R3: real] :
( ( filterlim @ nat @ A @ X5 @ ( topolo7230453075368039082e_nhds @ A @ L6 ) @ ( at_top @ nat ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ R3 )
=> ? [No: nat] :
! [N11: nat] :
( ( ord_less_eq @ nat @ No @ N11 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( X5 @ N11 ) @ L6 ) @ R3 ) ) ) ) ) ).
% metric_LIMSEQ_D
thf(fact_7362_metric__LIMSEQ__I,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X5: nat > A,L6: A] :
( ! [R2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R2 )
=> ? [No2: nat] :
! [N2: nat] :
( ( ord_less_eq @ nat @ No2 @ N2 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( X5 @ N2 ) @ L6 ) @ R2 ) ) )
=> ( filterlim @ nat @ A @ X5 @ ( topolo7230453075368039082e_nhds @ A @ L6 ) @ ( at_top @ nat ) ) ) ) ).
% metric_LIMSEQ_I
thf(fact_7363_lim__sequentially,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X5: nat > A,L6: A] :
( ( filterlim @ nat @ A @ X5 @ ( topolo7230453075368039082e_nhds @ A @ L6 ) @ ( at_top @ nat ) )
= ( ! [R5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R5 )
=> ? [No3: nat] :
! [N4: nat] :
( ( ord_less_eq @ nat @ No3 @ N4 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( X5 @ N4 ) @ L6 ) @ R5 ) ) ) ) ) ) ).
% lim_sequentially
thf(fact_7364_power__minus_H,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [X: A,N: nat] :
( ( nO_MATCH @ A @ A @ ( one_one @ A ) @ X )
=> ( ( power_power @ A @ ( uminus_uminus @ A @ X ) @ N )
= ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N ) @ ( power_power @ A @ X @ N ) ) ) ) ) ).
% power_minus'
thf(fact_7365_metric__Cauchy__iff2,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ( ( topolo3814608138187158403Cauchy @ A )
= ( ^ [X8: nat > A] :
! [J3: nat] :
? [M10: nat] :
! [M3: nat] :
( ( ord_less_eq @ nat @ M10 @ M3 )
=> ! [N4: nat] :
( ( ord_less_eq @ nat @ M10 @ N4 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( X8 @ M3 ) @ ( X8 @ N4 ) ) @ ( inverse_inverse @ real @ ( semiring_1_of_nat @ real @ ( suc @ J3 ) ) ) ) ) ) ) ) ) ).
% metric_Cauchy_iff2
thf(fact_7366_metric__LIM__compose2,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( real_V7819770556892013058_space @ A )
& ( topolo4958980785337419405_space @ B )
& ( topolo4958980785337419405_space @ C ) )
=> ! [F3: A > B,B3: B,A3: A,G2: B > C,C2: C] :
( ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ B3 ) @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) )
=> ( ( filterlim @ B @ C @ G2 @ ( topolo7230453075368039082e_nhds @ C @ C2 ) @ ( topolo174197925503356063within @ B @ B3 @ ( top_top @ ( set @ B ) ) ) )
=> ( ? [D5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D5 )
& ! [X3: A] :
( ( ( X3 != A3 )
& ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X3 @ A3 ) @ D5 ) )
=> ( ( F3 @ X3 )
!= B3 ) ) )
=> ( filterlim @ A @ C
@ ^ [X2: A] : ( G2 @ ( F3 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ C @ C2 )
@ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ).
% metric_LIM_compose2
thf(fact_7367_metric__isCont__LIM__compose2,axiom,
! [D: $tType,C: $tType,A: $tType] :
( ( ( real_V7819770556892013058_space @ A )
& ( topolo4958980785337419405_space @ C )
& ( topolo4958980785337419405_space @ D ) )
=> ! [A3: A,F3: A > C,G2: C > D,L: D] :
( ( topolo3448309680560233919inuous @ A @ C @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) @ F3 )
=> ( ( filterlim @ C @ D @ G2 @ ( topolo7230453075368039082e_nhds @ D @ L ) @ ( topolo174197925503356063within @ C @ ( F3 @ A3 ) @ ( top_top @ ( set @ C ) ) ) )
=> ( ? [D5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D5 )
& ! [X3: A] :
( ( ( X3 != A3 )
& ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X3 @ A3 ) @ D5 ) )
=> ( ( F3 @ X3 )
!= ( F3 @ A3 ) ) ) )
=> ( filterlim @ A @ D
@ ^ [X2: A] : ( G2 @ ( F3 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ D @ L )
@ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ).
% metric_isCont_LIM_compose2
thf(fact_7368_LIMSEQ__iff__nz,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X5: nat > A,L6: A] :
( ( filterlim @ nat @ A @ X5 @ ( topolo7230453075368039082e_nhds @ A @ L6 ) @ ( at_top @ nat ) )
= ( ! [R5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R5 )
=> ? [No3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ No3 )
& ! [N4: nat] :
( ( ord_less_eq @ nat @ No3 @ N4 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( X5 @ N4 ) @ L6 ) @ R5 ) ) ) ) ) ) ) ).
% LIMSEQ_iff_nz
thf(fact_7369_LIM__offset__zero__iff,axiom,
! [C: $tType,D: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( topolo4958980785337419405_space @ D )
& ( zero @ C ) )
=> ! [A3: A,F3: A > D,L6: D] :
( ( nO_MATCH @ C @ A @ ( zero_zero @ C ) @ A3 )
=> ( ( filterlim @ A @ D @ F3 @ ( topolo7230453075368039082e_nhds @ D @ L6 ) @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) )
= ( filterlim @ A @ D
@ ^ [H: A] : ( F3 @ ( plus_plus @ A @ A3 @ H ) )
@ ( topolo7230453075368039082e_nhds @ D @ L6 )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% LIM_offset_zero_iff
thf(fact_7370_filterlim__pow__at__bot__even,axiom,
! [N: nat,F3: real > real,F4: filter @ real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( filterlim @ real @ real @ F3 @ ( at_bot @ real ) @ F4 )
=> ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( filterlim @ real @ real
@ ^ [X2: real] : ( power_power @ real @ ( F3 @ X2 ) @ N )
@ ( at_top @ real )
@ F4 ) ) ) ) ).
% filterlim_pow_at_bot_even
thf(fact_7371_tendsto__exp__limit__at__right,axiom,
! [X: real] :
( filterlim @ real @ real
@ ^ [Y2: real] : ( powr @ real @ ( plus_plus @ real @ ( one_one @ real ) @ ( times_times @ real @ X @ Y2 ) ) @ ( divide_divide @ real @ ( one_one @ real ) @ Y2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( exp @ real @ X ) )
@ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) ) ).
% tendsto_exp_limit_at_right
thf(fact_7372_greaterThan__iff,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [I: A,K: A] :
( ( member @ A @ I @ ( set_ord_greaterThan @ A @ K ) )
= ( ord_less @ A @ K @ I ) ) ) ).
% greaterThan_iff
thf(fact_7373_trivial__limit__at__right__real,axiom,
! [A: $tType] :
( ( ( dense_order @ A )
& ( no_top @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [X: A] :
( ( topolo174197925503356063within @ A @ X @ ( set_ord_greaterThan @ A @ X ) )
!= ( bot_bot @ ( filter @ A ) ) ) ) ).
% trivial_limit_at_right_real
thf(fact_7374_trivial__limit__at__bot__linorder,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( at_bot @ A )
!= ( bot_bot @ ( filter @ A ) ) ) ) ).
% trivial_limit_at_bot_linorder
thf(fact_7375_greaterThan__non__empty,axiom,
! [A: $tType] :
( ( no_top @ A )
=> ! [X: A] :
( ( set_ord_greaterThan @ A @ X )
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% greaterThan_non_empty
thf(fact_7376_greaterThan__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( set_ord_greaterThan @ A )
= ( ^ [L2: A] : ( collect @ A @ ( ord_less @ A @ L2 ) ) ) ) ) ).
% greaterThan_def
thf(fact_7377_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_7378_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_7379_at__within__Icc__at__right,axiom,
! [A: $tType] :
( ( topolo2564578578187576103pology @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ A3 @ B3 )
=> ( ( topolo174197925503356063within @ A @ A3 @ ( set_or1337092689740270186AtMost @ A @ A3 @ B3 ) )
= ( topolo174197925503356063within @ A @ A3 @ ( set_ord_greaterThan @ A @ A3 ) ) ) ) ) ).
% at_within_Icc_at_right
thf(fact_7380_trivial__limit__at__right__top,axiom,
! [A: $tType] :
( ( ( order_top @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ( ( topolo174197925503356063within @ A @ ( top_top @ A ) @ ( set_ord_greaterThan @ A @ ( top_top @ A ) ) )
= ( bot_bot @ ( filter @ A ) ) ) ) ).
% trivial_limit_at_right_top
thf(fact_7381_filterlim__uminus__at__bot,axiom,
! [A: $tType,F3: A > real,F4: filter @ A] :
( ( filterlim @ A @ real @ F3 @ ( at_bot @ real ) @ F4 )
= ( filterlim @ A @ real
@ ^ [X2: A] : ( uminus_uminus @ real @ ( F3 @ X2 ) )
@ ( at_top @ real )
@ F4 ) ) ).
% filterlim_uminus_at_bot
thf(fact_7382_filterlim__uminus__at__top,axiom,
! [A: $tType,F3: A > real,F4: filter @ A] :
( ( filterlim @ A @ real @ F3 @ ( at_top @ real ) @ F4 )
= ( filterlim @ A @ real
@ ^ [X2: A] : ( uminus_uminus @ real @ ( F3 @ X2 ) )
@ ( at_bot @ real )
@ F4 ) ) ).
% filterlim_uminus_at_top
thf(fact_7383_filterlim__at__bot__mirror,axiom,
! [A: $tType,F3: real > A,F4: filter @ A] :
( ( filterlim @ real @ A @ F3 @ F4 @ ( at_bot @ real ) )
= ( filterlim @ real @ A
@ ^ [X2: real] : ( F3 @ ( uminus_uminus @ real @ X2 ) )
@ F4
@ ( at_top @ real ) ) ) ).
% filterlim_at_bot_mirror
thf(fact_7384_filterlim__at__top__mirror,axiom,
! [A: $tType,F3: real > A,F4: filter @ A] :
( ( filterlim @ real @ A @ F3 @ F4 @ ( at_top @ real ) )
= ( filterlim @ real @ A
@ ^ [X2: real] : ( F3 @ ( uminus_uminus @ real @ X2 ) )
@ F4
@ ( at_bot @ real ) ) ) ).
% filterlim_at_top_mirror
thf(fact_7385_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_7386_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_7387_filterlim__at__right__to__0,axiom,
! [A: $tType,F3: real > A,F4: filter @ A,A3: real] :
( ( filterlim @ real @ A @ F3 @ F4 @ ( topolo174197925503356063within @ real @ A3 @ ( set_ord_greaterThan @ real @ A3 ) ) )
= ( filterlim @ real @ A
@ ^ [X2: real] : ( F3 @ ( plus_plus @ real @ X2 @ A3 ) )
@ F4
@ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) ) ) ).
% filterlim_at_right_to_0
thf(fact_7388_filterlim__at__left__to__right,axiom,
! [A: $tType,F3: real > A,F4: filter @ A,A3: real] :
( ( filterlim @ real @ A @ F3 @ F4 @ ( topolo174197925503356063within @ real @ A3 @ ( set_ord_lessThan @ real @ A3 ) ) )
= ( filterlim @ real @ A
@ ^ [X2: real] : ( F3 @ ( uminus_uminus @ real @ X2 ) )
@ F4
@ ( topolo174197925503356063within @ real @ ( uminus_uminus @ real @ A3 ) @ ( set_ord_greaterThan @ real @ ( uminus_uminus @ real @ A3 ) ) ) ) ) ).
% filterlim_at_left_to_right
thf(fact_7389_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_7390_exp__at__bot,axiom,
filterlim @ real @ real @ ( exp @ real ) @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_bot @ real ) ).
% exp_at_bot
thf(fact_7391_filterlim__times__pos,axiom,
! [A: $tType,B: $tType] :
( ( ( linordered_field @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F3: B > A,P4: A,F12: filter @ B,C2: A,L: A] :
( ( filterlim @ B @ A @ F3 @ ( topolo174197925503356063within @ A @ P4 @ ( set_ord_greaterThan @ A @ P4 ) ) @ F12 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( L
= ( times_times @ A @ C2 @ P4 ) )
=> ( filterlim @ B @ A
@ ^ [X2: B] : ( times_times @ A @ C2 @ ( F3 @ X2 ) )
@ ( topolo174197925503356063within @ A @ L @ ( set_ord_greaterThan @ A @ L ) )
@ F12 ) ) ) ) ) ).
% filterlim_times_pos
thf(fact_7392_filterlim__at__right__to__top,axiom,
! [A: $tType,F3: real > A,F4: filter @ A] :
( ( filterlim @ real @ A @ F3 @ F4 @ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) )
= ( filterlim @ real @ A
@ ^ [X2: real] : ( F3 @ ( inverse_inverse @ real @ X2 ) )
@ F4
@ ( at_top @ real ) ) ) ).
% filterlim_at_right_to_top
thf(fact_7393_filterlim__at__top__to__right,axiom,
! [A: $tType,F3: real > A,F4: filter @ A] :
( ( filterlim @ real @ A @ F3 @ F4 @ ( at_top @ real ) )
= ( filterlim @ real @ A
@ ^ [X2: real] : ( F3 @ ( inverse_inverse @ real @ X2 ) )
@ F4
@ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) ) ) ).
% filterlim_at_top_to_right
thf(fact_7394_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_7395_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_7396_log__inj,axiom,
! [B3: real] :
( ( ord_less @ real @ ( one_one @ real ) @ B3 )
=> ( inj_on @ real @ real @ ( log @ B3 ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) ) ).
% log_inj
thf(fact_7397_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_7398_filterlim__tendsto__pos__mult__at__bot,axiom,
! [A: $tType,F3: A > real,C2: real,F4: filter @ A,G2: A > real] :
( ( filterlim @ A @ real @ F3 @ ( topolo7230453075368039082e_nhds @ real @ C2 ) @ F4 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ C2 )
=> ( ( filterlim @ A @ real @ G2 @ ( at_bot @ real ) @ F4 )
=> ( filterlim @ A @ real
@ ^ [X2: A] : ( times_times @ real @ ( F3 @ X2 ) @ ( G2 @ X2 ) )
@ ( at_bot @ real )
@ F4 ) ) ) ) ).
% filterlim_tendsto_pos_mult_at_bot
thf(fact_7399_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_7400_tendsto__at__botI__sequentially,axiom,
! [B: $tType] :
( ( topolo3112930676232923870pology @ B )
=> ! [F3: real > B,Y: B] :
( ! [X11: nat > real] :
( ( filterlim @ nat @ real @ X11 @ ( at_bot @ real ) @ ( at_top @ nat ) )
=> ( filterlim @ nat @ B
@ ^ [N4: nat] : ( F3 @ ( X11 @ N4 ) )
@ ( topolo7230453075368039082e_nhds @ B @ Y )
@ ( at_top @ nat ) ) )
=> ( filterlim @ real @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ Y ) @ ( at_bot @ real ) ) ) ) ).
% tendsto_at_botI_sequentially
thf(fact_7401_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_7402_filterlim__tendsto__neg__mult__at__bot,axiom,
! [A: $tType,F3: A > real,C2: real,F4: filter @ A,G2: A > real] :
( ( filterlim @ A @ real @ F3 @ ( topolo7230453075368039082e_nhds @ real @ C2 ) @ F4 )
=> ( ( ord_less @ real @ C2 @ ( zero_zero @ real ) )
=> ( ( filterlim @ A @ real @ G2 @ ( at_top @ real ) @ F4 )
=> ( filterlim @ A @ real
@ ^ [X2: A] : ( times_times @ real @ ( F3 @ X2 ) @ ( G2 @ X2 ) )
@ ( at_bot @ real )
@ F4 ) ) ) ) ).
% filterlim_tendsto_neg_mult_at_bot
thf(fact_7403_isCont__If__ge,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo1944317154257567458pology @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [A3: A,G2: A > B,F3: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A3 @ ( set_ord_lessThan @ A @ A3 ) ) @ G2 )
=> ( ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ ( G2 @ A3 ) ) @ ( topolo174197925503356063within @ A @ A3 @ ( set_ord_greaterThan @ A @ A3 ) ) )
=> ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) )
@ ^ [X2: A] : ( if @ B @ ( ord_less_eq @ A @ X2 @ A3 ) @ ( G2 @ X2 ) @ ( F3 @ X2 ) ) ) ) ) ) ).
% isCont_If_ge
thf(fact_7404_DERIV__pos__imp__increasing__at__bot,axiom,
! [B3: real,F3: real > real,Flim: real] :
( ! [X3: real] :
( ( ord_less_eq @ real @ X3 @ B3 )
=> ? [Y5: real] :
( ( has_field_derivative @ real @ F3 @ Y5 @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) )
& ( ord_less @ real @ ( zero_zero @ real ) @ Y5 ) ) )
=> ( ( filterlim @ real @ real @ F3 @ ( topolo7230453075368039082e_nhds @ real @ Flim ) @ ( at_bot @ real ) )
=> ( ord_less @ real @ Flim @ ( F3 @ B3 ) ) ) ) ).
% DERIV_pos_imp_increasing_at_bot
thf(fact_7405_filterlim__pow__at__bot__odd,axiom,
! [N: nat,F3: real > real,F4: filter @ real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( filterlim @ real @ real @ F3 @ ( at_bot @ real ) @ F4 )
=> ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( filterlim @ real @ real
@ ^ [X2: real] : ( power_power @ real @ ( F3 @ X2 ) @ N )
@ ( at_bot @ real )
@ F4 ) ) ) ) ).
% filterlim_pow_at_bot_odd
thf(fact_7406_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_7407_lhopital__left__at__top,axiom,
! [G2: real > real,X: real,G5: real > real,F3: real > real,F7: real > real,Y: real] :
( ( filterlim @ real @ real @ G2 @ ( at_top @ real ) @ ( topolo174197925503356063within @ real @ X @ ( set_ord_lessThan @ real @ X ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] :
( ( G5 @ X2 )
!= ( zero_zero @ real ) )
@ ( topolo174197925503356063within @ real @ X @ ( set_ord_lessThan @ real @ X ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] : ( has_field_derivative @ real @ F3 @ ( F7 @ X2 ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ X @ ( set_ord_lessThan @ real @ X ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] : ( has_field_derivative @ real @ G2 @ ( G5 @ X2 ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ X @ ( set_ord_lessThan @ real @ X ) ) )
=> ( ( filterlim @ real @ real
@ ^ [X2: real] : ( divide_divide @ real @ ( F7 @ X2 ) @ ( G5 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ Y )
@ ( topolo174197925503356063within @ real @ X @ ( set_ord_lessThan @ real @ X ) ) )
=> ( filterlim @ real @ real
@ ^ [X2: real] : ( divide_divide @ real @ ( F3 @ X2 ) @ ( G2 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ Y )
@ ( topolo174197925503356063within @ real @ X @ ( set_ord_lessThan @ real @ X ) ) ) ) ) ) ) ) ).
% lhopital_left_at_top
thf(fact_7408_lhopital__right__at__top,axiom,
! [G2: real > real,X: real,G5: real > real,F3: real > real,F7: real > real,Y: real] :
( ( filterlim @ real @ real @ G2 @ ( at_top @ real ) @ ( topolo174197925503356063within @ real @ X @ ( set_ord_greaterThan @ real @ X ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] :
( ( G5 @ X2 )
!= ( zero_zero @ real ) )
@ ( topolo174197925503356063within @ real @ X @ ( set_ord_greaterThan @ real @ X ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] : ( has_field_derivative @ real @ F3 @ ( F7 @ X2 ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ X @ ( set_ord_greaterThan @ real @ X ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] : ( has_field_derivative @ real @ G2 @ ( G5 @ X2 ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ X @ ( set_ord_greaterThan @ real @ X ) ) )
=> ( ( filterlim @ real @ real
@ ^ [X2: real] : ( divide_divide @ real @ ( F7 @ X2 ) @ ( G5 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ Y )
@ ( topolo174197925503356063within @ real @ X @ ( set_ord_greaterThan @ real @ X ) ) )
=> ( filterlim @ real @ real
@ ^ [X2: real] : ( divide_divide @ real @ ( F3 @ X2 ) @ ( G2 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ Y )
@ ( topolo174197925503356063within @ real @ X @ ( set_ord_greaterThan @ real @ X ) ) ) ) ) ) ) ) ).
% lhopital_right_at_top
thf(fact_7409_eventually__top,axiom,
! [A: $tType,P: A > $o] :
( ( eventually @ A @ P @ ( top_top @ ( filter @ A ) ) )
= ( ! [X8: A] : ( P @ X8 ) ) ) ).
% eventually_top
thf(fact_7410_eventually__const,axiom,
! [A: $tType,F4: filter @ A,P: $o] :
( ( F4
!= ( bot_bot @ ( filter @ A ) ) )
=> ( ( eventually @ A
@ ^ [X2: A] : P
@ F4 )
= P ) ) ).
% eventually_const
thf(fact_7411_eventually__at__right__field,axiom,
! [A: $tType] :
( ( ( linordered_field @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [P: A > $o,X: A] :
( ( eventually @ A @ P @ ( topolo174197925503356063within @ A @ X @ ( set_ord_greaterThan @ A @ X ) ) )
= ( ? [B5: A] :
( ( ord_less @ A @ X @ B5 )
& ! [Y2: A] :
( ( ord_less @ A @ X @ Y2 )
=> ( ( ord_less @ A @ Y2 @ B5 )
=> ( P @ Y2 ) ) ) ) ) ) ) ).
% eventually_at_right_field
thf(fact_7412_eventually__at__right,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [X: A,Y: A,P: A > $o] :
( ( ord_less @ A @ X @ Y )
=> ( ( eventually @ A @ P @ ( topolo174197925503356063within @ A @ X @ ( set_ord_greaterThan @ A @ X ) ) )
= ( ? [B5: A] :
( ( ord_less @ A @ X @ B5 )
& ! [Y2: A] :
( ( ord_less @ A @ X @ Y2 )
=> ( ( ord_less @ A @ Y2 @ B5 )
=> ( P @ Y2 ) ) ) ) ) ) ) ) ).
% eventually_at_right
thf(fact_7413_eventually__at__bot__dense,axiom,
! [A: $tType] :
( ( ( linorder @ A )
& ( no_bot @ A ) )
=> ! [P: A > $o] :
( ( eventually @ A @ P @ ( at_bot @ A ) )
= ( ? [N9: A] :
! [N4: A] :
( ( ord_less @ A @ N4 @ N9 )
=> ( P @ N4 ) ) ) ) ) ).
% eventually_at_bot_dense
thf(fact_7414_eventually__gt__at__bot,axiom,
! [A: $tType] :
( ( unboun7993243217541854897norder @ A )
=> ! [C2: A] :
( eventually @ A
@ ^ [X2: A] : ( ord_less @ A @ X2 @ C2 )
@ ( at_bot @ A ) ) ) ).
% eventually_gt_at_bot
thf(fact_7415_eventually__at__bot__not__equal,axiom,
! [A: $tType] :
( ( ( linorder @ A )
& ( no_bot @ A ) )
=> ! [C2: A] :
( eventually @ A
@ ^ [X2: A] : X2 != C2
@ ( at_bot @ A ) ) ) ).
% eventually_at_bot_not_equal
thf(fact_7416_eventually__at__bot__linorder,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > $o] :
( ( eventually @ A @ P @ ( at_bot @ A ) )
= ( ? [N9: A] :
! [N4: A] :
( ( ord_less_eq @ A @ N4 @ N9 )
=> ( P @ N4 ) ) ) ) ) ).
% eventually_at_bot_linorder
thf(fact_7417_eventually__le__at__bot,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [C2: A] :
( eventually @ A
@ ^ [X2: A] : ( ord_less_eq @ A @ X2 @ C2 )
@ ( at_bot @ A ) ) ) ).
% eventually_le_at_bot
thf(fact_7418_filterlim__at__within__not__equal,axiom,
! [B: $tType,A: $tType] :
( ( topological_t2_space @ B )
=> ! [F3: A > B,A3: B,S2: set @ B,F4: filter @ A,B3: B] :
( ( filterlim @ A @ B @ F3 @ ( topolo174197925503356063within @ B @ A3 @ S2 ) @ F4 )
=> ( eventually @ A
@ ^ [W2: A] :
( ( member @ B @ ( F3 @ W2 ) @ S2 )
& ( ( F3 @ W2 )
!= B3 ) )
@ F4 ) ) ) ).
% filterlim_at_within_not_equal
thf(fact_7419_sequentially__imp__eventually__within,axiom,
! [A: $tType] :
( ( topolo3112930676232923870pology @ A )
=> ! [S2: set @ A,A3: A,P: A > $o] :
( ! [F5: nat > A] :
( ( ! [N11: nat] :
( ( member @ A @ ( F5 @ N11 ) @ S2 )
& ( ( F5 @ N11 )
!= A3 ) )
& ( filterlim @ nat @ A @ F5 @ ( topolo7230453075368039082e_nhds @ A @ A3 ) @ ( at_top @ nat ) ) )
=> ( eventually @ nat
@ ^ [N4: nat] : ( P @ ( F5 @ N4 ) )
@ ( at_top @ nat ) ) )
=> ( eventually @ A @ P @ ( topolo174197925503356063within @ A @ A3 @ S2 ) ) ) ) ).
% sequentially_imp_eventually_within
thf(fact_7420_sequentially__imp__eventually__at,axiom,
! [A: $tType] :
( ( topolo3112930676232923870pology @ A )
=> ! [A3: A,P: A > $o] :
( ! [F5: nat > A] :
( ( ! [N11: nat] :
( ( F5 @ N11 )
!= A3 )
& ( filterlim @ nat @ A @ F5 @ ( topolo7230453075368039082e_nhds @ A @ A3 ) @ ( at_top @ nat ) ) )
=> ( eventually @ nat
@ ^ [N4: nat] : ( P @ ( F5 @ N4 ) )
@ ( at_top @ nat ) ) )
=> ( eventually @ A @ P @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% sequentially_imp_eventually_at
thf(fact_7421_filterlim__at__top__at__top,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( linorder @ B ) )
=> ! [Q: A > $o,F3: A > B,P: B > $o,G2: B > A] :
( ! [X3: A,Y4: A] :
( ( Q @ X3 )
=> ( ( Q @ Y4 )
=> ( ( ord_less_eq @ A @ X3 @ Y4 )
=> ( ord_less_eq @ B @ ( F3 @ X3 ) @ ( F3 @ Y4 ) ) ) ) )
=> ( ! [X3: B] :
( ( P @ X3 )
=> ( ( F3 @ ( G2 @ X3 ) )
= X3 ) )
=> ( ! [X3: B] :
( ( P @ X3 )
=> ( Q @ ( G2 @ X3 ) ) )
=> ( ( eventually @ A @ Q @ ( at_top @ A ) )
=> ( ( eventually @ B @ P @ ( at_top @ B ) )
=> ( filterlim @ A @ B @ F3 @ ( at_top @ B ) @ ( at_top @ A ) ) ) ) ) ) ) ) ).
% filterlim_at_top_at_top
thf(fact_7422_Lim__transform__eventually,axiom,
! [B: $tType,A: $tType] :
( ( topolo4958980785337419405_space @ B )
=> ! [F3: A > B,L: B,F4: filter @ A,G2: A > B] :
( ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ L ) @ F4 )
=> ( ( eventually @ A
@ ^ [X2: A] :
( ( F3 @ X2 )
= ( G2 @ X2 ) )
@ F4 )
=> ( filterlim @ A @ B @ G2 @ ( topolo7230453075368039082e_nhds @ B @ L ) @ F4 ) ) ) ) ).
% Lim_transform_eventually
thf(fact_7423_eventually__compose__filterlim,axiom,
! [A: $tType,B: $tType,P: A > $o,F4: filter @ A,F3: B > A,G4: filter @ B] :
( ( eventually @ A @ P @ F4 )
=> ( ( filterlim @ B @ A @ F3 @ F4 @ G4 )
=> ( eventually @ B
@ ^ [X2: B] : ( P @ ( F3 @ X2 ) )
@ G4 ) ) ) ).
% eventually_compose_filterlim
thf(fact_7424_filterlim__cong,axiom,
! [A: $tType,B: $tType,F12: filter @ A,F13: filter @ A,F23: filter @ B,F24: filter @ B,F3: B > A,G2: B > A] :
( ( F12 = F13 )
=> ( ( F23 = F24 )
=> ( ( eventually @ B
@ ^ [X2: B] :
( ( F3 @ X2 )
= ( G2 @ X2 ) )
@ F23 )
=> ( ( filterlim @ B @ A @ F3 @ F12 @ F23 )
= ( filterlim @ B @ A @ G2 @ F13 @ F24 ) ) ) ) ) ).
% filterlim_cong
thf(fact_7425_filterlim__iff,axiom,
! [B: $tType,A: $tType] :
( ( filterlim @ A @ B )
= ( ^ [F2: A > B,F25: filter @ B,F14: filter @ A] :
! [P3: B > $o] :
( ( eventually @ B @ P3 @ F25 )
=> ( eventually @ A
@ ^ [X2: A] : ( P3 @ ( F2 @ X2 ) )
@ F14 ) ) ) ) ).
% filterlim_iff
thf(fact_7426_eventually__nhds__iff__sequentially,axiom,
! [A: $tType] :
( ( topolo3112930676232923870pology @ A )
=> ! [P: A > $o,A3: A] :
( ( eventually @ A @ P @ ( topolo7230453075368039082e_nhds @ A @ A3 ) )
= ( ! [F2: nat > A] :
( ( filterlim @ nat @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ A3 ) @ ( at_top @ nat ) )
=> ( eventually @ nat
@ ^ [N4: nat] : ( P @ ( F2 @ N4 ) )
@ ( at_top @ nat ) ) ) ) ) ) ).
% eventually_nhds_iff_sequentially
thf(fact_7427_tendsto__eventually,axiom,
! [A: $tType,B: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [F3: B > A,L: A,Net: filter @ B] :
( ( eventually @ B
@ ^ [X2: B] :
( ( F3 @ X2 )
= L )
@ Net )
=> ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ Net ) ) ) ).
% tendsto_eventually
thf(fact_7428_tendsto__imp__eventually__ne,axiom,
! [A: $tType,B: $tType] :
( ( topological_t1_space @ A )
=> ! [F3: B > A,C2: A,F4: filter @ B,C8: A] :
( ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ C2 ) @ F4 )
=> ( ( C2 != C8 )
=> ( eventually @ B
@ ^ [Z4: B] :
( ( F3 @ Z4 )
!= C8 )
@ F4 ) ) ) ) ).
% tendsto_imp_eventually_ne
thf(fact_7429_tendsto__discrete,axiom,
! [A: $tType,B: $tType] :
( ( topolo8865339358273720382pology @ A )
=> ! [F3: B > A,Y: A,F4: filter @ B] :
( ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ Y ) @ F4 )
= ( eventually @ B
@ ^ [X2: B] :
( ( F3 @ X2 )
= Y )
@ F4 ) ) ) ).
% tendsto_discrete
thf(fact_7430_tendsto__cong,axiom,
! [A: $tType,B: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [F3: B > A,G2: B > A,F4: filter @ B,C2: A] :
( ( eventually @ B
@ ^ [X2: B] :
( ( F3 @ X2 )
= ( G2 @ X2 ) )
@ F4 )
=> ( ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ C2 ) @ F4 )
= ( filterlim @ B @ A @ G2 @ ( topolo7230453075368039082e_nhds @ A @ C2 ) @ F4 ) ) ) ) ).
% tendsto_cong
thf(fact_7431_filterlim__mono__eventually,axiom,
! [B: $tType,A: $tType,F3: A > B,F4: filter @ B,G4: filter @ A,F11: filter @ B,G6: filter @ A,F7: A > B] :
( ( filterlim @ A @ B @ F3 @ F4 @ G4 )
=> ( ( ord_less_eq @ ( filter @ B ) @ F4 @ F11 )
=> ( ( ord_less_eq @ ( filter @ A ) @ G6 @ G4 )
=> ( ( eventually @ A
@ ^ [X2: A] :
( ( F3 @ X2 )
= ( F7 @ X2 ) )
@ G6 )
=> ( filterlim @ A @ B @ F7 @ F11 @ G6 ) ) ) ) ) ).
% filterlim_mono_eventually
thf(fact_7432_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_7433_eventually__at__top__not__equal,axiom,
! [A: $tType] :
( ( ( linorder @ A )
& ( no_top @ A ) )
=> ! [C2: A] :
( eventually @ A
@ ^ [X2: A] : X2 != C2
@ ( at_top @ A ) ) ) ).
% eventually_at_top_not_equal
thf(fact_7434_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_7435_eventually__at__top__dense,axiom,
! [A: $tType] :
( ( ( linorder @ A )
& ( no_top @ A ) )
=> ! [P: A > $o] :
( ( eventually @ A @ P @ ( at_top @ A ) )
= ( ? [N9: A] :
! [N4: A] :
( ( ord_less @ A @ N9 @ N4 )
=> ( P @ N4 ) ) ) ) ) ).
% eventually_at_top_dense
thf(fact_7436_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_7437_eventually__at__top__linorder,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > $o] :
( ( eventually @ A @ P @ ( at_top @ A ) )
= ( ? [N9: A] :
! [N4: A] :
( ( ord_less_eq @ A @ N9 @ N4 )
=> ( P @ N4 ) ) ) ) ) ).
% eventually_at_top_linorder
thf(fact_7438_eventually__at__left__field,axiom,
! [A: $tType] :
( ( ( linordered_field @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [P: A > $o,X: A] :
( ( eventually @ A @ P @ ( topolo174197925503356063within @ A @ X @ ( set_ord_lessThan @ A @ X ) ) )
= ( ? [B5: A] :
( ( ord_less @ A @ B5 @ X )
& ! [Y2: A] :
( ( ord_less @ A @ B5 @ Y2 )
=> ( ( ord_less @ A @ Y2 @ X )
=> ( P @ Y2 ) ) ) ) ) ) ) ).
% eventually_at_left_field
thf(fact_7439_eventually__at__left,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [Y: A,X: A,P: A > $o] :
( ( ord_less @ A @ Y @ X )
=> ( ( eventually @ A @ P @ ( topolo174197925503356063within @ A @ X @ ( set_ord_lessThan @ A @ X ) ) )
= ( ? [B5: A] :
( ( ord_less @ A @ B5 @ X )
& ! [Y2: A] :
( ( ord_less @ A @ B5 @ Y2 )
=> ( ( ord_less @ A @ Y2 @ X )
=> ( P @ Y2 ) ) ) ) ) ) ) ) ).
% eventually_at_left
thf(fact_7440_eventually__nhds__in__open,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [S2: set @ A,X: A] :
( ( topolo1002775350975398744n_open @ A @ S2 )
=> ( ( member @ A @ X @ S2 )
=> ( eventually @ A
@ ^ [Y2: A] : ( member @ A @ Y2 @ S2 )
@ ( topolo7230453075368039082e_nhds @ A @ X ) ) ) ) ) ).
% eventually_nhds_in_open
thf(fact_7441_eventually__at__filter,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [P: A > $o,A3: A,S2: set @ A] :
( ( eventually @ A @ P @ ( topolo174197925503356063within @ A @ A3 @ S2 ) )
= ( eventually @ A
@ ^ [X2: A] :
( ( X2 != A3 )
=> ( ( member @ A @ X2 @ S2 )
=> ( P @ X2 ) ) )
@ ( topolo7230453075368039082e_nhds @ A @ A3 ) ) ) ) ).
% eventually_at_filter
thf(fact_7442_eventually__eventually,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [P: A > $o,X: A] :
( ( eventually @ A
@ ^ [Y2: A] : ( eventually @ A @ P @ ( topolo7230453075368039082e_nhds @ A @ Y2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ X ) )
= ( eventually @ A @ P @ ( topolo7230453075368039082e_nhds @ A @ X ) ) ) ) ).
% eventually_eventually
thf(fact_7443_t1__space__nhds,axiom,
! [A: $tType] :
( ( topological_t1_space @ A )
=> ! [X: A,Y: A] :
( ( X != Y )
=> ( eventually @ A
@ ^ [X2: A] : X2 != Y
@ ( topolo7230453075368039082e_nhds @ A @ X ) ) ) ) ).
% t1_space_nhds
thf(fact_7444_filter__leD,axiom,
! [A: $tType,F4: filter @ A,F11: filter @ A,P: A > $o] :
( ( ord_less_eq @ ( filter @ A ) @ F4 @ F11 )
=> ( ( eventually @ A @ P @ F11 )
=> ( eventually @ A @ P @ F4 ) ) ) ).
% filter_leD
thf(fact_7445_filter__leI,axiom,
! [A: $tType,F11: filter @ A,F4: filter @ A] :
( ! [P8: A > $o] :
( ( eventually @ A @ P8 @ F11 )
=> ( eventually @ A @ P8 @ F4 ) )
=> ( ord_less_eq @ ( filter @ A ) @ F4 @ F11 ) ) ).
% filter_leI
thf(fact_7446_le__filter__def,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( filter @ A ) )
= ( ^ [F8: filter @ A,F9: filter @ A] :
! [P3: A > $o] :
( ( eventually @ A @ P3 @ F9 )
=> ( eventually @ A @ P3 @ F8 ) ) ) ) ).
% le_filter_def
thf(fact_7447_has__field__derivative__cong__eventually,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,G2: A > A,X: A,S: set @ A,U2: A] :
( ( eventually @ A
@ ^ [X2: A] :
( ( F3 @ X2 )
= ( G2 @ X2 ) )
@ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( ( ( F3 @ X )
= ( G2 @ X ) )
=> ( ( has_field_derivative @ A @ F3 @ U2 @ ( topolo174197925503356063within @ A @ X @ S ) )
= ( has_field_derivative @ A @ G2 @ U2 @ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ) ).
% has_field_derivative_cong_eventually
thf(fact_7448_eventually__sup,axiom,
! [A: $tType,P: A > $o,F4: filter @ A,F11: filter @ A] :
( ( eventually @ A @ P @ ( sup_sup @ ( filter @ A ) @ F4 @ F11 ) )
= ( ( eventually @ A @ P @ F4 )
& ( eventually @ A @ P @ F11 ) ) ) ).
% eventually_sup
thf(fact_7449_eventuallyI,axiom,
! [A: $tType,P: A > $o,F4: filter @ A] :
( ! [X_1: A] : ( P @ X_1 )
=> ( eventually @ A @ P @ F4 ) ) ).
% eventuallyI
thf(fact_7450_filter__eq__iff,axiom,
! [A: $tType] :
( ( ^ [Y3: filter @ A,Z2: filter @ A] : Y3 = Z2 )
= ( ^ [F8: filter @ A,F9: filter @ A] :
! [P3: A > $o] :
( ( eventually @ A @ P3 @ F8 )
= ( eventually @ A @ P3 @ F9 ) ) ) ) ).
% filter_eq_iff
thf(fact_7451_eventually__mono,axiom,
! [A: $tType,P: A > $o,F4: filter @ A,Q: A > $o] :
( ( eventually @ A @ P @ F4 )
=> ( ! [X3: A] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( eventually @ A @ Q @ F4 ) ) ) ).
% eventually_mono
thf(fact_7452_not__eventuallyD,axiom,
! [A: $tType,P: A > $o,F4: filter @ A] :
( ~ ( eventually @ A @ P @ F4 )
=> ? [X3: A] :
~ ( P @ X3 ) ) ).
% not_eventuallyD
thf(fact_7453_always__eventually,axiom,
! [A: $tType,P: A > $o,F4: filter @ A] :
( ! [X_1: A] : ( P @ X_1 )
=> ( eventually @ A @ P @ F4 ) ) ).
% always_eventually
thf(fact_7454_not__eventually__impI,axiom,
! [A: $tType,P: A > $o,F4: filter @ A,Q: A > $o] :
( ( eventually @ A @ P @ F4 )
=> ( ~ ( eventually @ A @ Q @ F4 )
=> ~ ( eventually @ A
@ ^ [X2: A] :
( ( P @ X2 )
=> ( Q @ X2 ) )
@ F4 ) ) ) ).
% not_eventually_impI
thf(fact_7455_eventually__conj__iff,axiom,
! [A: $tType,P: A > $o,Q: A > $o,F4: filter @ A] :
( ( eventually @ A
@ ^ [X2: A] :
( ( P @ X2 )
& ( Q @ X2 ) )
@ F4 )
= ( ( eventually @ A @ P @ F4 )
& ( eventually @ A @ Q @ F4 ) ) ) ).
% eventually_conj_iff
thf(fact_7456_eventually__rev__mp,axiom,
! [A: $tType,P: A > $o,F4: filter @ A,Q: A > $o] :
( ( eventually @ A @ P @ F4 )
=> ( ( eventually @ A
@ ^ [X2: A] :
( ( P @ X2 )
=> ( Q @ X2 ) )
@ F4 )
=> ( eventually @ A @ Q @ F4 ) ) ) ).
% eventually_rev_mp
thf(fact_7457_eventually__subst,axiom,
! [A: $tType,P: A > $o,Q: A > $o,F4: filter @ A] :
( ( eventually @ A
@ ^ [N4: A] :
( ( P @ N4 )
= ( Q @ N4 ) )
@ F4 )
=> ( ( eventually @ A @ P @ F4 )
= ( eventually @ A @ Q @ F4 ) ) ) ).
% eventually_subst
thf(fact_7458_eventually__elim2,axiom,
! [A: $tType,P: A > $o,F4: filter @ A,Q: A > $o,R: A > $o] :
( ( eventually @ A @ P @ F4 )
=> ( ( eventually @ A @ Q @ F4 )
=> ( ! [I3: A] :
( ( P @ I3 )
=> ( ( Q @ I3 )
=> ( R @ I3 ) ) )
=> ( eventually @ A @ R @ F4 ) ) ) ) ).
% eventually_elim2
thf(fact_7459_eventually__conj,axiom,
! [A: $tType,P: A > $o,F4: filter @ A,Q: A > $o] :
( ( eventually @ A @ P @ F4 )
=> ( ( eventually @ A @ Q @ F4 )
=> ( eventually @ A
@ ^ [X2: A] :
( ( P @ X2 )
& ( Q @ X2 ) )
@ F4 ) ) ) ).
% eventually_conj
thf(fact_7460_eventually__True,axiom,
! [A: $tType,F4: filter @ A] :
( eventually @ A
@ ^ [X2: A] : $true
@ F4 ) ).
% eventually_True
thf(fact_7461_eventually__mp,axiom,
! [A: $tType,P: A > $o,Q: A > $o,F4: filter @ A] :
( ( eventually @ A
@ ^ [X2: A] :
( ( P @ X2 )
=> ( Q @ X2 ) )
@ F4 )
=> ( ( eventually @ A @ P @ F4 )
=> ( eventually @ A @ Q @ F4 ) ) ) ).
% eventually_mp
thf(fact_7462_eventually__frequently__const__simps_I3_J,axiom,
! [A: $tType,P: A > $o,C3: $o,F4: filter @ A] :
( ( eventually @ A
@ ^ [X2: A] :
( ( P @ X2 )
| C3 )
@ F4 )
= ( ( eventually @ A @ P @ F4 )
| C3 ) ) ).
% eventually_frequently_const_simps(3)
thf(fact_7463_eventually__frequently__const__simps_I4_J,axiom,
! [A: $tType,C3: $o,P: A > $o,F4: filter @ A] :
( ( eventually @ A
@ ^ [X2: A] :
( C3
| ( P @ X2 ) )
@ F4 )
= ( C3
| ( eventually @ A @ P @ F4 ) ) ) ).
% eventually_frequently_const_simps(4)
thf(fact_7464_eventually__frequently__const__simps_I6_J,axiom,
! [A: $tType,C3: $o,P: A > $o,F4: filter @ A] :
( ( eventually @ A
@ ^ [X2: A] :
( C3
=> ( P @ X2 ) )
@ F4 )
= ( C3
=> ( eventually @ A @ P @ F4 ) ) ) ).
% eventually_frequently_const_simps(6)
thf(fact_7465_has__derivative__transform__eventually,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F3: A > B,F7: A > B,X: A,S2: set @ A,G2: A > B] :
( ( has_derivative @ A @ B @ F3 @ F7 @ ( topolo174197925503356063within @ A @ X @ S2 ) )
=> ( ( eventually @ A
@ ^ [X9: A] :
( ( F3 @ X9 )
= ( G2 @ X9 ) )
@ ( topolo174197925503356063within @ A @ X @ S2 ) )
=> ( ( ( F3 @ X )
= ( G2 @ X ) )
=> ( ( member @ A @ X @ S2 )
=> ( has_derivative @ A @ B @ G2 @ F7 @ ( topolo174197925503356063within @ A @ X @ S2 ) ) ) ) ) ) ) ).
% has_derivative_transform_eventually
thf(fact_7466_trivial__limit__def,axiom,
! [A: $tType,F4: filter @ A] :
( ( F4
= ( bot_bot @ ( filter @ A ) ) )
= ( eventually @ A
@ ^ [X2: A] : $false
@ F4 ) ) ).
% trivial_limit_def
thf(fact_7467_eventually__const__iff,axiom,
! [A: $tType,P: $o,F4: filter @ A] :
( ( eventually @ A
@ ^ [X2: A] : P
@ F4 )
= ( P
| ( F4
= ( bot_bot @ ( filter @ A ) ) ) ) ) ).
% eventually_const_iff
thf(fact_7468_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_7469_eventually__happens_H,axiom,
! [A: $tType,F4: filter @ A,P: A > $o] :
( ( F4
!= ( bot_bot @ ( filter @ A ) ) )
=> ( ( eventually @ A @ P @ F4 )
=> ? [X_1: A] : ( P @ X_1 ) ) ) ).
% eventually_happens'
thf(fact_7470_eventually__happens,axiom,
! [A: $tType,P: A > $o,Net: filter @ A] :
( ( eventually @ A @ P @ Net )
=> ( ( Net
= ( bot_bot @ ( filter @ A ) ) )
| ? [X_1: A] : ( P @ X_1 ) ) ) ).
% eventually_happens
thf(fact_7471_eventually__bot,axiom,
! [A: $tType,P: A > $o] : ( eventually @ A @ P @ ( bot_bot @ ( filter @ A ) ) ) ).
% eventually_bot
thf(fact_7472_eventually__nhds__top,axiom,
! [A: $tType] :
( ( ( order_top @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [B3: A,P: A > $o] :
( ( ord_less @ A @ B3 @ ( top_top @ A ) )
=> ( ( eventually @ A @ P @ ( topolo7230453075368039082e_nhds @ A @ ( top_top @ A ) ) )
= ( ? [B5: A] :
( ( ord_less @ A @ B5 @ ( top_top @ A ) )
& ! [Z4: A] :
( ( ord_less @ A @ B5 @ Z4 )
=> ( P @ Z4 ) ) ) ) ) ) ) ).
% eventually_nhds_top
thf(fact_7473_eventually__Lim__ident__at,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [P: A > A > $o,X: A,X5: set @ A] :
( ( eventually @ A
@ ( P
@ ( topolo3827282254853284352ce_Lim @ A @ A @ ( topolo174197925503356063within @ A @ X @ X5 )
@ ^ [X2: A] : X2 ) )
@ ( topolo174197925503356063within @ A @ X @ X5 ) )
= ( eventually @ A @ ( P @ X ) @ ( topolo174197925503356063within @ A @ X @ X5 ) ) ) ) ).
% eventually_Lim_ident_at
thf(fact_7474_tendsto__sandwich,axiom,
! [A: $tType,B: $tType] :
( ( topolo2564578578187576103pology @ A )
=> ! [F3: B > A,G2: B > A,Net: filter @ B,H2: B > A,C2: A] :
( ( eventually @ B
@ ^ [N4: B] : ( ord_less_eq @ A @ ( F3 @ N4 ) @ ( G2 @ N4 ) )
@ Net )
=> ( ( eventually @ B
@ ^ [N4: B] : ( ord_less_eq @ A @ ( G2 @ N4 ) @ ( H2 @ N4 ) )
@ Net )
=> ( ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ C2 ) @ Net )
=> ( ( filterlim @ B @ A @ H2 @ ( topolo7230453075368039082e_nhds @ A @ C2 ) @ Net )
=> ( filterlim @ B @ A @ G2 @ ( topolo7230453075368039082e_nhds @ A @ C2 ) @ Net ) ) ) ) ) ) ).
% tendsto_sandwich
thf(fact_7475_order__tendstoD_I2_J,axiom,
! [A: $tType,B: $tType] :
( ( topolo2564578578187576103pology @ A )
=> ! [F3: B > A,Y: A,F4: filter @ B,A3: A] :
( ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ Y ) @ F4 )
=> ( ( ord_less @ A @ Y @ A3 )
=> ( eventually @ B
@ ^ [X2: B] : ( ord_less @ A @ ( F3 @ X2 ) @ A3 )
@ F4 ) ) ) ) ).
% order_tendstoD(2)
thf(fact_7476_order__tendstoD_I1_J,axiom,
! [A: $tType,B: $tType] :
( ( topolo2564578578187576103pology @ A )
=> ! [F3: B > A,Y: A,F4: filter @ B,A3: A] :
( ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ Y ) @ F4 )
=> ( ( ord_less @ A @ A3 @ Y )
=> ( eventually @ B
@ ^ [X2: B] : ( ord_less @ A @ A3 @ ( F3 @ X2 ) )
@ F4 ) ) ) ) ).
% order_tendstoD(1)
thf(fact_7477_order__tendstoI,axiom,
! [A: $tType,B: $tType] :
( ( topolo2564578578187576103pology @ A )
=> ! [Y: A,F3: B > A,F4: filter @ B] :
( ! [A6: A] :
( ( ord_less @ A @ A6 @ Y )
=> ( eventually @ B
@ ^ [X2: B] : ( ord_less @ A @ A6 @ ( F3 @ X2 ) )
@ F4 ) )
=> ( ! [A6: A] :
( ( ord_less @ A @ Y @ A6 )
=> ( eventually @ B
@ ^ [X2: B] : ( ord_less @ A @ ( F3 @ X2 ) @ A6 )
@ F4 ) )
=> ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ Y ) @ F4 ) ) ) ) ).
% order_tendstoI
thf(fact_7478_order__tendsto__iff,axiom,
! [B: $tType,A: $tType] :
( ( topolo2564578578187576103pology @ A )
=> ! [F3: B > A,X: A,F4: filter @ B] :
( ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ X ) @ F4 )
= ( ! [L2: A] :
( ( ord_less @ A @ L2 @ X )
=> ( eventually @ B
@ ^ [X2: B] : ( ord_less @ A @ L2 @ ( F3 @ X2 ) )
@ F4 ) )
& ! [U: A] :
( ( ord_less @ A @ X @ U )
=> ( eventually @ B
@ ^ [X2: B] : ( ord_less @ A @ ( F3 @ X2 ) @ U )
@ F4 ) ) ) ) ) ).
% order_tendsto_iff
thf(fact_7479_filterlim__at__top__mono,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F3: B > A,F4: filter @ B,G2: B > A] :
( ( filterlim @ B @ A @ F3 @ ( at_top @ A ) @ F4 )
=> ( ( eventually @ B
@ ^ [X2: B] : ( ord_less_eq @ A @ ( F3 @ X2 ) @ ( G2 @ X2 ) )
@ F4 )
=> ( filterlim @ B @ A @ G2 @ ( at_top @ A ) @ F4 ) ) ) ) ).
% filterlim_at_top_mono
thf(fact_7480_filterlim__at__top__ge,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ B )
=> ! [F3: A > B,F4: filter @ A,C2: B] :
( ( filterlim @ A @ B @ F3 @ ( at_top @ B ) @ F4 )
= ( ! [Z10: B] :
( ( ord_less_eq @ B @ C2 @ Z10 )
=> ( eventually @ A
@ ^ [X2: A] : ( ord_less_eq @ B @ Z10 @ ( F3 @ X2 ) )
@ F4 ) ) ) ) ) ).
% filterlim_at_top_ge
thf(fact_7481_filterlim__at__top,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ B )
=> ! [F3: A > B,F4: filter @ A] :
( ( filterlim @ A @ B @ F3 @ ( at_top @ B ) @ F4 )
= ( ! [Z10: B] :
( eventually @ A
@ ^ [X2: A] : ( ord_less_eq @ B @ Z10 @ ( F3 @ X2 ) )
@ F4 ) ) ) ) ).
% filterlim_at_top
thf(fact_7482_filterlim__at__top__dense,axiom,
! [A: $tType,B: $tType] :
( ( unboun7993243217541854897norder @ B )
=> ! [F3: A > B,F4: filter @ A] :
( ( filterlim @ A @ B @ F3 @ ( at_top @ B ) @ F4 )
= ( ! [Z10: B] :
( eventually @ A
@ ^ [X2: A] : ( ord_less @ B @ Z10 @ ( F3 @ X2 ) )
@ F4 ) ) ) ) ).
% filterlim_at_top_dense
thf(fact_7483_filterlim__at,axiom,
! [A: $tType,B: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [F3: B > A,B3: A,S2: set @ A,F4: filter @ B] :
( ( filterlim @ B @ A @ F3 @ ( topolo174197925503356063within @ A @ B3 @ S2 ) @ F4 )
= ( ( eventually @ B
@ ^ [X2: B] :
( ( member @ A @ ( F3 @ X2 ) @ S2 )
& ( ( F3 @ X2 )
!= B3 ) )
@ F4 )
& ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ B3 ) @ F4 ) ) ) ) ).
% filterlim_at
thf(fact_7484_eventually__at__right__less,axiom,
! [A: $tType] :
( ( ( no_top @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [X: A] : ( eventually @ A @ ( ord_less @ A @ X ) @ ( topolo174197925503356063within @ A @ X @ ( set_ord_greaterThan @ A @ X ) ) ) ) ).
% eventually_at_right_less
thf(fact_7485_has__field__derivative__cong__ev,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [X: A,Y: A,S: set @ A,F3: A > A,G2: A > A,U2: A,V2: A,T: set @ A] :
( ( X = Y )
=> ( ( eventually @ A
@ ^ [X2: A] :
( ( member @ A @ X2 @ S )
=> ( ( F3 @ X2 )
= ( G2 @ X2 ) ) )
@ ( topolo7230453075368039082e_nhds @ A @ X ) )
=> ( ( U2 = V2 )
=> ( ( S = T )
=> ( ( member @ A @ X @ S )
=> ( ( has_field_derivative @ A @ F3 @ U2 @ ( topolo174197925503356063within @ A @ X @ S ) )
= ( has_field_derivative @ A @ G2 @ V2 @ ( topolo174197925503356063within @ A @ Y @ T ) ) ) ) ) ) ) ) ) ).
% has_field_derivative_cong_ev
thf(fact_7486_tendsto__def,axiom,
! [B: $tType,A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [F3: B > A,L: A,F4: filter @ B] :
( ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ F4 )
= ( ! [S8: set @ A] :
( ( topolo1002775350975398744n_open @ A @ S8 )
=> ( ( member @ A @ L @ S8 )
=> ( eventually @ B
@ ^ [X2: B] : ( member @ A @ ( F3 @ X2 ) @ S8 )
@ F4 ) ) ) ) ) ) ).
% tendsto_def
thf(fact_7487_topological__tendstoD,axiom,
! [A: $tType,B: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [F3: B > A,L: A,F4: filter @ B,S: set @ A] :
( ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ F4 )
=> ( ( topolo1002775350975398744n_open @ A @ S )
=> ( ( member @ A @ L @ S )
=> ( eventually @ B
@ ^ [X2: B] : ( member @ A @ ( F3 @ X2 ) @ S )
@ F4 ) ) ) ) ) ).
% topological_tendstoD
thf(fact_7488_topological__tendstoI,axiom,
! [A: $tType,B: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [L: A,F3: B > A,F4: filter @ B] :
( ! [S6: set @ A] :
( ( topolo1002775350975398744n_open @ A @ S6 )
=> ( ( member @ A @ L @ S6 )
=> ( eventually @ B
@ ^ [X2: B] : ( member @ A @ ( F3 @ X2 ) @ S6 )
@ F4 ) ) )
=> ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ F4 ) ) ) ).
% topological_tendstoI
thf(fact_7489_filterlim__at__bot,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ B )
=> ! [F3: A > B,F4: filter @ A] :
( ( filterlim @ A @ B @ F3 @ ( at_bot @ B ) @ F4 )
= ( ! [Z10: B] :
( eventually @ A
@ ^ [X2: A] : ( ord_less_eq @ B @ ( F3 @ X2 ) @ Z10 )
@ F4 ) ) ) ) ).
% filterlim_at_bot
thf(fact_7490_filterlim__at__bot__le,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ B )
=> ! [F3: A > B,F4: filter @ A,C2: B] :
( ( filterlim @ A @ B @ F3 @ ( at_bot @ B ) @ F4 )
= ( ! [Z10: B] :
( ( ord_less_eq @ B @ Z10 @ C2 )
=> ( eventually @ A
@ ^ [X2: A] : ( ord_less_eq @ B @ ( F3 @ X2 ) @ Z10 )
@ F4 ) ) ) ) ) ).
% filterlim_at_bot_le
thf(fact_7491_filterlim__at__bot__dense,axiom,
! [A: $tType,B: $tType] :
( ( ( dense_linorder @ B )
& ( no_bot @ B ) )
=> ! [F3: A > B,F4: filter @ A] :
( ( filterlim @ A @ B @ F3 @ ( at_bot @ B ) @ F4 )
= ( ! [Z10: B] :
( eventually @ A
@ ^ [X2: A] : ( ord_less @ B @ ( F3 @ X2 ) @ Z10 )
@ F4 ) ) ) ) ).
% filterlim_at_bot_dense
thf(fact_7492_real__tendsto__sandwich,axiom,
! [B: $tType,F3: B > real,G2: B > real,Net: filter @ B,H2: B > real,C2: real] :
( ( eventually @ B
@ ^ [N4: B] : ( ord_less_eq @ real @ ( F3 @ N4 ) @ ( G2 @ N4 ) )
@ Net )
=> ( ( eventually @ B
@ ^ [N4: B] : ( ord_less_eq @ real @ ( G2 @ N4 ) @ ( H2 @ N4 ) )
@ Net )
=> ( ( filterlim @ B @ real @ F3 @ ( topolo7230453075368039082e_nhds @ real @ C2 ) @ Net )
=> ( ( filterlim @ B @ real @ H2 @ ( topolo7230453075368039082e_nhds @ real @ C2 ) @ Net )
=> ( filterlim @ B @ real @ G2 @ ( topolo7230453075368039082e_nhds @ real @ C2 ) @ Net ) ) ) ) ) ).
% real_tendsto_sandwich
thf(fact_7493_eventually__at,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [P: A > $o,A3: A,S: set @ A] :
( ( eventually @ A @ P @ ( topolo174197925503356063within @ A @ A3 @ S ) )
= ( ? [D4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D4 )
& ! [X2: A] :
( ( member @ A @ X2 @ S )
=> ( ( ( X2 != A3 )
& ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X2 @ A3 ) @ D4 ) )
=> ( P @ X2 ) ) ) ) ) ) ) ).
% eventually_at
thf(fact_7494_eventually__nhds__metric,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [P: A > $o,A3: A] :
( ( eventually @ A @ P @ ( topolo7230453075368039082e_nhds @ A @ A3 ) )
= ( ? [D4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D4 )
& ! [X2: A] :
( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X2 @ A3 ) @ D4 )
=> ( P @ X2 ) ) ) ) ) ) ).
% eventually_nhds_metric
thf(fact_7495_eventually__at__leftI,axiom,
! [A: $tType] :
( ( topolo2564578578187576103pology @ A )
=> ! [A3: A,B3: A,P: A > $o] :
( ! [X3: A] :
( ( member @ A @ X3 @ ( set_or5935395276787703475ssThan @ A @ A3 @ B3 ) )
=> ( P @ X3 ) )
=> ( ( ord_less @ A @ A3 @ B3 )
=> ( eventually @ A @ P @ ( topolo174197925503356063within @ A @ B3 @ ( set_ord_lessThan @ A @ B3 ) ) ) ) ) ) ).
% eventually_at_leftI
thf(fact_7496_eventually__at__rightI,axiom,
! [A: $tType] :
( ( topolo2564578578187576103pology @ A )
=> ! [A3: A,B3: A,P: A > $o] :
( ! [X3: A] :
( ( member @ A @ X3 @ ( set_or5935395276787703475ssThan @ A @ A3 @ B3 ) )
=> ( P @ X3 ) )
=> ( ( ord_less @ A @ A3 @ B3 )
=> ( eventually @ A @ P @ ( topolo174197925503356063within @ A @ A3 @ ( set_ord_greaterThan @ A @ A3 ) ) ) ) ) ) ).
% eventually_at_rightI
thf(fact_7497_eventually__at__to__0,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [P: A > $o,A3: A] :
( ( eventually @ A @ P @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) )
= ( eventually @ A
@ ^ [X2: A] : ( P @ ( plus_plus @ A @ X2 @ A3 ) )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% eventually_at_to_0
thf(fact_7498_decreasing__tendsto,axiom,
! [A: $tType,B: $tType] :
( ( topolo2564578578187576103pology @ A )
=> ! [L: A,F3: B > A,F4: filter @ B] :
( ( eventually @ B
@ ^ [N4: B] : ( ord_less_eq @ A @ L @ ( F3 @ N4 ) )
@ F4 )
=> ( ! [X3: A] :
( ( ord_less @ A @ L @ X3 )
=> ( eventually @ B
@ ^ [N4: B] : ( ord_less @ A @ ( F3 @ N4 ) @ X3 )
@ F4 ) )
=> ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ F4 ) ) ) ) ).
% decreasing_tendsto
thf(fact_7499_increasing__tendsto,axiom,
! [A: $tType,B: $tType] :
( ( topolo2564578578187576103pology @ A )
=> ! [F3: B > A,L: A,F4: filter @ B] :
( ( eventually @ B
@ ^ [N4: B] : ( ord_less_eq @ A @ ( F3 @ N4 ) @ L )
@ F4 )
=> ( ! [X3: A] :
( ( ord_less @ A @ X3 @ L )
=> ( eventually @ B
@ ^ [N4: B] : ( ord_less @ A @ X3 @ ( F3 @ N4 ) )
@ F4 ) )
=> ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ F4 ) ) ) ) ).
% increasing_tendsto
thf(fact_7500_filterlim__at__top__gt,axiom,
! [A: $tType,B: $tType] :
( ( unboun7993243217541854897norder @ B )
=> ! [F3: A > B,F4: filter @ A,C2: B] :
( ( filterlim @ A @ B @ F3 @ ( at_top @ B ) @ F4 )
= ( ! [Z10: B] :
( ( ord_less @ B @ C2 @ Z10 )
=> ( eventually @ A
@ ^ [X2: A] : ( ord_less_eq @ B @ Z10 @ ( F3 @ X2 ) )
@ F4 ) ) ) ) ) ).
% filterlim_at_top_gt
thf(fact_7501_filterlim__atI,axiom,
! [A: $tType,B: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [F3: B > A,C2: A,F4: filter @ B] :
( ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ C2 ) @ F4 )
=> ( ( eventually @ B
@ ^ [X2: B] :
( ( F3 @ X2 )
!= C2 )
@ F4 )
=> ( filterlim @ B @ A @ F3 @ ( topolo174197925503356063within @ A @ C2 @ ( top_top @ ( set @ A ) ) ) @ F4 ) ) ) ) ).
% filterlim_atI
thf(fact_7502_LIM__compose__eventually,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( topolo4958980785337419405_space @ A )
& ( topolo4958980785337419405_space @ B )
& ( topolo4958980785337419405_space @ C ) )
=> ! [F3: A > B,B3: B,A3: A,G2: B > C,C2: C] :
( ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ B3 ) @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) )
=> ( ( filterlim @ B @ C @ G2 @ ( topolo7230453075368039082e_nhds @ C @ C2 ) @ ( topolo174197925503356063within @ B @ B3 @ ( top_top @ ( set @ B ) ) ) )
=> ( ( eventually @ A
@ ^ [X2: A] :
( ( F3 @ X2 )
!= B3 )
@ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) )
=> ( filterlim @ A @ C
@ ^ [X2: A] : ( G2 @ ( F3 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ C @ C2 )
@ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ).
% LIM_compose_eventually
thf(fact_7503_tendsto__compose__eventually,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( ( topolo4958980785337419405_space @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [G2: A > B,M: B,L: A,F3: C > A,F4: filter @ C] :
( ( filterlim @ A @ B @ G2 @ ( topolo7230453075368039082e_nhds @ B @ M ) @ ( topolo174197925503356063within @ A @ L @ ( top_top @ ( set @ A ) ) ) )
=> ( ( filterlim @ C @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ F4 )
=> ( ( eventually @ C
@ ^ [X2: C] :
( ( F3 @ X2 )
!= L )
@ F4 )
=> ( filterlim @ C @ B
@ ^ [X2: C] : ( G2 @ ( F3 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ B @ M )
@ F4 ) ) ) ) ) ).
% tendsto_compose_eventually
thf(fact_7504_isCont__cong,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [F3: A > B,G2: A > B,X: A] :
( ( eventually @ A
@ ^ [X2: A] :
( ( F3 @ X2 )
= ( G2 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ X ) )
=> ( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) @ F3 )
= ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) @ G2 ) ) ) ) ).
% isCont_cong
thf(fact_7505_DERIV__cong__ev,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [X: A,Y: A,F3: A > A,G2: A > A,U2: A,V2: A] :
( ( X = Y )
=> ( ( eventually @ A
@ ^ [X2: A] :
( ( F3 @ X2 )
= ( G2 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ X ) )
=> ( ( U2 = V2 )
=> ( ( has_field_derivative @ A @ F3 @ U2 @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) )
= ( has_field_derivative @ A @ G2 @ V2 @ ( topolo174197925503356063within @ A @ Y @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ) ).
% DERIV_cong_ev
thf(fact_7506_filterlim__at__bot__lt,axiom,
! [A: $tType,B: $tType] :
( ( unboun7993243217541854897norder @ B )
=> ! [F3: A > B,F4: filter @ A,C2: B] :
( ( filterlim @ A @ B @ F3 @ ( at_bot @ B ) @ F4 )
= ( ! [Z10: B] :
( ( ord_less @ B @ Z10 @ C2 )
=> ( eventually @ A
@ ^ [X2: A] : ( ord_less_eq @ B @ ( F3 @ X2 ) @ Z10 )
@ F4 ) ) ) ) ) ).
% filterlim_at_bot_lt
thf(fact_7507_tendsto__upperbound,axiom,
! [B: $tType,A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [F3: B > A,X: A,F4: filter @ B,A3: A] :
( ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ X ) @ F4 )
=> ( ( eventually @ B
@ ^ [I4: B] : ( ord_less_eq @ A @ ( F3 @ I4 ) @ A3 )
@ F4 )
=> ( ( F4
!= ( bot_bot @ ( filter @ B ) ) )
=> ( ord_less_eq @ A @ X @ A3 ) ) ) ) ) ).
% tendsto_upperbound
thf(fact_7508_tendsto__lowerbound,axiom,
! [B: $tType,A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [F3: B > A,X: A,F4: filter @ B,A3: A] :
( ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ X ) @ F4 )
=> ( ( eventually @ B
@ ^ [I4: B] : ( ord_less_eq @ A @ A3 @ ( F3 @ I4 ) )
@ F4 )
=> ( ( F4
!= ( bot_bot @ ( filter @ B ) ) )
=> ( ord_less_eq @ A @ A3 @ X ) ) ) ) ) ).
% tendsto_lowerbound
thf(fact_7509_tendsto__le,axiom,
! [B: $tType,A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [F4: filter @ B,F3: B > A,X: A,G2: B > A,Y: A] :
( ( F4
!= ( bot_bot @ ( filter @ B ) ) )
=> ( ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ X ) @ F4 )
=> ( ( filterlim @ B @ A @ G2 @ ( topolo7230453075368039082e_nhds @ A @ Y ) @ F4 )
=> ( ( eventually @ B
@ ^ [X2: B] : ( ord_less_eq @ A @ ( G2 @ X2 ) @ ( F3 @ X2 ) )
@ F4 )
=> ( ord_less_eq @ A @ Y @ X ) ) ) ) ) ) ).
% tendsto_le
thf(fact_7510_metric__tendsto__imp__tendsto,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( ( real_V7819770556892013058_space @ B )
& ( real_V7819770556892013058_space @ A ) )
=> ! [F3: C > A,A3: A,F4: filter @ C,G2: C > B,B3: B] :
( ( filterlim @ C @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ A3 ) @ F4 )
=> ( ( eventually @ C
@ ^ [X2: C] : ( ord_less_eq @ real @ ( real_V557655796197034286t_dist @ B @ ( G2 @ X2 ) @ B3 ) @ ( real_V557655796197034286t_dist @ A @ ( F3 @ X2 ) @ A3 ) )
@ F4 )
=> ( filterlim @ C @ B @ G2 @ ( topolo7230453075368039082e_nhds @ B @ B3 ) @ F4 ) ) ) ) ).
% metric_tendsto_imp_tendsto
thf(fact_7511_greaterThan__0,axiom,
( ( set_ord_greaterThan @ nat @ ( zero_zero @ nat ) )
= ( image @ nat @ nat @ suc @ ( top_top @ ( set @ nat ) ) ) ) ).
% greaterThan_0
thf(fact_7512_eventually__floor__eq,axiom,
! [B: $tType,A: $tType] :
( ( ( archim2362893244070406136eiling @ B )
& ( topolo2564578578187576103pology @ B ) )
=> ! [F3: A > B,L: B,F4: filter @ A] :
( ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ L ) @ F4 )
=> ( ~ ( member @ B @ L @ ( ring_1_Ints @ B ) )
=> ( eventually @ A
@ ^ [X2: A] :
( ( archim6421214686448440834_floor @ B @ ( F3 @ X2 ) )
= ( archim6421214686448440834_floor @ B @ L ) )
@ F4 ) ) ) ) ).
% eventually_floor_eq
thf(fact_7513_eventually__ceiling__eq,axiom,
! [B: $tType,A: $tType] :
( ( ( archim2362893244070406136eiling @ B )
& ( topolo2564578578187576103pology @ B ) )
=> ! [F3: A > B,L: B,F4: filter @ A] :
( ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ L ) @ F4 )
=> ( ~ ( member @ B @ L @ ( ring_1_Ints @ B ) )
=> ( eventually @ A
@ ^ [X2: A] :
( ( archimedean_ceiling @ B @ ( F3 @ X2 ) )
= ( archimedean_ceiling @ B @ L ) )
@ F4 ) ) ) ) ).
% eventually_ceiling_eq
thf(fact_7514_eventually__at__right__to__0,axiom,
! [P: real > $o,A3: real] :
( ( eventually @ real @ P @ ( topolo174197925503356063within @ real @ A3 @ ( set_ord_greaterThan @ real @ A3 ) ) )
= ( eventually @ real
@ ^ [X2: real] : ( P @ ( plus_plus @ real @ X2 @ A3 ) )
@ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) ) ) ).
% eventually_at_right_to_0
thf(fact_7515_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_7516_eventually__at__left__to__right,axiom,
! [P: real > $o,A3: real] :
( ( eventually @ real @ P @ ( topolo174197925503356063within @ real @ A3 @ ( set_ord_lessThan @ real @ A3 ) ) )
= ( eventually @ real
@ ^ [X2: real] : ( P @ ( uminus_uminus @ real @ X2 ) )
@ ( topolo174197925503356063within @ real @ ( uminus_uminus @ real @ A3 ) @ ( set_ord_greaterThan @ real @ ( uminus_uminus @ real @ A3 ) ) ) ) ) ).
% eventually_at_left_to_right
thf(fact_7517_continuous__arcosh__strong,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [F4: filter @ A,F3: A > real] :
( ( topolo3448309680560233919inuous @ A @ real @ F4 @ F3 )
=> ( ( eventually @ A
@ ^ [X2: A] : ( ord_less_eq @ real @ ( one_one @ real ) @ ( F3 @ X2 ) )
@ F4 )
=> ( topolo3448309680560233919inuous @ A @ real @ F4
@ ^ [X2: A] : ( arcosh @ real @ ( F3 @ X2 ) ) ) ) ) ) ).
% continuous_arcosh_strong
thf(fact_7518_eventually__at__right__real,axiom,
! [A3: real,B3: real] :
( ( ord_less @ real @ A3 @ B3 )
=> ( eventually @ real
@ ^ [X2: real] : ( member @ real @ X2 @ ( set_or5935395276787703475ssThan @ real @ A3 @ B3 ) )
@ ( topolo174197925503356063within @ real @ A3 @ ( set_ord_greaterThan @ real @ A3 ) ) ) ) ).
% eventually_at_right_real
thf(fact_7519_eventually__at__left__real,axiom,
! [B3: real,A3: real] :
( ( ord_less @ real @ B3 @ A3 )
=> ( eventually @ real
@ ^ [X2: real] : ( member @ real @ X2 @ ( set_or5935395276787703475ssThan @ real @ B3 @ A3 ) )
@ ( topolo174197925503356063within @ real @ A3 @ ( set_ord_lessThan @ real @ A3 ) ) ) ) ).
% eventually_at_left_real
thf(fact_7520_eventually__at__le,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [P: A > $o,A3: A,S: set @ A] :
( ( eventually @ A @ P @ ( topolo174197925503356063within @ A @ A3 @ S ) )
= ( ? [D4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D4 )
& ! [X2: A] :
( ( member @ A @ X2 @ S )
=> ( ( ( X2 != A3 )
& ( ord_less_eq @ real @ ( real_V557655796197034286t_dist @ A @ X2 @ A3 ) @ D4 ) )
=> ( P @ X2 ) ) ) ) ) ) ) ).
% eventually_at_le
thf(fact_7521_tendsto__imp__filterlim__at__left,axiom,
! [B: $tType,A: $tType] :
( ( topolo2564578578187576103pology @ B )
=> ! [F3: A > B,L6: B,F4: filter @ A] :
( ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ L6 ) @ F4 )
=> ( ( eventually @ A
@ ^ [X2: A] : ( ord_less @ B @ ( F3 @ X2 ) @ L6 )
@ F4 )
=> ( filterlim @ A @ B @ F3 @ ( topolo174197925503356063within @ B @ L6 @ ( set_ord_lessThan @ B @ L6 ) ) @ F4 ) ) ) ) ).
% tendsto_imp_filterlim_at_left
thf(fact_7522_tendsto__imp__filterlim__at__right,axiom,
! [B: $tType,A: $tType] :
( ( topolo2564578578187576103pology @ B )
=> ! [F3: A > B,L6: B,F4: filter @ A] :
( ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ L6 ) @ F4 )
=> ( ( eventually @ A
@ ^ [X2: A] : ( ord_less @ B @ L6 @ ( F3 @ X2 ) )
@ F4 )
=> ( filterlim @ A @ B @ F3 @ ( topolo174197925503356063within @ B @ L6 @ ( set_ord_greaterThan @ B @ L6 ) ) @ F4 ) ) ) ) ).
% tendsto_imp_filterlim_at_right
thf(fact_7523_tendsto__iff,axiom,
! [B: $tType,A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [F3: B > A,L: A,F4: filter @ B] :
( ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ F4 )
= ( ! [E4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E4 )
=> ( eventually @ B
@ ^ [X2: B] : ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( F3 @ X2 ) @ L ) @ E4 )
@ F4 ) ) ) ) ) ).
% tendsto_iff
thf(fact_7524_tendstoI,axiom,
! [A: $tType,B: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [F3: B > A,L: A,F4: filter @ B] :
( ! [E: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E )
=> ( eventually @ B
@ ^ [X2: B] : ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( F3 @ X2 ) @ L ) @ E )
@ F4 ) )
=> ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ F4 ) ) ) ).
% tendstoI
thf(fact_7525_tendstoD,axiom,
! [A: $tType,B: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [F3: B > A,L: A,F4: filter @ B,E2: real] :
( ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ F4 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ E2 )
=> ( eventually @ B
@ ^ [X2: B] : ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( F3 @ X2 ) @ L ) @ E2 )
@ F4 ) ) ) ) ).
% tendstoD
thf(fact_7526_eventually__at__right__to__top,axiom,
! [P: real > $o] :
( ( eventually @ real @ P @ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) )
= ( eventually @ real
@ ^ [X2: real] : ( P @ ( inverse_inverse @ real @ X2 ) )
@ ( at_top @ real ) ) ) ).
% eventually_at_right_to_top
thf(fact_7527_eventually__at__top__to__right,axiom,
! [P: real > $o] :
( ( eventually @ real @ P @ ( at_top @ real ) )
= ( eventually @ real
@ ^ [X2: real] : ( P @ ( inverse_inverse @ real @ X2 ) )
@ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) ) ) ).
% eventually_at_top_to_right
thf(fact_7528_tendsto__arcosh__strong,axiom,
! [B: $tType,F3: B > real,A3: real,F4: filter @ B] :
( ( filterlim @ B @ real @ F3 @ ( topolo7230453075368039082e_nhds @ real @ A3 ) @ F4 )
=> ( ( ord_less_eq @ real @ ( one_one @ real ) @ A3 )
=> ( ( eventually @ B
@ ^ [X2: B] : ( ord_less_eq @ real @ ( one_one @ real ) @ ( F3 @ X2 ) )
@ F4 )
=> ( filterlim @ B @ real
@ ^ [X2: B] : ( arcosh @ real @ ( F3 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( arcosh @ real @ A3 ) )
@ F4 ) ) ) ) ).
% tendsto_arcosh_strong
thf(fact_7529_filterlim__at__top__at__left,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo1944317154257567458pology @ A )
& ( linorder @ B ) )
=> ! [Q: A > $o,F3: A > B,P: B > $o,G2: B > A,A3: A] :
( ! [X3: A,Y4: A] :
( ( Q @ X3 )
=> ( ( Q @ Y4 )
=> ( ( ord_less_eq @ A @ X3 @ Y4 )
=> ( ord_less_eq @ B @ ( F3 @ X3 ) @ ( F3 @ Y4 ) ) ) ) )
=> ( ! [X3: B] :
( ( P @ X3 )
=> ( ( F3 @ ( G2 @ X3 ) )
= X3 ) )
=> ( ! [X3: B] :
( ( P @ X3 )
=> ( Q @ ( G2 @ X3 ) ) )
=> ( ( eventually @ A @ Q @ ( topolo174197925503356063within @ A @ A3 @ ( set_ord_lessThan @ A @ A3 ) ) )
=> ( ! [B7: A] :
( ( Q @ B7 )
=> ( ord_less @ A @ B7 @ A3 ) )
=> ( ( eventually @ B @ P @ ( at_top @ B ) )
=> ( filterlim @ A @ B @ F3 @ ( at_top @ B ) @ ( topolo174197925503356063within @ A @ A3 @ ( set_ord_lessThan @ A @ A3 ) ) ) ) ) ) ) ) ) ) ).
% filterlim_at_top_at_left
thf(fact_7530_filterlim__at__bot__at__right,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo1944317154257567458pology @ A )
& ( linorder @ B ) )
=> ! [Q: A > $o,F3: A > B,P: B > $o,G2: B > A,A3: A] :
( ! [X3: A,Y4: A] :
( ( Q @ X3 )
=> ( ( Q @ Y4 )
=> ( ( ord_less_eq @ A @ X3 @ Y4 )
=> ( ord_less_eq @ B @ ( F3 @ X3 ) @ ( F3 @ Y4 ) ) ) ) )
=> ( ! [X3: B] :
( ( P @ X3 )
=> ( ( F3 @ ( G2 @ X3 ) )
= X3 ) )
=> ( ! [X3: B] :
( ( P @ X3 )
=> ( Q @ ( G2 @ X3 ) ) )
=> ( ( eventually @ A @ Q @ ( topolo174197925503356063within @ A @ A3 @ ( set_ord_greaterThan @ A @ A3 ) ) )
=> ( ! [B7: A] :
( ( Q @ B7 )
=> ( ord_less @ A @ A3 @ B7 ) )
=> ( ( eventually @ B @ P @ ( at_bot @ B ) )
=> ( filterlim @ A @ B @ F3 @ ( at_bot @ B ) @ ( topolo174197925503356063within @ A @ A3 @ ( set_ord_greaterThan @ A @ A3 ) ) ) ) ) ) ) ) ) ) ).
% filterlim_at_bot_at_right
thf(fact_7531_filterlim__at__withinI,axiom,
! [A: $tType,B: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [F3: B > A,C2: A,F4: filter @ B,A2: set @ A] :
( ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ C2 ) @ F4 )
=> ( ( eventually @ B
@ ^ [X2: B] : ( member @ A @ ( F3 @ X2 ) @ ( minus_minus @ ( set @ A ) @ A2 @ ( insert @ A @ C2 @ ( bot_bot @ ( set @ A ) ) ) ) )
@ F4 )
=> ( filterlim @ B @ A @ F3 @ ( topolo174197925503356063within @ A @ C2 @ A2 ) @ F4 ) ) ) ) ).
% filterlim_at_withinI
thf(fact_7532_tendsto__0__le,axiom,
! [B: $tType,C: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V822414075346904944vector @ B ) )
=> ! [F3: A > B,F4: filter @ A,G2: A > C,K4: real] :
( ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) ) @ F4 )
=> ( ( eventually @ A
@ ^ [X2: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ C @ ( G2 @ X2 ) ) @ ( times_times @ real @ ( real_V7770717601297561774m_norm @ B @ ( F3 @ X2 ) ) @ K4 ) )
@ F4 )
=> ( filterlim @ A @ C @ G2 @ ( topolo7230453075368039082e_nhds @ C @ ( zero_zero @ C ) ) @ F4 ) ) ) ) ).
% tendsto_0_le
thf(fact_7533_eventually__floor__less,axiom,
! [B: $tType,A: $tType] :
( ( ( archim2362893244070406136eiling @ B )
& ( topolo2564578578187576103pology @ B ) )
=> ! [F3: A > B,L: B,F4: filter @ A] :
( ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ L ) @ F4 )
=> ( ~ ( member @ B @ L @ ( ring_1_Ints @ B ) )
=> ( eventually @ A
@ ^ [X2: A] : ( ord_less @ B @ ( ring_1_of_int @ B @ ( archim6421214686448440834_floor @ B @ L ) ) @ ( F3 @ X2 ) )
@ F4 ) ) ) ) ).
% eventually_floor_less
thf(fact_7534_eventually__less__ceiling,axiom,
! [B: $tType,A: $tType] :
( ( ( archim2362893244070406136eiling @ B )
& ( topolo2564578578187576103pology @ B ) )
=> ! [F3: A > B,L: B,F4: filter @ A] :
( ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ L ) @ F4 )
=> ( ~ ( member @ B @ L @ ( ring_1_Ints @ B ) )
=> ( eventually @ A
@ ^ [X2: A] : ( ord_less @ B @ ( F3 @ X2 ) @ ( ring_1_of_int @ B @ ( archimedean_ceiling @ B @ L ) ) )
@ F4 ) ) ) ) ).
% eventually_less_ceiling
thf(fact_7535_tendsto__powr_H,axiom,
! [A: $tType,F3: A > real,A3: real,F4: filter @ A,G2: A > real,B3: real] :
( ( filterlim @ A @ real @ F3 @ ( topolo7230453075368039082e_nhds @ real @ A3 ) @ F4 )
=> ( ( filterlim @ A @ real @ G2 @ ( topolo7230453075368039082e_nhds @ real @ B3 ) @ F4 )
=> ( ( ( A3
!= ( zero_zero @ real ) )
| ( ( ord_less @ real @ ( zero_zero @ real ) @ B3 )
& ( eventually @ A
@ ^ [X2: A] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( F3 @ X2 ) )
@ F4 ) ) )
=> ( filterlim @ A @ real
@ ^ [X2: A] : ( powr @ real @ ( F3 @ X2 ) @ ( G2 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( powr @ real @ A3 @ B3 ) )
@ F4 ) ) ) ) ).
% tendsto_powr'
thf(fact_7536_tendsto__powr2,axiom,
! [A: $tType,F3: A > real,A3: real,F4: filter @ A,G2: A > real,B3: real] :
( ( filterlim @ A @ real @ F3 @ ( topolo7230453075368039082e_nhds @ real @ A3 ) @ F4 )
=> ( ( filterlim @ A @ real @ G2 @ ( topolo7230453075368039082e_nhds @ real @ B3 ) @ F4 )
=> ( ( eventually @ A
@ ^ [X2: A] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( F3 @ X2 ) )
@ F4 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ B3 )
=> ( filterlim @ A @ real
@ ^ [X2: A] : ( powr @ real @ ( F3 @ X2 ) @ ( G2 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( powr @ real @ A3 @ B3 ) )
@ F4 ) ) ) ) ) ).
% tendsto_powr2
thf(fact_7537_tendsto__zero__powrI,axiom,
! [A: $tType,F3: A > real,F4: filter @ A,G2: A > real,B3: real] :
( ( filterlim @ A @ real @ F3 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ F4 )
=> ( ( filterlim @ A @ real @ G2 @ ( topolo7230453075368039082e_nhds @ real @ B3 ) @ F4 )
=> ( ( eventually @ A
@ ^ [X2: A] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( F3 @ X2 ) )
@ F4 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ B3 )
=> ( filterlim @ A @ real
@ ^ [X2: A] : ( powr @ real @ ( F3 @ X2 ) @ ( G2 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ F4 ) ) ) ) ) ).
% tendsto_zero_powrI
thf(fact_7538_LIM__at__top__divide,axiom,
! [A: $tType,F3: A > real,A3: real,F4: filter @ A,G2: A > real] :
( ( filterlim @ A @ real @ F3 @ ( topolo7230453075368039082e_nhds @ real @ A3 ) @ F4 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ A3 )
=> ( ( filterlim @ A @ real @ G2 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ F4 )
=> ( ( eventually @ A
@ ^ [X2: A] : ( ord_less @ real @ ( zero_zero @ real ) @ ( G2 @ X2 ) )
@ F4 )
=> ( filterlim @ A @ real
@ ^ [X2: A] : ( divide_divide @ real @ ( F3 @ X2 ) @ ( G2 @ X2 ) )
@ ( at_top @ real )
@ F4 ) ) ) ) ) ).
% LIM_at_top_divide
thf(fact_7539_filterlim__inverse__at__top__iff,axiom,
! [A: $tType,F3: A > real,F4: filter @ A] :
( ( eventually @ A
@ ^ [X2: A] : ( ord_less @ real @ ( zero_zero @ real ) @ ( F3 @ X2 ) )
@ F4 )
=> ( ( filterlim @ A @ real
@ ^ [X2: A] : ( inverse_inverse @ real @ ( F3 @ X2 ) )
@ ( at_top @ real )
@ F4 )
= ( filterlim @ A @ real @ F3 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ F4 ) ) ) ).
% filterlim_inverse_at_top_iff
thf(fact_7540_filterlim__inverse__at__top,axiom,
! [A: $tType,F3: A > real,F4: filter @ A] :
( ( filterlim @ A @ real @ F3 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ F4 )
=> ( ( eventually @ A
@ ^ [X2: A] : ( ord_less @ real @ ( zero_zero @ real ) @ ( F3 @ X2 ) )
@ F4 )
=> ( filterlim @ A @ real
@ ^ [X2: A] : ( inverse_inverse @ real @ ( F3 @ X2 ) )
@ ( at_top @ real )
@ F4 ) ) ) ).
% filterlim_inverse_at_top
thf(fact_7541_filterlim__inverse__at__bot,axiom,
! [A: $tType,F3: A > real,F4: filter @ A] :
( ( filterlim @ A @ real @ F3 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ F4 )
=> ( ( eventually @ A
@ ^ [X2: A] : ( ord_less @ real @ ( F3 @ X2 ) @ ( zero_zero @ real ) )
@ F4 )
=> ( filterlim @ A @ real
@ ^ [X2: A] : ( inverse_inverse @ real @ ( F3 @ X2 ) )
@ ( at_bot @ real )
@ F4 ) ) ) ).
% filterlim_inverse_at_bot
thf(fact_7542_lhopital__at__top__at__top,axiom,
! [F3: real > real,A3: real,G2: real > real,F7: real > real,G5: real > real] :
( ( filterlim @ real @ real @ F3 @ ( at_top @ real ) @ ( topolo174197925503356063within @ real @ A3 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( filterlim @ real @ real @ G2 @ ( at_top @ real ) @ ( topolo174197925503356063within @ real @ A3 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] : ( has_field_derivative @ real @ F3 @ ( F7 @ X2 ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ A3 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] : ( has_field_derivative @ real @ G2 @ ( G5 @ X2 ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ A3 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( filterlim @ real @ real
@ ^ [X2: real] : ( divide_divide @ real @ ( F7 @ X2 ) @ ( G5 @ X2 ) )
@ ( at_top @ real )
@ ( topolo174197925503356063within @ real @ A3 @ ( top_top @ ( set @ real ) ) ) )
=> ( filterlim @ real @ real
@ ^ [X2: real] : ( divide_divide @ real @ ( F3 @ X2 ) @ ( G2 @ X2 ) )
@ ( at_top @ real )
@ ( topolo174197925503356063within @ real @ A3 @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ) ) ).
% lhopital_at_top_at_top
thf(fact_7543_lhopital,axiom,
! [F3: real > real,X: real,G2: real > real,G5: real > real,F7: real > real,F4: filter @ real] :
( ( filterlim @ real @ real @ F3 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
=> ( ( filterlim @ real @ real @ G2 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] :
( ( G2 @ X2 )
!= ( zero_zero @ real ) )
@ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] :
( ( G5 @ X2 )
!= ( zero_zero @ real ) )
@ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] : ( has_field_derivative @ real @ F3 @ ( F7 @ X2 ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] : ( has_field_derivative @ real @ G2 @ ( G5 @ X2 ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
=> ( ( filterlim @ real @ real
@ ^ [X2: real] : ( divide_divide @ real @ ( F7 @ X2 ) @ ( G5 @ X2 ) )
@ F4
@ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
=> ( filterlim @ real @ real
@ ^ [X2: real] : ( divide_divide @ real @ ( F3 @ X2 ) @ ( G2 @ X2 ) )
@ F4
@ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ) ) ) ) ).
% lhopital
thf(fact_7544_lhopital__right__at__top__at__top,axiom,
! [F3: real > real,A3: real,G2: real > real,F7: real > real,G5: real > real] :
( ( filterlim @ real @ real @ F3 @ ( at_top @ real ) @ ( topolo174197925503356063within @ real @ A3 @ ( set_ord_greaterThan @ real @ A3 ) ) )
=> ( ( filterlim @ real @ real @ G2 @ ( at_top @ real ) @ ( topolo174197925503356063within @ real @ A3 @ ( set_ord_greaterThan @ real @ A3 ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] : ( has_field_derivative @ real @ F3 @ ( F7 @ X2 ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ A3 @ ( set_ord_greaterThan @ real @ A3 ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] : ( has_field_derivative @ real @ G2 @ ( G5 @ X2 ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ A3 @ ( set_ord_greaterThan @ real @ A3 ) ) )
=> ( ( filterlim @ real @ real
@ ^ [X2: real] : ( divide_divide @ real @ ( F7 @ X2 ) @ ( G5 @ X2 ) )
@ ( at_top @ real )
@ ( topolo174197925503356063within @ real @ A3 @ ( set_ord_greaterThan @ real @ A3 ) ) )
=> ( filterlim @ real @ real
@ ^ [X2: real] : ( divide_divide @ real @ ( F3 @ X2 ) @ ( G2 @ X2 ) )
@ ( at_top @ real )
@ ( topolo174197925503356063within @ real @ A3 @ ( set_ord_greaterThan @ real @ A3 ) ) ) ) ) ) ) ) ).
% lhopital_right_at_top_at_top
thf(fact_7545_lhopital__at__top__at__bot,axiom,
! [F3: real > real,A3: real,G2: real > real,F7: real > real,G5: real > real] :
( ( filterlim @ real @ real @ F3 @ ( at_top @ real ) @ ( topolo174197925503356063within @ real @ A3 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( filterlim @ real @ real @ G2 @ ( at_bot @ real ) @ ( topolo174197925503356063within @ real @ A3 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] : ( has_field_derivative @ real @ F3 @ ( F7 @ X2 ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ A3 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] : ( has_field_derivative @ real @ G2 @ ( G5 @ X2 ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ A3 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( filterlim @ real @ real
@ ^ [X2: real] : ( divide_divide @ real @ ( F7 @ X2 ) @ ( G5 @ X2 ) )
@ ( at_bot @ real )
@ ( topolo174197925503356063within @ real @ A3 @ ( top_top @ ( set @ real ) ) ) )
=> ( filterlim @ real @ real
@ ^ [X2: real] : ( divide_divide @ real @ ( F3 @ X2 ) @ ( G2 @ X2 ) )
@ ( at_bot @ real )
@ ( topolo174197925503356063within @ real @ A3 @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ) ) ).
% lhopital_at_top_at_bot
thf(fact_7546_lhopital__left__at__top__at__top,axiom,
! [F3: real > real,A3: real,G2: real > real,F7: real > real,G5: real > real] :
( ( filterlim @ real @ real @ F3 @ ( at_top @ real ) @ ( topolo174197925503356063within @ real @ A3 @ ( set_ord_lessThan @ real @ A3 ) ) )
=> ( ( filterlim @ real @ real @ G2 @ ( at_top @ real ) @ ( topolo174197925503356063within @ real @ A3 @ ( set_ord_lessThan @ real @ A3 ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] : ( has_field_derivative @ real @ F3 @ ( F7 @ X2 ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ A3 @ ( set_ord_lessThan @ real @ A3 ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] : ( has_field_derivative @ real @ G2 @ ( G5 @ X2 ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ A3 @ ( set_ord_lessThan @ real @ A3 ) ) )
=> ( ( filterlim @ real @ real
@ ^ [X2: real] : ( divide_divide @ real @ ( F7 @ X2 ) @ ( G5 @ X2 ) )
@ ( at_top @ real )
@ ( topolo174197925503356063within @ real @ A3 @ ( set_ord_lessThan @ real @ A3 ) ) )
=> ( filterlim @ real @ real
@ ^ [X2: real] : ( divide_divide @ real @ ( F3 @ X2 ) @ ( G2 @ X2 ) )
@ ( at_top @ real )
@ ( topolo174197925503356063within @ real @ A3 @ ( set_ord_lessThan @ real @ A3 ) ) ) ) ) ) ) ) ).
% lhopital_left_at_top_at_top
thf(fact_7547_lhopital__at__top,axiom,
! [G2: real > real,X: real,G5: real > real,F3: real > real,F7: real > real,Y: real] :
( ( filterlim @ real @ real @ G2 @ ( at_top @ real ) @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] :
( ( G5 @ X2 )
!= ( zero_zero @ real ) )
@ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] : ( has_field_derivative @ real @ F3 @ ( F7 @ X2 ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] : ( has_field_derivative @ real @ G2 @ ( G5 @ X2 ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
=> ( ( filterlim @ real @ real
@ ^ [X2: real] : ( divide_divide @ real @ ( F7 @ X2 ) @ ( G5 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ Y )
@ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
=> ( filterlim @ real @ real
@ ^ [X2: real] : ( divide_divide @ real @ ( F3 @ X2 ) @ ( G2 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ Y )
@ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ) ) ).
% lhopital_at_top
thf(fact_7548_lhospital__at__top__at__top,axiom,
! [G2: real > real,G5: real > real,F3: real > real,F7: real > real,X: real] :
( ( filterlim @ real @ real @ G2 @ ( at_top @ real ) @ ( at_top @ real ) )
=> ( ( eventually @ real
@ ^ [X2: real] :
( ( G5 @ X2 )
!= ( zero_zero @ real ) )
@ ( at_top @ real ) )
=> ( ( eventually @ real
@ ^ [X2: real] : ( has_field_derivative @ real @ F3 @ ( F7 @ X2 ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
@ ( at_top @ real ) )
=> ( ( eventually @ real
@ ^ [X2: real] : ( has_field_derivative @ real @ G2 @ ( G5 @ X2 ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
@ ( at_top @ real ) )
=> ( ( filterlim @ real @ real
@ ^ [X2: real] : ( divide_divide @ real @ ( F7 @ X2 ) @ ( G5 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ X )
@ ( at_top @ real ) )
=> ( filterlim @ real @ real
@ ^ [X2: real] : ( divide_divide @ real @ ( F3 @ X2 ) @ ( G2 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ X )
@ ( at_top @ real ) ) ) ) ) ) ) ).
% lhospital_at_top_at_top
thf(fact_7549_lhopital__right,axiom,
! [F3: real > real,X: real,G2: real > real,G5: real > real,F7: real > real,F4: filter @ real] :
( ( filterlim @ real @ real @ F3 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( topolo174197925503356063within @ real @ X @ ( set_ord_greaterThan @ real @ X ) ) )
=> ( ( filterlim @ real @ real @ G2 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( topolo174197925503356063within @ real @ X @ ( set_ord_greaterThan @ real @ X ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] :
( ( G2 @ X2 )
!= ( zero_zero @ real ) )
@ ( topolo174197925503356063within @ real @ X @ ( set_ord_greaterThan @ real @ X ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] :
( ( G5 @ X2 )
!= ( zero_zero @ real ) )
@ ( topolo174197925503356063within @ real @ X @ ( set_ord_greaterThan @ real @ X ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] : ( has_field_derivative @ real @ F3 @ ( F7 @ X2 ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ X @ ( set_ord_greaterThan @ real @ X ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] : ( has_field_derivative @ real @ G2 @ ( G5 @ X2 ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ X @ ( set_ord_greaterThan @ real @ X ) ) )
=> ( ( filterlim @ real @ real
@ ^ [X2: real] : ( divide_divide @ real @ ( F7 @ X2 ) @ ( G5 @ X2 ) )
@ F4
@ ( topolo174197925503356063within @ real @ X @ ( set_ord_greaterThan @ real @ X ) ) )
=> ( filterlim @ real @ real
@ ^ [X2: real] : ( divide_divide @ real @ ( F3 @ X2 ) @ ( G2 @ X2 ) )
@ F4
@ ( topolo174197925503356063within @ real @ X @ ( set_ord_greaterThan @ real @ X ) ) ) ) ) ) ) ) ) ) ).
% lhopital_right
thf(fact_7550_lhopital__right__0,axiom,
! [F0: real > real,G0: real > real,G5: real > real,F7: real > real,F4: filter @ real] :
( ( filterlim @ real @ real @ F0 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) )
=> ( ( filterlim @ real @ real @ G0 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] :
( ( G0 @ X2 )
!= ( zero_zero @ real ) )
@ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] :
( ( G5 @ X2 )
!= ( zero_zero @ real ) )
@ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] : ( has_field_derivative @ real @ F0 @ ( F7 @ X2 ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] : ( has_field_derivative @ real @ G0 @ ( G5 @ X2 ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) )
=> ( ( filterlim @ real @ real
@ ^ [X2: real] : ( divide_divide @ real @ ( F7 @ X2 ) @ ( G5 @ X2 ) )
@ F4
@ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) )
=> ( filterlim @ real @ real
@ ^ [X2: real] : ( divide_divide @ real @ ( F0 @ X2 ) @ ( G0 @ X2 ) )
@ F4
@ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) ) ) ) ) ) ) ) ) ).
% lhopital_right_0
thf(fact_7551_lhopital__left,axiom,
! [F3: real > real,X: real,G2: real > real,G5: real > real,F7: real > real,F4: filter @ real] :
( ( filterlim @ real @ real @ F3 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( topolo174197925503356063within @ real @ X @ ( set_ord_lessThan @ real @ X ) ) )
=> ( ( filterlim @ real @ real @ G2 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( topolo174197925503356063within @ real @ X @ ( set_ord_lessThan @ real @ X ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] :
( ( G2 @ X2 )
!= ( zero_zero @ real ) )
@ ( topolo174197925503356063within @ real @ X @ ( set_ord_lessThan @ real @ X ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] :
( ( G5 @ X2 )
!= ( zero_zero @ real ) )
@ ( topolo174197925503356063within @ real @ X @ ( set_ord_lessThan @ real @ X ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] : ( has_field_derivative @ real @ F3 @ ( F7 @ X2 ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ X @ ( set_ord_lessThan @ real @ X ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] : ( has_field_derivative @ real @ G2 @ ( G5 @ X2 ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ X @ ( set_ord_lessThan @ real @ X ) ) )
=> ( ( filterlim @ real @ real
@ ^ [X2: real] : ( divide_divide @ real @ ( F7 @ X2 ) @ ( G5 @ X2 ) )
@ F4
@ ( topolo174197925503356063within @ real @ X @ ( set_ord_lessThan @ real @ X ) ) )
=> ( filterlim @ real @ real
@ ^ [X2: real] : ( divide_divide @ real @ ( F3 @ X2 ) @ ( G2 @ X2 ) )
@ F4
@ ( topolo174197925503356063within @ real @ X @ ( set_ord_lessThan @ real @ X ) ) ) ) ) ) ) ) ) ) ).
% lhopital_left
thf(fact_7552_lhopital__right__at__top__at__bot,axiom,
! [F3: real > real,A3: real,G2: real > real,F7: real > real,G5: real > real] :
( ( filterlim @ real @ real @ F3 @ ( at_top @ real ) @ ( topolo174197925503356063within @ real @ A3 @ ( set_ord_greaterThan @ real @ A3 ) ) )
=> ( ( filterlim @ real @ real @ G2 @ ( at_bot @ real ) @ ( topolo174197925503356063within @ real @ A3 @ ( set_ord_greaterThan @ real @ A3 ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] : ( has_field_derivative @ real @ F3 @ ( F7 @ X2 ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ A3 @ ( set_ord_greaterThan @ real @ A3 ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] : ( has_field_derivative @ real @ G2 @ ( G5 @ X2 ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ A3 @ ( set_ord_greaterThan @ real @ A3 ) ) )
=> ( ( filterlim @ real @ real
@ ^ [X2: real] : ( divide_divide @ real @ ( F7 @ X2 ) @ ( G5 @ X2 ) )
@ ( at_bot @ real )
@ ( topolo174197925503356063within @ real @ A3 @ ( set_ord_greaterThan @ real @ A3 ) ) )
=> ( filterlim @ real @ real
@ ^ [X2: real] : ( divide_divide @ real @ ( F3 @ X2 ) @ ( G2 @ X2 ) )
@ ( at_bot @ real )
@ ( topolo174197925503356063within @ real @ A3 @ ( set_ord_greaterThan @ real @ A3 ) ) ) ) ) ) ) ) ).
% lhopital_right_at_top_at_bot
thf(fact_7553_lhopital__left__at__top__at__bot,axiom,
! [F3: real > real,A3: real,G2: real > real,F7: real > real,G5: real > real] :
( ( filterlim @ real @ real @ F3 @ ( at_top @ real ) @ ( topolo174197925503356063within @ real @ A3 @ ( set_ord_lessThan @ real @ A3 ) ) )
=> ( ( filterlim @ real @ real @ G2 @ ( at_bot @ real ) @ ( topolo174197925503356063within @ real @ A3 @ ( set_ord_lessThan @ real @ A3 ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] : ( has_field_derivative @ real @ F3 @ ( F7 @ X2 ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ A3 @ ( set_ord_lessThan @ real @ A3 ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] : ( has_field_derivative @ real @ G2 @ ( G5 @ X2 ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ A3 @ ( set_ord_lessThan @ real @ A3 ) ) )
=> ( ( filterlim @ real @ real
@ ^ [X2: real] : ( divide_divide @ real @ ( F7 @ X2 ) @ ( G5 @ X2 ) )
@ ( at_bot @ real )
@ ( topolo174197925503356063within @ real @ A3 @ ( set_ord_lessThan @ real @ A3 ) ) )
=> ( filterlim @ real @ real
@ ^ [X2: real] : ( divide_divide @ real @ ( F3 @ X2 ) @ ( G2 @ X2 ) )
@ ( at_bot @ real )
@ ( topolo174197925503356063within @ real @ A3 @ ( set_ord_lessThan @ real @ A3 ) ) ) ) ) ) ) ) ).
% lhopital_left_at_top_at_bot
thf(fact_7554_lhopital__right__0__at__top,axiom,
! [G2: real > real,G5: real > real,F3: real > real,F7: real > real,X: real] :
( ( filterlim @ real @ real @ G2 @ ( at_top @ real ) @ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] :
( ( G5 @ X2 )
!= ( zero_zero @ real ) )
@ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] : ( has_field_derivative @ real @ F3 @ ( F7 @ X2 ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] : ( has_field_derivative @ real @ G2 @ ( G5 @ X2 ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) )
=> ( ( filterlim @ real @ real
@ ^ [X2: real] : ( divide_divide @ real @ ( F7 @ X2 ) @ ( G5 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ X )
@ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) )
=> ( filterlim @ real @ real
@ ^ [X2: real] : ( divide_divide @ real @ ( F3 @ X2 ) @ ( G2 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ X )
@ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) ) ) ) ) ) ) ).
% lhopital_right_0_at_top
thf(fact_7555_polyfun__extremal,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [C2: nat > A,K: nat,N: nat,B2: real] :
( ( ( C2 @ K )
!= ( zero_zero @ A ) )
=> ( ( ord_less_eq @ nat @ ( one_one @ nat ) @ K )
=> ( ( ord_less_eq @ nat @ K @ N )
=> ( eventually @ A
@ ^ [Z4: A] :
( ord_less_eq @ real @ B2
@ ( real_V7770717601297561774m_norm @ A
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( C2 @ I4 ) @ ( power_power @ A @ Z4 @ I4 ) )
@ ( set_ord_atMost @ nat @ N ) ) ) )
@ ( at_infinity @ A ) ) ) ) ) ) ).
% polyfun_extremal
thf(fact_7556_Bfun__metric__def,axiom,
! [B: $tType,A: $tType] :
( ( real_V7819770556892013058_space @ B )
=> ( ( bfun @ A @ B )
= ( ^ [F2: A > B,F8: filter @ A] :
? [Y2: B,K6: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K6 )
& ( eventually @ A
@ ^ [X2: A] : ( ord_less_eq @ real @ ( real_V557655796197034286t_dist @ B @ ( F2 @ X2 ) @ Y2 ) @ K6 )
@ F8 ) ) ) ) ) ).
% Bfun_metric_def
thf(fact_7557_eventually__sequentially__Suc,axiom,
! [P: nat > $o] :
( ( eventually @ nat
@ ^ [I4: nat] : ( P @ ( suc @ I4 ) )
@ ( at_top @ nat ) )
= ( eventually @ nat @ P @ ( at_top @ nat ) ) ) ).
% eventually_sequentially_Suc
thf(fact_7558_eventually__sequentially__seg,axiom,
! [P: nat > $o,K: nat] :
( ( eventually @ nat
@ ^ [N4: nat] : ( P @ ( plus_plus @ nat @ N4 @ K ) )
@ ( at_top @ nat ) )
= ( eventually @ nat @ P @ ( at_top @ nat ) ) ) ).
% eventually_sequentially_seg
thf(fact_7559_Bseq__eventually__mono,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F3: nat > A,G2: nat > B] :
( ( eventually @ nat
@ ^ [N4: nat] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( F3 @ N4 ) ) @ ( real_V7770717601297561774m_norm @ B @ ( G2 @ N4 ) ) )
@ ( at_top @ nat ) )
=> ( ( bfun @ nat @ B @ G2 @ ( at_top @ nat ) )
=> ( bfun @ nat @ A @ F3 @ ( at_top @ nat ) ) ) ) ) ).
% Bseq_eventually_mono
thf(fact_7560_le__sequentially,axiom,
! [F4: filter @ nat] :
( ( ord_less_eq @ ( filter @ nat ) @ F4 @ ( at_top @ nat ) )
= ( ! [N9: nat] : ( eventually @ nat @ ( ord_less_eq @ nat @ N9 ) @ F4 ) ) ) ).
% le_sequentially
thf(fact_7561_sequentially__offset,axiom,
! [P: nat > $o,K: nat] :
( ( eventually @ nat @ P @ ( at_top @ nat ) )
=> ( eventually @ nat
@ ^ [I4: nat] : ( P @ ( plus_plus @ nat @ I4 @ K ) )
@ ( at_top @ nat ) ) ) ).
% sequentially_offset
thf(fact_7562_summable__cong,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: nat > A,G2: nat > A] :
( ( eventually @ nat
@ ^ [X2: nat] :
( ( F3 @ X2 )
= ( G2 @ X2 ) )
@ ( at_top @ nat ) )
=> ( ( summable @ A @ F3 )
= ( summable @ A @ G2 ) ) ) ) ).
% summable_cong
thf(fact_7563_eventually__False__sequentially,axiom,
~ ( eventually @ nat
@ ^ [N4: nat] : $false
@ ( at_top @ nat ) ) ).
% eventually_False_sequentially
thf(fact_7564_eventually__sequentially,axiom,
! [P: nat > $o] :
( ( eventually @ nat @ P @ ( at_top @ nat ) )
= ( ? [N9: nat] :
! [N4: nat] :
( ( ord_less_eq @ nat @ N9 @ N4 )
=> ( P @ N4 ) ) ) ) ).
% eventually_sequentially
thf(fact_7565_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_7566_eventually__not__equal__at__infinity,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [A3: A] :
( eventually @ A
@ ^ [X2: A] : X2 != A3
@ ( at_infinity @ A ) ) ) ).
% eventually_not_equal_at_infinity
thf(fact_7567_Bfun__const,axiom,
! [B: $tType,A: $tType] :
( ( real_V7819770556892013058_space @ B )
=> ! [C2: B,F4: filter @ A] :
( bfun @ A @ B
@ ^ [Uu: A] : C2
@ F4 ) ) ).
% Bfun_const
thf(fact_7568_Bseq__minus__iff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X5: nat > A] :
( ( bfun @ nat @ A
@ ^ [N4: nat] : ( uminus_uminus @ A @ ( X5 @ N4 ) )
@ ( at_top @ nat ) )
= ( bfun @ nat @ A @ X5 @ ( at_top @ nat ) ) ) ) ).
% Bseq_minus_iff
thf(fact_7569_Bseq__subseq,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: nat > A,G2: nat > nat] :
( ( bfun @ nat @ A @ F3 @ ( at_top @ nat ) )
=> ( bfun @ nat @ A
@ ^ [X2: nat] : ( F3 @ ( G2 @ X2 ) )
@ ( at_top @ nat ) ) ) ) ).
% Bseq_subseq
thf(fact_7570_Bseq__Suc__iff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: nat > A] :
( ( bfun @ nat @ A
@ ^ [N4: nat] : ( F3 @ ( suc @ N4 ) )
@ ( at_top @ nat ) )
= ( bfun @ nat @ A @ F3 @ ( at_top @ nat ) ) ) ) ).
% Bseq_Suc_iff
thf(fact_7571_Bseq__mult,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: nat > A,G2: nat > A] :
( ( bfun @ nat @ A @ F3 @ ( at_top @ nat ) )
=> ( ( bfun @ nat @ A @ G2 @ ( at_top @ nat ) )
=> ( bfun @ nat @ A
@ ^ [X2: nat] : ( times_times @ A @ ( F3 @ X2 ) @ ( G2 @ X2 ) )
@ ( at_top @ nat ) ) ) ) ) ).
% Bseq_mult
thf(fact_7572_Bseq__add,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: nat > A,C2: A] :
( ( bfun @ nat @ A @ F3 @ ( at_top @ nat ) )
=> ( bfun @ nat @ A
@ ^ [X2: nat] : ( plus_plus @ A @ ( F3 @ X2 ) @ C2 )
@ ( at_top @ nat ) ) ) ) ).
% Bseq_add
thf(fact_7573_Bseq__offset,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X5: nat > A,K: nat] :
( ( bfun @ nat @ A
@ ^ [N4: nat] : ( X5 @ ( plus_plus @ nat @ N4 @ K ) )
@ ( at_top @ nat ) )
=> ( bfun @ nat @ A @ X5 @ ( at_top @ nat ) ) ) ) ).
% Bseq_offset
thf(fact_7574_Bseq__add__iff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: nat > A,C2: A] :
( ( bfun @ nat @ A
@ ^ [X2: nat] : ( plus_plus @ A @ ( F3 @ X2 ) @ C2 )
@ ( at_top @ nat ) )
= ( bfun @ nat @ A @ F3 @ ( at_top @ nat ) ) ) ) ).
% Bseq_add_iff
thf(fact_7575_Bseq__ignore__initial__segment,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X5: nat > A,K: nat] :
( ( bfun @ nat @ A @ X5 @ ( at_top @ nat ) )
=> ( bfun @ nat @ A
@ ^ [N4: nat] : ( X5 @ ( plus_plus @ nat @ N4 @ K ) )
@ ( at_top @ nat ) ) ) ) ).
% Bseq_ignore_initial_segment
thf(fact_7576_filterlim__at__infinity__imp__eventually__ne,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [F3: A > B,F4: filter @ A,C2: B] :
( ( filterlim @ A @ B @ F3 @ ( at_infinity @ B ) @ F4 )
=> ( eventually @ A
@ ^ [Z4: A] :
( ( F3 @ Z4 )
!= C2 )
@ F4 ) ) ) ).
% filterlim_at_infinity_imp_eventually_ne
thf(fact_7577_countable__basis__at__decseq,axiom,
! [A: $tType] :
( ( topolo3112930676232923870pology @ A )
=> ! [X: A] :
~ ! [A10: nat > ( set @ A )] :
( ! [I2: nat] : ( topolo1002775350975398744n_open @ A @ ( A10 @ I2 ) )
=> ( ! [I2: nat] : ( member @ A @ X @ ( A10 @ I2 ) )
=> ~ ! [S10: set @ A] :
( ( topolo1002775350975398744n_open @ A @ S10 )
=> ( ( member @ A @ X @ S10 )
=> ( eventually @ nat
@ ^ [I4: nat] : ( ord_less_eq @ ( set @ A ) @ ( A10 @ I4 ) @ S10 )
@ ( at_top @ nat ) ) ) ) ) ) ) ).
% countable_basis_at_decseq
thf(fact_7578_Bseq__cmult__iff,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [C2: A,F3: nat > A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( bfun @ nat @ A
@ ^ [X2: nat] : ( times_times @ A @ C2 @ ( F3 @ X2 ) )
@ ( at_top @ nat ) )
= ( bfun @ nat @ A @ F3 @ ( at_top @ nat ) ) ) ) ) ).
% Bseq_cmult_iff
thf(fact_7579_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_7580_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_7581_not__tendsto__and__filterlim__at__infinity,axiom,
! [A: $tType,B: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F4: filter @ B,F3: B > A,C2: A] :
( ( F4
!= ( bot_bot @ ( filter @ B ) ) )
=> ( ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ C2 ) @ F4 )
=> ~ ( filterlim @ B @ A @ F3 @ ( at_infinity @ A ) @ F4 ) ) ) ) ).
% not_tendsto_and_filterlim_at_infinity
thf(fact_7582_tendsto__add__filterlim__at__infinity_H,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [F3: A > B,F4: filter @ A,G2: A > B,C2: B] :
( ( filterlim @ A @ B @ F3 @ ( at_infinity @ B ) @ F4 )
=> ( ( filterlim @ A @ B @ G2 @ ( topolo7230453075368039082e_nhds @ B @ C2 ) @ F4 )
=> ( filterlim @ A @ B
@ ^ [X2: A] : ( plus_plus @ B @ ( F3 @ X2 ) @ ( G2 @ X2 ) )
@ ( at_infinity @ B )
@ F4 ) ) ) ) ).
% tendsto_add_filterlim_at_infinity'
thf(fact_7583_tendsto__add__filterlim__at__infinity,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [F3: A > B,C2: B,F4: filter @ A,G2: A > B] :
( ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ C2 ) @ F4 )
=> ( ( filterlim @ A @ B @ G2 @ ( at_infinity @ B ) @ F4 )
=> ( filterlim @ A @ B
@ ^ [X2: A] : ( plus_plus @ B @ ( F3 @ X2 ) @ ( G2 @ X2 ) )
@ ( at_infinity @ B )
@ F4 ) ) ) ) ).
% tendsto_add_filterlim_at_infinity
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_summable__comparison__test__ev,axiom,
! [A: $tType] :
( ( real_Vector_banach @ A )
=> ! [F3: nat > A,G2: nat > real] :
( ( eventually @ nat
@ ^ [N4: nat] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( F3 @ N4 ) ) @ ( G2 @ N4 ) )
@ ( at_top @ nat ) )
=> ( ( summable @ real @ G2 )
=> ( summable @ A @ F3 ) ) ) ) ).
% summable_comparison_test_ev
thf(fact_7586_BseqD,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X5: nat > A] :
( ( bfun @ nat @ A @ X5 @ ( at_top @ nat ) )
=> ? [K8: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K8 )
& ! [N11: nat] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( X5 @ N11 ) ) @ K8 ) ) ) ) ).
% BseqD
thf(fact_7587_BseqE,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X5: nat > A] :
( ( bfun @ nat @ A @ X5 @ ( at_top @ nat ) )
=> ~ ! [K8: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K8 )
=> ~ ! [N11: nat] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( X5 @ N11 ) ) @ K8 ) ) ) ) ).
% BseqE
thf(fact_7588_BseqI,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [K4: real,X5: nat > A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K4 )
=> ( ! [N2: nat] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( X5 @ N2 ) ) @ K4 )
=> ( bfun @ nat @ A @ X5 @ ( at_top @ nat ) ) ) ) ) ).
% BseqI
thf(fact_7589_Bseq__def,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X5: nat > A] :
( ( bfun @ nat @ A @ X5 @ ( at_top @ nat ) )
= ( ? [K6: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K6 )
& ! [N4: nat] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( X5 @ N4 ) ) @ K6 ) ) ) ) ) ).
% Bseq_def
thf(fact_7590_Bseq__iff1a,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X5: nat > A] :
( ( bfun @ nat @ A @ X5 @ ( at_top @ nat ) )
= ( ? [N9: nat] :
! [N4: nat] : ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( X5 @ N4 ) ) @ ( semiring_1_of_nat @ real @ ( suc @ N9 ) ) ) ) ) ) ).
% Bseq_iff1a
thf(fact_7591_Bseq__iff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X5: nat > A] :
( ( bfun @ nat @ A @ X5 @ ( at_top @ nat ) )
= ( ? [N9: nat] :
! [N4: nat] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( X5 @ N4 ) ) @ ( semiring_1_of_nat @ real @ ( suc @ N9 ) ) ) ) ) ) ).
% Bseq_iff
thf(fact_7592_Bseq__realpow,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ X @ ( one_one @ real ) )
=> ( bfun @ nat @ real @ ( power_power @ real @ X ) @ ( at_top @ nat ) ) ) ) ).
% Bseq_realpow
thf(fact_7593_tendsto__mult__filterlim__at__infinity,axiom,
! [A: $tType,B: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: B > A,C2: A,F4: filter @ B,G2: B > A] :
( ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ C2 ) @ F4 )
=> ( ( C2
!= ( zero_zero @ A ) )
=> ( ( filterlim @ B @ A @ G2 @ ( at_infinity @ A ) @ F4 )
=> ( filterlim @ B @ A
@ ^ [X2: B] : ( times_times @ A @ ( F3 @ X2 ) @ ( G2 @ X2 ) )
@ ( at_infinity @ A )
@ F4 ) ) ) ) ) ).
% tendsto_mult_filterlim_at_infinity
thf(fact_7594_filterlim__power__at__infinity,axiom,
! [B: $tType,A: $tType] :
( ( real_V8999393235501362500lgebra @ B )
=> ! [F3: A > B,F4: filter @ A,N: nat] :
( ( filterlim @ A @ B @ F3 @ ( at_infinity @ B ) @ F4 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( filterlim @ A @ B
@ ^ [X2: A] : ( power_power @ B @ ( F3 @ X2 ) @ N )
@ ( at_infinity @ B )
@ F4 ) ) ) ) ).
% filterlim_power_at_infinity
thf(fact_7595_tendsto__divide__0,axiom,
! [A: $tType,C: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [F3: C > A,C2: A,F4: filter @ C,G2: C > A] :
( ( filterlim @ C @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ C2 ) @ F4 )
=> ( ( filterlim @ C @ A @ G2 @ ( at_infinity @ A ) @ F4 )
=> ( filterlim @ C @ A
@ ^ [X2: C] : ( divide_divide @ A @ ( F3 @ X2 ) @ ( G2 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ F4 ) ) ) ) ).
% tendsto_divide_0
thf(fact_7596_filterlim__at__infinity__imp__norm__at__top,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [F3: A > B,F4: filter @ A] :
( ( filterlim @ A @ B @ F3 @ ( at_infinity @ B ) @ F4 )
=> ( filterlim @ A @ real
@ ^ [X2: A] : ( real_V7770717601297561774m_norm @ B @ ( F3 @ X2 ) )
@ ( at_top @ real )
@ F4 ) ) ) ).
% filterlim_at_infinity_imp_norm_at_top
thf(fact_7597_filterlim__norm__at__top__imp__at__infinity,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [F3: A > B,F4: filter @ A] :
( ( filterlim @ A @ real
@ ^ [X2: A] : ( real_V7770717601297561774m_norm @ B @ ( F3 @ X2 ) )
@ ( at_top @ real )
@ F4 )
=> ( filterlim @ A @ B @ F3 @ ( at_infinity @ B ) @ F4 ) ) ) ).
% filterlim_norm_at_top_imp_at_infinity
thf(fact_7598_filterlim__at__infinity__conv__norm__at__top,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [F3: A > B,G4: filter @ A] :
( ( filterlim @ A @ B @ F3 @ ( at_infinity @ B ) @ G4 )
= ( filterlim @ A @ real
@ ^ [X2: A] : ( real_V7770717601297561774m_norm @ B @ ( F3 @ X2 ) )
@ ( at_top @ real )
@ G4 ) ) ) ).
% filterlim_at_infinity_conv_norm_at_top
thf(fact_7599_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_7600_eventually__at__infinity__pos,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [P4: A > $o] :
( ( eventually @ A @ P4 @ ( at_infinity @ A ) )
= ( ? [B5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ B5 )
& ! [X2: A] :
( ( ord_less_eq @ real @ B5 @ ( real_V7770717601297561774m_norm @ A @ X2 ) )
=> ( P4 @ X2 ) ) ) ) ) ) ).
% eventually_at_infinity_pos
thf(fact_7601_BfunI,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [F3: A > B,K4: real,F4: filter @ A] :
( ( eventually @ A
@ ^ [X2: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( F3 @ X2 ) ) @ K4 )
@ F4 )
=> ( bfun @ A @ B @ F3 @ F4 ) ) ) ).
% BfunI
thf(fact_7602_filterlim__at__infinity__imp__filterlim__at__top,axiom,
! [A: $tType,F3: A > real,F4: filter @ A] :
( ( filterlim @ A @ real @ F3 @ ( at_infinity @ real ) @ F4 )
=> ( ( eventually @ A
@ ^ [X2: A] : ( ord_less @ real @ ( zero_zero @ real ) @ ( F3 @ X2 ) )
@ F4 )
=> ( filterlim @ A @ real @ F3 @ ( at_top @ real ) @ F4 ) ) ) ).
% filterlim_at_infinity_imp_filterlim_at_top
thf(fact_7603_filterlim__at__infinity__imp__filterlim__at__bot,axiom,
! [A: $tType,F3: A > real,F4: filter @ A] :
( ( filterlim @ A @ real @ F3 @ ( at_infinity @ real ) @ F4 )
=> ( ( eventually @ A
@ ^ [X2: A] : ( ord_less @ real @ ( F3 @ X2 ) @ ( zero_zero @ real ) )
@ F4 )
=> ( filterlim @ A @ real @ F3 @ ( at_bot @ real ) @ F4 ) ) ) ).
% filterlim_at_infinity_imp_filterlim_at_bot
thf(fact_7604_lim__infinity__imp__sequentially,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [F3: real > A,L: A] :
( ( filterlim @ real @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ ( at_infinity @ real ) )
=> ( filterlim @ nat @ A
@ ^ [N4: nat] : ( F3 @ ( semiring_1_of_nat @ real @ N4 ) )
@ ( topolo7230453075368039082e_nhds @ A @ L )
@ ( at_top @ nat ) ) ) ) ).
% lim_infinity_imp_sequentially
thf(fact_7605_filterlim__inverse__at__iff,axiom,
! [B: $tType,A: $tType] :
( ( real_V8999393235501362500lgebra @ B )
=> ! [G2: A > B,F4: filter @ A] :
( ( filterlim @ A @ B
@ ^ [X2: A] : ( inverse_inverse @ B @ ( G2 @ X2 ) )
@ ( topolo174197925503356063within @ B @ ( zero_zero @ B ) @ ( top_top @ ( set @ B ) ) )
@ F4 )
= ( filterlim @ A @ B @ G2 @ ( at_infinity @ B ) @ F4 ) ) ) ).
% filterlim_inverse_at_iff
thf(fact_7606_Bseq__iff3,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X5: nat > A] :
( ( bfun @ nat @ A @ X5 @ ( at_top @ nat ) )
= ( ? [K3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K3 )
& ? [N9: nat] :
! [N4: nat] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ ( X5 @ N4 ) @ ( uminus_uminus @ A @ ( X5 @ N9 ) ) ) ) @ K3 ) ) ) ) ) ).
% Bseq_iff3
thf(fact_7607_Bseq__iff2,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X5: nat > A] :
( ( bfun @ nat @ A @ X5 @ ( at_top @ nat ) )
= ( ? [K3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K3 )
& ? [X2: A] :
! [N4: nat] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ ( X5 @ N4 ) @ ( uminus_uminus @ A @ X2 ) ) ) @ K3 ) ) ) ) ) ).
% Bseq_iff2
thf(fact_7608_filterlim__divide__at__infinity,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,C2: A,F4: filter @ A,G2: A > A] :
( ( filterlim @ A @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ C2 ) @ F4 )
=> ( ( filterlim @ A @ A @ G2 @ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) @ F4 )
=> ( ( C2
!= ( zero_zero @ A ) )
=> ( filterlim @ A @ A
@ ^ [X2: A] : ( divide_divide @ A @ ( F3 @ X2 ) @ ( G2 @ X2 ) )
@ ( at_infinity @ A )
@ F4 ) ) ) ) ) ).
% filterlim_divide_at_infinity
thf(fact_7609_filterlim__at__infinity,axiom,
! [C: $tType,A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [C2: real,F3: C > A,F4: filter @ C] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ C2 )
=> ( ( filterlim @ C @ A @ F3 @ ( at_infinity @ A ) @ F4 )
= ( ! [R5: real] :
( ( ord_less @ real @ C2 @ R5 )
=> ( eventually @ C
@ ^ [X2: C] : ( ord_less_eq @ real @ R5 @ ( real_V7770717601297561774m_norm @ A @ ( F3 @ X2 ) ) )
@ F4 ) ) ) ) ) ) ).
% filterlim_at_infinity
thf(fact_7610_Bfun__inverse,axiom,
! [A: $tType,B: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [F3: B > A,A3: A,F4: filter @ B] :
( ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ A3 ) @ F4 )
=> ( ( A3
!= ( zero_zero @ A ) )
=> ( bfun @ B @ A
@ ^ [X2: B] : ( inverse_inverse @ A @ ( F3 @ X2 ) )
@ F4 ) ) ) ) ).
% Bfun_inverse
thf(fact_7611_filterlim__realpow__sequentially__gt1,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [X: A] :
( ( ord_less @ real @ ( one_one @ real ) @ ( real_V7770717601297561774m_norm @ A @ X ) )
=> ( filterlim @ nat @ A @ ( power_power @ A @ X ) @ ( at_infinity @ A ) @ ( at_top @ nat ) ) ) ) ).
% filterlim_realpow_sequentially_gt1
thf(fact_7612_lim__zero__infinity,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,L: A] :
( ( filterlim @ A @ A
@ ^ [X2: A] : ( F3 @ ( divide_divide @ A @ ( one_one @ A ) @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ L )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) )
=> ( filterlim @ A @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ ( at_infinity @ A ) ) ) ) ).
% lim_zero_infinity
thf(fact_7613_Bfun__def,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ( ( bfun @ A @ B )
= ( ^ [F2: A > B,F8: filter @ A] :
? [K6: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K6 )
& ( eventually @ A
@ ^ [X2: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( F2 @ X2 ) ) @ K6 )
@ F8 ) ) ) ) ) ).
% Bfun_def
thf(fact_7614_BfunE,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [F3: A > B,F4: filter @ A] :
( ( bfun @ A @ B @ F3 @ F4 )
=> ~ ! [B6: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ B6 )
=> ~ ( eventually @ A
@ ^ [X2: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( F3 @ X2 ) ) @ B6 )
@ F4 ) ) ) ) ).
% BfunE
thf(fact_7615_summable__Cauchy_H,axiom,
! [A: $tType] :
( ( real_Vector_banach @ A )
=> ! [F3: nat > A,G2: nat > real] :
( ( eventually @ nat
@ ^ [M3: nat] :
! [N4: nat] :
( ( ord_less_eq @ nat @ M3 @ N4 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_or7035219750837199246ssThan @ nat @ M3 @ N4 ) ) ) @ ( G2 @ M3 ) ) )
@ ( at_top @ nat ) )
=> ( ( filterlim @ nat @ real @ G2 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) )
=> ( summable @ A @ F3 ) ) ) ) ).
% summable_Cauchy'
thf(fact_7616_sequentially__imp__eventually__at__right,axiom,
! [A: $tType] :
( ( ( topolo3112930676232923870pology @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [A3: A,B3: A,P: A > $o] :
( ( ord_less @ A @ A3 @ B3 )
=> ( ! [F5: nat > A] :
( ! [N11: nat] : ( ord_less @ A @ A3 @ ( F5 @ N11 ) )
=> ( ! [N11: nat] : ( ord_less @ A @ ( F5 @ N11 ) @ B3 )
=> ( ( order_antimono @ nat @ A @ F5 )
=> ( ( filterlim @ nat @ A @ F5 @ ( topolo7230453075368039082e_nhds @ A @ A3 ) @ ( at_top @ nat ) )
=> ( eventually @ nat
@ ^ [N4: nat] : ( P @ ( F5 @ N4 ) )
@ ( at_top @ nat ) ) ) ) ) )
=> ( eventually @ A @ P @ ( topolo174197925503356063within @ A @ A3 @ ( set_ord_greaterThan @ A @ A3 ) ) ) ) ) ) ).
% sequentially_imp_eventually_at_right
thf(fact_7617_decseq__const,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [K: A] :
( order_antimono @ nat @ A
@ ^ [X2: nat] : K ) ) ).
% decseq_const
thf(fact_7618_eventually__all__finite,axiom,
! [B: $tType,A: $tType] :
( ( finite_finite @ B )
=> ! [P: A > B > $o,Net: filter @ A] :
( ! [Y4: B] :
( eventually @ A
@ ^ [X2: A] : ( P @ X2 @ Y4 )
@ Net )
=> ( eventually @ A
@ ^ [X2: A] :
! [X8: B] : ( P @ X2 @ X8 )
@ Net ) ) ) ).
% eventually_all_finite
thf(fact_7619_antimonoD,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [F3: A > B,X: A,Y: A] :
( ( order_antimono @ A @ B @ F3 )
=> ( ( ord_less_eq @ A @ X @ Y )
=> ( ord_less_eq @ B @ ( F3 @ Y ) @ ( F3 @ X ) ) ) ) ) ).
% antimonoD
thf(fact_7620_antimonoE,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [F3: A > B,X: A,Y: A] :
( ( order_antimono @ A @ B @ F3 )
=> ( ( ord_less_eq @ A @ X @ Y )
=> ( ord_less_eq @ B @ ( F3 @ Y ) @ ( F3 @ X ) ) ) ) ) ).
% antimonoE
thf(fact_7621_antimonoI,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [F3: A > B] :
( ! [X3: A,Y4: A] :
( ( ord_less_eq @ A @ X3 @ Y4 )
=> ( ord_less_eq @ B @ ( F3 @ Y4 ) @ ( F3 @ X3 ) ) )
=> ( order_antimono @ A @ B @ F3 ) ) ) ).
% antimonoI
thf(fact_7622_antimono__def,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ( ( order_antimono @ A @ B )
= ( ^ [F2: A > B] :
! [X2: A,Y2: A] :
( ( ord_less_eq @ A @ X2 @ Y2 )
=> ( ord_less_eq @ B @ ( F2 @ Y2 ) @ ( F2 @ X2 ) ) ) ) ) ) ).
% antimono_def
thf(fact_7623_finite__set__of__finite__funs,axiom,
! [B: $tType,A: $tType,A2: set @ A,B2: set @ B,D2: B] :
( ( finite_finite2 @ A @ A2 )
=> ( ( finite_finite2 @ B @ B2 )
=> ( finite_finite2 @ ( A > B )
@ ( collect @ ( A > B )
@ ^ [F2: A > B] :
! [X2: A] :
( ( ( member @ A @ X2 @ A2 )
=> ( member @ B @ ( F2 @ X2 ) @ B2 ) )
& ( ~ ( member @ A @ X2 @ A2 )
=> ( ( F2 @ X2 )
= D2 ) ) ) ) ) ) ) ).
% finite_set_of_finite_funs
thf(fact_7624_eventually__all__ge__at__top,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > $o] :
( ( eventually @ A @ P @ ( at_top @ A ) )
=> ( eventually @ A
@ ^ [X2: A] :
! [Y2: A] :
( ( ord_less_eq @ A @ X2 @ Y2 )
=> ( P @ Y2 ) )
@ ( at_top @ A ) ) ) ) ).
% eventually_all_ge_at_top
thf(fact_7625_min__of__antimono,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( linorder @ B ) )
=> ! [F3: A > B,X: A,Y: A] :
( ( order_antimono @ A @ B @ F3 )
=> ( ( ord_min @ B @ ( F3 @ X ) @ ( F3 @ Y ) )
= ( F3 @ ( ord_max @ A @ X @ Y ) ) ) ) ) ).
% min_of_antimono
thf(fact_7626_max__of__antimono,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( linorder @ B ) )
=> ! [F3: A > B,X: A,Y: A] :
( ( order_antimono @ A @ B @ F3 )
=> ( ( ord_max @ B @ ( F3 @ X ) @ ( F3 @ Y ) )
= ( F3 @ ( ord_min @ A @ X @ Y ) ) ) ) ) ).
% max_of_antimono
thf(fact_7627_tendsto__at__right__sequentially,axiom,
! [C: $tType,B: $tType] :
( ( ( topolo3112930676232923870pology @ B )
& ( topolo1944317154257567458pology @ B )
& ( topolo4958980785337419405_space @ C ) )
=> ! [A3: B,B3: B,X5: B > C,L6: C] :
( ( ord_less @ B @ A3 @ B3 )
=> ( ! [S6: nat > B] :
( ! [N11: nat] : ( ord_less @ B @ A3 @ ( S6 @ N11 ) )
=> ( ! [N11: nat] : ( ord_less @ B @ ( S6 @ N11 ) @ B3 )
=> ( ( order_antimono @ nat @ B @ S6 )
=> ( ( filterlim @ nat @ B @ S6 @ ( topolo7230453075368039082e_nhds @ B @ A3 ) @ ( at_top @ nat ) )
=> ( filterlim @ nat @ C
@ ^ [N4: nat] : ( X5 @ ( S6 @ N4 ) )
@ ( topolo7230453075368039082e_nhds @ C @ L6 )
@ ( at_top @ nat ) ) ) ) ) )
=> ( filterlim @ B @ C @ X5 @ ( topolo7230453075368039082e_nhds @ C @ L6 ) @ ( topolo174197925503356063within @ B @ A3 @ ( set_ord_greaterThan @ B @ A3 ) ) ) ) ) ) ).
% tendsto_at_right_sequentially
thf(fact_7628_summable__bounded__partials,axiom,
! [A: $tType] :
( ( ( real_V8037385150606011577_space @ A )
& ( real_V822414075346904944vector @ A ) )
=> ! [F3: nat > A,G2: nat > real] :
( ( eventually @ nat
@ ^ [X02: nat] :
! [A5: nat] :
( ( ord_less_eq @ nat @ X02 @ A5 )
=> ! [B5: nat] :
( ( ord_less @ nat @ A5 @ B5 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_or3652927894154168847AtMost @ nat @ A5 @ B5 ) ) ) @ ( G2 @ A5 ) ) ) )
@ ( at_top @ nat ) )
=> ( ( filterlim @ nat @ real @ G2 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) )
=> ( summable @ A @ F3 ) ) ) ) ).
% summable_bounded_partials
thf(fact_7629_Greatest__def,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( order_Greatest @ A )
= ( ^ [P3: A > $o] :
( the @ A
@ ^ [X2: A] :
( ( P3 @ X2 )
& ! [Y2: A] :
( ( P3 @ Y2 )
=> ( ord_less_eq @ A @ Y2 @ X2 ) ) ) ) ) ) ) ).
% Greatest_def
thf(fact_7630_greaterThanAtMost__iff,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [I: A,L: A,U2: A] :
( ( member @ A @ I @ ( set_or3652927894154168847AtMost @ A @ L @ U2 ) )
= ( ( ord_less @ A @ L @ I )
& ( ord_less_eq @ A @ I @ U2 ) ) ) ) ).
% greaterThanAtMost_iff
thf(fact_7631_greaterThanAtMost__empty,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L: A,K: A] :
( ( ord_less_eq @ A @ L @ K )
=> ( ( set_or3652927894154168847AtMost @ A @ K @ L )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% greaterThanAtMost_empty
thf(fact_7632_greaterThanAtMost__empty__iff2,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [K: A,L: A] :
( ( ( bot_bot @ ( set @ A ) )
= ( set_or3652927894154168847AtMost @ A @ K @ L ) )
= ( ~ ( ord_less @ A @ K @ L ) ) ) ) ).
% greaterThanAtMost_empty_iff2
thf(fact_7633_greaterThanAtMost__empty__iff,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [K: A,L: A] :
( ( ( set_or3652927894154168847AtMost @ A @ K @ L )
= ( bot_bot @ ( set @ A ) ) )
= ( ~ ( ord_less @ A @ K @ L ) ) ) ) ).
% greaterThanAtMost_empty_iff
thf(fact_7634_infinite__Ioc__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A3: A,B3: A] :
( ( ~ ( finite_finite2 @ A @ ( set_or3652927894154168847AtMost @ A @ A3 @ B3 ) ) )
= ( ord_less @ A @ A3 @ B3 ) ) ) ).
% infinite_Ioc_iff
thf(fact_7635_GreatestI__nat,axiom,
! [P: nat > $o,K: nat,B3: nat] :
( ( P @ K )
=> ( ! [Y4: nat] :
( ( P @ Y4 )
=> ( ord_less_eq @ nat @ Y4 @ B3 ) )
=> ( P @ ( order_Greatest @ nat @ P ) ) ) ) ).
% GreatestI_nat
thf(fact_7636_Greatest__le__nat,axiom,
! [P: nat > $o,K: nat,B3: nat] :
( ( P @ K )
=> ( ! [Y4: nat] :
( ( P @ Y4 )
=> ( ord_less_eq @ nat @ Y4 @ B3 ) )
=> ( ord_less_eq @ nat @ K @ ( order_Greatest @ nat @ P ) ) ) ) ).
% Greatest_le_nat
thf(fact_7637_GreatestI__ex__nat,axiom,
! [P: nat > $o,B3: nat] :
( ? [X_12: nat] : ( P @ X_12 )
=> ( ! [Y4: nat] :
( ( P @ Y4 )
=> ( ord_less_eq @ nat @ Y4 @ B3 ) )
=> ( P @ ( order_Greatest @ nat @ P ) ) ) ) ).
% GreatestI_ex_nat
thf(fact_7638_ivl__disj__un__two_I6_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,M: A,U2: A] :
( ( ord_less_eq @ A @ L @ M )
=> ( ( ord_less_eq @ A @ M @ U2 )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ L @ M ) @ ( set_or3652927894154168847AtMost @ A @ M @ U2 ) )
= ( set_or3652927894154168847AtMost @ A @ L @ U2 ) ) ) ) ) ).
% ivl_disj_un_two(6)
thf(fact_7639_infinite__Ioc,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ A3 @ B3 )
=> ~ ( finite_finite2 @ A @ ( set_or3652927894154168847AtMost @ A @ A3 @ B3 ) ) ) ) ).
% infinite_Ioc
thf(fact_7640_ivl__disj__un__two_I8_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,M: A,U2: A] :
( ( ord_less_eq @ A @ L @ M )
=> ( ( ord_less_eq @ A @ M @ U2 )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ L @ M ) @ ( set_or3652927894154168847AtMost @ A @ M @ U2 ) )
= ( set_or1337092689740270186AtMost @ A @ L @ U2 ) ) ) ) ) ).
% ivl_disj_un_two(8)
thf(fact_7641_open__left,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [S: set @ A,X: A,Y: A] :
( ( topolo1002775350975398744n_open @ A @ S )
=> ( ( member @ A @ X @ S )
=> ( ( ord_less @ A @ Y @ X )
=> ? [B7: A] :
( ( ord_less @ A @ B7 @ X )
& ( ord_less_eq @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ B7 @ X ) @ S ) ) ) ) ) ) ).
% open_left
thf(fact_7642_ivl__disj__un__one_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,U2: A] :
( ( ord_less_eq @ A @ L @ U2 )
=> ( ( sup_sup @ ( set @ A ) @ ( set_ord_atMost @ A @ L ) @ ( set_or3652927894154168847AtMost @ A @ L @ U2 ) )
= ( set_ord_atMost @ A @ U2 ) ) ) ) ).
% ivl_disj_un_one(3)
thf(fact_7643_ivl__disj__un__one_I5_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,U2: A] :
( ( ord_less_eq @ A @ L @ U2 )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ L @ U2 ) @ ( set_ord_greaterThan @ A @ U2 ) )
= ( set_ord_greaterThan @ A @ L ) ) ) ) ).
% ivl_disj_un_one(5)
thf(fact_7644_Greatest__equality,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [P: A > $o,X: A] :
( ( P @ X )
=> ( ! [Y4: A] :
( ( P @ Y4 )
=> ( ord_less_eq @ A @ Y4 @ X ) )
=> ( ( order_Greatest @ A @ P )
= X ) ) ) ) ).
% Greatest_equality
thf(fact_7645_GreatestI2__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [P: A > $o,X: A,Q: A > $o] :
( ( P @ X )
=> ( ! [Y4: A] :
( ( P @ Y4 )
=> ( ord_less_eq @ A @ Y4 @ X ) )
=> ( ! [X3: A] :
( ( P @ X3 )
=> ( ! [Y5: A] :
( ( P @ Y5 )
=> ( ord_less_eq @ A @ Y5 @ X3 ) )
=> ( Q @ X3 ) ) )
=> ( Q @ ( order_Greatest @ A @ P ) ) ) ) ) ) ).
% GreatestI2_order
thf(fact_7646_greaterThanAtMost__subseteq__atLeastAtMost__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A3: A,B3: A,C2: A,D2: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ A3 @ B3 ) @ ( set_or1337092689740270186AtMost @ A @ C2 @ D2 ) )
= ( ( ord_less @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ A @ C2 @ A3 )
& ( ord_less_eq @ A @ B3 @ D2 ) ) ) ) ) ).
% greaterThanAtMost_subseteq_atLeastAtMost_iff
thf(fact_7647_greaterThanAtMost__subseteq__atLeastLessThan__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A3: A,B3: A,C2: A,D2: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ A3 @ B3 ) @ ( set_or7035219750837199246ssThan @ A @ C2 @ D2 ) )
= ( ( ord_less @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ A @ C2 @ A3 )
& ( ord_less @ A @ B3 @ D2 ) ) ) ) ) ).
% greaterThanAtMost_subseteq_atLeastLessThan_iff
thf(fact_7648_ivl__disj__un__two__touch_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,M: A,U2: A] :
( ( ord_less @ A @ L @ M )
=> ( ( ord_less_eq @ A @ M @ U2 )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ L @ M ) @ ( set_or1337092689740270186AtMost @ A @ M @ U2 ) )
= ( set_or3652927894154168847AtMost @ A @ L @ U2 ) ) ) ) ) ).
% ivl_disj_un_two_touch(3)
thf(fact_7649_greaterThanLessThan__subseteq__greaterThanAtMost__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A3: A,B3: A,C2: A,D2: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ A3 @ B3 ) @ ( set_or3652927894154168847AtMost @ A @ C2 @ D2 ) )
= ( ( ord_less @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ A @ C2 @ A3 )
& ( ord_less_eq @ A @ B3 @ D2 ) ) ) ) ) ).
% greaterThanLessThan_subseteq_greaterThanAtMost_iff
thf(fact_7650_greaterThanAtMost__eq__atLeastAtMost__diff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( set_or3652927894154168847AtMost @ A )
= ( ^ [A5: A,B5: A] : ( minus_minus @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A5 @ B5 ) @ ( insert @ A @ A5 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% greaterThanAtMost_eq_atLeastAtMost_diff
thf(fact_7651_ivl__disj__un__two_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,M: A,U2: A] :
( ( ord_less_eq @ A @ L @ M )
=> ( ( ord_less @ A @ M @ U2 )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ L @ M ) @ ( set_or5935395276787703475ssThan @ A @ M @ U2 ) )
= ( set_or5935395276787703475ssThan @ A @ L @ U2 ) ) ) ) ) ).
% ivl_disj_un_two(2)
thf(fact_7652_ivl__disj__un__two__touch_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,M: A,U2: A] :
( ( ord_less @ A @ L @ M )
=> ( ( ord_less @ A @ M @ U2 )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ L @ M ) @ ( set_or7035219750837199246ssThan @ A @ M @ U2 ) )
= ( set_or5935395276787703475ssThan @ A @ L @ U2 ) ) ) ) ) ).
% ivl_disj_un_two_touch(1)
thf(fact_7653_ivl__disj__un__singleton_I5_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,U2: A] :
( ( ord_less_eq @ A @ L @ U2 )
=> ( ( sup_sup @ ( set @ A ) @ ( insert @ A @ L @ ( bot_bot @ ( set @ A ) ) ) @ ( set_or3652927894154168847AtMost @ A @ L @ U2 ) )
= ( set_or1337092689740270186AtMost @ A @ L @ U2 ) ) ) ) ).
% ivl_disj_un_singleton(5)
thf(fact_7654_ivl__disj__un__singleton_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,U2: A] :
( ( ord_less @ A @ L @ U2 )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ L @ U2 ) @ ( insert @ A @ U2 @ ( bot_bot @ ( set @ A ) ) ) )
= ( set_or3652927894154168847AtMost @ A @ L @ U2 ) ) ) ) ).
% ivl_disj_un_singleton(4)
thf(fact_7655_ivl__disj__un__two_I5_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,M: A,U2: A] :
( ( ord_less @ A @ L @ M )
=> ( ( ord_less_eq @ A @ M @ U2 )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ L @ M ) @ ( set_or1337092689740270186AtMost @ A @ M @ U2 ) )
= ( set_or3652927894154168847AtMost @ A @ L @ U2 ) ) ) ) ) ).
% ivl_disj_un_two(5)
thf(fact_7656_cauchy__filter__metric,axiom,
! [A: $tType] :
( ( ( real_V768167426530841204y_dist @ A )
& ( topolo7287701948861334536_space @ A ) )
=> ( ( topolo6773858410816713723filter @ A )
= ( ^ [F8: filter @ A] :
! [E4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E4 )
=> ? [P3: A > $o] :
( ( eventually @ A @ P3 @ F8 )
& ! [X2: A,Y2: A] :
( ( ( P3 @ X2 )
& ( P3 @ Y2 ) )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X2 @ Y2 ) @ E4 ) ) ) ) ) ) ) ).
% cauchy_filter_metric
thf(fact_7657_GMVT,axiom,
! [A3: real,B3: real,F3: real > real,G2: real > real] :
( ( ord_less @ real @ A3 @ B3 )
=> ( ! [X3: real] :
( ( ( ord_less_eq @ real @ A3 @ X3 )
& ( ord_less_eq @ real @ X3 @ B3 ) )
=> ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) @ F3 ) )
=> ( ! [X3: real] :
( ( ( ord_less @ real @ A3 @ X3 )
& ( ord_less @ real @ X3 @ B3 ) )
=> ( differentiable @ real @ real @ F3 @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) ) )
=> ( ! [X3: real] :
( ( ( ord_less_eq @ real @ A3 @ X3 )
& ( ord_less_eq @ real @ X3 @ B3 ) )
=> ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) @ G2 ) )
=> ( ! [X3: real] :
( ( ( ord_less @ real @ A3 @ X3 )
& ( ord_less @ real @ X3 @ B3 ) )
=> ( differentiable @ real @ real @ G2 @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) ) )
=> ? [G_c: real,F_c: real,C5: real] :
( ( has_field_derivative @ real @ G2 @ G_c @ ( topolo174197925503356063within @ real @ C5 @ ( top_top @ ( set @ real ) ) ) )
& ( has_field_derivative @ real @ F3 @ F_c @ ( topolo174197925503356063within @ real @ C5 @ ( top_top @ ( set @ real ) ) ) )
& ( ord_less @ real @ A3 @ C5 )
& ( ord_less @ real @ C5 @ B3 )
& ( ( times_times @ real @ ( minus_minus @ real @ ( F3 @ B3 ) @ ( F3 @ A3 ) ) @ G_c )
= ( times_times @ real @ ( minus_minus @ real @ ( G2 @ B3 ) @ ( G2 @ A3 ) ) @ F_c ) ) ) ) ) ) ) ) ).
% GMVT
thf(fact_7658_differentiable__cmult__right__iff,axiom,
! [A: $tType,B: $tType] :
( ( ( real_V822414075346904944vector @ B )
& ( real_V3459762299906320749_field @ A ) )
=> ! [Q3: B > A,C2: A,T: B] :
( ( differentiable @ B @ A
@ ^ [T2: B] : ( times_times @ A @ ( Q3 @ T2 ) @ C2 )
@ ( topolo174197925503356063within @ B @ T @ ( top_top @ ( set @ B ) ) ) )
= ( ( C2
= ( zero_zero @ A ) )
| ( differentiable @ B @ A @ Q3 @ ( topolo174197925503356063within @ B @ T @ ( top_top @ ( set @ B ) ) ) ) ) ) ) ).
% differentiable_cmult_right_iff
thf(fact_7659_differentiable__cmult__left__iff,axiom,
! [A: $tType,B: $tType] :
( ( ( real_V822414075346904944vector @ B )
& ( real_V3459762299906320749_field @ A ) )
=> ! [C2: A,Q3: B > A,T: B] :
( ( differentiable @ B @ A
@ ^ [T2: B] : ( times_times @ A @ C2 @ ( Q3 @ T2 ) )
@ ( topolo174197925503356063within @ B @ T @ ( top_top @ ( set @ B ) ) ) )
= ( ( C2
= ( zero_zero @ A ) )
| ( differentiable @ B @ A @ Q3 @ ( topolo174197925503356063within @ B @ T @ ( top_top @ ( set @ B ) ) ) ) ) ) ) ).
% differentiable_cmult_left_iff
thf(fact_7660_differentiable__sum,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( ( real_V822414075346904944vector @ B )
& ( real_V822414075346904944vector @ C ) )
=> ! [S2: set @ A,F3: A > B > C,Net: filter @ B] :
( ( finite_finite2 @ A @ S2 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ S2 )
=> ( differentiable @ B @ C @ ( F3 @ X3 ) @ Net ) )
=> ( differentiable @ B @ C
@ ^ [X2: B] :
( groups7311177749621191930dd_sum @ A @ C
@ ^ [A5: A] : ( F3 @ A5 @ X2 )
@ S2 )
@ Net ) ) ) ) ).
% differentiable_sum
thf(fact_7661_differentiable__compose,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F3: A > B,G2: C > A,X: C,S2: set @ C] :
( ( differentiable @ A @ B @ F3 @ ( topolo174197925503356063within @ A @ ( G2 @ X ) @ ( top_top @ ( set @ A ) ) ) )
=> ( ( differentiable @ C @ A @ G2 @ ( topolo174197925503356063within @ C @ X @ S2 ) )
=> ( differentiable @ C @ B
@ ^ [X2: C] : ( F3 @ ( G2 @ X2 ) )
@ ( topolo174197925503356063within @ C @ X @ S2 ) ) ) ) ) ).
% differentiable_compose
thf(fact_7662_differentiable__power,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V3459762299906320749_field @ B ) )
=> ! [F3: A > B,X: A,S2: set @ A,N: nat] :
( ( differentiable @ A @ B @ F3 @ ( topolo174197925503356063within @ A @ X @ S2 ) )
=> ( differentiable @ A @ B
@ ^ [X2: A] : ( power_power @ B @ ( F3 @ X2 ) @ N )
@ ( topolo174197925503356063within @ A @ X @ S2 ) ) ) ) ).
% differentiable_power
thf(fact_7663_differentiable__scaleR,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F3: A > real,X: A,S2: set @ A,G2: A > B] :
( ( differentiable @ A @ real @ F3 @ ( topolo174197925503356063within @ A @ X @ S2 ) )
=> ( ( differentiable @ A @ B @ G2 @ ( topolo174197925503356063within @ A @ X @ S2 ) )
=> ( differentiable @ A @ B
@ ^ [X2: A] : ( real_V8093663219630862766scaleR @ B @ ( F3 @ X2 ) @ ( G2 @ X2 ) )
@ ( topolo174197925503356063within @ A @ X @ S2 ) ) ) ) ) ).
% differentiable_scaleR
thf(fact_7664_differentiable__mult,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V4412858255891104859lgebra @ B ) )
=> ! [F3: A > B,X: A,S2: set @ A,G2: A > B] :
( ( differentiable @ A @ B @ F3 @ ( topolo174197925503356063within @ A @ X @ S2 ) )
=> ( ( differentiable @ A @ B @ G2 @ ( topolo174197925503356063within @ A @ X @ S2 ) )
=> ( differentiable @ A @ B
@ ^ [X2: A] : ( times_times @ B @ ( F3 @ X2 ) @ ( G2 @ X2 ) )
@ ( topolo174197925503356063within @ A @ X @ S2 ) ) ) ) ) ).
% differentiable_mult
thf(fact_7665_differentiable__in__compose,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F3: A > B,G2: C > A,X: C,S2: set @ C] :
( ( differentiable @ A @ B @ F3 @ ( topolo174197925503356063within @ A @ ( G2 @ X ) @ ( image @ C @ A @ G2 @ S2 ) ) )
=> ( ( differentiable @ C @ A @ G2 @ ( topolo174197925503356063within @ C @ X @ S2 ) )
=> ( differentiable @ C @ B
@ ^ [X2: C] : ( F3 @ ( G2 @ X2 ) )
@ ( topolo174197925503356063within @ C @ X @ S2 ) ) ) ) ) ).
% differentiable_in_compose
thf(fact_7666_differentiable__minus,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F3: A > B,F4: filter @ A] :
( ( differentiable @ A @ B @ F3 @ F4 )
=> ( differentiable @ A @ B
@ ^ [X2: A] : ( uminus_uminus @ B @ ( F3 @ X2 ) )
@ F4 ) ) ) ).
% differentiable_minus
thf(fact_7667_differentiable__ident,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F4: filter @ A] :
( differentiable @ A @ A
@ ^ [X2: A] : X2
@ F4 ) ) ).
% differentiable_ident
thf(fact_7668_differentiable__const,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [A3: B,F4: filter @ A] :
( differentiable @ A @ B
@ ^ [Z4: A] : A3
@ F4 ) ) ).
% differentiable_const
thf(fact_7669_differentiable__diff,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F3: A > B,F4: filter @ A,G2: A > B] :
( ( differentiable @ A @ B @ F3 @ F4 )
=> ( ( differentiable @ A @ B @ G2 @ F4 )
=> ( differentiable @ A @ B
@ ^ [X2: A] : ( minus_minus @ B @ ( F3 @ X2 ) @ ( G2 @ X2 ) )
@ F4 ) ) ) ) ).
% differentiable_diff
thf(fact_7670_differentiable__add,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F3: A > B,F4: filter @ A,G2: A > B] :
( ( differentiable @ A @ B @ F3 @ F4 )
=> ( ( differentiable @ A @ B @ G2 @ F4 )
=> ( differentiable @ A @ B
@ ^ [X2: A] : ( plus_plus @ B @ ( F3 @ X2 ) @ ( G2 @ X2 ) )
@ F4 ) ) ) ) ).
% differentiable_add
thf(fact_7671_atLeastPlusOneAtMost__greaterThanAtMost__int,axiom,
! [L: int,U2: int] :
( ( set_or1337092689740270186AtMost @ int @ ( plus_plus @ int @ L @ ( one_one @ int ) ) @ U2 )
= ( set_or3652927894154168847AtMost @ int @ L @ U2 ) ) ).
% atLeastPlusOneAtMost_greaterThanAtMost_int
thf(fact_7672_differentiable__divide,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V3459762299906320749_field @ B ) )
=> ! [F3: A > B,X: A,S2: set @ A,G2: A > B] :
( ( differentiable @ A @ B @ F3 @ ( topolo174197925503356063within @ A @ X @ S2 ) )
=> ( ( differentiable @ A @ B @ G2 @ ( topolo174197925503356063within @ A @ X @ S2 ) )
=> ( ( ( G2 @ X )
!= ( zero_zero @ B ) )
=> ( differentiable @ A @ B
@ ^ [X2: A] : ( divide_divide @ B @ ( F3 @ X2 ) @ ( G2 @ X2 ) )
@ ( topolo174197925503356063within @ A @ X @ S2 ) ) ) ) ) ) ).
% differentiable_divide
thf(fact_7673_differentiable__inverse,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V3459762299906320749_field @ B ) )
=> ! [F3: A > B,X: A,S2: set @ A] :
( ( differentiable @ A @ B @ F3 @ ( topolo174197925503356063within @ A @ X @ S2 ) )
=> ( ( ( F3 @ X )
!= ( zero_zero @ B ) )
=> ( differentiable @ A @ B
@ ^ [X2: A] : ( inverse_inverse @ B @ ( F3 @ X2 ) )
@ ( topolo174197925503356063within @ A @ X @ S2 ) ) ) ) ) ).
% differentiable_inverse
thf(fact_7674_atLeastPlusOneAtMost__greaterThanAtMost__integer,axiom,
! [L: code_integer,U2: code_integer] :
( ( set_or1337092689740270186AtMost @ code_integer @ ( plus_plus @ code_integer @ L @ ( one_one @ code_integer ) ) @ U2 )
= ( set_or3652927894154168847AtMost @ code_integer @ L @ U2 ) ) ).
% atLeastPlusOneAtMost_greaterThanAtMost_integer
thf(fact_7675_word__atLeastAtMost__Suc__greaterThanAtMost,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [M: word @ A,U2: word @ A] :
( ( M
!= ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) )
=> ( ( set_or3652927894154168847AtMost @ ( word @ A ) @ M @ U2 )
= ( set_or1337092689740270186AtMost @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ M @ ( one_one @ ( word @ A ) ) ) @ U2 ) ) ) ) ).
% word_atLeastAtMost_Suc_greaterThanAtMost
thf(fact_7676_word__range__minus__1_H,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [A3: word @ A,B3: word @ A] :
( ( A3
!= ( zero_zero @ ( word @ A ) ) )
=> ( ( set_or3652927894154168847AtMost @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ A3 @ ( one_one @ ( word @ A ) ) ) @ B3 )
= ( set_or1337092689740270186AtMost @ ( word @ A ) @ A3 @ B3 ) ) ) ) ).
% word_range_minus_1'
thf(fact_7677_MVT,axiom,
! [A3: real,B3: real,F3: real > real] :
( ( ord_less @ real @ A3 @ B3 )
=> ( ( topolo81223032696312382ous_on @ real @ real @ ( set_or1337092689740270186AtMost @ real @ A3 @ B3 ) @ F3 )
=> ( ! [X3: real] :
( ( ord_less @ real @ A3 @ X3 )
=> ( ( ord_less @ real @ X3 @ B3 )
=> ( differentiable @ real @ real @ F3 @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) ) ) )
=> ? [L3: real,Z3: real] :
( ( ord_less @ real @ A3 @ Z3 )
& ( ord_less @ real @ Z3 @ B3 )
& ( has_field_derivative @ real @ F3 @ L3 @ ( topolo174197925503356063within @ real @ Z3 @ ( top_top @ ( set @ real ) ) ) )
& ( ( minus_minus @ real @ ( F3 @ B3 ) @ ( F3 @ A3 ) )
= ( times_times @ real @ ( minus_minus @ real @ B3 @ A3 ) @ L3 ) ) ) ) ) ) ).
% MVT
thf(fact_7678_interval__cases,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [S: set @ A] :
( ! [A6: A,B7: A,X3: A] :
( ( member @ A @ A6 @ S )
=> ( ( member @ A @ B7 @ S )
=> ( ( ord_less_eq @ A @ A6 @ X3 )
=> ( ( ord_less_eq @ A @ X3 @ B7 )
=> ( member @ A @ X3 @ S ) ) ) ) )
=> ? [A6: A,B7: A] :
( ( S
= ( bot_bot @ ( set @ A ) ) )
| ( S
= ( top_top @ ( set @ A ) ) )
| ( S
= ( set_ord_lessThan @ A @ B7 ) )
| ( S
= ( set_ord_atMost @ A @ B7 ) )
| ( S
= ( set_ord_greaterThan @ A @ A6 ) )
| ( S
= ( set_ord_atLeast @ A @ A6 ) )
| ( S
= ( set_or5935395276787703475ssThan @ A @ A6 @ B7 ) )
| ( S
= ( set_or3652927894154168847AtMost @ A @ A6 @ B7 ) )
| ( S
= ( set_or7035219750837199246ssThan @ A @ A6 @ B7 ) )
| ( S
= ( set_or1337092689740270186AtMost @ A @ A6 @ B7 ) ) ) ) ) ).
% interval_cases
thf(fact_7679_atLeast__empty__triv,axiom,
! [A: $tType] :
( ( set_ord_atLeast @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) )
= ( top_top @ ( set @ ( set @ A ) ) ) ) ).
% atLeast_empty_triv
thf(fact_7680_atLeast__0,axiom,
( ( set_ord_atLeast @ nat @ ( zero_zero @ nat ) )
= ( top_top @ ( set @ nat ) ) ) ).
% atLeast_0
thf(fact_7681_continuous__on__compose2,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( ( topolo4958980785337419405_space @ C )
& ( topolo4958980785337419405_space @ B )
& ( topolo4958980785337419405_space @ A ) )
=> ! [T: set @ A,G2: A > B,S2: set @ C,F3: C > A] :
( ( topolo81223032696312382ous_on @ A @ B @ T @ G2 )
=> ( ( topolo81223032696312382ous_on @ C @ A @ S2 @ F3 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image @ C @ A @ F3 @ S2 ) @ T )
=> ( topolo81223032696312382ous_on @ C @ B @ S2
@ ^ [X2: C] : ( G2 @ ( F3 @ X2 ) ) ) ) ) ) ) ).
% continuous_on_compose2
thf(fact_7682_atLeast__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( set_ord_atLeast @ A )
= ( ^ [L2: A] : ( collect @ A @ ( ord_less_eq @ A @ L2 ) ) ) ) ) ).
% atLeast_def
thf(fact_7683_continuous__on__arcosh,axiom,
! [A2: set @ real] :
( ( ord_less_eq @ ( set @ real ) @ A2 @ ( set_ord_atLeast @ real @ ( one_one @ real ) ) )
=> ( topolo81223032696312382ous_on @ real @ real @ A2 @ ( arcosh @ real ) ) ) ).
% continuous_on_arcosh
thf(fact_7684_ex__gt__or__lt,axiom,
! [A: $tType] :
( ( condit5016429287641298734tinuum @ A )
=> ! [A3: A] :
? [B7: A] :
( ( ord_less @ A @ A3 @ B7 )
| ( ord_less @ A @ B7 @ A3 ) ) ) ).
% ex_gt_or_lt
thf(fact_7685_continuous__on__tan,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [S2: set @ A,F3: A > A] :
( ( topolo81223032696312382ous_on @ A @ A @ S2 @ F3 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ S2 )
=> ( ( cos @ A @ ( F3 @ X3 ) )
!= ( zero_zero @ A ) ) )
=> ( topolo81223032696312382ous_on @ A @ A @ S2
@ ^ [X2: A] : ( tan @ A @ ( F3 @ X2 ) ) ) ) ) ) ).
% continuous_on_tan
thf(fact_7686_continuous__on__dist,axiom,
! [A: $tType,D: $tType] :
( ( ( topolo4958980785337419405_space @ D )
& ( real_V7819770556892013058_space @ A ) )
=> ! [S2: set @ D,F3: D > A,G2: D > A] :
( ( topolo81223032696312382ous_on @ D @ A @ S2 @ F3 )
=> ( ( topolo81223032696312382ous_on @ D @ A @ S2 @ G2 )
=> ( topolo81223032696312382ous_on @ D @ real @ S2
@ ^ [X2: D] : ( real_V557655796197034286t_dist @ A @ ( F3 @ X2 ) @ ( G2 @ X2 ) ) ) ) ) ) ).
% continuous_on_dist
thf(fact_7687_atLeast__eq__UNIV__iff,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [X: A] :
( ( ( set_ord_atLeast @ A @ X )
= ( top_top @ ( set @ A ) ) )
= ( X
= ( bot_bot @ A ) ) ) ) ).
% atLeast_eq_UNIV_iff
thf(fact_7688_continuous__on__sing,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo4958980785337419405_space @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [X: A,F3: A > B] : ( topolo81223032696312382ous_on @ A @ B @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) @ F3 ) ) ).
% continuous_on_sing
thf(fact_7689_not__empty__eq__Ici__eq__empty,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [L: A] :
( ( bot_bot @ ( set @ A ) )
!= ( set_ord_atLeast @ A @ L ) ) ) ).
% not_empty_eq_Ici_eq_empty
thf(fact_7690_continuous__on__empty,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo4958980785337419405_space @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [F3: A > B] : ( topolo81223032696312382ous_on @ A @ B @ ( bot_bot @ ( set @ A ) ) @ F3 ) ) ).
% continuous_on_empty
thf(fact_7691_continuous__inj__imp__mono,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo8458572112393995274pology @ A )
& ( topolo1944317154257567458pology @ B ) )
=> ! [A3: A,X: A,B3: A,F3: A > B] :
( ( ord_less @ A @ A3 @ X )
=> ( ( ord_less @ A @ X @ B3 )
=> ( ( topolo81223032696312382ous_on @ A @ B @ ( set_or1337092689740270186AtMost @ A @ A3 @ B3 ) @ F3 )
=> ( ( inj_on @ A @ B @ F3 @ ( set_or1337092689740270186AtMost @ A @ A3 @ B3 ) )
=> ( ( ( ord_less @ B @ ( F3 @ A3 ) @ ( F3 @ X ) )
& ( ord_less @ B @ ( F3 @ X ) @ ( F3 @ B3 ) ) )
| ( ( ord_less @ B @ ( F3 @ B3 ) @ ( F3 @ X ) )
& ( ord_less @ B @ ( F3 @ X ) @ ( F3 @ A3 ) ) ) ) ) ) ) ) ) ).
% continuous_inj_imp_mono
thf(fact_7692_continuous__on__inverse,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo4958980785337419405_space @ A )
& ( real_V8999393235501362500lgebra @ B ) )
=> ! [S2: set @ A,F3: A > B] :
( ( topolo81223032696312382ous_on @ A @ B @ S2 @ F3 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ S2 )
=> ( ( F3 @ X3 )
!= ( zero_zero @ B ) ) )
=> ( topolo81223032696312382ous_on @ A @ B @ S2
@ ^ [X2: A] : ( inverse_inverse @ B @ ( F3 @ X2 ) ) ) ) ) ) ).
% continuous_on_inverse
thf(fact_7693_continuous__on__sgn,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo4958980785337419405_space @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [S2: set @ A,F3: A > B] :
( ( topolo81223032696312382ous_on @ A @ B @ S2 @ F3 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ S2 )
=> ( ( F3 @ X3 )
!= ( zero_zero @ B ) ) )
=> ( topolo81223032696312382ous_on @ A @ B @ S2
@ ^ [X2: A] : ( sgn_sgn @ B @ ( F3 @ X2 ) ) ) ) ) ) ).
% continuous_on_sgn
thf(fact_7694_continuous__on__powr,axiom,
! [C: $tType] :
( ( topolo4958980785337419405_space @ C )
=> ! [S2: set @ C,F3: C > real,G2: C > real] :
( ( topolo81223032696312382ous_on @ C @ real @ S2 @ F3 )
=> ( ( topolo81223032696312382ous_on @ C @ real @ S2 @ G2 )
=> ( ! [X3: C] :
( ( member @ C @ X3 @ S2 )
=> ( ( F3 @ X3 )
!= ( zero_zero @ real ) ) )
=> ( topolo81223032696312382ous_on @ C @ real @ S2
@ ^ [X2: C] : ( powr @ real @ ( F3 @ X2 ) @ ( G2 @ X2 ) ) ) ) ) ) ) ).
% continuous_on_powr
thf(fact_7695_continuous__on__ln,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [S2: set @ A,F3: A > real] :
( ( topolo81223032696312382ous_on @ A @ real @ S2 @ F3 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ S2 )
=> ( ( F3 @ X3 )
!= ( zero_zero @ real ) ) )
=> ( topolo81223032696312382ous_on @ A @ real @ S2
@ ^ [X2: A] : ( ln_ln @ real @ ( F3 @ X2 ) ) ) ) ) ) ).
% continuous_on_ln
thf(fact_7696_continuous__on__divide,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo4958980785337419405_space @ A )
& ( real_V3459762299906320749_field @ B ) )
=> ! [S2: set @ A,F3: A > B,G2: A > B] :
( ( topolo81223032696312382ous_on @ A @ B @ S2 @ F3 )
=> ( ( topolo81223032696312382ous_on @ A @ B @ S2 @ G2 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ S2 )
=> ( ( G2 @ X3 )
!= ( zero_zero @ B ) ) )
=> ( topolo81223032696312382ous_on @ A @ B @ S2
@ ^ [X2: A] : ( divide_divide @ B @ ( F3 @ X2 ) @ ( G2 @ X2 ) ) ) ) ) ) ) ).
% continuous_on_divide
thf(fact_7697_continuous__on__add,axiom,
! [B: $tType,D: $tType] :
( ( ( topolo4958980785337419405_space @ D )
& ( topolo6943815403480290642id_add @ B ) )
=> ! [S2: set @ D,F3: D > B,G2: D > B] :
( ( topolo81223032696312382ous_on @ D @ B @ S2 @ F3 )
=> ( ( topolo81223032696312382ous_on @ D @ B @ S2 @ G2 )
=> ( topolo81223032696312382ous_on @ D @ B @ S2
@ ^ [X2: D] : ( plus_plus @ B @ ( F3 @ X2 ) @ ( G2 @ X2 ) ) ) ) ) ) ).
% continuous_on_add
thf(fact_7698_continuous__on__diff,axiom,
! [B: $tType,D: $tType] :
( ( ( topolo4958980785337419405_space @ D )
& ( topolo1633459387980952147up_add @ B ) )
=> ! [S2: set @ D,F3: D > B,G2: D > B] :
( ( topolo81223032696312382ous_on @ D @ B @ S2 @ F3 )
=> ( ( topolo81223032696312382ous_on @ D @ B @ S2 @ G2 )
=> ( topolo81223032696312382ous_on @ D @ B @ S2
@ ^ [X2: D] : ( minus_minus @ B @ ( F3 @ X2 ) @ ( G2 @ X2 ) ) ) ) ) ) ).
% continuous_on_diff
thf(fact_7699_continuous__on__mult,axiom,
! [A: $tType,D: $tType] :
( ( ( topolo4958980785337419405_space @ D )
& ( real_V4412858255891104859lgebra @ A ) )
=> ! [S2: set @ D,F3: D > A,G2: D > A] :
( ( topolo81223032696312382ous_on @ D @ A @ S2 @ F3 )
=> ( ( topolo81223032696312382ous_on @ D @ A @ S2 @ G2 )
=> ( topolo81223032696312382ous_on @ D @ A @ S2
@ ^ [X2: D] : ( times_times @ A @ ( F3 @ X2 ) @ ( G2 @ X2 ) ) ) ) ) ) ).
% continuous_on_mult
thf(fact_7700_continuous__on__mult_H,axiom,
! [B: $tType,D: $tType] :
( ( ( topolo4958980785337419405_space @ D )
& ( topolo4211221413907600880p_mult @ B ) )
=> ! [A2: set @ D,F3: D > B,G2: D > B] :
( ( topolo81223032696312382ous_on @ D @ B @ A2 @ F3 )
=> ( ( topolo81223032696312382ous_on @ D @ B @ A2 @ G2 )
=> ( topolo81223032696312382ous_on @ D @ B @ A2
@ ^ [X2: D] : ( times_times @ B @ ( F3 @ X2 ) @ ( G2 @ X2 ) ) ) ) ) ) ).
% continuous_on_mult'
thf(fact_7701_continuous__on__mult__left,axiom,
! [A: $tType,B: $tType] :
( ( ( topolo4958980785337419405_space @ B )
& ( real_V4412858255891104859lgebra @ A ) )
=> ! [S2: set @ B,F3: B > A,C2: A] :
( ( topolo81223032696312382ous_on @ B @ A @ S2 @ F3 )
=> ( topolo81223032696312382ous_on @ B @ A @ S2
@ ^ [X2: B] : ( times_times @ A @ C2 @ ( F3 @ X2 ) ) ) ) ) ).
% continuous_on_mult_left
thf(fact_7702_continuous__on__mult__right,axiom,
! [A: $tType,B: $tType] :
( ( ( topolo4958980785337419405_space @ B )
& ( real_V4412858255891104859lgebra @ A ) )
=> ! [S2: set @ B,F3: B > A,C2: A] :
( ( topolo81223032696312382ous_on @ B @ A @ S2 @ F3 )
=> ( topolo81223032696312382ous_on @ B @ A @ S2
@ ^ [X2: B] : ( times_times @ A @ ( F3 @ X2 ) @ C2 ) ) ) ) ).
% continuous_on_mult_right
thf(fact_7703_continuous__on__Pair,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( topolo4958980785337419405_space @ A )
& ( topolo4958980785337419405_space @ B )
& ( topolo4958980785337419405_space @ C ) )
=> ! [S2: set @ A,F3: A > B,G2: A > C] :
( ( topolo81223032696312382ous_on @ A @ B @ S2 @ F3 )
=> ( ( topolo81223032696312382ous_on @ A @ C @ S2 @ G2 )
=> ( topolo81223032696312382ous_on @ A @ ( product_prod @ B @ C ) @ S2
@ ^ [X2: A] : ( product_Pair @ B @ C @ ( F3 @ X2 ) @ ( G2 @ X2 ) ) ) ) ) ) ).
% continuous_on_Pair
thf(fact_7704_continuous__on__rabs,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [S2: set @ A,F3: A > real] :
( ( topolo81223032696312382ous_on @ A @ real @ S2 @ F3 )
=> ( topolo81223032696312382ous_on @ A @ real @ S2
@ ^ [X2: A] : ( abs_abs @ real @ ( F3 @ X2 ) ) ) ) ) ).
% continuous_on_rabs
thf(fact_7705_continuous__on__max,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo4958980785337419405_space @ A )
& ( topolo1944317154257567458pology @ B ) )
=> ! [A2: set @ A,F3: A > B,G2: A > B] :
( ( topolo81223032696312382ous_on @ A @ B @ A2 @ F3 )
=> ( ( topolo81223032696312382ous_on @ A @ B @ A2 @ G2 )
=> ( topolo81223032696312382ous_on @ A @ B @ A2
@ ^ [X2: A] : ( ord_max @ B @ ( F3 @ X2 ) @ ( G2 @ X2 ) ) ) ) ) ) ).
% continuous_on_max
thf(fact_7706_continuous__on__arsinh_H,axiom,
! [A2: set @ real,F3: real > real] :
( ( topolo81223032696312382ous_on @ real @ real @ A2 @ F3 )
=> ( topolo81223032696312382ous_on @ real @ real @ A2
@ ^ [X2: real] : ( arsinh @ real @ ( F3 @ X2 ) ) ) ) ).
% continuous_on_arsinh'
thf(fact_7707_continuous__on__id,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [S2: set @ A] :
( topolo81223032696312382ous_on @ A @ A @ S2
@ ^ [X2: A] : X2 ) ) ).
% continuous_on_id
thf(fact_7708_continuous__on__const,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo4958980785337419405_space @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [S2: set @ A,C2: B] :
( topolo81223032696312382ous_on @ A @ B @ S2
@ ^ [X2: A] : C2 ) ) ).
% continuous_on_const
thf(fact_7709_continuous__on__arctan,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [S2: set @ A,F3: A > real] :
( ( topolo81223032696312382ous_on @ A @ real @ S2 @ F3 )
=> ( topolo81223032696312382ous_on @ A @ real @ S2
@ ^ [X2: A] : ( arctan @ ( F3 @ X2 ) ) ) ) ) ).
% continuous_on_arctan
thf(fact_7710_continuous__on__real__sqrt,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [S2: set @ A,F3: A > real] :
( ( topolo81223032696312382ous_on @ A @ real @ S2 @ F3 )
=> ( topolo81223032696312382ous_on @ A @ real @ S2
@ ^ [X2: A] : ( sqrt @ ( F3 @ X2 ) ) ) ) ) ).
% continuous_on_real_sqrt
thf(fact_7711_continuous__on__norm,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo4958980785337419405_space @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [S2: set @ A,F3: A > B] :
( ( topolo81223032696312382ous_on @ A @ B @ S2 @ F3 )
=> ( topolo81223032696312382ous_on @ A @ real @ S2
@ ^ [X2: A] : ( real_V7770717601297561774m_norm @ B @ ( F3 @ X2 ) ) ) ) ) ).
% continuous_on_norm
thf(fact_7712_continuous__on__of__real,axiom,
! [A: $tType,C: $tType] :
( ( ( topolo4958980785337419405_space @ C )
& ( real_V2191834092415804123ebra_1 @ A )
& ( real_V822414075346904944vector @ A ) )
=> ! [S2: set @ C,G2: C > real] :
( ( topolo81223032696312382ous_on @ C @ real @ S2 @ G2 )
=> ( topolo81223032696312382ous_on @ C @ A @ S2
@ ^ [X2: C] : ( real_Vector_of_real @ A @ ( G2 @ X2 ) ) ) ) ) ).
% continuous_on_of_real
thf(fact_7713_continuous__on__sin,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_Vector_banach @ B )
& ( real_V3459762299906320749_field @ B ) )
=> ! [S2: set @ A,F3: A > B] :
( ( topolo81223032696312382ous_on @ A @ B @ S2 @ F3 )
=> ( topolo81223032696312382ous_on @ A @ B @ S2
@ ^ [X2: A] : ( sin @ B @ ( F3 @ X2 ) ) ) ) ) ).
% continuous_on_sin
thf(fact_7714_continuous__on__cos,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_Vector_banach @ B )
& ( real_V3459762299906320749_field @ B ) )
=> ! [S2: set @ A,F3: A > B] :
( ( topolo81223032696312382ous_on @ A @ B @ S2 @ F3 )
=> ( topolo81223032696312382ous_on @ A @ B @ S2
@ ^ [X2: A] : ( cos @ B @ ( F3 @ X2 ) ) ) ) ) ).
% continuous_on_cos
thf(fact_7715_continuous__on__pochhammer_H,axiom,
! [A: $tType,C: $tType] :
( ( ( topolo4958980785337419405_space @ C )
& ( real_V3459762299906320749_field @ A ) )
=> ! [S2: set @ C,F3: C > A,N: nat] :
( ( topolo81223032696312382ous_on @ C @ A @ S2 @ F3 )
=> ( topolo81223032696312382ous_on @ C @ A @ S2
@ ^ [X2: C] : ( comm_s3205402744901411588hammer @ A @ ( F3 @ X2 ) @ N ) ) ) ) ).
% continuous_on_pochhammer'
thf(fact_7716_continuous__on__pochhammer,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [A2: set @ A,N: nat] :
( topolo81223032696312382ous_on @ A @ A @ A2
@ ^ [Z4: A] : ( comm_s3205402744901411588hammer @ A @ Z4 @ N ) ) ) ).
% continuous_on_pochhammer
thf(fact_7717_continuous__on__exp,axiom,
! [A: $tType,C: $tType] :
( ( ( topolo4958980785337419405_space @ C )
& ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [S2: set @ C,F3: C > A] :
( ( topolo81223032696312382ous_on @ C @ A @ S2 @ F3 )
=> ( topolo81223032696312382ous_on @ C @ A @ S2
@ ^ [X2: C] : ( exp @ A @ ( F3 @ X2 ) ) ) ) ) ).
% continuous_on_exp
thf(fact_7718_continuous__on__scaleR,axiom,
! [C: $tType,D: $tType] :
( ( ( topolo4958980785337419405_space @ D )
& ( real_V822414075346904944vector @ C ) )
=> ! [S2: set @ D,F3: D > real,G2: D > C] :
( ( topolo81223032696312382ous_on @ D @ real @ S2 @ F3 )
=> ( ( topolo81223032696312382ous_on @ D @ C @ S2 @ G2 )
=> ( topolo81223032696312382ous_on @ D @ C @ S2
@ ^ [X2: D] : ( real_V8093663219630862766scaleR @ C @ ( F3 @ X2 ) @ ( G2 @ X2 ) ) ) ) ) ) ).
% continuous_on_scaleR
thf(fact_7719_continuous__on__cosh,axiom,
! [A: $tType,C: $tType] :
( ( ( topolo4958980785337419405_space @ C )
& ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [A2: set @ C,F3: C > A] :
( ( topolo81223032696312382ous_on @ C @ A @ A2 @ F3 )
=> ( topolo81223032696312382ous_on @ C @ A @ A2
@ ^ [X2: C] : ( cosh @ A @ ( F3 @ X2 ) ) ) ) ) ).
% continuous_on_cosh
thf(fact_7720_continuous__on__sinh,axiom,
! [A: $tType,C: $tType] :
( ( ( topolo4958980785337419405_space @ C )
& ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [A2: set @ C,F3: C > A] :
( ( topolo81223032696312382ous_on @ C @ A @ A2 @ F3 )
=> ( topolo81223032696312382ous_on @ C @ A @ A2
@ ^ [X2: C] : ( sinh @ A @ ( F3 @ X2 ) ) ) ) ) ).
% continuous_on_sinh
thf(fact_7721_continuous__on__real__root,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [S2: set @ A,F3: A > real,N: nat] :
( ( topolo81223032696312382ous_on @ A @ real @ S2 @ F3 )
=> ( topolo81223032696312382ous_on @ A @ real @ S2
@ ^ [X2: A] : ( root @ N @ ( F3 @ X2 ) ) ) ) ) ).
% continuous_on_real_root
thf(fact_7722_continuous__on__power,axiom,
! [B: $tType,C: $tType] :
( ( ( topolo4958980785337419405_space @ C )
& ( power @ B )
& ( real_V4412858255891104859lgebra @ B ) )
=> ! [S2: set @ C,F3: C > B,N: nat] :
( ( topolo81223032696312382ous_on @ C @ B @ S2 @ F3 )
=> ( topolo81223032696312382ous_on @ C @ B @ S2
@ ^ [X2: C] : ( power_power @ B @ ( F3 @ X2 ) @ N ) ) ) ) ).
% continuous_on_power
thf(fact_7723_continuous__on__power_H,axiom,
! [B: $tType,C: $tType] :
( ( ( topolo4958980785337419405_space @ C )
& ( topolo1898628316856586783d_mult @ B ) )
=> ! [A2: set @ C,F3: C > B,G2: C > nat] :
( ( topolo81223032696312382ous_on @ C @ B @ A2 @ F3 )
=> ( ( topolo81223032696312382ous_on @ C @ nat @ A2 @ G2 )
=> ( topolo81223032696312382ous_on @ C @ B @ A2
@ ^ [X2: C] : ( power_power @ B @ ( F3 @ X2 ) @ ( G2 @ X2 ) ) ) ) ) ) ).
% continuous_on_power'
thf(fact_7724_continuous__on__prod,axiom,
! [A: $tType,C: $tType,D: $tType] :
( ( ( topolo4958980785337419405_space @ D )
& ( real_V4412858255891104859lgebra @ C )
& ( comm_ring_1 @ C ) )
=> ! [S: set @ A,S2: set @ D,F3: A > D > C] :
( ! [I3: A] :
( ( member @ A @ I3 @ S )
=> ( topolo81223032696312382ous_on @ D @ C @ S2 @ ( F3 @ I3 ) ) )
=> ( topolo81223032696312382ous_on @ D @ C @ S2
@ ^ [X2: D] :
( groups7121269368397514597t_prod @ A @ C
@ ^ [I4: A] : ( F3 @ I4 @ X2 )
@ S ) ) ) ) ).
% continuous_on_prod
thf(fact_7725_continuous__on__prod_H,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( ( topolo4958980785337419405_space @ B )
& ( topolo4987421752381908075d_mult @ C ) )
=> ! [I6: set @ A,S: set @ B,F3: A > B > C] :
( ! [I3: A] :
( ( member @ A @ I3 @ I6 )
=> ( topolo81223032696312382ous_on @ B @ C @ S @ ( F3 @ I3 ) ) )
=> ( topolo81223032696312382ous_on @ B @ C @ S
@ ^ [X2: B] :
( groups7121269368397514597t_prod @ A @ C
@ ^ [I4: A] : ( F3 @ I4 @ X2 )
@ I6 ) ) ) ) ).
% continuous_on_prod'
thf(fact_7726_continuous__on__sum,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( ( topolo4958980785337419405_space @ B )
& ( topolo5987344860129210374id_add @ C ) )
=> ! [I6: set @ A,S: set @ B,F3: A > B > C] :
( ! [I3: A] :
( ( member @ A @ I3 @ I6 )
=> ( topolo81223032696312382ous_on @ B @ C @ S @ ( F3 @ I3 ) ) )
=> ( topolo81223032696312382ous_on @ B @ C @ S
@ ^ [X2: B] :
( groups7311177749621191930dd_sum @ A @ C
@ ^ [I4: A] : ( F3 @ I4 @ X2 )
@ I6 ) ) ) ) ).
% continuous_on_sum
thf(fact_7727_continuous__on__minus,axiom,
! [B: $tType,C: $tType] :
( ( ( topolo4958980785337419405_space @ C )
& ( topolo1633459387980952147up_add @ B ) )
=> ! [S2: set @ C,F3: C > B] :
( ( topolo81223032696312382ous_on @ C @ B @ S2 @ F3 )
=> ( topolo81223032696312382ous_on @ C @ B @ S2
@ ^ [X2: C] : ( uminus_uminus @ B @ ( F3 @ X2 ) ) ) ) ) ).
% continuous_on_minus
thf(fact_7728_continuous__on__min,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo4958980785337419405_space @ A )
& ( topolo1944317154257567458pology @ B ) )
=> ! [A2: set @ A,F3: A > B,G2: A > B] :
( ( topolo81223032696312382ous_on @ A @ B @ A2 @ F3 )
=> ( ( topolo81223032696312382ous_on @ A @ B @ A2 @ G2 )
=> ( topolo81223032696312382ous_on @ A @ B @ A2
@ ^ [X2: A] : ( ord_min @ B @ ( F3 @ X2 ) @ ( G2 @ X2 ) ) ) ) ) ) ).
% continuous_on_min
thf(fact_7729_open__Collect__neq,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo4958980785337419405_space @ A )
& ( topological_t2_space @ B ) )
=> ! [F3: A > B,G2: A > B] :
( ( topolo81223032696312382ous_on @ A @ B @ ( top_top @ ( set @ A ) ) @ F3 )
=> ( ( topolo81223032696312382ous_on @ A @ B @ ( top_top @ ( set @ A ) ) @ G2 )
=> ( topolo1002775350975398744n_open @ A
@ ( collect @ A
@ ^ [X2: A] :
( ( F3 @ X2 )
!= ( G2 @ X2 ) ) ) ) ) ) ) ).
% open_Collect_neq
thf(fact_7730_continuous__on__open__Un,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo4958980785337419405_space @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [S2: set @ A,T: set @ A,F3: A > B] :
( ( topolo1002775350975398744n_open @ A @ S2 )
=> ( ( topolo1002775350975398744n_open @ A @ T )
=> ( ( topolo81223032696312382ous_on @ A @ B @ S2 @ F3 )
=> ( ( topolo81223032696312382ous_on @ A @ B @ T @ F3 )
=> ( topolo81223032696312382ous_on @ A @ B @ ( sup_sup @ ( set @ A ) @ S2 @ T ) @ F3 ) ) ) ) ) ) ).
% continuous_on_open_Un
thf(fact_7731_bounded__linear_Ocontinuous__on,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( ( topolo4958980785337419405_space @ C )
& ( real_V822414075346904944vector @ B )
& ( real_V822414075346904944vector @ A ) )
=> ! [F3: A > B,S2: set @ C,G2: C > A] :
( ( real_V3181309239436604168linear @ A @ B @ F3 )
=> ( ( topolo81223032696312382ous_on @ C @ A @ S2 @ G2 )
=> ( topolo81223032696312382ous_on @ C @ B @ S2
@ ^ [X2: C] : ( F3 @ ( G2 @ X2 ) ) ) ) ) ) ).
% bounded_linear.continuous_on
thf(fact_7732_continuous__on__tendsto__compose,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( ( topolo4958980785337419405_space @ B )
& ( topolo4958980785337419405_space @ A ) )
=> ! [S2: set @ A,F3: A > B,G2: C > A,L: A,F4: filter @ C] :
( ( topolo81223032696312382ous_on @ A @ B @ S2 @ F3 )
=> ( ( filterlim @ C @ A @ G2 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ F4 )
=> ( ( member @ A @ L @ S2 )
=> ( ( eventually @ C
@ ^ [X2: C] : ( member @ A @ ( G2 @ X2 ) @ S2 )
@ F4 )
=> ( filterlim @ C @ B
@ ^ [X2: C] : ( F3 @ ( G2 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( F3 @ L ) )
@ F4 ) ) ) ) ) ) ).
% continuous_on_tendsto_compose
thf(fact_7733_open__Collect__less,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo4958980785337419405_space @ A )
& ( topolo1944317154257567458pology @ B ) )
=> ! [F3: A > B,G2: A > B] :
( ( topolo81223032696312382ous_on @ A @ B @ ( top_top @ ( set @ A ) ) @ F3 )
=> ( ( topolo81223032696312382ous_on @ A @ B @ ( top_top @ ( set @ A ) ) @ G2 )
=> ( topolo1002775350975398744n_open @ A
@ ( collect @ A
@ ^ [X2: A] : ( ord_less @ B @ ( F3 @ X2 ) @ ( G2 @ X2 ) ) ) ) ) ) ) ).
% open_Collect_less
thf(fact_7734_continuous__on__cot,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [S2: set @ A,F3: A > A] :
( ( topolo81223032696312382ous_on @ A @ A @ S2 @ F3 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ S2 )
=> ( ( sin @ A @ ( F3 @ X3 ) )
!= ( zero_zero @ A ) ) )
=> ( topolo81223032696312382ous_on @ A @ A @ S2
@ ^ [X2: A] : ( cot @ A @ ( F3 @ X2 ) ) ) ) ) ) ).
% continuous_on_cot
thf(fact_7735_continuous__on__tanh,axiom,
! [A: $tType,C: $tType] :
( ( ( topolo4958980785337419405_space @ C )
& ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [A2: set @ C,F3: C > A] :
( ( topolo81223032696312382ous_on @ C @ A @ A2 @ F3 )
=> ( ! [X3: C] :
( ( member @ C @ X3 @ A2 )
=> ( ( cosh @ A @ ( F3 @ X3 ) )
!= ( zero_zero @ A ) ) )
=> ( topolo81223032696312382ous_on @ C @ A @ A2
@ ^ [X2: C] : ( tanh @ A @ ( F3 @ X2 ) ) ) ) ) ) ).
% continuous_on_tanh
thf(fact_7736_ivl__disj__un__one_I8_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,U2: A] :
( ( ord_less_eq @ A @ L @ U2 )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ L @ U2 ) @ ( set_ord_atLeast @ A @ U2 ) )
= ( set_ord_atLeast @ A @ L ) ) ) ) ).
% ivl_disj_un_one(8)
thf(fact_7737_Ici__subset__Ioi__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_ord_atLeast @ A @ A3 ) @ ( set_ord_greaterThan @ A @ B3 ) )
= ( ord_less @ A @ B3 @ A3 ) ) ) ).
% Ici_subset_Ioi_iff
thf(fact_7738_continuous__on__arcosh_H,axiom,
! [A2: set @ real,F3: real > real] :
( ( topolo81223032696312382ous_on @ real @ real @ A2 @ F3 )
=> ( ! [X3: real] :
( ( member @ real @ X3 @ A2 )
=> ( ord_less_eq @ real @ ( one_one @ real ) @ ( F3 @ X3 ) ) )
=> ( topolo81223032696312382ous_on @ real @ real @ A2
@ ^ [X2: real] : ( arcosh @ real @ ( F3 @ X2 ) ) ) ) ) ).
% continuous_on_arcosh'
thf(fact_7739_continuous__on__powr_H,axiom,
! [C: $tType] :
( ( topolo4958980785337419405_space @ C )
=> ! [S2: set @ C,F3: C > real,G2: C > real] :
( ( topolo81223032696312382ous_on @ C @ real @ S2 @ F3 )
=> ( ( topolo81223032696312382ous_on @ C @ real @ S2 @ G2 )
=> ( ! [X3: C] :
( ( member @ C @ X3 @ S2 )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( F3 @ X3 ) )
& ( ( ( F3 @ X3 )
= ( zero_zero @ real ) )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( G2 @ X3 ) ) ) ) )
=> ( topolo81223032696312382ous_on @ C @ real @ S2
@ ^ [X2: C] : ( powr @ real @ ( F3 @ X2 ) @ ( G2 @ X2 ) ) ) ) ) ) ) ).
% continuous_on_powr'
thf(fact_7740_continuous__on__log,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [S2: set @ A,F3: A > real,G2: A > real] :
( ( topolo81223032696312382ous_on @ A @ real @ S2 @ F3 )
=> ( ( topolo81223032696312382ous_on @ A @ real @ S2 @ G2 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ S2 )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( F3 @ X3 ) ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ S2 )
=> ( ( F3 @ X3 )
!= ( one_one @ real ) ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ S2 )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( G2 @ X3 ) ) )
=> ( topolo81223032696312382ous_on @ A @ real @ S2
@ ^ [X2: A] : ( log @ ( F3 @ X2 ) @ ( G2 @ X2 ) ) ) ) ) ) ) ) ) ).
% continuous_on_log
thf(fact_7741_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_7742_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_7743_continuous__on__arccos,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [S2: set @ A,F3: A > real] :
( ( topolo81223032696312382ous_on @ A @ real @ S2 @ F3 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ S2 )
=> ( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ ( F3 @ X3 ) )
& ( ord_less_eq @ real @ ( F3 @ X3 ) @ ( one_one @ real ) ) ) )
=> ( topolo81223032696312382ous_on @ A @ real @ S2
@ ^ [X2: A] : ( arccos @ ( F3 @ X2 ) ) ) ) ) ) ).
% continuous_on_arccos
thf(fact_7744_continuous__on__arcsin,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [S2: set @ A,F3: A > real] :
( ( topolo81223032696312382ous_on @ A @ real @ S2 @ F3 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ S2 )
=> ( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ ( F3 @ X3 ) )
& ( ord_less_eq @ real @ ( F3 @ X3 ) @ ( one_one @ real ) ) ) )
=> ( topolo81223032696312382ous_on @ A @ real @ S2
@ ^ [X2: A] : ( arcsin @ ( F3 @ X2 ) ) ) ) ) ) ).
% continuous_on_arcsin
thf(fact_7745_continuous__on__artanh,axiom,
! [A2: set @ real] :
( ( ord_less_eq @ ( set @ real ) @ A2 @ ( set_or5935395276787703475ssThan @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ ( one_one @ real ) ) )
=> ( topolo81223032696312382ous_on @ real @ real @ A2 @ ( artanh @ real ) ) ) ).
% continuous_on_artanh
thf(fact_7746_ivl__disj__un__singleton_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A] :
( ( sup_sup @ ( set @ A ) @ ( insert @ A @ L @ ( bot_bot @ ( set @ A ) ) ) @ ( set_ord_greaterThan @ A @ L ) )
= ( set_ord_atLeast @ A @ L ) ) ) ).
% ivl_disj_un_singleton(1)
thf(fact_7747_ivl__disj__un__one_I7_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,U2: A] :
( ( ord_less_eq @ A @ L @ U2 )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ L @ U2 ) @ ( set_ord_greaterThan @ A @ U2 ) )
= ( set_ord_atLeast @ A @ L ) ) ) ) ).
% ivl_disj_un_one(7)
thf(fact_7748_ivl__disj__un__one_I6_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,U2: A] :
( ( ord_less @ A @ L @ U2 )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ L @ U2 ) @ ( set_ord_atLeast @ A @ U2 ) )
= ( set_ord_greaterThan @ A @ L ) ) ) ) ).
% ivl_disj_un_one(6)
thf(fact_7749_continuous__on__artanh_H,axiom,
! [A2: set @ real,F3: real > real] :
( ( topolo81223032696312382ous_on @ real @ real @ A2 @ F3 )
=> ( ! [X3: real] :
( ( member @ real @ X3 @ A2 )
=> ( member @ real @ ( F3 @ X3 ) @ ( set_or5935395276787703475ssThan @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ ( one_one @ real ) ) ) )
=> ( topolo81223032696312382ous_on @ real @ real @ A2
@ ^ [X2: real] : ( artanh @ real @ ( F3 @ X2 ) ) ) ) ) ).
% continuous_on_artanh'
thf(fact_7750_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_7751_Rolle__deriv,axiom,
! [A3: real,B3: real,F3: real > real,F7: real > real > real] :
( ( ord_less @ real @ A3 @ B3 )
=> ( ( ( F3 @ A3 )
= ( F3 @ B3 ) )
=> ( ( topolo81223032696312382ous_on @ real @ real @ ( set_or1337092689740270186AtMost @ real @ A3 @ B3 ) @ F3 )
=> ( ! [X3: real] :
( ( ord_less @ real @ A3 @ X3 )
=> ( ( ord_less @ real @ X3 @ B3 )
=> ( has_derivative @ real @ real @ F3 @ ( F7 @ X3 ) @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) ) ) )
=> ? [Z3: real] :
( ( ord_less @ real @ A3 @ Z3 )
& ( ord_less @ real @ Z3 @ B3 )
& ( ( F7 @ Z3 )
= ( ^ [V3: real] : ( zero_zero @ real ) ) ) ) ) ) ) ) ).
% Rolle_deriv
thf(fact_7752_mvt,axiom,
! [A3: real,B3: real,F3: real > real,F7: real > real > real] :
( ( ord_less @ real @ A3 @ B3 )
=> ( ( topolo81223032696312382ous_on @ real @ real @ ( set_or1337092689740270186AtMost @ real @ A3 @ B3 ) @ F3 )
=> ( ! [X3: real] :
( ( ord_less @ real @ A3 @ X3 )
=> ( ( ord_less @ real @ X3 @ B3 )
=> ( has_derivative @ real @ real @ F3 @ ( F7 @ X3 ) @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) ) ) )
=> ~ ! [Xi: real] :
( ( ord_less @ real @ A3 @ Xi )
=> ( ( ord_less @ real @ Xi @ B3 )
=> ( ( minus_minus @ real @ ( F3 @ B3 ) @ ( F3 @ A3 ) )
!= ( F7 @ Xi @ ( minus_minus @ real @ B3 @ A3 ) ) ) ) ) ) ) ) ).
% mvt
thf(fact_7753_continuous__on__of__int__floor,axiom,
! [B: $tType,A: $tType] :
( ( ( archim2362893244070406136eiling @ A )
& ( topolo2564578578187576103pology @ A )
& ( ring_1 @ B )
& ( topolo4958980785337419405_space @ B ) )
=> ( topolo81223032696312382ous_on @ A @ B @ ( minus_minus @ ( set @ A ) @ ( top_top @ ( set @ A ) ) @ ( ring_1_Ints @ A ) )
@ ^ [X2: A] : ( ring_1_of_int @ B @ ( archim6421214686448440834_floor @ A @ X2 ) ) ) ) ).
% continuous_on_of_int_floor
thf(fact_7754_continuous__on__of__int__ceiling,axiom,
! [B: $tType,A: $tType] :
( ( ( archim2362893244070406136eiling @ A )
& ( topolo2564578578187576103pology @ A )
& ( ring_1 @ B )
& ( topolo4958980785337419405_space @ B ) )
=> ( topolo81223032696312382ous_on @ A @ B @ ( minus_minus @ ( set @ A ) @ ( top_top @ ( set @ A ) ) @ ( ring_1_Ints @ A ) )
@ ^ [X2: A] : ( ring_1_of_int @ B @ ( archimedean_ceiling @ A @ X2 ) ) ) ) ).
% continuous_on_of_int_ceiling
thf(fact_7755_continuous__on__Icc__at__leftD,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo1944317154257567458pology @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [A3: A,B3: A,F3: A > B] :
( ( topolo81223032696312382ous_on @ A @ B @ ( set_or1337092689740270186AtMost @ A @ A3 @ B3 ) @ F3 )
=> ( ( ord_less @ A @ A3 @ B3 )
=> ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ ( F3 @ B3 ) ) @ ( topolo174197925503356063within @ A @ B3 @ ( set_ord_lessThan @ A @ B3 ) ) ) ) ) ) ).
% continuous_on_Icc_at_leftD
thf(fact_7756_continuous__on__Icc__at__rightD,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo1944317154257567458pology @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [A3: A,B3: A,F3: A > B] :
( ( topolo81223032696312382ous_on @ A @ B @ ( set_or1337092689740270186AtMost @ A @ A3 @ B3 ) @ F3 )
=> ( ( ord_less @ A @ A3 @ B3 )
=> ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ ( F3 @ A3 ) ) @ ( topolo174197925503356063within @ A @ A3 @ ( set_ord_greaterThan @ A @ A3 ) ) ) ) ) ) ).
% continuous_on_Icc_at_rightD
thf(fact_7757_DERIV__pos__imp__increasing__open,axiom,
! [A3: real,B3: real,F3: real > real] :
( ( ord_less @ real @ A3 @ B3 )
=> ( ! [X3: real] :
( ( ord_less @ real @ A3 @ X3 )
=> ( ( ord_less @ real @ X3 @ B3 )
=> ? [Y5: real] :
( ( has_field_derivative @ real @ F3 @ Y5 @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) )
& ( ord_less @ real @ ( zero_zero @ real ) @ Y5 ) ) ) )
=> ( ( topolo81223032696312382ous_on @ real @ real @ ( set_or1337092689740270186AtMost @ real @ A3 @ B3 ) @ F3 )
=> ( ord_less @ real @ ( F3 @ A3 ) @ ( F3 @ B3 ) ) ) ) ) ).
% DERIV_pos_imp_increasing_open
thf(fact_7758_DERIV__neg__imp__decreasing__open,axiom,
! [A3: real,B3: real,F3: real > real] :
( ( ord_less @ real @ A3 @ B3 )
=> ( ! [X3: real] :
( ( ord_less @ real @ A3 @ X3 )
=> ( ( ord_less @ real @ X3 @ B3 )
=> ? [Y5: real] :
( ( has_field_derivative @ real @ F3 @ Y5 @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) )
& ( ord_less @ real @ Y5 @ ( zero_zero @ real ) ) ) ) )
=> ( ( topolo81223032696312382ous_on @ real @ real @ ( set_or1337092689740270186AtMost @ real @ A3 @ B3 ) @ F3 )
=> ( ord_less @ real @ ( F3 @ B3 ) @ ( F3 @ A3 ) ) ) ) ) ).
% DERIV_neg_imp_decreasing_open
thf(fact_7759_DERIV__isconst__end,axiom,
! [A3: real,B3: real,F3: real > real] :
( ( ord_less @ real @ A3 @ B3 )
=> ( ( topolo81223032696312382ous_on @ real @ real @ ( set_or1337092689740270186AtMost @ real @ A3 @ B3 ) @ F3 )
=> ( ! [X3: real] :
( ( ord_less @ real @ A3 @ X3 )
=> ( ( ord_less @ real @ X3 @ B3 )
=> ( has_field_derivative @ real @ F3 @ ( zero_zero @ real ) @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) ) ) )
=> ( ( F3 @ B3 )
= ( F3 @ A3 ) ) ) ) ) ).
% DERIV_isconst_end
thf(fact_7760_complete__interval,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [A3: A,B3: A,P: A > $o] :
( ( ord_less @ A @ A3 @ B3 )
=> ( ( P @ A3 )
=> ( ~ ( P @ B3 )
=> ? [C5: A] :
( ( ord_less_eq @ A @ A3 @ C5 )
& ( ord_less_eq @ A @ C5 @ B3 )
& ! [X6: A] :
( ( ( ord_less_eq @ A @ A3 @ X6 )
& ( ord_less @ A @ X6 @ C5 ) )
=> ( P @ X6 ) )
& ! [D5: A] :
( ! [X3: A] :
( ( ( ord_less_eq @ A @ A3 @ X3 )
& ( ord_less @ A @ X3 @ D5 ) )
=> ( P @ X3 ) )
=> ( ord_less_eq @ A @ D5 @ C5 ) ) ) ) ) ) ) ).
% complete_interval
thf(fact_7761_DERIV__isconst2,axiom,
! [A3: real,B3: real,F3: real > real,X: real] :
( ( ord_less @ real @ A3 @ B3 )
=> ( ( topolo81223032696312382ous_on @ real @ real @ ( set_or1337092689740270186AtMost @ real @ A3 @ B3 ) @ F3 )
=> ( ! [X3: real] :
( ( ord_less @ real @ A3 @ X3 )
=> ( ( ord_less @ real @ X3 @ B3 )
=> ( has_field_derivative @ real @ F3 @ ( zero_zero @ real ) @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) ) ) )
=> ( ( ord_less_eq @ real @ A3 @ X )
=> ( ( ord_less_eq @ real @ X @ B3 )
=> ( ( F3 @ X )
= ( F3 @ A3 ) ) ) ) ) ) ) ).
% DERIV_isconst2
thf(fact_7762_continuous__on__IccI,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo1944317154257567458pology @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [F3: A > B,A3: A,B3: A] :
( ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ ( F3 @ A3 ) ) @ ( topolo174197925503356063within @ A @ A3 @ ( set_ord_greaterThan @ A @ A3 ) ) )
=> ( ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ ( F3 @ B3 ) ) @ ( topolo174197925503356063within @ A @ B3 @ ( set_ord_lessThan @ A @ B3 ) ) )
=> ( ! [X3: A] :
( ( ord_less @ A @ A3 @ X3 )
=> ( ( ord_less @ A @ X3 @ B3 )
=> ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ ( F3 @ X3 ) ) @ ( topolo174197925503356063within @ A @ X3 @ ( top_top @ ( set @ A ) ) ) ) ) )
=> ( ( ord_less @ A @ A3 @ B3 )
=> ( topolo81223032696312382ous_on @ A @ B @ ( set_or1337092689740270186AtMost @ A @ A3 @ B3 ) @ F3 ) ) ) ) ) ) ).
% continuous_on_IccI
thf(fact_7763_Rolle,axiom,
! [A3: real,B3: real,F3: real > real] :
( ( ord_less @ real @ A3 @ B3 )
=> ( ( ( F3 @ A3 )
= ( F3 @ B3 ) )
=> ( ( topolo81223032696312382ous_on @ real @ real @ ( set_or1337092689740270186AtMost @ real @ A3 @ B3 ) @ F3 )
=> ( ! [X3: real] :
( ( ord_less @ real @ A3 @ X3 )
=> ( ( ord_less @ real @ X3 @ B3 )
=> ( differentiable @ real @ real @ F3 @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) ) ) )
=> ? [Z3: real] :
( ( ord_less @ real @ A3 @ Z3 )
& ( ord_less @ real @ Z3 @ B3 )
& ( has_field_derivative @ real @ F3 @ ( zero_zero @ real ) @ ( topolo174197925503356063within @ real @ Z3 @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ) ) ).
% Rolle
thf(fact_7764_int__of__integer__code,axiom,
( code_int_of_integer
= ( ^ [K3: code_integer] :
( if @ int @ ( ord_less @ code_integer @ K3 @ ( zero_zero @ code_integer ) ) @ ( uminus_uminus @ int @ ( code_int_of_integer @ ( uminus_uminus @ code_integer @ K3 ) ) )
@ ( if @ int
@ ( K3
= ( zero_zero @ code_integer ) )
@ ( zero_zero @ int )
@ ( product_case_prod @ code_integer @ code_integer @ int
@ ^ [L2: code_integer,J3: code_integer] :
( if @ int
@ ( J3
= ( zero_zero @ code_integer ) )
@ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( code_int_of_integer @ L2 ) )
@ ( plus_plus @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( code_int_of_integer @ L2 ) ) @ ( one_one @ int ) ) )
@ ( code_divmod_integer @ K3 @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% int_of_integer_code
thf(fact_7765_csqrt_Osimps_I1_J,axiom,
! [Z: complex] :
( ( re @ ( csqrt @ Z ) )
= ( sqrt @ ( divide_divide @ real @ ( plus_plus @ real @ ( real_V7770717601297561774m_norm @ complex @ Z ) @ ( re @ Z ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ).
% csqrt.simps(1)
thf(fact_7766_int__of__integer__of__nat,axiom,
! [N: nat] :
( ( code_int_of_integer @ ( semiring_1_of_nat @ code_integer @ N ) )
= ( semiring_1_of_nat @ int @ N ) ) ).
% int_of_integer_of_nat
thf(fact_7767_complex__Re__of__nat,axiom,
! [N: nat] :
( ( re @ ( semiring_1_of_nat @ complex @ N ) )
= ( semiring_1_of_nat @ real @ N ) ) ).
% complex_Re_of_nat
thf(fact_7768_continuous__on__of__real__o__iff,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [S: set @ A,G2: A > real] :
( ( topolo81223032696312382ous_on @ A @ complex @ S
@ ^ [X2: A] : ( real_Vector_of_real @ complex @ ( G2 @ X2 ) ) )
= ( topolo81223032696312382ous_on @ A @ real @ S @ G2 ) ) ) ).
% continuous_on_of_real_o_iff
thf(fact_7769_zero__integer_Orep__eq,axiom,
( ( code_int_of_integer @ ( zero_zero @ code_integer ) )
= ( zero_zero @ int ) ) ).
% zero_integer.rep_eq
thf(fact_7770_int__of__integer__numeral,axiom,
! [K: num] :
( ( code_int_of_integer @ ( numeral_numeral @ code_integer @ K ) )
= ( numeral_numeral @ int @ K ) ) ).
% int_of_integer_numeral
thf(fact_7771_one__integer_Orep__eq,axiom,
( ( code_int_of_integer @ ( one_one @ code_integer ) )
= ( one_one @ int ) ) ).
% one_integer.rep_eq
thf(fact_7772_complex__Re__numeral,axiom,
! [V2: num] :
( ( re @ ( numeral_numeral @ complex @ V2 ) )
= ( numeral_numeral @ real @ V2 ) ) ).
% complex_Re_numeral
thf(fact_7773_Re__sum,axiom,
! [A: $tType,F3: A > complex,S2: set @ A] :
( ( re @ ( groups7311177749621191930dd_sum @ A @ complex @ F3 @ S2 ) )
= ( groups7311177749621191930dd_sum @ A @ real
@ ^ [X2: A] : ( re @ ( F3 @ X2 ) )
@ S2 ) ) ).
% Re_sum
thf(fact_7774_Re__divide__of__nat,axiom,
! [Z: complex,N: nat] :
( ( re @ ( divide_divide @ complex @ Z @ ( semiring_1_of_nat @ complex @ N ) ) )
= ( divide_divide @ real @ ( re @ Z ) @ ( semiring_1_of_nat @ real @ N ) ) ) ).
% Re_divide_of_nat
thf(fact_7775_Re__divide__numeral,axiom,
! [Z: complex,W: num] :
( ( re @ ( divide_divide @ complex @ Z @ ( numeral_numeral @ complex @ W ) ) )
= ( divide_divide @ real @ ( re @ Z ) @ ( numeral_numeral @ real @ W ) ) ) ).
% Re_divide_numeral
thf(fact_7776_cos__Arg__i__mult__zero,axiom,
! [Y: complex] :
( ( Y
!= ( zero_zero @ complex ) )
=> ( ( ( re @ Y )
= ( zero_zero @ real ) )
=> ( ( cos @ real @ ( arg @ Y ) )
= ( zero_zero @ real ) ) ) ) ).
% cos_Arg_i_mult_zero
thf(fact_7777_continuous__on__cis,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [A2: set @ A,F3: A > real] :
( ( topolo81223032696312382ous_on @ A @ real @ A2 @ F3 )
=> ( topolo81223032696312382ous_on @ A @ complex @ A2
@ ^ [X2: A] : ( cis @ ( F3 @ X2 ) ) ) ) ) ).
% continuous_on_cis
thf(fact_7778_continuous__on__Re,axiom,
! [C: $tType] :
( ( topolo4958980785337419405_space @ C )
=> ! [S2: set @ C,G2: C > complex] :
( ( topolo81223032696312382ous_on @ C @ complex @ S2 @ G2 )
=> ( topolo81223032696312382ous_on @ C @ real @ S2
@ ^ [X2: C] : ( re @ ( G2 @ X2 ) ) ) ) ) ).
% continuous_on_Re
thf(fact_7779_summable__Re,axiom,
! [F3: nat > complex] :
( ( summable @ complex @ F3 )
=> ( summable @ real
@ ^ [X2: nat] : ( re @ ( F3 @ X2 ) ) ) ) ).
% summable_Re
thf(fact_7780_continuous__Re,axiom,
! [C: $tType] :
( ( topological_t2_space @ C )
=> ! [F4: filter @ C,G2: C > complex] :
( ( topolo3448309680560233919inuous @ C @ complex @ F4 @ G2 )
=> ( topolo3448309680560233919inuous @ C @ real @ F4
@ ^ [X2: C] : ( re @ ( G2 @ X2 ) ) ) ) ) ).
% continuous_Re
thf(fact_7781_zero__complex_Osimps_I1_J,axiom,
( ( re @ ( zero_zero @ complex ) )
= ( zero_zero @ real ) ) ).
% zero_complex.simps(1)
thf(fact_7782_one__complex_Osimps_I1_J,axiom,
( ( re @ ( one_one @ complex ) )
= ( one_one @ real ) ) ).
% one_complex.simps(1)
thf(fact_7783_imaginary__unit_Osimps_I1_J,axiom,
( ( re @ imaginary_unit )
= ( zero_zero @ real ) ) ).
% imaginary_unit.simps(1)
thf(fact_7784_sums__Re,axiom,
! [X5: nat > complex,A3: complex] :
( ( sums @ complex @ X5 @ A3 )
=> ( sums @ real
@ ^ [N4: nat] : ( re @ ( X5 @ N4 ) )
@ ( re @ A3 ) ) ) ).
% sums_Re
thf(fact_7785_Cauchy__Re,axiom,
! [X5: nat > complex] :
( ( topolo3814608138187158403Cauchy @ complex @ X5 )
=> ( topolo3814608138187158403Cauchy @ real
@ ^ [N4: nat] : ( re @ ( X5 @ N4 ) ) ) ) ).
% Cauchy_Re
thf(fact_7786_has__derivative__Re,axiom,
! [C: $tType] :
( ( real_V822414075346904944vector @ C )
=> ! [G2: C > complex,G5: C > complex,F4: filter @ C] :
( ( has_derivative @ C @ complex @ G2 @ G5 @ F4 )
=> ( has_derivative @ C @ real
@ ^ [X2: C] : ( re @ ( G2 @ X2 ) )
@ ^ [X2: C] : ( re @ ( G5 @ X2 ) )
@ F4 ) ) ) ).
% has_derivative_Re
thf(fact_7787_less__integer_Orep__eq,axiom,
( ( ord_less @ code_integer )
= ( ^ [X2: code_integer,Xa4: code_integer] : ( ord_less @ int @ ( code_int_of_integer @ X2 ) @ ( code_int_of_integer @ Xa4 ) ) ) ) ).
% less_integer.rep_eq
thf(fact_7788_integer__less__iff,axiom,
( ( ord_less @ code_integer )
= ( ^ [K3: code_integer,L2: code_integer] : ( ord_less @ int @ ( code_int_of_integer @ K3 ) @ ( code_int_of_integer @ L2 ) ) ) ) ).
% integer_less_iff
thf(fact_7789_int__of__integer__less__iff,axiom,
! [X: code_integer,Y: code_integer] :
( ( ord_less @ int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Y ) )
= ( ord_less @ code_integer @ X @ Y ) ) ).
% int_of_integer_less_iff
thf(fact_7790_Re__csqrt,axiom,
! [Z: complex] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( re @ ( csqrt @ Z ) ) ) ).
% Re_csqrt
thf(fact_7791_isCont__Re,axiom,
! [C: $tType] :
( ( topological_t2_space @ C )
=> ! [A3: C,G2: C > complex] :
( ( topolo3448309680560233919inuous @ C @ complex @ ( topolo174197925503356063within @ C @ A3 @ ( top_top @ ( set @ C ) ) ) @ G2 )
=> ( topolo3448309680560233919inuous @ C @ real @ ( topolo174197925503356063within @ C @ A3 @ ( top_top @ ( set @ C ) ) )
@ ^ [X2: C] : ( re @ ( G2 @ X2 ) ) ) ) ) ).
% isCont_Re
thf(fact_7792_cmod__plus__Re__le__0__iff,axiom,
! [Z: complex] :
( ( ord_less_eq @ real @ ( plus_plus @ real @ ( real_V7770717601297561774m_norm @ complex @ Z ) @ ( re @ Z ) ) @ ( zero_zero @ real ) )
= ( ( re @ Z )
= ( uminus_uminus @ real @ ( real_V7770717601297561774m_norm @ complex @ Z ) ) ) ) ).
% cmod_plus_Re_le_0_iff
thf(fact_7793_bin__last__integer_Orep__eq,axiom,
( bits_b8758750999018896077nteger
= ( ^ [X2: code_integer] :
~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( code_int_of_integer @ X2 ) ) ) ) ).
% bin_last_integer.rep_eq
thf(fact_7794_bin__rest__integer_Orep__eq,axiom,
! [X: code_integer] :
( ( code_int_of_integer @ ( bits_b2549910563261871055nteger @ X ) )
= ( divide_divide @ int @ ( code_int_of_integer @ X ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ).
% bin_rest_integer.rep_eq
thf(fact_7795_tendsto__Re,axiom,
! [C: $tType,G2: C > complex,A3: complex,F4: filter @ C] :
( ( filterlim @ C @ complex @ G2 @ ( topolo7230453075368039082e_nhds @ complex @ A3 ) @ F4 )
=> ( filterlim @ C @ real
@ ^ [X2: C] : ( re @ ( G2 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( re @ A3 ) )
@ F4 ) ) ).
% tendsto_Re
thf(fact_7796_cos__n__Re__cis__pow__n,axiom,
! [N: nat,A3: real] :
( ( cos @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ A3 ) )
= ( re @ ( power_power @ complex @ ( cis @ A3 ) @ N ) ) ) ).
% cos_n_Re_cis_pow_n
thf(fact_7797_Bit__integer_Orep__eq,axiom,
! [X: code_integer,Xa: $o] :
( ( code_int_of_integer @ ( bits_Bit_integer @ X @ Xa ) )
= ( plus_plus @ int @ ( zero_neq_one_of_bool @ int @ Xa ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( code_int_of_integer @ X ) ) ) ) ).
% Bit_integer.rep_eq
thf(fact_7798_num__of__integer__code,axiom,
( code_num_of_integer
= ( ^ [K3: code_integer] :
( if @ num @ ( ord_less_eq @ code_integer @ K3 @ ( one_one @ code_integer ) ) @ one2
@ ( product_case_prod @ code_integer @ code_integer @ num
@ ^ [L2: code_integer,J3: code_integer] :
( if @ num
@ ( J3
= ( zero_zero @ code_integer ) )
@ ( plus_plus @ num @ ( code_num_of_integer @ L2 ) @ ( code_num_of_integer @ L2 ) )
@ ( plus_plus @ num @ ( plus_plus @ num @ ( code_num_of_integer @ L2 ) @ ( code_num_of_integer @ L2 ) ) @ one2 ) )
@ ( code_divmod_integer @ K3 @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% num_of_integer_code
thf(fact_7799_csqrt_Ocode,axiom,
( csqrt
= ( ^ [Z4: complex] :
( complex2 @ ( sqrt @ ( divide_divide @ real @ ( plus_plus @ real @ ( real_V7770717601297561774m_norm @ complex @ Z4 ) @ ( re @ Z4 ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
@ ( times_times @ real
@ ( if @ real
@ ( ( im @ Z4 )
= ( zero_zero @ real ) )
@ ( one_one @ real )
@ ( sgn_sgn @ real @ ( im @ Z4 ) ) )
@ ( sqrt @ ( divide_divide @ real @ ( minus_minus @ real @ ( real_V7770717601297561774m_norm @ complex @ Z4 ) @ ( re @ Z4 ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% csqrt.code
thf(fact_7800_complex__Im__fact,axiom,
! [N: nat] :
( ( im @ ( semiring_char_0_fact @ complex @ N ) )
= ( zero_zero @ real ) ) ).
% complex_Im_fact
thf(fact_7801_complex__Im__of__int,axiom,
! [Z: int] :
( ( im @ ( ring_1_of_int @ complex @ Z ) )
= ( zero_zero @ real ) ) ).
% complex_Im_of_int
thf(fact_7802_Im__complex__of__real,axiom,
! [Z: real] :
( ( im @ ( real_Vector_of_real @ complex @ Z ) )
= ( zero_zero @ real ) ) ).
% Im_complex_of_real
thf(fact_7803_Im__power__real,axiom,
! [X: complex,N: nat] :
( ( ( im @ X )
= ( zero_zero @ real ) )
=> ( ( im @ ( power_power @ complex @ X @ N ) )
= ( zero_zero @ real ) ) ) ).
% Im_power_real
thf(fact_7804_complex__Im__numeral,axiom,
! [V2: num] :
( ( im @ ( numeral_numeral @ complex @ V2 ) )
= ( zero_zero @ real ) ) ).
% complex_Im_numeral
thf(fact_7805_complex__Im__of__nat,axiom,
! [N: nat] :
( ( im @ ( semiring_1_of_nat @ complex @ N ) )
= ( zero_zero @ real ) ) ).
% complex_Im_of_nat
thf(fact_7806_Im__sum,axiom,
! [A: $tType,F3: A > complex,S2: set @ A] :
( ( im @ ( groups7311177749621191930dd_sum @ A @ complex @ F3 @ S2 ) )
= ( groups7311177749621191930dd_sum @ A @ real
@ ^ [X2: A] : ( im @ ( F3 @ X2 ) )
@ S2 ) ) ).
% Im_sum
thf(fact_7807_Re__power__real,axiom,
! [X: complex,N: nat] :
( ( ( im @ X )
= ( zero_zero @ real ) )
=> ( ( re @ ( power_power @ complex @ X @ N ) )
= ( power_power @ real @ ( re @ X ) @ N ) ) ) ).
% Re_power_real
thf(fact_7808_Im__divide__numeral,axiom,
! [Z: complex,W: num] :
( ( im @ ( divide_divide @ complex @ Z @ ( numeral_numeral @ complex @ W ) ) )
= ( divide_divide @ real @ ( im @ Z ) @ ( numeral_numeral @ real @ W ) ) ) ).
% Im_divide_numeral
thf(fact_7809_Im__divide__of__nat,axiom,
! [Z: complex,N: nat] :
( ( im @ ( divide_divide @ complex @ Z @ ( semiring_1_of_nat @ complex @ N ) ) )
= ( divide_divide @ real @ ( im @ Z ) @ ( semiring_1_of_nat @ real @ N ) ) ) ).
% Im_divide_of_nat
thf(fact_7810_csqrt__of__real__nonneg,axiom,
! [X: complex] :
( ( ( im @ X )
= ( zero_zero @ real ) )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( re @ X ) )
=> ( ( csqrt @ X )
= ( real_Vector_of_real @ complex @ ( sqrt @ ( re @ X ) ) ) ) ) ) ).
% csqrt_of_real_nonneg
thf(fact_7811_csqrt__minus,axiom,
! [X: complex] :
( ( ( ord_less @ real @ ( im @ X ) @ ( zero_zero @ real ) )
| ( ( ( im @ X )
= ( zero_zero @ real ) )
& ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( re @ X ) ) ) )
=> ( ( csqrt @ ( uminus_uminus @ complex @ X ) )
= ( times_times @ complex @ imaginary_unit @ ( csqrt @ X ) ) ) ) ).
% csqrt_minus
thf(fact_7812_csqrt__of__real__nonpos,axiom,
! [X: complex] :
( ( ( im @ X )
= ( zero_zero @ real ) )
=> ( ( ord_less_eq @ real @ ( re @ X ) @ ( zero_zero @ real ) )
=> ( ( csqrt @ X )
= ( times_times @ complex @ imaginary_unit @ ( real_Vector_of_real @ complex @ ( sqrt @ ( abs_abs @ real @ ( re @ X ) ) ) ) ) ) ) ) ).
% csqrt_of_real_nonpos
thf(fact_7813_continuous__on__Im,axiom,
! [C: $tType] :
( ( topolo4958980785337419405_space @ C )
=> ! [S2: set @ C,G2: C > complex] :
( ( topolo81223032696312382ous_on @ C @ complex @ S2 @ G2 )
=> ( topolo81223032696312382ous_on @ C @ real @ S2
@ ^ [X2: C] : ( im @ ( G2 @ X2 ) ) ) ) ) ).
% continuous_on_Im
thf(fact_7814_continuous__complex__iff,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ( ( topolo3448309680560233919inuous @ A @ complex )
= ( ^ [F8: filter @ A,F2: A > complex] :
( ( topolo3448309680560233919inuous @ A @ real @ F8
@ ^ [X2: A] : ( re @ ( F2 @ X2 ) ) )
& ( topolo3448309680560233919inuous @ A @ real @ F8
@ ^ [X2: A] : ( im @ ( F2 @ X2 ) ) ) ) ) ) ) ).
% continuous_complex_iff
thf(fact_7815_complex__is__Int__iff,axiom,
! [Z: complex] :
( ( member @ complex @ Z @ ( ring_1_Ints @ complex ) )
= ( ( ( im @ Z )
= ( zero_zero @ real ) )
& ? [I4: int] :
( ( re @ Z )
= ( ring_1_of_int @ real @ I4 ) ) ) ) ).
% complex_is_Int_iff
thf(fact_7816_sums__complex__iff,axiom,
( ( sums @ complex )
= ( ^ [F2: nat > complex,X2: complex] :
( ( sums @ real
@ ^ [Y2: nat] : ( re @ ( F2 @ Y2 ) )
@ ( re @ X2 ) )
& ( sums @ real
@ ^ [Y2: nat] : ( im @ ( F2 @ Y2 ) )
@ ( im @ X2 ) ) ) ) ) ).
% sums_complex_iff
thf(fact_7817_has__derivative__Im,axiom,
! [C: $tType] :
( ( real_V822414075346904944vector @ C )
=> ! [G2: C > complex,G5: C > complex,F4: filter @ C] :
( ( has_derivative @ C @ complex @ G2 @ G5 @ F4 )
=> ( has_derivative @ C @ real
@ ^ [X2: C] : ( im @ ( G2 @ X2 ) )
@ ^ [X2: C] : ( im @ ( G5 @ X2 ) )
@ F4 ) ) ) ).
% has_derivative_Im
thf(fact_7818_sums__Im,axiom,
! [X5: nat > complex,A3: complex] :
( ( sums @ complex @ X5 @ A3 )
=> ( sums @ real
@ ^ [N4: nat] : ( im @ ( X5 @ N4 ) )
@ ( im @ A3 ) ) ) ).
% sums_Im
thf(fact_7819_Cauchy__Im,axiom,
! [X5: nat > complex] :
( ( topolo3814608138187158403Cauchy @ complex @ X5 )
=> ( topolo3814608138187158403Cauchy @ real
@ ^ [N4: nat] : ( im @ ( X5 @ N4 ) ) ) ) ).
% Cauchy_Im
thf(fact_7820_zero__complex_Osimps_I2_J,axiom,
( ( im @ ( zero_zero @ complex ) )
= ( zero_zero @ real ) ) ).
% zero_complex.simps(2)
thf(fact_7821_imaginary__unit_Osimps_I2_J,axiom,
( ( im @ imaginary_unit )
= ( one_one @ real ) ) ).
% imaginary_unit.simps(2)
thf(fact_7822_one__complex_Osimps_I2_J,axiom,
( ( im @ ( one_one @ complex ) )
= ( zero_zero @ real ) ) ).
% one_complex.simps(2)
thf(fact_7823_continuous__Im,axiom,
! [C: $tType] :
( ( topological_t2_space @ C )
=> ! [F4: filter @ C,G2: C > complex] :
( ( topolo3448309680560233919inuous @ C @ complex @ F4 @ G2 )
=> ( topolo3448309680560233919inuous @ C @ real @ F4
@ ^ [X2: C] : ( im @ ( G2 @ X2 ) ) ) ) ) ).
% continuous_Im
thf(fact_7824_summable__Im,axiom,
! [F3: nat > complex] :
( ( summable @ complex @ F3 )
=> ( summable @ real
@ ^ [X2: nat] : ( im @ ( F3 @ X2 ) ) ) ) ).
% summable_Im
thf(fact_7825_isCont__Im,axiom,
! [C: $tType] :
( ( topological_t2_space @ C )
=> ! [A3: C,G2: C > complex] :
( ( topolo3448309680560233919inuous @ C @ complex @ ( topolo174197925503356063within @ C @ A3 @ ( top_top @ ( set @ C ) ) ) @ G2 )
=> ( topolo3448309680560233919inuous @ C @ real @ ( topolo174197925503356063within @ C @ A3 @ ( top_top @ ( set @ C ) ) )
@ ^ [X2: C] : ( im @ ( G2 @ X2 ) ) ) ) ) ).
% isCont_Im
thf(fact_7826_summable__complex__iff,axiom,
( ( summable @ complex )
= ( ^ [F2: nat > complex] :
( ( summable @ real
@ ^ [X2: nat] : ( re @ ( F2 @ X2 ) ) )
& ( summable @ real
@ ^ [X2: nat] : ( im @ ( F2 @ X2 ) ) ) ) ) ) ).
% summable_complex_iff
thf(fact_7827_Im__eq__0,axiom,
! [Z: complex] :
( ( ( abs_abs @ real @ ( re @ Z ) )
= ( real_V7770717601297561774m_norm @ complex @ Z ) )
=> ( ( im @ Z )
= ( zero_zero @ real ) ) ) ).
% Im_eq_0
thf(fact_7828_cmod__eq__Im,axiom,
! [Z: complex] :
( ( ( re @ Z )
= ( zero_zero @ real ) )
=> ( ( real_V7770717601297561774m_norm @ complex @ Z )
= ( abs_abs @ real @ ( im @ Z ) ) ) ) ).
% cmod_eq_Im
thf(fact_7829_cmod__eq__Re,axiom,
! [Z: complex] :
( ( ( im @ Z )
= ( zero_zero @ real ) )
=> ( ( real_V7770717601297561774m_norm @ complex @ Z )
= ( abs_abs @ real @ ( re @ Z ) ) ) ) ).
% cmod_eq_Re
thf(fact_7830_csqrt__principal,axiom,
! [Z: complex] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( re @ ( csqrt @ Z ) ) )
| ( ( ( re @ ( csqrt @ Z ) )
= ( zero_zero @ real ) )
& ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( im @ ( csqrt @ Z ) ) ) ) ) ).
% csqrt_principal
thf(fact_7831_tendsto__Im,axiom,
! [C: $tType,G2: C > complex,A3: complex,F4: filter @ C] :
( ( filterlim @ C @ complex @ G2 @ ( topolo7230453075368039082e_nhds @ complex @ A3 ) @ F4 )
=> ( filterlim @ C @ real
@ ^ [X2: C] : ( im @ ( G2 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( im @ A3 ) )
@ F4 ) ) ).
% tendsto_Im
thf(fact_7832_sin__n__Im__cis__pow__n,axiom,
! [N: nat,A3: real] :
( ( sin @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ A3 ) )
= ( im @ ( power_power @ complex @ ( cis @ A3 ) @ N ) ) ) ).
% sin_n_Im_cis_pow_n
thf(fact_7833_tendsto__complex__iff,axiom,
! [A: $tType,F3: A > complex,X: complex,F4: filter @ A] :
( ( filterlim @ A @ complex @ F3 @ ( topolo7230453075368039082e_nhds @ complex @ X ) @ F4 )
= ( ( filterlim @ A @ real
@ ^ [X2: A] : ( re @ ( F3 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( re @ X ) )
@ F4 )
& ( filterlim @ A @ real
@ ^ [X2: A] : ( im @ ( F3 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( im @ X ) )
@ F4 ) ) ) ).
% tendsto_complex_iff
thf(fact_7834_cmod__power2,axiom,
! [Z: complex] :
( ( power_power @ real @ ( real_V7770717601297561774m_norm @ complex @ Z ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( plus_plus @ real @ ( power_power @ real @ ( re @ Z ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ ( im @ Z ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% cmod_power2
thf(fact_7835_Im__power2,axiom,
! [X: complex] :
( ( im @ ( power_power @ complex @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( times_times @ real @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( re @ X ) ) @ ( im @ X ) ) ) ).
% Im_power2
thf(fact_7836_Re__power2,axiom,
! [X: complex] :
( ( re @ ( power_power @ complex @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( minus_minus @ real @ ( power_power @ real @ ( re @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ ( im @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% Re_power2
thf(fact_7837_complex__eq__0,axiom,
! [Z: complex] :
( ( Z
= ( zero_zero @ complex ) )
= ( ( plus_plus @ real @ ( power_power @ real @ ( re @ Z ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ ( im @ Z ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( zero_zero @ real ) ) ) ).
% complex_eq_0
thf(fact_7838_norm__complex__def,axiom,
( ( real_V7770717601297561774m_norm @ complex )
= ( ^ [Z4: complex] : ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ ( re @ Z4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ ( im @ Z4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% norm_complex_def
thf(fact_7839_inverse__complex_Osimps_I1_J,axiom,
! [X: complex] :
( ( re @ ( inverse_inverse @ complex @ X ) )
= ( divide_divide @ real @ ( re @ X ) @ ( plus_plus @ real @ ( power_power @ real @ ( re @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ ( im @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% inverse_complex.simps(1)
thf(fact_7840_complex__neq__0,axiom,
! [Z: complex] :
( ( Z
!= ( zero_zero @ complex ) )
= ( ord_less @ real @ ( zero_zero @ real ) @ ( plus_plus @ real @ ( power_power @ real @ ( re @ Z ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ ( im @ Z ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% complex_neq_0
thf(fact_7841_Re__divide,axiom,
! [X: complex,Y: complex] :
( ( re @ ( divide_divide @ complex @ X @ Y ) )
= ( divide_divide @ real @ ( plus_plus @ real @ ( times_times @ real @ ( re @ X ) @ ( re @ Y ) ) @ ( times_times @ real @ ( im @ X ) @ ( im @ Y ) ) ) @ ( plus_plus @ real @ ( power_power @ real @ ( re @ Y ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ ( im @ Y ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% Re_divide
thf(fact_7842_csqrt__unique,axiom,
! [W: complex,Z: complex] :
( ( ( power_power @ complex @ W @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= Z )
=> ( ( ( ord_less @ real @ ( zero_zero @ real ) @ ( re @ W ) )
| ( ( ( re @ W )
= ( zero_zero @ real ) )
& ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( im @ W ) ) ) )
=> ( ( csqrt @ Z )
= W ) ) ) ).
% csqrt_unique
thf(fact_7843_csqrt__square,axiom,
! [B3: complex] :
( ( ( ord_less @ real @ ( zero_zero @ real ) @ ( re @ B3 ) )
| ( ( ( re @ B3 )
= ( zero_zero @ real ) )
& ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( im @ B3 ) ) ) )
=> ( ( csqrt @ ( power_power @ complex @ B3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= B3 ) ) ).
% csqrt_square
thf(fact_7844_inverse__complex_Osimps_I2_J,axiom,
! [X: complex] :
( ( im @ ( inverse_inverse @ complex @ X ) )
= ( divide_divide @ real @ ( uminus_uminus @ real @ ( im @ X ) ) @ ( plus_plus @ real @ ( power_power @ real @ ( re @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ ( im @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% inverse_complex.simps(2)
thf(fact_7845_Im__divide,axiom,
! [X: complex,Y: complex] :
( ( im @ ( divide_divide @ complex @ X @ Y ) )
= ( divide_divide @ real @ ( minus_minus @ real @ ( times_times @ real @ ( im @ X ) @ ( re @ Y ) ) @ ( times_times @ real @ ( re @ X ) @ ( im @ Y ) ) ) @ ( plus_plus @ real @ ( power_power @ real @ ( re @ Y ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ ( im @ Y ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% Im_divide
thf(fact_7846_complex__abs__le__norm,axiom,
! [Z: complex] : ( ord_less_eq @ real @ ( plus_plus @ real @ ( abs_abs @ real @ ( re @ Z ) ) @ ( abs_abs @ real @ ( im @ Z ) ) ) @ ( times_times @ real @ ( sqrt @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( real_V7770717601297561774m_norm @ complex @ Z ) ) ) ).
% complex_abs_le_norm
thf(fact_7847_complex__unit__circle,axiom,
! [Z: complex] :
( ( Z
!= ( zero_zero @ complex ) )
=> ( ( plus_plus @ real @ ( power_power @ real @ ( divide_divide @ real @ ( re @ Z ) @ ( real_V7770717601297561774m_norm @ complex @ Z ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ ( divide_divide @ real @ ( im @ Z ) @ ( real_V7770717601297561774m_norm @ complex @ Z ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( one_one @ real ) ) ) ).
% complex_unit_circle
thf(fact_7848_inverse__complex_Ocode,axiom,
( ( inverse_inverse @ complex )
= ( ^ [X2: complex] : ( complex2 @ ( divide_divide @ real @ ( re @ X2 ) @ ( plus_plus @ real @ ( power_power @ real @ ( re @ X2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ ( im @ X2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( divide_divide @ real @ ( uminus_uminus @ real @ ( im @ X2 ) ) @ ( plus_plus @ real @ ( power_power @ real @ ( re @ X2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ ( im @ X2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% inverse_complex.code
thf(fact_7849_Complex__divide,axiom,
( ( divide_divide @ complex )
= ( ^ [X2: complex,Y2: complex] : ( complex2 @ ( divide_divide @ real @ ( plus_plus @ real @ ( times_times @ real @ ( re @ X2 ) @ ( re @ Y2 ) ) @ ( times_times @ real @ ( im @ X2 ) @ ( im @ Y2 ) ) ) @ ( plus_plus @ real @ ( power_power @ real @ ( re @ Y2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ ( im @ Y2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( divide_divide @ real @ ( minus_minus @ real @ ( times_times @ real @ ( im @ X2 ) @ ( re @ Y2 ) ) @ ( times_times @ real @ ( re @ X2 ) @ ( im @ Y2 ) ) ) @ ( plus_plus @ real @ ( power_power @ real @ ( re @ Y2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ ( im @ Y2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% Complex_divide
thf(fact_7850_csqrt_Osimps_I2_J,axiom,
! [Z: complex] :
( ( im @ ( csqrt @ Z ) )
= ( times_times @ real
@ ( if @ real
@ ( ( im @ Z )
= ( zero_zero @ real ) )
@ ( one_one @ real )
@ ( sgn_sgn @ real @ ( im @ Z ) ) )
@ ( sqrt @ ( divide_divide @ real @ ( minus_minus @ real @ ( real_V7770717601297561774m_norm @ complex @ Z ) @ ( re @ Z ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ).
% csqrt.simps(2)
thf(fact_7851_Im__Reals__divide,axiom,
! [R3: complex,Z: complex] :
( ( member @ complex @ R3 @ ( real_Vector_Reals @ complex ) )
=> ( ( im @ ( divide_divide @ complex @ R3 @ Z ) )
= ( divide_divide @ real @ ( times_times @ real @ ( uminus_uminus @ real @ ( re @ R3 ) ) @ ( im @ Z ) ) @ ( power_power @ real @ ( real_V7770717601297561774m_norm @ complex @ Z ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% Im_Reals_divide
thf(fact_7852_nat__of__integer__code,axiom,
( code_nat_of_integer
= ( ^ [K3: code_integer] :
( if @ nat @ ( ord_less_eq @ code_integer @ K3 @ ( zero_zero @ code_integer ) ) @ ( zero_zero @ nat )
@ ( product_case_prod @ code_integer @ code_integer @ nat
@ ^ [L2: code_integer,J3: code_integer] :
( if @ nat
@ ( J3
= ( zero_zero @ code_integer ) )
@ ( plus_plus @ nat @ ( code_nat_of_integer @ L2 ) @ ( code_nat_of_integer @ L2 ) )
@ ( plus_plus @ nat @ ( plus_plus @ nat @ ( code_nat_of_integer @ L2 ) @ ( code_nat_of_integer @ L2 ) ) @ ( one_one @ nat ) ) )
@ ( code_divmod_integer @ K3 @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% nat_of_integer_code
thf(fact_7853_nat__of__integer__numeral,axiom,
! [N: num] :
( ( code_nat_of_integer @ ( numeral_numeral @ code_integer @ N ) )
= ( numeral_numeral @ nat @ N ) ) ).
% nat_of_integer_numeral
thf(fact_7854_nat__of__integer__code__post_I3_J,axiom,
! [K: num] :
( ( code_nat_of_integer @ ( numeral_numeral @ code_integer @ K ) )
= ( numeral_numeral @ nat @ K ) ) ).
% nat_of_integer_code_post(3)
thf(fact_7855_nat__of__integer__1,axiom,
( ( code_nat_of_integer @ ( one_one @ code_integer ) )
= ( one_one @ nat ) ) ).
% nat_of_integer_1
thf(fact_7856_of__nat__of__integer,axiom,
! [K: code_integer] :
( ( semiring_1_of_nat @ code_integer @ ( code_nat_of_integer @ K ) )
= ( ord_max @ code_integer @ ( zero_zero @ code_integer ) @ K ) ) ).
% of_nat_of_integer
thf(fact_7857_nat__of__integer__non__positive,axiom,
! [K: code_integer] :
( ( ord_less_eq @ code_integer @ K @ ( zero_zero @ code_integer ) )
=> ( ( code_nat_of_integer @ K )
= ( zero_zero @ nat ) ) ) ).
% nat_of_integer_non_positive
thf(fact_7858_real__eq__imaginary__iff,axiom,
! [Y: complex,X: complex] :
( ( member @ complex @ Y @ ( real_Vector_Reals @ complex ) )
=> ( ( member @ complex @ X @ ( real_Vector_Reals @ complex ) )
=> ( ( X
= ( times_times @ complex @ imaginary_unit @ Y ) )
= ( ( X
= ( zero_zero @ complex ) )
& ( Y
= ( zero_zero @ complex ) ) ) ) ) ) ).
% real_eq_imaginary_iff
thf(fact_7859_imaginary__eq__real__iff,axiom,
! [Y: complex,X: complex] :
( ( member @ complex @ Y @ ( real_Vector_Reals @ complex ) )
=> ( ( member @ complex @ X @ ( real_Vector_Reals @ complex ) )
=> ( ( ( times_times @ complex @ imaginary_unit @ Y )
= X )
= ( ( X
= ( zero_zero @ complex ) )
& ( Y
= ( zero_zero @ complex ) ) ) ) ) ) ).
% imaginary_eq_real_iff
thf(fact_7860_complex__is__Real__iff,axiom,
! [Z: complex] :
( ( member @ complex @ Z @ ( real_Vector_Reals @ complex ) )
= ( ( im @ Z )
= ( zero_zero @ real ) ) ) ).
% complex_is_Real_iff
thf(fact_7861_Reals__of__nat,axiom,
! [A: $tType] :
( ( real_V2191834092415804123ebra_1 @ A )
=> ! [N: nat] : ( member @ A @ ( semiring_1_of_nat @ A @ N ) @ ( real_Vector_Reals @ A ) ) ) ).
% Reals_of_nat
thf(fact_7862_nonzero__Reals__inverse,axiom,
! [A: $tType] :
( ( real_V5047593784448816457lgebra @ A )
=> ! [A3: A] :
( ( member @ A @ A3 @ ( real_Vector_Reals @ A ) )
=> ( ( A3
!= ( zero_zero @ A ) )
=> ( member @ A @ ( inverse_inverse @ A @ A3 ) @ ( real_Vector_Reals @ A ) ) ) ) ) ).
% nonzero_Reals_inverse
thf(fact_7863_Complex__in__Reals,axiom,
! [X: real] : ( member @ complex @ ( complex2 @ X @ ( zero_zero @ real ) ) @ ( real_Vector_Reals @ complex ) ) ).
% Complex_in_Reals
thf(fact_7864_nonzero__Reals__divide,axiom,
! [A: $tType] :
( ( real_V7773925162809079976_field @ A )
=> ! [A3: A,B3: A] :
( ( member @ A @ A3 @ ( real_Vector_Reals @ A ) )
=> ( ( member @ A @ B3 @ ( real_Vector_Reals @ A ) )
=> ( ( B3
!= ( zero_zero @ A ) )
=> ( member @ A @ ( divide_divide @ A @ A3 @ B3 ) @ ( real_Vector_Reals @ A ) ) ) ) ) ) ).
% nonzero_Reals_divide
thf(fact_7865_Reals__1,axiom,
! [B: $tType] :
( ( real_V2191834092415804123ebra_1 @ B )
=> ( member @ B @ ( one_one @ B ) @ ( real_Vector_Reals @ B ) ) ) ).
% Reals_1
thf(fact_7866_Reals__power,axiom,
! [A: $tType] :
( ( real_V2191834092415804123ebra_1 @ A )
=> ! [A3: A,N: nat] :
( ( member @ A @ A3 @ ( real_Vector_Reals @ A ) )
=> ( member @ A @ ( power_power @ A @ A3 @ N ) @ ( real_Vector_Reals @ A ) ) ) ) ).
% Reals_power
thf(fact_7867_Reals__numeral,axiom,
! [A: $tType] :
( ( real_V2191834092415804123ebra_1 @ A )
=> ! [W: num] : ( member @ A @ ( numeral_numeral @ A @ W ) @ ( real_Vector_Reals @ A ) ) ) ).
% Reals_numeral
thf(fact_7868_Reals__0,axiom,
! [A: $tType] :
( ( real_V2191834092415804123ebra_1 @ A )
=> ( member @ A @ ( zero_zero @ A ) @ ( real_Vector_Reals @ A ) ) ) ).
% Reals_0
thf(fact_7869_nat__of__integer__code__post_I1_J,axiom,
( ( code_nat_of_integer @ ( zero_zero @ code_integer ) )
= ( zero_zero @ nat ) ) ).
% nat_of_integer_code_post(1)
thf(fact_7870_Re__prod__Reals,axiom,
! [A: $tType,A2: set @ A,F3: A > complex] :
( ! [X3: A] :
( ( member @ A @ X3 @ A2 )
=> ( member @ complex @ ( F3 @ X3 ) @ ( real_Vector_Reals @ complex ) ) )
=> ( ( re @ ( groups7121269368397514597t_prod @ A @ complex @ F3 @ A2 ) )
= ( groups7121269368397514597t_prod @ A @ real
@ ^ [X2: A] : ( re @ ( F3 @ X2 ) )
@ A2 ) ) ) ).
% Re_prod_Reals
thf(fact_7871_nat__of__integer__less__iff,axiom,
! [X: code_integer,Y: code_integer] :
( ( ord_less_eq @ code_integer @ ( zero_zero @ code_integer ) @ X )
=> ( ( ord_less_eq @ code_integer @ ( zero_zero @ code_integer ) @ Y )
=> ( ( ord_less @ nat @ ( code_nat_of_integer @ X ) @ ( code_nat_of_integer @ Y ) )
= ( ord_less @ code_integer @ X @ Y ) ) ) ) ).
% nat_of_integer_less_iff
thf(fact_7872_image__atLeastZeroLessThan__integer,axiom,
! [U2: code_integer] :
( ( ord_less_eq @ code_integer @ ( zero_zero @ code_integer ) @ U2 )
=> ( ( set_or7035219750837199246ssThan @ code_integer @ ( zero_zero @ code_integer ) @ U2 )
= ( image @ nat @ code_integer @ ( semiring_1_of_nat @ code_integer ) @ ( set_ord_lessThan @ nat @ ( code_nat_of_integer @ U2 ) ) ) ) ) ).
% image_atLeastZeroLessThan_integer
thf(fact_7873_series__comparison__complex,axiom,
! [A: $tType] :
( ( real_Vector_banach @ A )
=> ! [G2: nat > complex,N3: nat,F3: nat > A] :
( ( summable @ complex @ G2 )
=> ( ! [N2: nat] : ( member @ complex @ ( G2 @ N2 ) @ ( real_Vector_Reals @ complex ) )
=> ( ! [N2: nat] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( re @ ( G2 @ N2 ) ) )
=> ( ! [N2: nat] :
( ( ord_less_eq @ nat @ N3 @ N2 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( F3 @ N2 ) ) @ ( real_V7770717601297561774m_norm @ complex @ ( G2 @ N2 ) ) ) )
=> ( summable @ A @ F3 ) ) ) ) ) ) ).
% series_comparison_complex
thf(fact_7874_Re__Reals__divide,axiom,
! [R3: complex,Z: complex] :
( ( member @ complex @ R3 @ ( real_Vector_Reals @ complex ) )
=> ( ( re @ ( divide_divide @ complex @ R3 @ Z ) )
= ( divide_divide @ real @ ( times_times @ real @ ( re @ R3 ) @ ( re @ Z ) ) @ ( power_power @ real @ ( real_V7770717601297561774m_norm @ complex @ Z ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% Re_Reals_divide
thf(fact_7875_integer__set__bit__code,axiom,
( bits_integer_set_bit
= ( ^ [X2: code_integer,N4: code_integer,B5: $o] : ( if @ code_integer @ ( ord_less @ code_integer @ N4 @ ( zero_zero @ code_integer ) ) @ ( undefined @ ( code_integer > code_integer > $o > code_integer ) @ X2 @ N4 @ B5 ) @ ( if @ code_integer @ B5 @ ( bit_se1065995026697491101ons_or @ code_integer @ X2 @ ( bit_se4730199178511100633sh_bit @ code_integer @ ( code_nat_of_integer @ N4 ) @ ( one_one @ code_integer ) ) ) @ ( bit_se5824344872417868541ns_and @ code_integer @ X2 @ ( bit_ri4277139882892585799ns_not @ code_integer @ ( bit_se4730199178511100633sh_bit @ code_integer @ ( code_nat_of_integer @ N4 ) @ ( one_one @ code_integer ) ) ) ) ) ) ) ) ).
% integer_set_bit_code
thf(fact_7876_uint32__shiftr__def,axiom,
( uint32_shiftr
= ( ^ [X2: uint32,N4: code_integer] :
( if @ uint32
@ ( ( ord_less @ code_integer @ N4 @ ( zero_zero @ code_integer ) )
| ( ord_less_eq @ code_integer @ ( numeral_numeral @ code_integer @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) @ N4 ) )
@ ( undefined @ ( ( nat > uint32 > uint32 ) > uint32 > code_integer > uint32 ) @ ( bit_se4197421643247451524op_bit @ uint32 ) @ X2 @ N4 )
@ ( bit_se4197421643247451524op_bit @ uint32 @ ( code_nat_of_integer @ N4 ) @ X2 ) ) ) ) ).
% uint32_shiftr_def
thf(fact_7877_integer__set__bit__def,axiom,
( bits_integer_set_bit
= ( ^ [X2: code_integer,N4: code_integer,B5: $o] : ( if @ code_integer @ ( ord_less @ code_integer @ N4 @ ( zero_zero @ code_integer ) ) @ ( undefined @ ( code_integer > code_integer > $o > code_integer ) @ X2 @ N4 @ B5 ) @ ( generi7602027413899671122et_bit @ code_integer @ X2 @ ( code_nat_of_integer @ N4 ) @ B5 ) ) ) ) ).
% integer_set_bit_def
thf(fact_7878_uint32__shiftl__def,axiom,
( uint32_shiftl
= ( ^ [X2: uint32,N4: code_integer] :
( if @ uint32
@ ( ( ord_less @ code_integer @ N4 @ ( zero_zero @ code_integer ) )
| ( ord_less_eq @ code_integer @ ( numeral_numeral @ code_integer @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) @ N4 ) )
@ ( undefined @ ( ( nat > uint32 > uint32 ) > uint32 > code_integer > uint32 ) @ ( bit_se4730199178511100633sh_bit @ uint32 ) @ X2 @ N4 )
@ ( bit_se4730199178511100633sh_bit @ uint32 @ ( code_nat_of_integer @ N4 ) @ X2 ) ) ) ) ).
% uint32_shiftl_def
thf(fact_7879_uint32__set__bit__def,axiom,
( uint32_set_bit
= ( ^ [X2: uint32,N4: code_integer,B5: $o] :
( if @ uint32
@ ( ( ord_less @ code_integer @ N4 @ ( zero_zero @ code_integer ) )
| ( ord_less @ code_integer @ ( numeral_numeral @ code_integer @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) @ N4 ) )
@ ( undefined @ ( ( uint32 > nat > $o > uint32 ) > uint32 > code_integer > $o > uint32 ) @ ( generi7602027413899671122et_bit @ uint32 ) @ X2 @ N4 @ B5 )
@ ( generi7602027413899671122et_bit @ uint32 @ X2 @ ( code_nat_of_integer @ N4 ) @ B5 ) ) ) ) ).
% uint32_set_bit_def
thf(fact_7880_uint32__test__bit__def,axiom,
( uint32_test_bit
= ( ^ [X2: uint32,N4: code_integer] :
( ( ( ( ord_less @ code_integer @ N4 @ ( zero_zero @ code_integer ) )
| ( ord_less @ code_integer @ ( numeral_numeral @ code_integer @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) @ N4 ) )
=> ( undefined @ ( ( uint32 > nat > $o ) > uint32 > code_integer > $o ) @ ( bit_se5641148757651400278ts_bit @ uint32 ) @ X2 @ N4 ) )
& ( ~ ( ( ord_less @ code_integer @ N4 @ ( zero_zero @ code_integer ) )
| ( ord_less @ code_integer @ ( numeral_numeral @ code_integer @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) @ N4 ) )
=> ( bit_se5641148757651400278ts_bit @ uint32 @ X2 @ ( code_nat_of_integer @ N4 ) ) ) ) ) ) ).
% uint32_test_bit_def
thf(fact_7881_uint32__shiftr__code,axiom,
! [N: code_integer,W: uint32] :
( ( ( ( ord_less @ code_integer @ N @ ( zero_zero @ code_integer ) )
| ( ord_less_eq @ code_integer @ ( numeral_numeral @ code_integer @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) @ N ) )
=> ( ( rep_uint322 @ ( uint32_shiftr @ W @ N ) )
= ( rep_uint322 @ ( undefined @ ( ( nat > uint32 > uint32 ) > uint32 > code_integer > uint32 ) @ ( bit_se4197421643247451524op_bit @ uint32 ) @ W @ N ) ) ) )
& ( ~ ( ( ord_less @ code_integer @ N @ ( zero_zero @ code_integer ) )
| ( ord_less_eq @ code_integer @ ( numeral_numeral @ code_integer @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) @ N ) )
=> ( ( rep_uint322 @ ( uint32_shiftr @ W @ N ) )
= ( bit_se4197421643247451524op_bit @ ( word @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ numeral_num1 ) ) ) ) ) ) @ ( code_nat_of_integer @ N ) @ ( rep_uint322 @ W ) ) ) ) ) ).
% uint32_shiftr_code
thf(fact_7882_uint32_Oless__iff__word__of,axiom,
( ( ord_less @ uint32 )
= ( ^ [P6: uint32,Q5: uint32] : ( ord_less @ ( word @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ numeral_num1 ) ) ) ) ) ) @ ( rep_uint322 @ P6 ) @ ( rep_uint322 @ Q5 ) ) ) ) ).
% uint32.less_iff_word_of
thf(fact_7883_less__uint32_Orep__eq,axiom,
( ( ord_less @ uint32 )
= ( ^ [X2: uint32,Xa4: uint32] : ( ord_less @ ( word @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ numeral_num1 ) ) ) ) ) ) @ ( rep_uint322 @ X2 ) @ ( rep_uint322 @ Xa4 ) ) ) ) ).
% less_uint32.rep_eq
thf(fact_7884_zero__uint32_Orep__eq,axiom,
( ( rep_uint322 @ ( zero_zero @ uint32 ) )
= ( zero_zero @ ( word @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ numeral_num1 ) ) ) ) ) ) ) ) ).
% zero_uint32.rep_eq
thf(fact_7885_one__uint32_Orep__eq,axiom,
( ( rep_uint322 @ ( one_one @ uint32 ) )
= ( one_one @ ( word @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ numeral_num1 ) ) ) ) ) ) ) ) ).
% one_uint32.rep_eq
thf(fact_7886_uint32_Oeven__iff__word__of,axiom,
! [P4: uint32] :
( ( dvd_dvd @ uint32 @ ( numeral_numeral @ uint32 @ ( bit0 @ one2 ) ) @ P4 )
= ( dvd_dvd @ ( word @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ numeral_num1 ) ) ) ) ) ) @ ( numeral_numeral @ ( word @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ numeral_num1 ) ) ) ) ) ) @ ( bit0 @ one2 ) ) @ ( rep_uint322 @ P4 ) ) ) ).
% uint32.even_iff_word_of
thf(fact_7887_uint32__test__bit__code,axiom,
( uint32_test_bit
= ( ^ [W2: uint32,N4: code_integer] :
( ( ( ( ord_less @ code_integer @ N4 @ ( zero_zero @ code_integer ) )
| ( ord_less @ code_integer @ ( numeral_numeral @ code_integer @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) @ N4 ) )
=> ( undefined @ ( ( uint32 > nat > $o ) > uint32 > code_integer > $o ) @ ( bit_se5641148757651400278ts_bit @ uint32 ) @ W2 @ N4 ) )
& ( ~ ( ( ord_less @ code_integer @ N4 @ ( zero_zero @ code_integer ) )
| ( ord_less @ code_integer @ ( numeral_numeral @ code_integer @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) @ N4 ) )
=> ( bit_se5641148757651400278ts_bit @ ( word @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ numeral_num1 ) ) ) ) ) ) @ ( rep_uint322 @ W2 ) @ ( code_nat_of_integer @ N4 ) ) ) ) ) ) ).
% uint32_test_bit_code
thf(fact_7888_uint32__set__bit__code,axiom,
! [N: code_integer,W: uint32,B3: $o] :
( ( ( ( ord_less @ code_integer @ N @ ( zero_zero @ code_integer ) )
| ( ord_less @ code_integer @ ( numeral_numeral @ code_integer @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) @ N ) )
=> ( ( rep_uint322 @ ( uint32_set_bit @ W @ N @ B3 ) )
= ( rep_uint322 @ ( undefined @ ( ( uint32 > nat > $o > uint32 ) > uint32 > code_integer > $o > uint32 ) @ ( generi7602027413899671122et_bit @ uint32 ) @ W @ N @ B3 ) ) ) )
& ( ~ ( ( ord_less @ code_integer @ N @ ( zero_zero @ code_integer ) )
| ( ord_less @ code_integer @ ( numeral_numeral @ code_integer @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) @ N ) )
=> ( ( rep_uint322 @ ( uint32_set_bit @ W @ N @ B3 ) )
= ( generi7602027413899671122et_bit @ ( word @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ numeral_num1 ) ) ) ) ) ) @ ( rep_uint322 @ W ) @ ( code_nat_of_integer @ N ) @ B3 ) ) ) ) ).
% uint32_set_bit_code
thf(fact_7889_uint32__shiftl__code,axiom,
! [N: code_integer,W: uint32] :
( ( ( ( ord_less @ code_integer @ N @ ( zero_zero @ code_integer ) )
| ( ord_less_eq @ code_integer @ ( numeral_numeral @ code_integer @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) @ N ) )
=> ( ( rep_uint322 @ ( uint32_shiftl @ W @ N ) )
= ( rep_uint322 @ ( undefined @ ( ( nat > uint32 > uint32 ) > uint32 > code_integer > uint32 ) @ ( bit_se4730199178511100633sh_bit @ uint32 ) @ W @ N ) ) ) )
& ( ~ ( ( ord_less @ code_integer @ N @ ( zero_zero @ code_integer ) )
| ( ord_less_eq @ code_integer @ ( numeral_numeral @ code_integer @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) @ N ) )
=> ( ( rep_uint322 @ ( uint32_shiftl @ W @ N ) )
= ( bit_se4730199178511100633sh_bit @ ( word @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ numeral_num1 ) ) ) ) ) ) @ ( code_nat_of_integer @ N ) @ ( rep_uint322 @ W ) ) ) ) ) ).
% uint32_shiftl_code
thf(fact_7890_Uint32__signed__code,axiom,
! [I: code_integer] :
( ( ( ( ord_less @ code_integer @ I @ ( 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 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ I ) )
=> ( ( rep_uint322 @ ( uint32_signed @ I ) )
= ( rep_uint322 @ ( undefined @ ( ( code_integer > uint32 ) > code_integer > uint32 ) @ uint322 @ I ) ) ) )
& ( ~ ( ( ord_less @ code_integer @ I @ ( 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 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ I ) )
=> ( ( rep_uint322 @ ( uint32_signed @ I ) )
= ( ring_1_of_int @ ( word @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ numeral_num1 ) ) ) ) ) ) @ ( code_I935103866777955880mbolic @ I ) ) ) ) ) ).
% Uint32_signed_code
thf(fact_7891_integer__shiftl__def,axiom,
( bits_integer_shiftl
= ( ^ [X2: code_integer,N4: code_integer] : ( if @ code_integer @ ( ord_less @ code_integer @ N4 @ ( zero_zero @ code_integer ) ) @ ( undefined @ ( code_integer > code_integer > code_integer ) @ X2 @ N4 ) @ ( bit_se4730199178511100633sh_bit @ code_integer @ ( code_nat_of_integer @ N4 ) @ X2 ) ) ) ) ).
% integer_shiftl_def
thf(fact_7892_one__uint32_Orsp,axiom,
( ( one_one @ ( word @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ numeral_num1 ) ) ) ) ) ) )
= ( one_one @ ( word @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ numeral_num1 ) ) ) ) ) ) ) ) ).
% one_uint32.rsp
thf(fact_7893_zero__uint32_Orsp,axiom,
( ( zero_zero @ ( word @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ numeral_num1 ) ) ) ) ) ) )
= ( zero_zero @ ( word @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ numeral_num1 ) ) ) ) ) ) ) ) ).
% zero_uint32.rsp
thf(fact_7894_integer__shiftl__code_I2_J,axiom,
! [X: code_integer] :
( ( bits_integer_shiftl @ X @ ( zero_zero @ code_integer ) )
= X ) ).
% integer_shiftl_code(2)
thf(fact_7895_int__of__integer__symbolic__aux__code_I1_J,axiom,
( ( code_I935103866777955880mbolic @ ( zero_zero @ code_integer ) )
= ( zero_zero @ int ) ) ).
% int_of_integer_symbolic_aux_code(1)
thf(fact_7896_uint32_Oset__bits__aux__code,axiom,
( set_bits_aux_uint32
= ( ^ [F2: nat > $o,N4: nat,W2: uint32] :
( if @ uint32
@ ( N4
= ( zero_zero @ nat ) )
@ W2
@ ( set_bits_aux_uint32 @ F2 @ ( minus_minus @ nat @ N4 @ ( one_one @ nat ) ) @ ( bit_se1065995026697491101ons_or @ uint32 @ ( bit_se4730199178511100633sh_bit @ uint32 @ ( one_one @ nat ) @ W2 ) @ ( if @ uint32 @ ( F2 @ ( minus_minus @ nat @ N4 @ ( one_one @ nat ) ) ) @ ( one_one @ uint32 ) @ ( zero_zero @ uint32 ) ) ) ) ) ) ) ).
% uint32.set_bits_aux_code
thf(fact_7897_shiftr__uint32__code,axiom,
( ( bit_se4197421643247451524op_bit @ uint32 )
= ( ^ [N4: nat,X2: uint32] : ( if @ uint32 @ ( ord_less @ nat @ N4 @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) @ ( uint32_shiftr @ X2 @ ( code_integer_of_nat @ N4 ) ) @ ( zero_zero @ uint32 ) ) ) ) ).
% shiftr_uint32_code
thf(fact_7898_shiftl__uint32__code,axiom,
( ( bit_se4730199178511100633sh_bit @ uint32 )
= ( ^ [N4: nat,X2: uint32] : ( if @ uint32 @ ( ord_less @ nat @ N4 @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) @ ( uint32_shiftl @ X2 @ ( code_integer_of_nat @ N4 ) ) @ ( zero_zero @ uint32 ) ) ) ) ).
% shiftl_uint32_code
thf(fact_7899_integer__of__nat_Orep__eq,axiom,
! [X: nat] :
( ( code_int_of_integer @ ( code_integer_of_nat @ X ) )
= ( semiring_1_of_nat @ int @ X ) ) ).
% integer_of_nat.rep_eq
thf(fact_7900_int__of__integer__integer__of__nat,axiom,
! [N: nat] :
( ( code_int_of_integer @ ( code_integer_of_nat @ N ) )
= ( semiring_1_of_nat @ int @ N ) ) ).
% int_of_integer_integer_of_nat
thf(fact_7901_integer__of__nat_Oabs__eq,axiom,
( code_integer_of_nat
= ( ^ [X2: nat] : ( code_integer_of_int @ ( semiring_1_of_nat @ int @ X2 ) ) ) ) ).
% integer_of_nat.abs_eq
thf(fact_7902_integer__of__nat__less__0__conv,axiom,
! [N: nat] :
~ ( ord_less @ code_integer @ ( code_integer_of_nat @ N ) @ ( zero_zero @ code_integer ) ) ).
% integer_of_nat_less_0_conv
thf(fact_7903_integer__of__nat__numeral,axiom,
! [N: num] :
( ( code_integer_of_nat @ ( numeral_numeral @ nat @ N ) )
= ( numeral_numeral @ code_integer @ N ) ) ).
% integer_of_nat_numeral
thf(fact_7904_integer__of__nat__0,axiom,
( ( code_integer_of_nat @ ( zero_zero @ nat ) )
= ( zero_zero @ code_integer ) ) ).
% integer_of_nat_0
thf(fact_7905_integer__of__nat__1,axiom,
( ( code_integer_of_nat @ ( one_one @ nat ) )
= ( one_one @ code_integer ) ) ).
% integer_of_nat_1
thf(fact_7906_test__bit__uint32__code,axiom,
( ( bit_se5641148757651400278ts_bit @ uint32 )
= ( ^ [X2: uint32,N4: nat] :
( ( ord_less @ nat @ N4 @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) )
& ( uint32_test_bit @ X2 @ ( code_integer_of_nat @ N4 ) ) ) ) ) ).
% test_bit_uint32_code
thf(fact_7907_set__bit__uint32__code,axiom,
( ( generi7602027413899671122et_bit @ uint32 )
= ( ^ [X2: uint32,N4: nat,B5: $o] : ( if @ uint32 @ ( ord_less @ nat @ N4 @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) @ ( uint32_set_bit @ X2 @ ( code_integer_of_nat @ N4 ) @ B5 ) @ X2 ) ) ) ).
% set_bit_uint32_code
thf(fact_7908_uint32__divmod__code,axiom,
( uint32_divmod
= ( ^ [X2: uint32,Y2: uint32] :
( if @ ( product_prod @ uint32 @ uint32 ) @ ( ord_less_eq @ uint32 @ ( numeral_numeral @ uint32 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ Y2 ) @ ( if @ ( product_prod @ uint32 @ uint32 ) @ ( ord_less @ uint32 @ X2 @ Y2 ) @ ( product_Pair @ uint32 @ uint32 @ ( zero_zero @ uint32 ) @ X2 ) @ ( product_Pair @ uint32 @ uint32 @ ( one_one @ uint32 ) @ ( minus_minus @ uint32 @ X2 @ Y2 ) ) )
@ ( if @ ( product_prod @ uint32 @ uint32 )
@ ( Y2
= ( zero_zero @ uint32 ) )
@ ( product_Pair @ uint32 @ uint32 @ ( div0_uint32 @ X2 ) @ ( mod0_uint32 @ X2 ) )
@ ( if @ ( product_prod @ uint32 @ uint32 ) @ ( ord_less_eq @ uint32 @ Y2 @ ( minus_minus @ uint32 @ X2 @ ( times_times @ uint32 @ ( bit_se4730199178511100633sh_bit @ uint32 @ ( one_one @ nat ) @ ( uint32_sdiv @ ( bit_se4197421643247451524op_bit @ uint32 @ ( one_one @ nat ) @ X2 ) @ Y2 ) ) @ Y2 ) ) ) @ ( product_Pair @ uint32 @ uint32 @ ( plus_plus @ uint32 @ ( bit_se4730199178511100633sh_bit @ uint32 @ ( one_one @ nat ) @ ( uint32_sdiv @ ( bit_se4197421643247451524op_bit @ uint32 @ ( one_one @ nat ) @ X2 ) @ Y2 ) ) @ ( one_one @ uint32 ) ) @ ( minus_minus @ uint32 @ ( minus_minus @ uint32 @ X2 @ ( times_times @ uint32 @ ( bit_se4730199178511100633sh_bit @ uint32 @ ( one_one @ nat ) @ ( uint32_sdiv @ ( bit_se4197421643247451524op_bit @ uint32 @ ( one_one @ nat ) @ X2 ) @ Y2 ) ) @ Y2 ) ) @ Y2 ) ) @ ( product_Pair @ uint32 @ uint32 @ ( bit_se4730199178511100633sh_bit @ uint32 @ ( one_one @ nat ) @ ( uint32_sdiv @ ( bit_se4197421643247451524op_bit @ uint32 @ ( one_one @ nat ) @ X2 ) @ Y2 ) ) @ ( minus_minus @ uint32 @ X2 @ ( times_times @ uint32 @ ( bit_se4730199178511100633sh_bit @ uint32 @ ( one_one @ nat ) @ ( uint32_sdiv @ ( bit_se4197421643247451524op_bit @ uint32 @ ( one_one @ nat ) @ X2 ) @ Y2 ) ) @ Y2 ) ) ) ) ) ) ) ) ).
% uint32_divmod_code
thf(fact_7909_sshiftr__uint32__code,axiom,
( signed489701013188660438uint32
= ( ^ [N4: nat,X2: uint32] : ( if @ uint32 @ ( ord_less @ nat @ N4 @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) @ ( uint32_sshiftr @ X2 @ ( code_integer_of_nat @ N4 ) ) @ ( if @ uint32 @ ( bit_se5641148757651400278ts_bit @ uint32 @ X2 @ ( numeral_numeral @ nat @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) ) @ ( uminus_uminus @ uint32 @ ( one_one @ uint32 ) ) @ ( zero_zero @ uint32 ) ) ) ) ) ).
% sshiftr_uint32_code
thf(fact_7910_uint32__divmod__def,axiom,
( uint32_divmod
= ( ^ [X2: uint32,Y2: uint32] :
( if @ ( product_prod @ uint32 @ uint32 )
@ ( Y2
= ( zero_zero @ uint32 ) )
@ ( product_Pair @ uint32 @ uint32 @ ( undefined @ ( ( uint32 > uint32 > uint32 ) > uint32 > uint32 > uint32 ) @ ( divide_divide @ uint32 ) @ X2 @ ( zero_zero @ uint32 ) ) @ ( undefined @ ( ( uint32 > uint32 > uint32 ) > uint32 > uint32 > uint32 ) @ ( modulo_modulo @ uint32 ) @ X2 @ ( zero_zero @ uint32 ) ) )
@ ( product_Pair @ uint32 @ uint32 @ ( divide_divide @ uint32 @ X2 @ Y2 ) @ ( modulo_modulo @ uint32 @ X2 @ Y2 ) ) ) ) ) ).
% uint32_divmod_def
thf(fact_7911_mod0__uint32__def,axiom,
( mod0_uint32
= ( ^ [X2: uint32] : ( undefined @ ( ( uint32 > uint32 > uint32 ) > uint32 > uint32 > uint32 ) @ ( modulo_modulo @ uint32 ) @ X2 @ ( zero_zero @ uint32 ) ) ) ) ).
% mod0_uint32_def
thf(fact_7912_div0__uint32__def,axiom,
( div0_uint32
= ( ^ [X2: uint32] : ( undefined @ ( ( uint32 > uint32 > uint32 ) > uint32 > uint32 > uint32 ) @ ( divide_divide @ uint32 ) @ X2 @ ( zero_zero @ uint32 ) ) ) ) ).
% div0_uint32_def
thf(fact_7913_uint32__sdiv__code,axiom,
! [Y: uint32,X: uint32] :
( ( ( Y
= ( zero_zero @ uint32 ) )
=> ( ( rep_uint322 @ ( uint32_sdiv @ X @ Y ) )
= ( rep_uint322 @ ( undefined @ ( ( uint32 > uint32 > uint32 ) > uint32 > uint32 > uint32 ) @ ( divide_divide @ uint32 ) @ X @ ( zero_zero @ uint32 ) ) ) ) )
& ( ( Y
!= ( zero_zero @ uint32 ) )
=> ( ( rep_uint322 @ ( uint32_sdiv @ X @ Y ) )
= ( signed7115095781618012415divide @ ( word @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ numeral_num1 ) ) ) ) ) ) @ ( rep_uint322 @ X ) @ ( rep_uint322 @ Y ) ) ) ) ) ).
% uint32_sdiv_code
thf(fact_7914_uint32__sshiftr__def,axiom,
( uint32_sshiftr
= ( ^ [X2: uint32,N4: code_integer] :
( if @ uint32
@ ( ( ord_less @ code_integer @ N4 @ ( zero_zero @ code_integer ) )
| ( ord_less_eq @ code_integer @ ( numeral_numeral @ code_integer @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) @ N4 ) )
@ ( undefined @ ( ( nat > uint32 > uint32 ) > code_integer > uint32 > uint32 ) @ signed489701013188660438uint32 @ N4 @ X2 )
@ ( signed489701013188660438uint32 @ ( code_nat_of_integer @ N4 ) @ X2 ) ) ) ) ).
% uint32_sshiftr_def
thf(fact_7915_uint32__sshiftr__code,axiom,
! [N: code_integer,W: uint32] :
( ( ( ( ord_less @ code_integer @ N @ ( zero_zero @ code_integer ) )
| ( ord_less_eq @ code_integer @ ( numeral_numeral @ code_integer @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) @ N ) )
=> ( ( rep_uint322 @ ( uint32_sshiftr @ W @ N ) )
= ( rep_uint322 @ ( undefined @ ( ( nat > uint32 > uint32 ) > code_integer > uint32 > uint32 ) @ signed489701013188660438uint32 @ N @ W ) ) ) )
& ( ~ ( ( ord_less @ code_integer @ N @ ( zero_zero @ code_integer ) )
| ( ord_less_eq @ code_integer @ ( numeral_numeral @ code_integer @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) @ N ) )
=> ( ( rep_uint322 @ ( uint32_sshiftr @ W @ N ) )
= ( signed_drop_bit @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ numeral_num1 ) ) ) ) ) @ ( code_nat_of_integer @ N ) @ ( rep_uint322 @ W ) ) ) ) ) ).
% uint32_sshiftr_code
thf(fact_7916_integer__shiftr__def,axiom,
( bits_integer_shiftr
= ( ^ [X2: code_integer,N4: code_integer] : ( if @ code_integer @ ( ord_less @ code_integer @ N4 @ ( zero_zero @ code_integer ) ) @ ( undefined @ ( code_integer > code_integer > code_integer ) @ X2 @ N4 ) @ ( bit_se4197421643247451524op_bit @ code_integer @ ( code_nat_of_integer @ N4 ) @ X2 ) ) ) ) ).
% integer_shiftr_def
thf(fact_7917_uint32_Oset__bits__code,axiom,
( ( bit_bi4170147762399595738t_bits @ uint32 )
= ( ^ [P3: nat > $o] : ( set_bits_aux_uint32 @ P3 @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) @ ( zero_zero @ uint32 ) ) ) ) ).
% uint32.set_bits_code
thf(fact_7918_integer__shiftr__code_I2_J,axiom,
! [X: code_integer] :
( ( bits_integer_shiftr @ X @ ( zero_zero @ code_integer ) )
= X ) ).
% integer_shiftr_code(2)
thf(fact_7919_set__bits__False__eq,axiom,
! [A: $tType] :
( ( bit_bi6583157726757044596ension @ A )
=> ( ( bit_bi4170147762399595738t_bits @ A
@ ^ [Uu: nat] : $false )
= ( zero_zero @ A ) ) ) ).
% set_bits_False_eq
thf(fact_7920_mod__uint32__code,axiom,
( ( modulo_modulo @ uint32 )
= ( ^ [X2: uint32,Y2: uint32] :
( if @ uint32
@ ( Y2
= ( zero_zero @ uint32 ) )
@ X2
@ ( uint32_mod @ X2 @ Y2 ) ) ) ) ).
% mod_uint32_code
thf(fact_7921_int__set__bits__K__False,axiom,
( ( bit_bi4170147762399595738t_bits @ int
@ ^ [Uu: nat] : $false )
= ( zero_zero @ int ) ) ).
% int_set_bits_K_False
thf(fact_7922_set__bits__K__False,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( bit_bi4170147762399595738t_bits @ ( word @ A )
@ ^ [Uu: nat] : $false )
= ( zero_zero @ ( word @ A ) ) ) ) ).
% set_bits_K_False
thf(fact_7923_int__set__bits__K__True,axiom,
( ( bit_bi4170147762399595738t_bits @ int
@ ^ [Uu: nat] : $true )
= ( uminus_uminus @ int @ ( one_one @ int ) ) ) ).
% int_set_bits_K_True
thf(fact_7924_bit__set__bits__word__iff,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [P: nat > $o,N: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( bit_bi4170147762399595738t_bits @ ( word @ A ) @ P ) @ N )
= ( ( ord_less @ nat @ N @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
& ( P @ N ) ) ) ) ).
% bit_set_bits_word_iff
thf(fact_7925_word__of__int__conv__set__bits,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( ring_1_of_int @ ( word @ A ) )
= ( ^ [I4: int] : ( bit_bi4170147762399595738t_bits @ ( word @ A ) @ ( bit_se5641148757651400278ts_bit @ int @ I4 ) ) ) ) ) ).
% word_of_int_conv_set_bits
thf(fact_7926_set__bits__conv__set__bits__aux,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( bit_bi4170147762399595738t_bits @ ( word @ A ) )
= ( ^ [F2: nat > $o] : ( code_T2661198915054445665ts_aux @ A @ F2 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( zero_zero @ ( word @ A ) ) ) ) ) ) ).
% set_bits_conv_set_bits_aux
thf(fact_7927_word__test__bit__set__bits,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [F3: nat > $o,N: nat] :
( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( bit_bi4170147762399595738t_bits @ ( word @ A ) @ F3 ) @ N )
= ( ( ord_less @ nat @ N @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
& ( F3 @ N ) ) ) ) ).
% word_test_bit_set_bits
thf(fact_7928_div__uint32__code,axiom,
( ( divide_divide @ uint32 )
= ( ^ [X2: uint32,Y2: uint32] :
( if @ uint32
@ ( Y2
= ( zero_zero @ uint32 ) )
@ ( zero_zero @ uint32 )
@ ( uint32_div @ X2 @ Y2 ) ) ) ) ).
% div_uint32_code
thf(fact_7929_bin__last__set__bits,axiom,
! [F3: nat > $o] :
( ( bit_wf_set_bits_int @ F3 )
=> ( ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_bi4170147762399595738t_bits @ int @ F3 ) ) )
= ( F3 @ ( zero_zero @ nat ) ) ) ) ).
% bin_last_set_bits
thf(fact_7930_wf__set__bits__int__Suc,axiom,
! [F3: nat > $o] :
( ( bit_wf_set_bits_int
@ ^ [N4: nat] : ( F3 @ ( suc @ N4 ) ) )
= ( bit_wf_set_bits_int @ F3 ) ) ).
% wf_set_bits_int_Suc
thf(fact_7931_wf__set__bits__int__const,axiom,
! [B3: $o] :
( bit_wf_set_bits_int
@ ^ [Uu: nat] : B3 ) ).
% wf_set_bits_int_const
thf(fact_7932_int__set__bits__unfold__BIT,axiom,
! [F3: nat > $o] :
( ( bit_wf_set_bits_int @ F3 )
=> ( ( bit_bi4170147762399595738t_bits @ int @ F3 )
= ( plus_plus @ int @ ( zero_neq_one_of_bool @ int @ ( F3 @ ( zero_zero @ nat ) ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_bi4170147762399595738t_bits @ int @ ( comp @ nat @ $o @ nat @ F3 @ suc ) ) ) ) ) ) ).
% int_set_bits_unfold_BIT
thf(fact_7933_msb__uint32__code,axiom,
( ( most_s684356279273892711sb_msb @ uint32 )
= ( ^ [X2: uint32] : ( uint32_test_bit @ X2 @ ( numeral_numeral @ code_integer @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ).
% msb_uint32_code
thf(fact_7934_bin__rest__set__bits,axiom,
! [F3: nat > $o] :
( ( bit_wf_set_bits_int @ F3 )
=> ( ( divide_divide @ int @ ( bit_bi4170147762399595738t_bits @ int @ F3 ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) )
= ( bit_bi4170147762399595738t_bits @ int @ ( comp @ nat @ $o @ nat @ F3 @ suc ) ) ) ) ).
% bin_rest_set_bits
thf(fact_7935_summable__inverse__divide,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: nat > A,C2: A] :
( ( summable @ A @ ( comp @ A @ A @ nat @ ( inverse_inverse @ A ) @ F3 ) )
=> ( summable @ A
@ ^ [N4: nat] : ( divide_divide @ A @ C2 @ ( F3 @ N4 ) ) ) ) ) ).
% summable_inverse_divide
thf(fact_7936_sum__comp__morphism,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( ( comm_monoid_add @ B )
& ( comm_monoid_add @ A ) )
=> ! [H2: B > A,G2: C > B,A2: set @ C] :
( ( ( H2 @ ( zero_zero @ B ) )
= ( zero_zero @ A ) )
=> ( ! [X3: B,Y4: B] :
( ( H2 @ ( plus_plus @ B @ X3 @ Y4 ) )
= ( plus_plus @ A @ ( H2 @ X3 ) @ ( H2 @ Y4 ) ) )
=> ( ( groups7311177749621191930dd_sum @ C @ A @ ( comp @ B @ A @ C @ H2 @ G2 ) @ A2 )
= ( H2 @ ( groups7311177749621191930dd_sum @ C @ B @ G2 @ A2 ) ) ) ) ) ) ).
% sum_comp_morphism
thf(fact_7937_sum_Oreindex__nontrivial,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A2: set @ B,H2: B > C,G2: C > A] :
( ( finite_finite2 @ B @ A2 )
=> ( ! [X3: B,Y4: B] :
( ( member @ B @ X3 @ A2 )
=> ( ( member @ B @ Y4 @ A2 )
=> ( ( X3 != Y4 )
=> ( ( ( H2 @ X3 )
= ( H2 @ Y4 ) )
=> ( ( G2 @ ( H2 @ X3 ) )
= ( zero_zero @ A ) ) ) ) ) )
=> ( ( groups7311177749621191930dd_sum @ C @ A @ G2 @ ( image @ B @ C @ H2 @ A2 ) )
= ( groups7311177749621191930dd_sum @ B @ A @ ( comp @ C @ A @ B @ G2 @ H2 ) @ A2 ) ) ) ) ) ).
% sum.reindex_nontrivial
thf(fact_7938_prod_Oreindex__nontrivial,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A2: set @ B,H2: B > C,G2: C > A] :
( ( finite_finite2 @ B @ A2 )
=> ( ! [X3: B,Y4: B] :
( ( member @ B @ X3 @ A2 )
=> ( ( member @ B @ Y4 @ A2 )
=> ( ( X3 != Y4 )
=> ( ( ( H2 @ X3 )
= ( H2 @ Y4 ) )
=> ( ( G2 @ ( H2 @ X3 ) )
= ( one_one @ A ) ) ) ) ) )
=> ( ( groups7121269368397514597t_prod @ C @ A @ G2 @ ( image @ B @ C @ H2 @ A2 ) )
= ( groups7121269368397514597t_prod @ B @ A @ ( comp @ C @ A @ B @ G2 @ H2 ) @ A2 ) ) ) ) ) ).
% prod.reindex_nontrivial
thf(fact_7939_sum__image__le,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( ordere6911136660526730532id_add @ B )
=> ! [I6: set @ C,G2: A > B,F3: C > A] :
( ( finite_finite2 @ C @ I6 )
=> ( ! [I3: C] :
( ( member @ C @ I3 @ I6 )
=> ( ord_less_eq @ B @ ( zero_zero @ B ) @ ( G2 @ ( F3 @ I3 ) ) ) )
=> ( ord_less_eq @ B @ ( groups7311177749621191930dd_sum @ A @ B @ G2 @ ( image @ C @ A @ F3 @ I6 ) ) @ ( groups7311177749621191930dd_sum @ C @ B @ ( comp @ A @ B @ C @ G2 @ F3 ) @ I6 ) ) ) ) ) ).
% sum_image_le
thf(fact_7940_sum_OatLeast0__lessThan__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G2: nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G2 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) )
= ( plus_plus @ A @ ( G2 @ ( zero_zero @ nat ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ nat @ A @ nat @ G2 @ suc ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ) ).
% sum.atLeast0_lessThan_Suc_shift
thf(fact_7941_sum_OatLeast0__atMost__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G2: nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G2 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) )
= ( plus_plus @ A @ ( G2 @ ( zero_zero @ nat ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ nat @ A @ nat @ G2 @ suc ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ) ).
% sum.atLeast0_atMost_Suc_shift
thf(fact_7942_prod_OatLeast0__lessThan__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G2: nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) )
= ( times_times @ A @ ( G2 @ ( zero_zero @ nat ) ) @ ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ nat @ A @ nat @ G2 @ suc ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ) ).
% prod.atLeast0_lessThan_Suc_shift
thf(fact_7943_prod_OatLeast0__atMost__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G2: nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) )
= ( times_times @ A @ ( G2 @ ( zero_zero @ nat ) ) @ ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ nat @ A @ nat @ G2 @ suc ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ) ).
% prod.atLeast0_atMost_Suc_shift
thf(fact_7944_sum_OatLeastLessThan__shift__0,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G2: nat > A,M: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G2 @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ nat @ A @ nat @ G2 @ ( plus_plus @ nat @ M ) ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ N @ M ) ) ) ) ) ).
% sum.atLeastLessThan_shift_0
thf(fact_7945_prod_OatLeastLessThan__shift__0,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G2: nat > A,M: nat,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) )
= ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ nat @ A @ nat @ G2 @ ( plus_plus @ nat @ M ) ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ N @ M ) ) ) ) ) ).
% prod.atLeastLessThan_shift_0
thf(fact_7946_tendsto__compose__at,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( topolo4958980785337419405_space @ B )
& ( topolo4958980785337419405_space @ C ) )
=> ! [F3: A > B,Y: B,F4: filter @ A,G2: B > C,Z: C] :
( ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ Y ) @ F4 )
=> ( ( filterlim @ B @ C @ G2 @ ( topolo7230453075368039082e_nhds @ C @ Z ) @ ( topolo174197925503356063within @ B @ Y @ ( top_top @ ( set @ B ) ) ) )
=> ( ( eventually @ A
@ ^ [W2: A] :
( ( ( F3 @ W2 )
= Y )
=> ( ( G2 @ Y )
= Z ) )
@ F4 )
=> ( filterlim @ A @ C @ ( comp @ B @ C @ A @ G2 @ F3 ) @ ( topolo7230453075368039082e_nhds @ C @ Z ) @ F4 ) ) ) ) ) ).
% tendsto_compose_at
thf(fact_7947_sum_OatLeast__lessThan__pred__shift,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G2: nat > A,M: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ( comp @ nat @ A @ nat @ G2
@ ^ [N4: nat] : ( minus_minus @ nat @ N4 @ ( suc @ ( zero_zero @ nat ) ) ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M ) @ ( suc @ N ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ G2 @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) ) ) ).
% sum.atLeast_lessThan_pred_shift
thf(fact_7948_sum_OatLeast__atMost__pred__shift,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G2: nat > A,M: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ( comp @ nat @ A @ nat @ G2
@ ^ [N4: nat] : ( minus_minus @ nat @ N4 @ ( suc @ ( zero_zero @ nat ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ ( suc @ N ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ G2 @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ).
% sum.atLeast_atMost_pred_shift
thf(fact_7949_prod_OatLeast__lessThan__pred__shift,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G2: nat > A,M: nat,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A
@ ( comp @ nat @ A @ nat @ G2
@ ^ [N4: nat] : ( minus_minus @ nat @ N4 @ ( suc @ ( zero_zero @ nat ) ) ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M ) @ ( suc @ N ) ) )
= ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) ) ) ).
% prod.atLeast_lessThan_pred_shift
thf(fact_7950_prod_OatLeast__atMost__pred__shift,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G2: nat > A,M: nat,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A
@ ( comp @ nat @ A @ nat @ G2
@ ^ [N4: nat] : ( minus_minus @ nat @ N4 @ ( suc @ ( zero_zero @ nat ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ ( suc @ N ) ) )
= ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ).
% prod.atLeast_atMost_pred_shift
thf(fact_7951_sum_OatLeastAtMost__shift__0,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [M: nat,N: nat,G2: nat > A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G2 @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ nat @ A @ nat @ G2 @ ( plus_plus @ nat @ M ) ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ N @ M ) ) ) ) ) ) ).
% sum.atLeastAtMost_shift_0
thf(fact_7952_prod_OatLeastAtMost__shift__0,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M: nat,N: nat,G2: nat > A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G2 @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ nat @ A @ nat @ G2 @ ( plus_plus @ nat @ M ) ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ N @ M ) ) ) ) ) ) ).
% prod.atLeastAtMost_shift_0
thf(fact_7953_lim__at__infinity__0,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,L: A] :
( ( filterlim @ A @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ ( at_infinity @ A ) )
= ( filterlim @ A @ A @ ( comp @ A @ A @ A @ F3 @ ( inverse_inverse @ A ) ) @ ( topolo7230453075368039082e_nhds @ A @ L ) @ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% lim_at_infinity_0
thf(fact_7954_tendsto__at__iff__sequentially,axiom,
! [C: $tType,A: $tType] :
( ( ( topolo3112930676232923870pology @ A )
& ( topolo4958980785337419405_space @ C ) )
=> ! [F3: A > C,A3: C,X: A,S2: set @ A] :
( ( filterlim @ A @ C @ F3 @ ( topolo7230453075368039082e_nhds @ C @ A3 ) @ ( topolo174197925503356063within @ A @ X @ S2 ) )
= ( ! [X8: nat > A] :
( ! [I4: nat] : ( member @ A @ ( X8 @ I4 ) @ ( minus_minus @ ( set @ A ) @ S2 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( ( filterlim @ nat @ A @ X8 @ ( topolo7230453075368039082e_nhds @ A @ X ) @ ( at_top @ nat ) )
=> ( filterlim @ nat @ C @ ( comp @ A @ C @ nat @ F3 @ X8 ) @ ( topolo7230453075368039082e_nhds @ C @ A3 ) @ ( at_top @ nat ) ) ) ) ) ) ) ).
% tendsto_at_iff_sequentially
thf(fact_7955_uint32__msb__test__bit,axiom,
( ( most_s684356279273892711sb_msb @ uint32 )
= ( ^ [X2: uint32] : ( bit_se5641148757651400278ts_bit @ uint32 @ X2 @ ( numeral_numeral @ nat @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ).
% uint32_msb_test_bit
thf(fact_7956_map__filter__on__comp,axiom,
! [A: $tType,C: $tType,B: $tType,G2: B > A,Y8: set @ B,X5: set @ A,F4: filter @ B,F3: A > C] :
( ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ G2 @ Y8 ) @ X5 )
=> ( ( eventually @ B
@ ^ [X2: B] : ( member @ B @ X2 @ Y8 )
@ F4 )
=> ( ( map_filter_on @ A @ C @ X5 @ F3 @ ( map_filter_on @ B @ A @ Y8 @ G2 @ F4 ) )
= ( map_filter_on @ B @ C @ Y8 @ ( comp @ A @ C @ B @ F3 @ G2 ) @ F4 ) ) ) ) ).
% map_filter_on_comp
thf(fact_7957_complex__mult__cnj,axiom,
! [Z: complex] :
( ( times_times @ complex @ Z @ ( cnj @ Z ) )
= ( real_Vector_of_real @ complex @ ( plus_plus @ real @ ( power_power @ real @ ( re @ Z ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ ( im @ Z ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% complex_mult_cnj
thf(fact_7958_complex__cnj__zero,axiom,
( ( cnj @ ( zero_zero @ complex ) )
= ( zero_zero @ complex ) ) ).
% complex_cnj_zero
thf(fact_7959_complex__cnj__zero__iff,axiom,
! [Z: complex] :
( ( ( cnj @ Z )
= ( zero_zero @ complex ) )
= ( Z
= ( zero_zero @ complex ) ) ) ).
% complex_cnj_zero_iff
thf(fact_7960_complex__cnj__one__iff,axiom,
! [Z: complex] :
( ( ( cnj @ Z )
= ( one_one @ complex ) )
= ( Z
= ( one_one @ complex ) ) ) ).
% complex_cnj_one_iff
thf(fact_7961_complex__cnj__one,axiom,
( ( cnj @ ( one_one @ complex ) )
= ( one_one @ complex ) ) ).
% complex_cnj_one
thf(fact_7962_cnj__sum,axiom,
! [A: $tType,F3: A > complex,S2: set @ A] :
( ( cnj @ ( groups7311177749621191930dd_sum @ A @ complex @ F3 @ S2 ) )
= ( groups7311177749621191930dd_sum @ A @ complex
@ ^ [X2: A] : ( cnj @ ( F3 @ X2 ) )
@ S2 ) ) ).
% cnj_sum
thf(fact_7963_cnj__prod,axiom,
! [A: $tType,F3: A > complex,S2: set @ A] :
( ( cnj @ ( groups7121269368397514597t_prod @ A @ complex @ F3 @ S2 ) )
= ( groups7121269368397514597t_prod @ A @ complex
@ ^ [X2: A] : ( cnj @ ( F3 @ X2 ) )
@ S2 ) ) ).
% cnj_prod
thf(fact_7964_complex__In__mult__cnj__zero,axiom,
! [Z: complex] :
( ( im @ ( times_times @ complex @ Z @ ( cnj @ Z ) ) )
= ( zero_zero @ real ) ) ).
% complex_In_mult_cnj_zero
thf(fact_7965_eventually__map__filter__on,axiom,
! [B: $tType,A: $tType,X5: set @ A,F4: filter @ A,P: B > $o,F3: A > B] :
( ( eventually @ A
@ ^ [X2: A] : ( member @ A @ X2 @ X5 )
@ F4 )
=> ( ( eventually @ B @ P @ ( map_filter_on @ A @ B @ X5 @ F3 @ F4 ) )
= ( eventually @ A
@ ^ [X2: A] :
( ( P @ ( F3 @ X2 ) )
& ( member @ A @ X2 @ X5 ) )
@ F4 ) ) ) ).
% eventually_map_filter_on
thf(fact_7966_continuous__on__cnj,axiom,
! [C: $tType] :
( ( topolo4958980785337419405_space @ C )
=> ! [S2: set @ C,G2: C > complex] :
( ( topolo81223032696312382ous_on @ C @ complex @ S2 @ G2 )
=> ( topolo81223032696312382ous_on @ C @ complex @ S2
@ ^ [X2: C] : ( cnj @ ( G2 @ X2 ) ) ) ) ) ).
% continuous_on_cnj
thf(fact_7967_has__derivative__cnj,axiom,
! [C: $tType] :
( ( real_V822414075346904944vector @ C )
=> ! [G2: C > complex,G5: C > complex,F4: filter @ C] :
( ( has_derivative @ C @ complex @ G2 @ G5 @ F4 )
=> ( has_derivative @ C @ complex
@ ^ [X2: C] : ( cnj @ ( G2 @ X2 ) )
@ ^ [X2: C] : ( cnj @ ( G5 @ X2 ) )
@ F4 ) ) ) ).
% has_derivative_cnj
thf(fact_7968_sums__cnj,axiom,
! [F3: nat > complex,L: complex] :
( ( sums @ complex
@ ^ [X2: nat] : ( cnj @ ( F3 @ X2 ) )
@ ( cnj @ L ) )
= ( sums @ complex @ F3 @ L ) ) ).
% sums_cnj
thf(fact_7969_continuous__cnj,axiom,
! [C: $tType] :
( ( topological_t2_space @ C )
=> ! [F4: filter @ C,G2: C > complex] :
( ( topolo3448309680560233919inuous @ C @ complex @ F4 @ G2 )
=> ( topolo3448309680560233919inuous @ C @ complex @ F4
@ ^ [X2: C] : ( cnj @ ( G2 @ X2 ) ) ) ) ) ).
% continuous_cnj
thf(fact_7970_differentiable__cnj__iff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: A > complex,X: A,A2: set @ A] :
( ( differentiable @ A @ complex
@ ^ [Z4: A] : ( cnj @ ( F3 @ Z4 ) )
@ ( topolo174197925503356063within @ A @ X @ A2 ) )
= ( differentiable @ A @ complex @ F3 @ ( topolo174197925503356063within @ A @ X @ A2 ) ) ) ) ).
% differentiable_cnj_iff
thf(fact_7971_tendsto__cnj,axiom,
! [C: $tType,G2: C > complex,A3: complex,F4: filter @ C] :
( ( filterlim @ C @ complex @ G2 @ ( topolo7230453075368039082e_nhds @ complex @ A3 ) @ F4 )
=> ( filterlim @ C @ complex
@ ^ [X2: C] : ( cnj @ ( G2 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ complex @ ( cnj @ A3 ) )
@ F4 ) ) ).
% tendsto_cnj
thf(fact_7972_lim__cnj,axiom,
! [A: $tType,F3: A > complex,L: complex,F4: filter @ A] :
( ( filterlim @ A @ complex
@ ^ [X2: A] : ( cnj @ ( F3 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ complex @ ( cnj @ L ) )
@ F4 )
= ( filterlim @ A @ complex @ F3 @ ( topolo7230453075368039082e_nhds @ complex @ L ) @ F4 ) ) ).
% lim_cnj
thf(fact_7973_isCont__cnj,axiom,
! [C: $tType] :
( ( topological_t2_space @ C )
=> ! [A3: C,G2: C > complex] :
( ( topolo3448309680560233919inuous @ C @ complex @ ( topolo174197925503356063within @ C @ A3 @ ( top_top @ ( set @ C ) ) ) @ G2 )
=> ( topolo3448309680560233919inuous @ C @ complex @ ( topolo174197925503356063within @ C @ A3 @ ( top_top @ ( set @ C ) ) )
@ ^ [X2: C] : ( cnj @ ( G2 @ X2 ) ) ) ) ) ).
% isCont_cnj
thf(fact_7974_Re__complex__div__eq__0,axiom,
! [A3: complex,B3: complex] :
( ( ( re @ ( divide_divide @ complex @ A3 @ B3 ) )
= ( zero_zero @ real ) )
= ( ( re @ ( times_times @ complex @ A3 @ ( cnj @ B3 ) ) )
= ( zero_zero @ real ) ) ) ).
% Re_complex_div_eq_0
thf(fact_7975_Im__complex__div__eq__0,axiom,
! [A3: complex,B3: complex] :
( ( ( im @ ( divide_divide @ complex @ A3 @ B3 ) )
= ( zero_zero @ real ) )
= ( ( im @ ( times_times @ complex @ A3 @ ( cnj @ B3 ) ) )
= ( zero_zero @ real ) ) ) ).
% Im_complex_div_eq_0
thf(fact_7976_Re__complex__div__lt__0,axiom,
! [A3: complex,B3: complex] :
( ( ord_less @ real @ ( re @ ( divide_divide @ complex @ A3 @ B3 ) ) @ ( zero_zero @ real ) )
= ( ord_less @ real @ ( re @ ( times_times @ complex @ A3 @ ( cnj @ B3 ) ) ) @ ( zero_zero @ real ) ) ) ).
% Re_complex_div_lt_0
thf(fact_7977_Re__complex__div__gt__0,axiom,
! [A3: complex,B3: complex] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( re @ ( divide_divide @ complex @ A3 @ B3 ) ) )
= ( ord_less @ real @ ( zero_zero @ real ) @ ( re @ ( times_times @ complex @ A3 @ ( cnj @ B3 ) ) ) ) ) ).
% Re_complex_div_gt_0
thf(fact_7978_Re__complex__div__ge__0,axiom,
! [A3: complex,B3: complex] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( re @ ( divide_divide @ complex @ A3 @ B3 ) ) )
= ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( re @ ( times_times @ complex @ A3 @ ( cnj @ B3 ) ) ) ) ) ).
% Re_complex_div_ge_0
thf(fact_7979_Re__complex__div__le__0,axiom,
! [A3: complex,B3: complex] :
( ( ord_less_eq @ real @ ( re @ ( divide_divide @ complex @ A3 @ B3 ) ) @ ( zero_zero @ real ) )
= ( ord_less_eq @ real @ ( re @ ( times_times @ complex @ A3 @ ( cnj @ B3 ) ) ) @ ( zero_zero @ real ) ) ) ).
% Re_complex_div_le_0
thf(fact_7980_Im__complex__div__gt__0,axiom,
! [A3: complex,B3: complex] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( im @ ( divide_divide @ complex @ A3 @ B3 ) ) )
= ( ord_less @ real @ ( zero_zero @ real ) @ ( im @ ( times_times @ complex @ A3 @ ( cnj @ B3 ) ) ) ) ) ).
% Im_complex_div_gt_0
thf(fact_7981_Im__complex__div__lt__0,axiom,
! [A3: complex,B3: complex] :
( ( ord_less @ real @ ( im @ ( divide_divide @ complex @ A3 @ B3 ) ) @ ( zero_zero @ real ) )
= ( ord_less @ real @ ( im @ ( times_times @ complex @ A3 @ ( cnj @ B3 ) ) ) @ ( zero_zero @ real ) ) ) ).
% Im_complex_div_lt_0
thf(fact_7982_Im__complex__div__le__0,axiom,
! [A3: complex,B3: complex] :
( ( ord_less_eq @ real @ ( im @ ( divide_divide @ complex @ A3 @ B3 ) ) @ ( zero_zero @ real ) )
= ( ord_less_eq @ real @ ( im @ ( times_times @ complex @ A3 @ ( cnj @ B3 ) ) ) @ ( zero_zero @ real ) ) ) ).
% Im_complex_div_le_0
thf(fact_7983_Im__complex__div__ge__0,axiom,
! [A3: complex,B3: complex] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( im @ ( divide_divide @ complex @ A3 @ B3 ) ) )
= ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( im @ ( times_times @ complex @ A3 @ ( cnj @ B3 ) ) ) ) ) ).
% Im_complex_div_ge_0
thf(fact_7984_summable__reindex,axiom,
! [F3: nat > real,G2: nat > nat] :
( ( summable @ real @ F3 )
=> ( ( inj_on @ nat @ nat @ G2 @ ( top_top @ ( set @ nat ) ) )
=> ( ! [X3: nat] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( F3 @ X3 ) )
=> ( summable @ real @ ( comp @ nat @ real @ nat @ F3 @ G2 ) ) ) ) ) ).
% summable_reindex
thf(fact_7985_sum_OatLeast__int__atMost__int__shift,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G2: int > A,M: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ int @ A @ G2 @ ( set_or1337092689740270186AtMost @ int @ ( semiring_1_of_nat @ int @ M ) @ ( semiring_1_of_nat @ int @ N ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ int @ A @ nat @ G2 @ ( semiring_1_of_nat @ int ) ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ).
% sum.atLeast_int_atMost_int_shift
thf(fact_7986_sum_OatLeast__int__lessThan__int__shift,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G2: int > A,M: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ int @ A @ G2 @ ( set_or7035219750837199246ssThan @ int @ ( semiring_1_of_nat @ int @ M ) @ ( semiring_1_of_nat @ int @ N ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ int @ A @ nat @ G2 @ ( semiring_1_of_nat @ int ) ) @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) ) ) ).
% sum.atLeast_int_lessThan_int_shift
thf(fact_7987_prod_OatLeast__int__atMost__int__shift,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G2: int > A,M: nat,N: nat] :
( ( groups7121269368397514597t_prod @ int @ A @ G2 @ ( set_or1337092689740270186AtMost @ int @ ( semiring_1_of_nat @ int @ M ) @ ( semiring_1_of_nat @ int @ N ) ) )
= ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ int @ A @ nat @ G2 @ ( semiring_1_of_nat @ int ) ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ).
% prod.atLeast_int_atMost_int_shift
thf(fact_7988_prod_OatLeast__int__lessThan__int__shift,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G2: int > A,M: nat,N: nat] :
( ( groups7121269368397514597t_prod @ int @ A @ G2 @ ( set_or7035219750837199246ssThan @ int @ ( semiring_1_of_nat @ int @ M ) @ ( semiring_1_of_nat @ int @ N ) ) )
= ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ int @ A @ nat @ G2 @ ( semiring_1_of_nat @ int ) ) @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) ) ) ).
% prod.atLeast_int_lessThan_int_shift
thf(fact_7989_complex__mod__mult__cnj,axiom,
! [Z: complex] :
( ( real_V7770717601297561774m_norm @ complex @ ( times_times @ complex @ Z @ ( cnj @ Z ) ) )
= ( power_power @ real @ ( real_V7770717601297561774m_norm @ complex @ Z ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% complex_mod_mult_cnj
thf(fact_7990_complex__div__gt__0,axiom,
! [A3: complex,B3: complex] :
( ( ( ord_less @ real @ ( zero_zero @ real ) @ ( re @ ( divide_divide @ complex @ A3 @ B3 ) ) )
= ( ord_less @ real @ ( zero_zero @ real ) @ ( re @ ( times_times @ complex @ A3 @ ( cnj @ B3 ) ) ) ) )
& ( ( ord_less @ real @ ( zero_zero @ real ) @ ( im @ ( divide_divide @ complex @ A3 @ B3 ) ) )
= ( ord_less @ real @ ( zero_zero @ real ) @ ( im @ ( times_times @ complex @ A3 @ ( cnj @ B3 ) ) ) ) ) ) ).
% complex_div_gt_0
thf(fact_7991_suminf__reindex__mono,axiom,
! [F3: nat > real,G2: nat > nat] :
( ( summable @ real @ F3 )
=> ( ( inj_on @ nat @ nat @ G2 @ ( top_top @ ( set @ nat ) ) )
=> ( ! [X3: nat] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( F3 @ X3 ) )
=> ( ord_less_eq @ real @ ( suminf @ real @ ( comp @ nat @ real @ nat @ F3 @ G2 ) ) @ ( suminf @ real @ F3 ) ) ) ) ) ).
% suminf_reindex_mono
thf(fact_7992_complex__norm__square,axiom,
! [Z: complex] :
( ( real_Vector_of_real @ complex @ ( power_power @ real @ ( real_V7770717601297561774m_norm @ complex @ Z ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( times_times @ complex @ Z @ ( cnj @ Z ) ) ) ).
% complex_norm_square
thf(fact_7993_complex__add__cnj,axiom,
! [Z: complex] :
( ( plus_plus @ complex @ Z @ ( cnj @ Z ) )
= ( real_Vector_of_real @ complex @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( re @ Z ) ) ) ) ).
% complex_add_cnj
thf(fact_7994_filterlim__at__top__iff__inverse__0,axiom,
! [A: $tType,F3: A > real,F4: filter @ A] :
( ( eventually @ A
@ ^ [X2: A] : ( ord_less @ real @ ( zero_zero @ real ) @ ( F3 @ X2 ) )
@ F4 )
=> ( ( filterlim @ A @ real @ F3 @ ( at_top @ real ) @ F4 )
= ( filterlim @ A @ real @ ( comp @ real @ real @ A @ ( inverse_inverse @ real ) @ F3 ) @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ F4 ) ) ) ).
% filterlim_at_top_iff_inverse_0
thf(fact_7995_suminf__reindex,axiom,
! [F3: nat > real,G2: nat > nat] :
( ( summable @ real @ F3 )
=> ( ( inj_on @ nat @ nat @ G2 @ ( top_top @ ( set @ nat ) ) )
=> ( ! [X3: nat] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( F3 @ X3 ) )
=> ( ! [X3: nat] :
( ~ ( member @ nat @ X3 @ ( image @ nat @ nat @ G2 @ ( top_top @ ( set @ nat ) ) ) )
=> ( ( F3 @ X3 )
= ( zero_zero @ real ) ) )
=> ( ( suminf @ real @ ( comp @ nat @ real @ nat @ F3 @ G2 ) )
= ( suminf @ real @ F3 ) ) ) ) ) ) ).
% suminf_reindex
thf(fact_7996_complex__diff__cnj,axiom,
! [Z: complex] :
( ( minus_minus @ complex @ Z @ ( cnj @ Z ) )
= ( times_times @ complex @ ( real_Vector_of_real @ complex @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( im @ Z ) ) ) @ imaginary_unit ) ) ).
% complex_diff_cnj
thf(fact_7997_complex__div__cnj,axiom,
( ( divide_divide @ complex )
= ( ^ [A5: complex,B5: complex] : ( divide_divide @ complex @ ( times_times @ complex @ A5 @ ( cnj @ B5 ) ) @ ( real_Vector_of_real @ complex @ ( power_power @ real @ ( real_V7770717601297561774m_norm @ complex @ B5 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% complex_div_cnj
thf(fact_7998_cnj__add__mult__eq__Re,axiom,
! [Z: complex,W: complex] :
( ( plus_plus @ complex @ ( times_times @ complex @ Z @ ( cnj @ W ) ) @ ( times_times @ complex @ ( cnj @ Z ) @ W ) )
= ( real_Vector_of_real @ complex @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( re @ ( times_times @ complex @ Z @ ( cnj @ W ) ) ) ) ) ) ).
% cnj_add_mult_eq_Re
thf(fact_7999_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_8000_divmod__integer__eq__cases,axiom,
( code_divmod_integer
= ( ^ [K3: code_integer,L2: code_integer] :
( if @ ( product_prod @ code_integer @ code_integer )
@ ( K3
= ( zero_zero @ code_integer ) )
@ ( product_Pair @ code_integer @ code_integer @ ( zero_zero @ code_integer ) @ ( zero_zero @ code_integer ) )
@ ( if @ ( product_prod @ code_integer @ code_integer )
@ ( L2
= ( zero_zero @ code_integer ) )
@ ( product_Pair @ code_integer @ code_integer @ ( zero_zero @ code_integer ) @ K3 )
@ ( comp @ code_integer @ ( ( product_prod @ code_integer @ code_integer ) > ( product_prod @ code_integer @ code_integer ) ) @ code_integer @ ( comp @ ( code_integer > code_integer ) @ ( ( product_prod @ code_integer @ code_integer ) > ( product_prod @ code_integer @ code_integer ) ) @ code_integer @ ( product_apsnd @ code_integer @ code_integer @ code_integer ) @ ( times_times @ code_integer ) ) @ ( sgn_sgn @ code_integer ) @ L2
@ ( if @ ( product_prod @ code_integer @ code_integer )
@ ( ( sgn_sgn @ code_integer @ K3 )
= ( sgn_sgn @ code_integer @ L2 ) )
@ ( code_divmod_abs @ K3 @ L2 )
@ ( product_case_prod @ code_integer @ code_integer @ ( product_prod @ code_integer @ code_integer )
@ ^ [R5: code_integer,S5: code_integer] :
( if @ ( product_prod @ code_integer @ code_integer )
@ ( S5
= ( zero_zero @ code_integer ) )
@ ( product_Pair @ code_integer @ code_integer @ ( uminus_uminus @ code_integer @ R5 ) @ ( zero_zero @ code_integer ) )
@ ( product_Pair @ code_integer @ code_integer @ ( minus_minus @ code_integer @ ( uminus_uminus @ code_integer @ R5 ) @ ( one_one @ code_integer ) ) @ ( minus_minus @ code_integer @ ( abs_abs @ code_integer @ L2 ) @ S5 ) ) )
@ ( code_divmod_abs @ K3 @ L2 ) ) ) ) ) ) ) ) ).
% divmod_integer_eq_cases
thf(fact_8001_divmod__abs__code_I6_J,axiom,
! [J: code_integer] :
( ( code_divmod_abs @ ( zero_zero @ code_integer ) @ J )
= ( product_Pair @ code_integer @ code_integer @ ( zero_zero @ code_integer ) @ ( zero_zero @ code_integer ) ) ) ).
% divmod_abs_code(6)
thf(fact_8002_divmod__abs__code_I5_J,axiom,
! [J: code_integer] :
( ( code_divmod_abs @ J @ ( zero_zero @ code_integer ) )
= ( product_Pair @ code_integer @ code_integer @ ( zero_zero @ code_integer ) @ ( abs_abs @ code_integer @ J ) ) ) ).
% divmod_abs_code(5)
thf(fact_8003_divmod__integer__code,axiom,
( code_divmod_integer
= ( ^ [K3: code_integer,L2: code_integer] :
( if @ ( product_prod @ code_integer @ code_integer )
@ ( K3
= ( zero_zero @ code_integer ) )
@ ( product_Pair @ code_integer @ code_integer @ ( zero_zero @ code_integer ) @ ( zero_zero @ code_integer ) )
@ ( if @ ( product_prod @ code_integer @ code_integer ) @ ( ord_less @ code_integer @ ( zero_zero @ code_integer ) @ L2 )
@ ( if @ ( product_prod @ code_integer @ code_integer ) @ ( ord_less @ code_integer @ ( zero_zero @ code_integer ) @ K3 ) @ ( code_divmod_abs @ K3 @ L2 )
@ ( product_case_prod @ code_integer @ code_integer @ ( product_prod @ code_integer @ code_integer )
@ ^ [R5: code_integer,S5: code_integer] :
( if @ ( product_prod @ code_integer @ code_integer )
@ ( S5
= ( zero_zero @ code_integer ) )
@ ( product_Pair @ code_integer @ code_integer @ ( uminus_uminus @ code_integer @ R5 ) @ ( zero_zero @ code_integer ) )
@ ( product_Pair @ code_integer @ code_integer @ ( minus_minus @ code_integer @ ( uminus_uminus @ code_integer @ R5 ) @ ( one_one @ code_integer ) ) @ ( minus_minus @ code_integer @ L2 @ S5 ) ) )
@ ( code_divmod_abs @ K3 @ L2 ) ) )
@ ( if @ ( product_prod @ code_integer @ code_integer )
@ ( L2
= ( zero_zero @ code_integer ) )
@ ( product_Pair @ code_integer @ code_integer @ ( zero_zero @ code_integer ) @ K3 )
@ ( product_apsnd @ code_integer @ code_integer @ code_integer @ ( uminus_uminus @ code_integer )
@ ( if @ ( product_prod @ code_integer @ code_integer ) @ ( ord_less @ code_integer @ K3 @ ( zero_zero @ code_integer ) ) @ ( code_divmod_abs @ K3 @ L2 )
@ ( product_case_prod @ code_integer @ code_integer @ ( product_prod @ code_integer @ code_integer )
@ ^ [R5: code_integer,S5: code_integer] :
( if @ ( product_prod @ code_integer @ code_integer )
@ ( S5
= ( zero_zero @ code_integer ) )
@ ( product_Pair @ code_integer @ code_integer @ ( uminus_uminus @ code_integer @ R5 ) @ ( zero_zero @ code_integer ) )
@ ( product_Pair @ code_integer @ code_integer @ ( minus_minus @ code_integer @ ( uminus_uminus @ code_integer @ R5 ) @ ( one_one @ code_integer ) ) @ ( minus_minus @ code_integer @ ( uminus_uminus @ code_integer @ L2 ) @ S5 ) ) )
@ ( code_divmod_abs @ K3 @ L2 ) ) ) ) ) ) ) ) ) ).
% divmod_integer_code
thf(fact_8004_bit__cut__integer__code,axiom,
( code_bit_cut_integer
= ( ^ [K3: code_integer] :
( if @ ( product_prod @ code_integer @ $o )
@ ( K3
= ( zero_zero @ code_integer ) )
@ ( product_Pair @ code_integer @ $o @ ( zero_zero @ code_integer ) @ $false )
@ ( product_case_prod @ code_integer @ code_integer @ ( product_prod @ code_integer @ $o )
@ ^ [R5: code_integer,S5: code_integer] :
( product_Pair @ code_integer @ $o @ ( if @ code_integer @ ( ord_less @ code_integer @ ( zero_zero @ code_integer ) @ K3 ) @ R5 @ ( minus_minus @ code_integer @ ( uminus_uminus @ code_integer @ R5 ) @ S5 ) )
@ ( S5
= ( one_one @ code_integer ) ) )
@ ( code_divmod_abs @ K3 @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% bit_cut_integer_code
thf(fact_8005_integer__of__uint32__code,axiom,
( integer_of_uint32
= ( ^ [N4: uint32] : ( if @ code_integer @ ( bit_se5641148757651400278ts_bit @ uint32 @ N4 @ ( numeral_numeral @ nat @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) ) @ ( bit_se1065995026697491101ons_or @ code_integer @ ( intege5370686899274169573signed @ ( bit_se5824344872417868541ns_and @ uint32 @ N4 @ ( numeral_numeral @ uint32 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numeral_numeral @ code_integer @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( intege5370686899274169573signed @ N4 ) ) ) ) ).
% integer_of_uint32_code
thf(fact_8006_integer__of__uint32__signed__def,axiom,
( intege5370686899274169573signed
= ( ^ [N4: uint32] : ( if @ code_integer @ ( bit_se5641148757651400278ts_bit @ uint32 @ N4 @ ( numeral_numeral @ nat @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) ) @ ( undefined @ ( ( uint32 > code_integer ) > uint32 > code_integer ) @ integer_of_uint32 @ N4 ) @ ( integer_of_uint32 @ N4 ) ) ) ) ).
% integer_of_uint32_signed_def
thf(fact_8007_bit__cut__integer__def,axiom,
( code_bit_cut_integer
= ( ^ [K3: code_integer] :
( product_Pair @ code_integer @ $o @ ( divide_divide @ code_integer @ K3 @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) )
@ ~ ( dvd_dvd @ code_integer @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) @ K3 ) ) ) ) ).
% bit_cut_integer_def
thf(fact_8008_integer__of__uint32__signed__code,axiom,
( intege5370686899274169573signed
= ( ^ [N4: uint32] : ( if @ code_integer @ ( bit_se5641148757651400278ts_bit @ uint32 @ N4 @ ( numeral_numeral @ nat @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) ) @ ( undefined @ ( ( uint32 > code_integer ) > uint32 > code_integer ) @ integer_of_uint32 @ N4 ) @ ( code_integer_of_int @ ( semiring_1_unsigned @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ numeral_num1 ) ) ) ) ) @ int @ ( rep_uint32 @ N4 ) ) ) ) ) ) ).
% integer_of_uint32_signed_code
thf(fact_8009_uint32__sdiv__def,axiom,
( uint32_sdiv
= ( ^ [X2: uint32,Y2: uint32] :
( if @ uint32
@ ( Y2
= ( zero_zero @ uint32 ) )
@ ( undefined @ ( ( uint32 > uint32 > uint32 ) > uint32 > uint32 > uint32 ) @ ( divide_divide @ uint32 ) @ X2 @ ( zero_zero @ uint32 ) )
@ ( abs_uint32 @ ( signed7115095781618012415divide @ ( word @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ numeral_num1 ) ) ) ) ) ) @ ( rep_uint322 @ X2 ) @ ( rep_uint322 @ Y2 ) ) ) ) ) ) ).
% uint32_sdiv_def
thf(fact_8010_one__uint32__def,axiom,
( ( one_one @ uint32 )
= ( abs_uint32 @ ( one_one @ ( word @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ numeral_num1 ) ) ) ) ) ) ) ) ) ).
% one_uint32_def
thf(fact_8011_zero__uint32__def,axiom,
( ( zero_zero @ uint32 )
= ( abs_uint32 @ ( zero_zero @ ( word @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ numeral_num1 ) ) ) ) ) ) ) ) ) ).
% zero_uint32_def
thf(fact_8012_less__uint32_Oabs__eq,axiom,
! [Xa: word @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ numeral_num1 ) ) ) ) ),X: word @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ numeral_num1 ) ) ) ) )] :
( ( ord_less @ uint32 @ ( abs_uint32 @ Xa ) @ ( abs_uint32 @ X ) )
= ( ord_less @ ( word @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ numeral_num1 ) ) ) ) ) ) @ Xa @ X ) ) ).
% less_uint32.abs_eq
thf(fact_8013_Rep__uint32_H__code,axiom,
( rep_uint32
= ( ^ [X2: uint32] : ( bit_bi4170147762399595738t_bits @ ( word @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ numeral_num1 ) ) ) ) ) ) @ ( bit_se5641148757651400278ts_bit @ uint32 @ X2 ) ) ) ) ).
% Rep_uint32'_code
thf(fact_8014_empty__natural,axiom,
! [C: $tType,B: $tType,D: $tType,A: $tType,F3: A > C,G2: D > B] :
( ( comp @ C @ ( set @ B ) @ A
@ ^ [Uu: C] : ( bot_bot @ ( set @ B ) )
@ F3 )
= ( comp @ ( set @ D ) @ ( set @ B ) @ A @ ( image @ D @ B @ G2 )
@ ^ [Uu: A] : ( bot_bot @ ( set @ D ) ) ) ) ).
% empty_natural
thf(fact_8015_even__sum__iff,axiom,
! [A: $tType,B: $tType] :
( ( semiring_parity @ A )
=> ! [A2: set @ B,F3: B > A] :
( ( finite_finite2 @ B @ A2 )
=> ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ A2 ) )
= ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) )
@ ( finite_card @ B
@ ( collect @ B
@ ^ [A5: B] :
( ( member @ B @ A5 @ A2 )
& ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( F3 @ A5 ) ) ) ) ) ) ) ) ) ).
% even_sum_iff
thf(fact_8016_card__UNIV__unit,axiom,
( ( finite_card @ product_unit @ ( top_top @ ( set @ product_unit ) ) )
= ( one_one @ nat ) ) ).
% card_UNIV_unit
thf(fact_8017_card__Collect__less__nat,axiom,
! [N: nat] :
( ( finite_card @ nat
@ ( collect @ nat
@ ^ [I4: nat] : ( ord_less @ nat @ I4 @ N ) ) )
= N ) ).
% card_Collect_less_nat
thf(fact_8018_card__Collect__le__nat,axiom,
! [N: nat] :
( ( finite_card @ nat
@ ( collect @ nat
@ ^ [I4: nat] : ( ord_less_eq @ nat @ I4 @ N ) ) )
= ( suc @ N ) ) ).
% card_Collect_le_nat
thf(fact_8019_card__UNIV__bool,axiom,
( ( finite_card @ $o @ ( top_top @ ( set @ $o ) ) )
= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ).
% card_UNIV_bool
thf(fact_8020_card_Oempty,axiom,
! [A: $tType] :
( ( finite_card @ A @ ( bot_bot @ ( set @ A ) ) )
= ( zero_zero @ nat ) ) ).
% card.empty
thf(fact_8021_card_Oinfinite,axiom,
! [A: $tType,A2: set @ A] :
( ~ ( finite_finite2 @ A @ A2 )
=> ( ( finite_card @ A @ A2 )
= ( zero_zero @ nat ) ) ) ).
% card.infinite
thf(fact_8022_prod__constant,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [Y: A,A2: set @ B] :
( ( groups7121269368397514597t_prod @ B @ A
@ ^ [X2: B] : Y
@ A2 )
= ( power_power @ A @ Y @ ( finite_card @ B @ A2 ) ) ) ) ).
% prod_constant
thf(fact_8023_card__0__eq,axiom,
! [A: $tType,A2: set @ A] :
( ( finite_finite2 @ A @ A2 )
=> ( ( ( finite_card @ A @ A2 )
= ( zero_zero @ nat ) )
= ( A2
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% card_0_eq
thf(fact_8024_card__insert__disjoint,axiom,
! [A: $tType,A2: set @ A,X: A] :
( ( finite_finite2 @ A @ A2 )
=> ( ~ ( member @ A @ X @ A2 )
=> ( ( finite_card @ A @ ( insert @ A @ X @ A2 ) )
= ( suc @ ( finite_card @ A @ A2 ) ) ) ) ) ).
% card_insert_disjoint
thf(fact_8025_sum__constant,axiom,
! [B: $tType,A: $tType] :
( ( semiring_1 @ A )
=> ! [Y: A,A2: set @ B] :
( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X2: B] : Y
@ A2 )
= ( times_times @ A @ ( semiring_1_of_nat @ A @ ( finite_card @ B @ A2 ) ) @ Y ) ) ) ).
% sum_constant
thf(fact_8026_card__Diff__insert,axiom,
! [A: $tType,A3: A,A2: set @ A,B2: set @ A] :
( ( member @ A @ A3 @ A2 )
=> ( ~ ( member @ A @ A3 @ B2 )
=> ( ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A2 @ ( insert @ A @ A3 @ B2 ) ) )
= ( minus_minus @ nat @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A2 @ B2 ) ) @ ( one_one @ nat ) ) ) ) ) ).
% card_Diff_insert
thf(fact_8027_card__atLeastAtMost__int,axiom,
! [L: int,U2: int] :
( ( finite_card @ int @ ( set_or1337092689740270186AtMost @ int @ L @ U2 ) )
= ( nat2 @ ( plus_plus @ int @ ( minus_minus @ int @ U2 @ L ) @ ( one_one @ int ) ) ) ) ).
% card_atLeastAtMost_int
thf(fact_8028_card__doubleton__eq__2__iff,axiom,
! [A: $tType,A3: A,B3: A] :
( ( ( finite_card @ A @ ( insert @ A @ A3 @ ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( A3 != B3 ) ) ).
% card_doubleton_eq_2_iff
thf(fact_8029_card__greaterThanLessThan__int,axiom,
! [L: int,U2: int] :
( ( finite_card @ int @ ( set_or5935395276787703475ssThan @ int @ L @ U2 ) )
= ( nat2 @ ( minus_minus @ int @ U2 @ ( plus_plus @ int @ L @ ( one_one @ int ) ) ) ) ) ).
% card_greaterThanLessThan_int
thf(fact_8030_card__eq__sum,axiom,
! [A: $tType] :
( ( finite_card @ A )
= ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [X2: A] : ( one_one @ nat ) ) ) ).
% card_eq_sum
thf(fact_8031_sum__multicount__gen,axiom,
! [A: $tType,B: $tType,S2: set @ A,T: set @ B,R: A > B > $o,K: B > nat] :
( ( finite_finite2 @ A @ S2 )
=> ( ( finite_finite2 @ B @ T )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ T )
=> ( ( finite_card @ A
@ ( collect @ A
@ ^ [I4: A] :
( ( member @ A @ I4 @ S2 )
& ( R @ I4 @ X3 ) ) ) )
= ( K @ X3 ) ) )
=> ( ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [I4: A] :
( finite_card @ B
@ ( collect @ B
@ ^ [J3: B] :
( ( member @ B @ J3 @ T )
& ( R @ I4 @ J3 ) ) ) )
@ S2 )
= ( groups7311177749621191930dd_sum @ B @ nat @ K @ T ) ) ) ) ) ).
% sum_multicount_gen
thf(fact_8032_card__less__sym__Diff,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( finite_finite2 @ A @ A2 )
=> ( ( finite_finite2 @ A @ B2 )
=> ( ( ord_less @ nat @ ( finite_card @ A @ A2 ) @ ( finite_card @ A @ B2 ) )
=> ( ord_less @ nat @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A2 @ B2 ) ) @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ B2 @ A2 ) ) ) ) ) ) ).
% card_less_sym_Diff
thf(fact_8033_card__map__elide,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat] :
( ( ord_less_eq @ nat @ N @ ( finite_card @ ( word @ A ) @ ( top_top @ ( set @ ( word @ A ) ) ) ) )
=> ( ( finite_card @ ( word @ A ) @ ( image @ nat @ ( word @ A ) @ ( semiring_1_of_nat @ ( word @ A ) ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) )
= ( finite_card @ nat @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ) ).
% card_map_elide
thf(fact_8034_card__lists__length__eq,axiom,
! [A: $tType,A2: set @ A,N: nat] :
( ( finite_finite2 @ A @ A2 )
=> ( ( finite_card @ ( list @ A )
@ ( collect @ ( list @ A )
@ ^ [Xs3: list @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs3 ) @ A2 )
& ( ( size_size @ ( list @ A ) @ Xs3 )
= N ) ) ) )
= ( power_power @ nat @ ( finite_card @ A @ A2 ) @ N ) ) ) ).
% card_lists_length_eq
thf(fact_8035_n__subsets,axiom,
! [A: $tType,A2: set @ A,K: nat] :
( ( finite_finite2 @ A @ A2 )
=> ( ( finite_card @ ( set @ A )
@ ( collect @ ( set @ A )
@ ^ [B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ B4 @ A2 )
& ( ( finite_card @ A @ B4 )
= K ) ) ) )
= ( binomial @ ( finite_card @ A @ A2 ) @ K ) ) ) ).
% n_subsets
thf(fact_8036_card__Un__le,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] : ( ord_less_eq @ nat @ ( finite_card @ A @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) ) @ ( plus_plus @ nat @ ( finite_card @ A @ A2 ) @ ( finite_card @ A @ B2 ) ) ) ).
% card_Un_le
thf(fact_8037_card__insert__le,axiom,
! [A: $tType,A2: set @ A,X: A] : ( ord_less_eq @ nat @ ( finite_card @ A @ A2 ) @ ( finite_card @ A @ ( insert @ A @ X @ A2 ) ) ) ).
% card_insert_le
thf(fact_8038_card__length,axiom,
! [A: $tType,Xs: list @ A] : ( ord_less_eq @ nat @ ( finite_card @ A @ ( set2 @ A @ Xs ) ) @ ( size_size @ ( list @ A ) @ Xs ) ) ).
% card_length
thf(fact_8039_card__insert__if,axiom,
! [A: $tType,A2: set @ A,X: A] :
( ( finite_finite2 @ A @ A2 )
=> ( ( ( member @ A @ X @ A2 )
=> ( ( finite_card @ A @ ( insert @ A @ X @ A2 ) )
= ( finite_card @ A @ A2 ) ) )
& ( ~ ( member @ A @ X @ A2 )
=> ( ( finite_card @ A @ ( insert @ A @ X @ A2 ) )
= ( suc @ ( finite_card @ A @ A2 ) ) ) ) ) ) ).
% card_insert_if
thf(fact_8040_card__Suc__eq__finite,axiom,
! [A: $tType,A2: set @ A,K: nat] :
( ( ( finite_card @ A @ A2 )
= ( suc @ K ) )
= ( ? [B5: A,B4: set @ A] :
( ( A2
= ( insert @ A @ B5 @ B4 ) )
& ~ ( member @ A @ B5 @ B4 )
& ( ( finite_card @ A @ B4 )
= K )
& ( finite_finite2 @ A @ B4 ) ) ) ) ).
% card_Suc_eq_finite
thf(fact_8041_card__atLeastZeroLessThan__int,axiom,
! [U2: int] :
( ( finite_card @ int @ ( set_or7035219750837199246ssThan @ int @ ( zero_zero @ int ) @ U2 ) )
= ( nat2 @ U2 ) ) ).
% card_atLeastZeroLessThan_int
thf(fact_8042_is__singleton__altdef,axiom,
! [A: $tType] :
( ( is_singleton @ A )
= ( ^ [A4: set @ A] :
( ( finite_card @ A @ A4 )
= ( one_one @ nat ) ) ) ) ).
% is_singleton_altdef
thf(fact_8043_card__2__iff_H,axiom,
! [A: $tType,S: set @ A] :
( ( ( finite_card @ A @ S )
= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( ? [X2: A] :
( ( member @ A @ X2 @ S )
& ? [Y2: A] :
( ( member @ A @ Y2 @ S )
& ( X2 != Y2 )
& ! [Z4: A] :
( ( member @ A @ Z4 @ S )
=> ( ( Z4 = X2 )
| ( Z4 = Y2 ) ) ) ) ) ) ) ).
% card_2_iff'
thf(fact_8044_card__1__singletonE,axiom,
! [A: $tType,A2: set @ A] :
( ( ( finite_card @ A @ A2 )
= ( one_one @ nat ) )
=> ~ ! [X3: A] :
( A2
!= ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% card_1_singletonE
thf(fact_8045_psubset__card__mono,axiom,
! [A: $tType,B2: set @ A,A2: set @ A] :
( ( finite_finite2 @ A @ B2 )
=> ( ( ord_less @ ( set @ A ) @ A2 @ B2 )
=> ( ord_less @ nat @ ( finite_card @ A @ A2 ) @ ( finite_card @ A @ B2 ) ) ) ) ).
% psubset_card_mono
thf(fact_8046_card__eq__0__iff,axiom,
! [A: $tType,A2: set @ A] :
( ( ( finite_card @ A @ A2 )
= ( zero_zero @ nat ) )
= ( ( A2
= ( bot_bot @ ( set @ A ) ) )
| ~ ( finite_finite2 @ A @ A2 ) ) ) ).
% card_eq_0_iff
thf(fact_8047_card__ge__0__finite,axiom,
! [A: $tType,A2: set @ A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( finite_card @ A @ A2 ) )
=> ( finite_finite2 @ A @ A2 ) ) ).
% card_ge_0_finite
thf(fact_8048_card__less,axiom,
! [M7: set @ nat,I: nat] :
( ( member @ nat @ ( zero_zero @ nat ) @ M7 )
=> ( ( finite_card @ nat
@ ( collect @ nat
@ ^ [K3: nat] :
( ( member @ nat @ K3 @ M7 )
& ( ord_less @ nat @ K3 @ ( suc @ I ) ) ) ) )
!= ( zero_zero @ nat ) ) ) ).
% card_less
thf(fact_8049_card__less__Suc,axiom,
! [M7: set @ nat,I: nat] :
( ( member @ nat @ ( zero_zero @ nat ) @ M7 )
=> ( ( suc
@ ( finite_card @ nat
@ ( collect @ nat
@ ^ [K3: nat] :
( ( member @ nat @ ( suc @ K3 ) @ M7 )
& ( ord_less @ nat @ K3 @ I ) ) ) ) )
= ( finite_card @ nat
@ ( collect @ nat
@ ^ [K3: nat] :
( ( member @ nat @ K3 @ M7 )
& ( ord_less @ nat @ K3 @ ( suc @ I ) ) ) ) ) ) ) ).
% card_less_Suc
thf(fact_8050_card__less__Suc2,axiom,
! [M7: set @ nat,I: nat] :
( ~ ( member @ nat @ ( zero_zero @ nat ) @ M7 )
=> ( ( finite_card @ nat
@ ( collect @ nat
@ ^ [K3: nat] :
( ( member @ nat @ ( suc @ K3 ) @ M7 )
& ( ord_less @ nat @ K3 @ I ) ) ) )
= ( finite_card @ nat
@ ( collect @ nat
@ ^ [K3: nat] :
( ( member @ nat @ K3 @ M7 )
& ( ord_less @ nat @ K3 @ ( suc @ I ) ) ) ) ) ) ) ).
% card_less_Suc2
thf(fact_8051_pigeonhole,axiom,
! [A: $tType,B: $tType,F3: B > A,A2: set @ B] :
( ( ord_less @ nat @ ( finite_card @ A @ ( image @ B @ A @ F3 @ A2 ) ) @ ( finite_card @ B @ A2 ) )
=> ~ ( inj_on @ B @ A @ F3 @ A2 ) ) ).
% pigeonhole
thf(fact_8052_sum__Suc,axiom,
! [A: $tType,F3: A > nat,A2: set @ A] :
( ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [X2: A] : ( suc @ ( F3 @ X2 ) )
@ A2 )
= ( plus_plus @ nat @ ( groups7311177749621191930dd_sum @ A @ nat @ F3 @ A2 ) @ ( finite_card @ A @ A2 ) ) ) ).
% sum_Suc
thf(fact_8053_sum__constant__scaleR,axiom,
! [C: $tType,A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [Y: A,A2: set @ C] :
( ( groups7311177749621191930dd_sum @ C @ A
@ ^ [X2: C] : Y
@ A2 )
= ( real_V8093663219630862766scaleR @ A @ ( semiring_1_of_nat @ real @ ( finite_card @ C @ A2 ) ) @ Y ) ) ) ).
% sum_constant_scaleR
thf(fact_8054_sum__multicount,axiom,
! [A: $tType,B: $tType,S: set @ A,T4: set @ B,R: A > B > $o,K: nat] :
( ( finite_finite2 @ A @ S )
=> ( ( finite_finite2 @ B @ T4 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ T4 )
=> ( ( finite_card @ A
@ ( collect @ A
@ ^ [I4: A] :
( ( member @ A @ I4 @ S )
& ( R @ I4 @ X3 ) ) ) )
= K ) )
=> ( ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [I4: A] :
( finite_card @ B
@ ( collect @ B
@ ^ [J3: B] :
( ( member @ B @ J3 @ T4 )
& ( R @ I4 @ J3 ) ) ) )
@ S )
= ( times_times @ nat @ K @ ( finite_card @ B @ T4 ) ) ) ) ) ) ).
% sum_multicount
thf(fact_8055_real__of__card,axiom,
! [A: $tType,A2: set @ A] :
( ( semiring_1_of_nat @ real @ ( finite_card @ A @ A2 ) )
= ( groups7311177749621191930dd_sum @ A @ real
@ ^ [X2: A] : ( one_one @ real )
@ A2 ) ) ).
% real_of_card
thf(fact_8056_sum__bounded__above,axiom,
! [B: $tType,A: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( semiring_1 @ A ) )
=> ! [A2: set @ B,F3: B > A,K4: A] :
( ! [I3: B] :
( ( member @ B @ I3 @ A2 )
=> ( ord_less_eq @ A @ ( F3 @ I3 ) @ K4 ) )
=> ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ A2 ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( finite_card @ B @ A2 ) ) @ K4 ) ) ) ) ).
% sum_bounded_above
thf(fact_8057_sum__bounded__below,axiom,
! [A: $tType,B: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( semiring_1 @ A ) )
=> ! [A2: set @ B,K4: A,F3: B > A] :
( ! [I3: B] :
( ( member @ B @ I3 @ A2 )
=> ( ord_less_eq @ A @ K4 @ ( F3 @ I3 ) ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( finite_card @ B @ A2 ) ) @ K4 ) @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ A2 ) ) ) ) ).
% sum_bounded_below
thf(fact_8058_card__gt__0__iff,axiom,
! [A: $tType,A2: set @ A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( finite_card @ A @ A2 ) )
= ( ( A2
!= ( bot_bot @ ( set @ A ) ) )
& ( finite_finite2 @ A @ A2 ) ) ) ).
% card_gt_0_iff
thf(fact_8059_card__le__Suc0__iff__eq,axiom,
! [A: $tType,A2: set @ A] :
( ( finite_finite2 @ A @ A2 )
=> ( ( ord_less_eq @ nat @ ( finite_card @ A @ A2 ) @ ( suc @ ( zero_zero @ nat ) ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ A2 )
=> ! [Y2: A] :
( ( member @ A @ Y2 @ A2 )
=> ( X2 = Y2 ) ) ) ) ) ) ).
% card_le_Suc0_iff_eq
thf(fact_8060_finite__UNIV__card__ge__0,axiom,
! [A: $tType] :
( ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ ( finite_card @ A @ ( top_top @ ( set @ A ) ) ) ) ) ).
% finite_UNIV_card_ge_0
thf(fact_8061_card__1__singleton__iff,axiom,
! [A: $tType,A2: set @ A] :
( ( ( finite_card @ A @ A2 )
= ( suc @ ( zero_zero @ nat ) ) )
= ( ? [X2: A] :
( A2
= ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% card_1_singleton_iff
thf(fact_8062_card__eq__SucD,axiom,
! [A: $tType,A2: set @ A,K: nat] :
( ( ( finite_card @ A @ A2 )
= ( suc @ K ) )
=> ? [B7: A,B6: set @ A] :
( ( A2
= ( insert @ A @ B7 @ B6 ) )
& ~ ( member @ A @ B7 @ B6 )
& ( ( finite_card @ A @ B6 )
= K )
& ( ( K
= ( zero_zero @ nat ) )
=> ( B6
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% card_eq_SucD
thf(fact_8063_card__Suc__eq,axiom,
! [A: $tType,A2: set @ A,K: nat] :
( ( ( finite_card @ A @ A2 )
= ( suc @ K ) )
= ( ? [B5: A,B4: set @ A] :
( ( A2
= ( insert @ A @ B5 @ B4 ) )
& ~ ( member @ A @ B5 @ B4 )
& ( ( finite_card @ A @ B4 )
= K )
& ( ( K
= ( zero_zero @ nat ) )
=> ( B4
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% card_Suc_eq
thf(fact_8064_card__le__Suc__iff,axiom,
! [A: $tType,N: nat,A2: set @ A] :
( ( ord_less_eq @ nat @ ( suc @ N ) @ ( finite_card @ A @ A2 ) )
= ( ? [A5: A,B4: set @ A] :
( ( A2
= ( insert @ A @ A5 @ B4 ) )
& ~ ( member @ A @ A5 @ B4 )
& ( ord_less_eq @ nat @ N @ ( finite_card @ A @ B4 ) )
& ( finite_finite2 @ A @ B4 ) ) ) ) ).
% card_le_Suc_iff
thf(fact_8065_card__1__singletonI,axiom,
! [A: $tType,S: set @ A,X: A] :
( ( finite_finite2 @ A @ S )
=> ( ( ( finite_card @ A @ S )
= ( one_one @ nat ) )
=> ( ( member @ A @ X @ S )
=> ( S
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% card_1_singletonI
thf(fact_8066_card__Diff1__le,axiom,
! [A: $tType,A2: set @ A,X: A] : ( ord_less_eq @ nat @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A2 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) @ ( finite_card @ A @ A2 ) ) ).
% card_Diff1_le
thf(fact_8067_card__psubset,axiom,
! [A: $tType,B2: set @ A,A2: set @ A] :
( ( finite_finite2 @ A @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ( ord_less @ nat @ ( finite_card @ A @ A2 ) @ ( finite_card @ A @ B2 ) )
=> ( ord_less @ ( set @ A ) @ A2 @ B2 ) ) ) ) ).
% card_psubset
thf(fact_8068_card__lists__length__le,axiom,
! [A: $tType,A2: set @ A,N: nat] :
( ( finite_finite2 @ A @ A2 )
=> ( ( finite_card @ ( list @ A )
@ ( collect @ ( list @ A )
@ ^ [Xs3: list @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs3 ) @ A2 )
& ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs3 ) @ N ) ) ) )
= ( groups7311177749621191930dd_sum @ nat @ nat @ ( power_power @ nat @ ( finite_card @ A @ A2 ) ) @ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% card_lists_length_le
thf(fact_8069_ex__bij__betw__nat__finite,axiom,
! [A: $tType,M7: set @ A] :
( ( finite_finite2 @ A @ M7 )
=> ? [H4: nat > A] : ( bij_betw @ nat @ A @ H4 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( finite_card @ A @ M7 ) ) @ M7 ) ) ).
% ex_bij_betw_nat_finite
thf(fact_8070_ex__bij__betw__nat__finite__1,axiom,
! [A: $tType,M7: set @ A] :
( ( finite_finite2 @ A @ M7 )
=> ? [H4: nat > A] : ( bij_betw @ nat @ A @ H4 @ ( set_or1337092689740270186AtMost @ nat @ ( one_one @ nat ) @ ( finite_card @ A @ M7 ) ) @ M7 ) ) ).
% ex_bij_betw_nat_finite_1
thf(fact_8071_card__roots__unity,axiom,
! [A: $tType] :
( ( ( real_V8999393235501362500lgebra @ A )
& ( idom @ A ) )
=> ! [N: nat] :
( ( ord_less_eq @ nat @ ( one_one @ nat ) @ N )
=> ( ord_less_eq @ nat
@ ( finite_card @ A
@ ( collect @ A
@ ^ [Z4: A] :
( ( power_power @ A @ Z4 @ N )
= ( one_one @ A ) ) ) )
@ N ) ) ) ).
% card_roots_unity
thf(fact_8072_ex__bij__betw__finite__nat,axiom,
! [A: $tType,M7: set @ A] :
( ( finite_finite2 @ A @ M7 )
=> ? [H4: A > nat] : ( bij_betw @ A @ nat @ H4 @ M7 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( finite_card @ A @ M7 ) ) ) ) ).
% ex_bij_betw_finite_nat
thf(fact_8073_subset__eq__atLeast0__lessThan__card,axiom,
! [N3: set @ nat,N: nat] :
( ( ord_less_eq @ ( set @ nat ) @ N3 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) )
=> ( ord_less_eq @ nat @ ( finite_card @ nat @ N3 ) @ N ) ) ).
% subset_eq_atLeast0_lessThan_card
thf(fact_8074_card__sum__le__nat__sum,axiom,
! [S: set @ nat] :
( ord_less_eq @ nat
@ ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [X2: nat] : X2
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( finite_card @ nat @ S ) ) )
@ ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [X2: nat] : X2
@ S ) ) ).
% card_sum_le_nat_sum
thf(fact_8075_conj__subset__def,axiom,
! [A: $tType,A2: set @ A,P: A > $o,Q: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ A2
@ ( collect @ A
@ ^ [X2: A] :
( ( P @ X2 )
& ( Q @ X2 ) ) ) )
= ( ( ord_less_eq @ ( set @ A ) @ A2 @ ( collect @ A @ P ) )
& ( ord_less_eq @ ( set @ A ) @ A2 @ ( collect @ A @ Q ) ) ) ) ).
% conj_subset_def
thf(fact_8076_card__nth__roots,axiom,
! [C2: complex,N: nat] :
( ( C2
!= ( zero_zero @ complex ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( finite_card @ complex
@ ( collect @ complex
@ ^ [Z4: complex] :
( ( power_power @ complex @ Z4 @ N )
= C2 ) ) )
= N ) ) ) ).
% card_nth_roots
thf(fact_8077_card__roots__unity__eq,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( finite_card @ complex
@ ( collect @ complex
@ ^ [Z4: complex] :
( ( power_power @ complex @ Z4 @ N )
= ( one_one @ complex ) ) ) )
= N ) ) ).
% card_roots_unity_eq
thf(fact_8078_card__2__iff,axiom,
! [A: $tType,S: set @ A] :
( ( ( finite_card @ A @ S )
= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( ? [X2: A,Y2: A] :
( ( S
= ( insert @ A @ X2 @ ( insert @ A @ Y2 @ ( bot_bot @ ( set @ A ) ) ) ) )
& ( X2 != Y2 ) ) ) ) ).
% card_2_iff
thf(fact_8079_card__3__iff,axiom,
! [A: $tType,S: set @ A] :
( ( ( finite_card @ A @ S )
= ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) )
= ( ? [X2: A,Y2: A,Z4: A] :
( ( S
= ( insert @ A @ X2 @ ( insert @ A @ Y2 @ ( insert @ A @ Z4 @ ( bot_bot @ ( set @ A ) ) ) ) ) )
& ( X2 != Y2 )
& ( Y2 != Z4 )
& ( X2 != Z4 ) ) ) ) ).
% card_3_iff
thf(fact_8080_card__range__greater__zero,axiom,
! [A: $tType,B: $tType,F3: B > A] :
( ( finite_finite2 @ A @ ( image @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) ) )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ ( finite_card @ A @ ( image @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) ) ) ) ) ).
% card_range_greater_zero
thf(fact_8081_odd__card__imp__not__empty,axiom,
! [A: $tType,A2: set @ A] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( finite_card @ A @ A2 ) )
=> ( A2
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% odd_card_imp_not_empty
thf(fact_8082_card__insert__disjoint_H,axiom,
! [A: $tType,A2: set @ A,X: A] :
( ( finite_finite2 @ A @ A2 )
=> ( ~ ( member @ A @ X @ A2 )
=> ( ( minus_minus @ nat @ ( finite_card @ A @ ( insert @ A @ X @ A2 ) ) @ ( suc @ ( zero_zero @ nat ) ) )
= ( finite_card @ A @ A2 ) ) ) ) ).
% card_insert_disjoint'
thf(fact_8083_card__Suc__Diff1,axiom,
! [A: $tType,A2: set @ A,X: A] :
( ( finite_finite2 @ A @ A2 )
=> ( ( member @ A @ X @ A2 )
=> ( ( suc @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A2 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) )
= ( finite_card @ A @ A2 ) ) ) ) ).
% card_Suc_Diff1
thf(fact_8084_card_Oinsert__remove,axiom,
! [A: $tType,A2: set @ A,X: A] :
( ( finite_finite2 @ A @ A2 )
=> ( ( finite_card @ A @ ( insert @ A @ X @ A2 ) )
= ( suc @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A2 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% card.insert_remove
thf(fact_8085_card_Oremove,axiom,
! [A: $tType,A2: set @ A,X: A] :
( ( finite_finite2 @ A @ A2 )
=> ( ( member @ A @ X @ A2 )
=> ( ( finite_card @ A @ A2 )
= ( suc @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A2 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ).
% card.remove
thf(fact_8086_card__Diff1__less__iff,axiom,
! [A: $tType,A2: set @ A,X: A] :
( ( ord_less @ nat @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A2 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) @ ( finite_card @ A @ A2 ) )
= ( ( finite_finite2 @ A @ A2 )
& ( member @ A @ X @ A2 ) ) ) ).
% card_Diff1_less_iff
thf(fact_8087_card__Diff2__less,axiom,
! [A: $tType,A2: set @ A,X: A,Y: A] :
( ( finite_finite2 @ A @ A2 )
=> ( ( member @ A @ X @ A2 )
=> ( ( member @ A @ Y @ A2 )
=> ( ord_less @ nat @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A2 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) @ ( insert @ A @ Y @ ( bot_bot @ ( set @ A ) ) ) ) ) @ ( finite_card @ A @ A2 ) ) ) ) ) ).
% card_Diff2_less
thf(fact_8088_card__Diff1__less,axiom,
! [A: $tType,A2: set @ A,X: A] :
( ( finite_finite2 @ A @ A2 )
=> ( ( member @ A @ X @ A2 )
=> ( ord_less @ nat @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A2 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) @ ( finite_card @ A @ A2 ) ) ) ) ).
% card_Diff1_less
thf(fact_8089_card__Diff__singleton__if,axiom,
! [A: $tType,X: A,A2: set @ A] :
( ( ( member @ A @ X @ A2 )
=> ( ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A2 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( minus_minus @ nat @ ( finite_card @ A @ A2 ) @ ( one_one @ nat ) ) ) )
& ( ~ ( member @ A @ X @ A2 )
=> ( ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A2 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( finite_card @ A @ A2 ) ) ) ) ).
% card_Diff_singleton_if
thf(fact_8090_card__Diff__singleton,axiom,
! [A: $tType,X: A,A2: set @ A] :
( ( member @ A @ X @ A2 )
=> ( ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A2 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( minus_minus @ nat @ ( finite_card @ A @ A2 ) @ ( one_one @ nat ) ) ) ) ).
% card_Diff_singleton
thf(fact_8091_sum__norm__bound,axiom,
! [A: $tType,B: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [S: set @ B,F3: B > A,K4: real] :
( ! [X3: B] :
( ( member @ B @ X3 @ S )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( F3 @ X3 ) ) @ K4 ) )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ S ) ) @ ( times_times @ real @ ( semiring_1_of_nat @ real @ ( finite_card @ B @ S ) ) @ K4 ) ) ) ) ).
% sum_norm_bound
thf(fact_8092_conj__comp__iff,axiom,
! [B: $tType,A: $tType,P: B > $o,Q: B > $o,G2: A > B] :
( ( comp @ B @ $o @ A
@ ^ [X2: B] :
( ( P @ X2 )
& ( Q @ X2 ) )
@ G2 )
= ( ^ [X2: A] :
( ( comp @ B @ $o @ A @ P @ G2 @ X2 )
& ( comp @ B @ $o @ A @ Q @ G2 @ X2 ) ) ) ) ).
% conj_comp_iff
thf(fact_8093_prod__le__power,axiom,
! [B: $tType,A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: set @ B,F3: B > A,N: A,K: nat] :
( ! [I3: B] :
( ( member @ B @ I3 @ A2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F3 @ I3 ) )
& ( ord_less_eq @ A @ ( F3 @ I3 ) @ N ) ) )
=> ( ( ord_less_eq @ nat @ ( finite_card @ B @ A2 ) @ K )
=> ( ( ord_less_eq @ A @ ( one_one @ A ) @ N )
=> ( ord_less_eq @ A @ ( groups7121269368397514597t_prod @ B @ A @ F3 @ A2 ) @ ( power_power @ A @ N @ K ) ) ) ) ) ) ).
% prod_le_power
thf(fact_8094_sum__bounded__above__strict,axiom,
! [B: $tType,A: $tType] :
( ( ( ordere8940638589300402666id_add @ A )
& ( semiring_1 @ A ) )
=> ! [A2: set @ B,F3: B > A,K4: A] :
( ! [I3: B] :
( ( member @ B @ I3 @ A2 )
=> ( ord_less @ A @ ( F3 @ I3 ) @ K4 ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( finite_card @ B @ A2 ) )
=> ( ord_less @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ A2 ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( finite_card @ B @ A2 ) ) @ K4 ) ) ) ) ) ).
% sum_bounded_above_strict
thf(fact_8095_sum__bounded__above__divide,axiom,
! [B: $tType,A: $tType] :
( ( linordered_field @ A )
=> ! [A2: set @ B,F3: B > A,K4: A] :
( ! [I3: B] :
( ( member @ B @ I3 @ A2 )
=> ( ord_less_eq @ A @ ( F3 @ I3 ) @ ( divide_divide @ A @ K4 @ ( semiring_1_of_nat @ A @ ( finite_card @ B @ A2 ) ) ) ) )
=> ( ( finite_finite2 @ B @ A2 )
=> ( ( A2
!= ( bot_bot @ ( set @ B ) ) )
=> ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ A2 ) @ K4 ) ) ) ) ) ).
% sum_bounded_above_divide
thf(fact_8096_card__insert__le__m1,axiom,
! [A: $tType,N: nat,Y: set @ A,X: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less_eq @ nat @ ( finite_card @ A @ Y ) @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) )
=> ( ord_less_eq @ nat @ ( finite_card @ A @ ( insert @ A @ X @ Y ) ) @ N ) ) ) ).
% card_insert_le_m1
thf(fact_8097_card__word__size,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [X: word @ A] :
( ( finite_card @ ( word @ A ) @ ( top_top @ ( set @ ( word @ A ) ) ) )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( size_size @ ( word @ A ) @ X ) ) ) ) ).
% card_word_size
thf(fact_8098_sum__fun__comp,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( semiring_1 @ C )
=> ! [S: set @ A,R: set @ B,G2: A > B,F3: B > C] :
( ( finite_finite2 @ A @ S )
=> ( ( finite_finite2 @ B @ R )
=> ( ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ G2 @ S ) @ R )
=> ( ( groups7311177749621191930dd_sum @ A @ C
@ ^ [X2: A] : ( F3 @ ( G2 @ X2 ) )
@ S )
= ( groups7311177749621191930dd_sum @ B @ C
@ ^ [Y2: B] :
( times_times @ C
@ ( semiring_1_of_nat @ C
@ ( finite_card @ A
@ ( collect @ A
@ ^ [X2: A] :
( ( member @ A @ X2 @ S )
& ( ( G2 @ X2 )
= Y2 ) ) ) ) )
@ ( F3 @ Y2 ) )
@ R ) ) ) ) ) ) ).
% sum_fun_comp
thf(fact_8099_prod__gen__delta,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S: set @ B,A3: B,B3: B > A,C2: A] :
( ( finite_finite2 @ B @ S )
=> ( ( ( member @ B @ A3 @ S )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K3: B] : ( if @ A @ ( K3 = A3 ) @ ( B3 @ K3 ) @ C2 )
@ S )
= ( times_times @ A @ ( B3 @ A3 ) @ ( power_power @ A @ C2 @ ( minus_minus @ nat @ ( finite_card @ B @ S ) @ ( one_one @ nat ) ) ) ) ) )
& ( ~ ( member @ B @ A3 @ S )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K3: B] : ( if @ A @ ( K3 = A3 ) @ ( B3 @ K3 ) @ C2 )
@ S )
= ( power_power @ A @ C2 @ ( finite_card @ B @ S ) ) ) ) ) ) ) ).
% prod_gen_delta
thf(fact_8100_polyfun__roots__card,axiom,
! [A: $tType] :
( ( ( real_V8999393235501362500lgebra @ A )
& ( idom @ A ) )
=> ! [C2: nat > A,K: nat,N: nat] :
( ( ( C2 @ K )
!= ( zero_zero @ A ) )
=> ( ( ord_less_eq @ nat @ K @ N )
=> ( ord_less_eq @ nat
@ ( finite_card @ A
@ ( collect @ A
@ ^ [Z4: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( C2 @ I4 ) @ ( power_power @ A @ Z4 @ I4 ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( zero_zero @ A ) ) ) )
@ N ) ) ) ) ).
% polyfun_roots_card
thf(fact_8101_finite__fun__UNIVD1,axiom,
! [B: $tType,A: $tType] :
( ( finite_finite2 @ ( A > B ) @ ( top_top @ ( set @ ( A > B ) ) ) )
=> ( ( ( finite_card @ B @ ( top_top @ ( set @ B ) ) )
!= ( suc @ ( zero_zero @ nat ) ) )
=> ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) ) ) ) ).
% finite_fun_UNIVD1
thf(fact_8102_card__map__elide2,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ! [N: nat] :
( ( ord_less_eq @ nat @ N @ ( finite_card @ ( word @ A ) @ ( top_top @ ( set @ ( word @ A ) ) ) ) )
=> ( ( finite_card @ ( word @ A ) @ ( image @ nat @ ( word @ A ) @ ( semiring_1_of_nat @ ( word @ A ) ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) )
= N ) ) ) ).
% card_map_elide2
thf(fact_8103_card__word,axiom,
! [A: $tType] :
( ( type_len @ A )
=> ( ( finite_card @ ( word @ A ) @ ( top_top @ ( set @ ( word @ A ) ) ) )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ).
% card_word
thf(fact_8104_polyfun__rootbound,axiom,
! [A: $tType] :
( ( ( real_V8999393235501362500lgebra @ A )
& ( idom @ A ) )
=> ! [C2: nat > A,K: nat,N: nat] :
( ( ( C2 @ K )
!= ( zero_zero @ A ) )
=> ( ( ord_less_eq @ nat @ K @ N )
=> ( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [Z4: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( C2 @ I4 ) @ ( power_power @ A @ Z4 @ I4 ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( zero_zero @ A ) ) ) )
& ( ord_less_eq @ nat
@ ( finite_card @ A
@ ( collect @ A
@ ^ [Z4: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( C2 @ I4 ) @ ( power_power @ A @ Z4 @ I4 ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( zero_zero @ A ) ) ) )
@ N ) ) ) ) ) ).
% polyfun_rootbound
thf(fact_8105_Cardinality_Ocard__set,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ( ( finite_card @ ( set @ A ) @ ( top_top @ ( set @ ( set @ A ) ) ) )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( finite_card @ A @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% Cardinality.card_set
thf(fact_8106_card__bit1,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ( ( finite_card @ ( numeral_bit1 @ A ) @ ( top_top @ ( set @ ( numeral_bit1 @ A ) ) ) )
= ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( finite_card @ A @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% card_bit1
thf(fact_8107_card__num0,axiom,
( ( finite_card @ numeral_num0 @ ( top_top @ ( set @ numeral_num0 ) ) )
= ( zero_zero @ nat ) ) ).
% card_num0
thf(fact_8108_card__num1,axiom,
( ( finite_card @ numeral_num1 @ ( top_top @ ( set @ numeral_num1 ) ) )
= ( one_one @ nat ) ) ).
% card_num1
thf(fact_8109_card__nat,axiom,
( ( finite_card @ nat @ ( top_top @ ( set @ nat ) ) )
= ( zero_zero @ nat ) ) ).
% card_nat
thf(fact_8110_card__bit0,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ( ( finite_card @ ( numeral_bit0 @ A ) @ ( top_top @ ( set @ ( numeral_bit0 @ A ) ) ) )
= ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( finite_card @ A @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% card_bit0
thf(fact_8111_card__literal,axiom,
( ( finite_card @ literal @ ( top_top @ ( set @ literal ) ) )
= ( zero_zero @ nat ) ) ).
% card_literal
thf(fact_8112_UNIV__bool,axiom,
( ( top_top @ ( set @ $o ) )
= ( insert @ $o @ $false @ ( insert @ $o @ $true @ ( bot_bot @ ( set @ $o ) ) ) ) ) ).
% UNIV_bool
thf(fact_8113_bit0_OCARD__eq,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ( ( finite_card @ ( numeral_bit0 @ A ) @ ( top_top @ ( set @ ( numeral_bit0 @ A ) ) ) )
= ( nat2 @ ( semiring_1_of_nat @ int @ ( finite_card @ ( numeral_bit0 @ A ) @ ( top_top @ ( set @ ( numeral_bit0 @ A ) ) ) ) ) ) ) ) ).
% bit0.CARD_eq
thf(fact_8114_bit1_OCARD__eq,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ( ( finite_card @ ( numeral_bit1 @ A ) @ ( top_top @ ( set @ ( numeral_bit1 @ A ) ) ) )
= ( nat2 @ ( semiring_1_of_nat @ int @ ( finite_card @ ( numeral_bit1 @ A ) @ ( top_top @ ( set @ ( numeral_bit1 @ A ) ) ) ) ) ) ) ) ).
% bit1.CARD_eq
thf(fact_8115_CARD__1,axiom,
! [A: $tType] :
( ( cARD_1 @ A )
=> ( ( finite_card @ A @ ( top_top @ ( set @ A ) ) )
= ( one_one @ nat ) ) ) ).
% CARD_1
thf(fact_8116_bit1_Osize0,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ( ord_less @ int @ ( zero_zero @ int ) @ ( semiring_1_of_nat @ int @ ( finite_card @ ( numeral_bit1 @ A ) @ ( top_top @ ( set @ ( numeral_bit1 @ A ) ) ) ) ) ) ) ).
% bit1.size0
thf(fact_8117_bit0_Osize0,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ( ord_less @ int @ ( zero_zero @ int ) @ ( semiring_1_of_nat @ int @ ( finite_card @ ( numeral_bit0 @ A ) @ ( top_top @ ( set @ ( numeral_bit0 @ A ) ) ) ) ) ) ) ).
% bit0.size0
thf(fact_8118_bit0_Osize1,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ( ord_less @ int @ ( one_one @ int ) @ ( semiring_1_of_nat @ int @ ( finite_card @ ( numeral_bit0 @ A ) @ ( top_top @ ( set @ ( numeral_bit0 @ A ) ) ) ) ) ) ) ).
% bit0.size1
thf(fact_8119_bit1_Osize1,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ( ord_less @ int @ ( one_one @ int ) @ ( semiring_1_of_nat @ int @ ( finite_card @ ( numeral_bit1 @ A ) @ ( top_top @ ( set @ ( numeral_bit1 @ A ) ) ) ) ) ) ) ).
% bit1.size1
thf(fact_8120_zero__less__card__finite,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ ( finite_card @ A @ ( top_top @ ( set @ A ) ) ) ) ) ).
% zero_less_card_finite
thf(fact_8121_card__UNIV__sum,axiom,
! [A: $tType,B: $tType] :
( ( ( ( ( finite_card @ A @ ( top_top @ ( set @ A ) ) )
!= ( zero_zero @ nat ) )
& ( ( finite_card @ B @ ( top_top @ ( set @ B ) ) )
!= ( zero_zero @ nat ) ) )
=> ( ( finite_card @ ( sum_sum @ A @ B ) @ ( top_top @ ( set @ ( sum_sum @ A @ B ) ) ) )
= ( plus_plus @ nat @ ( finite_card @ A @ ( top_top @ ( set @ A ) ) ) @ ( finite_card @ B @ ( top_top @ ( set @ B ) ) ) ) ) )
& ( ~ ( ( ( finite_card @ A @ ( top_top @ ( set @ A ) ) )
!= ( zero_zero @ nat ) )
& ( ( finite_card @ B @ ( top_top @ ( set @ B ) ) )
!= ( zero_zero @ nat ) ) )
=> ( ( finite_card @ ( sum_sum @ A @ B ) @ ( top_top @ ( set @ ( sum_sum @ A @ B ) ) ) )
= ( zero_zero @ nat ) ) ) ) ).
% card_UNIV_sum
thf(fact_8122_bit0__cases,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ! [X: numeral_bit0 @ A] :
~ ! [Z3: int] :
( ( X
= ( ring_1_of_int @ ( numeral_bit0 @ A ) @ Z3 ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z3 )
=> ~ ( ord_less @ int @ Z3 @ ( semiring_1_of_nat @ int @ ( finite_card @ ( numeral_bit0 @ A ) @ ( top_top @ ( set @ ( numeral_bit0 @ A ) ) ) ) ) ) ) ) ) ).
% bit0_cases
thf(fact_8123_bit1__cases,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ! [X: numeral_bit1 @ A] :
~ ! [Z3: int] :
( ( X
= ( ring_1_of_int @ ( numeral_bit1 @ A ) @ Z3 ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z3 )
=> ~ ( ord_less @ int @ Z3 @ ( semiring_1_of_nat @ int @ ( finite_card @ ( numeral_bit1 @ A ) @ ( top_top @ ( set @ ( numeral_bit1 @ A ) ) ) ) ) ) ) ) ) ).
% bit1_cases
thf(fact_8124_bit0__induct,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ! [P: ( numeral_bit0 @ A ) > $o,X: numeral_bit0 @ A] :
( ! [Z3: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z3 )
=> ( ( ord_less @ int @ Z3 @ ( semiring_1_of_nat @ int @ ( finite_card @ ( numeral_bit0 @ A ) @ ( top_top @ ( set @ ( numeral_bit0 @ A ) ) ) ) ) )
=> ( P @ ( ring_1_of_int @ ( numeral_bit0 @ A ) @ Z3 ) ) ) )
=> ( P @ X ) ) ) ).
% bit0_induct
thf(fact_8125_bit1__induct,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ! [P: ( numeral_bit1 @ A ) > $o,X: numeral_bit1 @ A] :
( ! [Z3: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z3 )
=> ( ( ord_less @ int @ Z3 @ ( semiring_1_of_nat @ int @ ( finite_card @ ( numeral_bit1 @ A ) @ ( top_top @ ( set @ ( numeral_bit1 @ A ) ) ) ) ) )
=> ( P @ ( ring_1_of_int @ ( numeral_bit1 @ A ) @ Z3 ) ) ) )
=> ( P @ X ) ) ) ).
% bit1_induct
thf(fact_8126_one__less__card,axiom,
! [A: $tType] :
( ( card2 @ A )
=> ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( finite_card @ A @ ( top_top @ ( set @ A ) ) ) ) ) ).
% one_less_card
thf(fact_8127_one__le__card__finite,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ( ord_less_eq @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( finite_card @ A @ ( top_top @ ( set @ A ) ) ) ) ) ).
% one_le_card_finite
thf(fact_8128_card__UNIV__option,axiom,
! [A: $tType] :
( ( ( ( finite_card @ A @ ( top_top @ ( set @ A ) ) )
= ( zero_zero @ nat ) )
=> ( ( finite_card @ ( option @ A ) @ ( top_top @ ( set @ ( option @ A ) ) ) )
= ( zero_zero @ nat ) ) )
& ( ( ( finite_card @ A @ ( top_top @ ( set @ A ) ) )
!= ( zero_zero @ nat ) )
=> ( ( finite_card @ ( option @ A ) @ ( top_top @ ( set @ ( option @ A ) ) ) )
= ( plus_plus @ nat @ ( finite_card @ A @ ( top_top @ ( set @ A ) ) ) @ ( one_one @ nat ) ) ) ) ) ).
% card_UNIV_option
thf(fact_8129_one__less__int__card,axiom,
! [A: $tType] :
( ( card2 @ A )
=> ( ord_less @ int @ ( one_one @ int ) @ ( semiring_1_of_nat @ int @ ( finite_card @ A @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% one_less_int_card
thf(fact_8130_two__le__card,axiom,
! [A: $tType] :
( ( card2 @ A )
=> ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( finite_card @ A @ ( top_top @ ( set @ A ) ) ) ) ) ).
% two_le_card
thf(fact_8131_card__fun,axiom,
! [A: $tType,B: $tType] :
( ( ( ( ( ( finite_card @ A @ ( top_top @ ( set @ A ) ) )
!= ( zero_zero @ nat ) )
& ( ( finite_card @ B @ ( top_top @ ( set @ B ) ) )
!= ( zero_zero @ nat ) ) )
| ( ( finite_card @ B @ ( top_top @ ( set @ B ) ) )
= ( one_one @ nat ) ) )
=> ( ( finite_card @ ( A > B ) @ ( top_top @ ( set @ ( A > B ) ) ) )
= ( power_power @ nat @ ( finite_card @ B @ ( top_top @ ( set @ B ) ) ) @ ( finite_card @ A @ ( top_top @ ( set @ A ) ) ) ) ) )
& ( ~ ( ( ( ( finite_card @ A @ ( top_top @ ( set @ A ) ) )
!= ( zero_zero @ nat ) )
& ( ( finite_card @ B @ ( top_top @ ( set @ B ) ) )
!= ( zero_zero @ nat ) ) )
| ( ( finite_card @ B @ ( top_top @ ( set @ B ) ) )
= ( one_one @ nat ) ) )
=> ( ( finite_card @ ( A > B ) @ ( top_top @ ( set @ ( A > B ) ) ) )
= ( zero_zero @ nat ) ) ) ) ).
% card_fun
thf(fact_8132_finite__UNIV__fun,axiom,
! [A: $tType,B: $tType] :
( ( finite_finite2 @ ( A > B ) @ ( top_top @ ( set @ ( A > B ) ) ) )
= ( ( ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) )
& ( finite_finite2 @ B @ ( top_top @ ( set @ B ) ) ) )
| ( ( finite_card @ B @ ( top_top @ ( set @ B ) ) )
= ( one_one @ nat ) ) ) ) ).
% finite_UNIV_fun
thf(fact_8133_card__UNIV__set,axiom,
! [A: $tType] :
( ( ( ( finite_card @ A @ ( top_top @ ( set @ A ) ) )
= ( zero_zero @ nat ) )
=> ( ( finite_card @ ( set @ A ) @ ( top_top @ ( set @ ( set @ A ) ) ) )
= ( zero_zero @ nat ) ) )
& ( ( ( finite_card @ A @ ( top_top @ ( set @ A ) ) )
!= ( zero_zero @ nat ) )
=> ( ( finite_card @ ( set @ A ) @ ( top_top @ ( set @ ( set @ A ) ) ) )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( finite_card @ A @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% card_UNIV_set
thf(fact_8134_inj__on__Abs__bit1,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ( inj_on @ int @ ( numeral_bit1 @ A ) @ ( numeral_Abs_bit12 @ A ) @ ( set_or7035219750837199246ssThan @ int @ ( zero_zero @ int ) @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ int @ ( finite_card @ A @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ) ).
% inj_on_Abs_bit1
thf(fact_8135_Abs__bit1__inject,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ! [X: int,Y: int] :
( ( member @ int @ X @ ( set_or7035219750837199246ssThan @ int @ ( zero_zero @ int ) @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ int @ ( finite_card @ A @ ( top_top @ ( set @ A ) ) ) ) ) ) ) )
=> ( ( member @ int @ Y @ ( set_or7035219750837199246ssThan @ int @ ( zero_zero @ int ) @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ int @ ( finite_card @ A @ ( top_top @ ( set @ A ) ) ) ) ) ) ) )
=> ( ( ( numeral_Abs_bit12 @ A @ X )
= ( numeral_Abs_bit12 @ A @ Y ) )
= ( X = Y ) ) ) ) ) ).
% Abs_bit1_inject
thf(fact_8136_one__bit1__def,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ( ( one_one @ ( numeral_bit1 @ A ) )
= ( numeral_Abs_bit12 @ A @ ( one_one @ int ) ) ) ) ).
% one_bit1_def
thf(fact_8137_zero__bit1__def,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ( ( zero_zero @ ( numeral_bit1 @ A ) )
= ( numeral_Abs_bit12 @ A @ ( zero_zero @ int ) ) ) ) ).
% zero_bit1_def
thf(fact_8138_bit1_Oof__nat__eq,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ( ( semiring_1_of_nat @ ( numeral_bit1 @ A ) )
= ( ^ [K3: nat] : ( numeral_Abs_bit12 @ A @ ( modulo_modulo @ int @ ( semiring_1_of_nat @ int @ K3 ) @ ( semiring_1_of_nat @ int @ ( finite_card @ ( numeral_bit1 @ A ) @ ( top_top @ ( set @ ( numeral_bit1 @ A ) ) ) ) ) ) ) ) ) ) ).
% bit1.of_nat_eq
thf(fact_8139_bit1_Oof__int__eq,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ( ( ring_1_of_int @ ( numeral_bit1 @ A ) )
= ( ^ [Z4: int] : ( numeral_Abs_bit12 @ A @ ( modulo_modulo @ int @ Z4 @ ( semiring_1_of_nat @ int @ ( finite_card @ ( numeral_bit1 @ A ) @ ( top_top @ ( set @ ( numeral_bit1 @ A ) ) ) ) ) ) ) ) ) ) ).
% bit1.of_int_eq
thf(fact_8140_bit1_OUNIV__eq,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ( ( top_top @ ( set @ ( numeral_bit1 @ A ) ) )
= ( image @ int @ ( numeral_bit1 @ A ) @ ( numeral_Abs_bit12 @ A ) @ ( set_or7035219750837199246ssThan @ int @ ( zero_zero @ int ) @ ( semiring_1_of_nat @ int @ ( finite_card @ ( numeral_bit1 @ A ) @ ( top_top @ ( set @ ( numeral_bit1 @ A ) ) ) ) ) ) ) ) ) ).
% bit1.UNIV_eq
thf(fact_8141_Abs__bit1__cases,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ! [X: numeral_bit1 @ A] :
~ ! [Y4: int] :
( ( X
= ( numeral_Abs_bit12 @ A @ Y4 ) )
=> ~ ( member @ int @ Y4 @ ( set_or7035219750837199246ssThan @ int @ ( zero_zero @ int ) @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ int @ ( finite_card @ A @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ) ) ).
% Abs_bit1_cases
thf(fact_8142_Abs__bit1__induct,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ! [P: ( numeral_bit1 @ A ) > $o,X: numeral_bit1 @ A] :
( ! [Y4: int] :
( ( member @ int @ Y4 @ ( set_or7035219750837199246ssThan @ int @ ( zero_zero @ int ) @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ int @ ( finite_card @ A @ ( top_top @ ( set @ A ) ) ) ) ) ) ) )
=> ( P @ ( numeral_Abs_bit12 @ A @ Y4 ) ) )
=> ( P @ X ) ) ) ).
% Abs_bit1_induct
thf(fact_8143_Abs__bit1__inverse,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ! [Y: int] :
( ( member @ int @ Y @ ( set_or7035219750837199246ssThan @ int @ ( zero_zero @ int ) @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ int @ ( finite_card @ A @ ( top_top @ ( set @ A ) ) ) ) ) ) ) )
=> ( ( numeral_Rep_bit1 @ A @ ( numeral_Abs_bit12 @ A @ Y ) )
= Y ) ) ) ).
% Abs_bit1_inverse
thf(fact_8144_Abs__bit1_H__def,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ( ( numeral_Abs_bit1 @ A )
= ( ^ [X2: int] : ( numeral_Abs_bit12 @ A @ ( modulo_modulo @ int @ X2 @ ( semiring_1_of_nat @ int @ ( finite_card @ ( numeral_bit1 @ A ) @ ( top_top @ ( set @ ( numeral_bit1 @ A ) ) ) ) ) ) ) ) ) ) ).
% Abs_bit1'_def
thf(fact_8145_Abs__bit1_H__code,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ! [X: int] :
( ( numeral_Rep_bit1 @ A @ ( numeral_Abs_bit1 @ A @ X ) )
= ( modulo_modulo @ int @ X @ ( semiring_1_of_nat @ int @ ( finite_card @ ( numeral_bit1 @ A ) @ ( top_top @ ( set @ ( numeral_bit1 @ A ) ) ) ) ) ) ) ) ).
% Abs_bit1'_code
thf(fact_8146_bit1_ORep__Abs__1,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ( ( numeral_Rep_bit1 @ A @ ( numeral_Abs_bit12 @ A @ ( one_one @ int ) ) )
= ( one_one @ int ) ) ) ).
% bit1.Rep_Abs_1
thf(fact_8147_bit1_ORep__Abs__0,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ( ( numeral_Rep_bit1 @ A @ ( numeral_Abs_bit12 @ A @ ( zero_zero @ int ) ) )
= ( zero_zero @ int ) ) ) ).
% bit1.Rep_Abs_0
thf(fact_8148_less__bit1__def,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ( ( ord_less @ ( numeral_bit1 @ A ) )
= ( ^ [A5: numeral_bit1 @ A,B5: numeral_bit1 @ A] : ( ord_less @ int @ ( numeral_Rep_bit1 @ A @ A5 ) @ ( numeral_Rep_bit1 @ A @ B5 ) ) ) ) ) ).
% less_bit1_def
thf(fact_8149_bit1_ORep__0,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ( ( numeral_Rep_bit1 @ A @ ( zero_zero @ ( numeral_bit1 @ A ) ) )
= ( zero_zero @ int ) ) ) ).
% bit1.Rep_0
thf(fact_8150_bit1_ORep__1,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ( ( numeral_Rep_bit1 @ A @ ( one_one @ ( numeral_bit1 @ A ) ) )
= ( one_one @ int ) ) ) ).
% bit1.Rep_1
thf(fact_8151_bit1_ORep__le__n,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ! [X: numeral_bit1 @ A] : ( ord_less_eq @ int @ ( numeral_Rep_bit1 @ A @ X ) @ ( semiring_1_of_nat @ int @ ( finite_card @ ( numeral_bit1 @ A ) @ ( top_top @ ( set @ ( numeral_bit1 @ A ) ) ) ) ) ) ) ).
% bit1.Rep_le_n
thf(fact_8152_bit1_ORep__less__n,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ! [X: numeral_bit1 @ A] : ( ord_less @ int @ ( numeral_Rep_bit1 @ A @ X ) @ ( semiring_1_of_nat @ int @ ( finite_card @ ( numeral_bit1 @ A ) @ ( top_top @ ( set @ ( numeral_bit1 @ A ) ) ) ) ) ) ) ).
% bit1.Rep_less_n
thf(fact_8153_bit1_ORep__mod,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ! [X: numeral_bit1 @ A] :
( ( modulo_modulo @ int @ ( numeral_Rep_bit1 @ A @ X ) @ ( semiring_1_of_nat @ int @ ( finite_card @ ( numeral_bit1 @ A ) @ ( top_top @ ( set @ ( numeral_bit1 @ A ) ) ) ) ) )
= ( numeral_Rep_bit1 @ A @ X ) ) ) ).
% bit1.Rep_mod
thf(fact_8154_bit1_ORep__numeral,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ! [W: num] :
( ( numeral_Rep_bit1 @ A @ ( numeral_numeral @ ( numeral_bit1 @ A ) @ W ) )
= ( modulo_modulo @ int @ ( numeral_numeral @ int @ W ) @ ( semiring_1_of_nat @ int @ ( finite_card @ ( numeral_bit1 @ A ) @ ( top_top @ ( set @ ( numeral_bit1 @ A ) ) ) ) ) ) ) ) ).
% bit1.Rep_numeral
thf(fact_8155_bit1_ORep__Abs__mod,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ! [M: int] :
( ( numeral_Rep_bit1 @ A @ ( numeral_Abs_bit12 @ A @ ( modulo_modulo @ int @ M @ ( semiring_1_of_nat @ int @ ( finite_card @ ( numeral_bit1 @ A ) @ ( top_top @ ( set @ ( numeral_bit1 @ A ) ) ) ) ) ) ) )
= ( modulo_modulo @ int @ M @ ( semiring_1_of_nat @ int @ ( finite_card @ ( numeral_bit1 @ A ) @ ( top_top @ ( set @ ( numeral_bit1 @ A ) ) ) ) ) ) ) ) ).
% bit1.Rep_Abs_mod
thf(fact_8156_bit1_Ominus__def,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ( ( uminus_uminus @ ( numeral_bit1 @ A ) )
= ( ^ [X2: numeral_bit1 @ A] : ( numeral_Abs_bit12 @ A @ ( modulo_modulo @ int @ ( uminus_uminus @ int @ ( numeral_Rep_bit1 @ A @ X2 ) ) @ ( semiring_1_of_nat @ int @ ( finite_card @ ( numeral_bit1 @ A ) @ ( top_top @ ( set @ ( numeral_bit1 @ A ) ) ) ) ) ) ) ) ) ) ).
% bit1.minus_def
thf(fact_8157_bit1_Oadd__def,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ( ( plus_plus @ ( numeral_bit1 @ A ) )
= ( ^ [X2: numeral_bit1 @ A,Y2: numeral_bit1 @ A] : ( numeral_Abs_bit12 @ A @ ( modulo_modulo @ int @ ( plus_plus @ int @ ( numeral_Rep_bit1 @ A @ X2 ) @ ( numeral_Rep_bit1 @ A @ Y2 ) ) @ ( semiring_1_of_nat @ int @ ( finite_card @ ( numeral_bit1 @ A ) @ ( top_top @ ( set @ ( numeral_bit1 @ A ) ) ) ) ) ) ) ) ) ) ).
% bit1.add_def
thf(fact_8158_bit1_Omult__def,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ( ( times_times @ ( numeral_bit1 @ A ) )
= ( ^ [X2: numeral_bit1 @ A,Y2: numeral_bit1 @ A] : ( numeral_Abs_bit12 @ A @ ( modulo_modulo @ int @ ( times_times @ int @ ( numeral_Rep_bit1 @ A @ X2 ) @ ( numeral_Rep_bit1 @ A @ Y2 ) ) @ ( semiring_1_of_nat @ int @ ( finite_card @ ( numeral_bit1 @ A ) @ ( top_top @ ( set @ ( numeral_bit1 @ A ) ) ) ) ) ) ) ) ) ) ).
% bit1.mult_def
thf(fact_8159_bit1_Odiff__def,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ( ( minus_minus @ ( numeral_bit1 @ A ) )
= ( ^ [X2: numeral_bit1 @ A,Y2: numeral_bit1 @ A] : ( numeral_Abs_bit12 @ A @ ( modulo_modulo @ int @ ( minus_minus @ int @ ( numeral_Rep_bit1 @ A @ X2 ) @ ( numeral_Rep_bit1 @ A @ Y2 ) ) @ ( semiring_1_of_nat @ int @ ( finite_card @ ( numeral_bit1 @ A ) @ ( top_top @ ( set @ ( numeral_bit1 @ A ) ) ) ) ) ) ) ) ) ) ).
% bit1.diff_def
thf(fact_8160_bit1_OAbs__inverse,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ! [M: int] :
( ( member @ int @ M @ ( set_or7035219750837199246ssThan @ int @ ( zero_zero @ int ) @ ( semiring_1_of_nat @ int @ ( finite_card @ ( numeral_bit1 @ A ) @ ( top_top @ ( set @ ( numeral_bit1 @ A ) ) ) ) ) ) )
=> ( ( numeral_Rep_bit1 @ A @ ( numeral_Abs_bit12 @ A @ M ) )
= M ) ) ) ).
% bit1.Abs_inverse
thf(fact_8161_Rep__bit1__induct,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ! [Y: int,P: int > $o] :
( ( member @ int @ Y @ ( set_or7035219750837199246ssThan @ int @ ( zero_zero @ int ) @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ int @ ( finite_card @ A @ ( top_top @ ( set @ A ) ) ) ) ) ) ) )
=> ( ! [X3: numeral_bit1 @ A] : ( P @ ( numeral_Rep_bit1 @ A @ X3 ) )
=> ( P @ Y ) ) ) ) ).
% Rep_bit1_induct
thf(fact_8162_Rep__bit1__cases,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ! [Y: int] :
( ( member @ int @ Y @ ( set_or7035219750837199246ssThan @ int @ ( zero_zero @ int ) @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ int @ ( finite_card @ A @ ( top_top @ ( set @ A ) ) ) ) ) ) ) )
=> ~ ! [X3: numeral_bit1 @ A] :
( Y
!= ( numeral_Rep_bit1 @ A @ X3 ) ) ) ) ).
% Rep_bit1_cases
thf(fact_8163_Rep__bit1,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ! [X: numeral_bit1 @ A] : ( member @ int @ ( numeral_Rep_bit1 @ A @ X ) @ ( set_or7035219750837199246ssThan @ int @ ( zero_zero @ int ) @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ int @ ( finite_card @ A @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ) ).
% Rep_bit1
thf(fact_8164_type__definition__bit1,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ( type_definition @ ( numeral_bit1 @ A ) @ int @ ( numeral_Rep_bit1 @ A ) @ ( numeral_Abs_bit12 @ A ) @ ( set_or7035219750837199246ssThan @ int @ ( zero_zero @ int ) @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ int @ ( finite_card @ A @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ) ).
% type_definition_bit1
thf(fact_8165_card__lists__distinct__length__eq,axiom,
! [A: $tType,A2: set @ A,K: nat] :
( ( finite_finite2 @ A @ A2 )
=> ( ( ord_less_eq @ nat @ K @ ( finite_card @ A @ A2 ) )
=> ( ( finite_card @ ( list @ A )
@ ( collect @ ( list @ A )
@ ^ [Xs3: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs3 )
= K )
& ( distinct @ A @ Xs3 )
& ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs3 ) @ A2 ) ) ) )
= ( groups7121269368397514597t_prod @ nat @ nat
@ ^ [X2: nat] : X2
@ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ ( minus_minus @ nat @ ( finite_card @ A @ A2 ) @ K ) @ ( one_one @ nat ) ) @ ( finite_card @ A @ A2 ) ) ) ) ) ) ).
% card_lists_distinct_length_eq
thf(fact_8166_distinct__union,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( distinct @ A @ ( union @ A @ Xs @ Ys ) )
= ( distinct @ A @ Ys ) ) ).
% distinct_union
thf(fact_8167_finite__lists__distinct__length__eq,axiom,
! [A: $tType,A2: set @ A,N: nat] :
( ( finite_finite2 @ A @ A2 )
=> ( finite_finite2 @ ( list @ A )
@ ( collect @ ( list @ A )
@ ^ [Xs3: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs3 )
= N )
& ( distinct @ A @ Xs3 )
& ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs3 ) @ A2 ) ) ) ) ) ).
% finite_lists_distinct_length_eq
thf(fact_8168_finite__set__image,axiom,
! [A: $tType,A2: set @ ( list @ A )] :
( ( finite_finite2 @ ( set @ A ) @ ( image @ ( list @ A ) @ ( set @ A ) @ ( set2 @ A ) @ A2 ) )
=> ( ! [Xs2: list @ A] :
( ( member @ ( list @ A ) @ Xs2 @ A2 )
=> ( distinct @ A @ Xs2 ) )
=> ( finite_finite2 @ ( list @ A ) @ A2 ) ) ) ).
% finite_set_image
thf(fact_8169_distinct__length__le,axiom,
! [A: $tType,Ys: list @ A,Xs: list @ A] :
( ( distinct @ A @ Ys )
=> ( ( ( set2 @ A @ Ys )
= ( set2 @ A @ Xs ) )
=> ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Ys ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ).
% distinct_length_le
thf(fact_8170_finite__distinct__list,axiom,
! [A: $tType,A2: set @ A] :
( ( finite_finite2 @ A @ A2 )
=> ? [Xs2: list @ A] :
( ( ( set2 @ A @ Xs2 )
= A2 )
& ( distinct @ A @ Xs2 ) ) ) ).
% finite_distinct_list
thf(fact_8171_distinct__finite__set,axiom,
! [A: $tType,X: set @ A] :
( finite_finite2 @ ( list @ A )
@ ( collect @ ( list @ A )
@ ^ [Ys3: list @ A] :
( ( ( set2 @ A @ Ys3 )
= X )
& ( distinct @ A @ Ys3 ) ) ) ) ).
% distinct_finite_set
thf(fact_8172_distinct__length__2__or__more,axiom,
! [A: $tType,A3: A,B3: A,Xs: list @ A] :
( ( distinct @ A @ ( cons @ A @ A3 @ ( cons @ A @ B3 @ Xs ) ) )
= ( ( A3 != B3 )
& ( distinct @ A @ ( cons @ A @ A3 @ Xs ) )
& ( distinct @ A @ ( cons @ A @ B3 @ Xs ) ) ) ) ).
% distinct_length_2_or_more
thf(fact_8173_distinct_Osimps_I2_J,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( distinct @ A @ ( cons @ A @ X @ Xs ) )
= ( ~ ( member @ A @ X @ ( set2 @ A @ Xs ) )
& ( distinct @ A @ Xs ) ) ) ).
% distinct.simps(2)
thf(fact_8174_distinct__product,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B] :
( ( distinct @ A @ Xs )
=> ( ( distinct @ B @ Ys )
=> ( distinct @ ( product_prod @ A @ B ) @ ( product @ A @ B @ Xs @ Ys ) ) ) ) ).
% distinct_product
thf(fact_8175_sorted__list__of__set_Odistinct__if__distinct__map,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A] :
( ( distinct @ A @ Xs )
=> ( distinct @ A @ Xs ) ) ) ).
% sorted_list_of_set.distinct_if_distinct_map
thf(fact_8176_nth__eq__iff__index__eq,axiom,
! [A: $tType,Xs: list @ A,I: nat,J: nat] :
( ( distinct @ A @ Xs )
=> ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ( nth @ A @ Xs @ I )
= ( nth @ A @ Xs @ J ) )
= ( I = J ) ) ) ) ) ).
% nth_eq_iff_index_eq
thf(fact_8177_distinct__conv__nth,axiom,
! [A: $tType] :
( ( distinct @ A )
= ( ^ [Xs3: list @ A] :
! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( size_size @ ( list @ A ) @ Xs3 ) )
=> ! [J3: nat] :
( ( ord_less @ nat @ J3 @ ( size_size @ ( list @ A ) @ Xs3 ) )
=> ( ( I4 != J3 )
=> ( ( nth @ A @ Xs3 @ I4 )
!= ( nth @ A @ Xs3 @ J3 ) ) ) ) ) ) ) ).
% distinct_conv_nth
% Type constructors (1270)
thf(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__linorder,axiom,
comple5582772986160207858norder @ product_unit ).
thf(tcon_Extended__Nat_Oenat___Complete__Lattices_Ocomplete__linorder_1,axiom,
comple5582772986160207858norder @ extended_enat ).
thf(tcon_Option_Ooption___Complete__Lattices_Ocomplete__linorder_2,axiom,
! [A11: $tType] :
( ( comple5582772986160207858norder @ A11 )
=> ( comple5582772986160207858norder @ ( option @ A11 ) ) ) ).
thf(tcon_Enum_Ofinite__3___Complete__Lattices_Ocomplete__linorder_3,axiom,
comple5582772986160207858norder @ finite_3 ).
thf(tcon_Enum_Ofinite__2___Complete__Lattices_Ocomplete__linorder_4,axiom,
comple5582772986160207858norder @ finite_2 ).
thf(tcon_Enum_Ofinite__1___Complete__Lattices_Ocomplete__linorder_5,axiom,
comple5582772986160207858norder @ finite_1 ).
thf(tcon_fun___Lattices_Obounded__semilattice__sup__bot,axiom,
! [A11: $tType,A14: $tType] :
( ( bounded_lattice @ A14 )
=> ( bounde4967611905675639751up_bot @ ( A11 > A14 ) ) ) ).
thf(tcon_fun___Boolean__Algebras_Oboolean__algebra,axiom,
! [A11: $tType,A14: $tType] :
( ( boolea8198339166811842893lgebra @ A14 )
=> ( boolea8198339166811842893lgebra @ ( A11 > A14 ) ) ) ).
thf(tcon_fun___Lattices_Obounded__lattice__top,axiom,
! [A11: $tType,A14: $tType] :
( ( bounded_lattice @ A14 )
=> ( bounded_lattice_top @ ( A11 > A14 ) ) ) ).
thf(tcon_fun___Lattices_Osemilattice__sup,axiom,
! [A11: $tType,A14: $tType] :
( ( semilattice_sup @ A14 )
=> ( semilattice_sup @ ( A11 > A14 ) ) ) ).
thf(tcon_fun___Lattices_Obounded__lattice,axiom,
! [A11: $tType,A14: $tType] :
( ( bounded_lattice @ A14 )
=> ( bounded_lattice @ ( A11 > A14 ) ) ) ).
thf(tcon_fun___Orderings_Oorder__top,axiom,
! [A11: $tType,A14: $tType] :
( ( order_top @ A14 )
=> ( order_top @ ( A11 > A14 ) ) ) ).
thf(tcon_fun___Orderings_Oorder__bot,axiom,
! [A11: $tType,A14: $tType] :
( ( order_bot @ A14 )
=> ( order_bot @ ( A11 > A14 ) ) ) ).
thf(tcon_fun___Orderings_Opreorder,axiom,
! [A11: $tType,A14: $tType] :
( ( preorder @ A14 )
=> ( preorder @ ( A11 > A14 ) ) ) ).
thf(tcon_fun___Finite__Set_Ofinite,axiom,
! [A11: $tType,A14: $tType] :
( ( ( finite_finite @ A11 )
& ( finite_finite @ A14 ) )
=> ( finite_finite @ ( A11 > A14 ) ) ) ).
thf(tcon_fun___Lattices_Olattice,axiom,
! [A11: $tType,A14: $tType] :
( ( lattice @ A14 )
=> ( lattice @ ( A11 > A14 ) ) ) ).
thf(tcon_fun___Orderings_Oorder,axiom,
! [A11: $tType,A14: $tType] :
( ( order @ A14 )
=> ( order @ ( A11 > A14 ) ) ) ).
thf(tcon_fun___Orderings_Otop,axiom,
! [A11: $tType,A14: $tType] :
( ( top @ A14 )
=> ( top @ ( A11 > A14 ) ) ) ).
thf(tcon_fun___Orderings_Oord,axiom,
! [A11: $tType,A14: $tType] :
( ( ord @ A14 )
=> ( ord @ ( A11 > A14 ) ) ) ).
thf(tcon_fun___Orderings_Obot,axiom,
! [A11: $tType,A14: $tType] :
( ( bot @ A14 )
=> ( bot @ ( A11 > A14 ) ) ) ).
thf(tcon_fun___Groups_Ouminus,axiom,
! [A11: $tType,A14: $tType] :
( ( uminus @ A14 )
=> ( uminus @ ( A11 > A14 ) ) ) ).
thf(tcon_fun___Groups_Ominus,axiom,
! [A11: $tType,A14: $tType] :
( ( minus @ A14 )
=> ( minus @ ( A11 > A14 ) ) ) ).
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_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___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___Bit__Operations_Oring__bit__operations,axiom,
bit_ri3973907225187159222ations @ int ).
thf(tcon_Int_Oint___Topological__Spaces_Oorder__topology,axiom,
topolo2564578578187576103pology @ int ).
thf(tcon_Int_Oint___Rings_Osemiring__1__no__zero__divisors,axiom,
semiri2026040879449505780visors @ int ).
thf(tcon_Int_Oint___Rings_Olinordered__nonzero__semiring,axiom,
linord181362715937106298miring @ int ).
thf(tcon_Int_Oint___Limits_Otopological__semigroup__mult,axiom,
topolo4211221413907600880p_mult @ int ).
thf(tcon_Int_Oint___Rings_Olinordered__semiring__strict,axiom,
linord8928482502909563296strict @ int ).
thf(tcon_Int_Oint___Rings_Osemiring__no__zero__divisors,axiom,
semiri3467727345109120633visors @ int ).
thf(tcon_Int_Oint___Groups_Oordered__ab__semigroup__add,axiom,
ordere6658533253407199908up_add @ int ).
thf(tcon_Int_Oint___Groups_Oordered__ab__group__add__abs,axiom,
ordere166539214618696060dd_abs @ int ).
thf(tcon_Int_Oint___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___Lattices_Osemilattice__sup_6,axiom,
semilattice_sup @ int ).
thf(tcon_Int_Oint___Groups_Oab__semigroup__mult,axiom,
ab_semigroup_mult @ int ).
thf(tcon_Int_Oint___Rings_Osemiring__1__cancel,axiom,
semiring_1_cancel @ int ).
thf(tcon_Int_Oint___Rings_Oalgebraic__semidom,axiom,
algebraic_semidom @ int ).
thf(tcon_Int_Oint___Groups_Ocomm__monoid__mult,axiom,
comm_monoid_mult @ int ).
thf(tcon_Int_Oint___Groups_Oab__semigroup__add,axiom,
ab_semigroup_add @ int ).
thf(tcon_Int_Oint___Generic__set__bit_Oset__bit,axiom,
generic_set_set_bit @ int ).
thf(tcon_Int_Oint___Rings_Oordered__semiring,axiom,
ordered_semiring @ int ).
thf(tcon_Int_Oint___Rings_Oordered__ring__abs,axiom,
ordered_ring_abs @ int ).
thf(tcon_Int_Oint___Parity_Osemiring__parity,axiom,
semiring_parity @ int ).
thf(tcon_Int_Oint___Groups_Ocomm__monoid__add,axiom,
comm_monoid_add @ int ).
thf(tcon_Int_Oint___Rings_Osemiring__modulo,axiom,
semiring_modulo @ int ).
thf(tcon_Int_Oint___Rings_Olinordered__ring,axiom,
linordered_ring @ int ).
thf(tcon_Int_Oint___Rings_Olinordered__idom,axiom,
linordered_idom @ int ).
thf(tcon_Int_Oint___Rings_Ocomm__semiring__1,axiom,
comm_semiring_1 @ int ).
thf(tcon_Int_Oint___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_7,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___Lattices_Olattice_8,axiom,
lattice @ int ).
thf(tcon_Int_Oint___Groups_Ogroup__add,axiom,
group_add @ int ).
thf(tcon_Int_Oint___GCD_Osemiring__gcd,axiom,
semiring_gcd @ int ).
thf(tcon_Int_Oint___Rings_Omult__zero,axiom,
mult_zero @ int ).
thf(tcon_Int_Oint___Rings_Ocomm__ring,axiom,
comm_ring @ int ).
thf(tcon_Int_Oint___Orderings_Oorder_9,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_10,axiom,
ord @ int ).
thf(tcon_Int_Oint___Groups_Ouminus_11,axiom,
uminus @ int ).
thf(tcon_Int_Oint___Rings_Oring__1,axiom,
ring_1 @ int ).
thf(tcon_Int_Oint___Rings_Oabs__if,axiom,
abs_if @ int ).
thf(tcon_Int_Oint___Groups_Ominus_12,axiom,
minus @ int ).
thf(tcon_Int_Oint___Power_Opower,axiom,
power @ int ).
thf(tcon_Int_Oint___Num_Onumeral,axiom,
numeral @ int ).
thf(tcon_Int_Oint___Groups_Ozero,axiom,
zero @ int ).
thf(tcon_Int_Oint___Groups_Oplus,axiom,
plus @ int ).
thf(tcon_Int_Oint___Rings_Oring,axiom,
ring @ int ).
thf(tcon_Int_Oint___Rings_Oidom,axiom,
idom @ int ).
thf(tcon_Int_Oint___Groups_Oone,axiom,
one @ int ).
thf(tcon_Int_Oint___Rings_Odvd,axiom,
dvd @ int ).
thf(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_13,axiom,
condit6923001295902523014norder @ nat ).
thf(tcon_Nat_Onat___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_14,axiom,
bit_un5681908812861735899ations @ nat ).
thf(tcon_Nat_Onat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_15,axiom,
semiri1453513574482234551roduct @ nat ).
thf(tcon_Nat_Onat___Euclidean__Division_Ounique__euclidean__semiring__with__nat_16,axiom,
euclid5411537665997757685th_nat @ nat ).
thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__monoid__add__imp__le_17,axiom,
ordere1937475149494474687imp_le @ nat ).
thf(tcon_Nat_Onat___Euclidean__Division_Oeuclidean__semiring__cancel_18,axiom,
euclid4440199948858584721cancel @ nat ).
thf(tcon_Nat_Onat___Divides_Ounique__euclidean__semiring__numeral_19,axiom,
unique1627219031080169319umeral @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors__cancel_20,axiom,
semiri6575147826004484403cancel @ nat ).
thf(tcon_Nat_Onat___Groups_Ostrict__ordered__ab__semigroup__add_21,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_22,axiom,
ordere580206878836729694up_add @ nat ).
thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le_23,axiom,
ordere2412721322843649153imp_le @ nat ).
thf(tcon_Nat_Onat___Bit__Operations_Osemiring__bit__operations_24,axiom,
bit_se359711467146920520ations @ nat ).
thf(tcon_Nat_Onat___Rings_Olinordered__comm__semiring__strict_25,axiom,
linord2810124833399127020strict @ nat ).
thf(tcon_Nat_Onat___Groups_Ostrict__ordered__comm__monoid__add_26,axiom,
strict7427464778891057005id_add @ nat ).
thf(tcon_Nat_Onat___Groups_Oordered__cancel__comm__monoid__add_27,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_28,axiom,
euclid3725896446679973847miring @ nat ).
thf(tcon_Nat_Onat___Topological__Spaces_Otopological__space_29,axiom,
topolo4958980785337419405_space @ nat ).
thf(tcon_Nat_Onat___Topological__Spaces_Olinorder__topology_30,axiom,
topolo1944317154257567458pology @ nat ).
thf(tcon_Nat_Onat___Topological__Spaces_Odiscrete__topology_31,axiom,
topolo8865339358273720382pology @ nat ).
thf(tcon_Nat_Onat___Limits_Otopological__comm__monoid__mult_32,axiom,
topolo4987421752381908075d_mult @ nat ).
thf(tcon_Nat_Onat___Limits_Otopological__comm__monoid__add_33,axiom,
topolo5987344860129210374id_add @ nat ).
thf(tcon_Nat_Onat___Topological__Spaces_Oorder__topology_34,axiom,
topolo2564578578187576103pology @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__1__no__zero__divisors_35,axiom,
semiri2026040879449505780visors @ nat ).
thf(tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring_36,axiom,
linord181362715937106298miring @ nat ).
thf(tcon_Nat_Onat___Limits_Otopological__semigroup__mult_37,axiom,
topolo4211221413907600880p_mult @ nat ).
thf(tcon_Nat_Onat___Rings_Olinordered__semiring__strict_38,axiom,
linord8928482502909563296strict @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors_39,axiom,
semiri3467727345109120633visors @ nat ).
thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add_40,axiom,
ordere6658533253407199908up_add @ nat ).
thf(tcon_Nat_Onat___Limits_Otopological__monoid__mult_41,axiom,
topolo1898628316856586783d_mult @ nat ).
thf(tcon_Nat_Onat___Groups_Oordered__comm__monoid__add_42,axiom,
ordere6911136660526730532id_add @ nat ).
thf(tcon_Nat_Onat___Groups_Ocancel__ab__semigroup__add_43,axiom,
cancel2418104881723323429up_add @ nat ).
thf(tcon_Nat_Onat___Limits_Otopological__monoid__add_44,axiom,
topolo6943815403480290642id_add @ nat ).
thf(tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add_45,axiom,
cancel1802427076303600483id_add @ nat ).
thf(tcon_Nat_Onat___Rings_Ocomm__semiring__1__cancel_46,axiom,
comm_s4317794764714335236cancel @ nat ).
thf(tcon_Nat_Onat___Bit__Operations_Osemiring__bits_47,axiom,
bit_semiring_bits @ nat ).
thf(tcon_Nat_Onat___Topological__Spaces_Ot2__space_48,axiom,
topological_t2_space @ nat ).
thf(tcon_Nat_Onat___Topological__Spaces_Ot1__space_49,axiom,
topological_t1_space @ nat ).
thf(tcon_Nat_Onat___Rings_Oordered__comm__semiring_50,axiom,
ordere2520102378445227354miring @ nat ).
thf(tcon_Nat_Onat___Groups_Ocancel__semigroup__add_51,axiom,
cancel_semigroup_add @ nat ).
thf(tcon_Nat_Onat___Rings_Olinordered__semiring_52,axiom,
linordered_semiring @ nat ).
thf(tcon_Nat_Onat___Rings_Oordered__semiring__0_53,axiom,
ordered_semiring_0 @ nat ).
thf(tcon_Nat_Onat___Rings_Olinordered__semidom_54,axiom,
linordered_semidom @ nat ).
thf(tcon_Nat_Onat___Lattices_Osemilattice__sup_55,axiom,
semilattice_sup @ nat ).
thf(tcon_Nat_Onat___Groups_Oab__semigroup__mult_56,axiom,
ab_semigroup_mult @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__1__cancel_57,axiom,
semiring_1_cancel @ nat ).
thf(tcon_Nat_Onat___Rings_Oalgebraic__semidom_58,axiom,
algebraic_semidom @ nat ).
thf(tcon_Nat_Onat___Groups_Ocomm__monoid__mult_59,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_60,axiom,
ab_semigroup_add @ nat ).
thf(tcon_Nat_Onat___Rings_Oordered__semiring_61,axiom,
ordered_semiring @ nat ).
thf(tcon_Nat_Onat___Parity_Osemiring__parity_62,axiom,
semiring_parity @ nat ).
thf(tcon_Nat_Onat___Groups_Ocomm__monoid__add_63,axiom,
comm_monoid_add @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__modulo_64,axiom,
semiring_modulo @ nat ).
thf(tcon_Nat_Onat___Rings_Ocomm__semiring__1_65,axiom,
comm_semiring_1 @ nat ).
thf(tcon_Nat_Onat___Groups_Osemigroup__mult_66,axiom,
semigroup_mult @ nat ).
thf(tcon_Nat_Onat___Rings_Osemidom__modulo_67,axiom,
semidom_modulo @ nat ).
thf(tcon_Nat_Onat___Rings_Osemidom__divide_68,axiom,
semidom_divide @ nat ).
thf(tcon_Nat_Onat___Num_Osemiring__numeral_69,axiom,
semiring_numeral @ nat ).
thf(tcon_Nat_Onat___Groups_Osemigroup__add_70,axiom,
semigroup_add @ nat ).
thf(tcon_Nat_Onat___Rings_Ozero__less__one_71,axiom,
zero_less_one @ nat ).
thf(tcon_Nat_Onat___Rings_Ocomm__semiring_72,axiom,
comm_semiring @ nat ).
thf(tcon_Nat_Onat___Orderings_Owellorder,axiom,
wellorder @ nat ).
thf(tcon_Nat_Onat___Orderings_Oorder__bot_73,axiom,
order_bot @ nat ).
thf(tcon_Nat_Onat___Nat_Osemiring__char__0_74,axiom,
semiring_char_0 @ nat ).
thf(tcon_Nat_Onat___Rings_Ozero__neq__one_75,axiom,
zero_neq_one @ nat ).
thf(tcon_Nat_Onat___Orderings_Opreorder_76,axiom,
preorder @ nat ).
thf(tcon_Nat_Onat___Orderings_Olinorder_77,axiom,
linorder @ nat ).
thf(tcon_Nat_Onat___Groups_Omonoid__mult_78,axiom,
monoid_mult @ nat ).
thf(tcon_Nat_Onat___Groups_Omonoid__add_79,axiom,
monoid_add @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__1_80,axiom,
semiring_1 @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__0_81,axiom,
semiring_0 @ nat ).
thf(tcon_Nat_Onat___Orderings_Ono__top_82,axiom,
no_top @ nat ).
thf(tcon_Nat_Onat___Lattices_Olattice_83,axiom,
lattice @ nat ).
thf(tcon_Nat_Onat___GCD_Osemiring__gcd_84,axiom,
semiring_gcd @ nat ).
thf(tcon_Nat_Onat___Rings_Omult__zero_85,axiom,
mult_zero @ nat ).
thf(tcon_Nat_Onat___Orderings_Oorder_86,axiom,
order @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring_87,axiom,
semiring @ nat ).
thf(tcon_Nat_Onat___Rings_Osemidom_88,axiom,
semidom @ nat ).
thf(tcon_Nat_Onat___Orderings_Oord_89,axiom,
ord @ nat ).
thf(tcon_Nat_Onat___Orderings_Obot_90,axiom,
bot @ nat ).
thf(tcon_Nat_Onat___Groups_Ominus_91,axiom,
minus @ nat ).
thf(tcon_Nat_Onat___Power_Opower_92,axiom,
power @ nat ).
thf(tcon_Nat_Onat___Num_Onumeral_93,axiom,
numeral @ nat ).
thf(tcon_Nat_Onat___Groups_Ozero_94,axiom,
zero @ nat ).
thf(tcon_Nat_Onat___Groups_Oplus_95,axiom,
plus @ nat ).
thf(tcon_Nat_Onat___Groups_Oone_96,axiom,
one @ nat ).
thf(tcon_Nat_Onat___Rings_Odvd_97,axiom,
dvd @ nat ).
thf(tcon_Nat_Onat___Nat_Osize,axiom,
size @ nat ).
thf(tcon_Num_Onum___Orderings_Opreorder_98,axiom,
preorder @ num ).
thf(tcon_Num_Onum___Orderings_Olinorder_99,axiom,
linorder @ num ).
thf(tcon_Num_Onum___Orderings_Oorder_100,axiom,
order @ num ).
thf(tcon_Num_Onum___Orderings_Oord_101,axiom,
ord @ num ).
thf(tcon_Num_Onum___Groups_Oplus_102,axiom,
plus @ num ).
thf(tcon_Num_Onum___Nat_Osize_103,axiom,
size @ num ).
thf(tcon_Rat_Orat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_104,axiom,
semiri1453513574482234551roduct @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__monoid__add__imp__le_105,axiom,
ordere1937475149494474687imp_le @ rat ).
thf(tcon_Rat_Orat___Rings_Osemiring__no__zero__divisors__cancel_106,axiom,
semiri6575147826004484403cancel @ rat ).
thf(tcon_Rat_Orat___Groups_Ostrict__ordered__ab__semigroup__add_107,axiom,
strict9044650504122735259up_add @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__cancel__ab__semigroup__add_108,axiom,
ordere580206878836729694up_add @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__add__imp__le_109,axiom,
ordere2412721322843649153imp_le @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__comm__semiring__strict_110,axiom,
linord2810124833399127020strict @ rat ).
thf(tcon_Rat_Orat___Groups_Ostrict__ordered__comm__monoid__add_111,axiom,
strict7427464778891057005id_add @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__cancel__comm__monoid__add_112,axiom,
ordere8940638589300402666id_add @ rat ).
thf(tcon_Rat_Orat___Archimedean__Field_Oarchimedean__field,axiom,
archim462609752435547400_field @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__semiring__1__strict_113,axiom,
linord715952674999750819strict @ rat ).
thf(tcon_Rat_Orat___Orderings_Ounbounded__dense__linorder,axiom,
unboun7993243217541854897norder @ rat ).
thf(tcon_Rat_Orat___Rings_Osemiring__1__no__zero__divisors_114,axiom,
semiri2026040879449505780visors @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__nonzero__semiring_115,axiom,
linord181362715937106298miring @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__semiring__strict_116,axiom,
linord8928482502909563296strict @ rat ).
thf(tcon_Rat_Orat___Rings_Osemiring__no__zero__divisors_117,axiom,
semiri3467727345109120633visors @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__add_118,axiom,
ordere6658533253407199908up_add @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__ab__group__add__abs_119,axiom,
ordere166539214618696060dd_abs @ rat ).
thf(tcon_Rat_Orat___Archimedean__Field_Ofloor__ceiling,axiom,
archim2362893244070406136eiling @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__comm__monoid__add_120,axiom,
ordere6911136660526730532id_add @ rat ).
thf(tcon_Rat_Orat___Groups_Olinordered__ab__group__add_121,axiom,
linord5086331880401160121up_add @ rat ).
thf(tcon_Rat_Orat___Groups_Ocancel__ab__semigroup__add_122,axiom,
cancel2418104881723323429up_add @ rat ).
thf(tcon_Rat_Orat___Rings_Oring__1__no__zero__divisors_123,axiom,
ring_15535105094025558882visors @ rat ).
thf(tcon_Rat_Orat___Groups_Ocancel__comm__monoid__add_124,axiom,
cancel1802427076303600483id_add @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__ring__strict_125,axiom,
linord4710134922213307826strict @ rat ).
thf(tcon_Rat_Orat___Rings_Ocomm__semiring__1__cancel_126,axiom,
comm_s4317794764714335236cancel @ rat ).
thf(tcon_Rat_Orat___Rings_Oordered__comm__semiring_127,axiom,
ordere2520102378445227354miring @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__semiring__1_128,axiom,
linord6961819062388156250ring_1 @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__ab__group__add_129,axiom,
ordered_ab_group_add @ rat ).
thf(tcon_Rat_Orat___Groups_Ocancel__semigroup__add_130,axiom,
cancel_semigroup_add @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__semiring_131,axiom,
linordered_semiring @ rat ).
thf(tcon_Rat_Orat___Rings_Oordered__semiring__0_132,axiom,
ordered_semiring_0 @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__semidom_133,axiom,
linordered_semidom @ rat ).
thf(tcon_Rat_Orat___Orderings_Odense__linorder,axiom,
dense_linorder @ rat ).
thf(tcon_Rat_Orat___Lattices_Osemilattice__sup_134,axiom,
semilattice_sup @ rat ).
thf(tcon_Rat_Orat___Groups_Oab__semigroup__mult_135,axiom,
ab_semigroup_mult @ rat ).
thf(tcon_Rat_Orat___Rings_Osemiring__1__cancel_136,axiom,
semiring_1_cancel @ rat ).
thf(tcon_Rat_Orat___Groups_Ocomm__monoid__mult_137,axiom,
comm_monoid_mult @ rat ).
thf(tcon_Rat_Orat___Groups_Oab__semigroup__add_138,axiom,
ab_semigroup_add @ rat ).
thf(tcon_Rat_Orat___Fields_Olinordered__field,axiom,
linordered_field @ rat ).
thf(tcon_Rat_Orat___Rings_Oordered__semiring_139,axiom,
ordered_semiring @ rat ).
thf(tcon_Rat_Orat___Rings_Oordered__ring__abs_140,axiom,
ordered_ring_abs @ rat ).
thf(tcon_Rat_Orat___Groups_Ocomm__monoid__add_141,axiom,
comm_monoid_add @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__ring_142,axiom,
linordered_ring @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__idom_143,axiom,
linordered_idom @ rat ).
thf(tcon_Rat_Orat___Rings_Ocomm__semiring__1_144,axiom,
comm_semiring_1 @ rat ).
thf(tcon_Rat_Orat___Orderings_Odense__order,axiom,
dense_order @ rat ).
thf(tcon_Rat_Orat___Groups_Osemigroup__mult_145,axiom,
semigroup_mult @ rat ).
thf(tcon_Rat_Orat___Rings_Osemidom__divide_146,axiom,
semidom_divide @ rat ).
thf(tcon_Rat_Orat___Num_Osemiring__numeral_147,axiom,
semiring_numeral @ rat ).
thf(tcon_Rat_Orat___Groups_Osemigroup__add_148,axiom,
semigroup_add @ rat ).
thf(tcon_Rat_Orat___Fields_Odivision__ring,axiom,
division_ring @ rat ).
thf(tcon_Rat_Orat___Rings_Ozero__less__one_149,axiom,
zero_less_one @ rat ).
thf(tcon_Rat_Orat___Rings_Ocomm__semiring_150,axiom,
comm_semiring @ rat ).
thf(tcon_Rat_Orat___Nat_Osemiring__char__0_151,axiom,
semiring_char_0 @ rat ).
thf(tcon_Rat_Orat___Groups_Oab__group__add_152,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_153,axiom,
zero_neq_one @ rat ).
thf(tcon_Rat_Orat___Rings_Oordered__ring_154,axiom,
ordered_ring @ rat ).
thf(tcon_Rat_Orat___Rings_Oidom__abs__sgn_155,axiom,
idom_abs_sgn @ rat ).
thf(tcon_Rat_Orat___Orderings_Opreorder_156,axiom,
preorder @ rat ).
thf(tcon_Rat_Orat___Orderings_Olinorder_157,axiom,
linorder @ rat ).
thf(tcon_Rat_Orat___Groups_Omonoid__mult_158,axiom,
monoid_mult @ rat ).
thf(tcon_Rat_Orat___Rings_Oidom__divide_159,axiom,
idom_divide @ rat ).
thf(tcon_Rat_Orat___Rings_Ocomm__ring__1_160,axiom,
comm_ring_1 @ rat ).
thf(tcon_Rat_Orat___Groups_Omonoid__add_161,axiom,
monoid_add @ rat ).
thf(tcon_Rat_Orat___Rings_Osemiring__1_162,axiom,
semiring_1 @ rat ).
thf(tcon_Rat_Orat___Rings_Osemiring__0_163,axiom,
semiring_0 @ rat ).
thf(tcon_Rat_Orat___Orderings_Ono__top_164,axiom,
no_top @ rat ).
thf(tcon_Rat_Orat___Orderings_Ono__bot_165,axiom,
no_bot @ rat ).
thf(tcon_Rat_Orat___Lattices_Olattice_166,axiom,
lattice @ rat ).
thf(tcon_Rat_Orat___Groups_Ogroup__add_167,axiom,
group_add @ rat ).
thf(tcon_Rat_Orat___Rings_Omult__zero_168,axiom,
mult_zero @ rat ).
thf(tcon_Rat_Orat___Rings_Ocomm__ring_169,axiom,
comm_ring @ rat ).
thf(tcon_Rat_Orat___Orderings_Oorder_170,axiom,
order @ rat ).
thf(tcon_Rat_Orat___Num_Oneg__numeral_171,axiom,
neg_numeral @ rat ).
thf(tcon_Rat_Orat___Nat_Oring__char__0_172,axiom,
ring_char_0 @ rat ).
thf(tcon_Rat_Orat___Rings_Osemiring_173,axiom,
semiring @ rat ).
thf(tcon_Rat_Orat___Fields_Oinverse,axiom,
inverse @ rat ).
thf(tcon_Rat_Orat___Rings_Osemidom_174,axiom,
semidom @ rat ).
thf(tcon_Rat_Orat___Orderings_Oord_175,axiom,
ord @ rat ).
thf(tcon_Rat_Orat___Groups_Ouminus_176,axiom,
uminus @ rat ).
thf(tcon_Rat_Orat___Rings_Oring__1_177,axiom,
ring_1 @ rat ).
thf(tcon_Rat_Orat___Rings_Oabs__if_178,axiom,
abs_if @ rat ).
thf(tcon_Rat_Orat___Groups_Ominus_179,axiom,
minus @ rat ).
thf(tcon_Rat_Orat___Fields_Ofield,axiom,
field @ rat ).
thf(tcon_Rat_Orat___Power_Opower_180,axiom,
power @ rat ).
thf(tcon_Rat_Orat___Num_Onumeral_181,axiom,
numeral @ rat ).
thf(tcon_Rat_Orat___Groups_Ozero_182,axiom,
zero @ rat ).
thf(tcon_Rat_Orat___Groups_Oplus_183,axiom,
plus @ rat ).
thf(tcon_Rat_Orat___Rings_Oring_184,axiom,
ring @ rat ).
thf(tcon_Rat_Orat___Rings_Oidom_185,axiom,
idom @ rat ).
thf(tcon_Rat_Orat___Groups_Oone_186,axiom,
one @ rat ).
thf(tcon_Rat_Orat___Rings_Odvd_187,axiom,
dvd @ rat ).
thf(tcon_Set_Oset___Lattices_Obounded__semilattice__sup__bot_188,axiom,
! [A11: $tType] : ( bounde4967611905675639751up_bot @ ( set @ A11 ) ) ).
thf(tcon_Set_Oset___Boolean__Algebras_Oboolean__algebra_189,axiom,
! [A11: $tType] : ( boolea8198339166811842893lgebra @ ( set @ A11 ) ) ).
thf(tcon_Set_Oset___Lattices_Obounded__lattice__top_190,axiom,
! [A11: $tType] : ( bounded_lattice_top @ ( set @ A11 ) ) ).
thf(tcon_Set_Oset___Lattices_Osemilattice__sup_191,axiom,
! [A11: $tType] : ( semilattice_sup @ ( set @ A11 ) ) ).
thf(tcon_Set_Oset___Lattices_Obounded__lattice_192,axiom,
! [A11: $tType] : ( bounded_lattice @ ( set @ A11 ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder__top_193,axiom,
! [A11: $tType] : ( order_top @ ( set @ A11 ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder__bot_194,axiom,
! [A11: $tType] : ( order_bot @ ( set @ A11 ) ) ).
thf(tcon_Set_Oset___Orderings_Opreorder_195,axiom,
! [A11: $tType] : ( preorder @ ( set @ A11 ) ) ).
thf(tcon_Set_Oset___Finite__Set_Ofinite_196,axiom,
! [A11: $tType] :
( ( finite_finite @ A11 )
=> ( finite_finite @ ( set @ A11 ) ) ) ).
thf(tcon_Set_Oset___Lattices_Olattice_197,axiom,
! [A11: $tType] : ( lattice @ ( set @ A11 ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder_198,axiom,
! [A11: $tType] : ( order @ ( set @ A11 ) ) ).
thf(tcon_Set_Oset___Orderings_Otop_199,axiom,
! [A11: $tType] : ( top @ ( set @ A11 ) ) ).
thf(tcon_Set_Oset___Orderings_Oord_200,axiom,
! [A11: $tType] : ( ord @ ( set @ A11 ) ) ).
thf(tcon_Set_Oset___Orderings_Obot_201,axiom,
! [A11: $tType] : ( bot @ ( set @ A11 ) ) ).
thf(tcon_Set_Oset___Groups_Ouminus_202,axiom,
! [A11: $tType] : ( uminus @ ( set @ A11 ) ) ).
thf(tcon_Set_Oset___Groups_Ominus_203,axiom,
! [A11: $tType] : ( minus @ ( set @ A11 ) ) ).
thf(tcon_HOL_Obool___Topological__Spaces_Otopological__space_204,axiom,
topolo4958980785337419405_space @ $o ).
thf(tcon_HOL_Obool___Topological__Spaces_Olinorder__topology_205,axiom,
topolo1944317154257567458pology @ $o ).
thf(tcon_HOL_Obool___Topological__Spaces_Odiscrete__topology_206,axiom,
topolo8865339358273720382pology @ $o ).
thf(tcon_HOL_Obool___Lattices_Obounded__semilattice__sup__bot_207,axiom,
bounde4967611905675639751up_bot @ $o ).
thf(tcon_HOL_Obool___Topological__Spaces_Oorder__topology_208,axiom,
topolo2564578578187576103pology @ $o ).
thf(tcon_HOL_Obool___Boolean__Algebras_Oboolean__algebra_209,axiom,
boolea8198339166811842893lgebra @ $o ).
thf(tcon_HOL_Obool___Lattices_Obounded__lattice__top_210,axiom,
bounded_lattice_top @ $o ).
thf(tcon_HOL_Obool___Topological__Spaces_Ot2__space_211,axiom,
topological_t2_space @ $o ).
thf(tcon_HOL_Obool___Topological__Spaces_Ot1__space_212,axiom,
topological_t1_space @ $o ).
thf(tcon_HOL_Obool___Lattices_Osemilattice__sup_213,axiom,
semilattice_sup @ $o ).
thf(tcon_HOL_Obool___Lattices_Obounded__lattice_214,axiom,
bounded_lattice @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder__top_215,axiom,
order_top @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder__bot_216,axiom,
order_bot @ $o ).
thf(tcon_HOL_Obool___Orderings_Opreorder_217,axiom,
preorder @ $o ).
thf(tcon_HOL_Obool___Orderings_Olinorder_218,axiom,
linorder @ $o ).
thf(tcon_HOL_Obool___Finite__Set_Ofinite_219,axiom,
finite_finite @ $o ).
thf(tcon_HOL_Obool___Lattices_Olattice_220,axiom,
lattice @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder_221,axiom,
order @ $o ).
thf(tcon_HOL_Obool___Orderings_Otop_222,axiom,
top @ $o ).
thf(tcon_HOL_Obool___Orderings_Oord_223,axiom,
ord @ $o ).
thf(tcon_HOL_Obool___Orderings_Obot_224,axiom,
bot @ $o ).
thf(tcon_HOL_Obool___Groups_Ouminus_225,axiom,
uminus @ $o ).
thf(tcon_HOL_Obool___Groups_Ominus_226,axiom,
minus @ $o ).
thf(tcon_List_Olist___Nat_Osize_227,axiom,
! [A11: $tType] : ( size @ ( list @ A11 ) ) ).
thf(tcon_Real_Oreal___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_228,axiom,
condit6923001295902523014norder @ real ).
thf(tcon_Real_Oreal___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_229,axiom,
semiri1453513574482234551roduct @ real ).
thf(tcon_Real_Oreal___Conditionally__Complete__Lattices_Olinear__continuum,axiom,
condit5016429287641298734tinuum @ real ).
thf(tcon_Real_Oreal___Groups_Oordered__ab__semigroup__monoid__add__imp__le_230,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_231,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_232,axiom,
strict9044650504122735259up_add @ real ).
thf(tcon_Real_Oreal___Groups_Oordered__cancel__ab__semigroup__add_233,axiom,
ordere580206878836729694up_add @ real ).
thf(tcon_Real_Oreal___Groups_Oordered__ab__semigroup__add__imp__le_234,axiom,
ordere2412721322843649153imp_le @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__comm__semiring__strict_235,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_236,axiom,
strict7427464778891057005id_add @ real ).
thf(tcon_Real_Oreal___Groups_Oordered__cancel__comm__monoid__add_237,axiom,
ordere8940638589300402666id_add @ real ).
thf(tcon_Real_Oreal___Topological__Spaces_Otopological__space_238,axiom,
topolo4958980785337419405_space @ real ).
thf(tcon_Real_Oreal___Topological__Spaces_Olinorder__topology_239,axiom,
topolo1944317154257567458pology @ real ).
thf(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__normed__field,axiom,
real_V3459762299906320749_field @ real ).
thf(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__div__algebra,axiom,
real_V5047593784448816457lgebra @ real ).
thf(tcon_Real_Oreal___Archimedean__Field_Oarchimedean__field_240,axiom,
archim462609752435547400_field @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__semiring__1__strict_241,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_242,axiom,
unboun7993243217541854897norder @ real ).
thf(tcon_Real_Oreal___Limits_Otopological__comm__monoid__add_243,axiom,
topolo5987344860129210374id_add @ real ).
thf(tcon_Real_Oreal___Topological__Spaces_Oorder__topology_244,axiom,
topolo2564578578187576103pology @ real ).
thf(tcon_Real_Oreal___Rings_Osemiring__1__no__zero__divisors_245,axiom,
semiri2026040879449505780visors @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__nonzero__semiring_246,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_247,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_248,axiom,
linord8928482502909563296strict @ real ).
thf(tcon_Real_Oreal___Rings_Osemiring__no__zero__divisors_249,axiom,
semiri3467727345109120633visors @ real ).
thf(tcon_Real_Oreal___Real__Vector__Spaces_Ometric__space,axiom,
real_V7819770556892013058_space @ real ).
thf(tcon_Real_Oreal___Limits_Otopological__ab__group__add,axiom,
topolo1287966508704411220up_add @ real ).
thf(tcon_Real_Oreal___Groups_Oordered__ab__semigroup__add_250,axiom,
ordere6658533253407199908up_add @ real ).
thf(tcon_Real_Oreal___Groups_Oordered__ab__group__add__abs_251,axiom,
ordere166539214618696060dd_abs @ real ).
thf(tcon_Real_Oreal___Archimedean__Field_Ofloor__ceiling_252,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_253,axiom,
ordere6911136660526730532id_add @ real ).
thf(tcon_Real_Oreal___Groups_Olinordered__ab__group__add_254,axiom,
linord5086331880401160121up_add @ real ).
thf(tcon_Real_Oreal___Groups_Ocancel__ab__semigroup__add_255,axiom,
cancel2418104881723323429up_add @ real ).
thf(tcon_Real_Oreal___Rings_Oring__1__no__zero__divisors_256,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_257,axiom,
topolo6943815403480290642id_add @ real ).
thf(tcon_Real_Oreal___Groups_Ocancel__comm__monoid__add_258,axiom,
cancel1802427076303600483id_add @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__ring__strict_259,axiom,
linord4710134922213307826strict @ real ).
thf(tcon_Real_Oreal___Rings_Ocomm__semiring__1__cancel_260,axiom,
comm_s4317794764714335236cancel @ real ).
thf(tcon_Real_Oreal___Limits_Otopological__group__add,axiom,
topolo1633459387980952147up_add @ real ).
thf(tcon_Real_Oreal___Topological__Spaces_Ot2__space_261,axiom,
topological_t2_space @ real ).
thf(tcon_Real_Oreal___Topological__Spaces_Ot1__space_262,axiom,
topological_t1_space @ real ).
thf(tcon_Real_Oreal___Rings_Oordered__comm__semiring_263,axiom,
ordere2520102378445227354miring @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__semiring__1_264,axiom,
linord6961819062388156250ring_1 @ real ).
thf(tcon_Real_Oreal___Groups_Oordered__ab__group__add_265,axiom,
ordered_ab_group_add @ real ).
thf(tcon_Real_Oreal___Groups_Ocancel__semigroup__add_266,axiom,
cancel_semigroup_add @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__semiring_267,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_268,axiom,
ordered_semiring_0 @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__semidom_269,axiom,
linordered_semidom @ real ).
thf(tcon_Real_Oreal___Orderings_Odense__linorder_270,axiom,
dense_linorder @ real ).
thf(tcon_Real_Oreal___Lattices_Osemilattice__sup_271,axiom,
semilattice_sup @ real ).
thf(tcon_Real_Oreal___Groups_Oab__semigroup__mult_272,axiom,
ab_semigroup_mult @ real ).
thf(tcon_Real_Oreal___Rings_Osemiring__1__cancel_273,axiom,
semiring_1_cancel @ real ).
thf(tcon_Real_Oreal___Groups_Ocomm__monoid__mult_274,axiom,
comm_monoid_mult @ real ).
thf(tcon_Real_Oreal___Groups_Oab__semigroup__add_275,axiom,
ab_semigroup_add @ real ).
thf(tcon_Real_Oreal___Fields_Olinordered__field_276,axiom,
linordered_field @ real ).
thf(tcon_Real_Oreal___Rings_Oordered__semiring_277,axiom,
ordered_semiring @ real ).
thf(tcon_Real_Oreal___Rings_Oordered__ring__abs_278,axiom,
ordered_ring_abs @ real ).
thf(tcon_Real_Oreal___Groups_Ocomm__monoid__add_279,axiom,
comm_monoid_add @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__ring_280,axiom,
linordered_ring @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__idom_281,axiom,
linordered_idom @ real ).
thf(tcon_Real_Oreal___Rings_Ocomm__semiring__1_282,axiom,
comm_semiring_1 @ real ).
thf(tcon_Real_Oreal___Orderings_Odense__order_283,axiom,
dense_order @ real ).
thf(tcon_Real_Oreal___Groups_Osemigroup__mult_284,axiom,
semigroup_mult @ real ).
thf(tcon_Real_Oreal___Rings_Osemidom__divide_285,axiom,
semidom_divide @ real ).
thf(tcon_Real_Oreal___Num_Osemiring__numeral_286,axiom,
semiring_numeral @ real ).
thf(tcon_Real_Oreal___Groups_Osemigroup__add_287,axiom,
semigroup_add @ real ).
thf(tcon_Real_Oreal___Fields_Odivision__ring_288,axiom,
division_ring @ real ).
thf(tcon_Real_Oreal___Rings_Ozero__less__one_289,axiom,
zero_less_one @ real ).
thf(tcon_Real_Oreal___Rings_Ocomm__semiring_290,axiom,
comm_semiring @ real ).
thf(tcon_Real_Oreal___Nat_Osemiring__char__0_291,axiom,
semiring_char_0 @ real ).
thf(tcon_Real_Oreal___Groups_Oab__group__add_292,axiom,
ab_group_add @ real ).
thf(tcon_Real_Oreal___Fields_Ofield__char__0_293,axiom,
field_char_0 @ real ).
thf(tcon_Real_Oreal___Rings_Ozero__neq__one_294,axiom,
zero_neq_one @ real ).
thf(tcon_Real_Oreal___Rings_Oordered__ring_295,axiom,
ordered_ring @ real ).
thf(tcon_Real_Oreal___Rings_Oidom__abs__sgn_296,axiom,
idom_abs_sgn @ real ).
thf(tcon_Real_Oreal___Orderings_Opreorder_297,axiom,
preorder @ real ).
thf(tcon_Real_Oreal___Orderings_Olinorder_298,axiom,
linorder @ real ).
thf(tcon_Real_Oreal___Groups_Omonoid__mult_299,axiom,
monoid_mult @ real ).
thf(tcon_Real_Oreal___Transcendental_Oln,axiom,
ln @ real ).
thf(tcon_Real_Oreal___Rings_Oidom__divide_300,axiom,
idom_divide @ real ).
thf(tcon_Real_Oreal___Rings_Ocomm__ring__1_301,axiom,
comm_ring_1 @ real ).
thf(tcon_Real_Oreal___Groups_Omonoid__add_302,axiom,
monoid_add @ real ).
thf(tcon_Real_Oreal___Rings_Osemiring__1_303,axiom,
semiring_1 @ real ).
thf(tcon_Real_Oreal___Rings_Osemiring__0_304,axiom,
semiring_0 @ real ).
thf(tcon_Real_Oreal___Orderings_Ono__top_305,axiom,
no_top @ real ).
thf(tcon_Real_Oreal___Orderings_Ono__bot_306,axiom,
no_bot @ real ).
thf(tcon_Real_Oreal___Lattices_Olattice_307,axiom,
lattice @ real ).
thf(tcon_Real_Oreal___Groups_Ogroup__add_308,axiom,
group_add @ real ).
thf(tcon_Real_Oreal___Rings_Omult__zero_309,axiom,
mult_zero @ real ).
thf(tcon_Real_Oreal___Rings_Ocomm__ring_310,axiom,
comm_ring @ real ).
thf(tcon_Real_Oreal___Orderings_Oorder_311,axiom,
order @ real ).
thf(tcon_Real_Oreal___Num_Oneg__numeral_312,axiom,
neg_numeral @ real ).
thf(tcon_Real_Oreal___Nat_Oring__char__0_313,axiom,
ring_char_0 @ real ).
thf(tcon_Real_Oreal___Rings_Osemiring_314,axiom,
semiring @ real ).
thf(tcon_Real_Oreal___Fields_Oinverse_315,axiom,
inverse @ real ).
thf(tcon_Real_Oreal___Rings_Osemidom_316,axiom,
semidom @ real ).
thf(tcon_Real_Oreal___Orderings_Oord_317,axiom,
ord @ real ).
thf(tcon_Real_Oreal___Groups_Ouminus_318,axiom,
uminus @ real ).
thf(tcon_Real_Oreal___Rings_Oring__1_319,axiom,
ring_1 @ real ).
thf(tcon_Real_Oreal___Rings_Oabs__if_320,axiom,
abs_if @ real ).
thf(tcon_Real_Oreal___Groups_Ominus_321,axiom,
minus @ real ).
thf(tcon_Real_Oreal___Fields_Ofield_322,axiom,
field @ real ).
thf(tcon_Real_Oreal___Power_Opower_323,axiom,
power @ real ).
thf(tcon_Real_Oreal___Num_Onumeral_324,axiom,
numeral @ real ).
thf(tcon_Real_Oreal___Groups_Ozero_325,axiom,
zero @ real ).
thf(tcon_Real_Oreal___Groups_Oplus_326,axiom,
plus @ real ).
thf(tcon_Real_Oreal___Rings_Oring_327,axiom,
ring @ real ).
thf(tcon_Real_Oreal___Rings_Oidom_328,axiom,
idom @ real ).
thf(tcon_Real_Oreal___Groups_Oone_329,axiom,
one @ real ).
thf(tcon_Real_Oreal___Rings_Odvd_330,axiom,
dvd @ real ).
thf(tcon_Word_Oword___Bit__Operations_Osemiring__bit__operations_331,axiom,
! [A11: $tType] :
( ( type_len @ A11 )
=> ( bit_se359711467146920520ations @ ( word @ A11 ) ) ) ).
thf(tcon_Word_Oword___Bit__Comprehension_Obit__comprehension_332,axiom,
! [A11: $tType] :
( ( type_len @ A11 )
=> ( bit_bi6583157726757044596ension @ ( word @ A11 ) ) ) ).
thf(tcon_Word_Oword___Bit__Operations_Oring__bit__operations_333,axiom,
! [A11: $tType] :
( ( type_len @ A11 )
=> ( bit_ri3973907225187159222ations @ ( word @ A11 ) ) ) ).
thf(tcon_Word_Oword___Groups_Ocancel__ab__semigroup__add_334,axiom,
! [A11: $tType] :
( ( type_len @ A11 )
=> ( cancel2418104881723323429up_add @ ( word @ A11 ) ) ) ).
thf(tcon_Word_Oword___Groups_Ocancel__comm__monoid__add_335,axiom,
! [A11: $tType] :
( ( type_len @ A11 )
=> ( cancel1802427076303600483id_add @ ( word @ A11 ) ) ) ).
thf(tcon_Word_Oword___Rings_Ocomm__semiring__1__cancel_336,axiom,
! [A11: $tType] :
( ( type_len @ A11 )
=> ( comm_s4317794764714335236cancel @ ( word @ A11 ) ) ) ).
thf(tcon_Word_Oword___Bit__Operations_Osemiring__bits_337,axiom,
! [A11: $tType] :
( ( type_len @ A11 )
=> ( bit_semiring_bits @ ( word @ A11 ) ) ) ).
thf(tcon_Word_Oword___Groups_Ocancel__semigroup__add_338,axiom,
! [A11: $tType] :
( ( type_len @ A11 )
=> ( cancel_semigroup_add @ ( word @ A11 ) ) ) ).
thf(tcon_Word_Oword___Least__significant__bit_Olsb_339,axiom,
! [A11: $tType] :
( ( type_len @ A11 )
=> ( least_6119777620449941438nt_lsb @ ( word @ A11 ) ) ) ).
thf(tcon_Word_Oword___Groups_Oab__semigroup__mult_340,axiom,
! [A11: $tType] :
( ( type_len @ A11 )
=> ( ab_semigroup_mult @ ( word @ A11 ) ) ) ).
thf(tcon_Word_Oword___Rings_Osemiring__1__cancel_341,axiom,
! [A11: $tType] :
( ( type_len @ A11 )
=> ( semiring_1_cancel @ ( word @ A11 ) ) ) ).
thf(tcon_Word_Oword___Groups_Ocomm__monoid__mult_342,axiom,
! [A11: $tType] :
( ( type_len @ A11 )
=> ( comm_monoid_mult @ ( word @ A11 ) ) ) ).
thf(tcon_Word_Oword___Groups_Oab__semigroup__add_343,axiom,
! [A11: $tType] :
( ( type_len @ A11 )
=> ( ab_semigroup_add @ ( word @ A11 ) ) ) ).
thf(tcon_Word_Oword___Generic__set__bit_Oset__bit_344,axiom,
! [A11: $tType] :
( ( type_len @ A11 )
=> ( generic_set_set_bit @ ( word @ A11 ) ) ) ).
thf(tcon_Word_Oword___Parity_Osemiring__parity_345,axiom,
! [A11: $tType] :
( ( type_len @ A11 )
=> ( semiring_parity @ ( word @ A11 ) ) ) ).
thf(tcon_Word_Oword___Groups_Ocomm__monoid__add_346,axiom,
! [A11: $tType] :
( ( type_len @ A11 )
=> ( comm_monoid_add @ ( word @ A11 ) ) ) ).
thf(tcon_Word_Oword___Rings_Osemiring__modulo_347,axiom,
! [A11: $tType] :
( ( type_len @ A11 )
=> ( semiring_modulo @ ( word @ A11 ) ) ) ).
thf(tcon_Word_Oword___Rings_Ocomm__semiring__1_348,axiom,
! [A11: $tType] :
( ( type_len @ A11 )
=> ( comm_semiring_1 @ ( word @ A11 ) ) ) ).
thf(tcon_Word_Oword___Groups_Osemigroup__mult_349,axiom,
! [A11: $tType] :
( ( type_len @ A11 )
=> ( semigroup_mult @ ( word @ A11 ) ) ) ).
thf(tcon_Word_Oword___Num_Osemiring__numeral_350,axiom,
! [A11: $tType] :
( ( type_len @ A11 )
=> ( semiring_numeral @ ( word @ A11 ) ) ) ).
thf(tcon_Word_Oword___Groups_Osemigroup__add_351,axiom,
! [A11: $tType] :
( ( type_len @ A11 )
=> ( semigroup_add @ ( word @ A11 ) ) ) ).
thf(tcon_Word_Oword___Rings_Ocomm__semiring_352,axiom,
! [A11: $tType] :
( ( type_len @ A11 )
=> ( comm_semiring @ ( word @ A11 ) ) ) ).
thf(tcon_Word_Oword___Orderings_Owellorder_353,axiom,
! [A11: $tType] :
( ( type_len @ A11 )
=> ( wellorder @ ( word @ A11 ) ) ) ).
thf(tcon_Word_Oword___Groups_Oab__group__add_354,axiom,
! [A11: $tType] :
( ( type_len @ A11 )
=> ( ab_group_add @ ( word @ A11 ) ) ) ).
thf(tcon_Word_Oword___Rings_Ozero__neq__one_355,axiom,
! [A11: $tType] :
( ( type_len @ A11 )
=> ( zero_neq_one @ ( word @ A11 ) ) ) ).
thf(tcon_Word_Oword___Parity_Oring__parity_356,axiom,
! [A11: $tType] :
( ( type_len @ A11 )
=> ( ring_parity @ ( word @ A11 ) ) ) ).
thf(tcon_Word_Oword___Orderings_Opreorder_357,axiom,
! [A11: $tType] :
( ( type_len @ A11 )
=> ( preorder @ ( word @ A11 ) ) ) ).
thf(tcon_Word_Oword___Orderings_Olinorder_358,axiom,
! [A11: $tType] :
( ( type_len @ A11 )
=> ( linorder @ ( word @ A11 ) ) ) ).
thf(tcon_Word_Oword___Groups_Omonoid__mult_359,axiom,
! [A11: $tType] :
( ( type_len @ A11 )
=> ( monoid_mult @ ( word @ A11 ) ) ) ).
thf(tcon_Word_Oword___Rings_Ocomm__ring__1_360,axiom,
! [A11: $tType] :
( ( type_len @ A11 )
=> ( comm_ring_1 @ ( word @ A11 ) ) ) ).
thf(tcon_Word_Oword___Groups_Omonoid__add_361,axiom,
! [A11: $tType] :
( ( type_len @ A11 )
=> ( monoid_add @ ( word @ A11 ) ) ) ).
thf(tcon_Word_Oword___Finite__Set_Ofinite_362,axiom,
! [A11: $tType] :
( ( type_len @ A11 )
=> ( finite_finite @ ( word @ A11 ) ) ) ).
thf(tcon_Word_Oword___Rings_Osemiring__1_363,axiom,
! [A11: $tType] :
( ( type_len @ A11 )
=> ( semiring_1 @ ( word @ A11 ) ) ) ).
thf(tcon_Word_Oword___Rings_Osemiring__0_364,axiom,
! [A11: $tType] :
( ( type_len @ A11 )
=> ( semiring_0 @ ( word @ A11 ) ) ) ).
thf(tcon_Word_Oword___Groups_Ogroup__add_365,axiom,
! [A11: $tType] :
( ( type_len @ A11 )
=> ( group_add @ ( word @ A11 ) ) ) ).
thf(tcon_Word_Oword___Rings_Omult__zero_366,axiom,
! [A11: $tType] :
( ( type_len @ A11 )
=> ( mult_zero @ ( word @ A11 ) ) ) ).
thf(tcon_Word_Oword___Rings_Ocomm__ring_367,axiom,
! [A11: $tType] :
( ( type_len @ A11 )
=> ( comm_ring @ ( word @ A11 ) ) ) ).
thf(tcon_Word_Oword___Orderings_Oorder_368,axiom,
! [A11: $tType] :
( ( type_len @ A11 )
=> ( order @ ( word @ A11 ) ) ) ).
thf(tcon_Word_Oword___Num_Oneg__numeral_369,axiom,
! [A11: $tType] :
( ( type_len @ A11 )
=> ( neg_numeral @ ( word @ A11 ) ) ) ).
thf(tcon_Word_Oword___Rings_Osemiring_370,axiom,
! [A11: $tType] :
( ( type_len @ A11 )
=> ( semiring @ ( word @ A11 ) ) ) ).
thf(tcon_Word_Oword___Orderings_Oord_371,axiom,
! [A11: $tType] :
( ( type_len @ A11 )
=> ( ord @ ( word @ A11 ) ) ) ).
thf(tcon_Word_Oword___Groups_Ouminus_372,axiom,
! [A11: $tType] :
( ( type_len @ A11 )
=> ( uminus @ ( word @ A11 ) ) ) ).
thf(tcon_Word_Oword___Rings_Oring__1_373,axiom,
! [A11: $tType] :
( ( type_len @ A11 )
=> ( ring_1 @ ( word @ A11 ) ) ) ).
thf(tcon_Word_Oword___Groups_Ominus_374,axiom,
! [A11: $tType] :
( ( type_len @ A11 )
=> ( minus @ ( word @ A11 ) ) ) ).
thf(tcon_Word_Oword___Power_Opower_375,axiom,
! [A11: $tType] :
( ( type_len @ A11 )
=> ( power @ ( word @ A11 ) ) ) ).
thf(tcon_Word_Oword___Num_Onumeral_376,axiom,
! [A11: $tType] :
( ( type_len @ A11 )
=> ( numeral @ ( word @ A11 ) ) ) ).
thf(tcon_Word_Oword___Groups_Ozero_377,axiom,
! [A11: $tType] :
( ( type_len @ A11 )
=> ( zero @ ( word @ A11 ) ) ) ).
thf(tcon_Word_Oword___Groups_Oplus_378,axiom,
! [A11: $tType] :
( ( type_len @ A11 )
=> ( plus @ ( word @ A11 ) ) ) ).
thf(tcon_Word_Oword___Rings_Oring_379,axiom,
! [A11: $tType] :
( ( type_len @ A11 )
=> ( ring @ ( word @ A11 ) ) ) ).
thf(tcon_Word_Oword___Groups_Oone_380,axiom,
! [A11: $tType] :
( ( type_len @ A11 )
=> ( one @ ( word @ A11 ) ) ) ).
thf(tcon_Word_Oword___Rings_Odvd_381,axiom,
! [A11: $tType] :
( ( type_len @ A11 )
=> ( dvd @ ( word @ A11 ) ) ) ).
thf(tcon_Word_Oword___Nat_Osize_382,axiom,
! [A11: $tType] :
( ( type_len @ A11 )
=> ( size @ ( word @ A11 ) ) ) ).
thf(tcon_Sum__Type_Osum___Finite__Set_Ofinite_383,axiom,
! [A11: $tType,A14: $tType] :
( ( ( finite_finite @ A11 )
& ( finite_finite @ A14 ) )
=> ( finite_finite @ ( sum_sum @ A11 @ A14 ) ) ) ).
thf(tcon_Sum__Type_Osum___Nat_Osize_384,axiom,
! [A11: $tType,A14: $tType] : ( size @ ( sum_sum @ A11 @ A14 ) ) ).
thf(tcon_Enum_Ofinite__1___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_385,axiom,
condit6923001295902523014norder @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Groups_Oordered__ab__semigroup__monoid__add__imp__le_386,axiom,
ordere1937475149494474687imp_le @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Rings_Osemiring__no__zero__divisors__cancel_387,axiom,
semiri6575147826004484403cancel @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Groups_Ostrict__ordered__ab__semigroup__add_388,axiom,
strict9044650504122735259up_add @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Groups_Oordered__cancel__comm__monoid__diff_389,axiom,
ordere1170586879665033532d_diff @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Groups_Oordered__cancel__ab__semigroup__add_390,axiom,
ordere580206878836729694up_add @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Groups_Oordered__ab__semigroup__add__imp__le_391,axiom,
ordere2412721322843649153imp_le @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Rings_Olinordered__comm__semiring__strict_392,axiom,
linord2810124833399127020strict @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Groups_Ostrict__ordered__comm__monoid__add_393,axiom,
strict7427464778891057005id_add @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Groups_Oordered__cancel__comm__monoid__add_394,axiom,
ordere8940638589300402666id_add @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Groups_Ocanonically__ordered__monoid__add_395,axiom,
canoni5634975068530333245id_add @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Lattices_Obounded__semilattice__sup__bot_396,axiom,
bounde4967611905675639751up_bot @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Rings_Olinordered__semiring__strict_397,axiom,
linord8928482502909563296strict @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Boolean__Algebras_Oboolean__algebra_398,axiom,
boolea8198339166811842893lgebra @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Rings_Osemiring__no__zero__divisors_399,axiom,
semiri3467727345109120633visors @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Groups_Oordered__ab__semigroup__add_400,axiom,
ordere6658533253407199908up_add @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Groups_Oordered__ab__group__add__abs_401,axiom,
ordere166539214618696060dd_abs @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Groups_Oordered__comm__monoid__add_402,axiom,
ordere6911136660526730532id_add @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Groups_Olinordered__ab__group__add_403,axiom,
linord5086331880401160121up_add @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Groups_Ocancel__ab__semigroup__add_404,axiom,
cancel2418104881723323429up_add @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Groups_Ocancel__comm__monoid__add_405,axiom,
cancel1802427076303600483id_add @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Rings_Olinordered__ring__strict_406,axiom,
linord4710134922213307826strict @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Lattices_Obounded__lattice__top_407,axiom,
bounded_lattice_top @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Rings_Oordered__comm__semiring_408,axiom,
ordere2520102378445227354miring @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Groups_Oordered__ab__group__add_409,axiom,
ordered_ab_group_add @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Groups_Ocancel__semigroup__add_410,axiom,
cancel_semigroup_add @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Rings_Olinordered__semiring_411,axiom,
linordered_semiring @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Rings_Oordered__semiring__0_412,axiom,
ordered_semiring_0 @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Orderings_Odense__linorder_413,axiom,
dense_linorder @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Lattices_Osemilattice__sup_414,axiom,
semilattice_sup @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Lattices_Obounded__lattice_415,axiom,
bounded_lattice @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Groups_Oab__semigroup__mult_416,axiom,
ab_semigroup_mult @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Groups_Ocomm__monoid__mult_417,axiom,
comm_monoid_mult @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Groups_Ocomm__monoid__diff_418,axiom,
comm_monoid_diff @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Groups_Oab__semigroup__add_419,axiom,
ab_semigroup_add @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Rings_Oordered__semiring_420,axiom,
ordered_semiring @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Rings_Oordered__ring__abs_421,axiom,
ordered_ring_abs @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Groups_Ocomm__monoid__add_422,axiom,
comm_monoid_add @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Rings_Olinordered__ring_423,axiom,
linordered_ring @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Orderings_Odense__order_424,axiom,
dense_order @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Groups_Osemigroup__mult_425,axiom,
semigroup_mult @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Groups_Osemigroup__add_426,axiom,
semigroup_add @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Rings_Ocomm__semiring_427,axiom,
comm_semiring @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Orderings_Owellorder_428,axiom,
wellorder @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Orderings_Oorder__top_429,axiom,
order_top @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Orderings_Oorder__bot_430,axiom,
order_bot @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Groups_Oab__group__add_431,axiom,
ab_group_add @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Rings_Oordered__ring_432,axiom,
ordered_ring @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Orderings_Opreorder_433,axiom,
preorder @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Orderings_Olinorder_434,axiom,
linorder @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Groups_Omonoid__mult_435,axiom,
monoid_mult @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Groups_Omonoid__add_436,axiom,
monoid_add @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Finite__Set_Ofinite_437,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_438,axiom,
semiring_0 @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Lattices_Olattice_439,axiom,
lattice @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Groups_Ogroup__add_440,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_441,axiom,
mult_zero @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Rings_Ocomm__ring_442,axiom,
comm_ring @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Orderings_Oorder_443,axiom,
order @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Rings_Osemiring_444,axiom,
semiring @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Fields_Oinverse_445,axiom,
inverse @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Orderings_Otop_446,axiom,
top @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Orderings_Oord_447,axiom,
ord @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Orderings_Obot_448,axiom,
bot @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Groups_Ouminus_449,axiom,
uminus @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Rings_Oabs__if_450,axiom,
abs_if @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Groups_Ominus_451,axiom,
minus @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Power_Opower_452,axiom,
power @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Groups_Ozero_453,axiom,
zero @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Groups_Oplus_454,axiom,
plus @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Rings_Oring_455,axiom,
ring @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Groups_Oone_456,axiom,
one @ finite_1 ).
thf(tcon_Enum_Ofinite__1___Rings_Odvd_457,axiom,
dvd @ finite_1 ).
thf(tcon_Enum_Ofinite__2___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_458,axiom,
condit6923001295902523014norder @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_459,axiom,
semiri1453513574482234551roduct @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Euclidean__Division_Oeuclidean__semiring__cancel_460,axiom,
euclid4440199948858584721cancel @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Rings_Osemiring__no__zero__divisors__cancel_461,axiom,
semiri6575147826004484403cancel @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Euclidean__Division_Oeuclidean__semiring_462,axiom,
euclid3725896446679973847miring @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Lattices_Obounded__semilattice__sup__bot_463,axiom,
bounde4967611905675639751up_bot @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Rings_Osemiring__1__no__zero__divisors_464,axiom,
semiri2026040879449505780visors @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Rings_Osemiring__no__zero__divisors_465,axiom,
semiri3467727345109120633visors @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Groups_Ocancel__ab__semigroup__add_466,axiom,
cancel2418104881723323429up_add @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Rings_Oring__1__no__zero__divisors_467,axiom,
ring_15535105094025558882visors @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Groups_Ocancel__comm__monoid__add_468,axiom,
cancel1802427076303600483id_add @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Rings_Ocomm__semiring__1__cancel_469,axiom,
comm_s4317794764714335236cancel @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Lattices_Obounded__lattice__top_470,axiom,
bounded_lattice_top @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Groups_Ocancel__semigroup__add_471,axiom,
cancel_semigroup_add @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Lattices_Osemilattice__sup_472,axiom,
semilattice_sup @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Lattices_Obounded__lattice_473,axiom,
bounded_lattice @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Groups_Oab__semigroup__mult_474,axiom,
ab_semigroup_mult @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Rings_Osemiring__1__cancel_475,axiom,
semiring_1_cancel @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Rings_Oalgebraic__semidom_476,axiom,
algebraic_semidom @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Groups_Ocomm__monoid__mult_477,axiom,
comm_monoid_mult @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Groups_Oab__semigroup__add_478,axiom,
ab_semigroup_add @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Groups_Ocomm__monoid__add_479,axiom,
comm_monoid_add @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Rings_Osemiring__modulo_480,axiom,
semiring_modulo @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Rings_Ocomm__semiring__1_481,axiom,
comm_semiring_1 @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Groups_Osemigroup__mult_482,axiom,
semigroup_mult @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Rings_Osemidom__modulo_483,axiom,
semidom_modulo @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Rings_Osemidom__divide_484,axiom,
semidom_divide @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Num_Osemiring__numeral_485,axiom,
semiring_numeral @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Groups_Osemigroup__add_486,axiom,
semigroup_add @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Fields_Odivision__ring_487,axiom,
division_ring @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Rings_Ocomm__semiring_488,axiom,
comm_semiring @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Orderings_Owellorder_489,axiom,
wellorder @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Orderings_Oorder__top_490,axiom,
order_top @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Orderings_Oorder__bot_491,axiom,
order_bot @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Groups_Oab__group__add_492,axiom,
ab_group_add @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Rings_Ozero__neq__one_493,axiom,
zero_neq_one @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Rings_Oidom__abs__sgn_494,axiom,
idom_abs_sgn @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Orderings_Opreorder_495,axiom,
preorder @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Orderings_Olinorder_496,axiom,
linorder @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Groups_Omonoid__mult_497,axiom,
monoid_mult @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Rings_Oidom__modulo_498,axiom,
idom_modulo @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Rings_Oidom__divide_499,axiom,
idom_divide @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Rings_Ocomm__ring__1_500,axiom,
comm_ring_1 @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Groups_Omonoid__add_501,axiom,
monoid_add @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Finite__Set_Ofinite_502,axiom,
finite_finite @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Type__Length_Olen0_503,axiom,
type_len0 @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Rings_Osemiring__1_504,axiom,
semiring_1 @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Rings_Osemiring__0_505,axiom,
semiring_0 @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Lattices_Olattice_506,axiom,
lattice @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Groups_Ogroup__add_507,axiom,
group_add @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Type__Length_Olen_508,axiom,
type_len @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Rings_Omult__zero_509,axiom,
mult_zero @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Rings_Ocomm__ring_510,axiom,
comm_ring @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Orderings_Oorder_511,axiom,
order @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Num_Oneg__numeral_512,axiom,
neg_numeral @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Rings_Osemiring_513,axiom,
semiring @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Fields_Oinverse_514,axiom,
inverse @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Rings_Osemidom_515,axiom,
semidom @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Orderings_Otop_516,axiom,
top @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Orderings_Oord_517,axiom,
ord @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Orderings_Obot_518,axiom,
bot @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Groups_Ouminus_519,axiom,
uminus @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Rings_Oring__1_520,axiom,
ring_1 @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Groups_Ominus_521,axiom,
minus @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Fields_Ofield_522,axiom,
field @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Power_Opower_523,axiom,
power @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Num_Onumeral_524,axiom,
numeral @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Groups_Ozero_525,axiom,
zero @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Groups_Oplus_526,axiom,
plus @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Rings_Oring_527,axiom,
ring @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Rings_Oidom_528,axiom,
idom @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Groups_Oone_529,axiom,
one @ finite_2 ).
thf(tcon_Enum_Ofinite__2___Rings_Odvd_530,axiom,
dvd @ finite_2 ).
thf(tcon_Enum_Ofinite__3___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_531,axiom,
condit6923001295902523014norder @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_532,axiom,
semiri1453513574482234551roduct @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Euclidean__Division_Oeuclidean__semiring__cancel_533,axiom,
euclid4440199948858584721cancel @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Rings_Osemiring__no__zero__divisors__cancel_534,axiom,
semiri6575147826004484403cancel @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Euclidean__Division_Oeuclidean__semiring_535,axiom,
euclid3725896446679973847miring @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Lattices_Obounded__semilattice__sup__bot_536,axiom,
bounde4967611905675639751up_bot @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Rings_Osemiring__1__no__zero__divisors_537,axiom,
semiri2026040879449505780visors @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Rings_Osemiring__no__zero__divisors_538,axiom,
semiri3467727345109120633visors @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Groups_Ocancel__ab__semigroup__add_539,axiom,
cancel2418104881723323429up_add @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Rings_Oring__1__no__zero__divisors_540,axiom,
ring_15535105094025558882visors @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Groups_Ocancel__comm__monoid__add_541,axiom,
cancel1802427076303600483id_add @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Rings_Ocomm__semiring__1__cancel_542,axiom,
comm_s4317794764714335236cancel @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Lattices_Obounded__lattice__top_543,axiom,
bounded_lattice_top @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Groups_Ocancel__semigroup__add_544,axiom,
cancel_semigroup_add @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Lattices_Osemilattice__sup_545,axiom,
semilattice_sup @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Lattices_Obounded__lattice_546,axiom,
bounded_lattice @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Groups_Oab__semigroup__mult_547,axiom,
ab_semigroup_mult @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Rings_Osemiring__1__cancel_548,axiom,
semiring_1_cancel @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Rings_Oalgebraic__semidom_549,axiom,
algebraic_semidom @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Groups_Ocomm__monoid__mult_550,axiom,
comm_monoid_mult @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Groups_Oab__semigroup__add_551,axiom,
ab_semigroup_add @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Groups_Ocomm__monoid__add_552,axiom,
comm_monoid_add @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Rings_Osemiring__modulo_553,axiom,
semiring_modulo @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Rings_Ocomm__semiring__1_554,axiom,
comm_semiring_1 @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Groups_Osemigroup__mult_555,axiom,
semigroup_mult @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Rings_Osemidom__modulo_556,axiom,
semidom_modulo @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Rings_Osemidom__divide_557,axiom,
semidom_divide @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Num_Osemiring__numeral_558,axiom,
semiring_numeral @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Groups_Osemigroup__add_559,axiom,
semigroup_add @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Fields_Odivision__ring_560,axiom,
division_ring @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Rings_Ocomm__semiring_561,axiom,
comm_semiring @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Orderings_Owellorder_562,axiom,
wellorder @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Orderings_Oorder__top_563,axiom,
order_top @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Orderings_Oorder__bot_564,axiom,
order_bot @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Groups_Oab__group__add_565,axiom,
ab_group_add @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Rings_Ozero__neq__one_566,axiom,
zero_neq_one @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Rings_Oidom__abs__sgn_567,axiom,
idom_abs_sgn @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Orderings_Opreorder_568,axiom,
preorder @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Orderings_Olinorder_569,axiom,
linorder @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Groups_Omonoid__mult_570,axiom,
monoid_mult @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Rings_Oidom__modulo_571,axiom,
idom_modulo @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Rings_Oidom__divide_572,axiom,
idom_divide @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Rings_Ocomm__ring__1_573,axiom,
comm_ring_1 @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Groups_Omonoid__add_574,axiom,
monoid_add @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Finite__Set_Ofinite_575,axiom,
finite_finite @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Type__Length_Olen0_576,axiom,
type_len0 @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Rings_Osemiring__1_577,axiom,
semiring_1 @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Rings_Osemiring__0_578,axiom,
semiring_0 @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Lattices_Olattice_579,axiom,
lattice @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Groups_Ogroup__add_580,axiom,
group_add @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Type__Length_Olen_581,axiom,
type_len @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Rings_Omult__zero_582,axiom,
mult_zero @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Rings_Ocomm__ring_583,axiom,
comm_ring @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Orderings_Oorder_584,axiom,
order @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Num_Oneg__numeral_585,axiom,
neg_numeral @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Rings_Osemiring_586,axiom,
semiring @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Fields_Oinverse_587,axiom,
inverse @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Rings_Osemidom_588,axiom,
semidom @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Orderings_Otop_589,axiom,
top @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Orderings_Oord_590,axiom,
ord @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Orderings_Obot_591,axiom,
bot @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Groups_Ouminus_592,axiom,
uminus @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Rings_Oring__1_593,axiom,
ring_1 @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Groups_Ominus_594,axiom,
minus @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Fields_Ofield_595,axiom,
field @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Power_Opower_596,axiom,
power @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Num_Onumeral_597,axiom,
numeral @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Groups_Ozero_598,axiom,
zero @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Groups_Oplus_599,axiom,
plus @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Rings_Oring_600,axiom,
ring @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Rings_Oidom_601,axiom,
idom @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Groups_Oone_602,axiom,
one @ finite_3 ).
thf(tcon_Enum_Ofinite__3___Rings_Odvd_603,axiom,
dvd @ finite_3 ).
thf(tcon_Filter_Ofilter___Lattices_Obounded__semilattice__sup__bot_604,axiom,
! [A11: $tType] : ( bounde4967611905675639751up_bot @ ( filter @ A11 ) ) ).
thf(tcon_Filter_Ofilter___Lattices_Obounded__lattice__top_605,axiom,
! [A11: $tType] : ( bounded_lattice_top @ ( filter @ A11 ) ) ).
thf(tcon_Filter_Ofilter___Lattices_Osemilattice__sup_606,axiom,
! [A11: $tType] : ( semilattice_sup @ ( filter @ A11 ) ) ).
thf(tcon_Filter_Ofilter___Lattices_Obounded__lattice_607,axiom,
! [A11: $tType] : ( bounded_lattice @ ( filter @ A11 ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Oorder__top_608,axiom,
! [A11: $tType] : ( order_top @ ( filter @ A11 ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Oorder__bot_609,axiom,
! [A11: $tType] : ( order_bot @ ( filter @ A11 ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Opreorder_610,axiom,
! [A11: $tType] : ( preorder @ ( filter @ A11 ) ) ).
thf(tcon_Filter_Ofilter___Lattices_Olattice_611,axiom,
! [A11: $tType] : ( lattice @ ( filter @ A11 ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Oorder_612,axiom,
! [A11: $tType] : ( order @ ( filter @ A11 ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Otop_613,axiom,
! [A11: $tType] : ( top @ ( filter @ A11 ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Oord_614,axiom,
! [A11: $tType] : ( ord @ ( filter @ A11 ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Obot_615,axiom,
! [A11: $tType] : ( bot @ ( filter @ A11 ) ) ).
thf(tcon_Option_Ooption___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_616,axiom,
! [A11: $tType] :
( ( comple5582772986160207858norder @ A11 )
=> ( condit6923001295902523014norder @ ( option @ A11 ) ) ) ).
thf(tcon_Option_Ooption___Lattices_Obounded__semilattice__sup__bot_617,axiom,
! [A11: $tType] :
( ( lattice @ A11 )
=> ( bounde4967611905675639751up_bot @ ( option @ A11 ) ) ) ).
thf(tcon_Option_Ooption___Lattices_Obounded__lattice__top_618,axiom,
! [A11: $tType] :
( ( bounded_lattice_top @ A11 )
=> ( bounded_lattice_top @ ( option @ A11 ) ) ) ).
thf(tcon_Option_Ooption___Lattices_Osemilattice__sup_619,axiom,
! [A11: $tType] :
( ( semilattice_sup @ A11 )
=> ( semilattice_sup @ ( option @ A11 ) ) ) ).
thf(tcon_Option_Ooption___Lattices_Obounded__lattice_620,axiom,
! [A11: $tType] :
( ( bounded_lattice_top @ A11 )
=> ( bounded_lattice @ ( option @ A11 ) ) ) ).
thf(tcon_Option_Ooption___Orderings_Owellorder_621,axiom,
! [A11: $tType] :
( ( wellorder @ A11 )
=> ( wellorder @ ( option @ A11 ) ) ) ).
thf(tcon_Option_Ooption___Orderings_Oorder__top_622,axiom,
! [A11: $tType] :
( ( order_top @ A11 )
=> ( order_top @ ( option @ A11 ) ) ) ).
thf(tcon_Option_Ooption___Orderings_Oorder__bot_623,axiom,
! [A11: $tType] :
( ( order @ A11 )
=> ( order_bot @ ( option @ A11 ) ) ) ).
thf(tcon_Option_Ooption___Orderings_Opreorder_624,axiom,
! [A11: $tType] :
( ( preorder @ A11 )
=> ( preorder @ ( option @ A11 ) ) ) ).
thf(tcon_Option_Ooption___Orderings_Olinorder_625,axiom,
! [A11: $tType] :
( ( linorder @ A11 )
=> ( linorder @ ( option @ A11 ) ) ) ).
thf(tcon_Option_Ooption___Finite__Set_Ofinite_626,axiom,
! [A11: $tType] :
( ( finite_finite @ A11 )
=> ( finite_finite @ ( option @ A11 ) ) ) ).
thf(tcon_Option_Ooption___Lattices_Olattice_627,axiom,
! [A11: $tType] :
( ( lattice @ A11 )
=> ( lattice @ ( option @ A11 ) ) ) ).
thf(tcon_Option_Ooption___Orderings_Oorder_628,axiom,
! [A11: $tType] :
( ( order @ A11 )
=> ( order @ ( option @ A11 ) ) ) ).
thf(tcon_Option_Ooption___Orderings_Otop_629,axiom,
! [A11: $tType] :
( ( order_top @ A11 )
=> ( top @ ( option @ A11 ) ) ) ).
thf(tcon_Option_Ooption___Orderings_Oord_630,axiom,
! [A11: $tType] :
( ( preorder @ A11 )
=> ( ord @ ( option @ A11 ) ) ) ).
thf(tcon_Option_Ooption___Orderings_Obot_631,axiom,
! [A11: $tType] :
( ( order @ A11 )
=> ( bot @ ( option @ A11 ) ) ) ).
thf(tcon_Option_Ooption___Nat_Osize_632,axiom,
! [A11: $tType] : ( size @ ( option @ A11 ) ) ).
thf(tcon_Uint32_Ouint32___Bit__Operations_Osemiring__bit__operations_633,axiom,
bit_se359711467146920520ations @ uint32 ).
thf(tcon_Uint32_Ouint32___Bit__Comprehension_Obit__comprehension_634,axiom,
bit_bi6583157726757044596ension @ uint32 ).
thf(tcon_Uint32_Ouint32___Bit__Operations_Oring__bit__operations_635,axiom,
bit_ri3973907225187159222ations @ uint32 ).
thf(tcon_Uint32_Ouint32___Groups_Ocancel__ab__semigroup__add_636,axiom,
cancel2418104881723323429up_add @ uint32 ).
thf(tcon_Uint32_Ouint32___Groups_Ocancel__comm__monoid__add_637,axiom,
cancel1802427076303600483id_add @ uint32 ).
thf(tcon_Uint32_Ouint32___Rings_Ocomm__semiring__1__cancel_638,axiom,
comm_s4317794764714335236cancel @ uint32 ).
thf(tcon_Uint32_Ouint32___Bit__Operations_Osemiring__bits_639,axiom,
bit_semiring_bits @ uint32 ).
thf(tcon_Uint32_Ouint32___Groups_Ocancel__semigroup__add_640,axiom,
cancel_semigroup_add @ uint32 ).
thf(tcon_Uint32_Ouint32___Least__significant__bit_Olsb_641,axiom,
least_6119777620449941438nt_lsb @ uint32 ).
thf(tcon_Uint32_Ouint32___Groups_Oab__semigroup__mult_642,axiom,
ab_semigroup_mult @ uint32 ).
thf(tcon_Uint32_Ouint32___Rings_Osemiring__1__cancel_643,axiom,
semiring_1_cancel @ uint32 ).
thf(tcon_Uint32_Ouint32___Groups_Ocomm__monoid__mult_644,axiom,
comm_monoid_mult @ uint32 ).
thf(tcon_Uint32_Ouint32___Groups_Oab__semigroup__add_645,axiom,
ab_semigroup_add @ uint32 ).
thf(tcon_Uint32_Ouint32___Generic__set__bit_Oset__bit_646,axiom,
generic_set_set_bit @ uint32 ).
thf(tcon_Uint32_Ouint32___Parity_Osemiring__parity_647,axiom,
semiring_parity @ uint32 ).
thf(tcon_Uint32_Ouint32___Groups_Ocomm__monoid__add_648,axiom,
comm_monoid_add @ uint32 ).
thf(tcon_Uint32_Ouint32___Rings_Osemiring__modulo_649,axiom,
semiring_modulo @ uint32 ).
thf(tcon_Uint32_Ouint32___Rings_Ocomm__semiring__1_650,axiom,
comm_semiring_1 @ uint32 ).
thf(tcon_Uint32_Ouint32___Groups_Osemigroup__mult_651,axiom,
semigroup_mult @ uint32 ).
thf(tcon_Uint32_Ouint32___Num_Osemiring__numeral_652,axiom,
semiring_numeral @ uint32 ).
thf(tcon_Uint32_Ouint32___Groups_Osemigroup__add_653,axiom,
semigroup_add @ uint32 ).
thf(tcon_Uint32_Ouint32___Rings_Ocomm__semiring_654,axiom,
comm_semiring @ uint32 ).
thf(tcon_Uint32_Ouint32___Groups_Oab__group__add_655,axiom,
ab_group_add @ uint32 ).
thf(tcon_Uint32_Ouint32___Rings_Ozero__neq__one_656,axiom,
zero_neq_one @ uint32 ).
thf(tcon_Uint32_Ouint32___Parity_Oring__parity_657,axiom,
ring_parity @ uint32 ).
thf(tcon_Uint32_Ouint32___Orderings_Opreorder_658,axiom,
preorder @ uint32 ).
thf(tcon_Uint32_Ouint32___Orderings_Olinorder_659,axiom,
linorder @ uint32 ).
thf(tcon_Uint32_Ouint32___Groups_Omonoid__mult_660,axiom,
monoid_mult @ uint32 ).
thf(tcon_Uint32_Ouint32___Rings_Ocomm__ring__1_661,axiom,
comm_ring_1 @ uint32 ).
thf(tcon_Uint32_Ouint32___Groups_Omonoid__add_662,axiom,
monoid_add @ uint32 ).
thf(tcon_Uint32_Ouint32___Rings_Osemiring__1_663,axiom,
semiring_1 @ uint32 ).
thf(tcon_Uint32_Ouint32___Rings_Osemiring__0_664,axiom,
semiring_0 @ uint32 ).
thf(tcon_Uint32_Ouint32___Groups_Ogroup__add_665,axiom,
group_add @ uint32 ).
thf(tcon_Uint32_Ouint32___Rings_Omult__zero_666,axiom,
mult_zero @ uint32 ).
thf(tcon_Uint32_Ouint32___Rings_Ocomm__ring_667,axiom,
comm_ring @ uint32 ).
thf(tcon_Uint32_Ouint32___Orderings_Oorder_668,axiom,
order @ uint32 ).
thf(tcon_Uint32_Ouint32___Num_Oneg__numeral_669,axiom,
neg_numeral @ uint32 ).
thf(tcon_Uint32_Ouint32___Rings_Osemiring_670,axiom,
semiring @ uint32 ).
thf(tcon_Uint32_Ouint32___Orderings_Oord_671,axiom,
ord @ uint32 ).
thf(tcon_Uint32_Ouint32___Groups_Ouminus_672,axiom,
uminus @ uint32 ).
thf(tcon_Uint32_Ouint32___Rings_Oring__1_673,axiom,
ring_1 @ uint32 ).
thf(tcon_Uint32_Ouint32___Groups_Ominus_674,axiom,
minus @ uint32 ).
thf(tcon_Uint32_Ouint32___Power_Opower_675,axiom,
power @ uint32 ).
thf(tcon_Uint32_Ouint32___Num_Onumeral_676,axiom,
numeral @ uint32 ).
thf(tcon_Uint32_Ouint32___Groups_Ozero_677,axiom,
zero @ uint32 ).
thf(tcon_Uint32_Ouint32___Groups_Oplus_678,axiom,
plus @ uint32 ).
thf(tcon_Uint32_Ouint32___Rings_Oring_679,axiom,
ring @ uint32 ).
thf(tcon_Uint32_Ouint32___Groups_Oone_680,axiom,
one @ uint32 ).
thf(tcon_Uint32_Ouint32___Rings_Odvd_681,axiom,
dvd @ uint32 ).
thf(tcon_Uint32_Ouint32___Nat_Osize_682,axiom,
size @ uint32 ).
thf(tcon_String_Oliteral___Groups_Osemigroup__add_683,axiom,
semigroup_add @ literal ).
thf(tcon_String_Oliteral___Orderings_Opreorder_684,axiom,
preorder @ literal ).
thf(tcon_String_Oliteral___Orderings_Olinorder_685,axiom,
linorder @ literal ).
thf(tcon_String_Oliteral___Groups_Omonoid__add_686,axiom,
monoid_add @ literal ).
thf(tcon_String_Oliteral___Orderings_Oorder_687,axiom,
order @ literal ).
thf(tcon_String_Oliteral___Orderings_Oord_688,axiom,
ord @ literal ).
thf(tcon_String_Oliteral___Groups_Ozero_689,axiom,
zero @ literal ).
thf(tcon_String_Oliteral___Groups_Oplus_690,axiom,
plus @ literal ).
thf(tcon_String_Oliteral___Nat_Osize_691,axiom,
size @ literal ).
thf(tcon_Assertions_Oassn___Lattices_Obounded__semilattice__sup__bot_692,axiom,
bounde4967611905675639751up_bot @ assn ).
thf(tcon_Assertions_Oassn___Boolean__Algebras_Oboolean__algebra_693,axiom,
boolea8198339166811842893lgebra @ assn ).
thf(tcon_Assertions_Oassn___Lattices_Obounded__lattice__top_694,axiom,
bounded_lattice_top @ assn ).
thf(tcon_Assertions_Oassn___Lattices_Osemilattice__sup_695,axiom,
semilattice_sup @ assn ).
thf(tcon_Assertions_Oassn___Lattices_Obounded__lattice_696,axiom,
bounded_lattice @ assn ).
thf(tcon_Assertions_Oassn___Groups_Oab__semigroup__mult_697,axiom,
ab_semigroup_mult @ assn ).
thf(tcon_Assertions_Oassn___Groups_Ocomm__monoid__mult_698,axiom,
comm_monoid_mult @ assn ).
thf(tcon_Assertions_Oassn___Groups_Osemigroup__mult_699,axiom,
semigroup_mult @ assn ).
thf(tcon_Assertions_Oassn___Orderings_Oorder__top_700,axiom,
order_top @ assn ).
thf(tcon_Assertions_Oassn___Orderings_Oorder__bot_701,axiom,
order_bot @ assn ).
thf(tcon_Assertions_Oassn___Orderings_Opreorder_702,axiom,
preorder @ assn ).
thf(tcon_Assertions_Oassn___Groups_Omonoid__mult_703,axiom,
monoid_mult @ assn ).
thf(tcon_Assertions_Oassn___Lattices_Olattice_704,axiom,
lattice @ assn ).
thf(tcon_Assertions_Oassn___Orderings_Oorder_705,axiom,
order @ assn ).
thf(tcon_Assertions_Oassn___Orderings_Otop_706,axiom,
top @ assn ).
thf(tcon_Assertions_Oassn___Orderings_Oord_707,axiom,
ord @ assn ).
thf(tcon_Assertions_Oassn___Orderings_Obot_708,axiom,
bot @ assn ).
thf(tcon_Assertions_Oassn___Groups_Ouminus_709,axiom,
uminus @ assn ).
thf(tcon_Assertions_Oassn___Groups_Ominus_710,axiom,
minus @ assn ).
thf(tcon_Assertions_Oassn___Power_Opower_711,axiom,
power @ assn ).
thf(tcon_Assertions_Oassn___Groups_Oone_712,axiom,
one @ assn ).
thf(tcon_Assertions_Oassn___Rings_Odvd_713,axiom,
dvd @ assn ).
thf(tcon_Complex_Ocomplex___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_714,axiom,
semiri1453513574482234551roduct @ complex ).
thf(tcon_Complex_Ocomplex___Topological__Spaces_Ofirst__countable__topology_715,axiom,
topolo3112930676232923870pology @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__div__algebra_716,axiom,
real_V8999393235501362500lgebra @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__algebra__1_717,axiom,
real_V2822296259951069270ebra_1 @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Osemiring__no__zero__divisors__cancel_718,axiom,
semiri6575147826004484403cancel @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__algebra_719,axiom,
real_V4412858255891104859lgebra @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__vector_720,axiom,
real_V822414075346904944vector @ complex ).
thf(tcon_Complex_Ocomplex___Topological__Spaces_Otopological__space_721,axiom,
topolo4958980785337419405_space @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__field_722,axiom,
real_V3459762299906320749_field @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__div__algebra_723,axiom,
real_V5047593784448816457lgebra @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Ouniformity__dist_724,axiom,
real_V768167426530841204y_dist @ complex ).
thf(tcon_Complex_Ocomplex___Limits_Otopological__comm__monoid__add_725,axiom,
topolo5987344860129210374id_add @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Osemiring__1__no__zero__divisors_726,axiom,
semiri2026040879449505780visors @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__algebra__1_727,axiom,
real_V2191834092415804123ebra_1 @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Ocomplete__space_728,axiom,
real_V8037385150606011577_space @ complex ).
thf(tcon_Complex_Ocomplex___Limits_Otopological__semigroup__mult_729,axiom,
topolo4211221413907600880p_mult @ complex ).
thf(tcon_Complex_Ocomplex___Topological__Spaces_Ouniform__space_730,axiom,
topolo7287701948861334536_space @ complex ).
thf(tcon_Complex_Ocomplex___Topological__Spaces_Operfect__space_731,axiom,
topolo8386298272705272623_space @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Osemiring__no__zero__divisors_732,axiom,
semiri3467727345109120633visors @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Ometric__space_733,axiom,
real_V7819770556892013058_space @ complex ).
thf(tcon_Complex_Ocomplex___Limits_Otopological__ab__group__add_734,axiom,
topolo1287966508704411220up_add @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__vector_735,axiom,
real_V4867850818363320053vector @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Ocancel__ab__semigroup__add_736,axiom,
cancel2418104881723323429up_add @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Oring__1__no__zero__divisors_737,axiom,
ring_15535105094025558882visors @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__field_738,axiom,
real_V7773925162809079976_field @ complex ).
thf(tcon_Complex_Ocomplex___Limits_Otopological__monoid__add_739,axiom,
topolo6943815403480290642id_add @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Ocancel__comm__monoid__add_740,axiom,
cancel1802427076303600483id_add @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Ocomm__semiring__1__cancel_741,axiom,
comm_s4317794764714335236cancel @ complex ).
thf(tcon_Complex_Ocomplex___Limits_Otopological__group__add_742,axiom,
topolo1633459387980952147up_add @ complex ).
thf(tcon_Complex_Ocomplex___Topological__Spaces_Ot2__space_743,axiom,
topological_t2_space @ complex ).
thf(tcon_Complex_Ocomplex___Topological__Spaces_Ot1__space_744,axiom,
topological_t1_space @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Ocancel__semigroup__add_745,axiom,
cancel_semigroup_add @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Obanach_746,axiom,
real_Vector_banach @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Oab__semigroup__mult_747,axiom,
ab_semigroup_mult @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Osemiring__1__cancel_748,axiom,
semiring_1_cancel @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Ocomm__monoid__mult_749,axiom,
comm_monoid_mult @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Oab__semigroup__add_750,axiom,
ab_semigroup_add @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Ocomm__monoid__add_751,axiom,
comm_monoid_add @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Ocomm__semiring__1_752,axiom,
comm_semiring_1 @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Osemigroup__mult_753,axiom,
semigroup_mult @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Osemidom__divide_754,axiom,
semidom_divide @ complex ).
thf(tcon_Complex_Ocomplex___Num_Osemiring__numeral_755,axiom,
semiring_numeral @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Osemigroup__add_756,axiom,
semigroup_add @ complex ).
thf(tcon_Complex_Ocomplex___Fields_Odivision__ring_757,axiom,
division_ring @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Ocomm__semiring_758,axiom,
comm_semiring @ complex ).
thf(tcon_Complex_Ocomplex___Nat_Osemiring__char__0_759,axiom,
semiring_char_0 @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Oab__group__add_760,axiom,
ab_group_add @ complex ).
thf(tcon_Complex_Ocomplex___Fields_Ofield__char__0_761,axiom,
field_char_0 @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Ozero__neq__one_762,axiom,
zero_neq_one @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Oidom__abs__sgn_763,axiom,
idom_abs_sgn @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Omonoid__mult_764,axiom,
monoid_mult @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Oidom__divide_765,axiom,
idom_divide @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Ocomm__ring__1_766,axiom,
comm_ring_1 @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Omonoid__add_767,axiom,
monoid_add @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Osemiring__1_768,axiom,
semiring_1 @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Osemiring__0_769,axiom,
semiring_0 @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Ogroup__add_770,axiom,
group_add @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Omult__zero_771,axiom,
mult_zero @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Ocomm__ring_772,axiom,
comm_ring @ complex ).
thf(tcon_Complex_Ocomplex___Num_Oneg__numeral_773,axiom,
neg_numeral @ complex ).
thf(tcon_Complex_Ocomplex___Nat_Oring__char__0_774,axiom,
ring_char_0 @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Osemiring_775,axiom,
semiring @ complex ).
thf(tcon_Complex_Ocomplex___Fields_Oinverse_776,axiom,
inverse @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Osemidom_777,axiom,
semidom @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Ouminus_778,axiom,
uminus @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Oring__1_779,axiom,
ring_1 @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Ominus_780,axiom,
minus @ complex ).
thf(tcon_Complex_Ocomplex___Fields_Ofield_781,axiom,
field @ complex ).
thf(tcon_Complex_Ocomplex___Power_Opower_782,axiom,
power @ complex ).
thf(tcon_Complex_Ocomplex___Num_Onumeral_783,axiom,
numeral @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Ozero_784,axiom,
zero @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Oplus_785,axiom,
plus @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Oring_786,axiom,
ring @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Oidom_787,axiom,
idom @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Oone_788,axiom,
one @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Odvd_789,axiom,
dvd @ complex ).
thf(tcon_Extended__Nat_Oenat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_790,axiom,
condit6923001295902523014norder @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Ostrict__ordered__ab__semigroup__add_791,axiom,
strict9044650504122735259up_add @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Ostrict__ordered__comm__monoid__add_792,axiom,
strict7427464778891057005id_add @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Ocanonically__ordered__monoid__add_793,axiom,
canoni5634975068530333245id_add @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Lattices_Obounded__semilattice__sup__bot_794,axiom,
bounde4967611905675639751up_bot @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Olinordered__nonzero__semiring_795,axiom,
linord181362715937106298miring @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Osemiring__no__zero__divisors_796,axiom,
semiri3467727345109120633visors @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Oordered__ab__semigroup__add_797,axiom,
ordere6658533253407199908up_add @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Oordered__comm__monoid__add_798,axiom,
ordere6911136660526730532id_add @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Lattices_Obounded__lattice__top_799,axiom,
bounded_lattice_top @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Oordered__comm__semiring_800,axiom,
ordere2520102378445227354miring @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Lattices_Osemilattice__sup_801,axiom,
semilattice_sup @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Lattices_Obounded__lattice_802,axiom,
bounded_lattice @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Oab__semigroup__mult_803,axiom,
ab_semigroup_mult @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Ocomm__monoid__mult_804,axiom,
comm_monoid_mult @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Oab__semigroup__add_805,axiom,
ab_semigroup_add @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Oordered__semiring_806,axiom,
ordered_semiring @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Ocomm__monoid__add_807,axiom,
comm_monoid_add @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Ocomm__semiring__1_808,axiom,
comm_semiring_1 @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Osemigroup__mult_809,axiom,
semigroup_mult @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Num_Osemiring__numeral_810,axiom,
semiring_numeral @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Osemigroup__add_811,axiom,
semigroup_add @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Ozero__less__one_812,axiom,
zero_less_one @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Ocomm__semiring_813,axiom,
comm_semiring @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Orderings_Owellorder_814,axiom,
wellorder @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Orderings_Oorder__top_815,axiom,
order_top @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Orderings_Oorder__bot_816,axiom,
order_bot @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Nat_Osemiring__char__0_817,axiom,
semiring_char_0 @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Ozero__neq__one_818,axiom,
zero_neq_one @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Orderings_Opreorder_819,axiom,
preorder @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Orderings_Olinorder_820,axiom,
linorder @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Omonoid__mult_821,axiom,
monoid_mult @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Omonoid__add_822,axiom,
monoid_add @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Osemiring__1_823,axiom,
semiring_1 @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Osemiring__0_824,axiom,
semiring_0 @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Lattices_Olattice_825,axiom,
lattice @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Omult__zero_826,axiom,
mult_zero @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Orderings_Oorder_827,axiom,
order @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Osemiring_828,axiom,
semiring @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Orderings_Otop_829,axiom,
top @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Orderings_Oord_830,axiom,
ord @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Orderings_Obot_831,axiom,
bot @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Ominus_832,axiom,
minus @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Power_Opower_833,axiom,
power @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Num_Onumeral_834,axiom,
numeral @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Ozero_835,axiom,
zero @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Oplus_836,axiom,
plus @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Oone_837,axiom,
one @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Odvd_838,axiom,
dvd @ extended_enat ).
thf(tcon_Multiset_Omultiset___Groups_Oordered__ab__semigroup__add_839,axiom,
! [A11: $tType] :
( ( preorder @ A11 )
=> ( ordere6658533253407199908up_add @ ( multiset @ A11 ) ) ) ).
thf(tcon_Multiset_Omultiset___Groups_Ocancel__ab__semigroup__add_840,axiom,
! [A11: $tType] : ( cancel2418104881723323429up_add @ ( multiset @ A11 ) ) ).
thf(tcon_Multiset_Omultiset___Groups_Ocancel__comm__monoid__add_841,axiom,
! [A11: $tType] : ( cancel1802427076303600483id_add @ ( multiset @ A11 ) ) ).
thf(tcon_Multiset_Omultiset___Groups_Ocancel__semigroup__add_842,axiom,
! [A11: $tType] : ( cancel_semigroup_add @ ( multiset @ A11 ) ) ).
thf(tcon_Multiset_Omultiset___Groups_Ocomm__monoid__diff_843,axiom,
! [A11: $tType] : ( comm_monoid_diff @ ( multiset @ A11 ) ) ).
thf(tcon_Multiset_Omultiset___Groups_Oab__semigroup__add_844,axiom,
! [A11: $tType] : ( ab_semigroup_add @ ( multiset @ A11 ) ) ).
thf(tcon_Multiset_Omultiset___Groups_Ocomm__monoid__add_845,axiom,
! [A11: $tType] : ( comm_monoid_add @ ( multiset @ A11 ) ) ).
thf(tcon_Multiset_Omultiset___Groups_Osemigroup__add_846,axiom,
! [A11: $tType] : ( semigroup_add @ ( multiset @ A11 ) ) ).
thf(tcon_Multiset_Omultiset___Orderings_Opreorder_847,axiom,
! [A11: $tType] :
( ( preorder @ A11 )
=> ( preorder @ ( multiset @ A11 ) ) ) ).
thf(tcon_Multiset_Omultiset___Groups_Omonoid__add_848,axiom,
! [A11: $tType] : ( monoid_add @ ( multiset @ A11 ) ) ).
thf(tcon_Multiset_Omultiset___Orderings_Oorder_849,axiom,
! [A11: $tType] :
( ( preorder @ A11 )
=> ( order @ ( multiset @ A11 ) ) ) ).
thf(tcon_Multiset_Omultiset___Orderings_Oord_850,axiom,
! [A11: $tType] :
( ( preorder @ A11 )
=> ( ord @ ( multiset @ A11 ) ) ) ).
thf(tcon_Multiset_Omultiset___Groups_Ominus_851,axiom,
! [A11: $tType] : ( minus @ ( multiset @ A11 ) ) ).
thf(tcon_Multiset_Omultiset___Groups_Ozero_852,axiom,
! [A11: $tType] : ( zero @ ( multiset @ A11 ) ) ).
thf(tcon_Multiset_Omultiset___Groups_Oplus_853,axiom,
! [A11: $tType] : ( plus @ ( multiset @ A11 ) ) ).
thf(tcon_Multiset_Omultiset___Nat_Osize_854,axiom,
! [A11: $tType] : ( size @ ( multiset @ A11 ) ) ).
thf(tcon_Numeral__Type_Obit0___Groups_Ocancel__ab__semigroup__add_855,axiom,
! [A11: $tType] :
( ( finite_finite @ A11 )
=> ( cancel2418104881723323429up_add @ ( numeral_bit0 @ A11 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Groups_Ocancel__comm__monoid__add_856,axiom,
! [A11: $tType] :
( ( finite_finite @ A11 )
=> ( cancel1802427076303600483id_add @ ( numeral_bit0 @ A11 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Rings_Ocomm__semiring__1__cancel_857,axiom,
! [A11: $tType] :
( ( finite_finite @ A11 )
=> ( comm_s4317794764714335236cancel @ ( numeral_bit0 @ A11 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Groups_Ocancel__semigroup__add_858,axiom,
! [A11: $tType] :
( ( finite_finite @ A11 )
=> ( cancel_semigroup_add @ ( numeral_bit0 @ A11 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Groups_Oab__semigroup__mult_859,axiom,
! [A11: $tType] :
( ( finite_finite @ A11 )
=> ( ab_semigroup_mult @ ( numeral_bit0 @ A11 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Rings_Osemiring__1__cancel_860,axiom,
! [A11: $tType] :
( ( finite_finite @ A11 )
=> ( semiring_1_cancel @ ( numeral_bit0 @ A11 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Groups_Ocomm__monoid__mult_861,axiom,
! [A11: $tType] :
( ( finite_finite @ A11 )
=> ( comm_monoid_mult @ ( numeral_bit0 @ A11 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Groups_Oab__semigroup__add_862,axiom,
! [A11: $tType] :
( ( finite_finite @ A11 )
=> ( ab_semigroup_add @ ( numeral_bit0 @ A11 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Groups_Ocomm__monoid__add_863,axiom,
! [A11: $tType] :
( ( finite_finite @ A11 )
=> ( comm_monoid_add @ ( numeral_bit0 @ A11 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Rings_Ocomm__semiring__1_864,axiom,
! [A11: $tType] :
( ( finite_finite @ A11 )
=> ( comm_semiring_1 @ ( numeral_bit0 @ A11 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Groups_Osemigroup__mult_865,axiom,
! [A11: $tType] :
( ( finite_finite @ A11 )
=> ( semigroup_mult @ ( numeral_bit0 @ A11 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Num_Osemiring__numeral_866,axiom,
! [A11: $tType] :
( ( finite_finite @ A11 )
=> ( semiring_numeral @ ( numeral_bit0 @ A11 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Groups_Osemigroup__add_867,axiom,
! [A11: $tType] :
( ( finite_finite @ A11 )
=> ( semigroup_add @ ( numeral_bit0 @ A11 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Rings_Ocomm__semiring_868,axiom,
! [A11: $tType] :
( ( finite_finite @ A11 )
=> ( comm_semiring @ ( numeral_bit0 @ A11 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Orderings_Owellorder_869,axiom,
! [A11: $tType] :
( ( finite_finite @ A11 )
=> ( wellorder @ ( numeral_bit0 @ A11 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Groups_Oab__group__add_870,axiom,
! [A11: $tType] :
( ( finite_finite @ A11 )
=> ( ab_group_add @ ( numeral_bit0 @ A11 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Rings_Ozero__neq__one_871,axiom,
! [A11: $tType] :
( ( finite_finite @ A11 )
=> ( zero_neq_one @ ( numeral_bit0 @ A11 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Orderings_Opreorder_872,axiom,
! [A11: $tType] :
( ( finite_finite @ A11 )
=> ( preorder @ ( numeral_bit0 @ A11 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Orderings_Olinorder_873,axiom,
! [A11: $tType] :
( ( finite_finite @ A11 )
=> ( linorder @ ( numeral_bit0 @ A11 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Groups_Omonoid__mult_874,axiom,
! [A11: $tType] :
( ( finite_finite @ A11 )
=> ( monoid_mult @ ( numeral_bit0 @ A11 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Rings_Ocomm__ring__1_875,axiom,
! [A11: $tType] :
( ( finite_finite @ A11 )
=> ( comm_ring_1 @ ( numeral_bit0 @ A11 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Groups_Omonoid__add_876,axiom,
! [A11: $tType] :
( ( finite_finite @ A11 )
=> ( monoid_add @ ( numeral_bit0 @ A11 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Finite__Set_Ofinite_877,axiom,
! [A11: $tType] :
( ( finite_finite @ A11 )
=> ( finite_finite @ ( numeral_bit0 @ A11 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Cardinality_Ocard2,axiom,
! [A11: $tType] :
( ( finite_finite @ A11 )
=> ( card2 @ ( numeral_bit0 @ A11 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Type__Length_Olen0_878,axiom,
! [A11: $tType] :
( ( type_len0 @ A11 )
=> ( type_len0 @ ( numeral_bit0 @ A11 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Rings_Osemiring__1_879,axiom,
! [A11: $tType] :
( ( finite_finite @ A11 )
=> ( semiring_1 @ ( numeral_bit0 @ A11 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Rings_Osemiring__0_880,axiom,
! [A11: $tType] :
( ( finite_finite @ A11 )
=> ( semiring_0 @ ( numeral_bit0 @ A11 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Groups_Ogroup__add_881,axiom,
! [A11: $tType] :
( ( finite_finite @ A11 )
=> ( group_add @ ( numeral_bit0 @ A11 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Type__Length_Olen_882,axiom,
! [A11: $tType] :
( ( type_len @ A11 )
=> ( type_len @ ( numeral_bit0 @ A11 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Rings_Omult__zero_883,axiom,
! [A11: $tType] :
( ( finite_finite @ A11 )
=> ( mult_zero @ ( numeral_bit0 @ A11 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Rings_Ocomm__ring_884,axiom,
! [A11: $tType] :
( ( finite_finite @ A11 )
=> ( comm_ring @ ( numeral_bit0 @ A11 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Orderings_Oorder_885,axiom,
! [A11: $tType] :
( ( finite_finite @ A11 )
=> ( order @ ( numeral_bit0 @ A11 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Num_Oneg__numeral_886,axiom,
! [A11: $tType] :
( ( finite_finite @ A11 )
=> ( neg_numeral @ ( numeral_bit0 @ A11 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Rings_Osemiring_887,axiom,
! [A11: $tType] :
( ( finite_finite @ A11 )
=> ( semiring @ ( numeral_bit0 @ A11 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Orderings_Oord_888,axiom,
! [A11: $tType] :
( ( finite_finite @ A11 )
=> ( ord @ ( numeral_bit0 @ A11 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Groups_Ouminus_889,axiom,
! [A11: $tType] :
( ( finite_finite @ A11 )
=> ( uminus @ ( numeral_bit0 @ A11 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Rings_Oring__1_890,axiom,
! [A11: $tType] :
( ( finite_finite @ A11 )
=> ( ring_1 @ ( numeral_bit0 @ A11 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Groups_Ominus_891,axiom,
! [A11: $tType] :
( ( finite_finite @ A11 )
=> ( minus @ ( numeral_bit0 @ A11 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Power_Opower_892,axiom,
! [A11: $tType] :
( ( finite_finite @ A11 )
=> ( power @ ( numeral_bit0 @ A11 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Num_Onumeral_893,axiom,
! [A11: $tType] :
( ( finite_finite @ A11 )
=> ( numeral @ ( numeral_bit0 @ A11 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Groups_Ozero_894,axiom,
! [A11: $tType] :
( ( finite_finite @ A11 )
=> ( zero @ ( numeral_bit0 @ A11 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Groups_Oplus_895,axiom,
! [A11: $tType] :
( ( finite_finite @ A11 )
=> ( plus @ ( numeral_bit0 @ A11 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Rings_Oring_896,axiom,
! [A11: $tType] :
( ( finite_finite @ A11 )
=> ( ring @ ( numeral_bit0 @ A11 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Groups_Oone_897,axiom,
! [A11: $tType] :
( ( finite_finite @ A11 )
=> ( one @ ( numeral_bit0 @ A11 ) ) ) ).
thf(tcon_Numeral__Type_Obit0___Rings_Odvd_898,axiom,
! [A11: $tType] :
( ( finite_finite @ A11 )
=> ( dvd @ ( numeral_bit0 @ A11 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Groups_Ocancel__ab__semigroup__add_899,axiom,
! [A11: $tType] :
( ( finite_finite @ A11 )
=> ( cancel2418104881723323429up_add @ ( numeral_bit1 @ A11 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Groups_Ocancel__comm__monoid__add_900,axiom,
! [A11: $tType] :
( ( finite_finite @ A11 )
=> ( cancel1802427076303600483id_add @ ( numeral_bit1 @ A11 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Rings_Ocomm__semiring__1__cancel_901,axiom,
! [A11: $tType] :
( ( finite_finite @ A11 )
=> ( comm_s4317794764714335236cancel @ ( numeral_bit1 @ A11 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Groups_Ocancel__semigroup__add_902,axiom,
! [A11: $tType] :
( ( finite_finite @ A11 )
=> ( cancel_semigroup_add @ ( numeral_bit1 @ A11 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Groups_Oab__semigroup__mult_903,axiom,
! [A11: $tType] :
( ( finite_finite @ A11 )
=> ( ab_semigroup_mult @ ( numeral_bit1 @ A11 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Rings_Osemiring__1__cancel_904,axiom,
! [A11: $tType] :
( ( finite_finite @ A11 )
=> ( semiring_1_cancel @ ( numeral_bit1 @ A11 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Groups_Ocomm__monoid__mult_905,axiom,
! [A11: $tType] :
( ( finite_finite @ A11 )
=> ( comm_monoid_mult @ ( numeral_bit1 @ A11 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Groups_Oab__semigroup__add_906,axiom,
! [A11: $tType] :
( ( finite_finite @ A11 )
=> ( ab_semigroup_add @ ( numeral_bit1 @ A11 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Groups_Ocomm__monoid__add_907,axiom,
! [A11: $tType] :
( ( finite_finite @ A11 )
=> ( comm_monoid_add @ ( numeral_bit1 @ A11 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Rings_Ocomm__semiring__1_908,axiom,
! [A11: $tType] :
( ( finite_finite @ A11 )
=> ( comm_semiring_1 @ ( numeral_bit1 @ A11 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Groups_Osemigroup__mult_909,axiom,
! [A11: $tType] :
( ( finite_finite @ A11 )
=> ( semigroup_mult @ ( numeral_bit1 @ A11 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Num_Osemiring__numeral_910,axiom,
! [A11: $tType] :
( ( finite_finite @ A11 )
=> ( semiring_numeral @ ( numeral_bit1 @ A11 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Groups_Osemigroup__add_911,axiom,
! [A11: $tType] :
( ( finite_finite @ A11 )
=> ( semigroup_add @ ( numeral_bit1 @ A11 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Rings_Ocomm__semiring_912,axiom,
! [A11: $tType] :
( ( finite_finite @ A11 )
=> ( comm_semiring @ ( numeral_bit1 @ A11 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Orderings_Owellorder_913,axiom,
! [A11: $tType] :
( ( finite_finite @ A11 )
=> ( wellorder @ ( numeral_bit1 @ A11 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Groups_Oab__group__add_914,axiom,
! [A11: $tType] :
( ( finite_finite @ A11 )
=> ( ab_group_add @ ( numeral_bit1 @ A11 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Rings_Ozero__neq__one_915,axiom,
! [A11: $tType] :
( ( finite_finite @ A11 )
=> ( zero_neq_one @ ( numeral_bit1 @ A11 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Orderings_Opreorder_916,axiom,
! [A11: $tType] :
( ( finite_finite @ A11 )
=> ( preorder @ ( numeral_bit1 @ A11 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Orderings_Olinorder_917,axiom,
! [A11: $tType] :
( ( finite_finite @ A11 )
=> ( linorder @ ( numeral_bit1 @ A11 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Groups_Omonoid__mult_918,axiom,
! [A11: $tType] :
( ( finite_finite @ A11 )
=> ( monoid_mult @ ( numeral_bit1 @ A11 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Rings_Ocomm__ring__1_919,axiom,
! [A11: $tType] :
( ( finite_finite @ A11 )
=> ( comm_ring_1 @ ( numeral_bit1 @ A11 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Groups_Omonoid__add_920,axiom,
! [A11: $tType] :
( ( finite_finite @ A11 )
=> ( monoid_add @ ( numeral_bit1 @ A11 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Finite__Set_Ofinite_921,axiom,
! [A11: $tType] :
( ( finite_finite @ A11 )
=> ( finite_finite @ ( numeral_bit1 @ A11 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Cardinality_Ocard2_922,axiom,
! [A11: $tType] :
( ( finite_finite @ A11 )
=> ( card2 @ ( numeral_bit1 @ A11 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Type__Length_Olen0_923,axiom,
! [A11: $tType] :
( ( type_len0 @ A11 )
=> ( type_len0 @ ( numeral_bit1 @ A11 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Rings_Osemiring__1_924,axiom,
! [A11: $tType] :
( ( finite_finite @ A11 )
=> ( semiring_1 @ ( numeral_bit1 @ A11 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Rings_Osemiring__0_925,axiom,
! [A11: $tType] :
( ( finite_finite @ A11 )
=> ( semiring_0 @ ( numeral_bit1 @ A11 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Groups_Ogroup__add_926,axiom,
! [A11: $tType] :
( ( finite_finite @ A11 )
=> ( group_add @ ( numeral_bit1 @ A11 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Type__Length_Olen_927,axiom,
! [A11: $tType] :
( ( type_len0 @ A11 )
=> ( type_len @ ( numeral_bit1 @ A11 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Rings_Omult__zero_928,axiom,
! [A11: $tType] :
( ( finite_finite @ A11 )
=> ( mult_zero @ ( numeral_bit1 @ A11 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Rings_Ocomm__ring_929,axiom,
! [A11: $tType] :
( ( finite_finite @ A11 )
=> ( comm_ring @ ( numeral_bit1 @ A11 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Orderings_Oorder_930,axiom,
! [A11: $tType] :
( ( finite_finite @ A11 )
=> ( order @ ( numeral_bit1 @ A11 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Num_Oneg__numeral_931,axiom,
! [A11: $tType] :
( ( finite_finite @ A11 )
=> ( neg_numeral @ ( numeral_bit1 @ A11 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Rings_Osemiring_932,axiom,
! [A11: $tType] :
( ( finite_finite @ A11 )
=> ( semiring @ ( numeral_bit1 @ A11 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Orderings_Oord_933,axiom,
! [A11: $tType] :
( ( finite_finite @ A11 )
=> ( ord @ ( numeral_bit1 @ A11 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Groups_Ouminus_934,axiom,
! [A11: $tType] :
( ( finite_finite @ A11 )
=> ( uminus @ ( numeral_bit1 @ A11 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Rings_Oring__1_935,axiom,
! [A11: $tType] :
( ( finite_finite @ A11 )
=> ( ring_1 @ ( numeral_bit1 @ A11 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Groups_Ominus_936,axiom,
! [A11: $tType] :
( ( finite_finite @ A11 )
=> ( minus @ ( numeral_bit1 @ A11 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Power_Opower_937,axiom,
! [A11: $tType] :
( ( finite_finite @ A11 )
=> ( power @ ( numeral_bit1 @ A11 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Num_Onumeral_938,axiom,
! [A11: $tType] :
( ( finite_finite @ A11 )
=> ( numeral @ ( numeral_bit1 @ A11 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Groups_Ozero_939,axiom,
! [A11: $tType] :
( ( finite_finite @ A11 )
=> ( zero @ ( numeral_bit1 @ A11 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Groups_Oplus_940,axiom,
! [A11: $tType] :
( ( finite_finite @ A11 )
=> ( plus @ ( numeral_bit1 @ A11 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Rings_Oring_941,axiom,
! [A11: $tType] :
( ( finite_finite @ A11 )
=> ( ring @ ( numeral_bit1 @ A11 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Groups_Oone_942,axiom,
! [A11: $tType] :
( ( finite_finite @ A11 )
=> ( one @ ( numeral_bit1 @ A11 ) ) ) ).
thf(tcon_Numeral__Type_Obit1___Rings_Odvd_943,axiom,
! [A11: $tType] :
( ( finite_finite @ A11 )
=> ( dvd @ ( numeral_bit1 @ A11 ) ) ) ).
thf(tcon_Numeral__Type_Onum0___Type__Length_Olen0_944,axiom,
type_len0 @ numeral_num0 ).
thf(tcon_Numeral__Type_Onum1___Groups_Ocancel__ab__semigroup__add_945,axiom,
cancel2418104881723323429up_add @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Groups_Ocancel__comm__monoid__add_946,axiom,
cancel1802427076303600483id_add @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Groups_Ocancel__semigroup__add_947,axiom,
cancel_semigroup_add @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Groups_Oab__semigroup__mult_948,axiom,
ab_semigroup_mult @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Groups_Ocomm__monoid__mult_949,axiom,
comm_monoid_mult @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Groups_Oab__semigroup__add_950,axiom,
ab_semigroup_add @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Groups_Ocomm__monoid__add_951,axiom,
comm_monoid_add @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Groups_Osemigroup__mult_952,axiom,
semigroup_mult @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Groups_Osemigroup__add_953,axiom,
semigroup_add @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Rings_Ocomm__semiring_954,axiom,
comm_semiring @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Orderings_Owellorder_955,axiom,
wellorder @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Groups_Oab__group__add_956,axiom,
ab_group_add @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Orderings_Opreorder_957,axiom,
preorder @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Orderings_Olinorder_958,axiom,
linorder @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Groups_Omonoid__mult_959,axiom,
monoid_mult @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Cardinality_OCARD__1,axiom,
cARD_1 @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Groups_Omonoid__add_960,axiom,
monoid_add @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Finite__Set_Ofinite_961,axiom,
finite_finite @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Type__Length_Olen0_962,axiom,
type_len0 @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Rings_Osemiring__0_963,axiom,
semiring_0 @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Groups_Ogroup__add_964,axiom,
group_add @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Type__Length_Olen_965,axiom,
type_len @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Rings_Omult__zero_966,axiom,
mult_zero @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Rings_Ocomm__ring_967,axiom,
comm_ring @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Orderings_Oorder_968,axiom,
order @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Rings_Osemiring_969,axiom,
semiring @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Orderings_Oord_970,axiom,
ord @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Groups_Ouminus_971,axiom,
uminus @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Groups_Ominus_972,axiom,
minus @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Power_Opower_973,axiom,
power @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Num_Onumeral_974,axiom,
numeral @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Groups_Ozero_975,axiom,
zero @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Groups_Oplus_976,axiom,
plus @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Rings_Oring_977,axiom,
ring @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Groups_Oone_978,axiom,
one @ numeral_num1 ).
thf(tcon_Numeral__Type_Onum1___Rings_Odvd_979,axiom,
dvd @ numeral_num1 ).
thf(tcon_Product__Type_Oprod___Topological__Spaces_Otopological__space_980,axiom,
! [A11: $tType,A14: $tType] :
( ( ( topolo4958980785337419405_space @ A11 )
& ( topolo4958980785337419405_space @ A14 ) )
=> ( topolo4958980785337419405_space @ ( product_prod @ A11 @ A14 ) ) ) ).
thf(tcon_Product__Type_Oprod___Topological__Spaces_Ot2__space_981,axiom,
! [A11: $tType,A14: $tType] :
( ( ( topological_t2_space @ A11 )
& ( topological_t2_space @ A14 ) )
=> ( topological_t2_space @ ( product_prod @ A11 @ A14 ) ) ) ).
thf(tcon_Product__Type_Oprod___Topological__Spaces_Ot1__space_982,axiom,
! [A11: $tType,A14: $tType] :
( ( ( topological_t1_space @ A11 )
& ( topological_t1_space @ A14 ) )
=> ( topological_t1_space @ ( product_prod @ A11 @ A14 ) ) ) ).
thf(tcon_Product__Type_Oprod___Finite__Set_Ofinite_983,axiom,
! [A11: $tType,A14: $tType] :
( ( ( finite_finite @ A11 )
& ( finite_finite @ A14 ) )
=> ( finite_finite @ ( product_prod @ A11 @ A14 ) ) ) ).
thf(tcon_Product__Type_Oprod___Nat_Osize_984,axiom,
! [A11: $tType,A14: $tType] : ( size @ ( product_prod @ A11 @ A14 ) ) ).
thf(tcon_Product__Type_Ounit___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_985,axiom,
condit6923001295902523014norder @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Obounded__semilattice__sup__bot_986,axiom,
bounde4967611905675639751up_bot @ product_unit ).
thf(tcon_Product__Type_Ounit___Boolean__Algebras_Oboolean__algebra_987,axiom,
boolea8198339166811842893lgebra @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Obounded__lattice__top_988,axiom,
bounded_lattice_top @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Osemilattice__sup_989,axiom,
semilattice_sup @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Obounded__lattice_990,axiom,
bounded_lattice @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Owellorder_991,axiom,
wellorder @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Oorder__top_992,axiom,
order_top @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Oorder__bot_993,axiom,
order_bot @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Opreorder_994,axiom,
preorder @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Olinorder_995,axiom,
linorder @ product_unit ).
thf(tcon_Product__Type_Ounit___Finite__Set_Ofinite_996,axiom,
finite_finite @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Olattice_997,axiom,
lattice @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Oorder_998,axiom,
order @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Otop_999,axiom,
top @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Oord_1000,axiom,
ord @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Obot_1001,axiom,
bot @ product_unit ).
thf(tcon_Product__Type_Ounit___Groups_Ouminus_1002,axiom,
uminus @ product_unit ).
thf(tcon_Product__Type_Ounit___Groups_Ominus_1003,axiom,
minus @ product_unit ).
thf(tcon_Code__Numeral_Ointeger___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_1004,axiom,
bit_un5681908812861735899ations @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_1005,axiom,
semiri1453513574482234551roduct @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__semiring__with__nat_1006,axiom,
euclid5411537665997757685th_nat @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__ring__with__nat_1007,axiom,
euclid8789492081693882211th_nat @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__monoid__add__imp__le_1008,axiom,
ordere1937475149494474687imp_le @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__semiring__cancel_1009,axiom,
euclid4440199948858584721cancel @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Divides_Ounique__euclidean__semiring__numeral_1010,axiom,
unique1627219031080169319umeral @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__ring__cancel_1011,axiom,
euclid8851590272496341667cancel @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__no__zero__divisors__cancel_1012,axiom,
semiri6575147826004484403cancel @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ostrict__ordered__ab__semigroup__add_1013,axiom,
strict9044650504122735259up_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__cancel__ab__semigroup__add_1014,axiom,
ordere580206878836729694up_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__add__imp__le_1015,axiom,
ordere2412721322843649153imp_le @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Bit__Operations_Osemiring__bit__operations_1016,axiom,
bit_se359711467146920520ations @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__comm__semiring__strict_1017,axiom,
linord2810124833399127020strict @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ostrict__ordered__comm__monoid__add_1018,axiom,
strict7427464778891057005id_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__cancel__comm__monoid__add_1019,axiom,
ordere8940638589300402666id_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__semiring_1020,axiom,
euclid3725896446679973847miring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__1__strict_1021,axiom,
linord715952674999750819strict @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Bit__Operations_Oring__bit__operations_1022,axiom,
bit_ri3973907225187159222ations @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1__no__zero__divisors_1023,axiom,
semiri2026040879449505780visors @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__nonzero__semiring_1024,axiom,
linord181362715937106298miring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__strict_1025,axiom,
linord8928482502909563296strict @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__no__zero__divisors_1026,axiom,
semiri3467727345109120633visors @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__add_1027,axiom,
ordere6658533253407199908up_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__group__add__abs_1028,axiom,
ordere166539214618696060dd_abs @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__comm__monoid__add_1029,axiom,
ordere6911136660526730532id_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Olinordered__ab__group__add_1030,axiom,
linord5086331880401160121up_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ocancel__ab__semigroup__add_1031,axiom,
cancel2418104881723323429up_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oring__1__no__zero__divisors_1032,axiom,
ring_15535105094025558882visors @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ocancel__comm__monoid__add_1033,axiom,
cancel1802427076303600483id_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__ring__strict_1034,axiom,
linord4710134922213307826strict @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__1__cancel_1035,axiom,
comm_s4317794764714335236cancel @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Bit__Operations_Osemiring__bits_1036,axiom,
bit_semiring_bits @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oordered__comm__semiring_1037,axiom,
ordere2520102378445227354miring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__1_1038,axiom,
linord6961819062388156250ring_1 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__group__add_1039,axiom,
ordered_ab_group_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ocancel__semigroup__add_1040,axiom,
cancel_semigroup_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring_1041,axiom,
linordered_semiring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Least__significant__bit_Olsb_1042,axiom,
least_6119777620449941438nt_lsb @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oordered__semiring__0_1043,axiom,
ordered_semiring_0 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semidom_1044,axiom,
linordered_semidom @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oab__semigroup__mult_1045,axiom,
ab_semigroup_mult @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1__cancel_1046,axiom,
semiring_1_cancel @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oalgebraic__semidom_1047,axiom,
algebraic_semidom @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ocomm__monoid__mult_1048,axiom,
comm_monoid_mult @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oab__semigroup__add_1049,axiom,
ab_semigroup_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Generic__set__bit_Oset__bit_1050,axiom,
generic_set_set_bit @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oordered__semiring_1051,axiom,
ordered_semiring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oordered__ring__abs_1052,axiom,
ordered_ring_abs @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Parity_Osemiring__parity_1053,axiom,
semiring_parity @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ocomm__monoid__add_1054,axiom,
comm_monoid_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__modulo_1055,axiom,
semiring_modulo @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__ring_1056,axiom,
linordered_ring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__idom_1057,axiom,
linordered_idom @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__1_1058,axiom,
comm_semiring_1 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Osemigroup__mult_1059,axiom,
semigroup_mult @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemidom__modulo_1060,axiom,
semidom_modulo @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemidom__divide_1061,axiom,
semidom_divide @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Num_Osemiring__numeral_1062,axiom,
semiring_numeral @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Osemigroup__add_1063,axiom,
semigroup_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ozero__less__one_1064,axiom,
zero_less_one @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring_1065,axiom,
comm_semiring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Nat_Osemiring__char__0_1066,axiom,
semiring_char_0 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oab__group__add_1067,axiom,
ab_group_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ozero__neq__one_1068,axiom,
zero_neq_one @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oordered__ring_1069,axiom,
ordered_ring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oidom__abs__sgn_1070,axiom,
idom_abs_sgn @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Parity_Oring__parity_1071,axiom,
ring_parity @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Orderings_Opreorder_1072,axiom,
preorder @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Orderings_Olinorder_1073,axiom,
linorder @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Omonoid__mult_1074,axiom,
monoid_mult @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oidom__modulo_1075,axiom,
idom_modulo @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oidom__divide_1076,axiom,
idom_divide @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__ring__1_1077,axiom,
comm_ring_1 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Omonoid__add_1078,axiom,
monoid_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1_1079,axiom,
semiring_1 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__0_1080,axiom,
semiring_0 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ogroup__add_1081,axiom,
group_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Omult__zero_1082,axiom,
mult_zero @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__ring_1083,axiom,
comm_ring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Orderings_Oorder_1084,axiom,
order @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Num_Oneg__numeral_1085,axiom,
neg_numeral @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Nat_Oring__char__0_1086,axiom,
ring_char_0 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring_1087,axiom,
semiring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemidom_1088,axiom,
semidom @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Orderings_Oord_1089,axiom,
ord @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ouminus_1090,axiom,
uminus @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oring__1_1091,axiom,
ring_1 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oabs__if_1092,axiom,
abs_if @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ominus_1093,axiom,
minus @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Power_Opower_1094,axiom,
power @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Num_Onumeral_1095,axiom,
numeral @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ozero_1096,axiom,
zero @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oplus_1097,axiom,
plus @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oring_1098,axiom,
ring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oidom_1099,axiom,
idom @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oone_1100,axiom,
one @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Odvd_1101,axiom,
dvd @ code_integer ).
thf(tcon_Heap__Time__Monad_OHeap___Nat_Osize_1102,axiom,
! [A11: $tType] : ( size @ ( heap_Time_Heap @ A11 ) ) ).
thf(tcon_VEBT__BuildupMemImp_OVEBTi___Nat_Osize_1103,axiom,
size @ vEBT_VEBTi ).
% Helper facts (4)
thf(help_If_3_1_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_T,axiom,
! [A: $tType,X: A,Y: A] :
( ( if @ A @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_T,axiom,
! [A: $tType,X: A,Y: A] :
( ( if @ A @ $true @ X @ Y )
= X ) ).
thf(help_fChoice_1_1_T,axiom,
! [A: $tType,P: A > $o] :
( ( P @ ( fChoice @ A @ P ) )
= ( ? [X8: A] : ( P @ X8 ) ) ) ).
% Conjectures (2)
thf(conj_0,hypothesis,
! [X6: nat] :
( ( member @ nat @ X6 @ ( set2 @ nat @ xs ) )
=> ( ord_less @ nat @ X6 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ n ) ) ) ).
thf(conj_1,conjecture,
( hoare_hoare_triple @ vEBT_VEBTi @ ( vEBT_Intf_vebt_assn @ n @ s @ t )
@ ( vEBT_Example_mfold @ nat @ vEBT_VEBTi
@ ^ [X2: nat,S5: vEBT_VEBTi] : ( vEBT_vebt_inserti @ S5 @ X2 )
@ xs
@ t )
@ ( vEBT_Intf_vebt_assn @ n @ ( sup_sup @ ( set @ nat ) @ s @ ( set2 @ nat @ xs ) ) ) ) ).
%------------------------------------------------------------------------------