TPTP Problem File: ITP234^2.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : ITP234^2 : TPTP v8.2.0. Released v8.0.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer problem VEBT_InsertCorrectness 00806_052000
% 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 : 0067_VEBT_InsertCorrectness_00806_052000 [Des22]
% Status : Theorem
% Rating : 0.67 v8.2.0, 0.33 v8.1.0
% Syntax : Number of formulae : 9528 (2680 unt; 704 typ; 0 def)
% Number of atoms : 29418 (9760 equ; 5 cnn)
% Maximal formula atoms : 71 ( 3 avg)
% Number of connectives : 175763 (2631 ~; 383 |;2471 &;156480 @)
% ( 0 <=>;13798 =>; 0 <=; 0 <~>)
% Maximal formula depth : 40 ( 8 avg)
% Number of types : 10 ( 9 usr)
% Number of type conns : 5216 (5216 >; 0 *; 0 +; 0 <<)
% Number of symbols : 699 ( 695 usr; 20 con; 0-9 aty)
% Number of variables : 32182 (2823 ^;27673 !; 964 ?;32182 :)
% ( 722 !>; 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
% 2022-02-17 21:34:24.347
%------------------------------------------------------------------------------
% Could-be-implicit typings (16)
thf(ty_t_VEBT__Definitions_OVEBT,type,
vEBT_VEBT: $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_Extended__Nat_Oenat,type,
extended_enat: $tType ).
thf(ty_t_Complex_Ocomplex,type,
complex: $tType ).
thf(ty_t_Sum__Type_Osum,type,
sum_sum: $tType > $tType > $tType ).
thf(ty_t_Option_Ooption,type,
option: $tType > $tType ).
thf(ty_t_Filter_Ofilter,type,
filter: $tType > $tType ).
thf(ty_t_String_Ochar,type,
char: $tType ).
thf(ty_t_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_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 (688)
thf(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : $o ).
thf(sy_cl_Nat_Osize,type,
size:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Odvd,type,
dvd:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oone,type,
one:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oidom,type,
idom:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oring,type,
ring:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oplus,type,
plus:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ozero,type,
zero:
!>[A: $tType] : $o ).
thf(sy_cl_Num_Onumeral,type,
numeral:
!>[A: $tType] : $o ).
thf(sy_cl_Power_Opower,type,
power:
!>[A: $tType] : $o ).
thf(sy_cl_Fields_Ofield,type,
field:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oabs__if,type,
abs_if:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oring__1,type,
ring_1:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ouminus,type,
uminus:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_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_GCD_Osemiring__Gcd,type,
semiring_Gcd:
!>[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_Finite__Set_Ofinite,type,
finite_finite:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Omonoid__add,type,
monoid_add:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Ocomm__ring__1,type,
comm_ring_1:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oidom__divide,type,
idom_divide:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oidom__modulo,type,
idom_modulo:
!>[A: $tType] : $o ).
thf(sy_cl_Transcendental_Oln,type,
ln:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Omonoid__mult,type,
monoid_mult:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Olinorder,type,
linorder:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Opreorder,type,
preorder:
!>[A: $tType] : $o ).
thf(sy_cl_Parity_Oring__parity,type,
ring_parity:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oidom__abs__sgn,type,
idom_abs_sgn:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oordered__ring,type,
ordered_ring:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Ozero__neq__one,type,
zero_neq_one:
!>[A: $tType] : $o ).
thf(sy_cl_Countable_Ocountable,type,
countable:
!>[A: $tType] : $o ).
thf(sy_cl_Enum_Ofinite__lattice,type,
finite_lattice:
!>[A: $tType] : $o ).
thf(sy_cl_Fields_Ofield__char__0,type,
field_char_0:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oab__group__add,type,
ab_group_add:
!>[A: $tType] : $o ).
thf(sy_cl_Nat_Osemiring__char__0,type,
semiring_char_0:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oorder__bot,type,
order_bot:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oorder__top,type,
order_top:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Owellorder,type,
wellorder:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Ocomm__semiring,type,
comm_semiring:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Ozero__less__one,type,
zero_less_one:
!>[A: $tType] : $o ).
thf(sy_cl_Fields_Odivision__ring,type,
division_ring:
!>[A: $tType] : $o ).
thf(sy_cl_Fields_Ofield__abs__sgn,type,
field_abs_sgn:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Osemigroup__add,type,
semigroup_add:
!>[A: $tType] : $o ).
thf(sy_cl_Num_Osemiring__numeral,type,
semiring_numeral:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemidom__divide,type,
semidom_divide:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemidom__modulo,type,
semidom_modulo:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Osemigroup__mult,type,
semigroup_mult:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Odense__order,type,
dense_order:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Ocomm__semiring__0,type,
comm_semiring_0:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Ocomm__semiring__1,type,
comm_semiring_1:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__idom,type,
linordered_idom:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__ring,type,
linordered_ring:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemiring__modulo,type,
semiring_modulo:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocomm__monoid__add,type,
comm_monoid_add:
!>[A: $tType] : $o ).
thf(sy_cl_Parity_Osemiring__parity,type,
semiring_parity:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oordered__ring__abs,type,
ordered_ring_abs:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oordered__semiring,type,
ordered_semiring:
!>[A: $tType] : $o ).
thf(sy_cl_Fields_Olinordered__field,type,
linordered_field:
!>[A: $tType] : $o ).
thf(sy_cl_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_Odistrib__lattice,type,
distrib_lattice:
!>[A: $tType] : $o ).
thf(sy_cl_Lattices_Osemilattice__inf,type,
semilattice_inf:
!>[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_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_Complete__Partial__Order_Occpo,type,
comple9053668089753744459l_ccpo:
!>[A: $tType] : $o ).
thf(sy_cl_Enum_Ofinite__distrib__lattice,type,
finite8700451911770168679attice:
!>[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_Topological__Spaces_Ot3__space,type,
topological_t3_space:
!>[A: $tType] : $o ).
thf(sy_cl_Topological__Spaces_Ot4__space,type,
topological_t4_space:
!>[A: $tType] : $o ).
thf(sy_cl_Bit__Operations_Osemiring__bits,type,
bit_semiring_bits:
!>[A: $tType] : $o ).
thf(sy_cl_Lattices_Obounded__lattice__bot,type,
bounded_lattice_bot:
!>[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_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_Complete__Lattices_Ocomplete__lattice,type,
comple6319245703460814977attice:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Olinordered__ab__semigroup__add,type,
linord4140545234300271783up_add:
!>[A: $tType] : $o ).
thf(sy_cl_Limits_Otopological__comm__monoid__add,type,
topolo5987344860129210374id_add:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Ounbounded__dense__linorder,type,
unboun7993243217541854897norder:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__semiring__1__strict,type,
linord715952674999750819strict:
!>[A: $tType] : $o ).
thf(sy_cl_Archimedean__Field_Oarchimedean__field,type,
archim462609752435547400_field:
!>[A: $tType] : $o ).
thf(sy_cl_Complete__Lattices_Ocomplete__linorder,type,
comple5582772986160207858norder:
!>[A: $tType] : $o ).
thf(sy_cl_Real__Vector__Spaces_Oreal__div__algebra,type,
real_V5047593784448816457lgebra:
!>[A: $tType] : $o ).
thf(sy_cl_Lattices_Obounded__semilattice__inf__top,type,
bounde4346867609351753570nf_top:
!>[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_Complete__Lattices_Ocomplete__distrib__lattice,type,
comple592849572758109894attice:
!>[A: $tType] : $o ).
thf(sy_cl_Real__Vector__Spaces_Oreal__normed__div__algebra,type,
real_V8999393235501362500lgebra:
!>[A: $tType] : $o ).
thf(sy_cl_Topological__Spaces_Ofirst__countable__topology,type,
topolo3112930676232923870pology:
!>[A: $tType] : $o ).
thf(sy_cl_Euclidean__Division_Oeuclidean__semiring__cancel,type,
euclid4440199948858584721cancel:
!>[A: $tType] : $o ).
thf(sy_cl_Euclidean__Division_Ounique__euclidean__semiring,type,
euclid3128863361964157862miring:
!>[A: $tType] : $o ).
thf(sy_cl_Topological__Spaces_Olinear__continuum__topology,type,
topolo8458572112393995274pology:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__ab__semigroup__monoid__add__imp__le,type,
ordere1937475149494474687imp_le:
!>[A: $tType] : $o ).
thf(sy_cl_Conditionally__Complete__Lattices_Olinear__continuum,type,
condit5016429287641298734tinuum:
!>[A: $tType] : $o ).
thf(sy_cl_Euclidean__Division_Ounique__euclidean__semiring__with__nat,type,
euclid5411537665997757685th_nat:
!>[A: $tType] : $o ).
thf(sy_cl_Countable__Complete__Lattices_Ocountable__complete__lattice,type,
counta3822494911875563373attice:
!>[A: $tType] : $o ).
thf(sy_cl_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct,type,
semiri1453513574482234551roduct:
!>[A: $tType] : $o ).
thf(sy_cl_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations,type,
bit_un5681908812861735899ations:
!>[A: $tType] : $o ).
thf(sy_cl_Conditionally__Complete__Lattices_Oconditionally__complete__lattice,type,
condit1219197933456340205attice:
!>[A: $tType] : $o ).
thf(sy_cl_Countable__Complete__Lattices_Ocountable__complete__distrib__lattice,type,
counta4013691401010221786attice:
!>[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_BNF__Cardinal__Order__Relation_Ocard__of,type,
bNF_Ca6860139660246222851ard_of:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) ) ).
thf(sy_c_BNF__Cardinal__Order__Relation_Ocard__order__on,type,
bNF_Ca8970107618336181345der_on:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_BNF__Cardinal__Order__Relation_OnatLeq,type,
bNF_Ca8665028551170535155natLeq: set @ ( product_prod @ nat @ nat ) ).
thf(sy_c_BNF__Cardinal__Order__Relation_OnatLess,type,
bNF_Ca8459412986667044542atLess: set @ ( product_prod @ nat @ nat ) ).
thf(sy_c_BNF__Cardinal__Order__Relation_OrelChain,type,
bNF_Ca3754400796208372196lChain:
!>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( A > B ) > $o ) ).
thf(sy_c_BNF__Def_OGr,type,
bNF_Gr:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B ) > ( set @ ( product_prod @ A @ B ) ) ) ).
thf(sy_c_BNF__Def_Orel__fun,type,
bNF_rel_fun:
!>[A: $tType,C: $tType,B: $tType,D: $tType] : ( ( A > C > $o ) > ( B > D > $o ) > ( A > B ) > ( C > D ) > $o ) ).
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_BNF__Wellorder__Constructions_Obsqr,type,
bNF_Wellorder_bsqr:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( product_prod @ A @ A ) @ ( product_prod @ A @ A ) ) ) ) ).
thf(sy_c_BNF__Wellorder__Constructions_OofilterIncl,type,
bNF_We413866401316099525erIncl:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) ) ).
thf(sy_c_BNF__Wellorder__Constructions_OordLeq,type,
bNF_Wellorder_ordLeq:
!>[A: $tType,A2: $tType] : ( set @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A2 @ A2 ) ) ) ) ).
thf(sy_c_BNF__Wellorder__Constructions_OordLess,type,
bNF_We4044943003108391690rdLess:
!>[A: $tType,A2: $tType] : ( set @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A2 @ A2 ) ) ) ) ).
thf(sy_c_BNF__Wellorder__Constructions_Oord__to__filter,type,
bNF_We8469521843155493636filter:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ A @ A ) ) > ( set @ A ) ) ).
thf(sy_c_BNF__Wellorder__Embedding_Ocompat,type,
bNF_Wellorder_compat:
!>[A: $tType,A2: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ A2 @ A2 ) ) > ( A > A2 ) > $o ) ).
thf(sy_c_BNF__Wellorder__Embedding_Oembed,type,
bNF_Wellorder_embed:
!>[A: $tType,A2: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ A2 @ A2 ) ) > ( A > A2 ) > $o ) ).
thf(sy_c_BNF__Wellorder__Embedding_OembedS,type,
bNF_Wellorder_embedS:
!>[A: $tType,A2: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ A2 @ A2 ) ) > ( A > A2 ) > $o ) ).
thf(sy_c_BNF__Wellorder__Embedding_Oiso,type,
bNF_Wellorder_iso:
!>[A: $tType,A2: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ A2 @ A2 ) ) > ( A > A2 ) > $o ) ).
thf(sy_c_BNF__Wellorder__Relation_Owo__rel,type,
bNF_Wellorder_wo_rel:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_BNF__Wellorder__Relation_Owo__rel_OisMinim,type,
bNF_We4791949203932849705sMinim:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ A ) > A > $o ) ).
thf(sy_c_BNF__Wellorder__Relation_Owo__rel_Omax2,type,
bNF_We1388413361240627857o_max2:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > A > A > A ) ).
thf(sy_c_BNF__Wellorder__Relation_Owo__rel_Ominim,type,
bNF_We6954850376910717587_minim:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ A ) > A ) ).
thf(sy_c_Basic__BNF__LFPs_Oprod_Osize__prod,type,
basic_BNF_size_prod:
!>[A: $tType,B: $tType] : ( ( A > nat ) > ( B > nat ) > ( product_prod @ A @ B ) > nat ) ).
thf(sy_c_Binomial_Obinomial,type,
binomial: nat > nat > nat ).
thf(sy_c_Binomial_Ogbinomial,type,
gbinomial:
!>[A: $tType] : ( A > nat > A ) ).
thf(sy_c_Bit__Operations_Oand__int__rel,type,
bit_and_int_rel: ( product_prod @ int @ int ) > ( product_prod @ int @ int ) > $o ).
thf(sy_c_Bit__Operations_Oconcat__bit,type,
bit_concat_bit: nat > int > int > int ).
thf(sy_c_Bit__Operations_Oring__bit__operations__class_Onot,type,
bit_ri4277139882892585799ns_not:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Bit__Operations_Oring__bit__operations__class_Osigned__take__bit,type,
bit_ri4674362597316999326ke_bit:
!>[A: $tType] : ( nat > A > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oand,type,
bit_se5824344872417868541ns_and:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Odrop__bit,type,
bit_se4197421643247451524op_bit:
!>[A: $tType] : ( nat > A > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oflip__bit,type,
bit_se8732182000553998342ip_bit:
!>[A: $tType] : ( nat > A > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Omask,type,
bit_se2239418461657761734s_mask:
!>[A: $tType] : ( nat > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oor,type,
bit_se1065995026697491101ons_or:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Opush__bit,type,
bit_se4730199178511100633sh_bit:
!>[A: $tType] : ( nat > A > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oset__bit,type,
bit_se5668285175392031749et_bit:
!>[A: $tType] : ( nat > A > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Otake__bit,type,
bit_se2584673776208193580ke_bit:
!>[A: $tType] : ( nat > A > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Ounset__bit,type,
bit_se2638667681897837118et_bit:
!>[A: $tType] : ( nat > A > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oxor,type,
bit_se5824344971392196577ns_xor:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bits__class_Obit,type,
bit_se5641148757651400278ts_bit:
!>[A: $tType] : ( A > nat > $o ) ).
thf(sy_c_Bit__Operations_Osemiring__bits__class_Opossible__bit,type,
bit_se6407376104438227557le_bit:
!>[A: $tType] : ( ( itself @ A ) > nat > $o ) ).
thf(sy_c_Bit__Operations_Otake__bit__num,type,
bit_take_bit_num: nat > num > ( option @ num ) ).
thf(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations__class_Oand__num,type,
bit_un7362597486090784418nd_num: num > num > ( option @ num ) ).
thf(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations__class_Oxor__num,type,
bit_un2480387367778600638or_num: num > num > ( option @ num ) ).
thf(sy_c_Boolean__Algebras_Oabstract__boolean__algebra,type,
boolea2506097494486148201lgebra:
!>[A: $tType] : ( ( A > A > A ) > ( A > A > A ) > ( A > A ) > A > A > $o ) ).
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_Complete__Lattices_OInf__class_OInf,type,
complete_Inf_Inf:
!>[A: $tType] : ( ( set @ A ) > A ) ).
thf(sy_c_Complete__Lattices_OSup__class_OSup,type,
complete_Sup_Sup:
!>[A: $tType] : ( ( set @ A ) > A ) ).
thf(sy_c_Complete__Partial__Order_Occpo_Oadmissible,type,
comple1908693960933563346ssible:
!>[A: $tType] : ( ( ( set @ A ) > A ) > ( A > A > $o ) > ( A > $o ) > $o ) ).
thf(sy_c_Complete__Partial__Order_Occpo__class_Ofixp,type,
comple115746919287870866o_fixp:
!>[A: $tType] : ( ( A > A ) > A ) ).
thf(sy_c_Complete__Partial__Order_Occpo__class_Oiteratesp,type,
comple7512665784863727008ratesp:
!>[A: $tType] : ( ( A > A ) > A > $o ) ).
thf(sy_c_Complete__Partial__Order_Ochain,type,
comple1602240252501008431_chain:
!>[A: $tType] : ( ( A > A > $o ) > ( set @ A ) > $o ) ).
thf(sy_c_Complete__Partial__Order_Omonotone,type,
comple7038119648293358887notone:
!>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( B > B > $o ) > ( A > B ) > $o ) ).
thf(sy_c_Complex_OArg,type,
arg: complex > real ).
thf(sy_c_Complex_Ocis,type,
cis: real > complex ).
thf(sy_c_Conditionally__Complete__Lattices_Opreorder__class_Obdd__above,type,
condit941137186595557371_above:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_Conditionally__Complete__Lattices_Opreorder__class_Obdd__below,type,
condit1013018076250108175_below:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_Countable__Set_Ocountable,type,
countable_countable:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_Countable__Set_Ofrom__nat__into,type,
counta4804993851260445106t_into:
!>[A: $tType] : ( ( set @ A ) > nat > A ) ).
thf(sy_c_Countable__Set_Oto__nat__on,type,
countable_to_nat_on:
!>[A: $tType] : ( ( set @ A ) > A > nat ) ).
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_Deriv_Ohas__vector__derivative,type,
has_ve8173657378732805170vative:
!>[B: $tType] : ( ( real > B ) > B > ( filter @ real ) > $o ) ).
thf(sy_c_Divides_Odivmod__nat,type,
divmod_nat: nat > nat > ( product_prod @ nat @ nat ) ).
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_Equiv__Relations_Ocongruent,type,
equiv_congruent:
!>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( A > B ) > $o ) ).
thf(sy_c_Equiv__Relations_Ocongruent2,type,
equiv_congruent2:
!>[A: $tType,B: $tType,C: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ B @ B ) ) > ( A > B > C ) > $o ) ).
thf(sy_c_Equiv__Relations_Oequiv,type,
equiv_equiv:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_Equiv__Relations_Oproj,type,
equiv_proj:
!>[B: $tType,A: $tType] : ( ( set @ ( product_prod @ B @ A ) ) > B > ( set @ A ) ) ).
thf(sy_c_Equiv__Relations_Oquotient,type,
equiv_quotient:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > ( set @ ( set @ A ) ) ) ).
thf(sy_c_Euclidean__Division_Oeuclidean__semiring__class_Oeuclidean__size,type,
euclid6346220572633701492n_size:
!>[A: $tType] : ( A > nat ) ).
thf(sy_c_Euclidean__Division_Ounique__euclidean__semiring__class_Odivision__segment,type,
euclid7384307370059645450egment:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Extended__Nat_OeSuc,type,
extended_eSuc: extended_enat > extended_enat ).
thf(sy_c_Extended__Nat_Oenat,type,
extended_enat2: nat > extended_enat ).
thf(sy_c_Extended__Nat_Oenat_Ocase__enat,type,
extended_case_enat:
!>[T: $tType] : ( ( nat > T ) > T > extended_enat > T ) ).
thf(sy_c_Extended__Nat_Oinfinity__class_Oinfinity,type,
extend4730790105801354508finity:
!>[A: $tType] : 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_Ofiltercomap,type,
filtercomap:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( filter @ B ) > ( filter @ A ) ) ).
thf(sy_c_Filter_Ofilterlim,type,
filterlim:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( filter @ B ) > ( filter @ A ) > $o ) ).
thf(sy_c_Filter_Ofiltermap,type,
filtermap:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( filter @ A ) > ( filter @ B ) ) ).
thf(sy_c_Filter_Ofinite__subsets__at__top,type,
finite5375528669736107172at_top:
!>[A: $tType] : ( ( set @ A ) > ( filter @ ( set @ A ) ) ) ).
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_Filter_Oprincipal,type,
principal:
!>[A: $tType] : ( ( set @ A ) > ( filter @ A ) ) ).
thf(sy_c_Filter_Oprod__filter,type,
prod_filter:
!>[A: $tType,B: $tType] : ( ( filter @ A ) > ( filter @ B ) > ( filter @ ( product_prod @ A @ B ) ) ) ).
thf(sy_c_Finite__Set_OFpow,type,
finite_Fpow:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( set @ A ) ) ) ).
thf(sy_c_Finite__Set_Ocard,type,
finite_card:
!>[B: $tType] : ( ( set @ B ) > nat ) ).
thf(sy_c_Finite__Set_Ocomp__fun__commute,type,
finite6289374366891150609ommute:
!>[A: $tType,B: $tType] : ( ( A > B > B ) > $o ) ).
thf(sy_c_Finite__Set_Ocomp__fun__commute__on,type,
finite4664212375090638736ute_on:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B > B ) > $o ) ).
thf(sy_c_Finite__Set_Ocomp__fun__idem__on,type,
finite673082921795544331dem_on:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B > B ) > $o ) ).
thf(sy_c_Finite__Set_Ocomp__fun__idem__on__axioms,type,
finite4980608107308702382axioms:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B > B ) > $o ) ).
thf(sy_c_Finite__Set_Ofinite,type,
finite_finite2:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_Finite__Set_Ofold,type,
finite_fold:
!>[A: $tType,B: $tType] : ( ( A > B > B ) > B > ( set @ A ) > B ) ).
thf(sy_c_Finite__Set_Ofold__graph,type,
finite_fold_graph:
!>[A: $tType,B: $tType] : ( ( A > B > B ) > B > ( set @ A ) > B > $o ) ).
thf(sy_c_Finite__Set_Ofolding__idem,type,
finite_folding_idem:
!>[A: $tType,B: $tType] : ( ( A > B > B ) > $o ) ).
thf(sy_c_Finite__Set_Ofolding__idem__on,type,
finite1890593828518410140dem_on:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B > B ) > $o ) ).
thf(sy_c_Finite__Set_Ofolding__idem__on__axioms,type,
finite6916993218817215295axioms:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B > B ) > $o ) ).
thf(sy_c_Finite__Set_Ofolding__on,type,
finite_folding_on:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B > B ) > $o ) ).
thf(sy_c_Finite__Set_Ofolding__on_OF,type,
finite_folding_F:
!>[A: $tType,B: $tType] : ( ( A > B > B ) > B > ( set @ A ) > B ) ).
thf(sy_c_Fun_Obij__betw,type,
bij_betw:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > ( set @ B ) > $o ) ).
thf(sy_c_Fun_Ocomp,type,
comp:
!>[B: $tType,C: $tType,A: $tType] : ( ( B > C ) > ( A > B ) > A > C ) ).
thf(sy_c_Fun_Ofun__upd,type,
fun_upd:
!>[A: $tType,B: $tType] : ( ( A > B ) > A > B > A > B ) ).
thf(sy_c_Fun_Oid,type,
id:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Fun_Oinj__on,type,
inj_on:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > $o ) ).
thf(sy_c_Fun_Omap__fun,type,
map_fun:
!>[C: $tType,A: $tType,B: $tType,D: $tType] : ( ( C > A ) > ( B > D ) > ( A > B ) > C > D ) ).
thf(sy_c_Fun_Ostrict__mono__on,type,
strict_mono_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_Fun__Def_Ois__measure,type,
fun_is_measure:
!>[A: $tType] : ( ( A > nat ) > $o ) ).
thf(sy_c_Fun__Def_Opair__leq,type,
fun_pair_leq: set @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) ).
thf(sy_c_Fun__Def_Opair__less,type,
fun_pair_less: set @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) ).
thf(sy_c_GCD_OGcd__class_OGcd,type,
gcd_Gcd:
!>[A: $tType] : ( ( set @ A ) > A ) ).
thf(sy_c_GCD_Obezw,type,
bezw: nat > nat > ( product_prod @ int @ int ) ).
thf(sy_c_GCD_Obezw__rel,type,
bezw_rel: ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) > $o ).
thf(sy_c_GCD_Osemiring__1__class_Osemiring__char,type,
semiri4206861660011772517g_char:
!>[A: $tType] : ( ( itself @ A ) > nat ) ).
thf(sy_c_GCD_Osemiring__gcd__class_OGcd__fin,type,
semiring_gcd_Gcd_fin:
!>[A: $tType] : ( ( set @ A ) > 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__Big_Ocomm__monoid__mult__class_Oprod_H,type,
groups1962203154675924110t_prod:
!>[C: $tType,A: $tType] : ( ( C > A ) > ( set @ C ) > 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_Groups__List_Omonoid__add__class_Osum__list,type,
groups8242544230860333062m_list:
!>[A: $tType] : ( ( list @ A ) > A ) ).
thf(sy_c_Groups__List_Omonoid__mult__class_Oprod__list,type,
groups5270119922927024881d_list:
!>[A: $tType] : ( ( list @ A ) > 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_If,type,
if:
!>[A: $tType] : ( $o > A > A > A ) ).
thf(sy_c_Inductive_Ocomplete__lattice__class_Olfp,type,
complete_lattice_lfp:
!>[A: $tType] : ( ( A > A ) > A ) ).
thf(sy_c_Infinite__Set_Owellorder__class_Oenumerate,type,
infini527867602293511546merate:
!>[A: $tType] : ( ( set @ A ) > nat > A ) ).
thf(sy_c_Int_OAbs__Integ,type,
abs_Integ: ( product_prod @ nat @ nat ) > int ).
thf(sy_c_Int_ORep__Integ,type,
rep_Integ: int > ( product_prod @ nat @ nat ) ).
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_Ointrel,type,
intrel: ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) > $o ).
thf(sy_c_Int_Onat,type,
nat2: int > nat ).
thf(sy_c_Int_Opcr__int,type,
pcr_int: ( product_prod @ nat @ nat ) > int > $o ).
thf(sy_c_Int_Opower__int,type,
power_int:
!>[A: $tType] : ( A > int > A ) ).
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_Oinf__class_Oinf,type,
inf_inf:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Lattices_Osemilattice__neutr__order,type,
semila1105856199041335345_order:
!>[A: $tType] : ( ( A > A > A ) > A > ( A > A > $o ) > ( A > A > $o ) > $o ) ).
thf(sy_c_Lattices_Osup__class_Osup,type,
sup_sup:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Lattices__Big_Olinorder_OMax,type,
lattices_Max:
!>[A: $tType] : ( ( A > A > $o ) > ( set @ A ) > A ) ).
thf(sy_c_Lattices__Big_Olinorder_OMin,type,
lattices_Min:
!>[A: $tType] : ( ( A > A > $o ) > ( set @ A ) > A ) ).
thf(sy_c_Lattices__Big_Olinorder__class_OMax,type,
lattic643756798349783984er_Max:
!>[A: $tType] : ( ( set @ A ) > A ) ).
thf(sy_c_Lattices__Big_Olinorder__class_OMin,type,
lattic643756798350308766er_Min:
!>[A: $tType] : ( ( set @ A ) > A ) ).
thf(sy_c_Lattices__Big_Oord__class_Oarg__min,type,
lattices_ord_arg_min:
!>[B: $tType,A: $tType] : ( ( B > A ) > ( B > $o ) > B ) ).
thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on,type,
lattic7623131987881927897min_on:
!>[B: $tType,A: $tType] : ( ( B > A ) > ( set @ B ) > B ) ).
thf(sy_c_Lattices__Big_Oord__class_Ois__arg__min,type,
lattic501386751177426532rg_min:
!>[B: $tType,A: $tType] : ( ( B > A ) > ( B > $o ) > B > $o ) ).
thf(sy_c_Lattices__Big_Osemilattice__inf__class_OInf__fin,type,
lattic7752659483105999362nf_fin:
!>[A: $tType] : ( ( set @ A ) > A ) ).
thf(sy_c_Lattices__Big_Osemilattice__order__set,type,
lattic4895041142388067077er_set:
!>[A: $tType] : ( ( A > A > A ) > ( A > A > $o ) > ( A > A > $o ) > $o ) ).
thf(sy_c_Lattices__Big_Osemilattice__sup__class_OSup__fin,type,
lattic5882676163264333800up_fin:
!>[A: $tType] : ( ( set @ A ) > A ) ).
thf(sy_c_Limits_OBfun,type,
bfun:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( filter @ A ) > $o ) ).
thf(sy_c_Limits_Oat__infinity,type,
at_infinity:
!>[A: $tType] : ( filter @ A ) ).
thf(sy_c_List_OBleast,type,
bleast:
!>[A: $tType] : ( ( set @ A ) > ( A > $o ) > A ) ).
thf(sy_c_List_Oabort__Bleast,type,
abort_Bleast:
!>[A: $tType] : ( ( set @ A ) > ( A > $o ) > A ) ).
thf(sy_c_List_Oappend,type,
append:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Oarg__min__list,type,
arg_min_list:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( list @ A ) > A ) ).
thf(sy_c_List_Oarg__min__list__rel,type,
arg_min_list_rel:
!>[A: $tType,B: $tType] : ( ( product_prod @ ( A > B ) @ ( list @ A ) ) > ( product_prod @ ( A > B ) @ ( list @ A ) ) > $o ) ).
thf(sy_c_List_Obind,type,
bind:
!>[A: $tType,B: $tType] : ( ( list @ A ) > ( A > ( list @ B ) ) > ( list @ B ) ) ).
thf(sy_c_List_Obutlast,type,
butlast:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Ocan__select,type,
can_select:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) > $o ) ).
thf(sy_c_List_Oconcat,type,
concat:
!>[A: $tType] : ( ( list @ ( list @ A ) ) > ( list @ A ) ) ).
thf(sy_c_List_Ocoset,type,
coset:
!>[A: $tType] : ( ( list @ A ) > ( set @ 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_Odistinct__adj,type,
distinct_adj:
!>[A: $tType] : ( ( list @ A ) > $o ) ).
thf(sy_c_List_Odrop,type,
drop:
!>[A: $tType] : ( nat > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_OdropWhile,type,
dropWhile:
!>[A: $tType] : ( ( A > $o ) > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Oenumerate,type,
enumerate:
!>[A: $tType] : ( nat > ( list @ A ) > ( list @ ( product_prod @ nat @ A ) ) ) ).
thf(sy_c_List_Oextract,type,
extract:
!>[A: $tType] : ( ( A > $o ) > ( list @ A ) > ( option @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) ) ) ).
thf(sy_c_List_Ofilter,type,
filter2:
!>[A: $tType] : ( ( A > $o ) > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Ofind,type,
find:
!>[A: $tType] : ( ( A > $o ) > ( list @ A ) > ( option @ A ) ) ).
thf(sy_c_List_Ofold,type,
fold:
!>[A: $tType,B: $tType] : ( ( A > B > B ) > ( list @ A ) > B > B ) ).
thf(sy_c_List_Ofolding__insort__key,type,
folding_insort_key:
!>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( A > A > $o ) > ( set @ B ) > ( B > A ) > $o ) ).
thf(sy_c_List_Ofoldl,type,
foldl:
!>[B: $tType,A: $tType] : ( ( B > A > B ) > B > ( list @ A ) > B ) ).
thf(sy_c_List_Ofoldr,type,
foldr:
!>[A: $tType,B: $tType] : ( ( A > B > B ) > ( list @ A ) > B > B ) ).
thf(sy_c_List_Ogen__length,type,
gen_length:
!>[A: $tType] : ( nat > ( list @ A ) > nat ) ).
thf(sy_c_List_Oinsert,type,
insert:
!>[A: $tType] : ( A > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Olast,type,
last:
!>[A: $tType] : ( ( list @ A ) > A ) ).
thf(sy_c_List_Olenlex,type,
lenlex:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) ) ).
thf(sy_c_List_Olex,type,
lex:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) ) ).
thf(sy_c_List_Olexn,type,
lexn:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > nat > ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) ) ).
thf(sy_c_List_Olexord,type,
lexord:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) ) ).
thf(sy_c_List_Olexordp,type,
lexordp:
!>[A: $tType] : ( ( A > A > $o ) > ( list @ A ) > ( list @ A ) > $o ) ).
thf(sy_c_List_Olinorder_Oinsort__key,type,
insort_key:
!>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( B > A ) > B > ( list @ B ) > ( list @ B ) ) ).
thf(sy_c_List_Olinorder_Osorted__key__list__of__set,type,
sorted8670434370408473282of_set:
!>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( B > A ) > ( set @ B ) > ( list @ B ) ) ).
thf(sy_c_List_Olinorder__class_Oinsort__insert__key,type,
linord329482645794927042rt_key:
!>[B: $tType,A: $tType] : ( ( B > A ) > B > ( list @ B ) > ( list @ B ) ) ).
thf(sy_c_List_Olinorder__class_Oinsort__key,type,
linorder_insort_key:
!>[B: $tType,A: $tType] : ( ( B > A ) > B > ( list @ B ) > ( list @ B ) ) ).
thf(sy_c_List_Olinorder__class_Osort__key,type,
linorder_sort_key:
!>[B: $tType,A: $tType] : ( ( B > A ) > ( list @ B ) > ( list @ B ) ) ).
thf(sy_c_List_Olinorder__class_Osorted__key__list__of__set,type,
linord144544945434240204of_set:
!>[B: $tType,A: $tType] : ( ( B > A ) > ( set @ B ) > ( list @ B ) ) ).
thf(sy_c_List_Olinorder__class_Osorted__list__of__set,type,
linord4507533701916653071of_set:
!>[A: $tType] : ( ( set @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Olinorder__class_Ostable__sort__key,type,
linord3483353639454293061rt_key:
!>[B: $tType,A: $tType] : ( ( ( B > A ) > ( list @ B ) > ( list @ B ) ) > $o ) ).
thf(sy_c_List_Olist_OCons,type,
cons:
!>[A: $tType] : ( A > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Olist_ONil,type,
nil:
!>[A: $tType] : ( list @ A ) ).
thf(sy_c_List_Olist_Ocase__list,type,
case_list:
!>[B: $tType,A: $tType] : ( B > ( A > ( list @ A ) > B ) > ( list @ A ) > B ) ).
thf(sy_c_List_Olist_Ohd,type,
hd:
!>[A: $tType] : ( ( list @ A ) > A ) ).
thf(sy_c_List_Olist_Olist__all,type,
list_all:
!>[A: $tType] : ( ( A > $o ) > ( list @ A ) > $o ) ).
thf(sy_c_List_Olist_Olist__all2,type,
list_all2:
!>[A: $tType,B: $tType] : ( ( A > B > $o ) > ( list @ A ) > ( list @ B ) > $o ) ).
thf(sy_c_List_Olist_Omap,type,
map:
!>[A: $tType,Aa: $tType] : ( ( A > Aa ) > ( list @ A ) > ( list @ Aa ) ) ).
thf(sy_c_List_Olist_Orec__list,type,
rec_list:
!>[C: $tType,A: $tType] : ( C > ( A > ( list @ A ) > C > C ) > ( list @ A ) > C ) ).
thf(sy_c_List_Olist_Oset,type,
set2:
!>[A: $tType] : ( ( list @ A ) > ( set @ A ) ) ).
thf(sy_c_List_Olist_Osize__list,type,
size_list:
!>[A: $tType] : ( ( A > nat ) > ( list @ A ) > nat ) ).
thf(sy_c_List_Olist_Otl,type,
tl:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Olist__ex,type,
list_ex:
!>[A: $tType] : ( ( A > $o ) > ( list @ A ) > $o ) ).
thf(sy_c_List_Olist__ex1,type,
list_ex1:
!>[A: $tType] : ( ( A > $o ) > ( list @ A ) > $o ) ).
thf(sy_c_List_Olist__update,type,
list_update:
!>[A: $tType] : ( ( list @ A ) > nat > A > ( list @ A ) ) ).
thf(sy_c_List_Olistrel,type,
listrel:
!>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > ( set @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) ) ).
thf(sy_c_List_Olistrel1,type,
listrel1:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) ) ).
thf(sy_c_List_Olistrel1p,type,
listrel1p:
!>[A: $tType] : ( ( A > A > $o ) > ( list @ A ) > ( list @ A ) > $o ) ).
thf(sy_c_List_Olistrelp,type,
listrelp:
!>[A: $tType,B: $tType] : ( ( A > B > $o ) > ( list @ A ) > ( list @ B ) > $o ) ).
thf(sy_c_List_Olists,type,
lists:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( list @ A ) ) ) ).
thf(sy_c_List_Olistset,type,
listset:
!>[A: $tType] : ( ( list @ ( set @ A ) ) > ( set @ ( list @ A ) ) ) ).
thf(sy_c_List_Omap__filter,type,
map_filter:
!>[A: $tType,B: $tType] : ( ( A > ( option @ B ) ) > ( list @ A ) > ( list @ B ) ) ).
thf(sy_c_List_Omap__project,type,
map_project:
!>[A: $tType,B: $tType] : ( ( A > ( option @ B ) ) > ( set @ A ) > ( set @ B ) ) ).
thf(sy_c_List_Omap__tailrec,type,
map_tailrec:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( list @ A ) > ( list @ B ) ) ).
thf(sy_c_List_Omap__tailrec__rev,type,
map_tailrec_rev:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( list @ A ) > ( list @ B ) > ( list @ B ) ) ).
thf(sy_c_List_Omaps,type,
maps:
!>[A: $tType,B: $tType] : ( ( A > ( list @ B ) ) > ( list @ A ) > ( list @ B ) ) ).
thf(sy_c_List_Omeasures,type,
measures:
!>[A: $tType] : ( ( list @ ( A > nat ) ) > ( set @ ( product_prod @ A @ A ) ) ) ).
thf(sy_c_List_Omin__list,type,
min_list:
!>[A: $tType] : ( ( list @ A ) > A ) ).
thf(sy_c_List_Omin__list__rel,type,
min_list_rel:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) > $o ) ).
thf(sy_c_List_On__lists,type,
n_lists:
!>[A: $tType] : ( nat > ( list @ A ) > ( list @ ( list @ A ) ) ) ).
thf(sy_c_List_Onth,type,
nth:
!>[A: $tType] : ( ( list @ A ) > nat > A ) ).
thf(sy_c_List_Onths,type,
nths:
!>[A: $tType] : ( ( list @ A ) > ( set @ nat ) > ( list @ A ) ) ).
thf(sy_c_List_Onull,type,
null:
!>[A: $tType] : ( ( list @ A ) > $o ) ).
thf(sy_c_List_Oord_Olexordp,type,
lexordp2:
!>[A: $tType] : ( ( A > A > $o ) > ( list @ A ) > ( list @ A ) > $o ) ).
thf(sy_c_List_Oord__class_Olexordp,type,
ord_lexordp:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) > $o ) ).
thf(sy_c_List_Opartition,type,
partition:
!>[A: $tType] : ( ( A > $o ) > ( list @ A ) > ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) ).
thf(sy_c_List_Oproduct,type,
product:
!>[A: $tType,B: $tType] : ( ( list @ A ) > ( list @ B ) > ( list @ ( product_prod @ A @ B ) ) ) ).
thf(sy_c_List_Oproduct__lists,type,
product_lists:
!>[A: $tType] : ( ( list @ ( list @ A ) ) > ( list @ ( list @ A ) ) ) ).
thf(sy_c_List_Oremdups,type,
remdups:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Oremdups__adj,type,
remdups_adj:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Oremdups__adj__rel,type,
remdups_adj_rel:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) > $o ) ).
thf(sy_c_List_Oremove1,type,
remove1:
!>[A: $tType] : ( A > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_OremoveAll,type,
removeAll:
!>[A: $tType] : ( A > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Oreplicate,type,
replicate:
!>[A: $tType] : ( nat > A > ( list @ A ) ) ).
thf(sy_c_List_Orev,type,
rev:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Orotate,type,
rotate:
!>[A: $tType] : ( nat > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Orotate1,type,
rotate1:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Oset__Cons,type,
set_Cons:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( list @ A ) ) > ( set @ ( list @ A ) ) ) ).
thf(sy_c_List_Oshuffles,type,
shuffles:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) > ( set @ ( list @ A ) ) ) ).
thf(sy_c_List_Oshuffles__rel,type,
shuffles_rel:
!>[A: $tType] : ( ( product_prod @ ( list @ A ) @ ( list @ A ) ) > ( product_prod @ ( list @ A ) @ ( list @ A ) ) > $o ) ).
thf(sy_c_List_Osorted__wrt,type,
sorted_wrt:
!>[A: $tType] : ( ( A > A > $o ) > ( list @ A ) > $o ) ).
thf(sy_c_List_Osorted__wrt__rel,type,
sorted_wrt_rel:
!>[A: $tType] : ( ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) > ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) > $o ) ).
thf(sy_c_List_Osplice,type,
splice:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Osplice__rel,type,
splice_rel:
!>[A: $tType] : ( ( product_prod @ ( list @ A ) @ ( list @ A ) ) > ( product_prod @ ( list @ A ) @ ( list @ A ) ) > $o ) ).
thf(sy_c_List_Osubseqs,type,
subseqs:
!>[A: $tType] : ( ( list @ A ) > ( list @ ( list @ A ) ) ) ).
thf(sy_c_List_Osuccessively,type,
successively:
!>[A: $tType] : ( ( A > A > $o ) > ( list @ A ) > $o ) ).
thf(sy_c_List_Osuccessively__rel,type,
successively_rel:
!>[A: $tType] : ( ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) > ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) > $o ) ).
thf(sy_c_List_Otake,type,
take:
!>[A: $tType] : ( nat > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_OtakeWhile,type,
takeWhile:
!>[A: $tType] : ( ( A > $o ) > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Otranspose,type,
transpose:
!>[A: $tType] : ( ( list @ ( list @ A ) ) > ( list @ ( list @ A ) ) ) ).
thf(sy_c_List_Otranspose__rel,type,
transpose_rel:
!>[A: $tType] : ( ( list @ ( list @ A ) ) > ( list @ ( list @ A ) ) > $o ) ).
thf(sy_c_List_Ounion,type,
union:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Oupt,type,
upt: nat > nat > ( list @ nat ) ).
thf(sy_c_List_Oupto,type,
upto: int > int > ( list @ int ) ).
thf(sy_c_List_Oupto__aux,type,
upto_aux: int > int > ( list @ int ) > ( list @ int ) ).
thf(sy_c_List_Oupto__rel,type,
upto_rel: ( product_prod @ int @ int ) > ( product_prod @ int @ int ) > $o ).
thf(sy_c_List_Ozip,type,
zip:
!>[A: $tType,B: $tType] : ( ( list @ A ) > ( list @ B ) > ( list @ ( product_prod @ A @ B ) ) ) ).
thf(sy_c_Map_Odom,type,
dom:
!>[A: $tType,B: $tType] : ( ( A > ( option @ B ) ) > ( set @ A ) ) ).
thf(sy_c_Map_Ograph,type,
graph:
!>[A: $tType,B: $tType] : ( ( A > ( option @ B ) ) > ( set @ ( product_prod @ A @ B ) ) ) ).
thf(sy_c_Map_Omap__add,type,
map_add:
!>[A: $tType,B: $tType] : ( ( A > ( option @ B ) ) > ( A > ( option @ B ) ) > A > ( option @ B ) ) ).
thf(sy_c_Map_Omap__of,type,
map_of:
!>[A: $tType,B: $tType] : ( ( list @ ( product_prod @ A @ B ) ) > A > ( option @ B ) ) ).
thf(sy_c_Map_Omap__upds,type,
map_upds:
!>[A: $tType,B: $tType] : ( ( A > ( option @ B ) ) > ( list @ A ) > ( list @ B ) > A > ( option @ B ) ) ).
thf(sy_c_Map_Oran,type,
ran:
!>[A: $tType,B: $tType] : ( ( A > ( option @ B ) ) > ( set @ B ) ) ).
thf(sy_c_Map_Orestrict__map,type,
restrict_map:
!>[A: $tType,B: $tType] : ( ( A > ( option @ B ) ) > ( set @ A ) > A > ( option @ B ) ) ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nat_Ocompow,type,
compow:
!>[A: $tType] : ( nat > A > A ) ).
thf(sy_c_Nat_Ofunpow,type,
funpow:
!>[A: $tType] : ( nat > ( A > A ) > A > A ) ).
thf(sy_c_Nat_Onat_Ocase__nat,type,
case_nat:
!>[A: $tType] : ( A > ( nat > A ) > nat > A ) ).
thf(sy_c_Nat_Onat_Opred,type,
pred: nat > nat ).
thf(sy_c_Nat_Oold_Onat_Orec__nat,type,
rec_nat:
!>[T: $tType] : ( T > ( nat > T > T ) > nat > T ) ).
thf(sy_c_Nat_Oold_Onat_Orec__set__nat,type,
rec_set_nat:
!>[T: $tType] : ( T > ( nat > T > T ) > nat > T > $o ) ).
thf(sy_c_Nat_Osemiring__1__class_ONats,type,
semiring_1_Nats:
!>[A: $tType] : ( set @ A ) ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat,type,
semiring_1_of_nat:
!>[A: $tType] : ( nat > A ) ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat__aux,type,
semiri8178284476397505188at_aux:
!>[A: $tType] : ( ( A > A ) > nat > A > A ) ).
thf(sy_c_Nat_Osize__class_Osize,type,
size_size:
!>[A: $tType] : ( A > nat ) ).
thf(sy_c_Nat__Bijection_Olist__decode,type,
nat_list_decode: nat > ( list @ nat ) ).
thf(sy_c_Nat__Bijection_Olist__decode__rel,type,
nat_list_decode_rel: nat > nat > $o ).
thf(sy_c_Nat__Bijection_Olist__encode,type,
nat_list_encode: ( list @ nat ) > nat ).
thf(sy_c_Nat__Bijection_Olist__encode__rel,type,
nat_list_encode_rel: ( list @ nat ) > ( list @ nat ) > $o ).
thf(sy_c_Nat__Bijection_Oprod__decode,type,
nat_prod_decode: nat > ( product_prod @ nat @ nat ) ).
thf(sy_c_Nat__Bijection_Oprod__decode__aux,type,
nat_prod_decode_aux: nat > nat > ( product_prod @ nat @ nat ) ).
thf(sy_c_Nat__Bijection_Oprod__decode__aux__rel,type,
nat_pr5047031295181774490ux_rel: ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) > $o ).
thf(sy_c_Nat__Bijection_Oprod__encode,type,
nat_prod_encode: ( product_prod @ nat @ nat ) > 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_Num_Oinc,type,
inc: num > num ).
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_Oneg__numeral__class_Osub,type,
neg_numeral_sub:
!>[A: $tType] : ( num > num > A ) ).
thf(sy_c_Num_Onum_OBit0,type,
bit0: num > num ).
thf(sy_c_Num_Onum_OBit1,type,
bit1: num > num ).
thf(sy_c_Num_Onum_OOne,type,
one2: num ).
thf(sy_c_Num_Onum_Osize__num,type,
size_num: num > nat ).
thf(sy_c_Num_Onum__of__nat,type,
num_of_nat: nat > num ).
thf(sy_c_Num_Onumeral__class_Onumeral,type,
numeral_numeral:
!>[A: $tType] : ( num > A ) ).
thf(sy_c_Num_Opow,type,
pow: num > num > num ).
thf(sy_c_Num_Opred__numeral,type,
pred_numeral: num > nat ).
thf(sy_c_Option_Ooption_ONone,type,
none:
!>[A: $tType] : ( option @ A ) ).
thf(sy_c_Option_Ooption_OSome,type,
some:
!>[A: $tType] : ( A > ( option @ A ) ) ).
thf(sy_c_Option_Ooption_Ocase__option,type,
case_option:
!>[B: $tType,A: $tType] : ( B > ( A > B ) > ( option @ A ) > B ) ).
thf(sy_c_Option_Ooption_Osize__option,type,
size_option:
!>[A: $tType] : ( ( A > nat ) > ( option @ A ) > nat ) ).
thf(sy_c_Option_Ooption_Othe,type,
the2:
!>[A: $tType] : ( ( option @ A ) > A ) ).
thf(sy_c_Order__Continuity_Ocountable__complete__lattice__class_Occlfp,type,
order_532582986084564980_cclfp:
!>[A: $tType] : ( ( A > A ) > A ) ).
thf(sy_c_Order__Continuity_Osup__continuous,type,
order_sup_continuous:
!>[A: $tType,B: $tType] : ( ( A > B ) > $o ) ).
thf(sy_c_Order__Relation_OaboveS,type,
order_aboveS:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > A > ( set @ A ) ) ).
thf(sy_c_Order__Relation_Olinear__order__on,type,
order_679001287576687338der_on:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_Order__Relation_Oofilter,type,
order_ofilter:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ A ) > $o ) ).
thf(sy_c_Order__Relation_Opartial__order__on,type,
order_7125193373082350890der_on:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_Order__Relation_Opreorder__on,type,
order_preorder_on:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_Order__Relation_Ounder,type,
order_under:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > A > ( set @ A ) ) ).
thf(sy_c_Order__Relation_OunderS,type,
order_underS:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > A > ( set @ A ) ) ).
thf(sy_c_Order__Relation_Owell__order__on,type,
order_well_order_on:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_Orderings_Obot__class_Obot,type,
bot_bot:
!>[A: $tType] : A ).
thf(sy_c_Orderings_Oord_OLeast,type,
least:
!>[A: $tType] : ( ( A > A > $o ) > ( A > $o ) > A ) ).
thf(sy_c_Orderings_Oord_Omax,type,
max:
!>[A: $tType] : ( ( A > A > $o ) > A > A > A ) ).
thf(sy_c_Orderings_Oord_Omin,type,
min:
!>[A: $tType] : ( ( A > A > $o ) > A > A > A ) ).
thf(sy_c_Orderings_Oord__class_OLeast,type,
ord_Least:
!>[A: $tType] : ( ( A > $o ) > A ) ).
thf(sy_c_Orderings_Oord__class_Oless,type,
ord_less:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Orderings_Oord__class_Oless__eq,type,
ord_less_eq:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Orderings_Oord__class_Omax,type,
ord_max:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Orderings_Oord__class_Omin,type,
ord_min:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Orderings_Oorder__class_OGreatest,type,
order_Greatest:
!>[A: $tType] : ( ( A > $o ) > A ) ).
thf(sy_c_Orderings_Oorder__class_Oantimono,type,
order_antimono:
!>[A: $tType,B: $tType] : ( ( A > B ) > $o ) ).
thf(sy_c_Orderings_Oorder__class_Omono,type,
order_mono:
!>[A: $tType,B: $tType] : ( ( A > B ) > $o ) ).
thf(sy_c_Orderings_Oorder__class_Ostrict__mono,type,
order_strict_mono:
!>[A: $tType,B: $tType] : ( ( A > B ) > $o ) ).
thf(sy_c_Orderings_Oordering__top,type,
ordering_top:
!>[A: $tType] : ( ( A > A > $o ) > ( A > A > $o ) > A > $o ) ).
thf(sy_c_Orderings_Otop__class_Otop,type,
top_top:
!>[A: $tType] : A ).
thf(sy_c_Partial__Function_Oflat__lub,type,
partial_flat_lub:
!>[A: $tType] : ( A > ( set @ A ) > A ) ).
thf(sy_c_Partial__Function_Ofun__ord,type,
partial_fun_ord:
!>[A: $tType,B: $tType,C: $tType] : ( ( A > B > $o ) > ( C > A ) > ( C > B ) > $o ) ).
thf(sy_c_Power_Opower_Opower,type,
power2:
!>[A: $tType] : ( A > ( A > A > A ) > A > nat > 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_OSigma,type,
product_Sigma:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > ( set @ B ) ) > ( set @ ( product_prod @ A @ B ) ) ) ).
thf(sy_c_Product__Type_Oprod_Ocase__prod,type,
product_case_prod:
!>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
thf(sy_c_Product__Type_Oprod_Ofst,type,
product_fst:
!>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > A ) ).
thf(sy_c_Product__Type_Oprod_Osnd,type,
product_snd:
!>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > B ) ).
thf(sy_c_Product__Type_Oproduct,type,
product_product:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( set @ B ) > ( set @ ( product_prod @ A @ B ) ) ) ).
thf(sy_c_Pure_Otype,type,
type2:
!>[A: $tType] : ( itself @ A ) ).
thf(sy_c_Rat_Ofield__char__0__class_ORats,type,
field_char_0_Rats:
!>[A: $tType] : ( set @ A ) ).
thf(sy_c_Real__Vector__Spaces_Obounded__bilinear,type,
real_V2442710119149674383linear:
!>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > $o ) ).
thf(sy_c_Real__Vector__Spaces_Obounded__linear,type,
real_V3181309239436604168linear:
!>[A: $tType,B: $tType] : ( ( A > B ) > $o ) ).
thf(sy_c_Real__Vector__Spaces_Oconstruct,type,
real_V4425403222259421789struct:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B ) > A > B ) ).
thf(sy_c_Real__Vector__Spaces_Odependent,type,
real_V358717886546972837endent:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_Real__Vector__Spaces_Odim,type,
real_Vector_dim:
!>[A: $tType] : ( ( set @ A ) > nat ) ).
thf(sy_c_Real__Vector__Spaces_Odist__class_Odist,type,
real_V557655796197034286t_dist:
!>[A: $tType] : ( A > A > real ) ).
thf(sy_c_Real__Vector__Spaces_Oextend__basis,type,
real_V4986007116245087402_basis:
!>[A: $tType] : ( ( set @ A ) > ( set @ A ) ) ).
thf(sy_c_Real__Vector__Spaces_Olinear,type,
real_Vector_linear:
!>[A: $tType,B: $tType] : ( ( A > B ) > $o ) ).
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_Orepresentation,type,
real_V7696804695334737415tation:
!>[A: $tType] : ( ( set @ A ) > A > A > real ) ).
thf(sy_c_Real__Vector__Spaces_OscaleR__class_OscaleR,type,
real_V8093663219630862766scaleR:
!>[A: $tType] : ( real > A > A ) ).
thf(sy_c_Real__Vector__Spaces_Ospan,type,
real_Vector_span:
!>[A: $tType] : ( ( set @ A ) > ( set @ A ) ) ).
thf(sy_c_Real__Vector__Spaces_Osubspace,type,
real_Vector_subspace:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_Relation_OField,type,
field2:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ A ) ) ).
thf(sy_c_Relation_OId,type,
id2:
!>[A: $tType] : ( set @ ( product_prod @ A @ A ) ) ).
thf(sy_c_Relation_OId__on,type,
id_on:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) ) ).
thf(sy_c_Relation_OImage,type,
image:
!>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > ( set @ A ) > ( set @ B ) ) ).
thf(sy_c_Relation_Oantisym,type,
antisym:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_Relation_Oconverse,type,
converse:
!>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > ( set @ ( product_prod @ B @ A ) ) ) ).
thf(sy_c_Relation_Oinv__image,type,
inv_image:
!>[B: $tType,A: $tType] : ( ( set @ ( product_prod @ B @ B ) ) > ( A > B ) > ( set @ ( product_prod @ A @ A ) ) ) ).
thf(sy_c_Relation_Oirrefl,type,
irrefl:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_Relation_Orefl__on,type,
refl_on:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_Relation_Orelcomp,type,
relcomp:
!>[A: $tType,B: $tType,C: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > ( set @ ( product_prod @ B @ C ) ) > ( set @ ( product_prod @ A @ C ) ) ) ).
thf(sy_c_Relation_Osingle__valued,type,
single_valued:
!>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > $o ) ).
thf(sy_c_Relation_Ototal__on,type,
total_on:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_Relation_Otrans,type,
trans:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_Relation_Otransp,type,
transp:
!>[A: $tType] : ( ( A > A > $o ) > $o ) ).
thf(sy_c_Rings_Oalgebraic__semidom__class_Ocoprime,type,
algebr8660921524188924756oprime:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Rings_Odivide__class_Odivide,type,
divide_divide:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Rings_Odvd__class_Odvd,type,
dvd_dvd:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Rings_Omodulo__class_Omodulo,type,
modulo_modulo:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool,type,
zero_neq_one_of_bool:
!>[A: $tType] : ( $o > A ) ).
thf(sy_c_Series_Osuminf,type,
suminf:
!>[A: $tType] : ( ( nat > A ) > A ) ).
thf(sy_c_Series_Osummable,type,
summable:
!>[A: $tType] : ( ( nat > A ) > $o ) ).
thf(sy_c_Series_Osums,type,
sums:
!>[A: $tType] : ( ( nat > A ) > A > $o ) ).
thf(sy_c_Set_OBall,type,
ball:
!>[A: $tType] : ( ( set @ A ) > ( A > $o ) > $o ) ).
thf(sy_c_Set_OBex,type,
bex:
!>[A: $tType] : ( ( set @ A ) > ( A > $o ) > $o ) ).
thf(sy_c_Set_OCollect,type,
collect:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
thf(sy_c_Set_OPow,type,
pow2:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( set @ A ) ) ) ).
thf(sy_c_Set_Odisjnt,type,
disjnt:
!>[A: $tType] : ( ( set @ A ) > ( set @ A ) > $o ) ).
thf(sy_c_Set_Ofilter,type,
filter3:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) > ( set @ A ) ) ).
thf(sy_c_Set_Oimage,type,
image2:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > ( set @ B ) ) ).
thf(sy_c_Set_Oinsert,type,
insert2:
!>[A: $tType] : ( A > ( set @ A ) > ( set @ A ) ) ).
thf(sy_c_Set_Ois__empty,type,
is_empty:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_Set_Ois__singleton,type,
is_singleton:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_Set_Opairwise,type,
pairwise:
!>[A: $tType] : ( ( A > A > $o ) > ( set @ A ) > $o ) ).
thf(sy_c_Set_Oremove,type,
remove:
!>[A: $tType] : ( A > ( set @ A ) > ( set @ A ) ) ).
thf(sy_c_Set_Othe__elem,type,
the_elem:
!>[A: $tType] : ( ( set @ A ) > A ) ).
thf(sy_c_Set_Ovimage,type,
vimage:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ B ) > ( set @ A ) ) ).
thf(sy_c_Set__Interval_Ofold__atLeastAtMost__nat,type,
set_fo6178422350223883121st_nat:
!>[A: $tType] : ( ( nat > A > A ) > nat > nat > A > A ) ).
thf(sy_c_Set__Interval_Ofold__atLeastAtMost__nat__rel,type,
set_fo1817059534552279752at_rel:
!>[A: $tType] : ( ( product_prod @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) ) > ( product_prod @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) ) > $o ) ).
thf(sy_c_Set__Interval_Oord__class_OatLeast,type,
set_ord_atLeast:
!>[A: $tType] : ( A > ( set @ A ) ) ).
thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost,type,
set_or1337092689740270186AtMost:
!>[A: $tType] : ( A > A > ( set @ A ) ) ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan,type,
set_or7035219750837199246ssThan:
!>[A: $tType] : ( A > A > ( set @ A ) ) ).
thf(sy_c_Set__Interval_Oord__class_OatMost,type,
set_ord_atMost:
!>[A: $tType] : ( A > ( set @ A ) ) ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThan,type,
set_ord_greaterThan:
!>[A: $tType] : ( A > ( set @ A ) ) ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost,type,
set_or3652927894154168847AtMost:
!>[A: $tType] : ( A > A > ( set @ A ) ) ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan,type,
set_or5935395276787703475ssThan:
!>[A: $tType] : ( A > A > ( set @ A ) ) ).
thf(sy_c_Set__Interval_Oord__class_OlessThan,type,
set_ord_lessThan:
!>[A: $tType] : ( A > ( set @ A ) ) ).
thf(sy_c_String_Ocomm__semiring__1__class_Oof__char,type,
comm_s6883823935334413003f_char:
!>[A: $tType] : ( char > A ) ).
thf(sy_c_String_Ounique__euclidean__semiring__with__bit__operations__class_Ochar__of,type,
unique5772411509450598832har_of:
!>[A: $tType] : ( A > char ) ).
thf(sy_c_Sum__Type_OPlus,type,
sum_Plus:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( set @ B ) > ( set @ ( sum_sum @ A @ B ) ) ) ).
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_Oclosed,type,
topolo7761053866217962861closed:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_Topological__Spaces_Otopological__space__class_Ocompact,type,
topolo2193935891317330818ompact:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_Topological__Spaces_Otopological__space__class_Oconnected,type,
topolo1966860045006549960nected:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
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_Topological__Spaces_Ouniform__space__class_Ocomplete,type,
topolo2479028161051973599mplete:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_Topological__Spaces_Ouniform__space__class_Ototally__bounded,type,
topolo6688025880775521714ounded:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_Topological__Spaces_Ouniformity__class_Ouniformity,type,
topolo7806501430040627800ormity:
!>[A: $tType] : ( filter @ ( product_prod @ A @ A ) ) ).
thf(sy_c_Transcendental_Oarcosh,type,
arcosh:
!>[A: $tType] : ( A > A ) ).
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_Transfer_Obi__total,type,
bi_total:
!>[A: $tType,B: $tType] : ( ( A > B > $o ) > $o ) ).
thf(sy_c_Transitive__Closure_Ontrancl,type,
transitive_ntrancl:
!>[A: $tType] : ( nat > ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ A @ A ) ) ) ).
thf(sy_c_Transitive__Closure_Ortrancl,type,
transitive_rtrancl:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ A @ A ) ) ) ).
thf(sy_c_Transitive__Closure_Otrancl,type,
transitive_trancl:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ A @ A ) ) ) ).
thf(sy_c_VEBT__Definitions_OVEBT_OLeaf,type,
vEBT_Leaf: $o > $o > vEBT_VEBT ).
thf(sy_c_VEBT__Definitions_OVEBT_ONode,type,
vEBT_Node: ( option @ ( product_prod @ nat @ nat ) ) > nat > ( list @ vEBT_VEBT ) > vEBT_VEBT > vEBT_VEBT ).
thf(sy_c_VEBT__Definitions_OVEBT_Osize__VEBT,type,
vEBT_size_VEBT: vEBT_VEBT > nat ).
thf(sy_c_VEBT__Definitions_OVEBT__internal_Oboth__member__options,type,
vEBT_V8194947554948674370ptions: vEBT_VEBT > nat > $o ).
thf(sy_c_VEBT__Definitions_OVEBT__internal_Oelim__dead,type,
vEBT_VEBT_elim_dead: vEBT_VEBT > extended_enat > vEBT_VEBT ).
thf(sy_c_VEBT__Definitions_OVEBT__internal_Ohigh,type,
vEBT_VEBT_high: nat > nat > nat ).
thf(sy_c_VEBT__Definitions_OVEBT__internal_Oin__children,type,
vEBT_V5917875025757280293ildren: nat > ( list @ vEBT_VEBT ) > nat > $o ).
thf(sy_c_VEBT__Definitions_OVEBT__internal_Olow,type,
vEBT_VEBT_low: nat > nat > nat ).
thf(sy_c_VEBT__Definitions_OVEBT__internal_Omembermima,type,
vEBT_VEBT_membermima: vEBT_VEBT > nat > $o ).
thf(sy_c_VEBT__Definitions_OVEBT__internal_Omembermima__rel,type,
vEBT_V4351362008482014158ma_rel: ( product_prod @ vEBT_VEBT @ nat ) > ( product_prod @ vEBT_VEBT @ nat ) > $o ).
thf(sy_c_VEBT__Definitions_OVEBT__internal_Onaive__member,type,
vEBT_V5719532721284313246member: vEBT_VEBT > nat > $o ).
thf(sy_c_VEBT__Definitions_OVEBT__internal_Onaive__member__rel,type,
vEBT_V5765760719290551771er_rel: ( product_prod @ vEBT_VEBT @ nat ) > ( product_prod @ vEBT_VEBT @ nat ) > $o ).
thf(sy_c_VEBT__Definitions_OVEBT__internal_Ovalid_H,type,
vEBT_VEBT_valid: vEBT_VEBT > nat > $o ).
thf(sy_c_VEBT__Definitions_OVEBT__internal_Ovalid_H__rel,type,
vEBT_VEBT_valid_rel: ( product_prod @ vEBT_VEBT @ nat ) > ( product_prod @ vEBT_VEBT @ nat ) > $o ).
thf(sy_c_VEBT__Definitions_Oinvar__vebt,type,
vEBT_invar_vebt: vEBT_VEBT > nat > $o ).
thf(sy_c_VEBT__Definitions_Oset__vebt,type,
vEBT_set_vebt: vEBT_VEBT > ( set @ nat ) ).
thf(sy_c_VEBT__Definitions_Ovebt__buildup,type,
vEBT_vebt_buildup: nat > vEBT_VEBT ).
thf(sy_c_VEBT__Definitions_Ovebt__buildup__rel,type,
vEBT_v4011308405150292612up_rel: nat > nat > $o ).
thf(sy_c_VEBT__Insert_Ovebt__insert,type,
vEBT_vebt_insert: vEBT_VEBT > nat > vEBT_VEBT ).
thf(sy_c_VEBT__Insert_Ovebt__insert__rel,type,
vEBT_vebt_insert_rel: ( product_prod @ vEBT_VEBT @ nat ) > ( product_prod @ vEBT_VEBT @ nat ) > $o ).
thf(sy_c_VEBT__Member_OVEBT__internal_Obit__concat,type,
vEBT_VEBT_bit_concat: nat > nat > nat > nat ).
thf(sy_c_VEBT__Member_OVEBT__internal_OminNull,type,
vEBT_VEBT_minNull: vEBT_VEBT > $o ).
thf(sy_c_VEBT__Member_OVEBT__internal_Oset__vebt_H,type,
vEBT_VEBT_set_vebt: vEBT_VEBT > ( set @ nat ) ).
thf(sy_c_VEBT__Member_Ovebt__member,type,
vEBT_vebt_member: vEBT_VEBT > nat > $o ).
thf(sy_c_VEBT__Member_Ovebt__member__rel,type,
vEBT_vebt_member_rel: ( product_prod @ vEBT_VEBT @ nat ) > ( product_prod @ vEBT_VEBT @ nat ) > $o ).
thf(sy_c_Wellfounded_Oaccp,type,
accp:
!>[A: $tType] : ( ( A > A > $o ) > A > $o ) ).
thf(sy_c_Wellfounded_Ofinite__psubset,type,
finite_psubset:
!>[A: $tType] : ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) ).
thf(sy_c_Wellfounded_Oless__than,type,
less_than: set @ ( product_prod @ nat @ nat ) ).
thf(sy_c_Wellfounded_Omax__ext,type,
max_ext:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) ) ).
thf(sy_c_Wellfounded_Omax__extp,type,
max_extp:
!>[A: $tType] : ( ( A > A > $o ) > ( set @ A ) > ( set @ A ) > $o ) ).
thf(sy_c_Wellfounded_Omeasure,type,
measure:
!>[A: $tType] : ( ( A > nat ) > ( set @ ( product_prod @ A @ A ) ) ) ).
thf(sy_c_Wellfounded_Omin__ext,type,
min_ext:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) ) ).
thf(sy_c_Wellfounded_Omlex__prod,type,
mlex_prod:
!>[A: $tType] : ( ( A > nat ) > ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ A @ A ) ) ) ).
thf(sy_c_Wellfounded_Opred__nat,type,
pred_nat: set @ ( product_prod @ nat @ nat ) ).
thf(sy_c_Wellfounded_Owf,type,
wf:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_Zorn_Ochain__subset,type,
chain_subset:
!>[A: $tType] : ( ( set @ ( set @ A ) ) > $o ) ).
thf(sy_c_Zorn_Ochains,type,
chains:
!>[A: $tType] : ( ( set @ ( set @ A ) ) > ( set @ ( set @ ( set @ A ) ) ) ) ).
thf(sy_c_Zorn_Opred__on_Ochain,type,
pred_chain:
!>[A: $tType] : ( ( set @ A ) > ( A > A > $o ) > ( set @ A ) > $o ) ).
thf(sy_c_Zorn_Opred__on_Omaxchain,type,
pred_maxchain:
!>[A: $tType] : ( ( set @ A ) > ( A > A > $o ) > ( set @ A ) > $o ) ).
thf(sy_c_Zorn_Opred__on_Osuc,type,
pred_suc:
!>[A: $tType] : ( ( set @ A ) > ( A > A > $o ) > ( set @ A ) > ( set @ 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_deg____,type,
deg: nat ).
thf(sy_v_i____,type,
i: nat ).
thf(sy_v_m____,type,
m: nat ).
thf(sy_v_ma____,type,
ma: nat ).
thf(sy_v_mi____,type,
mi: nat ).
thf(sy_v_na____,type,
na: nat ).
thf(sy_v_summary____,type,
summary: vEBT_VEBT ).
thf(sy_v_treeList____,type,
treeList: list @ vEBT_VEBT ).
thf(sy_v_xa____,type,
xa: nat ).
% Relevant facts (8187)
thf(fact_0_True,axiom,
( i
= ( vEBT_VEBT_high @ mi @ na ) ) ).
% True
thf(fact_1_insprop,axiom,
( ( nth @ vEBT_VEBT @ ( list_update @ vEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ mi @ na ) @ ( vEBT_vebt_insert @ ( nth @ vEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ mi @ na ) ) @ ( vEBT_VEBT_low @ mi @ na ) ) ) @ ( vEBT_VEBT_high @ mi @ na ) )
= ( vEBT_vebt_insert @ ( nth @ vEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ mi @ na ) ) @ ( vEBT_VEBT_low @ mi @ na ) ) ) ).
% insprop
thf(fact_2_bit__split__inv,axiom,
! [X: nat,D2: nat] :
( ( vEBT_VEBT_bit_concat @ ( vEBT_VEBT_high @ X @ D2 ) @ ( vEBT_VEBT_low @ X @ D2 ) @ D2 )
= X ) ).
% bit_split_inv
thf(fact_3_tc,axiom,
vEBT_VEBT_minNull @ ( nth @ vEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ mi @ na ) ) ).
% tc
thf(fact_4__C161_C,axiom,
~ ? [X_1: nat] : ( vEBT_vebt_member @ ( nth @ vEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ mi @ na ) ) @ X_1 ) ).
% "161"
thf(fact_5_list__update__id,axiom,
! [A: $tType,Xs: list @ A,I: nat] :
( ( list_update @ A @ Xs @ I @ ( nth @ A @ Xs @ I ) )
= Xs ) ).
% list_update_id
thf(fact_6_nth__list__update__neq,axiom,
! [A: $tType,I: nat,J: nat,Xs: list @ A,X: A] :
( ( I != J )
=> ( ( nth @ A @ ( list_update @ A @ Xs @ I @ X ) @ J )
= ( nth @ A @ Xs @ J ) ) ) ).
% nth_list_update_neq
thf(fact_7__C162_C,axiom,
~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ mi @ na ) ) @ X_1 ) ).
% "162"
thf(fact_8_list__update__overwrite,axiom,
! [A: $tType,Xs: list @ A,I: nat,X: A,Y: A] :
( ( list_update @ A @ ( list_update @ A @ Xs @ I @ X ) @ I @ Y )
= ( list_update @ A @ Xs @ I @ Y ) ) ).
% list_update_overwrite
thf(fact_9_nsprop,axiom,
( ~ ( vEBT_VEBT_minNull @ ( nth @ vEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ mi @ na ) ) )
=> ( summary
= ( vEBT_vebt_insert @ summary @ ( vEBT_VEBT_high @ mi @ na ) ) ) ) ).
% nsprop
thf(fact_10__C11_C,axiom,
! [X2: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X2 @ ( set2 @ vEBT_VEBT @ ( list_update @ vEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ mi @ na ) @ ( vEBT_vebt_insert @ ( nth @ vEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ mi @ na ) ) @ ( vEBT_VEBT_low @ mi @ na ) ) ) ) )
=> ( vEBT_invar_vebt @ X2 @ na ) ) ).
% "11"
thf(fact_11_False,axiom,
~ ( ord_less @ nat @ mi @ xa ) ).
% False
thf(fact_12_list__update__swap,axiom,
! [A: $tType,I: nat,I2: nat,Xs: list @ A,X: A,X3: A] :
( ( I != I2 )
=> ( ( list_update @ A @ ( list_update @ A @ Xs @ I @ X ) @ I2 @ X3 )
= ( list_update @ A @ ( list_update @ A @ Xs @ I2 @ X3 ) @ I @ X ) ) ) ).
% list_update_swap
thf(fact_13_in__children__def,axiom,
( vEBT_V5917875025757280293ildren
= ( ^ [N: nat,TreeList: list @ vEBT_VEBT,X4: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X4 @ N ) ) @ ( vEBT_VEBT_low @ X4 @ N ) ) ) ) ).
% in_children_def
thf(fact_14__C5_Ohyps_C_I7_J,axiom,
ord_less_eq @ nat @ mi @ ma ).
% "5.hyps"(7)
thf(fact_15__C12_C,axiom,
vEBT_invar_vebt @ ( if @ vEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth @ vEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ mi @ na ) ) ) @ ( vEBT_vebt_insert @ summary @ ( vEBT_VEBT_high @ mi @ na ) ) @ summary ) @ m ).
% "12"
thf(fact_16_not__min__Null__member,axiom,
! [T2: vEBT_VEBT] :
( ~ ( vEBT_VEBT_minNull @ T2 )
=> ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ T2 @ X_12 ) ) ).
% not_min_Null_member
thf(fact_17_min__Null__member,axiom,
! [T2: vEBT_VEBT,X: nat] :
( ( vEBT_VEBT_minNull @ T2 )
=> ~ ( vEBT_vebt_member @ T2 @ X ) ) ).
% min_Null_member
thf(fact_18__C1_C,axiom,
vEBT_invar_vebt @ summary @ m ).
% "1"
thf(fact_19_both__member__options__equiv__member,axiom,
! [T2: vEBT_VEBT,N2: nat,X: nat] :
( ( vEBT_invar_vebt @ T2 @ N2 )
=> ( ( vEBT_V8194947554948674370ptions @ T2 @ X )
= ( vEBT_vebt_member @ T2 @ X ) ) ) ).
% both_member_options_equiv_member
thf(fact_20_valid__member__both__member__options,axiom,
! [T2: vEBT_VEBT,N2: nat,X: nat] :
( ( vEBT_invar_vebt @ T2 @ N2 )
=> ( ( vEBT_V8194947554948674370ptions @ T2 @ X )
=> ( vEBT_vebt_member @ T2 @ X ) ) ) ).
% valid_member_both_member_options
thf(fact_21__C5_C,axiom,
( ( mi = ma )
=> ! [X2: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X2 @ ( set2 @ vEBT_VEBT @ treeList ) )
=> ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X_1 ) ) ) ).
% "5"
thf(fact_22_abcdef,axiom,
ord_less @ nat @ xa @ mi ).
% abcdef
thf(fact_23_member__correct,axiom,
! [T2: vEBT_VEBT,N2: nat,X: nat] :
( ( vEBT_invar_vebt @ T2 @ N2 )
=> ( ( vEBT_vebt_member @ T2 @ X )
= ( member @ nat @ X @ ( vEBT_set_vebt @ T2 ) ) ) ) ).
% member_correct
thf(fact_24__C0_C,axiom,
! [X2: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X2 @ ( set2 @ vEBT_VEBT @ treeList ) )
=> ( vEBT_invar_vebt @ X2 @ na ) ) ).
% "0"
thf(fact_25__C163_C,axiom,
~ ( vEBT_V8194947554948674370ptions @ summary @ i ) ).
% "163"
thf(fact_26_mimaxrel,axiom,
( ( xa != mi )
& ( xa != ma ) ) ).
% mimaxrel
thf(fact_27__C8_C,axiom,
( ( suc @ na )
= m ) ).
% "8"
thf(fact_28_set__vebt__set__vebt_H__valid,axiom,
! [T2: vEBT_VEBT,N2: nat] :
( ( vEBT_invar_vebt @ T2 @ N2 )
=> ( ( vEBT_set_vebt @ T2 )
= ( vEBT_VEBT_set_vebt @ T2 ) ) ) ).
% set_vebt_set_vebt'_valid
thf(fact_29_valid__eq,axiom,
vEBT_VEBT_valid = vEBT_invar_vebt ).
% valid_eq
thf(fact_30_valid__eq1,axiom,
! [T2: vEBT_VEBT,D2: nat] :
( ( vEBT_invar_vebt @ T2 @ D2 )
=> ( vEBT_VEBT_valid @ T2 @ D2 ) ) ).
% valid_eq1
thf(fact_31_valid__eq2,axiom,
! [T2: vEBT_VEBT,D2: nat] :
( ( vEBT_VEBT_valid @ T2 @ D2 )
=> ( vEBT_invar_vebt @ T2 @ D2 ) ) ).
% valid_eq2
thf(fact_32_order__refl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% order_refl
thf(fact_33_dual__order_Orefl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: A] : ( ord_less_eq @ A @ A3 @ A3 ) ) ).
% dual_order.refl
thf(fact_34_inthall,axiom,
! [A: $tType,Xs: list @ A,P: A > $o,N2: nat] :
( ! [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Xs ) )
=> ( P @ X5 ) )
=> ( ( ord_less @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( P @ ( nth @ A @ Xs @ N2 ) ) ) ) ).
% inthall
thf(fact_35_deg__not__0,axiom,
! [T2: vEBT_VEBT,N2: nat] :
( ( vEBT_invar_vebt @ T2 @ N2 )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) ) ).
% deg_not_0
thf(fact_36__C3_C,axiom,
( deg
= ( plus_plus @ nat @ na @ m ) ) ).
% "3"
thf(fact_37_nat__less__le,axiom,
( ( ord_less @ nat )
= ( ^ [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ N )
& ( M != N ) ) ) ) ).
% nat_less_le
thf(fact_38_less__imp__le__nat,axiom,
! [M2: nat,N2: nat] :
( ( ord_less @ nat @ M2 @ N2 )
=> ( ord_less_eq @ nat @ M2 @ N2 ) ) ).
% less_imp_le_nat
thf(fact_39_even__odd__cases,axiom,
! [X: nat] :
( ! [N3: nat] :
( X
!= ( plus_plus @ nat @ N3 @ N3 ) )
=> ~ ! [N3: nat] :
( X
!= ( plus_plus @ nat @ N3 @ ( suc @ N3 ) ) ) ) ).
% even_odd_cases
thf(fact_40_valid__tree__deg__neq__0,axiom,
! [T2: vEBT_VEBT] :
~ ( vEBT_invar_vebt @ T2 @ ( zero_zero @ nat ) ) ).
% valid_tree_deg_neq_0
thf(fact_41_valid__0__not,axiom,
! [T2: vEBT_VEBT] :
~ ( vEBT_invar_vebt @ T2 @ ( zero_zero @ nat ) ) ).
% valid_0_not
thf(fact_42_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_43_nat_Oinject,axiom,
! [X22: nat,Y2: nat] :
( ( ( suc @ X22 )
= ( suc @ Y2 ) )
= ( X22 = Y2 ) ) ).
% nat.inject
thf(fact_44_mem__Collect__eq,axiom,
! [A: $tType,A3: A,P: A > $o] :
( ( member @ A @ A3 @ ( collect @ A @ P ) )
= ( P @ A3 ) ) ).
% mem_Collect_eq
thf(fact_45_Collect__mem__eq,axiom,
! [A: $tType,A4: set @ A] :
( ( collect @ A
@ ^ [X4: A] : ( member @ A @ X4 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_46_Collect__cong,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X5: A] :
( ( P @ X5 )
= ( Q @ X5 ) )
=> ( ( collect @ A @ P )
= ( collect @ A @ Q ) ) ) ).
% Collect_cong
thf(fact_47_ext,axiom,
! [B: $tType,A: $tType,F2: A > B,G: A > B] :
( ! [X5: A] :
( ( F2 @ X5 )
= ( G @ X5 ) )
=> ( F2 = G ) ) ).
% ext
thf(fact_48_Suc__less__eq,axiom,
! [M2: nat,N2: nat] :
( ( ord_less @ nat @ ( suc @ M2 ) @ ( suc @ N2 ) )
= ( ord_less @ nat @ M2 @ N2 ) ) ).
% Suc_less_eq
thf(fact_49_Suc__mono,axiom,
! [M2: nat,N2: nat] :
( ( ord_less @ nat @ M2 @ N2 )
=> ( ord_less @ nat @ ( suc @ M2 ) @ ( suc @ N2 ) ) ) ).
% Suc_mono
thf(fact_50_lessI,axiom,
! [N2: nat] : ( ord_less @ nat @ N2 @ ( suc @ N2 ) ) ).
% lessI
thf(fact_51_less__nat__zero__code,axiom,
! [N2: nat] :
~ ( ord_less @ nat @ N2 @ ( zero_zero @ nat ) ) ).
% less_nat_zero_code
thf(fact_52_neq0__conv,axiom,
! [N2: nat] :
( ( N2
!= ( zero_zero @ nat ) )
= ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) ) ).
% neq0_conv
thf(fact_53_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_54_Suc__le__mono,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_eq @ nat @ ( suc @ N2 ) @ ( suc @ M2 ) )
= ( ord_less_eq @ nat @ N2 @ M2 ) ) ).
% Suc_le_mono
thf(fact_55_le0,axiom,
! [N2: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N2 ) ).
% le0
thf(fact_56_bot__nat__0_Oextremum,axiom,
! [A3: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ A3 ) ).
% bot_nat_0.extremum
thf(fact_57_add__Suc__right,axiom,
! [M2: nat,N2: nat] :
( ( plus_plus @ nat @ M2 @ ( suc @ N2 ) )
= ( suc @ ( plus_plus @ nat @ M2 @ N2 ) ) ) ).
% add_Suc_right
thf(fact_58_Nat_Oadd__0__right,axiom,
! [M2: nat] :
( ( plus_plus @ nat @ M2 @ ( zero_zero @ nat ) )
= M2 ) ).
% Nat.add_0_right
thf(fact_59_add__is__0,axiom,
! [M2: nat,N2: nat] :
( ( ( plus_plus @ nat @ M2 @ N2 )
= ( zero_zero @ nat ) )
= ( ( M2
= ( zero_zero @ nat ) )
& ( N2
= ( zero_zero @ nat ) ) ) ) ).
% add_is_0
thf(fact_60_nat__add__left__cancel__less,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ord_less @ nat @ ( plus_plus @ nat @ K @ M2 ) @ ( plus_plus @ nat @ K @ N2 ) )
= ( ord_less @ nat @ M2 @ N2 ) ) ).
% nat_add_left_cancel_less
thf(fact_61_nat__add__left__cancel__le,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ord_less_eq @ nat @ ( plus_plus @ nat @ K @ M2 ) @ ( plus_plus @ nat @ K @ N2 ) )
= ( ord_less_eq @ nat @ M2 @ N2 ) ) ).
% nat_add_left_cancel_le
thf(fact_62_length__list__update,axiom,
! [A: $tType,Xs: list @ A,I: nat,X: A] :
( ( size_size @ ( list @ A ) @ ( list_update @ A @ Xs @ I @ X ) )
= ( size_size @ ( list @ A ) @ Xs ) ) ).
% length_list_update
thf(fact_63_less__Suc0,axiom,
! [N2: nat] :
( ( ord_less @ nat @ N2 @ ( suc @ ( zero_zero @ nat ) ) )
= ( N2
= ( zero_zero @ nat ) ) ) ).
% less_Suc0
thf(fact_64_zero__less__Suc,axiom,
! [N2: nat] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( suc @ N2 ) ) ).
% zero_less_Suc
thf(fact_65_add__gr__0,axiom,
! [M2: nat,N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( plus_plus @ nat @ M2 @ N2 ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M2 )
| ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ).
% add_gr_0
thf(fact_66_list__update__beyond,axiom,
! [A: $tType,Xs: list @ A,I: nat,X: A] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ I )
=> ( ( list_update @ A @ Xs @ I @ X )
= Xs ) ) ).
% list_update_beyond
thf(fact_67_nth__list__update__eq,axiom,
! [A: $tType,I: nat,Xs: list @ A,X: A] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( nth @ A @ ( list_update @ A @ Xs @ I @ X ) @ I )
= X ) ) ).
% nth_list_update_eq
thf(fact_68_set__swap,axiom,
! [A: $tType,I: nat,Xs: list @ A,J: nat] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( set2 @ A @ ( list_update @ A @ ( list_update @ A @ Xs @ I @ ( nth @ A @ Xs @ J ) ) @ J @ ( nth @ A @ Xs @ I ) ) )
= ( set2 @ A @ Xs ) ) ) ) ).
% set_swap
thf(fact_69_less__natE,axiom,
! [M2: nat,N2: nat] :
( ( ord_less @ nat @ M2 @ N2 )
=> ~ ! [Q2: nat] :
( N2
!= ( suc @ ( plus_plus @ nat @ M2 @ Q2 ) ) ) ) ).
% less_natE
thf(fact_70_Ex__less__Suc2,axiom,
! [N2: nat,P: nat > $o] :
( ( ? [I3: nat] :
( ( ord_less @ nat @ I3 @ ( suc @ N2 ) )
& ( P @ I3 ) ) )
= ( ( P @ ( zero_zero @ nat ) )
| ? [I3: nat] :
( ( ord_less @ nat @ I3 @ N2 )
& ( P @ ( suc @ I3 ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_71_gr0__conv__Suc,axiom,
! [N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
= ( ? [M: nat] :
( N2
= ( suc @ M ) ) ) ) ).
% gr0_conv_Suc
thf(fact_72_All__less__Suc2,axiom,
! [N2: nat,P: nat > $o] :
( ( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( suc @ N2 ) )
=> ( P @ I3 ) ) )
= ( ( P @ ( zero_zero @ nat ) )
& ! [I3: nat] :
( ( ord_less @ nat @ I3 @ N2 )
=> ( P @ ( suc @ I3 ) ) ) ) ) ).
% All_less_Suc2
thf(fact_73_less__add__Suc1,axiom,
! [I: nat,M2: nat] : ( ord_less @ nat @ I @ ( suc @ ( plus_plus @ nat @ I @ M2 ) ) ) ).
% less_add_Suc1
thf(fact_74_less__add__Suc2,axiom,
! [I: nat,M2: nat] : ( ord_less @ nat @ I @ ( suc @ ( plus_plus @ nat @ M2 @ I ) ) ) ).
% less_add_Suc2
thf(fact_75_gr0__implies__Suc,axiom,
! [N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ? [M3: nat] :
( N2
= ( suc @ M3 ) ) ) ).
% gr0_implies_Suc
thf(fact_76_less__iff__Suc__add,axiom,
( ( ord_less @ nat )
= ( ^ [M: nat,N: nat] :
? [K2: nat] :
( N
= ( suc @ ( plus_plus @ nat @ M @ K2 ) ) ) ) ) ).
% less_iff_Suc_add
thf(fact_77_less__imp__Suc__add,axiom,
! [M2: nat,N2: nat] :
( ( ord_less @ nat @ M2 @ N2 )
=> ? [K3: nat] :
( N2
= ( suc @ ( plus_plus @ nat @ M2 @ K3 ) ) ) ) ).
% less_imp_Suc_add
thf(fact_78_Ex__list__of__length,axiom,
! [A: $tType,N2: nat] :
? [Xs2: list @ A] :
( ( size_size @ ( list @ A ) @ Xs2 )
= N2 ) ).
% Ex_list_of_length
thf(fact_79_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_80_less__Suc__eq__0__disj,axiom,
! [M2: nat,N2: nat] :
( ( ord_less @ nat @ M2 @ ( suc @ N2 ) )
= ( ( M2
= ( zero_zero @ nat ) )
| ? [J2: nat] :
( ( M2
= ( suc @ J2 ) )
& ( ord_less @ nat @ J2 @ N2 ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_81_less__imp__add__positive,axiom,
! [I: nat,J: nat] :
( ( ord_less @ nat @ I @ J )
=> ? [K3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ K3 )
& ( ( plus_plus @ nat @ I @ K3 )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_82_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_83_not0__implies__Suc,axiom,
! [N2: nat] :
( ( N2
!= ( zero_zero @ nat ) )
=> ? [M3: nat] :
( N2
= ( suc @ M3 ) ) ) ).
% not0_implies_Suc
thf(fact_84_add__eq__self__zero,axiom,
! [M2: nat,N2: nat] :
( ( ( plus_plus @ nat @ M2 @ N2 )
= M2 )
=> ( N2
= ( zero_zero @ nat ) ) ) ).
% add_eq_self_zero
thf(fact_85_add__Suc__shift,axiom,
! [M2: nat,N2: nat] :
( ( plus_plus @ nat @ ( suc @ M2 ) @ N2 )
= ( plus_plus @ nat @ M2 @ ( suc @ N2 ) ) ) ).
% add_Suc_shift
thf(fact_86_Zero__not__Suc,axiom,
! [M2: nat] :
( ( zero_zero @ nat )
!= ( suc @ M2 ) ) ).
% Zero_not_Suc
thf(fact_87_Zero__neq__Suc,axiom,
! [M2: nat] :
( ( zero_zero @ nat )
!= ( suc @ M2 ) ) ).
% Zero_neq_Suc
thf(fact_88_Suc__neq__Zero,axiom,
! [M2: nat] :
( ( suc @ M2 )
!= ( zero_zero @ nat ) ) ).
% Suc_neq_Zero
thf(fact_89_zero__induct,axiom,
! [P: nat > $o,K: nat] :
( ( P @ K )
=> ( ! [N3: nat] :
( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) )
=> ( P @ ( zero_zero @ nat ) ) ) ) ).
% zero_induct
thf(fact_90_n__not__Suc__n,axiom,
! [N2: nat] :
( N2
!= ( suc @ N2 ) ) ).
% n_not_Suc_n
thf(fact_91_diff__induct,axiom,
! [P: nat > nat > $o,M2: nat,N2: nat] :
( ! [X5: nat] : ( P @ X5 @ ( zero_zero @ nat ) )
=> ( ! [Y3: nat] : ( P @ ( zero_zero @ nat ) @ ( suc @ Y3 ) )
=> ( ! [X5: nat,Y3: nat] :
( ( P @ X5 @ Y3 )
=> ( P @ ( suc @ X5 ) @ ( suc @ Y3 ) ) )
=> ( P @ M2 @ N2 ) ) ) ) ).
% diff_induct
thf(fact_92_one__is__add,axiom,
! [M2: nat,N2: nat] :
( ( ( suc @ ( zero_zero @ nat ) )
= ( plus_plus @ nat @ M2 @ N2 ) )
= ( ( ( M2
= ( suc @ ( zero_zero @ nat ) ) )
& ( N2
= ( zero_zero @ nat ) ) )
| ( ( M2
= ( zero_zero @ nat ) )
& ( N2
= ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ).
% one_is_add
thf(fact_93_nat__induct,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ ( zero_zero @ nat ) )
=> ( ! [N3: nat] :
( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) )
=> ( P @ N2 ) ) ) ).
% nat_induct
thf(fact_94_Suc__inject,axiom,
! [X: nat,Y: nat] :
( ( ( suc @ X )
= ( suc @ Y ) )
=> ( X = Y ) ) ).
% Suc_inject
thf(fact_95_add__is__1,axiom,
! [M2: nat,N2: nat] :
( ( ( plus_plus @ nat @ M2 @ N2 )
= ( suc @ ( zero_zero @ nat ) ) )
= ( ( ( M2
= ( suc @ ( zero_zero @ nat ) ) )
& ( N2
= ( zero_zero @ nat ) ) )
| ( ( M2
= ( zero_zero @ nat ) )
& ( N2
= ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ).
% add_is_1
thf(fact_96_add__Suc,axiom,
! [M2: nat,N2: nat] :
( ( plus_plus @ nat @ ( suc @ M2 ) @ N2 )
= ( suc @ ( plus_plus @ nat @ M2 @ N2 ) ) ) ).
% add_Suc
thf(fact_97_old_Onat_Oexhaust,axiom,
! [Y: nat] :
( ( Y
!= ( zero_zero @ nat ) )
=> ~ ! [Nat3: nat] :
( Y
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_98_plus__nat_Oadd__0,axiom,
! [N2: nat] :
( ( plus_plus @ nat @ ( zero_zero @ nat ) @ N2 )
= N2 ) ).
% plus_nat.add_0
thf(fact_99_nat__arith_Osuc1,axiom,
! [A4: nat,K: nat,A3: nat] :
( ( A4
= ( plus_plus @ nat @ K @ A3 ) )
=> ( ( suc @ A4 )
= ( plus_plus @ nat @ K @ ( suc @ A3 ) ) ) ) ).
% nat_arith.suc1
thf(fact_100_nat_OdiscI,axiom,
! [Nat: nat,X22: nat] :
( ( Nat
= ( suc @ X22 ) )
=> ( Nat
!= ( zero_zero @ nat ) ) ) ).
% nat.discI
thf(fact_101_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( ( zero_zero @ nat )
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_102_old_Onat_Odistinct_I2_J,axiom,
! [Nat4: nat] :
( ( suc @ Nat4 )
!= ( zero_zero @ nat ) ) ).
% old.nat.distinct(2)
thf(fact_103_nat_Odistinct_I1_J,axiom,
! [X22: nat] :
( ( zero_zero @ nat )
!= ( suc @ X22 ) ) ).
% nat.distinct(1)
thf(fact_104_not__less__less__Suc__eq,axiom,
! [N2: nat,M2: nat] :
( ~ ( ord_less @ nat @ N2 @ M2 )
=> ( ( ord_less @ nat @ N2 @ ( suc @ M2 ) )
= ( N2 = M2 ) ) ) ).
% not_less_less_Suc_eq
thf(fact_105_strict__inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less @ nat @ I @ J )
=> ( ! [I4: nat] :
( ( J
= ( suc @ I4 ) )
=> ( P @ I4 ) )
=> ( ! [I4: nat] :
( ( ord_less @ nat @ I4 @ J )
=> ( ( P @ ( suc @ I4 ) )
=> ( P @ I4 ) ) )
=> ( P @ I ) ) ) ) ).
% strict_inc_induct
thf(fact_106_less__Suc__induct,axiom,
! [I: nat,J: nat,P: nat > nat > $o] :
( ( ord_less @ nat @ I @ J )
=> ( ! [I4: nat] : ( P @ I4 @ ( suc @ I4 ) )
=> ( ! [I4: nat,J3: nat,K3: nat] :
( ( ord_less @ nat @ I4 @ J3 )
=> ( ( ord_less @ nat @ J3 @ K3 )
=> ( ( P @ I4 @ J3 )
=> ( ( P @ J3 @ K3 )
=> ( P @ I4 @ K3 ) ) ) ) )
=> ( P @ I @ J ) ) ) ) ).
% less_Suc_induct
thf(fact_107_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_108_Suc__less__SucD,axiom,
! [M2: nat,N2: nat] :
( ( ord_less @ nat @ ( suc @ M2 ) @ ( suc @ N2 ) )
=> ( ord_less @ nat @ M2 @ N2 ) ) ).
% Suc_less_SucD
thf(fact_109_less__antisym,axiom,
! [N2: nat,M2: nat] :
( ~ ( ord_less @ nat @ N2 @ M2 )
=> ( ( ord_less @ nat @ N2 @ ( suc @ M2 ) )
=> ( M2 = N2 ) ) ) ).
% less_antisym
thf(fact_110_Suc__less__eq2,axiom,
! [N2: nat,M2: nat] :
( ( ord_less @ nat @ ( suc @ N2 ) @ M2 )
= ( ? [M4: nat] :
( ( M2
= ( suc @ M4 ) )
& ( ord_less @ nat @ N2 @ M4 ) ) ) ) ).
% Suc_less_eq2
thf(fact_111_All__less__Suc,axiom,
! [N2: nat,P: nat > $o] :
( ( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( suc @ N2 ) )
=> ( P @ I3 ) ) )
= ( ( P @ N2 )
& ! [I3: nat] :
( ( ord_less @ nat @ I3 @ N2 )
=> ( P @ I3 ) ) ) ) ).
% All_less_Suc
thf(fact_112_not__less__eq,axiom,
! [M2: nat,N2: nat] :
( ( ~ ( ord_less @ nat @ M2 @ N2 ) )
= ( ord_less @ nat @ N2 @ ( suc @ M2 ) ) ) ).
% not_less_eq
thf(fact_113_less__Suc__eq,axiom,
! [M2: nat,N2: nat] :
( ( ord_less @ nat @ M2 @ ( suc @ N2 ) )
= ( ( ord_less @ nat @ M2 @ N2 )
| ( M2 = N2 ) ) ) ).
% less_Suc_eq
thf(fact_114_Ex__less__Suc,axiom,
! [N2: nat,P: nat > $o] :
( ( ? [I3: nat] :
( ( ord_less @ nat @ I3 @ ( suc @ N2 ) )
& ( P @ I3 ) ) )
= ( ( P @ N2 )
| ? [I3: nat] :
( ( ord_less @ nat @ I3 @ N2 )
& ( P @ I3 ) ) ) ) ).
% Ex_less_Suc
thf(fact_115_less__SucI,axiom,
! [M2: nat,N2: nat] :
( ( ord_less @ nat @ M2 @ N2 )
=> ( ord_less @ nat @ M2 @ ( suc @ N2 ) ) ) ).
% less_SucI
thf(fact_116_less__SucE,axiom,
! [M2: nat,N2: nat] :
( ( ord_less @ nat @ M2 @ ( suc @ N2 ) )
=> ( ~ ( ord_less @ nat @ M2 @ N2 )
=> ( M2 = N2 ) ) ) ).
% less_SucE
thf(fact_117_Suc__lessI,axiom,
! [M2: nat,N2: nat] :
( ( ord_less @ nat @ M2 @ N2 )
=> ( ( ( suc @ M2 )
!= N2 )
=> ( ord_less @ nat @ ( suc @ M2 ) @ N2 ) ) ) ).
% Suc_lessI
thf(fact_118_Suc__lessE,axiom,
! [I: nat,K: nat] :
( ( ord_less @ nat @ ( suc @ I ) @ K )
=> ~ ! [J3: nat] :
( ( ord_less @ nat @ I @ J3 )
=> ( K
!= ( suc @ J3 ) ) ) ) ).
% Suc_lessE
thf(fact_119_Suc__lessD,axiom,
! [M2: nat,N2: nat] :
( ( ord_less @ nat @ ( suc @ M2 ) @ N2 )
=> ( ord_less @ nat @ M2 @ N2 ) ) ).
% Suc_lessD
thf(fact_120_Nat_OlessE,axiom,
! [I: nat,K: nat] :
( ( ord_less @ nat @ I @ K )
=> ( ( K
!= ( suc @ I ) )
=> ~ ! [J3: nat] :
( ( ord_less @ nat @ I @ J3 )
=> ( K
!= ( suc @ J3 ) ) ) ) ) ).
% Nat.lessE
thf(fact_121_less__add__eq__less,axiom,
! [K: nat,L: nat,M2: nat,N2: nat] :
( ( ord_less @ nat @ K @ L )
=> ( ( ( plus_plus @ nat @ M2 @ L )
= ( plus_plus @ nat @ K @ N2 ) )
=> ( ord_less @ nat @ M2 @ N2 ) ) ) ).
% less_add_eq_less
thf(fact_122_trans__less__add2,axiom,
! [I: nat,J: nat,M2: nat] :
( ( ord_less @ nat @ I @ J )
=> ( ord_less @ nat @ I @ ( plus_plus @ nat @ M2 @ J ) ) ) ).
% trans_less_add2
thf(fact_123_trans__less__add1,axiom,
! [I: nat,J: nat,M2: nat] :
( ( ord_less @ nat @ I @ J )
=> ( ord_less @ nat @ I @ ( plus_plus @ nat @ J @ M2 ) ) ) ).
% trans_less_add1
thf(fact_124_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_125_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less @ nat @ ( plus_plus @ nat @ J @ I ) @ I ) ).
% not_add_less2
thf(fact_126_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less @ nat @ ( plus_plus @ nat @ I @ J ) @ I ) ).
% not_add_less1
thf(fact_127_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_128_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_129_transitive__stepwise__le,axiom,
! [M2: nat,N2: nat,R: nat > nat > $o] :
( ( ord_less_eq @ nat @ M2 @ N2 )
=> ( ! [X5: nat] : ( R @ X5 @ X5 )
=> ( ! [X5: nat,Y3: nat,Z: nat] :
( ( R @ X5 @ Y3 )
=> ( ( R @ Y3 @ Z )
=> ( R @ X5 @ Z ) ) )
=> ( ! [N3: nat] : ( R @ N3 @ ( suc @ N3 ) )
=> ( R @ M2 @ N2 ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_130_nat__induct__at__least,axiom,
! [M2: nat,N2: nat,P: nat > $o] :
( ( ord_less_eq @ nat @ M2 @ N2 )
=> ( ( P @ M2 )
=> ( ! [N3: nat] :
( ( ord_less_eq @ nat @ M2 @ N3 )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) )
=> ( P @ N2 ) ) ) ) ).
% nat_induct_at_least
thf(fact_131_full__nat__induct,axiom,
! [P: nat > $o,N2: nat] :
( ! [N3: nat] :
( ! [M5: nat] :
( ( ord_less_eq @ nat @ ( suc @ M5 ) @ N3 )
=> ( P @ M5 ) )
=> ( P @ N3 ) )
=> ( P @ N2 ) ) ).
% full_nat_induct
thf(fact_132_not__less__eq__eq,axiom,
! [M2: nat,N2: nat] :
( ( ~ ( ord_less_eq @ nat @ M2 @ N2 ) )
= ( ord_less_eq @ nat @ ( suc @ N2 ) @ M2 ) ) ).
% not_less_eq_eq
thf(fact_133_Suc__n__not__le__n,axiom,
! [N2: nat] :
~ ( ord_less_eq @ nat @ ( suc @ N2 ) @ N2 ) ).
% Suc_n_not_le_n
thf(fact_134_le__Suc__eq,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq @ nat @ M2 @ ( suc @ N2 ) )
= ( ( ord_less_eq @ nat @ M2 @ N2 )
| ( M2
= ( suc @ N2 ) ) ) ) ).
% le_Suc_eq
thf(fact_135_Suc__le__D,axiom,
! [N2: nat,M6: nat] :
( ( ord_less_eq @ nat @ ( suc @ N2 ) @ M6 )
=> ? [M3: nat] :
( M6
= ( suc @ M3 ) ) ) ).
% Suc_le_D
thf(fact_136_le__SucI,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq @ nat @ M2 @ N2 )
=> ( ord_less_eq @ nat @ M2 @ ( suc @ N2 ) ) ) ).
% le_SucI
thf(fact_137_le__SucE,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq @ nat @ M2 @ ( suc @ N2 ) )
=> ( ~ ( ord_less_eq @ nat @ M2 @ N2 )
=> ( M2
= ( suc @ N2 ) ) ) ) ).
% le_SucE
thf(fact_138_Suc__leD,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq @ nat @ ( suc @ M2 ) @ N2 )
=> ( ord_less_eq @ nat @ M2 @ N2 ) ) ).
% Suc_leD
thf(fact_139_nat__le__iff__add,axiom,
( ( ord_less_eq @ nat )
= ( ^ [M: nat,N: nat] :
? [K2: nat] :
( N
= ( plus_plus @ nat @ M @ K2 ) ) ) ) ).
% nat_le_iff_add
thf(fact_140_trans__le__add2,axiom,
! [I: nat,J: nat,M2: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ord_less_eq @ nat @ I @ ( plus_plus @ nat @ M2 @ J ) ) ) ).
% trans_le_add2
thf(fact_141_trans__le__add1,axiom,
! [I: nat,J: nat,M2: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ord_less_eq @ nat @ I @ ( plus_plus @ nat @ J @ M2 ) ) ) ).
% trans_le_add1
thf(fact_142_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_143_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_144_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq @ nat @ K @ L )
=> ? [N3: nat] :
( L
= ( plus_plus @ nat @ K @ N3 ) ) ) ).
% le_Suc_ex
thf(fact_145_add__leD2,axiom,
! [M2: nat,K: nat,N2: nat] :
( ( ord_less_eq @ nat @ ( plus_plus @ nat @ M2 @ K ) @ N2 )
=> ( ord_less_eq @ nat @ K @ N2 ) ) ).
% add_leD2
thf(fact_146_add__leD1,axiom,
! [M2: nat,K: nat,N2: nat] :
( ( ord_less_eq @ nat @ ( plus_plus @ nat @ M2 @ K ) @ N2 )
=> ( ord_less_eq @ nat @ M2 @ N2 ) ) ).
% add_leD1
thf(fact_147_le__add2,axiom,
! [N2: nat,M2: nat] : ( ord_less_eq @ nat @ N2 @ ( plus_plus @ nat @ M2 @ N2 ) ) ).
% le_add2
thf(fact_148_le__add1,axiom,
! [N2: nat,M2: nat] : ( ord_less_eq @ nat @ N2 @ ( plus_plus @ nat @ N2 @ M2 ) ) ).
% le_add1
thf(fact_149_add__leE,axiom,
! [M2: nat,K: nat,N2: nat] :
( ( ord_less_eq @ nat @ ( plus_plus @ nat @ M2 @ K ) @ N2 )
=> ~ ( ( ord_less_eq @ nat @ M2 @ N2 )
=> ~ ( ord_less_eq @ nat @ K @ N2 ) ) ) ).
% add_leE
thf(fact_150_infinite__descent0__measure,axiom,
! [A: $tType,V: A > nat,P: A > $o,X: A] :
( ! [X5: A] :
( ( ( V @ X5 )
= ( zero_zero @ nat ) )
=> ( P @ X5 ) )
=> ( ! [X5: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( V @ X5 ) )
=> ( ~ ( P @ X5 )
=> ? [Y4: A] :
( ( ord_less @ nat @ ( V @ Y4 ) @ ( V @ X5 ) )
& ~ ( P @ Y4 ) ) ) )
=> ( P @ X ) ) ) ).
% infinite_descent0_measure
thf(fact_151_infinite__descent0,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ ( zero_zero @ nat ) )
=> ( ! [N3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( ~ ( P @ N3 )
=> ? [M5: nat] :
( ( ord_less @ nat @ M5 @ N3 )
& ~ ( P @ M5 ) ) ) )
=> ( P @ N2 ) ) ) ).
% infinite_descent0
thf(fact_152_gr__implies__not0,axiom,
! [M2: nat,N2: nat] :
( ( ord_less @ nat @ M2 @ N2 )
=> ( N2
!= ( zero_zero @ nat ) ) ) ).
% gr_implies_not0
thf(fact_153_less__zeroE,axiom,
! [N2: nat] :
~ ( ord_less @ nat @ N2 @ ( zero_zero @ nat ) ) ).
% less_zeroE
thf(fact_154_not__less0,axiom,
! [N2: nat] :
~ ( ord_less @ nat @ N2 @ ( zero_zero @ nat ) ) ).
% not_less0
thf(fact_155_not__gr0,axiom,
! [N2: nat] :
( ( ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) )
= ( N2
= ( zero_zero @ nat ) ) ) ).
% not_gr0
thf(fact_156_gr0I,axiom,
! [N2: nat] :
( ( N2
!= ( zero_zero @ nat ) )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) ) ).
% gr0I
thf(fact_157_bot__nat__0_Oextremum__strict,axiom,
! [A3: nat] :
~ ( ord_less @ nat @ A3 @ ( zero_zero @ nat ) ) ).
% bot_nat_0.extremum_strict
thf(fact_158_le__0__eq,axiom,
! [N2: nat] :
( ( ord_less_eq @ nat @ N2 @ ( zero_zero @ nat ) )
= ( N2
= ( zero_zero @ nat ) ) ) ).
% le_0_eq
thf(fact_159_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_160_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_161_less__eq__nat_Osimps_I1_J,axiom,
! [N2: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N2 ) ).
% less_eq_nat.simps(1)
thf(fact_162_ex__least__nat__less,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ N2 )
=> ( ~ ( P @ ( zero_zero @ nat ) )
=> ? [K3: nat] :
( ( ord_less @ nat @ K3 @ N2 )
& ! [I5: nat] :
( ( ord_less_eq @ nat @ I5 @ K3 )
=> ~ ( P @ I5 ) )
& ( P @ ( suc @ K3 ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_163_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_164_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_165_subset__code_I1_J,axiom,
! [A: $tType,Xs: list @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ B2 )
= ( ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
=> ( member @ A @ X4 @ B2 ) ) ) ) ).
% subset_code(1)
thf(fact_166_lift__Suc__mono__less__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [F2: nat > A,N2: nat,M2: nat] :
( ! [N3: nat] : ( ord_less @ A @ ( F2 @ N3 ) @ ( F2 @ ( suc @ N3 ) ) )
=> ( ( ord_less @ A @ ( F2 @ N2 ) @ ( F2 @ M2 ) )
= ( ord_less @ nat @ N2 @ M2 ) ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_167_lift__Suc__mono__less,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [F2: nat > A,N2: nat,N4: nat] :
( ! [N3: nat] : ( ord_less @ A @ ( F2 @ N3 ) @ ( F2 @ ( suc @ N3 ) ) )
=> ( ( ord_less @ nat @ N2 @ N4 )
=> ( ord_less @ A @ ( F2 @ N2 ) @ ( F2 @ N4 ) ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_168_lift__Suc__antimono__le,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [F2: nat > A,N2: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_eq @ A @ ( F2 @ ( suc @ N3 ) ) @ ( F2 @ N3 ) )
=> ( ( ord_less_eq @ nat @ N2 @ N4 )
=> ( ord_less_eq @ A @ ( F2 @ N4 ) @ ( F2 @ N2 ) ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_169_lift__Suc__mono__le,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [F2: nat > A,N2: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_eq @ A @ ( F2 @ N3 ) @ ( F2 @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq @ nat @ N2 @ N4 )
=> ( ord_less_eq @ A @ ( F2 @ N2 ) @ ( F2 @ N4 ) ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_170_le__imp__less__Suc,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq @ nat @ M2 @ N2 )
=> ( ord_less @ nat @ M2 @ ( suc @ N2 ) ) ) ).
% le_imp_less_Suc
thf(fact_171_less__eq__Suc__le,axiom,
( ( ord_less @ nat )
= ( ^ [N: nat] : ( ord_less_eq @ nat @ ( suc @ N ) ) ) ) ).
% less_eq_Suc_le
thf(fact_172_less__Suc__eq__le,axiom,
! [M2: nat,N2: nat] :
( ( ord_less @ nat @ M2 @ ( suc @ N2 ) )
= ( ord_less_eq @ nat @ M2 @ N2 ) ) ).
% less_Suc_eq_le
thf(fact_173_le__less__Suc__eq,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq @ nat @ M2 @ N2 )
=> ( ( ord_less @ nat @ N2 @ ( suc @ M2 ) )
= ( N2 = M2 ) ) ) ).
% le_less_Suc_eq
thf(fact_174_Suc__le__lessD,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq @ nat @ ( suc @ M2 ) @ N2 )
=> ( ord_less @ nat @ M2 @ N2 ) ) ).
% Suc_le_lessD
thf(fact_175_inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( P @ J )
=> ( ! [N3: nat] :
( ( ord_less_eq @ nat @ I @ N3 )
=> ( ( ord_less @ nat @ N3 @ J )
=> ( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) ) ) )
=> ( P @ I ) ) ) ) ).
% inc_induct
thf(fact_176_dec__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( P @ I )
=> ( ! [N3: nat] :
( ( ord_less_eq @ nat @ I @ N3 )
=> ( ( ord_less @ nat @ N3 @ J )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) ) )
=> ( P @ J ) ) ) ) ).
% dec_induct
thf(fact_177_Suc__le__eq,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq @ nat @ ( suc @ M2 ) @ N2 )
= ( ord_less @ nat @ M2 @ N2 ) ) ).
% Suc_le_eq
thf(fact_178_Suc__leI,axiom,
! [M2: nat,N2: nat] :
( ( ord_less @ nat @ M2 @ N2 )
=> ( ord_less_eq @ nat @ ( suc @ M2 ) @ N2 ) ) ).
% Suc_leI
thf(fact_179_mono__nat__linear__lb,axiom,
! [F2: nat > nat,M2: nat,K: nat] :
( ! [M3: nat,N3: nat] :
( ( ord_less @ nat @ M3 @ N3 )
=> ( ord_less @ nat @ ( F2 @ M3 ) @ ( F2 @ N3 ) ) )
=> ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( F2 @ M2 ) @ K ) @ ( F2 @ ( plus_plus @ nat @ M2 @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_180_ex__least__nat__le,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ N2 )
=> ( ~ ( P @ ( zero_zero @ nat ) )
=> ? [K3: nat] :
( ( ord_less_eq @ nat @ K3 @ N2 )
& ! [I5: nat] :
( ( ord_less @ nat @ I5 @ K3 )
=> ~ ( P @ I5 ) )
& ( P @ K3 ) ) ) ) ).
% ex_least_nat_le
thf(fact_181_list__eq__iff__nth__eq,axiom,
! [A: $tType] :
( ( ^ [Y5: list @ A,Z2: list @ A] : Y5 = Z2 )
= ( ^ [Xs3: list @ A,Ys3: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs3 )
= ( size_size @ ( list @ A ) @ Ys3 ) )
& ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs3 ) )
=> ( ( nth @ A @ Xs3 @ I3 )
= ( nth @ A @ Ys3 @ I3 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_182_Skolem__list__nth,axiom,
! [A: $tType,K: nat,P: nat > A > $o] :
( ( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ K )
=> ? [X6: A] : ( P @ I3 @ X6 ) ) )
= ( ? [Xs3: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs3 )
= K )
& ! [I3: nat] :
( ( ord_less @ nat @ I3 @ K )
=> ( P @ I3 @ ( nth @ A @ Xs3 @ I3 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_183_nth__equalityI,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ A ) @ Ys ) )
=> ( ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( nth @ A @ Xs @ I4 )
= ( nth @ A @ Ys @ I4 ) ) )
=> ( Xs = Ys ) ) ) ).
% nth_equalityI
thf(fact_184_set__update__subsetI,axiom,
! [A: $tType,Xs: list @ A,A4: set @ A,X: A,I: nat] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ A4 )
=> ( ( member @ A @ X @ A4 )
=> ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ ( list_update @ A @ Xs @ I @ X ) ) @ A4 ) ) ) ).
% set_update_subsetI
thf(fact_185_all__set__conv__all__nth,axiom,
! [A: $tType,Xs: list @ A,P: A > $o] :
( ( ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
=> ( P @ X4 ) ) )
= ( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( P @ ( nth @ A @ Xs @ I3 ) ) ) ) ) ).
% all_set_conv_all_nth
thf(fact_186_all__nth__imp__all__set,axiom,
! [A: $tType,Xs: list @ A,P: A > $o,X: A] :
( ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( P @ ( nth @ A @ Xs @ I4 ) ) )
=> ( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( P @ X ) ) ) ).
% all_nth_imp_all_set
thf(fact_187_in__set__conv__nth,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
= ( ? [I3: nat] :
( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs ) )
& ( ( nth @ A @ Xs @ I3 )
= X ) ) ) ) ).
% in_set_conv_nth
thf(fact_188_list__ball__nth,axiom,
! [A: $tType,N2: nat,Xs: list @ A,P: A > $o] :
( ( ord_less @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Xs ) )
=> ( P @ X5 ) )
=> ( P @ ( nth @ A @ Xs @ N2 ) ) ) ) ).
% list_ball_nth
thf(fact_189_nth__mem,axiom,
! [A: $tType,N2: nat,Xs: list @ A] :
( ( ord_less @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( member @ A @ ( nth @ A @ Xs @ N2 ) @ ( set2 @ A @ Xs ) ) ) ).
% nth_mem
thf(fact_190_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_191_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_192_ord__le__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A3: A,B3: A,F2: A > B,C2: B] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ( F2 @ B3 )
= C2 )
=> ( ! [X5: A,Y3: A] :
( ( ord_less_eq @ A @ X5 @ Y3 )
=> ( ord_less_eq @ B @ ( F2 @ X5 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq @ B @ ( F2 @ A3 ) @ C2 ) ) ) ) ) ).
% ord_le_eq_subst
thf(fact_193_ord__eq__le__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A3: A,F2: B > A,B3: B,C2: B] :
( ( A3
= ( F2 @ B3 ) )
=> ( ( ord_less_eq @ B @ B3 @ C2 )
=> ( ! [X5: B,Y3: B] :
( ( ord_less_eq @ B @ X5 @ Y3 )
=> ( ord_less_eq @ A @ ( F2 @ X5 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq @ A @ A3 @ ( F2 @ C2 ) ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_194_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_195_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_196_order__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C )
& ( order @ A ) )
=> ! [A3: A,B3: A,F2: A > C,C2: C] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ C @ ( F2 @ B3 ) @ C2 )
=> ( ! [X5: A,Y3: A] :
( ( ord_less_eq @ A @ X5 @ Y3 )
=> ( ord_less_eq @ C @ ( F2 @ X5 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq @ C @ ( F2 @ A3 ) @ C2 ) ) ) ) ) ).
% order_subst2
thf(fact_197_order__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A3: A,F2: B > A,B3: B,C2: B] :
( ( ord_less_eq @ A @ A3 @ ( F2 @ B3 ) )
=> ( ( ord_less_eq @ B @ B3 @ C2 )
=> ( ! [X5: B,Y3: B] :
( ( ord_less_eq @ B @ X5 @ Y3 )
=> ( ord_less_eq @ A @ ( F2 @ X5 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq @ A @ A3 @ ( F2 @ C2 ) ) ) ) ) ) ).
% order_subst1
thf(fact_198_Orderings_Oorder__eq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y5: A,Z2: A] : Y5 = Z2 )
= ( ^ [A5: A,B4: A] :
( ( ord_less_eq @ A @ A5 @ B4 )
& ( ord_less_eq @ A @ B4 @ A5 ) ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_199_le__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ( ( ord_less_eq @ ( A > B ) )
= ( ^ [F3: A > B,G2: A > B] :
! [X4: A] : ( ord_less_eq @ B @ ( F3 @ X4 ) @ ( G2 @ X4 ) ) ) ) ) ).
% le_fun_def
thf(fact_200_le__funI,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F2: A > B,G: A > B] :
( ! [X5: A] : ( ord_less_eq @ B @ ( F2 @ X5 ) @ ( G @ X5 ) )
=> ( ord_less_eq @ ( A > B ) @ F2 @ G ) ) ) ).
% le_funI
thf(fact_201_le__funE,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F2: A > B,G: A > B,X: A] :
( ( ord_less_eq @ ( A > B ) @ F2 @ G )
=> ( ord_less_eq @ B @ ( F2 @ X ) @ ( G @ X ) ) ) ) ).
% le_funE
thf(fact_202_le__funD,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F2: A > B,G: A > B,X: A] :
( ( ord_less_eq @ ( A > B ) @ F2 @ G )
=> ( ord_less_eq @ B @ ( F2 @ X ) @ ( G @ X ) ) ) ) ).
% le_funD
thf(fact_203_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_204_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_205_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_206_dual__order_Oeq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y5: A,Z2: A] : Y5 = Z2 )
= ( ^ [A5: A,B4: A] :
( ( ord_less_eq @ A @ B4 @ A5 )
& ( ord_less_eq @ A @ A5 @ B4 ) ) ) ) ) ).
% dual_order.eq_iff
thf(fact_207_linorder__wlog,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > A > $o,A3: A,B3: A] :
( ! [A6: A,B5: A] :
( ( ord_less_eq @ A @ A6 @ B5 )
=> ( P @ A6 @ B5 ) )
=> ( ! [A6: A,B5: A] :
( ( P @ B5 @ A6 )
=> ( P @ A6 @ B5 ) )
=> ( P @ A3 @ B3 ) ) ) ) ).
% linorder_wlog
thf(fact_208_order__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A,Z3: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less_eq @ A @ Y @ Z3 )
=> ( ord_less_eq @ A @ X @ Z3 ) ) ) ) ).
% order_trans
thf(fact_209_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_210_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_211_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_212_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_213_order__class_Oorder__eq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y5: A,Z2: A] : Y5 = Z2 )
= ( ^ [X4: A,Y6: A] :
( ( ord_less_eq @ A @ X4 @ Y6 )
& ( ord_less_eq @ A @ Y6 @ X4 ) ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_214_le__cases3,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A,Z3: A] :
( ( ( ord_less_eq @ A @ X @ Y )
=> ~ ( ord_less_eq @ A @ Y @ Z3 ) )
=> ( ( ( ord_less_eq @ A @ Y @ X )
=> ~ ( ord_less_eq @ A @ X @ Z3 ) )
=> ( ( ( ord_less_eq @ A @ X @ Z3 )
=> ~ ( ord_less_eq @ A @ Z3 @ Y ) )
=> ( ( ( ord_less_eq @ A @ Z3 @ Y )
=> ~ ( ord_less_eq @ A @ Y @ X ) )
=> ( ( ( ord_less_eq @ A @ Y @ Z3 )
=> ~ ( ord_less_eq @ A @ Z3 @ X ) )
=> ~ ( ( ord_less_eq @ A @ Z3 @ X )
=> ~ ( ord_less_eq @ A @ X @ Y ) ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_215_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_216_set__update__memI,axiom,
! [A: $tType,N2: nat,Xs: list @ A,X: A] :
( ( ord_less @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( member @ A @ X @ ( set2 @ A @ ( list_update @ A @ Xs @ N2 @ X ) ) ) ) ).
% set_update_memI
thf(fact_217_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_218_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_219_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_220_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_221_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_222_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_223_order__less__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C )
& ( order @ A ) )
=> ! [A3: A,B3: A,F2: A > C,C2: C] :
( ( ord_less @ A @ A3 @ B3 )
=> ( ( ord_less @ C @ ( F2 @ B3 ) @ C2 )
=> ( ! [X5: A,Y3: A] :
( ( ord_less @ A @ X5 @ Y3 )
=> ( ord_less @ C @ ( F2 @ X5 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less @ C @ ( F2 @ A3 ) @ C2 ) ) ) ) ) ).
% order_less_subst2
thf(fact_224_order__less__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A3: A,F2: B > A,B3: B,C2: B] :
( ( ord_less @ A @ A3 @ ( F2 @ B3 ) )
=> ( ( ord_less @ B @ B3 @ C2 )
=> ( ! [X5: B,Y3: B] :
( ( ord_less @ B @ X5 @ Y3 )
=> ( ord_less @ A @ ( F2 @ X5 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less @ A @ A3 @ ( F2 @ C2 ) ) ) ) ) ) ).
% order_less_subst1
thf(fact_225_order__less__irrefl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A] :
~ ( ord_less @ A @ X @ X ) ) ).
% order_less_irrefl
thf(fact_226_ord__less__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A3: A,B3: A,F2: A > B,C2: B] :
( ( ord_less @ A @ A3 @ B3 )
=> ( ( ( F2 @ B3 )
= C2 )
=> ( ! [X5: A,Y3: A] :
( ( ord_less @ A @ X5 @ Y3 )
=> ( ord_less @ B @ ( F2 @ X5 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less @ B @ ( F2 @ A3 ) @ C2 ) ) ) ) ) ).
% ord_less_eq_subst
thf(fact_227_ord__eq__less__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A3: A,F2: B > A,B3: B,C2: B] :
( ( A3
= ( F2 @ B3 ) )
=> ( ( ord_less @ B @ B3 @ C2 )
=> ( ! [X5: B,Y3: B] :
( ( ord_less @ B @ X5 @ Y3 )
=> ( ord_less @ A @ ( F2 @ X5 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less @ A @ A3 @ ( F2 @ C2 ) ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_228_order__less__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A,Z3: A] :
( ( ord_less @ A @ X @ Y )
=> ( ( ord_less @ A @ Y @ Z3 )
=> ( ord_less @ A @ X @ Z3 ) ) ) ) ).
% order_less_trans
thf(fact_229_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_230_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_231_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_232_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_233_measure__induct__rule,axiom,
! [B: $tType,A: $tType] :
( ( wellorder @ B )
=> ! [F2: A > B,P: A > $o,A3: A] :
( ! [X5: A] :
( ! [Y4: A] :
( ( ord_less @ B @ ( F2 @ Y4 ) @ ( F2 @ X5 ) )
=> ( P @ Y4 ) )
=> ( P @ X5 ) )
=> ( P @ A3 ) ) ) ).
% measure_induct_rule
thf(fact_234_measure__induct,axiom,
! [B: $tType,A: $tType] :
( ( wellorder @ B )
=> ! [F2: A > B,P: A > $o,A3: A] :
( ! [X5: A] :
( ! [Y4: A] :
( ( ord_less @ B @ ( F2 @ Y4 ) @ ( F2 @ X5 ) )
=> ( P @ Y4 ) )
=> ( P @ X5 ) )
=> ( P @ A3 ) ) ) ).
% measure_induct
thf(fact_235_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_236_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_237_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_238_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_239_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_240_linorder__less__wlog,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > A > $o,A3: A,B3: A] :
( ! [A6: A,B5: A] :
( ( ord_less @ A @ A6 @ B5 )
=> ( P @ A6 @ B5 ) )
=> ( ! [A6: A] : ( P @ A6 @ A6 )
=> ( ! [A6: A,B5: A] :
( ( P @ B5 @ A6 )
=> ( P @ A6 @ B5 ) )
=> ( P @ A3 @ B3 ) ) ) ) ) ).
% linorder_less_wlog
thf(fact_241_exists__least__iff,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ( ( ^ [P2: A > $o] :
? [X7: A] : ( P2 @ X7 ) )
= ( ^ [P3: A > $o] :
? [N: A] :
( ( P3 @ N )
& ! [M: A] :
( ( ord_less @ A @ M @ N )
=> ~ ( P3 @ M ) ) ) ) ) ) ).
% exists_least_iff
thf(fact_242_dual__order_Oirrefl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: A] :
~ ( ord_less @ A @ A3 @ A3 ) ) ).
% dual_order.irrefl
thf(fact_243_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_244_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_245_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_246_less__induct,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [P: A > $o,A3: A] :
( ! [X5: A] :
( ! [Y4: A] :
( ( ord_less @ A @ Y4 @ X5 )
=> ( P @ Y4 ) )
=> ( P @ X5 ) )
=> ( P @ A3 ) ) ) ).
% less_induct
thf(fact_247_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_248_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_249_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_250_less__imp__neq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ( X != Y ) ) ) ).
% less_imp_neq
thf(fact_251_dense,axiom,
! [A: $tType] :
( ( dense_order @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ? [Z: A] :
( ( ord_less @ A @ X @ Z )
& ( ord_less @ A @ Z @ Y ) ) ) ) ).
% dense
thf(fact_252_gt__ex,axiom,
! [A: $tType] :
( ( no_top @ A )
=> ! [X: A] :
? [X_12: A] : ( ord_less @ A @ X @ X_12 ) ) ).
% gt_ex
thf(fact_253_lt__ex,axiom,
! [A: $tType] :
( ( no_bot @ A )
=> ! [X: A] :
? [Y3: A] : ( ord_less @ A @ Y3 @ X ) ) ).
% lt_ex
thf(fact_254_nth__list__update,axiom,
! [A: $tType,I: nat,Xs: list @ A,J: nat,X: A] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ( I = J )
=> ( ( nth @ A @ ( list_update @ A @ Xs @ I @ X ) @ J )
= X ) )
& ( ( I != J )
=> ( ( nth @ A @ ( list_update @ A @ Xs @ I @ X ) @ J )
= ( nth @ A @ Xs @ J ) ) ) ) ) ).
% nth_list_update
thf(fact_255_list__update__same__conv,axiom,
! [A: $tType,I: nat,Xs: list @ A,X: A] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ( list_update @ A @ Xs @ I @ X )
= Xs )
= ( ( nth @ A @ Xs @ I )
= X ) ) ) ).
% list_update_same_conv
thf(fact_256_infinite__descent__measure,axiom,
! [A: $tType,P: A > $o,V: A > nat,X: A] :
( ! [X5: A] :
( ~ ( P @ X5 )
=> ? [Y4: A] :
( ( ord_less @ nat @ ( V @ Y4 ) @ ( V @ X5 ) )
& ~ ( P @ Y4 ) ) )
=> ( P @ X ) ) ).
% infinite_descent_measure
thf(fact_257_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_258_infinite__descent,axiom,
! [P: nat > $o,N2: nat] :
( ! [N3: nat] :
( ~ ( P @ N3 )
=> ? [M5: nat] :
( ( ord_less @ nat @ M5 @ N3 )
& ~ ( P @ M5 ) ) )
=> ( P @ N2 ) ) ).
% infinite_descent
thf(fact_259_nat__less__induct,axiom,
! [P: nat > $o,N2: nat] :
( ! [N3: nat] :
( ! [M5: nat] :
( ( ord_less @ nat @ M5 @ N3 )
=> ( P @ M5 ) )
=> ( P @ N3 ) )
=> ( P @ N2 ) ) ).
% nat_less_induct
thf(fact_260_less__irrefl__nat,axiom,
! [N2: nat] :
~ ( ord_less @ nat @ N2 @ N2 ) ).
% less_irrefl_nat
thf(fact_261_less__not__refl3,axiom,
! [S: nat,T2: nat] :
( ( ord_less @ nat @ S @ T2 )
=> ( S != T2 ) ) ).
% less_not_refl3
thf(fact_262_less__not__refl2,axiom,
! [N2: nat,M2: nat] :
( ( ord_less @ nat @ N2 @ M2 )
=> ( M2 != N2 ) ) ).
% less_not_refl2
thf(fact_263_less__not__refl,axiom,
! [N2: nat] :
~ ( ord_less @ nat @ N2 @ N2 ) ).
% less_not_refl
thf(fact_264_nat__neq__iff,axiom,
! [M2: nat,N2: nat] :
( ( M2 != N2 )
= ( ( ord_less @ nat @ M2 @ N2 )
| ( ord_less @ nat @ N2 @ M2 ) ) ) ).
% nat_neq_iff
thf(fact_265_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B3: nat] :
( ( P @ K )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq @ nat @ Y3 @ B3 ) )
=> ? [X5: nat] :
( ( P @ X5 )
& ! [Y4: nat] :
( ( P @ Y4 )
=> ( ord_less_eq @ nat @ Y4 @ X5 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_266_nat__le__linear,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq @ nat @ M2 @ N2 )
| ( ord_less_eq @ nat @ N2 @ M2 ) ) ).
% nat_le_linear
thf(fact_267_le__antisym,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq @ nat @ M2 @ N2 )
=> ( ( ord_less_eq @ nat @ N2 @ M2 )
=> ( M2 = N2 ) ) ) ).
% le_antisym
thf(fact_268_eq__imp__le,axiom,
! [M2: nat,N2: nat] :
( ( M2 = N2 )
=> ( ord_less_eq @ nat @ M2 @ N2 ) ) ).
% eq_imp_le
thf(fact_269_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_270_le__refl,axiom,
! [N2: nat] : ( ord_less_eq @ nat @ N2 @ N2 ) ).
% le_refl
thf(fact_271_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_272_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_273_order__less__le__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C )
& ( order @ A ) )
=> ! [A3: A,B3: A,F2: A > C,C2: C] :
( ( ord_less @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ C @ ( F2 @ B3 ) @ C2 )
=> ( ! [X5: A,Y3: A] :
( ( ord_less @ A @ X5 @ Y3 )
=> ( ord_less @ C @ ( F2 @ X5 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less @ C @ ( F2 @ A3 ) @ C2 ) ) ) ) ) ).
% order_less_le_subst2
thf(fact_274_order__less__le__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A3: A,F2: B > A,B3: B,C2: B] :
( ( ord_less @ A @ A3 @ ( F2 @ B3 ) )
=> ( ( ord_less_eq @ B @ B3 @ C2 )
=> ( ! [X5: B,Y3: B] :
( ( ord_less_eq @ B @ X5 @ Y3 )
=> ( ord_less_eq @ A @ ( F2 @ X5 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less @ A @ A3 @ ( F2 @ C2 ) ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_275_order__le__less__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C )
& ( order @ A ) )
=> ! [A3: A,B3: A,F2: A > C,C2: C] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ord_less @ C @ ( F2 @ B3 ) @ C2 )
=> ( ! [X5: A,Y3: A] :
( ( ord_less_eq @ A @ X5 @ Y3 )
=> ( ord_less_eq @ C @ ( F2 @ X5 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less @ C @ ( F2 @ A3 ) @ C2 ) ) ) ) ) ).
% order_le_less_subst2
thf(fact_276_order__le__less__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A3: A,F2: B > A,B3: B,C2: B] :
( ( ord_less_eq @ A @ A3 @ ( F2 @ B3 ) )
=> ( ( ord_less @ B @ B3 @ C2 )
=> ( ! [X5: B,Y3: B] :
( ( ord_less @ B @ X5 @ Y3 )
=> ( ord_less @ A @ ( F2 @ X5 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less @ A @ A3 @ ( F2 @ C2 ) ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_277_order__less__le__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A,Z3: A] :
( ( ord_less @ A @ X @ Y )
=> ( ( ord_less_eq @ A @ Y @ Z3 )
=> ( ord_less @ A @ X @ Z3 ) ) ) ) ).
% order_less_le_trans
thf(fact_278_order__le__less__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A,Z3: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less @ A @ Y @ Z3 )
=> ( ord_less @ A @ X @ Z3 ) ) ) ) ).
% order_le_less_trans
thf(fact_279_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_280_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_281_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_282_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_283_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_284_order__less__le,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less @ A )
= ( ^ [X4: A,Y6: A] :
( ( ord_less_eq @ A @ X4 @ Y6 )
& ( X4 != Y6 ) ) ) ) ) ).
% order_less_le
thf(fact_285_order__le__less,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [X4: A,Y6: A] :
( ( ord_less @ A @ X4 @ Y6 )
| ( X4 = Y6 ) ) ) ) ) ).
% order_le_less
thf(fact_286_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_287_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_288_dual__order_Ostrict__iff__not,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( ( ord_less @ A )
= ( ^ [B4: A,A5: A] :
( ( ord_less_eq @ A @ B4 @ A5 )
& ~ ( ord_less_eq @ A @ A5 @ B4 ) ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_289_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_290_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_291_dual__order_Ostrict__iff__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less @ A )
= ( ^ [B4: A,A5: A] :
( ( ord_less_eq @ A @ B4 @ A5 )
& ( A5 != B4 ) ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_292_dual__order_Oorder__iff__strict,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B4: A,A5: A] :
( ( ord_less @ A @ B4 @ A5 )
| ( A5 = B4 ) ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_293_dense__le__bounded,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [X: A,Y: A,Z3: A] :
( ( ord_less @ A @ X @ Y )
=> ( ! [W: A] :
( ( ord_less @ A @ X @ W )
=> ( ( ord_less @ A @ W @ Y )
=> ( ord_less_eq @ A @ W @ Z3 ) ) )
=> ( ord_less_eq @ A @ Y @ Z3 ) ) ) ) ).
% dense_le_bounded
thf(fact_294_dense__ge__bounded,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [Z3: A,X: A,Y: A] :
( ( ord_less @ A @ Z3 @ X )
=> ( ! [W: A] :
( ( ord_less @ A @ Z3 @ W )
=> ( ( ord_less @ A @ W @ X )
=> ( ord_less_eq @ A @ Y @ W ) ) )
=> ( ord_less_eq @ A @ Y @ Z3 ) ) ) ) ).
% dense_ge_bounded
thf(fact_295_order_Ostrict__iff__not,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( ( ord_less @ A )
= ( ^ [A5: A,B4: A] :
( ( ord_less_eq @ A @ A5 @ B4 )
& ~ ( ord_less_eq @ A @ B4 @ A5 ) ) ) ) ) ).
% order.strict_iff_not
thf(fact_296_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_297_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_298_order_Ostrict__iff__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less @ A )
= ( ^ [A5: A,B4: A] :
( ( ord_less_eq @ A @ A5 @ B4 )
& ( A5 != B4 ) ) ) ) ) ).
% order.strict_iff_order
thf(fact_299_order_Oorder__iff__strict,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A5: A,B4: A] :
( ( ord_less @ A @ A5 @ B4 )
| ( A5 = B4 ) ) ) ) ) ).
% order.order_iff_strict
thf(fact_300_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_301_less__le__not__le,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( ( ord_less @ A )
= ( ^ [X4: A,Y6: A] :
( ( ord_less_eq @ A @ X4 @ Y6 )
& ~ ( ord_less_eq @ A @ Y6 @ X4 ) ) ) ) ) ).
% less_le_not_le
thf(fact_302_dense__le,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [Y: A,Z3: A] :
( ! [X5: A] :
( ( ord_less @ A @ X5 @ Y )
=> ( ord_less_eq @ A @ X5 @ Z3 ) )
=> ( ord_less_eq @ A @ Y @ Z3 ) ) ) ).
% dense_le
thf(fact_303_dense__ge,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [Z3: A,Y: A] :
( ! [X5: A] :
( ( ord_less @ A @ Z3 @ X5 )
=> ( ord_less_eq @ A @ Y @ X5 ) )
=> ( ord_less_eq @ A @ Y @ Z3 ) ) ) ).
% dense_ge
thf(fact_304_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_305_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_306_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_307_leI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ~ ( ord_less @ A @ X @ Y )
=> ( ord_less_eq @ A @ Y @ X ) ) ) ).
% leI
thf(fact_308_leD,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ Y @ X )
=> ~ ( ord_less @ A @ X @ Y ) ) ) ).
% leD
thf(fact_309_less__mono__imp__le__mono,axiom,
! [F2: nat > nat,I: nat,J: nat] :
( ! [I4: nat,J3: nat] :
( ( ord_less @ nat @ I4 @ J3 )
=> ( ord_less @ nat @ ( F2 @ I4 ) @ ( F2 @ J3 ) ) )
=> ( ( ord_less_eq @ nat @ I @ J )
=> ( ord_less_eq @ nat @ ( F2 @ I ) @ ( F2 @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_310_le__neq__implies__less,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq @ nat @ M2 @ N2 )
=> ( ( M2 != N2 )
=> ( ord_less @ nat @ M2 @ N2 ) ) ) ).
% le_neq_implies_less
thf(fact_311_less__or__eq__imp__le,axiom,
! [M2: nat,N2: nat] :
( ( ( ord_less @ nat @ M2 @ N2 )
| ( M2 = N2 ) )
=> ( ord_less_eq @ nat @ M2 @ N2 ) ) ).
% less_or_eq_imp_le
thf(fact_312_le__eq__less__or__eq,axiom,
( ( ord_less_eq @ nat )
= ( ^ [M: nat,N: nat] :
( ( ord_less @ nat @ M @ N )
| ( M = N ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_313_buildup__gives__valid,axiom,
! [N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( vEBT_invar_vebt @ ( vEBT_vebt_buildup @ N2 ) @ N2 ) ) ).
% buildup_gives_valid
thf(fact_314_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_315_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_316_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_317_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_318_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_319_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_320_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_321_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_322_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_323_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_324_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_325_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_326_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_327_le__zero__eq,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [N2: A] :
( ( ord_less_eq @ A @ N2 @ ( zero_zero @ A ) )
= ( N2
= ( zero_zero @ A ) ) ) ) ).
% le_zero_eq
thf(fact_328_not__gr__zero,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [N2: A] :
( ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ N2 ) )
= ( N2
= ( zero_zero @ A ) ) ) ) ).
% not_gr_zero
thf(fact_329_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_330_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_331_add_Oright__neutral,axiom,
! [A: $tType] :
( ( monoid_add @ A )
=> ! [A3: A] :
( ( plus_plus @ A @ A3 @ ( zero_zero @ A ) )
= A3 ) ) ).
% add.right_neutral
thf(fact_332_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_333_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_334_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_335_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_336_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_337_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_338_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_339_add__0,axiom,
! [A: $tType] :
( ( monoid_add @ A )
=> ! [A3: A] :
( ( plus_plus @ A @ ( zero_zero @ A ) @ A3 )
= A3 ) ) ).
% add_0
thf(fact_340_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_341_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_342_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_343_less__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ( ( ord_less @ ( A > B ) )
= ( ^ [F3: A > B,G2: A > B] :
( ( ord_less_eq @ ( A > B ) @ F3 @ G2 )
& ~ ( ord_less_eq @ ( A > B ) @ G2 @ F3 ) ) ) ) ) ).
% less_fun_def
thf(fact_344_zero__reorient,axiom,
! [A: $tType] :
( ( zero @ A )
=> ! [X: A] :
( ( ( zero_zero @ A )
= X )
= ( X
= ( zero_zero @ A ) ) ) ) ).
% zero_reorient
thf(fact_345_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: $tType] :
( ( ab_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 ) ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_346_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_347_group__cancel_Oadd1,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A4: A,K: A,A3: A,B3: A] :
( ( A4
= ( plus_plus @ A @ K @ A3 ) )
=> ( ( plus_plus @ A @ A4 @ B3 )
= ( plus_plus @ A @ K @ ( plus_plus @ A @ A3 @ B3 ) ) ) ) ) ).
% group_cancel.add1
thf(fact_348_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_349_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_350_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_351_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_352_add_Ocommute,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A )
=> ( ( plus_plus @ A )
= ( ^ [A5: A,B4: A] : ( plus_plus @ A @ B4 @ A5 ) ) ) ) ).
% add.commute
thf(fact_353_add_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 ) ) ) ) ).
% add.left_commute
thf(fact_354_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_355_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_356_zero__le,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [X: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ X ) ) ).
% zero_le
thf(fact_357_gr__zeroI,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [N2: A] :
( ( N2
!= ( zero_zero @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ N2 ) ) ) ).
% gr_zeroI
thf(fact_358_not__less__zero,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [N2: A] :
~ ( ord_less @ A @ N2 @ ( zero_zero @ A ) ) ) ).
% not_less_zero
thf(fact_359_gr__implies__not__zero,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [M2: A,N2: A] :
( ( ord_less @ A @ M2 @ N2 )
=> ( N2
!= ( zero_zero @ A ) ) ) ) ).
% gr_implies_not_zero
thf(fact_360_zero__less__iff__neq__zero,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [N2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ N2 )
= ( N2
!= ( zero_zero @ A ) ) ) ) ).
% zero_less_iff_neq_zero
thf(fact_361_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_362_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_363_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_364_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_365_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_366_less__eqE,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ~ ! [C3: A] :
( B3
!= ( plus_plus @ A @ A3 @ C3 ) ) ) ) ).
% less_eqE
thf(fact_367_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_368_le__iff__add,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A5: A,B4: A] :
? [C4: A] :
( B4
= ( plus_plus @ A @ A5 @ C4 ) ) ) ) ) ).
% le_iff_add
thf(fact_369_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_370_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_371_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_372_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_373_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_374_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_375_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_376_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_377_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_378_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_379_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_380_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_381_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_382_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_383_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_384_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_385_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_386_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_387_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_388_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_389_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_390_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_391_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_392_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_393_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_394_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_395_canonically__ordered__monoid__add__class_OlessE,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ A3 @ B3 )
=> ~ ! [C3: A] :
( ( B3
= ( plus_plus @ A @ A3 @ C3 ) )
=> ( C3
= ( zero_zero @ A ) ) ) ) ) ).
% canonically_ordered_monoid_add_class.lessE
thf(fact_396_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_397_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_398_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_399_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_400_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_401_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_402_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_403_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_404_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_405_buildup__nothing__in__leaf,axiom,
! [N2: nat,X: nat] :
~ ( vEBT_V5719532721284313246member @ ( vEBT_vebt_buildup @ N2 ) @ X ) ).
% buildup_nothing_in_leaf
thf(fact_406_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_407_ex__has__greatest__nat__lemma,axiom,
! [A: $tType,P: A > $o,K: A,F2: A > nat,N2: nat] :
( ( P @ K )
=> ( ! [X5: A] :
( ( P @ X5 )
=> ? [Y4: A] :
( ( P @ Y4 )
& ~ ( ord_less_eq @ nat @ ( F2 @ Y4 ) @ ( F2 @ X5 ) ) ) )
=> ? [Y3: A] :
( ( P @ Y3 )
& ~ ( ord_less @ nat @ ( F2 @ Y3 ) @ ( plus_plus @ nat @ ( F2 @ K ) @ N2 ) ) ) ) ) ).
% ex_has_greatest_nat_lemma
thf(fact_408_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_409_buildup__gives__empty,axiom,
! [N2: nat] :
( ( vEBT_VEBT_set_vebt @ ( vEBT_vebt_buildup @ N2 ) )
= ( bot_bot @ ( set @ nat ) ) ) ).
% buildup_gives_empty
thf(fact_410_subsetI,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] :
( ! [X5: A] :
( ( member @ A @ X5 @ A4 )
=> ( member @ A @ X5 @ B2 ) )
=> ( ord_less_eq @ ( set @ A ) @ A4 @ B2 ) ) ).
% subsetI
thf(fact_411_psubsetI,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B2 )
=> ( ( A4 != B2 )
=> ( ord_less @ ( set @ A ) @ A4 @ B2 ) ) ) ).
% psubsetI
thf(fact_412_subset__antisym,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ A4 )
=> ( A4 = B2 ) ) ) ).
% subset_antisym
thf(fact_413_buildup__nothing__in__min__max,axiom,
! [N2: nat,X: nat] :
~ ( vEBT_VEBT_membermima @ ( vEBT_vebt_buildup @ N2 ) @ X ) ).
% buildup_nothing_in_min_max
thf(fact_414_set__vebt__finite,axiom,
! [T2: vEBT_VEBT,N2: nat] :
( ( vEBT_invar_vebt @ T2 @ N2 )
=> ( finite_finite2 @ nat @ ( vEBT_VEBT_set_vebt @ T2 ) ) ) ).
% set_vebt_finite
thf(fact_415_both__member__options__def,axiom,
( vEBT_V8194947554948674370ptions
= ( ^ [T3: vEBT_VEBT,X4: nat] :
( ( vEBT_V5719532721284313246member @ T3 @ X4 )
| ( vEBT_VEBT_membermima @ T3 @ X4 ) ) ) ) ).
% both_member_options_def
thf(fact_416_bot__apply,axiom,
! [C: $tType,D: $tType] :
( ( bot @ C )
=> ( ( bot_bot @ ( D > C ) )
= ( ^ [X4: D] : ( bot_bot @ C ) ) ) ) ).
% bot_apply
thf(fact_417_empty__Collect__eq,axiom,
! [A: $tType,P: A > $o] :
( ( ( bot_bot @ ( set @ A ) )
= ( collect @ A @ P ) )
= ( ! [X4: A] :
~ ( P @ X4 ) ) ) ).
% empty_Collect_eq
thf(fact_418_Collect__empty__eq,axiom,
! [A: $tType,P: A > $o] :
( ( ( collect @ A @ P )
= ( bot_bot @ ( set @ A ) ) )
= ( ! [X4: A] :
~ ( P @ X4 ) ) ) ).
% Collect_empty_eq
thf(fact_419_all__not__in__conv,axiom,
! [A: $tType,A4: set @ A] :
( ( ! [X4: A] :
~ ( member @ A @ X4 @ A4 ) )
= ( A4
= ( bot_bot @ ( set @ A ) ) ) ) ).
% all_not_in_conv
thf(fact_420_empty__iff,axiom,
! [A: $tType,C2: A] :
~ ( member @ A @ C2 @ ( bot_bot @ ( set @ A ) ) ) ).
% empty_iff
thf(fact_421_member__valid__both__member__options,axiom,
! [Tree: vEBT_VEBT,N2: nat,X: nat] :
( ( vEBT_invar_vebt @ Tree @ N2 )
=> ( ( vEBT_vebt_member @ Tree @ X )
=> ( ( vEBT_V5719532721284313246member @ Tree @ X )
| ( vEBT_VEBT_membermima @ Tree @ X ) ) ) ) ).
% member_valid_both_member_options
thf(fact_422_empty__subsetI,axiom,
! [A: $tType,A4: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A4 ) ).
% empty_subsetI
thf(fact_423_subset__empty,axiom,
! [A: $tType,A4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ ( bot_bot @ ( set @ A ) ) )
= ( A4
= ( bot_bot @ ( set @ A ) ) ) ) ).
% subset_empty
thf(fact_424_List_Ofinite__set,axiom,
! [A: $tType,Xs: list @ A] : ( finite_finite2 @ A @ ( set2 @ A @ Xs ) ) ).
% List.finite_set
thf(fact_425_ex__min__if__finite,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [S2: set @ A] :
( ( finite_finite2 @ A @ S2 )
=> ( ( S2
!= ( bot_bot @ ( set @ A ) ) )
=> ? [X5: A] :
( ( member @ A @ X5 @ S2 )
& ~ ? [Xa: A] :
( ( member @ A @ Xa @ S2 )
& ( ord_less @ A @ Xa @ X5 ) ) ) ) ) ) ).
% ex_min_if_finite
thf(fact_426_not__psubset__empty,axiom,
! [A: $tType,A4: set @ A] :
~ ( ord_less @ ( set @ A ) @ A4 @ ( bot_bot @ ( set @ A ) ) ) ).
% not_psubset_empty
thf(fact_427_psubset__trans,axiom,
! [A: $tType,A4: set @ A,B2: set @ A,C5: set @ A] :
( ( ord_less @ ( set @ A ) @ A4 @ B2 )
=> ( ( ord_less @ ( set @ A ) @ B2 @ C5 )
=> ( ord_less @ ( set @ A ) @ A4 @ C5 ) ) ) ).
% psubset_trans
thf(fact_428_psubsetD,axiom,
! [A: $tType,A4: set @ A,B2: set @ A,C2: A] :
( ( ord_less @ ( set @ A ) @ A4 @ B2 )
=> ( ( member @ A @ C2 @ A4 )
=> ( member @ A @ C2 @ B2 ) ) ) ).
% psubsetD
thf(fact_429_infinite__growing,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X8: set @ A] :
( ( X8
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ X8 )
=> ? [Xa: A] :
( ( member @ A @ Xa @ X8 )
& ( ord_less @ A @ X5 @ Xa ) ) )
=> ~ ( finite_finite2 @ A @ X8 ) ) ) ) ).
% infinite_growing
thf(fact_430_ex__in__conv,axiom,
! [A: $tType,A4: set @ A] :
( ( ? [X4: A] : ( member @ A @ X4 @ A4 ) )
= ( A4
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% ex_in_conv
thf(fact_431_equals0I,axiom,
! [A: $tType,A4: set @ A] :
( ! [Y3: A] :
~ ( member @ A @ Y3 @ A4 )
=> ( A4
= ( bot_bot @ ( set @ A ) ) ) ) ).
% equals0I
thf(fact_432_equals0D,axiom,
! [A: $tType,A4: set @ A,A3: A] :
( ( A4
= ( bot_bot @ ( set @ A ) ) )
=> ~ ( member @ A @ A3 @ A4 ) ) ).
% equals0D
thf(fact_433_emptyE,axiom,
! [A: $tType,A3: A] :
~ ( member @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ).
% emptyE
thf(fact_434_bot__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( bot @ B )
=> ( ( bot_bot @ ( A > B ) )
= ( ^ [X4: A] : ( bot_bot @ B ) ) ) ) ).
% bot_fun_def
thf(fact_435_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_436_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_437_bot_Oextremum,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [A3: A] : ( ord_less_eq @ A @ ( bot_bot @ A ) @ A3 ) ) ).
% bot.extremum
thf(fact_438_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_439_bot_Oextremum__strict,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [A3: A] :
~ ( ord_less @ A @ A3 @ ( bot_bot @ A ) ) ) ).
% bot.extremum_strict
thf(fact_440_finite__list,axiom,
! [A: $tType,A4: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ? [Xs2: list @ A] :
( ( set2 @ A @ Xs2 )
= A4 ) ) ).
% finite_list
thf(fact_441_linorder__neqE__linordered__idom,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A,Y: A] :
( ( X != Y )
=> ( ~ ( ord_less @ A @ X @ Y )
=> ( ord_less @ A @ Y @ X ) ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_442_linordered__field__no__ub,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X2: A] :
? [X_12: A] : ( ord_less @ A @ X2 @ X_12 ) ) ).
% linordered_field_no_ub
thf(fact_443_linordered__field__no__lb,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X2: A] :
? [Y3: A] : ( ord_less @ A @ Y3 @ X2 ) ) ).
% linordered_field_no_lb
thf(fact_444_subset__iff__psubset__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A7: set @ A,B6: set @ A] :
( ( ord_less @ ( set @ A ) @ A7 @ B6 )
| ( A7 = B6 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_445_subset__psubset__trans,axiom,
! [A: $tType,A4: set @ A,B2: set @ A,C5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B2 )
=> ( ( ord_less @ ( set @ A ) @ B2 @ C5 )
=> ( ord_less @ ( set @ A ) @ A4 @ C5 ) ) ) ).
% subset_psubset_trans
thf(fact_446_subset__not__subset__eq,axiom,
! [A: $tType] :
( ( ord_less @ ( set @ A ) )
= ( ^ [A7: set @ A,B6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A7 @ B6 )
& ~ ( ord_less_eq @ ( set @ A ) @ B6 @ A7 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_447_psubset__subset__trans,axiom,
! [A: $tType,A4: set @ A,B2: set @ A,C5: set @ A] :
( ( ord_less @ ( set @ A ) @ A4 @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ C5 )
=> ( ord_less @ ( set @ A ) @ A4 @ C5 ) ) ) ).
% psubset_subset_trans
thf(fact_448_psubset__imp__subset,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] :
( ( ord_less @ ( set @ A ) @ A4 @ B2 )
=> ( ord_less_eq @ ( set @ A ) @ A4 @ B2 ) ) ).
% psubset_imp_subset
thf(fact_449_Collect__mono__iff,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) )
= ( ! [X4: A] :
( ( P @ X4 )
=> ( Q @ X4 ) ) ) ) ).
% Collect_mono_iff
thf(fact_450_set__eq__subset,axiom,
! [A: $tType] :
( ( ^ [Y5: set @ A,Z2: set @ A] : Y5 = Z2 )
= ( ^ [A7: set @ A,B6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A7 @ B6 )
& ( ord_less_eq @ ( set @ A ) @ B6 @ A7 ) ) ) ) ).
% set_eq_subset
thf(fact_451_subset__trans,axiom,
! [A: $tType,A4: set @ A,B2: set @ A,C5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ C5 )
=> ( ord_less_eq @ ( set @ A ) @ A4 @ C5 ) ) ) ).
% subset_trans
thf(fact_452_Collect__mono,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X5: A] :
( ( P @ X5 )
=> ( Q @ X5 ) )
=> ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% Collect_mono
thf(fact_453_subset__refl,axiom,
! [A: $tType,A4: set @ A] : ( ord_less_eq @ ( set @ A ) @ A4 @ A4 ) ).
% subset_refl
thf(fact_454_subset__iff,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A7: set @ A,B6: set @ A] :
! [T3: A] :
( ( member @ A @ T3 @ A7 )
=> ( member @ A @ T3 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_455_psubset__eq,axiom,
! [A: $tType] :
( ( ord_less @ ( set @ A ) )
= ( ^ [A7: set @ A,B6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A7 @ B6 )
& ( A7 != B6 ) ) ) ) ).
% psubset_eq
thf(fact_456_equalityD2,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] :
( ( A4 = B2 )
=> ( ord_less_eq @ ( set @ A ) @ B2 @ A4 ) ) ).
% equalityD2
thf(fact_457_equalityD1,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] :
( ( A4 = B2 )
=> ( ord_less_eq @ ( set @ A ) @ A4 @ B2 ) ) ).
% equalityD1
thf(fact_458_subset__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A7: set @ A,B6: set @ A] :
! [X4: A] :
( ( member @ A @ X4 @ A7 )
=> ( member @ A @ X4 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_459_equalityE,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] :
( ( A4 = B2 )
=> ~ ( ( ord_less_eq @ ( set @ A ) @ A4 @ B2 )
=> ~ ( ord_less_eq @ ( set @ A ) @ B2 @ A4 ) ) ) ).
% equalityE
thf(fact_460_psubsetE,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] :
( ( ord_less @ ( set @ A ) @ A4 @ B2 )
=> ~ ( ( ord_less_eq @ ( set @ A ) @ A4 @ B2 )
=> ( ord_less_eq @ ( set @ A ) @ B2 @ A4 ) ) ) ).
% psubsetE
thf(fact_461_subsetD,axiom,
! [A: $tType,A4: set @ A,B2: set @ A,C2: A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B2 )
=> ( ( member @ A @ C2 @ A4 )
=> ( member @ A @ C2 @ B2 ) ) ) ).
% subsetD
thf(fact_462_in__mono,axiom,
! [A: $tType,A4: set @ A,B2: set @ A,X: A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B2 )
=> ( ( member @ A @ X @ A4 )
=> ( member @ A @ X @ B2 ) ) ) ).
% in_mono
thf(fact_463_ex__has__least__nat,axiom,
! [A: $tType,P: A > $o,K: A,M2: A > nat] :
( ( P @ K )
=> ? [X5: A] :
( ( P @ X5 )
& ! [Y4: A] :
( ( P @ Y4 )
=> ( ord_less_eq @ nat @ ( M2 @ X5 ) @ ( M2 @ Y4 ) ) ) ) ) ).
% ex_has_least_nat
thf(fact_464_Lattices__Big_Oex__has__greatest__nat,axiom,
! [A: $tType,P: A > $o,K: A,F2: A > nat,B3: nat] :
( ( P @ K )
=> ( ! [Y3: A] :
( ( P @ Y3 )
=> ( ord_less @ nat @ ( F2 @ Y3 ) @ B3 ) )
=> ? [X5: A] :
( ( P @ X5 )
& ! [Y4: A] :
( ( P @ Y4 )
=> ( ord_less_eq @ nat @ ( F2 @ Y4 ) @ ( F2 @ X5 ) ) ) ) ) ) ).
% Lattices_Big.ex_has_greatest_nat
thf(fact_465_finite__has__maximal,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A4: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ? [X5: A] :
( ( member @ A @ X5 @ A4 )
& ! [Xa: A] :
( ( member @ A @ Xa @ A4 )
=> ( ( ord_less_eq @ A @ X5 @ Xa )
=> ( X5 = Xa ) ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_466_finite__has__minimal,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A4: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ? [X5: A] :
( ( member @ A @ X5 @ A4 )
& ! [Xa: A] :
( ( member @ A @ Xa @ A4 )
=> ( ( ord_less_eq @ A @ Xa @ X5 )
=> ( X5 = Xa ) ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_467_finite__code,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ( ( finite_finite2 @ A )
= ( ^ [A7: set @ A] : $true ) ) ) ).
% finite_code
thf(fact_468_finite__nat__set__iff__bounded__le,axiom,
( ( finite_finite2 @ nat )
= ( ^ [N5: set @ nat] :
? [M: nat] :
! [X4: nat] :
( ( member @ nat @ X4 @ N5 )
=> ( ord_less_eq @ nat @ X4 @ M ) ) ) ) ).
% finite_nat_set_iff_bounded_le
thf(fact_469_infinite__nat__iff__unbounded__le,axiom,
! [S2: set @ nat] :
( ( ~ ( finite_finite2 @ nat @ S2 ) )
= ( ! [M: nat] :
? [N: nat] :
( ( ord_less_eq @ nat @ M @ N )
& ( member @ nat @ N @ S2 ) ) ) ) ).
% infinite_nat_iff_unbounded_le
thf(fact_470_finite__nat__set__iff__bounded,axiom,
( ( finite_finite2 @ nat )
= ( ^ [N5: set @ nat] :
? [M: nat] :
! [X4: nat] :
( ( member @ nat @ X4 @ N5 )
=> ( ord_less @ nat @ X4 @ M ) ) ) ) ).
% finite_nat_set_iff_bounded
thf(fact_471_infinite__nat__iff__unbounded,axiom,
! [S2: set @ nat] :
( ( ~ ( finite_finite2 @ nat @ S2 ) )
= ( ! [M: nat] :
? [N: nat] :
( ( ord_less @ nat @ M @ N )
& ( member @ nat @ N @ S2 ) ) ) ) ).
% infinite_nat_iff_unbounded
thf(fact_472_bounded__nat__set__is__finite,axiom,
! [N6: set @ nat,N2: nat] :
( ! [X5: nat] :
( ( member @ nat @ X5 @ N6 )
=> ( ord_less @ nat @ X5 @ N2 ) )
=> ( finite_finite2 @ nat @ N6 ) ) ).
% bounded_nat_set_is_finite
thf(fact_473_unbounded__k__infinite,axiom,
! [K: nat,S2: set @ nat] :
( ! [M3: nat] :
( ( ord_less @ nat @ K @ M3 )
=> ? [N7: nat] :
( ( ord_less @ nat @ M3 @ N7 )
& ( member @ nat @ N7 @ S2 ) ) )
=> ~ ( finite_finite2 @ nat @ S2 ) ) ).
% unbounded_k_infinite
thf(fact_474_finite__psubset__induct,axiom,
! [A: $tType,A4: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ A4 )
=> ( ! [A8: set @ A] :
( ( finite_finite2 @ A @ A8 )
=> ( ! [B7: set @ A] :
( ( ord_less @ ( set @ A ) @ B7 @ A8 )
=> ( P @ B7 ) )
=> ( P @ A8 ) ) )
=> ( P @ A4 ) ) ) ).
% finite_psubset_induct
thf(fact_475_arg__min__if__finite_I2_J,axiom,
! [B: $tType,A: $tType] :
( ( order @ B )
=> ! [S2: set @ A,F2: A > B] :
( ( finite_finite2 @ A @ S2 )
=> ( ( S2
!= ( bot_bot @ ( set @ A ) ) )
=> ~ ? [X2: A] :
( ( member @ A @ X2 @ S2 )
& ( ord_less @ B @ ( F2 @ X2 ) @ ( F2 @ ( lattic7623131987881927897min_on @ A @ B @ F2 @ S2 ) ) ) ) ) ) ) ).
% arg_min_if_finite(2)
thf(fact_476_bot__set__def,axiom,
! [A: $tType] :
( ( bot_bot @ ( set @ A ) )
= ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) ) ).
% bot_set_def
thf(fact_477_bot__nat__def,axiom,
( ( bot_bot @ nat )
= ( zero_zero @ nat ) ) ).
% bot_nat_def
thf(fact_478_finite__maxlen,axiom,
! [A: $tType,M7: set @ ( list @ A )] :
( ( finite_finite2 @ ( list @ A ) @ M7 )
=> ? [N3: nat] :
! [X2: list @ A] :
( ( member @ ( list @ A ) @ X2 @ M7 )
=> ( ord_less @ nat @ ( size_size @ ( list @ A ) @ X2 ) @ N3 ) ) ) ).
% finite_maxlen
thf(fact_479_finite,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ! [A4: set @ A] : ( finite_finite2 @ A @ A4 ) ) ).
% finite
thf(fact_480_finite__set__choice,axiom,
! [B: $tType,A: $tType,A4: set @ A,P: A > B > $o] :
( ( finite_finite2 @ A @ A4 )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ A4 )
=> ? [X_1: B] : ( P @ X5 @ X_1 ) )
=> ? [F4: A > B] :
! [X2: A] :
( ( member @ A @ X2 @ A4 )
=> ( P @ X2 @ ( F4 @ X2 ) ) ) ) ) ).
% finite_set_choice
thf(fact_481_bounded__Max__nat,axiom,
! [P: nat > $o,X: nat,M7: nat] :
( ( P @ X )
=> ( ! [X5: nat] :
( ( P @ X5 )
=> ( ord_less_eq @ nat @ X5 @ M7 ) )
=> ~ ! [M3: nat] :
( ( P @ M3 )
=> ~ ! [X2: nat] :
( ( P @ X2 )
=> ( ord_less_eq @ nat @ X2 @ M3 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_482_arg__min__if__finite_I1_J,axiom,
! [B: $tType,A: $tType] :
( ( order @ B )
=> ! [S2: set @ A,F2: A > B] :
( ( finite_finite2 @ A @ S2 )
=> ( ( S2
!= ( bot_bot @ ( set @ A ) ) )
=> ( member @ A @ ( lattic7623131987881927897min_on @ A @ B @ F2 @ S2 ) @ S2 ) ) ) ) ).
% arg_min_if_finite(1)
thf(fact_483_finite__has__maximal2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A4: set @ A,A3: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( member @ A @ A3 @ A4 )
=> ? [X5: A] :
( ( member @ A @ X5 @ A4 )
& ( ord_less_eq @ A @ A3 @ X5 )
& ! [Xa: A] :
( ( member @ A @ Xa @ A4 )
=> ( ( ord_less_eq @ A @ X5 @ Xa )
=> ( X5 = Xa ) ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_484_finite__has__minimal2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A4: set @ A,A3: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( member @ A @ A3 @ A4 )
=> ? [X5: A] :
( ( member @ A @ X5 @ A4 )
& ( ord_less_eq @ A @ X5 @ A3 )
& ! [Xa: A] :
( ( member @ A @ Xa @ A4 )
=> ( ( ord_less_eq @ A @ Xa @ X5 )
=> ( X5 = Xa ) ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_485_finite_OemptyI,axiom,
! [A: $tType] : ( finite_finite2 @ A @ ( bot_bot @ ( set @ A ) ) ) ).
% finite.emptyI
thf(fact_486_infinite__imp__nonempty,axiom,
! [A: $tType,S2: set @ A] :
( ~ ( finite_finite2 @ A @ S2 )
=> ( S2
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% infinite_imp_nonempty
thf(fact_487_finite__transitivity__chain,axiom,
! [A: $tType,A4: set @ A,R: A > A > $o] :
( ( finite_finite2 @ A @ A4 )
=> ( ! [X5: A] :
~ ( R @ X5 @ X5 )
=> ( ! [X5: A,Y3: A,Z: A] :
( ( R @ X5 @ Y3 )
=> ( ( R @ Y3 @ Z )
=> ( R @ X5 @ Z ) ) )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ A4 )
=> ? [Y4: A] :
( ( member @ A @ Y4 @ A4 )
& ( R @ X5 @ Y4 ) ) )
=> ( A4
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% finite_transitivity_chain
thf(fact_488_finite__subset,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B2 )
=> ( ( finite_finite2 @ A @ B2 )
=> ( finite_finite2 @ A @ A4 ) ) ) ).
% finite_subset
thf(fact_489_infinite__super,axiom,
! [A: $tType,S2: set @ A,T4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ S2 @ T4 )
=> ( ~ ( finite_finite2 @ A @ S2 )
=> ~ ( finite_finite2 @ A @ T4 ) ) ) ).
% infinite_super
thf(fact_490_rev__finite__subset,axiom,
! [A: $tType,B2: set @ A,A4: set @ A] :
( ( finite_finite2 @ A @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ A4 @ B2 )
=> ( finite_finite2 @ A @ A4 ) ) ) ).
% rev_finite_subset
thf(fact_491_arg__min__least,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ B )
=> ! [S2: set @ A,Y: A,F2: A > B] :
( ( finite_finite2 @ A @ S2 )
=> ( ( S2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( member @ A @ Y @ S2 )
=> ( ord_less_eq @ B @ ( F2 @ ( lattic7623131987881927897min_on @ A @ B @ F2 @ S2 ) ) @ ( F2 @ Y ) ) ) ) ) ) ).
% arg_min_least
thf(fact_492_Euclid__induct,axiom,
! [P: nat > nat > $o,A3: nat,B3: nat] :
( ! [A6: nat,B5: nat] :
( ( P @ A6 @ B5 )
= ( P @ B5 @ A6 ) )
=> ( ! [A6: nat] : ( P @ A6 @ ( zero_zero @ nat ) )
=> ( ! [A6: nat,B5: nat] :
( ( P @ A6 @ B5 )
=> ( P @ A6 @ ( plus_plus @ nat @ A6 @ B5 ) ) )
=> ( P @ A3 @ B3 ) ) ) ) ).
% Euclid_induct
thf(fact_493_nat__descend__induct,axiom,
! [N2: nat,P: nat > $o,M2: nat] :
( ! [K3: nat] :
( ( ord_less @ nat @ N2 @ K3 )
=> ( P @ K3 ) )
=> ( ! [K3: nat] :
( ( ord_less_eq @ nat @ K3 @ N2 )
=> ( ! [I5: nat] :
( ( ord_less @ nat @ K3 @ I5 )
=> ( P @ I5 ) )
=> ( P @ K3 ) ) )
=> ( P @ M2 ) ) ) ).
% nat_descend_induct
thf(fact_494_subset__emptyI,axiom,
! [A: $tType,A4: set @ A] :
( ! [X5: A] :
~ ( member @ A @ X5 @ A4 )
=> ( ord_less_eq @ ( set @ A ) @ A4 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% subset_emptyI
thf(fact_495_list__decode_Ocases,axiom,
! [X: nat] :
( ( X
!= ( zero_zero @ nat ) )
=> ~ ! [N3: nat] :
( X
!= ( suc @ N3 ) ) ) ).
% list_decode.cases
thf(fact_496_vebt__buildup_Ocases,axiom,
! [X: nat] :
( ( X
!= ( zero_zero @ nat ) )
=> ( ( X
!= ( suc @ ( zero_zero @ nat ) ) )
=> ~ ! [Va: nat] :
( X
!= ( suc @ ( suc @ Va ) ) ) ) ) ).
% vebt_buildup.cases
thf(fact_497_dependent__nat__choice,axiom,
! [A: $tType,P: nat > A > $o,Q: nat > A > A > $o] :
( ? [X_1: A] : ( P @ ( zero_zero @ nat ) @ X_1 )
=> ( ! [X5: A,N3: nat] :
( ( P @ N3 @ X5 )
=> ? [Y4: A] :
( ( P @ ( suc @ N3 ) @ Y4 )
& ( Q @ N3 @ X5 @ Y4 ) ) )
=> ? [F4: nat > A] :
! [N7: nat] :
( ( P @ N7 @ ( F4 @ N7 ) )
& ( Q @ N7 @ ( F4 @ N7 ) @ ( F4 @ ( suc @ N7 ) ) ) ) ) ) ).
% dependent_nat_choice
thf(fact_498_exists__least__lemma,axiom,
! [P: nat > $o] :
( ~ ( P @ ( zero_zero @ nat ) )
=> ( ? [X_1: nat] : ( P @ X_1 )
=> ? [N3: nat] :
( ~ ( P @ N3 )
& ( P @ ( suc @ N3 ) ) ) ) ) ).
% exists_least_lemma
thf(fact_499_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_500_verit__sum__simplify,axiom,
! [A: $tType] :
( ( cancel1802427076303600483id_add @ A )
=> ! [A3: A] :
( ( plus_plus @ A @ A3 @ ( zero_zero @ A ) )
= A3 ) ) ).
% verit_sum_simplify
thf(fact_501_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_502_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_503_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_504_verit__comp__simplify1_I2_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A] : ( ord_less_eq @ A @ A3 @ A3 ) ) ).
% verit_comp_simplify1(2)
thf(fact_505_verit__comp__simplify1_I1_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A] :
~ ( ord_less @ A @ A3 @ A3 ) ) ).
% verit_comp_simplify1(1)
thf(fact_506_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_507_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_508_verit__comp__simplify1_I3_J,axiom,
! [B: $tType] :
( ( linorder @ B )
=> ! [B8: B,A9: B] :
( ( ~ ( ord_less_eq @ B @ B8 @ A9 ) )
= ( ord_less @ B @ A9 @ B8 ) ) ) ).
% verit_comp_simplify1(3)
thf(fact_509_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_510_bot__empty__eq,axiom,
! [A: $tType] :
( ( bot_bot @ ( A > $o ) )
= ( ^ [X4: A] : ( member @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% bot_empty_eq
thf(fact_511_triangle__Suc,axiom,
! [N2: nat] :
( ( nat_triangle @ ( suc @ N2 ) )
= ( plus_plus @ nat @ ( nat_triangle @ N2 ) @ ( suc @ N2 ) ) ) ).
% triangle_Suc
thf(fact_512_complete__interval,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [A3: A,B3: A,P: A > $o] :
( ( ord_less @ A @ A3 @ B3 )
=> ( ( P @ A3 )
=> ( ~ ( P @ B3 )
=> ? [C3: A] :
( ( ord_less_eq @ A @ A3 @ C3 )
& ( ord_less_eq @ A @ C3 @ B3 )
& ! [X2: A] :
( ( ( ord_less_eq @ A @ A3 @ X2 )
& ( ord_less @ A @ X2 @ C3 ) )
=> ( P @ X2 ) )
& ! [D3: A] :
( ! [X5: A] :
( ( ( ord_less_eq @ A @ A3 @ X5 )
& ( ord_less @ A @ X5 @ D3 ) )
=> ( P @ X5 ) )
=> ( ord_less_eq @ A @ D3 @ C3 ) ) ) ) ) ) ) ).
% complete_interval
thf(fact_513_pinf_I6_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z: A] :
! [X2: A] :
( ( ord_less @ A @ Z @ X2 )
=> ~ ( ord_less_eq @ A @ X2 @ T2 ) ) ) ).
% pinf(6)
thf(fact_514_pinf_I8_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z: A] :
! [X2: A] :
( ( ord_less @ A @ Z @ X2 )
=> ( ord_less_eq @ A @ T2 @ X2 ) ) ) ).
% pinf(8)
thf(fact_515_minf_I6_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z: A] :
! [X2: A] :
( ( ord_less @ A @ X2 @ Z )
=> ( ord_less_eq @ A @ X2 @ T2 ) ) ) ).
% minf(6)
thf(fact_516_minf_I8_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z: A] :
! [X2: A] :
( ( ord_less @ A @ X2 @ Z )
=> ~ ( ord_less_eq @ A @ T2 @ X2 ) ) ) ).
% minf(8)
thf(fact_517_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_518__C10_C,axiom,
( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ mi @ ma ) ) @ deg @ treeList @ summary ) @ xa )
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ xa @ ( ord_max @ nat @ mi @ ma ) ) ) @ deg @ ( list_update @ vEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ mi @ na ) @ ( vEBT_vebt_insert @ ( nth @ vEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ mi @ na ) ) @ ( vEBT_VEBT_low @ mi @ na ) ) ) @ ( if @ vEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth @ vEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ mi @ na ) ) ) @ ( vEBT_vebt_insert @ summary @ ( vEBT_VEBT_high @ mi @ na ) ) @ summary ) ) ) ).
% "10"
thf(fact_519_deg__deg__n,axiom,
! [Info: option @ ( product_prod @ nat @ nat ),Deg: nat,TreeList2: list @ vEBT_VEBT,Summary: vEBT_VEBT,N2: nat] :
( ( vEBT_invar_vebt @ ( vEBT_Node @ Info @ Deg @ TreeList2 @ Summary ) @ N2 )
=> ( Deg = N2 ) ) ).
% deg_deg_n
thf(fact_520_deg__SUcn__Node,axiom,
! [Tree: vEBT_VEBT,N2: nat] :
( ( vEBT_invar_vebt @ Tree @ ( suc @ ( suc @ N2 ) ) )
=> ? [Info2: option @ ( product_prod @ nat @ nat ),TreeList3: list @ vEBT_VEBT,S3: vEBT_VEBT] :
( Tree
= ( vEBT_Node @ Info2 @ ( suc @ ( suc @ N2 ) ) @ TreeList3 @ S3 ) ) ) ).
% deg_SUcn_Node
thf(fact_521_mi__eq__ma__no__ch,axiom,
! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ Deg )
=> ( ( Mi = Ma )
=> ( ! [X2: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X2 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X_1 ) )
& ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_1 ) ) ) ) ).
% mi_eq_ma_no_ch
thf(fact_522_max__bot2,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [X: A] :
( ( ord_max @ A @ X @ ( bot_bot @ A ) )
= X ) ) ).
% max_bot2
thf(fact_523_max__bot,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [X: A] :
( ( ord_max @ A @ ( bot_bot @ A ) @ X )
= X ) ) ).
% max_bot
thf(fact_524_max__Suc__Suc,axiom,
! [M2: nat,N2: nat] :
( ( ord_max @ nat @ ( suc @ M2 ) @ ( suc @ N2 ) )
= ( suc @ ( ord_max @ nat @ M2 @ N2 ) ) ) ).
% max_Suc_Suc
thf(fact_525_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_526_max__nat_Oleft__neutral,axiom,
! [A3: nat] :
( ( ord_max @ nat @ ( zero_zero @ nat ) @ A3 )
= A3 ) ).
% max_nat.left_neutral
thf(fact_527_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_528_max__nat_Oright__neutral,axiom,
! [A3: nat] :
( ( ord_max @ nat @ A3 @ ( zero_zero @ nat ) )
= A3 ) ).
% max_nat.right_neutral
thf(fact_529_max__0L,axiom,
! [N2: nat] :
( ( ord_max @ nat @ ( zero_zero @ nat ) @ N2 )
= N2 ) ).
% max_0L
thf(fact_530_max__0R,axiom,
! [N2: nat] :
( ( ord_max @ nat @ N2 @ ( zero_zero @ nat ) )
= N2 ) ).
% max_0R
thf(fact_531_triangle__0,axiom,
( ( nat_triangle @ ( zero_zero @ nat ) )
= ( zero_zero @ nat ) ) ).
% triangle_0
thf(fact_532_VEBT__internal_Omembermima_Osimps_I3_J,axiom,
! [Mi: nat,Ma: nat,Va2: list @ vEBT_VEBT,Vb: vEBT_VEBT,X: nat] :
( ( vEBT_VEBT_membermima @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( zero_zero @ nat ) @ Va2 @ Vb ) @ X )
= ( ( X = Mi )
| ( X = Ma ) ) ) ).
% VEBT_internal.membermima.simps(3)
thf(fact_533_max__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( ord_max @ A )
= ( ^ [A5: A,B4: A] : ( if @ A @ ( ord_less_eq @ A @ A5 @ B4 ) @ B4 @ A5 ) ) ) ) ).
% max_def
thf(fact_534_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_535_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_536_max__add__distrib__left,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [X: A,Y: A,Z3: A] :
( ( plus_plus @ A @ ( ord_max @ A @ X @ Y ) @ Z3 )
= ( ord_max @ A @ ( plus_plus @ A @ X @ Z3 ) @ ( plus_plus @ A @ Y @ Z3 ) ) ) ) ).
% max_add_distrib_left
thf(fact_537_max__add__distrib__right,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [X: A,Y: A,Z3: A] :
( ( plus_plus @ A @ X @ ( ord_max @ A @ Y @ Z3 ) )
= ( ord_max @ A @ ( plus_plus @ A @ X @ Y ) @ ( plus_plus @ A @ X @ Z3 ) ) ) ) ).
% max_add_distrib_right
thf(fact_538_nat__add__max__left,axiom,
! [M2: nat,N2: nat,Q3: nat] :
( ( plus_plus @ nat @ ( ord_max @ nat @ M2 @ N2 ) @ Q3 )
= ( ord_max @ nat @ ( plus_plus @ nat @ M2 @ Q3 ) @ ( plus_plus @ nat @ N2 @ Q3 ) ) ) ).
% nat_add_max_left
thf(fact_539_nat__add__max__right,axiom,
! [M2: nat,N2: nat,Q3: nat] :
( ( plus_plus @ nat @ M2 @ ( ord_max @ nat @ N2 @ Q3 ) )
= ( ord_max @ nat @ ( plus_plus @ nat @ M2 @ N2 ) @ ( plus_plus @ nat @ M2 @ Q3 ) ) ) ).
% nat_add_max_right
thf(fact_540_VEBT__internal_Onaive__member_Osimps_I2_J,axiom,
! [Uu: option @ ( product_prod @ nat @ nat ),Uv: list @ vEBT_VEBT,Uw: vEBT_VEBT,Ux: nat] :
~ ( vEBT_V5719532721284313246member @ ( vEBT_Node @ Uu @ ( zero_zero @ nat ) @ Uv @ Uw ) @ Ux ) ).
% VEBT_internal.naive_member.simps(2)
thf(fact_541_minf_I11_J,axiom,
! [C: $tType,D: $tType] :
( ( ord @ C )
=> ! [F5: D] :
? [Z: C] :
! [X2: C] :
( ( ord_less @ C @ X2 @ Z )
=> ( F5 = F5 ) ) ) ).
% minf(11)
thf(fact_542_minf_I7_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z: A] :
! [X2: A] :
( ( ord_less @ A @ X2 @ Z )
=> ~ ( ord_less @ A @ T2 @ X2 ) ) ) ).
% minf(7)
thf(fact_543_minf_I5_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z: A] :
! [X2: A] :
( ( ord_less @ A @ X2 @ Z )
=> ( ord_less @ A @ X2 @ T2 ) ) ) ).
% minf(5)
thf(fact_544_minf_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z: A] :
! [X2: A] :
( ( ord_less @ A @ X2 @ Z )
=> ( X2 != T2 ) ) ) ).
% minf(4)
thf(fact_545_minf_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z: A] :
! [X2: A] :
( ( ord_less @ A @ X2 @ Z )
=> ( X2 != T2 ) ) ) ).
% minf(3)
thf(fact_546_minf_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > $o,P4: A > $o,Q: A > $o,Q4: A > $o] :
( ? [Z4: A] :
! [X5: A] :
( ( ord_less @ A @ X5 @ Z4 )
=> ( ( P @ X5 )
= ( P4 @ X5 ) ) )
=> ( ? [Z4: A] :
! [X5: A] :
( ( ord_less @ A @ X5 @ Z4 )
=> ( ( Q @ X5 )
= ( Q4 @ X5 ) ) )
=> ? [Z: A] :
! [X2: A] :
( ( ord_less @ A @ X2 @ Z )
=> ( ( ( P @ X2 )
| ( Q @ X2 ) )
= ( ( P4 @ X2 )
| ( Q4 @ X2 ) ) ) ) ) ) ) ).
% minf(2)
thf(fact_547_minf_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > $o,P4: A > $o,Q: A > $o,Q4: A > $o] :
( ? [Z4: A] :
! [X5: A] :
( ( ord_less @ A @ X5 @ Z4 )
=> ( ( P @ X5 )
= ( P4 @ X5 ) ) )
=> ( ? [Z4: A] :
! [X5: A] :
( ( ord_less @ A @ X5 @ Z4 )
=> ( ( Q @ X5 )
= ( Q4 @ X5 ) ) )
=> ? [Z: A] :
! [X2: A] :
( ( ord_less @ A @ X2 @ Z )
=> ( ( ( P @ X2 )
& ( Q @ X2 ) )
= ( ( P4 @ X2 )
& ( Q4 @ X2 ) ) ) ) ) ) ) ).
% minf(1)
thf(fact_548_pinf_I11_J,axiom,
! [C: $tType,D: $tType] :
( ( ord @ C )
=> ! [F5: D] :
? [Z: C] :
! [X2: C] :
( ( ord_less @ C @ Z @ X2 )
=> ( F5 = F5 ) ) ) ).
% pinf(11)
thf(fact_549_pinf_I7_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z: A] :
! [X2: A] :
( ( ord_less @ A @ Z @ X2 )
=> ( ord_less @ A @ T2 @ X2 ) ) ) ).
% pinf(7)
thf(fact_550_pinf_I5_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z: A] :
! [X2: A] :
( ( ord_less @ A @ Z @ X2 )
=> ~ ( ord_less @ A @ X2 @ T2 ) ) ) ).
% pinf(5)
thf(fact_551_pinf_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z: A] :
! [X2: A] :
( ( ord_less @ A @ Z @ X2 )
=> ( X2 != T2 ) ) ) ).
% pinf(4)
thf(fact_552_pinf_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z: A] :
! [X2: A] :
( ( ord_less @ A @ Z @ X2 )
=> ( X2 != T2 ) ) ) ).
% pinf(3)
thf(fact_553_pinf_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > $o,P4: A > $o,Q: A > $o,Q4: A > $o] :
( ? [Z4: A] :
! [X5: A] :
( ( ord_less @ A @ Z4 @ X5 )
=> ( ( P @ X5 )
= ( P4 @ X5 ) ) )
=> ( ? [Z4: A] :
! [X5: A] :
( ( ord_less @ A @ Z4 @ X5 )
=> ( ( Q @ X5 )
= ( Q4 @ X5 ) ) )
=> ? [Z: A] :
! [X2: A] :
( ( ord_less @ A @ Z @ X2 )
=> ( ( ( P @ X2 )
| ( Q @ X2 ) )
= ( ( P4 @ X2 )
| ( Q4 @ X2 ) ) ) ) ) ) ) ).
% pinf(2)
thf(fact_554_pinf_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > $o,P4: A > $o,Q: A > $o,Q4: A > $o] :
( ? [Z4: A] :
! [X5: A] :
( ( ord_less @ A @ Z4 @ X5 )
=> ( ( P @ X5 )
= ( P4 @ X5 ) ) )
=> ( ? [Z4: A] :
! [X5: A] :
( ( ord_less @ A @ Z4 @ X5 )
=> ( ( Q @ X5 )
= ( Q4 @ X5 ) ) )
=> ? [Z: A] :
! [X2: A] :
( ( ord_less @ A @ Z @ X2 )
=> ( ( ( P @ X2 )
& ( Q @ X2 ) )
= ( ( P4 @ X2 )
& ( Q4 @ X2 ) ) ) ) ) ) ) ).
% pinf(1)
thf(fact_555_ex__gt__or__lt,axiom,
! [A: $tType] :
( ( condit5016429287641298734tinuum @ A )
=> ! [A3: A] :
? [B5: A] :
( ( ord_less @ A @ A3 @ B5 )
| ( ord_less @ A @ B5 @ A3 ) ) ) ).
% ex_gt_or_lt
thf(fact_556_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_557_vebt__member_Osimps_I4_J,axiom,
! [V2: product_prod @ nat @ nat,Vb: list @ vEBT_VEBT,Vc: vEBT_VEBT,X: nat] :
~ ( vEBT_vebt_member @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V2 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb @ Vc ) @ X ) ).
% vebt_member.simps(4)
thf(fact_558_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_559_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_560_max__less__iff__conj,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A,Z3: A] :
( ( ord_less @ A @ ( ord_max @ A @ X @ Y ) @ Z3 )
= ( ( ord_less @ A @ X @ Z3 )
& ( ord_less @ A @ Y @ Z3 ) ) ) ) ).
% max_less_iff_conj
thf(fact_561_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_562_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_563_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_564_vebt__member_Osimps_I3_J,axiom,
! [V2: product_prod @ nat @ nat,Uy: list @ vEBT_VEBT,Uz: vEBT_VEBT,X: nat] :
~ ( vEBT_vebt_member @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V2 ) @ ( zero_zero @ nat ) @ Uy @ Uz ) @ X ) ).
% vebt_member.simps(3)
thf(fact_565_vebt__insert_Osimps_I3_J,axiom,
! [Info: option @ ( product_prod @ nat @ nat ),Ts: list @ vEBT_VEBT,S: vEBT_VEBT,X: nat] :
( ( vEBT_vebt_insert @ ( vEBT_Node @ Info @ ( suc @ ( zero_zero @ nat ) ) @ Ts @ S ) @ X )
= ( vEBT_Node @ Info @ ( suc @ ( zero_zero @ nat ) ) @ Ts @ S ) ) ).
% vebt_insert.simps(3)
thf(fact_566_VEBT__internal_OminNull_Osimps_I5_J,axiom,
! [Uz: product_prod @ nat @ nat,Va2: nat,Vb: list @ vEBT_VEBT,Vc: vEBT_VEBT] :
~ ( vEBT_VEBT_minNull @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ Uz ) @ Va2 @ Vb @ Vc ) ) ).
% VEBT_internal.minNull.simps(5)
thf(fact_567_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_568_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_569_max_Oabsorb__iff2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A5: A,B4: A] :
( ( ord_max @ A @ A5 @ B4 )
= B4 ) ) ) ) ).
% max.absorb_iff2
thf(fact_570_max_Oabsorb__iff1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B4: A,A5: A] :
( ( ord_max @ A @ A5 @ B4 )
= A5 ) ) ) ) ).
% max.absorb_iff1
thf(fact_571_le__max__iff__disj,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Z3: A,X: A,Y: A] :
( ( ord_less_eq @ A @ Z3 @ ( ord_max @ A @ X @ Y ) )
= ( ( ord_less_eq @ A @ Z3 @ X )
| ( ord_less_eq @ A @ Z3 @ Y ) ) ) ) ).
% le_max_iff_disj
thf(fact_572_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_573_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_574_max_Oorder__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B4: A,A5: A] :
( A5
= ( ord_max @ A @ A5 @ B4 ) ) ) ) ) ).
% max.order_iff
thf(fact_575_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_576_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_577_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_578_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_579_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_580_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_581_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_582_max_Ostrict__order__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less @ A )
= ( ^ [B4: A,A5: A] :
( ( A5
= ( ord_max @ A @ A5 @ B4 ) )
& ( A5 != B4 ) ) ) ) ) ).
% max.strict_order_iff
thf(fact_583_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_584_less__max__iff__disj,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Z3: A,X: A,Y: A] :
( ( ord_less @ A @ Z3 @ ( ord_max @ A @ X @ Y ) )
= ( ( ord_less @ A @ Z3 @ X )
| ( ord_less @ A @ Z3 @ Y ) ) ) ) ).
% less_max_iff_disj
thf(fact_585_vebt__insert_Osimps_I2_J,axiom,
! [Info: option @ ( product_prod @ nat @ nat ),Ts: list @ vEBT_VEBT,S: vEBT_VEBT,X: nat] :
( ( vEBT_vebt_insert @ ( vEBT_Node @ Info @ ( zero_zero @ nat ) @ Ts @ S ) @ X )
= ( vEBT_Node @ Info @ ( zero_zero @ nat ) @ Ts @ S ) ) ).
% vebt_insert.simps(2)
thf(fact_586_option_Osize_I4_J,axiom,
! [A: $tType,X22: A] :
( ( size_size @ ( option @ A ) @ ( some @ A @ X22 ) )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% option.size(4)
thf(fact_587_nth__enumerate__eq,axiom,
! [A: $tType,M2: nat,Xs: list @ A,N2: nat] :
( ( ord_less @ nat @ M2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( nth @ ( product_prod @ nat @ A ) @ ( enumerate @ A @ N2 @ Xs ) @ M2 )
= ( product_Pair @ nat @ A @ ( plus_plus @ nat @ N2 @ M2 ) @ ( nth @ A @ Xs @ M2 ) ) ) ) ).
% nth_enumerate_eq
thf(fact_588_find__Some__iff2,axiom,
! [A: $tType,X: A,P: A > $o,Xs: list @ A] :
( ( ( some @ A @ X )
= ( find @ A @ P @ Xs ) )
= ( ? [I3: nat] :
( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs ) )
& ( P @ ( nth @ A @ Xs @ I3 ) )
& ( X
= ( nth @ A @ Xs @ I3 ) )
& ! [J2: nat] :
( ( ord_less @ nat @ J2 @ I3 )
=> ~ ( P @ ( nth @ A @ Xs @ J2 ) ) ) ) ) ) ).
% find_Some_iff2
thf(fact_589_find__Some__iff,axiom,
! [A: $tType,P: A > $o,Xs: list @ A,X: A] :
( ( ( find @ A @ P @ Xs )
= ( some @ A @ X ) )
= ( ? [I3: nat] :
( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs ) )
& ( P @ ( nth @ A @ Xs @ I3 ) )
& ( X
= ( nth @ A @ Xs @ I3 ) )
& ! [J2: nat] :
( ( ord_less @ nat @ J2 @ I3 )
=> ~ ( P @ ( nth @ A @ Xs @ J2 ) ) ) ) ) ) ).
% find_Some_iff
thf(fact_590__C7_C,axiom,
( ( mi != ma )
=> ! [I5: nat] :
( ( ord_less @ nat @ I5 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ m ) )
=> ( ( ( ( vEBT_VEBT_high @ ma @ na )
= I5 )
=> ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ treeList @ I5 ) @ ( vEBT_VEBT_low @ ma @ na ) ) )
& ! [Y4: nat] :
( ( ( ( vEBT_VEBT_high @ Y4 @ na )
= I5 )
& ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ treeList @ I5 ) @ ( vEBT_VEBT_low @ Y4 @ na ) ) )
=> ( ( ord_less @ nat @ mi @ Y4 )
& ( ord_less_eq @ nat @ Y4 @ ma ) ) ) ) ) ) ).
% "7"
thf(fact_591_listrel1__iff__update,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( listrel1 @ A @ R2 ) )
= ( ? [Y6: A,N: nat] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( nth @ A @ Xs @ N ) @ Y6 ) @ R2 )
& ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
& ( Ys
= ( list_update @ A @ Xs @ N @ Y6 ) ) ) ) ) ).
% listrel1_iff_update
thf(fact_592_option_Osize__gen_I2_J,axiom,
! [A: $tType,X: A > nat,X22: A] :
( ( size_option @ A @ X @ ( some @ A @ X22 ) )
= ( plus_plus @ nat @ ( X @ X22 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% option.size_gen(2)
thf(fact_593_intind,axiom,
! [A: $tType,I: nat,N2: nat,P: A > $o,X: A] :
( ( ord_less @ nat @ I @ N2 )
=> ( ( P @ X )
=> ( P @ ( nth @ A @ ( replicate @ A @ N2 @ X ) @ I ) ) ) ) ).
% intind
thf(fact_594_vebt__insert_Osimps_I4_J,axiom,
! [V2: nat,TreeList2: list @ vEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
( ( vEBT_vebt_insert @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ V2 ) ) @ TreeList2 @ Summary ) @ X )
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ X @ X ) ) @ ( suc @ ( suc @ V2 ) ) @ TreeList2 @ Summary ) ) ).
% vebt_insert.simps(4)
thf(fact_595_gen__length__def,axiom,
! [A: $tType] :
( ( gen_length @ A )
= ( ^ [N: nat,Xs3: list @ A] : ( plus_plus @ nat @ N @ ( size_size @ ( list @ A ) @ Xs3 ) ) ) ) ).
% gen_length_def
thf(fact_596__C2_C,axiom,
( ( size_size @ ( list @ vEBT_VEBT ) @ treeList )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ m ) ) ).
% "2"
thf(fact_597__C5_Ohyps_C_I8_J,axiom,
ord_less @ nat @ ma @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ deg ) ).
% "5.hyps"(8)
thf(fact_598__C5_Oprems_C,axiom,
ord_less @ nat @ xa @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ deg ) ).
% "5.prems"
thf(fact_599__092_060open_062i_A_060_A2_A_094_Am_092_060close_062,axiom,
ord_less @ nat @ i @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ m ) ).
% \<open>i < 2 ^ m\<close>
thf(fact_600__C5_OIH_C_I1_J,axiom,
! [X2: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X2 @ ( set2 @ vEBT_VEBT @ treeList ) )
=> ( ( vEBT_invar_vebt @ X2 @ na )
& ! [Xa: nat] :
( ( ord_less @ nat @ Xa @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ na ) )
=> ( vEBT_invar_vebt @ ( vEBT_vebt_insert @ X2 @ Xa ) @ na ) ) ) ) ).
% "5.IH"(1)
thf(fact_601__C4_C,axiom,
! [I5: nat] :
( ( ord_less @ nat @ I5 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ m ) )
=> ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ treeList @ I5 ) @ X6 ) )
= ( vEBT_V8194947554948674370ptions @ summary @ I5 ) ) ) ).
% "4"
thf(fact_602_high__bound__aux,axiom,
! [Ma: nat,N2: nat,M2: nat] :
( ( ord_less @ nat @ Ma @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( plus_plus @ nat @ N2 @ M2 ) ) )
=> ( ord_less @ nat @ ( vEBT_VEBT_high @ Ma @ N2 ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M2 ) ) ) ).
% high_bound_aux
thf(fact_603_member__bound,axiom,
! [Tree: vEBT_VEBT,X: nat,N2: nat] :
( ( vEBT_vebt_member @ Tree @ X )
=> ( ( vEBT_invar_vebt @ Tree @ N2 )
=> ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ).
% member_bound
thf(fact_604_numeral__le__iff,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [M2: num,N2: num] :
( ( ord_less_eq @ A @ ( numeral_numeral @ A @ M2 ) @ ( numeral_numeral @ A @ N2 ) )
= ( ord_less_eq @ num @ M2 @ N2 ) ) ) ).
% numeral_le_iff
thf(fact_605_numeral__less__iff,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [M2: num,N2: num] :
( ( ord_less @ A @ ( numeral_numeral @ A @ M2 ) @ ( numeral_numeral @ A @ N2 ) )
= ( ord_less @ num @ M2 @ N2 ) ) ) ).
% numeral_less_iff
thf(fact_606_numeral__plus__numeral,axiom,
! [A: $tType] :
( ( numeral @ A )
=> ! [M2: num,N2: num] :
( ( plus_plus @ A @ ( numeral_numeral @ A @ M2 ) @ ( numeral_numeral @ A @ N2 ) )
= ( numeral_numeral @ A @ ( plus_plus @ num @ M2 @ N2 ) ) ) ) ).
% numeral_plus_numeral
thf(fact_607_add__numeral__left,axiom,
! [A: $tType] :
( ( numeral @ A )
=> ! [V2: num,W2: num,Z3: A] :
( ( plus_plus @ A @ ( numeral_numeral @ A @ V2 ) @ ( plus_plus @ A @ ( numeral_numeral @ A @ W2 ) @ Z3 ) )
= ( plus_plus @ A @ ( numeral_numeral @ A @ ( plus_plus @ num @ V2 @ W2 ) ) @ Z3 ) ) ) ).
% add_numeral_left
thf(fact_608_insert__simp__mima,axiom,
! [X: nat,Mi: nat,Ma: nat,Deg: nat,TreeList2: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( ( X = Mi )
| ( X = Ma ) )
=> ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) ) ) ) ).
% insert_simp_mima
thf(fact_609_valid__insert__both__member__options__add,axiom,
! [T2: vEBT_VEBT,N2: nat,X: nat] :
( ( vEBT_invar_vebt @ T2 @ N2 )
=> ( ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) )
=> ( vEBT_V8194947554948674370ptions @ ( vEBT_vebt_insert @ T2 @ X ) @ X ) ) ) ).
% valid_insert_both_member_options_add
thf(fact_610_valid__insert__both__member__options__pres,axiom,
! [T2: vEBT_VEBT,N2: nat,X: nat,Y: nat] :
( ( vEBT_invar_vebt @ T2 @ N2 )
=> ( ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) )
=> ( ( ord_less @ nat @ Y @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) )
=> ( ( vEBT_V8194947554948674370ptions @ T2 @ X )
=> ( vEBT_V8194947554948674370ptions @ ( vEBT_vebt_insert @ T2 @ Y ) @ X ) ) ) ) ) ).
% valid_insert_both_member_options_pres
thf(fact_611_post__member__pre__member,axiom,
! [T2: vEBT_VEBT,N2: nat,X: nat,Y: nat] :
( ( vEBT_invar_vebt @ T2 @ N2 )
=> ( ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) )
=> ( ( ord_less @ nat @ Y @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) )
=> ( ( vEBT_vebt_member @ ( vEBT_vebt_insert @ T2 @ X ) @ Y )
=> ( ( vEBT_vebt_member @ T2 @ Y )
| ( X = Y ) ) ) ) ) ) ).
% post_member_pre_member
thf(fact_612_replicate__eq__replicate,axiom,
! [A: $tType,M2: nat,X: A,N2: nat,Y: A] :
( ( ( replicate @ A @ M2 @ X )
= ( replicate @ A @ N2 @ Y ) )
= ( ( M2 = N2 )
& ( ( M2
!= ( zero_zero @ nat ) )
=> ( X = Y ) ) ) ) ).
% replicate_eq_replicate
thf(fact_613_length__replicate,axiom,
! [A: $tType,N2: nat,X: A] :
( ( size_size @ ( list @ A ) @ ( replicate @ A @ N2 @ X ) )
= N2 ) ).
% length_replicate
thf(fact_614_length__enumerate,axiom,
! [A: $tType,N2: nat,Xs: list @ A] :
( ( size_size @ ( list @ ( product_prod @ nat @ A ) ) @ ( enumerate @ A @ N2 @ Xs ) )
= ( size_size @ ( list @ A ) @ Xs ) ) ).
% length_enumerate
thf(fact_615_mi__ma__2__deg,axiom,
! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list @ vEBT_VEBT,Summary: vEBT_VEBT,N2: nat] :
( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ N2 )
=> ( ( ord_less_eq @ nat @ Mi @ Ma )
& ( ord_less @ nat @ Ma @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg ) ) ) ) ).
% mi_ma_2_deg
thf(fact_616__C5_OIH_C_I2_J,axiom,
! [X: nat] :
( ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ m ) )
=> ( vEBT_invar_vebt @ ( vEBT_vebt_insert @ summary @ X ) @ m ) ) ).
% "5.IH"(2)
thf(fact_617_Suc__numeral,axiom,
! [N2: num] :
( ( suc @ ( numeral_numeral @ nat @ N2 ) )
= ( numeral_numeral @ nat @ ( plus_plus @ num @ N2 @ one2 ) ) ) ).
% Suc_numeral
thf(fact_618_max__number__of_I1_J,axiom,
! [A: $tType] :
( ( ( numeral @ A )
& ( ord @ A ) )
=> ! [U: num,V2: num] :
( ( ( ord_less_eq @ A @ ( numeral_numeral @ A @ U ) @ ( numeral_numeral @ A @ V2 ) )
=> ( ( ord_max @ A @ ( numeral_numeral @ A @ U ) @ ( numeral_numeral @ A @ V2 ) )
= ( numeral_numeral @ A @ V2 ) ) )
& ( ~ ( ord_less_eq @ A @ ( numeral_numeral @ A @ U ) @ ( numeral_numeral @ A @ V2 ) )
=> ( ( ord_max @ A @ ( numeral_numeral @ A @ U ) @ ( numeral_numeral @ A @ V2 ) )
= ( numeral_numeral @ A @ U ) ) ) ) ) ).
% max_number_of(1)
thf(fact_619_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_620_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_621__C6_C,axiom,
( ( ord_less_eq @ nat @ mi @ ma )
& ( ord_less @ nat @ ma @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ deg ) ) ) ).
% "6"
thf(fact_622_myIHs,axiom,
! [X: vEBT_VEBT,Xa2: nat] :
( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ treeList ) )
=> ( ( vEBT_invar_vebt @ X @ na )
=> ( ( ord_less @ nat @ Xa2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ na ) )
=> ( vEBT_invar_vebt @ ( vEBT_vebt_insert @ X @ Xa2 ) @ na ) ) ) ) ).
% myIHs
thf(fact_623_in__set__replicate,axiom,
! [A: $tType,X: A,N2: nat,Y: A] :
( ( member @ A @ X @ ( set2 @ A @ ( replicate @ A @ N2 @ Y ) ) )
= ( ( X = Y )
& ( N2
!= ( zero_zero @ nat ) ) ) ) ).
% in_set_replicate
thf(fact_624_Bex__set__replicate,axiom,
! [A: $tType,N2: nat,A3: A,P: A > $o] :
( ( ? [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ ( replicate @ A @ N2 @ A3 ) ) )
& ( P @ X4 ) ) )
= ( ( P @ A3 )
& ( N2
!= ( zero_zero @ nat ) ) ) ) ).
% Bex_set_replicate
thf(fact_625_Ball__set__replicate,axiom,
! [A: $tType,N2: nat,A3: A,P: A > $o] :
( ( ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ ( replicate @ A @ N2 @ A3 ) ) )
=> ( P @ X4 ) ) )
= ( ( P @ A3 )
| ( N2
= ( zero_zero @ nat ) ) ) ) ).
% Ball_set_replicate
thf(fact_626_nth__replicate,axiom,
! [A: $tType,I: nat,N2: nat,X: A] :
( ( ord_less @ nat @ I @ N2 )
=> ( ( nth @ A @ ( replicate @ A @ N2 @ X ) @ I )
= X ) ) ).
% nth_replicate
thf(fact_627_highlowprop,axiom,
( ( ord_less @ nat @ ( vEBT_VEBT_high @ mi @ na ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ m ) )
& ( ord_less @ nat @ ( vEBT_VEBT_low @ mi @ na ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ na ) ) ) ).
% highlowprop
thf(fact_628_add__2__eq__Suc,axiom,
! [N2: nat] :
( ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
= ( suc @ ( suc @ N2 ) ) ) ).
% add_2_eq_Suc
thf(fact_629_add__2__eq__Suc_H,axiom,
! [N2: nat] :
( ( plus_plus @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( suc @ ( suc @ N2 ) ) ) ).
% add_2_eq_Suc'
thf(fact_630_listrel1__mono,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),S: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ S )
=> ( ord_less_eq @ ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) @ ( listrel1 @ A @ R2 ) @ ( listrel1 @ A @ S ) ) ) ).
% listrel1_mono
thf(fact_631_numeral__Bit0,axiom,
! [A: $tType] :
( ( numeral @ A )
=> ! [N2: num] :
( ( numeral_numeral @ A @ ( bit0 @ N2 ) )
= ( plus_plus @ A @ ( numeral_numeral @ A @ N2 ) @ ( numeral_numeral @ A @ N2 ) ) ) ) ).
% numeral_Bit0
thf(fact_632_listrel1__eq__len,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( listrel1 @ A @ R2 ) )
=> ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ A ) @ Ys ) ) ) ).
% listrel1_eq_len
thf(fact_633_numeral__2__eq__2,axiom,
( ( numeral_numeral @ nat @ ( bit0 @ one2 ) )
= ( suc @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% numeral_2_eq_2
thf(fact_634_find__None__iff,axiom,
! [A: $tType,P: A > $o,Xs: list @ A] :
( ( ( find @ A @ P @ Xs )
= ( none @ A ) )
= ( ~ ? [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
& ( P @ X4 ) ) ) ) ).
% find_None_iff
thf(fact_635_find__None__iff2,axiom,
! [A: $tType,P: A > $o,Xs: list @ A] :
( ( ( none @ A )
= ( find @ A @ P @ Xs ) )
= ( ~ ? [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
& ( P @ X4 ) ) ) ) ).
% find_None_iff2
thf(fact_636_VEBT__internal_Oexp__split__high__low_I1_J,axiom,
! [X: nat,N2: nat,M2: nat] :
( ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( plus_plus @ nat @ N2 @ M2 ) ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M2 )
=> ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ N2 ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M2 ) ) ) ) ) ).
% VEBT_internal.exp_split_high_low(1)
thf(fact_637_option_Osize__gen_I1_J,axiom,
! [A: $tType,X: A > nat] :
( ( size_option @ A @ X @ ( none @ A ) )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% option.size_gen(1)
thf(fact_638_VEBT__internal_Oexp__split__high__low_I2_J,axiom,
! [X: nat,N2: nat,M2: nat] :
( ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( plus_plus @ nat @ N2 @ M2 ) ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M2 )
=> ( ord_less @ nat @ ( vEBT_VEBT_low @ X @ N2 ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ) ).
% VEBT_internal.exp_split_high_low(2)
thf(fact_639_less__2__cases__iff,axiom,
! [N2: nat] :
( ( ord_less @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( ( N2
= ( zero_zero @ nat ) )
| ( N2
= ( suc @ ( zero_zero @ nat ) ) ) ) ) ).
% less_2_cases_iff
thf(fact_640_less__2__cases,axiom,
! [N2: nat] :
( ( ord_less @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
=> ( ( N2
= ( zero_zero @ nat ) )
| ( N2
= ( suc @ ( zero_zero @ nat ) ) ) ) ) ).
% less_2_cases
thf(fact_641_invar__vebt_Ointros_I2_J,axiom,
! [TreeList2: list @ vEBT_VEBT,N2: nat,Summary: vEBT_VEBT,M2: nat,Deg: nat] :
( ! [X5: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X5 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ( vEBT_invar_vebt @ X5 @ N2 ) )
=> ( ( vEBT_invar_vebt @ Summary @ M2 )
=> ( ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M2 ) )
=> ( ( M2 = N2 )
=> ( ( Deg
= ( plus_plus @ nat @ N2 @ M2 ) )
=> ( ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_12 )
=> ( ! [X5: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X5 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_12 ) )
=> ( vEBT_invar_vebt @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg @ TreeList2 @ Summary ) @ Deg ) ) ) ) ) ) ) ) ).
% invar_vebt.intros(2)
thf(fact_642_numeral__1__eq__Suc__0,axiom,
( ( numeral_numeral @ nat @ one2 )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% numeral_1_eq_Suc_0
thf(fact_643_Suc__nat__number__of__add,axiom,
! [V2: num,N2: nat] :
( ( suc @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ V2 ) @ N2 ) )
= ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( plus_plus @ num @ V2 @ one2 ) ) @ N2 ) ) ).
% Suc_nat_number_of_add
thf(fact_644_zero__neq__numeral,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [N2: num] :
( ( zero_zero @ A )
!= ( numeral_numeral @ A @ N2 ) ) ) ).
% zero_neq_numeral
thf(fact_645_fold__atLeastAtMost__nat_Ocases,axiom,
! [A: $tType,X: product_prod @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) )] :
~ ! [F4: nat > A > A,A6: nat,B5: nat,Acc: A] :
( X
!= ( product_Pair @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) @ F4 @ ( product_Pair @ nat @ ( product_prod @ nat @ A ) @ A6 @ ( product_Pair @ nat @ A @ B5 @ Acc ) ) ) ) ).
% fold_atLeastAtMost_nat.cases
thf(fact_646_invar__vebt_Ointros_I3_J,axiom,
! [TreeList2: list @ vEBT_VEBT,N2: nat,Summary: vEBT_VEBT,M2: nat,Deg: nat] :
( ! [X5: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X5 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ( vEBT_invar_vebt @ X5 @ N2 ) )
=> ( ( vEBT_invar_vebt @ Summary @ M2 )
=> ( ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M2 ) )
=> ( ( M2
= ( suc @ N2 ) )
=> ( ( Deg
= ( plus_plus @ nat @ N2 @ M2 ) )
=> ( ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_12 )
=> ( ! [X5: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X5 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_12 ) )
=> ( vEBT_invar_vebt @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg @ TreeList2 @ Summary ) @ Deg ) ) ) ) ) ) ) ) ).
% invar_vebt.intros(3)
thf(fact_647_num_Osize_I4_J,axiom,
( ( size_size @ num @ one2 )
= ( zero_zero @ nat ) ) ).
% num.size(4)
thf(fact_648_option_Osize_I3_J,axiom,
! [A: $tType] :
( ( size_size @ ( option @ A ) @ ( none @ A ) )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% option.size(3)
thf(fact_649_find__cong,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,P: A > $o,Q: A > $o] :
( ( Xs = Ys )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Ys ) )
=> ( ( P @ X5 )
= ( Q @ X5 ) ) )
=> ( ( find @ A @ P @ Xs )
= ( find @ A @ Q @ Ys ) ) ) ) ).
% find_cong
thf(fact_650_not__numeral__le__zero,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [N2: num] :
~ ( ord_less_eq @ A @ ( numeral_numeral @ A @ N2 ) @ ( zero_zero @ A ) ) ) ).
% not_numeral_le_zero
thf(fact_651_zero__le__numeral,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [N2: num] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ N2 ) ) ) ).
% zero_le_numeral
thf(fact_652_not__numeral__less__zero,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [N2: num] :
~ ( ord_less @ A @ ( numeral_numeral @ A @ N2 ) @ ( zero_zero @ A ) ) ) ).
% not_numeral_less_zero
thf(fact_653_zero__less__numeral,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [N2: num] : ( ord_less @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ N2 ) ) ) ).
% zero_less_numeral
thf(fact_654_replicate__eqI,axiom,
! [A: $tType,Xs: list @ A,N2: nat,X: A] :
( ( ( size_size @ ( list @ A ) @ Xs )
= N2 )
=> ( ! [Y3: A] :
( ( member @ A @ Y3 @ ( set2 @ A @ Xs ) )
=> ( Y3 = X ) )
=> ( Xs
= ( replicate @ A @ N2 @ X ) ) ) ) ).
% replicate_eqI
thf(fact_655_replicate__length__same,axiom,
! [A: $tType,Xs: list @ A,X: A] :
( ! [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Xs ) )
=> ( X5 = X ) )
=> ( ( replicate @ A @ ( size_size @ ( list @ A ) @ Xs ) @ X )
= Xs ) ) ).
% replicate_length_same
thf(fact_656_vebt__member_Osimps_I2_J,axiom,
! [Uu: nat,Uv: list @ vEBT_VEBT,Uw: vEBT_VEBT,X: nat] :
~ ( vEBT_vebt_member @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu @ Uv @ Uw ) @ X ) ).
% vebt_member.simps(2)
thf(fact_657_VEBT__internal_OminNull_Osimps_I4_J,axiom,
! [Uw: nat,Ux: list @ vEBT_VEBT,Uy: vEBT_VEBT] : ( vEBT_VEBT_minNull @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uw @ Ux @ Uy ) ) ).
% VEBT_internal.minNull.simps(4)
thf(fact_658_invar__vebt_Ointros_I4_J,axiom,
! [TreeList2: list @ vEBT_VEBT,N2: nat,Summary: vEBT_VEBT,M2: nat,Deg: nat,Mi: nat,Ma: nat] :
( ! [X5: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X5 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ( vEBT_invar_vebt @ X5 @ N2 ) )
=> ( ( vEBT_invar_vebt @ Summary @ M2 )
=> ( ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M2 ) )
=> ( ( M2 = N2 )
=> ( ( Deg
= ( plus_plus @ nat @ N2 @ M2 ) )
=> ( ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M2 ) )
=> ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I4 ) @ X6 ) )
= ( vEBT_V8194947554948674370ptions @ Summary @ I4 ) ) )
=> ( ( ( Mi = Ma )
=> ! [X5: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X5 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_12 ) ) )
=> ( ( ord_less_eq @ nat @ Mi @ Ma )
=> ( ( ord_less @ nat @ Ma @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg ) )
=> ( ( ( Mi != Ma )
=> ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M2 ) )
=> ( ( ( ( vEBT_VEBT_high @ Ma @ N2 )
= I4 )
=> ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I4 ) @ ( vEBT_VEBT_low @ Ma @ N2 ) ) )
& ! [X5: nat] :
( ( ( ( vEBT_VEBT_high @ X5 @ N2 )
= I4 )
& ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I4 ) @ ( vEBT_VEBT_low @ X5 @ N2 ) ) )
=> ( ( ord_less @ nat @ Mi @ X5 )
& ( ord_less_eq @ nat @ X5 @ Ma ) ) ) ) ) )
=> ( vEBT_invar_vebt @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ Deg ) ) ) ) ) ) ) ) ) ) ) ).
% invar_vebt.intros(4)
thf(fact_659_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_660_option_Osize__neq,axiom,
! [A: $tType,X: option @ A] :
( ( size_size @ ( option @ A ) @ X )
!= ( zero_zero @ nat ) ) ).
% option.size_neq
thf(fact_661_VEBT__internal_Omembermima_Osimps_I2_J,axiom,
! [Ux: list @ vEBT_VEBT,Uy: vEBT_VEBT,Uz: nat] :
~ ( vEBT_VEBT_membermima @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( zero_zero @ nat ) @ Ux @ Uy ) @ Uz ) ).
% VEBT_internal.membermima.simps(2)
thf(fact_662_invar__vebt_Ointros_I5_J,axiom,
! [TreeList2: list @ vEBT_VEBT,N2: nat,Summary: vEBT_VEBT,M2: nat,Deg: nat,Mi: nat,Ma: nat] :
( ! [X5: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X5 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ( vEBT_invar_vebt @ X5 @ N2 ) )
=> ( ( vEBT_invar_vebt @ Summary @ M2 )
=> ( ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M2 ) )
=> ( ( M2
= ( suc @ N2 ) )
=> ( ( Deg
= ( plus_plus @ nat @ N2 @ M2 ) )
=> ( ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M2 ) )
=> ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I4 ) @ X6 ) )
= ( vEBT_V8194947554948674370ptions @ Summary @ I4 ) ) )
=> ( ( ( Mi = Ma )
=> ! [X5: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X5 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_12 ) ) )
=> ( ( ord_less_eq @ nat @ Mi @ Ma )
=> ( ( ord_less @ nat @ Ma @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg ) )
=> ( ( ( Mi != Ma )
=> ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M2 ) )
=> ( ( ( ( vEBT_VEBT_high @ Ma @ N2 )
= I4 )
=> ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I4 ) @ ( vEBT_VEBT_low @ Ma @ N2 ) ) )
& ! [X5: nat] :
( ( ( ( vEBT_VEBT_high @ X5 @ N2 )
= I4 )
& ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I4 ) @ ( vEBT_VEBT_low @ X5 @ N2 ) ) )
=> ( ( ord_less @ nat @ Mi @ X5 )
& ( ord_less_eq @ nat @ X5 @ Ma ) ) ) ) ) )
=> ( vEBT_invar_vebt @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ Deg ) ) ) ) ) ) ) ) ) ) ) ).
% invar_vebt.intros(5)
thf(fact_663_length__code,axiom,
! [A: $tType] :
( ( size_size @ ( list @ A ) )
= ( gen_length @ A @ ( zero_zero @ nat ) ) ) ).
% length_code
thf(fact_664_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_665_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_666_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_667_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_668_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_669_power__mono__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,B3: A,N2: nat] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B3 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( ord_less_eq @ A @ ( power_power @ A @ A3 @ N2 ) @ ( power_power @ A @ B3 @ N2 ) )
= ( ord_less_eq @ A @ A3 @ B3 ) ) ) ) ) ) ).
% power_mono_iff
thf(fact_670_insert__simp__norm,axiom,
! [X: nat,Deg: nat,TreeList2: list @ vEBT_VEBT,Mi: nat,Ma: nat,Summary: vEBT_VEBT] :
( ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( ( ord_less @ nat @ Mi @ X )
=> ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ( X != Ma )
=> ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ ( ord_max @ nat @ X @ Ma ) ) ) @ Deg @ ( list_update @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_insert @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( if @ vEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ Summary ) ) ) ) ) ) ) ).
% insert_simp_norm
thf(fact_671_insert__simp__excp,axiom,
! [Mi: nat,Deg: nat,TreeList2: list @ vEBT_VEBT,X: nat,Ma: nat,Summary: vEBT_VEBT] :
( ( ord_less @ nat @ ( vEBT_VEBT_high @ Mi @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( ( ord_less @ nat @ X @ Mi )
=> ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ( X != Ma )
=> ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ X @ ( ord_max @ nat @ Mi @ Ma ) ) ) @ Deg @ ( list_update @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Mi @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_insert @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Mi @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Mi @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( if @ vEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Mi @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary @ ( vEBT_VEBT_high @ Mi @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ Summary ) ) ) ) ) ) ) ).
% insert_simp_excp
thf(fact_672_power__eq__0__iff,axiom,
! [A: $tType] :
( ( semiri2026040879449505780visors @ A )
=> ! [A3: A,N2: nat] :
( ( ( power_power @ A @ A3 @ N2 )
= ( zero_zero @ A ) )
= ( ( A3
= ( zero_zero @ A ) )
& ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ) ).
% power_eq_0_iff
thf(fact_673_member__inv,axiom,
! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list @ vEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
( ( vEBT_vebt_member @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
=> ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
& ( ( X = Mi )
| ( X = Ma )
| ( ( ord_less @ nat @ X @ Ma )
& ( ord_less @ nat @ Mi @ X )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
& ( vEBT_vebt_member @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ).
% member_inv
thf(fact_674_pow__sum,axiom,
! [A3: nat,B3: nat] :
( ( divide_divide @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( plus_plus @ nat @ A3 @ B3 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ A3 ) )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ B3 ) ) ).
% pow_sum
thf(fact_675_high__def,axiom,
( vEBT_VEBT_high
= ( ^ [X4: nat,N: nat] : ( divide_divide @ nat @ X4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% high_def
thf(fact_676__C9_C,axiom,
( ( divide_divide @ nat @ deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= na ) ).
% "9"
thf(fact_677_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_678_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_679_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_680_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_681_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_682_div__0,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [A3: A] :
( ( divide_divide @ A @ ( zero_zero @ A ) @ A3 )
= ( zero_zero @ A ) ) ) ).
% div_0
thf(fact_683_power__0__Suc,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [N2: nat] :
( ( power_power @ A @ ( zero_zero @ A ) @ ( suc @ N2 ) )
= ( zero_zero @ A ) ) ) ).
% power_0_Suc
thf(fact_684_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_685_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_686_nat__power__eq__Suc__0__iff,axiom,
! [X: nat,M2: nat] :
( ( ( power_power @ nat @ X @ M2 )
= ( suc @ ( zero_zero @ nat ) ) )
= ( ( M2
= ( zero_zero @ nat ) )
| ( X
= ( suc @ ( zero_zero @ nat ) ) ) ) ) ).
% nat_power_eq_Suc_0_iff
thf(fact_687_power__Suc__0,axiom,
! [N2: nat] :
( ( power_power @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N2 )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% power_Suc_0
thf(fact_688_nat__zero__less__power__iff,axiom,
! [X: nat,N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( power_power @ nat @ X @ N2 ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ X )
| ( N2
= ( zero_zero @ nat ) ) ) ) ).
% nat_zero_less_power_iff
thf(fact_689_both__member__options__ding,axiom,
! [Info: option @ ( product_prod @ nat @ nat ),Deg: nat,TreeList2: list @ vEBT_VEBT,Summary: vEBT_VEBT,N2: nat,X: nat] :
( ( vEBT_invar_vebt @ ( vEBT_Node @ Info @ Deg @ TreeList2 @ Summary ) @ N2 )
=> ( ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg ) )
=> ( ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
=> ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ Info @ Deg @ TreeList2 @ Summary ) @ X ) ) ) ) ).
% both_member_options_ding
thf(fact_690_less__by__empty,axiom,
! [A: $tType,A4: set @ ( product_prod @ A @ A ),B2: set @ ( product_prod @ A @ A )] :
( ( A4
= ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ A4 @ B2 ) ) ).
% less_by_empty
thf(fact_691_add__divide__distrib,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A,B3: A,C2: A] :
( ( divide_divide @ A @ ( plus_plus @ A @ A3 @ B3 ) @ C2 )
= ( plus_plus @ A @ ( divide_divide @ A @ A3 @ C2 ) @ ( divide_divide @ A @ B3 @ C2 ) ) ) ) ).
% add_divide_distrib
thf(fact_692_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_693_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_694_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_695_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_696_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_697_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_698_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_699_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_700_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_701_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_702_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_703_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_704_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_705_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_706_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_707_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_708_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_709_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_710_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_711_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_712_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_713_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_714_frac__less2,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y: A,W2: A,Z3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ W2 )
=> ( ( ord_less @ A @ W2 @ Z3 )
=> ( ord_less @ A @ ( divide_divide @ A @ X @ Z3 ) @ ( divide_divide @ A @ Y @ W2 ) ) ) ) ) ) ) ).
% frac_less2
thf(fact_715_frac__less,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y: A,W2: A,Z3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less @ A @ X @ Y )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ W2 )
=> ( ( ord_less_eq @ A @ W2 @ Z3 )
=> ( ord_less @ A @ ( divide_divide @ A @ X @ Z3 ) @ ( divide_divide @ A @ Y @ W2 ) ) ) ) ) ) ) ).
% frac_less
thf(fact_716_frac__le,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [Y: A,X: A,W2: A,Z3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
=> ( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ W2 )
=> ( ( ord_less_eq @ A @ W2 @ Z3 )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ X @ Z3 ) @ ( divide_divide @ A @ Y @ W2 ) ) ) ) ) ) ) ).
% frac_le
thf(fact_717_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_718_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_719_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_720_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_721_power__not__zero,axiom,
! [A: $tType] :
( ( semiri2026040879449505780visors @ A )
=> ! [A3: A,N2: nat] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( power_power @ A @ A3 @ N2 )
!= ( zero_zero @ A ) ) ) ) ).
% power_not_zero
thf(fact_722_power__mono,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,B3: A,N2: nat] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ord_less_eq @ A @ ( power_power @ A @ A3 @ N2 ) @ ( power_power @ A @ B3 @ N2 ) ) ) ) ) ).
% power_mono
thf(fact_723_zero__le__power,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,N2: nat] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A3 @ N2 ) ) ) ) ).
% zero_le_power
thf(fact_724_zero__less__power,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,N2: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A3 @ N2 ) ) ) ) ).
% zero_less_power
thf(fact_725_nat__power__less__imp__less,axiom,
! [I: nat,M2: nat,N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ I )
=> ( ( ord_less @ nat @ ( power_power @ nat @ I @ M2 ) @ ( power_power @ nat @ I @ N2 ) )
=> ( ord_less @ nat @ M2 @ N2 ) ) ) ).
% nat_power_less_imp_less
thf(fact_726_power__less__imp__less__base,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,N2: nat,B3: A] :
( ( ord_less @ A @ ( power_power @ A @ A3 @ N2 ) @ ( power_power @ A @ B3 @ N2 ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B3 )
=> ( ord_less @ A @ A3 @ B3 ) ) ) ) ).
% power_less_imp_less_base
thf(fact_727_power__inject__base,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,N2: nat,B3: A] :
( ( ( power_power @ A @ A3 @ ( suc @ N2 ) )
= ( power_power @ A @ B3 @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B3 )
=> ( A3 = B3 ) ) ) ) ) ).
% power_inject_base
thf(fact_728_power__le__imp__le__base,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,N2: nat,B3: A] :
( ( ord_less_eq @ A @ ( power_power @ A @ A3 @ ( suc @ N2 ) ) @ ( power_power @ A @ B3 @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B3 )
=> ( ord_less_eq @ A @ A3 @ B3 ) ) ) ) ).
% power_le_imp_le_base
thf(fact_729_zero__power,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( power_power @ A @ ( zero_zero @ A ) @ N2 )
= ( zero_zero @ A ) ) ) ) ).
% zero_power
thf(fact_730_power__gt__expt,axiom,
! [N2: nat,K: nat] :
( ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N2 )
=> ( ord_less @ nat @ K @ ( power_power @ nat @ N2 @ K ) ) ) ).
% power_gt_expt
thf(fact_731_nat__one__le__power,axiom,
! [I: nat,N2: nat] :
( ( ord_less_eq @ nat @ ( suc @ ( zero_zero @ nat ) ) @ I )
=> ( ord_less_eq @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( power_power @ nat @ I @ N2 ) ) ) ).
% nat_one_le_power
thf(fact_732_power__eq__imp__eq__base,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,N2: nat,B3: A] :
( ( ( power_power @ A @ A3 @ N2 )
= ( power_power @ A @ B3 @ N2 ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B3 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( A3 = B3 ) ) ) ) ) ) ).
% power_eq_imp_eq_base
thf(fact_733_power__eq__iff__eq__base,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N2: nat,A3: A,B3: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B3 )
=> ( ( ( power_power @ A @ A3 @ N2 )
= ( power_power @ A @ B3 @ N2 ) )
= ( A3 = B3 ) ) ) ) ) ) ).
% power_eq_iff_eq_base
thf(fact_734_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_735_less__exp,axiom,
! [N2: nat] : ( ord_less @ nat @ N2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ).
% less_exp
thf(fact_736_self__le__ge2__pow,axiom,
! [K: nat,M2: nat] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K )
=> ( ord_less_eq @ nat @ M2 @ ( power_power @ nat @ K @ M2 ) ) ) ).
% self_le_ge2_pow
thf(fact_737_power2__nat__le__eq__le,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq @ nat @ ( power_power @ nat @ M2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( ord_less_eq @ nat @ M2 @ N2 ) ) ).
% power2_nat_le_eq_le
thf(fact_738_power2__nat__le__imp__le,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq @ nat @ ( power_power @ nat @ M2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ N2 )
=> ( ord_less_eq @ nat @ M2 @ N2 ) ) ).
% power2_nat_le_imp_le
thf(fact_739_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_740_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_741_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_742_power__strict__mono,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,B3: A,N2: nat] :
( ( ord_less @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ord_less @ A @ ( power_power @ A @ A3 @ N2 ) @ ( power_power @ A @ B3 @ N2 ) ) ) ) ) ) ).
% power_strict_mono
thf(fact_743_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_744_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_745_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_746_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_747_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_748_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_749_both__member__options__from__chilf__to__complete__tree,axiom,
! [X: nat,Deg: nat,TreeList2: list @ vEBT_VEBT,Mi: nat,Ma: nat,Summary: vEBT_VEBT] :
( ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( ( ord_less_eq @ nat @ ( one_one @ nat ) @ Deg )
=> ( ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
=> ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X ) ) ) ) ).
% both_member_options_from_chilf_to_complete_tree
thf(fact_750_add__self__div__2,axiom,
! [M2: nat] :
( ( divide_divide @ nat @ ( plus_plus @ nat @ M2 @ M2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= M2 ) ).
% add_self_div_2
thf(fact_751_div2__Suc__Suc,axiom,
! [M2: nat] :
( ( divide_divide @ nat @ ( suc @ ( suc @ M2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( suc @ ( divide_divide @ nat @ M2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% div2_Suc_Suc
thf(fact_752_both__member__options__from__complete__tree__to__child,axiom,
! [Deg: nat,Mi: nat,Ma: nat,TreeList2: list @ vEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
( ( ord_less_eq @ nat @ ( one_one @ nat ) @ Deg )
=> ( ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
=> ( ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
| ( X = Mi )
| ( X = Ma ) ) ) ) ).
% both_member_options_from_complete_tree_to_child
thf(fact_753_div__less,axiom,
! [M2: nat,N2: nat] :
( ( ord_less @ nat @ M2 @ N2 )
=> ( ( divide_divide @ nat @ M2 @ N2 )
= ( zero_zero @ nat ) ) ) ).
% div_less
thf(fact_754_div__by__Suc__0,axiom,
! [M2: nat] :
( ( divide_divide @ nat @ M2 @ ( suc @ ( zero_zero @ nat ) ) )
= M2 ) ).
% div_by_Suc_0
thf(fact_755_set__n__deg__not__0,axiom,
! [TreeList2: list @ vEBT_VEBT,N2: nat,M2: nat] :
( ! [X5: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X5 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ( vEBT_invar_vebt @ X5 @ N2 ) )
=> ( ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M2 ) )
=> ( ord_less_eq @ nat @ ( one_one @ nat ) @ N2 ) ) ) ).
% set_n_deg_not_0
thf(fact_756_div__2__gt__zero,axiom,
! [N2: nat] :
( ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N2 )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% div_2_gt_zero
thf(fact_757_Suc__n__div__2__gt__zero,axiom,
! [N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ ( divide_divide @ nat @ ( suc @ N2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% Suc_n_div_2_gt_zero
thf(fact_758_div__exp__eq,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A3: A,M2: nat,N2: nat] :
( ( divide_divide @ A @ ( divide_divide @ A @ A3 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M2 ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) )
= ( divide_divide @ A @ A3 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( plus_plus @ nat @ M2 @ N2 ) ) ) ) ) ).
% div_exp_eq
thf(fact_759_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_760_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_761_div__by__1,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [A3: A] :
( ( divide_divide @ A @ A3 @ ( one_one @ A ) )
= A3 ) ) ).
% div_by_1
thf(fact_762_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_763_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_764_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_765_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_766_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_767_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_768_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_769_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_770_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_771_power__inject__exp,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,M2: nat,N2: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ A3 )
=> ( ( ( power_power @ A @ A3 @ M2 )
= ( power_power @ A @ A3 @ N2 ) )
= ( M2 = N2 ) ) ) ) ).
% power_inject_exp
thf(fact_772_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_773_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_774_less__one,axiom,
! [N2: nat] :
( ( ord_less @ nat @ N2 @ ( one_one @ nat ) )
= ( N2
= ( zero_zero @ nat ) ) ) ).
% less_one
thf(fact_775_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_776_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_777_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_778_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_779_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_780_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_781_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_782_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_783_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_784_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_785_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_786_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_787_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_788_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_789_power__strict__decreasing__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [B3: A,M2: nat,N2: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ B3 )
=> ( ( ord_less @ A @ B3 @ ( one_one @ A ) )
=> ( ( ord_less @ A @ ( power_power @ A @ B3 @ M2 ) @ ( power_power @ A @ B3 @ N2 ) )
= ( ord_less @ nat @ N2 @ M2 ) ) ) ) ) ).
% power_strict_decreasing_iff
thf(fact_790_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_791_Suc__1,axiom,
( ( suc @ ( one_one @ nat ) )
= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ).
% Suc_1
thf(fact_792_one__plus__numeral,axiom,
! [A: $tType] :
( ( numeral @ A )
=> ! [N2: num] :
( ( plus_plus @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ N2 ) )
= ( numeral_numeral @ A @ ( plus_plus @ num @ one2 @ N2 ) ) ) ) ).
% one_plus_numeral
thf(fact_793_numeral__plus__one,axiom,
! [A: $tType] :
( ( numeral @ A )
=> ! [N2: num] :
( ( plus_plus @ A @ ( numeral_numeral @ A @ N2 ) @ ( one_one @ A ) )
= ( numeral_numeral @ A @ ( plus_plus @ num @ N2 @ one2 ) ) ) ) ).
% numeral_plus_one
thf(fact_794_numeral__le__one__iff,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [N2: num] :
( ( ord_less_eq @ A @ ( numeral_numeral @ A @ N2 ) @ ( one_one @ A ) )
= ( ord_less_eq @ num @ N2 @ one2 ) ) ) ).
% numeral_le_one_iff
thf(fact_795_one__less__numeral__iff,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [N2: num] :
( ( ord_less @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ N2 ) )
= ( ord_less @ num @ one2 @ N2 ) ) ) ).
% one_less_numeral_iff
thf(fact_796_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_797_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_798_power__decreasing__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [B3: A,M2: nat,N2: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ B3 )
=> ( ( ord_less @ A @ B3 @ ( one_one @ A ) )
=> ( ( ord_less_eq @ A @ ( power_power @ A @ B3 @ M2 ) @ ( power_power @ A @ B3 @ N2 ) )
= ( ord_less_eq @ nat @ N2 @ M2 ) ) ) ) ) ).
% power_decreasing_iff
thf(fact_799_one__reorient,axiom,
! [A: $tType] :
( ( one @ A )
=> ! [X: A] :
( ( ( one_one @ A )
= X )
= ( X
= ( one_one @ A ) ) ) ) ).
% one_reorient
thf(fact_800_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_801_zero__neq__one,axiom,
! [A: $tType] :
( ( zero_neq_one @ A )
=> ( ( zero_zero @ A )
!= ( one_one @ A ) ) ) ).
% zero_neq_one
thf(fact_802_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_803_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_804_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_805_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_806_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_807_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_808_not__one__less__zero,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ~ ( ord_less @ A @ ( one_one @ A ) @ ( zero_zero @ A ) ) ) ).
% not_one_less_zero
thf(fact_809_zero__less__one,axiom,
! [A: $tType] :
( ( zero_less_one @ A )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( one_one @ A ) ) ) ).
% zero_less_one
thf(fact_810_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_811_one__le__numeral,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [N2: num] : ( ord_less_eq @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ N2 ) ) ) ).
% one_le_numeral
thf(fact_812_not__numeral__less__one,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [N2: num] :
~ ( ord_less @ A @ ( numeral_numeral @ A @ N2 ) @ ( one_one @ A ) ) ) ).
% not_numeral_less_one
thf(fact_813_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_814_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_815_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_816_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_817_one__le__power,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,N2: nat] :
( ( ord_less_eq @ A @ ( one_one @ A ) @ A3 )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ ( power_power @ A @ A3 @ N2 ) ) ) ) ).
% one_le_power
thf(fact_818_div__eq__dividend__iff,axiom,
! [M2: nat,N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ M2 )
=> ( ( ( divide_divide @ nat @ M2 @ N2 )
= M2 )
= ( N2
= ( one_one @ nat ) ) ) ) ).
% div_eq_dividend_iff
thf(fact_819_div__less__dividend,axiom,
! [N2: nat,M2: nat] :
( ( ord_less @ nat @ ( one_one @ nat ) @ N2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M2 )
=> ( ord_less @ nat @ ( divide_divide @ nat @ M2 @ N2 ) @ M2 ) ) ) ).
% div_less_dividend
thf(fact_820_power__0,axiom,
! [A: $tType] :
( ( power @ A )
=> ! [A3: A] :
( ( power_power @ A @ A3 @ ( zero_zero @ nat ) )
= ( one_one @ A ) ) ) ).
% power_0
thf(fact_821_One__nat__def,axiom,
( ( one_one @ nat )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% One_nat_def
thf(fact_822_Suc__eq__plus1,axiom,
( suc
= ( ^ [N: nat] : ( plus_plus @ nat @ N @ ( one_one @ nat ) ) ) ) ).
% Suc_eq_plus1
thf(fact_823_plus__1__eq__Suc,axiom,
( ( plus_plus @ nat @ ( one_one @ nat ) )
= suc ) ).
% plus_1_eq_Suc
thf(fact_824_Suc__eq__plus1__left,axiom,
( suc
= ( plus_plus @ nat @ ( one_one @ nat ) ) ) ).
% Suc_eq_plus1_left
thf(fact_825_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_826_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_827_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_828_power__le__one,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,N2: 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 @ N2 ) @ ( one_one @ A ) ) ) ) ) ).
% power_le_one
thf(fact_829_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_830_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_831_power__0__left,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [N2: nat] :
( ( ( N2
= ( zero_zero @ nat ) )
=> ( ( power_power @ A @ ( zero_zero @ A ) @ N2 )
= ( one_one @ A ) ) )
& ( ( N2
!= ( zero_zero @ nat ) )
=> ( ( power_power @ A @ ( zero_zero @ A ) @ N2 )
= ( zero_zero @ A ) ) ) ) ) ).
% power_0_left
thf(fact_832_power__gt1,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,N2: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ A3 )
=> ( ord_less @ A @ ( one_one @ A ) @ ( power_power @ A @ A3 @ ( suc @ N2 ) ) ) ) ) ).
% power_gt1
thf(fact_833_power__less__imp__less__exp,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,M2: nat,N2: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ A3 )
=> ( ( ord_less @ A @ ( power_power @ A @ A3 @ M2 ) @ ( power_power @ A @ A3 @ N2 ) )
=> ( ord_less @ nat @ M2 @ N2 ) ) ) ) ).
% power_less_imp_less_exp
thf(fact_834_power__strict__increasing,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N2: nat,N6: nat,A3: A] :
( ( ord_less @ nat @ N2 @ N6 )
=> ( ( ord_less @ A @ ( one_one @ A ) @ A3 )
=> ( ord_less @ A @ ( power_power @ A @ A3 @ N2 ) @ ( power_power @ A @ A3 @ N6 ) ) ) ) ) ).
% power_strict_increasing
thf(fact_835_power__increasing,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N2: nat,N6: nat,A3: A] :
( ( ord_less_eq @ nat @ N2 @ N6 )
=> ( ( ord_less_eq @ A @ ( one_one @ A ) @ A3 )
=> ( ord_less_eq @ A @ ( power_power @ A @ A3 @ N2 ) @ ( power_power @ A @ A3 @ N6 ) ) ) ) ) ).
% power_increasing
thf(fact_836_nat__induct__non__zero,axiom,
! [N2: nat,P: nat > $o] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( P @ ( one_one @ nat ) )
=> ( ! [N3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) )
=> ( P @ N2 ) ) ) ) ).
% nat_induct_non_zero
thf(fact_837_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_838_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_839_power__Suc__le__self,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,N2: 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 @ N2 ) ) @ A3 ) ) ) ) ).
% power_Suc_le_self
thf(fact_840_power__Suc__less__one,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,N2: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ A3 @ ( one_one @ A ) )
=> ( ord_less @ A @ ( power_power @ A @ A3 @ ( suc @ N2 ) ) @ ( one_one @ A ) ) ) ) ) ).
% power_Suc_less_one
thf(fact_841_power__strict__decreasing,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N2: nat,N6: nat,A3: A] :
( ( ord_less @ nat @ N2 @ N6 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ A3 @ ( one_one @ A ) )
=> ( ord_less @ A @ ( power_power @ A @ A3 @ N6 ) @ ( power_power @ A @ A3 @ N2 ) ) ) ) ) ) ).
% power_strict_decreasing
thf(fact_842_power__decreasing,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N2: nat,N6: nat,A3: A] :
( ( ord_less_eq @ nat @ N2 @ N6 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ A3 @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( power_power @ A @ A3 @ N6 ) @ ( power_power @ A @ A3 @ N2 ) ) ) ) ) ) ).
% power_decreasing
thf(fact_843_power__le__imp__le__exp,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,M2: nat,N2: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ ( power_power @ A @ A3 @ M2 ) @ ( power_power @ A @ A3 @ N2 ) )
=> ( ord_less_eq @ nat @ M2 @ N2 ) ) ) ) ).
% power_le_imp_le_exp
thf(fact_844_self__le__power,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,N2: nat] :
( ( ord_less_eq @ A @ ( one_one @ A ) @ A3 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ord_less_eq @ A @ A3 @ ( power_power @ A @ A3 @ N2 ) ) ) ) ) ).
% self_le_power
thf(fact_845_one__less__power,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,N2: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ A3 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ord_less @ A @ ( one_one @ A ) @ ( power_power @ A @ A3 @ N2 ) ) ) ) ) ).
% one_less_power
thf(fact_846_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_847_div__le__mono,axiom,
! [M2: nat,N2: nat,K: nat] :
( ( ord_less_eq @ nat @ M2 @ N2 )
=> ( ord_less_eq @ nat @ ( divide_divide @ nat @ M2 @ K ) @ ( divide_divide @ nat @ N2 @ K ) ) ) ).
% div_le_mono
thf(fact_848_div__le__dividend,axiom,
! [M2: nat,N2: nat] : ( ord_less_eq @ nat @ ( divide_divide @ nat @ M2 @ N2 ) @ M2 ) ).
% div_le_dividend
thf(fact_849_ex__power__ivl2,axiom,
! [B3: nat,K: nat] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ B3 )
=> ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K )
=> ? [N3: nat] :
( ( ord_less @ nat @ ( power_power @ nat @ B3 @ N3 ) @ K )
& ( ord_less_eq @ nat @ K @ ( power_power @ nat @ B3 @ ( plus_plus @ nat @ N3 @ ( one_one @ nat ) ) ) ) ) ) ) ).
% ex_power_ivl2
thf(fact_850_ex__power__ivl1,axiom,
! [B3: nat,K: nat] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ B3 )
=> ( ( ord_less_eq @ nat @ ( one_one @ nat ) @ K )
=> ? [N3: nat] :
( ( ord_less_eq @ nat @ ( power_power @ nat @ B3 @ N3 ) @ K )
& ( ord_less @ nat @ K @ ( power_power @ nat @ B3 @ ( plus_plus @ nat @ N3 @ ( one_one @ nat ) ) ) ) ) ) ) ).
% ex_power_ivl1
thf(fact_851_Euclidean__Division_Odiv__eq__0__iff,axiom,
! [M2: nat,N2: nat] :
( ( ( divide_divide @ nat @ M2 @ N2 )
= ( zero_zero @ nat ) )
= ( ( ord_less @ nat @ M2 @ N2 )
| ( N2
= ( zero_zero @ nat ) ) ) ) ).
% Euclidean_Division.div_eq_0_iff
thf(fact_852_Suc__div__le__mono,axiom,
! [M2: nat,N2: nat] : ( ord_less_eq @ nat @ ( divide_divide @ nat @ M2 @ N2 ) @ ( divide_divide @ nat @ ( suc @ M2 ) @ N2 ) ) ).
% Suc_div_le_mono
thf(fact_853_div__le__mono2,axiom,
! [M2: nat,N2: nat,K: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ M2 )
=> ( ( ord_less_eq @ nat @ M2 @ N2 )
=> ( ord_less_eq @ nat @ ( divide_divide @ nat @ K @ N2 ) @ ( divide_divide @ nat @ K @ M2 ) ) ) ) ).
% div_le_mono2
thf(fact_854_div__greater__zero__iff,axiom,
! [M2: nat,N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( divide_divide @ nat @ M2 @ N2 ) )
= ( ( ord_less_eq @ nat @ N2 @ M2 )
& ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ).
% div_greater_zero_iff
thf(fact_855_exp__add__not__zero__imp__left,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [M2: nat,N2: nat] :
( ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( plus_plus @ nat @ M2 @ N2 ) )
!= ( zero_zero @ A ) )
=> ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M2 )
!= ( zero_zero @ A ) ) ) ) ).
% exp_add_not_zero_imp_left
thf(fact_856_exp__add__not__zero__imp__right,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [M2: nat,N2: nat] :
( ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( plus_plus @ nat @ M2 @ N2 ) )
!= ( zero_zero @ A ) )
=> ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 )
!= ( zero_zero @ A ) ) ) ) ).
% exp_add_not_zero_imp_right
thf(fact_857_nat__induct2,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ ( zero_zero @ nat ) )
=> ( ( P @ ( one_one @ nat ) )
=> ( ! [N3: nat] :
( ( P @ N3 )
=> ( P @ ( plus_plus @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
=> ( P @ N2 ) ) ) ) ).
% nat_induct2
thf(fact_858_bit__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 ) ) ) ).
% bit_concat_def
thf(fact_859_low__inv,axiom,
! [X: nat,N2: nat,Y: nat] :
( ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) )
=> ( ( vEBT_VEBT_low @ ( plus_plus @ nat @ ( times_times @ nat @ Y @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) @ X ) @ N2 )
= X ) ) ).
% low_inv
thf(fact_860_high__inv,axiom,
! [X: nat,N2: nat,Y: nat] :
( ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) )
=> ( ( vEBT_VEBT_high @ ( plus_plus @ nat @ ( times_times @ nat @ Y @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) @ X ) @ N2 )
= Y ) ) ).
% high_inv
thf(fact_861_invar__vebt_Ocases,axiom,
! [A1: vEBT_VEBT,A22: nat] :
( ( vEBT_invar_vebt @ A1 @ A22 )
=> ( ( ? [A6: $o,B5: $o] :
( A1
= ( vEBT_Leaf @ A6 @ B5 ) )
=> ( A22
!= ( suc @ ( zero_zero @ nat ) ) ) )
=> ( ! [TreeList3: list @ vEBT_VEBT,N3: nat,Summary2: vEBT_VEBT,M3: nat,Deg2: nat] :
( ( A1
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg2 @ TreeList3 @ Summary2 ) )
=> ( ( A22 = Deg2 )
=> ( ! [X2: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X2 @ ( set2 @ vEBT_VEBT @ TreeList3 ) )
=> ( vEBT_invar_vebt @ X2 @ N3 ) )
=> ( ( vEBT_invar_vebt @ Summary2 @ M3 )
=> ( ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M3 ) )
=> ( ( M3 = N3 )
=> ( ( Deg2
= ( plus_plus @ nat @ N3 @ M3 ) )
=> ( ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X_1 )
=> ~ ! [X2: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X2 @ ( set2 @ vEBT_VEBT @ TreeList3 ) )
=> ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X_1 ) ) ) ) ) ) ) ) ) )
=> ( ! [TreeList3: list @ vEBT_VEBT,N3: nat,Summary2: vEBT_VEBT,M3: nat,Deg2: nat] :
( ( A1
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg2 @ TreeList3 @ Summary2 ) )
=> ( ( A22 = Deg2 )
=> ( ! [X2: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X2 @ ( set2 @ vEBT_VEBT @ TreeList3 ) )
=> ( vEBT_invar_vebt @ X2 @ N3 ) )
=> ( ( vEBT_invar_vebt @ Summary2 @ M3 )
=> ( ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M3 ) )
=> ( ( M3
= ( suc @ N3 ) )
=> ( ( Deg2
= ( plus_plus @ nat @ N3 @ M3 ) )
=> ( ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X_1 )
=> ~ ! [X2: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X2 @ ( set2 @ vEBT_VEBT @ TreeList3 ) )
=> ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X_1 ) ) ) ) ) ) ) ) ) )
=> ( ! [TreeList3: list @ vEBT_VEBT,N3: nat,Summary2: vEBT_VEBT,M3: nat,Deg2: nat,Mi2: nat,Ma2: nat] :
( ( A1
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ Deg2 @ TreeList3 @ Summary2 ) )
=> ( ( A22 = Deg2 )
=> ( ! [X2: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X2 @ ( set2 @ vEBT_VEBT @ TreeList3 ) )
=> ( vEBT_invar_vebt @ X2 @ N3 ) )
=> ( ( vEBT_invar_vebt @ Summary2 @ M3 )
=> ( ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M3 ) )
=> ( ( M3 = N3 )
=> ( ( Deg2
= ( plus_plus @ nat @ N3 @ M3 ) )
=> ( ! [I5: nat] :
( ( ord_less @ nat @ I5 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M3 ) )
=> ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList3 @ I5 ) @ X6 ) )
= ( vEBT_V8194947554948674370ptions @ Summary2 @ I5 ) ) )
=> ( ( ( Mi2 = Ma2 )
=> ! [X2: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X2 @ ( set2 @ vEBT_VEBT @ TreeList3 ) )
=> ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X_1 ) ) )
=> ( ( ord_less_eq @ nat @ Mi2 @ Ma2 )
=> ( ( ord_less @ nat @ Ma2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
=> ~ ( ( Mi2 != Ma2 )
=> ! [I5: nat] :
( ( ord_less @ nat @ I5 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M3 ) )
=> ( ( ( ( vEBT_VEBT_high @ Ma2 @ N3 )
= I5 )
=> ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList3 @ I5 ) @ ( vEBT_VEBT_low @ Ma2 @ N3 ) ) )
& ! [X2: nat] :
( ( ( ( vEBT_VEBT_high @ X2 @ N3 )
= I5 )
& ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList3 @ I5 ) @ ( vEBT_VEBT_low @ X2 @ N3 ) ) )
=> ( ( ord_less @ nat @ Mi2 @ X2 )
& ( ord_less_eq @ nat @ X2 @ Ma2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
=> ~ ! [TreeList3: list @ vEBT_VEBT,N3: nat,Summary2: vEBT_VEBT,M3: nat,Deg2: nat,Mi2: nat,Ma2: nat] :
( ( A1
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ Deg2 @ TreeList3 @ Summary2 ) )
=> ( ( A22 = Deg2 )
=> ( ! [X2: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X2 @ ( set2 @ vEBT_VEBT @ TreeList3 ) )
=> ( vEBT_invar_vebt @ X2 @ N3 ) )
=> ( ( vEBT_invar_vebt @ Summary2 @ M3 )
=> ( ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M3 ) )
=> ( ( M3
= ( suc @ N3 ) )
=> ( ( Deg2
= ( plus_plus @ nat @ N3 @ M3 ) )
=> ( ! [I5: nat] :
( ( ord_less @ nat @ I5 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M3 ) )
=> ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList3 @ I5 ) @ X6 ) )
= ( vEBT_V8194947554948674370ptions @ Summary2 @ I5 ) ) )
=> ( ( ( Mi2 = Ma2 )
=> ! [X2: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X2 @ ( set2 @ vEBT_VEBT @ TreeList3 ) )
=> ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X_1 ) ) )
=> ( ( ord_less_eq @ nat @ Mi2 @ Ma2 )
=> ( ( ord_less @ nat @ Ma2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
=> ~ ( ( Mi2 != Ma2 )
=> ! [I5: nat] :
( ( ord_less @ nat @ I5 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M3 ) )
=> ( ( ( ( vEBT_VEBT_high @ Ma2 @ N3 )
= I5 )
=> ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList3 @ I5 ) @ ( vEBT_VEBT_low @ Ma2 @ N3 ) ) )
& ! [X2: nat] :
( ( ( ( vEBT_VEBT_high @ X2 @ N3 )
= I5 )
& ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList3 @ I5 ) @ ( vEBT_VEBT_low @ X2 @ N3 ) ) )
=> ( ( ord_less @ nat @ Mi2 @ X2 )
& ( ord_less_eq @ nat @ X2 @ Ma2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% invar_vebt.cases
thf(fact_862_invar__vebt_Osimps,axiom,
( vEBT_invar_vebt
= ( ^ [A12: vEBT_VEBT,A23: nat] :
( ( ? [A5: $o,B4: $o] :
( A12
= ( vEBT_Leaf @ A5 @ B4 ) )
& ( A23
= ( suc @ ( zero_zero @ nat ) ) ) )
| ? [TreeList: list @ vEBT_VEBT,N: nat,Summary3: vEBT_VEBT] :
( ( A12
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ A23 @ TreeList @ Summary3 ) )
& ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ( vEBT_invar_vebt @ X4 @ N ) )
& ( vEBT_invar_vebt @ Summary3 @ N )
& ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
& ( A23
= ( plus_plus @ nat @ N @ N ) )
& ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X6 )
& ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X6 ) ) )
| ? [TreeList: list @ vEBT_VEBT,N: nat,Summary3: vEBT_VEBT] :
( ( A12
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ A23 @ TreeList @ Summary3 ) )
& ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ( vEBT_invar_vebt @ X4 @ N ) )
& ( vEBT_invar_vebt @ Summary3 @ ( suc @ N ) )
& ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ N ) ) )
& ( A23
= ( plus_plus @ nat @ N @ ( suc @ N ) ) )
& ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X6 )
& ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X6 ) ) )
| ? [TreeList: list @ vEBT_VEBT,N: nat,Summary3: vEBT_VEBT,Mi3: nat,Ma3: nat] :
( ( A12
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi3 @ Ma3 ) ) @ A23 @ TreeList @ Summary3 ) )
& ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ( vEBT_invar_vebt @ X4 @ N ) )
& ( vEBT_invar_vebt @ Summary3 @ N )
& ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
& ( A23
= ( plus_plus @ nat @ N @ N ) )
& ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
=> ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList @ I3 ) @ X6 ) )
= ( vEBT_V8194947554948674370ptions @ Summary3 @ I3 ) ) )
& ( ( Mi3 = Ma3 )
=> ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X6 ) ) )
& ( ord_less_eq @ nat @ Mi3 @ Ma3 )
& ( ord_less @ nat @ Ma3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ A23 ) )
& ( ( Mi3 != Ma3 )
=> ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
=> ( ( ( ( vEBT_VEBT_high @ Ma3 @ N )
= I3 )
=> ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList @ I3 ) @ ( vEBT_VEBT_low @ Ma3 @ N ) ) )
& ! [X4: nat] :
( ( ( ( vEBT_VEBT_high @ X4 @ N )
= I3 )
& ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList @ I3 ) @ ( vEBT_VEBT_low @ X4 @ N ) ) )
=> ( ( ord_less @ nat @ Mi3 @ X4 )
& ( ord_less_eq @ nat @ X4 @ Ma3 ) ) ) ) ) ) )
| ? [TreeList: list @ vEBT_VEBT,N: nat,Summary3: vEBT_VEBT,Mi3: nat,Ma3: nat] :
( ( A12
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi3 @ Ma3 ) ) @ A23 @ TreeList @ Summary3 ) )
& ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ( vEBT_invar_vebt @ X4 @ N ) )
& ( vEBT_invar_vebt @ Summary3 @ ( suc @ N ) )
& ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ N ) ) )
& ( A23
= ( plus_plus @ nat @ N @ ( suc @ N ) ) )
& ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ N ) ) )
=> ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList @ I3 ) @ X6 ) )
= ( vEBT_V8194947554948674370ptions @ Summary3 @ I3 ) ) )
& ( ( Mi3 = Ma3 )
=> ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X6 ) ) )
& ( ord_less_eq @ nat @ Mi3 @ Ma3 )
& ( ord_less @ nat @ Ma3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ A23 ) )
& ( ( Mi3 != Ma3 )
=> ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ N ) ) )
=> ( ( ( ( vEBT_VEBT_high @ Ma3 @ N )
= I3 )
=> ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList @ I3 ) @ ( vEBT_VEBT_low @ Ma3 @ N ) ) )
& ! [X4: nat] :
( ( ( ( vEBT_VEBT_high @ X4 @ N )
= I3 )
& ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList @ I3 ) @ ( vEBT_VEBT_low @ X4 @ N ) ) )
=> ( ( ord_less @ nat @ Mi3 @ X4 )
& ( ord_less_eq @ nat @ X4 @ Ma3 ) ) ) ) ) ) ) ) ) ) ).
% invar_vebt.simps
thf(fact_863_enat__ord__number_I1_J,axiom,
! [M2: num,N2: num] :
( ( ord_less_eq @ extended_enat @ ( numeral_numeral @ extended_enat @ M2 ) @ ( numeral_numeral @ extended_enat @ N2 ) )
= ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ M2 ) @ ( numeral_numeral @ nat @ N2 ) ) ) ).
% enat_ord_number(1)
thf(fact_864_enat__ord__number_I2_J,axiom,
! [M2: num,N2: num] :
( ( ord_less @ extended_enat @ ( numeral_numeral @ extended_enat @ M2 ) @ ( numeral_numeral @ extended_enat @ N2 ) )
= ( ord_less @ nat @ ( numeral_numeral @ nat @ M2 ) @ ( numeral_numeral @ nat @ N2 ) ) ) ).
% enat_ord_number(2)
thf(fact_865_pos2,axiom,
ord_less @ nat @ ( zero_zero @ nat ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ).
% pos2
thf(fact_866_Leaf__0__not,axiom,
! [A3: $o,B3: $o] :
~ ( vEBT_invar_vebt @ ( vEBT_Leaf @ A3 @ B3 ) @ ( zero_zero @ nat ) ) ).
% Leaf_0_not
thf(fact_867_deg__1__Leafy,axiom,
! [T2: vEBT_VEBT,N2: nat] :
( ( vEBT_invar_vebt @ T2 @ N2 )
=> ( ( N2
= ( one_one @ nat ) )
=> ? [A6: $o,B5: $o] :
( T2
= ( vEBT_Leaf @ A6 @ B5 ) ) ) ) ).
% deg_1_Leafy
thf(fact_868_deg__1__Leaf,axiom,
! [T2: vEBT_VEBT] :
( ( vEBT_invar_vebt @ T2 @ ( one_one @ nat ) )
=> ? [A6: $o,B5: $o] :
( T2
= ( vEBT_Leaf @ A6 @ B5 ) ) ) ).
% deg_1_Leaf
thf(fact_869_deg1Leaf,axiom,
! [T2: vEBT_VEBT] :
( ( vEBT_invar_vebt @ T2 @ ( one_one @ nat ) )
= ( ? [A5: $o,B4: $o] :
( T2
= ( vEBT_Leaf @ A5 @ B4 ) ) ) ) ).
% deg1Leaf
thf(fact_870_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_871_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_872_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_873_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_874_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_875_mult_Oright__neutral,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A3: A] :
( ( times_times @ A @ A3 @ ( one_one @ A ) )
= A3 ) ) ).
% mult.right_neutral
thf(fact_876_mult__1,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A3: A] :
( ( times_times @ A @ ( one_one @ A ) @ A3 )
= A3 ) ) ).
% mult_1
thf(fact_877_times__divide__eq__left,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [B3: A,C2: A,A3: A] :
( ( times_times @ A @ ( divide_divide @ A @ B3 @ C2 ) @ A3 )
= ( divide_divide @ A @ ( times_times @ A @ B3 @ A3 ) @ C2 ) ) ) ).
% times_divide_eq_left
thf(fact_878_divide__divide__eq__left,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A3: A,B3: A,C2: A] :
( ( divide_divide @ A @ ( divide_divide @ A @ A3 @ B3 ) @ C2 )
= ( divide_divide @ A @ A3 @ ( times_times @ A @ B3 @ C2 ) ) ) ) ).
% divide_divide_eq_left
thf(fact_879_divide__divide__eq__right,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A3: A,B3: A,C2: A] :
( ( divide_divide @ A @ A3 @ ( divide_divide @ A @ B3 @ C2 ) )
= ( divide_divide @ A @ ( times_times @ A @ A3 @ C2 ) @ B3 ) ) ) ).
% divide_divide_eq_right
thf(fact_880_times__divide__eq__right,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A,B3: A,C2: A] :
( ( times_times @ A @ A3 @ ( divide_divide @ A @ B3 @ C2 ) )
= ( divide_divide @ A @ ( times_times @ A @ A3 @ B3 ) @ C2 ) ) ) ).
% times_divide_eq_right
thf(fact_881_mult__cancel2,axiom,
! [M2: nat,K: nat,N2: nat] :
( ( ( times_times @ nat @ M2 @ K )
= ( times_times @ nat @ N2 @ K ) )
= ( ( M2 = N2 )
| ( K
= ( zero_zero @ nat ) ) ) ) ).
% mult_cancel2
thf(fact_882_mult__cancel1,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ( times_times @ nat @ K @ M2 )
= ( times_times @ nat @ K @ N2 ) )
= ( ( M2 = N2 )
| ( K
= ( zero_zero @ nat ) ) ) ) ).
% mult_cancel1
thf(fact_883_mult__0__right,axiom,
! [M2: nat] :
( ( times_times @ nat @ M2 @ ( zero_zero @ nat ) )
= ( zero_zero @ nat ) ) ).
% mult_0_right
thf(fact_884_mult__is__0,axiom,
! [M2: nat,N2: nat] :
( ( ( times_times @ nat @ M2 @ N2 )
= ( zero_zero @ nat ) )
= ( ( M2
= ( zero_zero @ nat ) )
| ( N2
= ( zero_zero @ nat ) ) ) ) ).
% mult_is_0
thf(fact_885_nat__1__eq__mult__iff,axiom,
! [M2: nat,N2: nat] :
( ( ( one_one @ nat )
= ( times_times @ nat @ M2 @ N2 ) )
= ( ( M2
= ( one_one @ nat ) )
& ( N2
= ( one_one @ nat ) ) ) ) ).
% nat_1_eq_mult_iff
thf(fact_886_nat__mult__eq__1__iff,axiom,
! [M2: nat,N2: nat] :
( ( ( times_times @ nat @ M2 @ N2 )
= ( one_one @ nat ) )
= ( ( M2
= ( one_one @ nat ) )
& ( N2
= ( one_one @ nat ) ) ) ) ).
% nat_mult_eq_1_iff
thf(fact_887_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_888_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_889_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_890_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_891_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_892_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_893_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_894_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_895_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_896_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_897_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_898_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_899_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_900_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_901_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_902_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_903_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_904_one__eq__mult__iff,axiom,
! [M2: nat,N2: nat] :
( ( ( suc @ ( zero_zero @ nat ) )
= ( times_times @ nat @ M2 @ N2 ) )
= ( ( M2
= ( suc @ ( zero_zero @ nat ) ) )
& ( N2
= ( suc @ ( zero_zero @ nat ) ) ) ) ) ).
% one_eq_mult_iff
thf(fact_905_mult__eq__1__iff,axiom,
! [M2: nat,N2: nat] :
( ( ( times_times @ nat @ M2 @ N2 )
= ( suc @ ( zero_zero @ nat ) ) )
= ( ( M2
= ( suc @ ( zero_zero @ nat ) ) )
& ( N2
= ( suc @ ( zero_zero @ nat ) ) ) ) ) ).
% mult_eq_1_iff
thf(fact_906_mult__less__cancel2,axiom,
! [M2: nat,K: nat,N2: nat] :
( ( ord_less @ nat @ ( times_times @ nat @ M2 @ K ) @ ( times_times @ nat @ N2 @ K ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
& ( ord_less @ nat @ M2 @ N2 ) ) ) ).
% mult_less_cancel2
thf(fact_907_nat__0__less__mult__iff,axiom,
! [M2: nat,N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( times_times @ nat @ M2 @ N2 ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M2 )
& ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ).
% nat_0_less_mult_iff
thf(fact_908_mult__Suc__right,axiom,
! [M2: nat,N2: nat] :
( ( times_times @ nat @ M2 @ ( suc @ N2 ) )
= ( plus_plus @ nat @ M2 @ ( times_times @ nat @ M2 @ N2 ) ) ) ).
% mult_Suc_right
thf(fact_909_le__divide__eq__numeral1_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B3: A,W2: num] :
( ( ord_less_eq @ A @ A3 @ ( divide_divide @ A @ B3 @ ( numeral_numeral @ A @ W2 ) ) )
= ( ord_less_eq @ A @ ( times_times @ A @ A3 @ ( numeral_numeral @ A @ W2 ) ) @ B3 ) ) ) ).
% le_divide_eq_numeral1(1)
thf(fact_910_divide__le__eq__numeral1_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B3: A,W2: num,A3: A] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ B3 @ ( numeral_numeral @ A @ W2 ) ) @ A3 )
= ( ord_less_eq @ A @ B3 @ ( times_times @ A @ A3 @ ( numeral_numeral @ A @ W2 ) ) ) ) ) ).
% divide_le_eq_numeral1(1)
thf(fact_911_divide__eq__eq__numeral1_I1_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B3: A,W2: num,A3: A] :
( ( ( divide_divide @ A @ B3 @ ( numeral_numeral @ A @ W2 ) )
= A3 )
= ( ( ( ( numeral_numeral @ A @ W2 )
!= ( zero_zero @ A ) )
=> ( B3
= ( times_times @ A @ A3 @ ( numeral_numeral @ A @ W2 ) ) ) )
& ( ( ( numeral_numeral @ A @ W2 )
= ( zero_zero @ A ) )
=> ( A3
= ( zero_zero @ A ) ) ) ) ) ) ).
% divide_eq_eq_numeral1(1)
thf(fact_912_eq__divide__eq__numeral1_I1_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A,B3: A,W2: num] :
( ( A3
= ( divide_divide @ A @ B3 @ ( numeral_numeral @ A @ W2 ) ) )
= ( ( ( ( numeral_numeral @ A @ W2 )
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ A3 @ ( numeral_numeral @ A @ W2 ) )
= B3 ) )
& ( ( ( numeral_numeral @ A @ W2 )
= ( zero_zero @ A ) )
=> ( A3
= ( zero_zero @ A ) ) ) ) ) ) ).
% eq_divide_eq_numeral1(1)
thf(fact_913_less__divide__eq__numeral1_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B3: A,W2: num] :
( ( ord_less @ A @ A3 @ ( divide_divide @ A @ B3 @ ( numeral_numeral @ A @ W2 ) ) )
= ( ord_less @ A @ ( times_times @ A @ A3 @ ( numeral_numeral @ A @ W2 ) ) @ B3 ) ) ) ).
% less_divide_eq_numeral1(1)
thf(fact_914_divide__less__eq__numeral1_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B3: A,W2: num,A3: A] :
( ( ord_less @ A @ ( divide_divide @ A @ B3 @ ( numeral_numeral @ A @ W2 ) ) @ A3 )
= ( ord_less @ A @ B3 @ ( times_times @ A @ A3 @ ( numeral_numeral @ A @ W2 ) ) ) ) ) ).
% divide_less_eq_numeral1(1)
thf(fact_915_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_916_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_917_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_918_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_919_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_920_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_921_one__le__mult__iff,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( times_times @ nat @ M2 @ N2 ) )
= ( ( ord_less_eq @ nat @ ( suc @ ( zero_zero @ nat ) ) @ M2 )
& ( ord_less_eq @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N2 ) ) ) ).
% one_le_mult_iff
thf(fact_922_mult__le__cancel2,axiom,
! [M2: nat,K: nat,N2: nat] :
( ( ord_less_eq @ nat @ ( times_times @ nat @ M2 @ K ) @ ( times_times @ nat @ N2 @ K ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ord_less_eq @ nat @ M2 @ N2 ) ) ) ).
% mult_le_cancel2
thf(fact_923_div__mult__self__is__m,axiom,
! [N2: nat,M2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( divide_divide @ nat @ ( times_times @ nat @ M2 @ N2 ) @ N2 )
= M2 ) ) ).
% div_mult_self_is_m
thf(fact_924_div__mult__self1__is__m,axiom,
! [N2: nat,M2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( divide_divide @ nat @ ( times_times @ nat @ N2 @ M2 ) @ N2 )
= M2 ) ) ).
% div_mult_self1_is_m
thf(fact_925_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: $tType] :
( ( ab_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 ) ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_926_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_927_mult_Ocommute,axiom,
! [A: $tType] :
( ( ab_semigroup_mult @ A )
=> ( ( times_times @ A )
= ( ^ [A5: A,B4: A] : ( times_times @ A @ B4 @ A5 ) ) ) ) ).
% mult.commute
thf(fact_928_mult_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 ) ) ) ) ).
% mult.left_commute
thf(fact_929_VEBT_Osize_I4_J,axiom,
! [X21: $o,X222: $o] :
( ( size_size @ vEBT_VEBT @ ( vEBT_Leaf @ X21 @ X222 ) )
= ( zero_zero @ nat ) ) ).
% VEBT.size(4)
thf(fact_930_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_931_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_932_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_933_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_934_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_935_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_936_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_937_crossproduct__noteq,axiom,
! [A: $tType] :
( ( semiri1453513574482234551roduct @ A )
=> ! [A3: A,B3: A,C2: A,D2: A] :
( ( ( A3 != B3 )
& ( C2 != D2 ) )
= ( ( plus_plus @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B3 @ D2 ) )
!= ( plus_plus @ A @ ( times_times @ A @ A3 @ D2 ) @ ( times_times @ A @ B3 @ C2 ) ) ) ) ) ).
% crossproduct_noteq
thf(fact_938_crossproduct__eq,axiom,
! [A: $tType] :
( ( semiri1453513574482234551roduct @ A )
=> ! [W2: A,Y: A,X: A,Z3: A] :
( ( ( plus_plus @ A @ ( times_times @ A @ W2 @ Y ) @ ( times_times @ A @ X @ Z3 ) )
= ( plus_plus @ A @ ( times_times @ A @ W2 @ Z3 ) @ ( times_times @ A @ X @ Y ) ) )
= ( ( W2 = X )
| ( Y = Z3 ) ) ) ) ).
% crossproduct_eq
thf(fact_939_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_940_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_941_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_942_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_943_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_944_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_945_times__divide__times__eq,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [X: A,Y: A,Z3: A,W2: A] :
( ( times_times @ A @ ( divide_divide @ A @ X @ Y ) @ ( divide_divide @ A @ Z3 @ W2 ) )
= ( divide_divide @ A @ ( times_times @ A @ X @ Z3 ) @ ( times_times @ A @ Y @ W2 ) ) ) ) ).
% times_divide_times_eq
thf(fact_946_divide__divide__times__eq,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [X: A,Y: A,Z3: A,W2: A] :
( ( divide_divide @ A @ ( divide_divide @ A @ X @ Y ) @ ( divide_divide @ A @ Z3 @ W2 ) )
= ( divide_divide @ A @ ( times_times @ A @ X @ W2 ) @ ( times_times @ A @ Y @ Z3 ) ) ) ) ).
% divide_divide_times_eq
thf(fact_947_divide__divide__eq__left_H,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A,B3: A,C2: A] :
( ( divide_divide @ A @ ( divide_divide @ A @ A3 @ B3 ) @ C2 )
= ( divide_divide @ A @ A3 @ ( times_times @ A @ C2 @ B3 ) ) ) ) ).
% divide_divide_eq_left'
thf(fact_948_Suc__mult__cancel1,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ( times_times @ nat @ ( suc @ K ) @ M2 )
= ( times_times @ nat @ ( suc @ K ) @ N2 ) )
= ( M2 = N2 ) ) ).
% Suc_mult_cancel1
thf(fact_949_mult__0,axiom,
! [N2: nat] :
( ( times_times @ nat @ ( zero_zero @ nat ) @ N2 )
= ( zero_zero @ nat ) ) ).
% mult_0
thf(fact_950_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_951_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_952_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_953_le__square,axiom,
! [M2: nat] : ( ord_less_eq @ nat @ M2 @ ( times_times @ nat @ M2 @ M2 ) ) ).
% le_square
thf(fact_954_le__cube,axiom,
! [M2: nat] : ( ord_less_eq @ nat @ M2 @ ( times_times @ nat @ M2 @ ( times_times @ nat @ M2 @ M2 ) ) ) ).
% le_cube
thf(fact_955_add__mult__distrib2,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( times_times @ nat @ K @ ( plus_plus @ nat @ M2 @ N2 ) )
= ( plus_plus @ nat @ ( times_times @ nat @ K @ M2 ) @ ( times_times @ nat @ K @ N2 ) ) ) ).
% add_mult_distrib2
thf(fact_956_add__mult__distrib,axiom,
! [M2: nat,N2: nat,K: nat] :
( ( times_times @ nat @ ( plus_plus @ nat @ M2 @ N2 ) @ K )
= ( plus_plus @ nat @ ( times_times @ nat @ M2 @ K ) @ ( times_times @ nat @ N2 @ K ) ) ) ).
% add_mult_distrib
thf(fact_957_nat__mult__1,axiom,
! [N2: nat] :
( ( times_times @ nat @ ( one_one @ nat ) @ N2 )
= N2 ) ).
% nat_mult_1
thf(fact_958_nat__mult__1__right,axiom,
! [N2: nat] :
( ( times_times @ nat @ N2 @ ( one_one @ nat ) )
= N2 ) ).
% nat_mult_1_right
thf(fact_959_VEBT__internal_Omembermima_Osimps_I1_J,axiom,
! [Uu: $o,Uv: $o,Uw: nat] :
~ ( vEBT_VEBT_membermima @ ( vEBT_Leaf @ Uu @ Uv ) @ Uw ) ).
% VEBT_internal.membermima.simps(1)
thf(fact_960_VEBT__internal_OminNull_Osimps_I1_J,axiom,
vEBT_VEBT_minNull @ ( vEBT_Leaf @ $false @ $false ) ).
% VEBT_internal.minNull.simps(1)
thf(fact_961_VEBT__internal_OminNull_Osimps_I2_J,axiom,
! [Uv: $o] :
~ ( vEBT_VEBT_minNull @ ( vEBT_Leaf @ $true @ Uv ) ) ).
% VEBT_internal.minNull.simps(2)
thf(fact_962_VEBT__internal_OminNull_Osimps_I3_J,axiom,
! [Uu: $o] :
~ ( vEBT_VEBT_minNull @ ( vEBT_Leaf @ Uu @ $true ) ) ).
% VEBT_internal.minNull.simps(3)
thf(fact_963_nat__mult__max__left,axiom,
! [M2: nat,N2: nat,Q3: nat] :
( ( times_times @ nat @ ( ord_max @ nat @ M2 @ N2 ) @ Q3 )
= ( ord_max @ nat @ ( times_times @ nat @ M2 @ Q3 ) @ ( times_times @ nat @ N2 @ Q3 ) ) ) ).
% nat_mult_max_left
thf(fact_964_nat__mult__max__right,axiom,
! [M2: nat,N2: nat,Q3: nat] :
( ( times_times @ nat @ M2 @ ( ord_max @ nat @ N2 @ Q3 ) )
= ( ord_max @ nat @ ( times_times @ nat @ M2 @ N2 ) @ ( times_times @ nat @ M2 @ Q3 ) ) ) ).
% nat_mult_max_right
thf(fact_965_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_966_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_967_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_968_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_969_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_970_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_971_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_972_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_973_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_974_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_975_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_976_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_977_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_978_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_979_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_980_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_981_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_982_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_983_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_984_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_985_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_986_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_987_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_988_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_989_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_990_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_991_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_992_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_993_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_994_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_995_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_996_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_997_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_998_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_999_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_1000_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_1001_add__scale__eq__noteq,axiom,
! [A: $tType] :
( ( semiri1453513574482234551roduct @ A )
=> ! [R2: A,A3: A,B3: A,C2: A,D2: A] :
( ( R2
!= ( zero_zero @ A ) )
=> ( ( ( A3 = B3 )
& ( C2 != D2 ) )
=> ( ( plus_plus @ A @ A3 @ ( times_times @ A @ R2 @ C2 ) )
!= ( plus_plus @ A @ B3 @ ( times_times @ A @ R2 @ D2 ) ) ) ) ) ) ).
% add_scale_eq_noteq
thf(fact_1002_less__1__mult,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [M2: A,N2: A] :
( ( ord_less @ A @ ( one_one @ A ) @ M2 )
=> ( ( ord_less @ A @ ( one_one @ A ) @ N2 )
=> ( ord_less @ A @ ( one_one @ A ) @ ( times_times @ A @ M2 @ N2 ) ) ) ) ) ).
% less_1_mult
thf(fact_1003_frac__eq__eq,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [Y: A,Z3: A,X: A,W2: A] :
( ( Y
!= ( zero_zero @ A ) )
=> ( ( Z3
!= ( zero_zero @ A ) )
=> ( ( ( divide_divide @ A @ X @ Y )
= ( divide_divide @ A @ W2 @ Z3 ) )
= ( ( times_times @ A @ X @ Z3 )
= ( times_times @ A @ W2 @ Y ) ) ) ) ) ) ).
% frac_eq_eq
thf(fact_1004_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_1005_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_1006_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_1007_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_1008_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_1009_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_1010_power__Suc,axiom,
! [A: $tType] :
( ( power @ A )
=> ! [A3: A,N2: nat] :
( ( power_power @ A @ A3 @ ( suc @ N2 ) )
= ( times_times @ A @ A3 @ ( power_power @ A @ A3 @ N2 ) ) ) ) ).
% power_Suc
thf(fact_1011_power__Suc2,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A3: A,N2: nat] :
( ( power_power @ A @ A3 @ ( suc @ N2 ) )
= ( times_times @ A @ ( power_power @ A @ A3 @ N2 ) @ A3 ) ) ) ).
% power_Suc2
thf(fact_1012_Suc__mult__less__cancel1,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ord_less @ nat @ ( times_times @ nat @ ( suc @ K ) @ M2 ) @ ( times_times @ nat @ ( suc @ K ) @ N2 ) )
= ( ord_less @ nat @ M2 @ N2 ) ) ).
% Suc_mult_less_cancel1
thf(fact_1013_power__add,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A3: A,M2: nat,N2: nat] :
( ( power_power @ A @ A3 @ ( plus_plus @ nat @ M2 @ N2 ) )
= ( times_times @ A @ ( power_power @ A @ A3 @ M2 ) @ ( power_power @ A @ A3 @ N2 ) ) ) ) ).
% power_add
thf(fact_1014_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_1015_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_1016_vebt__buildup_Osimps_I1_J,axiom,
( ( vEBT_vebt_buildup @ ( zero_zero @ nat ) )
= ( vEBT_Leaf @ $false @ $false ) ) ).
% vebt_buildup.simps(1)
thf(fact_1017_Suc__mult__le__cancel1,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ord_less_eq @ nat @ ( times_times @ nat @ ( suc @ K ) @ M2 ) @ ( times_times @ nat @ ( suc @ K ) @ N2 ) )
= ( ord_less_eq @ nat @ M2 @ N2 ) ) ).
% Suc_mult_le_cancel1
thf(fact_1018_mult__Suc,axiom,
! [M2: nat,N2: nat] :
( ( times_times @ nat @ ( suc @ M2 ) @ N2 )
= ( plus_plus @ nat @ N2 @ ( times_times @ nat @ M2 @ N2 ) ) ) ).
% mult_Suc
thf(fact_1019_VEBT__internal_Ovalid_H_Osimps_I1_J,axiom,
! [Uu: $o,Uv: $o,D2: nat] :
( ( vEBT_VEBT_valid @ ( vEBT_Leaf @ Uu @ Uv ) @ D2 )
= ( D2
= ( one_one @ nat ) ) ) ).
% VEBT_internal.valid'.simps(1)
thf(fact_1020_mult__eq__self__implies__10,axiom,
! [M2: nat,N2: nat] :
( ( M2
= ( times_times @ nat @ M2 @ N2 ) )
=> ( ( N2
= ( one_one @ nat ) )
| ( M2
= ( zero_zero @ nat ) ) ) ) ).
% mult_eq_self_implies_10
thf(fact_1021_less__mult__imp__div__less,axiom,
! [M2: nat,I: nat,N2: nat] :
( ( ord_less @ nat @ M2 @ ( times_times @ nat @ I @ N2 ) )
=> ( ord_less @ nat @ ( divide_divide @ nat @ M2 @ N2 ) @ I ) ) ).
% less_mult_imp_div_less
thf(fact_1022_div__times__less__eq__dividend,axiom,
! [M2: nat,N2: nat] : ( ord_less_eq @ nat @ ( times_times @ nat @ ( divide_divide @ nat @ M2 @ N2 ) @ N2 ) @ M2 ) ).
% div_times_less_eq_dividend
thf(fact_1023_times__div__less__eq__dividend,axiom,
! [N2: nat,M2: nat] : ( ord_less_eq @ nat @ ( times_times @ nat @ N2 @ ( divide_divide @ nat @ M2 @ N2 ) ) @ M2 ) ).
% times_div_less_eq_dividend
thf(fact_1024_power__odd__eq,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A3: A,N2: nat] :
( ( power_power @ A @ A3 @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) )
= ( times_times @ A @ A3 @ ( power_power @ A @ ( power_power @ A @ A3 @ N2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% power_odd_eq
thf(fact_1025_Suc__double__not__eq__double,axiom,
! [M2: nat,N2: nat] :
( ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M2 ) )
!= ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ).
% Suc_double_not_eq_double
thf(fact_1026_double__not__eq__Suc__double,axiom,
! [M2: nat,N2: nat] :
( ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M2 )
!= ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) ).
% double_not_eq_Suc_double
thf(fact_1027_VEBT__internal_Onaive__member_Ocases,axiom,
! [X: product_prod @ vEBT_VEBT @ nat] :
( ! [A6: $o,B5: $o,X5: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A6 @ B5 ) @ X5 ) )
=> ( ! [Uu2: option @ ( product_prod @ nat @ nat ),Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT,Ux2: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Uu2 @ ( zero_zero @ nat ) @ Uv2 @ Uw2 ) @ Ux2 ) )
=> ~ ! [Uy2: option @ ( product_prod @ nat @ nat ),V3: nat,TreeList3: list @ vEBT_VEBT,S3: vEBT_VEBT,X5: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList3 @ S3 ) @ X5 ) ) ) ) ).
% VEBT_internal.naive_member.cases
thf(fact_1028_invar__vebt_Ointros_I1_J,axiom,
! [A3: $o,B3: $o] : ( vEBT_invar_vebt @ ( vEBT_Leaf @ A3 @ B3 ) @ ( suc @ ( zero_zero @ nat ) ) ) ).
% invar_vebt.intros(1)
thf(fact_1029_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_1030_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_1031_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_1032_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_1033_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_1034_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_1035_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_1036_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_1037_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_1038_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_1039_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_1040_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_1041_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_1042_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_1043_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_1044_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_1045_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_1046_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_1047_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_1048_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_1049_vebt__insert_Osimps_I1_J,axiom,
! [X: nat,A3: $o,B3: $o] :
( ( ( X
= ( zero_zero @ nat ) )
=> ( ( vEBT_vebt_insert @ ( vEBT_Leaf @ A3 @ B3 ) @ X )
= ( vEBT_Leaf @ $true @ B3 ) ) )
& ( ( X
!= ( zero_zero @ nat ) )
=> ( ( ( X
= ( one_one @ nat ) )
=> ( ( vEBT_vebt_insert @ ( vEBT_Leaf @ A3 @ B3 ) @ X )
= ( vEBT_Leaf @ A3 @ $true ) ) )
& ( ( X
!= ( one_one @ nat ) )
=> ( ( vEBT_vebt_insert @ ( vEBT_Leaf @ A3 @ B3 ) @ X )
= ( vEBT_Leaf @ A3 @ B3 ) ) ) ) ) ) ).
% vebt_insert.simps(1)
thf(fact_1050_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_1051_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_1052_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_1053_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_1054_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_1055_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_1056_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_1057_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_1058_mult__imp__div__pos__less,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [Y: A,X: A,Z3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Y )
=> ( ( ord_less @ A @ X @ ( times_times @ A @ Z3 @ Y ) )
=> ( ord_less @ A @ ( divide_divide @ A @ X @ Y ) @ Z3 ) ) ) ) ).
% mult_imp_div_pos_less
thf(fact_1059_mult__imp__less__div__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [Y: A,Z3: A,X: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Y )
=> ( ( ord_less @ A @ ( times_times @ A @ Z3 @ Y ) @ X )
=> ( ord_less @ A @ Z3 @ ( divide_divide @ A @ X @ Y ) ) ) ) ) ).
% mult_imp_less_div_pos
thf(fact_1060_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_1061_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_1062_vebt__buildup_Osimps_I2_J,axiom,
( ( vEBT_vebt_buildup @ ( suc @ ( zero_zero @ nat ) ) )
= ( vEBT_Leaf @ $false @ $false ) ) ).
% vebt_buildup.simps(2)
thf(fact_1063_divide__eq__eq__numeral_I1_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B3: A,C2: A,W2: num] :
( ( ( divide_divide @ A @ B3 @ C2 )
= ( numeral_numeral @ A @ W2 ) )
= ( ( ( C2
!= ( zero_zero @ A ) )
=> ( B3
= ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ C2 ) ) )
& ( ( C2
= ( zero_zero @ A ) )
=> ( ( numeral_numeral @ A @ W2 )
= ( zero_zero @ A ) ) ) ) ) ) ).
% divide_eq_eq_numeral(1)
thf(fact_1064_eq__divide__eq__numeral_I1_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [W2: num,B3: A,C2: A] :
( ( ( numeral_numeral @ A @ W2 )
= ( divide_divide @ A @ B3 @ C2 ) )
= ( ( ( C2
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ C2 )
= B3 ) )
& ( ( C2
= ( zero_zero @ A ) )
=> ( ( numeral_numeral @ A @ W2 )
= ( zero_zero @ A ) ) ) ) ) ) ).
% eq_divide_eq_numeral(1)
thf(fact_1065_vebt__member_Osimps_I1_J,axiom,
! [A3: $o,B3: $o,X: nat] :
( ( vEBT_vebt_member @ ( vEBT_Leaf @ A3 @ B3 ) @ X )
= ( ( ( X
= ( zero_zero @ nat ) )
=> A3 )
& ( ( X
!= ( zero_zero @ nat ) )
=> ( ( ( X
= ( one_one @ nat ) )
=> B3 )
& ( X
= ( one_one @ nat ) ) ) ) ) ) ).
% vebt_member.simps(1)
thf(fact_1066_divide__add__eq__iff,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z3: A,X: A,Y: A] :
( ( Z3
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( divide_divide @ A @ X @ Z3 ) @ Y )
= ( divide_divide @ A @ ( plus_plus @ A @ X @ ( times_times @ A @ Y @ Z3 ) ) @ Z3 ) ) ) ) ).
% divide_add_eq_iff
thf(fact_1067_add__divide__eq__iff,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z3: A,X: A,Y: A] :
( ( Z3
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ X @ ( divide_divide @ A @ Y @ Z3 ) )
= ( divide_divide @ A @ ( plus_plus @ A @ ( times_times @ A @ X @ Z3 ) @ Y ) @ Z3 ) ) ) ) ).
% add_divide_eq_iff
thf(fact_1068_add__num__frac,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [Y: A,Z3: A,X: A] :
( ( Y
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ Z3 @ ( divide_divide @ A @ X @ Y ) )
= ( divide_divide @ A @ ( plus_plus @ A @ X @ ( times_times @ A @ Z3 @ Y ) ) @ Y ) ) ) ) ).
% add_num_frac
thf(fact_1069_add__frac__num,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [Y: A,X: A,Z3: A] :
( ( Y
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( divide_divide @ A @ X @ Y ) @ Z3 )
= ( divide_divide @ A @ ( plus_plus @ A @ X @ ( times_times @ A @ Z3 @ Y ) ) @ Y ) ) ) ) ).
% add_frac_num
thf(fact_1070_add__frac__eq,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [Y: A,Z3: A,X: A,W2: A] :
( ( Y
!= ( zero_zero @ A ) )
=> ( ( Z3
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( divide_divide @ A @ X @ Y ) @ ( divide_divide @ A @ W2 @ Z3 ) )
= ( divide_divide @ A @ ( plus_plus @ A @ ( times_times @ A @ X @ Z3 ) @ ( times_times @ A @ W2 @ Y ) ) @ ( times_times @ A @ Y @ Z3 ) ) ) ) ) ) ).
% add_frac_eq
thf(fact_1071_add__divide__eq__if__simps_I1_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z3: A,A3: A,B3: A] :
( ( ( Z3
= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ A3 @ ( divide_divide @ A @ B3 @ Z3 ) )
= A3 ) )
& ( ( Z3
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ A3 @ ( divide_divide @ A @ B3 @ Z3 ) )
= ( divide_divide @ A @ ( plus_plus @ A @ ( times_times @ A @ A3 @ Z3 ) @ B3 ) @ Z3 ) ) ) ) ) ).
% add_divide_eq_if_simps(1)
thf(fact_1072_add__divide__eq__if__simps_I2_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z3: A,A3: A,B3: A] :
( ( ( Z3
= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( divide_divide @ A @ A3 @ Z3 ) @ B3 )
= B3 ) )
& ( ( Z3
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( divide_divide @ A @ A3 @ Z3 ) @ B3 )
= ( divide_divide @ A @ ( plus_plus @ A @ A3 @ ( times_times @ A @ B3 @ Z3 ) ) @ Z3 ) ) ) ) ) ).
% add_divide_eq_if_simps(2)
thf(fact_1073_power__gt1__lemma,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,N2: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ A3 )
=> ( ord_less @ A @ ( one_one @ A ) @ ( times_times @ A @ A3 @ ( power_power @ A @ A3 @ N2 ) ) ) ) ) ).
% power_gt1_lemma
thf(fact_1074_power__less__power__Suc,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,N2: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ A3 )
=> ( ord_less @ A @ ( power_power @ A @ A3 @ N2 ) @ ( times_times @ A @ A3 @ ( power_power @ A @ A3 @ N2 ) ) ) ) ) ).
% power_less_power_Suc
thf(fact_1075_VEBT__internal_Onaive__member_Osimps_I1_J,axiom,
! [A3: $o,B3: $o,X: nat] :
( ( vEBT_V5719532721284313246member @ ( vEBT_Leaf @ A3 @ B3 ) @ X )
= ( ( ( X
= ( zero_zero @ nat ) )
=> A3 )
& ( ( X
!= ( zero_zero @ nat ) )
=> ( ( ( X
= ( one_one @ nat ) )
=> B3 )
& ( X
= ( one_one @ nat ) ) ) ) ) ) ).
% VEBT_internal.naive_member.simps(1)
thf(fact_1076_one__less__mult,axiom,
! [N2: nat,M2: nat] :
( ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N2 )
=> ( ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ M2 )
=> ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( times_times @ nat @ M2 @ N2 ) ) ) ) ).
% one_less_mult
thf(fact_1077_n__less__m__mult__n,axiom,
! [N2: nat,M2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ M2 )
=> ( ord_less @ nat @ N2 @ ( times_times @ nat @ M2 @ N2 ) ) ) ) ).
% n_less_m_mult_n
thf(fact_1078_n__less__n__mult__m,axiom,
! [N2: nat,M2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ M2 )
=> ( ord_less @ nat @ N2 @ ( times_times @ nat @ N2 @ M2 ) ) ) ) ).
% n_less_n_mult_m
thf(fact_1079_VEBT__internal_OminNull_Oelims_I3_J,axiom,
! [X: vEBT_VEBT] :
( ~ ( vEBT_VEBT_minNull @ X )
=> ( ! [Uv2: $o] :
( X
!= ( vEBT_Leaf @ $true @ Uv2 ) )
=> ( ! [Uu2: $o] :
( X
!= ( vEBT_Leaf @ Uu2 @ $true ) )
=> ~ ! [Uz2: product_prod @ nat @ nat,Va3: nat,Vb2: list @ vEBT_VEBT,Vc2: vEBT_VEBT] :
( X
!= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) ) ) ) ) ).
% VEBT_internal.minNull.elims(3)
thf(fact_1080_div__less__iff__less__mult,axiom,
! [Q3: nat,M2: nat,N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ Q3 )
=> ( ( ord_less @ nat @ ( divide_divide @ nat @ M2 @ Q3 ) @ N2 )
= ( ord_less @ nat @ M2 @ ( times_times @ nat @ N2 @ Q3 ) ) ) ) ).
% div_less_iff_less_mult
thf(fact_1081_VEBT__internal_OminNull_Oelims_I2_J,axiom,
! [X: vEBT_VEBT] :
( ( vEBT_VEBT_minNull @ X )
=> ( ( X
!= ( vEBT_Leaf @ $false @ $false ) )
=> ~ ! [Uw2: nat,Ux2: list @ vEBT_VEBT,Uy2: vEBT_VEBT] :
( X
!= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uw2 @ Ux2 @ Uy2 ) ) ) ) ).
% VEBT_internal.minNull.elims(2)
thf(fact_1082_realpow__pos__nth2,axiom,
! [A3: real,N2: nat] :
( ( ord_less @ real @ ( zero_zero @ real ) @ A3 )
=> ? [R3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R3 )
& ( ( power_power @ real @ R3 @ ( suc @ N2 ) )
= A3 ) ) ) ).
% realpow_pos_nth2
thf(fact_1083_field__le__mult__one__interval,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y: A] :
( ! [Z: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Z )
=> ( ( ord_less @ A @ Z @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ Z @ X ) @ Y ) ) )
=> ( ord_less_eq @ A @ X @ Y ) ) ) ).
% field_le_mult_one_interval
thf(fact_1084_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_1085_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_1086_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_1087_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_1088_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_1089_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_1090_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_1091_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_1092_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_1093_mult__imp__le__div__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [Y: A,Z3: A,X: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Y )
=> ( ( ord_less_eq @ A @ ( times_times @ A @ Z3 @ Y ) @ X )
=> ( ord_less_eq @ A @ Z3 @ ( divide_divide @ A @ X @ Y ) ) ) ) ) ).
% mult_imp_le_div_pos
thf(fact_1094_mult__imp__div__pos__le,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [Y: A,X: A,Z3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Y )
=> ( ( ord_less_eq @ A @ X @ ( times_times @ A @ Z3 @ Y ) )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ X @ Y ) @ Z3 ) ) ) ) ).
% mult_imp_div_pos_le
thf(fact_1095_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_1096_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_1097_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_1098_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_1099_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_1100_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_1101_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_1102_convex__bound__le,axiom,
! [A: $tType] :
( ( linord6961819062388156250ring_1 @ A )
=> ! [X: A,A3: A,Y: A,U: A,V2: A] :
( ( ord_less_eq @ A @ X @ A3 )
=> ( ( ord_less_eq @ A @ Y @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ U )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ V2 )
=> ( ( ( plus_plus @ A @ U @ V2 )
= ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ ( times_times @ A @ U @ X ) @ ( times_times @ A @ V2 @ Y ) ) @ A3 ) ) ) ) ) ) ) ).
% convex_bound_le
thf(fact_1103_divide__less__eq__numeral_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B3: A,C2: A,W2: num] :
( ( ord_less @ A @ ( divide_divide @ A @ B3 @ C2 ) @ ( numeral_numeral @ A @ W2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ B3 @ ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ C2 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ C2 ) @ B3 ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ W2 ) ) ) ) ) ) ) ) ).
% divide_less_eq_numeral(1)
thf(fact_1104_less__divide__eq__numeral_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [W2: num,B3: A,C2: A] :
( ( ord_less @ A @ ( numeral_numeral @ A @ W2 ) @ ( divide_divide @ A @ B3 @ C2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ 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 @ W2 ) @ C2 ) ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( numeral_numeral @ A @ W2 ) @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% less_divide_eq_numeral(1)
thf(fact_1105_power__Suc__less,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,N2: 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 @ N2 ) ) @ ( power_power @ A @ A3 @ N2 ) ) ) ) ) ).
% power_Suc_less
thf(fact_1106_mult__2,axiom,
! [A: $tType] :
( ( semiring_numeral @ A )
=> ! [Z3: A] :
( ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Z3 )
= ( plus_plus @ A @ Z3 @ Z3 ) ) ) ).
% mult_2
thf(fact_1107_mult__2__right,axiom,
! [A: $tType] :
( ( semiring_numeral @ A )
=> ! [Z3: A] :
( ( times_times @ A @ Z3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( plus_plus @ A @ Z3 @ Z3 ) ) ) ).
% mult_2_right
thf(fact_1108_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_1109_div__nat__eqI,axiom,
! [N2: nat,Q3: nat,M2: nat] :
( ( ord_less_eq @ nat @ ( times_times @ nat @ N2 @ Q3 ) @ M2 )
=> ( ( ord_less @ nat @ M2 @ ( times_times @ nat @ N2 @ ( suc @ Q3 ) ) )
=> ( ( divide_divide @ nat @ M2 @ N2 )
= Q3 ) ) ) ).
% div_nat_eqI
thf(fact_1110_less__eq__div__iff__mult__less__eq,axiom,
! [Q3: nat,M2: nat,N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ Q3 )
=> ( ( ord_less_eq @ nat @ M2 @ ( divide_divide @ nat @ N2 @ Q3 ) )
= ( ord_less_eq @ nat @ ( times_times @ nat @ M2 @ Q3 ) @ N2 ) ) ) ).
% less_eq_div_iff_mult_less_eq
thf(fact_1111_split__div,axiom,
! [P: nat > $o,M2: nat,N2: nat] :
( ( P @ ( divide_divide @ nat @ M2 @ N2 ) )
= ( ( ( N2
= ( zero_zero @ nat ) )
=> ( P @ ( zero_zero @ nat ) ) )
& ( ( N2
!= ( zero_zero @ nat ) )
=> ! [I3: nat,J2: nat] :
( ( ord_less @ nat @ J2 @ N2 )
=> ( ( M2
= ( plus_plus @ nat @ ( times_times @ nat @ N2 @ I3 ) @ J2 ) )
=> ( P @ I3 ) ) ) ) ) ) ).
% split_div
thf(fact_1112_dividend__less__div__times,axiom,
! [N2: nat,M2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ord_less @ nat @ M2 @ ( plus_plus @ nat @ N2 @ ( times_times @ nat @ ( divide_divide @ nat @ M2 @ N2 ) @ N2 ) ) ) ) ).
% dividend_less_div_times
thf(fact_1113_dividend__less__times__div,axiom,
! [N2: nat,M2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ord_less @ nat @ M2 @ ( plus_plus @ nat @ N2 @ ( times_times @ nat @ N2 @ ( divide_divide @ nat @ M2 @ N2 ) ) ) ) ) ).
% dividend_less_times_div
thf(fact_1114_VEBT__internal_OminNull_Oelims_I1_J,axiom,
! [X: vEBT_VEBT,Y: $o] :
( ( ( vEBT_VEBT_minNull @ X )
= Y )
=> ( ( ( X
= ( vEBT_Leaf @ $false @ $false ) )
=> ~ Y )
=> ( ( ? [Uv2: $o] :
( X
= ( vEBT_Leaf @ $true @ Uv2 ) )
=> Y )
=> ( ( ? [Uu2: $o] :
( X
= ( vEBT_Leaf @ Uu2 @ $true ) )
=> Y )
=> ( ( ? [Uw2: nat,Ux2: list @ vEBT_VEBT,Uy2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uw2 @ Ux2 @ Uy2 ) )
=> ~ Y )
=> ~ ( ? [Uz2: product_prod @ nat @ nat,Va3: nat,Vb2: list @ vEBT_VEBT,Vc2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) )
=> Y ) ) ) ) ) ) ).
% VEBT_internal.minNull.elims(1)
thf(fact_1115_convex__bound__lt,axiom,
! [A: $tType] :
( ( linord715952674999750819strict @ A )
=> ! [X: A,A3: A,Y: A,U: A,V2: A] :
( ( ord_less @ A @ X @ A3 )
=> ( ( ord_less @ A @ Y @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ U )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ V2 )
=> ( ( ( plus_plus @ A @ U @ V2 )
= ( one_one @ A ) )
=> ( ord_less @ A @ ( plus_plus @ A @ ( times_times @ A @ U @ X ) @ ( times_times @ A @ V2 @ Y ) ) @ A3 ) ) ) ) ) ) ) ).
% convex_bound_lt
thf(fact_1116_divide__le__eq__numeral_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B3: A,C2: A,W2: num] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ B3 @ C2 ) @ ( numeral_numeral @ A @ W2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ B3 @ ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ 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 @ W2 ) @ C2 ) @ B3 ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ W2 ) ) ) ) ) ) ) ) ).
% divide_le_eq_numeral(1)
thf(fact_1117_le__divide__eq__numeral_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [W2: num,B3: A,C2: A] :
( ( ord_less_eq @ A @ ( numeral_numeral @ A @ W2 ) @ ( divide_divide @ A @ B3 @ C2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ 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 @ W2 ) @ C2 ) ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( numeral_numeral @ A @ W2 ) @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% le_divide_eq_numeral(1)
thf(fact_1118_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_1119_split__div_H,axiom,
! [P: nat > $o,M2: nat,N2: nat] :
( ( P @ ( divide_divide @ nat @ M2 @ N2 ) )
= ( ( ( N2
= ( zero_zero @ nat ) )
& ( P @ ( zero_zero @ nat ) ) )
| ? [Q5: nat] :
( ( ord_less_eq @ nat @ ( times_times @ nat @ N2 @ Q5 ) @ M2 )
& ( ord_less @ nat @ M2 @ ( times_times @ nat @ N2 @ ( suc @ Q5 ) ) )
& ( P @ Q5 ) ) ) ) ).
% split_div'
thf(fact_1120_vebt__member_Ocases,axiom,
! [X: product_prod @ vEBT_VEBT @ nat] :
( ! [A6: $o,B5: $o,X5: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A6 @ B5 ) @ X5 ) )
=> ( ! [Uu2: nat,Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT,X5: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) @ X5 ) )
=> ( ! [V3: product_prod @ nat @ nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT,X5: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Uy2 @ Uz2 ) @ X5 ) )
=> ( ! [V3: product_prod @ nat @ nat,Vb2: list @ vEBT_VEBT,Vc2: vEBT_VEBT,X5: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb2 @ Vc2 ) @ X5 ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList3: list @ vEBT_VEBT,Summary2: vEBT_VEBT,X5: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList3 @ Summary2 ) @ X5 ) ) ) ) ) ) ).
% vebt_member.cases
thf(fact_1121_vebt__insert_Ocases,axiom,
! [X: product_prod @ vEBT_VEBT @ nat] :
( ! [A6: $o,B5: $o,X5: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A6 @ B5 ) @ X5 ) )
=> ( ! [Info2: option @ ( product_prod @ nat @ nat ),Ts2: list @ vEBT_VEBT,S3: vEBT_VEBT,X5: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Info2 @ ( zero_zero @ nat ) @ Ts2 @ S3 ) @ X5 ) )
=> ( ! [Info2: option @ ( product_prod @ nat @ nat ),Ts2: list @ vEBT_VEBT,S3: vEBT_VEBT,X5: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Info2 @ ( suc @ ( zero_zero @ nat ) ) @ Ts2 @ S3 ) @ X5 ) )
=> ( ! [V3: nat,TreeList3: list @ vEBT_VEBT,Summary2: vEBT_VEBT,X5: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ V3 ) ) @ TreeList3 @ Summary2 ) @ X5 ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList3: list @ vEBT_VEBT,Summary2: vEBT_VEBT,X5: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList3 @ Summary2 ) @ X5 ) ) ) ) ) ) ).
% vebt_insert.cases
thf(fact_1122_VEBT__internal_Omembermima_Ocases,axiom,
! [X: product_prod @ vEBT_VEBT @ nat] :
( ! [Uu2: $o,Uv2: $o,Uw2: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Uw2 ) )
=> ( ! [Ux2: list @ vEBT_VEBT,Uy2: vEBT_VEBT,Uz2: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( zero_zero @ nat ) @ Ux2 @ Uy2 ) @ Uz2 ) )
=> ( ! [Mi2: nat,Ma2: nat,Va3: list @ vEBT_VEBT,Vb2: vEBT_VEBT,X5: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ Va3 @ Vb2 ) @ X5 ) )
=> ( ! [Mi2: nat,Ma2: nat,V3: nat,TreeList3: list @ vEBT_VEBT,Vc2: vEBT_VEBT,X5: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList3 @ Vc2 ) @ X5 ) )
=> ~ ! [V3: nat,TreeList3: list @ vEBT_VEBT,Vd: vEBT_VEBT,X5: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V3 ) @ TreeList3 @ Vd ) @ X5 ) ) ) ) ) ) ).
% VEBT_internal.membermima.cases
thf(fact_1123_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_1124_zero__le__even__power_H,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,N2: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A3 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ).
% zero_le_even_power'
thf(fact_1125_nat__bit__induct,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ ( zero_zero @ nat ) )
=> ( ! [N3: nat] :
( ( P @ N3 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( P @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) ) ) )
=> ( ! [N3: nat] :
( ( P @ N3 )
=> ( P @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) ) ) )
=> ( P @ N2 ) ) ) ) ).
% nat_bit_induct
thf(fact_1126_arith__geo__mean,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [U: A,X: A,Y: A] :
( ( ( power_power @ A @ U @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( times_times @ A @ X @ Y ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
=> ( ord_less_eq @ A @ U @ ( divide_divide @ A @ ( plus_plus @ A @ X @ Y ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% arith_geo_mean
thf(fact_1127_triangle__def,axiom,
( nat_triangle
= ( ^ [N: nat] : ( divide_divide @ nat @ ( times_times @ nat @ N @ ( suc @ N ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% triangle_def
thf(fact_1128_realpow__pos__nth,axiom,
! [N2: nat,A3: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ A3 )
=> ? [R3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R3 )
& ( ( power_power @ real @ R3 @ N2 )
= A3 ) ) ) ) ).
% realpow_pos_nth
thf(fact_1129_realpow__pos__nth__unique,axiom,
! [N2: nat,A3: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ A3 )
=> ? [X5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X5 )
& ( ( power_power @ real @ X5 @ N2 )
= A3 )
& ! [Y4: real] :
( ( ( ord_less @ real @ ( zero_zero @ real ) @ Y4 )
& ( ( power_power @ real @ Y4 @ N2 )
= A3 ) )
=> ( Y4 = X5 ) ) ) ) ) ).
% realpow_pos_nth_unique
thf(fact_1130_odd__0__le__power__imp__0__le,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,N2: nat] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A3 @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% odd_0_le_power_imp_0_le
thf(fact_1131_odd__power__less__zero,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,N2: nat] :
( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( power_power @ A @ A3 @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) @ ( zero_zero @ A ) ) ) ) ).
% odd_power_less_zero
thf(fact_1132_nat__mult__le__cancel__disj,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ord_less_eq @ nat @ ( times_times @ nat @ K @ M2 ) @ ( times_times @ nat @ K @ N2 ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ord_less_eq @ nat @ M2 @ N2 ) ) ) ).
% nat_mult_le_cancel_disj
thf(fact_1133_nat__mult__div__cancel__disj,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ( K
= ( zero_zero @ nat ) )
=> ( ( divide_divide @ nat @ ( times_times @ nat @ K @ M2 ) @ ( times_times @ nat @ K @ N2 ) )
= ( zero_zero @ nat ) ) )
& ( ( K
!= ( zero_zero @ nat ) )
=> ( ( divide_divide @ nat @ ( times_times @ nat @ K @ M2 ) @ ( times_times @ nat @ K @ N2 ) )
= ( divide_divide @ nat @ M2 @ N2 ) ) ) ) ).
% nat_mult_div_cancel_disj
thf(fact_1134_nat__mult__less__cancel__disj,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ord_less @ nat @ ( times_times @ nat @ K @ M2 ) @ ( times_times @ nat @ K @ N2 ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
& ( ord_less @ nat @ M2 @ N2 ) ) ) ).
% nat_mult_less_cancel_disj
thf(fact_1135_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_1136_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_1137_nat__mult__div__cancel1,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ( divide_divide @ nat @ ( times_times @ nat @ K @ M2 ) @ ( times_times @ nat @ K @ N2 ) )
= ( divide_divide @ nat @ M2 @ N2 ) ) ) ).
% nat_mult_div_cancel1
thf(fact_1138_nat__mult__le__cancel1,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ( ord_less_eq @ nat @ ( times_times @ nat @ K @ M2 ) @ ( times_times @ nat @ K @ N2 ) )
= ( ord_less_eq @ nat @ M2 @ N2 ) ) ) ).
% nat_mult_le_cancel1
thf(fact_1139_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_1140_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_1141_nat__mult__eq__cancel__disj,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ( times_times @ nat @ K @ M2 )
= ( times_times @ nat @ K @ N2 ) )
= ( ( K
= ( zero_zero @ nat ) )
| ( M2 = N2 ) ) ) ).
% nat_mult_eq_cancel_disj
thf(fact_1142_left__add__mult__distrib,axiom,
! [I: nat,U: nat,J: nat,K: nat] :
( ( plus_plus @ nat @ ( times_times @ nat @ I @ U ) @ ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ K ) )
= ( plus_plus @ nat @ ( times_times @ nat @ ( plus_plus @ nat @ I @ J ) @ U ) @ K ) ) ).
% left_add_mult_distrib
thf(fact_1143_nat__mult__less__cancel1,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ( ord_less @ nat @ ( times_times @ nat @ K @ M2 ) @ ( times_times @ nat @ K @ N2 ) )
= ( ord_less @ nat @ M2 @ N2 ) ) ) ).
% nat_mult_less_cancel1
thf(fact_1144_nat__mult__eq__cancel1,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ( ( times_times @ nat @ K @ M2 )
= ( times_times @ nat @ K @ N2 ) )
= ( M2 = N2 ) ) ) ).
% nat_mult_eq_cancel1
thf(fact_1145_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_1146_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_1147_mult__le__cancel__iff1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z3: A,X: A,Y: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Z3 )
=> ( ( ord_less_eq @ A @ ( times_times @ A @ X @ Z3 ) @ ( times_times @ A @ Y @ Z3 ) )
= ( ord_less_eq @ A @ X @ Y ) ) ) ) ).
% mult_le_cancel_iff1
thf(fact_1148_mult__le__cancel__iff2,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z3: A,X: A,Y: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Z3 )
=> ( ( ord_less_eq @ A @ ( times_times @ A @ Z3 @ X ) @ ( times_times @ A @ Z3 @ Y ) )
= ( ord_less_eq @ A @ X @ Y ) ) ) ) ).
% mult_le_cancel_iff2
thf(fact_1149_divides__aux__eq,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [Q3: A,R2: A] :
( ( unique5940410009612947441es_aux @ A @ ( product_Pair @ A @ A @ Q3 @ R2 ) )
= ( R2
= ( zero_zero @ A ) ) ) ) ).
% divides_aux_eq
thf(fact_1150_low__def,axiom,
( vEBT_VEBT_low
= ( ^ [X4: nat,N: nat] : ( modulo_modulo @ nat @ X4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% low_def
thf(fact_1151_even__succ__div__exp,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A3: A,N2: nat] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ ( one_one @ A ) @ A3 ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) )
= ( divide_divide @ A @ A3 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ) ) ).
% even_succ_div_exp
thf(fact_1152_set__decode__Suc,axiom,
! [N2: nat,X: nat] :
( ( member @ nat @ ( suc @ N2 ) @ ( nat_set_decode @ X ) )
= ( member @ nat @ N2 @ ( nat_set_decode @ ( divide_divide @ nat @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% set_decode_Suc
thf(fact_1153_vebt__insert_Oelims,axiom,
! [X: vEBT_VEBT,Xa2: nat,Y: vEBT_VEBT] :
( ( ( vEBT_vebt_insert @ X @ Xa2 )
= Y )
=> ( ! [A6: $o,B5: $o] :
( ( X
= ( vEBT_Leaf @ A6 @ B5 ) )
=> ~ ( ( ( Xa2
= ( zero_zero @ nat ) )
=> ( Y
= ( vEBT_Leaf @ $true @ B5 ) ) )
& ( ( Xa2
!= ( zero_zero @ nat ) )
=> ( ( ( Xa2
= ( one_one @ nat ) )
=> ( Y
= ( vEBT_Leaf @ A6 @ $true ) ) )
& ( ( Xa2
!= ( one_one @ nat ) )
=> ( Y
= ( vEBT_Leaf @ A6 @ B5 ) ) ) ) ) ) )
=> ( ! [Info2: option @ ( product_prod @ nat @ nat ),Ts2: list @ vEBT_VEBT,S3: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Info2 @ ( zero_zero @ nat ) @ Ts2 @ S3 ) )
=> ( Y
!= ( vEBT_Node @ Info2 @ ( zero_zero @ nat ) @ Ts2 @ S3 ) ) )
=> ( ! [Info2: option @ ( product_prod @ nat @ nat ),Ts2: list @ vEBT_VEBT,S3: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Info2 @ ( suc @ ( zero_zero @ nat ) ) @ Ts2 @ S3 ) )
=> ( Y
!= ( vEBT_Node @ Info2 @ ( suc @ ( zero_zero @ nat ) ) @ Ts2 @ S3 ) ) )
=> ( ! [V3: nat,TreeList3: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ V3 ) ) @ TreeList3 @ Summary2 ) )
=> ( Y
!= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Xa2 @ Xa2 ) ) @ ( suc @ ( suc @ V3 ) ) @ TreeList3 @ Summary2 ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList3: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList3 @ Summary2 ) )
=> ( Y
!= ( if @ vEBT_VEBT
@ ( ( ord_less @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) )
& ~ ( ( Xa2 = Mi2 )
| ( Xa2 = Ma2 ) ) )
@ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ ( if @ nat @ ( ord_less @ nat @ Xa2 @ Mi2 ) @ Xa2 @ Mi2 ) @ ( ord_max @ nat @ ( if @ nat @ ( ord_less @ nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ Ma2 ) ) ) @ ( suc @ ( suc @ Va ) ) @ ( list_update @ vEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_insert @ ( nth @ vEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( ord_less @ nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( if @ vEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth @ vEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ Summary2 ) )
@ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList3 @ Summary2 ) ) ) ) ) ) ) ) ) ).
% vebt_insert.elims
thf(fact_1154_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_1155_mult__less__iff1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z3: A,X: A,Y: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Z3 )
=> ( ( ord_less @ A @ ( times_times @ A @ X @ Z3 ) @ ( times_times @ A @ Y @ Z3 ) )
= ( ord_less @ A @ X @ Y ) ) ) ) ).
% mult_less_iff1
thf(fact_1156_set__vebt_H__def,axiom,
( vEBT_VEBT_set_vebt
= ( ^ [T3: vEBT_VEBT] : ( collect @ nat @ ( vEBT_vebt_member @ T3 ) ) ) ) ).
% set_vebt'_def
thf(fact_1157_nat__dvd__1__iff__1,axiom,
! [M2: nat] :
( ( dvd_dvd @ nat @ M2 @ ( one_one @ nat ) )
= ( M2
= ( one_one @ nat ) ) ) ).
% nat_dvd_1_iff_1
thf(fact_1158_finite__Collect__disjI,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X4: A] :
( ( P @ X4 )
| ( Q @ X4 ) ) ) )
= ( ( finite_finite2 @ A @ ( collect @ A @ P ) )
& ( finite_finite2 @ A @ ( collect @ A @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_1159_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
@ ^ [X4: A] :
( ( P @ X4 )
& ( Q @ X4 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_1160_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_1161_dvd__0__right,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A3: A] : ( dvd_dvd @ A @ A3 @ ( zero_zero @ A ) ) ) ).
% dvd_0_right
thf(fact_1162_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_1163_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_1164_dvd__1__iff__1,axiom,
! [M2: nat] :
( ( dvd_dvd @ nat @ M2 @ ( suc @ ( zero_zero @ nat ) ) )
= ( M2
= ( suc @ ( zero_zero @ nat ) ) ) ) ).
% dvd_1_iff_1
thf(fact_1165_dvd__1__left,axiom,
! [K: nat] : ( dvd_dvd @ nat @ ( suc @ ( zero_zero @ nat ) ) @ K ) ).
% dvd_1_left
thf(fact_1166_mod__self,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ! [A3: A] :
( ( modulo_modulo @ A @ A3 @ A3 )
= ( zero_zero @ A ) ) ) ).
% mod_self
thf(fact_1167_mod__by__0,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ! [A3: A] :
( ( modulo_modulo @ A @ A3 @ ( zero_zero @ A ) )
= A3 ) ) ).
% mod_by_0
thf(fact_1168_mod__0,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ! [A3: A] :
( ( modulo_modulo @ A @ ( zero_zero @ A ) @ A3 )
= ( zero_zero @ A ) ) ) ).
% mod_0
thf(fact_1169_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_1170_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_1171_mod__add__self2,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A3: A,B3: A] :
( ( modulo_modulo @ A @ ( plus_plus @ A @ A3 @ B3 ) @ B3 )
= ( modulo_modulo @ A @ A3 @ B3 ) ) ) ).
% mod_add_self2
thf(fact_1172_mod__add__self1,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [B3: A,A3: A] :
( ( modulo_modulo @ A @ ( plus_plus @ A @ B3 @ A3 ) @ B3 )
= ( modulo_modulo @ A @ A3 @ B3 ) ) ) ).
% mod_add_self1
thf(fact_1173_nat__mult__dvd__cancel__disj,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( dvd_dvd @ nat @ ( times_times @ nat @ K @ M2 ) @ ( times_times @ nat @ K @ N2 ) )
= ( ( K
= ( zero_zero @ nat ) )
| ( dvd_dvd @ nat @ M2 @ N2 ) ) ) ).
% nat_mult_dvd_cancel_disj
thf(fact_1174_mod__less,axiom,
! [M2: nat,N2: nat] :
( ( ord_less @ nat @ M2 @ N2 )
=> ( ( modulo_modulo @ nat @ M2 @ N2 )
= M2 ) ) ).
% mod_less
thf(fact_1175_finite__Collect__subsets,axiom,
! [A: $tType,A4: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( finite_finite2 @ ( set @ A )
@ ( collect @ ( set @ A )
@ ^ [B6: set @ A] : ( ord_less_eq @ ( set @ A ) @ B6 @ A4 ) ) ) ) ).
% finite_Collect_subsets
thf(fact_1176_finite__Collect__less__nat,axiom,
! [K: nat] :
( finite_finite2 @ nat
@ ( collect @ nat
@ ^ [N: nat] : ( ord_less @ nat @ N @ K ) ) ) ).
% finite_Collect_less_nat
thf(fact_1177_finite__Collect__le__nat,axiom,
! [K: nat] :
( finite_finite2 @ nat
@ ( collect @ nat
@ ^ [N: nat] : ( ord_less_eq @ nat @ N @ K ) ) ) ).
% finite_Collect_le_nat
thf(fact_1178_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_1179_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_1180_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_1181_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_1182_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_1183_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_1184_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_1185_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_1186_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_1187_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_1188_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_1189_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_1190_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_1191_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_1192_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_1193_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_1194_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_1195_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_1196_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_1197_mod__mult__self4,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [B3: A,C2: A,A3: A] :
( ( modulo_modulo @ A @ ( plus_plus @ A @ ( times_times @ A @ B3 @ C2 ) @ A3 ) @ B3 )
= ( modulo_modulo @ A @ A3 @ B3 ) ) ) ).
% mod_mult_self4
thf(fact_1198_mod__mult__self3,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [C2: A,B3: A,A3: A] :
( ( modulo_modulo @ A @ ( plus_plus @ A @ ( times_times @ A @ C2 @ B3 ) @ A3 ) @ B3 )
= ( modulo_modulo @ A @ A3 @ B3 ) ) ) ).
% mod_mult_self3
thf(fact_1199_mod__mult__self2,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A3: A,B3: A,C2: A] :
( ( modulo_modulo @ A @ ( plus_plus @ A @ A3 @ ( times_times @ A @ B3 @ C2 ) ) @ B3 )
= ( modulo_modulo @ A @ A3 @ B3 ) ) ) ).
% mod_mult_self2
thf(fact_1200_mod__mult__self1,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A3: A,C2: A,B3: A] :
( ( modulo_modulo @ A @ ( plus_plus @ A @ A3 @ ( times_times @ A @ C2 @ B3 ) ) @ B3 )
= ( modulo_modulo @ A @ A3 @ B3 ) ) ) ).
% mod_mult_self1
thf(fact_1201_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_1202_mod__by__Suc__0,axiom,
! [M2: nat] :
( ( modulo_modulo @ nat @ M2 @ ( suc @ ( zero_zero @ nat ) ) )
= ( zero_zero @ nat ) ) ).
% mod_by_Suc_0
thf(fact_1203_set__decode__zero,axiom,
( ( nat_set_decode @ ( zero_zero @ nat ) )
= ( bot_bot @ ( set @ nat ) ) ) ).
% set_decode_zero
thf(fact_1204_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_1205_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_1206_even__Suc,axiom,
! [N2: nat] :
( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ N2 ) )
= ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) ).
% even_Suc
thf(fact_1207_even__Suc__Suc__iff,axiom,
! [N2: nat] :
( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ N2 ) ) )
= ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ).
% even_Suc_Suc_iff
thf(fact_1208_pow__divides__pow__iff,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [N2: nat,A3: A,B3: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( dvd_dvd @ A @ ( power_power @ A @ A3 @ N2 ) @ ( power_power @ A @ B3 @ N2 ) )
= ( dvd_dvd @ A @ A3 @ B3 ) ) ) ) ).
% pow_divides_pow_iff
thf(fact_1209_Suc__mod__mult__self4,axiom,
! [N2: nat,K: nat,M2: nat] :
( ( modulo_modulo @ nat @ ( suc @ ( plus_plus @ nat @ ( times_times @ nat @ N2 @ K ) @ M2 ) ) @ N2 )
= ( modulo_modulo @ nat @ ( suc @ M2 ) @ N2 ) ) ).
% Suc_mod_mult_self4
thf(fact_1210_Suc__mod__mult__self3,axiom,
! [K: nat,N2: nat,M2: nat] :
( ( modulo_modulo @ nat @ ( suc @ ( plus_plus @ nat @ ( times_times @ nat @ K @ N2 ) @ M2 ) ) @ N2 )
= ( modulo_modulo @ nat @ ( suc @ M2 ) @ N2 ) ) ).
% Suc_mod_mult_self3
thf(fact_1211_Suc__mod__mult__self2,axiom,
! [M2: nat,N2: nat,K: nat] :
( ( modulo_modulo @ nat @ ( suc @ ( plus_plus @ nat @ M2 @ ( times_times @ nat @ N2 @ K ) ) ) @ N2 )
= ( modulo_modulo @ nat @ ( suc @ M2 ) @ N2 ) ) ).
% Suc_mod_mult_self2
thf(fact_1212_Suc__mod__mult__self1,axiom,
! [M2: nat,K: nat,N2: nat] :
( ( modulo_modulo @ nat @ ( suc @ ( plus_plus @ nat @ M2 @ ( times_times @ nat @ K @ N2 ) ) ) @ N2 )
= ( modulo_modulo @ nat @ ( suc @ M2 ) @ N2 ) ) ).
% Suc_mod_mult_self1
thf(fact_1213_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_1214_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_1215_odd__Suc__div__two,axiom,
! [N2: nat] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
=> ( ( divide_divide @ nat @ ( suc @ N2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( suc @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% odd_Suc_div_two
thf(fact_1216_even__Suc__div__two,axiom,
! [N2: nat] :
( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
=> ( ( divide_divide @ nat @ ( suc @ N2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% even_Suc_div_two
thf(fact_1217_mod2__Suc__Suc,axiom,
! [M2: nat] :
( ( modulo_modulo @ nat @ ( suc @ ( suc @ M2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( modulo_modulo @ nat @ M2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% mod2_Suc_Suc
thf(fact_1218_Suc__times__numeral__mod__eq,axiom,
! [K: num,N2: nat] :
( ( ( numeral_numeral @ nat @ K )
!= ( one_one @ nat ) )
=> ( ( modulo_modulo @ nat @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ K ) @ N2 ) ) @ ( numeral_numeral @ nat @ K ) )
= ( one_one @ nat ) ) ) ).
% Suc_times_numeral_mod_eq
thf(fact_1219_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_1220_zero__le__power__eq__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,W2: num] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ W2 ) ) )
= ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ W2 ) )
| ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ W2 ) )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 ) ) ) ) ) ).
% zero_le_power_eq_numeral
thf(fact_1221_power__less__zero__eq__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,W2: num] :
( ( ord_less @ A @ ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ W2 ) ) @ ( zero_zero @ A ) )
= ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ W2 ) )
& ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ) ).
% power_less_zero_eq_numeral
thf(fact_1222_power__less__zero__eq,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,N2: nat] :
( ( ord_less @ A @ ( power_power @ A @ A3 @ N2 ) @ ( zero_zero @ A ) )
= ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
& ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ) ).
% power_less_zero_eq
thf(fact_1223_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_1224_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_1225_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_1226_not__mod2__eq__Suc__0__eq__0,axiom,
! [N2: nat] :
( ( ( modulo_modulo @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
!= ( suc @ ( zero_zero @ nat ) ) )
= ( ( modulo_modulo @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( zero_zero @ nat ) ) ) ).
% not_mod2_eq_Suc_0_eq_0
thf(fact_1227_add__self__mod__2,axiom,
! [M2: nat] :
( ( modulo_modulo @ nat @ ( plus_plus @ nat @ M2 @ M2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( zero_zero @ nat ) ) ).
% add_self_mod_2
thf(fact_1228_zero__less__power__eq__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,W2: num] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ W2 ) ) )
= ( ( ( numeral_numeral @ nat @ W2 )
= ( zero_zero @ nat ) )
| ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ W2 ) )
& ( A3
!= ( zero_zero @ A ) ) )
| ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ W2 ) )
& ( ord_less @ A @ ( zero_zero @ A ) @ A3 ) ) ) ) ) ).
% zero_less_power_eq_numeral
thf(fact_1229_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_1230_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_1231_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_1232_even__power,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A3: A,N2: nat] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( power_power @ A @ A3 @ N2 ) )
= ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
& ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ) ).
% even_power
thf(fact_1233_mod2__gr__0,axiom,
! [M2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( modulo_modulo @ nat @ M2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( ( modulo_modulo @ nat @ M2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( one_one @ nat ) ) ) ).
% mod2_gr_0
thf(fact_1234_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_1235_power__le__zero__eq__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,W2: num] :
( ( ord_less_eq @ A @ ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ W2 ) ) @ ( zero_zero @ A ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( numeral_numeral @ nat @ W2 ) )
& ( ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ W2 ) )
& ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) ) )
| ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ W2 ) )
& ( A3
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% power_le_zero_eq_numeral
thf(fact_1236_even__succ__mod__exp,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A3: A,N2: nat] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( modulo_modulo @ A @ ( plus_plus @ A @ ( one_one @ A ) @ A3 ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) )
= ( plus_plus @ A @ ( one_one @ A ) @ ( modulo_modulo @ A @ A3 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ) ) ) ).
% even_succ_mod_exp
thf(fact_1237_strict__subset__divisors__dvd,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ ( set @ A )
@ ( collect @ A
@ ^ [C4: A] : ( dvd_dvd @ A @ C4 @ A3 ) )
@ ( collect @ A
@ ^ [C4: A] : ( dvd_dvd @ A @ C4 @ B3 ) ) )
= ( ( dvd_dvd @ A @ A3 @ B3 )
& ~ ( dvd_dvd @ A @ B3 @ A3 ) ) ) ) ).
% strict_subset_divisors_dvd
thf(fact_1238_less__set__def,axiom,
! [A: $tType] :
( ( ord_less @ ( set @ A ) )
= ( ^ [A7: set @ A,B6: set @ A] :
( ord_less @ ( A > $o )
@ ^ [X4: A] : ( member @ A @ X4 @ A7 )
@ ^ [X4: A] : ( member @ A @ X4 @ B6 ) ) ) ) ).
% less_set_def
thf(fact_1239_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_1240_dvd__eq__mod__eq__0,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ( ( dvd_dvd @ A )
= ( ^ [A5: A,B4: A] :
( ( modulo_modulo @ A @ B4 @ A5 )
= ( zero_zero @ A ) ) ) ) ) ).
% dvd_eq_mod_eq_0
thf(fact_1241_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_1242_empty__def,axiom,
! [A: $tType] :
( ( bot_bot @ ( set @ A ) )
= ( collect @ A
@ ^ [X4: A] : $false ) ) ).
% empty_def
thf(fact_1243_dvd__refl,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: A] : ( dvd_dvd @ A @ A3 @ A3 ) ) ).
% dvd_refl
thf(fact_1244_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_1245_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_1246_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_1247_not__finite__existsD,axiom,
! [A: $tType,P: A > $o] :
( ~ ( finite_finite2 @ A @ ( collect @ A @ P ) )
=> ? [X_12: A] : ( P @ X_12 ) ) ).
% not_finite_existsD
thf(fact_1248_pigeonhole__infinite__rel,axiom,
! [B: $tType,A: $tType,A4: set @ A,B2: set @ B,R: A > B > $o] :
( ~ ( finite_finite2 @ A @ A4 )
=> ( ( finite_finite2 @ B @ B2 )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ A4 )
=> ? [Xa: B] :
( ( member @ B @ Xa @ B2 )
& ( R @ X5 @ Xa ) ) )
=> ? [X5: B] :
( ( member @ B @ X5 @ B2 )
& ~ ( finite_finite2 @ A
@ ( collect @ A
@ ^ [A5: A] :
( ( member @ A @ A5 @ A4 )
& ( R @ A5 @ X5 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_1249_pred__subset__eq,axiom,
! [A: $tType,R: set @ A,S2: set @ A] :
( ( ord_less_eq @ ( A > $o )
@ ^ [X4: A] : ( member @ A @ X4 @ R )
@ ^ [X4: A] : ( member @ A @ X4 @ S2 ) )
= ( ord_less_eq @ ( set @ A ) @ R @ S2 ) ) ).
% pred_subset_eq
thf(fact_1250_less__eq__set__def,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A7: set @ A,B6: set @ A] :
( ord_less_eq @ ( A > $o )
@ ^ [X4: A] : ( member @ A @ X4 @ A7 )
@ ^ [X4: A] : ( member @ A @ X4 @ B6 ) ) ) ) ).
% less_eq_set_def
thf(fact_1251_Collect__subset,axiom,
! [A: $tType,A4: set @ A,P: A > $o] :
( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [X4: A] :
( ( member @ A @ X4 @ A4 )
& ( P @ X4 ) ) )
@ A4 ) ).
% Collect_subset
thf(fact_1252_subset__divisors__dvd,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [C4: A] : ( dvd_dvd @ A @ C4 @ A3 ) )
@ ( collect @ A
@ ^ [C4: A] : ( dvd_dvd @ A @ C4 @ B3 ) ) )
= ( dvd_dvd @ A @ A3 @ B3 ) ) ) ).
% subset_divisors_dvd
thf(fact_1253_Collect__restrict,axiom,
! [A: $tType,X8: set @ A,P: A > $o] :
( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [X4: A] :
( ( member @ A @ X4 @ X8 )
& ( P @ X4 ) ) )
@ X8 ) ).
% Collect_restrict
thf(fact_1254_prop__restrict,axiom,
! [A: $tType,X: A,Z5: set @ A,X8: set @ A,P: A > $o] :
( ( member @ A @ X @ Z5 )
=> ( ( ord_less_eq @ ( set @ A ) @ Z5
@ ( collect @ A
@ ^ [X4: A] :
( ( member @ A @ X4 @ X8 )
& ( P @ X4 ) ) ) )
=> ( P @ X ) ) ) ).
% prop_restrict
thf(fact_1255_lambda__zero,axiom,
! [A: $tType] :
( ( mult_zero @ A )
=> ( ( ^ [H: A] : ( zero_zero @ A ) )
= ( times_times @ A @ ( zero_zero @ A ) ) ) ) ).
% lambda_zero
thf(fact_1256_lambda__one,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ( ( ^ [X4: A] : X4 )
= ( times_times @ A @ ( one_one @ A ) ) ) ) ).
% lambda_one
thf(fact_1257_max__def__raw,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( ord_max @ A )
= ( ^ [A5: A,B4: A] : ( if @ A @ ( ord_less_eq @ A @ A5 @ B4 ) @ B4 @ A5 ) ) ) ) ).
% max_def_raw
thf(fact_1258_finite__divisors__nat,axiom,
! [M2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ M2 )
=> ( finite_finite2 @ nat
@ ( collect @ nat
@ ^ [D4: nat] : ( dvd_dvd @ nat @ D4 @ M2 ) ) ) ) ).
% finite_divisors_nat
thf(fact_1259_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_1260_bot__empty__eq2,axiom,
! [B: $tType,A: $tType] :
( ( bot_bot @ ( A > B > $o ) )
= ( ^ [X4: A,Y6: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y6 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ) ) ).
% bot_empty_eq2
thf(fact_1261_finite__M__bounded__by__nat,axiom,
! [P: nat > $o,I: nat] :
( finite_finite2 @ nat
@ ( collect @ nat
@ ^ [K2: nat] :
( ( P @ K2 )
& ( ord_less @ nat @ K2 @ I ) ) ) ) ).
% finite_M_bounded_by_nat
thf(fact_1262_finite__less__ub,axiom,
! [F2: nat > nat,U: nat] :
( ! [N3: nat] : ( ord_less_eq @ nat @ N3 @ ( F2 @ N3 ) )
=> ( finite_finite2 @ nat
@ ( collect @ nat
@ ^ [N: nat] : ( ord_less_eq @ nat @ ( F2 @ N ) @ U ) ) ) ) ).
% finite_less_ub
thf(fact_1263_mod__greater__zero__iff__not__dvd,axiom,
! [M2: nat,N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( modulo_modulo @ nat @ M2 @ N2 ) )
= ( ~ ( dvd_dvd @ nat @ N2 @ M2 ) ) ) ).
% mod_greater_zero_iff_not_dvd
thf(fact_1264_mod__add__right__eq,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A3: A,B3: A,C2: A] :
( ( modulo_modulo @ A @ ( plus_plus @ A @ A3 @ ( modulo_modulo @ A @ B3 @ C2 ) ) @ C2 )
= ( modulo_modulo @ A @ ( plus_plus @ A @ A3 @ B3 ) @ C2 ) ) ) ).
% mod_add_right_eq
thf(fact_1265_mod__add__left__eq,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A3: A,C2: A,B3: A] :
( ( modulo_modulo @ A @ ( plus_plus @ A @ ( modulo_modulo @ A @ A3 @ C2 ) @ B3 ) @ C2 )
= ( modulo_modulo @ A @ ( plus_plus @ A @ A3 @ B3 ) @ C2 ) ) ) ).
% mod_add_left_eq
thf(fact_1266_mod__add__cong,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A3: A,C2: A,A9: A,B3: A,B8: A] :
( ( ( modulo_modulo @ A @ A3 @ C2 )
= ( modulo_modulo @ A @ A9 @ C2 ) )
=> ( ( ( modulo_modulo @ A @ B3 @ C2 )
= ( modulo_modulo @ A @ B8 @ C2 ) )
=> ( ( modulo_modulo @ A @ ( plus_plus @ A @ A3 @ B3 ) @ C2 )
= ( modulo_modulo @ A @ ( plus_plus @ A @ A9 @ B8 ) @ C2 ) ) ) ) ) ).
% mod_add_cong
thf(fact_1267_mod__add__eq,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A3: A,C2: A,B3: A] :
( ( modulo_modulo @ A @ ( plus_plus @ A @ ( modulo_modulo @ A @ A3 @ C2 ) @ ( modulo_modulo @ A @ B3 @ C2 ) ) @ C2 )
= ( modulo_modulo @ A @ ( plus_plus @ A @ A3 @ B3 ) @ C2 ) ) ) ).
% mod_add_eq
thf(fact_1268_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_1269_dvd__field__iff,axiom,
! [A: $tType] :
( ( field @ A )
=> ( ( dvd_dvd @ A )
= ( ^ [A5: A,B4: A] :
( ( A5
= ( zero_zero @ A ) )
=> ( B4
= ( zero_zero @ A ) ) ) ) ) ) ).
% dvd_field_iff
thf(fact_1270_mod__Suc__Suc__eq,axiom,
! [M2: nat,N2: nat] :
( ( modulo_modulo @ nat @ ( suc @ ( suc @ ( modulo_modulo @ nat @ M2 @ N2 ) ) ) @ N2 )
= ( modulo_modulo @ nat @ ( suc @ ( suc @ M2 ) ) @ N2 ) ) ).
% mod_Suc_Suc_eq
thf(fact_1271_mod__Suc__eq,axiom,
! [M2: nat,N2: nat] :
( ( modulo_modulo @ nat @ ( suc @ ( modulo_modulo @ nat @ M2 @ N2 ) ) @ N2 )
= ( modulo_modulo @ nat @ ( suc @ M2 ) @ N2 ) ) ).
% mod_Suc_eq
thf(fact_1272_dvdE,axiom,
! [A: $tType] :
( ( dvd @ A )
=> ! [B3: A,A3: A] :
( ( dvd_dvd @ A @ B3 @ A3 )
=> ~ ! [K3: A] :
( A3
!= ( times_times @ A @ B3 @ K3 ) ) ) ) ).
% dvdE
thf(fact_1273_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_1274_dvd__def,axiom,
! [A: $tType] :
( ( dvd @ A )
=> ( ( dvd_dvd @ A )
= ( ^ [B4: A,A5: A] :
? [K2: A] :
( A5
= ( times_times @ A @ B4 @ K2 ) ) ) ) ) ).
% dvd_def
thf(fact_1275_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_1276_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_1277_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_1278_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_1279_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_1280_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_1281_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_1282_one__dvd,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: A] : ( dvd_dvd @ A @ ( one_one @ A ) @ A3 ) ) ).
% one_dvd
thf(fact_1283_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_1284_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_1285_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_1286_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_1287_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_1288_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_1289_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_1290_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_1291_mod__less__eq__dividend,axiom,
! [M2: nat,N2: nat] : ( ord_less_eq @ nat @ ( modulo_modulo @ nat @ M2 @ N2 ) @ M2 ) ).
% mod_less_eq_dividend
thf(fact_1292_gcd__nat_Oextremum__uniqueI,axiom,
! [A3: nat] :
( ( dvd_dvd @ nat @ ( zero_zero @ nat ) @ A3 )
=> ( A3
= ( zero_zero @ nat ) ) ) ).
% gcd_nat.extremum_uniqueI
thf(fact_1293_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_1294_gcd__nat_Oextremum__unique,axiom,
! [A3: nat] :
( ( dvd_dvd @ nat @ ( zero_zero @ nat ) @ A3 )
= ( A3
= ( zero_zero @ nat ) ) ) ).
% gcd_nat.extremum_unique
thf(fact_1295_gcd__nat_Oextremum__strict,axiom,
! [A3: nat] :
~ ( ( dvd_dvd @ nat @ ( zero_zero @ nat ) @ A3 )
& ( ( zero_zero @ nat )
!= A3 ) ) ).
% gcd_nat.extremum_strict
thf(fact_1296_gcd__nat_Oextremum,axiom,
! [A3: nat] : ( dvd_dvd @ nat @ A3 @ ( zero_zero @ nat ) ) ).
% gcd_nat.extremum
thf(fact_1297_numeral__code_I2_J,axiom,
! [A: $tType] :
( ( numeral @ A )
=> ! [N2: num] :
( ( numeral_numeral @ A @ ( bit0 @ N2 ) )
= ( plus_plus @ A @ ( numeral_numeral @ A @ N2 ) @ ( numeral_numeral @ A @ N2 ) ) ) ) ).
% numeral_code(2)
thf(fact_1298_set__vebt__def,axiom,
( vEBT_set_vebt
= ( ^ [T3: vEBT_VEBT] : ( collect @ nat @ ( vEBT_V8194947554948674370ptions @ T3 ) ) ) ) ).
% set_vebt_def
thf(fact_1299_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_1300_subset__decode__imp__le,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq @ ( set @ nat ) @ ( nat_set_decode @ M2 ) @ ( nat_set_decode @ N2 ) )
=> ( ord_less_eq @ nat @ M2 @ N2 ) ) ).
% subset_decode_imp_le
thf(fact_1301_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_1302_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_1303_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_1304_mod__eqE,axiom,
! [A: $tType] :
( ( euclid8851590272496341667cancel @ A )
=> ! [A3: A,C2: A,B3: A] :
( ( ( modulo_modulo @ A @ A3 @ C2 )
= ( modulo_modulo @ A @ B3 @ C2 ) )
=> ~ ! [D5: A] :
( B3
!= ( plus_plus @ A @ A3 @ ( times_times @ A @ C2 @ D5 ) ) ) ) ) ).
% mod_eqE
thf(fact_1305_div__add1__eq,axiom,
! [A: $tType] :
( ( euclid3128863361964157862miring @ A )
=> ! [A3: A,B3: A,C2: A] :
( ( divide_divide @ A @ ( plus_plus @ A @ A3 @ B3 ) @ C2 )
= ( plus_plus @ A @ ( plus_plus @ A @ ( divide_divide @ A @ A3 @ C2 ) @ ( divide_divide @ A @ B3 @ C2 ) ) @ ( divide_divide @ A @ ( plus_plus @ A @ ( modulo_modulo @ A @ A3 @ C2 ) @ ( modulo_modulo @ A @ B3 @ C2 ) ) @ C2 ) ) ) ) ).
% div_add1_eq
thf(fact_1306_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_1307_minf_I10_J,axiom,
! [B: $tType] :
( ( ( plus @ B )
& ( linorder @ B )
& ( dvd @ B ) )
=> ! [D2: B,S: B] :
? [Z: B] :
! [X2: B] :
( ( ord_less @ B @ X2 @ Z )
=> ( ( ~ ( dvd_dvd @ B @ D2 @ ( plus_plus @ B @ X2 @ S ) ) )
= ( ~ ( dvd_dvd @ B @ D2 @ ( plus_plus @ B @ X2 @ S ) ) ) ) ) ) ).
% minf(10)
thf(fact_1308_minf_I9_J,axiom,
! [B: $tType] :
( ( ( plus @ B )
& ( linorder @ B )
& ( dvd @ B ) )
=> ! [D2: B,S: B] :
? [Z: B] :
! [X2: B] :
( ( ord_less @ B @ X2 @ Z )
=> ( ( dvd_dvd @ B @ D2 @ ( plus_plus @ B @ X2 @ S ) )
= ( dvd_dvd @ B @ D2 @ ( plus_plus @ B @ X2 @ S ) ) ) ) ) ).
% minf(9)
thf(fact_1309_pinf_I10_J,axiom,
! [B: $tType] :
( ( ( plus @ B )
& ( linorder @ B )
& ( dvd @ B ) )
=> ! [D2: B,S: B] :
? [Z: B] :
! [X2: B] :
( ( ord_less @ B @ Z @ X2 )
=> ( ( ~ ( dvd_dvd @ B @ D2 @ ( plus_plus @ B @ X2 @ S ) ) )
= ( ~ ( dvd_dvd @ B @ D2 @ ( plus_plus @ B @ X2 @ S ) ) ) ) ) ) ).
% pinf(10)
thf(fact_1310_pinf_I9_J,axiom,
! [B: $tType] :
( ( ( plus @ B )
& ( linorder @ B )
& ( dvd @ B ) )
=> ! [D2: B,S: B] :
? [Z: B] :
! [X2: B] :
( ( ord_less @ B @ Z @ X2 )
=> ( ( dvd_dvd @ B @ D2 @ ( plus_plus @ B @ X2 @ S ) )
= ( dvd_dvd @ B @ D2 @ ( plus_plus @ B @ X2 @ S ) ) ) ) ) ).
% pinf(9)
thf(fact_1311_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_1312_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_1313_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_1314_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_1315_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_1316_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_1317_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_1318_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_1319_mod__Suc,axiom,
! [M2: nat,N2: nat] :
( ( ( ( suc @ ( modulo_modulo @ nat @ M2 @ N2 ) )
= N2 )
=> ( ( modulo_modulo @ nat @ ( suc @ M2 ) @ N2 )
= ( zero_zero @ nat ) ) )
& ( ( ( suc @ ( modulo_modulo @ nat @ M2 @ N2 ) )
!= N2 )
=> ( ( modulo_modulo @ nat @ ( suc @ M2 ) @ N2 )
= ( suc @ ( modulo_modulo @ nat @ M2 @ N2 ) ) ) ) ) ).
% mod_Suc
thf(fact_1320_finite__set__decode,axiom,
! [N2: nat] : ( finite_finite2 @ nat @ ( nat_set_decode @ N2 ) ) ).
% finite_set_decode
thf(fact_1321_mod__induct,axiom,
! [P: nat > $o,N2: nat,P5: nat,M2: nat] :
( ( P @ N2 )
=> ( ( ord_less @ nat @ N2 @ P5 )
=> ( ( ord_less @ nat @ M2 @ P5 )
=> ( ! [N3: nat] :
( ( ord_less @ nat @ N3 @ P5 )
=> ( ( P @ N3 )
=> ( P @ ( modulo_modulo @ nat @ ( suc @ N3 ) @ P5 ) ) ) )
=> ( P @ M2 ) ) ) ) ) ).
% mod_induct
thf(fact_1322_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_1323_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_1324_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_1325_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_1326_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_1327_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_1328_gcd__nat__induct,axiom,
! [P: nat > nat > $o,M2: nat,N2: nat] :
( ! [M3: nat] : ( P @ M3 @ ( zero_zero @ nat ) )
=> ( ! [M3: nat,N3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( ( P @ N3 @ ( modulo_modulo @ nat @ M3 @ N3 ) )
=> ( P @ M3 @ N3 ) ) )
=> ( P @ M2 @ N2 ) ) ) ).
% gcd_nat_induct
thf(fact_1329_mod__less__divisor,axiom,
! [N2: nat,M2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ord_less @ nat @ ( modulo_modulo @ nat @ M2 @ N2 ) @ N2 ) ) ).
% mod_less_divisor
thf(fact_1330_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_1331_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_1332_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_1333_div__plus__div__distrib__dvd__right,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [C2: A,B3: A,A3: A] :
( ( 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_plus_div_distrib_dvd_right
thf(fact_1334_div__plus__div__distrib__dvd__left,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [C2: A,A3: A,B3: A] :
( ( dvd_dvd @ A @ C2 @ A3 )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ A3 @ B3 ) @ C2 )
= ( plus_plus @ A @ ( divide_divide @ A @ A3 @ C2 ) @ ( divide_divide @ A @ B3 @ C2 ) ) ) ) ) ).
% div_plus_div_distrib_dvd_left
thf(fact_1335_mod__Suc__le__divisor,axiom,
! [M2: nat,N2: nat] : ( ord_less_eq @ nat @ ( modulo_modulo @ nat @ M2 @ ( suc @ N2 ) ) @ N2 ) ).
% mod_Suc_le_divisor
thf(fact_1336_mod__eq__0D,axiom,
! [M2: nat,D2: nat] :
( ( ( modulo_modulo @ nat @ M2 @ D2 )
= ( zero_zero @ nat ) )
=> ? [Q2: nat] :
( M2
= ( times_times @ nat @ D2 @ Q2 ) ) ) ).
% mod_eq_0D
thf(fact_1337_dvd__power__le,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [X: A,Y: A,N2: nat,M2: nat] :
( ( dvd_dvd @ A @ X @ Y )
=> ( ( ord_less_eq @ nat @ N2 @ M2 )
=> ( dvd_dvd @ A @ ( power_power @ A @ X @ N2 ) @ ( power_power @ A @ Y @ M2 ) ) ) ) ) ).
% dvd_power_le
thf(fact_1338_power__le__dvd,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A3: A,N2: nat,B3: A,M2: nat] :
( ( dvd_dvd @ A @ ( power_power @ A @ A3 @ N2 ) @ B3 )
=> ( ( ord_less_eq @ nat @ M2 @ N2 )
=> ( dvd_dvd @ A @ ( power_power @ A @ A3 @ M2 ) @ B3 ) ) ) ) ).
% power_le_dvd
thf(fact_1339_le__imp__power__dvd,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [M2: nat,N2: nat,A3: A] :
( ( ord_less_eq @ nat @ M2 @ N2 )
=> ( dvd_dvd @ A @ ( power_power @ A @ A3 @ M2 ) @ ( power_power @ A @ A3 @ N2 ) ) ) ) ).
% le_imp_power_dvd
thf(fact_1340_nat__mod__eq__iff,axiom,
! [X: nat,N2: nat,Y: nat] :
( ( ( modulo_modulo @ nat @ X @ N2 )
= ( modulo_modulo @ nat @ Y @ N2 ) )
= ( ? [Q1: nat,Q22: nat] :
( ( plus_plus @ nat @ X @ ( times_times @ nat @ N2 @ Q1 ) )
= ( plus_plus @ nat @ Y @ ( times_times @ nat @ N2 @ Q22 ) ) ) ) ) ).
% nat_mod_eq_iff
thf(fact_1341_dvd__pos__nat,axiom,
! [N2: nat,M2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( dvd_dvd @ nat @ M2 @ N2 )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ M2 ) ) ) ).
% dvd_pos_nat
thf(fact_1342_nat__dvd__not__less,axiom,
! [M2: nat,N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ M2 )
=> ( ( ord_less @ nat @ M2 @ N2 )
=> ~ ( dvd_dvd @ nat @ N2 @ M2 ) ) ) ).
% nat_dvd_not_less
thf(fact_1343_bezout__add__nat,axiom,
! [A3: nat,B3: nat] :
? [D5: nat,X5: nat,Y3: nat] :
( ( dvd_dvd @ nat @ D5 @ A3 )
& ( dvd_dvd @ nat @ D5 @ B3 )
& ( ( ( times_times @ nat @ A3 @ X5 )
= ( plus_plus @ nat @ ( times_times @ nat @ B3 @ Y3 ) @ D5 ) )
| ( ( times_times @ nat @ B3 @ X5 )
= ( plus_plus @ nat @ ( times_times @ nat @ A3 @ Y3 ) @ D5 ) ) ) ) ).
% bezout_add_nat
thf(fact_1344_bezout__lemma__nat,axiom,
! [D2: nat,A3: nat,B3: nat,X: nat,Y: nat] :
( ( dvd_dvd @ nat @ D2 @ A3 )
=> ( ( dvd_dvd @ nat @ D2 @ B3 )
=> ( ( ( ( times_times @ nat @ A3 @ X )
= ( plus_plus @ nat @ ( times_times @ nat @ B3 @ Y ) @ D2 ) )
| ( ( times_times @ nat @ B3 @ X )
= ( plus_plus @ nat @ ( times_times @ nat @ A3 @ Y ) @ D2 ) ) )
=> ? [X5: nat,Y3: nat] :
( ( dvd_dvd @ nat @ D2 @ A3 )
& ( dvd_dvd @ nat @ D2 @ ( plus_plus @ nat @ A3 @ B3 ) )
& ( ( ( times_times @ nat @ A3 @ X5 )
= ( plus_plus @ nat @ ( times_times @ nat @ ( plus_plus @ nat @ A3 @ B3 ) @ Y3 ) @ D2 ) )
| ( ( times_times @ nat @ ( plus_plus @ nat @ A3 @ B3 ) @ X5 )
= ( plus_plus @ nat @ ( times_times @ nat @ A3 @ Y3 ) @ D2 ) ) ) ) ) ) ) ).
% bezout_lemma_nat
thf(fact_1345_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_1346_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_1347_finite__lists__length__eq,axiom,
! [A: $tType,A4: set @ A,N2: nat] :
( ( finite_finite2 @ A @ A4 )
=> ( finite_finite2 @ ( list @ A )
@ ( collect @ ( list @ A )
@ ^ [Xs3: list @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs3 ) @ A4 )
& ( ( size_size @ ( list @ A ) @ Xs3 )
= N2 ) ) ) ) ) ).
% finite_lists_length_eq
thf(fact_1348_vebt__buildup_Osimps_I3_J,axiom,
! [Va2: nat] :
( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ Va2 ) ) )
=> ( ( vEBT_vebt_buildup @ ( suc @ ( suc @ Va2 ) ) )
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ Va2 ) ) @ ( replicate @ vEBT_VEBT @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) )
& ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ Va2 ) ) )
=> ( ( vEBT_vebt_buildup @ ( suc @ ( suc @ Va2 ) ) )
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ Va2 ) ) @ ( replicate @ vEBT_VEBT @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( suc @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ).
% vebt_buildup.simps(3)
thf(fact_1349_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_1350_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_1351_cong__exp__iff__simps_I2_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [N2: num,Q3: num] :
( ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit0 @ N2 ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q3 ) ) )
= ( zero_zero @ A ) )
= ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ N2 ) @ ( numeral_numeral @ A @ Q3 ) )
= ( zero_zero @ A ) ) ) ) ).
% cong_exp_iff_simps(2)
thf(fact_1352_cong__exp__iff__simps_I1_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [N2: num] :
( ( modulo_modulo @ A @ ( numeral_numeral @ A @ N2 ) @ ( numeral_numeral @ A @ one2 ) )
= ( zero_zero @ A ) ) ) ).
% cong_exp_iff_simps(1)
thf(fact_1353_finite__lists__length__le,axiom,
! [A: $tType,A4: set @ A,N2: nat] :
( ( finite_finite2 @ A @ A4 )
=> ( finite_finite2 @ ( list @ A )
@ ( collect @ ( list @ A )
@ ^ [Xs3: list @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs3 ) @ A4 )
& ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs3 ) @ N2 ) ) ) ) ) ).
% finite_lists_length_le
thf(fact_1354_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_1355_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_1356_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_1357_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_1358_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_1359_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_1360_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_1361_div__mult1__eq,axiom,
! [A: $tType] :
( ( euclid3128863361964157862miring @ A )
=> ! [A3: A,B3: A,C2: A] :
( ( divide_divide @ A @ ( times_times @ A @ A3 @ B3 ) @ C2 )
= ( plus_plus @ A @ ( times_times @ A @ A3 @ ( divide_divide @ A @ B3 @ C2 ) ) @ ( divide_divide @ A @ ( times_times @ A @ A3 @ ( modulo_modulo @ A @ B3 @ C2 ) ) @ C2 ) ) ) ) ).
% div_mult1_eq
thf(fact_1362_unit__dvdE,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B3: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ~ ( ( A3
!= ( zero_zero @ A ) )
=> ! [C3: A] :
( B3
!= ( times_times @ A @ A3 @ C3 ) ) ) ) ) ).
% unit_dvdE
thf(fact_1363_unity__coeff__ex,axiom,
! [A: $tType] :
( ( ( dvd @ A )
& ( semiring_0 @ A ) )
=> ! [P: A > $o,L: A] :
( ( ? [X4: A] : ( P @ ( times_times @ A @ L @ X4 ) ) )
= ( ? [X4: A] :
( ( dvd_dvd @ A @ L @ ( plus_plus @ A @ X4 @ ( zero_zero @ A ) ) )
& ( P @ X4 ) ) ) ) ) ).
% unity_coeff_ex
thf(fact_1364_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_1365_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_1366_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_1367_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_1368_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_1369_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_1370_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_1371_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_1372_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_1373_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_1374_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_1375_mod__le__divisor,axiom,
! [N2: nat,M2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ord_less_eq @ nat @ ( modulo_modulo @ nat @ M2 @ N2 ) @ N2 ) ) ).
% mod_le_divisor
thf(fact_1376_is__unit__power__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,N2: nat] :
( ( dvd_dvd @ A @ ( power_power @ A @ A3 @ N2 ) @ ( one_one @ A ) )
= ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
| ( N2
= ( zero_zero @ nat ) ) ) ) ) ).
% is_unit_power_iff
thf(fact_1377_div__less__mono,axiom,
! [A4: nat,B2: nat,N2: nat] :
( ( ord_less @ nat @ A4 @ B2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( ( modulo_modulo @ nat @ A4 @ N2 )
= ( zero_zero @ nat ) )
=> ( ( ( modulo_modulo @ nat @ B2 @ N2 )
= ( zero_zero @ nat ) )
=> ( ord_less @ nat @ ( divide_divide @ nat @ A4 @ N2 ) @ ( divide_divide @ nat @ B2 @ N2 ) ) ) ) ) ) ).
% div_less_mono
thf(fact_1378_mod__eq__nat1E,axiom,
! [M2: nat,Q3: nat,N2: nat] :
( ( ( modulo_modulo @ nat @ M2 @ Q3 )
= ( modulo_modulo @ nat @ N2 @ Q3 ) )
=> ( ( ord_less_eq @ nat @ N2 @ M2 )
=> ~ ! [S3: nat] :
( M2
!= ( plus_plus @ nat @ N2 @ ( times_times @ nat @ Q3 @ S3 ) ) ) ) ) ).
% mod_eq_nat1E
thf(fact_1379_mod__eq__nat2E,axiom,
! [M2: nat,Q3: nat,N2: nat] :
( ( ( modulo_modulo @ nat @ M2 @ Q3 )
= ( modulo_modulo @ nat @ N2 @ Q3 ) )
=> ( ( ord_less_eq @ nat @ M2 @ N2 )
=> ~ ! [S3: nat] :
( N2
!= ( plus_plus @ nat @ M2 @ ( times_times @ nat @ Q3 @ S3 ) ) ) ) ) ).
% mod_eq_nat2E
thf(fact_1380_nat__mod__eq__lemma,axiom,
! [X: nat,N2: nat,Y: nat] :
( ( ( modulo_modulo @ nat @ X @ N2 )
= ( modulo_modulo @ nat @ Y @ N2 ) )
=> ( ( ord_less_eq @ nat @ Y @ X )
=> ? [Q2: nat] :
( X
= ( plus_plus @ nat @ Y @ ( times_times @ nat @ N2 @ Q2 ) ) ) ) ) ).
% nat_mod_eq_lemma
thf(fact_1381_dvd__imp__le,axiom,
! [K: nat,N2: nat] :
( ( dvd_dvd @ nat @ K @ N2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ord_less_eq @ nat @ K @ N2 ) ) ) ).
% dvd_imp_le
thf(fact_1382_div__mod__decomp,axiom,
! [A4: nat,N2: nat] :
( A4
= ( plus_plus @ nat @ ( times_times @ nat @ ( divide_divide @ nat @ A4 @ N2 ) @ N2 ) @ ( modulo_modulo @ nat @ A4 @ N2 ) ) ) ).
% div_mod_decomp
thf(fact_1383_mod__mult2__eq,axiom,
! [M2: nat,N2: nat,Q3: nat] :
( ( modulo_modulo @ nat @ M2 @ ( times_times @ nat @ N2 @ Q3 ) )
= ( plus_plus @ nat @ ( times_times @ nat @ N2 @ ( modulo_modulo @ nat @ ( divide_divide @ nat @ M2 @ N2 ) @ Q3 ) ) @ ( modulo_modulo @ nat @ M2 @ N2 ) ) ) ).
% mod_mult2_eq
thf(fact_1384_dvd__mult__cancel,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( dvd_dvd @ nat @ ( times_times @ nat @ K @ M2 ) @ ( times_times @ nat @ K @ N2 ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( dvd_dvd @ nat @ M2 @ N2 ) ) ) ).
% dvd_mult_cancel
thf(fact_1385_nat__mult__dvd__cancel1,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ( dvd_dvd @ nat @ ( times_times @ nat @ K @ M2 ) @ ( times_times @ nat @ K @ N2 ) )
= ( dvd_dvd @ nat @ M2 @ N2 ) ) ) ).
% nat_mult_dvd_cancel1
thf(fact_1386_bezout__add__strong__nat,axiom,
! [A3: nat,B3: nat] :
( ( A3
!= ( zero_zero @ nat ) )
=> ? [D5: nat,X5: nat,Y3: nat] :
( ( dvd_dvd @ nat @ D5 @ A3 )
& ( dvd_dvd @ nat @ D5 @ B3 )
& ( ( times_times @ nat @ A3 @ X5 )
= ( plus_plus @ nat @ ( times_times @ nat @ B3 @ Y3 ) @ D5 ) ) ) ) ).
% bezout_add_strong_nat
thf(fact_1387_vebt__buildup_Oelims,axiom,
! [X: nat,Y: vEBT_VEBT] :
( ( ( vEBT_vebt_buildup @ X )
= Y )
=> ( ( ( X
= ( zero_zero @ nat ) )
=> ( Y
!= ( vEBT_Leaf @ $false @ $false ) ) )
=> ( ( ( X
= ( suc @ ( zero_zero @ nat ) ) )
=> ( Y
!= ( vEBT_Leaf @ $false @ $false ) ) )
=> ~ ! [Va: nat] :
( ( X
= ( suc @ ( suc @ Va ) ) )
=> ~ ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ Va ) ) )
=> ( Y
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ Va ) ) @ ( replicate @ vEBT_VEBT @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) )
& ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ Va ) ) )
=> ( Y
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ Va ) ) @ ( replicate @ vEBT_VEBT @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( suc @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_buildup.elims
thf(fact_1388_even__zero,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( zero_zero @ A ) ) ) ).
% even_zero
thf(fact_1389_is__unitE,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,C2: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ~ ( ( A3
!= ( zero_zero @ A ) )
=> ! [B5: A] :
( ( B5
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ B5 @ ( one_one @ A ) )
=> ( ( ( divide_divide @ A @ ( one_one @ A ) @ A3 )
= B5 )
=> ( ( ( divide_divide @ A @ ( one_one @ A ) @ B5 )
= A3 )
=> ( ( ( times_times @ A @ A3 @ B5 )
= ( one_one @ A ) )
=> ( ( divide_divide @ A @ C2 @ A3 )
!= ( times_times @ A @ C2 @ B5 ) ) ) ) ) ) ) ) ) ) ).
% is_unitE
thf(fact_1390_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_1391_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_1392_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_1393_dvd__power__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [X: A,M2: nat,N2: nat] :
( ( X
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ ( power_power @ A @ X @ M2 ) @ ( power_power @ A @ X @ N2 ) )
= ( ( dvd_dvd @ A @ X @ ( one_one @ A ) )
| ( ord_less_eq @ nat @ M2 @ N2 ) ) ) ) ) ).
% dvd_power_iff
thf(fact_1394_dvd__power,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [N2: nat,X: A] :
( ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
| ( X
= ( one_one @ A ) ) )
=> ( dvd_dvd @ A @ X @ ( power_power @ A @ X @ N2 ) ) ) ) ).
% dvd_power
thf(fact_1395_split__mod,axiom,
! [P: nat > $o,M2: nat,N2: nat] :
( ( P @ ( modulo_modulo @ nat @ M2 @ N2 ) )
= ( ( ( N2
= ( zero_zero @ nat ) )
=> ( P @ M2 ) )
& ( ( N2
!= ( zero_zero @ nat ) )
=> ! [I3: nat,J2: nat] :
( ( ord_less @ nat @ J2 @ N2 )
=> ( ( M2
= ( plus_plus @ nat @ ( times_times @ nat @ N2 @ I3 ) @ J2 ) )
=> ( P @ J2 ) ) ) ) ) ) ).
% split_mod
thf(fact_1396_dvd__mult__cancel1,axiom,
! [M2: nat,N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ M2 )
=> ( ( dvd_dvd @ nat @ ( times_times @ nat @ M2 @ N2 ) @ M2 )
= ( N2
= ( one_one @ nat ) ) ) ) ).
% dvd_mult_cancel1
thf(fact_1397_dvd__mult__cancel2,axiom,
! [M2: nat,N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ M2 )
=> ( ( dvd_dvd @ nat @ ( times_times @ nat @ N2 @ M2 ) @ M2 )
= ( N2
= ( one_one @ nat ) ) ) ) ).
% dvd_mult_cancel2
thf(fact_1398_power__dvd__imp__le,axiom,
! [I: nat,M2: nat,N2: nat] :
( ( dvd_dvd @ nat @ ( power_power @ nat @ I @ M2 ) @ ( power_power @ nat @ I @ N2 ) )
=> ( ( ord_less @ nat @ ( one_one @ nat ) @ I )
=> ( ord_less_eq @ nat @ M2 @ N2 ) ) ) ).
% power_dvd_imp_le
thf(fact_1399_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_1400_product__nth,axiom,
! [A: $tType,B: $tType,N2: nat,Xs: list @ A,Ys: list @ B] :
( ( ord_less @ nat @ N2 @ ( times_times @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ B ) @ Ys ) ) )
=> ( ( nth @ ( product_prod @ A @ B ) @ ( product @ A @ B @ Xs @ Ys ) @ N2 )
= ( product_Pair @ A @ B @ ( nth @ A @ Xs @ ( divide_divide @ nat @ N2 @ ( size_size @ ( list @ B ) @ Ys ) ) ) @ ( nth @ B @ Ys @ ( modulo_modulo @ nat @ N2 @ ( size_size @ ( list @ B ) @ Ys ) ) ) ) ) ) ).
% product_nth
thf(fact_1401_power__mono__odd,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N2: nat,A3: A,B3: A] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
=> ( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ord_less_eq @ A @ ( power_power @ A @ A3 @ N2 ) @ ( power_power @ A @ B3 @ N2 ) ) ) ) ) ).
% power_mono_odd
thf(fact_1402_Suc__times__mod__eq,axiom,
! [M2: nat,N2: nat] :
( ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ M2 )
=> ( ( modulo_modulo @ nat @ ( suc @ ( times_times @ nat @ M2 @ N2 ) ) @ M2 )
= ( one_one @ nat ) ) ) ).
% Suc_times_mod_eq
thf(fact_1403_odd__pos,axiom,
! [N2: nat] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) ) ).
% odd_pos
thf(fact_1404_dvd__power__iff__le,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K )
=> ( ( dvd_dvd @ nat @ ( power_power @ nat @ K @ M2 ) @ ( power_power @ nat @ K @ N2 ) )
= ( ord_less_eq @ nat @ M2 @ N2 ) ) ) ).
% dvd_power_iff_le
thf(fact_1405_even__unset__bit__iff,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M2: nat,A3: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se2638667681897837118et_bit @ A @ M2 @ A3 ) )
= ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
| ( M2
= ( zero_zero @ nat ) ) ) ) ) ).
% even_unset_bit_iff
thf(fact_1406_even__set__bit__iff,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M2: nat,A3: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5668285175392031749et_bit @ A @ M2 @ A3 ) )
= ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
& ( M2
!= ( zero_zero @ nat ) ) ) ) ) ).
% even_set_bit_iff
thf(fact_1407_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_1408_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_1409_VEBT__internal_Onaive__member_Osimps_I3_J,axiom,
! [Uy: option @ ( product_prod @ nat @ nat ),V2: nat,TreeList2: list @ vEBT_VEBT,S: vEBT_VEBT,X: nat] :
( ( vEBT_V5719532721284313246member @ ( vEBT_Node @ Uy @ ( suc @ V2 ) @ TreeList2 @ S ) @ X )
= ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ V2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_V5719532721284313246member @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ V2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ ( suc @ V2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ V2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ).
% VEBT_internal.naive_member.simps(3)
thf(fact_1410_div__exp__mod__exp__eq,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A3: A,N2: nat,M2: nat] :
( ( modulo_modulo @ A @ ( divide_divide @ A @ A3 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M2 ) )
= ( divide_divide @ A @ ( modulo_modulo @ A @ A3 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( plus_plus @ nat @ N2 @ M2 ) ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ).
% div_exp_mod_exp_eq
thf(fact_1411_oddE,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A3: A] :
( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
=> ~ ! [B5: A] :
( A3
!= ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B5 ) @ ( one_one @ A ) ) ) ) ) ).
% oddE
thf(fact_1412_zero__le__power__eq,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,N2: nat] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A3 @ N2 ) )
= ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
| ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 ) ) ) ) ) ).
% zero_le_power_eq
thf(fact_1413_zero__le__odd__power,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N2: nat,A3: A] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A3 @ N2 ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 ) ) ) ) ).
% zero_le_odd_power
thf(fact_1414_zero__le__even__power,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N2: nat,A3: A] :
( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A3 @ N2 ) ) ) ) ).
% zero_le_even_power
thf(fact_1415_verit__le__mono__div,axiom,
! [A4: nat,B2: nat,N2: nat] :
( ( ord_less @ nat @ A4 @ B2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ord_less_eq @ nat
@ ( plus_plus @ nat @ ( divide_divide @ nat @ A4 @ N2 )
@ ( if @ nat
@ ( ( modulo_modulo @ nat @ B2 @ N2 )
= ( zero_zero @ nat ) )
@ ( one_one @ nat )
@ ( zero_zero @ nat ) ) )
@ ( divide_divide @ nat @ B2 @ N2 ) ) ) ) ).
% verit_le_mono_div
thf(fact_1416_VEBT__internal_Omembermima_Osimps_I5_J,axiom,
! [V2: nat,TreeList2: list @ vEBT_VEBT,Vd2: vEBT_VEBT,X: nat] :
( ( vEBT_VEBT_membermima @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V2 ) @ TreeList2 @ Vd2 ) @ X )
= ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ V2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ V2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ ( suc @ V2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ V2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ).
% VEBT_internal.membermima.simps(5)
thf(fact_1417_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_1418_vebt__member_Osimps_I5_J,axiom,
! [Mi: nat,Ma: nat,Va2: nat,TreeList2: list @ vEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
( ( vEBT_vebt_member @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary ) @ X )
= ( ( X != Mi )
=> ( ( X != Ma )
=> ( ~ ( ord_less @ nat @ X @ Mi )
& ( ~ ( ord_less @ nat @ X @ Mi )
=> ( ~ ( ord_less @ nat @ Ma @ X )
& ( ~ ( ord_less @ nat @ Ma @ X )
=> ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_vebt_member @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ) ) ) ) ).
% vebt_member.simps(5)
thf(fact_1419_VEBT__internal_Omembermima_Osimps_I4_J,axiom,
! [Mi: nat,Ma: nat,V2: nat,TreeList2: list @ vEBT_VEBT,Vc: vEBT_VEBT,X: nat] :
( ( vEBT_VEBT_membermima @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ V2 ) @ TreeList2 @ Vc ) @ X )
= ( ( X = Mi )
| ( X = Ma )
| ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ V2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ V2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ ( suc @ V2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ V2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ).
% VEBT_internal.membermima.simps(4)
thf(fact_1420_zero__less__power__eq,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,N2: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A3 @ N2 ) )
= ( ( N2
= ( zero_zero @ nat ) )
| ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
& ( A3
!= ( zero_zero @ A ) ) )
| ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
& ( ord_less @ A @ ( zero_zero @ A ) @ A3 ) ) ) ) ) ).
% zero_less_power_eq
thf(fact_1421_VEBT__internal_Onaive__member_Oelims_I1_J,axiom,
! [X: vEBT_VEBT,Xa2: nat,Y: $o] :
( ( ( vEBT_V5719532721284313246member @ X @ Xa2 )
= Y )
=> ( ! [A6: $o,B5: $o] :
( ( X
= ( vEBT_Leaf @ A6 @ B5 ) )
=> ( Y
= ( ~ ( ( ( Xa2
= ( zero_zero @ nat ) )
=> A6 )
& ( ( Xa2
!= ( zero_zero @ nat ) )
=> ( ( ( Xa2
= ( one_one @ nat ) )
=> B5 )
& ( Xa2
= ( one_one @ nat ) ) ) ) ) ) ) )
=> ( ( ? [Uu2: option @ ( product_prod @ nat @ nat ),Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT] :
( X
= ( vEBT_Node @ Uu2 @ ( zero_zero @ nat ) @ Uv2 @ Uw2 ) )
=> Y )
=> ~ ! [Uy2: option @ ( product_prod @ nat @ nat ),V3: nat,TreeList3: list @ vEBT_VEBT] :
( ? [S3: vEBT_VEBT] :
( X
= ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList3 @ S3 ) )
=> ( Y
= ( ~ ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) )
=> ( vEBT_V5719532721284313246member @ ( nth @ vEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.naive_member.elims(1)
thf(fact_1422_VEBT__internal_Onaive__member_Oelims_I2_J,axiom,
! [X: vEBT_VEBT,Xa2: nat] :
( ( vEBT_V5719532721284313246member @ X @ Xa2 )
=> ( ! [A6: $o,B5: $o] :
( ( X
= ( vEBT_Leaf @ A6 @ B5 ) )
=> ~ ( ( ( Xa2
= ( zero_zero @ nat ) )
=> A6 )
& ( ( Xa2
!= ( zero_zero @ nat ) )
=> ( ( ( Xa2
= ( one_one @ nat ) )
=> B5 )
& ( Xa2
= ( one_one @ nat ) ) ) ) ) )
=> ~ ! [Uy2: option @ ( product_prod @ nat @ nat ),V3: nat,TreeList3: list @ vEBT_VEBT] :
( ? [S3: vEBT_VEBT] :
( X
= ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList3 @ S3 ) )
=> ~ ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) )
=> ( vEBT_V5719532721284313246member @ ( nth @ vEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) ) ) ) ) ) ).
% VEBT_internal.naive_member.elims(2)
thf(fact_1423_VEBT__internal_Onaive__member_Oelims_I3_J,axiom,
! [X: vEBT_VEBT,Xa2: nat] :
( ~ ( vEBT_V5719532721284313246member @ X @ Xa2 )
=> ( ! [A6: $o,B5: $o] :
( ( X
= ( vEBT_Leaf @ A6 @ B5 ) )
=> ( ( ( Xa2
= ( zero_zero @ nat ) )
=> A6 )
& ( ( Xa2
!= ( zero_zero @ nat ) )
=> ( ( ( Xa2
= ( one_one @ nat ) )
=> B5 )
& ( Xa2
= ( one_one @ nat ) ) ) ) ) )
=> ( ! [Uu2: option @ ( product_prod @ nat @ nat ),Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT] :
( X
!= ( vEBT_Node @ Uu2 @ ( zero_zero @ nat ) @ Uv2 @ Uw2 ) )
=> ~ ! [Uy2: option @ ( product_prod @ nat @ nat ),V3: nat,TreeList3: list @ vEBT_VEBT] :
( ? [S3: vEBT_VEBT] :
( X
= ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList3 @ S3 ) )
=> ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) )
=> ( vEBT_V5719532721284313246member @ ( nth @ vEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) ) ) ) ) ) ) ).
% VEBT_internal.naive_member.elims(3)
thf(fact_1424_mod__double__modulus,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M2: A,X: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ M2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( ( modulo_modulo @ A @ X @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M2 ) )
= ( modulo_modulo @ A @ X @ M2 ) )
| ( ( modulo_modulo @ A @ X @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M2 ) )
= ( plus_plus @ A @ ( modulo_modulo @ A @ X @ M2 ) @ M2 ) ) ) ) ) ) ).
% mod_double_modulus
thf(fact_1425_VEBT__internal_Omembermima_Oelims_I2_J,axiom,
! [X: vEBT_VEBT,Xa2: nat] :
( ( vEBT_VEBT_membermima @ X @ Xa2 )
=> ( ! [Mi2: nat,Ma2: nat] :
( ? [Va3: list @ vEBT_VEBT,Vb2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ Va3 @ Vb2 ) )
=> ~ ( ( Xa2 = Mi2 )
| ( Xa2 = Ma2 ) ) )
=> ( ! [Mi2: nat,Ma2: nat,V3: nat,TreeList3: list @ vEBT_VEBT] :
( ? [Vc2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList3 @ Vc2 ) )
=> ~ ( ( Xa2 = Mi2 )
| ( Xa2 = Ma2 )
| ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) )
=> ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) ) ) ) )
=> ~ ! [V3: nat,TreeList3: list @ vEBT_VEBT] :
( ? [Vd: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V3 ) @ TreeList3 @ Vd ) )
=> ~ ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) )
=> ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) ) ) ) ) ) ) ).
% VEBT_internal.membermima.elims(2)
thf(fact_1426_unset__bit__Suc,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N2: nat,A3: A] :
( ( bit_se2638667681897837118et_bit @ A @ ( suc @ N2 ) @ 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 @ N2 @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% unset_bit_Suc
thf(fact_1427_set__bit__Suc,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N2: nat,A3: A] :
( ( bit_se5668285175392031749et_bit @ A @ ( suc @ N2 ) @ 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 @ N2 @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% set_bit_Suc
thf(fact_1428_power__le__zero__eq,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,N2: nat] :
( ( ord_less_eq @ A @ ( power_power @ A @ A3 @ N2 ) @ ( zero_zero @ A ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
& ( ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
& ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) ) )
| ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
& ( A3
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% power_le_zero_eq
thf(fact_1429_vebt__member_Oelims_I2_J,axiom,
! [X: vEBT_VEBT,Xa2: nat] :
( ( vEBT_vebt_member @ X @ Xa2 )
=> ( ! [A6: $o,B5: $o] :
( ( X
= ( vEBT_Leaf @ A6 @ B5 ) )
=> ~ ( ( ( Xa2
= ( zero_zero @ nat ) )
=> A6 )
& ( ( Xa2
!= ( zero_zero @ nat ) )
=> ( ( ( Xa2
= ( one_one @ nat ) )
=> B5 )
& ( Xa2
= ( one_one @ nat ) ) ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList3: list @ vEBT_VEBT] :
( ? [Summary2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList3 @ Summary2 ) )
=> ~ ( ( Xa2 != Mi2 )
=> ( ( Xa2 != Ma2 )
=> ( ~ ( ord_less @ nat @ Xa2 @ Mi2 )
& ( ~ ( ord_less @ nat @ Xa2 @ Mi2 )
=> ( ~ ( ord_less @ nat @ Ma2 @ Xa2 )
& ( ~ ( ord_less @ nat @ Ma2 @ Xa2 )
=> ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) )
=> ( vEBT_vebt_member @ ( nth @ vEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_member.elims(2)
thf(fact_1430_VEBT__internal_Omembermima_Oelims_I1_J,axiom,
! [X: vEBT_VEBT,Xa2: nat,Y: $o] :
( ( ( vEBT_VEBT_membermima @ X @ Xa2 )
= Y )
=> ( ( ? [Uu2: $o,Uv2: $o] :
( X
= ( vEBT_Leaf @ Uu2 @ Uv2 ) )
=> Y )
=> ( ( ? [Ux2: list @ vEBT_VEBT,Uy2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( zero_zero @ nat ) @ Ux2 @ Uy2 ) )
=> Y )
=> ( ! [Mi2: nat,Ma2: nat] :
( ? [Va3: list @ vEBT_VEBT,Vb2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ Va3 @ Vb2 ) )
=> ( Y
= ( ~ ( ( Xa2 = Mi2 )
| ( Xa2 = Ma2 ) ) ) ) )
=> ( ! [Mi2: nat,Ma2: nat,V3: nat,TreeList3: list @ vEBT_VEBT] :
( ? [Vc2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList3 @ Vc2 ) )
=> ( Y
= ( ~ ( ( Xa2 = Mi2 )
| ( Xa2 = Ma2 )
| ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) )
=> ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) ) ) ) ) ) )
=> ~ ! [V3: nat,TreeList3: list @ vEBT_VEBT] :
( ? [Vd: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V3 ) @ TreeList3 @ Vd ) )
=> ( Y
= ( ~ ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) )
=> ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.membermima.elims(1)
thf(fact_1431_VEBT__internal_Omembermima_Oelims_I3_J,axiom,
! [X: vEBT_VEBT,Xa2: nat] :
( ~ ( vEBT_VEBT_membermima @ X @ Xa2 )
=> ( ! [Uu2: $o,Uv2: $o] :
( X
!= ( vEBT_Leaf @ Uu2 @ Uv2 ) )
=> ( ! [Ux2: list @ vEBT_VEBT,Uy2: vEBT_VEBT] :
( X
!= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( zero_zero @ nat ) @ Ux2 @ Uy2 ) )
=> ( ! [Mi2: nat,Ma2: nat] :
( ? [Va3: list @ vEBT_VEBT,Vb2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ Va3 @ Vb2 ) )
=> ( ( Xa2 = Mi2 )
| ( Xa2 = Ma2 ) ) )
=> ( ! [Mi2: nat,Ma2: nat,V3: nat,TreeList3: list @ vEBT_VEBT] :
( ? [Vc2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList3 @ Vc2 ) )
=> ( ( Xa2 = Mi2 )
| ( Xa2 = Ma2 )
| ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) )
=> ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) ) ) ) )
=> ~ ! [V3: nat,TreeList3: list @ vEBT_VEBT] :
( ? [Vd: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V3 ) @ TreeList3 @ Vd ) )
=> ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) )
=> ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.membermima.elims(3)
thf(fact_1432_vebt__insert_Osimps_I5_J,axiom,
! [Mi: nat,Ma: nat,Va2: nat,TreeList2: list @ vEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary ) @ X )
= ( if @ vEBT_VEBT
@ ( ( ord_less @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
& ~ ( ( X = Mi )
| ( X = Ma ) ) )
@ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ ( if @ nat @ ( ord_less @ nat @ X @ Mi ) @ X @ Mi ) @ ( ord_max @ nat @ ( if @ nat @ ( ord_less @ nat @ X @ Mi ) @ Mi @ X ) @ Ma ) ) ) @ ( suc @ ( suc @ Va2 ) ) @ ( list_update @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_insert @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( ord_less @ nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( if @ vEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ Summary ) )
@ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary ) ) ) ).
% vebt_insert.simps(5)
thf(fact_1433_vebt__member_Oelims_I3_J,axiom,
! [X: vEBT_VEBT,Xa2: nat] :
( ~ ( vEBT_vebt_member @ X @ Xa2 )
=> ( ! [A6: $o,B5: $o] :
( ( X
= ( vEBT_Leaf @ A6 @ B5 ) )
=> ( ( ( Xa2
= ( zero_zero @ nat ) )
=> A6 )
& ( ( Xa2
!= ( zero_zero @ nat ) )
=> ( ( ( Xa2
= ( one_one @ nat ) )
=> B5 )
& ( Xa2
= ( one_one @ nat ) ) ) ) ) )
=> ( ! [Uu2: nat,Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT] :
( X
!= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) )
=> ( ! [V3: product_prod @ nat @ nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( X
!= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Uy2 @ Uz2 ) )
=> ( ! [V3: product_prod @ nat @ nat,Vb2: list @ vEBT_VEBT,Vc2: vEBT_VEBT] :
( X
!= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb2 @ Vc2 ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList3: list @ vEBT_VEBT] :
( ? [Summary2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList3 @ Summary2 ) )
=> ( ( Xa2 != Mi2 )
=> ( ( Xa2 != Ma2 )
=> ( ~ ( ord_less @ nat @ Xa2 @ Mi2 )
& ( ~ ( ord_less @ nat @ Xa2 @ Mi2 )
=> ( ~ ( ord_less @ nat @ Ma2 @ Xa2 )
& ( ~ ( ord_less @ nat @ Ma2 @ Xa2 )
=> ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) )
=> ( vEBT_vebt_member @ ( nth @ vEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_member.elims(3)
thf(fact_1434_vebt__member_Oelims_I1_J,axiom,
! [X: vEBT_VEBT,Xa2: nat,Y: $o] :
( ( ( vEBT_vebt_member @ X @ Xa2 )
= Y )
=> ( ! [A6: $o,B5: $o] :
( ( X
= ( vEBT_Leaf @ A6 @ B5 ) )
=> ( Y
= ( ~ ( ( ( Xa2
= ( zero_zero @ nat ) )
=> A6 )
& ( ( Xa2
!= ( zero_zero @ nat ) )
=> ( ( ( Xa2
= ( one_one @ nat ) )
=> B5 )
& ( Xa2
= ( one_one @ nat ) ) ) ) ) ) ) )
=> ( ( ? [Uu2: nat,Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) )
=> Y )
=> ( ( ? [V3: product_prod @ nat @ nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Uy2 @ Uz2 ) )
=> Y )
=> ( ( ? [V3: product_prod @ nat @ nat,Vb2: list @ vEBT_VEBT,Vc2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb2 @ Vc2 ) )
=> Y )
=> ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList3: list @ vEBT_VEBT] :
( ? [Summary2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList3 @ Summary2 ) )
=> ( Y
= ( ~ ( ( Xa2 != Mi2 )
=> ( ( Xa2 != Ma2 )
=> ( ~ ( ord_less @ nat @ Xa2 @ Mi2 )
& ( ~ ( ord_less @ nat @ Xa2 @ Mi2 )
=> ( ~ ( ord_less @ nat @ Ma2 @ Xa2 )
& ( ~ ( ord_less @ nat @ Ma2 @ Xa2 )
=> ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) )
=> ( vEBT_vebt_member @ ( nth @ vEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_member.elims(1)
thf(fact_1435_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_1436_finite__nth__roots,axiom,
! [N2: nat,C2: complex] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( finite_finite2 @ complex
@ ( collect @ complex
@ ^ [Z6: complex] :
( ( power_power @ complex @ Z6 @ N2 )
= C2 ) ) ) ) ).
% finite_nth_roots
thf(fact_1437_finite__roots__unity,axiom,
! [A: $tType] :
( ( ( real_V8999393235501362500lgebra @ A )
& ( idom @ A ) )
=> ! [N2: nat] :
( ( ord_less_eq @ nat @ ( one_one @ nat ) @ N2 )
=> ( finite_finite2 @ A
@ ( collect @ A
@ ^ [Z6: A] :
( ( power_power @ A @ Z6 @ N2 )
= ( one_one @ A ) ) ) ) ) ) ).
% finite_roots_unity
thf(fact_1438_vebt__insert_Opelims,axiom,
! [X: vEBT_VEBT,Xa2: nat,Y: vEBT_VEBT] :
( ( ( vEBT_vebt_insert @ X @ Xa2 )
= Y )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_insert_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa2 ) )
=> ( ! [A6: $o,B5: $o] :
( ( X
= ( vEBT_Leaf @ A6 @ B5 ) )
=> ( ( ( ( Xa2
= ( zero_zero @ nat ) )
=> ( Y
= ( vEBT_Leaf @ $true @ B5 ) ) )
& ( ( Xa2
!= ( zero_zero @ nat ) )
=> ( ( ( Xa2
= ( one_one @ nat ) )
=> ( Y
= ( vEBT_Leaf @ A6 @ $true ) ) )
& ( ( Xa2
!= ( one_one @ nat ) )
=> ( Y
= ( vEBT_Leaf @ A6 @ B5 ) ) ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_insert_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A6 @ B5 ) @ Xa2 ) ) ) )
=> ( ! [Info2: option @ ( product_prod @ nat @ nat ),Ts2: list @ vEBT_VEBT,S3: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Info2 @ ( zero_zero @ nat ) @ Ts2 @ S3 ) )
=> ( ( Y
= ( vEBT_Node @ Info2 @ ( zero_zero @ nat ) @ Ts2 @ S3 ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_insert_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Info2 @ ( zero_zero @ nat ) @ Ts2 @ S3 ) @ Xa2 ) ) ) )
=> ( ! [Info2: option @ ( product_prod @ nat @ nat ),Ts2: list @ vEBT_VEBT,S3: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Info2 @ ( suc @ ( zero_zero @ nat ) ) @ Ts2 @ S3 ) )
=> ( ( Y
= ( vEBT_Node @ Info2 @ ( suc @ ( zero_zero @ nat ) ) @ Ts2 @ S3 ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_insert_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Info2 @ ( suc @ ( zero_zero @ nat ) ) @ Ts2 @ S3 ) @ Xa2 ) ) ) )
=> ( ! [V3: nat,TreeList3: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ V3 ) ) @ TreeList3 @ Summary2 ) )
=> ( ( Y
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Xa2 @ Xa2 ) ) @ ( suc @ ( suc @ V3 ) ) @ TreeList3 @ Summary2 ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_insert_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ V3 ) ) @ TreeList3 @ Summary2 ) @ Xa2 ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList3: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList3 @ Summary2 ) )
=> ( ( Y
= ( if @ vEBT_VEBT
@ ( ( ord_less @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) )
& ~ ( ( Xa2 = Mi2 )
| ( Xa2 = Ma2 ) ) )
@ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ ( if @ nat @ ( ord_less @ nat @ Xa2 @ Mi2 ) @ Xa2 @ Mi2 ) @ ( ord_max @ nat @ ( if @ nat @ ( ord_less @ nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ Ma2 ) ) ) @ ( suc @ ( suc @ Va ) ) @ ( list_update @ vEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_insert @ ( nth @ vEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( ord_less @ nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( if @ vEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth @ vEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ Summary2 ) )
@ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList3 @ Summary2 ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_insert_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList3 @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ).
% vebt_insert.pelims
thf(fact_1439_flip__bit__Suc,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N2: nat,A3: A] :
( ( bit_se8732182000553998342ip_bit @ A @ ( suc @ N2 ) @ 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 @ N2 @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% flip_bit_Suc
thf(fact_1440_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
@ ^ [I3: B] :
( ( member @ B @ I3 @ I6 )
& ( ( X @ I3 )
!= ( one_one @ A ) ) ) ) )
=> ( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [I3: B] :
( ( member @ B @ I3 @ I6 )
& ( ( Y @ I3 )
!= ( one_one @ A ) ) ) ) )
=> ( finite_finite2 @ B
@ ( collect @ B
@ ^ [I3: B] :
( ( member @ B @ I3 @ I6 )
& ( ( times_times @ A @ ( X @ I3 ) @ ( Y @ I3 ) )
!= ( one_one @ A ) ) ) ) ) ) ) ) ).
% prod.finite_Collect_op
thf(fact_1441_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
@ ^ [I3: B] :
( ( member @ B @ I3 @ I6 )
& ( ( X @ I3 )
!= ( zero_zero @ A ) ) ) ) )
=> ( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [I3: B] :
( ( member @ B @ I3 @ I6 )
& ( ( Y @ I3 )
!= ( zero_zero @ A ) ) ) ) )
=> ( finite_finite2 @ B
@ ( collect @ B
@ ^ [I3: B] :
( ( member @ B @ I3 @ I6 )
& ( ( plus_plus @ A @ ( X @ I3 ) @ ( Y @ I3 ) )
!= ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% sum.finite_Collect_op
thf(fact_1442_vebt__member_Opelims_I1_J,axiom,
! [X: vEBT_VEBT,Xa2: nat,Y: $o] :
( ( ( vEBT_vebt_member @ X @ Xa2 )
= Y )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa2 ) )
=> ( ! [A6: $o,B5: $o] :
( ( X
= ( vEBT_Leaf @ A6 @ B5 ) )
=> ( ( Y
= ( ( ( Xa2
= ( zero_zero @ nat ) )
=> A6 )
& ( ( Xa2
!= ( zero_zero @ nat ) )
=> ( ( ( Xa2
= ( one_one @ nat ) )
=> B5 )
& ( Xa2
= ( one_one @ nat ) ) ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A6 @ B5 ) @ Xa2 ) ) ) )
=> ( ! [Uu2: nat,Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) )
=> ( ~ Y
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) @ Xa2 ) ) ) )
=> ( ! [V3: product_prod @ nat @ nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Uy2 @ Uz2 ) )
=> ( ~ Y
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Uy2 @ Uz2 ) @ Xa2 ) ) ) )
=> ( ! [V3: product_prod @ nat @ nat,Vb2: list @ vEBT_VEBT,Vc2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb2 @ Vc2 ) )
=> ( ~ Y
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb2 @ Vc2 ) @ Xa2 ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList3: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList3 @ Summary2 ) )
=> ( ( Y
= ( ( Xa2 != Mi2 )
=> ( ( Xa2 != Ma2 )
=> ( ~ ( ord_less @ nat @ Xa2 @ Mi2 )
& ( ~ ( ord_less @ nat @ Xa2 @ Mi2 )
=> ( ~ ( ord_less @ nat @ Ma2 @ Xa2 )
& ( ~ ( ord_less @ nat @ Ma2 @ Xa2 )
=> ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) )
=> ( vEBT_vebt_member @ ( nth @ vEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) ) ) ) ) ) ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList3 @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ).
% vebt_member.pelims(1)
thf(fact_1443_vebt__member_Opelims_I3_J,axiom,
! [X: vEBT_VEBT,Xa2: nat] :
( ~ ( vEBT_vebt_member @ X @ Xa2 )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa2 ) )
=> ( ! [A6: $o,B5: $o] :
( ( X
= ( vEBT_Leaf @ A6 @ B5 ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A6 @ B5 ) @ Xa2 ) )
=> ( ( ( Xa2
= ( zero_zero @ nat ) )
=> A6 )
& ( ( Xa2
!= ( zero_zero @ nat ) )
=> ( ( ( Xa2
= ( one_one @ nat ) )
=> B5 )
& ( Xa2
= ( one_one @ nat ) ) ) ) ) ) )
=> ( ! [Uu2: nat,Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) @ Xa2 ) ) )
=> ( ! [V3: product_prod @ nat @ nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Uy2 @ Uz2 ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Uy2 @ Uz2 ) @ Xa2 ) ) )
=> ( ! [V3: product_prod @ nat @ nat,Vb2: list @ vEBT_VEBT,Vc2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb2 @ Vc2 ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb2 @ Vc2 ) @ Xa2 ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList3: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList3 @ Summary2 ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList3 @ Summary2 ) @ Xa2 ) )
=> ( ( Xa2 != Mi2 )
=> ( ( Xa2 != Ma2 )
=> ( ~ ( ord_less @ nat @ Xa2 @ Mi2 )
& ( ~ ( ord_less @ nat @ Xa2 @ Mi2 )
=> ( ~ ( ord_less @ nat @ Ma2 @ Xa2 )
& ( ~ ( ord_less @ nat @ Ma2 @ Xa2 )
=> ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) )
=> ( vEBT_vebt_member @ ( nth @ vEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_member.pelims(3)
thf(fact_1444_vebt__member_Opelims_I2_J,axiom,
! [X: vEBT_VEBT,Xa2: nat] :
( ( vEBT_vebt_member @ X @ Xa2 )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa2 ) )
=> ( ! [A6: $o,B5: $o] :
( ( X
= ( vEBT_Leaf @ A6 @ B5 ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A6 @ B5 ) @ Xa2 ) )
=> ~ ( ( ( Xa2
= ( zero_zero @ nat ) )
=> A6 )
& ( ( Xa2
!= ( zero_zero @ nat ) )
=> ( ( ( Xa2
= ( one_one @ nat ) )
=> B5 )
& ( Xa2
= ( one_one @ nat ) ) ) ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList3: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList3 @ Summary2 ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList3 @ Summary2 ) @ Xa2 ) )
=> ~ ( ( Xa2 != Mi2 )
=> ( ( Xa2 != Ma2 )
=> ( ~ ( ord_less @ nat @ Xa2 @ Mi2 )
& ( ~ ( ord_less @ nat @ Xa2 @ Mi2 )
=> ( ~ ( ord_less @ nat @ Ma2 @ Xa2 )
& ( ~ ( ord_less @ nat @ Ma2 @ Xa2 )
=> ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) )
=> ( vEBT_vebt_member @ ( nth @ vEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_member.pelims(2)
thf(fact_1445_dvd__antisym,axiom,
! [M2: nat,N2: nat] :
( ( dvd_dvd @ nat @ M2 @ N2 )
=> ( ( dvd_dvd @ nat @ N2 @ M2 )
=> ( M2 = N2 ) ) ) ).
% dvd_antisym
thf(fact_1446_bot2E,axiom,
! [A: $tType,B: $tType,X: A,Y: B] :
~ ( bot_bot @ ( A > B > $o ) @ X @ Y ) ).
% bot2E
thf(fact_1447_even__flip__bit__iff,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M2: nat,A3: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se8732182000553998342ip_bit @ A @ M2 @ A3 ) )
= ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
!= ( M2
= ( zero_zero @ nat ) ) ) ) ) ).
% even_flip_bit_iff
thf(fact_1448_VEBT__internal_Onaive__member_Opelims_I1_J,axiom,
! [X: vEBT_VEBT,Xa2: nat,Y: $o] :
( ( ( vEBT_V5719532721284313246member @ X @ Xa2 )
= Y )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa2 ) )
=> ( ! [A6: $o,B5: $o] :
( ( X
= ( vEBT_Leaf @ A6 @ B5 ) )
=> ( ( Y
= ( ( ( Xa2
= ( zero_zero @ nat ) )
=> A6 )
& ( ( Xa2
!= ( zero_zero @ nat ) )
=> ( ( ( Xa2
= ( one_one @ nat ) )
=> B5 )
& ( Xa2
= ( one_one @ nat ) ) ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A6 @ B5 ) @ Xa2 ) ) ) )
=> ( ! [Uu2: option @ ( product_prod @ nat @ nat ),Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Uu2 @ ( zero_zero @ nat ) @ Uv2 @ Uw2 ) )
=> ( ~ Y
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Uu2 @ ( zero_zero @ nat ) @ Uv2 @ Uw2 ) @ Xa2 ) ) ) )
=> ~ ! [Uy2: option @ ( product_prod @ nat @ nat ),V3: nat,TreeList3: list @ vEBT_VEBT,S3: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList3 @ S3 ) )
=> ( ( Y
= ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) )
=> ( vEBT_V5719532721284313246member @ ( nth @ vEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList3 @ S3 ) @ Xa2 ) ) ) ) ) ) ) ) ).
% VEBT_internal.naive_member.pelims(1)
thf(fact_1449_VEBT__internal_Onaive__member_Opelims_I2_J,axiom,
! [X: vEBT_VEBT,Xa2: nat] :
( ( vEBT_V5719532721284313246member @ X @ Xa2 )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa2 ) )
=> ( ! [A6: $o,B5: $o] :
( ( X
= ( vEBT_Leaf @ A6 @ B5 ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A6 @ B5 ) @ Xa2 ) )
=> ~ ( ( ( Xa2
= ( zero_zero @ nat ) )
=> A6 )
& ( ( Xa2
!= ( zero_zero @ nat ) )
=> ( ( ( Xa2
= ( one_one @ nat ) )
=> B5 )
& ( Xa2
= ( one_one @ nat ) ) ) ) ) ) )
=> ~ ! [Uy2: option @ ( product_prod @ nat @ nat ),V3: nat,TreeList3: list @ vEBT_VEBT,S3: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList3 @ S3 ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList3 @ S3 ) @ Xa2 ) )
=> ~ ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) )
=> ( vEBT_V5719532721284313246member @ ( nth @ vEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) ) ) ) ) ) ) ) ).
% VEBT_internal.naive_member.pelims(2)
thf(fact_1450_VEBT__internal_Onaive__member_Opelims_I3_J,axiom,
! [X: vEBT_VEBT,Xa2: nat] :
( ~ ( vEBT_V5719532721284313246member @ X @ Xa2 )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa2 ) )
=> ( ! [A6: $o,B5: $o] :
( ( X
= ( vEBT_Leaf @ A6 @ B5 ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A6 @ B5 ) @ Xa2 ) )
=> ( ( ( Xa2
= ( zero_zero @ nat ) )
=> A6 )
& ( ( Xa2
!= ( zero_zero @ nat ) )
=> ( ( ( Xa2
= ( one_one @ nat ) )
=> B5 )
& ( Xa2
= ( one_one @ nat ) ) ) ) ) ) )
=> ( ! [Uu2: option @ ( product_prod @ nat @ nat ),Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Uu2 @ ( zero_zero @ nat ) @ Uv2 @ Uw2 ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Uu2 @ ( zero_zero @ nat ) @ Uv2 @ Uw2 ) @ Xa2 ) ) )
=> ~ ! [Uy2: option @ ( product_prod @ nat @ nat ),V3: nat,TreeList3: list @ vEBT_VEBT,S3: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList3 @ S3 ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList3 @ S3 ) @ Xa2 ) )
=> ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) )
=> ( vEBT_V5719532721284313246member @ ( nth @ vEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.naive_member.pelims(3)
thf(fact_1451_VEBT__internal_Omembermima_Opelims_I1_J,axiom,
! [X: vEBT_VEBT,Xa2: nat,Y: $o] :
( ( ( vEBT_VEBT_membermima @ X @ Xa2 )
= Y )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa2 ) )
=> ( ! [Uu2: $o,Uv2: $o] :
( ( X
= ( vEBT_Leaf @ Uu2 @ Uv2 ) )
=> ( ~ Y
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa2 ) ) ) )
=> ( ! [Ux2: list @ vEBT_VEBT,Uy2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( zero_zero @ nat ) @ Ux2 @ Uy2 ) )
=> ( ~ Y
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( zero_zero @ nat ) @ Ux2 @ Uy2 ) @ Xa2 ) ) ) )
=> ( ! [Mi2: nat,Ma2: nat,Va3: list @ vEBT_VEBT,Vb2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ Va3 @ Vb2 ) )
=> ( ( Y
= ( ( Xa2 = Mi2 )
| ( Xa2 = Ma2 ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ Va3 @ Vb2 ) @ Xa2 ) ) ) )
=> ( ! [Mi2: nat,Ma2: nat,V3: nat,TreeList3: list @ vEBT_VEBT,Vc2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList3 @ Vc2 ) )
=> ( ( Y
= ( ( Xa2 = Mi2 )
| ( Xa2 = Ma2 )
| ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) )
=> ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList3 @ Vc2 ) @ Xa2 ) ) ) )
=> ~ ! [V3: nat,TreeList3: list @ vEBT_VEBT,Vd: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V3 ) @ TreeList3 @ Vd ) )
=> ( ( Y
= ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) )
=> ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V3 ) @ TreeList3 @ Vd ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.membermima.pelims(1)
thf(fact_1452_VEBT__internal_Omembermima_Opelims_I3_J,axiom,
! [X: vEBT_VEBT,Xa2: nat] :
( ~ ( vEBT_VEBT_membermima @ X @ Xa2 )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa2 ) )
=> ( ! [Uu2: $o,Uv2: $o] :
( ( X
= ( vEBT_Leaf @ Uu2 @ Uv2 ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa2 ) ) )
=> ( ! [Ux2: list @ vEBT_VEBT,Uy2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( zero_zero @ nat ) @ Ux2 @ Uy2 ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( zero_zero @ nat ) @ Ux2 @ Uy2 ) @ Xa2 ) ) )
=> ( ! [Mi2: nat,Ma2: nat,Va3: list @ vEBT_VEBT,Vb2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ Va3 @ Vb2 ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ Va3 @ Vb2 ) @ Xa2 ) )
=> ( ( Xa2 = Mi2 )
| ( Xa2 = Ma2 ) ) ) )
=> ( ! [Mi2: nat,Ma2: nat,V3: nat,TreeList3: list @ vEBT_VEBT,Vc2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList3 @ Vc2 ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList3 @ Vc2 ) @ Xa2 ) )
=> ( ( Xa2 = Mi2 )
| ( Xa2 = Ma2 )
| ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) )
=> ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) ) ) ) ) )
=> ~ ! [V3: nat,TreeList3: list @ vEBT_VEBT,Vd: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V3 ) @ TreeList3 @ Vd ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V3 ) @ TreeList3 @ Vd ) @ Xa2 ) )
=> ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) )
=> ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.membermima.pelims(3)
thf(fact_1453_VEBT__internal_Omembermima_Opelims_I2_J,axiom,
! [X: vEBT_VEBT,Xa2: nat] :
( ( vEBT_VEBT_membermima @ X @ Xa2 )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa2 ) )
=> ( ! [Mi2: nat,Ma2: nat,Va3: list @ vEBT_VEBT,Vb2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ Va3 @ Vb2 ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ Va3 @ Vb2 ) @ Xa2 ) )
=> ~ ( ( Xa2 = Mi2 )
| ( Xa2 = Ma2 ) ) ) )
=> ( ! [Mi2: nat,Ma2: nat,V3: nat,TreeList3: list @ vEBT_VEBT,Vc2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList3 @ Vc2 ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList3 @ Vc2 ) @ Xa2 ) )
=> ~ ( ( Xa2 = Mi2 )
| ( Xa2 = Ma2 )
| ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) )
=> ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) ) ) ) ) )
=> ~ ! [V3: nat,TreeList3: list @ vEBT_VEBT,Vd: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V3 ) @ TreeList3 @ Vd ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V3 ) @ TreeList3 @ Vd ) @ Xa2 ) )
=> ~ ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) )
=> ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.membermima.pelims(2)
thf(fact_1454_arcosh__1,axiom,
! [A: $tType] :
( ( ln @ A )
=> ( ( arcosh @ A @ ( one_one @ A ) )
= ( zero_zero @ A ) ) ) ).
% arcosh_1
thf(fact_1455_arsinh__0,axiom,
! [A: $tType] :
( ( ln @ A )
=> ( ( arsinh @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% arsinh_0
thf(fact_1456_artanh__0,axiom,
! [A: $tType] :
( ( ( real_V3459762299906320749_field @ A )
& ( ln @ A ) )
=> ( ( artanh @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% artanh_0
thf(fact_1457_vebt__buildup_Opelims,axiom,
! [X: nat,Y: vEBT_VEBT] :
( ( ( vEBT_vebt_buildup @ X )
= Y )
=> ( ( accp @ nat @ vEBT_v4011308405150292612up_rel @ X )
=> ( ( ( X
= ( zero_zero @ nat ) )
=> ( ( Y
= ( vEBT_Leaf @ $false @ $false ) )
=> ~ ( accp @ nat @ vEBT_v4011308405150292612up_rel @ ( zero_zero @ nat ) ) ) )
=> ( ( ( X
= ( suc @ ( zero_zero @ nat ) ) )
=> ( ( Y
= ( vEBT_Leaf @ $false @ $false ) )
=> ~ ( accp @ nat @ vEBT_v4011308405150292612up_rel @ ( suc @ ( zero_zero @ nat ) ) ) ) )
=> ~ ! [Va: nat] :
( ( X
= ( suc @ ( suc @ Va ) ) )
=> ( ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ Va ) ) )
=> ( Y
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ Va ) ) @ ( replicate @ vEBT_VEBT @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) )
& ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ Va ) ) )
=> ( Y
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ Va ) ) @ ( replicate @ vEBT_VEBT @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( suc @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) )
=> ~ ( accp @ nat @ vEBT_v4011308405150292612up_rel @ ( suc @ ( suc @ Va ) ) ) ) ) ) ) ) ) ).
% vebt_buildup.pelims
thf(fact_1458_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_1459_signed__take__bit__Suc,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N2: nat,A3: A] :
( ( bit_ri4674362597316999326ke_bit @ A @ ( suc @ N2 ) @ 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 @ N2 @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% signed_take_bit_Suc
thf(fact_1460_even__mult__exp__div__exp__iff,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A3: A,M2: nat,N2: nat] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ ( times_times @ A @ A3 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M2 ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) ) )
= ( ( ord_less @ nat @ N2 @ M2 )
| ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 )
= ( zero_zero @ A ) )
| ( ( ord_less_eq @ nat @ M2 @ N2 )
& ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A3 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ N2 @ M2 ) ) ) ) ) ) ) ) ).
% even_mult_exp_div_exp_iff
thf(fact_1461_even__set__encode__iff,axiom,
! [A4: set @ nat] :
( ( finite_finite2 @ nat @ A4 )
=> ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( nat_set_encode @ A4 ) )
= ( ~ ( member @ nat @ ( zero_zero @ nat ) @ A4 ) ) ) ) ).
% even_set_encode_iff
thf(fact_1462_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_1463_one__mod__2__pow__eq,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [N2: nat] :
( ( modulo_modulo @ A @ ( one_one @ A ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) )
= ( zero_neq_one_of_bool @ A @ ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ) ).
% one_mod_2_pow_eq
thf(fact_1464_diff__self,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A] :
( ( minus_minus @ A @ A3 @ A3 )
= ( zero_zero @ A ) ) ) ).
% diff_self
thf(fact_1465_diff__0__right,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A] :
( ( minus_minus @ A @ A3 @ ( zero_zero @ A ) )
= A3 ) ) ).
% diff_0_right
thf(fact_1466_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_1467_diff__zero,axiom,
! [A: $tType] :
( ( cancel1802427076303600483id_add @ A )
=> ! [A3: A] :
( ( minus_minus @ A @ A3 @ ( zero_zero @ A ) )
= A3 ) ) ).
% diff_zero
thf(fact_1468_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_1469_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_1470_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_1471_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_1472_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_1473_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_1474_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_1475_diff__Suc__Suc,axiom,
! [M2: nat,N2: nat] :
( ( minus_minus @ nat @ ( suc @ M2 ) @ ( suc @ N2 ) )
= ( minus_minus @ nat @ M2 @ N2 ) ) ).
% diff_Suc_Suc
thf(fact_1476_Suc__diff__diff,axiom,
! [M2: nat,N2: nat,K: nat] :
( ( minus_minus @ nat @ ( minus_minus @ nat @ ( suc @ M2 ) @ N2 ) @ ( suc @ K ) )
= ( minus_minus @ nat @ ( minus_minus @ nat @ M2 @ N2 ) @ K ) ) ).
% Suc_diff_diff
thf(fact_1477_diff__0__eq__0,axiom,
! [N2: nat] :
( ( minus_minus @ nat @ ( zero_zero @ nat ) @ N2 )
= ( zero_zero @ nat ) ) ).
% diff_0_eq_0
thf(fact_1478_diff__self__eq__0,axiom,
! [M2: nat] :
( ( minus_minus @ nat @ M2 @ M2 )
= ( zero_zero @ nat ) ) ).
% diff_self_eq_0
thf(fact_1479_diff__diff__cancel,axiom,
! [I: nat,N2: nat] :
( ( ord_less_eq @ nat @ I @ N2 )
=> ( ( minus_minus @ nat @ N2 @ ( minus_minus @ nat @ N2 @ I ) )
= I ) ) ).
% diff_diff_cancel
thf(fact_1480_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_1481_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_1482_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_1483_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_1484_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_1485_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_1486_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_1487_signed__take__bit__of__0,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N2: nat] :
( ( bit_ri4674362597316999326ke_bit @ A @ N2 @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% signed_take_bit_of_0
thf(fact_1488_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_1489_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_1490_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_1491_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_1492_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_1493_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_1494_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_1495_signed__take__bit__Suc__bit0,axiom,
! [N2: nat,K: num] :
( ( bit_ri4674362597316999326ke_bit @ int @ ( suc @ N2 ) @ ( numeral_numeral @ int @ ( bit0 @ K ) ) )
= ( times_times @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N2 @ ( numeral_numeral @ int @ K ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ).
% signed_take_bit_Suc_bit0
thf(fact_1496_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_1497_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_1498_zero__less__diff,axiom,
! [N2: nat,M2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ N2 @ M2 ) )
= ( ord_less @ nat @ M2 @ N2 ) ) ).
% zero_less_diff
thf(fact_1499_diff__is__0__eq_H,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq @ nat @ M2 @ N2 )
=> ( ( minus_minus @ nat @ M2 @ N2 )
= ( zero_zero @ nat ) ) ) ).
% diff_is_0_eq'
thf(fact_1500_diff__is__0__eq,axiom,
! [M2: nat,N2: nat] :
( ( ( minus_minus @ nat @ M2 @ N2 )
= ( zero_zero @ nat ) )
= ( ord_less_eq @ nat @ M2 @ N2 ) ) ).
% diff_is_0_eq
thf(fact_1501_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_1502_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_1503_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_1504_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_1505_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_1506_diff__Suc__1,axiom,
! [N2: nat] :
( ( minus_minus @ nat @ ( suc @ N2 ) @ ( one_one @ nat ) )
= N2 ) ).
% diff_Suc_1
thf(fact_1507_Suc__0__mod__eq,axiom,
! [N2: nat] :
( ( modulo_modulo @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N2 )
= ( zero_neq_one_of_bool @ nat
@ ( N2
!= ( suc @ ( zero_zero @ nat ) ) ) ) ) ).
% Suc_0_mod_eq
thf(fact_1508_signed__take__bit__Suc__1,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N2: nat] :
( ( bit_ri4674362597316999326ke_bit @ A @ ( suc @ N2 ) @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% signed_take_bit_Suc_1
thf(fact_1509_set__encode__empty,axiom,
( ( nat_set_encode @ ( bot_bot @ ( set @ nat ) ) )
= ( zero_zero @ nat ) ) ).
% set_encode_empty
thf(fact_1510_set__encode__inverse,axiom,
! [A4: set @ nat] :
( ( finite_finite2 @ nat @ A4 )
=> ( ( nat_set_decode @ ( nat_set_encode @ A4 ) )
= A4 ) ) ).
% set_encode_inverse
thf(fact_1511_Suc__pred,axiom,
! [N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( suc @ ( minus_minus @ nat @ N2 @ ( suc @ ( zero_zero @ nat ) ) ) )
= N2 ) ) ).
% Suc_pred
thf(fact_1512_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_1513_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_1514_Suc__diff__1,axiom,
! [N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( suc @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) )
= N2 ) ) ).
% Suc_diff_1
thf(fact_1515_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_1516_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_1517_odd__Suc__minus__one,axiom,
! [N2: nat] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
=> ( ( suc @ ( minus_minus @ nat @ N2 @ ( suc @ ( zero_zero @ nat ) ) ) )
= N2 ) ) ).
% odd_Suc_minus_one
thf(fact_1518_even__diff__nat,axiom,
! [M2: nat,N2: nat] :
( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ M2 @ N2 ) )
= ( ( ord_less @ nat @ M2 @ N2 )
| ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( plus_plus @ nat @ M2 @ N2 ) ) ) ) ).
% even_diff_nat
thf(fact_1519_semiring__parity__class_Oeven__mask__iff,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [N2: nat] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) @ ( one_one @ A ) ) )
= ( N2
= ( zero_zero @ nat ) ) ) ) ).
% semiring_parity_class.even_mask_iff
thf(fact_1520_bits__1__div__exp,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [N2: nat] :
( ( divide_divide @ A @ ( one_one @ A ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) )
= ( zero_neq_one_of_bool @ A
@ ( N2
= ( zero_zero @ nat ) ) ) ) ) ).
% bits_1_div_exp
thf(fact_1521_one__div__2__pow__eq,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [N2: nat] :
( ( divide_divide @ A @ ( one_one @ A ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) )
= ( zero_neq_one_of_bool @ A
@ ( N2
= ( zero_zero @ nat ) ) ) ) ) ).
% one_div_2_pow_eq
thf(fact_1522_of__bool__eq__iff,axiom,
! [A: $tType] :
( ( zero_neq_one @ A )
=> ! [P5: $o,Q3: $o] :
( ( ( zero_neq_one_of_bool @ A @ P5 )
= ( zero_neq_one_of_bool @ A @ Q3 ) )
= ( P5 = Q3 ) ) ) ).
% of_bool_eq_iff
thf(fact_1523_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_1524_diff__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 ) ) ) ).
% diff_right_commute
thf(fact_1525_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_1526_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_1527_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_1528_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_1529_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_1530_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_1531_eq__iff__diff__eq__0,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ( ( ^ [Y5: A,Z2: A] : Y5 = Z2 )
= ( ^ [A5: A,B4: A] :
( ( minus_minus @ A @ A5 @ B4 )
= ( zero_zero @ A ) ) ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_1532_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_1533_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_1534_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_1535_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_1536_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_1537_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_1538_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_1539_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_1540_group__cancel_Osub1,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A4: A,K: A,A3: A,B3: A] :
( ( A4
= ( plus_plus @ A @ K @ A3 ) )
=> ( ( minus_minus @ A @ A4 @ B3 )
= ( plus_plus @ A @ K @ ( minus_minus @ A @ A3 @ B3 ) ) ) ) ) ).
% group_cancel.sub1
thf(fact_1541_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_1542_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_1543_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_1544_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_1545_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_1546_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_1547_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_1548_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_1549_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_1550_diff__divide__distrib,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A,B3: A,C2: A] :
( ( divide_divide @ A @ ( minus_minus @ A @ A3 @ B3 ) @ C2 )
= ( minus_minus @ A @ ( divide_divide @ A @ A3 @ C2 ) @ ( divide_divide @ A @ B3 @ C2 ) ) ) ) ).
% diff_divide_distrib
thf(fact_1551_dvd__diff,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [X: A,Y: A,Z3: A] :
( ( dvd_dvd @ A @ X @ Y )
=> ( ( dvd_dvd @ A @ X @ Z3 )
=> ( dvd_dvd @ A @ X @ ( minus_minus @ A @ Y @ Z3 ) ) ) ) ) ).
% dvd_diff
thf(fact_1552_zero__induct__lemma,axiom,
! [P: nat > $o,K: nat,I: nat] :
( ( P @ K )
=> ( ! [N3: nat] :
( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) )
=> ( P @ ( minus_minus @ nat @ K @ I ) ) ) ) ).
% zero_induct_lemma
thf(fact_1553_minus__nat_Odiff__0,axiom,
! [M2: nat] :
( ( minus_minus @ nat @ M2 @ ( zero_zero @ nat ) )
= M2 ) ).
% minus_nat.diff_0
thf(fact_1554_diffs0__imp__equal,axiom,
! [M2: nat,N2: nat] :
( ( ( minus_minus @ nat @ M2 @ N2 )
= ( zero_zero @ nat ) )
=> ( ( ( minus_minus @ nat @ N2 @ M2 )
= ( zero_zero @ nat ) )
=> ( M2 = N2 ) ) ) ).
% diffs0_imp_equal
thf(fact_1555_diff__less__mono2,axiom,
! [M2: nat,N2: nat,L: nat] :
( ( ord_less @ nat @ M2 @ N2 )
=> ( ( ord_less @ nat @ M2 @ L )
=> ( ord_less @ nat @ ( minus_minus @ nat @ L @ N2 ) @ ( minus_minus @ nat @ L @ M2 ) ) ) ) ).
% diff_less_mono2
thf(fact_1556_less__imp__diff__less,axiom,
! [J: nat,K: nat,N2: nat] :
( ( ord_less @ nat @ J @ K )
=> ( ord_less @ nat @ ( minus_minus @ nat @ J @ N2 ) @ K ) ) ).
% less_imp_diff_less
thf(fact_1557_eq__diff__iff,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ord_less_eq @ nat @ K @ M2 )
=> ( ( ord_less_eq @ nat @ K @ N2 )
=> ( ( ( minus_minus @ nat @ M2 @ K )
= ( minus_minus @ nat @ N2 @ K ) )
= ( M2 = N2 ) ) ) ) ).
% eq_diff_iff
thf(fact_1558_le__diff__iff,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ord_less_eq @ nat @ K @ M2 )
=> ( ( ord_less_eq @ nat @ K @ N2 )
=> ( ( ord_less_eq @ nat @ ( minus_minus @ nat @ M2 @ K ) @ ( minus_minus @ nat @ N2 @ K ) )
= ( ord_less_eq @ nat @ M2 @ N2 ) ) ) ) ).
% le_diff_iff
thf(fact_1559_Nat_Odiff__diff__eq,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ord_less_eq @ nat @ K @ M2 )
=> ( ( ord_less_eq @ nat @ K @ N2 )
=> ( ( minus_minus @ nat @ ( minus_minus @ nat @ M2 @ K ) @ ( minus_minus @ nat @ N2 @ K ) )
= ( minus_minus @ nat @ M2 @ N2 ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_1560_diff__le__mono,axiom,
! [M2: nat,N2: nat,L: nat] :
( ( ord_less_eq @ nat @ M2 @ N2 )
=> ( ord_less_eq @ nat @ ( minus_minus @ nat @ M2 @ L ) @ ( minus_minus @ nat @ N2 @ L ) ) ) ).
% diff_le_mono
thf(fact_1561_diff__le__self,axiom,
! [M2: nat,N2: nat] : ( ord_less_eq @ nat @ ( minus_minus @ nat @ M2 @ N2 ) @ M2 ) ).
% diff_le_self
thf(fact_1562_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_1563_diff__le__mono2,axiom,
! [M2: nat,N2: nat,L: nat] :
( ( ord_less_eq @ nat @ M2 @ N2 )
=> ( ord_less_eq @ nat @ ( minus_minus @ nat @ L @ N2 ) @ ( minus_minus @ nat @ L @ M2 ) ) ) ).
% diff_le_mono2
thf(fact_1564_Nat_Odiff__cancel,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( minus_minus @ nat @ ( plus_plus @ nat @ K @ M2 ) @ ( plus_plus @ nat @ K @ N2 ) )
= ( minus_minus @ nat @ M2 @ N2 ) ) ).
% Nat.diff_cancel
thf(fact_1565_diff__cancel2,axiom,
! [M2: nat,K: nat,N2: nat] :
( ( minus_minus @ nat @ ( plus_plus @ nat @ M2 @ K ) @ ( plus_plus @ nat @ N2 @ K ) )
= ( minus_minus @ nat @ M2 @ N2 ) ) ).
% diff_cancel2
thf(fact_1566_diff__add__inverse,axiom,
! [N2: nat,M2: nat] :
( ( minus_minus @ nat @ ( plus_plus @ nat @ N2 @ M2 ) @ N2 )
= M2 ) ).
% diff_add_inverse
thf(fact_1567_diff__add__inverse2,axiom,
! [M2: nat,N2: nat] :
( ( minus_minus @ nat @ ( plus_plus @ nat @ M2 @ N2 ) @ N2 )
= M2 ) ).
% diff_add_inverse2
thf(fact_1568_diff__mult__distrib2,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( times_times @ nat @ K @ ( minus_minus @ nat @ M2 @ N2 ) )
= ( minus_minus @ nat @ ( times_times @ nat @ K @ M2 ) @ ( times_times @ nat @ K @ N2 ) ) ) ).
% diff_mult_distrib2
thf(fact_1569_diff__mult__distrib,axiom,
! [M2: nat,N2: nat,K: nat] :
( ( times_times @ nat @ ( minus_minus @ nat @ M2 @ N2 ) @ K )
= ( minus_minus @ nat @ ( times_times @ nat @ M2 @ K ) @ ( times_times @ nat @ N2 @ K ) ) ) ).
% diff_mult_distrib
thf(fact_1570_max__diff__distrib__left,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [X: A,Y: A,Z3: A] :
( ( minus_minus @ A @ ( ord_max @ A @ X @ Y ) @ Z3 )
= ( ord_max @ A @ ( minus_minus @ A @ X @ Z3 ) @ ( minus_minus @ A @ Y @ Z3 ) ) ) ) ).
% max_diff_distrib_left
thf(fact_1571_dvd__diff__nat,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( dvd_dvd @ nat @ K @ M2 )
=> ( ( dvd_dvd @ nat @ K @ N2 )
=> ( dvd_dvd @ nat @ K @ ( minus_minus @ nat @ M2 @ N2 ) ) ) ) ).
% dvd_diff_nat
thf(fact_1572_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_1573_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_1574_split__of__bool__asm,axiom,
! [A: $tType] :
( ( zero_neq_one @ A )
=> ! [P: A > $o,P5: $o] :
( ( P @ ( zero_neq_one_of_bool @ A @ P5 ) )
= ( ~ ( ( P5
& ~ ( P @ ( one_one @ A ) ) )
| ( ~ P5
& ~ ( P @ ( zero_zero @ A ) ) ) ) ) ) ) ).
% split_of_bool_asm
thf(fact_1575_split__of__bool,axiom,
! [A: $tType] :
( ( zero_neq_one @ A )
=> ! [P: A > $o,P5: $o] :
( ( P @ ( zero_neq_one_of_bool @ A @ P5 ) )
= ( ( P5
=> ( P @ ( one_one @ A ) ) )
& ( ~ P5
=> ( P @ ( zero_zero @ A ) ) ) ) ) ) ).
% split_of_bool
thf(fact_1576_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_1577_le__iff__diff__le__0,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A5: A,B4: A] : ( ord_less_eq @ A @ ( minus_minus @ A @ A5 @ B4 ) @ ( zero_zero @ A ) ) ) ) ) ).
% le_iff_diff_le_0
thf(fact_1578_less__iff__diff__less__0,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ( ( ord_less @ A )
= ( ^ [A5: A,B4: A] : ( ord_less @ A @ ( minus_minus @ A @ A5 @ B4 ) @ ( zero_zero @ A ) ) ) ) ) ).
% less_iff_diff_less_0
thf(fact_1579_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_1580_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_1581_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_1582_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_1583_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_1584_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_1585_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_1586_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_1587_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_1588_diff__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 ) ) ) ).
% diff_add
thf(fact_1589_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_1590_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_1591_add__le__add__imp__diff__le,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [I: A,K: A,N2: A,J: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ N2 )
=> ( ( ord_less_eq @ A @ N2 @ ( plus_plus @ A @ J @ K ) )
=> ( ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ N2 )
=> ( ( ord_less_eq @ A @ N2 @ ( plus_plus @ A @ J @ K ) )
=> ( ord_less_eq @ A @ ( minus_minus @ A @ N2 @ K ) @ J ) ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_1592_add__le__imp__le__diff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [I: A,K: A,N2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ N2 )
=> ( ord_less_eq @ A @ I @ ( minus_minus @ A @ N2 @ K ) ) ) ) ).
% add_le_imp_le_diff
thf(fact_1593_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_1594_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_1595_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_1596_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_1597_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_1598_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_1599_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_1600_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_1601_diff__less__Suc,axiom,
! [M2: nat,N2: nat] : ( ord_less @ nat @ ( minus_minus @ nat @ M2 @ N2 ) @ ( suc @ M2 ) ) ).
% diff_less_Suc
thf(fact_1602_Suc__diff__Suc,axiom,
! [N2: nat,M2: nat] :
( ( ord_less @ nat @ N2 @ M2 )
=> ( ( suc @ ( minus_minus @ nat @ M2 @ ( suc @ N2 ) ) )
= ( minus_minus @ nat @ M2 @ N2 ) ) ) ).
% Suc_diff_Suc
thf(fact_1603_diff__less,axiom,
! [N2: nat,M2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M2 )
=> ( ord_less @ nat @ ( minus_minus @ nat @ M2 @ N2 ) @ M2 ) ) ) ).
% diff_less
thf(fact_1604_Suc__diff__le,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_eq @ nat @ N2 @ M2 )
=> ( ( minus_minus @ nat @ ( suc @ M2 ) @ N2 )
= ( suc @ ( minus_minus @ nat @ M2 @ N2 ) ) ) ) ).
% Suc_diff_le
thf(fact_1605_less__diff__iff,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ord_less_eq @ nat @ K @ M2 )
=> ( ( ord_less_eq @ nat @ K @ N2 )
=> ( ( ord_less @ nat @ ( minus_minus @ nat @ M2 @ K ) @ ( minus_minus @ nat @ N2 @ K ) )
= ( ord_less @ nat @ M2 @ N2 ) ) ) ) ).
% less_diff_iff
thf(fact_1606_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_1607_diff__add__0,axiom,
! [N2: nat,M2: nat] :
( ( minus_minus @ nat @ N2 @ ( plus_plus @ nat @ N2 @ M2 ) )
= ( zero_zero @ nat ) ) ).
% diff_add_0
thf(fact_1608_set__encode__eq,axiom,
! [A4: set @ nat,B2: set @ nat] :
( ( finite_finite2 @ nat @ A4 )
=> ( ( finite_finite2 @ nat @ B2 )
=> ( ( ( nat_set_encode @ A4 )
= ( nat_set_encode @ B2 ) )
= ( A4 = B2 ) ) ) ) ).
% set_encode_eq
thf(fact_1609_add__diff__inverse__nat,axiom,
! [M2: nat,N2: nat] :
( ~ ( ord_less @ nat @ M2 @ N2 )
=> ( ( plus_plus @ nat @ N2 @ ( minus_minus @ nat @ M2 @ N2 ) )
= M2 ) ) ).
% add_diff_inverse_nat
thf(fact_1610_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_1611_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_1612_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_1613_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_1614_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_1615_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_1616_diff__Suc__eq__diff__pred,axiom,
! [M2: nat,N2: nat] :
( ( minus_minus @ nat @ M2 @ ( suc @ N2 ) )
= ( minus_minus @ nat @ ( minus_minus @ nat @ M2 @ ( one_one @ nat ) ) @ N2 ) ) ).
% diff_Suc_eq_diff_pred
thf(fact_1617_dvd__minus__self,axiom,
! [M2: nat,N2: nat] :
( ( dvd_dvd @ nat @ M2 @ ( minus_minus @ nat @ N2 @ M2 ) )
= ( ( ord_less @ nat @ N2 @ M2 )
| ( dvd_dvd @ nat @ M2 @ N2 ) ) ) ).
% dvd_minus_self
thf(fact_1618_less__eq__dvd__minus,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq @ nat @ M2 @ N2 )
=> ( ( dvd_dvd @ nat @ M2 @ N2 )
= ( dvd_dvd @ nat @ M2 @ ( minus_minus @ nat @ N2 @ M2 ) ) ) ) ).
% less_eq_dvd_minus
thf(fact_1619_dvd__diffD1,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( dvd_dvd @ nat @ K @ ( minus_minus @ nat @ M2 @ N2 ) )
=> ( ( dvd_dvd @ nat @ K @ M2 )
=> ( ( ord_less_eq @ nat @ N2 @ M2 )
=> ( dvd_dvd @ nat @ K @ N2 ) ) ) ) ).
% dvd_diffD1
thf(fact_1620_dvd__diffD,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( dvd_dvd @ nat @ K @ ( minus_minus @ nat @ M2 @ N2 ) )
=> ( ( dvd_dvd @ nat @ K @ N2 )
=> ( ( ord_less_eq @ nat @ N2 @ M2 )
=> ( dvd_dvd @ nat @ K @ M2 ) ) ) ) ).
% dvd_diffD
thf(fact_1621_mod__geq,axiom,
! [M2: nat,N2: nat] :
( ~ ( ord_less @ nat @ M2 @ N2 )
=> ( ( modulo_modulo @ nat @ M2 @ N2 )
= ( modulo_modulo @ nat @ ( minus_minus @ nat @ M2 @ N2 ) @ N2 ) ) ) ).
% mod_geq
thf(fact_1622_mod__if,axiom,
( ( modulo_modulo @ nat )
= ( ^ [M: nat,N: nat] : ( if @ nat @ ( ord_less @ nat @ M @ N ) @ M @ ( modulo_modulo @ nat @ ( minus_minus @ nat @ M @ N ) @ N ) ) ) ) ).
% mod_if
thf(fact_1623_le__mod__geq,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_eq @ nat @ N2 @ M2 )
=> ( ( modulo_modulo @ nat @ M2 @ N2 )
= ( modulo_modulo @ nat @ ( minus_minus @ nat @ M2 @ N2 ) @ N2 ) ) ) ).
% le_mod_geq
thf(fact_1624_nat__minus__add__max,axiom,
! [N2: nat,M2: nat] :
( ( plus_plus @ nat @ ( minus_minus @ nat @ N2 @ M2 ) @ M2 )
= ( ord_max @ nat @ N2 @ M2 ) ) ).
% nat_minus_add_max
thf(fact_1625_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_1626_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_1627_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_1628_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_1629_divide__diff__eq__iff,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z3: A,X: A,Y: A] :
( ( Z3
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( divide_divide @ A @ X @ Z3 ) @ Y )
= ( divide_divide @ A @ ( minus_minus @ A @ X @ ( times_times @ A @ Y @ Z3 ) ) @ Z3 ) ) ) ) ).
% divide_diff_eq_iff
thf(fact_1630_diff__divide__eq__iff,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z3: A,X: A,Y: A] :
( ( Z3
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ X @ ( divide_divide @ A @ Y @ Z3 ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ X @ Z3 ) @ Y ) @ Z3 ) ) ) ) ).
% diff_divide_eq_iff
thf(fact_1631_diff__frac__eq,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [Y: A,Z3: A,X: A,W2: A] :
( ( Y
!= ( zero_zero @ A ) )
=> ( ( Z3
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( divide_divide @ A @ X @ Y ) @ ( divide_divide @ A @ W2 @ Z3 ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ X @ Z3 ) @ ( times_times @ A @ W2 @ Y ) ) @ ( times_times @ A @ Y @ Z3 ) ) ) ) ) ) ).
% diff_frac_eq
thf(fact_1632_add__divide__eq__if__simps_I4_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z3: A,A3: A,B3: A] :
( ( ( Z3
= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ A3 @ ( divide_divide @ A @ B3 @ Z3 ) )
= A3 ) )
& ( ( Z3
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ A3 @ ( divide_divide @ A @ B3 @ Z3 ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ A3 @ Z3 ) @ B3 ) @ Z3 ) ) ) ) ) ).
% add_divide_eq_if_simps(4)
thf(fact_1633_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_1634_inf__period_I4_J,axiom,
! [A: $tType] :
( ( ( comm_ring @ A )
& ( dvd @ A ) )
=> ! [D2: A,D6: A,T2: A] :
( ( dvd_dvd @ A @ D2 @ D6 )
=> ! [X2: A,K4: A] :
( ( ~ ( dvd_dvd @ A @ D2 @ ( plus_plus @ A @ X2 @ T2 ) ) )
= ( ~ ( dvd_dvd @ A @ D2 @ ( plus_plus @ A @ ( minus_minus @ A @ X2 @ ( times_times @ A @ K4 @ D6 ) ) @ T2 ) ) ) ) ) ) ).
% inf_period(4)
thf(fact_1635_inf__period_I3_J,axiom,
! [A: $tType] :
( ( ( comm_ring @ A )
& ( dvd @ A ) )
=> ! [D2: A,D6: A,T2: A] :
( ( dvd_dvd @ A @ D2 @ D6 )
=> ! [X2: A,K4: A] :
( ( dvd_dvd @ A @ D2 @ ( plus_plus @ A @ X2 @ T2 ) )
= ( dvd_dvd @ A @ D2 @ ( plus_plus @ A @ ( minus_minus @ A @ X2 @ ( times_times @ A @ K4 @ D6 ) ) @ T2 ) ) ) ) ) ).
% inf_period(3)
thf(fact_1636_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_1637_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_1638_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_1639_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_1640_int__power__div__base,axiom,
! [M2: nat,K: int] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ M2 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ K )
=> ( ( divide_divide @ int @ ( power_power @ int @ K @ M2 ) @ K )
= ( power_power @ int @ K @ ( minus_minus @ nat @ M2 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ).
% int_power_div_base
thf(fact_1641_diff__Suc__less,axiom,
! [N2: nat,I: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ord_less @ nat @ ( minus_minus @ nat @ N2 @ ( suc @ I ) ) @ N2 ) ) ).
% diff_Suc_less
thf(fact_1642_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_1643_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_1644_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_1645_nat__diff__add__eq2,axiom,
! [I: nat,J: nat,U: nat,M2: nat,N2: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( minus_minus @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ I @ U ) @ M2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ N2 ) )
= ( minus_minus @ nat @ M2 @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ J @ I ) @ U ) @ N2 ) ) ) ) ).
% nat_diff_add_eq2
thf(fact_1646_nat__diff__add__eq1,axiom,
! [J: nat,I: nat,U: nat,M2: nat,N2: nat] :
( ( ord_less_eq @ nat @ J @ I )
=> ( ( minus_minus @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ I @ U ) @ M2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ N2 ) )
= ( minus_minus @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ I @ J ) @ U ) @ M2 ) @ N2 ) ) ) ).
% nat_diff_add_eq1
thf(fact_1647_nat__le__add__iff2,axiom,
! [I: nat,J: nat,U: nat,M2: nat,N2: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ I @ U ) @ M2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ N2 ) )
= ( ord_less_eq @ nat @ M2 @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ J @ I ) @ U ) @ N2 ) ) ) ) ).
% nat_le_add_iff2
thf(fact_1648_nat__le__add__iff1,axiom,
! [J: nat,I: nat,U: nat,M2: nat,N2: nat] :
( ( ord_less_eq @ nat @ J @ I )
=> ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ I @ U ) @ M2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ N2 ) )
= ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ I @ J ) @ U ) @ M2 ) @ N2 ) ) ) ).
% nat_le_add_iff1
thf(fact_1649_nat__eq__add__iff2,axiom,
! [I: nat,J: nat,U: nat,M2: nat,N2: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( ( plus_plus @ nat @ ( times_times @ nat @ I @ U ) @ M2 )
= ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ N2 ) )
= ( M2
= ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ J @ I ) @ U ) @ N2 ) ) ) ) ).
% nat_eq_add_iff2
thf(fact_1650_nat__eq__add__iff1,axiom,
! [J: nat,I: nat,U: nat,M2: nat,N2: nat] :
( ( ord_less_eq @ nat @ J @ I )
=> ( ( ( plus_plus @ nat @ ( times_times @ nat @ I @ U ) @ M2 )
= ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ N2 ) )
= ( ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ I @ J ) @ U ) @ M2 )
= N2 ) ) ) ).
% nat_eq_add_iff1
thf(fact_1651_mod__eq__dvd__iff__nat,axiom,
! [N2: nat,M2: nat,Q3: nat] :
( ( ord_less_eq @ nat @ N2 @ M2 )
=> ( ( ( modulo_modulo @ nat @ M2 @ Q3 )
= ( modulo_modulo @ nat @ N2 @ Q3 ) )
= ( dvd_dvd @ nat @ Q3 @ ( minus_minus @ nat @ M2 @ N2 ) ) ) ) ).
% mod_eq_dvd_iff_nat
thf(fact_1652_set__encode__inf,axiom,
! [A4: set @ nat] :
( ~ ( finite_finite2 @ nat @ A4 )
=> ( ( nat_set_encode @ A4 )
= ( zero_zero @ nat ) ) ) ).
% set_encode_inf
thf(fact_1653_exp__div__exp__eq,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [M2: nat,N2: nat] :
( ( divide_divide @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M2 ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) )
= ( times_times @ A
@ ( zero_neq_one_of_bool @ A
@ ( ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M2 )
!= ( zero_zero @ A ) )
& ( ord_less_eq @ nat @ N2 @ M2 ) ) )
@ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ M2 @ N2 ) ) ) ) ) ).
% exp_div_exp_eq
thf(fact_1654_frac__le__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [Y: A,Z3: A,X: A,W2: A] :
( ( Y
!= ( zero_zero @ A ) )
=> ( ( Z3
!= ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( divide_divide @ A @ X @ Y ) @ ( divide_divide @ A @ W2 @ Z3 ) )
= ( ord_less_eq @ A @ ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ X @ Z3 ) @ ( times_times @ A @ W2 @ Y ) ) @ ( times_times @ A @ Y @ Z3 ) ) @ ( zero_zero @ A ) ) ) ) ) ) ).
% frac_le_eq
thf(fact_1655_frac__less__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [Y: A,Z3: A,X: A,W2: A] :
( ( Y
!= ( zero_zero @ A ) )
=> ( ( Z3
!= ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( divide_divide @ A @ X @ Y ) @ ( divide_divide @ A @ W2 @ Z3 ) )
= ( ord_less @ A @ ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ X @ Z3 ) @ ( times_times @ A @ W2 @ Y ) ) @ ( times_times @ A @ Y @ Z3 ) ) @ ( zero_zero @ A ) ) ) ) ) ) ).
% frac_less_eq
thf(fact_1656_power__diff,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [A3: A,N2: nat,M2: nat] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( ord_less_eq @ nat @ N2 @ M2 )
=> ( ( power_power @ A @ A3 @ ( minus_minus @ nat @ M2 @ N2 ) )
= ( divide_divide @ A @ ( power_power @ A @ A3 @ M2 ) @ ( power_power @ A @ A3 @ N2 ) ) ) ) ) ) ).
% power_diff
thf(fact_1657_Suc__diff__eq__diff__pred,axiom,
! [N2: nat,M2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( minus_minus @ nat @ ( suc @ M2 ) @ N2 )
= ( minus_minus @ nat @ M2 @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) ) ) ) ).
% Suc_diff_eq_diff_pred
thf(fact_1658_Suc__pred_H,axiom,
! [N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( N2
= ( suc @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) ) ) ) ).
% Suc_pred'
thf(fact_1659_div__geq,axiom,
! [N2: nat,M2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ~ ( ord_less @ nat @ M2 @ N2 )
=> ( ( divide_divide @ nat @ M2 @ N2 )
= ( suc @ ( divide_divide @ nat @ ( minus_minus @ nat @ M2 @ N2 ) @ N2 ) ) ) ) ) ).
% div_geq
thf(fact_1660_div__if,axiom,
( ( divide_divide @ nat )
= ( ^ [M: nat,N: nat] :
( if @ nat
@ ( ( ord_less @ nat @ M @ N )
| ( N
= ( zero_zero @ nat ) ) )
@ ( zero_zero @ nat )
@ ( suc @ ( divide_divide @ nat @ ( minus_minus @ nat @ M @ N ) @ N ) ) ) ) ) ).
% div_if
thf(fact_1661_add__eq__if,axiom,
( ( plus_plus @ nat )
= ( ^ [M: nat,N: nat] :
( if @ nat
@ ( M
= ( zero_zero @ nat ) )
@ N
@ ( suc @ ( plus_plus @ nat @ ( minus_minus @ nat @ M @ ( one_one @ nat ) ) @ N ) ) ) ) ) ).
% add_eq_if
thf(fact_1662_nat__less__add__iff1,axiom,
! [J: nat,I: nat,U: nat,M2: nat,N2: nat] :
( ( ord_less_eq @ nat @ J @ I )
=> ( ( ord_less @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ I @ U ) @ M2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ N2 ) )
= ( ord_less @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ I @ J ) @ U ) @ M2 ) @ N2 ) ) ) ).
% nat_less_add_iff1
thf(fact_1663_nat__less__add__iff2,axiom,
! [I: nat,J: nat,U: nat,M2: nat,N2: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( ord_less @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ I @ U ) @ M2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ N2 ) )
= ( ord_less @ nat @ M2 @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ J @ I ) @ U ) @ N2 ) ) ) ) ).
% nat_less_add_iff2
thf(fact_1664_mult__eq__if,axiom,
( ( times_times @ nat )
= ( ^ [M: nat,N: nat] :
( if @ nat
@ ( M
= ( zero_zero @ nat ) )
@ ( zero_zero @ nat )
@ ( plus_plus @ nat @ N @ ( times_times @ nat @ ( minus_minus @ nat @ M @ ( one_one @ nat ) ) @ N ) ) ) ) ) ).
% mult_eq_if
thf(fact_1665_dvd__minus__add,axiom,
! [Q3: nat,N2: nat,R2: nat,M2: nat] :
( ( ord_less_eq @ nat @ Q3 @ N2 )
=> ( ( ord_less_eq @ nat @ Q3 @ ( times_times @ nat @ R2 @ M2 ) )
=> ( ( dvd_dvd @ nat @ M2 @ ( minus_minus @ nat @ N2 @ Q3 ) )
= ( dvd_dvd @ nat @ M2 @ ( plus_plus @ nat @ N2 @ ( minus_minus @ nat @ ( times_times @ nat @ R2 @ M2 ) @ Q3 ) ) ) ) ) ) ).
% dvd_minus_add
thf(fact_1666_mod__nat__eqI,axiom,
! [R2: nat,N2: nat,M2: nat] :
( ( ord_less @ nat @ R2 @ N2 )
=> ( ( ord_less_eq @ nat @ R2 @ M2 )
=> ( ( dvd_dvd @ nat @ N2 @ ( minus_minus @ nat @ M2 @ R2 ) )
=> ( ( modulo_modulo @ nat @ M2 @ N2 )
= R2 ) ) ) ) ).
% mod_nat_eqI
thf(fact_1667_scaling__mono,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [U: A,V2: A,R2: A,S: A] :
( ( ord_less_eq @ A @ U @ V2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ R2 )
=> ( ( ord_less_eq @ A @ R2 @ S )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ U @ ( divide_divide @ A @ ( times_times @ A @ R2 @ ( minus_minus @ A @ V2 @ U ) ) @ S ) ) @ V2 ) ) ) ) ) ).
% scaling_mono
thf(fact_1668_exp__not__zero__imp__exp__diff__not__zero,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [N2: nat,M2: nat] :
( ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 )
!= ( zero_zero @ A ) )
=> ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ N2 @ M2 ) )
!= ( zero_zero @ A ) ) ) ) ).
% exp_not_zero_imp_exp_diff_not_zero
thf(fact_1669_power__diff__power__eq,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A3: A,N2: nat,M2: nat] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( ( ord_less_eq @ nat @ N2 @ M2 )
=> ( ( divide_divide @ A @ ( power_power @ A @ A3 @ M2 ) @ ( power_power @ A @ A3 @ N2 ) )
= ( power_power @ A @ A3 @ ( minus_minus @ nat @ M2 @ N2 ) ) ) )
& ( ~ ( ord_less_eq @ nat @ N2 @ M2 )
=> ( ( divide_divide @ A @ ( power_power @ A @ A3 @ M2 ) @ ( power_power @ A @ A3 @ N2 ) )
= ( divide_divide @ A @ ( one_one @ A ) @ ( power_power @ A @ A3 @ ( minus_minus @ nat @ N2 @ M2 ) ) ) ) ) ) ) ) ).
% power_diff_power_eq
thf(fact_1670_power__eq__if,axiom,
! [A: $tType] :
( ( power @ A )
=> ( ( power_power @ A )
= ( ^ [P6: A,M: nat] :
( if @ A
@ ( M
= ( zero_zero @ nat ) )
@ ( one_one @ A )
@ ( times_times @ A @ P6 @ ( power_power @ A @ P6 @ ( minus_minus @ nat @ M @ ( one_one @ nat ) ) ) ) ) ) ) ) ).
% power_eq_if
thf(fact_1671_power__minus__mult,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [N2: nat,A3: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( times_times @ A @ ( power_power @ A @ A3 @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) ) @ A3 )
= ( power_power @ A @ A3 @ N2 ) ) ) ) ).
% power_minus_mult
thf(fact_1672_diff__le__diff__pow,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K )
=> ( ord_less_eq @ nat @ ( minus_minus @ nat @ M2 @ N2 ) @ ( minus_minus @ nat @ ( power_power @ nat @ K @ M2 ) @ ( power_power @ nat @ K @ N2 ) ) ) ) ).
% diff_le_diff_pow
thf(fact_1673_le__div__geq,axiom,
! [N2: nat,M2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( ord_less_eq @ nat @ N2 @ M2 )
=> ( ( divide_divide @ nat @ M2 @ N2 )
= ( suc @ ( divide_divide @ nat @ ( minus_minus @ nat @ M2 @ N2 ) @ N2 ) ) ) ) ) ).
% le_div_geq
thf(fact_1674_num_Osize__gen_I1_J,axiom,
( ( size_num @ one2 )
= ( zero_zero @ nat ) ) ).
% num.size_gen(1)
thf(fact_1675_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,B5: $o] :
( ( P @ A6 )
=> ( ( ( divide_divide @ A @ ( plus_plus @ A @ ( zero_neq_one_of_bool @ A @ B5 ) @ ( 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 @ B5 ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A6 ) ) ) ) )
=> ( P @ A3 ) ) ) ) ).
% bits_induct
thf(fact_1676_exp__mod__exp,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [M2: nat,N2: nat] :
( ( modulo_modulo @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M2 ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) )
= ( times_times @ A @ ( zero_neq_one_of_bool @ A @ ( ord_less @ nat @ M2 @ N2 ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M2 ) ) ) ) ).
% exp_mod_exp
thf(fact_1677_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_1678_mult__exp__mod__exp__eq,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [M2: nat,N2: nat,A3: A] :
( ( ord_less_eq @ nat @ M2 @ N2 )
=> ( ( modulo_modulo @ A @ ( times_times @ A @ A3 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M2 ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) )
= ( times_times @ A @ ( modulo_modulo @ A @ A3 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ N2 @ M2 ) ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M2 ) ) ) ) ) ).
% mult_exp_mod_exp_eq
thf(fact_1679_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_1680_even__mask__div__iff_H,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [M2: nat,N2: nat] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M2 ) @ ( one_one @ A ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) ) )
= ( ord_less_eq @ nat @ M2 @ N2 ) ) ) ).
% even_mask_div_iff'
thf(fact_1681_even__mod__4__div__2,axiom,
! [N2: nat] :
( ( ( modulo_modulo @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) )
= ( suc @ ( zero_zero @ nat ) ) )
=> ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( minus_minus @ nat @ N2 @ ( suc @ ( zero_zero @ nat ) ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% even_mod_4_div_2
thf(fact_1682_even__mask__div__iff,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [M2: nat,N2: nat] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M2 ) @ ( one_one @ A ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) ) )
= ( ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 )
= ( zero_zero @ A ) )
| ( ord_less_eq @ nat @ M2 @ N2 ) ) ) ) ).
% even_mask_div_iff
thf(fact_1683_divmod__step__eq,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [L: num,R2: A,Q3: A] :
( ( ( ord_less_eq @ A @ ( numeral_numeral @ A @ L ) @ R2 )
=> ( ( unique1321980374590559556d_step @ A @ L @ ( product_Pair @ A @ A @ Q3 @ R2 ) )
= ( product_Pair @ A @ A @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Q3 ) @ ( one_one @ A ) ) @ ( minus_minus @ A @ R2 @ ( numeral_numeral @ A @ L ) ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( numeral_numeral @ A @ L ) @ R2 )
=> ( ( unique1321980374590559556d_step @ A @ L @ ( product_Pair @ A @ A @ Q3 @ R2 ) )
= ( product_Pair @ A @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Q3 ) @ R2 ) ) ) ) ) ).
% divmod_step_eq
thf(fact_1684_inrange,axiom,
! [T2: vEBT_VEBT,N2: nat] :
( ( vEBT_invar_vebt @ T2 @ N2 )
=> ( ord_less_eq @ ( set @ nat ) @ ( vEBT_VEBT_set_vebt @ T2 ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) @ ( one_one @ nat ) ) ) ) ) ).
% inrange
thf(fact_1685_artanh__def,axiom,
! [A: $tType] :
( ( ( real_V3459762299906320749_field @ A )
& ( ln @ A ) )
=> ( ( artanh @ A )
= ( ^ [X4: A] : ( divide_divide @ A @ ( ln_ln @ A @ ( divide_divide @ A @ ( plus_plus @ A @ ( one_one @ A ) @ X4 ) @ ( minus_minus @ A @ ( one_one @ A ) @ X4 ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% artanh_def
thf(fact_1686_Suc__if__eq,axiom,
! [A: $tType,F2: nat > A,H2: nat > A,G: A,N2: nat] :
( ! [N3: nat] :
( ( F2 @ ( suc @ N3 ) )
= ( H2 @ N3 ) )
=> ( ( ( F2 @ ( zero_zero @ nat ) )
= G )
=> ( ( ( N2
= ( zero_zero @ nat ) )
=> ( ( F2 @ N2 )
= G ) )
& ( ( N2
!= ( zero_zero @ nat ) )
=> ( ( F2 @ N2 )
= ( H2 @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) ) ) ) ) ) ) ).
% Suc_if_eq
thf(fact_1687_signed__take__bit__rec,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( ( bit_ri4674362597316999326ke_bit @ A )
= ( ^ [N: nat,A5: A] :
( if @ A
@ ( N
= ( 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 @ N @ ( one_one @ nat ) ) @ ( divide_divide @ A @ A5 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% signed_take_bit_rec
thf(fact_1688_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_1689_take__bit__rec,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se2584673776208193580ke_bit @ A )
= ( ^ [N: nat,A5: A] :
( if @ A
@ ( N
= ( zero_zero @ nat ) )
@ ( zero_zero @ A )
@ ( plus_plus @ A @ ( times_times @ A @ ( bit_se2584673776208193580ke_bit @ A @ ( minus_minus @ nat @ N @ ( 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_1690_odd__mod__4__div__2,axiom,
! [N2: nat] :
( ( ( modulo_modulo @ nat @ N2 @ ( 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 @ N2 @ ( suc @ ( zero_zero @ nat ) ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% odd_mod_4_div_2
thf(fact_1691_add_Oinverse__inverse,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A] :
( ( uminus_uminus @ A @ ( uminus_uminus @ A @ A3 ) )
= A3 ) ) ).
% add.inverse_inverse
thf(fact_1692_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_1693_Diff__cancel,axiom,
! [A: $tType,A4: set @ A] :
( ( minus_minus @ ( set @ A ) @ A4 @ A4 )
= ( bot_bot @ ( set @ A ) ) ) ).
% Diff_cancel
thf(fact_1694_empty__Diff,axiom,
! [A: $tType,A4: set @ A] :
( ( minus_minus @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A4 )
= ( bot_bot @ ( set @ A ) ) ) ).
% empty_Diff
thf(fact_1695_Diff__empty,axiom,
! [A: $tType,A4: set @ A] :
( ( minus_minus @ ( set @ A ) @ A4 @ ( bot_bot @ ( set @ A ) ) )
= A4 ) ).
% Diff_empty
thf(fact_1696_finite__Diff2,axiom,
! [A: $tType,B2: set @ A,A4: set @ A] :
( ( finite_finite2 @ A @ B2 )
=> ( ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ A4 @ B2 ) )
= ( finite_finite2 @ A @ A4 ) ) ) ).
% finite_Diff2
thf(fact_1697_finite__Diff,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ A4 @ B2 ) ) ) ).
% finite_Diff
thf(fact_1698_Compl__subset__Compl__iff,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A4 ) @ ( uminus_uminus @ ( set @ A ) @ B2 ) )
= ( ord_less_eq @ ( set @ A ) @ B2 @ A4 ) ) ).
% Compl_subset_Compl_iff
thf(fact_1699_Compl__anti__mono,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B2 )
=> ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ B2 ) @ ( uminus_uminus @ ( set @ A ) @ A4 ) ) ) ).
% Compl_anti_mono
thf(fact_1700_zle__diff1__eq,axiom,
! [W2: int,Z3: int] :
( ( ord_less_eq @ int @ W2 @ ( minus_minus @ int @ Z3 @ ( one_one @ int ) ) )
= ( ord_less @ int @ W2 @ Z3 ) ) ).
% zle_diff1_eq
thf(fact_1701_zle__add1__eq__le,axiom,
! [W2: int,Z3: int] :
( ( ord_less @ int @ W2 @ ( plus_plus @ int @ Z3 @ ( one_one @ int ) ) )
= ( ord_less_eq @ int @ W2 @ Z3 ) ) ).
% zle_add1_eq_le
thf(fact_1702_finite__interval__int2,axiom,
! [A3: int,B3: int] :
( finite_finite2 @ int
@ ( collect @ int
@ ^ [I3: int] :
( ( ord_less_eq @ int @ A3 @ I3 )
& ( ord_less @ int @ I3 @ B3 ) ) ) ) ).
% finite_interval_int2
thf(fact_1703_finite__interval__int3,axiom,
! [A3: int,B3: int] :
( finite_finite2 @ int
@ ( collect @ int
@ ^ [I3: int] :
( ( ord_less @ int @ A3 @ I3 )
& ( ord_less_eq @ int @ I3 @ B3 ) ) ) ) ).
% finite_interval_int3
thf(fact_1704_finite__interval__int4,axiom,
! [A3: int,B3: int] :
( finite_finite2 @ int
@ ( collect @ int
@ ^ [I3: int] :
( ( ord_less @ int @ A3 @ I3 )
& ( ord_less @ int @ I3 @ B3 ) ) ) ) ).
% finite_interval_int4
thf(fact_1705_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_1706_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_1707_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_1708_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_1709_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_1710_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_1711_add_Oinverse__neutral,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ( ( uminus_uminus @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% add.inverse_neutral
thf(fact_1712_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_1713_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_1714_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_1715_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_1716_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_1717_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_1718_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_1719_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_1720_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_1721_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_1722_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_1723_Diff__eq__empty__iff,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] :
( ( ( minus_minus @ ( set @ A ) @ A4 @ B2 )
= ( bot_bot @ ( set @ A ) ) )
= ( ord_less_eq @ ( set @ A ) @ A4 @ B2 ) ) ).
% Diff_eq_empty_iff
thf(fact_1724_atLeastAtMost__iff,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [I: A,L: A,U: A] :
( ( member @ A @ I @ ( set_or1337092689740270186AtMost @ A @ L @ U ) )
= ( ( ord_less_eq @ A @ L @ I )
& ( ord_less_eq @ A @ I @ U ) ) ) ) ).
% atLeastAtMost_iff
thf(fact_1725_Icc__eq__Icc,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L: A,H2: A,L3: A,H3: A] :
( ( ( set_or1337092689740270186AtMost @ A @ L @ H2 )
= ( set_or1337092689740270186AtMost @ A @ L3 @ H3 ) )
= ( ( ( L = L3 )
& ( H2 = H3 ) )
| ( ~ ( ord_less_eq @ A @ L @ H2 )
& ~ ( ord_less_eq @ A @ L3 @ H3 ) ) ) ) ) ).
% Icc_eq_Icc
thf(fact_1726_take__bit__of__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N2: nat] :
( ( bit_se2584673776208193580ke_bit @ A @ N2 @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% take_bit_of_0
thf(fact_1727_finite__atLeastAtMost,axiom,
! [L: nat,U: nat] : ( finite_finite2 @ nat @ ( set_or1337092689740270186AtMost @ nat @ L @ U ) ) ).
% finite_atLeastAtMost
thf(fact_1728_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_1729_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_1730_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_1731_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_1732_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_1733_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_1734_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_1735_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_1736_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_1737_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_1738_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_1739_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_1740_add__neg__numeral__simps_I3_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [M2: num,N2: num] :
( ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M2 ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N2 ) ) )
= ( uminus_uminus @ A @ ( plus_plus @ A @ ( numeral_numeral @ A @ M2 ) @ ( numeral_numeral @ A @ N2 ) ) ) ) ) ).
% add_neg_numeral_simps(3)
thf(fact_1741_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_1742_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_1743_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_1744_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_1745_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_1746_atLeastatMost__subset__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_or1337092689740270186AtMost @ A @ C2 @ D2 ) )
= ( ~ ( ord_less_eq @ A @ A3 @ B3 )
| ( ( ord_less_eq @ A @ C2 @ A3 )
& ( ord_less_eq @ A @ B3 @ D2 ) ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_1747_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_1748_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_1749_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_1750_take__bit__Suc__1,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [N2: nat] :
( ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N2 ) @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% take_bit_Suc_1
thf(fact_1751_ln__one,axiom,
! [A: $tType] :
( ( ln @ A )
=> ( ( ln_ln @ A @ ( one_one @ A ) )
= ( zero_zero @ A ) ) ) ).
% ln_one
thf(fact_1752_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_1753_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_1754_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_1755_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_1756_max__number__of_I2_J,axiom,
! [A: $tType] :
( ( ( uminus @ A )
& ( numeral @ A )
& ( ord @ A ) )
=> ! [U: num,V2: num] :
( ( ( ord_less_eq @ A @ ( numeral_numeral @ A @ U ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) )
=> ( ( ord_max @ A @ ( numeral_numeral @ A @ U ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( numeral_numeral @ A @ U ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) )
=> ( ( ord_max @ A @ ( numeral_numeral @ A @ U ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) )
= ( numeral_numeral @ A @ U ) ) ) ) ) ).
% max_number_of(2)
thf(fact_1757_max__number__of_I3_J,axiom,
! [A: $tType] :
( ( ( uminus @ A )
& ( numeral @ A )
& ( ord @ A ) )
=> ! [U: num,V2: num] :
( ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( numeral_numeral @ A @ V2 ) )
=> ( ( ord_max @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( numeral_numeral @ A @ V2 ) )
= ( numeral_numeral @ A @ V2 ) ) )
& ( ~ ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( numeral_numeral @ A @ V2 ) )
=> ( ( ord_max @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( numeral_numeral @ A @ V2 ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) ) ) ) ) ).
% max_number_of(3)
thf(fact_1758_max__number__of_I4_J,axiom,
! [A: $tType] :
( ( ( uminus @ A )
& ( numeral @ A )
& ( ord @ A ) )
=> ! [U: num,V2: num] :
( ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) )
=> ( ( ord_max @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) )
=> ( ( ord_max @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) ) ) ) ) ).
% max_number_of(4)
thf(fact_1759_take__bit__of__1__eq__0__iff,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [N2: nat] :
( ( ( bit_se2584673776208193580ke_bit @ A @ N2 @ ( one_one @ A ) )
= ( zero_zero @ A ) )
= ( N2
= ( zero_zero @ nat ) ) ) ) ).
% take_bit_of_1_eq_0_iff
thf(fact_1760_semiring__norm_I168_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [V2: num,W2: num,Y: A] :
( ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) @ ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ Y ) )
= ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( plus_plus @ num @ V2 @ W2 ) ) ) @ Y ) ) ) ).
% semiring_norm(168)
thf(fact_1761_neg__numeral__le__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M2: num,N2: num] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M2 ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N2 ) ) )
= ( ord_less_eq @ num @ N2 @ M2 ) ) ) ).
% neg_numeral_le_iff
thf(fact_1762_neg__numeral__less__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M2: num,N2: num] :
( ( ord_less @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M2 ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N2 ) ) )
= ( ord_less @ num @ N2 @ M2 ) ) ) ).
% neg_numeral_less_iff
thf(fact_1763_take__bit__of__Suc__0,axiom,
! [N2: nat] :
( ( bit_se2584673776208193580ke_bit @ nat @ N2 @ ( suc @ ( zero_zero @ nat ) ) )
= ( zero_neq_one_of_bool @ nat @ ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ).
% take_bit_of_Suc_0
thf(fact_1764_not__neg__one__le__neg__numeral__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M2: num] :
( ( ~ ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M2 ) ) ) )
= ( M2 != one2 ) ) ) ).
% not_neg_one_le_neg_numeral_iff
thf(fact_1765_le__divide__eq__numeral1_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B3: A,W2: num] :
( ( ord_less_eq @ A @ A3 @ ( divide_divide @ A @ B3 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) )
= ( ord_less_eq @ A @ B3 @ ( times_times @ A @ A3 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) ) ) ) ).
% le_divide_eq_numeral1(2)
thf(fact_1766_divide__le__eq__numeral1_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B3: A,W2: num,A3: A] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ B3 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) @ A3 )
= ( ord_less_eq @ A @ ( times_times @ A @ A3 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) @ B3 ) ) ) ).
% divide_le_eq_numeral1(2)
thf(fact_1767_divide__eq__eq__numeral1_I2_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B3: A,W2: num,A3: A] :
( ( ( divide_divide @ A @ B3 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) )
= A3 )
= ( ( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) )
!= ( zero_zero @ A ) )
=> ( B3
= ( times_times @ A @ A3 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) ) )
& ( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) )
= ( zero_zero @ A ) )
=> ( A3
= ( zero_zero @ A ) ) ) ) ) ) ).
% divide_eq_eq_numeral1(2)
thf(fact_1768_eq__divide__eq__numeral1_I2_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A,B3: A,W2: num] :
( ( A3
= ( divide_divide @ A @ B3 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) )
= ( ( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) )
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ A3 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) )
= B3 ) )
& ( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) )
= ( zero_zero @ A ) )
=> ( A3
= ( zero_zero @ A ) ) ) ) ) ) ).
% eq_divide_eq_numeral1(2)
thf(fact_1769_neg__numeral__less__neg__one__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M2: num] :
( ( ord_less @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M2 ) ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( M2 != one2 ) ) ) ).
% neg_numeral_less_neg_one_iff
thf(fact_1770_less__divide__eq__numeral1_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B3: A,W2: num] :
( ( ord_less @ A @ A3 @ ( divide_divide @ A @ B3 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) )
= ( ord_less @ A @ B3 @ ( times_times @ A @ A3 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) ) ) ) ).
% less_divide_eq_numeral1(2)
thf(fact_1771_divide__less__eq__numeral1_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B3: A,W2: num,A3: A] :
( ( ord_less @ A @ ( divide_divide @ A @ B3 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) @ A3 )
= ( ord_less @ A @ ( times_times @ A @ A3 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) @ B3 ) ) ) ).
% divide_less_eq_numeral1(2)
thf(fact_1772_take__bit__of__1,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N2: nat] :
( ( bit_se2584673776208193580ke_bit @ A @ N2 @ ( one_one @ A ) )
= ( zero_neq_one_of_bool @ A @ ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ) ).
% take_bit_of_1
thf(fact_1773_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_1774_even__take__bit__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N2: nat,A3: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se2584673776208193580ke_bit @ A @ N2 @ A3 ) )
= ( ( N2
= ( zero_zero @ nat ) )
| ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) ) ) ) ).
% even_take_bit_eq
thf(fact_1775_Suc__div__eq__add3__div__numeral,axiom,
! [M2: nat,V2: num] :
( ( divide_divide @ nat @ ( suc @ ( suc @ ( suc @ M2 ) ) ) @ ( numeral_numeral @ nat @ V2 ) )
= ( divide_divide @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) @ M2 ) @ ( numeral_numeral @ nat @ V2 ) ) ) ).
% Suc_div_eq_add3_div_numeral
thf(fact_1776_div__Suc__eq__div__add3,axiom,
! [M2: nat,N2: nat] :
( ( divide_divide @ nat @ M2 @ ( suc @ ( suc @ ( suc @ N2 ) ) ) )
= ( divide_divide @ nat @ M2 @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) @ N2 ) ) ) ).
% div_Suc_eq_div_add3
thf(fact_1777_Suc__mod__eq__add3__mod__numeral,axiom,
! [M2: nat,V2: num] :
( ( modulo_modulo @ nat @ ( suc @ ( suc @ ( suc @ M2 ) ) ) @ ( numeral_numeral @ nat @ V2 ) )
= ( modulo_modulo @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) @ M2 ) @ ( numeral_numeral @ nat @ V2 ) ) ) ).
% Suc_mod_eq_add3_mod_numeral
thf(fact_1778_mod__Suc__eq__mod__add3,axiom,
! [M2: nat,N2: nat] :
( ( modulo_modulo @ nat @ M2 @ ( suc @ ( suc @ ( suc @ N2 ) ) ) )
= ( modulo_modulo @ nat @ M2 @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) @ N2 ) ) ) ).
% mod_Suc_eq_mod_add3
thf(fact_1779_signed__take__bit__Suc__minus__bit0,axiom,
! [N2: nat,K: num] :
( ( bit_ri4674362597316999326ke_bit @ int @ ( suc @ N2 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ K ) ) ) )
= ( times_times @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N2 @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ).
% signed_take_bit_Suc_minus_bit0
thf(fact_1780_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_1781_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_1782_signed__take__bit__Suc__minus__bit1,axiom,
! [N2: nat,K: num] :
( ( bit_ri4674362597316999326ke_bit @ int @ ( suc @ N2 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ K ) ) ) )
= ( plus_plus @ int @ ( times_times @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N2 @ ( 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_1783_signed__take__bit__Suc__bit1,axiom,
! [N2: nat,K: num] :
( ( bit_ri4674362597316999326ke_bit @ int @ ( suc @ N2 ) @ ( numeral_numeral @ int @ ( bit1 @ K ) ) )
= ( plus_plus @ int @ ( times_times @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N2 @ ( numeral_numeral @ int @ K ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( one_one @ int ) ) ) ).
% signed_take_bit_Suc_bit1
thf(fact_1784_take__bit__of__exp,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [M2: nat,N2: nat] :
( ( bit_se2584673776208193580ke_bit @ A @ M2 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) )
= ( times_times @ A @ ( zero_neq_one_of_bool @ A @ ( ord_less @ nat @ N2 @ M2 ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ).
% take_bit_of_exp
thf(fact_1785_take__bit__of__2,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [N2: nat] :
( ( bit_se2584673776208193580ke_bit @ A @ N2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( times_times @ A @ ( zero_neq_one_of_bool @ A @ ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% take_bit_of_2
thf(fact_1786_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_1787_minus__int__code_I1_J,axiom,
! [K: int] :
( ( minus_minus @ int @ K @ ( zero_zero @ int ) )
= K ) ).
% minus_int_code(1)
thf(fact_1788_int__distrib_I3_J,axiom,
! [Z1: int,Z22: int,W2: int] :
( ( times_times @ int @ ( minus_minus @ int @ Z1 @ Z22 ) @ W2 )
= ( minus_minus @ int @ ( times_times @ int @ Z1 @ W2 ) @ ( times_times @ int @ Z22 @ W2 ) ) ) ).
% int_distrib(3)
thf(fact_1789_int__distrib_I4_J,axiom,
! [W2: int,Z1: int,Z22: int] :
( ( times_times @ int @ W2 @ ( minus_minus @ int @ Z1 @ Z22 ) )
= ( minus_minus @ int @ ( times_times @ int @ W2 @ Z1 ) @ ( times_times @ int @ W2 @ Z22 ) ) ) ).
% int_distrib(4)
thf(fact_1790_int__less__induct,axiom,
! [I: int,K: int,P: int > $o] :
( ( ord_less @ int @ I @ K )
=> ( ( P @ ( minus_minus @ int @ K @ ( one_one @ int ) ) )
=> ( ! [I4: int] :
( ( ord_less @ int @ I4 @ K )
=> ( ( P @ I4 )
=> ( P @ ( minus_minus @ int @ I4 @ ( one_one @ int ) ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_less_induct
thf(fact_1791_zdvd__mult__cancel,axiom,
! [K: int,M2: int,N2: int] :
( ( dvd_dvd @ int @ ( times_times @ int @ K @ M2 ) @ ( times_times @ int @ K @ N2 ) )
=> ( ( K
!= ( zero_zero @ int ) )
=> ( dvd_dvd @ int @ M2 @ N2 ) ) ) ).
% zdvd_mult_cancel
thf(fact_1792_zdvd__imp__le,axiom,
! [Z3: int,N2: int] :
( ( dvd_dvd @ int @ Z3 @ N2 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ N2 )
=> ( ord_less_eq @ int @ Z3 @ N2 ) ) ) ).
% zdvd_imp_le
thf(fact_1793_zdvd__not__zless,axiom,
! [M2: int,N2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ M2 )
=> ( ( ord_less @ int @ M2 @ N2 )
=> ~ ( dvd_dvd @ int @ N2 @ M2 ) ) ) ).
% zdvd_not_zless
thf(fact_1794_zdvd__antisym__nonneg,axiom,
! [M2: int,N2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ M2 )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ N2 )
=> ( ( dvd_dvd @ int @ M2 @ N2 )
=> ( ( dvd_dvd @ int @ N2 @ M2 )
=> ( M2 = N2 ) ) ) ) ) ).
% zdvd_antisym_nonneg
thf(fact_1795_zdvd__reduce,axiom,
! [K: int,N2: int,M2: int] :
( ( dvd_dvd @ int @ K @ ( plus_plus @ int @ N2 @ ( times_times @ int @ K @ M2 ) ) )
= ( dvd_dvd @ int @ K @ N2 ) ) ).
% zdvd_reduce
thf(fact_1796_zdvd__period,axiom,
! [A3: int,D2: int,X: int,T2: int,C2: int] :
( ( dvd_dvd @ int @ A3 @ D2 )
=> ( ( dvd_dvd @ int @ A3 @ ( plus_plus @ int @ X @ T2 ) )
= ( dvd_dvd @ int @ A3 @ ( plus_plus @ int @ ( plus_plus @ int @ X @ ( times_times @ int @ C2 @ D2 ) ) @ T2 ) ) ) ) ).
% zdvd_period
thf(fact_1797_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_1798_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_1799_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_1800_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_1801_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_1802_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_1803_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_1804_add1__zle__eq,axiom,
! [W2: int,Z3: int] :
( ( ord_less_eq @ int @ ( plus_plus @ int @ W2 @ ( one_one @ int ) ) @ Z3 )
= ( ord_less @ int @ W2 @ Z3 ) ) ).
% add1_zle_eq
thf(fact_1805_int__gr__induct,axiom,
! [K: int,I: int,P: int > $o] :
( ( ord_less @ int @ K @ I )
=> ( ( P @ ( plus_plus @ int @ K @ ( one_one @ int ) ) )
=> ( ! [I4: int] :
( ( ord_less @ int @ K @ I4 )
=> ( ( P @ I4 )
=> ( P @ ( plus_plus @ int @ I4 @ ( one_one @ int ) ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_gr_induct
thf(fact_1806_le__imp__0__less,axiom,
! [Z3: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z3 )
=> ( ord_less @ int @ ( zero_zero @ int ) @ ( plus_plus @ int @ ( one_one @ int ) @ Z3 ) ) ) ).
% le_imp_0_less
thf(fact_1807_zless__add1__eq,axiom,
! [W2: int,Z3: int] :
( ( ord_less @ int @ W2 @ ( plus_plus @ int @ Z3 @ ( one_one @ int ) ) )
= ( ( ord_less @ int @ W2 @ Z3 )
| ( W2 = Z3 ) ) ) ).
% zless_add1_eq
thf(fact_1808_odd__less__0__iff,axiom,
! [Z3: int] :
( ( ord_less @ int @ ( plus_plus @ int @ ( plus_plus @ int @ ( one_one @ int ) @ Z3 ) @ Z3 ) @ ( zero_zero @ int ) )
= ( ord_less @ int @ Z3 @ ( zero_zero @ int ) ) ) ).
% odd_less_0_iff
thf(fact_1809_pos__zmult__eq__1__iff,axiom,
! [M2: int,N2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ M2 )
=> ( ( ( times_times @ int @ M2 @ N2 )
= ( one_one @ int ) )
= ( ( M2
= ( one_one @ int ) )
& ( N2
= ( one_one @ int ) ) ) ) ) ).
% pos_zmult_eq_1_iff
thf(fact_1810_zless__imp__add1__zle,axiom,
! [W2: int,Z3: int] :
( ( ord_less @ int @ W2 @ Z3 )
=> ( ord_less_eq @ int @ ( plus_plus @ int @ W2 @ ( one_one @ int ) ) @ Z3 ) ) ).
% zless_imp_add1_zle
thf(fact_1811_int__one__le__iff__zero__less,axiom,
! [Z3: int] :
( ( ord_less_eq @ int @ ( one_one @ int ) @ Z3 )
= ( ord_less @ int @ ( zero_zero @ int ) @ Z3 ) ) ).
% int_one_le_iff_zero_less
thf(fact_1812_odd__nonzero,axiom,
! [Z3: int] :
( ( plus_plus @ int @ ( plus_plus @ int @ ( one_one @ int ) @ Z3 ) @ Z3 )
!= ( zero_zero @ int ) ) ).
% odd_nonzero
thf(fact_1813_pos__zmult__eq__1__iff__lemma,axiom,
! [M2: int,N2: int] :
( ( ( times_times @ int @ M2 @ N2 )
= ( one_one @ int ) )
=> ( ( M2
= ( one_one @ int ) )
| ( M2
= ( uminus_uminus @ int @ ( one_one @ int ) ) ) ) ) ).
% pos_zmult_eq_1_iff_lemma
thf(fact_1814_zmult__eq__1__iff,axiom,
! [M2: int,N2: int] :
( ( ( times_times @ int @ M2 @ N2 )
= ( one_one @ int ) )
= ( ( ( M2
= ( one_one @ int ) )
& ( N2
= ( one_one @ int ) ) )
| ( ( M2
= ( uminus_uminus @ int @ ( one_one @ int ) ) )
& ( N2
= ( uminus_uminus @ int @ ( one_one @ int ) ) ) ) ) ) ).
% zmult_eq_1_iff
thf(fact_1815_times__int__code_I2_J,axiom,
! [L: int] :
( ( times_times @ int @ ( zero_zero @ int ) @ L )
= ( zero_zero @ int ) ) ).
% times_int_code(2)
thf(fact_1816_times__int__code_I1_J,axiom,
! [K: int] :
( ( times_times @ int @ K @ ( zero_zero @ int ) )
= ( zero_zero @ int ) ) ).
% times_int_code(1)
thf(fact_1817_int__distrib_I1_J,axiom,
! [Z1: int,Z22: int,W2: int] :
( ( times_times @ int @ ( plus_plus @ int @ Z1 @ Z22 ) @ W2 )
= ( plus_plus @ int @ ( times_times @ int @ Z1 @ W2 ) @ ( times_times @ int @ Z22 @ W2 ) ) ) ).
% int_distrib(1)
thf(fact_1818_int__distrib_I2_J,axiom,
! [W2: int,Z1: int,Z22: int] :
( ( times_times @ int @ W2 @ ( plus_plus @ int @ Z1 @ Z22 ) )
= ( plus_plus @ int @ ( times_times @ int @ W2 @ Z1 ) @ ( times_times @ int @ W2 @ Z22 ) ) ) ).
% int_distrib(2)
thf(fact_1819_uminus__int__code_I1_J,axiom,
( ( uminus_uminus @ int @ ( zero_zero @ int ) )
= ( zero_zero @ int ) ) ).
% uminus_int_code(1)
thf(fact_1820_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_1821_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_1822_take__bit__add,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [N2: nat,A3: A,B3: A] :
( ( bit_se2584673776208193580ke_bit @ A @ N2 @ ( plus_plus @ A @ ( bit_se2584673776208193580ke_bit @ A @ N2 @ A3 ) @ ( bit_se2584673776208193580ke_bit @ A @ N2 @ B3 ) ) )
= ( bit_se2584673776208193580ke_bit @ A @ N2 @ ( plus_plus @ A @ A3 @ B3 ) ) ) ) ).
% take_bit_add
thf(fact_1823_take__bit__tightened,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N2: nat,A3: A,B3: A,M2: nat] :
( ( ( bit_se2584673776208193580ke_bit @ A @ N2 @ A3 )
= ( bit_se2584673776208193580ke_bit @ A @ N2 @ B3 ) )
=> ( ( ord_less_eq @ nat @ M2 @ N2 )
=> ( ( bit_se2584673776208193580ke_bit @ A @ M2 @ A3 )
= ( bit_se2584673776208193580ke_bit @ A @ M2 @ B3 ) ) ) ) ) ).
% take_bit_tightened
thf(fact_1824_take__bit__tightened__less__eq__nat,axiom,
! [M2: nat,N2: nat,Q3: nat] :
( ( ord_less_eq @ nat @ M2 @ N2 )
=> ( ord_less_eq @ nat @ ( bit_se2584673776208193580ke_bit @ nat @ M2 @ Q3 ) @ ( bit_se2584673776208193580ke_bit @ nat @ N2 @ Q3 ) ) ) ).
% take_bit_tightened_less_eq_nat
thf(fact_1825_take__bit__nat__less__eq__self,axiom,
! [N2: nat,M2: nat] : ( ord_less_eq @ nat @ ( bit_se2584673776208193580ke_bit @ nat @ N2 @ M2 ) @ M2 ) ).
% take_bit_nat_less_eq_self
thf(fact_1826_take__bit__tightened__less__eq__int,axiom,
! [M2: nat,N2: nat,K: int] :
( ( ord_less_eq @ nat @ M2 @ N2 )
=> ( ord_less_eq @ int @ ( bit_se2584673776208193580ke_bit @ int @ M2 @ K ) @ ( bit_se2584673776208193580ke_bit @ int @ N2 @ K ) ) ) ).
% take_bit_tightened_less_eq_int
thf(fact_1827_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_1828_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_1829_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_1830_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_1831_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_1832_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_1833_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_1834_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_1835_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_1836_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_1837_group__cancel_Oneg1,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A4: A,K: A,A3: A] :
( ( A4
= ( plus_plus @ A @ K @ A3 ) )
=> ( ( uminus_uminus @ A @ A4 )
= ( plus_plus @ A @ ( uminus_uminus @ A @ K ) @ ( uminus_uminus @ A @ A3 ) ) ) ) ) ).
% group_cancel.neg1
thf(fact_1838_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_1839_minus__divide__right,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A3: A,B3: A] :
( ( uminus_uminus @ A @ ( divide_divide @ A @ A3 @ B3 ) )
= ( divide_divide @ A @ A3 @ ( uminus_uminus @ A @ B3 ) ) ) ) ).
% minus_divide_right
thf(fact_1840_minus__divide__divide,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A3: A,B3: A] :
( ( divide_divide @ A @ ( uminus_uminus @ A @ A3 ) @ ( uminus_uminus @ A @ B3 ) )
= ( divide_divide @ A @ A3 @ B3 ) ) ) ).
% minus_divide_divide
thf(fact_1841_minus__divide__left,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A,B3: A] :
( ( uminus_uminus @ A @ ( divide_divide @ A @ A3 @ B3 ) )
= ( divide_divide @ A @ ( uminus_uminus @ A @ A3 ) @ B3 ) ) ) ).
% minus_divide_left
thf(fact_1842_Diff__infinite__finite,axiom,
! [A: $tType,T4: set @ A,S2: set @ A] :
( ( finite_finite2 @ A @ T4 )
=> ( ~ ( finite_finite2 @ A @ S2 )
=> ~ ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ S2 @ T4 ) ) ) ) ).
% Diff_infinite_finite
thf(fact_1843_double__diff,axiom,
! [A: $tType,A4: set @ A,B2: set @ A,C5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ C5 )
=> ( ( minus_minus @ ( set @ A ) @ B2 @ ( minus_minus @ ( set @ A ) @ C5 @ A4 ) )
= A4 ) ) ) ).
% double_diff
thf(fact_1844_Diff__subset,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A4 @ B2 ) @ A4 ) ).
% Diff_subset
thf(fact_1845_Diff__mono,axiom,
! [A: $tType,A4: set @ A,C5: set @ A,D6: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ C5 )
=> ( ( ord_less_eq @ ( set @ A ) @ D6 @ B2 )
=> ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A4 @ B2 ) @ ( minus_minus @ ( set @ A ) @ C5 @ D6 ) ) ) ) ).
% Diff_mono
thf(fact_1846_psubset__imp__ex__mem,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] :
( ( ord_less @ ( set @ A ) @ A4 @ B2 )
=> ? [B5: A] : ( member @ A @ B5 @ ( minus_minus @ ( set @ A ) @ B2 @ A4 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_1847_signed__take__bit__eq__take__bit__shift,axiom,
( ( bit_ri4674362597316999326ke_bit @ int )
= ( ^ [N: nat,K2: int] : ( minus_minus @ int @ ( bit_se2584673776208193580ke_bit @ int @ ( suc @ N ) @ ( plus_plus @ int @ K2 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% signed_take_bit_eq_take_bit_shift
thf(fact_1848_take__bit__Suc__minus__bit0,axiom,
! [N2: nat,K: num] :
( ( bit_se2584673776208193580ke_bit @ int @ ( suc @ N2 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ K ) ) ) )
= ( times_times @ int @ ( bit_se2584673776208193580ke_bit @ int @ N2 @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ).
% take_bit_Suc_minus_bit0
thf(fact_1849_take__bit__int__less__eq,axiom,
! [N2: nat,K: int] :
( ( ord_less_eq @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) @ K )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ord_less_eq @ int @ ( bit_se2584673776208193580ke_bit @ int @ N2 @ K ) @ ( minus_minus @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ) ).
% take_bit_int_less_eq
thf(fact_1850_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_1851_signed__take__bit__eq__iff__take__bit__eq,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N2: nat,A3: A,B3: A] :
( ( ( bit_ri4674362597316999326ke_bit @ A @ N2 @ A3 )
= ( bit_ri4674362597316999326ke_bit @ A @ N2 @ B3 ) )
= ( ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N2 ) @ A3 )
= ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N2 ) @ B3 ) ) ) ) ).
% signed_take_bit_eq_iff_take_bit_eq
thf(fact_1852_signed__take__bit__take__bit,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [M2: nat,N2: nat,A3: A] :
( ( bit_ri4674362597316999326ke_bit @ A @ M2 @ ( bit_se2584673776208193580ke_bit @ A @ N2 @ A3 ) )
= ( if @ ( A > A ) @ ( ord_less_eq @ nat @ N2 @ M2 ) @ ( bit_se2584673776208193580ke_bit @ A @ N2 ) @ ( bit_ri4674362597316999326ke_bit @ A @ M2 ) @ A3 ) ) ) ).
% signed_take_bit_take_bit
thf(fact_1853_all__nat__less,axiom,
! [N2: nat,P: nat > $o] :
( ( ! [M: nat] :
( ( ord_less_eq @ nat @ M @ N2 )
=> ( P @ M ) ) )
= ( ! [X4: nat] :
( ( member @ nat @ X4 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N2 ) )
=> ( P @ X4 ) ) ) ) ).
% all_nat_less
thf(fact_1854_ex__nat__less,axiom,
! [N2: nat,P: nat > $o] :
( ( ? [M: nat] :
( ( ord_less_eq @ nat @ M @ N2 )
& ( P @ M ) ) )
= ( ? [X4: nat] :
( ( member @ nat @ X4 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N2 ) )
& ( P @ X4 ) ) ) ) ).
% ex_nat_less
thf(fact_1855_not__numeral__le__neg__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M2: num,N2: num] :
~ ( ord_less_eq @ A @ ( numeral_numeral @ A @ M2 ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N2 ) ) ) ) ).
% not_numeral_le_neg_numeral
thf(fact_1856_neg__numeral__le__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M2: num,N2: num] : ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M2 ) ) @ ( numeral_numeral @ A @ N2 ) ) ) ).
% neg_numeral_le_numeral
thf(fact_1857_zero__neq__neg__numeral,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [N2: num] :
( ( zero_zero @ A )
!= ( uminus_uminus @ A @ ( numeral_numeral @ A @ N2 ) ) ) ) ).
% zero_neq_neg_numeral
thf(fact_1858_not__numeral__less__neg__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M2: num,N2: num] :
~ ( ord_less @ A @ ( numeral_numeral @ A @ M2 ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N2 ) ) ) ) ).
% not_numeral_less_neg_numeral
thf(fact_1859_neg__numeral__less__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M2: num,N2: num] : ( ord_less @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M2 ) ) @ ( numeral_numeral @ A @ N2 ) ) ) ).
% neg_numeral_less_numeral
thf(fact_1860_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_1861_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_1862_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_1863_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_1864_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_1865_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_1866_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_1867_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_1868_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_1869_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_1870_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_1871_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_1872_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_1873_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_1874_diff__conv__add__uminus,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ( ( minus_minus @ A )
= ( ^ [A5: A,B4: A] : ( plus_plus @ A @ A5 @ ( uminus_uminus @ A @ B4 ) ) ) ) ) ).
% diff_conv_add_uminus
thf(fact_1875_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ( ( minus_minus @ A )
= ( ^ [A5: A,B4: A] : ( plus_plus @ A @ A5 @ ( uminus_uminus @ A @ B4 ) ) ) ) ) ).
% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_1876_take__bit__unset__bit__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N2: nat,M2: nat,A3: A] :
( ( ( ord_less_eq @ nat @ N2 @ M2 )
=> ( ( bit_se2584673776208193580ke_bit @ A @ N2 @ ( bit_se2638667681897837118et_bit @ A @ M2 @ A3 ) )
= ( bit_se2584673776208193580ke_bit @ A @ N2 @ A3 ) ) )
& ( ~ ( ord_less_eq @ nat @ N2 @ M2 )
=> ( ( bit_se2584673776208193580ke_bit @ A @ N2 @ ( bit_se2638667681897837118et_bit @ A @ M2 @ A3 ) )
= ( bit_se2638667681897837118et_bit @ A @ M2 @ ( bit_se2584673776208193580ke_bit @ A @ N2 @ A3 ) ) ) ) ) ) ).
% take_bit_unset_bit_eq
thf(fact_1877_take__bit__set__bit__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N2: nat,M2: nat,A3: A] :
( ( ( ord_less_eq @ nat @ N2 @ M2 )
=> ( ( bit_se2584673776208193580ke_bit @ A @ N2 @ ( bit_se5668285175392031749et_bit @ A @ M2 @ A3 ) )
= ( bit_se2584673776208193580ke_bit @ A @ N2 @ A3 ) ) )
& ( ~ ( ord_less_eq @ nat @ N2 @ M2 )
=> ( ( bit_se2584673776208193580ke_bit @ A @ N2 @ ( bit_se5668285175392031749et_bit @ A @ M2 @ A3 ) )
= ( bit_se5668285175392031749et_bit @ A @ M2 @ ( bit_se2584673776208193580ke_bit @ A @ N2 @ A3 ) ) ) ) ) ) ).
% take_bit_set_bit_eq
thf(fact_1878_take__bit__flip__bit__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N2: nat,M2: nat,A3: A] :
( ( ( ord_less_eq @ nat @ N2 @ M2 )
=> ( ( bit_se2584673776208193580ke_bit @ A @ N2 @ ( bit_se8732182000553998342ip_bit @ A @ M2 @ A3 ) )
= ( bit_se2584673776208193580ke_bit @ A @ N2 @ A3 ) ) )
& ( ~ ( ord_less_eq @ nat @ N2 @ M2 )
=> ( ( bit_se2584673776208193580ke_bit @ A @ N2 @ ( bit_se8732182000553998342ip_bit @ A @ M2 @ A3 ) )
= ( bit_se8732182000553998342ip_bit @ A @ M2 @ ( bit_se2584673776208193580ke_bit @ A @ N2 @ A3 ) ) ) ) ) ) ).
% take_bit_flip_bit_eq
thf(fact_1879_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_1880_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_1881_subset__Compl__self__eq,axiom,
! [A: $tType,A4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ ( uminus_uminus @ ( set @ A ) @ A4 ) )
= ( A4
= ( bot_bot @ ( set @ A ) ) ) ) ).
% subset_Compl_self_eq
thf(fact_1882_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_1883_take__bit__signed__take__bit,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [M2: nat,N2: nat,A3: A] :
( ( ord_less_eq @ nat @ M2 @ ( suc @ N2 ) )
=> ( ( bit_se2584673776208193580ke_bit @ A @ M2 @ ( bit_ri4674362597316999326ke_bit @ A @ N2 @ A3 ) )
= ( bit_se2584673776208193580ke_bit @ A @ M2 @ A3 ) ) ) ) ).
% take_bit_signed_take_bit
thf(fact_1884_neg__numeral__le__zero,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N2: num] : ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N2 ) ) @ ( zero_zero @ A ) ) ) ).
% neg_numeral_le_zero
thf(fact_1885_not__zero__le__neg__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N2: num] :
~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N2 ) ) ) ) ).
% not_zero_le_neg_numeral
thf(fact_1886_neg__numeral__less__zero,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N2: num] : ( ord_less @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N2 ) ) @ ( zero_zero @ A ) ) ) ).
% neg_numeral_less_zero
thf(fact_1887_not__zero__less__neg__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N2: num] :
~ ( ord_less @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N2 ) ) ) ) ).
% not_zero_less_neg_numeral
thf(fact_1888_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_1889_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_1890_numeral__Bit1,axiom,
! [A: $tType] :
( ( numeral @ A )
=> ! [N2: num] :
( ( numeral_numeral @ A @ ( bit1 @ N2 ) )
= ( plus_plus @ A @ ( plus_plus @ A @ ( numeral_numeral @ A @ N2 ) @ ( numeral_numeral @ A @ N2 ) ) @ ( one_one @ A ) ) ) ) ).
% numeral_Bit1
thf(fact_1891_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_1892_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_1893_not__one__le__neg__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M2: num] :
~ ( ord_less_eq @ A @ ( one_one @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M2 ) ) ) ) ).
% not_one_le_neg_numeral
thf(fact_1894_not__numeral__le__neg__one,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M2: num] :
~ ( ord_less_eq @ A @ ( numeral_numeral @ A @ M2 ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% not_numeral_le_neg_one
thf(fact_1895_neg__numeral__le__neg__one,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M2: num] : ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M2 ) ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% neg_numeral_le_neg_one
thf(fact_1896_neg__one__le__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M2: num] : ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( numeral_numeral @ A @ M2 ) ) ) ).
% neg_one_le_numeral
thf(fact_1897_neg__numeral__le__one,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M2: num] : ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M2 ) ) @ ( one_one @ A ) ) ) ).
% neg_numeral_le_one
thf(fact_1898_not__neg__one__less__neg__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M2: num] :
~ ( ord_less @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M2 ) ) ) ) ).
% not_neg_one_less_neg_numeral
thf(fact_1899_not__one__less__neg__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M2: num] :
~ ( ord_less @ A @ ( one_one @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M2 ) ) ) ) ).
% not_one_less_neg_numeral
thf(fact_1900_not__numeral__less__neg__one,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M2: num] :
~ ( ord_less @ A @ ( numeral_numeral @ A @ M2 ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% not_numeral_less_neg_one
thf(fact_1901_neg__one__less__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M2: num] : ( ord_less @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( numeral_numeral @ A @ M2 ) ) ) ).
% neg_one_less_numeral
thf(fact_1902_neg__numeral__less__one,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M2: num] : ( ord_less @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M2 ) ) @ ( one_one @ A ) ) ) ).
% neg_numeral_less_one
thf(fact_1903_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_1904_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_1905_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_1906_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_1907_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_1908_eval__nat__numeral_I3_J,axiom,
! [N2: num] :
( ( numeral_numeral @ nat @ ( bit1 @ N2 ) )
= ( suc @ ( numeral_numeral @ nat @ ( bit0 @ N2 ) ) ) ) ).
% eval_nat_numeral(3)
thf(fact_1909_signed__take__bit__int__less__eq,axiom,
! [N2: nat,K: int] :
( ( ord_less_eq @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) @ K )
=> ( ord_less_eq @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N2 @ K ) @ ( minus_minus @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( suc @ N2 ) ) ) ) ) ).
% signed_take_bit_int_less_eq
thf(fact_1910_signed__take__bit__int__greater__eq,axiom,
! [K: int,N2: nat] :
( ( ord_less @ int @ K @ ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) ) )
=> ( ord_less_eq @ int @ ( plus_plus @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( suc @ N2 ) ) ) @ ( bit_ri4674362597316999326ke_bit @ int @ N2 @ K ) ) ) ).
% signed_take_bit_int_greater_eq
thf(fact_1911_take__bit__Suc__bit1,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [N2: nat,K: num] :
( ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N2 ) @ ( numeral_numeral @ A @ ( bit1 @ K ) ) )
= ( plus_plus @ A @ ( times_times @ A @ ( bit_se2584673776208193580ke_bit @ A @ N2 @ ( numeral_numeral @ A @ K ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) ) ) ) ).
% take_bit_Suc_bit1
thf(fact_1912_take__bit__Suc__minus__1__eq,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N2: nat] :
( ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N2 ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( suc @ N2 ) ) @ ( one_one @ A ) ) ) ) ).
% take_bit_Suc_minus_1_eq
thf(fact_1913_numeral__code_I3_J,axiom,
! [A: $tType] :
( ( numeral @ A )
=> ! [N2: num] :
( ( numeral_numeral @ A @ ( bit1 @ N2 ) )
= ( plus_plus @ A @ ( plus_plus @ A @ ( numeral_numeral @ A @ N2 ) @ ( numeral_numeral @ A @ N2 ) ) @ ( one_one @ A ) ) ) ) ).
% numeral_code(3)
thf(fact_1914_subset__eq__atLeast0__atMost__finite,axiom,
! [N6: set @ nat,N2: nat] :
( ( ord_less_eq @ ( set @ nat ) @ N6 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N2 ) )
=> ( finite_finite2 @ nat @ N6 ) ) ).
% subset_eq_atLeast0_atMost_finite
thf(fact_1915_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_1916_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_1917_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_1918_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_1919_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_1920_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_1921_divide__eq__eq__numeral_I2_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B3: A,C2: A,W2: num] :
( ( ( divide_divide @ A @ B3 @ C2 )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) )
= ( ( ( C2
!= ( zero_zero @ A ) )
=> ( B3
= ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ C2 ) ) )
& ( ( C2
= ( zero_zero @ A ) )
=> ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) )
= ( zero_zero @ A ) ) ) ) ) ) ).
% divide_eq_eq_numeral(2)
thf(fact_1922_eq__divide__eq__numeral_I2_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [W2: num,B3: A,C2: A] :
( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) )
= ( divide_divide @ A @ B3 @ C2 ) )
= ( ( ( C2
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ C2 )
= B3 ) )
& ( ( C2
= ( zero_zero @ A ) )
=> ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) )
= ( zero_zero @ A ) ) ) ) ) ) ).
% eq_divide_eq_numeral(2)
thf(fact_1923_cong__exp__iff__simps_I3_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [N2: num,Q3: num] :
( ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit1 @ N2 ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q3 ) ) )
!= ( zero_zero @ A ) ) ) ).
% cong_exp_iff_simps(3)
thf(fact_1924_add__divide__eq__if__simps_I3_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z3: A,A3: A,B3: A] :
( ( ( Z3
= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ A3 @ Z3 ) ) @ B3 )
= B3 ) )
& ( ( Z3
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ A3 @ Z3 ) ) @ B3 )
= ( divide_divide @ A @ ( plus_plus @ A @ ( uminus_uminus @ A @ A3 ) @ ( times_times @ A @ B3 @ Z3 ) ) @ Z3 ) ) ) ) ) ).
% add_divide_eq_if_simps(3)
thf(fact_1925_minus__divide__add__eq__iff,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z3: A,X: A,Y: A] :
( ( Z3
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ X @ Z3 ) ) @ Y )
= ( divide_divide @ A @ ( plus_plus @ A @ ( uminus_uminus @ A @ X ) @ ( times_times @ A @ Y @ Z3 ) ) @ Z3 ) ) ) ) ).
% minus_divide_add_eq_iff
thf(fact_1926_add__divide__eq__if__simps_I6_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z3: A,A3: A,B3: A] :
( ( ( Z3
= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ A3 @ Z3 ) ) @ B3 )
= ( uminus_uminus @ A @ B3 ) ) )
& ( ( Z3
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ A3 @ Z3 ) ) @ B3 )
= ( divide_divide @ A @ ( minus_minus @ A @ ( uminus_uminus @ A @ A3 ) @ ( times_times @ A @ B3 @ Z3 ) ) @ Z3 ) ) ) ) ) ).
% add_divide_eq_if_simps(6)
thf(fact_1927_add__divide__eq__if__simps_I5_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z3: A,A3: A,B3: A] :
( ( ( Z3
= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( divide_divide @ A @ A3 @ Z3 ) @ B3 )
= ( uminus_uminus @ A @ B3 ) ) )
& ( ( Z3
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( divide_divide @ A @ A3 @ Z3 ) @ B3 )
= ( divide_divide @ A @ ( minus_minus @ A @ A3 @ ( times_times @ A @ B3 @ Z3 ) ) @ Z3 ) ) ) ) ) ).
% add_divide_eq_if_simps(5)
thf(fact_1928_minus__divide__diff__eq__iff,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z3: A,X: A,Y: A] :
( ( Z3
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ X @ Z3 ) ) @ Y )
= ( divide_divide @ A @ ( minus_minus @ A @ ( uminus_uminus @ A @ X ) @ ( times_times @ A @ Y @ Z3 ) ) @ Z3 ) ) ) ) ).
% minus_divide_diff_eq_iff
thf(fact_1929_numeral__3__eq__3,axiom,
( ( numeral_numeral @ nat @ ( bit1 @ one2 ) )
= ( suc @ ( suc @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ).
% numeral_3_eq_3
thf(fact_1930_Suc3__eq__add__3,axiom,
! [N2: nat] :
( ( suc @ ( suc @ ( suc @ N2 ) ) )
= ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) @ N2 ) ) ).
% Suc3_eq_add_3
thf(fact_1931_take__bit__Suc__bit0,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [N2: nat,K: num] :
( ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N2 ) @ ( numeral_numeral @ A @ ( bit0 @ K ) ) )
= ( times_times @ A @ ( bit_se2584673776208193580ke_bit @ A @ N2 @ ( numeral_numeral @ A @ K ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% take_bit_Suc_bit0
thf(fact_1932_take__bit__nat__eq__self,axiom,
! [M2: nat,N2: nat] :
( ( ord_less @ nat @ M2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) )
=> ( ( bit_se2584673776208193580ke_bit @ nat @ N2 @ M2 )
= M2 ) ) ).
% take_bit_nat_eq_self
thf(fact_1933_take__bit__nat__less__exp,axiom,
! [N2: nat,M2: nat] : ( ord_less @ nat @ ( bit_se2584673776208193580ke_bit @ nat @ N2 @ M2 ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ).
% take_bit_nat_less_exp
thf(fact_1934_take__bit__nat__eq__self__iff,axiom,
! [N2: nat,M2: nat] :
( ( ( bit_se2584673776208193580ke_bit @ nat @ N2 @ M2 )
= M2 )
= ( ord_less @ nat @ M2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) ).
% take_bit_nat_eq_self_iff
thf(fact_1935_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_1936_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_1937_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_1938_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_1939_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_1940_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_1941_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_1942_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_1943_divide__less__eq__numeral_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B3: A,C2: A,W2: num] :
( ( ord_less @ A @ ( divide_divide @ A @ B3 @ C2 ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ B3 @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ 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 @ W2 ) ) @ C2 ) @ B3 ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) ) ) ) ) ) ) ).
% divide_less_eq_numeral(2)
thf(fact_1944_less__divide__eq__numeral_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [W2: num,B3: A,C2: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ ( divide_divide @ A @ B3 @ C2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ 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 @ W2 ) ) @ C2 ) ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% less_divide_eq_numeral(2)
thf(fact_1945_cong__exp__iff__simps_I7_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [Q3: num,N2: num] :
( ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ one2 ) @ ( numeral_numeral @ A @ ( bit0 @ Q3 ) ) )
= ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit1 @ N2 ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q3 ) ) ) )
= ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ N2 ) @ ( numeral_numeral @ A @ Q3 ) )
= ( zero_zero @ A ) ) ) ) ).
% cong_exp_iff_simps(7)
thf(fact_1946_cong__exp__iff__simps_I11_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M2: num,Q3: num] :
( ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit1 @ M2 ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q3 ) ) )
= ( modulo_modulo @ A @ ( numeral_numeral @ A @ one2 ) @ ( numeral_numeral @ A @ ( bit0 @ Q3 ) ) ) )
= ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ M2 ) @ ( numeral_numeral @ A @ Q3 ) )
= ( zero_zero @ A ) ) ) ) ).
% cong_exp_iff_simps(11)
thf(fact_1947_Suc__div__eq__add3__div,axiom,
! [M2: nat,N2: nat] :
( ( divide_divide @ nat @ ( suc @ ( suc @ ( suc @ M2 ) ) ) @ N2 )
= ( divide_divide @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) @ M2 ) @ N2 ) ) ).
% Suc_div_eq_add3_div
thf(fact_1948_neg__one__power__add__eq__neg__one__power__diff,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [K: nat,N2: nat] :
( ( ord_less_eq @ nat @ K @ N2 )
=> ( ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( plus_plus @ nat @ N2 @ K ) )
= ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( minus_minus @ nat @ N2 @ K ) ) ) ) ) ).
% neg_one_power_add_eq_neg_one_power_diff
thf(fact_1949_Suc__mod__eq__add3__mod,axiom,
! [M2: nat,N2: nat] :
( ( modulo_modulo @ nat @ ( suc @ ( suc @ ( suc @ M2 ) ) ) @ N2 )
= ( modulo_modulo @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) @ M2 ) @ N2 ) ) ).
% Suc_mod_eq_add3_mod
thf(fact_1950_take__bit__eq__0__iff,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N2: nat,A3: A] :
( ( ( bit_se2584673776208193580ke_bit @ A @ N2 @ A3 )
= ( zero_zero @ A ) )
= ( dvd_dvd @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) @ A3 ) ) ) ).
% take_bit_eq_0_iff
thf(fact_1951_take__bit__nat__less__self__iff,axiom,
! [N2: nat,M2: nat] :
( ( ord_less @ nat @ ( bit_se2584673776208193580ke_bit @ nat @ N2 @ M2 ) @ M2 )
= ( ord_less_eq @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) @ M2 ) ) ).
% take_bit_nat_less_self_iff
thf(fact_1952_divide__le__eq__numeral_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B3: A,C2: A,W2: num] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ B3 @ C2 ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ B3 @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ 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 @ W2 ) ) @ C2 ) @ B3 ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) ) ) ) ) ) ) ).
% divide_le_eq_numeral(2)
thf(fact_1953_le__divide__eq__numeral_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [W2: num,B3: A,C2: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ ( 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 @ W2 ) ) @ 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 @ W2 ) ) @ C2 ) ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% le_divide_eq_numeral(2)
thf(fact_1954_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_1955_power__minus1__odd,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [N2: nat] :
( ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% power_minus1_odd
thf(fact_1956_take__bit__Suc,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N2: nat,A3: A] :
( ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N2 ) @ A3 )
= ( plus_plus @ A @ ( times_times @ A @ ( bit_se2584673776208193580ke_bit @ A @ N2 @ ( 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_1957_stable__imp__take__bit__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A,N2: nat] :
( ( ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= A3 )
=> ( ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
=> ( ( bit_se2584673776208193580ke_bit @ A @ N2 @ A3 )
= ( zero_zero @ A ) ) )
& ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
=> ( ( bit_se2584673776208193580ke_bit @ A @ N2 @ A3 )
= ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) @ ( one_one @ A ) ) ) ) ) ) ) ).
% stable_imp_take_bit_eq
thf(fact_1958_mod__exhaust__less__4,axiom,
! [M2: nat] :
( ( ( modulo_modulo @ nat @ M2 @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) )
= ( zero_zero @ nat ) )
| ( ( modulo_modulo @ nat @ M2 @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) )
= ( one_one @ nat ) )
| ( ( modulo_modulo @ nat @ M2 @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) )
= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
| ( ( modulo_modulo @ nat @ M2 @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) )
= ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) ) ) ).
% mod_exhaust_less_4
thf(fact_1959_divmod__algorithm__code_I8_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M2: num,N2: num] :
( ( ( ord_less @ num @ M2 @ N2 )
=> ( ( unique8689654367752047608divmod @ A @ ( bit1 @ M2 ) @ ( bit1 @ N2 ) )
= ( product_Pair @ A @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ ( bit1 @ M2 ) ) ) ) )
& ( ~ ( ord_less @ num @ M2 @ N2 )
=> ( ( unique8689654367752047608divmod @ A @ ( bit1 @ M2 ) @ ( bit1 @ N2 ) )
= ( unique1321980374590559556d_step @ A @ ( bit1 @ N2 ) @ ( unique8689654367752047608divmod @ A @ ( bit1 @ M2 ) @ ( bit0 @ ( bit1 @ N2 ) ) ) ) ) ) ) ) ).
% divmod_algorithm_code(8)
thf(fact_1960_divmod__algorithm__code_I7_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M2: num,N2: num] :
( ( ( ord_less_eq @ num @ M2 @ N2 )
=> ( ( unique8689654367752047608divmod @ A @ ( bit0 @ M2 ) @ ( bit1 @ N2 ) )
= ( product_Pair @ A @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ ( bit0 @ M2 ) ) ) ) )
& ( ~ ( ord_less_eq @ num @ M2 @ N2 )
=> ( ( unique8689654367752047608divmod @ A @ ( bit0 @ M2 ) @ ( bit1 @ N2 ) )
= ( unique1321980374590559556d_step @ A @ ( bit1 @ N2 ) @ ( unique8689654367752047608divmod @ A @ ( bit0 @ M2 ) @ ( bit0 @ ( bit1 @ N2 ) ) ) ) ) ) ) ) ).
% divmod_algorithm_code(7)
thf(fact_1961_divmod__step__def,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ( ( unique1321980374590559556d_step @ A )
= ( ^ [L2: num] :
( product_case_prod @ A @ A @ ( product_prod @ A @ A )
@ ^ [Q5: A,R4: A] : ( if @ ( product_prod @ A @ A ) @ ( ord_less_eq @ A @ ( numeral_numeral @ A @ L2 ) @ R4 ) @ ( product_Pair @ A @ A @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Q5 ) @ ( one_one @ A ) ) @ ( minus_minus @ A @ R4 @ ( numeral_numeral @ A @ L2 ) ) ) @ ( product_Pair @ A @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Q5 ) @ R4 ) ) ) ) ) ) ).
% divmod_step_def
thf(fact_1962_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_1963_divmod__divmod__step,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ( ( unique8689654367752047608divmod @ A )
= ( ^ [M: num,N: num] : ( if @ ( product_prod @ A @ A ) @ ( ord_less @ num @ M @ N ) @ ( product_Pair @ A @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ M ) ) @ ( unique1321980374590559556d_step @ A @ N @ ( unique8689654367752047608divmod @ A @ M @ ( bit0 @ N ) ) ) ) ) ) ) ).
% divmod_divmod_step
thf(fact_1964_set__encode__insert,axiom,
! [A4: set @ nat,N2: nat] :
( ( finite_finite2 @ nat @ A4 )
=> ( ~ ( member @ nat @ N2 @ A4 )
=> ( ( nat_set_encode @ ( insert2 @ nat @ N2 @ A4 ) )
= ( plus_plus @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) @ ( nat_set_encode @ A4 ) ) ) ) ) ).
% set_encode_insert
thf(fact_1965_insert__absorb2,axiom,
! [A: $tType,X: A,A4: set @ A] :
( ( insert2 @ A @ X @ ( insert2 @ A @ X @ A4 ) )
= ( insert2 @ A @ X @ A4 ) ) ).
% insert_absorb2
thf(fact_1966_insert__iff,axiom,
! [A: $tType,A3: A,B3: A,A4: set @ A] :
( ( member @ A @ A3 @ ( insert2 @ A @ B3 @ A4 ) )
= ( ( A3 = B3 )
| ( member @ A @ A3 @ A4 ) ) ) ).
% insert_iff
thf(fact_1967_insertCI,axiom,
! [A: $tType,A3: A,B2: set @ A,B3: A] :
( ( ~ ( member @ A @ A3 @ B2 )
=> ( A3 = B3 ) )
=> ( member @ A @ A3 @ ( insert2 @ A @ B3 @ B2 ) ) ) ).
% insertCI
thf(fact_1968_Diff__idemp,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] :
( ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A4 @ B2 ) @ B2 )
= ( minus_minus @ ( set @ A ) @ A4 @ B2 ) ) ).
% Diff_idemp
thf(fact_1969_Diff__iff,axiom,
! [A: $tType,C2: A,A4: set @ A,B2: set @ A] :
( ( member @ A @ C2 @ ( minus_minus @ ( set @ A ) @ A4 @ B2 ) )
= ( ( member @ A @ C2 @ A4 )
& ~ ( member @ A @ C2 @ B2 ) ) ) ).
% Diff_iff
thf(fact_1970_DiffI,axiom,
! [A: $tType,C2: A,A4: set @ A,B2: set @ A] :
( ( member @ A @ C2 @ A4 )
=> ( ~ ( member @ A @ C2 @ B2 )
=> ( member @ A @ C2 @ ( minus_minus @ ( set @ A ) @ A4 @ B2 ) ) ) ) ).
% DiffI
thf(fact_1971_ComplI,axiom,
! [A: $tType,C2: A,A4: set @ A] :
( ~ ( member @ A @ C2 @ A4 )
=> ( member @ A @ C2 @ ( uminus_uminus @ ( set @ A ) @ A4 ) ) ) ).
% ComplI
thf(fact_1972_Compl__iff,axiom,
! [A: $tType,C2: A,A4: set @ A] :
( ( member @ A @ C2 @ ( uminus_uminus @ ( set @ A ) @ A4 ) )
= ( ~ ( member @ A @ C2 @ A4 ) ) ) ).
% Compl_iff
thf(fact_1973_Compl__eq__Compl__iff,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] :
( ( ( uminus_uminus @ ( set @ A ) @ A4 )
= ( uminus_uminus @ ( set @ A ) @ B2 ) )
= ( A4 = B2 ) ) ).
% Compl_eq_Compl_iff
thf(fact_1974_finite__atLeastAtMost__int,axiom,
! [L: int,U: int] : ( finite_finite2 @ int @ ( set_or1337092689740270186AtMost @ int @ L @ U ) ) ).
% finite_atLeastAtMost_int
thf(fact_1975_singletonI,axiom,
! [A: $tType,A3: A] : ( member @ A @ A3 @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% singletonI
thf(fact_1976_finite__insert,axiom,
! [A: $tType,A3: A,A4: set @ A] :
( ( finite_finite2 @ A @ ( insert2 @ A @ A3 @ A4 ) )
= ( finite_finite2 @ A @ A4 ) ) ).
% finite_insert
thf(fact_1977_insert__subset,axiom,
! [A: $tType,X: A,A4: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( insert2 @ A @ X @ A4 ) @ B2 )
= ( ( member @ A @ X @ B2 )
& ( ord_less_eq @ ( set @ A ) @ A4 @ B2 ) ) ) ).
% insert_subset
thf(fact_1978_insert__Diff1,axiom,
! [A: $tType,X: A,B2: set @ A,A4: set @ A] :
( ( member @ A @ X @ B2 )
=> ( ( minus_minus @ ( set @ A ) @ ( insert2 @ A @ X @ A4 ) @ B2 )
= ( minus_minus @ ( set @ A ) @ A4 @ B2 ) ) ) ).
% insert_Diff1
thf(fact_1979_Diff__insert0,axiom,
! [A: $tType,X: A,A4: set @ A,B2: set @ A] :
( ~ ( member @ A @ X @ A4 )
=> ( ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ B2 ) )
= ( minus_minus @ ( set @ A ) @ A4 @ B2 ) ) ) ).
% Diff_insert0
thf(fact_1980_singleton__conv,axiom,
! [A: $tType,A3: A] :
( ( collect @ A
@ ^ [X4: A] : X4 = A3 )
= ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% singleton_conv
thf(fact_1981_singleton__conv2,axiom,
! [A: $tType,A3: A] :
( ( collect @ A
@ ( ^ [Y5: A,Z2: A] : Y5 = Z2
@ A3 ) )
= ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% singleton_conv2
thf(fact_1982_finite__interval__int1,axiom,
! [A3: int,B3: int] :
( finite_finite2 @ int
@ ( collect @ int
@ ^ [I3: int] :
( ( ord_less_eq @ int @ A3 @ I3 )
& ( ord_less_eq @ int @ I3 @ B3 ) ) ) ) ).
% finite_interval_int1
thf(fact_1983_singleton__insert__inj__eq_H,axiom,
! [A: $tType,A3: A,A4: set @ A,B3: A] :
( ( ( insert2 @ A @ A3 @ A4 )
= ( insert2 @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( ( A3 = B3 )
& ( ord_less_eq @ ( set @ A ) @ A4 @ ( insert2 @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_1984_singleton__insert__inj__eq,axiom,
! [A: $tType,B3: A,A3: A,A4: set @ A] :
( ( ( insert2 @ A @ B3 @ ( bot_bot @ ( set @ A ) ) )
= ( insert2 @ A @ A3 @ A4 ) )
= ( ( A3 = B3 )
& ( ord_less_eq @ ( set @ A ) @ A4 @ ( insert2 @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_1985_atLeastAtMost__singleton__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A,B3: A,C2: A] :
( ( ( set_or1337092689740270186AtMost @ A @ A3 @ B3 )
= ( insert2 @ A @ C2 @ ( bot_bot @ ( set @ A ) ) ) )
= ( ( A3 = B3 )
& ( B3 = C2 ) ) ) ) ).
% atLeastAtMost_singleton_iff
thf(fact_1986_atLeastAtMost__singleton,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A] :
( ( set_or1337092689740270186AtMost @ A @ A3 @ A3 )
= ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% atLeastAtMost_singleton
thf(fact_1987_insert__Diff__single,axiom,
! [A: $tType,A3: A,A4: set @ A] :
( ( insert2 @ A @ A3 @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( insert2 @ A @ A3 @ A4 ) ) ).
% insert_Diff_single
thf(fact_1988_finite__Diff__insert,axiom,
! [A: $tType,A4: set @ A,A3: A,B2: set @ A] :
( ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ A3 @ B2 ) ) )
= ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ A4 @ B2 ) ) ) ).
% finite_Diff_insert
thf(fact_1989_pred__numeral__simps_I1_J,axiom,
( ( pred_numeral @ one2 )
= ( zero_zero @ nat ) ) ).
% pred_numeral_simps(1)
thf(fact_1990_eq__numeral__Suc,axiom,
! [K: num,N2: nat] :
( ( ( numeral_numeral @ nat @ K )
= ( suc @ N2 ) )
= ( ( pred_numeral @ K )
= N2 ) ) ).
% eq_numeral_Suc
thf(fact_1991_Suc__eq__numeral,axiom,
! [N2: nat,K: num] :
( ( ( suc @ N2 )
= ( numeral_numeral @ nat @ K ) )
= ( N2
= ( pred_numeral @ K ) ) ) ).
% Suc_eq_numeral
thf(fact_1992_subset__Compl__singleton,axiom,
! [A: $tType,A4: set @ A,B3: A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ ( uminus_uminus @ ( set @ A ) @ ( insert2 @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( ~ ( member @ A @ B3 @ A4 ) ) ) ).
% subset_Compl_singleton
thf(fact_1993_less__numeral__Suc,axiom,
! [K: num,N2: nat] :
( ( ord_less @ nat @ ( numeral_numeral @ nat @ K ) @ ( suc @ N2 ) )
= ( ord_less @ nat @ ( pred_numeral @ K ) @ N2 ) ) ).
% less_numeral_Suc
thf(fact_1994_less__Suc__numeral,axiom,
! [N2: nat,K: num] :
( ( ord_less @ nat @ ( suc @ N2 ) @ ( numeral_numeral @ nat @ K ) )
= ( ord_less @ nat @ N2 @ ( pred_numeral @ K ) ) ) ).
% less_Suc_numeral
thf(fact_1995_le__numeral__Suc,axiom,
! [K: num,N2: nat] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ K ) @ ( suc @ N2 ) )
= ( ord_less_eq @ nat @ ( pred_numeral @ K ) @ N2 ) ) ).
% le_numeral_Suc
thf(fact_1996_le__Suc__numeral,axiom,
! [N2: nat,K: num] :
( ( ord_less_eq @ nat @ ( suc @ N2 ) @ ( numeral_numeral @ nat @ K ) )
= ( ord_less_eq @ nat @ N2 @ ( pred_numeral @ K ) ) ) ).
% le_Suc_numeral
thf(fact_1997_diff__Suc__numeral,axiom,
! [N2: nat,K: num] :
( ( minus_minus @ nat @ ( suc @ N2 ) @ ( numeral_numeral @ nat @ K ) )
= ( minus_minus @ nat @ N2 @ ( pred_numeral @ K ) ) ) ).
% diff_Suc_numeral
thf(fact_1998_diff__numeral__Suc,axiom,
! [K: num,N2: nat] :
( ( minus_minus @ nat @ ( numeral_numeral @ nat @ K ) @ ( suc @ N2 ) )
= ( minus_minus @ nat @ ( pred_numeral @ K ) @ N2 ) ) ).
% diff_numeral_Suc
thf(fact_1999_max__Suc__numeral,axiom,
! [N2: nat,K: num] :
( ( ord_max @ nat @ ( suc @ N2 ) @ ( numeral_numeral @ nat @ K ) )
= ( suc @ ( ord_max @ nat @ N2 @ ( pred_numeral @ K ) ) ) ) ).
% max_Suc_numeral
thf(fact_2000_max__numeral__Suc,axiom,
! [K: num,N2: nat] :
( ( ord_max @ nat @ ( numeral_numeral @ nat @ K ) @ ( suc @ N2 ) )
= ( suc @ ( ord_max @ nat @ ( pred_numeral @ K ) @ N2 ) ) ) ).
% max_numeral_Suc
thf(fact_2001_divmod__algorithm__code_I2_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M2: num] :
( ( unique8689654367752047608divmod @ A @ M2 @ one2 )
= ( product_Pair @ A @ A @ ( numeral_numeral @ A @ M2 ) @ ( zero_zero @ A ) ) ) ) ).
% divmod_algorithm_code(2)
thf(fact_2002_set__replicate,axiom,
! [A: $tType,N2: nat,X: A] :
( ( N2
!= ( zero_zero @ nat ) )
=> ( ( set2 @ A @ ( replicate @ A @ N2 @ X ) )
= ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% set_replicate
thf(fact_2003_divmod__algorithm__code_I3_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [N2: num] :
( ( unique8689654367752047608divmod @ A @ one2 @ ( bit0 @ N2 ) )
= ( product_Pair @ A @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ one2 ) ) ) ) ).
% divmod_algorithm_code(3)
thf(fact_2004_divmod__algorithm__code_I4_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [N2: num] :
( ( unique8689654367752047608divmod @ A @ one2 @ ( bit1 @ N2 ) )
= ( product_Pair @ A @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ one2 ) ) ) ) ).
% divmod_algorithm_code(4)
thf(fact_2005_divmod__algorithm__code_I6_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M2: num,N2: num] :
( ( unique8689654367752047608divmod @ A @ ( bit1 @ M2 ) @ ( bit0 @ N2 ) )
= ( product_case_prod @ A @ A @ ( product_prod @ A @ A )
@ ^ [Q5: A,R4: A] : ( product_Pair @ A @ A @ Q5 @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ R4 ) @ ( one_one @ A ) ) )
@ ( unique8689654367752047608divmod @ A @ M2 @ N2 ) ) ) ) ).
% divmod_algorithm_code(6)
thf(fact_2006_set__diff__eq,axiom,
! [A: $tType] :
( ( minus_minus @ ( set @ A ) )
= ( ^ [A7: set @ A,B6: set @ A] :
( collect @ A
@ ^ [X4: A] :
( ( member @ A @ X4 @ A7 )
& ~ ( member @ A @ X4 @ B6 ) ) ) ) ) ).
% set_diff_eq
thf(fact_2007_minus__set__def,axiom,
! [A: $tType] :
( ( minus_minus @ ( set @ A ) )
= ( ^ [A7: set @ A,B6: set @ A] :
( collect @ A
@ ( minus_minus @ ( A > $o )
@ ^ [X4: A] : ( member @ A @ X4 @ A7 )
@ ^ [X4: A] : ( member @ A @ X4 @ B6 ) ) ) ) ) ).
% minus_set_def
thf(fact_2008_insert__Diff__if,axiom,
! [A: $tType,X: A,B2: set @ A,A4: set @ A] :
( ( ( member @ A @ X @ B2 )
=> ( ( minus_minus @ ( set @ A ) @ ( insert2 @ A @ X @ A4 ) @ B2 )
= ( minus_minus @ ( set @ A ) @ A4 @ B2 ) ) )
& ( ~ ( member @ A @ X @ B2 )
=> ( ( minus_minus @ ( set @ A ) @ ( insert2 @ A @ X @ A4 ) @ B2 )
= ( insert2 @ A @ X @ ( minus_minus @ ( set @ A ) @ A4 @ B2 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_2009_DiffD2,axiom,
! [A: $tType,C2: A,A4: set @ A,B2: set @ A] :
( ( member @ A @ C2 @ ( minus_minus @ ( set @ A ) @ A4 @ B2 ) )
=> ~ ( member @ A @ C2 @ B2 ) ) ).
% DiffD2
thf(fact_2010_DiffD1,axiom,
! [A: $tType,C2: A,A4: set @ A,B2: set @ A] :
( ( member @ A @ C2 @ ( minus_minus @ ( set @ A ) @ A4 @ B2 ) )
=> ( member @ A @ C2 @ A4 ) ) ).
% DiffD1
thf(fact_2011_DiffE,axiom,
! [A: $tType,C2: A,A4: set @ A,B2: set @ A] :
( ( member @ A @ C2 @ ( minus_minus @ ( set @ A ) @ A4 @ B2 ) )
=> ~ ( ( member @ A @ C2 @ A4 )
=> ( member @ A @ C2 @ B2 ) ) ) ).
% DiffE
thf(fact_2012_int__induct,axiom,
! [P: int > $o,K: int,I: int] :
( ( P @ K )
=> ( ! [I4: int] :
( ( ord_less_eq @ int @ K @ I4 )
=> ( ( P @ I4 )
=> ( P @ ( plus_plus @ int @ I4 @ ( one_one @ int ) ) ) ) )
=> ( ! [I4: int] :
( ( ord_less_eq @ int @ I4 @ K )
=> ( ( P @ I4 )
=> ( P @ ( minus_minus @ int @ I4 @ ( one_one @ int ) ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_induct
thf(fact_2013_int__le__induct,axiom,
! [I: int,K: int,P: int > $o] :
( ( ord_less_eq @ int @ I @ K )
=> ( ( P @ K )
=> ( ! [I4: int] :
( ( ord_less_eq @ int @ I4 @ K )
=> ( ( P @ I4 )
=> ( P @ ( minus_minus @ int @ I4 @ ( one_one @ int ) ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_le_induct
thf(fact_2014_zdvd__zdiffD,axiom,
! [K: int,M2: int,N2: int] :
( ( dvd_dvd @ int @ K @ ( minus_minus @ int @ M2 @ N2 ) )
=> ( ( dvd_dvd @ int @ K @ N2 )
=> ( dvd_dvd @ int @ K @ M2 ) ) ) ).
% zdvd_zdiffD
thf(fact_2015_ComplD,axiom,
! [A: $tType,C2: A,A4: set @ A] :
( ( member @ A @ C2 @ ( uminus_uminus @ ( set @ A ) @ A4 ) )
=> ~ ( member @ A @ C2 @ A4 ) ) ).
% ComplD
thf(fact_2016_double__complement,axiom,
! [A: $tType,A4: set @ A] :
( ( uminus_uminus @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A4 ) )
= A4 ) ).
% double_complement
thf(fact_2017_uminus__set__def,axiom,
! [A: $tType] :
( ( uminus_uminus @ ( set @ A ) )
= ( ^ [A7: set @ A] :
( collect @ A
@ ( uminus_uminus @ ( A > $o )
@ ^ [X4: A] : ( member @ A @ X4 @ A7 ) ) ) ) ) ).
% uminus_set_def
thf(fact_2018_Collect__neg__eq,axiom,
! [A: $tType,P: A > $o] :
( ( collect @ A
@ ^ [X4: A] :
~ ( P @ X4 ) )
= ( uminus_uminus @ ( set @ A ) @ ( collect @ A @ P ) ) ) ).
% Collect_neg_eq
thf(fact_2019_Compl__eq,axiom,
! [A: $tType] :
( ( uminus_uminus @ ( set @ A ) )
= ( ^ [A7: set @ A] :
( collect @ A
@ ^ [X4: A] :
~ ( member @ A @ X4 @ A7 ) ) ) ) ).
% Compl_eq
thf(fact_2020_insert__compr,axiom,
! [A: $tType] :
( ( insert2 @ A )
= ( ^ [A5: A,B6: set @ A] :
( collect @ A
@ ^ [X4: A] :
( ( X4 = A5 )
| ( member @ A @ X4 @ B6 ) ) ) ) ) ).
% insert_compr
thf(fact_2021_insert__Collect,axiom,
! [A: $tType,A3: A,P: A > $o] :
( ( insert2 @ A @ A3 @ ( collect @ A @ P ) )
= ( collect @ A
@ ^ [U2: A] :
( ( U2 != A3 )
=> ( P @ U2 ) ) ) ) ).
% insert_Collect
thf(fact_2022_mk__disjoint__insert,axiom,
! [A: $tType,A3: A,A4: set @ A] :
( ( member @ A @ A3 @ A4 )
=> ? [B9: set @ A] :
( ( A4
= ( insert2 @ A @ A3 @ B9 ) )
& ~ ( member @ A @ A3 @ B9 ) ) ) ).
% mk_disjoint_insert
thf(fact_2023_insert__commute,axiom,
! [A: $tType,X: A,Y: A,A4: set @ A] :
( ( insert2 @ A @ X @ ( insert2 @ A @ Y @ A4 ) )
= ( insert2 @ A @ Y @ ( insert2 @ A @ X @ A4 ) ) ) ).
% insert_commute
thf(fact_2024_insert__eq__iff,axiom,
! [A: $tType,A3: A,A4: set @ A,B3: A,B2: set @ A] :
( ~ ( member @ A @ A3 @ A4 )
=> ( ~ ( member @ A @ B3 @ B2 )
=> ( ( ( insert2 @ A @ A3 @ A4 )
= ( insert2 @ A @ B3 @ B2 ) )
= ( ( ( A3 = B3 )
=> ( A4 = B2 ) )
& ( ( A3 != B3 )
=> ? [C6: set @ A] :
( ( A4
= ( insert2 @ A @ B3 @ C6 ) )
& ~ ( member @ A @ B3 @ C6 )
& ( B2
= ( insert2 @ A @ A3 @ C6 ) )
& ~ ( member @ A @ A3 @ C6 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_2025_insert__absorb,axiom,
! [A: $tType,A3: A,A4: set @ A] :
( ( member @ A @ A3 @ A4 )
=> ( ( insert2 @ A @ A3 @ A4 )
= A4 ) ) ).
% insert_absorb
thf(fact_2026_insert__ident,axiom,
! [A: $tType,X: A,A4: set @ A,B2: set @ A] :
( ~ ( member @ A @ X @ A4 )
=> ( ~ ( member @ A @ X @ B2 )
=> ( ( ( insert2 @ A @ X @ A4 )
= ( insert2 @ A @ X @ B2 ) )
= ( A4 = B2 ) ) ) ) ).
% insert_ident
thf(fact_2027_Set_Oset__insert,axiom,
! [A: $tType,X: A,A4: set @ A] :
( ( member @ A @ X @ A4 )
=> ~ ! [B9: set @ A] :
( ( A4
= ( insert2 @ A @ X @ B9 ) )
=> ( member @ A @ X @ B9 ) ) ) ).
% Set.set_insert
thf(fact_2028_insertI2,axiom,
! [A: $tType,A3: A,B2: set @ A,B3: A] :
( ( member @ A @ A3 @ B2 )
=> ( member @ A @ A3 @ ( insert2 @ A @ B3 @ B2 ) ) ) ).
% insertI2
thf(fact_2029_insertI1,axiom,
! [A: $tType,A3: A,B2: set @ A] : ( member @ A @ A3 @ ( insert2 @ A @ A3 @ B2 ) ) ).
% insertI1
thf(fact_2030_insertE,axiom,
! [A: $tType,A3: A,B3: A,A4: set @ A] :
( ( member @ A @ A3 @ ( insert2 @ A @ B3 @ A4 ) )
=> ( ( A3 != B3 )
=> ( member @ A @ A3 @ A4 ) ) ) ).
% insertE
thf(fact_2031_int__ge__induct,axiom,
! [K: int,I: int,P: int > $o] :
( ( ord_less_eq @ int @ K @ I )
=> ( ( P @ K )
=> ( ! [I4: int] :
( ( ord_less_eq @ int @ K @ I4 )
=> ( ( P @ I4 )
=> ( P @ ( plus_plus @ int @ I4 @ ( one_one @ int ) ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_ge_induct
thf(fact_2032_singletonD,axiom,
! [A: $tType,B3: A,A3: A] :
( ( member @ A @ B3 @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) )
=> ( B3 = A3 ) ) ).
% singletonD
thf(fact_2033_singleton__iff,axiom,
! [A: $tType,B3: A,A3: A] :
( ( member @ A @ B3 @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( B3 = A3 ) ) ).
% singleton_iff
thf(fact_2034_doubleton__eq__iff,axiom,
! [A: $tType,A3: A,B3: A,C2: A,D2: A] :
( ( ( insert2 @ A @ A3 @ ( insert2 @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( insert2 @ A @ C2 @ ( insert2 @ A @ D2 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( ( ( A3 = C2 )
& ( B3 = D2 ) )
| ( ( A3 = D2 )
& ( B3 = C2 ) ) ) ) ).
% doubleton_eq_iff
thf(fact_2035_insert__not__empty,axiom,
! [A: $tType,A3: A,A4: set @ A] :
( ( insert2 @ A @ A3 @ A4 )
!= ( bot_bot @ ( set @ A ) ) ) ).
% insert_not_empty
thf(fact_2036_singleton__inject,axiom,
! [A: $tType,A3: A,B3: A] :
( ( ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) )
= ( insert2 @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) )
=> ( A3 = B3 ) ) ).
% singleton_inject
thf(fact_2037_Compl__insert,axiom,
! [A: $tType,X: A,A4: set @ A] :
( ( uminus_uminus @ ( set @ A ) @ ( insert2 @ A @ X @ A4 ) )
= ( minus_minus @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A4 ) @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Compl_insert
thf(fact_2038_Diff__insert,axiom,
! [A: $tType,A4: set @ A,A3: A,B2: set @ A] :
( ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ A3 @ B2 ) )
= ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A4 @ B2 ) @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Diff_insert
thf(fact_2039_insert__Diff,axiom,
! [A: $tType,A3: A,A4: set @ A] :
( ( member @ A @ A3 @ A4 )
=> ( ( insert2 @ A @ A3 @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) )
= A4 ) ) ).
% insert_Diff
thf(fact_2040_Diff__insert2,axiom,
! [A: $tType,A4: set @ A,A3: A,B2: set @ A] :
( ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ A3 @ B2 ) )
= ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) @ B2 ) ) ).
% Diff_insert2
thf(fact_2041_Diff__insert__absorb,axiom,
! [A: $tType,X: A,A4: set @ A] :
( ~ ( member @ A @ X @ A4 )
=> ( ( minus_minus @ ( set @ A ) @ ( insert2 @ A @ X @ A4 ) @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= A4 ) ) ).
% Diff_insert_absorb
thf(fact_2042_finite_OinsertI,axiom,
! [A: $tType,A4: set @ A,A3: A] :
( ( finite_finite2 @ A @ A4 )
=> ( finite_finite2 @ A @ ( insert2 @ A @ A3 @ A4 ) ) ) ).
% finite.insertI
thf(fact_2043_insert__mono,axiom,
! [A: $tType,C5: set @ A,D6: set @ A,A3: A] :
( ( ord_less_eq @ ( set @ A ) @ C5 @ D6 )
=> ( ord_less_eq @ ( set @ A ) @ ( insert2 @ A @ A3 @ C5 ) @ ( insert2 @ A @ A3 @ D6 ) ) ) ).
% insert_mono
thf(fact_2044_subset__insert,axiom,
! [A: $tType,X: A,A4: set @ A,B2: set @ A] :
( ~ ( member @ A @ X @ A4 )
=> ( ( ord_less_eq @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ B2 ) )
= ( ord_less_eq @ ( set @ A ) @ A4 @ B2 ) ) ) ).
% subset_insert
thf(fact_2045_subset__insertI,axiom,
! [A: $tType,B2: set @ A,A3: A] : ( ord_less_eq @ ( set @ A ) @ B2 @ ( insert2 @ A @ A3 @ B2 ) ) ).
% subset_insertI
thf(fact_2046_subset__insertI2,axiom,
! [A: $tType,A4: set @ A,B2: set @ A,B3: A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B2 )
=> ( ord_less_eq @ ( set @ A ) @ A4 @ ( insert2 @ A @ B3 @ B2 ) ) ) ).
% subset_insertI2
thf(fact_2047_insert__subsetI,axiom,
! [A: $tType,X: A,A4: set @ A,X8: set @ A] :
( ( member @ A @ X @ A4 )
=> ( ( ord_less_eq @ ( set @ A ) @ X8 @ A4 )
=> ( ord_less_eq @ ( set @ A ) @ ( insert2 @ A @ X @ X8 ) @ A4 ) ) ) ).
% insert_subsetI
thf(fact_2048_subset__Diff__insert,axiom,
! [A: $tType,A4: set @ A,B2: set @ A,X: A,C5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ ( minus_minus @ ( set @ A ) @ B2 @ ( insert2 @ A @ X @ C5 ) ) )
= ( ( ord_less_eq @ ( set @ A ) @ A4 @ ( minus_minus @ ( set @ A ) @ B2 @ C5 ) )
& ~ ( member @ A @ X @ A4 ) ) ) ).
% subset_Diff_insert
thf(fact_2049_Collect__conv__if,axiom,
! [A: $tType,P: A > $o,A3: A] :
( ( ( P @ A3 )
=> ( ( collect @ A
@ ^ [X4: A] :
( ( X4 = A3 )
& ( P @ X4 ) ) )
= ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) )
& ( ~ ( P @ A3 )
=> ( ( collect @ A
@ ^ [X4: A] :
( ( X4 = A3 )
& ( P @ X4 ) ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Collect_conv_if
thf(fact_2050_Collect__conv__if2,axiom,
! [A: $tType,P: A > $o,A3: A] :
( ( ( P @ A3 )
=> ( ( collect @ A
@ ^ [X4: A] :
( ( A3 = X4 )
& ( P @ X4 ) ) )
= ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) )
& ( ~ ( P @ A3 )
=> ( ( collect @ A
@ ^ [X4: A] :
( ( A3 = X4 )
& ( P @ X4 ) ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Collect_conv_if2
thf(fact_2051_finite_Ocases,axiom,
! [A: $tType,A3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ~ ! [A8: set @ A] :
( ? [A6: A] :
( A3
= ( insert2 @ A @ A6 @ A8 ) )
=> ~ ( finite_finite2 @ A @ A8 ) ) ) ) ).
% finite.cases
thf(fact_2052_finite_Osimps,axiom,
! [A: $tType] :
( ( finite_finite2 @ A )
= ( ^ [A5: set @ A] :
( ( A5
= ( bot_bot @ ( set @ A ) ) )
| ? [A7: set @ A,B4: A] :
( ( A5
= ( insert2 @ A @ B4 @ A7 ) )
& ( finite_finite2 @ A @ A7 ) ) ) ) ) ).
% finite.simps
thf(fact_2053_finite__induct,axiom,
! [A: $tType,F5: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ F5 )
=> ( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [X5: A,F6: set @ A] :
( ( finite_finite2 @ A @ F6 )
=> ( ~ ( member @ A @ X5 @ F6 )
=> ( ( P @ F6 )
=> ( P @ ( insert2 @ A @ X5 @ F6 ) ) ) ) )
=> ( P @ F5 ) ) ) ) ).
% finite_induct
thf(fact_2054_finite__ne__induct,axiom,
! [A: $tType,F5: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ F5 )
=> ( ( F5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X5: A] : ( P @ ( insert2 @ A @ X5 @ ( bot_bot @ ( set @ A ) ) ) )
=> ( ! [X5: A,F6: set @ A] :
( ( finite_finite2 @ A @ F6 )
=> ( ( F6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ~ ( member @ A @ X5 @ F6 )
=> ( ( P @ F6 )
=> ( P @ ( insert2 @ A @ X5 @ F6 ) ) ) ) ) )
=> ( P @ F5 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_2055_infinite__finite__induct,axiom,
! [A: $tType,P: ( set @ A ) > $o,A4: set @ A] :
( ! [A8: set @ A] :
( ~ ( finite_finite2 @ A @ A8 )
=> ( P @ A8 ) )
=> ( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [X5: A,F6: set @ A] :
( ( finite_finite2 @ A @ F6 )
=> ( ~ ( member @ A @ X5 @ F6 )
=> ( ( P @ F6 )
=> ( P @ ( insert2 @ A @ X5 @ F6 ) ) ) ) )
=> ( P @ A4 ) ) ) ) ).
% infinite_finite_induct
thf(fact_2056_infinite__remove,axiom,
! [A: $tType,S2: set @ A,A3: A] :
( ~ ( finite_finite2 @ A @ S2 )
=> ~ ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ S2 @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% infinite_remove
thf(fact_2057_infinite__coinduct,axiom,
! [A: $tType,X8: ( set @ A ) > $o,A4: set @ A] :
( ( X8 @ A4 )
=> ( ! [A8: set @ A] :
( ( X8 @ A8 )
=> ? [X2: A] :
( ( member @ A @ X2 @ A8 )
& ( ( X8 @ ( minus_minus @ ( set @ A ) @ A8 @ ( insert2 @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) )
| ~ ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ A8 @ ( insert2 @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) )
=> ~ ( finite_finite2 @ A @ A4 ) ) ) ).
% infinite_coinduct
thf(fact_2058_finite__empty__induct,axiom,
! [A: $tType,A4: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ A4 )
=> ( ( P @ A4 )
=> ( ! [A6: A,A8: set @ A] :
( ( finite_finite2 @ A @ A8 )
=> ( ( member @ A @ A6 @ A8 )
=> ( ( P @ A8 )
=> ( P @ ( minus_minus @ ( set @ A ) @ A8 @ ( insert2 @ A @ A6 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) )
=> ( P @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% finite_empty_induct
thf(fact_2059_subset__singleton__iff,axiom,
! [A: $tType,X8: set @ A,A3: A] :
( ( ord_less_eq @ ( set @ A ) @ X8 @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( ( X8
= ( bot_bot @ ( set @ A ) ) )
| ( X8
= ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% subset_singleton_iff
thf(fact_2060_subset__singletonD,axiom,
! [A: $tType,A4: set @ A,X: A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
=> ( ( A4
= ( bot_bot @ ( set @ A ) ) )
| ( A4
= ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% subset_singletonD
thf(fact_2061_Diff__single__insert,axiom,
! [A: $tType,A4: set @ A,X: A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) @ B2 )
=> ( ord_less_eq @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ B2 ) ) ) ).
% Diff_single_insert
thf(fact_2062_subset__insert__iff,axiom,
! [A: $tType,A4: set @ A,X: A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ B2 ) )
= ( ( ( member @ A @ X @ A4 )
=> ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) @ B2 ) )
& ( ~ ( member @ A @ X @ A4 )
=> ( ord_less_eq @ ( set @ A ) @ A4 @ B2 ) ) ) ) ).
% subset_insert_iff
thf(fact_2063_atLeastAtMost__singleton_H,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A,B3: A] :
( ( A3 = B3 )
=> ( ( set_or1337092689740270186AtMost @ A @ A3 @ B3 )
= ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% atLeastAtMost_singleton'
thf(fact_2064_numeral__eq__Suc,axiom,
( ( numeral_numeral @ nat )
= ( ^ [K2: num] : ( suc @ ( pred_numeral @ K2 ) ) ) ) ).
% numeral_eq_Suc
thf(fact_2065_finite__ranking__induct,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [S2: set @ B,P: ( set @ B ) > $o,F2: B > A] :
( ( finite_finite2 @ B @ S2 )
=> ( ( P @ ( bot_bot @ ( set @ B ) ) )
=> ( ! [X5: B,S4: set @ B] :
( ( finite_finite2 @ B @ S4 )
=> ( ! [Y4: B] :
( ( member @ B @ Y4 @ S4 )
=> ( ord_less_eq @ A @ ( F2 @ Y4 ) @ ( F2 @ X5 ) ) )
=> ( ( P @ S4 )
=> ( P @ ( insert2 @ B @ X5 @ S4 ) ) ) ) )
=> ( P @ S2 ) ) ) ) ) ).
% finite_ranking_induct
thf(fact_2066_finite__linorder__max__induct,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ A4 )
=> ( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [B5: A,A8: set @ A] :
( ( finite_finite2 @ A @ A8 )
=> ( ! [X2: A] :
( ( member @ A @ X2 @ A8 )
=> ( ord_less @ A @ X2 @ B5 ) )
=> ( ( P @ A8 )
=> ( P @ ( insert2 @ A @ B5 @ A8 ) ) ) ) )
=> ( P @ A4 ) ) ) ) ) ).
% finite_linorder_max_induct
thf(fact_2067_finite__linorder__min__induct,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ A4 )
=> ( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [B5: A,A8: set @ A] :
( ( finite_finite2 @ A @ A8 )
=> ( ! [X2: A] :
( ( member @ A @ X2 @ A8 )
=> ( ord_less @ A @ B5 @ X2 ) )
=> ( ( P @ A8 )
=> ( P @ ( insert2 @ A @ B5 @ A8 ) ) ) ) )
=> ( P @ A4 ) ) ) ) ) ).
% finite_linorder_min_induct
thf(fact_2068_finite__subset__induct_H,axiom,
! [A: $tType,F5: set @ A,A4: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ F5 )
=> ( ( ord_less_eq @ ( set @ A ) @ F5 @ A4 )
=> ( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [A6: A,F6: set @ A] :
( ( finite_finite2 @ A @ F6 )
=> ( ( member @ A @ A6 @ A4 )
=> ( ( ord_less_eq @ ( set @ A ) @ F6 @ A4 )
=> ( ~ ( member @ A @ A6 @ F6 )
=> ( ( P @ F6 )
=> ( P @ ( insert2 @ A @ A6 @ F6 ) ) ) ) ) ) )
=> ( P @ F5 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_2069_finite__subset__induct,axiom,
! [A: $tType,F5: set @ A,A4: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ F5 )
=> ( ( ord_less_eq @ ( set @ A ) @ F5 @ A4 )
=> ( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [A6: A,F6: set @ A] :
( ( finite_finite2 @ A @ F6 )
=> ( ( member @ A @ A6 @ A4 )
=> ( ~ ( member @ A @ A6 @ F6 )
=> ( ( P @ F6 )
=> ( P @ ( insert2 @ A @ A6 @ F6 ) ) ) ) ) )
=> ( P @ F5 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_2070_finite__remove__induct,axiom,
! [A: $tType,B2: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ B2 )
=> ( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [A8: set @ A] :
( ( finite_finite2 @ A @ A8 )
=> ( ( A8
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ A ) @ A8 @ B2 )
=> ( ! [X2: A] :
( ( member @ A @ X2 @ A8 )
=> ( P @ ( minus_minus @ ( set @ A ) @ A8 @ ( insert2 @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) )
=> ( P @ A8 ) ) ) ) )
=> ( P @ B2 ) ) ) ) ).
% finite_remove_induct
thf(fact_2071_remove__induct,axiom,
! [A: $tType,P: ( set @ A ) > $o,B2: set @ A] :
( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ( ~ ( finite_finite2 @ A @ B2 )
=> ( P @ B2 ) )
=> ( ! [A8: set @ A] :
( ( finite_finite2 @ A @ A8 )
=> ( ( A8
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ A ) @ A8 @ B2 )
=> ( ! [X2: A] :
( ( member @ A @ X2 @ A8 )
=> ( P @ ( minus_minus @ ( set @ A ) @ A8 @ ( insert2 @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) )
=> ( P @ A8 ) ) ) ) )
=> ( P @ B2 ) ) ) ) ).
% remove_induct
thf(fact_2072_atLeast0__atMost__Suc,axiom,
! [N2: nat] :
( ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( suc @ N2 ) )
= ( insert2 @ nat @ ( suc @ N2 ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ).
% atLeast0_atMost_Suc
thf(fact_2073_finite__induct__select,axiom,
! [A: $tType,S2: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ S2 )
=> ( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [T5: set @ A] :
( ( ord_less @ ( set @ A ) @ T5 @ S2 )
=> ( ( P @ T5 )
=> ? [X2: A] :
( ( member @ A @ X2 @ ( minus_minus @ ( set @ A ) @ S2 @ T5 ) )
& ( P @ ( insert2 @ A @ X2 @ T5 ) ) ) ) )
=> ( P @ S2 ) ) ) ) ).
% finite_induct_select
thf(fact_2074_psubset__insert__iff,axiom,
! [A: $tType,A4: set @ A,X: A,B2: set @ A] :
( ( ord_less @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ B2 ) )
= ( ( ( member @ A @ X @ B2 )
=> ( ord_less @ ( set @ A ) @ A4 @ B2 ) )
& ( ~ ( member @ A @ X @ B2 )
=> ( ( ( member @ A @ X @ A4 )
=> ( ord_less @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) @ B2 ) )
& ( ~ ( member @ A @ X @ A4 )
=> ( ord_less_eq @ ( set @ A ) @ A4 @ B2 ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_2075_atLeastAtMost__insertL,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq @ nat @ M2 @ N2 )
=> ( ( insert2 @ nat @ M2 @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M2 ) @ N2 ) )
= ( set_or1337092689740270186AtMost @ nat @ M2 @ N2 ) ) ) ).
% atLeastAtMost_insertL
thf(fact_2076_atLeastAtMostSuc__conv,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq @ nat @ M2 @ ( suc @ N2 ) )
=> ( ( set_or1337092689740270186AtMost @ nat @ M2 @ ( suc @ N2 ) )
= ( insert2 @ nat @ ( suc @ N2 ) @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N2 ) ) ) ) ).
% atLeastAtMostSuc_conv
thf(fact_2077_Icc__eq__insert__lb__nat,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq @ nat @ M2 @ N2 )
=> ( ( set_or1337092689740270186AtMost @ nat @ M2 @ N2 )
= ( insert2 @ nat @ M2 @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M2 ) @ N2 ) ) ) ) ).
% Icc_eq_insert_lb_nat
thf(fact_2078_set__update__subset__insert,axiom,
! [A: $tType,Xs: list @ A,I: nat,X: A] : ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ ( list_update @ A @ Xs @ I @ X ) ) @ ( insert2 @ A @ X @ ( set2 @ A @ Xs ) ) ) ).
% set_update_subset_insert
thf(fact_2079_set__replicate__Suc,axiom,
! [A: $tType,N2: nat,X: A] :
( ( set2 @ A @ ( replicate @ A @ ( suc @ N2 ) @ X ) )
= ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ).
% set_replicate_Suc
thf(fact_2080_set__replicate__conv__if,axiom,
! [A: $tType,N2: nat,X: A] :
( ( ( N2
= ( zero_zero @ nat ) )
=> ( ( set2 @ A @ ( replicate @ A @ N2 @ X ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( N2
!= ( zero_zero @ nat ) )
=> ( ( set2 @ A @ ( replicate @ A @ N2 @ X ) )
= ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% set_replicate_conv_if
thf(fact_2081_set__decode__plus__power__2,axiom,
! [N2: nat,Z3: nat] :
( ~ ( member @ nat @ N2 @ ( nat_set_decode @ Z3 ) )
=> ( ( nat_set_decode @ ( plus_plus @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) @ Z3 ) )
= ( insert2 @ nat @ N2 @ ( nat_set_decode @ Z3 ) ) ) ) ).
% set_decode_plus_power_2
thf(fact_2082_divmod__step__nat__def,axiom,
( ( unique1321980374590559556d_step @ nat )
= ( ^ [L2: num] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [Q5: nat,R4: nat] : ( if @ ( product_prod @ nat @ nat ) @ ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ L2 ) @ R4 ) @ ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Q5 ) @ ( one_one @ nat ) ) @ ( minus_minus @ nat @ R4 @ ( numeral_numeral @ nat @ L2 ) ) ) @ ( product_Pair @ nat @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Q5 ) @ R4 ) ) ) ) ) ).
% divmod_step_nat_def
thf(fact_2083_take__bit__Suc__minus__bit1,axiom,
! [N2: nat,K: num] :
( ( bit_se2584673776208193580ke_bit @ int @ ( suc @ N2 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ K ) ) ) )
= ( plus_plus @ int @ ( times_times @ int @ ( bit_se2584673776208193580ke_bit @ int @ N2 @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( inc @ K ) ) ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( one_one @ int ) ) ) ).
% take_bit_Suc_minus_bit1
thf(fact_2084_divmod__nat__if,axiom,
( divmod_nat
= ( ^ [M: nat,N: nat] :
( if @ ( product_prod @ nat @ nat )
@ ( ( N
= ( zero_zero @ nat ) )
| ( ord_less @ nat @ M @ N ) )
@ ( product_Pair @ nat @ nat @ ( zero_zero @ nat ) @ M )
@ ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [Q5: nat] : ( product_Pair @ nat @ nat @ ( suc @ Q5 ) )
@ ( divmod_nat @ ( minus_minus @ nat @ M @ N ) @ N ) ) ) ) ) ).
% divmod_nat_if
thf(fact_2085_mask__numeral,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N2: num] :
( ( bit_se2239418461657761734s_mask @ A @ ( numeral_numeral @ nat @ N2 ) )
= ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se2239418461657761734s_mask @ A @ ( pred_numeral @ N2 ) ) ) ) ) ) ).
% mask_numeral
thf(fact_2086_of__int__code__if,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ( ( ring_1_of_int @ A )
= ( ^ [K2: int] :
( if @ A
@ ( K2
= ( zero_zero @ int ) )
@ ( zero_zero @ A )
@ ( if @ A @ ( ord_less @ int @ K2 @ ( zero_zero @ int ) ) @ ( uminus_uminus @ A @ ( ring_1_of_int @ A @ ( uminus_uminus @ int @ K2 ) ) )
@ ( if @ A
@ ( ( modulo_modulo @ int @ K2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) )
= ( zero_zero @ int ) )
@ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( ring_1_of_int @ A @ ( divide_divide @ int @ K2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) )
@ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( ring_1_of_int @ A @ ( divide_divide @ int @ K2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) @ ( one_one @ A ) ) ) ) ) ) ) ) ).
% of_int_code_if
thf(fact_2087_concat__bit__Suc,axiom,
! [N2: nat,K: int,L: int] :
( ( bit_concat_bit @ ( suc @ N2 ) @ 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 @ N2 @ ( divide_divide @ int @ K @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ L ) ) ) ) ).
% concat_bit_Suc
thf(fact_2088_of__int__less__neg__numeral__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: int,X: num,N2: nat] :
( ( ord_less @ A @ ( ring_1_of_int @ A @ A3 ) @ ( power_power @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ X ) ) @ N2 ) )
= ( ord_less @ int @ A3 @ ( power_power @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ X ) ) @ N2 ) ) ) ) ).
% of_int_less_neg_numeral_power_cancel_iff
thf(fact_2089_neg__numeral__power__less__of__int__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: num,N2: nat,A3: int] :
( ( ord_less @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ X ) ) @ N2 ) @ ( ring_1_of_int @ A @ A3 ) )
= ( ord_less @ int @ ( power_power @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ X ) ) @ N2 ) @ A3 ) ) ) ).
% neg_numeral_power_less_of_int_cancel_iff
thf(fact_2090_of__int__eq__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [W2: int,Z3: int] :
( ( ( ring_1_of_int @ A @ W2 )
= ( ring_1_of_int @ A @ Z3 ) )
= ( W2 = Z3 ) ) ) ).
% of_int_eq_iff
thf(fact_2091_mask__nat__positive__iff,axiom,
! [N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( bit_se2239418461657761734s_mask @ nat @ N2 ) )
= ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) ) ).
% mask_nat_positive_iff
thf(fact_2092_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_2093_concat__bit__0,axiom,
! [K: int,L: int] :
( ( bit_concat_bit @ ( zero_zero @ nat ) @ K @ L )
= L ) ).
% concat_bit_0
thf(fact_2094_of__int__eq__0__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [Z3: int] :
( ( ( ring_1_of_int @ A @ Z3 )
= ( zero_zero @ A ) )
= ( Z3
= ( zero_zero @ int ) ) ) ) ).
% of_int_eq_0_iff
thf(fact_2095_of__int__0__eq__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [Z3: int] :
( ( ( zero_zero @ A )
= ( ring_1_of_int @ A @ Z3 ) )
= ( Z3
= ( zero_zero @ int ) ) ) ) ).
% of_int_0_eq_iff
thf(fact_2096_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_2097_of__int__le__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [W2: int,Z3: int] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ W2 ) @ ( ring_1_of_int @ A @ Z3 ) )
= ( ord_less_eq @ int @ W2 @ Z3 ) ) ) ).
% of_int_le_iff
thf(fact_2098_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_2099_of__int__eq__numeral__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [Z3: int,N2: num] :
( ( ( ring_1_of_int @ A @ Z3 )
= ( numeral_numeral @ A @ N2 ) )
= ( Z3
= ( numeral_numeral @ int @ N2 ) ) ) ) ).
% of_int_eq_numeral_iff
thf(fact_2100_of__int__less__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [W2: int,Z3: int] :
( ( ord_less @ A @ ( ring_1_of_int @ A @ W2 ) @ ( ring_1_of_int @ A @ Z3 ) )
= ( ord_less @ int @ W2 @ Z3 ) ) ) ).
% of_int_less_iff
thf(fact_2101_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_2102_of__int__eq__1__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [Z3: int] :
( ( ( ring_1_of_int @ A @ Z3 )
= ( one_one @ A ) )
= ( Z3
= ( one_one @ int ) ) ) ) ).
% of_int_eq_1_iff
thf(fact_2103_of__int__mult,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [W2: int,Z3: int] :
( ( ring_1_of_int @ A @ ( times_times @ int @ W2 @ Z3 ) )
= ( times_times @ A @ ( ring_1_of_int @ A @ W2 ) @ ( ring_1_of_int @ A @ Z3 ) ) ) ) ).
% of_int_mult
thf(fact_2104_of__int__add,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [W2: int,Z3: int] :
( ( ring_1_of_int @ A @ ( plus_plus @ int @ W2 @ Z3 ) )
= ( plus_plus @ A @ ( ring_1_of_int @ A @ W2 ) @ ( ring_1_of_int @ A @ Z3 ) ) ) ) ).
% of_int_add
thf(fact_2105_of__int__minus,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [Z3: int] :
( ( ring_1_of_int @ A @ ( uminus_uminus @ int @ Z3 ) )
= ( uminus_uminus @ A @ ( ring_1_of_int @ A @ Z3 ) ) ) ) ).
% of_int_minus
thf(fact_2106_mask__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se2239418461657761734s_mask @ A @ ( zero_zero @ nat ) )
= ( zero_zero @ A ) ) ) ).
% mask_0
thf(fact_2107_mask__eq__0__iff,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N2: nat] :
( ( ( bit_se2239418461657761734s_mask @ A @ N2 )
= ( zero_zero @ A ) )
= ( N2
= ( zero_zero @ nat ) ) ) ) ).
% mask_eq_0_iff
thf(fact_2108_of__int__diff,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [W2: int,Z3: int] :
( ( ring_1_of_int @ A @ ( minus_minus @ int @ W2 @ Z3 ) )
= ( minus_minus @ A @ ( ring_1_of_int @ A @ W2 ) @ ( ring_1_of_int @ A @ Z3 ) ) ) ) ).
% of_int_diff
thf(fact_2109_of__int__power,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [Z3: int,N2: nat] :
( ( ring_1_of_int @ A @ ( power_power @ int @ Z3 @ N2 ) )
= ( power_power @ A @ ( ring_1_of_int @ A @ Z3 ) @ N2 ) ) ) ).
% of_int_power
thf(fact_2110_of__int__eq__of__int__power__cancel__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [B3: int,W2: nat,X: int] :
( ( ( power_power @ A @ ( ring_1_of_int @ A @ B3 ) @ W2 )
= ( ring_1_of_int @ A @ X ) )
= ( ( power_power @ int @ B3 @ W2 )
= X ) ) ) ).
% of_int_eq_of_int_power_cancel_iff
thf(fact_2111_of__int__power__eq__of__int__cancel__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [X: int,B3: int,W2: nat] :
( ( ( ring_1_of_int @ A @ X )
= ( power_power @ A @ ( ring_1_of_int @ A @ B3 ) @ W2 ) )
= ( X
= ( power_power @ int @ B3 @ W2 ) ) ) ) ).
% of_int_power_eq_of_int_cancel_iff
thf(fact_2112_mask__Suc__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se2239418461657761734s_mask @ A @ ( suc @ ( zero_zero @ nat ) ) )
= ( one_one @ A ) ) ) ).
% mask_Suc_0
thf(fact_2113_of__int__0__le__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z3: int] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( ring_1_of_int @ A @ Z3 ) )
= ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z3 ) ) ) ).
% of_int_0_le_iff
thf(fact_2114_of__int__le__0__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z3: int] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Z3 ) @ ( zero_zero @ A ) )
= ( ord_less_eq @ int @ Z3 @ ( zero_zero @ int ) ) ) ) ).
% of_int_le_0_iff
thf(fact_2115_of__int__0__less__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z3: int] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( ring_1_of_int @ A @ Z3 ) )
= ( ord_less @ int @ ( zero_zero @ int ) @ Z3 ) ) ) ).
% of_int_0_less_iff
thf(fact_2116_of__int__less__0__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z3: int] :
( ( ord_less @ A @ ( ring_1_of_int @ A @ Z3 ) @ ( zero_zero @ A ) )
= ( ord_less @ int @ Z3 @ ( zero_zero @ int ) ) ) ) ).
% of_int_less_0_iff
thf(fact_2117_of__int__numeral__le__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N2: num,Z3: int] :
( ( ord_less_eq @ A @ ( numeral_numeral @ A @ N2 ) @ ( ring_1_of_int @ A @ Z3 ) )
= ( ord_less_eq @ int @ ( numeral_numeral @ int @ N2 ) @ Z3 ) ) ) ).
% of_int_numeral_le_iff
thf(fact_2118_of__int__le__numeral__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z3: int,N2: num] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Z3 ) @ ( numeral_numeral @ A @ N2 ) )
= ( ord_less_eq @ int @ Z3 @ ( numeral_numeral @ int @ N2 ) ) ) ) ).
% of_int_le_numeral_iff
thf(fact_2119_of__int__less__numeral__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z3: int,N2: num] :
( ( ord_less @ A @ ( ring_1_of_int @ A @ Z3 ) @ ( numeral_numeral @ A @ N2 ) )
= ( ord_less @ int @ Z3 @ ( numeral_numeral @ int @ N2 ) ) ) ) ).
% of_int_less_numeral_iff
thf(fact_2120_of__int__numeral__less__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N2: num,Z3: int] :
( ( ord_less @ A @ ( numeral_numeral @ A @ N2 ) @ ( ring_1_of_int @ A @ Z3 ) )
= ( ord_less @ int @ ( numeral_numeral @ int @ N2 ) @ Z3 ) ) ) ).
% of_int_numeral_less_iff
thf(fact_2121_of__int__1__le__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z3: int] :
( ( ord_less_eq @ A @ ( one_one @ A ) @ ( ring_1_of_int @ A @ Z3 ) )
= ( ord_less_eq @ int @ ( one_one @ int ) @ Z3 ) ) ) ).
% of_int_1_le_iff
thf(fact_2122_of__int__le__1__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z3: int] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Z3 ) @ ( one_one @ A ) )
= ( ord_less_eq @ int @ Z3 @ ( one_one @ int ) ) ) ) ).
% of_int_le_1_iff
thf(fact_2123_of__int__less__1__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z3: int] :
( ( ord_less @ A @ ( ring_1_of_int @ A @ Z3 ) @ ( one_one @ A ) )
= ( ord_less @ int @ Z3 @ ( one_one @ int ) ) ) ) ).
% of_int_less_1_iff
thf(fact_2124_of__int__1__less__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z3: int] :
( ( ord_less @ A @ ( one_one @ A ) @ ( ring_1_of_int @ A @ Z3 ) )
= ( ord_less @ int @ ( one_one @ int ) @ Z3 ) ) ) ).
% of_int_1_less_iff
thf(fact_2125_of__int__power__le__of__int__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: int,B3: int,W2: nat] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ X ) @ ( power_power @ A @ ( ring_1_of_int @ A @ B3 ) @ W2 ) )
= ( ord_less_eq @ int @ X @ ( power_power @ int @ B3 @ W2 ) ) ) ) ).
% of_int_power_le_of_int_cancel_iff
thf(fact_2126_of__int__le__of__int__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [B3: int,W2: nat,X: int] :
( ( ord_less_eq @ A @ ( power_power @ A @ ( ring_1_of_int @ A @ B3 ) @ W2 ) @ ( ring_1_of_int @ A @ X ) )
= ( ord_less_eq @ int @ ( power_power @ int @ B3 @ W2 ) @ X ) ) ) ).
% of_int_le_of_int_power_cancel_iff
thf(fact_2127_numeral__power__eq__of__int__cancel__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [X: num,N2: nat,Y: int] :
( ( ( power_power @ A @ ( numeral_numeral @ A @ X ) @ N2 )
= ( ring_1_of_int @ A @ Y ) )
= ( ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N2 )
= Y ) ) ) ).
% numeral_power_eq_of_int_cancel_iff
thf(fact_2128_of__int__eq__numeral__power__cancel__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [Y: int,X: num,N2: nat] :
( ( ( ring_1_of_int @ A @ Y )
= ( power_power @ A @ ( numeral_numeral @ A @ X ) @ N2 ) )
= ( Y
= ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N2 ) ) ) ) ).
% of_int_eq_numeral_power_cancel_iff
thf(fact_2129_add__neg__numeral__special_I6_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [M2: num] :
( ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M2 ) ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( inc @ M2 ) ) ) ) ) ).
% add_neg_numeral_special(6)
thf(fact_2130_add__neg__numeral__special_I5_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [N2: num] :
( ( plus_plus @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N2 ) ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( inc @ N2 ) ) ) ) ) ).
% add_neg_numeral_special(5)
thf(fact_2131_of__int__less__of__int__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [B3: int,W2: nat,X: int] :
( ( ord_less @ A @ ( power_power @ A @ ( ring_1_of_int @ A @ B3 ) @ W2 ) @ ( ring_1_of_int @ A @ X ) )
= ( ord_less @ int @ ( power_power @ int @ B3 @ W2 ) @ X ) ) ) ).
% of_int_less_of_int_power_cancel_iff
thf(fact_2132_of__int__power__less__of__int__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: int,B3: int,W2: nat] :
( ( ord_less @ A @ ( ring_1_of_int @ A @ X ) @ ( power_power @ A @ ( ring_1_of_int @ A @ B3 ) @ W2 ) )
= ( ord_less @ int @ X @ ( power_power @ int @ B3 @ W2 ) ) ) ) ).
% of_int_power_less_of_int_cancel_iff
thf(fact_2133_of__int__le__numeral__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: int,X: num,N2: nat] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ A3 ) @ ( power_power @ A @ ( numeral_numeral @ A @ X ) @ N2 ) )
= ( ord_less_eq @ int @ A3 @ ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N2 ) ) ) ) ).
% of_int_le_numeral_power_cancel_iff
thf(fact_2134_numeral__power__le__of__int__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: num,N2: nat,A3: int] :
( ( ord_less_eq @ A @ ( power_power @ A @ ( numeral_numeral @ A @ X ) @ N2 ) @ ( ring_1_of_int @ A @ A3 ) )
= ( ord_less_eq @ int @ ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N2 ) @ A3 ) ) ) ).
% numeral_power_le_of_int_cancel_iff
thf(fact_2135_numeral__power__less__of__int__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: num,N2: nat,A3: int] :
( ( ord_less @ A @ ( power_power @ A @ ( numeral_numeral @ A @ X ) @ N2 ) @ ( ring_1_of_int @ A @ A3 ) )
= ( ord_less @ int @ ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N2 ) @ A3 ) ) ) ).
% numeral_power_less_of_int_cancel_iff
thf(fact_2136_of__int__less__numeral__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: int,X: num,N2: nat] :
( ( ord_less @ A @ ( ring_1_of_int @ A @ A3 ) @ ( power_power @ A @ ( numeral_numeral @ A @ X ) @ N2 ) )
= ( ord_less @ int @ A3 @ ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N2 ) ) ) ) ).
% of_int_less_numeral_power_cancel_iff
thf(fact_2137_neg__numeral__power__eq__of__int__cancel__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [X: num,N2: nat,Y: int] :
( ( ( power_power @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ X ) ) @ N2 )
= ( ring_1_of_int @ A @ Y ) )
= ( ( power_power @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ X ) ) @ N2 )
= Y ) ) ) ).
% neg_numeral_power_eq_of_int_cancel_iff
thf(fact_2138_of__int__eq__neg__numeral__power__cancel__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [Y: int,X: num,N2: nat] :
( ( ( ring_1_of_int @ A @ Y )
= ( power_power @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ X ) ) @ N2 ) )
= ( Y
= ( power_power @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ X ) ) @ N2 ) ) ) ) ).
% of_int_eq_neg_numeral_power_cancel_iff
thf(fact_2139_of__int__le__neg__numeral__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: int,X: num,N2: nat] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ A3 ) @ ( power_power @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ X ) ) @ N2 ) )
= ( ord_less_eq @ int @ A3 @ ( power_power @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ X ) ) @ N2 ) ) ) ) ).
% of_int_le_neg_numeral_power_cancel_iff
thf(fact_2140_neg__numeral__power__le__of__int__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: num,N2: nat,A3: int] :
( ( ord_less_eq @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ X ) ) @ N2 ) @ ( ring_1_of_int @ A @ A3 ) )
= ( ord_less_eq @ int @ ( power_power @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ X ) ) @ N2 ) @ A3 ) ) ) ).
% neg_numeral_power_le_of_int_cancel_iff
thf(fact_2141_ex__le__of__int,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X: A] :
? [Z: int] : ( ord_less_eq @ A @ X @ ( ring_1_of_int @ A @ Z ) ) ) ).
% ex_le_of_int
thf(fact_2142_ex__of__int__less,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X: A] :
? [Z: int] : ( ord_less @ A @ ( ring_1_of_int @ A @ Z ) @ X ) ) ).
% ex_of_int_less
thf(fact_2143_ex__less__of__int,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X: A] :
? [Z: int] : ( ord_less @ A @ X @ ( ring_1_of_int @ A @ Z ) ) ) ).
% ex_less_of_int
thf(fact_2144_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_2145_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_2146_less__eq__mask,axiom,
! [N2: nat] : ( ord_less_eq @ nat @ N2 @ ( bit_se2239418461657761734s_mask @ nat @ N2 ) ) ).
% less_eq_mask
thf(fact_2147_concat__bit__assoc,axiom,
! [N2: nat,K: int,M2: nat,L: int,R2: int] :
( ( bit_concat_bit @ N2 @ K @ ( bit_concat_bit @ M2 @ L @ R2 ) )
= ( bit_concat_bit @ ( plus_plus @ nat @ M2 @ N2 ) @ ( bit_concat_bit @ N2 @ K @ L ) @ R2 ) ) ).
% concat_bit_assoc
thf(fact_2148_atLeastAtMostPlus1__int__conv,axiom,
! [M2: int,N2: int] :
( ( ord_less_eq @ int @ M2 @ ( plus_plus @ int @ ( one_one @ int ) @ N2 ) )
=> ( ( set_or1337092689740270186AtMost @ int @ M2 @ ( plus_plus @ int @ ( one_one @ int ) @ N2 ) )
= ( insert2 @ int @ ( plus_plus @ int @ ( one_one @ int ) @ N2 ) @ ( set_or1337092689740270186AtMost @ int @ M2 @ N2 ) ) ) ) ).
% atLeastAtMostPlus1_int_conv
thf(fact_2149_simp__from__to,axiom,
( ( set_or1337092689740270186AtMost @ int )
= ( ^ [I3: int,J2: int] : ( if @ ( set @ int ) @ ( ord_less @ int @ J2 @ I3 ) @ ( bot_bot @ ( set @ int ) ) @ ( insert2 @ int @ I3 @ ( set_or1337092689740270186AtMost @ int @ ( plus_plus @ int @ I3 @ ( one_one @ int ) ) @ J2 ) ) ) ) ) ).
% simp_from_to
thf(fact_2150_less__mask,axiom,
! [N2: nat] :
( ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N2 )
=> ( ord_less @ nat @ N2 @ ( bit_se2239418461657761734s_mask @ nat @ N2 ) ) ) ).
% less_mask
thf(fact_2151_of__int__nonneg,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z3: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z3 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( ring_1_of_int @ A @ Z3 ) ) ) ) ).
% of_int_nonneg
thf(fact_2152_of__int__pos,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z3: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ Z3 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( ring_1_of_int @ A @ Z3 ) ) ) ) ).
% of_int_pos
thf(fact_2153_floor__exists1,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X: A] :
? [X5: int] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ X5 ) @ X )
& ( ord_less @ A @ X @ ( ring_1_of_int @ A @ ( plus_plus @ int @ X5 @ ( one_one @ int ) ) ) )
& ! [Y4: int] :
( ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Y4 ) @ X )
& ( ord_less @ A @ X @ ( ring_1_of_int @ A @ ( plus_plus @ int @ Y4 @ ( one_one @ int ) ) ) ) )
=> ( Y4 = X5 ) ) ) ) ).
% floor_exists1
thf(fact_2154_floor__exists,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X: A] :
? [Z: int] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Z ) @ X )
& ( ord_less @ A @ X @ ( ring_1_of_int @ A @ ( plus_plus @ int @ Z @ ( one_one @ int ) ) ) ) ) ) ).
% floor_exists
thf(fact_2155_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_2156_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_2157_Suc__mask__eq__exp,axiom,
! [N2: nat] :
( ( suc @ ( bit_se2239418461657761734s_mask @ nat @ N2 ) )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ).
% Suc_mask_eq_exp
thf(fact_2158_mask__nat__less__exp,axiom,
! [N2: nat] : ( ord_less @ nat @ ( bit_se2239418461657761734s_mask @ nat @ N2 ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ).
% mask_nat_less_exp
thf(fact_2159_semiring__bit__operations__class_Oeven__mask__iff,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N2: nat] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se2239418461657761734s_mask @ A @ N2 ) )
= ( N2
= ( zero_zero @ nat ) ) ) ) ).
% semiring_bit_operations_class.even_mask_iff
thf(fact_2160_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_2161_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_2162_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_2163_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_2164_Sum__Icc__int,axiom,
! [M2: int,N2: int] :
( ( ord_less_eq @ int @ M2 @ N2 )
=> ( ( groups7311177749621191930dd_sum @ int @ int
@ ^ [X4: int] : X4
@ ( set_or1337092689740270186AtMost @ int @ M2 @ N2 ) )
= ( divide_divide @ int @ ( minus_minus @ int @ ( times_times @ int @ N2 @ ( plus_plus @ int @ N2 @ ( one_one @ int ) ) ) @ ( times_times @ int @ M2 @ ( minus_minus @ int @ M2 @ ( one_one @ int ) ) ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ).
% Sum_Icc_int
thf(fact_2165_and__int_Oelims,axiom,
! [X: int,Xa2: int,Y: int] :
( ( ( bit_se5824344872417868541ns_and @ int @ X @ Xa2 )
= Y )
=> ( ( ( ( member @ int @ X @ ( insert2 @ int @ ( zero_zero @ int ) @ ( insert2 @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) )
& ( member @ int @ Xa2 @ ( insert2 @ int @ ( zero_zero @ int ) @ ( insert2 @ 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 ) ) @ Xa2 ) ) ) ) ) )
& ( ~ ( ( member @ int @ X @ ( insert2 @ int @ ( zero_zero @ int ) @ ( insert2 @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) )
& ( member @ int @ Xa2 @ ( insert2 @ int @ ( zero_zero @ int ) @ ( insert2 @ 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 ) ) @ Xa2 ) ) )
@ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( divide_divide @ int @ X @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ Xa2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% and_int.elims
thf(fact_2166_and__int_Osimps,axiom,
( ( bit_se5824344872417868541ns_and @ int )
= ( ^ [K2: int,L2: int] :
( if @ int
@ ( ( member @ int @ K2 @ ( insert2 @ int @ ( zero_zero @ int ) @ ( insert2 @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) )
& ( member @ int @ L2 @ ( insert2 @ int @ ( zero_zero @ int ) @ ( insert2 @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) )
@ ( uminus_uminus @ int
@ ( zero_neq_one_of_bool @ int
@ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K2 )
& ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ L2 ) ) ) )
@ ( plus_plus @ int
@ ( zero_neq_one_of_bool @ int
@ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K2 )
& ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ L2 ) ) )
@ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( divide_divide @ int @ K2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ).
% and_int.simps
thf(fact_2167_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_2168_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_2169_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_2170_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_2171_sum_Oneutral__const,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A4: set @ B] :
( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [Uu3: B] : ( zero_zero @ A )
@ A4 )
= ( zero_zero @ A ) ) ) ).
% sum.neutral_const
thf(fact_2172_sum_Oempty,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: B > A] :
( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( bot_bot @ ( set @ B ) ) )
= ( zero_zero @ A ) ) ) ).
% sum.empty
thf(fact_2173_sum_Oinfinite,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A4: set @ B,G: B > A] :
( ~ ( finite_finite2 @ B @ A4 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ A4 )
= ( zero_zero @ A ) ) ) ) ).
% sum.infinite
thf(fact_2174_sum__eq__0__iff,axiom,
! [A: $tType,B: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [F5: set @ B,F2: B > A] :
( ( finite_finite2 @ B @ F5 )
=> ( ( ( groups7311177749621191930dd_sum @ B @ A @ F2 @ F5 )
= ( zero_zero @ A ) )
= ( ! [X4: B] :
( ( member @ B @ X4 @ F5 )
=> ( ( F2 @ X4 )
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% sum_eq_0_iff
thf(fact_2175_round__0,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ( ( archimedean_round @ A @ ( zero_zero @ A ) )
= ( zero_zero @ int ) ) ) ).
% round_0
thf(fact_2176_sum_Odelta,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add @ A )
=> ! [S2: set @ B,A3: B,B3: B > A] :
( ( finite_finite2 @ B @ S2 )
=> ( ( ( member @ B @ A3 @ S2 )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [K2: B] : ( if @ A @ ( K2 = A3 ) @ ( B3 @ K2 ) @ ( zero_zero @ A ) )
@ S2 )
= ( B3 @ A3 ) ) )
& ( ~ ( member @ B @ A3 @ S2 )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [K2: B] : ( if @ A @ ( K2 = A3 ) @ ( B3 @ K2 ) @ ( zero_zero @ A ) )
@ S2 )
= ( zero_zero @ A ) ) ) ) ) ) ).
% sum.delta
thf(fact_2177_sum_Odelta_H,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add @ A )
=> ! [S2: set @ B,A3: B,B3: B > A] :
( ( finite_finite2 @ B @ S2 )
=> ( ( ( member @ B @ A3 @ S2 )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [K2: B] : ( if @ A @ ( A3 = K2 ) @ ( B3 @ K2 ) @ ( zero_zero @ A ) )
@ S2 )
= ( B3 @ A3 ) ) )
& ( ~ ( member @ B @ A3 @ S2 )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [K2: B] : ( if @ A @ ( A3 = K2 ) @ ( B3 @ K2 ) @ ( zero_zero @ A ) )
@ S2 )
= ( zero_zero @ A ) ) ) ) ) ) ).
% sum.delta'
thf(fact_2178_sum_Oinsert,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A4: set @ B,X: B,G: B > A] :
( ( finite_finite2 @ B @ A4 )
=> ( ~ ( member @ B @ X @ A4 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( insert2 @ B @ X @ A4 ) )
= ( plus_plus @ A @ ( G @ X ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ A4 ) ) ) ) ) ) ).
% sum.insert
thf(fact_2179_of__int__sum,axiom,
! [A: $tType,B: $tType] :
( ( ring_1 @ A )
=> ! [F2: B > int,A4: set @ B] :
( ( ring_1_of_int @ A @ ( groups7311177749621191930dd_sum @ B @ int @ F2 @ A4 ) )
= ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X4: B] : ( ring_1_of_int @ A @ ( F2 @ X4 ) )
@ A4 ) ) ) ).
% of_int_sum
thf(fact_2180_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_2181_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_2182_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_2183_sum_Oneutral,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A4: set @ B,G: B > A] :
( ! [X5: B] :
( ( member @ B @ X5 @ A4 )
=> ( ( G @ X5 )
= ( zero_zero @ A ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ A4 )
= ( zero_zero @ A ) ) ) ) ).
% sum.neutral
thf(fact_2184_sum_Onot__neutral__contains__not__neutral,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: B > A,A4: set @ B] :
( ( ( groups7311177749621191930dd_sum @ B @ A @ G @ A4 )
!= ( zero_zero @ A ) )
=> ~ ! [A6: B] :
( ( member @ B @ A6 @ A4 )
=> ( ( G @ A6 )
= ( zero_zero @ A ) ) ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_2185_sum__mono,axiom,
! [A: $tType,B: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [K5: set @ B,F2: B > A,G: B > A] :
( ! [I4: B] :
( ( member @ B @ I4 @ K5 )
=> ( ord_less_eq @ A @ ( F2 @ I4 ) @ ( G @ I4 ) ) )
=> ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ K5 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ K5 ) ) ) ) ).
% sum_mono
thf(fact_2186_sum_Odistrib,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: B > A,H2: B > A,A4: set @ B] :
( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X4: B] : ( plus_plus @ A @ ( G @ X4 ) @ ( H2 @ X4 ) )
@ A4 )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ G @ A4 ) @ ( groups7311177749621191930dd_sum @ B @ A @ H2 @ A4 ) ) ) ) ).
% sum.distrib
thf(fact_2187_sum_Oswap__restrict,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( comm_monoid_add @ A )
=> ! [A4: set @ B,B2: set @ C,G: B > C > A,R: B > C > $o] :
( ( finite_finite2 @ B @ A4 )
=> ( ( finite_finite2 @ C @ B2 )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X4: B] :
( groups7311177749621191930dd_sum @ C @ A @ ( G @ X4 )
@ ( collect @ C
@ ^ [Y6: C] :
( ( member @ C @ Y6 @ B2 )
& ( R @ X4 @ Y6 ) ) ) )
@ A4 )
= ( groups7311177749621191930dd_sum @ C @ A
@ ^ [Y6: C] :
( groups7311177749621191930dd_sum @ B @ A
@ ^ [X4: B] : ( G @ X4 @ Y6 )
@ ( collect @ B
@ ^ [X4: B] :
( ( member @ B @ X4 @ A4 )
& ( R @ X4 @ Y6 ) ) ) )
@ B2 ) ) ) ) ) ).
% sum.swap_restrict
thf(fact_2188_sum__nonpos,axiom,
! [B: $tType,A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A4: set @ B,F2: B > A] :
( ! [X5: B] :
( ( member @ B @ X5 @ A4 )
=> ( ord_less_eq @ A @ ( F2 @ X5 ) @ ( zero_zero @ A ) ) )
=> ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A4 ) @ ( zero_zero @ A ) ) ) ) ).
% sum_nonpos
thf(fact_2189_sum__nonneg,axiom,
! [A: $tType,B: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A4: set @ B,F2: B > A] :
( ! [X5: B] :
( ( member @ B @ X5 @ A4 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ X5 ) ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A4 ) ) ) ) ).
% sum_nonneg
thf(fact_2190_sum__mono__inv,axiom,
! [A: $tType,I7: $tType] :
( ( ordere8940638589300402666id_add @ A )
=> ! [F2: I7 > A,I6: set @ I7,G: I7 > A,I: I7] :
( ( ( groups7311177749621191930dd_sum @ I7 @ A @ F2 @ I6 )
= ( groups7311177749621191930dd_sum @ I7 @ A @ G @ I6 ) )
=> ( ! [I4: I7] :
( ( member @ I7 @ I4 @ I6 )
=> ( ord_less_eq @ A @ ( F2 @ I4 ) @ ( G @ I4 ) ) )
=> ( ( member @ I7 @ I @ I6 )
=> ( ( finite_finite2 @ I7 @ I6 )
=> ( ( F2 @ I )
= ( G @ I ) ) ) ) ) ) ) ).
% sum_mono_inv
thf(fact_2191_sum_Ointer__filter,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A4: set @ B,G: B > A,P: B > $o] :
( ( finite_finite2 @ B @ A4 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G
@ ( collect @ B
@ ^ [X4: B] :
( ( member @ B @ X4 @ A4 )
& ( P @ X4 ) ) ) )
= ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X4: B] : ( if @ A @ ( P @ X4 ) @ ( G @ X4 ) @ ( zero_zero @ A ) )
@ A4 ) ) ) ) ).
% sum.inter_filter
thf(fact_2192_sum__nonneg__eq__0__iff,axiom,
! [A: $tType,B: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A4: set @ B,F2: B > A] :
( ( finite_finite2 @ B @ A4 )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ A4 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ X5 ) ) )
=> ( ( ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A4 )
= ( zero_zero @ A ) )
= ( ! [X4: B] :
( ( member @ B @ X4 @ A4 )
=> ( ( F2 @ X4 )
= ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
thf(fact_2193_sum__le__included,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [S: set @ B,T2: set @ C,G: C > A,I: C > B,F2: B > A] :
( ( finite_finite2 @ B @ S )
=> ( ( finite_finite2 @ C @ T2 )
=> ( ! [X5: C] :
( ( member @ C @ X5 @ T2 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( G @ X5 ) ) )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ S )
=> ? [Xa: C] :
( ( member @ C @ Xa @ T2 )
& ( ( I @ Xa )
= X5 )
& ( ord_less_eq @ A @ ( F2 @ X5 ) @ ( G @ Xa ) ) ) )
=> ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ S ) @ ( groups7311177749621191930dd_sum @ C @ A @ G @ T2 ) ) ) ) ) ) ) ).
% sum_le_included
thf(fact_2194_sum__strict__mono__ex1,axiom,
! [A: $tType,I7: $tType] :
( ( ordere8940638589300402666id_add @ A )
=> ! [A4: set @ I7,F2: I7 > A,G: I7 > A] :
( ( finite_finite2 @ I7 @ A4 )
=> ( ! [X5: I7] :
( ( member @ I7 @ X5 @ A4 )
=> ( ord_less_eq @ A @ ( F2 @ X5 ) @ ( G @ X5 ) ) )
=> ( ? [X2: I7] :
( ( member @ I7 @ X2 @ A4 )
& ( ord_less @ A @ ( F2 @ X2 ) @ ( G @ X2 ) ) )
=> ( ord_less @ A @ ( groups7311177749621191930dd_sum @ I7 @ A @ F2 @ A4 ) @ ( groups7311177749621191930dd_sum @ I7 @ A @ G @ A4 ) ) ) ) ) ) ).
% sum_strict_mono_ex1
thf(fact_2195_sum_Orelated,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [R: A > A > $o,S2: set @ B,H2: B > A,G: B > A] :
( ( R @ ( zero_zero @ A ) @ ( zero_zero @ A ) )
=> ( ! [X1: A,Y1: A,X23: A,Y22: A] :
( ( ( R @ X1 @ X23 )
& ( R @ Y1 @ Y22 ) )
=> ( R @ ( plus_plus @ A @ X1 @ Y1 ) @ ( plus_plus @ A @ X23 @ Y22 ) ) )
=> ( ( finite_finite2 @ B @ S2 )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ S2 )
=> ( R @ ( H2 @ X5 ) @ ( G @ X5 ) ) )
=> ( R @ ( groups7311177749621191930dd_sum @ B @ A @ H2 @ S2 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ S2 ) ) ) ) ) ) ) ).
% sum.related
thf(fact_2196_sum__strict__mono,axiom,
! [A: $tType,B: $tType] :
( ( strict7427464778891057005id_add @ A )
=> ! [A4: set @ B,F2: B > A,G: B > A] :
( ( finite_finite2 @ B @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ A4 )
=> ( ord_less @ A @ ( F2 @ X5 ) @ ( G @ X5 ) ) )
=> ( ord_less @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A4 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ A4 ) ) ) ) ) ) ).
% sum_strict_mono
thf(fact_2197_sum_Oinsert__if,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A4: set @ B,X: B,G: B > A] :
( ( finite_finite2 @ B @ A4 )
=> ( ( ( member @ B @ X @ A4 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( insert2 @ B @ X @ A4 ) )
= ( groups7311177749621191930dd_sum @ B @ A @ G @ A4 ) ) )
& ( ~ ( member @ B @ X @ A4 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( insert2 @ B @ X @ A4 ) )
= ( plus_plus @ A @ ( G @ X ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ A4 ) ) ) ) ) ) ) ).
% sum.insert_if
thf(fact_2198_sum_Oreindex__bij__witness__not__neutral,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comm_monoid_add @ A )
=> ! [S5: set @ B,T6: set @ C,S2: set @ B,I: C > B,J: B > C,T4: set @ C,G: B > A,H2: C > A] :
( ( finite_finite2 @ B @ S5 )
=> ( ( finite_finite2 @ C @ T6 )
=> ( ! [A6: B] :
( ( member @ B @ A6 @ ( minus_minus @ ( set @ B ) @ S2 @ S5 ) )
=> ( ( I @ ( J @ A6 ) )
= A6 ) )
=> ( ! [A6: B] :
( ( member @ B @ A6 @ ( minus_minus @ ( set @ B ) @ S2 @ S5 ) )
=> ( member @ C @ ( J @ A6 ) @ ( minus_minus @ ( set @ C ) @ T4 @ T6 ) ) )
=> ( ! [B5: C] :
( ( member @ C @ B5 @ ( minus_minus @ ( set @ C ) @ T4 @ T6 ) )
=> ( ( J @ ( I @ B5 ) )
= B5 ) )
=> ( ! [B5: C] :
( ( member @ C @ B5 @ ( minus_minus @ ( set @ C ) @ T4 @ T6 ) )
=> ( member @ B @ ( I @ B5 ) @ ( minus_minus @ ( set @ B ) @ S2 @ S5 ) ) )
=> ( ! [A6: B] :
( ( member @ B @ A6 @ S5 )
=> ( ( G @ A6 )
= ( zero_zero @ A ) ) )
=> ( ! [B5: C] :
( ( member @ C @ B5 @ T6 )
=> ( ( H2 @ B5 )
= ( zero_zero @ A ) ) )
=> ( ! [A6: B] :
( ( member @ B @ A6 @ S2 )
=> ( ( H2 @ ( J @ A6 ) )
= ( G @ A6 ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ S2 )
= ( groups7311177749621191930dd_sum @ C @ A @ H2 @ T4 ) ) ) ) ) ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness_not_neutral
thf(fact_2199_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_2200_sum__nonneg__0,axiom,
! [B: $tType,A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [S: set @ B,F2: B > A,I: B] :
( ( finite_finite2 @ B @ S )
=> ( ! [I4: B] :
( ( member @ B @ I4 @ S )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ I4 ) ) )
=> ( ( ( groups7311177749621191930dd_sum @ B @ A @ F2 @ S )
= ( zero_zero @ A ) )
=> ( ( member @ B @ I @ S )
=> ( ( F2 @ I )
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% sum_nonneg_0
thf(fact_2201_sum__nonneg__leq__bound,axiom,
! [B: $tType,A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [S: set @ B,F2: B > A,B2: A,I: B] :
( ( finite_finite2 @ B @ S )
=> ( ! [I4: B] :
( ( member @ B @ I4 @ S )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ I4 ) ) )
=> ( ( ( groups7311177749621191930dd_sum @ B @ A @ F2 @ S )
= B2 )
=> ( ( member @ B @ I @ S )
=> ( ord_less_eq @ A @ ( F2 @ I ) @ B2 ) ) ) ) ) ) ).
% sum_nonneg_leq_bound
thf(fact_2202_sum_Osetdiff__irrelevant,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A4: set @ B,G: B > A] :
( ( finite_finite2 @ B @ A4 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G
@ ( minus_minus @ ( set @ B ) @ A4
@ ( collect @ B
@ ^ [X4: B] :
( ( G @ X4 )
= ( zero_zero @ A ) ) ) ) )
= ( groups7311177749621191930dd_sum @ B @ A @ G @ A4 ) ) ) ) ).
% sum.setdiff_irrelevant
thf(fact_2203_sum__pos2,axiom,
! [A: $tType,B: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [I6: set @ B,I: B,F2: B > A] :
( ( finite_finite2 @ B @ I6 )
=> ( ( member @ B @ I @ I6 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ ( F2 @ I ) )
=> ( ! [I4: B] :
( ( member @ B @ I4 @ I6 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ I4 ) ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ I6 ) ) ) ) ) ) ) ).
% sum_pos2
thf(fact_2204_sum__pos,axiom,
! [A: $tType,B: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [I6: set @ B,F2: B > A] :
( ( finite_finite2 @ B @ I6 )
=> ( ( I6
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [I4: B] :
( ( member @ B @ I4 @ I6 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( F2 @ I4 ) ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ I6 ) ) ) ) ) ) ).
% sum_pos
thf(fact_2205_sum_Omono__neutral__cong__right,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [T4: set @ B,S2: set @ B,G: B > A,H2: B > A] :
( ( finite_finite2 @ B @ T4 )
=> ( ( ord_less_eq @ ( set @ B ) @ S2 @ T4 )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ ( minus_minus @ ( set @ B ) @ T4 @ S2 ) )
=> ( ( G @ X5 )
= ( zero_zero @ A ) ) )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ S2 )
=> ( ( G @ X5 )
= ( H2 @ X5 ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ T4 )
= ( groups7311177749621191930dd_sum @ B @ A @ H2 @ S2 ) ) ) ) ) ) ) ).
% sum.mono_neutral_cong_right
thf(fact_2206_sum_Omono__neutral__cong__left,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [T4: set @ B,S2: set @ B,H2: B > A,G: B > A] :
( ( finite_finite2 @ B @ T4 )
=> ( ( ord_less_eq @ ( set @ B ) @ S2 @ T4 )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ ( minus_minus @ ( set @ B ) @ T4 @ S2 ) )
=> ( ( H2 @ X5 )
= ( zero_zero @ A ) ) )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ S2 )
=> ( ( G @ X5 )
= ( H2 @ X5 ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ S2 )
= ( groups7311177749621191930dd_sum @ B @ A @ H2 @ T4 ) ) ) ) ) ) ) ).
% sum.mono_neutral_cong_left
thf(fact_2207_sum_Omono__neutral__right,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [T4: set @ B,S2: set @ B,G: B > A] :
( ( finite_finite2 @ B @ T4 )
=> ( ( ord_less_eq @ ( set @ B ) @ S2 @ T4 )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ ( minus_minus @ ( set @ B ) @ T4 @ S2 ) )
=> ( ( G @ X5 )
= ( zero_zero @ A ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ T4 )
= ( groups7311177749621191930dd_sum @ B @ A @ G @ S2 ) ) ) ) ) ) ).
% sum.mono_neutral_right
thf(fact_2208_sum_Omono__neutral__left,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [T4: set @ B,S2: set @ B,G: B > A] :
( ( finite_finite2 @ B @ T4 )
=> ( ( ord_less_eq @ ( set @ B ) @ S2 @ T4 )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ ( minus_minus @ ( set @ B ) @ T4 @ S2 ) )
=> ( ( G @ X5 )
= ( zero_zero @ A ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ S2 )
= ( groups7311177749621191930dd_sum @ B @ A @ G @ T4 ) ) ) ) ) ) ).
% sum.mono_neutral_left
thf(fact_2209_sum_Osame__carrierI,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [C5: set @ B,A4: set @ B,B2: set @ B,G: B > A,H2: B > A] :
( ( finite_finite2 @ B @ C5 )
=> ( ( ord_less_eq @ ( set @ B ) @ A4 @ C5 )
=> ( ( ord_less_eq @ ( set @ B ) @ B2 @ C5 )
=> ( ! [A6: B] :
( ( member @ B @ A6 @ ( minus_minus @ ( set @ B ) @ C5 @ A4 ) )
=> ( ( G @ A6 )
= ( zero_zero @ A ) ) )
=> ( ! [B5: B] :
( ( member @ B @ B5 @ ( minus_minus @ ( set @ B ) @ C5 @ B2 ) )
=> ( ( H2 @ B5 )
= ( zero_zero @ A ) ) )
=> ( ( ( groups7311177749621191930dd_sum @ B @ A @ G @ C5 )
= ( groups7311177749621191930dd_sum @ B @ A @ H2 @ C5 ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ A4 )
= ( groups7311177749621191930dd_sum @ B @ A @ H2 @ B2 ) ) ) ) ) ) ) ) ) ).
% sum.same_carrierI
thf(fact_2210_sum_Osame__carrier,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [C5: set @ B,A4: set @ B,B2: set @ B,G: B > A,H2: B > A] :
( ( finite_finite2 @ B @ C5 )
=> ( ( ord_less_eq @ ( set @ B ) @ A4 @ C5 )
=> ( ( ord_less_eq @ ( set @ B ) @ B2 @ C5 )
=> ( ! [A6: B] :
( ( member @ B @ A6 @ ( minus_minus @ ( set @ B ) @ C5 @ A4 ) )
=> ( ( G @ A6 )
= ( zero_zero @ A ) ) )
=> ( ! [B5: B] :
( ( member @ B @ B5 @ ( minus_minus @ ( set @ B ) @ C5 @ B2 ) )
=> ( ( H2 @ B5 )
= ( zero_zero @ A ) ) )
=> ( ( ( groups7311177749621191930dd_sum @ B @ A @ G @ A4 )
= ( groups7311177749621191930dd_sum @ B @ A @ H2 @ B2 ) )
= ( ( groups7311177749621191930dd_sum @ B @ A @ G @ C5 )
= ( groups7311177749621191930dd_sum @ B @ A @ H2 @ C5 ) ) ) ) ) ) ) ) ) ).
% sum.same_carrier
thf(fact_2211_sum_Osubset__diff,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [B2: set @ B,A4: set @ B,G: B > A] :
( ( ord_less_eq @ ( set @ B ) @ B2 @ A4 )
=> ( ( finite_finite2 @ B @ A4 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ A4 )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ A4 @ B2 ) ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ B2 ) ) ) ) ) ) ).
% sum.subset_diff
thf(fact_2212_sum__diff,axiom,
! [A: $tType,B: $tType] :
( ( ab_group_add @ A )
=> ! [A4: set @ B,B2: set @ B,F2: B > A] :
( ( finite_finite2 @ B @ A4 )
=> ( ( ord_less_eq @ ( set @ B ) @ B2 @ A4 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ F2 @ ( minus_minus @ ( set @ B ) @ A4 @ B2 ) )
= ( minus_minus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A4 ) @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ B2 ) ) ) ) ) ) ).
% sum_diff
thf(fact_2213_sum__mono2,axiom,
! [A: $tType,B: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [B2: set @ B,A4: set @ B,F2: B > A] :
( ( finite_finite2 @ B @ B2 )
=> ( ( ord_less_eq @ ( set @ B ) @ A4 @ B2 )
=> ( ! [B5: B] :
( ( member @ B @ B5 @ ( minus_minus @ ( set @ B ) @ B2 @ A4 ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ B5 ) ) )
=> ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A4 ) @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ B2 ) ) ) ) ) ) ).
% sum_mono2
thf(fact_2214_sum_Oremove,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A4: set @ B,X: B,G: B > A] :
( ( finite_finite2 @ B @ A4 )
=> ( ( member @ B @ X @ A4 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ A4 )
= ( plus_plus @ A @ ( G @ X ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ A4 @ ( insert2 @ B @ X @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ) ).
% sum.remove
thf(fact_2215_sum_Oinsert__remove,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A4: set @ B,G: B > A,X: B] :
( ( finite_finite2 @ B @ A4 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( insert2 @ B @ X @ A4 ) )
= ( plus_plus @ A @ ( G @ X ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ A4 @ ( insert2 @ B @ X @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ).
% sum.insert_remove
thf(fact_2216_sum__diff1,axiom,
! [A: $tType,B: $tType] :
( ( ab_group_add @ A )
=> ! [A4: set @ B,A3: B,F2: B > A] :
( ( finite_finite2 @ B @ A4 )
=> ( ( ( member @ B @ A3 @ A4 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ F2 @ ( minus_minus @ ( set @ B ) @ A4 @ ( insert2 @ B @ A3 @ ( bot_bot @ ( set @ B ) ) ) ) )
= ( minus_minus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A4 ) @ ( F2 @ A3 ) ) ) )
& ( ~ ( member @ B @ A3 @ A4 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ F2 @ ( minus_minus @ ( set @ B ) @ A4 @ ( insert2 @ B @ A3 @ ( bot_bot @ ( set @ B ) ) ) ) )
= ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A4 ) ) ) ) ) ) ).
% sum_diff1
thf(fact_2217_sum_Odelta__remove,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [S2: set @ B,A3: B,B3: B > A,C2: B > A] :
( ( finite_finite2 @ B @ S2 )
=> ( ( ( member @ B @ A3 @ S2 )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [K2: B] : ( if @ A @ ( K2 = A3 ) @ ( B3 @ K2 ) @ ( C2 @ K2 ) )
@ S2 )
= ( plus_plus @ A @ ( B3 @ A3 ) @ ( groups7311177749621191930dd_sum @ B @ A @ C2 @ ( minus_minus @ ( set @ B ) @ S2 @ ( insert2 @ B @ A3 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) )
& ( ~ ( member @ B @ A3 @ S2 )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [K2: B] : ( if @ A @ ( K2 = A3 ) @ ( B3 @ K2 ) @ ( C2 @ K2 ) )
@ S2 )
= ( groups7311177749621191930dd_sum @ B @ A @ C2 @ ( minus_minus @ ( set @ B ) @ S2 @ ( insert2 @ B @ A3 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ) ).
% sum.delta_remove
thf(fact_2218_sum__strict__mono2,axiom,
! [B: $tType,A: $tType] :
( ( ordere8940638589300402666id_add @ B )
=> ! [B2: set @ A,A4: set @ A,B3: A,F2: A > B] :
( ( finite_finite2 @ A @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ A4 @ B2 )
=> ( ( member @ A @ B3 @ ( minus_minus @ ( set @ A ) @ B2 @ A4 ) )
=> ( ( ord_less @ B @ ( zero_zero @ B ) @ ( F2 @ B3 ) )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ B2 )
=> ( ord_less_eq @ B @ ( zero_zero @ B ) @ ( F2 @ X5 ) ) )
=> ( ord_less @ B @ ( groups7311177749621191930dd_sum @ A @ B @ F2 @ A4 ) @ ( groups7311177749621191930dd_sum @ A @ B @ F2 @ B2 ) ) ) ) ) ) ) ) ).
% sum_strict_mono2
thf(fact_2219_member__le__sum,axiom,
! [B: $tType,C: $tType] :
( ( ( ordere6911136660526730532id_add @ B )
& ( semiring_1 @ B ) )
=> ! [I: C,A4: set @ C,F2: C > B] :
( ( member @ C @ I @ A4 )
=> ( ! [X5: C] :
( ( member @ C @ X5 @ ( minus_minus @ ( set @ C ) @ A4 @ ( insert2 @ C @ I @ ( bot_bot @ ( set @ C ) ) ) ) )
=> ( ord_less_eq @ B @ ( zero_zero @ B ) @ ( F2 @ X5 ) ) )
=> ( ( finite_finite2 @ C @ A4 )
=> ( ord_less_eq @ B @ ( F2 @ I ) @ ( groups7311177749621191930dd_sum @ C @ B @ F2 @ A4 ) ) ) ) ) ) ).
% member_le_sum
thf(fact_2220_and__int_Opelims,axiom,
! [X: int,Xa2: int,Y: int] :
( ( ( bit_se5824344872417868541ns_and @ int @ X @ Xa2 )
= Y )
=> ( ( accp @ ( product_prod @ int @ int ) @ bit_and_int_rel @ ( product_Pair @ int @ int @ X @ Xa2 ) )
=> ~ ( ( ( ( ( member @ int @ X @ ( insert2 @ int @ ( zero_zero @ int ) @ ( insert2 @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) )
& ( member @ int @ Xa2 @ ( insert2 @ int @ ( zero_zero @ int ) @ ( insert2 @ 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 ) ) @ Xa2 ) ) ) ) ) )
& ( ~ ( ( member @ int @ X @ ( insert2 @ int @ ( zero_zero @ int ) @ ( insert2 @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) )
& ( member @ int @ Xa2 @ ( insert2 @ int @ ( zero_zero @ int ) @ ( insert2 @ 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 ) ) @ Xa2 ) ) )
@ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( divide_divide @ int @ X @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ Xa2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) )
=> ~ ( accp @ ( product_prod @ int @ int ) @ bit_and_int_rel @ ( product_Pair @ int @ int @ X @ Xa2 ) ) ) ) ) ).
% and_int.pelims
thf(fact_2221_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 @ ( insert2 @ int @ ( zero_zero @ int ) @ ( insert2 @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) )
& ( member @ int @ L @ ( insert2 @ int @ ( zero_zero @ int ) @ ( insert2 @ 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 @ ( insert2 @ int @ ( zero_zero @ int ) @ ( insert2 @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) )
& ( member @ int @ L @ ( insert2 @ int @ ( zero_zero @ int ) @ ( insert2 @ 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_2222_signed__take__bit__eq__take__bit__minus,axiom,
( ( bit_ri4674362597316999326ke_bit @ int )
= ( ^ [N: nat,K2: int] : ( minus_minus @ int @ ( bit_se2584673776208193580ke_bit @ int @ ( suc @ N ) @ K2 ) @ ( times_times @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( suc @ N ) ) @ ( zero_neq_one_of_bool @ int @ ( bit_se5641148757651400278ts_bit @ int @ K2 @ N ) ) ) ) ) ) ).
% signed_take_bit_eq_take_bit_minus
thf(fact_2223_Sum__Icc__nat,axiom,
! [M2: nat,N2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [X4: nat] : X4
@ ( set_or1337092689740270186AtMost @ nat @ M2 @ N2 ) )
= ( divide_divide @ nat @ ( minus_minus @ nat @ ( times_times @ nat @ N2 @ ( plus_plus @ nat @ N2 @ ( one_one @ nat ) ) ) @ ( times_times @ nat @ M2 @ ( minus_minus @ nat @ M2 @ ( one_one @ nat ) ) ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% Sum_Icc_nat
thf(fact_2224_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_2225_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_2226_arith__series__nat,axiom,
! [A3: nat,D2: nat,N2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [I3: nat] : ( plus_plus @ nat @ A3 @ ( times_times @ nat @ I3 @ D2 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N2 ) )
= ( divide_divide @ nat @ ( times_times @ nat @ ( suc @ N2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ A3 ) @ ( times_times @ nat @ N2 @ D2 ) ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% arith_series_nat
thf(fact_2227_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_2228_ceiling__zero,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ( ( archimedean_ceiling @ A @ ( zero_zero @ A ) )
= ( zero_zero @ int ) ) ) ).
% ceiling_zero
thf(fact_2229_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_2230_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_2231_bit__numeral__Bit0__Suc__iff,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [M2: num,N2: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( numeral_numeral @ A @ ( bit0 @ M2 ) ) @ ( suc @ N2 ) )
= ( bit_se5641148757651400278ts_bit @ A @ ( numeral_numeral @ A @ M2 ) @ N2 ) ) ) ).
% bit_numeral_Bit0_Suc_iff
thf(fact_2232_bit__numeral__Bit1__Suc__iff,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [M2: num,N2: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( numeral_numeral @ A @ ( bit1 @ M2 ) ) @ ( suc @ N2 ) )
= ( bit_se5641148757651400278ts_bit @ A @ ( numeral_numeral @ A @ M2 ) @ N2 ) ) ) ).
% bit_numeral_Bit1_Suc_iff
thf(fact_2233_ceiling__add__of__int,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Z3: int] :
( ( archimedean_ceiling @ A @ ( plus_plus @ A @ X @ ( ring_1_of_int @ A @ Z3 ) ) )
= ( plus_plus @ int @ ( archimedean_ceiling @ A @ X ) @ Z3 ) ) ) ).
% ceiling_add_of_int
thf(fact_2234_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_2235_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_2236_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_2237_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_2238_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_2239_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_2240_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_2241_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_2242_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_2243_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_2244_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_2245_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_2246_bit__minus__numeral__Bit0__Suc__iff,axiom,
! [W2: num,N2: nat] :
( ( bit_se5641148757651400278ts_bit @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ W2 ) ) ) @ ( suc @ N2 ) )
= ( bit_se5641148757651400278ts_bit @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ W2 ) ) @ N2 ) ) ).
% bit_minus_numeral_Bit0_Suc_iff
thf(fact_2247_bit__minus__numeral__Bit1__Suc__iff,axiom,
! [W2: num,N2: nat] :
( ( bit_se5641148757651400278ts_bit @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ W2 ) ) ) @ ( suc @ N2 ) )
= ( ~ ( bit_se5641148757651400278ts_bit @ int @ ( numeral_numeral @ int @ W2 ) @ N2 ) ) ) ).
% bit_minus_numeral_Bit1_Suc_iff
thf(fact_2248_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_2249_and__Suc__0__eq,axiom,
! [N2: nat] :
( ( bit_se5824344872417868541ns_and @ nat @ N2 @ ( suc @ ( zero_zero @ nat ) ) )
= ( modulo_modulo @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% and_Suc_0_eq
thf(fact_2250_Suc__0__and__eq,axiom,
! [N2: nat] :
( ( bit_se5824344872417868541ns_and @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N2 )
= ( modulo_modulo @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% Suc_0_and_eq
thf(fact_2251_sum_Ocl__ivl__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [N2: nat,M2: nat,G: nat > A] :
( ( ( ord_less @ nat @ ( suc @ N2 ) @ M2 )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ ( suc @ N2 ) ) )
= ( zero_zero @ A ) ) )
& ( ~ ( ord_less @ nat @ ( suc @ N2 ) @ M2 )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ ( suc @ N2 ) ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ) ) ).
% sum.cl_ivl_Suc
thf(fact_2252_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_2253_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_2254_sum__zero__power,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A4: set @ nat,C2: nat > A] :
( ( ( ( finite_finite2 @ nat @ A4 )
& ( member @ nat @ ( zero_zero @ nat ) @ A4 ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( C2 @ I3 ) @ ( power_power @ A @ ( zero_zero @ A ) @ I3 ) )
@ A4 )
= ( C2 @ ( zero_zero @ nat ) ) ) )
& ( ~ ( ( finite_finite2 @ nat @ A4 )
& ( member @ nat @ ( zero_zero @ nat ) @ A4 ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( C2 @ I3 ) @ ( power_power @ A @ ( zero_zero @ A ) @ I3 ) )
@ A4 )
= ( zero_zero @ A ) ) ) ) ) ).
% sum_zero_power
thf(fact_2255_bit__mod__2__iff,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A3: A,N2: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ N2 )
= ( ( N2
= ( zero_zero @ nat ) )
& ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) ) ) ) ).
% bit_mod_2_iff
thf(fact_2256_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_2257_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_2258_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_2259_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_2260_sum__zero__power_H,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A4: set @ nat,C2: nat > A,D2: nat > A] :
( ( ( ( finite_finite2 @ nat @ A4 )
& ( member @ nat @ ( zero_zero @ nat ) @ A4 ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( divide_divide @ A @ ( times_times @ A @ ( C2 @ I3 ) @ ( power_power @ A @ ( zero_zero @ A ) @ I3 ) ) @ ( D2 @ I3 ) )
@ A4 )
= ( divide_divide @ A @ ( C2 @ ( zero_zero @ nat ) ) @ ( D2 @ ( zero_zero @ nat ) ) ) ) )
& ( ~ ( ( finite_finite2 @ nat @ A4 )
& ( member @ nat @ ( zero_zero @ nat ) @ A4 ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( divide_divide @ A @ ( times_times @ A @ ( C2 @ I3 ) @ ( power_power @ A @ ( zero_zero @ A ) @ I3 ) ) @ ( D2 @ I3 ) )
@ A4 )
= ( zero_zero @ A ) ) ) ) ) ).
% sum_zero_power'
thf(fact_2261_bit__disjunctive__add__iff,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A3: A,B3: A,N2: nat] :
( ! [N3: nat] :
( ~ ( bit_se5641148757651400278ts_bit @ A @ A3 @ N3 )
| ~ ( bit_se5641148757651400278ts_bit @ A @ B3 @ N3 ) )
=> ( ( bit_se5641148757651400278ts_bit @ A @ ( plus_plus @ A @ A3 @ B3 ) @ N2 )
= ( ( bit_se5641148757651400278ts_bit @ A @ A3 @ N2 )
| ( bit_se5641148757651400278ts_bit @ A @ B3 @ N2 ) ) ) ) ) ).
% bit_disjunctive_add_iff
thf(fact_2262_not__bit__1__Suc,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N2: nat] :
~ ( bit_se5641148757651400278ts_bit @ A @ ( one_one @ A ) @ ( suc @ N2 ) ) ) ).
% not_bit_1_Suc
thf(fact_2263_bit__1__iff,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [N2: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( one_one @ A ) @ N2 )
= ( N2
= ( zero_zero @ nat ) ) ) ) ).
% bit_1_iff
thf(fact_2264_sum__cong__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A4: set @ nat,F2: nat > A,G: nat > A] :
( ~ ( member @ nat @ ( zero_zero @ nat ) @ A4 )
=> ( ! [X5: nat] :
( ( member @ nat @ ( suc @ X5 ) @ A4 )
=> ( ( F2 @ ( suc @ X5 ) )
= ( G @ ( suc @ X5 ) ) ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ A4 )
= ( groups7311177749621191930dd_sum @ nat @ A @ G @ A4 ) ) ) ) ) ).
% sum_cong_Suc
thf(fact_2265_bit__take__bit__iff,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M2: nat,A3: A,N2: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( bit_se2584673776208193580ke_bit @ A @ M2 @ A3 ) @ N2 )
= ( ( ord_less @ nat @ N2 @ M2 )
& ( bit_se5641148757651400278ts_bit @ A @ A3 @ N2 ) ) ) ) ).
% bit_take_bit_iff
thf(fact_2266_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_2267_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_2268_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_2269_bit__of__bool__iff,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [B3: $o,N2: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( zero_neq_one_of_bool @ A @ B3 ) @ N2 )
= ( B3
& ( N2
= ( zero_zero @ nat ) ) ) ) ) ).
% bit_of_bool_iff
thf(fact_2270_sum__subtractf__nat,axiom,
! [A: $tType,A4: set @ A,G: A > nat,F2: A > nat] :
( ! [X5: A] :
( ( member @ A @ X5 @ A4 )
=> ( ord_less_eq @ nat @ ( G @ X5 ) @ ( F2 @ X5 ) ) )
=> ( ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [X4: A] : ( minus_minus @ nat @ ( F2 @ X4 ) @ ( G @ X4 ) )
@ A4 )
= ( minus_minus @ nat @ ( groups7311177749621191930dd_sum @ A @ nat @ F2 @ A4 ) @ ( groups7311177749621191930dd_sum @ A @ nat @ G @ A4 ) ) ) ) ).
% sum_subtractf_nat
thf(fact_2271_sum_Oshift__bounds__cl__Suc__ivl,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,M2: nat,N2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M2 ) @ ( suc @ N2 ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_or1337092689740270186AtMost @ nat @ M2 @ N2 ) ) ) ) ).
% sum.shift_bounds_cl_Suc_ivl
thf(fact_2272_sum_Oshift__bounds__cl__nat__ivl,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,M2: nat,K: nat,N2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ M2 @ K ) @ ( plus_plus @ nat @ N2 @ K ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( G @ ( plus_plus @ nat @ I3 @ K ) )
@ ( set_or1337092689740270186AtMost @ nat @ M2 @ N2 ) ) ) ) ).
% sum.shift_bounds_cl_nat_ivl
thf(fact_2273_sum__eq__Suc0__iff,axiom,
! [A: $tType,A4: set @ A,F2: A > nat] :
( ( finite_finite2 @ A @ A4 )
=> ( ( ( groups7311177749621191930dd_sum @ A @ nat @ F2 @ A4 )
= ( suc @ ( zero_zero @ nat ) ) )
= ( ? [X4: A] :
( ( member @ A @ X4 @ A4 )
& ( ( F2 @ X4 )
= ( suc @ ( zero_zero @ nat ) ) )
& ! [Y6: A] :
( ( member @ A @ Y6 @ A4 )
=> ( ( X4 != Y6 )
=> ( ( F2 @ Y6 )
= ( zero_zero @ nat ) ) ) ) ) ) ) ) ).
% sum_eq_Suc0_iff
thf(fact_2274_sum__SucD,axiom,
! [A: $tType,F2: A > nat,A4: set @ A,N2: nat] :
( ( ( groups7311177749621191930dd_sum @ A @ nat @ F2 @ A4 )
= ( suc @ N2 ) )
=> ? [X5: A] :
( ( member @ A @ X5 @ A4 )
& ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F2 @ X5 ) ) ) ) ).
% sum_SucD
thf(fact_2275_sum__eq__1__iff,axiom,
! [A: $tType,A4: set @ A,F2: A > nat] :
( ( finite_finite2 @ A @ A4 )
=> ( ( ( groups7311177749621191930dd_sum @ A @ nat @ F2 @ A4 )
= ( one_one @ nat ) )
= ( ? [X4: A] :
( ( member @ A @ X4 @ A4 )
& ( ( F2 @ X4 )
= ( one_one @ nat ) )
& ! [Y6: A] :
( ( member @ A @ Y6 @ A4 )
=> ( ( X4 != Y6 )
=> ( ( F2 @ Y6 )
= ( zero_zero @ nat ) ) ) ) ) ) ) ) ).
% sum_eq_1_iff
thf(fact_2276_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_2277_ceiling__le__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Z3: int] :
( ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ X ) @ Z3 )
= ( ord_less_eq @ A @ X @ ( ring_1_of_int @ A @ Z3 ) ) ) ) ).
% ceiling_le_iff
thf(fact_2278_less__ceiling__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z3: int,X: A] :
( ( ord_less @ int @ Z3 @ ( archimedean_ceiling @ A @ X ) )
= ( ord_less @ A @ ( ring_1_of_int @ A @ Z3 ) @ X ) ) ) ).
% less_ceiling_iff
thf(fact_2279_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_2280_sum__power__add,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [X: A,M2: nat,I6: set @ nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( power_power @ A @ X @ ( plus_plus @ nat @ M2 @ I3 ) )
@ I6 )
= ( times_times @ A @ ( power_power @ A @ X @ M2 ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ I6 ) ) ) ) ).
% sum_power_add
thf(fact_2281_sum_OatLeastAtMost__rev,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,N2: nat,M2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ N2 @ M2 ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( G @ ( minus_minus @ nat @ ( plus_plus @ nat @ M2 @ N2 ) @ I3 ) )
@ ( set_or1337092689740270186AtMost @ nat @ N2 @ M2 ) ) ) ) ).
% sum.atLeastAtMost_rev
thf(fact_2282_sum__nth__roots,axiom,
! [N2: nat,C2: complex] :
( ( ord_less @ nat @ ( one_one @ nat ) @ N2 )
=> ( ( groups7311177749621191930dd_sum @ complex @ complex
@ ^ [X4: complex] : X4
@ ( collect @ complex
@ ^ [Z6: complex] :
( ( power_power @ complex @ Z6 @ N2 )
= C2 ) ) )
= ( zero_zero @ complex ) ) ) ).
% sum_nth_roots
thf(fact_2283_sum__roots__unity,axiom,
! [N2: nat] :
( ( ord_less @ nat @ ( one_one @ nat ) @ N2 )
=> ( ( groups7311177749621191930dd_sum @ complex @ complex
@ ^ [X4: complex] : X4
@ ( collect @ complex
@ ^ [Z6: complex] :
( ( power_power @ complex @ Z6 @ N2 )
= ( one_one @ complex ) ) ) )
= ( zero_zero @ complex ) ) ) ).
% sum_roots_unity
thf(fact_2284_of__int__ceiling__le__add__one,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [R2: A] : ( ord_less_eq @ A @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ A @ R2 ) ) @ ( plus_plus @ A @ R2 @ ( one_one @ A ) ) ) ) ).
% of_int_ceiling_le_add_one
thf(fact_2285_of__int__ceiling__diff__one__le,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [R2: A] : ( ord_less_eq @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ A @ R2 ) ) @ ( one_one @ A ) ) @ R2 ) ) ).
% of_int_ceiling_diff_one_le
thf(fact_2286_sum__diff__nat,axiom,
! [A: $tType,B2: set @ A,A4: set @ A,F2: A > nat] :
( ( finite_finite2 @ A @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ A4 )
=> ( ( groups7311177749621191930dd_sum @ A @ nat @ F2 @ ( minus_minus @ ( set @ A ) @ A4 @ B2 ) )
= ( minus_minus @ nat @ ( groups7311177749621191930dd_sum @ A @ nat @ F2 @ A4 ) @ ( groups7311177749621191930dd_sum @ A @ nat @ F2 @ B2 ) ) ) ) ) ).
% sum_diff_nat
thf(fact_2287_sum__diff1__nat,axiom,
! [A: $tType,A3: A,A4: set @ A,F2: A > nat] :
( ( ( member @ A @ A3 @ A4 )
=> ( ( groups7311177749621191930dd_sum @ A @ nat @ F2 @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( minus_minus @ nat @ ( groups7311177749621191930dd_sum @ A @ nat @ F2 @ A4 ) @ ( F2 @ A3 ) ) ) )
& ( ~ ( member @ A @ A3 @ A4 )
=> ( ( groups7311177749621191930dd_sum @ A @ nat @ F2 @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( groups7311177749621191930dd_sum @ A @ nat @ F2 @ A4 ) ) ) ) ).
% sum_diff1_nat
thf(fact_2288_sum__shift__lb__Suc0__0,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [F2: nat > A,K: nat] :
( ( ( F2 @ ( zero_zero @ nat ) )
= ( zero_zero @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ ( zero_zero @ nat ) ) @ K ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ K ) ) ) ) ) ).
% sum_shift_lb_Suc0_0
thf(fact_2289_sum_OatLeast0__atMost__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,N2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( suc @ N2 ) ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ).
% sum.atLeast0_atMost_Suc
thf(fact_2290_sum_Onat__ivl__Suc_H,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [M2: nat,N2: nat,G: nat > A] :
( ( ord_less_eq @ nat @ M2 @ ( suc @ N2 ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ ( suc @ N2 ) ) )
= ( plus_plus @ A @ ( G @ ( suc @ N2 ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N2 ) ) ) ) ) ) ).
% sum.nat_ivl_Suc'
thf(fact_2291_sum_OatLeast__Suc__atMost,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [M2: nat,N2: nat,G: nat > A] :
( ( ord_less_eq @ nat @ M2 @ N2 )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N2 ) )
= ( plus_plus @ A @ ( G @ M2 ) @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M2 ) @ N2 ) ) ) ) ) ) ).
% sum.atLeast_Suc_atMost
thf(fact_2292_bit__imp__take__bit__positive,axiom,
! [N2: nat,M2: nat,K: int] :
( ( ord_less @ nat @ N2 @ M2 )
=> ( ( bit_se5641148757651400278ts_bit @ int @ K @ N2 )
=> ( ord_less @ int @ ( zero_zero @ int ) @ ( bit_se2584673776208193580ke_bit @ int @ M2 @ K ) ) ) ) ).
% bit_imp_take_bit_positive
thf(fact_2293_bit__concat__bit__iff,axiom,
! [M2: nat,K: int,L: int,N2: nat] :
( ( bit_se5641148757651400278ts_bit @ int @ ( bit_concat_bit @ M2 @ K @ L ) @ N2 )
= ( ( ( ord_less @ nat @ N2 @ M2 )
& ( bit_se5641148757651400278ts_bit @ int @ K @ N2 ) )
| ( ( ord_less_eq @ nat @ M2 @ N2 )
& ( bit_se5641148757651400278ts_bit @ int @ L @ ( minus_minus @ nat @ N2 @ M2 ) ) ) ) ) ).
% bit_concat_bit_iff
thf(fact_2294_sum_OSuc__reindex__ivl,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [M2: nat,N2: nat,G: nat > A] :
( ( ord_less_eq @ nat @ M2 @ N2 )
=> ( ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N2 ) ) @ ( G @ ( suc @ N2 ) ) )
= ( plus_plus @ A @ ( G @ M2 )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_or1337092689740270186AtMost @ nat @ M2 @ N2 ) ) ) ) ) ) ).
% sum.Suc_reindex_ivl
thf(fact_2295_sum__Suc__diff,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [M2: nat,N2: nat,F2: nat > A] :
( ( ord_less_eq @ nat @ M2 @ ( suc @ N2 ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( minus_minus @ A @ ( F2 @ ( suc @ I3 ) ) @ ( F2 @ I3 ) )
@ ( set_or1337092689740270186AtMost @ nat @ M2 @ N2 ) )
= ( minus_minus @ A @ ( F2 @ ( suc @ N2 ) ) @ ( F2 @ M2 ) ) ) ) ) ).
% sum_Suc_diff
thf(fact_2296_exp__eq__0__imp__not__bit,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [N2: nat,A3: A] :
( ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 )
= ( zero_zero @ A ) )
=> ~ ( bit_se5641148757651400278ts_bit @ A @ A3 @ N2 ) ) ) ).
% exp_eq_0_imp_not_bit
thf(fact_2297_bit__Suc,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A3: A,N2: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ A3 @ ( suc @ N2 ) )
= ( bit_se5641148757651400278ts_bit @ A @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ N2 ) ) ) ).
% bit_Suc
thf(fact_2298_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_2299_ceiling__unique,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z3: int,X: A] :
( ( ord_less @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ Z3 ) @ ( one_one @ A ) ) @ X )
=> ( ( ord_less_eq @ A @ X @ ( ring_1_of_int @ A @ Z3 ) )
=> ( ( archimedean_ceiling @ A @ X )
= Z3 ) ) ) ) ).
% ceiling_unique
thf(fact_2300_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_2301_ceiling__split,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [P: int > $o,T2: A] :
( ( P @ ( archimedean_ceiling @ A @ T2 ) )
= ( ! [I3: int] :
( ( ( ord_less @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ I3 ) @ ( one_one @ A ) ) @ T2 )
& ( ord_less_eq @ A @ T2 @ ( ring_1_of_int @ A @ I3 ) ) )
=> ( P @ I3 ) ) ) ) ) ).
% ceiling_split
thf(fact_2302_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_2303_ceiling__less__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Z3: int] :
( ( ord_less @ int @ ( archimedean_ceiling @ A @ X ) @ Z3 )
= ( ord_less_eq @ A @ X @ ( minus_minus @ A @ ( ring_1_of_int @ A @ Z3 ) @ ( one_one @ A ) ) ) ) ) ).
% ceiling_less_iff
thf(fact_2304_le__ceiling__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z3: int,X: A] :
( ( ord_less_eq @ int @ Z3 @ ( archimedean_ceiling @ A @ X ) )
= ( ord_less @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ Z3 ) @ ( one_one @ A ) ) @ X ) ) ) ).
% le_ceiling_iff
thf(fact_2305_int__bit__bound,axiom,
! [K: int] :
~ ! [N3: nat] :
( ! [M5: nat] :
( ( ord_less_eq @ nat @ N3 @ M5 )
=> ( ( bit_se5641148757651400278ts_bit @ int @ K @ M5 )
= ( bit_se5641148757651400278ts_bit @ int @ K @ N3 ) ) )
=> ~ ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( ( bit_se5641148757651400278ts_bit @ int @ K @ ( minus_minus @ nat @ N3 @ ( one_one @ nat ) ) )
= ( ~ ( bit_se5641148757651400278ts_bit @ int @ K @ N3 ) ) ) ) ) ).
% int_bit_bound
thf(fact_2306_sum_Oub__add__nat,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [M2: nat,N2: nat,G: nat > A,P5: nat] :
( ( ord_less_eq @ nat @ M2 @ ( plus_plus @ nat @ N2 @ ( one_one @ nat ) ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ ( plus_plus @ nat @ N2 @ P5 ) ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N2 ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ N2 @ ( one_one @ nat ) ) @ ( plus_plus @ nat @ N2 @ P5 ) ) ) ) ) ) ) ).
% sum.ub_add_nat
thf(fact_2307_sum__count__set,axiom,
! [A: $tType,Xs: list @ A,X8: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ X8 )
=> ( ( finite_finite2 @ A @ X8 )
=> ( ( groups7311177749621191930dd_sum @ A @ nat @ ( count_list @ A @ Xs ) @ X8 )
= ( size_size @ ( list @ A ) @ Xs ) ) ) ) ).
% sum_count_set
thf(fact_2308_and__exp__eq__0__iff__not__bit,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A,N2: nat] :
( ( ( bit_se5824344872417868541ns_and @ A @ A3 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) )
= ( zero_zero @ A ) )
= ( ~ ( bit_se5641148757651400278ts_bit @ A @ A3 @ N2 ) ) ) ) ).
% and_exp_eq_0_iff_not_bit
thf(fact_2309_ceiling__divide__upper,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Q3: A,P5: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Q3 )
=> ( ord_less_eq @ A @ P5 @ ( times_times @ A @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ A @ ( divide_divide @ A @ P5 @ Q3 ) ) ) @ Q3 ) ) ) ) ).
% ceiling_divide_upper
thf(fact_2310_sum__natinterval__diff,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [M2: nat,N2: nat,F2: nat > A] :
( ( ( ord_less_eq @ nat @ M2 @ N2 )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K2: nat] : ( minus_minus @ A @ ( F2 @ K2 ) @ ( F2 @ ( plus_plus @ nat @ K2 @ ( one_one @ nat ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ M2 @ N2 ) )
= ( minus_minus @ A @ ( F2 @ M2 ) @ ( F2 @ ( plus_plus @ nat @ N2 @ ( one_one @ nat ) ) ) ) ) )
& ( ~ ( ord_less_eq @ nat @ M2 @ N2 )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K2: nat] : ( minus_minus @ A @ ( F2 @ K2 ) @ ( F2 @ ( plus_plus @ nat @ K2 @ ( one_one @ nat ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ M2 @ N2 ) )
= ( zero_zero @ A ) ) ) ) ) ).
% sum_natinterval_diff
thf(fact_2311_sum__telescope_H_H,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [M2: nat,N2: nat,F2: nat > A] :
( ( ord_less_eq @ nat @ M2 @ N2 )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K2: nat] : ( minus_minus @ A @ ( F2 @ K2 ) @ ( F2 @ ( minus_minus @ nat @ K2 @ ( one_one @ nat ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M2 ) @ N2 ) )
= ( minus_minus @ A @ ( F2 @ N2 ) @ ( F2 @ M2 ) ) ) ) ) ).
% sum_telescope''
thf(fact_2312_even__bit__succ__iff,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A3: A,N2: nat] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
=> ( ( bit_se5641148757651400278ts_bit @ A @ ( plus_plus @ A @ ( one_one @ A ) @ A3 ) @ N2 )
= ( ( bit_se5641148757651400278ts_bit @ A @ A3 @ N2 )
| ( N2
= ( zero_zero @ nat ) ) ) ) ) ) ).
% even_bit_succ_iff
thf(fact_2313_odd__bit__iff__bit__pred,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A3: A,N2: nat] :
( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
=> ( ( bit_se5641148757651400278ts_bit @ A @ A3 @ N2 )
= ( ( bit_se5641148757651400278ts_bit @ A @ ( minus_minus @ A @ A3 @ ( one_one @ A ) ) @ N2 )
| ( N2
= ( zero_zero @ nat ) ) ) ) ) ) ).
% odd_bit_iff_bit_pred
thf(fact_2314_ceiling__divide__lower,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Q3: A,P5: 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 @ P5 @ Q3 ) ) ) @ ( one_one @ A ) ) @ Q3 ) @ P5 ) ) ) ).
% ceiling_divide_lower
thf(fact_2315_ceiling__eq,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [N2: int,X: A] :
( ( ord_less @ A @ ( ring_1_of_int @ A @ N2 ) @ X )
=> ( ( ord_less_eq @ A @ X @ ( plus_plus @ A @ ( ring_1_of_int @ A @ N2 ) @ ( one_one @ A ) ) )
=> ( ( archimedean_ceiling @ A @ X )
= ( plus_plus @ int @ N2 @ ( one_one @ int ) ) ) ) ) ) ).
% ceiling_eq
thf(fact_2316_mask__eq__sum__exp,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [N2: nat] :
( ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) @ ( one_one @ A ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
@ ( collect @ nat
@ ^ [Q5: nat] : ( ord_less @ nat @ Q5 @ N2 ) ) ) ) ) ).
% mask_eq_sum_exp
thf(fact_2317_sum__gp__multiplied,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [M2: nat,N2: nat,X: A] :
( ( ord_less_eq @ nat @ M2 @ N2 )
=> ( ( times_times @ A @ ( minus_minus @ A @ ( one_one @ A ) @ X ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N2 ) ) )
= ( minus_minus @ A @ ( power_power @ A @ X @ M2 ) @ ( power_power @ A @ X @ ( suc @ N2 ) ) ) ) ) ) ).
% sum_gp_multiplied
thf(fact_2318_sum_Oin__pairs,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,M2: nat,N2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M2 ) @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( plus_plus @ A @ ( G @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I3 ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ M2 @ N2 ) ) ) ) ).
% sum.in_pairs
thf(fact_2319_and__int_Opinduct,axiom,
! [A0: int,A1: int,P: int > int > $o] :
( ( accp @ ( product_prod @ int @ int ) @ bit_and_int_rel @ ( product_Pair @ int @ int @ A0 @ A1 ) )
=> ( ! [K3: int,L4: int] :
( ( accp @ ( product_prod @ int @ int ) @ bit_and_int_rel @ ( product_Pair @ int @ int @ K3 @ L4 ) )
=> ( ( ~ ( ( member @ int @ K3 @ ( insert2 @ int @ ( zero_zero @ int ) @ ( insert2 @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) )
& ( member @ int @ L4 @ ( insert2 @ int @ ( zero_zero @ int ) @ ( insert2 @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) )
=> ( P @ ( divide_divide @ int @ K3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L4 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) )
=> ( P @ K3 @ L4 ) ) )
=> ( P @ A0 @ A1 ) ) ) ).
% and_int.pinduct
thf(fact_2320_bit__sum__mult__2__cases,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A,B3: A,N2: nat] :
( ! [J3: nat] :
~ ( bit_se5641148757651400278ts_bit @ A @ A3 @ ( suc @ J3 ) )
=> ( ( bit_se5641148757651400278ts_bit @ A @ ( plus_plus @ A @ A3 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B3 ) ) @ N2 )
= ( ( ( N2
= ( zero_zero @ nat ) )
=> ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) )
& ( ( N2
!= ( zero_zero @ nat ) )
=> ( bit_se5641148757651400278ts_bit @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B3 ) @ N2 ) ) ) ) ) ) ).
% bit_sum_mult_2_cases
thf(fact_2321_bit__rec,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ( ( bit_se5641148757651400278ts_bit @ A )
= ( ^ [A5: A,N: nat] :
( ( ( N
= ( zero_zero @ nat ) )
=> ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A5 ) )
& ( ( N
!= ( zero_zero @ nat ) )
=> ( bit_se5641148757651400278ts_bit @ A @ ( divide_divide @ A @ A5 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ) ) ) ).
% bit_rec
thf(fact_2322_and__nat__unfold,axiom,
( ( bit_se5824344872417868541ns_and @ nat )
= ( ^ [M: nat,N: nat] :
( if @ nat
@ ( ( M
= ( zero_zero @ nat ) )
| ( N
= ( zero_zero @ nat ) ) )
@ ( zero_zero @ nat )
@ ( plus_plus @ nat @ ( times_times @ nat @ ( modulo_modulo @ nat @ M @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ nat @ ( divide_divide @ nat @ M @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ).
% and_nat_unfold
thf(fact_2323_and__nat__rec,axiom,
( ( bit_se5824344872417868541ns_and @ nat )
= ( ^ [M: nat,N: nat] :
( plus_plus @ nat
@ ( zero_neq_one_of_bool @ nat
@ ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M )
& ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) )
@ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ nat @ ( divide_divide @ nat @ M @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% and_nat_rec
thf(fact_2324_mask__eq__sum__exp__nat,axiom,
! [N2: nat] :
( ( minus_minus @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) @ ( suc @ ( zero_zero @ nat ) ) )
= ( groups7311177749621191930dd_sum @ nat @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
@ ( collect @ nat
@ ^ [Q5: nat] : ( ord_less @ nat @ Q5 @ N2 ) ) ) ) ).
% mask_eq_sum_exp_nat
thf(fact_2325_gauss__sum__nat,axiom,
! [N2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [X4: nat] : X4
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N2 ) )
= ( divide_divide @ nat @ ( times_times @ nat @ N2 @ ( suc @ N2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% gauss_sum_nat
thf(fact_2326_take__bit__Suc__from__most,axiom,
! [N2: nat,K: int] :
( ( bit_se2584673776208193580ke_bit @ int @ ( suc @ N2 ) @ K )
= ( plus_plus @ int @ ( times_times @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) @ ( zero_neq_one_of_bool @ int @ ( bit_se5641148757651400278ts_bit @ int @ K @ N2 ) ) ) @ ( bit_se2584673776208193580ke_bit @ int @ N2 @ K ) ) ) ).
% take_bit_Suc_from_most
thf(fact_2327_upto_Opinduct,axiom,
! [A0: int,A1: int,P: int > int > $o] :
( ( accp @ ( product_prod @ int @ int ) @ upto_rel @ ( product_Pair @ int @ int @ A0 @ A1 ) )
=> ( ! [I4: int,J3: int] :
( ( accp @ ( product_prod @ int @ int ) @ upto_rel @ ( product_Pair @ int @ int @ I4 @ J3 ) )
=> ( ( ( ord_less_eq @ int @ I4 @ J3 )
=> ( P @ ( plus_plus @ int @ I4 @ ( one_one @ int ) ) @ J3 ) )
=> ( P @ I4 @ J3 ) ) )
=> ( P @ A0 @ A1 ) ) ) ).
% upto.pinduct
thf(fact_2328_sum__gp,axiom,
! [A: $tType] :
( ( ( division_ring @ A )
& ( comm_ring @ A ) )
=> ! [N2: nat,M2: nat,X: A] :
( ( ( ord_less @ nat @ N2 @ M2 )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N2 ) )
= ( zero_zero @ A ) ) )
& ( ~ ( ord_less @ nat @ N2 @ M2 )
=> ( ( ( X
= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N2 ) )
= ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ ( plus_plus @ nat @ N2 @ ( one_one @ nat ) ) @ M2 ) ) ) )
& ( ( X
!= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N2 ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( power_power @ A @ X @ M2 ) @ ( power_power @ A @ X @ ( suc @ N2 ) ) ) @ ( minus_minus @ A @ ( one_one @ A ) @ X ) ) ) ) ) ) ) ) ).
% sum_gp
thf(fact_2329_gauss__sum__from__Suc__0,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [N2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ ( semiring_1_of_nat @ A ) @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N2 ) )
= ( divide_divide @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N2 ) @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N2 ) @ ( one_one @ A ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% gauss_sum_from_Suc_0
thf(fact_2330_sum__gp__offset,axiom,
! [A: $tType] :
( ( ( division_ring @ A )
& ( comm_ring @ A ) )
=> ! [X: A,M2: nat,N2: nat] :
( ( ( X
= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_or1337092689740270186AtMost @ nat @ M2 @ ( plus_plus @ nat @ M2 @ N2 ) ) )
= ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N2 ) @ ( one_one @ A ) ) ) )
& ( ( X
!= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_or1337092689740270186AtMost @ nat @ M2 @ ( plus_plus @ nat @ M2 @ N2 ) ) )
= ( divide_divide @ A @ ( times_times @ A @ ( power_power @ A @ X @ M2 ) @ ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ X @ ( suc @ N2 ) ) ) ) @ ( minus_minus @ A @ ( one_one @ A ) @ X ) ) ) ) ) ) ).
% sum_gp_offset
thf(fact_2331_double__gauss__sum__from__Suc__0,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [N2: 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 ) ) @ N2 ) ) )
= ( times_times @ A @ ( semiring_1_of_nat @ A @ N2 ) @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N2 ) @ ( one_one @ A ) ) ) ) ) ).
% double_gauss_sum_from_Suc_0
thf(fact_2332_arith__series,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [A3: A,D2: A,N2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( plus_plus @ A @ A3 @ ( times_times @ A @ ( semiring_1_of_nat @ A @ I3 ) @ D2 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N2 ) )
= ( divide_divide @ A @ ( times_times @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N2 ) @ ( one_one @ A ) ) @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N2 ) @ D2 ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% arith_series
thf(fact_2333_of__nat__eq__iff,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [M2: nat,N2: nat] :
( ( ( semiring_1_of_nat @ A @ M2 )
= ( semiring_1_of_nat @ A @ N2 ) )
= ( M2 = N2 ) ) ) ).
% of_nat_eq_iff
thf(fact_2334_numeral__le__real__of__nat__iff,axiom,
! [N2: num,M2: nat] :
( ( ord_less_eq @ real @ ( numeral_numeral @ real @ N2 ) @ ( semiring_1_of_nat @ real @ M2 ) )
= ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ N2 ) @ M2 ) ) ).
% numeral_le_real_of_nat_iff
thf(fact_2335_int__eq__iff__numeral,axiom,
! [M2: nat,V2: num] :
( ( ( semiring_1_of_nat @ int @ M2 )
= ( numeral_numeral @ int @ V2 ) )
= ( M2
= ( numeral_numeral @ nat @ V2 ) ) ) ).
% int_eq_iff_numeral
thf(fact_2336_negative__eq__positive,axiom,
! [N2: nat,M2: nat] :
( ( ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N2 ) )
= ( semiring_1_of_nat @ int @ M2 ) )
= ( ( N2
= ( zero_zero @ nat ) )
& ( M2
= ( zero_zero @ nat ) ) ) ) ).
% negative_eq_positive
thf(fact_2337_of__int__of__nat__eq,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [N2: nat] :
( ( ring_1_of_int @ A @ ( semiring_1_of_nat @ int @ N2 ) )
= ( semiring_1_of_nat @ A @ N2 ) ) ) ).
% of_int_of_nat_eq
thf(fact_2338_negative__zle,axiom,
! [N2: nat,M2: nat] : ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N2 ) ) @ ( semiring_1_of_nat @ int @ M2 ) ) ).
% negative_zle
thf(fact_2339_int__dvd__int__iff,axiom,
! [M2: nat,N2: nat] :
( ( dvd_dvd @ int @ ( semiring_1_of_nat @ int @ M2 ) @ ( semiring_1_of_nat @ int @ N2 ) )
= ( dvd_dvd @ nat @ M2 @ N2 ) ) ).
% int_dvd_int_iff
thf(fact_2340_of__nat__0,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ( ( semiring_1_of_nat @ A @ ( zero_zero @ nat ) )
= ( zero_zero @ A ) ) ) ).
% of_nat_0
thf(fact_2341_of__nat__0__eq__iff,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [N2: nat] :
( ( ( zero_zero @ A )
= ( semiring_1_of_nat @ A @ N2 ) )
= ( ( zero_zero @ nat )
= N2 ) ) ) ).
% of_nat_0_eq_iff
thf(fact_2342_of__nat__eq__0__iff,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [M2: nat] :
( ( ( semiring_1_of_nat @ A @ M2 )
= ( zero_zero @ A ) )
= ( M2
= ( zero_zero @ nat ) ) ) ) ).
% of_nat_eq_0_iff
thf(fact_2343_of__nat__less__iff,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [M2: nat,N2: nat] :
( ( ord_less @ A @ ( semiring_1_of_nat @ A @ M2 ) @ ( semiring_1_of_nat @ A @ N2 ) )
= ( ord_less @ nat @ M2 @ N2 ) ) ) ).
% of_nat_less_iff
thf(fact_2344_of__nat__le__iff,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [M2: nat,N2: nat] :
( ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ M2 ) @ ( semiring_1_of_nat @ A @ N2 ) )
= ( ord_less_eq @ nat @ M2 @ N2 ) ) ) ).
% of_nat_le_iff
thf(fact_2345_numeral__less__real__of__nat__iff,axiom,
! [W2: num,N2: nat] :
( ( ord_less @ real @ ( numeral_numeral @ real @ W2 ) @ ( semiring_1_of_nat @ real @ N2 ) )
= ( ord_less @ nat @ ( numeral_numeral @ nat @ W2 ) @ N2 ) ) ).
% numeral_less_real_of_nat_iff
thf(fact_2346_real__of__nat__less__numeral__iff,axiom,
! [N2: nat,W2: num] :
( ( ord_less @ real @ ( semiring_1_of_nat @ real @ N2 ) @ ( numeral_numeral @ real @ W2 ) )
= ( ord_less @ nat @ N2 @ ( numeral_numeral @ nat @ W2 ) ) ) ).
% real_of_nat_less_numeral_iff
thf(fact_2347_of__nat__add,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [M2: nat,N2: nat] :
( ( semiring_1_of_nat @ A @ ( plus_plus @ nat @ M2 @ N2 ) )
= ( plus_plus @ A @ ( semiring_1_of_nat @ A @ M2 ) @ ( semiring_1_of_nat @ A @ N2 ) ) ) ) ).
% of_nat_add
thf(fact_2348_of__nat__mult,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [M2: nat,N2: nat] :
( ( semiring_1_of_nat @ A @ ( times_times @ nat @ M2 @ N2 ) )
= ( times_times @ A @ ( semiring_1_of_nat @ A @ M2 ) @ ( semiring_1_of_nat @ A @ N2 ) ) ) ) ).
% of_nat_mult
thf(fact_2349_of__nat__eq__1__iff,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [N2: nat] :
( ( ( semiring_1_of_nat @ A @ N2 )
= ( one_one @ A ) )
= ( N2
= ( one_one @ nat ) ) ) ) ).
% of_nat_eq_1_iff
thf(fact_2350_of__nat__1__eq__iff,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [N2: nat] :
( ( ( one_one @ A )
= ( semiring_1_of_nat @ A @ N2 ) )
= ( N2
= ( one_one @ nat ) ) ) ) ).
% of_nat_1_eq_iff
thf(fact_2351_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_2352_negative__zless,axiom,
! [N2: nat,M2: nat] : ( ord_less @ int @ ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ ( suc @ N2 ) ) ) @ ( semiring_1_of_nat @ int @ M2 ) ) ).
% negative_zless
thf(fact_2353_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_2354_of__nat__sum,axiom,
! [A: $tType,B: $tType] :
( ( semiring_1 @ A )
=> ! [F2: B > nat,A4: set @ B] :
( ( semiring_1_of_nat @ A @ ( groups7311177749621191930dd_sum @ B @ nat @ F2 @ A4 ) )
= ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X4: B] : ( semiring_1_of_nat @ A @ ( F2 @ X4 ) )
@ A4 ) ) ) ).
% of_nat_sum
thf(fact_2355_of__nat__le__0__iff,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [M2: nat] :
( ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ M2 ) @ ( zero_zero @ A ) )
= ( M2
= ( zero_zero @ nat ) ) ) ) ).
% of_nat_le_0_iff
thf(fact_2356_of__nat__Suc,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [M2: nat] :
( ( semiring_1_of_nat @ A @ ( suc @ M2 ) )
= ( plus_plus @ A @ ( one_one @ A ) @ ( semiring_1_of_nat @ A @ M2 ) ) ) ) ).
% of_nat_Suc
thf(fact_2357_of__nat__0__less__iff,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [N2: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( semiring_1_of_nat @ A @ N2 ) )
= ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ).
% of_nat_0_less_iff
thf(fact_2358_of__nat__less__of__nat__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [B3: nat,W2: nat,X: nat] :
( ( ord_less @ A @ ( power_power @ A @ ( semiring_1_of_nat @ A @ B3 ) @ W2 ) @ ( semiring_1_of_nat @ A @ X ) )
= ( ord_less @ nat @ ( power_power @ nat @ B3 @ W2 ) @ X ) ) ) ).
% of_nat_less_of_nat_power_cancel_iff
thf(fact_2359_of__nat__power__less__of__nat__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X: nat,B3: nat,W2: nat] :
( ( ord_less @ A @ ( semiring_1_of_nat @ A @ X ) @ ( power_power @ A @ ( semiring_1_of_nat @ A @ B3 ) @ W2 ) )
= ( ord_less @ nat @ X @ ( power_power @ nat @ B3 @ W2 ) ) ) ) ).
% of_nat_power_less_of_nat_cancel_iff
thf(fact_2360_of__nat__power__le__of__nat__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X: nat,B3: nat,W2: nat] :
( ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ X ) @ ( power_power @ A @ ( semiring_1_of_nat @ A @ B3 ) @ W2 ) )
= ( ord_less_eq @ nat @ X @ ( power_power @ nat @ B3 @ W2 ) ) ) ) ).
% of_nat_power_le_of_nat_cancel_iff
thf(fact_2361_of__nat__le__of__nat__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [B3: nat,W2: nat,X: nat] :
( ( ord_less_eq @ A @ ( power_power @ A @ ( semiring_1_of_nat @ A @ B3 ) @ W2 ) @ ( semiring_1_of_nat @ A @ X ) )
= ( ord_less_eq @ nat @ ( power_power @ nat @ B3 @ W2 ) @ X ) ) ) ).
% of_nat_le_of_nat_power_cancel_iff
thf(fact_2362_of__nat__zero__less__power__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X: nat,N2: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( power_power @ A @ ( semiring_1_of_nat @ A @ X ) @ N2 ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ X )
| ( N2
= ( zero_zero @ nat ) ) ) ) ) ).
% of_nat_zero_less_power_iff
thf(fact_2363_of__nat__less__numeral__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X: nat,I: num,N2: nat] :
( ( ord_less @ A @ ( semiring_1_of_nat @ A @ X ) @ ( power_power @ A @ ( numeral_numeral @ A @ I ) @ N2 ) )
= ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ I ) @ N2 ) ) ) ) ).
% of_nat_less_numeral_power_cancel_iff
thf(fact_2364_numeral__power__less__of__nat__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [I: num,N2: nat,X: nat] :
( ( ord_less @ A @ ( power_power @ A @ ( numeral_numeral @ A @ I ) @ N2 ) @ ( semiring_1_of_nat @ A @ X ) )
= ( ord_less @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ I ) @ N2 ) @ X ) ) ) ).
% numeral_power_less_of_nat_cancel_iff
thf(fact_2365_of__nat__le__numeral__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X: nat,I: num,N2: nat] :
( ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ X ) @ ( power_power @ A @ ( numeral_numeral @ A @ I ) @ N2 ) )
= ( ord_less_eq @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ I ) @ N2 ) ) ) ) ).
% of_nat_le_numeral_power_cancel_iff
thf(fact_2366_numeral__power__le__of__nat__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [I: num,N2: nat,X: nat] :
( ( ord_less_eq @ A @ ( power_power @ A @ ( numeral_numeral @ A @ I ) @ N2 ) @ ( semiring_1_of_nat @ A @ X ) )
= ( ord_less_eq @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ I ) @ N2 ) @ X ) ) ) ).
% numeral_power_le_of_nat_cancel_iff
thf(fact_2367_real__arch__simple,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X: A] :
? [N3: nat] : ( ord_less_eq @ A @ X @ ( semiring_1_of_nat @ A @ N3 ) ) ) ).
% real_arch_simple
thf(fact_2368_reals__Archimedean2,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X: A] :
? [N3: nat] : ( ord_less @ A @ X @ ( semiring_1_of_nat @ A @ N3 ) ) ) ).
% reals_Archimedean2
thf(fact_2369_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_2370_nat__less__real__le,axiom,
( ( ord_less @ nat )
= ( ^ [N: nat,M: nat] : ( ord_less_eq @ real @ ( plus_plus @ real @ ( semiring_1_of_nat @ real @ N ) @ ( one_one @ real ) ) @ ( semiring_1_of_nat @ real @ M ) ) ) ) ).
% nat_less_real_le
thf(fact_2371_int__cases2,axiom,
! [Z3: int] :
( ! [N3: nat] :
( Z3
!= ( semiring_1_of_nat @ int @ N3 ) )
=> ~ ! [N3: nat] :
( Z3
!= ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N3 ) ) ) ) ).
% int_cases2
thf(fact_2372_int__diff__cases,axiom,
! [Z3: int] :
~ ! [M3: nat,N3: nat] :
( Z3
!= ( minus_minus @ int @ ( semiring_1_of_nat @ int @ M3 ) @ ( semiring_1_of_nat @ int @ N3 ) ) ) ).
% int_diff_cases
thf(fact_2373_of__nat__less__of__int__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N2: nat,X: int] :
( ( ord_less @ A @ ( semiring_1_of_nat @ A @ N2 ) @ ( ring_1_of_int @ A @ X ) )
= ( ord_less @ int @ ( semiring_1_of_nat @ int @ N2 ) @ X ) ) ) ).
% of_nat_less_of_int_iff
thf(fact_2374_bit__Suc__0__iff,axiom,
! [N2: nat] :
( ( bit_se5641148757651400278ts_bit @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N2 )
= ( N2
= ( zero_zero @ nat ) ) ) ).
% bit_Suc_0_iff
thf(fact_2375_not__bit__Suc__0__Suc,axiom,
! [N2: nat] :
~ ( bit_se5641148757651400278ts_bit @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( suc @ N2 ) ) ).
% not_bit_Suc_0_Suc
thf(fact_2376_of__nat__0__le__iff,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [N2: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( semiring_1_of_nat @ A @ N2 ) ) ) ).
% of_nat_0_le_iff
thf(fact_2377_of__nat__less__0__iff,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [M2: nat] :
~ ( ord_less @ A @ ( semiring_1_of_nat @ A @ M2 ) @ ( zero_zero @ A ) ) ) ).
% of_nat_less_0_iff
thf(fact_2378_of__nat__neq__0,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [N2: nat] :
( ( semiring_1_of_nat @ A @ ( suc @ N2 ) )
!= ( zero_zero @ A ) ) ) ).
% of_nat_neq_0
thf(fact_2379_less__imp__of__nat__less,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [M2: nat,N2: nat] :
( ( ord_less @ nat @ M2 @ N2 )
=> ( ord_less @ A @ ( semiring_1_of_nat @ A @ M2 ) @ ( semiring_1_of_nat @ A @ N2 ) ) ) ) ).
% less_imp_of_nat_less
thf(fact_2380_of__nat__less__imp__less,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [M2: nat,N2: nat] :
( ( ord_less @ A @ ( semiring_1_of_nat @ A @ M2 ) @ ( semiring_1_of_nat @ A @ N2 ) )
=> ( ord_less @ nat @ M2 @ N2 ) ) ) ).
% of_nat_less_imp_less
thf(fact_2381_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_2382_nat__le__real__less,axiom,
( ( ord_less_eq @ nat )
= ( ^ [N: nat,M: nat] : ( ord_less @ real @ ( semiring_1_of_nat @ real @ N ) @ ( plus_plus @ real @ ( semiring_1_of_nat @ real @ M ) @ ( one_one @ real ) ) ) ) ) ).
% nat_le_real_less
thf(fact_2383_int__ops_I1_J,axiom,
( ( semiring_1_of_nat @ int @ ( zero_zero @ nat ) )
= ( zero_zero @ int ) ) ).
% int_ops(1)
thf(fact_2384_nat__int__comparison_I2_J,axiom,
( ( ord_less @ nat )
= ( ^ [A5: nat,B4: nat] : ( ord_less @ int @ ( semiring_1_of_nat @ int @ A5 ) @ ( semiring_1_of_nat @ int @ B4 ) ) ) ) ).
% nat_int_comparison(2)
thf(fact_2385_int__cases,axiom,
! [Z3: int] :
( ! [N3: nat] :
( Z3
!= ( semiring_1_of_nat @ int @ N3 ) )
=> ~ ! [N3: nat] :
( Z3
!= ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ ( suc @ N3 ) ) ) ) ) ).
% int_cases
thf(fact_2386_int__of__nat__induct,axiom,
! [P: int > $o,Z3: int] :
( ! [N3: nat] : ( P @ ( semiring_1_of_nat @ int @ N3 ) )
=> ( ! [N3: nat] : ( P @ ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ ( suc @ N3 ) ) ) )
=> ( P @ Z3 ) ) ) ).
% int_of_nat_induct
thf(fact_2387_zle__int,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ M2 ) @ ( semiring_1_of_nat @ int @ N2 ) )
= ( ord_less_eq @ nat @ M2 @ N2 ) ) ).
% zle_int
thf(fact_2388_nat__int__comparison_I3_J,axiom,
( ( ord_less_eq @ nat )
= ( ^ [A5: nat,B4: nat] : ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ A5 ) @ ( semiring_1_of_nat @ int @ B4 ) ) ) ) ).
% nat_int_comparison(3)
thf(fact_2389_nonneg__int__cases,axiom,
! [K: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
=> ~ ! [N3: nat] :
( K
!= ( semiring_1_of_nat @ int @ N3 ) ) ) ).
% nonneg_int_cases
thf(fact_2390_zero__le__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
=> ? [N3: nat] :
( K
= ( semiring_1_of_nat @ int @ N3 ) ) ) ).
% zero_le_imp_eq_int
thf(fact_2391_zadd__int__left,axiom,
! [M2: nat,N2: nat,Z3: int] :
( ( plus_plus @ int @ ( semiring_1_of_nat @ int @ M2 ) @ ( plus_plus @ int @ ( semiring_1_of_nat @ int @ N2 ) @ Z3 ) )
= ( plus_plus @ int @ ( semiring_1_of_nat @ int @ ( plus_plus @ nat @ M2 @ N2 ) ) @ Z3 ) ) ).
% zadd_int_left
thf(fact_2392_int__plus,axiom,
! [N2: nat,M2: nat] :
( ( semiring_1_of_nat @ int @ ( plus_plus @ nat @ N2 @ M2 ) )
= ( plus_plus @ int @ ( semiring_1_of_nat @ int @ N2 ) @ ( semiring_1_of_nat @ int @ M2 ) ) ) ).
% int_plus
thf(fact_2393_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_2394_zle__iff__zadd,axiom,
( ( ord_less_eq @ int )
= ( ^ [W3: int,Z6: int] :
? [N: nat] :
( Z6
= ( plus_plus @ int @ W3 @ ( semiring_1_of_nat @ int @ N ) ) ) ) ) ).
% zle_iff_zadd
thf(fact_2395_not__int__zless__negative,axiom,
! [N2: nat,M2: nat] :
~ ( ord_less @ int @ ( semiring_1_of_nat @ int @ N2 ) @ ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ M2 ) ) ) ).
% not_int_zless_negative
thf(fact_2396_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_2397_nat__less__as__int,axiom,
( ( ord_less @ nat )
= ( ^ [A5: nat,B4: nat] : ( ord_less @ int @ ( semiring_1_of_nat @ int @ A5 ) @ ( semiring_1_of_nat @ int @ B4 ) ) ) ) ).
% nat_less_as_int
thf(fact_2398_nat__leq__as__int,axiom,
( ( ord_less_eq @ nat )
= ( ^ [A5: nat,B4: nat] : ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ A5 ) @ ( semiring_1_of_nat @ int @ B4 ) ) ) ) ).
% nat_leq_as_int
thf(fact_2399_not__bit__Suc__0__numeral,axiom,
! [N2: num] :
~ ( bit_se5641148757651400278ts_bit @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( numeral_numeral @ nat @ N2 ) ) ).
% not_bit_Suc_0_numeral
thf(fact_2400_ex__less__of__nat__mult,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ X )
=> ? [N3: nat] : ( ord_less @ A @ Y @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N3 ) @ X ) ) ) ) ).
% ex_less_of_nat_mult
thf(fact_2401_of__nat__diff,axiom,
! [A: $tType] :
( ( semiring_1_cancel @ A )
=> ! [N2: nat,M2: nat] :
( ( ord_less_eq @ nat @ N2 @ M2 )
=> ( ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ M2 @ N2 ) )
= ( minus_minus @ A @ ( semiring_1_of_nat @ A @ M2 ) @ ( semiring_1_of_nat @ A @ N2 ) ) ) ) ) ).
% of_nat_diff
thf(fact_2402_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 )
=> ( ! [M3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ M3 )
=> ( ord_less_eq @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ M3 ) @ X ) @ C2 ) )
=> ( X
= ( zero_zero @ real ) ) ) ) ) ).
% real_archimedian_rdiv_eq_0
thf(fact_2403_int__cases4,axiom,
! [M2: int] :
( ! [N3: nat] :
( M2
!= ( semiring_1_of_nat @ int @ N3 ) )
=> ~ ! [N3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( M2
!= ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N3 ) ) ) ) ) ).
% int_cases4
thf(fact_2404_int__zle__neg,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ N2 ) @ ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ M2 ) ) )
= ( ( N2
= ( zero_zero @ nat ) )
& ( M2
= ( zero_zero @ nat ) ) ) ) ).
% int_zle_neg
thf(fact_2405_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_2406_int__Suc,axiom,
! [N2: nat] :
( ( semiring_1_of_nat @ int @ ( suc @ N2 ) )
= ( plus_plus @ int @ ( semiring_1_of_nat @ int @ N2 ) @ ( one_one @ int ) ) ) ).
% int_Suc
thf(fact_2407_zless__iff__Suc__zadd,axiom,
( ( ord_less @ int )
= ( ^ [W3: int,Z6: int] :
? [N: nat] :
( Z6
= ( plus_plus @ int @ W3 @ ( semiring_1_of_nat @ int @ ( suc @ N ) ) ) ) ) ) ).
% zless_iff_Suc_zadd
thf(fact_2408_negative__zle__0,axiom,
! [N2: nat] : ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N2 ) ) @ ( zero_zero @ int ) ) ).
% negative_zle_0
thf(fact_2409_nonpos__int__cases,axiom,
! [K: int] :
( ( ord_less_eq @ int @ K @ ( zero_zero @ int ) )
=> ~ ! [N3: nat] :
( K
!= ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N3 ) ) ) ) ).
% nonpos_int_cases
thf(fact_2410_int__sum,axiom,
! [B: $tType,F2: B > nat,A4: set @ B] :
( ( semiring_1_of_nat @ int @ ( groups7311177749621191930dd_sum @ B @ nat @ F2 @ A4 ) )
= ( groups7311177749621191930dd_sum @ B @ int
@ ^ [X4: B] : ( semiring_1_of_nat @ int @ ( F2 @ X4 ) )
@ A4 ) ) ).
% int_sum
thf(fact_2411_mod__mult2__eq_H,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [A3: A,M2: nat,N2: nat] :
( ( modulo_modulo @ A @ A3 @ ( times_times @ A @ ( semiring_1_of_nat @ A @ M2 ) @ ( semiring_1_of_nat @ A @ N2 ) ) )
= ( plus_plus @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ M2 ) @ ( modulo_modulo @ A @ ( divide_divide @ A @ A3 @ ( semiring_1_of_nat @ A @ M2 ) ) @ ( semiring_1_of_nat @ A @ N2 ) ) ) @ ( modulo_modulo @ A @ A3 @ ( semiring_1_of_nat @ A @ M2 ) ) ) ) ) ).
% mod_mult2_eq'
thf(fact_2412_zero__less__imp__eq__int,axiom,
! [K: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ K )
=> ? [N3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
& ( K
= ( semiring_1_of_nat @ int @ N3 ) ) ) ) ).
% zero_less_imp_eq_int
thf(fact_2413_pos__int__cases,axiom,
! [K: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ K )
=> ~ ! [N3: nat] :
( ( K
= ( semiring_1_of_nat @ int @ N3 ) )
=> ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 ) ) ) ).
% pos_int_cases
thf(fact_2414_int__cases3,axiom,
! [K: int] :
( ( K
!= ( zero_zero @ int ) )
=> ( ! [N3: nat] :
( ( K
= ( semiring_1_of_nat @ int @ N3 ) )
=> ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 ) )
=> ~ ! [N3: nat] :
( ( K
= ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N3 ) ) )
=> ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 ) ) ) ) ).
% int_cases3
thf(fact_2415_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_2416_not__zle__0__negative,axiom,
! [N2: nat] :
~ ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ ( suc @ N2 ) ) ) ) ).
% not_zle_0_negative
thf(fact_2417_negD,axiom,
! [X: int] :
( ( ord_less @ int @ X @ ( zero_zero @ int ) )
=> ? [N3: nat] :
( X
= ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ ( suc @ N3 ) ) ) ) ) ).
% negD
thf(fact_2418_negative__zless__0,axiom,
! [N2: nat] : ( ord_less @ int @ ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ ( suc @ N2 ) ) ) @ ( zero_zero @ int ) ) ).
% negative_zless_0
thf(fact_2419_nat__approx__posE,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [E2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ E2 )
=> ~ ! [N3: nat] :
~ ( ord_less @ A @ ( divide_divide @ A @ ( one_one @ A ) @ ( semiring_1_of_nat @ A @ ( suc @ N3 ) ) ) @ E2 ) ) ) ).
% nat_approx_posE
thf(fact_2420_of__nat__less__two__power,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N2: nat] : ( ord_less @ A @ ( semiring_1_of_nat @ A @ N2 ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) ) ) ).
% of_nat_less_two_power
thf(fact_2421_inverse__of__nat__le,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [N2: nat,M2: nat] :
( ( ord_less_eq @ nat @ N2 @ M2 )
=> ( ( N2
!= ( zero_zero @ nat ) )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ ( one_one @ A ) @ ( semiring_1_of_nat @ A @ M2 ) ) @ ( divide_divide @ A @ ( one_one @ A ) @ ( semiring_1_of_nat @ A @ N2 ) ) ) ) ) ) ).
% inverse_of_nat_le
thf(fact_2422_neg__int__cases,axiom,
! [K: int] :
( ( ord_less @ int @ K @ ( zero_zero @ int ) )
=> ~ ! [N3: nat] :
( ( K
= ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N3 ) ) )
=> ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 ) ) ) ).
% neg_int_cases
thf(fact_2423_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_2424_double__arith__series,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A3: A,D2: A,N2: nat] :
( ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( plus_plus @ A @ A3 @ ( times_times @ A @ ( semiring_1_of_nat @ A @ I3 ) @ D2 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N2 ) ) )
= ( times_times @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N2 ) @ ( one_one @ A ) ) @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N2 ) @ D2 ) ) ) ) ) ).
% double_arith_series
thf(fact_2425_double__gauss__sum,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [N2: nat] :
( ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( semiring_1_of_nat @ A ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N2 ) ) )
= ( times_times @ A @ ( semiring_1_of_nat @ A @ N2 ) @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N2 ) @ ( one_one @ A ) ) ) ) ) ).
% double_gauss_sum
thf(fact_2426_gauss__sum,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [N2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ ( semiring_1_of_nat @ A ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N2 ) )
= ( divide_divide @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N2 ) @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N2 ) @ ( one_one @ A ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% gauss_sum
thf(fact_2427_of__nat__code__if,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ( ( semiring_1_of_nat @ A )
= ( ^ [N: nat] :
( if @ A
@ ( N
= ( zero_zero @ nat ) )
@ ( zero_zero @ A )
@ ( product_case_prod @ nat @ nat @ A
@ ^ [M: nat,Q5: nat] :
( if @ A
@ ( Q5
= ( zero_zero @ nat ) )
@ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ A @ M ) )
@ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ A @ M ) ) @ ( one_one @ A ) ) )
@ ( divmod_nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% of_nat_code_if
thf(fact_2428_lemma__termdiff2,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [H2: A,Z3: A,N2: nat] :
( ( H2
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( divide_divide @ A @ ( minus_minus @ A @ ( power_power @ A @ ( plus_plus @ A @ Z3 @ H2 ) @ N2 ) @ ( power_power @ A @ Z3 @ N2 ) ) @ H2 ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N2 ) @ ( power_power @ A @ Z3 @ ( minus_minus @ nat @ N2 @ ( 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 @ Z3 @ H2 ) @ Q5 ) @ ( power_power @ A @ Z3 @ ( minus_minus @ nat @ ( minus_minus @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ Q5 ) ) )
@ ( set_ord_lessThan @ nat @ ( minus_minus @ nat @ ( minus_minus @ nat @ N2 @ ( suc @ ( zero_zero @ nat ) ) ) @ P6 ) ) )
@ ( set_ord_lessThan @ nat @ ( minus_minus @ nat @ N2 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ) ) ).
% lemma_termdiff2
thf(fact_2429_lemma__termdiff3,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [H2: A,Z3: A,K5: real,N2: nat] :
( ( H2
!= ( zero_zero @ A ) )
=> ( ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ Z3 ) @ K5 )
=> ( ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ Z3 @ H2 ) ) @ K5 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( divide_divide @ A @ ( minus_minus @ A @ ( power_power @ A @ ( plus_plus @ A @ Z3 @ H2 ) @ N2 ) @ ( power_power @ A @ Z3 @ N2 ) ) @ H2 ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N2 ) @ ( power_power @ A @ Z3 @ ( minus_minus @ nat @ N2 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) @ ( times_times @ real @ ( times_times @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N2 ) @ ( semiring_1_of_nat @ real @ ( minus_minus @ nat @ N2 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) @ ( power_power @ real @ K5 @ ( minus_minus @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( real_V7770717601297561774m_norm @ A @ H2 ) ) ) ) ) ) ) ).
% lemma_termdiff3
thf(fact_2430_pochhammer__double,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [Z3: A,N2: nat] :
( ( comm_s3205402744901411588hammer @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Z3 ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) )
= ( 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 ) ) @ N2 ) ) ) @ ( comm_s3205402744901411588hammer @ A @ Z3 @ N2 ) ) @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ Z3 @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ N2 ) ) ) ) ).
% pochhammer_double
thf(fact_2431_of__nat__code,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ( ( semiring_1_of_nat @ A )
= ( ^ [N: nat] :
( semiri8178284476397505188at_aux @ A
@ ^ [I3: A] : ( plus_plus @ A @ I3 @ ( one_one @ A ) )
@ N
@ ( zero_zero @ A ) ) ) ) ) ).
% of_nat_code
thf(fact_2432_ceiling__log__nat__eq__powr__iff,axiom,
! [B3: nat,K: nat,N2: 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 @ N2 ) @ ( one_one @ int ) ) )
= ( ( 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 ) ) ) ) ) ) ) ) ).
% ceiling_log_nat_eq_powr_iff
thf(fact_2433_lessThan__eq__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( ( set_ord_lessThan @ A @ X )
= ( set_ord_lessThan @ A @ Y ) )
= ( X = Y ) ) ) ).
% lessThan_eq_iff
thf(fact_2434_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_2435_finite__lessThan,axiom,
! [K: nat] : ( finite_finite2 @ nat @ ( set_ord_lessThan @ nat @ K ) ) ).
% finite_lessThan
thf(fact_2436_lessThan__subset__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_ord_lessThan @ A @ X ) @ ( set_ord_lessThan @ A @ Y ) )
= ( ord_less_eq @ A @ X @ Y ) ) ) ).
% lessThan_subset_iff
thf(fact_2437_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_2438_pochhammer__Suc0,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A3: A] :
( ( comm_s3205402744901411588hammer @ A @ A3 @ ( suc @ ( zero_zero @ nat ) ) )
= A3 ) ) ).
% pochhammer_Suc0
thf(fact_2439_lessThan__0,axiom,
( ( set_ord_lessThan @ nat @ ( zero_zero @ nat ) )
= ( bot_bot @ ( set @ nat ) ) ) ).
% lessThan_0
thf(fact_2440_single__Diff__lessThan,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [K: A] :
( ( minus_minus @ ( set @ A ) @ ( insert2 @ A @ K @ ( bot_bot @ ( set @ A ) ) ) @ ( set_ord_lessThan @ A @ K ) )
= ( insert2 @ A @ K @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% single_Diff_lessThan
thf(fact_2441_sum_OlessThan__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,N2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_ord_lessThan @ nat @ ( suc @ N2 ) ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_ord_lessThan @ nat @ N2 ) ) @ ( G @ N2 ) ) ) ) ).
% sum.lessThan_Suc
thf(fact_2442_int__int__eq,axiom,
! [M2: nat,N2: nat] :
( ( ( semiring_1_of_nat @ int @ M2 )
= ( semiring_1_of_nat @ int @ N2 ) )
= ( M2 = N2 ) ) ).
% int_int_eq
thf(fact_2443_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_2444_infinite__Iio,axiom,
! [A: $tType] :
( ( ( linorder @ A )
& ( no_bot @ A ) )
=> ! [A3: A] :
~ ( finite_finite2 @ A @ ( set_ord_lessThan @ A @ A3 ) ) ) ).
% infinite_Iio
thf(fact_2445_lessThan__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( set_ord_lessThan @ A )
= ( ^ [U2: A] :
( collect @ A
@ ^ [X4: A] : ( ord_less @ A @ X4 @ U2 ) ) ) ) ) ).
% lessThan_def
thf(fact_2446_Iio__eq__empty__iff,axiom,
! [A: $tType] :
( ( ( linorder @ A )
& ( order_bot @ A ) )
=> ! [N2: A] :
( ( ( set_ord_lessThan @ A @ N2 )
= ( bot_bot @ ( set @ A ) ) )
= ( N2
= ( bot_bot @ A ) ) ) ) ).
% Iio_eq_empty_iff
thf(fact_2447_lessThan__strict__subset__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [M2: A,N2: A] :
( ( ord_less @ ( set @ A ) @ ( set_ord_lessThan @ A @ M2 ) @ ( set_ord_lessThan @ A @ N2 ) )
= ( ord_less @ A @ M2 @ N2 ) ) ) ).
% lessThan_strict_subset_iff
thf(fact_2448_pochhammer__pos,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X: A,N2: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ X )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( comm_s3205402744901411588hammer @ A @ X @ N2 ) ) ) ) ).
% pochhammer_pos
thf(fact_2449_pochhammer__eq__0__mono,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: A,N2: nat,M2: nat] :
( ( ( comm_s3205402744901411588hammer @ A @ A3 @ N2 )
= ( zero_zero @ A ) )
=> ( ( ord_less_eq @ nat @ N2 @ M2 )
=> ( ( comm_s3205402744901411588hammer @ A @ A3 @ M2 )
= ( zero_zero @ A ) ) ) ) ) ).
% pochhammer_eq_0_mono
thf(fact_2450_pochhammer__neq__0__mono,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: A,M2: nat,N2: nat] :
( ( ( comm_s3205402744901411588hammer @ A @ A3 @ M2 )
!= ( zero_zero @ A ) )
=> ( ( ord_less_eq @ nat @ N2 @ M2 )
=> ( ( comm_s3205402744901411588hammer @ A @ A3 @ N2 )
!= ( zero_zero @ A ) ) ) ) ) ).
% pochhammer_neq_0_mono
thf(fact_2451_lessThan__Suc,axiom,
! [K: nat] :
( ( set_ord_lessThan @ nat @ ( suc @ K ) )
= ( insert2 @ nat @ K @ ( set_ord_lessThan @ nat @ K ) ) ) ).
% lessThan_Suc
thf(fact_2452_lessThan__empty__iff,axiom,
! [N2: nat] :
( ( ( set_ord_lessThan @ nat @ N2 )
= ( bot_bot @ ( set @ nat ) ) )
= ( N2
= ( zero_zero @ nat ) ) ) ).
% lessThan_empty_iff
thf(fact_2453_finite__nat__bounded,axiom,
! [S2: set @ nat] :
( ( finite_finite2 @ nat @ S2 )
=> ? [K3: nat] : ( ord_less_eq @ ( set @ nat ) @ S2 @ ( set_ord_lessThan @ nat @ K3 ) ) ) ).
% finite_nat_bounded
thf(fact_2454_finite__nat__iff__bounded,axiom,
( ( finite_finite2 @ nat )
= ( ^ [S6: set @ nat] :
? [K2: nat] : ( ord_less_eq @ ( set @ nat ) @ S6 @ ( set_ord_lessThan @ nat @ K2 ) ) ) ) ).
% finite_nat_iff_bounded
thf(fact_2455_pochhammer__nonneg,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X: A,N2: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ X )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( comm_s3205402744901411588hammer @ A @ X @ N2 ) ) ) ) ).
% pochhammer_nonneg
thf(fact_2456_of__nat__aux_Osimps_I2_J,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [Inc: A > A,N2: nat,I: A] :
( ( semiri8178284476397505188at_aux @ A @ Inc @ ( suc @ N2 ) @ I )
= ( semiri8178284476397505188at_aux @ A @ Inc @ N2 @ ( Inc @ I ) ) ) ) ).
% of_nat_aux.simps(2)
thf(fact_2457_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_2458_pochhammer__0__left,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [N2: nat] :
( ( ( N2
= ( zero_zero @ nat ) )
=> ( ( comm_s3205402744901411588hammer @ A @ ( zero_zero @ A ) @ N2 )
= ( one_one @ A ) ) )
& ( ( N2
!= ( zero_zero @ nat ) )
=> ( ( comm_s3205402744901411588hammer @ A @ ( zero_zero @ A ) @ N2 )
= ( zero_zero @ A ) ) ) ) ) ).
% pochhammer_0_left
thf(fact_2459_lessThan__nat__numeral,axiom,
! [K: num] :
( ( set_ord_lessThan @ nat @ ( numeral_numeral @ nat @ K ) )
= ( insert2 @ nat @ ( pred_numeral @ K ) @ ( set_ord_lessThan @ nat @ ( pred_numeral @ K ) ) ) ) ).
% lessThan_nat_numeral
thf(fact_2460_sum_Onat__diff__reindex,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,N2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( G @ ( minus_minus @ nat @ N2 @ ( suc @ I3 ) ) )
@ ( set_ord_lessThan @ nat @ N2 ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_ord_lessThan @ nat @ N2 ) ) ) ) ).
% sum.nat_diff_reindex
thf(fact_2461_sum__diff__distrib,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [Q: A > nat,P: A > nat,N2: A] :
( ! [X5: A] : ( ord_less_eq @ nat @ ( Q @ X5 ) @ ( P @ X5 ) )
=> ( ( minus_minus @ nat @ ( groups7311177749621191930dd_sum @ A @ nat @ P @ ( set_ord_lessThan @ A @ N2 ) ) @ ( groups7311177749621191930dd_sum @ A @ nat @ Q @ ( set_ord_lessThan @ A @ N2 ) ) )
= ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [X4: A] : ( minus_minus @ nat @ ( P @ X4 ) @ ( Q @ X4 ) )
@ ( set_ord_lessThan @ A @ N2 ) ) ) ) ) ).
% sum_diff_distrib
thf(fact_2462_log__of__power__le,axiom,
! [M2: nat,B3: real,N2: nat] :
( ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ M2 ) @ ( power_power @ real @ B3 @ N2 ) )
=> ( ( ord_less @ real @ ( one_one @ real ) @ B3 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M2 )
=> ( ord_less_eq @ real @ ( log @ B3 @ ( semiring_1_of_nat @ real @ M2 ) ) @ ( semiring_1_of_nat @ real @ N2 ) ) ) ) ) ).
% log_of_power_le
thf(fact_2463_log__of__power__less,axiom,
! [M2: nat,B3: real,N2: nat] :
( ( ord_less @ real @ ( semiring_1_of_nat @ real @ M2 ) @ ( power_power @ real @ B3 @ N2 ) )
=> ( ( ord_less @ real @ ( one_one @ real ) @ B3 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M2 )
=> ( ord_less @ real @ ( log @ B3 @ ( semiring_1_of_nat @ real @ M2 ) ) @ ( semiring_1_of_nat @ real @ N2 ) ) ) ) ) ).
% log_of_power_less
thf(fact_2464_pochhammer__rec,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A3: A,N2: nat] :
( ( comm_s3205402744901411588hammer @ A @ A3 @ ( suc @ N2 ) )
= ( times_times @ A @ A3 @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) @ N2 ) ) ) ) ).
% pochhammer_rec
thf(fact_2465_pochhammer__rec_H,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [Z3: A,N2: nat] :
( ( comm_s3205402744901411588hammer @ A @ Z3 @ ( suc @ N2 ) )
= ( times_times @ A @ ( plus_plus @ A @ Z3 @ ( semiring_1_of_nat @ A @ N2 ) ) @ ( comm_s3205402744901411588hammer @ A @ Z3 @ N2 ) ) ) ) ).
% pochhammer_rec'
thf(fact_2466_pochhammer__Suc,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A3: A,N2: nat] :
( ( comm_s3205402744901411588hammer @ A @ A3 @ ( suc @ N2 ) )
= ( times_times @ A @ ( comm_s3205402744901411588hammer @ A @ A3 @ N2 ) @ ( plus_plus @ A @ A3 @ ( semiring_1_of_nat @ A @ N2 ) ) ) ) ) ).
% pochhammer_Suc
thf(fact_2467_pochhammer__of__nat__eq__0__lemma,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [N2: nat,K: nat] :
( ( ord_less @ nat @ N2 @ K )
=> ( ( comm_s3205402744901411588hammer @ A @ ( uminus_uminus @ A @ ( semiring_1_of_nat @ A @ N2 ) ) @ K )
= ( zero_zero @ A ) ) ) ) ).
% pochhammer_of_nat_eq_0_lemma
thf(fact_2468_pochhammer__of__nat__eq__0__iff,axiom,
! [A: $tType] :
( ( ( ring_char_0 @ A )
& ( idom @ A ) )
=> ! [N2: nat,K: nat] :
( ( ( comm_s3205402744901411588hammer @ A @ ( uminus_uminus @ A @ ( semiring_1_of_nat @ A @ N2 ) ) @ K )
= ( zero_zero @ A ) )
= ( ord_less @ nat @ N2 @ K ) ) ) ).
% pochhammer_of_nat_eq_0_iff
thf(fact_2469_pochhammer__eq__0__iff,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: A,N2: nat] :
( ( ( comm_s3205402744901411588hammer @ A @ A3 @ N2 )
= ( zero_zero @ A ) )
= ( ? [K2: nat] :
( ( ord_less @ nat @ K2 @ N2 )
& ( A3
= ( uminus_uminus @ A @ ( semiring_1_of_nat @ A @ K2 ) ) ) ) ) ) ) ).
% pochhammer_eq_0_iff
thf(fact_2470_pochhammer__of__nat__eq__0__lemma_H,axiom,
! [A: $tType] :
( ( ( ring_char_0 @ A )
& ( idom @ A ) )
=> ! [K: nat,N2: nat] :
( ( ord_less_eq @ nat @ K @ N2 )
=> ( ( comm_s3205402744901411588hammer @ A @ ( uminus_uminus @ A @ ( semiring_1_of_nat @ A @ N2 ) ) @ K )
!= ( zero_zero @ A ) ) ) ) ).
% pochhammer_of_nat_eq_0_lemma'
thf(fact_2471_pochhammer__product_H,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [Z3: A,N2: nat,M2: nat] :
( ( comm_s3205402744901411588hammer @ A @ Z3 @ ( plus_plus @ nat @ N2 @ M2 ) )
= ( times_times @ A @ ( comm_s3205402744901411588hammer @ A @ Z3 @ N2 ) @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ Z3 @ ( semiring_1_of_nat @ A @ N2 ) ) @ M2 ) ) ) ) ).
% pochhammer_product'
thf(fact_2472_sum_OlessThan__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,N2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_ord_lessThan @ nat @ ( suc @ N2 ) ) )
= ( plus_plus @ A @ ( G @ ( zero_zero @ nat ) )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_ord_lessThan @ nat @ N2 ) ) ) ) ) ).
% sum.lessThan_Suc_shift
thf(fact_2473_sum__lessThan__telescope,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [F2: nat > A,M2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [N: nat] : ( minus_minus @ A @ ( F2 @ ( suc @ N ) ) @ ( F2 @ N ) )
@ ( set_ord_lessThan @ nat @ M2 ) )
= ( minus_minus @ A @ ( F2 @ M2 ) @ ( F2 @ ( zero_zero @ nat ) ) ) ) ) ).
% sum_lessThan_telescope
thf(fact_2474_sum__lessThan__telescope_H,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [F2: nat > A,M2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [N: nat] : ( minus_minus @ A @ ( F2 @ N ) @ ( F2 @ ( suc @ N ) ) )
@ ( set_ord_lessThan @ nat @ M2 ) )
= ( minus_minus @ A @ ( F2 @ ( zero_zero @ nat ) ) @ ( F2 @ M2 ) ) ) ) ).
% sum_lessThan_telescope'
thf(fact_2475_sum_OatLeast1__atMost__eq,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,N2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N2 ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K2: nat] : ( G @ ( suc @ K2 ) )
@ ( set_ord_lessThan @ nat @ N2 ) ) ) ) ).
% sum.atLeast1_atMost_eq
thf(fact_2476_le__log2__of__power,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_eq @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) @ M2 )
=> ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ N2 ) @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ M2 ) ) ) ) ).
% le_log2_of_power
thf(fact_2477_one__diff__power__eq,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [X: A,N2: nat] :
( ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ X @ N2 ) )
= ( times_times @ A @ ( minus_minus @ A @ ( one_one @ A ) @ X ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_ord_lessThan @ nat @ N2 ) ) ) ) ) ).
% one_diff_power_eq
thf(fact_2478_power__diff__1__eq,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [X: A,N2: nat] :
( ( minus_minus @ A @ ( power_power @ A @ X @ N2 ) @ ( 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 @ N2 ) ) ) ) ) ).
% power_diff_1_eq
thf(fact_2479_geometric__sum,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [X: A,N2: nat] :
( ( X
!= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_ord_lessThan @ nat @ N2 ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( power_power @ A @ X @ N2 ) @ ( one_one @ A ) ) @ ( minus_minus @ A @ X @ ( one_one @ A ) ) ) ) ) ) ).
% geometric_sum
thf(fact_2480_pochhammer__product,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [M2: nat,N2: nat,Z3: A] :
( ( ord_less_eq @ nat @ M2 @ N2 )
=> ( ( comm_s3205402744901411588hammer @ A @ Z3 @ N2 )
= ( times_times @ A @ ( comm_s3205402744901411588hammer @ A @ Z3 @ M2 ) @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ Z3 @ ( semiring_1_of_nat @ A @ M2 ) ) @ ( minus_minus @ nat @ N2 @ M2 ) ) ) ) ) ) ).
% pochhammer_product
thf(fact_2481_less__log2__of__power,axiom,
! [N2: nat,M2: nat] :
( ( ord_less @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) @ M2 )
=> ( ord_less @ real @ ( semiring_1_of_nat @ real @ N2 ) @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ M2 ) ) ) ) ).
% less_log2_of_power
thf(fact_2482_sum__gp__strict,axiom,
! [A: $tType] :
( ( ( division_ring @ A )
& ( comm_ring @ A ) )
=> ! [X: A,N2: nat] :
( ( ( X
= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_ord_lessThan @ nat @ N2 ) )
= ( semiring_1_of_nat @ A @ N2 ) ) )
& ( ( X
!= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_ord_lessThan @ nat @ N2 ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ X @ N2 ) ) @ ( minus_minus @ A @ ( one_one @ A ) @ X ) ) ) ) ) ) ).
% sum_gp_strict
thf(fact_2483_lemma__termdiff1,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [Z3: A,H2: A,M2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [P6: nat] : ( minus_minus @ A @ ( times_times @ A @ ( power_power @ A @ ( plus_plus @ A @ Z3 @ H2 ) @ ( minus_minus @ nat @ M2 @ P6 ) ) @ ( power_power @ A @ Z3 @ P6 ) ) @ ( power_power @ A @ Z3 @ M2 ) )
@ ( set_ord_lessThan @ nat @ M2 ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [P6: nat] : ( times_times @ A @ ( power_power @ A @ Z3 @ P6 ) @ ( minus_minus @ A @ ( power_power @ A @ ( plus_plus @ A @ Z3 @ H2 ) @ ( minus_minus @ nat @ M2 @ P6 ) ) @ ( power_power @ A @ Z3 @ ( minus_minus @ nat @ M2 @ P6 ) ) ) )
@ ( set_ord_lessThan @ nat @ M2 ) ) ) ) ).
% lemma_termdiff1
thf(fact_2484_diff__power__eq__sum,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [X: A,N2: nat,Y: A] :
( ( minus_minus @ A @ ( power_power @ A @ X @ ( suc @ N2 ) ) @ ( power_power @ A @ Y @ ( suc @ N2 ) ) )
= ( 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 @ N2 @ P6 ) ) )
@ ( set_ord_lessThan @ nat @ ( suc @ N2 ) ) ) ) ) ) ).
% diff_power_eq_sum
thf(fact_2485_power__diff__sumr2,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [X: A,N2: nat,Y: A] :
( ( minus_minus @ A @ ( power_power @ A @ X @ N2 ) @ ( power_power @ A @ Y @ N2 ) )
= ( times_times @ A @ ( minus_minus @ A @ X @ Y )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( power_power @ A @ Y @ ( minus_minus @ nat @ N2 @ ( suc @ I3 ) ) ) @ ( power_power @ A @ X @ I3 ) )
@ ( set_ord_lessThan @ nat @ N2 ) ) ) ) ) ).
% power_diff_sumr2
thf(fact_2486_pochhammer__absorb__comp,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [R2: A,K: nat] :
( ( times_times @ A @ ( minus_minus @ A @ R2 @ ( semiring_1_of_nat @ A @ K ) ) @ ( comm_s3205402744901411588hammer @ A @ ( uminus_uminus @ A @ R2 ) @ K ) )
= ( times_times @ A @ R2 @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ ( uminus_uminus @ A @ R2 ) @ ( one_one @ A ) ) @ K ) ) ) ) ).
% pochhammer_absorb_comp
thf(fact_2487_log2__of__power__less,axiom,
! [M2: nat,N2: nat] :
( ( ord_less @ nat @ M2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M2 )
=> ( ord_less @ real @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ M2 ) ) @ ( semiring_1_of_nat @ real @ N2 ) ) ) ) ).
% log2_of_power_less
thf(fact_2488_log2__of__power__le,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq @ nat @ M2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M2 )
=> ( ord_less_eq @ real @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ M2 ) ) @ ( semiring_1_of_nat @ real @ N2 ) ) ) ) ).
% log2_of_power_le
thf(fact_2489_real__sum__nat__ivl__bounded2,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N2: nat,F2: nat > A,K5: A,K: nat] :
( ! [P7: nat] :
( ( ord_less @ nat @ P7 @ N2 )
=> ( ord_less_eq @ A @ ( F2 @ P7 ) @ K5 ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ K5 )
=> ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_ord_lessThan @ nat @ ( minus_minus @ nat @ N2 @ K ) ) ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N2 ) @ K5 ) ) ) ) ) ).
% real_sum_nat_ivl_bounded2
thf(fact_2490_one__diff__power__eq_H,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [X: A,N2: nat] :
( ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ X @ N2 ) )
= ( times_times @ A @ ( minus_minus @ A @ ( one_one @ A ) @ X )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( power_power @ A @ X @ ( minus_minus @ nat @ N2 @ ( suc @ I3 ) ) )
@ ( set_ord_lessThan @ nat @ N2 ) ) ) ) ) ).
% one_diff_power_eq'
thf(fact_2491_sum__split__even__odd,axiom,
! [F2: nat > real,G: nat > real,N2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I3: nat] : ( if @ real @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I3 ) @ ( F2 @ I3 ) @ ( G @ I3 ) )
@ ( set_ord_lessThan @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I3: nat] : ( F2 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I3 ) )
@ ( set_ord_lessThan @ nat @ N2 ) )
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I3: nat] : ( G @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I3 ) @ ( one_one @ nat ) ) )
@ ( set_ord_lessThan @ nat @ N2 ) ) ) ) ).
% sum_split_even_odd
thf(fact_2492_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_2493_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_2494_ceiling__log__nat__eq__if,axiom,
! [B3: nat,N2: nat,K: 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 ) ) ) )
=> ( ( 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 @ N2 ) @ ( one_one @ int ) ) ) ) ) ) ).
% ceiling_log_nat_eq_if
thf(fact_2495_ceiling__log2__div2,axiom,
! [N2: nat] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
=> ( ( archimedean_ceiling @ real @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ N2 ) ) )
= ( 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 @ N2 @ ( one_one @ nat ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) ) @ ( one_one @ int ) ) ) ) ).
% ceiling_log2_div2
thf(fact_2496_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_2497_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_2498_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_2499_norm__zero,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ( ( real_V7770717601297561774m_norm @ A @ ( zero_zero @ A ) )
= ( zero_zero @ real ) ) ) ).
% norm_zero
thf(fact_2500_lemma__NBseq__def2,axiom,
! [A: $tType,B: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [X8: A > B] :
( ( ? [K6: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K6 )
& ! [N: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( X8 @ N ) ) @ K6 ) ) )
= ( ? [N5: nat] :
! [N: A] : ( ord_less @ real @ ( real_V7770717601297561774m_norm @ B @ ( X8 @ N ) ) @ ( semiring_1_of_nat @ real @ ( suc @ N5 ) ) ) ) ) ) ).
% lemma_NBseq_def2
thf(fact_2501_lemma__NBseq__def,axiom,
! [A: $tType,B: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [X8: A > B] :
( ( ? [K6: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K6 )
& ! [N: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( X8 @ N ) ) @ K6 ) ) )
= ( ? [N5: nat] :
! [N: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( X8 @ N ) ) @ ( semiring_1_of_nat @ real @ ( suc @ N5 ) ) ) ) ) ) ).
% lemma_NBseq_def
thf(fact_2502_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_2503_norm__diff__ineq,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [A3: A,B3: A] : ( ord_less_eq @ real @ ( minus_minus @ real @ ( real_V7770717601297561774m_norm @ A @ A3 ) @ ( real_V7770717601297561774m_norm @ A @ B3 ) ) @ ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ A3 @ B3 ) ) ) ) ).
% norm_diff_ineq
thf(fact_2504_norm__uminus__minus,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X: A,Y: A] :
( ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( uminus_uminus @ A @ X ) @ Y ) )
= ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ X @ Y ) ) ) ) ).
% norm_uminus_minus
thf(fact_2505_power__eq__imp__eq__norm,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [W2: A,N2: nat,Z3: A] :
( ( ( power_power @ A @ W2 @ N2 )
= ( power_power @ A @ Z3 @ N2 ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( real_V7770717601297561774m_norm @ A @ W2 )
= ( real_V7770717601297561774m_norm @ A @ Z3 ) ) ) ) ) ).
% power_eq_imp_eq_norm
thf(fact_2506_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_2507_norm__add__less,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X: A,R2: real,Y: A,S: real] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ R2 )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ Y ) @ S )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ X @ Y ) ) @ ( plus_plus @ real @ R2 @ S ) ) ) ) ) ).
% norm_add_less
thf(fact_2508_norm__add__leD,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [A3: A,B3: A,C2: real] :
( ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ A3 @ B3 ) ) @ C2 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ B3 ) @ ( plus_plus @ real @ ( real_V7770717601297561774m_norm @ A @ A3 ) @ C2 ) ) ) ) ).
% norm_add_leD
thf(fact_2509_norm__triangle__le,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X: A,Y: A,E2: real] :
( ( ord_less_eq @ real @ ( plus_plus @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ ( real_V7770717601297561774m_norm @ A @ Y ) ) @ E2 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ X @ Y ) ) @ E2 ) ) ) ).
% norm_triangle_le
thf(fact_2510_norm__triangle__ineq,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X: A,Y: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ X @ Y ) ) @ ( plus_plus @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ ( real_V7770717601297561774m_norm @ A @ Y ) ) ) ) ).
% norm_triangle_ineq
thf(fact_2511_norm__triangle__mono,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [A3: A,R2: real,B3: A,S: real] :
( ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ A3 ) @ R2 )
=> ( ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ B3 ) @ S )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ A3 @ B3 ) ) @ ( plus_plus @ real @ R2 @ S ) ) ) ) ) ).
% norm_triangle_mono
thf(fact_2512_power__eq__1__iff,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [W2: A,N2: nat] :
( ( ( power_power @ A @ W2 @ N2 )
= ( one_one @ A ) )
=> ( ( ( real_V7770717601297561774m_norm @ A @ W2 )
= ( one_one @ real ) )
| ( N2
= ( zero_zero @ nat ) ) ) ) ) ).
% power_eq_1_iff
thf(fact_2513_norm__diff__triangle__ineq,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [A3: A,B3: A,C2: A,D2: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( plus_plus @ A @ A3 @ B3 ) @ ( plus_plus @ A @ C2 @ D2 ) ) ) @ ( plus_plus @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ A3 @ C2 ) ) @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ B3 @ D2 ) ) ) ) ) ).
% norm_diff_triangle_ineq
thf(fact_2514_sum__bounds__lt__plus1,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [F2: nat > A,Mm: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K2: nat] : ( F2 @ ( suc @ K2 ) )
@ ( set_ord_lessThan @ nat @ Mm ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_or1337092689740270186AtMost @ nat @ ( one_one @ nat ) @ Mm ) ) ) ) ).
% sum_bounds_lt_plus1
thf(fact_2515_sumr__cos__zero__one,axiom,
! [N2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M: nat] : ( times_times @ real @ ( cos_coeff @ M ) @ ( power_power @ real @ ( zero_zero @ real ) @ M ) )
@ ( set_ord_lessThan @ nat @ ( suc @ N2 ) ) )
= ( one_one @ real ) ) ).
% sumr_cos_zero_one
thf(fact_2516_pochhammer__times__pochhammer__half,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [Z3: A,N2: nat] :
( ( times_times @ A @ ( comm_s3205402744901411588hammer @ A @ Z3 @ ( suc @ N2 ) ) @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ Z3 @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ ( suc @ N2 ) ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [K2: nat] : ( plus_plus @ A @ Z3 @ ( divide_divide @ A @ ( semiring_1_of_nat @ A @ K2 ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) @ ( one_one @ nat ) ) ) ) ) ) ).
% pochhammer_times_pochhammer_half
thf(fact_2517_pochhammer__code,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ( ( comm_s3205402744901411588hammer @ A )
= ( ^ [A5: A,N: nat] :
( if @ A
@ ( N
= ( 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 @ N @ ( one_one @ nat ) )
@ ( one_one @ A ) ) ) ) ) ) ).
% pochhammer_code
thf(fact_2518_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_2519_geometric__deriv__sums,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [Z3: A] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ Z3 ) @ ( one_one @ real ) )
=> ( sums @ A
@ ^ [N: nat] : ( times_times @ A @ ( semiring_1_of_nat @ A @ ( suc @ N ) ) @ ( power_power @ A @ Z3 @ N ) )
@ ( divide_divide @ A @ ( one_one @ A ) @ ( power_power @ A @ ( minus_minus @ A @ ( one_one @ A ) @ Z3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% geometric_deriv_sums
thf(fact_2520_powr__eq__0__iff,axiom,
! [A: $tType] :
( ( ln @ A )
=> ! [W2: A,Z3: A] :
( ( ( powr @ A @ W2 @ Z3 )
= ( zero_zero @ A ) )
= ( W2
= ( zero_zero @ A ) ) ) ) ).
% powr_eq_0_iff
thf(fact_2521_powr__0,axiom,
! [A: $tType] :
( ( ln @ A )
=> ! [Z3: A] :
( ( powr @ A @ ( zero_zero @ A ) @ Z3 )
= ( zero_zero @ A ) ) ) ).
% powr_0
thf(fact_2522_of__nat__prod,axiom,
! [A: $tType,B: $tType] :
( ( comm_semiring_1 @ A )
=> ! [F2: B > nat,A4: set @ B] :
( ( semiring_1_of_nat @ A @ ( groups7121269368397514597t_prod @ B @ nat @ F2 @ A4 ) )
= ( groups7121269368397514597t_prod @ B @ A
@ ^ [X4: B] : ( semiring_1_of_nat @ A @ ( F2 @ X4 ) )
@ A4 ) ) ) ).
% of_nat_prod
thf(fact_2523_of__int__prod,axiom,
! [A: $tType,B: $tType] :
( ( comm_ring_1 @ A )
=> ! [F2: B > int,A4: set @ B] :
( ( ring_1_of_int @ A @ ( groups7121269368397514597t_prod @ B @ int @ F2 @ A4 ) )
= ( groups7121269368397514597t_prod @ B @ A
@ ^ [X4: B] : ( ring_1_of_int @ A @ ( F2 @ X4 ) )
@ A4 ) ) ) ).
% of_int_prod
thf(fact_2524_prod__zero__iff,axiom,
! [A: $tType,B: $tType] :
( ( semidom @ A )
=> ! [A4: set @ B,F2: B > A] :
( ( finite_finite2 @ B @ A4 )
=> ( ( ( groups7121269368397514597t_prod @ B @ A @ F2 @ A4 )
= ( zero_zero @ A ) )
= ( ? [X4: B] :
( ( member @ B @ X4 @ A4 )
& ( ( F2 @ X4 )
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% prod_zero_iff
thf(fact_2525_prod_Oempty,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: B > A] :
( ( groups7121269368397514597t_prod @ B @ A @ G @ ( bot_bot @ ( set @ B ) ) )
= ( one_one @ A ) ) ) ).
% prod.empty
thf(fact_2526_prod_Oinfinite,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A4: set @ B,G: B > A] :
( ~ ( finite_finite2 @ B @ A4 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ A4 )
= ( one_one @ A ) ) ) ) ).
% prod.infinite
thf(fact_2527_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_2528_dvd__prodI,axiom,
! [A: $tType,B: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A4: set @ B,A3: B,F2: B > A] :
( ( finite_finite2 @ B @ A4 )
=> ( ( member @ B @ A3 @ A4 )
=> ( dvd_dvd @ A @ ( F2 @ A3 ) @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ A4 ) ) ) ) ) ).
% dvd_prodI
thf(fact_2529_dvd__prod__eqI,axiom,
! [A: $tType,B: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A4: set @ B,A3: B,B3: A,F2: B > A] :
( ( finite_finite2 @ B @ A4 )
=> ( ( member @ B @ A3 @ A4 )
=> ( ( B3
= ( F2 @ A3 ) )
=> ( dvd_dvd @ A @ B3 @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ A4 ) ) ) ) ) ) ).
% dvd_prod_eqI
thf(fact_2530_cos__coeff__0,axiom,
( ( cos_coeff @ ( zero_zero @ nat ) )
= ( one_one @ real ) ) ).
% cos_coeff_0
thf(fact_2531_prod_Odelta,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S2: set @ B,A3: B,B3: B > A] :
( ( finite_finite2 @ B @ S2 )
=> ( ( ( member @ B @ A3 @ S2 )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K2: B] : ( if @ A @ ( K2 = A3 ) @ ( B3 @ K2 ) @ ( one_one @ A ) )
@ S2 )
= ( B3 @ A3 ) ) )
& ( ~ ( member @ B @ A3 @ S2 )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K2: B] : ( if @ A @ ( K2 = A3 ) @ ( B3 @ K2 ) @ ( one_one @ A ) )
@ S2 )
= ( one_one @ A ) ) ) ) ) ) ).
% prod.delta
thf(fact_2532_prod_Odelta_H,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S2: set @ B,A3: B,B3: B > A] :
( ( finite_finite2 @ B @ S2 )
=> ( ( ( member @ B @ A3 @ S2 )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K2: B] : ( if @ A @ ( A3 = K2 ) @ ( B3 @ K2 ) @ ( one_one @ A ) )
@ S2 )
= ( B3 @ A3 ) ) )
& ( ~ ( member @ B @ A3 @ S2 )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K2: B] : ( if @ A @ ( A3 = K2 ) @ ( B3 @ K2 ) @ ( one_one @ A ) )
@ S2 )
= ( one_one @ A ) ) ) ) ) ) ).
% prod.delta'
thf(fact_2533_prod_Oinsert,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A4: set @ B,X: B,G: B > A] :
( ( finite_finite2 @ B @ A4 )
=> ( ~ ( member @ B @ X @ A4 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( insert2 @ B @ X @ A4 ) )
= ( times_times @ A @ ( G @ X ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ A4 ) ) ) ) ) ) ).
% prod.insert
thf(fact_2534_prod_OlessThan__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,N2: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_lessThan @ nat @ ( suc @ N2 ) ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_lessThan @ nat @ N2 ) ) @ ( G @ N2 ) ) ) ) ).
% prod.lessThan_Suc
thf(fact_2535_powser__sums__zero__iff,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [A3: nat > A,X: A] :
( ( sums @ A
@ ^ [N: nat] : ( times_times @ A @ ( A3 @ N ) @ ( power_power @ A @ ( zero_zero @ A ) @ N ) )
@ X )
= ( ( A3 @ ( zero_zero @ nat ) )
= X ) ) ) ).
% powser_sums_zero_iff
thf(fact_2536_prod_Ocl__ivl__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [N2: nat,M2: nat,G: nat > A] :
( ( ( ord_less @ nat @ ( suc @ N2 ) @ M2 )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ ( suc @ N2 ) ) )
= ( one_one @ A ) ) )
& ( ~ ( ord_less @ nat @ ( suc @ N2 ) @ M2 )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ ( suc @ N2 ) ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ) ) ).
% prod.cl_ivl_Suc
thf(fact_2537_prod_Oswap__restrict,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A4: set @ B,B2: set @ C,G: B > C > A,R: B > C > $o] :
( ( finite_finite2 @ B @ A4 )
=> ( ( finite_finite2 @ C @ B2 )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [X4: B] :
( groups7121269368397514597t_prod @ C @ A @ ( G @ X4 )
@ ( collect @ C
@ ^ [Y6: C] :
( ( member @ C @ Y6 @ B2 )
& ( R @ X4 @ Y6 ) ) ) )
@ A4 )
= ( groups7121269368397514597t_prod @ C @ A
@ ^ [Y6: C] :
( groups7121269368397514597t_prod @ B @ A
@ ^ [X4: B] : ( G @ X4 @ Y6 )
@ ( collect @ B
@ ^ [X4: B] :
( ( member @ B @ X4 @ A4 )
& ( R @ X4 @ Y6 ) ) ) )
@ B2 ) ) ) ) ) ).
% prod.swap_restrict
thf(fact_2538_prod__mono,axiom,
! [A: $tType,B: $tType] :
( ( linordered_semidom @ A )
=> ! [A4: set @ B,F2: B > A,G: B > A] :
( ! [I4: B] :
( ( member @ B @ I4 @ A4 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ I4 ) )
& ( ord_less_eq @ A @ ( F2 @ I4 ) @ ( G @ I4 ) ) ) )
=> ( ord_less_eq @ A @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ A4 ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ A4 ) ) ) ) ).
% prod_mono
thf(fact_2539_prod__nonneg,axiom,
! [A: $tType,B: $tType] :
( ( linordered_semidom @ A )
=> ! [A4: set @ B,F2: B > A] :
( ! [X5: B] :
( ( member @ B @ X5 @ A4 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ X5 ) ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ A4 ) ) ) ) ).
% prod_nonneg
thf(fact_2540_prod__pos,axiom,
! [A: $tType,B: $tType] :
( ( linordered_semidom @ A )
=> ! [A4: set @ B,F2: B > A] :
( ! [X5: B] :
( ( member @ B @ X5 @ A4 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( F2 @ X5 ) ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ A4 ) ) ) ) ).
% prod_pos
thf(fact_2541_prod__ge__1,axiom,
! [A: $tType,B: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [A4: set @ B,F2: B > A] :
( ! [X5: B] :
( ( member @ B @ X5 @ A4 )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ ( F2 @ X5 ) ) )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ A4 ) ) ) ) ).
% prod_ge_1
thf(fact_2542_prod__zero,axiom,
! [B: $tType,A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A4: set @ B,F2: B > A] :
( ( finite_finite2 @ B @ A4 )
=> ( ? [X2: B] :
( ( member @ B @ X2 @ A4 )
& ( ( F2 @ X2 )
= ( zero_zero @ A ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ F2 @ A4 )
= ( zero_zero @ A ) ) ) ) ) ).
% prod_zero
thf(fact_2543_prod__atLeastAtMost__code,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [F2: nat > A,A3: nat,B3: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ F2 @ ( set_or1337092689740270186AtMost @ nat @ A3 @ B3 ) )
= ( set_fo6178422350223883121st_nat @ A
@ ^ [A5: nat] : ( times_times @ A @ ( F2 @ A5 ) )
@ A3
@ B3
@ ( one_one @ A ) ) ) ) ).
% prod_atLeastAtMost_code
thf(fact_2544_prod_Ointer__filter,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A4: set @ B,G: B > A,P: B > $o] :
( ( finite_finite2 @ B @ A4 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G
@ ( collect @ B
@ ^ [X4: B] :
( ( member @ B @ X4 @ A4 )
& ( P @ X4 ) ) ) )
= ( groups7121269368397514597t_prod @ B @ A
@ ^ [X4: B] : ( if @ A @ ( P @ X4 ) @ ( G @ X4 ) @ ( one_one @ A ) )
@ A4 ) ) ) ) ).
% prod.inter_filter
thf(fact_2545_prod_Oshift__bounds__cl__Suc__ivl,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,M2: nat,N2: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M2 ) @ ( suc @ N2 ) ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_or1337092689740270186AtMost @ nat @ M2 @ N2 ) ) ) ) ).
% prod.shift_bounds_cl_Suc_ivl
thf(fact_2546_prod_Oshift__bounds__cl__nat__ivl,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,M2: nat,K: nat,N2: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ M2 @ K ) @ ( plus_plus @ nat @ N2 @ K ) ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( G @ ( plus_plus @ nat @ I3 @ K ) )
@ ( set_or1337092689740270186AtMost @ nat @ M2 @ N2 ) ) ) ) ).
% prod.shift_bounds_cl_nat_ivl
thf(fact_2547_prod__le__1,axiom,
! [B: $tType,A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [A4: set @ B,F2: B > A] :
( ! [X5: B] :
( ( member @ B @ X5 @ A4 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ X5 ) )
& ( ord_less_eq @ A @ ( F2 @ X5 ) @ ( one_one @ A ) ) ) )
=> ( ord_less_eq @ A @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ A4 ) @ ( one_one @ A ) ) ) ) ).
% prod_le_1
thf(fact_2548_prod_Orelated,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [R: A > A > $o,S2: set @ B,H2: B > A,G: B > A] :
( ( R @ ( one_one @ A ) @ ( one_one @ A ) )
=> ( ! [X1: A,Y1: A,X23: A,Y22: A] :
( ( ( R @ X1 @ X23 )
& ( R @ Y1 @ Y22 ) )
=> ( R @ ( times_times @ A @ X1 @ Y1 ) @ ( times_times @ A @ X23 @ Y22 ) ) )
=> ( ( finite_finite2 @ B @ S2 )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ S2 )
=> ( R @ ( H2 @ X5 ) @ ( G @ X5 ) ) )
=> ( R @ ( groups7121269368397514597t_prod @ B @ A @ H2 @ S2 ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ S2 ) ) ) ) ) ) ) ).
% prod.related
thf(fact_2549_prod_Oinsert__if,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A4: set @ B,X: B,G: B > A] :
( ( finite_finite2 @ B @ A4 )
=> ( ( ( member @ B @ X @ A4 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( insert2 @ B @ X @ A4 ) )
= ( groups7121269368397514597t_prod @ B @ A @ G @ A4 ) ) )
& ( ~ ( member @ B @ X @ A4 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( insert2 @ B @ X @ A4 ) )
= ( times_times @ A @ ( G @ X ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ A4 ) ) ) ) ) ) ) ).
% prod.insert_if
thf(fact_2550_prod__dvd__prod__subset,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [B2: set @ B,A4: set @ B,F2: B > A] :
( ( finite_finite2 @ B @ B2 )
=> ( ( ord_less_eq @ ( set @ B ) @ A4 @ B2 )
=> ( dvd_dvd @ A @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ A4 ) @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ B2 ) ) ) ) ) ).
% prod_dvd_prod_subset
thf(fact_2551_prod__dvd__prod__subset2,axiom,
! [A: $tType,B: $tType] :
( ( comm_semiring_1 @ A )
=> ! [B2: set @ B,A4: set @ B,F2: B > A,G: B > A] :
( ( finite_finite2 @ B @ B2 )
=> ( ( ord_less_eq @ ( set @ B ) @ A4 @ B2 )
=> ( ! [A6: B] :
( ( member @ B @ A6 @ A4 )
=> ( dvd_dvd @ A @ ( F2 @ A6 ) @ ( G @ A6 ) ) )
=> ( dvd_dvd @ A @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ A4 ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ B2 ) ) ) ) ) ) ).
% prod_dvd_prod_subset2
thf(fact_2552_prod_Oreindex__bij__witness__not__neutral,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S5: set @ B,T6: set @ C,S2: set @ B,I: C > B,J: B > C,T4: set @ C,G: B > A,H2: C > A] :
( ( finite_finite2 @ B @ S5 )
=> ( ( finite_finite2 @ C @ T6 )
=> ( ! [A6: B] :
( ( member @ B @ A6 @ ( minus_minus @ ( set @ B ) @ S2 @ S5 ) )
=> ( ( I @ ( J @ A6 ) )
= A6 ) )
=> ( ! [A6: B] :
( ( member @ B @ A6 @ ( minus_minus @ ( set @ B ) @ S2 @ S5 ) )
=> ( member @ C @ ( J @ A6 ) @ ( minus_minus @ ( set @ C ) @ T4 @ T6 ) ) )
=> ( ! [B5: C] :
( ( member @ C @ B5 @ ( minus_minus @ ( set @ C ) @ T4 @ T6 ) )
=> ( ( J @ ( I @ B5 ) )
= B5 ) )
=> ( ! [B5: C] :
( ( member @ C @ B5 @ ( minus_minus @ ( set @ C ) @ T4 @ T6 ) )
=> ( member @ B @ ( I @ B5 ) @ ( minus_minus @ ( set @ B ) @ S2 @ S5 ) ) )
=> ( ! [A6: B] :
( ( member @ B @ A6 @ S5 )
=> ( ( G @ A6 )
= ( one_one @ A ) ) )
=> ( ! [B5: C] :
( ( member @ C @ B5 @ T6 )
=> ( ( H2 @ B5 )
= ( one_one @ A ) ) )
=> ( ! [A6: B] :
( ( member @ B @ A6 @ S2 )
=> ( ( H2 @ ( J @ A6 ) )
= ( G @ A6 ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ S2 )
= ( groups7121269368397514597t_prod @ C @ A @ H2 @ T4 ) ) ) ) ) ) ) ) ) ) ) ) ).
% prod.reindex_bij_witness_not_neutral
thf(fact_2553_powr__add,axiom,
! [A: $tType] :
( ( ( real_V3459762299906320749_field @ A )
& ( ln @ A ) )
=> ! [X: A,A3: A,B3: A] :
( ( powr @ A @ X @ ( plus_plus @ A @ A3 @ B3 ) )
= ( times_times @ A @ ( powr @ A @ X @ A3 ) @ ( powr @ A @ X @ B3 ) ) ) ) ).
% powr_add
thf(fact_2554_prod_Osetdiff__irrelevant,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A4: set @ B,G: B > A] :
( ( finite_finite2 @ B @ A4 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G
@ ( minus_minus @ ( set @ B ) @ A4
@ ( collect @ B
@ ^ [X4: B] :
( ( G @ X4 )
= ( one_one @ A ) ) ) ) )
= ( groups7121269368397514597t_prod @ B @ A @ G @ A4 ) ) ) ) ).
% prod.setdiff_irrelevant
thf(fact_2555_prod_Onat__diff__reindex,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,N2: nat] :
( ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( G @ ( minus_minus @ nat @ N2 @ ( suc @ I3 ) ) )
@ ( set_ord_lessThan @ nat @ N2 ) )
= ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_lessThan @ nat @ N2 ) ) ) ) ).
% prod.nat_diff_reindex
thf(fact_2556_prod_OatLeastAtMost__rev,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,N2: nat,M2: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ N2 @ M2 ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( G @ ( minus_minus @ nat @ ( plus_plus @ nat @ M2 @ N2 ) @ I3 ) )
@ ( set_or1337092689740270186AtMost @ nat @ N2 @ M2 ) ) ) ) ).
% prod.atLeastAtMost_rev
thf(fact_2557_less__1__prod2,axiom,
! [B: $tType,A: $tType] :
( ( linordered_idom @ B )
=> ! [I6: set @ A,I: A,F2: A > B] :
( ( finite_finite2 @ A @ I6 )
=> ( ( member @ A @ I @ I6 )
=> ( ( ord_less @ B @ ( one_one @ B ) @ ( F2 @ I ) )
=> ( ! [I4: A] :
( ( member @ A @ I4 @ I6 )
=> ( ord_less_eq @ B @ ( one_one @ B ) @ ( F2 @ I4 ) ) )
=> ( ord_less @ B @ ( one_one @ B ) @ ( groups7121269368397514597t_prod @ A @ B @ F2 @ I6 ) ) ) ) ) ) ) ).
% less_1_prod2
thf(fact_2558_less__1__prod,axiom,
! [B: $tType,A: $tType] :
( ( linordered_idom @ B )
=> ! [I6: set @ A,F2: A > B] :
( ( finite_finite2 @ A @ I6 )
=> ( ( I6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [I4: A] :
( ( member @ A @ I4 @ I6 )
=> ( ord_less @ B @ ( one_one @ B ) @ ( F2 @ I4 ) ) )
=> ( ord_less @ B @ ( one_one @ B ) @ ( groups7121269368397514597t_prod @ A @ B @ F2 @ I6 ) ) ) ) ) ) ).
% less_1_prod
thf(fact_2559_prod_Osubset__diff,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [B2: set @ B,A4: set @ B,G: B > A] :
( ( ord_less_eq @ ( set @ B ) @ B2 @ A4 )
=> ( ( finite_finite2 @ B @ A4 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ A4 )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ A4 @ B2 ) ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ B2 ) ) ) ) ) ) ).
% prod.subset_diff
thf(fact_2560_prod_Osame__carrier,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [C5: set @ B,A4: set @ B,B2: set @ B,G: B > A,H2: B > A] :
( ( finite_finite2 @ B @ C5 )
=> ( ( ord_less_eq @ ( set @ B ) @ A4 @ C5 )
=> ( ( ord_less_eq @ ( set @ B ) @ B2 @ C5 )
=> ( ! [A6: B] :
( ( member @ B @ A6 @ ( minus_minus @ ( set @ B ) @ C5 @ A4 ) )
=> ( ( G @ A6 )
= ( one_one @ A ) ) )
=> ( ! [B5: B] :
( ( member @ B @ B5 @ ( minus_minus @ ( set @ B ) @ C5 @ B2 ) )
=> ( ( H2 @ B5 )
= ( one_one @ A ) ) )
=> ( ( ( groups7121269368397514597t_prod @ B @ A @ G @ A4 )
= ( groups7121269368397514597t_prod @ B @ A @ H2 @ B2 ) )
= ( ( groups7121269368397514597t_prod @ B @ A @ G @ C5 )
= ( groups7121269368397514597t_prod @ B @ A @ H2 @ C5 ) ) ) ) ) ) ) ) ) ).
% prod.same_carrier
thf(fact_2561_prod_Osame__carrierI,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [C5: set @ B,A4: set @ B,B2: set @ B,G: B > A,H2: B > A] :
( ( finite_finite2 @ B @ C5 )
=> ( ( ord_less_eq @ ( set @ B ) @ A4 @ C5 )
=> ( ( ord_less_eq @ ( set @ B ) @ B2 @ C5 )
=> ( ! [A6: B] :
( ( member @ B @ A6 @ ( minus_minus @ ( set @ B ) @ C5 @ A4 ) )
=> ( ( G @ A6 )
= ( one_one @ A ) ) )
=> ( ! [B5: B] :
( ( member @ B @ B5 @ ( minus_minus @ ( set @ B ) @ C5 @ B2 ) )
=> ( ( H2 @ B5 )
= ( one_one @ A ) ) )
=> ( ( ( groups7121269368397514597t_prod @ B @ A @ G @ C5 )
= ( groups7121269368397514597t_prod @ B @ A @ H2 @ C5 ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ A4 )
= ( groups7121269368397514597t_prod @ B @ A @ H2 @ B2 ) ) ) ) ) ) ) ) ) ).
% prod.same_carrierI
thf(fact_2562_prod_Omono__neutral__left,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [T4: set @ B,S2: set @ B,G: B > A] :
( ( finite_finite2 @ B @ T4 )
=> ( ( ord_less_eq @ ( set @ B ) @ S2 @ T4 )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ ( minus_minus @ ( set @ B ) @ T4 @ S2 ) )
=> ( ( G @ X5 )
= ( one_one @ A ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ S2 )
= ( groups7121269368397514597t_prod @ B @ A @ G @ T4 ) ) ) ) ) ) ).
% prod.mono_neutral_left
thf(fact_2563_prod_Omono__neutral__right,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [T4: set @ B,S2: set @ B,G: B > A] :
( ( finite_finite2 @ B @ T4 )
=> ( ( ord_less_eq @ ( set @ B ) @ S2 @ T4 )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ ( minus_minus @ ( set @ B ) @ T4 @ S2 ) )
=> ( ( G @ X5 )
= ( one_one @ A ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ T4 )
= ( groups7121269368397514597t_prod @ B @ A @ G @ S2 ) ) ) ) ) ) ).
% prod.mono_neutral_right
thf(fact_2564_prod_Omono__neutral__cong__left,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [T4: set @ B,S2: set @ B,H2: B > A,G: B > A] :
( ( finite_finite2 @ B @ T4 )
=> ( ( ord_less_eq @ ( set @ B ) @ S2 @ T4 )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ ( minus_minus @ ( set @ B ) @ T4 @ S2 ) )
=> ( ( H2 @ X5 )
= ( one_one @ A ) ) )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ S2 )
=> ( ( G @ X5 )
= ( H2 @ X5 ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ S2 )
= ( groups7121269368397514597t_prod @ B @ A @ H2 @ T4 ) ) ) ) ) ) ) ).
% prod.mono_neutral_cong_left
thf(fact_2565_prod_Omono__neutral__cong__right,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [T4: set @ B,S2: set @ B,G: B > A,H2: B > A] :
( ( finite_finite2 @ B @ T4 )
=> ( ( ord_less_eq @ ( set @ B ) @ S2 @ T4 )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ ( minus_minus @ ( set @ B ) @ T4 @ S2 ) )
=> ( ( G @ X5 )
= ( one_one @ A ) ) )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ S2 )
=> ( ( G @ X5 )
= ( H2 @ X5 ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ T4 )
= ( groups7121269368397514597t_prod @ B @ A @ H2 @ S2 ) ) ) ) ) ) ) ).
% prod.mono_neutral_cong_right
thf(fact_2566_prod_OatLeast0__atMost__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,N2: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( suc @ N2 ) ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ).
% prod.atLeast0_atMost_Suc
thf(fact_2567_powser__sums__zero,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [A3: nat > A] :
( sums @ A
@ ^ [N: nat] : ( times_times @ A @ ( A3 @ N ) @ ( power_power @ A @ ( zero_zero @ A ) @ N ) )
@ ( A3 @ ( zero_zero @ nat ) ) ) ) ).
% powser_sums_zero
thf(fact_2568_prod_OatLeast__Suc__atMost,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M2: nat,N2: nat,G: nat > A] :
( ( ord_less_eq @ nat @ M2 @ N2 )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N2 ) )
= ( times_times @ A @ ( G @ M2 ) @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M2 ) @ N2 ) ) ) ) ) ) ).
% prod.atLeast_Suc_atMost
thf(fact_2569_prod_Onat__ivl__Suc_H,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M2: nat,N2: nat,G: nat > A] :
( ( ord_less_eq @ nat @ M2 @ ( suc @ N2 ) )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ ( suc @ N2 ) ) )
= ( times_times @ A @ ( G @ ( suc @ N2 ) ) @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N2 ) ) ) ) ) ) ).
% prod.nat_ivl_Suc'
thf(fact_2570_prod_OlessThan__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,N2: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_lessThan @ nat @ ( suc @ N2 ) ) )
= ( times_times @ A @ ( G @ ( zero_zero @ nat ) )
@ ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_ord_lessThan @ nat @ N2 ) ) ) ) ) ).
% prod.lessThan_Suc_shift
thf(fact_2571_prod_OSuc__reindex__ivl,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M2: nat,N2: nat,G: nat > A] :
( ( ord_less_eq @ nat @ M2 @ N2 )
=> ( ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N2 ) ) @ ( G @ ( suc @ N2 ) ) )
= ( times_times @ A @ ( G @ M2 )
@ ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_or1337092689740270186AtMost @ nat @ M2 @ N2 ) ) ) ) ) ) ).
% prod.Suc_reindex_ivl
thf(fact_2572_prod_OatLeast1__atMost__eq,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,N2: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N2 ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [K2: nat] : ( G @ ( suc @ K2 ) )
@ ( set_ord_lessThan @ nat @ N2 ) ) ) ) ).
% prod.atLeast1_atMost_eq
thf(fact_2573_prod__mono__strict,axiom,
! [A: $tType,B: $tType] :
( ( linordered_semidom @ A )
=> ! [A4: set @ B,F2: B > A,G: B > A] :
( ( finite_finite2 @ B @ A4 )
=> ( ! [I4: B] :
( ( member @ B @ I4 @ A4 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ I4 ) )
& ( ord_less @ A @ ( F2 @ I4 ) @ ( G @ I4 ) ) ) )
=> ( ( A4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ord_less @ A @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ A4 ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ A4 ) ) ) ) ) ) ).
% prod_mono_strict
thf(fact_2574_even__prod__iff,axiom,
! [A: $tType,B: $tType] :
( ( semiring_parity @ A )
=> ! [A4: set @ B,F2: B > A] :
( ( finite_finite2 @ B @ A4 )
=> ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ A4 ) )
= ( ? [X4: B] :
( ( member @ B @ X4 @ A4 )
& ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( F2 @ X4 ) ) ) ) ) ) ) ).
% even_prod_iff
thf(fact_2575_prod_Oinsert__remove,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A4: set @ B,G: B > A,X: B] :
( ( finite_finite2 @ B @ A4 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( insert2 @ B @ X @ A4 ) )
= ( times_times @ A @ ( G @ X ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ A4 @ ( insert2 @ B @ X @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ).
% prod.insert_remove
thf(fact_2576_prod_Oremove,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A4: set @ B,X: B,G: B > A] :
( ( finite_finite2 @ B @ A4 )
=> ( ( member @ B @ X @ A4 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ A4 )
= ( times_times @ A @ ( G @ X ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ A4 @ ( insert2 @ B @ X @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ) ).
% prod.remove
thf(fact_2577_prod_Oub__add__nat,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M2: nat,N2: nat,G: nat > A,P5: nat] :
( ( ord_less_eq @ nat @ M2 @ ( plus_plus @ nat @ N2 @ ( one_one @ nat ) ) )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ ( plus_plus @ nat @ N2 @ P5 ) ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N2 ) ) @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ N2 @ ( one_one @ nat ) ) @ ( plus_plus @ nat @ N2 @ P5 ) ) ) ) ) ) ) ).
% prod.ub_add_nat
thf(fact_2578_prod_Odelta__remove,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S2: set @ B,A3: B,B3: B > A,C2: B > A] :
( ( finite_finite2 @ B @ S2 )
=> ( ( ( member @ B @ A3 @ S2 )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K2: B] : ( if @ A @ ( K2 = A3 ) @ ( B3 @ K2 ) @ ( C2 @ K2 ) )
@ S2 )
= ( times_times @ A @ ( B3 @ A3 ) @ ( groups7121269368397514597t_prod @ B @ A @ C2 @ ( minus_minus @ ( set @ B ) @ S2 @ ( insert2 @ B @ A3 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) )
& ( ~ ( member @ B @ A3 @ S2 )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K2: B] : ( if @ A @ ( K2 = A3 ) @ ( B3 @ K2 ) @ ( C2 @ K2 ) )
@ S2 )
= ( groups7121269368397514597t_prod @ B @ A @ C2 @ ( minus_minus @ ( set @ B ) @ S2 @ ( insert2 @ B @ A3 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ) ).
% prod.delta_remove
thf(fact_2579_fold__atLeastAtMost__nat_Oelims,axiom,
! [A: $tType,X: nat > A > A,Xa2: nat,Xb: nat,Xc: A,Y: A] :
( ( ( set_fo6178422350223883121st_nat @ A @ X @ Xa2 @ Xb @ Xc )
= Y )
=> ( ( ( ord_less @ nat @ Xb @ Xa2 )
=> ( Y = Xc ) )
& ( ~ ( ord_less @ nat @ Xb @ Xa2 )
=> ( Y
= ( set_fo6178422350223883121st_nat @ A @ X @ ( plus_plus @ nat @ Xa2 @ ( one_one @ nat ) ) @ Xb @ ( X @ Xa2 @ Xc ) ) ) ) ) ) ).
% fold_atLeastAtMost_nat.elims
thf(fact_2580_fold__atLeastAtMost__nat_Osimps,axiom,
! [A: $tType] :
( ( set_fo6178422350223883121st_nat @ A )
= ( ^ [F3: nat > A > A,A5: nat,B4: nat,Acc2: A] : ( if @ A @ ( ord_less @ nat @ B4 @ A5 ) @ Acc2 @ ( set_fo6178422350223883121st_nat @ A @ F3 @ ( plus_plus @ nat @ A5 @ ( one_one @ nat ) ) @ B4 @ ( F3 @ A5 @ Acc2 ) ) ) ) ) ).
% fold_atLeastAtMost_nat.simps
thf(fact_2581_prod__mono2,axiom,
! [B: $tType,A: $tType] :
( ( linordered_idom @ B )
=> ! [B2: set @ A,A4: set @ A,F2: A > B] :
( ( finite_finite2 @ A @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ A4 @ B2 )
=> ( ! [B5: A] :
( ( member @ A @ B5 @ ( minus_minus @ ( set @ A ) @ B2 @ A4 ) )
=> ( ord_less_eq @ B @ ( one_one @ B ) @ ( F2 @ B5 ) ) )
=> ( ! [A6: A] :
( ( member @ A @ A6 @ A4 )
=> ( ord_less_eq @ B @ ( zero_zero @ B ) @ ( F2 @ A6 ) ) )
=> ( ord_less_eq @ B @ ( groups7121269368397514597t_prod @ A @ B @ F2 @ A4 ) @ ( groups7121269368397514597t_prod @ A @ B @ F2 @ B2 ) ) ) ) ) ) ) ).
% prod_mono2
thf(fact_2582_prod__diff1,axiom,
! [A: $tType,B: $tType] :
( ( semidom_divide @ A )
=> ! [A4: set @ B,F2: B > A,A3: B] :
( ( finite_finite2 @ B @ A4 )
=> ( ( ( F2 @ A3 )
!= ( zero_zero @ A ) )
=> ( ( ( member @ B @ A3 @ A4 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ F2 @ ( minus_minus @ ( set @ B ) @ A4 @ ( insert2 @ B @ A3 @ ( bot_bot @ ( set @ B ) ) ) ) )
= ( divide_divide @ A @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ A4 ) @ ( F2 @ A3 ) ) ) )
& ( ~ ( member @ B @ A3 @ A4 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ F2 @ ( minus_minus @ ( set @ B ) @ A4 @ ( insert2 @ B @ A3 @ ( bot_bot @ ( set @ B ) ) ) ) )
= ( groups7121269368397514597t_prod @ B @ A @ F2 @ A4 ) ) ) ) ) ) ) ).
% prod_diff1
thf(fact_2583_pochhammer__Suc__prod,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A3: A,N2: nat] :
( ( comm_s3205402744901411588hammer @ A @ A3 @ ( suc @ N2 ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( plus_plus @ A @ A3 @ ( semiring_1_of_nat @ A @ I3 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ) ).
% pochhammer_Suc_prod
thf(fact_2584_pochhammer__prod__rev,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ( ( comm_s3205402744901411588hammer @ A )
= ( ^ [A5: A,N: nat] :
( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( plus_plus @ A @ A5 @ ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ N @ I3 ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( one_one @ nat ) @ N ) ) ) ) ) ).
% pochhammer_prod_rev
thf(fact_2585_prod_Oin__pairs,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,M2: nat,N2: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M2 ) @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( G @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I3 ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ M2 @ N2 ) ) ) ) ).
% prod.in_pairs
thf(fact_2586_sum__atLeastAtMost__code,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [F2: nat > A,A3: nat,B3: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_or1337092689740270186AtMost @ nat @ A3 @ B3 ) )
= ( set_fo6178422350223883121st_nat @ A
@ ^ [A5: nat] : ( plus_plus @ A @ ( F2 @ A5 ) )
@ A3
@ B3
@ ( zero_zero @ A ) ) ) ) ).
% sum_atLeastAtMost_code
thf(fact_2587_pochhammer__Suc__prod__rev,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A3: A,N2: nat] :
( ( comm_s3205402744901411588hammer @ A @ A3 @ ( suc @ N2 ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( plus_plus @ A @ A3 @ ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ N2 @ I3 ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ) ).
% pochhammer_Suc_prod_rev
thf(fact_2588_power__half__series,axiom,
( sums @ real
@ ^ [N: nat] : ( power_power @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( suc @ N ) )
@ ( one_one @ real ) ) ).
% power_half_series
thf(fact_2589_sums__zero,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ( sums @ A
@ ^ [N: nat] : ( zero_zero @ A )
@ ( zero_zero @ A ) ) ) ).
% sums_zero
thf(fact_2590_sums__If__finite__set_H,axiom,
! [A: $tType] :
( ( ( topolo1287966508704411220up_add @ A )
& ( topological_t2_space @ A ) )
=> ! [G: nat > A,S2: A,A4: set @ nat,S5: A,F2: nat > A] :
( ( sums @ A @ G @ S2 )
=> ( ( finite_finite2 @ nat @ A4 )
=> ( ( S5
= ( plus_plus @ A @ S2
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [N: nat] : ( minus_minus @ A @ ( F2 @ N ) @ ( G @ N ) )
@ A4 ) ) )
=> ( sums @ A
@ ^ [N: nat] : ( if @ A @ ( member @ nat @ N @ A4 ) @ ( F2 @ N ) @ ( G @ N ) )
@ S5 ) ) ) ) ) ).
% sums_If_finite_set'
thf(fact_2591_sums__iff__shift_H,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A,N2: nat,S: A] :
( ( sums @ A
@ ^ [I3: nat] : ( F2 @ ( plus_plus @ nat @ I3 @ N2 ) )
@ ( minus_minus @ A @ S @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_ord_lessThan @ nat @ N2 ) ) ) )
= ( sums @ A @ F2 @ S ) ) ) ).
% sums_iff_shift'
thf(fact_2592_sums__split__initial__segment,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A,S: A,N2: nat] :
( ( sums @ A @ F2 @ S )
=> ( sums @ A
@ ^ [I3: nat] : ( F2 @ ( plus_plus @ nat @ I3 @ N2 ) )
@ ( minus_minus @ A @ S @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_ord_lessThan @ nat @ N2 ) ) ) ) ) ) ).
% sums_split_initial_segment
thf(fact_2593_sums__iff__shift,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A,N2: nat,S: A] :
( ( sums @ A
@ ^ [I3: nat] : ( F2 @ ( plus_plus @ nat @ I3 @ N2 ) )
@ S )
= ( sums @ A @ F2 @ ( plus_plus @ A @ S @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_ord_lessThan @ nat @ N2 ) ) ) ) ) ) ).
% sums_iff_shift
thf(fact_2594_prod__eq__1__iff,axiom,
! [A: $tType,A4: set @ A,F2: A > nat] :
( ( finite_finite2 @ A @ A4 )
=> ( ( ( groups7121269368397514597t_prod @ A @ nat @ F2 @ A4 )
= ( one_one @ nat ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ A4 )
=> ( ( F2 @ X4 )
= ( one_one @ nat ) ) ) ) ) ) ).
% prod_eq_1_iff
thf(fact_2595_prod__pos__nat__iff,axiom,
! [A: $tType,A4: set @ A,F2: A > nat] :
( ( finite_finite2 @ A @ A4 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( groups7121269368397514597t_prod @ A @ nat @ F2 @ A4 ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ A4 )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F2 @ X4 ) ) ) ) ) ) ).
% prod_pos_nat_iff
thf(fact_2596_int__prod,axiom,
! [B: $tType,F2: B > nat,A4: set @ B] :
( ( semiring_1_of_nat @ int @ ( groups7121269368397514597t_prod @ B @ nat @ F2 @ A4 ) )
= ( groups7121269368397514597t_prod @ B @ int
@ ^ [X4: B] : ( semiring_1_of_nat @ int @ ( F2 @ X4 ) )
@ A4 ) ) ).
% int_prod
thf(fact_2597_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
@ ^ [X4: int] : X4
@ ( set_or1337092689740270186AtMost @ int @ ( semiring_1_of_nat @ int @ I ) @ ( semiring_1_of_nat @ int @ J ) ) ) ) ).
% prod_int_eq
thf(fact_2598_ln__prod,axiom,
! [A: $tType,I6: set @ A,F2: A > real] :
( ( finite_finite2 @ A @ I6 )
=> ( ! [I4: A] :
( ( member @ A @ I4 @ I6 )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( F2 @ I4 ) ) )
=> ( ( ln_ln @ real @ ( groups7121269368397514597t_prod @ A @ real @ F2 @ I6 ) )
= ( groups7311177749621191930dd_sum @ A @ real
@ ^ [X4: A] : ( ln_ln @ real @ ( F2 @ X4 ) )
@ I6 ) ) ) ) ).
% ln_prod
thf(fact_2599_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
@ ^ [X4: int] : X4
@ ( 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_2600_sums__le,axiom,
! [A: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F2: nat > A,G: nat > A,S: A,T2: A] :
( ! [N3: nat] : ( ord_less_eq @ A @ ( F2 @ N3 ) @ ( G @ N3 ) )
=> ( ( sums @ A @ F2 @ S )
=> ( ( sums @ A @ G @ T2 )
=> ( ord_less_eq @ A @ S @ T2 ) ) ) ) ) ).
% sums_le
thf(fact_2601_sums__0,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [F2: nat > A] :
( ! [N3: nat] :
( ( F2 @ N3 )
= ( zero_zero @ A ) )
=> ( sums @ A @ F2 @ ( zero_zero @ A ) ) ) ) ).
% sums_0
thf(fact_2602_sums__single,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [I: nat,F2: nat > A] :
( sums @ A
@ ^ [R4: nat] : ( if @ A @ ( R4 = I ) @ ( F2 @ R4 ) @ ( zero_zero @ A ) )
@ ( F2 @ I ) ) ) ).
% sums_single
thf(fact_2603_sums__add,axiom,
! [A: $tType] :
( ( ( topolo5987344860129210374id_add @ A )
& ( topological_t2_space @ A ) )
=> ! [F2: nat > A,A3: A,G: nat > A,B3: A] :
( ( sums @ A @ F2 @ A3 )
=> ( ( sums @ A @ G @ B3 )
=> ( sums @ A
@ ^ [N: nat] : ( plus_plus @ A @ ( F2 @ N ) @ ( G @ N ) )
@ ( plus_plus @ A @ A3 @ B3 ) ) ) ) ) ).
% sums_add
thf(fact_2604_sums__mult__iff,axiom,
! [A: $tType] :
( ( ( field @ A )
& ( real_V4412858255891104859lgebra @ A ) )
=> ! [C2: A,F2: nat > A,D2: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( sums @ A
@ ^ [N: nat] : ( times_times @ A @ C2 @ ( F2 @ N ) )
@ ( times_times @ A @ C2 @ D2 ) )
= ( sums @ A @ F2 @ D2 ) ) ) ) ).
% sums_mult_iff
thf(fact_2605_sums__mult2__iff,axiom,
! [A: $tType] :
( ( ( field @ A )
& ( real_V4412858255891104859lgebra @ A ) )
=> ! [C2: A,F2: nat > A,D2: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( sums @ A
@ ^ [N: nat] : ( times_times @ A @ ( F2 @ N ) @ C2 )
@ ( times_times @ A @ D2 @ C2 ) )
= ( sums @ A @ F2 @ D2 ) ) ) ) ).
% sums_mult2_iff
thf(fact_2606_sums__mult__D,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [C2: A,F2: nat > A,A3: A] :
( ( sums @ A
@ ^ [N: nat] : ( times_times @ A @ C2 @ ( F2 @ N ) )
@ A3 )
=> ( ( C2
!= ( zero_zero @ A ) )
=> ( sums @ A @ F2 @ ( divide_divide @ A @ A3 @ C2 ) ) ) ) ) ).
% sums_mult_D
thf(fact_2607_sums__Suc__imp,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A,S: A] :
( ( ( F2 @ ( zero_zero @ nat ) )
= ( zero_zero @ A ) )
=> ( ( sums @ A
@ ^ [N: nat] : ( F2 @ ( suc @ N ) )
@ S )
=> ( sums @ A @ F2 @ S ) ) ) ) ).
% sums_Suc_imp
thf(fact_2608_sums__Suc__iff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A,S: A] :
( ( sums @ A
@ ^ [N: nat] : ( F2 @ ( suc @ N ) )
@ S )
= ( sums @ A @ F2 @ ( plus_plus @ A @ S @ ( F2 @ ( zero_zero @ nat ) ) ) ) ) ) ).
% sums_Suc_iff
thf(fact_2609_sums__Suc,axiom,
! [A: $tType] :
( ( ( topolo5987344860129210374id_add @ A )
& ( topological_t2_space @ A ) )
=> ! [F2: nat > A,L: A] :
( ( sums @ A
@ ^ [N: nat] : ( F2 @ ( suc @ N ) )
@ L )
=> ( sums @ A @ F2 @ ( plus_plus @ A @ L @ ( F2 @ ( zero_zero @ nat ) ) ) ) ) ) ).
% sums_Suc
thf(fact_2610_sums__zero__iff__shift,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [N2: nat,F2: nat > A,S: A] :
( ! [I4: nat] :
( ( ord_less @ nat @ I4 @ N2 )
=> ( ( F2 @ I4 )
= ( zero_zero @ A ) ) )
=> ( ( sums @ A
@ ^ [I3: nat] : ( F2 @ ( plus_plus @ nat @ I3 @ N2 ) )
@ S )
= ( sums @ A @ F2 @ S ) ) ) ) ).
% sums_zero_iff_shift
thf(fact_2611_sums__finite,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [N6: set @ nat,F2: nat > A] :
( ( finite_finite2 @ nat @ N6 )
=> ( ! [N3: nat] :
( ~ ( member @ nat @ N3 @ N6 )
=> ( ( F2 @ N3 )
= ( zero_zero @ A ) ) )
=> ( sums @ A @ F2 @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ N6 ) ) ) ) ) ).
% sums_finite
thf(fact_2612_sums__If__finite,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [P: nat > $o,F2: nat > A] :
( ( finite_finite2 @ nat @ ( collect @ nat @ P ) )
=> ( sums @ A
@ ^ [R4: nat] : ( if @ A @ ( P @ R4 ) @ ( F2 @ R4 ) @ ( zero_zero @ A ) )
@ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( collect @ nat @ P ) ) ) ) ) ).
% sums_If_finite
thf(fact_2613_sums__If__finite__set,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [A4: set @ nat,F2: nat > A] :
( ( finite_finite2 @ nat @ A4 )
=> ( sums @ A
@ ^ [R4: nat] : ( if @ A @ ( member @ nat @ R4 @ A4 ) @ ( F2 @ R4 ) @ ( zero_zero @ A ) )
@ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ A4 ) ) ) ) ).
% sums_If_finite_set
thf(fact_2614_powser__sums__if,axiom,
! [A: $tType] :
( ( ( ring_1 @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [M2: nat,Z3: A] :
( sums @ A
@ ^ [N: nat] : ( times_times @ A @ ( if @ A @ ( N = M2 ) @ ( one_one @ A ) @ ( zero_zero @ A ) ) @ ( power_power @ A @ Z3 @ N ) )
@ ( power_power @ A @ Z3 @ M2 ) ) ) ).
% powser_sums_if
thf(fact_2615_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
@ ^ [N: nat] : ( times_times @ real @ ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ N ) @ ( divide_divide @ real @ ( one_one @ real ) @ ( semiring_1_of_nat @ real @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) ) ) ) @ ( power_power @ real @ ( minus_minus @ real @ X @ ( one_one @ real ) ) @ ( suc @ N ) ) ) ) ) ) ) ).
% ln_series
thf(fact_2616_arcosh__def,axiom,
! [A: $tType] :
( ( ln @ A )
=> ( ( arcosh @ A )
= ( ^ [X4: A] : ( ln_ln @ A @ ( plus_plus @ A @ X4 @ ( powr @ A @ ( minus_minus @ A @ ( power_power @ A @ X4 @ ( 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_2617_arsinh__def,axiom,
! [A: $tType] :
( ( ln @ A )
=> ( ( arsinh @ A )
= ( ^ [X4: A] : ( ln_ln @ A @ ( plus_plus @ A @ X4 @ ( powr @ A @ ( plus_plus @ A @ ( power_power @ A @ X4 @ ( 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_2618_floor__log__nat__eq__powr__iff,axiom,
! [B3: nat,K: nat,N2: 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 @ N2 ) )
= ( ( 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 ) ) ) ) ) ) ) ) ).
% floor_log_nat_eq_powr_iff
thf(fact_2619_gchoose__row__sum__weighted,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [R2: A,M2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K2: nat] : ( times_times @ A @ ( gbinomial @ A @ R2 @ K2 ) @ ( minus_minus @ A @ ( divide_divide @ A @ R2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( semiring_1_of_nat @ A @ K2 ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ M2 ) )
= ( times_times @ A @ ( divide_divide @ A @ ( semiring_1_of_nat @ A @ ( suc @ M2 ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( gbinomial @ A @ R2 @ ( suc @ M2 ) ) ) ) ) ).
% gchoose_row_sum_weighted
thf(fact_2620_central__binomial__lower__bound,axiom,
! [N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ord_less_eq @ real @ ( divide_divide @ real @ ( power_power @ real @ ( numeral_numeral @ real @ ( bit0 @ ( bit0 @ one2 ) ) ) @ N2 ) @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ N2 ) ) ) @ ( semiring_1_of_nat @ real @ ( binomial @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) @ N2 ) ) ) ) ).
% central_binomial_lower_bound
thf(fact_2621_binomial__Suc__n,axiom,
! [N2: nat] :
( ( binomial @ ( suc @ N2 ) @ N2 )
= ( suc @ N2 ) ) ).
% binomial_Suc_n
thf(fact_2622_suminf__zero,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topological_t2_space @ A ) )
=> ( ( suminf @ A
@ ^ [N: nat] : ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% suminf_zero
thf(fact_2623_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_2624_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_2625_floor__zero,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ( ( archim6421214686448440834_floor @ A @ ( zero_zero @ A ) )
= ( zero_zero @ int ) ) ) ).
% floor_zero
thf(fact_2626_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_2627_binomial__0__Suc,axiom,
! [K: nat] :
( ( binomial @ ( zero_zero @ nat ) @ ( suc @ K ) )
= ( zero_zero @ nat ) ) ).
% binomial_0_Suc
thf(fact_2628_binomial__1,axiom,
! [N2: nat] :
( ( binomial @ N2 @ ( suc @ ( zero_zero @ nat ) ) )
= N2 ) ).
% binomial_1
thf(fact_2629_binomial__eq__0__iff,axiom,
! [N2: nat,K: nat] :
( ( ( binomial @ N2 @ K )
= ( zero_zero @ nat ) )
= ( ord_less @ nat @ N2 @ K ) ) ).
% binomial_eq_0_iff
thf(fact_2630_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_2631_of__real__add,axiom,
! [A: $tType] :
( ( real_V2191834092415804123ebra_1 @ A )
=> ! [X: real,Y: real] :
( ( real_Vector_of_real @ A @ ( plus_plus @ real @ X @ Y ) )
= ( plus_plus @ A @ ( real_Vector_of_real @ A @ X ) @ ( real_Vector_of_real @ A @ Y ) ) ) ) ).
% of_real_add
thf(fact_2632_binomial__Suc__Suc,axiom,
! [N2: nat,K: nat] :
( ( binomial @ ( suc @ N2 ) @ ( suc @ K ) )
= ( plus_plus @ nat @ ( binomial @ N2 @ K ) @ ( binomial @ N2 @ ( suc @ K ) ) ) ) ).
% binomial_Suc_Suc
thf(fact_2633_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_2634_binomial__n__0,axiom,
! [N2: nat] :
( ( binomial @ N2 @ ( zero_zero @ nat ) )
= ( one_one @ nat ) ) ).
% binomial_n_0
thf(fact_2635_zero__less__binomial__iff,axiom,
! [N2: nat,K: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( binomial @ N2 @ K ) )
= ( ord_less_eq @ nat @ K @ N2 ) ) ).
% zero_less_binomial_iff
thf(fact_2636_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_2637_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_2638_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_2639_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_2640_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_2641_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_2642_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_2643_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_2644_powser__zero,axiom,
! [A: $tType] :
( ( real_V2822296259951069270ebra_1 @ A )
=> ! [F2: nat > A] :
( ( suminf @ A
@ ^ [N: nat] : ( times_times @ A @ ( F2 @ N ) @ ( power_power @ A @ ( zero_zero @ A ) @ N ) ) )
= ( F2 @ ( zero_zero @ nat ) ) ) ) ).
% powser_zero
thf(fact_2645_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_2646_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_2647_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_2648_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_2649_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_2650_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_2651_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_2652_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_2653_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_2654_binomial__eq__0,axiom,
! [N2: nat,K: nat] :
( ( ord_less @ nat @ N2 @ K )
=> ( ( binomial @ N2 @ K )
= ( zero_zero @ nat ) ) ) ).
% binomial_eq_0
thf(fact_2655_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_2656_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_2657_Suc__times__binomial,axiom,
! [K: nat,N2: nat] :
( ( times_times @ nat @ ( suc @ K ) @ ( binomial @ ( suc @ N2 ) @ ( suc @ K ) ) )
= ( times_times @ nat @ ( suc @ N2 ) @ ( binomial @ N2 @ K ) ) ) ).
% Suc_times_binomial
thf(fact_2658_Suc__times__binomial__eq,axiom,
! [N2: nat,K: nat] :
( ( times_times @ nat @ ( suc @ N2 ) @ ( binomial @ N2 @ K ) )
= ( times_times @ nat @ ( binomial @ ( suc @ N2 ) @ ( suc @ K ) ) @ ( suc @ K ) ) ) ).
% Suc_times_binomial_eq
thf(fact_2659_binomial__symmetric,axiom,
! [K: nat,N2: nat] :
( ( ord_less_eq @ nat @ K @ N2 )
=> ( ( binomial @ N2 @ K )
= ( binomial @ N2 @ ( minus_minus @ nat @ N2 @ K ) ) ) ) ).
% binomial_symmetric
thf(fact_2660_choose__mult__lemma,axiom,
! [M2: nat,R2: nat,K: nat] :
( ( times_times @ nat @ ( binomial @ ( plus_plus @ nat @ ( plus_plus @ nat @ M2 @ R2 ) @ K ) @ ( plus_plus @ nat @ M2 @ K ) ) @ ( binomial @ ( plus_plus @ nat @ M2 @ K ) @ K ) )
= ( times_times @ nat @ ( binomial @ ( plus_plus @ nat @ ( plus_plus @ nat @ M2 @ R2 ) @ K ) @ K ) @ ( binomial @ ( plus_plus @ nat @ M2 @ R2 ) @ M2 ) ) ) ).
% choose_mult_lemma
thf(fact_2661_binomial__le__pow,axiom,
! [R2: nat,N2: nat] :
( ( ord_less_eq @ nat @ R2 @ N2 )
=> ( ord_less_eq @ nat @ ( binomial @ N2 @ R2 ) @ ( power_power @ nat @ N2 @ R2 ) ) ) ).
% binomial_le_pow
thf(fact_2662_zero__less__binomial,axiom,
! [K: nat,N2: nat] :
( ( ord_less_eq @ nat @ K @ N2 )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ ( binomial @ N2 @ K ) ) ) ).
% zero_less_binomial
thf(fact_2663_le__floor__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z3: int,X: A] :
( ( ord_less_eq @ int @ Z3 @ ( archim6421214686448440834_floor @ A @ X ) )
= ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Z3 ) @ X ) ) ) ).
% le_floor_iff
thf(fact_2664_floor__less__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Z3: int] :
( ( ord_less @ int @ ( archim6421214686448440834_floor @ A @ X ) @ Z3 )
= ( ord_less @ A @ X @ ( ring_1_of_int @ A @ Z3 ) ) ) ) ).
% floor_less_iff
thf(fact_2665_Suc__times__binomial__add,axiom,
! [A3: nat,B3: nat] :
( ( times_times @ nat @ ( suc @ A3 ) @ ( binomial @ ( suc @ ( plus_plus @ nat @ A3 @ B3 ) ) @ ( suc @ A3 ) ) )
= ( times_times @ nat @ ( suc @ B3 ) @ ( binomial @ ( suc @ ( plus_plus @ nat @ A3 @ B3 ) ) @ A3 ) ) ) ).
% Suc_times_binomial_add
thf(fact_2666_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_2667_int__add__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z3: int,X: A] :
( ( plus_plus @ int @ Z3 @ ( archim6421214686448440834_floor @ A @ X ) )
= ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ ( ring_1_of_int @ A @ Z3 ) @ X ) ) ) ) ).
% int_add_floor
thf(fact_2668_floor__add__int,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Z3: int] :
( ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X ) @ Z3 )
= ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ X @ ( ring_1_of_int @ A @ Z3 ) ) ) ) ) ).
% floor_add_int
thf(fact_2669_choose__mult,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ord_less_eq @ nat @ K @ M2 )
=> ( ( ord_less_eq @ nat @ M2 @ N2 )
=> ( ( times_times @ nat @ ( binomial @ N2 @ M2 ) @ ( binomial @ M2 @ K ) )
= ( times_times @ nat @ ( binomial @ N2 @ K ) @ ( binomial @ ( minus_minus @ nat @ N2 @ K ) @ ( minus_minus @ nat @ M2 @ K ) ) ) ) ) ) ).
% choose_mult
thf(fact_2670_binomial__Suc__Suc__eq__times,axiom,
! [N2: nat,K: nat] :
( ( binomial @ ( suc @ N2 ) @ ( suc @ K ) )
= ( divide_divide @ nat @ ( times_times @ nat @ ( suc @ N2 ) @ ( binomial @ N2 @ K ) ) @ ( suc @ K ) ) ) ).
% binomial_Suc_Suc_eq_times
thf(fact_2671_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_2672_gbinomial__of__nat__symmetric,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K: nat,N2: nat] :
( ( ord_less_eq @ nat @ K @ N2 )
=> ( ( gbinomial @ A @ ( semiring_1_of_nat @ A @ N2 ) @ K )
= ( gbinomial @ A @ ( semiring_1_of_nat @ A @ N2 ) @ ( minus_minus @ nat @ N2 @ K ) ) ) ) ) ).
% gbinomial_of_nat_symmetric
thf(fact_2673_suminf__finite,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topological_t2_space @ A ) )
=> ! [N6: set @ nat,F2: nat > A] :
( ( finite_finite2 @ nat @ N6 )
=> ( ! [N3: nat] :
( ~ ( member @ nat @ N3 @ N6 )
=> ( ( F2 @ N3 )
= ( zero_zero @ A ) ) )
=> ( ( suminf @ A @ F2 )
= ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ N6 ) ) ) ) ) ).
% suminf_finite
thf(fact_2674_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_2675_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_2676_binomial__absorption,axiom,
! [K: nat,N2: nat] :
( ( times_times @ nat @ ( suc @ K ) @ ( binomial @ N2 @ ( suc @ K ) ) )
= ( times_times @ nat @ N2 @ ( binomial @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) @ K ) ) ) ).
% binomial_absorption
thf(fact_2677_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_2678_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_2679_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_2680_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_2681_floor__split,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [P: int > $o,T2: A] :
( ( P @ ( archim6421214686448440834_floor @ A @ T2 ) )
= ( ! [I3: int] :
( ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ I3 ) @ T2 )
& ( ord_less @ A @ T2 @ ( plus_plus @ A @ ( ring_1_of_int @ A @ I3 ) @ ( one_one @ A ) ) ) )
=> ( P @ I3 ) ) ) ) ) ).
% floor_split
thf(fact_2682_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_2683_floor__unique,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z3: int,X: A] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Z3 ) @ X )
=> ( ( ord_less @ A @ X @ ( plus_plus @ A @ ( ring_1_of_int @ A @ Z3 ) @ ( one_one @ A ) ) )
=> ( ( archim6421214686448440834_floor @ A @ X )
= Z3 ) ) ) ) ).
% floor_unique
thf(fact_2684_binomial__ge__n__over__k__pow__k,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [K: nat,N2: nat] :
( ( ord_less_eq @ nat @ K @ N2 )
=> ( ord_less_eq @ A @ ( power_power @ A @ ( divide_divide @ A @ ( semiring_1_of_nat @ A @ N2 ) @ ( semiring_1_of_nat @ A @ K ) ) @ K ) @ ( semiring_1_of_nat @ A @ ( binomial @ N2 @ K ) ) ) ) ) ).
% binomial_ge_n_over_k_pow_k
thf(fact_2685_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_2686_less__floor__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z3: int,X: A] :
( ( ord_less @ int @ Z3 @ ( archim6421214686448440834_floor @ A @ X ) )
= ( ord_less_eq @ A @ ( plus_plus @ A @ ( ring_1_of_int @ A @ Z3 ) @ ( one_one @ A ) ) @ X ) ) ) ).
% less_floor_iff
thf(fact_2687_floor__le__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Z3: int] :
( ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X ) @ Z3 )
= ( ord_less @ A @ X @ ( plus_plus @ A @ ( ring_1_of_int @ A @ Z3 ) @ ( one_one @ A ) ) ) ) ) ).
% floor_le_iff
thf(fact_2688_binomial__mono,axiom,
! [K: nat,K7: nat,N2: nat] :
( ( ord_less_eq @ nat @ K @ K7 )
=> ( ( ord_less_eq @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K7 ) @ N2 )
=> ( ord_less_eq @ nat @ ( binomial @ N2 @ K ) @ ( binomial @ N2 @ K7 ) ) ) ) ).
% binomial_mono
thf(fact_2689_binomial__maximum_H,axiom,
! [N2: nat,K: nat] : ( ord_less_eq @ nat @ ( binomial @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) @ K ) @ ( binomial @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) @ N2 ) ) ).
% binomial_maximum'
thf(fact_2690_binomial__maximum,axiom,
! [N2: nat,K: nat] : ( ord_less_eq @ nat @ ( binomial @ N2 @ K ) @ ( binomial @ N2 @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% binomial_maximum
thf(fact_2691_binomial__antimono,axiom,
! [K: nat,K7: nat,N2: nat] :
( ( ord_less_eq @ nat @ K @ K7 )
=> ( ( ord_less_eq @ nat @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ K )
=> ( ( ord_less_eq @ nat @ K7 @ N2 )
=> ( ord_less_eq @ nat @ ( binomial @ N2 @ K7 ) @ ( binomial @ N2 @ K ) ) ) ) ) ).
% binomial_antimono
thf(fact_2692_binomial__le__pow2,axiom,
! [N2: nat,K: nat] : ( ord_less_eq @ nat @ ( binomial @ N2 @ K ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ).
% binomial_le_pow2
thf(fact_2693_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_2694_choose__reduce__nat,axiom,
! [N2: nat,K: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ( binomial @ N2 @ K )
= ( plus_plus @ nat @ ( binomial @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) @ ( minus_minus @ nat @ K @ ( one_one @ nat ) ) ) @ ( binomial @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) @ K ) ) ) ) ) ).
% choose_reduce_nat
thf(fact_2695_times__binomial__minus1__eq,axiom,
! [K: nat,N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ( times_times @ nat @ K @ ( binomial @ N2 @ K ) )
= ( times_times @ nat @ N2 @ ( binomial @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) @ ( minus_minus @ nat @ K @ ( one_one @ nat ) ) ) ) ) ) ).
% times_binomial_minus1_eq
thf(fact_2696_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_2697_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_2698_gbinomial__trinomial__revision,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K: nat,M2: nat,A3: A] :
( ( ord_less_eq @ nat @ K @ M2 )
=> ( ( times_times @ A @ ( gbinomial @ A @ A3 @ M2 ) @ ( gbinomial @ A @ ( semiring_1_of_nat @ A @ M2 ) @ K ) )
= ( times_times @ A @ ( gbinomial @ A @ A3 @ K ) @ ( gbinomial @ A @ ( minus_minus @ A @ A3 @ ( semiring_1_of_nat @ A @ K ) ) @ ( minus_minus @ nat @ M2 @ K ) ) ) ) ) ) ).
% gbinomial_trinomial_revision
thf(fact_2699_floor__divide__lower,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Q3: A,P5: 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 @ P5 @ Q3 ) ) ) @ Q3 ) @ P5 ) ) ) ).
% floor_divide_lower
thf(fact_2700_binomial__less__binomial__Suc,axiom,
! [K: nat,N2: nat] :
( ( ord_less @ nat @ K @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
=> ( ord_less @ nat @ ( binomial @ N2 @ K ) @ ( binomial @ N2 @ ( suc @ K ) ) ) ) ).
% binomial_less_binomial_Suc
thf(fact_2701_binomial__strict__antimono,axiom,
! [K: nat,K7: nat,N2: nat] :
( ( ord_less @ nat @ K @ K7 )
=> ( ( ord_less_eq @ nat @ N2 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K ) )
=> ( ( ord_less_eq @ nat @ K7 @ N2 )
=> ( ord_less @ nat @ ( binomial @ N2 @ K7 ) @ ( binomial @ N2 @ K ) ) ) ) ) ).
% binomial_strict_antimono
thf(fact_2702_binomial__strict__mono,axiom,
! [K: nat,K7: nat,N2: nat] :
( ( ord_less @ nat @ K @ K7 )
=> ( ( ord_less_eq @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K7 ) @ N2 )
=> ( ord_less @ nat @ ( binomial @ N2 @ K ) @ ( binomial @ N2 @ K7 ) ) ) ) ).
% binomial_strict_mono
thf(fact_2703_central__binomial__odd,axiom,
! [N2: nat] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
=> ( ( binomial @ N2 @ ( suc @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
= ( binomial @ N2 @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% central_binomial_odd
thf(fact_2704_binomial__addition__formula,axiom,
! [N2: nat,K: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( binomial @ N2 @ ( suc @ K ) )
= ( plus_plus @ nat @ ( binomial @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) @ ( suc @ K ) ) @ ( binomial @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) @ K ) ) ) ) ).
% binomial_addition_formula
thf(fact_2705_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_2706_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_2707_floor__divide__upper,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Q3: A,P5: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Q3 )
=> ( ord_less @ A @ P5 @ ( times_times @ A @ ( plus_plus @ A @ ( ring_1_of_int @ A @ ( archim6421214686448440834_floor @ A @ ( divide_divide @ A @ P5 @ Q3 ) ) ) @ ( one_one @ A ) ) @ Q3 ) ) ) ) ).
% floor_divide_upper
thf(fact_2708_round__def,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ( ( archimedean_round @ A )
= ( ^ [X4: A] : ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ X4 @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% round_def
thf(fact_2709_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_2710_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_2711_gbinomial__sum__up__index,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K: nat,N2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [J2: nat] : ( gbinomial @ A @ ( semiring_1_of_nat @ A @ J2 ) @ K )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N2 ) )
= ( gbinomial @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N2 ) @ ( one_one @ A ) ) @ ( plus_plus @ nat @ K @ ( one_one @ nat ) ) ) ) ) ).
% gbinomial_sum_up_index
thf(fact_2712_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_2713_floor__log2__div2,axiom,
! [N2: nat] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
=> ( ( archim6421214686448440834_floor @ real @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ N2 ) ) )
= ( plus_plus @ int @ ( archim6421214686448440834_floor @ real @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( one_one @ int ) ) ) ) ).
% floor_log2_div2
thf(fact_2714_floor__log__nat__eq__if,axiom,
! [B3: nat,N2: nat,K: 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 ) ) ) )
=> ( ( 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 @ N2 ) ) ) ) ) ).
% floor_log_nat_eq_if
thf(fact_2715_pi__series,axiom,
( ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ ( bit0 @ one2 ) ) ) )
= ( suminf @ real
@ ^ [K2: nat] : ( divide_divide @ real @ ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ K2 ) @ ( one_one @ real ) ) @ ( semiring_1_of_nat @ real @ ( plus_plus @ nat @ ( times_times @ nat @ K2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) ) ) ).
% pi_series
thf(fact_2716_gbinomial__partial__row__sum,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: A,M2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K2: nat] : ( times_times @ A @ ( gbinomial @ A @ A3 @ K2 ) @ ( minus_minus @ A @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( semiring_1_of_nat @ A @ K2 ) ) )
@ ( set_ord_atMost @ nat @ M2 ) )
= ( times_times @ A @ ( divide_divide @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ M2 ) @ ( one_one @ A ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( gbinomial @ A @ A3 @ ( plus_plus @ nat @ M2 @ ( one_one @ nat ) ) ) ) ) ) ).
% gbinomial_partial_row_sum
thf(fact_2717_choose__even__sum,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( if @ A @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I3 ) @ ( semiring_1_of_nat @ A @ ( binomial @ N2 @ I3 ) ) @ ( zero_zero @ A ) )
@ ( set_ord_atMost @ nat @ N2 ) ) )
= ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ).
% choose_even_sum
thf(fact_2718_choose__odd__sum,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] :
( if @ A
@ ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I3 )
@ ( semiring_1_of_nat @ A @ ( binomial @ N2 @ I3 ) )
@ ( zero_zero @ A ) )
@ ( set_ord_atMost @ nat @ N2 ) ) )
= ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ).
% choose_odd_sum
thf(fact_2719_round__altdef,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ( ( archimedean_round @ A )
= ( ^ [X4: A] : ( if @ int @ ( ord_less_eq @ A @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( archimedean_frac @ A @ X4 ) ) @ ( archimedean_ceiling @ A @ X4 ) @ ( archim6421214686448440834_floor @ A @ X4 ) ) ) ) ) ).
% round_altdef
thf(fact_2720_gbinomial__r__part__sum,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [M2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ ( gbinomial @ A @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ A @ M2 ) ) @ ( one_one @ A ) ) ) @ ( set_ord_atMost @ nat @ M2 ) )
= ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M2 ) ) ) ) ).
% gbinomial_r_part_sum
thf(fact_2721_atMost__eq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X: A,Y: A] :
( ( ( set_ord_atMost @ A @ X )
= ( set_ord_atMost @ A @ Y ) )
= ( X = Y ) ) ) ).
% atMost_eq_iff
thf(fact_2722_atMost__iff,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [I: A,K: A] :
( ( member @ A @ I @ ( set_ord_atMost @ A @ K ) )
= ( ord_less_eq @ A @ I @ K ) ) ) ).
% atMost_iff
thf(fact_2723_finite__atMost,axiom,
! [K: nat] : ( finite_finite2 @ nat @ ( set_ord_atMost @ nat @ K ) ) ).
% finite_atMost
thf(fact_2724_atMost__subset__iff,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_ord_atMost @ A @ X ) @ ( set_ord_atMost @ A @ Y ) )
= ( ord_less_eq @ A @ X @ Y ) ) ) ).
% atMost_subset_iff
thf(fact_2725_frac__of__int,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z3: int] :
( ( archimedean_frac @ A @ ( ring_1_of_int @ A @ Z3 ) )
= ( zero_zero @ A ) ) ) ).
% frac_of_int
thf(fact_2726_Icc__subset__Iic__iff,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [L: A,H2: A,H3: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ L @ H2 ) @ ( set_ord_atMost @ A @ H3 ) )
= ( ~ ( ord_less_eq @ A @ L @ H2 )
| ( ord_less_eq @ A @ H2 @ H3 ) ) ) ) ).
% Icc_subset_Iic_iff
thf(fact_2727_sum_OatMost__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,N2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_ord_atMost @ nat @ ( suc @ N2 ) ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_ord_atMost @ nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ).
% sum.atMost_Suc
thf(fact_2728_prod_OatMost__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,N2: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_atMost @ nat @ ( suc @ N2 ) ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_atMost @ nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ).
% prod.atMost_Suc
thf(fact_2729_atMost__0,axiom,
( ( set_ord_atMost @ nat @ ( zero_zero @ nat ) )
= ( insert2 @ nat @ ( zero_zero @ nat ) @ ( bot_bot @ ( set @ nat ) ) ) ) ).
% atMost_0
thf(fact_2730_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_2731_infinite__Iic,axiom,
! [A: $tType] :
( ( ( linorder @ A )
& ( no_bot @ A ) )
=> ! [A3: A] :
~ ( finite_finite2 @ A @ ( set_ord_atMost @ A @ A3 ) ) ) ).
% infinite_Iic
thf(fact_2732_not__Iic__eq__Icc,axiom,
! [A: $tType] :
( ( no_bot @ A )
=> ! [H3: A,L: A,H2: A] :
( ( set_ord_atMost @ A @ H3 )
!= ( set_or1337092689740270186AtMost @ A @ L @ H2 ) ) ) ).
% not_Iic_eq_Icc
thf(fact_2733_atMost__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( set_ord_atMost @ A )
= ( ^ [U2: A] :
( collect @ A
@ ^ [X4: A] : ( ord_less_eq @ A @ X4 @ U2 ) ) ) ) ) ).
% atMost_def
thf(fact_2734_atMost__atLeast0,axiom,
( ( set_ord_atMost @ nat )
= ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) ) ) ).
% atMost_atLeast0
thf(fact_2735_lessThan__Suc__atMost,axiom,
! [K: nat] :
( ( set_ord_lessThan @ nat @ ( suc @ K ) )
= ( set_ord_atMost @ nat @ K ) ) ).
% lessThan_Suc_atMost
thf(fact_2736_atMost__Suc,axiom,
! [K: nat] :
( ( set_ord_atMost @ nat @ ( suc @ K ) )
= ( insert2 @ nat @ ( suc @ K ) @ ( set_ord_atMost @ nat @ K ) ) ) ).
% atMost_Suc
thf(fact_2737_not__Iic__le__Icc,axiom,
! [A: $tType] :
( ( no_bot @ A )
=> ! [H2: A,L3: A,H3: A] :
~ ( ord_less_eq @ ( set @ A ) @ ( set_ord_atMost @ A @ H2 ) @ ( set_or1337092689740270186AtMost @ A @ L3 @ H3 ) ) ) ).
% not_Iic_le_Icc
thf(fact_2738_finite__nat__iff__bounded__le,axiom,
( ( finite_finite2 @ nat )
= ( ^ [S6: set @ nat] :
? [K2: nat] : ( ord_less_eq @ ( set @ nat ) @ S6 @ ( set_ord_atMost @ nat @ K2 ) ) ) ) ).
% finite_nat_iff_bounded_le
thf(fact_2739_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_2740_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_2741_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_2742_atMost__nat__numeral,axiom,
! [K: num] :
( ( set_ord_atMost @ nat @ ( numeral_numeral @ nat @ K ) )
= ( insert2 @ nat @ ( numeral_numeral @ nat @ K ) @ ( set_ord_atMost @ nat @ ( pred_numeral @ K ) ) ) ) ).
% atMost_nat_numeral
thf(fact_2743_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_2744_sum__choose__upper,axiom,
! [M2: nat,N2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [K2: nat] : ( binomial @ K2 @ M2 )
@ ( set_ord_atMost @ nat @ N2 ) )
= ( binomial @ ( suc @ N2 ) @ ( suc @ M2 ) ) ) ).
% sum_choose_upper
thf(fact_2745_sum_OatMost__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,N2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_ord_atMost @ nat @ ( suc @ N2 ) ) )
= ( plus_plus @ A @ ( G @ ( zero_zero @ nat ) )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_ord_atMost @ nat @ N2 ) ) ) ) ) ).
% sum.atMost_Suc_shift
thf(fact_2746_sum__telescope,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [F2: nat > A,I: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( minus_minus @ A @ ( F2 @ I3 ) @ ( F2 @ ( suc @ I3 ) ) )
@ ( set_ord_atMost @ nat @ I ) )
= ( minus_minus @ A @ ( F2 @ ( zero_zero @ nat ) ) @ ( F2 @ ( suc @ I ) ) ) ) ) ).
% sum_telescope
thf(fact_2747_polyfun__eq__coeffs,axiom,
! [A: $tType] :
( ( ( real_V8999393235501362500lgebra @ A )
& ( idom @ A ) )
=> ! [C2: nat > A,N2: nat,D2: nat > A] :
( ( ! [X4: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( C2 @ I3 ) @ ( power_power @ A @ X4 @ I3 ) )
@ ( set_ord_atMost @ nat @ N2 ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( D2 @ I3 ) @ ( power_power @ A @ X4 @ I3 ) )
@ ( set_ord_atMost @ nat @ N2 ) ) ) )
= ( ! [I3: nat] :
( ( ord_less_eq @ nat @ I3 @ N2 )
=> ( ( C2 @ I3 )
= ( D2 @ I3 ) ) ) ) ) ) ).
% polyfun_eq_coeffs
thf(fact_2748_prod_OatMost__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,N2: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_atMost @ nat @ ( suc @ N2 ) ) )
= ( times_times @ A @ ( G @ ( zero_zero @ nat ) )
@ ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_ord_atMost @ nat @ N2 ) ) ) ) ) ).
% prod.atMost_Suc_shift
thf(fact_2749_sum_Onested__swap_H,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A3: nat > nat > A,N2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( groups7311177749621191930dd_sum @ nat @ A @ ( A3 @ I3 ) @ ( set_ord_lessThan @ nat @ I3 ) )
@ ( set_ord_atMost @ nat @ N2 ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [J2: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( A3 @ I3 @ J2 )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ J2 ) @ N2 ) )
@ ( set_ord_lessThan @ nat @ N2 ) ) ) ) ).
% sum.nested_swap'
thf(fact_2750_prod_Onested__swap_H,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: nat > nat > A,N2: nat] :
( ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( groups7121269368397514597t_prod @ nat @ A @ ( A3 @ I3 ) @ ( set_ord_lessThan @ nat @ I3 ) )
@ ( set_ord_atMost @ nat @ N2 ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [J2: nat] :
( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( A3 @ I3 @ J2 )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ J2 ) @ N2 ) )
@ ( set_ord_lessThan @ nat @ N2 ) ) ) ) ).
% prod.nested_swap'
thf(fact_2751_sum__choose__lower,axiom,
! [R2: nat,N2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [K2: nat] : ( binomial @ ( plus_plus @ nat @ R2 @ K2 ) @ K2 )
@ ( set_ord_atMost @ nat @ N2 ) )
= ( binomial @ ( suc @ ( plus_plus @ nat @ R2 @ N2 ) ) @ N2 ) ) ).
% sum_choose_lower
thf(fact_2752_choose__rising__sum_I2_J,axiom,
! [N2: nat,M2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [J2: nat] : ( binomial @ ( plus_plus @ nat @ N2 @ J2 ) @ N2 )
@ ( set_ord_atMost @ nat @ M2 ) )
= ( binomial @ ( plus_plus @ nat @ ( plus_plus @ nat @ N2 @ M2 ) @ ( one_one @ nat ) ) @ M2 ) ) ).
% choose_rising_sum(2)
thf(fact_2753_choose__rising__sum_I1_J,axiom,
! [N2: nat,M2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [J2: nat] : ( binomial @ ( plus_plus @ nat @ N2 @ J2 ) @ N2 )
@ ( set_ord_atMost @ nat @ M2 ) )
= ( binomial @ ( plus_plus @ nat @ ( plus_plus @ nat @ N2 @ M2 ) @ ( one_one @ nat ) ) @ ( plus_plus @ nat @ N2 @ ( one_one @ nat ) ) ) ) ).
% choose_rising_sum(1)
thf(fact_2754_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_2755_zero__polynom__imp__zero__coeffs,axiom,
! [A: $tType] :
( ( ( ab_semigroup_mult @ A )
& ( real_V8999393235501362500lgebra @ A ) )
=> ! [C2: nat > A,N2: nat,K: nat] :
( ! [W: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( C2 @ I3 ) @ ( power_power @ A @ W @ I3 ) )
@ ( set_ord_atMost @ nat @ N2 ) )
= ( zero_zero @ A ) )
=> ( ( ord_less_eq @ nat @ K @ N2 )
=> ( ( C2 @ K )
= ( zero_zero @ A ) ) ) ) ) ).
% zero_polynom_imp_zero_coeffs
thf(fact_2756_polyfun__eq__0,axiom,
! [A: $tType] :
( ( ( real_V8999393235501362500lgebra @ A )
& ( idom @ A ) )
=> ! [C2: nat > A,N2: nat] :
( ( ! [X4: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( C2 @ I3 ) @ ( power_power @ A @ X4 @ I3 ) )
@ ( set_ord_atMost @ nat @ N2 ) )
= ( zero_zero @ A ) ) )
= ( ! [I3: nat] :
( ( ord_less_eq @ nat @ I3 @ N2 )
=> ( ( C2 @ I3 )
= ( zero_zero @ A ) ) ) ) ) ) ).
% polyfun_eq_0
thf(fact_2757_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_2758_sum_OatMost__shift,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,N2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_ord_atMost @ nat @ N2 ) )
= ( plus_plus @ A @ ( G @ ( zero_zero @ nat ) )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_ord_lessThan @ nat @ N2 ) ) ) ) ) ).
% sum.atMost_shift
thf(fact_2759_sum__up__index__split,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [F2: nat > A,M2: nat,N2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_ord_atMost @ nat @ ( plus_plus @ nat @ M2 @ N2 ) ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_ord_atMost @ nat @ M2 ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M2 ) @ ( plus_plus @ nat @ M2 @ N2 ) ) ) ) ) ) ).
% sum_up_index_split
thf(fact_2760_prod_OatMost__shift,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,N2: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_atMost @ nat @ N2 ) )
= ( times_times @ A @ ( G @ ( zero_zero @ nat ) )
@ ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_ord_lessThan @ nat @ N2 ) ) ) ) ) ).
% prod.atMost_shift
thf(fact_2761_atLeast1__atMost__eq__remove0,axiom,
! [N2: nat] :
( ( set_or1337092689740270186AtMost @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N2 )
= ( minus_minus @ ( set @ nat ) @ ( set_ord_atMost @ nat @ N2 ) @ ( insert2 @ nat @ ( zero_zero @ nat ) @ ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% atLeast1_atMost_eq_remove0
thf(fact_2762_gbinomial__parallel__sum,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: A,N2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K2: nat] : ( gbinomial @ A @ ( plus_plus @ A @ A3 @ ( semiring_1_of_nat @ A @ K2 ) ) @ K2 )
@ ( set_ord_atMost @ nat @ N2 ) )
= ( gbinomial @ A @ ( plus_plus @ A @ ( plus_plus @ A @ A3 @ ( semiring_1_of_nat @ A @ N2 ) ) @ ( one_one @ A ) ) @ N2 ) ) ) ).
% gbinomial_parallel_sum
thf(fact_2763_sum_Otriangle__reindex__eq,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > nat > A,N2: nat] :
( ( groups7311177749621191930dd_sum @ ( product_prod @ nat @ nat ) @ A @ ( product_case_prod @ nat @ nat @ A @ G )
@ ( collect @ ( product_prod @ nat @ nat )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [I3: nat,J2: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ I3 @ J2 ) @ N2 ) ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K2: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( G @ I3 @ ( minus_minus @ nat @ K2 @ I3 ) )
@ ( set_ord_atMost @ nat @ K2 ) )
@ ( set_ord_atMost @ nat @ N2 ) ) ) ) ).
% sum.triangle_reindex_eq
thf(fact_2764_prod_Otriangle__reindex__eq,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > nat > A,N2: nat] :
( ( groups7121269368397514597t_prod @ ( product_prod @ nat @ nat ) @ A @ ( product_case_prod @ nat @ nat @ A @ G )
@ ( collect @ ( product_prod @ nat @ nat )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [I3: nat,J2: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ I3 @ J2 ) @ N2 ) ) ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [K2: nat] :
( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( G @ I3 @ ( minus_minus @ nat @ K2 @ I3 ) )
@ ( set_ord_atMost @ nat @ K2 ) )
@ ( set_ord_atMost @ nat @ N2 ) ) ) ) ).
% prod.triangle_reindex_eq
thf(fact_2765_sum__choose__diagonal,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq @ nat @ M2 @ N2 )
=> ( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [K2: nat] : ( binomial @ ( minus_minus @ nat @ N2 @ K2 ) @ ( minus_minus @ nat @ M2 @ K2 ) )
@ ( set_ord_atMost @ nat @ M2 ) )
= ( binomial @ ( suc @ N2 ) @ M2 ) ) ) ).
% sum_choose_diagonal
thf(fact_2766_vandermonde,axiom,
! [M2: nat,N2: nat,R2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [K2: nat] : ( times_times @ nat @ ( binomial @ M2 @ K2 ) @ ( binomial @ N2 @ ( minus_minus @ nat @ R2 @ K2 ) ) )
@ ( set_ord_atMost @ nat @ R2 ) )
= ( binomial @ ( plus_plus @ nat @ M2 @ N2 ) @ R2 ) ) ).
% vandermonde
thf(fact_2767_sum__gp__basic,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [X: A,N2: nat] :
( ( times_times @ A @ ( minus_minus @ A @ ( one_one @ A ) @ X ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_ord_atMost @ nat @ N2 ) ) )
= ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ X @ ( suc @ N2 ) ) ) ) ) ).
% sum_gp_basic
thf(fact_2768_polyfun__roots__finite,axiom,
! [A: $tType] :
( ( ( real_V8999393235501362500lgebra @ A )
& ( idom @ A ) )
=> ! [C2: nat > A,K: nat,N2: nat] :
( ( ( C2 @ K )
!= ( zero_zero @ A ) )
=> ( ( ord_less_eq @ nat @ K @ N2 )
=> ( finite_finite2 @ A
@ ( collect @ A
@ ^ [Z6: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( C2 @ I3 ) @ ( power_power @ A @ Z6 @ I3 ) )
@ ( set_ord_atMost @ nat @ N2 ) )
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% polyfun_roots_finite
thf(fact_2769_polyfun__finite__roots,axiom,
! [A: $tType] :
( ( ( real_V8999393235501362500lgebra @ A )
& ( idom @ A ) )
=> ! [C2: nat > A,N2: nat] :
( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X4: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( C2 @ I3 ) @ ( power_power @ A @ X4 @ I3 ) )
@ ( set_ord_atMost @ nat @ N2 ) )
= ( zero_zero @ A ) ) ) )
= ( ? [I3: nat] :
( ( ord_less_eq @ nat @ I3 @ N2 )
& ( ( C2 @ I3 )
!= ( zero_zero @ A ) ) ) ) ) ) ).
% polyfun_finite_roots
thf(fact_2770_polyfun__linear__factor__root,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [C2: nat > A,A3: A,N2: nat] :
( ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( C2 @ I3 ) @ ( power_power @ A @ A3 @ I3 ) )
@ ( set_ord_atMost @ nat @ N2 ) )
= ( zero_zero @ A ) )
=> ~ ! [B5: nat > A] :
~ ! [Z4: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( C2 @ I3 ) @ ( power_power @ A @ Z4 @ I3 ) )
@ ( set_ord_atMost @ nat @ N2 ) )
= ( times_times @ A @ ( minus_minus @ A @ Z4 @ A3 )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( B5 @ I3 ) @ ( power_power @ A @ Z4 @ I3 ) )
@ ( set_ord_lessThan @ nat @ N2 ) ) ) ) ) ) ).
% polyfun_linear_factor_root
thf(fact_2771_polyfun__linear__factor,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [C2: nat > A,N2: nat,A3: A] :
? [B5: nat > A] :
! [Z4: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( C2 @ I3 ) @ ( power_power @ A @ Z4 @ I3 ) )
@ ( set_ord_atMost @ nat @ N2 ) )
= ( plus_plus @ A
@ ( times_times @ A @ ( minus_minus @ A @ Z4 @ A3 )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( B5 @ I3 ) @ ( power_power @ A @ Z4 @ I3 ) )
@ ( set_ord_lessThan @ nat @ N2 ) ) )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( C2 @ I3 ) @ ( power_power @ A @ A3 @ I3 ) )
@ ( set_ord_atMost @ nat @ N2 ) ) ) ) ) ).
% polyfun_linear_factor
thf(fact_2772_sum__power__shift,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [M2: nat,N2: nat,X: A] :
( ( ord_less_eq @ nat @ M2 @ N2 )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N2 ) )
= ( times_times @ A @ ( power_power @ A @ X @ M2 ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_ord_atMost @ nat @ ( minus_minus @ nat @ N2 @ M2 ) ) ) ) ) ) ) ).
% sum_power_shift
thf(fact_2773_sum_Otriangle__reindex,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > nat > A,N2: nat] :
( ( groups7311177749621191930dd_sum @ ( product_prod @ nat @ nat ) @ A @ ( product_case_prod @ nat @ nat @ A @ G )
@ ( collect @ ( product_prod @ nat @ nat )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [I3: nat,J2: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ I3 @ J2 ) @ N2 ) ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K2: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( G @ I3 @ ( minus_minus @ nat @ K2 @ I3 ) )
@ ( set_ord_atMost @ nat @ K2 ) )
@ ( set_ord_lessThan @ nat @ N2 ) ) ) ) ).
% sum.triangle_reindex
thf(fact_2774_prod_Otriangle__reindex,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > nat > A,N2: nat] :
( ( groups7121269368397514597t_prod @ ( product_prod @ nat @ nat ) @ A @ ( product_case_prod @ nat @ nat @ A @ G )
@ ( collect @ ( product_prod @ nat @ nat )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [I3: nat,J2: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ I3 @ J2 ) @ N2 ) ) ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [K2: nat] :
( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( G @ I3 @ ( minus_minus @ nat @ K2 @ I3 ) )
@ ( set_ord_atMost @ nat @ K2 ) )
@ ( set_ord_lessThan @ nat @ N2 ) ) ) ) ).
% prod.triangle_reindex
thf(fact_2775_binomial,axiom,
! [A3: nat,B3: nat,N2: nat] :
( ( power_power @ nat @ ( plus_plus @ nat @ A3 @ B3 ) @ N2 )
= ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [K2: nat] : ( times_times @ nat @ ( times_times @ nat @ ( semiring_1_of_nat @ nat @ ( binomial @ N2 @ K2 ) ) @ ( power_power @ nat @ A3 @ K2 ) ) @ ( power_power @ nat @ B3 @ ( minus_minus @ nat @ N2 @ K2 ) ) )
@ ( set_ord_atMost @ nat @ N2 ) ) ) ).
% binomial
thf(fact_2776_sum_Oin__pairs__0,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,N2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_ord_atMost @ nat @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( plus_plus @ A @ ( G @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I3 ) ) ) )
@ ( set_ord_atMost @ nat @ N2 ) ) ) ) ).
% sum.in_pairs_0
thf(fact_2777_polynomial__product,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [M2: nat,A3: nat > A,N2: nat,B3: nat > A,X: A] :
( ! [I4: nat] :
( ( ord_less @ nat @ M2 @ I4 )
=> ( ( A3 @ I4 )
= ( zero_zero @ A ) ) )
=> ( ! [J3: nat] :
( ( ord_less @ nat @ N2 @ J3 )
=> ( ( B3 @ J3 )
= ( zero_zero @ A ) ) )
=> ( ( times_times @ A
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( A3 @ I3 ) @ ( power_power @ A @ X @ I3 ) )
@ ( set_ord_atMost @ nat @ M2 ) )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [J2: nat] : ( times_times @ A @ ( B3 @ J2 ) @ ( power_power @ A @ X @ J2 ) )
@ ( set_ord_atMost @ nat @ N2 ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [R4: nat] :
( times_times @ A
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K2: nat] : ( times_times @ A @ ( A3 @ K2 ) @ ( B3 @ ( minus_minus @ nat @ R4 @ K2 ) ) )
@ ( set_ord_atMost @ nat @ R4 ) )
@ ( power_power @ A @ X @ R4 ) )
@ ( set_ord_atMost @ nat @ ( plus_plus @ nat @ M2 @ N2 ) ) ) ) ) ) ) ).
% polynomial_product
thf(fact_2778_prod_Oin__pairs__0,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,N2: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_atMost @ nat @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( G @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I3 ) ) ) )
@ ( set_ord_atMost @ nat @ N2 ) ) ) ) ).
% prod.in_pairs_0
thf(fact_2779_polyfun__eq__const,axiom,
! [A: $tType] :
( ( ( real_V8999393235501362500lgebra @ A )
& ( idom @ A ) )
=> ! [C2: nat > A,N2: nat,K: A] :
( ( ! [X4: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( C2 @ I3 ) @ ( power_power @ A @ X4 @ I3 ) )
@ ( set_ord_atMost @ nat @ N2 ) )
= K ) )
= ( ( ( C2 @ ( zero_zero @ nat ) )
= K )
& ! [X4: nat] :
( ( member @ nat @ X4 @ ( set_or1337092689740270186AtMost @ nat @ ( one_one @ nat ) @ N2 ) )
=> ( ( C2 @ X4 )
= ( zero_zero @ A ) ) ) ) ) ) ).
% polyfun_eq_const
thf(fact_2780_binomial__ring,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A3: A,B3: A,N2: nat] :
( ( power_power @ A @ ( plus_plus @ A @ A3 @ B3 ) @ N2 )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K2: nat] : ( times_times @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( binomial @ N2 @ K2 ) ) @ ( power_power @ A @ A3 @ K2 ) ) @ ( power_power @ A @ B3 @ ( minus_minus @ nat @ N2 @ K2 ) ) )
@ ( set_ord_atMost @ nat @ N2 ) ) ) ) ).
% binomial_ring
thf(fact_2781_pochhammer__binomial__sum,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [A3: A,B3: A,N2: nat] :
( ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ A3 @ B3 ) @ N2 )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K2: nat] : ( times_times @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( binomial @ N2 @ K2 ) ) @ ( comm_s3205402744901411588hammer @ A @ A3 @ K2 ) ) @ ( comm_s3205402744901411588hammer @ A @ B3 @ ( minus_minus @ nat @ N2 @ K2 ) ) )
@ ( set_ord_atMost @ nat @ N2 ) ) ) ) ).
% pochhammer_binomial_sum
thf(fact_2782_polynomial__product__nat,axiom,
! [M2: nat,A3: nat > nat,N2: nat,B3: nat > nat,X: nat] :
( ! [I4: nat] :
( ( ord_less @ nat @ M2 @ I4 )
=> ( ( A3 @ I4 )
= ( zero_zero @ nat ) ) )
=> ( ! [J3: nat] :
( ( ord_less @ nat @ N2 @ J3 )
=> ( ( B3 @ J3 )
= ( zero_zero @ nat ) ) )
=> ( ( times_times @ nat
@ ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [I3: nat] : ( times_times @ nat @ ( A3 @ I3 ) @ ( power_power @ nat @ X @ I3 ) )
@ ( set_ord_atMost @ nat @ M2 ) )
@ ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [J2: nat] : ( times_times @ nat @ ( B3 @ J2 ) @ ( power_power @ nat @ X @ J2 ) )
@ ( set_ord_atMost @ nat @ N2 ) ) )
= ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [R4: nat] :
( times_times @ nat
@ ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [K2: nat] : ( times_times @ nat @ ( A3 @ K2 ) @ ( B3 @ ( minus_minus @ nat @ R4 @ K2 ) ) )
@ ( set_ord_atMost @ nat @ R4 ) )
@ ( power_power @ nat @ X @ R4 ) )
@ ( set_ord_atMost @ nat @ ( plus_plus @ nat @ M2 @ N2 ) ) ) ) ) ) ).
% polynomial_product_nat
thf(fact_2783_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_2784_sum_Ozero__middle,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [P5: nat,K: nat,G: nat > A,H2: nat > A] :
( ( ord_less_eq @ nat @ ( one_one @ nat ) @ P5 )
=> ( ( ord_less_eq @ nat @ K @ P5 )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [J2: nat] : ( if @ A @ ( ord_less @ nat @ J2 @ K ) @ ( G @ J2 ) @ ( if @ A @ ( J2 = K ) @ ( zero_zero @ A ) @ ( H2 @ ( minus_minus @ nat @ J2 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) )
@ ( set_ord_atMost @ nat @ P5 ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [J2: nat] : ( if @ A @ ( ord_less @ nat @ J2 @ K ) @ ( G @ J2 ) @ ( H2 @ J2 ) )
@ ( set_ord_atMost @ nat @ ( minus_minus @ nat @ P5 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ) ) ).
% sum.zero_middle
thf(fact_2785_prod_Ozero__middle,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [P5: nat,K: nat,G: nat > A,H2: nat > A] :
( ( ord_less_eq @ nat @ ( one_one @ nat ) @ P5 )
=> ( ( ord_less_eq @ nat @ K @ P5 )
=> ( ( groups7121269368397514597t_prod @ nat @ A
@ ^ [J2: nat] : ( if @ A @ ( ord_less @ nat @ J2 @ K ) @ ( G @ J2 ) @ ( if @ A @ ( J2 = K ) @ ( one_one @ A ) @ ( H2 @ ( minus_minus @ nat @ J2 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) )
@ ( set_ord_atMost @ nat @ P5 ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [J2: nat] : ( if @ A @ ( ord_less @ nat @ J2 @ K ) @ ( G @ J2 ) @ ( H2 @ J2 ) )
@ ( set_ord_atMost @ nat @ ( minus_minus @ nat @ P5 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ) ) ).
% prod.zero_middle
thf(fact_2786_gbinomial__partial__sum__poly,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [M2: nat,A3: A,X: A,Y: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K2: nat] : ( times_times @ A @ ( times_times @ A @ ( gbinomial @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ M2 ) @ A3 ) @ K2 ) @ ( power_power @ A @ X @ K2 ) ) @ ( power_power @ A @ Y @ ( minus_minus @ nat @ M2 @ K2 ) ) )
@ ( set_ord_atMost @ nat @ M2 ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K2: nat] : ( times_times @ A @ ( times_times @ A @ ( gbinomial @ A @ ( uminus_uminus @ A @ A3 ) @ K2 ) @ ( power_power @ A @ ( uminus_uminus @ A @ X ) @ K2 ) ) @ ( power_power @ A @ ( plus_plus @ A @ X @ Y ) @ ( minus_minus @ nat @ M2 @ K2 ) ) )
@ ( set_ord_atMost @ nat @ M2 ) ) ) ) ).
% gbinomial_partial_sum_poly
thf(fact_2787_root__polyfun,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [N2: nat,Z3: A,A3: A] :
( ( ord_less_eq @ nat @ ( one_one @ nat ) @ N2 )
=> ( ( ( power_power @ A @ Z3 @ N2 )
= A3 )
= ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] :
( times_times @ A
@ ( if @ A
@ ( I3
= ( zero_zero @ nat ) )
@ ( uminus_uminus @ A @ A3 )
@ ( if @ A @ ( I3 = N2 ) @ ( one_one @ A ) @ ( zero_zero @ A ) ) )
@ ( power_power @ A @ Z3 @ I3 ) )
@ ( set_ord_atMost @ nat @ N2 ) )
= ( zero_zero @ A ) ) ) ) ) ).
% root_polyfun
thf(fact_2788_sum__gp0,axiom,
! [A: $tType] :
( ( ( division_ring @ A )
& ( comm_ring @ A ) )
=> ! [X: A,N2: nat] :
( ( ( X
= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_ord_atMost @ nat @ N2 ) )
= ( semiring_1_of_nat @ A @ ( plus_plus @ nat @ N2 @ ( one_one @ nat ) ) ) ) )
& ( ( X
!= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_ord_atMost @ nat @ N2 ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ X @ ( suc @ N2 ) ) ) @ ( minus_minus @ A @ ( one_one @ A ) @ X ) ) ) ) ) ) ).
% sum_gp0
thf(fact_2789_choose__alternating__linear__sum,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [N2: nat] :
( ( N2
!= ( one_one @ nat ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ I3 ) @ ( semiring_1_of_nat @ A @ I3 ) ) @ ( semiring_1_of_nat @ A @ ( binomial @ N2 @ I3 ) ) )
@ ( set_ord_atMost @ nat @ N2 ) )
= ( zero_zero @ A ) ) ) ) ).
% choose_alternating_linear_sum
thf(fact_2790_gbinomial__sum__nat__pow2,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [M2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K2: nat] : ( divide_divide @ A @ ( gbinomial @ A @ ( semiring_1_of_nat @ A @ ( plus_plus @ nat @ M2 @ K2 ) ) @ K2 ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ K2 ) )
@ ( set_ord_atMost @ nat @ M2 ) )
= ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M2 ) ) ) ).
% gbinomial_sum_nat_pow2
thf(fact_2791_gbinomial__partial__sum__poly__xpos,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [M2: nat,A3: A,X: A,Y: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K2: nat] : ( times_times @ A @ ( times_times @ A @ ( gbinomial @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ M2 ) @ A3 ) @ K2 ) @ ( power_power @ A @ X @ K2 ) ) @ ( power_power @ A @ Y @ ( minus_minus @ nat @ M2 @ K2 ) ) )
@ ( set_ord_atMost @ nat @ M2 ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K2: nat] : ( times_times @ A @ ( times_times @ A @ ( gbinomial @ A @ ( minus_minus @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ K2 ) @ A3 ) @ ( one_one @ A ) ) @ K2 ) @ ( power_power @ A @ X @ K2 ) ) @ ( power_power @ A @ ( plus_plus @ A @ X @ Y ) @ ( minus_minus @ nat @ M2 @ K2 ) ) )
@ ( set_ord_atMost @ nat @ M2 ) ) ) ) ).
% gbinomial_partial_sum_poly_xpos
thf(fact_2792_polyfun__diff__alt,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [N2: nat,A3: nat > A,X: A,Y: A] :
( ( ord_less_eq @ nat @ ( one_one @ nat ) @ N2 )
=> ( ( minus_minus @ A
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( A3 @ I3 ) @ ( power_power @ A @ X @ I3 ) )
@ ( set_ord_atMost @ nat @ N2 ) )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( A3 @ I3 ) @ ( power_power @ A @ Y @ I3 ) )
@ ( set_ord_atMost @ nat @ N2 ) ) )
= ( times_times @ A @ ( minus_minus @ A @ X @ Y )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [J2: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K2: nat] : ( times_times @ A @ ( times_times @ A @ ( A3 @ ( plus_plus @ nat @ ( plus_plus @ nat @ J2 @ K2 ) @ ( one_one @ nat ) ) ) @ ( power_power @ A @ Y @ K2 ) ) @ ( power_power @ A @ X @ J2 ) )
@ ( set_ord_lessThan @ nat @ ( minus_minus @ nat @ N2 @ J2 ) ) )
@ ( set_ord_lessThan @ nat @ N2 ) ) ) ) ) ) ).
% polyfun_diff_alt
thf(fact_2793_binomial__r__part__sum,axiom,
! [M2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat @ ( binomial @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M2 ) @ ( one_one @ nat ) ) ) @ ( set_ord_atMost @ nat @ M2 ) )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M2 ) ) ) ).
% binomial_r_part_sum
thf(fact_2794_choose__alternating__sum,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ I3 ) @ ( semiring_1_of_nat @ A @ ( binomial @ N2 @ I3 ) ) )
@ ( set_ord_atMost @ nat @ N2 ) )
= ( zero_zero @ A ) ) ) ) ).
% choose_alternating_sum
thf(fact_2795_polyfun__extremal__lemma,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [E2: real,C2: nat > A,N2: nat] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E2 )
=> ? [M8: real] :
! [Z4: A] :
( ( ord_less_eq @ real @ M8 @ ( real_V7770717601297561774m_norm @ A @ Z4 ) )
=> ( ord_less_eq @ real
@ ( real_V7770717601297561774m_norm @ A
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( C2 @ I3 ) @ ( power_power @ A @ Z4 @ I3 ) )
@ ( set_ord_atMost @ nat @ N2 ) ) )
@ ( times_times @ real @ E2 @ ( power_power @ real @ ( real_V7770717601297561774m_norm @ A @ Z4 ) @ ( suc @ N2 ) ) ) ) ) ) ) ).
% polyfun_extremal_lemma
thf(fact_2796_polyfun__diff,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [N2: nat,A3: nat > A,X: A,Y: A] :
( ( ord_less_eq @ nat @ ( one_one @ nat ) @ N2 )
=> ( ( minus_minus @ A
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( A3 @ I3 ) @ ( power_power @ A @ X @ I3 ) )
@ ( set_ord_atMost @ nat @ N2 ) )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( A3 @ I3 ) @ ( power_power @ A @ Y @ I3 ) )
@ ( set_ord_atMost @ nat @ N2 ) ) )
= ( times_times @ A @ ( minus_minus @ A @ X @ Y )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [J2: nat] :
( times_times @ A
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( A3 @ I3 ) @ ( power_power @ A @ Y @ ( minus_minus @ nat @ ( minus_minus @ nat @ I3 @ J2 ) @ ( one_one @ nat ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ J2 ) @ N2 ) )
@ ( power_power @ A @ X @ J2 ) )
@ ( set_ord_lessThan @ nat @ N2 ) ) ) ) ) ) ).
% polyfun_diff
thf(fact_2797_sin__cos__npi,axiom,
! [N2: nat] :
( ( sin @ real @ ( divide_divide @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) @ pi ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
= ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ N2 ) ) ).
% sin_cos_npi
thf(fact_2798_cos__pi__eq__zero,axiom,
! [M2: nat] :
( ( cos @ real @ ( divide_divide @ real @ ( times_times @ real @ pi @ ( semiring_1_of_nat @ real @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M2 ) ) ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
= ( zero_zero @ real ) ) ).
% cos_pi_eq_zero
thf(fact_2799_gbinomial__code,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( ( gbinomial @ A )
= ( ^ [A5: A,K2: nat] :
( if @ A
@ ( K2
= ( 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 @ K2 @ ( one_one @ nat ) )
@ ( one_one @ A ) )
@ ( semiring_char_0_fact @ A @ K2 ) ) ) ) ) ) ).
% gbinomial_code
thf(fact_2800_sum__pos__lt__pair,axiom,
! [F2: nat > real,K: nat] :
( ( summable @ real @ F2 )
=> ( ! [D5: nat] : ( ord_less @ real @ ( zero_zero @ real ) @ ( plus_plus @ real @ ( F2 @ ( plus_plus @ nat @ K @ ( times_times @ nat @ ( suc @ ( suc @ ( zero_zero @ nat ) ) ) @ D5 ) ) ) @ ( F2 @ ( plus_plus @ nat @ K @ ( plus_plus @ nat @ ( times_times @ nat @ ( suc @ ( suc @ ( zero_zero @ nat ) ) ) @ D5 ) @ ( one_one @ nat ) ) ) ) ) )
=> ( ord_less @ real @ ( groups7311177749621191930dd_sum @ nat @ real @ F2 @ ( set_ord_lessThan @ nat @ K ) ) @ ( suminf @ real @ F2 ) ) ) ) ).
% sum_pos_lt_pair
thf(fact_2801_binomial__code,axiom,
( binomial
= ( ^ [N: nat,K2: nat] : ( if @ nat @ ( ord_less @ nat @ N @ K2 ) @ ( zero_zero @ nat ) @ ( if @ nat @ ( ord_less @ nat @ N @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K2 ) ) @ ( binomial @ N @ ( minus_minus @ nat @ N @ K2 ) ) @ ( divide_divide @ nat @ ( set_fo6178422350223883121st_nat @ nat @ ( times_times @ nat ) @ ( plus_plus @ nat @ ( minus_minus @ nat @ N @ K2 ) @ ( one_one @ nat ) ) @ N @ ( one_one @ nat ) ) @ ( semiring_char_0_fact @ nat @ K2 ) ) ) ) ) ) ).
% binomial_code
thf(fact_2802_modulo__int__unfold,axiom,
! [L: int,K: int,N2: nat,M2: nat] :
( ( ( ( ( sgn_sgn @ int @ L )
= ( zero_zero @ int ) )
| ( ( sgn_sgn @ int @ K )
= ( zero_zero @ int ) )
| ( N2
= ( zero_zero @ nat ) ) )
=> ( ( modulo_modulo @ int @ ( times_times @ int @ ( sgn_sgn @ int @ K ) @ ( semiring_1_of_nat @ int @ M2 ) ) @ ( times_times @ int @ ( sgn_sgn @ int @ L ) @ ( semiring_1_of_nat @ int @ N2 ) ) )
= ( times_times @ int @ ( sgn_sgn @ int @ K ) @ ( semiring_1_of_nat @ int @ M2 ) ) ) )
& ( ~ ( ( ( sgn_sgn @ int @ L )
= ( zero_zero @ int ) )
| ( ( sgn_sgn @ int @ K )
= ( zero_zero @ int ) )
| ( N2
= ( 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 @ M2 ) ) @ ( times_times @ int @ ( sgn_sgn @ int @ L ) @ ( semiring_1_of_nat @ int @ N2 ) ) )
= ( times_times @ int @ ( sgn_sgn @ int @ L ) @ ( semiring_1_of_nat @ int @ ( modulo_modulo @ nat @ M2 @ N2 ) ) ) ) )
& ( ( ( sgn_sgn @ int @ K )
!= ( sgn_sgn @ int @ L ) )
=> ( ( modulo_modulo @ int @ ( times_times @ int @ ( sgn_sgn @ int @ K ) @ ( semiring_1_of_nat @ int @ M2 ) ) @ ( times_times @ int @ ( sgn_sgn @ int @ L ) @ ( semiring_1_of_nat @ int @ N2 ) ) )
= ( times_times @ int @ ( sgn_sgn @ int @ L )
@ ( minus_minus @ int
@ ( semiring_1_of_nat @ int
@ ( times_times @ nat @ N2
@ ( zero_neq_one_of_bool @ nat
@ ~ ( dvd_dvd @ nat @ N2 @ M2 ) ) ) )
@ ( semiring_1_of_nat @ int @ ( modulo_modulo @ nat @ M2 @ N2 ) ) ) ) ) ) ) ) ) ).
% modulo_int_unfold
thf(fact_2803_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_2804_of__nat__id,axiom,
( ( semiring_1_of_nat @ nat )
= ( ^ [N: nat] : N ) ) ).
% of_nat_id
thf(fact_2805_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_2806_sgn__0,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ( ( sgn_sgn @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% sgn_0
thf(fact_2807_sgn__zero,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ( ( sgn_sgn @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% sgn_zero
thf(fact_2808_sgn__1,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ( ( sgn_sgn @ A @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% sgn_1
thf(fact_2809_sgn__divide,axiom,
! [A: $tType] :
( ( field_abs_sgn @ A )
=> ! [A3: A,B3: A] :
( ( sgn_sgn @ A @ ( divide_divide @ A @ A3 @ B3 ) )
= ( divide_divide @ A @ ( sgn_sgn @ A @ A3 ) @ ( sgn_sgn @ A @ B3 ) ) ) ) ).
% sgn_divide
thf(fact_2810_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_2811_summable__zero,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ( summable @ A
@ ^ [N: nat] : ( zero_zero @ A ) ) ) ).
% summable_zero
thf(fact_2812_summable__single,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [I: nat,F2: nat > A] :
( summable @ A
@ ^ [R4: nat] : ( if @ A @ ( R4 = I ) @ ( F2 @ R4 ) @ ( zero_zero @ A ) ) ) ) ).
% summable_single
thf(fact_2813_summable__iff__shift,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A,K: nat] :
( ( summable @ A
@ ^ [N: nat] : ( F2 @ ( plus_plus @ nat @ N @ K ) ) )
= ( summable @ A @ F2 ) ) ) ).
% summable_iff_shift
thf(fact_2814_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_2815_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_2816_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_2817_divide__sgn,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B3: A] :
( ( divide_divide @ A @ A3 @ ( sgn_sgn @ A @ B3 ) )
= ( times_times @ A @ A3 @ ( sgn_sgn @ A @ B3 ) ) ) ) ).
% divide_sgn
thf(fact_2818_fact__0,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ( ( semiring_char_0_fact @ A @ ( zero_zero @ nat ) )
= ( one_one @ A ) ) ) ).
% fact_0
thf(fact_2819_summable__cmult__iff,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [C2: A,F2: nat > A] :
( ( summable @ A
@ ^ [N: nat] : ( times_times @ A @ C2 @ ( F2 @ N ) ) )
= ( ( C2
= ( zero_zero @ A ) )
| ( summable @ A @ F2 ) ) ) ) ).
% summable_cmult_iff
thf(fact_2820_summable__divide__iff,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: nat > A,C2: A] :
( ( summable @ A
@ ^ [N: nat] : ( divide_divide @ A @ ( F2 @ N ) @ C2 ) )
= ( ( C2
= ( zero_zero @ A ) )
| ( summable @ A @ F2 ) ) ) ) ).
% summable_divide_iff
thf(fact_2821_summable__If__finite__set,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [A4: set @ nat,F2: nat > A] :
( ( finite_finite2 @ nat @ A4 )
=> ( summable @ A
@ ^ [R4: nat] : ( if @ A @ ( member @ nat @ R4 @ A4 ) @ ( F2 @ R4 ) @ ( zero_zero @ A ) ) ) ) ) ).
% summable_If_finite_set
thf(fact_2822_summable__If__finite,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [P: nat > $o,F2: nat > A] :
( ( finite_finite2 @ nat @ ( collect @ nat @ P ) )
=> ( summable @ A
@ ^ [R4: nat] : ( if @ A @ ( P @ R4 ) @ ( F2 @ R4 ) @ ( zero_zero @ A ) ) ) ) ) ).
% summable_If_finite
thf(fact_2823_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_2824_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_2825_fact__Suc,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [N2: nat] :
( ( semiring_char_0_fact @ A @ ( suc @ N2 ) )
= ( times_times @ A @ ( semiring_1_of_nat @ A @ ( suc @ N2 ) ) @ ( semiring_char_0_fact @ A @ N2 ) ) ) ) ).
% fact_Suc
thf(fact_2826_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_2827_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_2828_dvd__mult__sgn__iff,axiom,
! [L: int,K: int,R2: int] :
( ( dvd_dvd @ int @ L @ ( times_times @ int @ K @ ( sgn_sgn @ int @ R2 ) ) )
= ( ( dvd_dvd @ int @ L @ K )
| ( R2
= ( zero_zero @ int ) ) ) ) ).
% dvd_mult_sgn_iff
thf(fact_2829_dvd__sgn__mult__iff,axiom,
! [L: int,R2: int,K: int] :
( ( dvd_dvd @ int @ L @ ( times_times @ int @ ( sgn_sgn @ int @ R2 ) @ K ) )
= ( ( dvd_dvd @ int @ L @ K )
| ( R2
= ( zero_zero @ int ) ) ) ) ).
% dvd_sgn_mult_iff
thf(fact_2830_mult__sgn__dvd__iff,axiom,
! [L: int,R2: int,K: int] :
( ( dvd_dvd @ int @ ( times_times @ int @ L @ ( sgn_sgn @ int @ R2 ) ) @ K )
= ( ( dvd_dvd @ int @ L @ K )
& ( ( R2
= ( zero_zero @ int ) )
=> ( K
= ( zero_zero @ int ) ) ) ) ) ).
% mult_sgn_dvd_iff
thf(fact_2831_sgn__mult__dvd__iff,axiom,
! [R2: int,L: int,K: int] :
( ( dvd_dvd @ int @ ( times_times @ int @ ( sgn_sgn @ int @ R2 ) @ L ) @ K )
= ( ( dvd_dvd @ int @ L @ K )
& ( ( R2
= ( zero_zero @ int ) )
=> ( K
= ( zero_zero @ int ) ) ) ) ) ).
% sgn_mult_dvd_iff
thf(fact_2832_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_2833_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_2834_sgn__of__nat,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N2: nat] :
( ( sgn_sgn @ A @ ( semiring_1_of_nat @ A @ N2 ) )
= ( zero_neq_one_of_bool @ A @ ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ) ).
% sgn_of_nat
thf(fact_2835_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_2836_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_2837_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_2838_sin__add,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A,Y: A] :
( ( sin @ A @ ( plus_plus @ A @ X @ Y ) )
= ( plus_plus @ A @ ( times_times @ A @ ( sin @ A @ X ) @ ( cos @ A @ Y ) ) @ ( times_times @ A @ ( cos @ A @ X ) @ ( sin @ A @ Y ) ) ) ) ) ).
% sin_add
thf(fact_2839_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_2840_fact__ge__self,axiom,
! [N2: nat] : ( ord_less_eq @ nat @ N2 @ ( semiring_char_0_fact @ nat @ N2 ) ) ).
% fact_ge_self
thf(fact_2841_fact__mono__nat,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq @ nat @ M2 @ N2 )
=> ( ord_less_eq @ nat @ ( semiring_char_0_fact @ nat @ M2 ) @ ( semiring_char_0_fact @ nat @ N2 ) ) ) ).
% fact_mono_nat
thf(fact_2842_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_2843_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_2844_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_2845_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_2846_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_2847_fact__nonzero,axiom,
! [A: $tType] :
( ( ( semiring_char_0 @ A )
& ( semiri3467727345109120633visors @ A ) )
=> ! [N2: nat] :
( ( semiring_char_0_fact @ A @ N2 )
!= ( zero_zero @ A ) ) ) ).
% fact_nonzero
thf(fact_2848_summable__const__iff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [C2: A] :
( ( summable @ A
@ ^ [Uu3: nat] : C2 )
= ( C2
= ( zero_zero @ A ) ) ) ) ).
% summable_const_iff
thf(fact_2849_summable__comparison__test,axiom,
! [A: $tType] :
( ( real_Vector_banach @ A )
=> ! [F2: nat > A,G: nat > real] :
( ? [N8: nat] :
! [N3: nat] :
( ( ord_less_eq @ nat @ N8 @ N3 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( F2 @ N3 ) ) @ ( G @ N3 ) ) )
=> ( ( summable @ real @ G )
=> ( summable @ A @ F2 ) ) ) ) ).
% summable_comparison_test
thf(fact_2850_summable__comparison__test_H,axiom,
! [A: $tType] :
( ( real_Vector_banach @ A )
=> ! [G: nat > real,N6: nat,F2: nat > A] :
( ( summable @ real @ G )
=> ( ! [N3: nat] :
( ( ord_less_eq @ nat @ N6 @ N3 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( F2 @ N3 ) ) @ ( G @ N3 ) ) )
=> ( summable @ A @ F2 ) ) ) ) ).
% summable_comparison_test'
thf(fact_2851_summable__add,axiom,
! [A: $tType] :
( ( ( topolo5987344860129210374id_add @ A )
& ( topological_t2_space @ A ) )
=> ! [F2: nat > A,G: nat > A] :
( ( summable @ A @ F2 )
=> ( ( summable @ A @ G )
=> ( summable @ A
@ ^ [N: nat] : ( plus_plus @ A @ ( F2 @ N ) @ ( G @ N ) ) ) ) ) ) ).
% summable_add
thf(fact_2852_cos__diff,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A,Y: A] :
( ( cos @ A @ ( minus_minus @ A @ X @ Y ) )
= ( plus_plus @ A @ ( times_times @ A @ ( cos @ A @ X ) @ ( cos @ A @ Y ) ) @ ( times_times @ A @ ( sin @ A @ X ) @ ( sin @ A @ Y ) ) ) ) ) ).
% cos_diff
thf(fact_2853_cos__add,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A,Y: A] :
( ( cos @ A @ ( plus_plus @ A @ X @ Y ) )
= ( minus_minus @ A @ ( times_times @ A @ ( cos @ A @ X ) @ ( cos @ A @ Y ) ) @ ( times_times @ A @ ( sin @ A @ X ) @ ( sin @ A @ Y ) ) ) ) ) ).
% cos_add
thf(fact_2854_summable__Suc__iff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A] :
( ( summable @ A
@ ^ [N: nat] : ( F2 @ ( suc @ N ) ) )
= ( summable @ A @ F2 ) ) ) ).
% summable_Suc_iff
thf(fact_2855_summable__ignore__initial__segment,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A,K: nat] :
( ( summable @ A @ F2 )
=> ( summable @ A
@ ^ [N: nat] : ( F2 @ ( plus_plus @ nat @ N @ K ) ) ) ) ) ).
% summable_ignore_initial_segment
thf(fact_2856_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_2857_suminf__le,axiom,
! [A: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F2: nat > A,G: nat > A] :
( ! [N3: nat] : ( ord_less_eq @ A @ ( F2 @ N3 ) @ ( G @ N3 ) )
=> ( ( summable @ A @ F2 )
=> ( ( summable @ A @ G )
=> ( ord_less_eq @ A @ ( suminf @ A @ F2 ) @ ( suminf @ A @ G ) ) ) ) ) ) ).
% suminf_le
thf(fact_2858_summable__finite,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [N6: set @ nat,F2: nat > A] :
( ( finite_finite2 @ nat @ N6 )
=> ( ! [N3: nat] :
( ~ ( member @ nat @ N3 @ N6 )
=> ( ( F2 @ N3 )
= ( zero_zero @ A ) ) )
=> ( summable @ A @ F2 ) ) ) ) ).
% summable_finite
thf(fact_2859_fact__less__mono__nat,axiom,
! [M2: nat,N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ M2 )
=> ( ( ord_less @ nat @ M2 @ N2 )
=> ( ord_less @ nat @ ( semiring_char_0_fact @ nat @ M2 ) @ ( semiring_char_0_fact @ nat @ N2 ) ) ) ) ).
% fact_less_mono_nat
thf(fact_2860_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_2861_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_2862_fact__ge__zero,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N2: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( semiring_char_0_fact @ A @ N2 ) ) ) ).
% fact_ge_zero
thf(fact_2863_fact__gt__zero,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N2: nat] : ( ord_less @ A @ ( zero_zero @ A ) @ ( semiring_char_0_fact @ A @ N2 ) ) ) ).
% fact_gt_zero
thf(fact_2864_fact__not__neg,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N2: nat] :
~ ( ord_less @ A @ ( semiring_char_0_fact @ A @ N2 ) @ ( zero_zero @ A ) ) ) ).
% fact_not_neg
thf(fact_2865_int__sgnE,axiom,
! [K: int] :
~ ! [N3: nat,L4: int] :
( K
!= ( times_times @ int @ ( sgn_sgn @ int @ L4 ) @ ( semiring_1_of_nat @ int @ N3 ) ) ) ).
% int_sgnE
thf(fact_2866_fact__ge__1,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N2: nat] : ( ord_less_eq @ A @ ( one_one @ A ) @ ( semiring_char_0_fact @ A @ N2 ) ) ) ).
% fact_ge_1
thf(fact_2867_fact__mono,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [M2: nat,N2: nat] :
( ( ord_less_eq @ nat @ M2 @ N2 )
=> ( ord_less_eq @ A @ ( semiring_char_0_fact @ A @ M2 ) @ ( semiring_char_0_fact @ A @ N2 ) ) ) ) ).
% fact_mono
thf(fact_2868_summable__mult__D,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [C2: A,F2: nat > A] :
( ( summable @ A
@ ^ [N: nat] : ( times_times @ A @ C2 @ ( F2 @ N ) ) )
=> ( ( C2
!= ( zero_zero @ A ) )
=> ( summable @ A @ F2 ) ) ) ) ).
% summable_mult_D
thf(fact_2869_fact__dvd,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N2: nat,M2: nat] :
( ( ord_less_eq @ nat @ N2 @ M2 )
=> ( dvd_dvd @ A @ ( semiring_char_0_fact @ A @ N2 ) @ ( semiring_char_0_fact @ A @ M2 ) ) ) ) ).
% fact_dvd
thf(fact_2870_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_2871_suminf__add,axiom,
! [A: $tType] :
( ( ( topolo5987344860129210374id_add @ A )
& ( topological_t2_space @ A ) )
=> ! [F2: nat > A,G: nat > A] :
( ( summable @ A @ F2 )
=> ( ( summable @ A @ G )
=> ( ( plus_plus @ A @ ( suminf @ A @ F2 ) @ ( suminf @ A @ G ) )
= ( suminf @ A
@ ^ [N: nat] : ( plus_plus @ A @ ( F2 @ N ) @ ( G @ N ) ) ) ) ) ) ) ).
% suminf_add
thf(fact_2872_suminf__nonneg,axiom,
! [A: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F2: nat > A] :
( ( summable @ A @ F2 )
=> ( ! [N3: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ N3 ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( suminf @ A @ F2 ) ) ) ) ) ).
% suminf_nonneg
thf(fact_2873_suminf__eq__zero__iff,axiom,
! [A: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F2: nat > A] :
( ( summable @ A @ F2 )
=> ( ! [N3: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ N3 ) )
=> ( ( ( suminf @ A @ F2 )
= ( zero_zero @ A ) )
= ( ! [N: nat] :
( ( F2 @ N )
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% suminf_eq_zero_iff
thf(fact_2874_suminf__pos,axiom,
! [A: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F2: nat > A] :
( ( summable @ A @ F2 )
=> ( ! [N3: nat] : ( ord_less @ A @ ( zero_zero @ A ) @ ( F2 @ N3 ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( suminf @ A @ F2 ) ) ) ) ) ).
% suminf_pos
thf(fact_2875_fact__ge__Suc__0__nat,axiom,
! [N2: nat] : ( ord_less_eq @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( semiring_char_0_fact @ nat @ N2 ) ) ).
% fact_ge_Suc_0_nat
thf(fact_2876_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_2877_dvd__fact,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq @ nat @ ( one_one @ nat ) @ M2 )
=> ( ( ord_less_eq @ nat @ M2 @ N2 )
=> ( dvd_dvd @ nat @ M2 @ ( semiring_char_0_fact @ nat @ N2 ) ) ) ) ).
% dvd_fact
thf(fact_2878_summable__0__powser,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [F2: nat > A] :
( summable @ A
@ ^ [N: nat] : ( times_times @ A @ ( F2 @ N ) @ ( power_power @ A @ ( zero_zero @ A ) @ N ) ) ) ) ).
% summable_0_powser
thf(fact_2879_summable__zero__power_H,axiom,
! [A: $tType] :
( ( ( ring_1 @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [F2: nat > A] :
( summable @ A
@ ^ [N: nat] : ( times_times @ A @ ( F2 @ N ) @ ( power_power @ A @ ( zero_zero @ A ) @ N ) ) ) ) ).
% summable_zero_power'
thf(fact_2880_summable__powser__split__head,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [F2: nat > A,Z3: A] :
( ( summable @ A
@ ^ [N: nat] : ( times_times @ A @ ( F2 @ ( suc @ N ) ) @ ( power_power @ A @ Z3 @ N ) ) )
= ( summable @ A
@ ^ [N: nat] : ( times_times @ A @ ( F2 @ N ) @ ( power_power @ A @ Z3 @ N ) ) ) ) ) ).
% summable_powser_split_head
thf(fact_2881_powser__split__head_I3_J,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V8999393235501362500lgebra @ A ) )
=> ! [F2: nat > A,Z3: A] :
( ( summable @ A
@ ^ [N: nat] : ( times_times @ A @ ( F2 @ N ) @ ( power_power @ A @ Z3 @ N ) ) )
=> ( summable @ A
@ ^ [N: nat] : ( times_times @ A @ ( F2 @ ( suc @ N ) ) @ ( power_power @ A @ Z3 @ N ) ) ) ) ) ).
% powser_split_head(3)
thf(fact_2882_summable__powser__ignore__initial__segment,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [F2: nat > A,M2: nat,Z3: A] :
( ( summable @ A
@ ^ [N: nat] : ( times_times @ A @ ( F2 @ ( plus_plus @ nat @ N @ M2 ) ) @ ( power_power @ A @ Z3 @ N ) ) )
= ( summable @ A
@ ^ [N: nat] : ( times_times @ A @ ( F2 @ N ) @ ( power_power @ A @ Z3 @ N ) ) ) ) ) ).
% summable_powser_ignore_initial_segment
thf(fact_2883_fact__less__mono,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [M2: nat,N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ M2 )
=> ( ( ord_less @ nat @ M2 @ N2 )
=> ( ord_less @ A @ ( semiring_char_0_fact @ A @ M2 ) @ ( semiring_char_0_fact @ A @ N2 ) ) ) ) ) ).
% fact_less_mono
thf(fact_2884_sin__times__sin,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [W2: A,Z3: A] :
( ( times_times @ A @ ( sin @ A @ W2 ) @ ( sin @ A @ Z3 ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( cos @ A @ ( minus_minus @ A @ W2 @ Z3 ) ) @ ( cos @ A @ ( plus_plus @ A @ W2 @ Z3 ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% sin_times_sin
thf(fact_2885_sin__times__cos,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [W2: A,Z3: A] :
( ( times_times @ A @ ( sin @ A @ W2 ) @ ( cos @ A @ Z3 ) )
= ( divide_divide @ A @ ( plus_plus @ A @ ( sin @ A @ ( plus_plus @ A @ W2 @ Z3 ) ) @ ( sin @ A @ ( minus_minus @ A @ W2 @ Z3 ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% sin_times_cos
thf(fact_2886_cos__times__sin,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [W2: A,Z3: A] :
( ( times_times @ A @ ( cos @ A @ W2 ) @ ( sin @ A @ Z3 ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( sin @ A @ ( plus_plus @ A @ W2 @ Z3 ) ) @ ( sin @ A @ ( minus_minus @ A @ W2 @ Z3 ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% cos_times_sin
thf(fact_2887_sin__plus__sin,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [W2: A,Z3: A] :
( ( plus_plus @ A @ ( sin @ A @ W2 ) @ ( sin @ A @ Z3 ) )
= ( times_times @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( sin @ A @ ( divide_divide @ A @ ( plus_plus @ A @ W2 @ Z3 ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) @ ( cos @ A @ ( divide_divide @ A @ ( minus_minus @ A @ W2 @ Z3 ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% sin_plus_sin
thf(fact_2888_sin__diff__sin,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [W2: A,Z3: A] :
( ( minus_minus @ A @ ( sin @ A @ W2 ) @ ( sin @ A @ Z3 ) )
= ( times_times @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( sin @ A @ ( divide_divide @ A @ ( minus_minus @ A @ W2 @ Z3 ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) @ ( cos @ A @ ( divide_divide @ A @ ( plus_plus @ A @ W2 @ Z3 ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% sin_diff_sin
thf(fact_2889_cos__diff__cos,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [W2: A,Z3: A] :
( ( minus_minus @ A @ ( cos @ A @ W2 ) @ ( cos @ A @ Z3 ) )
= ( times_times @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( sin @ A @ ( divide_divide @ A @ ( plus_plus @ A @ W2 @ Z3 ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) @ ( sin @ A @ ( divide_divide @ A @ ( minus_minus @ A @ Z3 @ W2 ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% cos_diff_cos
thf(fact_2890_fact__fact__dvd__fact,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [K: nat,N2: nat] : ( dvd_dvd @ A @ ( times_times @ A @ ( semiring_char_0_fact @ A @ K ) @ ( semiring_char_0_fact @ A @ N2 ) ) @ ( semiring_char_0_fact @ A @ ( plus_plus @ nat @ K @ N2 ) ) ) ) ).
% fact_fact_dvd_fact
thf(fact_2891_fact__mod,axiom,
! [A: $tType] :
( ( ( linordered_semidom @ A )
& ( semidom_modulo @ A ) )
=> ! [M2: nat,N2: nat] :
( ( ord_less_eq @ nat @ M2 @ N2 )
=> ( ( modulo_modulo @ A @ ( semiring_char_0_fact @ A @ N2 ) @ ( semiring_char_0_fact @ A @ M2 ) )
= ( zero_zero @ A ) ) ) ) ).
% fact_mod
thf(fact_2892_fact__le__power,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N2: nat] : ( ord_less_eq @ A @ ( semiring_char_0_fact @ A @ N2 ) @ ( semiring_1_of_nat @ A @ ( power_power @ nat @ N2 @ N2 ) ) ) ) ).
% fact_le_power
thf(fact_2893_summable__norm__comparison__test,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A,G: nat > real] :
( ? [N8: nat] :
! [N3: nat] :
( ( ord_less_eq @ nat @ N8 @ N3 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( F2 @ N3 ) ) @ ( G @ N3 ) ) )
=> ( ( summable @ real @ G )
=> ( summable @ real
@ ^ [N: nat] : ( real_V7770717601297561774m_norm @ A @ ( F2 @ N ) ) ) ) ) ) ).
% summable_norm_comparison_test
thf(fact_2894_suminf__pos2,axiom,
! [A: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F2: nat > A,I: nat] :
( ( summable @ A @ F2 )
=> ( ! [N3: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ N3 ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ ( F2 @ I ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( suminf @ A @ F2 ) ) ) ) ) ) ).
% suminf_pos2
thf(fact_2895_suminf__pos__iff,axiom,
! [A: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F2: nat > A] :
( ( summable @ A @ F2 )
=> ( ! [N3: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ N3 ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ ( suminf @ A @ F2 ) )
= ( ? [I3: nat] : ( ord_less @ A @ ( zero_zero @ A ) @ ( F2 @ I3 ) ) ) ) ) ) ) ).
% suminf_pos_iff
thf(fact_2896_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_2897_suminf__le__const,axiom,
! [A: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F2: nat > A,X: A] :
( ( summable @ A @ F2 )
=> ( ! [N3: nat] : ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_ord_lessThan @ nat @ N3 ) ) @ X )
=> ( ord_less_eq @ A @ ( suminf @ A @ F2 ) @ X ) ) ) ) ).
% suminf_le_const
thf(fact_2898_fact__diff__Suc,axiom,
! [N2: nat,M2: nat] :
( ( ord_less @ nat @ N2 @ ( suc @ M2 ) )
=> ( ( semiring_char_0_fact @ nat @ ( minus_minus @ nat @ ( suc @ M2 ) @ N2 ) )
= ( times_times @ nat @ ( minus_minus @ nat @ ( suc @ M2 ) @ N2 ) @ ( semiring_char_0_fact @ nat @ ( minus_minus @ nat @ M2 @ N2 ) ) ) ) ) ).
% fact_diff_Suc
thf(fact_2899_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_2900_sgn__if,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( ( sgn_sgn @ A )
= ( ^ [X4: A] :
( if @ A
@ ( X4
= ( zero_zero @ A ) )
@ ( zero_zero @ A )
@ ( if @ A @ ( ord_less @ A @ ( zero_zero @ A ) @ X4 ) @ ( one_one @ A ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ) ) ).
% sgn_if
thf(fact_2901_zsgn__def,axiom,
( ( sgn_sgn @ int )
= ( ^ [I3: int] :
( if @ int
@ ( I3
= ( zero_zero @ int ) )
@ ( zero_zero @ int )
@ ( if @ int @ ( ord_less @ int @ ( zero_zero @ int ) @ I3 ) @ ( one_one @ int ) @ ( uminus_uminus @ int @ ( one_one @ int ) ) ) ) ) ) ).
% zsgn_def
thf(fact_2902_fact__div__fact__le__pow,axiom,
! [R2: nat,N2: nat] :
( ( ord_less_eq @ nat @ R2 @ N2 )
=> ( ord_less_eq @ nat @ ( divide_divide @ nat @ ( semiring_char_0_fact @ nat @ N2 ) @ ( semiring_char_0_fact @ nat @ ( minus_minus @ nat @ N2 @ R2 ) ) ) @ ( power_power @ nat @ N2 @ R2 ) ) ) ).
% fact_div_fact_le_pow
thf(fact_2903_binomial__fact__lemma,axiom,
! [K: nat,N2: nat] :
( ( ord_less_eq @ nat @ K @ N2 )
=> ( ( times_times @ nat @ ( times_times @ nat @ ( semiring_char_0_fact @ nat @ K ) @ ( semiring_char_0_fact @ nat @ ( minus_minus @ nat @ N2 @ K ) ) ) @ ( binomial @ N2 @ K ) )
= ( semiring_char_0_fact @ nat @ N2 ) ) ) ).
% binomial_fact_lemma
thf(fact_2904_summableI__nonneg__bounded,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( ordere6911136660526730532id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F2: nat > A,X: A] :
( ! [N3: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ N3 ) )
=> ( ! [N3: nat] : ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_ord_lessThan @ nat @ N3 ) ) @ X )
=> ( summable @ A @ F2 ) ) ) ) ).
% summableI_nonneg_bounded
thf(fact_2905_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_2906_bounded__imp__summable,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( linord2810124833399127020strict @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [A3: nat > A,B2: A] :
( ! [N3: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( A3 @ N3 ) )
=> ( ! [N3: nat] : ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ A3 @ ( set_ord_atMost @ nat @ N3 ) ) @ B2 )
=> ( summable @ A @ A3 ) ) ) ) ).
% bounded_imp_summable
thf(fact_2907_suminf__split__head,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A] :
( ( summable @ A @ F2 )
=> ( ( suminf @ A
@ ^ [N: nat] : ( F2 @ ( suc @ N ) ) )
= ( minus_minus @ A @ ( suminf @ A @ F2 ) @ ( F2 @ ( zero_zero @ nat ) ) ) ) ) ) ).
% suminf_split_head
thf(fact_2908_choose__dvd,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [K: nat,N2: nat] :
( ( ord_less_eq @ nat @ K @ N2 )
=> ( dvd_dvd @ A @ ( times_times @ A @ ( semiring_char_0_fact @ A @ K ) @ ( semiring_char_0_fact @ A @ ( minus_minus @ nat @ N2 @ K ) ) ) @ ( semiring_char_0_fact @ A @ N2 ) ) ) ) ).
% choose_dvd
thf(fact_2909_fact__eq__fact__times,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_eq @ nat @ N2 @ M2 )
=> ( ( semiring_char_0_fact @ nat @ M2 )
= ( times_times @ nat @ ( semiring_char_0_fact @ nat @ N2 )
@ ( groups7121269368397514597t_prod @ nat @ nat
@ ^ [X4: nat] : X4
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ N2 ) @ M2 ) ) ) ) ) ).
% fact_eq_fact_times
thf(fact_2910_sin__expansion__lemma,axiom,
! [X: real,M2: nat] :
( ( sin @ real @ ( plus_plus @ real @ X @ ( divide_divide @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ ( suc @ M2 ) ) @ pi ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) )
= ( cos @ real @ ( plus_plus @ real @ X @ ( divide_divide @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ M2 ) @ pi ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ).
% sin_expansion_lemma
thf(fact_2911_sum__le__suminf,axiom,
! [A: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F2: nat > A,I6: set @ nat] :
( ( summable @ A @ F2 )
=> ( ( finite_finite2 @ nat @ I6 )
=> ( ! [N3: nat] :
( ( member @ nat @ N3 @ ( uminus_uminus @ ( set @ nat ) @ I6 ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ N3 ) ) )
=> ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ I6 ) @ ( suminf @ A @ F2 ) ) ) ) ) ) ).
% sum_le_suminf
thf(fact_2912_cos__expansion__lemma,axiom,
! [X: real,M2: nat] :
( ( cos @ real @ ( plus_plus @ real @ X @ ( divide_divide @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ ( suc @ M2 ) ) @ 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 @ M2 ) @ pi ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% cos_expansion_lemma
thf(fact_2913_binomial__altdef__nat,axiom,
! [K: nat,N2: nat] :
( ( ord_less_eq @ nat @ K @ N2 )
=> ( ( binomial @ N2 @ K )
= ( divide_divide @ nat @ ( semiring_char_0_fact @ nat @ N2 ) @ ( times_times @ nat @ ( semiring_char_0_fact @ nat @ K ) @ ( semiring_char_0_fact @ nat @ ( minus_minus @ nat @ N2 @ K ) ) ) ) ) ) ).
% binomial_altdef_nat
thf(fact_2914_suminf__split__initial__segment,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A,K: nat] :
( ( summable @ A @ F2 )
=> ( ( suminf @ A @ F2 )
= ( plus_plus @ A
@ ( suminf @ A
@ ^ [N: nat] : ( F2 @ ( plus_plus @ nat @ N @ K ) ) )
@ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_ord_lessThan @ nat @ K ) ) ) ) ) ) ).
% suminf_split_initial_segment
thf(fact_2915_suminf__minus__initial__segment,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A,K: nat] :
( ( summable @ A @ F2 )
=> ( ( suminf @ A
@ ^ [N: nat] : ( F2 @ ( plus_plus @ nat @ N @ K ) ) )
= ( minus_minus @ A @ ( suminf @ A @ F2 ) @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_ord_lessThan @ nat @ K ) ) ) ) ) ) ).
% suminf_minus_initial_segment
thf(fact_2916_fact__div__fact,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_eq @ nat @ N2 @ M2 )
=> ( ( divide_divide @ nat @ ( semiring_char_0_fact @ nat @ M2 ) @ ( semiring_char_0_fact @ nat @ N2 ) )
= ( groups7121269368397514597t_prod @ nat @ nat
@ ^ [X4: nat] : X4
@ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ N2 @ ( one_one @ nat ) ) @ M2 ) ) ) ) ).
% fact_div_fact
thf(fact_2917_sum__less__suminf,axiom,
! [A: $tType] :
( ( ( ordere8940638589300402666id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F2: nat > A,N2: nat] :
( ( summable @ A @ F2 )
=> ( ! [M3: nat] :
( ( ord_less_eq @ nat @ N2 @ M3 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( F2 @ M3 ) ) )
=> ( ord_less @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_ord_lessThan @ nat @ N2 ) ) @ ( suminf @ A @ F2 ) ) ) ) ) ).
% sum_less_suminf
thf(fact_2918_powser__split__head_I1_J,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V8999393235501362500lgebra @ A ) )
=> ! [F2: nat > A,Z3: A] :
( ( summable @ A
@ ^ [N: nat] : ( times_times @ A @ ( F2 @ N ) @ ( power_power @ A @ Z3 @ N ) ) )
=> ( ( suminf @ A
@ ^ [N: nat] : ( times_times @ A @ ( F2 @ N ) @ ( power_power @ A @ Z3 @ N ) ) )
= ( plus_plus @ A @ ( F2 @ ( zero_zero @ nat ) )
@ ( times_times @ A
@ ( suminf @ A
@ ^ [N: nat] : ( times_times @ A @ ( F2 @ ( suc @ N ) ) @ ( power_power @ A @ Z3 @ N ) ) )
@ Z3 ) ) ) ) ) ).
% powser_split_head(1)
thf(fact_2919_powser__split__head_I2_J,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V8999393235501362500lgebra @ A ) )
=> ! [F2: nat > A,Z3: A] :
( ( summable @ A
@ ^ [N: nat] : ( times_times @ A @ ( F2 @ N ) @ ( power_power @ A @ Z3 @ N ) ) )
=> ( ( times_times @ A
@ ( suminf @ A
@ ^ [N: nat] : ( times_times @ A @ ( F2 @ ( suc @ N ) ) @ ( power_power @ A @ Z3 @ N ) ) )
@ Z3 )
= ( minus_minus @ A
@ ( suminf @ A
@ ^ [N: nat] : ( times_times @ A @ ( F2 @ N ) @ ( power_power @ A @ Z3 @ N ) ) )
@ ( F2 @ ( zero_zero @ nat ) ) ) ) ) ) ).
% powser_split_head(2)
thf(fact_2920_summable__partial__sum__bound,axiom,
! [A: $tType] :
( ( real_Vector_banach @ A )
=> ! [F2: nat > A,E2: real] :
( ( summable @ A @ F2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ E2 )
=> ~ ! [N9: nat] :
~ ! [M5: nat] :
( ( ord_less_eq @ nat @ N9 @ M5 )
=> ! [N7: nat] : ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_or1337092689740270186AtMost @ nat @ M5 @ N7 ) ) ) @ E2 ) ) ) ) ) ).
% summable_partial_sum_bound
thf(fact_2921_suminf__exist__split,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [R2: real,F2: nat > A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R2 )
=> ( ( summable @ A @ F2 )
=> ? [N9: nat] :
! [N7: nat] :
( ( ord_less_eq @ nat @ N9 @ N7 )
=> ( ord_less @ real
@ ( real_V7770717601297561774m_norm @ A
@ ( suminf @ A
@ ^ [I3: nat] : ( F2 @ ( plus_plus @ nat @ I3 @ N7 ) ) ) )
@ R2 ) ) ) ) ) ).
% suminf_exist_split
thf(fact_2922_sin__pi__divide__n__ge__0,axiom,
! [N2: nat] :
( ( N2
!= ( zero_zero @ nat ) )
=> ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( sin @ real @ ( divide_divide @ real @ pi @ ( semiring_1_of_nat @ real @ N2 ) ) ) ) ) ).
% sin_pi_divide_n_ge_0
thf(fact_2923_sin__paired,axiom,
! [X: real] :
( sums @ real
@ ^ [N: nat] : ( times_times @ real @ ( divide_divide @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ N ) @ ( semiring_char_0_fact @ real @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ ( one_one @ nat ) ) ) ) @ ( power_power @ real @ X @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ ( one_one @ nat ) ) ) )
@ ( sin @ real @ X ) ) ).
% sin_paired
thf(fact_2924_fact__num__eq__if,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ( ( semiring_char_0_fact @ A )
= ( ^ [M: nat] :
( if @ A
@ ( M
= ( zero_zero @ nat ) )
@ ( one_one @ A )
@ ( times_times @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_char_0_fact @ A @ ( minus_minus @ nat @ M @ ( one_one @ nat ) ) ) ) ) ) ) ) ).
% fact_num_eq_if
thf(fact_2925_fact__reduce,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( semiring_char_0_fact @ A @ N2 )
= ( times_times @ A @ ( semiring_1_of_nat @ A @ N2 ) @ ( semiring_char_0_fact @ A @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) ) ) ) ) ) ).
% fact_reduce
thf(fact_2926_fact__binomial,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K: nat,N2: nat] :
( ( ord_less_eq @ nat @ K @ N2 )
=> ( ( times_times @ A @ ( semiring_char_0_fact @ A @ K ) @ ( semiring_1_of_nat @ A @ ( binomial @ N2 @ K ) ) )
= ( divide_divide @ A @ ( semiring_char_0_fact @ A @ N2 ) @ ( semiring_char_0_fact @ A @ ( minus_minus @ nat @ N2 @ K ) ) ) ) ) ) ).
% fact_binomial
thf(fact_2927_binomial__fact,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K: nat,N2: nat] :
( ( ord_less_eq @ nat @ K @ N2 )
=> ( ( semiring_1_of_nat @ A @ ( binomial @ N2 @ K ) )
= ( divide_divide @ A @ ( semiring_char_0_fact @ A @ N2 ) @ ( times_times @ A @ ( semiring_char_0_fact @ A @ K ) @ ( semiring_char_0_fact @ A @ ( minus_minus @ nat @ N2 @ K ) ) ) ) ) ) ) ).
% binomial_fact
thf(fact_2928_cos__times__cos,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [W2: A,Z3: A] :
( ( times_times @ A @ ( cos @ A @ W2 ) @ ( cos @ A @ Z3 ) )
= ( divide_divide @ A @ ( plus_plus @ A @ ( cos @ A @ ( minus_minus @ A @ W2 @ Z3 ) ) @ ( cos @ A @ ( plus_plus @ A @ W2 @ Z3 ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% cos_times_cos
thf(fact_2929_cos__plus__cos,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [W2: A,Z3: A] :
( ( plus_plus @ A @ ( cos @ A @ W2 ) @ ( cos @ A @ Z3 ) )
= ( times_times @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( cos @ A @ ( divide_divide @ A @ ( plus_plus @ A @ W2 @ Z3 ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) @ ( cos @ A @ ( divide_divide @ A @ ( minus_minus @ A @ W2 @ Z3 ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% cos_plus_cos
thf(fact_2930_summable__ratio__test,axiom,
! [A: $tType] :
( ( real_Vector_banach @ A )
=> ! [C2: real,N6: nat,F2: nat > A] :
( ( ord_less @ real @ C2 @ ( one_one @ real ) )
=> ( ! [N3: nat] :
( ( ord_less_eq @ nat @ N6 @ N3 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( F2 @ ( suc @ N3 ) ) ) @ ( times_times @ real @ C2 @ ( real_V7770717601297561774m_norm @ A @ ( F2 @ N3 ) ) ) ) )
=> ( summable @ A @ F2 ) ) ) ) ).
% summable_ratio_test
thf(fact_2931_sum__less__suminf2,axiom,
! [A: $tType] :
( ( ( ordere8940638589300402666id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F2: nat > A,N2: nat,I: nat] :
( ( summable @ A @ F2 )
=> ( ! [M3: nat] :
( ( ord_less_eq @ nat @ N2 @ M3 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ M3 ) ) )
=> ( ( ord_less_eq @ nat @ N2 @ I )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ ( F2 @ I ) )
=> ( ord_less @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_ord_lessThan @ nat @ N2 ) ) @ ( suminf @ A @ F2 ) ) ) ) ) ) ) ).
% sum_less_suminf2
thf(fact_2932_gbinomial__pochhammer_H,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( ( gbinomial @ A )
= ( ^ [A5: A,K2: nat] : ( divide_divide @ A @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ ( minus_minus @ A @ A5 @ ( semiring_1_of_nat @ A @ K2 ) ) @ ( one_one @ A ) ) @ K2 ) @ ( semiring_char_0_fact @ A @ K2 ) ) ) ) ) ).
% gbinomial_pochhammer'
thf(fact_2933_Maclaurin__zero,axiom,
! [A: $tType] :
( ( zero @ A )
=> ! [X: real,N2: nat,Diff: nat > A > real] :
( ( X
= ( zero_zero @ real ) )
=> ( ( N2
!= ( zero_zero @ nat ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M @ ( zero_zero @ A ) ) @ ( semiring_char_0_fact @ real @ M ) ) @ ( power_power @ real @ X @ M ) )
@ ( set_ord_lessThan @ nat @ N2 ) )
= ( Diff @ ( zero_zero @ nat ) @ ( zero_zero @ A ) ) ) ) ) ) ).
% Maclaurin_zero
thf(fact_2934_Maclaurin__cos__expansion2,axiom,
! [X: real,N2: nat] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ? [T7: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ T7 )
& ( ord_less @ real @ T7 @ X )
& ( ( cos @ real @ X )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M: nat] : ( times_times @ real @ ( cos_coeff @ M ) @ ( power_power @ real @ X @ M ) )
@ ( set_ord_lessThan @ nat @ N2 ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( cos @ real @ ( plus_plus @ real @ T7 @ ( times_times @ real @ ( times_times @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( semiring_1_of_nat @ real @ N2 ) ) @ pi ) ) ) @ ( semiring_char_0_fact @ real @ N2 ) ) @ ( power_power @ real @ X @ N2 ) ) ) ) ) ) ) ).
% Maclaurin_cos_expansion2
thf(fact_2935_Maclaurin__minus__cos__expansion,axiom,
! [N2: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( ord_less @ real @ X @ ( zero_zero @ real ) )
=> ? [T7: real] :
( ( ord_less @ real @ X @ T7 )
& ( ord_less @ real @ T7 @ ( zero_zero @ real ) )
& ( ( cos @ real @ X )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M: nat] : ( times_times @ real @ ( cos_coeff @ M ) @ ( power_power @ real @ X @ M ) )
@ ( set_ord_lessThan @ nat @ N2 ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( cos @ real @ ( plus_plus @ real @ T7 @ ( times_times @ real @ ( times_times @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( semiring_1_of_nat @ real @ N2 ) ) @ pi ) ) ) @ ( semiring_char_0_fact @ real @ N2 ) ) @ ( power_power @ real @ X @ N2 ) ) ) ) ) ) ) ).
% Maclaurin_minus_cos_expansion
thf(fact_2936_sin__pi__divide__n__gt__0,axiom,
! [N2: nat] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( sin @ real @ ( divide_divide @ real @ pi @ ( semiring_1_of_nat @ real @ N2 ) ) ) ) ) ).
% sin_pi_divide_n_gt_0
thf(fact_2937_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
@ ^ [I3: nat] : ( minus_minus @ A @ A3 @ ( semiring_1_of_nat @ A @ I3 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ K ) )
@ ( semiring_char_0_fact @ A @ ( suc @ K ) ) ) ) ) ).
% gbinomial_Suc
thf(fact_2938_divide__int__unfold,axiom,
! [L: int,K: int,N2: nat,M2: nat] :
( ( ( ( ( sgn_sgn @ int @ L )
= ( zero_zero @ int ) )
| ( ( sgn_sgn @ int @ K )
= ( zero_zero @ int ) )
| ( N2
= ( zero_zero @ nat ) ) )
=> ( ( divide_divide @ int @ ( times_times @ int @ ( sgn_sgn @ int @ K ) @ ( semiring_1_of_nat @ int @ M2 ) ) @ ( times_times @ int @ ( sgn_sgn @ int @ L ) @ ( semiring_1_of_nat @ int @ N2 ) ) )
= ( zero_zero @ int ) ) )
& ( ~ ( ( ( sgn_sgn @ int @ L )
= ( zero_zero @ int ) )
| ( ( sgn_sgn @ int @ K )
= ( zero_zero @ int ) )
| ( N2
= ( 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 @ M2 ) ) @ ( times_times @ int @ ( sgn_sgn @ int @ L ) @ ( semiring_1_of_nat @ int @ N2 ) ) )
= ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ M2 @ N2 ) ) ) )
& ( ( ( sgn_sgn @ int @ K )
!= ( sgn_sgn @ int @ L ) )
=> ( ( divide_divide @ int @ ( times_times @ int @ ( sgn_sgn @ int @ K ) @ ( semiring_1_of_nat @ int @ M2 ) ) @ ( times_times @ int @ ( sgn_sgn @ int @ L ) @ ( semiring_1_of_nat @ int @ N2 ) ) )
= ( uminus_uminus @ int
@ ( semiring_1_of_nat @ int
@ ( plus_plus @ nat @ ( divide_divide @ nat @ M2 @ N2 )
@ ( zero_neq_one_of_bool @ nat
@ ~ ( dvd_dvd @ nat @ N2 @ M2 ) ) ) ) ) ) ) ) ) ) ).
% divide_int_unfold
thf(fact_2939_Maclaurin__sin__expansion3,axiom,
! [N2: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ? [T7: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ T7 )
& ( ord_less @ real @ T7 @ X )
& ( ( sin @ real @ X )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M: nat] : ( times_times @ real @ ( sin_coeff @ M ) @ ( power_power @ real @ X @ M ) )
@ ( set_ord_lessThan @ nat @ N2 ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( sin @ real @ ( plus_plus @ real @ T7 @ ( times_times @ real @ ( times_times @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( semiring_1_of_nat @ real @ N2 ) ) @ pi ) ) ) @ ( semiring_char_0_fact @ real @ N2 ) ) @ ( power_power @ real @ X @ N2 ) ) ) ) ) ) ) ).
% Maclaurin_sin_expansion3
thf(fact_2940_listrel1p__def,axiom,
! [A: $tType] :
( ( listrel1p @ A )
= ( ^ [R4: A > A > $o,Xs3: list @ A,Ys3: list @ A] : ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs3 @ Ys3 ) @ ( listrel1 @ A @ ( collect @ ( product_prod @ A @ A ) @ ( product_case_prod @ A @ A @ $o @ R4 ) ) ) ) ) ) ).
% listrel1p_def
thf(fact_2941_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
@ ^ [N: nat] :
( if @ A
@ ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ P6 )
& ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
@ ( 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 @ N ) ) ) ) @ ( semiring_char_0_fact @ real @ P6 ) ) ) @ ( power_power @ A @ X @ N ) ) @ ( power_power @ A @ Y @ ( minus_minus @ nat @ P6 @ N ) ) )
@ ( zero_zero @ A ) )
@ ( set_ord_atMost @ nat @ P6 ) )
@ ( times_times @ A @ ( sin @ A @ X ) @ ( sin @ A @ Y ) ) ) ) ).
% sin_x_sin_y
thf(fact_2942_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
@ ^ [N: 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 @ N ) ) ) ) @ ( semiring_char_0_fact @ real @ P6 ) ) @ ( power_power @ A @ X @ N ) ) @ ( power_power @ A @ Y @ ( minus_minus @ nat @ P6 @ N ) ) ) @ ( zero_zero @ A ) )
@ ( set_ord_atMost @ nat @ P6 ) )
@ ( cos @ A @ ( plus_plus @ A @ X @ Y ) ) ) ) ).
% sums_cos_x_plus_y
thf(fact_2943_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
@ ^ [N: nat] :
( if @ A
@ ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ P6 )
& ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
@ ( 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 @ N ) ) ) ) @ ( semiring_char_0_fact @ real @ P6 ) ) @ ( power_power @ A @ X @ N ) ) @ ( power_power @ A @ Y @ ( minus_minus @ nat @ P6 @ N ) ) )
@ ( zero_zero @ A ) )
@ ( set_ord_atMost @ nat @ P6 ) )
@ ( times_times @ A @ ( cos @ A @ X ) @ ( cos @ A @ Y ) ) ) ) ).
% cos_x_cos_y
thf(fact_2944_sin__coeff__def,axiom,
( sin_coeff
= ( ^ [N: nat] : ( if @ real @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ ( zero_zero @ real ) @ ( divide_divide @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ ( divide_divide @ nat @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( semiring_char_0_fact @ real @ N ) ) ) ) ) ).
% sin_coeff_def
thf(fact_2945_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_2946_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_2947_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_2948_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_2949_scaleR__eq__iff,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [B3: A,U: real,A3: A] :
( ( ( plus_plus @ A @ B3 @ ( real_V8093663219630862766scaleR @ A @ U @ A3 ) )
= ( plus_plus @ A @ A3 @ ( real_V8093663219630862766scaleR @ A @ U @ B3 ) ) )
= ( ( A3 = B3 )
| ( U
= ( one_one @ real ) ) ) ) ) ).
% scaleR_eq_iff
thf(fact_2950_sin__coeff__0,axiom,
( ( sin_coeff @ ( zero_zero @ nat ) )
= ( zero_zero @ real ) ) ).
% sin_coeff_0
thf(fact_2951_scaleR__collapse,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [U: real,A3: A] :
( ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ ( minus_minus @ real @ ( one_one @ real ) @ U ) @ A3 ) @ ( real_V8093663219630862766scaleR @ A @ U @ A3 ) )
= A3 ) ) ).
% scaleR_collapse
thf(fact_2952_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_2953_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_2954_scaleR__right__distrib,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [A3: real,X: A,Y: A] :
( ( real_V8093663219630862766scaleR @ A @ A3 @ ( plus_plus @ A @ X @ Y ) )
= ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ A3 @ X ) @ ( real_V8093663219630862766scaleR @ A @ A3 @ Y ) ) ) ) ).
% scaleR_right_distrib
thf(fact_2955_scaleR__left_Oadd,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [X: real,Y: real,Xa2: A] :
( ( real_V8093663219630862766scaleR @ A @ ( plus_plus @ real @ X @ Y ) @ Xa2 )
= ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ X @ Xa2 ) @ ( real_V8093663219630862766scaleR @ A @ Y @ Xa2 ) ) ) ) ).
% scaleR_left.add
thf(fact_2956_scaleR__left__distrib,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [A3: real,B3: real,X: A] :
( ( real_V8093663219630862766scaleR @ A @ ( plus_plus @ real @ A3 @ B3 ) @ X )
= ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ A3 @ X ) @ ( real_V8093663219630862766scaleR @ A @ B3 @ X ) ) ) ) ).
% scaleR_left_distrib
thf(fact_2957_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_2958_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_2959_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_2960_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_2961_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_2962_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_2963_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_2964_eq__vector__fraction__iff,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [X: A,U: real,V2: real,A3: A] :
( ( X
= ( real_V8093663219630862766scaleR @ A @ ( divide_divide @ real @ U @ V2 ) @ A3 ) )
= ( ( ( V2
= ( zero_zero @ real ) )
=> ( X
= ( zero_zero @ A ) ) )
& ( ( V2
!= ( zero_zero @ real ) )
=> ( ( real_V8093663219630862766scaleR @ A @ V2 @ X )
= ( real_V8093663219630862766scaleR @ A @ U @ A3 ) ) ) ) ) ) ).
% eq_vector_fraction_iff
thf(fact_2965_vector__fraction__eq__iff,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [U: real,V2: real,A3: A,X: A] :
( ( ( real_V8093663219630862766scaleR @ A @ ( divide_divide @ real @ U @ V2 ) @ A3 )
= X )
= ( ( ( V2
= ( zero_zero @ real ) )
=> ( X
= ( zero_zero @ A ) ) )
& ( ( V2
!= ( zero_zero @ real ) )
=> ( ( real_V8093663219630862766scaleR @ A @ U @ A3 )
= ( real_V8093663219630862766scaleR @ A @ V2 @ X ) ) ) ) ) ) ).
% vector_fraction_eq_iff
thf(fact_2966_Real__Vector__Spaces_Ole__add__iff2,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [A3: real,E2: A,C2: A,B3: real,D2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ A3 @ E2 ) @ C2 ) @ ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ B3 @ E2 ) @ D2 ) )
= ( ord_less_eq @ A @ C2 @ ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ ( minus_minus @ real @ B3 @ A3 ) @ E2 ) @ D2 ) ) ) ) ).
% Real_Vector_Spaces.le_add_iff2
thf(fact_2967_Real__Vector__Spaces_Ole__add__iff1,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [A3: real,E2: A,C2: A,B3: real,D2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ A3 @ E2 ) @ C2 ) @ ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ B3 @ E2 ) @ D2 ) )
= ( ord_less_eq @ A @ ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ ( minus_minus @ real @ A3 @ B3 ) @ E2 ) @ C2 ) @ D2 ) ) ) ).
% Real_Vector_Spaces.le_add_iff1
thf(fact_2968_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_2969_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_2970_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_2971_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_2972_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_2973_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_2974_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_2975_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_2976_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_2977_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_2978_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_2979_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_2980_sin__coeff__Suc,axiom,
! [N2: nat] :
( ( sin_coeff @ ( suc @ N2 ) )
= ( divide_divide @ real @ ( cos_coeff @ N2 ) @ ( semiring_1_of_nat @ real @ ( suc @ N2 ) ) ) ) ).
% sin_coeff_Suc
thf(fact_2981_cos__coeff__Suc,axiom,
! [N2: nat] :
( ( cos_coeff @ ( suc @ N2 ) )
= ( divide_divide @ real @ ( uminus_uminus @ real @ ( sin_coeff @ N2 ) ) @ ( semiring_1_of_nat @ real @ ( suc @ N2 ) ) ) ) ).
% cos_coeff_Suc
thf(fact_2982_summable__arctan__series,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ X ) @ ( one_one @ real ) )
=> ( summable @ real
@ ^ [K2: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ K2 ) @ ( times_times @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( semiring_1_of_nat @ real @ ( plus_plus @ nat @ ( times_times @ nat @ K2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) @ ( power_power @ real @ X @ ( plus_plus @ nat @ ( times_times @ nat @ K2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) ) ) ) ).
% summable_arctan_series
thf(fact_2983_diffs__equiv,axiom,
! [A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( ring_1 @ A ) )
=> ! [C2: nat > A,X: A] :
( ( summable @ A
@ ^ [N: nat] : ( times_times @ A @ ( diffs @ A @ C2 @ N ) @ ( power_power @ A @ X @ N ) ) )
=> ( sums @ A
@ ^ [N: nat] : ( times_times @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ ( C2 @ N ) ) @ ( power_power @ A @ X @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) ) )
@ ( suminf @ A
@ ^ [N: nat] : ( times_times @ A @ ( diffs @ A @ C2 @ N ) @ ( power_power @ A @ X @ N ) ) ) ) ) ) ).
% diffs_equiv
thf(fact_2984_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_2985_tan__half,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ( ( tan @ A )
= ( ^ [X4: A] : ( divide_divide @ A @ ( sin @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X4 ) ) @ ( plus_plus @ A @ ( cos @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X4 ) ) @ ( one_one @ A ) ) ) ) ) ) ).
% tan_half
thf(fact_2986_int__ge__less__than2__def,axiom,
( int_ge_less_than2
= ( ^ [D4: int] :
( collect @ ( product_prod @ int @ int )
@ ( product_case_prod @ int @ int @ $o
@ ^ [Z7: int,Z6: int] :
( ( ord_less_eq @ int @ D4 @ Z6 )
& ( ord_less @ int @ Z7 @ Z6 ) ) ) ) ) ) ).
% int_ge_less_than2_def
thf(fact_2987_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_2988_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_2989_abs__0,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ( ( abs_abs @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% abs_0
thf(fact_2990_abs__zero,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ( ( abs_abs @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% abs_zero
thf(fact_2991_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_2992_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_2993_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_2994_abs__1,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ( ( abs_abs @ A @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% abs_1
thf(fact_2995_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_2996_abs__divide,axiom,
! [A: $tType] :
( ( field_abs_sgn @ A )
=> ! [A3: A,B3: A] :
( ( abs_abs @ A @ ( divide_divide @ A @ A3 @ B3 ) )
= ( divide_divide @ A @ ( abs_abs @ A @ A3 ) @ ( abs_abs @ A @ B3 ) ) ) ) ).
% abs_divide
thf(fact_2997_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_2998_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_2999_abs__dvd__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M2: A,K: A] :
( ( dvd_dvd @ A @ ( abs_abs @ A @ M2 ) @ K )
= ( dvd_dvd @ A @ M2 @ K ) ) ) ).
% abs_dvd_iff
thf(fact_3000_dvd__abs__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M2: A,K: A] :
( ( dvd_dvd @ A @ M2 @ ( abs_abs @ A @ K ) )
= ( dvd_dvd @ A @ M2 @ K ) ) ) ).
% dvd_abs_iff
thf(fact_3001_abs__of__nat,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N2: nat] :
( ( abs_abs @ A @ ( semiring_1_of_nat @ A @ N2 ) )
= ( semiring_1_of_nat @ A @ N2 ) ) ) ).
% abs_of_nat
thf(fact_3002_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_3003_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_3004_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_3005_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_3006_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_3007_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_3008_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_3009_sum__abs,axiom,
! [B: $tType,A: $tType] :
( ( ordere166539214618696060dd_abs @ B )
=> ! [F2: A > B,A4: set @ A] :
( ord_less_eq @ B @ ( abs_abs @ B @ ( groups7311177749621191930dd_sum @ A @ B @ F2 @ A4 ) )
@ ( groups7311177749621191930dd_sum @ A @ B
@ ^ [I3: A] : ( abs_abs @ B @ ( F2 @ I3 ) )
@ A4 ) ) ) ).
% sum_abs
thf(fact_3010_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_3011_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_3012_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_3013_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_3014_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_3015_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_3016_sum__abs__ge__zero,axiom,
! [B: $tType,A: $tType] :
( ( ordere166539214618696060dd_abs @ B )
=> ! [F2: A > B,A4: set @ A] :
( ord_less_eq @ B @ ( zero_zero @ B )
@ ( groups7311177749621191930dd_sum @ A @ B
@ ^ [I3: A] : ( abs_abs @ B @ ( F2 @ I3 ) )
@ A4 ) ) ) ).
% sum_abs_ge_zero
thf(fact_3017_zero__less__power__abs__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,N2: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( power_power @ A @ ( abs_abs @ A @ A3 ) @ N2 ) )
= ( ( A3
!= ( zero_zero @ A ) )
| ( N2
= ( zero_zero @ nat ) ) ) ) ) ).
% zero_less_power_abs_iff
thf(fact_3018_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_3019_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_3020_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_3021_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_3022_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_3023_abs__one,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( ( abs_abs @ A @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% abs_one
thf(fact_3024_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_3025_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_3026_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_3027_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_3028_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_3029_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_3030_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_3031_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_3032_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_3033_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_3034_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_3035_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_3036_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_3037_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_3038_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_3039_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_3040_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_3041_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_3042_linordered__idom__class_Oabs__sgn,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( ( abs_abs @ A )
= ( ^ [K2: A] : ( times_times @ A @ K2 @ ( sgn_sgn @ A @ K2 ) ) ) ) ) ).
% linordered_idom_class.abs_sgn
thf(fact_3043_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_3044_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_3045_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_3046_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_3047_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_3048_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_3049_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_3050_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_3051_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_3052_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_3053_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_3054_zero__le__power__abs,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,N2: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ ( abs_abs @ A @ A3 ) @ N2 ) ) ) ).
% zero_le_power_abs
thf(fact_3055_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_3056_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_3057_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_3058_abs__diff__le__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A,A3: A,R2: A] :
( ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ X @ A3 ) ) @ R2 )
= ( ( ord_less_eq @ A @ ( minus_minus @ A @ A3 @ R2 ) @ X )
& ( ord_less_eq @ A @ X @ ( plus_plus @ A @ A3 @ R2 ) ) ) ) ) ).
% abs_diff_le_iff
thf(fact_3059_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_3060_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_3061_abs__diff__less__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A,A3: A,R2: A] :
( ( ord_less @ A @ ( abs_abs @ A @ ( minus_minus @ A @ X @ A3 ) ) @ R2 )
= ( ( ord_less @ A @ ( minus_minus @ A @ A3 @ R2 ) @ X )
& ( ord_less @ A @ X @ ( plus_plus @ A @ A3 @ R2 ) ) ) ) ) ).
% abs_diff_less_iff
thf(fact_3062_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_3063_summable__rabs__comparison__test,axiom,
! [F2: nat > real,G: nat > real] :
( ? [N8: nat] :
! [N3: nat] :
( ( ord_less_eq @ nat @ N8 @ N3 )
=> ( ord_less_eq @ real @ ( abs_abs @ real @ ( F2 @ N3 ) ) @ ( G @ N3 ) ) )
=> ( ( summable @ real @ G )
=> ( summable @ real
@ ^ [N: nat] : ( abs_abs @ real @ ( F2 @ N ) ) ) ) ) ).
% summable_rabs_comparison_test
thf(fact_3064_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_3065_of__int__leD,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N2: int,X: A] :
( ( ord_less_eq @ A @ ( abs_abs @ A @ ( ring_1_of_int @ A @ N2 ) ) @ X )
=> ( ( N2
= ( zero_zero @ int ) )
| ( ord_less_eq @ A @ ( one_one @ A ) @ X ) ) ) ) ).
% of_int_leD
thf(fact_3066_of__int__lessD,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N2: int,X: A] :
( ( ord_less @ A @ ( abs_abs @ A @ ( ring_1_of_int @ A @ N2 ) ) @ X )
=> ( ( N2
= ( zero_zero @ int ) )
| ( ord_less @ A @ ( one_one @ A ) @ X ) ) ) ) ).
% of_int_lessD
thf(fact_3067_sgn__power__injE,axiom,
! [A3: real,N2: nat,X: real,B3: real] :
( ( ( times_times @ real @ ( sgn_sgn @ real @ A3 ) @ ( power_power @ real @ ( abs_abs @ real @ A3 ) @ N2 ) )
= X )
=> ( ( X
= ( times_times @ real @ ( sgn_sgn @ real @ B3 ) @ ( power_power @ real @ ( abs_abs @ real @ B3 ) @ N2 ) ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( A3 = B3 ) ) ) ) ).
% sgn_power_injE
thf(fact_3068_round__diff__minimal,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z3: A,M2: int] : ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ Z3 @ ( ring_1_of_int @ A @ ( archimedean_round @ A @ Z3 ) ) ) ) @ ( abs_abs @ A @ ( minus_minus @ A @ Z3 @ ( ring_1_of_int @ A @ M2 ) ) ) ) ) ).
% round_diff_minimal
thf(fact_3069_diffs__def,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ( ( diffs @ A )
= ( ^ [C4: nat > A,N: nat] : ( times_times @ A @ ( semiring_1_of_nat @ A @ ( suc @ N ) ) @ ( C4 @ ( suc @ N ) ) ) ) ) ) ).
% diffs_def
thf(fact_3070_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_3071_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_3072_abs__sqrt__wlog,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [P: A > A > $o,X: A] :
( ! [X5: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X5 )
=> ( P @ X5 @ ( power_power @ A @ X5 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
=> ( P @ ( abs_abs @ A @ X ) @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% abs_sqrt_wlog
thf(fact_3073_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_3074_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_3075_power__mono__even,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N2: nat,A3: A,B3: A] :
( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
=> ( ( ord_less_eq @ A @ ( abs_abs @ A @ A3 ) @ ( abs_abs @ A @ B3 ) )
=> ( ord_less_eq @ A @ ( power_power @ A @ A3 @ N2 ) @ ( power_power @ A @ B3 @ N2 ) ) ) ) ) ).
% power_mono_even
thf(fact_3076_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] :
( ! [I4: A] :
( ( member @ A @ I4 @ I6 )
=> ( ord_less_eq @ B @ ( zero_zero @ B ) @ ( X @ I4 ) ) )
=> ( ( ( groups7311177749621191930dd_sum @ A @ B @ X @ I6 )
= ( one_one @ B ) )
=> ( ! [I4: A] :
( ( member @ A @ I4 @ I6 )
=> ( ord_less_eq @ B @ ( abs_abs @ B @ ( minus_minus @ B @ ( A3 @ I4 ) @ B3 ) ) @ Delta ) )
=> ( ord_less_eq @ B
@ ( abs_abs @ B
@ ( minus_minus @ B
@ ( groups7311177749621191930dd_sum @ A @ B
@ ^ [I3: A] : ( times_times @ B @ ( A3 @ I3 ) @ ( X @ I3 ) )
@ I6 )
@ B3 ) )
@ Delta ) ) ) ) ) ).
% convex_sum_bound_le
thf(fact_3077_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_3078_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_3079_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_3080_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_3081_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_3082_round__unique_H,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,N2: int] :
( ( ord_less @ A @ ( abs_abs @ A @ ( minus_minus @ A @ X @ ( ring_1_of_int @ A @ N2 ) ) ) @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
=> ( ( archimedean_round @ A @ X )
= N2 ) ) ) ).
% round_unique'
thf(fact_3083_monoseq__arctan__series,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ X ) @ ( one_one @ real ) )
=> ( topological_monoseq @ real
@ ^ [N: nat] : ( times_times @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( semiring_1_of_nat @ real @ ( plus_plus @ nat @ ( times_times @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) @ ( power_power @ real @ X @ ( plus_plus @ nat @ ( times_times @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) ) ) ).
% monoseq_arctan_series
thf(fact_3084_arctan__series,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ X ) @ ( one_one @ real ) )
=> ( ( arctan @ X )
= ( suminf @ real
@ ^ [K2: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ K2 ) @ ( times_times @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( semiring_1_of_nat @ real @ ( plus_plus @ nat @ ( times_times @ nat @ K2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) @ ( power_power @ real @ X @ ( plus_plus @ nat @ ( times_times @ nat @ K2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) ) ) ) ) ).
% arctan_series
thf(fact_3085_int__ge__less__than__def,axiom,
( int_ge_less_than
= ( ^ [D4: int] :
( collect @ ( product_prod @ int @ int )
@ ( product_case_prod @ int @ int @ $o
@ ^ [Z7: int,Z6: int] :
( ( ord_less_eq @ int @ D4 @ Z7 )
& ( ord_less @ int @ Z7 @ Z6 ) ) ) ) ) ) ).
% int_ge_less_than_def
thf(fact_3086_Maclaurin__exp__lt,axiom,
! [X: real,N2: nat] :
( ( X
!= ( zero_zero @ real ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ? [T7: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( abs_abs @ real @ T7 ) )
& ( ord_less @ real @ ( abs_abs @ real @ T7 ) @ ( abs_abs @ real @ X ) )
& ( ( exp @ real @ X )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M: nat] : ( divide_divide @ real @ ( power_power @ real @ X @ M ) @ ( semiring_char_0_fact @ real @ M ) )
@ ( set_ord_lessThan @ nat @ N2 ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( exp @ real @ T7 ) @ ( semiring_char_0_fact @ real @ N2 ) ) @ ( power_power @ real @ X @ N2 ) ) ) ) ) ) ) ).
% Maclaurin_exp_lt
thf(fact_3087_xor__Suc__0__eq,axiom,
! [N2: nat] :
( ( bit_se5824344971392196577ns_xor @ nat @ N2 @ ( suc @ ( zero_zero @ nat ) ) )
= ( minus_minus @ nat @ ( plus_plus @ nat @ N2 @ ( zero_neq_one_of_bool @ nat @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) )
@ ( zero_neq_one_of_bool @ nat
@ ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ).
% xor_Suc_0_eq
thf(fact_3088_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_3089_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_3090_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_3091_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_3092_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_3093_zdvd1__eq,axiom,
! [X: int] :
( ( dvd_dvd @ int @ X @ ( one_one @ int ) )
= ( ( abs_abs @ int @ X )
= ( one_one @ int ) ) ) ).
% zdvd1_eq
thf(fact_3094_zabs__less__one__iff,axiom,
! [Z3: int] :
( ( ord_less @ int @ ( abs_abs @ int @ Z3 ) @ ( one_one @ int ) )
= ( Z3
= ( zero_zero @ int ) ) ) ).
% zabs_less_one_iff
thf(fact_3095_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_3096_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_3097_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_3098_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_3099_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_3100_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_3101_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_3102_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_3103_exp__add__commuting,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X: A,Y: A] :
( ( ( times_times @ A @ X @ Y )
= ( times_times @ A @ Y @ X ) )
=> ( ( exp @ A @ ( plus_plus @ A @ X @ Y ) )
= ( times_times @ A @ ( exp @ A @ X ) @ ( exp @ A @ Y ) ) ) ) ) ).
% exp_add_commuting
thf(fact_3104_mult__exp__exp,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A,Y: A] :
( ( times_times @ A @ ( exp @ A @ X ) @ ( exp @ A @ Y ) )
= ( exp @ A @ ( plus_plus @ A @ X @ Y ) ) ) ) ).
% mult_exp_exp
thf(fact_3105_abs__zmult__eq__1,axiom,
! [M2: int,N2: int] :
( ( ( abs_abs @ int @ ( times_times @ int @ M2 @ N2 ) )
= ( one_one @ int ) )
=> ( ( abs_abs @ int @ M2 )
= ( one_one @ int ) ) ) ).
% abs_zmult_eq_1
thf(fact_3106_infinite__int__iff__unbounded__le,axiom,
! [S2: set @ int] :
( ( ~ ( finite_finite2 @ int @ S2 ) )
= ( ! [M: int] :
? [N: int] :
( ( ord_less_eq @ int @ M @ ( abs_abs @ int @ N ) )
& ( member @ int @ N @ S2 ) ) ) ) ).
% infinite_int_iff_unbounded_le
thf(fact_3107_infinite__int__iff__unbounded,axiom,
! [S2: set @ int] :
( ( ~ ( finite_finite2 @ int @ S2 ) )
= ( ! [M: int] :
? [N: int] :
( ( ord_less @ int @ M @ ( abs_abs @ int @ N ) )
& ( member @ int @ N @ S2 ) ) ) ) ).
% infinite_int_iff_unbounded
thf(fact_3108_zabs__def,axiom,
( ( abs_abs @ int )
= ( ^ [I3: int] : ( if @ int @ ( ord_less @ int @ I3 @ ( zero_zero @ int ) ) @ ( uminus_uminus @ int @ I3 ) @ I3 ) ) ) ).
% zabs_def
thf(fact_3109_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_3110_exp__sum,axiom,
! [B: $tType,A: $tType] :
( ( ( comm_monoid_mult @ B )
& ( real_Vector_banach @ B )
& ( real_V2822296259951069270ebra_1 @ B ) )
=> ! [I6: set @ A,F2: A > B] :
( ( finite_finite2 @ A @ I6 )
=> ( ( exp @ B @ ( groups7311177749621191930dd_sum @ A @ B @ F2 @ I6 ) )
= ( groups7121269368397514597t_prod @ A @ B
@ ^ [X4: A] : ( exp @ B @ ( F2 @ X4 ) )
@ I6 ) ) ) ) ).
% exp_sum
thf(fact_3111_zdvd__mult__cancel1,axiom,
! [M2: int,N2: int] :
( ( M2
!= ( zero_zero @ int ) )
=> ( ( dvd_dvd @ int @ ( times_times @ int @ M2 @ N2 ) @ M2 )
= ( ( abs_abs @ int @ N2 )
= ( one_one @ int ) ) ) ) ).
% zdvd_mult_cancel1
thf(fact_3112_powr__def,axiom,
! [A: $tType] :
( ( ln @ A )
=> ( ( powr @ A )
= ( ^ [X4: A,A5: A] :
( if @ A
@ ( X4
= ( zero_zero @ A ) )
@ ( zero_zero @ A )
@ ( exp @ A @ ( times_times @ A @ A5 @ ( ln_ln @ A @ X4 ) ) ) ) ) ) ) ).
% powr_def
thf(fact_3113_exp__divide__power__eq,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [N2: nat,X: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( power_power @ A @ ( exp @ A @ ( divide_divide @ A @ X @ ( semiring_1_of_nat @ A @ N2 ) ) ) @ N2 )
= ( exp @ A @ X ) ) ) ) ).
% exp_divide_power_eq
thf(fact_3114_nat__intermed__int__val,axiom,
! [M2: nat,N2: nat,F2: nat > int,K: int] :
( ! [I4: nat] :
( ( ( ord_less_eq @ nat @ M2 @ I4 )
& ( ord_less @ nat @ I4 @ N2 ) )
=> ( ord_less_eq @ int @ ( abs_abs @ int @ ( minus_minus @ int @ ( F2 @ ( suc @ I4 ) ) @ ( F2 @ I4 ) ) ) @ ( one_one @ int ) ) )
=> ( ( ord_less_eq @ nat @ M2 @ N2 )
=> ( ( ord_less_eq @ int @ ( F2 @ M2 ) @ K )
=> ( ( ord_less_eq @ int @ K @ ( F2 @ N2 ) )
=> ? [I4: nat] :
( ( ord_less_eq @ nat @ M2 @ I4 )
& ( ord_less_eq @ nat @ I4 @ N2 )
& ( ( F2 @ I4 )
= K ) ) ) ) ) ) ).
% nat_intermed_int_val
thf(fact_3115_nat__ivt__aux,axiom,
! [N2: nat,F2: nat > int,K: int] :
( ! [I4: nat] :
( ( ord_less @ nat @ I4 @ N2 )
=> ( ord_less_eq @ int @ ( abs_abs @ int @ ( minus_minus @ int @ ( F2 @ ( suc @ I4 ) ) @ ( F2 @ I4 ) ) ) @ ( one_one @ int ) ) )
=> ( ( ord_less_eq @ int @ ( F2 @ ( zero_zero @ nat ) ) @ K )
=> ( ( ord_less_eq @ int @ K @ ( F2 @ N2 ) )
=> ? [I4: nat] :
( ( ord_less_eq @ nat @ I4 @ N2 )
& ( ( F2 @ I4 )
= K ) ) ) ) ) ).
% nat_ivt_aux
thf(fact_3116_xor__nat__unfold,axiom,
( ( bit_se5824344971392196577ns_xor @ nat )
= ( ^ [M: nat,N: nat] :
( if @ nat
@ ( M
= ( zero_zero @ nat ) )
@ N
@ ( if @ nat
@ ( N
= ( zero_zero @ nat ) )
@ M
@ ( plus_plus @ nat @ ( modulo_modulo @ nat @ ( plus_plus @ nat @ ( modulo_modulo @ nat @ M @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ nat @ N @ ( 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 @ M @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% xor_nat_unfold
thf(fact_3117_nat0__intermed__int__val,axiom,
! [N2: nat,F2: nat > int,K: int] :
( ! [I4: nat] :
( ( ord_less @ nat @ I4 @ N2 )
=> ( ord_less_eq @ int @ ( abs_abs @ int @ ( minus_minus @ int @ ( F2 @ ( plus_plus @ nat @ I4 @ ( one_one @ nat ) ) ) @ ( F2 @ I4 ) ) ) @ ( one_one @ int ) ) )
=> ( ( ord_less_eq @ int @ ( F2 @ ( zero_zero @ nat ) ) @ K )
=> ( ( ord_less_eq @ int @ K @ ( F2 @ N2 ) )
=> ? [I4: nat] :
( ( ord_less_eq @ nat @ I4 @ N2 )
& ( ( F2 @ I4 )
= K ) ) ) ) ) ).
% nat0_intermed_int_val
thf(fact_3118_xor__nat__rec,axiom,
( ( bit_se5824344971392196577ns_xor @ nat )
= ( ^ [M: nat,N: nat] :
( plus_plus @ nat
@ ( zero_neq_one_of_bool @ nat
@ ( ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) )
!= ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) )
@ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( bit_se5824344971392196577ns_xor @ nat @ ( divide_divide @ nat @ M @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% xor_nat_rec
thf(fact_3119_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_3120_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_3121_exp__ge__one__plus__x__over__n__power__n,axiom,
! [N2: nat,X: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( semiring_1_of_nat @ real @ N2 ) ) @ X )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ord_less_eq @ real @ ( power_power @ real @ ( plus_plus @ real @ ( one_one @ real ) @ ( divide_divide @ real @ X @ ( semiring_1_of_nat @ real @ N2 ) ) ) @ N2 ) @ ( exp @ real @ X ) ) ) ) ).
% exp_ge_one_plus_x_over_n_power_n
thf(fact_3122_exp__ge__one__minus__x__over__n__power__n,axiom,
! [X: real,N2: nat] :
( ( ord_less_eq @ real @ X @ ( semiring_1_of_nat @ real @ N2 ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ord_less_eq @ real @ ( power_power @ real @ ( minus_minus @ real @ ( one_one @ real ) @ ( divide_divide @ real @ X @ ( semiring_1_of_nat @ real @ N2 ) ) ) @ N2 ) @ ( exp @ real @ ( uminus_uminus @ real @ X ) ) ) ) ) ).
% exp_ge_one_minus_x_over_n_power_n
thf(fact_3123_Suc__0__xor__eq,axiom,
! [N2: nat] :
( ( bit_se5824344971392196577ns_xor @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N2 )
= ( minus_minus @ nat @ ( plus_plus @ nat @ N2 @ ( zero_neq_one_of_bool @ nat @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) )
@ ( zero_neq_one_of_bool @ nat
@ ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ).
% Suc_0_xor_eq
thf(fact_3124_monoI1,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X8: nat > A] :
( ! [M3: nat,N3: nat] :
( ( ord_less_eq @ nat @ M3 @ N3 )
=> ( ord_less_eq @ A @ ( X8 @ M3 ) @ ( X8 @ N3 ) ) )
=> ( topological_monoseq @ A @ X8 ) ) ) ).
% monoI1
thf(fact_3125_monoI2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X8: nat > A] :
( ! [M3: nat,N3: nat] :
( ( ord_less_eq @ nat @ M3 @ N3 )
=> ( ord_less_eq @ A @ ( X8 @ N3 ) @ ( X8 @ M3 ) ) )
=> ( topological_monoseq @ A @ X8 ) ) ) ).
% monoI2
thf(fact_3126_monoseq__def,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( topological_monoseq @ A )
= ( ^ [X6: nat > A] :
( ! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ord_less_eq @ A @ ( X6 @ M ) @ ( X6 @ N ) ) )
| ! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ord_less_eq @ A @ ( X6 @ N ) @ ( X6 @ M ) ) ) ) ) ) ) ).
% monoseq_def
thf(fact_3127_monoseq__Suc,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( topological_monoseq @ A )
= ( ^ [X6: nat > A] :
( ! [N: nat] : ( ord_less_eq @ A @ ( X6 @ N ) @ ( X6 @ ( suc @ N ) ) )
| ! [N: nat] : ( ord_less_eq @ A @ ( X6 @ ( suc @ N ) ) @ ( X6 @ N ) ) ) ) ) ) ).
% monoseq_Suc
thf(fact_3128_mono__SucI2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X8: nat > A] :
( ! [N3: nat] : ( ord_less_eq @ A @ ( X8 @ ( suc @ N3 ) ) @ ( X8 @ N3 ) )
=> ( topological_monoseq @ A @ X8 ) ) ) ).
% mono_SucI2
thf(fact_3129_mono__SucI1,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X8: nat > A] :
( ! [N3: nat] : ( ord_less_eq @ A @ ( X8 @ N3 ) @ ( X8 @ ( suc @ N3 ) ) )
=> ( topological_monoseq @ A @ X8 ) ) ) ).
% mono_SucI1
thf(fact_3130_exp__first__two__terms,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ( ( exp @ A )
= ( ^ [X4: A] :
( plus_plus @ A @ ( plus_plus @ A @ ( one_one @ A ) @ X4 )
@ ( suminf @ A
@ ^ [N: nat] : ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ ( plus_plus @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( power_power @ A @ X4 @ ( plus_plus @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% exp_first_two_terms
thf(fact_3131_divide__int__def,axiom,
( ( divide_divide @ int )
= ( ^ [K2: int,L2: int] :
( if @ int
@ ( L2
= ( zero_zero @ int ) )
@ ( zero_zero @ int )
@ ( if @ int
@ ( ( sgn_sgn @ int @ K2 )
= ( sgn_sgn @ int @ L2 ) )
@ ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ ( nat2 @ ( abs_abs @ int @ K2 ) ) @ ( nat2 @ ( abs_abs @ int @ L2 ) ) ) )
@ ( uminus_uminus @ int
@ ( semiring_1_of_nat @ int
@ ( plus_plus @ nat @ ( divide_divide @ nat @ ( nat2 @ ( abs_abs @ int @ K2 ) ) @ ( nat2 @ ( abs_abs @ int @ L2 ) ) )
@ ( zero_neq_one_of_bool @ nat
@ ~ ( dvd_dvd @ int @ L2 @ K2 ) ) ) ) ) ) ) ) ) ).
% divide_int_def
thf(fact_3132_exp__first__terms,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [K: nat] :
( ( exp @ A )
= ( ^ [X4: A] :
( plus_plus @ A
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [N: nat] : ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ A @ X4 @ N ) )
@ ( set_ord_lessThan @ nat @ K ) )
@ ( suminf @ A
@ ^ [N: nat] : ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ ( plus_plus @ nat @ N @ K ) ) ) @ ( power_power @ A @ X4 @ ( plus_plus @ nat @ N @ K ) ) ) ) ) ) ) ) ).
% exp_first_terms
thf(fact_3133_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_3134_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_3135_inverse__inverse__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A] :
( ( inverse_inverse @ A @ ( inverse_inverse @ A @ A3 ) )
= A3 ) ) ).
% inverse_inverse_eq
thf(fact_3136_inverse__eq__iff__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A,B3: A] :
( ( ( inverse_inverse @ A @ A3 )
= ( inverse_inverse @ A @ B3 ) )
= ( A3 = B3 ) ) ) ).
% inverse_eq_iff_eq
thf(fact_3137_inverse__zero,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ( ( inverse_inverse @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% inverse_zero
thf(fact_3138_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_3139_inverse__mult__distrib,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A3: A,B3: A] :
( ( inverse_inverse @ A @ ( times_times @ A @ A3 @ B3 ) )
= ( times_times @ A @ ( inverse_inverse @ A @ A3 ) @ ( inverse_inverse @ A @ B3 ) ) ) ) ).
% inverse_mult_distrib
thf(fact_3140_inverse__1,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ( ( inverse_inverse @ A @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% inverse_1
thf(fact_3141_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_3142_inverse__divide,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A3: A,B3: A] :
( ( inverse_inverse @ A @ ( divide_divide @ A @ A3 @ B3 ) )
= ( divide_divide @ A @ B3 @ A3 ) ) ) ).
% inverse_divide
thf(fact_3143_inverse__minus__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A] :
( ( inverse_inverse @ A @ ( uminus_uminus @ A @ A3 ) )
= ( uminus_uminus @ A @ ( inverse_inverse @ A @ A3 ) ) ) ) ).
% inverse_minus_eq
thf(fact_3144_abs__inverse,axiom,
! [A: $tType] :
( ( field_abs_sgn @ A )
=> ! [A3: A] :
( ( abs_abs @ A @ ( inverse_inverse @ A @ A3 ) )
= ( inverse_inverse @ A @ ( abs_abs @ A @ A3 ) ) ) ) ).
% abs_inverse
thf(fact_3145_inverse__sgn,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A] :
( ( inverse_inverse @ A @ ( sgn_sgn @ A @ A3 ) )
= ( sgn_sgn @ A @ A3 ) ) ) ).
% inverse_sgn
thf(fact_3146_sgn__inverse,axiom,
! [A: $tType] :
( ( field_abs_sgn @ A )
=> ! [A3: A] :
( ( sgn_sgn @ A @ ( inverse_inverse @ A @ A3 ) )
= ( inverse_inverse @ A @ ( sgn_sgn @ A @ A3 ) ) ) ) ).
% sgn_inverse
thf(fact_3147_nat__int,axiom,
! [N2: nat] :
( ( nat2 @ ( semiring_1_of_nat @ int @ N2 ) )
= N2 ) ).
% nat_int
thf(fact_3148_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_3149_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_3150_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_3151_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_3152_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_3153_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_3154_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_3155_nat__numeral,axiom,
! [K: num] :
( ( nat2 @ ( numeral_numeral @ int @ K ) )
= ( numeral_numeral @ nat @ K ) ) ).
% nat_numeral
thf(fact_3156_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_3157_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_3158_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_3159_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_3160_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_3161_nat__1,axiom,
( ( nat2 @ ( one_one @ int ) )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% nat_1
thf(fact_3162_nat__le__0,axiom,
! [Z3: int] :
( ( ord_less_eq @ int @ Z3 @ ( zero_zero @ int ) )
=> ( ( nat2 @ Z3 )
= ( zero_zero @ nat ) ) ) ).
% nat_le_0
thf(fact_3163_nat__0__iff,axiom,
! [I: int] :
( ( ( nat2 @ I )
= ( zero_zero @ nat ) )
= ( ord_less_eq @ int @ I @ ( zero_zero @ int ) ) ) ).
% nat_0_iff
thf(fact_3164_zless__nat__conj,axiom,
! [W2: int,Z3: int] :
( ( ord_less @ nat @ ( nat2 @ W2 ) @ ( nat2 @ Z3 ) )
= ( ( ord_less @ int @ ( zero_zero @ int ) @ Z3 )
& ( ord_less @ int @ W2 @ Z3 ) ) ) ).
% zless_nat_conj
thf(fact_3165_nat__neg__numeral,axiom,
! [K: num] :
( ( nat2 @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) )
= ( zero_zero @ nat ) ) ).
% nat_neg_numeral
thf(fact_3166_nat__zminus__int,axiom,
! [N2: nat] :
( ( nat2 @ ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N2 ) ) )
= ( zero_zero @ nat ) ) ).
% nat_zminus_int
thf(fact_3167_int__nat__eq,axiom,
! [Z3: int] :
( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z3 )
=> ( ( semiring_1_of_nat @ int @ ( nat2 @ Z3 ) )
= Z3 ) )
& ( ~ ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z3 )
=> ( ( semiring_1_of_nat @ int @ ( nat2 @ Z3 ) )
= ( zero_zero @ int ) ) ) ) ).
% int_nat_eq
thf(fact_3168_zero__less__nat__eq,axiom,
! [Z3: int] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( nat2 @ Z3 ) )
= ( ord_less @ int @ ( zero_zero @ int ) @ Z3 ) ) ).
% zero_less_nat_eq
thf(fact_3169_of__nat__nat,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [Z3: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z3 )
=> ( ( semiring_1_of_nat @ A @ ( nat2 @ Z3 ) )
= ( ring_1_of_int @ A @ Z3 ) ) ) ) ).
% of_nat_nat
thf(fact_3170_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_3171_numeral__power__eq__nat__cancel__iff,axiom,
! [X: num,N2: nat,Y: int] :
( ( ( power_power @ nat @ ( numeral_numeral @ nat @ X ) @ N2 )
= ( nat2 @ Y ) )
= ( ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N2 )
= Y ) ) ).
% numeral_power_eq_nat_cancel_iff
thf(fact_3172_nat__eq__numeral__power__cancel__iff,axiom,
! [Y: int,X: num,N2: nat] :
( ( ( nat2 @ Y )
= ( power_power @ nat @ ( numeral_numeral @ nat @ X ) @ N2 ) )
= ( Y
= ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N2 ) ) ) ).
% nat_eq_numeral_power_cancel_iff
thf(fact_3173_dvd__nat__abs__iff,axiom,
! [N2: nat,K: int] :
( ( dvd_dvd @ nat @ N2 @ ( nat2 @ ( abs_abs @ int @ K ) ) )
= ( dvd_dvd @ int @ ( semiring_1_of_nat @ int @ N2 ) @ K ) ) ).
% dvd_nat_abs_iff
thf(fact_3174_nat__abs__dvd__iff,axiom,
! [K: int,N2: nat] :
( ( dvd_dvd @ nat @ ( nat2 @ ( abs_abs @ int @ K ) ) @ N2 )
= ( dvd_dvd @ int @ K @ ( semiring_1_of_nat @ int @ N2 ) ) ) ).
% nat_abs_dvd_iff
thf(fact_3175_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_3176_one__less__nat__eq,axiom,
! [Z3: int] :
( ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( nat2 @ Z3 ) )
= ( ord_less @ int @ ( one_one @ int ) @ Z3 ) ) ).
% one_less_nat_eq
thf(fact_3177_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_3178_numeral__power__less__nat__cancel__iff,axiom,
! [X: num,N2: nat,A3: int] :
( ( ord_less @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ X ) @ N2 ) @ ( nat2 @ A3 ) )
= ( ord_less @ int @ ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N2 ) @ A3 ) ) ).
% numeral_power_less_nat_cancel_iff
thf(fact_3179_nat__less__numeral__power__cancel__iff,axiom,
! [A3: int,X: num,N2: nat] :
( ( ord_less @ nat @ ( nat2 @ A3 ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ X ) @ N2 ) )
= ( ord_less @ int @ A3 @ ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N2 ) ) ) ).
% nat_less_numeral_power_cancel_iff
thf(fact_3180_nat__le__numeral__power__cancel__iff,axiom,
! [A3: int,X: num,N2: nat] :
( ( ord_less_eq @ nat @ ( nat2 @ A3 ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ X ) @ N2 ) )
= ( ord_less_eq @ int @ A3 @ ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N2 ) ) ) ).
% nat_le_numeral_power_cancel_iff
thf(fact_3181_numeral__power__le__nat__cancel__iff,axiom,
! [X: num,N2: nat,A3: int] :
( ( ord_less_eq @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ X ) @ N2 ) @ ( nat2 @ A3 ) )
= ( ord_less_eq @ int @ ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N2 ) @ A3 ) ) ).
% numeral_power_le_nat_cancel_iff
thf(fact_3182_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_3183_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_3184_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_3185_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_3186_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_3187_mult__commute__imp__mult__inverse__commute,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Y: A,X: A] :
( ( ( times_times @ A @ Y @ X )
= ( times_times @ A @ X @ Y ) )
=> ( ( times_times @ A @ ( inverse_inverse @ A @ Y ) @ X )
= ( times_times @ A @ X @ ( inverse_inverse @ A @ Y ) ) ) ) ) ).
% mult_commute_imp_mult_inverse_commute
thf(fact_3188_inverse__eq__imp__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A,B3: A] :
( ( ( inverse_inverse @ A @ A3 )
= ( inverse_inverse @ A @ B3 ) )
=> ( A3 = B3 ) ) ) ).
% inverse_eq_imp_eq
thf(fact_3189_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_3190_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_3191_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_3192_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_3193_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_3194_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_3195_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_3196_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_3197_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_3198_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_3199_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_3200_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_3201_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_3202_divide__inverse__commute,axiom,
! [A: $tType] :
( ( field @ A )
=> ( ( divide_divide @ A )
= ( ^ [A5: A,B4: A] : ( times_times @ A @ ( inverse_inverse @ A @ B4 ) @ A5 ) ) ) ) ).
% divide_inverse_commute
thf(fact_3203_divide__inverse,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ( ( divide_divide @ A )
= ( ^ [A5: A,B4: A] : ( times_times @ A @ A5 @ ( inverse_inverse @ A @ B4 ) ) ) ) ) ).
% divide_inverse
thf(fact_3204_field__class_Ofield__divide__inverse,axiom,
! [A: $tType] :
( ( field @ A )
=> ( ( divide_divide @ A )
= ( ^ [A5: A,B4: A] : ( times_times @ A @ A5 @ ( inverse_inverse @ A @ B4 ) ) ) ) ) ).
% field_class.field_divide_inverse
thf(fact_3205_inverse__eq__divide,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ( ( inverse_inverse @ A )
= ( divide_divide @ A @ ( one_one @ A ) ) ) ) ).
% inverse_eq_divide
thf(fact_3206_power__mult__power__inverse__commute,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X: A,M2: nat,N2: nat] :
( ( times_times @ A @ ( power_power @ A @ X @ M2 ) @ ( power_power @ A @ ( inverse_inverse @ A @ X ) @ N2 ) )
= ( times_times @ A @ ( power_power @ A @ ( inverse_inverse @ A @ X ) @ N2 ) @ ( power_power @ A @ X @ M2 ) ) ) ) ).
% power_mult_power_inverse_commute
thf(fact_3207_power__mult__inverse__distrib,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X: A,M2: nat] :
( ( times_times @ A @ ( power_power @ A @ X @ M2 ) @ ( inverse_inverse @ A @ X ) )
= ( times_times @ A @ ( inverse_inverse @ A @ X ) @ ( power_power @ A @ X @ M2 ) ) ) ) ).
% power_mult_inverse_distrib
thf(fact_3208_mult__inverse__of__nat__commute,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Xa2: nat,X: A] :
( ( times_times @ A @ ( inverse_inverse @ A @ ( semiring_1_of_nat @ A @ Xa2 ) ) @ X )
= ( times_times @ A @ X @ ( inverse_inverse @ A @ ( semiring_1_of_nat @ A @ Xa2 ) ) ) ) ) ).
% mult_inverse_of_nat_commute
thf(fact_3209_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_3210_mult__inverse__of__int__commute,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Xa2: int,X: A] :
( ( times_times @ A @ ( inverse_inverse @ A @ ( ring_1_of_int @ A @ Xa2 ) ) @ X )
= ( times_times @ A @ X @ ( inverse_inverse @ A @ ( ring_1_of_int @ A @ Xa2 ) ) ) ) ) ).
% mult_inverse_of_int_commute
thf(fact_3211_nat__zero__as__int,axiom,
( ( zero_zero @ nat )
= ( nat2 @ ( zero_zero @ int ) ) ) ).
% nat_zero_as_int
thf(fact_3212_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_3213_ex__nat,axiom,
( ( ^ [P2: nat > $o] :
? [X7: nat] : ( P2 @ X7 ) )
= ( ^ [P3: nat > $o] :
? [X4: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X4 )
& ( P3 @ ( nat2 @ X4 ) ) ) ) ) ).
% ex_nat
thf(fact_3214_all__nat,axiom,
( ( ^ [P2: nat > $o] :
! [X7: nat] : ( P2 @ X7 ) )
= ( ^ [P3: nat > $o] :
! [X4: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X4 )
=> ( P3 @ ( nat2 @ X4 ) ) ) ) ) ).
% all_nat
thf(fact_3215_eq__nat__nat__iff,axiom,
! [Z3: int,Z8: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z3 )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z8 )
=> ( ( ( nat2 @ Z3 )
= ( nat2 @ Z8 ) )
= ( Z3 = Z8 ) ) ) ) ).
% eq_nat_nat_iff
thf(fact_3216_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_3217_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_3218_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_3219_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_3220_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_3221_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_3222_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_3223_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_3224_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_3225_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_3226_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_3227_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_3228_nat__mono__iff,axiom,
! [Z3: int,W2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ Z3 )
=> ( ( ord_less @ nat @ ( nat2 @ W2 ) @ ( nat2 @ Z3 ) )
= ( ord_less @ int @ W2 @ Z3 ) ) ) ).
% nat_mono_iff
thf(fact_3229_of__nat__ceiling,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [R2: A] : ( ord_less_eq @ A @ R2 @ ( semiring_1_of_nat @ A @ ( nat2 @ ( archimedean_ceiling @ A @ R2 ) ) ) ) ) ).
% of_nat_ceiling
thf(fact_3230_zless__nat__eq__int__zless,axiom,
! [M2: nat,Z3: int] :
( ( ord_less @ nat @ M2 @ ( nat2 @ Z3 ) )
= ( ord_less @ int @ ( semiring_1_of_nat @ int @ M2 ) @ Z3 ) ) ).
% zless_nat_eq_int_zless
thf(fact_3231_nat__le__iff,axiom,
! [X: int,N2: nat] :
( ( ord_less_eq @ nat @ ( nat2 @ X ) @ N2 )
= ( ord_less_eq @ int @ X @ ( semiring_1_of_nat @ int @ N2 ) ) ) ).
% nat_le_iff
thf(fact_3232_int__eq__iff,axiom,
! [M2: nat,Z3: int] :
( ( ( semiring_1_of_nat @ int @ M2 )
= Z3 )
= ( ( M2
= ( nat2 @ Z3 ) )
& ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z3 ) ) ) ).
% int_eq_iff
thf(fact_3233_nat__0__le,axiom,
! [Z3: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z3 )
=> ( ( semiring_1_of_nat @ int @ ( nat2 @ Z3 ) )
= Z3 ) ) ).
% nat_0_le
thf(fact_3234_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_3235_nat__abs__mult__distrib,axiom,
! [W2: int,Z3: int] :
( ( nat2 @ ( abs_abs @ int @ ( times_times @ int @ W2 @ Z3 ) ) )
= ( times_times @ nat @ ( nat2 @ ( abs_abs @ int @ W2 ) ) @ ( nat2 @ ( abs_abs @ int @ Z3 ) ) ) ) ).
% nat_abs_mult_distrib
thf(fact_3236_nat__plus__as__int,axiom,
( ( plus_plus @ nat )
= ( ^ [A5: nat,B4: nat] : ( nat2 @ ( plus_plus @ int @ ( semiring_1_of_nat @ int @ A5 ) @ ( semiring_1_of_nat @ int @ B4 ) ) ) ) ) ).
% nat_plus_as_int
thf(fact_3237_cosh__def,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ( ( cosh @ A )
= ( ^ [X4: A] : ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( plus_plus @ A @ ( exp @ A @ X4 ) @ ( exp @ A @ ( uminus_uminus @ A @ X4 ) ) ) ) ) ) ) ).
% cosh_def
thf(fact_3238_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_3239_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_3240_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_3241_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_3242_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_3243_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_3244_reals__Archimedean,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ X )
=> ? [N3: nat] : ( ord_less @ A @ ( inverse_inverse @ A @ ( semiring_1_of_nat @ A @ ( suc @ N3 ) ) ) @ X ) ) ) ).
% reals_Archimedean
thf(fact_3245_of__nat__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [R2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ R2 )
=> ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ ( nat2 @ ( archim6421214686448440834_floor @ A @ R2 ) ) ) @ R2 ) ) ) ).
% of_nat_floor
thf(fact_3246_real__vector__affinity__eq,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [M2: real,X: A,C2: A,Y: A] :
( ( M2
!= ( zero_zero @ real ) )
=> ( ( ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ M2 @ X ) @ C2 )
= Y )
= ( X
= ( minus_minus @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ M2 ) @ Y ) @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ M2 ) @ C2 ) ) ) ) ) ) ).
% real_vector_affinity_eq
thf(fact_3247_real__vector__eq__affinity,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [M2: real,Y: A,X: A,C2: A] :
( ( M2
!= ( zero_zero @ real ) )
=> ( ( Y
= ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ M2 @ X ) @ C2 ) )
= ( ( minus_minus @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ M2 ) @ Y ) @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ M2 ) @ C2 ) )
= X ) ) ) ) ).
% real_vector_eq_affinity
thf(fact_3248_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_3249_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_3250_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_3251_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_3252_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_3253_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_3254_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_3255_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_3256_nat__less__eq__zless,axiom,
! [W2: int,Z3: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ W2 )
=> ( ( ord_less @ nat @ ( nat2 @ W2 ) @ ( nat2 @ Z3 ) )
= ( ord_less @ int @ W2 @ Z3 ) ) ) ).
% nat_less_eq_zless
thf(fact_3257_nat__le__eq__zle,axiom,
! [W2: int,Z3: int] :
( ( ( ord_less @ int @ ( zero_zero @ int ) @ W2 )
| ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z3 ) )
=> ( ( ord_less_eq @ nat @ ( nat2 @ W2 ) @ ( nat2 @ Z3 ) )
= ( ord_less_eq @ int @ W2 @ Z3 ) ) ) ).
% nat_le_eq_zle
thf(fact_3258_nat__eq__iff,axiom,
! [W2: int,M2: nat] :
( ( ( nat2 @ W2 )
= M2 )
= ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ W2 )
=> ( W2
= ( semiring_1_of_nat @ int @ M2 ) ) )
& ( ~ ( ord_less_eq @ int @ ( zero_zero @ int ) @ W2 )
=> ( M2
= ( zero_zero @ nat ) ) ) ) ) ).
% nat_eq_iff
thf(fact_3259_nat__eq__iff2,axiom,
! [M2: nat,W2: int] :
( ( M2
= ( nat2 @ W2 ) )
= ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ W2 )
=> ( W2
= ( semiring_1_of_nat @ int @ M2 ) ) )
& ( ~ ( ord_less_eq @ int @ ( zero_zero @ int ) @ W2 )
=> ( M2
= ( zero_zero @ nat ) ) ) ) ) ).
% nat_eq_iff2
thf(fact_3260_split__nat,axiom,
! [P: nat > $o,I: int] :
( ( P @ ( nat2 @ I ) )
= ( ! [N: nat] :
( ( I
= ( semiring_1_of_nat @ int @ N ) )
=> ( P @ N ) )
& ( ( ord_less @ int @ I @ ( zero_zero @ int ) )
=> ( P @ ( zero_zero @ nat ) ) ) ) ) ).
% split_nat
thf(fact_3261_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_3262_le__nat__iff,axiom,
! [K: int,N2: nat] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
=> ( ( ord_less_eq @ nat @ N2 @ ( nat2 @ K ) )
= ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ N2 ) @ K ) ) ) ).
% le_nat_iff
thf(fact_3263_nat__add__distrib,axiom,
! [Z3: int,Z8: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z3 )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z8 )
=> ( ( nat2 @ ( plus_plus @ int @ Z3 @ Z8 ) )
= ( plus_plus @ nat @ ( nat2 @ Z3 ) @ ( nat2 @ Z8 ) ) ) ) ) ).
% nat_add_distrib
thf(fact_3264_nat__mult__distrib,axiom,
! [Z3: int,Z8: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z3 )
=> ( ( nat2 @ ( times_times @ int @ Z3 @ Z8 ) )
= ( times_times @ nat @ ( nat2 @ Z3 ) @ ( nat2 @ Z8 ) ) ) ) ).
% nat_mult_distrib
thf(fact_3265_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_3266_forall__pos__mono__1,axiom,
! [P: real > $o,E2: real] :
( ! [D5: real,E: real] :
( ( ord_less @ real @ D5 @ E )
=> ( ( P @ D5 )
=> ( P @ E ) ) )
=> ( ! [N3: nat] : ( P @ ( inverse_inverse @ real @ ( semiring_1_of_nat @ real @ ( suc @ N3 ) ) ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ E2 )
=> ( P @ E2 ) ) ) ) ).
% forall_pos_mono_1
thf(fact_3267_real__arch__inverse,axiom,
! [E2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E2 )
= ( ? [N: nat] :
( ( N
!= ( zero_zero @ nat ) )
& ( ord_less @ real @ ( zero_zero @ real ) @ ( inverse_inverse @ real @ ( semiring_1_of_nat @ real @ N ) ) )
& ( ord_less @ real @ ( inverse_inverse @ real @ ( semiring_1_of_nat @ real @ N ) ) @ E2 ) ) ) ) ).
% real_arch_inverse
thf(fact_3268_forall__pos__mono,axiom,
! [P: real > $o,E2: real] :
( ! [D5: real,E: real] :
( ( ord_less @ real @ D5 @ E )
=> ( ( P @ D5 )
=> ( P @ E ) ) )
=> ( ! [N3: nat] :
( ( N3
!= ( zero_zero @ nat ) )
=> ( P @ ( inverse_inverse @ real @ ( semiring_1_of_nat @ real @ N3 ) ) ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ E2 )
=> ( P @ E2 ) ) ) ) ).
% forall_pos_mono
thf(fact_3269_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_3270_nat__diff__distrib,axiom,
! [Z8: int,Z3: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z8 )
=> ( ( ord_less_eq @ int @ Z8 @ Z3 )
=> ( ( nat2 @ ( minus_minus @ int @ Z3 @ Z8 ) )
= ( minus_minus @ nat @ ( nat2 @ Z3 ) @ ( nat2 @ Z8 ) ) ) ) ) ).
% nat_diff_distrib
thf(fact_3271_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_3272_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_3273_nat__power__eq,axiom,
! [Z3: int,N2: nat] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z3 )
=> ( ( nat2 @ ( power_power @ int @ Z3 @ N2 ) )
= ( power_power @ nat @ ( nat2 @ Z3 ) @ N2 ) ) ) ).
% nat_power_eq
thf(fact_3274_floor__eq3,axiom,
! [N2: nat,X: real] :
( ( ord_less @ real @ ( semiring_1_of_nat @ real @ N2 ) @ X )
=> ( ( ord_less @ real @ X @ ( semiring_1_of_nat @ real @ ( suc @ N2 ) ) )
=> ( ( nat2 @ ( archim6421214686448440834_floor @ real @ X ) )
= N2 ) ) ) ).
% floor_eq3
thf(fact_3275_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_3276_ex__inverse__of__nat__less,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ X )
=> ? [N3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
& ( ord_less @ A @ ( inverse_inverse @ A @ ( semiring_1_of_nat @ A @ N3 ) ) @ X ) ) ) ) ).
% ex_inverse_of_nat_less
thf(fact_3277_power__diff__conv__inverse,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X: A,M2: nat,N2: nat] :
( ( X
!= ( zero_zero @ A ) )
=> ( ( ord_less_eq @ nat @ M2 @ N2 )
=> ( ( power_power @ A @ X @ ( minus_minus @ nat @ N2 @ M2 ) )
= ( times_times @ A @ ( power_power @ A @ X @ N2 ) @ ( power_power @ A @ ( inverse_inverse @ A @ X ) @ M2 ) ) ) ) ) ) ).
% power_diff_conv_inverse
thf(fact_3278_nat__2,axiom,
( ( nat2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) )
= ( suc @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% nat_2
thf(fact_3279_cosh__converges,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X: A] :
( sums @ A
@ ^ [N: nat] : ( if @ A @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ A @ X @ N ) ) @ ( zero_zero @ A ) )
@ ( cosh @ A @ X ) ) ) ).
% cosh_converges
thf(fact_3280_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_3281_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_3282_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_3283_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_3284_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_3285_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_3286_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_3287_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_3288_Suc__nat__eq__nat__zadd1,axiom,
! [Z3: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z3 )
=> ( ( suc @ ( nat2 @ Z3 ) )
= ( nat2 @ ( plus_plus @ int @ ( one_one @ int ) @ Z3 ) ) ) ) ).
% Suc_nat_eq_nat_zadd1
thf(fact_3289_nat__less__iff,axiom,
! [W2: int,M2: nat] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ W2 )
=> ( ( ord_less @ nat @ ( nat2 @ W2 ) @ M2 )
= ( ord_less @ int @ W2 @ ( semiring_1_of_nat @ int @ M2 ) ) ) ) ).
% nat_less_iff
thf(fact_3290_nat__mult__distrib__neg,axiom,
! [Z3: int,Z8: int] :
( ( ord_less_eq @ int @ Z3 @ ( zero_zero @ int ) )
=> ( ( nat2 @ ( times_times @ int @ Z3 @ Z8 ) )
= ( times_times @ nat @ ( nat2 @ ( uminus_uminus @ int @ Z3 ) ) @ ( nat2 @ ( uminus_uminus @ int @ Z8 ) ) ) ) ) ).
% nat_mult_distrib_neg
thf(fact_3291_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_3292_floor__eq4,axiom,
! [N2: nat,X: real] :
( ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ N2 ) @ X )
=> ( ( ord_less @ real @ X @ ( semiring_1_of_nat @ real @ ( suc @ N2 ) ) )
=> ( ( nat2 @ ( archim6421214686448440834_floor @ real @ X ) )
= N2 ) ) ) ).
% floor_eq4
thf(fact_3293_diff__nat__eq__if,axiom,
! [Z8: int,Z3: int] :
( ( ( ord_less @ int @ Z8 @ ( zero_zero @ int ) )
=> ( ( minus_minus @ nat @ ( nat2 @ Z3 ) @ ( nat2 @ Z8 ) )
= ( nat2 @ Z3 ) ) )
& ( ~ ( ord_less @ int @ Z8 @ ( zero_zero @ int ) )
=> ( ( minus_minus @ nat @ ( nat2 @ Z3 ) @ ( nat2 @ Z8 ) )
= ( if @ nat @ ( ord_less @ int @ ( minus_minus @ int @ Z3 @ Z8 ) @ ( zero_zero @ int ) ) @ ( zero_zero @ nat ) @ ( nat2 @ ( minus_minus @ int @ Z3 @ Z8 ) ) ) ) ) ) ).
% diff_nat_eq_if
thf(fact_3294_of__int__of__nat,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ( ( ring_1_of_int @ A )
= ( ^ [K2: int] : ( if @ A @ ( ord_less @ int @ K2 @ ( zero_zero @ int ) ) @ ( uminus_uminus @ A @ ( semiring_1_of_nat @ A @ ( nat2 @ ( uminus_uminus @ int @ K2 ) ) ) ) @ ( semiring_1_of_nat @ A @ ( nat2 @ K2 ) ) ) ) ) ) ).
% of_int_of_nat
thf(fact_3295_nat__dvd__iff,axiom,
! [Z3: int,M2: nat] :
( ( dvd_dvd @ nat @ ( nat2 @ Z3 ) @ M2 )
= ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z3 )
=> ( dvd_dvd @ int @ Z3 @ ( semiring_1_of_nat @ int @ M2 ) ) )
& ( ~ ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z3 )
=> ( M2
= ( zero_zero @ nat ) ) ) ) ) ).
% nat_dvd_iff
thf(fact_3296_dbl__dec__def,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( ( neg_numeral_dbl_dec @ A )
= ( ^ [X4: A] : ( minus_minus @ A @ ( plus_plus @ A @ X4 @ X4 ) @ ( one_one @ A ) ) ) ) ) ).
% dbl_dec_def
thf(fact_3297_cosh__field__def,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ( ( cosh @ A )
= ( ^ [Z6: A] : ( divide_divide @ A @ ( plus_plus @ A @ ( exp @ A @ Z6 ) @ ( exp @ A @ ( uminus_uminus @ A @ Z6 ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% cosh_field_def
thf(fact_3298_exp__series__add__commuting,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X: A,Y: A,N2: nat] :
( ( ( times_times @ A @ X @ Y )
= ( times_times @ A @ Y @ X ) )
=> ( ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ N2 ) ) @ ( power_power @ A @ ( plus_plus @ A @ X @ Y ) @ N2 ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ I3 ) ) @ ( power_power @ A @ X @ I3 ) ) @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ ( minus_minus @ nat @ N2 @ I3 ) ) ) @ ( power_power @ A @ Y @ ( minus_minus @ nat @ N2 @ I3 ) ) ) )
@ ( set_ord_atMost @ nat @ N2 ) ) ) ) ) ).
% exp_series_add_commuting
thf(fact_3299_exp__first__term,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ( ( exp @ A )
= ( ^ [X4: A] :
( plus_plus @ A @ ( one_one @ A )
@ ( suminf @ A
@ ^ [N: nat] : ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ ( suc @ N ) ) ) @ ( power_power @ A @ X4 @ ( suc @ N ) ) ) ) ) ) ) ) ).
% exp_first_term
thf(fact_3300_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_3301_sinh__converges,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X: A] :
( sums @ A
@ ^ [N: nat] : ( if @ A @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ ( zero_zero @ A ) @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ A @ X @ N ) ) )
@ ( sinh @ A @ X ) ) ) ).
% sinh_converges
thf(fact_3302_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_3303_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_3304_the__elem__eq,axiom,
! [A: $tType,X: A] :
( ( the_elem @ A @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= X ) ).
% the_elem_eq
thf(fact_3305_bit__horner__sum__bit__iff,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [Bs: list @ $o,N2: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( groups4207007520872428315er_sum @ $o @ A @ ( zero_neq_one_of_bool @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Bs ) @ N2 )
= ( ( ord_less @ nat @ N2 @ ( size_size @ ( list @ $o ) @ Bs ) )
& ( nth @ $o @ Bs @ N2 ) ) ) ) ).
% bit_horner_sum_bit_iff
thf(fact_3306_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_3307_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_3308_sinh__add,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A,Y: A] :
( ( sinh @ A @ ( plus_plus @ A @ X @ Y ) )
= ( plus_plus @ A @ ( times_times @ A @ ( sinh @ A @ X ) @ ( cosh @ A @ Y ) ) @ ( times_times @ A @ ( cosh @ A @ X ) @ ( sinh @ A @ Y ) ) ) ) ) ).
% sinh_add
thf(fact_3309_cosh__add,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A,Y: A] :
( ( cosh @ A @ ( plus_plus @ A @ X @ Y ) )
= ( plus_plus @ A @ ( times_times @ A @ ( cosh @ A @ X ) @ ( cosh @ A @ Y ) ) @ ( times_times @ A @ ( sinh @ A @ X ) @ ( sinh @ A @ Y ) ) ) ) ) ).
% cosh_add
thf(fact_3310_cosh__plus__sinh,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X: A] :
( ( plus_plus @ A @ ( cosh @ A @ X ) @ ( sinh @ A @ X ) )
= ( exp @ A @ X ) ) ) ).
% cosh_plus_sinh
thf(fact_3311_sinh__plus__cosh,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X: A] :
( ( plus_plus @ A @ ( sinh @ A @ X ) @ ( cosh @ A @ X ) )
= ( exp @ A @ X ) ) ) ).
% sinh_plus_cosh
thf(fact_3312_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 ) @ ( insert2 @ A @ ( one_one @ A ) @ ( insert2 @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% sinh_zero_iff
thf(fact_3313_tanh__altdef,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ( ( tanh @ A )
= ( ^ [X4: A] : ( divide_divide @ A @ ( minus_minus @ A @ ( exp @ A @ X4 ) @ ( exp @ A @ ( uminus_uminus @ A @ X4 ) ) ) @ ( plus_plus @ A @ ( exp @ A @ X4 ) @ ( exp @ A @ ( uminus_uminus @ A @ X4 ) ) ) ) ) ) ) ).
% tanh_altdef
thf(fact_3314_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_3315_sum__diff1_H__aux,axiom,
! [B: $tType,A: $tType] :
( ( ab_group_add @ B )
=> ! [F5: set @ A,I6: set @ A,F2: A > B,I: A] :
( ( finite_finite2 @ A @ F5 )
=> ( ( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [I3: A] :
( ( member @ A @ I3 @ I6 )
& ( ( F2 @ I3 )
!= ( zero_zero @ B ) ) ) )
@ F5 )
=> ( ( ( member @ A @ I @ I6 )
=> ( ( groups1027152243600224163dd_sum @ A @ B @ F2 @ ( minus_minus @ ( set @ A ) @ I6 @ ( insert2 @ A @ I @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( minus_minus @ B @ ( groups1027152243600224163dd_sum @ A @ B @ F2 @ I6 ) @ ( F2 @ I ) ) ) )
& ( ~ ( member @ A @ I @ I6 )
=> ( ( groups1027152243600224163dd_sum @ A @ B @ F2 @ ( minus_minus @ ( set @ A ) @ I6 @ ( insert2 @ A @ I @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( groups1027152243600224163dd_sum @ A @ B @ F2 @ I6 ) ) ) ) ) ) ) ).
% sum_diff1'_aux
thf(fact_3316_or__nat__unfold,axiom,
( ( bit_se1065995026697491101ons_or @ nat )
= ( ^ [M: nat,N: nat] :
( if @ nat
@ ( M
= ( zero_zero @ nat ) )
@ N
@ ( if @ nat
@ ( N
= ( zero_zero @ nat ) )
@ M
@ ( plus_plus @ nat @ ( ord_max @ nat @ ( modulo_modulo @ nat @ M @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ nat @ ( divide_divide @ nat @ M @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% or_nat_unfold
thf(fact_3317_Sum__Ico__nat,axiom,
! [M2: nat,N2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [X4: nat] : X4
@ ( set_or7035219750837199246ssThan @ nat @ M2 @ N2 ) )
= ( divide_divide @ nat @ ( minus_minus @ nat @ ( times_times @ nat @ N2 @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) ) @ ( times_times @ nat @ M2 @ ( minus_minus @ nat @ M2 @ ( one_one @ nat ) ) ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% Sum_Ico_nat
thf(fact_3318_VEBT_Osize_I3_J,axiom,
! [X11: option @ ( product_prod @ nat @ nat ),X12: nat,X13: list @ vEBT_VEBT,X14: vEBT_VEBT] :
( ( size_size @ vEBT_VEBT @ ( vEBT_Node @ X11 @ X12 @ X13 @ X14 ) )
= ( plus_plus @ nat @ ( plus_plus @ nat @ ( size_list @ vEBT_VEBT @ ( size_size @ vEBT_VEBT ) @ X13 ) @ ( size_size @ vEBT_VEBT @ X14 ) ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% VEBT.size(3)
thf(fact_3319_Cauchy__iff2,axiom,
( ( topolo3814608138187158403Cauchy @ real )
= ( ^ [X6: nat > real] :
! [J2: nat] :
? [M9: nat] :
! [M: nat] :
( ( ord_less_eq @ nat @ M9 @ M )
=> ! [N: nat] :
( ( ord_less_eq @ nat @ M9 @ N )
=> ( ord_less @ real @ ( abs_abs @ real @ ( minus_minus @ real @ ( X6 @ M ) @ ( X6 @ N ) ) ) @ ( inverse_inverse @ real @ ( semiring_1_of_nat @ real @ ( suc @ J2 ) ) ) ) ) ) ) ) ).
% Cauchy_iff2
thf(fact_3320_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_3321_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_3322_finite__atLeastLessThan,axiom,
! [L: nat,U: nat] : ( finite_finite2 @ nat @ ( set_or7035219750837199246ssThan @ nat @ L @ U ) ) ).
% finite_atLeastLessThan
thf(fact_3323_atLeastLessThan__iff,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [I: A,L: A,U: A] :
( ( member @ A @ I @ ( set_or7035219750837199246ssThan @ A @ L @ U ) )
= ( ( ord_less_eq @ A @ L @ I )
& ( ord_less @ A @ I @ U ) ) ) ) ).
% atLeastLessThan_iff
thf(fact_3324_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_3325_ivl__subset,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [I: A,J: A,M2: A,N2: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ I @ J ) @ ( set_or7035219750837199246ssThan @ A @ M2 @ N2 ) )
= ( ( ord_less_eq @ A @ J @ I )
| ( ( ord_less_eq @ A @ M2 @ I )
& ( ord_less_eq @ A @ J @ N2 ) ) ) ) ) ).
% ivl_subset
thf(fact_3326_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_3327_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_3328_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_3329_ivl__diff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [I: A,N2: A,M2: A] :
( ( ord_less_eq @ A @ I @ N2 )
=> ( ( minus_minus @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ I @ M2 ) @ ( set_or7035219750837199246ssThan @ A @ I @ N2 ) )
= ( set_or7035219750837199246ssThan @ A @ N2 @ M2 ) ) ) ) ).
% ivl_diff
thf(fact_3330_lessThan__minus__lessThan,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [N2: A,M2: A] :
( ( minus_minus @ ( set @ A ) @ ( set_ord_lessThan @ A @ N2 ) @ ( set_ord_lessThan @ A @ M2 ) )
= ( set_or7035219750837199246ssThan @ A @ M2 @ N2 ) ) ) ).
% lessThan_minus_lessThan
thf(fact_3331_sum_Oempty_H,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add @ A )
=> ! [P5: B > A] :
( ( groups1027152243600224163dd_sum @ B @ A @ P5 @ ( bot_bot @ ( set @ B ) ) )
= ( zero_zero @ A ) ) ) ).
% sum.empty'
thf(fact_3332_sum_Oeq__sum,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [I6: set @ B,P5: B > A] :
( ( finite_finite2 @ B @ I6 )
=> ( ( groups1027152243600224163dd_sum @ B @ A @ P5 @ I6 )
= ( groups7311177749621191930dd_sum @ B @ A @ P5 @ I6 ) ) ) ) ).
% sum.eq_sum
thf(fact_3333_atLeastLessThan__singleton,axiom,
! [M2: nat] :
( ( set_or7035219750837199246ssThan @ nat @ M2 @ ( suc @ M2 ) )
= ( insert2 @ nat @ M2 @ ( bot_bot @ ( set @ nat ) ) ) ) ).
% atLeastLessThan_singleton
thf(fact_3334_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_3335_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_3336_sum_Oinsert_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [I6: set @ B,P5: B > A,I: B] :
( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X4: B] :
( ( member @ B @ X4 @ I6 )
& ( ( P5 @ X4 )
!= ( zero_zero @ A ) ) ) ) )
=> ( ( ( member @ B @ I @ I6 )
=> ( ( groups1027152243600224163dd_sum @ B @ A @ P5 @ ( insert2 @ B @ I @ I6 ) )
= ( groups1027152243600224163dd_sum @ B @ A @ P5 @ I6 ) ) )
& ( ~ ( member @ B @ I @ I6 )
=> ( ( groups1027152243600224163dd_sum @ B @ A @ P5 @ ( insert2 @ B @ I @ I6 ) )
= ( plus_plus @ A @ ( P5 @ I ) @ ( groups1027152243600224163dd_sum @ B @ A @ P5 @ I6 ) ) ) ) ) ) ) ).
% sum.insert'
thf(fact_3337_sum_Oop__ivl__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [N2: nat,M2: nat,G: nat > A] :
( ( ( ord_less @ nat @ N2 @ M2 )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M2 @ ( suc @ N2 ) ) )
= ( zero_zero @ A ) ) )
& ( ~ ( ord_less @ nat @ N2 @ M2 )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M2 @ ( suc @ N2 ) ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N2 ) ) @ ( G @ N2 ) ) ) ) ) ) ).
% sum.op_ivl_Suc
thf(fact_3338_prod_Oop__ivl__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [N2: nat,M2: nat,G: nat > A] :
( ( ( ord_less @ nat @ N2 @ M2 )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M2 @ ( suc @ N2 ) ) )
= ( one_one @ A ) ) )
& ( ~ ( ord_less @ nat @ N2 @ M2 )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M2 @ ( suc @ N2 ) ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N2 ) ) @ ( G @ N2 ) ) ) ) ) ) ).
% prod.op_ivl_Suc
thf(fact_3339_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_3340_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_3341_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_3342_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_3343_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_3344_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_3345_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_3346_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_3347_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_3348_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_3349_sum_Onon__neutral_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: B > A,I6: set @ B] :
( ( groups1027152243600224163dd_sum @ B @ A @ G
@ ( collect @ B
@ ^ [X4: B] :
( ( member @ B @ X4 @ I6 )
& ( ( G @ X4 )
!= ( zero_zero @ A ) ) ) ) )
= ( groups1027152243600224163dd_sum @ B @ A @ G @ I6 ) ) ) ).
% sum.non_neutral'
thf(fact_3350_atLeastLessThan__subset__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B3: A,C2: A,D2: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ A3 @ B3 ) @ ( set_or7035219750837199246ssThan @ A @ C2 @ D2 ) )
=> ( ( ord_less_eq @ A @ B3 @ A3 )
| ( ( ord_less_eq @ A @ C2 @ A3 )
& ( ord_less_eq @ A @ B3 @ D2 ) ) ) ) ) ).
% atLeastLessThan_subset_iff
thf(fact_3351_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_3352_ex__nat__less__eq,axiom,
! [N2: nat,P: nat > $o] :
( ( ? [M: nat] :
( ( ord_less @ nat @ M @ N2 )
& ( P @ M ) ) )
= ( ? [X4: nat] :
( ( member @ nat @ X4 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N2 ) )
& ( P @ X4 ) ) ) ) ).
% ex_nat_less_eq
thf(fact_3353_all__nat__less__eq,axiom,
! [N2: nat,P: nat > $o] :
( ( ! [M: nat] :
( ( ord_less @ nat @ M @ N2 )
=> ( P @ M ) ) )
= ( ! [X4: nat] :
( ( member @ nat @ X4 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N2 ) )
=> ( P @ X4 ) ) ) ) ).
% all_nat_less_eq
thf(fact_3354_disjunctive__add,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A,B3: A] :
( ! [N3: nat] :
( ~ ( bit_se5641148757651400278ts_bit @ A @ A3 @ N3 )
| ~ ( bit_se5641148757651400278ts_bit @ A @ B3 @ N3 ) )
=> ( ( plus_plus @ A @ A3 @ B3 )
= ( bit_se1065995026697491101ons_or @ A @ A3 @ B3 ) ) ) ) ).
% disjunctive_add
thf(fact_3355_atLeastLessThanSuc__atLeastAtMost,axiom,
! [L: nat,U: nat] :
( ( set_or7035219750837199246ssThan @ nat @ L @ ( suc @ U ) )
= ( set_or1337092689740270186AtMost @ nat @ L @ U ) ) ).
% atLeastLessThanSuc_atLeastAtMost
thf(fact_3356_lessThan__atLeast0,axiom,
( ( set_ord_lessThan @ nat )
= ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) ) ) ).
% lessThan_atLeast0
thf(fact_3357_atLeastLessThan0,axiom,
! [M2: nat] :
( ( set_or7035219750837199246ssThan @ nat @ M2 @ ( zero_zero @ nat ) )
= ( bot_bot @ ( set @ nat ) ) ) ).
% atLeastLessThan0
thf(fact_3358_sum_Oshift__bounds__Suc__ivl,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,M2: nat,N2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M2 ) @ ( suc @ N2 ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_or7035219750837199246ssThan @ nat @ M2 @ N2 ) ) ) ) ).
% sum.shift_bounds_Suc_ivl
thf(fact_3359_sum_Oshift__bounds__nat__ivl,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,M2: nat,K: nat,N2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( plus_plus @ nat @ M2 @ K ) @ ( plus_plus @ nat @ N2 @ K ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( G @ ( plus_plus @ nat @ I3 @ K ) )
@ ( set_or7035219750837199246ssThan @ nat @ M2 @ N2 ) ) ) ) ).
% sum.shift_bounds_nat_ivl
thf(fact_3360_prod_Oshift__bounds__Suc__ivl,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,M2: nat,N2: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M2 ) @ ( suc @ N2 ) ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_or7035219750837199246ssThan @ nat @ M2 @ N2 ) ) ) ) ).
% prod.shift_bounds_Suc_ivl
thf(fact_3361_prod_Oshift__bounds__nat__ivl,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,M2: nat,K: nat,N2: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( plus_plus @ nat @ M2 @ K ) @ ( plus_plus @ nat @ N2 @ K ) ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( G @ ( plus_plus @ nat @ I3 @ K ) )
@ ( set_or7035219750837199246ssThan @ nat @ M2 @ N2 ) ) ) ) ).
% prod.shift_bounds_nat_ivl
thf(fact_3362_sum_Odistrib__triv_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [I6: set @ B,G: B > A,H2: B > A] :
( ( finite_finite2 @ B @ I6 )
=> ( ( groups1027152243600224163dd_sum @ B @ A
@ ^ [I3: B] : ( plus_plus @ A @ ( G @ I3 ) @ ( H2 @ I3 ) )
@ I6 )
= ( plus_plus @ A @ ( groups1027152243600224163dd_sum @ B @ A @ G @ I6 ) @ ( groups1027152243600224163dd_sum @ B @ A @ H2 @ I6 ) ) ) ) ) ).
% sum.distrib_triv'
thf(fact_3363_sum_Oivl__cong,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( comm_monoid_add @ A ) )
=> ! [A3: B,C2: B,B3: B,D2: B,G: B > A,H2: B > A] :
( ( A3 = C2 )
=> ( ( B3 = D2 )
=> ( ! [X5: B] :
( ( ord_less_eq @ B @ C2 @ X5 )
=> ( ( ord_less @ B @ X5 @ D2 )
=> ( ( G @ X5 )
= ( H2 @ X5 ) ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( set_or7035219750837199246ssThan @ B @ A3 @ B3 ) )
= ( groups7311177749621191930dd_sum @ B @ A @ H2 @ ( set_or7035219750837199246ssThan @ B @ C2 @ D2 ) ) ) ) ) ) ) ).
% sum.ivl_cong
thf(fact_3364_prod_Oivl__cong,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( comm_monoid_mult @ A ) )
=> ! [A3: B,C2: B,B3: B,D2: B,G: B > A,H2: B > A] :
( ( A3 = C2 )
=> ( ( B3 = D2 )
=> ( ! [X5: B] :
( ( ord_less_eq @ B @ C2 @ X5 )
=> ( ( ord_less @ B @ X5 @ D2 )
=> ( ( G @ X5 )
= ( H2 @ X5 ) ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( set_or7035219750837199246ssThan @ B @ A3 @ B3 ) )
= ( groups7121269368397514597t_prod @ B @ A @ H2 @ ( set_or7035219750837199246ssThan @ B @ C2 @ D2 ) ) ) ) ) ) ) ).
% prod.ivl_cong
thf(fact_3365_sum_OatLeastLessThan__concat,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [M2: nat,N2: nat,P5: nat,G: nat > A] :
( ( ord_less_eq @ nat @ M2 @ N2 )
=> ( ( ord_less_eq @ nat @ N2 @ P5 )
=> ( ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N2 ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ N2 @ P5 ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M2 @ P5 ) ) ) ) ) ) ).
% sum.atLeastLessThan_concat
thf(fact_3366_sum__diff__nat__ivl,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [M2: nat,N2: nat,P5: nat,F2: nat > A] :
( ( ord_less_eq @ nat @ M2 @ N2 )
=> ( ( ord_less_eq @ nat @ N2 @ P5 )
=> ( ( minus_minus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_or7035219750837199246ssThan @ nat @ M2 @ P5 ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N2 ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_or7035219750837199246ssThan @ nat @ N2 @ P5 ) ) ) ) ) ) ).
% sum_diff_nat_ivl
thf(fact_3367_size__list__estimation,axiom,
! [A: $tType,X: A,Xs: list @ A,Y: nat,F2: A > nat] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ( ord_less @ nat @ Y @ ( F2 @ X ) )
=> ( ord_less @ nat @ Y @ ( size_list @ A @ F2 @ Xs ) ) ) ) ).
% size_list_estimation
thf(fact_3368_size__list__pointwise,axiom,
! [A: $tType,Xs: list @ A,F2: A > nat,G: A > nat] :
( ! [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Xs ) )
=> ( ord_less_eq @ nat @ ( F2 @ X5 ) @ ( G @ X5 ) ) )
=> ( ord_less_eq @ nat @ ( size_list @ A @ F2 @ Xs ) @ ( size_list @ A @ G @ Xs ) ) ) ).
% size_list_pointwise
thf(fact_3369_size__list__estimation_H,axiom,
! [A: $tType,X: A,Xs: list @ A,Y: nat,F2: A > nat] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ( ord_less_eq @ nat @ Y @ ( F2 @ X ) )
=> ( ord_less_eq @ nat @ Y @ ( size_list @ A @ F2 @ Xs ) ) ) ) ).
% size_list_estimation'
thf(fact_3370_prod_OatLeastLessThan__concat,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M2: nat,N2: nat,P5: nat,G: nat > A] :
( ( ord_less_eq @ nat @ M2 @ N2 )
=> ( ( ord_less_eq @ nat @ N2 @ P5 )
=> ( ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N2 ) ) @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ N2 @ P5 ) ) )
= ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M2 @ P5 ) ) ) ) ) ) ).
% prod.atLeastLessThan_concat
thf(fact_3371_atLeast0__lessThan__Suc,axiom,
! [N2: nat] :
( ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( suc @ N2 ) )
= ( insert2 @ nat @ N2 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ).
% atLeast0_lessThan_Suc
thf(fact_3372_subset__eq__atLeast0__lessThan__finite,axiom,
! [N6: set @ nat,N2: nat] :
( ( ord_less_eq @ ( set @ nat ) @ N6 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N2 ) )
=> ( finite_finite2 @ nat @ N6 ) ) ).
% subset_eq_atLeast0_lessThan_finite
thf(fact_3373_sum_Omono__neutral__left_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [S2: set @ B,T4: set @ B,G: B > A] :
( ( ord_less_eq @ ( set @ B ) @ S2 @ T4 )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ ( minus_minus @ ( set @ B ) @ T4 @ S2 ) )
=> ( ( G @ X5 )
= ( zero_zero @ A ) ) )
=> ( ( groups1027152243600224163dd_sum @ B @ A @ G @ S2 )
= ( groups1027152243600224163dd_sum @ B @ A @ G @ T4 ) ) ) ) ) ).
% sum.mono_neutral_left'
thf(fact_3374_sum_Omono__neutral__right_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [S2: set @ B,T4: set @ B,G: B > A] :
( ( ord_less_eq @ ( set @ B ) @ S2 @ T4 )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ ( minus_minus @ ( set @ B ) @ T4 @ S2 ) )
=> ( ( G @ X5 )
= ( zero_zero @ A ) ) )
=> ( ( groups1027152243600224163dd_sum @ B @ A @ G @ T4 )
= ( groups1027152243600224163dd_sum @ B @ A @ G @ S2 ) ) ) ) ) ).
% sum.mono_neutral_right'
thf(fact_3375_sum_Omono__neutral__cong__left_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [S2: set @ B,T4: set @ B,H2: B > A,G: B > A] :
( ( ord_less_eq @ ( set @ B ) @ S2 @ T4 )
=> ( ! [I4: B] :
( ( member @ B @ I4 @ ( minus_minus @ ( set @ B ) @ T4 @ S2 ) )
=> ( ( H2 @ I4 )
= ( zero_zero @ A ) ) )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ S2 )
=> ( ( G @ X5 )
= ( H2 @ X5 ) ) )
=> ( ( groups1027152243600224163dd_sum @ B @ A @ G @ S2 )
= ( groups1027152243600224163dd_sum @ B @ A @ H2 @ T4 ) ) ) ) ) ) ).
% sum.mono_neutral_cong_left'
thf(fact_3376_sum_Omono__neutral__cong__right_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [S2: set @ B,T4: set @ B,G: B > A,H2: B > A] :
( ( ord_less_eq @ ( set @ B ) @ S2 @ T4 )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ ( minus_minus @ ( set @ B ) @ T4 @ S2 ) )
=> ( ( G @ X5 )
= ( zero_zero @ A ) ) )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ S2 )
=> ( ( G @ X5 )
= ( H2 @ X5 ) ) )
=> ( ( groups1027152243600224163dd_sum @ B @ A @ G @ T4 )
= ( groups1027152243600224163dd_sum @ B @ A @ H2 @ S2 ) ) ) ) ) ) ).
% sum.mono_neutral_cong_right'
thf(fact_3377_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_3378_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_3379_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_3380_sum__shift__lb__Suc0__0__upt,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [F2: nat > A,K: nat] :
( ( ( F2 @ ( zero_zero @ nat ) )
= ( zero_zero @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ ( zero_zero @ nat ) ) @ K ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ K ) ) ) ) ) ).
% sum_shift_lb_Suc0_0_upt
thf(fact_3381_sum_OatLeast0__lessThan__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,N2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( suc @ N2 ) ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N2 ) ) @ ( G @ N2 ) ) ) ) ).
% sum.atLeast0_lessThan_Suc
thf(fact_3382_sum_OatLeast__Suc__lessThan,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [M2: nat,N2: nat,G: nat > A] :
( ( ord_less @ nat @ M2 @ N2 )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N2 ) )
= ( plus_plus @ A @ ( G @ M2 ) @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M2 ) @ N2 ) ) ) ) ) ) ).
% sum.atLeast_Suc_lessThan
thf(fact_3383_sum_OatLeastLessThan__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A3: nat,B3: nat,G: nat > A] :
( ( ord_less_eq @ nat @ A3 @ B3 )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ A3 @ ( suc @ B3 ) ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ A3 @ B3 ) ) @ ( G @ B3 ) ) ) ) ) ).
% sum.atLeastLessThan_Suc
thf(fact_3384_sum_Odistrib_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [I6: set @ B,G: B > A,H2: B > A] :
( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X4: B] :
( ( member @ B @ X4 @ I6 )
& ( ( G @ X4 )
!= ( zero_zero @ A ) ) ) ) )
=> ( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X4: B] :
( ( member @ B @ X4 @ I6 )
& ( ( H2 @ X4 )
!= ( zero_zero @ A ) ) ) ) )
=> ( ( groups1027152243600224163dd_sum @ B @ A
@ ^ [I3: B] : ( plus_plus @ A @ ( G @ I3 ) @ ( H2 @ I3 ) )
@ I6 )
= ( plus_plus @ A @ ( groups1027152243600224163dd_sum @ B @ A @ G @ I6 ) @ ( groups1027152243600224163dd_sum @ B @ A @ H2 @ I6 ) ) ) ) ) ) ).
% sum.distrib'
thf(fact_3385_atLeastLessThan__eq__atLeastAtMost__diff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( set_or7035219750837199246ssThan @ A )
= ( ^ [A5: A,B4: A] : ( minus_minus @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A5 @ B4 ) @ ( insert2 @ A @ B4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% atLeastLessThan_eq_atLeastAtMost_diff
thf(fact_3386_prod_OatLeast0__lessThan__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,N2: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( suc @ N2 ) ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N2 ) ) @ ( G @ N2 ) ) ) ) ).
% prod.atLeast0_lessThan_Suc
thf(fact_3387_prod_OatLeast__Suc__lessThan,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M2: nat,N2: nat,G: nat > A] :
( ( ord_less @ nat @ M2 @ N2 )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N2 ) )
= ( times_times @ A @ ( G @ M2 ) @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M2 ) @ N2 ) ) ) ) ) ) ).
% prod.atLeast_Suc_lessThan
thf(fact_3388_prod_OatLeastLessThan__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: nat,B3: nat,G: nat > A] :
( ( ord_less_eq @ nat @ A3 @ B3 )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ A3 @ ( suc @ B3 ) ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ A3 @ B3 ) ) @ ( G @ B3 ) ) ) ) ) ).
% prod.atLeastLessThan_Suc
thf(fact_3389_sum_OG__def,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ( ( groups1027152243600224163dd_sum @ B @ A )
= ( ^ [P6: B > A,I8: set @ B] :
( if @ A
@ ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X4: B] :
( ( member @ B @ X4 @ I8 )
& ( ( P6 @ X4 )
!= ( zero_zero @ A ) ) ) ) )
@ ( groups7311177749621191930dd_sum @ B @ A @ P6
@ ( collect @ B
@ ^ [X4: B] :
( ( member @ B @ X4 @ I8 )
& ( ( P6 @ X4 )
!= ( zero_zero @ A ) ) ) ) )
@ ( zero_zero @ A ) ) ) ) ) ).
% sum.G_def
thf(fact_3390_sum_Olast__plus,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [M2: nat,N2: nat,G: nat > A] :
( ( ord_less_eq @ nat @ M2 @ N2 )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N2 ) )
= ( plus_plus @ A @ ( G @ N2 ) @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N2 ) ) ) ) ) ) ).
% sum.last_plus
thf(fact_3391_prod_Olast__plus,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M2: nat,N2: nat,G: nat > A] :
( ( ord_less_eq @ nat @ M2 @ N2 )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N2 ) )
= ( times_times @ A @ ( G @ N2 ) @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N2 ) ) ) ) ) ) ).
% prod.last_plus
thf(fact_3392_sum__Suc__diff_H,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [M2: nat,N2: nat,F2: nat > A] :
( ( ord_less_eq @ nat @ M2 @ N2 )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( minus_minus @ A @ ( F2 @ ( suc @ I3 ) ) @ ( F2 @ I3 ) )
@ ( set_or7035219750837199246ssThan @ nat @ M2 @ N2 ) )
= ( minus_minus @ A @ ( F2 @ N2 ) @ ( F2 @ M2 ) ) ) ) ) ).
% sum_Suc_diff'
thf(fact_3393_sum_OatLeastLessThan__rev,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,N2: nat,M2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ N2 @ M2 ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( G @ ( minus_minus @ nat @ ( plus_plus @ nat @ M2 @ N2 ) @ ( suc @ I3 ) ) )
@ ( set_or7035219750837199246ssThan @ nat @ N2 @ M2 ) ) ) ) ).
% sum.atLeastLessThan_rev
thf(fact_3394_atLeastLessThanSuc,axiom,
! [M2: nat,N2: nat] :
( ( ( ord_less_eq @ nat @ M2 @ N2 )
=> ( ( set_or7035219750837199246ssThan @ nat @ M2 @ ( suc @ N2 ) )
= ( insert2 @ nat @ N2 @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N2 ) ) ) )
& ( ~ ( ord_less_eq @ nat @ M2 @ N2 )
=> ( ( set_or7035219750837199246ssThan @ nat @ M2 @ ( suc @ N2 ) )
= ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% atLeastLessThanSuc
thf(fact_3395_sum_Onested__swap,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A3: nat > nat > A,N2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( groups7311177749621191930dd_sum @ nat @ A @ ( A3 @ I3 ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ I3 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N2 ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [J2: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( A3 @ I3 @ J2 )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ J2 ) @ N2 ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ) ).
% sum.nested_swap
thf(fact_3396_prod_OatLeastLessThan__rev,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,N2: nat,M2: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ N2 @ M2 ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( G @ ( minus_minus @ nat @ ( plus_plus @ nat @ M2 @ N2 ) @ ( suc @ I3 ) ) )
@ ( set_or7035219750837199246ssThan @ nat @ N2 @ M2 ) ) ) ) ).
% prod.atLeastLessThan_rev
thf(fact_3397_prod_Onested__swap,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: nat > nat > A,N2: nat] :
( ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( groups7121269368397514597t_prod @ nat @ A @ ( A3 @ I3 ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ I3 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N2 ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [J2: nat] :
( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( A3 @ I3 @ J2 )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ J2 ) @ N2 ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ) ).
% prod.nested_swap
thf(fact_3398_sum_Onat__group,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,K: nat,N2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [M: nat] : ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( times_times @ nat @ M @ K ) @ ( plus_plus @ nat @ ( times_times @ nat @ M @ K ) @ K ) ) )
@ ( set_ord_lessThan @ nat @ N2 ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_ord_lessThan @ nat @ ( times_times @ nat @ N2 @ K ) ) ) ) ) ).
% sum.nat_group
thf(fact_3399_prod_Onat__group,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,K: nat,N2: nat] :
( ( groups7121269368397514597t_prod @ nat @ A
@ ^ [M: nat] : ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( times_times @ nat @ M @ K ) @ ( plus_plus @ nat @ ( times_times @ nat @ M @ K ) @ K ) ) )
@ ( set_ord_lessThan @ nat @ N2 ) )
= ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_lessThan @ nat @ ( times_times @ nat @ N2 @ K ) ) ) ) ) ).
% prod.nat_group
thf(fact_3400_prod__Suc__Suc__fact,axiom,
! [N2: nat] :
( ( groups7121269368397514597t_prod @ nat @ nat @ suc @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N2 ) )
= ( semiring_char_0_fact @ nat @ N2 ) ) ).
% prod_Suc_Suc_fact
thf(fact_3401_prod__Suc__fact,axiom,
! [N2: nat] :
( ( groups7121269368397514597t_prod @ nat @ nat @ suc @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N2 ) )
= ( semiring_char_0_fact @ nat @ N2 ) ) ).
% prod_Suc_fact
thf(fact_3402_sum_Ohead__if,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [N2: nat,M2: nat,G: nat > A] :
( ( ( ord_less @ nat @ N2 @ M2 )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N2 ) )
= ( zero_zero @ A ) ) )
& ( ~ ( ord_less @ nat @ N2 @ M2 )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N2 ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N2 ) ) @ ( G @ N2 ) ) ) ) ) ) ).
% sum.head_if
thf(fact_3403_prod_Ohead__if,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [N2: nat,M2: nat,G: nat > A] :
( ( ( ord_less @ nat @ N2 @ M2 )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N2 ) )
= ( one_one @ A ) ) )
& ( ~ ( ord_less @ nat @ N2 @ M2 )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N2 ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N2 ) ) @ ( G @ N2 ) ) ) ) ) ) ).
% prod.head_if
thf(fact_3404_fact__prod__Suc,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ( ( semiring_char_0_fact @ A )
= ( ^ [N: nat] : ( semiring_1_of_nat @ A @ ( groups7121269368397514597t_prod @ nat @ nat @ suc @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ) ) ).
% fact_prod_Suc
thf(fact_3405_sum_OatLeastLessThan__rev__at__least__Suc__atMost,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,N2: nat,M2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ N2 @ M2 ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( G @ ( minus_minus @ nat @ ( plus_plus @ nat @ M2 @ N2 ) @ I3 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ N2 ) @ M2 ) ) ) ) ).
% sum.atLeastLessThan_rev_at_least_Suc_atMost
thf(fact_3406_prod_OatLeastLessThan__rev__at__least__Suc__atMost,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,N2: nat,M2: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ N2 @ M2 ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( G @ ( minus_minus @ nat @ ( plus_plus @ nat @ M2 @ N2 ) @ I3 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ N2 ) @ M2 ) ) ) ) ).
% prod.atLeastLessThan_rev_at_least_Suc_atMost
thf(fact_3407_pochhammer__prod,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ( ( comm_s3205402744901411588hammer @ A )
= ( ^ [A5: A,N: nat] :
( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( plus_plus @ A @ A5 @ ( semiring_1_of_nat @ A @ I3 ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ) ).
% pochhammer_prod
thf(fact_3408_atLeastLessThan__nat__numeral,axiom,
! [M2: nat,K: num] :
( ( ( ord_less_eq @ nat @ M2 @ ( pred_numeral @ K ) )
=> ( ( set_or7035219750837199246ssThan @ nat @ M2 @ ( numeral_numeral @ nat @ K ) )
= ( insert2 @ nat @ ( pred_numeral @ K ) @ ( set_or7035219750837199246ssThan @ nat @ M2 @ ( pred_numeral @ K ) ) ) ) )
& ( ~ ( ord_less_eq @ nat @ M2 @ ( pred_numeral @ K ) )
=> ( ( set_or7035219750837199246ssThan @ nat @ M2 @ ( numeral_numeral @ nat @ K ) )
= ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% atLeastLessThan_nat_numeral
thf(fact_3409_fact__prod__rev,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ( ( semiring_char_0_fact @ A )
= ( ^ [N: nat] : ( semiring_1_of_nat @ A @ ( groups7121269368397514597t_prod @ nat @ nat @ ( minus_minus @ nat @ N ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ) ) ).
% fact_prod_rev
thf(fact_3410_mask__Suc__exp,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N2: nat] :
( ( bit_se2239418461657761734s_mask @ A @ ( suc @ N2 ) )
= ( bit_se1065995026697491101ons_or @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) @ ( bit_se2239418461657761734s_mask @ A @ N2 ) ) ) ) ).
% mask_Suc_exp
thf(fact_3411_summable__Cauchy,axiom,
! [A: $tType] :
( ( real_Vector_banach @ A )
=> ( ( summable @ A )
= ( ^ [F3: nat > A] :
! [E3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E3 )
=> ? [N5: nat] :
! [M: nat] :
( ( ord_less_eq @ nat @ N5 @ M )
=> ! [N: nat] : ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) ) @ E3 ) ) ) ) ) ) ).
% summable_Cauchy
thf(fact_3412_CauchyD,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X8: nat > A,E2: real] :
( ( topolo3814608138187158403Cauchy @ A @ X8 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ E2 )
=> ? [M8: nat] :
! [M5: nat] :
( ( ord_less_eq @ nat @ M8 @ M5 )
=> ! [N7: nat] :
( ( ord_less_eq @ nat @ M8 @ N7 )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( X8 @ M5 ) @ ( X8 @ N7 ) ) ) @ E2 ) ) ) ) ) ) ).
% CauchyD
thf(fact_3413_CauchyI,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X8: nat > A] :
( ! [E: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E )
=> ? [M10: nat] :
! [M3: nat] :
( ( ord_less_eq @ nat @ M10 @ M3 )
=> ! [N3: nat] :
( ( ord_less_eq @ nat @ M10 @ N3 )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( X8 @ M3 ) @ ( X8 @ N3 ) ) ) @ E ) ) ) )
=> ( topolo3814608138187158403Cauchy @ A @ X8 ) ) ) ).
% CauchyI
thf(fact_3414_Cauchy__iff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ( ( topolo3814608138187158403Cauchy @ A )
= ( ^ [X6: nat > A] :
! [E3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E3 )
=> ? [M9: nat] :
! [M: nat] :
( ( ord_less_eq @ nat @ M9 @ M )
=> ! [N: nat] :
( ( ord_less_eq @ nat @ M9 @ N )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( X6 @ M ) @ ( X6 @ N ) ) ) @ E3 ) ) ) ) ) ) ) ).
% Cauchy_iff
thf(fact_3415_sums__group,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [F2: nat > A,S: A,K: nat] :
( ( sums @ A @ F2 @ S )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( sums @ A
@ ^ [N: nat] : ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_or7035219750837199246ssThan @ nat @ ( times_times @ nat @ N @ K ) @ ( plus_plus @ nat @ ( times_times @ nat @ N @ K ) @ K ) ) )
@ S ) ) ) ) ).
% sums_group
thf(fact_3416_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_3417_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_3418_mask__Suc__double,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N2: nat] :
( ( bit_se2239418461657761734s_mask @ A @ ( suc @ N2 ) )
= ( bit_se1065995026697491101ons_or @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se2239418461657761734s_mask @ A @ N2 ) ) ) ) ) ).
% mask_Suc_double
thf(fact_3419_atLeast1__lessThan__eq__remove0,axiom,
! [N2: nat] :
( ( set_or7035219750837199246ssThan @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N2 )
= ( minus_minus @ ( set @ nat ) @ ( set_ord_lessThan @ nat @ N2 ) @ ( insert2 @ nat @ ( zero_zero @ nat ) @ ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% atLeast1_lessThan_eq_remove0
thf(fact_3420_fact__split,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [K: nat,N2: nat] :
( ( ord_less_eq @ nat @ K @ N2 )
=> ( ( semiring_char_0_fact @ A @ N2 )
= ( times_times @ A @ ( semiring_1_of_nat @ A @ ( groups7121269368397514597t_prod @ nat @ nat @ suc @ ( set_or7035219750837199246ssThan @ nat @ ( minus_minus @ nat @ N2 @ K ) @ N2 ) ) ) @ ( semiring_char_0_fact @ A @ ( minus_minus @ nat @ N2 @ K ) ) ) ) ) ) ).
% fact_split
thf(fact_3421_binomial__altdef__of__nat,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K: nat,N2: nat] :
( ( ord_less_eq @ nat @ K @ N2 )
=> ( ( semiring_1_of_nat @ A @ ( binomial @ N2 @ K ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( divide_divide @ A @ ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ N2 @ I3 ) ) @ ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ K @ I3 ) ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ K ) ) ) ) ) ).
% binomial_altdef_of_nat
thf(fact_3422_gbinomial__altdef__of__nat,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( ( gbinomial @ A )
= ( ^ [A5: A,K2: nat] :
( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( divide_divide @ A @ ( minus_minus @ A @ A5 @ ( semiring_1_of_nat @ A @ I3 ) ) @ ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ K2 @ I3 ) ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ K2 ) ) ) ) ) ).
% gbinomial_altdef_of_nat
thf(fact_3423_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
@ ^ [I3: nat] : ( minus_minus @ A @ A3 @ ( semiring_1_of_nat @ A @ I3 ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ K ) ) ) ) ).
% gbinomial_mult_fact'
thf(fact_3424_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
@ ^ [I3: nat] : ( minus_minus @ A @ A3 @ ( semiring_1_of_nat @ A @ I3 ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ K ) ) ) ) ).
% gbinomial_mult_fact
thf(fact_3425_gbinomial__prod__rev,axiom,
! [A: $tType] :
( ( ( semiring_char_0 @ A )
& ( semidom_divide @ A ) )
=> ( ( gbinomial @ A )
= ( ^ [A5: A,K2: nat] :
( divide_divide @ A
@ ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( minus_minus @ A @ A5 @ ( semiring_1_of_nat @ A @ I3 ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ K2 ) )
@ ( semiring_char_0_fact @ A @ K2 ) ) ) ) ) ).
% gbinomial_prod_rev
thf(fact_3426_sum__diff1_H,axiom,
! [B: $tType,A: $tType] :
( ( ab_group_add @ B )
=> ! [I6: set @ A,F2: A > B,I: A] :
( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [I3: A] :
( ( member @ A @ I3 @ I6 )
& ( ( F2 @ I3 )
!= ( zero_zero @ B ) ) ) ) )
=> ( ( ( member @ A @ I @ I6 )
=> ( ( groups1027152243600224163dd_sum @ A @ B @ F2 @ ( minus_minus @ ( set @ A ) @ I6 @ ( insert2 @ A @ I @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( minus_minus @ B @ ( groups1027152243600224163dd_sum @ A @ B @ F2 @ I6 ) @ ( F2 @ I ) ) ) )
& ( ~ ( member @ A @ I @ I6 )
=> ( ( groups1027152243600224163dd_sum @ A @ B @ F2 @ ( minus_minus @ ( set @ A ) @ I6 @ ( insert2 @ A @ I @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( groups1027152243600224163dd_sum @ A @ B @ F2 @ I6 ) ) ) ) ) ) ).
% sum_diff1'
thf(fact_3427_Suc__0__or__eq,axiom,
! [N2: nat] :
( ( bit_se1065995026697491101ons_or @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N2 )
= ( plus_plus @ nat @ N2 @ ( zero_neq_one_of_bool @ nat @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ).
% Suc_0_or_eq
thf(fact_3428_or__Suc__0__eq,axiom,
! [N2: nat] :
( ( bit_se1065995026697491101ons_or @ nat @ N2 @ ( suc @ ( zero_zero @ nat ) ) )
= ( plus_plus @ nat @ N2 @ ( zero_neq_one_of_bool @ nat @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ).
% or_Suc_0_eq
thf(fact_3429_or__nat__rec,axiom,
( ( bit_se1065995026697491101ons_or @ nat )
= ( ^ [M: nat,N: nat] :
( plus_plus @ nat
@ ( zero_neq_one_of_bool @ nat
@ ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M )
| ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) )
@ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ nat @ ( divide_divide @ nat @ M @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% or_nat_rec
thf(fact_3430_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_3431_horner__sum__eq__sum,axiom,
! [A: $tType,B: $tType] :
( ( comm_semiring_1 @ A )
=> ( ( groups4207007520872428315er_sum @ B @ A )
= ( ^ [F3: B > A,A5: A,Xs3: list @ B] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [N: nat] : ( times_times @ A @ ( F3 @ ( nth @ B @ Xs3 @ N ) ) @ ( power_power @ A @ A5 @ N ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ B ) @ Xs3 ) ) ) ) ) ) ).
% horner_sum_eq_sum
thf(fact_3432_Chebyshev__sum__upper,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N2: nat,A3: nat > A,B3: nat > A] :
( ! [I4: nat,J3: nat] :
( ( ord_less_eq @ nat @ I4 @ J3 )
=> ( ( ord_less @ nat @ J3 @ N2 )
=> ( ord_less_eq @ A @ ( A3 @ I4 ) @ ( A3 @ J3 ) ) ) )
=> ( ! [I4: nat,J3: nat] :
( ( ord_less_eq @ nat @ I4 @ J3 )
=> ( ( ord_less @ nat @ J3 @ N2 )
=> ( ord_less_eq @ A @ ( B3 @ J3 ) @ ( B3 @ I4 ) ) ) )
=> ( ord_less_eq @ A
@ ( times_times @ A @ ( semiring_1_of_nat @ A @ N2 )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K2: nat] : ( times_times @ A @ ( A3 @ K2 ) @ ( B3 @ K2 ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N2 ) ) )
@ ( times_times @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ A3 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N2 ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ B3 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ) ) ) ) ).
% Chebyshev_sum_upper
thf(fact_3433_Chebyshev__sum__upper__nat,axiom,
! [N2: nat,A3: nat > nat,B3: nat > nat] :
( ! [I4: nat,J3: nat] :
( ( ord_less_eq @ nat @ I4 @ J3 )
=> ( ( ord_less @ nat @ J3 @ N2 )
=> ( ord_less_eq @ nat @ ( A3 @ I4 ) @ ( A3 @ J3 ) ) ) )
=> ( ! [I4: nat,J3: nat] :
( ( ord_less_eq @ nat @ I4 @ J3 )
=> ( ( ord_less @ nat @ J3 @ N2 )
=> ( ord_less_eq @ nat @ ( B3 @ J3 ) @ ( B3 @ I4 ) ) ) )
=> ( ord_less_eq @ nat
@ ( times_times @ nat @ N2
@ ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [I3: nat] : ( times_times @ nat @ ( A3 @ I3 ) @ ( B3 @ I3 ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N2 ) ) )
@ ( times_times @ nat @ ( groups7311177749621191930dd_sum @ nat @ nat @ A3 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N2 ) ) @ ( groups7311177749621191930dd_sum @ nat @ nat @ B3 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ) ) ) ).
% Chebyshev_sum_upper_nat
thf(fact_3434_VEBT_Osize__gen_I1_J,axiom,
! [X11: option @ ( product_prod @ nat @ nat ),X12: nat,X13: list @ vEBT_VEBT,X14: vEBT_VEBT] :
( ( vEBT_size_VEBT @ ( vEBT_Node @ X11 @ X12 @ X13 @ X14 ) )
= ( plus_plus @ nat @ ( plus_plus @ nat @ ( size_list @ vEBT_VEBT @ vEBT_size_VEBT @ X13 ) @ ( vEBT_size_VEBT @ X14 ) ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% VEBT.size_gen(1)
thf(fact_3435_is__singleton__the__elem,axiom,
! [A: $tType] :
( ( is_singleton @ A )
= ( ^ [A7: set @ A] :
( A7
= ( insert2 @ A @ ( the_elem @ A @ A7 ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% is_singleton_the_elem
thf(fact_3436_finite__atLeastLessThan__int,axiom,
! [L: int,U: int] : ( finite_finite2 @ int @ ( set_or7035219750837199246ssThan @ int @ L @ U ) ) ).
% finite_atLeastLessThan_int
thf(fact_3437_is__singletonI,axiom,
! [A: $tType,X: A] : ( is_singleton @ A @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ).
% is_singletonI
thf(fact_3438_finite__atLeastZeroLessThan__int,axiom,
! [U: int] : ( finite_finite2 @ int @ ( set_or7035219750837199246ssThan @ int @ ( zero_zero @ int ) @ U ) ) ).
% finite_atLeastZeroLessThan_int
thf(fact_3439_is__singletonI_H,axiom,
! [A: $tType,A4: set @ A] :
( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X5: A,Y3: A] :
( ( member @ A @ X5 @ A4 )
=> ( ( member @ A @ Y3 @ A4 )
=> ( X5 = Y3 ) ) )
=> ( is_singleton @ A @ A4 ) ) ) ).
% is_singletonI'
thf(fact_3440_atLeastLessThanPlusOne__atLeastAtMost__int,axiom,
! [L: int,U: int] :
( ( set_or7035219750837199246ssThan @ int @ L @ ( plus_plus @ int @ U @ ( one_one @ int ) ) )
= ( set_or1337092689740270186AtMost @ int @ L @ U ) ) ).
% atLeastLessThanPlusOne_atLeastAtMost_int
thf(fact_3441_is__singletonE,axiom,
! [A: $tType,A4: set @ A] :
( ( is_singleton @ A @ A4 )
=> ~ ! [X5: A] :
( A4
!= ( insert2 @ A @ X5 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% is_singletonE
thf(fact_3442_is__singleton__def,axiom,
! [A: $tType] :
( ( is_singleton @ A )
= ( ^ [A7: set @ A] :
? [X4: A] :
( A7
= ( insert2 @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% is_singleton_def
thf(fact_3443_VEBT_Osize__gen_I2_J,axiom,
! [X21: $o,X222: $o] :
( ( vEBT_size_VEBT @ ( vEBT_Leaf @ X21 @ X222 ) )
= ( zero_zero @ nat ) ) ).
% VEBT.size_gen(2)
thf(fact_3444_length__subseqs,axiom,
! [A: $tType,Xs: list @ A] :
( ( size_size @ ( list @ ( list @ A ) ) @ ( subseqs @ A @ Xs ) )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ).
% length_subseqs
thf(fact_3445_even__sum__iff,axiom,
! [A: $tType,B: $tType] :
( ( semiring_parity @ A )
=> ! [A4: set @ B,F2: B > A] :
( ( finite_finite2 @ B @ A4 )
=> ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A4 ) )
= ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) )
@ ( finite_card @ B
@ ( collect @ B
@ ^ [A5: B] :
( ( member @ B @ A5 @ A4 )
& ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( F2 @ A5 ) ) ) ) ) ) ) ) ) ).
% even_sum_iff
thf(fact_3446_case__nat__add__eq__if,axiom,
! [A: $tType,A3: A,F2: nat > A,V2: num,N2: nat] :
( ( case_nat @ A @ A3 @ F2 @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ V2 ) @ N2 ) )
= ( F2 @ ( plus_plus @ nat @ ( pred_numeral @ V2 ) @ N2 ) ) ) ).
% case_nat_add_eq_if
thf(fact_3447_rec__nat__add__eq__if,axiom,
! [A: $tType,A3: A,F2: nat > A > A,V2: num,N2: nat] :
( ( rec_nat @ A @ A3 @ F2 @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ V2 ) @ N2 ) )
= ( F2 @ ( plus_plus @ nat @ ( pred_numeral @ V2 ) @ N2 ) @ ( rec_nat @ A @ A3 @ F2 @ ( plus_plus @ nat @ ( pred_numeral @ V2 ) @ N2 ) ) ) ) ).
% rec_nat_add_eq_if
thf(fact_3448_signed__take__bit__code,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( ( bit_ri4674362597316999326ke_bit @ A )
= ( ^ [N: nat,A5: A] : ( if @ A @ ( bit_se5641148757651400278ts_bit @ A @ ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N ) @ A5 ) @ N ) @ ( plus_plus @ A @ ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N ) @ A5 ) @ ( bit_se4730199178511100633sh_bit @ A @ ( suc @ N ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) @ ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N ) @ A5 ) ) ) ) ) ).
% signed_take_bit_code
thf(fact_3449_card__lessThan,axiom,
! [U: nat] :
( ( finite_card @ nat @ ( set_ord_lessThan @ nat @ U ) )
= U ) ).
% card_lessThan
thf(fact_3450_card__Collect__less__nat,axiom,
! [N2: nat] :
( ( finite_card @ nat
@ ( collect @ nat
@ ^ [I3: nat] : ( ord_less @ nat @ I3 @ N2 ) ) )
= N2 ) ).
% card_Collect_less_nat
thf(fact_3451_push__bit__of__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N2: nat] :
( ( bit_se4730199178511100633sh_bit @ A @ N2 @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% push_bit_of_0
thf(fact_3452_push__bit__eq__0__iff,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [N2: nat,A3: A] :
( ( ( bit_se4730199178511100633sh_bit @ A @ N2 @ A3 )
= ( zero_zero @ A ) )
= ( A3
= ( zero_zero @ A ) ) ) ) ).
% push_bit_eq_0_iff
thf(fact_3453_card__atMost,axiom,
! [U: nat] :
( ( finite_card @ nat @ ( set_ord_atMost @ nat @ U ) )
= ( suc @ U ) ) ).
% card_atMost
thf(fact_3454_card__atLeastLessThan,axiom,
! [L: nat,U: nat] :
( ( finite_card @ nat @ ( set_or7035219750837199246ssThan @ nat @ L @ U ) )
= ( minus_minus @ nat @ U @ L ) ) ).
% card_atLeastLessThan
thf(fact_3455_push__bit__push__bit,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M2: nat,N2: nat,A3: A] :
( ( bit_se4730199178511100633sh_bit @ A @ M2 @ ( bit_se4730199178511100633sh_bit @ A @ N2 @ A3 ) )
= ( bit_se4730199178511100633sh_bit @ A @ ( plus_plus @ nat @ M2 @ N2 ) @ A3 ) ) ) ).
% push_bit_push_bit
thf(fact_3456_old_Onat_Osimps_I7_J,axiom,
! [T: $tType,F1: T,F22: nat > T > T,Nat: nat] :
( ( rec_nat @ T @ F1 @ F22 @ ( suc @ Nat ) )
= ( F22 @ Nat @ ( rec_nat @ T @ F1 @ F22 @ Nat ) ) ) ).
% old.nat.simps(7)
thf(fact_3457_old_Onat_Osimps_I6_J,axiom,
! [T: $tType,F1: T,F22: nat > T > T] :
( ( rec_nat @ T @ F1 @ F22 @ ( zero_zero @ nat ) )
= F1 ) ).
% old.nat.simps(6)
thf(fact_3458_card__Collect__le__nat,axiom,
! [N2: nat] :
( ( finite_card @ nat
@ ( collect @ nat
@ ^ [I3: nat] : ( ord_less_eq @ nat @ I3 @ N2 ) ) )
= ( suc @ N2 ) ) ).
% card_Collect_le_nat
thf(fact_3459_card_Oempty,axiom,
! [A: $tType] :
( ( finite_card @ A @ ( bot_bot @ ( set @ A ) ) )
= ( zero_zero @ nat ) ) ).
% card.empty
thf(fact_3460_card_Oinfinite,axiom,
! [A: $tType,A4: set @ A] :
( ~ ( finite_finite2 @ A @ A4 )
=> ( ( finite_card @ A @ A4 )
= ( zero_zero @ nat ) ) ) ).
% card.infinite
thf(fact_3461_card__atLeastAtMost,axiom,
! [L: nat,U: nat] :
( ( finite_card @ nat @ ( set_or1337092689740270186AtMost @ nat @ L @ U ) )
= ( minus_minus @ nat @ ( suc @ U ) @ L ) ) ).
% card_atLeastAtMost
thf(fact_3462_card__atLeastLessThan__int,axiom,
! [L: int,U: int] :
( ( finite_card @ int @ ( set_or7035219750837199246ssThan @ int @ L @ U ) )
= ( nat2 @ ( minus_minus @ int @ U @ L ) ) ) ).
% card_atLeastLessThan_int
thf(fact_3463_card__0__eq,axiom,
! [A: $tType,A4: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( ( finite_card @ A @ A4 )
= ( zero_zero @ nat ) )
= ( A4
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% card_0_eq
thf(fact_3464_card__insert__disjoint,axiom,
! [A: $tType,A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ~ ( member @ A @ X @ A4 )
=> ( ( finite_card @ A @ ( insert2 @ A @ X @ A4 ) )
= ( suc @ ( finite_card @ A @ A4 ) ) ) ) ) ).
% card_insert_disjoint
thf(fact_3465_push__bit__Suc__numeral,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N2: nat,K: num] :
( ( bit_se4730199178511100633sh_bit @ A @ ( suc @ N2 ) @ ( numeral_numeral @ A @ K ) )
= ( bit_se4730199178511100633sh_bit @ A @ N2 @ ( numeral_numeral @ A @ ( bit0 @ K ) ) ) ) ) ).
% push_bit_Suc_numeral
thf(fact_3466_push__bit__Suc__minus__numeral,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N2: nat,K: num] :
( ( bit_se4730199178511100633sh_bit @ A @ ( suc @ N2 ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ K ) ) )
= ( bit_se4730199178511100633sh_bit @ A @ N2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( bit0 @ K ) ) ) ) ) ) ).
% push_bit_Suc_minus_numeral
thf(fact_3467_card__Diff__insert,axiom,
! [A: $tType,A3: A,A4: set @ A,B2: set @ A] :
( ( member @ A @ A3 @ A4 )
=> ( ~ ( member @ A @ A3 @ B2 )
=> ( ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ A3 @ B2 ) ) )
= ( minus_minus @ nat @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A4 @ B2 ) ) @ ( one_one @ nat ) ) ) ) ) ).
% card_Diff_insert
thf(fact_3468_card__atLeastAtMost__int,axiom,
! [L: int,U: int] :
( ( finite_card @ int @ ( set_or1337092689740270186AtMost @ int @ L @ U ) )
= ( nat2 @ ( plus_plus @ int @ ( minus_minus @ int @ U @ L ) @ ( one_one @ int ) ) ) ) ).
% card_atLeastAtMost_int
thf(fact_3469_push__bit__Suc,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N2: nat,A3: A] :
( ( bit_se4730199178511100633sh_bit @ A @ ( suc @ N2 ) @ A3 )
= ( bit_se4730199178511100633sh_bit @ A @ N2 @ ( times_times @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% push_bit_Suc
thf(fact_3470_push__bit__of__Suc__0,axiom,
! [N2: nat] :
( ( bit_se4730199178511100633sh_bit @ nat @ N2 @ ( suc @ ( zero_zero @ nat ) ) )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ).
% push_bit_of_Suc_0
thf(fact_3471_even__push__bit__iff,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N2: nat,A3: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se4730199178511100633sh_bit @ A @ N2 @ A3 ) )
= ( ( N2
!= ( zero_zero @ nat ) )
| ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) ) ) ) ).
% even_push_bit_iff
thf(fact_3472_push__bit__add,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N2: nat,A3: A,B3: A] :
( ( bit_se4730199178511100633sh_bit @ A @ N2 @ ( plus_plus @ A @ A3 @ B3 ) )
= ( plus_plus @ A @ ( bit_se4730199178511100633sh_bit @ A @ N2 @ A3 ) @ ( bit_se4730199178511100633sh_bit @ A @ N2 @ B3 ) ) ) ) ).
% push_bit_add
thf(fact_3473_subseqs__refl,axiom,
! [A: $tType,Xs: list @ A] : ( member @ ( list @ A ) @ Xs @ ( set2 @ ( list @ A ) @ ( subseqs @ A @ Xs ) ) ) ).
% subseqs_refl
thf(fact_3474_nat_Ocase__distrib,axiom,
! [B: $tType,A: $tType,H2: A > B,F1: A,F22: nat > A,Nat: nat] :
( ( H2 @ ( case_nat @ A @ F1 @ F22 @ Nat ) )
= ( case_nat @ B @ ( H2 @ F1 )
@ ^ [X4: nat] : ( H2 @ ( F22 @ X4 ) )
@ Nat ) ) ).
% nat.case_distrib
thf(fact_3475_n__subsets,axiom,
! [A: $tType,A4: set @ A,K: nat] :
( ( finite_finite2 @ A @ A4 )
=> ( ( finite_card @ ( set @ A )
@ ( collect @ ( set @ A )
@ ^ [B6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ B6 @ A4 )
& ( ( finite_card @ A @ B6 )
= K ) ) ) )
= ( binomial @ ( finite_card @ A @ A4 ) @ K ) ) ) ).
% n_subsets
thf(fact_3476_old_Onat_Osimps_I5_J,axiom,
! [A: $tType,F1: A,F22: nat > A,X22: nat] :
( ( case_nat @ A @ F1 @ F22 @ ( suc @ X22 ) )
= ( F22 @ X22 ) ) ).
% old.nat.simps(5)
thf(fact_3477_old_Onat_Osimps_I4_J,axiom,
! [A: $tType,F1: A,F22: nat > A] :
( ( case_nat @ A @ F1 @ F22 @ ( zero_zero @ nat ) )
= F1 ) ).
% old.nat.simps(4)
thf(fact_3478_infinite__arbitrarily__large,axiom,
! [A: $tType,A4: set @ A,N2: nat] :
( ~ ( finite_finite2 @ A @ A4 )
=> ? [B9: set @ A] :
( ( finite_finite2 @ A @ B9 )
& ( ( finite_card @ A @ B9 )
= N2 )
& ( ord_less_eq @ ( set @ A ) @ B9 @ A4 ) ) ) ).
% infinite_arbitrarily_large
thf(fact_3479_card__subset__eq,axiom,
! [A: $tType,B2: set @ A,A4: set @ A] :
( ( finite_finite2 @ A @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ A4 @ B2 )
=> ( ( ( finite_card @ A @ A4 )
= ( finite_card @ A @ B2 ) )
=> ( A4 = B2 ) ) ) ) ).
% card_subset_eq
thf(fact_3480_card__le__if__inj__on__rel,axiom,
! [B: $tType,A: $tType,B2: set @ A,A4: set @ B,R2: B > A > $o] :
( ( finite_finite2 @ A @ B2 )
=> ( ! [A6: B] :
( ( member @ B @ A6 @ A4 )
=> ? [B10: A] :
( ( member @ A @ B10 @ B2 )
& ( R2 @ A6 @ B10 ) ) )
=> ( ! [A13: B,A24: B,B5: A] :
( ( member @ B @ A13 @ A4 )
=> ( ( member @ B @ A24 @ A4 )
=> ( ( member @ A @ B5 @ B2 )
=> ( ( R2 @ A13 @ B5 )
=> ( ( R2 @ A24 @ B5 )
=> ( A13 = A24 ) ) ) ) ) )
=> ( ord_less_eq @ nat @ ( finite_card @ B @ A4 ) @ ( finite_card @ A @ B2 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_3481_card__insert__le,axiom,
! [A: $tType,A4: set @ A,X: A] : ( ord_less_eq @ nat @ ( finite_card @ A @ A4 ) @ ( finite_card @ A @ ( insert2 @ A @ X @ A4 ) ) ) ).
% card_insert_le
thf(fact_3482_card__atLeastZeroLessThan__int,axiom,
! [U: int] :
( ( finite_card @ int @ ( set_or7035219750837199246ssThan @ int @ ( zero_zero @ int ) @ U ) )
= ( nat2 @ U ) ) ).
% card_atLeastZeroLessThan_int
thf(fact_3483_nat_Odisc__eq__case_I1_J,axiom,
! [Nat: nat] :
( ( Nat
= ( zero_zero @ nat ) )
= ( case_nat @ $o @ $true
@ ^ [Uu3: nat] : $false
@ Nat ) ) ).
% nat.disc_eq_case(1)
thf(fact_3484_nat_Odisc__eq__case_I2_J,axiom,
! [Nat: nat] :
( ( Nat
!= ( zero_zero @ nat ) )
= ( case_nat @ $o @ $false
@ ^ [Uu3: nat] : $true
@ Nat ) ) ).
% nat.disc_eq_case(2)
thf(fact_3485_push__bit__take__bit,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M2: nat,N2: nat,A3: A] :
( ( bit_se4730199178511100633sh_bit @ A @ M2 @ ( bit_se2584673776208193580ke_bit @ A @ N2 @ A3 ) )
= ( bit_se2584673776208193580ke_bit @ A @ ( plus_plus @ nat @ M2 @ N2 ) @ ( bit_se4730199178511100633sh_bit @ A @ M2 @ A3 ) ) ) ) ).
% push_bit_take_bit
thf(fact_3486_sum__multicount__gen,axiom,
! [A: $tType,B: $tType,S: set @ A,T2: set @ B,R: A > B > $o,K: B > nat] :
( ( finite_finite2 @ A @ S )
=> ( ( finite_finite2 @ B @ T2 )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ T2 )
=> ( ( finite_card @ A
@ ( collect @ A
@ ^ [I3: A] :
( ( member @ A @ I3 @ S )
& ( R @ I3 @ X5 ) ) ) )
= ( K @ X5 ) ) )
=> ( ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [I3: A] :
( finite_card @ B
@ ( collect @ B
@ ^ [J2: B] :
( ( member @ B @ J2 @ T2 )
& ( R @ I3 @ J2 ) ) ) )
@ S )
= ( groups7311177749621191930dd_sum @ B @ nat @ K @ T2 ) ) ) ) ) ).
% sum_multicount_gen
thf(fact_3487_card__lists__length__eq,axiom,
! [A: $tType,A4: set @ A,N2: nat] :
( ( finite_finite2 @ A @ A4 )
=> ( ( finite_card @ ( list @ A )
@ ( collect @ ( list @ A )
@ ^ [Xs3: list @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs3 ) @ A4 )
& ( ( size_size @ ( list @ A ) @ Xs3 )
= N2 ) ) ) )
= ( power_power @ nat @ ( finite_card @ A @ A4 ) @ N2 ) ) ) ).
% card_lists_length_eq
thf(fact_3488_is__singleton__altdef,axiom,
! [A: $tType] :
( ( is_singleton @ A )
= ( ^ [A7: set @ A] :
( ( finite_card @ A @ A7 )
= ( one_one @ nat ) ) ) ) ).
% is_singleton_altdef
thf(fact_3489_card__2__iff_H,axiom,
! [A: $tType,S2: set @ A] :
( ( ( finite_card @ A @ S2 )
= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( ? [X4: A] :
( ( member @ A @ X4 @ S2 )
& ? [Y6: A] :
( ( member @ A @ Y6 @ S2 )
& ( X4 != Y6 )
& ! [Z6: A] :
( ( member @ A @ Z6 @ S2 )
=> ( ( Z6 = X4 )
| ( Z6 = Y6 ) ) ) ) ) ) ) ).
% card_2_iff'
thf(fact_3490_card__eq__0__iff,axiom,
! [A: $tType,A4: set @ A] :
( ( ( finite_card @ A @ A4 )
= ( zero_zero @ nat ) )
= ( ( A4
= ( bot_bot @ ( set @ A ) ) )
| ~ ( finite_finite2 @ A @ A4 ) ) ) ).
% card_eq_0_iff
thf(fact_3491_card__ge__0__finite,axiom,
! [A: $tType,A4: set @ A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( finite_card @ A @ A4 ) )
=> ( finite_finite2 @ A @ A4 ) ) ).
% card_ge_0_finite
thf(fact_3492_card__Suc__eq__finite,axiom,
! [A: $tType,A4: set @ A,K: nat] :
( ( ( finite_card @ A @ A4 )
= ( suc @ K ) )
= ( ? [B4: A,B6: set @ A] :
( ( A4
= ( insert2 @ A @ B4 @ B6 ) )
& ~ ( member @ A @ B4 @ B6 )
& ( ( finite_card @ A @ B6 )
= K )
& ( finite_finite2 @ A @ B6 ) ) ) ) ).
% card_Suc_eq_finite
thf(fact_3493_card__insert__if,axiom,
! [A: $tType,A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( ( member @ A @ X @ A4 )
=> ( ( finite_card @ A @ ( insert2 @ A @ X @ A4 ) )
= ( finite_card @ A @ A4 ) ) )
& ( ~ ( member @ A @ X @ A4 )
=> ( ( finite_card @ A @ ( insert2 @ A @ X @ A4 ) )
= ( suc @ ( finite_card @ A @ A4 ) ) ) ) ) ) ).
% card_insert_if
thf(fact_3494_card__mono,axiom,
! [A: $tType,B2: set @ A,A4: set @ A] :
( ( finite_finite2 @ A @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ A4 @ B2 )
=> ( ord_less_eq @ nat @ ( finite_card @ A @ A4 ) @ ( finite_card @ A @ B2 ) ) ) ) ).
% card_mono
thf(fact_3495_card__seteq,axiom,
! [A: $tType,B2: set @ A,A4: set @ A] :
( ( finite_finite2 @ A @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ A4 @ B2 )
=> ( ( ord_less_eq @ nat @ ( finite_card @ A @ B2 ) @ ( finite_card @ A @ A4 ) )
=> ( A4 = B2 ) ) ) ) ).
% card_seteq
thf(fact_3496_obtain__subset__with__card__n,axiom,
! [A: $tType,N2: nat,S2: set @ A] :
( ( ord_less_eq @ nat @ N2 @ ( finite_card @ A @ S2 ) )
=> ~ ! [T5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ T5 @ S2 )
=> ( ( ( finite_card @ A @ T5 )
= N2 )
=> ~ ( finite_finite2 @ A @ T5 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_3497_finite__if__finite__subsets__card__bdd,axiom,
! [A: $tType,F5: set @ A,C5: nat] :
( ! [G3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ G3 @ F5 )
=> ( ( finite_finite2 @ A @ G3 )
=> ( ord_less_eq @ nat @ ( finite_card @ A @ G3 ) @ C5 ) ) )
=> ( ( finite_finite2 @ A @ F5 )
& ( ord_less_eq @ nat @ ( finite_card @ A @ F5 ) @ C5 ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_3498_card__less__sym__Diff,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( finite_finite2 @ A @ B2 )
=> ( ( ord_less @ nat @ ( finite_card @ A @ A4 ) @ ( finite_card @ A @ B2 ) )
=> ( ord_less @ nat @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A4 @ B2 ) ) @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ B2 @ A4 ) ) ) ) ) ) ).
% card_less_sym_Diff
thf(fact_3499_card__le__sym__Diff,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( finite_finite2 @ A @ B2 )
=> ( ( ord_less_eq @ nat @ ( finite_card @ A @ A4 ) @ ( finite_card @ A @ B2 ) )
=> ( ord_less_eq @ nat @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A4 @ B2 ) ) @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ B2 @ A4 ) ) ) ) ) ) ).
% card_le_sym_Diff
thf(fact_3500_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_3501_card__1__singletonE,axiom,
! [A: $tType,A4: set @ A] :
( ( ( finite_card @ A @ A4 )
= ( one_one @ nat ) )
=> ~ ! [X5: A] :
( A4
!= ( insert2 @ A @ X5 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% card_1_singletonE
thf(fact_3502_psubset__card__mono,axiom,
! [A: $tType,B2: set @ A,A4: set @ A] :
( ( finite_finite2 @ A @ B2 )
=> ( ( ord_less @ ( set @ A ) @ A4 @ B2 )
=> ( ord_less @ nat @ ( finite_card @ A @ A4 ) @ ( finite_card @ A @ B2 ) ) ) ) ).
% psubset_card_mono
thf(fact_3503_bit__push__bit__iff__int,axiom,
! [M2: nat,K: int,N2: nat] :
( ( bit_se5641148757651400278ts_bit @ int @ ( bit_se4730199178511100633sh_bit @ int @ M2 @ K ) @ N2 )
= ( ( ord_less_eq @ nat @ M2 @ N2 )
& ( bit_se5641148757651400278ts_bit @ int @ K @ ( minus_minus @ nat @ N2 @ M2 ) ) ) ) ).
% bit_push_bit_iff_int
thf(fact_3504_card__less__Suc2,axiom,
! [M7: set @ nat,I: nat] :
( ~ ( member @ nat @ ( zero_zero @ nat ) @ M7 )
=> ( ( finite_card @ nat
@ ( collect @ nat
@ ^ [K2: nat] :
( ( member @ nat @ ( suc @ K2 ) @ M7 )
& ( ord_less @ nat @ K2 @ I ) ) ) )
= ( finite_card @ nat
@ ( collect @ nat
@ ^ [K2: nat] :
( ( member @ nat @ K2 @ M7 )
& ( ord_less @ nat @ K2 @ ( suc @ I ) ) ) ) ) ) ) ).
% card_less_Suc2
thf(fact_3505_card__less__Suc,axiom,
! [M7: set @ nat,I: nat] :
( ( member @ nat @ ( zero_zero @ nat ) @ M7 )
=> ( ( suc
@ ( finite_card @ nat
@ ( collect @ nat
@ ^ [K2: nat] :
( ( member @ nat @ ( suc @ K2 ) @ M7 )
& ( ord_less @ nat @ K2 @ I ) ) ) ) )
= ( finite_card @ nat
@ ( collect @ nat
@ ^ [K2: nat] :
( ( member @ nat @ K2 @ M7 )
& ( ord_less @ nat @ K2 @ ( suc @ I ) ) ) ) ) ) ) ).
% card_less_Suc
thf(fact_3506_card__less,axiom,
! [M7: set @ nat,I: nat] :
( ( member @ nat @ ( zero_zero @ nat ) @ M7 )
=> ( ( finite_card @ nat
@ ( collect @ nat
@ ^ [K2: nat] :
( ( member @ nat @ K2 @ M7 )
& ( ord_less @ nat @ K2 @ ( suc @ I ) ) ) ) )
!= ( zero_zero @ nat ) ) ) ).
% card_less
thf(fact_3507_bit__push__bit__iff__nat,axiom,
! [M2: nat,Q3: nat,N2: nat] :
( ( bit_se5641148757651400278ts_bit @ nat @ ( bit_se4730199178511100633sh_bit @ nat @ M2 @ Q3 ) @ N2 )
= ( ( ord_less_eq @ nat @ M2 @ N2 )
& ( bit_se5641148757651400278ts_bit @ nat @ Q3 @ ( minus_minus @ nat @ N2 @ M2 ) ) ) ) ).
% bit_push_bit_iff_nat
thf(fact_3508_sum__Suc,axiom,
! [A: $tType,F2: A > nat,A4: set @ A] :
( ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [X4: A] : ( suc @ ( F2 @ X4 ) )
@ A4 )
= ( plus_plus @ nat @ ( groups7311177749621191930dd_sum @ A @ nat @ F2 @ A4 ) @ ( finite_card @ A @ A4 ) ) ) ).
% sum_Suc
thf(fact_3509_subset__card__intvl__is__intvl,axiom,
! [A4: set @ nat,K: nat] :
( ( ord_less_eq @ ( set @ nat ) @ A4 @ ( set_or7035219750837199246ssThan @ nat @ K @ ( plus_plus @ nat @ K @ ( finite_card @ nat @ A4 ) ) ) )
=> ( A4
= ( set_or7035219750837199246ssThan @ nat @ K @ ( plus_plus @ nat @ K @ ( finite_card @ nat @ A4 ) ) ) ) ) ).
% subset_card_intvl_is_intvl
thf(fact_3510_sum__multicount,axiom,
! [A: $tType,B: $tType,S2: set @ A,T4: set @ B,R: A > B > $o,K: nat] :
( ( finite_finite2 @ A @ S2 )
=> ( ( finite_finite2 @ B @ T4 )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ T4 )
=> ( ( finite_card @ A
@ ( collect @ A
@ ^ [I3: A] :
( ( member @ A @ I3 @ S2 )
& ( R @ I3 @ X5 ) ) ) )
= K ) )
=> ( ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [I3: A] :
( finite_card @ B
@ ( collect @ B
@ ^ [J2: B] :
( ( member @ B @ J2 @ T4 )
& ( R @ I3 @ J2 ) ) ) )
@ S2 )
= ( times_times @ nat @ K @ ( finite_card @ B @ T4 ) ) ) ) ) ) ).
% sum_multicount
thf(fact_3511_less__eq__nat_Osimps_I2_J,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq @ nat @ ( suc @ M2 ) @ N2 )
= ( case_nat @ $o @ $false @ ( ord_less_eq @ nat @ M2 ) @ N2 ) ) ).
% less_eq_nat.simps(2)
thf(fact_3512_max__Suc2,axiom,
! [M2: nat,N2: nat] :
( ( ord_max @ nat @ M2 @ ( suc @ N2 ) )
= ( case_nat @ nat @ ( suc @ N2 )
@ ^ [M4: nat] : ( suc @ ( ord_max @ nat @ M4 @ N2 ) )
@ M2 ) ) ).
% max_Suc2
thf(fact_3513_max__Suc1,axiom,
! [N2: nat,M2: nat] :
( ( ord_max @ nat @ ( suc @ N2 ) @ M2 )
= ( case_nat @ nat @ ( suc @ N2 )
@ ^ [M4: nat] : ( suc @ ( ord_max @ nat @ N2 @ M4 ) )
@ M2 ) ) ).
% max_Suc1
thf(fact_3514_sum__bounded__below,axiom,
! [A: $tType,B: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( semiring_1 @ A ) )
=> ! [A4: set @ B,K5: A,F2: B > A] :
( ! [I4: B] :
( ( member @ B @ I4 @ A4 )
=> ( ord_less_eq @ A @ K5 @ ( F2 @ I4 ) ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( finite_card @ B @ A4 ) ) @ K5 ) @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A4 ) ) ) ) ).
% sum_bounded_below
thf(fact_3515_sum__bounded__above,axiom,
! [B: $tType,A: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( semiring_1 @ A ) )
=> ! [A4: set @ B,F2: B > A,K5: A] :
( ! [I4: B] :
( ( member @ B @ I4 @ A4 )
=> ( ord_less_eq @ A @ ( F2 @ I4 ) @ K5 ) )
=> ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A4 ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( finite_card @ B @ A4 ) ) @ K5 ) ) ) ) ).
% sum_bounded_above
thf(fact_3516_card__gt__0__iff,axiom,
! [A: $tType,A4: set @ A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( finite_card @ A @ A4 ) )
= ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
& ( finite_finite2 @ A @ A4 ) ) ) ).
% card_gt_0_iff
thf(fact_3517_card__1__singleton__iff,axiom,
! [A: $tType,A4: set @ A] :
( ( ( finite_card @ A @ A4 )
= ( suc @ ( zero_zero @ nat ) ) )
= ( ? [X4: A] :
( A4
= ( insert2 @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% card_1_singleton_iff
thf(fact_3518_card__eq__SucD,axiom,
! [A: $tType,A4: set @ A,K: nat] :
( ( ( finite_card @ A @ A4 )
= ( suc @ K ) )
=> ? [B5: A,B9: set @ A] :
( ( A4
= ( insert2 @ A @ B5 @ B9 ) )
& ~ ( member @ A @ B5 @ B9 )
& ( ( finite_card @ A @ B9 )
= K )
& ( ( K
= ( zero_zero @ nat ) )
=> ( B9
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% card_eq_SucD
thf(fact_3519_card__Suc__eq,axiom,
! [A: $tType,A4: set @ A,K: nat] :
( ( ( finite_card @ A @ A4 )
= ( suc @ K ) )
= ( ? [B4: A,B6: set @ A] :
( ( A4
= ( insert2 @ A @ B4 @ B6 ) )
& ~ ( member @ A @ B4 @ B6 )
& ( ( finite_card @ A @ B6 )
= K )
& ( ( K
= ( zero_zero @ nat ) )
=> ( B6
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% card_Suc_eq
thf(fact_3520_card__le__Suc0__iff__eq,axiom,
! [A: $tType,A4: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( ord_less_eq @ nat @ ( finite_card @ A @ A4 ) @ ( suc @ ( zero_zero @ nat ) ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ A4 )
=> ! [Y6: A] :
( ( member @ A @ Y6 @ A4 )
=> ( X4 = Y6 ) ) ) ) ) ) ).
% card_le_Suc0_iff_eq
thf(fact_3521_card__le__Suc__iff,axiom,
! [A: $tType,N2: nat,A4: set @ A] :
( ( ord_less_eq @ nat @ ( suc @ N2 ) @ ( finite_card @ A @ A4 ) )
= ( ? [A5: A,B6: set @ A] :
( ( A4
= ( insert2 @ A @ A5 @ B6 ) )
& ~ ( member @ A @ A5 @ B6 )
& ( ord_less_eq @ nat @ N2 @ ( finite_card @ A @ B6 ) )
& ( finite_finite2 @ A @ B6 ) ) ) ) ).
% card_le_Suc_iff
thf(fact_3522_card__Diff1__le,axiom,
! [A: $tType,A4: set @ A,X: A] : ( ord_less_eq @ nat @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) @ ( finite_card @ A @ A4 ) ) ).
% card_Diff1_le
thf(fact_3523_card__Diff__subset,axiom,
! [A: $tType,B2: set @ A,A4: set @ A] :
( ( finite_finite2 @ A @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ A4 )
=> ( ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A4 @ B2 ) )
= ( minus_minus @ nat @ ( finite_card @ A @ A4 ) @ ( finite_card @ A @ B2 ) ) ) ) ) ).
% card_Diff_subset
thf(fact_3524_card__psubset,axiom,
! [A: $tType,B2: set @ A,A4: set @ A] :
( ( finite_finite2 @ A @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ A4 @ B2 )
=> ( ( ord_less @ nat @ ( finite_card @ A @ A4 ) @ ( finite_card @ A @ B2 ) )
=> ( ord_less @ ( set @ A ) @ A4 @ B2 ) ) ) ) ).
% card_psubset
thf(fact_3525_diff__card__le__card__Diff,axiom,
! [A: $tType,B2: set @ A,A4: set @ A] :
( ( finite_finite2 @ A @ B2 )
=> ( ord_less_eq @ nat @ ( minus_minus @ nat @ ( finite_card @ A @ A4 ) @ ( finite_card @ A @ B2 ) ) @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A4 @ B2 ) ) ) ) ).
% diff_card_le_card_Diff
thf(fact_3526_card__lists__length__le,axiom,
! [A: $tType,A4: set @ A,N2: nat] :
( ( finite_finite2 @ A @ A4 )
=> ( ( finite_card @ ( list @ A )
@ ( collect @ ( list @ A )
@ ^ [Xs3: list @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs3 ) @ A4 )
& ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs3 ) @ N2 ) ) ) )
= ( groups7311177749621191930dd_sum @ nat @ nat @ ( power_power @ nat @ ( finite_card @ A @ A4 ) ) @ ( set_ord_atMost @ nat @ N2 ) ) ) ) ).
% card_lists_length_le
thf(fact_3527_bit__iff__and__push__bit__not__eq__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se5641148757651400278ts_bit @ A )
= ( ^ [A5: A,N: nat] :
( ( bit_se5824344872417868541ns_and @ A @ A5 @ ( bit_se4730199178511100633sh_bit @ A @ N @ ( one_one @ A ) ) )
!= ( zero_zero @ A ) ) ) ) ) ).
% bit_iff_and_push_bit_not_eq_0
thf(fact_3528_card__roots__unity,axiom,
! [A: $tType] :
( ( ( real_V8999393235501362500lgebra @ A )
& ( idom @ A ) )
=> ! [N2: nat] :
( ( ord_less_eq @ nat @ ( one_one @ nat ) @ N2 )
=> ( ord_less_eq @ nat
@ ( finite_card @ A
@ ( collect @ A
@ ^ [Z6: A] :
( ( power_power @ A @ Z6 @ N2 )
= ( one_one @ A ) ) ) )
@ N2 ) ) ) ).
% card_roots_unity
thf(fact_3529_subset__eq__atLeast0__lessThan__card,axiom,
! [N6: set @ nat,N2: nat] :
( ( ord_less_eq @ ( set @ nat ) @ N6 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N2 ) )
=> ( ord_less_eq @ nat @ ( finite_card @ nat @ N6 ) @ N2 ) ) ).
% subset_eq_atLeast0_lessThan_card
thf(fact_3530_card__sum__le__nat__sum,axiom,
! [S2: set @ nat] :
( ord_less_eq @ nat
@ ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [X4: nat] : X4
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( finite_card @ nat @ S2 ) ) )
@ ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [X4: nat] : X4
@ S2 ) ) ).
% card_sum_le_nat_sum
thf(fact_3531_card__nth__roots,axiom,
! [C2: complex,N2: nat] :
( ( C2
!= ( zero_zero @ complex ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( finite_card @ complex
@ ( collect @ complex
@ ^ [Z6: complex] :
( ( power_power @ complex @ Z6 @ N2 )
= C2 ) ) )
= N2 ) ) ) ).
% card_nth_roots
thf(fact_3532_card__roots__unity__eq,axiom,
! [N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( finite_card @ complex
@ ( collect @ complex
@ ^ [Z6: complex] :
( ( power_power @ complex @ Z6 @ N2 )
= ( one_one @ complex ) ) ) )
= N2 ) ) ).
% card_roots_unity_eq
thf(fact_3533_diff__Suc,axiom,
! [M2: nat,N2: nat] :
( ( minus_minus @ nat @ M2 @ ( suc @ N2 ) )
= ( case_nat @ nat @ ( zero_zero @ nat )
@ ^ [K2: nat] : K2
@ ( minus_minus @ nat @ M2 @ N2 ) ) ) ).
% diff_Suc
thf(fact_3534_card__2__iff,axiom,
! [A: $tType,S2: set @ A] :
( ( ( finite_card @ A @ S2 )
= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( ? [X4: A,Y6: A] :
( ( S2
= ( insert2 @ A @ X4 @ ( insert2 @ A @ Y6 @ ( bot_bot @ ( set @ A ) ) ) ) )
& ( X4 != Y6 ) ) ) ) ).
% card_2_iff
thf(fact_3535_card__3__iff,axiom,
! [A: $tType,S2: set @ A] :
( ( ( finite_card @ A @ S2 )
= ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) )
= ( ? [X4: A,Y6: A,Z6: A] :
( ( S2
= ( insert2 @ A @ X4 @ ( insert2 @ A @ Y6 @ ( insert2 @ A @ Z6 @ ( bot_bot @ ( set @ A ) ) ) ) ) )
& ( X4 != Y6 )
& ( Y6 != Z6 )
& ( X4 != Z6 ) ) ) ) ).
% card_3_iff
thf(fact_3536_odd__card__imp__not__empty,axiom,
! [A: $tType,A4: set @ A] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( finite_card @ A @ A4 ) )
=> ( A4
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% odd_card_imp_not_empty
thf(fact_3537_card__Suc__Diff1,axiom,
! [A: $tType,A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( member @ A @ X @ A4 )
=> ( ( suc @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) )
= ( finite_card @ A @ A4 ) ) ) ) ).
% card_Suc_Diff1
thf(fact_3538_card_Oinsert__remove,axiom,
! [A: $tType,A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( finite_card @ A @ ( insert2 @ A @ X @ A4 ) )
= ( suc @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% card.insert_remove
thf(fact_3539_card_Oremove,axiom,
! [A: $tType,A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( member @ A @ X @ A4 )
=> ( ( finite_card @ A @ A4 )
= ( suc @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ).
% card.remove
thf(fact_3540_card__Diff1__less,axiom,
! [A: $tType,A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( member @ A @ X @ A4 )
=> ( ord_less @ nat @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) @ ( finite_card @ A @ A4 ) ) ) ) ).
% card_Diff1_less
thf(fact_3541_card__Diff2__less,axiom,
! [A: $tType,A4: set @ A,X: A,Y: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( member @ A @ X @ A4 )
=> ( ( member @ A @ Y @ A4 )
=> ( ord_less @ nat @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) @ ( insert2 @ A @ Y @ ( bot_bot @ ( set @ A ) ) ) ) ) @ ( finite_card @ A @ A4 ) ) ) ) ) ).
% card_Diff2_less
thf(fact_3542_card__Diff1__less__iff,axiom,
! [A: $tType,A4: set @ A,X: A] :
( ( ord_less @ nat @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) @ ( finite_card @ A @ A4 ) )
= ( ( finite_finite2 @ A @ A4 )
& ( member @ A @ X @ A4 ) ) ) ).
% card_Diff1_less_iff
thf(fact_3543_card__Diff__singleton,axiom,
! [A: $tType,X: A,A4: set @ A] :
( ( member @ A @ X @ A4 )
=> ( ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( minus_minus @ nat @ ( finite_card @ A @ A4 ) @ ( one_one @ nat ) ) ) ) ).
% card_Diff_singleton
thf(fact_3544_card__Diff__singleton__if,axiom,
! [A: $tType,X: A,A4: set @ A] :
( ( ( member @ A @ X @ A4 )
=> ( ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( minus_minus @ nat @ ( finite_card @ A @ A4 ) @ ( one_one @ nat ) ) ) )
& ( ~ ( member @ A @ X @ A4 )
=> ( ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( finite_card @ A @ A4 ) ) ) ) ).
% card_Diff_singleton_if
thf(fact_3545_prod__le__power,axiom,
! [B: $tType,A: $tType] :
( ( linordered_semidom @ A )
=> ! [A4: set @ B,F2: B > A,N2: A,K: nat] :
( ! [I4: B] :
( ( member @ B @ I4 @ A4 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ I4 ) )
& ( ord_less_eq @ A @ ( F2 @ I4 ) @ N2 ) ) )
=> ( ( ord_less_eq @ nat @ ( finite_card @ B @ A4 ) @ K )
=> ( ( ord_less_eq @ A @ ( one_one @ A ) @ N2 )
=> ( ord_less_eq @ A @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ A4 ) @ ( power_power @ A @ N2 @ K ) ) ) ) ) ) ).
% prod_le_power
thf(fact_3546_sum__bounded__above__strict,axiom,
! [B: $tType,A: $tType] :
( ( ( ordere8940638589300402666id_add @ A )
& ( semiring_1 @ A ) )
=> ! [A4: set @ B,F2: B > A,K5: A] :
( ! [I4: B] :
( ( member @ B @ I4 @ A4 )
=> ( ord_less @ A @ ( F2 @ I4 ) @ K5 ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( finite_card @ B @ A4 ) )
=> ( ord_less @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A4 ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( finite_card @ B @ A4 ) ) @ K5 ) ) ) ) ) ).
% sum_bounded_above_strict
thf(fact_3547_sum__bounded__above__divide,axiom,
! [B: $tType,A: $tType] :
( ( linordered_field @ A )
=> ! [A4: set @ B,F2: B > A,K5: A] :
( ! [I4: B] :
( ( member @ B @ I4 @ A4 )
=> ( ord_less_eq @ A @ ( F2 @ I4 ) @ ( divide_divide @ A @ K5 @ ( semiring_1_of_nat @ A @ ( finite_card @ B @ A4 ) ) ) ) )
=> ( ( finite_finite2 @ B @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A4 ) @ K5 ) ) ) ) ) ).
% sum_bounded_above_divide
thf(fact_3548_card__insert__le__m1,axiom,
! [A: $tType,N2: nat,Y: set @ A,X: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( ord_less_eq @ nat @ ( finite_card @ A @ Y ) @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) )
=> ( ord_less_eq @ nat @ ( finite_card @ A @ ( insert2 @ A @ X @ Y ) ) @ N2 ) ) ) ).
% card_insert_le_m1
thf(fact_3549_polyfun__roots__card,axiom,
! [A: $tType] :
( ( ( real_V8999393235501362500lgebra @ A )
& ( idom @ A ) )
=> ! [C2: nat > A,K: nat,N2: nat] :
( ( ( C2 @ K )
!= ( zero_zero @ A ) )
=> ( ( ord_less_eq @ nat @ K @ N2 )
=> ( ord_less_eq @ nat
@ ( finite_card @ A
@ ( collect @ A
@ ^ [Z6: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( C2 @ I3 ) @ ( power_power @ A @ Z6 @ I3 ) )
@ ( set_ord_atMost @ nat @ N2 ) )
= ( zero_zero @ A ) ) ) )
@ N2 ) ) ) ) ).
% polyfun_roots_card
thf(fact_3550_prod__gen__delta,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S2: set @ B,A3: B,B3: B > A,C2: A] :
( ( finite_finite2 @ B @ S2 )
=> ( ( ( member @ B @ A3 @ S2 )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K2: B] : ( if @ A @ ( K2 = A3 ) @ ( B3 @ K2 ) @ C2 )
@ S2 )
= ( times_times @ A @ ( B3 @ A3 ) @ ( power_power @ A @ C2 @ ( minus_minus @ nat @ ( finite_card @ B @ S2 ) @ ( one_one @ nat ) ) ) ) ) )
& ( ~ ( member @ B @ A3 @ S2 )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K2: B] : ( if @ A @ ( K2 = A3 ) @ ( B3 @ K2 ) @ C2 )
@ S2 )
= ( power_power @ A @ C2 @ ( finite_card @ B @ S2 ) ) ) ) ) ) ) ).
% prod_gen_delta
thf(fact_3551_take__bit__sum,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ( ( bit_se2584673776208193580ke_bit @ A )
= ( ^ [N: nat,A5: A] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K2: nat] : ( bit_se4730199178511100633sh_bit @ A @ K2 @ ( zero_neq_one_of_bool @ A @ ( bit_se5641148757651400278ts_bit @ A @ A5 @ K2 ) ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ) ).
% take_bit_sum
thf(fact_3552_polyfun__rootbound,axiom,
! [A: $tType] :
( ( ( real_V8999393235501362500lgebra @ A )
& ( idom @ A ) )
=> ! [C2: nat > A,K: nat,N2: nat] :
( ( ( C2 @ K )
!= ( zero_zero @ A ) )
=> ( ( ord_less_eq @ nat @ K @ N2 )
=> ( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [Z6: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( C2 @ I3 ) @ ( power_power @ A @ Z6 @ I3 ) )
@ ( set_ord_atMost @ nat @ N2 ) )
= ( zero_zero @ A ) ) ) )
& ( ord_less_eq @ nat
@ ( finite_card @ A
@ ( collect @ A
@ ^ [Z6: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( C2 @ I3 ) @ ( power_power @ A @ Z6 @ I3 ) )
@ ( set_ord_atMost @ nat @ N2 ) )
= ( zero_zero @ A ) ) ) )
@ N2 ) ) ) ) ) ).
% polyfun_rootbound
thf(fact_3553_Nitpick_Ocase__nat__unfold,axiom,
! [A: $tType] :
( ( case_nat @ A )
= ( ^ [X4: A,F3: nat > A,N: nat] :
( if @ A
@ ( N
= ( zero_zero @ nat ) )
@ X4
@ ( F3 @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ) ).
% Nitpick.case_nat_unfold
thf(fact_3554_card__lists__distinct__length__eq,axiom,
! [A: $tType,A4: set @ A,K: nat] :
( ( finite_finite2 @ A @ A4 )
=> ( ( ord_less_eq @ nat @ K @ ( finite_card @ A @ A4 ) )
=> ( ( 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 ) @ A4 ) ) ) )
= ( groups7121269368397514597t_prod @ nat @ nat
@ ^ [X4: nat] : X4
@ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ ( minus_minus @ nat @ ( finite_card @ A @ A4 ) @ K ) @ ( one_one @ nat ) ) @ ( finite_card @ A @ A4 ) ) ) ) ) ) ).
% card_lists_distinct_length_eq
thf(fact_3555_length__mul__elem,axiom,
! [A: $tType,Xs: list @ ( list @ A ),N2: nat] :
( ! [X5: list @ A] :
( ( member @ ( list @ A ) @ X5 @ ( set2 @ ( list @ A ) @ Xs ) )
=> ( ( size_size @ ( list @ A ) @ X5 )
= N2 ) )
=> ( ( size_size @ ( list @ A ) @ ( concat @ A @ Xs ) )
= ( times_times @ nat @ ( size_size @ ( list @ ( list @ A ) ) @ Xs ) @ N2 ) ) ) ).
% length_mul_elem
thf(fact_3556_card__lists__distinct__length__eq_H,axiom,
! [A: $tType,K: nat,A4: set @ A] :
( ( ord_less @ nat @ K @ ( finite_card @ A @ A4 ) )
=> ( ( 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 ) @ A4 ) ) ) )
= ( groups7121269368397514597t_prod @ nat @ nat
@ ^ [X4: nat] : X4
@ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ ( minus_minus @ nat @ ( finite_card @ A @ A4 ) @ K ) @ ( one_one @ nat ) ) @ ( finite_card @ A @ A4 ) ) ) ) ) ).
% card_lists_distinct_length_eq'
thf(fact_3557_rec__nat__0__imp,axiom,
! [A: $tType,F2: nat > A,F1: A,F22: nat > A > A] :
( ( F2
= ( rec_nat @ A @ F1 @ F22 ) )
=> ( ( F2 @ ( zero_zero @ nat ) )
= F1 ) ) ).
% rec_nat_0_imp
thf(fact_3558_distinct__swap,axiom,
! [A: $tType,I: nat,Xs: list @ A,J: nat] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( distinct @ A @ ( list_update @ A @ ( list_update @ A @ Xs @ I @ ( nth @ A @ Xs @ J ) ) @ J @ ( nth @ A @ Xs @ I ) ) )
= ( distinct @ A @ Xs ) ) ) ) ).
% distinct_swap
thf(fact_3559_finite__lists__distinct__length__eq,axiom,
! [A: $tType,A4: set @ A,N2: nat] :
( ( finite_finite2 @ A @ A4 )
=> ( finite_finite2 @ ( list @ A )
@ ( collect @ ( list @ A )
@ ^ [Xs3: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs3 )
= N2 )
& ( distinct @ A @ Xs3 )
& ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs3 ) @ A4 ) ) ) ) ) ).
% finite_lists_distinct_length_eq
thf(fact_3560_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_3561_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_3562_distinct__enumerate,axiom,
! [A: $tType,N2: nat,Xs: list @ A] : ( distinct @ ( product_prod @ nat @ A ) @ ( enumerate @ A @ N2 @ Xs ) ) ).
% distinct_enumerate
thf(fact_3563_subseqs__distinctD,axiom,
! [A: $tType,Ys: list @ A,Xs: list @ A] :
( ( member @ ( list @ A ) @ Ys @ ( set2 @ ( list @ A ) @ ( subseqs @ A @ Xs ) ) )
=> ( ( distinct @ A @ Xs )
=> ( distinct @ A @ Ys ) ) ) ).
% subseqs_distinctD
thf(fact_3564_finite__distinct__list,axiom,
! [A: $tType,A4: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ? [Xs2: list @ A] :
( ( ( set2 @ A @ Xs2 )
= A4 )
& ( distinct @ A @ Xs2 ) ) ) ).
% finite_distinct_list
thf(fact_3565_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_3566_distinct__conv__nth,axiom,
! [A: $tType] :
( ( distinct @ A )
= ( ^ [Xs3: list @ A] :
! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs3 ) )
=> ! [J2: nat] :
( ( ord_less @ nat @ J2 @ ( size_size @ ( list @ A ) @ Xs3 ) )
=> ( ( I3 != J2 )
=> ( ( nth @ A @ Xs3 @ I3 )
!= ( nth @ A @ Xs3 @ J2 ) ) ) ) ) ) ) ).
% distinct_conv_nth
thf(fact_3567_card__distinct,axiom,
! [A: $tType,Xs: list @ A] :
( ( ( finite_card @ A @ ( set2 @ A @ Xs ) )
= ( size_size @ ( list @ A ) @ Xs ) )
=> ( distinct @ A @ Xs ) ) ).
% card_distinct
thf(fact_3568_distinct__card,axiom,
! [A: $tType,Xs: list @ A] :
( ( distinct @ A @ Xs )
=> ( ( finite_card @ A @ ( set2 @ A @ Xs ) )
= ( size_size @ ( list @ A ) @ Xs ) ) ) ).
% distinct_card
thf(fact_3569_distinct__Ex1,axiom,
! [A: $tType,Xs: list @ A,X: A] :
( ( distinct @ A @ Xs )
=> ( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ? [X5: nat] :
( ( ord_less @ nat @ X5 @ ( size_size @ ( list @ A ) @ Xs ) )
& ( ( nth @ A @ Xs @ X5 )
= X )
& ! [Y4: nat] :
( ( ( ord_less @ nat @ Y4 @ ( size_size @ ( list @ A ) @ Xs ) )
& ( ( nth @ A @ Xs @ Y4 )
= X ) )
=> ( Y4 = X5 ) ) ) ) ) ).
% distinct_Ex1
thf(fact_3570_subset__CollectI,axiom,
! [A: $tType,B2: set @ A,A4: set @ A,Q: A > $o,P: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ B2 @ A4 )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ B2 )
=> ( ( Q @ X5 )
=> ( P @ X5 ) ) )
=> ( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [X4: A] :
( ( member @ A @ X4 @ B2 )
& ( Q @ X4 ) ) )
@ ( collect @ A
@ ^ [X4: A] :
( ( member @ A @ X4 @ A4 )
& ( P @ X4 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_3571_subset__Collect__iff,axiom,
! [A: $tType,B2: set @ A,A4: set @ A,P: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ B2 @ A4 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2
@ ( collect @ A
@ ^ [X4: A] :
( ( member @ A @ X4 @ A4 )
& ( P @ X4 ) ) ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ B2 )
=> ( P @ X4 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_3572_distinct__list__update,axiom,
! [A: $tType,Xs: list @ A,A3: A,I: nat] :
( ( distinct @ A @ Xs )
=> ( ~ ( member @ A @ A3 @ ( minus_minus @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( insert2 @ A @ ( nth @ A @ Xs @ I ) @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( distinct @ A @ ( list_update @ A @ Xs @ I @ A3 ) ) ) ) ).
% distinct_list_update
thf(fact_3573_set__update__distinct,axiom,
! [A: $tType,Xs: list @ A,N2: nat,X: A] :
( ( distinct @ A @ Xs )
=> ( ( ord_less @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( set2 @ A @ ( list_update @ A @ Xs @ N2 @ X ) )
= ( insert2 @ A @ X @ ( minus_minus @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( insert2 @ A @ ( nth @ A @ Xs @ N2 ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% set_update_distinct
thf(fact_3574_rec__nat__Suc__imp,axiom,
! [A: $tType,F2: nat > A,F1: A,F22: nat > A > A,N2: nat] :
( ( F2
= ( rec_nat @ A @ F1 @ F22 ) )
=> ( ( F2 @ ( suc @ N2 ) )
= ( F22 @ N2 @ ( F2 @ N2 ) ) ) ) ).
% rec_nat_Suc_imp
thf(fact_3575_set__n__lists,axiom,
! [A: $tType,N2: nat,Xs: list @ A] :
( ( set2 @ ( list @ A ) @ ( n_lists @ A @ N2 @ Xs ) )
= ( collect @ ( list @ A )
@ ^ [Ys3: list @ A] :
( ( ( size_size @ ( list @ A ) @ Ys3 )
= N2 )
& ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Ys3 ) @ ( set2 @ A @ Xs ) ) ) ) ) ).
% set_n_lists
thf(fact_3576_distinct__union,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( distinct @ A @ ( union @ A @ Xs @ Ys ) )
= ( distinct @ A @ Ys ) ) ).
% distinct_union
thf(fact_3577_nat_Osplit__sels_I1_J,axiom,
! [A: $tType,P: A > $o,F1: A,F22: nat > A,Nat: nat] :
( ( P @ ( case_nat @ A @ F1 @ F22 @ Nat ) )
= ( ( ( Nat
= ( zero_zero @ nat ) )
=> ( P @ F1 ) )
& ( ( Nat
= ( suc @ ( pred @ Nat ) ) )
=> ( P @ ( F22 @ ( pred @ Nat ) ) ) ) ) ) ).
% nat.split_sels(1)
thf(fact_3578_nat_Osplit__sels_I2_J,axiom,
! [A: $tType,P: A > $o,F1: A,F22: nat > A,Nat: nat] :
( ( P @ ( case_nat @ A @ F1 @ F22 @ Nat ) )
= ( ~ ( ( ( Nat
= ( zero_zero @ nat ) )
& ~ ( P @ F1 ) )
| ( ( Nat
= ( suc @ ( pred @ Nat ) ) )
& ~ ( P @ ( F22 @ ( pred @ Nat ) ) ) ) ) ) ) ).
% nat.split_sels(2)
thf(fact_3579_card__Pow,axiom,
! [A: $tType,A4: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( finite_card @ ( set @ A ) @ ( pow2 @ A @ A4 ) )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( finite_card @ A @ A4 ) ) ) ) ).
% card_Pow
thf(fact_3580_Pow__singleton__iff,axiom,
! [A: $tType,X8: set @ A,Y7: set @ A] :
( ( ( pow2 @ A @ X8 )
= ( insert2 @ ( set @ A ) @ Y7 @ ( bot_bot @ ( set @ ( set @ A ) ) ) ) )
= ( ( X8
= ( bot_bot @ ( set @ A ) ) )
& ( Y7
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Pow_singleton_iff
thf(fact_3581_Pow__empty,axiom,
! [A: $tType] :
( ( pow2 @ A @ ( bot_bot @ ( set @ A ) ) )
= ( insert2 @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ ( bot_bot @ ( set @ ( set @ A ) ) ) ) ) ).
% Pow_empty
thf(fact_3582_PowI,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B2 )
=> ( member @ ( set @ A ) @ A4 @ ( pow2 @ A @ B2 ) ) ) ).
% PowI
thf(fact_3583_Pow__iff,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] :
( ( member @ ( set @ A ) @ A4 @ ( pow2 @ A @ B2 ) )
= ( ord_less_eq @ ( set @ A ) @ A4 @ B2 ) ) ).
% Pow_iff
thf(fact_3584_finite__Pow__iff,axiom,
! [A: $tType,A4: set @ A] :
( ( finite_finite2 @ ( set @ A ) @ ( pow2 @ A @ A4 ) )
= ( finite_finite2 @ A @ A4 ) ) ).
% finite_Pow_iff
thf(fact_3585_Pow__bottom,axiom,
! [A: $tType,B2: set @ A] : ( member @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ ( pow2 @ A @ B2 ) ) ).
% Pow_bottom
thf(fact_3586_Pow__mono,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B2 )
=> ( ord_less_eq @ ( set @ ( set @ A ) ) @ ( pow2 @ A @ A4 ) @ ( pow2 @ A @ B2 ) ) ) ).
% Pow_mono
thf(fact_3587_PowD,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] :
( ( member @ ( set @ A ) @ A4 @ ( pow2 @ A @ B2 ) )
=> ( ord_less_eq @ ( set @ A ) @ A4 @ B2 ) ) ).
% PowD
thf(fact_3588_Pow__top,axiom,
! [A: $tType,A4: set @ A] : ( member @ ( set @ A ) @ A4 @ ( pow2 @ A @ A4 ) ) ).
% Pow_top
thf(fact_3589_Pow__not__empty,axiom,
! [A: $tType,A4: set @ A] :
( ( pow2 @ A @ A4 )
!= ( bot_bot @ ( set @ ( set @ A ) ) ) ) ).
% Pow_not_empty
thf(fact_3590_Pow__def,axiom,
! [A: $tType] :
( ( pow2 @ A )
= ( ^ [A7: set @ A] :
( collect @ ( set @ A )
@ ^ [B6: set @ A] : ( ord_less_eq @ ( set @ A ) @ B6 @ A7 ) ) ) ) ).
% Pow_def
thf(fact_3591_length__n__lists__elem,axiom,
! [A: $tType,Ys: list @ A,N2: nat,Xs: list @ A] :
( ( member @ ( list @ A ) @ Ys @ ( set2 @ ( list @ A ) @ ( n_lists @ A @ N2 @ Xs ) ) )
=> ( ( size_size @ ( list @ A ) @ Ys )
= N2 ) ) ).
% length_n_lists_elem
thf(fact_3592_binomial__def,axiom,
( binomial
= ( ^ [N: nat,K2: nat] :
( finite_card @ ( set @ nat )
@ ( collect @ ( set @ nat )
@ ^ [K6: set @ nat] :
( ( member @ ( set @ nat ) @ K6 @ ( pow2 @ nat @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) )
& ( ( finite_card @ nat @ K6 )
= K2 ) ) ) ) ) ) ).
% binomial_def
thf(fact_3593_length__n__lists,axiom,
! [A: $tType,N2: nat,Xs: list @ A] :
( ( size_size @ ( list @ ( list @ A ) ) @ ( n_lists @ A @ N2 @ Xs ) )
= ( power_power @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ N2 ) ) ).
% length_n_lists
thf(fact_3594_pred__def,axiom,
( pred
= ( case_nat @ nat @ ( zero_zero @ nat )
@ ^ [X24: nat] : X24 ) ) ).
% pred_def
thf(fact_3595_bezw__0,axiom,
! [X: nat] :
( ( bezw @ X @ ( zero_zero @ nat ) )
= ( product_Pair @ int @ int @ ( one_one @ int ) @ ( zero_zero @ int ) ) ) ).
% bezw_0
thf(fact_3596_prod__decode__aux_Osimps,axiom,
( nat_prod_decode_aux
= ( ^ [K2: nat,M: nat] : ( if @ ( product_prod @ nat @ nat ) @ ( ord_less_eq @ nat @ M @ K2 ) @ ( product_Pair @ nat @ nat @ M @ ( minus_minus @ nat @ K2 @ M ) ) @ ( nat_prod_decode_aux @ ( suc @ K2 ) @ ( minus_minus @ nat @ M @ ( suc @ K2 ) ) ) ) ) ) ).
% prod_decode_aux.simps
thf(fact_3597_prod__decode__aux_Oelims,axiom,
! [X: nat,Xa2: nat,Y: product_prod @ nat @ nat] :
( ( ( nat_prod_decode_aux @ X @ Xa2 )
= Y )
=> ( ( ( ord_less_eq @ nat @ Xa2 @ X )
=> ( Y
= ( product_Pair @ nat @ nat @ Xa2 @ ( minus_minus @ nat @ X @ Xa2 ) ) ) )
& ( ~ ( ord_less_eq @ nat @ Xa2 @ X )
=> ( Y
= ( nat_prod_decode_aux @ ( suc @ X ) @ ( minus_minus @ nat @ Xa2 @ ( suc @ X ) ) ) ) ) ) ) ).
% prod_decode_aux.elims
thf(fact_3598_finite__enumerate,axiom,
! [S2: set @ nat] :
( ( finite_finite2 @ nat @ S2 )
=> ? [R3: nat > nat] :
( ( strict_mono_on @ nat @ nat @ R3 @ ( set_ord_lessThan @ nat @ ( finite_card @ nat @ S2 ) ) )
& ! [N7: nat] :
( ( ord_less @ nat @ N7 @ ( finite_card @ nat @ S2 ) )
=> ( member @ nat @ ( R3 @ N7 ) @ S2 ) ) ) ) ).
% finite_enumerate
thf(fact_3599_nth__rotate1,axiom,
! [A: $tType,N2: nat,Xs: list @ A] :
( ( ord_less @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( nth @ A @ ( rotate1 @ A @ Xs ) @ N2 )
= ( nth @ A @ Xs @ ( modulo_modulo @ nat @ ( suc @ N2 ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ) ).
% nth_rotate1
thf(fact_3600_set__rotate1,axiom,
! [A: $tType,Xs: list @ A] :
( ( set2 @ A @ ( rotate1 @ A @ Xs ) )
= ( set2 @ A @ Xs ) ) ).
% set_rotate1
thf(fact_3601_length__rotate1,axiom,
! [A: $tType,Xs: list @ A] :
( ( size_size @ ( list @ A ) @ ( rotate1 @ A @ Xs ) )
= ( size_size @ ( list @ A ) @ Xs ) ) ).
% length_rotate1
thf(fact_3602_distinct1__rotate,axiom,
! [A: $tType,Xs: list @ A] :
( ( distinct @ A @ ( rotate1 @ A @ Xs ) )
= ( distinct @ A @ Xs ) ) ).
% distinct1_rotate
thf(fact_3603_rotate1__length01,axiom,
! [A: $tType,Xs: list @ A] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) )
=> ( ( rotate1 @ A @ Xs )
= Xs ) ) ).
% rotate1_length01
thf(fact_3604_strict__mono__onD,axiom,
! [B: $tType,A: $tType] :
( ( ( ord @ A )
& ( ord @ B ) )
=> ! [F2: A > B,A4: set @ A,R2: A,S: A] :
( ( strict_mono_on @ A @ B @ F2 @ A4 )
=> ( ( member @ A @ R2 @ A4 )
=> ( ( member @ A @ S @ A4 )
=> ( ( ord_less @ A @ R2 @ S )
=> ( ord_less @ B @ ( F2 @ R2 ) @ ( F2 @ S ) ) ) ) ) ) ) ).
% strict_mono_onD
thf(fact_3605_strict__mono__onI,axiom,
! [B: $tType,A: $tType] :
( ( ( ord @ A )
& ( ord @ B ) )
=> ! [A4: set @ A,F2: A > B] :
( ! [R3: A,S3: A] :
( ( member @ A @ R3 @ A4 )
=> ( ( member @ A @ S3 @ A4 )
=> ( ( ord_less @ A @ R3 @ S3 )
=> ( ord_less @ B @ ( F2 @ R3 ) @ ( F2 @ S3 ) ) ) ) )
=> ( strict_mono_on @ A @ B @ F2 @ A4 ) ) ) ).
% strict_mono_onI
thf(fact_3606_strict__mono__on__def,axiom,
! [B: $tType,A: $tType] :
( ( ( ord @ A )
& ( ord @ B ) )
=> ( ( strict_mono_on @ A @ B )
= ( ^ [F3: A > B,A7: set @ A] :
! [R4: A,S7: A] :
( ( ( member @ A @ R4 @ A7 )
& ( member @ A @ S7 @ A7 )
& ( ord_less @ A @ R4 @ S7 ) )
=> ( ord_less @ B @ ( F3 @ R4 ) @ ( F3 @ S7 ) ) ) ) ) ) ).
% strict_mono_on_def
thf(fact_3607_strict__mono__on__leD,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( preorder @ B ) )
=> ! [F2: A > B,A4: set @ A,X: A,Y: A] :
( ( strict_mono_on @ A @ B @ F2 @ A4 )
=> ( ( member @ A @ X @ A4 )
=> ( ( member @ A @ Y @ A4 )
=> ( ( ord_less_eq @ A @ X @ Y )
=> ( ord_less_eq @ B @ ( F2 @ X ) @ ( F2 @ Y ) ) ) ) ) ) ) ).
% strict_mono_on_leD
thf(fact_3608_card__UNION,axiom,
! [A: $tType,A4: set @ ( set @ A )] :
( ( finite_finite2 @ ( set @ A ) @ A4 )
=> ( ! [X5: set @ A] :
( ( member @ ( set @ A ) @ X5 @ A4 )
=> ( finite_finite2 @ A @ X5 ) )
=> ( ( finite_card @ A @ ( complete_Sup_Sup @ ( set @ A ) @ A4 ) )
= ( nat2
@ ( groups7311177749621191930dd_sum @ ( set @ ( set @ A ) ) @ int
@ ^ [I8: set @ ( set @ A )] : ( times_times @ int @ ( power_power @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( plus_plus @ nat @ ( finite_card @ ( set @ A ) @ I8 ) @ ( one_one @ nat ) ) ) @ ( semiring_1_of_nat @ int @ ( finite_card @ A @ ( complete_Inf_Inf @ ( set @ A ) @ I8 ) ) ) )
@ ( collect @ ( set @ ( set @ A ) )
@ ^ [I8: set @ ( set @ A )] :
( ( ord_less_eq @ ( set @ ( set @ A ) ) @ I8 @ A4 )
& ( I8
!= ( bot_bot @ ( set @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ).
% card_UNION
thf(fact_3609_set__remove1__eq,axiom,
! [A: $tType,Xs: list @ A,X: A] :
( ( distinct @ A @ Xs )
=> ( ( set2 @ A @ ( remove1 @ A @ X @ Xs ) )
= ( minus_minus @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% set_remove1_eq
thf(fact_3610_drop__bit__rec,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se4197421643247451524op_bit @ A )
= ( ^ [N: nat,A5: A] :
( if @ A
@ ( N
= ( zero_zero @ nat ) )
@ A5
@ ( bit_se4197421643247451524op_bit @ A @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ ( divide_divide @ A @ A5 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% drop_bit_rec
thf(fact_3611_root__powr__inverse,axiom,
! [N2: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( root @ N2 @ X )
= ( powr @ real @ X @ ( divide_divide @ real @ ( one_one @ real ) @ ( semiring_1_of_nat @ real @ N2 ) ) ) ) ) ) ).
% root_powr_inverse
thf(fact_3612_Sup__lessThan,axiom,
! [A: $tType] :
( ( ( comple6319245703460814977attice @ A )
& ( dense_linorder @ A ) )
=> ! [Y: A] :
( ( complete_Sup_Sup @ A @ ( set_ord_lessThan @ A @ Y ) )
= Y ) ) ).
% Sup_lessThan
thf(fact_3613_finite__Inter,axiom,
! [A: $tType,M7: set @ ( set @ A )] :
( ? [X2: set @ A] :
( ( member @ ( set @ A ) @ X2 @ M7 )
& ( finite_finite2 @ A @ X2 ) )
=> ( finite_finite2 @ A @ ( complete_Inf_Inf @ ( set @ A ) @ M7 ) ) ) ).
% finite_Inter
thf(fact_3614_drop__bit__of__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N2: nat] :
( ( bit_se4197421643247451524op_bit @ A @ N2 @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% drop_bit_of_0
thf(fact_3615_Sup__atMost,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [Y: A] :
( ( complete_Sup_Sup @ A @ ( set_ord_atMost @ A @ Y ) )
= Y ) ) ).
% Sup_atMost
thf(fact_3616_drop__bit__drop__bit,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M2: nat,N2: nat,A3: A] :
( ( bit_se4197421643247451524op_bit @ A @ M2 @ ( bit_se4197421643247451524op_bit @ A @ N2 @ A3 ) )
= ( bit_se4197421643247451524op_bit @ A @ ( plus_plus @ nat @ M2 @ N2 ) @ A3 ) ) ) ).
% drop_bit_drop_bit
thf(fact_3617_in__set__remove1,axiom,
! [A: $tType,A3: A,B3: A,Xs: list @ A] :
( ( A3 != B3 )
=> ( ( member @ A @ A3 @ ( set2 @ A @ ( remove1 @ A @ B3 @ Xs ) ) )
= ( member @ A @ A3 @ ( set2 @ A @ Xs ) ) ) ) ).
% in_set_remove1
thf(fact_3618_Sup__atLeastAtMost,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( complete_Sup_Sup @ A @ ( set_or1337092689740270186AtMost @ A @ X @ Y ) )
= Y ) ) ) ).
% Sup_atLeastAtMost
thf(fact_3619_cSup__atLeastAtMost,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ Y @ X )
=> ( ( complete_Sup_Sup @ A @ ( set_or1337092689740270186AtMost @ A @ Y @ X ) )
= X ) ) ) ).
% cSup_atLeastAtMost
thf(fact_3620_cSup__singleton,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X: A] :
( ( complete_Sup_Sup @ A @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= X ) ) ).
% cSup_singleton
thf(fact_3621_Inf__atLeastAtMost,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( complete_Inf_Inf @ A @ ( set_or1337092689740270186AtMost @ A @ X @ Y ) )
= X ) ) ) ).
% Inf_atLeastAtMost
thf(fact_3622_cInf__atLeastAtMost,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ Y @ X )
=> ( ( complete_Inf_Inf @ A @ ( set_or1337092689740270186AtMost @ A @ Y @ X ) )
= Y ) ) ) ).
% cInf_atLeastAtMost
thf(fact_3623_cInf__singleton,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X: A] :
( ( complete_Inf_Inf @ A @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= X ) ) ).
% cInf_singleton
thf(fact_3624_cSup__atLeastLessThan,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( dense_linorder @ A ) )
=> ! [Y: A,X: A] :
( ( ord_less @ A @ Y @ X )
=> ( ( complete_Sup_Sup @ A @ ( set_or7035219750837199246ssThan @ A @ Y @ X ) )
= X ) ) ) ).
% cSup_atLeastLessThan
thf(fact_3625_Sup__atLeastLessThan,axiom,
! [A: $tType] :
( ( ( comple6319245703460814977attice @ A )
& ( dense_linorder @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ( ( complete_Sup_Sup @ A @ ( set_or7035219750837199246ssThan @ A @ X @ Y ) )
= Y ) ) ) ).
% Sup_atLeastLessThan
thf(fact_3626_cInf__atLeastLessThan,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [Y: A,X: A] :
( ( ord_less @ A @ Y @ X )
=> ( ( complete_Inf_Inf @ A @ ( set_or7035219750837199246ssThan @ A @ Y @ X ) )
= Y ) ) ) ).
% cInf_atLeastLessThan
thf(fact_3627_Inf__atLeastLessThan,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ( ( complete_Inf_Inf @ A @ ( set_or7035219750837199246ssThan @ A @ X @ Y ) )
= X ) ) ) ).
% Inf_atLeastLessThan
thf(fact_3628_Inf__atMost,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [X: A] :
( ( complete_Inf_Inf @ A @ ( set_ord_atMost @ A @ X ) )
= ( bot_bot @ A ) ) ) ).
% Inf_atMost
thf(fact_3629_real__root__Suc__0,axiom,
! [X: real] :
( ( root @ ( suc @ ( zero_zero @ nat ) ) @ X )
= X ) ).
% real_root_Suc_0
thf(fact_3630_root__0,axiom,
! [X: real] :
( ( root @ ( zero_zero @ nat ) @ X )
= ( zero_zero @ real ) ) ).
% root_0
thf(fact_3631_real__root__eq__iff,axiom,
! [N2: nat,X: real,Y: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( ( root @ N2 @ X )
= ( root @ N2 @ Y ) )
= ( X = Y ) ) ) ).
% real_root_eq_iff
thf(fact_3632_drop__bit__of__bool,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N2: nat,B3: $o] :
( ( bit_se4197421643247451524op_bit @ A @ N2 @ ( zero_neq_one_of_bool @ A @ B3 ) )
= ( zero_neq_one_of_bool @ A
@ ( ( N2
= ( zero_zero @ nat ) )
& B3 ) ) ) ) ).
% drop_bit_of_bool
thf(fact_3633_finite__Union,axiom,
! [A: $tType,A4: set @ ( set @ A )] :
( ( finite_finite2 @ ( set @ A ) @ A4 )
=> ( ! [M8: set @ A] :
( ( member @ ( set @ A ) @ M8 @ A4 )
=> ( finite_finite2 @ A @ M8 ) )
=> ( finite_finite2 @ A @ ( complete_Sup_Sup @ ( set @ A ) @ A4 ) ) ) ) ).
% finite_Union
thf(fact_3634_drop__bit__of__Suc__0,axiom,
! [N2: nat] :
( ( bit_se4197421643247451524op_bit @ nat @ N2 @ ( suc @ ( zero_zero @ nat ) ) )
= ( zero_neq_one_of_bool @ nat
@ ( N2
= ( zero_zero @ nat ) ) ) ) ).
% drop_bit_of_Suc_0
thf(fact_3635_drop__bit__Suc__minus__bit0,axiom,
! [N2: nat,K: num] :
( ( bit_se4197421643247451524op_bit @ int @ ( suc @ N2 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ K ) ) ) )
= ( bit_se4197421643247451524op_bit @ int @ N2 @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) ) ) ).
% drop_bit_Suc_minus_bit0
thf(fact_3636_drop__bit__Suc__bit0,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [N2: nat,K: num] :
( ( bit_se4197421643247451524op_bit @ A @ ( suc @ N2 ) @ ( numeral_numeral @ A @ ( bit0 @ K ) ) )
= ( bit_se4197421643247451524op_bit @ A @ N2 @ ( numeral_numeral @ A @ K ) ) ) ) ).
% drop_bit_Suc_bit0
thf(fact_3637_drop__bit__Suc__bit1,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [N2: nat,K: num] :
( ( bit_se4197421643247451524op_bit @ A @ ( suc @ N2 ) @ ( numeral_numeral @ A @ ( bit1 @ K ) ) )
= ( bit_se4197421643247451524op_bit @ A @ N2 @ ( numeral_numeral @ A @ K ) ) ) ) ).
% drop_bit_Suc_bit1
thf(fact_3638_real__root__eq__0__iff,axiom,
! [N2: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( ( root @ N2 @ X )
= ( zero_zero @ real ) )
= ( X
= ( zero_zero @ real ) ) ) ) ).
% real_root_eq_0_iff
thf(fact_3639_real__root__less__iff,axiom,
! [N2: nat,X: real,Y: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( ord_less @ real @ ( root @ N2 @ X ) @ ( root @ N2 @ Y ) )
= ( ord_less @ real @ X @ Y ) ) ) ).
% real_root_less_iff
thf(fact_3640_real__root__le__iff,axiom,
! [N2: nat,X: real,Y: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( ord_less_eq @ real @ ( root @ N2 @ X ) @ ( root @ N2 @ Y ) )
= ( ord_less_eq @ real @ X @ Y ) ) ) ).
% real_root_le_iff
thf(fact_3641_real__root__one,axiom,
! [N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( root @ N2 @ ( one_one @ real ) )
= ( one_one @ real ) ) ) ).
% real_root_one
thf(fact_3642_real__root__eq__1__iff,axiom,
! [N2: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( ( root @ N2 @ X )
= ( one_one @ real ) )
= ( X
= ( one_one @ real ) ) ) ) ).
% real_root_eq_1_iff
thf(fact_3643_drop__bit__of__1,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N2: nat] :
( ( bit_se4197421643247451524op_bit @ A @ N2 @ ( one_one @ A ) )
= ( zero_neq_one_of_bool @ A
@ ( N2
= ( zero_zero @ nat ) ) ) ) ) ).
% drop_bit_of_1
thf(fact_3644_real__root__gt__0__iff,axiom,
! [N2: nat,Y: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( root @ N2 @ Y ) )
= ( ord_less @ real @ ( zero_zero @ real ) @ Y ) ) ) ).
% real_root_gt_0_iff
thf(fact_3645_real__root__lt__0__iff,axiom,
! [N2: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( ord_less @ real @ ( root @ N2 @ X ) @ ( zero_zero @ real ) )
= ( ord_less @ real @ X @ ( zero_zero @ real ) ) ) ) ).
% real_root_lt_0_iff
thf(fact_3646_real__root__ge__0__iff,axiom,
! [N2: nat,Y: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( root @ N2 @ Y ) )
= ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y ) ) ) ).
% real_root_ge_0_iff
thf(fact_3647_real__root__le__0__iff,axiom,
! [N2: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( ord_less_eq @ real @ ( root @ N2 @ X ) @ ( zero_zero @ real ) )
= ( ord_less_eq @ real @ X @ ( zero_zero @ real ) ) ) ) ).
% real_root_le_0_iff
thf(fact_3648_real__root__gt__1__iff,axiom,
! [N2: nat,Y: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( ord_less @ real @ ( one_one @ real ) @ ( root @ N2 @ Y ) )
= ( ord_less @ real @ ( one_one @ real ) @ Y ) ) ) ).
% real_root_gt_1_iff
thf(fact_3649_real__root__lt__1__iff,axiom,
! [N2: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( ord_less @ real @ ( root @ N2 @ X ) @ ( one_one @ real ) )
= ( ord_less @ real @ X @ ( one_one @ real ) ) ) ) ).
% real_root_lt_1_iff
thf(fact_3650_real__root__ge__1__iff,axiom,
! [N2: nat,Y: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( ord_less_eq @ real @ ( one_one @ real ) @ ( root @ N2 @ Y ) )
= ( ord_less_eq @ real @ ( one_one @ real ) @ Y ) ) ) ).
% real_root_ge_1_iff
thf(fact_3651_real__root__le__1__iff,axiom,
! [N2: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( ord_less_eq @ real @ ( root @ N2 @ X ) @ ( one_one @ real ) )
= ( ord_less_eq @ real @ X @ ( one_one @ real ) ) ) ) ).
% real_root_le_1_iff
thf(fact_3652_drop__bit__Suc__minus__bit1,axiom,
! [N2: nat,K: num] :
( ( bit_se4197421643247451524op_bit @ int @ ( suc @ N2 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ K ) ) ) )
= ( bit_se4197421643247451524op_bit @ int @ N2 @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( inc @ K ) ) ) ) ) ).
% drop_bit_Suc_minus_bit1
thf(fact_3653_real__root__pow__pos2,axiom,
! [N2: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( power_power @ real @ ( root @ N2 @ X ) @ N2 )
= X ) ) ) ).
% real_root_pow_pos2
thf(fact_3654_remove1__commute,axiom,
! [A: $tType,X: A,Y: A,Zs: list @ A] :
( ( remove1 @ A @ X @ ( remove1 @ A @ Y @ Zs ) )
= ( remove1 @ A @ Y @ ( remove1 @ A @ X @ Zs ) ) ) ).
% remove1_commute
thf(fact_3655_cSup__eq,axiom,
! [A: $tType] :
( ( ( condit1219197933456340205attice @ A )
& ( no_bot @ A ) )
=> ! [X8: set @ A,A3: A] :
( ! [X5: A] :
( ( member @ A @ X5 @ X8 )
=> ( ord_less_eq @ A @ X5 @ A3 ) )
=> ( ! [Y3: A] :
( ! [X2: A] :
( ( member @ A @ X2 @ X8 )
=> ( ord_less_eq @ A @ X2 @ Y3 ) )
=> ( ord_less_eq @ A @ A3 @ Y3 ) )
=> ( ( complete_Sup_Sup @ A @ X8 )
= A3 ) ) ) ) ).
% cSup_eq
thf(fact_3656_cSup__eq__maximum,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [Z3: A,X8: set @ A] :
( ( member @ A @ Z3 @ X8 )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ X8 )
=> ( ord_less_eq @ A @ X5 @ Z3 ) )
=> ( ( complete_Sup_Sup @ A @ X8 )
= Z3 ) ) ) ) ).
% cSup_eq_maximum
thf(fact_3657_cInf__eq,axiom,
! [A: $tType] :
( ( ( condit1219197933456340205attice @ A )
& ( no_top @ A ) )
=> ! [X8: set @ A,A3: A] :
( ! [X5: A] :
( ( member @ A @ X5 @ X8 )
=> ( ord_less_eq @ A @ A3 @ X5 ) )
=> ( ! [Y3: A] :
( ! [X2: A] :
( ( member @ A @ X2 @ X8 )
=> ( ord_less_eq @ A @ Y3 @ X2 ) )
=> ( ord_less_eq @ A @ Y3 @ A3 ) )
=> ( ( complete_Inf_Inf @ A @ X8 )
= A3 ) ) ) ) ).
% cInf_eq
thf(fact_3658_cInf__eq__minimum,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [Z3: A,X8: set @ A] :
( ( member @ A @ Z3 @ X8 )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ X8 )
=> ( ord_less_eq @ A @ Z3 @ X5 ) )
=> ( ( complete_Inf_Inf @ A @ X8 )
= Z3 ) ) ) ) ).
% cInf_eq_minimum
thf(fact_3659_notin__set__remove1,axiom,
! [A: $tType,X: A,Xs: list @ A,Y: A] :
( ~ ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ~ ( member @ A @ X @ ( set2 @ A @ ( remove1 @ A @ Y @ Xs ) ) ) ) ).
% notin_set_remove1
thf(fact_3660_remove1__idem,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ~ ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ( remove1 @ A @ X @ Xs )
= Xs ) ) ).
% remove1_idem
thf(fact_3661_distinct__remove1,axiom,
! [A: $tType,Xs: list @ A,X: A] :
( ( distinct @ A @ Xs )
=> ( distinct @ A @ ( remove1 @ A @ X @ Xs ) ) ) ).
% distinct_remove1
thf(fact_3662_cInf__greatest,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X8: set @ A,Z3: A] :
( ( X8
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ X8 )
=> ( ord_less_eq @ A @ Z3 @ X5 ) )
=> ( ord_less_eq @ A @ Z3 @ ( complete_Inf_Inf @ A @ X8 ) ) ) ) ) ).
% cInf_greatest
thf(fact_3663_cInf__eq__non__empty,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X8: set @ A,A3: A] :
( ( X8
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ X8 )
=> ( ord_less_eq @ A @ A3 @ X5 ) )
=> ( ! [Y3: A] :
( ! [X2: A] :
( ( member @ A @ X2 @ X8 )
=> ( ord_less_eq @ A @ Y3 @ X2 ) )
=> ( ord_less_eq @ A @ Y3 @ A3 ) )
=> ( ( complete_Inf_Inf @ A @ X8 )
= A3 ) ) ) ) ) ).
% cInf_eq_non_empty
thf(fact_3664_cInf__le__finite,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X8: set @ A,X: A] :
( ( finite_finite2 @ A @ X8 )
=> ( ( member @ A @ X @ X8 )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ X8 ) @ X ) ) ) ) ).
% cInf_le_finite
thf(fact_3665_cInf__lessD,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [X8: set @ A,Z3: A] :
( ( X8
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less @ A @ ( complete_Inf_Inf @ A @ X8 ) @ Z3 )
=> ? [X5: A] :
( ( member @ A @ X5 @ X8 )
& ( ord_less @ A @ X5 @ Z3 ) ) ) ) ) ).
% cInf_lessD
thf(fact_3666_finite__imp__less__Inf,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [X8: set @ A,X: A,A3: A] :
( ( finite_finite2 @ A @ X8 )
=> ( ( member @ A @ X @ X8 )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ X8 )
=> ( ord_less @ A @ A3 @ X5 ) )
=> ( ord_less @ A @ A3 @ ( complete_Inf_Inf @ A @ X8 ) ) ) ) ) ) ).
% finite_imp_less_Inf
thf(fact_3667_cSup__least,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X8: set @ A,Z3: A] :
( ( X8
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ X8 )
=> ( ord_less_eq @ A @ X5 @ Z3 ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ X8 ) @ Z3 ) ) ) ) ).
% cSup_least
thf(fact_3668_cSup__eq__non__empty,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X8: set @ A,A3: A] :
( ( X8
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ X8 )
=> ( ord_less_eq @ A @ X5 @ A3 ) )
=> ( ! [Y3: A] :
( ! [X2: A] :
( ( member @ A @ X2 @ X8 )
=> ( ord_less_eq @ A @ X2 @ Y3 ) )
=> ( ord_less_eq @ A @ A3 @ Y3 ) )
=> ( ( complete_Sup_Sup @ A @ X8 )
= A3 ) ) ) ) ) ).
% cSup_eq_non_empty
thf(fact_3669_le__cSup__finite,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X8: set @ A,X: A] :
( ( finite_finite2 @ A @ X8 )
=> ( ( member @ A @ X @ X8 )
=> ( ord_less_eq @ A @ X @ ( complete_Sup_Sup @ A @ X8 ) ) ) ) ) ).
% le_cSup_finite
thf(fact_3670_less__cSupE,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [Y: A,X8: set @ A] :
( ( ord_less @ A @ Y @ ( complete_Sup_Sup @ A @ X8 ) )
=> ( ( X8
!= ( bot_bot @ ( set @ A ) ) )
=> ~ ! [X5: A] :
( ( member @ A @ X5 @ X8 )
=> ~ ( ord_less @ A @ Y @ X5 ) ) ) ) ) ).
% less_cSupE
thf(fact_3671_less__cSupD,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [X8: set @ A,Z3: A] :
( ( X8
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less @ A @ Z3 @ ( complete_Sup_Sup @ A @ X8 ) )
=> ? [X5: A] :
( ( member @ A @ X5 @ X8 )
& ( ord_less @ A @ Z3 @ X5 ) ) ) ) ) ).
% less_cSupD
thf(fact_3672_finite__imp__Sup__less,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [X8: set @ A,X: A,A3: A] :
( ( finite_finite2 @ A @ X8 )
=> ( ( member @ A @ X @ X8 )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ X8 )
=> ( ord_less @ A @ X5 @ A3 ) )
=> ( ord_less @ A @ ( complete_Sup_Sup @ A @ X8 ) @ A3 ) ) ) ) ) ).
% finite_imp_Sup_less
thf(fact_3673_card__Union__le__sum__card,axiom,
! [A: $tType,U3: set @ ( set @ A )] : ( ord_less_eq @ nat @ ( finite_card @ A @ ( complete_Sup_Sup @ ( set @ A ) @ U3 ) ) @ ( groups7311177749621191930dd_sum @ ( set @ A ) @ nat @ ( finite_card @ A ) @ U3 ) ) ).
% card_Union_le_sum_card
thf(fact_3674_take__bit__eq__self__iff__drop__bit__eq__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N2: nat,A3: A] :
( ( ( bit_se2584673776208193580ke_bit @ A @ N2 @ A3 )
= A3 )
= ( ( bit_se4197421643247451524op_bit @ A @ N2 @ A3 )
= ( zero_zero @ A ) ) ) ) ).
% take_bit_eq_self_iff_drop_bit_eq_0
thf(fact_3675_finite__UnionD,axiom,
! [A: $tType,A4: set @ ( set @ A )] :
( ( finite_finite2 @ A @ ( complete_Sup_Sup @ ( set @ A ) @ A4 ) )
=> ( finite_finite2 @ ( set @ A ) @ A4 ) ) ).
% finite_UnionD
thf(fact_3676_take__bit__drop__bit,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M2: nat,N2: nat,A3: A] :
( ( bit_se2584673776208193580ke_bit @ A @ M2 @ ( bit_se4197421643247451524op_bit @ A @ N2 @ A3 ) )
= ( bit_se4197421643247451524op_bit @ A @ N2 @ ( bit_se2584673776208193580ke_bit @ A @ ( plus_plus @ nat @ M2 @ N2 ) @ A3 ) ) ) ) ).
% take_bit_drop_bit
thf(fact_3677_set__remove1__subset,axiom,
! [A: $tType,X: A,Xs: list @ A] : ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ ( remove1 @ A @ X @ Xs ) ) @ ( set2 @ A @ Xs ) ) ).
% set_remove1_subset
thf(fact_3678_real__root__less__mono,axiom,
! [N2: nat,X: real,Y: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( ord_less @ real @ X @ Y )
=> ( ord_less @ real @ ( root @ N2 @ X ) @ ( root @ N2 @ Y ) ) ) ) ).
% real_root_less_mono
thf(fact_3679_real__root__le__mono,axiom,
! [N2: nat,X: real,Y: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( ord_less_eq @ real @ X @ Y )
=> ( ord_less_eq @ real @ ( root @ N2 @ X ) @ ( root @ N2 @ Y ) ) ) ) ).
% real_root_le_mono
thf(fact_3680_real__root__power,axiom,
! [N2: nat,X: real,K: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( root @ N2 @ ( power_power @ real @ X @ K ) )
= ( power_power @ real @ ( root @ N2 @ X ) @ K ) ) ) ).
% real_root_power
thf(fact_3681_real__root__abs,axiom,
! [N2: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( root @ N2 @ ( abs_abs @ real @ X ) )
= ( abs_abs @ real @ ( root @ N2 @ X ) ) ) ) ).
% real_root_abs
thf(fact_3682_finite__less__Inf__iff,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [X8: set @ A,A3: A] :
( ( finite_finite2 @ A @ X8 )
=> ( ( X8
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less @ A @ A3 @ ( complete_Inf_Inf @ A @ X8 ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ X8 )
=> ( ord_less @ A @ A3 @ X4 ) ) ) ) ) ) ) ).
% finite_less_Inf_iff
thf(fact_3683_finite__Sup__less__iff,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [X8: set @ A,A3: A] :
( ( finite_finite2 @ A @ X8 )
=> ( ( X8
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less @ A @ ( complete_Sup_Sup @ A @ X8 ) @ A3 )
= ( ! [X4: A] :
( ( member @ A @ X4 @ X8 )
=> ( ord_less @ A @ X4 @ A3 ) ) ) ) ) ) ) ).
% finite_Sup_less_iff
thf(fact_3684_cInf__abs__ge,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( linordered_idom @ A ) )
=> ! [S2: set @ A,A3: A] :
( ( S2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ S2 )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ X5 ) @ A3 ) )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ ( complete_Inf_Inf @ A @ S2 ) ) @ A3 ) ) ) ) ).
% cInf_abs_ge
thf(fact_3685_cSup__abs__le,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( linordered_idom @ A ) )
=> ! [S2: set @ A,A3: A] :
( ( S2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ S2 )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ X5 ) @ A3 ) )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ ( complete_Sup_Sup @ A @ S2 ) ) @ A3 ) ) ) ) ).
% cSup_abs_le
thf(fact_3686_sgn__root,axiom,
! [N2: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( sgn_sgn @ real @ ( root @ N2 @ X ) )
= ( sgn_sgn @ real @ X ) ) ) ).
% sgn_root
thf(fact_3687_card__Union__le__sum__card__weak,axiom,
! [A: $tType,U3: set @ ( set @ A )] :
( ! [X5: set @ A] :
( ( member @ ( set @ A ) @ X5 @ U3 )
=> ( finite_finite2 @ A @ X5 ) )
=> ( ord_less_eq @ nat @ ( finite_card @ A @ ( complete_Sup_Sup @ ( set @ A ) @ U3 ) ) @ ( groups7311177749621191930dd_sum @ ( set @ A ) @ nat @ ( finite_card @ A ) @ U3 ) ) ) ).
% card_Union_le_sum_card_weak
thf(fact_3688_bits__ident,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N2: nat,A3: A] :
( ( plus_plus @ A @ ( bit_se4730199178511100633sh_bit @ A @ N2 @ ( bit_se4197421643247451524op_bit @ A @ N2 @ A3 ) ) @ ( bit_se2584673776208193580ke_bit @ A @ N2 @ A3 ) )
= A3 ) ) ).
% bits_ident
thf(fact_3689_real__root__gt__zero,axiom,
! [N2: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( root @ N2 @ X ) ) ) ) ).
% real_root_gt_zero
thf(fact_3690_real__root__strict__decreasing,axiom,
! [N2: nat,N6: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( ord_less @ nat @ N2 @ N6 )
=> ( ( ord_less @ real @ ( one_one @ real ) @ X )
=> ( ord_less @ real @ ( root @ N6 @ X ) @ ( root @ N2 @ X ) ) ) ) ) ).
% real_root_strict_decreasing
thf(fact_3691_root__abs__power,axiom,
! [N2: nat,Y: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( abs_abs @ real @ ( root @ N2 @ ( power_power @ real @ Y @ N2 ) ) )
= ( abs_abs @ real @ Y ) ) ) ).
% root_abs_power
thf(fact_3692_cInf__asclose,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( linordered_idom @ A ) )
=> ! [S2: set @ A,L: A,E2: A] :
( ( S2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ S2 )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ X5 @ L ) ) @ E2 ) )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ ( complete_Inf_Inf @ A @ S2 ) @ L ) ) @ E2 ) ) ) ) ).
% cInf_asclose
thf(fact_3693_cSup__asclose,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( linordered_idom @ A ) )
=> ! [S2: set @ A,L: A,E2: A] :
( ( S2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ S2 )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ X5 @ L ) ) @ E2 ) )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ ( complete_Sup_Sup @ A @ S2 ) @ L ) ) @ E2 ) ) ) ) ).
% cSup_asclose
thf(fact_3694_Sup__insert__finite,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [S2: set @ A,X: A] :
( ( finite_finite2 @ A @ S2 )
=> ( ( ( S2
= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Sup_Sup @ A @ ( insert2 @ A @ X @ S2 ) )
= X ) )
& ( ( S2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Sup_Sup @ A @ ( insert2 @ A @ X @ S2 ) )
= ( ord_max @ A @ X @ ( complete_Sup_Sup @ A @ S2 ) ) ) ) ) ) ) ).
% Sup_insert_finite
thf(fact_3695_finite__subset__Union,axiom,
! [A: $tType,A4: set @ A,B11: set @ ( set @ A )] :
( ( finite_finite2 @ A @ A4 )
=> ( ( ord_less_eq @ ( set @ A ) @ A4 @ ( complete_Sup_Sup @ ( set @ A ) @ B11 ) )
=> ~ ! [F7: set @ ( set @ A )] :
( ( finite_finite2 @ ( set @ A ) @ F7 )
=> ( ( ord_less_eq @ ( set @ ( set @ A ) ) @ F7 @ B11 )
=> ~ ( ord_less_eq @ ( set @ A ) @ A4 @ ( complete_Sup_Sup @ ( set @ A ) @ F7 ) ) ) ) ) ) ).
% finite_subset_Union
thf(fact_3696_real__root__pos__pos,axiom,
! [N2: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( root @ N2 @ X ) ) ) ) ).
% real_root_pos_pos
thf(fact_3697_real__root__strict__increasing,axiom,
! [N2: nat,N6: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( ord_less @ nat @ N2 @ N6 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ X @ ( one_one @ real ) )
=> ( ord_less @ real @ ( root @ N2 @ X ) @ ( root @ N6 @ X ) ) ) ) ) ) ).
% real_root_strict_increasing
thf(fact_3698_real__root__decreasing,axiom,
! [N2: nat,N6: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( ord_less_eq @ nat @ N2 @ N6 )
=> ( ( ord_less_eq @ real @ ( one_one @ real ) @ X )
=> ( ord_less_eq @ real @ ( root @ N6 @ X ) @ ( root @ N2 @ X ) ) ) ) ) ).
% real_root_decreasing
thf(fact_3699_real__root__pow__pos,axiom,
! [N2: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( power_power @ real @ ( root @ N2 @ X ) @ N2 )
= X ) ) ) ).
% real_root_pow_pos
thf(fact_3700_real__root__pos__unique,axiom,
! [N2: nat,Y: real,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y )
=> ( ( ( power_power @ real @ Y @ N2 )
= X )
=> ( ( root @ N2 @ X )
= Y ) ) ) ) ).
% real_root_pos_unique
thf(fact_3701_real__root__power__cancel,axiom,
! [N2: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( root @ N2 @ ( power_power @ real @ X @ N2 ) )
= X ) ) ) ).
% real_root_power_cancel
thf(fact_3702_length__remove1,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ( size_size @ ( list @ A ) @ ( remove1 @ A @ X @ Xs ) )
= ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) ) ) )
& ( ~ ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ( size_size @ ( list @ A ) @ ( remove1 @ A @ X @ Xs ) )
= ( size_size @ ( list @ A ) @ Xs ) ) ) ) ).
% length_remove1
thf(fact_3703_drop__bit__Suc,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N2: nat,A3: A] :
( ( bit_se4197421643247451524op_bit @ A @ ( suc @ N2 ) @ A3 )
= ( bit_se4197421643247451524op_bit @ A @ N2 @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% drop_bit_Suc
thf(fact_3704_real__root__increasing,axiom,
! [N2: nat,N6: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( ord_less_eq @ nat @ N2 @ N6 )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ X @ ( one_one @ real ) )
=> ( ord_less_eq @ real @ ( root @ N2 @ X ) @ ( root @ N6 @ X ) ) ) ) ) ) ).
% real_root_increasing
thf(fact_3705_root__sgn__power,axiom,
! [N2: nat,Y: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( root @ N2 @ ( times_times @ real @ ( sgn_sgn @ real @ Y ) @ ( power_power @ real @ ( abs_abs @ real @ Y ) @ N2 ) ) )
= Y ) ) ).
% root_sgn_power
thf(fact_3706_sgn__power__root,axiom,
! [N2: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( times_times @ real @ ( sgn_sgn @ real @ ( root @ N2 @ X ) ) @ ( power_power @ real @ ( abs_abs @ real @ ( root @ N2 @ X ) ) @ N2 ) )
= X ) ) ).
% sgn_power_root
thf(fact_3707_ln__root,axiom,
! [N2: nat,B3: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ B3 )
=> ( ( ln_ln @ real @ ( root @ N2 @ B3 ) )
= ( divide_divide @ real @ ( ln_ln @ real @ B3 ) @ ( semiring_1_of_nat @ real @ N2 ) ) ) ) ) ).
% ln_root
thf(fact_3708_log__root,axiom,
! [N2: nat,A3: real,B3: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ A3 )
=> ( ( log @ B3 @ ( root @ N2 @ A3 ) )
= ( divide_divide @ real @ ( log @ B3 @ A3 ) @ ( semiring_1_of_nat @ real @ N2 ) ) ) ) ) ).
% log_root
thf(fact_3709_log__base__root,axiom,
! [N2: nat,B3: real,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ B3 )
=> ( ( log @ ( root @ N2 @ B3 ) @ X )
= ( times_times @ real @ ( semiring_1_of_nat @ real @ N2 ) @ ( log @ B3 @ X ) ) ) ) ) ).
% log_base_root
thf(fact_3710_split__root,axiom,
! [P: real > $o,N2: nat,X: real] :
( ( P @ ( root @ N2 @ X ) )
= ( ( ( N2
= ( zero_zero @ nat ) )
=> ( P @ ( zero_zero @ real ) ) )
& ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ! [Y6: real] :
( ( ( times_times @ real @ ( sgn_sgn @ real @ Y6 ) @ ( power_power @ real @ ( abs_abs @ real @ Y6 ) @ N2 ) )
= X )
=> ( P @ Y6 ) ) ) ) ) ).
% split_root
thf(fact_3711_ccpo__Sup__singleton,axiom,
! [A: $tType] :
( ( comple9053668089753744459l_ccpo @ A )
=> ! [X: A] :
( ( complete_Sup_Sup @ A @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= X ) ) ).
% ccpo_Sup_singleton
thf(fact_3712_ccSup__empty,axiom,
! [A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ( ( complete_Sup_Sup @ A @ ( bot_bot @ ( set @ A ) ) )
= ( bot_bot @ A ) ) ) ).
% ccSup_empty
thf(fact_3713_Sup__empty,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ( ( complete_Sup_Sup @ A @ ( bot_bot @ ( set @ A ) ) )
= ( bot_bot @ A ) ) ) ).
% Sup_empty
thf(fact_3714_Inf__eq__bot__iff,axiom,
! [A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [A4: set @ A] :
( ( ( complete_Inf_Inf @ A @ A4 )
= ( bot_bot @ A ) )
= ( ! [X4: A] :
( ( ord_less @ A @ ( bot_bot @ A ) @ X4 )
=> ? [Y6: A] :
( ( member @ A @ Y6 @ A4 )
& ( ord_less @ A @ Y6 @ X4 ) ) ) ) ) ) ).
% Inf_eq_bot_iff
thf(fact_3715_Sup__bot__conv_I2_J,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A4: set @ A] :
( ( ( bot_bot @ A )
= ( complete_Sup_Sup @ A @ A4 ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ A4 )
=> ( X4
= ( bot_bot @ A ) ) ) ) ) ) ).
% Sup_bot_conv(2)
thf(fact_3716_Sup__bot__conv_I1_J,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A4: set @ A] :
( ( ( complete_Sup_Sup @ A @ A4 )
= ( bot_bot @ A ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ A4 )
=> ( X4
= ( bot_bot @ A ) ) ) ) ) ) ).
% Sup_bot_conv(1)
thf(fact_3717_Sup__nat__empty,axiom,
( ( complete_Sup_Sup @ nat @ ( bot_bot @ ( set @ nat ) ) )
= ( zero_zero @ nat ) ) ).
% Sup_nat_empty
thf(fact_3718_Inf__nat__def1,axiom,
! [K5: set @ nat] :
( ( K5
!= ( bot_bot @ ( set @ nat ) ) )
=> ( member @ nat @ ( complete_Inf_Inf @ nat @ K5 ) @ K5 ) ) ).
% Inf_nat_def1
thf(fact_3719_Sup__eqI,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A4: set @ A,X: A] :
( ! [Y3: A] :
( ( member @ A @ Y3 @ A4 )
=> ( ord_less_eq @ A @ Y3 @ X ) )
=> ( ! [Y3: A] :
( ! [Z4: A] :
( ( member @ A @ Z4 @ A4 )
=> ( ord_less_eq @ A @ Z4 @ Y3 ) )
=> ( ord_less_eq @ A @ X @ Y3 ) )
=> ( ( complete_Sup_Sup @ A @ A4 )
= X ) ) ) ) ).
% Sup_eqI
thf(fact_3720_Sup__mono,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A4: set @ A,B2: set @ A] :
( ! [A6: A] :
( ( member @ A @ A6 @ A4 )
=> ? [X2: A] :
( ( member @ A @ X2 @ B2 )
& ( ord_less_eq @ A @ A6 @ X2 ) ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A4 ) @ ( complete_Sup_Sup @ A @ B2 ) ) ) ) ).
% Sup_mono
thf(fact_3721_Sup__least,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A4: set @ A,Z3: A] :
( ! [X5: A] :
( ( member @ A @ X5 @ A4 )
=> ( ord_less_eq @ A @ X5 @ Z3 ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A4 ) @ Z3 ) ) ) ).
% Sup_least
thf(fact_3722_Sup__upper,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [X: A,A4: set @ A] :
( ( member @ A @ X @ A4 )
=> ( ord_less_eq @ A @ X @ ( complete_Sup_Sup @ A @ A4 ) ) ) ) ).
% Sup_upper
thf(fact_3723_Sup__le__iff,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A4: set @ A,B3: A] :
( ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A4 ) @ B3 )
= ( ! [X4: A] :
( ( member @ A @ X4 @ A4 )
=> ( ord_less_eq @ A @ X4 @ B3 ) ) ) ) ) ).
% Sup_le_iff
thf(fact_3724_Sup__upper2,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [U: A,A4: set @ A,V2: A] :
( ( member @ A @ U @ A4 )
=> ( ( ord_less_eq @ A @ V2 @ U )
=> ( ord_less_eq @ A @ V2 @ ( complete_Sup_Sup @ A @ A4 ) ) ) ) ) ).
% Sup_upper2
thf(fact_3725_less__Sup__iff,axiom,
! [A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [A3: A,S2: set @ A] :
( ( ord_less @ A @ A3 @ ( complete_Sup_Sup @ A @ S2 ) )
= ( ? [X4: A] :
( ( member @ A @ X4 @ S2 )
& ( ord_less @ A @ A3 @ X4 ) ) ) ) ) ).
% less_Sup_iff
thf(fact_3726_Inf__eqI,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A4: set @ A,X: A] :
( ! [I4: A] :
( ( member @ A @ I4 @ A4 )
=> ( ord_less_eq @ A @ X @ I4 ) )
=> ( ! [Y3: A] :
( ! [I5: A] :
( ( member @ A @ I5 @ A4 )
=> ( ord_less_eq @ A @ Y3 @ I5 ) )
=> ( ord_less_eq @ A @ Y3 @ X ) )
=> ( ( complete_Inf_Inf @ A @ A4 )
= X ) ) ) ) ).
% Inf_eqI
thf(fact_3727_Inf__mono,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [B2: set @ A,A4: set @ A] :
( ! [B5: A] :
( ( member @ A @ B5 @ B2 )
=> ? [X2: A] :
( ( member @ A @ X2 @ A4 )
& ( ord_less_eq @ A @ X2 @ B5 ) ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A4 ) @ ( complete_Inf_Inf @ A @ B2 ) ) ) ) ).
% Inf_mono
thf(fact_3728_Inf__lower,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [X: A,A4: set @ A] :
( ( member @ A @ X @ A4 )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A4 ) @ X ) ) ) ).
% Inf_lower
thf(fact_3729_Inf__lower2,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [U: A,A4: set @ A,V2: A] :
( ( member @ A @ U @ A4 )
=> ( ( ord_less_eq @ A @ U @ V2 )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A4 ) @ V2 ) ) ) ) ).
% Inf_lower2
thf(fact_3730_le__Inf__iff,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [B3: A,A4: set @ A] :
( ( ord_less_eq @ A @ B3 @ ( complete_Inf_Inf @ A @ A4 ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ A4 )
=> ( ord_less_eq @ A @ B3 @ X4 ) ) ) ) ) ).
% le_Inf_iff
thf(fact_3731_Inf__greatest,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A4: set @ A,Z3: A] :
( ! [X5: A] :
( ( member @ A @ X5 @ A4 )
=> ( ord_less_eq @ A @ Z3 @ X5 ) )
=> ( ord_less_eq @ A @ Z3 @ ( complete_Inf_Inf @ A @ A4 ) ) ) ) ).
% Inf_greatest
thf(fact_3732_Inf__less__iff,axiom,
! [A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [S2: set @ A,A3: A] :
( ( ord_less @ A @ ( complete_Inf_Inf @ A @ S2 ) @ A3 )
= ( ? [X4: A] :
( ( member @ A @ X4 @ S2 )
& ( ord_less @ A @ X4 @ A3 ) ) ) ) ) ).
% Inf_less_iff
thf(fact_3733_empty__Union__conv,axiom,
! [A: $tType,A4: set @ ( set @ A )] :
( ( ( bot_bot @ ( set @ A ) )
= ( complete_Sup_Sup @ ( set @ A ) @ A4 ) )
= ( ! [X4: set @ A] :
( ( member @ ( set @ A ) @ X4 @ A4 )
=> ( X4
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% empty_Union_conv
thf(fact_3734_Union__empty__conv,axiom,
! [A: $tType,A4: set @ ( set @ A )] :
( ( ( complete_Sup_Sup @ ( set @ A ) @ A4 )
= ( bot_bot @ ( set @ A ) ) )
= ( ! [X4: set @ A] :
( ( member @ ( set @ A ) @ X4 @ A4 )
=> ( X4
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% Union_empty_conv
thf(fact_3735_Union__subsetI,axiom,
! [A: $tType,A4: set @ ( set @ A ),B2: set @ ( set @ A )] :
( ! [X5: set @ A] :
( ( member @ ( set @ A ) @ X5 @ A4 )
=> ? [Y4: set @ A] :
( ( member @ ( set @ A ) @ Y4 @ B2 )
& ( ord_less_eq @ ( set @ A ) @ X5 @ Y4 ) ) )
=> ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ A4 ) @ ( complete_Sup_Sup @ ( set @ A ) @ B2 ) ) ) ).
% Union_subsetI
thf(fact_3736_Union__upper,axiom,
! [A: $tType,B2: set @ A,A4: set @ ( set @ A )] :
( ( member @ ( set @ A ) @ B2 @ A4 )
=> ( ord_less_eq @ ( set @ A ) @ B2 @ ( complete_Sup_Sup @ ( set @ A ) @ A4 ) ) ) ).
% Union_upper
thf(fact_3737_Union__least,axiom,
! [A: $tType,A4: set @ ( set @ A ),C5: set @ A] :
( ! [X9: set @ A] :
( ( member @ ( set @ A ) @ X9 @ A4 )
=> ( ord_less_eq @ ( set @ A ) @ X9 @ C5 ) )
=> ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ A4 ) @ C5 ) ) ).
% Union_least
thf(fact_3738_Inter__greatest,axiom,
! [A: $tType,A4: set @ ( set @ A ),C5: set @ A] :
( ! [X9: set @ A] :
( ( member @ ( set @ A ) @ X9 @ A4 )
=> ( ord_less_eq @ ( set @ A ) @ C5 @ X9 ) )
=> ( ord_less_eq @ ( set @ A ) @ C5 @ ( complete_Inf_Inf @ ( set @ A ) @ A4 ) ) ) ).
% Inter_greatest
thf(fact_3739_Inter__lower,axiom,
! [A: $tType,B2: set @ A,A4: set @ ( set @ A )] :
( ( member @ ( set @ A ) @ B2 @ A4 )
=> ( ord_less_eq @ ( set @ A ) @ ( complete_Inf_Inf @ ( set @ A ) @ A4 ) @ B2 ) ) ).
% Inter_lower
thf(fact_3740_le__Sup__iff,axiom,
! [A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [X: A,A4: set @ A] :
( ( ord_less_eq @ A @ X @ ( complete_Sup_Sup @ A @ A4 ) )
= ( ! [Y6: A] :
( ( ord_less @ A @ Y6 @ X )
=> ? [X4: A] :
( ( member @ A @ X4 @ A4 )
& ( ord_less @ A @ Y6 @ X4 ) ) ) ) ) ) ).
% le_Sup_iff
thf(fact_3741_Inf__le__iff,axiom,
! [A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [A4: set @ A,X: A] :
( ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A4 ) @ X )
= ( ! [Y6: A] :
( ( ord_less @ A @ X @ Y6 )
=> ? [X4: A] :
( ( member @ A @ X4 @ A4 )
& ( ord_less @ A @ X4 @ Y6 ) ) ) ) ) ) ).
% Inf_le_iff
thf(fact_3742_less__eq__Sup,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A4: set @ A,U: A] :
( ! [V3: A] :
( ( member @ A @ V3 @ A4 )
=> ( ord_less_eq @ A @ U @ V3 ) )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ord_less_eq @ A @ U @ ( complete_Sup_Sup @ A @ A4 ) ) ) ) ) ).
% less_eq_Sup
thf(fact_3743_Sup__subset__mono,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A4: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B2 )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A4 ) @ ( complete_Sup_Sup @ A @ B2 ) ) ) ) ).
% Sup_subset_mono
thf(fact_3744_Inf__less__eq,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A4: set @ A,U: A] :
( ! [V3: A] :
( ( member @ A @ V3 @ A4 )
=> ( ord_less_eq @ A @ V3 @ U ) )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A4 ) @ U ) ) ) ) ).
% Inf_less_eq
thf(fact_3745_Inf__superset__mono,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [B2: set @ A,A4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ B2 @ A4 )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A4 ) @ ( complete_Inf_Inf @ A @ B2 ) ) ) ) ).
% Inf_superset_mono
thf(fact_3746_Union__mono,axiom,
! [A: $tType,A4: set @ ( set @ A ),B2: set @ ( set @ A )] :
( ( ord_less_eq @ ( set @ ( set @ A ) ) @ A4 @ B2 )
=> ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ A4 ) @ ( complete_Sup_Sup @ ( set @ A ) @ B2 ) ) ) ).
% Union_mono
thf(fact_3747_Inter__anti__mono,axiom,
! [A: $tType,B2: set @ ( set @ A ),A4: set @ ( set @ A )] :
( ( ord_less_eq @ ( set @ ( set @ A ) ) @ B2 @ A4 )
=> ( ord_less_eq @ ( set @ A ) @ ( complete_Inf_Inf @ ( set @ A ) @ A4 ) @ ( complete_Inf_Inf @ ( set @ A ) @ B2 ) ) ) ).
% Inter_anti_mono
thf(fact_3748_Union__empty,axiom,
! [A: $tType] :
( ( complete_Sup_Sup @ ( set @ A ) @ ( bot_bot @ ( set @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Union_empty
thf(fact_3749_Inter__subset,axiom,
! [A: $tType,A4: set @ ( set @ A ),B2: set @ A] :
( ! [X9: set @ A] :
( ( member @ ( set @ A ) @ X9 @ A4 )
=> ( ord_less_eq @ ( set @ A ) @ X9 @ B2 ) )
=> ( ( A4
!= ( bot_bot @ ( set @ ( set @ A ) ) ) )
=> ( ord_less_eq @ ( set @ A ) @ ( complete_Inf_Inf @ ( set @ A ) @ A4 ) @ B2 ) ) ) ).
% Inter_subset
thf(fact_3750_Inf__le__Sup,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A4: set @ A] :
( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A4 ) @ ( complete_Sup_Sup @ A @ A4 ) ) ) ) ).
% Inf_le_Sup
thf(fact_3751_card__partition,axiom,
! [A: $tType,C5: set @ ( set @ A ),K: nat] :
( ( finite_finite2 @ ( set @ A ) @ C5 )
=> ( ( finite_finite2 @ A @ ( complete_Sup_Sup @ ( set @ A ) @ C5 ) )
=> ( ! [C3: set @ A] :
( ( member @ ( set @ A ) @ C3 @ C5 )
=> ( ( finite_card @ A @ C3 )
= K ) )
=> ( ! [C1: set @ A,C22: set @ A] :
( ( member @ ( set @ A ) @ C1 @ C5 )
=> ( ( member @ ( set @ A ) @ C22 @ C5 )
=> ( ( C1 != C22 )
=> ( ( inf_inf @ ( set @ A ) @ C1 @ C22 )
= ( bot_bot @ ( set @ A ) ) ) ) ) )
=> ( ( times_times @ nat @ K @ ( finite_card @ ( set @ A ) @ C5 ) )
= ( finite_card @ A @ ( complete_Sup_Sup @ ( set @ A ) @ C5 ) ) ) ) ) ) ) ).
% card_partition
thf(fact_3752_totally__bounded__Union,axiom,
! [A: $tType] :
( ( topolo7287701948861334536_space @ A )
=> ! [M7: set @ ( set @ A )] :
( ( finite_finite2 @ ( set @ A ) @ M7 )
=> ( ! [S4: set @ A] :
( ( member @ ( set @ A ) @ S4 @ M7 )
=> ( topolo6688025880775521714ounded @ A @ S4 ) )
=> ( topolo6688025880775521714ounded @ A @ ( complete_Sup_Sup @ ( set @ A ) @ M7 ) ) ) ) ) ).
% totally_bounded_Union
thf(fact_3753_card__greaterThanLessThan__int,axiom,
! [L: int,U: int] :
( ( finite_card @ int @ ( set_or5935395276787703475ssThan @ int @ L @ U ) )
= ( nat2 @ ( minus_minus @ int @ U @ ( plus_plus @ int @ L @ ( one_one @ int ) ) ) ) ) ).
% card_greaterThanLessThan_int
thf(fact_3754_horner__sum__eq__sum__funpow,axiom,
! [A: $tType,B: $tType] :
( ( comm_semiring_0 @ A )
=> ( ( groups4207007520872428315er_sum @ B @ A )
= ( ^ [F3: B > A,A5: A,Xs3: list @ B] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [N: nat] : ( compow @ ( A > A ) @ N @ ( times_times @ A @ A5 ) @ ( F3 @ ( nth @ B @ Xs3 @ N ) ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ B ) @ Xs3 ) ) ) ) ) ) ).
% horner_sum_eq_sum_funpow
thf(fact_3755_IntI,axiom,
! [A: $tType,C2: A,A4: set @ A,B2: set @ A] :
( ( member @ A @ C2 @ A4 )
=> ( ( member @ A @ C2 @ B2 )
=> ( member @ A @ C2 @ ( inf_inf @ ( set @ A ) @ A4 @ B2 ) ) ) ) ).
% IntI
thf(fact_3756_Int__iff,axiom,
! [A: $tType,C2: A,A4: set @ A,B2: set @ A] :
( ( member @ A @ C2 @ ( inf_inf @ ( set @ A ) @ A4 @ B2 ) )
= ( ( member @ A @ C2 @ A4 )
& ( member @ A @ C2 @ B2 ) ) ) ).
% Int_iff
thf(fact_3757_inf_Obounded__iff,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,B3: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ ( inf_inf @ A @ B3 @ C2 ) )
= ( ( ord_less_eq @ A @ A3 @ B3 )
& ( ord_less_eq @ A @ A3 @ C2 ) ) ) ) ).
% inf.bounded_iff
thf(fact_3758_le__inf__iff,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X: A,Y: A,Z3: A] :
( ( ord_less_eq @ A @ X @ ( inf_inf @ A @ Y @ Z3 ) )
= ( ( ord_less_eq @ A @ X @ Y )
& ( ord_less_eq @ A @ X @ Z3 ) ) ) ) ).
% le_inf_iff
thf(fact_3759_inf__bot__left,axiom,
! [A: $tType] :
( ( bounded_lattice_bot @ A )
=> ! [X: A] :
( ( inf_inf @ A @ ( bot_bot @ A ) @ X )
= ( bot_bot @ A ) ) ) ).
% inf_bot_left
thf(fact_3760_inf__bot__right,axiom,
! [A: $tType] :
( ( bounded_lattice_bot @ A )
=> ! [X: A] :
( ( inf_inf @ A @ X @ ( bot_bot @ A ) )
= ( bot_bot @ A ) ) ) ).
% inf_bot_right
thf(fact_3761_boolean__algebra_Oconj__zero__right,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A] :
( ( inf_inf @ A @ X @ ( bot_bot @ A ) )
= ( bot_bot @ A ) ) ) ).
% boolean_algebra.conj_zero_right
thf(fact_3762_boolean__algebra_Oconj__zero__left,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A] :
( ( inf_inf @ A @ ( bot_bot @ A ) @ X )
= ( bot_bot @ A ) ) ) ).
% boolean_algebra.conj_zero_left
thf(fact_3763_Suc__funpow,axiom,
! [N2: nat] :
( ( compow @ ( nat > nat ) @ N2 @ suc )
= ( plus_plus @ nat @ N2 ) ) ).
% Suc_funpow
thf(fact_3764_finite__Int,axiom,
! [A: $tType,F5: set @ A,G4: set @ A] :
( ( ( finite_finite2 @ A @ F5 )
| ( finite_finite2 @ A @ G4 ) )
=> ( finite_finite2 @ A @ ( inf_inf @ ( set @ A ) @ F5 @ G4 ) ) ) ).
% finite_Int
thf(fact_3765_Int__subset__iff,axiom,
! [A: $tType,C5: set @ A,A4: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ C5 @ ( inf_inf @ ( set @ A ) @ A4 @ B2 ) )
= ( ( ord_less_eq @ ( set @ A ) @ C5 @ A4 )
& ( ord_less_eq @ ( set @ A ) @ C5 @ B2 ) ) ) ).
% Int_subset_iff
thf(fact_3766_Int__insert__left__if0,axiom,
! [A: $tType,A3: A,C5: set @ A,B2: set @ A] :
( ~ ( member @ A @ A3 @ C5 )
=> ( ( inf_inf @ ( set @ A ) @ ( insert2 @ A @ A3 @ B2 ) @ C5 )
= ( inf_inf @ ( set @ A ) @ B2 @ C5 ) ) ) ).
% Int_insert_left_if0
thf(fact_3767_Int__insert__left__if1,axiom,
! [A: $tType,A3: A,C5: set @ A,B2: set @ A] :
( ( member @ A @ A3 @ C5 )
=> ( ( inf_inf @ ( set @ A ) @ ( insert2 @ A @ A3 @ B2 ) @ C5 )
= ( insert2 @ A @ A3 @ ( inf_inf @ ( set @ A ) @ B2 @ C5 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_3768_insert__inter__insert,axiom,
! [A: $tType,A3: A,A4: set @ A,B2: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( insert2 @ A @ A3 @ A4 ) @ ( insert2 @ A @ A3 @ B2 ) )
= ( insert2 @ A @ A3 @ ( inf_inf @ ( set @ A ) @ A4 @ B2 ) ) ) ).
% insert_inter_insert
thf(fact_3769_Int__insert__right__if0,axiom,
! [A: $tType,A3: A,A4: set @ A,B2: set @ A] :
( ~ ( member @ A @ A3 @ A4 )
=> ( ( inf_inf @ ( set @ A ) @ A4 @ ( insert2 @ A @ A3 @ B2 ) )
= ( inf_inf @ ( set @ A ) @ A4 @ B2 ) ) ) ).
% Int_insert_right_if0
thf(fact_3770_Int__insert__right__if1,axiom,
! [A: $tType,A3: A,A4: set @ A,B2: set @ A] :
( ( member @ A @ A3 @ A4 )
=> ( ( inf_inf @ ( set @ A ) @ A4 @ ( insert2 @ A @ A3 @ B2 ) )
= ( insert2 @ A @ A3 @ ( inf_inf @ ( set @ A ) @ A4 @ B2 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_3771_greaterThanLessThan__iff,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [I: A,L: A,U: A] :
( ( member @ A @ I @ ( set_or5935395276787703475ssThan @ A @ L @ U ) )
= ( ( ord_less @ A @ L @ I )
& ( ord_less @ A @ I @ U ) ) ) ) ).
% greaterThanLessThan_iff
thf(fact_3772_funpow__0,axiom,
! [A: $tType,F2: A > A,X: A] :
( ( compow @ ( A > A ) @ ( zero_zero @ nat ) @ F2 @ X )
= X ) ).
% funpow_0
thf(fact_3773_Pow__Int__eq,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] :
( ( pow2 @ A @ ( inf_inf @ ( set @ A ) @ A4 @ B2 ) )
= ( inf_inf @ ( set @ ( set @ A ) ) @ ( pow2 @ A @ A4 ) @ ( pow2 @ A @ B2 ) ) ) ).
% Pow_Int_eq
thf(fact_3774_totally__bounded__empty,axiom,
! [A: $tType] :
( ( topolo7287701948861334536_space @ A )
=> ( topolo6688025880775521714ounded @ A @ ( bot_bot @ ( set @ A ) ) ) ) ).
% totally_bounded_empty
thf(fact_3775_inf__compl__bot__left1,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,Y: A] :
( ( inf_inf @ A @ ( uminus_uminus @ A @ X ) @ ( inf_inf @ A @ X @ Y ) )
= ( bot_bot @ A ) ) ) ).
% inf_compl_bot_left1
thf(fact_3776_inf__compl__bot__left2,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,Y: A] :
( ( inf_inf @ A @ X @ ( inf_inf @ A @ ( uminus_uminus @ A @ X ) @ Y ) )
= ( bot_bot @ A ) ) ) ).
% inf_compl_bot_left2
thf(fact_3777_inf__compl__bot__right,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,Y: A] :
( ( inf_inf @ A @ X @ ( inf_inf @ A @ Y @ ( uminus_uminus @ A @ X ) ) )
= ( bot_bot @ A ) ) ) ).
% inf_compl_bot_right
thf(fact_3778_boolean__algebra_Oconj__cancel__left,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A] :
( ( inf_inf @ A @ ( uminus_uminus @ A @ X ) @ X )
= ( bot_bot @ A ) ) ) ).
% boolean_algebra.conj_cancel_left
thf(fact_3779_boolean__algebra_Oconj__cancel__right,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A] :
( ( inf_inf @ A @ X @ ( uminus_uminus @ A @ X ) )
= ( bot_bot @ A ) ) ) ).
% boolean_algebra.conj_cancel_right
thf(fact_3780_insert__disjoint_I1_J,axiom,
! [A: $tType,A3: A,A4: set @ A,B2: set @ A] :
( ( ( inf_inf @ ( set @ A ) @ ( insert2 @ A @ A3 @ A4 ) @ B2 )
= ( bot_bot @ ( set @ A ) ) )
= ( ~ ( member @ A @ A3 @ B2 )
& ( ( inf_inf @ ( set @ A ) @ A4 @ B2 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% insert_disjoint(1)
thf(fact_3781_insert__disjoint_I2_J,axiom,
! [A: $tType,A3: A,A4: set @ A,B2: set @ A] :
( ( ( bot_bot @ ( set @ A ) )
= ( inf_inf @ ( set @ A ) @ ( insert2 @ A @ A3 @ A4 ) @ B2 ) )
= ( ~ ( member @ A @ A3 @ B2 )
& ( ( bot_bot @ ( set @ A ) )
= ( inf_inf @ ( set @ A ) @ A4 @ B2 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_3782_disjoint__insert_I1_J,axiom,
! [A: $tType,B2: set @ A,A3: A,A4: set @ A] :
( ( ( inf_inf @ ( set @ A ) @ B2 @ ( insert2 @ A @ A3 @ A4 ) )
= ( bot_bot @ ( set @ A ) ) )
= ( ~ ( member @ A @ A3 @ B2 )
& ( ( inf_inf @ ( set @ A ) @ B2 @ A4 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% disjoint_insert(1)
thf(fact_3783_disjoint__insert_I2_J,axiom,
! [A: $tType,A4: set @ A,B3: A,B2: set @ A] :
( ( ( bot_bot @ ( set @ A ) )
= ( inf_inf @ ( set @ A ) @ A4 @ ( insert2 @ A @ B3 @ B2 ) ) )
= ( ~ ( member @ A @ B3 @ A4 )
& ( ( bot_bot @ ( set @ A ) )
= ( inf_inf @ ( set @ A ) @ A4 @ B2 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_3784_Diff__disjoint,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] :
( ( inf_inf @ ( set @ A ) @ A4 @ ( minus_minus @ ( set @ A ) @ B2 @ A4 ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Diff_disjoint
thf(fact_3785_finite__greaterThanLessThan__int,axiom,
! [L: int,U: int] : ( finite_finite2 @ int @ ( set_or5935395276787703475ssThan @ int @ L @ U ) ) ).
% finite_greaterThanLessThan_int
thf(fact_3786_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_3787_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_3788_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_3789_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_3790_Sup__greaterThanLessThan,axiom,
! [A: $tType] :
( ( ( comple6319245703460814977attice @ A )
& ( dense_linorder @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ( ( complete_Sup_Sup @ A @ ( set_or5935395276787703475ssThan @ A @ X @ Y ) )
= Y ) ) ) ).
% Sup_greaterThanLessThan
thf(fact_3791_cSup__greaterThanLessThan,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( dense_linorder @ A ) )
=> ! [Y: A,X: A] :
( ( ord_less @ A @ Y @ X )
=> ( ( complete_Sup_Sup @ A @ ( set_or5935395276787703475ssThan @ A @ Y @ X ) )
= X ) ) ) ).
% cSup_greaterThanLessThan
thf(fact_3792_Inf__greaterThanLessThan,axiom,
! [A: $tType] :
( ( ( comple6319245703460814977attice @ A )
& ( dense_linorder @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ( ( complete_Inf_Inf @ A @ ( set_or5935395276787703475ssThan @ A @ X @ Y ) )
= X ) ) ) ).
% Inf_greaterThanLessThan
thf(fact_3793_cInf__greaterThanLessThan,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( dense_linorder @ A ) )
=> ! [Y: A,X: A] :
( ( ord_less @ A @ Y @ X )
=> ( ( complete_Inf_Inf @ A @ ( set_or5935395276787703475ssThan @ A @ Y @ X ) )
= Y ) ) ) ).
% cInf_greaterThanLessThan
thf(fact_3794_Compl__disjoint2,axiom,
! [A: $tType,A4: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A4 ) @ A4 )
= ( bot_bot @ ( set @ A ) ) ) ).
% Compl_disjoint2
thf(fact_3795_Compl__disjoint,axiom,
! [A: $tType,A4: set @ A] :
( ( inf_inf @ ( set @ A ) @ A4 @ ( uminus_uminus @ ( set @ A ) @ A4 ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Compl_disjoint
thf(fact_3796_Diff__Compl,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] :
( ( minus_minus @ ( set @ A ) @ A4 @ ( uminus_uminus @ ( set @ A ) @ B2 ) )
= ( inf_inf @ ( set @ A ) @ A4 @ B2 ) ) ).
% Diff_Compl
thf(fact_3797_sum__of__bool__mult__eq,axiom,
! [A: $tType,B: $tType] :
( ( semiring_1 @ A )
=> ! [A4: set @ B,P: B > $o,F2: B > A] :
( ( finite_finite2 @ B @ A4 )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X4: B] : ( times_times @ A @ ( zero_neq_one_of_bool @ A @ ( P @ X4 ) ) @ ( F2 @ X4 ) )
@ A4 )
= ( groups7311177749621191930dd_sum @ B @ A @ F2 @ ( inf_inf @ ( set @ B ) @ A4 @ ( collect @ B @ P ) ) ) ) ) ) ).
% sum_of_bool_mult_eq
thf(fact_3798_sum__mult__of__bool__eq,axiom,
! [A: $tType,B: $tType] :
( ( semiring_1 @ A )
=> ! [A4: set @ B,F2: B > A,P: B > $o] :
( ( finite_finite2 @ B @ A4 )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X4: B] : ( times_times @ A @ ( F2 @ X4 ) @ ( zero_neq_one_of_bool @ A @ ( P @ X4 ) ) )
@ A4 )
= ( groups7311177749621191930dd_sum @ B @ A @ F2 @ ( inf_inf @ ( set @ B ) @ A4 @ ( collect @ B @ P ) ) ) ) ) ) ).
% sum_mult_of_bool_eq
thf(fact_3799_sum__of__bool__eq,axiom,
! [A: $tType,B: $tType] :
( ( semiring_1 @ A )
=> ! [A4: set @ B,P: B > $o] :
( ( finite_finite2 @ B @ A4 )
=> ( ( finite_finite2 @ B @ A4 )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X4: B] : ( zero_neq_one_of_bool @ A @ ( P @ X4 ) )
@ A4 )
= ( semiring_1_of_nat @ A @ ( finite_card @ B @ ( inf_inf @ ( set @ B ) @ A4 @ ( collect @ B @ P ) ) ) ) ) ) ) ) ).
% sum_of_bool_eq
thf(fact_3800_Sup__inter__less__eq,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A4: set @ A,B2: set @ A] : ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( inf_inf @ ( set @ A ) @ A4 @ B2 ) ) @ ( inf_inf @ A @ ( complete_Sup_Sup @ A @ A4 ) @ ( complete_Sup_Sup @ A @ B2 ) ) ) ) ).
% Sup_inter_less_eq
thf(fact_3801_Union__Int__subset,axiom,
! [A: $tType,A4: set @ ( set @ A ),B2: set @ ( set @ A )] : ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( inf_inf @ ( set @ ( set @ A ) ) @ A4 @ B2 ) ) @ ( inf_inf @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ A4 ) @ ( complete_Sup_Sup @ ( set @ A ) @ B2 ) ) ) ).
% Union_Int_subset
thf(fact_3802_less__infI1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,X: A,B3: A] :
( ( ord_less @ A @ A3 @ X )
=> ( ord_less @ A @ ( inf_inf @ A @ A3 @ B3 ) @ X ) ) ) ).
% less_infI1
thf(fact_3803_less__infI2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [B3: A,X: A,A3: A] :
( ( ord_less @ A @ B3 @ X )
=> ( ord_less @ A @ ( inf_inf @ A @ A3 @ B3 ) @ X ) ) ) ).
% less_infI2
thf(fact_3804_inf_Oabsorb3,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ A3 @ B3 )
=> ( ( inf_inf @ A @ A3 @ B3 )
= A3 ) ) ) ).
% inf.absorb3
thf(fact_3805_inf_Oabsorb4,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [B3: A,A3: A] :
( ( ord_less @ A @ B3 @ A3 )
=> ( ( inf_inf @ A @ A3 @ B3 )
= B3 ) ) ) ).
% inf.absorb4
thf(fact_3806_inf_Ostrict__boundedE,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,B3: A,C2: A] :
( ( ord_less @ A @ A3 @ ( inf_inf @ A @ B3 @ C2 ) )
=> ~ ( ( ord_less @ A @ A3 @ B3 )
=> ~ ( ord_less @ A @ A3 @ C2 ) ) ) ) ).
% inf.strict_boundedE
thf(fact_3807_inf_Ostrict__order__iff,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( ( ord_less @ A )
= ( ^ [A5: A,B4: A] :
( ( A5
= ( inf_inf @ A @ A5 @ B4 ) )
& ( A5 != B4 ) ) ) ) ) ).
% inf.strict_order_iff
thf(fact_3808_inf_Ostrict__coboundedI1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,C2: A,B3: A] :
( ( ord_less @ A @ A3 @ C2 )
=> ( ord_less @ A @ ( inf_inf @ A @ A3 @ B3 ) @ C2 ) ) ) ).
% inf.strict_coboundedI1
thf(fact_3809_inf_Ostrict__coboundedI2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [B3: A,C2: A,A3: A] :
( ( ord_less @ A @ B3 @ C2 )
=> ( ord_less @ A @ ( inf_inf @ A @ A3 @ B3 ) @ C2 ) ) ) ).
% inf.strict_coboundedI2
thf(fact_3810_funpow__swap1,axiom,
! [A: $tType,F2: A > A,N2: nat,X: A] :
( ( F2 @ ( compow @ ( A > A ) @ N2 @ F2 @ X ) )
= ( compow @ ( A > A ) @ N2 @ F2 @ ( F2 @ X ) ) ) ).
% funpow_swap1
thf(fact_3811_IntE,axiom,
! [A: $tType,C2: A,A4: set @ A,B2: set @ A] :
( ( member @ A @ C2 @ ( inf_inf @ ( set @ A ) @ A4 @ B2 ) )
=> ~ ( ( member @ A @ C2 @ A4 )
=> ~ ( member @ A @ C2 @ B2 ) ) ) ).
% IntE
thf(fact_3812_IntD1,axiom,
! [A: $tType,C2: A,A4: set @ A,B2: set @ A] :
( ( member @ A @ C2 @ ( inf_inf @ ( set @ A ) @ A4 @ B2 ) )
=> ( member @ A @ C2 @ A4 ) ) ).
% IntD1
thf(fact_3813_IntD2,axiom,
! [A: $tType,C2: A,A4: set @ A,B2: set @ A] :
( ( member @ A @ C2 @ ( inf_inf @ ( set @ A ) @ A4 @ B2 ) )
=> ( member @ A @ C2 @ B2 ) ) ).
% IntD2
thf(fact_3814_Int__assoc,axiom,
! [A: $tType,A4: set @ A,B2: set @ A,C5: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A4 @ B2 ) @ C5 )
= ( inf_inf @ ( set @ A ) @ A4 @ ( inf_inf @ ( set @ A ) @ B2 @ C5 ) ) ) ).
% Int_assoc
thf(fact_3815_Int__absorb,axiom,
! [A: $tType,A4: set @ A] :
( ( inf_inf @ ( set @ A ) @ A4 @ A4 )
= A4 ) ).
% Int_absorb
thf(fact_3816_Int__commute,axiom,
! [A: $tType] :
( ( inf_inf @ ( set @ A ) )
= ( ^ [A7: set @ A,B6: set @ A] : ( inf_inf @ ( set @ A ) @ B6 @ A7 ) ) ) ).
% Int_commute
thf(fact_3817_Int__left__absorb,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] :
( ( inf_inf @ ( set @ A ) @ A4 @ ( inf_inf @ ( set @ A ) @ A4 @ B2 ) )
= ( inf_inf @ ( set @ A ) @ A4 @ B2 ) ) ).
% Int_left_absorb
thf(fact_3818_Int__left__commute,axiom,
! [A: $tType,A4: set @ A,B2: set @ A,C5: set @ A] :
( ( inf_inf @ ( set @ A ) @ A4 @ ( inf_inf @ ( set @ A ) @ B2 @ C5 ) )
= ( inf_inf @ ( set @ A ) @ B2 @ ( inf_inf @ ( set @ A ) @ A4 @ C5 ) ) ) ).
% Int_left_commute
thf(fact_3819_Collect__conj__eq,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ( collect @ A
@ ^ [X4: A] :
( ( P @ X4 )
& ( Q @ X4 ) ) )
= ( inf_inf @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% Collect_conj_eq
thf(fact_3820_Int__Collect,axiom,
! [A: $tType,X: A,A4: set @ A,P: A > $o] :
( ( member @ A @ X @ ( inf_inf @ ( set @ A ) @ A4 @ ( collect @ A @ P ) ) )
= ( ( member @ A @ X @ A4 )
& ( P @ X ) ) ) ).
% Int_Collect
thf(fact_3821_Int__def,axiom,
! [A: $tType] :
( ( inf_inf @ ( set @ A ) )
= ( ^ [A7: set @ A,B6: set @ A] :
( collect @ A
@ ^ [X4: A] :
( ( member @ A @ X4 @ A7 )
& ( member @ A @ X4 @ B6 ) ) ) ) ) ).
% Int_def
thf(fact_3822_funpow__mult,axiom,
! [A: $tType,N2: nat,M2: nat,F2: A > A] :
( ( compow @ ( A > A ) @ N2 @ ( compow @ ( A > A ) @ M2 @ F2 ) )
= ( compow @ ( A > A ) @ ( times_times @ nat @ M2 @ N2 ) @ F2 ) ) ).
% funpow_mult
thf(fact_3823_Diff__Int__distrib2,axiom,
! [A: $tType,A4: set @ A,B2: set @ A,C5: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A4 @ B2 ) @ C5 )
= ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A4 @ C5 ) @ ( inf_inf @ ( set @ A ) @ B2 @ C5 ) ) ) ).
% Diff_Int_distrib2
thf(fact_3824_Diff__Int__distrib,axiom,
! [A: $tType,C5: set @ A,A4: set @ A,B2: set @ A] :
( ( inf_inf @ ( set @ A ) @ C5 @ ( minus_minus @ ( set @ A ) @ A4 @ B2 ) )
= ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ C5 @ A4 ) @ ( inf_inf @ ( set @ A ) @ C5 @ B2 ) ) ) ).
% Diff_Int_distrib
thf(fact_3825_Diff__Diff__Int,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] :
( ( minus_minus @ ( set @ A ) @ A4 @ ( minus_minus @ ( set @ A ) @ A4 @ B2 ) )
= ( inf_inf @ ( set @ A ) @ A4 @ B2 ) ) ).
% Diff_Diff_Int
thf(fact_3826_Diff__Int2,axiom,
! [A: $tType,A4: set @ A,C5: set @ A,B2: set @ A] :
( ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A4 @ C5 ) @ ( inf_inf @ ( set @ A ) @ B2 @ C5 ) )
= ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A4 @ C5 ) @ B2 ) ) ).
% Diff_Int2
thf(fact_3827_Int__Diff,axiom,
! [A: $tType,A4: set @ A,B2: set @ A,C5: set @ A] :
( ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A4 @ B2 ) @ C5 )
= ( inf_inf @ ( set @ A ) @ A4 @ ( minus_minus @ ( set @ A ) @ B2 @ C5 ) ) ) ).
% Int_Diff
thf(fact_3828_Int__insert__left,axiom,
! [A: $tType,A3: A,C5: set @ A,B2: set @ A] :
( ( ( member @ A @ A3 @ C5 )
=> ( ( inf_inf @ ( set @ A ) @ ( insert2 @ A @ A3 @ B2 ) @ C5 )
= ( insert2 @ A @ A3 @ ( inf_inf @ ( set @ A ) @ B2 @ C5 ) ) ) )
& ( ~ ( member @ A @ A3 @ C5 )
=> ( ( inf_inf @ ( set @ A ) @ ( insert2 @ A @ A3 @ B2 ) @ C5 )
= ( inf_inf @ ( set @ A ) @ B2 @ C5 ) ) ) ) ).
% Int_insert_left
thf(fact_3829_Int__insert__right,axiom,
! [A: $tType,A3: A,A4: set @ A,B2: set @ A] :
( ( ( member @ A @ A3 @ A4 )
=> ( ( inf_inf @ ( set @ A ) @ A4 @ ( insert2 @ A @ A3 @ B2 ) )
= ( insert2 @ A @ A3 @ ( inf_inf @ ( set @ A ) @ A4 @ B2 ) ) ) )
& ( ~ ( member @ A @ A3 @ A4 )
=> ( ( inf_inf @ ( set @ A ) @ A4 @ ( insert2 @ A @ A3 @ B2 ) )
= ( inf_inf @ ( set @ A ) @ A4 @ B2 ) ) ) ) ).
% Int_insert_right
thf(fact_3830_disjoint__iff__not__equal,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] :
( ( ( inf_inf @ ( set @ A ) @ A4 @ B2 )
= ( bot_bot @ ( set @ A ) ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ A4 )
=> ! [Y6: A] :
( ( member @ A @ Y6 @ B2 )
=> ( X4 != Y6 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_3831_Int__empty__right,axiom,
! [A: $tType,A4: set @ A] :
( ( inf_inf @ ( set @ A ) @ A4 @ ( bot_bot @ ( set @ A ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Int_empty_right
thf(fact_3832_Int__empty__left,axiom,
! [A: $tType,B2: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ B2 )
= ( bot_bot @ ( set @ A ) ) ) ).
% Int_empty_left
thf(fact_3833_disjoint__iff,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] :
( ( ( inf_inf @ ( set @ A ) @ A4 @ B2 )
= ( bot_bot @ ( set @ A ) ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ A4 )
=> ~ ( member @ A @ X4 @ B2 ) ) ) ) ).
% disjoint_iff
thf(fact_3834_Int__emptyI,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] :
( ! [X5: A] :
( ( member @ A @ X5 @ A4 )
=> ~ ( member @ A @ X5 @ B2 ) )
=> ( ( inf_inf @ ( set @ A ) @ A4 @ B2 )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% Int_emptyI
thf(fact_3835_Int__Collect__mono,axiom,
! [A: $tType,A4: set @ A,B2: set @ A,P: A > $o,Q: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B2 )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ A4 )
=> ( ( P @ X5 )
=> ( Q @ X5 ) ) )
=> ( ord_less_eq @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A4 @ ( collect @ A @ P ) ) @ ( inf_inf @ ( set @ A ) @ B2 @ ( collect @ A @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_3836_Int__greatest,axiom,
! [A: $tType,C5: set @ A,A4: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ C5 @ A4 )
=> ( ( ord_less_eq @ ( set @ A ) @ C5 @ B2 )
=> ( ord_less_eq @ ( set @ A ) @ C5 @ ( inf_inf @ ( set @ A ) @ A4 @ B2 ) ) ) ) ).
% Int_greatest
thf(fact_3837_Int__absorb2,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B2 )
=> ( ( inf_inf @ ( set @ A ) @ A4 @ B2 )
= A4 ) ) ).
% Int_absorb2
thf(fact_3838_Int__absorb1,axiom,
! [A: $tType,B2: set @ A,A4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ B2 @ A4 )
=> ( ( inf_inf @ ( set @ A ) @ A4 @ B2 )
= B2 ) ) ).
% Int_absorb1
thf(fact_3839_Int__lower2,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A4 @ B2 ) @ B2 ) ).
% Int_lower2
thf(fact_3840_Int__lower1,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A4 @ B2 ) @ A4 ) ).
% Int_lower1
thf(fact_3841_Int__mono,axiom,
! [A: $tType,A4: set @ A,C5: set @ A,B2: set @ A,D6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ C5 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ D6 )
=> ( ord_less_eq @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A4 @ B2 ) @ ( inf_inf @ ( set @ A ) @ C5 @ D6 ) ) ) ) ).
% Int_mono
thf(fact_3842_ivl__disj__int__one_I1_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L: A,U: A] :
( ( inf_inf @ ( set @ A ) @ ( set_ord_atMost @ A @ L ) @ ( set_or5935395276787703475ssThan @ A @ L @ U ) )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% ivl_disj_int_one(1)
thf(fact_3843_ivl__disj__int__two_I5_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L: A,M2: A,U: A] :
( ( inf_inf @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ L @ M2 ) @ ( set_or1337092689740270186AtMost @ A @ M2 @ U ) )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% ivl_disj_int_two(5)
thf(fact_3844_ivl__disj__int__two_I4_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L: A,M2: A,U: A] :
( ( inf_inf @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ L @ M2 ) @ ( set_or5935395276787703475ssThan @ A @ M2 @ U ) )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% ivl_disj_int_two(4)
thf(fact_3845_inf_OcoboundedI2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [B3: A,C2: A,A3: A] :
( ( ord_less_eq @ A @ B3 @ C2 )
=> ( ord_less_eq @ A @ ( inf_inf @ A @ A3 @ B3 ) @ C2 ) ) ) ).
% inf.coboundedI2
thf(fact_3846_inf_OcoboundedI1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,C2: A,B3: A] :
( ( ord_less_eq @ A @ A3 @ C2 )
=> ( ord_less_eq @ A @ ( inf_inf @ A @ A3 @ B3 ) @ C2 ) ) ) ).
% inf.coboundedI1
thf(fact_3847_inf_Oabsorb__iff2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B4: A,A5: A] :
( ( inf_inf @ A @ A5 @ B4 )
= B4 ) ) ) ) ).
% inf.absorb_iff2
thf(fact_3848_inf_Oabsorb__iff1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A5: A,B4: A] :
( ( inf_inf @ A @ A5 @ B4 )
= A5 ) ) ) ) ).
% inf.absorb_iff1
thf(fact_3849_inf_Ocobounded2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,B3: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ A3 @ B3 ) @ B3 ) ) ).
% inf.cobounded2
thf(fact_3850_inf_Ocobounded1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,B3: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ A3 @ B3 ) @ A3 ) ) ).
% inf.cobounded1
thf(fact_3851_inf_Oorder__iff,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A5: A,B4: A] :
( A5
= ( inf_inf @ A @ A5 @ B4 ) ) ) ) ) ).
% inf.order_iff
thf(fact_3852_inf__greatest,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X: A,Y: A,Z3: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less_eq @ A @ X @ Z3 )
=> ( ord_less_eq @ A @ X @ ( inf_inf @ A @ Y @ Z3 ) ) ) ) ) ).
% inf_greatest
thf(fact_3853_inf_OboundedI,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,B3: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ A @ A3 @ C2 )
=> ( ord_less_eq @ A @ A3 @ ( inf_inf @ A @ B3 @ C2 ) ) ) ) ) ).
% inf.boundedI
thf(fact_3854_inf_OboundedE,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,B3: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ ( inf_inf @ A @ B3 @ C2 ) )
=> ~ ( ( ord_less_eq @ A @ A3 @ B3 )
=> ~ ( ord_less_eq @ A @ A3 @ C2 ) ) ) ) ).
% inf.boundedE
thf(fact_3855_inf__absorb2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ Y @ X )
=> ( ( inf_inf @ A @ X @ Y )
= Y ) ) ) ).
% inf_absorb2
thf(fact_3856_inf__absorb1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( inf_inf @ A @ X @ Y )
= X ) ) ) ).
% inf_absorb1
thf(fact_3857_inf_Oabsorb2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [B3: A,A3: A] :
( ( ord_less_eq @ A @ B3 @ A3 )
=> ( ( inf_inf @ A @ A3 @ B3 )
= B3 ) ) ) ).
% inf.absorb2
thf(fact_3858_inf_Oabsorb1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( inf_inf @ A @ A3 @ B3 )
= A3 ) ) ) ).
% inf.absorb1
thf(fact_3859_le__iff__inf,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [X4: A,Y6: A] :
( ( inf_inf @ A @ X4 @ Y6 )
= X4 ) ) ) ) ).
% le_iff_inf
thf(fact_3860_inf__unique,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [F2: A > A > A,X: A,Y: A] :
( ! [X5: A,Y3: A] : ( ord_less_eq @ A @ ( F2 @ X5 @ Y3 ) @ X5 )
=> ( ! [X5: A,Y3: A] : ( ord_less_eq @ A @ ( F2 @ X5 @ Y3 ) @ Y3 )
=> ( ! [X5: A,Y3: A,Z: A] :
( ( ord_less_eq @ A @ X5 @ Y3 )
=> ( ( ord_less_eq @ A @ X5 @ Z )
=> ( ord_less_eq @ A @ X5 @ ( F2 @ Y3 @ Z ) ) ) )
=> ( ( inf_inf @ A @ X @ Y )
= ( F2 @ X @ Y ) ) ) ) ) ) ).
% inf_unique
thf(fact_3861_inf_OorderI,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,B3: A] :
( ( A3
= ( inf_inf @ A @ A3 @ B3 ) )
=> ( ord_less_eq @ A @ A3 @ B3 ) ) ) ).
% inf.orderI
thf(fact_3862_inf_OorderE,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( A3
= ( inf_inf @ A @ A3 @ B3 ) ) ) ) ).
% inf.orderE
thf(fact_3863_le__infI2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [B3: A,X: A,A3: A] :
( ( ord_less_eq @ A @ B3 @ X )
=> ( ord_less_eq @ A @ ( inf_inf @ A @ A3 @ B3 ) @ X ) ) ) ).
% le_infI2
thf(fact_3864_le__infI1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,X: A,B3: A] :
( ( ord_less_eq @ A @ A3 @ X )
=> ( ord_less_eq @ A @ ( inf_inf @ A @ A3 @ B3 ) @ X ) ) ) ).
% le_infI1
thf(fact_3865_inf__mono,axiom,
! [A: $tType] :
( ( semilattice_inf @ 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 @ ( inf_inf @ A @ A3 @ B3 ) @ ( inf_inf @ A @ C2 @ D2 ) ) ) ) ) ).
% inf_mono
thf(fact_3866_le__infI,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X: A,A3: A,B3: A] :
( ( ord_less_eq @ A @ X @ A3 )
=> ( ( ord_less_eq @ A @ X @ B3 )
=> ( ord_less_eq @ A @ X @ ( inf_inf @ A @ A3 @ B3 ) ) ) ) ) ).
% le_infI
thf(fact_3867_le__infE,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X: A,A3: A,B3: A] :
( ( ord_less_eq @ A @ X @ ( inf_inf @ A @ A3 @ B3 ) )
=> ~ ( ( ord_less_eq @ A @ X @ A3 )
=> ~ ( ord_less_eq @ A @ X @ B3 ) ) ) ) ).
% le_infE
thf(fact_3868_inf__le2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X: A,Y: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ X @ Y ) @ Y ) ) ).
% inf_le2
thf(fact_3869_inf__le1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X: A,Y: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ X @ Y ) @ X ) ) ).
% inf_le1
thf(fact_3870_inf__sup__ord_I1_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X: A,Y: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ X @ Y ) @ X ) ) ).
% inf_sup_ord(1)
thf(fact_3871_inf__sup__ord_I2_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X: A,Y: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ X @ Y ) @ Y ) ) ).
% inf_sup_ord(2)
thf(fact_3872_ivl__disj__int__two_I1_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L: A,M2: A,U: A] :
( ( inf_inf @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ L @ M2 ) @ ( set_or7035219750837199246ssThan @ A @ M2 @ U ) )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% ivl_disj_int_two(1)
thf(fact_3873_inf__cancel__left1,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,A3: A,B3: A] :
( ( inf_inf @ A @ ( inf_inf @ A @ X @ A3 ) @ ( inf_inf @ A @ ( uminus_uminus @ A @ X ) @ B3 ) )
= ( bot_bot @ A ) ) ) ).
% inf_cancel_left1
thf(fact_3874_inf__cancel__left2,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,A3: A,B3: A] :
( ( inf_inf @ A @ ( inf_inf @ A @ ( uminus_uminus @ A @ X ) @ A3 ) @ ( inf_inf @ A @ X @ B3 ) )
= ( bot_bot @ A ) ) ) ).
% inf_cancel_left2
thf(fact_3875_Sup__inf__eq__bot__iff,axiom,
! [A: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [B2: set @ A,A3: A] :
( ( ( inf_inf @ A @ ( complete_Sup_Sup @ A @ B2 ) @ A3 )
= ( bot_bot @ A ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ B2 )
=> ( ( inf_inf @ A @ X4 @ A3 )
= ( bot_bot @ A ) ) ) ) ) ) ).
% Sup_inf_eq_bot_iff
thf(fact_3876_Int__Diff__disjoint,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A4 @ B2 ) @ ( minus_minus @ ( set @ A ) @ A4 @ B2 ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Int_Diff_disjoint
thf(fact_3877_Diff__triv,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] :
( ( ( inf_inf @ ( set @ A ) @ A4 @ B2 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( minus_minus @ ( set @ A ) @ A4 @ B2 )
= A4 ) ) ).
% Diff_triv
thf(fact_3878_Union__disjoint,axiom,
! [A: $tType,C5: set @ ( set @ A ),A4: set @ A] :
( ( ( inf_inf @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ C5 ) @ A4 )
= ( bot_bot @ ( set @ A ) ) )
= ( ! [X4: set @ A] :
( ( member @ ( set @ A ) @ X4 @ C5 )
=> ( ( inf_inf @ ( set @ A ) @ X4 @ A4 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% Union_disjoint
thf(fact_3879_ivl__disj__int__two_I3_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L: A,M2: A,U: A] :
( ( inf_inf @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ L @ M2 ) @ ( set_or7035219750837199246ssThan @ A @ M2 @ U ) )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% ivl_disj_int_two(3)
thf(fact_3880_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_3881_Diff__eq,axiom,
! [A: $tType] :
( ( minus_minus @ ( set @ A ) )
= ( ^ [A7: set @ A,B6: set @ A] : ( inf_inf @ ( set @ A ) @ A7 @ ( uminus_uminus @ ( set @ A ) @ B6 ) ) ) ) ).
% Diff_eq
thf(fact_3882_totally__bounded__subset,axiom,
! [A: $tType] :
( ( topolo7287701948861334536_space @ A )
=> ! [S2: set @ A,T4: set @ A] :
( ( topolo6688025880775521714ounded @ A @ S2 )
=> ( ( ord_less_eq @ ( set @ A ) @ T4 @ S2 )
=> ( topolo6688025880775521714ounded @ A @ T4 ) ) ) ) ).
% totally_bounded_subset
thf(fact_3883_inf__shunt,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,Y: A] :
( ( ( inf_inf @ A @ X @ Y )
= ( bot_bot @ A ) )
= ( ord_less_eq @ A @ X @ ( uminus_uminus @ A @ Y ) ) ) ) ).
% inf_shunt
thf(fact_3884_finite__Inf__in,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A4: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X5: A,Y3: A] :
( ( member @ A @ X5 @ A4 )
=> ( ( member @ A @ Y3 @ A4 )
=> ( member @ A @ ( inf_inf @ A @ X5 @ Y3 ) @ A4 ) ) )
=> ( member @ A @ ( complete_Inf_Inf @ A @ A4 ) @ A4 ) ) ) ) ) ).
% finite_Inf_in
thf(fact_3885_ivl__disj__int__two_I7_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L: A,M2: A,U: A] :
( ( inf_inf @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ L @ M2 ) @ ( set_or1337092689740270186AtMost @ A @ M2 @ U ) )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% ivl_disj_int_two(7)
thf(fact_3886_ivl__disj__int__one_I4_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L: A,U: A] :
( ( inf_inf @ ( set @ A ) @ ( set_ord_lessThan @ A @ L ) @ ( set_or1337092689740270186AtMost @ A @ L @ U ) )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% ivl_disj_int_one(4)
thf(fact_3887_ivl__disj__int__one_I2_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L: A,U: A] :
( ( inf_inf @ ( set @ A ) @ ( set_ord_lessThan @ A @ L ) @ ( set_or7035219750837199246ssThan @ A @ L @ U ) )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% ivl_disj_int_one(2)
thf(fact_3888_disjoint__eq__subset__Compl,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] :
( ( ( inf_inf @ ( set @ A ) @ A4 @ B2 )
= ( bot_bot @ ( set @ A ) ) )
= ( ord_less_eq @ ( set @ A ) @ A4 @ ( uminus_uminus @ ( set @ A ) @ B2 ) ) ) ).
% disjoint_eq_subset_Compl
thf(fact_3889_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_3890_insert__partition,axiom,
! [A: $tType,X: set @ A,F5: set @ ( set @ A )] :
( ~ ( member @ ( set @ A ) @ X @ F5 )
=> ( ! [X5: set @ A] :
( ( member @ ( set @ A ) @ X5 @ ( insert2 @ ( set @ A ) @ X @ F5 ) )
=> ! [Xa3: set @ A] :
( ( member @ ( set @ A ) @ Xa3 @ ( insert2 @ ( set @ A ) @ X @ F5 ) )
=> ( ( X5 != Xa3 )
=> ( ( inf_inf @ ( set @ A ) @ X5 @ Xa3 )
= ( bot_bot @ ( set @ A ) ) ) ) ) )
=> ( ( inf_inf @ ( set @ A ) @ X @ ( complete_Sup_Sup @ ( set @ A ) @ F5 ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% insert_partition
thf(fact_3891_sum_Ointer__restrict,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A4: set @ B,G: B > A,B2: set @ B] :
( ( finite_finite2 @ B @ A4 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( inf_inf @ ( set @ B ) @ A4 @ B2 ) )
= ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X4: B] : ( if @ A @ ( member @ B @ X4 @ B2 ) @ ( G @ X4 ) @ ( zero_zero @ A ) )
@ A4 ) ) ) ) ).
% sum.inter_restrict
thf(fact_3892_prod_Ointer__restrict,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A4: set @ B,G: B > A,B2: set @ B] :
( ( finite_finite2 @ B @ A4 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( inf_inf @ ( set @ B ) @ A4 @ B2 ) )
= ( groups7121269368397514597t_prod @ B @ A
@ ^ [X4: B] : ( if @ A @ ( member @ B @ X4 @ B2 ) @ ( G @ X4 ) @ ( one_one @ A ) )
@ A4 ) ) ) ) ).
% prod.inter_restrict
thf(fact_3893_atLeastPlusOneLessThan__greaterThanLessThan__int,axiom,
! [L: int,U: int] :
( ( set_or7035219750837199246ssThan @ int @ ( plus_plus @ int @ L @ ( one_one @ int ) ) @ U )
= ( set_or5935395276787703475ssThan @ int @ L @ U ) ) ).
% atLeastPlusOneLessThan_greaterThanLessThan_int
thf(fact_3894_of__nat__def,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ( ( semiring_1_of_nat @ A )
= ( ^ [N: nat] : ( compow @ ( A > A ) @ N @ ( plus_plus @ A @ ( one_one @ A ) ) @ ( zero_zero @ A ) ) ) ) ) ).
% of_nat_def
thf(fact_3895_numeral__add__unfold__funpow,axiom,
! [A: $tType] :
( ( semiring_numeral @ A )
=> ! [K: num,A3: A] :
( ( plus_plus @ A @ ( numeral_numeral @ A @ K ) @ A3 )
= ( compow @ ( A > A ) @ ( numeral_numeral @ nat @ K ) @ ( plus_plus @ A @ ( one_one @ A ) ) @ A3 ) ) ) ).
% numeral_add_unfold_funpow
thf(fact_3896_sum_Omono__neutral__cong,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [T4: set @ B,S2: set @ B,H2: B > A,G: B > A] :
( ( finite_finite2 @ B @ T4 )
=> ( ( finite_finite2 @ B @ S2 )
=> ( ! [I4: B] :
( ( member @ B @ I4 @ ( minus_minus @ ( set @ B ) @ T4 @ S2 ) )
=> ( ( H2 @ I4 )
= ( zero_zero @ A ) ) )
=> ( ! [I4: B] :
( ( member @ B @ I4 @ ( minus_minus @ ( set @ B ) @ S2 @ T4 ) )
=> ( ( G @ I4 )
= ( zero_zero @ A ) ) )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ ( inf_inf @ ( set @ B ) @ S2 @ T4 ) )
=> ( ( G @ X5 )
= ( H2 @ X5 ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ S2 )
= ( groups7311177749621191930dd_sum @ B @ A @ H2 @ T4 ) ) ) ) ) ) ) ) ).
% sum.mono_neutral_cong
thf(fact_3897_Iio__Int__singleton,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X: A,K: A] :
( ( ( ord_less @ A @ X @ K )
=> ( ( inf_inf @ ( set @ A ) @ ( set_ord_lessThan @ A @ K ) @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
& ( ~ ( ord_less @ A @ X @ K )
=> ( ( inf_inf @ ( set @ A ) @ ( set_ord_lessThan @ A @ K ) @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% Iio_Int_singleton
thf(fact_3898_sum_OInt__Diff,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A4: set @ B,G: B > A,B2: set @ B] :
( ( finite_finite2 @ B @ A4 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ A4 )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ G @ ( inf_inf @ ( set @ B ) @ A4 @ B2 ) ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ A4 @ B2 ) ) ) ) ) ) ).
% sum.Int_Diff
thf(fact_3899_prod_OInt__Diff,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A4: set @ B,G: B > A,B2: set @ B] :
( ( finite_finite2 @ B @ A4 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ A4 )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ G @ ( inf_inf @ ( set @ B ) @ A4 @ B2 ) ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ A4 @ B2 ) ) ) ) ) ) ).
% prod.Int_Diff
thf(fact_3900_prod_Omono__neutral__cong,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [T4: set @ B,S2: set @ B,H2: B > A,G: B > A] :
( ( finite_finite2 @ B @ T4 )
=> ( ( finite_finite2 @ B @ S2 )
=> ( ! [I4: B] :
( ( member @ B @ I4 @ ( minus_minus @ ( set @ B ) @ T4 @ S2 ) )
=> ( ( H2 @ I4 )
= ( one_one @ A ) ) )
=> ( ! [I4: B] :
( ( member @ B @ I4 @ ( minus_minus @ ( set @ B ) @ S2 @ T4 ) )
=> ( ( G @ I4 )
= ( one_one @ A ) ) )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ ( inf_inf @ ( set @ B ) @ S2 @ T4 ) )
=> ( ( G @ X5 )
= ( H2 @ X5 ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ S2 )
= ( groups7121269368397514597t_prod @ B @ A @ H2 @ T4 ) ) ) ) ) ) ) ) ).
% prod.mono_neutral_cong
thf(fact_3901_card__Diff__subset__Int,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] :
( ( finite_finite2 @ A @ ( inf_inf @ ( set @ A ) @ A4 @ B2 ) )
=> ( ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A4 @ B2 ) )
= ( minus_minus @ nat @ ( finite_card @ A @ A4 ) @ ( finite_card @ A @ ( inf_inf @ ( set @ A ) @ A4 @ B2 ) ) ) ) ) ).
% card_Diff_subset_Int
thf(fact_3902_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_3903_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_3904_atLeastAtMost__diff__ends,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B3: A] :
( ( minus_minus @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A3 @ B3 ) @ ( insert2 @ A @ A3 @ ( insert2 @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( set_or5935395276787703475ssThan @ A @ A3 @ B3 ) ) ) ).
% atLeastAtMost_diff_ends
thf(fact_3905_sum_OIf__cases,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A4: set @ B,P: B > $o,H2: B > A,G: B > A] :
( ( finite_finite2 @ B @ A4 )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X4: B] : ( if @ A @ ( P @ X4 ) @ ( H2 @ X4 ) @ ( G @ X4 ) )
@ A4 )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ H2 @ ( inf_inf @ ( set @ B ) @ A4 @ ( collect @ B @ P ) ) ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ ( inf_inf @ ( set @ B ) @ A4 @ ( uminus_uminus @ ( set @ B ) @ ( collect @ B @ P ) ) ) ) ) ) ) ) ).
% sum.If_cases
thf(fact_3906_prod_OIf__cases,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A4: set @ B,P: B > $o,H2: B > A,G: B > A] :
( ( finite_finite2 @ B @ A4 )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [X4: B] : ( if @ A @ ( P @ X4 ) @ ( H2 @ X4 ) @ ( G @ X4 ) )
@ A4 )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ H2 @ ( inf_inf @ ( set @ B ) @ A4 @ ( collect @ B @ P ) ) ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ ( inf_inf @ ( set @ B ) @ A4 @ ( uminus_uminus @ ( set @ B ) @ ( collect @ B @ P ) ) ) ) ) ) ) ) ).
% prod.If_cases
thf(fact_3907_numeral__unfold__funpow,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ( ( numeral_numeral @ A )
= ( ^ [K2: num] : ( compow @ ( A > A ) @ ( numeral_numeral @ nat @ K2 ) @ ( plus_plus @ A @ ( one_one @ A ) ) @ ( zero_zero @ A ) ) ) ) ) ).
% numeral_unfold_funpow
thf(fact_3908_dvd__partition,axiom,
! [A: $tType,C5: set @ ( set @ A ),K: nat] :
( ( finite_finite2 @ A @ ( complete_Sup_Sup @ ( set @ A ) @ C5 ) )
=> ( ! [X5: set @ A] :
( ( member @ ( set @ A ) @ X5 @ C5 )
=> ( dvd_dvd @ nat @ K @ ( finite_card @ A @ X5 ) ) )
=> ( ! [X5: set @ A] :
( ( member @ ( set @ A ) @ X5 @ C5 )
=> ! [Xa3: set @ A] :
( ( member @ ( set @ A ) @ Xa3 @ C5 )
=> ( ( X5 != Xa3 )
=> ( ( inf_inf @ ( set @ A ) @ X5 @ Xa3 )
= ( bot_bot @ ( set @ A ) ) ) ) ) )
=> ( dvd_dvd @ nat @ K @ ( finite_card @ A @ ( complete_Sup_Sup @ ( set @ A ) @ C5 ) ) ) ) ) ) ).
% dvd_partition
thf(fact_3909_sum__div__partition,axiom,
! [B: $tType,A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A4: set @ B,F2: B > A,B3: A] :
( ( finite_finite2 @ B @ A4 )
=> ( ( divide_divide @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A4 ) @ B3 )
= ( plus_plus @ A
@ ( groups7311177749621191930dd_sum @ B @ A
@ ^ [A5: B] : ( divide_divide @ A @ ( F2 @ A5 ) @ B3 )
@ ( inf_inf @ ( set @ B ) @ A4
@ ( collect @ B
@ ^ [A5: B] : ( dvd_dvd @ A @ B3 @ ( F2 @ A5 ) ) ) ) )
@ ( divide_divide @ A
@ ( groups7311177749621191930dd_sum @ B @ A @ F2
@ ( inf_inf @ ( set @ B ) @ A4
@ ( collect @ B
@ ^ [A5: B] :
~ ( dvd_dvd @ A @ B3 @ ( F2 @ A5 ) ) ) ) )
@ B3 ) ) ) ) ) ).
% sum_div_partition
thf(fact_3910_distinct__concat,axiom,
! [A: $tType,Xs: list @ ( list @ A )] :
( ( distinct @ ( list @ A ) @ Xs )
=> ( ! [Ys4: list @ A] :
( ( member @ ( list @ A ) @ Ys4 @ ( set2 @ ( list @ A ) @ Xs ) )
=> ( distinct @ A @ Ys4 ) )
=> ( ! [Ys4: list @ A,Zs2: list @ A] :
( ( member @ ( list @ A ) @ Ys4 @ ( set2 @ ( list @ A ) @ Xs ) )
=> ( ( member @ ( list @ A ) @ Zs2 @ ( set2 @ ( list @ A ) @ Xs ) )
=> ( ( Ys4 != Zs2 )
=> ( ( inf_inf @ ( set @ A ) @ ( set2 @ A @ Ys4 ) @ ( set2 @ A @ Zs2 ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) )
=> ( distinct @ A @ ( concat @ A @ Xs ) ) ) ) ) ).
% distinct_concat
thf(fact_3911_relpowp__bot,axiom,
! [A: $tType,N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( compow @ ( A > A > $o ) @ N2 @ ( bot_bot @ ( A > A > $o ) ) )
= ( bot_bot @ ( A > A > $o ) ) ) ) ).
% relpowp_bot
thf(fact_3912_relpowp__fun__conv,axiom,
! [A: $tType] :
( ( compow @ ( A > A > $o ) )
= ( ^ [N: nat,P3: A > A > $o,X4: A,Y6: A] :
? [F3: nat > A] :
( ( ( F3 @ ( zero_zero @ nat ) )
= X4 )
& ( ( F3 @ N )
= Y6 )
& ! [I3: nat] :
( ( ord_less @ nat @ I3 @ N )
=> ( P3 @ ( F3 @ I3 ) @ ( F3 @ ( suc @ I3 ) ) ) ) ) ) ) ).
% relpowp_fun_conv
thf(fact_3913_Nat_Ofunpow__code__def,axiom,
! [A: $tType] :
( ( funpow @ A )
= ( compow @ ( A > A ) ) ) ).
% Nat.funpow_code_def
thf(fact_3914_card__disjoint__shuffles,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( ( inf_inf @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ Ys ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_card @ ( list @ A ) @ ( shuffles @ A @ Xs @ Ys ) )
= ( binomial @ ( plus_plus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ A ) @ Ys ) ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ).
% card_disjoint_shuffles
thf(fact_3915_finite__greaterThanLessThan,axiom,
! [L: nat,U: nat] : ( finite_finite2 @ nat @ ( set_or5935395276787703475ssThan @ nat @ L @ U ) ) ).
% finite_greaterThanLessThan
thf(fact_3916_finite__shuffles,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] : ( finite_finite2 @ ( list @ A ) @ ( shuffles @ A @ Xs @ Ys ) ) ).
% finite_shuffles
thf(fact_3917_card__greaterThanLessThan,axiom,
! [L: nat,U: nat] :
( ( finite_card @ nat @ ( set_or5935395276787703475ssThan @ nat @ L @ U ) )
= ( minus_minus @ nat @ U @ ( suc @ L ) ) ) ).
% card_greaterThanLessThan
thf(fact_3918_shuffles__commutes,axiom,
! [A: $tType] :
( ( shuffles @ A )
= ( ^ [Xs3: list @ A,Ys3: list @ A] : ( shuffles @ A @ Ys3 @ Xs3 ) ) ) ).
% shuffles_commutes
thf(fact_3919_inf__set__def,axiom,
! [A: $tType] :
( ( inf_inf @ ( set @ A ) )
= ( ^ [A7: set @ A,B6: set @ A] :
( collect @ A
@ ( inf_inf @ ( A > $o )
@ ^ [X4: A] : ( member @ A @ X4 @ A7 )
@ ^ [X4: A] : ( member @ A @ X4 @ B6 ) ) ) ) ) ).
% inf_set_def
thf(fact_3920_atLeastSucLessThan__greaterThanLessThan,axiom,
! [L: nat,U: nat] :
( ( set_or7035219750837199246ssThan @ nat @ ( suc @ L ) @ U )
= ( set_or5935395276787703475ssThan @ nat @ L @ U ) ) ).
% atLeastSucLessThan_greaterThanLessThan
thf(fact_3921_length__shuffles,axiom,
! [A: $tType,Zs: list @ A,Xs: list @ A,Ys: list @ A] :
( ( member @ ( list @ A ) @ Zs @ ( shuffles @ A @ Xs @ Ys ) )
=> ( ( size_size @ ( list @ A ) @ Zs )
= ( plus_plus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ A ) @ Ys ) ) ) ) ).
% length_shuffles
thf(fact_3922_relpowp__Suc__I2,axiom,
! [A: $tType,P: A > A > $o,X: A,Y: A,N2: nat,Z3: A] :
( ( P @ X @ Y )
=> ( ( compow @ ( A > A > $o ) @ N2 @ P @ Y @ Z3 )
=> ( compow @ ( A > A > $o ) @ ( suc @ N2 ) @ P @ X @ Z3 ) ) ) ).
% relpowp_Suc_I2
thf(fact_3923_relpowp__Suc__E2,axiom,
! [A: $tType,N2: nat,P: A > A > $o,X: A,Z3: A] :
( ( compow @ ( A > A > $o ) @ ( suc @ N2 ) @ P @ X @ Z3 )
=> ~ ! [Y3: A] :
( ( P @ X @ Y3 )
=> ~ ( compow @ ( A > A > $o ) @ N2 @ P @ Y3 @ Z3 ) ) ) ).
% relpowp_Suc_E2
thf(fact_3924_relpowp__Suc__D2,axiom,
! [A: $tType,N2: nat,P: A > A > $o,X: A,Z3: A] :
( ( compow @ ( A > A > $o ) @ ( suc @ N2 ) @ P @ X @ Z3 )
=> ? [Y3: A] :
( ( P @ X @ Y3 )
& ( compow @ ( A > A > $o ) @ N2 @ P @ Y3 @ Z3 ) ) ) ).
% relpowp_Suc_D2
thf(fact_3925_relpowp__Suc__I,axiom,
! [A: $tType,N2: nat,P: A > A > $o,X: A,Y: A,Z3: A] :
( ( compow @ ( A > A > $o ) @ N2 @ P @ X @ Y )
=> ( ( P @ Y @ Z3 )
=> ( compow @ ( A > A > $o ) @ ( suc @ N2 ) @ P @ X @ Z3 ) ) ) ).
% relpowp_Suc_I
thf(fact_3926_relpowp__Suc__E,axiom,
! [A: $tType,N2: nat,P: A > A > $o,X: A,Z3: A] :
( ( compow @ ( A > A > $o ) @ ( suc @ N2 ) @ P @ X @ Z3 )
=> ~ ! [Y3: A] :
( ( compow @ ( A > A > $o ) @ N2 @ P @ X @ Y3 )
=> ~ ( P @ Y3 @ Z3 ) ) ) ).
% relpowp_Suc_E
thf(fact_3927_relpowp_Osimps_I1_J,axiom,
! [A: $tType,R: A > A > $o] :
( ( compow @ ( A > A > $o ) @ ( zero_zero @ nat ) @ R )
= ( ^ [Y5: A,Z2: A] : Y5 = Z2 ) ) ).
% relpowp.simps(1)
thf(fact_3928_relpowp__0__E,axiom,
! [A: $tType,P: A > A > $o,X: A,Y: A] :
( ( compow @ ( A > A > $o ) @ ( zero_zero @ nat ) @ P @ X @ Y )
=> ( X = Y ) ) ).
% relpowp_0_E
thf(fact_3929_relpowp__0__I,axiom,
! [A: $tType,P: A > A > $o,X: A] : ( compow @ ( A > A > $o ) @ ( zero_zero @ nat ) @ P @ X @ X ) ).
% relpowp_0_I
thf(fact_3930_distinct__disjoint__shuffles,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,Zs: list @ A] :
( ( distinct @ A @ Xs )
=> ( ( distinct @ A @ Ys )
=> ( ( ( inf_inf @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ Ys ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( member @ ( list @ A ) @ Zs @ ( shuffles @ A @ Xs @ Ys ) )
=> ( distinct @ A @ Zs ) ) ) ) ) ).
% distinct_disjoint_shuffles
thf(fact_3931_relpowp__E2,axiom,
! [A: $tType,N2: nat,P: A > A > $o,X: A,Z3: A] :
( ( compow @ ( A > A > $o ) @ N2 @ P @ X @ Z3 )
=> ( ( ( N2
= ( zero_zero @ nat ) )
=> ( X != Z3 ) )
=> ~ ! [Y3: A,M3: nat] :
( ( N2
= ( suc @ M3 ) )
=> ( ( P @ X @ Y3 )
=> ~ ( compow @ ( A > A > $o ) @ M3 @ P @ Y3 @ Z3 ) ) ) ) ) ).
% relpowp_E2
thf(fact_3932_relpowp__E,axiom,
! [A: $tType,N2: nat,P: A > A > $o,X: A,Z3: A] :
( ( compow @ ( A > A > $o ) @ N2 @ P @ X @ Z3 )
=> ( ( ( N2
= ( zero_zero @ nat ) )
=> ( X != Z3 ) )
=> ~ ! [Y3: A,M3: nat] :
( ( N2
= ( suc @ M3 ) )
=> ( ( compow @ ( A > A > $o ) @ M3 @ P @ X @ Y3 )
=> ~ ( P @ Y3 @ Z3 ) ) ) ) ) ).
% relpowp_E
thf(fact_3933_distinct__product__lists,axiom,
! [A: $tType,Xss: list @ ( list @ A )] :
( ! [X5: list @ A] :
( ( member @ ( list @ A ) @ X5 @ ( set2 @ ( list @ A ) @ Xss ) )
=> ( distinct @ A @ X5 ) )
=> ( distinct @ ( list @ A ) @ ( product_lists @ A @ Xss ) ) ) ).
% distinct_product_lists
thf(fact_3934_Fpow__Pow__finite,axiom,
! [A: $tType] :
( ( finite_Fpow @ A )
= ( ^ [A7: set @ A] : ( inf_inf @ ( set @ ( set @ A ) ) @ ( pow2 @ A @ A7 ) @ ( collect @ ( set @ A ) @ ( finite_finite2 @ A ) ) ) ) ) ).
% Fpow_Pow_finite
thf(fact_3935_max__nat_Osemilattice__neutr__order__axioms,axiom,
( semila1105856199041335345_order @ nat @ ( ord_max @ nat ) @ ( zero_zero @ nat )
@ ^ [X4: nat,Y6: nat] : ( ord_less_eq @ nat @ Y6 @ X4 )
@ ^ [X4: nat,Y6: nat] : ( ord_less @ nat @ Y6 @ X4 ) ) ).
% max_nat.semilattice_neutr_order_axioms
thf(fact_3936_set__removeAll,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( set2 @ A @ ( removeAll @ A @ X @ Xs ) )
= ( minus_minus @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% set_removeAll
thf(fact_3937_removeAll__id,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ~ ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ( removeAll @ A @ X @ Xs )
= Xs ) ) ).
% removeAll_id
thf(fact_3938_distinct__removeAll,axiom,
! [A: $tType,Xs: list @ A,X: A] :
( ( distinct @ A @ Xs )
=> ( distinct @ A @ ( removeAll @ A @ X @ Xs ) ) ) ).
% distinct_removeAll
thf(fact_3939_empty__in__Fpow,axiom,
! [A: $tType,A4: set @ A] : ( member @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ ( finite_Fpow @ A @ A4 ) ) ).
% empty_in_Fpow
thf(fact_3940_Fpow__not__empty,axiom,
! [A: $tType,A4: set @ A] :
( ( finite_Fpow @ A @ A4 )
!= ( bot_bot @ ( set @ ( set @ A ) ) ) ) ).
% Fpow_not_empty
thf(fact_3941_length__removeAll__less__eq,axiom,
! [A: $tType,X: A,Xs: list @ A] : ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ ( removeAll @ A @ X @ Xs ) ) @ ( size_size @ ( list @ A ) @ Xs ) ) ).
% length_removeAll_less_eq
thf(fact_3942_distinct__remove1__removeAll,axiom,
! [A: $tType,Xs: list @ A,X: A] :
( ( distinct @ A @ Xs )
=> ( ( remove1 @ A @ X @ Xs )
= ( removeAll @ A @ X @ Xs ) ) ) ).
% distinct_remove1_removeAll
thf(fact_3943_Fpow__mono,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B2 )
=> ( ord_less_eq @ ( set @ ( set @ A ) ) @ ( finite_Fpow @ A @ A4 ) @ ( finite_Fpow @ A @ B2 ) ) ) ).
% Fpow_mono
thf(fact_3944_length__removeAll__less,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ord_less @ nat @ ( size_size @ ( list @ A ) @ ( removeAll @ A @ X @ Xs ) ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ).
% length_removeAll_less
thf(fact_3945_Fpow__subset__Pow,axiom,
! [A: $tType,A4: set @ A] : ( ord_less_eq @ ( set @ ( set @ A ) ) @ ( finite_Fpow @ A @ A4 ) @ ( pow2 @ A @ A4 ) ) ).
% Fpow_subset_Pow
thf(fact_3946_Fpow__def,axiom,
! [A: $tType] :
( ( finite_Fpow @ A )
= ( ^ [A7: set @ A] :
( collect @ ( set @ A )
@ ^ [X6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ X6 @ A7 )
& ( finite_finite2 @ A @ X6 ) ) ) ) ) ).
% Fpow_def
thf(fact_3947_in__set__product__lists__length,axiom,
! [A: $tType,Xs: list @ A,Xss: list @ ( list @ A )] :
( ( member @ ( list @ A ) @ Xs @ ( set2 @ ( list @ A ) @ ( product_lists @ A @ Xss ) ) )
=> ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ ( list @ A ) ) @ Xss ) ) ) ).
% in_set_product_lists_length
thf(fact_3948_distinct__concat__iff,axiom,
! [A: $tType,Xs: list @ ( list @ A )] :
( ( distinct @ A @ ( concat @ A @ Xs ) )
= ( ( distinct @ ( list @ A ) @ ( removeAll @ ( list @ A ) @ ( nil @ A ) @ Xs ) )
& ! [Ys3: list @ A] :
( ( member @ ( list @ A ) @ Ys3 @ ( set2 @ ( list @ A ) @ Xs ) )
=> ( distinct @ A @ Ys3 ) )
& ! [Ys3: list @ A,Zs3: list @ A] :
( ( ( member @ ( list @ A ) @ Ys3 @ ( set2 @ ( list @ A ) @ Xs ) )
& ( member @ ( list @ A ) @ Zs3 @ ( set2 @ ( list @ A ) @ Xs ) )
& ( Ys3 != Zs3 ) )
=> ( ( inf_inf @ ( set @ A ) @ ( set2 @ A @ Ys3 ) @ ( set2 @ A @ Zs3 ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% distinct_concat_iff
thf(fact_3949_times__int_Oabs__eq,axiom,
! [Xa2: product_prod @ nat @ nat,X: product_prod @ nat @ nat] :
( ( times_times @ int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X ) )
= ( abs_Integ
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
@ ^ [X4: nat,Y6: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V5: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ X4 @ U2 ) @ ( times_times @ nat @ Y6 @ V5 ) ) @ ( plus_plus @ nat @ ( times_times @ nat @ X4 @ V5 ) @ ( times_times @ nat @ Y6 @ U2 ) ) ) )
@ Xa2
@ X ) ) ) ).
% times_int.abs_eq
thf(fact_3950_Gcd__remove0__nat,axiom,
! [M7: set @ nat] :
( ( finite_finite2 @ nat @ M7 )
=> ( ( gcd_Gcd @ nat @ M7 )
= ( gcd_Gcd @ nat @ ( minus_minus @ ( set @ nat ) @ M7 @ ( insert2 @ nat @ ( zero_zero @ nat ) @ ( bot_bot @ ( set @ nat ) ) ) ) ) ) ) ).
% Gcd_remove0_nat
thf(fact_3951_add__neg__numeral__special_I4_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [N2: num] :
( ( plus_plus @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( numeral_numeral @ A @ N2 ) )
= ( neg_numeral_sub @ A @ N2 @ one2 ) ) ) ).
% add_neg_numeral_special(4)
thf(fact_3952_list__update__nonempty,axiom,
! [A: $tType,Xs: list @ A,K: nat,X: A] :
( ( ( list_update @ A @ Xs @ K @ X )
= ( nil @ A ) )
= ( Xs
= ( nil @ A ) ) ) ).
% list_update_nonempty
thf(fact_3953_concat__replicate__trivial,axiom,
! [A: $tType,I: nat] :
( ( concat @ A @ ( replicate @ ( list @ A ) @ I @ ( nil @ A ) ) )
= ( nil @ A ) ) ).
% concat_replicate_trivial
thf(fact_3954_Nil__in__shuffles,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( member @ ( list @ A ) @ ( nil @ A ) @ ( shuffles @ A @ Xs @ Ys ) )
= ( ( Xs
= ( nil @ A ) )
& ( Ys
= ( nil @ A ) ) ) ) ).
% Nil_in_shuffles
thf(fact_3955_enumerate__simps_I1_J,axiom,
! [A: $tType,N2: nat] :
( ( enumerate @ A @ N2 @ ( nil @ A ) )
= ( nil @ ( product_prod @ nat @ A ) ) ) ).
% enumerate_simps(1)
thf(fact_3956_rotate1__is__Nil__conv,axiom,
! [A: $tType,Xs: list @ A] :
( ( ( rotate1 @ A @ Xs )
= ( nil @ A ) )
= ( Xs
= ( nil @ A ) ) ) ).
% rotate1_is_Nil_conv
thf(fact_3957_set__empty,axiom,
! [A: $tType,Xs: list @ A] :
( ( ( set2 @ A @ Xs )
= ( bot_bot @ ( set @ A ) ) )
= ( Xs
= ( nil @ A ) ) ) ).
% set_empty
thf(fact_3958_set__empty2,axiom,
! [A: $tType,Xs: list @ A] :
( ( ( bot_bot @ ( set @ A ) )
= ( set2 @ A @ Xs ) )
= ( Xs
= ( nil @ A ) ) ) ).
% set_empty2
thf(fact_3959_length__0__conv,axiom,
! [A: $tType,Xs: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( zero_zero @ nat ) )
= ( Xs
= ( nil @ A ) ) ) ).
% length_0_conv
thf(fact_3960_sub__num__simps_I1_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( ( neg_numeral_sub @ A @ one2 @ one2 )
= ( zero_zero @ A ) ) ) ).
% sub_num_simps(1)
thf(fact_3961_Gcd__empty,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ( ( gcd_Gcd @ A @ ( bot_bot @ ( set @ A ) ) )
= ( zero_zero @ A ) ) ) ).
% Gcd_empty
thf(fact_3962_replicate__empty,axiom,
! [A: $tType,N2: nat,X: A] :
( ( ( replicate @ A @ N2 @ X )
= ( nil @ A ) )
= ( N2
= ( zero_zero @ nat ) ) ) ).
% replicate_empty
thf(fact_3963_empty__replicate,axiom,
! [A: $tType,N2: nat,X: A] :
( ( ( nil @ A )
= ( replicate @ A @ N2 @ X ) )
= ( N2
= ( zero_zero @ nat ) ) ) ).
% empty_replicate
thf(fact_3964_horner__sum__simps_I1_J,axiom,
! [B: $tType,A: $tType] :
( ( comm_semiring_0 @ A )
=> ! [F2: B > A,A3: A] :
( ( groups4207007520872428315er_sum @ B @ A @ F2 @ A3 @ ( nil @ B ) )
= ( zero_zero @ A ) ) ) ).
% horner_sum_simps(1)
thf(fact_3965_Nil__eq__concat__conv,axiom,
! [A: $tType,Xss: list @ ( list @ A )] :
( ( ( nil @ A )
= ( concat @ A @ Xss ) )
= ( ! [X4: list @ A] :
( ( member @ ( list @ A ) @ X4 @ ( set2 @ ( list @ A ) @ Xss ) )
=> ( X4
= ( nil @ A ) ) ) ) ) ).
% Nil_eq_concat_conv
thf(fact_3966_concat__eq__Nil__conv,axiom,
! [A: $tType,Xss: list @ ( list @ A )] :
( ( ( concat @ A @ Xss )
= ( nil @ A ) )
= ( ! [X4: list @ A] :
( ( member @ ( list @ A ) @ X4 @ ( set2 @ ( list @ A ) @ Xss ) )
=> ( X4
= ( nil @ A ) ) ) ) ) ).
% concat_eq_Nil_conv
thf(fact_3967_length__greater__0__conv,axiom,
! [A: $tType,Xs: list @ A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Xs ) )
= ( Xs
!= ( nil @ A ) ) ) ).
% length_greater_0_conv
thf(fact_3968_semiring__norm_I167_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [V2: num,W2: num,Y: A] :
( ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) @ ( plus_plus @ A @ ( numeral_numeral @ A @ W2 ) @ Y ) )
= ( plus_plus @ A @ ( neg_numeral_sub @ A @ W2 @ V2 ) @ Y ) ) ) ).
% semiring_norm(167)
thf(fact_3969_semiring__norm_I166_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [V2: num,W2: num,Y: A] :
( ( plus_plus @ A @ ( numeral_numeral @ A @ V2 ) @ ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ Y ) )
= ( plus_plus @ A @ ( neg_numeral_sub @ A @ V2 @ W2 ) @ Y ) ) ) ).
% semiring_norm(166)
thf(fact_3970_add__neg__numeral__simps_I1_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [M2: num,N2: num] :
( ( plus_plus @ A @ ( numeral_numeral @ A @ M2 ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N2 ) ) )
= ( neg_numeral_sub @ A @ M2 @ N2 ) ) ) ).
% add_neg_numeral_simps(1)
thf(fact_3971_add__neg__numeral__simps_I2_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [M2: num,N2: num] :
( ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M2 ) ) @ ( numeral_numeral @ A @ N2 ) )
= ( neg_numeral_sub @ A @ N2 @ M2 ) ) ) ).
% add_neg_numeral_simps(2)
thf(fact_3972_Gcd__0__iff,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [A4: set @ A] :
( ( ( gcd_Gcd @ A @ A4 )
= ( zero_zero @ A ) )
= ( ord_less_eq @ ( set @ A ) @ A4 @ ( insert2 @ A @ ( zero_zero @ A ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% Gcd_0_iff
thf(fact_3973_add__neg__numeral__special_I1_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [M2: num] :
( ( plus_plus @ A @ ( one_one @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M2 ) ) )
= ( neg_numeral_sub @ A @ one2 @ M2 ) ) ) ).
% add_neg_numeral_special(1)
thf(fact_3974_add__neg__numeral__special_I2_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [M2: num] :
( ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M2 ) ) @ ( one_one @ A ) )
= ( neg_numeral_sub @ A @ one2 @ M2 ) ) ) ).
% add_neg_numeral_special(2)
thf(fact_3975_add__neg__numeral__special_I3_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [M2: num] :
( ( plus_plus @ A @ ( numeral_numeral @ A @ M2 ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( neg_numeral_sub @ A @ M2 @ one2 ) ) ) ).
% add_neg_numeral_special(3)
thf(fact_3976_remove1_Osimps_I1_J,axiom,
! [A: $tType,X: A] :
( ( remove1 @ A @ X @ ( nil @ A ) )
= ( nil @ A ) ) ).
% remove1.simps(1)
thf(fact_3977_list__update__code_I1_J,axiom,
! [A: $tType,I: nat,Y: A] :
( ( list_update @ A @ ( nil @ A ) @ I @ Y )
= ( nil @ A ) ) ).
% list_update_code(1)
thf(fact_3978_list__update_Osimps_I1_J,axiom,
! [A: $tType,I: nat,V2: A] :
( ( list_update @ A @ ( nil @ A ) @ I @ V2 )
= ( nil @ A ) ) ).
% list_update.simps(1)
thf(fact_3979_concat_Osimps_I1_J,axiom,
! [A: $tType] :
( ( concat @ A @ ( nil @ ( list @ A ) ) )
= ( nil @ A ) ) ).
% concat.simps(1)
thf(fact_3980_product_Osimps_I1_J,axiom,
! [B: $tType,A: $tType,Uu: list @ B] :
( ( product @ A @ B @ ( nil @ A ) @ Uu )
= ( nil @ ( product_prod @ A @ B ) ) ) ).
% product.simps(1)
thf(fact_3981_int_Oabs__induct,axiom,
! [P: int > $o,X: int] :
( ! [Y3: product_prod @ nat @ nat] : ( P @ ( abs_Integ @ Y3 ) )
=> ( P @ X ) ) ).
% int.abs_induct
thf(fact_3982_Nil__in__shufflesI,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( Xs
= ( nil @ A ) )
=> ( ( Ys
= ( nil @ A ) )
=> ( member @ ( list @ A ) @ ( nil @ A ) @ ( shuffles @ A @ Xs @ Ys ) ) ) ) ).
% Nil_in_shufflesI
thf(fact_3983_distinct_Osimps_I1_J,axiom,
! [A: $tType] : ( distinct @ A @ ( nil @ A ) ) ).
% distinct.simps(1)
thf(fact_3984_shuffles_Osimps_I2_J,axiom,
! [A: $tType,Xs: list @ A] :
( ( shuffles @ A @ Xs @ ( nil @ A ) )
= ( insert2 @ ( list @ A ) @ Xs @ ( bot_bot @ ( set @ ( list @ A ) ) ) ) ) ).
% shuffles.simps(2)
thf(fact_3985_shuffles_Osimps_I1_J,axiom,
! [A: $tType,Ys: list @ A] :
( ( shuffles @ A @ ( nil @ A ) @ Ys )
= ( insert2 @ ( list @ A ) @ Ys @ ( bot_bot @ ( set @ ( list @ A ) ) ) ) ) ).
% shuffles.simps(1)
thf(fact_3986_removeAll_Osimps_I1_J,axiom,
! [A: $tType,X: A] :
( ( removeAll @ A @ X @ ( nil @ A ) )
= ( nil @ A ) ) ).
% removeAll.simps(1)
thf(fact_3987_eq__Abs__Integ,axiom,
! [Z3: int] :
~ ! [X5: nat,Y3: nat] :
( Z3
!= ( abs_Integ @ ( product_Pair @ nat @ nat @ X5 @ Y3 ) ) ) ).
% eq_Abs_Integ
thf(fact_3988_rotate1_Osimps_I1_J,axiom,
! [A: $tType] :
( ( rotate1 @ A @ ( nil @ A ) )
= ( nil @ A ) ) ).
% rotate1.simps(1)
thf(fact_3989_gen__length__code_I1_J,axiom,
! [A: $tType,N2: nat] :
( ( gen_length @ A @ N2 @ ( nil @ A ) )
= N2 ) ).
% gen_length_code(1)
thf(fact_3990_empty__set,axiom,
! [A: $tType] :
( ( bot_bot @ ( set @ A ) )
= ( set2 @ A @ ( nil @ A ) ) ) ).
% empty_set
thf(fact_3991_list_Osize_I3_J,axiom,
! [A: $tType] :
( ( size_size @ ( list @ A ) @ ( nil @ A ) )
= ( zero_zero @ nat ) ) ).
% list.size(3)
thf(fact_3992_replicate__0,axiom,
! [A: $tType,X: A] :
( ( replicate @ A @ ( zero_zero @ nat ) @ X )
= ( nil @ A ) ) ).
% replicate_0
thf(fact_3993_not__Nil__listrel1,axiom,
! [A: $tType,Xs: list @ A,R2: set @ ( product_prod @ A @ A )] :
~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Xs ) @ ( listrel1 @ A @ R2 ) ) ).
% not_Nil_listrel1
thf(fact_3994_not__listrel1__Nil,axiom,
! [A: $tType,Xs: list @ A,R2: set @ ( product_prod @ A @ A )] :
~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ ( nil @ A ) ) @ ( listrel1 @ A @ R2 ) ) ).
% not_listrel1_Nil
thf(fact_3995_list_Osize__gen_I1_J,axiom,
! [A: $tType,X: A > nat] :
( ( size_list @ A @ X @ ( nil @ A ) )
= ( zero_zero @ nat ) ) ).
% list.size_gen(1)
thf(fact_3996_find_Osimps_I1_J,axiom,
! [A: $tType,Uu: A > $o] :
( ( find @ A @ Uu @ ( nil @ A ) )
= ( none @ A ) ) ).
% find.simps(1)
thf(fact_3997_nat_Oabs__eq,axiom,
! [X: product_prod @ nat @ nat] :
( ( nat2 @ ( abs_Integ @ X ) )
= ( product_case_prod @ nat @ nat @ nat @ ( minus_minus @ nat ) @ X ) ) ).
% nat.abs_eq
thf(fact_3998_count__list_Osimps_I1_J,axiom,
! [A: $tType,Y: A] :
( ( count_list @ A @ ( nil @ A ) @ Y )
= ( zero_zero @ nat ) ) ).
% count_list.simps(1)
thf(fact_3999_sub__non__positive,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N2: num,M2: num] :
( ( ord_less_eq @ A @ ( neg_numeral_sub @ A @ N2 @ M2 ) @ ( zero_zero @ A ) )
= ( ord_less_eq @ num @ N2 @ M2 ) ) ) ).
% sub_non_positive
thf(fact_4000_sub__non__negative,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N2: num,M2: num] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( neg_numeral_sub @ A @ N2 @ M2 ) )
= ( ord_less_eq @ num @ M2 @ N2 ) ) ) ).
% sub_non_negative
thf(fact_4001_sub__positive,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N2: num,M2: num] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( neg_numeral_sub @ A @ N2 @ M2 ) )
= ( ord_less @ num @ M2 @ N2 ) ) ) ).
% sub_positive
thf(fact_4002_sub__negative,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N2: num,M2: num] :
( ( ord_less @ A @ ( neg_numeral_sub @ A @ N2 @ M2 ) @ ( zero_zero @ A ) )
= ( ord_less @ num @ N2 @ M2 ) ) ) ).
% sub_negative
thf(fact_4003_zero__int__def,axiom,
( ( zero_zero @ int )
= ( abs_Integ @ ( product_Pair @ nat @ nat @ ( zero_zero @ nat ) @ ( zero_zero @ nat ) ) ) ) ).
% zero_int_def
thf(fact_4004_int__def,axiom,
( ( semiring_1_of_nat @ int )
= ( ^ [N: nat] : ( abs_Integ @ ( product_Pair @ nat @ nat @ N @ ( zero_zero @ nat ) ) ) ) ) ).
% int_def
thf(fact_4005_uminus__int_Oabs__eq,axiom,
! [X: product_prod @ nat @ nat] :
( ( uminus_uminus @ int @ ( abs_Integ @ X ) )
= ( abs_Integ
@ ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [X4: nat,Y6: nat] : ( product_Pair @ nat @ nat @ Y6 @ X4 )
@ X ) ) ) ).
% uminus_int.abs_eq
thf(fact_4006_one__int__def,axiom,
( ( one_one @ int )
= ( abs_Integ @ ( product_Pair @ nat @ nat @ ( one_one @ nat ) @ ( zero_zero @ nat ) ) ) ) ).
% one_int_def
thf(fact_4007_of__int_Oabs__eq,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [X: product_prod @ nat @ nat] :
( ( ring_1_of_int @ A @ ( abs_Integ @ X ) )
= ( product_case_prod @ nat @ nat @ A
@ ^ [I3: nat,J2: nat] : ( minus_minus @ A @ ( semiring_1_of_nat @ A @ I3 ) @ ( semiring_1_of_nat @ A @ J2 ) )
@ X ) ) ) ).
% of_int.abs_eq
thf(fact_4008_less__int_Oabs__eq,axiom,
! [Xa2: product_prod @ nat @ nat,X: product_prod @ nat @ nat] :
( ( ord_less @ int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X ) )
= ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [X4: nat,Y6: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U2: nat,V5: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ X4 @ V5 ) @ ( plus_plus @ nat @ U2 @ Y6 ) ) )
@ Xa2
@ X ) ) ).
% less_int.abs_eq
thf(fact_4009_less__eq__int_Oabs__eq,axiom,
! [Xa2: product_prod @ nat @ nat,X: product_prod @ nat @ nat] :
( ( ord_less_eq @ int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X ) )
= ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [X4: nat,Y6: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U2: nat,V5: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ X4 @ V5 ) @ ( plus_plus @ nat @ U2 @ Y6 ) ) )
@ Xa2
@ X ) ) ).
% less_eq_int.abs_eq
thf(fact_4010_plus__int_Oabs__eq,axiom,
! [Xa2: product_prod @ nat @ nat,X: product_prod @ nat @ nat] :
( ( plus_plus @ int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X ) )
= ( abs_Integ
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
@ ^ [X4: nat,Y6: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V5: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X4 @ U2 ) @ ( plus_plus @ nat @ Y6 @ V5 ) ) )
@ Xa2
@ X ) ) ) ).
% plus_int.abs_eq
thf(fact_4011_minus__int_Oabs__eq,axiom,
! [Xa2: product_prod @ nat @ nat,X: product_prod @ nat @ nat] :
( ( minus_minus @ int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X ) )
= ( abs_Integ
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
@ ^ [X4: nat,Y6: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V5: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X4 @ V5 ) @ ( plus_plus @ nat @ Y6 @ U2 ) ) )
@ Xa2
@ X ) ) ) ).
% minus_int.abs_eq
thf(fact_4012_Pow__set_I1_J,axiom,
! [A: $tType] :
( ( pow2 @ A @ ( set2 @ A @ ( nil @ A ) ) )
= ( insert2 @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ ( bot_bot @ ( set @ ( set @ A ) ) ) ) ) ).
% Pow_set(1)
thf(fact_4013_semiring__char__def,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ( ( semiri4206861660011772517g_char @ A )
= ( ^ [Uu4: itself @ A] :
( gcd_Gcd @ nat
@ ( collect @ nat
@ ^ [N: nat] :
( ( semiring_1_of_nat @ A @ N )
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% semiring_char_def
thf(fact_4014_num__of__nat_Osimps_I2_J,axiom,
! [N2: nat] :
( ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( num_of_nat @ ( suc @ N2 ) )
= ( inc @ ( num_of_nat @ N2 ) ) ) )
& ( ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( num_of_nat @ ( suc @ N2 ) )
= one2 ) ) ) ).
% num_of_nat.simps(2)
thf(fact_4015_prod_Oinsert_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [I6: set @ B,P5: B > A,I: B] :
( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X4: B] :
( ( member @ B @ X4 @ I6 )
& ( ( P5 @ X4 )
!= ( one_one @ A ) ) ) ) )
=> ( ( ( member @ B @ I @ I6 )
=> ( ( groups1962203154675924110t_prod @ B @ A @ P5 @ ( insert2 @ B @ I @ I6 ) )
= ( groups1962203154675924110t_prod @ B @ A @ P5 @ I6 ) ) )
& ( ~ ( member @ B @ I @ I6 )
=> ( ( groups1962203154675924110t_prod @ B @ A @ P5 @ ( insert2 @ B @ I @ I6 ) )
= ( times_times @ A @ ( P5 @ I ) @ ( groups1962203154675924110t_prod @ B @ A @ P5 @ I6 ) ) ) ) ) ) ) ).
% prod.insert'
thf(fact_4016_sorted__list__of__set_Osorted__key__list__of__set__remove,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( linord4507533701916653071of_set @ A @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( remove1 @ A @ X @ ( linord4507533701916653071of_set @ A @ A4 ) ) ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_remove
thf(fact_4017_sorted__list__of__set_Osorted__key__list__of__set__empty,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( linord4507533701916653071of_set @ A @ ( bot_bot @ ( set @ A ) ) )
= ( nil @ A ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_empty
thf(fact_4018_sorted__list__of__set_Ofold__insort__key_Oinfinite,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A] :
( ~ ( finite_finite2 @ A @ A4 )
=> ( ( linord4507533701916653071of_set @ A @ A4 )
= ( nil @ A ) ) ) ) ).
% sorted_list_of_set.fold_insort_key.infinite
thf(fact_4019_sorted__list__of__set_Oset__sorted__key__list__of__set,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( set2 @ A @ ( linord4507533701916653071of_set @ A @ A4 ) )
= A4 ) ) ) ).
% sorted_list_of_set.set_sorted_key_list_of_set
thf(fact_4020_prod_Oempty_H,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [P5: B > A] :
( ( groups1962203154675924110t_prod @ B @ A @ P5 @ ( bot_bot @ ( set @ B ) ) )
= ( one_one @ A ) ) ) ).
% prod.empty'
thf(fact_4021_sorted__list__of__set_Olength__sorted__key__list__of__set,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A] :
( ( size_size @ ( list @ A ) @ ( linord4507533701916653071of_set @ A @ A4 ) )
= ( finite_card @ A @ A4 ) ) ) ).
% sorted_list_of_set.length_sorted_key_list_of_set
thf(fact_4022_prod_Oeq__sum,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [I6: set @ B,P5: B > A] :
( ( finite_finite2 @ B @ I6 )
=> ( ( groups1962203154675924110t_prod @ B @ A @ P5 @ I6 )
= ( groups7121269368397514597t_prod @ B @ A @ P5 @ I6 ) ) ) ) ).
% prod.eq_sum
thf(fact_4023_sorted__list__of__set_Osorted__key__list__of__set__eq__Nil__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( ( linord4507533701916653071of_set @ A @ A4 )
= ( nil @ A ) )
= ( A4
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_eq_Nil_iff
thf(fact_4024_sorted__list__of__set_Osorted__key__list__of__set__inject,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,B2: set @ A] :
( ( ( linord4507533701916653071of_set @ A @ A4 )
= ( linord4507533701916653071of_set @ A @ B2 ) )
=> ( ( finite_finite2 @ A @ A4 )
=> ( ( finite_finite2 @ A @ B2 )
=> ( A4 = B2 ) ) ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_inject
thf(fact_4025_sorted__list__of__set_Odistinct__sorted__key__list__of__set,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A] : ( distinct @ A @ ( linord4507533701916653071of_set @ A @ A4 ) ) ) ).
% sorted_list_of_set.distinct_sorted_key_list_of_set
thf(fact_4026_num__of__nat_Osimps_I1_J,axiom,
( ( num_of_nat @ ( zero_zero @ nat ) )
= one2 ) ).
% num_of_nat.simps(1)
thf(fact_4027_prod_Odistrib__triv_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [I6: set @ B,G: B > A,H2: B > A] :
( ( finite_finite2 @ B @ I6 )
=> ( ( groups1962203154675924110t_prod @ B @ A
@ ^ [I3: B] : ( times_times @ A @ ( G @ I3 ) @ ( H2 @ I3 ) )
@ I6 )
= ( times_times @ A @ ( groups1962203154675924110t_prod @ B @ A @ G @ I6 ) @ ( groups1962203154675924110t_prod @ B @ A @ H2 @ I6 ) ) ) ) ) ).
% prod.distrib_triv'
thf(fact_4028_prod_Omono__neutral__cong__right_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S2: set @ B,T4: set @ B,G: B > A,H2: B > A] :
( ( ord_less_eq @ ( set @ B ) @ S2 @ T4 )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ ( minus_minus @ ( set @ B ) @ T4 @ S2 ) )
=> ( ( G @ X5 )
= ( one_one @ A ) ) )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ S2 )
=> ( ( G @ X5 )
= ( H2 @ X5 ) ) )
=> ( ( groups1962203154675924110t_prod @ B @ A @ G @ T4 )
= ( groups1962203154675924110t_prod @ B @ A @ H2 @ S2 ) ) ) ) ) ) ).
% prod.mono_neutral_cong_right'
thf(fact_4029_prod_Omono__neutral__cong__left_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S2: set @ B,T4: set @ B,H2: B > A,G: B > A] :
( ( ord_less_eq @ ( set @ B ) @ S2 @ T4 )
=> ( ! [I4: B] :
( ( member @ B @ I4 @ ( minus_minus @ ( set @ B ) @ T4 @ S2 ) )
=> ( ( H2 @ I4 )
= ( one_one @ A ) ) )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ S2 )
=> ( ( G @ X5 )
= ( H2 @ X5 ) ) )
=> ( ( groups1962203154675924110t_prod @ B @ A @ G @ S2 )
= ( groups1962203154675924110t_prod @ B @ A @ H2 @ T4 ) ) ) ) ) ) ).
% prod.mono_neutral_cong_left'
thf(fact_4030_prod_Omono__neutral__right_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S2: set @ B,T4: set @ B,G: B > A] :
( ( ord_less_eq @ ( set @ B ) @ S2 @ T4 )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ ( minus_minus @ ( set @ B ) @ T4 @ S2 ) )
=> ( ( G @ X5 )
= ( one_one @ A ) ) )
=> ( ( groups1962203154675924110t_prod @ B @ A @ G @ T4 )
= ( groups1962203154675924110t_prod @ B @ A @ G @ S2 ) ) ) ) ) ).
% prod.mono_neutral_right'
thf(fact_4031_prod_Omono__neutral__left_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S2: set @ B,T4: set @ B,G: B > A] :
( ( ord_less_eq @ ( set @ B ) @ S2 @ T4 )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ ( minus_minus @ ( set @ B ) @ T4 @ S2 ) )
=> ( ( G @ X5 )
= ( one_one @ A ) ) )
=> ( ( groups1962203154675924110t_prod @ B @ A @ G @ S2 )
= ( groups1962203154675924110t_prod @ B @ A @ G @ T4 ) ) ) ) ) ).
% prod.mono_neutral_left'
thf(fact_4032_numeral__num__of__nat,axiom,
! [N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( numeral_numeral @ nat @ ( num_of_nat @ N2 ) )
= N2 ) ) ).
% numeral_num_of_nat
thf(fact_4033_num__of__nat__One,axiom,
! [N2: nat] :
( ( ord_less_eq @ nat @ N2 @ ( one_one @ nat ) )
=> ( ( num_of_nat @ N2 )
= one2 ) ) ).
% num_of_nat_One
thf(fact_4034_prod_Odistrib_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [I6: set @ B,G: B > A,H2: B > A] :
( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X4: B] :
( ( member @ B @ X4 @ I6 )
& ( ( G @ X4 )
!= ( one_one @ A ) ) ) ) )
=> ( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X4: B] :
( ( member @ B @ X4 @ I6 )
& ( ( H2 @ X4 )
!= ( one_one @ A ) ) ) ) )
=> ( ( groups1962203154675924110t_prod @ B @ A
@ ^ [I3: B] : ( times_times @ A @ ( G @ I3 ) @ ( H2 @ I3 ) )
@ I6 )
= ( times_times @ A @ ( groups1962203154675924110t_prod @ B @ A @ G @ I6 ) @ ( groups1962203154675924110t_prod @ B @ A @ H2 @ I6 ) ) ) ) ) ) ).
% prod.distrib'
thf(fact_4035_prod_OG__def,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ( ( groups1962203154675924110t_prod @ B @ A )
= ( ^ [P6: B > A,I8: set @ B] :
( if @ A
@ ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X4: B] :
( ( member @ B @ X4 @ I8 )
& ( ( P6 @ X4 )
!= ( one_one @ A ) ) ) ) )
@ ( groups7121269368397514597t_prod @ B @ A @ P6
@ ( collect @ B
@ ^ [X4: B] :
( ( member @ B @ X4 @ I8 )
& ( ( P6 @ X4 )
!= ( one_one @ A ) ) ) ) )
@ ( one_one @ A ) ) ) ) ) ).
% prod.G_def
thf(fact_4036_numeral__num__of__nat__unfold,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [N2: nat] :
( ( ( N2
= ( zero_zero @ nat ) )
=> ( ( numeral_numeral @ A @ ( num_of_nat @ N2 ) )
= ( one_one @ A ) ) )
& ( ( N2
!= ( zero_zero @ nat ) )
=> ( ( numeral_numeral @ A @ ( num_of_nat @ N2 ) )
= ( semiring_1_of_nat @ A @ N2 ) ) ) ) ) ).
% numeral_num_of_nat_unfold
thf(fact_4037_num__of__nat__double,axiom,
! [N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( num_of_nat @ ( plus_plus @ nat @ N2 @ N2 ) )
= ( bit0 @ ( num_of_nat @ N2 ) ) ) ) ).
% num_of_nat_double
thf(fact_4038_num__of__nat__plus__distrib,axiom,
! [M2: nat,N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ M2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( num_of_nat @ ( plus_plus @ nat @ M2 @ N2 ) )
= ( plus_plus @ num @ ( num_of_nat @ M2 ) @ ( num_of_nat @ N2 ) ) ) ) ) ).
% num_of_nat_plus_distrib
thf(fact_4039_nth__sorted__list__of__set__greaterThanLessThan,axiom,
! [N2: nat,J: nat,I: nat] :
( ( ord_less @ nat @ N2 @ ( minus_minus @ nat @ J @ ( suc @ I ) ) )
=> ( ( nth @ nat @ ( linord4507533701916653071of_set @ nat @ ( set_or5935395276787703475ssThan @ nat @ I @ J ) ) @ N2 )
= ( suc @ ( plus_plus @ nat @ I @ N2 ) ) ) ) ).
% nth_sorted_list_of_set_greaterThanLessThan
thf(fact_4040_listset_Osimps_I1_J,axiom,
! [A: $tType] :
( ( listset @ A @ ( nil @ ( set @ A ) ) )
= ( insert2 @ ( list @ A ) @ ( nil @ A ) @ ( bot_bot @ ( set @ ( list @ A ) ) ) ) ) ).
% listset.simps(1)
thf(fact_4041_less__eq__int_Orep__eq,axiom,
( ( ord_less_eq @ int )
= ( ^ [X4: int,Xa4: int] :
( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [Y6: nat,Z6: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U2: nat,V5: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ Y6 @ V5 ) @ ( plus_plus @ nat @ U2 @ Z6 ) ) )
@ ( rep_Integ @ X4 )
@ ( rep_Integ @ Xa4 ) ) ) ) ).
% less_eq_int.rep_eq
thf(fact_4042_less__int_Orep__eq,axiom,
( ( ord_less @ int )
= ( ^ [X4: int,Xa4: int] :
( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [Y6: nat,Z6: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U2: nat,V5: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ Y6 @ V5 ) @ ( plus_plus @ nat @ U2 @ Z6 ) ) )
@ ( rep_Integ @ X4 )
@ ( rep_Integ @ Xa4 ) ) ) ) ).
% less_int.rep_eq
thf(fact_4043_sorted__list__of__set_Osorted__key__list__of__set__insert__remove,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( linord4507533701916653071of_set @ A @ ( insert2 @ A @ X @ A4 ) )
= ( linorder_insort_key @ A @ A
@ ^ [X4: A] : X4
@ X
@ ( linord4507533701916653071of_set @ A @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_insert_remove
thf(fact_4044_remove1__insort__key,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [X: B,F2: B > A,Xs: list @ B] :
( ( remove1 @ B @ X @ ( linorder_insort_key @ B @ A @ F2 @ X @ Xs ) )
= Xs ) ) ).
% remove1_insort_key
thf(fact_4045_length__insort,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F2: B > A,X: B,Xs: list @ B] :
( ( size_size @ ( list @ B ) @ ( linorder_insort_key @ B @ A @ F2 @ X @ Xs ) )
= ( suc @ ( size_size @ ( list @ B ) @ Xs ) ) ) ) ).
% length_insort
thf(fact_4046_sorted__list__of__set_Osorted__key__list__of__set__insert,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ~ ( member @ A @ X @ A4 )
=> ( ( linord4507533701916653071of_set @ A @ ( insert2 @ A @ X @ A4 ) )
= ( linorder_insort_key @ A @ A
@ ^ [X4: A] : X4
@ X
@ ( linord4507533701916653071of_set @ A @ A4 ) ) ) ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_insert
thf(fact_4047_insort__not__Nil,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F2: B > A,A3: B,Xs: list @ B] :
( ( linorder_insort_key @ B @ A @ F2 @ A3 @ Xs )
!= ( nil @ B ) ) ) ).
% insort_not_Nil
thf(fact_4048_insort__left__comm,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A,Xs: list @ A] :
( ( linorder_insort_key @ A @ A
@ ^ [X4: A] : X4
@ X
@ ( linorder_insort_key @ A @ A
@ ^ [X4: A] : X4
@ Y
@ Xs ) )
= ( linorder_insort_key @ A @ A
@ ^ [X4: A] : X4
@ Y
@ ( linorder_insort_key @ A @ A
@ ^ [X4: A] : X4
@ X
@ Xs ) ) ) ) ).
% insort_left_comm
thf(fact_4049_insort__key__left__comm,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F2: B > A,X: B,Y: B,Xs: list @ B] :
( ( ( F2 @ X )
!= ( F2 @ Y ) )
=> ( ( linorder_insort_key @ B @ A @ F2 @ Y @ ( linorder_insort_key @ B @ A @ F2 @ X @ Xs ) )
= ( linorder_insort_key @ B @ A @ F2 @ X @ ( linorder_insort_key @ B @ A @ F2 @ Y @ Xs ) ) ) ) ) ).
% insort_key_left_comm
thf(fact_4050_set__insort__key,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F2: B > A,X: B,Xs: list @ B] :
( ( set2 @ B @ ( linorder_insort_key @ B @ A @ F2 @ X @ Xs ) )
= ( insert2 @ B @ X @ ( set2 @ B @ Xs ) ) ) ) ).
% set_insort_key
thf(fact_4051_distinct__insort,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F2: B > A,X: B,Xs: list @ B] :
( ( distinct @ B @ ( linorder_insort_key @ B @ A @ F2 @ X @ Xs ) )
= ( ~ ( member @ B @ X @ ( set2 @ B @ Xs ) )
& ( distinct @ B @ Xs ) ) ) ) ).
% distinct_insort
thf(fact_4052_nat_Orep__eq,axiom,
( nat2
= ( ^ [X4: int] : ( product_case_prod @ nat @ nat @ nat @ ( minus_minus @ nat ) @ ( rep_Integ @ X4 ) ) ) ) ).
% nat.rep_eq
thf(fact_4053_sorted__list__of__set_Ofold__insort__key_Oremove,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( member @ A @ X @ A4 )
=> ( ( linord4507533701916653071of_set @ A @ A4 )
= ( linorder_insort_key @ A @ A
@ ^ [X4: A] : X4
@ X
@ ( linord4507533701916653071of_set @ A @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ).
% sorted_list_of_set.fold_insort_key.remove
thf(fact_4054_of__int_Orep__eq,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ( ( ring_1_of_int @ A )
= ( ^ [X4: int] :
( product_case_prod @ nat @ nat @ A
@ ^ [I3: nat,J2: nat] : ( minus_minus @ A @ ( semiring_1_of_nat @ A @ I3 ) @ ( semiring_1_of_nat @ A @ J2 ) )
@ ( rep_Integ @ X4 ) ) ) ) ) ).
% of_int.rep_eq
thf(fact_4055_nth__sorted__list__of__set__greaterThanAtMost,axiom,
! [N2: nat,J: nat,I: nat] :
( ( ord_less @ nat @ N2 @ ( minus_minus @ nat @ J @ I ) )
=> ( ( nth @ nat @ ( linord4507533701916653071of_set @ nat @ ( set_or3652927894154168847AtMost @ nat @ I @ J ) ) @ N2 )
= ( suc @ ( plus_plus @ nat @ I @ N2 ) ) ) ) ).
% nth_sorted_list_of_set_greaterThanAtMost
thf(fact_4056_uminus__int__def,axiom,
( ( uminus_uminus @ int )
= ( map_fun @ int @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ int @ rep_Integ @ abs_Integ
@ ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [X4: nat,Y6: nat] : ( product_Pair @ nat @ nat @ Y6 @ X4 ) ) ) ) ).
% uminus_int_def
thf(fact_4057_prod__encode__def,axiom,
( nat_prod_encode
= ( product_case_prod @ nat @ nat @ nat
@ ^ [M: nat,N: nat] : ( plus_plus @ nat @ ( nat_triangle @ ( plus_plus @ nat @ M @ N ) ) @ M ) ) ) ).
% prod_encode_def
thf(fact_4058_sorted__list__of__set__def,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( linord4507533701916653071of_set @ A )
= ( linord144544945434240204of_set @ A @ A
@ ^ [X4: A] : X4 ) ) ) ).
% sorted_list_of_set_def
thf(fact_4059_finite__greaterThanAtMost,axiom,
! [L: nat,U: nat] : ( finite_finite2 @ nat @ ( set_or3652927894154168847AtMost @ nat @ L @ U ) ) ).
% finite_greaterThanAtMost
thf(fact_4060_greaterThanAtMost__iff,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [I: A,L: A,U: A] :
( ( member @ A @ I @ ( set_or3652927894154168847AtMost @ A @ L @ U ) )
= ( ( ord_less @ A @ L @ I )
& ( ord_less_eq @ A @ I @ U ) ) ) ) ).
% greaterThanAtMost_iff
thf(fact_4061_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_4062_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_4063_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_4064_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_4065_Sup__greaterThanAtMost,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ( ( complete_Sup_Sup @ A @ ( set_or3652927894154168847AtMost @ A @ X @ Y ) )
= Y ) ) ) ).
% Sup_greaterThanAtMost
thf(fact_4066_cSup__greaterThanAtMost,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [Y: A,X: A] :
( ( ord_less @ A @ Y @ X )
=> ( ( complete_Sup_Sup @ A @ ( set_or3652927894154168847AtMost @ A @ Y @ X ) )
= X ) ) ) ).
% cSup_greaterThanAtMost
thf(fact_4067_cInf__greaterThanAtMost,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( dense_linorder @ A ) )
=> ! [Y: A,X: A] :
( ( ord_less @ A @ Y @ X )
=> ( ( complete_Inf_Inf @ A @ ( set_or3652927894154168847AtMost @ A @ Y @ X ) )
= Y ) ) ) ).
% cInf_greaterThanAtMost
thf(fact_4068_Inf__greaterThanAtMost,axiom,
! [A: $tType] :
( ( ( comple6319245703460814977attice @ A )
& ( dense_linorder @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ( ( complete_Inf_Inf @ A @ ( set_or3652927894154168847AtMost @ A @ X @ Y ) )
= X ) ) ) ).
% Inf_greaterThanAtMost
thf(fact_4069_card__greaterThanAtMost,axiom,
! [L: nat,U: nat] :
( ( finite_card @ nat @ ( set_or3652927894154168847AtMost @ nat @ L @ U ) )
= ( minus_minus @ nat @ U @ L ) ) ).
% card_greaterThanAtMost
thf(fact_4070_Ioc__inj,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B3: A,C2: A,D2: A] :
( ( ( set_or3652927894154168847AtMost @ A @ A3 @ B3 )
= ( set_or3652927894154168847AtMost @ A @ C2 @ D2 ) )
= ( ( ( ord_less_eq @ A @ B3 @ A3 )
& ( ord_less_eq @ A @ D2 @ C2 ) )
| ( ( A3 = C2 )
& ( B3 = D2 ) ) ) ) ) ).
% Ioc_inj
thf(fact_4071_atLeastSucAtMost__greaterThanAtMost,axiom,
! [L: nat,U: nat] :
( ( set_or1337092689740270186AtMost @ nat @ ( suc @ L ) @ U )
= ( set_or3652927894154168847AtMost @ nat @ L @ U ) ) ).
% atLeastSucAtMost_greaterThanAtMost
thf(fact_4072_Ioc__subset__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B3: A,C2: A,D2: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ A3 @ B3 ) @ ( set_or3652927894154168847AtMost @ A @ C2 @ D2 ) )
= ( ( ord_less_eq @ A @ B3 @ A3 )
| ( ( ord_less_eq @ A @ C2 @ A3 )
& ( ord_less_eq @ A @ B3 @ D2 ) ) ) ) ) ).
% Ioc_subset_iff
thf(fact_4073_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_4074_ivl__disj__int__two_I6_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L: A,M2: A,U: A] :
( ( inf_inf @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ L @ M2 ) @ ( set_or3652927894154168847AtMost @ A @ M2 @ U ) )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% ivl_disj_int_two(6)
thf(fact_4075_Ioc__disjoint,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B3: A,C2: A,D2: A] :
( ( ( inf_inf @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ A3 @ B3 ) @ ( set_or3652927894154168847AtMost @ A @ C2 @ D2 ) )
= ( bot_bot @ ( set @ A ) ) )
= ( ( ord_less_eq @ A @ B3 @ A3 )
| ( ord_less_eq @ A @ D2 @ C2 )
| ( ord_less_eq @ A @ B3 @ C2 )
| ( ord_less_eq @ A @ D2 @ A3 ) ) ) ) ).
% Ioc_disjoint
thf(fact_4076_ivl__disj__int__two_I8_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L: A,M2: A,U: A] :
( ( inf_inf @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ L @ M2 ) @ ( set_or3652927894154168847AtMost @ A @ M2 @ U ) )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% ivl_disj_int_two(8)
thf(fact_4077_ivl__disj__int__one_I3_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L: A,U: A] :
( ( inf_inf @ ( set @ A ) @ ( set_ord_atMost @ A @ L ) @ ( set_or3652927894154168847AtMost @ A @ L @ U ) )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% ivl_disj_int_one(3)
thf(fact_4078_ivl__disj__int__two_I2_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L: A,M2: A,U: A] :
( ( inf_inf @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ L @ M2 ) @ ( set_or5935395276787703475ssThan @ A @ M2 @ U ) )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% ivl_disj_int_two(2)
thf(fact_4079_le__prod__encode__1,axiom,
! [A3: nat,B3: nat] : ( ord_less_eq @ nat @ A3 @ ( nat_prod_encode @ ( product_Pair @ nat @ nat @ A3 @ B3 ) ) ) ).
% le_prod_encode_1
thf(fact_4080_le__prod__encode__2,axiom,
! [B3: nat,A3: nat] : ( ord_less_eq @ nat @ B3 @ ( nat_prod_encode @ ( product_Pair @ nat @ nat @ A3 @ B3 ) ) ) ).
% le_prod_encode_2
thf(fact_4081_sum_Ohead,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [M2: nat,N2: nat,G: nat > A] :
( ( ord_less_eq @ nat @ M2 @ N2 )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N2 ) )
= ( plus_plus @ A @ ( G @ M2 ) @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or3652927894154168847AtMost @ nat @ M2 @ N2 ) ) ) ) ) ) ).
% sum.head
thf(fact_4082_prod_Ohead,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M2: nat,N2: nat,G: nat > A] :
( ( ord_less_eq @ nat @ M2 @ N2 )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N2 ) )
= ( times_times @ A @ ( G @ M2 ) @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or3652927894154168847AtMost @ nat @ M2 @ N2 ) ) ) ) ) ) ).
% prod.head
thf(fact_4083_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_4084_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_4085_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_4086_greaterThanAtMost__eq__atLeastAtMost__diff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( set_or3652927894154168847AtMost @ A )
= ( ^ [A5: A,B4: A] : ( minus_minus @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A5 @ B4 ) @ ( insert2 @ A @ A5 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% greaterThanAtMost_eq_atLeastAtMost_diff
thf(fact_4087_prod__encode__prod__decode__aux,axiom,
! [K: nat,M2: nat] :
( ( nat_prod_encode @ ( nat_prod_decode_aux @ K @ M2 ) )
= ( plus_plus @ nat @ ( nat_triangle @ K ) @ M2 ) ) ).
% prod_encode_prod_decode_aux
thf(fact_4088_times__int__def,axiom,
( ( times_times @ int )
= ( map_fun @ int @ ( product_prod @ nat @ nat ) @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) ) @ ( int > int ) @ rep_Integ @ ( map_fun @ int @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ int @ rep_Integ @ abs_Integ )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
@ ^ [X4: nat,Y6: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V5: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ X4 @ U2 ) @ ( times_times @ nat @ Y6 @ V5 ) ) @ ( plus_plus @ nat @ ( times_times @ nat @ X4 @ V5 ) @ ( times_times @ nat @ Y6 @ U2 ) ) ) ) ) ) ) ).
% times_int_def
thf(fact_4089_minus__int__def,axiom,
( ( minus_minus @ int )
= ( map_fun @ int @ ( product_prod @ nat @ nat ) @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) ) @ ( int > int ) @ rep_Integ @ ( map_fun @ int @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ int @ rep_Integ @ abs_Integ )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
@ ^ [X4: nat,Y6: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V5: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X4 @ V5 ) @ ( plus_plus @ nat @ Y6 @ U2 ) ) ) ) ) ) ).
% minus_int_def
thf(fact_4090_plus__int__def,axiom,
( ( plus_plus @ int )
= ( map_fun @ int @ ( product_prod @ nat @ nat ) @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) ) @ ( int > int ) @ rep_Integ @ ( map_fun @ int @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ int @ rep_Integ @ abs_Integ )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
@ ^ [X4: nat,Y6: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V5: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X4 @ U2 ) @ ( plus_plus @ nat @ Y6 @ V5 ) ) ) ) ) ) ).
% plus_int_def
thf(fact_4091_image__minus__const__atLeastLessThan__nat,axiom,
! [C2: nat,Y: nat,X: nat] :
( ( ( ord_less @ nat @ C2 @ Y )
=> ( ( image2 @ nat @ nat
@ ^ [I3: nat] : ( minus_minus @ nat @ I3 @ C2 )
@ ( set_or7035219750837199246ssThan @ nat @ X @ Y ) )
= ( set_or7035219750837199246ssThan @ nat @ ( minus_minus @ nat @ X @ C2 ) @ ( minus_minus @ nat @ Y @ C2 ) ) ) )
& ( ~ ( ord_less @ nat @ C2 @ Y )
=> ( ( ( ord_less @ nat @ X @ Y )
=> ( ( image2 @ nat @ nat
@ ^ [I3: nat] : ( minus_minus @ nat @ I3 @ C2 )
@ ( set_or7035219750837199246ssThan @ nat @ X @ Y ) )
= ( insert2 @ nat @ ( zero_zero @ nat ) @ ( bot_bot @ ( set @ nat ) ) ) ) )
& ( ~ ( ord_less @ nat @ X @ Y )
=> ( ( image2 @ nat @ nat
@ ^ [I3: nat] : ( minus_minus @ nat @ I3 @ C2 )
@ ( set_or7035219750837199246ssThan @ nat @ X @ Y ) )
= ( bot_bot @ ( set @ nat ) ) ) ) ) ) ) ).
% image_minus_const_atLeastLessThan_nat
thf(fact_4092_image__eqI,axiom,
! [A: $tType,B: $tType,B3: A,F2: B > A,X: B,A4: set @ B] :
( ( B3
= ( F2 @ X ) )
=> ( ( member @ B @ X @ A4 )
=> ( member @ A @ B3 @ ( image2 @ B @ A @ F2 @ A4 ) ) ) ) ).
% image_eqI
thf(fact_4093_finite__greaterThanAtMost__int,axiom,
! [L: int,U: int] : ( finite_finite2 @ int @ ( set_or3652927894154168847AtMost @ int @ L @ U ) ) ).
% finite_greaterThanAtMost_int
thf(fact_4094_image__ident,axiom,
! [A: $tType,Y7: set @ A] :
( ( image2 @ A @ A
@ ^ [X4: A] : X4
@ Y7 )
= Y7 ) ).
% image_ident
thf(fact_4095_image__empty,axiom,
! [B: $tType,A: $tType,F2: B > A] :
( ( image2 @ B @ A @ F2 @ ( bot_bot @ ( set @ B ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% image_empty
thf(fact_4096_empty__is__image,axiom,
! [A: $tType,B: $tType,F2: B > A,A4: set @ B] :
( ( ( bot_bot @ ( set @ A ) )
= ( image2 @ B @ A @ F2 @ A4 ) )
= ( A4
= ( bot_bot @ ( set @ B ) ) ) ) ).
% empty_is_image
thf(fact_4097_image__is__empty,axiom,
! [A: $tType,B: $tType,F2: B > A,A4: set @ B] :
( ( ( image2 @ B @ A @ F2 @ A4 )
= ( bot_bot @ ( set @ A ) ) )
= ( A4
= ( bot_bot @ ( set @ B ) ) ) ) ).
% image_is_empty
thf(fact_4098_finite__imageI,axiom,
! [B: $tType,A: $tType,F5: set @ A,H2: A > B] :
( ( finite_finite2 @ A @ F5 )
=> ( finite_finite2 @ B @ ( image2 @ A @ B @ H2 @ F5 ) ) ) ).
% finite_imageI
thf(fact_4099_insert__image,axiom,
! [B: $tType,A: $tType,X: A,A4: set @ A,F2: A > B] :
( ( member @ A @ X @ A4 )
=> ( ( insert2 @ B @ ( F2 @ X ) @ ( image2 @ A @ B @ F2 @ A4 ) )
= ( image2 @ A @ B @ F2 @ A4 ) ) ) ).
% insert_image
thf(fact_4100_image__insert,axiom,
! [A: $tType,B: $tType,F2: B > A,A3: B,B2: set @ B] :
( ( image2 @ B @ A @ F2 @ ( insert2 @ B @ A3 @ B2 ) )
= ( insert2 @ A @ ( F2 @ A3 ) @ ( image2 @ B @ A @ F2 @ B2 ) ) ) ).
% image_insert
thf(fact_4101_image__add__0,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [S2: set @ A] :
( ( image2 @ A @ A @ ( plus_plus @ A @ ( zero_zero @ A ) ) @ S2 )
= S2 ) ) ).
% image_add_0
thf(fact_4102_image__add__atLeastAtMost,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [K: A,I: A,J: A] :
( ( image2 @ A @ A @ ( plus_plus @ A @ K ) @ ( set_or1337092689740270186AtMost @ A @ I @ J ) )
= ( set_or1337092689740270186AtMost @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ K ) ) ) ) ).
% image_add_atLeastAtMost
thf(fact_4103_image__diff__atLeastAtMost,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [D2: A,A3: A,B3: A] :
( ( image2 @ A @ A @ ( minus_minus @ A @ D2 ) @ ( set_or1337092689740270186AtMost @ A @ A3 @ B3 ) )
= ( set_or1337092689740270186AtMost @ A @ ( minus_minus @ A @ D2 @ B3 ) @ ( minus_minus @ A @ D2 @ A3 ) ) ) ) ).
% image_diff_atLeastAtMost
thf(fact_4104_image__uminus__atLeastAtMost,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [X: A,Y: A] :
( ( image2 @ A @ A @ ( uminus_uminus @ A ) @ ( set_or1337092689740270186AtMost @ A @ X @ Y ) )
= ( set_or1337092689740270186AtMost @ A @ ( uminus_uminus @ A @ Y ) @ ( uminus_uminus @ A @ X ) ) ) ) ).
% image_uminus_atLeastAtMost
thf(fact_4105_image__add__atLeastLessThan,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [K: A,I: A,J: A] :
( ( image2 @ A @ A @ ( plus_plus @ A @ K ) @ ( set_or7035219750837199246ssThan @ A @ I @ J ) )
= ( set_or7035219750837199246ssThan @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ K ) ) ) ) ).
% image_add_atLeastLessThan
thf(fact_4106_image__add__atMost,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [C2: A,A3: A] :
( ( image2 @ A @ A @ ( plus_plus @ A @ C2 ) @ ( set_ord_atMost @ A @ A3 ) )
= ( set_ord_atMost @ A @ ( plus_plus @ A @ C2 @ A3 ) ) ) ) ).
% image_add_atMost
thf(fact_4107_image__add__greaterThanAtMost,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [C2: A,A3: A,B3: A] :
( ( image2 @ A @ A @ ( plus_plus @ A @ C2 ) @ ( set_or3652927894154168847AtMost @ A @ A3 @ B3 ) )
= ( set_or3652927894154168847AtMost @ A @ ( plus_plus @ A @ C2 @ A3 ) @ ( plus_plus @ A @ C2 @ B3 ) ) ) ) ).
% image_add_greaterThanAtMost
thf(fact_4108_image__uminus__greaterThanLessThan,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [X: A,Y: A] :
( ( image2 @ A @ A @ ( uminus_uminus @ A ) @ ( set_or5935395276787703475ssThan @ A @ X @ Y ) )
= ( set_or5935395276787703475ssThan @ A @ ( uminus_uminus @ A @ Y ) @ ( uminus_uminus @ A @ X ) ) ) ) ).
% image_uminus_greaterThanLessThan
thf(fact_4109_image__Suc__atLeastAtMost,axiom,
! [I: nat,J: nat] :
( ( image2 @ nat @ nat @ suc @ ( set_or1337092689740270186AtMost @ nat @ I @ J ) )
= ( set_or1337092689740270186AtMost @ nat @ ( suc @ I ) @ ( suc @ J ) ) ) ).
% image_Suc_atLeastAtMost
thf(fact_4110_image__Suc__atLeastLessThan,axiom,
! [I: nat,J: nat] :
( ( image2 @ nat @ nat @ suc @ ( set_or7035219750837199246ssThan @ nat @ I @ J ) )
= ( set_or7035219750837199246ssThan @ nat @ ( suc @ I ) @ ( suc @ J ) ) ) ).
% image_Suc_atLeastLessThan
thf(fact_4111_ccSUP__bot,axiom,
! [B: $tType,A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A4: set @ B] :
( ( complete_Sup_Sup @ A
@ ( image2 @ B @ A
@ ^ [X4: B] : ( bot_bot @ A )
@ A4 ) )
= ( bot_bot @ A ) ) ) ).
% ccSUP_bot
thf(fact_4112_SUP__bot,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A4: set @ B] :
( ( complete_Sup_Sup @ A
@ ( image2 @ B @ A
@ ^ [X4: B] : ( bot_bot @ A )
@ A4 ) )
= ( bot_bot @ A ) ) ) ).
% SUP_bot
thf(fact_4113_SUP__bot__conv_I1_J,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [B2: B > A,A4: set @ B] :
( ( ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ B2 @ A4 ) )
= ( bot_bot @ A ) )
= ( ! [X4: B] :
( ( member @ B @ X4 @ A4 )
=> ( ( B2 @ X4 )
= ( bot_bot @ A ) ) ) ) ) ) ).
% SUP_bot_conv(1)
thf(fact_4114_SUP__bot__conv_I2_J,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [B2: B > A,A4: set @ B] :
( ( ( bot_bot @ A )
= ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ B2 @ A4 ) ) )
= ( ! [X4: B] :
( ( member @ B @ X4 @ A4 )
=> ( ( B2 @ X4 )
= ( bot_bot @ A ) ) ) ) ) ) ).
% SUP_bot_conv(2)
thf(fact_4115_ccSUP__const,axiom,
! [B: $tType,A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A4: set @ B,F2: A] :
( ( A4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Sup_Sup @ A
@ ( image2 @ B @ A
@ ^ [I3: B] : F2
@ A4 ) )
= F2 ) ) ) ).
% ccSUP_const
thf(fact_4116_SUP__const,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A4: set @ B,F2: A] :
( ( A4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Sup_Sup @ A
@ ( image2 @ B @ A
@ ^ [I3: B] : F2
@ A4 ) )
= F2 ) ) ) ).
% SUP_const
thf(fact_4117_cSUP__const,axiom,
! [B: $tType,A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A4: set @ B,C2: A] :
( ( A4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Sup_Sup @ A
@ ( image2 @ B @ A
@ ^ [X4: B] : C2
@ A4 ) )
= C2 ) ) ) ).
% cSUP_const
thf(fact_4118_image__add__atLeastAtMost_H,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [K: A,I: A,J: A] :
( ( image2 @ A @ A
@ ^ [N: A] : ( plus_plus @ A @ N @ K )
@ ( set_or1337092689740270186AtMost @ A @ I @ J ) )
= ( set_or1337092689740270186AtMost @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ K ) ) ) ) ).
% image_add_atLeastAtMost'
thf(fact_4119_ccINF__const,axiom,
! [B: $tType,A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A4: set @ B,F2: A] :
( ( A4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Inf_Inf @ A
@ ( image2 @ B @ A
@ ^ [I3: B] : F2
@ A4 ) )
= F2 ) ) ) ).
% ccINF_const
thf(fact_4120_INF__const,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A4: set @ B,F2: A] :
( ( A4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Inf_Inf @ A
@ ( image2 @ B @ A
@ ^ [I3: B] : F2
@ A4 ) )
= F2 ) ) ) ).
% INF_const
thf(fact_4121_cINF__const,axiom,
! [B: $tType,A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A4: set @ B,C2: A] :
( ( A4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Inf_Inf @ A
@ ( image2 @ B @ A
@ ^ [X4: B] : C2
@ A4 ) )
= C2 ) ) ) ).
% cINF_const
thf(fact_4122_image__minus__const__atLeastAtMost_H,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [D2: A,A3: A,B3: A] :
( ( image2 @ A @ A
@ ^ [T3: A] : ( minus_minus @ A @ T3 @ D2 )
@ ( set_or1337092689740270186AtMost @ A @ A3 @ B3 ) )
= ( set_or1337092689740270186AtMost @ A @ ( minus_minus @ A @ A3 @ D2 ) @ ( minus_minus @ A @ B3 @ D2 ) ) ) ) ).
% image_minus_const_atLeastAtMost'
thf(fact_4123_image__add__atLeastLessThan_H,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [K: A,I: A,J: A] :
( ( image2 @ A @ A
@ ^ [N: A] : ( plus_plus @ A @ N @ K )
@ ( set_or7035219750837199246ssThan @ A @ I @ J ) )
= ( set_or7035219750837199246ssThan @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ K ) ) ) ) ).
% image_add_atLeastLessThan'
thf(fact_4124_card__greaterThanAtMost__int,axiom,
! [L: int,U: int] :
( ( finite_card @ int @ ( set_or3652927894154168847AtMost @ int @ L @ U ) )
= ( nat2 @ ( minus_minus @ int @ U @ L ) ) ) ).
% card_greaterThanAtMost_int
thf(fact_4125_image__diff__atLeastLessThan,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [C2: A,A3: A,B3: A] :
( ( image2 @ A @ A @ ( minus_minus @ A @ C2 ) @ ( set_or7035219750837199246ssThan @ A @ A3 @ B3 ) )
= ( set_or3652927894154168847AtMost @ A @ ( minus_minus @ A @ C2 @ B3 ) @ ( minus_minus @ A @ C2 @ A3 ) ) ) ) ).
% image_diff_atLeastLessThan
thf(fact_4126_image__minus__const__greaterThanAtMost,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [C2: A,A3: A,B3: A] :
( ( image2 @ A @ A @ ( minus_minus @ A @ C2 ) @ ( set_or3652927894154168847AtMost @ A @ A3 @ B3 ) )
= ( set_or7035219750837199246ssThan @ A @ ( minus_minus @ A @ C2 @ B3 ) @ ( minus_minus @ A @ C2 @ A3 ) ) ) ) ).
% image_minus_const_greaterThanAtMost
thf(fact_4127_image__uminus__greaterThanAtMost,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [X: A,Y: A] :
( ( image2 @ A @ A @ ( uminus_uminus @ A ) @ ( set_or3652927894154168847AtMost @ A @ X @ Y ) )
= ( set_or7035219750837199246ssThan @ A @ ( uminus_uminus @ A @ Y ) @ ( uminus_uminus @ A @ X ) ) ) ) ).
% image_uminus_greaterThanAtMost
thf(fact_4128_image__uminus__atLeastLessThan,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [X: A,Y: A] :
( ( image2 @ A @ A @ ( uminus_uminus @ A ) @ ( set_or7035219750837199246ssThan @ A @ X @ Y ) )
= ( set_or3652927894154168847AtMost @ A @ ( uminus_uminus @ A @ Y ) @ ( uminus_uminus @ A @ X ) ) ) ) ).
% image_uminus_atLeastLessThan
thf(fact_4129_INF__eq__bot__iff,axiom,
! [B: $tType,A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [F2: B > A,A4: set @ B] :
( ( ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F2 @ A4 ) )
= ( bot_bot @ A ) )
= ( ! [X4: A] :
( ( ord_less @ A @ ( bot_bot @ A ) @ X4 )
=> ? [Y6: B] :
( ( member @ B @ Y6 @ A4 )
& ( ord_less @ A @ ( F2 @ Y6 ) @ X4 ) ) ) ) ) ) ).
% INF_eq_bot_iff
thf(fact_4130_ccSUP__empty,axiom,
! [B: $tType,A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [F2: B > A] :
( ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F2 @ ( bot_bot @ ( set @ B ) ) ) )
= ( bot_bot @ A ) ) ) ).
% ccSUP_empty
thf(fact_4131_image__mult__atLeastAtMost,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [D2: A,A3: A,B3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ D2 )
=> ( ( image2 @ 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_4132_image__divide__atLeastAtMost,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [D2: A,A3: A,B3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ D2 )
=> ( ( image2 @ A @ A
@ ^ [C4: A] : ( divide_divide @ A @ C4 @ D2 )
@ ( set_or1337092689740270186AtMost @ A @ A3 @ B3 ) )
= ( set_or1337092689740270186AtMost @ A @ ( divide_divide @ A @ A3 @ D2 ) @ ( divide_divide @ A @ B3 @ D2 ) ) ) ) ) ).
% image_divide_atLeastAtMost
thf(fact_4133_image__Pow__surj,axiom,
! [B: $tType,A: $tType,F2: B > A,A4: set @ B,B2: set @ A] :
( ( ( image2 @ B @ A @ F2 @ A4 )
= B2 )
=> ( ( image2 @ ( set @ B ) @ ( set @ A ) @ ( image2 @ B @ A @ F2 ) @ ( pow2 @ B @ A4 ) )
= ( pow2 @ A @ B2 ) ) ) ).
% image_Pow_surj
thf(fact_4134_all__subset__image,axiom,
! [A: $tType,B: $tType,F2: B > A,A4: set @ B,P: ( set @ A ) > $o] :
( ( ! [B6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ B6 @ ( image2 @ B @ A @ F2 @ A4 ) )
=> ( P @ B6 ) ) )
= ( ! [B6: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ B6 @ A4 )
=> ( P @ ( image2 @ B @ A @ F2 @ B6 ) ) ) ) ) ).
% all_subset_image
thf(fact_4135_image__mono,axiom,
! [B: $tType,A: $tType,A4: set @ A,B2: set @ A,F2: A > B] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B2 )
=> ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F2 @ A4 ) @ ( image2 @ A @ B @ F2 @ B2 ) ) ) ).
% image_mono
thf(fact_4136_image__subsetI,axiom,
! [A: $tType,B: $tType,A4: set @ A,F2: A > B,B2: set @ B] :
( ! [X5: A] :
( ( member @ A @ X5 @ A4 )
=> ( member @ B @ ( F2 @ X5 ) @ B2 ) )
=> ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F2 @ A4 ) @ B2 ) ) ).
% image_subsetI
thf(fact_4137_subset__imageE,axiom,
! [A: $tType,B: $tType,B2: set @ A,F2: B > A,A4: set @ B] :
( ( ord_less_eq @ ( set @ A ) @ B2 @ ( image2 @ B @ A @ F2 @ A4 ) )
=> ~ ! [C7: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ C7 @ A4 )
=> ( B2
!= ( image2 @ B @ A @ F2 @ C7 ) ) ) ) ).
% subset_imageE
thf(fact_4138_image__subset__iff,axiom,
! [A: $tType,B: $tType,F2: B > A,A4: set @ B,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( image2 @ B @ A @ F2 @ A4 ) @ B2 )
= ( ! [X4: B] :
( ( member @ B @ X4 @ A4 )
=> ( member @ A @ ( F2 @ X4 ) @ B2 ) ) ) ) ).
% image_subset_iff
thf(fact_4139_subset__image__iff,axiom,
! [A: $tType,B: $tType,B2: set @ A,F2: B > A,A4: set @ B] :
( ( ord_less_eq @ ( set @ A ) @ B2 @ ( image2 @ B @ A @ F2 @ A4 ) )
= ( ? [AA: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ AA @ A4 )
& ( B2
= ( image2 @ B @ A @ F2 @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_4140_imageE,axiom,
! [A: $tType,B: $tType,B3: A,F2: B > A,A4: set @ B] :
( ( member @ A @ B3 @ ( image2 @ B @ A @ F2 @ A4 ) )
=> ~ ! [X5: B] :
( ( B3
= ( F2 @ X5 ) )
=> ~ ( member @ B @ X5 @ A4 ) ) ) ).
% imageE
thf(fact_4141_image__image,axiom,
! [A: $tType,B: $tType,C: $tType,F2: B > A,G: C > B,A4: set @ C] :
( ( image2 @ B @ A @ F2 @ ( image2 @ C @ B @ G @ A4 ) )
= ( image2 @ C @ A
@ ^ [X4: C] : ( F2 @ ( G @ X4 ) )
@ A4 ) ) ).
% image_image
thf(fact_4142_Compr__image__eq,axiom,
! [A: $tType,B: $tType,F2: B > A,A4: set @ B,P: A > $o] :
( ( collect @ A
@ ^ [X4: A] :
( ( member @ A @ X4 @ ( image2 @ B @ A @ F2 @ A4 ) )
& ( P @ X4 ) ) )
= ( image2 @ B @ A @ F2
@ ( collect @ B
@ ^ [X4: B] :
( ( member @ B @ X4 @ A4 )
& ( P @ ( F2 @ X4 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_4143_rev__image__eqI,axiom,
! [B: $tType,A: $tType,X: A,A4: set @ A,B3: B,F2: A > B] :
( ( member @ A @ X @ A4 )
=> ( ( B3
= ( F2 @ X ) )
=> ( member @ B @ B3 @ ( image2 @ A @ B @ F2 @ A4 ) ) ) ) ).
% rev_image_eqI
thf(fact_4144_ball__imageD,axiom,
! [A: $tType,B: $tType,F2: B > A,A4: set @ B,P: A > $o] :
( ! [X5: A] :
( ( member @ A @ X5 @ ( image2 @ B @ A @ F2 @ A4 ) )
=> ( P @ X5 ) )
=> ! [X2: B] :
( ( member @ B @ X2 @ A4 )
=> ( P @ ( F2 @ X2 ) ) ) ) ).
% ball_imageD
thf(fact_4145_image__cong,axiom,
! [B: $tType,A: $tType,M7: set @ A,N6: set @ A,F2: A > B,G: A > B] :
( ( M7 = N6 )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ N6 )
=> ( ( F2 @ X5 )
= ( G @ X5 ) ) )
=> ( ( image2 @ A @ B @ F2 @ M7 )
= ( image2 @ A @ B @ G @ N6 ) ) ) ) ).
% image_cong
thf(fact_4146_bex__imageD,axiom,
! [A: $tType,B: $tType,F2: B > A,A4: set @ B,P: A > $o] :
( ? [X2: A] :
( ( member @ A @ X2 @ ( image2 @ B @ A @ F2 @ A4 ) )
& ( P @ X2 ) )
=> ? [X5: B] :
( ( member @ B @ X5 @ A4 )
& ( P @ ( F2 @ X5 ) ) ) ) ).
% bex_imageD
thf(fact_4147_image__iff,axiom,
! [A: $tType,B: $tType,Z3: A,F2: B > A,A4: set @ B] :
( ( member @ A @ Z3 @ ( image2 @ B @ A @ F2 @ A4 ) )
= ( ? [X4: B] :
( ( member @ B @ X4 @ A4 )
& ( Z3
= ( F2 @ X4 ) ) ) ) ) ).
% image_iff
thf(fact_4148_imageI,axiom,
! [B: $tType,A: $tType,X: A,A4: set @ A,F2: A > B] :
( ( member @ A @ X @ A4 )
=> ( member @ B @ ( F2 @ X ) @ ( image2 @ A @ B @ F2 @ A4 ) ) ) ).
% imageI
thf(fact_4149_image__Collect__subsetI,axiom,
! [A: $tType,B: $tType,P: A > $o,F2: A > B,B2: set @ B] :
( ! [X5: A] :
( ( P @ X5 )
=> ( member @ B @ ( F2 @ X5 ) @ B2 ) )
=> ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F2 @ ( collect @ A @ P ) ) @ B2 ) ) ).
% image_Collect_subsetI
thf(fact_4150_pigeonhole__infinite,axiom,
! [B: $tType,A: $tType,A4: set @ A,F2: A > B] :
( ~ ( finite_finite2 @ A @ A4 )
=> ( ( finite_finite2 @ B @ ( image2 @ A @ B @ F2 @ A4 ) )
=> ? [X5: A] :
( ( member @ A @ X5 @ A4 )
& ~ ( finite_finite2 @ A
@ ( collect @ A
@ ^ [A5: A] :
( ( member @ A @ A5 @ A4 )
& ( ( F2 @ A5 )
= ( F2 @ X5 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_4151_image__Pow__mono,axiom,
! [B: $tType,A: $tType,F2: B > A,A4: set @ B,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( image2 @ B @ A @ F2 @ A4 ) @ B2 )
=> ( ord_less_eq @ ( set @ ( set @ A ) ) @ ( image2 @ ( set @ B ) @ ( set @ A ) @ ( image2 @ B @ A @ F2 ) @ ( pow2 @ B @ A4 ) ) @ ( pow2 @ A @ B2 ) ) ) ).
% image_Pow_mono
thf(fact_4152_UNION__singleton__eq__range,axiom,
! [A: $tType,B: $tType,F2: B > A,A4: set @ B] :
( ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [X4: B] : ( insert2 @ A @ ( F2 @ X4 ) @ ( bot_bot @ ( set @ A ) ) )
@ A4 ) )
= ( image2 @ B @ A @ F2 @ A4 ) ) ).
% UNION_singleton_eq_range
thf(fact_4153_image__Fpow__mono,axiom,
! [B: $tType,A: $tType,F2: B > A,A4: set @ B,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( image2 @ B @ A @ F2 @ A4 ) @ B2 )
=> ( ord_less_eq @ ( set @ ( set @ A ) ) @ ( image2 @ ( set @ B ) @ ( set @ A ) @ ( image2 @ B @ A @ F2 ) @ ( finite_Fpow @ B @ A4 ) ) @ ( finite_Fpow @ A @ B2 ) ) ) ).
% image_Fpow_mono
thf(fact_4154_SUP__eq,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A4: set @ B,B2: set @ C,F2: B > A,G: C > A] :
( ! [I4: B] :
( ( member @ B @ I4 @ A4 )
=> ? [X2: C] :
( ( member @ C @ X2 @ B2 )
& ( ord_less_eq @ A @ ( F2 @ I4 ) @ ( G @ X2 ) ) ) )
=> ( ! [J3: C] :
( ( member @ C @ J3 @ B2 )
=> ? [X2: B] :
( ( member @ B @ X2 @ A4 )
& ( ord_less_eq @ A @ ( G @ J3 ) @ ( F2 @ X2 ) ) ) )
=> ( ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F2 @ A4 ) )
= ( complete_Sup_Sup @ A @ ( image2 @ C @ A @ G @ B2 ) ) ) ) ) ) ).
% SUP_eq
thf(fact_4155_INF__eq,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A4: set @ B,B2: set @ C,G: C > A,F2: B > A] :
( ! [I4: B] :
( ( member @ B @ I4 @ A4 )
=> ? [X2: C] :
( ( member @ C @ X2 @ B2 )
& ( ord_less_eq @ A @ ( G @ X2 ) @ ( F2 @ I4 ) ) ) )
=> ( ! [J3: C] :
( ( member @ C @ J3 @ B2 )
=> ? [X2: B] :
( ( member @ B @ X2 @ A4 )
& ( ord_less_eq @ A @ ( F2 @ X2 ) @ ( G @ J3 ) ) ) )
=> ( ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F2 @ A4 ) )
= ( complete_Inf_Inf @ A @ ( image2 @ C @ A @ G @ B2 ) ) ) ) ) ) ).
% INF_eq
thf(fact_4156_zero__notin__Suc__image,axiom,
! [A4: set @ nat] :
~ ( member @ nat @ ( zero_zero @ nat ) @ ( image2 @ nat @ nat @ suc @ A4 ) ) ).
% zero_notin_Suc_image
thf(fact_4157_all__finite__subset__image,axiom,
! [A: $tType,B: $tType,F2: B > A,A4: set @ B,P: ( set @ A ) > $o] :
( ( ! [B6: set @ A] :
( ( ( finite_finite2 @ A @ B6 )
& ( ord_less_eq @ ( set @ A ) @ B6 @ ( image2 @ B @ A @ F2 @ A4 ) ) )
=> ( P @ B6 ) ) )
= ( ! [B6: set @ B] :
( ( ( finite_finite2 @ B @ B6 )
& ( ord_less_eq @ ( set @ B ) @ B6 @ A4 ) )
=> ( P @ ( image2 @ B @ A @ F2 @ B6 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_4158_ex__finite__subset__image,axiom,
! [A: $tType,B: $tType,F2: B > A,A4: set @ B,P: ( set @ A ) > $o] :
( ( ? [B6: set @ A] :
( ( finite_finite2 @ A @ B6 )
& ( ord_less_eq @ ( set @ A ) @ B6 @ ( image2 @ B @ A @ F2 @ A4 ) )
& ( P @ B6 ) ) )
= ( ? [B6: set @ B] :
( ( finite_finite2 @ B @ B6 )
& ( ord_less_eq @ ( set @ B ) @ B6 @ A4 )
& ( P @ ( image2 @ B @ A @ F2 @ B6 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_4159_finite__subset__image,axiom,
! [A: $tType,B: $tType,B2: set @ A,F2: B > A,A4: set @ B] :
( ( finite_finite2 @ A @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ ( image2 @ B @ A @ F2 @ A4 ) )
=> ? [C7: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ C7 @ A4 )
& ( finite_finite2 @ B @ C7 )
& ( B2
= ( image2 @ B @ A @ F2 @ C7 ) ) ) ) ) ).
% finite_subset_image
thf(fact_4160_finite__surj,axiom,
! [A: $tType,B: $tType,A4: set @ A,B2: set @ B,F2: A > B] :
( ( finite_finite2 @ A @ A4 )
=> ( ( ord_less_eq @ ( set @ B ) @ B2 @ ( image2 @ A @ B @ F2 @ A4 ) )
=> ( finite_finite2 @ B @ B2 ) ) ) ).
% finite_surj
thf(fact_4161_translation__Int,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A3: A,S: set @ A,T2: set @ A] :
( ( image2 @ A @ A @ ( plus_plus @ A @ A3 ) @ ( inf_inf @ ( set @ A ) @ S @ T2 ) )
= ( inf_inf @ ( set @ A ) @ ( image2 @ A @ A @ ( plus_plus @ A @ A3 ) @ S ) @ ( image2 @ A @ A @ ( plus_plus @ A @ A3 ) @ T2 ) ) ) ) ).
% translation_Int
thf(fact_4162_SUP__eq__const,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [I6: set @ B,F2: B > A,X: A] :
( ( I6
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [I4: B] :
( ( member @ B @ I4 @ I6 )
=> ( ( F2 @ I4 )
= X ) )
=> ( ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F2 @ I6 ) )
= X ) ) ) ) ).
% SUP_eq_const
thf(fact_4163_translation__diff,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A3: A,S: set @ A,T2: set @ A] :
( ( image2 @ A @ A @ ( plus_plus @ A @ A3 ) @ ( minus_minus @ ( set @ A ) @ S @ T2 ) )
= ( minus_minus @ ( set @ A ) @ ( image2 @ A @ A @ ( plus_plus @ A @ A3 ) @ S ) @ ( image2 @ A @ A @ ( plus_plus @ A @ A3 ) @ T2 ) ) ) ) ).
% translation_diff
thf(fact_4164_INF__eq__const,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [I6: set @ B,F2: B > A,X: A] :
( ( I6
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [I4: B] :
( ( member @ B @ I4 @ I6 )
=> ( ( F2 @ I4 )
= X ) )
=> ( ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F2 @ I6 ) )
= X ) ) ) ) ).
% INF_eq_const
thf(fact_4165_finite__image__absD,axiom,
! [A: $tType] :
( ( linordered_ring @ A )
=> ! [S2: set @ A] :
( ( finite_finite2 @ A @ ( image2 @ A @ A @ ( abs_abs @ A ) @ S2 ) )
=> ( finite_finite2 @ A @ S2 ) ) ) ).
% finite_image_absD
thf(fact_4166_image__Int__subset,axiom,
! [A: $tType,B: $tType,F2: B > A,A4: set @ B,B2: set @ B] : ( ord_less_eq @ ( set @ A ) @ ( image2 @ B @ A @ F2 @ ( inf_inf @ ( set @ B ) @ A4 @ B2 ) ) @ ( inf_inf @ ( set @ A ) @ ( image2 @ B @ A @ F2 @ A4 ) @ ( image2 @ B @ A @ F2 @ B2 ) ) ) ).
% image_Int_subset
thf(fact_4167_image__diff__subset,axiom,
! [A: $tType,B: $tType,F2: B > A,A4: set @ B,B2: set @ B] : ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ ( image2 @ B @ A @ F2 @ A4 ) @ ( image2 @ B @ A @ F2 @ B2 ) ) @ ( image2 @ B @ A @ F2 @ ( minus_minus @ ( set @ B ) @ A4 @ B2 ) ) ) ).
% image_diff_subset
thf(fact_4168_translation__Compl,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A3: A,T2: set @ A] :
( ( image2 @ A @ A @ ( plus_plus @ A @ A3 ) @ ( uminus_uminus @ ( set @ A ) @ T2 ) )
= ( uminus_uminus @ ( set @ A ) @ ( image2 @ A @ A @ ( plus_plus @ A @ A3 ) @ T2 ) ) ) ) ).
% translation_Compl
thf(fact_4169_SUP__eqI,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A4: set @ B,F2: B > A,X: A] :
( ! [I4: B] :
( ( member @ B @ I4 @ A4 )
=> ( ord_less_eq @ A @ ( F2 @ I4 ) @ X ) )
=> ( ! [Y3: A] :
( ! [I5: B] :
( ( member @ B @ I5 @ A4 )
=> ( ord_less_eq @ A @ ( F2 @ I5 ) @ Y3 ) )
=> ( ord_less_eq @ A @ X @ Y3 ) )
=> ( ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F2 @ A4 ) )
= X ) ) ) ) ).
% SUP_eqI
thf(fact_4170_SUP__mono,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A4: set @ B,B2: set @ C,F2: B > A,G: C > A] :
( ! [N3: B] :
( ( member @ B @ N3 @ A4 )
=> ? [X2: C] :
( ( member @ C @ X2 @ B2 )
& ( ord_less_eq @ A @ ( F2 @ N3 ) @ ( G @ X2 ) ) ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F2 @ A4 ) ) @ ( complete_Sup_Sup @ A @ ( image2 @ C @ A @ G @ B2 ) ) ) ) ) ).
% SUP_mono
thf(fact_4171_SUP__least,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A4: set @ B,F2: B > A,U: A] :
( ! [I4: B] :
( ( member @ B @ I4 @ A4 )
=> ( ord_less_eq @ A @ ( F2 @ I4 ) @ U ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F2 @ A4 ) ) @ U ) ) ) ).
% SUP_least
thf(fact_4172_SUP__mono_H,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F2: B > A,G: B > A,A4: set @ B] :
( ! [X5: B] : ( ord_less_eq @ A @ ( F2 @ X5 ) @ ( G @ X5 ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F2 @ A4 ) ) @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ G @ A4 ) ) ) ) ) ).
% SUP_mono'
thf(fact_4173_SUP__upper,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [I: B,A4: set @ B,F2: B > A] :
( ( member @ B @ I @ A4 )
=> ( ord_less_eq @ A @ ( F2 @ I ) @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F2 @ A4 ) ) ) ) ) ).
% SUP_upper
thf(fact_4174_SUP__le__iff,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F2: B > A,A4: set @ B,U: A] :
( ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F2 @ A4 ) ) @ U )
= ( ! [X4: B] :
( ( member @ B @ X4 @ A4 )
=> ( ord_less_eq @ A @ ( F2 @ X4 ) @ U ) ) ) ) ) ).
% SUP_le_iff
thf(fact_4175_SUP__upper2,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [I: B,A4: set @ B,U: A,F2: B > A] :
( ( member @ B @ I @ A4 )
=> ( ( ord_less_eq @ A @ U @ ( F2 @ I ) )
=> ( ord_less_eq @ A @ U @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F2 @ A4 ) ) ) ) ) ) ).
% SUP_upper2
thf(fact_4176_less__SUP__iff,axiom,
! [A: $tType,B: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [A3: A,F2: B > A,A4: set @ B] :
( ( ord_less @ A @ A3 @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F2 @ A4 ) ) )
= ( ? [X4: B] :
( ( member @ B @ X4 @ A4 )
& ( ord_less @ A @ A3 @ ( F2 @ X4 ) ) ) ) ) ) ).
% less_SUP_iff
thf(fact_4177_SUP__lessD,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F2: B > A,A4: set @ B,Y: A,I: B] :
( ( ord_less @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F2 @ A4 ) ) @ Y )
=> ( ( member @ B @ I @ A4 )
=> ( ord_less @ A @ ( F2 @ I ) @ Y ) ) ) ) ).
% SUP_lessD
thf(fact_4178_INF__eqI,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A4: set @ B,X: A,F2: B > A] :
( ! [I4: B] :
( ( member @ B @ I4 @ A4 )
=> ( ord_less_eq @ A @ X @ ( F2 @ I4 ) ) )
=> ( ! [Y3: A] :
( ! [I5: B] :
( ( member @ B @ I5 @ A4 )
=> ( ord_less_eq @ A @ Y3 @ ( F2 @ I5 ) ) )
=> ( ord_less_eq @ A @ Y3 @ X ) )
=> ( ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F2 @ A4 ) )
= X ) ) ) ) ).
% INF_eqI
thf(fact_4179_INF__mono,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [B2: set @ B,A4: set @ C,F2: C > A,G: B > A] :
( ! [M3: B] :
( ( member @ B @ M3 @ B2 )
=> ? [X2: C] :
( ( member @ C @ X2 @ A4 )
& ( ord_less_eq @ A @ ( F2 @ X2 ) @ ( G @ M3 ) ) ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image2 @ C @ A @ F2 @ A4 ) ) @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ G @ B2 ) ) ) ) ) ).
% INF_mono
thf(fact_4180_INF__lower,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [I: B,A4: set @ B,F2: B > A] :
( ( member @ B @ I @ A4 )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F2 @ A4 ) ) @ ( F2 @ I ) ) ) ) ).
% INF_lower
thf(fact_4181_INF__mono_H,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F2: B > A,G: B > A,A4: set @ B] :
( ! [X5: B] : ( ord_less_eq @ A @ ( F2 @ X5 ) @ ( G @ X5 ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F2 @ A4 ) ) @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ G @ A4 ) ) ) ) ) ).
% INF_mono'
thf(fact_4182_INF__lower2,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [I: B,A4: set @ B,F2: B > A,U: A] :
( ( member @ B @ I @ A4 )
=> ( ( ord_less_eq @ A @ ( F2 @ I ) @ U )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F2 @ A4 ) ) @ U ) ) ) ) ).
% INF_lower2
thf(fact_4183_le__INF__iff,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [U: A,F2: B > A,A4: set @ B] :
( ( ord_less_eq @ A @ U @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F2 @ A4 ) ) )
= ( ! [X4: B] :
( ( member @ B @ X4 @ A4 )
=> ( ord_less_eq @ A @ U @ ( F2 @ X4 ) ) ) ) ) ) ).
% le_INF_iff
thf(fact_4184_INF__greatest,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A4: set @ B,U: A,F2: B > A] :
( ! [I4: B] :
( ( member @ B @ I4 @ A4 )
=> ( ord_less_eq @ A @ U @ ( F2 @ I4 ) ) )
=> ( ord_less_eq @ A @ U @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F2 @ A4 ) ) ) ) ) ).
% INF_greatest
thf(fact_4185_INF__less__iff,axiom,
! [A: $tType,B: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [F2: B > A,A4: set @ B,A3: A] :
( ( ord_less @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F2 @ A4 ) ) @ A3 )
= ( ? [X4: B] :
( ( member @ B @ X4 @ A4 )
& ( ord_less @ A @ ( F2 @ X4 ) @ A3 ) ) ) ) ) ).
% INF_less_iff
thf(fact_4186_less__INF__D,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [Y: A,F2: B > A,A4: set @ B,I: B] :
( ( ord_less @ A @ Y @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F2 @ A4 ) ) )
=> ( ( member @ B @ I @ A4 )
=> ( ord_less @ A @ Y @ ( F2 @ I ) ) ) ) ) ).
% less_INF_D
thf(fact_4187_nat__seg__image__imp__finite,axiom,
! [A: $tType,A4: set @ A,F2: nat > A,N2: nat] :
( ( A4
= ( image2 @ nat @ A @ F2
@ ( collect @ nat
@ ^ [I3: nat] : ( ord_less @ nat @ I3 @ N2 ) ) ) )
=> ( finite_finite2 @ A @ A4 ) ) ).
% nat_seg_image_imp_finite
thf(fact_4188_finite__conv__nat__seg__image,axiom,
! [A: $tType] :
( ( finite_finite2 @ A )
= ( ^ [A7: set @ A] :
? [N: nat,F3: nat > A] :
( A7
= ( image2 @ nat @ A @ F3
@ ( collect @ nat
@ ^ [I3: nat] : ( ord_less @ nat @ I3 @ N ) ) ) ) ) ) ).
% finite_conv_nat_seg_image
thf(fact_4189_image__constant__conv,axiom,
! [B: $tType,A: $tType,A4: set @ B,C2: A] :
( ( ( A4
= ( bot_bot @ ( set @ B ) ) )
=> ( ( image2 @ B @ A
@ ^ [X4: B] : C2
@ A4 )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( A4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( image2 @ B @ A
@ ^ [X4: B] : C2
@ A4 )
= ( insert2 @ A @ C2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% image_constant_conv
thf(fact_4190_image__constant,axiom,
! [A: $tType,B: $tType,X: A,A4: set @ A,C2: B] :
( ( member @ A @ X @ A4 )
=> ( ( image2 @ A @ B
@ ^ [X4: A] : C2
@ A4 )
= ( insert2 @ B @ C2 @ ( bot_bot @ ( set @ B ) ) ) ) ) ).
% image_constant
thf(fact_4191_sum_Oimage__gen,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [S2: set @ B,H2: B > A,G: B > C] :
( ( finite_finite2 @ B @ S2 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ H2 @ S2 )
= ( groups7311177749621191930dd_sum @ C @ A
@ ^ [Y6: C] :
( groups7311177749621191930dd_sum @ B @ A @ H2
@ ( collect @ B
@ ^ [X4: B] :
( ( member @ B @ X4 @ S2 )
& ( ( G @ X4 )
= Y6 ) ) ) )
@ ( image2 @ B @ C @ G @ S2 ) ) ) ) ) ).
% sum.image_gen
thf(fact_4192_prod_Oimage__gen,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S2: set @ B,H2: B > A,G: B > C] :
( ( finite_finite2 @ B @ S2 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ H2 @ S2 )
= ( groups7121269368397514597t_prod @ C @ A
@ ^ [Y6: C] :
( groups7121269368397514597t_prod @ B @ A @ H2
@ ( collect @ B
@ ^ [X4: B] :
( ( member @ B @ X4 @ S2 )
& ( ( G @ X4 )
= Y6 ) ) ) )
@ ( image2 @ B @ C @ G @ S2 ) ) ) ) ) ).
% prod.image_gen
thf(fact_4193_the__elem__image__unique,axiom,
! [B: $tType,A: $tType,A4: set @ A,F2: A > B,X: A] :
( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [Y3: A] :
( ( member @ A @ Y3 @ A4 )
=> ( ( F2 @ Y3 )
= ( F2 @ X ) ) )
=> ( ( the_elem @ B @ ( image2 @ A @ B @ F2 @ A4 ) )
= ( F2 @ X ) ) ) ) ).
% the_elem_image_unique
thf(fact_4194_le__SUP__iff,axiom,
! [B: $tType,A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [X: A,F2: B > A,A4: set @ B] :
( ( ord_less_eq @ A @ X @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F2 @ A4 ) ) )
= ( ! [Y6: A] :
( ( ord_less @ A @ Y6 @ X )
=> ? [X4: B] :
( ( member @ B @ X4 @ A4 )
& ( ord_less @ A @ Y6 @ ( F2 @ X4 ) ) ) ) ) ) ) ).
% le_SUP_iff
thf(fact_4195_INF__le__iff,axiom,
! [B: $tType,A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [F2: B > A,A4: set @ B,X: A] :
( ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F2 @ A4 ) ) @ X )
= ( ! [Y6: A] :
( ( ord_less @ A @ X @ Y6 )
=> ? [X4: B] :
( ( member @ B @ X4 @ A4 )
& ( ord_less @ A @ ( F2 @ X4 ) @ Y6 ) ) ) ) ) ) ).
% INF_le_iff
thf(fact_4196_SUP__eq__iff,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [I6: set @ B,C2: A,F2: B > A] :
( ( I6
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [I4: B] :
( ( member @ B @ I4 @ I6 )
=> ( ord_less_eq @ A @ C2 @ ( F2 @ I4 ) ) )
=> ( ( ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F2 @ I6 ) )
= C2 )
= ( ! [X4: B] :
( ( member @ B @ X4 @ I6 )
=> ( ( F2 @ X4 )
= C2 ) ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_4197_cSUP__least,axiom,
! [B: $tType,A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A4: set @ B,F2: B > A,M7: A] :
( ( A4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ A4 )
=> ( ord_less_eq @ A @ ( F2 @ X5 ) @ M7 ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F2 @ A4 ) ) @ M7 ) ) ) ) ).
% cSUP_least
thf(fact_4198_INF__eq__iff,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [I6: set @ B,F2: B > A,C2: A] :
( ( I6
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [I4: B] :
( ( member @ B @ I4 @ I6 )
=> ( ord_less_eq @ A @ ( F2 @ I4 ) @ C2 ) )
=> ( ( ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F2 @ I6 ) )
= C2 )
= ( ! [X4: B] :
( ( member @ B @ X4 @ I6 )
=> ( ( F2 @ X4 )
= C2 ) ) ) ) ) ) ) ).
% INF_eq_iff
thf(fact_4199_cINF__greatest,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A4: set @ B,M2: A,F2: B > A] :
( ( A4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ A4 )
=> ( ord_less_eq @ A @ M2 @ ( F2 @ X5 ) ) )
=> ( ord_less_eq @ A @ M2 @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F2 @ A4 ) ) ) ) ) ) ).
% cINF_greatest
thf(fact_4200_card__image__le,axiom,
! [B: $tType,A: $tType,A4: set @ A,F2: A > B] :
( ( finite_finite2 @ A @ A4 )
=> ( ord_less_eq @ nat @ ( finite_card @ B @ ( image2 @ A @ B @ F2 @ A4 ) ) @ ( finite_card @ A @ A4 ) ) ) ).
% card_image_le
thf(fact_4201_SUP__subset__mono,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A4: set @ B,B2: set @ B,F2: B > A,G: B > A] :
( ( ord_less_eq @ ( set @ B ) @ A4 @ B2 )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ A4 )
=> ( ord_less_eq @ A @ ( F2 @ X5 ) @ ( G @ X5 ) ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F2 @ A4 ) ) @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ G @ B2 ) ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_4202_INF__superset__mono,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [B2: set @ B,A4: set @ B,F2: B > A,G: B > A] :
( ( ord_less_eq @ ( set @ B ) @ B2 @ A4 )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ B2 )
=> ( ord_less_eq @ A @ ( F2 @ X5 ) @ ( G @ X5 ) ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F2 @ A4 ) ) @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ G @ B2 ) ) ) ) ) ) ).
% INF_superset_mono
thf(fact_4203_SUP__empty,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F2: B > A] :
( ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F2 @ ( bot_bot @ ( set @ B ) ) ) )
= ( bot_bot @ A ) ) ) ).
% SUP_empty
thf(fact_4204_SUP__constant,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A4: set @ B,C2: A] :
( ( ( A4
= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Sup_Sup @ A
@ ( image2 @ B @ A
@ ^ [Y6: B] : C2
@ A4 ) )
= ( bot_bot @ A ) ) )
& ( ( A4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Sup_Sup @ A
@ ( image2 @ B @ A
@ ^ [Y6: B] : C2
@ A4 ) )
= C2 ) ) ) ) ).
% SUP_constant
thf(fact_4205_sum_Ogroup,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [S2: set @ B,T4: set @ C,G: B > C,H2: B > A] :
( ( finite_finite2 @ B @ S2 )
=> ( ( finite_finite2 @ C @ T4 )
=> ( ( ord_less_eq @ ( set @ C ) @ ( image2 @ B @ C @ G @ S2 ) @ T4 )
=> ( ( groups7311177749621191930dd_sum @ C @ A
@ ^ [Y6: C] :
( groups7311177749621191930dd_sum @ B @ A @ H2
@ ( collect @ B
@ ^ [X4: B] :
( ( member @ B @ X4 @ S2 )
& ( ( G @ X4 )
= Y6 ) ) ) )
@ T4 )
= ( groups7311177749621191930dd_sum @ B @ A @ H2 @ S2 ) ) ) ) ) ) ).
% sum.group
thf(fact_4206_INF__inf__const2,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [I6: set @ B,F2: B > A,X: A] :
( ( I6
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Inf_Inf @ A
@ ( image2 @ B @ A
@ ^ [I3: B] : ( inf_inf @ A @ ( F2 @ I3 ) @ X )
@ I6 ) )
= ( inf_inf @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F2 @ I6 ) ) @ X ) ) ) ) ).
% INF_inf_const2
thf(fact_4207_INF__inf__const1,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [I6: set @ B,X: A,F2: B > A] :
( ( I6
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Inf_Inf @ A
@ ( image2 @ B @ A
@ ^ [I3: B] : ( inf_inf @ A @ X @ ( F2 @ I3 ) )
@ I6 ) )
= ( inf_inf @ A @ X @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F2 @ I6 ) ) ) ) ) ) ).
% INF_inf_const1
thf(fact_4208_prod_Ogroup,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S2: set @ B,T4: set @ C,G: B > C,H2: B > A] :
( ( finite_finite2 @ B @ S2 )
=> ( ( finite_finite2 @ C @ T4 )
=> ( ( ord_less_eq @ ( set @ C ) @ ( image2 @ B @ C @ G @ S2 ) @ T4 )
=> ( ( groups7121269368397514597t_prod @ C @ A
@ ^ [Y6: C] :
( groups7121269368397514597t_prod @ B @ A @ H2
@ ( collect @ B
@ ^ [X4: B] :
( ( member @ B @ X4 @ S2 )
& ( ( G @ X4 )
= Y6 ) ) ) )
@ T4 )
= ( groups7121269368397514597t_prod @ B @ A @ H2 @ S2 ) ) ) ) ) ) ).
% prod.group
thf(fact_4209_atLeastPlusOneAtMost__greaterThanAtMost__int,axiom,
! [L: int,U: int] :
( ( set_or1337092689740270186AtMost @ int @ ( plus_plus @ int @ L @ ( one_one @ int ) ) @ U )
= ( set_or3652927894154168847AtMost @ int @ L @ U ) ) ).
% atLeastPlusOneAtMost_greaterThanAtMost_int
thf(fact_4210_INF__le__SUP,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A4: set @ B,F2: B > A] :
( ( A4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F2 @ A4 ) ) @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F2 @ A4 ) ) ) ) ) ).
% INF_le_SUP
thf(fact_4211_surj__card__le,axiom,
! [B: $tType,A: $tType,A4: set @ A,B2: set @ B,F2: A > B] :
( ( finite_finite2 @ A @ A4 )
=> ( ( ord_less_eq @ ( set @ B ) @ B2 @ ( image2 @ A @ B @ F2 @ A4 ) )
=> ( ord_less_eq @ nat @ ( finite_card @ B @ B2 ) @ ( finite_card @ A @ A4 ) ) ) ) ).
% surj_card_le
thf(fact_4212_image__Suc__lessThan,axiom,
! [N2: nat] :
( ( image2 @ nat @ nat @ suc @ ( set_ord_lessThan @ nat @ N2 ) )
= ( set_or1337092689740270186AtMost @ nat @ ( one_one @ nat ) @ N2 ) ) ).
% image_Suc_lessThan
thf(fact_4213_image__Suc__atMost,axiom,
! [N2: nat] :
( ( image2 @ nat @ nat @ suc @ ( set_ord_atMost @ nat @ N2 ) )
= ( set_or1337092689740270186AtMost @ nat @ ( one_one @ nat ) @ ( suc @ N2 ) ) ) ).
% image_Suc_atMost
thf(fact_4214_atLeast0__atMost__Suc__eq__insert__0,axiom,
! [N2: nat] :
( ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( suc @ N2 ) )
= ( insert2 @ nat @ ( zero_zero @ nat ) @ ( image2 @ nat @ nat @ suc @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ) ).
% atLeast0_atMost_Suc_eq_insert_0
thf(fact_4215_atLeast0__lessThan__Suc__eq__insert__0,axiom,
! [N2: nat] :
( ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( suc @ N2 ) )
= ( insert2 @ nat @ ( zero_zero @ nat ) @ ( image2 @ nat @ nat @ suc @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ) ).
% atLeast0_lessThan_Suc_eq_insert_0
thf(fact_4216_lessThan__Suc__eq__insert__0,axiom,
! [N2: nat] :
( ( set_ord_lessThan @ nat @ ( suc @ N2 ) )
= ( insert2 @ nat @ ( zero_zero @ nat ) @ ( image2 @ nat @ nat @ suc @ ( set_ord_lessThan @ nat @ N2 ) ) ) ) ).
% lessThan_Suc_eq_insert_0
thf(fact_4217_atMost__Suc__eq__insert__0,axiom,
! [N2: nat] :
( ( set_ord_atMost @ nat @ ( suc @ N2 ) )
= ( insert2 @ nat @ ( zero_zero @ nat ) @ ( image2 @ nat @ nat @ suc @ ( set_ord_atMost @ nat @ N2 ) ) ) ) ).
% atMost_Suc_eq_insert_0
thf(fact_4218_image__mult__atLeastAtMost__if,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,X: A,Y: A] :
( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( image2 @ 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 )
=> ( ( image2 @ 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 )
=> ( ( image2 @ A @ A @ ( times_times @ A @ C2 ) @ ( set_or1337092689740270186AtMost @ A @ X @ Y ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% image_mult_atLeastAtMost_if
thf(fact_4219_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 )
=> ( ( image2 @ A @ A
@ ^ [X4: A] : ( times_times @ A @ X4 @ 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 )
=> ( ( image2 @ A @ A
@ ^ [X4: A] : ( times_times @ A @ X4 @ 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 )
=> ( ( image2 @ A @ A
@ ^ [X4: A] : ( times_times @ A @ X4 @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ X @ Y ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% image_mult_atLeastAtMost_if'
thf(fact_4220_image__affinity__atLeastAtMost,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B3: A,M2: A,C2: A] :
( ( ( ( set_or1337092689740270186AtMost @ A @ A3 @ B3 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( image2 @ A @ A
@ ^ [X4: A] : ( plus_plus @ A @ ( times_times @ A @ M2 @ X4 ) @ 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 ) @ M2 )
=> ( ( image2 @ A @ A
@ ^ [X4: A] : ( plus_plus @ A @ ( times_times @ A @ M2 @ X4 ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A3 @ B3 ) )
= ( set_or1337092689740270186AtMost @ A @ ( plus_plus @ A @ ( times_times @ A @ M2 @ A3 ) @ C2 ) @ ( plus_plus @ A @ ( times_times @ A @ M2 @ B3 ) @ C2 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ M2 )
=> ( ( image2 @ A @ A
@ ^ [X4: A] : ( plus_plus @ A @ ( times_times @ A @ M2 @ X4 ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A3 @ B3 ) )
= ( set_or1337092689740270186AtMost @ A @ ( plus_plus @ A @ ( times_times @ A @ M2 @ B3 ) @ C2 ) @ ( plus_plus @ A @ ( times_times @ A @ M2 @ A3 ) @ C2 ) ) ) ) ) ) ) ) ).
% image_affinity_atLeastAtMost
thf(fact_4221_image__affinity__atLeastAtMost__diff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B3: A,M2: A,C2: A] :
( ( ( ( set_or1337092689740270186AtMost @ A @ A3 @ B3 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( image2 @ A @ A
@ ^ [X4: A] : ( minus_minus @ A @ ( times_times @ A @ M2 @ X4 ) @ 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 ) @ M2 )
=> ( ( image2 @ A @ A
@ ^ [X4: A] : ( minus_minus @ A @ ( times_times @ A @ M2 @ X4 ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A3 @ B3 ) )
= ( set_or1337092689740270186AtMost @ A @ ( minus_minus @ A @ ( times_times @ A @ M2 @ A3 ) @ C2 ) @ ( minus_minus @ A @ ( times_times @ A @ M2 @ B3 ) @ C2 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ M2 )
=> ( ( image2 @ A @ A
@ ^ [X4: A] : ( minus_minus @ A @ ( times_times @ A @ M2 @ X4 ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A3 @ B3 ) )
= ( set_or1337092689740270186AtMost @ A @ ( minus_minus @ A @ ( times_times @ A @ M2 @ B3 ) @ C2 ) @ ( minus_minus @ A @ ( times_times @ A @ M2 @ A3 ) @ C2 ) ) ) ) ) ) ) ) ).
% image_affinity_atLeastAtMost_diff
thf(fact_4222_image__affinity__atLeastAtMost__div,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B3: A,M2: A,C2: A] :
( ( ( ( set_or1337092689740270186AtMost @ A @ A3 @ B3 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( image2 @ A @ A
@ ^ [X4: A] : ( plus_plus @ A @ ( divide_divide @ A @ X4 @ M2 ) @ 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 ) @ M2 )
=> ( ( image2 @ A @ A
@ ^ [X4: A] : ( plus_plus @ A @ ( divide_divide @ A @ X4 @ M2 ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A3 @ B3 ) )
= ( set_or1337092689740270186AtMost @ A @ ( plus_plus @ A @ ( divide_divide @ A @ A3 @ M2 ) @ C2 ) @ ( plus_plus @ A @ ( divide_divide @ A @ B3 @ M2 ) @ C2 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ M2 )
=> ( ( image2 @ A @ A
@ ^ [X4: A] : ( plus_plus @ A @ ( divide_divide @ A @ X4 @ M2 ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A3 @ B3 ) )
= ( set_or1337092689740270186AtMost @ A @ ( plus_plus @ A @ ( divide_divide @ A @ B3 @ M2 ) @ C2 ) @ ( plus_plus @ A @ ( divide_divide @ A @ A3 @ M2 ) @ C2 ) ) ) ) ) ) ) ) ).
% image_affinity_atLeastAtMost_div
thf(fact_4223_image__affinity__atLeastAtMost__div__diff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B3: A,M2: A,C2: A] :
( ( ( ( set_or1337092689740270186AtMost @ A @ A3 @ B3 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( image2 @ A @ A
@ ^ [X4: A] : ( minus_minus @ A @ ( divide_divide @ A @ X4 @ M2 ) @ 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 ) @ M2 )
=> ( ( image2 @ A @ A
@ ^ [X4: A] : ( minus_minus @ A @ ( divide_divide @ A @ X4 @ M2 ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A3 @ B3 ) )
= ( set_or1337092689740270186AtMost @ A @ ( minus_minus @ A @ ( divide_divide @ A @ A3 @ M2 ) @ C2 ) @ ( minus_minus @ A @ ( divide_divide @ A @ B3 @ M2 ) @ C2 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ M2 )
=> ( ( image2 @ A @ A
@ ^ [X4: A] : ( minus_minus @ A @ ( divide_divide @ A @ X4 @ M2 ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A3 @ B3 ) )
= ( set_or1337092689740270186AtMost @ A @ ( minus_minus @ A @ ( divide_divide @ A @ B3 @ M2 ) @ C2 ) @ ( minus_minus @ A @ ( divide_divide @ A @ A3 @ M2 ) @ C2 ) ) ) ) ) ) ) ) ).
% image_affinity_atLeastAtMost_div_diff
thf(fact_4224_sum__fun__comp,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( semiring_1 @ C )
=> ! [S2: set @ A,R: set @ B,G: A > B,F2: B > C] :
( ( finite_finite2 @ A @ S2 )
=> ( ( finite_finite2 @ B @ R )
=> ( ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ G @ S2 ) @ R )
=> ( ( groups7311177749621191930dd_sum @ A @ C
@ ^ [X4: A] : ( F2 @ ( G @ X4 ) )
@ S2 )
= ( groups7311177749621191930dd_sum @ B @ C
@ ^ [Y6: B] :
( times_times @ C
@ ( semiring_1_of_nat @ C
@ ( finite_card @ A
@ ( collect @ A
@ ^ [X4: A] :
( ( member @ A @ X4 @ S2 )
& ( ( G @ X4 )
= Y6 ) ) ) ) )
@ ( F2 @ Y6 ) )
@ R ) ) ) ) ) ) ).
% sum_fun_comp
thf(fact_4225_INF__nat__binary,axiom,
! [A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A4: A,B2: A] :
( ( inf_inf @ A @ A4
@ ( complete_Inf_Inf @ A
@ ( image2 @ nat @ A
@ ^ [X4: nat] : B2
@ ( collect @ nat @ ( ord_less @ nat @ ( zero_zero @ nat ) ) ) ) ) )
= ( inf_inf @ A @ A4 @ B2 ) ) ) ).
% INF_nat_binary
thf(fact_4226_ring__1__class_Oof__int__def,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ( ( ring_1_of_int @ A )
= ( map_fun @ int @ ( product_prod @ nat @ nat ) @ A @ A @ rep_Integ @ ( id @ A )
@ ( product_case_prod @ nat @ nat @ A
@ ^ [I3: nat,J2: nat] : ( minus_minus @ A @ ( semiring_1_of_nat @ A @ I3 ) @ ( semiring_1_of_nat @ A @ J2 ) ) ) ) ) ) ).
% ring_1_class.of_int_def
thf(fact_4227_sorted__key__list__of__set__def,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ( ( linord144544945434240204of_set @ B @ A )
= ( ^ [F3: B > A] : ( finite_folding_F @ B @ ( list @ B ) @ ( linorder_insort_key @ B @ A @ F3 ) @ ( nil @ B ) ) ) ) ) ).
% sorted_key_list_of_set_def
thf(fact_4228_nth__image,axiom,
! [A: $tType,L: nat,Xs: list @ A] :
( ( ord_less_eq @ nat @ L @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( image2 @ nat @ A @ ( nth @ A @ Xs ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ L ) )
= ( set2 @ A @ ( take @ A @ L @ Xs ) ) ) ) ).
% nth_image
thf(fact_4229_of__nat__eq__id,axiom,
( ( semiring_1_of_nat @ nat )
= ( id @ nat ) ) ).
% of_nat_eq_id
thf(fact_4230_id__funpow,axiom,
! [A: $tType,N2: nat] :
( ( compow @ ( A > A ) @ N2 @ ( id @ A ) )
= ( id @ A ) ) ).
% id_funpow
thf(fact_4231_finite__UN,axiom,
! [B: $tType,A: $tType,A4: set @ A,B2: A > ( set @ B )] :
( ( finite_finite2 @ A @ A4 )
=> ( ( finite_finite2 @ B @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ B2 @ A4 ) ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ A4 )
=> ( finite_finite2 @ B @ ( B2 @ X4 ) ) ) ) ) ) ).
% finite_UN
thf(fact_4232_take0,axiom,
! [A: $tType] :
( ( take @ A @ ( zero_zero @ nat ) )
= ( ^ [Xs3: list @ A] : ( nil @ A ) ) ) ).
% take0
thf(fact_4233_take__eq__Nil,axiom,
! [A: $tType,N2: nat,Xs: list @ A] :
( ( ( take @ A @ N2 @ Xs )
= ( nil @ A ) )
= ( ( N2
= ( zero_zero @ nat ) )
| ( Xs
= ( nil @ A ) ) ) ) ).
% take_eq_Nil
thf(fact_4234_take__eq__Nil2,axiom,
! [A: $tType,N2: nat,Xs: list @ A] :
( ( ( nil @ A )
= ( take @ A @ N2 @ Xs ) )
= ( ( N2
= ( zero_zero @ nat ) )
| ( Xs
= ( nil @ A ) ) ) ) ).
% take_eq_Nil2
thf(fact_4235_take__all__iff,axiom,
! [A: $tType,N2: nat,Xs: list @ A] :
( ( ( take @ A @ N2 @ Xs )
= Xs )
= ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ N2 ) ) ).
% take_all_iff
thf(fact_4236_take__all,axiom,
! [A: $tType,Xs: list @ A,N2: nat] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ N2 )
=> ( ( take @ A @ N2 @ Xs )
= Xs ) ) ).
% take_all
thf(fact_4237_nth__take,axiom,
! [A: $tType,I: nat,N2: nat,Xs: list @ A] :
( ( ord_less @ nat @ I @ N2 )
=> ( ( nth @ A @ ( take @ A @ N2 @ Xs ) @ I )
= ( nth @ A @ Xs @ I ) ) ) ).
% nth_take
thf(fact_4238_push__bit__0__id,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se4730199178511100633sh_bit @ A @ ( zero_zero @ nat ) )
= ( id @ A ) ) ) ).
% push_bit_0_id
thf(fact_4239_drop__bit__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se4197421643247451524op_bit @ A @ ( zero_zero @ nat ) )
= ( id @ A ) ) ) ).
% drop_bit_0
thf(fact_4240_take__update__cancel,axiom,
! [A: $tType,N2: nat,M2: nat,Xs: list @ A,Y: A] :
( ( ord_less_eq @ nat @ N2 @ M2 )
=> ( ( take @ A @ N2 @ ( list_update @ A @ Xs @ M2 @ Y ) )
= ( take @ A @ N2 @ Xs ) ) ) ).
% take_update_cancel
thf(fact_4241_UN__constant,axiom,
! [B: $tType,A: $tType,A4: set @ B,C2: set @ A] :
( ( ( A4
= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [Y6: B] : C2
@ A4 ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( A4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [Y6: B] : C2
@ A4 ) )
= C2 ) ) ) ).
% UN_constant
thf(fact_4242_finite__UN__I,axiom,
! [B: $tType,A: $tType,A4: set @ A,B2: A > ( set @ B )] :
( ( finite_finite2 @ A @ A4 )
=> ( ! [A6: A] :
( ( member @ A @ A6 @ A4 )
=> ( finite_finite2 @ B @ ( B2 @ A6 ) ) )
=> ( finite_finite2 @ B @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ B2 @ A4 ) ) ) ) ) ).
% finite_UN_I
thf(fact_4243_finite__INT,axiom,
! [B: $tType,A: $tType,I6: set @ A,A4: A > ( set @ B )] :
( ? [X2: A] :
( ( member @ A @ X2 @ I6 )
& ( finite_finite2 @ B @ ( A4 @ X2 ) ) )
=> ( finite_finite2 @ B @ ( complete_Inf_Inf @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ A4 @ I6 ) ) ) ) ).
% finite_INT
thf(fact_4244_UN__simps_I1_J,axiom,
! [A: $tType,B: $tType,C5: set @ B,A3: A,B2: B > ( set @ A )] :
( ( ( C5
= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [X4: B] : ( insert2 @ A @ A3 @ ( B2 @ X4 ) )
@ C5 ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( C5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [X4: B] : ( insert2 @ A @ A3 @ ( B2 @ X4 ) )
@ C5 ) )
= ( insert2 @ A @ A3 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B2 @ C5 ) ) ) ) ) ) ).
% UN_simps(1)
thf(fact_4245_UN__singleton,axiom,
! [A: $tType,A4: set @ A] :
( ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ A @ ( set @ A )
@ ^ [X4: A] : ( insert2 @ A @ X4 @ ( bot_bot @ ( set @ A ) ) )
@ A4 ) )
= A4 ) ).
% UN_singleton
thf(fact_4246_set__concat,axiom,
! [A: $tType,Xs: list @ ( list @ A )] :
( ( set2 @ A @ ( concat @ A @ Xs ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ ( list @ A ) @ ( set @ A ) @ ( set2 @ A ) @ ( set2 @ ( list @ A ) @ Xs ) ) ) ) ).
% set_concat
thf(fact_4247_take__Nil,axiom,
! [A: $tType,N2: nat] :
( ( take @ A @ N2 @ ( nil @ A ) )
= ( nil @ A ) ) ).
% take_Nil
thf(fact_4248_distinct__take,axiom,
! [A: $tType,Xs: list @ A,I: nat] :
( ( distinct @ A @ Xs )
=> ( distinct @ A @ ( take @ A @ I @ Xs ) ) ) ).
% distinct_take
thf(fact_4249_folding__on_OF_Ocong,axiom,
! [B: $tType,A: $tType] :
( ( finite_folding_F @ A @ B )
= ( finite_folding_F @ A @ B ) ) ).
% folding_on.F.cong
thf(fact_4250_take__equalityI,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ! [I4: nat] :
( ( take @ A @ I4 @ Xs )
= ( take @ A @ I4 @ Ys ) )
=> ( Xs = Ys ) ) ).
% take_equalityI
thf(fact_4251_take__update__swap,axiom,
! [A: $tType,M2: nat,Xs: list @ A,N2: nat,X: A] :
( ( take @ A @ M2 @ ( list_update @ A @ Xs @ N2 @ X ) )
= ( list_update @ A @ ( take @ A @ M2 @ Xs ) @ N2 @ X ) ) ).
% take_update_swap
thf(fact_4252_in__set__takeD,axiom,
! [A: $tType,X: A,N2: nat,Xs: list @ A] :
( ( member @ A @ X @ ( set2 @ A @ ( take @ A @ N2 @ Xs ) ) )
=> ( member @ A @ X @ ( set2 @ A @ Xs ) ) ) ).
% in_set_takeD
thf(fact_4253_take__0,axiom,
! [A: $tType,Xs: list @ A] :
( ( take @ A @ ( zero_zero @ nat ) @ Xs )
= ( nil @ A ) ) ).
% take_0
thf(fact_4254_set__take__subset,axiom,
! [A: $tType,N2: nat,Xs: list @ A] : ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ ( take @ A @ N2 @ Xs ) ) @ ( set2 @ A @ Xs ) ) ).
% set_take_subset
thf(fact_4255_funpow__simps__right_I1_J,axiom,
! [A: $tType,F2: A > A] :
( ( compow @ ( A > A ) @ ( zero_zero @ nat ) @ F2 )
= ( id @ A ) ) ).
% funpow_simps_right(1)
thf(fact_4256_less__int__def,axiom,
( ( ord_less @ int )
= ( map_fun @ int @ ( product_prod @ nat @ nat ) @ ( ( product_prod @ nat @ nat ) > $o ) @ ( int > $o ) @ rep_Integ @ ( map_fun @ int @ ( product_prod @ nat @ nat ) @ $o @ $o @ rep_Integ @ ( id @ $o ) )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [X4: nat,Y6: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U2: nat,V5: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ X4 @ V5 ) @ ( plus_plus @ nat @ U2 @ Y6 ) ) ) ) ) ) ).
% less_int_def
thf(fact_4257_less__eq__int__def,axiom,
( ( ord_less_eq @ int )
= ( map_fun @ int @ ( product_prod @ nat @ nat ) @ ( ( product_prod @ nat @ nat ) > $o ) @ ( int > $o ) @ rep_Integ @ ( map_fun @ int @ ( product_prod @ nat @ nat ) @ $o @ $o @ rep_Integ @ ( id @ $o ) )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [X4: nat,Y6: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U2: nat,V5: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ X4 @ V5 ) @ ( plus_plus @ nat @ U2 @ Y6 ) ) ) ) ) ) ).
% less_eq_int_def
thf(fact_4258_INF__filter__not__bot,axiom,
! [I7: $tType,A: $tType,B2: set @ I7,F5: I7 > ( filter @ A )] :
( ! [X9: set @ I7] :
( ( ord_less_eq @ ( set @ I7 ) @ X9 @ B2 )
=> ( ( finite_finite2 @ I7 @ X9 )
=> ( ( complete_Inf_Inf @ ( filter @ A ) @ ( image2 @ I7 @ ( filter @ A ) @ F5 @ X9 ) )
!= ( bot_bot @ ( filter @ A ) ) ) ) )
=> ( ( complete_Inf_Inf @ ( filter @ A ) @ ( image2 @ I7 @ ( filter @ A ) @ F5 @ B2 ) )
!= ( bot_bot @ ( filter @ A ) ) ) ) ).
% INF_filter_not_bot
thf(fact_4259_finite__int__iff__bounded__le,axiom,
( ( finite_finite2 @ int )
= ( ^ [S6: set @ int] :
? [K2: int] : ( ord_less_eq @ ( set @ int ) @ ( image2 @ int @ int @ ( abs_abs @ int ) @ S6 ) @ ( set_ord_atMost @ int @ K2 ) ) ) ) ).
% finite_int_iff_bounded_le
thf(fact_4260_finite__int__iff__bounded,axiom,
( ( finite_finite2 @ int )
= ( ^ [S6: set @ int] :
? [K2: int] : ( ord_less_eq @ ( set @ int ) @ ( image2 @ int @ int @ ( abs_abs @ int ) @ S6 ) @ ( set_ord_lessThan @ int @ K2 ) ) ) ) ).
% finite_int_iff_bounded
thf(fact_4261_UNION__empty__conv_I2_J,axiom,
! [A: $tType,B: $tType,B2: B > ( set @ A ),A4: set @ B] :
( ( ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B2 @ A4 ) )
= ( bot_bot @ ( set @ A ) ) )
= ( ! [X4: B] :
( ( member @ B @ X4 @ A4 )
=> ( ( B2 @ X4 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% UNION_empty_conv(2)
thf(fact_4262_UNION__empty__conv_I1_J,axiom,
! [A: $tType,B: $tType,B2: B > ( set @ A ),A4: set @ B] :
( ( ( bot_bot @ ( set @ A ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B2 @ A4 ) ) )
= ( ! [X4: B] :
( ( member @ B @ X4 @ A4 )
=> ( ( B2 @ X4 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% UNION_empty_conv(1)
thf(fact_4263_UN__empty,axiom,
! [B: $tType,A: $tType,B2: B > ( set @ A )] :
( ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B2 @ ( bot_bot @ ( set @ B ) ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% UN_empty
thf(fact_4264_UN__empty2,axiom,
! [B: $tType,A: $tType,A4: set @ B] :
( ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [X4: B] : ( bot_bot @ ( set @ A ) )
@ A4 ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% UN_empty2
thf(fact_4265_UN__subset__iff,axiom,
! [A: $tType,B: $tType,A4: B > ( set @ A ),I6: set @ B,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ A4 @ I6 ) ) @ B2 )
= ( ! [X4: B] :
( ( member @ B @ X4 @ I6 )
=> ( ord_less_eq @ ( set @ A ) @ ( A4 @ X4 ) @ B2 ) ) ) ) ).
% UN_subset_iff
thf(fact_4266_UN__upper,axiom,
! [B: $tType,A: $tType,A3: A,A4: set @ A,B2: A > ( set @ B )] :
( ( member @ A @ A3 @ A4 )
=> ( ord_less_eq @ ( set @ B ) @ ( B2 @ A3 ) @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ B2 @ A4 ) ) ) ) ).
% UN_upper
thf(fact_4267_UN__least,axiom,
! [A: $tType,B: $tType,A4: set @ A,B2: A > ( set @ B ),C5: set @ B] :
( ! [X5: A] :
( ( member @ A @ X5 @ A4 )
=> ( ord_less_eq @ ( set @ B ) @ ( B2 @ X5 ) @ C5 ) )
=> ( ord_less_eq @ ( set @ B ) @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ B2 @ A4 ) ) @ C5 ) ) ).
% UN_least
thf(fact_4268_UN__mono,axiom,
! [B: $tType,A: $tType,A4: set @ A,B2: set @ A,F2: A > ( set @ B ),G: A > ( set @ B )] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B2 )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ A4 )
=> ( ord_less_eq @ ( set @ B ) @ ( F2 @ X5 ) @ ( G @ X5 ) ) )
=> ( ord_less_eq @ ( set @ B ) @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ F2 @ A4 ) ) @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ G @ B2 ) ) ) ) ) ).
% UN_mono
thf(fact_4269_INT__subset__iff,axiom,
! [A: $tType,B: $tType,B2: set @ A,A4: B > ( set @ A ),I6: set @ B] :
( ( ord_less_eq @ ( set @ A ) @ B2 @ ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ A4 @ I6 ) ) )
= ( ! [X4: B] :
( ( member @ B @ X4 @ I6 )
=> ( ord_less_eq @ ( set @ A ) @ B2 @ ( A4 @ X4 ) ) ) ) ) ).
% INT_subset_iff
thf(fact_4270_INT__anti__mono,axiom,
! [B: $tType,A: $tType,A4: set @ A,B2: set @ A,F2: A > ( set @ B ),G: A > ( set @ B )] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B2 )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ A4 )
=> ( ord_less_eq @ ( set @ B ) @ ( F2 @ X5 ) @ ( G @ X5 ) ) )
=> ( ord_less_eq @ ( set @ B ) @ ( complete_Inf_Inf @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ F2 @ B2 ) ) @ ( complete_Inf_Inf @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ G @ A4 ) ) ) ) ) ).
% INT_anti_mono
thf(fact_4271_INT__greatest,axiom,
! [B: $tType,A: $tType,A4: set @ A,C5: set @ B,B2: A > ( set @ B )] :
( ! [X5: A] :
( ( member @ A @ X5 @ A4 )
=> ( ord_less_eq @ ( set @ B ) @ C5 @ ( B2 @ X5 ) ) )
=> ( ord_less_eq @ ( set @ B ) @ C5 @ ( complete_Inf_Inf @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ B2 @ A4 ) ) ) ) ).
% INT_greatest
thf(fact_4272_INT__lower,axiom,
! [B: $tType,A: $tType,A3: A,A4: set @ A,B2: A > ( set @ B )] :
( ( member @ A @ A3 @ A4 )
=> ( ord_less_eq @ ( set @ B ) @ ( complete_Inf_Inf @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ B2 @ A4 ) ) @ ( B2 @ A3 ) ) ) ).
% INT_lower
thf(fact_4273_nat__def,axiom,
( nat2
= ( map_fun @ int @ ( product_prod @ nat @ nat ) @ nat @ nat @ rep_Integ @ ( id @ nat ) @ ( product_case_prod @ nat @ nat @ nat @ ( minus_minus @ nat ) ) ) ) ).
% nat_def
thf(fact_4274_set__take__subset__set__take,axiom,
! [A: $tType,M2: nat,N2: nat,Xs: list @ A] :
( ( ord_less_eq @ nat @ M2 @ N2 )
=> ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ ( take @ A @ M2 @ Xs ) ) @ ( set2 @ A @ ( take @ A @ N2 @ Xs ) ) ) ) ).
% set_take_subset_set_take
thf(fact_4275_in__image__insert__iff,axiom,
! [A: $tType,B2: set @ ( set @ A ),X: A,A4: set @ A] :
( ! [C7: set @ A] :
( ( member @ ( set @ A ) @ C7 @ B2 )
=> ~ ( member @ A @ X @ C7 ) )
=> ( ( member @ ( set @ A ) @ A4 @ ( image2 @ ( set @ A ) @ ( set @ A ) @ ( insert2 @ A @ X ) @ B2 ) )
= ( ( member @ A @ X @ A4 )
& ( member @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) @ B2 ) ) ) ) ).
% in_image_insert_iff
thf(fact_4276_image__int__atLeastAtMost,axiom,
! [A3: nat,B3: nat] :
( ( image2 @ 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_4277_UN__extend__simps_I1_J,axiom,
! [A: $tType,B: $tType,C5: set @ B,A3: A,B2: B > ( set @ A )] :
( ( ( C5
= ( bot_bot @ ( set @ B ) ) )
=> ( ( insert2 @ A @ A3 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B2 @ C5 ) ) )
= ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) )
& ( ( C5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( insert2 @ A @ A3 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B2 @ C5 ) ) )
= ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [X4: B] : ( insert2 @ A @ A3 @ ( B2 @ X4 ) )
@ C5 ) ) ) ) ) ).
% UN_extend_simps(1)
thf(fact_4278_image__int__atLeastLessThan,axiom,
! [A3: nat,B3: nat] :
( ( image2 @ 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_4279_INT__extend__simps_I2_J,axiom,
! [C: $tType,D: $tType,C5: set @ D,A4: set @ C,B2: D > ( set @ C )] :
( ( ( C5
= ( bot_bot @ ( set @ D ) ) )
=> ( ( inf_inf @ ( set @ C ) @ A4 @ ( complete_Inf_Inf @ ( set @ C ) @ ( image2 @ D @ ( set @ C ) @ B2 @ C5 ) ) )
= A4 ) )
& ( ( C5
!= ( bot_bot @ ( set @ D ) ) )
=> ( ( inf_inf @ ( set @ C ) @ A4 @ ( complete_Inf_Inf @ ( set @ C ) @ ( image2 @ D @ ( set @ C ) @ B2 @ C5 ) ) )
= ( complete_Inf_Inf @ ( set @ C )
@ ( image2 @ D @ ( set @ C )
@ ^ [X4: D] : ( inf_inf @ ( set @ C ) @ A4 @ ( B2 @ X4 ) )
@ C5 ) ) ) ) ) ).
% INT_extend_simps(2)
thf(fact_4280_INT__extend__simps_I1_J,axiom,
! [B: $tType,A: $tType,C5: set @ A,A4: A > ( set @ B ),B2: set @ B] :
( ( ( C5
= ( bot_bot @ ( set @ A ) ) )
=> ( ( inf_inf @ ( set @ B ) @ ( complete_Inf_Inf @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ A4 @ C5 ) ) @ B2 )
= B2 ) )
& ( ( C5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( inf_inf @ ( set @ B ) @ ( complete_Inf_Inf @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ A4 @ C5 ) ) @ B2 )
= ( complete_Inf_Inf @ ( set @ B )
@ ( image2 @ A @ ( set @ B )
@ ^ [X4: A] : ( inf_inf @ ( set @ B ) @ ( A4 @ X4 ) @ B2 )
@ C5 ) ) ) ) ) ).
% INT_extend_simps(1)
thf(fact_4281_Int__Inter__eq_I2_J,axiom,
! [A: $tType,B11: set @ ( set @ A ),A4: set @ A] :
( ( ( B11
= ( bot_bot @ ( set @ ( set @ A ) ) ) )
=> ( ( inf_inf @ ( set @ A ) @ ( complete_Inf_Inf @ ( set @ A ) @ B11 ) @ A4 )
= A4 ) )
& ( ( B11
!= ( bot_bot @ ( set @ ( set @ A ) ) ) )
=> ( ( inf_inf @ ( set @ A ) @ ( complete_Inf_Inf @ ( set @ A ) @ B11 ) @ A4 )
= ( complete_Inf_Inf @ ( set @ A )
@ ( image2 @ ( set @ A ) @ ( set @ A )
@ ^ [B6: set @ A] : ( inf_inf @ ( set @ A ) @ B6 @ A4 )
@ B11 ) ) ) ) ) ).
% Int_Inter_eq(2)
thf(fact_4282_Int__Inter__eq_I1_J,axiom,
! [A: $tType,B11: set @ ( set @ A ),A4: set @ A] :
( ( ( B11
= ( bot_bot @ ( set @ ( set @ A ) ) ) )
=> ( ( inf_inf @ ( set @ A ) @ A4 @ ( complete_Inf_Inf @ ( set @ A ) @ B11 ) )
= A4 ) )
& ( ( B11
!= ( bot_bot @ ( set @ ( set @ A ) ) ) )
=> ( ( inf_inf @ ( set @ A ) @ A4 @ ( complete_Inf_Inf @ ( set @ A ) @ B11 ) )
= ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ ( set @ A ) @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A4 ) @ B11 ) ) ) ) ) ).
% Int_Inter_eq(1)
thf(fact_4283_nth__take__lemma,axiom,
! [A: $tType,K: nat,Xs: list @ A,Ys: list @ A] :
( ( ord_less_eq @ nat @ K @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ord_less_eq @ nat @ K @ ( size_size @ ( list @ A ) @ Ys ) )
=> ( ! [I4: nat] :
( ( ord_less @ nat @ I4 @ K )
=> ( ( nth @ A @ Xs @ I4 )
= ( nth @ A @ Ys @ I4 ) ) )
=> ( ( take @ A @ K @ Xs )
= ( take @ A @ K @ Ys ) ) ) ) ) ).
% nth_take_lemma
thf(fact_4284_INT__extend__simps_I4_J,axiom,
! [G5: $tType,H4: $tType,C5: set @ H4,A4: set @ G5,B2: H4 > ( set @ G5 )] :
( ( ( C5
= ( bot_bot @ ( set @ H4 ) ) )
=> ( ( minus_minus @ ( set @ G5 ) @ A4 @ ( complete_Sup_Sup @ ( set @ G5 ) @ ( image2 @ H4 @ ( set @ G5 ) @ B2 @ C5 ) ) )
= A4 ) )
& ( ( C5
!= ( bot_bot @ ( set @ H4 ) ) )
=> ( ( minus_minus @ ( set @ G5 ) @ A4 @ ( complete_Sup_Sup @ ( set @ G5 ) @ ( image2 @ H4 @ ( set @ G5 ) @ B2 @ C5 ) ) )
= ( complete_Inf_Inf @ ( set @ G5 )
@ ( image2 @ H4 @ ( set @ G5 )
@ ^ [X4: H4] : ( minus_minus @ ( set @ G5 ) @ A4 @ ( B2 @ X4 ) )
@ C5 ) ) ) ) ) ).
% INT_extend_simps(4)
thf(fact_4285_UN__le__add__shift__strict,axiom,
! [A: $tType,M7: nat > ( set @ A ),K: nat,N2: nat] :
( ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ nat @ ( set @ A )
@ ^ [I3: nat] : ( M7 @ ( plus_plus @ nat @ I3 @ K ) )
@ ( set_ord_lessThan @ nat @ N2 ) ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ M7 @ ( set_or7035219750837199246ssThan @ nat @ K @ ( plus_plus @ nat @ N2 @ K ) ) ) ) ) ).
% UN_le_add_shift_strict
thf(fact_4286_UN__le__add__shift,axiom,
! [A: $tType,M7: nat > ( set @ A ),K: nat,N2: nat] :
( ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ nat @ ( set @ A )
@ ^ [I3: nat] : ( M7 @ ( plus_plus @ nat @ I3 @ K ) )
@ ( set_ord_atMost @ nat @ N2 ) ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ M7 @ ( set_or1337092689740270186AtMost @ nat @ K @ ( plus_plus @ nat @ N2 @ K ) ) ) ) ) ).
% UN_le_add_shift
thf(fact_4287_subset__subseqs,axiom,
! [A: $tType,X8: set @ A,Xs: list @ A] :
( ( ord_less_eq @ ( set @ A ) @ X8 @ ( set2 @ A @ Xs ) )
=> ( member @ ( set @ A ) @ X8 @ ( image2 @ ( list @ A ) @ ( set @ A ) @ ( set2 @ A ) @ ( set2 @ ( list @ A ) @ ( subseqs @ A @ Xs ) ) ) ) ) ).
% subset_subseqs
thf(fact_4288_subseqs__powset,axiom,
! [A: $tType,Xs: list @ A] :
( ( image2 @ ( list @ A ) @ ( set @ A ) @ ( set2 @ A ) @ ( set2 @ ( list @ A ) @ ( subseqs @ A @ Xs ) ) )
= ( pow2 @ A @ ( set2 @ A @ Xs ) ) ) ).
% subseqs_powset
thf(fact_4289_image__add__int__atLeastLessThan,axiom,
! [L: int,U: int] :
( ( image2 @ int @ int
@ ^ [X4: int] : ( plus_plus @ int @ X4 @ L )
@ ( set_or7035219750837199246ssThan @ int @ ( zero_zero @ int ) @ ( minus_minus @ int @ U @ L ) ) )
= ( set_or7035219750837199246ssThan @ int @ L @ U ) ) ).
% image_add_int_atLeastLessThan
thf(fact_4290_sum_OUNION__disjoint,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [I6: set @ B,A4: B > ( set @ C ),G: C > A] :
( ( finite_finite2 @ B @ I6 )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ I6 )
=> ( finite_finite2 @ C @ ( A4 @ X5 ) ) )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ I6 )
=> ! [Xa3: B] :
( ( member @ B @ Xa3 @ I6 )
=> ( ( X5 != Xa3 )
=> ( ( inf_inf @ ( set @ C ) @ ( A4 @ X5 ) @ ( A4 @ Xa3 ) )
= ( bot_bot @ ( set @ C ) ) ) ) ) )
=> ( ( groups7311177749621191930dd_sum @ C @ A @ G @ ( complete_Sup_Sup @ ( set @ C ) @ ( image2 @ B @ ( set @ C ) @ A4 @ I6 ) ) )
= ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X4: B] : ( groups7311177749621191930dd_sum @ C @ A @ G @ ( A4 @ X4 ) )
@ I6 ) ) ) ) ) ) ).
% sum.UNION_disjoint
thf(fact_4291_prod_OUNION__disjoint,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [I6: set @ B,A4: B > ( set @ C ),G: C > A] :
( ( finite_finite2 @ B @ I6 )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ I6 )
=> ( finite_finite2 @ C @ ( A4 @ X5 ) ) )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ I6 )
=> ! [Xa3: B] :
( ( member @ B @ Xa3 @ I6 )
=> ( ( X5 != Xa3 )
=> ( ( inf_inf @ ( set @ C ) @ ( A4 @ X5 ) @ ( A4 @ Xa3 ) )
= ( bot_bot @ ( set @ C ) ) ) ) ) )
=> ( ( groups7121269368397514597t_prod @ C @ A @ G @ ( complete_Sup_Sup @ ( set @ C ) @ ( image2 @ B @ ( set @ C ) @ A4 @ I6 ) ) )
= ( groups7121269368397514597t_prod @ B @ A
@ ^ [X4: B] : ( groups7121269368397514597t_prod @ C @ A @ G @ ( A4 @ X4 ) )
@ I6 ) ) ) ) ) ) ).
% prod.UNION_disjoint
thf(fact_4292_card__UN__le,axiom,
! [B: $tType,A: $tType,I6: set @ A,A4: A > ( set @ B )] :
( ( finite_finite2 @ A @ I6 )
=> ( ord_less_eq @ nat @ ( finite_card @ B @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ A4 @ I6 ) ) )
@ ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [I3: A] : ( finite_card @ B @ ( A4 @ I3 ) )
@ I6 ) ) ) ).
% card_UN_le
thf(fact_4293_image__atLeastZeroLessThan__int,axiom,
! [U: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ U )
=> ( ( set_or7035219750837199246ssThan @ int @ ( zero_zero @ int ) @ U )
= ( image2 @ nat @ int @ ( semiring_1_of_nat @ int ) @ ( set_ord_lessThan @ nat @ ( nat2 @ U ) ) ) ) ) ).
% image_atLeastZeroLessThan_int
thf(fact_4294_card__UN__disjoint,axiom,
! [B: $tType,A: $tType,I6: set @ A,A4: A > ( set @ B )] :
( ( finite_finite2 @ A @ I6 )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ I6 )
=> ( finite_finite2 @ B @ ( A4 @ X5 ) ) )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ I6 )
=> ! [Xa3: A] :
( ( member @ A @ Xa3 @ I6 )
=> ( ( X5 != Xa3 )
=> ( ( inf_inf @ ( set @ B ) @ ( A4 @ X5 ) @ ( A4 @ Xa3 ) )
= ( bot_bot @ ( set @ B ) ) ) ) ) )
=> ( ( finite_card @ B @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ A4 @ I6 ) ) )
= ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [I3: A] : ( finite_card @ B @ ( A4 @ I3 ) )
@ I6 ) ) ) ) ) ).
% card_UN_disjoint
thf(fact_4295_UN__image__subset,axiom,
! [C: $tType,A: $tType,B: $tType,F2: B > ( set @ A ),G: C > ( set @ B ),X: C,X8: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ F2 @ ( G @ X ) ) ) @ X8 )
= ( ord_less_eq @ ( set @ B ) @ ( G @ X )
@ ( collect @ B
@ ^ [X4: B] : ( ord_less_eq @ ( set @ A ) @ ( F2 @ X4 ) @ X8 ) ) ) ) ).
% UN_image_subset
thf(fact_4296_folding__on_Oremove,axiom,
! [B: $tType,A: $tType,S2: set @ A,F2: A > B > B,A4: set @ A,X: A,Z3: B] :
( ( finite_folding_on @ A @ B @ S2 @ F2 )
=> ( ( ord_less_eq @ ( set @ A ) @ A4 @ S2 )
=> ( ( finite_finite2 @ A @ A4 )
=> ( ( member @ A @ X @ A4 )
=> ( ( finite_folding_F @ A @ B @ F2 @ Z3 @ A4 )
= ( F2 @ X @ ( finite_folding_F @ A @ B @ F2 @ Z3 @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ).
% folding_on.remove
thf(fact_4297_folding__on_Oinsert__remove,axiom,
! [B: $tType,A: $tType,S2: set @ A,F2: A > B > B,X: A,A4: set @ A,Z3: B] :
( ( finite_folding_on @ A @ B @ S2 @ F2 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( insert2 @ A @ X @ A4 ) @ S2 )
=> ( ( finite_finite2 @ A @ A4 )
=> ( ( finite_folding_F @ A @ B @ F2 @ Z3 @ ( insert2 @ A @ X @ A4 ) )
= ( F2 @ X @ ( finite_folding_F @ A @ B @ F2 @ Z3 @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ).
% folding_on.insert_remove
thf(fact_4298_folding__idem__on_Oinsert__idem,axiom,
! [B: $tType,A: $tType,S2: set @ A,F2: A > B > B,X: A,A4: set @ A,Z3: B] :
( ( finite1890593828518410140dem_on @ A @ B @ S2 @ F2 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( insert2 @ A @ X @ A4 ) @ S2 )
=> ( ( finite_finite2 @ A @ A4 )
=> ( ( finite_folding_F @ A @ B @ F2 @ Z3 @ ( insert2 @ A @ X @ A4 ) )
= ( F2 @ X @ ( finite_folding_F @ A @ B @ F2 @ Z3 @ A4 ) ) ) ) ) ) ).
% folding_idem_on.insert_idem
thf(fact_4299_INF__filter__bot__base,axiom,
! [B: $tType,A: $tType,I6: set @ A,F5: A > ( filter @ B )] :
( ! [I4: A] :
( ( member @ A @ I4 @ I6 )
=> ! [J3: A] :
( ( member @ A @ J3 @ I6 )
=> ? [X2: A] :
( ( member @ A @ X2 @ I6 )
& ( ord_less_eq @ ( filter @ B ) @ ( F5 @ X2 ) @ ( inf_inf @ ( filter @ B ) @ ( F5 @ I4 ) @ ( F5 @ J3 ) ) ) ) ) )
=> ( ( ( complete_Inf_Inf @ ( filter @ B ) @ ( image2 @ A @ ( filter @ B ) @ F5 @ I6 ) )
= ( bot_bot @ ( filter @ B ) ) )
= ( ? [X4: A] :
( ( member @ A @ X4 @ I6 )
& ( ( F5 @ X4 )
= ( bot_bot @ ( filter @ B ) ) ) ) ) ) ) ).
% INF_filter_bot_base
thf(fact_4300_Inf__filter__not__bot,axiom,
! [A: $tType,B2: set @ ( filter @ A )] :
( ! [X9: set @ ( filter @ A )] :
( ( ord_less_eq @ ( set @ ( filter @ A ) ) @ X9 @ B2 )
=> ( ( finite_finite2 @ ( filter @ A ) @ X9 )
=> ( ( complete_Inf_Inf @ ( filter @ A ) @ X9 )
!= ( bot_bot @ ( filter @ A ) ) ) ) )
=> ( ( complete_Inf_Inf @ ( filter @ A ) @ B2 )
!= ( bot_bot @ ( filter @ A ) ) ) ) ).
% Inf_filter_not_bot
thf(fact_4301_folding__idem__on_Oaxioms_I1_J,axiom,
! [B: $tType,A: $tType,S2: set @ A,F2: A > B > B] :
( ( finite1890593828518410140dem_on @ A @ B @ S2 @ F2 )
=> ( finite_folding_on @ A @ B @ S2 @ F2 ) ) ).
% folding_idem_on.axioms(1)
thf(fact_4302_folding__on_Oempty,axiom,
! [A: $tType,B: $tType,S2: set @ A,F2: A > B > B,Z3: B] :
( ( finite_folding_on @ A @ B @ S2 @ F2 )
=> ( ( finite_folding_F @ A @ B @ F2 @ Z3 @ ( bot_bot @ ( set @ A ) ) )
= Z3 ) ) ).
% folding_on.empty
thf(fact_4303_folding__on_Oinfinite,axiom,
! [A: $tType,B: $tType,S2: set @ A,F2: A > B > B,A4: set @ A,Z3: B] :
( ( finite_folding_on @ A @ B @ S2 @ F2 )
=> ( ~ ( finite_finite2 @ A @ A4 )
=> ( ( finite_folding_F @ A @ B @ F2 @ Z3 @ A4 )
= Z3 ) ) ) ).
% folding_on.infinite
thf(fact_4304_conj__subset__def,axiom,
! [A: $tType,A4: set @ A,P: A > $o,Q: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ A4
@ ( collect @ A
@ ^ [X4: A] :
( ( P @ X4 )
& ( Q @ X4 ) ) ) )
= ( ( ord_less_eq @ ( set @ A ) @ A4 @ ( collect @ A @ P ) )
& ( ord_less_eq @ ( set @ A ) @ A4 @ ( collect @ A @ Q ) ) ) ) ).
% conj_subset_def
thf(fact_4305_card__def,axiom,
! [B: $tType] :
( ( finite_card @ B )
= ( finite_folding_F @ B @ nat
@ ^ [Uu3: B] : suc
@ ( zero_zero @ nat ) ) ) ).
% card_def
thf(fact_4306_folding__on_Oinsert,axiom,
! [B: $tType,A: $tType,S2: set @ A,F2: A > B > B,X: A,A4: set @ A,Z3: B] :
( ( finite_folding_on @ A @ B @ S2 @ F2 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( insert2 @ A @ X @ A4 ) @ S2 )
=> ( ( finite_finite2 @ A @ A4 )
=> ( ~ ( member @ A @ X @ A4 )
=> ( ( finite_folding_F @ A @ B @ F2 @ Z3 @ ( insert2 @ A @ X @ A4 ) )
= ( F2 @ X @ ( finite_folding_F @ A @ B @ F2 @ Z3 @ A4 ) ) ) ) ) ) ) ).
% folding_on.insert
thf(fact_4307_length__remdups__concat,axiom,
! [A: $tType,Xss: list @ ( list @ A )] :
( ( size_size @ ( list @ A ) @ ( remdups @ A @ ( concat @ A @ Xss ) ) )
= ( finite_card @ A @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ ( list @ A ) @ ( set @ A ) @ ( set2 @ A ) @ ( set2 @ ( list @ A ) @ Xss ) ) ) ) ) ).
% length_remdups_concat
thf(fact_4308_finite__mono__strict__prefix__implies__finite__fixpoint,axiom,
! [A: $tType,F2: nat > ( set @ A ),S2: set @ A] :
( ! [I4: nat] : ( ord_less_eq @ ( set @ A ) @ ( F2 @ I4 ) @ S2 )
=> ( ( finite_finite2 @ A @ S2 )
=> ( ? [N8: nat] :
( ! [N3: nat] :
( ( ord_less_eq @ nat @ N3 @ N8 )
=> ! [M3: nat] :
( ( ord_less_eq @ nat @ M3 @ N8 )
=> ( ( ord_less @ nat @ M3 @ N3 )
=> ( ord_less @ ( set @ A ) @ ( F2 @ M3 ) @ ( F2 @ N3 ) ) ) ) )
& ! [N3: nat] :
( ( ord_less_eq @ nat @ N8 @ N3 )
=> ( ( F2 @ N8 )
= ( F2 @ N3 ) ) ) )
=> ( ( F2 @ ( finite_card @ A @ S2 ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ F2 @ ( top_top @ ( set @ nat ) ) ) ) ) ) ) ) ).
% finite_mono_strict_prefix_implies_finite_fixpoint
thf(fact_4309_lex__take__index,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( lex @ A @ R2 ) )
=> ~ ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ord_less @ nat @ I4 @ ( size_size @ ( list @ A ) @ Ys ) )
=> ( ( ( take @ A @ I4 @ Xs )
= ( take @ A @ I4 @ Ys ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( nth @ A @ Xs @ I4 ) @ ( nth @ A @ Ys @ I4 ) ) @ R2 ) ) ) ) ) ).
% lex_take_index
thf(fact_4310_INT__simps_I4_J,axiom,
! [G5: $tType,H4: $tType,C5: set @ H4,A4: set @ G5,B2: H4 > ( set @ G5 )] :
( ( ( C5
= ( bot_bot @ ( set @ H4 ) ) )
=> ( ( complete_Inf_Inf @ ( set @ G5 )
@ ( image2 @ H4 @ ( set @ G5 )
@ ^ [X4: H4] : ( minus_minus @ ( set @ G5 ) @ A4 @ ( B2 @ X4 ) )
@ C5 ) )
= ( top_top @ ( set @ G5 ) ) ) )
& ( ( C5
!= ( bot_bot @ ( set @ H4 ) ) )
=> ( ( complete_Inf_Inf @ ( set @ G5 )
@ ( image2 @ H4 @ ( set @ G5 )
@ ^ [X4: H4] : ( minus_minus @ ( set @ G5 ) @ A4 @ ( B2 @ X4 ) )
@ C5 ) )
= ( minus_minus @ ( set @ G5 ) @ A4 @ ( complete_Sup_Sup @ ( set @ G5 ) @ ( image2 @ H4 @ ( set @ G5 ) @ B2 @ C5 ) ) ) ) ) ) ).
% INT_simps(4)
thf(fact_4311_top__apply,axiom,
! [C: $tType,D: $tType] :
( ( top @ C )
=> ( ( top_top @ ( D > C ) )
= ( ^ [X4: D] : ( top_top @ C ) ) ) ) ).
% top_apply
thf(fact_4312_atMost__UNIV__triv,axiom,
! [A: $tType] :
( ( set_ord_atMost @ ( set @ A ) @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ ( set @ A ) ) ) ) ).
% atMost_UNIV_triv
thf(fact_4313_UNIV__I,axiom,
! [A: $tType,X: A] : ( member @ A @ X @ ( top_top @ ( set @ A ) ) ) ).
% UNIV_I
thf(fact_4314_card__UNIV__unit,axiom,
( ( finite_card @ product_unit @ ( top_top @ ( set @ product_unit ) ) )
= ( one_one @ nat ) ) ).
% card_UNIV_unit
thf(fact_4315_finite__option__UNIV,axiom,
! [A: $tType] :
( ( finite_finite2 @ ( option @ A ) @ ( top_top @ ( set @ ( option @ A ) ) ) )
= ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) ) ) ).
% finite_option_UNIV
thf(fact_4316_finite__Plus__UNIV__iff,axiom,
! [A: $tType,B: $tType] :
( ( finite_finite2 @ ( sum_sum @ A @ B ) @ ( top_top @ ( set @ ( sum_sum @ A @ B ) ) ) )
= ( ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) )
& ( finite_finite2 @ B @ ( top_top @ ( set @ B ) ) ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_4317_Int__UNIV,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] :
( ( ( inf_inf @ ( set @ A ) @ A4 @ B2 )
= ( top_top @ ( set @ A ) ) )
= ( ( A4
= ( top_top @ ( set @ A ) ) )
& ( B2
= ( top_top @ ( set @ A ) ) ) ) ) ).
% Int_UNIV
thf(fact_4318_range__mult,axiom,
! [A3: real] :
( ( ( A3
= ( zero_zero @ real ) )
=> ( ( image2 @ real @ real @ ( times_times @ real @ A3 ) @ ( top_top @ ( set @ real ) ) )
= ( insert2 @ real @ ( zero_zero @ real ) @ ( bot_bot @ ( set @ real ) ) ) ) )
& ( ( A3
!= ( zero_zero @ real ) )
=> ( ( image2 @ real @ real @ ( times_times @ real @ A3 ) @ ( top_top @ ( set @ real ) ) )
= ( top_top @ ( set @ real ) ) ) ) ) ).
% range_mult
thf(fact_4319_max__top2,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [X: A] :
( ( ord_max @ A @ X @ ( top_top @ A ) )
= ( top_top @ A ) ) ) ).
% max_top2
thf(fact_4320_max__top,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [X: A] :
( ( ord_max @ A @ ( top_top @ A ) @ X )
= ( top_top @ A ) ) ) ).
% max_top
thf(fact_4321_Pow__UNIV,axiom,
! [A: $tType] :
( ( pow2 @ A @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ ( set @ A ) ) ) ) ).
% Pow_UNIV
thf(fact_4322_remdups__eq__nil__iff,axiom,
! [A: $tType,X: list @ A] :
( ( ( remdups @ A @ X )
= ( nil @ A ) )
= ( X
= ( nil @ A ) ) ) ).
% remdups_eq_nil_iff
thf(fact_4323_remdups__eq__nil__right__iff,axiom,
! [A: $tType,X: list @ A] :
( ( ( nil @ A )
= ( remdups @ A @ X ) )
= ( X
= ( nil @ A ) ) ) ).
% remdups_eq_nil_right_iff
thf(fact_4324_set__remdups,axiom,
! [A: $tType,Xs: list @ A] :
( ( set2 @ A @ ( remdups @ A @ Xs ) )
= ( set2 @ A @ Xs ) ) ).
% set_remdups
thf(fact_4325_length__remdups__eq,axiom,
! [A: $tType,Xs: list @ A] :
( ( ( size_size @ ( list @ A ) @ ( remdups @ A @ Xs ) )
= ( size_size @ ( list @ A ) @ Xs ) )
= ( ( remdups @ A @ Xs )
= Xs ) ) ).
% length_remdups_eq
thf(fact_4326_distinct__remdups,axiom,
! [A: $tType,Xs: list @ A] : ( distinct @ A @ ( remdups @ A @ Xs ) ) ).
% distinct_remdups
thf(fact_4327_remdups__id__iff__distinct,axiom,
! [A: $tType,Xs: list @ A] :
( ( ( remdups @ A @ Xs )
= Xs )
= ( distinct @ A @ Xs ) ) ).
% remdups_id_iff_distinct
thf(fact_4328_Collect__const,axiom,
! [A: $tType,P: $o] :
( ( P
=> ( ( collect @ A
@ ^ [S7: A] : P )
= ( top_top @ ( set @ A ) ) ) )
& ( ~ P
=> ( ( collect @ A
@ ^ [S7: A] : P )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Collect_const
thf(fact_4329_finite__Collect__not,axiom,
! [A: $tType,P: A > $o] :
( ( finite_finite2 @ A @ ( collect @ A @ P ) )
=> ( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X4: A] :
~ ( P @ X4 ) ) )
= ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) ) ) ) ).
% finite_Collect_not
thf(fact_4330_surj__plus,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A3: A] :
( ( image2 @ A @ A @ ( plus_plus @ A @ A3 ) @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ A ) ) ) ) ).
% surj_plus
thf(fact_4331_range__add,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A] :
( ( image2 @ A @ A @ ( plus_plus @ A @ A3 ) @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ A ) ) ) ) ).
% range_add
thf(fact_4332_Sup__eq__top__iff,axiom,
! [A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [A4: set @ A] :
( ( ( complete_Sup_Sup @ A @ A4 )
= ( top_top @ A ) )
= ( ! [X4: A] :
( ( ord_less @ A @ X4 @ ( top_top @ A ) )
=> ? [Y6: A] :
( ( member @ A @ Y6 @ A4 )
& ( ord_less @ A @ X4 @ Y6 ) ) ) ) ) ) ).
% Sup_eq_top_iff
thf(fact_4333_boolean__algebra_Ocompl__one,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ( ( uminus_uminus @ A @ ( top_top @ A ) )
= ( bot_bot @ A ) ) ) ).
% boolean_algebra.compl_one
thf(fact_4334_boolean__algebra_Ocompl__zero,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ( ( uminus_uminus @ A @ ( bot_bot @ A ) )
= ( top_top @ A ) ) ) ).
% boolean_algebra.compl_zero
thf(fact_4335_Inf__UNIV,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ( ( complete_Inf_Inf @ A @ ( top_top @ ( set @ A ) ) )
= ( bot_bot @ A ) ) ) ).
% Inf_UNIV
thf(fact_4336_ccInf__empty,axiom,
! [A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ( ( complete_Inf_Inf @ A @ ( bot_bot @ ( set @ A ) ) )
= ( top_top @ A ) ) ) ).
% ccInf_empty
thf(fact_4337_Inf__empty,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ( ( complete_Inf_Inf @ A @ ( bot_bot @ ( set @ A ) ) )
= ( top_top @ A ) ) ) ).
% Inf_empty
thf(fact_4338_Diff__UNIV,axiom,
! [A: $tType,A4: set @ A] :
( ( minus_minus @ ( set @ A ) @ A4 @ ( top_top @ ( set @ A ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Diff_UNIV
thf(fact_4339_surj__fn,axiom,
! [A: $tType,F2: A > A,N2: nat] :
( ( ( image2 @ A @ A @ F2 @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ A ) ) )
=> ( ( image2 @ A @ A @ ( compow @ ( A > A ) @ N2 @ F2 ) @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ A ) ) ) ) ).
% surj_fn
thf(fact_4340_finite__compl,axiom,
! [A: $tType,A4: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( finite_finite2 @ A @ ( uminus_uminus @ ( set @ A ) @ A4 ) )
= ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) ) ) ) ).
% finite_compl
thf(fact_4341_length__remdups__leq,axiom,
! [A: $tType,Xs: list @ A] : ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ ( remdups @ A @ Xs ) ) @ ( size_size @ ( list @ A ) @ Xs ) ) ).
% length_remdups_leq
thf(fact_4342_SUP__eq__top__iff,axiom,
! [B: $tType,A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [F2: B > A,A4: set @ B] :
( ( ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F2 @ A4 ) )
= ( top_top @ A ) )
= ( ! [X4: A] :
( ( ord_less @ A @ X4 @ ( top_top @ A ) )
=> ? [Y6: B] :
( ( member @ B @ Y6 @ A4 )
& ( ord_less @ A @ X4 @ ( F2 @ Y6 ) ) ) ) ) ) ) ).
% SUP_eq_top_iff
thf(fact_4343_range__constant,axiom,
! [B: $tType,A: $tType,X: A] :
( ( image2 @ B @ A
@ ^ [Uu3: B] : X
@ ( top_top @ ( set @ B ) ) )
= ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ).
% range_constant
thf(fact_4344_ccINF__empty,axiom,
! [B: $tType,A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [F2: B > A] :
( ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F2 @ ( bot_bot @ ( set @ B ) ) ) )
= ( top_top @ A ) ) ) ).
% ccINF_empty
thf(fact_4345_INT__constant,axiom,
! [B: $tType,A: $tType,A4: set @ B,C2: set @ A] :
( ( ( A4
= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Inf_Inf @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [Y6: B] : C2
@ A4 ) )
= ( top_top @ ( set @ A ) ) ) )
& ( ( A4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Inf_Inf @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [Y6: B] : C2
@ A4 ) )
= C2 ) ) ) ).
% INT_constant
thf(fact_4346_Inf__atMostLessThan,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [X: A] :
( ( ord_less @ A @ ( top_top @ A ) @ X )
=> ( ( complete_Inf_Inf @ A @ ( set_ord_lessThan @ A @ X ) )
= ( bot_bot @ A ) ) ) ) ).
% Inf_atMostLessThan
thf(fact_4347_INT__simps_I2_J,axiom,
! [C: $tType,D: $tType,C5: set @ D,A4: set @ C,B2: D > ( set @ C )] :
( ( ( C5
= ( bot_bot @ ( set @ D ) ) )
=> ( ( complete_Inf_Inf @ ( set @ C )
@ ( image2 @ D @ ( set @ C )
@ ^ [X4: D] : ( inf_inf @ ( set @ C ) @ A4 @ ( B2 @ X4 ) )
@ C5 ) )
= ( top_top @ ( set @ C ) ) ) )
& ( ( C5
!= ( bot_bot @ ( set @ D ) ) )
=> ( ( complete_Inf_Inf @ ( set @ C )
@ ( image2 @ D @ ( set @ C )
@ ^ [X4: D] : ( inf_inf @ ( set @ C ) @ A4 @ ( B2 @ X4 ) )
@ C5 ) )
= ( inf_inf @ ( set @ C ) @ A4 @ ( complete_Inf_Inf @ ( set @ C ) @ ( image2 @ D @ ( set @ C ) @ B2 @ C5 ) ) ) ) ) ) ).
% INT_simps(2)
thf(fact_4348_INT__simps_I1_J,axiom,
! [A: $tType,B: $tType,C5: set @ A,A4: A > ( set @ B ),B2: set @ B] :
( ( ( C5
= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Inf_Inf @ ( set @ B )
@ ( image2 @ A @ ( set @ B )
@ ^ [X4: A] : ( inf_inf @ ( set @ B ) @ ( A4 @ X4 ) @ B2 )
@ C5 ) )
= ( top_top @ ( set @ B ) ) ) )
& ( ( C5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Inf_Inf @ ( set @ B )
@ ( image2 @ A @ ( set @ B )
@ ^ [X4: A] : ( inf_inf @ ( set @ B ) @ ( A4 @ X4 ) @ B2 )
@ C5 ) )
= ( inf_inf @ ( set @ B ) @ ( complete_Inf_Inf @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ A4 @ C5 ) ) @ B2 ) ) ) ) ).
% INT_simps(1)
thf(fact_4349_INT__simps_I3_J,axiom,
! [E4: $tType,F: $tType,C5: set @ E4,A4: E4 > ( set @ F ),B2: set @ F] :
( ( ( C5
= ( bot_bot @ ( set @ E4 ) ) )
=> ( ( complete_Inf_Inf @ ( set @ F )
@ ( image2 @ E4 @ ( set @ F )
@ ^ [X4: E4] : ( minus_minus @ ( set @ F ) @ ( A4 @ X4 ) @ B2 )
@ C5 ) )
= ( top_top @ ( set @ F ) ) ) )
& ( ( C5
!= ( bot_bot @ ( set @ E4 ) ) )
=> ( ( complete_Inf_Inf @ ( set @ F )
@ ( image2 @ E4 @ ( set @ F )
@ ^ [X4: E4] : ( minus_minus @ ( set @ F ) @ ( A4 @ X4 ) @ B2 )
@ C5 ) )
= ( minus_minus @ ( set @ F ) @ ( complete_Inf_Inf @ ( set @ F ) @ ( image2 @ E4 @ ( set @ F ) @ A4 @ C5 ) ) @ B2 ) ) ) ) ).
% INT_simps(3)
thf(fact_4350_range__eqI,axiom,
! [A: $tType,B: $tType,B3: A,F2: B > A,X: B] :
( ( B3
= ( F2 @ X ) )
=> ( member @ A @ B3 @ ( image2 @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) ) ) ) ).
% range_eqI
thf(fact_4351_rangeI,axiom,
! [A: $tType,B: $tType,F2: B > A,X: B] : ( member @ A @ ( F2 @ X ) @ ( image2 @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) ) ) ).
% rangeI
thf(fact_4352_empty__not__UNIV,axiom,
! [A: $tType] :
( ( bot_bot @ ( set @ A ) )
!= ( top_top @ ( set @ A ) ) ) ).
% empty_not_UNIV
thf(fact_4353_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_4354_top__greatest,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [A3: A] : ( ord_less_eq @ A @ A3 @ ( top_top @ A ) ) ) ).
% top_greatest
thf(fact_4355_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_4356_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_4357_subset__UNIV,axiom,
! [A: $tType,A4: set @ A] : ( ord_less_eq @ ( set @ A ) @ A4 @ ( top_top @ ( set @ A ) ) ) ).
% subset_UNIV
thf(fact_4358_atMost__eq__UNIV__iff,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [X: A] :
( ( ( set_ord_atMost @ A @ X )
= ( top_top @ ( set @ A ) ) )
= ( X
= ( top_top @ A ) ) ) ) ).
% atMost_eq_UNIV_iff
thf(fact_4359_not__UNIV__eq__Iic,axiom,
! [A: $tType] :
( ( no_top @ A )
=> ! [H3: A] :
( ( top_top @ ( set @ A ) )
!= ( set_ord_atMost @ A @ H3 ) ) ) ).
% not_UNIV_eq_Iic
thf(fact_4360_remdups_Osimps_I1_J,axiom,
! [A: $tType] :
( ( remdups @ A @ ( nil @ A ) )
= ( nil @ A ) ) ).
% remdups.simps(1)
thf(fact_4361_nat__not__finite,axiom,
~ ( finite_finite2 @ nat @ ( top_top @ ( set @ nat ) ) ) ).
% nat_not_finite
thf(fact_4362_infinite__UNIV__nat,axiom,
~ ( finite_finite2 @ nat @ ( top_top @ ( set @ nat ) ) ) ).
% infinite_UNIV_nat
thf(fact_4363_top_Oextremum__strict,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [A3: A] :
~ ( ord_less @ A @ ( top_top @ A ) @ A3 ) ) ).
% top.extremum_strict
thf(fact_4364_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_4365_finite__Prod__UNIV,axiom,
! [B: $tType,A: $tType] :
( ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) )
=> ( ( finite_finite2 @ B @ ( top_top @ ( set @ B ) ) )
=> ( finite_finite2 @ ( product_prod @ A @ B ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) ) ) ).
% finite_Prod_UNIV
thf(fact_4366_finite__prod,axiom,
! [A: $tType,B: $tType] :
( ( finite_finite2 @ ( product_prod @ A @ B ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) )
= ( ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) )
& ( finite_finite2 @ B @ ( top_top @ ( set @ B ) ) ) ) ) ).
% finite_prod
thf(fact_4367_finite__fun__UNIVD2,axiom,
! [A: $tType,B: $tType] :
( ( finite_finite2 @ ( A > B ) @ ( top_top @ ( set @ ( A > B ) ) ) )
=> ( finite_finite2 @ B @ ( top_top @ ( set @ B ) ) ) ) ).
% finite_fun_UNIVD2
thf(fact_4368_remdups__remdups,axiom,
! [A: $tType,Xs: list @ A] :
( ( remdups @ A @ ( remdups @ A @ Xs ) )
= ( remdups @ A @ Xs ) ) ).
% remdups_remdups
thf(fact_4369_UNIV__eq__I,axiom,
! [A: $tType,A4: set @ A] :
( ! [X5: A] : ( member @ A @ X5 @ A4 )
=> ( ( top_top @ ( set @ A ) )
= A4 ) ) ).
% UNIV_eq_I
thf(fact_4370_UNIV__witness,axiom,
! [A: $tType] :
? [X5: A] : ( member @ A @ X5 @ ( top_top @ ( set @ A ) ) ) ).
% UNIV_witness
thf(fact_4371_UNIV__def,axiom,
! [A: $tType] :
( ( top_top @ ( set @ A ) )
= ( collect @ A
@ ^ [X4: A] : $true ) ) ).
% UNIV_def
thf(fact_4372_Finite__Set_Ofinite__set,axiom,
! [A: $tType] :
( ( finite_finite2 @ ( set @ A ) @ ( top_top @ ( set @ ( set @ A ) ) ) )
= ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) ) ) ).
% Finite_Set.finite_set
thf(fact_4373_finite__UNIV,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) ) ) ).
% finite_UNIV
thf(fact_4374_ex__new__if__finite,axiom,
! [A: $tType,A4: set @ A] :
( ~ ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ A4 )
=> ? [A6: A] :
~ ( member @ A @ A6 @ A4 ) ) ) ).
% ex_new_if_finite
thf(fact_4375_infinite__UNIV__char__0,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ~ ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) ) ) ).
% infinite_UNIV_char_0
thf(fact_4376_not__UNIV__eq__Icc,axiom,
! [A: $tType] :
( ( no_top @ A )
=> ! [L3: A,H3: A] :
( ( top_top @ ( set @ A ) )
!= ( set_or1337092689740270186AtMost @ A @ L3 @ H3 ) ) ) ).
% not_UNIV_eq_Icc
thf(fact_4377_Int__UNIV__right,axiom,
! [A: $tType,A4: set @ A] :
( ( inf_inf @ ( set @ A ) @ A4 @ ( top_top @ ( set @ A ) ) )
= A4 ) ).
% Int_UNIV_right
thf(fact_4378_Int__UNIV__left,axiom,
! [A: $tType,B2: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( top_top @ ( set @ A ) ) @ B2 )
= B2 ) ).
% Int_UNIV_left
thf(fact_4379_distinct__remdups__id,axiom,
! [A: $tType,Xs: list @ A] :
( ( distinct @ A @ Xs )
=> ( ( remdups @ A @ Xs )
= Xs ) ) ).
% distinct_remdups_id
thf(fact_4380_insert__UNIV,axiom,
! [A: $tType,X: A] :
( ( insert2 @ A @ X @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ A ) ) ) ).
% insert_UNIV
thf(fact_4381_infinite__UNIV__int,axiom,
~ ( finite_finite2 @ int @ ( top_top @ ( set @ int ) ) ) ).
% infinite_UNIV_int
thf(fact_4382_range__composition,axiom,
! [A: $tType,C: $tType,B: $tType,F2: C > A,G: B > C] :
( ( image2 @ B @ A
@ ^ [X4: B] : ( F2 @ ( G @ X4 ) )
@ ( top_top @ ( set @ B ) ) )
= ( image2 @ C @ A @ F2 @ ( image2 @ B @ C @ G @ ( top_top @ ( set @ B ) ) ) ) ) ).
% range_composition
thf(fact_4383_rangeE,axiom,
! [A: $tType,B: $tType,B3: A,F2: B > A] :
( ( member @ A @ B3 @ ( image2 @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) ) )
=> ~ ! [X5: B] :
( B3
!= ( F2 @ X5 ) ) ) ).
% rangeE
thf(fact_4384_card_Ofolding__on__axioms,axiom,
! [A: $tType] :
( finite_folding_on @ A @ nat @ ( top_top @ ( set @ A ) )
@ ^ [Uu3: A] : suc ) ).
% card.folding_on_axioms
thf(fact_4385_UN__lessThan__UNIV,axiom,
( ( complete_Sup_Sup @ ( set @ nat ) @ ( image2 @ nat @ ( set @ nat ) @ ( set_ord_lessThan @ nat ) @ ( top_top @ ( set @ nat ) ) ) )
= ( top_top @ ( set @ nat ) ) ) ).
% UN_lessThan_UNIV
thf(fact_4386_UN__atMost__UNIV,axiom,
( ( complete_Sup_Sup @ ( set @ nat ) @ ( image2 @ nat @ ( set @ nat ) @ ( set_ord_atMost @ nat ) @ ( top_top @ ( set @ nat ) ) ) )
= ( top_top @ ( set @ nat ) ) ) ).
% UN_atMost_UNIV
thf(fact_4387_sorted__list__of__set_Ofold__insort__key_Ofolding__on__axioms,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( finite_folding_on @ A @ ( list @ A ) @ ( top_top @ ( set @ A ) )
@ ( linorder_insort_key @ A @ A
@ ^ [X4: A] : X4 ) ) ) ).
% sorted_list_of_set.fold_insort_key.folding_on_axioms
thf(fact_4388_infinite__UNIV__listI,axiom,
! [A: $tType] :
~ ( finite_finite2 @ ( list @ A ) @ ( top_top @ ( set @ ( list @ A ) ) ) ) ).
% infinite_UNIV_listI
thf(fact_4389_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_4390_range__subsetD,axiom,
! [B: $tType,A: $tType,F2: B > A,B2: set @ A,I: B] :
( ( ord_less_eq @ ( set @ A ) @ ( image2 @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) ) @ B2 )
=> ( member @ A @ ( F2 @ I ) @ B2 ) ) ).
% range_subsetD
thf(fact_4391_perfect__space__class_OUNIV__not__singleton,axiom,
! [A: $tType] :
( ( topolo8386298272705272623_space @ A )
=> ! [X: A] :
( ( top_top @ ( set @ A ) )
!= ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% perfect_space_class.UNIV_not_singleton
thf(fact_4392_not__UNIV__le__Icc,axiom,
! [A: $tType] :
( ( no_top @ A )
=> ! [L: A,H2: A] :
~ ( ord_less_eq @ ( set @ A ) @ ( top_top @ ( set @ A ) ) @ ( set_or1337092689740270186AtMost @ A @ L @ H2 ) ) ) ).
% not_UNIV_le_Icc
thf(fact_4393_card__eq__UNIV__imp__eq__UNIV,axiom,
! [A: $tType,A4: set @ A] :
( ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) )
=> ( ( ( finite_card @ A @ A4 )
= ( finite_card @ A @ ( top_top @ ( set @ A ) ) ) )
=> ( A4
= ( top_top @ ( set @ A ) ) ) ) ) ).
% card_eq_UNIV_imp_eq_UNIV
thf(fact_4394_finite__range__Some,axiom,
! [A: $tType] :
( ( finite_finite2 @ ( option @ A ) @ ( image2 @ A @ ( option @ A ) @ ( some @ A ) @ ( top_top @ ( set @ A ) ) ) )
= ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) ) ) ).
% finite_range_Some
thf(fact_4395_not__UNIV__le__Iic,axiom,
! [A: $tType] :
( ( no_top @ A )
=> ! [H2: A] :
~ ( ord_less_eq @ ( set @ A ) @ ( top_top @ ( set @ A ) ) @ ( set_ord_atMost @ A @ H2 ) ) ) ).
% not_UNIV_le_Iic
thf(fact_4396_Inter__UNIV,axiom,
! [A: $tType] :
( ( complete_Inf_Inf @ ( set @ A ) @ ( top_top @ ( set @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Inter_UNIV
thf(fact_4397_Compl__UNIV__eq,axiom,
! [A: $tType] :
( ( uminus_uminus @ ( set @ A ) @ ( top_top @ ( set @ A ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Compl_UNIV_eq
thf(fact_4398_Compl__empty__eq,axiom,
! [A: $tType] :
( ( uminus_uminus @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) )
= ( top_top @ ( set @ A ) ) ) ).
% Compl_empty_eq
thf(fact_4399_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_4400_Inter__empty,axiom,
! [A: $tType] :
( ( complete_Inf_Inf @ ( set @ A ) @ ( bot_bot @ ( set @ ( set @ A ) ) ) )
= ( top_top @ ( set @ A ) ) ) ).
% Inter_empty
thf(fact_4401_finite__range__imageI,axiom,
! [C: $tType,A: $tType,B: $tType,G: B > A,F2: A > C] :
( ( finite_finite2 @ A @ ( image2 @ B @ A @ G @ ( top_top @ ( set @ B ) ) ) )
=> ( finite_finite2 @ C
@ ( image2 @ B @ C
@ ^ [X4: B] : ( F2 @ ( G @ X4 ) )
@ ( top_top @ ( set @ B ) ) ) ) ) ).
% finite_range_imageI
thf(fact_4402_remove1__remdups,axiom,
! [A: $tType,Xs: list @ A,X: A] :
( ( distinct @ A @ Xs )
=> ( ( remove1 @ A @ X @ ( remdups @ A @ Xs ) )
= ( remdups @ A @ ( remove1 @ A @ X @ Xs ) ) ) ) ).
% remove1_remdups
thf(fact_4403_Nil2__notin__lex,axiom,
! [A: $tType,Xs: list @ A,R2: set @ ( product_prod @ A @ A )] :
~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ ( nil @ A ) ) @ ( lex @ A @ R2 ) ) ).
% Nil2_notin_lex
thf(fact_4404_Nil__notin__lex,axiom,
! [A: $tType,Ys: list @ A,R2: set @ ( product_prod @ A @ A )] :
~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Ys ) @ ( lex @ A @ R2 ) ) ).
% Nil_notin_lex
thf(fact_4405_range__eq__singletonD,axiom,
! [B: $tType,A: $tType,F2: B > A,A3: A,X: B] :
( ( ( image2 @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) )
= ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) )
=> ( ( F2 @ X )
= A3 ) ) ).
% range_eq_singletonD
thf(fact_4406_INF__empty,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F2: B > A] :
( ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F2 @ ( bot_bot @ ( set @ B ) ) ) )
= ( top_top @ A ) ) ) ).
% INF_empty
thf(fact_4407_INF__constant,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A4: set @ B,C2: A] :
( ( ( A4
= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Inf_Inf @ A
@ ( image2 @ B @ A
@ ^ [Y6: B] : C2
@ A4 ) )
= ( top_top @ A ) ) )
& ( ( A4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Inf_Inf @ A
@ ( image2 @ B @ A
@ ^ [Y6: B] : C2
@ A4 ) )
= C2 ) ) ) ) ).
% INF_constant
thf(fact_4408_surj__Compl__image__subset,axiom,
! [A: $tType,B: $tType,F2: B > A,A4: set @ B] :
( ( ( image2 @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
=> ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ ( image2 @ B @ A @ F2 @ A4 ) ) @ ( image2 @ B @ A @ F2 @ ( uminus_uminus @ ( set @ B ) @ A4 ) ) ) ) ).
% surj_Compl_image_subset
thf(fact_4409_length__remdups__card__conv,axiom,
! [A: $tType,Xs: list @ A] :
( ( size_size @ ( list @ A ) @ ( remdups @ A @ Xs ) )
= ( finite_card @ A @ ( set2 @ A @ Xs ) ) ) ).
% length_remdups_card_conv
thf(fact_4410_INT__empty,axiom,
! [B: $tType,A: $tType,B2: B > ( set @ A )] :
( ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B2 @ ( bot_bot @ ( set @ B ) ) ) )
= ( top_top @ ( set @ A ) ) ) ).
% INT_empty
thf(fact_4411_int__in__range__abs,axiom,
! [N2: nat] : ( member @ int @ ( semiring_1_of_nat @ int @ N2 ) @ ( image2 @ int @ int @ ( abs_abs @ int ) @ ( top_top @ ( set @ int ) ) ) ) ).
% int_in_range_abs
thf(fact_4412_inf__top_Osemilattice__neutr__order__axioms,axiom,
! [A: $tType] :
( ( bounde4346867609351753570nf_top @ A )
=> ( semila1105856199041335345_order @ A @ ( inf_inf @ A ) @ ( top_top @ A ) @ ( ord_less_eq @ A ) @ ( ord_less @ A ) ) ) ).
% inf_top.semilattice_neutr_order_axioms
thf(fact_4413_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_4414_UNIV__nat__eq,axiom,
( ( top_top @ ( set @ nat ) )
= ( insert2 @ nat @ ( zero_zero @ nat ) @ ( image2 @ nat @ nat @ suc @ ( top_top @ ( set @ nat ) ) ) ) ) ).
% UNIV_nat_eq
thf(fact_4415_INT__extend__simps_I3_J,axiom,
! [F: $tType,E4: $tType,C5: set @ E4,A4: E4 > ( set @ F ),B2: set @ F] :
( ( ( C5
= ( bot_bot @ ( set @ E4 ) ) )
=> ( ( minus_minus @ ( set @ F ) @ ( complete_Inf_Inf @ ( set @ F ) @ ( image2 @ E4 @ ( set @ F ) @ A4 @ C5 ) ) @ B2 )
= ( minus_minus @ ( set @ F ) @ ( top_top @ ( set @ F ) ) @ B2 ) ) )
& ( ( C5
!= ( bot_bot @ ( set @ E4 ) ) )
=> ( ( minus_minus @ ( set @ F ) @ ( complete_Inf_Inf @ ( set @ F ) @ ( image2 @ E4 @ ( set @ F ) @ A4 @ C5 ) ) @ B2 )
= ( complete_Inf_Inf @ ( set @ F )
@ ( image2 @ E4 @ ( set @ F )
@ ^ [X4: E4] : ( minus_minus @ ( set @ F ) @ ( A4 @ X4 ) @ B2 )
@ C5 ) ) ) ) ) ).
% INT_extend_simps(3)
thf(fact_4416_UN__UN__finite__eq,axiom,
! [A: $tType,A4: nat > ( set @ A )] :
( ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ nat @ ( set @ A )
@ ^ [N: nat] : ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ A4 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) )
@ ( top_top @ ( set @ nat ) ) ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ A4 @ ( top_top @ ( set @ nat ) ) ) ) ) ).
% UN_UN_finite_eq
thf(fact_4417_card__range__greater__zero,axiom,
! [A: $tType,B: $tType,F2: B > A] :
( ( finite_finite2 @ A @ ( image2 @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) ) )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ ( finite_card @ A @ ( image2 @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) ) ) ) ) ).
% card_range_greater_zero
thf(fact_4418_UN__finite__subset,axiom,
! [A: $tType,A4: nat > ( set @ A ),C5: set @ A] :
( ! [N3: nat] : ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ A4 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N3 ) ) ) @ C5 )
=> ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ A4 @ ( top_top @ ( set @ nat ) ) ) ) @ C5 ) ) ).
% UN_finite_subset
thf(fact_4419_UN__finite2__eq,axiom,
! [A: $tType,A4: nat > ( set @ A ),B2: nat > ( set @ A ),K: nat] :
( ! [N3: nat] :
( ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ A4 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N3 ) ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ B2 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( plus_plus @ nat @ N3 @ K ) ) ) ) )
=> ( ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ A4 @ ( top_top @ ( set @ nat ) ) ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ B2 @ ( top_top @ ( set @ nat ) ) ) ) ) ) ).
% UN_finite2_eq
thf(fact_4420_range__mod,axiom,
! [N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( image2 @ nat @ nat
@ ^ [M: nat] : ( modulo_modulo @ nat @ M @ N2 )
@ ( top_top @ ( set @ nat ) ) )
= ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ).
% range_mod
thf(fact_4421_UN__finite2__subset,axiom,
! [A: $tType,A4: nat > ( set @ A ),B2: nat > ( set @ A ),K: nat] :
( ! [N3: nat] : ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ A4 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N3 ) ) ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ B2 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( plus_plus @ nat @ N3 @ K ) ) ) ) )
=> ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ A4 @ ( top_top @ ( set @ nat ) ) ) ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ B2 @ ( top_top @ ( set @ nat ) ) ) ) ) ) ).
% UN_finite2_subset
thf(fact_4422_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,B4: B] : P ) )
= ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) )
& ( ~ P
=> ( ( collect @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [A5: A,B4: B] : P ) )
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ) ) ).
% Collect_const_case_prod
thf(fact_4423_Sup__finite__empty,axiom,
! [A: $tType] :
( ( finite_lattice @ A )
=> ( ( complete_Sup_Sup @ A @ ( bot_bot @ ( set @ A ) ) )
= ( complete_Inf_Inf @ A @ ( top_top @ ( set @ A ) ) ) ) ) ).
% Sup_finite_empty
thf(fact_4424_Inf__finite__empty,axiom,
! [A: $tType] :
( ( finite_lattice @ A )
=> ( ( complete_Inf_Inf @ A @ ( bot_bot @ ( set @ A ) ) )
= ( complete_Sup_Sup @ A @ ( top_top @ ( set @ A ) ) ) ) ) ).
% Inf_finite_empty
thf(fact_4425_cclfp__def,axiom,
! [A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ( ( order_532582986084564980_cclfp @ A )
= ( ^ [F3: A > A] :
( complete_Sup_Sup @ A
@ ( image2 @ nat @ A
@ ^ [I3: nat] : ( compow @ ( A > A ) @ I3 @ F3 @ ( bot_bot @ A ) )
@ ( top_top @ ( set @ nat ) ) ) ) ) ) ) ).
% cclfp_def
thf(fact_4426_top__set__def,axiom,
! [A: $tType] :
( ( top_top @ ( set @ A ) )
= ( collect @ A @ ( top_top @ ( A > $o ) ) ) ) ).
% top_set_def
thf(fact_4427_bot__finite__def,axiom,
! [A: $tType] :
( ( finite_lattice @ A )
=> ( ( bot_bot @ A )
= ( complete_Inf_Inf @ A @ ( top_top @ ( set @ A ) ) ) ) ) ).
% bot_finite_def
thf(fact_4428_root__def,axiom,
( root
= ( ^ [N: nat,X4: real] :
( if @ real
@ ( N
= ( zero_zero @ nat ) )
@ ( zero_zero @ real )
@ ( the_inv_into @ real @ real @ ( top_top @ ( set @ real ) )
@ ^ [Y6: real] : ( times_times @ real @ ( sgn_sgn @ real @ Y6 ) @ ( power_power @ real @ ( abs_abs @ real @ Y6 ) @ N ) )
@ X4 ) ) ) ) ).
% root_def
thf(fact_4429_mlex__eq,axiom,
! [A: $tType] :
( ( mlex_prod @ A )
= ( ^ [F3: A > nat,R5: set @ ( product_prod @ A @ A )] :
( collect @ ( product_prod @ A @ A )
@ ( product_case_prod @ A @ A @ $o
@ ^ [X4: A,Y6: A] :
( ( ord_less @ nat @ ( F3 @ X4 ) @ ( F3 @ Y6 ) )
| ( ( ord_less_eq @ nat @ ( F3 @ X4 ) @ ( F3 @ Y6 ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y6 ) @ R5 ) ) ) ) ) ) ) ).
% mlex_eq
thf(fact_4430_DERIV__real__root__generic,axiom,
! [N2: nat,X: real,D6: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( X
!= ( zero_zero @ real ) )
=> ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( D6
= ( inverse_inverse @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N2 ) @ ( power_power @ real @ ( root @ N2 @ X ) @ ( minus_minus @ nat @ N2 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ) )
=> ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
=> ( ( ord_less @ real @ X @ ( zero_zero @ real ) )
=> ( D6
= ( uminus_uminus @ real @ ( inverse_inverse @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N2 ) @ ( power_power @ real @ ( root @ N2 @ X ) @ ( minus_minus @ nat @ N2 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ) ) )
=> ( ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
=> ( D6
= ( inverse_inverse @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N2 ) @ ( power_power @ real @ ( root @ N2 @ X ) @ ( minus_minus @ nat @ N2 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) )
=> ( has_field_derivative @ real @ ( root @ N2 ) @ D6 @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ) ) ).
% DERIV_real_root_generic
thf(fact_4431_DERIV__even__real__root,axiom,
! [N2: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
=> ( ( ord_less @ real @ X @ ( zero_zero @ real ) )
=> ( has_field_derivative @ real @ ( root @ N2 ) @ ( inverse_inverse @ real @ ( times_times @ real @ ( uminus_uminus @ real @ ( semiring_1_of_nat @ real @ N2 ) ) @ ( power_power @ real @ ( root @ N2 @ X ) @ ( minus_minus @ nat @ N2 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ).
% DERIV_even_real_root
thf(fact_4432_top1I,axiom,
! [A: $tType,X: A] : ( top_top @ ( A > $o ) @ X ) ).
% top1I
thf(fact_4433_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_4434_at__discrete,axiom,
! [A: $tType] :
( ( topolo8865339358273720382pology @ A )
=> ( ( topolo174197925503356063within @ A )
= ( ^ [X4: A,S6: set @ A] : ( bot_bot @ ( filter @ A ) ) ) ) ) ).
% at_discrete
thf(fact_4435_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_4436_at__le,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [S: set @ A,T2: set @ A,X: A] :
( ( ord_less_eq @ ( set @ A ) @ S @ T2 )
=> ( ord_less_eq @ ( filter @ A ) @ ( topolo174197925503356063within @ A @ X @ S ) @ ( topolo174197925503356063within @ A @ X @ T2 ) ) ) ) ).
% at_le
thf(fact_4437_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_4438_DERIV__cos__add,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [K: A,Xa2: A] :
( has_field_derivative @ A
@ ^ [X4: A] : ( cos @ A @ ( plus_plus @ A @ X4 @ K ) )
@ ( uminus_uminus @ A @ ( sin @ A @ ( plus_plus @ A @ Xa2 @ K ) ) )
@ ( topolo174197925503356063within @ A @ Xa2 @ ( top_top @ ( set @ A ) ) ) ) ) ).
% DERIV_cos_add
thf(fact_4439_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_4440_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_4441_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_4442_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_4443_has__field__derivative__tanh,axiom,
! [A10: $tType] :
( ( ( real_Vector_banach @ A10 )
& ( real_V3459762299906320749_field @ A10 ) )
=> ! [G: A10 > A10,X: A10,Db: A10,S: set @ A10] :
( ( ( cosh @ A10 @ ( G @ X ) )
!= ( zero_zero @ A10 ) )
=> ( ( has_field_derivative @ A10 @ G @ Db @ ( topolo174197925503356063within @ A10 @ X @ S ) )
=> ( has_field_derivative @ A10
@ ^ [X4: A10] : ( tanh @ A10 @ ( G @ X4 ) )
@ ( times_times @ A10 @ ( minus_minus @ A10 @ ( one_one @ A10 ) @ ( power_power @ A10 @ ( tanh @ A10 @ ( G @ X ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ Db )
@ ( topolo174197925503356063within @ A10 @ X @ S ) ) ) ) ) ).
% has_field_derivative_tanh
thf(fact_4444_DERIV__power__series_H,axiom,
! [R: real,F2: nat > real,X0: real] :
( ! [X5: real] :
( ( member @ real @ X5 @ ( set_or5935395276787703475ssThan @ real @ ( uminus_uminus @ real @ R ) @ R ) )
=> ( summable @ real
@ ^ [N: nat] : ( times_times @ real @ ( times_times @ real @ ( F2 @ N ) @ ( semiring_1_of_nat @ real @ ( suc @ N ) ) ) @ ( power_power @ real @ X5 @ N ) ) ) )
=> ( ( member @ real @ X0 @ ( set_or5935395276787703475ssThan @ real @ ( uminus_uminus @ real @ R ) @ R ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ R )
=> ( has_field_derivative @ real
@ ^ [X4: real] :
( suminf @ real
@ ^ [N: nat] : ( times_times @ real @ ( F2 @ N ) @ ( power_power @ real @ X4 @ ( suc @ N ) ) ) )
@ ( suminf @ real
@ ^ [N: nat] : ( times_times @ real @ ( times_times @ real @ ( F2 @ N ) @ ( semiring_1_of_nat @ real @ ( suc @ N ) ) ) @ ( power_power @ real @ X0 @ N ) ) )
@ ( topolo174197925503356063within @ real @ X0 @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ).
% DERIV_power_series'
thf(fact_4445_DERIV__real__root,axiom,
! [N2: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( has_field_derivative @ real @ ( root @ N2 ) @ ( inverse_inverse @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N2 ) @ ( power_power @ real @ ( root @ N2 @ X ) @ ( minus_minus @ nat @ N2 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) ) ) ) ).
% DERIV_real_root
thf(fact_4446_Maclaurin__all__le,axiom,
! [Diff: nat > real > real,F2: real > real,X: real,N2: nat] :
( ( ( Diff @ ( zero_zero @ nat ) )
= F2 )
=> ( ! [M3: nat,X5: real] : ( has_field_derivative @ real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ X5 ) @ ( topolo174197925503356063within @ real @ X5 @ ( top_top @ ( set @ real ) ) ) )
=> ? [T7: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ T7 ) @ ( abs_abs @ real @ X ) )
& ( ( F2 @ X )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M @ ( zero_zero @ real ) ) @ ( semiring_char_0_fact @ real @ M ) ) @ ( power_power @ real @ X @ M ) )
@ ( set_ord_lessThan @ nat @ N2 ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N2 @ T7 ) @ ( semiring_char_0_fact @ real @ N2 ) ) @ ( power_power @ real @ X @ N2 ) ) ) ) ) ) ) ).
% Maclaurin_all_le
thf(fact_4447_Maclaurin__all__le__objl,axiom,
! [Diff: nat > real > real,F2: real > real,X: real,N2: nat] :
( ( ( ( Diff @ ( zero_zero @ nat ) )
= F2 )
& ! [M3: nat,X5: real] : ( has_field_derivative @ real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ X5 ) @ ( topolo174197925503356063within @ real @ X5 @ ( top_top @ ( set @ real ) ) ) ) )
=> ? [T7: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ T7 ) @ ( abs_abs @ real @ X ) )
& ( ( F2 @ X )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M @ ( zero_zero @ real ) ) @ ( semiring_char_0_fact @ real @ M ) ) @ ( power_power @ real @ X @ M ) )
@ ( set_ord_lessThan @ nat @ N2 ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N2 @ T7 ) @ ( semiring_char_0_fact @ real @ N2 ) ) @ ( power_power @ real @ X @ N2 ) ) ) ) ) ) ).
% Maclaurin_all_le_objl
thf(fact_4448_DERIV__odd__real__root,axiom,
! [N2: nat,X: real] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
=> ( ( X
!= ( zero_zero @ real ) )
=> ( has_field_derivative @ real @ ( root @ N2 ) @ ( inverse_inverse @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N2 ) @ ( power_power @ real @ ( root @ N2 @ X ) @ ( minus_minus @ nat @ N2 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) ) ) ) ).
% DERIV_odd_real_root
thf(fact_4449_Maclaurin__minus,axiom,
! [H2: real,N2: nat,Diff: nat > real > real,F2: real > real] :
( ( ord_less @ real @ H2 @ ( zero_zero @ real ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( ( Diff @ ( zero_zero @ nat ) )
= F2 )
=> ( ! [M3: nat,T7: real] :
( ( ( ord_less @ nat @ M3 @ N2 )
& ( ord_less_eq @ real @ H2 @ T7 )
& ( ord_less_eq @ real @ T7 @ ( zero_zero @ real ) ) )
=> ( has_field_derivative @ real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ T7 ) @ ( topolo174197925503356063within @ real @ T7 @ ( top_top @ ( set @ real ) ) ) ) )
=> ? [T7: real] :
( ( ord_less @ real @ H2 @ T7 )
& ( ord_less @ real @ T7 @ ( zero_zero @ real ) )
& ( ( F2 @ H2 )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M @ ( zero_zero @ real ) ) @ ( semiring_char_0_fact @ real @ M ) ) @ ( power_power @ real @ H2 @ M ) )
@ ( set_ord_lessThan @ nat @ N2 ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N2 @ T7 ) @ ( semiring_char_0_fact @ real @ N2 ) ) @ ( power_power @ real @ H2 @ N2 ) ) ) ) ) ) ) ) ) ).
% Maclaurin_minus
thf(fact_4450_Maclaurin2,axiom,
! [H2: real,Diff: nat > real > real,F2: real > real,N2: nat] :
( ( ord_less @ real @ ( zero_zero @ real ) @ H2 )
=> ( ( ( Diff @ ( zero_zero @ nat ) )
= F2 )
=> ( ! [M3: nat,T7: real] :
( ( ( ord_less @ nat @ M3 @ N2 )
& ( ord_less_eq @ real @ ( zero_zero @ real ) @ T7 )
& ( ord_less_eq @ real @ T7 @ H2 ) )
=> ( has_field_derivative @ real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ T7 ) @ ( topolo174197925503356063within @ real @ T7 @ ( top_top @ ( set @ real ) ) ) ) )
=> ? [T7: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ T7 )
& ( ord_less_eq @ real @ T7 @ H2 )
& ( ( F2 @ H2 )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M @ ( zero_zero @ real ) ) @ ( semiring_char_0_fact @ real @ M ) ) @ ( power_power @ real @ H2 @ M ) )
@ ( set_ord_lessThan @ nat @ N2 ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N2 @ T7 ) @ ( semiring_char_0_fact @ real @ N2 ) ) @ ( power_power @ real @ H2 @ N2 ) ) ) ) ) ) ) ) ).
% Maclaurin2
thf(fact_4451_Maclaurin,axiom,
! [H2: real,N2: nat,Diff: nat > real > real,F2: real > real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ H2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( ( Diff @ ( zero_zero @ nat ) )
= F2 )
=> ( ! [M3: nat,T7: real] :
( ( ( ord_less @ nat @ M3 @ N2 )
& ( ord_less_eq @ real @ ( zero_zero @ real ) @ T7 )
& ( ord_less_eq @ real @ T7 @ H2 ) )
=> ( has_field_derivative @ real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ T7 ) @ ( topolo174197925503356063within @ real @ T7 @ ( top_top @ ( set @ real ) ) ) ) )
=> ? [T7: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ T7 )
& ( ord_less @ real @ T7 @ H2 )
& ( ( F2 @ H2 )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M @ ( zero_zero @ real ) ) @ ( semiring_char_0_fact @ real @ M ) ) @ ( power_power @ real @ H2 @ M ) )
@ ( set_ord_lessThan @ nat @ N2 ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N2 @ T7 ) @ ( semiring_char_0_fact @ real @ N2 ) ) @ ( power_power @ real @ H2 @ N2 ) ) ) ) ) ) ) ) ) ).
% Maclaurin
thf(fact_4452_Maclaurin__all__lt,axiom,
! [Diff: nat > real > real,F2: real > real,N2: nat,X: real] :
( ( ( Diff @ ( zero_zero @ nat ) )
= F2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( X
!= ( zero_zero @ real ) )
=> ( ! [M3: nat,X5: real] : ( has_field_derivative @ real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ X5 ) @ ( topolo174197925503356063within @ real @ X5 @ ( top_top @ ( set @ real ) ) ) )
=> ? [T7: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( abs_abs @ real @ T7 ) )
& ( ord_less @ real @ ( abs_abs @ real @ T7 ) @ ( abs_abs @ real @ X ) )
& ( ( F2 @ X )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M @ ( zero_zero @ real ) ) @ ( semiring_char_0_fact @ real @ M ) ) @ ( power_power @ real @ X @ M ) )
@ ( set_ord_lessThan @ nat @ N2 ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N2 @ T7 ) @ ( semiring_char_0_fact @ real @ N2 ) ) @ ( power_power @ real @ X @ N2 ) ) ) ) ) ) ) ) ) ).
% Maclaurin_all_lt
thf(fact_4453_Maclaurin__bi__le,axiom,
! [Diff: nat > real > real,F2: real > real,N2: nat,X: real] :
( ( ( Diff @ ( zero_zero @ nat ) )
= F2 )
=> ( ! [M3: nat,T7: real] :
( ( ( ord_less @ nat @ M3 @ N2 )
& ( ord_less_eq @ real @ ( abs_abs @ real @ T7 ) @ ( abs_abs @ real @ X ) ) )
=> ( has_field_derivative @ real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ T7 ) @ ( topolo174197925503356063within @ real @ T7 @ ( top_top @ ( set @ real ) ) ) ) )
=> ? [T7: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ T7 ) @ ( abs_abs @ real @ X ) )
& ( ( F2 @ X )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M @ ( zero_zero @ real ) ) @ ( semiring_char_0_fact @ real @ M ) ) @ ( power_power @ real @ X @ M ) )
@ ( set_ord_lessThan @ nat @ N2 ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N2 @ T7 ) @ ( semiring_char_0_fact @ real @ N2 ) ) @ ( power_power @ real @ X @ N2 ) ) ) ) ) ) ) ).
% Maclaurin_bi_le
thf(fact_4454_Taylor,axiom,
! [N2: nat,Diff: nat > real > real,F2: real > real,A3: real,B3: real,C2: real,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( ( Diff @ ( zero_zero @ nat ) )
= F2 )
=> ( ! [M3: nat,T7: real] :
( ( ( ord_less @ nat @ M3 @ N2 )
& ( ord_less_eq @ real @ A3 @ T7 )
& ( ord_less_eq @ real @ T7 @ B3 ) )
=> ( has_field_derivative @ real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ T7 ) @ ( topolo174197925503356063within @ real @ T7 @ ( 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 )
=> ? [T7: real] :
( ( ( ord_less @ real @ X @ C2 )
=> ( ( ord_less @ real @ X @ T7 )
& ( ord_less @ real @ T7 @ C2 ) ) )
& ( ~ ( ord_less @ real @ X @ C2 )
=> ( ( ord_less @ real @ C2 @ T7 )
& ( ord_less @ real @ T7 @ X ) ) )
& ( ( F2 @ X )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M @ C2 ) @ ( semiring_char_0_fact @ real @ M ) ) @ ( power_power @ real @ ( minus_minus @ real @ X @ C2 ) @ M ) )
@ ( set_ord_lessThan @ nat @ N2 ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N2 @ T7 ) @ ( semiring_char_0_fact @ real @ N2 ) ) @ ( power_power @ real @ ( minus_minus @ real @ X @ C2 ) @ N2 ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% Taylor
thf(fact_4455_Taylor__up,axiom,
! [N2: nat,Diff: nat > real > real,F2: real > real,A3: real,B3: real,C2: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( ( Diff @ ( zero_zero @ nat ) )
= F2 )
=> ( ! [M3: nat,T7: real] :
( ( ( ord_less @ nat @ M3 @ N2 )
& ( ord_less_eq @ real @ A3 @ T7 )
& ( ord_less_eq @ real @ T7 @ B3 ) )
=> ( has_field_derivative @ real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ T7 ) @ ( topolo174197925503356063within @ real @ T7 @ ( top_top @ ( set @ real ) ) ) ) )
=> ( ( ord_less_eq @ real @ A3 @ C2 )
=> ( ( ord_less @ real @ C2 @ B3 )
=> ? [T7: real] :
( ( ord_less @ real @ C2 @ T7 )
& ( ord_less @ real @ T7 @ B3 )
& ( ( F2 @ B3 )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M @ C2 ) @ ( semiring_char_0_fact @ real @ M ) ) @ ( power_power @ real @ ( minus_minus @ real @ B3 @ C2 ) @ M ) )
@ ( set_ord_lessThan @ nat @ N2 ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N2 @ T7 ) @ ( semiring_char_0_fact @ real @ N2 ) ) @ ( power_power @ real @ ( minus_minus @ real @ B3 @ C2 ) @ N2 ) ) ) ) ) ) ) ) ) ) ).
% Taylor_up
thf(fact_4456_Taylor__down,axiom,
! [N2: nat,Diff: nat > real > real,F2: real > real,A3: real,B3: real,C2: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( ( Diff @ ( zero_zero @ nat ) )
= F2 )
=> ( ! [M3: nat,T7: real] :
( ( ( ord_less @ nat @ M3 @ N2 )
& ( ord_less_eq @ real @ A3 @ T7 )
& ( ord_less_eq @ real @ T7 @ B3 ) )
=> ( has_field_derivative @ real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ T7 ) @ ( topolo174197925503356063within @ real @ T7 @ ( top_top @ ( set @ real ) ) ) ) )
=> ( ( ord_less @ real @ A3 @ C2 )
=> ( ( ord_less_eq @ real @ C2 @ B3 )
=> ? [T7: real] :
( ( ord_less @ real @ A3 @ T7 )
& ( ord_less @ real @ T7 @ C2 )
& ( ( F2 @ A3 )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M @ C2 ) @ ( semiring_char_0_fact @ real @ M ) ) @ ( power_power @ real @ ( minus_minus @ real @ A3 @ C2 ) @ M ) )
@ ( set_ord_lessThan @ nat @ N2 ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N2 @ T7 ) @ ( semiring_char_0_fact @ real @ N2 ) ) @ ( power_power @ real @ ( minus_minus @ real @ A3 @ C2 ) @ N2 ) ) ) ) ) ) ) ) ) ) ).
% Taylor_down
thf(fact_4457_Maclaurin__lemma2,axiom,
! [N2: nat,H2: real,Diff: nat > real > real,K: nat,B2: real] :
( ! [M3: nat,T7: real] :
( ( ( ord_less @ nat @ M3 @ N2 )
& ( ord_less_eq @ real @ ( zero_zero @ real ) @ T7 )
& ( ord_less_eq @ real @ T7 @ H2 ) )
=> ( has_field_derivative @ real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ T7 ) @ ( topolo174197925503356063within @ real @ T7 @ ( top_top @ ( set @ real ) ) ) ) )
=> ( ( N2
= ( suc @ K ) )
=> ! [M5: nat,T8: real] :
( ( ( ord_less @ nat @ M5 @ N2 )
& ( ord_less_eq @ real @ ( zero_zero @ real ) @ T8 )
& ( ord_less_eq @ real @ T8 @ H2 ) )
=> ( has_field_derivative @ real
@ ^ [U2: real] :
( minus_minus @ real @ ( Diff @ M5 @ U2 )
@ ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [P6: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ ( plus_plus @ nat @ M5 @ P6 ) @ ( zero_zero @ real ) ) @ ( semiring_char_0_fact @ real @ P6 ) ) @ ( power_power @ real @ U2 @ P6 ) )
@ ( set_ord_lessThan @ nat @ ( minus_minus @ nat @ N2 @ M5 ) ) )
@ ( times_times @ real @ B2 @ ( divide_divide @ real @ ( power_power @ real @ U2 @ ( minus_minus @ nat @ N2 @ M5 ) ) @ ( semiring_char_0_fact @ real @ ( minus_minus @ nat @ N2 @ M5 ) ) ) ) ) )
@ ( minus_minus @ real @ ( Diff @ ( suc @ M5 ) @ T8 )
@ ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [P6: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ ( plus_plus @ nat @ ( suc @ M5 ) @ P6 ) @ ( zero_zero @ real ) ) @ ( semiring_char_0_fact @ real @ P6 ) ) @ ( power_power @ real @ T8 @ P6 ) )
@ ( set_ord_lessThan @ nat @ ( minus_minus @ nat @ N2 @ ( suc @ M5 ) ) ) )
@ ( times_times @ real @ B2 @ ( divide_divide @ real @ ( power_power @ real @ T8 @ ( minus_minus @ nat @ N2 @ ( suc @ M5 ) ) ) @ ( semiring_char_0_fact @ real @ ( minus_minus @ nat @ N2 @ ( suc @ M5 ) ) ) ) ) ) )
@ ( topolo174197925503356063within @ real @ T8 @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ).
% Maclaurin_lemma2
thf(fact_4458_DERIV__arctan__series,axiom,
! [X: real] :
( ( ord_less @ real @ ( abs_abs @ real @ X ) @ ( one_one @ real ) )
=> ( has_field_derivative @ real
@ ^ [X10: real] :
( suminf @ real
@ ^ [K2: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ K2 ) @ ( times_times @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( semiring_1_of_nat @ real @ ( plus_plus @ nat @ ( times_times @ nat @ K2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) @ ( power_power @ real @ X10 @ ( plus_plus @ nat @ ( times_times @ nat @ K2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) ) )
@ ( suminf @ real
@ ^ [K2: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ K2 ) @ ( power_power @ real @ X @ ( times_times @ nat @ K2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) ) ) ).
% DERIV_arctan_series
thf(fact_4459_DERIV__inverse__fun,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,D2: A,X: A,S: set @ A] :
( ( has_field_derivative @ A @ F2 @ D2 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( ( ( F2 @ X )
!= ( zero_zero @ A ) )
=> ( has_field_derivative @ A
@ ^ [X4: A] : ( inverse_inverse @ A @ ( F2 @ X4 ) )
@ ( uminus_uminus @ A @ ( times_times @ A @ D2 @ ( inverse_inverse @ A @ ( power_power @ A @ ( F2 @ X ) @ ( suc @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ).
% DERIV_inverse_fun
thf(fact_4460_DERIV__quotient,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,D2: A,X: A,S: set @ A,G: A > A,E2: A] :
( ( has_field_derivative @ A @ F2 @ D2 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( ( has_field_derivative @ A @ G @ E2 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( ( ( G @ X )
!= ( zero_zero @ A ) )
=> ( has_field_derivative @ A
@ ^ [Y6: A] : ( divide_divide @ A @ ( F2 @ Y6 ) @ ( G @ Y6 ) )
@ ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ D2 @ ( G @ X ) ) @ ( times_times @ A @ E2 @ ( F2 @ X ) ) ) @ ( power_power @ A @ ( G @ X ) @ ( suc @ ( suc @ ( zero_zero @ nat ) ) ) ) )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ) ).
% DERIV_quotient
thf(fact_4461_DERIV__pow,axiom,
! [N2: nat,X: real,S: set @ real] :
( has_field_derivative @ real
@ ^ [X4: real] : ( power_power @ real @ X4 @ N2 )
@ ( times_times @ real @ ( semiring_1_of_nat @ real @ N2 ) @ ( power_power @ real @ X @ ( minus_minus @ nat @ N2 @ ( suc @ ( zero_zero @ nat ) ) ) ) )
@ ( topolo174197925503356063within @ real @ X @ S ) ) ).
% DERIV_pow
thf(fact_4462_DERIV__power,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,D6: A,X: A,S: set @ A,N2: nat] :
( ( has_field_derivative @ A @ F2 @ D6 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( has_field_derivative @ A
@ ^ [X4: A] : ( power_power @ A @ ( F2 @ X4 ) @ N2 )
@ ( times_times @ A @ ( semiring_1_of_nat @ A @ N2 ) @ ( times_times @ A @ D6 @ ( power_power @ A @ ( F2 @ X ) @ ( minus_minus @ nat @ N2 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ).
% DERIV_power
thf(fact_4463_DERIV__const,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [K: A,F5: filter @ A] :
( has_field_derivative @ A
@ ^ [X4: A] : K
@ ( zero_zero @ A )
@ F5 ) ) ).
% DERIV_const
thf(fact_4464_field__differentiable__add,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,F8: A,F5: filter @ A,G: A > A,G6: A] :
( ( has_field_derivative @ A @ F2 @ F8 @ F5 )
=> ( ( has_field_derivative @ A @ G @ G6 @ F5 )
=> ( has_field_derivative @ A
@ ^ [Z6: A] : ( plus_plus @ A @ ( F2 @ Z6 ) @ ( G @ Z6 ) )
@ ( plus_plus @ A @ F8 @ G6 )
@ F5 ) ) ) ) ).
% field_differentiable_add
thf(fact_4465_has__field__derivative__subset,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,Y: A,X: A,S: set @ A,T2: set @ A] :
( ( has_field_derivative @ A @ F2 @ Y @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( ( ord_less_eq @ ( set @ A ) @ T2 @ S )
=> ( has_field_derivative @ A @ F2 @ Y @ ( topolo174197925503356063within @ A @ X @ T2 ) ) ) ) ) ).
% has_field_derivative_subset
thf(fact_4466_DERIV__subset,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,F8: A,X: A,S: set @ A,T2: set @ A] :
( ( has_field_derivative @ A @ F2 @ F8 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( ( ord_less_eq @ ( set @ A ) @ T2 @ S )
=> ( has_field_derivative @ A @ F2 @ F8 @ ( topolo174197925503356063within @ A @ X @ T2 ) ) ) ) ) ).
% DERIV_subset
thf(fact_4467_DERIV__add,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,D6: A,X: A,S: set @ A,G: A > A,E5: A] :
( ( has_field_derivative @ A @ F2 @ D6 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( ( has_field_derivative @ A @ G @ E5 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( has_field_derivative @ A
@ ^ [X4: A] : ( plus_plus @ A @ ( F2 @ X4 ) @ ( G @ X4 ) )
@ ( plus_plus @ A @ D6 @ E5 )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ).
% DERIV_add
thf(fact_4468_DERIV__mult_H,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,D6: A,X: A,S: set @ A,G: A > A,E5: A] :
( ( has_field_derivative @ A @ F2 @ D6 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( ( has_field_derivative @ A @ G @ E5 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( has_field_derivative @ A
@ ^ [X4: A] : ( times_times @ A @ ( F2 @ X4 ) @ ( G @ X4 ) )
@ ( plus_plus @ A @ ( times_times @ A @ ( F2 @ X ) @ E5 ) @ ( times_times @ A @ D6 @ ( G @ X ) ) )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ).
% DERIV_mult'
thf(fact_4469_DERIV__mult,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,Da: A,X: A,S: set @ A,G: A > A,Db: A] :
( ( has_field_derivative @ A @ F2 @ Da @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( ( has_field_derivative @ A @ G @ Db @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( has_field_derivative @ A
@ ^ [X4: A] : ( times_times @ A @ ( F2 @ X4 ) @ ( G @ X4 ) )
@ ( plus_plus @ A @ ( times_times @ A @ Da @ ( G @ X ) ) @ ( times_times @ A @ Db @ ( F2 @ X ) ) )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ).
% DERIV_mult
thf(fact_4470_DERIV__at__within__shift,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,Y: A,Z3: A,X: A,S2: set @ A] :
( ( has_field_derivative @ A @ F2 @ Y @ ( topolo174197925503356063within @ A @ ( plus_plus @ A @ Z3 @ X ) @ ( image2 @ A @ A @ ( plus_plus @ A @ Z3 ) @ S2 ) ) )
= ( has_field_derivative @ A
@ ^ [X4: A] : ( F2 @ ( plus_plus @ A @ Z3 @ X4 ) )
@ Y
@ ( topolo174197925503356063within @ A @ X @ S2 ) ) ) ) ).
% DERIV_at_within_shift
thf(fact_4471_DERIV__shift,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,Y: A,X: A,Z3: A] :
( ( has_field_derivative @ A @ F2 @ Y @ ( topolo174197925503356063within @ A @ ( plus_plus @ A @ X @ Z3 ) @ ( top_top @ ( set @ A ) ) ) )
= ( has_field_derivative @ A
@ ^ [X4: A] : ( F2 @ ( plus_plus @ A @ X4 @ Z3 ) )
@ Y
@ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% DERIV_shift
thf(fact_4472_DERIV__divide,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,D6: A,X: A,S: set @ A,G: A > A,E5: A] :
( ( has_field_derivative @ A @ F2 @ D6 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( ( has_field_derivative @ A @ G @ E5 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( ( ( G @ X )
!= ( zero_zero @ A ) )
=> ( has_field_derivative @ A
@ ^ [X4: A] : ( divide_divide @ A @ ( F2 @ X4 ) @ ( G @ X4 ) )
@ ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ D6 @ ( G @ X ) ) @ ( times_times @ A @ ( F2 @ X ) @ E5 ) ) @ ( times_times @ A @ ( G @ X ) @ ( G @ X ) ) )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ) ).
% DERIV_divide
thf(fact_4473_DERIV__inverse_H,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,D6: A,X: A,S: set @ A] :
( ( has_field_derivative @ A @ F2 @ D6 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( ( ( F2 @ X )
!= ( zero_zero @ A ) )
=> ( has_field_derivative @ A
@ ^ [X4: A] : ( inverse_inverse @ A @ ( F2 @ X4 ) )
@ ( uminus_uminus @ A @ ( times_times @ A @ ( times_times @ A @ ( inverse_inverse @ A @ ( F2 @ X ) ) @ D6 ) @ ( inverse_inverse @ A @ ( F2 @ X ) ) ) )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ).
% DERIV_inverse'
thf(fact_4474_DERIV__power__Suc,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,D6: A,X: A,S: set @ A,N2: nat] :
( ( has_field_derivative @ A @ F2 @ D6 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( has_field_derivative @ A
@ ^ [X4: A] : ( power_power @ A @ ( F2 @ X4 ) @ ( suc @ N2 ) )
@ ( times_times @ A @ ( plus_plus @ A @ ( one_one @ A ) @ ( semiring_1_of_nat @ A @ N2 ) ) @ ( times_times @ A @ D6 @ ( power_power @ A @ ( F2 @ X ) @ N2 ) ) )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ).
% DERIV_power_Suc
thf(fact_4475_DERIV__inverse,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [X: A,S: set @ A] :
( ( X
!= ( zero_zero @ A ) )
=> ( has_field_derivative @ A @ ( inverse_inverse @ A ) @ ( uminus_uminus @ A @ ( power_power @ A @ ( inverse_inverse @ A @ X ) @ ( suc @ ( suc @ ( zero_zero @ nat ) ) ) ) ) @ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ).
% DERIV_inverse
thf(fact_4476_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_4477_mlex__leq,axiom,
! [A: $tType,F2: A > nat,X: A,Y: A,R: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ nat @ ( F2 @ X ) @ ( F2 @ Y ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( mlex_prod @ A @ F2 @ R ) ) ) ) ).
% mlex_leq
thf(fact_4478_mlex__less,axiom,
! [A: $tType,F2: A > nat,X: A,Y: A,R: set @ ( product_prod @ A @ A )] :
( ( ord_less @ nat @ ( F2 @ X ) @ ( F2 @ Y ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( mlex_prod @ A @ F2 @ R ) ) ) ).
% mlex_less
thf(fact_4479_mlex__iff,axiom,
! [A: $tType,X: A,Y: A,F2: A > nat,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( mlex_prod @ A @ F2 @ R ) )
= ( ( ord_less @ nat @ ( F2 @ X ) @ ( F2 @ Y ) )
| ( ( ( F2 @ X )
= ( F2 @ Y ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R ) ) ) ) ).
% mlex_iff
thf(fact_4480_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_4481_in__measure,axiom,
! [A: $tType,X: A,Y: A,F2: A > nat] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( measure @ A @ F2 ) )
= ( ord_less @ nat @ ( F2 @ X ) @ ( F2 @ Y ) ) ) ).
% in_measure
thf(fact_4482_in__finite__psubset,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] :
( ( member @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ A4 @ B2 ) @ ( finite_psubset @ A ) )
= ( ( ord_less @ ( set @ A ) @ A4 @ B2 )
& ( finite_finite2 @ A @ B2 ) ) ) ).
% in_finite_psubset
thf(fact_4483_termdiffs__aux,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [C2: nat > A,K5: A,X: A] :
( ( summable @ A
@ ^ [N: nat] : ( times_times @ A @ ( diffs @ A @ ( diffs @ A @ C2 ) @ N ) @ ( power_power @ A @ K5 @ N ) ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ ( real_V7770717601297561774m_norm @ A @ K5 ) )
=> ( filterlim @ A @ A
@ ^ [H: A] :
( suminf @ A
@ ^ [N: nat] : ( times_times @ A @ ( C2 @ N ) @ ( minus_minus @ A @ ( divide_divide @ A @ ( minus_minus @ A @ ( power_power @ A @ ( plus_plus @ A @ X @ H ) @ N ) @ ( power_power @ A @ X @ N ) ) @ H ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ ( power_power @ A @ X @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% termdiffs_aux
thf(fact_4484_Gcd__eq__Max,axiom,
! [M7: set @ nat] :
( ( finite_finite2 @ nat @ M7 )
=> ( ( M7
!= ( bot_bot @ ( set @ nat ) ) )
=> ( ~ ( member @ nat @ ( zero_zero @ nat ) @ M7 )
=> ( ( gcd_Gcd @ nat @ M7 )
= ( lattic643756798349783984er_Max @ nat
@ ( complete_Inf_Inf @ ( set @ nat )
@ ( image2 @ nat @ ( set @ nat )
@ ^ [M: nat] :
( collect @ nat
@ ^ [D4: nat] : ( dvd_dvd @ nat @ D4 @ M ) )
@ M7 ) ) ) ) ) ) ) ).
% Gcd_eq_Max
thf(fact_4485_Max__singleton,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A] :
( ( lattic643756798349783984er_Max @ A @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= X ) ) ).
% Max_singleton
thf(fact_4486_Max__divisors__self__nat,axiom,
! [N2: nat] :
( ( N2
!= ( zero_zero @ nat ) )
=> ( ( lattic643756798349783984er_Max @ nat
@ ( collect @ nat
@ ^ [D4: nat] : ( dvd_dvd @ nat @ D4 @ N2 ) ) )
= N2 ) ) ).
% Max_divisors_self_nat
thf(fact_4487_Max_Obounded__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ ( lattic643756798349783984er_Max @ A @ A4 ) @ X )
= ( ! [X4: A] :
( ( member @ A @ X4 @ A4 )
=> ( ord_less_eq @ A @ X4 @ X ) ) ) ) ) ) ) ).
% Max.bounded_iff
thf(fact_4488_Max__less__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less @ A @ ( lattic643756798349783984er_Max @ A @ A4 ) @ X )
= ( ! [X4: A] :
( ( member @ A @ X4 @ A4 )
=> ( ord_less @ A @ X4 @ X ) ) ) ) ) ) ) ).
% Max_less_iff
thf(fact_4489_tendsto__mult__left__iff,axiom,
! [A: $tType,B: $tType] :
( ( ( field @ A )
& ( topolo4211221413907600880p_mult @ A ) )
=> ! [C2: A,F2: B > A,L: A,F5: filter @ B] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( filterlim @ B @ A
@ ^ [X4: B] : ( times_times @ A @ C2 @ ( F2 @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( times_times @ A @ C2 @ L ) )
@ F5 )
= ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ F5 ) ) ) ) ).
% tendsto_mult_left_iff
thf(fact_4490_tendsto__mult__right__iff,axiom,
! [A: $tType,B: $tType] :
( ( ( field @ A )
& ( topolo4211221413907600880p_mult @ A ) )
=> ! [C2: A,F2: B > A,L: A,F5: filter @ B] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( filterlim @ B @ A
@ ^ [X4: B] : ( times_times @ A @ ( F2 @ X4 ) @ C2 )
@ ( topolo7230453075368039082e_nhds @ A @ ( times_times @ A @ L @ C2 ) )
@ F5 )
= ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ F5 ) ) ) ) ).
% tendsto_mult_right_iff
thf(fact_4491_power__tendsto__0__iff,axiom,
! [A: $tType,N2: nat,F2: A > real,F5: filter @ A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( filterlim @ A @ real
@ ^ [X4: A] : ( power_power @ real @ ( F2 @ X4 ) @ N2 )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ F5 )
= ( filterlim @ A @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ F5 ) ) ) ).
% power_tendsto_0_iff
thf(fact_4492_Max__const,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ B,C2: A] :
( ( finite_finite2 @ B @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( lattic643756798349783984er_Max @ A
@ ( image2 @ B @ A
@ ^ [Uu3: B] : C2
@ A4 ) )
= C2 ) ) ) ) ).
% Max_const
thf(fact_4493_Max__insert,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798349783984er_Max @ A @ ( insert2 @ A @ X @ A4 ) )
= ( ord_max @ A @ X @ ( lattic643756798349783984er_Max @ A @ A4 ) ) ) ) ) ) ).
% Max_insert
thf(fact_4494_tendsto__inverse,axiom,
! [A: $tType,B: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [F2: B > A,A3: A,F5: filter @ B] :
( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ A3 ) @ F5 )
=> ( ( A3
!= ( zero_zero @ A ) )
=> ( filterlim @ B @ A
@ ^ [X4: B] : ( inverse_inverse @ A @ ( F2 @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( inverse_inverse @ A @ A3 ) )
@ F5 ) ) ) ) ).
% tendsto_inverse
thf(fact_4495_tendsto__add__const__iff,axiom,
! [A: $tType,B: $tType] :
( ( topolo1633459387980952147up_add @ A )
=> ! [C2: A,F2: B > A,D2: A,F5: filter @ B] :
( ( filterlim @ B @ A
@ ^ [X4: B] : ( plus_plus @ A @ C2 @ ( F2 @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( plus_plus @ A @ C2 @ D2 ) )
@ F5 )
= ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ D2 ) @ F5 ) ) ) ).
% tendsto_add_const_iff
thf(fact_4496_tendsto__add,axiom,
! [A: $tType,B: $tType] :
( ( topolo6943815403480290642id_add @ A )
=> ! [F2: B > A,A3: A,F5: filter @ B,G: B > A,B3: A] :
( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ A3 ) @ F5 )
=> ( ( filterlim @ B @ A @ G @ ( topolo7230453075368039082e_nhds @ A @ B3 ) @ F5 )
=> ( filterlim @ B @ A
@ ^ [X4: B] : ( plus_plus @ A @ ( F2 @ X4 ) @ ( G @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( plus_plus @ A @ A3 @ B3 ) )
@ F5 ) ) ) ) ).
% tendsto_add
thf(fact_4497_tendsto__norm__zero,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [F2: A > B,F5: filter @ A] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) ) @ F5 )
=> ( filterlim @ A @ real
@ ^ [X4: A] : ( real_V7770717601297561774m_norm @ B @ ( F2 @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ F5 ) ) ) ).
% tendsto_norm_zero
thf(fact_4498_tendsto__norm__zero__iff,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [F2: A > B,F5: filter @ A] :
( ( filterlim @ A @ real
@ ^ [X4: A] : ( real_V7770717601297561774m_norm @ B @ ( F2 @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ F5 )
= ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) ) @ F5 ) ) ) ).
% tendsto_norm_zero_iff
thf(fact_4499_tendsto__norm__zero__cancel,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [F2: A > B,F5: filter @ A] :
( ( filterlim @ A @ real
@ ^ [X4: A] : ( real_V7770717601297561774m_norm @ B @ ( F2 @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ F5 )
=> ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) ) @ F5 ) ) ) ).
% tendsto_norm_zero_cancel
thf(fact_4500_tendsto__divide,axiom,
! [A: $tType,B: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: B > A,A3: A,F5: filter @ B,G: B > A,B3: A] :
( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ A3 ) @ F5 )
=> ( ( filterlim @ B @ A @ G @ ( topolo7230453075368039082e_nhds @ A @ B3 ) @ F5 )
=> ( ( B3
!= ( zero_zero @ A ) )
=> ( filterlim @ B @ A
@ ^ [X4: B] : ( divide_divide @ A @ ( F2 @ X4 ) @ ( G @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( divide_divide @ A @ A3 @ B3 ) )
@ F5 ) ) ) ) ) ).
% tendsto_divide
thf(fact_4501_tendsto__divide__zero,axiom,
! [A: $tType,B: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: B > A,F5: filter @ B,C2: A] :
( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ F5 )
=> ( filterlim @ B @ A
@ ^ [X4: B] : ( divide_divide @ A @ ( F2 @ X4 ) @ C2 )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ F5 ) ) ) ).
% tendsto_divide_zero
thf(fact_4502_LIM__zero,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [F2: A > B,L: B,F5: filter @ A] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L ) @ F5 )
=> ( filterlim @ A @ B
@ ^ [X4: A] : ( minus_minus @ B @ ( F2 @ X4 ) @ L )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ F5 ) ) ) ).
% LIM_zero
thf(fact_4503_LIM__zero__iff,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [F2: A > B,L: B,F5: filter @ A] :
( ( filterlim @ A @ B
@ ^ [X4: A] : ( minus_minus @ B @ ( F2 @ X4 ) @ L )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ F5 )
= ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L ) @ F5 ) ) ) ).
% LIM_zero_iff
thf(fact_4504_Lim__transform,axiom,
! [A: $tType,B: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [G: B > A,A3: A,F5: filter @ B,F2: B > A] :
( ( filterlim @ B @ A @ G @ ( topolo7230453075368039082e_nhds @ A @ A3 ) @ F5 )
=> ( ( filterlim @ B @ A
@ ^ [X4: B] : ( minus_minus @ A @ ( F2 @ X4 ) @ ( G @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ F5 )
=> ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ A3 ) @ F5 ) ) ) ) ).
% Lim_transform
thf(fact_4505_Lim__transform2,axiom,
! [A: $tType,B: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: B > A,A3: A,F5: filter @ B,G: B > A] :
( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ A3 ) @ F5 )
=> ( ( filterlim @ B @ A
@ ^ [X4: B] : ( minus_minus @ A @ ( F2 @ X4 ) @ ( G @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ F5 )
=> ( filterlim @ B @ A @ G @ ( topolo7230453075368039082e_nhds @ A @ A3 ) @ F5 ) ) ) ) ).
% Lim_transform2
thf(fact_4506_LIM__zero__cancel,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [F2: A > B,L: B,F5: filter @ A] :
( ( filterlim @ A @ B
@ ^ [X4: A] : ( minus_minus @ B @ ( F2 @ X4 ) @ L )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ F5 )
=> ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L ) @ F5 ) ) ) ).
% LIM_zero_cancel
thf(fact_4507_Lim__transform__eq,axiom,
! [A: $tType,B: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: B > A,G: B > A,F5: filter @ B,A3: A] :
( ( filterlim @ B @ A
@ ^ [X4: B] : ( minus_minus @ A @ ( F2 @ X4 ) @ ( G @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ F5 )
=> ( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ A3 ) @ F5 )
= ( filterlim @ B @ A @ G @ ( topolo7230453075368039082e_nhds @ A @ A3 ) @ F5 ) ) ) ) ).
% Lim_transform_eq
thf(fact_4508_tendsto__add__zero,axiom,
! [B: $tType,D: $tType] :
( ( topolo6943815403480290642id_add @ B )
=> ! [F2: D > B,F5: filter @ D,G: D > B] :
( ( filterlim @ D @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) ) @ F5 )
=> ( ( filterlim @ D @ B @ G @ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) ) @ F5 )
=> ( filterlim @ D @ B
@ ^ [X4: D] : ( plus_plus @ B @ ( F2 @ X4 ) @ ( G @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ F5 ) ) ) ) ).
% tendsto_add_zero
thf(fact_4509_tendsto__mult__zero,axiom,
! [A: $tType,D: $tType] :
( ( real_V4412858255891104859lgebra @ A )
=> ! [F2: D > A,F5: filter @ D,G: D > A] :
( ( filterlim @ D @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ F5 )
=> ( ( filterlim @ D @ A @ G @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ F5 )
=> ( filterlim @ D @ A
@ ^ [X4: D] : ( times_times @ A @ ( F2 @ X4 ) @ ( G @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ F5 ) ) ) ) ).
% tendsto_mult_zero
thf(fact_4510_tendsto__mult__left__zero,axiom,
! [A: $tType,D: $tType] :
( ( real_V4412858255891104859lgebra @ A )
=> ! [F2: D > A,F5: filter @ D,C2: A] :
( ( filterlim @ D @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ F5 )
=> ( filterlim @ D @ A
@ ^ [X4: D] : ( times_times @ A @ ( F2 @ X4 ) @ C2 )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ F5 ) ) ) ).
% tendsto_mult_left_zero
thf(fact_4511_tendsto__mult__right__zero,axiom,
! [A: $tType,D: $tType] :
( ( real_V4412858255891104859lgebra @ A )
=> ! [F2: D > A,F5: filter @ D,C2: A] :
( ( filterlim @ D @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ F5 )
=> ( filterlim @ D @ A
@ ^ [X4: D] : ( times_times @ A @ C2 @ ( F2 @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ F5 ) ) ) ).
% tendsto_mult_right_zero
thf(fact_4512_tendsto__sgn,axiom,
! [A: $tType,B: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: B > A,L: A,F5: filter @ B] :
( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ F5 )
=> ( ( L
!= ( zero_zero @ A ) )
=> ( filterlim @ B @ A
@ ^ [X4: B] : ( sgn_sgn @ A @ ( F2 @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( sgn_sgn @ A @ L ) )
@ F5 ) ) ) ) ).
% tendsto_sgn
thf(fact_4513_tendsto__null__sum,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( topolo5987344860129210374id_add @ C )
=> ! [I6: set @ B,F2: A > B > C,F5: filter @ A] :
( ! [I4: B] :
( ( member @ B @ I4 @ I6 )
=> ( filterlim @ A @ C
@ ^ [X4: A] : ( F2 @ X4 @ I4 )
@ ( topolo7230453075368039082e_nhds @ C @ ( zero_zero @ C ) )
@ F5 ) )
=> ( filterlim @ A @ C
@ ^ [I3: A] : ( groups7311177749621191930dd_sum @ B @ C @ ( F2 @ I3 ) @ I6 )
@ ( topolo7230453075368039082e_nhds @ C @ ( zero_zero @ C ) )
@ F5 ) ) ) ).
% tendsto_null_sum
thf(fact_4514_tendsto__const__iff,axiom,
! [B: $tType,A: $tType] :
( ( topological_t2_space @ A )
=> ! [F5: filter @ B,A3: A,B3: A] :
( ( F5
!= ( bot_bot @ ( filter @ B ) ) )
=> ( ( filterlim @ B @ A
@ ^ [X4: B] : A3
@ ( topolo7230453075368039082e_nhds @ A @ B3 )
@ F5 )
= ( A3 = B3 ) ) ) ) ).
% tendsto_const_iff
thf(fact_4515_tendsto__unique,axiom,
! [B: $tType,A: $tType] :
( ( topological_t2_space @ A )
=> ! [F5: filter @ B,F2: B > A,A3: A,B3: A] :
( ( F5
!= ( bot_bot @ ( filter @ B ) ) )
=> ( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ A3 ) @ F5 )
=> ( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ B3 ) @ F5 )
=> ( A3 = B3 ) ) ) ) ) ).
% tendsto_unique
thf(fact_4516_tendsto__bot,axiom,
! [A: $tType,B: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [F2: B > A,A3: A] : ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ A3 ) @ ( bot_bot @ ( filter @ B ) ) ) ) ).
% tendsto_bot
thf(fact_4517_nhds__neq__bot,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [A3: A] :
( ( topolo7230453075368039082e_nhds @ A @ A3 )
!= ( bot_bot @ ( filter @ A ) ) ) ) ).
% nhds_neq_bot
thf(fact_4518_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
@ ^ [X4: A] : K
@ ( topolo7230453075368039082e_nhds @ Aa @ ( zero_zero @ Aa ) )
@ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% LIM_not_zero
thf(fact_4519_tendsto__tan,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [F2: A > A,A3: A,F5: filter @ A] :
( ( filterlim @ A @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ A3 ) @ F5 )
=> ( ( ( cos @ A @ A3 )
!= ( zero_zero @ A ) )
=> ( filterlim @ A @ A
@ ^ [X4: A] : ( tan @ A @ ( F2 @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( tan @ A @ A3 ) )
@ F5 ) ) ) ) ).
% tendsto_tan
thf(fact_4520_tendsto__cot,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [F2: A > A,A3: A,F5: filter @ A] :
( ( filterlim @ A @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ A3 ) @ F5 )
=> ( ( ( sin @ A @ A3 )
!= ( zero_zero @ A ) )
=> ( filterlim @ A @ A
@ ^ [X4: A] : ( cot @ A @ ( F2 @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( cot @ A @ A3 ) )
@ F5 ) ) ) ) ).
% tendsto_cot
thf(fact_4521_tendsto__tanh,axiom,
! [A: $tType,C: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [F2: C > A,A3: A,F5: filter @ C] :
( ( filterlim @ C @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ A3 ) @ F5 )
=> ( ( ( cosh @ A @ A3 )
!= ( zero_zero @ A ) )
=> ( filterlim @ C @ A
@ ^ [X4: C] : ( tanh @ A @ ( F2 @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( tanh @ A @ A3 ) )
@ F5 ) ) ) ) ).
% tendsto_tanh
thf(fact_4522_tendsto__within__subset,axiom,
! [B: $tType,A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [F2: A > B,L: filter @ B,X: A,S2: set @ A,T4: set @ A] :
( ( filterlim @ A @ B @ F2 @ L @ ( topolo174197925503356063within @ A @ X @ S2 ) )
=> ( ( ord_less_eq @ ( set @ A ) @ T4 @ S2 )
=> ( filterlim @ A @ B @ F2 @ L @ ( topolo174197925503356063within @ A @ X @ T4 ) ) ) ) ) ).
% tendsto_within_subset
thf(fact_4523_LIM__isCont__iff,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [F2: A > B,A3: A] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ ( F2 @ A3 ) ) @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) )
= ( filterlim @ A @ B
@ ^ [H: A] : ( F2 @ ( plus_plus @ A @ A3 @ H ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( F2 @ A3 ) )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% LIM_isCont_iff
thf(fact_4524_LIM__offset__zero,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [F2: A > B,L5: B,A3: A] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L5 ) @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) )
=> ( filterlim @ A @ B
@ ^ [H: A] : ( F2 @ ( plus_plus @ A @ A3 @ H ) )
@ ( topolo7230453075368039082e_nhds @ B @ L5 )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% LIM_offset_zero
thf(fact_4525_LIM__offset__zero__cancel,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [F2: A > B,A3: A,L5: B] :
( ( filterlim @ A @ B
@ ^ [H: A] : ( F2 @ ( plus_plus @ A @ A3 @ H ) )
@ ( topolo7230453075368039082e_nhds @ B @ L5 )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) )
=> ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L5 ) @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% LIM_offset_zero_cancel
thf(fact_4526_tendsto__null__power,axiom,
! [B: $tType,A: $tType] :
( ( real_V2822296259951069270ebra_1 @ B )
=> ! [F2: A > B,F5: filter @ A,N2: nat] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) ) @ F5 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( filterlim @ A @ B
@ ^ [X4: A] : ( power_power @ B @ ( F2 @ X4 ) @ N2 )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ F5 ) ) ) ) ).
% tendsto_null_power
thf(fact_4527_LIM__offset,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [F2: A > B,L5: B,A3: A,K: A] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L5 ) @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) )
=> ( filterlim @ A @ B
@ ^ [X4: A] : ( F2 @ ( plus_plus @ A @ X4 @ K ) )
@ ( topolo7230453075368039082e_nhds @ B @ L5 )
@ ( topolo174197925503356063within @ A @ ( minus_minus @ A @ A3 @ K ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% LIM_offset
thf(fact_4528_Max_OcoboundedI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,A3: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( member @ A @ A3 @ A4 )
=> ( ord_less_eq @ A @ A3 @ ( lattic643756798349783984er_Max @ A @ A4 ) ) ) ) ) ).
% Max.coboundedI
thf(fact_4529_Max__eq__if,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,B2: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( finite_finite2 @ A @ B2 )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ A4 )
=> ? [Xa: A] :
( ( member @ A @ Xa @ B2 )
& ( ord_less_eq @ A @ X5 @ Xa ) ) )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ B2 )
=> ? [Xa: A] :
( ( member @ A @ Xa @ A4 )
& ( ord_less_eq @ A @ X5 @ Xa ) ) )
=> ( ( lattic643756798349783984er_Max @ A @ A4 )
= ( lattic643756798349783984er_Max @ A @ B2 ) ) ) ) ) ) ) ).
% Max_eq_if
thf(fact_4530_Max__eqI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ! [Y3: A] :
( ( member @ A @ Y3 @ A4 )
=> ( ord_less_eq @ A @ Y3 @ X ) )
=> ( ( member @ A @ X @ A4 )
=> ( ( lattic643756798349783984er_Max @ A @ A4 )
= X ) ) ) ) ) ).
% Max_eqI
thf(fact_4531_Max__ge,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( member @ A @ X @ A4 )
=> ( ord_less_eq @ A @ X @ ( lattic643756798349783984er_Max @ A @ A4 ) ) ) ) ) ).
% Max_ge
thf(fact_4532_Max__in,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( member @ A @ ( lattic643756798349783984er_Max @ A @ A4 ) @ A4 ) ) ) ) ).
% Max_in
thf(fact_4533_Max_Oin__idem,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( member @ A @ X @ A4 )
=> ( ( ord_max @ A @ X @ ( lattic643756798349783984er_Max @ A @ A4 ) )
= ( lattic643756798349783984er_Max @ A @ A4 ) ) ) ) ) ).
% Max.in_idem
thf(fact_4534_DERIV__LIM__iff,axiom,
! [A: $tType] :
( ( ( inverse @ A )
& ( real_V822414075346904944vector @ A ) )
=> ! [F2: A > A,A3: A,D6: A] :
( ( filterlim @ A @ A
@ ^ [H: A] : ( divide_divide @ A @ ( minus_minus @ A @ ( F2 @ ( plus_plus @ A @ A3 @ H ) ) @ ( F2 @ A3 ) ) @ H )
@ ( topolo7230453075368039082e_nhds @ A @ D6 )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) )
= ( filterlim @ A @ A
@ ^ [X4: A] : ( divide_divide @ A @ ( minus_minus @ A @ ( F2 @ X4 ) @ ( F2 @ A3 ) ) @ ( minus_minus @ A @ X4 @ A3 ) )
@ ( topolo7230453075368039082e_nhds @ A @ D6 )
@ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% DERIV_LIM_iff
thf(fact_4535_Max_OboundedI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [A6: A] :
( ( member @ A @ A6 @ A4 )
=> ( ord_less_eq @ A @ A6 @ X ) )
=> ( ord_less_eq @ A @ ( lattic643756798349783984er_Max @ A @ A4 ) @ X ) ) ) ) ) ).
% Max.boundedI
thf(fact_4536_Max_OboundedE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ ( lattic643756798349783984er_Max @ A @ A4 ) @ X )
=> ! [A11: A] :
( ( member @ A @ A11 @ A4 )
=> ( ord_less_eq @ A @ A11 @ X ) ) ) ) ) ) ).
% Max.boundedE
thf(fact_4537_eq__Max__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,M2: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( M2
= ( lattic643756798349783984er_Max @ A @ A4 ) )
= ( ( member @ A @ M2 @ A4 )
& ! [X4: A] :
( ( member @ A @ X4 @ A4 )
=> ( ord_less_eq @ A @ X4 @ M2 ) ) ) ) ) ) ) ).
% eq_Max_iff
thf(fact_4538_Max__ge__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ X @ ( lattic643756798349783984er_Max @ A @ A4 ) )
= ( ? [X4: A] :
( ( member @ A @ X4 @ A4 )
& ( ord_less_eq @ A @ X @ X4 ) ) ) ) ) ) ) ).
% Max_ge_iff
thf(fact_4539_Max__eq__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,M2: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ( lattic643756798349783984er_Max @ A @ A4 )
= M2 )
= ( ( member @ A @ M2 @ A4 )
& ! [X4: A] :
( ( member @ A @ X4 @ A4 )
=> ( ord_less_eq @ A @ X4 @ M2 ) ) ) ) ) ) ) ).
% Max_eq_iff
thf(fact_4540_Max__gr__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less @ A @ X @ ( lattic643756798349783984er_Max @ A @ A4 ) )
= ( ? [X4: A] :
( ( member @ A @ X4 @ A4 )
& ( ord_less @ A @ X @ X4 ) ) ) ) ) ) ) ).
% Max_gr_iff
thf(fact_4541_Max__insert2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,A3: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ! [B5: A] :
( ( member @ A @ B5 @ A4 )
=> ( ord_less_eq @ A @ B5 @ A3 ) )
=> ( ( lattic643756798349783984er_Max @ A @ ( insert2 @ A @ A3 @ A4 ) )
= A3 ) ) ) ) ).
% Max_insert2
thf(fact_4542_cSup__eq__Max,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [X8: set @ A] :
( ( finite_finite2 @ A @ X8 )
=> ( ( X8
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Sup_Sup @ A @ X8 )
= ( lattic643756798349783984er_Max @ A @ X8 ) ) ) ) ) ).
% cSup_eq_Max
thf(fact_4543_Max__Sup,axiom,
! [A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [A4: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798349783984er_Max @ A @ A4 )
= ( complete_Sup_Sup @ A @ A4 ) ) ) ) ) ).
% Max_Sup
thf(fact_4544_DERIV__D,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,D6: A,X: A] :
( ( has_field_derivative @ A @ F2 @ D6 @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) )
=> ( filterlim @ A @ A
@ ^ [H: A] : ( divide_divide @ A @ ( minus_minus @ A @ ( F2 @ ( plus_plus @ A @ X @ H ) ) @ ( F2 @ X ) ) @ H )
@ ( topolo7230453075368039082e_nhds @ A @ D6 )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% DERIV_D
thf(fact_4545_DERIV__def,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,D6: A,X: A] :
( ( has_field_derivative @ A @ F2 @ D6 @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) )
= ( filterlim @ A @ A
@ ^ [H: A] : ( divide_divide @ A @ ( minus_minus @ A @ ( F2 @ ( plus_plus @ A @ X @ H ) ) @ ( F2 @ X ) ) @ H )
@ ( topolo7230453075368039082e_nhds @ A @ D6 )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% DERIV_def
thf(fact_4546_lim__exp__minus__1,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ( filterlim @ A @ A
@ ^ [Z6: A] : ( divide_divide @ A @ ( minus_minus @ A @ ( exp @ A @ Z6 ) @ ( one_one @ A ) ) @ Z6 )
@ ( topolo7230453075368039082e_nhds @ A @ ( one_one @ A ) )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ).
% lim_exp_minus_1
thf(fact_4547_lemma__termdiff4,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [K: real,F2: A > B,K5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K )
=> ( ! [H5: A] :
( ( H5
!= ( zero_zero @ A ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ H5 ) @ K )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( F2 @ H5 ) ) @ ( times_times @ real @ K5 @ ( real_V7770717601297561774m_norm @ A @ H5 ) ) ) ) )
=> ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) ) @ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% lemma_termdiff4
thf(fact_4548_filterlim__at__to__0,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: A > B,F5: filter @ B,A3: A] :
( ( filterlim @ A @ B @ F2 @ F5 @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) )
= ( filterlim @ A @ B
@ ^ [X4: A] : ( F2 @ ( plus_plus @ A @ X4 @ A3 ) )
@ F5
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% filterlim_at_to_0
thf(fact_4549_Max_Osubset__imp,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B2 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B2 )
=> ( ord_less_eq @ A @ ( lattic643756798349783984er_Max @ A @ A4 ) @ ( lattic643756798349783984er_Max @ A @ B2 ) ) ) ) ) ) ).
% Max.subset_imp
thf(fact_4550_Max__mono,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [M7: set @ A,N6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ M7 @ N6 )
=> ( ( M7
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ N6 )
=> ( ord_less_eq @ A @ ( lattic643756798349783984er_Max @ A @ M7 ) @ ( lattic643756798349783984er_Max @ A @ N6 ) ) ) ) ) ) ).
% Max_mono
thf(fact_4551_hom__Max__commute,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [H2: A > A,N6: set @ A] :
( ! [X5: A,Y3: A] :
( ( H2 @ ( ord_max @ A @ X5 @ Y3 ) )
= ( ord_max @ A @ ( H2 @ X5 ) @ ( H2 @ Y3 ) ) )
=> ( ( finite_finite2 @ A @ N6 )
=> ( ( N6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( H2 @ ( lattic643756798349783984er_Max @ A @ N6 ) )
= ( lattic643756798349783984er_Max @ A @ ( image2 @ A @ A @ H2 @ N6 ) ) ) ) ) ) ) ).
% hom_Max_commute
thf(fact_4552_Max_Osubset,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,B2: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( B2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ A4 )
=> ( ( ord_max @ A @ ( lattic643756798349783984er_Max @ A @ B2 ) @ ( lattic643756798349783984er_Max @ A @ A4 ) )
= ( lattic643756798349783984er_Max @ A @ A4 ) ) ) ) ) ) ).
% Max.subset
thf(fact_4553_Max_Oclosed,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X5: A,Y3: A] : ( member @ A @ ( ord_max @ A @ X5 @ Y3 ) @ ( insert2 @ A @ X5 @ ( insert2 @ A @ Y3 @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( member @ A @ ( lattic643756798349783984er_Max @ A @ A4 ) @ A4 ) ) ) ) ) ).
% Max.closed
thf(fact_4554_Max_Oinsert__not__elem,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ~ ( member @ A @ X @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798349783984er_Max @ A @ ( insert2 @ A @ X @ A4 ) )
= ( ord_max @ A @ X @ ( lattic643756798349783984er_Max @ A @ A4 ) ) ) ) ) ) ) ).
% Max.insert_not_elem
thf(fact_4555_card__le__Suc__Max,axiom,
! [S2: set @ nat] :
( ( finite_finite2 @ nat @ S2 )
=> ( ord_less_eq @ nat @ ( finite_card @ nat @ S2 ) @ ( suc @ ( lattic643756798349783984er_Max @ nat @ S2 ) ) ) ) ).
% card_le_Suc_Max
thf(fact_4556_Sup__nat__def,axiom,
( ( complete_Sup_Sup @ nat )
= ( ^ [X6: set @ nat] :
( if @ nat
@ ( X6
= ( bot_bot @ ( set @ nat ) ) )
@ ( zero_zero @ nat )
@ ( lattic643756798349783984er_Max @ nat @ X6 ) ) ) ) ).
% Sup_nat_def
thf(fact_4557_divide__nat__def,axiom,
( ( divide_divide @ nat )
= ( ^ [M: nat,N: nat] :
( if @ nat
@ ( N
= ( zero_zero @ nat ) )
@ ( zero_zero @ nat )
@ ( lattic643756798349783984er_Max @ nat
@ ( collect @ nat
@ ^ [K2: nat] : ( ord_less_eq @ nat @ ( times_times @ nat @ K2 @ N ) @ M ) ) ) ) ) ) ).
% divide_nat_def
thf(fact_4558_Max__add__commute,axiom,
! [B: $tType,A: $tType] :
( ( linord4140545234300271783up_add @ A )
=> ! [S2: set @ B,F2: B > A,K: A] :
( ( finite_finite2 @ B @ S2 )
=> ( ( S2
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( lattic643756798349783984er_Max @ A
@ ( image2 @ B @ A
@ ^ [X4: B] : ( plus_plus @ A @ ( F2 @ X4 ) @ K )
@ S2 ) )
= ( plus_plus @ A @ ( lattic643756798349783984er_Max @ A @ ( image2 @ B @ A @ F2 @ S2 ) ) @ K ) ) ) ) ) ).
% Max_add_commute
thf(fact_4559_powser__limit__0,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [S: real,A3: nat > A,F2: A > A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ S )
=> ( ! [X5: A] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X5 ) @ S )
=> ( sums @ A
@ ^ [N: nat] : ( times_times @ A @ ( A3 @ N ) @ ( power_power @ A @ X5 @ N ) )
@ ( F2 @ X5 ) ) )
=> ( filterlim @ A @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ ( A3 @ ( zero_zero @ nat ) ) ) @ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% powser_limit_0
thf(fact_4560_powser__limit__0__strong,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [S: real,A3: nat > A,F2: A > A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ S )
=> ( ! [X5: A] :
( ( X5
!= ( zero_zero @ A ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X5 ) @ S )
=> ( sums @ A
@ ^ [N: nat] : ( times_times @ A @ ( A3 @ N ) @ ( power_power @ A @ X5 @ N ) )
@ ( F2 @ X5 ) ) ) )
=> ( filterlim @ A @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ ( A3 @ ( zero_zero @ nat ) ) ) @ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% powser_limit_0_strong
thf(fact_4561_lemma__termdiff5,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_Vector_banach @ B ) )
=> ! [K: real,F2: nat > real,G: A > nat > B] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K )
=> ( ( summable @ real @ F2 )
=> ( ! [H5: A,N3: nat] :
( ( H5
!= ( zero_zero @ A ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ H5 ) @ K )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( G @ H5 @ N3 ) ) @ ( times_times @ real @ ( F2 @ N3 ) @ ( real_V7770717601297561774m_norm @ A @ H5 ) ) ) ) )
=> ( filterlim @ A @ B
@ ^ [H: A] : ( suminf @ B @ ( G @ H ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ).
% lemma_termdiff5
thf(fact_4562_Max_Oinsert__remove,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( ( ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798349783984er_Max @ A @ ( insert2 @ A @ X @ A4 ) )
= X ) )
& ( ( ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798349783984er_Max @ A @ ( insert2 @ A @ X @ A4 ) )
= ( ord_max @ A @ X @ ( lattic643756798349783984er_Max @ A @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ).
% Max.insert_remove
thf(fact_4563_Max_Oremove,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( member @ A @ X @ A4 )
=> ( ( ( ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798349783984er_Max @ A @ A4 )
= X ) )
& ( ( ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798349783984er_Max @ A @ A4 )
= ( ord_max @ A @ X @ ( lattic643756798349783984er_Max @ A @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ) ).
% Max.remove
thf(fact_4564_sum__le__card__Max,axiom,
! [A: $tType,A4: set @ A,F2: A > nat] :
( ( finite_finite2 @ A @ A4 )
=> ( ord_less_eq @ nat @ ( groups7311177749621191930dd_sum @ A @ nat @ F2 @ A4 ) @ ( times_times @ nat @ ( finite_card @ A @ A4 ) @ ( lattic643756798349783984er_Max @ nat @ ( image2 @ A @ nat @ F2 @ A4 ) ) ) ) ) ).
% sum_le_card_Max
thf(fact_4565_finite__psubset__def,axiom,
! [A: $tType] :
( ( finite_psubset @ A )
= ( collect @ ( product_prod @ ( set @ A ) @ ( set @ A ) )
@ ( product_case_prod @ ( set @ A ) @ ( set @ A ) @ $o
@ ^ [A7: set @ A,B6: set @ A] :
( ( ord_less @ ( set @ A ) @ A7 @ B6 )
& ( finite_finite2 @ A @ B6 ) ) ) ) ) ).
% finite_psubset_def
thf(fact_4566_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 ) )
=> ! [N7: nat] :
( member @ real
@ ( suminf @ real
@ ^ [I3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I3 ) @ ( A3 @ I3 ) ) )
@ ( set_or1337092689740270186AtMost @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I3 ) @ ( A3 @ I3 ) )
@ ( set_ord_lessThan @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N7 ) @ ( one_one @ nat ) ) ) )
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I3 ) @ ( A3 @ I3 ) )
@ ( set_ord_lessThan @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N7 ) ) ) ) ) ) ) ) ).
% summable_Leibniz(3)
thf(fact_4567_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 ) ) )
=> ! [N7: nat] :
( member @ real
@ ( suminf @ real
@ ^ [I3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I3 ) @ ( A3 @ I3 ) ) )
@ ( set_or1337092689740270186AtMost @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I3 ) @ ( A3 @ I3 ) )
@ ( set_ord_lessThan @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N7 ) ) )
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I3 ) @ ( A3 @ I3 ) )
@ ( set_ord_lessThan @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N7 ) @ ( one_one @ nat ) ) ) ) ) ) ) ) ) ).
% summable_Leibniz(2)
thf(fact_4568_summable__Leibniz_H_I4_J,axiom,
! [A3: nat > real,N2: nat] :
( ( filterlim @ nat @ real @ A3 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) )
=> ( ! [N3: nat] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( A3 @ N3 ) )
=> ( ! [N3: nat] : ( ord_less_eq @ real @ ( A3 @ ( suc @ N3 ) ) @ ( A3 @ N3 ) )
=> ( ord_less_eq @ real
@ ( suminf @ real
@ ^ [I3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I3 ) @ ( A3 @ I3 ) ) )
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I3 ) @ ( A3 @ I3 ) )
@ ( set_ord_lessThan @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) @ ( one_one @ nat ) ) ) ) ) ) ) ) ).
% summable_Leibniz'(4)
thf(fact_4569_summable__Leibniz_H_I5_J,axiom,
! [A3: nat > real] :
( ( filterlim @ nat @ real @ A3 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) )
=> ( ! [N3: nat] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( A3 @ N3 ) )
=> ( ! [N3: nat] : ( ord_less_eq @ real @ ( A3 @ ( suc @ N3 ) ) @ ( A3 @ N3 ) )
=> ( filterlim @ nat @ real
@ ^ [N: nat] :
( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I3 ) @ ( A3 @ I3 ) )
@ ( set_ord_lessThan @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ ( one_one @ nat ) ) ) )
@ ( topolo7230453075368039082e_nhds @ real
@ ( suminf @ real
@ ^ [I3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I3 ) @ ( A3 @ I3 ) ) ) )
@ ( at_top @ nat ) ) ) ) ) ).
% summable_Leibniz'(5)
thf(fact_4570_trivial__limit__sequentially,axiom,
( ( at_top @ nat )
!= ( bot_bot @ ( filter @ nat ) ) ) ).
% trivial_limit_sequentially
thf(fact_4571_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
@ ^ [N: nat] : ( times_times @ A @ C2 @ ( A3 @ N ) )
@ ( 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_4572_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
@ ^ [N: nat] : ( times_times @ A @ ( A3 @ N ) @ 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_4573_tendsto__zero__divide__iff,axiom,
! [A: $tType] :
( ( ( field @ A )
& ( topolo4211221413907600880p_mult @ A ) )
=> ! [C2: A,A3: nat > A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( filterlim @ nat @ A
@ ^ [N: nat] : ( divide_divide @ A @ ( A3 @ N ) @ 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_4574_filterlim__Suc,axiom,
filterlim @ nat @ nat @ suc @ ( at_top @ nat ) @ ( at_top @ nat ) ).
% filterlim_Suc
thf(fact_4575_filterlim__sequentially__Suc,axiom,
! [A: $tType,F2: nat > A,F5: filter @ A] :
( ( filterlim @ nat @ A
@ ^ [X4: nat] : ( F2 @ ( suc @ X4 ) )
@ F5
@ ( at_top @ nat ) )
= ( filterlim @ nat @ A @ F2 @ F5 @ ( at_top @ nat ) ) ) ).
% filterlim_sequentially_Suc
thf(fact_4576_trivial__limit__at__top__linorder,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( at_top @ A )
!= ( bot_bot @ ( filter @ A ) ) ) ) ).
% trivial_limit_at_top_linorder
thf(fact_4577_approx__from__above__dense__linorder,axiom,
! [A: $tType] :
( ( ( dense_linorder @ A )
& ( topolo3112930676232923870pology @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ? [U4: nat > A] :
( ! [N7: nat] : ( ord_less @ A @ X @ ( U4 @ N7 ) )
& ( filterlim @ nat @ A @ U4 @ ( topolo7230453075368039082e_nhds @ A @ X ) @ ( at_top @ nat ) ) ) ) ) ).
% approx_from_above_dense_linorder
thf(fact_4578_approx__from__below__dense__linorder,axiom,
! [A: $tType] :
( ( ( dense_linorder @ A )
& ( topolo3112930676232923870pology @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [Y: A,X: A] :
( ( ord_less @ A @ Y @ X )
=> ? [U4: nat > A] :
( ! [N7: nat] : ( ord_less @ A @ ( U4 @ N7 ) @ X )
& ( filterlim @ nat @ A @ U4 @ ( topolo7230453075368039082e_nhds @ A @ X ) @ ( at_top @ nat ) ) ) ) ) ).
% approx_from_below_dense_linorder
thf(fact_4579_LIMSEQ__Suc,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [F2: nat > A,L: A] :
( ( filterlim @ nat @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ ( at_top @ nat ) )
=> ( filterlim @ nat @ A
@ ^ [N: nat] : ( F2 @ ( suc @ N ) )
@ ( topolo7230453075368039082e_nhds @ A @ L )
@ ( at_top @ nat ) ) ) ) ).
% LIMSEQ_Suc
thf(fact_4580_LIMSEQ__imp__Suc,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [F2: nat > A,L: A] :
( ( filterlim @ nat @ A
@ ^ [N: nat] : ( F2 @ ( suc @ N ) )
@ ( topolo7230453075368039082e_nhds @ A @ L )
@ ( at_top @ nat ) )
=> ( filterlim @ nat @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ ( at_top @ nat ) ) ) ) ).
% LIMSEQ_imp_Suc
thf(fact_4581_LIMSEQ__offset,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [F2: nat > A,K: nat,A3: A] :
( ( filterlim @ nat @ A
@ ^ [N: nat] : ( F2 @ ( plus_plus @ nat @ N @ K ) )
@ ( topolo7230453075368039082e_nhds @ A @ A3 )
@ ( at_top @ nat ) )
=> ( filterlim @ nat @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ A3 ) @ ( at_top @ nat ) ) ) ) ).
% LIMSEQ_offset
thf(fact_4582_LIMSEQ__ignore__initial__segment,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [F2: nat > A,A3: A,K: nat] :
( ( filterlim @ nat @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ A3 ) @ ( at_top @ nat ) )
=> ( filterlim @ nat @ A
@ ^ [N: nat] : ( F2 @ ( plus_plus @ nat @ N @ K ) )
@ ( topolo7230453075368039082e_nhds @ A @ A3 )
@ ( at_top @ nat ) ) ) ) ).
% LIMSEQ_ignore_initial_segment
thf(fact_4583_LIMSEQ__le__const2,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [X8: nat > A,X: A,A3: A] :
( ( filterlim @ nat @ A @ X8 @ ( topolo7230453075368039082e_nhds @ A @ X ) @ ( at_top @ nat ) )
=> ( ? [N8: nat] :
! [N3: nat] :
( ( ord_less_eq @ nat @ N8 @ N3 )
=> ( ord_less_eq @ A @ ( X8 @ N3 ) @ A3 ) )
=> ( ord_less_eq @ A @ X @ A3 ) ) ) ) ).
% LIMSEQ_le_const2
thf(fact_4584_LIMSEQ__le__const,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [X8: nat > A,X: A,A3: A] :
( ( filterlim @ nat @ A @ X8 @ ( topolo7230453075368039082e_nhds @ A @ X ) @ ( at_top @ nat ) )
=> ( ? [N8: nat] :
! [N3: nat] :
( ( ord_less_eq @ nat @ N8 @ N3 )
=> ( ord_less_eq @ A @ A3 @ ( X8 @ N3 ) ) )
=> ( ord_less_eq @ A @ A3 @ X ) ) ) ) ).
% LIMSEQ_le_const
thf(fact_4585_Lim__bounded2,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [F2: nat > A,L: A,N6: nat,C5: A] :
( ( filterlim @ nat @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ ( at_top @ nat ) )
=> ( ! [N3: nat] :
( ( ord_less_eq @ nat @ N6 @ N3 )
=> ( ord_less_eq @ A @ C5 @ ( F2 @ N3 ) ) )
=> ( ord_less_eq @ A @ C5 @ L ) ) ) ) ).
% Lim_bounded2
thf(fact_4586_Lim__bounded,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [F2: nat > A,L: A,M7: nat,C5: A] :
( ( filterlim @ nat @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ ( at_top @ nat ) )
=> ( ! [N3: nat] :
( ( ord_less_eq @ nat @ M7 @ N3 )
=> ( ord_less_eq @ A @ ( F2 @ N3 ) @ C5 ) )
=> ( ord_less_eq @ A @ L @ C5 ) ) ) ) ).
% Lim_bounded
thf(fact_4587_LIMSEQ__le,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [X8: nat > A,X: A,Y7: nat > A,Y: A] :
( ( filterlim @ nat @ A @ X8 @ ( topolo7230453075368039082e_nhds @ A @ X ) @ ( at_top @ nat ) )
=> ( ( filterlim @ nat @ A @ Y7 @ ( topolo7230453075368039082e_nhds @ A @ Y ) @ ( at_top @ nat ) )
=> ( ? [N8: nat] :
! [N3: nat] :
( ( ord_less_eq @ nat @ N8 @ N3 )
=> ( ord_less_eq @ A @ ( X8 @ N3 ) @ ( Y7 @ N3 ) ) )
=> ( ord_less_eq @ A @ X @ Y ) ) ) ) ) ).
% LIMSEQ_le
thf(fact_4588_lim__mono,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [N6: nat,X8: nat > A,Y7: nat > A,X: A,Y: A] :
( ! [N3: nat] :
( ( ord_less_eq @ nat @ N6 @ N3 )
=> ( ord_less_eq @ A @ ( X8 @ N3 ) @ ( Y7 @ N3 ) ) )
=> ( ( filterlim @ nat @ A @ X8 @ ( topolo7230453075368039082e_nhds @ A @ X ) @ ( at_top @ nat ) )
=> ( ( filterlim @ nat @ A @ Y7 @ ( topolo7230453075368039082e_nhds @ A @ Y ) @ ( at_top @ nat ) )
=> ( ord_less_eq @ A @ X @ Y ) ) ) ) ) ).
% lim_mono
thf(fact_4589_Sup__lim,axiom,
! [A: $tType] :
( ( ( comple5582772986160207858norder @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [B3: nat > A,S: set @ A,A3: A] :
( ! [N3: nat] : ( member @ A @ ( B3 @ N3 ) @ S )
=> ( ( filterlim @ nat @ A @ B3 @ ( topolo7230453075368039082e_nhds @ A @ A3 ) @ ( at_top @ nat ) )
=> ( ord_less_eq @ A @ A3 @ ( complete_Sup_Sup @ A @ S ) ) ) ) ) ).
% Sup_lim
thf(fact_4590_Inf__lim,axiom,
! [A: $tType] :
( ( ( comple5582772986160207858norder @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [B3: nat > A,S: set @ A,A3: A] :
( ! [N3: nat] : ( member @ A @ ( B3 @ N3 ) @ S )
=> ( ( filterlim @ nat @ A @ B3 @ ( topolo7230453075368039082e_nhds @ A @ A3 ) @ ( at_top @ nat ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ S ) @ A3 ) ) ) ) ).
% Inf_lim
thf(fact_4591_Inf__as__limit,axiom,
! [A: $tType] :
( ( ( comple5582772986160207858norder @ A )
& ( topolo3112930676232923870pology @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [A4: set @ A] :
( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ? [U4: nat > A] :
( ! [N7: nat] : ( member @ A @ ( U4 @ N7 ) @ A4 )
& ( filterlim @ nat @ A @ U4 @ ( topolo7230453075368039082e_nhds @ A @ ( complete_Inf_Inf @ A @ A4 ) ) @ ( at_top @ nat ) ) ) ) ) ).
% Inf_as_limit
thf(fact_4592_summable__LIMSEQ__zero,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A] :
( ( summable @ A @ F2 )
=> ( filterlim @ nat @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ ( at_top @ nat ) ) ) ) ).
% summable_LIMSEQ_zero
thf(fact_4593_mult__nat__right__at__top,axiom,
! [C2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ C2 )
=> ( filterlim @ nat @ nat
@ ^ [X4: nat] : ( times_times @ nat @ X4 @ C2 )
@ ( at_top @ nat )
@ ( at_top @ nat ) ) ) ).
% mult_nat_right_at_top
thf(fact_4594_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_4595_monoseq__le,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [A3: nat > A,X: A] :
( ( topological_monoseq @ A @ A3 )
=> ( ( filterlim @ nat @ A @ A3 @ ( topolo7230453075368039082e_nhds @ A @ X ) @ ( at_top @ nat ) )
=> ( ( ! [N7: nat] : ( ord_less_eq @ A @ ( A3 @ N7 ) @ X )
& ! [M5: nat,N7: nat] :
( ( ord_less_eq @ nat @ M5 @ N7 )
=> ( ord_less_eq @ A @ ( A3 @ M5 ) @ ( A3 @ N7 ) ) ) )
| ( ! [N7: nat] : ( ord_less_eq @ A @ X @ ( A3 @ N7 ) )
& ! [M5: nat,N7: nat] :
( ( ord_less_eq @ nat @ M5 @ N7 )
=> ( ord_less_eq @ A @ ( A3 @ N7 ) @ ( A3 @ M5 ) ) ) ) ) ) ) ) ).
% monoseq_le
thf(fact_4596_lim__const__over__n,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [A3: A] :
( filterlim @ nat @ A
@ ^ [N: nat] : ( divide_divide @ A @ A3 @ ( semiring_1_of_nat @ A @ N ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ ( at_top @ nat ) ) ) ).
% lim_const_over_n
thf(fact_4597_lim__inverse__n,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ( filterlim @ nat @ A
@ ^ [N: nat] : ( inverse_inverse @ A @ ( semiring_1_of_nat @ A @ N ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ ( at_top @ nat ) ) ) ).
% lim_inverse_n
thf(fact_4598_LIMSEQ__linear,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [X8: nat > A,X: A,L: nat] :
( ( filterlim @ nat @ A @ X8 @ ( topolo7230453075368039082e_nhds @ A @ X ) @ ( at_top @ nat ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ L )
=> ( filterlim @ nat @ A
@ ^ [N: nat] : ( X8 @ ( times_times @ nat @ N @ L ) )
@ ( topolo7230453075368039082e_nhds @ A @ X )
@ ( at_top @ nat ) ) ) ) ) ).
% LIMSEQ_linear
thf(fact_4599_telescope__summable_H,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A,C2: A] :
( ( filterlim @ nat @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ C2 ) @ ( at_top @ nat ) )
=> ( summable @ A
@ ^ [N: nat] : ( minus_minus @ A @ ( F2 @ N ) @ ( F2 @ ( suc @ N ) ) ) ) ) ) ).
% telescope_summable'
thf(fact_4600_telescope__summable,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A,C2: A] :
( ( filterlim @ nat @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ C2 ) @ ( at_top @ nat ) )
=> ( summable @ A
@ ^ [N: nat] : ( minus_minus @ A @ ( F2 @ ( suc @ N ) ) @ ( F2 @ N ) ) ) ) ) ).
% telescope_summable
thf(fact_4601_nested__sequence__unique,axiom,
! [F2: nat > real,G: nat > real] :
( ! [N3: nat] : ( ord_less_eq @ real @ ( F2 @ N3 ) @ ( F2 @ ( suc @ N3 ) ) )
=> ( ! [N3: nat] : ( ord_less_eq @ real @ ( G @ ( suc @ N3 ) ) @ ( G @ N3 ) )
=> ( ! [N3: nat] : ( ord_less_eq @ real @ ( F2 @ N3 ) @ ( G @ N3 ) )
=> ( ( filterlim @ nat @ real
@ ^ [N: nat] : ( minus_minus @ real @ ( F2 @ N ) @ ( G @ N ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ ( at_top @ nat ) )
=> ? [L4: real] :
( ! [N7: nat] : ( ord_less_eq @ real @ ( F2 @ N7 ) @ L4 )
& ( filterlim @ nat @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ L4 ) @ ( at_top @ nat ) )
& ! [N7: nat] : ( ord_less_eq @ real @ L4 @ ( G @ N7 ) )
& ( filterlim @ nat @ real @ G @ ( topolo7230453075368039082e_nhds @ real @ L4 ) @ ( at_top @ nat ) ) ) ) ) ) ) ).
% nested_sequence_unique
thf(fact_4602_LIMSEQ__inverse__zero,axiom,
! [X8: nat > real] :
( ! [R3: real] :
? [N8: nat] :
! [N3: nat] :
( ( ord_less_eq @ nat @ N8 @ N3 )
=> ( ord_less @ real @ R3 @ ( X8 @ N3 ) ) )
=> ( filterlim @ nat @ real
@ ^ [N: nat] : ( inverse_inverse @ real @ ( X8 @ N ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ ( at_top @ nat ) ) ) ).
% LIMSEQ_inverse_zero
thf(fact_4603_LIMSEQ__inverse__real__of__nat,axiom,
( filterlim @ nat @ real
@ ^ [N: nat] : ( inverse_inverse @ real @ ( semiring_1_of_nat @ real @ ( suc @ N ) ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ ( at_top @ nat ) ) ).
% LIMSEQ_inverse_real_of_nat
thf(fact_4604_LIMSEQ__inverse__real__of__nat__add,axiom,
! [R2: real] :
( filterlim @ nat @ real
@ ^ [N: nat] : ( plus_plus @ real @ R2 @ ( inverse_inverse @ real @ ( semiring_1_of_nat @ real @ ( suc @ N ) ) ) )
@ ( topolo7230453075368039082e_nhds @ real @ R2 )
@ ( at_top @ nat ) ) ).
% LIMSEQ_inverse_real_of_nat_add
thf(fact_4605_increasing__LIMSEQ,axiom,
! [F2: nat > real,L: real] :
( ! [N3: nat] : ( ord_less_eq @ real @ ( F2 @ N3 ) @ ( F2 @ ( suc @ N3 ) ) )
=> ( ! [N3: nat] : ( ord_less_eq @ real @ ( F2 @ N3 ) @ L )
=> ( ! [E: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E )
=> ? [N7: nat] : ( ord_less_eq @ real @ L @ ( plus_plus @ real @ ( F2 @ N7 ) @ E ) ) )
=> ( filterlim @ nat @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ L ) @ ( at_top @ nat ) ) ) ) ) ).
% increasing_LIMSEQ
thf(fact_4606_lim__1__over__n,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ( filterlim @ nat @ A
@ ^ [N: nat] : ( divide_divide @ A @ ( one_one @ A ) @ ( semiring_1_of_nat @ A @ N ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ ( at_top @ nat ) ) ) ).
% lim_1_over_n
thf(fact_4607_LIMSEQ__n__over__Suc__n,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ( filterlim @ nat @ A
@ ^ [N: nat] : ( divide_divide @ A @ ( semiring_1_of_nat @ A @ N ) @ ( semiring_1_of_nat @ A @ ( suc @ N ) ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( one_one @ A ) )
@ ( at_top @ nat ) ) ) ).
% LIMSEQ_n_over_Suc_n
thf(fact_4608_LIMSEQ__Suc__n__over__n,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ( filterlim @ nat @ A
@ ^ [N: nat] : ( divide_divide @ A @ ( semiring_1_of_nat @ A @ ( suc @ N ) ) @ ( semiring_1_of_nat @ A @ N ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( one_one @ A ) )
@ ( at_top @ nat ) ) ) ).
% LIMSEQ_Suc_n_over_n
thf(fact_4609_telescope__sums_H,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A,C2: A] :
( ( filterlim @ nat @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ C2 ) @ ( at_top @ nat ) )
=> ( sums @ A
@ ^ [N: nat] : ( minus_minus @ A @ ( F2 @ N ) @ ( F2 @ ( suc @ N ) ) )
@ ( minus_minus @ A @ ( F2 @ ( zero_zero @ nat ) ) @ C2 ) ) ) ) ).
% telescope_sums'
thf(fact_4610_telescope__sums,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A,C2: A] :
( ( filterlim @ nat @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ C2 ) @ ( at_top @ nat ) )
=> ( sums @ A
@ ^ [N: nat] : ( minus_minus @ A @ ( F2 @ ( suc @ N ) ) @ ( F2 @ N ) )
@ ( minus_minus @ A @ C2 @ ( F2 @ ( zero_zero @ nat ) ) ) ) ) ) ).
% telescope_sums
thf(fact_4611_sums__def_H,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ( ( sums @ A )
= ( ^ [F3: nat > A,S7: A] :
( filterlim @ nat @ A
@ ^ [N: nat] : ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
@ ( topolo7230453075368039082e_nhds @ A @ S7 )
@ ( at_top @ nat ) ) ) ) ) ).
% sums_def'
thf(fact_4612_LIMSEQ__inverse__real__of__nat__add__minus,axiom,
! [R2: real] :
( filterlim @ nat @ real
@ ^ [N: nat] : ( plus_plus @ real @ R2 @ ( uminus_uminus @ real @ ( inverse_inverse @ real @ ( semiring_1_of_nat @ real @ ( suc @ N ) ) ) ) )
@ ( topolo7230453075368039082e_nhds @ real @ R2 )
@ ( at_top @ nat ) ) ).
% LIMSEQ_inverse_real_of_nat_add_minus
thf(fact_4613_LIMSEQ__D,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X8: nat > A,L5: A,R2: real] :
( ( filterlim @ nat @ A @ X8 @ ( topolo7230453075368039082e_nhds @ A @ L5 ) @ ( at_top @ nat ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ R2 )
=> ? [No: nat] :
! [N7: nat] :
( ( ord_less_eq @ nat @ No @ N7 )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( X8 @ N7 ) @ L5 ) ) @ R2 ) ) ) ) ) ).
% LIMSEQ_D
thf(fact_4614_LIMSEQ__I,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X8: nat > A,L5: A] :
( ! [R3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R3 )
=> ? [No2: nat] :
! [N3: nat] :
( ( ord_less_eq @ nat @ No2 @ N3 )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( X8 @ N3 ) @ L5 ) ) @ R3 ) ) )
=> ( filterlim @ nat @ A @ X8 @ ( topolo7230453075368039082e_nhds @ A @ L5 ) @ ( at_top @ nat ) ) ) ) ).
% LIMSEQ_I
thf(fact_4615_LIMSEQ__iff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X8: nat > A,L5: A] :
( ( filterlim @ nat @ A @ X8 @ ( topolo7230453075368039082e_nhds @ A @ L5 ) @ ( at_top @ nat ) )
= ( ! [R4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R4 )
=> ? [No3: nat] :
! [N: nat] :
( ( ord_less_eq @ nat @ No3 @ N )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( X8 @ N ) @ L5 ) ) @ R4 ) ) ) ) ) ) ).
% LIMSEQ_iff
thf(fact_4616_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_4617_tendsto__power__zero,axiom,
! [A: $tType,B: $tType] :
( ( real_V2822296259951069270ebra_1 @ A )
=> ! [F2: B > nat,F5: filter @ B,X: A] :
( ( filterlim @ B @ nat @ F2 @ ( at_top @ nat ) @ F5 )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ ( one_one @ real ) )
=> ( filterlim @ B @ A
@ ^ [Y6: B] : ( power_power @ A @ X @ ( F2 @ Y6 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ F5 ) ) ) ) ).
% tendsto_power_zero
thf(fact_4618_LIMSEQ__inverse__real__of__nat__add__minus__mult,axiom,
! [R2: real] :
( filterlim @ nat @ real
@ ^ [N: nat] : ( times_times @ real @ R2 @ ( plus_plus @ real @ ( one_one @ real ) @ ( uminus_uminus @ real @ ( inverse_inverse @ real @ ( semiring_1_of_nat @ real @ ( suc @ N ) ) ) ) ) )
@ ( topolo7230453075368039082e_nhds @ real @ R2 )
@ ( at_top @ nat ) ) ).
% LIMSEQ_inverse_real_of_nat_add_minus_mult
thf(fact_4619_LIMSEQ__norm__0,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A] :
( ! [N3: nat] : ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( F2 @ N3 ) ) @ ( divide_divide @ real @ ( one_one @ real ) @ ( semiring_1_of_nat @ real @ ( suc @ N3 ) ) ) )
=> ( filterlim @ nat @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ ( at_top @ nat ) ) ) ) ).
% LIMSEQ_norm_0
thf(fact_4620_field__derivative__lim__unique,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,Df: A,Z3: A,S: nat > A,A3: A] :
( ( has_field_derivative @ A @ F2 @ Df @ ( topolo174197925503356063within @ A @ Z3 @ ( top_top @ ( set @ A ) ) ) )
=> ( ( filterlim @ nat @ A @ S @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ ( at_top @ nat ) )
=> ( ! [N3: nat] :
( ( S @ N3 )
!= ( zero_zero @ A ) )
=> ( ( filterlim @ nat @ A
@ ^ [N: nat] : ( divide_divide @ A @ ( minus_minus @ A @ ( F2 @ ( plus_plus @ A @ Z3 @ ( S @ N ) ) ) @ ( F2 @ Z3 ) ) @ ( S @ N ) )
@ ( topolo7230453075368039082e_nhds @ A @ A3 )
@ ( at_top @ nat ) )
=> ( Df = A3 ) ) ) ) ) ) ).
% field_derivative_lim_unique
thf(fact_4621_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
@ ^ [N: nat] : ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ ( power_power @ A @ X @ N ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ ( at_top @ nat ) ) ) ) ).
% powser_times_n_limit_0
thf(fact_4622_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
@ ^ [N: nat] : ( divide_divide @ A @ ( semiring_1_of_nat @ A @ N ) @ ( power_power @ A @ X @ N ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ ( at_top @ nat ) ) ) ) ).
% lim_n_over_pown
thf(fact_4623_summable,axiom,
! [A3: nat > real] :
( ( filterlim @ nat @ real @ A3 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) )
=> ( ! [N3: nat] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( A3 @ N3 ) )
=> ( ! [N3: nat] : ( ord_less_eq @ real @ ( A3 @ ( suc @ N3 ) ) @ ( A3 @ N3 ) )
=> ( summable @ real
@ ^ [N: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ N ) @ ( A3 @ N ) ) ) ) ) ) ).
% summable
thf(fact_4624_zeroseq__arctan__series,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ X ) @ ( one_one @ real ) )
=> ( filterlim @ nat @ real
@ ^ [N: nat] : ( times_times @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( semiring_1_of_nat @ real @ ( plus_plus @ nat @ ( times_times @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) @ ( power_power @ real @ X @ ( plus_plus @ nat @ ( times_times @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ ( at_top @ nat ) ) ) ).
% zeroseq_arctan_series
thf(fact_4625_summable__Leibniz_H_I2_J,axiom,
! [A3: nat > real,N2: nat] :
( ( filterlim @ nat @ real @ A3 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) )
=> ( ! [N3: nat] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( A3 @ N3 ) )
=> ( ! [N3: nat] : ( ord_less_eq @ real @ ( A3 @ ( suc @ N3 ) ) @ ( A3 @ N3 ) )
=> ( ord_less_eq @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I3 ) @ ( A3 @ I3 ) )
@ ( set_ord_lessThan @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) )
@ ( suminf @ real
@ ^ [I3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I3 ) @ ( A3 @ I3 ) ) ) ) ) ) ) ).
% summable_Leibniz'(2)
thf(fact_4626_summable__Leibniz_H_I3_J,axiom,
! [A3: nat > real] :
( ( filterlim @ nat @ real @ A3 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) )
=> ( ! [N3: nat] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( A3 @ N3 ) )
=> ( ! [N3: nat] : ( ord_less_eq @ real @ ( A3 @ ( suc @ N3 ) ) @ ( A3 @ N3 ) )
=> ( filterlim @ nat @ real
@ ^ [N: nat] :
( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I3 ) @ ( A3 @ I3 ) )
@ ( set_ord_lessThan @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) )
@ ( topolo7230453075368039082e_nhds @ real
@ ( suminf @ real
@ ^ [I3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I3 ) @ ( A3 @ I3 ) ) ) )
@ ( at_top @ nat ) ) ) ) ) ).
% summable_Leibniz'(3)
thf(fact_4627_sums__alternating__upper__lower,axiom,
! [A3: nat > real] :
( ! [N3: nat] : ( ord_less_eq @ real @ ( A3 @ ( suc @ N3 ) ) @ ( A3 @ N3 ) )
=> ( ! [N3: nat] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( A3 @ N3 ) )
=> ( ( filterlim @ nat @ real @ A3 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) )
=> ? [L4: real] :
( ! [N7: nat] :
( ord_less_eq @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I3 ) @ ( A3 @ I3 ) )
@ ( set_ord_lessThan @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N7 ) ) )
@ L4 )
& ( filterlim @ nat @ real
@ ^ [N: nat] :
( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I3 ) @ ( A3 @ I3 ) )
@ ( set_ord_lessThan @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) )
@ ( topolo7230453075368039082e_nhds @ real @ L4 )
@ ( at_top @ nat ) )
& ! [N7: nat] :
( ord_less_eq @ real @ L4
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I3 ) @ ( A3 @ I3 ) )
@ ( set_ord_lessThan @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N7 ) @ ( one_one @ nat ) ) ) ) )
& ( filterlim @ nat @ real
@ ^ [N: nat] :
( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I3 ) @ ( A3 @ I3 ) )
@ ( set_ord_lessThan @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ ( one_one @ nat ) ) ) )
@ ( topolo7230453075368039082e_nhds @ real @ L4 )
@ ( at_top @ nat ) ) ) ) ) ) ).
% sums_alternating_upper_lower
thf(fact_4628_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
@ ^ [N: nat] :
( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I3 ) @ ( A3 @ I3 ) )
@ ( set_ord_lessThan @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ ( one_one @ nat ) ) ) )
@ ( topolo7230453075368039082e_nhds @ real
@ ( suminf @ real
@ ^ [I3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I3 ) @ ( A3 @ I3 ) ) ) )
@ ( at_top @ nat ) ) ) ) ).
% summable_Leibniz(5)
thf(fact_4629_lim__zero__infinity,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,L: A] :
( ( filterlim @ A @ A
@ ^ [X4: A] : ( F2 @ ( divide_divide @ A @ ( one_one @ A ) @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ A @ L )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) )
=> ( filterlim @ A @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ ( at_infinity @ A ) ) ) ) ).
% lim_zero_infinity
thf(fact_4630_LIM__offset__zero__iff,axiom,
! [C: $tType,D: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( topolo4958980785337419405_space @ D )
& ( zero @ C ) )
=> ! [A3: A,F2: A > D,L5: D] :
( ( nO_MATCH @ C @ A @ ( zero_zero @ C ) @ A3 )
=> ( ( filterlim @ A @ D @ F2 @ ( topolo7230453075368039082e_nhds @ D @ L5 ) @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) )
= ( filterlim @ A @ D
@ ^ [H: A] : ( F2 @ ( plus_plus @ A @ A3 @ H ) )
@ ( topolo7230453075368039082e_nhds @ D @ L5 )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% LIM_offset_zero_iff
thf(fact_4631_Bfun__inverse,axiom,
! [A: $tType,B: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [F2: B > A,A3: A,F5: filter @ B] :
( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ A3 ) @ F5 )
=> ( ( A3
!= ( zero_zero @ A ) )
=> ( bfun @ B @ A
@ ^ [X4: B] : ( inverse_inverse @ A @ ( F2 @ X4 ) )
@ F5 ) ) ) ) ).
% Bfun_inverse
thf(fact_4632_filterlim__divide__at__infinity,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,C2: A,F5: filter @ A,G: A > A] :
( ( filterlim @ A @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ C2 ) @ F5 )
=> ( ( filterlim @ A @ A @ G @ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) @ F5 )
=> ( ( C2
!= ( zero_zero @ A ) )
=> ( filterlim @ A @ A
@ ^ [X4: A] : ( divide_divide @ A @ ( F2 @ X4 ) @ ( G @ X4 ) )
@ ( at_infinity @ A )
@ F5 ) ) ) ) ) ).
% filterlim_divide_at_infinity
thf(fact_4633_Bseq__add__iff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A,C2: A] :
( ( bfun @ nat @ A
@ ^ [X4: nat] : ( plus_plus @ A @ ( F2 @ X4 ) @ C2 )
@ ( at_top @ nat ) )
= ( bfun @ nat @ A @ F2 @ ( at_top @ nat ) ) ) ) ).
% Bseq_add_iff
thf(fact_4634_Bseq__add,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A,C2: A] :
( ( bfun @ nat @ A @ F2 @ ( at_top @ nat ) )
=> ( bfun @ nat @ A
@ ^ [X4: nat] : ( plus_plus @ A @ ( F2 @ X4 ) @ C2 )
@ ( at_top @ nat ) ) ) ) ).
% Bseq_add
thf(fact_4635_Bseq__Suc__iff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A] :
( ( bfun @ nat @ A
@ ^ [N: nat] : ( F2 @ ( suc @ N ) )
@ ( at_top @ nat ) )
= ( bfun @ nat @ A @ F2 @ ( at_top @ nat ) ) ) ) ).
% Bseq_Suc_iff
thf(fact_4636_Bseq__ignore__initial__segment,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X8: nat > A,K: nat] :
( ( bfun @ nat @ A @ X8 @ ( at_top @ nat ) )
=> ( bfun @ nat @ A
@ ^ [N: nat] : ( X8 @ ( plus_plus @ nat @ N @ K ) )
@ ( at_top @ nat ) ) ) ) ).
% Bseq_ignore_initial_segment
thf(fact_4637_Bseq__offset,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X8: nat > A,K: nat] :
( ( bfun @ nat @ A
@ ^ [N: nat] : ( X8 @ ( plus_plus @ nat @ N @ K ) )
@ ( at_top @ nat ) )
=> ( bfun @ nat @ A @ X8 @ ( at_top @ nat ) ) ) ) ).
% Bseq_offset
thf(fact_4638_Bseq__cmult__iff,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [C2: A,F2: nat > A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( bfun @ nat @ A
@ ^ [X4: nat] : ( times_times @ A @ C2 @ ( F2 @ X4 ) )
@ ( at_top @ nat ) )
= ( bfun @ nat @ A @ F2 @ ( at_top @ nat ) ) ) ) ) ).
% Bseq_cmult_iff
thf(fact_4639_not__tendsto__and__filterlim__at__infinity,axiom,
! [A: $tType,B: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F5: filter @ B,F2: B > A,C2: A] :
( ( F5
!= ( bot_bot @ ( filter @ B ) ) )
=> ( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ C2 ) @ F5 )
=> ~ ( filterlim @ B @ A @ F2 @ ( at_infinity @ A ) @ F5 ) ) ) ) ).
% not_tendsto_and_filterlim_at_infinity
thf(fact_4640_scale__right__distrib__NO__MATCH,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [X: A,Y: A,A3: real] :
( ( nO_MATCH @ A @ real @ ( divide_divide @ A @ X @ Y ) @ A3 )
=> ( ( real_V8093663219630862766scaleR @ A @ A3 @ ( plus_plus @ A @ X @ Y ) )
= ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ A3 @ X ) @ ( real_V8093663219630862766scaleR @ A @ A3 @ Y ) ) ) ) ) ).
% scale_right_distrib_NO_MATCH
thf(fact_4641_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_4642_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_4643_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_4644_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_4645_filterlim__pow__at__top,axiom,
! [A: $tType,N2: nat,F2: A > real,F5: filter @ A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( filterlim @ A @ real @ F2 @ ( at_top @ real ) @ F5 )
=> ( filterlim @ A @ real
@ ^ [X4: A] : ( power_power @ real @ ( F2 @ X4 ) @ N2 )
@ ( at_top @ real )
@ F5 ) ) ) ).
% filterlim_pow_at_top
thf(fact_4646_tendsto__add__filterlim__at__infinity_H,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [F2: A > B,F5: filter @ A,G: A > B,C2: B] :
( ( filterlim @ A @ B @ F2 @ ( at_infinity @ B ) @ F5 )
=> ( ( filterlim @ A @ B @ G @ ( topolo7230453075368039082e_nhds @ B @ C2 ) @ F5 )
=> ( filterlim @ A @ B
@ ^ [X4: A] : ( plus_plus @ B @ ( F2 @ X4 ) @ ( G @ X4 ) )
@ ( at_infinity @ B )
@ F5 ) ) ) ) ).
% tendsto_add_filterlim_at_infinity'
thf(fact_4647_tendsto__add__filterlim__at__infinity,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [F2: A > B,C2: B,F5: filter @ A,G: A > B] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ C2 ) @ F5 )
=> ( ( filterlim @ A @ B @ G @ ( at_infinity @ B ) @ F5 )
=> ( filterlim @ A @ B
@ ^ [X4: A] : ( plus_plus @ B @ ( F2 @ X4 ) @ ( G @ X4 ) )
@ ( at_infinity @ B )
@ F5 ) ) ) ) ).
% tendsto_add_filterlim_at_infinity
thf(fact_4648_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_4649_Bseq__iff1a,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X8: nat > A] :
( ( bfun @ nat @ A @ X8 @ ( at_top @ nat ) )
= ( ? [N5: nat] :
! [N: nat] : ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( X8 @ N ) ) @ ( semiring_1_of_nat @ real @ ( suc @ N5 ) ) ) ) ) ) ).
% Bseq_iff1a
thf(fact_4650_Bseq__iff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X8: nat > A] :
( ( bfun @ nat @ A @ X8 @ ( at_top @ nat ) )
= ( ? [N5: nat] :
! [N: nat] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( X8 @ N ) ) @ ( semiring_1_of_nat @ real @ ( suc @ N5 ) ) ) ) ) ) ).
% Bseq_iff
thf(fact_4651_tendsto__mult__filterlim__at__infinity,axiom,
! [A: $tType,B: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: B > A,C2: A,F5: filter @ B,G: B > A] :
( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ C2 ) @ F5 )
=> ( ( C2
!= ( zero_zero @ A ) )
=> ( ( filterlim @ B @ A @ G @ ( at_infinity @ A ) @ F5 )
=> ( filterlim @ B @ A
@ ^ [X4: B] : ( times_times @ A @ ( F2 @ X4 ) @ ( G @ X4 ) )
@ ( at_infinity @ A )
@ F5 ) ) ) ) ) ).
% tendsto_mult_filterlim_at_infinity
thf(fact_4652_tendsto__divide__0,axiom,
! [A: $tType,C: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [F2: C > A,C2: A,F5: filter @ C,G: C > A] :
( ( filterlim @ C @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ C2 ) @ F5 )
=> ( ( filterlim @ C @ A @ G @ ( at_infinity @ A ) @ F5 )
=> ( filterlim @ C @ A
@ ^ [X4: C] : ( divide_divide @ A @ ( F2 @ X4 ) @ ( G @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ F5 ) ) ) ) ).
% tendsto_divide_0
thf(fact_4653_filterlim__power__at__infinity,axiom,
! [B: $tType,A: $tType] :
( ( real_V8999393235501362500lgebra @ B )
=> ! [F2: A > B,F5: filter @ A,N2: nat] :
( ( filterlim @ A @ B @ F2 @ ( at_infinity @ B ) @ F5 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( filterlim @ A @ B
@ ^ [X4: A] : ( power_power @ B @ ( F2 @ X4 ) @ N2 )
@ ( at_infinity @ B )
@ F5 ) ) ) ) ).
% filterlim_power_at_infinity
thf(fact_4654_scale__left__distrib__NO__MATCH,axiom,
! [C: $tType,A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [X: A,Y: A,C2: C,A3: real,B3: real] :
( ( nO_MATCH @ A @ C @ ( divide_divide @ A @ X @ Y ) @ C2 )
=> ( ( real_V8093663219630862766scaleR @ A @ ( plus_plus @ real @ A3 @ B3 ) @ X )
= ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ A3 @ X ) @ ( real_V8093663219630862766scaleR @ A @ B3 @ X ) ) ) ) ) ).
% scale_left_distrib_NO_MATCH
thf(fact_4655_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_4656_filterlim__inverse__at__iff,axiom,
! [B: $tType,A: $tType] :
( ( real_V8999393235501362500lgebra @ B )
=> ! [G: A > B,F5: filter @ A] :
( ( filterlim @ A @ B
@ ^ [X4: A] : ( inverse_inverse @ B @ ( G @ X4 ) )
@ ( topolo174197925503356063within @ B @ ( zero_zero @ B ) @ ( top_top @ ( set @ B ) ) )
@ F5 )
= ( filterlim @ A @ B @ G @ ( at_infinity @ B ) @ F5 ) ) ) ).
% filterlim_inverse_at_iff
thf(fact_4657_Bseq__iff2,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X8: nat > A] :
( ( bfun @ nat @ A @ X8 @ ( at_top @ nat ) )
= ( ? [K2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K2 )
& ? [X4: A] :
! [N: nat] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ ( X8 @ N ) @ ( uminus_uminus @ A @ X4 ) ) ) @ K2 ) ) ) ) ) ).
% Bseq_iff2
thf(fact_4658_Bseq__iff3,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X8: nat > A] :
( ( bfun @ nat @ A @ X8 @ ( at_top @ nat ) )
= ( ? [K2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K2 )
& ? [N5: nat] :
! [N: nat] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ ( X8 @ N ) @ ( uminus_uminus @ A @ ( X8 @ N5 ) ) ) ) @ K2 ) ) ) ) ) ).
% Bseq_iff3
thf(fact_4659_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_4660_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_4661_polyfun__extremal,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [C2: nat > A,K: nat,N2: nat,B2: real] :
( ( ( C2 @ K )
!= ( zero_zero @ A ) )
=> ( ( ord_less_eq @ nat @ ( one_one @ nat ) @ K )
=> ( ( ord_less_eq @ nat @ K @ N2 )
=> ( eventually @ A
@ ^ [Z6: A] :
( ord_less_eq @ real @ B2
@ ( real_V7770717601297561774m_norm @ A
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( C2 @ I3 ) @ ( power_power @ A @ Z6 @ I3 ) )
@ ( set_ord_atMost @ nat @ N2 ) ) ) )
@ ( at_infinity @ A ) ) ) ) ) ) ).
% polyfun_extremal
thf(fact_4662_eventually__sequentially__Suc,axiom,
! [P: nat > $o] :
( ( eventually @ nat
@ ^ [I3: nat] : ( P @ ( suc @ I3 ) )
@ ( at_top @ nat ) )
= ( eventually @ nat @ P @ ( at_top @ nat ) ) ) ).
% eventually_sequentially_Suc
thf(fact_4663_eventually__sequentially__seg,axiom,
! [P: nat > $o,K: nat] :
( ( eventually @ nat
@ ^ [N: nat] : ( P @ ( plus_plus @ nat @ N @ K ) )
@ ( at_top @ nat ) )
= ( eventually @ nat @ P @ ( at_top @ nat ) ) ) ).
% eventually_sequentially_seg
thf(fact_4664_eventually__const,axiom,
! [A: $tType,F5: filter @ A,P: $o] :
( ( F5
!= ( bot_bot @ ( filter @ A ) ) )
=> ( ( eventually @ A
@ ^ [X4: A] : P
@ F5 )
= P ) ) ).
% eventually_const
thf(fact_4665_eventually__bot,axiom,
! [A: $tType,P: A > $o] : ( eventually @ A @ P @ ( bot_bot @ ( filter @ A ) ) ) ).
% eventually_bot
thf(fact_4666_eventually__happens,axiom,
! [A: $tType,P: A > $o,Net: filter @ A] :
( ( eventually @ A @ P @ Net )
=> ( ( Net
= ( bot_bot @ ( filter @ A ) ) )
| ? [X_12: A] : ( P @ X_12 ) ) ) ).
% eventually_happens
thf(fact_4667_eventually__happens_H,axiom,
! [A: $tType,F5: filter @ A,P: A > $o] :
( ( F5
!= ( bot_bot @ ( filter @ A ) ) )
=> ( ( eventually @ A @ P @ F5 )
=> ? [X_12: A] : ( P @ X_12 ) ) ) ).
% eventually_happens'
thf(fact_4668_False__imp__not__eventually,axiom,
! [A: $tType,P: A > $o,Net: filter @ A] :
( ! [X5: A] :
~ ( P @ X5 )
=> ( ( Net
!= ( bot_bot @ ( filter @ A ) ) )
=> ~ ( eventually @ A @ P @ Net ) ) ) ).
% False_imp_not_eventually
thf(fact_4669_eventually__const__iff,axiom,
! [A: $tType,P: $o,F5: filter @ A] :
( ( eventually @ A
@ ^ [X4: A] : P
@ F5 )
= ( P
| ( F5
= ( bot_bot @ ( filter @ A ) ) ) ) ) ).
% eventually_const_iff
thf(fact_4670_trivial__limit__def,axiom,
! [A: $tType,F5: filter @ A] :
( ( F5
= ( bot_bot @ ( filter @ A ) ) )
= ( eventually @ A
@ ^ [X4: A] : $false
@ F5 ) ) ).
% trivial_limit_def
thf(fact_4671_eventually__at__top__linorder,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > $o] :
( ( eventually @ A @ P @ ( at_top @ A ) )
= ( ? [N5: A] :
! [N: A] :
( ( ord_less_eq @ A @ N5 @ N )
=> ( P @ N ) ) ) ) ) ).
% eventually_at_top_linorder
thf(fact_4672_eventually__at__top__linorderI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [C2: A,P: A > $o] :
( ! [X5: A] :
( ( ord_less_eq @ A @ C2 @ X5 )
=> ( P @ X5 ) )
=> ( eventually @ A @ P @ ( at_top @ A ) ) ) ) ).
% eventually_at_top_linorderI
thf(fact_4673_eventually__at__top__dense,axiom,
! [A: $tType] :
( ( ( linorder @ A )
& ( no_top @ A ) )
=> ! [P: A > $o] :
( ( eventually @ A @ P @ ( at_top @ A ) )
= ( ? [N5: A] :
! [N: A] :
( ( ord_less @ A @ N5 @ N )
=> ( P @ N ) ) ) ) ) ).
% eventually_at_top_dense
thf(fact_4674_eventually__sequentiallyI,axiom,
! [C2: nat,P: nat > $o] :
( ! [X5: nat] :
( ( ord_less_eq @ nat @ C2 @ X5 )
=> ( P @ X5 ) )
=> ( eventually @ nat @ P @ ( at_top @ nat ) ) ) ).
% eventually_sequentiallyI
thf(fact_4675_eventually__sequentially,axiom,
! [P: nat > $o] :
( ( eventually @ nat @ P @ ( at_top @ nat ) )
= ( ? [N5: nat] :
! [N: nat] :
( ( ord_less_eq @ nat @ N5 @ N )
=> ( P @ N ) ) ) ) ).
% eventually_sequentially
thf(fact_4676_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_4677_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_4678_le__sequentially,axiom,
! [F5: filter @ nat] :
( ( ord_less_eq @ ( filter @ nat ) @ F5 @ ( at_top @ nat ) )
= ( ! [N5: nat] : ( eventually @ nat @ ( ord_less_eq @ nat @ N5 ) @ F5 ) ) ) ).
% le_sequentially
thf(fact_4679_sequentially__offset,axiom,
! [P: nat > $o,K: nat] :
( ( eventually @ nat @ P @ ( at_top @ nat ) )
=> ( eventually @ nat
@ ^ [I3: nat] : ( P @ ( plus_plus @ nat @ I3 @ K ) )
@ ( at_top @ nat ) ) ) ).
% sequentially_offset
thf(fact_4680_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 ) ) )
= ( ? [B4: A] :
( ( ord_less @ A @ B4 @ ( top_top @ A ) )
& ! [Z6: A] :
( ( ord_less @ A @ B4 @ Z6 )
=> ( P @ Z6 ) ) ) ) ) ) ) ).
% eventually_nhds_top
thf(fact_4681_filterlim__at__top__at__top,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( linorder @ B ) )
=> ! [Q: A > $o,F2: A > B,P: B > $o,G: B > A] :
( ! [X5: A,Y3: A] :
( ( Q @ X5 )
=> ( ( Q @ Y3 )
=> ( ( ord_less_eq @ A @ X5 @ Y3 )
=> ( ord_less_eq @ B @ ( F2 @ X5 ) @ ( F2 @ Y3 ) ) ) ) )
=> ( ! [X5: B] :
( ( P @ X5 )
=> ( ( F2 @ ( G @ X5 ) )
= X5 ) )
=> ( ! [X5: B] :
( ( P @ X5 )
=> ( Q @ ( G @ X5 ) ) )
=> ( ( eventually @ A @ Q @ ( at_top @ A ) )
=> ( ( eventually @ B @ P @ ( at_top @ B ) )
=> ( filterlim @ A @ B @ F2 @ ( at_top @ B ) @ ( at_top @ A ) ) ) ) ) ) ) ) ).
% filterlim_at_top_at_top
thf(fact_4682_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 ) ) )
= ( ? [B4: A] :
( ( ord_less @ A @ B4 @ X )
& ! [Y6: A] :
( ( ord_less @ A @ B4 @ Y6 )
=> ( ( ord_less @ A @ Y6 @ X )
=> ( P @ Y6 ) ) ) ) ) ) ) ).
% eventually_at_left_field
thf(fact_4683_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 ) ) )
= ( ? [B4: A] :
( ( ord_less @ A @ B4 @ X )
& ! [Y6: A] :
( ( ord_less @ A @ B4 @ Y6 )
=> ( ( ord_less @ A @ Y6 @ X )
=> ( P @ Y6 ) ) ) ) ) ) ) ) ).
% eventually_at_left
thf(fact_4684_tendsto__sandwich,axiom,
! [A: $tType,B: $tType] :
( ( topolo2564578578187576103pology @ A )
=> ! [F2: B > A,G: B > A,Net: filter @ B,H2: B > A,C2: A] :
( ( eventually @ B
@ ^ [N: B] : ( ord_less_eq @ A @ ( F2 @ N ) @ ( G @ N ) )
@ Net )
=> ( ( eventually @ B
@ ^ [N: B] : ( ord_less_eq @ A @ ( G @ N ) @ ( H2 @ N ) )
@ Net )
=> ( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ C2 ) @ Net )
=> ( ( filterlim @ B @ A @ H2 @ ( topolo7230453075368039082e_nhds @ A @ C2 ) @ Net )
=> ( filterlim @ B @ A @ G @ ( topolo7230453075368039082e_nhds @ A @ C2 ) @ Net ) ) ) ) ) ) ).
% tendsto_sandwich
thf(fact_4685_order__tendsto__iff,axiom,
! [B: $tType,A: $tType] :
( ( topolo2564578578187576103pology @ A )
=> ! [F2: B > A,X: A,F5: filter @ B] :
( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ X ) @ F5 )
= ( ! [L2: A] :
( ( ord_less @ A @ L2 @ X )
=> ( eventually @ B
@ ^ [X4: B] : ( ord_less @ A @ L2 @ ( F2 @ X4 ) )
@ F5 ) )
& ! [U2: A] :
( ( ord_less @ A @ X @ U2 )
=> ( eventually @ B
@ ^ [X4: B] : ( ord_less @ A @ ( F2 @ X4 ) @ U2 )
@ F5 ) ) ) ) ) ).
% order_tendsto_iff
thf(fact_4686_order__tendstoI,axiom,
! [A: $tType,B: $tType] :
( ( topolo2564578578187576103pology @ A )
=> ! [Y: A,F2: B > A,F5: filter @ B] :
( ! [A6: A] :
( ( ord_less @ A @ A6 @ Y )
=> ( eventually @ B
@ ^ [X4: B] : ( ord_less @ A @ A6 @ ( F2 @ X4 ) )
@ F5 ) )
=> ( ! [A6: A] :
( ( ord_less @ A @ Y @ A6 )
=> ( eventually @ B
@ ^ [X4: B] : ( ord_less @ A @ ( F2 @ X4 ) @ A6 )
@ F5 ) )
=> ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ Y ) @ F5 ) ) ) ) ).
% order_tendstoI
thf(fact_4687_order__tendstoD_I1_J,axiom,
! [A: $tType,B: $tType] :
( ( topolo2564578578187576103pology @ A )
=> ! [F2: B > A,Y: A,F5: filter @ B,A3: A] :
( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ Y ) @ F5 )
=> ( ( ord_less @ A @ A3 @ Y )
=> ( eventually @ B
@ ^ [X4: B] : ( ord_less @ A @ A3 @ ( F2 @ X4 ) )
@ F5 ) ) ) ) ).
% order_tendstoD(1)
thf(fact_4688_order__tendstoD_I2_J,axiom,
! [A: $tType,B: $tType] :
( ( topolo2564578578187576103pology @ A )
=> ! [F2: B > A,Y: A,F5: filter @ B,A3: A] :
( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ Y ) @ F5 )
=> ( ( ord_less @ A @ Y @ A3 )
=> ( eventually @ B
@ ^ [X4: B] : ( ord_less @ A @ ( F2 @ X4 ) @ A3 )
@ F5 ) ) ) ) ).
% order_tendstoD(2)
thf(fact_4689_filterlim__at__top__mono,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F2: B > A,F5: filter @ B,G: B > A] :
( ( filterlim @ B @ A @ F2 @ ( at_top @ A ) @ F5 )
=> ( ( eventually @ B
@ ^ [X4: B] : ( ord_less_eq @ A @ ( F2 @ X4 ) @ ( G @ X4 ) )
@ F5 )
=> ( filterlim @ B @ A @ G @ ( at_top @ A ) @ F5 ) ) ) ) ).
% filterlim_at_top_mono
thf(fact_4690_filterlim__at__top__ge,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ B )
=> ! [F2: A > B,F5: filter @ A,C2: B] :
( ( filterlim @ A @ B @ F2 @ ( at_top @ B ) @ F5 )
= ( ! [Z9: B] :
( ( ord_less_eq @ B @ C2 @ Z9 )
=> ( eventually @ A
@ ^ [X4: A] : ( ord_less_eq @ B @ Z9 @ ( F2 @ X4 ) )
@ F5 ) ) ) ) ) ).
% filterlim_at_top_ge
thf(fact_4691_filterlim__at__top,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ B )
=> ! [F2: A > B,F5: filter @ A] :
( ( filterlim @ A @ B @ F2 @ ( at_top @ B ) @ F5 )
= ( ! [Z9: B] :
( eventually @ A
@ ^ [X4: A] : ( ord_less_eq @ B @ Z9 @ ( F2 @ X4 ) )
@ F5 ) ) ) ) ).
% filterlim_at_top
thf(fact_4692_filterlim__at__top__dense,axiom,
! [A: $tType,B: $tType] :
( ( unboun7993243217541854897norder @ B )
=> ! [F2: A > B,F5: filter @ A] :
( ( filterlim @ A @ B @ F2 @ ( at_top @ B ) @ F5 )
= ( ! [Z9: B] :
( eventually @ A
@ ^ [X4: A] : ( ord_less @ B @ Z9 @ ( F2 @ X4 ) )
@ F5 ) ) ) ) ).
% filterlim_at_top_dense
thf(fact_4693_eventually__Inf__base,axiom,
! [A: $tType,B2: set @ ( filter @ A ),P: A > $o] :
( ( B2
!= ( bot_bot @ ( set @ ( filter @ A ) ) ) )
=> ( ! [F6: filter @ A] :
( ( member @ ( filter @ A ) @ F6 @ B2 )
=> ! [G3: filter @ A] :
( ( member @ ( filter @ A ) @ G3 @ B2 )
=> ? [X2: filter @ A] :
( ( member @ ( filter @ A ) @ X2 @ B2 )
& ( ord_less_eq @ ( filter @ A ) @ X2 @ ( inf_inf @ ( filter @ A ) @ F6 @ G3 ) ) ) ) )
=> ( ( eventually @ A @ P @ ( complete_Inf_Inf @ ( filter @ A ) @ B2 ) )
= ( ? [X4: filter @ A] :
( ( member @ ( filter @ A ) @ X4 @ B2 )
& ( eventually @ A @ P @ X4 ) ) ) ) ) ) ).
% eventually_Inf_base
thf(fact_4694_eventually__INF__finite,axiom,
! [B: $tType,A: $tType,A4: set @ A,P: B > $o,F5: A > ( filter @ B )] :
( ( finite_finite2 @ A @ A4 )
=> ( ( eventually @ B @ P @ ( complete_Inf_Inf @ ( filter @ B ) @ ( image2 @ A @ ( filter @ B ) @ F5 @ A4 ) ) )
= ( ? [Q6: A > B > $o] :
( ! [X4: A] :
( ( member @ A @ X4 @ A4 )
=> ( eventually @ B @ ( Q6 @ X4 ) @ ( F5 @ X4 ) ) )
& ! [Y6: B] :
( ! [X4: A] :
( ( member @ A @ X4 @ A4 )
=> ( Q6 @ X4 @ Y6 ) )
=> ( P @ Y6 ) ) ) ) ) ) ).
% eventually_INF_finite
thf(fact_4695_eventually__at__leftI,axiom,
! [A: $tType] :
( ( topolo2564578578187576103pology @ A )
=> ! [A3: A,B3: A,P: A > $o] :
( ! [X5: A] :
( ( member @ A @ X5 @ ( set_or5935395276787703475ssThan @ A @ A3 @ B3 ) )
=> ( P @ X5 ) )
=> ( ( ord_less @ A @ A3 @ B3 )
=> ( eventually @ A @ P @ ( topolo174197925503356063within @ A @ B3 @ ( set_ord_lessThan @ A @ B3 ) ) ) ) ) ) ).
% eventually_at_leftI
thf(fact_4696_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
@ ^ [X4: A] : ( P @ ( plus_plus @ A @ X4 @ A3 ) )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% eventually_at_to_0
thf(fact_4697_increasing__tendsto,axiom,
! [A: $tType,B: $tType] :
( ( topolo2564578578187576103pology @ A )
=> ! [F2: B > A,L: A,F5: filter @ B] :
( ( eventually @ B
@ ^ [N: B] : ( ord_less_eq @ A @ ( F2 @ N ) @ L )
@ F5 )
=> ( ! [X5: A] :
( ( ord_less @ A @ X5 @ L )
=> ( eventually @ B
@ ^ [N: B] : ( ord_less @ A @ X5 @ ( F2 @ N ) )
@ F5 ) )
=> ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ F5 ) ) ) ) ).
% increasing_tendsto
thf(fact_4698_decreasing__tendsto,axiom,
! [A: $tType,B: $tType] :
( ( topolo2564578578187576103pology @ A )
=> ! [L: A,F2: B > A,F5: filter @ B] :
( ( eventually @ B
@ ^ [N: B] : ( ord_less_eq @ A @ L @ ( F2 @ N ) )
@ F5 )
=> ( ! [X5: A] :
( ( ord_less @ A @ L @ X5 )
=> ( eventually @ B
@ ^ [N: B] : ( ord_less @ A @ ( F2 @ N ) @ X5 )
@ F5 ) )
=> ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ F5 ) ) ) ) ).
% decreasing_tendsto
thf(fact_4699_filterlim__at__top__gt,axiom,
! [A: $tType,B: $tType] :
( ( unboun7993243217541854897norder @ B )
=> ! [F2: A > B,F5: filter @ A,C2: B] :
( ( filterlim @ A @ B @ F2 @ ( at_top @ B ) @ F5 )
= ( ! [Z9: B] :
( ( ord_less @ B @ C2 @ Z9 )
=> ( eventually @ A
@ ^ [X4: A] : ( ord_less_eq @ B @ Z9 @ ( F2 @ X4 ) )
@ F5 ) ) ) ) ) ).
% filterlim_at_top_gt
thf(fact_4700_tendsto__le,axiom,
! [B: $tType,A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [F5: filter @ B,F2: B > A,X: A,G: B > A,Y: A] :
( ( F5
!= ( bot_bot @ ( filter @ B ) ) )
=> ( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ X ) @ F5 )
=> ( ( filterlim @ B @ A @ G @ ( topolo7230453075368039082e_nhds @ A @ Y ) @ F5 )
=> ( ( eventually @ B
@ ^ [X4: B] : ( ord_less_eq @ A @ ( G @ X4 ) @ ( F2 @ X4 ) )
@ F5 )
=> ( ord_less_eq @ A @ Y @ X ) ) ) ) ) ) ).
% tendsto_le
thf(fact_4701_tendsto__lowerbound,axiom,
! [B: $tType,A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [F2: B > A,X: A,F5: filter @ B,A3: A] :
( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ X ) @ F5 )
=> ( ( eventually @ B
@ ^ [I3: B] : ( ord_less_eq @ A @ A3 @ ( F2 @ I3 ) )
@ F5 )
=> ( ( F5
!= ( bot_bot @ ( filter @ B ) ) )
=> ( ord_less_eq @ A @ A3 @ X ) ) ) ) ) ).
% tendsto_lowerbound
thf(fact_4702_tendsto__upperbound,axiom,
! [B: $tType,A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [F2: B > A,X: A,F5: filter @ B,A3: A] :
( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ X ) @ F5 )
=> ( ( eventually @ B
@ ^ [I3: B] : ( ord_less_eq @ A @ ( F2 @ I3 ) @ A3 )
@ F5 )
=> ( ( F5
!= ( bot_bot @ ( filter @ B ) ) )
=> ( ord_less_eq @ A @ X @ A3 ) ) ) ) ) ).
% tendsto_upperbound
thf(fact_4703_eventually__INF,axiom,
! [A: $tType,B: $tType,P: A > $o,F5: B > ( filter @ A ),B2: set @ B] :
( ( eventually @ A @ P @ ( complete_Inf_Inf @ ( filter @ A ) @ ( image2 @ B @ ( filter @ A ) @ F5 @ B2 ) ) )
= ( ? [X6: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ X6 @ B2 )
& ( finite_finite2 @ B @ X6 )
& ( eventually @ A @ P @ ( complete_Inf_Inf @ ( filter @ A ) @ ( image2 @ B @ ( filter @ A ) @ F5 @ X6 ) ) ) ) ) ) ).
% eventually_INF
thf(fact_4704_tendsto__imp__filterlim__at__left,axiom,
! [B: $tType,A: $tType] :
( ( topolo2564578578187576103pology @ B )
=> ! [F2: A > B,L5: B,F5: filter @ A] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L5 ) @ F5 )
=> ( ( eventually @ A
@ ^ [X4: A] : ( ord_less @ B @ ( F2 @ X4 ) @ L5 )
@ F5 )
=> ( filterlim @ A @ B @ F2 @ ( topolo174197925503356063within @ B @ L5 @ ( set_ord_lessThan @ B @ L5 ) ) @ F5 ) ) ) ) ).
% tendsto_imp_filterlim_at_left
thf(fact_4705_eventually__Inf,axiom,
! [A: $tType,P: A > $o,B2: set @ ( filter @ A )] :
( ( eventually @ A @ P @ ( complete_Inf_Inf @ ( filter @ A ) @ B2 ) )
= ( ? [X6: set @ ( filter @ A )] :
( ( ord_less_eq @ ( set @ ( filter @ A ) ) @ X6 @ B2 )
& ( finite_finite2 @ ( filter @ A ) @ X6 )
& ( eventually @ A @ P @ ( complete_Inf_Inf @ ( filter @ A ) @ X6 ) ) ) ) ) ).
% eventually_Inf
thf(fact_4706_filterlim__at__top__at__left,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo1944317154257567458pology @ A )
& ( linorder @ B ) )
=> ! [Q: A > $o,F2: A > B,P: B > $o,G: B > A,A3: A] :
( ! [X5: A,Y3: A] :
( ( Q @ X5 )
=> ( ( Q @ Y3 )
=> ( ( ord_less_eq @ A @ X5 @ Y3 )
=> ( ord_less_eq @ B @ ( F2 @ X5 ) @ ( F2 @ Y3 ) ) ) ) )
=> ( ! [X5: B] :
( ( P @ X5 )
=> ( ( F2 @ ( G @ X5 ) )
= X5 ) )
=> ( ! [X5: B] :
( ( P @ X5 )
=> ( Q @ ( G @ X5 ) ) )
=> ( ( eventually @ A @ Q @ ( topolo174197925503356063within @ A @ A3 @ ( set_ord_lessThan @ A @ A3 ) ) )
=> ( ! [B5: A] :
( ( Q @ B5 )
=> ( ord_less @ A @ B5 @ A3 ) )
=> ( ( eventually @ B @ P @ ( at_top @ B ) )
=> ( filterlim @ A @ B @ F2 @ ( at_top @ B ) @ ( topolo174197925503356063within @ A @ A3 @ ( set_ord_lessThan @ A @ A3 ) ) ) ) ) ) ) ) ) ) ).
% filterlim_at_top_at_left
thf(fact_4707_eventually__INF__base,axiom,
! [B: $tType,A: $tType,B2: set @ A,F5: A > ( filter @ B ),P: B > $o] :
( ( B2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [A6: A] :
( ( member @ A @ A6 @ B2 )
=> ! [B5: A] :
( ( member @ A @ B5 @ B2 )
=> ? [X2: A] :
( ( member @ A @ X2 @ B2 )
& ( ord_less_eq @ ( filter @ B ) @ ( F5 @ X2 ) @ ( inf_inf @ ( filter @ B ) @ ( F5 @ A6 ) @ ( F5 @ B5 ) ) ) ) ) )
=> ( ( eventually @ B @ P @ ( complete_Inf_Inf @ ( filter @ B ) @ ( image2 @ A @ ( filter @ B ) @ F5 @ B2 ) ) )
= ( ? [X4: A] :
( ( member @ A @ X4 @ B2 )
& ( eventually @ B @ P @ ( F5 @ X4 ) ) ) ) ) ) ) ).
% eventually_INF_base
thf(fact_4708_tendsto__0__le,axiom,
! [B: $tType,C: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,F5: filter @ A,G: A > C,K5: real] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) ) @ F5 )
=> ( ( eventually @ A
@ ^ [X4: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ C @ ( G @ X4 ) ) @ ( times_times @ real @ ( real_V7770717601297561774m_norm @ B @ ( F2 @ X4 ) ) @ K5 ) )
@ F5 )
=> ( filterlim @ A @ C @ G @ ( topolo7230453075368039082e_nhds @ C @ ( zero_zero @ C ) ) @ F5 ) ) ) ) ).
% tendsto_0_le
thf(fact_4709_filterlim__at__withinI,axiom,
! [A: $tType,B: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [F2: B > A,C2: A,F5: filter @ B,A4: set @ A] :
( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ C2 ) @ F5 )
=> ( ( eventually @ B
@ ^ [X4: B] : ( member @ A @ ( F2 @ X4 ) @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ C2 @ ( bot_bot @ ( set @ A ) ) ) ) )
@ F5 )
=> ( filterlim @ B @ A @ F2 @ ( topolo174197925503356063within @ A @ C2 @ A4 ) @ F5 ) ) ) ) ).
% filterlim_at_withinI
thf(fact_4710_summable__bounded__partials,axiom,
! [A: $tType] :
( ( ( real_V8037385150606011577_space @ A )
& ( real_V822414075346904944vector @ A ) )
=> ! [F2: nat > A,G: nat > real] :
( ( eventually @ nat
@ ^ [X02: nat] :
! [A5: nat] :
( ( ord_less_eq @ nat @ X02 @ A5 )
=> ! [B4: nat] :
( ( ord_less @ nat @ A5 @ B4 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_or3652927894154168847AtMost @ nat @ A5 @ B4 ) ) ) @ ( G @ A5 ) ) ) )
@ ( at_top @ nat ) )
=> ( ( filterlim @ nat @ real @ G @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) )
=> ( summable @ A @ F2 ) ) ) ) ).
% summable_bounded_partials
thf(fact_4711_summable__Cauchy_H,axiom,
! [A: $tType] :
( ( real_Vector_banach @ A )
=> ! [F2: nat > A,G: nat > real] :
( ( eventually @ nat
@ ^ [M: nat] :
! [N: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) ) @ ( G @ M ) ) )
@ ( at_top @ nat ) )
=> ( ( filterlim @ nat @ real @ G @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) )
=> ( summable @ A @ F2 ) ) ) ) ).
% summable_Cauchy'
thf(fact_4712_filterlim__pow__at__bot__even,axiom,
! [N2: nat,F2: real > real,F5: filter @ real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( filterlim @ real @ real @ F2 @ ( at_bot @ real ) @ F5 )
=> ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
=> ( filterlim @ real @ real
@ ^ [X4: real] : ( power_power @ real @ ( F2 @ X4 ) @ N2 )
@ ( at_top @ real )
@ F5 ) ) ) ) ).
% filterlim_pow_at_bot_even
thf(fact_4713_trivial__limit__at__bot__linorder,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( at_bot @ A )
!= ( bot_bot @ ( filter @ A ) ) ) ) ).
% trivial_limit_at_bot_linorder
thf(fact_4714_eventually__at__bot__linorder,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > $o] :
( ( eventually @ A @ P @ ( at_bot @ A ) )
= ( ? [N5: A] :
! [N: A] :
( ( ord_less_eq @ A @ N @ N5 )
=> ( P @ N ) ) ) ) ) ).
% eventually_at_bot_linorder
thf(fact_4715_eventually__at__bot__dense,axiom,
! [A: $tType] :
( ( ( linorder @ A )
& ( no_bot @ A ) )
=> ! [P: A > $o] :
( ( eventually @ A @ P @ ( at_bot @ A ) )
= ( ? [N5: A] :
! [N: A] :
( ( ord_less @ A @ N @ N5 )
=> ( P @ N ) ) ) ) ) ).
% eventually_at_bot_dense
thf(fact_4716_eventually__all__ge__at__top,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > $o] :
( ( eventually @ A @ P @ ( at_top @ A ) )
=> ( eventually @ A
@ ^ [X4: A] :
! [Y6: A] :
( ( ord_less_eq @ A @ X4 @ Y6 )
=> ( P @ Y6 ) )
@ ( at_top @ A ) ) ) ) ).
% eventually_all_ge_at_top
thf(fact_4717_finite__set__of__finite__funs,axiom,
! [B: $tType,A: $tType,A4: set @ A,B2: set @ B,D2: B] :
( ( finite_finite2 @ A @ A4 )
=> ( ( finite_finite2 @ B @ B2 )
=> ( finite_finite2 @ ( A > B )
@ ( collect @ ( A > B )
@ ^ [F3: A > B] :
! [X4: A] :
( ( ( member @ A @ X4 @ A4 )
=> ( member @ B @ ( F3 @ X4 ) @ B2 ) )
& ( ~ ( member @ A @ X4 @ A4 )
=> ( ( F3 @ X4 )
= D2 ) ) ) ) ) ) ) ).
% finite_set_of_finite_funs
thf(fact_4718_eventually__le__at__bot,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [C2: A] :
( eventually @ A
@ ^ [X4: A] : ( ord_less_eq @ A @ X4 @ C2 )
@ ( at_bot @ A ) ) ) ).
% eventually_le_at_bot
thf(fact_4719_eventually__gt__at__bot,axiom,
! [A: $tType] :
( ( unboun7993243217541854897norder @ A )
=> ! [C2: A] :
( eventually @ A
@ ^ [X4: A] : ( ord_less @ A @ X4 @ C2 )
@ ( at_bot @ A ) ) ) ).
% eventually_gt_at_bot
thf(fact_4720_filterlim__at__bot,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ B )
=> ! [F2: A > B,F5: filter @ A] :
( ( filterlim @ A @ B @ F2 @ ( at_bot @ B ) @ F5 )
= ( ! [Z9: B] :
( eventually @ A
@ ^ [X4: A] : ( ord_less_eq @ B @ ( F2 @ X4 ) @ Z9 )
@ F5 ) ) ) ) ).
% filterlim_at_bot
thf(fact_4721_filterlim__at__bot__le,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ B )
=> ! [F2: A > B,F5: filter @ A,C2: B] :
( ( filterlim @ A @ B @ F2 @ ( at_bot @ B ) @ F5 )
= ( ! [Z9: B] :
( ( ord_less_eq @ B @ Z9 @ C2 )
=> ( eventually @ A
@ ^ [X4: A] : ( ord_less_eq @ B @ ( F2 @ X4 ) @ Z9 )
@ F5 ) ) ) ) ) ).
% filterlim_at_bot_le
thf(fact_4722_filterlim__at__bot__dense,axiom,
! [A: $tType,B: $tType] :
( ( ( dense_linorder @ B )
& ( no_bot @ B ) )
=> ! [F2: A > B,F5: filter @ A] :
( ( filterlim @ A @ B @ F2 @ ( at_bot @ B ) @ F5 )
= ( ! [Z9: B] :
( eventually @ A
@ ^ [X4: A] : ( ord_less @ B @ ( F2 @ X4 ) @ Z9 )
@ F5 ) ) ) ) ).
% filterlim_at_bot_dense
thf(fact_4723_filterlim__at__bot__lt,axiom,
! [A: $tType,B: $tType] :
( ( unboun7993243217541854897norder @ B )
=> ! [F2: A > B,F5: filter @ A,C2: B] :
( ( filterlim @ A @ B @ F2 @ ( at_bot @ B ) @ F5 )
= ( ! [Z9: B] :
( ( ord_less @ B @ Z9 @ C2 )
=> ( eventually @ A
@ ^ [X4: A] : ( ord_less_eq @ B @ ( F2 @ X4 ) @ Z9 )
@ F5 ) ) ) ) ) ).
% filterlim_at_bot_lt
thf(fact_4724_filterlim__pow__at__bot__odd,axiom,
! [N2: nat,F2: real > real,F5: filter @ real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( filterlim @ real @ real @ F2 @ ( at_bot @ real ) @ F5 )
=> ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
=> ( filterlim @ real @ real
@ ^ [X4: real] : ( power_power @ real @ ( F2 @ X4 ) @ N2 )
@ ( at_bot @ real )
@ F5 ) ) ) ) ).
% filterlim_pow_at_bot_odd
thf(fact_4725_field__has__derivative__at,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,D6: A,X: A] :
( ( has_derivative @ A @ A @ F2 @ ( times_times @ A @ D6 ) @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) )
= ( filterlim @ A @ A
@ ^ [H: A] : ( divide_divide @ A @ ( minus_minus @ A @ ( F2 @ ( plus_plus @ A @ X @ H ) ) @ ( F2 @ X ) ) @ H )
@ ( topolo7230453075368039082e_nhds @ A @ D6 )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% field_has_derivative_at
thf(fact_4726_dual__Min,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( lattices_Min @ A
@ ^ [X4: A,Y6: A] : ( ord_less_eq @ A @ Y6 @ X4 ) )
= ( lattic643756798349783984er_Max @ A ) ) ) ).
% dual_Min
thf(fact_4727_tendsto__at__iff__sequentially,axiom,
! [C: $tType,A: $tType] :
( ( ( topolo3112930676232923870pology @ A )
& ( topolo4958980785337419405_space @ C ) )
=> ! [F2: A > C,A3: C,X: A,S: set @ A] :
( ( filterlim @ A @ C @ F2 @ ( topolo7230453075368039082e_nhds @ C @ A3 ) @ ( topolo174197925503356063within @ A @ X @ S ) )
= ( ! [X6: nat > A] :
( ! [I3: nat] : ( member @ A @ ( X6 @ I3 ) @ ( minus_minus @ ( set @ A ) @ S @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( ( filterlim @ nat @ A @ X6 @ ( topolo7230453075368039082e_nhds @ A @ X ) @ ( at_top @ nat ) )
=> ( filterlim @ nat @ C @ ( comp @ A @ C @ nat @ F2 @ X6 ) @ ( topolo7230453075368039082e_nhds @ C @ A3 ) @ ( at_top @ nat ) ) ) ) ) ) ) ).
% tendsto_at_iff_sequentially
thf(fact_4728_has__derivative__scaleR,axiom,
! [C: $tType,D: $tType] :
( ( ( real_V822414075346904944vector @ D )
& ( real_V822414075346904944vector @ C ) )
=> ! [F2: D > real,F8: D > real,X: D,S: set @ D,G: D > C,G6: D > C] :
( ( has_derivative @ D @ real @ F2 @ F8 @ ( topolo174197925503356063within @ D @ X @ S ) )
=> ( ( has_derivative @ D @ C @ G @ G6 @ ( topolo174197925503356063within @ D @ X @ S ) )
=> ( has_derivative @ D @ C
@ ^ [X4: D] : ( real_V8093663219630862766scaleR @ C @ ( F2 @ X4 ) @ ( G @ X4 ) )
@ ^ [H: D] : ( plus_plus @ C @ ( real_V8093663219630862766scaleR @ C @ ( F2 @ X ) @ ( G6 @ H ) ) @ ( real_V8093663219630862766scaleR @ C @ ( F8 @ H ) @ ( G @ X ) ) )
@ ( topolo174197925503356063within @ D @ X @ S ) ) ) ) ) ).
% has_derivative_scaleR
thf(fact_4729_has__derivative__add,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,F8: A > B,F5: filter @ A,G: A > B,G6: A > B] :
( ( has_derivative @ A @ B @ F2 @ F8 @ F5 )
=> ( ( has_derivative @ A @ B @ G @ G6 @ F5 )
=> ( has_derivative @ A @ B
@ ^ [X4: A] : ( plus_plus @ B @ ( F2 @ X4 ) @ ( G @ X4 ) )
@ ^ [X4: A] : ( plus_plus @ B @ ( F8 @ X4 ) @ ( G6 @ X4 ) )
@ F5 ) ) ) ) ).
% has_derivative_add
thf(fact_4730_has__derivative__const,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [C2: B,F5: filter @ A] :
( has_derivative @ A @ B
@ ^ [X4: A] : C2
@ ^ [X4: A] : ( zero_zero @ B )
@ F5 ) ) ).
% has_derivative_const
thf(fact_4731_folding__on_Ointro,axiom,
! [B: $tType,A: $tType,S2: set @ A,F2: A > B > B] :
( ! [X5: A,Y3: A] :
( ( member @ A @ X5 @ S2 )
=> ( ( member @ A @ Y3 @ S2 )
=> ( ( comp @ B @ B @ B @ ( F2 @ Y3 ) @ ( F2 @ X5 ) )
= ( comp @ B @ B @ B @ ( F2 @ X5 ) @ ( F2 @ Y3 ) ) ) ) )
=> ( finite_folding_on @ A @ B @ S2 @ F2 ) ) ).
% folding_on.intro
thf(fact_4732_folding__on_Ocomp__fun__commute__on,axiom,
! [B: $tType,A: $tType,S2: set @ A,F2: A > B > B,X: A,Y: A] :
( ( finite_folding_on @ A @ B @ S2 @ F2 )
=> ( ( member @ A @ X @ S2 )
=> ( ( member @ A @ Y @ S2 )
=> ( ( comp @ B @ B @ B @ ( F2 @ Y ) @ ( F2 @ X ) )
= ( comp @ B @ B @ B @ ( F2 @ X ) @ ( F2 @ Y ) ) ) ) ) ) ).
% folding_on.comp_fun_commute_on
thf(fact_4733_folding__on__def,axiom,
! [B: $tType,A: $tType] :
( ( finite_folding_on @ A @ B )
= ( ^ [S6: set @ A,F3: A > B > B] :
! [X4: A,Y6: A] :
( ( member @ A @ X4 @ S6 )
=> ( ( member @ A @ Y6 @ S6 )
=> ( ( comp @ B @ B @ B @ ( F3 @ Y6 ) @ ( F3 @ X4 ) )
= ( comp @ B @ B @ B @ ( F3 @ X4 ) @ ( F3 @ Y6 ) ) ) ) ) ) ) ).
% folding_on_def
thf(fact_4734_folding__idem__on_Ocomp__fun__idem__on,axiom,
! [B: $tType,A: $tType,S2: set @ A,F2: A > B > B,X: A,Y: A] :
( ( finite1890593828518410140dem_on @ A @ B @ S2 @ F2 )
=> ( ( member @ A @ X @ S2 )
=> ( ( member @ A @ Y @ S2 )
=> ( ( comp @ B @ B @ B @ ( F2 @ X ) @ ( F2 @ X ) )
= ( F2 @ X ) ) ) ) ) ).
% folding_idem_on.comp_fun_idem_on
thf(fact_4735_comp__funpow,axiom,
! [B: $tType,A: $tType,N2: nat,F2: A > A] :
( ( compow @ ( ( B > A ) > B > A ) @ N2 @ ( comp @ A @ A @ B @ F2 ) )
= ( comp @ A @ A @ B @ ( compow @ ( A > A ) @ N2 @ F2 ) ) ) ).
% comp_funpow
thf(fact_4736_card_Ocomp__fun__commute__on,axiom,
( ( comp @ nat @ nat @ nat @ suc @ suc )
= ( comp @ nat @ nat @ nat @ suc @ suc ) ) ).
% card.comp_fun_commute_on
thf(fact_4737_linorder_OMin_Ocong,axiom,
! [A: $tType] :
( ( lattices_Min @ A )
= ( lattices_Min @ A ) ) ).
% linorder.Min.cong
thf(fact_4738_has__derivative__subset,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,F8: A > B,X: A,S: set @ A,T2: set @ A] :
( ( has_derivative @ A @ B @ F2 @ F8 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( ( ord_less_eq @ ( set @ A ) @ T2 @ S )
=> ( has_derivative @ A @ B @ F2 @ F8 @ ( topolo174197925503356063within @ A @ X @ T2 ) ) ) ) ) ).
% has_derivative_subset
thf(fact_4739_funpow_Osimps_I2_J,axiom,
! [A: $tType,N2: nat,F2: A > A] :
( ( compow @ ( A > A ) @ ( suc @ N2 ) @ F2 )
= ( comp @ A @ A @ A @ F2 @ ( compow @ ( A > A ) @ N2 @ F2 ) ) ) ).
% funpow.simps(2)
thf(fact_4740_funpow__Suc__right,axiom,
! [A: $tType,N2: nat,F2: A > A] :
( ( compow @ ( A > A ) @ ( suc @ N2 ) @ F2 )
= ( comp @ A @ A @ A @ ( compow @ ( A > A ) @ N2 @ F2 ) @ F2 ) ) ).
% funpow_Suc_right
thf(fact_4741_funpow__add,axiom,
! [A: $tType,M2: nat,N2: nat,F2: A > A] :
( ( compow @ ( A > A ) @ ( plus_plus @ nat @ M2 @ N2 ) @ F2 )
= ( comp @ A @ A @ A @ ( compow @ ( A > A ) @ M2 @ F2 ) @ ( compow @ ( A > A ) @ N2 @ F2 ) ) ) ).
% funpow_add
thf(fact_4742_sum__comp__morphism,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( ( comm_monoid_add @ B )
& ( comm_monoid_add @ A ) )
=> ! [H2: B > A,G: C > B,A4: set @ C] :
( ( ( H2 @ ( zero_zero @ B ) )
= ( zero_zero @ A ) )
=> ( ! [X5: B,Y3: B] :
( ( H2 @ ( plus_plus @ B @ X5 @ Y3 ) )
= ( plus_plus @ A @ ( H2 @ X5 ) @ ( H2 @ Y3 ) ) )
=> ( ( groups7311177749621191930dd_sum @ C @ A @ ( comp @ B @ A @ C @ H2 @ G ) @ A4 )
= ( H2 @ ( groups7311177749621191930dd_sum @ C @ B @ G @ A4 ) ) ) ) ) ) ).
% sum_comp_morphism
thf(fact_4743_sum_OatLeast__Suc__atMost__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,M2: nat,N2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M2 ) @ ( suc @ N2 ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ nat @ A @ nat @ G @ suc ) @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N2 ) ) ) ) ).
% sum.atLeast_Suc_atMost_Suc_shift
thf(fact_4744_sum_OatLeast__Suc__lessThan__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,M2: nat,N2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M2 ) @ ( suc @ N2 ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ nat @ A @ nat @ G @ suc ) @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N2 ) ) ) ) ).
% sum.atLeast_Suc_lessThan_Suc_shift
thf(fact_4745_has__derivative__zero__unique,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F5: A > B,X: A] :
( ( has_derivative @ A @ B
@ ^ [X4: A] : ( zero_zero @ B )
@ F5
@ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) )
=> ( F5
= ( ^ [H: A] : ( zero_zero @ B ) ) ) ) ) ).
% has_derivative_zero_unique
thf(fact_4746_has__derivative__mult,axiom,
! [A: $tType,D: $tType] :
( ( ( real_V822414075346904944vector @ D )
& ( real_V4412858255891104859lgebra @ A ) )
=> ! [F2: D > A,F8: D > A,X: D,S: set @ D,G: D > A,G6: D > A] :
( ( has_derivative @ D @ A @ F2 @ F8 @ ( topolo174197925503356063within @ D @ X @ S ) )
=> ( ( has_derivative @ D @ A @ G @ G6 @ ( topolo174197925503356063within @ D @ X @ S ) )
=> ( has_derivative @ D @ A
@ ^ [X4: D] : ( times_times @ A @ ( F2 @ X4 ) @ ( G @ X4 ) )
@ ^ [H: D] : ( plus_plus @ A @ ( times_times @ A @ ( F2 @ X ) @ ( G6 @ H ) ) @ ( times_times @ A @ ( F8 @ H ) @ ( G @ X ) ) )
@ ( topolo174197925503356063within @ D @ X @ S ) ) ) ) ) ).
% has_derivative_mult
thf(fact_4747_sum_OatLeastAtMost__shift__bounds,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,M2: nat,K: nat,N2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ M2 @ K ) @ ( plus_plus @ nat @ N2 @ K ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ nat @ A @ nat @ G @ ( plus_plus @ nat @ K ) ) @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N2 ) ) ) ) ).
% sum.atLeastAtMost_shift_bounds
thf(fact_4748_sum_OatLeastLessThan__shift__bounds,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,M2: nat,K: nat,N2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( plus_plus @ nat @ M2 @ K ) @ ( plus_plus @ nat @ N2 @ K ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ nat @ A @ nat @ G @ ( plus_plus @ nat @ K ) ) @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N2 ) ) ) ) ).
% sum.atLeastLessThan_shift_bounds
thf(fact_4749_has__derivative__in__compose2,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [T2: set @ A,G: A > B,G6: A > A > B,F2: C > A,S: set @ C,X: C,F8: C > A] :
( ! [X5: A] :
( ( member @ A @ X5 @ T2 )
=> ( has_derivative @ A @ B @ G @ ( G6 @ X5 ) @ ( topolo174197925503356063within @ A @ X5 @ T2 ) ) )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image2 @ C @ A @ F2 @ S ) @ T2 )
=> ( ( member @ C @ X @ S )
=> ( ( has_derivative @ C @ A @ F2 @ F8 @ ( topolo174197925503356063within @ C @ X @ S ) )
=> ( has_derivative @ C @ B
@ ^ [X4: C] : ( G @ ( F2 @ X4 ) )
@ ^ [Y6: C] : ( G6 @ ( F2 @ X ) @ ( F8 @ Y6 ) )
@ ( topolo174197925503356063within @ C @ X @ S ) ) ) ) ) ) ) ).
% has_derivative_in_compose2
thf(fact_4750_prod_OatLeast__Suc__atMost__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,M2: nat,N2: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M2 ) @ ( suc @ N2 ) ) )
= ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ nat @ A @ nat @ G @ suc ) @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N2 ) ) ) ) ).
% prod.atLeast_Suc_atMost_Suc_shift
thf(fact_4751_prod_OatLeast__Suc__lessThan__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,M2: nat,N2: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M2 ) @ ( suc @ N2 ) ) )
= ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ nat @ A @ nat @ G @ suc ) @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N2 ) ) ) ) ).
% prod.atLeast_Suc_lessThan_Suc_shift
thf(fact_4752_prod_OatLeastAtMost__shift__bounds,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,M2: nat,K: nat,N2: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ M2 @ K ) @ ( plus_plus @ nat @ N2 @ K ) ) )
= ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ nat @ A @ nat @ G @ ( plus_plus @ nat @ K ) ) @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N2 ) ) ) ) ).
% prod.atLeastAtMost_shift_bounds
thf(fact_4753_prod_OatLeastLessThan__shift__bounds,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,M2: nat,K: nat,N2: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( plus_plus @ nat @ M2 @ K ) @ ( plus_plus @ nat @ N2 @ K ) ) )
= ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ nat @ A @ nat @ G @ ( plus_plus @ nat @ K ) ) @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N2 ) ) ) ) ).
% prod.atLeastLessThan_shift_bounds
thf(fact_4754_bit__drop__bit__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N2: nat,A3: A] :
( ( bit_se5641148757651400278ts_bit @ A @ ( bit_se4197421643247451524op_bit @ A @ N2 @ A3 ) )
= ( comp @ nat @ $o @ nat @ ( bit_se5641148757651400278ts_bit @ A @ A3 ) @ ( plus_plus @ nat @ N2 ) ) ) ) ).
% bit_drop_bit_eq
thf(fact_4755_sum_Oreindex__nontrivial,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A4: set @ B,H2: B > C,G: C > A] :
( ( finite_finite2 @ B @ A4 )
=> ( ! [X5: B,Y3: B] :
( ( member @ B @ X5 @ A4 )
=> ( ( member @ B @ Y3 @ A4 )
=> ( ( X5 != Y3 )
=> ( ( ( H2 @ X5 )
= ( H2 @ Y3 ) )
=> ( ( G @ ( H2 @ X5 ) )
= ( zero_zero @ A ) ) ) ) ) )
=> ( ( groups7311177749621191930dd_sum @ C @ A @ G @ ( image2 @ B @ C @ H2 @ A4 ) )
= ( groups7311177749621191930dd_sum @ B @ A @ ( comp @ C @ A @ B @ G @ H2 ) @ A4 ) ) ) ) ) ).
% sum.reindex_nontrivial
thf(fact_4756_prod_Oreindex__nontrivial,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A4: set @ B,H2: B > C,G: C > A] :
( ( finite_finite2 @ B @ A4 )
=> ( ! [X5: B,Y3: B] :
( ( member @ B @ X5 @ A4 )
=> ( ( member @ B @ Y3 @ A4 )
=> ( ( X5 != Y3 )
=> ( ( ( H2 @ X5 )
= ( H2 @ Y3 ) )
=> ( ( G @ ( H2 @ X5 ) )
= ( one_one @ A ) ) ) ) ) )
=> ( ( groups7121269368397514597t_prod @ C @ A @ G @ ( image2 @ B @ C @ H2 @ A4 ) )
= ( groups7121269368397514597t_prod @ B @ A @ ( comp @ C @ A @ B @ G @ H2 ) @ A4 ) ) ) ) ) ).
% prod.reindex_nontrivial
thf(fact_4757_DERIV__at__within__shift__lemma,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,Y: A,Z3: A,X: A,S2: set @ A] :
( ( has_field_derivative @ A @ F2 @ Y @ ( topolo174197925503356063within @ A @ ( plus_plus @ A @ Z3 @ X ) @ ( image2 @ A @ A @ ( plus_plus @ A @ Z3 ) @ S2 ) ) )
=> ( has_field_derivative @ A @ ( comp @ A @ A @ A @ F2 @ ( plus_plus @ A @ Z3 ) ) @ Y @ ( topolo174197925503356063within @ A @ X @ S2 ) ) ) ) ).
% DERIV_at_within_shift_lemma
thf(fact_4758_sum__image__le,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( ordere6911136660526730532id_add @ B )
=> ! [I6: set @ C,G: A > B,F2: C > A] :
( ( finite_finite2 @ C @ I6 )
=> ( ! [I4: C] :
( ( member @ C @ I4 @ I6 )
=> ( ord_less_eq @ B @ ( zero_zero @ B ) @ ( G @ ( F2 @ I4 ) ) ) )
=> ( ord_less_eq @ B @ ( groups7311177749621191930dd_sum @ A @ B @ G @ ( image2 @ C @ A @ F2 @ I6 ) ) @ ( groups7311177749621191930dd_sum @ C @ B @ ( comp @ A @ B @ C @ G @ F2 ) @ I6 ) ) ) ) ) ).
% sum_image_le
thf(fact_4759_has__derivative__divide_H,axiom,
! [A: $tType,C: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V3459762299906320749_field @ A ) )
=> ! [F2: C > A,F8: C > A,X: C,S2: set @ C,G: C > A,G6: C > A] :
( ( has_derivative @ C @ A @ F2 @ F8 @ ( topolo174197925503356063within @ C @ X @ S2 ) )
=> ( ( has_derivative @ C @ A @ G @ G6 @ ( topolo174197925503356063within @ C @ X @ S2 ) )
=> ( ( ( G @ X )
!= ( zero_zero @ A ) )
=> ( has_derivative @ C @ A
@ ^ [X4: C] : ( divide_divide @ A @ ( F2 @ X4 ) @ ( G @ X4 ) )
@ ^ [H: C] : ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ ( F8 @ H ) @ ( G @ X ) ) @ ( times_times @ A @ ( F2 @ X ) @ ( G6 @ H ) ) ) @ ( times_times @ A @ ( G @ X ) @ ( G @ X ) ) )
@ ( topolo174197925503356063within @ C @ X @ S2 ) ) ) ) ) ) ).
% has_derivative_divide'
thf(fact_4760_filterlim__shift,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: A > B,F5: filter @ B,A3: A,D2: A] :
( ( filterlim @ A @ B @ F2 @ F5 @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) )
=> ( filterlim @ A @ B @ ( comp @ A @ B @ A @ F2 @ ( plus_plus @ A @ D2 ) ) @ F5 @ ( topolo174197925503356063within @ A @ ( minus_minus @ A @ A3 @ D2 ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% filterlim_shift
thf(fact_4761_filterlim__shift__iff,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: A > B,D2: A,F5: filter @ B,A3: A] :
( ( filterlim @ A @ B @ ( comp @ A @ B @ A @ F2 @ ( plus_plus @ A @ D2 ) ) @ F5 @ ( topolo174197925503356063within @ A @ ( minus_minus @ A @ A3 @ D2 ) @ ( top_top @ ( set @ A ) ) ) )
= ( filterlim @ A @ B @ F2 @ F5 @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% filterlim_shift_iff
thf(fact_4762_has__derivative__inverse,axiom,
! [A: $tType,C: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V8999393235501362500lgebra @ A ) )
=> ! [F2: C > A,X: C,F8: C > A,S2: set @ C] :
( ( ( F2 @ X )
!= ( zero_zero @ A ) )
=> ( ( has_derivative @ C @ A @ F2 @ F8 @ ( topolo174197925503356063within @ C @ X @ S2 ) )
=> ( has_derivative @ C @ A
@ ^ [X4: C] : ( inverse_inverse @ A @ ( F2 @ X4 ) )
@ ^ [H: C] : ( uminus_uminus @ A @ ( times_times @ A @ ( times_times @ A @ ( inverse_inverse @ A @ ( F2 @ X ) ) @ ( F8 @ H ) ) @ ( inverse_inverse @ A @ ( F2 @ X ) ) ) )
@ ( topolo174197925503356063within @ C @ X @ S2 ) ) ) ) ) ).
% has_derivative_inverse
thf(fact_4763_has__derivative__inverse_H,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [X: A,S2: 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 @ S2 ) ) ) ) ).
% has_derivative_inverse'
thf(fact_4764_sum_OatLeast0__atMost__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,N2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( suc @ N2 ) ) )
= ( plus_plus @ A @ ( G @ ( zero_zero @ nat ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ nat @ A @ nat @ G @ suc ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ) ) ).
% sum.atLeast0_atMost_Suc_shift
thf(fact_4765_sum_OatLeast0__lessThan__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,N2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( suc @ N2 ) ) )
= ( plus_plus @ A @ ( G @ ( zero_zero @ nat ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ nat @ A @ nat @ G @ suc ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ) ) ).
% sum.atLeast0_lessThan_Suc_shift
thf(fact_4766_prod_OatLeast0__atMost__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,N2: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( suc @ N2 ) ) )
= ( times_times @ A @ ( G @ ( zero_zero @ nat ) ) @ ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ nat @ A @ nat @ G @ suc ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ) ) ).
% prod.atLeast0_atMost_Suc_shift
thf(fact_4767_prod_OatLeast0__lessThan__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,N2: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( suc @ N2 ) ) )
= ( times_times @ A @ ( G @ ( zero_zero @ nat ) ) @ ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ nat @ A @ nat @ G @ suc ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ) ) ).
% prod.atLeast0_lessThan_Suc_shift
thf(fact_4768_sum_OatLeastLessThan__shift__0,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,M2: nat,N2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N2 ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ nat @ A @ nat @ G @ ( plus_plus @ nat @ M2 ) ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ N2 @ M2 ) ) ) ) ) ).
% sum.atLeastLessThan_shift_0
thf(fact_4769_prod_OatLeastLessThan__shift__0,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,M2: nat,N2: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N2 ) )
= ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ nat @ A @ nat @ G @ ( plus_plus @ nat @ M2 ) ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ N2 @ M2 ) ) ) ) ) ).
% prod.atLeastLessThan_shift_0
thf(fact_4770_sum_OatLeast__atMost__pred__shift,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,M2: nat,N2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ( comp @ nat @ A @ nat @ G
@ ^ [N: nat] : ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M2 ) @ ( suc @ N2 ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N2 ) ) ) ) ).
% sum.atLeast_atMost_pred_shift
thf(fact_4771_sum_OatLeast__lessThan__pred__shift,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,M2: nat,N2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ( comp @ nat @ A @ nat @ G
@ ^ [N: nat] : ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M2 ) @ ( suc @ N2 ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N2 ) ) ) ) ).
% sum.atLeast_lessThan_pred_shift
thf(fact_4772_prod_OatLeast__atMost__pred__shift,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,M2: nat,N2: nat] :
( ( groups7121269368397514597t_prod @ nat @ A
@ ( comp @ nat @ A @ nat @ G
@ ^ [N: nat] : ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M2 ) @ ( suc @ N2 ) ) )
= ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N2 ) ) ) ) ).
% prod.atLeast_atMost_pred_shift
thf(fact_4773_prod_OatLeast__lessThan__pred__shift,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,M2: nat,N2: nat] :
( ( groups7121269368397514597t_prod @ nat @ A
@ ( comp @ nat @ A @ nat @ G
@ ^ [N: nat] : ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M2 ) @ ( suc @ N2 ) ) )
= ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N2 ) ) ) ) ).
% prod.atLeast_lessThan_pred_shift
thf(fact_4774_sum_OatLeast__int__atMost__int__shift,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: int > A,M2: nat,N2: nat] :
( ( groups7311177749621191930dd_sum @ int @ A @ G @ ( set_or1337092689740270186AtMost @ int @ ( semiring_1_of_nat @ int @ M2 ) @ ( semiring_1_of_nat @ int @ N2 ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ int @ A @ nat @ G @ ( semiring_1_of_nat @ int ) ) @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N2 ) ) ) ) ).
% sum.atLeast_int_atMost_int_shift
thf(fact_4775_prod_OatLeast__int__atMost__int__shift,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: int > A,M2: nat,N2: nat] :
( ( groups7121269368397514597t_prod @ int @ A @ G @ ( set_or1337092689740270186AtMost @ int @ ( semiring_1_of_nat @ int @ M2 ) @ ( semiring_1_of_nat @ int @ N2 ) ) )
= ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ int @ A @ nat @ G @ ( semiring_1_of_nat @ int ) ) @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N2 ) ) ) ) ).
% prod.atLeast_int_atMost_int_shift
thf(fact_4776_sum_OatLeast__int__lessThan__int__shift,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: int > A,M2: nat,N2: nat] :
( ( groups7311177749621191930dd_sum @ int @ A @ G @ ( set_or7035219750837199246ssThan @ int @ ( semiring_1_of_nat @ int @ M2 ) @ ( semiring_1_of_nat @ int @ N2 ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ int @ A @ nat @ G @ ( semiring_1_of_nat @ int ) ) @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N2 ) ) ) ) ).
% sum.atLeast_int_lessThan_int_shift
thf(fact_4777_sum_OatLeastAtMost__shift__0,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [M2: nat,N2: nat,G: nat > A] :
( ( ord_less_eq @ nat @ M2 @ N2 )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N2 ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ nat @ A @ nat @ G @ ( plus_plus @ nat @ M2 ) ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ N2 @ M2 ) ) ) ) ) ) ).
% sum.atLeastAtMost_shift_0
thf(fact_4778_prod_OatLeastAtMost__shift__0,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M2: nat,N2: nat,G: nat > A] :
( ( ord_less_eq @ nat @ M2 @ N2 )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N2 ) )
= ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ nat @ A @ nat @ G @ ( plus_plus @ nat @ M2 ) ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ N2 @ M2 ) ) ) ) ) ) ).
% prod.atLeastAtMost_shift_0
thf(fact_4779_prod_OatLeast__int__lessThan__int__shift,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: int > A,M2: nat,N2: nat] :
( ( groups7121269368397514597t_prod @ int @ A @ G @ ( set_or7035219750837199246ssThan @ int @ ( semiring_1_of_nat @ int @ M2 ) @ ( semiring_1_of_nat @ int @ N2 ) ) )
= ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ int @ A @ nat @ G @ ( semiring_1_of_nat @ int ) ) @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N2 ) ) ) ) ).
% prod.atLeast_int_lessThan_int_shift
thf(fact_4780_has__derivative__divide,axiom,
! [A: $tType,C: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V8999393235501362500lgebra @ A ) )
=> ! [F2: C > A,F8: C > A,X: C,S2: set @ C,G: C > A,G6: C > A] :
( ( has_derivative @ C @ A @ F2 @ F8 @ ( topolo174197925503356063within @ C @ X @ S2 ) )
=> ( ( has_derivative @ C @ A @ G @ G6 @ ( topolo174197925503356063within @ C @ X @ S2 ) )
=> ( ( ( G @ X )
!= ( zero_zero @ A ) )
=> ( has_derivative @ C @ A
@ ^ [X4: C] : ( divide_divide @ A @ ( F2 @ X4 ) @ ( G @ X4 ) )
@ ^ [H: C] : ( plus_plus @ A @ ( times_times @ A @ ( uminus_uminus @ A @ ( F2 @ X ) ) @ ( times_times @ A @ ( times_times @ A @ ( inverse_inverse @ A @ ( G @ X ) ) @ ( G6 @ H ) ) @ ( inverse_inverse @ A @ ( G @ X ) ) ) ) @ ( divide_divide @ A @ ( F8 @ H ) @ ( G @ X ) ) )
@ ( topolo174197925503356063within @ C @ X @ S2 ) ) ) ) ) ) ).
% has_derivative_divide
thf(fact_4781_has__derivative__prod,axiom,
! [B: $tType,I7: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V3459762299906320749_field @ B ) )
=> ! [I6: set @ I7,F2: I7 > A > B,F8: I7 > A > B,X: A,S2: set @ A] :
( ! [I4: I7] :
( ( member @ I7 @ I4 @ I6 )
=> ( has_derivative @ A @ B @ ( F2 @ I4 ) @ ( F8 @ I4 ) @ ( topolo174197925503356063within @ A @ X @ S2 ) ) )
=> ( has_derivative @ A @ B
@ ^ [X4: A] :
( groups7121269368397514597t_prod @ I7 @ B
@ ^ [I3: I7] : ( F2 @ I3 @ X4 )
@ I6 )
@ ^ [Y6: A] :
( groups7311177749621191930dd_sum @ I7 @ B
@ ^ [I3: I7] :
( times_times @ B @ ( F8 @ I3 @ Y6 )
@ ( groups7121269368397514597t_prod @ I7 @ B
@ ^ [J2: I7] : ( F2 @ J2 @ X )
@ ( minus_minus @ ( set @ I7 ) @ I6 @ ( insert2 @ I7 @ I3 @ ( bot_bot @ ( set @ I7 ) ) ) ) ) )
@ I6 )
@ ( topolo174197925503356063within @ A @ X @ S2 ) ) ) ) ).
% has_derivative_prod
thf(fact_4782_lim__at__infinity__0,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,L: A] :
( ( filterlim @ A @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ ( at_infinity @ A ) )
= ( filterlim @ A @ A @ ( comp @ A @ A @ A @ F2 @ ( inverse_inverse @ A ) ) @ ( topolo7230453075368039082e_nhds @ A @ L ) @ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% lim_at_infinity_0
thf(fact_4783_has__derivative__at2,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,F8: A > B,X: A] :
( ( has_derivative @ A @ B @ F2 @ F8 @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) )
= ( ( real_V3181309239436604168linear @ A @ B @ F8 )
& ( filterlim @ A @ B
@ ^ [Y6: A] : ( real_V8093663219630862766scaleR @ B @ ( divide_divide @ real @ ( one_one @ real ) @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ Y6 @ X ) ) ) @ ( minus_minus @ B @ ( F2 @ Y6 ) @ ( plus_plus @ B @ ( F2 @ X ) @ ( F8 @ ( minus_minus @ A @ Y6 @ X ) ) ) ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% has_derivative_at2
thf(fact_4784_has__derivative__at,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,D6: A > B,X: A] :
( ( has_derivative @ A @ B @ F2 @ D6 @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) )
= ( ( real_V3181309239436604168linear @ A @ B @ D6 )
& ( filterlim @ A @ real
@ ^ [H: A] : ( divide_divide @ real @ ( real_V7770717601297561774m_norm @ B @ ( minus_minus @ B @ ( minus_minus @ B @ ( F2 @ ( plus_plus @ A @ X @ H ) ) @ ( F2 @ X ) ) @ ( D6 @ 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_4785_has__derivative__within,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,F8: A > B,X: A,S: set @ A] :
( ( has_derivative @ A @ B @ F2 @ F8 @ ( topolo174197925503356063within @ A @ X @ S ) )
= ( ( real_V3181309239436604168linear @ A @ B @ F8 )
& ( filterlim @ A @ B
@ ^ [Y6: A] : ( real_V8093663219630862766scaleR @ B @ ( divide_divide @ real @ ( one_one @ real ) @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ Y6 @ X ) ) ) @ ( minus_minus @ B @ ( F2 @ Y6 ) @ ( plus_plus @ B @ ( F2 @ X ) @ ( F8 @ ( minus_minus @ A @ Y6 @ X ) ) ) ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ).
% has_derivative_within
thf(fact_4786_empty__natural,axiom,
! [C: $tType,B: $tType,D: $tType,A: $tType,F2: A > C,G: D > B] :
( ( comp @ C @ ( set @ B ) @ A
@ ^ [Uu3: C] : ( bot_bot @ ( set @ B ) )
@ F2 )
= ( comp @ ( set @ D ) @ ( set @ B ) @ A @ ( image2 @ D @ B @ G )
@ ^ [Uu3: A] : ( bot_bot @ ( set @ D ) ) ) ) ).
% empty_natural
thf(fact_4787_sorted__list__of__set_Ofold__insort__key_Ocomp__fun__commute__on,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Y: A,X: A] :
( ( comp @ ( list @ A ) @ ( list @ A ) @ ( list @ A )
@ ( linorder_insort_key @ A @ A
@ ^ [X4: A] : X4
@ Y )
@ ( linorder_insort_key @ A @ A
@ ^ [X4: A] : X4
@ X ) )
= ( comp @ ( list @ A ) @ ( list @ A ) @ ( list @ A )
@ ( linorder_insort_key @ A @ A
@ ^ [X4: A] : X4
@ X )
@ ( linorder_insort_key @ A @ A
@ ^ [X4: A] : X4
@ Y ) ) ) ) ).
% sorted_list_of_set.fold_insort_key.comp_fun_commute_on
thf(fact_4788_bounded__linear__zero,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ( real_V3181309239436604168linear @ A @ B
@ ^ [X4: A] : ( zero_zero @ B ) ) ) ).
% bounded_linear_zero
thf(fact_4789_bounded__linear__add,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,G: A > B] :
( ( real_V3181309239436604168linear @ A @ B @ F2 )
=> ( ( real_V3181309239436604168linear @ A @ B @ G )
=> ( real_V3181309239436604168linear @ A @ B
@ ^ [X4: A] : ( plus_plus @ B @ ( F2 @ X4 ) @ ( G @ X4 ) ) ) ) ) ) ).
% bounded_linear_add
thf(fact_4790_bounded__linear_Otendsto__zero,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,G: C > A,F5: filter @ C] :
( ( real_V3181309239436604168linear @ A @ B @ F2 )
=> ( ( filterlim @ C @ A @ G @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ F5 )
=> ( filterlim @ C @ B
@ ^ [X4: C] : ( F2 @ ( G @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ F5 ) ) ) ) ).
% bounded_linear.tendsto_zero
thf(fact_4791_prod_OUnion__comp,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [B2: set @ ( set @ B ),G: B > A] :
( ! [X5: set @ B] :
( ( member @ ( set @ B ) @ X5 @ B2 )
=> ( finite_finite2 @ B @ X5 ) )
=> ( ! [A14: set @ B] :
( ( member @ ( set @ B ) @ A14 @ B2 )
=> ! [A25: set @ B] :
( ( member @ ( set @ B ) @ A25 @ B2 )
=> ( ( A14 != A25 )
=> ! [X5: B] :
( ( member @ B @ X5 @ A14 )
=> ( ( member @ B @ X5 @ A25 )
=> ( ( G @ X5 )
= ( one_one @ A ) ) ) ) ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( complete_Sup_Sup @ ( set @ B ) @ B2 ) )
= ( comp @ ( ( set @ B ) > A ) @ ( ( set @ ( set @ B ) ) > A ) @ ( B > A ) @ ( groups7121269368397514597t_prod @ ( set @ B ) @ A ) @ ( groups7121269368397514597t_prod @ B @ A ) @ G @ B2 ) ) ) ) ) ).
% prod.Union_comp
thf(fact_4792_has__derivative__within__singleton__iff,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,G: A > B,X: A] :
( ( has_derivative @ A @ B @ F2 @ G @ ( topolo174197925503356063within @ A @ X @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( real_V3181309239436604168linear @ A @ B @ G ) ) ) ).
% has_derivative_within_singleton_iff
thf(fact_4793_infinite__int__iff__infinite__nat__abs,axiom,
! [S2: set @ int] :
( ( ~ ( finite_finite2 @ int @ S2 ) )
= ( ~ ( finite_finite2 @ nat @ ( image2 @ int @ nat @ ( comp @ int @ nat @ int @ nat2 @ ( abs_abs @ int ) ) @ S2 ) ) ) ) ).
% infinite_int_iff_infinite_nat_abs
thf(fact_4794_prod_OUnion__disjoint,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [C5: set @ ( set @ B ),G: B > A] :
( ! [X5: set @ B] :
( ( member @ ( set @ B ) @ X5 @ C5 )
=> ( finite_finite2 @ B @ X5 ) )
=> ( ! [X5: set @ B] :
( ( member @ ( set @ B ) @ X5 @ C5 )
=> ! [Xa3: set @ B] :
( ( member @ ( set @ B ) @ Xa3 @ C5 )
=> ( ( X5 != Xa3 )
=> ( ( inf_inf @ ( set @ B ) @ X5 @ Xa3 )
= ( bot_bot @ ( set @ B ) ) ) ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( complete_Sup_Sup @ ( set @ B ) @ C5 ) )
= ( comp @ ( ( set @ B ) > A ) @ ( ( set @ ( set @ B ) ) > A ) @ ( B > A ) @ ( groups7121269368397514597t_prod @ ( set @ B ) @ A ) @ ( groups7121269368397514597t_prod @ B @ A ) @ G @ C5 ) ) ) ) ) ).
% prod.Union_disjoint
thf(fact_4795_bounded__linear__intro,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,K5: real] :
( ! [X5: A,Y3: A] :
( ( F2 @ ( plus_plus @ A @ X5 @ Y3 ) )
= ( plus_plus @ B @ ( F2 @ X5 ) @ ( F2 @ Y3 ) ) )
=> ( ! [R3: real,X5: A] :
( ( F2 @ ( real_V8093663219630862766scaleR @ A @ R3 @ X5 ) )
= ( real_V8093663219630862766scaleR @ B @ R3 @ ( F2 @ X5 ) ) )
=> ( ! [X5: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( F2 @ X5 ) ) @ ( times_times @ real @ ( real_V7770717601297561774m_norm @ A @ X5 ) @ K5 ) )
=> ( real_V3181309239436604168linear @ A @ B @ F2 ) ) ) ) ) ).
% bounded_linear_intro
thf(fact_4796_sum_OUnion__comp,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [B2: set @ ( set @ B ),G: B > A] :
( ! [X5: set @ B] :
( ( member @ ( set @ B ) @ X5 @ B2 )
=> ( finite_finite2 @ B @ X5 ) )
=> ( ! [A14: set @ B] :
( ( member @ ( set @ B ) @ A14 @ B2 )
=> ! [A25: set @ B] :
( ( member @ ( set @ B ) @ A25 @ B2 )
=> ( ( A14 != A25 )
=> ! [X5: B] :
( ( member @ B @ X5 @ A14 )
=> ( ( member @ B @ X5 @ A25 )
=> ( ( G @ X5 )
= ( zero_zero @ A ) ) ) ) ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( complete_Sup_Sup @ ( set @ B ) @ B2 ) )
= ( comp @ ( ( set @ B ) > A ) @ ( ( set @ ( set @ B ) ) > A ) @ ( B > A ) @ ( groups7311177749621191930dd_sum @ ( set @ B ) @ A ) @ ( groups7311177749621191930dd_sum @ B @ A ) @ G @ B2 ) ) ) ) ) ).
% sum.Union_comp
thf(fact_4797_sum_OUnion__disjoint,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [C5: set @ ( set @ B ),G: B > A] :
( ! [X5: set @ B] :
( ( member @ ( set @ B ) @ X5 @ C5 )
=> ( finite_finite2 @ B @ X5 ) )
=> ( ! [X5: set @ B] :
( ( member @ ( set @ B ) @ X5 @ C5 )
=> ! [Xa3: set @ B] :
( ( member @ ( set @ B ) @ Xa3 @ C5 )
=> ( ( X5 != Xa3 )
=> ( ( inf_inf @ ( set @ B ) @ X5 @ Xa3 )
= ( bot_bot @ ( set @ B ) ) ) ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( complete_Sup_Sup @ ( set @ B ) @ C5 ) )
= ( comp @ ( ( set @ B ) > A ) @ ( ( set @ ( set @ B ) ) > A ) @ ( B > A ) @ ( groups7311177749621191930dd_sum @ ( set @ B ) @ A ) @ ( groups7311177749621191930dd_sum @ B @ A ) @ G @ C5 ) ) ) ) ) ).
% sum.Union_disjoint
thf(fact_4798_has__derivative__at__within,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,F8: A > B,X: A,S: set @ A] :
( ( has_derivative @ A @ B @ F2 @ F8 @ ( topolo174197925503356063within @ A @ X @ S ) )
= ( ( real_V3181309239436604168linear @ A @ B @ F8 )
& ( filterlim @ A @ B
@ ^ [Y6: A] : ( real_V8093663219630862766scaleR @ B @ ( inverse_inverse @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ Y6 @ X ) ) ) @ ( minus_minus @ B @ ( minus_minus @ B @ ( F2 @ Y6 ) @ ( F2 @ X ) ) @ ( F8 @ ( minus_minus @ A @ Y6 @ X ) ) ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ).
% has_derivative_at_within
thf(fact_4799_has__derivativeI,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F8: A > B,X: A,F2: A > B,S: set @ A] :
( ( real_V3181309239436604168linear @ A @ B @ F8 )
=> ( ( filterlim @ A @ B
@ ^ [Y6: A] : ( real_V8093663219630862766scaleR @ B @ ( inverse_inverse @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ Y6 @ X ) ) ) @ ( minus_minus @ B @ ( minus_minus @ B @ ( F2 @ Y6 ) @ ( F2 @ X ) ) @ ( F8 @ ( minus_minus @ A @ Y6 @ X ) ) ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( has_derivative @ A @ B @ F2 @ F8 @ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ).
% has_derivativeI
thf(fact_4800_has__derivative__iff__Ex,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,F8: A > B,X: A] :
( ( has_derivative @ A @ B @ F2 @ F8 @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) )
= ( ( real_V3181309239436604168linear @ A @ B @ F8 )
& ? [E3: A > B] :
( ! [H: A] :
( ( F2 @ ( plus_plus @ A @ X @ H ) )
= ( plus_plus @ B @ ( plus_plus @ B @ ( F2 @ X ) @ ( F8 @ H ) ) @ ( E3 @ H ) ) )
& ( filterlim @ A @ real
@ ^ [H: A] : ( divide_divide @ real @ ( real_V7770717601297561774m_norm @ B @ ( E3 @ 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_4801_has__derivative__def,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ( ( has_derivative @ A @ B )
= ( ^ [F3: A > B,F9: A > B,F10: filter @ A] :
( ( real_V3181309239436604168linear @ A @ B @ F9 )
& ( filterlim @ A @ B
@ ^ [Y6: A] :
( real_V8093663219630862766scaleR @ B
@ ( inverse_inverse @ real
@ ( real_V7770717601297561774m_norm @ A
@ ( minus_minus @ A @ Y6
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F10
@ ^ [X4: A] : X4 ) ) ) )
@ ( minus_minus @ B
@ ( minus_minus @ B @ ( F3 @ Y6 )
@ ( F3
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F10
@ ^ [X4: A] : X4 ) ) )
@ ( F9
@ ( minus_minus @ A @ Y6
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F10
@ ^ [X4: A] : X4 ) ) ) ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ F10 ) ) ) ) ) ).
% has_derivative_def
thf(fact_4802_has__derivative__at__within__iff__Ex,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [X: A,S2: set @ A,F2: A > B,F8: A > B] :
( ( member @ A @ X @ S2 )
=> ( ( topolo1002775350975398744n_open @ A @ S2 )
=> ( ( has_derivative @ A @ B @ F2 @ F8 @ ( topolo174197925503356063within @ A @ X @ S2 ) )
= ( ( real_V3181309239436604168linear @ A @ B @ F8 )
& ? [E3: A > B] :
( ! [H: A] :
( ( member @ A @ ( plus_plus @ A @ X @ H ) @ S2 )
=> ( ( F2 @ ( plus_plus @ A @ X @ H ) )
= ( plus_plus @ B @ ( plus_plus @ B @ ( F2 @ X ) @ ( F8 @ H ) ) @ ( E3 @ H ) ) ) )
& ( filterlim @ A @ real
@ ^ [H: A] : ( divide_divide @ real @ ( real_V7770717601297561774m_norm @ B @ ( E3 @ 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_4803_map__filter__on__comp,axiom,
! [A: $tType,C: $tType,B: $tType,G: B > A,Y7: set @ B,X8: set @ A,F5: filter @ B,F2: A > C] :
( ( ord_less_eq @ ( set @ A ) @ ( image2 @ B @ A @ G @ Y7 ) @ X8 )
=> ( ( eventually @ B
@ ^ [X4: B] : ( member @ B @ X4 @ Y7 )
@ F5 )
=> ( ( map_filter_on @ A @ C @ X8 @ F2 @ ( map_filter_on @ B @ A @ Y7 @ G @ F5 ) )
= ( map_filter_on @ B @ C @ Y7 @ ( comp @ A @ C @ B @ F2 @ G ) @ F5 ) ) ) ) ).
% map_filter_on_comp
thf(fact_4804_open__empty,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ( topolo1002775350975398744n_open @ A @ ( bot_bot @ ( set @ A ) ) ) ) ).
% open_empty
thf(fact_4805_open__Inter,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [S2: set @ ( set @ A )] :
( ( finite_finite2 @ ( set @ A ) @ S2 )
=> ( ! [X5: set @ A] :
( ( member @ ( set @ A ) @ X5 @ S2 )
=> ( topolo1002775350975398744n_open @ A @ X5 ) )
=> ( topolo1002775350975398744n_open @ A @ ( complete_Inf_Inf @ ( set @ A ) @ S2 ) ) ) ) ) ).
% open_Inter
thf(fact_4806_open__INT,axiom,
! [A: $tType,B: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [A4: set @ B,B2: B > ( set @ A )] :
( ( finite_finite2 @ B @ A4 )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ A4 )
=> ( topolo1002775350975398744n_open @ A @ ( B2 @ X5 ) ) )
=> ( topolo1002775350975398744n_open @ A @ ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B2 @ A4 ) ) ) ) ) ) ).
% open_INT
thf(fact_4807_openI,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [S2: set @ A] :
( ! [X5: A] :
( ( member @ A @ X5 @ S2 )
=> ? [T9: set @ A] :
( ( topolo1002775350975398744n_open @ A @ T9 )
& ( member @ A @ X5 @ T9 )
& ( ord_less_eq @ ( set @ A ) @ T9 @ S2 ) ) )
=> ( topolo1002775350975398744n_open @ A @ S2 ) ) ) ).
% openI
thf(fact_4808_open__subopen,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ( ( topolo1002775350975398744n_open @ A )
= ( ^ [S6: set @ A] :
! [X4: A] :
( ( member @ A @ X4 @ S6 )
=> ? [T10: set @ A] :
( ( topolo1002775350975398744n_open @ A @ T10 )
& ( member @ A @ X4 @ T10 )
& ( ord_less_eq @ ( set @ A ) @ T10 @ S6 ) ) ) ) ) ) ).
% open_subopen
thf(fact_4809_first__countable__basis,axiom,
! [A: $tType] :
( ( topolo3112930676232923870pology @ A )
=> ! [X: A] :
? [A8: nat > ( set @ A )] :
( ! [I5: nat] :
( ( member @ A @ X @ ( A8 @ I5 ) )
& ( topolo1002775350975398744n_open @ A @ ( A8 @ I5 ) ) )
& ! [S8: set @ A] :
( ( ( topolo1002775350975398744n_open @ A @ S8 )
& ( member @ A @ X @ S8 ) )
=> ? [I4: nat] : ( ord_less_eq @ ( set @ A ) @ ( A8 @ I4 ) @ S8 ) ) ) ) ).
% first_countable_basis
thf(fact_4810_Sup__notin__open,axiom,
! [A: $tType] :
( ( topolo8458572112393995274pology @ A )
=> ! [A4: set @ A,X: A] :
( ( topolo1002775350975398744n_open @ A @ A4 )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ A4 )
=> ( ord_less @ A @ X5 @ X ) )
=> ~ ( member @ A @ ( complete_Sup_Sup @ A @ A4 ) @ A4 ) ) ) ) ).
% Sup_notin_open
thf(fact_4811_Inf__notin__open,axiom,
! [A: $tType] :
( ( topolo8458572112393995274pology @ A )
=> ! [A4: set @ A,X: A] :
( ( topolo1002775350975398744n_open @ A @ A4 )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ A4 )
=> ( ord_less @ A @ X @ X5 ) )
=> ~ ( member @ A @ ( complete_Inf_Inf @ A @ A4 ) @ A4 ) ) ) ) ).
% Inf_notin_open
thf(fact_4812_not__open__singleton,axiom,
! [A: $tType] :
( ( topolo8386298272705272623_space @ A )
=> ! [X: A] :
~ ( topolo1002775350975398744n_open @ A @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% not_open_singleton
thf(fact_4813_hausdorff,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [X: A,Y: A] :
( ( X != Y )
=> ? [U5: set @ A,V6: set @ A] :
( ( topolo1002775350975398744n_open @ A @ U5 )
& ( topolo1002775350975398744n_open @ A @ V6 )
& ( member @ A @ X @ U5 )
& ( member @ A @ Y @ V6 )
& ( ( inf_inf @ ( set @ A ) @ U5 @ V6 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% hausdorff
thf(fact_4814_separation__t2,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [X: A,Y: A] :
( ( X != Y )
= ( ? [U6: set @ A,V7: set @ A] :
( ( topolo1002775350975398744n_open @ A @ U6 )
& ( topolo1002775350975398744n_open @ A @ V7 )
& ( member @ A @ X @ U6 )
& ( member @ A @ Y @ V7 )
& ( ( inf_inf @ ( set @ A ) @ U6 @ V7 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% separation_t2
thf(fact_4815_at__within__open__subset,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [A3: A,S2: set @ A,T4: set @ A] :
( ( member @ A @ A3 @ S2 )
=> ( ( topolo1002775350975398744n_open @ A @ S2 )
=> ( ( ord_less_eq @ ( set @ A ) @ S2 @ T4 )
=> ( ( topolo174197925503356063within @ A @ A3 @ T4 )
= ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ).
% at_within_open_subset
thf(fact_4816_open__right,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [S2: set @ A,X: A,Y: A] :
( ( topolo1002775350975398744n_open @ A @ S2 )
=> ( ( member @ A @ X @ S2 )
=> ( ( ord_less @ A @ X @ Y )
=> ? [B5: A] :
( ( ord_less @ A @ X @ B5 )
& ( ord_less_eq @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ X @ B5 ) @ S2 ) ) ) ) ) ) ).
% open_right
thf(fact_4817_open__left,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [S2: set @ A,X: A,Y: A] :
( ( topolo1002775350975398744n_open @ A @ S2 )
=> ( ( member @ A @ X @ S2 )
=> ( ( ord_less @ A @ Y @ X )
=> ? [B5: A] :
( ( ord_less @ A @ B5 @ X )
& ( ord_less_eq @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ B5 @ X ) @ S2 ) ) ) ) ) ) ).
% open_left
thf(fact_4818_Lim__ident__at,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [X: A,S: set @ A] :
( ( ( topolo174197925503356063within @ A @ X @ S )
!= ( bot_bot @ ( filter @ A ) ) )
=> ( ( topolo3827282254853284352ce_Lim @ A @ A @ ( topolo174197925503356063within @ A @ X @ S )
@ ^ [X4: A] : X4 )
= X ) ) ) ).
% Lim_ident_at
thf(fact_4819_countable__basis__at__decseq,axiom,
! [A: $tType] :
( ( topolo3112930676232923870pology @ A )
=> ! [X: A] :
~ ! [A8: nat > ( set @ A )] :
( ! [I5: nat] : ( topolo1002775350975398744n_open @ A @ ( A8 @ I5 ) )
=> ( ! [I5: nat] : ( member @ A @ X @ ( A8 @ I5 ) )
=> ~ ! [S8: set @ A] :
( ( topolo1002775350975398744n_open @ A @ S8 )
=> ( ( member @ A @ X @ S8 )
=> ( eventually @ nat
@ ^ [I3: nat] : ( ord_less_eq @ ( set @ A ) @ ( A8 @ I3 ) @ S8 )
@ ( at_top @ nat ) ) ) ) ) ) ) ).
% countable_basis_at_decseq
thf(fact_4820_lim__explicit,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [F2: nat > A,F0: A] :
( ( filterlim @ nat @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ F0 ) @ ( at_top @ nat ) )
= ( ! [S6: set @ A] :
( ( topolo1002775350975398744n_open @ A @ S6 )
=> ( ( member @ A @ F0 @ S6 )
=> ? [N5: nat] :
! [N: nat] :
( ( ord_less_eq @ nat @ N5 @ N )
=> ( member @ A @ ( F2 @ N ) @ S6 ) ) ) ) ) ) ) ).
% lim_explicit
thf(fact_4821_tendsto__Lim,axiom,
! [A: $tType,B: $tType] :
( ( topological_t2_space @ B )
=> ! [Net: filter @ A,F2: A > B,L: B] :
( ( Net
!= ( bot_bot @ ( filter @ A ) ) )
=> ( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L ) @ Net )
=> ( ( topolo3827282254853284352ce_Lim @ A @ B @ Net @ F2 )
= L ) ) ) ) ).
% tendsto_Lim
thf(fact_4822_at__within__nhd,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [X: A,S2: set @ A,T4: set @ A,U3: set @ A] :
( ( member @ A @ X @ S2 )
=> ( ( topolo1002775350975398744n_open @ A @ S2 )
=> ( ( ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ T4 @ S2 ) @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ U3 @ S2 ) @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( ( topolo174197925503356063within @ A @ X @ T4 )
= ( topolo174197925503356063within @ A @ X @ U3 ) ) ) ) ) ) ).
% at_within_nhd
thf(fact_4823_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 @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% at_eq_bot_iff
thf(fact_4824_tendsto__offset__zero__iff,axiom,
! [C: $tType,D: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( topolo4958980785337419405_space @ D )
& ( zero @ C ) )
=> ! [A3: A,S2: set @ A,F2: A > D,L5: D] :
( ( nO_MATCH @ C @ A @ ( zero_zero @ C ) @ A3 )
=> ( ( member @ A @ A3 @ S2 )
=> ( ( topolo1002775350975398744n_open @ A @ S2 )
=> ( ( filterlim @ A @ D @ F2 @ ( topolo7230453075368039082e_nhds @ D @ L5 ) @ ( topolo174197925503356063within @ A @ A3 @ S2 ) )
= ( filterlim @ A @ D
@ ^ [H: A] : ( F2 @ ( plus_plus @ A @ A3 @ H ) )
@ ( topolo7230453075368039082e_nhds @ D @ L5 )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ) ).
% tendsto_offset_zero_iff
thf(fact_4825_eventually__filtercomap__at__topological,axiom,
! [A: $tType,B: $tType] :
( ( topolo4958980785337419405_space @ B )
=> ! [P: A > $o,F2: A > B,A4: B,B2: set @ B] :
( ( eventually @ A @ P @ ( filtercomap @ A @ B @ F2 @ ( topolo174197925503356063within @ B @ A4 @ B2 ) ) )
= ( ? [S6: set @ B] :
( ( topolo1002775350975398744n_open @ B @ S6 )
& ( member @ B @ A4 @ S6 )
& ! [X4: A] :
( ( member @ B @ ( F2 @ X4 ) @ ( minus_minus @ ( set @ B ) @ ( inf_inf @ ( set @ B ) @ S6 @ B2 ) @ ( insert2 @ B @ A4 @ ( bot_bot @ ( set @ B ) ) ) ) )
=> ( P @ X4 ) ) ) ) ) ) ).
% eventually_filtercomap_at_topological
thf(fact_4826_at__within__eq,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ( ( topolo174197925503356063within @ A )
= ( ^ [X4: A,S7: set @ A] :
( complete_Inf_Inf @ ( filter @ A )
@ ( image2 @ ( set @ A ) @ ( filter @ A )
@ ^ [S6: set @ A] : ( principal @ A @ ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ S6 @ S7 ) @ ( insert2 @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) ) )
@ ( collect @ ( set @ A )
@ ^ [S6: set @ A] :
( ( topolo1002775350975398744n_open @ A @ S6 )
& ( member @ A @ X4 @ S6 ) ) ) ) ) ) ) ) ).
% at_within_eq
thf(fact_4827_compactE__image,axiom,
! [A: $tType,B: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [S2: set @ A,C5: set @ B,F2: B > ( set @ A )] :
( ( topolo2193935891317330818ompact @ A @ S2 )
=> ( ! [T5: B] :
( ( member @ B @ T5 @ C5 )
=> ( topolo1002775350975398744n_open @ A @ ( F2 @ T5 ) ) )
=> ( ( ord_less_eq @ ( set @ A ) @ S2 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ F2 @ C5 ) ) )
=> ~ ! [C8: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ C8 @ C5 )
=> ( ( finite_finite2 @ B @ C8 )
=> ~ ( ord_less_eq @ ( set @ A ) @ S2 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ F2 @ C8 ) ) ) ) ) ) ) ) ) ).
% compactE_image
thf(fact_4828_filtercomap__bot,axiom,
! [B: $tType,A: $tType,F2: A > B] :
( ( filtercomap @ A @ B @ F2 @ ( bot_bot @ ( filter @ B ) ) )
= ( bot_bot @ ( filter @ A ) ) ) ).
% filtercomap_bot
thf(fact_4829_principal__le__iff,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( filter @ A ) @ ( principal @ A @ A4 ) @ ( principal @ A @ B2 ) )
= ( ord_less_eq @ ( set @ A ) @ A4 @ B2 ) ) ).
% principal_le_iff
thf(fact_4830_compact__empty,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ( topolo2193935891317330818ompact @ A @ ( bot_bot @ ( set @ A ) ) ) ) ).
% compact_empty
thf(fact_4831_principal__eq__bot__iff,axiom,
! [A: $tType,X8: set @ A] :
( ( ( principal @ A @ X8 )
= ( bot_bot @ ( filter @ A ) ) )
= ( X8
= ( bot_bot @ ( set @ A ) ) ) ) ).
% principal_eq_bot_iff
thf(fact_4832_bot__eq__principal__empty,axiom,
! [A: $tType] :
( ( bot_bot @ ( filter @ A ) )
= ( principal @ A @ ( bot_bot @ ( set @ A ) ) ) ) ).
% bot_eq_principal_empty
thf(fact_4833_filtercomap__neq__bot,axiom,
! [A: $tType,B: $tType,F5: filter @ A,F2: B > A] :
( ! [P8: A > $o] :
( ( eventually @ A @ P8 @ F5 )
=> ? [X2: B] : ( P8 @ ( F2 @ X2 ) ) )
=> ( ( filtercomap @ B @ A @ F2 @ F5 )
!= ( bot_bot @ ( filter @ B ) ) ) ) ).
% filtercomap_neq_bot
thf(fact_4834_compact__attains__sup,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [S2: set @ A] :
( ( topolo2193935891317330818ompact @ A @ S2 )
=> ( ( S2
!= ( bot_bot @ ( set @ A ) ) )
=> ? [X5: A] :
( ( member @ A @ X5 @ S2 )
& ! [Xa: A] :
( ( member @ A @ Xa @ S2 )
=> ( ord_less_eq @ A @ Xa @ X5 ) ) ) ) ) ) ).
% compact_attains_sup
thf(fact_4835_compact__attains__inf,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [S2: set @ A] :
( ( topolo2193935891317330818ompact @ A @ S2 )
=> ( ( S2
!= ( bot_bot @ ( set @ A ) ) )
=> ? [X5: A] :
( ( member @ A @ X5 @ S2 )
& ! [Xa: A] :
( ( member @ A @ Xa @ S2 )
=> ( ord_less_eq @ A @ X5 @ Xa ) ) ) ) ) ) ).
% compact_attains_inf
thf(fact_4836_nhds__discrete,axiom,
! [A: $tType] :
( ( topolo8865339358273720382pology @ A )
=> ( ( topolo7230453075368039082e_nhds @ A )
= ( ^ [X4: A] : ( principal @ A @ ( insert2 @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% nhds_discrete
thf(fact_4837_eventually__filtercomap__at__top__linorder,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ A )
=> ! [P: B > $o,F2: B > A] :
( ( eventually @ B @ P @ ( filtercomap @ B @ A @ F2 @ ( at_top @ A ) ) )
= ( ? [N5: A] :
! [X4: B] :
( ( ord_less_eq @ A @ N5 @ ( F2 @ X4 ) )
=> ( P @ X4 ) ) ) ) ) ).
% eventually_filtercomap_at_top_linorder
thf(fact_4838_eventually__filtercomap__at__top__dense,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( no_top @ A ) )
=> ! [P: B > $o,F2: B > A] :
( ( eventually @ B @ P @ ( filtercomap @ B @ A @ F2 @ ( at_top @ A ) ) )
= ( ? [N5: A] :
! [X4: B] :
( ( ord_less @ A @ N5 @ ( F2 @ X4 ) )
=> ( P @ X4 ) ) ) ) ) ).
% eventually_filtercomap_at_top_dense
thf(fact_4839_filtercomap__neq__bot__surj,axiom,
! [A: $tType,B: $tType,F5: filter @ A,F2: B > A] :
( ( F5
!= ( bot_bot @ ( filter @ A ) ) )
=> ( ( ( image2 @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
=> ( ( filtercomap @ B @ A @ F2 @ F5 )
!= ( bot_bot @ ( filter @ B ) ) ) ) ) ).
% filtercomap_neq_bot_surj
thf(fact_4840_eventually__filtercomap__at__bot__linorder,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ A )
=> ! [P: B > $o,F2: B > A] :
( ( eventually @ B @ P @ ( filtercomap @ B @ A @ F2 @ ( at_bot @ A ) ) )
= ( ? [N5: A] :
! [X4: B] :
( ( ord_less_eq @ A @ ( F2 @ X4 ) @ N5 )
=> ( P @ X4 ) ) ) ) ) ).
% eventually_filtercomap_at_bot_linorder
thf(fact_4841_eventually__filtercomap__at__bot__dense,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( no_bot @ A ) )
=> ! [P: B > $o,F2: B > A] :
( ( eventually @ B @ P @ ( filtercomap @ B @ A @ F2 @ ( at_bot @ A ) ) )
= ( ? [N5: A] :
! [X4: B] :
( ( ord_less @ A @ ( F2 @ X4 ) @ N5 )
=> ( P @ X4 ) ) ) ) ) ).
% eventually_filtercomap_at_bot_dense
thf(fact_4842_tendsto__principal__singleton,axiom,
! [A: $tType,B: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [F2: B > A,X: B] : ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ ( F2 @ X ) ) @ ( principal @ B @ ( insert2 @ B @ X @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ).
% tendsto_principal_singleton
thf(fact_4843_nhds__discrete__open,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [X: A] :
( ( topolo1002775350975398744n_open @ A @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
=> ( ( topolo7230453075368039082e_nhds @ A @ X )
= ( principal @ A @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% nhds_discrete_open
thf(fact_4844_filterlim__base__iff,axiom,
! [A: $tType,C: $tType,B: $tType,D: $tType,I6: set @ A,F5: A > ( set @ B ),F2: B > C,G4: D > ( set @ C ),J4: set @ D] :
( ( I6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [I4: A] :
( ( member @ A @ I4 @ I6 )
=> ! [J3: A] :
( ( member @ A @ J3 @ I6 )
=> ( ( ord_less_eq @ ( set @ B ) @ ( F5 @ I4 ) @ ( F5 @ J3 ) )
| ( ord_less_eq @ ( set @ B ) @ ( F5 @ J3 ) @ ( F5 @ I4 ) ) ) ) )
=> ( ( filterlim @ B @ C @ F2
@ ( complete_Inf_Inf @ ( filter @ C )
@ ( image2 @ D @ ( filter @ C )
@ ^ [J2: D] : ( principal @ C @ ( G4 @ J2 ) )
@ J4 ) )
@ ( complete_Inf_Inf @ ( filter @ B )
@ ( image2 @ A @ ( filter @ B )
@ ^ [I3: A] : ( principal @ B @ ( F5 @ I3 ) )
@ I6 ) ) )
= ( ! [X4: D] :
( ( member @ D @ X4 @ J4 )
=> ? [Y6: A] :
( ( member @ A @ Y6 @ I6 )
& ! [Z6: B] :
( ( member @ B @ Z6 @ ( F5 @ Y6 ) )
=> ( member @ C @ ( F2 @ Z6 ) @ ( G4 @ X4 ) ) ) ) ) ) ) ) ) ).
% filterlim_base_iff
thf(fact_4845_INF__principal__finite,axiom,
! [B: $tType,A: $tType,X8: set @ A,F2: A > ( set @ B )] :
( ( finite_finite2 @ A @ X8 )
=> ( ( complete_Inf_Inf @ ( filter @ B )
@ ( image2 @ A @ ( filter @ B )
@ ^ [X4: A] : ( principal @ B @ ( F2 @ X4 ) )
@ X8 ) )
= ( principal @ B @ ( complete_Inf_Inf @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ F2 @ X8 ) ) ) ) ) ).
% INF_principal_finite
thf(fact_4846_at__within__def,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ( ( topolo174197925503356063within @ A )
= ( ^ [A5: A,S7: set @ A] : ( inf_inf @ ( filter @ A ) @ ( topolo7230453075368039082e_nhds @ A @ A5 ) @ ( principal @ A @ ( minus_minus @ ( set @ A ) @ S7 @ ( insert2 @ A @ A5 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ).
% at_within_def
thf(fact_4847_at__left__eq,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [Y: A,X: A] :
( ( ord_less @ A @ Y @ X )
=> ( ( topolo174197925503356063within @ A @ X @ ( set_ord_lessThan @ A @ X ) )
= ( complete_Inf_Inf @ ( filter @ A )
@ ( image2 @ A @ ( filter @ A )
@ ^ [A5: A] : ( principal @ A @ ( set_or5935395276787703475ssThan @ A @ A5 @ X ) )
@ ( set_ord_lessThan @ A @ X ) ) ) ) ) ) ).
% at_left_eq
thf(fact_4848_compact__eq__Heine__Borel,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ( ( topolo2193935891317330818ompact @ A )
= ( ^ [S6: set @ A] :
! [C6: set @ ( set @ A )] :
( ( ! [X4: set @ A] :
( ( member @ ( set @ A ) @ X4 @ C6 )
=> ( topolo1002775350975398744n_open @ A @ X4 ) )
& ( ord_less_eq @ ( set @ A ) @ S6 @ ( complete_Sup_Sup @ ( set @ A ) @ C6 ) ) )
=> ? [D7: set @ ( set @ A )] :
( ( ord_less_eq @ ( set @ ( set @ A ) ) @ D7 @ C6 )
& ( finite_finite2 @ ( set @ A ) @ D7 )
& ( ord_less_eq @ ( set @ A ) @ S6 @ ( complete_Sup_Sup @ ( set @ A ) @ D7 ) ) ) ) ) ) ) ).
% compact_eq_Heine_Borel
thf(fact_4849_compactI,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [S: set @ A] :
( ! [C7: set @ ( set @ A )] :
( ! [X2: set @ A] :
( ( member @ ( set @ A ) @ X2 @ C7 )
=> ( topolo1002775350975398744n_open @ A @ X2 ) )
=> ( ( ord_less_eq @ ( set @ A ) @ S @ ( complete_Sup_Sup @ ( set @ A ) @ C7 ) )
=> ? [C9: set @ ( set @ A )] :
( ( ord_less_eq @ ( set @ ( set @ A ) ) @ C9 @ C7 )
& ( finite_finite2 @ ( set @ A ) @ C9 )
& ( ord_less_eq @ ( set @ A ) @ S @ ( complete_Sup_Sup @ ( set @ A ) @ C9 ) ) ) ) )
=> ( topolo2193935891317330818ompact @ A @ S ) ) ) ).
% compactI
thf(fact_4850_compactE,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [S2: set @ A,T11: set @ ( set @ A )] :
( ( topolo2193935891317330818ompact @ A @ S2 )
=> ( ( ord_less_eq @ ( set @ A ) @ S2 @ ( complete_Sup_Sup @ ( set @ A ) @ T11 ) )
=> ( ! [B9: set @ A] :
( ( member @ ( set @ A ) @ B9 @ T11 )
=> ( topolo1002775350975398744n_open @ A @ B9 ) )
=> ~ ! [T12: set @ ( set @ A )] :
( ( ord_less_eq @ ( set @ ( set @ A ) ) @ T12 @ T11 )
=> ( ( finite_finite2 @ ( set @ A ) @ T12 )
=> ~ ( ord_less_eq @ ( set @ A ) @ S2 @ ( complete_Sup_Sup @ ( set @ A ) @ T12 ) ) ) ) ) ) ) ) ).
% compactE
thf(fact_4851_at__within__order,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [X: A,S: set @ A] :
( ( ( top_top @ ( set @ A ) )
!= ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
=> ( ( topolo174197925503356063within @ A @ X @ S )
= ( inf_inf @ ( filter @ A )
@ ( complete_Inf_Inf @ ( filter @ A )
@ ( image2 @ A @ ( filter @ A )
@ ^ [A5: A] : ( principal @ A @ ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ ( set_ord_lessThan @ A @ A5 ) @ S ) @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
@ ( set_ord_greaterThan @ A @ X ) ) )
@ ( complete_Inf_Inf @ ( filter @ A )
@ ( image2 @ A @ ( filter @ A )
@ ^ [A5: A] : ( principal @ A @ ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ ( set_ord_greaterThan @ A @ A5 ) @ S ) @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
@ ( set_ord_lessThan @ A @ X ) ) ) ) ) ) ) ).
% at_within_order
thf(fact_4852_sequentially__imp__eventually__at__left,axiom,
! [A: $tType] :
( ( ( topolo3112930676232923870pology @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [B3: A,A3: A,P: A > $o] :
( ( ord_less @ A @ B3 @ A3 )
=> ( ! [F4: nat > A] :
( ! [N7: nat] : ( ord_less @ A @ B3 @ ( F4 @ N7 ) )
=> ( ! [N7: nat] : ( ord_less @ A @ ( F4 @ N7 ) @ A3 )
=> ( ( order_mono @ nat @ A @ F4 )
=> ( ( filterlim @ nat @ A @ F4 @ ( topolo7230453075368039082e_nhds @ A @ A3 ) @ ( at_top @ nat ) )
=> ( eventually @ nat
@ ^ [N: nat] : ( P @ ( F4 @ N ) )
@ ( at_top @ nat ) ) ) ) ) )
=> ( eventually @ A @ P @ ( topolo174197925503356063within @ A @ A3 @ ( set_ord_lessThan @ A @ A3 ) ) ) ) ) ) ).
% sequentially_imp_eventually_at_left
thf(fact_4853_LIMSEQ__iff__nz,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X8: nat > A,L5: A] :
( ( filterlim @ nat @ A @ X8 @ ( topolo7230453075368039082e_nhds @ A @ L5 ) @ ( at_top @ nat ) )
= ( ! [R4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R4 )
=> ? [No3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ No3 )
& ! [N: nat] :
( ( ord_less_eq @ nat @ No3 @ N )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( X8 @ N ) @ L5 ) @ R4 ) ) ) ) ) ) ) ).
% LIMSEQ_iff_nz
thf(fact_4854_greaterThan__eq__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( ( set_ord_greaterThan @ A @ X )
= ( set_ord_greaterThan @ A @ Y ) )
= ( X = Y ) ) ) ).
% greaterThan_eq_iff
thf(fact_4855_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_4856_dist__add__cancel2,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [B3: A,A3: A,C2: A] :
( ( real_V557655796197034286t_dist @ A @ ( plus_plus @ A @ B3 @ A3 ) @ ( plus_plus @ A @ C2 @ A3 ) )
= ( real_V557655796197034286t_dist @ A @ B3 @ C2 ) ) ) ).
% dist_add_cancel2
thf(fact_4857_dist__add__cancel,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [A3: A,B3: A,C2: A] :
( ( real_V557655796197034286t_dist @ A @ ( plus_plus @ A @ A3 @ B3 ) @ ( plus_plus @ A @ A3 @ C2 ) )
= ( real_V557655796197034286t_dist @ A @ B3 @ C2 ) ) ) ).
% dist_add_cancel
thf(fact_4858_Inf__greaterThan,axiom,
! [A: $tType] :
( ( ( comple6319245703460814977attice @ A )
& ( dense_linorder @ A ) )
=> ! [X: A] :
( ( complete_Inf_Inf @ A @ ( set_ord_greaterThan @ A @ X ) )
= X ) ) ).
% Inf_greaterThan
thf(fact_4859_greaterThan__subset__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_ord_greaterThan @ A @ X ) @ ( set_ord_greaterThan @ A @ Y ) )
= ( ord_less_eq @ A @ Y @ X ) ) ) ).
% greaterThan_subset_iff
thf(fact_4860_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_4861_Compl__atMost,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [K: A] :
( ( uminus_uminus @ ( set @ A ) @ ( set_ord_atMost @ A @ K ) )
= ( set_ord_greaterThan @ A @ K ) ) ) ).
% Compl_atMost
thf(fact_4862_Compl__greaterThan,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [K: A] :
( ( uminus_uminus @ ( set @ A ) @ ( set_ord_greaterThan @ A @ K ) )
= ( set_ord_atMost @ A @ K ) ) ) ).
% Compl_greaterThan
thf(fact_4863_Sup__greaterThanAtLeast,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [X: A] :
( ( ord_less @ A @ X @ ( top_top @ A ) )
=> ( ( complete_Sup_Sup @ A @ ( set_ord_greaterThan @ A @ X ) )
= ( top_top @ A ) ) ) ) ).
% Sup_greaterThanAtLeast
thf(fact_4864_image__uminus__lessThan,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [X: A] :
( ( image2 @ A @ A @ ( uminus_uminus @ A ) @ ( set_ord_lessThan @ A @ X ) )
= ( set_ord_greaterThan @ A @ ( uminus_uminus @ A @ X ) ) ) ) ).
% image_uminus_lessThan
thf(fact_4865_image__uminus__greaterThan,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [X: A] :
( ( image2 @ A @ A @ ( uminus_uminus @ A ) @ ( set_ord_greaterThan @ A @ X ) )
= ( set_ord_lessThan @ A @ ( uminus_uminus @ A @ X ) ) ) ) ).
% image_uminus_greaterThan
thf(fact_4866_mono__funpow,axiom,
! [A: $tType] :
( ( ( lattice @ A )
& ( order_bot @ A ) )
=> ! [Q: A > A] :
( ( order_mono @ A @ A @ Q )
=> ( order_mono @ nat @ A
@ ^ [I3: nat] : ( compow @ ( A > A ) @ I3 @ Q @ ( bot_bot @ A ) ) ) ) ) ).
% mono_funpow
thf(fact_4867_mono__pow,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F2: A > A,N2: nat] :
( ( order_mono @ A @ A @ F2 )
=> ( order_mono @ A @ A @ ( compow @ ( A > A ) @ N2 @ F2 ) ) ) ) ).
% mono_pow
thf(fact_4868_monoD,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [F2: A > B,X: A,Y: A] :
( ( order_mono @ A @ B @ F2 )
=> ( ( ord_less_eq @ A @ X @ Y )
=> ( ord_less_eq @ B @ ( F2 @ X ) @ ( F2 @ Y ) ) ) ) ) ).
% monoD
thf(fact_4869_monoE,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [F2: A > B,X: A,Y: A] :
( ( order_mono @ A @ B @ F2 )
=> ( ( ord_less_eq @ A @ X @ Y )
=> ( ord_less_eq @ B @ ( F2 @ X ) @ ( F2 @ Y ) ) ) ) ) ).
% monoE
thf(fact_4870_monoI,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [F2: A > B] :
( ! [X5: A,Y3: A] :
( ( ord_less_eq @ A @ X5 @ Y3 )
=> ( ord_less_eq @ B @ ( F2 @ X5 ) @ ( F2 @ Y3 ) ) )
=> ( order_mono @ A @ B @ F2 ) ) ) ).
% monoI
thf(fact_4871_mono__def,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ( ( order_mono @ A @ B )
= ( ^ [F3: A > B] :
! [X4: A,Y6: A] :
( ( ord_less_eq @ A @ X4 @ Y6 )
=> ( ord_less_eq @ B @ ( F3 @ X4 ) @ ( F3 @ Y6 ) ) ) ) ) ) ).
% mono_def
thf(fact_4872_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_4873_infinite__Ioi,axiom,
! [A: $tType] :
( ( ( linorder @ A )
& ( no_top @ A ) )
=> ! [A3: A] :
~ ( finite_finite2 @ A @ ( set_ord_greaterThan @ A @ A3 ) ) ) ).
% infinite_Ioi
thf(fact_4874_mono__strict__invE,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( order @ B ) )
=> ! [F2: A > B,X: A,Y: A] :
( ( order_mono @ A @ B @ F2 )
=> ( ( ord_less @ B @ ( F2 @ X ) @ ( F2 @ Y ) )
=> ( ord_less @ A @ X @ Y ) ) ) ) ).
% mono_strict_invE
thf(fact_4875_mono__add,axiom,
! [A: $tType] :
( ( ordere6658533253407199908up_add @ A )
=> ! [A3: A] : ( order_mono @ A @ A @ ( plus_plus @ A @ A3 ) ) ) ).
% mono_add
thf(fact_4876_mono__Suc,axiom,
order_mono @ nat @ nat @ suc ).
% mono_Suc
thf(fact_4877_greaterThan__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( set_ord_greaterThan @ A )
= ( ^ [L2: A] : ( collect @ A @ ( ord_less @ A @ L2 ) ) ) ) ) ).
% greaterThan_def
thf(fact_4878_incseq__Suc__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( order_mono @ nat @ A )
= ( ^ [F3: nat > A] :
! [N: nat] : ( ord_less_eq @ A @ ( F3 @ N ) @ ( F3 @ ( suc @ N ) ) ) ) ) ) ).
% incseq_Suc_iff
thf(fact_4879_incseq__SucI,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X8: nat > A] :
( ! [N3: nat] : ( ord_less_eq @ A @ ( X8 @ N3 ) @ ( X8 @ ( suc @ N3 ) ) )
=> ( order_mono @ nat @ A @ X8 ) ) ) ).
% incseq_SucI
thf(fact_4880_incseq__SucD,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A4: nat > A,I: nat] :
( ( order_mono @ nat @ A @ A4 )
=> ( ord_less_eq @ A @ ( A4 @ I ) @ ( A4 @ ( suc @ I ) ) ) ) ) ).
% incseq_SucD
thf(fact_4881_incseq__def,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( order_mono @ nat @ A )
= ( ^ [X6: nat > A] :
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ord_less_eq @ A @ ( X6 @ M ) @ ( X6 @ N ) ) ) ) ) ) ).
% incseq_def
thf(fact_4882_incseqD,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [F2: nat > A,I: nat,J: nat] :
( ( order_mono @ nat @ A @ F2 )
=> ( ( ord_less_eq @ nat @ I @ J )
=> ( ord_less_eq @ A @ ( F2 @ I ) @ ( F2 @ J ) ) ) ) ) ).
% incseqD
thf(fact_4883_mono__invE,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( order @ B ) )
=> ! [F2: A > B,X: A,Y: A] :
( ( order_mono @ A @ B @ F2 )
=> ( ( ord_less @ B @ ( F2 @ X ) @ ( F2 @ Y ) )
=> ( ord_less_eq @ A @ X @ Y ) ) ) ) ).
% mono_invE
thf(fact_4884_mono__inf,axiom,
! [B: $tType,A: $tType] :
( ( ( semilattice_inf @ A )
& ( semilattice_inf @ B ) )
=> ! [F2: A > B,A4: A,B2: A] :
( ( order_mono @ A @ B @ F2 )
=> ( ord_less_eq @ B @ ( F2 @ ( inf_inf @ A @ A4 @ B2 ) ) @ ( inf_inf @ B @ ( F2 @ A4 ) @ ( F2 @ B2 ) ) ) ) ) ).
% mono_inf
thf(fact_4885_norm__conv__dist,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ( ( real_V7770717601297561774m_norm @ A )
= ( ^ [X4: A] : ( real_V557655796197034286t_dist @ A @ X4 @ ( zero_zero @ A ) ) ) ) ) ).
% norm_conv_dist
thf(fact_4886_funpow__mono,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [F2: A > A,A4: A,B2: A,N2: nat] :
( ( order_mono @ A @ A @ F2 )
=> ( ( ord_less_eq @ A @ A4 @ B2 )
=> ( ord_less_eq @ A @ ( compow @ ( A > A ) @ N2 @ F2 @ A4 ) @ ( compow @ ( A > A ) @ N2 @ F2 @ B2 ) ) ) ) ) ).
% funpow_mono
thf(fact_4887_lessThan__Int__lessThan,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B3: A] :
( ( inf_inf @ ( set @ A ) @ ( set_ord_greaterThan @ A @ A3 ) @ ( set_ord_greaterThan @ A @ B3 ) )
= ( set_ord_greaterThan @ A @ ( ord_max @ A @ A3 @ B3 ) ) ) ) ).
% lessThan_Int_lessThan
thf(fact_4888_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_4889_cclfp__lowerbound,axiom,
! [A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [F2: A > A,A4: A] :
( ( order_mono @ A @ A @ F2 )
=> ( ( ord_less_eq @ A @ ( F2 @ A4 ) @ A4 )
=> ( ord_less_eq @ A @ ( order_532582986084564980_cclfp @ A @ F2 ) @ A4 ) ) ) ) ).
% cclfp_lowerbound
thf(fact_4890_mono__times__nat,axiom,
! [N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( order_mono @ nat @ nat @ ( times_times @ nat @ N2 ) ) ) ).
% mono_times_nat
thf(fact_4891_mono__mult,axiom,
! [A: $tType] :
( ( ordered_semiring @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( order_mono @ A @ A @ ( times_times @ A @ A3 ) ) ) ) ).
% mono_mult
thf(fact_4892_mono__image__least,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [F2: A > B,M2: A,N2: A,M6: B,N4: B] :
( ( order_mono @ A @ B @ F2 )
=> ( ( ( image2 @ A @ B @ F2 @ ( set_or7035219750837199246ssThan @ A @ M2 @ N2 ) )
= ( set_or7035219750837199246ssThan @ B @ M6 @ N4 ) )
=> ( ( ord_less @ A @ M2 @ N2 )
=> ( ( F2 @ M2 )
= M6 ) ) ) ) ) ).
% mono_image_least
thf(fact_4893_Kleene__iter__gpfp,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [F2: A > A,P5: A,K: nat] :
( ( order_mono @ A @ A @ F2 )
=> ( ( ord_less_eq @ A @ P5 @ ( F2 @ P5 ) )
=> ( ord_less_eq @ A @ P5 @ ( compow @ ( A > A ) @ K @ F2 @ ( top_top @ A ) ) ) ) ) ) ).
% Kleene_iter_gpfp
thf(fact_4894_Kleene__iter__lpfp,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [F2: A > A,P5: A,K: nat] :
( ( order_mono @ A @ A @ F2 )
=> ( ( ord_less_eq @ A @ ( F2 @ P5 ) @ P5 )
=> ( ord_less_eq @ A @ ( compow @ ( A > A ) @ K @ F2 @ ( bot_bot @ A ) ) @ P5 ) ) ) ) ).
% Kleene_iter_lpfp
thf(fact_4895_funpow__mono2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [F2: A > A,I: nat,J: nat,X: A,Y: A] :
( ( order_mono @ A @ A @ F2 )
=> ( ( ord_less_eq @ nat @ I @ J )
=> ( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less_eq @ A @ X @ ( F2 @ X ) )
=> ( ord_less_eq @ A @ ( compow @ ( A > A ) @ I @ F2 @ X ) @ ( compow @ ( A > A ) @ J @ F2 @ Y ) ) ) ) ) ) ) ).
% funpow_mono2
thf(fact_4896_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 ) ) )
= ( ? [B4: A] :
( ( ord_less @ A @ X @ B4 )
& ! [Y6: A] :
( ( ord_less @ A @ X @ Y6 )
=> ( ( ord_less @ A @ Y6 @ B4 )
=> ( P @ Y6 ) ) ) ) ) ) ) ) ).
% eventually_at_right
thf(fact_4897_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 ) ) )
= ( ? [B4: A] :
( ( ord_less @ A @ X @ B4 )
& ! [Y6: A] :
( ( ord_less @ A @ X @ Y6 )
=> ( ( ord_less @ A @ Y6 @ B4 )
=> ( P @ Y6 ) ) ) ) ) ) ) ).
% eventually_at_right_field
thf(fact_4898_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_4899_ivl__disj__int__one_I7_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L: A,U: A] :
( ( inf_inf @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ L @ U ) @ ( set_ord_greaterThan @ A @ U ) )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% ivl_disj_int_one(7)
thf(fact_4900_mono__SUP,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( ( comple6319245703460814977attice @ A )
& ( comple6319245703460814977attice @ B ) )
=> ! [F2: A > B,A4: C > A,I6: set @ C] :
( ( order_mono @ A @ B @ F2 )
=> ( ord_less_eq @ B
@ ( complete_Sup_Sup @ B
@ ( image2 @ C @ B
@ ^ [X4: C] : ( F2 @ ( A4 @ X4 ) )
@ I6 ) )
@ ( F2 @ ( complete_Sup_Sup @ A @ ( image2 @ C @ A @ A4 @ I6 ) ) ) ) ) ) ).
% mono_SUP
thf(fact_4901_mono__Sup,axiom,
! [B: $tType,A: $tType] :
( ( ( comple6319245703460814977attice @ A )
& ( comple6319245703460814977attice @ B ) )
=> ! [F2: A > B,A4: set @ A] :
( ( order_mono @ A @ B @ F2 )
=> ( ord_less_eq @ B @ ( complete_Sup_Sup @ B @ ( image2 @ A @ B @ F2 @ A4 ) ) @ ( F2 @ ( complete_Sup_Sup @ A @ A4 ) ) ) ) ) ).
% mono_Sup
thf(fact_4902_mono__INF,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( ( comple6319245703460814977attice @ A )
& ( comple6319245703460814977attice @ B ) )
=> ! [F2: A > B,A4: C > A,I6: set @ C] :
( ( order_mono @ A @ B @ F2 )
=> ( ord_less_eq @ B @ ( F2 @ ( complete_Inf_Inf @ A @ ( image2 @ C @ A @ A4 @ I6 ) ) )
@ ( complete_Inf_Inf @ B
@ ( image2 @ C @ B
@ ^ [X4: C] : ( F2 @ ( A4 @ X4 ) )
@ I6 ) ) ) ) ) ).
% mono_INF
thf(fact_4903_mono__Inf,axiom,
! [B: $tType,A: $tType] :
( ( ( comple6319245703460814977attice @ A )
& ( comple6319245703460814977attice @ B ) )
=> ! [F2: A > B,A4: set @ A] :
( ( order_mono @ A @ B @ F2 )
=> ( ord_less_eq @ B @ ( F2 @ ( complete_Inf_Inf @ A @ A4 ) ) @ ( complete_Inf_Inf @ B @ ( image2 @ A @ B @ F2 @ A4 ) ) ) ) ) ).
% mono_Inf
thf(fact_4904_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_4905_ivl__disj__int__one_I5_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L: A,U: A] :
( ( inf_inf @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ L @ U ) @ ( set_ord_greaterThan @ A @ U ) )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% ivl_disj_int_one(5)
thf(fact_4906_greaterThanLessThan__eq,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( set_or5935395276787703475ssThan @ A )
= ( ^ [A5: A,B4: A] : ( inf_inf @ ( set @ A ) @ ( set_ord_greaterThan @ A @ A5 ) @ ( set_ord_lessThan @ A @ B4 ) ) ) ) ) ).
% greaterThanLessThan_eq
thf(fact_4907_greaterThanLessThan__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( set_or5935395276787703475ssThan @ A )
= ( ^ [L2: A,U2: A] : ( inf_inf @ ( set @ A ) @ ( set_ord_greaterThan @ A @ L2 ) @ ( set_ord_lessThan @ A @ U2 ) ) ) ) ) ).
% greaterThanLessThan_def
thf(fact_4908_greaterThanAtMost__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( set_or3652927894154168847AtMost @ A )
= ( ^ [L2: A,U2: A] : ( inf_inf @ ( set @ A ) @ ( set_ord_greaterThan @ A @ L2 ) @ ( set_ord_atMost @ A @ U2 ) ) ) ) ) ).
% greaterThanAtMost_def
thf(fact_4909_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_4910_incseq__le,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [X8: nat > A,L5: A,N2: nat] :
( ( order_mono @ nat @ A @ X8 )
=> ( ( filterlim @ nat @ A @ X8 @ ( topolo7230453075368039082e_nhds @ A @ L5 ) @ ( at_top @ nat ) )
=> ( ord_less_eq @ A @ ( X8 @ N2 ) @ L5 ) ) ) ) ).
% incseq_le
thf(fact_4911_funpow__increasing,axiom,
! [A: $tType] :
( ( ( lattice @ A )
& ( order_top @ A ) )
=> ! [M2: nat,N2: nat,F2: A > A] :
( ( ord_less_eq @ nat @ M2 @ N2 )
=> ( ( order_mono @ A @ A @ F2 )
=> ( ord_less_eq @ A @ ( compow @ ( A > A ) @ N2 @ F2 @ ( top_top @ A ) ) @ ( compow @ ( A > A ) @ M2 @ F2 @ ( top_top @ A ) ) ) ) ) ) ).
% funpow_increasing
thf(fact_4912_funpow__decreasing,axiom,
! [A: $tType] :
( ( ( lattice @ A )
& ( order_bot @ A ) )
=> ! [M2: nat,N2: nat,F2: A > A] :
( ( ord_less_eq @ nat @ M2 @ N2 )
=> ( ( order_mono @ A @ A @ F2 )
=> ( ord_less_eq @ A @ ( compow @ ( A > A ) @ M2 @ F2 @ ( bot_bot @ A ) ) @ ( compow @ ( A > A ) @ N2 @ F2 @ ( bot_bot @ A ) ) ) ) ) ) ).
% funpow_decreasing
thf(fact_4913_mono__Max__commute,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( linorder @ B ) )
=> ! [F2: A > B,A4: set @ A] :
( ( order_mono @ A @ B @ F2 )
=> ( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( F2 @ ( lattic643756798349783984er_Max @ A @ A4 ) )
= ( lattic643756798349783984er_Max @ B @ ( image2 @ A @ B @ F2 @ A4 ) ) ) ) ) ) ) ).
% mono_Max_commute
thf(fact_4914_metric__CauchyI,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X8: nat > A] :
( ! [E: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E )
=> ? [M10: nat] :
! [M3: nat] :
( ( ord_less_eq @ nat @ M10 @ M3 )
=> ! [N3: nat] :
( ( ord_less_eq @ nat @ M10 @ N3 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( X8 @ M3 ) @ ( X8 @ N3 ) ) @ E ) ) ) )
=> ( topolo3814608138187158403Cauchy @ A @ X8 ) ) ) ).
% metric_CauchyI
thf(fact_4915_metric__CauchyD,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X8: nat > A,E2: real] :
( ( topolo3814608138187158403Cauchy @ A @ X8 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ E2 )
=> ? [M8: nat] :
! [M5: nat] :
( ( ord_less_eq @ nat @ M8 @ M5 )
=> ! [N7: nat] :
( ( ord_less_eq @ nat @ M8 @ N7 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( X8 @ M5 ) @ ( X8 @ N7 ) ) @ E2 ) ) ) ) ) ) ).
% metric_CauchyD
thf(fact_4916_Cauchy__altdef2,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ( ( topolo3814608138187158403Cauchy @ A )
= ( ^ [S7: nat > A] :
! [E3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E3 )
=> ? [N5: nat] :
! [N: nat] :
( ( ord_less_eq @ nat @ N5 @ N )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( S7 @ N ) @ ( S7 @ N5 ) ) @ E3 ) ) ) ) ) ) ).
% Cauchy_altdef2
thf(fact_4917_Cauchy__def,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ( ( topolo3814608138187158403Cauchy @ A )
= ( ^ [X6: nat > A] :
! [E3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E3 )
=> ? [M9: nat] :
! [M: nat] :
( ( ord_less_eq @ nat @ M9 @ M )
=> ! [N: nat] :
( ( ord_less_eq @ nat @ M9 @ N )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( X6 @ M ) @ ( X6 @ N ) ) @ E3 ) ) ) ) ) ) ) ).
% Cauchy_def
thf(fact_4918_less__separate,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ? [A6: A,B5: A] :
( ( member @ A @ X @ ( set_ord_lessThan @ A @ A6 ) )
& ( member @ A @ Y @ ( set_ord_greaterThan @ A @ B5 ) )
& ( ( inf_inf @ ( set @ A ) @ ( set_ord_lessThan @ A @ A6 ) @ ( set_ord_greaterThan @ A @ B5 ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% less_separate
thf(fact_4919_eventually__at__rightI,axiom,
! [A: $tType] :
( ( topolo2564578578187576103pology @ A )
=> ! [A3: A,B3: A,P: A > $o] :
( ! [X5: A] :
( ( member @ A @ X5 @ ( set_or5935395276787703475ssThan @ A @ A3 @ B3 ) )
=> ( P @ X5 ) )
=> ( ( ord_less @ A @ A3 @ B3 )
=> ( eventually @ A @ P @ ( topolo174197925503356063within @ A @ A3 @ ( set_ord_greaterThan @ A @ A3 ) ) ) ) ) ) ).
% eventually_at_rightI
thf(fact_4920_greaterThan__0,axiom,
( ( set_ord_greaterThan @ nat @ ( zero_zero @ nat ) )
= ( image2 @ nat @ nat @ suc @ ( top_top @ ( set @ nat ) ) ) ) ).
% greaterThan_0
thf(fact_4921_Cauchy__altdef,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ( ( topolo3814608138187158403Cauchy @ A )
= ( ^ [F3: nat > A] :
! [E3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E3 )
=> ? [M9: nat] :
! [M: nat] :
( ( ord_less_eq @ nat @ M9 @ M )
=> ! [N: nat] :
( ( ord_less @ nat @ M @ N )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( F3 @ M ) @ ( F3 @ N ) ) @ E3 ) ) ) ) ) ) ) ).
% Cauchy_altdef
thf(fact_4922_CauchyI_H,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X8: nat > A] :
( ! [E: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E )
=> ? [M10: nat] :
! [M3: nat] :
( ( ord_less_eq @ nat @ M10 @ M3 )
=> ! [N3: nat] :
( ( ord_less @ nat @ M3 @ N3 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( X8 @ M3 ) @ ( X8 @ N3 ) ) @ E ) ) ) )
=> ( topolo3814608138187158403Cauchy @ A @ X8 ) ) ) ).
% CauchyI'
thf(fact_4923_greaterThan__Suc,axiom,
! [K: nat] :
( ( set_ord_greaterThan @ nat @ ( suc @ K ) )
= ( minus_minus @ ( set @ nat ) @ ( set_ord_greaterThan @ nat @ K ) @ ( insert2 @ nat @ ( suc @ K ) @ ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% greaterThan_Suc
thf(fact_4924_filterlim__times__pos,axiom,
! [A: $tType,B: $tType] :
( ( ( linordered_field @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F2: B > A,P5: A,F12: filter @ B,C2: A,L: A] :
( ( filterlim @ B @ A @ F2 @ ( topolo174197925503356063within @ A @ P5 @ ( set_ord_greaterThan @ A @ P5 ) ) @ F12 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( L
= ( times_times @ A @ C2 @ P5 ) )
=> ( filterlim @ B @ A
@ ^ [X4: B] : ( times_times @ A @ C2 @ ( F2 @ X4 ) )
@ ( topolo174197925503356063within @ A @ L @ ( set_ord_greaterThan @ A @ L ) )
@ F12 ) ) ) ) ) ).
% filterlim_times_pos
thf(fact_4925_tendsto__imp__filterlim__at__right,axiom,
! [B: $tType,A: $tType] :
( ( topolo2564578578187576103pology @ B )
=> ! [F2: A > B,L5: B,F5: filter @ A] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L5 ) @ F5 )
=> ( ( eventually @ A
@ ^ [X4: A] : ( ord_less @ B @ L5 @ ( F2 @ X4 ) )
@ F5 )
=> ( filterlim @ A @ B @ F2 @ ( topolo174197925503356063within @ B @ L5 @ ( set_ord_greaterThan @ B @ L5 ) ) @ F5 ) ) ) ) ).
% tendsto_imp_filterlim_at_right
thf(fact_4926_mono__ge2__power__minus__self,axiom,
! [K: nat] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K )
=> ( order_mono @ nat @ nat
@ ^ [M: nat] : ( minus_minus @ nat @ ( power_power @ nat @ K @ M ) @ M ) ) ) ).
% mono_ge2_power_minus_self
thf(fact_4927_lim__sequentially,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X8: nat > A,L5: A] :
( ( filterlim @ nat @ A @ X8 @ ( topolo7230453075368039082e_nhds @ A @ L5 ) @ ( at_top @ nat ) )
= ( ! [R4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R4 )
=> ? [No3: nat] :
! [N: nat] :
( ( ord_less_eq @ nat @ No3 @ N )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( X8 @ N ) @ L5 ) @ R4 ) ) ) ) ) ) ).
% lim_sequentially
thf(fact_4928_metric__LIMSEQ__I,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X8: nat > A,L5: A] :
( ! [R3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R3 )
=> ? [No2: nat] :
! [N3: nat] :
( ( ord_less_eq @ nat @ No2 @ N3 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( X8 @ N3 ) @ L5 ) @ R3 ) ) )
=> ( filterlim @ nat @ A @ X8 @ ( topolo7230453075368039082e_nhds @ A @ L5 ) @ ( at_top @ nat ) ) ) ) ).
% metric_LIMSEQ_I
thf(fact_4929_metric__LIMSEQ__D,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X8: nat > A,L5: A,R2: real] :
( ( filterlim @ nat @ A @ X8 @ ( topolo7230453075368039082e_nhds @ A @ L5 ) @ ( at_top @ nat ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ R2 )
=> ? [No: nat] :
! [N7: nat] :
( ( ord_less_eq @ nat @ No @ N7 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( X8 @ N7 ) @ L5 ) @ R2 ) ) ) ) ) ).
% metric_LIMSEQ_D
thf(fact_4930_metric__Cauchy__iff2,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ( ( topolo3814608138187158403Cauchy @ A )
= ( ^ [X6: nat > A] :
! [J2: nat] :
? [M9: nat] :
! [M: nat] :
( ( ord_less_eq @ nat @ M9 @ M )
=> ! [N: nat] :
( ( ord_less_eq @ nat @ M9 @ N )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( X6 @ M ) @ ( X6 @ N ) ) @ ( inverse_inverse @ real @ ( semiring_1_of_nat @ real @ ( suc @ J2 ) ) ) ) ) ) ) ) ) ).
% metric_Cauchy_iff2
thf(fact_4931_filterlim__at__bot__at__right,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo1944317154257567458pology @ A )
& ( linorder @ B ) )
=> ! [Q: A > $o,F2: A > B,P: B > $o,G: B > A,A3: A] :
( ! [X5: A,Y3: A] :
( ( Q @ X5 )
=> ( ( Q @ Y3 )
=> ( ( ord_less_eq @ A @ X5 @ Y3 )
=> ( ord_less_eq @ B @ ( F2 @ X5 ) @ ( F2 @ Y3 ) ) ) ) )
=> ( ! [X5: B] :
( ( P @ X5 )
=> ( ( F2 @ ( G @ X5 ) )
= X5 ) )
=> ( ! [X5: B] :
( ( P @ X5 )
=> ( Q @ ( G @ X5 ) ) )
=> ( ( eventually @ A @ Q @ ( topolo174197925503356063within @ A @ A3 @ ( set_ord_greaterThan @ A @ A3 ) ) )
=> ( ! [B5: A] :
( ( Q @ B5 )
=> ( ord_less @ A @ A3 @ B5 ) )
=> ( ( eventually @ B @ P @ ( at_bot @ B ) )
=> ( filterlim @ A @ B @ F2 @ ( at_bot @ B ) @ ( topolo174197925503356063within @ A @ A3 @ ( set_ord_greaterThan @ A @ A3 ) ) ) ) ) ) ) ) ) ) ).
% filterlim_at_bot_at_right
thf(fact_4932_finite__mono__remains__stable__implies__strict__prefix,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [F2: nat > A] :
( ( finite_finite2 @ A @ ( image2 @ nat @ A @ F2 @ ( top_top @ ( set @ nat ) ) ) )
=> ( ( order_mono @ nat @ A @ F2 )
=> ( ! [N3: nat] :
( ( ( F2 @ N3 )
= ( F2 @ ( suc @ N3 ) ) )
=> ( ( F2 @ ( suc @ N3 ) )
= ( F2 @ ( suc @ ( suc @ N3 ) ) ) ) )
=> ? [N9: nat] :
( ! [N7: nat] :
( ( ord_less_eq @ nat @ N7 @ N9 )
=> ! [M5: nat] :
( ( ord_less_eq @ nat @ M5 @ N9 )
=> ( ( ord_less @ nat @ M5 @ N7 )
=> ( ord_less @ A @ ( F2 @ M5 ) @ ( F2 @ N7 ) ) ) ) )
& ! [N7: nat] :
( ( ord_less_eq @ nat @ N9 @ N7 )
=> ( ( F2 @ N9 )
= ( F2 @ N7 ) ) ) ) ) ) ) ) ).
% finite_mono_remains_stable_implies_strict_prefix
thf(fact_4933_INT__greaterThan__UNIV,axiom,
( ( complete_Inf_Inf @ ( set @ nat ) @ ( image2 @ nat @ ( set @ nat ) @ ( set_ord_greaterThan @ nat ) @ ( top_top @ ( set @ nat ) ) ) )
= ( bot_bot @ ( set @ nat ) ) ) ).
% INT_greaterThan_UNIV
thf(fact_4934_tendsto__at__left__sequentially,axiom,
! [A: $tType,B: $tType] :
( ( ( topolo3112930676232923870pology @ B )
& ( topolo1944317154257567458pology @ B )
& ( topolo4958980785337419405_space @ A ) )
=> ! [B3: B,A3: B,X8: B > A,L5: A] :
( ( ord_less @ B @ B3 @ A3 )
=> ( ! [S4: nat > B] :
( ! [N7: nat] : ( ord_less @ B @ ( S4 @ N7 ) @ A3 )
=> ( ! [N7: nat] : ( ord_less @ B @ B3 @ ( S4 @ N7 ) )
=> ( ( order_mono @ nat @ B @ S4 )
=> ( ( filterlim @ nat @ B @ S4 @ ( topolo7230453075368039082e_nhds @ B @ A3 ) @ ( at_top @ nat ) )
=> ( filterlim @ nat @ A
@ ^ [N: nat] : ( X8 @ ( S4 @ N ) )
@ ( topolo7230453075368039082e_nhds @ A @ L5 )
@ ( at_top @ nat ) ) ) ) ) )
=> ( filterlim @ B @ A @ X8 @ ( topolo7230453075368039082e_nhds @ A @ L5 ) @ ( topolo174197925503356063within @ B @ A3 @ ( set_ord_lessThan @ B @ A3 ) ) ) ) ) ) ).
% tendsto_at_left_sequentially
thf(fact_4935_at__right__eq,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ( ( topolo174197925503356063within @ A @ X @ ( set_ord_greaterThan @ A @ X ) )
= ( complete_Inf_Inf @ ( filter @ A )
@ ( image2 @ A @ ( filter @ A )
@ ^ [A5: A] : ( principal @ A @ ( set_or5935395276787703475ssThan @ A @ X @ A5 ) )
@ ( set_ord_greaterThan @ A @ X ) ) ) ) ) ) ).
% at_right_eq
thf(fact_4936_totally__bounded__metric,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ( ( topolo6688025880775521714ounded @ A )
= ( ^ [S6: set @ A] :
! [E3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E3 )
=> ? [K2: set @ A] :
( ( finite_finite2 @ A @ K2 )
& ( ord_less_eq @ ( set @ A ) @ S6
@ ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ A @ ( set @ A )
@ ^ [X4: A] :
( collect @ A
@ ^ [Y6: A] : ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X4 @ Y6 ) @ E3 ) )
@ K2 ) ) ) ) ) ) ) ) ).
% totally_bounded_metric
thf(fact_4937_interval__cases,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [S2: set @ A] :
( ! [A6: A,B5: A,X5: A] :
( ( member @ A @ A6 @ S2 )
=> ( ( member @ A @ B5 @ S2 )
=> ( ( ord_less_eq @ A @ A6 @ X5 )
=> ( ( ord_less_eq @ A @ X5 @ B5 )
=> ( member @ A @ X5 @ S2 ) ) ) ) )
=> ? [A6: A,B5: A] :
( ( S2
= ( bot_bot @ ( set @ A ) ) )
| ( S2
= ( top_top @ ( set @ A ) ) )
| ( S2
= ( set_ord_lessThan @ A @ B5 ) )
| ( S2
= ( set_ord_atMost @ A @ B5 ) )
| ( S2
= ( set_ord_greaterThan @ A @ A6 ) )
| ( S2
= ( set_ord_atLeast @ A @ A6 ) )
| ( S2
= ( set_or5935395276787703475ssThan @ A @ A6 @ B5 ) )
| ( S2
= ( set_or3652927894154168847AtMost @ A @ A6 @ B5 ) )
| ( S2
= ( set_or7035219750837199246ssThan @ A @ A6 @ B5 ) )
| ( S2
= ( set_or1337092689740270186AtMost @ A @ A6 @ B5 ) ) ) ) ) ).
% interval_cases
thf(fact_4938_sequentially__imp__eventually__at__right,axiom,
! [A: $tType] :
( ( ( topolo3112930676232923870pology @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [A3: A,B3: A,P: A > $o] :
( ( ord_less @ A @ A3 @ B3 )
=> ( ! [F4: nat > A] :
( ! [N7: nat] : ( ord_less @ A @ A3 @ ( F4 @ N7 ) )
=> ( ! [N7: nat] : ( ord_less @ A @ ( F4 @ N7 ) @ B3 )
=> ( ( order_antimono @ nat @ A @ F4 )
=> ( ( filterlim @ nat @ A @ F4 @ ( topolo7230453075368039082e_nhds @ A @ A3 ) @ ( at_top @ nat ) )
=> ( eventually @ nat
@ ^ [N: nat] : ( P @ ( F4 @ N ) )
@ ( at_top @ nat ) ) ) ) ) )
=> ( eventually @ A @ P @ ( topolo174197925503356063within @ A @ A3 @ ( set_ord_greaterThan @ A @ A3 ) ) ) ) ) ) ).
% sequentially_imp_eventually_at_right
thf(fact_4939_tendsto__at__right__sequentially,axiom,
! [C: $tType,B: $tType] :
( ( ( topolo3112930676232923870pology @ B )
& ( topolo1944317154257567458pology @ B )
& ( topolo4958980785337419405_space @ C ) )
=> ! [A3: B,B3: B,X8: B > C,L5: C] :
( ( ord_less @ B @ A3 @ B3 )
=> ( ! [S4: nat > B] :
( ! [N7: nat] : ( ord_less @ B @ A3 @ ( S4 @ N7 ) )
=> ( ! [N7: nat] : ( ord_less @ B @ ( S4 @ N7 ) @ B3 )
=> ( ( order_antimono @ nat @ B @ S4 )
=> ( ( filterlim @ nat @ B @ S4 @ ( topolo7230453075368039082e_nhds @ B @ A3 ) @ ( at_top @ nat ) )
=> ( filterlim @ nat @ C
@ ^ [N: nat] : ( X8 @ ( S4 @ N ) )
@ ( topolo7230453075368039082e_nhds @ C @ L5 )
@ ( at_top @ nat ) ) ) ) ) )
=> ( filterlim @ B @ C @ X8 @ ( topolo7230453075368039082e_nhds @ C @ L5 ) @ ( topolo174197925503356063within @ B @ A3 @ ( set_ord_greaterThan @ B @ A3 ) ) ) ) ) ) ).
% tendsto_at_right_sequentially
thf(fact_4940_atLeast__eq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X: A,Y: A] :
( ( ( set_ord_atLeast @ A @ X )
= ( set_ord_atLeast @ A @ Y ) )
= ( X = Y ) ) ) ).
% atLeast_eq_iff
thf(fact_4941_atLeast__0,axiom,
( ( set_ord_atLeast @ nat @ ( zero_zero @ nat ) )
= ( top_top @ ( set @ nat ) ) ) ).
% atLeast_0
thf(fact_4942_atLeast__iff,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [I: A,K: A] :
( ( member @ A @ I @ ( set_ord_atLeast @ A @ K ) )
= ( ord_less_eq @ A @ K @ I ) ) ) ).
% atLeast_iff
thf(fact_4943_Inf__atLeast,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [X: A] :
( ( complete_Inf_Inf @ A @ ( set_ord_atLeast @ A @ X ) )
= X ) ) ).
% Inf_atLeast
thf(fact_4944_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_4945_atLeast__subset__iff,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_ord_atLeast @ A @ X ) @ ( set_ord_atLeast @ A @ Y ) )
= ( ord_less_eq @ A @ Y @ X ) ) ) ).
% atLeast_subset_iff
thf(fact_4946_image__add__atLeast,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [K: A,I: A] :
( ( image2 @ A @ A @ ( plus_plus @ A @ K ) @ ( set_ord_atLeast @ A @ I ) )
= ( set_ord_atLeast @ A @ ( plus_plus @ A @ K @ I ) ) ) ) ).
% image_add_atLeast
thf(fact_4947_Sup__atLeast,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [X: A] :
( ( complete_Sup_Sup @ A @ ( set_ord_atLeast @ A @ X ) )
= ( top_top @ A ) ) ) ).
% Sup_atLeast
thf(fact_4948_Compl__atLeast,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [K: A] :
( ( uminus_uminus @ ( set @ A ) @ ( set_ord_atLeast @ A @ K ) )
= ( set_ord_lessThan @ A @ K ) ) ) ).
% Compl_atLeast
thf(fact_4949_Compl__lessThan,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [K: A] :
( ( uminus_uminus @ ( set @ A ) @ ( set_ord_lessThan @ A @ K ) )
= ( set_ord_atLeast @ A @ K ) ) ) ).
% Compl_lessThan
thf(fact_4950_Icc__subset__Ici__iff,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [L: A,H2: A,L3: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ L @ H2 ) @ ( set_ord_atLeast @ A @ L3 ) )
= ( ~ ( ord_less_eq @ A @ L @ H2 )
| ( ord_less_eq @ A @ L3 @ L ) ) ) ) ).
% Icc_subset_Ici_iff
thf(fact_4951_image__minus__const__atLeast,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [C2: A,A3: A] :
( ( image2 @ A @ A @ ( minus_minus @ A @ C2 ) @ ( set_ord_atLeast @ A @ A3 ) )
= ( set_ord_atMost @ A @ ( minus_minus @ A @ C2 @ A3 ) ) ) ) ).
% image_minus_const_atLeast
thf(fact_4952_image__minus__const__AtMost,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [C2: A,B3: A] :
( ( image2 @ A @ A @ ( minus_minus @ A @ C2 ) @ ( set_ord_atMost @ A @ B3 ) )
= ( set_ord_atLeast @ A @ ( minus_minus @ A @ C2 @ B3 ) ) ) ) ).
% image_minus_const_AtMost
thf(fact_4953_image__uminus__atMost,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [X: A] :
( ( image2 @ A @ A @ ( uminus_uminus @ A ) @ ( set_ord_atMost @ A @ X ) )
= ( set_ord_atLeast @ A @ ( uminus_uminus @ A @ X ) ) ) ) ).
% image_uminus_atMost
thf(fact_4954_image__uminus__atLeast,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [X: A] :
( ( image2 @ A @ A @ ( uminus_uminus @ A ) @ ( set_ord_atLeast @ A @ X ) )
= ( set_ord_atMost @ A @ ( uminus_uminus @ A @ X ) ) ) ) ).
% image_uminus_atLeast
thf(fact_4955_Int__atLeastAtMostL2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B3: A,C2: A] :
( ( inf_inf @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A3 @ B3 ) @ ( set_ord_atLeast @ A @ C2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( ord_max @ A @ A3 @ C2 ) @ B3 ) ) ) ).
% Int_atLeastAtMostL2
thf(fact_4956_Int__atLeastAtMostR2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,C2: A,D2: A] :
( ( inf_inf @ ( set @ A ) @ ( set_ord_atLeast @ A @ A3 ) @ ( set_or1337092689740270186AtMost @ A @ C2 @ D2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( ord_max @ A @ A3 @ C2 ) @ D2 ) ) ) ).
% Int_atLeastAtMostR2
thf(fact_4957_atLeast__Suc__greaterThan,axiom,
! [K: nat] :
( ( set_ord_atLeast @ nat @ ( suc @ K ) )
= ( set_ord_greaterThan @ nat @ K ) ) ).
% atLeast_Suc_greaterThan
thf(fact_4958_atLeast__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( set_ord_atLeast @ A )
= ( ^ [L2: A] : ( collect @ A @ ( ord_less_eq @ A @ L2 ) ) ) ) ) ).
% atLeast_def
thf(fact_4959_not__Ici__eq__Icc,axiom,
! [A: $tType] :
( ( no_top @ A )
=> ! [L3: A,L: A,H2: A] :
( ( set_ord_atLeast @ A @ L3 )
!= ( set_or1337092689740270186AtMost @ A @ L @ H2 ) ) ) ).
% not_Ici_eq_Icc
thf(fact_4960_not__UNIV__eq__Ici,axiom,
! [A: $tType] :
( ( no_bot @ A )
=> ! [L3: A] :
( ( top_top @ ( set @ A ) )
!= ( set_ord_atLeast @ A @ L3 ) ) ) ).
% not_UNIV_eq_Ici
thf(fact_4961_not__Iic__eq__Ici,axiom,
! [A: $tType] :
( ( no_top @ A )
=> ! [H2: A,L3: A] :
( ( set_ord_atMost @ A @ H2 )
!= ( set_ord_atLeast @ A @ L3 ) ) ) ).
% not_Iic_eq_Ici
thf(fact_4962_infinite__Ici,axiom,
! [A: $tType] :
( ( ( linorder @ A )
& ( no_top @ A ) )
=> ! [A3: A] :
~ ( finite_finite2 @ A @ ( set_ord_atLeast @ A @ A3 ) ) ) ).
% infinite_Ici
thf(fact_4963_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_4964_mono__Int,axiom,
! [B: $tType,A: $tType,F2: ( set @ A ) > ( set @ B ),A4: set @ A,B2: set @ A] :
( ( order_mono @ ( set @ A ) @ ( set @ B ) @ F2 )
=> ( ord_less_eq @ ( set @ B ) @ ( F2 @ ( inf_inf @ ( set @ A ) @ A4 @ B2 ) ) @ ( inf_inf @ ( set @ B ) @ ( F2 @ A4 ) @ ( F2 @ B2 ) ) ) ) ).
% mono_Int
thf(fact_4965_antimonoD,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [F2: A > B,X: A,Y: A] :
( ( order_antimono @ A @ B @ F2 )
=> ( ( ord_less_eq @ A @ X @ Y )
=> ( ord_less_eq @ B @ ( F2 @ Y ) @ ( F2 @ X ) ) ) ) ) ).
% antimonoD
thf(fact_4966_antimonoE,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [F2: A > B,X: A,Y: A] :
( ( order_antimono @ A @ B @ F2 )
=> ( ( ord_less_eq @ A @ X @ Y )
=> ( ord_less_eq @ B @ ( F2 @ Y ) @ ( F2 @ X ) ) ) ) ) ).
% antimonoE
thf(fact_4967_antimonoI,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [F2: A > B] :
( ! [X5: A,Y3: A] :
( ( ord_less_eq @ A @ X5 @ Y3 )
=> ( ord_less_eq @ B @ ( F2 @ Y3 ) @ ( F2 @ X5 ) ) )
=> ( order_antimono @ A @ B @ F2 ) ) ) ).
% antimonoI
thf(fact_4968_antimono__def,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ( ( order_antimono @ A @ B )
= ( ^ [F3: A > B] :
! [X4: A,Y6: A] :
( ( ord_less_eq @ A @ X4 @ Y6 )
=> ( ord_less_eq @ B @ ( F3 @ Y6 ) @ ( F3 @ X4 ) ) ) ) ) ) ).
% antimono_def
thf(fact_4969_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_4970_not__UNIV__le__Ici,axiom,
! [A: $tType] :
( ( no_bot @ A )
=> ! [L: A] :
~ ( ord_less_eq @ ( set @ A ) @ ( top_top @ ( set @ A ) ) @ ( set_ord_atLeast @ A @ L ) ) ) ).
% not_UNIV_le_Ici
thf(fact_4971_not__Ici__le__Icc,axiom,
! [A: $tType] :
( ( no_top @ A )
=> ! [L: A,L3: A,H3: A] :
~ ( ord_less_eq @ ( set @ A ) @ ( set_ord_atLeast @ A @ L ) @ ( set_or1337092689740270186AtMost @ A @ L3 @ H3 ) ) ) ).
% not_Ici_le_Icc
thf(fact_4972_not__Iic__le__Ici,axiom,
! [A: $tType] :
( ( no_bot @ A )
=> ! [H2: A,L3: A] :
~ ( ord_less_eq @ ( set @ A ) @ ( set_ord_atMost @ A @ H2 ) @ ( set_ord_atLeast @ A @ L3 ) ) ) ).
% not_Iic_le_Ici
thf(fact_4973_not__Ici__le__Iic,axiom,
! [A: $tType] :
( ( no_top @ A )
=> ! [L: A,H3: A] :
~ ( ord_less_eq @ ( set @ A ) @ ( set_ord_atLeast @ A @ L ) @ ( set_ord_atMost @ A @ H3 ) ) ) ).
% not_Ici_le_Iic
thf(fact_4974_Ioi__le__Ico,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: A] : ( ord_less_eq @ ( set @ A ) @ ( set_ord_greaterThan @ A @ A3 ) @ ( set_ord_atLeast @ A @ A3 ) ) ) ).
% Ioi_le_Ico
thf(fact_4975_decseq__Suc__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( order_antimono @ nat @ A )
= ( ^ [F3: nat > A] :
! [N: nat] : ( ord_less_eq @ A @ ( F3 @ ( suc @ N ) ) @ ( F3 @ N ) ) ) ) ) ).
% decseq_Suc_iff
thf(fact_4976_decseq__SucI,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X8: nat > A] :
( ! [N3: nat] : ( ord_less_eq @ A @ ( X8 @ ( suc @ N3 ) ) @ ( X8 @ N3 ) )
=> ( order_antimono @ nat @ A @ X8 ) ) ) ).
% decseq_SucI
thf(fact_4977_decseq__SucD,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A4: nat > A,I: nat] :
( ( order_antimono @ nat @ A @ A4 )
=> ( ord_less_eq @ A @ ( A4 @ ( suc @ I ) ) @ ( A4 @ I ) ) ) ) ).
% decseq_SucD
thf(fact_4978_decseqD,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [F2: nat > A,I: nat,J: nat] :
( ( order_antimono @ nat @ A @ F2 )
=> ( ( ord_less_eq @ nat @ I @ J )
=> ( ord_less_eq @ A @ ( F2 @ J ) @ ( F2 @ I ) ) ) ) ) ).
% decseqD
thf(fact_4979_decseq__def,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( order_antimono @ nat @ A )
= ( ^ [X6: nat > A] :
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ord_less_eq @ A @ ( X6 @ N ) @ ( X6 @ M ) ) ) ) ) ) ).
% decseq_def
thf(fact_4980_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_4981_ivl__disj__int__one_I8_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L: A,U: A] :
( ( inf_inf @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ L @ U ) @ ( set_ord_atLeast @ A @ U ) )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% ivl_disj_int_one(8)
thf(fact_4982_atLeastLessThan__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( set_or7035219750837199246ssThan @ A )
= ( ^ [L2: A,U2: A] : ( inf_inf @ ( set @ A ) @ ( set_ord_atLeast @ A @ L2 ) @ ( set_ord_lessThan @ A @ U2 ) ) ) ) ) ).
% atLeastLessThan_def
thf(fact_4983_atLeastAtMost__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( set_or1337092689740270186AtMost @ A )
= ( ^ [L2: A,U2: A] : ( inf_inf @ ( set @ A ) @ ( set_ord_atLeast @ A @ L2 ) @ ( set_ord_atMost @ A @ U2 ) ) ) ) ) ).
% atLeastAtMost_def
thf(fact_4984_ivl__disj__int__one_I6_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L: A,U: A] :
( ( inf_inf @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ L @ U ) @ ( set_ord_atLeast @ A @ U ) )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% ivl_disj_int_one(6)
thf(fact_4985_antimono__funpow,axiom,
! [A: $tType] :
( ( ( lattice @ A )
& ( order_top @ A ) )
=> ! [Q: A > A] :
( ( order_mono @ A @ A @ Q )
=> ( order_antimono @ nat @ A
@ ^ [I3: nat] : ( compow @ ( A > A ) @ I3 @ Q @ ( top_top @ A ) ) ) ) ) ).
% antimono_funpow
thf(fact_4986_atMost__Int__atLeast,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [N2: A] :
( ( inf_inf @ ( set @ A ) @ ( set_ord_atMost @ A @ N2 ) @ ( set_ord_atLeast @ A @ N2 ) )
= ( insert2 @ A @ N2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% atMost_Int_atLeast
thf(fact_4987_decseq__ge,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [X8: nat > A,L5: A,N2: nat] :
( ( order_antimono @ nat @ A @ X8 )
=> ( ( filterlim @ nat @ A @ X8 @ ( topolo7230453075368039082e_nhds @ A @ L5 ) @ ( at_top @ nat ) )
=> ( ord_less_eq @ A @ L5 @ ( X8 @ N2 ) ) ) ) ) ).
% decseq_ge
thf(fact_4988_UN__atLeast__UNIV,axiom,
( ( complete_Sup_Sup @ ( set @ nat ) @ ( image2 @ nat @ ( set @ nat ) @ ( set_ord_atLeast @ nat ) @ ( top_top @ ( set @ nat ) ) ) )
= ( top_top @ ( set @ nat ) ) ) ).
% UN_atLeast_UNIV
thf(fact_4989_atLeast__Suc,axiom,
! [K: nat] :
( ( set_ord_atLeast @ nat @ ( suc @ K ) )
= ( minus_minus @ ( set @ nat ) @ ( set_ord_atLeast @ nat @ K ) @ ( insert2 @ nat @ K @ ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% atLeast_Suc
thf(fact_4990_remdups__adj__altdef,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( ( remdups_adj @ A @ Xs )
= Ys )
= ( ? [F3: nat > nat] :
( ( order_mono @ nat @ nat @ F3 )
& ( ( image2 @ nat @ nat @ F3 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Xs ) ) )
= ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Ys ) ) )
& ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( nth @ A @ Xs @ I3 )
= ( nth @ A @ Ys @ ( F3 @ I3 ) ) ) )
& ! [I3: nat] :
( ( ord_less @ nat @ ( plus_plus @ nat @ I3 @ ( one_one @ nat ) ) @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ( nth @ A @ Xs @ I3 )
= ( nth @ A @ Xs @ ( plus_plus @ nat @ I3 @ ( one_one @ nat ) ) ) )
= ( ( F3 @ I3 )
= ( F3 @ ( plus_plus @ nat @ I3 @ ( one_one @ nat ) ) ) ) ) ) ) ) ) ).
% remdups_adj_altdef
thf(fact_4991_range__abs__Nats,axiom,
( ( image2 @ int @ int @ ( abs_abs @ int ) @ ( top_top @ ( set @ int ) ) )
= ( semiring_1_Nats @ int ) ) ).
% range_abs_Nats
thf(fact_4992_compact__imp__fip__image,axiom,
! [B: $tType,A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [S: set @ A,I6: set @ B,F2: B > ( set @ A )] :
( ( topolo2193935891317330818ompact @ A @ S )
=> ( ! [I4: B] :
( ( member @ B @ I4 @ I6 )
=> ( topolo7761053866217962861closed @ A @ ( F2 @ I4 ) ) )
=> ( ! [I9: set @ B] :
( ( finite_finite2 @ B @ I9 )
=> ( ( ord_less_eq @ ( set @ B ) @ I9 @ I6 )
=> ( ( inf_inf @ ( set @ A ) @ S @ ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ F2 @ I9 ) ) )
!= ( bot_bot @ ( set @ A ) ) ) ) )
=> ( ( inf_inf @ ( set @ A ) @ S @ ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ F2 @ I6 ) ) )
!= ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% compact_imp_fip_image
thf(fact_4993_closed__empty,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ( topolo7761053866217962861closed @ A @ ( bot_bot @ ( set @ A ) ) ) ) ).
% closed_empty
thf(fact_4994_remdups__adj__Nil__iff,axiom,
! [A: $tType,Xs: list @ A] :
( ( ( remdups_adj @ A @ Xs )
= ( nil @ A ) )
= ( Xs
= ( nil @ A ) ) ) ).
% remdups_adj_Nil_iff
thf(fact_4995_remdups__adj__set,axiom,
! [A: $tType,Xs: list @ A] :
( ( set2 @ A @ ( remdups_adj @ A @ Xs ) )
= ( set2 @ A @ Xs ) ) ).
% remdups_adj_set
thf(fact_4996_closed__singleton,axiom,
! [A: $tType] :
( ( topological_t1_space @ A )
=> ! [A3: A] : ( topolo7761053866217962861closed @ A @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% closed_singleton
thf(fact_4997_closed__Union,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [S2: set @ ( set @ A )] :
( ( finite_finite2 @ ( set @ A ) @ S2 )
=> ( ! [X5: set @ A] :
( ( member @ ( set @ A ) @ X5 @ S2 )
=> ( topolo7761053866217962861closed @ A @ X5 ) )
=> ( topolo7761053866217962861closed @ A @ ( complete_Sup_Sup @ ( set @ A ) @ S2 ) ) ) ) ) ).
% closed_Union
thf(fact_4998_closed__UN,axiom,
! [A: $tType,B: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [A4: set @ B,B2: B > ( set @ A )] :
( ( finite_finite2 @ B @ A4 )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ A4 )
=> ( topolo7761053866217962861closed @ A @ ( B2 @ X5 ) ) )
=> ( topolo7761053866217962861closed @ A @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B2 @ A4 ) ) ) ) ) ) ).
% closed_UN
thf(fact_4999_of__nat__in__Nats,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [N2: nat] : ( member @ A @ ( semiring_1_of_nat @ A @ N2 ) @ ( semiring_1_Nats @ A ) ) ) ).
% of_nat_in_Nats
thf(fact_5000_Nats__induct,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [X: A,P: A > $o] :
( ( member @ A @ X @ ( semiring_1_Nats @ A ) )
=> ( ! [N3: nat] : ( P @ ( semiring_1_of_nat @ A @ N3 ) )
=> ( P @ X ) ) ) ) ).
% Nats_induct
thf(fact_5001_Nats__cases,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [X: A] :
( ( member @ A @ X @ ( semiring_1_Nats @ A ) )
=> ~ ! [N3: nat] :
( X
!= ( semiring_1_of_nat @ A @ N3 ) ) ) ) ).
% Nats_cases
thf(fact_5002_Nats__0,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ( member @ A @ ( zero_zero @ A ) @ ( semiring_1_Nats @ A ) ) ) ).
% Nats_0
thf(fact_5003_Nats__numeral,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [W2: num] : ( member @ A @ ( numeral_numeral @ A @ W2 ) @ ( semiring_1_Nats @ A ) ) ) ).
% Nats_numeral
thf(fact_5004_Nats__mult,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [A3: A,B3: A] :
( ( member @ A @ A3 @ ( semiring_1_Nats @ A ) )
=> ( ( member @ A @ B3 @ ( semiring_1_Nats @ A ) )
=> ( member @ A @ ( times_times @ A @ A3 @ B3 ) @ ( semiring_1_Nats @ A ) ) ) ) ) ).
% Nats_mult
thf(fact_5005_finite__imp__closed,axiom,
! [A: $tType] :
( ( topological_t1_space @ A )
=> ! [S2: set @ A] :
( ( finite_finite2 @ A @ S2 )
=> ( topolo7761053866217962861closed @ A @ S2 ) ) ) ).
% finite_imp_closed
thf(fact_5006_remdups__adj_Osimps_I1_J,axiom,
! [A: $tType] :
( ( remdups_adj @ A @ ( nil @ A ) )
= ( nil @ A ) ) ).
% remdups_adj.simps(1)
thf(fact_5007_remdups__adj__distinct,axiom,
! [A: $tType,Xs: list @ A] :
( ( distinct @ A @ Xs )
=> ( ( remdups_adj @ A @ Xs )
= Xs ) ) ).
% remdups_adj_distinct
thf(fact_5008_Nats__add,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [A3: A,B3: A] :
( ( member @ A @ A3 @ ( semiring_1_Nats @ A ) )
=> ( ( member @ A @ B3 @ ( semiring_1_Nats @ A ) )
=> ( member @ A @ ( plus_plus @ A @ A3 @ B3 ) @ ( semiring_1_Nats @ A ) ) ) ) ) ).
% Nats_add
thf(fact_5009_Nats__1,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ( member @ A @ ( one_one @ A ) @ ( semiring_1_Nats @ A ) ) ) ).
% Nats_1
thf(fact_5010_remdups__adj__length,axiom,
! [A: $tType,Xs: list @ A] : ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ ( remdups_adj @ A @ Xs ) ) @ ( size_size @ ( list @ A ) @ Xs ) ) ).
% remdups_adj_length
thf(fact_5011_Nats__diff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,B3: A] :
( ( member @ A @ A3 @ ( semiring_1_Nats @ A ) )
=> ( ( member @ A @ B3 @ ( semiring_1_Nats @ A ) )
=> ( ( ord_less_eq @ A @ B3 @ A3 )
=> ( member @ A @ ( minus_minus @ A @ A3 @ B3 ) @ ( semiring_1_Nats @ A ) ) ) ) ) ) ).
% Nats_diff
thf(fact_5012_t3__space,axiom,
! [A: $tType] :
( ( topological_t3_space @ A )
=> ! [S2: set @ A,Y: A] :
( ( topolo7761053866217962861closed @ A @ S2 )
=> ( ~ ( member @ A @ Y @ S2 )
=> ? [U5: set @ A,V6: set @ A] :
( ( topolo1002775350975398744n_open @ A @ U5 )
& ( topolo1002775350975398744n_open @ A @ V6 )
& ( member @ A @ Y @ U5 )
& ( ord_less_eq @ ( set @ A ) @ S2 @ V6 )
& ( ( inf_inf @ ( set @ A ) @ U5 @ V6 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% t3_space
thf(fact_5013_t4__space,axiom,
! [A: $tType] :
( ( topological_t4_space @ A )
=> ! [S2: set @ A,T4: set @ A] :
( ( topolo7761053866217962861closed @ A @ S2 )
=> ( ( topolo7761053866217962861closed @ A @ T4 )
=> ( ( ( inf_inf @ ( set @ A ) @ S2 @ T4 )
= ( bot_bot @ ( set @ A ) ) )
=> ? [U5: set @ A,V6: set @ A] :
( ( topolo1002775350975398744n_open @ A @ U5 )
& ( topolo1002775350975398744n_open @ A @ V6 )
& ( ord_less_eq @ ( set @ A ) @ S2 @ U5 )
& ( ord_less_eq @ ( set @ A ) @ T4 @ V6 )
& ( ( inf_inf @ ( set @ A ) @ U5 @ V6 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% t4_space
thf(fact_5014_remdups__adj__adjacent,axiom,
! [A: $tType,I: nat,Xs: list @ A] :
( ( ord_less @ nat @ ( suc @ I ) @ ( size_size @ ( list @ A ) @ ( remdups_adj @ A @ Xs ) ) )
=> ( ( nth @ A @ ( remdups_adj @ A @ Xs ) @ I )
!= ( nth @ A @ ( remdups_adj @ A @ Xs ) @ ( suc @ I ) ) ) ) ).
% remdups_adj_adjacent
thf(fact_5015_nhds__closed,axiom,
! [A: $tType] :
( ( topological_t3_space @ A )
=> ! [X: A,A4: set @ A] :
( ( member @ A @ X @ A4 )
=> ( ( topolo1002775350975398744n_open @ A @ A4 )
=> ? [A15: set @ A] :
( ( member @ A @ X @ A15 )
& ( topolo7761053866217962861closed @ A @ A15 )
& ( ord_less_eq @ ( set @ A ) @ A15 @ A4 )
& ( eventually @ A
@ ^ [Y6: A] : ( member @ A @ Y6 @ A15 )
@ ( topolo7230453075368039082e_nhds @ A @ X ) ) ) ) ) ) ).
% nhds_closed
thf(fact_5016_Lim__in__closed__set,axiom,
! [B: $tType,A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [S2: set @ A,F2: B > A,F5: filter @ B,L: A] :
( ( topolo7761053866217962861closed @ A @ S2 )
=> ( ( eventually @ B
@ ^ [X4: B] : ( member @ A @ ( F2 @ X4 ) @ S2 )
@ F5 )
=> ( ( F5
!= ( bot_bot @ ( filter @ B ) ) )
=> ( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ F5 )
=> ( member @ A @ L @ S2 ) ) ) ) ) ) ).
% Lim_in_closed_set
thf(fact_5017_Nats__def,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ( ( semiring_1_Nats @ A )
= ( image2 @ nat @ A @ ( semiring_1_of_nat @ A ) @ ( top_top @ ( set @ nat ) ) ) ) ) ).
% Nats_def
thf(fact_5018_remdups__adj__length__ge1,axiom,
! [A: $tType,Xs: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ( ord_less_eq @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( size_size @ ( list @ A ) @ ( remdups_adj @ A @ Xs ) ) ) ) ).
% remdups_adj_length_ge1
thf(fact_5019_compact__imp__fip,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [S2: set @ A,F5: set @ ( set @ A )] :
( ( topolo2193935891317330818ompact @ A @ S2 )
=> ( ! [T5: set @ A] :
( ( member @ ( set @ A ) @ T5 @ F5 )
=> ( topolo7761053866217962861closed @ A @ T5 ) )
=> ( ! [F11: set @ ( set @ A )] :
( ( finite_finite2 @ ( set @ A ) @ F11 )
=> ( ( ord_less_eq @ ( set @ ( set @ A ) ) @ F11 @ F5 )
=> ( ( inf_inf @ ( set @ A ) @ S2 @ ( complete_Inf_Inf @ ( set @ A ) @ F11 ) )
!= ( bot_bot @ ( set @ A ) ) ) ) )
=> ( ( inf_inf @ ( set @ A ) @ S2 @ ( complete_Inf_Inf @ ( set @ A ) @ F5 ) )
!= ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% compact_imp_fip
thf(fact_5020_compact__fip,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ( ( topolo2193935891317330818ompact @ A )
= ( ^ [U6: set @ A] :
! [A7: set @ ( set @ A )] :
( ! [X4: set @ A] :
( ( member @ ( set @ A ) @ X4 @ A7 )
=> ( topolo7761053866217962861closed @ A @ X4 ) )
=> ( ! [B6: set @ ( set @ A )] :
( ( ord_less_eq @ ( set @ ( set @ A ) ) @ B6 @ A7 )
=> ( ( finite_finite2 @ ( set @ A ) @ B6 )
=> ( ( inf_inf @ ( set @ A ) @ U6 @ ( complete_Inf_Inf @ ( set @ A ) @ B6 ) )
!= ( bot_bot @ ( set @ A ) ) ) ) )
=> ( ( inf_inf @ ( set @ A ) @ U6 @ ( complete_Inf_Inf @ ( set @ A ) @ A7 ) )
!= ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% compact_fip
thf(fact_5021_inj__sgn__power,axiom,
! [N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( inj_on @ real @ real
@ ^ [Y6: real] : ( times_times @ real @ ( sgn_sgn @ real @ Y6 ) @ ( power_power @ real @ ( abs_abs @ real @ Y6 ) @ N2 ) )
@ ( top_top @ ( set @ real ) ) ) ) ).
% inj_sgn_power
thf(fact_5022_continuous__on__IccI,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo1944317154257567458pology @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [F2: A > B,A3: A,B3: A] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ ( F2 @ A3 ) ) @ ( topolo174197925503356063within @ A @ A3 @ ( set_ord_greaterThan @ A @ A3 ) ) )
=> ( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ ( F2 @ B3 ) ) @ ( topolo174197925503356063within @ A @ B3 @ ( set_ord_lessThan @ A @ B3 ) ) )
=> ( ! [X5: A] :
( ( ord_less @ A @ A3 @ X5 )
=> ( ( ord_less @ A @ X5 @ B3 )
=> ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ ( F2 @ X5 ) ) @ ( topolo174197925503356063within @ A @ X5 @ ( top_top @ ( set @ A ) ) ) ) ) )
=> ( ( ord_less @ A @ A3 @ B3 )
=> ( topolo81223032696312382ous_on @ A @ B @ ( set_or1337092689740270186AtMost @ A @ A3 @ B3 ) @ F2 ) ) ) ) ) ) ).
% continuous_on_IccI
thf(fact_5023_measure__function__int,axiom,
fun_is_measure @ int @ ( comp @ int @ nat @ int @ nat2 @ ( abs_abs @ int ) ) ).
% measure_function_int
thf(fact_5024_inj__on__empty,axiom,
! [B: $tType,A: $tType,F2: A > B] : ( inj_on @ A @ B @ F2 @ ( bot_bot @ ( set @ A ) ) ) ).
% inj_on_empty
thf(fact_5025_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_5026_inj__divide__right,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A3: A] :
( ( inj_on @ A @ A
@ ^ [B4: A] : ( divide_divide @ A @ B4 @ A3 )
@ ( top_top @ ( set @ A ) ) )
= ( A3
!= ( zero_zero @ A ) ) ) ) ).
% inj_divide_right
thf(fact_5027_inj__on__insert,axiom,
! [B: $tType,A: $tType,F2: A > B,A3: A,A4: set @ A] :
( ( inj_on @ A @ B @ F2 @ ( insert2 @ A @ A3 @ A4 ) )
= ( ( inj_on @ A @ B @ F2 @ A4 )
& ~ ( member @ B @ ( F2 @ A3 ) @ ( image2 @ A @ B @ F2 @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% inj_on_insert
thf(fact_5028_finite__inverse__image__gen,axiom,
! [A: $tType,B: $tType,A4: set @ A,F2: B > A,D6: set @ B] :
( ( finite_finite2 @ A @ A4 )
=> ( ( inj_on @ B @ A @ F2 @ D6 )
=> ( finite_finite2 @ B
@ ( collect @ B
@ ^ [J2: B] :
( ( member @ B @ J2 @ D6 )
& ( member @ A @ ( F2 @ J2 ) @ A4 ) ) ) ) ) ) ).
% finite_inverse_image_gen
thf(fact_5029_inj__on__add_H,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [A3: A,A4: set @ A] :
( inj_on @ A @ A
@ ^ [B4: A] : ( plus_plus @ A @ B4 @ A3 )
@ A4 ) ) ).
% inj_on_add'
thf(fact_5030_continuous__on__add,axiom,
! [B: $tType,D: $tType] :
( ( ( topolo4958980785337419405_space @ D )
& ( topolo6943815403480290642id_add @ B ) )
=> ! [S: set @ D,F2: D > B,G: D > B] :
( ( topolo81223032696312382ous_on @ D @ B @ S @ F2 )
=> ( ( topolo81223032696312382ous_on @ D @ B @ S @ G )
=> ( topolo81223032696312382ous_on @ D @ B @ S
@ ^ [X4: D] : ( plus_plus @ B @ ( F2 @ X4 ) @ ( G @ X4 ) ) ) ) ) ) ).
% continuous_on_add
thf(fact_5031_sorted__list__of__set_Oinj__on,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( inj_on @ A @ A
@ ^ [X4: A] : X4
@ ( top_top @ ( set @ A ) ) ) ) ).
% sorted_list_of_set.inj_on
thf(fact_5032_continuous__inj__imp__mono,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo8458572112393995274pology @ A )
& ( topolo1944317154257567458pology @ B ) )
=> ! [A3: A,X: A,B3: A,F2: A > B] :
( ( ord_less @ A @ A3 @ X )
=> ( ( ord_less @ A @ X @ B3 )
=> ( ( topolo81223032696312382ous_on @ A @ B @ ( set_or1337092689740270186AtMost @ A @ A3 @ B3 ) @ F2 )
=> ( ( inj_on @ A @ B @ F2 @ ( set_or1337092689740270186AtMost @ A @ A3 @ B3 ) )
=> ( ( ( ord_less @ B @ ( F2 @ A3 ) @ ( F2 @ X ) )
& ( ord_less @ B @ ( F2 @ X ) @ ( F2 @ B3 ) ) )
| ( ( ord_less @ B @ ( F2 @ B3 ) @ ( F2 @ X ) )
& ( ord_less @ B @ ( F2 @ X ) @ ( F2 @ A3 ) ) ) ) ) ) ) ) ) ).
% continuous_inj_imp_mono
thf(fact_5033_inj__on__add,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [A3: A,A4: set @ A] : ( inj_on @ A @ A @ ( plus_plus @ A @ A3 ) @ A4 ) ) ).
% inj_on_add
thf(fact_5034_continuous__on__empty,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo4958980785337419405_space @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [F2: A > B] : ( topolo81223032696312382ous_on @ A @ B @ ( bot_bot @ ( set @ A ) ) @ F2 ) ) ).
% continuous_on_empty
thf(fact_5035_inj__on__subset,axiom,
! [B: $tType,A: $tType,F2: A > B,A4: set @ A,B2: set @ A] :
( ( inj_on @ A @ B @ F2 @ A4 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ A4 )
=> ( inj_on @ A @ B @ F2 @ B2 ) ) ) ).
% inj_on_subset
thf(fact_5036_subset__inj__on,axiom,
! [B: $tType,A: $tType,F2: A > B,B2: set @ A,A4: set @ A] :
( ( inj_on @ A @ B @ F2 @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ A4 @ B2 )
=> ( inj_on @ A @ B @ F2 @ A4 ) ) ) ).
% subset_inj_on
thf(fact_5037_continuous__on__subset,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo4958980785337419405_space @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [S: set @ A,F2: A > B,T2: set @ A] :
( ( topolo81223032696312382ous_on @ A @ B @ S @ F2 )
=> ( ( ord_less_eq @ ( set @ A ) @ T2 @ S )
=> ( topolo81223032696312382ous_on @ A @ B @ T2 @ F2 ) ) ) ) ).
% continuous_on_subset
thf(fact_5038_inj__on__image__Fpow,axiom,
! [B: $tType,A: $tType,F2: A > B,A4: set @ A] :
( ( inj_on @ A @ B @ F2 @ A4 )
=> ( inj_on @ ( set @ A ) @ ( set @ B ) @ ( image2 @ A @ B @ F2 ) @ ( finite_Fpow @ A @ A4 ) ) ) ).
% inj_on_image_Fpow
thf(fact_5039_IVT_H,axiom,
! [A: $tType,B: $tType] :
( ( ( topolo1944317154257567458pology @ B )
& ( topolo8458572112393995274pology @ A ) )
=> ! [F2: A > B,A3: A,Y: B,B3: A] :
( ( ord_less_eq @ B @ ( F2 @ A3 ) @ Y )
=> ( ( ord_less_eq @ B @ Y @ ( F2 @ B3 ) )
=> ( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( topolo81223032696312382ous_on @ A @ B @ ( set_or1337092689740270186AtMost @ A @ A3 @ B3 ) @ F2 )
=> ? [X5: A] :
( ( ord_less_eq @ A @ A3 @ X5 )
& ( ord_less_eq @ A @ X5 @ B3 )
& ( ( F2 @ X5 )
= Y ) ) ) ) ) ) ) ).
% IVT'
thf(fact_5040_IVT2_H,axiom,
! [A: $tType,B: $tType] :
( ( ( topolo1944317154257567458pology @ B )
& ( topolo8458572112393995274pology @ A ) )
=> ! [F2: A > B,B3: A,Y: B,A3: A] :
( ( ord_less_eq @ B @ ( F2 @ B3 ) @ Y )
=> ( ( ord_less_eq @ B @ Y @ ( F2 @ A3 ) )
=> ( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( topolo81223032696312382ous_on @ A @ B @ ( set_or1337092689740270186AtMost @ A @ A3 @ B3 ) @ F2 )
=> ? [X5: A] :
( ( ord_less_eq @ A @ A3 @ X5 )
& ( ord_less_eq @ A @ X5 @ B3 )
& ( ( F2 @ X5 )
= Y ) ) ) ) ) ) ) ).
% IVT2'
thf(fact_5041_linorder__inj__onI,axiom,
! [B: $tType,A: $tType] :
( ( order @ A )
=> ! [A4: set @ A,F2: A > B] :
( ! [X5: A,Y3: A] :
( ( ord_less @ A @ X5 @ Y3 )
=> ( ( member @ A @ X5 @ A4 )
=> ( ( member @ A @ Y3 @ A4 )
=> ( ( F2 @ X5 )
!= ( F2 @ Y3 ) ) ) ) )
=> ( ! [X5: A,Y3: A] :
( ( member @ A @ X5 @ A4 )
=> ( ( member @ A @ Y3 @ A4 )
=> ( ( ord_less_eq @ A @ X5 @ Y3 )
| ( ord_less_eq @ A @ Y3 @ X5 ) ) ) )
=> ( inj_on @ A @ B @ F2 @ A4 ) ) ) ) ).
% linorder_inj_onI
thf(fact_5042_continuous__on__sing,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo4958980785337419405_space @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [X: A,F2: A > B] : ( topolo81223032696312382ous_on @ A @ B @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) @ F2 ) ) ).
% continuous_on_sing
thf(fact_5043_inj__on__mult,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [A3: A,A4: set @ A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( inj_on @ A @ A @ ( times_times @ A @ A3 ) @ A4 ) ) ) ).
% inj_on_mult
thf(fact_5044_linorder__injI,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ A )
=> ! [F2: A > B] :
( ! [X5: A,Y3: A] :
( ( ord_less @ A @ X5 @ Y3 )
=> ( ( F2 @ X5 )
!= ( F2 @ Y3 ) ) )
=> ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) ) ) ) ).
% linorder_injI
thf(fact_5045_inj__add__left,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [A3: A] : ( inj_on @ A @ A @ ( plus_plus @ A @ A3 ) @ ( top_top @ ( set @ A ) ) ) ) ).
% inj_add_left
thf(fact_5046_finite__image__iff,axiom,
! [B: $tType,A: $tType,F2: A > B,A4: set @ A] :
( ( inj_on @ A @ B @ F2 @ A4 )
=> ( ( finite_finite2 @ B @ ( image2 @ A @ B @ F2 @ A4 ) )
= ( finite_finite2 @ A @ A4 ) ) ) ).
% finite_image_iff
thf(fact_5047_finite__imageD,axiom,
! [A: $tType,B: $tType,F2: B > A,A4: set @ B] :
( ( finite_finite2 @ A @ ( image2 @ B @ A @ F2 @ A4 ) )
=> ( ( inj_on @ B @ A @ F2 @ A4 )
=> ( finite_finite2 @ B @ A4 ) ) ) ).
% finite_imageD
thf(fact_5048_subset__image__inj,axiom,
! [A: $tType,B: $tType,S2: set @ A,F2: B > A,T4: set @ B] :
( ( ord_less_eq @ ( set @ A ) @ S2 @ ( image2 @ B @ A @ F2 @ T4 ) )
= ( ? [U6: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ U6 @ T4 )
& ( inj_on @ B @ A @ F2 @ U6 )
& ( S2
= ( image2 @ B @ A @ F2 @ U6 ) ) ) ) ) ).
% subset_image_inj
thf(fact_5049_inj__on__image__eq__iff,axiom,
! [B: $tType,A: $tType,F2: A > B,C5: set @ A,A4: set @ A,B2: set @ A] :
( ( inj_on @ A @ B @ F2 @ C5 )
=> ( ( ord_less_eq @ ( set @ A ) @ A4 @ C5 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ C5 )
=> ( ( ( image2 @ A @ B @ F2 @ A4 )
= ( image2 @ A @ B @ F2 @ B2 ) )
= ( A4 = B2 ) ) ) ) ) ).
% inj_on_image_eq_iff
thf(fact_5050_inj__on__image__mem__iff,axiom,
! [B: $tType,A: $tType,F2: A > B,B2: set @ A,A3: A,A4: set @ A] :
( ( inj_on @ A @ B @ F2 @ B2 )
=> ( ( member @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ A4 @ B2 )
=> ( ( member @ B @ ( F2 @ A3 ) @ ( image2 @ A @ B @ F2 @ A4 ) )
= ( member @ A @ A3 @ A4 ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_5051_card__image,axiom,
! [B: $tType,A: $tType,F2: A > B,A4: set @ A] :
( ( inj_on @ A @ B @ F2 @ A4 )
=> ( ( finite_card @ B @ ( image2 @ A @ B @ F2 @ A4 ) )
= ( finite_card @ A @ A4 ) ) ) ).
% card_image
thf(fact_5052_inj__on__strict__subset,axiom,
! [B: $tType,A: $tType,F2: A > B,B2: set @ A,A4: set @ A] :
( ( inj_on @ A @ B @ F2 @ B2 )
=> ( ( ord_less @ ( set @ A ) @ A4 @ B2 )
=> ( ord_less @ ( set @ B ) @ ( image2 @ A @ B @ F2 @ A4 ) @ ( image2 @ A @ B @ F2 @ B2 ) ) ) ) ).
% inj_on_strict_subset
thf(fact_5053_inj__fn,axiom,
! [A: $tType,F2: A > A,N2: nat] :
( ( inj_on @ A @ A @ F2 @ ( top_top @ ( set @ A ) ) )
=> ( inj_on @ A @ A @ ( compow @ ( A > A ) @ N2 @ F2 ) @ ( top_top @ ( set @ A ) ) ) ) ).
% inj_fn
thf(fact_5054_continuous__on__divide,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo4958980785337419405_space @ A )
& ( real_V3459762299906320749_field @ B ) )
=> ! [S: set @ A,F2: A > B,G: A > B] :
( ( topolo81223032696312382ous_on @ A @ B @ S @ F2 )
=> ( ( topolo81223032696312382ous_on @ A @ B @ S @ G )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ S )
=> ( ( G @ X5 )
!= ( zero_zero @ B ) ) )
=> ( topolo81223032696312382ous_on @ A @ B @ S
@ ^ [X4: A] : ( divide_divide @ B @ ( F2 @ X4 ) @ ( G @ X4 ) ) ) ) ) ) ) ).
% continuous_on_divide
thf(fact_5055_continuous__on__compose2,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( ( topolo4958980785337419405_space @ C )
& ( topolo4958980785337419405_space @ B )
& ( topolo4958980785337419405_space @ A ) )
=> ! [T2: set @ A,G: A > B,S: set @ C,F2: C > A] :
( ( topolo81223032696312382ous_on @ A @ B @ T2 @ G )
=> ( ( topolo81223032696312382ous_on @ C @ A @ S @ F2 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image2 @ C @ A @ F2 @ S ) @ T2 )
=> ( topolo81223032696312382ous_on @ C @ B @ S
@ ^ [X4: C] : ( G @ ( F2 @ X4 ) ) ) ) ) ) ) ).
% continuous_on_compose2
thf(fact_5056_inj__on__Inter,axiom,
! [B: $tType,A: $tType,S2: set @ ( set @ A ),F2: A > B] :
( ( S2
!= ( bot_bot @ ( set @ ( set @ A ) ) ) )
=> ( ! [A8: set @ A] :
( ( member @ ( set @ A ) @ A8 @ S2 )
=> ( inj_on @ A @ B @ F2 @ A8 ) )
=> ( inj_on @ A @ B @ F2 @ ( complete_Inf_Inf @ ( set @ A ) @ S2 ) ) ) ) ).
% inj_on_Inter
thf(fact_5057_continuous__on__inverse,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo4958980785337419405_space @ A )
& ( real_V8999393235501362500lgebra @ B ) )
=> ! [S: set @ A,F2: A > B] :
( ( topolo81223032696312382ous_on @ A @ B @ S @ F2 )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ S )
=> ( ( F2 @ X5 )
!= ( zero_zero @ B ) ) )
=> ( topolo81223032696312382ous_on @ A @ B @ S
@ ^ [X4: A] : ( inverse_inverse @ B @ ( F2 @ X4 ) ) ) ) ) ) ).
% continuous_on_inverse
thf(fact_5058_continuous__on__sgn,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo4958980785337419405_space @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [S: set @ A,F2: A > B] :
( ( topolo81223032696312382ous_on @ A @ B @ S @ F2 )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ S )
=> ( ( F2 @ X5 )
!= ( zero_zero @ B ) ) )
=> ( topolo81223032696312382ous_on @ A @ B @ S
@ ^ [X4: A] : ( sgn_sgn @ B @ ( F2 @ X4 ) ) ) ) ) ) ).
% continuous_on_sgn
thf(fact_5059_finite__inverse__image,axiom,
! [A: $tType,B: $tType,A4: set @ A,F2: B > A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( inj_on @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) )
=> ( finite_finite2 @ B
@ ( collect @ B
@ ^ [J2: B] : ( member @ A @ ( F2 @ J2 ) @ A4 ) ) ) ) ) ).
% finite_inverse_image
thf(fact_5060_measure__size,axiom,
! [A: $tType] :
( ( size @ A )
=> ( fun_is_measure @ A @ ( size_size @ A ) ) ) ).
% measure_size
thf(fact_5061_continuous__onI__mono,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo1944317154257567458pology @ A )
& ( dense_order @ B )
& ( topolo1944317154257567458pology @ B ) )
=> ! [F2: A > B,A4: set @ A] :
( ( topolo1002775350975398744n_open @ B @ ( image2 @ A @ B @ F2 @ A4 ) )
=> ( ! [X5: A,Y3: A] :
( ( member @ A @ X5 @ A4 )
=> ( ( member @ A @ Y3 @ A4 )
=> ( ( ord_less_eq @ A @ X5 @ Y3 )
=> ( ord_less_eq @ B @ ( F2 @ X5 ) @ ( F2 @ Y3 ) ) ) ) )
=> ( topolo81223032696312382ous_on @ A @ B @ A4 @ F2 ) ) ) ) ).
% continuous_onI_mono
thf(fact_5062_finite__UNIV__inj__surj,axiom,
! [A: $tType,F2: A > A] :
( ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) )
=> ( ( inj_on @ A @ A @ F2 @ ( top_top @ ( set @ A ) ) )
=> ( ( image2 @ A @ A @ F2 @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ A ) ) ) ) ) ).
% finite_UNIV_inj_surj
thf(fact_5063_finite__UNIV__surj__inj,axiom,
! [A: $tType,F2: A > A] :
( ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) )
=> ( ( ( image2 @ A @ A @ F2 @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ A ) ) )
=> ( inj_on @ A @ A @ F2 @ ( top_top @ ( set @ A ) ) ) ) ) ).
% finite_UNIV_surj_inj
thf(fact_5064_inj__image__subset__iff,axiom,
! [B: $tType,A: $tType,F2: A > B,A4: set @ A,B2: set @ A] :
( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F2 @ A4 ) @ ( image2 @ A @ B @ F2 @ B2 ) )
= ( ord_less_eq @ ( set @ A ) @ A4 @ B2 ) ) ) ).
% inj_image_subset_iff
thf(fact_5065_inj__on__iff__surj,axiom,
! [A: $tType,B: $tType,A4: set @ A,A16: set @ B] :
( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ? [F3: A > B] :
( ( inj_on @ A @ B @ F3 @ A4 )
& ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F3 @ A4 ) @ A16 ) ) )
= ( ? [G2: B > A] :
( ( image2 @ B @ A @ G2 @ A16 )
= A4 ) ) ) ) ).
% inj_on_iff_surj
thf(fact_5066_endo__inj__surj,axiom,
! [A: $tType,A4: set @ A,F2: A > A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image2 @ A @ A @ F2 @ A4 ) @ A4 )
=> ( ( inj_on @ A @ A @ F2 @ A4 )
=> ( ( image2 @ A @ A @ F2 @ A4 )
= A4 ) ) ) ) ).
% endo_inj_surj
thf(fact_5067_inj__on__finite,axiom,
! [B: $tType,A: $tType,F2: A > B,A4: set @ A,B2: set @ B] :
( ( inj_on @ A @ B @ F2 @ A4 )
=> ( ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F2 @ A4 ) @ B2 )
=> ( ( finite_finite2 @ B @ B2 )
=> ( finite_finite2 @ A @ A4 ) ) ) ) ).
% inj_on_finite
thf(fact_5068_finite__surj__inj,axiom,
! [A: $tType,A4: set @ A,F2: A > A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( ord_less_eq @ ( set @ A ) @ A4 @ ( image2 @ A @ A @ F2 @ A4 ) )
=> ( inj_on @ A @ A @ F2 @ A4 ) ) ) ).
% finite_surj_inj
thf(fact_5069_inj__on__image__Int,axiom,
! [B: $tType,A: $tType,F2: A > B,C5: set @ A,A4: set @ A,B2: set @ A] :
( ( inj_on @ A @ B @ F2 @ C5 )
=> ( ( ord_less_eq @ ( set @ A ) @ A4 @ C5 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ C5 )
=> ( ( image2 @ A @ B @ F2 @ ( inf_inf @ ( set @ A ) @ A4 @ B2 ) )
= ( inf_inf @ ( set @ B ) @ ( image2 @ A @ B @ F2 @ A4 ) @ ( image2 @ A @ B @ F2 @ B2 ) ) ) ) ) ) ).
% inj_on_image_Int
thf(fact_5070_eq__card__imp__inj__on,axiom,
! [B: $tType,A: $tType,A4: set @ A,F2: A > B] :
( ( finite_finite2 @ A @ A4 )
=> ( ( ( finite_card @ B @ ( image2 @ A @ B @ F2 @ A4 ) )
= ( finite_card @ A @ A4 ) )
=> ( inj_on @ A @ B @ F2 @ A4 ) ) ) ).
% eq_card_imp_inj_on
thf(fact_5071_inj__on__iff__eq__card,axiom,
! [B: $tType,A: $tType,A4: set @ A,F2: A > B] :
( ( finite_finite2 @ A @ A4 )
=> ( ( inj_on @ A @ B @ F2 @ A4 )
= ( ( finite_card @ B @ ( image2 @ A @ B @ F2 @ A4 ) )
= ( finite_card @ A @ A4 ) ) ) ) ).
% inj_on_iff_eq_card
thf(fact_5072_inj__on__image__set__diff,axiom,
! [B: $tType,A: $tType,F2: A > B,C5: set @ A,A4: set @ A,B2: set @ A] :
( ( inj_on @ A @ B @ F2 @ C5 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A4 @ B2 ) @ C5 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ C5 )
=> ( ( image2 @ A @ B @ F2 @ ( minus_minus @ ( set @ A ) @ A4 @ B2 ) )
= ( minus_minus @ ( set @ B ) @ ( image2 @ A @ B @ F2 @ A4 ) @ ( image2 @ A @ B @ F2 @ B2 ) ) ) ) ) ) ).
% inj_on_image_set_diff
thf(fact_5073_pigeonhole,axiom,
! [A: $tType,B: $tType,F2: B > A,A4: set @ B] :
( ( ord_less @ nat @ ( finite_card @ A @ ( image2 @ B @ A @ F2 @ A4 ) ) @ ( finite_card @ B @ A4 ) )
=> ~ ( inj_on @ B @ A @ F2 @ A4 ) ) ).
% pigeonhole
thf(fact_5074_continuous__attains__inf,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo4958980785337419405_space @ A )
& ( topolo1944317154257567458pology @ B ) )
=> ! [S: set @ A,F2: A > B] :
( ( topolo2193935891317330818ompact @ A @ S )
=> ( ( S
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( topolo81223032696312382ous_on @ A @ B @ S @ F2 )
=> ? [X5: A] :
( ( member @ A @ X5 @ S )
& ! [Xa: A] :
( ( member @ A @ Xa @ S )
=> ( ord_less_eq @ B @ ( F2 @ X5 ) @ ( F2 @ Xa ) ) ) ) ) ) ) ) ).
% continuous_attains_inf
thf(fact_5075_continuous__attains__sup,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo4958980785337419405_space @ A )
& ( topolo1944317154257567458pology @ B ) )
=> ! [S: set @ A,F2: A > B] :
( ( topolo2193935891317330818ompact @ A @ S )
=> ( ( S
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( topolo81223032696312382ous_on @ A @ B @ S @ F2 )
=> ? [X5: A] :
( ( member @ A @ X5 @ S )
& ! [Xa: A] :
( ( member @ A @ Xa @ S )
=> ( ord_less_eq @ B @ ( F2 @ Xa ) @ ( F2 @ X5 ) ) ) ) ) ) ) ) ).
% continuous_attains_sup
thf(fact_5076_the__inv__into__into,axiom,
! [B: $tType,A: $tType,F2: A > B,A4: set @ A,X: B,B2: set @ A] :
( ( inj_on @ A @ B @ F2 @ A4 )
=> ( ( member @ B @ X @ ( image2 @ A @ B @ F2 @ A4 ) )
=> ( ( ord_less_eq @ ( set @ A ) @ A4 @ B2 )
=> ( member @ A @ ( the_inv_into @ A @ B @ A4 @ F2 @ X ) @ B2 ) ) ) ) ).
% the_inv_into_into
thf(fact_5077_open__Collect__less,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo4958980785337419405_space @ A )
& ( topolo1944317154257567458pology @ B ) )
=> ! [F2: A > B,G: A > B] :
( ( topolo81223032696312382ous_on @ A @ B @ ( top_top @ ( set @ A ) ) @ F2 )
=> ( ( topolo81223032696312382ous_on @ A @ B @ ( top_top @ ( set @ A ) ) @ G )
=> ( topolo1002775350975398744n_open @ A
@ ( collect @ A
@ ^ [X4: A] : ( ord_less @ B @ ( F2 @ X4 ) @ ( G @ X4 ) ) ) ) ) ) ) ).
% open_Collect_less
thf(fact_5078_closed__Collect__le,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo4958980785337419405_space @ A )
& ( topolo1944317154257567458pology @ B ) )
=> ! [F2: A > B,G: A > B] :
( ( topolo81223032696312382ous_on @ A @ B @ ( top_top @ ( set @ A ) ) @ F2 )
=> ( ( topolo81223032696312382ous_on @ A @ B @ ( top_top @ ( set @ A ) ) @ G )
=> ( topolo7761053866217962861closed @ A
@ ( collect @ A
@ ^ [X4: A] : ( ord_less_eq @ B @ ( F2 @ X4 ) @ ( G @ X4 ) ) ) ) ) ) ) ).
% closed_Collect_le
thf(fact_5079_continuous__on__tan,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [S: set @ A,F2: A > A] :
( ( topolo81223032696312382ous_on @ A @ A @ S @ F2 )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ S )
=> ( ( cos @ A @ ( F2 @ X5 ) )
!= ( zero_zero @ A ) ) )
=> ( topolo81223032696312382ous_on @ A @ A @ S
@ ^ [X4: A] : ( tan @ A @ ( F2 @ X4 ) ) ) ) ) ) ).
% continuous_on_tan
thf(fact_5080_inj__on__UNION__chain,axiom,
! [C: $tType,B: $tType,A: $tType,I6: set @ A,A4: A > ( set @ B ),F2: B > C] :
( ! [I4: A,J3: A] :
( ( member @ A @ I4 @ I6 )
=> ( ( member @ A @ J3 @ I6 )
=> ( ( ord_less_eq @ ( set @ B ) @ ( A4 @ I4 ) @ ( A4 @ J3 ) )
| ( ord_less_eq @ ( set @ B ) @ ( A4 @ J3 ) @ ( A4 @ I4 ) ) ) ) )
=> ( ! [I4: A] :
( ( member @ A @ I4 @ I6 )
=> ( inj_on @ B @ C @ F2 @ ( A4 @ I4 ) ) )
=> ( inj_on @ B @ C @ F2 @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ A4 @ I6 ) ) ) ) ) ).
% inj_on_UNION_chain
thf(fact_5081_inj__on__INTER,axiom,
! [C: $tType,B: $tType,A: $tType,I6: set @ A,F2: B > C,A4: A > ( set @ B )] :
( ( I6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [I4: A] :
( ( member @ A @ I4 @ I6 )
=> ( inj_on @ B @ C @ F2 @ ( A4 @ I4 ) ) )
=> ( inj_on @ B @ C @ F2 @ ( complete_Inf_Inf @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ A4 @ I6 ) ) ) ) ) ).
% inj_on_INTER
thf(fact_5082_continuous__on__cot,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [S: set @ A,F2: A > A] :
( ( topolo81223032696312382ous_on @ A @ A @ S @ F2 )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ S )
=> ( ( sin @ A @ ( F2 @ X5 ) )
!= ( zero_zero @ A ) ) )
=> ( topolo81223032696312382ous_on @ A @ A @ S
@ ^ [X4: A] : ( cot @ A @ ( F2 @ X4 ) ) ) ) ) ) ).
% continuous_on_cot
thf(fact_5083_continuous__on__tanh,axiom,
! [A: $tType,C: $tType] :
( ( ( topolo4958980785337419405_space @ C )
& ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [A4: set @ C,F2: C > A] :
( ( topolo81223032696312382ous_on @ C @ A @ A4 @ F2 )
=> ( ! [X5: C] :
( ( member @ C @ X5 @ A4 )
=> ( ( cosh @ A @ ( F2 @ X5 ) )
!= ( zero_zero @ A ) ) )
=> ( topolo81223032696312382ous_on @ C @ A @ A4
@ ^ [X4: C] : ( tanh @ A @ ( F2 @ X4 ) ) ) ) ) ) ).
% continuous_on_tanh
thf(fact_5084_surjective__iff__injective__gen,axiom,
! [B: $tType,A: $tType,S2: set @ A,T4: set @ B,F2: A > B] :
( ( finite_finite2 @ A @ S2 )
=> ( ( finite_finite2 @ B @ T4 )
=> ( ( ( finite_card @ A @ S2 )
= ( finite_card @ B @ T4 ) )
=> ( ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F2 @ S2 ) @ T4 )
=> ( ( ! [X4: B] :
( ( member @ B @ X4 @ T4 )
=> ? [Y6: A] :
( ( member @ A @ Y6 @ S2 )
& ( ( F2 @ Y6 )
= X4 ) ) ) )
= ( inj_on @ A @ B @ F2 @ S2 ) ) ) ) ) ) ).
% surjective_iff_injective_gen
thf(fact_5085_card__bij__eq,axiom,
! [A: $tType,B: $tType,F2: A > B,A4: set @ A,B2: set @ B,G: B > A] :
( ( inj_on @ A @ B @ F2 @ A4 )
=> ( ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F2 @ A4 ) @ B2 )
=> ( ( inj_on @ B @ A @ G @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image2 @ B @ A @ G @ B2 ) @ A4 )
=> ( ( finite_finite2 @ A @ A4 )
=> ( ( finite_finite2 @ B @ B2 )
=> ( ( finite_card @ A @ A4 )
= ( finite_card @ B @ B2 ) ) ) ) ) ) ) ) ).
% card_bij_eq
thf(fact_5086_inj__image__Compl__subset,axiom,
! [B: $tType,A: $tType,F2: A > B,A4: set @ A] :
( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
=> ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F2 @ ( uminus_uminus @ ( set @ A ) @ A4 ) ) @ ( uminus_uminus @ ( set @ B ) @ ( image2 @ A @ B @ F2 @ A4 ) ) ) ) ).
% inj_image_Compl_subset
thf(fact_5087_continuous__on__closed__Union,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( ( topolo4958980785337419405_space @ B )
& ( topolo4958980785337419405_space @ C ) )
=> ! [I6: set @ A,U3: A > ( set @ B ),F2: B > C] :
( ( finite_finite2 @ A @ I6 )
=> ( ! [I4: A] :
( ( member @ A @ I4 @ I6 )
=> ( topolo7761053866217962861closed @ B @ ( U3 @ I4 ) ) )
=> ( ! [I4: A] :
( ( member @ A @ I4 @ I6 )
=> ( topolo81223032696312382ous_on @ B @ C @ ( U3 @ I4 ) @ F2 ) )
=> ( topolo81223032696312382ous_on @ B @ C @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ U3 @ I6 ) ) @ F2 ) ) ) ) ) ).
% continuous_on_closed_Union
thf(fact_5088_image__INT,axiom,
! [B: $tType,A: $tType,C: $tType,F2: A > B,C5: set @ A,A4: set @ C,B2: C > ( set @ A ),J: C] :
( ( inj_on @ A @ B @ F2 @ C5 )
=> ( ! [X5: C] :
( ( member @ C @ X5 @ A4 )
=> ( ord_less_eq @ ( set @ A ) @ ( B2 @ X5 ) @ C5 ) )
=> ( ( member @ C @ J @ A4 )
=> ( ( image2 @ A @ B @ F2 @ ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ C @ ( set @ A ) @ B2 @ A4 ) ) )
= ( complete_Inf_Inf @ ( set @ B )
@ ( image2 @ C @ ( set @ B )
@ ^ [X4: C] : ( image2 @ A @ B @ F2 @ ( B2 @ X4 ) )
@ A4 ) ) ) ) ) ) ).
% image_INT
thf(fact_5089_DERIV__atLeastAtMost__imp__continuous__on,axiom,
! [A: $tType] :
( ( ( ord @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [A3: A,B3: A,F2: A > A] :
( ! [X5: A] :
( ( ord_less_eq @ A @ A3 @ X5 )
=> ( ( ord_less_eq @ A @ X5 @ B3 )
=> ? [Y4: A] : ( has_field_derivative @ A @ F2 @ Y4 @ ( topolo174197925503356063within @ A @ X5 @ ( top_top @ ( set @ A ) ) ) ) ) )
=> ( topolo81223032696312382ous_on @ A @ A @ ( set_or1337092689740270186AtMost @ A @ A3 @ B3 ) @ F2 ) ) ) ).
% DERIV_atLeastAtMost_imp_continuous_on
thf(fact_5090_inj__on__iff__card__le,axiom,
! [A: $tType,B: $tType,A4: set @ A,B2: set @ B] :
( ( finite_finite2 @ A @ A4 )
=> ( ( finite_finite2 @ B @ B2 )
=> ( ( ? [F3: A > B] :
( ( inj_on @ A @ B @ F3 @ A4 )
& ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F3 @ A4 ) @ B2 ) ) )
= ( ord_less_eq @ nat @ ( finite_card @ A @ A4 ) @ ( finite_card @ B @ B2 ) ) ) ) ) ).
% inj_on_iff_card_le
thf(fact_5091_card__inj__on__le,axiom,
! [A: $tType,B: $tType,F2: A > B,A4: set @ A,B2: set @ B] :
( ( inj_on @ A @ B @ F2 @ A4 )
=> ( ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F2 @ A4 ) @ B2 )
=> ( ( finite_finite2 @ B @ B2 )
=> ( ord_less_eq @ nat @ ( finite_card @ A @ A4 ) @ ( finite_card @ B @ B2 ) ) ) ) ) ).
% card_inj_on_le
thf(fact_5092_card__le__inj,axiom,
! [B: $tType,A: $tType,A4: set @ A,B2: set @ B] :
( ( finite_finite2 @ A @ A4 )
=> ( ( finite_finite2 @ B @ B2 )
=> ( ( ord_less_eq @ nat @ ( finite_card @ A @ A4 ) @ ( finite_card @ B @ B2 ) )
=> ? [F4: A > B] :
( ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F4 @ A4 ) @ B2 )
& ( inj_on @ A @ B @ F4 @ A4 ) ) ) ) ) ).
% card_le_inj
thf(fact_5093_continuous__on__Icc__at__leftD,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo1944317154257567458pology @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [A3: A,B3: A,F2: A > B] :
( ( topolo81223032696312382ous_on @ A @ B @ ( set_or1337092689740270186AtMost @ A @ A3 @ B3 ) @ F2 )
=> ( ( ord_less @ A @ A3 @ B3 )
=> ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ ( F2 @ B3 ) ) @ ( topolo174197925503356063within @ A @ B3 @ ( set_ord_lessThan @ A @ B3 ) ) ) ) ) ) ).
% continuous_on_Icc_at_leftD
thf(fact_5094_continuous__on__Icc__at__rightD,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo1944317154257567458pology @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [A3: A,B3: A,F2: A > B] :
( ( topolo81223032696312382ous_on @ A @ B @ ( set_or1337092689740270186AtMost @ A @ A3 @ B3 ) @ F2 )
=> ( ( ord_less @ A @ A3 @ B3 )
=> ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ ( F2 @ A3 ) ) @ ( topolo174197925503356063within @ A @ A3 @ ( set_ord_greaterThan @ A @ A3 ) ) ) ) ) ) ).
% continuous_on_Icc_at_rightD
thf(fact_5095_ex__subset__image__inj,axiom,
! [A: $tType,B: $tType,F2: B > A,S2: set @ B,P: ( set @ A ) > $o] :
( ( ? [T10: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ T10 @ ( image2 @ B @ A @ F2 @ S2 ) )
& ( P @ T10 ) ) )
= ( ? [T10: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ T10 @ S2 )
& ( inj_on @ B @ A @ F2 @ T10 )
& ( P @ ( image2 @ B @ A @ F2 @ T10 ) ) ) ) ) ).
% ex_subset_image_inj
thf(fact_5096_all__subset__image__inj,axiom,
! [A: $tType,B: $tType,F2: B > A,S2: set @ B,P: ( set @ A ) > $o] :
( ( ! [T10: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ T10 @ ( image2 @ B @ A @ F2 @ S2 ) )
=> ( P @ T10 ) ) )
= ( ! [T10: set @ B] :
( ( ( ord_less_eq @ ( set @ B ) @ T10 @ S2 )
& ( inj_on @ B @ A @ F2 @ T10 ) )
=> ( P @ ( image2 @ B @ A @ F2 @ T10 ) ) ) ) ) ).
% all_subset_image_inj
thf(fact_5097_set__list__bind,axiom,
! [A: $tType,B: $tType,Xs: list @ B,F2: B > ( list @ A )] :
( ( set2 @ A @ ( bind @ B @ A @ Xs @ F2 ) )
= ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [X4: B] : ( set2 @ A @ ( F2 @ X4 ) )
@ ( set2 @ B @ Xs ) ) ) ) ).
% set_list_bind
thf(fact_5098_bind__simps_I1_J,axiom,
! [B: $tType,A: $tType,F2: B > ( list @ A )] :
( ( bind @ B @ A @ ( nil @ B ) @ F2 )
= ( nil @ A ) ) ).
% bind_simps(1)
thf(fact_5099_inj__Suc,axiom,
! [N6: set @ nat] : ( inj_on @ nat @ nat @ suc @ N6 ) ).
% inj_Suc
thf(fact_5100_inj__on__of__nat,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [N6: set @ nat] : ( inj_on @ nat @ A @ ( semiring_1_of_nat @ A ) @ N6 ) ) ).
% inj_on_of_nat
thf(fact_5101_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_5102_inj__singleton,axiom,
! [A: $tType,A4: set @ A] :
( inj_on @ A @ ( set @ A )
@ ^ [X4: A] : ( insert2 @ A @ X4 @ ( bot_bot @ ( set @ A ) ) )
@ A4 ) ).
% inj_singleton
thf(fact_5103_inj__on__diff__nat,axiom,
! [N6: set @ nat,K: nat] :
( ! [N3: nat] :
( ( member @ nat @ N3 @ N6 )
=> ( ord_less_eq @ nat @ K @ N3 ) )
=> ( inj_on @ nat @ nat
@ ^ [N: nat] : ( minus_minus @ nat @ N @ K )
@ N6 ) ) ).
% inj_on_diff_nat
thf(fact_5104_inj__on__set__encode,axiom,
inj_on @ ( set @ nat ) @ nat @ nat_set_encode @ ( collect @ ( set @ nat ) @ ( finite_finite2 @ nat ) ) ).
% inj_on_set_encode
thf(fact_5105_list__bind__cong,axiom,
! [B: $tType,A: $tType,Xs: list @ A,Ys: list @ A,F2: A > ( list @ B ),G: A > ( list @ B )] :
( ( Xs = Ys )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Xs ) )
=> ( ( F2 @ X5 )
= ( G @ X5 ) ) )
=> ( ( bind @ A @ B @ Xs @ F2 )
= ( bind @ A @ B @ Ys @ G ) ) ) ) ).
% list_bind_cong
thf(fact_5106_inj__graph,axiom,
! [B: $tType,A: $tType] :
( inj_on @ ( A > B ) @ ( set @ ( product_prod @ A @ B ) )
@ ^ [F3: A > B] :
( collect @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [X4: A,Y6: B] :
( Y6
= ( F3 @ X4 ) ) ) )
@ ( top_top @ ( set @ ( A > B ) ) ) ) ).
% inj_graph
thf(fact_5107_range__inj__infinite,axiom,
! [A: $tType,F2: nat > A] :
( ( inj_on @ nat @ A @ F2 @ ( top_top @ ( set @ nat ) ) )
=> ~ ( finite_finite2 @ A @ ( image2 @ nat @ A @ F2 @ ( top_top @ ( set @ nat ) ) ) ) ) ).
% range_inj_infinite
thf(fact_5108_finite__imp__nat__seg__image__inj__on,axiom,
! [A: $tType,A4: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ? [N3: nat,F4: nat > A] :
( ( A4
= ( image2 @ nat @ A @ F4
@ ( collect @ nat
@ ^ [I3: nat] : ( ord_less @ nat @ I3 @ N3 ) ) ) )
& ( inj_on @ nat @ A @ F4
@ ( collect @ nat
@ ^ [I3: nat] : ( ord_less @ nat @ I3 @ N3 ) ) ) ) ) ).
% finite_imp_nat_seg_image_inj_on
thf(fact_5109_finite__imp__inj__to__nat__seg,axiom,
! [A: $tType,A4: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ? [F4: A > nat,N3: nat] :
( ( ( image2 @ A @ nat @ F4 @ A4 )
= ( collect @ nat
@ ^ [I3: nat] : ( ord_less @ nat @ I3 @ N3 ) ) )
& ( inj_on @ A @ nat @ F4 @ A4 ) ) ) ).
% finite_imp_inj_to_nat_seg
thf(fact_5110_inj__on__nth,axiom,
! [A: $tType,Xs: list @ A,I6: set @ nat] :
( ( distinct @ A @ Xs )
=> ( ! [X5: nat] :
( ( member @ nat @ X5 @ I6 )
=> ( ord_less @ nat @ X5 @ ( size_size @ ( list @ A ) @ Xs ) ) )
=> ( inj_on @ nat @ A @ ( nth @ A @ Xs ) @ I6 ) ) ) ).
% inj_on_nth
thf(fact_5111_infinite__countable__subset,axiom,
! [A: $tType,S2: set @ A] :
( ~ ( finite_finite2 @ A @ S2 )
=> ? [F4: nat > A] :
( ( inj_on @ nat @ A @ F4 @ ( top_top @ ( set @ nat ) ) )
& ( ord_less_eq @ ( set @ A ) @ ( image2 @ nat @ A @ F4 @ ( top_top @ ( set @ nat ) ) ) @ S2 ) ) ) ).
% infinite_countable_subset
thf(fact_5112_infinite__iff__countable__subset,axiom,
! [A: $tType,S2: set @ A] :
( ( ~ ( finite_finite2 @ A @ S2 ) )
= ( ? [F3: nat > A] :
( ( inj_on @ nat @ A @ F3 @ ( top_top @ ( set @ nat ) ) )
& ( ord_less_eq @ ( set @ A ) @ ( image2 @ nat @ A @ F3 @ ( top_top @ ( set @ nat ) ) ) @ S2 ) ) ) ) ).
% infinite_iff_countable_subset
thf(fact_5113_inj__on__funpow__least,axiom,
! [A: $tType,N2: nat,F2: A > A,S: A] :
( ( ( compow @ ( A > A ) @ N2 @ F2 @ S )
= S )
=> ( ! [M3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ M3 )
=> ( ( ord_less @ nat @ M3 @ N2 )
=> ( ( compow @ ( A > A ) @ M3 @ F2 @ S )
!= S ) ) )
=> ( inj_on @ nat @ A
@ ^ [K2: nat] : ( compow @ ( A > A ) @ K2 @ F2 @ S )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ) ).
% inj_on_funpow_least
thf(fact_5114_has__derivative__power__int_H,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [X: A,N2: int,S2: set @ A] :
( ( X
!= ( zero_zero @ A ) )
=> ( has_derivative @ A @ A
@ ^ [X4: A] : ( power_int @ A @ X4 @ N2 )
@ ^ [Y6: A] : ( times_times @ A @ Y6 @ ( times_times @ A @ ( ring_1_of_int @ A @ N2 ) @ ( power_int @ A @ X @ ( minus_minus @ int @ N2 @ ( one_one @ int ) ) ) ) )
@ ( topolo174197925503356063within @ A @ X @ S2 ) ) ) ) ).
% has_derivative_power_int'
thf(fact_5115_has__derivative__power__int,axiom,
! [A: $tType,C: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V3459762299906320749_field @ A ) )
=> ! [F2: C > A,X: C,F8: C > A,S2: set @ C,N2: int] :
( ( ( F2 @ X )
!= ( zero_zero @ A ) )
=> ( ( has_derivative @ C @ A @ F2 @ F8 @ ( topolo174197925503356063within @ C @ X @ S2 ) )
=> ( has_derivative @ C @ A
@ ^ [X4: C] : ( power_int @ A @ ( F2 @ X4 ) @ N2 )
@ ^ [H: C] : ( times_times @ A @ ( F8 @ H ) @ ( times_times @ A @ ( ring_1_of_int @ A @ N2 ) @ ( power_int @ A @ ( F2 @ X ) @ ( minus_minus @ int @ N2 @ ( one_one @ int ) ) ) ) )
@ ( topolo174197925503356063within @ C @ X @ S2 ) ) ) ) ) ).
% has_derivative_power_int
thf(fact_5116_DERIV__power__int,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,D2: A,X: A,S: set @ A,N2: int] :
( ( has_field_derivative @ A @ F2 @ D2 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( ( ( F2 @ X )
!= ( zero_zero @ A ) )
=> ( has_field_derivative @ A
@ ^ [X4: A] : ( power_int @ A @ ( F2 @ X4 ) @ N2 )
@ ( times_times @ A @ ( times_times @ A @ ( ring_1_of_int @ A @ N2 ) @ ( power_int @ A @ ( F2 @ X ) @ ( minus_minus @ int @ N2 @ ( one_one @ int ) ) ) ) @ D2 )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ).
% DERIV_power_int
thf(fact_5117_power__int__1__left,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [N2: int] :
( ( power_int @ A @ ( one_one @ A ) @ N2 )
= ( one_one @ A ) ) ) ).
% power_int_1_left
thf(fact_5118_power__int__mult__numeral,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X: A,M2: num,N2: num] :
( ( power_int @ A @ ( power_int @ A @ X @ ( numeral_numeral @ int @ M2 ) ) @ ( numeral_numeral @ int @ N2 ) )
= ( power_int @ A @ X @ ( numeral_numeral @ int @ ( times_times @ num @ M2 @ N2 ) ) ) ) ) ).
% power_int_mult_numeral
thf(fact_5119_power__int__1__right,axiom,
! [A: $tType] :
( ( ( inverse @ A )
& ( monoid_mult @ A ) )
=> ! [Y: A] :
( ( power_int @ A @ Y @ ( one_one @ int ) )
= Y ) ) ).
% power_int_1_right
thf(fact_5120_power__int__sgn,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,N2: int] :
( ( sgn_sgn @ A @ ( power_int @ A @ A3 @ N2 ) )
= ( power_int @ A @ ( sgn_sgn @ A @ A3 ) @ N2 ) ) ) ).
% power_int_sgn
thf(fact_5121_power__int__mult__distrib__numeral1,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [W2: num,Y: A,M2: int] :
( ( power_int @ A @ ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ Y ) @ M2 )
= ( times_times @ A @ ( power_int @ A @ ( numeral_numeral @ A @ W2 ) @ M2 ) @ ( power_int @ A @ Y @ M2 ) ) ) ) ).
% power_int_mult_distrib_numeral1
thf(fact_5122_power__int__mult__distrib__numeral2,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [X: A,W2: num,M2: int] :
( ( power_int @ A @ ( times_times @ A @ X @ ( numeral_numeral @ A @ W2 ) ) @ M2 )
= ( times_times @ A @ ( power_int @ A @ X @ M2 ) @ ( power_int @ A @ ( numeral_numeral @ A @ W2 ) @ M2 ) ) ) ) ).
% power_int_mult_distrib_numeral2
thf(fact_5123_power__int__0__left,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [M2: int] :
( ( M2
!= ( zero_zero @ int ) )
=> ( ( power_int @ A @ ( zero_zero @ A ) @ M2 )
= ( zero_zero @ A ) ) ) ) ).
% power_int_0_left
thf(fact_5124_power__int__eq__0__iff,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X: A,N2: int] :
( ( ( power_int @ A @ X @ N2 )
= ( zero_zero @ A ) )
= ( ( X
= ( zero_zero @ A ) )
& ( N2
!= ( zero_zero @ int ) ) ) ) ) ).
% power_int_eq_0_iff
thf(fact_5125_power__int__0__right,axiom,
! [B: $tType] :
( ( ( inverse @ B )
& ( power @ B ) )
=> ! [X: B] :
( ( power_int @ B @ X @ ( zero_zero @ int ) )
= ( one_one @ B ) ) ) ).
% power_int_0_right
thf(fact_5126_abs__power__int__minus,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,N2: int] :
( ( abs_abs @ A @ ( power_int @ A @ ( uminus_uminus @ A @ A3 ) @ N2 ) )
= ( abs_abs @ A @ ( power_int @ A @ A3 @ N2 ) ) ) ) ).
% abs_power_int_minus
thf(fact_5127_power__int__of__nat,axiom,
! [A: $tType] :
( ( ( inverse @ A )
& ( power @ A ) )
=> ! [X: A,N2: nat] :
( ( power_int @ A @ X @ ( semiring_1_of_nat @ int @ N2 ) )
= ( power_power @ A @ X @ N2 ) ) ) ).
% power_int_of_nat
thf(fact_5128_power__int__numeral,axiom,
! [A: $tType] :
( ( ( inverse @ A )
& ( power @ A ) )
=> ! [X: A,N2: num] :
( ( power_int @ A @ X @ ( numeral_numeral @ int @ N2 ) )
= ( power_power @ A @ X @ ( numeral_numeral @ nat @ N2 ) ) ) ) ).
% power_int_numeral
thf(fact_5129_power__int__minus1__right,axiom,
! [A: $tType] :
( ( ( inverse @ A )
& ( monoid_mult @ A ) )
=> ! [Y: A] :
( ( power_int @ A @ Y @ ( uminus_uminus @ int @ ( one_one @ int ) ) )
= ( inverse_inverse @ A @ Y ) ) ) ).
% power_int_minus1_right
thf(fact_5130_power__int__add__numeral,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X: A,M2: num,N2: num] :
( ( times_times @ A @ ( power_int @ A @ X @ ( numeral_numeral @ int @ M2 ) ) @ ( power_int @ A @ X @ ( numeral_numeral @ int @ N2 ) ) )
= ( power_int @ A @ X @ ( numeral_numeral @ int @ ( plus_plus @ num @ M2 @ N2 ) ) ) ) ) ).
% power_int_add_numeral
thf(fact_5131_power__int__add__numeral2,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X: A,M2: num,N2: num,B3: A] :
( ( times_times @ A @ ( power_int @ A @ X @ ( numeral_numeral @ int @ M2 ) ) @ ( times_times @ A @ ( power_int @ A @ X @ ( numeral_numeral @ int @ N2 ) ) @ B3 ) )
= ( times_times @ A @ ( power_int @ A @ X @ ( numeral_numeral @ int @ ( plus_plus @ num @ M2 @ N2 ) ) ) @ B3 ) ) ) ).
% power_int_add_numeral2
thf(fact_5132_power__int__mono__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B3: A,N2: int] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B3 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ N2 )
=> ( ( ord_less_eq @ A @ ( power_int @ A @ A3 @ N2 ) @ ( power_int @ A @ B3 @ N2 ) )
= ( ord_less_eq @ A @ A3 @ B3 ) ) ) ) ) ) ).
% power_int_mono_iff
thf(fact_5133_zero__le__power__int,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,N2: int] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_int @ A @ X @ N2 ) ) ) ) ).
% zero_le_power_int
thf(fact_5134_zero__less__power__int,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,N2: int] :
( ( ord_less @ A @ ( zero_zero @ A ) @ X )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( power_int @ A @ X @ N2 ) ) ) ) ).
% zero_less_power_int
thf(fact_5135_power__int__inverse,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X: A,N2: int] :
( ( power_int @ A @ ( inverse_inverse @ A @ X ) @ N2 )
= ( inverse_inverse @ A @ ( power_int @ A @ X @ N2 ) ) ) ) ).
% power_int_inverse
thf(fact_5136_power__int__divide__distrib,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [X: A,Y: A,M2: int] :
( ( power_int @ A @ ( divide_divide @ A @ X @ Y ) @ M2 )
= ( divide_divide @ A @ ( power_int @ A @ X @ M2 ) @ ( power_int @ A @ Y @ M2 ) ) ) ) ).
% power_int_divide_distrib
thf(fact_5137_power__int__commutes,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X: A,N2: int] :
( ( times_times @ A @ ( power_int @ A @ X @ N2 ) @ X )
= ( times_times @ A @ X @ ( power_int @ A @ X @ N2 ) ) ) ) ).
% power_int_commutes
thf(fact_5138_power__int__mult__distrib,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [X: A,Y: A,M2: int] :
( ( power_int @ A @ ( times_times @ A @ X @ Y ) @ M2 )
= ( times_times @ A @ ( power_int @ A @ X @ M2 ) @ ( power_int @ A @ Y @ M2 ) ) ) ) ).
% power_int_mult_distrib
thf(fact_5139_power__int__mult,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X: A,M2: int,N2: int] :
( ( power_int @ A @ X @ ( times_times @ int @ M2 @ N2 ) )
= ( power_int @ A @ ( power_int @ A @ X @ M2 ) @ N2 ) ) ) ).
% power_int_mult
thf(fact_5140_power__int__abs,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,N2: int] :
( ( abs_abs @ A @ ( power_int @ A @ A3 @ N2 ) )
= ( power_int @ A @ ( abs_abs @ A @ A3 ) @ N2 ) ) ) ).
% power_int_abs
thf(fact_5141_power__int__one__over,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X: A,N2: int] :
( ( power_int @ A @ ( divide_divide @ A @ ( one_one @ A ) @ X ) @ N2 )
= ( divide_divide @ A @ ( one_one @ A ) @ ( power_int @ A @ X @ N2 ) ) ) ) ).
% power_int_one_over
thf(fact_5142_power__int__not__zero,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X: A,N2: int] :
( ( ( X
!= ( zero_zero @ A ) )
| ( N2
= ( zero_zero @ int ) ) )
=> ( ( power_int @ A @ X @ N2 )
!= ( zero_zero @ A ) ) ) ) ).
% power_int_not_zero
thf(fact_5143_power__int__minus,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X: A,N2: int] :
( ( power_int @ A @ X @ ( uminus_uminus @ int @ N2 ) )
= ( inverse_inverse @ A @ ( power_int @ A @ X @ N2 ) ) ) ) ).
% power_int_minus
thf(fact_5144_continuous__on__power__int,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo4958980785337419405_space @ A )
& ( real_V8999393235501362500lgebra @ B ) )
=> ! [S: set @ A,F2: A > B,N2: int] :
( ( topolo81223032696312382ous_on @ A @ B @ S @ F2 )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ S )
=> ( ( F2 @ X5 )
!= ( zero_zero @ B ) ) )
=> ( topolo81223032696312382ous_on @ A @ B @ S
@ ^ [X4: A] : ( power_int @ B @ ( F2 @ X4 ) @ N2 ) ) ) ) ) ).
% continuous_on_power_int
thf(fact_5145_power__int__0__left__If,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [M2: int] :
( ( ( M2
= ( zero_zero @ int ) )
=> ( ( power_int @ A @ ( zero_zero @ A ) @ M2 )
= ( one_one @ A ) ) )
& ( ( M2
!= ( zero_zero @ int ) )
=> ( ( power_int @ A @ ( zero_zero @ A ) @ M2 )
= ( zero_zero @ A ) ) ) ) ) ).
% power_int_0_left_If
thf(fact_5146_power__int__increasing,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [N2: int,N6: int,A3: A] :
( ( ord_less_eq @ int @ N2 @ N6 )
=> ( ( ord_less_eq @ A @ ( one_one @ A ) @ A3 )
=> ( ord_less_eq @ A @ ( power_int @ A @ A3 @ N2 ) @ ( power_int @ A @ A3 @ N6 ) ) ) ) ) ).
% power_int_increasing
thf(fact_5147_power__int__strict__increasing,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [N2: int,N6: int,A3: A] :
( ( ord_less @ int @ N2 @ N6 )
=> ( ( ord_less @ A @ ( one_one @ A ) @ A3 )
=> ( ord_less @ A @ ( power_int @ A @ A3 @ N2 ) @ ( power_int @ A @ A3 @ N6 ) ) ) ) ) ).
% power_int_strict_increasing
thf(fact_5148_power__int__diff,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [X: A,M2: int,N2: int] :
( ( ( X
!= ( zero_zero @ A ) )
| ( M2 != N2 ) )
=> ( ( power_int @ A @ X @ ( minus_minus @ int @ M2 @ N2 ) )
= ( divide_divide @ A @ ( power_int @ A @ X @ M2 ) @ ( power_int @ A @ X @ N2 ) ) ) ) ) ).
% power_int_diff
thf(fact_5149_tendsto__power__int,axiom,
! [A: $tType,B: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [F2: B > A,A3: A,F5: filter @ B,N2: int] :
( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ A3 ) @ F5 )
=> ( ( A3
!= ( zero_zero @ A ) )
=> ( filterlim @ B @ A
@ ^ [X4: B] : ( power_int @ A @ ( F2 @ X4 ) @ N2 )
@ ( topolo7230453075368039082e_nhds @ A @ ( power_int @ A @ A3 @ N2 ) )
@ F5 ) ) ) ) ).
% tendsto_power_int
thf(fact_5150_power__int__strict__decreasing,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [N2: int,N6: int,A3: A] :
( ( ord_less @ int @ N2 @ N6 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ A3 @ ( one_one @ A ) )
=> ( ord_less @ A @ ( power_int @ A @ A3 @ N6 ) @ ( power_int @ A @ A3 @ N2 ) ) ) ) ) ) ).
% power_int_strict_decreasing
thf(fact_5151_power__int__mono,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y: A,N2: int] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ N2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ord_less_eq @ A @ ( power_int @ A @ X @ N2 ) @ ( power_int @ A @ Y @ N2 ) ) ) ) ) ) ).
% power_int_mono
thf(fact_5152_power__int__strict__antimono,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B3: A,N2: int] :
( ( ord_less @ A @ A3 @ B3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ int @ N2 @ ( zero_zero @ int ) )
=> ( ord_less @ A @ ( power_int @ A @ B3 @ N2 ) @ ( power_int @ A @ A3 @ N2 ) ) ) ) ) ) ).
% power_int_strict_antimono
thf(fact_5153_one__le__power__int,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,N2: int] :
( ( ord_less_eq @ A @ ( one_one @ A ) @ X )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ N2 )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ ( power_int @ A @ X @ N2 ) ) ) ) ) ).
% one_le_power_int
thf(fact_5154_one__less__power__int,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,N2: int] :
( ( ord_less @ A @ ( one_one @ A ) @ A3 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ N2 )
=> ( ord_less @ A @ ( one_one @ A ) @ ( power_int @ A @ A3 @ N2 ) ) ) ) ) ).
% one_less_power_int
thf(fact_5155_power__int__add,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X: A,M2: int,N2: int] :
( ( ( X
!= ( zero_zero @ A ) )
| ( ( plus_plus @ int @ M2 @ N2 )
!= ( zero_zero @ int ) ) )
=> ( ( power_int @ A @ X @ ( plus_plus @ int @ M2 @ N2 ) )
= ( times_times @ A @ ( power_int @ A @ X @ M2 ) @ ( power_int @ A @ X @ N2 ) ) ) ) ) ).
% power_int_add
thf(fact_5156_power__int__antimono,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B3: A,N2: int] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ int @ N2 @ ( zero_zero @ int ) )
=> ( ord_less_eq @ A @ ( power_int @ A @ B3 @ N2 ) @ ( power_int @ A @ A3 @ N2 ) ) ) ) ) ) ).
% power_int_antimono
thf(fact_5157_power__int__strict__mono,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B3: A,N2: int] :
( ( ord_less @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ N2 )
=> ( ord_less @ A @ ( power_int @ A @ A3 @ N2 ) @ ( power_int @ A @ B3 @ N2 ) ) ) ) ) ) ).
% power_int_strict_mono
thf(fact_5158_power__int__decreasing,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [N2: int,N6: int,A3: A] :
( ( ord_less_eq @ int @ N2 @ N6 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ A3 @ ( one_one @ A ) )
=> ( ( ( A3
!= ( zero_zero @ A ) )
| ( N6
!= ( zero_zero @ int ) )
| ( N2
= ( zero_zero @ int ) ) )
=> ( ord_less_eq @ A @ ( power_int @ A @ A3 @ N6 ) @ ( power_int @ A @ A3 @ N2 ) ) ) ) ) ) ) ).
% power_int_decreasing
thf(fact_5159_power__int__le__one,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,N2: int] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ N2 )
=> ( ( ord_less_eq @ A @ X @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( power_int @ A @ X @ N2 ) @ ( one_one @ A ) ) ) ) ) ) ).
% power_int_le_one
thf(fact_5160_power__int__le__imp__le__exp,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,M2: int,N2: int] :
( ( ord_less @ A @ ( one_one @ A ) @ X )
=> ( ( ord_less_eq @ A @ ( power_int @ A @ X @ M2 ) @ ( power_int @ A @ X @ N2 ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ N2 )
=> ( ord_less_eq @ int @ M2 @ N2 ) ) ) ) ) ).
% power_int_le_imp_le_exp
thf(fact_5161_power__int__le__imp__less__exp,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,M2: int,N2: int] :
( ( ord_less @ A @ ( one_one @ A ) @ X )
=> ( ( ord_less @ A @ ( power_int @ A @ X @ M2 ) @ ( power_int @ A @ X @ N2 ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ N2 )
=> ( ord_less @ int @ M2 @ N2 ) ) ) ) ) ).
% power_int_le_imp_less_exp
thf(fact_5162_power__int__minus__mult,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [X: A,N2: int] :
( ( ( X
!= ( zero_zero @ A ) )
| ( N2
!= ( zero_zero @ int ) ) )
=> ( ( times_times @ A @ ( power_int @ A @ X @ ( minus_minus @ int @ N2 @ ( one_one @ int ) ) ) @ X )
= ( power_int @ A @ X @ N2 ) ) ) ) ).
% power_int_minus_mult
thf(fact_5163_power__int__add__1_H,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X: A,M2: int] :
( ( ( X
!= ( zero_zero @ A ) )
| ( M2
!= ( uminus_uminus @ int @ ( one_one @ int ) ) ) )
=> ( ( power_int @ A @ X @ ( plus_plus @ int @ M2 @ ( one_one @ int ) ) )
= ( times_times @ A @ X @ ( power_int @ A @ X @ M2 ) ) ) ) ) ).
% power_int_add_1'
thf(fact_5164_power__int__add__1,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X: A,M2: int] :
( ( ( X
!= ( zero_zero @ A ) )
| ( M2
!= ( uminus_uminus @ int @ ( one_one @ int ) ) ) )
=> ( ( power_int @ A @ X @ ( plus_plus @ int @ M2 @ ( one_one @ int ) ) )
= ( times_times @ A @ ( power_int @ A @ X @ M2 ) @ X ) ) ) ) ).
% power_int_add_1
thf(fact_5165_power__int__def,axiom,
! [A: $tType] :
( ( ( inverse @ A )
& ( power @ A ) )
=> ( ( power_int @ A )
= ( ^ [X4: A,N: int] : ( if @ A @ ( ord_less_eq @ int @ ( zero_zero @ int ) @ N ) @ ( power_power @ A @ X4 @ ( nat2 @ N ) ) @ ( power_power @ A @ ( inverse_inverse @ A @ X4 ) @ ( nat2 @ ( uminus_uminus @ int @ N ) ) ) ) ) ) ) ).
% power_int_def
thf(fact_5166_power__int__numeral__neg__numeral,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [M2: num,N2: num] :
( ( power_int @ A @ ( numeral_numeral @ A @ M2 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N2 ) ) )
= ( inverse_inverse @ A @ ( numeral_numeral @ A @ ( pow @ M2 @ N2 ) ) ) ) ) ).
% power_int_numeral_neg_numeral
thf(fact_5167_pred__nat__def,axiom,
( pred_nat
= ( collect @ ( product_prod @ nat @ nat )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [M: nat,N: nat] :
( N
= ( suc @ M ) ) ) ) ) ).
% pred_nat_def
thf(fact_5168_eventually__less__ceiling,axiom,
! [B: $tType,A: $tType] :
( ( ( archim2362893244070406136eiling @ B )
& ( topolo2564578578187576103pology @ B ) )
=> ! [F2: A > B,L: B,F5: filter @ A] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L ) @ F5 )
=> ( ~ ( member @ B @ L @ ( ring_1_Ints @ B ) )
=> ( eventually @ A
@ ^ [X4: A] : ( ord_less @ B @ ( F2 @ X4 ) @ ( ring_1_of_int @ B @ ( archimedean_ceiling @ B @ L ) ) )
@ F5 ) ) ) ) ).
% eventually_less_ceiling
thf(fact_5169_Ints__sum,axiom,
! [A: $tType,B: $tType] :
( ( ring_1 @ B )
=> ! [A4: set @ A,F2: A > B] :
( ! [X5: A] :
( ( member @ A @ X5 @ A4 )
=> ( member @ B @ ( F2 @ X5 ) @ ( ring_1_Ints @ B ) ) )
=> ( member @ B @ ( groups7311177749621191930dd_sum @ A @ B @ F2 @ A4 ) @ ( ring_1_Ints @ B ) ) ) ) ).
% Ints_sum
thf(fact_5170_Ints__prod,axiom,
! [A: $tType,B: $tType] :
( ( ( comm_monoid_mult @ B )
& ( ring_1 @ B ) )
=> ! [A4: set @ A,F2: A > B] :
( ! [X5: A] :
( ( member @ A @ X5 @ A4 )
=> ( member @ B @ ( F2 @ X5 ) @ ( ring_1_Ints @ B ) ) )
=> ( member @ B @ ( groups7121269368397514597t_prod @ A @ B @ F2 @ A4 ) @ ( ring_1_Ints @ B ) ) ) ) ).
% Ints_prod
thf(fact_5171_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_5172_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_5173_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_5174_Ints__power,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [A3: A,N2: nat] :
( ( member @ A @ A3 @ ( ring_1_Ints @ A ) )
=> ( member @ A @ ( power_power @ A @ A3 @ N2 ) @ ( ring_1_Ints @ A ) ) ) ) ).
% Ints_power
thf(fact_5175_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_5176_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_5177_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_5178_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_5179_Ints__1,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ( member @ A @ ( one_one @ A ) @ ( ring_1_Ints @ A ) ) ) ).
% Ints_1
thf(fact_5180_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_5181_Ints__numeral,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [N2: num] : ( member @ A @ ( numeral_numeral @ A @ N2 ) @ ( ring_1_Ints @ A ) ) ) ).
% Ints_numeral
thf(fact_5182_Ints__0,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ( member @ A @ ( zero_zero @ A ) @ ( ring_1_Ints @ A ) ) ) ).
% Ints_0
thf(fact_5183_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_5184_Ints__of__nat,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [N2: nat] : ( member @ A @ ( semiring_1_of_nat @ A @ N2 ) @ ( ring_1_Ints @ A ) ) ) ).
% Ints_of_nat
thf(fact_5185_Ints__of__int,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [Z3: int] : ( member @ A @ ( ring_1_of_int @ A @ Z3 ) @ ( ring_1_Ints @ A ) ) ) ).
% Ints_of_int
thf(fact_5186_Ints__induct,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [Q3: A,P: A > $o] :
( ( member @ A @ Q3 @ ( ring_1_Ints @ A ) )
=> ( ! [Z: int] : ( P @ ( ring_1_of_int @ A @ Z ) )
=> ( P @ Q3 ) ) ) ) ).
% Ints_induct
thf(fact_5187_Ints__cases,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [Q3: A] :
( ( member @ A @ Q3 @ ( ring_1_Ints @ A ) )
=> ~ ! [Z: int] :
( Q3
!= ( ring_1_of_int @ A @ Z ) ) ) ) ).
% Ints_cases
thf(fact_5188_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_5189_Nats__subset__Ints,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ( ord_less_eq @ ( set @ A ) @ ( semiring_1_Nats @ A ) @ ( ring_1_Ints @ A ) ) ) ).
% Nats_subset_Ints
thf(fact_5190_finite__int__segment,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [A3: A,B3: A] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X4: A] :
( ( member @ A @ X4 @ ( ring_1_Ints @ A ) )
& ( ord_less_eq @ A @ A3 @ X4 )
& ( ord_less_eq @ A @ X4 @ B3 ) ) ) ) ) ).
% finite_int_segment
thf(fact_5191_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_5192_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_5193_finite__abs__int__segment,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [A3: A] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [K2: A] :
( ( member @ A @ K2 @ ( ring_1_Ints @ A ) )
& ( ord_less_eq @ A @ ( abs_abs @ A @ K2 ) @ A3 ) ) ) ) ) ).
% finite_abs_int_segment
thf(fact_5194_Nats__altdef2,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( ( semiring_1_Nats @ A )
= ( collect @ A
@ ^ [N: A] :
( ( member @ A @ N @ ( ring_1_Ints @ A ) )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ N ) ) ) ) ) ).
% Nats_altdef2
thf(fact_5195_Ints__def,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ( ( ring_1_Ints @ A )
= ( image2 @ int @ A @ ( ring_1_of_int @ A ) @ ( top_top @ ( set @ int ) ) ) ) ) ).
% Ints_def
thf(fact_5196_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_5197_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_5198_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_5199_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_5200_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_5201_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_5202_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_5203_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_5204_eventually__floor__less,axiom,
! [B: $tType,A: $tType] :
( ( ( archim2362893244070406136eiling @ B )
& ( topolo2564578578187576103pology @ B ) )
=> ! [F2: A > B,L: B,F5: filter @ A] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L ) @ F5 )
=> ( ~ ( member @ B @ L @ ( ring_1_Ints @ B ) )
=> ( eventually @ A
@ ^ [X4: A] : ( ord_less @ B @ ( ring_1_of_int @ B @ ( archim6421214686448440834_floor @ B @ L ) ) @ ( F2 @ X4 ) )
@ F5 ) ) ) ) ).
% eventually_floor_less
thf(fact_5205_folding__idem__on_Ointro,axiom,
! [B: $tType,A: $tType,S2: set @ A,F2: A > B > B] :
( ( finite_folding_on @ A @ B @ S2 @ F2 )
=> ( ( finite6916993218817215295axioms @ A @ B @ S2 @ F2 )
=> ( finite1890593828518410140dem_on @ A @ B @ S2 @ F2 ) ) ) ).
% folding_idem_on.intro
thf(fact_5206_folding__idem__on__def,axiom,
! [B: $tType,A: $tType] :
( ( finite1890593828518410140dem_on @ A @ B )
= ( ^ [S6: set @ A,F3: A > B > B] :
( ( finite_folding_on @ A @ B @ S6 @ F3 )
& ( finite6916993218817215295axioms @ A @ B @ S6 @ F3 ) ) ) ) ).
% folding_idem_on_def
thf(fact_5207_lenlex__conv,axiom,
! [A: $tType] :
( ( lenlex @ A )
= ( ^ [R4: set @ ( product_prod @ A @ A )] :
( collect @ ( product_prod @ ( list @ A ) @ ( list @ A ) )
@ ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ $o
@ ^ [Xs3: list @ A,Ys3: list @ A] :
( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Xs3 ) @ ( size_size @ ( list @ A ) @ Ys3 ) )
| ( ( ( size_size @ ( list @ A ) @ Xs3 )
= ( size_size @ ( list @ A ) @ Ys3 ) )
& ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs3 @ Ys3 ) @ ( lex @ A @ R4 ) ) ) ) ) ) ) ) ).
% lenlex_conv
thf(fact_5208_Nil__lenlex__iff1,axiom,
! [A: $tType,Ns: list @ A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Ns ) @ ( lenlex @ A @ R2 ) )
= ( Ns
!= ( nil @ A ) ) ) ).
% Nil_lenlex_iff1
thf(fact_5209_folding__idem__on__axioms__def,axiom,
! [B: $tType,A: $tType] :
( ( finite6916993218817215295axioms @ A @ B )
= ( ^ [S6: set @ A,F3: A > B > B] :
! [X4: A,Y6: A] :
( ( member @ A @ X4 @ S6 )
=> ( ( member @ A @ Y6 @ S6 )
=> ( ( comp @ B @ B @ B @ ( F3 @ X4 ) @ ( F3 @ X4 ) )
= ( F3 @ X4 ) ) ) ) ) ) ).
% folding_idem_on_axioms_def
thf(fact_5210_folding__idem__on__axioms_Ointro,axiom,
! [B: $tType,A: $tType,S2: set @ A,F2: A > B > B] :
( ! [X5: A,Y3: A] :
( ( member @ A @ X5 @ S2 )
=> ( ( member @ A @ Y3 @ S2 )
=> ( ( comp @ B @ B @ B @ ( F2 @ X5 ) @ ( F2 @ X5 ) )
= ( F2 @ X5 ) ) ) )
=> ( finite6916993218817215295axioms @ A @ B @ S2 @ F2 ) ) ).
% folding_idem_on_axioms.intro
thf(fact_5211_lenlex__irreflexive,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),Xs: list @ A] :
( ! [X5: A] :
~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X5 @ X5 ) @ R2 )
=> ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Xs ) @ ( lenlex @ A @ R2 ) ) ) ).
% lenlex_irreflexive
thf(fact_5212_Nil__lenlex__iff2,axiom,
! [A: $tType,Ns: list @ A,R2: set @ ( product_prod @ A @ A )] :
~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ns @ ( nil @ A ) ) @ ( lenlex @ A @ R2 ) ) ).
% Nil_lenlex_iff2
thf(fact_5213_folding__idem__on_Oaxioms_I2_J,axiom,
! [B: $tType,A: $tType,S2: set @ A,F2: A > B > B] :
( ( finite1890593828518410140dem_on @ A @ B @ S2 @ F2 )
=> ( finite6916993218817215295axioms @ A @ B @ S2 @ F2 ) ) ).
% folding_idem_on.axioms(2)
thf(fact_5214_lenlex__length,axiom,
! [A: $tType,Ms: list @ A,Ns: list @ A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ms @ Ns ) @ ( lenlex @ A @ R2 ) )
=> ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Ms ) @ ( size_size @ ( list @ A ) @ Ns ) ) ) ).
% lenlex_length
thf(fact_5215_VEBT__internal_Ovalid_H_Oelims_I3_J,axiom,
! [X: vEBT_VEBT,Xa2: nat] :
( ~ ( vEBT_VEBT_valid @ X @ Xa2 )
=> ( ( ? [Uu2: $o,Uv2: $o] :
( X
= ( vEBT_Leaf @ Uu2 @ Uv2 ) )
=> ( Xa2
= ( one_one @ nat ) ) )
=> ~ ! [Mima: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList3: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) )
=> ( ( Deg2 = Xa2 )
& ! [X5: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X5 @ ( set2 @ vEBT_VEBT @ TreeList3 ) )
=> ( vEBT_VEBT_valid @ X5 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
& ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
& ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( case_option @ $o @ ( product_prod @ nat @ nat )
@ ( ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X6 )
& ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList3 ) )
=> ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X6 ) ) )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [Mi3: nat,Ma3: nat] :
( ( ord_less_eq @ nat @ Mi3 @ Ma3 )
& ( ord_less @ nat @ Ma3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
& ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
=> ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList3 @ I3 ) @ X6 ) )
= ( vEBT_V8194947554948674370ptions @ Summary2 @ I3 ) ) )
& ( ( Mi3 = Ma3 )
=> ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList3 ) )
=> ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X6 ) ) )
& ( ( Mi3 != Ma3 )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList3 @ Ma3 )
& ! [X4: nat] :
( ( ord_less @ nat @ X4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList3 @ X4 )
=> ( ( ord_less @ nat @ Mi3 @ X4 )
& ( ord_less_eq @ nat @ X4 @ Ma3 ) ) ) ) ) ) ) )
@ Mima ) ) ) ) ) ).
% VEBT_internal.valid'.elims(3)
thf(fact_5216_VEBT__internal_Ovalid_H_Oelims_I2_J,axiom,
! [X: vEBT_VEBT,Xa2: nat] :
( ( vEBT_VEBT_valid @ X @ Xa2 )
=> ( ( ? [Uu2: $o,Uv2: $o] :
( X
= ( vEBT_Leaf @ Uu2 @ Uv2 ) )
=> ( Xa2
!= ( one_one @ nat ) ) )
=> ~ ! [Mima: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList3: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) )
=> ~ ( ( Deg2 = Xa2 )
& ! [X2: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X2 @ ( set2 @ vEBT_VEBT @ TreeList3 ) )
=> ( vEBT_VEBT_valid @ X2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
& ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
& ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( case_option @ $o @ ( product_prod @ nat @ nat )
@ ( ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X6 )
& ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList3 ) )
=> ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X6 ) ) )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [Mi3: nat,Ma3: nat] :
( ( ord_less_eq @ nat @ Mi3 @ Ma3 )
& ( ord_less @ nat @ Ma3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
& ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
=> ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList3 @ I3 ) @ X6 ) )
= ( vEBT_V8194947554948674370ptions @ Summary2 @ I3 ) ) )
& ( ( Mi3 = Ma3 )
=> ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList3 ) )
=> ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X6 ) ) )
& ( ( Mi3 != Ma3 )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList3 @ Ma3 )
& ! [X4: nat] :
( ( ord_less @ nat @ X4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList3 @ X4 )
=> ( ( ord_less @ nat @ Mi3 @ X4 )
& ( ord_less_eq @ nat @ X4 @ Ma3 ) ) ) ) ) ) ) )
@ Mima ) ) ) ) ) ).
% VEBT_internal.valid'.elims(2)
thf(fact_5217_VEBT__internal_Ovalid_H_Oelims_I1_J,axiom,
! [X: vEBT_VEBT,Xa2: nat,Y: $o] :
( ( ( vEBT_VEBT_valid @ X @ Xa2 )
= Y )
=> ( ( ? [Uu2: $o,Uv2: $o] :
( X
= ( vEBT_Leaf @ Uu2 @ Uv2 ) )
=> ( Y
= ( Xa2
!= ( one_one @ nat ) ) ) )
=> ~ ! [Mima: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList3: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) )
=> ( Y
= ( ~ ( ( Deg2 = Xa2 )
& ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList3 ) )
=> ( vEBT_VEBT_valid @ X4 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
& ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
& ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( case_option @ $o @ ( product_prod @ nat @ nat )
@ ( ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X6 )
& ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList3 ) )
=> ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X6 ) ) )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [Mi3: nat,Ma3: nat] :
( ( ord_less_eq @ nat @ Mi3 @ Ma3 )
& ( ord_less @ nat @ Ma3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
& ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
=> ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList3 @ I3 ) @ X6 ) )
= ( vEBT_V8194947554948674370ptions @ Summary2 @ I3 ) ) )
& ( ( Mi3 = Ma3 )
=> ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList3 ) )
=> ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X6 ) ) )
& ( ( Mi3 != Ma3 )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList3 @ Ma3 )
& ! [X4: nat] :
( ( ord_less @ nat @ X4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList3 @ X4 )
=> ( ( ord_less @ nat @ Mi3 @ X4 )
& ( ord_less_eq @ nat @ X4 @ Ma3 ) ) ) ) ) ) ) )
@ Mima ) ) ) ) ) ) ) ).
% VEBT_internal.valid'.elims(1)
thf(fact_5218_ball__empty,axiom,
! [A: $tType,P: A > $o,X2: A] :
( ( member @ A @ X2 @ ( bot_bot @ ( set @ A ) ) )
=> ( P @ X2 ) ) ).
% ball_empty
thf(fact_5219_finite__Collect__bounded__ex,axiom,
! [B: $tType,A: $tType,P: A > $o,Q: B > A > $o] :
( ( finite_finite2 @ A @ ( collect @ A @ P ) )
=> ( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X4: B] :
? [Y6: A] :
( ( P @ Y6 )
& ( Q @ X4 @ Y6 ) ) ) )
= ( ! [Y6: A] :
( ( P @ Y6 )
=> ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X4: B] : ( Q @ X4 @ Y6 ) ) ) ) ) ) ) ).
% finite_Collect_bounded_ex
thf(fact_5220_ball__UNIV,axiom,
! [A: $tType,P: A > $o] :
( ( ! [X4: A] :
( ( member @ A @ X4 @ ( top_top @ ( set @ A ) ) )
=> ( P @ X4 ) ) )
= ( ! [X6: A] : ( P @ X6 ) ) ) ).
% ball_UNIV
thf(fact_5221_Setcompr__eq__image,axiom,
! [A: $tType,B: $tType,F2: B > A,A4: set @ B] :
( ( collect @ A
@ ^ [Uu3: A] :
? [X4: B] :
( ( Uu3
= ( F2 @ X4 ) )
& ( member @ B @ X4 @ A4 ) ) )
= ( image2 @ B @ A @ F2 @ A4 ) ) ).
% Setcompr_eq_image
thf(fact_5222_setcompr__eq__image,axiom,
! [A: $tType,B: $tType,F2: B > A,P: B > $o] :
( ( collect @ A
@ ^ [Uu3: A] :
? [X4: B] :
( ( Uu3
= ( F2 @ X4 ) )
& ( P @ X4 ) ) )
= ( image2 @ B @ A @ F2 @ ( collect @ B @ P ) ) ) ).
% setcompr_eq_image
thf(fact_5223_Ball__def,axiom,
! [A: $tType] :
( ( ball @ A )
= ( ^ [A7: set @ A,P3: A > $o] :
! [X4: A] :
( ( member @ A @ X4 @ A7 )
=> ( P3 @ X4 ) ) ) ) ).
% Ball_def
thf(fact_5224_finite__image__set,axiom,
! [A: $tType,B: $tType,P: A > $o,F2: A > B] :
( ( finite_finite2 @ A @ ( collect @ A @ P ) )
=> ( finite_finite2 @ B
@ ( collect @ B
@ ^ [Uu3: B] :
? [X4: A] :
( ( Uu3
= ( F2 @ X4 ) )
& ( P @ X4 ) ) ) ) ) ).
% finite_image_set
thf(fact_5225_finite__image__set2,axiom,
! [A: $tType,B: $tType,C: $tType,P: A > $o,Q: B > $o,F2: A > B > C] :
( ( finite_finite2 @ A @ ( collect @ A @ P ) )
=> ( ( finite_finite2 @ B @ ( collect @ B @ Q ) )
=> ( finite_finite2 @ C
@ ( collect @ C
@ ^ [Uu3: C] :
? [X4: A,Y6: B] :
( ( Uu3
= ( F2 @ X4 @ Y6 ) )
& ( P @ X4 )
& ( Q @ Y6 ) ) ) ) ) ) ).
% finite_image_set2
thf(fact_5226_finite_Omono,axiom,
! [A: $tType] :
( order_mono @ ( ( set @ A ) > $o ) @ ( ( set @ A ) > $o )
@ ^ [P6: ( set @ A ) > $o,X4: set @ A] :
( ( X4
= ( bot_bot @ ( set @ A ) ) )
| ? [A7: set @ A,A5: A] :
( ( X4
= ( insert2 @ A @ A5 @ A7 ) )
& ( P6 @ A7 ) ) ) ) ).
% finite.mono
thf(fact_5227_closed__subdiagonal,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ( topolo7761053866217962861closed @ ( product_prod @ A @ A )
@ ( collect @ ( product_prod @ A @ A )
@ ^ [Uu3: product_prod @ A @ A] :
? [X4: A,Y6: A] :
( ( Uu3
= ( product_Pair @ A @ A @ X4 @ Y6 ) )
& ( ord_less_eq @ A @ X4 @ Y6 ) ) ) ) ) ).
% closed_subdiagonal
thf(fact_5228_closed__superdiagonal,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ( topolo7761053866217962861closed @ ( product_prod @ A @ A )
@ ( collect @ ( product_prod @ A @ A )
@ ^ [Uu3: product_prod @ A @ A] :
? [X4: A,Y6: A] :
( ( Uu3
= ( product_Pair @ A @ A @ X4 @ Y6 ) )
& ( ord_less_eq @ A @ Y6 @ X4 ) ) ) ) ) ).
% closed_superdiagonal
thf(fact_5229_open__subdiagonal,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ( topolo1002775350975398744n_open @ ( product_prod @ A @ A )
@ ( collect @ ( product_prod @ A @ A )
@ ^ [Uu3: product_prod @ A @ A] :
? [X4: A,Y6: A] :
( ( Uu3
= ( product_Pair @ A @ A @ X4 @ Y6 ) )
& ( ord_less @ A @ X4 @ Y6 ) ) ) ) ) ).
% open_subdiagonal
thf(fact_5230_open__superdiagonal,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ( topolo1002775350975398744n_open @ ( product_prod @ A @ A )
@ ( collect @ ( product_prod @ A @ A )
@ ^ [Uu3: product_prod @ A @ A] :
? [X4: A,Y6: A] :
( ( Uu3
= ( product_Pair @ A @ A @ X4 @ Y6 ) )
& ( ord_less @ A @ Y6 @ X4 ) ) ) ) ) ).
% open_superdiagonal
thf(fact_5231_full__SetCompr__eq,axiom,
! [A: $tType,B: $tType,F2: B > A] :
( ( collect @ A
@ ^ [U2: A] :
? [X4: B] :
( U2
= ( F2 @ X4 ) ) )
= ( image2 @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) ) ) ).
% full_SetCompr_eq
thf(fact_5232_eventually__ball__finite,axiom,
! [A: $tType,B: $tType,A4: set @ A,P: B > A > $o,Net: filter @ B] :
( ( finite_finite2 @ A @ A4 )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ A4 )
=> ( eventually @ B
@ ^ [Y6: B] : ( P @ Y6 @ X5 )
@ Net ) )
=> ( eventually @ B
@ ^ [X4: B] :
! [Y6: A] :
( ( member @ A @ Y6 @ A4 )
=> ( P @ X4 @ Y6 ) )
@ Net ) ) ) ).
% eventually_ball_finite
thf(fact_5233_eventually__ball__finite__distrib,axiom,
! [B: $tType,A: $tType,A4: set @ A,P: B > A > $o,Net: filter @ B] :
( ( finite_finite2 @ A @ A4 )
=> ( ( eventually @ B
@ ^ [X4: B] :
! [Y6: A] :
( ( member @ A @ Y6 @ A4 )
=> ( P @ X4 @ Y6 ) )
@ Net )
= ( ! [X4: A] :
( ( member @ A @ X4 @ A4 )
=> ( eventually @ B
@ ^ [Y6: B] : ( P @ Y6 @ X4 )
@ Net ) ) ) ) ) ).
% eventually_ball_finite_distrib
thf(fact_5234_Sup__eq__Inf,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ( ( complete_Sup_Sup @ A )
= ( ^ [A7: set @ A] :
( complete_Inf_Inf @ A
@ ( collect @ A
@ ^ [B4: A] :
! [X4: A] :
( ( member @ A @ X4 @ A7 )
=> ( ord_less_eq @ A @ X4 @ B4 ) ) ) ) ) ) ) ).
% Sup_eq_Inf
thf(fact_5235_Inf__eq__Sup,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ( ( complete_Inf_Inf @ A )
= ( ^ [A7: set @ A] :
( complete_Sup_Sup @ A
@ ( collect @ A
@ ^ [B4: A] :
! [X4: A] :
( ( member @ A @ X4 @ A7 )
=> ( ord_less_eq @ A @ B4 @ X4 ) ) ) ) ) ) ) ).
% Inf_eq_Sup
thf(fact_5236_set__conv__nth,axiom,
! [A: $tType] :
( ( set2 @ A )
= ( ^ [Xs3: list @ A] :
( collect @ A
@ ^ [Uu3: A] :
? [I3: nat] :
( ( Uu3
= ( nth @ A @ Xs3 @ I3 ) )
& ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs3 ) ) ) ) ) ) ).
% set_conv_nth
thf(fact_5237_VEBT__internal_Ovalid_H_Osimps_I2_J,axiom,
! [Mima2: option @ ( product_prod @ nat @ nat ),Deg: nat,TreeList2: list @ vEBT_VEBT,Summary: vEBT_VEBT,Deg3: nat] :
( ( vEBT_VEBT_valid @ ( vEBT_Node @ Mima2 @ Deg @ TreeList2 @ Summary ) @ Deg3 )
= ( ( Deg = Deg3 )
& ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ( vEBT_VEBT_valid @ X4 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
& ( vEBT_VEBT_valid @ Summary @ ( minus_minus @ nat @ Deg @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
& ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( case_option @ $o @ ( product_prod @ nat @ nat )
@ ( ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X6 )
& ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X6 ) ) )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [Mi3: nat,Ma3: nat] :
( ( ord_less_eq @ nat @ Mi3 @ Ma3 )
& ( ord_less @ nat @ Ma3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg ) )
& ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
=> ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I3 ) @ X6 ) )
= ( vEBT_V8194947554948674370ptions @ Summary @ I3 ) ) )
& ( ( Mi3 = Ma3 )
=> ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X6 ) ) )
& ( ( Mi3 != Ma3 )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList2 @ Ma3 )
& ! [X4: nat] :
( ( ord_less @ nat @ X4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg ) )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList2 @ X4 )
=> ( ( ord_less @ nat @ Mi3 @ X4 )
& ( ord_less_eq @ nat @ X4 @ Ma3 ) ) ) ) ) ) ) )
@ Mima2 ) ) ) ).
% VEBT_internal.valid'.simps(2)
thf(fact_5238_funpow__inj__finite,axiom,
! [A: $tType,P5: A > A,X: A] :
( ( inj_on @ A @ A @ P5 @ ( top_top @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [Y6: A] :
? [N: nat] :
( Y6
= ( compow @ ( A > A ) @ N @ P5 @ X ) ) ) )
=> ~ ! [N3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( ( compow @ ( A > A ) @ N3 @ P5 @ X )
!= X ) ) ) ) ).
% funpow_inj_finite
thf(fact_5239_VEBT__internal_Ovalid_H_Opelims_I1_J,axiom,
! [X: vEBT_VEBT,Xa2: nat,Y: $o] :
( ( ( vEBT_VEBT_valid @ X @ Xa2 )
= Y )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_VEBT_valid_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa2 ) )
=> ( ! [Uu2: $o,Uv2: $o] :
( ( X
= ( vEBT_Leaf @ Uu2 @ Uv2 ) )
=> ( ( Y
= ( Xa2
= ( one_one @ nat ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_VEBT_valid_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa2 ) ) ) )
=> ~ ! [Mima: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList3: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) )
=> ( ( Y
= ( ( Deg2 = Xa2 )
& ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList3 ) )
=> ( vEBT_VEBT_valid @ X4 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
& ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
& ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( case_option @ $o @ ( product_prod @ nat @ nat )
@ ( ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X6 )
& ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList3 ) )
=> ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X6 ) ) )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [Mi3: nat,Ma3: nat] :
( ( ord_less_eq @ nat @ Mi3 @ Ma3 )
& ( ord_less @ nat @ Ma3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
& ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
=> ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList3 @ I3 ) @ X6 ) )
= ( vEBT_V8194947554948674370ptions @ Summary2 @ I3 ) ) )
& ( ( Mi3 = Ma3 )
=> ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList3 ) )
=> ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X6 ) ) )
& ( ( Mi3 != Ma3 )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList3 @ Ma3 )
& ! [X4: nat] :
( ( ord_less @ nat @ X4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList3 @ X4 )
=> ( ( ord_less @ nat @ Mi3 @ X4 )
& ( ord_less_eq @ nat @ X4 @ Ma3 ) ) ) ) ) ) ) )
@ Mima ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_VEBT_valid_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ).
% VEBT_internal.valid'.pelims(1)
thf(fact_5240_VEBT__internal_Ovalid_H_Opelims_I2_J,axiom,
! [X: vEBT_VEBT,Xa2: nat] :
( ( vEBT_VEBT_valid @ X @ Xa2 )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_VEBT_valid_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa2 ) )
=> ( ! [Uu2: $o,Uv2: $o] :
( ( X
= ( vEBT_Leaf @ Uu2 @ Uv2 ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_VEBT_valid_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa2 ) )
=> ( Xa2
!= ( one_one @ nat ) ) ) )
=> ~ ! [Mima: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList3: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_VEBT_valid_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) @ Xa2 ) )
=> ~ ( ( Deg2 = Xa2 )
& ! [X2: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X2 @ ( set2 @ vEBT_VEBT @ TreeList3 ) )
=> ( vEBT_VEBT_valid @ X2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
& ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
& ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( case_option @ $o @ ( product_prod @ nat @ nat )
@ ( ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X6 )
& ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList3 ) )
=> ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X6 ) ) )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [Mi3: nat,Ma3: nat] :
( ( ord_less_eq @ nat @ Mi3 @ Ma3 )
& ( ord_less @ nat @ Ma3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
& ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
=> ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList3 @ I3 ) @ X6 ) )
= ( vEBT_V8194947554948674370ptions @ Summary2 @ I3 ) ) )
& ( ( Mi3 = Ma3 )
=> ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList3 ) )
=> ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X6 ) ) )
& ( ( Mi3 != Ma3 )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList3 @ Ma3 )
& ! [X4: nat] :
( ( ord_less @ nat @ X4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList3 @ X4 )
=> ( ( ord_less @ nat @ Mi3 @ X4 )
& ( ord_less_eq @ nat @ X4 @ Ma3 ) ) ) ) ) ) ) )
@ Mima ) ) ) ) ) ) ) ).
% VEBT_internal.valid'.pelims(2)
thf(fact_5241_VEBT__internal_Ovalid_H_Opelims_I3_J,axiom,
! [X: vEBT_VEBT,Xa2: nat] :
( ~ ( vEBT_VEBT_valid @ X @ Xa2 )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_VEBT_valid_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa2 ) )
=> ( ! [Uu2: $o,Uv2: $o] :
( ( X
= ( vEBT_Leaf @ Uu2 @ Uv2 ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_VEBT_valid_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa2 ) )
=> ( Xa2
= ( one_one @ nat ) ) ) )
=> ~ ! [Mima: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList3: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_VEBT_valid_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) @ Xa2 ) )
=> ( ( Deg2 = Xa2 )
& ! [X5: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X5 @ ( set2 @ vEBT_VEBT @ TreeList3 ) )
=> ( vEBT_VEBT_valid @ X5 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
& ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
& ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( case_option @ $o @ ( product_prod @ nat @ nat )
@ ( ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X6 )
& ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList3 ) )
=> ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X6 ) ) )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [Mi3: nat,Ma3: nat] :
( ( ord_less_eq @ nat @ Mi3 @ Ma3 )
& ( ord_less @ nat @ Ma3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
& ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
=> ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList3 @ I3 ) @ X6 ) )
= ( vEBT_V8194947554948674370ptions @ Summary2 @ I3 ) ) )
& ( ( Mi3 = Ma3 )
=> ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList3 ) )
=> ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X6 ) ) )
& ( ( Mi3 != Ma3 )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList3 @ Ma3 )
& ! [X4: nat] :
( ( ord_less @ nat @ X4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList3 @ X4 )
=> ( ( ord_less @ nat @ Mi3 @ X4 )
& ( ord_less_eq @ nat @ X4 @ Ma3 ) ) ) ) ) ) ) )
@ Mima ) ) ) ) ) ) ) ).
% VEBT_internal.valid'.pelims(3)
thf(fact_5242_mono__compose,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ C ) )
=> ! [Q: A > B > C,F2: D > B] :
( ( order_mono @ A @ ( B > C ) @ Q )
=> ( order_mono @ A @ ( D > C )
@ ^ [I3: A,X4: D] : ( Q @ I3 @ ( F2 @ X4 ) ) ) ) ) ).
% mono_compose
thf(fact_5243_finite__Inf__Sup,axiom,
! [A: $tType] :
( ( finite8700451911770168679attice @ A )
=> ! [A4: set @ ( set @ A )] :
( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image2 @ ( set @ A ) @ A @ ( complete_Sup_Sup @ A ) @ A4 ) )
@ ( complete_Sup_Sup @ A
@ ( image2 @ ( set @ A ) @ A @ ( complete_Inf_Inf @ A )
@ ( collect @ ( set @ A )
@ ^ [Uu3: set @ A] :
? [F3: ( set @ A ) > A] :
( ( Uu3
= ( image2 @ ( set @ A ) @ A @ F3 @ A4 ) )
& ! [X4: set @ A] :
( ( member @ ( set @ A ) @ X4 @ A4 )
=> ( member @ A @ ( F3 @ X4 ) @ X4 ) ) ) ) ) ) ) ) ).
% finite_Inf_Sup
thf(fact_5244_Sup__Inf__le,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A4: set @ ( set @ A )] :
( ord_less_eq @ A
@ ( complete_Sup_Sup @ A
@ ( image2 @ ( set @ A ) @ A @ ( complete_Inf_Inf @ A )
@ ( collect @ ( set @ A )
@ ^ [Uu3: set @ A] :
? [F3: ( set @ A ) > A] :
( ( Uu3
= ( image2 @ ( set @ A ) @ A @ F3 @ A4 ) )
& ! [X4: set @ A] :
( ( member @ ( set @ A ) @ X4 @ A4 )
=> ( member @ A @ ( F3 @ X4 ) @ X4 ) ) ) ) ) )
@ ( complete_Inf_Inf @ A @ ( image2 @ ( set @ A ) @ A @ ( complete_Sup_Sup @ A ) @ A4 ) ) ) ) ).
% Sup_Inf_le
thf(fact_5245_Inf__Sup__le,axiom,
! [A: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [A4: set @ ( set @ A )] :
( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image2 @ ( set @ A ) @ A @ ( complete_Sup_Sup @ A ) @ A4 ) )
@ ( complete_Sup_Sup @ A
@ ( image2 @ ( set @ A ) @ A @ ( complete_Inf_Inf @ A )
@ ( collect @ ( set @ A )
@ ^ [Uu3: set @ A] :
? [F3: ( set @ A ) > A] :
( ( Uu3
= ( image2 @ ( set @ A ) @ A @ F3 @ A4 ) )
& ! [X4: set @ A] :
( ( member @ ( set @ A ) @ X4 @ A4 )
=> ( member @ A @ ( F3 @ X4 ) @ X4 ) ) ) ) ) ) ) ) ).
% Inf_Sup_le
thf(fact_5246_Pow__Compl,axiom,
! [A: $tType,A4: set @ A] :
( ( pow2 @ A @ ( uminus_uminus @ ( set @ A ) @ A4 ) )
= ( collect @ ( set @ A )
@ ^ [Uu3: set @ A] :
? [B6: set @ A] :
( ( Uu3
= ( uminus_uminus @ ( set @ A ) @ B6 ) )
& ( member @ ( set @ A ) @ A4 @ ( pow2 @ A @ B6 ) ) ) ) ) ).
% Pow_Compl
thf(fact_5247_Union__maximal__sets,axiom,
! [A: $tType,F13: set @ ( set @ A )] :
( ( finite_finite2 @ ( set @ A ) @ F13 )
=> ( ( complete_Sup_Sup @ ( set @ A )
@ ( collect @ ( set @ A )
@ ^ [T10: set @ A] :
( ( member @ ( set @ A ) @ T10 @ F13 )
& ! [X4: set @ A] :
( ( member @ ( set @ A ) @ X4 @ F13 )
=> ~ ( ord_less @ ( set @ A ) @ T10 @ X4 ) ) ) ) )
= ( complete_Sup_Sup @ ( set @ A ) @ F13 ) ) ) ).
% Union_maximal_sets
thf(fact_5248_Nats__altdef1,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ( ( semiring_1_Nats @ A )
= ( collect @ A
@ ^ [Uu3: A] :
? [N: int] :
( ( Uu3
= ( ring_1_of_int @ A @ N ) )
& ( ord_less_eq @ int @ ( zero_zero @ int ) @ N ) ) ) ) ) ).
% Nats_altdef1
thf(fact_5249_iteratesp_Omono,axiom,
! [A: $tType] :
( ( comple9053668089753744459l_ccpo @ A )
=> ! [F2: A > A] :
( order_mono @ ( A > $o ) @ ( A > $o )
@ ^ [P6: A > $o,X4: A] :
( ? [Y6: A] :
( ( X4
= ( F2 @ Y6 ) )
& ( P6 @ Y6 ) )
| ? [M9: set @ A] :
( ( X4
= ( complete_Sup_Sup @ A @ M9 ) )
& ( comple1602240252501008431_chain @ A @ ( ord_less_eq @ A ) @ M9 )
& ! [Y6: A] :
( ( member @ A @ Y6 @ M9 )
=> ( P6 @ Y6 ) ) ) ) ) ) ).
% iteratesp.mono
thf(fact_5250_Ball__Collect,axiom,
! [A: $tType] :
( ( ball @ A )
= ( ^ [A7: set @ A,P3: A > $o] : ( ord_less_eq @ ( set @ A ) @ A7 @ ( collect @ A @ P3 ) ) ) ) ).
% Ball_Collect
thf(fact_5251_UN__le__eq__Un0,axiom,
! [A: $tType,M7: nat > ( set @ A ),N2: nat] :
( ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ M7 @ ( set_ord_atMost @ nat @ N2 ) ) )
= ( sup_sup @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ M7 @ ( set_or1337092689740270186AtMost @ nat @ ( one_one @ nat ) @ N2 ) ) ) @ ( M7 @ ( zero_zero @ nat ) ) ) ) ).
% UN_le_eq_Un0
thf(fact_5252_UnCI,axiom,
! [A: $tType,C2: A,B2: set @ A,A4: set @ A] :
( ( ~ ( member @ A @ C2 @ B2 )
=> ( member @ A @ C2 @ A4 ) )
=> ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A4 @ B2 ) ) ) ).
% UnCI
thf(fact_5253_Un__iff,axiom,
! [A: $tType,C2: A,A4: set @ A,B2: set @ A] :
( ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A4 @ B2 ) )
= ( ( member @ A @ C2 @ A4 )
| ( member @ A @ C2 @ B2 ) ) ) ).
% Un_iff
thf(fact_5254_le__sup__iff,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X: A,Y: A,Z3: A] :
( ( ord_less_eq @ A @ ( sup_sup @ A @ X @ Y ) @ Z3 )
= ( ( ord_less_eq @ A @ X @ Z3 )
& ( ord_less_eq @ A @ Y @ Z3 ) ) ) ) ).
% le_sup_iff
thf(fact_5255_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_5256_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_5257_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_5258_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_5259_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_5260_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_5261_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_5262_sup__bot__right,axiom,
! [A: $tType] :
( ( bounde4967611905675639751up_bot @ A )
=> ! [X: A] :
( ( sup_sup @ A @ X @ ( bot_bot @ A ) )
= X ) ) ).
% sup_bot_right
thf(fact_5263_sup__bot__left,axiom,
! [A: $tType] :
( ( bounde4967611905675639751up_bot @ A )
=> ! [X: A] :
( ( sup_sup @ A @ ( bot_bot @ A ) @ X )
= X ) ) ).
% sup_bot_left
thf(fact_5264_Un__empty,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] :
( ( ( sup_sup @ ( set @ A ) @ A4 @ B2 )
= ( bot_bot @ ( set @ A ) ) )
= ( ( A4
= ( bot_bot @ ( set @ A ) ) )
& ( B2
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Un_empty
thf(fact_5265_finite__Un,axiom,
! [A: $tType,F5: set @ A,G4: set @ A] :
( ( finite_finite2 @ A @ ( sup_sup @ ( set @ A ) @ F5 @ G4 ) )
= ( ( finite_finite2 @ A @ F5 )
& ( finite_finite2 @ A @ G4 ) ) ) ).
% finite_Un
thf(fact_5266_Un__subset__iff,axiom,
! [A: $tType,A4: set @ A,B2: set @ A,C5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A4 @ B2 ) @ C5 )
= ( ( ord_less_eq @ ( set @ A ) @ A4 @ C5 )
& ( ord_less_eq @ ( set @ A ) @ B2 @ C5 ) ) ) ).
% Un_subset_iff
thf(fact_5267_Un__insert__right,axiom,
! [A: $tType,A4: set @ A,A3: A,B2: set @ A] :
( ( sup_sup @ ( set @ A ) @ A4 @ ( insert2 @ A @ A3 @ B2 ) )
= ( insert2 @ A @ A3 @ ( sup_sup @ ( set @ A ) @ A4 @ B2 ) ) ) ).
% Un_insert_right
thf(fact_5268_Un__insert__left,axiom,
! [A: $tType,A3: A,B2: set @ A,C5: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( insert2 @ A @ A3 @ B2 ) @ C5 )
= ( insert2 @ A @ A3 @ ( sup_sup @ ( set @ A ) @ B2 @ C5 ) ) ) ).
% Un_insert_left
thf(fact_5269_Int__Un__eq_I4_J,axiom,
! [A: $tType,T4: set @ A,S2: set @ A] :
( ( sup_sup @ ( set @ A ) @ T4 @ ( inf_inf @ ( set @ A ) @ S2 @ T4 ) )
= T4 ) ).
% Int_Un_eq(4)
thf(fact_5270_Int__Un__eq_I3_J,axiom,
! [A: $tType,S2: set @ A,T4: set @ A] :
( ( sup_sup @ ( set @ A ) @ S2 @ ( inf_inf @ ( set @ A ) @ S2 @ T4 ) )
= S2 ) ).
% Int_Un_eq(3)
thf(fact_5271_Int__Un__eq_I2_J,axiom,
! [A: $tType,S2: set @ A,T4: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ S2 @ T4 ) @ T4 )
= T4 ) ).
% Int_Un_eq(2)
thf(fact_5272_Int__Un__eq_I1_J,axiom,
! [A: $tType,S2: set @ A,T4: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ S2 @ T4 ) @ S2 )
= S2 ) ).
% Int_Un_eq(1)
thf(fact_5273_Un__Int__eq_I4_J,axiom,
! [A: $tType,T4: set @ A,S2: set @ A] :
( ( inf_inf @ ( set @ A ) @ T4 @ ( sup_sup @ ( set @ A ) @ S2 @ T4 ) )
= T4 ) ).
% Un_Int_eq(4)
thf(fact_5274_Un__Int__eq_I3_J,axiom,
! [A: $tType,S2: set @ A,T4: set @ A] :
( ( inf_inf @ ( set @ A ) @ S2 @ ( sup_sup @ ( set @ A ) @ S2 @ T4 ) )
= S2 ) ).
% Un_Int_eq(3)
thf(fact_5275_Un__Int__eq_I2_J,axiom,
! [A: $tType,S2: set @ A,T4: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ S2 @ T4 ) @ T4 )
= T4 ) ).
% Un_Int_eq(2)
thf(fact_5276_Un__Int__eq_I1_J,axiom,
! [A: $tType,S2: set @ A,T4: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ S2 @ T4 ) @ S2 )
= S2 ) ).
% Un_Int_eq(1)
thf(fact_5277_Un__Diff__cancel2,axiom,
! [A: $tType,B2: set @ A,A4: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ B2 @ A4 ) @ A4 )
= ( sup_sup @ ( set @ A ) @ B2 @ A4 ) ) ).
% Un_Diff_cancel2
thf(fact_5278_Un__Diff__cancel,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] :
( ( sup_sup @ ( set @ A ) @ A4 @ ( minus_minus @ ( set @ A ) @ B2 @ A4 ) )
= ( sup_sup @ ( set @ A ) @ A4 @ B2 ) ) ).
% Un_Diff_cancel
thf(fact_5279_Compl__Diff__eq,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] :
( ( uminus_uminus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A4 @ B2 ) )
= ( sup_sup @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A4 ) @ B2 ) ) ).
% Compl_Diff_eq
thf(fact_5280_if__image__distrib,axiom,
! [A: $tType,B: $tType,P: B > $o,F2: B > A,G: B > A,S2: set @ B] :
( ( image2 @ B @ A
@ ^ [X4: B] : ( if @ A @ ( P @ X4 ) @ ( F2 @ X4 ) @ ( G @ X4 ) )
@ S2 )
= ( sup_sup @ ( set @ A ) @ ( image2 @ B @ A @ F2 @ ( inf_inf @ ( set @ B ) @ S2 @ ( collect @ B @ P ) ) )
@ ( image2 @ B @ A @ G
@ ( inf_inf @ ( set @ B ) @ S2
@ ( collect @ B
@ ^ [X4: B] :
~ ( P @ X4 ) ) ) ) ) ) ).
% if_image_distrib
thf(fact_5281_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_5282_UN__simps_I2_J,axiom,
! [C: $tType,D: $tType,C5: set @ C,A4: C > ( set @ D ),B2: set @ D] :
( ( ( C5
= ( bot_bot @ ( set @ C ) ) )
=> ( ( complete_Sup_Sup @ ( set @ D )
@ ( image2 @ C @ ( set @ D )
@ ^ [X4: C] : ( sup_sup @ ( set @ D ) @ ( A4 @ X4 ) @ B2 )
@ C5 ) )
= ( bot_bot @ ( set @ D ) ) ) )
& ( ( C5
!= ( bot_bot @ ( set @ C ) ) )
=> ( ( complete_Sup_Sup @ ( set @ D )
@ ( image2 @ C @ ( set @ D )
@ ^ [X4: C] : ( sup_sup @ ( set @ D ) @ ( A4 @ X4 ) @ B2 )
@ C5 ) )
= ( sup_sup @ ( set @ D ) @ ( complete_Sup_Sup @ ( set @ D ) @ ( image2 @ C @ ( set @ D ) @ A4 @ C5 ) ) @ B2 ) ) ) ) ).
% UN_simps(2)
thf(fact_5283_UN__simps_I3_J,axiom,
! [E4: $tType,F: $tType,C5: set @ F,A4: set @ E4,B2: F > ( set @ E4 )] :
( ( ( C5
= ( bot_bot @ ( set @ F ) ) )
=> ( ( complete_Sup_Sup @ ( set @ E4 )
@ ( image2 @ F @ ( set @ E4 )
@ ^ [X4: F] : ( sup_sup @ ( set @ E4 ) @ A4 @ ( B2 @ X4 ) )
@ C5 ) )
= ( bot_bot @ ( set @ E4 ) ) ) )
& ( ( C5
!= ( bot_bot @ ( set @ F ) ) )
=> ( ( complete_Sup_Sup @ ( set @ E4 )
@ ( image2 @ F @ ( set @ E4 )
@ ^ [X4: F] : ( sup_sup @ ( set @ E4 ) @ A4 @ ( B2 @ X4 ) )
@ C5 ) )
= ( sup_sup @ ( set @ E4 ) @ A4 @ ( complete_Sup_Sup @ ( set @ E4 ) @ ( image2 @ F @ ( set @ E4 ) @ B2 @ C5 ) ) ) ) ) ) ).
% UN_simps(3)
thf(fact_5284_image__Un,axiom,
! [A: $tType,B: $tType,F2: B > A,A4: set @ B,B2: set @ B] :
( ( image2 @ B @ A @ F2 @ ( sup_sup @ ( set @ B ) @ A4 @ B2 ) )
= ( sup_sup @ ( set @ A ) @ ( image2 @ B @ A @ F2 @ A4 ) @ ( image2 @ B @ A @ F2 @ B2 ) ) ) ).
% image_Un
thf(fact_5285_Un__Pow__subset,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] : ( ord_less_eq @ ( set @ ( set @ A ) ) @ ( sup_sup @ ( set @ ( set @ A ) ) @ ( pow2 @ A @ A4 ) @ ( pow2 @ A @ B2 ) ) @ ( pow2 @ A @ ( sup_sup @ ( set @ A ) @ A4 @ B2 ) ) ) ).
% Un_Pow_subset
thf(fact_5286_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_5287_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_5288_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_5289_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_5290_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_5291_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_5292_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_5293_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_5294_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_5295_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_5296_sup__least,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [Y: A,X: A,Z3: A] :
( ( ord_less_eq @ A @ Y @ X )
=> ( ( ord_less_eq @ A @ Z3 @ X )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ Y @ Z3 ) @ X ) ) ) ) ).
% sup_least
thf(fact_5297_le__iff__sup,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [X4: A,Y6: A] :
( ( sup_sup @ A @ X4 @ Y6 )
= Y6 ) ) ) ) ).
% le_iff_sup
thf(fact_5298_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_5299_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_5300_sup__unique,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [F2: A > A > A,X: A,Y: A] :
( ! [X5: A,Y3: A] : ( ord_less_eq @ A @ X5 @ ( F2 @ X5 @ Y3 ) )
=> ( ! [X5: A,Y3: A] : ( ord_less_eq @ A @ Y3 @ ( F2 @ X5 @ Y3 ) )
=> ( ! [X5: A,Y3: A,Z: A] :
( ( ord_less_eq @ A @ Y3 @ X5 )
=> ( ( ord_less_eq @ A @ Z @ X5 )
=> ( ord_less_eq @ A @ ( F2 @ Y3 @ Z ) @ X5 ) ) )
=> ( ( sup_sup @ A @ X @ Y )
= ( F2 @ X @ Y ) ) ) ) ) ) ).
% sup_unique
thf(fact_5301_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_5302_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_5303_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_5304_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_5305_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_5306_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_5307_sup_Oorder__iff,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B4: A,A5: A] :
( A5
= ( sup_sup @ A @ A5 @ B4 ) ) ) ) ) ).
% sup.order_iff
thf(fact_5308_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_5309_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_5310_sup_Oabsorb__iff1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B4: A,A5: A] :
( ( sup_sup @ A @ A5 @ B4 )
= A5 ) ) ) ) ).
% sup.absorb_iff1
thf(fact_5311_sup_Oabsorb__iff2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A5: A,B4: A] :
( ( sup_sup @ A @ A5 @ B4 )
= B4 ) ) ) ) ).
% sup.absorb_iff2
thf(fact_5312_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_5313_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_5314_Un__mono,axiom,
! [A: $tType,A4: set @ A,C5: set @ A,B2: set @ A,D6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ C5 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ D6 )
=> ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A4 @ B2 ) @ ( sup_sup @ ( set @ A ) @ C5 @ D6 ) ) ) ) ).
% Un_mono
thf(fact_5315_Un__least,axiom,
! [A: $tType,A4: set @ A,C5: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ C5 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ C5 )
=> ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A4 @ B2 ) @ C5 ) ) ) ).
% Un_least
thf(fact_5316_Un__upper1,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] : ( ord_less_eq @ ( set @ A ) @ A4 @ ( sup_sup @ ( set @ A ) @ A4 @ B2 ) ) ).
% Un_upper1
thf(fact_5317_Un__upper2,axiom,
! [A: $tType,B2: set @ A,A4: set @ A] : ( ord_less_eq @ ( set @ A ) @ B2 @ ( sup_sup @ ( set @ A ) @ A4 @ B2 ) ) ).
% Un_upper2
thf(fact_5318_Un__absorb1,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B2 )
=> ( ( sup_sup @ ( set @ A ) @ A4 @ B2 )
= B2 ) ) ).
% Un_absorb1
thf(fact_5319_Un__absorb2,axiom,
! [A: $tType,B2: set @ A,A4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ B2 @ A4 )
=> ( ( sup_sup @ ( set @ A ) @ A4 @ B2 )
= A4 ) ) ).
% Un_absorb2
thf(fact_5320_subset__UnE,axiom,
! [A: $tType,C5: set @ A,A4: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ C5 @ ( sup_sup @ ( set @ A ) @ A4 @ B2 ) )
=> ~ ! [A15: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A15 @ A4 )
=> ! [B12: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ B12 @ B2 )
=> ( C5
!= ( sup_sup @ ( set @ A ) @ A15 @ B12 ) ) ) ) ) ).
% subset_UnE
thf(fact_5321_subset__Un__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A7: set @ A,B6: set @ A] :
( ( sup_sup @ ( set @ A ) @ A7 @ B6 )
= B6 ) ) ) ).
% subset_Un_eq
thf(fact_5322_chain__subset,axiom,
! [A: $tType,Ord: A > A > $o,A4: set @ A,B2: set @ A] :
( ( comple1602240252501008431_chain @ A @ Ord @ A4 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ A4 )
=> ( comple1602240252501008431_chain @ A @ Ord @ B2 ) ) ) ).
% chain_subset
thf(fact_5323_Un__empty__left,axiom,
! [A: $tType,B2: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ B2 )
= B2 ) ).
% Un_empty_left
thf(fact_5324_Un__empty__right,axiom,
! [A: $tType,A4: set @ A] :
( ( sup_sup @ ( set @ A ) @ A4 @ ( bot_bot @ ( set @ A ) ) )
= A4 ) ).
% Un_empty_right
thf(fact_5325_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_5326_chain__empty,axiom,
! [A: $tType,Ord: A > A > $o] : ( comple1602240252501008431_chain @ A @ Ord @ ( bot_bot @ ( set @ A ) ) ) ).
% chain_empty
thf(fact_5327_finite__UnI,axiom,
! [A: $tType,F5: set @ A,G4: set @ A] :
( ( finite_finite2 @ A @ F5 )
=> ( ( finite_finite2 @ A @ G4 )
=> ( finite_finite2 @ A @ ( sup_sup @ ( set @ A ) @ F5 @ G4 ) ) ) ) ).
% finite_UnI
thf(fact_5328_Un__infinite,axiom,
! [A: $tType,S2: set @ A,T4: set @ A] :
( ~ ( finite_finite2 @ A @ S2 )
=> ~ ( finite_finite2 @ A @ ( sup_sup @ ( set @ A ) @ S2 @ T4 ) ) ) ).
% Un_infinite
thf(fact_5329_infinite__Un,axiom,
! [A: $tType,S2: set @ A,T4: set @ A] :
( ( ~ ( finite_finite2 @ A @ ( sup_sup @ ( set @ A ) @ S2 @ T4 ) ) )
= ( ~ ( finite_finite2 @ A @ S2 )
| ~ ( finite_finite2 @ A @ T4 ) ) ) ).
% infinite_Un
thf(fact_5330_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_5331_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_5332_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_5333_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_5334_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_5335_sup_Ostrict__order__iff,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( ord_less @ A )
= ( ^ [B4: A,A5: A] :
( ( A5
= ( sup_sup @ A @ A5 @ B4 ) )
& ( A5 != B4 ) ) ) ) ) ).
% sup.strict_order_iff
thf(fact_5336_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_5337_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_5338_UnE,axiom,
! [A: $tType,C2: A,A4: set @ A,B2: set @ A] :
( ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A4 @ B2 ) )
=> ( ~ ( member @ A @ C2 @ A4 )
=> ( member @ A @ C2 @ B2 ) ) ) ).
% UnE
thf(fact_5339_UnI1,axiom,
! [A: $tType,C2: A,A4: set @ A,B2: set @ A] :
( ( member @ A @ C2 @ A4 )
=> ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A4 @ B2 ) ) ) ).
% UnI1
thf(fact_5340_UnI2,axiom,
! [A: $tType,C2: A,B2: set @ A,A4: set @ A] :
( ( member @ A @ C2 @ B2 )
=> ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A4 @ B2 ) ) ) ).
% UnI2
thf(fact_5341_bex__Un,axiom,
! [A: $tType,A4: set @ A,B2: set @ A,P: A > $o] :
( ( ? [X4: A] :
( ( member @ A @ X4 @ ( sup_sup @ ( set @ A ) @ A4 @ B2 ) )
& ( P @ X4 ) ) )
= ( ? [X4: A] :
( ( member @ A @ X4 @ A4 )
& ( P @ X4 ) )
| ? [X4: A] :
( ( member @ A @ X4 @ B2 )
& ( P @ X4 ) ) ) ) ).
% bex_Un
thf(fact_5342_ball__Un,axiom,
! [A: $tType,A4: set @ A,B2: set @ A,P: A > $o] :
( ( ! [X4: A] :
( ( member @ A @ X4 @ ( sup_sup @ ( set @ A ) @ A4 @ B2 ) )
=> ( P @ X4 ) ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ A4 )
=> ( P @ X4 ) )
& ! [X4: A] :
( ( member @ A @ X4 @ B2 )
=> ( P @ X4 ) ) ) ) ).
% ball_Un
thf(fact_5343_Un__assoc,axiom,
! [A: $tType,A4: set @ A,B2: set @ A,C5: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A4 @ B2 ) @ C5 )
= ( sup_sup @ ( set @ A ) @ A4 @ ( sup_sup @ ( set @ A ) @ B2 @ C5 ) ) ) ).
% Un_assoc
thf(fact_5344_Un__absorb,axiom,
! [A: $tType,A4: set @ A] :
( ( sup_sup @ ( set @ A ) @ A4 @ A4 )
= A4 ) ).
% Un_absorb
thf(fact_5345_Un__commute,axiom,
! [A: $tType] :
( ( sup_sup @ ( set @ A ) )
= ( ^ [A7: set @ A,B6: set @ A] : ( sup_sup @ ( set @ A ) @ B6 @ A7 ) ) ) ).
% Un_commute
thf(fact_5346_Un__left__absorb,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] :
( ( sup_sup @ ( set @ A ) @ A4 @ ( sup_sup @ ( set @ A ) @ A4 @ B2 ) )
= ( sup_sup @ ( set @ A ) @ A4 @ B2 ) ) ).
% Un_left_absorb
thf(fact_5347_Un__left__commute,axiom,
! [A: $tType,A4: set @ A,B2: set @ A,C5: set @ A] :
( ( sup_sup @ ( set @ A ) @ A4 @ ( sup_sup @ ( set @ A ) @ B2 @ C5 ) )
= ( sup_sup @ ( set @ A ) @ B2 @ ( sup_sup @ ( set @ A ) @ A4 @ C5 ) ) ) ).
% Un_left_commute
thf(fact_5348_Un__Diff,axiom,
! [A: $tType,A4: set @ A,B2: set @ A,C5: set @ A] :
( ( minus_minus @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A4 @ B2 ) @ C5 )
= ( sup_sup @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A4 @ C5 ) @ ( minus_minus @ ( set @ A ) @ B2 @ C5 ) ) ) ).
% Un_Diff
thf(fact_5349_Un__Int__crazy,axiom,
! [A: $tType,A4: set @ A,B2: set @ A,C5: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A4 @ B2 ) @ ( inf_inf @ ( set @ A ) @ B2 @ C5 ) ) @ ( inf_inf @ ( set @ A ) @ C5 @ A4 ) )
= ( inf_inf @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A4 @ B2 ) @ ( sup_sup @ ( set @ A ) @ B2 @ C5 ) ) @ ( sup_sup @ ( set @ A ) @ C5 @ A4 ) ) ) ).
% Un_Int_crazy
thf(fact_5350_Int__Un__distrib,axiom,
! [A: $tType,A4: set @ A,B2: set @ A,C5: set @ A] :
( ( inf_inf @ ( set @ A ) @ A4 @ ( sup_sup @ ( set @ A ) @ B2 @ C5 ) )
= ( sup_sup @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A4 @ B2 ) @ ( inf_inf @ ( set @ A ) @ A4 @ C5 ) ) ) ).
% Int_Un_distrib
thf(fact_5351_Un__Int__distrib,axiom,
! [A: $tType,A4: set @ A,B2: set @ A,C5: set @ A] :
( ( sup_sup @ ( set @ A ) @ A4 @ ( inf_inf @ ( set @ A ) @ B2 @ C5 ) )
= ( inf_inf @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A4 @ B2 ) @ ( sup_sup @ ( set @ A ) @ A4 @ C5 ) ) ) ).
% Un_Int_distrib
thf(fact_5352_Int__Un__distrib2,axiom,
! [A: $tType,B2: set @ A,C5: set @ A,A4: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ B2 @ C5 ) @ A4 )
= ( sup_sup @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ B2 @ A4 ) @ ( inf_inf @ ( set @ A ) @ C5 @ A4 ) ) ) ).
% Int_Un_distrib2
thf(fact_5353_Un__Int__distrib2,axiom,
! [A: $tType,B2: set @ A,C5: set @ A,A4: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ B2 @ C5 ) @ A4 )
= ( inf_inf @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ B2 @ A4 ) @ ( sup_sup @ ( set @ A ) @ C5 @ A4 ) ) ) ).
% Un_Int_distrib2
thf(fact_5354_Un__UNIV__right,axiom,
! [A: $tType,A4: set @ A] :
( ( sup_sup @ ( set @ A ) @ A4 @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ A ) ) ) ).
% Un_UNIV_right
thf(fact_5355_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_5356_Collect__imp__eq,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ( collect @ A
@ ^ [X4: A] :
( ( P @ X4 )
=> ( Q @ X4 ) ) )
= ( sup_sup @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ ( collect @ A @ P ) ) @ ( collect @ A @ Q ) ) ) ).
% Collect_imp_eq
thf(fact_5357_Un__def,axiom,
! [A: $tType] :
( ( sup_sup @ ( set @ A ) )
= ( ^ [A7: set @ A,B6: set @ A] :
( collect @ A
@ ^ [X4: A] :
( ( member @ A @ X4 @ A7 )
| ( member @ A @ X4 @ B6 ) ) ) ) ) ).
% Un_def
thf(fact_5358_Collect__disj__eq,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ( collect @ A
@ ^ [X4: A] :
( ( P @ X4 )
| ( Q @ X4 ) ) )
= ( sup_sup @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% Collect_disj_eq
thf(fact_5359_insert__def,axiom,
! [A: $tType] :
( ( insert2 @ A )
= ( ^ [A5: A] :
( sup_sup @ ( set @ A )
@ ( collect @ A
@ ^ [X4: A] : X4 = A5 ) ) ) ) ).
% insert_def
thf(fact_5360_distrib__sup__le,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X: A,Y: A,Z3: A] : ( ord_less_eq @ A @ ( sup_sup @ A @ X @ ( inf_inf @ A @ Y @ Z3 ) ) @ ( inf_inf @ A @ ( sup_sup @ A @ X @ Y ) @ ( sup_sup @ A @ X @ Z3 ) ) ) ) ).
% distrib_sup_le
thf(fact_5361_distrib__inf__le,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X: A,Y: A,Z3: A] : ( ord_less_eq @ A @ ( sup_sup @ A @ ( inf_inf @ A @ X @ Y ) @ ( inf_inf @ A @ X @ Z3 ) ) @ ( inf_inf @ A @ X @ ( sup_sup @ A @ Y @ Z3 ) ) ) ) ).
% distrib_inf_le
thf(fact_5362_mono__sup,axiom,
! [B: $tType,A: $tType] :
( ( ( semilattice_sup @ A )
& ( semilattice_sup @ B ) )
=> ! [F2: A > B,A4: A,B2: A] :
( ( order_mono @ A @ B @ F2 )
=> ( ord_less_eq @ B @ ( sup_sup @ B @ ( F2 @ A4 ) @ ( F2 @ B2 ) ) @ ( F2 @ ( sup_sup @ A @ A4 @ B2 ) ) ) ) ) ).
% mono_sup
thf(fact_5363_ivl__disj__un__two__touch_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,M2: A,U: A] :
( ( ord_less_eq @ A @ L @ M2 )
=> ( ( ord_less_eq @ A @ M2 @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ L @ M2 ) @ ( set_or1337092689740270186AtMost @ A @ M2 @ U ) )
= ( set_or1337092689740270186AtMost @ A @ L @ U ) ) ) ) ) ).
% ivl_disj_un_two_touch(4)
thf(fact_5364_insert__is__Un,axiom,
! [A: $tType] :
( ( insert2 @ A )
= ( ^ [A5: A] : ( sup_sup @ ( set @ A ) @ ( insert2 @ A @ A5 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% insert_is_Un
thf(fact_5365_Un__singleton__iff,axiom,
! [A: $tType,A4: set @ A,B2: set @ A,X: A] :
( ( ( sup_sup @ ( set @ A ) @ A4 @ B2 )
= ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( ( ( A4
= ( bot_bot @ ( set @ A ) ) )
& ( B2
= ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
| ( ( A4
= ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
& ( B2
= ( bot_bot @ ( set @ A ) ) ) )
| ( ( A4
= ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
& ( B2
= ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_5366_singleton__Un__iff,axiom,
! [A: $tType,X: A,A4: set @ A,B2: set @ A] :
( ( ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) )
= ( sup_sup @ ( set @ A ) @ A4 @ B2 ) )
= ( ( ( A4
= ( bot_bot @ ( set @ A ) ) )
& ( B2
= ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
| ( ( A4
= ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
& ( B2
= ( bot_bot @ ( set @ A ) ) ) )
| ( ( A4
= ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
& ( B2
= ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_5367_ivl__disj__un__two_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,M2: A,U: A] :
( ( ord_less_eq @ A @ L @ M2 )
=> ( ( ord_less_eq @ A @ M2 @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ L @ M2 ) @ ( set_or7035219750837199246ssThan @ A @ M2 @ U ) )
= ( set_or7035219750837199246ssThan @ A @ L @ U ) ) ) ) ) ).
% ivl_disj_un_two(3)
thf(fact_5368_Un__Int__assoc__eq,axiom,
! [A: $tType,A4: set @ A,B2: set @ A,C5: set @ A] :
( ( ( sup_sup @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A4 @ B2 ) @ C5 )
= ( inf_inf @ ( set @ A ) @ A4 @ ( sup_sup @ ( set @ A ) @ B2 @ C5 ) ) )
= ( ord_less_eq @ ( set @ A ) @ C5 @ A4 ) ) ).
% Un_Int_assoc_eq
thf(fact_5369_Diff__subset__conv,axiom,
! [A: $tType,A4: set @ A,B2: set @ A,C5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A4 @ B2 ) @ C5 )
= ( ord_less_eq @ ( set @ A ) @ A4 @ ( sup_sup @ ( set @ A ) @ B2 @ C5 ) ) ) ).
% Diff_subset_conv
thf(fact_5370_Diff__partition,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B2 )
=> ( ( sup_sup @ ( set @ A ) @ A4 @ ( minus_minus @ ( set @ A ) @ B2 @ A4 ) )
= B2 ) ) ).
% Diff_partition
thf(fact_5371_Diff__Un,axiom,
! [A: $tType,A4: set @ A,B2: set @ A,C5: set @ A] :
( ( minus_minus @ ( set @ A ) @ A4 @ ( sup_sup @ ( set @ A ) @ B2 @ C5 ) )
= ( inf_inf @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A4 @ B2 ) @ ( minus_minus @ ( set @ A ) @ A4 @ C5 ) ) ) ).
% Diff_Un
thf(fact_5372_Diff__Int,axiom,
! [A: $tType,A4: set @ A,B2: set @ A,C5: set @ A] :
( ( minus_minus @ ( set @ A ) @ A4 @ ( inf_inf @ ( set @ A ) @ B2 @ C5 ) )
= ( sup_sup @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A4 @ B2 ) @ ( minus_minus @ ( set @ A ) @ A4 @ C5 ) ) ) ).
% Diff_Int
thf(fact_5373_Int__Diff__Un,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A4 @ B2 ) @ ( minus_minus @ ( set @ A ) @ A4 @ B2 ) )
= A4 ) ).
% Int_Diff_Un
thf(fact_5374_Un__Diff__Int,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A4 @ B2 ) @ ( inf_inf @ ( set @ A ) @ A4 @ B2 ) )
= A4 ) ).
% Un_Diff_Int
thf(fact_5375_ivl__disj__un__two_I6_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,M2: A,U: A] :
( ( ord_less_eq @ A @ L @ M2 )
=> ( ( ord_less_eq @ A @ M2 @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ L @ M2 ) @ ( set_or3652927894154168847AtMost @ A @ M2 @ U ) )
= ( set_or3652927894154168847AtMost @ A @ L @ U ) ) ) ) ) ).
% ivl_disj_un_two(6)
thf(fact_5376_Compl__partition2,axiom,
! [A: $tType,A4: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A4 ) @ A4 )
= ( top_top @ ( set @ A ) ) ) ).
% Compl_partition2
thf(fact_5377_Compl__partition,axiom,
! [A: $tType,A4: set @ A] :
( ( sup_sup @ ( set @ A ) @ A4 @ ( uminus_uminus @ ( set @ A ) @ A4 ) )
= ( top_top @ ( set @ A ) ) ) ).
% Compl_partition
thf(fact_5378_ccpo__Sup__least,axiom,
! [A: $tType] :
( ( comple9053668089753744459l_ccpo @ A )
=> ! [A4: set @ A,Z3: A] :
( ( comple1602240252501008431_chain @ A @ ( ord_less_eq @ A ) @ A4 )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ A4 )
=> ( ord_less_eq @ A @ X5 @ Z3 ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A4 ) @ Z3 ) ) ) ) ).
% ccpo_Sup_least
thf(fact_5379_ccpo__Sup__upper,axiom,
! [A: $tType] :
( ( comple9053668089753744459l_ccpo @ A )
=> ! [A4: set @ A,X: A] :
( ( comple1602240252501008431_chain @ A @ ( ord_less_eq @ A ) @ A4 )
=> ( ( member @ A @ X @ A4 )
=> ( ord_less_eq @ A @ X @ ( complete_Sup_Sup @ A @ A4 ) ) ) ) ) ).
% ccpo_Sup_upper
thf(fact_5380_Compl__Int,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] :
( ( uminus_uminus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A4 @ B2 ) )
= ( sup_sup @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A4 ) @ ( uminus_uminus @ ( set @ A ) @ B2 ) ) ) ).
% Compl_Int
thf(fact_5381_Compl__Un,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] :
( ( uminus_uminus @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A4 @ B2 ) )
= ( inf_inf @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A4 ) @ ( uminus_uminus @ ( set @ A ) @ B2 ) ) ) ).
% Compl_Un
thf(fact_5382_set__shuffles,axiom,
! [A: $tType,Zs: list @ A,Xs: list @ A,Ys: list @ A] :
( ( member @ ( list @ A ) @ Zs @ ( shuffles @ A @ Xs @ Ys ) )
=> ( ( set2 @ A @ Zs )
= ( sup_sup @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ Ys ) ) ) ) ).
% set_shuffles
thf(fact_5383_mono__Un,axiom,
! [B: $tType,A: $tType,F2: ( set @ A ) > ( set @ B ),A4: set @ A,B2: set @ A] :
( ( order_mono @ ( set @ A ) @ ( set @ B ) @ F2 )
=> ( ord_less_eq @ ( set @ B ) @ ( sup_sup @ ( set @ B ) @ ( F2 @ A4 ) @ ( F2 @ B2 ) ) @ ( F2 @ ( sup_sup @ ( set @ A ) @ A4 @ B2 ) ) ) ) ).
% mono_Un
thf(fact_5384_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_5385_shunt1,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,Y: A,Z3: A] :
( ( ord_less_eq @ A @ ( inf_inf @ A @ X @ Y ) @ Z3 )
= ( ord_less_eq @ A @ X @ ( sup_sup @ A @ ( uminus_uminus @ A @ Y ) @ Z3 ) ) ) ) ).
% shunt1
thf(fact_5386_shunt2,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,Y: A,Z3: A] :
( ( ord_less_eq @ A @ ( inf_inf @ A @ X @ ( uminus_uminus @ A @ Y ) ) @ Z3 )
= ( ord_less_eq @ A @ X @ ( sup_sup @ A @ Y @ Z3 ) ) ) ) ).
% shunt2
thf(fact_5387_sup__neg__inf,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [P5: A,Q3: A,R2: A] :
( ( ord_less_eq @ A @ P5 @ ( sup_sup @ A @ Q3 @ R2 ) )
= ( ord_less_eq @ A @ ( inf_inf @ A @ P5 @ ( uminus_uminus @ A @ Q3 ) ) @ R2 ) ) ) ).
% sup_neg_inf
thf(fact_5388_boolean__algebra_Ocomplement__unique,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [A3: A,X: A,Y: A] :
( ( ( inf_inf @ A @ A3 @ X )
= ( bot_bot @ A ) )
=> ( ( ( sup_sup @ A @ A3 @ X )
= ( top_top @ A ) )
=> ( ( ( inf_inf @ A @ A3 @ Y )
= ( bot_bot @ A ) )
=> ( ( ( sup_sup @ A @ A3 @ Y )
= ( top_top @ A ) )
=> ( X = Y ) ) ) ) ) ) ).
% boolean_algebra.complement_unique
thf(fact_5389_finite__Sup__in,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A4: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X5: A,Y3: A] :
( ( member @ A @ X5 @ A4 )
=> ( ( member @ A @ Y3 @ A4 )
=> ( member @ A @ ( sup_sup @ A @ X5 @ Y3 ) @ A4 ) ) )
=> ( member @ A @ ( complete_Sup_Sup @ A @ A4 ) @ A4 ) ) ) ) ) ).
% finite_Sup_in
thf(fact_5390_less__eq__Inf__inter,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A4: set @ A,B2: set @ A] : ( ord_less_eq @ A @ ( sup_sup @ A @ ( complete_Inf_Inf @ A @ A4 ) @ ( complete_Inf_Inf @ A @ B2 ) ) @ ( complete_Inf_Inf @ A @ ( inf_inf @ ( set @ A ) @ A4 @ B2 ) ) ) ) ).
% less_eq_Inf_inter
thf(fact_5391_ivl__disj__un__two_I7_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,M2: A,U: A] :
( ( ord_less_eq @ A @ L @ M2 )
=> ( ( ord_less_eq @ A @ M2 @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ L @ M2 ) @ ( set_or1337092689740270186AtMost @ A @ M2 @ U ) )
= ( set_or1337092689740270186AtMost @ A @ L @ U ) ) ) ) ) ).
% ivl_disj_un_two(7)
thf(fact_5392_ivl__disj__un__one_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,U: A] :
( ( ord_less_eq @ A @ L @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_ord_lessThan @ A @ L ) @ ( set_or7035219750837199246ssThan @ A @ L @ U ) )
= ( set_ord_lessThan @ A @ U ) ) ) ) ).
% ivl_disj_un_one(2)
thf(fact_5393_Union__image__empty,axiom,
! [B: $tType,A: $tType,A4: set @ A,F2: B > ( set @ A )] :
( ( sup_sup @ ( set @ A ) @ A4 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ F2 @ ( bot_bot @ ( set @ B ) ) ) ) )
= A4 ) ).
% Union_image_empty
thf(fact_5394_card__Un__le,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] : ( ord_less_eq @ nat @ ( finite_card @ A @ ( sup_sup @ ( set @ A ) @ A4 @ B2 ) ) @ ( plus_plus @ nat @ ( finite_card @ A @ A4 ) @ ( finite_card @ A @ B2 ) ) ) ).
% card_Un_le
thf(fact_5395_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_5396_ivl__disj__un__two_I8_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,M2: A,U: A] :
( ( ord_less_eq @ A @ L @ M2 )
=> ( ( ord_less_eq @ A @ M2 @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ L @ M2 ) @ ( set_or3652927894154168847AtMost @ A @ M2 @ U ) )
= ( set_or1337092689740270186AtMost @ A @ L @ U ) ) ) ) ) ).
% ivl_disj_un_two(8)
thf(fact_5397_ivl__disj__un__one_I8_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,U: A] :
( ( ord_less_eq @ A @ L @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ L @ U ) @ ( set_ord_atLeast @ A @ U ) )
= ( set_ord_atLeast @ A @ L ) ) ) ) ).
% ivl_disj_un_one(8)
thf(fact_5398_ivl__disj__un__one_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,U: A] :
( ( ord_less_eq @ A @ L @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_ord_atMost @ A @ L ) @ ( set_or3652927894154168847AtMost @ A @ L @ U ) )
= ( set_ord_atMost @ A @ U ) ) ) ) ).
% ivl_disj_un_one(3)
thf(fact_5399_ivl__disj__un__one_I5_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,U: A] :
( ( ord_less_eq @ A @ L @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ L @ U ) @ ( set_ord_greaterThan @ A @ U ) )
= ( set_ord_greaterThan @ A @ L ) ) ) ) ).
% ivl_disj_un_one(5)
thf(fact_5400_chain__singleton,axiom,
! [A: $tType] :
( ( comple9053668089753744459l_ccpo @ A )
=> ! [X: A] : ( comple1602240252501008431_chain @ A @ ( ord_less_eq @ A ) @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% chain_singleton
thf(fact_5401_Pow__insert,axiom,
! [A: $tType,A3: A,A4: set @ A] :
( ( pow2 @ A @ ( insert2 @ A @ A3 @ A4 ) )
= ( sup_sup @ ( set @ ( set @ A ) ) @ ( pow2 @ A @ A4 ) @ ( image2 @ ( set @ A ) @ ( set @ A ) @ ( insert2 @ A @ A3 ) @ ( pow2 @ A @ A4 ) ) ) ) ).
% Pow_insert
thf(fact_5402_UN__extend__simps_I2_J,axiom,
! [D: $tType,C: $tType,C5: set @ C,A4: C > ( set @ D ),B2: set @ D] :
( ( ( C5
= ( bot_bot @ ( set @ C ) ) )
=> ( ( sup_sup @ ( set @ D ) @ ( complete_Sup_Sup @ ( set @ D ) @ ( image2 @ C @ ( set @ D ) @ A4 @ C5 ) ) @ B2 )
= B2 ) )
& ( ( C5
!= ( bot_bot @ ( set @ C ) ) )
=> ( ( sup_sup @ ( set @ D ) @ ( complete_Sup_Sup @ ( set @ D ) @ ( image2 @ C @ ( set @ D ) @ A4 @ C5 ) ) @ B2 )
= ( complete_Sup_Sup @ ( set @ D )
@ ( image2 @ C @ ( set @ D )
@ ^ [X4: C] : ( sup_sup @ ( set @ D ) @ ( A4 @ X4 ) @ B2 )
@ C5 ) ) ) ) ) ).
% UN_extend_simps(2)
thf(fact_5403_UN__extend__simps_I3_J,axiom,
! [E4: $tType,F: $tType,C5: set @ F,A4: set @ E4,B2: F > ( set @ E4 )] :
( ( ( C5
= ( bot_bot @ ( set @ F ) ) )
=> ( ( sup_sup @ ( set @ E4 ) @ A4 @ ( complete_Sup_Sup @ ( set @ E4 ) @ ( image2 @ F @ ( set @ E4 ) @ B2 @ C5 ) ) )
= A4 ) )
& ( ( C5
!= ( bot_bot @ ( set @ F ) ) )
=> ( ( sup_sup @ ( set @ E4 ) @ A4 @ ( complete_Sup_Sup @ ( set @ E4 ) @ ( image2 @ F @ ( set @ E4 ) @ B2 @ C5 ) ) )
= ( complete_Sup_Sup @ ( set @ E4 )
@ ( image2 @ F @ ( set @ E4 )
@ ^ [X4: F] : ( sup_sup @ ( set @ E4 ) @ A4 @ ( B2 @ X4 ) )
@ C5 ) ) ) ) ) ).
% UN_extend_simps(3)
thf(fact_5404_Inter__Un__subset,axiom,
! [A: $tType,A4: set @ ( set @ A ),B2: set @ ( set @ A )] : ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ ( complete_Inf_Inf @ ( set @ A ) @ A4 ) @ ( complete_Inf_Inf @ ( set @ A ) @ B2 ) ) @ ( complete_Inf_Inf @ ( set @ A ) @ ( inf_inf @ ( set @ ( set @ A ) ) @ A4 @ B2 ) ) ) ).
% Inter_Un_subset
thf(fact_5405_boolean__algebra__class_Oboolean__algebra_Ocompl__unique,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,Y: A] :
( ( ( inf_inf @ A @ X @ Y )
= ( bot_bot @ A ) )
=> ( ( ( sup_sup @ A @ X @ Y )
= ( top_top @ A ) )
=> ( ( uminus_uminus @ A @ X )
= Y ) ) ) ) ).
% boolean_algebra_class.boolean_algebra.compl_unique
thf(fact_5406_ivl__disj__un__two__touch_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,M2: A,U: A] :
( ( ord_less_eq @ A @ L @ M2 )
=> ( ( ord_less @ A @ M2 @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ L @ M2 ) @ ( set_or7035219750837199246ssThan @ A @ M2 @ U ) )
= ( set_or7035219750837199246ssThan @ A @ L @ U ) ) ) ) ) ).
% ivl_disj_un_two_touch(2)
thf(fact_5407_sum_Ounion__inter,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A4: set @ B,B2: set @ B,G: B > A] :
( ( finite_finite2 @ B @ A4 )
=> ( ( finite_finite2 @ B @ B2 )
=> ( ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ G @ ( sup_sup @ ( set @ B ) @ A4 @ B2 ) ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ ( inf_inf @ ( set @ B ) @ A4 @ B2 ) ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ G @ A4 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ B2 ) ) ) ) ) ) ).
% sum.union_inter
thf(fact_5408_prod_Ounion__inter,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A4: set @ B,B2: set @ B,G: B > A] :
( ( finite_finite2 @ B @ A4 )
=> ( ( finite_finite2 @ B @ B2 )
=> ( ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ G @ ( sup_sup @ ( set @ B ) @ A4 @ B2 ) ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ ( inf_inf @ ( set @ B ) @ A4 @ B2 ) ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ G @ A4 ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ B2 ) ) ) ) ) ) ).
% prod.union_inter
thf(fact_5409_card__Un__Int,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( finite_finite2 @ A @ B2 )
=> ( ( plus_plus @ nat @ ( finite_card @ A @ A4 ) @ ( finite_card @ A @ B2 ) )
= ( plus_plus @ nat @ ( finite_card @ A @ ( sup_sup @ ( set @ A ) @ A4 @ B2 ) ) @ ( finite_card @ A @ ( inf_inf @ ( set @ A ) @ A4 @ B2 ) ) ) ) ) ) ).
% card_Un_Int
thf(fact_5410_ivl__disj__un__two__touch_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,M2: A,U: A] :
( ( ord_less @ A @ L @ M2 )
=> ( ( ord_less_eq @ A @ M2 @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ L @ M2 ) @ ( set_or1337092689740270186AtMost @ A @ M2 @ U ) )
= ( set_or3652927894154168847AtMost @ A @ L @ U ) ) ) ) ) ).
% ivl_disj_un_two_touch(3)
thf(fact_5411_ivl__disj__un__two_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,M2: A,U: A] :
( ( ord_less @ A @ L @ M2 )
=> ( ( ord_less_eq @ A @ M2 @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ L @ M2 ) @ ( set_or7035219750837199246ssThan @ A @ M2 @ U ) )
= ( set_or5935395276787703475ssThan @ A @ L @ U ) ) ) ) ) ).
% ivl_disj_un_two(1)
thf(fact_5412_ivl__disj__un__one_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,U: A] :
( ( ord_less_eq @ A @ L @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_ord_lessThan @ A @ L ) @ ( set_or1337092689740270186AtMost @ A @ L @ U ) )
= ( set_ord_atMost @ A @ U ) ) ) ) ).
% ivl_disj_un_one(4)
thf(fact_5413_ivl__disj__un__singleton_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [U: A] :
( ( sup_sup @ ( set @ A ) @ ( set_ord_lessThan @ A @ U ) @ ( insert2 @ A @ U @ ( bot_bot @ ( set @ A ) ) ) )
= ( set_ord_atMost @ A @ U ) ) ) ).
% ivl_disj_un_singleton(2)
thf(fact_5414_ivl__disj__un__two_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,M2: A,U: A] :
( ( ord_less_eq @ A @ L @ M2 )
=> ( ( ord_less @ A @ M2 @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ L @ M2 ) @ ( set_or5935395276787703475ssThan @ A @ M2 @ U ) )
= ( set_or5935395276787703475ssThan @ A @ L @ U ) ) ) ) ) ).
% ivl_disj_un_two(2)
thf(fact_5415_Max_Ounion,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,B2: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B2 )
=> ( ( B2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798349783984er_Max @ A @ ( sup_sup @ ( set @ A ) @ A4 @ B2 ) )
= ( ord_max @ A @ ( lattic643756798349783984er_Max @ A @ A4 ) @ ( lattic643756798349783984er_Max @ A @ B2 ) ) ) ) ) ) ) ) ).
% Max.union
thf(fact_5416_ivl__disj__un__one_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,U: A] :
( ( ord_less @ A @ L @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_ord_atMost @ A @ L ) @ ( set_or5935395276787703475ssThan @ A @ L @ U ) )
= ( set_ord_lessThan @ A @ U ) ) ) ) ).
% ivl_disj_un_one(1)
thf(fact_5417_ivl__disj__un__one_I7_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,U: A] :
( ( ord_less_eq @ A @ L @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ L @ U ) @ ( set_ord_greaterThan @ A @ U ) )
= ( set_ord_atLeast @ A @ L ) ) ) ) ).
% ivl_disj_un_one(7)
thf(fact_5418_ivl__disj__un__singleton_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A] :
( ( sup_sup @ ( set @ A ) @ ( insert2 @ A @ L @ ( bot_bot @ ( set @ A ) ) ) @ ( set_ord_greaterThan @ A @ L ) )
= ( set_ord_atLeast @ A @ L ) ) ) ).
% ivl_disj_un_singleton(1)
thf(fact_5419_ivl__disj__un__two__touch_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,M2: A,U: A] :
( ( ord_less @ A @ L @ M2 )
=> ( ( ord_less @ A @ M2 @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ L @ M2 ) @ ( set_or7035219750837199246ssThan @ A @ M2 @ U ) )
= ( set_or5935395276787703475ssThan @ A @ L @ U ) ) ) ) ) ).
% ivl_disj_un_two_touch(1)
thf(fact_5420_SUP__nat__binary,axiom,
! [A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A4: A,B2: A] :
( ( sup_sup @ A @ A4
@ ( complete_Sup_Sup @ A
@ ( image2 @ nat @ A
@ ^ [X4: nat] : B2
@ ( collect @ nat @ ( ord_less @ nat @ ( zero_zero @ nat ) ) ) ) ) )
= ( sup_sup @ A @ A4 @ B2 ) ) ) ).
% SUP_nat_binary
thf(fact_5421_ivl__disj__un__one_I6_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,U: A] :
( ( ord_less @ A @ L @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ L @ U ) @ ( set_ord_atLeast @ A @ U ) )
= ( set_ord_greaterThan @ A @ L ) ) ) ) ).
% ivl_disj_un_one(6)
thf(fact_5422_inj__on__disjoint__Un,axiom,
! [B: $tType,A: $tType,F2: A > B,A4: set @ A,G: A > B,B2: set @ A] :
( ( inj_on @ A @ B @ F2 @ A4 )
=> ( ( inj_on @ A @ B @ G @ B2 )
=> ( ( ( inf_inf @ ( set @ B ) @ ( image2 @ A @ B @ F2 @ A4 ) @ ( image2 @ A @ B @ G @ B2 ) )
= ( bot_bot @ ( set @ B ) ) )
=> ( inj_on @ A @ B
@ ^ [X4: A] : ( if @ B @ ( member @ A @ X4 @ A4 ) @ ( F2 @ X4 ) @ ( G @ X4 ) )
@ ( sup_sup @ ( set @ A ) @ A4 @ B2 ) ) ) ) ) ).
% inj_on_disjoint_Un
thf(fact_5423_sup__bot_Osemilattice__neutr__order__axioms,axiom,
! [A: $tType] :
( ( bounde4967611905675639751up_bot @ A )
=> ( semila1105856199041335345_order @ A @ ( sup_sup @ A ) @ ( bot_bot @ A )
@ ^ [X4: A,Y6: A] : ( ord_less_eq @ A @ Y6 @ X4 )
@ ^ [X4: A,Y6: A] : ( ord_less @ A @ Y6 @ X4 ) ) ) ).
% sup_bot.semilattice_neutr_order_axioms
thf(fact_5424_sum_Ounion__inter__neutral,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A4: set @ B,B2: set @ B,G: B > A] :
( ( finite_finite2 @ B @ A4 )
=> ( ( finite_finite2 @ B @ B2 )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ ( inf_inf @ ( set @ B ) @ A4 @ B2 ) )
=> ( ( G @ X5 )
= ( zero_zero @ A ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( sup_sup @ ( set @ B ) @ A4 @ B2 ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ G @ A4 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ B2 ) ) ) ) ) ) ) ).
% sum.union_inter_neutral
thf(fact_5425_sum__Un,axiom,
! [A: $tType,B: $tType] :
( ( ab_group_add @ A )
=> ! [A4: set @ B,B2: set @ B,F2: B > A] :
( ( finite_finite2 @ B @ A4 )
=> ( ( finite_finite2 @ B @ B2 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ F2 @ ( sup_sup @ ( set @ B ) @ A4 @ B2 ) )
= ( minus_minus @ A @ ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A4 ) @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ B2 ) ) @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ ( inf_inf @ ( set @ B ) @ A4 @ B2 ) ) ) ) ) ) ) ).
% sum_Un
thf(fact_5426_sum_Ounion__disjoint,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A4: set @ B,B2: set @ B,G: B > A] :
( ( finite_finite2 @ B @ A4 )
=> ( ( finite_finite2 @ B @ B2 )
=> ( ( ( inf_inf @ ( set @ B ) @ A4 @ B2 )
= ( bot_bot @ ( set @ B ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( sup_sup @ ( set @ B ) @ A4 @ B2 ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ G @ A4 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ B2 ) ) ) ) ) ) ) ).
% sum.union_disjoint
thf(fact_5427_prod_Ounion__inter__neutral,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A4: set @ B,B2: set @ B,G: B > A] :
( ( finite_finite2 @ B @ A4 )
=> ( ( finite_finite2 @ B @ B2 )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ ( inf_inf @ ( set @ B ) @ A4 @ B2 ) )
=> ( ( G @ X5 )
= ( one_one @ A ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( sup_sup @ ( set @ B ) @ A4 @ B2 ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ G @ A4 ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ B2 ) ) ) ) ) ) ) ).
% prod.union_inter_neutral
thf(fact_5428_prod_Ounion__disjoint,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A4: set @ B,B2: set @ B,G: B > A] :
( ( finite_finite2 @ B @ A4 )
=> ( ( finite_finite2 @ B @ B2 )
=> ( ( ( inf_inf @ ( set @ B ) @ A4 @ B2 )
= ( bot_bot @ ( set @ B ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( sup_sup @ ( set @ B ) @ A4 @ B2 ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ G @ A4 ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ B2 ) ) ) ) ) ) ) ).
% prod.union_disjoint
thf(fact_5429_ivl__disj__un__singleton_I6_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,U: A] :
( ( ord_less_eq @ A @ L @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ L @ U ) @ ( insert2 @ A @ U @ ( bot_bot @ ( set @ A ) ) ) )
= ( set_or1337092689740270186AtMost @ A @ L @ U ) ) ) ) ).
% ivl_disj_un_singleton(6)
thf(fact_5430_sum_Ounion__diff2,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A4: set @ B,B2: set @ B,G: B > A] :
( ( finite_finite2 @ B @ A4 )
=> ( ( finite_finite2 @ B @ B2 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( sup_sup @ ( set @ B ) @ A4 @ B2 ) )
= ( plus_plus @ A @ ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ A4 @ B2 ) ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ B2 @ A4 ) ) ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ ( inf_inf @ ( set @ B ) @ A4 @ B2 ) ) ) ) ) ) ) ).
% sum.union_diff2
thf(fact_5431_sum__Un2,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add @ B )
=> ! [A4: set @ A,B2: set @ A,F2: A > B] :
( ( finite_finite2 @ A @ ( sup_sup @ ( set @ A ) @ A4 @ B2 ) )
=> ( ( groups7311177749621191930dd_sum @ A @ B @ F2 @ ( sup_sup @ ( set @ A ) @ A4 @ B2 ) )
= ( plus_plus @ B @ ( plus_plus @ B @ ( groups7311177749621191930dd_sum @ A @ B @ F2 @ ( minus_minus @ ( set @ A ) @ A4 @ B2 ) ) @ ( groups7311177749621191930dd_sum @ A @ B @ F2 @ ( minus_minus @ ( set @ A ) @ B2 @ A4 ) ) ) @ ( groups7311177749621191930dd_sum @ A @ B @ F2 @ ( inf_inf @ ( set @ A ) @ A4 @ B2 ) ) ) ) ) ) ).
% sum_Un2
thf(fact_5432_prod_Ounion__diff2,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A4: set @ B,B2: set @ B,G: B > A] :
( ( finite_finite2 @ B @ A4 )
=> ( ( finite_finite2 @ B @ B2 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( sup_sup @ ( set @ B ) @ A4 @ B2 ) )
= ( times_times @ A @ ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ A4 @ B2 ) ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ B2 @ A4 ) ) ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ ( inf_inf @ ( set @ B ) @ A4 @ B2 ) ) ) ) ) ) ) ).
% prod.union_diff2
thf(fact_5433_card__Un__disjoint,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( finite_finite2 @ A @ B2 )
=> ( ( ( inf_inf @ ( set @ A ) @ A4 @ B2 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_card @ A @ ( sup_sup @ ( set @ A ) @ A4 @ B2 ) )
= ( plus_plus @ nat @ ( finite_card @ A @ A4 ) @ ( finite_card @ A @ B2 ) ) ) ) ) ) ).
% card_Un_disjoint
thf(fact_5434_inj__on__Un,axiom,
! [A: $tType,B: $tType,F2: A > B,A4: set @ A,B2: set @ A] :
( ( inj_on @ A @ B @ F2 @ ( sup_sup @ ( set @ A ) @ A4 @ B2 ) )
= ( ( inj_on @ A @ B @ F2 @ A4 )
& ( inj_on @ A @ B @ F2 @ B2 )
& ( ( inf_inf @ ( set @ B ) @ ( image2 @ A @ B @ F2 @ ( minus_minus @ ( set @ A ) @ A4 @ B2 ) ) @ ( image2 @ A @ B @ F2 @ ( minus_minus @ ( set @ A ) @ B2 @ A4 ) ) )
= ( bot_bot @ ( set @ B ) ) ) ) ) ).
% inj_on_Un
thf(fact_5435_ivl__disj__un__singleton_I5_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,U: A] :
( ( ord_less_eq @ A @ L @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( insert2 @ A @ L @ ( bot_bot @ ( set @ A ) ) ) @ ( set_or3652927894154168847AtMost @ A @ L @ U ) )
= ( set_or1337092689740270186AtMost @ A @ L @ U ) ) ) ) ).
% ivl_disj_un_singleton(5)
thf(fact_5436_ivl__disj__un__two_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,M2: A,U: A] :
( ( ord_less_eq @ A @ L @ M2 )
=> ( ( ord_less @ A @ M2 @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ L @ M2 ) @ ( set_or5935395276787703475ssThan @ A @ M2 @ U ) )
= ( set_or7035219750837199246ssThan @ A @ L @ U ) ) ) ) ) ).
% ivl_disj_un_two(4)
thf(fact_5437_ivl__disj__un__singleton_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,U: A] :
( ( ord_less @ A @ L @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( insert2 @ A @ L @ ( bot_bot @ ( set @ A ) ) ) @ ( set_or5935395276787703475ssThan @ A @ L @ U ) )
= ( set_or7035219750837199246ssThan @ A @ L @ U ) ) ) ) ).
% ivl_disj_un_singleton(3)
thf(fact_5438_sum__Un__nat,axiom,
! [A: $tType,A4: set @ A,B2: set @ A,F2: A > nat] :
( ( finite_finite2 @ A @ A4 )
=> ( ( finite_finite2 @ A @ B2 )
=> ( ( groups7311177749621191930dd_sum @ A @ nat @ F2 @ ( sup_sup @ ( set @ A ) @ A4 @ B2 ) )
= ( minus_minus @ nat @ ( plus_plus @ nat @ ( groups7311177749621191930dd_sum @ A @ nat @ F2 @ A4 ) @ ( groups7311177749621191930dd_sum @ A @ nat @ F2 @ B2 ) ) @ ( groups7311177749621191930dd_sum @ A @ nat @ F2 @ ( inf_inf @ ( set @ A ) @ A4 @ B2 ) ) ) ) ) ) ).
% sum_Un_nat
thf(fact_5439_ivl__disj__un__two_I5_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,M2: A,U: A] :
( ( ord_less @ A @ L @ M2 )
=> ( ( ord_less_eq @ A @ M2 @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ L @ M2 ) @ ( set_or1337092689740270186AtMost @ A @ M2 @ U ) )
= ( set_or3652927894154168847AtMost @ A @ L @ U ) ) ) ) ) ).
% ivl_disj_un_two(5)
thf(fact_5440_ivl__disj__un__singleton_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,U: A] :
( ( ord_less @ A @ L @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ L @ U ) @ ( insert2 @ A @ U @ ( bot_bot @ ( set @ A ) ) ) )
= ( set_or3652927894154168847AtMost @ A @ L @ U ) ) ) ) ).
% ivl_disj_un_singleton(4)
thf(fact_5441_prod__Un,axiom,
! [A: $tType,B: $tType] :
( ( field @ A )
=> ! [A4: set @ B,B2: set @ B,F2: B > A] :
( ( finite_finite2 @ B @ A4 )
=> ( ( finite_finite2 @ B @ B2 )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ ( inf_inf @ ( set @ B ) @ A4 @ B2 ) )
=> ( ( F2 @ X5 )
!= ( zero_zero @ A ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ F2 @ ( sup_sup @ ( set @ B ) @ A4 @ B2 ) )
= ( divide_divide @ A @ ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ A4 ) @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ B2 ) ) @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ ( inf_inf @ ( set @ B ) @ A4 @ B2 ) ) ) ) ) ) ) ) ).
% prod_Un
thf(fact_5442_in__chain__finite,axiom,
! [A: $tType] :
( ( comple9053668089753744459l_ccpo @ A )
=> ! [A4: set @ A] :
( ( comple1602240252501008431_chain @ A @ ( ord_less_eq @ A ) @ A4 )
=> ( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( member @ A @ ( complete_Sup_Sup @ A @ A4 ) @ A4 ) ) ) ) ) ).
% in_chain_finite
thf(fact_5443_If__the__inv__into__in__Func,axiom,
! [B: $tType,A: $tType,G: A > B,C5: set @ A,B2: set @ A,X: A] :
( ( inj_on @ A @ B @ G @ C5 )
=> ( ( ord_less_eq @ ( set @ A ) @ C5 @ ( sup_sup @ ( set @ A ) @ B2 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( member @ ( B > A )
@ ^ [I3: B] : ( if @ A @ ( member @ B @ I3 @ ( image2 @ A @ B @ G @ C5 ) ) @ ( the_inv_into @ A @ B @ C5 @ G @ I3 ) @ X )
@ ( bNF_Wellorder_Func @ B @ A @ ( top_top @ ( set @ B ) ) @ ( sup_sup @ ( set @ A ) @ B2 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% If_the_inv_into_in_Func
thf(fact_5444_Pow__fold,axiom,
! [A: $tType,A4: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( pow2 @ A @ A4 )
= ( finite_fold @ A @ ( set @ ( set @ A ) )
@ ^ [X4: A,A7: set @ ( set @ A )] : ( sup_sup @ ( set @ ( set @ A ) ) @ A7 @ ( image2 @ ( set @ A ) @ ( set @ A ) @ ( insert2 @ A @ X4 ) @ A7 ) )
@ ( insert2 @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ ( bot_bot @ ( set @ ( set @ A ) ) ) )
@ A4 ) ) ) ).
% Pow_fold
thf(fact_5445_fold__empty,axiom,
! [B: $tType,A: $tType,F2: B > A > A,Z3: A] :
( ( finite_fold @ B @ A @ F2 @ Z3 @ ( bot_bot @ ( set @ B ) ) )
= Z3 ) ).
% fold_empty
thf(fact_5446_fold__infinite,axiom,
! [A: $tType,B: $tType,A4: set @ A,F2: A > B > B,Z3: B] :
( ~ ( finite_finite2 @ A @ A4 )
=> ( ( finite_fold @ A @ B @ F2 @ Z3 @ A4 )
= Z3 ) ) ).
% fold_infinite
thf(fact_5447_sup__int__def,axiom,
( ( sup_sup @ int )
= ( ord_max @ int ) ) ).
% sup_int_def
thf(fact_5448_sup__nat__def,axiom,
( ( sup_sup @ nat )
= ( ord_max @ nat ) ) ).
% sup_nat_def
thf(fact_5449_sup__set__def,axiom,
! [A: $tType] :
( ( sup_sup @ ( set @ A ) )
= ( ^ [A7: set @ A,B6: set @ A] :
( collect @ A
@ ( sup_sup @ ( A > $o )
@ ^ [X4: A] : ( member @ A @ X4 @ A7 )
@ ^ [X4: A] : ( member @ A @ X4 @ B6 ) ) ) ) ) ).
% sup_set_def
thf(fact_5450_fold__closed__eq,axiom,
! [B: $tType,A: $tType,A4: set @ A,B2: set @ B,F2: A > B > B,G: A > B > B,Z3: B] :
( ! [A6: A,B5: B] :
( ( member @ A @ A6 @ A4 )
=> ( ( member @ B @ B5 @ B2 )
=> ( ( F2 @ A6 @ B5 )
= ( G @ A6 @ B5 ) ) ) )
=> ( ! [A6: A,B5: B] :
( ( member @ A @ A6 @ A4 )
=> ( ( member @ B @ B5 @ B2 )
=> ( member @ B @ ( G @ A6 @ B5 ) @ B2 ) ) )
=> ( ( member @ B @ Z3 @ B2 )
=> ( ( finite_fold @ A @ B @ F2 @ Z3 @ A4 )
= ( finite_fold @ A @ B @ G @ Z3 @ A4 ) ) ) ) ) ).
% fold_closed_eq
thf(fact_5451_folding__on_Oeq__fold,axiom,
! [B: $tType,A: $tType,S2: set @ A,F2: A > B > B,Z3: B,A4: set @ A] :
( ( finite_folding_on @ A @ B @ S2 @ F2 )
=> ( ( finite_folding_F @ A @ B @ F2 @ Z3 @ A4 )
= ( finite_fold @ A @ B @ F2 @ Z3 @ A4 ) ) ) ).
% folding_on.eq_fold
thf(fact_5452_union__fold__insert,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( sup_sup @ ( set @ A ) @ A4 @ B2 )
= ( finite_fold @ A @ ( set @ A ) @ ( insert2 @ A ) @ B2 @ A4 ) ) ) ).
% union_fold_insert
thf(fact_5453_sup__Sup__fold__sup,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A4: set @ A,B2: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( sup_sup @ A @ ( complete_Sup_Sup @ A @ A4 ) @ B2 )
= ( finite_fold @ A @ A @ ( sup_sup @ A ) @ B2 @ A4 ) ) ) ) ).
% sup_Sup_fold_sup
thf(fact_5454_inf__Inf__fold__inf,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A4: set @ A,B2: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( inf_inf @ A @ ( complete_Inf_Inf @ A @ A4 ) @ B2 )
= ( finite_fold @ A @ A @ ( inf_inf @ A ) @ B2 @ A4 ) ) ) ) ).
% inf_Inf_fold_inf
thf(fact_5455_fold__image,axiom,
! [C: $tType,B: $tType,A: $tType,G: A > B,A4: set @ A,F2: B > C > C,Z3: C] :
( ( inj_on @ A @ B @ G @ A4 )
=> ( ( finite_fold @ B @ C @ F2 @ Z3 @ ( image2 @ A @ B @ G @ A4 ) )
= ( finite_fold @ A @ C @ ( comp @ B @ ( C > C ) @ A @ F2 @ G ) @ Z3 @ A4 ) ) ) ).
% fold_image
thf(fact_5456_Sup__fold__sup,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A4: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( complete_Sup_Sup @ A @ A4 )
= ( finite_fold @ A @ A @ ( sup_sup @ A ) @ ( bot_bot @ A ) @ A4 ) ) ) ) ).
% Sup_fold_sup
thf(fact_5457_Inf__fold__inf,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A4: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( complete_Inf_Inf @ A @ A4 )
= ( finite_fold @ A @ A @ ( inf_inf @ A ) @ ( top_top @ A ) @ A4 ) ) ) ) ).
% Inf_fold_inf
thf(fact_5458_sum_Oeq__fold,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ( ( groups7311177749621191930dd_sum @ B @ A )
= ( ^ [G2: B > A] : ( finite_fold @ B @ A @ ( comp @ A @ ( A > A ) @ B @ ( plus_plus @ A ) @ G2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% sum.eq_fold
thf(fact_5459_Max_Oeq__fold,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( lattic643756798349783984er_Max @ A @ ( insert2 @ A @ X @ A4 ) )
= ( finite_fold @ A @ A @ ( ord_max @ A ) @ X @ A4 ) ) ) ) ).
% Max.eq_fold
thf(fact_5460_image__fold__insert,axiom,
! [B: $tType,A: $tType,A4: set @ A,F2: A > B] :
( ( finite_finite2 @ A @ A4 )
=> ( ( image2 @ A @ B @ F2 @ A4 )
= ( finite_fold @ A @ ( set @ B )
@ ^ [K2: A] : ( insert2 @ B @ ( F2 @ K2 ) )
@ ( bot_bot @ ( set @ B ) )
@ A4 ) ) ) ).
% image_fold_insert
thf(fact_5461_sup__SUP__fold__sup,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A4: set @ B,B2: A,F2: B > A] :
( ( finite_finite2 @ B @ A4 )
=> ( ( sup_sup @ A @ B2 @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F2 @ A4 ) ) )
= ( finite_fold @ B @ A @ ( comp @ A @ ( A > A ) @ B @ ( sup_sup @ A ) @ F2 ) @ B2 @ A4 ) ) ) ) ).
% sup_SUP_fold_sup
thf(fact_5462_inf__INF__fold__inf,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A4: set @ B,B2: A,F2: B > A] :
( ( finite_finite2 @ B @ A4 )
=> ( ( inf_inf @ A @ B2 @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F2 @ A4 ) ) )
= ( finite_fold @ B @ A @ ( comp @ A @ ( A > A ) @ B @ ( inf_inf @ A ) @ F2 ) @ B2 @ A4 ) ) ) ) ).
% inf_INF_fold_inf
thf(fact_5463_SUP__fold__sup,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A4: set @ B,F2: B > A] :
( ( finite_finite2 @ B @ A4 )
=> ( ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F2 @ A4 ) )
= ( finite_fold @ B @ A @ ( comp @ A @ ( A > A ) @ B @ ( sup_sup @ A ) @ F2 ) @ ( bot_bot @ A ) @ A4 ) ) ) ) ).
% SUP_fold_sup
thf(fact_5464_INF__fold__inf,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A4: set @ B,F2: B > A] :
( ( finite_finite2 @ B @ A4 )
=> ( ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F2 @ A4 ) )
= ( finite_fold @ B @ A @ ( comp @ A @ ( A > A ) @ B @ ( inf_inf @ A ) @ F2 ) @ ( top_top @ A ) @ A4 ) ) ) ) ).
% INF_fold_inf
thf(fact_5465_Set__filter__fold,axiom,
! [A: $tType,A4: set @ A,P: A > $o] :
( ( finite_finite2 @ A @ A4 )
=> ( ( filter3 @ A @ P @ A4 )
= ( finite_fold @ A @ ( set @ A )
@ ^ [X4: A,A17: set @ A] : ( if @ ( set @ A ) @ ( P @ X4 ) @ ( insert2 @ A @ X4 @ A17 ) @ A17 )
@ ( bot_bot @ ( set @ A ) )
@ A4 ) ) ) ).
% Set_filter_fold
thf(fact_5466_Rats__eq__int__div__nat,axiom,
( ( field_char_0_Rats @ real )
= ( collect @ real
@ ^ [Uu3: real] :
? [I3: int,N: nat] :
( ( Uu3
= ( divide_divide @ real @ ( ring_1_of_int @ real @ I3 ) @ ( semiring_1_of_nat @ real @ N ) ) )
& ( N
!= ( zero_zero @ nat ) ) ) ) ) ).
% Rats_eq_int_div_nat
thf(fact_5467_Func__map__surj,axiom,
! [C: $tType,A: $tType,D: $tType,B: $tType,F1: B > A,A18: set @ B,B1: set @ A,F22: C > D,B22: set @ C,A26: set @ D] :
( ( ( image2 @ B @ A @ F1 @ A18 )
= B1 )
=> ( ( inj_on @ C @ D @ F22 @ B22 )
=> ( ( ord_less_eq @ ( set @ D ) @ ( image2 @ C @ D @ F22 @ B22 ) @ A26 )
=> ( ( ( B22
= ( bot_bot @ ( set @ C ) ) )
=> ( A26
= ( bot_bot @ ( set @ D ) ) ) )
=> ( ( bNF_Wellorder_Func @ C @ A @ B22 @ B1 )
= ( image2 @ ( D > B ) @ ( C > A ) @ ( bNF_We4925052301507509544nc_map @ C @ B @ A @ D @ B22 @ F1 @ F22 ) @ ( bNF_Wellorder_Func @ D @ B @ A26 @ A18 ) ) ) ) ) ) ) ).
% Func_map_surj
thf(fact_5468_member__filter,axiom,
! [A: $tType,X: A,P: A > $o,A4: set @ A] :
( ( member @ A @ X @ ( filter3 @ A @ P @ A4 ) )
= ( ( member @ A @ X @ A4 )
& ( P @ X ) ) ) ).
% member_filter
thf(fact_5469_Rats__add,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: A,B3: A] :
( ( member @ A @ A3 @ ( field_char_0_Rats @ A ) )
=> ( ( member @ A @ B3 @ ( field_char_0_Rats @ A ) )
=> ( member @ A @ ( plus_plus @ A @ A3 @ B3 ) @ ( field_char_0_Rats @ A ) ) ) ) ) ).
% Rats_add
thf(fact_5470_Rats__0,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( member @ A @ ( zero_zero @ A ) @ ( field_char_0_Rats @ A ) ) ) ).
% Rats_0
thf(fact_5471_Rats__infinite,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ~ ( finite_finite2 @ A @ ( field_char_0_Rats @ A ) ) ) ).
% Rats_infinite
thf(fact_5472_Set_Ofilter__def,axiom,
! [A: $tType] :
( ( filter3 @ A )
= ( ^ [P3: A > $o,A7: set @ A] :
( collect @ A
@ ^ [A5: A] :
( ( member @ A @ A5 @ A7 )
& ( P3 @ A5 ) ) ) ) ) ).
% Set.filter_def
thf(fact_5473_finite__filter,axiom,
! [A: $tType,S2: set @ A,P: A > $o] :
( ( finite_finite2 @ A @ S2 )
=> ( finite_finite2 @ A @ ( filter3 @ A @ P @ S2 ) ) ) ).
% finite_filter
thf(fact_5474_Ball__fold,axiom,
! [A: $tType,A4: set @ A,P: A > $o] :
( ( finite_finite2 @ A @ A4 )
=> ( ( ! [X4: A] :
( ( member @ A @ X4 @ A4 )
=> ( P @ X4 ) ) )
= ( finite_fold @ A @ $o
@ ^ [K2: A,S7: $o] :
( S7
& ( P @ K2 ) )
@ $true
@ A4 ) ) ) ).
% Ball_fold
thf(fact_5475_Ints__subset__Rats,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( ord_less_eq @ ( set @ A ) @ ( ring_1_Ints @ A ) @ ( field_char_0_Rats @ A ) ) ) ).
% Ints_subset_Rats
thf(fact_5476_Nats__subset__Rats,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( ord_less_eq @ ( set @ A ) @ ( semiring_1_Nats @ A ) @ ( field_char_0_Rats @ A ) ) ) ).
% Nats_subset_Rats
thf(fact_5477_fold__union__pair,axiom,
! [B: $tType,A: $tType,B2: set @ A,X: B,A4: set @ ( product_prod @ B @ A )] :
( ( finite_finite2 @ A @ B2 )
=> ( ( sup_sup @ ( set @ ( product_prod @ B @ A ) )
@ ( complete_Sup_Sup @ ( set @ ( product_prod @ B @ A ) )
@ ( image2 @ A @ ( set @ ( product_prod @ B @ A ) )
@ ^ [Y6: A] : ( insert2 @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X @ Y6 ) @ ( bot_bot @ ( set @ ( product_prod @ B @ A ) ) ) )
@ B2 ) )
@ A4 )
= ( finite_fold @ A @ ( set @ ( product_prod @ B @ A ) )
@ ^ [Y6: A] : ( insert2 @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X @ Y6 ) )
@ A4
@ B2 ) ) ) ).
% fold_union_pair
thf(fact_5478_card_Oeq__fold,axiom,
! [A: $tType] :
( ( finite_card @ A )
= ( finite_fold @ A @ nat
@ ^ [Uu3: A] : suc
@ ( zero_zero @ nat ) ) ) ).
% card.eq_fold
thf(fact_5479_sorted__list__of__set_Ofold__insort__key_Oeq__fold,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( linord4507533701916653071of_set @ A )
= ( finite_fold @ A @ ( list @ A )
@ ( linorder_insort_key @ A @ A
@ ^ [X4: A] : X4 )
@ ( nil @ A ) ) ) ) ).
% sorted_list_of_set.fold_insort_key.eq_fold
thf(fact_5480_inter__Set__filter,axiom,
! [A: $tType,B2: set @ A,A4: set @ A] :
( ( finite_finite2 @ A @ B2 )
=> ( ( inf_inf @ ( set @ A ) @ A4 @ B2 )
= ( filter3 @ A
@ ^ [X4: A] : ( member @ A @ X4 @ A4 )
@ B2 ) ) ) ).
% inter_Set_filter
thf(fact_5481_Func__map,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,G: A > B,A26: set @ A,A18: set @ B,F1: B > C,B1: set @ C,F22: D > A,B22: set @ D] :
( ( member @ ( A > B ) @ G @ ( bNF_Wellorder_Func @ A @ B @ A26 @ A18 ) )
=> ( ( ord_less_eq @ ( set @ C ) @ ( image2 @ B @ C @ F1 @ A18 ) @ B1 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image2 @ D @ A @ F22 @ B22 ) @ A26 )
=> ( member @ ( D > C ) @ ( bNF_We4925052301507509544nc_map @ D @ B @ C @ A @ B22 @ F1 @ F22 @ G ) @ ( bNF_Wellorder_Func @ D @ C @ B22 @ B1 ) ) ) ) ) ).
% Func_map
thf(fact_5482_Func__is__emp,axiom,
! [A: $tType,B: $tType,A4: set @ A,B2: set @ B] :
( ( ( bNF_Wellorder_Func @ A @ B @ A4 @ B2 )
= ( bot_bot @ ( set @ ( A > B ) ) ) )
= ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
& ( B2
= ( bot_bot @ ( set @ B ) ) ) ) ) ).
% Func_is_emp
thf(fact_5483_Func__non__emp,axiom,
! [A: $tType,B: $tType,B2: set @ A,A4: set @ B] :
( ( B2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( bNF_Wellorder_Func @ B @ A @ A4 @ B2 )
!= ( bot_bot @ ( set @ ( B > A ) ) ) ) ) ).
% Func_non_emp
thf(fact_5484_UNION__fun__upd,axiom,
! [B: $tType,A: $tType,A4: B > ( set @ A ),I: B,B2: set @ A,J4: set @ B] :
( ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ ( fun_upd @ B @ ( set @ A ) @ A4 @ I @ B2 ) @ J4 ) )
= ( sup_sup @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ A4 @ ( minus_minus @ ( set @ B ) @ J4 @ ( insert2 @ B @ I @ ( bot_bot @ ( set @ B ) ) ) ) ) ) @ ( if @ ( set @ A ) @ ( member @ B @ I @ J4 ) @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% UNION_fun_upd
thf(fact_5485_comp__fun__commute__on_Ofold__set__union__disj,axiom,
! [B: $tType,A: $tType,S2: set @ A,F2: A > B > B,A4: set @ A,B2: set @ A,Z3: B] :
( ( finite4664212375090638736ute_on @ A @ B @ S2 @ F2 )
=> ( ( ord_less_eq @ ( set @ A ) @ A4 @ S2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ S2 )
=> ( ( finite_finite2 @ A @ A4 )
=> ( ( finite_finite2 @ A @ B2 )
=> ( ( ( inf_inf @ ( set @ A ) @ A4 @ B2 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_fold @ A @ B @ F2 @ Z3 @ ( sup_sup @ ( set @ A ) @ A4 @ B2 ) )
= ( finite_fold @ A @ B @ F2 @ ( finite_fold @ A @ B @ F2 @ Z3 @ A4 ) @ B2 ) ) ) ) ) ) ) ) ).
% comp_fun_commute_on.fold_set_union_disj
thf(fact_5486_comp__fun__commute__on_Ofold__rec,axiom,
! [B: $tType,A: $tType,S2: set @ A,F2: A > B > B,A4: set @ A,X: A,Z3: B] :
( ( finite4664212375090638736ute_on @ A @ B @ S2 @ F2 )
=> ( ( ord_less_eq @ ( set @ A ) @ A4 @ S2 )
=> ( ( finite_finite2 @ A @ A4 )
=> ( ( member @ A @ X @ A4 )
=> ( ( finite_fold @ A @ B @ F2 @ Z3 @ A4 )
= ( F2 @ X @ ( finite_fold @ A @ B @ F2 @ Z3 @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ).
% comp_fun_commute_on.fold_rec
thf(fact_5487_comp__fun__commute__on__def,axiom,
! [B: $tType,A: $tType] :
( ( finite4664212375090638736ute_on @ A @ B )
= ( ^ [S6: set @ A,F3: A > B > B] :
! [X4: A,Y6: A] :
( ( member @ A @ X4 @ S6 )
=> ( ( member @ A @ Y6 @ S6 )
=> ( ( comp @ B @ B @ B @ ( F3 @ Y6 ) @ ( F3 @ X4 ) )
= ( comp @ B @ B @ B @ ( F3 @ X4 ) @ ( F3 @ Y6 ) ) ) ) ) ) ) ).
% comp_fun_commute_on_def
thf(fact_5488_comp__fun__commute__on_Ocomp__fun__commute__on,axiom,
! [B: $tType,A: $tType,S2: set @ A,F2: A > B > B,X: A,Y: A] :
( ( finite4664212375090638736ute_on @ A @ B @ S2 @ F2 )
=> ( ( member @ A @ X @ S2 )
=> ( ( member @ A @ Y @ S2 )
=> ( ( comp @ B @ B @ B @ ( F2 @ Y ) @ ( F2 @ X ) )
= ( comp @ B @ B @ B @ ( F2 @ X ) @ ( F2 @ Y ) ) ) ) ) ) ).
% comp_fun_commute_on.comp_fun_commute_on
thf(fact_5489_comp__fun__commute__on_Ocommute__left__comp,axiom,
! [A: $tType,B: $tType,C: $tType,S2: set @ A,F2: A > B > B,X: A,Y: A,G: C > B] :
( ( finite4664212375090638736ute_on @ A @ B @ S2 @ F2 )
=> ( ( member @ A @ X @ S2 )
=> ( ( member @ A @ Y @ S2 )
=> ( ( comp @ B @ B @ C @ ( F2 @ Y ) @ ( comp @ B @ B @ C @ ( F2 @ X ) @ G ) )
= ( comp @ B @ B @ C @ ( F2 @ X ) @ ( comp @ B @ B @ C @ ( F2 @ Y ) @ G ) ) ) ) ) ) ).
% comp_fun_commute_on.commute_left_comp
thf(fact_5490_comp__fun__commute__on_Ointro,axiom,
! [B: $tType,A: $tType,S2: set @ A,F2: A > B > B] :
( ! [X5: A,Y3: A] :
( ( member @ A @ X5 @ S2 )
=> ( ( member @ A @ Y3 @ S2 )
=> ( ( comp @ B @ B @ B @ ( F2 @ Y3 ) @ ( F2 @ X5 ) )
= ( comp @ B @ B @ B @ ( F2 @ X5 ) @ ( F2 @ Y3 ) ) ) ) )
=> ( finite4664212375090638736ute_on @ A @ B @ S2 @ F2 ) ) ).
% comp_fun_commute_on.intro
thf(fact_5491_comp__fun__commute__on_Ofun__left__comm,axiom,
! [A: $tType,B: $tType,S2: set @ A,F2: A > B > B,X: A,Y: A,Z3: B] :
( ( finite4664212375090638736ute_on @ A @ B @ S2 @ F2 )
=> ( ( member @ A @ X @ S2 )
=> ( ( member @ A @ Y @ S2 )
=> ( ( F2 @ Y @ ( F2 @ X @ Z3 ) )
= ( F2 @ X @ ( F2 @ Y @ Z3 ) ) ) ) ) ) ).
% comp_fun_commute_on.fun_left_comm
thf(fact_5492_finite__update__induct,axiom,
! [B: $tType,A: $tType,F2: A > B,C2: B,P: ( A > B ) > $o] :
( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [A5: A] :
( ( F2 @ A5 )
!= C2 ) ) )
=> ( ( P
@ ^ [A5: A] : C2 )
=> ( ! [A6: A,B5: B,F4: A > B] :
( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [C4: A] :
( ( F4 @ C4 )
!= C2 ) ) )
=> ( ( ( F4 @ A6 )
= C2 )
=> ( ( B5 != C2 )
=> ( ( P @ F4 )
=> ( P @ ( fun_upd @ A @ B @ F4 @ A6 @ B5 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ).
% finite_update_induct
thf(fact_5493_comp__fun__commute__on_Ocomp__fun__commute__on__funpow,axiom,
! [B: $tType,A: $tType,S2: set @ A,F2: A > B > B,G: A > nat] :
( ( finite4664212375090638736ute_on @ A @ B @ S2 @ F2 )
=> ( finite4664212375090638736ute_on @ A @ B @ S2
@ ^ [X4: A] : ( compow @ ( B > B ) @ ( G @ X4 ) @ ( F2 @ X4 ) ) ) ) ).
% comp_fun_commute_on.comp_fun_commute_on_funpow
thf(fact_5494_Finite__Set_Ofold__cong,axiom,
! [B: $tType,A: $tType,S2: set @ A,F2: A > B > B,G: A > B > B,A4: set @ A,S: B,T2: B,B2: set @ A] :
( ( finite4664212375090638736ute_on @ A @ B @ S2 @ F2 )
=> ( ( finite4664212375090638736ute_on @ A @ B @ S2 @ G )
=> ( ( ord_less_eq @ ( set @ A ) @ A4 @ S2 )
=> ( ( finite_finite2 @ A @ A4 )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ A4 )
=> ( ( F2 @ X5 )
= ( G @ X5 ) ) )
=> ( ( S = T2 )
=> ( ( A4 = B2 )
=> ( ( finite_fold @ A @ B @ F2 @ S @ A4 )
= ( finite_fold @ A @ B @ G @ T2 @ B2 ) ) ) ) ) ) ) ) ) ).
% Finite_Set.fold_cong
thf(fact_5495_fun__upd__image,axiom,
! [A: $tType,B: $tType,X: B,A4: set @ B,F2: B > A,Y: A] :
( ( ( member @ B @ X @ A4 )
=> ( ( image2 @ B @ A @ ( fun_upd @ B @ A @ F2 @ X @ Y ) @ A4 )
= ( insert2 @ A @ Y @ ( image2 @ B @ A @ F2 @ ( minus_minus @ ( set @ B ) @ A4 @ ( insert2 @ B @ X @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) )
& ( ~ ( member @ B @ X @ A4 )
=> ( ( image2 @ B @ A @ ( fun_upd @ B @ A @ F2 @ X @ Y ) @ A4 )
= ( image2 @ B @ A @ F2 @ A4 ) ) ) ) ).
% fun_upd_image
thf(fact_5496_comp__fun__commute__on_Ocomp__comp__fun__commute__on,axiom,
! [B: $tType,A: $tType,C: $tType,S2: set @ A,F2: A > B > B,G: C > A,R: set @ C] :
( ( finite4664212375090638736ute_on @ A @ B @ S2 @ F2 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image2 @ C @ A @ G @ ( top_top @ ( set @ C ) ) ) @ S2 )
=> ( finite4664212375090638736ute_on @ C @ B @ R @ ( comp @ A @ ( B > B ) @ C @ F2 @ G ) ) ) ) ).
% comp_fun_commute_on.comp_comp_fun_commute_on
thf(fact_5497_comp__fun__commute__on_Ofold__fun__left__comm,axiom,
! [B: $tType,A: $tType,S2: set @ A,F2: A > B > B,X: A,A4: set @ A,Z3: B] :
( ( finite4664212375090638736ute_on @ A @ B @ S2 @ F2 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( insert2 @ A @ X @ A4 ) @ S2 )
=> ( ( finite_finite2 @ A @ A4 )
=> ( ( F2 @ X @ ( finite_fold @ A @ B @ F2 @ Z3 @ A4 ) )
= ( finite_fold @ A @ B @ F2 @ ( F2 @ X @ Z3 ) @ A4 ) ) ) ) ) ).
% comp_fun_commute_on.fold_fun_left_comm
thf(fact_5498_comp__fun__commute__on_Ofold__insert2,axiom,
! [B: $tType,A: $tType,S2: set @ A,F2: A > B > B,X: A,A4: set @ A,Z3: B] :
( ( finite4664212375090638736ute_on @ A @ B @ S2 @ F2 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( insert2 @ A @ X @ A4 ) @ S2 )
=> ( ( finite_finite2 @ A @ A4 )
=> ( ~ ( member @ A @ X @ A4 )
=> ( ( finite_fold @ A @ B @ F2 @ Z3 @ ( insert2 @ A @ X @ A4 ) )
= ( finite_fold @ A @ B @ F2 @ ( F2 @ X @ Z3 ) @ A4 ) ) ) ) ) ) ).
% comp_fun_commute_on.fold_insert2
thf(fact_5499_comp__fun__commute__on_Ofold__insert,axiom,
! [B: $tType,A: $tType,S2: set @ A,F2: A > B > B,X: A,A4: set @ A,Z3: B] :
( ( finite4664212375090638736ute_on @ A @ B @ S2 @ F2 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( insert2 @ A @ X @ A4 ) @ S2 )
=> ( ( finite_finite2 @ A @ A4 )
=> ( ~ ( member @ A @ X @ A4 )
=> ( ( finite_fold @ A @ B @ F2 @ Z3 @ ( insert2 @ A @ X @ A4 ) )
= ( F2 @ X @ ( finite_fold @ A @ B @ F2 @ Z3 @ A4 ) ) ) ) ) ) ) ).
% comp_fun_commute_on.fold_insert
thf(fact_5500_comp__fun__commute__on_Ofold__insert__remove,axiom,
! [B: $tType,A: $tType,S2: set @ A,F2: A > B > B,X: A,A4: set @ A,Z3: B] :
( ( finite4664212375090638736ute_on @ A @ B @ S2 @ F2 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( insert2 @ A @ X @ A4 ) @ S2 )
=> ( ( finite_finite2 @ A @ A4 )
=> ( ( finite_fold @ A @ B @ F2 @ Z3 @ ( insert2 @ A @ X @ A4 ) )
= ( F2 @ X @ ( finite_fold @ A @ B @ F2 @ Z3 @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ).
% comp_fun_commute_on.fold_insert_remove
thf(fact_5501_Id__on__fold,axiom,
! [A: $tType,A4: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( id_on @ A @ A4 )
= ( finite_fold @ A @ ( set @ ( product_prod @ A @ A ) )
@ ^ [X4: A] : ( insert2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ X4 ) )
@ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) )
@ A4 ) ) ) ).
% Id_on_fold
thf(fact_5502_Id__on__def,axiom,
! [A: $tType] :
( ( id_on @ A )
= ( ^ [A7: set @ A] :
( complete_Sup_Sup @ ( set @ ( product_prod @ A @ A ) )
@ ( image2 @ A @ ( set @ ( product_prod @ A @ A ) )
@ ^ [X4: A] : ( insert2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ X4 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
@ A7 ) ) ) ) ).
% Id_on_def
thf(fact_5503_comp__fun__commute__product__fold,axiom,
! [A: $tType,B: $tType,B2: set @ A] :
( ( finite_finite2 @ A @ B2 )
=> ( finite6289374366891150609ommute @ B @ ( set @ ( product_prod @ B @ A ) )
@ ^ [X4: B,Z6: set @ ( product_prod @ B @ A )] :
( finite_fold @ A @ ( set @ ( product_prod @ B @ A ) )
@ ^ [Y6: A] : ( insert2 @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X4 @ Y6 ) )
@ Z6
@ B2 ) ) ) ).
% comp_fun_commute_product_fold
thf(fact_5504_Id__on__empty,axiom,
! [A: $tType] :
( ( id_on @ A @ ( bot_bot @ ( set @ A ) ) )
= ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ).
% Id_on_empty
thf(fact_5505_comp__fun__commute__const,axiom,
! [B: $tType,A: $tType,F2: B > B] :
( finite6289374366891150609ommute @ A @ B
@ ^ [Uu3: A] : F2 ) ).
% comp_fun_commute_const
thf(fact_5506_comp__fun__commute_Ointro,axiom,
! [B: $tType,A: $tType,F2: A > B > B] :
( ! [Y3: A,X5: A] :
( ( comp @ B @ B @ B @ ( F2 @ Y3 ) @ ( F2 @ X5 ) )
= ( comp @ B @ B @ B @ ( F2 @ X5 ) @ ( F2 @ Y3 ) ) )
=> ( finite6289374366891150609ommute @ A @ B @ F2 ) ) ).
% comp_fun_commute.intro
thf(fact_5507_comp__fun__commute_Ocomp__fun__commute,axiom,
! [B: $tType,A: $tType,F2: A > B > B,Y: A,X: A] :
( ( finite6289374366891150609ommute @ A @ B @ F2 )
=> ( ( comp @ B @ B @ B @ ( F2 @ Y ) @ ( F2 @ X ) )
= ( comp @ B @ B @ B @ ( F2 @ X ) @ ( F2 @ Y ) ) ) ) ).
% comp_fun_commute.comp_fun_commute
thf(fact_5508_comp__fun__commute__def,axiom,
! [B: $tType,A: $tType] :
( ( finite6289374366891150609ommute @ A @ B )
= ( ^ [F3: A > B > B] :
! [Y6: A,X4: A] :
( ( comp @ B @ B @ B @ ( F3 @ Y6 ) @ ( F3 @ X4 ) )
= ( comp @ B @ B @ B @ ( F3 @ X4 ) @ ( F3 @ Y6 ) ) ) ) ) ).
% comp_fun_commute_def
thf(fact_5509_comp__fun__commute_Ocomp__comp__fun__commute,axiom,
! [B: $tType,A: $tType,C: $tType,F2: A > B > B,G: C > A] :
( ( finite6289374366891150609ommute @ A @ B @ F2 )
=> ( finite6289374366891150609ommute @ C @ B @ ( comp @ A @ ( B > B ) @ C @ F2 @ G ) ) ) ).
% comp_fun_commute.comp_comp_fun_commute
thf(fact_5510_comp__fun__commute__filter__fold,axiom,
! [A: $tType,P: A > $o] :
( finite6289374366891150609ommute @ A @ ( set @ A )
@ ^ [X4: A,A17: set @ A] : ( if @ ( set @ A ) @ ( P @ X4 ) @ ( insert2 @ A @ X4 @ A17 ) @ A17 ) ) ).
% comp_fun_commute_filter_fold
thf(fact_5511_comp__fun__commute_Ocomp__fun__commute__funpow,axiom,
! [B: $tType,A: $tType,F2: A > B > B,G: A > nat] :
( ( finite6289374366891150609ommute @ A @ B @ F2 )
=> ( finite6289374366891150609ommute @ A @ B
@ ^ [X4: A] : ( compow @ ( B > B ) @ ( G @ X4 ) @ ( F2 @ X4 ) ) ) ) ).
% comp_fun_commute.comp_fun_commute_funpow
thf(fact_5512_comp__fun__commute__def_H,axiom,
! [B: $tType,A: $tType] :
( ( finite6289374366891150609ommute @ A @ B )
= ( finite4664212375090638736ute_on @ A @ B @ ( top_top @ ( set @ A ) ) ) ) ).
% comp_fun_commute_def'
thf(fact_5513_finite__range__updI,axiom,
! [A: $tType,B: $tType,F2: B > ( option @ A ),A3: B,B3: A] :
( ( finite_finite2 @ ( option @ A ) @ ( image2 @ B @ ( option @ A ) @ F2 @ ( top_top @ ( set @ B ) ) ) )
=> ( finite_finite2 @ ( option @ A ) @ ( image2 @ B @ ( option @ A ) @ ( fun_upd @ B @ ( option @ A ) @ F2 @ A3 @ ( some @ A @ B3 ) ) @ ( top_top @ ( set @ B ) ) ) ) ) ).
% finite_range_updI
thf(fact_5514_relpow__finite__bounded1,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),K: nat] :
( ( finite_finite2 @ ( product_prod @ A @ A ) @ R )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ K @ R )
@ ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ A ) )
@ ( image2 @ nat @ ( set @ ( product_prod @ A @ A ) )
@ ^ [N: nat] : ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R )
@ ( collect @ nat
@ ^ [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
& ( ord_less_eq @ nat @ N @ ( finite_card @ ( product_prod @ A @ A ) @ R ) ) ) ) ) ) ) ) ) ).
% relpow_finite_bounded1
thf(fact_5515_set__nths,axiom,
! [A: $tType,Xs: list @ A,I6: set @ nat] :
( ( set2 @ A @ ( nths @ A @ Xs @ I6 ) )
= ( collect @ A
@ ^ [Uu3: A] :
? [I3: nat] :
( ( Uu3
= ( nth @ A @ Xs @ I3 ) )
& ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs ) )
& ( member @ nat @ I3 @ I6 ) ) ) ) ).
% set_nths
thf(fact_5516_nths__nil,axiom,
! [A: $tType,A4: set @ nat] :
( ( nths @ A @ ( nil @ A ) @ A4 )
= ( nil @ A ) ) ).
% nths_nil
thf(fact_5517_nths__upt__eq__take,axiom,
! [A: $tType,L: list @ A,N2: nat] :
( ( nths @ A @ L @ ( set_ord_lessThan @ nat @ N2 ) )
= ( take @ A @ N2 @ L ) ) ).
% nths_upt_eq_take
thf(fact_5518_nths__empty,axiom,
! [A: $tType,Xs: list @ A] :
( ( nths @ A @ Xs @ ( bot_bot @ ( set @ nat ) ) )
= ( nil @ A ) ) ).
% nths_empty
thf(fact_5519_finite__relpow,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),N2: nat] :
( ( finite_finite2 @ ( product_prod @ A @ A ) @ R )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( finite_finite2 @ ( product_prod @ A @ A ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N2 @ R ) ) ) ) ).
% finite_relpow
thf(fact_5520_notin__set__nthsI,axiom,
! [A: $tType,X: A,Xs: list @ A,I6: set @ nat] :
( ~ ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ~ ( member @ A @ X @ ( set2 @ A @ ( nths @ A @ Xs @ I6 ) ) ) ) ).
% notin_set_nthsI
thf(fact_5521_in__set__nthsD,axiom,
! [A: $tType,X: A,Xs: list @ A,I6: set @ nat] :
( ( member @ A @ X @ ( set2 @ A @ ( nths @ A @ Xs @ I6 ) ) )
=> ( member @ A @ X @ ( set2 @ A @ Xs ) ) ) ).
% in_set_nthsD
thf(fact_5522_distinct__nthsI,axiom,
! [A: $tType,Xs: list @ A,I6: set @ nat] :
( ( distinct @ A @ Xs )
=> ( distinct @ A @ ( nths @ A @ Xs @ I6 ) ) ) ).
% distinct_nthsI
thf(fact_5523_relpow__Suc__I2,axiom,
! [A: $tType,X: A,Y: A,R: set @ ( product_prod @ A @ A ),Z3: A,N2: nat] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z3 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N2 @ R ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z3 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( suc @ N2 ) @ R ) ) ) ) ).
% relpow_Suc_I2
thf(fact_5524_relpow__Suc__E2,axiom,
! [A: $tType,X: A,Z3: A,N2: nat,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z3 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( suc @ N2 ) @ R ) )
=> ~ ! [Y3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y3 ) @ R )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Z3 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N2 @ R ) ) ) ) ).
% relpow_Suc_E2
thf(fact_5525_relpow__Suc__D2,axiom,
! [A: $tType,X: A,Z3: A,N2: nat,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z3 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( suc @ N2 ) @ R ) )
=> ? [Y3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y3 ) @ R )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Z3 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N2 @ R ) ) ) ) ).
% relpow_Suc_D2
thf(fact_5526_relpow__Suc__I,axiom,
! [A: $tType,X: A,Y: A,N2: nat,R: set @ ( product_prod @ A @ A ),Z3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N2 @ R ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z3 ) @ R )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z3 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( suc @ N2 ) @ R ) ) ) ) ).
% relpow_Suc_I
thf(fact_5527_relpow__Suc__E,axiom,
! [A: $tType,X: A,Z3: A,N2: nat,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z3 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( suc @ N2 ) @ R ) )
=> ~ ! [Y3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y3 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N2 @ R ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Z3 ) @ R ) ) ) ).
% relpow_Suc_E
thf(fact_5528_relpow__0__E,axiom,
! [A: $tType,X: A,Y: A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( zero_zero @ nat ) @ R ) )
=> ( X = Y ) ) ).
% relpow_0_E
thf(fact_5529_relpow__0__I,axiom,
! [A: $tType,X: A,R: set @ ( product_prod @ A @ A )] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ X ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( zero_zero @ nat ) @ R ) ) ).
% relpow_0_I
thf(fact_5530_set__nths__subset,axiom,
! [A: $tType,Xs: list @ A,I6: set @ nat] : ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ ( nths @ A @ Xs @ I6 ) ) @ ( set2 @ A @ Xs ) ) ).
% set_nths_subset
thf(fact_5531_nths__all,axiom,
! [A: $tType,Xs: list @ A,I6: set @ nat] :
( ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( member @ nat @ I4 @ I6 ) )
=> ( ( nths @ A @ Xs @ I6 )
= Xs ) ) ).
% nths_all
thf(fact_5532_comp__fun__commute__insort,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( finite6289374366891150609ommute @ A @ ( list @ A )
@ ( linorder_insort_key @ A @ A
@ ^ [X4: A] : X4 ) ) ) ).
% comp_fun_commute_insort
thf(fact_5533_relpow__E,axiom,
! [A: $tType,X: A,Z3: A,N2: nat,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z3 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N2 @ R ) )
=> ( ( ( N2
= ( zero_zero @ nat ) )
=> ( X != Z3 ) )
=> ~ ! [Y3: A,M3: nat] :
( ( N2
= ( suc @ M3 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y3 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ M3 @ R ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Z3 ) @ R ) ) ) ) ) ).
% relpow_E
thf(fact_5534_relpow__E2,axiom,
! [A: $tType,X: A,Z3: A,N2: nat,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z3 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N2 @ R ) )
=> ( ( ( N2
= ( zero_zero @ nat ) )
=> ( X != Z3 ) )
=> ~ ! [Y3: A,M3: nat] :
( ( N2
= ( suc @ M3 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y3 ) @ R )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Z3 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ M3 @ R ) ) ) ) ) ) ).
% relpow_E2
thf(fact_5535_relpow__empty,axiom,
! [A: $tType,N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N2 @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
= ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ).
% relpow_empty
thf(fact_5536_fold__atLeastAtMost__nat,axiom,
! [A: $tType,F2: nat > A > A,A3: nat,B3: nat,Acc3: A] :
( ( finite6289374366891150609ommute @ nat @ A @ F2 )
=> ( ( set_fo6178422350223883121st_nat @ A @ F2 @ A3 @ B3 @ Acc3 )
= ( finite_fold @ nat @ A @ F2 @ Acc3 @ ( set_or1337092689740270186AtMost @ nat @ A3 @ B3 ) ) ) ) ).
% fold_atLeastAtMost_nat
thf(fact_5537_relpow__fun__conv,axiom,
! [A: $tType,A3: A,B3: A,N2: nat,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B3 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N2 @ R ) )
= ( ? [F3: nat > A] :
( ( ( F3 @ ( zero_zero @ nat ) )
= A3 )
& ( ( F3 @ N2 )
= B3 )
& ! [I3: nat] :
( ( ord_less @ nat @ I3 @ N2 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( F3 @ I3 ) @ ( F3 @ ( suc @ I3 ) ) ) @ R ) ) ) ) ) ).
% relpow_fun_conv
thf(fact_5538_length__nths,axiom,
! [A: $tType,Xs: list @ A,I6: set @ nat] :
( ( size_size @ ( list @ A ) @ ( nths @ A @ Xs @ I6 ) )
= ( finite_card @ nat
@ ( collect @ nat
@ ^ [I3: nat] :
( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs ) )
& ( member @ nat @ I3 @ I6 ) ) ) ) ) ).
% length_nths
thf(fact_5539_relpow__finite__bounded,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),K: nat] :
( ( finite_finite2 @ ( product_prod @ A @ A ) @ R )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ K @ R )
@ ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ A ) )
@ ( image2 @ nat @ ( set @ ( product_prod @ A @ A ) )
@ ^ [N: nat] : ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R )
@ ( collect @ nat
@ ^ [N: nat] : ( ord_less_eq @ nat @ N @ ( finite_card @ ( product_prod @ A @ A ) @ R ) ) ) ) ) ) ) ).
% relpow_finite_bounded
thf(fact_5540_comp__fun__commute__Pow__fold,axiom,
! [A: $tType] :
( finite6289374366891150609ommute @ A @ ( set @ ( set @ A ) )
@ ^ [X4: A,A7: set @ ( set @ A )] : ( sup_sup @ ( set @ ( set @ A ) ) @ A7 @ ( image2 @ ( set @ A ) @ ( set @ A ) @ ( insert2 @ A @ X4 ) @ A7 ) ) ) ).
% comp_fun_commute_Pow_fold
thf(fact_5541_ntrancl__def,axiom,
! [A: $tType] :
( ( transitive_ntrancl @ A )
= ( ^ [N: nat,R5: set @ ( product_prod @ A @ A )] :
( complete_Sup_Sup @ ( set @ ( product_prod @ A @ A ) )
@ ( image2 @ nat @ ( set @ ( product_prod @ A @ A ) )
@ ^ [I3: nat] : ( compow @ ( set @ ( product_prod @ A @ A ) ) @ I3 @ R5 )
@ ( collect @ nat
@ ^ [I3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ I3 )
& ( ord_less_eq @ nat @ I3 @ ( suc @ N ) ) ) ) ) ) ) ) ).
% ntrancl_def
thf(fact_5542_trancl__finite__eq__relpow,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( finite_finite2 @ ( product_prod @ A @ A ) @ R )
=> ( ( transitive_trancl @ A @ R )
= ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ A ) )
@ ( image2 @ nat @ ( set @ ( product_prod @ A @ A ) )
@ ^ [N: nat] : ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R )
@ ( collect @ nat
@ ^ [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
& ( ord_less_eq @ nat @ N @ ( finite_card @ ( product_prod @ A @ A ) @ R ) ) ) ) ) ) ) ) ).
% trancl_finite_eq_relpow
thf(fact_5543_restrict__upd__same,axiom,
! [B: $tType,A: $tType,M2: A > ( option @ B ),X: A,Y: B] :
( ( restrict_map @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ M2 @ X @ ( some @ B @ Y ) ) @ ( uminus_uminus @ ( set @ A ) @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( restrict_map @ A @ B @ M2 @ ( uminus_uminus @ ( set @ A ) @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% restrict_upd_same
thf(fact_5544_trancl__empty,axiom,
! [A: $tType] :
( ( transitive_trancl @ A @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
= ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ).
% trancl_empty
thf(fact_5545_finite__trancl,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( finite_finite2 @ ( product_prod @ A @ A ) @ ( transitive_trancl @ A @ R2 ) )
= ( finite_finite2 @ ( product_prod @ A @ A ) @ R2 ) ) ).
% finite_trancl
thf(fact_5546_ntrancl__Zero,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( transitive_ntrancl @ A @ ( zero_zero @ nat ) @ R )
= R ) ).
% ntrancl_Zero
thf(fact_5547_restrict__map__to__empty,axiom,
! [B: $tType,A: $tType,M2: A > ( option @ B )] :
( ( restrict_map @ A @ B @ M2 @ ( bot_bot @ ( set @ A ) ) )
= ( ^ [X4: A] : ( none @ B ) ) ) ).
% restrict_map_to_empty
thf(fact_5548_restrict__fun__upd,axiom,
! [B: $tType,A: $tType,X: A,D6: set @ A,M2: A > ( option @ B ),Y: option @ B] :
( ( ( member @ A @ X @ D6 )
=> ( ( restrict_map @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ M2 @ X @ Y ) @ D6 )
= ( fun_upd @ A @ ( option @ B ) @ ( restrict_map @ A @ B @ M2 @ ( minus_minus @ ( set @ A ) @ D6 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) @ X @ Y ) ) )
& ( ~ ( member @ A @ X @ D6 )
=> ( ( restrict_map @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ M2 @ X @ Y ) @ D6 )
= ( restrict_map @ A @ B @ M2 @ D6 ) ) ) ) ).
% restrict_fun_upd
thf(fact_5549_fun__upd__restrict__conv,axiom,
! [A: $tType,B: $tType,X: A,D6: set @ A,M2: A > ( option @ B ),Y: option @ B] :
( ( member @ A @ X @ D6 )
=> ( ( fun_upd @ A @ ( option @ B ) @ ( restrict_map @ A @ B @ M2 @ D6 ) @ X @ Y )
= ( fun_upd @ A @ ( option @ B ) @ ( restrict_map @ A @ B @ M2 @ ( minus_minus @ ( set @ A ) @ D6 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) @ X @ Y ) ) ) ).
% fun_upd_restrict_conv
thf(fact_5550_fun__upd__None__restrict,axiom,
! [B: $tType,A: $tType,X: A,D6: set @ A,M2: A > ( option @ B )] :
( ( ( member @ A @ X @ D6 )
=> ( ( fun_upd @ A @ ( option @ B ) @ ( restrict_map @ A @ B @ M2 @ D6 ) @ X @ ( none @ B ) )
= ( restrict_map @ A @ B @ M2 @ ( minus_minus @ ( set @ A ) @ D6 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) )
& ( ~ ( member @ A @ X @ D6 )
=> ( ( fun_upd @ A @ ( option @ B ) @ ( restrict_map @ A @ B @ M2 @ D6 ) @ X @ ( none @ B ) )
= ( restrict_map @ A @ B @ M2 @ D6 ) ) ) ) ).
% fun_upd_None_restrict
thf(fact_5551_trancl__set__ntrancl,axiom,
! [A: $tType,Xs: list @ ( product_prod @ A @ A )] :
( ( transitive_trancl @ A @ ( set2 @ ( product_prod @ A @ A ) @ Xs ) )
= ( transitive_ntrancl @ A @ ( minus_minus @ nat @ ( finite_card @ ( product_prod @ A @ A ) @ ( set2 @ ( product_prod @ A @ A ) @ Xs ) ) @ ( one_one @ nat ) ) @ ( set2 @ ( product_prod @ A @ A ) @ Xs ) ) ) ).
% trancl_set_ntrancl
thf(fact_5552_finite__trancl__ntranl,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( finite_finite2 @ ( product_prod @ A @ A ) @ R )
=> ( ( transitive_trancl @ A @ R )
= ( transitive_ntrancl @ A @ ( minus_minus @ nat @ ( finite_card @ ( product_prod @ A @ A ) @ R ) @ ( one_one @ nat ) ) @ R ) ) ) ).
% finite_trancl_ntranl
thf(fact_5553_trancl__power,axiom,
! [A: $tType,P5: product_prod @ A @ A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ P5 @ ( transitive_trancl @ A @ R ) )
= ( ? [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
& ( member @ ( product_prod @ A @ A ) @ P5 @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R ) ) ) ) ) ).
% trancl_power
thf(fact_5554_less__eq,axiom,
! [M2: nat,N2: nat] :
( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ M2 @ N2 ) @ ( transitive_trancl @ nat @ pred_nat ) )
= ( ord_less @ nat @ M2 @ N2 ) ) ).
% less_eq
thf(fact_5555_fun__upd__restrict,axiom,
! [A: $tType,B: $tType,M2: A > ( option @ B ),D6: set @ A,X: A,Y: option @ B] :
( ( fun_upd @ A @ ( option @ B ) @ ( restrict_map @ A @ B @ M2 @ D6 ) @ X @ Y )
= ( fun_upd @ A @ ( option @ B ) @ ( restrict_map @ A @ B @ M2 @ ( minus_minus @ ( set @ A ) @ D6 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) @ X @ Y ) ) ).
% fun_upd_restrict
thf(fact_5556_restrict__complement__singleton__eq,axiom,
! [A: $tType,B: $tType,F2: A > ( option @ B ),X: A] :
( ( restrict_map @ A @ B @ F2 @ ( uminus_uminus @ ( set @ A ) @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( fun_upd @ A @ ( option @ B ) @ F2 @ X @ ( none @ B ) ) ) ).
% restrict_complement_singleton_eq
thf(fact_5557_restrict__map__upds,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B,D6: set @ A,M2: A > ( option @ B )] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ D6 )
=> ( ( restrict_map @ A @ B @ ( map_upds @ A @ B @ M2 @ Xs @ Ys ) @ D6 )
= ( map_upds @ A @ B @ ( restrict_map @ A @ B @ M2 @ ( minus_minus @ ( set @ A ) @ D6 @ ( set2 @ A @ Xs ) ) ) @ Xs @ Ys ) ) ) ) ).
% restrict_map_upds
thf(fact_5558_rtrancl__finite__eq__relpow,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( finite_finite2 @ ( product_prod @ A @ A ) @ R )
=> ( ( transitive_rtrancl @ A @ R )
= ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ A ) )
@ ( image2 @ nat @ ( set @ ( product_prod @ A @ A ) )
@ ^ [N: nat] : ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R )
@ ( collect @ nat
@ ^ [N: nat] : ( ord_less_eq @ nat @ N @ ( finite_card @ ( product_prod @ A @ A ) @ R ) ) ) ) ) ) ) ).
% rtrancl_finite_eq_relpow
thf(fact_5559_minus__fold__remove,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( minus_minus @ ( set @ A ) @ B2 @ A4 )
= ( finite_fold @ A @ ( set @ A ) @ ( remove @ A ) @ B2 @ A4 ) ) ) ).
% minus_fold_remove
thf(fact_5560_member__remove,axiom,
! [A: $tType,X: A,Y: A,A4: set @ A] :
( ( member @ A @ X @ ( remove @ A @ Y @ A4 ) )
= ( ( member @ A @ X @ A4 )
& ( X != Y ) ) ) ).
% member_remove
thf(fact_5561_map__upds__apply__nontin,axiom,
! [B: $tType,A: $tType,X: A,Xs: list @ A,F2: A > ( option @ B ),Ys: list @ B] :
( ~ ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ( map_upds @ A @ B @ F2 @ Xs @ Ys @ X )
= ( F2 @ X ) ) ) ).
% map_upds_apply_nontin
thf(fact_5562_map__upds__list__update2__drop,axiom,
! [A: $tType,B: $tType,Xs: list @ A,I: nat,M2: A > ( option @ B ),Ys: list @ B,Y: B] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ I )
=> ( ( map_upds @ A @ B @ M2 @ Xs @ ( list_update @ B @ Ys @ I @ Y ) )
= ( map_upds @ A @ B @ M2 @ Xs @ Ys ) ) ) ).
% map_upds_list_update2_drop
thf(fact_5563_map__upds__twist,axiom,
! [A: $tType,B: $tType,A3: A,As: list @ A,M2: A > ( option @ B ),B3: B,Bs: list @ B] :
( ~ ( member @ A @ A3 @ ( set2 @ A @ As ) )
=> ( ( map_upds @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ M2 @ A3 @ ( some @ B @ B3 ) ) @ As @ Bs )
= ( fun_upd @ A @ ( option @ B ) @ ( map_upds @ A @ B @ M2 @ As @ Bs ) @ A3 @ ( some @ B @ B3 ) ) ) ) ).
% map_upds_twist
thf(fact_5564_listrel1__rtrancl__subset__rtrancl__listrel1,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] : ( ord_less_eq @ ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) @ ( listrel1 @ A @ ( transitive_rtrancl @ A @ R2 ) ) @ ( transitive_rtrancl @ ( list @ A ) @ ( listrel1 @ A @ R2 ) ) ) ).
% listrel1_rtrancl_subset_rtrancl_listrel1
thf(fact_5565_rtrancl__listrel1__eq__len,axiom,
! [A: $tType,X: list @ A,Y: list @ A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Y ) @ ( transitive_rtrancl @ ( list @ A ) @ ( listrel1 @ A @ R2 ) ) )
=> ( ( size_size @ ( list @ A ) @ X )
= ( size_size @ ( list @ A ) @ Y ) ) ) ).
% rtrancl_listrel1_eq_len
thf(fact_5566_pred__nat__trancl__eq__le,axiom,
! [M2: nat,N2: nat] :
( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ M2 @ N2 ) @ ( transitive_rtrancl @ nat @ pred_nat ) )
= ( ord_less_eq @ nat @ M2 @ N2 ) ) ).
% pred_nat_trancl_eq_le
thf(fact_5567_remove__code_I1_J,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( remove @ A @ X @ ( set2 @ A @ Xs ) )
= ( set2 @ A @ ( removeAll @ A @ X @ Xs ) ) ) ).
% remove_code(1)
thf(fact_5568_remove__def,axiom,
! [A: $tType] :
( ( remove @ A )
= ( ^ [X4: A,A7: set @ A] : ( minus_minus @ ( set @ A ) @ A7 @ ( insert2 @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% remove_def
thf(fact_5569_map__upd__upds__conv__if,axiom,
! [A: $tType,B: $tType,X: A,Ys: list @ B,Xs: list @ A,F2: A > ( option @ B ),Y: B] :
( ( ( member @ A @ X @ ( set2 @ A @ ( take @ A @ ( size_size @ ( list @ B ) @ Ys ) @ Xs ) ) )
=> ( ( map_upds @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ F2 @ X @ ( some @ B @ Y ) ) @ Xs @ Ys )
= ( map_upds @ A @ B @ F2 @ Xs @ Ys ) ) )
& ( ~ ( member @ A @ X @ ( set2 @ A @ ( take @ A @ ( size_size @ ( list @ B ) @ Ys ) @ Xs ) ) )
=> ( ( map_upds @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ F2 @ X @ ( some @ B @ Y ) ) @ Xs @ Ys )
= ( fun_upd @ A @ ( option @ B ) @ ( map_upds @ A @ B @ F2 @ Xs @ Ys ) @ X @ ( some @ B @ Y ) ) ) ) ) ).
% map_upd_upds_conv_if
thf(fact_5570_isCont__If__ge,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo1944317154257567458pology @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [A3: A,G: A > B,F2: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A3 @ ( set_ord_lessThan @ A @ A3 ) ) @ G )
=> ( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ ( G @ A3 ) ) @ ( topolo174197925503356063within @ A @ A3 @ ( set_ord_greaterThan @ A @ A3 ) ) )
=> ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) )
@ ^ [X4: A] : ( if @ B @ ( ord_less_eq @ A @ X4 @ A3 ) @ ( G @ X4 ) @ ( F2 @ X4 ) ) ) ) ) ) ).
% isCont_If_ge
thf(fact_5571_finite__subsets__at__top__finite,axiom,
! [A: $tType,A4: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( finite5375528669736107172at_top @ A @ A4 )
= ( principal @ ( set @ A ) @ ( insert2 @ ( set @ A ) @ A4 @ ( bot_bot @ ( set @ ( set @ A ) ) ) ) ) ) ) ).
% finite_subsets_at_top_finite
thf(fact_5572_Max_Oeq__fold_H,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( lattic643756798349783984er_Max @ A )
= ( ^ [A7: set @ A] :
( the2 @ A
@ ( finite_fold @ A @ ( option @ A )
@ ^ [X4: A,Y6: option @ A] : ( some @ A @ ( case_option @ A @ A @ X4 @ ( ord_max @ A @ X4 ) @ Y6 ) )
@ ( none @ A )
@ A7 ) ) ) ) ) ).
% Max.eq_fold'
thf(fact_5573_eventually__finite__subsets__at__top__weakI,axiom,
! [A: $tType,A4: set @ A,P: ( set @ A ) > $o] :
( ! [X9: set @ A] :
( ( finite_finite2 @ A @ X9 )
=> ( ( ord_less_eq @ ( set @ A ) @ X9 @ A4 )
=> ( P @ X9 ) ) )
=> ( eventually @ ( set @ A ) @ P @ ( finite5375528669736107172at_top @ A @ A4 ) ) ) ).
% eventually_finite_subsets_at_top_weakI
thf(fact_5574_continuous__bot,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [F2: A > B] : ( topolo3448309680560233919inuous @ A @ B @ ( bot_bot @ ( filter @ A ) ) @ F2 ) ) ).
% continuous_bot
thf(fact_5575_continuous__trivial__limit,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [Net: filter @ A,F2: A > B] :
( ( Net
= ( bot_bot @ ( filter @ A ) ) )
=> ( topolo3448309680560233919inuous @ A @ B @ Net @ F2 ) ) ) ).
% continuous_trivial_limit
thf(fact_5576_finite__subsets__at__top__neq__bot,axiom,
! [A: $tType,A4: set @ A] :
( ( finite5375528669736107172at_top @ A @ A4 )
!= ( bot_bot @ ( filter @ ( set @ A ) ) ) ) ).
% finite_subsets_at_top_neq_bot
thf(fact_5577_continuous__add,axiom,
! [B: $tType,D: $tType] :
( ( ( topological_t2_space @ D )
& ( topolo6943815403480290642id_add @ B ) )
=> ! [F5: filter @ D,F2: D > B,G: D > B] :
( ( topolo3448309680560233919inuous @ D @ B @ F5 @ F2 )
=> ( ( topolo3448309680560233919inuous @ D @ B @ F5 @ G )
=> ( topolo3448309680560233919inuous @ D @ B @ F5
@ ^ [X4: D] : ( plus_plus @ B @ ( F2 @ X4 ) @ ( G @ X4 ) ) ) ) ) ) ).
% continuous_add
thf(fact_5578_eventually__finite__subsets__at__top__finite,axiom,
! [A: $tType,A4: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ A4 )
=> ( ( eventually @ ( set @ A ) @ P @ ( finite5375528669736107172at_top @ A @ A4 ) )
= ( P @ A4 ) ) ) ).
% eventually_finite_subsets_at_top_finite
thf(fact_5579_IVT,axiom,
! [A: $tType,B: $tType] :
( ( ( topolo1944317154257567458pology @ B )
& ( topolo8458572112393995274pology @ A ) )
=> ! [F2: A > B,A3: A,Y: B,B3: A] :
( ( ord_less_eq @ B @ ( F2 @ A3 ) @ Y )
=> ( ( ord_less_eq @ B @ Y @ ( F2 @ B3 ) )
=> ( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ! [X5: A] :
( ( ( ord_less_eq @ A @ A3 @ X5 )
& ( ord_less_eq @ A @ X5 @ B3 ) )
=> ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ X5 @ ( top_top @ ( set @ A ) ) ) @ F2 ) )
=> ? [X5: A] :
( ( ord_less_eq @ A @ A3 @ X5 )
& ( ord_less_eq @ A @ X5 @ B3 )
& ( ( F2 @ X5 )
= Y ) ) ) ) ) ) ) ).
% IVT
thf(fact_5580_IVT2,axiom,
! [A: $tType,B: $tType] :
( ( ( topolo1944317154257567458pology @ B )
& ( topolo8458572112393995274pology @ A ) )
=> ! [F2: A > B,B3: A,Y: B,A3: A] :
( ( ord_less_eq @ B @ ( F2 @ B3 ) @ Y )
=> ( ( ord_less_eq @ B @ Y @ ( F2 @ A3 ) )
=> ( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ! [X5: A] :
( ( ( ord_less_eq @ A @ A3 @ X5 )
& ( ord_less_eq @ A @ X5 @ B3 ) )
=> ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ X5 @ ( top_top @ ( set @ A ) ) ) @ F2 ) )
=> ? [X5: A] :
( ( ord_less_eq @ A @ A3 @ X5 )
& ( ord_less_eq @ A @ X5 @ B3 )
& ( ( F2 @ X5 )
= Y ) ) ) ) ) ) ) ).
% IVT2
thf(fact_5581_continuous__at__within__divide,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_V3459762299906320749_field @ B ) )
=> ! [A3: A,S: set @ A,F2: A > B,G: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A3 @ S ) @ F2 )
=> ( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A3 @ S ) @ G )
=> ( ( ( G @ A3 )
!= ( zero_zero @ B ) )
=> ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A3 @ S )
@ ^ [X4: A] : ( divide_divide @ B @ ( F2 @ X4 ) @ ( G @ X4 ) ) ) ) ) ) ) ).
% continuous_at_within_divide
thf(fact_5582_isCont__add,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( topolo6943815403480290642id_add @ B ) )
=> ! [A3: A,F2: A > B,G: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) @ F2 )
=> ( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) @ G )
=> ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) )
@ ^ [X4: A] : ( plus_plus @ B @ ( F2 @ X4 ) @ ( G @ X4 ) ) ) ) ) ) ).
% isCont_add
thf(fact_5583_continuous__at__within__inverse,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_V8999393235501362500lgebra @ B ) )
=> ! [A3: A,S: set @ A,F2: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A3 @ S ) @ F2 )
=> ( ( ( F2 @ A3 )
!= ( zero_zero @ B ) )
=> ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A3 @ S )
@ ^ [X4: A] : ( inverse_inverse @ B @ ( F2 @ X4 ) ) ) ) ) ) ).
% continuous_at_within_inverse
thf(fact_5584_continuous__at__within__sgn,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [A3: A,S: set @ A,F2: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A3 @ S ) @ F2 )
=> ( ( ( F2 @ A3 )
!= ( zero_zero @ B ) )
=> ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A3 @ S )
@ ^ [X4: A] : ( sgn_sgn @ B @ ( F2 @ X4 ) ) ) ) ) ) ).
% continuous_at_within_sgn
thf(fact_5585_continuous__at__within__power__int,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_V8999393235501362500lgebra @ B ) )
=> ! [A3: A,S: set @ A,F2: A > B,N2: int] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A3 @ S ) @ F2 )
=> ( ( ( F2 @ A3 )
!= ( zero_zero @ B ) )
=> ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A3 @ S )
@ ^ [X4: A] : ( power_int @ B @ ( F2 @ X4 ) @ N2 ) ) ) ) ) ).
% continuous_at_within_power_int
thf(fact_5586_continuous__divide,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_V3459762299906320749_field @ B ) )
=> ! [F5: filter @ A,F2: A > B,G: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ F5 @ F2 )
=> ( ( topolo3448309680560233919inuous @ A @ B @ F5 @ G )
=> ( ( ( G
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F5
@ ^ [X4: A] : X4 ) )
!= ( zero_zero @ B ) )
=> ( topolo3448309680560233919inuous @ A @ B @ F5
@ ^ [X4: A] : ( divide_divide @ B @ ( F2 @ X4 ) @ ( G @ X4 ) ) ) ) ) ) ) ).
% continuous_divide
thf(fact_5587_continuous__inverse,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_V8999393235501362500lgebra @ B ) )
=> ! [F5: filter @ A,F2: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ F5 @ F2 )
=> ( ( ( F2
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F5
@ ^ [X4: A] : X4 ) )
!= ( zero_zero @ B ) )
=> ( topolo3448309680560233919inuous @ A @ B @ F5
@ ^ [X4: A] : ( inverse_inverse @ B @ ( F2 @ X4 ) ) ) ) ) ) ).
% continuous_inverse
thf(fact_5588_continuous__sgn,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F5: filter @ A,F2: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ F5 @ F2 )
=> ( ( ( F2
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F5
@ ^ [X4: A] : X4 ) )
!= ( zero_zero @ B ) )
=> ( topolo3448309680560233919inuous @ A @ B @ F5
@ ^ [X4: A] : ( sgn_sgn @ B @ ( F2 @ X4 ) ) ) ) ) ) ).
% continuous_sgn
thf(fact_5589_continuous__power__int,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_V8999393235501362500lgebra @ B ) )
=> ! [F5: filter @ A,F2: A > B,N2: int] :
( ( topolo3448309680560233919inuous @ A @ B @ F5 @ F2 )
=> ( ( ( F2
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F5
@ ^ [X4: A] : X4 ) )
!= ( zero_zero @ B ) )
=> ( topolo3448309680560233919inuous @ A @ B @ F5
@ ^ [X4: A] : ( power_int @ B @ ( F2 @ X4 ) @ N2 ) ) ) ) ) ).
% continuous_power_int
thf(fact_5590_isCont__eq__Lb,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [A3: real,B3: real,F2: real > A] :
( ( ord_less_eq @ real @ A3 @ B3 )
=> ( ! [X5: real] :
( ( ( ord_less_eq @ real @ A3 @ X5 )
& ( ord_less_eq @ real @ X5 @ B3 ) )
=> ( topolo3448309680560233919inuous @ real @ A @ ( topolo174197925503356063within @ real @ X5 @ ( top_top @ ( set @ real ) ) ) @ F2 ) )
=> ? [M8: A] :
( ! [X2: real] :
( ( ( ord_less_eq @ real @ A3 @ X2 )
& ( ord_less_eq @ real @ X2 @ B3 ) )
=> ( ord_less_eq @ A @ M8 @ ( F2 @ X2 ) ) )
& ? [X5: real] :
( ( ord_less_eq @ real @ A3 @ X5 )
& ( ord_less_eq @ real @ X5 @ B3 )
& ( ( F2 @ X5 )
= M8 ) ) ) ) ) ) ).
% isCont_eq_Lb
thf(fact_5591_isCont__eq__Ub,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [A3: real,B3: real,F2: real > A] :
( ( ord_less_eq @ real @ A3 @ B3 )
=> ( ! [X5: real] :
( ( ( ord_less_eq @ real @ A3 @ X5 )
& ( ord_less_eq @ real @ X5 @ B3 ) )
=> ( topolo3448309680560233919inuous @ real @ A @ ( topolo174197925503356063within @ real @ X5 @ ( top_top @ ( set @ real ) ) ) @ F2 ) )
=> ? [M8: A] :
( ! [X2: real] :
( ( ( ord_less_eq @ real @ A3 @ X2 )
& ( ord_less_eq @ real @ X2 @ B3 ) )
=> ( ord_less_eq @ A @ ( F2 @ X2 ) @ M8 ) )
& ? [X5: real] :
( ( ord_less_eq @ real @ A3 @ X5 )
& ( ord_less_eq @ real @ X5 @ B3 )
& ( ( F2 @ X5 )
= M8 ) ) ) ) ) ) ).
% isCont_eq_Ub
thf(fact_5592_isCont__bounded,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [A3: real,B3: real,F2: real > A] :
( ( ord_less_eq @ real @ A3 @ B3 )
=> ( ! [X5: real] :
( ( ( ord_less_eq @ real @ A3 @ X5 )
& ( ord_less_eq @ real @ X5 @ B3 ) )
=> ( topolo3448309680560233919inuous @ real @ A @ ( topolo174197925503356063within @ real @ X5 @ ( top_top @ ( set @ real ) ) ) @ F2 ) )
=> ? [M8: A] :
! [X2: real] :
( ( ( ord_less_eq @ real @ A3 @ X2 )
& ( ord_less_eq @ real @ X2 @ B3 ) )
=> ( ord_less_eq @ A @ ( F2 @ X2 ) @ M8 ) ) ) ) ) ).
% isCont_bounded
thf(fact_5593_Max_Oinfinite,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A] :
( ~ ( finite_finite2 @ A @ A4 )
=> ( ( lattic643756798349783984er_Max @ A @ A4 )
= ( the2 @ A @ ( none @ A ) ) ) ) ) ).
% Max.infinite
thf(fact_5594_isCont__divide,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_V3459762299906320749_field @ B ) )
=> ! [A3: A,F2: A > B,G: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) @ F2 )
=> ( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) @ G )
=> ( ( ( G @ A3 )
!= ( zero_zero @ B ) )
=> ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) )
@ ^ [X4: A] : ( divide_divide @ B @ ( F2 @ X4 ) @ ( G @ X4 ) ) ) ) ) ) ) ).
% isCont_divide
thf(fact_5595_eventually__finite__subsets__at__top,axiom,
! [A: $tType,P: ( set @ A ) > $o,A4: set @ A] :
( ( eventually @ ( set @ A ) @ P @ ( finite5375528669736107172at_top @ A @ A4 ) )
= ( ? [X6: set @ A] :
( ( finite_finite2 @ A @ X6 )
& ( ord_less_eq @ ( set @ A ) @ X6 @ A4 )
& ! [Y8: set @ A] :
( ( ( finite_finite2 @ A @ Y8 )
& ( ord_less_eq @ ( set @ A ) @ X6 @ Y8 )
& ( ord_less_eq @ ( set @ A ) @ Y8 @ A4 ) )
=> ( P @ Y8 ) ) ) ) ) ).
% eventually_finite_subsets_at_top
thf(fact_5596_isCont__sgn,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [A3: A,F2: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) @ F2 )
=> ( ( ( F2 @ A3 )
!= ( zero_zero @ B ) )
=> ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) )
@ ^ [X4: A] : ( sgn_sgn @ B @ ( F2 @ X4 ) ) ) ) ) ) ).
% isCont_sgn
thf(fact_5597_continuous__within__tan,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A,S: set @ A,F2: A > A] :
( ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ X @ S ) @ F2 )
=> ( ( ( cos @ A @ ( F2 @ X ) )
!= ( zero_zero @ A ) )
=> ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ X @ S )
@ ^ [X4: A] : ( tan @ A @ ( F2 @ X4 ) ) ) ) ) ) ).
% continuous_within_tan
thf(fact_5598_continuous__within__cot,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A,S: set @ A,F2: A > A] :
( ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ X @ S ) @ F2 )
=> ( ( ( sin @ A @ ( F2 @ X ) )
!= ( zero_zero @ A ) )
=> ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ X @ S )
@ ^ [X4: A] : ( cot @ A @ ( F2 @ X4 ) ) ) ) ) ) ).
% continuous_within_cot
thf(fact_5599_continuous__at__within__tanh,axiom,
! [A: $tType,C: $tType] :
( ( ( topological_t2_space @ C )
& ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: C,A4: set @ C,F2: C > A] :
( ( topolo3448309680560233919inuous @ C @ A @ ( topolo174197925503356063within @ C @ X @ A4 ) @ F2 )
=> ( ( ( cosh @ A @ ( F2 @ X ) )
!= ( zero_zero @ A ) )
=> ( topolo3448309680560233919inuous @ C @ A @ ( topolo174197925503356063within @ C @ X @ A4 )
@ ^ [X4: C] : ( tanh @ A @ ( F2 @ X4 ) ) ) ) ) ) ).
% continuous_at_within_tanh
thf(fact_5600_continuous__tan,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [F5: filter @ A,F2: A > A] :
( ( topolo3448309680560233919inuous @ A @ A @ F5 @ F2 )
=> ( ( ( cos @ A
@ ( F2
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F5
@ ^ [X4: A] : X4 ) ) )
!= ( zero_zero @ A ) )
=> ( topolo3448309680560233919inuous @ A @ A @ F5
@ ^ [X4: A] : ( tan @ A @ ( F2 @ X4 ) ) ) ) ) ) ).
% continuous_tan
thf(fact_5601_continuous__cot,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [F5: filter @ A,F2: A > A] :
( ( topolo3448309680560233919inuous @ A @ A @ F5 @ F2 )
=> ( ( ( sin @ A
@ ( F2
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F5
@ ^ [X4: A] : X4 ) ) )
!= ( zero_zero @ A ) )
=> ( topolo3448309680560233919inuous @ A @ A @ F5
@ ^ [X4: A] : ( cot @ A @ ( F2 @ X4 ) ) ) ) ) ) ).
% continuous_cot
thf(fact_5602_continuous__tanh,axiom,
! [A: $tType,C: $tType] :
( ( ( topological_t2_space @ C )
& ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [F5: filter @ C,F2: C > A] :
( ( topolo3448309680560233919inuous @ C @ A @ F5 @ F2 )
=> ( ( ( cosh @ A
@ ( F2
@ ( topolo3827282254853284352ce_Lim @ C @ C @ F5
@ ^ [X4: C] : X4 ) ) )
!= ( zero_zero @ A ) )
=> ( topolo3448309680560233919inuous @ C @ A @ F5
@ ^ [X4: C] : ( tanh @ A @ ( F2 @ X4 ) ) ) ) ) ) ).
% continuous_tanh
thf(fact_5603_isCont__has__Ub,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [A3: real,B3: real,F2: real > A] :
( ( ord_less_eq @ real @ A3 @ B3 )
=> ( ! [X5: real] :
( ( ( ord_less_eq @ real @ A3 @ X5 )
& ( ord_less_eq @ real @ X5 @ B3 ) )
=> ( topolo3448309680560233919inuous @ real @ A @ ( topolo174197925503356063within @ real @ X5 @ ( top_top @ ( set @ real ) ) ) @ F2 ) )
=> ? [M8: A] :
( ! [X2: real] :
( ( ( ord_less_eq @ real @ A3 @ X2 )
& ( ord_less_eq @ real @ X2 @ B3 ) )
=> ( ord_less_eq @ A @ ( F2 @ X2 ) @ M8 ) )
& ! [N8: A] :
( ( ord_less @ A @ N8 @ M8 )
=> ? [X5: real] :
( ( ord_less_eq @ real @ A3 @ X5 )
& ( ord_less_eq @ real @ X5 @ B3 )
& ( ord_less @ A @ N8 @ ( F2 @ X5 ) ) ) ) ) ) ) ) ).
% isCont_has_Ub
thf(fact_5604_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_5605_finite__subsets__at__top__def,axiom,
! [A: $tType] :
( ( finite5375528669736107172at_top @ A )
= ( ^ [A7: set @ A] :
( complete_Inf_Inf @ ( filter @ ( set @ A ) )
@ ( image2 @ ( set @ A ) @ ( filter @ ( set @ A ) )
@ ^ [X6: set @ A] :
( principal @ ( set @ A )
@ ( collect @ ( set @ A )
@ ^ [Y8: set @ A] :
( ( finite_finite2 @ A @ Y8 )
& ( ord_less_eq @ ( set @ A ) @ X6 @ Y8 )
& ( ord_less_eq @ ( set @ A ) @ Y8 @ A7 ) ) ) )
@ ( collect @ ( set @ A )
@ ^ [X6: set @ A] :
( ( finite_finite2 @ A @ X6 )
& ( ord_less_eq @ ( set @ A ) @ X6 @ A7 ) ) ) ) ) ) ) ).
% finite_subsets_at_top_def
thf(fact_5606_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_5607_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_5608_isCont__tan_H,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [A3: A,F2: A > A] :
( ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) @ F2 )
=> ( ( ( cos @ A @ ( F2 @ A3 ) )
!= ( zero_zero @ A ) )
=> ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) )
@ ^ [X4: A] : ( tan @ A @ ( F2 @ X4 ) ) ) ) ) ) ).
% isCont_tan'
thf(fact_5609_isCont__cot_H,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [A3: A,F2: A > A] :
( ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) @ F2 )
=> ( ( ( sin @ A @ ( F2 @ A3 ) )
!= ( zero_zero @ A ) )
=> ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) )
@ ^ [X4: A] : ( cot @ A @ ( F2 @ X4 ) ) ) ) ) ) ).
% isCont_cot'
thf(fact_5610_isCont__iff,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [X: A,F2: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) @ F2 )
= ( filterlim @ A @ B
@ ^ [H: A] : ( F2 @ ( plus_plus @ A @ X @ H ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( F2 @ X ) )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% isCont_iff
thf(fact_5611_filterlim__finite__subsets__at__top,axiom,
! [A: $tType,B: $tType,F2: A > ( set @ B ),A4: set @ B,F5: filter @ A] :
( ( filterlim @ A @ ( set @ B ) @ F2 @ ( finite5375528669736107172at_top @ B @ A4 ) @ F5 )
= ( ! [X6: set @ B] :
( ( ( finite_finite2 @ B @ X6 )
& ( ord_less_eq @ ( set @ B ) @ X6 @ A4 ) )
=> ( eventually @ A
@ ^ [Y6: A] :
( ( finite_finite2 @ B @ ( F2 @ Y6 ) )
& ( ord_less_eq @ ( set @ B ) @ X6 @ ( F2 @ Y6 ) )
& ( ord_less_eq @ ( set @ B ) @ ( F2 @ Y6 ) @ A4 ) )
@ F5 ) ) ) ) ).
% filterlim_finite_subsets_at_top
thf(fact_5612_continuous__at__Sup__antimono,axiom,
! [B: $tType,A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( topolo1944317154257567458pology @ A )
& ( condit6923001295902523014norder @ B )
& ( topolo1944317154257567458pology @ B ) )
=> ! [F2: A > B,S2: set @ A] :
( ( order_antimono @ A @ B @ F2 )
=> ( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ ( complete_Sup_Sup @ A @ S2 ) @ ( set_ord_lessThan @ A @ ( complete_Sup_Sup @ A @ S2 ) ) ) @ F2 )
=> ( ( S2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit941137186595557371_above @ A @ S2 )
=> ( ( F2 @ ( complete_Sup_Sup @ A @ S2 ) )
= ( complete_Inf_Inf @ B @ ( image2 @ A @ B @ F2 @ S2 ) ) ) ) ) ) ) ) ).
% continuous_at_Sup_antimono
thf(fact_5613_continuous__at__Inf__antimono,axiom,
! [B: $tType,A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( topolo1944317154257567458pology @ A )
& ( condit6923001295902523014norder @ B )
& ( topolo1944317154257567458pology @ B ) )
=> ! [F2: A > B,S2: set @ A] :
( ( order_antimono @ A @ B @ F2 )
=> ( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ ( complete_Inf_Inf @ A @ S2 ) @ ( set_ord_greaterThan @ A @ ( complete_Inf_Inf @ A @ S2 ) ) ) @ F2 )
=> ( ( S2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit1013018076250108175_below @ A @ S2 )
=> ( ( F2 @ ( complete_Inf_Inf @ A @ S2 ) )
= ( complete_Sup_Sup @ B @ ( image2 @ A @ B @ F2 @ S2 ) ) ) ) ) ) ) ) ).
% continuous_at_Inf_antimono
thf(fact_5614_Sup__fin_Oeq__fold_H,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( lattic5882676163264333800up_fin @ A )
= ( ^ [A7: set @ A] :
( the2 @ A
@ ( finite_fold @ A @ ( option @ A )
@ ^ [X4: A,Y6: option @ A] : ( some @ A @ ( case_option @ A @ A @ X4 @ ( sup_sup @ A @ X4 ) @ Y6 ) )
@ ( none @ A )
@ A7 ) ) ) ) ) ).
% Sup_fin.eq_fold'
thf(fact_5615_bdd__belowI,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A4: set @ A,M2: A] :
( ! [X5: A] :
( ( member @ A @ X5 @ A4 )
=> ( ord_less_eq @ A @ M2 @ X5 ) )
=> ( condit1013018076250108175_below @ A @ A4 ) ) ) ).
% bdd_belowI
thf(fact_5616_bdd__below_OI,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A4: set @ A,M7: A] :
( ! [X5: A] :
( ( member @ A @ X5 @ A4 )
=> ( ord_less_eq @ A @ M7 @ X5 ) )
=> ( condit1013018076250108175_below @ A @ A4 ) ) ) ).
% bdd_below.I
thf(fact_5617_bdd__above_OI,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A4: set @ A,M7: A] :
( ! [X5: A] :
( ( member @ A @ X5 @ A4 )
=> ( ord_less_eq @ A @ X5 @ M7 ) )
=> ( condit941137186595557371_above @ A @ A4 ) ) ) ).
% bdd_above.I
thf(fact_5618_bdd__below__empty,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( condit1013018076250108175_below @ A @ ( bot_bot @ ( set @ A ) ) ) ) ).
% bdd_below_empty
thf(fact_5619_bdd__above__empty,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( condit941137186595557371_above @ A @ ( bot_bot @ ( set @ A ) ) ) ) ).
% bdd_above_empty
thf(fact_5620_inf__Sup__absorb,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [A4: set @ A,A3: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( member @ A @ A3 @ A4 )
=> ( ( inf_inf @ A @ A3 @ ( lattic5882676163264333800up_fin @ A @ A4 ) )
= A3 ) ) ) ) ).
% inf_Sup_absorb
thf(fact_5621_Sup__fin_Osingleton,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X: A] :
( ( lattic5882676163264333800up_fin @ A @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= X ) ) ).
% Sup_fin.singleton
thf(fact_5622_bdd__below__UN,axiom,
! [A: $tType,B: $tType] :
( ( lattice @ A )
=> ! [I6: set @ B,A4: B > ( set @ A )] :
( ( finite_finite2 @ B @ I6 )
=> ( ( condit1013018076250108175_below @ A @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ A4 @ I6 ) ) )
= ( ! [X4: B] :
( ( member @ B @ X4 @ I6 )
=> ( condit1013018076250108175_below @ A @ ( A4 @ X4 ) ) ) ) ) ) ) ).
% bdd_below_UN
thf(fact_5623_bdd__above__UN,axiom,
! [A: $tType,B: $tType] :
( ( lattice @ A )
=> ! [I6: set @ B,A4: B > ( set @ A )] :
( ( finite_finite2 @ B @ I6 )
=> ( ( condit941137186595557371_above @ A @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ A4 @ I6 ) ) )
= ( ! [X4: B] :
( ( member @ B @ X4 @ I6 )
=> ( condit941137186595557371_above @ A @ ( A4 @ X4 ) ) ) ) ) ) ) ).
% bdd_above_UN
thf(fact_5624_Sup__fin_Oinsert,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic5882676163264333800up_fin @ A @ ( insert2 @ A @ X @ A4 ) )
= ( sup_sup @ A @ X @ ( lattic5882676163264333800up_fin @ A @ A4 ) ) ) ) ) ) ).
% Sup_fin.insert
thf(fact_5625_Sup__fin__Max,axiom,
! [A: $tType] :
( ( ( semilattice_sup @ A )
& ( linorder @ A ) )
=> ( ( lattic5882676163264333800up_fin @ A )
= ( lattic643756798349783984er_Max @ A ) ) ) ).
% Sup_fin_Max
thf(fact_5626_bdd__above__nat,axiom,
( ( condit941137186595557371_above @ nat )
= ( finite_finite2 @ nat ) ) ).
% bdd_above_nat
thf(fact_5627_bdd__above__finite,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [A4: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( condit941137186595557371_above @ A @ A4 ) ) ) ).
% bdd_above_finite
thf(fact_5628_bdd__below__finite,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [A4: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( condit1013018076250108175_below @ A @ A4 ) ) ) ).
% bdd_below_finite
thf(fact_5629_bdd__above_Ounfold,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( ( condit941137186595557371_above @ A )
= ( ^ [A7: set @ A] :
? [M9: A] :
! [X4: A] :
( ( member @ A @ X4 @ A7 )
=> ( ord_less_eq @ A @ X4 @ M9 ) ) ) ) ) ).
% bdd_above.unfold
thf(fact_5630_bdd__above_OE,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A4: set @ A] :
( ( condit941137186595557371_above @ A @ A4 )
=> ~ ! [M8: A] :
~ ! [X2: A] :
( ( member @ A @ X2 @ A4 )
=> ( ord_less_eq @ A @ X2 @ M8 ) ) ) ) ).
% bdd_above.E
thf(fact_5631_bdd__below_Ounfold,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( ( condit1013018076250108175_below @ A )
= ( ^ [A7: set @ A] :
? [M9: A] :
! [X4: A] :
( ( member @ A @ X4 @ A7 )
=> ( ord_less_eq @ A @ M9 @ X4 ) ) ) ) ) ).
% bdd_below.unfold
thf(fact_5632_bdd__below_OE,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A4: set @ A] :
( ( condit1013018076250108175_below @ A @ A4 )
=> ~ ! [M8: A] :
~ ! [X2: A] :
( ( member @ A @ X2 @ A4 )
=> ( ord_less_eq @ A @ M8 @ X2 ) ) ) ) ).
% bdd_below.E
thf(fact_5633_bdd__above__mono,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [B2: set @ A,A4: set @ A] :
( ( condit941137186595557371_above @ A @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ A4 @ B2 )
=> ( condit941137186595557371_above @ A @ A4 ) ) ) ) ).
% bdd_above_mono
thf(fact_5634_bdd__below__mono,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [B2: set @ A,A4: set @ A] :
( ( condit1013018076250108175_below @ A @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ A4 @ B2 )
=> ( condit1013018076250108175_below @ A @ A4 ) ) ) ) ).
% bdd_below_mono
thf(fact_5635_cSup__upper2,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X: A,X8: set @ A,Y: A] :
( ( member @ A @ X @ X8 )
=> ( ( ord_less_eq @ A @ Y @ X )
=> ( ( condit941137186595557371_above @ A @ X8 )
=> ( ord_less_eq @ A @ Y @ ( complete_Sup_Sup @ A @ X8 ) ) ) ) ) ) ).
% cSup_upper2
thf(fact_5636_cSup__upper,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X: A,X8: set @ A] :
( ( member @ A @ X @ X8 )
=> ( ( condit941137186595557371_above @ A @ X8 )
=> ( ord_less_eq @ A @ X @ ( complete_Sup_Sup @ A @ X8 ) ) ) ) ) ).
% cSup_upper
thf(fact_5637_cInf__lower2,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X: A,X8: set @ A,Y: A] :
( ( member @ A @ X @ X8 )
=> ( ( ord_less_eq @ A @ X @ Y )
=> ( ( condit1013018076250108175_below @ A @ X8 )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ X8 ) @ Y ) ) ) ) ) ).
% cInf_lower2
thf(fact_5638_cInf__lower,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X: A,X8: set @ A] :
( ( member @ A @ X @ X8 )
=> ( ( condit1013018076250108175_below @ A @ X8 )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ X8 ) @ X ) ) ) ) ).
% cInf_lower
thf(fact_5639_cInf__le__cSup,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A4: set @ A] :
( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit941137186595557371_above @ A @ A4 )
=> ( ( condit1013018076250108175_below @ A @ A4 )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A4 ) @ ( complete_Sup_Sup @ A @ A4 ) ) ) ) ) ) ).
% cInf_le_cSup
thf(fact_5640_bdd__belowI2,axiom,
! [A: $tType,B: $tType] :
( ( preorder @ A )
=> ! [A4: set @ B,M2: A,F2: B > A] :
( ! [X5: B] :
( ( member @ B @ X5 @ A4 )
=> ( ord_less_eq @ A @ M2 @ ( F2 @ X5 ) ) )
=> ( condit1013018076250108175_below @ A @ ( image2 @ B @ A @ F2 @ A4 ) ) ) ) ).
% bdd_belowI2
thf(fact_5641_bdd__below_OI2,axiom,
! [A: $tType,B: $tType] :
( ( preorder @ A )
=> ! [A4: set @ B,M7: A,F2: B > A] :
( ! [X5: B] :
( ( member @ B @ X5 @ A4 )
=> ( ord_less_eq @ A @ M7 @ ( F2 @ X5 ) ) )
=> ( condit1013018076250108175_below @ A @ ( image2 @ B @ A @ F2 @ A4 ) ) ) ) ).
% bdd_below.I2
thf(fact_5642_bdd__above_OI2,axiom,
! [A: $tType,B: $tType] :
( ( preorder @ A )
=> ! [A4: set @ B,F2: B > A,M7: A] :
( ! [X5: B] :
( ( member @ B @ X5 @ A4 )
=> ( ord_less_eq @ A @ ( F2 @ X5 ) @ M7 ) )
=> ( condit941137186595557371_above @ A @ ( image2 @ B @ A @ F2 @ A4 ) ) ) ) ).
% bdd_above.I2
thf(fact_5643_Sup__fin_OcoboundedI,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A4: set @ A,A3: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( member @ A @ A3 @ A4 )
=> ( ord_less_eq @ A @ A3 @ ( lattic5882676163264333800up_fin @ A @ A4 ) ) ) ) ) ).
% Sup_fin.coboundedI
thf(fact_5644_Sup__fin_Oin__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( member @ A @ X @ A4 )
=> ( ( sup_sup @ A @ X @ ( lattic5882676163264333800up_fin @ A @ A4 ) )
= ( lattic5882676163264333800up_fin @ A @ A4 ) ) ) ) ) ).
% Sup_fin.in_idem
thf(fact_5645_cINF__lower2,axiom,
! [B: $tType,A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [F2: B > A,A4: set @ B,X: B,U: A] :
( ( condit1013018076250108175_below @ A @ ( image2 @ B @ A @ F2 @ A4 ) )
=> ( ( member @ B @ X @ A4 )
=> ( ( ord_less_eq @ A @ ( F2 @ X ) @ U )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F2 @ A4 ) ) @ U ) ) ) ) ) ).
% cINF_lower2
thf(fact_5646_cINF__lower,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [F2: B > A,A4: set @ B,X: B] :
( ( condit1013018076250108175_below @ A @ ( image2 @ B @ A @ F2 @ A4 ) )
=> ( ( member @ B @ X @ A4 )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F2 @ A4 ) ) @ ( F2 @ X ) ) ) ) ) ).
% cINF_lower
thf(fact_5647_cInf__mono,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [B2: set @ A,A4: set @ A] :
( ( B2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit1013018076250108175_below @ A @ A4 )
=> ( ! [B5: A] :
( ( member @ A @ B5 @ B2 )
=> ? [X2: A] :
( ( member @ A @ X2 @ A4 )
& ( ord_less_eq @ A @ X2 @ B5 ) ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A4 ) @ ( complete_Inf_Inf @ A @ B2 ) ) ) ) ) ) ).
% cInf_mono
thf(fact_5648_le__cInf__iff,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [S2: set @ A,A3: A] :
( ( S2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit1013018076250108175_below @ A @ S2 )
=> ( ( ord_less_eq @ A @ A3 @ ( complete_Inf_Inf @ A @ S2 ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ S2 )
=> ( ord_less_eq @ A @ A3 @ X4 ) ) ) ) ) ) ) ).
% le_cInf_iff
thf(fact_5649_cSUP__upper2,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [F2: B > A,A4: set @ B,X: B,U: A] :
( ( condit941137186595557371_above @ A @ ( image2 @ B @ A @ F2 @ A4 ) )
=> ( ( member @ B @ X @ A4 )
=> ( ( ord_less_eq @ A @ U @ ( F2 @ X ) )
=> ( ord_less_eq @ A @ U @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F2 @ A4 ) ) ) ) ) ) ) ).
% cSUP_upper2
thf(fact_5650_cSUP__upper,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X: B,A4: set @ B,F2: B > A] :
( ( member @ B @ X @ A4 )
=> ( ( condit941137186595557371_above @ A @ ( image2 @ B @ A @ F2 @ A4 ) )
=> ( ord_less_eq @ A @ ( F2 @ X ) @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F2 @ A4 ) ) ) ) ) ) ).
% cSUP_upper
thf(fact_5651_cInf__less__iff,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [X8: set @ A,Y: A] :
( ( X8
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit1013018076250108175_below @ A @ X8 )
=> ( ( ord_less @ A @ ( complete_Inf_Inf @ A @ X8 ) @ Y )
= ( ? [X4: A] :
( ( member @ A @ X4 @ X8 )
& ( ord_less @ A @ X4 @ Y ) ) ) ) ) ) ) ).
% cInf_less_iff
thf(fact_5652_cSup__mono,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [B2: set @ A,A4: set @ A] :
( ( B2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit941137186595557371_above @ A @ A4 )
=> ( ! [B5: A] :
( ( member @ A @ B5 @ B2 )
=> ? [X2: A] :
( ( member @ A @ X2 @ A4 )
& ( ord_less_eq @ A @ B5 @ X2 ) ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ B2 ) @ ( complete_Sup_Sup @ A @ A4 ) ) ) ) ) ) ).
% cSup_mono
thf(fact_5653_cSup__le__iff,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [S2: set @ A,A3: A] :
( ( S2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit941137186595557371_above @ A @ S2 )
=> ( ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ S2 ) @ A3 )
= ( ! [X4: A] :
( ( member @ A @ X4 @ S2 )
=> ( ord_less_eq @ A @ X4 @ A3 ) ) ) ) ) ) ) ).
% cSup_le_iff
thf(fact_5654_less__cSup__iff,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [X8: set @ A,Y: A] :
( ( X8
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit941137186595557371_above @ A @ X8 )
=> ( ( ord_less @ A @ Y @ ( complete_Sup_Sup @ A @ X8 ) )
= ( ? [X4: A] :
( ( member @ A @ X4 @ X8 )
& ( ord_less @ A @ Y @ X4 ) ) ) ) ) ) ) ).
% less_cSup_iff
thf(fact_5655_less__cINF__D,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [F2: B > A,A4: set @ B,Y: A,I: B] :
( ( condit1013018076250108175_below @ A @ ( image2 @ B @ A @ F2 @ A4 ) )
=> ( ( ord_less @ A @ Y @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F2 @ A4 ) ) )
=> ( ( member @ B @ I @ A4 )
=> ( ord_less @ A @ Y @ ( F2 @ I ) ) ) ) ) ) ).
% less_cINF_D
thf(fact_5656_cSUP__lessD,axiom,
! [B: $tType,A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [F2: B > A,A4: set @ B,Y: A,I: B] :
( ( condit941137186595557371_above @ A @ ( image2 @ B @ A @ F2 @ A4 ) )
=> ( ( ord_less @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F2 @ A4 ) ) @ Y )
=> ( ( member @ B @ I @ A4 )
=> ( ord_less @ A @ ( F2 @ I ) @ Y ) ) ) ) ) ).
% cSUP_lessD
thf(fact_5657_Sup__fin_Obounded__iff,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ ( lattic5882676163264333800up_fin @ A @ A4 ) @ X )
= ( ! [X4: A] :
( ( member @ A @ X4 @ A4 )
=> ( ord_less_eq @ A @ X4 @ X ) ) ) ) ) ) ) ).
% Sup_fin.bounded_iff
thf(fact_5658_Sup__fin_OboundedI,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [A6: A] :
( ( member @ A @ A6 @ A4 )
=> ( ord_less_eq @ A @ A6 @ X ) )
=> ( ord_less_eq @ A @ ( lattic5882676163264333800up_fin @ A @ A4 ) @ X ) ) ) ) ) ).
% Sup_fin.boundedI
thf(fact_5659_Sup__fin_OboundedE,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ ( lattic5882676163264333800up_fin @ A @ A4 ) @ X )
=> ! [A11: A] :
( ( member @ A @ A11 @ A4 )
=> ( ord_less_eq @ A @ A11 @ X ) ) ) ) ) ) ).
% Sup_fin.boundedE
thf(fact_5660_cSup__eq__Sup__fin,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X8: set @ A] :
( ( finite_finite2 @ A @ X8 )
=> ( ( X8
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Sup_Sup @ A @ X8 )
= ( lattic5882676163264333800up_fin @ A @ X8 ) ) ) ) ) ).
% cSup_eq_Sup_fin
thf(fact_5661_Sup__fin__Sup,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A4: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic5882676163264333800up_fin @ A @ A4 )
= ( complete_Sup_Sup @ A @ A4 ) ) ) ) ) ).
% Sup_fin_Sup
thf(fact_5662_Sup__fin_Oinfinite,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A4: set @ A] :
( ~ ( finite_finite2 @ A @ A4 )
=> ( ( lattic5882676163264333800up_fin @ A @ A4 )
= ( the2 @ A @ ( none @ A ) ) ) ) ) ).
% Sup_fin.infinite
thf(fact_5663_le__cINF__iff,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A4: set @ B,F2: B > A,U: A] :
( ( A4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit1013018076250108175_below @ A @ ( image2 @ B @ A @ F2 @ A4 ) )
=> ( ( ord_less_eq @ A @ U @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F2 @ A4 ) ) )
= ( ! [X4: B] :
( ( member @ B @ X4 @ A4 )
=> ( ord_less_eq @ A @ U @ ( F2 @ X4 ) ) ) ) ) ) ) ) ).
% le_cINF_iff
thf(fact_5664_cINF__mono,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [B2: set @ B,F2: C > A,A4: set @ C,G: B > A] :
( ( B2
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit1013018076250108175_below @ A @ ( image2 @ C @ A @ F2 @ A4 ) )
=> ( ! [M3: B] :
( ( member @ B @ M3 @ B2 )
=> ? [X2: C] :
( ( member @ C @ X2 @ A4 )
& ( ord_less_eq @ A @ ( F2 @ X2 ) @ ( G @ M3 ) ) ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image2 @ C @ A @ F2 @ A4 ) ) @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ G @ B2 ) ) ) ) ) ) ) ).
% cINF_mono
thf(fact_5665_cInf__superset__mono,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A4: set @ A,B2: set @ A] :
( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit1013018076250108175_below @ A @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ A4 @ B2 )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ B2 ) @ ( complete_Inf_Inf @ A @ A4 ) ) ) ) ) ) ).
% cInf_superset_mono
thf(fact_5666_cSUP__le__iff,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A4: set @ B,F2: B > A,U: A] :
( ( A4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit941137186595557371_above @ A @ ( image2 @ B @ A @ F2 @ A4 ) )
=> ( ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F2 @ A4 ) ) @ U )
= ( ! [X4: B] :
( ( member @ B @ X4 @ A4 )
=> ( ord_less_eq @ A @ ( F2 @ X4 ) @ U ) ) ) ) ) ) ) ).
% cSUP_le_iff
thf(fact_5667_cSUP__mono,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A4: set @ B,G: C > A,B2: set @ C,F2: B > A] :
( ( A4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit941137186595557371_above @ A @ ( image2 @ C @ A @ G @ B2 ) )
=> ( ! [N3: B] :
( ( member @ B @ N3 @ A4 )
=> ? [X2: C] :
( ( member @ C @ X2 @ B2 )
& ( ord_less_eq @ A @ ( F2 @ N3 ) @ ( G @ X2 ) ) ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F2 @ A4 ) ) @ ( complete_Sup_Sup @ A @ ( image2 @ C @ A @ G @ B2 ) ) ) ) ) ) ) ).
% cSUP_mono
thf(fact_5668_cSup__subset__mono,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A4: set @ A,B2: set @ A] :
( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit941137186595557371_above @ A @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ A4 @ B2 )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A4 ) @ ( complete_Sup_Sup @ A @ B2 ) ) ) ) ) ) ).
% cSup_subset_mono
thf(fact_5669_cInf__insert__If,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X8: set @ A,A3: A] :
( ( condit1013018076250108175_below @ A @ X8 )
=> ( ( ( X8
= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Inf_Inf @ A @ ( insert2 @ A @ A3 @ X8 ) )
= A3 ) )
& ( ( X8
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Inf_Inf @ A @ ( insert2 @ A @ A3 @ X8 ) )
= ( inf_inf @ A @ A3 @ ( complete_Inf_Inf @ A @ X8 ) ) ) ) ) ) ) ).
% cInf_insert_If
thf(fact_5670_cInf__insert,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X8: set @ A,A3: A] :
( ( X8
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit1013018076250108175_below @ A @ X8 )
=> ( ( complete_Inf_Inf @ A @ ( insert2 @ A @ A3 @ X8 ) )
= ( inf_inf @ A @ A3 @ ( complete_Inf_Inf @ A @ X8 ) ) ) ) ) ) ).
% cInf_insert
thf(fact_5671_cSup__insert__If,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X8: set @ A,A3: A] :
( ( condit941137186595557371_above @ A @ X8 )
=> ( ( ( X8
= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Sup_Sup @ A @ ( insert2 @ A @ A3 @ X8 ) )
= A3 ) )
& ( ( X8
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Sup_Sup @ A @ ( insert2 @ A @ A3 @ X8 ) )
= ( sup_sup @ A @ A3 @ ( complete_Sup_Sup @ A @ X8 ) ) ) ) ) ) ) ).
% cSup_insert_If
thf(fact_5672_cSup__insert,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X8: set @ A,A3: A] :
( ( X8
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit941137186595557371_above @ A @ X8 )
=> ( ( complete_Sup_Sup @ A @ ( insert2 @ A @ A3 @ X8 ) )
= ( sup_sup @ A @ A3 @ ( complete_Sup_Sup @ A @ X8 ) ) ) ) ) ) ).
% cSup_insert
thf(fact_5673_cInf__union__distrib,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A4: set @ A,B2: set @ A] :
( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit1013018076250108175_below @ A @ A4 )
=> ( ( B2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit1013018076250108175_below @ A @ B2 )
=> ( ( complete_Inf_Inf @ A @ ( sup_sup @ ( set @ A ) @ A4 @ B2 ) )
= ( inf_inf @ A @ ( complete_Inf_Inf @ A @ A4 ) @ ( complete_Inf_Inf @ A @ B2 ) ) ) ) ) ) ) ) ).
% cInf_union_distrib
thf(fact_5674_cSup__union__distrib,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A4: set @ A,B2: set @ A] :
( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit941137186595557371_above @ A @ A4 )
=> ( ( B2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit941137186595557371_above @ A @ B2 )
=> ( ( complete_Sup_Sup @ A @ ( sup_sup @ ( set @ A ) @ A4 @ B2 ) )
= ( sup_sup @ A @ ( complete_Sup_Sup @ A @ A4 ) @ ( complete_Sup_Sup @ A @ B2 ) ) ) ) ) ) ) ) ).
% cSup_union_distrib
thf(fact_5675_cINF__less__iff,axiom,
! [A: $tType,B: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [A4: set @ B,F2: B > A,A3: A] :
( ( A4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit1013018076250108175_below @ A @ ( image2 @ B @ A @ F2 @ A4 ) )
=> ( ( ord_less @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F2 @ A4 ) ) @ A3 )
= ( ? [X4: B] :
( ( member @ B @ X4 @ A4 )
& ( ord_less @ A @ ( F2 @ X4 ) @ A3 ) ) ) ) ) ) ) ).
% cINF_less_iff
thf(fact_5676_less__cSUP__iff,axiom,
! [A: $tType,B: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [A4: set @ B,F2: B > A,A3: A] :
( ( A4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit941137186595557371_above @ A @ ( image2 @ B @ A @ F2 @ A4 ) )
=> ( ( ord_less @ A @ A3 @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F2 @ A4 ) ) )
= ( ? [X4: B] :
( ( member @ B @ X4 @ A4 )
& ( ord_less @ A @ A3 @ ( F2 @ X4 ) ) ) ) ) ) ) ) ).
% less_cSUP_iff
thf(fact_5677_cINF__inf__distrib,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A4: set @ B,F2: B > A,G: B > A] :
( ( A4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit1013018076250108175_below @ A @ ( image2 @ B @ A @ F2 @ A4 ) )
=> ( ( condit1013018076250108175_below @ A @ ( image2 @ B @ A @ G @ A4 ) )
=> ( ( inf_inf @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F2 @ A4 ) ) @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ G @ A4 ) ) )
= ( complete_Inf_Inf @ A
@ ( image2 @ B @ A
@ ^ [A5: B] : ( inf_inf @ A @ ( F2 @ A5 ) @ ( G @ A5 ) )
@ A4 ) ) ) ) ) ) ) ).
% cINF_inf_distrib
thf(fact_5678_conditionally__complete__lattice__class_OSUP__sup__distrib,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A4: set @ B,F2: B > A,G: B > A] :
( ( A4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit941137186595557371_above @ A @ ( image2 @ B @ A @ F2 @ A4 ) )
=> ( ( condit941137186595557371_above @ A @ ( image2 @ B @ A @ G @ A4 ) )
=> ( ( sup_sup @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F2 @ A4 ) ) @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ G @ A4 ) ) )
= ( complete_Sup_Sup @ A
@ ( image2 @ B @ A
@ ^ [A5: B] : ( sup_sup @ A @ ( F2 @ A5 ) @ ( G @ A5 ) )
@ A4 ) ) ) ) ) ) ) ).
% conditionally_complete_lattice_class.SUP_sup_distrib
thf(fact_5679_Sup__fin_Osubset__imp,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A4: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B2 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B2 )
=> ( ord_less_eq @ A @ ( lattic5882676163264333800up_fin @ A @ A4 ) @ ( lattic5882676163264333800up_fin @ A @ B2 ) ) ) ) ) ) ).
% Sup_fin.subset_imp
thf(fact_5680_Sup__fin_Ohom__commute,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [H2: A > A,N6: set @ A] :
( ! [X5: A,Y3: A] :
( ( H2 @ ( sup_sup @ A @ X5 @ Y3 ) )
= ( sup_sup @ A @ ( H2 @ X5 ) @ ( H2 @ Y3 ) ) )
=> ( ( finite_finite2 @ A @ N6 )
=> ( ( N6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( H2 @ ( lattic5882676163264333800up_fin @ A @ N6 ) )
= ( lattic5882676163264333800up_fin @ A @ ( image2 @ A @ A @ H2 @ N6 ) ) ) ) ) ) ) ).
% Sup_fin.hom_commute
thf(fact_5681_Sup__fin_Osubset,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A4: set @ A,B2: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( B2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ A4 )
=> ( ( sup_sup @ A @ ( lattic5882676163264333800up_fin @ A @ B2 ) @ ( lattic5882676163264333800up_fin @ A @ A4 ) )
= ( lattic5882676163264333800up_fin @ A @ A4 ) ) ) ) ) ) ).
% Sup_fin.subset
thf(fact_5682_Sup__fin_Oclosed,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A4: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X5: A,Y3: A] : ( member @ A @ ( sup_sup @ A @ X5 @ Y3 ) @ ( insert2 @ A @ X5 @ ( insert2 @ A @ Y3 @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( member @ A @ ( lattic5882676163264333800up_fin @ A @ A4 ) @ A4 ) ) ) ) ) ).
% Sup_fin.closed
thf(fact_5683_Sup__fin_Oinsert__not__elem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ~ ( member @ A @ X @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic5882676163264333800up_fin @ A @ ( insert2 @ A @ X @ A4 ) )
= ( sup_sup @ A @ X @ ( lattic5882676163264333800up_fin @ A @ A4 ) ) ) ) ) ) ) ).
% Sup_fin.insert_not_elem
thf(fact_5684_Sup__fin_Ounion,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A4: set @ A,B2: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B2 )
=> ( ( B2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic5882676163264333800up_fin @ A @ ( sup_sup @ ( set @ A ) @ A4 @ B2 ) )
= ( sup_sup @ A @ ( lattic5882676163264333800up_fin @ A @ A4 ) @ ( lattic5882676163264333800up_fin @ A @ B2 ) ) ) ) ) ) ) ) ).
% Sup_fin.union
thf(fact_5685_Sup__fin_Oeq__fold,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( lattic5882676163264333800up_fin @ A @ ( insert2 @ A @ X @ A4 ) )
= ( finite_fold @ A @ A @ ( sup_sup @ A ) @ X @ A4 ) ) ) ) ).
% Sup_fin.eq_fold
thf(fact_5686_cINF__superset__mono,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A4: set @ B,G: B > A,B2: set @ B,F2: B > A] :
( ( A4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit1013018076250108175_below @ A @ ( image2 @ B @ A @ G @ B2 ) )
=> ( ( ord_less_eq @ ( set @ B ) @ A4 @ B2 )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ B2 )
=> ( ord_less_eq @ A @ ( G @ X5 ) @ ( F2 @ X5 ) ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ G @ B2 ) ) @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F2 @ A4 ) ) ) ) ) ) ) ) ).
% cINF_superset_mono
thf(fact_5687_cSUP__subset__mono,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A4: set @ B,G: B > A,B2: set @ B,F2: B > A] :
( ( A4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit941137186595557371_above @ A @ ( image2 @ B @ A @ G @ B2 ) )
=> ( ( ord_less_eq @ ( set @ B ) @ A4 @ B2 )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ A4 )
=> ( ord_less_eq @ A @ ( F2 @ X5 ) @ ( G @ X5 ) ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F2 @ A4 ) ) @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ G @ B2 ) ) ) ) ) ) ) ) ).
% cSUP_subset_mono
thf(fact_5688_less__eq__cInf__inter,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A4: set @ A,B2: set @ A] :
( ( condit1013018076250108175_below @ A @ A4 )
=> ( ( condit1013018076250108175_below @ A @ B2 )
=> ( ( ( inf_inf @ ( set @ A ) @ A4 @ B2 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ord_less_eq @ A @ ( inf_inf @ A @ ( complete_Inf_Inf @ A @ A4 ) @ ( complete_Inf_Inf @ A @ B2 ) ) @ ( complete_Inf_Inf @ A @ ( inf_inf @ ( set @ A ) @ A4 @ B2 ) ) ) ) ) ) ) ).
% less_eq_cInf_inter
thf(fact_5689_cINF__insert,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A4: set @ B,F2: B > A,A3: B] :
( ( A4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit1013018076250108175_below @ A @ ( image2 @ B @ A @ F2 @ A4 ) )
=> ( ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F2 @ ( insert2 @ B @ A3 @ A4 ) ) )
= ( inf_inf @ A @ ( F2 @ A3 ) @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F2 @ A4 ) ) ) ) ) ) ) ).
% cINF_insert
thf(fact_5690_cSup__inter__less__eq,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A4: set @ A,B2: set @ A] :
( ( condit941137186595557371_above @ A @ A4 )
=> ( ( condit941137186595557371_above @ A @ B2 )
=> ( ( ( inf_inf @ ( set @ A ) @ A4 @ B2 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( inf_inf @ ( set @ A ) @ A4 @ B2 ) ) @ ( sup_sup @ A @ ( complete_Sup_Sup @ A @ A4 ) @ ( complete_Sup_Sup @ A @ B2 ) ) ) ) ) ) ) ).
% cSup_inter_less_eq
thf(fact_5691_cSUP__insert,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A4: set @ B,F2: B > A,A3: B] :
( ( A4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit941137186595557371_above @ A @ ( image2 @ B @ A @ F2 @ A4 ) )
=> ( ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F2 @ ( insert2 @ B @ A3 @ A4 ) ) )
= ( sup_sup @ A @ ( F2 @ A3 ) @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F2 @ A4 ) ) ) ) ) ) ) ).
% cSUP_insert
thf(fact_5692_cINF__union,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A4: set @ B,F2: B > A,B2: set @ B] :
( ( A4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit1013018076250108175_below @ A @ ( image2 @ B @ A @ F2 @ A4 ) )
=> ( ( B2
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit1013018076250108175_below @ A @ ( image2 @ B @ A @ F2 @ B2 ) )
=> ( ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F2 @ ( sup_sup @ ( set @ B ) @ A4 @ B2 ) ) )
= ( inf_inf @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F2 @ A4 ) ) @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F2 @ B2 ) ) ) ) ) ) ) ) ) ).
% cINF_union
thf(fact_5693_cSUP__union,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A4: set @ B,F2: B > A,B2: set @ B] :
( ( A4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit941137186595557371_above @ A @ ( image2 @ B @ A @ F2 @ A4 ) )
=> ( ( B2
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit941137186595557371_above @ A @ ( image2 @ B @ A @ F2 @ B2 ) )
=> ( ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F2 @ ( sup_sup @ ( set @ B ) @ A4 @ B2 ) ) )
= ( sup_sup @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F2 @ A4 ) ) @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F2 @ B2 ) ) ) ) ) ) ) ) ) ).
% cSUP_union
thf(fact_5694_inf__Sup1__distrib,axiom,
! [A: $tType] :
( ( distrib_lattice @ A )
=> ! [A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( inf_inf @ A @ X @ ( lattic5882676163264333800up_fin @ A @ A4 ) )
= ( lattic5882676163264333800up_fin @ A
@ ( collect @ A
@ ^ [Uu3: A] :
? [A5: A] :
( ( Uu3
= ( inf_inf @ A @ X @ A5 ) )
& ( member @ A @ A5 @ A4 ) ) ) ) ) ) ) ) ).
% inf_Sup1_distrib
thf(fact_5695_inf__Sup2__distrib,axiom,
! [A: $tType] :
( ( distrib_lattice @ A )
=> ! [A4: set @ A,B2: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B2 )
=> ( ( B2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( inf_inf @ A @ ( lattic5882676163264333800up_fin @ A @ A4 ) @ ( lattic5882676163264333800up_fin @ A @ B2 ) )
= ( lattic5882676163264333800up_fin @ A
@ ( collect @ A
@ ^ [Uu3: A] :
? [A5: A,B4: A] :
( ( Uu3
= ( inf_inf @ A @ A5 @ B4 ) )
& ( member @ A @ A5 @ A4 )
& ( member @ A @ B4 @ B2 ) ) ) ) ) ) ) ) ) ) ).
% inf_Sup2_distrib
thf(fact_5696_cInf__cSup,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [S2: set @ A] :
( ( S2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit1013018076250108175_below @ A @ S2 )
=> ( ( complete_Inf_Inf @ A @ S2 )
= ( complete_Sup_Sup @ A
@ ( collect @ A
@ ^ [X4: A] :
! [Y6: A] :
( ( member @ A @ Y6 @ S2 )
=> ( ord_less_eq @ A @ X4 @ Y6 ) ) ) ) ) ) ) ) ).
% cInf_cSup
thf(fact_5697_cSup__cInf,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [S2: set @ A] :
( ( S2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit941137186595557371_above @ A @ S2 )
=> ( ( complete_Sup_Sup @ A @ S2 )
= ( complete_Inf_Inf @ A
@ ( collect @ A
@ ^ [X4: A] :
! [Y6: A] :
( ( member @ A @ Y6 @ S2 )
=> ( ord_less_eq @ A @ Y6 @ X4 ) ) ) ) ) ) ) ) ).
% cSup_cInf
thf(fact_5698_mono__cInf,axiom,
! [B: $tType,A: $tType] :
( ( ( condit1219197933456340205attice @ A )
& ( condit1219197933456340205attice @ B ) )
=> ! [F2: A > B,A4: set @ A] :
( ( order_mono @ A @ B @ F2 )
=> ( ( condit1013018076250108175_below @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ord_less_eq @ B @ ( F2 @ ( complete_Inf_Inf @ A @ A4 ) ) @ ( complete_Inf_Inf @ B @ ( image2 @ A @ B @ F2 @ A4 ) ) ) ) ) ) ) ).
% mono_cInf
thf(fact_5699_mono__cINF,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( ( condit1219197933456340205attice @ A )
& ( condit1219197933456340205attice @ B ) )
=> ! [F2: A > B,A4: C > A,I6: set @ C] :
( ( order_mono @ A @ B @ F2 )
=> ( ( condit1013018076250108175_below @ A @ ( image2 @ C @ A @ A4 @ I6 ) )
=> ( ( I6
!= ( bot_bot @ ( set @ C ) ) )
=> ( ord_less_eq @ B @ ( F2 @ ( complete_Inf_Inf @ A @ ( image2 @ C @ A @ A4 @ I6 ) ) )
@ ( complete_Inf_Inf @ B
@ ( image2 @ C @ B
@ ^ [X4: C] : ( F2 @ ( A4 @ X4 ) )
@ I6 ) ) ) ) ) ) ) ).
% mono_cINF
thf(fact_5700_cINF__UNION,axiom,
! [B: $tType,D: $tType,C: $tType] :
( ( condit1219197933456340205attice @ B )
=> ! [A4: set @ C,B2: C > ( set @ D ),F2: D > B] :
( ( A4
!= ( bot_bot @ ( set @ C ) ) )
=> ( ! [X5: C] :
( ( member @ C @ X5 @ A4 )
=> ( ( B2 @ X5 )
!= ( bot_bot @ ( set @ D ) ) ) )
=> ( ( condit1013018076250108175_below @ B
@ ( complete_Sup_Sup @ ( set @ B )
@ ( image2 @ C @ ( set @ B )
@ ^ [X4: C] : ( image2 @ D @ B @ F2 @ ( B2 @ X4 ) )
@ A4 ) ) )
=> ( ( complete_Inf_Inf @ B @ ( image2 @ D @ B @ F2 @ ( complete_Sup_Sup @ ( set @ D ) @ ( image2 @ C @ ( set @ D ) @ B2 @ A4 ) ) ) )
= ( complete_Inf_Inf @ B
@ ( image2 @ C @ B
@ ^ [X4: C] : ( complete_Inf_Inf @ B @ ( image2 @ D @ B @ F2 @ ( B2 @ X4 ) ) )
@ A4 ) ) ) ) ) ) ) ).
% cINF_UNION
thf(fact_5701_mono__cSup,axiom,
! [B: $tType,A: $tType] :
( ( ( condit1219197933456340205attice @ A )
& ( condit1219197933456340205attice @ B ) )
=> ! [F2: A > B,A4: set @ A] :
( ( order_mono @ A @ B @ F2 )
=> ( ( condit941137186595557371_above @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ord_less_eq @ B @ ( complete_Sup_Sup @ B @ ( image2 @ A @ B @ F2 @ A4 ) ) @ ( F2 @ ( complete_Sup_Sup @ A @ A4 ) ) ) ) ) ) ) ).
% mono_cSup
thf(fact_5702_mono__cSUP,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( ( condit1219197933456340205attice @ A )
& ( condit1219197933456340205attice @ B ) )
=> ! [F2: A > B,A4: C > A,I6: set @ C] :
( ( order_mono @ A @ B @ F2 )
=> ( ( condit941137186595557371_above @ A @ ( image2 @ C @ A @ A4 @ I6 ) )
=> ( ( I6
!= ( bot_bot @ ( set @ C ) ) )
=> ( ord_less_eq @ B
@ ( complete_Sup_Sup @ B
@ ( image2 @ C @ B
@ ^ [X4: C] : ( F2 @ ( A4 @ X4 ) )
@ I6 ) )
@ ( F2 @ ( complete_Sup_Sup @ A @ ( image2 @ C @ A @ A4 @ I6 ) ) ) ) ) ) ) ) ).
% mono_cSUP
thf(fact_5703_cSUP__UNION,axiom,
! [B: $tType,D: $tType,C: $tType] :
( ( condit1219197933456340205attice @ B )
=> ! [A4: set @ C,B2: C > ( set @ D ),F2: D > B] :
( ( A4
!= ( bot_bot @ ( set @ C ) ) )
=> ( ! [X5: C] :
( ( member @ C @ X5 @ A4 )
=> ( ( B2 @ X5 )
!= ( bot_bot @ ( set @ D ) ) ) )
=> ( ( condit941137186595557371_above @ B
@ ( complete_Sup_Sup @ ( set @ B )
@ ( image2 @ C @ ( set @ B )
@ ^ [X4: C] : ( image2 @ D @ B @ F2 @ ( B2 @ X4 ) )
@ A4 ) ) )
=> ( ( complete_Sup_Sup @ B @ ( image2 @ D @ B @ F2 @ ( complete_Sup_Sup @ ( set @ D ) @ ( image2 @ C @ ( set @ D ) @ B2 @ A4 ) ) ) )
= ( complete_Sup_Sup @ B
@ ( image2 @ C @ B
@ ^ [X4: C] : ( complete_Sup_Sup @ B @ ( image2 @ D @ B @ F2 @ ( B2 @ X4 ) ) )
@ A4 ) ) ) ) ) ) ) ).
% cSUP_UNION
thf(fact_5704_Sup__fin_Oinsert__remove,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( ( ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic5882676163264333800up_fin @ A @ ( insert2 @ A @ X @ A4 ) )
= X ) )
& ( ( ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic5882676163264333800up_fin @ A @ ( insert2 @ A @ X @ A4 ) )
= ( sup_sup @ A @ X @ ( lattic5882676163264333800up_fin @ A @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ).
% Sup_fin.insert_remove
thf(fact_5705_Sup__fin_Oremove,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( member @ A @ X @ A4 )
=> ( ( ( ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic5882676163264333800up_fin @ A @ A4 )
= X ) )
& ( ( ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic5882676163264333800up_fin @ A @ A4 )
= ( sup_sup @ A @ X @ ( lattic5882676163264333800up_fin @ A @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ) ).
% Sup_fin.remove
thf(fact_5706_continuous__at__Inf__mono,axiom,
! [B: $tType,A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( topolo1944317154257567458pology @ A )
& ( condit6923001295902523014norder @ B )
& ( topolo1944317154257567458pology @ B ) )
=> ! [F2: A > B,S2: set @ A] :
( ( order_mono @ A @ B @ F2 )
=> ( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ ( complete_Inf_Inf @ A @ S2 ) @ ( set_ord_greaterThan @ A @ ( complete_Inf_Inf @ A @ S2 ) ) ) @ F2 )
=> ( ( S2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit1013018076250108175_below @ A @ S2 )
=> ( ( F2 @ ( complete_Inf_Inf @ A @ S2 ) )
= ( complete_Inf_Inf @ B @ ( image2 @ A @ B @ F2 @ S2 ) ) ) ) ) ) ) ) ).
% continuous_at_Inf_mono
thf(fact_5707_continuous__at__Sup__mono,axiom,
! [B: $tType,A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( topolo1944317154257567458pology @ A )
& ( condit6923001295902523014norder @ B )
& ( topolo1944317154257567458pology @ B ) )
=> ! [F2: A > B,S2: set @ A] :
( ( order_mono @ A @ B @ F2 )
=> ( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ ( complete_Sup_Sup @ A @ S2 ) @ ( set_ord_lessThan @ A @ ( complete_Sup_Sup @ A @ S2 ) ) ) @ F2 )
=> ( ( S2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit941137186595557371_above @ A @ S2 )
=> ( ( F2 @ ( complete_Sup_Sup @ A @ S2 ) )
= ( complete_Sup_Sup @ B @ ( image2 @ A @ B @ F2 @ S2 ) ) ) ) ) ) ) ) ).
% continuous_at_Sup_mono
thf(fact_5708_Inf__fin_Oeq__fold_H,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( ( lattic7752659483105999362nf_fin @ A )
= ( ^ [A7: set @ A] :
( the2 @ A
@ ( finite_fold @ A @ ( option @ A )
@ ^ [X4: A,Y6: option @ A] : ( some @ A @ ( case_option @ A @ A @ X4 @ ( inf_inf @ A @ X4 ) @ Y6 ) )
@ ( none @ A )
@ A7 ) ) ) ) ) ).
% Inf_fin.eq_fold'
thf(fact_5709_take__bit__numeral__numeral,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M2: num,N2: num] :
( ( bit_se2584673776208193580ke_bit @ A @ ( numeral_numeral @ nat @ M2 ) @ ( numeral_numeral @ A @ N2 ) )
= ( case_option @ A @ num @ ( zero_zero @ A ) @ ( numeral_numeral @ A ) @ ( bit_take_bit_num @ ( numeral_numeral @ nat @ M2 ) @ N2 ) ) ) ) ).
% take_bit_numeral_numeral
thf(fact_5710_comp__fun__idem__on_Ofold__insert__idem2,axiom,
! [B: $tType,A: $tType,S2: set @ A,F2: A > B > B,X: A,A4: set @ A,Z3: B] :
( ( finite673082921795544331dem_on @ A @ B @ S2 @ F2 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( insert2 @ A @ X @ A4 ) @ S2 )
=> ( ( finite_finite2 @ A @ A4 )
=> ( ( finite_fold @ A @ B @ F2 @ Z3 @ ( insert2 @ A @ X @ A4 ) )
= ( finite_fold @ A @ B @ F2 @ ( F2 @ X @ Z3 ) @ A4 ) ) ) ) ) ).
% comp_fun_idem_on.fold_insert_idem2
thf(fact_5711_take__bit__num__simps_I1_J,axiom,
! [M2: num] :
( ( bit_take_bit_num @ ( zero_zero @ nat ) @ M2 )
= ( none @ num ) ) ).
% take_bit_num_simps(1)
thf(fact_5712_Inf__fin_Osingleton,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X: A] :
( ( lattic7752659483105999362nf_fin @ A @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= X ) ) ).
% Inf_fin.singleton
thf(fact_5713_sup__Inf__absorb,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [A4: set @ A,A3: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( member @ A @ A3 @ A4 )
=> ( ( sup_sup @ A @ ( lattic7752659483105999362nf_fin @ A @ A4 ) @ A3 )
= A3 ) ) ) ) ).
% sup_Inf_absorb
thf(fact_5714_take__bit__num__simps_I2_J,axiom,
! [N2: nat] :
( ( bit_take_bit_num @ ( suc @ N2 ) @ one2 )
= ( some @ num @ one2 ) ) ).
% take_bit_num_simps(2)
thf(fact_5715_take__bit__num__simps_I3_J,axiom,
! [N2: nat,M2: num] :
( ( bit_take_bit_num @ ( suc @ N2 ) @ ( bit0 @ M2 ) )
= ( case_option @ ( option @ num ) @ num @ ( none @ num )
@ ^ [Q5: num] : ( some @ num @ ( bit0 @ Q5 ) )
@ ( bit_take_bit_num @ N2 @ M2 ) ) ) ).
% take_bit_num_simps(3)
thf(fact_5716_take__bit__num__simps_I4_J,axiom,
! [N2: nat,M2: num] :
( ( bit_take_bit_num @ ( suc @ N2 ) @ ( bit1 @ M2 ) )
= ( some @ num @ ( case_option @ num @ num @ one2 @ bit1 @ ( bit_take_bit_num @ N2 @ M2 ) ) ) ) ).
% take_bit_num_simps(4)
thf(fact_5717_Inf__fin_Oinsert,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic7752659483105999362nf_fin @ A @ ( insert2 @ A @ X @ A4 ) )
= ( inf_inf @ A @ X @ ( lattic7752659483105999362nf_fin @ A @ A4 ) ) ) ) ) ) ).
% Inf_fin.insert
thf(fact_5718_comp__fun__idem__on_Ofun__left__idem,axiom,
! [A: $tType,B: $tType,S2: set @ A,F2: A > B > B,X: A,Z3: B] :
( ( finite673082921795544331dem_on @ A @ B @ S2 @ F2 )
=> ( ( member @ A @ X @ S2 )
=> ( ( F2 @ X @ ( F2 @ X @ Z3 ) )
= ( F2 @ X @ Z3 ) ) ) ) ).
% comp_fun_idem_on.fun_left_idem
thf(fact_5719_comp__fun__idem__on_Ocomp__fun__idem__on,axiom,
! [B: $tType,A: $tType,S2: set @ A,F2: A > B > B,X: A] :
( ( finite673082921795544331dem_on @ A @ B @ S2 @ F2 )
=> ( ( member @ A @ X @ S2 )
=> ( ( comp @ B @ B @ B @ ( F2 @ X ) @ ( F2 @ X ) )
= ( F2 @ X ) ) ) ) ).
% comp_fun_idem_on.comp_fun_idem_on
thf(fact_5720_comp__fun__idem__on_Oaxioms_I1_J,axiom,
! [B: $tType,A: $tType,S2: set @ A,F2: A > B > B] :
( ( finite673082921795544331dem_on @ A @ B @ S2 @ F2 )
=> ( finite4664212375090638736ute_on @ A @ B @ S2 @ F2 ) ) ).
% comp_fun_idem_on.axioms(1)
thf(fact_5721_Inf__fin_OcoboundedI,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A4: set @ A,A3: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( member @ A @ A3 @ A4 )
=> ( ord_less_eq @ A @ ( lattic7752659483105999362nf_fin @ A @ A4 ) @ A3 ) ) ) ) ).
% Inf_fin.coboundedI
thf(fact_5722_Inf__fin_Oin__idem,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( member @ A @ X @ A4 )
=> ( ( inf_inf @ A @ X @ ( lattic7752659483105999362nf_fin @ A @ A4 ) )
= ( lattic7752659483105999362nf_fin @ A @ A4 ) ) ) ) ) ).
% Inf_fin.in_idem
thf(fact_5723_Inf__fin_OboundedE,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ X @ ( lattic7752659483105999362nf_fin @ A @ A4 ) )
=> ! [A11: A] :
( ( member @ A @ A11 @ A4 )
=> ( ord_less_eq @ A @ X @ A11 ) ) ) ) ) ) ).
% Inf_fin.boundedE
thf(fact_5724_Inf__fin_OboundedI,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [A6: A] :
( ( member @ A @ A6 @ A4 )
=> ( ord_less_eq @ A @ X @ A6 ) )
=> ( ord_less_eq @ A @ X @ ( lattic7752659483105999362nf_fin @ A @ A4 ) ) ) ) ) ) ).
% Inf_fin.boundedI
thf(fact_5725_Inf__fin_Obounded__iff,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ X @ ( lattic7752659483105999362nf_fin @ A @ A4 ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ A4 )
=> ( ord_less_eq @ A @ X @ X4 ) ) ) ) ) ) ) ).
% Inf_fin.bounded_iff
thf(fact_5726_Inf__fin__Inf,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A4: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic7752659483105999362nf_fin @ A @ A4 )
= ( complete_Inf_Inf @ A @ A4 ) ) ) ) ) ).
% Inf_fin_Inf
thf(fact_5727_cInf__eq__Inf__fin,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X8: set @ A] :
( ( finite_finite2 @ A @ X8 )
=> ( ( X8
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Inf_Inf @ A @ X8 )
= ( lattic7752659483105999362nf_fin @ A @ X8 ) ) ) ) ) ).
% cInf_eq_Inf_fin
thf(fact_5728_Inf__fin_Oinfinite,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A4: set @ A] :
( ~ ( finite_finite2 @ A @ A4 )
=> ( ( lattic7752659483105999362nf_fin @ A @ A4 )
= ( the2 @ A @ ( none @ A ) ) ) ) ) ).
% Inf_fin.infinite
thf(fact_5729_take__bit__num__eq__None__imp,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M2: nat,N2: num] :
( ( ( bit_take_bit_num @ M2 @ N2 )
= ( none @ num ) )
=> ( ( bit_se2584673776208193580ke_bit @ A @ M2 @ ( numeral_numeral @ A @ N2 ) )
= ( zero_zero @ A ) ) ) ) ).
% take_bit_num_eq_None_imp
thf(fact_5730_Inf__fin_Osubset__imp,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A4: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B2 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B2 )
=> ( ord_less_eq @ A @ ( lattic7752659483105999362nf_fin @ A @ B2 ) @ ( lattic7752659483105999362nf_fin @ A @ A4 ) ) ) ) ) ) ).
% Inf_fin.subset_imp
thf(fact_5731_Inf__fin_Ohom__commute,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [H2: A > A,N6: set @ A] :
( ! [X5: A,Y3: A] :
( ( H2 @ ( inf_inf @ A @ X5 @ Y3 ) )
= ( inf_inf @ A @ ( H2 @ X5 ) @ ( H2 @ Y3 ) ) )
=> ( ( finite_finite2 @ A @ N6 )
=> ( ( N6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( H2 @ ( lattic7752659483105999362nf_fin @ A @ N6 ) )
= ( lattic7752659483105999362nf_fin @ A @ ( image2 @ A @ A @ H2 @ N6 ) ) ) ) ) ) ) ).
% Inf_fin.hom_commute
thf(fact_5732_Inf__fin_Osubset,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A4: set @ A,B2: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( B2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ A4 )
=> ( ( inf_inf @ A @ ( lattic7752659483105999362nf_fin @ A @ B2 ) @ ( lattic7752659483105999362nf_fin @ A @ A4 ) )
= ( lattic7752659483105999362nf_fin @ A @ A4 ) ) ) ) ) ) ).
% Inf_fin.subset
thf(fact_5733_Inf__fin_Oinsert__not__elem,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ~ ( member @ A @ X @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic7752659483105999362nf_fin @ A @ ( insert2 @ A @ X @ A4 ) )
= ( inf_inf @ A @ X @ ( lattic7752659483105999362nf_fin @ A @ A4 ) ) ) ) ) ) ) ).
% Inf_fin.insert_not_elem
thf(fact_5734_Inf__fin_Oclosed,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A4: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X5: A,Y3: A] : ( member @ A @ ( inf_inf @ A @ X5 @ Y3 ) @ ( insert2 @ A @ X5 @ ( insert2 @ A @ Y3 @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( member @ A @ ( lattic7752659483105999362nf_fin @ A @ A4 ) @ A4 ) ) ) ) ) ).
% Inf_fin.closed
thf(fact_5735_Inf__fin_Ounion,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A4: set @ A,B2: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B2 )
=> ( ( B2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic7752659483105999362nf_fin @ A @ ( sup_sup @ ( set @ A ) @ A4 @ B2 ) )
= ( inf_inf @ A @ ( lattic7752659483105999362nf_fin @ A @ A4 ) @ ( lattic7752659483105999362nf_fin @ A @ B2 ) ) ) ) ) ) ) ) ).
% Inf_fin.union
thf(fact_5736_Inf__fin__le__Sup__fin,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [A4: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ord_less_eq @ A @ ( lattic7752659483105999362nf_fin @ A @ A4 ) @ ( lattic5882676163264333800up_fin @ A @ A4 ) ) ) ) ) ).
% Inf_fin_le_Sup_fin
thf(fact_5737_Inf__fin_Oeq__fold,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( lattic7752659483105999362nf_fin @ A @ ( insert2 @ A @ X @ A4 ) )
= ( finite_fold @ A @ A @ ( inf_inf @ A ) @ X @ A4 ) ) ) ) ).
% Inf_fin.eq_fold
thf(fact_5738_sup__Inf2__distrib,axiom,
! [A: $tType] :
( ( distrib_lattice @ A )
=> ! [A4: set @ A,B2: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B2 )
=> ( ( B2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( sup_sup @ A @ ( lattic7752659483105999362nf_fin @ A @ A4 ) @ ( lattic7752659483105999362nf_fin @ A @ B2 ) )
= ( lattic7752659483105999362nf_fin @ A
@ ( collect @ A
@ ^ [Uu3: A] :
? [A5: A,B4: A] :
( ( Uu3
= ( sup_sup @ A @ A5 @ B4 ) )
& ( member @ A @ A5 @ A4 )
& ( member @ A @ B4 @ B2 ) ) ) ) ) ) ) ) ) ) ).
% sup_Inf2_distrib
thf(fact_5739_sup__Inf1__distrib,axiom,
! [A: $tType] :
( ( distrib_lattice @ A )
=> ! [A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( sup_sup @ A @ X @ ( lattic7752659483105999362nf_fin @ A @ A4 ) )
= ( lattic7752659483105999362nf_fin @ A
@ ( collect @ A
@ ^ [Uu3: A] :
? [A5: A] :
( ( Uu3
= ( sup_sup @ A @ X @ A5 ) )
& ( member @ A @ A5 @ A4 ) ) ) ) ) ) ) ) ).
% sup_Inf1_distrib
thf(fact_5740_Inf__fin_Oremove,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( member @ A @ X @ A4 )
=> ( ( ( ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic7752659483105999362nf_fin @ A @ A4 )
= X ) )
& ( ( ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic7752659483105999362nf_fin @ A @ A4 )
= ( inf_inf @ A @ X @ ( lattic7752659483105999362nf_fin @ A @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ) ).
% Inf_fin.remove
thf(fact_5741_Inf__fin_Oinsert__remove,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( ( ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic7752659483105999362nf_fin @ A @ ( insert2 @ A @ X @ A4 ) )
= X ) )
& ( ( ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic7752659483105999362nf_fin @ A @ ( insert2 @ A @ X @ A4 ) )
= ( inf_inf @ A @ X @ ( lattic7752659483105999362nf_fin @ A @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ).
% Inf_fin.insert_remove
thf(fact_5742_take__bit__num__def,axiom,
( bit_take_bit_num
= ( ^ [N: nat,M: num] :
( if @ ( option @ num )
@ ( ( bit_se2584673776208193580ke_bit @ nat @ N @ ( numeral_numeral @ nat @ M ) )
= ( zero_zero @ nat ) )
@ ( none @ num )
@ ( some @ num @ ( num_of_nat @ ( bit_se2584673776208193580ke_bit @ nat @ N @ ( numeral_numeral @ nat @ M ) ) ) ) ) ) ) ).
% take_bit_num_def
thf(fact_5743_comp__fun__idem__on_Ocomp__comp__fun__idem__on,axiom,
! [B: $tType,A: $tType,C: $tType,S2: set @ A,F2: A > B > B,G: C > A,R: set @ C] :
( ( finite673082921795544331dem_on @ A @ B @ S2 @ F2 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image2 @ C @ A @ G @ ( top_top @ ( set @ C ) ) ) @ S2 )
=> ( finite673082921795544331dem_on @ C @ B @ R @ ( comp @ A @ ( B > B ) @ C @ F2 @ G ) ) ) ) ).
% comp_fun_idem_on.comp_comp_fun_idem_on
thf(fact_5744_comp__fun__idem__on_Ofold__insert__idem,axiom,
! [B: $tType,A: $tType,S2: set @ A,F2: A > B > B,X: A,A4: set @ A,Z3: B] :
( ( finite673082921795544331dem_on @ A @ B @ S2 @ F2 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( insert2 @ A @ X @ A4 ) @ S2 )
=> ( ( finite_finite2 @ A @ A4 )
=> ( ( finite_fold @ A @ B @ F2 @ Z3 @ ( insert2 @ A @ X @ A4 ) )
= ( F2 @ X @ ( finite_fold @ A @ B @ F2 @ Z3 @ A4 ) ) ) ) ) ) ).
% comp_fun_idem_on.fold_insert_idem
thf(fact_5745_dual__min,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( min @ A
@ ^ [X4: A,Y6: A] : ( ord_less_eq @ A @ Y6 @ X4 ) )
= ( ord_max @ A ) ) ) ).
% dual_min
thf(fact_5746_card__Min__le__sum,axiom,
! [A: $tType,A4: set @ A,F2: A > nat] :
( ( finite_finite2 @ A @ A4 )
=> ( ord_less_eq @ nat @ ( times_times @ nat @ ( finite_card @ A @ A4 ) @ ( lattic643756798350308766er_Min @ nat @ ( image2 @ A @ nat @ F2 @ A4 ) ) ) @ ( groups7311177749621191930dd_sum @ A @ nat @ F2 @ A4 ) ) ) ).
% card_Min_le_sum
thf(fact_5747_total__on__singleton,axiom,
! [A: $tType,X: A] : ( total_on @ A @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) @ ( insert2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ X ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ).
% total_on_singleton
thf(fact_5748_Min__singleton,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A] :
( ( lattic643756798350308766er_Min @ A @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= X ) ) ).
% Min_singleton
thf(fact_5749_Min_Obounded__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ X @ ( lattic643756798350308766er_Min @ A @ A4 ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ A4 )
=> ( ord_less_eq @ A @ X @ X4 ) ) ) ) ) ) ) ).
% Min.bounded_iff
thf(fact_5750_Min__gr__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less @ A @ X @ ( lattic643756798350308766er_Min @ A @ A4 ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ A4 )
=> ( ord_less @ A @ X @ X4 ) ) ) ) ) ) ) ).
% Min_gr_iff
thf(fact_5751_Min__const,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ B,C2: A] :
( ( finite_finite2 @ B @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( lattic643756798350308766er_Min @ A
@ ( image2 @ B @ A
@ ^ [Uu3: B] : C2
@ A4 ) )
= C2 ) ) ) ) ).
% Min_const
thf(fact_5752_minus__Max__eq__Min,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [S2: set @ A] :
( ( finite_finite2 @ A @ S2 )
=> ( ( S2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( uminus_uminus @ A @ ( lattic643756798349783984er_Max @ A @ S2 ) )
= ( lattic643756798350308766er_Min @ A @ ( image2 @ A @ A @ ( uminus_uminus @ A ) @ S2 ) ) ) ) ) ) ).
% minus_Max_eq_Min
thf(fact_5753_minus__Min__eq__Max,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [S2: set @ A] :
( ( finite_finite2 @ A @ S2 )
=> ( ( S2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( uminus_uminus @ A @ ( lattic643756798350308766er_Min @ A @ S2 ) )
= ( lattic643756798349783984er_Max @ A @ ( image2 @ A @ A @ ( uminus_uminus @ A ) @ S2 ) ) ) ) ) ) ).
% minus_Min_eq_Max
thf(fact_5754_Inf__fin__Min,axiom,
! [A: $tType] :
( ( ( semilattice_inf @ A )
& ( linorder @ A ) )
=> ( ( lattic7752659483105999362nf_fin @ A )
= ( lattic643756798350308766er_Min @ A ) ) ) ).
% Inf_fin_Min
thf(fact_5755_Min__le,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( member @ A @ X @ A4 )
=> ( ord_less_eq @ A @ ( lattic643756798350308766er_Min @ A @ A4 ) @ X ) ) ) ) ).
% Min_le
thf(fact_5756_Min__eqI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ! [Y3: A] :
( ( member @ A @ Y3 @ A4 )
=> ( ord_less_eq @ A @ X @ Y3 ) )
=> ( ( member @ A @ X @ A4 )
=> ( ( lattic643756798350308766er_Min @ A @ A4 )
= X ) ) ) ) ) ).
% Min_eqI
thf(fact_5757_Min_OcoboundedI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,A3: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( member @ A @ A3 @ A4 )
=> ( ord_less_eq @ A @ ( lattic643756798350308766er_Min @ A @ A4 ) @ A3 ) ) ) ) ).
% Min.coboundedI
thf(fact_5758_Min__in,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( member @ A @ ( lattic643756798350308766er_Min @ A @ A4 ) @ A4 ) ) ) ) ).
% Min_in
thf(fact_5759_ord_Omin_Ocong,axiom,
! [A: $tType] :
( ( min @ A )
= ( min @ A ) ) ).
% ord.min.cong
thf(fact_5760_ord_Omin__def,axiom,
! [A: $tType] :
( ( min @ A )
= ( ^ [Less_eq: A > A > $o,A5: A,B4: A] : ( if @ A @ ( Less_eq @ A5 @ B4 ) @ A5 @ B4 ) ) ) ).
% ord.min_def
thf(fact_5761_total__on__empty,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] : ( total_on @ A @ ( bot_bot @ ( set @ A ) ) @ R2 ) ).
% total_on_empty
thf(fact_5762_total__lenlex,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( total_on @ A @ ( top_top @ ( set @ A ) ) @ R2 )
=> ( total_on @ ( list @ A ) @ ( top_top @ ( set @ ( list @ A ) ) ) @ ( lenlex @ A @ R2 ) ) ) ).
% total_lenlex
thf(fact_5763_Min_OboundedI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [A6: A] :
( ( member @ A @ A6 @ A4 )
=> ( ord_less_eq @ A @ X @ A6 ) )
=> ( ord_less_eq @ A @ X @ ( lattic643756798350308766er_Min @ A @ A4 ) ) ) ) ) ) ).
% Min.boundedI
thf(fact_5764_Min_OboundedE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ X @ ( lattic643756798350308766er_Min @ A @ A4 ) )
=> ! [A11: A] :
( ( member @ A @ A11 @ A4 )
=> ( ord_less_eq @ A @ X @ A11 ) ) ) ) ) ) ).
% Min.boundedE
thf(fact_5765_eq__Min__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,M2: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( M2
= ( lattic643756798350308766er_Min @ A @ A4 ) )
= ( ( member @ A @ M2 @ A4 )
& ! [X4: A] :
( ( member @ A @ X4 @ A4 )
=> ( ord_less_eq @ A @ M2 @ X4 ) ) ) ) ) ) ) ).
% eq_Min_iff
thf(fact_5766_Min__le__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ ( lattic643756798350308766er_Min @ A @ A4 ) @ X )
= ( ? [X4: A] :
( ( member @ A @ X4 @ A4 )
& ( ord_less_eq @ A @ X4 @ X ) ) ) ) ) ) ) ).
% Min_le_iff
thf(fact_5767_Min__eq__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,M2: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ( lattic643756798350308766er_Min @ A @ A4 )
= M2 )
= ( ( member @ A @ M2 @ A4 )
& ! [X4: A] :
( ( member @ A @ X4 @ A4 )
=> ( ord_less_eq @ A @ M2 @ X4 ) ) ) ) ) ) ) ).
% Min_eq_iff
thf(fact_5768_Min__less__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less @ A @ ( lattic643756798350308766er_Min @ A @ A4 ) @ X )
= ( ? [X4: A] :
( ( member @ A @ X4 @ A4 )
& ( ord_less @ A @ X4 @ X ) ) ) ) ) ) ) ).
% Min_less_iff
thf(fact_5769_Min__insert2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,A3: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ! [B5: A] :
( ( member @ A @ B5 @ A4 )
=> ( ord_less_eq @ A @ A3 @ B5 ) )
=> ( ( lattic643756798350308766er_Min @ A @ ( insert2 @ A @ A3 @ A4 ) )
= A3 ) ) ) ) ).
% Min_insert2
thf(fact_5770_cInf__eq__Min,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [X8: set @ A] :
( ( finite_finite2 @ A @ X8 )
=> ( ( X8
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Inf_Inf @ A @ X8 )
= ( lattic643756798350308766er_Min @ A @ X8 ) ) ) ) ) ).
% cInf_eq_Min
thf(fact_5771_Min__Inf,axiom,
! [A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [A4: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798350308766er_Min @ A @ A4 )
= ( complete_Inf_Inf @ A @ A4 ) ) ) ) ) ).
% Min_Inf
thf(fact_5772_Min_Oinfinite,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A] :
( ~ ( finite_finite2 @ A @ A4 )
=> ( ( lattic643756798350308766er_Min @ A @ A4 )
= ( the2 @ A @ ( none @ A ) ) ) ) ) ).
% Min.infinite
thf(fact_5773_Min_Osubset__imp,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B2 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B2 )
=> ( ord_less_eq @ A @ ( lattic643756798350308766er_Min @ A @ B2 ) @ ( lattic643756798350308766er_Min @ A @ A4 ) ) ) ) ) ) ).
% Min.subset_imp
thf(fact_5774_Min__antimono,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [M7: set @ A,N6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ M7 @ N6 )
=> ( ( M7
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ N6 )
=> ( ord_less_eq @ A @ ( lattic643756798350308766er_Min @ A @ N6 ) @ ( lattic643756798350308766er_Min @ A @ M7 ) ) ) ) ) ) ).
% Min_antimono
thf(fact_5775_mono__Min__commute,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( linorder @ B ) )
=> ! [F2: A > B,A4: set @ A] :
( ( order_mono @ A @ B @ F2 )
=> ( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( F2 @ ( lattic643756798350308766er_Min @ A @ A4 ) )
= ( lattic643756798350308766er_Min @ B @ ( image2 @ A @ B @ F2 @ A4 ) ) ) ) ) ) ) ).
% mono_Min_commute
thf(fact_5776_Min__add__commute,axiom,
! [B: $tType,A: $tType] :
( ( linord4140545234300271783up_add @ A )
=> ! [S2: set @ B,F2: B > A,K: A] :
( ( finite_finite2 @ B @ S2 )
=> ( ( S2
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( lattic643756798350308766er_Min @ A
@ ( image2 @ B @ A
@ ^ [X4: B] : ( plus_plus @ A @ ( F2 @ X4 ) @ K )
@ S2 ) )
= ( plus_plus @ A @ ( lattic643756798350308766er_Min @ A @ ( image2 @ B @ A @ F2 @ S2 ) ) @ K ) ) ) ) ) ).
% Min_add_commute
thf(fact_5777_dual__Max,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( lattices_Max @ A
@ ^ [X4: A,Y6: A] : ( ord_less_eq @ A @ Y6 @ X4 ) )
= ( lattic643756798350308766er_Min @ A ) ) ) ).
% dual_Max
thf(fact_5778_f__arg__min__list__f,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ B )
=> ! [Xs: list @ A,F2: A > B] :
( ( Xs
!= ( nil @ A ) )
=> ( ( F2 @ ( arg_min_list @ A @ B @ F2 @ Xs ) )
= ( lattic643756798350308766er_Min @ B @ ( image2 @ A @ B @ F2 @ ( set2 @ A @ Xs ) ) ) ) ) ) ).
% f_arg_min_list_f
thf(fact_5779_Min_Oeq__fold_H,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( lattic643756798350308766er_Min @ A )
= ( ^ [A7: set @ A] :
( the2 @ A
@ ( finite_fold @ A @ ( option @ A )
@ ^ [X4: A,Y6: option @ A] : ( some @ A @ ( case_option @ A @ A @ X4 @ ( ord_min @ A @ X4 ) @ Y6 ) )
@ ( none @ A )
@ A7 ) ) ) ) ) ).
% Min.eq_fold'
thf(fact_5780_min__Suc__Suc,axiom,
! [M2: nat,N2: nat] :
( ( ord_min @ nat @ ( suc @ M2 ) @ ( suc @ N2 ) )
= ( suc @ ( ord_min @ nat @ M2 @ N2 ) ) ) ).
% min_Suc_Suc
thf(fact_5781_min__0L,axiom,
! [N2: nat] :
( ( ord_min @ nat @ ( zero_zero @ nat ) @ N2 )
= ( zero_zero @ nat ) ) ).
% min_0L
thf(fact_5782_min__0R,axiom,
! [N2: nat] :
( ( ord_min @ nat @ N2 @ ( zero_zero @ nat ) )
= ( zero_zero @ nat ) ) ).
% min_0R
thf(fact_5783_take__take,axiom,
! [A: $tType,N2: nat,M2: nat,Xs: list @ A] :
( ( take @ A @ N2 @ ( take @ A @ M2 @ Xs ) )
= ( take @ A @ ( ord_min @ nat @ N2 @ M2 ) @ Xs ) ) ).
% take_take
thf(fact_5784_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_5785_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_5786_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_5787_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_5788_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_5789_min__less__iff__conj,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Z3: A,X: A,Y: A] :
( ( ord_less @ A @ Z3 @ ( ord_min @ A @ X @ Y ) )
= ( ( ord_less @ A @ Z3 @ X )
& ( ord_less @ A @ Z3 @ Y ) ) ) ) ).
% min_less_iff_conj
thf(fact_5790_min__top2,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [X: A] :
( ( ord_min @ A @ X @ ( top_top @ A ) )
= X ) ) ).
% min_top2
thf(fact_5791_min__top,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [X: A] :
( ( ord_min @ A @ ( top_top @ A ) @ X )
= X ) ) ).
% min_top
thf(fact_5792_min__bot2,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [X: A] :
( ( ord_min @ A @ X @ ( bot_bot @ A ) )
= ( bot_bot @ A ) ) ) ).
% min_bot2
thf(fact_5793_min__bot,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [X: A] :
( ( ord_min @ A @ ( bot_bot @ A ) @ X )
= ( bot_bot @ A ) ) ) ).
% min_bot
thf(fact_5794_length__take,axiom,
! [A: $tType,N2: nat,Xs: list @ A] :
( ( size_size @ ( list @ A ) @ ( take @ A @ N2 @ Xs ) )
= ( ord_min @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ N2 ) ) ).
% length_take
thf(fact_5795_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_5796_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_5797_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_5798_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_5799_take__replicate,axiom,
! [A: $tType,I: nat,K: nat,X: A] :
( ( take @ A @ I @ ( replicate @ A @ K @ X ) )
= ( replicate @ A @ ( ord_min @ nat @ I @ K ) @ X ) ) ).
% take_replicate
thf(fact_5800_min__number__of_I1_J,axiom,
! [A: $tType] :
( ( ( numeral @ A )
& ( ord @ A ) )
=> ! [U: num,V2: num] :
( ( ( ord_less_eq @ A @ ( numeral_numeral @ A @ U ) @ ( numeral_numeral @ A @ V2 ) )
=> ( ( ord_min @ A @ ( numeral_numeral @ A @ U ) @ ( numeral_numeral @ A @ V2 ) )
= ( numeral_numeral @ A @ U ) ) )
& ( ~ ( ord_less_eq @ A @ ( numeral_numeral @ A @ U ) @ ( numeral_numeral @ A @ V2 ) )
=> ( ( ord_min @ A @ ( numeral_numeral @ A @ U ) @ ( numeral_numeral @ A @ V2 ) )
= ( numeral_numeral @ A @ V2 ) ) ) ) ) ).
% min_number_of(1)
thf(fact_5801_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_5802_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_5803_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_5804_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_5805_min__numeral__Suc,axiom,
! [K: num,N2: nat] :
( ( ord_min @ nat @ ( numeral_numeral @ nat @ K ) @ ( suc @ N2 ) )
= ( suc @ ( ord_min @ nat @ ( pred_numeral @ K ) @ N2 ) ) ) ).
% min_numeral_Suc
thf(fact_5806_min__Suc__numeral,axiom,
! [N2: nat,K: num] :
( ( ord_min @ nat @ ( suc @ N2 ) @ ( numeral_numeral @ nat @ K ) )
= ( suc @ ( ord_min @ nat @ N2 @ ( pred_numeral @ K ) ) ) ) ).
% min_Suc_numeral
thf(fact_5807_Int__atMost,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B3: A] :
( ( inf_inf @ ( set @ A ) @ ( set_ord_atMost @ A @ A3 ) @ ( set_ord_atMost @ A @ B3 ) )
= ( set_ord_atMost @ A @ ( ord_min @ A @ A3 @ B3 ) ) ) ) ).
% Int_atMost
thf(fact_5808_min__number__of_I2_J,axiom,
! [A: $tType] :
( ( ( uminus @ A )
& ( numeral @ A )
& ( ord @ A ) )
=> ! [U: num,V2: num] :
( ( ( ord_less_eq @ A @ ( numeral_numeral @ A @ U ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) )
=> ( ( ord_min @ A @ ( numeral_numeral @ A @ U ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) )
= ( numeral_numeral @ A @ U ) ) )
& ( ~ ( ord_less_eq @ A @ ( numeral_numeral @ A @ U ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) )
=> ( ( ord_min @ A @ ( numeral_numeral @ A @ U ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) ) ) ) ) ).
% min_number_of(2)
thf(fact_5809_min__number__of_I3_J,axiom,
! [A: $tType] :
( ( ( uminus @ A )
& ( numeral @ A )
& ( ord @ A ) )
=> ! [U: num,V2: num] :
( ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( numeral_numeral @ A @ V2 ) )
=> ( ( ord_min @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( numeral_numeral @ A @ V2 ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( numeral_numeral @ A @ V2 ) )
=> ( ( ord_min @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( numeral_numeral @ A @ V2 ) )
= ( numeral_numeral @ A @ V2 ) ) ) ) ) ).
% min_number_of(3)
thf(fact_5810_min__number__of_I4_J,axiom,
! [A: $tType] :
( ( ( uminus @ A )
& ( numeral @ A )
& ( ord @ A ) )
=> ! [U: num,V2: num] :
( ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) )
=> ( ( ord_min @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) )
=> ( ( ord_min @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) ) ) ) ) ).
% min_number_of(4)
thf(fact_5811_Int__atLeastAtMost,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B3: A,C2: A,D2: A] :
( ( inf_inf @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A3 @ B3 ) @ ( set_or1337092689740270186AtMost @ A @ C2 @ D2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( ord_max @ A @ A3 @ C2 ) @ ( ord_min @ A @ B3 @ D2 ) ) ) ) ).
% Int_atLeastAtMost
thf(fact_5812_Int__atLeastAtMostL1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B3: A,D2: A] :
( ( inf_inf @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A3 @ B3 ) @ ( set_ord_atMost @ A @ D2 ) )
= ( set_or1337092689740270186AtMost @ A @ A3 @ ( ord_min @ A @ B3 @ D2 ) ) ) ) ).
% Int_atLeastAtMostL1
thf(fact_5813_Int__atLeastAtMostR1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B3: A,C2: A,D2: A] :
( ( inf_inf @ ( set @ A ) @ ( set_ord_atMost @ A @ B3 ) @ ( set_or1337092689740270186AtMost @ A @ C2 @ D2 ) )
= ( set_or1337092689740270186AtMost @ A @ C2 @ ( ord_min @ A @ B3 @ D2 ) ) ) ) ).
% Int_atLeastAtMostR1
thf(fact_5814_Int__atLeastLessThan,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B3: A,C2: A,D2: A] :
( ( inf_inf @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ A3 @ B3 ) @ ( set_or7035219750837199246ssThan @ A @ C2 @ D2 ) )
= ( set_or7035219750837199246ssThan @ A @ ( ord_max @ A @ A3 @ C2 ) @ ( ord_min @ A @ B3 @ D2 ) ) ) ) ).
% Int_atLeastLessThan
thf(fact_5815_Int__greaterThanLessThan,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B3: A,C2: A,D2: A] :
( ( inf_inf @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ A3 @ B3 ) @ ( set_or5935395276787703475ssThan @ A @ C2 @ D2 ) )
= ( set_or5935395276787703475ssThan @ A @ ( ord_max @ A @ A3 @ C2 ) @ ( ord_min @ A @ B3 @ D2 ) ) ) ) ).
% Int_greaterThanLessThan
thf(fact_5816_Int__greaterThanAtMost,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B3: A,C2: A,D2: A] :
( ( inf_inf @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ A3 @ B3 ) @ ( set_or3652927894154168847AtMost @ A @ C2 @ D2 ) )
= ( set_or3652927894154168847AtMost @ A @ ( ord_max @ A @ A3 @ C2 ) @ ( ord_min @ A @ B3 @ D2 ) ) ) ) ).
% Int_greaterThanAtMost
thf(fact_5817_Min__insert,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798350308766er_Min @ A @ ( insert2 @ A @ X @ A4 ) )
= ( ord_min @ A @ X @ ( lattic643756798350308766er_Min @ A @ A4 ) ) ) ) ) ) ).
% Min_insert
thf(fact_5818_Min_Oin__idem,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( member @ A @ X @ A4 )
=> ( ( ord_min @ A @ X @ ( lattic643756798350308766er_Min @ A @ A4 ) )
= ( lattic643756798350308766er_Min @ A @ A4 ) ) ) ) ) ).
% Min.in_idem
thf(fact_5819_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_5820_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_5821_greaterThan__Int__greaterThan,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B3: A] :
( ( inf_inf @ ( set @ A ) @ ( set_ord_lessThan @ A @ A3 ) @ ( set_ord_lessThan @ A @ B3 ) )
= ( set_ord_lessThan @ A @ ( ord_min @ A @ A3 @ B3 ) ) ) ) ).
% greaterThan_Int_greaterThan
thf(fact_5822_min__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( ord_min @ A )
= ( ^ [A5: A,B4: A] : ( if @ A @ ( ord_less_eq @ A @ A5 @ B4 ) @ A5 @ B4 ) ) ) ) ).
% min_def
thf(fact_5823_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_5824_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_5825_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_5826_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_5827_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_5828_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_5829_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_5830_min_Oorder__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A5: A,B4: A] :
( A5
= ( ord_min @ A @ A5 @ B4 ) ) ) ) ) ).
% min.order_iff
thf(fact_5831_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_5832_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_5833_min_Oabsorb__iff1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A5: A,B4: A] :
( ( ord_min @ A @ A5 @ B4 )
= A5 ) ) ) ) ).
% min.absorb_iff1
thf(fact_5834_min_Oabsorb__iff2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B4: A,A5: A] :
( ( ord_min @ A @ A5 @ B4 )
= B4 ) ) ) ) ).
% min.absorb_iff2
thf(fact_5835_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_5836_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_5837_min__le__iff__disj,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A,Z3: A] :
( ( ord_less_eq @ A @ ( ord_min @ A @ X @ Y ) @ Z3 )
= ( ( ord_less_eq @ A @ X @ Z3 )
| ( ord_less_eq @ A @ Y @ Z3 ) ) ) ) ).
% min_le_iff_disj
thf(fact_5838_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_5839_min__less__iff__disj,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A,Z3: A] :
( ( ord_less @ A @ ( ord_min @ A @ X @ Y ) @ Z3 )
= ( ( ord_less @ A @ X @ Z3 )
| ( ord_less @ A @ Y @ Z3 ) ) ) ) ).
% min_less_iff_disj
thf(fact_5840_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_5841_min_Ostrict__order__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less @ A )
= ( ^ [A5: A,B4: A] :
( ( A5
= ( ord_min @ A @ A5 @ B4 ) )
& ( A5 != B4 ) ) ) ) ) ).
% min.strict_order_iff
thf(fact_5842_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_5843_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_5844_linorder_OMax_Ocong,axiom,
! [A: $tType] :
( ( lattices_Max @ A )
= ( lattices_Max @ A ) ) ).
% linorder.Max.cong
thf(fact_5845_min__diff__distrib__left,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [X: A,Y: A,Z3: A] :
( ( minus_minus @ A @ ( ord_min @ A @ X @ Y ) @ Z3 )
= ( ord_min @ A @ ( minus_minus @ A @ X @ Z3 ) @ ( minus_minus @ A @ Y @ Z3 ) ) ) ) ).
% min_diff_distrib_left
thf(fact_5846_min__add__distrib__left,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [X: A,Y: A,Z3: A] :
( ( plus_plus @ A @ ( ord_min @ A @ X @ Y ) @ Z3 )
= ( ord_min @ A @ ( plus_plus @ A @ X @ Z3 ) @ ( plus_plus @ A @ Y @ Z3 ) ) ) ) ).
% min_add_distrib_left
thf(fact_5847_min__add__distrib__right,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [X: A,Y: A,Z3: A] :
( ( plus_plus @ A @ X @ ( ord_min @ A @ Y @ Z3 ) )
= ( ord_min @ A @ ( plus_plus @ A @ X @ Y ) @ ( plus_plus @ A @ X @ Z3 ) ) ) ) ).
% min_add_distrib_right
thf(fact_5848_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_5849_min__diff,axiom,
! [M2: nat,I: nat,N2: nat] :
( ( ord_min @ nat @ ( minus_minus @ nat @ M2 @ I ) @ ( minus_minus @ nat @ N2 @ I ) )
= ( minus_minus @ nat @ ( ord_min @ nat @ M2 @ N2 ) @ I ) ) ).
% min_diff
thf(fact_5850_min__def__raw,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( ord_min @ A )
= ( ^ [A5: A,B4: A] : ( if @ A @ ( ord_less_eq @ A @ A5 @ B4 ) @ A5 @ B4 ) ) ) ) ).
% min_def_raw
thf(fact_5851_nat__mult__min__left,axiom,
! [M2: nat,N2: nat,Q3: nat] :
( ( times_times @ nat @ ( ord_min @ nat @ M2 @ N2 ) @ Q3 )
= ( ord_min @ nat @ ( times_times @ nat @ M2 @ Q3 ) @ ( times_times @ nat @ N2 @ Q3 ) ) ) ).
% nat_mult_min_left
thf(fact_5852_nat__mult__min__right,axiom,
! [M2: nat,N2: nat,Q3: nat] :
( ( times_times @ nat @ M2 @ ( ord_min @ nat @ N2 @ Q3 ) )
= ( ord_min @ nat @ ( times_times @ nat @ M2 @ N2 ) @ ( times_times @ nat @ M2 @ Q3 ) ) ) ).
% nat_mult_min_right
thf(fact_5853_max__mult__distrib__left,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [P5: A,X: A,Y: A] :
( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ P5 )
=> ( ( times_times @ A @ P5 @ ( ord_max @ A @ X @ Y ) )
= ( ord_max @ A @ ( times_times @ A @ P5 @ X ) @ ( times_times @ A @ P5 @ Y ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ P5 )
=> ( ( times_times @ A @ P5 @ ( ord_max @ A @ X @ Y ) )
= ( ord_min @ A @ ( times_times @ A @ P5 @ X ) @ ( times_times @ A @ P5 @ Y ) ) ) ) ) ) ).
% max_mult_distrib_left
thf(fact_5854_min__mult__distrib__left,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [P5: A,X: A,Y: A] :
( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ P5 )
=> ( ( times_times @ A @ P5 @ ( ord_min @ A @ X @ Y ) )
= ( ord_min @ A @ ( times_times @ A @ P5 @ X ) @ ( times_times @ A @ P5 @ Y ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ P5 )
=> ( ( times_times @ A @ P5 @ ( ord_min @ A @ X @ Y ) )
= ( ord_max @ A @ ( times_times @ A @ P5 @ X ) @ ( times_times @ A @ P5 @ Y ) ) ) ) ) ) ).
% min_mult_distrib_left
thf(fact_5855_max__mult__distrib__right,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [P5: A,X: A,Y: A] :
( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ P5 )
=> ( ( times_times @ A @ ( ord_max @ A @ X @ Y ) @ P5 )
= ( ord_max @ A @ ( times_times @ A @ X @ P5 ) @ ( times_times @ A @ Y @ P5 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ P5 )
=> ( ( times_times @ A @ ( ord_max @ A @ X @ Y ) @ P5 )
= ( ord_min @ A @ ( times_times @ A @ X @ P5 ) @ ( times_times @ A @ Y @ P5 ) ) ) ) ) ) ).
% max_mult_distrib_right
thf(fact_5856_min__mult__distrib__right,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [P5: A,X: A,Y: A] :
( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ P5 )
=> ( ( times_times @ A @ ( ord_min @ A @ X @ Y ) @ P5 )
= ( ord_min @ A @ ( times_times @ A @ X @ P5 ) @ ( times_times @ A @ Y @ P5 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ P5 )
=> ( ( times_times @ A @ ( ord_min @ A @ X @ Y ) @ P5 )
= ( ord_max @ A @ ( times_times @ A @ X @ P5 ) @ ( times_times @ A @ Y @ P5 ) ) ) ) ) ) ).
% min_mult_distrib_right
thf(fact_5857_max__divide__distrib__right,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [P5: A,X: A,Y: A] :
( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ P5 )
=> ( ( divide_divide @ A @ ( ord_max @ A @ X @ Y ) @ P5 )
= ( ord_max @ A @ ( divide_divide @ A @ X @ P5 ) @ ( divide_divide @ A @ Y @ P5 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ P5 )
=> ( ( divide_divide @ A @ ( ord_max @ A @ X @ Y ) @ P5 )
= ( ord_min @ A @ ( divide_divide @ A @ X @ P5 ) @ ( divide_divide @ A @ Y @ P5 ) ) ) ) ) ) ).
% max_divide_distrib_right
thf(fact_5858_min__divide__distrib__right,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [P5: A,X: A,Y: A] :
( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ P5 )
=> ( ( divide_divide @ A @ ( ord_min @ A @ X @ Y ) @ P5 )
= ( ord_min @ A @ ( divide_divide @ A @ X @ P5 ) @ ( divide_divide @ A @ Y @ P5 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ P5 )
=> ( ( divide_divide @ A @ ( ord_min @ A @ X @ Y ) @ P5 )
= ( ord_max @ A @ ( divide_divide @ A @ X @ P5 ) @ ( divide_divide @ A @ Y @ P5 ) ) ) ) ) ) ).
% min_divide_distrib_right
thf(fact_5859_min__Suc2,axiom,
! [M2: nat,N2: nat] :
( ( ord_min @ nat @ M2 @ ( suc @ N2 ) )
= ( case_nat @ nat @ ( zero_zero @ nat )
@ ^ [M4: nat] : ( suc @ ( ord_min @ nat @ M4 @ N2 ) )
@ M2 ) ) ).
% min_Suc2
thf(fact_5860_min__Suc1,axiom,
! [N2: nat,M2: nat] :
( ( ord_min @ nat @ ( suc @ N2 ) @ M2 )
= ( case_nat @ nat @ ( zero_zero @ nat )
@ ^ [M4: nat] : ( suc @ ( ord_min @ nat @ N2 @ M4 ) )
@ M2 ) ) ).
% min_Suc1
thf(fact_5861_Inf__insert__finite,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [S2: set @ A,X: A] :
( ( finite_finite2 @ A @ S2 )
=> ( ( ( S2
= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Inf_Inf @ A @ ( insert2 @ A @ X @ S2 ) )
= X ) )
& ( ( S2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Inf_Inf @ A @ ( insert2 @ A @ X @ S2 ) )
= ( ord_min @ A @ X @ ( complete_Inf_Inf @ A @ S2 ) ) ) ) ) ) ) ).
% Inf_insert_finite
thf(fact_5862_arg__min__list__in,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ B )
=> ! [Xs: list @ A,F2: A > B] :
( ( Xs
!= ( nil @ A ) )
=> ( member @ A @ ( arg_min_list @ A @ B @ F2 @ Xs ) @ ( set2 @ A @ Xs ) ) ) ) ).
% arg_min_list_in
thf(fact_5863_hom__Min__commute,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [H2: A > A,N6: set @ A] :
( ! [X5: A,Y3: A] :
( ( H2 @ ( ord_min @ A @ X5 @ Y3 ) )
= ( ord_min @ A @ ( H2 @ X5 ) @ ( H2 @ Y3 ) ) )
=> ( ( finite_finite2 @ A @ N6 )
=> ( ( N6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( H2 @ ( lattic643756798350308766er_Min @ A @ N6 ) )
= ( lattic643756798350308766er_Min @ A @ ( image2 @ A @ A @ H2 @ N6 ) ) ) ) ) ) ) ).
% hom_Min_commute
thf(fact_5864_Min_Osubset,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,B2: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( B2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ A4 )
=> ( ( ord_min @ A @ ( lattic643756798350308766er_Min @ A @ B2 ) @ ( lattic643756798350308766er_Min @ A @ A4 ) )
= ( lattic643756798350308766er_Min @ A @ A4 ) ) ) ) ) ) ).
% Min.subset
thf(fact_5865_Min_Oclosed,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X5: A,Y3: A] : ( member @ A @ ( ord_min @ A @ X5 @ Y3 ) @ ( insert2 @ A @ X5 @ ( insert2 @ A @ Y3 @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( member @ A @ ( lattic643756798350308766er_Min @ A @ A4 ) @ A4 ) ) ) ) ) ).
% Min.closed
thf(fact_5866_Min_Oinsert__not__elem,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ~ ( member @ A @ X @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798350308766er_Min @ A @ ( insert2 @ A @ X @ A4 ) )
= ( ord_min @ A @ X @ ( lattic643756798350308766er_Min @ A @ A4 ) ) ) ) ) ) ) ).
% Min.insert_not_elem
thf(fact_5867_Min_Ounion,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,B2: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B2 )
=> ( ( B2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798350308766er_Min @ A @ ( sup_sup @ ( set @ A ) @ A4 @ B2 ) )
= ( ord_min @ A @ ( lattic643756798350308766er_Min @ A @ A4 ) @ ( lattic643756798350308766er_Min @ A @ B2 ) ) ) ) ) ) ) ) ).
% Min.union
thf(fact_5868_Min_Oeq__fold,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( lattic643756798350308766er_Min @ A @ ( insert2 @ A @ X @ A4 ) )
= ( finite_fold @ A @ A @ ( ord_min @ A ) @ X @ A4 ) ) ) ) ).
% Min.eq_fold
thf(fact_5869_Min_Oinsert__remove,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( ( ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798350308766er_Min @ A @ ( insert2 @ A @ X @ A4 ) )
= X ) )
& ( ( ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798350308766er_Min @ A @ ( insert2 @ A @ X @ A4 ) )
= ( ord_min @ A @ X @ ( lattic643756798350308766er_Min @ A @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ).
% Min.insert_remove
thf(fact_5870_Min_Oremove,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( member @ A @ X @ A4 )
=> ( ( ( ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798350308766er_Min @ A @ A4 )
= X ) )
& ( ( ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798350308766er_Min @ A @ A4 )
= ( ord_min @ A @ X @ ( lattic643756798350308766er_Min @ A @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ) ).
% Min.remove
thf(fact_5871_lexord__take__index__conv,axiom,
! [A: $tType,X: list @ A,Y: list @ A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Y ) @ ( lexord @ A @ R2 ) )
= ( ( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ X ) @ ( size_size @ ( list @ A ) @ Y ) )
& ( ( take @ A @ ( size_size @ ( list @ A ) @ X ) @ Y )
= X ) )
| ? [I3: nat] :
( ( ord_less @ nat @ I3 @ ( ord_min @ nat @ ( size_size @ ( list @ A ) @ X ) @ ( size_size @ ( list @ A ) @ Y ) ) )
& ( ( take @ A @ I3 @ X )
= ( take @ A @ I3 @ Y ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( nth @ A @ X @ I3 ) @ ( nth @ A @ Y @ I3 ) ) @ R2 ) ) ) ) ).
% lexord_take_index_conv
thf(fact_5872_min__list__Min,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( min_list @ A @ Xs )
= ( lattic643756798350308766er_Min @ A @ ( set2 @ A @ Xs ) ) ) ) ) ).
% min_list_Min
thf(fact_5873_total__lexord,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( total_on @ A @ ( top_top @ ( set @ A ) ) @ R2 )
=> ( total_on @ ( list @ A ) @ ( top_top @ ( set @ ( list @ A ) ) ) @ ( lexord @ A @ R2 ) ) ) ).
% total_lexord
thf(fact_5874_inf__nat__def,axiom,
( ( inf_inf @ nat )
= ( ord_min @ nat ) ) ).
% inf_nat_def
thf(fact_5875_inf__int__def,axiom,
( ( inf_inf @ int )
= ( ord_min @ int ) ) ).
% inf_int_def
thf(fact_5876_lexord__irreflexive,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),Xs: list @ A] :
( ! [X5: A] :
~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X5 @ X5 ) @ R2 )
=> ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Xs ) @ ( lexord @ A @ R2 ) ) ) ).
% lexord_irreflexive
thf(fact_5877_lexord__linear,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),X: list @ A,Y: list @ A] :
( ! [A6: A,B5: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A6 @ B5 ) @ R2 )
| ( A6 = B5 )
| ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B5 @ A6 ) @ R2 ) )
=> ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Y ) @ ( lexord @ A @ R2 ) )
| ( X = Y )
| ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Y @ X ) @ ( lexord @ A @ R2 ) ) ) ) ).
% lexord_linear
thf(fact_5878_lexord__Nil__right,axiom,
! [A: $tType,X: list @ A,R2: set @ ( product_prod @ A @ A )] :
~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ ( nil @ A ) ) @ ( lexord @ A @ R2 ) ) ).
% lexord_Nil_right
thf(fact_5879_lexord__partial__trans,axiom,
! [A: $tType,Xs: list @ A,R2: set @ ( product_prod @ A @ A ),Ys: list @ A,Zs: list @ A] :
( ! [X5: A,Y3: A,Z: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Xs ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X5 @ Y3 ) @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Z ) @ R2 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X5 @ Z ) @ R2 ) ) ) )
=> ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( lexord @ A @ R2 ) )
=> ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ys @ Zs ) @ ( lexord @ A @ R2 ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Zs ) @ ( lexord @ A @ R2 ) ) ) ) ) ).
% lexord_partial_trans
thf(fact_5880_lexord__lex,axiom,
! [A: $tType,X: list @ A,Y: list @ A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Y ) @ ( lex @ A @ R2 ) )
= ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Y ) @ ( lexord @ A @ R2 ) )
& ( ( size_size @ ( list @ A ) @ X )
= ( size_size @ ( list @ A ) @ Y ) ) ) ) ).
% lexord_lex
thf(fact_5881_List_Olexordp__def,axiom,
! [A: $tType] :
( ( lexordp @ A )
= ( ^ [R4: A > A > $o,Xs3: list @ A,Ys3: list @ A] : ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs3 @ Ys3 ) @ ( lexord @ A @ ( collect @ ( product_prod @ A @ A ) @ ( product_case_prod @ A @ A @ $o @ R4 ) ) ) ) ) ) ).
% List.lexordp_def
thf(fact_5882_set__zip,axiom,
! [B: $tType,A: $tType,Xs: list @ A,Ys: list @ B] :
( ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys ) )
= ( collect @ ( product_prod @ A @ B )
@ ^ [Uu3: product_prod @ A @ B] :
? [I3: nat] :
( ( Uu3
= ( product_Pair @ A @ B @ ( nth @ A @ Xs @ I3 ) @ ( nth @ B @ Ys @ I3 ) ) )
& ( ord_less @ nat @ I3 @ ( ord_min @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ B ) @ Ys ) ) ) ) ) ) ).
% set_zip
thf(fact_5883_dual__max,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( max @ A
@ ^ [X4: A,Y6: A] : ( ord_less_eq @ A @ Y6 @ X4 ) )
= ( ord_min @ A ) ) ) ).
% dual_max
thf(fact_5884_zip__eq__Nil__iff,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B] :
( ( ( zip @ A @ B @ Xs @ Ys )
= ( nil @ ( product_prod @ A @ B ) ) )
= ( ( Xs
= ( nil @ A ) )
| ( Ys
= ( nil @ B ) ) ) ) ).
% zip_eq_Nil_iff
thf(fact_5885_Nil__eq__zip__iff,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B] :
( ( ( nil @ ( product_prod @ A @ B ) )
= ( zip @ A @ B @ Xs @ Ys ) )
= ( ( Xs
= ( nil @ A ) )
| ( Ys
= ( nil @ B ) ) ) ) ).
% Nil_eq_zip_iff
thf(fact_5886_zip__Nil,axiom,
! [B: $tType,A: $tType,Ys: list @ B] :
( ( zip @ A @ B @ ( nil @ A ) @ Ys )
= ( nil @ ( product_prod @ A @ B ) ) ) ).
% zip_Nil
thf(fact_5887_zip__replicate,axiom,
! [A: $tType,B: $tType,I: nat,X: A,J: nat,Y: B] :
( ( zip @ A @ B @ ( replicate @ A @ I @ X ) @ ( replicate @ B @ J @ Y ) )
= ( replicate @ ( product_prod @ A @ B ) @ ( ord_min @ nat @ I @ J ) @ ( product_Pair @ A @ B @ X @ Y ) ) ) ).
% zip_replicate
thf(fact_5888_length__zip,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B] :
( ( size_size @ ( list @ ( product_prod @ A @ B ) ) @ ( zip @ A @ B @ Xs @ Ys ) )
= ( ord_min @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ B ) @ Ys ) ) ) ).
% length_zip
thf(fact_5889_nth__zip,axiom,
! [A: $tType,B: $tType,I: nat,Xs: list @ A,Ys: list @ B] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ord_less @ nat @ I @ ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( nth @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys ) @ I )
= ( product_Pair @ A @ B @ ( nth @ A @ Xs @ I ) @ ( nth @ B @ Ys @ I ) ) ) ) ) ).
% nth_zip
thf(fact_5890_zip_Osimps_I1_J,axiom,
! [B: $tType,A: $tType,Xs: list @ A] :
( ( zip @ A @ B @ Xs @ ( nil @ B ) )
= ( nil @ ( product_prod @ A @ B ) ) ) ).
% zip.simps(1)
thf(fact_5891_zip__update,axiom,
! [A: $tType,B: $tType,Xs: list @ A,I: nat,X: A,Ys: list @ B,Y: B] :
( ( zip @ A @ B @ ( list_update @ A @ Xs @ I @ X ) @ ( list_update @ B @ Ys @ I @ Y ) )
= ( list_update @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys ) @ I @ ( product_Pair @ A @ B @ X @ Y ) ) ) ).
% zip_update
thf(fact_5892_ord_Omax__def,axiom,
! [A: $tType] :
( ( max @ A )
= ( ^ [Less_eq: A > A > $o,A5: A,B4: A] : ( if @ A @ ( Less_eq @ A5 @ B4 ) @ B4 @ A5 ) ) ) ).
% ord.max_def
thf(fact_5893_ord_Omax_Ocong,axiom,
! [A: $tType] :
( ( max @ A )
= ( max @ A ) ) ).
% ord.max.cong
thf(fact_5894_take__zip,axiom,
! [A: $tType,B: $tType,N2: nat,Xs: list @ A,Ys: list @ B] :
( ( take @ ( product_prod @ A @ B ) @ N2 @ ( zip @ A @ B @ Xs @ Ys ) )
= ( zip @ A @ B @ ( take @ A @ N2 @ Xs ) @ ( take @ B @ N2 @ Ys ) ) ) ).
% take_zip
thf(fact_5895_distinct__zipI2,axiom,
! [B: $tType,A: $tType,Ys: list @ A,Xs: list @ B] :
( ( distinct @ A @ Ys )
=> ( distinct @ ( product_prod @ B @ A ) @ ( zip @ B @ A @ Xs @ Ys ) ) ) ).
% distinct_zipI2
thf(fact_5896_distinct__zipI1,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B] :
( ( distinct @ A @ Xs )
=> ( distinct @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys ) ) ) ).
% distinct_zipI1
thf(fact_5897_set__zip__rightD,axiom,
! [A: $tType,B: $tType,X: A,Y: B,Xs: list @ A,Ys: list @ B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys ) ) )
=> ( member @ B @ Y @ ( set2 @ B @ Ys ) ) ) ).
% set_zip_rightD
thf(fact_5898_set__zip__leftD,axiom,
! [B: $tType,A: $tType,X: A,Y: B,Xs: list @ A,Ys: list @ B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys ) ) )
=> ( member @ A @ X @ ( set2 @ A @ Xs ) ) ) ).
% set_zip_leftD
thf(fact_5899_in__set__zipE,axiom,
! [A: $tType,B: $tType,X: A,Y: B,Xs: list @ A,Ys: list @ B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys ) ) )
=> ~ ( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ~ ( member @ B @ Y @ ( set2 @ B @ Ys ) ) ) ) ).
% in_set_zipE
thf(fact_5900_zip__same,axiom,
! [A: $tType,A3: A,B3: A,Xs: list @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B3 ) @ ( set2 @ ( product_prod @ A @ A ) @ ( zip @ A @ A @ Xs @ Xs ) ) )
= ( ( member @ A @ A3 @ ( set2 @ A @ Xs ) )
& ( A3 = B3 ) ) ) ).
% zip_same
thf(fact_5901_zip__obtain__same__length,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B,P: ( list @ ( product_prod @ A @ B ) ) > $o] :
( ! [Zs2: list @ A,Ws: list @ B,N3: nat] :
( ( ( size_size @ ( list @ A ) @ Zs2 )
= ( size_size @ ( list @ B ) @ Ws ) )
=> ( ( N3
= ( ord_min @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ B ) @ Ys ) ) )
=> ( ( Zs2
= ( take @ A @ N3 @ Xs ) )
=> ( ( Ws
= ( take @ B @ N3 @ Ys ) )
=> ( P @ ( zip @ A @ B @ Zs2 @ Ws ) ) ) ) ) )
=> ( P @ ( zip @ A @ B @ Xs @ Ys ) ) ) ).
% zip_obtain_same_length
thf(fact_5902_list__eq__iff__zip__eq,axiom,
! [A: $tType] :
( ( ^ [Y5: list @ A,Z2: list @ A] : Y5 = Z2 )
= ( ^ [Xs3: list @ A,Ys3: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs3 )
= ( size_size @ ( list @ A ) @ Ys3 ) )
& ! [X4: product_prod @ A @ A] :
( ( member @ ( product_prod @ A @ A ) @ X4 @ ( set2 @ ( product_prod @ A @ A ) @ ( zip @ A @ A @ Xs3 @ Ys3 ) ) )
=> ( product_case_prod @ A @ A @ $o
@ ^ [Y5: A,Z2: A] : Y5 = Z2
@ X4 ) ) ) ) ) ).
% list_eq_iff_zip_eq
thf(fact_5903_in__set__impl__in__set__zip1,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B,X: A] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ~ ! [Y3: B] :
~ ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y3 ) @ ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys ) ) ) ) ) ).
% in_set_impl_in_set_zip1
thf(fact_5904_in__set__impl__in__set__zip2,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B,Y: B] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( member @ B @ Y @ ( set2 @ B @ Ys ) )
=> ~ ! [X5: A] :
~ ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X5 @ Y ) @ ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys ) ) ) ) ) ).
% in_set_impl_in_set_zip2
thf(fact_5905_concat__injective,axiom,
! [A: $tType,Xs: list @ ( list @ A ),Ys: list @ ( list @ A )] :
( ( ( concat @ A @ Xs )
= ( concat @ A @ Ys ) )
=> ( ( ( size_size @ ( list @ ( list @ A ) ) @ Xs )
= ( size_size @ ( list @ ( list @ A ) ) @ Ys ) )
=> ( ! [X5: product_prod @ ( list @ A ) @ ( list @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ X5 @ ( set2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( zip @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) ) )
=> ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ $o
@ ^ [Y6: list @ A,Z6: list @ A] :
( ( size_size @ ( list @ A ) @ Y6 )
= ( size_size @ ( list @ A ) @ Z6 ) )
@ X5 ) )
=> ( Xs = Ys ) ) ) ) ).
% concat_injective
thf(fact_5906_concat__eq__concat__iff,axiom,
! [A: $tType,Xs: list @ ( list @ A ),Ys: list @ ( list @ A )] :
( ! [X5: product_prod @ ( list @ A ) @ ( list @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ X5 @ ( set2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( zip @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) ) )
=> ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ $o
@ ^ [Y6: list @ A,Z6: list @ A] :
( ( size_size @ ( list @ A ) @ Y6 )
= ( size_size @ ( list @ A ) @ Z6 ) )
@ X5 ) )
=> ( ( ( size_size @ ( list @ ( list @ A ) ) @ Xs )
= ( size_size @ ( list @ ( list @ A ) ) @ Ys ) )
=> ( ( ( concat @ A @ Xs )
= ( concat @ A @ Ys ) )
= ( Xs = Ys ) ) ) ) ).
% concat_eq_concat_iff
thf(fact_5907_listrel__iff__zip,axiom,
! [B: $tType,A: $tType,Xs: list @ A,Ys: list @ B,R2: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs @ Ys ) @ ( listrel @ A @ B @ R2 ) )
= ( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys ) )
& ! [X4: product_prod @ A @ B] :
( ( member @ ( product_prod @ A @ B ) @ X4 @ ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys ) ) )
=> ( product_case_prod @ A @ B @ $o
@ ^ [Y6: A,Z6: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Y6 @ Z6 ) @ R2 )
@ X4 ) ) ) ) ).
% listrel_iff_zip
thf(fact_5908_map__of__zip__nth,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B,I: nat] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( distinct @ A @ Xs )
=> ( ( ord_less @ nat @ I @ ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( map_of @ A @ B @ ( zip @ A @ B @ Xs @ Ys ) @ ( nth @ A @ Xs @ I ) )
= ( some @ B @ ( nth @ B @ Ys @ I ) ) ) ) ) ) ).
% map_of_zip_nth
thf(fact_5909_remdups__adj__singleton__iff,axiom,
! [A: $tType,Xs: list @ A] :
( ( ( size_size @ ( list @ A ) @ ( remdups_adj @ A @ Xs ) )
= ( suc @ ( zero_zero @ nat ) ) )
= ( ( Xs
!= ( nil @ A ) )
& ( Xs
= ( replicate @ A @ ( size_size @ ( list @ A ) @ Xs ) @ ( hd @ A @ Xs ) ) ) ) ) ).
% remdups_adj_singleton_iff
thf(fact_5910_hd__remdups__adj,axiom,
! [A: $tType,Xs: list @ A] :
( ( hd @ A @ ( remdups_adj @ A @ Xs ) )
= ( hd @ A @ Xs ) ) ).
% hd_remdups_adj
thf(fact_5911_hd__replicate,axiom,
! [A: $tType,N2: nat,X: A] :
( ( N2
!= ( zero_zero @ nat ) )
=> ( ( hd @ A @ ( replicate @ A @ N2 @ X ) )
= X ) ) ).
% hd_replicate
thf(fact_5912_listrel__rtrancl__refl,axiom,
! [A: $tType,Xs: list @ A,R2: set @ ( product_prod @ A @ A )] : ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Xs ) @ ( listrel @ A @ A @ ( transitive_rtrancl @ A @ R2 ) ) ) ).
% listrel_rtrancl_refl
thf(fact_5913_hd__take,axiom,
! [A: $tType,J: nat,Xs: list @ A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ J )
=> ( ( hd @ A @ ( take @ A @ J @ Xs ) )
= ( hd @ A @ Xs ) ) ) ).
% hd_take
thf(fact_5914_map__of__zip__is__None,axiom,
! [B: $tType,A: $tType,Xs: list @ A,Ys: list @ B,X: A] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( ( map_of @ A @ B @ ( zip @ A @ B @ Xs @ Ys ) @ X )
= ( none @ B ) )
= ( ~ ( member @ A @ X @ ( set2 @ A @ Xs ) ) ) ) ) ).
% map_of_zip_is_None
thf(fact_5915_hd__zip,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B] :
( ( Xs
!= ( nil @ A ) )
=> ( ( Ys
!= ( nil @ B ) )
=> ( ( hd @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys ) )
= ( product_Pair @ A @ B @ ( hd @ A @ Xs ) @ ( hd @ B @ Ys ) ) ) ) ) ).
% hd_zip
thf(fact_5916_hd__concat,axiom,
! [A: $tType,Xs: list @ ( list @ A )] :
( ( Xs
!= ( nil @ ( list @ A ) ) )
=> ( ( ( hd @ ( list @ A ) @ Xs )
!= ( nil @ A ) )
=> ( ( hd @ A @ ( concat @ A @ Xs ) )
= ( hd @ A @ ( hd @ ( list @ A ) @ Xs ) ) ) ) ) ).
% hd_concat
thf(fact_5917_hd__in__set,axiom,
! [A: $tType,Xs: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ( member @ A @ ( hd @ A @ Xs ) @ ( set2 @ A @ Xs ) ) ) ).
% hd_in_set
thf(fact_5918_list_Oset__sel_I1_J,axiom,
! [A: $tType,A3: list @ A] :
( ( A3
!= ( nil @ A ) )
=> ( member @ A @ ( hd @ A @ A3 ) @ ( set2 @ A @ A3 ) ) ) ).
% list.set_sel(1)
thf(fact_5919_listrel_ONil,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ B )] : ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( nil @ A ) @ ( nil @ B ) ) @ ( listrel @ A @ B @ R2 ) ) ).
% listrel.Nil
thf(fact_5920_listrel__Nil1,axiom,
! [A: $tType,B: $tType,Xs: list @ B,R2: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( nil @ A ) @ Xs ) @ ( listrel @ A @ B @ R2 ) )
=> ( Xs
= ( nil @ B ) ) ) ).
% listrel_Nil1
thf(fact_5921_listrel__Nil2,axiom,
! [B: $tType,A: $tType,Xs: list @ A,R2: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs @ ( nil @ B ) ) @ ( listrel @ A @ B @ R2 ) )
=> ( Xs
= ( nil @ A ) ) ) ).
% listrel_Nil2
thf(fact_5922_listrel__eq__len,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B,R2: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs @ Ys ) @ ( listrel @ A @ B @ R2 ) )
=> ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys ) ) ) ).
% listrel_eq_len
thf(fact_5923_listrel__rtrancl__trans,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,R2: set @ ( product_prod @ A @ A ),Zs: list @ A] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( listrel @ A @ A @ ( transitive_rtrancl @ A @ R2 ) ) )
=> ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ys @ Zs ) @ ( listrel @ A @ A @ ( transitive_rtrancl @ A @ R2 ) ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Zs ) @ ( listrel @ A @ A @ ( transitive_rtrancl @ A @ R2 ) ) ) ) ) ).
% listrel_rtrancl_trans
thf(fact_5924_listrel__mono,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ A @ B )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R2 @ S )
=> ( ord_less_eq @ ( set @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) @ ( listrel @ A @ B @ R2 ) @ ( listrel @ A @ B @ S ) ) ) ).
% listrel_mono
thf(fact_5925_hd__conv__nth,axiom,
! [A: $tType,Xs: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( hd @ A @ Xs )
= ( nth @ A @ Xs @ ( zero_zero @ nat ) ) ) ) ).
% hd_conv_nth
thf(fact_5926_map__of__zip__inject,axiom,
! [B: $tType,A: $tType,Ys: list @ A,Xs: list @ B,Zs: list @ A] :
( ( ( size_size @ ( list @ A ) @ Ys )
= ( size_size @ ( list @ B ) @ Xs ) )
=> ( ( ( size_size @ ( list @ A ) @ Zs )
= ( size_size @ ( list @ B ) @ Xs ) )
=> ( ( distinct @ B @ Xs )
=> ( ( ( map_of @ B @ A @ ( zip @ B @ A @ Xs @ Ys ) )
= ( map_of @ B @ A @ ( zip @ B @ A @ Xs @ Zs ) ) )
=> ( Ys = Zs ) ) ) ) ) ).
% map_of_zip_inject
thf(fact_5927_listrel__reflcl__if__listrel1,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( listrel1 @ A @ R2 ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( listrel @ A @ A @ ( transitive_rtrancl @ A @ R2 ) ) ) ) ).
% listrel_reflcl_if_listrel1
thf(fact_5928_finite__range__map__of,axiom,
! [A: $tType,B: $tType,Xys: list @ ( product_prod @ B @ A )] : ( finite_finite2 @ ( option @ A ) @ ( image2 @ B @ ( option @ A ) @ ( map_of @ B @ A @ Xys ) @ ( top_top @ ( set @ B ) ) ) ) ).
% finite_range_map_of
thf(fact_5929_listrel__rtrancl__eq__rtrancl__listrel1,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( listrel @ A @ A @ ( transitive_rtrancl @ A @ R2 ) )
= ( transitive_rtrancl @ ( list @ A ) @ ( listrel1 @ A @ R2 ) ) ) ).
% listrel_rtrancl_eq_rtrancl_listrel1
thf(fact_5930_rtrancl__listrel1__if__listrel,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( listrel @ A @ A @ R2 ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( transitive_rtrancl @ ( list @ A ) @ ( listrel1 @ A @ R2 ) ) ) ) ).
% rtrancl_listrel1_if_listrel
thf(fact_5931_map__of__zip__is__Some,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B,X: A] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( member @ A @ X @ ( set2 @ A @ Xs ) )
= ( ? [Y6: B] :
( ( map_of @ A @ B @ ( zip @ A @ B @ Xs @ Ys ) @ X )
= ( some @ B @ Y6 ) ) ) ) ) ).
% map_of_zip_is_Some
thf(fact_5932_listrel__subset__rtrancl__listrel1,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] : ( ord_less_eq @ ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) @ ( listrel @ A @ A @ R2 ) @ ( transitive_rtrancl @ ( list @ A ) @ ( listrel1 @ A @ R2 ) ) ) ).
% listrel_subset_rtrancl_listrel1
thf(fact_5933_map__of__zip__upd,axiom,
! [A: $tType,B: $tType,Ys: list @ B,Xs: list @ A,Zs: list @ B,X: A,Y: B,Z3: B] :
( ( ( size_size @ ( list @ B ) @ Ys )
= ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ( size_size @ ( list @ B ) @ Zs )
= ( size_size @ ( list @ A ) @ Xs ) )
=> ( ~ ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ( ( fun_upd @ A @ ( option @ B ) @ ( map_of @ A @ B @ ( zip @ A @ B @ Xs @ Ys ) ) @ X @ ( some @ B @ Y ) )
= ( fun_upd @ A @ ( option @ B ) @ ( map_of @ A @ B @ ( zip @ A @ B @ Xs @ Zs ) ) @ X @ ( some @ B @ Z3 ) ) )
=> ( ( map_of @ A @ B @ ( zip @ A @ B @ Xs @ Ys ) )
= ( map_of @ A @ B @ ( zip @ A @ B @ Xs @ Zs ) ) ) ) ) ) ) ).
% map_of_zip_upd
thf(fact_5934_listrel__iff__nth,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B,R2: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs @ Ys ) @ ( listrel @ A @ B @ R2 ) )
= ( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys ) )
& ! [N: nat] :
( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ ( nth @ A @ Xs @ N ) @ ( nth @ B @ Ys @ N ) ) @ R2 ) ) ) ) ).
% listrel_iff_nth
thf(fact_5935_ran__map__of__zip,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( distinct @ A @ Xs )
=> ( ( ran @ A @ B @ ( map_of @ A @ B @ ( zip @ A @ B @ Xs @ Ys ) ) )
= ( set2 @ B @ Ys ) ) ) ) ).
% ran_map_of_zip
thf(fact_5936_listrel__def,axiom,
! [B: $tType,A: $tType] :
( ( listrel @ A @ B )
= ( ^ [R4: set @ ( product_prod @ A @ B )] :
( collect @ ( product_prod @ ( list @ A ) @ ( list @ B ) )
@ ( product_case_prod @ ( list @ A ) @ ( list @ B ) @ $o
@ ( listrelp @ A @ B
@ ^ [X4: A,Y6: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y6 ) @ R4 ) ) ) ) ) ) ).
% listrel_def
thf(fact_5937_Nitpick_Osize__list__simp_I1_J,axiom,
! [A: $tType] :
( ( size_list @ A )
= ( ^ [F3: A > nat,Xs3: list @ A] :
( if @ nat
@ ( Xs3
= ( nil @ A ) )
@ ( zero_zero @ nat )
@ ( suc @ ( plus_plus @ nat @ ( F3 @ ( hd @ A @ Xs3 ) ) @ ( size_list @ A @ F3 @ ( tl @ A @ Xs3 ) ) ) ) ) ) ) ).
% Nitpick.size_list_simp(1)
thf(fact_5938_ran__empty,axiom,
! [B: $tType,A: $tType] :
( ( ran @ B @ A
@ ^ [X4: B] : ( none @ A ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% ran_empty
thf(fact_5939_length__tl,axiom,
! [A: $tType,Xs: list @ A] :
( ( size_size @ ( list @ A ) @ ( tl @ A @ Xs ) )
= ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) ) ) ).
% length_tl
thf(fact_5940_tl__replicate,axiom,
! [A: $tType,N2: nat,X: A] :
( ( tl @ A @ ( replicate @ A @ N2 @ X ) )
= ( replicate @ A @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) @ X ) ) ).
% tl_replicate
thf(fact_5941_list_Oset__sel_I2_J,axiom,
! [A: $tType,A3: list @ A,X: A] :
( ( A3
!= ( nil @ A ) )
=> ( ( member @ A @ X @ ( set2 @ A @ ( tl @ A @ A3 ) ) )
=> ( member @ A @ X @ ( set2 @ A @ A3 ) ) ) ) ).
% list.set_sel(2)
thf(fact_5942_listrelp_ONil,axiom,
! [A: $tType,B: $tType,R2: A > B > $o] : ( listrelp @ A @ B @ R2 @ ( nil @ A ) @ ( nil @ B ) ) ).
% listrelp.Nil
thf(fact_5943_list_Osel_I2_J,axiom,
! [A: $tType] :
( ( tl @ A @ ( nil @ A ) )
= ( nil @ A ) ) ).
% list.sel(2)
thf(fact_5944_distinct__tl,axiom,
! [A: $tType,Xs: list @ A] :
( ( distinct @ A @ Xs )
=> ( distinct @ A @ ( tl @ A @ Xs ) ) ) ).
% distinct_tl
thf(fact_5945_take__tl,axiom,
! [A: $tType,N2: nat,Xs: list @ A] :
( ( take @ A @ N2 @ ( tl @ A @ Xs ) )
= ( tl @ A @ ( take @ A @ ( suc @ N2 ) @ Xs ) ) ) ).
% take_tl
thf(fact_5946_list_Oexpand,axiom,
! [A: $tType,List: list @ A,List2: list @ A] :
( ( ( List
= ( nil @ A ) )
= ( List2
= ( nil @ A ) ) )
=> ( ( ( List
!= ( nil @ A ) )
=> ( ( List2
!= ( nil @ A ) )
=> ( ( ( hd @ A @ List )
= ( hd @ A @ List2 ) )
& ( ( tl @ A @ List )
= ( tl @ A @ List2 ) ) ) ) )
=> ( List = List2 ) ) ) ).
% list.expand
thf(fact_5947_tl__take,axiom,
! [A: $tType,N2: nat,Xs: list @ A] :
( ( tl @ A @ ( take @ A @ N2 @ Xs ) )
= ( take @ A @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) @ ( tl @ A @ Xs ) ) ) ).
% tl_take
thf(fact_5948_Nitpick_Osize__list__simp_I2_J,axiom,
! [A: $tType] :
( ( size_size @ ( list @ A ) )
= ( ^ [Xs3: list @ A] :
( if @ nat
@ ( Xs3
= ( nil @ A ) )
@ ( zero_zero @ nat )
@ ( suc @ ( size_size @ ( list @ A ) @ ( tl @ A @ Xs3 ) ) ) ) ) ) ).
% Nitpick.size_list_simp(2)
thf(fact_5949_nth__tl,axiom,
! [A: $tType,N2: nat,Xs: list @ A] :
( ( ord_less @ nat @ N2 @ ( size_size @ ( list @ A ) @ ( tl @ A @ Xs ) ) )
=> ( ( nth @ A @ ( tl @ A @ Xs ) @ N2 )
= ( nth @ A @ Xs @ ( suc @ N2 ) ) ) ) ).
% nth_tl
thf(fact_5950_listrelp__listrel__eq,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ B )] :
( ( listrelp @ A @ B
@ ^ [X4: A,Y6: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y6 ) @ R2 ) )
= ( ^ [X4: list @ A,Y6: list @ B] : ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ X4 @ Y6 ) @ ( listrel @ A @ B @ R2 ) ) ) ) ).
% listrelp_listrel_eq
thf(fact_5951_ran__map__upd__Some,axiom,
! [B: $tType,A: $tType,M2: B > ( option @ A ),X: B,Y: A,Z3: A] :
( ( ( M2 @ X )
= ( some @ A @ Y ) )
=> ( ( inj_on @ B @ ( option @ A ) @ M2 @ ( dom @ B @ A @ M2 ) )
=> ( ~ ( member @ A @ Z3 @ ( ran @ B @ A @ M2 ) )
=> ( ( ran @ B @ A @ ( fun_upd @ B @ ( option @ A ) @ M2 @ X @ ( some @ A @ Z3 ) ) )
= ( sup_sup @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ ( ran @ B @ A @ M2 ) @ ( insert2 @ A @ Y @ ( bot_bot @ ( set @ A ) ) ) ) @ ( insert2 @ A @ Z3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% ran_map_upd_Some
thf(fact_5952_hd__rotate__conv__nth,axiom,
! [A: $tType,Xs: list @ A,N2: nat] :
( ( Xs
!= ( nil @ A ) )
=> ( ( hd @ A @ ( rotate @ A @ N2 @ Xs ) )
= ( nth @ A @ Xs @ ( modulo_modulo @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ) ).
% hd_rotate_conv_nth
thf(fact_5953_listrel1__subset__listrel,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),R6: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ R6 )
=> ( ( refl_on @ A @ ( top_top @ ( set @ A ) ) @ R6 )
=> ( ord_less_eq @ ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) @ ( listrel1 @ A @ R2 ) @ ( listrel @ A @ A @ R6 ) ) ) ) ).
% listrel1_subset_listrel
thf(fact_5954_rotate0,axiom,
! [A: $tType] :
( ( rotate @ A @ ( zero_zero @ nat ) )
= ( id @ ( list @ A ) ) ) ).
% rotate0
thf(fact_5955_rotate__is__Nil__conv,axiom,
! [A: $tType,N2: nat,Xs: list @ A] :
( ( ( rotate @ A @ N2 @ Xs )
= ( nil @ A ) )
= ( Xs
= ( nil @ A ) ) ) ).
% rotate_is_Nil_conv
thf(fact_5956_set__rotate,axiom,
! [A: $tType,N2: nat,Xs: list @ A] :
( ( set2 @ A @ ( rotate @ A @ N2 @ Xs ) )
= ( set2 @ A @ Xs ) ) ).
% set_rotate
thf(fact_5957_length__rotate,axiom,
! [A: $tType,N2: nat,Xs: list @ A] :
( ( size_size @ ( list @ A ) @ ( rotate @ A @ N2 @ Xs ) )
= ( size_size @ ( list @ A ) @ Xs ) ) ).
% length_rotate
thf(fact_5958_distinct__rotate,axiom,
! [A: $tType,N2: nat,Xs: list @ A] :
( ( distinct @ A @ ( rotate @ A @ N2 @ Xs ) )
= ( distinct @ A @ Xs ) ) ).
% distinct_rotate
thf(fact_5959_dom__eq__empty__conv,axiom,
! [B: $tType,A: $tType,F2: A > ( option @ B )] :
( ( ( dom @ A @ B @ F2 )
= ( bot_bot @ ( set @ A ) ) )
= ( F2
= ( ^ [X4: A] : ( none @ B ) ) ) ) ).
% dom_eq_empty_conv
thf(fact_5960_rotate__Suc,axiom,
! [A: $tType,N2: nat,Xs: list @ A] :
( ( rotate @ A @ ( suc @ N2 ) @ Xs )
= ( rotate1 @ A @ ( rotate @ A @ N2 @ Xs ) ) ) ).
% rotate_Suc
thf(fact_5961_dom__empty,axiom,
! [B: $tType,A: $tType] :
( ( dom @ A @ B
@ ^ [X4: A] : ( none @ B ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% dom_empty
thf(fact_5962_rotate__length01,axiom,
! [A: $tType,Xs: list @ A,N2: nat] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) )
=> ( ( rotate @ A @ N2 @ Xs )
= Xs ) ) ).
% rotate_length01
thf(fact_5963_rotate__id,axiom,
! [A: $tType,N2: nat,Xs: list @ A] :
( ( ( modulo_modulo @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs ) )
= ( zero_zero @ nat ) )
=> ( ( rotate @ A @ N2 @ Xs )
= Xs ) ) ).
% rotate_id
thf(fact_5964_dom__map__of__zip,axiom,
! [B: $tType,A: $tType,Xs: list @ A,Ys: list @ B] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( dom @ A @ B @ ( map_of @ A @ B @ ( zip @ A @ B @ Xs @ Ys ) ) )
= ( set2 @ A @ Xs ) ) ) ).
% dom_map_of_zip
thf(fact_5965_dom__fun__upd,axiom,
! [B: $tType,A: $tType,Y: option @ B,F2: A > ( option @ B ),X: A] :
( ( ( Y
= ( none @ B ) )
=> ( ( dom @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ F2 @ X @ Y ) )
= ( minus_minus @ ( set @ A ) @ ( dom @ A @ B @ F2 ) @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) )
& ( ( Y
!= ( none @ B ) )
=> ( ( dom @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ F2 @ X @ Y ) )
= ( insert2 @ A @ X @ ( dom @ A @ B @ F2 ) ) ) ) ) ).
% dom_fun_upd
thf(fact_5966_dom__map__upds,axiom,
! [B: $tType,A: $tType,M2: A > ( option @ B ),Xs: list @ A,Ys: list @ B] :
( ( dom @ A @ B @ ( map_upds @ A @ B @ M2 @ Xs @ Ys ) )
= ( sup_sup @ ( set @ A ) @ ( set2 @ A @ ( take @ A @ ( size_size @ ( list @ B ) @ Ys ) @ Xs ) ) @ ( dom @ A @ B @ M2 ) ) ) ).
% dom_map_upds
thf(fact_5967_rotate1__rotate__swap,axiom,
! [A: $tType,N2: nat,Xs: list @ A] :
( ( rotate1 @ A @ ( rotate @ A @ N2 @ Xs ) )
= ( rotate @ A @ N2 @ ( rotate1 @ A @ Xs ) ) ) ).
% rotate1_rotate_swap
thf(fact_5968_rotate__rotate,axiom,
! [A: $tType,M2: nat,N2: nat,Xs: list @ A] :
( ( rotate @ A @ M2 @ ( rotate @ A @ N2 @ Xs ) )
= ( rotate @ A @ ( plus_plus @ nat @ M2 @ N2 ) @ Xs ) ) ).
% rotate_rotate
thf(fact_5969_rotate__def,axiom,
! [A: $tType] :
( ( rotate @ A )
= ( ^ [N: nat] : ( compow @ ( ( list @ A ) > ( list @ A ) ) @ N @ ( rotate1 @ A ) ) ) ) ).
% rotate_def
thf(fact_5970_finite__dom__map__of,axiom,
! [B: $tType,A: $tType,L: list @ ( product_prod @ A @ B )] : ( finite_finite2 @ A @ ( dom @ A @ B @ ( map_of @ A @ B @ L ) ) ) ).
% finite_dom_map_of
thf(fact_5971_finite__ran,axiom,
! [B: $tType,A: $tType,P5: A > ( option @ B )] :
( ( finite_finite2 @ A @ ( dom @ A @ B @ P5 ) )
=> ( finite_finite2 @ B @ ( ran @ A @ B @ P5 ) ) ) ).
% finite_ran
thf(fact_5972_refl__on__empty,axiom,
! [A: $tType] : ( refl_on @ A @ ( bot_bot @ ( set @ A ) ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ).
% refl_on_empty
thf(fact_5973_rotate__conv__mod,axiom,
! [A: $tType] :
( ( rotate @ A )
= ( ^ [N: nat,Xs3: list @ A] : ( rotate @ A @ ( modulo_modulo @ nat @ N @ ( size_size @ ( list @ A ) @ Xs3 ) ) @ Xs3 ) ) ) ).
% rotate_conv_mod
thf(fact_5974_finite__map__freshness,axiom,
! [A: $tType,B: $tType,F2: A > ( option @ B )] :
( ( finite_finite2 @ A @ ( dom @ A @ B @ F2 ) )
=> ( ~ ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) )
=> ? [X5: A] :
( ( F2 @ X5 )
= ( none @ B ) ) ) ) ).
% finite_map_freshness
thf(fact_5975_rotate__add,axiom,
! [A: $tType,M2: nat,N2: nat] :
( ( rotate @ A @ ( plus_plus @ nat @ M2 @ N2 ) )
= ( comp @ ( list @ A ) @ ( list @ A ) @ ( list @ A ) @ ( rotate @ A @ M2 ) @ ( rotate @ A @ N2 ) ) ) ).
% rotate_add
thf(fact_5976_finite__set__of__finite__maps,axiom,
! [B: $tType,A: $tType,A4: set @ A,B2: set @ B] :
( ( finite_finite2 @ A @ A4 )
=> ( ( finite_finite2 @ B @ B2 )
=> ( finite_finite2 @ ( A > ( option @ B ) )
@ ( collect @ ( A > ( option @ B ) )
@ ^ [M: A > ( option @ B )] :
( ( ( dom @ A @ B @ M )
= A4 )
& ( ord_less_eq @ ( set @ B ) @ ( ran @ A @ B @ M ) @ B2 ) ) ) ) ) ) ).
% finite_set_of_finite_maps
thf(fact_5977_finite__Map__induct,axiom,
! [B: $tType,A: $tType,M2: A > ( option @ B ),P: ( A > ( option @ B ) ) > $o] :
( ( finite_finite2 @ A @ ( dom @ A @ B @ M2 ) )
=> ( ( P
@ ^ [X4: A] : ( none @ B ) )
=> ( ! [K3: A,V3: B,M3: A > ( option @ B )] :
( ( finite_finite2 @ A @ ( dom @ A @ B @ M3 ) )
=> ( ~ ( member @ A @ K3 @ ( dom @ A @ B @ M3 ) )
=> ( ( P @ M3 )
=> ( P @ ( fun_upd @ A @ ( option @ B ) @ M3 @ K3 @ ( some @ B @ V3 ) ) ) ) ) )
=> ( P @ M2 ) ) ) ) ).
% finite_Map_induct
thf(fact_5978_dom__eq__singleton__conv,axiom,
! [A: $tType,B: $tType,F2: A > ( option @ B ),X: A] :
( ( ( dom @ A @ B @ F2 )
= ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( ? [V5: B] :
( F2
= ( fun_upd @ A @ ( option @ B )
@ ^ [X4: A] : ( none @ B )
@ X
@ ( some @ B @ V5 ) ) ) ) ) ).
% dom_eq_singleton_conv
thf(fact_5979_refl__on__singleton,axiom,
! [A: $tType,X: A] : ( refl_on @ A @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) @ ( insert2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ X ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ).
% refl_on_singleton
thf(fact_5980_nth__rotate,axiom,
! [A: $tType,N2: nat,Xs: list @ A,M2: nat] :
( ( ord_less @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( nth @ A @ ( rotate @ A @ M2 @ Xs ) @ N2 )
= ( nth @ A @ Xs @ ( modulo_modulo @ nat @ ( plus_plus @ nat @ M2 @ N2 ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ) ).
% nth_rotate
thf(fact_5981_sorted__list__of__set__nonempty,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( linord4507533701916653071of_set @ A @ A4 )
= ( cons @ A @ ( lattic643756798350308766er_Min @ A @ A4 ) @ ( linord4507533701916653071of_set @ A @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ ( lattic643756798350308766er_Min @ A @ A4 ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ).
% sorted_list_of_set_nonempty
thf(fact_5982_slice__eq__mask,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N2: nat,M2: nat,A3: A] :
( ( bit_se4730199178511100633sh_bit @ A @ N2 @ ( bit_se2584673776208193580ke_bit @ A @ M2 @ ( bit_se4197421643247451524op_bit @ A @ N2 @ A3 ) ) )
= ( bit_se5824344872417868541ns_and @ A @ A3 @ ( bit_se5824344872417868541ns_and @ A @ ( bit_se2239418461657761734s_mask @ A @ ( plus_plus @ nat @ M2 @ N2 ) ) @ ( bit_ri4277139882892585799ns_not @ A @ ( bit_se2239418461657761734s_mask @ A @ N2 ) ) ) ) ) ) ).
% slice_eq_mask
thf(fact_5983_bij__betw__roots__unity,axiom,
! [N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( bij_betw @ nat @ complex
@ ^ [K2: nat] : ( cis @ ( divide_divide @ real @ ( times_times @ real @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) @ ( semiring_1_of_nat @ real @ K2 ) ) @ ( semiring_1_of_nat @ real @ N2 ) ) )
@ ( set_ord_lessThan @ nat @ N2 )
@ ( collect @ complex
@ ^ [Z6: complex] :
( ( power_power @ complex @ Z6 @ N2 )
= ( one_one @ complex ) ) ) ) ) ).
% bij_betw_roots_unity
thf(fact_5984_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_5985_list_Osimps_I15_J,axiom,
! [A: $tType,X21: A,X222: list @ A] :
( ( set2 @ A @ ( cons @ A @ X21 @ X222 ) )
= ( insert2 @ A @ X21 @ ( set2 @ A @ X222 ) ) ) ).
% list.simps(15)
thf(fact_5986_bij__betw__add,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [A3: A,A4: set @ A,B2: set @ A] :
( ( bij_betw @ A @ A @ ( plus_plus @ A @ A3 ) @ A4 @ B2 )
= ( ( image2 @ A @ A @ ( plus_plus @ A @ A3 ) @ A4 )
= B2 ) ) ) ).
% bij_betw_add
thf(fact_5987_nth__Cons__Suc,axiom,
! [A: $tType,X: A,Xs: list @ A,N2: nat] :
( ( nth @ A @ ( cons @ A @ X @ Xs ) @ ( suc @ N2 ) )
= ( nth @ A @ Xs @ N2 ) ) ).
% nth_Cons_Suc
thf(fact_5988_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_5989_zip__Cons__Cons,axiom,
! [A: $tType,B: $tType,X: A,Xs: list @ A,Y: B,Ys: list @ B] :
( ( zip @ A @ B @ ( cons @ A @ X @ Xs ) @ ( cons @ B @ Y @ Ys ) )
= ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ ( zip @ A @ B @ Xs @ Ys ) ) ) ).
% zip_Cons_Cons
thf(fact_5990_take__Suc__Cons,axiom,
! [A: $tType,N2: nat,X: A,Xs: list @ A] :
( ( take @ A @ ( suc @ N2 ) @ ( cons @ A @ X @ Xs ) )
= ( cons @ A @ X @ ( take @ A @ N2 @ Xs ) ) ) ).
% take_Suc_Cons
thf(fact_5991_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_5992_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_5993_remdups__adj__Cons__alt,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( cons @ A @ X @ ( tl @ A @ ( remdups_adj @ A @ ( cons @ A @ X @ Xs ) ) ) )
= ( remdups_adj @ A @ ( cons @ A @ X @ Xs ) ) ) ).
% remdups_adj_Cons_alt
thf(fact_5994_bij__betw__of__nat,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [N6: set @ nat,A4: set @ A] :
( ( bij_betw @ nat @ A @ ( semiring_1_of_nat @ A ) @ N6 @ A4 )
= ( ( image2 @ nat @ A @ ( semiring_1_of_nat @ A ) @ N6 )
= A4 ) ) ) ).
% bij_betw_of_nat
thf(fact_5995_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_5996_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_5997_nths__singleton,axiom,
! [A: $tType,A4: set @ nat,X: A] :
( ( ( member @ nat @ ( zero_zero @ nat ) @ A4 )
=> ( ( nths @ A @ ( cons @ A @ X @ ( nil @ A ) ) @ A4 )
= ( cons @ A @ X @ ( nil @ A ) ) ) )
& ( ~ ( member @ nat @ ( zero_zero @ nat ) @ A4 )
=> ( ( nths @ A @ ( cons @ A @ X @ ( nil @ A ) ) @ A4 )
= ( nil @ A ) ) ) ) ).
% nths_singleton
thf(fact_5998_Cons__listrel1__Cons,axiom,
! [A: $tType,X: A,Xs: list @ A,Y: A,Ys: list @ A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y @ Ys ) ) @ ( listrel1 @ A @ R2 ) )
= ( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R2 )
& ( Xs = Ys ) )
| ( ( X = Y )
& ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( listrel1 @ A @ R2 ) ) ) ) ) ).
% Cons_listrel1_Cons
thf(fact_5999_horner__sum__simps_I2_J,axiom,
! [A: $tType,B: $tType] :
( ( comm_semiring_0 @ A )
=> ! [F2: B > A,A3: A,X: B,Xs: list @ B] :
( ( groups4207007520872428315er_sum @ B @ A @ F2 @ A3 @ ( cons @ B @ X @ Xs ) )
= ( plus_plus @ A @ ( F2 @ X ) @ ( times_times @ A @ A3 @ ( groups4207007520872428315er_sum @ B @ A @ F2 @ A3 @ Xs ) ) ) ) ) ).
% horner_sum_simps(2)
thf(fact_6000_lexord__cons__cons,axiom,
! [A: $tType,A3: A,X: list @ A,B3: A,Y: list @ A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ A3 @ X ) @ ( cons @ A @ B3 @ Y ) ) @ ( lexord @ A @ R2 ) )
= ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B3 ) @ R2 )
| ( ( A3 = B3 )
& ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Y ) @ ( lexord @ A @ R2 ) ) ) ) ) ).
% lexord_cons_cons
thf(fact_6001_lexord__Nil__left,axiom,
! [A: $tType,Y: list @ A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Y ) @ ( lexord @ A @ R2 ) )
= ( ? [A5: A,X4: list @ A] :
( Y
= ( cons @ A @ A5 @ X4 ) ) ) ) ).
% lexord_Nil_left
thf(fact_6002_list_Ocollapse,axiom,
! [A: $tType,List: list @ A] :
( ( List
!= ( nil @ A ) )
=> ( ( cons @ A @ ( hd @ A @ List ) @ ( tl @ A @ List ) )
= List ) ) ).
% list.collapse
thf(fact_6003_hd__Cons__tl,axiom,
! [A: $tType,Xs: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( cons @ A @ ( hd @ A @ Xs ) @ ( tl @ A @ Xs ) )
= Xs ) ) ).
% hd_Cons_tl
thf(fact_6004_enumerate__simps_I2_J,axiom,
! [B: $tType,N2: nat,X: B,Xs: list @ B] :
( ( enumerate @ B @ N2 @ ( cons @ B @ X @ Xs ) )
= ( cons @ ( product_prod @ nat @ B ) @ ( product_Pair @ nat @ B @ N2 @ X ) @ ( enumerate @ B @ ( suc @ N2 ) @ Xs ) ) ) ).
% enumerate_simps(2)
thf(fact_6005_n__lists__Nil,axiom,
! [A: $tType,N2: nat] :
( ( ( N2
= ( zero_zero @ nat ) )
=> ( ( n_lists @ A @ N2 @ ( nil @ A ) )
= ( cons @ ( list @ A ) @ ( nil @ A ) @ ( nil @ ( list @ A ) ) ) ) )
& ( ( N2
!= ( zero_zero @ nat ) )
=> ( ( n_lists @ A @ N2 @ ( nil @ A ) )
= ( nil @ ( list @ A ) ) ) ) ) ).
% n_lists_Nil
thf(fact_6006_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_6007_take__Cons__numeral,axiom,
! [A: $tType,V2: num,X: A,Xs: list @ A] :
( ( take @ A @ ( numeral_numeral @ nat @ V2 ) @ ( cons @ A @ X @ Xs ) )
= ( cons @ A @ X @ ( take @ A @ ( minus_minus @ nat @ ( numeral_numeral @ nat @ V2 ) @ ( one_one @ nat ) ) @ Xs ) ) ) ).
% take_Cons_numeral
thf(fact_6008_Cons__in__lex,axiom,
! [A: $tType,X: A,Xs: list @ A,Y: A,Ys: list @ A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y @ Ys ) ) @ ( lex @ A @ R2 ) )
= ( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R2 )
& ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ A ) @ Ys ) ) )
| ( ( X = Y )
& ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( lex @ A @ R2 ) ) ) ) ) ).
% Cons_in_lex
thf(fact_6009_nth__Cons__pos,axiom,
! [A: $tType,N2: nat,X: A,Xs: list @ A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( nth @ A @ ( cons @ A @ X @ Xs ) @ N2 )
= ( nth @ A @ Xs @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) ) ) ) ).
% nth_Cons_pos
thf(fact_6010_inj__on__Cons1,axiom,
! [A: $tType,X: A,A4: set @ ( list @ A )] : ( inj_on @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X ) @ A4 ) ).
% inj_on_Cons1
thf(fact_6011_zip__eq__ConsE,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B,Xy: product_prod @ A @ B,Xys: list @ ( product_prod @ A @ B )] :
( ( ( zip @ A @ B @ Xs @ Ys )
= ( cons @ ( product_prod @ A @ B ) @ Xy @ Xys ) )
=> ~ ! [X5: A,Xs4: list @ A] :
( ( Xs
= ( cons @ A @ X5 @ Xs4 ) )
=> ! [Y3: B,Ys5: list @ B] :
( ( Ys
= ( cons @ B @ Y3 @ Ys5 ) )
=> ( ( Xy
= ( product_Pair @ A @ B @ X5 @ Y3 ) )
=> ( Xys
!= ( zip @ A @ B @ Xs4 @ Ys5 ) ) ) ) ) ) ).
% zip_eq_ConsE
thf(fact_6012_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_6013_list__update_Osimps_I2_J,axiom,
! [A: $tType,X: A,Xs: list @ A,I: nat,V2: A] :
( ( list_update @ A @ ( cons @ A @ X @ Xs ) @ I @ V2 )
= ( case_nat @ ( list @ A ) @ ( cons @ A @ V2 @ Xs )
@ ^ [J2: nat] : ( cons @ A @ X @ ( list_update @ A @ Xs @ J2 @ V2 ) )
@ I ) ) ).
% list_update.simps(2)
thf(fact_6014_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_6015_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_6016_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_6017_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_6018_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_6019_replicate__Suc,axiom,
! [A: $tType,N2: nat,X: A] :
( ( replicate @ A @ ( suc @ N2 ) @ X )
= ( cons @ A @ X @ ( replicate @ A @ N2 @ X ) ) ) ).
% replicate_Suc
thf(fact_6020_list__update__code_I3_J,axiom,
! [A: $tType,X: A,Xs: list @ A,I: nat,Y: A] :
( ( list_update @ A @ ( cons @ A @ X @ Xs ) @ ( suc @ I ) @ Y )
= ( cons @ A @ X @ ( list_update @ A @ Xs @ I @ Y ) ) ) ).
% list_update_code(3)
thf(fact_6021_map__tailrec__rev_Ocases,axiom,
! [A: $tType,B: $tType,X: product_prod @ ( A > B ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) )] :
( ! [F4: A > B,Bs2: list @ B] :
( X
!= ( product_Pair @ ( A > B ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ F4 @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( nil @ A ) @ Bs2 ) ) )
=> ~ ! [F4: A > B,A6: A,As2: list @ A,Bs2: list @ B] :
( X
!= ( product_Pair @ ( A > B ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ F4 @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( cons @ A @ A6 @ As2 ) @ Bs2 ) ) ) ) ).
% map_tailrec_rev.cases
thf(fact_6022_shuffles_Ocases,axiom,
! [A: $tType,X: product_prod @ ( list @ A ) @ ( list @ A )] :
( ! [Ys4: list @ A] :
( X
!= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Ys4 ) )
=> ( ! [Xs2: list @ A] :
( X
!= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ ( nil @ A ) ) )
=> ~ ! [X5: A,Xs2: list @ A,Y3: A,Ys4: list @ A] :
( X
!= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X5 @ Xs2 ) @ ( cons @ A @ Y3 @ Ys4 ) ) ) ) ) ).
% shuffles.cases
thf(fact_6023_splice_Ocases,axiom,
! [A: $tType,X: product_prod @ ( list @ A ) @ ( list @ A )] :
( ! [Ys4: list @ A] :
( X
!= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Ys4 ) )
=> ~ ! [X5: A,Xs2: list @ A,Ys4: list @ A] :
( X
!= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X5 @ Xs2 ) @ Ys4 ) ) ) ).
% splice.cases
thf(fact_6024_distinct__singleton,axiom,
! [A: $tType,X: A] : ( distinct @ A @ ( cons @ A @ X @ ( nil @ A ) ) ) ).
% distinct_singleton
thf(fact_6025_bij__betw__empty2,axiom,
! [B: $tType,A: $tType,F2: A > B,A4: set @ A] :
( ( bij_betw @ A @ B @ F2 @ A4 @ ( bot_bot @ ( set @ B ) ) )
=> ( A4
= ( bot_bot @ ( set @ A ) ) ) ) ).
% bij_betw_empty2
thf(fact_6026_bij__betw__empty1,axiom,
! [A: $tType,B: $tType,F2: A > B,A4: set @ B] :
( ( bij_betw @ A @ B @ F2 @ ( bot_bot @ ( set @ A ) ) @ A4 )
=> ( A4
= ( bot_bot @ ( set @ B ) ) ) ) ).
% bij_betw_empty1
thf(fact_6027_shuffles_Osimps_I3_J,axiom,
! [A: $tType,X: A,Xs: list @ A,Y: A,Ys: list @ A] :
( ( shuffles @ A @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y @ Ys ) )
= ( sup_sup @ ( set @ ( list @ A ) ) @ ( image2 @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X ) @ ( shuffles @ A @ Xs @ ( cons @ A @ Y @ Ys ) ) ) @ ( image2 @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ Y ) @ ( shuffles @ A @ ( cons @ A @ X @ Xs ) @ Ys ) ) ) ) ).
% shuffles.simps(3)
thf(fact_6028_bij__betw__subset,axiom,
! [A: $tType,B: $tType,F2: A > B,A4: set @ A,A16: set @ B,B2: set @ A,B13: set @ B] :
( ( bij_betw @ A @ B @ F2 @ A4 @ A16 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ A4 )
=> ( ( ( image2 @ A @ B @ F2 @ B2 )
= B13 )
=> ( bij_betw @ A @ B @ F2 @ B2 @ B13 ) ) ) ) ).
% bij_betw_subset
thf(fact_6029_bij__betw__byWitness,axiom,
! [A: $tType,B: $tType,A4: set @ A,F8: B > A,F2: A > B,A16: set @ B] :
( ! [X5: A] :
( ( member @ A @ X5 @ A4 )
=> ( ( F8 @ ( F2 @ X5 ) )
= X5 ) )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ A16 )
=> ( ( F2 @ ( F8 @ X5 ) )
= X5 ) )
=> ( ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F2 @ A4 ) @ A16 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image2 @ B @ A @ F8 @ A16 ) @ A4 )
=> ( bij_betw @ A @ B @ F2 @ A4 @ A16 ) ) ) ) ) ).
% bij_betw_byWitness
thf(fact_6030_Cons__shuffles__subset1,axiom,
! [A: $tType,X: A,Xs: list @ A,Ys: list @ A] : ( ord_less_eq @ ( set @ ( list @ A ) ) @ ( image2 @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X ) @ ( shuffles @ A @ Xs @ Ys ) ) @ ( shuffles @ A @ ( cons @ A @ X @ Xs ) @ Ys ) ) ).
% Cons_shuffles_subset1
thf(fact_6031_Cons__shuffles__subset2,axiom,
! [A: $tType,Y: A,Xs: list @ A,Ys: list @ A] : ( ord_less_eq @ ( set @ ( list @ A ) ) @ ( image2 @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ Y ) @ ( shuffles @ A @ Xs @ Ys ) ) @ ( shuffles @ A @ Xs @ ( cons @ A @ Y @ Ys ) ) ) ).
% Cons_shuffles_subset2
thf(fact_6032_bij__betw__same__card,axiom,
! [A: $tType,B: $tType,F2: A > B,A4: set @ A,B2: set @ B] :
( ( bij_betw @ A @ B @ F2 @ A4 @ B2 )
=> ( ( finite_card @ A @ A4 )
= ( finite_card @ B @ B2 ) ) ) ).
% bij_betw_same_card
thf(fact_6033_bij__betw__funpow,axiom,
! [A: $tType,F2: A > A,S2: set @ A,N2: nat] :
( ( bij_betw @ A @ A @ F2 @ S2 @ S2 )
=> ( bij_betw @ A @ A @ ( compow @ ( A > A ) @ N2 @ F2 ) @ S2 @ S2 ) ) ).
% bij_betw_funpow
thf(fact_6034_bij__betw__iff__card,axiom,
! [A: $tType,B: $tType,A4: set @ A,B2: set @ B] :
( ( finite_finite2 @ A @ A4 )
=> ( ( finite_finite2 @ B @ B2 )
=> ( ( ? [F3: A > B] : ( bij_betw @ A @ B @ F3 @ A4 @ B2 ) )
= ( ( finite_card @ A @ A4 )
= ( finite_card @ B @ B2 ) ) ) ) ) ).
% bij_betw_iff_card
thf(fact_6035_finite__same__card__bij,axiom,
! [A: $tType,B: $tType,A4: set @ A,B2: set @ B] :
( ( finite_finite2 @ A @ A4 )
=> ( ( finite_finite2 @ B @ B2 )
=> ( ( ( finite_card @ A @ A4 )
= ( finite_card @ B @ B2 ) )
=> ? [H5: A > B] : ( bij_betw @ A @ B @ H5 @ A4 @ B2 ) ) ) ) ).
% finite_same_card_bij
thf(fact_6036_bij__fn,axiom,
! [A: $tType,F2: A > A,N2: nat] :
( ( bij_betw @ A @ A @ F2 @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ A ) ) )
=> ( bij_betw @ A @ A @ ( compow @ ( A > A ) @ N2 @ F2 ) @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ A ) ) ) ) ).
% bij_fn
thf(fact_6037_successively_Ocases,axiom,
! [A: $tType,X: product_prod @ ( A > A > $o ) @ ( list @ A )] :
( ! [P8: A > A > $o] :
( X
!= ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ P8 @ ( nil @ A ) ) )
=> ( ! [P8: A > A > $o,X5: A] :
( X
!= ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ P8 @ ( cons @ A @ X5 @ ( nil @ A ) ) ) )
=> ~ ! [P8: A > A > $o,X5: A,Y3: A,Xs2: list @ A] :
( X
!= ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ P8 @ ( cons @ A @ X5 @ ( cons @ A @ Y3 @ Xs2 ) ) ) ) ) ) ).
% successively.cases
thf(fact_6038_arg__min__list_Ocases,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ B )
=> ! [X: product_prod @ ( A > B ) @ ( list @ A )] :
( ! [F4: A > B,X5: A] :
( X
!= ( product_Pair @ ( A > B ) @ ( list @ A ) @ F4 @ ( cons @ A @ X5 @ ( nil @ A ) ) ) )
=> ( ! [F4: A > B,X5: A,Y3: A,Zs2: list @ A] :
( X
!= ( product_Pair @ ( A > B ) @ ( list @ A ) @ F4 @ ( cons @ A @ X5 @ ( cons @ A @ Y3 @ Zs2 ) ) ) )
=> ~ ! [A6: A > B] :
( X
!= ( product_Pair @ ( A > B ) @ ( list @ A ) @ A6 @ ( nil @ A ) ) ) ) ) ) ).
% arg_min_list.cases
thf(fact_6039_sorted__wrt_Ocases,axiom,
! [A: $tType,X: product_prod @ ( A > A > $o ) @ ( list @ A )] :
( ! [P8: A > A > $o] :
( X
!= ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ P8 @ ( nil @ A ) ) )
=> ~ ! [P8: A > A > $o,X5: A,Ys4: list @ A] :
( X
!= ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ P8 @ ( cons @ A @ X5 @ Ys4 ) ) ) ) ).
% sorted_wrt.cases
thf(fact_6040_bij__betw__finite,axiom,
! [A: $tType,B: $tType,F2: A > B,A4: set @ A,B2: set @ B] :
( ( bij_betw @ A @ B @ F2 @ A4 @ B2 )
=> ( ( finite_finite2 @ A @ A4 )
= ( finite_finite2 @ B @ B2 ) ) ) ).
% bij_betw_finite
thf(fact_6041_list__nonempty__induct,axiom,
! [A: $tType,Xs: list @ A,P: ( list @ A ) > $o] :
( ( Xs
!= ( nil @ A ) )
=> ( ! [X5: A] : ( P @ ( cons @ A @ X5 @ ( nil @ A ) ) )
=> ( ! [X5: A,Xs2: list @ A] :
( ( Xs2
!= ( nil @ A ) )
=> ( ( P @ Xs2 )
=> ( P @ ( cons @ A @ X5 @ Xs2 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_6042_list__induct2_H,axiom,
! [A: $tType,B: $tType,P: ( list @ A ) > ( list @ B ) > $o,Xs: list @ A,Ys: list @ B] :
( ( P @ ( nil @ A ) @ ( nil @ B ) )
=> ( ! [X5: A,Xs2: list @ A] : ( P @ ( cons @ A @ X5 @ Xs2 ) @ ( nil @ B ) )
=> ( ! [Y3: B,Ys4: list @ B] : ( P @ ( nil @ A ) @ ( cons @ B @ Y3 @ Ys4 ) )
=> ( ! [X5: A,Xs2: list @ A,Y3: B,Ys4: list @ B] :
( ( P @ Xs2 @ Ys4 )
=> ( P @ ( cons @ A @ X5 @ Xs2 ) @ ( cons @ B @ Y3 @ Ys4 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_6043_neq__Nil__conv,axiom,
! [A: $tType,Xs: list @ A] :
( ( Xs
!= ( nil @ A ) )
= ( ? [Y6: A,Ys3: list @ A] :
( Xs
= ( cons @ A @ Y6 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_6044_remdups__adj_Ocases,axiom,
! [A: $tType,X: list @ A] :
( ( X
!= ( nil @ A ) )
=> ( ! [X5: A] :
( X
!= ( cons @ A @ X5 @ ( nil @ A ) ) )
=> ~ ! [X5: A,Y3: A,Xs2: list @ A] :
( X
!= ( cons @ A @ X5 @ ( cons @ A @ Y3 @ Xs2 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_6045_transpose_Ocases,axiom,
! [A: $tType,X: list @ ( list @ A )] :
( ( X
!= ( nil @ ( list @ A ) ) )
=> ( ! [Xss2: list @ ( list @ A )] :
( X
!= ( cons @ ( list @ A ) @ ( nil @ A ) @ Xss2 ) )
=> ~ ! [X5: A,Xs2: list @ A,Xss2: list @ ( list @ A )] :
( X
!= ( cons @ ( list @ A ) @ ( cons @ A @ X5 @ Xs2 ) @ Xss2 ) ) ) ) ).
% transpose.cases
thf(fact_6046_min__list_Ocases,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X: list @ A] :
( ! [X5: A,Xs2: list @ A] :
( X
!= ( cons @ A @ X5 @ Xs2 ) )
=> ( X
= ( nil @ A ) ) ) ) ).
% min_list.cases
thf(fact_6047_list_Oexhaust,axiom,
! [A: $tType,Y: list @ A] :
( ( Y
!= ( nil @ A ) )
=> ~ ! [X212: A,X223: list @ A] :
( Y
!= ( cons @ A @ X212 @ X223 ) ) ) ).
% list.exhaust
thf(fact_6048_list_OdiscI,axiom,
! [A: $tType,List: list @ A,X21: A,X222: list @ A] :
( ( List
= ( cons @ A @ X21 @ X222 ) )
=> ( List
!= ( nil @ A ) ) ) ).
% list.discI
thf(fact_6049_list_Odistinct_I1_J,axiom,
! [A: $tType,X21: A,X222: list @ A] :
( ( nil @ A )
!= ( cons @ A @ X21 @ X222 ) ) ).
% list.distinct(1)
thf(fact_6050_removeAll_Osimps_I2_J,axiom,
! [A: $tType,X: A,Y: A,Xs: list @ A] :
( ( ( X = Y )
=> ( ( removeAll @ A @ X @ ( cons @ A @ Y @ Xs ) )
= ( removeAll @ A @ X @ Xs ) ) )
& ( ( X != Y )
=> ( ( removeAll @ A @ X @ ( cons @ A @ Y @ Xs ) )
= ( cons @ A @ Y @ ( removeAll @ A @ X @ Xs ) ) ) ) ) ).
% removeAll.simps(2)
thf(fact_6051_Cons__in__shuffles__rightI,axiom,
! [A: $tType,Zs: list @ A,Xs: list @ A,Ys: list @ A,Z3: A] :
( ( member @ ( list @ A ) @ Zs @ ( shuffles @ A @ Xs @ Ys ) )
=> ( member @ ( list @ A ) @ ( cons @ A @ Z3 @ Zs ) @ ( shuffles @ A @ Xs @ ( cons @ A @ Z3 @ Ys ) ) ) ) ).
% Cons_in_shuffles_rightI
thf(fact_6052_Cons__in__shuffles__leftI,axiom,
! [A: $tType,Zs: list @ A,Xs: list @ A,Ys: list @ A,Z3: A] :
( ( member @ ( list @ A ) @ Zs @ ( shuffles @ A @ Xs @ Ys ) )
=> ( member @ ( list @ A ) @ ( cons @ A @ Z3 @ Zs ) @ ( shuffles @ A @ ( cons @ A @ Z3 @ Xs ) @ Ys ) ) ) ).
% Cons_in_shuffles_leftI
thf(fact_6053_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_6054_not__Cons__self2,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( cons @ A @ X @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_6055_remove1_Osimps_I2_J,axiom,
! [A: $tType,X: A,Y: A,Xs: list @ A] :
( ( ( X = Y )
=> ( ( remove1 @ A @ X @ ( cons @ A @ Y @ Xs ) )
= Xs ) )
& ( ( X != Y )
=> ( ( remove1 @ A @ X @ ( cons @ A @ Y @ Xs ) )
= ( cons @ A @ Y @ ( remove1 @ A @ X @ Xs ) ) ) ) ) ).
% remove1.simps(2)
thf(fact_6056_remdups__adj_Osimps_I3_J,axiom,
! [A: $tType,X: A,Y: A,Xs: list @ A] :
( ( ( X = Y )
=> ( ( remdups_adj @ A @ ( cons @ A @ X @ ( cons @ A @ Y @ Xs ) ) )
= ( remdups_adj @ A @ ( cons @ A @ X @ Xs ) ) ) )
& ( ( X != Y )
=> ( ( remdups_adj @ A @ ( cons @ A @ X @ ( cons @ A @ Y @ Xs ) ) )
= ( cons @ A @ X @ ( remdups_adj @ A @ ( cons @ A @ Y @ Xs ) ) ) ) ) ) ).
% remdups_adj.simps(3)
thf(fact_6057_not__add__distrib,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [A3: A,B3: A] :
( ( bit_ri4277139882892585799ns_not @ A @ ( plus_plus @ A @ A3 @ B3 ) )
= ( minus_minus @ A @ ( bit_ri4277139882892585799ns_not @ A @ A3 ) @ B3 ) ) ) ).
% not_add_distrib
thf(fact_6058_not__diff__distrib,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [A3: A,B3: A] :
( ( bit_ri4277139882892585799ns_not @ A @ ( minus_minus @ A @ A3 @ B3 ) )
= ( plus_plus @ A @ ( bit_ri4277139882892585799ns_not @ A @ A3 ) @ B3 ) ) ) ).
% not_diff_distrib
thf(fact_6059_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_6060_list__induct2,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B,P: ( list @ A ) > ( list @ B ) > $o] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( P @ ( nil @ A ) @ ( nil @ B ) )
=> ( ! [X5: A,Xs2: list @ A,Y3: B,Ys4: list @ B] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
= ( size_size @ ( list @ B ) @ Ys4 ) )
=> ( ( P @ Xs2 @ Ys4 )
=> ( P @ ( cons @ A @ X5 @ Xs2 ) @ ( cons @ B @ Y3 @ Ys4 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_6061_list__induct3,axiom,
! [B: $tType,A: $tType,C: $tType,Xs: list @ A,Ys: list @ B,Zs: list @ C,P: ( list @ A ) > ( list @ B ) > ( list @ C ) > $o] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( ( size_size @ ( list @ B ) @ Ys )
= ( size_size @ ( list @ C ) @ Zs ) )
=> ( ( P @ ( nil @ A ) @ ( nil @ B ) @ ( nil @ C ) )
=> ( ! [X5: A,Xs2: list @ A,Y3: B,Ys4: list @ B,Z: C,Zs2: list @ C] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
= ( size_size @ ( list @ B ) @ Ys4 ) )
=> ( ( ( size_size @ ( list @ B ) @ Ys4 )
= ( size_size @ ( list @ C ) @ Zs2 ) )
=> ( ( P @ Xs2 @ Ys4 @ Zs2 )
=> ( P @ ( cons @ A @ X5 @ Xs2 ) @ ( cons @ B @ Y3 @ Ys4 ) @ ( cons @ C @ Z @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_6062_list__induct4,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,Xs: list @ A,Ys: list @ B,Zs: list @ C,Ws2: list @ D,P: ( list @ A ) > ( list @ B ) > ( list @ C ) > ( list @ D ) > $o] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( ( size_size @ ( list @ B ) @ Ys )
= ( size_size @ ( list @ C ) @ Zs ) )
=> ( ( ( size_size @ ( list @ C ) @ Zs )
= ( size_size @ ( list @ D ) @ Ws2 ) )
=> ( ( P @ ( nil @ A ) @ ( nil @ B ) @ ( nil @ C ) @ ( nil @ D ) )
=> ( ! [X5: A,Xs2: list @ A,Y3: B,Ys4: list @ B,Z: C,Zs2: list @ C,W: D,Ws: list @ D] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
= ( size_size @ ( list @ B ) @ Ys4 ) )
=> ( ( ( size_size @ ( list @ B ) @ Ys4 )
= ( size_size @ ( list @ C ) @ Zs2 ) )
=> ( ( ( size_size @ ( list @ C ) @ Zs2 )
= ( size_size @ ( list @ D ) @ Ws ) )
=> ( ( P @ Xs2 @ Ys4 @ Zs2 @ Ws )
=> ( P @ ( cons @ A @ X5 @ Xs2 ) @ ( cons @ B @ Y3 @ Ys4 ) @ ( cons @ C @ Z @ Zs2 ) @ ( cons @ D @ W @ Ws ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws2 ) ) ) ) ) ) ).
% list_induct4
thf(fact_6063_length__Suc__conv,axiom,
! [A: $tType,Xs: list @ A,N2: nat] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( suc @ N2 ) )
= ( ? [Y6: A,Ys3: list @ A] :
( ( Xs
= ( cons @ A @ Y6 @ Ys3 ) )
& ( ( size_size @ ( list @ A ) @ Ys3 )
= N2 ) ) ) ) ).
% length_Suc_conv
thf(fact_6064_Suc__length__conv,axiom,
! [A: $tType,N2: nat,Xs: list @ A] :
( ( ( suc @ N2 )
= ( size_size @ ( list @ A ) @ Xs ) )
= ( ? [Y6: A,Ys3: list @ A] :
( ( Xs
= ( cons @ A @ Y6 @ Ys3 ) )
& ( ( size_size @ ( list @ A ) @ Ys3 )
= N2 ) ) ) ) ).
% Suc_length_conv
thf(fact_6065_length__Cons,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( size_size @ ( list @ A ) @ ( cons @ A @ X @ Xs ) )
= ( suc @ ( size_size @ ( list @ A ) @ Xs ) ) ) ).
% length_Cons
thf(fact_6066_insort__key_Osimps_I2_J,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F2: B > A,X: B,Y: B,Ys: list @ B] :
( ( ( ord_less_eq @ A @ ( F2 @ X ) @ ( F2 @ Y ) )
=> ( ( linorder_insort_key @ B @ A @ F2 @ X @ ( cons @ B @ Y @ Ys ) )
= ( cons @ B @ X @ ( cons @ B @ Y @ Ys ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( F2 @ X ) @ ( F2 @ Y ) )
=> ( ( linorder_insort_key @ B @ A @ F2 @ X @ ( cons @ B @ Y @ Ys ) )
= ( cons @ B @ Y @ ( linorder_insort_key @ B @ A @ F2 @ X @ Ys ) ) ) ) ) ) ).
% insort_key.simps(2)
thf(fact_6067_list__update__code_I2_J,axiom,
! [A: $tType,X: A,Xs: list @ A,Y: A] :
( ( list_update @ A @ ( cons @ A @ X @ Xs ) @ ( zero_zero @ nat ) @ Y )
= ( cons @ A @ Y @ Xs ) ) ).
% list_update_code(2)
thf(fact_6068_listrelp_OCons,axiom,
! [A: $tType,B: $tType,R2: A > B > $o,X: A,Y: B,Xs: list @ A,Ys: list @ B] :
( ( R2 @ X @ Y )
=> ( ( listrelp @ A @ B @ R2 @ Xs @ Ys )
=> ( listrelp @ A @ B @ R2 @ ( cons @ A @ X @ Xs ) @ ( cons @ B @ Y @ Ys ) ) ) ) ).
% listrelp.Cons
thf(fact_6069_list_Osel_I3_J,axiom,
! [A: $tType,X21: A,X222: list @ A] :
( ( tl @ A @ ( cons @ A @ X21 @ X222 ) )
= X222 ) ).
% list.sel(3)
thf(fact_6070_list_Osel_I1_J,axiom,
! [A: $tType,X21: A,X222: list @ A] :
( ( hd @ A @ ( cons @ A @ X21 @ X222 ) )
= X21 ) ).
% list.sel(1)
thf(fact_6071_tl__Nil,axiom,
! [A: $tType,Xs: list @ A] :
( ( ( tl @ A @ Xs )
= ( nil @ A ) )
= ( ( Xs
= ( nil @ A ) )
| ? [X4: A] :
( Xs
= ( cons @ A @ X4 @ ( nil @ A ) ) ) ) ) ).
% tl_Nil
thf(fact_6072_Nil__tl,axiom,
! [A: $tType,Xs: list @ A] :
( ( ( nil @ A )
= ( tl @ A @ Xs ) )
= ( ( Xs
= ( nil @ A ) )
| ? [X4: A] :
( Xs
= ( cons @ A @ X4 @ ( nil @ A ) ) ) ) ) ).
% Nil_tl
thf(fact_6073_remdups__adj_Oelims,axiom,
! [A: $tType,X: list @ A,Y: list @ A] :
( ( ( remdups_adj @ A @ X )
= Y )
=> ( ( ( X
= ( nil @ A ) )
=> ( Y
!= ( nil @ A ) ) )
=> ( ! [X5: A] :
( ( X
= ( cons @ A @ X5 @ ( nil @ A ) ) )
=> ( Y
!= ( cons @ A @ X5 @ ( nil @ A ) ) ) )
=> ~ ! [X5: A,Y3: A,Xs2: list @ A] :
( ( X
= ( cons @ A @ X5 @ ( cons @ A @ Y3 @ Xs2 ) ) )
=> ~ ( ( ( X5 = Y3 )
=> ( Y
= ( remdups_adj @ A @ ( cons @ A @ X5 @ Xs2 ) ) ) )
& ( ( X5 != Y3 )
=> ( Y
= ( cons @ A @ X5 @ ( remdups_adj @ A @ ( cons @ A @ Y3 @ Xs2 ) ) ) ) ) ) ) ) ) ) ).
% remdups_adj.elims
thf(fact_6074_remdups__adj_Osimps_I2_J,axiom,
! [A: $tType,X: A] :
( ( remdups_adj @ A @ ( cons @ A @ X @ ( nil @ A ) ) )
= ( cons @ A @ X @ ( nil @ A ) ) ) ).
% remdups_adj.simps(2)
thf(fact_6075_shufflesE,axiom,
! [A: $tType,Zs: list @ A,Xs: list @ A,Ys: list @ A] :
( ( member @ ( list @ A ) @ Zs @ ( shuffles @ A @ Xs @ Ys ) )
=> ( ( ( Zs = Xs )
=> ( Ys
!= ( nil @ A ) ) )
=> ( ( ( Zs = Ys )
=> ( Xs
!= ( nil @ A ) ) )
=> ( ! [X5: A,Xs4: list @ A] :
( ( Xs
= ( cons @ A @ X5 @ Xs4 ) )
=> ! [Z: A,Zs4: list @ A] :
( ( Zs
= ( cons @ A @ Z @ Zs4 ) )
=> ( ( X5 = Z )
=> ~ ( member @ ( list @ A ) @ Zs4 @ ( shuffles @ A @ Xs4 @ Ys ) ) ) ) )
=> ~ ! [Y3: A,Ys5: list @ A] :
( ( Ys
= ( cons @ A @ Y3 @ Ys5 ) )
=> ! [Z: A,Zs4: list @ A] :
( ( Zs
= ( cons @ A @ Z @ Zs4 ) )
=> ( ( Y3 = Z )
=> ~ ( member @ ( list @ A ) @ Zs4 @ ( shuffles @ A @ Xs @ Ys5 ) ) ) ) ) ) ) ) ) ).
% shufflesE
thf(fact_6076_insort__key_Osimps_I1_J,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F2: B > A,X: B] :
( ( linorder_insort_key @ B @ A @ F2 @ X @ ( nil @ B ) )
= ( cons @ B @ X @ ( nil @ B ) ) ) ) ).
% insort_key.simps(1)
thf(fact_6077_remdups_Osimps_I2_J,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ( remdups @ A @ ( cons @ A @ X @ Xs ) )
= ( remdups @ A @ Xs ) ) )
& ( ~ ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ( remdups @ A @ ( cons @ A @ X @ Xs ) )
= ( cons @ A @ X @ ( remdups @ A @ Xs ) ) ) ) ) ).
% remdups.simps(2)
thf(fact_6078_listrel1I2,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,R2: set @ ( product_prod @ A @ A ),X: A] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( listrel1 @ A @ R2 ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ X @ Ys ) ) @ ( listrel1 @ A @ R2 ) ) ) ).
% listrel1I2
thf(fact_6079_find_Osimps_I2_J,axiom,
! [A: $tType,P: A > $o,X: A,Xs: list @ A] :
( ( ( P @ X )
=> ( ( find @ A @ P @ ( cons @ A @ X @ Xs ) )
= ( some @ A @ X ) ) )
& ( ~ ( P @ X )
=> ( ( find @ A @ P @ ( cons @ A @ X @ Xs ) )
= ( find @ A @ P @ Xs ) ) ) ) ).
% find.simps(2)
thf(fact_6080_listrelp_Osimps,axiom,
! [B: $tType,A: $tType] :
( ( listrelp @ A @ B )
= ( ^ [R4: A > B > $o,A12: list @ A,A23: list @ B] :
( ( ( A12
= ( nil @ A ) )
& ( A23
= ( nil @ B ) ) )
| ? [X4: A,Y6: B,Xs3: list @ A,Ys3: list @ B] :
( ( A12
= ( cons @ A @ X4 @ Xs3 ) )
& ( A23
= ( cons @ B @ Y6 @ Ys3 ) )
& ( R4 @ X4 @ Y6 )
& ( listrelp @ A @ B @ R4 @ Xs3 @ Ys3 ) ) ) ) ) ).
% listrelp.simps
thf(fact_6081_listrelp_Ocases,axiom,
! [A: $tType,B: $tType,R2: A > B > $o,A1: list @ A,A22: list @ B] :
( ( listrelp @ A @ B @ R2 @ A1 @ A22 )
=> ( ( ( A1
= ( nil @ A ) )
=> ( A22
!= ( nil @ B ) ) )
=> ~ ! [X5: A,Y3: B,Xs2: list @ A] :
( ( A1
= ( cons @ A @ X5 @ Xs2 ) )
=> ! [Ys4: list @ B] :
( ( A22
= ( cons @ B @ Y3 @ Ys4 ) )
=> ( ( R2 @ X5 @ Y3 )
=> ~ ( listrelp @ A @ B @ R2 @ Xs2 @ Ys4 ) ) ) ) ) ) ).
% listrelp.cases
thf(fact_6082_arg__min__list_Osimps_I1_J,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ B )
=> ! [F2: A > B,X: A] :
( ( arg_min_list @ A @ B @ F2 @ ( cons @ A @ X @ ( nil @ A ) ) )
= X ) ) ).
% arg_min_list.simps(1)
thf(fact_6083_take__Cons,axiom,
! [A: $tType,N2: nat,X: A,Xs: list @ A] :
( ( take @ A @ N2 @ ( cons @ A @ X @ Xs ) )
= ( case_nat @ ( list @ A ) @ ( nil @ A )
@ ^ [M: nat] : ( cons @ A @ X @ ( take @ A @ M @ Xs ) )
@ N2 ) ) ).
% take_Cons
thf(fact_6084_bij__betw__comp__iff2,axiom,
! [C: $tType,A: $tType,B: $tType,F8: A > B,A16: set @ A,A19: set @ B,F2: C > A,A4: set @ C] :
( ( bij_betw @ A @ B @ F8 @ A16 @ A19 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image2 @ C @ A @ F2 @ A4 ) @ A16 )
=> ( ( bij_betw @ C @ A @ F2 @ A4 @ A16 )
= ( bij_betw @ C @ B @ ( comp @ A @ B @ C @ F8 @ F2 ) @ A4 @ A19 ) ) ) ) ).
% bij_betw_comp_iff2
thf(fact_6085_Cons__in__subseqsD,axiom,
! [A: $tType,Y: A,Ys: list @ A,Xs: list @ A] :
( ( member @ ( list @ A ) @ ( cons @ A @ Y @ Ys ) @ ( set2 @ ( list @ A ) @ ( subseqs @ A @ Xs ) ) )
=> ( member @ ( list @ A ) @ Ys @ ( set2 @ ( list @ A ) @ ( subseqs @ A @ Xs ) ) ) ) ).
% Cons_in_subseqsD
thf(fact_6086_gen__length__code_I2_J,axiom,
! [B: $tType,N2: nat,X: B,Xs: list @ B] :
( ( gen_length @ B @ N2 @ ( cons @ B @ X @ Xs ) )
= ( gen_length @ B @ ( suc @ N2 ) @ Xs ) ) ).
% gen_length_code(2)
thf(fact_6087_notIn__Un__bij__betw3,axiom,
! [A: $tType,B: $tType,B3: A,A4: set @ A,F2: A > B,A16: set @ B] :
( ~ ( member @ A @ B3 @ A4 )
=> ( ~ ( member @ B @ ( F2 @ B3 ) @ A16 )
=> ( ( bij_betw @ A @ B @ F2 @ A4 @ A16 )
= ( bij_betw @ A @ B @ F2 @ ( sup_sup @ ( set @ A ) @ A4 @ ( insert2 @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( sup_sup @ ( set @ B ) @ A16 @ ( insert2 @ B @ ( F2 @ B3 ) @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ).
% notIn_Un_bij_betw3
thf(fact_6088_notIn__Un__bij__betw,axiom,
! [A: $tType,B: $tType,B3: A,A4: set @ A,F2: A > B,A16: set @ B] :
( ~ ( member @ A @ B3 @ A4 )
=> ( ~ ( member @ B @ ( F2 @ B3 ) @ A16 )
=> ( ( bij_betw @ A @ B @ F2 @ A4 @ A16 )
=> ( bij_betw @ A @ B @ F2 @ ( sup_sup @ ( set @ A ) @ A4 @ ( insert2 @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( sup_sup @ ( set @ B ) @ A16 @ ( insert2 @ B @ ( F2 @ B3 ) @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ).
% notIn_Un_bij_betw
thf(fact_6089_bij__betw__partition,axiom,
! [A: $tType,B: $tType,F2: A > B,A4: set @ A,C5: set @ A,B2: set @ B,D6: set @ B] :
( ( bij_betw @ A @ B @ F2 @ ( sup_sup @ ( set @ A ) @ A4 @ C5 ) @ ( sup_sup @ ( set @ B ) @ B2 @ D6 ) )
=> ( ( bij_betw @ A @ B @ F2 @ C5 @ D6 )
=> ( ( ( inf_inf @ ( set @ A ) @ A4 @ C5 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( ( inf_inf @ ( set @ B ) @ B2 @ D6 )
= ( bot_bot @ ( set @ B ) ) )
=> ( bij_betw @ A @ B @ F2 @ A4 @ B2 ) ) ) ) ) ).
% bij_betw_partition
thf(fact_6090_bij__betw__combine,axiom,
! [A: $tType,B: $tType,F2: A > B,A4: set @ A,B2: set @ B,C5: set @ A,D6: set @ B] :
( ( bij_betw @ A @ B @ F2 @ A4 @ B2 )
=> ( ( bij_betw @ A @ B @ F2 @ C5 @ D6 )
=> ( ( ( inf_inf @ ( set @ B ) @ B2 @ D6 )
= ( bot_bot @ ( set @ B ) ) )
=> ( bij_betw @ A @ B @ F2 @ ( sup_sup @ ( set @ A ) @ A4 @ C5 ) @ ( sup_sup @ ( set @ B ) @ B2 @ D6 ) ) ) ) ) ).
% bij_betw_combine
thf(fact_6091_nth__Cons,axiom,
! [A: $tType,X: A,Xs: list @ A,N2: nat] :
( ( nth @ A @ ( cons @ A @ X @ Xs ) @ N2 )
= ( case_nat @ A @ X @ ( nth @ A @ Xs ) @ N2 ) ) ).
% nth_Cons
thf(fact_6092_ord_Olexordp_Omono,axiom,
! [A: $tType,Less: A > A > $o] :
( order_mono @ ( ( list @ A ) > ( list @ A ) > $o ) @ ( ( list @ A ) > ( list @ A ) > $o )
@ ^ [P6: ( list @ A ) > ( list @ A ) > $o,X15: list @ A,X24: list @ A] :
( ? [Y6: A,Ys3: list @ A] :
( ( X15
= ( nil @ A ) )
& ( X24
= ( cons @ A @ Y6 @ Ys3 ) ) )
| ? [X4: A,Y6: A,Xs3: list @ A,Ys3: list @ A] :
( ( X15
= ( cons @ A @ X4 @ Xs3 ) )
& ( X24
= ( cons @ A @ Y6 @ Ys3 ) )
& ( Less @ X4 @ Y6 ) )
| ? [X4: A,Y6: A,Xs3: list @ A,Ys3: list @ A] :
( ( X15
= ( cons @ A @ X4 @ Xs3 ) )
& ( X24
= ( cons @ A @ Y6 @ Ys3 ) )
& ~ ( Less @ X4 @ Y6 )
& ~ ( Less @ Y6 @ X4 )
& ( P6 @ Xs3 @ Ys3 ) ) ) ) ).
% ord.lexordp.mono
thf(fact_6093_arg__min__list_Osimps_I2_J,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ B )
=> ! [F2: A > B,X: A,Y: A,Zs: list @ A] :
( ( arg_min_list @ A @ B @ F2 @ ( cons @ A @ X @ ( cons @ A @ Y @ Zs ) ) )
= ( if @ A @ ( ord_less_eq @ B @ ( F2 @ X ) @ ( F2 @ ( arg_min_list @ A @ B @ F2 @ ( cons @ A @ Y @ Zs ) ) ) ) @ X @ ( arg_min_list @ A @ B @ F2 @ ( cons @ A @ Y @ Zs ) ) ) ) ) ).
% arg_min_list.simps(2)
thf(fact_6094_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_6095_bij__betw__disjoint__Un,axiom,
! [A: $tType,B: $tType,F2: A > B,A4: set @ A,C5: set @ B,G: A > B,B2: set @ A,D6: set @ B] :
( ( bij_betw @ A @ B @ F2 @ A4 @ C5 )
=> ( ( bij_betw @ A @ B @ G @ B2 @ D6 )
=> ( ( ( inf_inf @ ( set @ A ) @ A4 @ B2 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( ( inf_inf @ ( set @ B ) @ C5 @ D6 )
= ( bot_bot @ ( set @ B ) ) )
=> ( bij_betw @ A @ B
@ ^ [X4: A] : ( if @ B @ ( member @ A @ X4 @ A4 ) @ ( F2 @ X4 ) @ ( G @ X4 ) )
@ ( sup_sup @ ( set @ A ) @ A4 @ B2 )
@ ( sup_sup @ ( set @ B ) @ C5 @ D6 ) ) ) ) ) ) ).
% bij_betw_disjoint_Un
thf(fact_6096_bij__betw__UNION__chain,axiom,
! [B: $tType,C: $tType,A: $tType,I6: set @ A,A4: A > ( set @ B ),F2: B > C,A16: A > ( set @ C )] :
( ! [I4: A,J3: A] :
( ( member @ A @ I4 @ I6 )
=> ( ( member @ A @ J3 @ I6 )
=> ( ( ord_less_eq @ ( set @ B ) @ ( A4 @ I4 ) @ ( A4 @ J3 ) )
| ( ord_less_eq @ ( set @ B ) @ ( A4 @ J3 ) @ ( A4 @ I4 ) ) ) ) )
=> ( ! [I4: A] :
( ( member @ A @ I4 @ I6 )
=> ( bij_betw @ B @ C @ F2 @ ( A4 @ I4 ) @ ( A16 @ I4 ) ) )
=> ( bij_betw @ B @ C @ F2 @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ A4 @ I6 ) ) @ ( complete_Sup_Sup @ ( set @ C ) @ ( image2 @ A @ ( set @ C ) @ A16 @ I6 ) ) ) ) ) ).
% bij_betw_UNION_chain
thf(fact_6097_Suc__le__length__iff,axiom,
! [A: $tType,N2: nat,Xs: list @ A] :
( ( ord_less_eq @ nat @ ( suc @ N2 ) @ ( size_size @ ( list @ A ) @ Xs ) )
= ( ? [X4: A,Ys3: list @ A] :
( ( Xs
= ( cons @ A @ X4 @ Ys3 ) )
& ( ord_less_eq @ nat @ N2 @ ( size_size @ ( list @ A ) @ Ys3 ) ) ) ) ) ).
% Suc_le_length_iff
thf(fact_6098_insort__is__Cons,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ B,F2: B > A,A3: B] :
( ! [X5: B] :
( ( member @ B @ X5 @ ( set2 @ B @ Xs ) )
=> ( ord_less_eq @ A @ ( F2 @ A3 ) @ ( F2 @ X5 ) ) )
=> ( ( linorder_insort_key @ B @ A @ F2 @ A3 @ Xs )
= ( cons @ B @ A3 @ Xs ) ) ) ) ).
% insort_is_Cons
thf(fact_6099_listrel1I1,axiom,
! [A: $tType,X: A,Y: A,R2: set @ ( product_prod @ A @ A ),Xs: list @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R2 )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y @ Xs ) ) @ ( listrel1 @ A @ R2 ) ) ) ).
% listrel1I1
thf(fact_6100_Cons__listrel1E1,axiom,
! [A: $tType,X: A,Xs: list @ A,Ys: list @ A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X @ Xs ) @ Ys ) @ ( listrel1 @ A @ R2 ) )
=> ( ! [Y3: A] :
( ( Ys
= ( cons @ A @ Y3 @ Xs ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y3 ) @ R2 ) )
=> ~ ! [Zs2: list @ A] :
( ( Ys
= ( cons @ A @ X @ Zs2 ) )
=> ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Zs2 ) @ ( listrel1 @ A @ R2 ) ) ) ) ) ).
% Cons_listrel1E1
thf(fact_6101_Cons__listrel1E2,axiom,
! [A: $tType,Xs: list @ A,Y: A,Ys: list @ A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ ( cons @ A @ Y @ Ys ) ) @ ( listrel1 @ A @ R2 ) )
=> ( ! [X5: A] :
( ( Xs
= ( cons @ A @ X5 @ Ys ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X5 @ Y ) @ R2 ) )
=> ~ ! [Zs2: list @ A] :
( ( Xs
= ( cons @ A @ Y @ Zs2 ) )
=> ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Zs2 @ Ys ) @ ( listrel1 @ A @ R2 ) ) ) ) ) ).
% Cons_listrel1E2
thf(fact_6102_listrel_OCons,axiom,
! [B: $tType,A: $tType,X: A,Y: B,R2: set @ ( product_prod @ A @ B ),Xs: list @ A,Ys: list @ B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ R2 )
=> ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs @ Ys ) @ ( listrel @ A @ B @ R2 ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( cons @ A @ X @ Xs ) @ ( cons @ B @ Y @ Ys ) ) @ ( listrel @ A @ B @ R2 ) ) ) ) ).
% listrel.Cons
thf(fact_6103_listrel__Cons1,axiom,
! [B: $tType,A: $tType,Y: A,Ys: list @ A,Xs: list @ B,R2: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( cons @ A @ Y @ Ys ) @ Xs ) @ ( listrel @ A @ B @ R2 ) )
=> ~ ! [Y3: B,Ys4: list @ B] :
( ( Xs
= ( cons @ B @ Y3 @ Ys4 ) )
=> ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Y @ Y3 ) @ R2 )
=> ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Ys @ Ys4 ) @ ( listrel @ A @ B @ R2 ) ) ) ) ) ).
% listrel_Cons1
thf(fact_6104_listrel__Cons2,axiom,
! [B: $tType,A: $tType,Xs: list @ A,Y: B,Ys: list @ B,R2: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs @ ( cons @ B @ Y @ Ys ) ) @ ( listrel @ A @ B @ R2 ) )
=> ~ ! [X5: A,Xs2: list @ A] :
( ( Xs
= ( cons @ A @ X5 @ Xs2 ) )
=> ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X5 @ Y ) @ R2 )
=> ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs2 @ Ys ) @ ( listrel @ A @ B @ R2 ) ) ) ) ) ).
% listrel_Cons2
thf(fact_6105_list_Oexhaust__sel,axiom,
! [A: $tType,List: list @ A] :
( ( List
!= ( nil @ A ) )
=> ( List
= ( cons @ A @ ( hd @ A @ List ) @ ( tl @ A @ List ) ) ) ) ).
% list.exhaust_sel
thf(fact_6106_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_6107_infinite__imp__bij__betw2,axiom,
! [A: $tType,A4: set @ A,A3: A] :
( ~ ( finite_finite2 @ A @ A4 )
=> ? [H5: A > A] : ( bij_betw @ A @ A @ H5 @ A4 @ ( sup_sup @ ( set @ A ) @ A4 @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% infinite_imp_bij_betw2
thf(fact_6108_infinite__imp__bij__betw,axiom,
! [A: $tType,A4: set @ A,A3: A] :
( ~ ( finite_finite2 @ A @ A4 )
=> ? [H5: A > A] : ( bij_betw @ A @ A @ H5 @ A4 @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% infinite_imp_bij_betw
thf(fact_6109_subseqs_Osimps_I1_J,axiom,
! [A: $tType] :
( ( subseqs @ A @ ( nil @ A ) )
= ( cons @ ( list @ A ) @ ( nil @ A ) @ ( nil @ ( list @ A ) ) ) ) ).
% subseqs.simps(1)
thf(fact_6110_ex__bij__betw__nat__finite,axiom,
! [A: $tType,M7: set @ A] :
( ( finite_finite2 @ A @ M7 )
=> ? [H5: nat > A] : ( bij_betw @ nat @ A @ H5 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( finite_card @ A @ M7 ) ) @ M7 ) ) ).
% ex_bij_betw_nat_finite
thf(fact_6111_ex__bij__betw__nat__finite__1,axiom,
! [A: $tType,M7: set @ A] :
( ( finite_finite2 @ A @ M7 )
=> ? [H5: nat > A] : ( bij_betw @ nat @ A @ H5 @ ( set_or1337092689740270186AtMost @ nat @ ( one_one @ nat ) @ ( finite_card @ A @ M7 ) ) @ M7 ) ) ).
% ex_bij_betw_nat_finite_1
thf(fact_6112_product__lists_Osimps_I1_J,axiom,
! [A: $tType] :
( ( product_lists @ A @ ( nil @ ( list @ A ) ) )
= ( cons @ ( list @ A ) @ ( nil @ A ) @ ( nil @ ( list @ A ) ) ) ) ).
% product_lists.simps(1)
thf(fact_6113_the__elem__set,axiom,
! [A: $tType,X: A] :
( ( the_elem @ A @ ( set2 @ A @ ( cons @ A @ X @ ( nil @ A ) ) ) )
= X ) ).
% the_elem_set
thf(fact_6114_rtrancl__listrel1__ConsI1,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,R2: set @ ( product_prod @ A @ A ),X: A] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( transitive_rtrancl @ ( list @ A ) @ ( listrel1 @ A @ R2 ) ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ X @ Ys ) ) @ ( transitive_rtrancl @ ( list @ A ) @ ( listrel1 @ A @ R2 ) ) ) ) ).
% rtrancl_listrel1_ConsI1
thf(fact_6115_lexordp_Omono,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( order_mono @ ( ( list @ A ) > ( list @ A ) > $o ) @ ( ( list @ A ) > ( list @ A ) > $o )
@ ^ [P6: ( list @ A ) > ( list @ A ) > $o,X15: list @ A,X24: list @ A] :
( ? [Y6: A,Ys3: list @ A] :
( ( X15
= ( nil @ A ) )
& ( X24
= ( cons @ A @ Y6 @ Ys3 ) ) )
| ? [X4: A,Y6: A,Xs3: list @ A,Ys3: list @ A] :
( ( X15
= ( cons @ A @ X4 @ Xs3 ) )
& ( X24
= ( cons @ A @ Y6 @ Ys3 ) )
& ( ord_less @ A @ X4 @ Y6 ) )
| ? [X4: A,Y6: A,Xs3: list @ A,Ys3: list @ A] :
( ( X15
= ( cons @ A @ X4 @ Xs3 ) )
& ( X24
= ( cons @ A @ Y6 @ Ys3 ) )
& ~ ( ord_less @ A @ X4 @ Y6 )
& ~ ( ord_less @ A @ Y6 @ X4 )
& ( P6 @ Xs3 @ Ys3 ) ) ) ) ) ).
% lexordp.mono
thf(fact_6116_sum_Oreindex__bij__betw__not__neutral,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comm_monoid_add @ A )
=> ! [S5: set @ B,T6: set @ C,H2: B > C,S2: set @ B,T4: set @ C,G: C > A] :
( ( finite_finite2 @ B @ S5 )
=> ( ( finite_finite2 @ C @ T6 )
=> ( ( bij_betw @ B @ C @ H2 @ ( minus_minus @ ( set @ B ) @ S2 @ S5 ) @ ( minus_minus @ ( set @ C ) @ T4 @ T6 ) )
=> ( ! [A6: B] :
( ( member @ B @ A6 @ S5 )
=> ( ( G @ ( H2 @ A6 ) )
= ( zero_zero @ A ) ) )
=> ( ! [B5: C] :
( ( member @ C @ B5 @ T6 )
=> ( ( G @ B5 )
= ( zero_zero @ A ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X4: B] : ( G @ ( H2 @ X4 ) )
@ S2 )
= ( groups7311177749621191930dd_sum @ C @ A @ G @ T4 ) ) ) ) ) ) ) ) ).
% sum.reindex_bij_betw_not_neutral
thf(fact_6117_prod_Oreindex__bij__betw__not__neutral,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S5: set @ B,T6: set @ C,H2: B > C,S2: set @ B,T4: set @ C,G: C > A] :
( ( finite_finite2 @ B @ S5 )
=> ( ( finite_finite2 @ C @ T6 )
=> ( ( bij_betw @ B @ C @ H2 @ ( minus_minus @ ( set @ B ) @ S2 @ S5 ) @ ( minus_minus @ ( set @ C ) @ T4 @ T6 ) )
=> ( ! [A6: B] :
( ( member @ B @ A6 @ S5 )
=> ( ( G @ ( H2 @ A6 ) )
= ( one_one @ A ) ) )
=> ( ! [B5: C] :
( ( member @ C @ B5 @ T6 )
=> ( ( G @ B5 )
= ( one_one @ A ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [X4: B] : ( G @ ( H2 @ X4 ) )
@ S2 )
= ( groups7121269368397514597t_prod @ C @ A @ G @ T4 ) ) ) ) ) ) ) ) ).
% prod.reindex_bij_betw_not_neutral
thf(fact_6118_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_6119_take__bit__not__mask__eq__0,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [M2: nat,N2: nat] :
( ( ord_less_eq @ nat @ M2 @ N2 )
=> ( ( bit_se2584673776208193580ke_bit @ A @ M2 @ ( bit_ri4277139882892585799ns_not @ A @ ( bit_se2239418461657761734s_mask @ A @ N2 ) ) )
= ( zero_zero @ A ) ) ) ) ).
% take_bit_not_mask_eq_0
thf(fact_6120_nth__Cons_H,axiom,
! [A: $tType,N2: nat,X: A,Xs: list @ A] :
( ( ( N2
= ( zero_zero @ nat ) )
=> ( ( nth @ A @ ( cons @ A @ X @ Xs ) @ N2 )
= X ) )
& ( ( N2
!= ( zero_zero @ nat ) )
=> ( ( nth @ A @ ( cons @ A @ X @ Xs ) @ N2 )
= ( nth @ A @ Xs @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) ) ) ) ) ).
% nth_Cons'
thf(fact_6121_push__bit__mask__eq,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [M2: nat,N2: nat] :
( ( bit_se4730199178511100633sh_bit @ A @ M2 @ ( bit_se2239418461657761734s_mask @ A @ N2 ) )
= ( bit_se5824344872417868541ns_and @ A @ ( bit_se2239418461657761734s_mask @ A @ ( plus_plus @ nat @ N2 @ M2 ) ) @ ( bit_ri4277139882892585799ns_not @ A @ ( bit_se2239418461657761734s_mask @ A @ M2 ) ) ) ) ) ).
% push_bit_mask_eq
thf(fact_6122_listrel_Osimps,axiom,
! [B: $tType,A: $tType,A1: list @ A,A22: list @ B,R2: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ A1 @ A22 ) @ ( listrel @ A @ B @ R2 ) )
= ( ( ( A1
= ( nil @ A ) )
& ( A22
= ( nil @ B ) ) )
| ? [X4: A,Y6: B,Xs3: list @ A,Ys3: list @ B] :
( ( A1
= ( cons @ A @ X4 @ Xs3 ) )
& ( A22
= ( cons @ B @ Y6 @ Ys3 ) )
& ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y6 ) @ R2 )
& ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs3 @ Ys3 ) @ ( listrel @ A @ B @ R2 ) ) ) ) ) ).
% listrel.simps
thf(fact_6123_listrel_Ocases,axiom,
! [B: $tType,A: $tType,A1: list @ A,A22: list @ B,R2: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ A1 @ A22 ) @ ( listrel @ A @ B @ R2 ) )
=> ( ( ( A1
= ( nil @ A ) )
=> ( A22
!= ( nil @ B ) ) )
=> ~ ! [X5: A,Y3: B,Xs2: list @ A] :
( ( A1
= ( cons @ A @ X5 @ Xs2 ) )
=> ! [Ys4: list @ B] :
( ( A22
= ( cons @ B @ Y3 @ Ys4 ) )
=> ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X5 @ Y3 ) @ R2 )
=> ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs2 @ Ys4 ) @ ( listrel @ A @ B @ R2 ) ) ) ) ) ) ) ).
% listrel.cases
thf(fact_6124_remdups__adj__replicate,axiom,
! [A: $tType,N2: nat,X: A] :
( ( ( N2
= ( zero_zero @ nat ) )
=> ( ( remdups_adj @ A @ ( replicate @ A @ N2 @ X ) )
= ( nil @ A ) ) )
& ( ( N2
!= ( zero_zero @ nat ) )
=> ( ( remdups_adj @ A @ ( replicate @ A @ N2 @ X ) )
= ( cons @ A @ X @ ( nil @ A ) ) ) ) ) ).
% remdups_adj_replicate
thf(fact_6125_remdups__adj__singleton,axiom,
! [A: $tType,Xs: list @ A,X: A] :
( ( ( remdups_adj @ A @ Xs )
= ( cons @ A @ X @ ( nil @ A ) ) )
=> ( Xs
= ( replicate @ A @ ( size_size @ ( list @ A ) @ Xs ) @ X ) ) ) ).
% remdups_adj_singleton
thf(fact_6126_Cons__in__shuffles__iff,axiom,
! [A: $tType,Z3: A,Zs: list @ A,Xs: list @ A,Ys: list @ A] :
( ( member @ ( list @ A ) @ ( cons @ A @ Z3 @ Zs ) @ ( shuffles @ A @ Xs @ Ys ) )
= ( ( ( Xs
!= ( nil @ A ) )
& ( ( hd @ A @ Xs )
= Z3 )
& ( member @ ( list @ A ) @ Zs @ ( shuffles @ A @ ( tl @ A @ Xs ) @ Ys ) ) )
| ( ( Ys
!= ( nil @ A ) )
& ( ( hd @ A @ Ys )
= Z3 )
& ( member @ ( list @ A ) @ Zs @ ( shuffles @ A @ Xs @ ( tl @ A @ Ys ) ) ) ) ) ) ).
% Cons_in_shuffles_iff
thf(fact_6127_bij__betw__nth,axiom,
! [A: $tType,Xs: list @ A,A4: set @ nat,B2: set @ A] :
( ( distinct @ A @ Xs )
=> ( ( A4
= ( set_ord_lessThan @ nat @ ( size_size @ ( list @ A ) @ Xs ) ) )
=> ( ( B2
= ( set2 @ A @ Xs ) )
=> ( bij_betw @ nat @ A @ ( nth @ A @ Xs ) @ A4 @ B2 ) ) ) ) ).
% bij_betw_nth
thf(fact_6128_list_Osize__gen_I2_J,axiom,
! [A: $tType,X: A > nat,X21: A,X222: list @ A] :
( ( size_list @ A @ X @ ( cons @ A @ X21 @ X222 ) )
= ( plus_plus @ nat @ ( plus_plus @ nat @ ( X @ X21 ) @ ( size_list @ A @ X @ X222 ) ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% list.size_gen(2)
thf(fact_6129_ex__bij__betw__strict__mono__card,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [M7: set @ A] :
( ( finite_finite2 @ A @ M7 )
=> ~ ! [H5: nat > A] :
( ( bij_betw @ nat @ A @ H5 @ ( set_ord_lessThan @ nat @ ( finite_card @ A @ M7 ) ) @ M7 )
=> ~ ( strict_mono_on @ nat @ A @ H5 @ ( set_ord_lessThan @ nat @ ( finite_card @ A @ M7 ) ) ) ) ) ) ).
% ex_bij_betw_strict_mono_card
thf(fact_6130_sum_OatLeastAtMost__reindex,axiom,
! [B: $tType,A: $tType] :
( ( ( comm_monoid_add @ A )
& ( ord @ B ) )
=> ! [H2: nat > B,M2: nat,N2: nat,G: B > A] :
( ( bij_betw @ nat @ B @ H2 @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N2 ) @ ( set_or1337092689740270186AtMost @ B @ ( H2 @ M2 ) @ ( H2 @ N2 ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( set_or1337092689740270186AtMost @ B @ ( H2 @ M2 ) @ ( H2 @ N2 ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ B @ A @ nat @ G @ H2 ) @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N2 ) ) ) ) ) ).
% sum.atLeastAtMost_reindex
thf(fact_6131_n__lists_Osimps_I1_J,axiom,
! [A: $tType,Xs: list @ A] :
( ( n_lists @ A @ ( zero_zero @ nat ) @ Xs )
= ( cons @ ( list @ A ) @ ( nil @ A ) @ ( nil @ ( list @ A ) ) ) ) ).
% n_lists.simps(1)
thf(fact_6132_sum_OatLeastLessThan__reindex,axiom,
! [B: $tType,A: $tType] :
( ( ( comm_monoid_add @ A )
& ( ord @ B ) )
=> ! [H2: nat > B,M2: nat,N2: nat,G: B > A] :
( ( bij_betw @ nat @ B @ H2 @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N2 ) @ ( set_or7035219750837199246ssThan @ B @ ( H2 @ M2 ) @ ( H2 @ N2 ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( set_or7035219750837199246ssThan @ B @ ( H2 @ M2 ) @ ( H2 @ N2 ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ B @ A @ nat @ G @ H2 ) @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N2 ) ) ) ) ) ).
% sum.atLeastLessThan_reindex
thf(fact_6133_prod_OatLeastAtMost__reindex,axiom,
! [B: $tType,A: $tType] :
( ( ( comm_monoid_mult @ A )
& ( ord @ B ) )
=> ! [H2: nat > B,M2: nat,N2: nat,G: B > A] :
( ( bij_betw @ nat @ B @ H2 @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N2 ) @ ( set_or1337092689740270186AtMost @ B @ ( H2 @ M2 ) @ ( H2 @ N2 ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( set_or1337092689740270186AtMost @ B @ ( H2 @ M2 ) @ ( H2 @ N2 ) ) )
= ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ B @ A @ nat @ G @ H2 ) @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N2 ) ) ) ) ) ).
% prod.atLeastAtMost_reindex
thf(fact_6134_prod_OatLeastLessThan__reindex,axiom,
! [B: $tType,A: $tType] :
( ( ( comm_monoid_mult @ A )
& ( ord @ B ) )
=> ! [H2: nat > B,M2: nat,N2: nat,G: B > A] :
( ( bij_betw @ nat @ B @ H2 @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N2 ) @ ( set_or7035219750837199246ssThan @ B @ ( H2 @ M2 ) @ ( H2 @ N2 ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( set_or7035219750837199246ssThan @ B @ ( H2 @ M2 ) @ ( H2 @ N2 ) ) )
= ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ B @ A @ nat @ G @ H2 ) @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N2 ) ) ) ) ) ).
% prod.atLeastLessThan_reindex
thf(fact_6135_sorted__list__of__set__greaterThanAtMost,axiom,
! [I: nat,J: nat] :
( ( ord_less_eq @ nat @ ( suc @ I ) @ J )
=> ( ( linord4507533701916653071of_set @ nat @ ( set_or3652927894154168847AtMost @ nat @ I @ J ) )
= ( cons @ nat @ ( suc @ I ) @ ( linord4507533701916653071of_set @ nat @ ( set_or3652927894154168847AtMost @ nat @ ( suc @ I ) @ J ) ) ) ) ) ).
% sorted_list_of_set_greaterThanAtMost
thf(fact_6136_shuffles_Oelims,axiom,
! [A: $tType,X: list @ A,Xa2: list @ A,Y: set @ ( list @ A )] :
( ( ( shuffles @ A @ X @ Xa2 )
= Y )
=> ( ( ( X
= ( nil @ A ) )
=> ( Y
!= ( insert2 @ ( list @ A ) @ Xa2 @ ( bot_bot @ ( set @ ( list @ A ) ) ) ) ) )
=> ( ( ( Xa2
= ( nil @ A ) )
=> ( Y
!= ( insert2 @ ( list @ A ) @ X @ ( bot_bot @ ( set @ ( list @ A ) ) ) ) ) )
=> ~ ! [X5: A,Xs2: list @ A] :
( ( X
= ( cons @ A @ X5 @ Xs2 ) )
=> ! [Y3: A,Ys4: list @ A] :
( ( Xa2
= ( cons @ A @ Y3 @ Ys4 ) )
=> ( Y
!= ( sup_sup @ ( set @ ( list @ A ) ) @ ( image2 @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X5 ) @ ( shuffles @ A @ Xs2 @ ( cons @ A @ Y3 @ Ys4 ) ) ) @ ( image2 @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ Y3 ) @ ( shuffles @ A @ ( cons @ A @ X5 @ Xs2 ) @ Ys4 ) ) ) ) ) ) ) ) ) ).
% shuffles.elims
thf(fact_6137_sorted__list__of__set__greaterThanLessThan,axiom,
! [I: nat,J: nat] :
( ( ord_less @ nat @ ( suc @ I ) @ J )
=> ( ( linord4507533701916653071of_set @ nat @ ( set_or5935395276787703475ssThan @ nat @ I @ J ) )
= ( cons @ nat @ ( suc @ I ) @ ( linord4507533701916653071of_set @ nat @ ( set_or5935395276787703475ssThan @ nat @ ( suc @ I ) @ J ) ) ) ) ) ).
% sorted_list_of_set_greaterThanLessThan
thf(fact_6138_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_6139_nth__equal__first__eq,axiom,
! [A: $tType,X: A,Xs: list @ A,N2: nat] :
( ~ ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ( ord_less_eq @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ( nth @ A @ ( cons @ A @ X @ Xs ) @ N2 )
= X )
= ( N2
= ( zero_zero @ nat ) ) ) ) ) ).
% nth_equal_first_eq
thf(fact_6140_nth__non__equal__first__eq,axiom,
! [A: $tType,X: A,Y: A,Xs: list @ A,N2: nat] :
( ( X != Y )
=> ( ( ( nth @ A @ ( cons @ A @ X @ Xs ) @ N2 )
= Y )
= ( ( ( nth @ A @ Xs @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) )
= Y )
& ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ) ).
% nth_non_equal_first_eq
thf(fact_6141_take__Cons_H,axiom,
! [A: $tType,N2: nat,X: A,Xs: list @ A] :
( ( ( N2
= ( zero_zero @ nat ) )
=> ( ( take @ A @ N2 @ ( cons @ A @ X @ Xs ) )
= ( nil @ A ) ) )
& ( ( N2
!= ( zero_zero @ nat ) )
=> ( ( take @ A @ N2 @ ( cons @ A @ X @ Xs ) )
= ( cons @ A @ X @ ( take @ A @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) @ Xs ) ) ) ) ) ).
% take_Cons'
thf(fact_6142_Cons__replicate__eq,axiom,
! [A: $tType,X: A,Xs: list @ A,N2: nat,Y: A] :
( ( ( cons @ A @ X @ Xs )
= ( replicate @ A @ N2 @ Y ) )
= ( ( X = Y )
& ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
& ( Xs
= ( replicate @ A @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) @ X ) ) ) ) ).
% Cons_replicate_eq
thf(fact_6143_Cons__lenlex__iff,axiom,
! [A: $tType,M2: A,Ms: list @ A,N2: A,Ns: list @ A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ M2 @ Ms ) @ ( cons @ A @ N2 @ Ns ) ) @ ( lenlex @ A @ R2 ) )
= ( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Ms ) @ ( size_size @ ( list @ A ) @ Ns ) )
| ( ( ( size_size @ ( list @ A ) @ Ms )
= ( size_size @ ( list @ A ) @ Ns ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ M2 @ N2 ) @ R2 ) )
| ( ( M2 = N2 )
& ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ms @ Ns ) @ ( lenlex @ A @ R2 ) ) ) ) ) ).
% Cons_lenlex_iff
thf(fact_6144_take__Suc,axiom,
! [A: $tType,Xs: list @ A,N2: nat] :
( ( Xs
!= ( nil @ A ) )
=> ( ( take @ A @ ( suc @ N2 ) @ Xs )
= ( cons @ A @ ( hd @ A @ Xs ) @ ( take @ A @ N2 @ ( tl @ A @ Xs ) ) ) ) ) ).
% take_Suc
thf(fact_6145_rtrancl__listrel1__ConsI2,axiom,
! [A: $tType,X: A,Y: A,R2: set @ ( product_prod @ A @ A ),Xs: list @ A,Ys: list @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( transitive_rtrancl @ A @ R2 ) )
=> ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( transitive_rtrancl @ ( list @ A ) @ ( listrel1 @ A @ R2 ) ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y @ Ys ) ) @ ( transitive_rtrancl @ ( list @ A ) @ ( listrel1 @ A @ R2 ) ) ) ) ) ).
% rtrancl_listrel1_ConsI2
thf(fact_6146_bit__not__iff__eq,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [A3: A,N2: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( bit_ri4277139882892585799ns_not @ A @ A3 ) @ N2 )
= ( ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 )
!= ( zero_zero @ A ) )
& ~ ( bit_se5641148757651400278ts_bit @ A @ A3 @ N2 ) ) ) ) ).
% bit_not_iff_eq
thf(fact_6147_Pow__set_I2_J,axiom,
! [B: $tType,X: B,Xs: list @ B] :
( ( pow2 @ B @ ( set2 @ B @ ( cons @ B @ X @ Xs ) ) )
= ( sup_sup @ ( set @ ( set @ B ) ) @ ( pow2 @ B @ ( set2 @ B @ Xs ) ) @ ( image2 @ ( set @ B ) @ ( set @ B ) @ ( insert2 @ B @ X ) @ ( pow2 @ B @ ( set2 @ B @ Xs ) ) ) ) ) ).
% Pow_set(2)
thf(fact_6148_set__Cons__sing__Nil,axiom,
! [A: $tType,A4: set @ A] :
( ( set_Cons @ A @ A4 @ ( insert2 @ ( list @ A ) @ ( nil @ A ) @ ( bot_bot @ ( set @ ( list @ A ) ) ) ) )
= ( image2 @ A @ ( list @ A )
@ ^ [X4: A] : ( cons @ A @ X4 @ ( nil @ A ) )
@ A4 ) ) ).
% set_Cons_sing_Nil
thf(fact_6149_Schroeder__Bernstein,axiom,
! [A: $tType,B: $tType,F2: A > B,A4: set @ A,B2: set @ B,G: B > A] :
( ( inj_on @ A @ B @ F2 @ A4 )
=> ( ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F2 @ A4 ) @ B2 )
=> ( ( inj_on @ B @ A @ G @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image2 @ B @ A @ G @ B2 ) @ A4 )
=> ? [H5: A > B] : ( bij_betw @ A @ B @ H5 @ A4 @ B2 ) ) ) ) ) ).
% Schroeder_Bernstein
thf(fact_6150_bij__betw__Suc,axiom,
! [M7: set @ nat,N6: set @ nat] :
( ( bij_betw @ nat @ nat @ suc @ M7 @ N6 )
= ( ( image2 @ nat @ nat @ suc @ M7 )
= N6 ) ) ).
% bij_betw_Suc
thf(fact_6151_le__rel__bool__arg__iff,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( ord_less_eq @ ( $o > A ) )
= ( ^ [X6: $o > A,Y8: $o > A] :
( ( ord_less_eq @ A @ ( X6 @ $false ) @ ( Y8 @ $false ) )
& ( ord_less_eq @ A @ ( X6 @ $true ) @ ( Y8 @ $true ) ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_6152_ex__bij__betw__finite__nat,axiom,
! [A: $tType,M7: set @ A] :
( ( finite_finite2 @ A @ M7 )
=> ? [H5: A > nat] : ( bij_betw @ A @ nat @ H5 @ M7 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( finite_card @ A @ M7 ) ) ) ) ).
% ex_bij_betw_finite_nat
thf(fact_6153_listset_Osimps_I2_J,axiom,
! [A: $tType,A4: set @ A,As3: list @ ( set @ A )] :
( ( listset @ A @ ( cons @ ( set @ A ) @ A4 @ As3 ) )
= ( set_Cons @ A @ A4 @ ( listset @ A @ As3 ) ) ) ).
% listset.simps(2)
thf(fact_6154_set__Cons__def,axiom,
! [A: $tType] :
( ( set_Cons @ A )
= ( ^ [A7: set @ A,XS: set @ ( list @ A )] :
( collect @ ( list @ A )
@ ^ [Z6: list @ A] :
? [X4: A,Xs3: list @ A] :
( ( Z6
= ( cons @ A @ X4 @ Xs3 ) )
& ( member @ A @ X4 @ A7 )
& ( member @ ( list @ A ) @ Xs3 @ XS ) ) ) ) ) ).
% set_Cons_def
thf(fact_6155_bij__betw__nth__root__unity,axiom,
! [C2: complex,N2: nat] :
( ( C2
!= ( zero_zero @ complex ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( bij_betw @ complex @ complex @ ( times_times @ complex @ ( times_times @ complex @ ( real_Vector_of_real @ complex @ ( root @ N2 @ ( real_V7770717601297561774m_norm @ complex @ C2 ) ) ) @ ( cis @ ( divide_divide @ real @ ( arg @ C2 ) @ ( semiring_1_of_nat @ real @ N2 ) ) ) ) )
@ ( collect @ complex
@ ^ [Z6: complex] :
( ( power_power @ complex @ Z6 @ N2 )
= ( one_one @ complex ) ) )
@ ( collect @ complex
@ ^ [Z6: complex] :
( ( power_power @ complex @ Z6 @ N2 )
= C2 ) ) ) ) ) ).
% bij_betw_nth_root_unity
thf(fact_6156_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_6157_concat__inth,axiom,
! [A: $tType,Xs: list @ A,X: A,Ys: list @ A] :
( ( nth @ A @ ( append @ A @ Xs @ ( append @ A @ ( cons @ A @ X @ ( nil @ A ) ) @ Ys ) ) @ ( size_size @ ( list @ A ) @ Xs ) )
= X ) ).
% concat_inth
thf(fact_6158_map__upds__append1,axiom,
! [B: $tType,A: $tType,Xs: list @ A,Ys: list @ B,M2: A > ( option @ B ),X: A] :
( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( map_upds @ A @ B @ M2 @ ( append @ A @ Xs @ ( cons @ A @ X @ ( nil @ A ) ) ) @ Ys )
= ( fun_upd @ A @ ( option @ B ) @ ( map_upds @ A @ B @ M2 @ Xs @ Ys ) @ X @ ( some @ B @ ( nth @ B @ Ys @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ) ) ).
% map_upds_append1
thf(fact_6159_same__append__eq,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,Zs: list @ A] :
( ( ( append @ A @ Xs @ Ys )
= ( append @ A @ Xs @ Zs ) )
= ( Ys = Zs ) ) ).
% same_append_eq
thf(fact_6160_append__same__eq,axiom,
! [A: $tType,Ys: list @ A,Xs: list @ A,Zs: list @ A] :
( ( ( append @ A @ Ys @ Xs )
= ( append @ A @ Zs @ Xs ) )
= ( Ys = Zs ) ) ).
% append_same_eq
thf(fact_6161_append__assoc,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,Zs: list @ A] :
( ( append @ A @ ( append @ A @ Xs @ Ys ) @ Zs )
= ( append @ A @ Xs @ ( append @ A @ Ys @ Zs ) ) ) ).
% append_assoc
thf(fact_6162_append_Oassoc,axiom,
! [A: $tType,A3: list @ A,B3: list @ A,C2: list @ A] :
( ( append @ A @ ( append @ A @ A3 @ B3 ) @ C2 )
= ( append @ A @ A3 @ ( append @ A @ B3 @ C2 ) ) ) ).
% append.assoc
thf(fact_6163_append_Oright__neutral,axiom,
! [A: $tType,A3: list @ A] :
( ( append @ A @ A3 @ ( nil @ A ) )
= A3 ) ).
% append.right_neutral
thf(fact_6164_append__Nil2,axiom,
! [A: $tType,Xs: list @ A] :
( ( append @ A @ Xs @ ( nil @ A ) )
= Xs ) ).
% append_Nil2
thf(fact_6165_append__self__conv,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( ( append @ A @ Xs @ Ys )
= Xs )
= ( Ys
= ( nil @ A ) ) ) ).
% append_self_conv
thf(fact_6166_self__append__conv,axiom,
! [A: $tType,Y: list @ A,Ys: list @ A] :
( ( Y
= ( append @ A @ Y @ Ys ) )
= ( Ys
= ( nil @ A ) ) ) ).
% self_append_conv
thf(fact_6167_append__self__conv2,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( ( append @ A @ Xs @ Ys )
= Ys )
= ( Xs
= ( nil @ A ) ) ) ).
% append_self_conv2
thf(fact_6168_self__append__conv2,axiom,
! [A: $tType,Y: list @ A,Xs: list @ A] :
( ( Y
= ( append @ A @ Xs @ Y ) )
= ( Xs
= ( nil @ A ) ) ) ).
% self_append_conv2
thf(fact_6169_Nil__is__append__conv,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( ( nil @ A )
= ( append @ A @ Xs @ Ys ) )
= ( ( Xs
= ( nil @ A ) )
& ( Ys
= ( nil @ A ) ) ) ) ).
% Nil_is_append_conv
thf(fact_6170_append__is__Nil__conv,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( ( append @ A @ Xs @ Ys )
= ( nil @ A ) )
= ( ( Xs
= ( nil @ A ) )
& ( Ys
= ( nil @ A ) ) ) ) ).
% append_is_Nil_conv
thf(fact_6171_append__eq__append__conv,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,Us: list @ A,Vs: list @ A] :
( ( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ A ) @ Ys ) )
| ( ( size_size @ ( list @ A ) @ Us )
= ( size_size @ ( list @ A ) @ Vs ) ) )
=> ( ( ( append @ A @ Xs @ Us )
= ( append @ A @ Ys @ Vs ) )
= ( ( Xs = Ys )
& ( Us = Vs ) ) ) ) ).
% append_eq_append_conv
thf(fact_6172_concat__append,axiom,
! [A: $tType,Xs: list @ ( list @ A ),Ys: list @ ( list @ A )] :
( ( concat @ A @ ( append @ ( list @ A ) @ Xs @ Ys ) )
= ( append @ A @ ( concat @ A @ Xs ) @ ( concat @ A @ Ys ) ) ) ).
% concat_append
thf(fact_6173_removeAll__append,axiom,
! [A: $tType,X: A,Xs: list @ A,Ys: list @ A] :
( ( removeAll @ A @ X @ ( append @ A @ Xs @ Ys ) )
= ( append @ A @ ( removeAll @ A @ X @ Xs ) @ ( removeAll @ A @ X @ Ys ) ) ) ).
% removeAll_append
thf(fact_6174_append1__eq__conv,axiom,
! [A: $tType,Xs: list @ A,X: A,Ys: list @ A,Y: A] :
( ( ( append @ A @ Xs @ ( cons @ A @ X @ ( nil @ A ) ) )
= ( append @ A @ Ys @ ( cons @ A @ Y @ ( nil @ A ) ) ) )
= ( ( Xs = Ys )
& ( X = Y ) ) ) ).
% append1_eq_conv
thf(fact_6175_length__append,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( size_size @ ( list @ A ) @ ( append @ A @ Xs @ Ys ) )
= ( plus_plus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ A ) @ Ys ) ) ) ).
% length_append
thf(fact_6176_set__append,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( set2 @ A @ ( append @ A @ Xs @ Ys ) )
= ( sup_sup @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ Ys ) ) ) ).
% set_append
thf(fact_6177_zip__append,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Us: list @ B,Ys: list @ A,Vs: list @ B] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Us ) )
=> ( ( zip @ A @ B @ ( append @ A @ Xs @ Ys ) @ ( append @ B @ Us @ Vs ) )
= ( append @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Us ) @ ( zip @ A @ B @ Ys @ Vs ) ) ) ) ).
% zip_append
thf(fact_6178_hd__append2,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( hd @ A @ ( append @ A @ Xs @ Ys ) )
= ( hd @ A @ Xs ) ) ) ).
% hd_append2
thf(fact_6179_tl__append2,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( tl @ A @ ( append @ A @ Xs @ Ys ) )
= ( append @ A @ ( tl @ A @ Xs ) @ Ys ) ) ) ).
% tl_append2
thf(fact_6180_fun__upds__append2__drop,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B,M2: A > ( option @ B ),Zs: list @ B] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( map_upds @ A @ B @ M2 @ Xs @ ( append @ B @ Ys @ Zs ) )
= ( map_upds @ A @ B @ M2 @ Xs @ Ys ) ) ) ).
% fun_upds_append2_drop
thf(fact_6181_fun__upds__append__drop,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B,M2: A > ( option @ B ),Zs: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( map_upds @ A @ B @ M2 @ ( append @ A @ Xs @ Zs ) @ Ys )
= ( map_upds @ A @ B @ M2 @ Xs @ Ys ) ) ) ).
% fun_upds_append_drop
thf(fact_6182_size__list__append,axiom,
! [A: $tType,F2: A > nat,Xs: list @ A,Ys: list @ A] :
( ( size_list @ A @ F2 @ ( append @ A @ Xs @ Ys ) )
= ( plus_plus @ nat @ ( size_list @ A @ F2 @ Xs ) @ ( size_list @ A @ F2 @ Ys ) ) ) ).
% size_list_append
thf(fact_6183_bind__simps_I2_J,axiom,
! [A: $tType,B: $tType,X: B,Xs: list @ B,F2: B > ( list @ A )] :
( ( bind @ B @ A @ ( cons @ B @ X @ Xs ) @ F2 )
= ( append @ A @ ( F2 @ X ) @ ( bind @ B @ A @ Xs @ F2 ) ) ) ).
% bind_simps(2)
thf(fact_6184_nth__append__length,axiom,
! [A: $tType,Xs: list @ A,X: A,Ys: list @ A] :
( ( nth @ A @ ( append @ A @ Xs @ ( cons @ A @ X @ Ys ) ) @ ( size_size @ ( list @ A ) @ Xs ) )
= X ) ).
% nth_append_length
thf(fact_6185_nth__append__length__plus,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,N2: nat] :
( ( nth @ A @ ( append @ A @ Xs @ Ys ) @ ( plus_plus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ N2 ) )
= ( nth @ A @ Ys @ N2 ) ) ).
% nth_append_length_plus
thf(fact_6186_take__append,axiom,
! [A: $tType,N2: nat,Xs: list @ A,Ys: list @ A] :
( ( take @ A @ N2 @ ( append @ A @ Xs @ Ys ) )
= ( append @ A @ ( take @ A @ N2 @ Xs ) @ ( take @ A @ ( minus_minus @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs ) ) @ Ys ) ) ) ).
% take_append
thf(fact_6187_list__update__length,axiom,
! [A: $tType,Xs: list @ A,X: A,Ys: list @ A,Y: A] :
( ( list_update @ A @ ( append @ A @ Xs @ ( cons @ A @ X @ Ys ) ) @ ( size_size @ ( list @ A ) @ Xs ) @ Y )
= ( append @ A @ Xs @ ( cons @ A @ Y @ Ys ) ) ) ).
% list_update_length
thf(fact_6188_sorted__list__of__set__lessThan__Suc,axiom,
! [K: nat] :
( ( linord4507533701916653071of_set @ nat @ ( set_ord_lessThan @ nat @ ( suc @ K ) ) )
= ( append @ nat @ ( linord4507533701916653071of_set @ nat @ ( set_ord_lessThan @ nat @ K ) ) @ ( cons @ nat @ K @ ( nil @ nat ) ) ) ) ).
% sorted_list_of_set_lessThan_Suc
thf(fact_6189_sorted__list__of__set__atMost__Suc,axiom,
! [K: nat] :
( ( linord4507533701916653071of_set @ nat @ ( set_ord_atMost @ nat @ ( suc @ K ) ) )
= ( append @ nat @ ( linord4507533701916653071of_set @ nat @ ( set_ord_atMost @ nat @ K ) ) @ ( cons @ nat @ ( suc @ K ) @ ( nil @ nat ) ) ) ) ).
% sorted_list_of_set_atMost_Suc
thf(fact_6190_distinct__append,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( distinct @ A @ ( append @ A @ Xs @ Ys ) )
= ( ( distinct @ A @ Xs )
& ( distinct @ A @ Ys )
& ( ( inf_inf @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ Ys ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% distinct_append
thf(fact_6191_replicate__app__Cons__same,axiom,
! [A: $tType,N2: nat,X: A,Xs: list @ A] :
( ( append @ A @ ( replicate @ A @ N2 @ X ) @ ( cons @ A @ X @ Xs ) )
= ( cons @ A @ X @ ( append @ A @ ( replicate @ A @ N2 @ X ) @ Xs ) ) ) ).
% replicate_app_Cons_same
thf(fact_6192_split__list__first__prop__iff,axiom,
! [A: $tType,Xs: list @ A,P: A > $o] :
( ( ? [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
& ( P @ X4 ) ) )
= ( ? [Ys3: list @ A,X4: A] :
( ? [Zs3: list @ A] :
( Xs
= ( append @ A @ Ys3 @ ( cons @ A @ X4 @ Zs3 ) ) )
& ( P @ X4 )
& ! [Y6: A] :
( ( member @ A @ Y6 @ ( set2 @ A @ Ys3 ) )
=> ~ ( P @ Y6 ) ) ) ) ) ).
% split_list_first_prop_iff
thf(fact_6193_split__list__last__prop__iff,axiom,
! [A: $tType,Xs: list @ A,P: A > $o] :
( ( ? [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
& ( P @ X4 ) ) )
= ( ? [Ys3: list @ A,X4: A,Zs3: list @ A] :
( ( Xs
= ( append @ A @ Ys3 @ ( cons @ A @ X4 @ Zs3 ) ) )
& ( P @ X4 )
& ! [Y6: A] :
( ( member @ A @ Y6 @ ( set2 @ A @ Zs3 ) )
=> ~ ( P @ Y6 ) ) ) ) ) ).
% split_list_last_prop_iff
thf(fact_6194_in__set__conv__decomp__first,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
= ( ? [Ys3: list @ A,Zs3: list @ A] :
( ( Xs
= ( append @ A @ Ys3 @ ( cons @ A @ X @ Zs3 ) ) )
& ~ ( member @ A @ X @ ( set2 @ A @ Ys3 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_6195_in__set__conv__decomp__last,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
= ( ? [Ys3: list @ A,Zs3: list @ A] :
( ( Xs
= ( append @ A @ Ys3 @ ( cons @ A @ X @ Zs3 ) ) )
& ~ ( member @ A @ X @ ( set2 @ A @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_6196_split__list__first__propE,axiom,
! [A: $tType,Xs: list @ A,P: A > $o] :
( ? [X2: A] :
( ( member @ A @ X2 @ ( set2 @ A @ Xs ) )
& ( P @ X2 ) )
=> ~ ! [Ys4: list @ A,X5: A] :
( ? [Zs2: list @ A] :
( Xs
= ( append @ A @ Ys4 @ ( cons @ A @ X5 @ Zs2 ) ) )
=> ( ( P @ X5 )
=> ~ ! [Xa: A] :
( ( member @ A @ Xa @ ( set2 @ A @ Ys4 ) )
=> ~ ( P @ Xa ) ) ) ) ) ).
% split_list_first_propE
thf(fact_6197_split__list__last__propE,axiom,
! [A: $tType,Xs: list @ A,P: A > $o] :
( ? [X2: A] :
( ( member @ A @ X2 @ ( set2 @ A @ Xs ) )
& ( P @ X2 ) )
=> ~ ! [Ys4: list @ A,X5: A,Zs2: list @ A] :
( ( Xs
= ( append @ A @ Ys4 @ ( cons @ A @ X5 @ Zs2 ) ) )
=> ( ( P @ X5 )
=> ~ ! [Xa: A] :
( ( member @ A @ Xa @ ( set2 @ A @ Zs2 ) )
=> ~ ( P @ Xa ) ) ) ) ) ).
% split_list_last_propE
thf(fact_6198_split__list__first__prop,axiom,
! [A: $tType,Xs: list @ A,P: A > $o] :
( ? [X2: A] :
( ( member @ A @ X2 @ ( set2 @ A @ Xs ) )
& ( P @ X2 ) )
=> ? [Ys4: list @ A,X5: A] :
( ? [Zs2: list @ A] :
( Xs
= ( append @ A @ Ys4 @ ( cons @ A @ X5 @ Zs2 ) ) )
& ( P @ X5 )
& ! [Xa: A] :
( ( member @ A @ Xa @ ( set2 @ A @ Ys4 ) )
=> ~ ( P @ Xa ) ) ) ) ).
% split_list_first_prop
thf(fact_6199_split__list__last__prop,axiom,
! [A: $tType,Xs: list @ A,P: A > $o] :
( ? [X2: A] :
( ( member @ A @ X2 @ ( set2 @ A @ Xs ) )
& ( P @ X2 ) )
=> ? [Ys4: list @ A,X5: A,Zs2: list @ A] :
( ( Xs
= ( append @ A @ Ys4 @ ( cons @ A @ X5 @ Zs2 ) ) )
& ( P @ X5 )
& ! [Xa: A] :
( ( member @ A @ Xa @ ( set2 @ A @ Zs2 ) )
=> ~ ( P @ Xa ) ) ) ) ).
% split_list_last_prop
thf(fact_6200_in__set__conv__decomp,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
= ( ? [Ys3: list @ A,Zs3: list @ A] :
( Xs
= ( append @ A @ Ys3 @ ( cons @ A @ X @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_6201_append__Cons__eq__iff,axiom,
! [A: $tType,X: A,Xs: list @ A,Ys: list @ A,Xs5: list @ A,Ys6: list @ A] :
( ~ ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ~ ( member @ A @ X @ ( set2 @ A @ Ys ) )
=> ( ( ( append @ A @ Xs @ ( cons @ A @ X @ Ys ) )
= ( append @ A @ Xs5 @ ( cons @ A @ X @ Ys6 ) ) )
= ( ( Xs = Xs5 )
& ( Ys = Ys6 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_6202_split__list__propE,axiom,
! [A: $tType,Xs: list @ A,P: A > $o] :
( ? [X2: A] :
( ( member @ A @ X2 @ ( set2 @ A @ Xs ) )
& ( P @ X2 ) )
=> ~ ! [Ys4: list @ A,X5: A] :
( ? [Zs2: list @ A] :
( Xs
= ( append @ A @ Ys4 @ ( cons @ A @ X5 @ Zs2 ) ) )
=> ~ ( P @ X5 ) ) ) ).
% split_list_propE
thf(fact_6203_split__list__first,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ? [Ys4: list @ A,Zs2: list @ A] :
( ( Xs
= ( append @ A @ Ys4 @ ( cons @ A @ X @ Zs2 ) ) )
& ~ ( member @ A @ X @ ( set2 @ A @ Ys4 ) ) ) ) ).
% split_list_first
thf(fact_6204_split__list__prop,axiom,
! [A: $tType,Xs: list @ A,P: A > $o] :
( ? [X2: A] :
( ( member @ A @ X2 @ ( set2 @ A @ Xs ) )
& ( P @ X2 ) )
=> ? [Ys4: list @ A,X5: A] :
( ? [Zs2: list @ A] :
( Xs
= ( append @ A @ Ys4 @ ( cons @ A @ X5 @ Zs2 ) ) )
& ( P @ X5 ) ) ) ).
% split_list_prop
thf(fact_6205_split__list__last,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ? [Ys4: list @ A,Zs2: list @ A] :
( ( Xs
= ( append @ A @ Ys4 @ ( cons @ A @ X @ Zs2 ) ) )
& ~ ( member @ A @ X @ ( set2 @ A @ Zs2 ) ) ) ) ).
% split_list_last
thf(fact_6206_split__list,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ? [Ys4: list @ A,Zs2: list @ A] :
( Xs
= ( append @ A @ Ys4 @ ( cons @ A @ X @ Zs2 ) ) ) ) ).
% split_list
thf(fact_6207_concat__eq__appendD,axiom,
! [A: $tType,Xss: list @ ( list @ A ),Ys: list @ A,Zs: list @ A] :
( ( ( concat @ A @ Xss )
= ( append @ A @ Ys @ Zs ) )
=> ( ( Xss
!= ( nil @ ( list @ A ) ) )
=> ? [Xss1: list @ ( list @ A ),Xs2: list @ A,Xs4: list @ A,Xss22: list @ ( list @ A )] :
( ( Xss
= ( append @ ( list @ A ) @ Xss1 @ ( cons @ ( list @ A ) @ ( append @ A @ Xs2 @ Xs4 ) @ Xss22 ) ) )
& ( Ys
= ( append @ A @ ( concat @ A @ Xss1 ) @ Xs2 ) )
& ( Zs
= ( append @ A @ Xs4 @ ( concat @ A @ Xss22 ) ) ) ) ) ) ).
% concat_eq_appendD
thf(fact_6208_concat__eq__append__conv,axiom,
! [A: $tType,Xss: list @ ( list @ A ),Ys: list @ A,Zs: list @ A] :
( ( ( concat @ A @ Xss )
= ( append @ A @ Ys @ Zs ) )
= ( ( ( Xss
= ( nil @ ( list @ A ) ) )
=> ( ( Ys
= ( nil @ A ) )
& ( Zs
= ( nil @ A ) ) ) )
& ( ( Xss
!= ( nil @ ( list @ A ) ) )
=> ? [Xss12: list @ ( list @ A ),Xs3: list @ A,Xs6: list @ A,Xss23: list @ ( list @ A )] :
( ( Xss
= ( append @ ( list @ A ) @ Xss12 @ ( cons @ ( list @ A ) @ ( append @ A @ Xs3 @ Xs6 ) @ Xss23 ) ) )
& ( Ys
= ( append @ A @ ( concat @ A @ Xss12 ) @ Xs3 ) )
& ( Zs
= ( append @ A @ Xs6 @ ( concat @ A @ Xss23 ) ) ) ) ) ) ) ).
% concat_eq_append_conv
thf(fact_6209_rev__nonempty__induct,axiom,
! [A: $tType,Xs: list @ A,P: ( list @ A ) > $o] :
( ( Xs
!= ( nil @ A ) )
=> ( ! [X5: A] : ( P @ ( cons @ A @ X5 @ ( nil @ A ) ) )
=> ( ! [X5: A,Xs2: list @ A] :
( ( Xs2
!= ( nil @ A ) )
=> ( ( P @ Xs2 )
=> ( P @ ( append @ A @ Xs2 @ ( cons @ A @ X5 @ ( nil @ A ) ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_6210_append__eq__Cons__conv,axiom,
! [A: $tType,Ys: list @ A,Zs: list @ A,X: A,Xs: list @ A] :
( ( ( append @ A @ Ys @ Zs )
= ( cons @ A @ X @ Xs ) )
= ( ( ( Ys
= ( nil @ A ) )
& ( Zs
= ( cons @ A @ X @ Xs ) ) )
| ? [Ys7: list @ A] :
( ( Ys
= ( cons @ A @ X @ Ys7 ) )
& ( ( append @ A @ Ys7 @ Zs )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_6211_Cons__eq__append__conv,axiom,
! [A: $tType,X: A,Xs: list @ A,Ys: list @ A,Zs: list @ A] :
( ( ( cons @ A @ X @ Xs )
= ( append @ A @ Ys @ Zs ) )
= ( ( ( Ys
= ( nil @ A ) )
& ( ( cons @ A @ X @ Xs )
= Zs ) )
| ? [Ys7: list @ A] :
( ( ( cons @ A @ X @ Ys7 )
= Ys )
& ( Xs
= ( append @ A @ Ys7 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_6212_rev__exhaust,axiom,
! [A: $tType,Xs: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ~ ! [Ys4: list @ A,Y3: A] :
( Xs
!= ( append @ A @ Ys4 @ ( cons @ A @ Y3 @ ( nil @ A ) ) ) ) ) ).
% rev_exhaust
thf(fact_6213_rev__induct,axiom,
! [A: $tType,P: ( list @ A ) > $o,Xs: list @ A] :
( ( P @ ( nil @ A ) )
=> ( ! [X5: A,Xs2: list @ A] :
( ( P @ Xs2 )
=> ( P @ ( append @ A @ Xs2 @ ( cons @ A @ X5 @ ( nil @ A ) ) ) ) )
=> ( P @ Xs ) ) ) ).
% rev_induct
thf(fact_6214_append__Cons,axiom,
! [A: $tType,X: A,Xs: list @ A,Ys: list @ A] :
( ( append @ A @ ( cons @ A @ X @ Xs ) @ Ys )
= ( cons @ A @ X @ ( append @ A @ Xs @ Ys ) ) ) ).
% append_Cons
thf(fact_6215_Cons__eq__appendI,axiom,
! [A: $tType,X: A,Xs1: list @ A,Ys: list @ A,Xs: list @ A,Zs: list @ A] :
( ( ( cons @ A @ X @ Xs1 )
= Ys )
=> ( ( Xs
= ( append @ A @ Xs1 @ Zs ) )
=> ( ( cons @ A @ X @ Xs )
= ( append @ A @ Ys @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_6216_concat_Osimps_I2_J,axiom,
! [A: $tType,X: list @ A,Xs: list @ ( list @ A )] :
( ( concat @ A @ ( cons @ ( list @ A ) @ X @ Xs ) )
= ( append @ A @ X @ ( concat @ A @ Xs ) ) ) ).
% concat.simps(2)
thf(fact_6217_rotate__append,axiom,
! [A: $tType,L: list @ A,Q3: list @ A] :
( ( rotate @ A @ ( size_size @ ( list @ A ) @ L ) @ ( append @ A @ L @ Q3 ) )
= ( append @ A @ Q3 @ L ) ) ).
% rotate_append
thf(fact_6218_hd__append,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( ( Xs
= ( nil @ A ) )
=> ( ( hd @ A @ ( append @ A @ Xs @ Ys ) )
= ( hd @ A @ Ys ) ) )
& ( ( Xs
!= ( nil @ A ) )
=> ( ( hd @ A @ ( append @ A @ Xs @ Ys ) )
= ( hd @ A @ Xs ) ) ) ) ).
% hd_append
thf(fact_6219_longest__common__prefix,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
? [Ps: list @ A,Xs4: list @ A,Ys5: list @ A] :
( ( Xs
= ( append @ A @ Ps @ Xs4 ) )
& ( Ys
= ( append @ A @ Ps @ Ys5 ) )
& ( ( Xs4
= ( nil @ A ) )
| ( Ys5
= ( nil @ A ) )
| ( ( hd @ A @ Xs4 )
!= ( hd @ A @ Ys5 ) ) ) ) ).
% longest_common_prefix
thf(fact_6220_remove1__append,axiom,
! [A: $tType,X: A,Xs: list @ A,Ys: list @ A] :
( ( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ( remove1 @ A @ X @ ( append @ A @ Xs @ Ys ) )
= ( append @ A @ ( remove1 @ A @ X @ Xs ) @ Ys ) ) )
& ( ~ ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ( remove1 @ A @ X @ ( append @ A @ Xs @ Ys ) )
= ( append @ A @ Xs @ ( remove1 @ A @ X @ Ys ) ) ) ) ) ).
% remove1_append
thf(fact_6221_remdups__append2,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( remdups @ A @ ( append @ A @ Xs @ ( remdups @ A @ Ys ) ) )
= ( remdups @ A @ ( append @ A @ Xs @ Ys ) ) ) ).
% remdups_append2
thf(fact_6222_enumerate__append__eq,axiom,
! [A: $tType,N2: nat,Xs: list @ A,Ys: list @ A] :
( ( enumerate @ A @ N2 @ ( append @ A @ Xs @ Ys ) )
= ( append @ ( product_prod @ nat @ A ) @ ( enumerate @ A @ N2 @ Xs ) @ ( enumerate @ A @ ( plus_plus @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs ) ) @ Ys ) ) ) ).
% enumerate_append_eq
thf(fact_6223_append__eq__append__conv2,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,Zs: list @ A,Ts: list @ A] :
( ( ( append @ A @ Xs @ Ys )
= ( append @ A @ Zs @ Ts ) )
= ( ? [Us2: list @ A] :
( ( ( Xs
= ( append @ A @ Zs @ Us2 ) )
& ( ( append @ A @ Us2 @ Ys )
= Ts ) )
| ( ( ( append @ A @ Xs @ Us2 )
= Zs )
& ( Ys
= ( append @ A @ Us2 @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_6224_append__eq__appendI,axiom,
! [A: $tType,Xs: list @ A,Xs1: list @ A,Zs: list @ A,Ys: list @ A,Us: list @ A] :
( ( ( append @ A @ Xs @ Xs1 )
= Zs )
=> ( ( Ys
= ( append @ A @ Xs1 @ Us ) )
=> ( ( append @ A @ Xs @ Ys )
= ( append @ A @ Zs @ Us ) ) ) ) ).
% append_eq_appendI
thf(fact_6225_append__Nil,axiom,
! [A: $tType,Ys: list @ A] :
( ( append @ A @ ( nil @ A ) @ Ys )
= Ys ) ).
% append_Nil
thf(fact_6226_append_Oleft__neutral,axiom,
! [A: $tType,A3: list @ A] :
( ( append @ A @ ( nil @ A ) @ A3 )
= A3 ) ).
% append.left_neutral
thf(fact_6227_eq__Nil__appendI,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( Xs = Ys )
=> ( Xs
= ( append @ A @ ( nil @ A ) @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_6228_replicate__add,axiom,
! [A: $tType,N2: nat,M2: nat,X: A] :
( ( replicate @ A @ ( plus_plus @ nat @ N2 @ M2 ) @ X )
= ( append @ A @ ( replicate @ A @ N2 @ X ) @ ( replicate @ A @ M2 @ X ) ) ) ).
% replicate_add
thf(fact_6229_append__replicate__commute,axiom,
! [A: $tType,N2: nat,X: A,K: nat] :
( ( append @ A @ ( replicate @ A @ N2 @ X ) @ ( replicate @ A @ K @ X ) )
= ( append @ A @ ( replicate @ A @ K @ X ) @ ( replicate @ A @ N2 @ X ) ) ) ).
% append_replicate_commute
thf(fact_6230_lex__append__leftI,axiom,
! [A: $tType,Ys: list @ A,Zs: list @ A,R2: set @ ( product_prod @ A @ A ),Xs: list @ A] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ys @ Zs ) @ ( lex @ A @ R2 ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs @ Ys ) @ ( append @ A @ Xs @ Zs ) ) @ ( lex @ A @ R2 ) ) ) ).
% lex_append_leftI
thf(fact_6231_lexord__append__leftI,axiom,
! [A: $tType,U: list @ A,V2: list @ A,R2: set @ ( product_prod @ A @ A ),X: list @ A] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ U @ V2 ) @ ( lexord @ A @ R2 ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ X @ U ) @ ( append @ A @ X @ V2 ) ) @ ( lexord @ A @ R2 ) ) ) ).
% lexord_append_leftI
thf(fact_6232_append__listrel1I,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,R2: set @ ( product_prod @ A @ A ),Us: list @ A,Vs: list @ A] :
( ( ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( listrel1 @ A @ R2 ) )
& ( Us = Vs ) )
| ( ( Xs = Ys )
& ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Us @ Vs ) @ ( listrel1 @ A @ R2 ) ) ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs @ Us ) @ ( append @ A @ Ys @ Vs ) ) @ ( listrel1 @ A @ R2 ) ) ) ).
% append_listrel1I
thf(fact_6233_comm__append__are__replicate,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( ( append @ A @ Xs @ Ys )
= ( append @ A @ Ys @ Xs ) )
=> ? [M3: nat,N3: nat,Zs2: list @ A] :
( ( ( concat @ A @ ( replicate @ ( list @ A ) @ M3 @ Zs2 ) )
= Xs )
& ( ( concat @ A @ ( replicate @ ( list @ A ) @ N3 @ Zs2 ) )
= Ys ) ) ) ).
% comm_append_are_replicate
thf(fact_6234_same__length__different,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( Xs != Ys )
=> ( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ A ) @ Ys ) )
=> ? [Pre: list @ A,X5: A,Xs4: list @ A,Y3: A,Ys5: list @ A] :
( ( X5 != Y3 )
& ( Xs
= ( append @ A @ Pre @ ( append @ A @ ( cons @ A @ X5 @ ( nil @ A ) ) @ Xs4 ) ) )
& ( Ys
= ( append @ A @ Pre @ ( append @ A @ ( cons @ A @ Y3 @ ( nil @ A ) ) @ Ys5 ) ) ) ) ) ) ).
% same_length_different
thf(fact_6235_not__distinct__decomp,axiom,
! [A: $tType,Ws2: list @ A] :
( ~ ( distinct @ A @ Ws2 )
=> ? [Xs2: list @ A,Ys4: list @ A,Zs2: list @ A,Y3: A] :
( Ws2
= ( append @ A @ Xs2 @ ( append @ A @ ( cons @ A @ Y3 @ ( nil @ A ) ) @ ( append @ A @ Ys4 @ ( append @ A @ ( cons @ A @ Y3 @ ( nil @ A ) ) @ Zs2 ) ) ) ) ) ) ).
% not_distinct_decomp
thf(fact_6236_not__distinct__conv__prefix,axiom,
! [A: $tType,As: list @ A] :
( ( ~ ( distinct @ A @ As ) )
= ( ? [Xs3: list @ A,Y6: A,Ys3: list @ A] :
( ( member @ A @ Y6 @ ( set2 @ A @ Xs3 ) )
& ( distinct @ A @ Xs3 )
& ( As
= ( append @ A @ Xs3 @ ( cons @ A @ Y6 @ Ys3 ) ) ) ) ) ) ).
% not_distinct_conv_prefix
thf(fact_6237_replicate__append__same,axiom,
! [A: $tType,I: nat,X: A] :
( ( append @ A @ ( replicate @ A @ I @ X ) @ ( cons @ A @ X @ ( nil @ A ) ) )
= ( cons @ A @ X @ ( replicate @ A @ I @ X ) ) ) ).
% replicate_append_same
thf(fact_6238_list__update__append1,axiom,
! [A: $tType,I: nat,Xs: list @ A,Ys: list @ A,X: A] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( list_update @ A @ ( append @ A @ Xs @ Ys ) @ I @ X )
= ( append @ A @ ( list_update @ A @ Xs @ I @ X ) @ Ys ) ) ) ).
% list_update_append1
thf(fact_6239_remdups__adj__append__two,axiom,
! [A: $tType,Xs: list @ A,X: A,Y: A] :
( ( remdups_adj @ A @ ( append @ A @ Xs @ ( cons @ A @ X @ ( cons @ A @ Y @ ( nil @ A ) ) ) ) )
= ( append @ A @ ( remdups_adj @ A @ ( append @ A @ Xs @ ( cons @ A @ X @ ( nil @ A ) ) ) ) @ ( if @ ( list @ A ) @ ( X = Y ) @ ( nil @ A ) @ ( cons @ A @ Y @ ( nil @ A ) ) ) ) ) ).
% remdups_adj_append_two
thf(fact_6240_remove1__split,axiom,
! [A: $tType,A3: A,Xs: list @ A,Ys: list @ A] :
( ( member @ A @ A3 @ ( set2 @ A @ Xs ) )
=> ( ( ( remove1 @ A @ A3 @ Xs )
= Ys )
= ( ? [Ls: list @ A,Rs: list @ A] :
( ( Xs
= ( append @ A @ Ls @ ( cons @ A @ A3 @ Rs ) ) )
& ~ ( member @ A @ A3 @ ( set2 @ A @ Ls ) )
& ( Ys
= ( append @ A @ Ls @ Rs ) ) ) ) ) ) ).
% remove1_split
thf(fact_6241_lexord__append__leftD,axiom,
! [A: $tType,X: list @ A,U: list @ A,V2: list @ A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ X @ U ) @ ( append @ A @ X @ V2 ) ) @ ( lexord @ A @ R2 ) )
=> ( ! [A6: A] :
~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A6 @ A6 ) @ R2 )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ U @ V2 ) @ ( lexord @ A @ R2 ) ) ) ) ).
% lexord_append_leftD
thf(fact_6242_lexord__append__rightI,axiom,
! [A: $tType,Y: list @ A,X: list @ A,R2: set @ ( product_prod @ A @ A )] :
( ? [B10: A,Z4: list @ A] :
( Y
= ( cons @ A @ B10 @ Z4 ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ ( append @ A @ X @ Y ) ) @ ( lexord @ A @ R2 ) ) ) ).
% lexord_append_rightI
thf(fact_6243_lexord__sufE,axiom,
! [A: $tType,Xs: list @ A,Zs: list @ A,Ys: list @ A,Qs: list @ A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs @ Zs ) @ ( append @ A @ Ys @ Qs ) ) @ ( lexord @ A @ R2 ) )
=> ( ( Xs != Ys )
=> ( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ A ) @ Ys ) )
=> ( ( ( size_size @ ( list @ A ) @ Zs )
= ( size_size @ ( list @ A ) @ Qs ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( lexord @ A @ R2 ) ) ) ) ) ) ).
% lexord_sufE
thf(fact_6244_lex__append__leftD,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),Xs: list @ A,Ys: list @ A,Zs: list @ A] :
( ! [X5: A] :
~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X5 @ X5 ) @ R2 )
=> ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs @ Ys ) @ ( append @ A @ Xs @ Zs ) ) @ ( lex @ A @ R2 ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ys @ Zs ) @ ( lex @ A @ R2 ) ) ) ) ).
% lex_append_leftD
thf(fact_6245_lex__append__left__iff,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),Xs: list @ A,Ys: list @ A,Zs: list @ A] :
( ! [X5: A] :
~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X5 @ X5 ) @ R2 )
=> ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs @ Ys ) @ ( append @ A @ Xs @ Zs ) ) @ ( lex @ A @ R2 ) )
= ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ys @ Zs ) @ ( lex @ A @ R2 ) ) ) ) ).
% lex_append_left_iff
thf(fact_6246_rotate1_Osimps_I2_J,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( rotate1 @ A @ ( cons @ A @ X @ Xs ) )
= ( append @ A @ Xs @ ( cons @ A @ X @ ( nil @ A ) ) ) ) ).
% rotate1.simps(2)
thf(fact_6247_lex__append__rightI,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,R2: set @ ( product_prod @ A @ A ),Vs: list @ A,Us: list @ A] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( lex @ A @ R2 ) )
=> ( ( ( size_size @ ( list @ A ) @ Vs )
= ( size_size @ ( list @ A ) @ Us ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs @ Us ) @ ( append @ A @ Ys @ Vs ) ) @ ( lex @ A @ R2 ) ) ) ) ).
% lex_append_rightI
thf(fact_6248_lenlex__append1,axiom,
! [A: $tType,Us: list @ A,Xs: list @ A,R: set @ ( product_prod @ A @ A ),Vs: list @ A,Ys: list @ A] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Us @ Xs ) @ ( lenlex @ A @ R ) )
=> ( ( ( size_size @ ( list @ A ) @ Vs )
= ( size_size @ ( list @ A ) @ Ys ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Us @ Vs ) @ ( append @ A @ Xs @ Ys ) ) @ ( lenlex @ A @ R ) ) ) ) ).
% lenlex_append1
thf(fact_6249_nths__append,axiom,
! [A: $tType,L: list @ A,L3: list @ A,A4: set @ nat] :
( ( nths @ A @ ( append @ A @ L @ L3 ) @ A4 )
= ( append @ A @ ( nths @ A @ L @ A4 )
@ ( nths @ A @ L3
@ ( collect @ nat
@ ^ [J2: nat] : ( member @ nat @ ( plus_plus @ nat @ J2 @ ( size_size @ ( list @ A ) @ L ) ) @ A4 ) ) ) ) ) ).
% nths_append
thf(fact_6250_length__append__singleton,axiom,
! [A: $tType,Xs: list @ A,X: A] :
( ( size_size @ ( list @ A ) @ ( append @ A @ Xs @ ( cons @ A @ X @ ( nil @ A ) ) ) )
= ( suc @ ( size_size @ ( list @ A ) @ Xs ) ) ) ).
% length_append_singleton
thf(fact_6251_length__Suc__conv__rev,axiom,
! [A: $tType,Xs: list @ A,N2: nat] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( suc @ N2 ) )
= ( ? [Y6: A,Ys3: list @ A] :
( ( Xs
= ( append @ A @ Ys3 @ ( cons @ A @ Y6 @ ( nil @ A ) ) ) )
& ( ( size_size @ ( list @ A ) @ Ys3 )
= N2 ) ) ) ) ).
% length_Suc_conv_rev
thf(fact_6252_nth__append,axiom,
! [A: $tType,N2: nat,Xs: list @ A,Ys: list @ A] :
( ( ( ord_less @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( nth @ A @ ( append @ A @ Xs @ Ys ) @ N2 )
= ( nth @ A @ Xs @ N2 ) ) )
& ( ~ ( ord_less @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( nth @ A @ ( append @ A @ Xs @ Ys ) @ N2 )
= ( nth @ A @ Ys @ ( minus_minus @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ) ) ).
% nth_append
thf(fact_6253_list__update__append,axiom,
! [A: $tType,N2: nat,Xs: list @ A,Ys: list @ A,X: A] :
( ( ( ord_less @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( list_update @ A @ ( append @ A @ Xs @ Ys ) @ N2 @ X )
= ( append @ A @ ( list_update @ A @ Xs @ N2 @ X ) @ Ys ) ) )
& ( ~ ( ord_less @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( list_update @ A @ ( append @ A @ Xs @ Ys ) @ N2 @ X )
= ( append @ A @ Xs @ ( list_update @ A @ Ys @ ( minus_minus @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs ) ) @ X ) ) ) ) ) ).
% list_update_append
thf(fact_6254_listrel1E,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( listrel1 @ A @ R2 ) )
=> ~ ! [X5: A,Y3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X5 @ Y3 ) @ R2 )
=> ! [Us3: list @ A,Vs2: list @ A] :
( ( Xs
= ( append @ A @ Us3 @ ( cons @ A @ X5 @ Vs2 ) ) )
=> ( Ys
!= ( append @ A @ Us3 @ ( cons @ A @ Y3 @ Vs2 ) ) ) ) ) ) ).
% listrel1E
thf(fact_6255_listrel1I,axiom,
! [A: $tType,X: A,Y: A,R2: set @ ( product_prod @ A @ A ),Xs: list @ A,Us: list @ A,Vs: list @ A,Ys: list @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R2 )
=> ( ( Xs
= ( append @ A @ Us @ ( cons @ A @ X @ Vs ) ) )
=> ( ( Ys
= ( append @ A @ Us @ ( cons @ A @ Y @ Vs ) ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( listrel1 @ A @ R2 ) ) ) ) ) ).
% listrel1I
thf(fact_6256_lexord__append__left__rightI,axiom,
! [A: $tType,A3: A,B3: A,R2: set @ ( product_prod @ A @ A ),U: list @ A,X: list @ A,Y: list @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B3 ) @ R2 )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ U @ ( cons @ A @ A3 @ X ) ) @ ( append @ A @ U @ ( cons @ A @ B3 @ Y ) ) ) @ ( lexord @ A @ R2 ) ) ) ).
% lexord_append_left_rightI
thf(fact_6257_lexord__same__pref__iff,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,Zs: list @ A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs @ Ys ) @ ( append @ A @ Xs @ Zs ) ) @ ( lexord @ A @ R2 ) )
= ( ? [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ X4 ) @ R2 ) )
| ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ys @ Zs ) @ ( lexord @ A @ R2 ) ) ) ) ).
% lexord_same_pref_iff
thf(fact_6258_lexord__sufI,axiom,
! [A: $tType,U: list @ A,W2: list @ A,R2: set @ ( product_prod @ A @ A ),V2: list @ A,Z3: list @ A] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ U @ W2 ) @ ( lexord @ A @ R2 ) )
=> ( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ W2 ) @ ( size_size @ ( list @ A ) @ U ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ U @ V2 ) @ ( append @ A @ W2 @ Z3 ) ) @ ( lexord @ A @ R2 ) ) ) ) ).
% lexord_sufI
thf(fact_6259_remdups__adj__append,axiom,
! [A: $tType,Xs_1: list @ A,X: A,Xs_2: list @ A] :
( ( remdups_adj @ A @ ( append @ A @ Xs_1 @ ( cons @ A @ X @ Xs_2 ) ) )
= ( append @ A @ ( remdups_adj @ A @ ( append @ A @ Xs_1 @ ( cons @ A @ X @ ( nil @ A ) ) ) ) @ ( tl @ A @ ( remdups_adj @ A @ ( cons @ A @ X @ Xs_2 ) ) ) ) ) ).
% remdups_adj_append
thf(fact_6260_snoc__listrel1__snoc__iff,axiom,
! [A: $tType,Xs: list @ A,X: A,Ys: list @ A,Y: A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs @ ( cons @ A @ X @ ( nil @ A ) ) ) @ ( append @ A @ Ys @ ( cons @ A @ Y @ ( nil @ A ) ) ) ) @ ( listrel1 @ A @ R2 ) )
= ( ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( listrel1 @ A @ R2 ) )
& ( X = Y ) )
| ( ( Xs = Ys )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R2 ) ) ) ) ).
% snoc_listrel1_snoc_iff
thf(fact_6261_horner__sum__append,axiom,
! [A: $tType,B: $tType] :
( ( comm_semiring_1 @ A )
=> ! [F2: B > A,A3: A,Xs: list @ B,Ys: list @ B] :
( ( groups4207007520872428315er_sum @ B @ A @ F2 @ A3 @ ( append @ B @ Xs @ Ys ) )
= ( plus_plus @ A @ ( groups4207007520872428315er_sum @ B @ A @ F2 @ A3 @ Xs ) @ ( times_times @ A @ ( power_power @ A @ A3 @ ( size_size @ ( list @ B ) @ Xs ) ) @ ( groups4207007520872428315er_sum @ B @ A @ F2 @ A3 @ Ys ) ) ) ) ) ).
% horner_sum_append
thf(fact_6262_nths__Cons,axiom,
! [A: $tType,X: A,L: list @ A,A4: set @ nat] :
( ( nths @ A @ ( cons @ A @ X @ L ) @ A4 )
= ( append @ A @ ( if @ ( list @ A ) @ ( member @ nat @ ( zero_zero @ nat ) @ A4 ) @ ( cons @ A @ X @ ( nil @ A ) ) @ ( nil @ A ) )
@ ( nths @ A @ L
@ ( collect @ nat
@ ^ [J2: nat] : ( member @ nat @ ( suc @ J2 ) @ A4 ) ) ) ) ) ).
% nths_Cons
thf(fact_6263_rotate1__hd__tl,axiom,
! [A: $tType,Xs: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( rotate1 @ A @ Xs )
= ( append @ A @ ( tl @ A @ Xs ) @ ( cons @ A @ ( hd @ A @ Xs ) @ ( nil @ A ) ) ) ) ) ).
% rotate1_hd_tl
thf(fact_6264_comm__append__is__replicate,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( Ys
!= ( nil @ A ) )
=> ( ( ( append @ A @ Xs @ Ys )
= ( append @ A @ Ys @ Xs ) )
=> ? [N3: nat,Zs2: list @ A] :
( ( ord_less @ nat @ ( one_one @ nat ) @ N3 )
& ( ( concat @ A @ ( replicate @ ( list @ A ) @ N3 @ Zs2 ) )
= ( append @ A @ Xs @ Ys ) ) ) ) ) ) ).
% comm_append_is_replicate
thf(fact_6265_listrel1__def,axiom,
! [A: $tType] :
( ( listrel1 @ A )
= ( ^ [R4: set @ ( product_prod @ A @ A )] :
( collect @ ( product_prod @ ( list @ A ) @ ( list @ A ) )
@ ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ $o
@ ^ [Xs3: list @ A,Ys3: list @ A] :
? [Us2: list @ A,Z6: A,Z7: A,Vs3: list @ A] :
( ( Xs3
= ( append @ A @ Us2 @ ( cons @ A @ Z6 @ Vs3 ) ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z6 @ Z7 ) @ R4 )
& ( Ys3
= ( append @ A @ Us2 @ ( cons @ A @ Z7 @ Vs3 ) ) ) ) ) ) ) ) ).
% listrel1_def
thf(fact_6266_lexord__def,axiom,
! [A: $tType] :
( ( lexord @ A )
= ( ^ [R4: set @ ( product_prod @ A @ A )] :
( collect @ ( product_prod @ ( list @ A ) @ ( list @ A ) )
@ ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ $o
@ ^ [X4: list @ A,Y6: list @ A] :
? [A5: A,V5: list @ A] :
( ( Y6
= ( append @ A @ X4 @ ( cons @ A @ A5 @ V5 ) ) )
| ? [U2: list @ A,B4: A,C4: A,W3: list @ A,Z6: list @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B4 @ C4 ) @ R4 )
& ( X4
= ( append @ A @ U2 @ ( cons @ A @ B4 @ W3 ) ) )
& ( Y6
= ( append @ A @ U2 @ ( cons @ A @ C4 @ Z6 ) ) ) ) ) ) ) ) ) ).
% lexord_def
thf(fact_6267_take__Suc__conv__app__nth,axiom,
! [A: $tType,I: nat,Xs: list @ A] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( take @ A @ ( suc @ I ) @ Xs )
= ( append @ A @ ( take @ A @ I @ Xs ) @ ( cons @ A @ ( nth @ A @ Xs @ I ) @ ( nil @ A ) ) ) ) ) ).
% take_Suc_conv_app_nth
thf(fact_6268_lex__conv,axiom,
! [A: $tType] :
( ( lex @ A )
= ( ^ [R4: set @ ( product_prod @ A @ A )] :
( collect @ ( product_prod @ ( list @ A ) @ ( list @ A ) )
@ ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ $o
@ ^ [Xs3: list @ A,Ys3: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs3 )
= ( size_size @ ( list @ A ) @ Ys3 ) )
& ? [Xys2: list @ A,X4: A,Y6: A,Xs6: list @ A,Ys7: list @ A] :
( ( Xs3
= ( append @ A @ Xys2 @ ( cons @ A @ X4 @ Xs6 ) ) )
& ( Ys3
= ( append @ A @ Xys2 @ ( cons @ A @ Y6 @ Ys7 ) ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y6 ) @ R4 ) ) ) ) ) ) ) ).
% lex_conv
thf(fact_6269_nth__repl,axiom,
! [A: $tType,M2: nat,Xs: list @ A,N2: nat,X: A] :
( ( ord_less @ nat @ M2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ord_less @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( M2 != N2 )
=> ( ( nth @ A @ ( append @ A @ ( take @ A @ N2 @ Xs ) @ ( append @ A @ ( cons @ A @ X @ ( nil @ A ) ) @ ( drop @ A @ ( plus_plus @ nat @ N2 @ ( one_one @ nat ) ) @ Xs ) ) ) @ M2 )
= ( nth @ A @ Xs @ M2 ) ) ) ) ) ).
% nth_repl
thf(fact_6270_pos__n__replace,axiom,
! [A: $tType,N2: nat,Xs: list @ A,Y: A] :
( ( ord_less @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ A ) @ ( append @ A @ ( take @ A @ N2 @ Xs ) @ ( append @ A @ ( cons @ A @ Y @ ( nil @ A ) ) @ ( drop @ A @ ( suc @ N2 ) @ Xs ) ) ) ) ) ) ).
% pos_n_replace
thf(fact_6271_drop0,axiom,
! [A: $tType] :
( ( drop @ A @ ( zero_zero @ nat ) )
= ( ^ [X4: list @ A] : X4 ) ) ).
% drop0
thf(fact_6272_drop__drop,axiom,
! [A: $tType,N2: nat,M2: nat,Xs: list @ A] :
( ( drop @ A @ N2 @ ( drop @ A @ M2 @ Xs ) )
= ( drop @ A @ ( plus_plus @ nat @ N2 @ M2 ) @ Xs ) ) ).
% drop_drop
thf(fact_6273_drop__Suc__Cons,axiom,
! [A: $tType,N2: nat,X: A,Xs: list @ A] :
( ( drop @ A @ ( suc @ N2 ) @ ( cons @ A @ X @ Xs ) )
= ( drop @ A @ N2 @ Xs ) ) ).
% drop_Suc_Cons
thf(fact_6274_length__drop,axiom,
! [A: $tType,N2: nat,Xs: list @ A] :
( ( size_size @ ( list @ A ) @ ( drop @ A @ N2 @ Xs ) )
= ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ N2 ) ) ).
% length_drop
thf(fact_6275_drop__update__cancel,axiom,
! [A: $tType,N2: nat,M2: nat,Xs: list @ A,X: A] :
( ( ord_less @ nat @ N2 @ M2 )
=> ( ( drop @ A @ M2 @ ( list_update @ A @ Xs @ N2 @ X ) )
= ( drop @ A @ M2 @ Xs ) ) ) ).
% drop_update_cancel
thf(fact_6276_append__take__drop__id,axiom,
! [A: $tType,N2: nat,Xs: list @ A] :
( ( append @ A @ ( take @ A @ N2 @ Xs ) @ ( drop @ A @ N2 @ Xs ) )
= Xs ) ).
% append_take_drop_id
thf(fact_6277_drop__replicate,axiom,
! [A: $tType,I: nat,K: nat,X: A] :
( ( drop @ A @ I @ ( replicate @ A @ K @ X ) )
= ( replicate @ A @ ( minus_minus @ nat @ K @ I ) @ X ) ) ).
% drop_replicate
thf(fact_6278_drop__all,axiom,
! [A: $tType,Xs: list @ A,N2: nat] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ N2 )
=> ( ( drop @ A @ N2 @ Xs )
= ( nil @ A ) ) ) ).
% drop_all
thf(fact_6279_drop__eq__Nil,axiom,
! [A: $tType,N2: nat,Xs: list @ A] :
( ( ( drop @ A @ N2 @ Xs )
= ( nil @ A ) )
= ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ N2 ) ) ).
% drop_eq_Nil
thf(fact_6280_drop__eq__Nil2,axiom,
! [A: $tType,N2: nat,Xs: list @ A] :
( ( ( nil @ A )
= ( drop @ A @ N2 @ Xs ) )
= ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ N2 ) ) ).
% drop_eq_Nil2
thf(fact_6281_drop__append,axiom,
! [A: $tType,N2: nat,Xs: list @ A,Ys: list @ A] :
( ( drop @ A @ N2 @ ( append @ A @ Xs @ Ys ) )
= ( append @ A @ ( drop @ A @ N2 @ Xs ) @ ( drop @ A @ ( minus_minus @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs ) ) @ Ys ) ) ) ).
% drop_append
thf(fact_6282_drop__Cons__numeral,axiom,
! [A: $tType,V2: num,X: A,Xs: list @ A] :
( ( drop @ A @ ( numeral_numeral @ nat @ V2 ) @ ( cons @ A @ X @ Xs ) )
= ( drop @ A @ ( minus_minus @ nat @ ( numeral_numeral @ nat @ V2 ) @ ( one_one @ nat ) ) @ Xs ) ) ).
% drop_Cons_numeral
thf(fact_6283_nth__drop,axiom,
! [A: $tType,N2: nat,Xs: list @ A,I: nat] :
( ( ord_less_eq @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( nth @ A @ ( drop @ A @ N2 @ Xs ) @ I )
= ( nth @ A @ Xs @ ( plus_plus @ nat @ N2 @ I ) ) ) ) ).
% nth_drop
thf(fact_6284_drop__zip,axiom,
! [A: $tType,B: $tType,N2: nat,Xs: list @ A,Ys: list @ B] :
( ( drop @ ( product_prod @ A @ B ) @ N2 @ ( zip @ A @ B @ Xs @ Ys ) )
= ( zip @ A @ B @ ( drop @ A @ N2 @ Xs ) @ ( drop @ B @ N2 @ Ys ) ) ) ).
% drop_zip
thf(fact_6285_take__drop,axiom,
! [A: $tType,N2: nat,M2: nat,Xs: list @ A] :
( ( take @ A @ N2 @ ( drop @ A @ M2 @ Xs ) )
= ( drop @ A @ M2 @ ( take @ A @ ( plus_plus @ nat @ N2 @ M2 ) @ Xs ) ) ) ).
% take_drop
thf(fact_6286_drop__take,axiom,
! [A: $tType,N2: nat,M2: nat,Xs: list @ A] :
( ( drop @ A @ N2 @ ( take @ A @ M2 @ Xs ) )
= ( take @ A @ ( minus_minus @ nat @ M2 @ N2 ) @ ( drop @ A @ N2 @ Xs ) ) ) ).
% drop_take
thf(fact_6287_in__set__dropD,axiom,
! [A: $tType,X: A,N2: nat,Xs: list @ A] :
( ( member @ A @ X @ ( set2 @ A @ ( drop @ A @ N2 @ Xs ) ) )
=> ( member @ A @ X @ ( set2 @ A @ Xs ) ) ) ).
% in_set_dropD
thf(fact_6288_drop__Nil,axiom,
! [A: $tType,N2: nat] :
( ( drop @ A @ N2 @ ( nil @ A ) )
= ( nil @ A ) ) ).
% drop_Nil
thf(fact_6289_distinct__drop,axiom,
! [A: $tType,Xs: list @ A,I: nat] :
( ( distinct @ A @ Xs )
=> ( distinct @ A @ ( drop @ A @ I @ Xs ) ) ) ).
% distinct_drop
thf(fact_6290_drop__0,axiom,
! [A: $tType,Xs: list @ A] :
( ( drop @ A @ ( zero_zero @ nat ) @ Xs )
= Xs ) ).
% drop_0
thf(fact_6291_tl__drop,axiom,
! [A: $tType,N2: nat,Xs: list @ A] :
( ( tl @ A @ ( drop @ A @ N2 @ Xs ) )
= ( drop @ A @ N2 @ ( tl @ A @ Xs ) ) ) ).
% tl_drop
thf(fact_6292_drop__Suc,axiom,
! [A: $tType,N2: nat,Xs: list @ A] :
( ( drop @ A @ ( suc @ N2 ) @ Xs )
= ( drop @ A @ N2 @ ( tl @ A @ Xs ) ) ) ).
% drop_Suc
thf(fact_6293_set__drop__subset,axiom,
! [A: $tType,N2: nat,Xs: list @ A] : ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ ( drop @ A @ N2 @ Xs ) ) @ ( set2 @ A @ Xs ) ) ).
% set_drop_subset
thf(fact_6294_nth__via__drop,axiom,
! [A: $tType,N2: nat,Xs: list @ A,Y: A,Ys: list @ A] :
( ( ( drop @ A @ N2 @ Xs )
= ( cons @ A @ Y @ Ys ) )
=> ( ( nth @ A @ Xs @ N2 )
= Y ) ) ).
% nth_via_drop
thf(fact_6295_drop__eq__nths,axiom,
! [A: $tType] :
( ( drop @ A )
= ( ^ [N: nat,Xs3: list @ A] : ( nths @ A @ Xs3 @ ( collect @ nat @ ( ord_less_eq @ nat @ N ) ) ) ) ) ).
% drop_eq_nths
thf(fact_6296_set__drop__subset__set__drop,axiom,
! [A: $tType,N2: nat,M2: nat,Xs: list @ A] :
( ( ord_less_eq @ nat @ N2 @ M2 )
=> ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ ( drop @ A @ M2 @ Xs ) ) @ ( set2 @ A @ ( drop @ A @ N2 @ Xs ) ) ) ) ).
% set_drop_subset_set_drop
thf(fact_6297_append__eq__conv__conj,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,Zs: list @ A] :
( ( ( append @ A @ Xs @ Ys )
= Zs )
= ( ( Xs
= ( take @ A @ ( size_size @ ( list @ A ) @ Xs ) @ Zs ) )
& ( Ys
= ( drop @ A @ ( size_size @ ( list @ A ) @ Xs ) @ Zs ) ) ) ) ).
% append_eq_conv_conj
thf(fact_6298_take__add,axiom,
! [A: $tType,I: nat,J: nat,Xs: list @ A] :
( ( take @ A @ ( plus_plus @ nat @ I @ J ) @ Xs )
= ( append @ A @ ( take @ A @ I @ Xs ) @ ( take @ A @ J @ ( drop @ A @ I @ Xs ) ) ) ) ).
% take_add
thf(fact_6299_drop__update__swap,axiom,
! [A: $tType,M2: nat,N2: nat,Xs: list @ A,X: A] :
( ( ord_less_eq @ nat @ M2 @ N2 )
=> ( ( drop @ A @ M2 @ ( list_update @ A @ Xs @ N2 @ X ) )
= ( list_update @ A @ ( drop @ A @ M2 @ Xs ) @ ( minus_minus @ nat @ N2 @ M2 ) @ X ) ) ) ).
% drop_update_swap
thf(fact_6300_drop__Cons,axiom,
! [A: $tType,N2: nat,X: A,Xs: list @ A] :
( ( drop @ A @ N2 @ ( cons @ A @ X @ Xs ) )
= ( case_nat @ ( list @ A ) @ ( cons @ A @ X @ Xs )
@ ^ [M: nat] : ( drop @ A @ M @ Xs )
@ N2 ) ) ).
% drop_Cons
thf(fact_6301_nths__drop,axiom,
! [A: $tType,N2: nat,Xs: list @ A,I6: set @ nat] :
( ( nths @ A @ ( drop @ A @ N2 @ Xs ) @ I6 )
= ( nths @ A @ Xs @ ( image2 @ nat @ nat @ ( plus_plus @ nat @ N2 ) @ I6 ) ) ) ).
% nths_drop
thf(fact_6302_drop__Cons_H,axiom,
! [A: $tType,N2: nat,X: A,Xs: list @ A] :
( ( ( N2
= ( zero_zero @ nat ) )
=> ( ( drop @ A @ N2 @ ( cons @ A @ X @ Xs ) )
= ( cons @ A @ X @ Xs ) ) )
& ( ( N2
!= ( zero_zero @ nat ) )
=> ( ( drop @ A @ N2 @ ( cons @ A @ X @ Xs ) )
= ( drop @ A @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) @ Xs ) ) ) ) ).
% drop_Cons'
thf(fact_6303_append__eq__append__conv__if,axiom,
! [A: $tType,Xs_1: list @ A,Xs_2: list @ A,Ys_1: list @ A,Ys_2: list @ A] :
( ( ( append @ A @ Xs_1 @ Xs_2 )
= ( append @ A @ Ys_1 @ Ys_2 ) )
= ( ( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs_1 ) @ ( size_size @ ( list @ A ) @ Ys_1 ) )
=> ( ( Xs_1
= ( take @ A @ ( size_size @ ( list @ A ) @ Xs_1 ) @ Ys_1 ) )
& ( Xs_2
= ( append @ A @ ( drop @ A @ ( size_size @ ( list @ A ) @ Xs_1 ) @ Ys_1 ) @ Ys_2 ) ) ) )
& ( ~ ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs_1 ) @ ( size_size @ ( list @ A ) @ Ys_1 ) )
=> ( ( ( take @ A @ ( size_size @ ( list @ A ) @ Ys_1 ) @ Xs_1 )
= Ys_1 )
& ( ( append @ A @ ( drop @ A @ ( size_size @ ( list @ A ) @ Ys_1 ) @ Xs_1 ) @ Xs_2 )
= Ys_2 ) ) ) ) ) ).
% append_eq_append_conv_if
thf(fact_6304_hd__drop__conv__nth,axiom,
! [A: $tType,N2: nat,Xs: list @ A] :
( ( ord_less @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( hd @ A @ ( drop @ A @ N2 @ Xs ) )
= ( nth @ A @ Xs @ N2 ) ) ) ).
% hd_drop_conv_nth
thf(fact_6305_zip__append2,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B,Zs: list @ B] :
( ( zip @ A @ B @ Xs @ ( append @ B @ Ys @ Zs ) )
= ( append @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ ( take @ A @ ( size_size @ ( list @ B ) @ Ys ) @ Xs ) @ Ys ) @ ( zip @ A @ B @ ( drop @ A @ ( size_size @ ( list @ B ) @ Ys ) @ Xs ) @ Zs ) ) ) ).
% zip_append2
thf(fact_6306_zip__append1,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ A,Zs: list @ B] :
( ( zip @ A @ B @ ( append @ A @ Xs @ Ys ) @ Zs )
= ( append @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ ( take @ B @ ( size_size @ ( list @ A ) @ Xs ) @ Zs ) ) @ ( zip @ A @ B @ Ys @ ( drop @ B @ ( size_size @ ( list @ A ) @ Xs ) @ Zs ) ) ) ) ).
% zip_append1
thf(fact_6307_Cons__nth__drop__Suc,axiom,
! [A: $tType,I: nat,Xs: list @ A] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( cons @ A @ ( nth @ A @ Xs @ I ) @ ( drop @ A @ ( suc @ I ) @ Xs ) )
= ( drop @ A @ I @ Xs ) ) ) ).
% Cons_nth_drop_Suc
thf(fact_6308_rotate__drop__take,axiom,
! [A: $tType] :
( ( rotate @ A )
= ( ^ [N: nat,Xs3: list @ A] : ( append @ A @ ( drop @ A @ ( modulo_modulo @ nat @ N @ ( size_size @ ( list @ A ) @ Xs3 ) ) @ Xs3 ) @ ( take @ A @ ( modulo_modulo @ nat @ N @ ( size_size @ ( list @ A ) @ Xs3 ) ) @ Xs3 ) ) ) ) ).
% rotate_drop_take
thf(fact_6309_set__take__disj__set__drop__if__distinct,axiom,
! [A: $tType,Vs: list @ A,I: nat,J: nat] :
( ( distinct @ A @ Vs )
=> ( ( ord_less_eq @ nat @ I @ J )
=> ( ( inf_inf @ ( set @ A ) @ ( set2 @ A @ ( take @ A @ I @ Vs ) ) @ ( set2 @ A @ ( drop @ A @ J @ Vs ) ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% set_take_disj_set_drop_if_distinct
thf(fact_6310_id__take__nth__drop,axiom,
! [A: $tType,I: nat,Xs: list @ A] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( Xs
= ( append @ A @ ( take @ A @ I @ Xs ) @ ( cons @ A @ ( nth @ A @ Xs @ I ) @ ( drop @ A @ ( suc @ I ) @ Xs ) ) ) ) ) ).
% id_take_nth_drop
thf(fact_6311_upd__conv__take__nth__drop,axiom,
! [A: $tType,I: nat,Xs: list @ A,A3: A] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( list_update @ A @ Xs @ I @ A3 )
= ( append @ A @ ( take @ A @ I @ Xs ) @ ( cons @ A @ A3 @ ( drop @ A @ ( suc @ I ) @ Xs ) ) ) ) ) ).
% upd_conv_take_nth_drop
thf(fact_6312_take__hd__drop,axiom,
! [A: $tType,N2: nat,Xs: list @ A] :
( ( ord_less @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( append @ A @ ( take @ A @ N2 @ Xs ) @ ( cons @ A @ ( hd @ A @ ( drop @ A @ N2 @ Xs ) ) @ ( nil @ A ) ) )
= ( take @ A @ ( suc @ N2 ) @ Xs ) ) ) ).
% take_hd_drop
thf(fact_6313_lexn__conv,axiom,
! [A: $tType] :
( ( lexn @ A )
= ( ^ [R4: set @ ( product_prod @ A @ A ),N: nat] :
( collect @ ( product_prod @ ( list @ A ) @ ( list @ A ) )
@ ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ $o
@ ^ [Xs3: list @ A,Ys3: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs3 )
= N )
& ( ( size_size @ ( list @ A ) @ Ys3 )
= N )
& ? [Xys2: list @ A,X4: A,Y6: A,Xs6: list @ A,Ys7: list @ A] :
( ( Xs3
= ( append @ A @ Xys2 @ ( cons @ A @ X4 @ Xs6 ) ) )
& ( Ys3
= ( append @ A @ Xys2 @ ( cons @ A @ Y6 @ Ys7 ) ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y6 ) @ R4 ) ) ) ) ) ) ) ).
% lexn_conv
thf(fact_6314_list__encode_Oelims,axiom,
! [X: list @ nat,Y: nat] :
( ( ( nat_list_encode @ X )
= Y )
=> ( ( ( X
= ( nil @ nat ) )
=> ( Y
!= ( zero_zero @ nat ) ) )
=> ~ ! [X5: nat,Xs2: list @ nat] :
( ( X
= ( cons @ nat @ X5 @ Xs2 ) )
=> ( Y
!= ( suc @ ( nat_prod_encode @ ( product_Pair @ nat @ nat @ X5 @ ( nat_list_encode @ Xs2 ) ) ) ) ) ) ) ) ).
% list_encode.elims
thf(fact_6315_lexn_Osimps_I1_J,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( lexn @ A @ R2 @ ( zero_zero @ nat ) )
= ( bot_bot @ ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) ) ) ).
% lexn.simps(1)
thf(fact_6316_list__encode_Osimps_I1_J,axiom,
( ( nat_list_encode @ ( nil @ nat ) )
= ( zero_zero @ nat ) ) ).
% list_encode.simps(1)
thf(fact_6317_lexn__length,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,R2: set @ ( product_prod @ A @ A ),N2: nat] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( lexn @ A @ R2 @ N2 ) )
=> ( ( ( size_size @ ( list @ A ) @ Xs )
= N2 )
& ( ( size_size @ ( list @ A ) @ Ys )
= N2 ) ) ) ).
% lexn_length
thf(fact_6318_lex__def,axiom,
! [A: $tType] :
( ( lex @ A )
= ( ^ [R4: set @ ( product_prod @ A @ A )] : ( complete_Sup_Sup @ ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) @ ( image2 @ nat @ ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) @ ( lexn @ A @ R4 ) @ ( top_top @ ( set @ nat ) ) ) ) ) ) ).
% lex_def
thf(fact_6319_list__encode_Osimps_I2_J,axiom,
! [X: nat,Xs: list @ nat] :
( ( nat_list_encode @ ( cons @ nat @ X @ Xs ) )
= ( suc @ ( nat_prod_encode @ ( product_Pair @ nat @ nat @ X @ ( nat_list_encode @ Xs ) ) ) ) ) ).
% list_encode.simps(2)
thf(fact_6320_upto__aux__rec,axiom,
( upto_aux
= ( ^ [I3: int,J2: int,Js: list @ int] : ( if @ ( list @ int ) @ ( ord_less @ int @ J2 @ I3 ) @ Js @ ( upto_aux @ I3 @ ( minus_minus @ int @ J2 @ ( one_one @ int ) ) @ ( cons @ int @ J2 @ Js ) ) ) ) ) ).
% upto_aux_rec
thf(fact_6321_extract__Some__iff,axiom,
! [A: $tType,P: A > $o,Xs: list @ A,Ys: list @ A,Y: A,Zs: list @ A] :
( ( ( extract @ A @ P @ Xs )
= ( some @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) @ ( product_Pair @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) @ Ys @ ( product_Pair @ A @ ( list @ A ) @ Y @ Zs ) ) ) )
= ( ( Xs
= ( append @ A @ Ys @ ( cons @ A @ Y @ Zs ) ) )
& ( P @ Y )
& ~ ? [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Ys ) )
& ( P @ X4 ) ) ) ) ).
% extract_Some_iff
thf(fact_6322_extract__None__iff,axiom,
! [A: $tType,P: A > $o,Xs: list @ A] :
( ( ( extract @ A @ P @ Xs )
= ( none @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) ) )
= ( ~ ? [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
& ( P @ X4 ) ) ) ) ).
% extract_None_iff
thf(fact_6323_extract__Nil__code,axiom,
! [A: $tType,P: A > $o] :
( ( extract @ A @ P @ ( nil @ A ) )
= ( none @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) ) ) ).
% extract_Nil_code
thf(fact_6324_extract__SomeE,axiom,
! [A: $tType,P: A > $o,Xs: list @ A,Ys: list @ A,Y: A,Zs: list @ A] :
( ( ( extract @ A @ P @ Xs )
= ( some @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) @ ( product_Pair @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) @ Ys @ ( product_Pair @ A @ ( list @ A ) @ Y @ Zs ) ) ) )
=> ( ( Xs
= ( append @ A @ Ys @ ( cons @ A @ Y @ Zs ) ) )
& ( P @ Y )
& ~ ? [X2: A] :
( ( member @ A @ X2 @ ( set2 @ A @ Ys ) )
& ( P @ X2 ) ) ) ) ).
% extract_SomeE
thf(fact_6325_upto_Opsimps,axiom,
! [I: int,J: int] :
( ( accp @ ( product_prod @ int @ int ) @ upto_rel @ ( product_Pair @ int @ int @ I @ J ) )
=> ( ( ( ord_less_eq @ int @ I @ J )
=> ( ( upto @ I @ J )
= ( cons @ int @ I @ ( upto @ ( plus_plus @ int @ I @ ( one_one @ int ) ) @ J ) ) ) )
& ( ~ ( ord_less_eq @ int @ I @ J )
=> ( ( upto @ I @ J )
= ( nil @ int ) ) ) ) ) ).
% upto.psimps
thf(fact_6326_upto_Opelims,axiom,
! [X: int,Xa2: int,Y: list @ int] :
( ( ( upto @ X @ Xa2 )
= Y )
=> ( ( accp @ ( product_prod @ int @ int ) @ upto_rel @ ( product_Pair @ int @ int @ X @ Xa2 ) )
=> ~ ( ( ( ( ord_less_eq @ int @ X @ Xa2 )
=> ( Y
= ( cons @ int @ X @ ( upto @ ( plus_plus @ int @ X @ ( one_one @ int ) ) @ Xa2 ) ) ) )
& ( ~ ( ord_less_eq @ int @ X @ Xa2 )
=> ( Y
= ( nil @ int ) ) ) )
=> ~ ( accp @ ( product_prod @ int @ int ) @ upto_rel @ ( product_Pair @ int @ int @ X @ Xa2 ) ) ) ) ) ).
% upto.pelims
thf(fact_6327_upto__Nil,axiom,
! [I: int,J: int] :
( ( ( upto @ I @ J )
= ( nil @ int ) )
= ( ord_less @ int @ J @ I ) ) ).
% upto_Nil
thf(fact_6328_upto__Nil2,axiom,
! [I: int,J: int] :
( ( ( nil @ int )
= ( upto @ I @ J ) )
= ( ord_less @ int @ J @ I ) ) ).
% upto_Nil2
thf(fact_6329_upto__empty,axiom,
! [J: int,I: int] :
( ( ord_less @ int @ J @ I )
=> ( ( upto @ I @ J )
= ( nil @ int ) ) ) ).
% upto_empty
thf(fact_6330_upto__single,axiom,
! [I: int] :
( ( upto @ I @ I )
= ( cons @ int @ I @ ( nil @ int ) ) ) ).
% upto_single
thf(fact_6331_nth__upto,axiom,
! [I: int,K: nat,J: int] :
( ( ord_less_eq @ int @ ( plus_plus @ int @ I @ ( semiring_1_of_nat @ int @ K ) ) @ J )
=> ( ( nth @ int @ ( upto @ I @ J ) @ K )
= ( plus_plus @ int @ I @ ( semiring_1_of_nat @ int @ K ) ) ) ) ).
% nth_upto
thf(fact_6332_length__upto,axiom,
! [I: int,J: int] :
( ( size_size @ ( list @ int ) @ ( upto @ I @ J ) )
= ( nat2 @ ( plus_plus @ int @ ( minus_minus @ int @ J @ I ) @ ( one_one @ int ) ) ) ) ).
% length_upto
thf(fact_6333_upto__rec__numeral_I1_J,axiom,
! [M2: num,N2: num] :
( ( ( ord_less_eq @ int @ ( numeral_numeral @ int @ M2 ) @ ( numeral_numeral @ int @ N2 ) )
=> ( ( upto @ ( numeral_numeral @ int @ M2 ) @ ( numeral_numeral @ int @ N2 ) )
= ( cons @ int @ ( numeral_numeral @ int @ M2 ) @ ( upto @ ( plus_plus @ int @ ( numeral_numeral @ int @ M2 ) @ ( one_one @ int ) ) @ ( numeral_numeral @ int @ N2 ) ) ) ) )
& ( ~ ( ord_less_eq @ int @ ( numeral_numeral @ int @ M2 ) @ ( numeral_numeral @ int @ N2 ) )
=> ( ( upto @ ( numeral_numeral @ int @ M2 ) @ ( numeral_numeral @ int @ N2 ) )
= ( nil @ int ) ) ) ) ).
% upto_rec_numeral(1)
thf(fact_6334_upto__rec__numeral_I2_J,axiom,
! [M2: num,N2: num] :
( ( ( ord_less_eq @ int @ ( numeral_numeral @ int @ M2 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N2 ) ) )
=> ( ( upto @ ( numeral_numeral @ int @ M2 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N2 ) ) )
= ( cons @ int @ ( numeral_numeral @ int @ M2 ) @ ( upto @ ( plus_plus @ int @ ( numeral_numeral @ int @ M2 ) @ ( one_one @ int ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N2 ) ) ) ) ) )
& ( ~ ( ord_less_eq @ int @ ( numeral_numeral @ int @ M2 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N2 ) ) )
=> ( ( upto @ ( numeral_numeral @ int @ M2 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N2 ) ) )
= ( nil @ int ) ) ) ) ).
% upto_rec_numeral(2)
thf(fact_6335_upto__rec__numeral_I3_J,axiom,
! [M2: num,N2: num] :
( ( ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M2 ) ) @ ( numeral_numeral @ int @ N2 ) )
=> ( ( upto @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M2 ) ) @ ( numeral_numeral @ int @ N2 ) )
= ( cons @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M2 ) ) @ ( upto @ ( plus_plus @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M2 ) ) @ ( one_one @ int ) ) @ ( numeral_numeral @ int @ N2 ) ) ) ) )
& ( ~ ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M2 ) ) @ ( numeral_numeral @ int @ N2 ) )
=> ( ( upto @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M2 ) ) @ ( numeral_numeral @ int @ N2 ) )
= ( nil @ int ) ) ) ) ).
% upto_rec_numeral(3)
thf(fact_6336_upto__rec__numeral_I4_J,axiom,
! [M2: num,N2: num] :
( ( ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M2 ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N2 ) ) )
=> ( ( upto @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M2 ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N2 ) ) )
= ( cons @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M2 ) ) @ ( upto @ ( plus_plus @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M2 ) ) @ ( one_one @ int ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N2 ) ) ) ) ) )
& ( ~ ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M2 ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N2 ) ) )
=> ( ( upto @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M2 ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N2 ) ) )
= ( nil @ int ) ) ) ) ).
% upto_rec_numeral(4)
thf(fact_6337_upto__code,axiom,
( upto
= ( ^ [I3: int,J2: int] : ( upto_aux @ I3 @ J2 @ ( nil @ int ) ) ) ) ).
% upto_code
thf(fact_6338_upto__aux__def,axiom,
( upto_aux
= ( ^ [I3: int,J2: int] : ( append @ int @ ( upto @ I3 @ J2 ) ) ) ) ).
% upto_aux_def
thf(fact_6339_distinct__upto,axiom,
! [I: int,J: int] : ( distinct @ int @ ( upto @ I @ J ) ) ).
% distinct_upto
thf(fact_6340_atLeastAtMost__upto,axiom,
( ( set_or1337092689740270186AtMost @ int )
= ( ^ [I3: int,J2: int] : ( set2 @ int @ ( upto @ I3 @ J2 ) ) ) ) ).
% atLeastAtMost_upto
thf(fact_6341_upto__split2,axiom,
! [I: int,J: int,K: int] :
( ( ord_less_eq @ int @ I @ J )
=> ( ( ord_less_eq @ int @ J @ K )
=> ( ( upto @ I @ K )
= ( append @ int @ ( upto @ I @ J ) @ ( upto @ ( plus_plus @ int @ J @ ( one_one @ int ) ) @ K ) ) ) ) ) ).
% upto_split2
thf(fact_6342_upto__split1,axiom,
! [I: int,J: int,K: int] :
( ( ord_less_eq @ int @ I @ J )
=> ( ( ord_less_eq @ int @ J @ K )
=> ( ( upto @ I @ K )
= ( append @ int @ ( upto @ I @ ( minus_minus @ int @ J @ ( one_one @ int ) ) ) @ ( upto @ J @ K ) ) ) ) ) ).
% upto_split1
thf(fact_6343_atLeastLessThan__upto,axiom,
( ( set_or7035219750837199246ssThan @ int )
= ( ^ [I3: int,J2: int] : ( set2 @ int @ ( upto @ I3 @ ( minus_minus @ int @ J2 @ ( one_one @ int ) ) ) ) ) ) ).
% atLeastLessThan_upto
thf(fact_6344_greaterThanAtMost__upto,axiom,
( ( set_or3652927894154168847AtMost @ int )
= ( ^ [I3: int,J2: int] : ( set2 @ int @ ( upto @ ( plus_plus @ int @ I3 @ ( one_one @ int ) ) @ J2 ) ) ) ) ).
% greaterThanAtMost_upto
thf(fact_6345_upto_Osimps,axiom,
( upto
= ( ^ [I3: int,J2: int] : ( if @ ( list @ int ) @ ( ord_less_eq @ int @ I3 @ J2 ) @ ( cons @ int @ I3 @ ( upto @ ( plus_plus @ int @ I3 @ ( one_one @ int ) ) @ J2 ) ) @ ( nil @ int ) ) ) ) ).
% upto.simps
thf(fact_6346_upto_Oelims,axiom,
! [X: int,Xa2: int,Y: list @ int] :
( ( ( upto @ X @ Xa2 )
= Y )
=> ( ( ( ord_less_eq @ int @ X @ Xa2 )
=> ( Y
= ( cons @ int @ X @ ( upto @ ( plus_plus @ int @ X @ ( one_one @ int ) ) @ Xa2 ) ) ) )
& ( ~ ( ord_less_eq @ int @ X @ Xa2 )
=> ( Y
= ( nil @ int ) ) ) ) ) ).
% upto.elims
thf(fact_6347_upto__rec1,axiom,
! [I: int,J: int] :
( ( ord_less_eq @ int @ I @ J )
=> ( ( upto @ I @ J )
= ( cons @ int @ I @ ( upto @ ( plus_plus @ int @ I @ ( one_one @ int ) ) @ J ) ) ) ) ).
% upto_rec1
thf(fact_6348_upto__rec2,axiom,
! [I: int,J: int] :
( ( ord_less_eq @ int @ I @ J )
=> ( ( upto @ I @ J )
= ( append @ int @ ( upto @ I @ ( minus_minus @ int @ J @ ( one_one @ int ) ) ) @ ( cons @ int @ J @ ( nil @ int ) ) ) ) ) ).
% upto_rec2
thf(fact_6349_greaterThanLessThan__upto,axiom,
( ( set_or5935395276787703475ssThan @ int )
= ( ^ [I3: int,J2: int] : ( set2 @ int @ ( upto @ ( plus_plus @ int @ I3 @ ( one_one @ int ) ) @ ( minus_minus @ int @ J2 @ ( one_one @ int ) ) ) ) ) ) ).
% greaterThanLessThan_upto
thf(fact_6350_upto__split3,axiom,
! [I: int,J: int,K: int] :
( ( ord_less_eq @ int @ I @ J )
=> ( ( ord_less_eq @ int @ J @ K )
=> ( ( upto @ I @ K )
= ( append @ int @ ( upto @ I @ ( minus_minus @ int @ J @ ( one_one @ int ) ) ) @ ( cons @ int @ J @ ( upto @ ( plus_plus @ int @ J @ ( one_one @ int ) ) @ K ) ) ) ) ) ) ).
% upto_split3
thf(fact_6351_zip__Cons,axiom,
! [B: $tType,A: $tType,Xs: list @ A,Y: B,Ys: list @ B] :
( ( zip @ A @ B @ Xs @ ( cons @ B @ Y @ Ys ) )
= ( case_list @ ( list @ ( product_prod @ A @ B ) ) @ A @ ( nil @ ( product_prod @ A @ B ) )
@ ^ [Z6: A,Zs3: list @ A] : ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Z6 @ Y ) @ ( zip @ A @ B @ Zs3 @ Ys ) )
@ Xs ) ) ).
% zip_Cons
thf(fact_6352_zip__Cons1,axiom,
! [A: $tType,B: $tType,X: A,Xs: list @ A,Ys: list @ B] :
( ( zip @ A @ B @ ( cons @ A @ X @ Xs ) @ Ys )
= ( case_list @ ( list @ ( product_prod @ A @ B ) ) @ B @ ( nil @ ( product_prod @ A @ B ) )
@ ^ [Y6: B,Ys3: list @ B] : ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y6 ) @ ( zip @ A @ B @ Xs @ Ys3 ) )
@ Ys ) ) ).
% zip_Cons1
thf(fact_6353_list_Osimps_I4_J,axiom,
! [A: $tType,B: $tType,F1: B,F22: A > ( list @ A ) > B] :
( ( case_list @ B @ A @ F1 @ F22 @ ( nil @ A ) )
= F1 ) ).
% list.simps(4)
thf(fact_6354_list_Ocase__distrib,axiom,
! [B: $tType,C: $tType,A: $tType,H2: B > C,F1: B,F22: A > ( list @ A ) > B,List: list @ A] :
( ( H2 @ ( case_list @ B @ A @ F1 @ F22 @ List ) )
= ( case_list @ C @ A @ ( H2 @ F1 )
@ ^ [X15: A,X24: list @ A] : ( H2 @ ( F22 @ X15 @ X24 ) )
@ List ) ) ).
% list.case_distrib
thf(fact_6355_list_Osimps_I5_J,axiom,
! [B: $tType,A: $tType,F1: B,F22: A > ( list @ A ) > B,X21: A,X222: list @ A] :
( ( case_list @ B @ A @ F1 @ F22 @ ( cons @ A @ X21 @ X222 ) )
= ( F22 @ X21 @ X222 ) ) ).
% list.simps(5)
thf(fact_6356_tl__def,axiom,
! [A: $tType] :
( ( tl @ A )
= ( case_list @ ( list @ A ) @ A @ ( nil @ A )
@ ^ [X213: A,X224: list @ A] : X224 ) ) ).
% tl_def
thf(fact_6357_tl__append,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( tl @ A @ ( append @ A @ Xs @ Ys ) )
= ( case_list @ ( list @ A ) @ A @ ( tl @ A @ Ys )
@ ^ [Z6: A,Zs3: list @ A] : ( append @ A @ Zs3 @ Ys )
@ Xs ) ) ).
% tl_append
thf(fact_6358_remdups__adj__Cons,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( remdups_adj @ A @ ( cons @ A @ X @ Xs ) )
= ( case_list @ ( list @ A ) @ A @ ( cons @ A @ X @ ( nil @ A ) )
@ ^ [Y6: A,Xs3: list @ A] : ( if @ ( list @ A ) @ ( X = Y6 ) @ ( cons @ A @ Y6 @ Xs3 ) @ ( cons @ A @ X @ ( cons @ A @ Y6 @ Xs3 ) ) )
@ ( remdups_adj @ A @ Xs ) ) ) ).
% remdups_adj_Cons
thf(fact_6359_list_Ocase__eq__if,axiom,
! [A: $tType,B: $tType] :
( ( case_list @ B @ A )
= ( ^ [F14: B,F23: A > ( list @ A ) > B,List3: list @ A] :
( if @ B
@ ( List3
= ( nil @ A ) )
@ F14
@ ( F23 @ ( hd @ A @ List3 ) @ ( tl @ A @ List3 ) ) ) ) ) ).
% list.case_eq_if
thf(fact_6360_min__list_Osimps,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X: A,Xs: list @ A] :
( ( min_list @ A @ ( cons @ A @ X @ Xs ) )
= ( case_list @ A @ A @ X
@ ^ [A5: A,List3: list @ A] : ( ord_min @ A @ X @ ( min_list @ A @ Xs ) )
@ Xs ) ) ) ).
% min_list.simps
thf(fact_6361_list_Osplit__sel__asm,axiom,
! [B: $tType,A: $tType,P: B > $o,F1: B,F22: A > ( list @ A ) > B,List: list @ A] :
( ( P @ ( case_list @ B @ A @ F1 @ F22 @ List ) )
= ( ~ ( ( ( List
= ( nil @ A ) )
& ~ ( P @ F1 ) )
| ( ( List
= ( cons @ A @ ( hd @ A @ List ) @ ( tl @ A @ List ) ) )
& ~ ( P @ ( F22 @ ( hd @ A @ List ) @ ( tl @ A @ List ) ) ) ) ) ) ) ).
% list.split_sel_asm
thf(fact_6362_list_Osplit__sel,axiom,
! [B: $tType,A: $tType,P: B > $o,F1: B,F22: A > ( list @ A ) > B,List: list @ A] :
( ( P @ ( case_list @ B @ A @ F1 @ F22 @ List ) )
= ( ( ( List
= ( nil @ A ) )
=> ( P @ F1 ) )
& ( ( List
= ( cons @ A @ ( hd @ A @ List ) @ ( tl @ A @ List ) ) )
=> ( P @ ( F22 @ ( hd @ A @ List ) @ ( tl @ A @ List ) ) ) ) ) ) ).
% list.split_sel
thf(fact_6363_lenlex__append2,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),Us: list @ A,Xs: list @ A,Ys: list @ A] :
( ( irrefl @ A @ R )
=> ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Us @ Xs ) @ ( append @ A @ Us @ Ys ) ) @ ( lenlex @ A @ R ) )
= ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( lenlex @ A @ R ) ) ) ) ).
% lenlex_append2
thf(fact_6364_min__list_Oelims,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X: list @ A,Y: A] :
( ( ( min_list @ A @ X )
= Y )
=> ( ! [X5: A,Xs2: list @ A] :
( ( X
= ( cons @ A @ X5 @ Xs2 ) )
=> ( Y
!= ( case_list @ A @ A @ X5
@ ^ [A5: A,List3: list @ A] : ( ord_min @ A @ X5 @ ( min_list @ A @ Xs2 ) )
@ Xs2 ) ) )
=> ~ ( ( X
= ( nil @ A ) )
=> ( Y
!= ( undefined @ A ) ) ) ) ) ) ).
% min_list.elims
thf(fact_6365_lexord__same__pref__if__irrefl,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),Xs: list @ A,Ys: list @ A,Zs: list @ A] :
( ( irrefl @ A @ R2 )
=> ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs @ Ys ) @ ( append @ A @ Xs @ Zs ) ) @ ( lexord @ A @ R2 ) )
= ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ys @ Zs ) @ ( lexord @ A @ R2 ) ) ) ) ).
% lexord_same_pref_if_irrefl
thf(fact_6366_list_Odisc__eq__case_I2_J,axiom,
! [A: $tType,List: list @ A] :
( ( List
!= ( nil @ A ) )
= ( case_list @ $o @ A @ $false
@ ^ [Uu3: A,Uv3: list @ A] : $true
@ List ) ) ).
% list.disc_eq_case(2)
thf(fact_6367_list_Odisc__eq__case_I1_J,axiom,
! [A: $tType,List: list @ A] :
( ( List
= ( nil @ A ) )
= ( case_list @ $o @ A @ $true
@ ^ [Uu3: A,Uv3: list @ A] : $false
@ List ) ) ).
% list.disc_eq_case(1)
thf(fact_6368_irrefl__lex,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( irrefl @ A @ R2 )
=> ( irrefl @ ( list @ A ) @ ( lex @ A @ R2 ) ) ) ).
% irrefl_lex
thf(fact_6369_lexord__irrefl,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( irrefl @ A @ R )
=> ( irrefl @ ( list @ A ) @ ( lexord @ A @ R ) ) ) ).
% lexord_irrefl
thf(fact_6370_hd__def,axiom,
! [A: $tType] :
( ( hd @ A )
= ( case_list @ A @ A @ ( undefined @ A )
@ ^ [X213: A,X224: list @ A] : X213 ) ) ).
% hd_def
thf(fact_6371_lexl__not__refl,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),X: list @ A] :
( ( irrefl @ A @ R2 )
=> ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ X ) @ ( lex @ A @ R2 ) ) ) ).
% lexl_not_refl
thf(fact_6372_Func__empty,axiom,
! [B: $tType,A: $tType,B2: set @ B] :
( ( bNF_Wellorder_Func @ A @ B @ ( bot_bot @ ( set @ A ) ) @ B2 )
= ( insert2 @ ( A > B )
@ ^ [X4: A] : ( undefined @ B )
@ ( bot_bot @ ( set @ ( A > B ) ) ) ) ) ).
% Func_empty
thf(fact_6373_arg__min__list_Oelims,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ B )
=> ! [X: A > B,Xa2: list @ A,Y: A] :
( ( ( arg_min_list @ A @ B @ X @ Xa2 )
= Y )
=> ( ! [X5: A] :
( ( Xa2
= ( cons @ A @ X5 @ ( nil @ A ) ) )
=> ( Y != X5 ) )
=> ( ! [X5: A,Y3: A,Zs2: list @ A] :
( ( Xa2
= ( cons @ A @ X5 @ ( cons @ A @ Y3 @ Zs2 ) ) )
=> ( Y
!= ( if @ A @ ( ord_less_eq @ B @ ( X @ X5 ) @ ( X @ ( arg_min_list @ A @ B @ X @ ( cons @ A @ Y3 @ Zs2 ) ) ) ) @ X5 @ ( arg_min_list @ A @ B @ X @ ( cons @ A @ Y3 @ Zs2 ) ) ) ) )
=> ~ ( ( Xa2
= ( nil @ A ) )
=> ( Y
!= ( undefined @ A ) ) ) ) ) ) ) ).
% arg_min_list.elims
thf(fact_6374_bit_Oabstract__boolean__algebra__axioms,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( boolea2506097494486148201lgebra @ A @ ( bit_se5824344872417868541ns_and @ A ) @ ( bit_se1065995026697491101ons_or @ A ) @ ( bit_ri4277139882892585799ns_not @ A ) @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% bit.abstract_boolean_algebra_axioms
thf(fact_6375_list__encode_Opelims,axiom,
! [X: list @ nat,Y: nat] :
( ( ( nat_list_encode @ X )
= Y )
=> ( ( accp @ ( list @ nat ) @ nat_list_encode_rel @ X )
=> ( ( ( X
= ( nil @ nat ) )
=> ( ( Y
= ( zero_zero @ nat ) )
=> ~ ( accp @ ( list @ nat ) @ nat_list_encode_rel @ ( nil @ nat ) ) ) )
=> ~ ! [X5: nat,Xs2: list @ nat] :
( ( X
= ( cons @ nat @ X5 @ Xs2 ) )
=> ( ( Y
= ( suc @ ( nat_prod_encode @ ( product_Pair @ nat @ nat @ X5 @ ( nat_list_encode @ Xs2 ) ) ) ) )
=> ~ ( accp @ ( list @ nat ) @ nat_list_encode_rel @ ( cons @ nat @ X5 @ Xs2 ) ) ) ) ) ) ) ).
% list_encode.pelims
thf(fact_6376_boolean__algebra_Oabstract__boolean__algebra__axioms,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ( boolea2506097494486148201lgebra @ A @ ( inf_inf @ A ) @ ( sup_sup @ A ) @ ( uminus_uminus @ A ) @ ( bot_bot @ A ) @ ( top_top @ A ) ) ) ).
% boolean_algebra.abstract_boolean_algebra_axioms
thf(fact_6377_min__list_Opelims,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X: list @ A,Y: A] :
( ( ( min_list @ A @ X )
= Y )
=> ( ( accp @ ( list @ A ) @ ( min_list_rel @ A ) @ X )
=> ( ! [X5: A,Xs2: list @ A] :
( ( X
= ( cons @ A @ X5 @ Xs2 ) )
=> ( ( Y
= ( case_list @ A @ A @ X5
@ ^ [A5: A,List3: list @ A] : ( ord_min @ A @ X5 @ ( min_list @ A @ Xs2 ) )
@ Xs2 ) )
=> ~ ( accp @ ( list @ A ) @ ( min_list_rel @ A ) @ ( cons @ A @ X5 @ Xs2 ) ) ) )
=> ~ ( ( X
= ( nil @ A ) )
=> ( ( Y
= ( undefined @ A ) )
=> ~ ( accp @ ( list @ A ) @ ( min_list_rel @ A ) @ ( nil @ A ) ) ) ) ) ) ) ) ).
% min_list.pelims
thf(fact_6378_remdups__adj_Opelims,axiom,
! [A: $tType,X: list @ A,Y: list @ A] :
( ( ( remdups_adj @ A @ X )
= Y )
=> ( ( accp @ ( list @ A ) @ ( remdups_adj_rel @ A ) @ X )
=> ( ( ( X
= ( nil @ A ) )
=> ( ( Y
= ( nil @ A ) )
=> ~ ( accp @ ( list @ A ) @ ( remdups_adj_rel @ A ) @ ( nil @ A ) ) ) )
=> ( ! [X5: A] :
( ( X
= ( cons @ A @ X5 @ ( nil @ A ) ) )
=> ( ( Y
= ( cons @ A @ X5 @ ( nil @ A ) ) )
=> ~ ( accp @ ( list @ A ) @ ( remdups_adj_rel @ A ) @ ( cons @ A @ X5 @ ( nil @ A ) ) ) ) )
=> ~ ! [X5: A,Y3: A,Xs2: list @ A] :
( ( X
= ( cons @ A @ X5 @ ( cons @ A @ Y3 @ Xs2 ) ) )
=> ( ( ( ( X5 = Y3 )
=> ( Y
= ( remdups_adj @ A @ ( cons @ A @ X5 @ Xs2 ) ) ) )
& ( ( X5 != Y3 )
=> ( Y
= ( cons @ A @ X5 @ ( remdups_adj @ A @ ( cons @ A @ Y3 @ Xs2 ) ) ) ) ) )
=> ~ ( accp @ ( list @ A ) @ ( remdups_adj_rel @ A ) @ ( cons @ A @ X5 @ ( cons @ A @ Y3 @ Xs2 ) ) ) ) ) ) ) ) ) ).
% remdups_adj.pelims
thf(fact_6379_to__nat__on__finite,axiom,
! [A: $tType,S2: set @ A] :
( ( finite_finite2 @ A @ S2 )
=> ( bij_betw @ A @ nat @ ( countable_to_nat_on @ A @ S2 ) @ S2 @ ( set_ord_lessThan @ nat @ ( finite_card @ A @ S2 ) ) ) ) ).
% to_nat_on_finite
thf(fact_6380_upt__rec__numeral,axiom,
! [M2: num,N2: num] :
( ( ( ord_less @ nat @ ( numeral_numeral @ nat @ M2 ) @ ( numeral_numeral @ nat @ N2 ) )
=> ( ( upt @ ( numeral_numeral @ nat @ M2 ) @ ( numeral_numeral @ nat @ N2 ) )
= ( cons @ nat @ ( numeral_numeral @ nat @ M2 ) @ ( upt @ ( suc @ ( numeral_numeral @ nat @ M2 ) ) @ ( numeral_numeral @ nat @ N2 ) ) ) ) )
& ( ~ ( ord_less @ nat @ ( numeral_numeral @ nat @ M2 ) @ ( numeral_numeral @ nat @ N2 ) )
=> ( ( upt @ ( numeral_numeral @ nat @ M2 ) @ ( numeral_numeral @ nat @ N2 ) )
= ( nil @ nat ) ) ) ) ).
% upt_rec_numeral
thf(fact_6381_remdups__upt,axiom,
! [M2: nat,N2: nat] :
( ( remdups @ nat @ ( upt @ M2 @ N2 ) )
= ( upt @ M2 @ N2 ) ) ).
% remdups_upt
thf(fact_6382_tl__upt,axiom,
! [M2: nat,N2: nat] :
( ( tl @ nat @ ( upt @ M2 @ N2 ) )
= ( upt @ ( suc @ M2 ) @ N2 ) ) ).
% tl_upt
thf(fact_6383_hd__upt,axiom,
! [I: nat,J: nat] :
( ( ord_less @ nat @ I @ J )
=> ( ( hd @ nat @ ( upt @ I @ J ) )
= I ) ) ).
% hd_upt
thf(fact_6384_drop__upt,axiom,
! [M2: nat,I: nat,J: nat] :
( ( drop @ nat @ M2 @ ( upt @ I @ J ) )
= ( upt @ ( plus_plus @ nat @ I @ M2 ) @ J ) ) ).
% drop_upt
thf(fact_6385_length__upt,axiom,
! [I: nat,J: nat] :
( ( size_size @ ( list @ nat ) @ ( upt @ I @ J ) )
= ( minus_minus @ nat @ J @ I ) ) ).
% length_upt
thf(fact_6386_take__upt,axiom,
! [I: nat,M2: nat,N2: nat] :
( ( ord_less_eq @ nat @ ( plus_plus @ nat @ I @ M2 ) @ N2 )
=> ( ( take @ nat @ M2 @ ( upt @ I @ N2 ) )
= ( upt @ I @ ( plus_plus @ nat @ I @ M2 ) ) ) ) ).
% take_upt
thf(fact_6387_upt__conv__Nil,axiom,
! [J: nat,I: nat] :
( ( ord_less_eq @ nat @ J @ I )
=> ( ( upt @ I @ J )
= ( nil @ nat ) ) ) ).
% upt_conv_Nil
thf(fact_6388_sorted__list__of__set__range,axiom,
! [M2: nat,N2: nat] :
( ( linord4507533701916653071of_set @ nat @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N2 ) )
= ( upt @ M2 @ N2 ) ) ).
% sorted_list_of_set_range
thf(fact_6389_upt__eq__Nil__conv,axiom,
! [I: nat,J: nat] :
( ( ( upt @ I @ J )
= ( nil @ nat ) )
= ( ( J
= ( zero_zero @ nat ) )
| ( ord_less_eq @ nat @ J @ I ) ) ) ).
% upt_eq_Nil_conv
thf(fact_6390_nth__upt,axiom,
! [I: nat,K: nat,J: nat] :
( ( ord_less @ nat @ ( plus_plus @ nat @ I @ K ) @ J )
=> ( ( nth @ nat @ ( upt @ I @ J ) @ K )
= ( plus_plus @ nat @ I @ K ) ) ) ).
% nth_upt
thf(fact_6391_upt__0,axiom,
! [I: nat] :
( ( upt @ I @ ( zero_zero @ nat ) )
= ( nil @ nat ) ) ).
% upt_0
thf(fact_6392_upt__conv__Cons__Cons,axiom,
! [M2: nat,N2: nat,Ns: list @ nat,Q3: nat] :
( ( ( cons @ nat @ M2 @ ( cons @ nat @ N2 @ Ns ) )
= ( upt @ M2 @ Q3 ) )
= ( ( cons @ nat @ N2 @ Ns )
= ( upt @ ( suc @ M2 ) @ Q3 ) ) ) ).
% upt_conv_Cons_Cons
thf(fact_6393_greaterThanAtMost__upt,axiom,
( ( set_or3652927894154168847AtMost @ nat )
= ( ^ [N: nat,M: nat] : ( set2 @ nat @ ( upt @ ( suc @ N ) @ ( suc @ M ) ) ) ) ) ).
% greaterThanAtMost_upt
thf(fact_6394_distinct__upt,axiom,
! [I: nat,J: nat] : ( distinct @ nat @ ( upt @ I @ J ) ) ).
% distinct_upt
thf(fact_6395_atLeastAtMost__upt,axiom,
( ( set_or1337092689740270186AtMost @ nat )
= ( ^ [N: nat,M: nat] : ( set2 @ nat @ ( upt @ N @ ( suc @ M ) ) ) ) ) ).
% atLeastAtMost_upt
thf(fact_6396_atLeastLessThan__upt,axiom,
( ( set_or7035219750837199246ssThan @ nat )
= ( ^ [I3: nat,J2: nat] : ( set2 @ nat @ ( upt @ I3 @ J2 ) ) ) ) ).
% atLeastLessThan_upt
thf(fact_6397_atLeast__upt,axiom,
( ( set_ord_lessThan @ nat )
= ( ^ [N: nat] : ( set2 @ nat @ ( upt @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% atLeast_upt
thf(fact_6398_greaterThanLessThan__upt,axiom,
( ( set_or5935395276787703475ssThan @ nat )
= ( ^ [N: nat,M: nat] : ( set2 @ nat @ ( upt @ ( suc @ N ) @ M ) ) ) ) ).
% greaterThanLessThan_upt
thf(fact_6399_atMost__upto,axiom,
( ( set_ord_atMost @ nat )
= ( ^ [N: nat] : ( set2 @ nat @ ( upt @ ( zero_zero @ nat ) @ ( suc @ N ) ) ) ) ) ).
% atMost_upto
thf(fact_6400_upt__conv__Cons,axiom,
! [I: nat,J: nat] :
( ( ord_less @ nat @ I @ J )
=> ( ( upt @ I @ J )
= ( cons @ nat @ I @ ( upt @ ( suc @ I ) @ J ) ) ) ) ).
% upt_conv_Cons
thf(fact_6401_enumerate__eq__zip,axiom,
! [A: $tType] :
( ( enumerate @ A )
= ( ^ [N: nat,Xs3: list @ A] : ( zip @ nat @ A @ ( upt @ N @ ( plus_plus @ nat @ N @ ( size_size @ ( list @ A ) @ Xs3 ) ) ) @ Xs3 ) ) ) ).
% enumerate_eq_zip
thf(fact_6402_upt__add__eq__append,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( upt @ I @ ( plus_plus @ nat @ J @ K ) )
= ( append @ nat @ ( upt @ I @ J ) @ ( upt @ J @ ( plus_plus @ nat @ J @ K ) ) ) ) ) ).
% upt_add_eq_append
thf(fact_6403_upt__eq__Cons__conv,axiom,
! [I: nat,J: nat,X: nat,Xs: list @ nat] :
( ( ( upt @ I @ J )
= ( cons @ nat @ X @ Xs ) )
= ( ( ord_less @ nat @ I @ J )
& ( I = X )
& ( ( upt @ ( plus_plus @ nat @ I @ ( one_one @ nat ) ) @ J )
= Xs ) ) ) ).
% upt_eq_Cons_conv
thf(fact_6404_upt__rec,axiom,
( upt
= ( ^ [I3: nat,J2: nat] : ( if @ ( list @ nat ) @ ( ord_less @ nat @ I3 @ J2 ) @ ( cons @ nat @ I3 @ ( upt @ ( suc @ I3 ) @ J2 ) ) @ ( nil @ nat ) ) ) ) ).
% upt_rec
thf(fact_6405_upt__Suc,axiom,
! [I: nat,J: nat] :
( ( ( ord_less_eq @ nat @ I @ J )
=> ( ( upt @ I @ ( suc @ J ) )
= ( append @ nat @ ( upt @ I @ J ) @ ( cons @ nat @ J @ ( nil @ nat ) ) ) ) )
& ( ~ ( ord_less_eq @ nat @ I @ J )
=> ( ( upt @ I @ ( suc @ J ) )
= ( nil @ nat ) ) ) ) ).
% upt_Suc
thf(fact_6406_upt__Suc__append,axiom,
! [I: nat,J: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( upt @ I @ ( suc @ J ) )
= ( append @ nat @ ( upt @ I @ J ) @ ( cons @ nat @ J @ ( nil @ nat ) ) ) ) ) ).
% upt_Suc_append
thf(fact_6407_horner__sum__bit__eq__take__bit,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A,N2: nat] :
( ( groups4207007520872428315er_sum @ $o @ A @ ( zero_neq_one_of_bool @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( map @ nat @ $o @ ( bit_se5641148757651400278ts_bit @ A @ A3 ) @ ( upt @ ( zero_zero @ nat ) @ N2 ) ) )
= ( bit_se2584673776208193580ke_bit @ A @ N2 @ A3 ) ) ) ).
% horner_sum_bit_eq_take_bit
thf(fact_6408_to__nat__on__def,axiom,
! [A: $tType] :
( ( countable_to_nat_on @ A )
= ( ^ [S6: set @ A] :
( fChoice @ ( A > nat )
@ ^ [F3: A > nat] :
( ( ( finite_finite2 @ A @ S6 )
=> ( bij_betw @ A @ nat @ F3 @ S6 @ ( set_ord_lessThan @ nat @ ( finite_card @ A @ S6 ) ) ) )
& ( ~ ( finite_finite2 @ A @ S6 )
=> ( bij_betw @ A @ nat @ F3 @ S6 @ ( top_top @ ( set @ nat ) ) ) ) ) ) ) ) ).
% to_nat_on_def
thf(fact_6409_map__ident,axiom,
! [A: $tType] :
( ( map @ A @ A
@ ^ [X4: A] : X4 )
= ( ^ [Xs3: list @ A] : Xs3 ) ) ).
% map_ident
thf(fact_6410_list_Omap__disc__iff,axiom,
! [B: $tType,A: $tType,F2: A > B,A3: list @ A] :
( ( ( map @ A @ B @ F2 @ A3 )
= ( nil @ B ) )
= ( A3
= ( nil @ A ) ) ) ).
% list.map_disc_iff
thf(fact_6411_Nil__is__map__conv,axiom,
! [A: $tType,B: $tType,F2: B > A,Xs: list @ B] :
( ( ( nil @ A )
= ( map @ B @ A @ F2 @ Xs ) )
= ( Xs
= ( nil @ B ) ) ) ).
% Nil_is_map_conv
thf(fact_6412_map__is__Nil__conv,axiom,
! [A: $tType,B: $tType,F2: B > A,Xs: list @ B] :
( ( ( map @ B @ A @ F2 @ Xs )
= ( nil @ A ) )
= ( Xs
= ( nil @ B ) ) ) ).
% map_is_Nil_conv
thf(fact_6413_map__map,axiom,
! [B: $tType,A: $tType,C: $tType,F2: B > A,G: C > B,Xs: list @ C] :
( ( map @ B @ A @ F2 @ ( map @ C @ B @ G @ Xs ) )
= ( map @ C @ A @ ( comp @ B @ A @ C @ F2 @ G ) @ Xs ) ) ).
% map_map
thf(fact_6414_List_Omap_Ocompositionality,axiom,
! [B: $tType,C: $tType,A: $tType,F2: B > C,G: A > B,List: list @ A] :
( ( map @ B @ C @ F2 @ ( map @ A @ B @ G @ List ) )
= ( map @ A @ C @ ( comp @ B @ C @ A @ F2 @ G ) @ List ) ) ).
% List.map.compositionality
thf(fact_6415_list_Omap__comp,axiom,
! [B: $tType,C: $tType,A: $tType,G: B > C,F2: A > B,V2: list @ A] :
( ( map @ B @ C @ G @ ( map @ A @ B @ F2 @ V2 ) )
= ( map @ A @ C @ ( comp @ B @ C @ A @ G @ F2 ) @ V2 ) ) ).
% list.map_comp
thf(fact_6416_map__eq__conv,axiom,
! [A: $tType,B: $tType,F2: B > A,Xs: list @ B,G: B > A] :
( ( ( map @ B @ A @ F2 @ Xs )
= ( map @ B @ A @ G @ Xs ) )
= ( ! [X4: B] :
( ( member @ B @ X4 @ ( set2 @ B @ Xs ) )
=> ( ( F2 @ X4 )
= ( G @ X4 ) ) ) ) ) ).
% map_eq_conv
thf(fact_6417_length__map,axiom,
! [A: $tType,B: $tType,F2: B > A,Xs: list @ B] :
( ( size_size @ ( list @ A ) @ ( map @ B @ A @ F2 @ Xs ) )
= ( size_size @ ( list @ B ) @ Xs ) ) ).
% length_map
thf(fact_6418_map__append,axiom,
! [A: $tType,B: $tType,F2: B > A,Xs: list @ B,Ys: list @ B] :
( ( map @ B @ A @ F2 @ ( append @ B @ Xs @ Ys ) )
= ( append @ A @ ( map @ B @ A @ F2 @ Xs ) @ ( map @ B @ A @ F2 @ Ys ) ) ) ).
% map_append
thf(fact_6419_map__replicate,axiom,
! [A: $tType,B: $tType,F2: B > A,N2: nat,X: B] :
( ( map @ B @ A @ F2 @ ( replicate @ B @ N2 @ X ) )
= ( replicate @ A @ N2 @ ( F2 @ X ) ) ) ).
% map_replicate
thf(fact_6420_list_Oset__map,axiom,
! [B: $tType,A: $tType,F2: A > B,V2: list @ A] :
( ( set2 @ B @ ( map @ A @ B @ F2 @ V2 ) )
= ( image2 @ A @ B @ F2 @ ( set2 @ A @ V2 ) ) ) ).
% list.set_map
thf(fact_6421_inj__map__eq__map,axiom,
! [B: $tType,A: $tType,F2: A > B,Xs: list @ A,Ys: list @ A] :
( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
=> ( ( ( map @ A @ B @ F2 @ Xs )
= ( map @ A @ B @ F2 @ Ys ) )
= ( Xs = Ys ) ) ) ).
% inj_map_eq_map
thf(fact_6422_map__fun__upd,axiom,
! [B: $tType,A: $tType,Y: A,Xs: list @ A,F2: A > B,V2: B] :
( ~ ( member @ A @ Y @ ( set2 @ A @ Xs ) )
=> ( ( map @ A @ B @ ( fun_upd @ A @ B @ F2 @ Y @ V2 ) @ Xs )
= ( map @ A @ B @ F2 @ Xs ) ) ) ).
% map_fun_upd
thf(fact_6423_map__comp__map,axiom,
! [B: $tType,C: $tType,A: $tType,F2: C > B,G: A > C] :
( ( comp @ ( list @ C ) @ ( list @ B ) @ ( list @ A ) @ ( map @ C @ B @ F2 ) @ ( map @ A @ C @ G ) )
= ( map @ A @ B @ ( comp @ C @ B @ A @ F2 @ G ) ) ) ).
% map_comp_map
thf(fact_6424_List_Omap_Ocomp,axiom,
! [C: $tType,B: $tType,A: $tType,F2: B > C,G: A > B] :
( ( comp @ ( list @ B ) @ ( list @ C ) @ ( list @ A ) @ ( map @ B @ C @ F2 ) @ ( map @ A @ B @ G ) )
= ( map @ A @ C @ ( comp @ B @ C @ A @ F2 @ G ) ) ) ).
% List.map.comp
thf(fact_6425_size__list__map,axiom,
! [A: $tType,B: $tType,F2: A > nat,G: B > A,Xs: list @ B] :
( ( size_list @ A @ F2 @ ( map @ B @ A @ G @ Xs ) )
= ( size_list @ B @ ( comp @ A @ nat @ B @ F2 @ G ) @ Xs ) ) ).
% size_list_map
thf(fact_6426_list_Omap__id0,axiom,
! [A: $tType] :
( ( map @ A @ A @ ( id @ A ) )
= ( id @ ( list @ A ) ) ) ).
% list.map_id0
thf(fact_6427_nth__map,axiom,
! [B: $tType,A: $tType,N2: nat,Xs: list @ A,F2: A > B] :
( ( ord_less @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( nth @ B @ ( map @ A @ B @ F2 @ Xs ) @ N2 )
= ( F2 @ ( nth @ A @ Xs @ N2 ) ) ) ) ).
% nth_map
thf(fact_6428_concat__map__singleton,axiom,
! [A: $tType,B: $tType,F2: B > A,Xs: list @ B] :
( ( concat @ A
@ ( map @ B @ ( list @ A )
@ ^ [X4: B] : ( cons @ A @ ( F2 @ X4 ) @ ( nil @ A ) )
@ Xs ) )
= ( map @ B @ A @ F2 @ Xs ) ) ).
% concat_map_singleton
thf(fact_6429_inj__map,axiom,
! [B: $tType,A: $tType,F2: A > B] :
( ( inj_on @ ( list @ A ) @ ( list @ B ) @ ( map @ A @ B @ F2 ) @ ( top_top @ ( set @ ( list @ A ) ) ) )
= ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) ) ) ).
% inj_map
thf(fact_6430_inj__mapI,axiom,
! [B: $tType,A: $tType,F2: A > B] :
( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
=> ( inj_on @ ( list @ A ) @ ( list @ B ) @ ( map @ A @ B @ F2 ) @ ( top_top @ ( set @ ( list @ A ) ) ) ) ) ).
% inj_mapI
thf(fact_6431_enumerate__map__upt,axiom,
! [A: $tType,N2: nat,F2: nat > A,M2: nat] :
( ( enumerate @ A @ N2 @ ( map @ nat @ A @ F2 @ ( upt @ N2 @ M2 ) ) )
= ( map @ nat @ ( product_prod @ nat @ A )
@ ^ [K2: nat] : ( product_Pair @ nat @ A @ K2 @ ( F2 @ K2 ) )
@ ( upt @ N2 @ M2 ) ) ) ).
% enumerate_map_upt
thf(fact_6432_ex__map__conv,axiom,
! [A: $tType,B: $tType,Ys: list @ B,F2: A > B] :
( ( ? [Xs3: list @ A] :
( Ys
= ( map @ A @ B @ F2 @ Xs3 ) ) )
= ( ! [X4: B] :
( ( member @ B @ X4 @ ( set2 @ B @ Ys ) )
=> ? [Y6: A] :
( X4
= ( F2 @ Y6 ) ) ) ) ) ).
% ex_map_conv
thf(fact_6433_map__cong,axiom,
! [B: $tType,A: $tType,Xs: list @ A,Ys: list @ A,F2: A > B,G: A > B] :
( ( Xs = Ys )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Ys ) )
=> ( ( F2 @ X5 )
= ( G @ X5 ) ) )
=> ( ( map @ A @ B @ F2 @ Xs )
= ( map @ A @ B @ G @ Ys ) ) ) ) ).
% map_cong
thf(fact_6434_map__idI,axiom,
! [A: $tType,Xs: list @ A,F2: A > A] :
( ! [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Xs ) )
=> ( ( F2 @ X5 )
= X5 ) )
=> ( ( map @ A @ A @ F2 @ Xs )
= Xs ) ) ).
% map_idI
thf(fact_6435_map__ext,axiom,
! [B: $tType,A: $tType,Xs: list @ A,F2: A > B,G: A > B] :
( ! [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Xs ) )
=> ( ( F2 @ X5 )
= ( G @ X5 ) ) )
=> ( ( map @ A @ B @ F2 @ Xs )
= ( map @ A @ B @ G @ Xs ) ) ) ).
% map_ext
thf(fact_6436_list_Oinj__map__strong,axiom,
! [B: $tType,A: $tType,X: list @ A,Xa2: list @ A,F2: A > B,Fa: A > B] :
( ! [Z: A,Za: A] :
( ( member @ A @ Z @ ( set2 @ A @ X ) )
=> ( ( member @ A @ Za @ ( set2 @ A @ Xa2 ) )
=> ( ( ( F2 @ Z )
= ( Fa @ Za ) )
=> ( Z = Za ) ) ) )
=> ( ( ( map @ A @ B @ F2 @ X )
= ( map @ A @ B @ Fa @ Xa2 ) )
=> ( X = Xa2 ) ) ) ).
% list.inj_map_strong
thf(fact_6437_list_Omap__cong0,axiom,
! [B: $tType,A: $tType,X: list @ A,F2: A > B,G: A > B] :
( ! [Z: A] :
( ( member @ A @ Z @ ( set2 @ A @ X ) )
=> ( ( F2 @ Z )
= ( G @ Z ) ) )
=> ( ( map @ A @ B @ F2 @ X )
= ( map @ A @ B @ G @ X ) ) ) ).
% list.map_cong0
thf(fact_6438_list_Omap__cong,axiom,
! [B: $tType,A: $tType,X: list @ A,Ya: list @ A,F2: A > B,G: A > B] :
( ( X = Ya )
=> ( ! [Z: A] :
( ( member @ A @ Z @ ( set2 @ A @ Ya ) )
=> ( ( F2 @ Z )
= ( G @ Z ) ) )
=> ( ( map @ A @ B @ F2 @ X )
= ( map @ A @ B @ G @ Ya ) ) ) ) ).
% list.map_cong
thf(fact_6439_map__eq__imp__length__eq,axiom,
! [A: $tType,B: $tType,C: $tType,F2: B > A,Xs: list @ B,G: C > A,Ys: list @ C] :
( ( ( map @ B @ A @ F2 @ Xs )
= ( map @ C @ A @ G @ Ys ) )
=> ( ( size_size @ ( list @ B ) @ Xs )
= ( size_size @ ( list @ C ) @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_6440_map__replicate__const,axiom,
! [B: $tType,A: $tType,K: A,Lst: list @ B] :
( ( map @ B @ A
@ ^ [X4: B] : K
@ Lst )
= ( replicate @ A @ ( size_size @ ( list @ B ) @ Lst ) @ K ) ) ).
% map_replicate_const
thf(fact_6441_append__eq__map__conv,axiom,
! [A: $tType,B: $tType,Ys: list @ A,Zs: list @ A,F2: B > A,Xs: list @ B] :
( ( ( append @ A @ Ys @ Zs )
= ( map @ B @ A @ F2 @ Xs ) )
= ( ? [Us2: list @ B,Vs3: list @ B] :
( ( Xs
= ( append @ B @ Us2 @ Vs3 ) )
& ( Ys
= ( map @ B @ A @ F2 @ Us2 ) )
& ( Zs
= ( map @ B @ A @ F2 @ Vs3 ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_6442_map__eq__append__conv,axiom,
! [A: $tType,B: $tType,F2: B > A,Xs: list @ B,Ys: list @ A,Zs: list @ A] :
( ( ( map @ B @ A @ F2 @ Xs )
= ( append @ A @ Ys @ Zs ) )
= ( ? [Us2: list @ B,Vs3: list @ B] :
( ( Xs
= ( append @ B @ Us2 @ Vs3 ) )
& ( Ys
= ( map @ B @ A @ F2 @ Us2 ) )
& ( Zs
= ( map @ B @ A @ F2 @ Vs3 ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_6443_image__set,axiom,
! [A: $tType,B: $tType,F2: B > A,Xs: list @ B] :
( ( image2 @ B @ A @ F2 @ ( set2 @ B @ Xs ) )
= ( set2 @ A @ ( map @ B @ A @ F2 @ Xs ) ) ) ).
% image_set
thf(fact_6444_map__zip__map2,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,F2: ( product_prod @ B @ C ) > A,Xs: list @ B,G: D > C,Ys: list @ D] :
( ( map @ ( product_prod @ B @ C ) @ A @ F2 @ ( zip @ B @ C @ Xs @ ( map @ D @ C @ G @ Ys ) ) )
= ( map @ ( product_prod @ B @ D ) @ A
@ ( product_case_prod @ B @ D @ A
@ ^ [X4: B,Y6: D] : ( F2 @ ( product_Pair @ B @ C @ X4 @ ( G @ Y6 ) ) ) )
@ ( zip @ B @ D @ Xs @ Ys ) ) ) ).
% map_zip_map2
thf(fact_6445_zip__map__map,axiom,
! [B: $tType,A: $tType,C: $tType,D: $tType,F2: C > A,Xs: list @ C,G: D > B,Ys: list @ D] :
( ( zip @ A @ B @ ( map @ C @ A @ F2 @ Xs ) @ ( map @ D @ B @ G @ Ys ) )
= ( map @ ( product_prod @ C @ D ) @ ( product_prod @ A @ B )
@ ( product_case_prod @ C @ D @ ( product_prod @ A @ B )
@ ^ [X4: C,Y6: D] : ( product_Pair @ A @ B @ ( F2 @ X4 ) @ ( G @ Y6 ) ) )
@ ( zip @ C @ D @ Xs @ Ys ) ) ) ).
% zip_map_map
thf(fact_6446_map__zip__map,axiom,
! [B: $tType,A: $tType,D: $tType,C: $tType,F2: ( product_prod @ B @ C ) > A,G: D > B,Xs: list @ D,Ys: list @ C] :
( ( map @ ( product_prod @ B @ C ) @ A @ F2 @ ( zip @ B @ C @ ( map @ D @ B @ G @ Xs ) @ Ys ) )
= ( map @ ( product_prod @ D @ C ) @ A
@ ( product_case_prod @ D @ C @ A
@ ^ [X4: D,Y6: C] : ( F2 @ ( product_Pair @ B @ C @ ( G @ X4 ) @ Y6 ) ) )
@ ( zip @ D @ C @ Xs @ Ys ) ) ) ).
% map_zip_map
thf(fact_6447_zip__map2,axiom,
! [B: $tType,A: $tType,C: $tType,Xs: list @ A,F2: C > B,Ys: list @ C] :
( ( zip @ A @ B @ Xs @ ( map @ C @ B @ F2 @ Ys ) )
= ( map @ ( product_prod @ A @ C ) @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ C @ ( product_prod @ A @ B )
@ ^ [X4: A,Y6: C] : ( product_Pair @ A @ B @ X4 @ ( F2 @ Y6 ) ) )
@ ( zip @ A @ C @ Xs @ Ys ) ) ) ).
% zip_map2
thf(fact_6448_zip__map1,axiom,
! [A: $tType,C: $tType,B: $tType,F2: C > A,Xs: list @ C,Ys: list @ B] :
( ( zip @ A @ B @ ( map @ C @ A @ F2 @ Xs ) @ Ys )
= ( map @ ( product_prod @ C @ B ) @ ( product_prod @ A @ B )
@ ( product_case_prod @ C @ B @ ( product_prod @ A @ B )
@ ^ [X4: C] : ( product_Pair @ A @ B @ ( F2 @ X4 ) ) )
@ ( zip @ C @ B @ Xs @ Ys ) ) ) ).
% zip_map1
thf(fact_6449_some__in__eq,axiom,
! [A: $tType,A4: set @ A] :
( ( member @ A
@ ( fChoice @ A
@ ^ [X4: A] : ( member @ A @ X4 @ A4 ) )
@ A4 )
= ( A4
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% some_in_eq
thf(fact_6450_map__eq__Cons__conv,axiom,
! [A: $tType,B: $tType,F2: B > A,Xs: list @ B,Y: A,Ys: list @ A] :
( ( ( map @ B @ A @ F2 @ Xs )
= ( cons @ A @ Y @ Ys ) )
= ( ? [Z6: B,Zs3: list @ B] :
( ( Xs
= ( cons @ B @ Z6 @ Zs3 ) )
& ( ( F2 @ Z6 )
= Y )
& ( ( map @ B @ A @ F2 @ Zs3 )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_6451_Cons__eq__map__conv,axiom,
! [A: $tType,B: $tType,X: A,Xs: list @ A,F2: B > A,Ys: list @ B] :
( ( ( cons @ A @ X @ Xs )
= ( map @ B @ A @ F2 @ Ys ) )
= ( ? [Z6: B,Zs3: list @ B] :
( ( Ys
= ( cons @ B @ Z6 @ Zs3 ) )
& ( X
= ( F2 @ Z6 ) )
& ( Xs
= ( map @ B @ A @ F2 @ Zs3 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_6452_map__eq__Cons__D,axiom,
! [B: $tType,A: $tType,F2: B > A,Xs: list @ B,Y: A,Ys: list @ A] :
( ( ( map @ B @ A @ F2 @ Xs )
= ( cons @ A @ Y @ Ys ) )
=> ? [Z: B,Zs2: list @ B] :
( ( Xs
= ( cons @ B @ Z @ Zs2 ) )
& ( ( F2 @ Z )
= Y )
& ( ( map @ B @ A @ F2 @ Zs2 )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_6453_Cons__eq__map__D,axiom,
! [A: $tType,B: $tType,X: A,Xs: list @ A,F2: B > A,Ys: list @ B] :
( ( ( cons @ A @ X @ Xs )
= ( map @ B @ A @ F2 @ Ys ) )
=> ? [Z: B,Zs2: list @ B] :
( ( Ys
= ( cons @ B @ Z @ Zs2 ) )
& ( X
= ( F2 @ Z ) )
& ( Xs
= ( map @ B @ A @ F2 @ Zs2 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_6454_list_Osimps_I9_J,axiom,
! [B: $tType,A: $tType,F2: A > B,X21: A,X222: list @ A] :
( ( map @ A @ B @ F2 @ ( cons @ A @ X21 @ X222 ) )
= ( cons @ B @ ( F2 @ X21 ) @ ( map @ A @ B @ F2 @ X222 ) ) ) ).
% list.simps(9)
thf(fact_6455_list_Osize__gen__o__map,axiom,
! [B: $tType,A: $tType,F2: B > nat,G: A > B] :
( ( comp @ ( list @ B ) @ nat @ ( list @ A ) @ ( size_list @ B @ F2 ) @ ( map @ A @ B @ G ) )
= ( size_list @ A @ ( comp @ B @ nat @ A @ F2 @ G ) ) ) ).
% list.size_gen_o_map
thf(fact_6456_list_Omap__id,axiom,
! [A: $tType,T2: list @ A] :
( ( map @ A @ A @ ( id @ A ) @ T2 )
= T2 ) ).
% list.map_id
thf(fact_6457_List_Omap_Oidentity,axiom,
! [A: $tType] :
( ( map @ A @ A
@ ^ [X4: A] : X4 )
= ( id @ ( list @ A ) ) ) ).
% List.map.identity
thf(fact_6458_map__tl,axiom,
! [A: $tType,B: $tType,F2: B > A,Xs: list @ B] :
( ( map @ B @ A @ F2 @ ( tl @ B @ Xs ) )
= ( tl @ A @ ( map @ B @ A @ F2 @ Xs ) ) ) ).
% map_tl
thf(fact_6459_list_Omap__ident,axiom,
! [A: $tType,T2: list @ A] :
( ( map @ A @ A
@ ^ [X4: A] : X4
@ T2 )
= T2 ) ).
% list.map_ident
thf(fact_6460_remdups__map__remdups,axiom,
! [A: $tType,B: $tType,F2: B > A,Xs: list @ B] :
( ( remdups @ A @ ( map @ B @ A @ F2 @ ( remdups @ B @ Xs ) ) )
= ( remdups @ A @ ( map @ B @ A @ F2 @ Xs ) ) ) ).
% remdups_map_remdups
thf(fact_6461_rotate1__map,axiom,
! [A: $tType,B: $tType,F2: B > A,Xs: list @ B] :
( ( rotate1 @ A @ ( map @ B @ A @ F2 @ Xs ) )
= ( map @ B @ A @ F2 @ ( rotate1 @ B @ Xs ) ) ) ).
% rotate1_map
thf(fact_6462_map__concat,axiom,
! [A: $tType,B: $tType,F2: B > A,Xs: list @ ( list @ B )] :
( ( map @ B @ A @ F2 @ ( concat @ B @ Xs ) )
= ( concat @ A @ ( map @ ( list @ B ) @ ( list @ A ) @ ( map @ B @ A @ F2 ) @ Xs ) ) ) ).
% map_concat
thf(fact_6463_nths__map,axiom,
! [A: $tType,B: $tType,F2: B > A,Xs: list @ B,I6: set @ nat] :
( ( nths @ A @ ( map @ B @ A @ F2 @ Xs ) @ I6 )
= ( map @ B @ A @ F2 @ ( nths @ B @ Xs @ I6 ) ) ) ).
% nths_map
thf(fact_6464_map__update,axiom,
! [A: $tType,B: $tType,F2: B > A,Xs: list @ B,K: nat,Y: B] :
( ( map @ B @ A @ F2 @ ( list_update @ B @ Xs @ K @ Y ) )
= ( list_update @ A @ ( map @ B @ A @ F2 @ Xs ) @ K @ ( F2 @ Y ) ) ) ).
% map_update
thf(fact_6465_take__map,axiom,
! [A: $tType,B: $tType,N2: nat,F2: B > A,Xs: list @ B] :
( ( take @ A @ N2 @ ( map @ B @ A @ F2 @ Xs ) )
= ( map @ B @ A @ F2 @ ( take @ B @ N2 @ Xs ) ) ) ).
% take_map
thf(fact_6466_map2__map__map,axiom,
! [B: $tType,A: $tType,C: $tType,D: $tType,H2: B > C > A,F2: D > B,Xs: list @ D,G: D > C] :
( ( map @ ( product_prod @ B @ C ) @ A @ ( product_case_prod @ B @ C @ A @ H2 ) @ ( zip @ B @ C @ ( map @ D @ B @ F2 @ Xs ) @ ( map @ D @ C @ G @ Xs ) ) )
= ( map @ D @ A
@ ^ [X4: D] : ( H2 @ ( F2 @ X4 ) @ ( G @ X4 ) )
@ Xs ) ) ).
% map2_map_map
thf(fact_6467_rotate__map,axiom,
! [A: $tType,B: $tType,N2: nat,F2: B > A,Xs: list @ B] :
( ( rotate @ A @ N2 @ ( map @ B @ A @ F2 @ Xs ) )
= ( map @ B @ A @ F2 @ ( rotate @ B @ N2 @ Xs ) ) ) ).
% rotate_map
thf(fact_6468_map__injective,axiom,
! [A: $tType,B: $tType,F2: B > A,Xs: list @ B,Ys: list @ B] :
( ( ( map @ B @ A @ F2 @ Xs )
= ( map @ B @ A @ F2 @ Ys ) )
=> ( ( inj_on @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) )
=> ( Xs = Ys ) ) ) ).
% map_injective
thf(fact_6469_drop__map,axiom,
! [A: $tType,B: $tType,N2: nat,F2: B > A,Xs: list @ B] :
( ( drop @ A @ N2 @ ( map @ B @ A @ F2 @ Xs ) )
= ( map @ B @ A @ F2 @ ( drop @ B @ N2 @ Xs ) ) ) ).
% drop_map
thf(fact_6470_list_Omap__sel_I1_J,axiom,
! [B: $tType,A: $tType,A3: list @ A,F2: A > B] :
( ( A3
!= ( nil @ A ) )
=> ( ( hd @ B @ ( map @ A @ B @ F2 @ A3 ) )
= ( F2 @ ( hd @ A @ A3 ) ) ) ) ).
% list.map_sel(1)
thf(fact_6471_hd__map,axiom,
! [B: $tType,A: $tType,Xs: list @ A,F2: A > B] :
( ( Xs
!= ( nil @ A ) )
=> ( ( hd @ B @ ( map @ A @ B @ F2 @ Xs ) )
= ( F2 @ ( hd @ A @ Xs ) ) ) ) ).
% hd_map
thf(fact_6472_list_Omap__sel_I2_J,axiom,
! [B: $tType,A: $tType,A3: list @ A,F2: A > B] :
( ( A3
!= ( nil @ A ) )
=> ( ( tl @ B @ ( map @ A @ B @ F2 @ A3 ) )
= ( map @ A @ B @ F2 @ ( tl @ A @ A3 ) ) ) ) ).
% list.map_sel(2)
thf(fact_6473_list_Osimps_I8_J,axiom,
! [A: $tType,B: $tType,F2: A > B] :
( ( map @ A @ B @ F2 @ ( nil @ A ) )
= ( nil @ B ) ) ).
% list.simps(8)
thf(fact_6474_map__inj__on,axiom,
! [A: $tType,B: $tType,F2: B > A,Xs: list @ B,Ys: list @ B] :
( ( ( map @ B @ A @ F2 @ Xs )
= ( map @ B @ A @ F2 @ Ys ) )
=> ( ( inj_on @ B @ A @ F2 @ ( sup_sup @ ( set @ B ) @ ( set2 @ B @ Xs ) @ ( set2 @ B @ Ys ) ) )
=> ( Xs = Ys ) ) ) ).
% map_inj_on
thf(fact_6475_inj__on__map__eq__map,axiom,
! [B: $tType,A: $tType,F2: A > B,Xs: list @ A,Ys: list @ A] :
( ( inj_on @ A @ B @ F2 @ ( sup_sup @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ Ys ) ) )
=> ( ( ( map @ A @ B @ F2 @ Xs )
= ( map @ A @ B @ F2 @ Ys ) )
= ( Xs = Ys ) ) ) ).
% inj_on_map_eq_map
thf(fact_6476_distinct__map,axiom,
! [A: $tType,B: $tType,F2: B > A,Xs: list @ B] :
( ( distinct @ A @ ( map @ B @ A @ F2 @ Xs ) )
= ( ( distinct @ B @ Xs )
& ( inj_on @ B @ A @ F2 @ ( set2 @ B @ Xs ) ) ) ) ).
% distinct_map
thf(fact_6477_remdups__adj__map__injective,axiom,
! [B: $tType,A: $tType,F2: A > B,Xs: list @ A] :
( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
=> ( ( remdups_adj @ B @ ( map @ A @ B @ F2 @ Xs ) )
= ( map @ A @ B @ F2 @ ( remdups_adj @ A @ Xs ) ) ) ) ).
% remdups_adj_map_injective
thf(fact_6478_map__removeAll__inj,axiom,
! [B: $tType,A: $tType,F2: A > B,X: A,Xs: list @ A] :
( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
=> ( ( map @ A @ B @ F2 @ ( removeAll @ A @ X @ Xs ) )
= ( removeAll @ B @ ( F2 @ X ) @ ( map @ A @ B @ F2 @ Xs ) ) ) ) ).
% map_removeAll_inj
thf(fact_6479_map__replicate__trivial,axiom,
! [A: $tType,X: A,I: nat] :
( ( map @ nat @ A
@ ^ [I3: nat] : X
@ ( upt @ ( zero_zero @ nat ) @ I ) )
= ( replicate @ A @ I @ X ) ) ).
% map_replicate_trivial
thf(fact_6480_distinct__insort__key,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F2: B > A,X: B,Xs: list @ B] :
( ( distinct @ A @ ( map @ B @ A @ F2 @ ( linorder_insort_key @ B @ A @ F2 @ X @ Xs ) ) )
= ( ~ ( member @ A @ ( F2 @ X ) @ ( image2 @ B @ A @ F2 @ ( set2 @ B @ Xs ) ) )
& ( distinct @ A @ ( map @ B @ A @ F2 @ Xs ) ) ) ) ) ).
% distinct_insort_key
thf(fact_6481_map__removeAll__inj__on,axiom,
! [B: $tType,A: $tType,F2: A > B,X: A,Xs: list @ A] :
( ( inj_on @ A @ B @ F2 @ ( insert2 @ A @ X @ ( set2 @ A @ Xs ) ) )
=> ( ( map @ A @ B @ F2 @ ( removeAll @ A @ X @ Xs ) )
= ( removeAll @ B @ ( F2 @ X ) @ ( map @ A @ B @ F2 @ Xs ) ) ) ) ).
% map_removeAll_inj_on
thf(fact_6482_map__upt__Suc,axiom,
! [A: $tType,F2: nat > A,N2: nat] :
( ( map @ nat @ A @ F2 @ ( upt @ ( zero_zero @ nat ) @ ( suc @ N2 ) ) )
= ( cons @ A @ ( F2 @ ( zero_zero @ nat ) )
@ ( map @ nat @ A
@ ^ [I3: nat] : ( F2 @ ( suc @ I3 ) )
@ ( upt @ ( zero_zero @ nat ) @ N2 ) ) ) ) ).
% map_upt_Suc
thf(fact_6483_map__nth,axiom,
! [A: $tType,Xs: list @ A] :
( ( map @ nat @ A @ ( nth @ A @ Xs ) @ ( upt @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Xs ) ) )
= Xs ) ).
% map_nth
thf(fact_6484_inj__mapD,axiom,
! [B: $tType,A: $tType,F2: A > B] :
( ( inj_on @ ( list @ A ) @ ( list @ B ) @ ( map @ A @ B @ F2 ) @ ( top_top @ ( set @ ( list @ A ) ) ) )
=> ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) ) ) ).
% inj_mapD
thf(fact_6485_nth__map__upt,axiom,
! [A: $tType,I: nat,N2: nat,M2: nat,F2: nat > A] :
( ( ord_less @ nat @ I @ ( minus_minus @ nat @ N2 @ M2 ) )
=> ( ( nth @ A @ ( map @ nat @ A @ F2 @ ( upt @ M2 @ N2 ) ) @ I )
= ( F2 @ ( plus_plus @ nat @ M2 @ I ) ) ) ) ).
% nth_map_upt
thf(fact_6486_map__of__zip__map,axiom,
! [B: $tType,A: $tType,Xs: list @ A,F2: A > B] :
( ( map_of @ A @ B @ ( zip @ A @ B @ Xs @ ( map @ A @ B @ F2 @ Xs ) ) )
= ( ^ [X4: A] : ( if @ ( option @ B ) @ ( member @ A @ X4 @ ( set2 @ A @ Xs ) ) @ ( some @ B @ ( F2 @ X4 ) ) @ ( none @ B ) ) ) ) ).
% map_of_zip_map
thf(fact_6487_arg__min__SOME__Min,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ B )
=> ! [S2: set @ A,F2: A > B] :
( ( finite_finite2 @ A @ S2 )
=> ( ( lattic7623131987881927897min_on @ A @ B @ F2 @ S2 )
= ( fChoice @ A
@ ^ [Y6: A] :
( ( member @ A @ Y6 @ S2 )
& ( ( F2 @ Y6 )
= ( lattic643756798350308766er_Min @ B @ ( image2 @ A @ B @ F2 @ S2 ) ) ) ) ) ) ) ) ).
% arg_min_SOME_Min
thf(fact_6488_inj__on__mapI,axiom,
! [B: $tType,A: $tType,F2: A > B,A4: set @ ( list @ A )] :
( ( inj_on @ A @ B @ F2 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ ( list @ A ) @ ( set @ A ) @ ( set2 @ A ) @ A4 ) ) )
=> ( inj_on @ ( list @ A ) @ ( list @ B ) @ ( map @ A @ B @ F2 ) @ A4 ) ) ).
% inj_on_mapI
thf(fact_6489_map__upt__eqI,axiom,
! [A: $tType,Xs: list @ A,N2: nat,M2: nat,F2: nat > A] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( minus_minus @ nat @ N2 @ M2 ) )
=> ( ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( nth @ A @ Xs @ I4 )
= ( F2 @ ( plus_plus @ nat @ M2 @ I4 ) ) ) )
=> ( ( map @ nat @ A @ F2 @ ( upt @ M2 @ N2 ) )
= Xs ) ) ) ).
% map_upt_eqI
thf(fact_6490_transpose__rectangle,axiom,
! [A: $tType,Xs: list @ ( list @ A ),N2: nat] :
( ( ( Xs
= ( nil @ ( list @ A ) ) )
=> ( N2
= ( zero_zero @ nat ) ) )
=> ( ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( size_size @ ( list @ ( list @ A ) ) @ Xs ) )
=> ( ( size_size @ ( list @ A ) @ ( nth @ ( list @ A ) @ Xs @ I4 ) )
= N2 ) )
=> ( ( transpose @ A @ Xs )
= ( map @ nat @ ( list @ A )
@ ^ [I3: nat] :
( map @ nat @ A
@ ^ [J2: nat] : ( nth @ A @ ( nth @ ( list @ A ) @ Xs @ J2 ) @ I3 )
@ ( upt @ ( zero_zero @ nat ) @ ( size_size @ ( list @ ( list @ A ) ) @ Xs ) ) )
@ ( upt @ ( zero_zero @ nat ) @ N2 ) ) ) ) ) ).
% transpose_rectangle
thf(fact_6491_prod_Oset__conv__list,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: B > A,Xs: list @ B] :
( ( groups7121269368397514597t_prod @ B @ A @ G @ ( set2 @ B @ Xs ) )
= ( groups5270119922927024881d_list @ A @ ( map @ B @ A @ G @ ( remdups @ B @ Xs ) ) ) ) ) ).
% prod.set_conv_list
thf(fact_6492_map__Suc__upt,axiom,
! [M2: nat,N2: nat] :
( ( map @ nat @ nat @ suc @ ( upt @ M2 @ N2 ) )
= ( upt @ ( suc @ M2 ) @ ( suc @ N2 ) ) ) ).
% map_Suc_upt
thf(fact_6493_transpose_Osimps_I3_J,axiom,
! [A: $tType,X: A,Xs: list @ A,Xss: list @ ( list @ A )] :
( ( transpose @ A @ ( cons @ ( list @ A ) @ ( cons @ A @ X @ Xs ) @ Xss ) )
= ( cons @ ( list @ A )
@ ( cons @ A @ X
@ ( concat @ A
@ ( map @ ( list @ A ) @ ( list @ A )
@ ( case_list @ ( list @ A ) @ A @ ( nil @ A )
@ ^ [H: A,T3: list @ A] : ( cons @ A @ H @ ( nil @ A ) ) )
@ Xss ) ) )
@ ( transpose @ A
@ ( cons @ ( list @ A ) @ Xs
@ ( concat @ ( list @ A )
@ ( map @ ( list @ A ) @ ( list @ ( list @ A ) )
@ ( case_list @ ( list @ ( list @ A ) ) @ A @ ( nil @ ( list @ A ) )
@ ^ [H: A,T3: list @ A] : ( cons @ ( list @ A ) @ T3 @ ( nil @ ( list @ A ) ) ) )
@ Xss ) ) ) ) ) ) ).
% transpose.simps(3)
thf(fact_6494_transpose_Oelims,axiom,
! [A: $tType,X: list @ ( list @ A ),Y: list @ ( list @ A )] :
( ( ( transpose @ A @ X )
= Y )
=> ( ( ( X
= ( nil @ ( list @ A ) ) )
=> ( Y
!= ( nil @ ( list @ A ) ) ) )
=> ( ! [Xss2: list @ ( list @ A )] :
( ( X
= ( cons @ ( list @ A ) @ ( nil @ A ) @ Xss2 ) )
=> ( Y
!= ( transpose @ A @ Xss2 ) ) )
=> ~ ! [X5: A,Xs2: list @ A,Xss2: list @ ( list @ A )] :
( ( X
= ( cons @ ( list @ A ) @ ( cons @ A @ X5 @ Xs2 ) @ Xss2 ) )
=> ( Y
!= ( cons @ ( list @ A )
@ ( cons @ A @ X5
@ ( concat @ A
@ ( map @ ( list @ A ) @ ( list @ A )
@ ( case_list @ ( list @ A ) @ A @ ( nil @ A )
@ ^ [H: A,T3: list @ A] : ( cons @ A @ H @ ( nil @ A ) ) )
@ Xss2 ) ) )
@ ( transpose @ A
@ ( cons @ ( list @ A ) @ Xs2
@ ( concat @ ( list @ A )
@ ( map @ ( list @ A ) @ ( list @ ( list @ A ) )
@ ( case_list @ ( list @ ( list @ A ) ) @ A @ ( nil @ ( list @ A ) )
@ ^ [H: A,T3: list @ A] : ( cons @ ( list @ A ) @ T3 @ ( nil @ ( list @ A ) ) ) )
@ Xss2 ) ) ) ) ) ) ) ) ) ) ).
% transpose.elims
thf(fact_6495_product__concat__map,axiom,
! [B: $tType,A: $tType] :
( ( product @ A @ B )
= ( ^ [Xs3: list @ A,Ys3: list @ B] :
( concat @ ( product_prod @ A @ B )
@ ( map @ A @ ( list @ ( product_prod @ A @ B ) )
@ ^ [X4: A] : ( map @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 ) @ Ys3 )
@ Xs3 ) ) ) ) ).
% product_concat_map
thf(fact_6496_zip__assoc,axiom,
! [B: $tType,A: $tType,C: $tType,Xs: list @ A,Ys: list @ B,Zs: list @ C] :
( ( zip @ A @ ( product_prod @ B @ C ) @ Xs @ ( zip @ B @ C @ Ys @ Zs ) )
= ( map @ ( product_prod @ ( product_prod @ A @ B ) @ C ) @ ( product_prod @ A @ ( product_prod @ B @ C ) )
@ ( product_case_prod @ ( product_prod @ A @ B ) @ C @ ( product_prod @ A @ ( product_prod @ B @ C ) )
@ ( product_case_prod @ A @ B @ ( C > ( product_prod @ A @ ( product_prod @ B @ C ) ) )
@ ^ [X4: A,Y6: B,Z6: C] : ( product_Pair @ A @ ( product_prod @ B @ C ) @ X4 @ ( product_Pair @ B @ C @ Y6 @ Z6 ) ) ) )
@ ( zip @ ( product_prod @ A @ B ) @ C @ ( zip @ A @ B @ Xs @ Ys ) @ Zs ) ) ) ).
% zip_assoc
thf(fact_6497_zip__same__conv__map,axiom,
! [A: $tType,Xs: list @ A] :
( ( zip @ A @ A @ Xs @ Xs )
= ( map @ A @ ( product_prod @ A @ A )
@ ^ [X4: A] : ( product_Pair @ A @ A @ X4 @ X4 )
@ Xs ) ) ).
% zip_same_conv_map
thf(fact_6498_n__lists_Osimps_I2_J,axiom,
! [A: $tType,N2: nat,Xs: list @ A] :
( ( n_lists @ A @ ( suc @ N2 ) @ Xs )
= ( concat @ ( list @ A )
@ ( map @ ( list @ A ) @ ( list @ ( list @ A ) )
@ ^ [Ys3: list @ A] :
( map @ A @ ( list @ A )
@ ^ [Y6: A] : ( cons @ A @ Y6 @ Ys3 )
@ Xs )
@ ( n_lists @ A @ N2 @ Xs ) ) ) ) ).
% n_lists.simps(2)
thf(fact_6499_product__lists_Osimps_I2_J,axiom,
! [A: $tType,Xs: list @ A,Xss: list @ ( list @ A )] :
( ( product_lists @ A @ ( cons @ ( list @ A ) @ Xs @ Xss ) )
= ( concat @ ( list @ A )
@ ( map @ A @ ( list @ ( list @ A ) )
@ ^ [X4: A] : ( map @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X4 ) @ ( product_lists @ A @ Xss ) )
@ Xs ) ) ) ).
% product_lists.simps(2)
thf(fact_6500_transpose__map__map,axiom,
! [A: $tType,B: $tType,F2: B > A,Xs: list @ ( list @ B )] :
( ( transpose @ A @ ( map @ ( list @ B ) @ ( list @ A ) @ ( map @ B @ A @ F2 ) @ Xs ) )
= ( map @ ( list @ B ) @ ( list @ A ) @ ( map @ B @ A @ F2 ) @ ( transpose @ B @ Xs ) ) ) ).
% transpose_map_map
thf(fact_6501_List_Obind__def,axiom,
! [B: $tType,A: $tType] :
( ( bind @ A @ B )
= ( ^ [Xs3: list @ A,F3: A > ( list @ B )] : ( concat @ B @ ( map @ A @ ( list @ B ) @ F3 @ Xs3 ) ) ) ) ).
% List.bind_def
thf(fact_6502_transpose_Osimps_I1_J,axiom,
! [A: $tType] :
( ( transpose @ A @ ( nil @ ( list @ A ) ) )
= ( nil @ ( list @ A ) ) ) ).
% transpose.simps(1)
thf(fact_6503_inj__split__Cons,axiom,
! [A: $tType,X8: 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,N: A] : ( cons @ A @ N @ Xs3 ) )
@ X8 ) ).
% inj_split_Cons
thf(fact_6504_transpose_Osimps_I2_J,axiom,
! [A: $tType,Xss: list @ ( list @ A )] :
( ( transpose @ A @ ( cons @ ( list @ A ) @ ( nil @ A ) @ Xss ) )
= ( transpose @ A @ Xss ) ) ).
% transpose.simps(2)
thf(fact_6505_prod__list__zero__iff,axiom,
! [A: $tType] :
( ( ( semiring_1 @ A )
& ( semiri3467727345109120633visors @ A ) )
=> ! [Xs: list @ A] :
( ( ( groups5270119922927024881d_list @ A @ Xs )
= ( zero_zero @ A ) )
= ( member @ A @ ( zero_zero @ A ) @ ( set2 @ A @ Xs ) ) ) ) ).
% prod_list_zero_iff
thf(fact_6506_distinct__set__subseqs,axiom,
! [A: $tType,Xs: list @ A] :
( ( distinct @ A @ Xs )
=> ( distinct @ ( set @ A ) @ ( map @ ( list @ A ) @ ( set @ A ) @ ( set2 @ A ) @ ( subseqs @ A @ Xs ) ) ) ) ).
% distinct_set_subseqs
thf(fact_6507_map__add__upt,axiom,
! [N2: nat,M2: nat] :
( ( map @ nat @ nat
@ ^ [I3: nat] : ( plus_plus @ nat @ I3 @ N2 )
@ ( upt @ ( zero_zero @ nat ) @ M2 ) )
= ( upt @ N2 @ ( plus_plus @ nat @ M2 @ N2 ) ) ) ).
% map_add_upt
thf(fact_6508_zip__left__commute,axiom,
! [B: $tType,A: $tType,C: $tType,Xs: list @ A,Ys: list @ B,Zs: list @ C] :
( ( zip @ A @ ( product_prod @ B @ C ) @ Xs @ ( zip @ B @ C @ Ys @ Zs ) )
= ( map @ ( product_prod @ B @ ( product_prod @ A @ C ) ) @ ( product_prod @ A @ ( product_prod @ B @ C ) )
@ ( product_case_prod @ B @ ( product_prod @ A @ C ) @ ( product_prod @ A @ ( product_prod @ B @ C ) )
@ ^ [Y6: B] :
( product_case_prod @ A @ C @ ( product_prod @ A @ ( product_prod @ B @ C ) )
@ ^ [X4: A,Z6: C] : ( product_Pair @ A @ ( product_prod @ B @ C ) @ X4 @ ( product_Pair @ B @ C @ Y6 @ Z6 ) ) ) )
@ ( zip @ B @ ( product_prod @ A @ C ) @ Ys @ ( zip @ A @ C @ Xs @ Zs ) ) ) ) ).
% zip_left_commute
thf(fact_6509_zip__commute,axiom,
! [B: $tType,A: $tType] :
( ( zip @ A @ B )
= ( ^ [Xs3: list @ A,Ys3: list @ B] :
( map @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B )
@ ( product_case_prod @ B @ A @ ( product_prod @ A @ B )
@ ^ [X4: B,Y6: A] : ( product_Pair @ A @ B @ Y6 @ X4 ) )
@ ( zip @ B @ A @ Ys3 @ Xs3 ) ) ) ) ).
% zip_commute
thf(fact_6510_comp__fun__commute__relcomp__fold,axiom,
! [A: $tType,B: $tType,C: $tType,S2: set @ ( product_prod @ A @ B )] :
( ( finite_finite2 @ ( product_prod @ A @ B ) @ S2 )
=> ( finite6289374366891150609ommute @ ( product_prod @ C @ A ) @ ( set @ ( product_prod @ C @ B ) )
@ ( product_case_prod @ C @ A @ ( ( set @ ( product_prod @ C @ B ) ) > ( set @ ( product_prod @ C @ B ) ) )
@ ^ [X4: C,Y6: A,A7: set @ ( product_prod @ C @ B )] :
( finite_fold @ ( product_prod @ A @ B ) @ ( set @ ( product_prod @ C @ B ) )
@ ( product_case_prod @ A @ B @ ( ( set @ ( product_prod @ C @ B ) ) > ( set @ ( product_prod @ C @ B ) ) )
@ ^ [W3: A,Z6: B,A17: set @ ( product_prod @ C @ B )] : ( if @ ( set @ ( product_prod @ C @ B ) ) @ ( Y6 = W3 ) @ ( insert2 @ ( product_prod @ C @ B ) @ ( product_Pair @ C @ B @ X4 @ Z6 ) @ A17 ) @ A17 ) )
@ A7
@ S2 ) ) ) ) ).
% comp_fun_commute_relcomp_fold
thf(fact_6511_zip__replicate1,axiom,
! [A: $tType,B: $tType,N2: nat,X: A,Ys: list @ B] :
( ( zip @ A @ B @ ( replicate @ A @ N2 @ X ) @ Ys )
= ( map @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X ) @ ( take @ B @ N2 @ Ys ) ) ) ).
% zip_replicate1
thf(fact_6512_subseqs_Osimps_I2_J,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( subseqs @ A @ ( cons @ A @ X @ Xs ) )
= ( append @ ( list @ A ) @ ( map @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X ) @ ( subseqs @ A @ Xs ) ) @ ( subseqs @ A @ Xs ) ) ) ).
% subseqs.simps(2)
thf(fact_6513_map__decr__upt,axiom,
! [M2: nat,N2: nat] :
( ( map @ nat @ nat
@ ^ [N: nat] : ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) )
@ ( upt @ ( suc @ M2 ) @ ( suc @ N2 ) ) )
= ( upt @ M2 @ N2 ) ) ).
% map_decr_upt
thf(fact_6514_transpose__empty,axiom,
! [A: $tType,Xs: list @ ( list @ A )] :
( ( ( transpose @ A @ Xs )
= ( nil @ ( list @ A ) ) )
= ( ! [X4: list @ A] :
( ( member @ ( list @ A ) @ X4 @ ( set2 @ ( list @ A ) @ Xs ) )
=> ( X4
= ( nil @ A ) ) ) ) ) ).
% transpose_empty
thf(fact_6515_zip__replicate2,axiom,
! [B: $tType,A: $tType,Xs: list @ A,N2: nat,Y: B] :
( ( zip @ A @ B @ Xs @ ( replicate @ B @ N2 @ Y ) )
= ( map @ A @ ( product_prod @ A @ B )
@ ^ [X4: A] : ( product_Pair @ A @ B @ X4 @ Y )
@ ( take @ A @ N2 @ Xs ) ) ) ).
% zip_replicate2
thf(fact_6516_Id__on__set,axiom,
! [A: $tType,Xs: list @ A] :
( ( id_on @ A @ ( set2 @ A @ Xs ) )
= ( set2 @ ( product_prod @ A @ A )
@ ( map @ A @ ( product_prod @ A @ A )
@ ^ [X4: A] : ( product_Pair @ A @ A @ X4 @ X4 )
@ Xs ) ) ) ).
% Id_on_set
thf(fact_6517_product_Osimps_I2_J,axiom,
! [A: $tType,B: $tType,X: A,Xs: list @ A,Ys: list @ B] :
( ( product @ A @ B @ ( cons @ A @ X @ Xs ) @ Ys )
= ( append @ ( product_prod @ A @ B ) @ ( map @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X ) @ Ys ) @ ( product @ A @ B @ Xs @ Ys ) ) ) ).
% product.simps(2)
thf(fact_6518_extract__Cons__code,axiom,
! [A: $tType,P: A > $o,X: A,Xs: list @ A] :
( ( ( P @ X )
=> ( ( extract @ A @ P @ ( cons @ A @ X @ Xs ) )
= ( some @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) @ ( product_Pair @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) @ ( nil @ A ) @ ( product_Pair @ A @ ( list @ A ) @ X @ Xs ) ) ) ) )
& ( ~ ( P @ X )
=> ( ( extract @ A @ P @ ( cons @ A @ X @ Xs ) )
= ( case_option @ ( option @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) ) @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) @ ( none @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) )
@ ( product_case_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) @ ( option @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) )
@ ^ [Ys3: list @ A] :
( product_case_prod @ A @ ( list @ A ) @ ( option @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) )
@ ^ [Y6: A,Zs3: list @ A] : ( some @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) @ ( product_Pair @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) @ ( cons @ A @ X @ Ys3 ) @ ( product_Pair @ A @ ( list @ A ) @ Y6 @ Zs3 ) ) ) ) )
@ ( extract @ A @ P @ Xs ) ) ) ) ) ).
% extract_Cons_code
thf(fact_6519_map__of__map__restrict,axiom,
! [B: $tType,A: $tType,F2: A > B,Ks: list @ A] :
( ( map_of @ A @ B
@ ( map @ A @ ( product_prod @ A @ B )
@ ^ [K2: A] : ( product_Pair @ A @ B @ K2 @ ( F2 @ K2 ) )
@ Ks ) )
= ( restrict_map @ A @ B @ ( comp @ B @ ( option @ B ) @ A @ ( some @ B ) @ F2 ) @ ( set2 @ A @ Ks ) ) ) ).
% map_of_map_restrict
thf(fact_6520_map__of__map__keys,axiom,
! [B: $tType,A: $tType,Xs: list @ A,M2: A > ( option @ B )] :
( ( ( set2 @ A @ Xs )
= ( dom @ A @ B @ M2 ) )
=> ( ( map_of @ A @ B
@ ( map @ A @ ( product_prod @ A @ B )
@ ^ [K2: A] : ( product_Pair @ A @ B @ K2 @ ( the2 @ B @ ( M2 @ K2 ) ) )
@ Xs ) )
= M2 ) ) ).
% map_of_map_keys
thf(fact_6521_enumerate__replicate__eq,axiom,
! [A: $tType,N2: nat,M2: nat,A3: A] :
( ( enumerate @ A @ N2 @ ( replicate @ A @ M2 @ A3 ) )
= ( map @ nat @ ( product_prod @ nat @ A )
@ ^ [Q5: nat] : ( product_Pair @ nat @ A @ Q5 @ A3 )
@ ( upt @ N2 @ ( plus_plus @ nat @ N2 @ M2 ) ) ) ) ).
% enumerate_replicate_eq
thf(fact_6522_prod_Odistinct__set__conv__list,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [Xs: list @ B,G: B > A] :
( ( distinct @ B @ Xs )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( set2 @ B @ Xs ) )
= ( groups5270119922927024881d_list @ A @ ( map @ B @ A @ G @ Xs ) ) ) ) ) ).
% prod.distinct_set_conv_list
thf(fact_6523_transpose_Opsimps_I3_J,axiom,
! [A: $tType,X: A,Xs: list @ A,Xss: list @ ( list @ A )] :
( ( accp @ ( list @ ( list @ A ) ) @ ( transpose_rel @ A ) @ ( cons @ ( list @ A ) @ ( cons @ A @ X @ Xs ) @ Xss ) )
=> ( ( transpose @ A @ ( cons @ ( list @ A ) @ ( cons @ A @ X @ Xs ) @ Xss ) )
= ( cons @ ( list @ A )
@ ( cons @ A @ X
@ ( concat @ A
@ ( map @ ( list @ A ) @ ( list @ A )
@ ( case_list @ ( list @ A ) @ A @ ( nil @ A )
@ ^ [H: A,T3: list @ A] : ( cons @ A @ H @ ( nil @ A ) ) )
@ Xss ) ) )
@ ( transpose @ A
@ ( cons @ ( list @ A ) @ Xs
@ ( concat @ ( list @ A )
@ ( map @ ( list @ A ) @ ( list @ ( list @ A ) )
@ ( case_list @ ( list @ ( list @ A ) ) @ A @ ( nil @ ( list @ A ) )
@ ^ [H: A,T3: list @ A] : ( cons @ ( list @ A ) @ T3 @ ( nil @ ( list @ A ) ) ) )
@ Xss ) ) ) ) ) ) ) ).
% transpose.psimps(3)
thf(fact_6524_transpose_Opelims,axiom,
! [A: $tType,X: list @ ( list @ A ),Y: list @ ( list @ A )] :
( ( ( transpose @ A @ X )
= Y )
=> ( ( accp @ ( list @ ( list @ A ) ) @ ( transpose_rel @ A ) @ X )
=> ( ( ( X
= ( nil @ ( list @ A ) ) )
=> ( ( Y
= ( nil @ ( list @ A ) ) )
=> ~ ( accp @ ( list @ ( list @ A ) ) @ ( transpose_rel @ A ) @ ( nil @ ( list @ A ) ) ) ) )
=> ( ! [Xss2: list @ ( list @ A )] :
( ( X
= ( cons @ ( list @ A ) @ ( nil @ A ) @ Xss2 ) )
=> ( ( Y
= ( transpose @ A @ Xss2 ) )
=> ~ ( accp @ ( list @ ( list @ A ) ) @ ( transpose_rel @ A ) @ ( cons @ ( list @ A ) @ ( nil @ A ) @ Xss2 ) ) ) )
=> ~ ! [X5: A,Xs2: list @ A,Xss2: list @ ( list @ A )] :
( ( X
= ( cons @ ( list @ A ) @ ( cons @ A @ X5 @ Xs2 ) @ Xss2 ) )
=> ( ( Y
= ( cons @ ( list @ A )
@ ( cons @ A @ X5
@ ( concat @ A
@ ( map @ ( list @ A ) @ ( list @ A )
@ ( case_list @ ( list @ A ) @ A @ ( nil @ A )
@ ^ [H: A,T3: list @ A] : ( cons @ A @ H @ ( nil @ A ) ) )
@ Xss2 ) ) )
@ ( transpose @ A
@ ( cons @ ( list @ A ) @ Xs2
@ ( concat @ ( list @ A )
@ ( map @ ( list @ A ) @ ( list @ ( list @ A ) )
@ ( case_list @ ( list @ ( list @ A ) ) @ A @ ( nil @ ( list @ A ) )
@ ^ [H: A,T3: list @ A] : ( cons @ ( list @ A ) @ T3 @ ( nil @ ( list @ A ) ) ) )
@ Xss2 ) ) ) ) ) )
=> ~ ( accp @ ( list @ ( list @ A ) ) @ ( transpose_rel @ A ) @ ( cons @ ( list @ A ) @ ( cons @ A @ X5 @ Xs2 ) @ Xss2 ) ) ) ) ) ) ) ) ).
% transpose.pelims
thf(fact_6525_transpose_Opsimps_I1_J,axiom,
! [A: $tType] :
( ( accp @ ( list @ ( list @ A ) ) @ ( transpose_rel @ A ) @ ( nil @ ( list @ A ) ) )
=> ( ( transpose @ A @ ( nil @ ( list @ A ) ) )
= ( nil @ ( list @ A ) ) ) ) ).
% transpose.psimps(1)
thf(fact_6526_transpose_Opsimps_I2_J,axiom,
! [A: $tType,Xss: list @ ( list @ A )] :
( ( accp @ ( list @ ( list @ A ) ) @ ( transpose_rel @ A ) @ ( cons @ ( list @ A ) @ ( nil @ A ) @ Xss ) )
=> ( ( transpose @ A @ ( cons @ ( list @ A ) @ ( nil @ A ) @ Xss ) )
= ( transpose @ A @ Xss ) ) ) ).
% transpose.psimps(2)
thf(fact_6527_transpose_Opinduct,axiom,
! [A: $tType,A0: list @ ( list @ A ),P: ( list @ ( list @ A ) ) > $o] :
( ( accp @ ( list @ ( list @ A ) ) @ ( transpose_rel @ A ) @ A0 )
=> ( ( ( accp @ ( list @ ( list @ A ) ) @ ( transpose_rel @ A ) @ ( nil @ ( list @ A ) ) )
=> ( P @ ( nil @ ( list @ A ) ) ) )
=> ( ! [Xss2: list @ ( list @ A )] :
( ( accp @ ( list @ ( list @ A ) ) @ ( transpose_rel @ A ) @ ( cons @ ( list @ A ) @ ( nil @ A ) @ Xss2 ) )
=> ( ( P @ Xss2 )
=> ( P @ ( cons @ ( list @ A ) @ ( nil @ A ) @ Xss2 ) ) ) )
=> ( ! [X5: A,Xs2: list @ A,Xss2: list @ ( list @ A )] :
( ( accp @ ( list @ ( list @ A ) ) @ ( transpose_rel @ A ) @ ( cons @ ( list @ A ) @ ( cons @ A @ X5 @ Xs2 ) @ Xss2 ) )
=> ( ( P
@ ( cons @ ( list @ A ) @ Xs2
@ ( concat @ ( list @ A )
@ ( map @ ( list @ A ) @ ( list @ ( list @ A ) )
@ ( case_list @ ( list @ ( list @ A ) ) @ A @ ( nil @ ( list @ A ) )
@ ^ [H: A,T3: list @ A] : ( cons @ ( list @ A ) @ T3 @ ( nil @ ( list @ A ) ) ) )
@ Xss2 ) ) ) )
=> ( P @ ( cons @ ( list @ A ) @ ( cons @ A @ X5 @ Xs2 ) @ Xss2 ) ) ) )
=> ( P @ A0 ) ) ) ) ) ).
% transpose.pinduct
thf(fact_6528_product__code,axiom,
! [B: $tType,A: $tType,Xs: list @ A,Ys: list @ B] :
( ( product_product @ A @ B @ ( set2 @ A @ Xs ) @ ( set2 @ B @ Ys ) )
= ( set2 @ ( product_prod @ A @ B )
@ ( concat @ ( product_prod @ A @ B )
@ ( map @ A @ ( list @ ( product_prod @ A @ B ) )
@ ^ [X4: A] : ( map @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 ) @ Ys )
@ Xs ) ) ) ) ).
% product_code
thf(fact_6529_transpose__aux__filter__tail,axiom,
! [A: $tType,Xss: list @ ( list @ A )] :
( ( concat @ ( list @ A )
@ ( map @ ( list @ A ) @ ( list @ ( list @ A ) )
@ ( case_list @ ( list @ ( list @ A ) ) @ A @ ( nil @ ( list @ A ) )
@ ^ [H: A,T3: list @ A] : ( cons @ ( list @ A ) @ T3 @ ( nil @ ( list @ A ) ) ) )
@ Xss ) )
= ( map @ ( list @ A ) @ ( list @ A ) @ ( tl @ A )
@ ( filter2 @ ( list @ A )
@ ^ [Ys3: list @ A] :
( Ys3
!= ( nil @ A ) )
@ Xss ) ) ) ).
% transpose_aux_filter_tail
thf(fact_6530_filter__filter,axiom,
! [A: $tType,P: A > $o,Q: A > $o,Xs: list @ A] :
( ( filter2 @ A @ P @ ( filter2 @ A @ Q @ Xs ) )
= ( filter2 @ A
@ ^ [X4: A] :
( ( Q @ X4 )
& ( P @ X4 ) )
@ Xs ) ) ).
% filter_filter
thf(fact_6531_filter__True,axiom,
! [A: $tType,Xs: list @ A,P: A > $o] :
( ! [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Xs ) )
=> ( P @ X5 ) )
=> ( ( filter2 @ A @ P @ Xs )
= Xs ) ) ).
% filter_True
thf(fact_6532_filter__append,axiom,
! [A: $tType,P: A > $o,Xs: list @ A,Ys: list @ A] :
( ( filter2 @ A @ P @ ( append @ A @ Xs @ Ys ) )
= ( append @ A @ ( filter2 @ A @ P @ Xs ) @ ( filter2 @ A @ P @ Ys ) ) ) ).
% filter_append
thf(fact_6533_remove1__filter__not,axiom,
! [A: $tType,P: A > $o,X: A,Xs: list @ A] :
( ~ ( P @ X )
=> ( ( remove1 @ A @ X @ ( filter2 @ A @ P @ Xs ) )
= ( filter2 @ A @ P @ Xs ) ) ) ).
% remove1_filter_not
thf(fact_6534_removeAll__filter__not,axiom,
! [A: $tType,P: A > $o,X: A,Xs: list @ A] :
( ~ ( P @ X )
=> ( ( removeAll @ A @ X @ ( filter2 @ A @ P @ Xs ) )
= ( filter2 @ A @ P @ Xs ) ) ) ).
% removeAll_filter_not
thf(fact_6535_set__filter,axiom,
! [A: $tType,P: A > $o,Xs: list @ A] :
( ( set2 @ A @ ( filter2 @ A @ P @ Xs ) )
= ( collect @ A
@ ^ [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
& ( P @ X4 ) ) ) ) ).
% set_filter
thf(fact_6536_filter__False,axiom,
! [A: $tType,Xs: list @ A,P: A > $o] :
( ! [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Xs ) )
=> ~ ( P @ X5 ) )
=> ( ( filter2 @ A @ P @ Xs )
= ( nil @ A ) ) ) ).
% filter_False
thf(fact_6537_length__filter__map,axiom,
! [A: $tType,B: $tType,P: A > $o,F2: B > A,Xs: list @ B] :
( ( size_size @ ( list @ A ) @ ( filter2 @ A @ P @ ( map @ B @ A @ F2 @ Xs ) ) )
= ( size_size @ ( list @ B ) @ ( filter2 @ B @ ( comp @ A @ $o @ B @ P @ F2 ) @ Xs ) ) ) ).
% length_filter_map
thf(fact_6538_distinct__map__filter,axiom,
! [A: $tType,B: $tType,F2: B > A,Xs: list @ B,P: B > $o] :
( ( distinct @ A @ ( map @ B @ A @ F2 @ Xs ) )
=> ( distinct @ A @ ( map @ B @ A @ F2 @ ( filter2 @ B @ P @ Xs ) ) ) ) ).
% distinct_map_filter
thf(fact_6539_filter__map,axiom,
! [A: $tType,B: $tType,P: A > $o,F2: B > A,Xs: list @ B] :
( ( filter2 @ A @ P @ ( map @ B @ A @ F2 @ Xs ) )
= ( map @ B @ A @ F2 @ ( filter2 @ B @ ( comp @ A @ $o @ B @ P @ F2 ) @ Xs ) ) ) ).
% filter_map
thf(fact_6540_filter__concat,axiom,
! [A: $tType,P5: A > $o,Xs: list @ ( list @ A )] :
( ( filter2 @ A @ P5 @ ( concat @ A @ Xs ) )
= ( concat @ A @ ( map @ ( list @ A ) @ ( list @ A ) @ ( filter2 @ A @ P5 ) @ Xs ) ) ) ).
% filter_concat
thf(fact_6541_distinct__filter,axiom,
! [A: $tType,Xs: list @ A,P: A > $o] :
( ( distinct @ A @ Xs )
=> ( distinct @ A @ ( filter2 @ A @ P @ Xs ) ) ) ).
% distinct_filter
thf(fact_6542_empty__filter__conv,axiom,
! [A: $tType,P: A > $o,Xs: list @ A] :
( ( ( nil @ A )
= ( filter2 @ A @ P @ Xs ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
=> ~ ( P @ X4 ) ) ) ) ).
% empty_filter_conv
thf(fact_6543_filter__empty__conv,axiom,
! [A: $tType,P: A > $o,Xs: list @ A] :
( ( ( filter2 @ A @ P @ Xs )
= ( nil @ A ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
=> ~ ( P @ X4 ) ) ) ) ).
% filter_empty_conv
thf(fact_6544_replicate__length__filter,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( replicate @ A
@ ( size_size @ ( list @ A )
@ ( filter2 @ A
@ ( ^ [Y5: A,Z2: A] : Y5 = Z2
@ X )
@ Xs ) )
@ X )
= ( filter2 @ A
@ ( ^ [Y5: A,Z2: A] : Y5 = Z2
@ X )
@ Xs ) ) ).
% replicate_length_filter
thf(fact_6545_inter__set__filter,axiom,
! [A: $tType,A4: set @ A,Xs: list @ A] :
( ( inf_inf @ ( set @ A ) @ A4 @ ( set2 @ A @ Xs ) )
= ( set2 @ A
@ ( filter2 @ A
@ ^ [X4: A] : ( member @ A @ X4 @ A4 )
@ Xs ) ) ) ).
% inter_set_filter
thf(fact_6546_filter__set,axiom,
! [A: $tType,P: A > $o,Xs: list @ A] :
( ( filter3 @ A @ P @ ( set2 @ A @ Xs ) )
= ( set2 @ A @ ( filter2 @ A @ P @ Xs ) ) ) ).
% filter_set
thf(fact_6547_filter__id__conv,axiom,
! [A: $tType,P: A > $o,Xs: list @ A] :
( ( ( filter2 @ A @ P @ Xs )
= Xs )
= ( ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
=> ( P @ X4 ) ) ) ) ).
% filter_id_conv
thf(fact_6548_filter__cong,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,P: A > $o,Q: A > $o] :
( ( Xs = Ys )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Ys ) )
=> ( ( P @ X5 )
= ( Q @ X5 ) ) )
=> ( ( filter2 @ A @ P @ Xs )
= ( filter2 @ A @ Q @ Ys ) ) ) ) ).
% filter_cong
thf(fact_6549_sum__length__filter__compl,axiom,
! [A: $tType,P: A > $o,Xs: list @ A] :
( ( plus_plus @ nat @ ( size_size @ ( list @ A ) @ ( filter2 @ A @ P @ Xs ) )
@ ( size_size @ ( list @ A )
@ ( filter2 @ A
@ ^ [X4: A] :
~ ( P @ X4 )
@ Xs ) ) )
= ( size_size @ ( list @ A ) @ Xs ) ) ).
% sum_length_filter_compl
thf(fact_6550_filter__is__subset,axiom,
! [A: $tType,P: A > $o,Xs: list @ A] : ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ ( filter2 @ A @ P @ Xs ) ) @ ( set2 @ A @ Xs ) ) ).
% filter_is_subset
thf(fact_6551_length__filter__le,axiom,
! [A: $tType,P: A > $o,Xs: list @ A] : ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ ( filter2 @ A @ P @ Xs ) ) @ ( size_size @ ( list @ A ) @ Xs ) ) ).
% length_filter_le
thf(fact_6552_remdups__filter,axiom,
! [A: $tType,P: A > $o,Xs: list @ A] :
( ( remdups @ A @ ( filter2 @ A @ P @ Xs ) )
= ( filter2 @ A @ P @ ( remdups @ A @ Xs ) ) ) ).
% remdups_filter
thf(fact_6553_filter__replicate,axiom,
! [A: $tType,P: A > $o,X: A,N2: nat] :
( ( ( P @ X )
=> ( ( filter2 @ A @ P @ ( replicate @ A @ N2 @ X ) )
= ( replicate @ A @ N2 @ X ) ) )
& ( ~ ( P @ X )
=> ( ( filter2 @ A @ P @ ( replicate @ A @ N2 @ X ) )
= ( nil @ A ) ) ) ) ).
% filter_replicate
thf(fact_6554_filter_Osimps_I1_J,axiom,
! [A: $tType,P: A > $o] :
( ( filter2 @ A @ P @ ( nil @ A ) )
= ( nil @ A ) ) ).
% filter.simps(1)
thf(fact_6555_filter__insort__triv,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [P: B > $o,X: B,F2: B > A,Xs: list @ B] :
( ~ ( P @ X )
=> ( ( filter2 @ B @ P @ ( linorder_insort_key @ B @ A @ F2 @ X @ Xs ) )
= ( filter2 @ B @ P @ Xs ) ) ) ) ).
% filter_insort_triv
thf(fact_6556_filter__remove1,axiom,
! [A: $tType,Q: A > $o,X: A,Xs: list @ A] :
( ( filter2 @ A @ Q @ ( remove1 @ A @ X @ Xs ) )
= ( remove1 @ A @ X @ ( filter2 @ A @ Q @ Xs ) ) ) ).
% filter_remove1
thf(fact_6557_removeAll__filter__not__eq,axiom,
! [A: $tType] :
( ( removeAll @ A )
= ( ^ [X4: A] :
( filter2 @ A
@ ^ [Y6: A] : X4 != Y6 ) ) ) ).
% removeAll_filter_not_eq
thf(fact_6558_partition__in__shuffles,axiom,
! [A: $tType,Xs: list @ A,P: A > $o] :
( member @ ( list @ A ) @ Xs
@ ( shuffles @ A @ ( filter2 @ A @ P @ Xs )
@ ( filter2 @ A
@ ^ [X4: A] :
~ ( P @ X4 )
@ Xs ) ) ) ).
% partition_in_shuffles
thf(fact_6559_filter_Osimps_I2_J,axiom,
! [A: $tType,P: A > $o,X: A,Xs: list @ A] :
( ( ( P @ X )
=> ( ( filter2 @ A @ P @ ( cons @ A @ X @ Xs ) )
= ( cons @ A @ X @ ( filter2 @ A @ P @ Xs ) ) ) )
& ( ~ ( P @ X )
=> ( ( filter2 @ A @ P @ ( cons @ A @ X @ Xs ) )
= ( filter2 @ A @ P @ Xs ) ) ) ) ).
% filter.simps(2)
thf(fact_6560_filter__shuffles,axiom,
! [A: $tType,P: A > $o,Xs: list @ A,Ys: list @ A] :
( ( image2 @ ( list @ A ) @ ( list @ A ) @ ( filter2 @ A @ P ) @ ( shuffles @ A @ Xs @ Ys ) )
= ( shuffles @ A @ ( filter2 @ A @ P @ Xs ) @ ( filter2 @ A @ P @ Ys ) ) ) ).
% filter_shuffles
thf(fact_6561_length__filter__less,axiom,
! [A: $tType,X: A,Xs: list @ A,P: A > $o] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ~ ( P @ X )
=> ( ord_less @ nat @ ( size_size @ ( list @ A ) @ ( filter2 @ A @ P @ Xs ) ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ).
% length_filter_less
thf(fact_6562_filter__eq__Cons__iff,axiom,
! [A: $tType,P: A > $o,Ys: list @ A,X: A,Xs: list @ A] :
( ( ( filter2 @ A @ P @ Ys )
= ( cons @ A @ X @ Xs ) )
= ( ? [Us2: list @ A,Vs3: list @ A] :
( ( Ys
= ( append @ A @ Us2 @ ( cons @ A @ X @ Vs3 ) ) )
& ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Us2 ) )
=> ~ ( P @ X4 ) )
& ( P @ X )
& ( Xs
= ( filter2 @ A @ P @ Vs3 ) ) ) ) ) ).
% filter_eq_Cons_iff
thf(fact_6563_Cons__eq__filter__iff,axiom,
! [A: $tType,X: A,Xs: list @ A,P: A > $o,Ys: list @ A] :
( ( ( cons @ A @ X @ Xs )
= ( filter2 @ A @ P @ Ys ) )
= ( ? [Us2: list @ A,Vs3: list @ A] :
( ( Ys
= ( append @ A @ Us2 @ ( cons @ A @ X @ Vs3 ) ) )
& ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Us2 ) )
=> ~ ( P @ X4 ) )
& ( P @ X )
& ( Xs
= ( filter2 @ A @ P @ Vs3 ) ) ) ) ) ).
% Cons_eq_filter_iff
thf(fact_6564_filter__eq__ConsD,axiom,
! [A: $tType,P: A > $o,Ys: list @ A,X: A,Xs: list @ A] :
( ( ( filter2 @ A @ P @ Ys )
= ( cons @ A @ X @ Xs ) )
=> ? [Us3: list @ A,Vs2: list @ A] :
( ( Ys
= ( append @ A @ Us3 @ ( cons @ A @ X @ Vs2 ) ) )
& ! [X2: A] :
( ( member @ A @ X2 @ ( set2 @ A @ Us3 ) )
=> ~ ( P @ X2 ) )
& ( P @ X )
& ( Xs
= ( filter2 @ A @ P @ Vs2 ) ) ) ) ).
% filter_eq_ConsD
thf(fact_6565_Cons__eq__filterD,axiom,
! [A: $tType,X: A,Xs: list @ A,P: A > $o,Ys: list @ A] :
( ( ( cons @ A @ X @ Xs )
= ( filter2 @ A @ P @ Ys ) )
=> ? [Us3: list @ A,Vs2: list @ A] :
( ( Ys
= ( append @ A @ Us3 @ ( cons @ A @ X @ Vs2 ) ) )
& ! [X2: A] :
( ( member @ A @ X2 @ ( set2 @ A @ Us3 ) )
=> ~ ( P @ X2 ) )
& ( P @ X )
& ( Xs
= ( filter2 @ A @ P @ Vs2 ) ) ) ) ).
% Cons_eq_filterD
thf(fact_6566_inj__on__filter__key__eq,axiom,
! [B: $tType,A: $tType,F2: A > B,Y: A,Xs: list @ A] :
( ( inj_on @ A @ B @ F2 @ ( insert2 @ A @ Y @ ( set2 @ A @ Xs ) ) )
=> ( ( filter2 @ A
@ ^ [X4: A] :
( ( F2 @ Y )
= ( F2 @ X4 ) )
@ Xs )
= ( filter2 @ A
@ ( ^ [Y5: A,Z2: A] : Y5 = Z2
@ Y )
@ Xs ) ) ) ).
% inj_on_filter_key_eq
thf(fact_6567_filter__in__nths,axiom,
! [A: $tType,Xs: list @ A,S: set @ nat] :
( ( distinct @ A @ Xs )
=> ( ( filter2 @ A
@ ^ [X4: A] : ( member @ A @ X4 @ ( set2 @ A @ ( nths @ A @ Xs @ S ) ) )
@ Xs )
= ( nths @ A @ Xs @ S ) ) ) ).
% filter_in_nths
thf(fact_6568_set__minus__filter__out,axiom,
! [A: $tType,Xs: list @ A,Y: A] :
( ( minus_minus @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( insert2 @ A @ Y @ ( bot_bot @ ( set @ A ) ) ) )
= ( set2 @ A
@ ( filter2 @ A
@ ^ [X4: A] : X4 != Y
@ Xs ) ) ) ).
% set_minus_filter_out
thf(fact_6569_filter__shuffles__disjoint1_I2_J,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,Zs: list @ A] :
( ( ( inf_inf @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ Ys ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( member @ ( list @ A ) @ Zs @ ( shuffles @ A @ Xs @ Ys ) )
=> ( ( filter2 @ A
@ ^ [X4: A] :
~ ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
@ Zs )
= Ys ) ) ) ).
% filter_shuffles_disjoint1(2)
thf(fact_6570_filter__shuffles__disjoint1_I1_J,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,Zs: list @ A] :
( ( ( inf_inf @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ Ys ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( member @ ( list @ A ) @ Zs @ ( shuffles @ A @ Xs @ Ys ) )
=> ( ( filter2 @ A
@ ^ [X4: A] : ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
@ Zs )
= Xs ) ) ) ).
% filter_shuffles_disjoint1(1)
thf(fact_6571_filter__shuffles__disjoint2_I2_J,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,Zs: list @ A] :
( ( ( inf_inf @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ Ys ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( member @ ( list @ A ) @ Zs @ ( shuffles @ A @ Xs @ Ys ) )
=> ( ( filter2 @ A
@ ^ [X4: A] :
~ ( member @ A @ X4 @ ( set2 @ A @ Ys ) )
@ Zs )
= Xs ) ) ) ).
% filter_shuffles_disjoint2(2)
thf(fact_6572_filter__shuffles__disjoint2_I1_J,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,Zs: list @ A] :
( ( ( inf_inf @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ Ys ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( member @ ( list @ A ) @ Zs @ ( shuffles @ A @ Xs @ Ys ) )
=> ( ( filter2 @ A
@ ^ [X4: A] : ( member @ A @ X4 @ ( set2 @ A @ Ys ) )
@ Zs )
= Ys ) ) ) ).
% filter_shuffles_disjoint2(1)
thf(fact_6573_filter__eq__nths,axiom,
! [A: $tType] :
( ( filter2 @ A )
= ( ^ [P3: A > $o,Xs3: list @ A] :
( nths @ A @ Xs3
@ ( collect @ nat
@ ^ [I3: nat] :
( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs3 ) )
& ( P3 @ ( nth @ A @ Xs3 @ I3 ) ) ) ) ) ) ) ).
% filter_eq_nths
thf(fact_6574_length__filter__conv__card,axiom,
! [A: $tType,P5: A > $o,Xs: list @ A] :
( ( size_size @ ( list @ A ) @ ( filter2 @ A @ P5 @ Xs ) )
= ( finite_card @ nat
@ ( collect @ nat
@ ^ [I3: nat] :
( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs ) )
& ( P5 @ ( nth @ A @ Xs @ I3 ) ) ) ) ) ) ).
% length_filter_conv_card
thf(fact_6575_distinct__length__filter,axiom,
! [A: $tType,Xs: list @ A,P: A > $o] :
( ( distinct @ A @ Xs )
=> ( ( size_size @ ( list @ A ) @ ( filter2 @ A @ P @ Xs ) )
= ( finite_card @ A @ ( inf_inf @ ( set @ A ) @ ( collect @ A @ P ) @ ( set2 @ A @ Xs ) ) ) ) ) ).
% distinct_length_filter
thf(fact_6576_transpose__aux__filter__head,axiom,
! [A: $tType,Xss: list @ ( list @ A )] :
( ( concat @ A
@ ( map @ ( list @ A ) @ ( list @ A )
@ ( case_list @ ( list @ A ) @ A @ ( nil @ A )
@ ^ [H: A,T3: list @ A] : ( cons @ A @ H @ ( nil @ A ) ) )
@ Xss ) )
= ( map @ ( list @ A ) @ A @ ( hd @ A )
@ ( filter2 @ ( list @ A )
@ ^ [Ys3: list @ A] :
( Ys3
!= ( nil @ A ) )
@ Xss ) ) ) ).
% transpose_aux_filter_head
thf(fact_6577_nth__transpose,axiom,
! [A: $tType,I: nat,Xs: list @ ( list @ A )] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ ( list @ A ) ) @ ( transpose @ A @ Xs ) ) )
=> ( ( nth @ ( list @ A ) @ ( transpose @ A @ Xs ) @ I )
= ( map @ ( list @ A ) @ A
@ ^ [Xs3: list @ A] : ( nth @ A @ Xs3 @ I )
@ ( filter2 @ ( list @ A )
@ ^ [Ys3: list @ A] : ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Ys3 ) )
@ Xs ) ) ) ) ).
% nth_transpose
thf(fact_6578_transpose__max__length,axiom,
! [A: $tType,Xs: list @ ( list @ A )] :
( ( foldr @ ( list @ A ) @ nat
@ ^ [Xs3: list @ A] : ( ord_max @ nat @ ( size_size @ ( list @ A ) @ Xs3 ) )
@ ( transpose @ A @ Xs )
@ ( zero_zero @ nat ) )
= ( size_size @ ( list @ ( list @ A ) )
@ ( filter2 @ ( list @ A )
@ ^ [X4: list @ A] :
( X4
!= ( nil @ A ) )
@ Xs ) ) ) ).
% transpose_max_length
thf(fact_6579_map__filter__def,axiom,
! [B: $tType,A: $tType] :
( ( map_filter @ A @ B )
= ( ^ [F3: A > ( option @ B ),Xs3: list @ A] :
( map @ A @ B @ ( comp @ ( option @ B ) @ B @ A @ ( the2 @ B ) @ F3 )
@ ( filter2 @ A
@ ^ [X4: A] :
( ( F3 @ X4 )
!= ( none @ B ) )
@ Xs3 ) ) ) ) ).
% map_filter_def
thf(fact_6580_foldr__append,axiom,
! [B: $tType,A: $tType,F2: B > A > A,Xs: list @ B,Ys: list @ B,A3: A] :
( ( foldr @ B @ A @ F2 @ ( append @ B @ Xs @ Ys ) @ A3 )
= ( foldr @ B @ A @ F2 @ Xs @ ( foldr @ B @ A @ F2 @ Ys @ A3 ) ) ) ).
% foldr_append
thf(fact_6581_foldr__replicate,axiom,
! [A: $tType,B: $tType,F2: B > A > A,N2: nat,X: B] :
( ( foldr @ B @ A @ F2 @ ( replicate @ B @ N2 @ X ) )
= ( compow @ ( A > A ) @ N2 @ ( F2 @ X ) ) ) ).
% foldr_replicate
thf(fact_6582_foldr__cong,axiom,
! [B: $tType,A: $tType,A3: A,B3: A,L: list @ B,K: list @ B,F2: B > A > A,G: B > A > A] :
( ( A3 = B3 )
=> ( ( L = K )
=> ( ! [A6: A,X5: B] :
( ( member @ B @ X5 @ ( set2 @ B @ L ) )
=> ( ( F2 @ X5 @ A6 )
= ( G @ X5 @ A6 ) ) )
=> ( ( foldr @ B @ A @ F2 @ L @ A3 )
= ( foldr @ B @ A @ G @ K @ B3 ) ) ) ) ) ).
% foldr_cong
thf(fact_6583_map__filter__simps_I2_J,axiom,
! [B: $tType,A: $tType,F2: B > ( option @ A )] :
( ( map_filter @ B @ A @ F2 @ ( nil @ B ) )
= ( nil @ A ) ) ).
% map_filter_simps(2)
thf(fact_6584_foldr__Cons,axiom,
! [B: $tType,A: $tType,F2: A > B > B,X: A,Xs: list @ A] :
( ( foldr @ A @ B @ F2 @ ( cons @ A @ X @ Xs ) )
= ( comp @ B @ B @ B @ ( F2 @ X ) @ ( foldr @ A @ B @ F2 @ Xs ) ) ) ).
% foldr_Cons
thf(fact_6585_foldr__Nil,axiom,
! [A: $tType,B: $tType,F2: A > B > B] :
( ( foldr @ A @ B @ F2 @ ( nil @ A ) )
= ( id @ B ) ) ).
% foldr_Nil
thf(fact_6586_foldr__map,axiom,
! [C: $tType,B: $tType,A: $tType,G: B > A > A,F2: C > B,Xs: list @ C,A3: A] :
( ( foldr @ B @ A @ G @ ( map @ C @ B @ F2 @ Xs ) @ A3 )
= ( foldr @ C @ A @ ( comp @ B @ ( A > A ) @ C @ G @ F2 ) @ Xs @ A3 ) ) ).
% foldr_map
thf(fact_6587_foldr__filter,axiom,
! [A: $tType,B: $tType,F2: B > A > A,P: B > $o,Xs: list @ B] :
( ( foldr @ B @ A @ F2 @ ( filter2 @ B @ P @ Xs ) )
= ( foldr @ B @ A
@ ^ [X4: B] : ( if @ ( A > A ) @ ( P @ X4 ) @ ( F2 @ X4 ) @ ( id @ A ) )
@ Xs ) ) ).
% foldr_filter
thf(fact_6588_map__filter__simps_I1_J,axiom,
! [A: $tType,B: $tType,F2: B > ( option @ A ),X: B,Xs: list @ B] :
( ( map_filter @ B @ A @ F2 @ ( cons @ B @ X @ Xs ) )
= ( case_option @ ( list @ A ) @ A @ ( map_filter @ B @ A @ F2 @ Xs )
@ ^ [Y6: A] : ( cons @ A @ Y6 @ ( map_filter @ B @ A @ F2 @ Xs ) )
@ ( F2 @ X ) ) ) ).
% map_filter_simps(1)
thf(fact_6589_horner__sum__foldr,axiom,
! [A: $tType,B: $tType] :
( ( comm_semiring_0 @ A )
=> ( ( groups4207007520872428315er_sum @ B @ A )
= ( ^ [F3: B > A,A5: A,Xs3: list @ B] :
( foldr @ B @ A
@ ^ [X4: B,B4: A] : ( plus_plus @ A @ ( F3 @ X4 ) @ ( times_times @ A @ A5 @ B4 ) )
@ Xs3
@ ( zero_zero @ A ) ) ) ) ) ).
% horner_sum_foldr
thf(fact_6590_length__transpose,axiom,
! [A: $tType,Xs: list @ ( list @ A )] :
( ( size_size @ ( list @ ( list @ A ) ) @ ( transpose @ A @ Xs ) )
= ( foldr @ ( list @ A ) @ nat
@ ^ [Xs3: list @ A] : ( ord_max @ nat @ ( size_size @ ( list @ A ) @ Xs3 ) )
@ Xs
@ ( zero_zero @ nat ) ) ) ).
% length_transpose
thf(fact_6591_transpose__aux__max,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Xss: list @ ( list @ B )] :
( ( ord_max @ nat @ ( suc @ ( size_size @ ( list @ A ) @ Xs ) )
@ ( foldr @ ( list @ B ) @ nat
@ ^ [Xs3: list @ B] : ( ord_max @ nat @ ( size_size @ ( list @ B ) @ Xs3 ) )
@ Xss
@ ( zero_zero @ nat ) ) )
= ( suc
@ ( ord_max @ nat @ ( size_size @ ( list @ A ) @ Xs )
@ ( foldr @ ( list @ B ) @ nat
@ ^ [X4: list @ B] : ( ord_max @ nat @ ( minus_minus @ nat @ ( size_size @ ( list @ B ) @ X4 ) @ ( suc @ ( zero_zero @ nat ) ) ) )
@ ( filter2 @ ( list @ B )
@ ^ [Ys3: list @ B] :
( Ys3
!= ( nil @ B ) )
@ Xss )
@ ( zero_zero @ nat ) ) ) ) ) ).
% transpose_aux_max
thf(fact_6592_map__filter__map__filter,axiom,
! [A: $tType,B: $tType,F2: B > A,P: B > $o,Xs: list @ B] :
( ( map @ B @ A @ F2 @ ( filter2 @ B @ P @ Xs ) )
= ( map_filter @ B @ A
@ ^ [X4: B] : ( if @ ( option @ A ) @ ( P @ X4 ) @ ( some @ A @ ( F2 @ X4 ) ) @ ( none @ A ) )
@ Xs ) ) ).
% map_filter_map_filter
thf(fact_6593_sum__list__map__eq__sum__count2,axiom,
! [A: $tType,Xs: list @ A,X8: set @ A,F2: A > nat] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ X8 )
=> ( ( finite_finite2 @ A @ X8 )
=> ( ( groups8242544230860333062m_list @ nat @ ( map @ A @ nat @ F2 @ Xs ) )
= ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [X4: A] : ( times_times @ nat @ ( count_list @ A @ Xs @ X4 ) @ ( F2 @ X4 ) )
@ X8 ) ) ) ) ).
% sum_list_map_eq_sum_count2
thf(fact_6594_relcomp__fold,axiom,
! [C: $tType,B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S2: set @ ( product_prod @ B @ C )] :
( ( finite_finite2 @ ( product_prod @ A @ B ) @ R )
=> ( ( finite_finite2 @ ( product_prod @ B @ C ) @ S2 )
=> ( ( relcomp @ A @ B @ C @ R @ S2 )
= ( finite_fold @ ( product_prod @ A @ B ) @ ( set @ ( product_prod @ A @ C ) )
@ ( product_case_prod @ A @ B @ ( ( set @ ( product_prod @ A @ C ) ) > ( set @ ( product_prod @ A @ C ) ) )
@ ^ [X4: A,Y6: B,A7: set @ ( product_prod @ A @ C )] :
( finite_fold @ ( product_prod @ B @ C ) @ ( set @ ( product_prod @ A @ C ) )
@ ( product_case_prod @ B @ C @ ( ( set @ ( product_prod @ A @ C ) ) > ( set @ ( product_prod @ A @ C ) ) )
@ ^ [W3: B,Z6: C,A17: set @ ( product_prod @ A @ C )] : ( if @ ( set @ ( product_prod @ A @ C ) ) @ ( Y6 = W3 ) @ ( insert2 @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ X4 @ Z6 ) @ A17 ) @ A17 ) )
@ A7
@ S2 ) )
@ ( bot_bot @ ( set @ ( product_prod @ A @ C ) ) )
@ R ) ) ) ) ).
% relcomp_fold
thf(fact_6595_relcomp__empty2,axiom,
! [C: $tType,B: $tType,A: $tType,R: set @ ( product_prod @ A @ C )] :
( ( relcomp @ A @ C @ B @ R @ ( bot_bot @ ( set @ ( product_prod @ C @ B ) ) ) )
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% relcomp_empty2
thf(fact_6596_relcomp__empty1,axiom,
! [C: $tType,B: $tType,A: $tType,R: set @ ( product_prod @ C @ B )] :
( ( relcomp @ A @ C @ B @ ( bot_bot @ ( set @ ( product_prod @ A @ C ) ) ) @ R )
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% relcomp_empty1
thf(fact_6597_finite__relcomp,axiom,
! [A: $tType,C: $tType,B: $tType,R: set @ ( product_prod @ A @ B ),S2: set @ ( product_prod @ B @ C )] :
( ( finite_finite2 @ ( product_prod @ A @ B ) @ R )
=> ( ( finite_finite2 @ ( product_prod @ B @ C ) @ S2 )
=> ( finite_finite2 @ ( product_prod @ A @ C ) @ ( relcomp @ A @ B @ C @ R @ S2 ) ) ) ) ).
% finite_relcomp
thf(fact_6598_sum__list_ONil,axiom,
! [A: $tType] :
( ( monoid_add @ A )
=> ( ( groups8242544230860333062m_list @ A @ ( nil @ A ) )
= ( zero_zero @ A ) ) ) ).
% sum_list.Nil
thf(fact_6599_sum__list__eq__0__iff,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [Ns: list @ A] :
( ( ( groups8242544230860333062m_list @ A @ Ns )
= ( zero_zero @ A ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Ns ) )
=> ( X4
= ( zero_zero @ A ) ) ) ) ) ) ).
% sum_list_eq_0_iff
thf(fact_6600_sum__list_OCons,axiom,
! [A: $tType] :
( ( monoid_add @ A )
=> ! [X: A,Xs: list @ A] :
( ( groups8242544230860333062m_list @ A @ ( cons @ A @ X @ Xs ) )
= ( plus_plus @ A @ X @ ( groups8242544230860333062m_list @ A @ Xs ) ) ) ) ).
% sum_list.Cons
thf(fact_6601_sum__list__append,axiom,
! [A: $tType] :
( ( monoid_add @ A )
=> ! [Xs: list @ A,Ys: list @ A] :
( ( groups8242544230860333062m_list @ A @ ( append @ A @ Xs @ Ys ) )
= ( plus_plus @ A @ ( groups8242544230860333062m_list @ A @ Xs ) @ ( groups8242544230860333062m_list @ A @ Ys ) ) ) ) ).
% sum_list_append
thf(fact_6602_sum__list__0,axiom,
! [B: $tType,A: $tType] :
( ( monoid_add @ A )
=> ! [Xs: list @ B] :
( ( groups8242544230860333062m_list @ A
@ ( map @ B @ A
@ ^ [X4: B] : ( zero_zero @ A )
@ Xs ) )
= ( zero_zero @ A ) ) ) ).
% sum_list_0
thf(fact_6603_sum__list__upt,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq @ nat @ M2 @ N2 )
=> ( ( groups8242544230860333062m_list @ nat @ ( upt @ M2 @ N2 ) )
= ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [X4: nat] : X4
@ ( set_or7035219750837199246ssThan @ nat @ M2 @ N2 ) ) ) ) ).
% sum_list_upt
thf(fact_6604_length__concat,axiom,
! [B: $tType,Xss: list @ ( list @ B )] :
( ( size_size @ ( list @ B ) @ ( concat @ B @ Xss ) )
= ( groups8242544230860333062m_list @ nat @ ( map @ ( list @ B ) @ nat @ ( size_size @ ( list @ B ) ) @ Xss ) ) ) ).
% length_concat
thf(fact_6605_sum__list__addf,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [F2: B > A,G: B > A,Xs: list @ B] :
( ( groups8242544230860333062m_list @ A
@ ( map @ B @ A
@ ^ [X4: B] : ( plus_plus @ A @ ( F2 @ X4 ) @ ( G @ X4 ) )
@ Xs ) )
= ( plus_plus @ A @ ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ F2 @ Xs ) ) @ ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ G @ Xs ) ) ) ) ) ).
% sum_list_addf
thf(fact_6606_member__le__sum__list,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [X: A,Xs: list @ A] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ord_less_eq @ A @ X @ ( groups8242544230860333062m_list @ A @ Xs ) ) ) ) ).
% member_le_sum_list
thf(fact_6607_relpow_Osimps_I2_J,axiom,
! [A: $tType,N2: nat,R: set @ ( product_prod @ A @ A )] :
( ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( suc @ N2 ) @ R )
= ( relcomp @ A @ A @ A @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N2 @ R ) @ R ) ) ).
% relpow.simps(2)
thf(fact_6608_relpow__add,axiom,
! [A: $tType,M2: nat,N2: nat,R: set @ ( product_prod @ A @ A )] :
( ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( plus_plus @ nat @ M2 @ N2 ) @ R )
= ( relcomp @ A @ A @ A @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ M2 @ R ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N2 @ R ) ) ) ).
% relpow_add
thf(fact_6609_union__comp__emptyL,axiom,
! [A: $tType,A4: set @ ( product_prod @ A @ A ),C5: set @ ( product_prod @ A @ A ),B2: set @ ( product_prod @ A @ A )] :
( ( ( relcomp @ A @ A @ A @ A4 @ C5 )
= ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
=> ( ( ( relcomp @ A @ A @ A @ B2 @ C5 )
= ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
=> ( ( relcomp @ A @ A @ A @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ A4 @ B2 ) @ C5 )
= ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ) ).
% union_comp_emptyL
thf(fact_6610_union__comp__emptyR,axiom,
! [A: $tType,A4: set @ ( product_prod @ A @ A ),B2: set @ ( product_prod @ A @ A ),C5: set @ ( product_prod @ A @ A )] :
( ( ( relcomp @ A @ A @ A @ A4 @ B2 )
= ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
=> ( ( ( relcomp @ A @ A @ A @ A4 @ C5 )
= ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
=> ( ( relcomp @ A @ A @ A @ A4 @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ B2 @ C5 ) )
= ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ) ).
% union_comp_emptyR
thf(fact_6611_Groups__List_Osum__list__nonneg,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [Xs: list @ A] :
( ! [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Xs ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ X5 ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( groups8242544230860333062m_list @ A @ Xs ) ) ) ) ).
% Groups_List.sum_list_nonneg
thf(fact_6612_sum__list__nonneg__eq__0__iff,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [Xs: list @ A] :
( ! [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Xs ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ X5 ) )
=> ( ( ( groups8242544230860333062m_list @ A @ Xs )
= ( zero_zero @ A ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
=> ( X4
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% sum_list_nonneg_eq_0_iff
thf(fact_6613_sum__list__nonpos,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [Xs: list @ A] :
( ! [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Xs ) )
=> ( ord_less_eq @ A @ X5 @ ( zero_zero @ A ) ) )
=> ( ord_less_eq @ A @ ( groups8242544230860333062m_list @ A @ Xs ) @ ( zero_zero @ A ) ) ) ) ).
% sum_list_nonpos
thf(fact_6614_sum__list__abs,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [Xs: list @ A] : ( ord_less_eq @ A @ ( abs_abs @ A @ ( groups8242544230860333062m_list @ A @ Xs ) ) @ ( groups8242544230860333062m_list @ A @ ( map @ A @ A @ ( abs_abs @ A ) @ Xs ) ) ) ) ).
% sum_list_abs
thf(fact_6615_sum__list_Oeq__foldr,axiom,
! [A: $tType] :
( ( monoid_add @ A )
=> ( ( groups8242544230860333062m_list @ A )
= ( ^ [Xs3: list @ A] : ( foldr @ A @ A @ ( plus_plus @ A ) @ Xs3 @ ( zero_zero @ A ) ) ) ) ) ).
% sum_list.eq_foldr
thf(fact_6616_sum__list__filter__le__nat,axiom,
! [A: $tType,F2: A > nat,P: A > $o,Xs: list @ A] : ( ord_less_eq @ nat @ ( groups8242544230860333062m_list @ nat @ ( map @ A @ nat @ F2 @ ( filter2 @ A @ P @ Xs ) ) ) @ ( groups8242544230860333062m_list @ nat @ ( map @ A @ nat @ F2 @ Xs ) ) ) ).
% sum_list_filter_le_nat
thf(fact_6617_sum__list__mono,axiom,
! [B: $tType,A: $tType] :
( ( ( monoid_add @ B )
& ( ordere6658533253407199908up_add @ B ) )
=> ! [Xs: list @ A,F2: A > B,G: A > B] :
( ! [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Xs ) )
=> ( ord_less_eq @ B @ ( F2 @ X5 ) @ ( G @ X5 ) ) )
=> ( ord_less_eq @ B @ ( groups8242544230860333062m_list @ B @ ( map @ A @ B @ F2 @ Xs ) ) @ ( groups8242544230860333062m_list @ B @ ( map @ A @ B @ G @ Xs ) ) ) ) ) ).
% sum_list_mono
thf(fact_6618_sum__list__map__filter_H,axiom,
! [A: $tType,B: $tType] :
( ( monoid_add @ A )
=> ! [F2: B > A,P: B > $o,Xs: list @ B] :
( ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ F2 @ ( filter2 @ B @ P @ Xs ) ) )
= ( groups8242544230860333062m_list @ A
@ ( map @ B @ A
@ ^ [X4: B] : ( if @ A @ ( P @ X4 ) @ ( F2 @ X4 ) @ ( zero_zero @ A ) )
@ Xs ) ) ) ) ).
% sum_list_map_filter'
thf(fact_6619_distinct__sum__list__conv__Sum,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [Xs: list @ A] :
( ( distinct @ A @ Xs )
=> ( ( groups8242544230860333062m_list @ A @ Xs )
= ( groups7311177749621191930dd_sum @ A @ A
@ ^ [X4: A] : X4
@ ( set2 @ A @ Xs ) ) ) ) ) ).
% distinct_sum_list_conv_Sum
thf(fact_6620_concat__conv__foldr,axiom,
! [A: $tType] :
( ( concat @ A )
= ( ^ [Xss3: list @ ( list @ A )] : ( foldr @ ( list @ A ) @ ( list @ A ) @ ( append @ A ) @ Xss3 @ ( nil @ A ) ) ) ) ).
% concat_conv_foldr
thf(fact_6621_sum__list__strict__mono,axiom,
! [B: $tType,A: $tType] :
( ( ( monoid_add @ B )
& ( strict9044650504122735259up_add @ B ) )
=> ! [Xs: list @ A,F2: A > B,G: A > B] :
( ( Xs
!= ( nil @ A ) )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Xs ) )
=> ( ord_less @ B @ ( F2 @ X5 ) @ ( G @ X5 ) ) )
=> ( ord_less @ B @ ( groups8242544230860333062m_list @ B @ ( map @ A @ B @ F2 @ Xs ) ) @ ( groups8242544230860333062m_list @ B @ ( map @ A @ B @ G @ Xs ) ) ) ) ) ) ).
% sum_list_strict_mono
thf(fact_6622_elem__le__sum__list,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [K: nat,Ns: list @ A] :
( ( ord_less @ nat @ K @ ( size_size @ ( list @ A ) @ Ns ) )
=> ( ord_less_eq @ A @ ( nth @ A @ Ns @ K ) @ ( groups8242544230860333062m_list @ A @ Ns ) ) ) ) ).
% elem_le_sum_list
thf(fact_6623_sum__list__map__filter,axiom,
! [A: $tType,B: $tType] :
( ( monoid_add @ A )
=> ! [Xs: list @ B,P: B > $o,F2: B > A] :
( ! [X5: B] :
( ( member @ B @ X5 @ ( set2 @ B @ Xs ) )
=> ( ~ ( P @ X5 )
=> ( ( F2 @ X5 )
= ( zero_zero @ A ) ) ) )
=> ( ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ F2 @ ( filter2 @ B @ P @ Xs ) ) )
= ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ F2 @ Xs ) ) ) ) ) ).
% sum_list_map_filter
thf(fact_6624_sum__list__distinct__conv__sum__set,axiom,
! [C: $tType,B: $tType] :
( ( comm_monoid_add @ C )
=> ! [Xs: list @ B,F2: B > C] :
( ( distinct @ B @ Xs )
=> ( ( groups8242544230860333062m_list @ C @ ( map @ B @ C @ F2 @ Xs ) )
= ( groups7311177749621191930dd_sum @ B @ C @ F2 @ ( set2 @ B @ Xs ) ) ) ) ) ).
% sum_list_distinct_conv_sum_set
thf(fact_6625_sum_Odistinct__set__conv__list,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [Xs: list @ B,G: B > A] :
( ( distinct @ B @ Xs )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( set2 @ B @ Xs ) )
= ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ G @ Xs ) ) ) ) ) ).
% sum.distinct_set_conv_list
thf(fact_6626_sum__list__map__remove1,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [X: B,Xs: list @ B,F2: B > A] :
( ( member @ B @ X @ ( set2 @ B @ Xs ) )
=> ( ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ F2 @ Xs ) )
= ( plus_plus @ A @ ( F2 @ X ) @ ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ F2 @ ( remove1 @ B @ X @ Xs ) ) ) ) ) ) ) ).
% sum_list_map_remove1
thf(fact_6627_sum__code,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: B > A,Xs: list @ B] :
( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( set2 @ B @ Xs ) )
= ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ G @ ( remdups @ B @ Xs ) ) ) ) ) ).
% sum_code
thf(fact_6628_size__list__conv__sum__list,axiom,
! [B: $tType] :
( ( size_list @ B )
= ( ^ [F3: B > nat,Xs3: list @ B] : ( plus_plus @ nat @ ( groups8242544230860333062m_list @ nat @ ( map @ B @ nat @ F3 @ Xs3 ) ) @ ( size_size @ ( list @ B ) @ Xs3 ) ) ) ) ).
% size_list_conv_sum_list
thf(fact_6629_sum__list__triv,axiom,
! [C: $tType,B: $tType] :
( ( semiring_1 @ B )
=> ! [R2: B,Xs: list @ C] :
( ( groups8242544230860333062m_list @ B
@ ( map @ C @ B
@ ^ [X4: C] : R2
@ Xs ) )
= ( times_times @ B @ ( semiring_1_of_nat @ B @ ( size_size @ ( list @ C ) @ Xs ) ) @ R2 ) ) ) ).
% sum_list_triv
thf(fact_6630_sum__list__Suc,axiom,
! [A: $tType,F2: A > nat,Xs: list @ A] :
( ( groups8242544230860333062m_list @ nat
@ ( map @ A @ nat
@ ^ [X4: A] : ( suc @ ( F2 @ X4 ) )
@ Xs ) )
= ( plus_plus @ nat @ ( groups8242544230860333062m_list @ nat @ ( map @ A @ nat @ F2 @ Xs ) ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ).
% sum_list_Suc
thf(fact_6631_sum__list__sum__nth,axiom,
! [B: $tType] :
( ( comm_monoid_add @ B )
=> ( ( groups8242544230860333062m_list @ B )
= ( ^ [Xs3: list @ B] : ( groups7311177749621191930dd_sum @ nat @ B @ ( nth @ B @ Xs3 ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ B ) @ Xs3 ) ) ) ) ) ) ).
% sum_list_sum_nth
thf(fact_6632_card__length__sum__list__rec,axiom,
! [M2: nat,N6: nat] :
( ( ord_less_eq @ nat @ ( one_one @ nat ) @ M2 )
=> ( ( finite_card @ ( list @ nat )
@ ( collect @ ( list @ nat )
@ ^ [L2: list @ nat] :
( ( ( size_size @ ( list @ nat ) @ L2 )
= M2 )
& ( ( groups8242544230860333062m_list @ nat @ L2 )
= N6 ) ) ) )
= ( plus_plus @ nat
@ ( finite_card @ ( list @ nat )
@ ( collect @ ( list @ nat )
@ ^ [L2: list @ nat] :
( ( ( size_size @ ( list @ nat ) @ L2 )
= ( minus_minus @ nat @ M2 @ ( one_one @ nat ) ) )
& ( ( groups8242544230860333062m_list @ nat @ L2 )
= N6 ) ) ) )
@ ( finite_card @ ( list @ nat )
@ ( collect @ ( list @ nat )
@ ^ [L2: list @ nat] :
( ( ( size_size @ ( list @ nat ) @ L2 )
= M2 )
& ( ( plus_plus @ nat @ ( groups8242544230860333062m_list @ nat @ L2 ) @ ( one_one @ nat ) )
= N6 ) ) ) ) ) ) ) ).
% card_length_sum_list_rec
thf(fact_6633_card__length__sum__list,axiom,
! [M2: nat,N6: nat] :
( ( finite_card @ ( list @ nat )
@ ( collect @ ( list @ nat )
@ ^ [L2: list @ nat] :
( ( ( size_size @ ( list @ nat ) @ L2 )
= M2 )
& ( ( groups8242544230860333062m_list @ nat @ L2 )
= N6 ) ) ) )
= ( binomial @ ( minus_minus @ nat @ ( plus_plus @ nat @ N6 @ M2 ) @ ( one_one @ nat ) ) @ N6 ) ) ).
% card_length_sum_list
thf(fact_6634_sum__list__map__eq__sum__count,axiom,
! [A: $tType,F2: A > nat,Xs: list @ A] :
( ( groups8242544230860333062m_list @ nat @ ( map @ A @ nat @ F2 @ Xs ) )
= ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [X4: A] : ( times_times @ nat @ ( count_list @ A @ Xs @ X4 ) @ ( F2 @ X4 ) )
@ ( set2 @ A @ Xs ) ) ) ).
% sum_list_map_eq_sum_count
thf(fact_6635_sum__list__update,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [K: nat,Xs: list @ A,X: A] :
( ( ord_less @ nat @ K @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( groups8242544230860333062m_list @ A @ ( list_update @ A @ Xs @ K @ X ) )
= ( minus_minus @ A @ ( plus_plus @ A @ ( groups8242544230860333062m_list @ A @ Xs ) @ X ) @ ( nth @ A @ Xs @ K ) ) ) ) ) ).
% sum_list_update
thf(fact_6636_length__product__lists,axiom,
! [B: $tType,Xss: list @ ( list @ B )] :
( ( size_size @ ( list @ ( list @ B ) ) @ ( product_lists @ B @ Xss ) )
= ( foldr @ nat @ nat @ ( times_times @ nat ) @ ( map @ ( list @ B ) @ nat @ ( size_size @ ( list @ B ) ) @ Xss ) @ ( one_one @ nat ) ) ) ).
% length_product_lists
thf(fact_6637_min__ext__compat,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),S2: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( relcomp @ A @ A @ A @ R @ S2 ) @ R )
=> ( ord_less_eq @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( relcomp @ ( set @ A ) @ ( set @ A ) @ ( set @ A ) @ ( min_ext @ A @ R ) @ ( sup_sup @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( min_ext @ A @ S2 ) @ ( insert2 @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ ( bot_bot @ ( set @ A ) ) ) @ ( bot_bot @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) ) ) ) ) @ ( min_ext @ A @ R ) ) ) ).
% min_ext_compat
thf(fact_6638_sorted__wrt__less__sum__mono__lowerbound,axiom,
! [B: $tType] :
( ( ordere6911136660526730532id_add @ B )
=> ! [F2: nat > B,Ns: list @ nat] :
( ! [X5: nat,Y3: nat] :
( ( ord_less_eq @ nat @ X5 @ Y3 )
=> ( ord_less_eq @ B @ ( F2 @ X5 ) @ ( F2 @ Y3 ) ) )
=> ( ( sorted_wrt @ nat @ ( ord_less @ nat ) @ Ns )
=> ( ord_less_eq @ B @ ( groups7311177749621191930dd_sum @ nat @ B @ F2 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ nat ) @ Ns ) ) ) @ ( groups8242544230860333062m_list @ B @ ( map @ nat @ B @ F2 @ Ns ) ) ) ) ) ) ).
% sorted_wrt_less_sum_mono_lowerbound
thf(fact_6639_sorted__wrt__map,axiom,
! [A: $tType,B: $tType,R: A > A > $o,F2: B > A,Xs: list @ B] :
( ( sorted_wrt @ A @ R @ ( map @ B @ A @ F2 @ Xs ) )
= ( sorted_wrt @ B
@ ^ [X4: B,Y6: B] : ( R @ ( F2 @ X4 ) @ ( F2 @ Y6 ) )
@ Xs ) ) ).
% sorted_wrt_map
thf(fact_6640_sorted__map,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F2: B > A,Xs: list @ B] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F2 @ Xs ) )
= ( sorted_wrt @ B
@ ^ [X4: B,Y6: B] : ( ord_less_eq @ A @ ( F2 @ X4 ) @ ( F2 @ Y6 ) )
@ Xs ) ) ) ).
% sorted_map
thf(fact_6641_sorted__wrt__upt,axiom,
! [M2: nat,N2: nat] : ( sorted_wrt @ nat @ ( ord_less @ nat ) @ ( upt @ M2 @ N2 ) ) ).
% sorted_wrt_upt
thf(fact_6642_sorted__upt,axiom,
! [M2: nat,N2: nat] : ( sorted_wrt @ nat @ ( ord_less_eq @ nat ) @ ( upt @ M2 @ N2 ) ) ).
% sorted_upt
thf(fact_6643_sorted2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A,Zs: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( cons @ A @ X @ ( cons @ A @ Y @ Zs ) ) )
= ( ( ord_less_eq @ A @ X @ Y )
& ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( cons @ A @ Y @ Zs ) ) ) ) ) ).
% sorted2
thf(fact_6644_sorted__remdups__adj,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( remdups_adj @ A @ Xs ) ) ) ) ).
% sorted_remdups_adj
thf(fact_6645_strict__sorted__imp__sorted,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A] :
( ( sorted_wrt @ A @ ( ord_less @ A ) @ Xs )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs ) ) ) ).
% strict_sorted_imp_sorted
thf(fact_6646_sorted__remove1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,A3: A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( remove1 @ A @ A3 @ Xs ) ) ) ) ).
% sorted_remove1
thf(fact_6647_sorted__list__of__set_Osorted__sorted__key__list__of__set,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A] : ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( linord4507533701916653071of_set @ A @ A4 ) ) ) ).
% sorted_list_of_set.sorted_sorted_key_list_of_set
thf(fact_6648_sorted__nths,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,I6: set @ nat] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( nths @ A @ Xs @ I6 ) ) ) ) ).
% sorted_nths
thf(fact_6649_sorted__replicate,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [N2: nat,X: A] : ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( replicate @ A @ N2 @ X ) ) ) ).
% sorted_replicate
thf(fact_6650_sorted__insort,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Xs: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A )
@ ( linorder_insort_key @ A @ A
@ ^ [X4: A] : X4
@ X
@ Xs ) )
= ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs ) ) ) ).
% sorted_insort
thf(fact_6651_sorted__wrt__true,axiom,
! [A: $tType,Xs: list @ A] :
( sorted_wrt @ A
@ ^ [Uu3: A,Uv3: A] : $true
@ Xs ) ).
% sorted_wrt_true
thf(fact_6652_sorted__list__of__set_Ostrict__sorted__key__list__of__set,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A] : ( sorted_wrt @ A @ ( ord_less @ A ) @ ( linord4507533701916653071of_set @ A @ A4 ) ) ) ).
% sorted_list_of_set.strict_sorted_key_list_of_set
thf(fact_6653_sorted0,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( nil @ A ) ) ) ).
% sorted0
thf(fact_6654_strict__sorted__simps_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( sorted_wrt @ A @ ( ord_less @ A ) @ ( nil @ A ) ) ) ).
% strict_sorted_simps(1)
thf(fact_6655_sorted__wrt_Osimps_I1_J,axiom,
! [A: $tType,P: A > A > $o] : ( sorted_wrt @ A @ P @ ( nil @ A ) ) ).
% sorted_wrt.simps(1)
thf(fact_6656_sorted__wrt1,axiom,
! [A: $tType,P: A > A > $o,X: A] : ( sorted_wrt @ A @ P @ ( cons @ A @ X @ ( nil @ A ) ) ) ).
% sorted_wrt1
thf(fact_6657_sorted__wrt__drop,axiom,
! [A: $tType,F2: A > A > $o,Xs: list @ A,N2: nat] :
( ( sorted_wrt @ A @ F2 @ Xs )
=> ( sorted_wrt @ A @ F2 @ ( drop @ A @ N2 @ Xs ) ) ) ).
% sorted_wrt_drop
thf(fact_6658_sorted__drop,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,N2: nat] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( drop @ A @ N2 @ Xs ) ) ) ) ).
% sorted_drop
thf(fact_6659_sorted__wrt__take,axiom,
! [A: $tType,F2: A > A > $o,Xs: list @ A,N2: nat] :
( ( sorted_wrt @ A @ F2 @ Xs )
=> ( sorted_wrt @ A @ F2 @ ( take @ A @ N2 @ Xs ) ) ) ).
% sorted_wrt_take
thf(fact_6660_sorted__take,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,N2: nat] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( take @ A @ N2 @ Xs ) ) ) ) ).
% sorted_take
thf(fact_6661_sorted__remdups,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( remdups @ A @ Xs ) ) ) ) ).
% sorted_remdups
thf(fact_6662_sorted__tl,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( tl @ A @ Xs ) ) ) ) ).
% sorted_tl
thf(fact_6663_sorted__wrt__append,axiom,
! [A: $tType,P: A > A > $o,Xs: list @ A,Ys: list @ A] :
( ( sorted_wrt @ A @ P @ ( append @ A @ Xs @ Ys ) )
= ( ( sorted_wrt @ A @ P @ Xs )
& ( sorted_wrt @ A @ P @ Ys )
& ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
=> ! [Y6: A] :
( ( member @ A @ Y6 @ ( set2 @ A @ Ys ) )
=> ( P @ X4 @ Y6 ) ) ) ) ) ).
% sorted_wrt_append
thf(fact_6664_strict__sorted__equal,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,Ys: list @ A] :
( ( sorted_wrt @ A @ ( ord_less @ A ) @ Xs )
=> ( ( sorted_wrt @ A @ ( ord_less @ A ) @ Ys )
=> ( ( ( set2 @ A @ Ys )
= ( set2 @ A @ Xs ) )
=> ( Ys = Xs ) ) ) ) ) ).
% strict_sorted_equal
thf(fact_6665_sorted__wrt__mono__rel,axiom,
! [A: $tType,Xs: list @ A,P: A > A > $o,Q: A > A > $o] :
( ! [X5: A,Y3: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Xs ) )
=> ( ( member @ A @ Y3 @ ( set2 @ A @ Xs ) )
=> ( ( P @ X5 @ Y3 )
=> ( Q @ X5 @ Y3 ) ) ) )
=> ( ( sorted_wrt @ A @ P @ Xs )
=> ( sorted_wrt @ A @ Q @ Xs ) ) ) ).
% sorted_wrt_mono_rel
thf(fact_6666_sorted__same,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [G: ( list @ A ) > A,Xs: list @ A] :
( sorted_wrt @ A @ ( ord_less_eq @ A )
@ ( filter2 @ A
@ ^ [X4: A] :
( X4
= ( G @ Xs ) )
@ Xs ) ) ) ).
% sorted_same
thf(fact_6667_sorted__wrt__filter,axiom,
! [A: $tType,F2: A > A > $o,Xs: list @ A,P: A > $o] :
( ( sorted_wrt @ A @ F2 @ Xs )
=> ( sorted_wrt @ A @ F2 @ ( filter2 @ A @ P @ Xs ) ) ) ).
% sorted_wrt_filter
thf(fact_6668_sorted1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A] : ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( cons @ A @ X @ ( nil @ A ) ) ) ) ).
% sorted1
thf(fact_6669_sorted__simps_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Ys: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( cons @ A @ X @ Ys ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Ys ) )
=> ( ord_less_eq @ A @ X @ X4 ) )
& ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Ys ) ) ) ) ).
% sorted_simps(2)
thf(fact_6670_strict__sorted__simps_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Ys: list @ A] :
( ( sorted_wrt @ A @ ( ord_less @ A ) @ ( cons @ A @ X @ Ys ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Ys ) )
=> ( ord_less @ A @ X @ X4 ) )
& ( sorted_wrt @ A @ ( ord_less @ A ) @ Ys ) ) ) ) ).
% strict_sorted_simps(2)
thf(fact_6671_strict__sorted__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: list @ A] :
( ( sorted_wrt @ A @ ( ord_less @ A ) @ L )
= ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ L )
& ( distinct @ A @ L ) ) ) ) ).
% strict_sorted_iff
thf(fact_6672_sorted__append,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,Ys: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( append @ A @ Xs @ Ys ) )
= ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
& ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Ys )
& ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
=> ! [Y6: A] :
( ( member @ A @ Y6 @ ( set2 @ A @ Ys ) )
=> ( ord_less_eq @ A @ X4 @ Y6 ) ) ) ) ) ) ).
% sorted_append
thf(fact_6673_sorted__wrt_Oelims_I3_J,axiom,
! [A: $tType,X: A > A > $o,Xa2: list @ A] :
( ~ ( sorted_wrt @ A @ X @ Xa2 )
=> ~ ! [X5: A,Ys4: list @ A] :
( ( Xa2
= ( cons @ A @ X5 @ Ys4 ) )
=> ( ! [Xa3: A] :
( ( member @ A @ Xa3 @ ( set2 @ A @ Ys4 ) )
=> ( X @ X5 @ Xa3 ) )
& ( sorted_wrt @ A @ X @ Ys4 ) ) ) ) ).
% sorted_wrt.elims(3)
thf(fact_6674_sorted__wrt_Osimps_I2_J,axiom,
! [A: $tType,P: A > A > $o,X: A,Ys: list @ A] :
( ( sorted_wrt @ A @ P @ ( cons @ A @ X @ Ys ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Ys ) )
=> ( P @ X @ X4 ) )
& ( sorted_wrt @ A @ P @ Ys ) ) ) ).
% sorted_wrt.simps(2)
thf(fact_6675_sorted__distinct__set__unique,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,Ys: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
=> ( ( distinct @ A @ Xs )
=> ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Ys )
=> ( ( distinct @ A @ Ys )
=> ( ( ( set2 @ A @ Xs )
= ( set2 @ A @ Ys ) )
=> ( Xs = Ys ) ) ) ) ) ) ) ).
% sorted_distinct_set_unique
thf(fact_6676_sorted__wrt__nth__less,axiom,
! [A: $tType,P: A > A > $o,Xs: list @ A,I: nat,J: nat] :
( ( sorted_wrt @ A @ P @ Xs )
=> ( ( ord_less @ nat @ I @ J )
=> ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( P @ ( nth @ A @ Xs @ I ) @ ( nth @ A @ Xs @ J ) ) ) ) ) ).
% sorted_wrt_nth_less
thf(fact_6677_sorted__wrt__iff__nth__less,axiom,
! [A: $tType] :
( ( sorted_wrt @ A )
= ( ^ [P3: A > A > $o,Xs3: list @ A] :
! [I3: nat,J2: nat] :
( ( ord_less @ nat @ I3 @ J2 )
=> ( ( ord_less @ nat @ J2 @ ( size_size @ ( list @ A ) @ Xs3 ) )
=> ( P3 @ ( nth @ A @ Xs3 @ I3 ) @ ( nth @ A @ Xs3 @ J2 ) ) ) ) ) ) ).
% sorted_wrt_iff_nth_less
thf(fact_6678_sorted__wrt01,axiom,
! [A: $tType,Xs: list @ A,P: A > A > $o] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) )
=> ( sorted_wrt @ A @ P @ Xs ) ) ).
% sorted_wrt01
thf(fact_6679_sorted__filter,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F2: B > A,Xs: list @ B,P: B > $o] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F2 @ Xs ) )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F2 @ ( filter2 @ B @ P @ Xs ) ) ) ) ) ).
% sorted_filter
thf(fact_6680_sorted__insort__key,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F2: B > A,X: B,Xs: list @ B] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F2 @ ( linorder_insort_key @ B @ A @ F2 @ X @ Xs ) ) )
= ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F2 @ Xs ) ) ) ) ).
% sorted_insort_key
thf(fact_6681_sorted__map__remove1,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F2: B > A,Xs: list @ B,X: B] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F2 @ Xs ) )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F2 @ ( remove1 @ B @ X @ Xs ) ) ) ) ) ).
% sorted_map_remove1
thf(fact_6682_sorted__map__same,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F2: B > A,G: ( list @ B ) > A,Xs: list @ B] :
( sorted_wrt @ A @ ( ord_less_eq @ A )
@ ( map @ B @ A @ F2
@ ( filter2 @ B
@ ^ [X4: B] :
( ( F2 @ X4 )
= ( G @ Xs ) )
@ Xs ) ) ) ) ).
% sorted_map_same
thf(fact_6683_sorted__iff__nth__mono__less,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
= ( ! [I3: nat,J2: nat] :
( ( ord_less @ nat @ I3 @ J2 )
=> ( ( ord_less @ nat @ J2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ord_less_eq @ A @ ( nth @ A @ Xs @ I3 ) @ ( nth @ A @ Xs @ J2 ) ) ) ) ) ) ) ).
% sorted_iff_nth_mono_less
thf(fact_6684_sorted01,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs ) ) ) ).
% sorted01
thf(fact_6685_sorted__wrt_Oelims_I1_J,axiom,
! [A: $tType,X: A > A > $o,Xa2: list @ A,Y: $o] :
( ( ( sorted_wrt @ A @ X @ Xa2 )
= Y )
=> ( ( ( Xa2
= ( nil @ A ) )
=> ~ Y )
=> ~ ! [X5: A,Ys4: list @ A] :
( ( Xa2
= ( cons @ A @ X5 @ Ys4 ) )
=> ( Y
= ( ~ ( ! [Y6: A] :
( ( member @ A @ Y6 @ ( set2 @ A @ Ys4 ) )
=> ( X @ X5 @ Y6 ) )
& ( sorted_wrt @ A @ X @ Ys4 ) ) ) ) ) ) ) ).
% sorted_wrt.elims(1)
thf(fact_6686_sorted__wrt_Oelims_I2_J,axiom,
! [A: $tType,X: A > A > $o,Xa2: list @ A] :
( ( sorted_wrt @ A @ X @ Xa2 )
=> ( ( Xa2
!= ( nil @ A ) )
=> ~ ! [X5: A,Ys4: list @ A] :
( ( Xa2
= ( cons @ A @ X5 @ Ys4 ) )
=> ~ ( ! [Xa: A] :
( ( member @ A @ Xa @ ( set2 @ A @ Ys4 ) )
=> ( X @ X5 @ Xa ) )
& ( sorted_wrt @ A @ X @ Ys4 ) ) ) ) ) ).
% sorted_wrt.elims(2)
thf(fact_6687_finite__sorted__distinct__unique,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ? [X5: list @ A] :
( ( ( set2 @ A @ X5 )
= A4 )
& ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ X5 )
& ( distinct @ A @ X5 )
& ! [Y4: list @ A] :
( ( ( ( set2 @ A @ Y4 )
= A4 )
& ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Y4 )
& ( distinct @ A @ Y4 ) )
=> ( Y4 = X5 ) ) ) ) ) ).
% finite_sorted_distinct_unique
thf(fact_6688_sorted__list__of__set_Oidem__if__sorted__distinct,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
=> ( ( distinct @ A @ Xs )
=> ( ( linord4507533701916653071of_set @ A @ ( set2 @ A @ Xs ) )
= Xs ) ) ) ) ).
% sorted_list_of_set.idem_if_sorted_distinct
thf(fact_6689_filter__insort,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F2: B > A,Xs: list @ B,P: B > $o,X: B] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F2 @ Xs ) )
=> ( ( P @ X )
=> ( ( filter2 @ B @ P @ ( linorder_insort_key @ B @ A @ F2 @ X @ Xs ) )
= ( linorder_insort_key @ B @ A @ F2 @ X @ ( filter2 @ B @ P @ Xs ) ) ) ) ) ) ).
% filter_insort
thf(fact_6690_insort__remove1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,Xs: list @ A] :
( ( member @ A @ A3 @ ( set2 @ A @ Xs ) )
=> ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
=> ( ( linorder_insort_key @ A @ A
@ ^ [X4: A] : X4
@ A3
@ ( remove1 @ A @ A3 @ Xs ) )
= Xs ) ) ) ) ).
% insort_remove1
thf(fact_6691_sorted__iff__nth__Suc,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
= ( ! [I3: nat] :
( ( ord_less @ nat @ ( suc @ I3 ) @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ord_less_eq @ A @ ( nth @ A @ Xs @ I3 ) @ ( nth @ A @ Xs @ ( suc @ I3 ) ) ) ) ) ) ) ).
% sorted_iff_nth_Suc
thf(fact_6692_sorted__list__of__set_Ofinite__set__strict__sorted,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ~ ! [L4: list @ A] :
( ( sorted_wrt @ A @ ( ord_less @ A ) @ L4 )
=> ( ( ( set2 @ A @ L4 )
= A4 )
=> ( ( size_size @ ( list @ A ) @ L4 )
!= ( finite_card @ A @ A4 ) ) ) ) ) ) ).
% sorted_list_of_set.finite_set_strict_sorted
thf(fact_6693_sorted__nth__mono,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,I: nat,J: nat] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
=> ( ( ord_less_eq @ nat @ I @ J )
=> ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ord_less_eq @ A @ ( nth @ A @ Xs @ I ) @ ( nth @ A @ Xs @ J ) ) ) ) ) ) ).
% sorted_nth_mono
thf(fact_6694_sorted__iff__nth__mono,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
= ( ! [I3: nat,J2: nat] :
( ( ord_less_eq @ nat @ I3 @ J2 )
=> ( ( ord_less @ nat @ J2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ord_less_eq @ A @ ( nth @ A @ Xs @ I3 ) @ ( nth @ A @ Xs @ J2 ) ) ) ) ) ) ) ).
% sorted_iff_nth_mono
thf(fact_6695_sorted__wrt__less__idx,axiom,
! [Ns: list @ nat,I: nat] :
( ( sorted_wrt @ nat @ ( ord_less @ nat ) @ Ns )
=> ( ( ord_less @ nat @ I @ ( size_size @ ( list @ nat ) @ Ns ) )
=> ( ord_less_eq @ nat @ I @ ( nth @ nat @ Ns @ I ) ) ) ) ).
% sorted_wrt_less_idx
thf(fact_6696_sorted__find__Min,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,P: A > $o] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
=> ( ? [X2: A] :
( ( member @ A @ X2 @ ( set2 @ A @ Xs ) )
& ( P @ X2 ) )
=> ( ( find @ A @ P @ Xs )
= ( some @ A
@ ( lattic643756798350308766er_Min @ A
@ ( collect @ A
@ ^ [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
& ( P @ X4 ) ) ) ) ) ) ) ) ) ).
% sorted_find_Min
thf(fact_6697_map__sorted__distinct__set__unique,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F2: B > A,Xs: list @ B,Ys: list @ B] :
( ( inj_on @ B @ A @ F2 @ ( sup_sup @ ( set @ B ) @ ( set2 @ B @ Xs ) @ ( set2 @ B @ Ys ) ) )
=> ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F2 @ Xs ) )
=> ( ( distinct @ A @ ( map @ B @ A @ F2 @ Xs ) )
=> ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F2 @ Ys ) )
=> ( ( distinct @ A @ ( map @ B @ A @ F2 @ Ys ) )
=> ( ( ( set2 @ B @ Xs )
= ( set2 @ B @ Ys ) )
=> ( Xs = Ys ) ) ) ) ) ) ) ) ).
% map_sorted_distinct_set_unique
thf(fact_6698_sorted__list__of__set_Osorted__key__list__of__set__unique,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,L: list @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( ( sorted_wrt @ A @ ( ord_less @ A ) @ L )
& ( ( set2 @ A @ L )
= A4 )
& ( ( size_size @ ( list @ A ) @ L )
= ( finite_card @ A @ A4 ) ) )
= ( ( linord4507533701916653071of_set @ A @ A4 )
= L ) ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_unique
thf(fact_6699_sorted__insort__is__snoc,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,A3: A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Xs ) )
=> ( ord_less_eq @ A @ X5 @ A3 ) )
=> ( ( linorder_insort_key @ A @ A
@ ^ [X4: A] : X4
@ A3
@ Xs )
= ( append @ A @ Xs @ ( cons @ A @ A3 @ ( nil @ A ) ) ) ) ) ) ) ).
% sorted_insort_is_snoc
thf(fact_6700_insort__key__remove1,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [A3: B,Xs: list @ B,F2: B > A] :
( ( member @ B @ A3 @ ( set2 @ B @ Xs ) )
=> ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F2 @ Xs ) )
=> ( ( ( hd @ B
@ ( filter2 @ B
@ ^ [X4: B] :
( ( F2 @ A3 )
= ( F2 @ X4 ) )
@ Xs ) )
= A3 )
=> ( ( linorder_insort_key @ B @ A @ F2 @ A3 @ ( remove1 @ B @ A3 @ Xs ) )
= Xs ) ) ) ) ) ).
% insort_key_remove1
thf(fact_6701_max__ext__compat,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),S2: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( relcomp @ A @ A @ A @ R @ S2 ) @ R )
=> ( ord_less_eq @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( relcomp @ ( set @ A ) @ ( set @ A ) @ ( set @ A ) @ ( max_ext @ A @ R ) @ ( sup_sup @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( max_ext @ A @ S2 ) @ ( insert2 @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ ( bot_bot @ ( set @ A ) ) ) @ ( bot_bot @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) ) ) ) ) @ ( max_ext @ A @ R ) ) ) ).
% max_ext_compat
thf(fact_6702_nth__nth__transpose__sorted,axiom,
! [A: $tType,Xs: list @ ( list @ A ),I: nat,J: nat] :
( ( sorted_wrt @ nat @ ( ord_less_eq @ nat ) @ ( rev @ nat @ ( map @ ( list @ A ) @ nat @ ( size_size @ ( list @ A ) ) @ Xs ) ) )
=> ( ( ord_less @ nat @ I @ ( size_size @ ( list @ ( list @ A ) ) @ ( transpose @ A @ Xs ) ) )
=> ( ( ord_less @ nat @ J
@ ( size_size @ ( list @ ( list @ A ) )
@ ( filter2 @ ( list @ A )
@ ^ [Ys3: list @ A] : ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Ys3 ) )
@ Xs ) ) )
=> ( ( nth @ A @ ( nth @ ( list @ A ) @ ( transpose @ A @ Xs ) @ I ) @ J )
= ( nth @ A @ ( nth @ ( list @ A ) @ Xs @ J ) @ I ) ) ) ) ) ).
% nth_nth_transpose_sorted
thf(fact_6703_rev__rev__ident,axiom,
! [A: $tType,Xs: list @ A] :
( ( rev @ A @ ( rev @ A @ Xs ) )
= Xs ) ).
% rev_rev_ident
thf(fact_6704_rev__is__rev__conv,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( ( rev @ A @ Xs )
= ( rev @ A @ Ys ) )
= ( Xs = Ys ) ) ).
% rev_is_rev_conv
thf(fact_6705_rev__is__Nil__conv,axiom,
! [A: $tType,Xs: list @ A] :
( ( ( rev @ A @ Xs )
= ( nil @ A ) )
= ( Xs
= ( nil @ A ) ) ) ).
% rev_is_Nil_conv
thf(fact_6706_Nil__is__rev__conv,axiom,
! [A: $tType,Xs: list @ A] :
( ( ( nil @ A )
= ( rev @ A @ Xs ) )
= ( Xs
= ( nil @ A ) ) ) ).
% Nil_is_rev_conv
thf(fact_6707_set__rev,axiom,
! [A: $tType,Xs: list @ A] :
( ( set2 @ A @ ( rev @ A @ Xs ) )
= ( set2 @ A @ Xs ) ) ).
% set_rev
thf(fact_6708_length__rev,axiom,
! [A: $tType,Xs: list @ A] :
( ( size_size @ ( list @ A ) @ ( rev @ A @ Xs ) )
= ( size_size @ ( list @ A ) @ Xs ) ) ).
% length_rev
thf(fact_6709_rev__append,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( rev @ A @ ( append @ A @ Xs @ Ys ) )
= ( append @ A @ ( rev @ A @ Ys ) @ ( rev @ A @ Xs ) ) ) ).
% rev_append
thf(fact_6710_distinct__rev,axiom,
! [A: $tType,Xs: list @ A] :
( ( distinct @ A @ ( rev @ A @ Xs ) )
= ( distinct @ A @ Xs ) ) ).
% distinct_rev
thf(fact_6711_rev__replicate,axiom,
! [A: $tType,N2: nat,X: A] :
( ( rev @ A @ ( replicate @ A @ N2 @ X ) )
= ( replicate @ A @ N2 @ X ) ) ).
% rev_replicate
thf(fact_6712_remdups__adj__rev,axiom,
! [A: $tType,Xs: list @ A] :
( ( remdups_adj @ A @ ( rev @ A @ Xs ) )
= ( rev @ A @ ( remdups_adj @ A @ Xs ) ) ) ).
% remdups_adj_rev
thf(fact_6713_inj__on__rev,axiom,
! [A: $tType,A4: set @ ( list @ A )] : ( inj_on @ ( list @ A ) @ ( list @ A ) @ ( rev @ A ) @ A4 ) ).
% inj_on_rev
thf(fact_6714_rev__singleton__conv,axiom,
! [A: $tType,Xs: list @ A,X: A] :
( ( ( rev @ A @ Xs )
= ( cons @ A @ X @ ( nil @ A ) ) )
= ( Xs
= ( cons @ A @ X @ ( nil @ A ) ) ) ) ).
% rev_singleton_conv
thf(fact_6715_singleton__rev__conv,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( ( cons @ A @ X @ ( nil @ A ) )
= ( rev @ A @ Xs ) )
= ( ( cons @ A @ X @ ( nil @ A ) )
= Xs ) ) ).
% singleton_rev_conv
thf(fact_6716_rev__eq__Cons__iff,axiom,
! [A: $tType,Xs: list @ A,Y: A,Ys: list @ A] :
( ( ( rev @ A @ Xs )
= ( cons @ A @ Y @ Ys ) )
= ( Xs
= ( append @ A @ ( rev @ A @ Ys ) @ ( cons @ A @ Y @ ( nil @ A ) ) ) ) ) ).
% rev_eq_Cons_iff
thf(fact_6717_rev__map,axiom,
! [A: $tType,B: $tType,F2: B > A,Xs: list @ B] :
( ( rev @ A @ ( map @ B @ A @ F2 @ Xs ) )
= ( map @ B @ A @ F2 @ ( rev @ B @ Xs ) ) ) ).
% rev_map
thf(fact_6718_rev__concat,axiom,
! [A: $tType,Xs: list @ ( list @ A )] :
( ( rev @ A @ ( concat @ A @ Xs ) )
= ( concat @ A @ ( map @ ( list @ A ) @ ( list @ A ) @ ( rev @ A ) @ ( rev @ ( list @ A ) @ Xs ) ) ) ) ).
% rev_concat
thf(fact_6719_sorted__wrt__rev,axiom,
! [A: $tType,P: A > A > $o,Xs: list @ A] :
( ( sorted_wrt @ A @ P @ ( rev @ A @ Xs ) )
= ( sorted_wrt @ A
@ ^ [X4: A,Y6: A] : ( P @ Y6 @ X4 )
@ Xs ) ) ).
% sorted_wrt_rev
thf(fact_6720_sorted__wrt__upto,axiom,
! [I: int,J: int] : ( sorted_wrt @ int @ ( ord_less @ int ) @ ( upto @ I @ J ) ) ).
% sorted_wrt_upto
thf(fact_6721_sorted__upto,axiom,
! [M2: int,N2: int] : ( sorted_wrt @ int @ ( ord_less_eq @ int ) @ ( upto @ M2 @ N2 ) ) ).
% sorted_upto
thf(fact_6722_rev__filter,axiom,
! [A: $tType,P: A > $o,Xs: list @ A] :
( ( rev @ A @ ( filter2 @ A @ P @ Xs ) )
= ( filter2 @ A @ P @ ( rev @ A @ Xs ) ) ) ).
% rev_filter
thf(fact_6723_zip__rev,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( zip @ A @ B @ ( rev @ A @ Xs ) @ ( rev @ B @ Ys ) )
= ( rev @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys ) ) ) ) ).
% zip_rev
thf(fact_6724_rev_Osimps_I1_J,axiom,
! [A: $tType] :
( ( rev @ A @ ( nil @ A ) )
= ( nil @ A ) ) ).
% rev.simps(1)
thf(fact_6725_rev__swap,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( ( rev @ A @ Xs )
= Ys )
= ( Xs
= ( rev @ A @ Ys ) ) ) ).
% rev_swap
thf(fact_6726_rev_Osimps_I2_J,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( rev @ A @ ( cons @ A @ X @ Xs ) )
= ( append @ A @ ( rev @ A @ Xs ) @ ( cons @ A @ X @ ( nil @ A ) ) ) ) ).
% rev.simps(2)
thf(fact_6727_drop__rev,axiom,
! [A: $tType,N2: nat,Xs: list @ A] :
( ( drop @ A @ N2 @ ( rev @ A @ Xs ) )
= ( rev @ A @ ( take @ A @ ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ N2 ) @ Xs ) ) ) ).
% drop_rev
thf(fact_6728_rev__drop,axiom,
! [A: $tType,I: nat,Xs: list @ A] :
( ( rev @ A @ ( drop @ A @ I @ Xs ) )
= ( take @ A @ ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ I ) @ ( rev @ A @ Xs ) ) ) ).
% rev_drop
thf(fact_6729_rev__take,axiom,
! [A: $tType,I: nat,Xs: list @ A] :
( ( rev @ A @ ( take @ A @ I @ Xs ) )
= ( drop @ A @ ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ I ) @ ( rev @ A @ Xs ) ) ) ).
% rev_take
thf(fact_6730_take__rev,axiom,
! [A: $tType,N2: nat,Xs: list @ A] :
( ( take @ A @ N2 @ ( rev @ A @ Xs ) )
= ( rev @ A @ ( drop @ A @ ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ N2 ) @ Xs ) ) ) ).
% take_rev
thf(fact_6731_rotate__rev,axiom,
! [A: $tType,N2: nat,Xs: list @ A] :
( ( rotate @ A @ N2 @ ( rev @ A @ Xs ) )
= ( rev @ A @ ( rotate @ A @ ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( modulo_modulo @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs ) ) ) @ Xs ) ) ) ).
% rotate_rev
thf(fact_6732_rev__nth,axiom,
! [A: $tType,N2: nat,Xs: list @ A] :
( ( ord_less @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( nth @ A @ ( rev @ A @ Xs ) @ N2 )
= ( nth @ A @ Xs @ ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( suc @ N2 ) ) ) ) ) ).
% rev_nth
thf(fact_6733_rev__update,axiom,
! [A: $tType,K: nat,Xs: list @ A,Y: A] :
( ( ord_less @ nat @ K @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( rev @ A @ ( list_update @ A @ Xs @ K @ Y ) )
= ( list_update @ A @ ( rev @ A @ Xs ) @ ( minus_minus @ nat @ ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ K ) @ ( one_one @ nat ) ) @ Y ) ) ) ).
% rev_update
thf(fact_6734_sorted__transpose,axiom,
! [A: $tType,Xs: list @ ( list @ A )] : ( sorted_wrt @ nat @ ( ord_less_eq @ nat ) @ ( rev @ nat @ ( map @ ( list @ A ) @ nat @ ( size_size @ ( list @ A ) ) @ ( transpose @ A @ Xs ) ) ) ) ).
% sorted_transpose
thf(fact_6735_sorted__rev__iff__nth__Suc,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( rev @ A @ Xs ) )
= ( ! [I3: nat] :
( ( ord_less @ nat @ ( suc @ I3 ) @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ord_less_eq @ A @ ( nth @ A @ Xs @ ( suc @ I3 ) ) @ ( nth @ A @ Xs @ I3 ) ) ) ) ) ) ).
% sorted_rev_iff_nth_Suc
thf(fact_6736_sorted__rev__iff__nth__mono,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( rev @ A @ Xs ) )
= ( ! [I3: nat,J2: nat] :
( ( ord_less_eq @ nat @ I3 @ J2 )
=> ( ( ord_less @ nat @ J2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ord_less_eq @ A @ ( nth @ A @ Xs @ J2 ) @ ( nth @ A @ Xs @ I3 ) ) ) ) ) ) ) ).
% sorted_rev_iff_nth_mono
thf(fact_6737_sorted__rev__nth__mono,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,I: nat,J: nat] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( rev @ A @ Xs ) )
=> ( ( ord_less_eq @ nat @ I @ J )
=> ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ord_less_eq @ A @ ( nth @ A @ Xs @ J ) @ ( nth @ A @ Xs @ I ) ) ) ) ) ) ).
% sorted_rev_nth_mono
thf(fact_6738_max__ext_Ocases,axiom,
! [A: $tType,A1: set @ A,A22: set @ A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ A1 @ A22 ) @ ( max_ext @ A @ R ) )
=> ~ ( ( finite_finite2 @ A @ A1 )
=> ( ( finite_finite2 @ A @ A22 )
=> ( ( A22
!= ( bot_bot @ ( set @ A ) ) )
=> ~ ! [X2: A] :
( ( member @ A @ X2 @ A1 )
=> ? [Xa3: A] :
( ( member @ A @ Xa3 @ A22 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Xa3 ) @ R ) ) ) ) ) ) ) ).
% max_ext.cases
thf(fact_6739_max__ext_Osimps,axiom,
! [A: $tType,A1: set @ A,A22: set @ A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ A1 @ A22 ) @ ( max_ext @ A @ R ) )
= ( ( finite_finite2 @ A @ A1 )
& ( finite_finite2 @ A @ A22 )
& ( A22
!= ( bot_bot @ ( set @ A ) ) )
& ! [X4: A] :
( ( member @ A @ X4 @ A1 )
=> ? [Y6: A] :
( ( member @ A @ Y6 @ A22 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y6 ) @ R ) ) ) ) ) ).
% max_ext.simps
thf(fact_6740_max__ext_Omax__extI,axiom,
! [A: $tType,X8: set @ A,Y7: set @ A,R: set @ ( product_prod @ A @ A )] :
( ( finite_finite2 @ A @ X8 )
=> ( ( finite_finite2 @ A @ Y7 )
=> ( ( Y7
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ X8 )
=> ? [Xa: A] :
( ( member @ A @ Xa @ Y7 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X5 @ Xa ) @ R ) ) )
=> ( member @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ X8 @ Y7 ) @ ( max_ext @ A @ R ) ) ) ) ) ) ).
% max_ext.max_extI
thf(fact_6741_foldr__max__sorted,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,Y: A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( rev @ A @ Xs ) )
=> ( ( ( Xs
= ( nil @ A ) )
=> ( ( foldr @ A @ A @ ( ord_max @ A ) @ Xs @ Y )
= Y ) )
& ( ( Xs
!= ( nil @ A ) )
=> ( ( foldr @ A @ A @ ( ord_max @ A ) @ Xs @ Y )
= ( ord_max @ A @ ( nth @ A @ Xs @ ( zero_zero @ nat ) ) @ Y ) ) ) ) ) ) ).
% foldr_max_sorted
thf(fact_6742_length__transpose__sorted,axiom,
! [A: $tType,Xs: list @ ( list @ A )] :
( ( sorted_wrt @ nat @ ( ord_less_eq @ nat ) @ ( rev @ nat @ ( map @ ( list @ A ) @ nat @ ( size_size @ ( list @ A ) ) @ Xs ) ) )
=> ( ( ( Xs
= ( nil @ ( list @ A ) ) )
=> ( ( size_size @ ( list @ ( list @ A ) ) @ ( transpose @ A @ Xs ) )
= ( zero_zero @ nat ) ) )
& ( ( Xs
!= ( nil @ ( list @ A ) ) )
=> ( ( size_size @ ( list @ ( list @ A ) ) @ ( transpose @ A @ Xs ) )
= ( size_size @ ( list @ A ) @ ( nth @ ( list @ A ) @ Xs @ ( zero_zero @ nat ) ) ) ) ) ) ) ).
% length_transpose_sorted
thf(fact_6743_transpose__column__length,axiom,
! [A: $tType,Xs: list @ ( list @ A ),I: nat] :
( ( sorted_wrt @ nat @ ( ord_less_eq @ nat ) @ ( rev @ nat @ ( map @ ( list @ A ) @ nat @ ( size_size @ ( list @ A ) ) @ Xs ) ) )
=> ( ( ord_less @ nat @ I @ ( size_size @ ( list @ ( list @ A ) ) @ Xs ) )
=> ( ( size_size @ ( list @ ( list @ A ) )
@ ( filter2 @ ( list @ A )
@ ^ [Ys3: list @ A] : ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Ys3 ) )
@ ( transpose @ A @ Xs ) ) )
= ( size_size @ ( list @ A ) @ ( nth @ ( list @ A ) @ Xs @ I ) ) ) ) ) ).
% transpose_column_length
thf(fact_6744_transpose__column,axiom,
! [A: $tType,Xs: list @ ( list @ A ),I: nat] :
( ( sorted_wrt @ nat @ ( ord_less_eq @ nat ) @ ( rev @ nat @ ( map @ ( list @ A ) @ nat @ ( size_size @ ( list @ A ) ) @ Xs ) ) )
=> ( ( ord_less @ nat @ I @ ( size_size @ ( list @ ( list @ A ) ) @ Xs ) )
=> ( ( map @ ( list @ A ) @ A
@ ^ [Ys3: list @ A] : ( nth @ A @ Ys3 @ I )
@ ( filter2 @ ( list @ A )
@ ^ [Ys3: list @ A] : ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Ys3 ) )
@ ( transpose @ A @ Xs ) ) )
= ( nth @ ( list @ A ) @ Xs @ I ) ) ) ) ).
% transpose_column
thf(fact_6745_folding__insort__key_Ofinite__set__strict__sorted,axiom,
! [A: $tType,B: $tType,Less_eq2: A > A > $o,Less: A > A > $o,S2: set @ B,F2: B > A,A4: set @ B] :
( ( folding_insort_key @ A @ B @ Less_eq2 @ Less @ S2 @ F2 )
=> ( ( ord_less_eq @ ( set @ B ) @ A4 @ S2 )
=> ( ( finite_finite2 @ B @ A4 )
=> ~ ! [L4: list @ B] :
( ( sorted_wrt @ A @ Less @ ( map @ B @ A @ F2 @ L4 ) )
=> ( ( ( set2 @ B @ L4 )
= A4 )
=> ( ( size_size @ ( list @ B ) @ L4 )
!= ( finite_card @ B @ A4 ) ) ) ) ) ) ) ).
% folding_insort_key.finite_set_strict_sorted
thf(fact_6746_transpose__transpose,axiom,
! [A: $tType,Xs: list @ ( list @ A )] :
( ( sorted_wrt @ nat @ ( ord_less_eq @ nat ) @ ( rev @ nat @ ( map @ ( list @ A ) @ nat @ ( size_size @ ( list @ A ) ) @ Xs ) ) )
=> ( ( transpose @ A @ ( transpose @ A @ Xs ) )
= ( takeWhile @ ( list @ A )
@ ^ [X4: list @ A] :
( X4
!= ( nil @ A ) )
@ Xs ) ) ) ).
% transpose_transpose
thf(fact_6747_takeWhile__idem,axiom,
! [A: $tType,P: A > $o,Xs: list @ A] :
( ( takeWhile @ A @ P @ ( takeWhile @ A @ P @ Xs ) )
= ( takeWhile @ A @ P @ Xs ) ) ).
% takeWhile_idem
thf(fact_6748_takeWhile__eq__all__conv,axiom,
! [A: $tType,P: A > $o,Xs: list @ A] :
( ( ( takeWhile @ A @ P @ Xs )
= Xs )
= ( ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
=> ( P @ X4 ) ) ) ) ).
% takeWhile_eq_all_conv
thf(fact_6749_takeWhile__append1,axiom,
! [A: $tType,X: A,Xs: list @ A,P: A > $o,Ys: list @ A] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ~ ( P @ X )
=> ( ( takeWhile @ A @ P @ ( append @ A @ Xs @ Ys ) )
= ( takeWhile @ A @ P @ Xs ) ) ) ) ).
% takeWhile_append1
thf(fact_6750_takeWhile__append2,axiom,
! [A: $tType,Xs: list @ A,P: A > $o,Ys: list @ A] :
( ! [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Xs ) )
=> ( P @ X5 ) )
=> ( ( takeWhile @ A @ P @ ( append @ A @ Xs @ Ys ) )
= ( append @ A @ Xs @ ( takeWhile @ A @ P @ Ys ) ) ) ) ).
% takeWhile_append2
thf(fact_6751_takeWhile__replicate,axiom,
! [A: $tType,P: A > $o,X: A,N2: nat] :
( ( ( P @ X )
=> ( ( takeWhile @ A @ P @ ( replicate @ A @ N2 @ X ) )
= ( replicate @ A @ N2 @ X ) ) )
& ( ~ ( P @ X )
=> ( ( takeWhile @ A @ P @ ( replicate @ A @ N2 @ X ) )
= ( nil @ A ) ) ) ) ).
% takeWhile_replicate
thf(fact_6752_length__concat__rev,axiom,
! [A: $tType,Xs: list @ ( list @ A )] :
( ( size_size @ ( list @ A ) @ ( concat @ A @ ( rev @ ( list @ A ) @ Xs ) ) )
= ( size_size @ ( list @ A ) @ ( concat @ A @ Xs ) ) ) ).
% length_concat_rev
thf(fact_6753_sorted__takeWhile,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,P: A > $o] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( takeWhile @ A @ P @ Xs ) ) ) ) ).
% sorted_takeWhile
thf(fact_6754_takeWhile__tail,axiom,
! [A: $tType,P: A > $o,X: A,Xs: list @ A,L: list @ A] :
( ~ ( P @ X )
=> ( ( takeWhile @ A @ P @ ( append @ A @ Xs @ ( cons @ A @ X @ L ) ) )
= ( takeWhile @ A @ P @ Xs ) ) ) ).
% takeWhile_tail
thf(fact_6755_takeWhile_Osimps_I1_J,axiom,
! [A: $tType,P: A > $o] :
( ( takeWhile @ A @ P @ ( nil @ A ) )
= ( nil @ A ) ) ).
% takeWhile.simps(1)
thf(fact_6756_takeWhile_Osimps_I2_J,axiom,
! [A: $tType,P: A > $o,X: A,Xs: list @ A] :
( ( ( P @ X )
=> ( ( takeWhile @ A @ P @ ( cons @ A @ X @ Xs ) )
= ( cons @ A @ X @ ( takeWhile @ A @ P @ Xs ) ) ) )
& ( ~ ( P @ X )
=> ( ( takeWhile @ A @ P @ ( cons @ A @ X @ Xs ) )
= ( nil @ A ) ) ) ) ).
% takeWhile.simps(2)
thf(fact_6757_takeWhile__eq__Nil__iff,axiom,
! [A: $tType,P: A > $o,Xs: list @ A] :
( ( ( takeWhile @ A @ P @ Xs )
= ( nil @ A ) )
= ( ( Xs
= ( nil @ A ) )
| ~ ( P @ ( hd @ A @ Xs ) ) ) ) ).
% takeWhile_eq_Nil_iff
thf(fact_6758_set__takeWhileD,axiom,
! [A: $tType,X: A,P: A > $o,Xs: list @ A] :
( ( member @ A @ X @ ( set2 @ A @ ( takeWhile @ A @ P @ Xs ) ) )
=> ( ( member @ A @ X @ ( set2 @ A @ Xs ) )
& ( P @ X ) ) ) ).
% set_takeWhileD
thf(fact_6759_takeWhile__cong,axiom,
! [A: $tType,L: list @ A,K: list @ A,P: A > $o,Q: A > $o] :
( ( L = K )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ L ) )
=> ( ( P @ X5 )
= ( Q @ X5 ) ) )
=> ( ( takeWhile @ A @ P @ L )
= ( takeWhile @ A @ Q @ K ) ) ) ) ).
% takeWhile_cong
thf(fact_6760_takeWhile__eq__take,axiom,
! [A: $tType] :
( ( takeWhile @ A )
= ( ^ [P3: A > $o,Xs3: list @ A] : ( take @ A @ ( size_size @ ( list @ A ) @ ( takeWhile @ A @ P3 @ Xs3 ) ) @ Xs3 ) ) ) ).
% takeWhile_eq_take
thf(fact_6761_distinct__takeWhile,axiom,
! [A: $tType,Xs: list @ A,P: A > $o] :
( ( distinct @ A @ Xs )
=> ( distinct @ A @ ( takeWhile @ A @ P @ Xs ) ) ) ).
% distinct_takeWhile
thf(fact_6762_folding__insort__key_Oinj__on,axiom,
! [A: $tType,B: $tType,Less_eq2: A > A > $o,Less: A > A > $o,S2: set @ B,F2: B > A] :
( ( folding_insort_key @ A @ B @ Less_eq2 @ Less @ S2 @ F2 )
=> ( inj_on @ B @ A @ F2 @ S2 ) ) ).
% folding_insort_key.inj_on
thf(fact_6763_length__takeWhile__le,axiom,
! [A: $tType,P: A > $o,Xs: list @ A] : ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ ( takeWhile @ A @ P @ Xs ) ) @ ( size_size @ ( list @ A ) @ Xs ) ) ).
% length_takeWhile_le
thf(fact_6764_takeWhile__map,axiom,
! [A: $tType,B: $tType,P: A > $o,F2: B > A,Xs: list @ B] :
( ( takeWhile @ A @ P @ ( map @ B @ A @ F2 @ Xs ) )
= ( map @ B @ A @ F2 @ ( takeWhile @ B @ ( comp @ A @ $o @ B @ P @ F2 ) @ Xs ) ) ) ).
% takeWhile_map
thf(fact_6765_folding__insort__key_Odistinct__if__distinct__map,axiom,
! [A: $tType,B: $tType,Less_eq2: A > A > $o,Less: A > A > $o,S2: set @ B,F2: B > A,Xs: list @ B] :
( ( folding_insort_key @ A @ B @ Less_eq2 @ Less @ S2 @ F2 )
=> ( ( distinct @ A @ ( map @ B @ A @ F2 @ Xs ) )
=> ( distinct @ B @ Xs ) ) ) ).
% folding_insort_key.distinct_if_distinct_map
thf(fact_6766_takeWhile__nth,axiom,
! [A: $tType,J: nat,P: A > $o,Xs: list @ A] :
( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ ( takeWhile @ A @ P @ Xs ) ) )
=> ( ( nth @ A @ ( takeWhile @ A @ P @ Xs ) @ J )
= ( nth @ A @ Xs @ J ) ) ) ).
% takeWhile_nth
thf(fact_6767_nth__length__takeWhile,axiom,
! [A: $tType,P: A > $o,Xs: list @ A] :
( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ ( takeWhile @ A @ P @ Xs ) ) @ ( size_size @ ( list @ A ) @ Xs ) )
=> ~ ( P @ ( nth @ A @ Xs @ ( size_size @ ( list @ A ) @ ( takeWhile @ A @ P @ Xs ) ) ) ) ) ).
% nth_length_takeWhile
thf(fact_6768_takeWhile__append,axiom,
! [A: $tType,Xs: list @ A,P: A > $o,Ys: list @ A] :
( ( ! [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Xs ) )
=> ( P @ X5 ) )
=> ( ( takeWhile @ A @ P @ ( append @ A @ Xs @ Ys ) )
= ( append @ A @ Xs @ ( takeWhile @ A @ P @ Ys ) ) ) )
& ( ~ ! [X2: A] :
( ( member @ A @ X2 @ ( set2 @ A @ Xs ) )
=> ( P @ X2 ) )
=> ( ( takeWhile @ A @ P @ ( append @ A @ Xs @ Ys ) )
= ( takeWhile @ A @ P @ Xs ) ) ) ) ).
% takeWhile_append
thf(fact_6769_length__takeWhile__less__P__nth,axiom,
! [A: $tType,J: nat,P: A > $o,Xs: list @ A] :
( ! [I4: nat] :
( ( ord_less @ nat @ I4 @ J )
=> ( P @ ( nth @ A @ Xs @ I4 ) ) )
=> ( ( ord_less_eq @ nat @ J @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ord_less_eq @ nat @ J @ ( size_size @ ( list @ A ) @ ( takeWhile @ A @ P @ Xs ) ) ) ) ) ).
% length_takeWhile_less_P_nth
thf(fact_6770_takeWhile__eq__take__P__nth,axiom,
! [A: $tType,N2: nat,Xs: list @ A,P: A > $o] :
( ! [I4: nat] :
( ( ord_less @ nat @ I4 @ N2 )
=> ( ( ord_less @ nat @ I4 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( P @ ( nth @ A @ Xs @ I4 ) ) ) )
=> ( ( ( ord_less @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ~ ( P @ ( nth @ A @ Xs @ N2 ) ) )
=> ( ( takeWhile @ A @ P @ Xs )
= ( take @ A @ N2 @ Xs ) ) ) ) ).
% takeWhile_eq_take_P_nth
thf(fact_6771_sorted__list__of__set_Ofolding__insort__key__axioms,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( folding_insort_key @ A @ A @ ( ord_less_eq @ A ) @ ( ord_less @ A ) @ ( top_top @ ( set @ A ) )
@ ^ [X4: A] : X4 ) ) ).
% sorted_list_of_set.folding_insort_key_axioms
thf(fact_6772_filter__equals__takeWhile__sorted__rev,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F2: B > A,Xs: list @ B,T2: A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( rev @ A @ ( map @ B @ A @ F2 @ Xs ) ) )
=> ( ( filter2 @ B
@ ^ [X4: B] : ( ord_less @ A @ T2 @ ( F2 @ X4 ) )
@ Xs )
= ( takeWhile @ B
@ ^ [X4: B] : ( ord_less @ A @ T2 @ ( F2 @ X4 ) )
@ Xs ) ) ) ) ).
% filter_equals_takeWhile_sorted_rev
thf(fact_6773_folding__insort__key_Osorted__key__list__of__set__unique,axiom,
! [A: $tType,B: $tType,Less_eq2: A > A > $o,Less: A > A > $o,S2: set @ B,F2: B > A,A4: set @ B,L: list @ B] :
( ( folding_insort_key @ A @ B @ Less_eq2 @ Less @ S2 @ F2 )
=> ( ( ord_less_eq @ ( set @ B ) @ A4 @ S2 )
=> ( ( finite_finite2 @ B @ A4 )
=> ( ( ( sorted_wrt @ A @ Less @ ( map @ B @ A @ F2 @ L ) )
& ( ( set2 @ B @ L )
= A4 )
& ( ( size_size @ ( list @ B ) @ L )
= ( finite_card @ B @ A4 ) ) )
= ( ( sorted8670434370408473282of_set @ A @ B @ Less_eq2 @ F2 @ A4 )
= L ) ) ) ) ) ).
% folding_insort_key.sorted_key_list_of_set_unique
thf(fact_6774_folding__insort__key_Osorted__key__list__of__set__remove,axiom,
! [A: $tType,B: $tType,Less_eq2: A > A > $o,Less: A > A > $o,S2: set @ B,F2: B > A,X: B,A4: set @ B] :
( ( folding_insort_key @ A @ B @ Less_eq2 @ Less @ S2 @ F2 )
=> ( ( ord_less_eq @ ( set @ B ) @ ( insert2 @ B @ X @ A4 ) @ S2 )
=> ( ( finite_finite2 @ B @ A4 )
=> ( ( sorted8670434370408473282of_set @ A @ B @ Less_eq2 @ F2 @ ( minus_minus @ ( set @ B ) @ A4 @ ( insert2 @ B @ X @ ( bot_bot @ ( set @ B ) ) ) ) )
= ( remove1 @ B @ X @ ( sorted8670434370408473282of_set @ A @ B @ Less_eq2 @ F2 @ A4 ) ) ) ) ) ) ).
% folding_insort_key.sorted_key_list_of_set_remove
thf(fact_6775_linorder_Osorted__key__list__of__set_Ocong,axiom,
! [B: $tType,A: $tType] :
( ( sorted8670434370408473282of_set @ A @ B )
= ( sorted8670434370408473282of_set @ A @ B ) ) ).
% linorder.sorted_key_list_of_set.cong
thf(fact_6776_folding__insort__key_Osorted__key__list__of__set__inject,axiom,
! [A: $tType,B: $tType,Less_eq2: A > A > $o,Less: A > A > $o,S2: set @ B,F2: B > A,A4: set @ B,B2: set @ B] :
( ( folding_insort_key @ A @ B @ Less_eq2 @ Less @ S2 @ F2 )
=> ( ( ord_less_eq @ ( set @ B ) @ A4 @ S2 )
=> ( ( ord_less_eq @ ( set @ B ) @ B2 @ S2 )
=> ( ( ( sorted8670434370408473282of_set @ A @ B @ Less_eq2 @ F2 @ A4 )
= ( sorted8670434370408473282of_set @ A @ B @ Less_eq2 @ F2 @ B2 ) )
=> ( ( finite_finite2 @ B @ A4 )
=> ( ( finite_finite2 @ B @ B2 )
=> ( A4 = B2 ) ) ) ) ) ) ) ).
% folding_insort_key.sorted_key_list_of_set_inject
thf(fact_6777_folding__insort__key_Osorted__key__list__of__set__empty,axiom,
! [A: $tType,B: $tType,Less_eq2: A > A > $o,Less: A > A > $o,S2: set @ B,F2: B > A] :
( ( folding_insort_key @ A @ B @ Less_eq2 @ Less @ S2 @ F2 )
=> ( ( sorted8670434370408473282of_set @ A @ B @ Less_eq2 @ F2 @ ( bot_bot @ ( set @ B ) ) )
= ( nil @ B ) ) ) ).
% folding_insort_key.sorted_key_list_of_set_empty
thf(fact_6778_folding__insort__key_Oset__sorted__key__list__of__set,axiom,
! [A: $tType,B: $tType,Less_eq2: A > A > $o,Less: A > A > $o,S2: set @ B,F2: B > A,A4: set @ B] :
( ( folding_insort_key @ A @ B @ Less_eq2 @ Less @ S2 @ F2 )
=> ( ( ord_less_eq @ ( set @ B ) @ A4 @ S2 )
=> ( ( finite_finite2 @ B @ A4 )
=> ( ( set2 @ B @ ( sorted8670434370408473282of_set @ A @ B @ Less_eq2 @ F2 @ A4 ) )
= A4 ) ) ) ) ).
% folding_insort_key.set_sorted_key_list_of_set
thf(fact_6779_folding__insort__key_Olength__sorted__key__list__of__set,axiom,
! [A: $tType,B: $tType,Less_eq2: A > A > $o,Less: A > A > $o,S2: set @ B,F2: B > A,A4: set @ B] :
( ( folding_insort_key @ A @ B @ Less_eq2 @ Less @ S2 @ F2 )
=> ( ( ord_less_eq @ ( set @ B ) @ A4 @ S2 )
=> ( ( size_size @ ( list @ B ) @ ( sorted8670434370408473282of_set @ A @ B @ Less_eq2 @ F2 @ A4 ) )
= ( finite_card @ B @ A4 ) ) ) ) ).
% folding_insort_key.length_sorted_key_list_of_set
thf(fact_6780_folding__insort__key_Odistinct__sorted__key__list__of__set,axiom,
! [A: $tType,B: $tType,Less_eq2: A > A > $o,Less: A > A > $o,S2: set @ B,F2: B > A,A4: set @ B] :
( ( folding_insort_key @ A @ B @ Less_eq2 @ Less @ S2 @ F2 )
=> ( ( ord_less_eq @ ( set @ B ) @ A4 @ S2 )
=> ( distinct @ A @ ( map @ B @ A @ F2 @ ( sorted8670434370408473282of_set @ A @ B @ Less_eq2 @ F2 @ A4 ) ) ) ) ) ).
% folding_insort_key.distinct_sorted_key_list_of_set
thf(fact_6781_folding__insort__key_Osorted__sorted__key__list__of__set,axiom,
! [A: $tType,B: $tType,Less_eq2: A > A > $o,Less: A > A > $o,S2: set @ B,F2: B > A,A4: set @ B] :
( ( folding_insort_key @ A @ B @ Less_eq2 @ Less @ S2 @ F2 )
=> ( ( ord_less_eq @ ( set @ B ) @ A4 @ S2 )
=> ( sorted_wrt @ A @ Less_eq2 @ ( map @ B @ A @ F2 @ ( sorted8670434370408473282of_set @ A @ B @ Less_eq2 @ F2 @ A4 ) ) ) ) ) ).
% folding_insort_key.sorted_sorted_key_list_of_set
thf(fact_6782_folding__insort__key_Ostrict__sorted__key__list__of__set,axiom,
! [A: $tType,B: $tType,Less_eq2: A > A > $o,Less: A > A > $o,S2: set @ B,F2: B > A,A4: set @ B] :
( ( folding_insort_key @ A @ B @ Less_eq2 @ Less @ S2 @ F2 )
=> ( ( ord_less_eq @ ( set @ B ) @ A4 @ S2 )
=> ( sorted_wrt @ A @ Less @ ( map @ B @ A @ F2 @ ( sorted8670434370408473282of_set @ A @ B @ Less_eq2 @ F2 @ A4 ) ) ) ) ) ).
% folding_insort_key.strict_sorted_key_list_of_set
thf(fact_6783_folding__insort__key_Osorted__key__list__of__set__eq__Nil__iff,axiom,
! [A: $tType,B: $tType,Less_eq2: A > A > $o,Less: A > A > $o,S2: set @ B,F2: B > A,A4: set @ B] :
( ( folding_insort_key @ A @ B @ Less_eq2 @ Less @ S2 @ F2 )
=> ( ( ord_less_eq @ ( set @ B ) @ A4 @ S2 )
=> ( ( finite_finite2 @ B @ A4 )
=> ( ( ( sorted8670434370408473282of_set @ A @ B @ Less_eq2 @ F2 @ A4 )
= ( nil @ B ) )
= ( A4
= ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ).
% folding_insort_key.sorted_key_list_of_set_eq_Nil_iff
thf(fact_6784_folding__insort__key_Oidem__if__sorted__distinct,axiom,
! [A: $tType,B: $tType,Less_eq2: A > A > $o,Less: A > A > $o,S2: set @ B,F2: B > A,Xs: list @ B] :
( ( folding_insort_key @ A @ B @ Less_eq2 @ Less @ S2 @ F2 )
=> ( ( ord_less_eq @ ( set @ B ) @ ( set2 @ B @ Xs ) @ S2 )
=> ( ( sorted_wrt @ A @ Less_eq2 @ ( map @ B @ A @ F2 @ Xs ) )
=> ( ( distinct @ B @ Xs )
=> ( ( sorted8670434370408473282of_set @ A @ B @ Less_eq2 @ F2 @ ( set2 @ B @ Xs ) )
= Xs ) ) ) ) ) ).
% folding_insort_key.idem_if_sorted_distinct
thf(fact_6785_folding__insort__key_Osorted__key__list__of__set__insert__remove,axiom,
! [A: $tType,B: $tType,Less_eq2: A > A > $o,Less: A > A > $o,S2: set @ B,F2: B > A,X: B,A4: set @ B] :
( ( folding_insort_key @ A @ B @ Less_eq2 @ Less @ S2 @ F2 )
=> ( ( ord_less_eq @ ( set @ B ) @ ( insert2 @ B @ X @ A4 ) @ S2 )
=> ( ( finite_finite2 @ B @ A4 )
=> ( ( sorted8670434370408473282of_set @ A @ B @ Less_eq2 @ F2 @ ( insert2 @ B @ X @ A4 ) )
= ( insort_key @ A @ B @ Less_eq2 @ F2 @ X @ ( sorted8670434370408473282of_set @ A @ B @ Less_eq2 @ F2 @ ( minus_minus @ ( set @ B ) @ A4 @ ( insert2 @ B @ X @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ) ).
% folding_insort_key.sorted_key_list_of_set_insert_remove
thf(fact_6786_folding__insort__key_Osorted__key__list__of__set__insert,axiom,
! [A: $tType,B: $tType,Less_eq2: A > A > $o,Less: A > A > $o,S2: set @ B,F2: B > A,X: B,A4: set @ B] :
( ( folding_insort_key @ A @ B @ Less_eq2 @ Less @ S2 @ F2 )
=> ( ( ord_less_eq @ ( set @ B ) @ ( insert2 @ B @ X @ A4 ) @ S2 )
=> ( ( finite_finite2 @ B @ A4 )
=> ( ~ ( member @ B @ X @ A4 )
=> ( ( sorted8670434370408473282of_set @ A @ B @ Less_eq2 @ F2 @ ( insert2 @ B @ X @ A4 ) )
= ( insort_key @ A @ B @ Less_eq2 @ F2 @ X @ ( sorted8670434370408473282of_set @ A @ B @ Less_eq2 @ F2 @ A4 ) ) ) ) ) ) ) ).
% folding_insort_key.sorted_key_list_of_set_insert
thf(fact_6787_linorder_Oinsort__key_Ocong,axiom,
! [B: $tType,A: $tType] :
( ( insort_key @ A @ B )
= ( insort_key @ A @ B ) ) ).
% linorder.insort_key.cong
thf(fact_6788_folding__insort__key_Oinsort__key__commute,axiom,
! [A: $tType,B: $tType,Less_eq2: A > A > $o,Less: A > A > $o,S2: set @ B,F2: B > A,X: B,Y: B] :
( ( folding_insort_key @ A @ B @ Less_eq2 @ Less @ S2 @ F2 )
=> ( ( member @ B @ X @ S2 )
=> ( ( member @ B @ Y @ S2 )
=> ( ( comp @ ( list @ B ) @ ( list @ B ) @ ( list @ B ) @ ( insort_key @ A @ B @ Less_eq2 @ F2 @ Y ) @ ( insort_key @ A @ B @ Less_eq2 @ F2 @ X ) )
= ( comp @ ( list @ B ) @ ( list @ B ) @ ( list @ B ) @ ( insort_key @ A @ B @ Less_eq2 @ F2 @ X ) @ ( insort_key @ A @ B @ Less_eq2 @ F2 @ Y ) ) ) ) ) ) ).
% folding_insort_key.insort_key_commute
thf(fact_6789_extract__def,axiom,
! [A: $tType] :
( ( extract @ A )
= ( ^ [P3: A > $o,Xs3: list @ A] :
( case_list @ ( option @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) ) @ A @ ( none @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) )
@ ^ [Y6: A,Ys3: list @ A] : ( some @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) @ ( product_Pair @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) @ ( takeWhile @ A @ ( comp @ $o @ $o @ A @ (~) @ P3 ) @ Xs3 ) @ ( product_Pair @ A @ ( list @ A ) @ Y6 @ Ys3 ) ) )
@ ( dropWhile @ A @ ( comp @ $o @ $o @ A @ (~) @ P3 ) @ Xs3 ) ) ) ) ).
% extract_def
thf(fact_6790_lists__length__Suc__eq,axiom,
! [A: $tType,A4: set @ A,N2: nat] :
( ( collect @ ( list @ A )
@ ^ [Xs3: list @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs3 ) @ A4 )
& ( ( size_size @ ( list @ A ) @ Xs3 )
= ( suc @ N2 ) ) ) )
= ( image2 @ ( product_prod @ ( list @ A ) @ A ) @ ( list @ A )
@ ( product_case_prod @ ( list @ A ) @ A @ ( list @ A )
@ ^ [Xs3: list @ A,N: A] : ( cons @ A @ N @ Xs3 ) )
@ ( product_Sigma @ ( list @ A ) @ A
@ ( collect @ ( list @ A )
@ ^ [Xs3: list @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs3 ) @ A4 )
& ( ( size_size @ ( list @ A ) @ Xs3 )
= N2 ) ) )
@ ^ [Uu3: list @ A] : A4 ) ) ) ).
% lists_length_Suc_eq
thf(fact_6791_dropWhile__idem,axiom,
! [A: $tType,P: A > $o,Xs: list @ A] :
( ( dropWhile @ A @ P @ ( dropWhile @ A @ P @ Xs ) )
= ( dropWhile @ A @ P @ Xs ) ) ).
% dropWhile_idem
thf(fact_6792_Sigma__empty1,axiom,
! [B: $tType,A: $tType,B2: A > ( set @ B )] :
( ( product_Sigma @ A @ B @ ( bot_bot @ ( set @ A ) ) @ B2 )
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% Sigma_empty1
thf(fact_6793_dropWhile__eq__Nil__conv,axiom,
! [A: $tType,P: A > $o,Xs: list @ A] :
( ( ( dropWhile @ A @ P @ Xs )
= ( nil @ A ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
=> ( P @ X4 ) ) ) ) ).
% dropWhile_eq_Nil_conv
thf(fact_6794_dropWhile__append2,axiom,
! [A: $tType,Xs: list @ A,P: A > $o,Ys: list @ A] :
( ! [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Xs ) )
=> ( P @ X5 ) )
=> ( ( dropWhile @ A @ P @ ( append @ A @ Xs @ Ys ) )
= ( dropWhile @ A @ P @ Ys ) ) ) ).
% dropWhile_append2
thf(fact_6795_dropWhile__append1,axiom,
! [A: $tType,X: A,Xs: list @ A,P: A > $o,Ys: list @ A] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ~ ( P @ X )
=> ( ( dropWhile @ A @ P @ ( append @ A @ Xs @ Ys ) )
= ( append @ A @ ( dropWhile @ A @ P @ Xs ) @ Ys ) ) ) ) ).
% dropWhile_append1
thf(fact_6796_dropWhile__replicate,axiom,
! [A: $tType,P: A > $o,X: A,N2: nat] :
( ( ( P @ X )
=> ( ( dropWhile @ A @ P @ ( replicate @ A @ N2 @ X ) )
= ( nil @ A ) ) )
& ( ~ ( P @ X )
=> ( ( dropWhile @ A @ P @ ( replicate @ A @ N2 @ X ) )
= ( replicate @ A @ N2 @ X ) ) ) ) ).
% dropWhile_replicate
thf(fact_6797_takeWhile__dropWhile__id,axiom,
! [A: $tType,P: A > $o,Xs: list @ A] :
( ( append @ A @ ( takeWhile @ A @ P @ Xs ) @ ( dropWhile @ A @ P @ Xs ) )
= Xs ) ).
% takeWhile_dropWhile_id
thf(fact_6798_Sigma__empty2,axiom,
! [B: $tType,A: $tType,A4: set @ A] :
( ( product_Sigma @ A @ B @ A4
@ ^ [Uu3: A] : ( bot_bot @ ( set @ B ) ) )
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% Sigma_empty2
thf(fact_6799_Times__empty,axiom,
! [A: $tType,B: $tType,A4: set @ A,B2: set @ B] :
( ( ( product_Sigma @ A @ B @ A4
@ ^ [Uu3: A] : B2 )
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) )
= ( ( A4
= ( bot_bot @ ( set @ A ) ) )
| ( B2
= ( bot_bot @ ( set @ B ) ) ) ) ) ).
% Times_empty
thf(fact_6800_finite__SigmaI,axiom,
! [B: $tType,A: $tType,A4: set @ A,B2: A > ( set @ B )] :
( ( finite_finite2 @ A @ A4 )
=> ( ! [A6: A] :
( ( member @ A @ A6 @ A4 )
=> ( finite_finite2 @ B @ ( B2 @ A6 ) ) )
=> ( finite_finite2 @ ( product_prod @ A @ B ) @ ( product_Sigma @ A @ B @ A4 @ B2 ) ) ) ) ).
% finite_SigmaI
thf(fact_6801_set__product,axiom,
! [B: $tType,A: $tType,Xs: list @ A,Ys: list @ B] :
( ( set2 @ ( product_prod @ A @ B ) @ ( product @ A @ B @ Xs @ Ys ) )
= ( product_Sigma @ A @ B @ ( set2 @ A @ Xs )
@ ^ [Uu3: A] : ( set2 @ B @ Ys ) ) ) ).
% set_product
thf(fact_6802_card__SigmaI,axiom,
! [B: $tType,A: $tType,A4: set @ A,B2: A > ( set @ B )] :
( ( finite_finite2 @ A @ A4 )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ A4 )
=> ( finite_finite2 @ B @ ( B2 @ X5 ) ) )
=> ( ( finite_card @ ( product_prod @ A @ B ) @ ( product_Sigma @ A @ B @ A4 @ B2 ) )
= ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [A5: A] : ( finite_card @ B @ ( B2 @ A5 ) )
@ A4 ) ) ) ) ).
% card_SigmaI
thf(fact_6803_sorted__dropWhile,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,P: A > $o] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( dropWhile @ A @ P @ Xs ) ) ) ) ).
% sorted_dropWhile
thf(fact_6804_dropWhile__map,axiom,
! [A: $tType,B: $tType,P: A > $o,F2: B > A,Xs: list @ B] :
( ( dropWhile @ A @ P @ ( map @ B @ A @ F2 @ Xs ) )
= ( map @ B @ A @ F2 @ ( dropWhile @ B @ ( comp @ A @ $o @ B @ P @ F2 ) @ Xs ) ) ) ).
% dropWhile_map
thf(fact_6805_times__eq__iff,axiom,
! [A: $tType,B: $tType,A4: set @ A,B2: set @ B,C5: set @ A,D6: set @ B] :
( ( ( product_Sigma @ A @ B @ A4
@ ^ [Uu3: A] : B2 )
= ( product_Sigma @ A @ B @ C5
@ ^ [Uu3: A] : D6 ) )
= ( ( ( A4 = C5 )
& ( B2 = D6 ) )
| ( ( ( A4
= ( bot_bot @ ( set @ A ) ) )
| ( B2
= ( bot_bot @ ( set @ B ) ) ) )
& ( ( C5
= ( bot_bot @ ( set @ A ) ) )
| ( D6
= ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ).
% times_eq_iff
thf(fact_6806_Sigma__empty__iff,axiom,
! [B: $tType,A: $tType,I6: set @ A,X8: A > ( set @ B )] :
( ( ( product_Sigma @ A @ B @ I6 @ X8 )
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ I6 )
=> ( ( X8 @ X4 )
= ( bot_bot @ ( set @ B ) ) ) ) ) ) ).
% Sigma_empty_iff
thf(fact_6807_Restr__subset,axiom,
! [A: $tType,A4: set @ A,B2: set @ A,R2: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B2 )
=> ( ( inf_inf @ ( set @ ( product_prod @ A @ A ) )
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R2
@ ( product_Sigma @ A @ A @ B2
@ ^ [Uu3: A] : B2 ) )
@ ( product_Sigma @ A @ A @ A4
@ ^ [Uu3: A] : A4 ) )
= ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R2
@ ( product_Sigma @ A @ A @ A4
@ ^ [Uu3: A] : A4 ) ) ) ) ).
% Restr_subset
thf(fact_6808_Sigma__mono,axiom,
! [B: $tType,A: $tType,A4: set @ A,C5: set @ A,B2: A > ( set @ B ),D6: A > ( set @ B )] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ C5 )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ A4 )
=> ( ord_less_eq @ ( set @ B ) @ ( B2 @ X5 ) @ ( D6 @ X5 ) ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( product_Sigma @ A @ B @ A4 @ B2 ) @ ( product_Sigma @ A @ B @ C5 @ D6 ) ) ) ) ).
% Sigma_mono
thf(fact_6809_Times__subset__cancel2,axiom,
! [A: $tType,B: $tType,X: A,C5: set @ A,A4: set @ B,B2: set @ B] :
( ( member @ A @ X @ C5 )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ B @ A ) )
@ ( product_Sigma @ B @ A @ A4
@ ^ [Uu3: B] : C5 )
@ ( product_Sigma @ B @ A @ B2
@ ^ [Uu3: B] : C5 ) )
= ( ord_less_eq @ ( set @ B ) @ A4 @ B2 ) ) ) ).
% Times_subset_cancel2
thf(fact_6810_length__dropWhile__le,axiom,
! [A: $tType,P: A > $o,Xs: list @ A] : ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ ( dropWhile @ A @ P @ Xs ) ) @ ( size_size @ ( list @ A ) @ Xs ) ) ).
% length_dropWhile_le
thf(fact_6811_distinct__dropWhile,axiom,
! [A: $tType,Xs: list @ A,P: A > $o] :
( ( distinct @ A @ Xs )
=> ( distinct @ A @ ( dropWhile @ A @ P @ Xs ) ) ) ).
% distinct_dropWhile
thf(fact_6812_set__dropWhileD,axiom,
! [A: $tType,X: A,P: A > $o,Xs: list @ A] :
( ( member @ A @ X @ ( set2 @ A @ ( dropWhile @ A @ P @ Xs ) ) )
=> ( member @ A @ X @ ( set2 @ A @ Xs ) ) ) ).
% set_dropWhileD
thf(fact_6813_dropWhile__cong,axiom,
! [A: $tType,L: list @ A,K: list @ A,P: A > $o,Q: A > $o] :
( ( L = K )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ L ) )
=> ( ( P @ X5 )
= ( Q @ X5 ) ) )
=> ( ( dropWhile @ A @ P @ L )
= ( dropWhile @ A @ Q @ K ) ) ) ) ).
% dropWhile_cong
thf(fact_6814_dropWhile__eq__self__iff,axiom,
! [A: $tType,P: A > $o,Xs: list @ A] :
( ( ( dropWhile @ A @ P @ Xs )
= Xs )
= ( ( Xs
= ( nil @ A ) )
| ~ ( P @ ( hd @ A @ Xs ) ) ) ) ).
% dropWhile_eq_self_iff
thf(fact_6815_hd__dropWhile,axiom,
! [A: $tType,P: A > $o,Xs: list @ A] :
( ( ( dropWhile @ A @ P @ Xs )
!= ( nil @ A ) )
=> ~ ( P @ ( hd @ A @ ( dropWhile @ A @ P @ Xs ) ) ) ) ).
% hd_dropWhile
thf(fact_6816_dropWhile_Osimps_I1_J,axiom,
! [A: $tType,P: A > $o] :
( ( dropWhile @ A @ P @ ( nil @ A ) )
= ( nil @ A ) ) ).
% dropWhile.simps(1)
thf(fact_6817_finite__cartesian__product,axiom,
! [B: $tType,A: $tType,A4: set @ A,B2: set @ B] :
( ( finite_finite2 @ A @ A4 )
=> ( ( finite_finite2 @ B @ B2 )
=> ( finite_finite2 @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ A4
@ ^ [Uu3: A] : B2 ) ) ) ) ).
% finite_cartesian_product
thf(fact_6818_dropWhile_Osimps_I2_J,axiom,
! [A: $tType,P: A > $o,X: A,Xs: list @ A] :
( ( ( P @ X )
=> ( ( dropWhile @ A @ P @ ( cons @ A @ X @ Xs ) )
= ( dropWhile @ A @ P @ Xs ) ) )
& ( ~ ( P @ X )
=> ( ( dropWhile @ A @ P @ ( cons @ A @ X @ Xs ) )
= ( cons @ A @ X @ Xs ) ) ) ) ).
% dropWhile.simps(2)
thf(fact_6819_dropWhile__append3,axiom,
! [A: $tType,P: A > $o,Y: A,Xs: list @ A,Ys: list @ A] :
( ~ ( P @ Y )
=> ( ( dropWhile @ A @ P @ ( append @ A @ Xs @ ( cons @ A @ Y @ Ys ) ) )
= ( append @ A @ ( dropWhile @ A @ P @ Xs ) @ ( cons @ A @ Y @ Ys ) ) ) ) ).
% dropWhile_append3
thf(fact_6820_remdups__adj__Cons_H,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( remdups_adj @ A @ ( cons @ A @ X @ Xs ) )
= ( cons @ A @ X
@ ( remdups_adj @ A
@ ( dropWhile @ A
@ ^ [Y6: A] : Y6 = X
@ Xs ) ) ) ) ).
% remdups_adj_Cons'
thf(fact_6821_times__subset__iff,axiom,
! [A: $tType,B: $tType,A4: set @ A,C5: set @ B,B2: set @ A,D6: set @ B] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) )
@ ( product_Sigma @ A @ B @ A4
@ ^ [Uu3: A] : C5 )
@ ( product_Sigma @ A @ B @ B2
@ ^ [Uu3: A] : D6 ) )
= ( ( A4
= ( bot_bot @ ( set @ A ) ) )
| ( C5
= ( bot_bot @ ( set @ B ) ) )
| ( ( ord_less_eq @ ( set @ A ) @ A4 @ B2 )
& ( ord_less_eq @ ( set @ B ) @ C5 @ D6 ) ) ) ) ).
% times_subset_iff
thf(fact_6822_finite__cartesian__product__iff,axiom,
! [A: $tType,B: $tType,A4: set @ A,B2: set @ B] :
( ( finite_finite2 @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ A4
@ ^ [Uu3: A] : B2 ) )
= ( ( A4
= ( bot_bot @ ( set @ A ) ) )
| ( B2
= ( bot_bot @ ( set @ B ) ) )
| ( ( finite_finite2 @ A @ A4 )
& ( finite_finite2 @ B @ B2 ) ) ) ) ).
% finite_cartesian_product_iff
thf(fact_6823_finite__cartesian__productD2,axiom,
! [A: $tType,B: $tType,A4: set @ A,B2: set @ B] :
( ( finite_finite2 @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ A4
@ ^ [Uu3: A] : B2 ) )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( finite_finite2 @ B @ B2 ) ) ) ).
% finite_cartesian_productD2
thf(fact_6824_finite__cartesian__productD1,axiom,
! [B: $tType,A: $tType,A4: set @ A,B2: set @ B] :
( ( finite_finite2 @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ A4
@ ^ [Uu3: A] : B2 ) )
=> ( ( B2
!= ( bot_bot @ ( set @ B ) ) )
=> ( finite_finite2 @ A @ A4 ) ) ) ).
% finite_cartesian_productD1
thf(fact_6825_finite__SigmaI2,axiom,
! [B: $tType,A: $tType,A4: set @ A,B2: A > ( set @ B )] :
( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X4: A] :
( ( member @ A @ X4 @ A4 )
& ( ( B2 @ X4 )
!= ( bot_bot @ ( set @ B ) ) ) ) ) )
=> ( ! [A6: A] :
( ( member @ A @ A6 @ A4 )
=> ( finite_finite2 @ B @ ( B2 @ A6 ) ) )
=> ( finite_finite2 @ ( product_prod @ A @ B ) @ ( product_Sigma @ A @ B @ A4 @ B2 ) ) ) ) ).
% finite_SigmaI2
thf(fact_6826_dropWhile__eq__Cons__conv,axiom,
! [A: $tType,P: A > $o,Xs: list @ A,Y: A,Ys: list @ A] :
( ( ( dropWhile @ A @ P @ Xs )
= ( cons @ A @ Y @ Ys ) )
= ( ( Xs
= ( append @ A @ ( takeWhile @ A @ P @ Xs ) @ ( cons @ A @ Y @ Ys ) ) )
& ~ ( P @ Y ) ) ) ).
% dropWhile_eq_Cons_conv
thf(fact_6827_Sigma__interval__disjoint,axiom,
! [A: $tType,B: $tType] :
( ( order @ A )
=> ! [A4: set @ B,V2: B > A,W2: A] :
( ( inf_inf @ ( set @ ( product_prod @ B @ A ) )
@ ( product_Sigma @ B @ A @ A4
@ ^ [I3: B] : ( set_ord_atMost @ A @ ( V2 @ I3 ) ) )
@ ( product_Sigma @ B @ A @ A4
@ ^ [I3: B] : ( set_or3652927894154168847AtMost @ A @ ( V2 @ I3 ) @ W2 ) ) )
= ( bot_bot @ ( set @ ( product_prod @ B @ A ) ) ) ) ) ).
% Sigma_interval_disjoint
thf(fact_6828_takeWhile__eq__filter,axiom,
! [A: $tType,P: A > $o,Xs: list @ A] :
( ! [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ ( dropWhile @ A @ P @ Xs ) ) )
=> ~ ( P @ X5 ) )
=> ( ( takeWhile @ A @ P @ Xs )
= ( filter2 @ A @ P @ Xs ) ) ) ).
% takeWhile_eq_filter
thf(fact_6829_dropWhile__eq__drop,axiom,
! [A: $tType] :
( ( dropWhile @ A )
= ( ^ [P3: A > $o,Xs3: list @ A] : ( drop @ A @ ( size_size @ ( list @ A ) @ ( takeWhile @ A @ P3 @ Xs3 ) ) @ Xs3 ) ) ) ).
% dropWhile_eq_drop
thf(fact_6830_dropWhile__append,axiom,
! [A: $tType,Xs: list @ A,P: A > $o,Ys: list @ A] :
( ( ! [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Xs ) )
=> ( P @ X5 ) )
=> ( ( dropWhile @ A @ P @ ( append @ A @ Xs @ Ys ) )
= ( dropWhile @ A @ P @ Ys ) ) )
& ( ~ ! [X2: A] :
( ( member @ A @ X2 @ ( set2 @ A @ Xs ) )
=> ( P @ X2 ) )
=> ( ( dropWhile @ A @ P @ ( append @ A @ Xs @ Ys ) )
= ( append @ A @ ( dropWhile @ A @ P @ Xs ) @ Ys ) ) ) ) ).
% dropWhile_append
thf(fact_6831_find__dropWhile,axiom,
! [A: $tType] :
( ( find @ A )
= ( ^ [P3: A > $o,Xs3: list @ A] :
( case_list @ ( option @ A ) @ A @ ( none @ A )
@ ^ [X4: A,Xa4: list @ A] : ( some @ A @ X4 )
@ ( dropWhile @ A @ ( comp @ $o @ $o @ A @ (~) @ P3 ) @ Xs3 ) ) ) ) ).
% find_dropWhile
thf(fact_6832_card__cartesian__product__singleton,axiom,
! [A: $tType,B: $tType,X: A,A4: set @ B] :
( ( finite_card @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) )
@ ^ [Uu3: A] : A4 ) )
= ( finite_card @ B @ A4 ) ) ).
% card_cartesian_product_singleton
thf(fact_6833_sum_OSigma,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A4: set @ B,B2: B > ( set @ C ),G: B > C > A] :
( ( finite_finite2 @ B @ A4 )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ A4 )
=> ( finite_finite2 @ C @ ( B2 @ X5 ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X4: B] : ( groups7311177749621191930dd_sum @ C @ A @ ( G @ X4 ) @ ( B2 @ X4 ) )
@ A4 )
= ( groups7311177749621191930dd_sum @ ( product_prod @ B @ C ) @ A @ ( product_case_prod @ B @ C @ A @ G ) @ ( product_Sigma @ B @ C @ A4 @ B2 ) ) ) ) ) ) ).
% sum.Sigma
thf(fact_6834_prod_OSigma,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A4: set @ B,B2: B > ( set @ C ),G: B > C > A] :
( ( finite_finite2 @ B @ A4 )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ A4 )
=> ( finite_finite2 @ C @ ( B2 @ X5 ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [X4: B] : ( groups7121269368397514597t_prod @ C @ A @ ( G @ X4 ) @ ( B2 @ X4 ) )
@ A4 )
= ( groups7121269368397514597t_prod @ ( product_prod @ B @ C ) @ A @ ( product_case_prod @ B @ C @ A @ G ) @ ( product_Sigma @ B @ C @ A4 @ B2 ) ) ) ) ) ) ).
% prod.Sigma
thf(fact_6835_remdups__adj__append__dropWhile,axiom,
! [A: $tType,Xs: list @ A,Y: A,Ys: list @ A] :
( ( remdups_adj @ A @ ( append @ A @ Xs @ ( cons @ A @ Y @ Ys ) ) )
= ( append @ A @ ( remdups_adj @ A @ ( append @ A @ Xs @ ( cons @ A @ Y @ ( nil @ A ) ) ) )
@ ( remdups_adj @ A
@ ( dropWhile @ A
@ ^ [X4: A] : X4 = Y
@ Ys ) ) ) ) ).
% remdups_adj_append_dropWhile
thf(fact_6836_tl__remdups__adj,axiom,
! [A: $tType,Ys: list @ A] :
( ( Ys
!= ( nil @ A ) )
=> ( ( tl @ A @ ( remdups_adj @ A @ Ys ) )
= ( remdups_adj @ A
@ ( dropWhile @ A
@ ^ [X4: A] :
( X4
= ( hd @ A @ Ys ) )
@ ( tl @ A @ Ys ) ) ) ) ) ).
% tl_remdups_adj
thf(fact_6837_dropWhile__nth,axiom,
! [A: $tType,J: nat,P: A > $o,Xs: list @ A] :
( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ ( dropWhile @ A @ P @ Xs ) ) )
=> ( ( nth @ A @ ( dropWhile @ A @ P @ Xs ) @ J )
= ( nth @ A @ Xs @ ( plus_plus @ nat @ J @ ( size_size @ ( list @ A ) @ ( takeWhile @ A @ P @ Xs ) ) ) ) ) ) ).
% dropWhile_nth
thf(fact_6838_Sigma__def,axiom,
! [B: $tType,A: $tType] :
( ( product_Sigma @ A @ B )
= ( ^ [A7: set @ A,B6: A > ( set @ B )] :
( complete_Sup_Sup @ ( set @ ( product_prod @ A @ B ) )
@ ( image2 @ A @ ( set @ ( product_prod @ A @ B ) )
@ ^ [X4: A] :
( complete_Sup_Sup @ ( set @ ( product_prod @ A @ B ) )
@ ( image2 @ B @ ( set @ ( product_prod @ A @ B ) )
@ ^ [Y6: B] : ( insert2 @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y6 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) )
@ ( B6 @ X4 ) ) )
@ A7 ) ) ) ) ).
% Sigma_def
thf(fact_6839_product__fold,axiom,
! [B: $tType,A: $tType,A4: set @ A,B2: set @ B] :
( ( finite_finite2 @ A @ A4 )
=> ( ( finite_finite2 @ B @ B2 )
=> ( ( product_Sigma @ A @ B @ A4
@ ^ [Uu3: A] : B2 )
= ( finite_fold @ A @ ( set @ ( product_prod @ A @ B ) )
@ ^ [X4: A,Z6: set @ ( product_prod @ A @ B )] :
( finite_fold @ B @ ( set @ ( product_prod @ A @ B ) )
@ ^ [Y6: B] : ( insert2 @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y6 ) )
@ Z6
@ B2 )
@ ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) )
@ A4 ) ) ) ) ).
% product_fold
thf(fact_6840_dropWhile__neq__rev,axiom,
! [A: $tType,Xs: list @ A,X: A] :
( ( distinct @ A @ Xs )
=> ( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ( dropWhile @ A
@ ^ [Y6: A] : Y6 != X
@ ( rev @ A @ Xs ) )
= ( cons @ A @ X
@ ( rev @ A
@ ( takeWhile @ A
@ ^ [Y6: A] : Y6 != X
@ Xs ) ) ) ) ) ) ).
% dropWhile_neq_rev
thf(fact_6841_takeWhile__neq__rev,axiom,
! [A: $tType,Xs: list @ A,X: A] :
( ( distinct @ A @ Xs )
=> ( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ( takeWhile @ A
@ ^ [Y6: A] : Y6 != X
@ ( rev @ A @ Xs ) )
= ( rev @ A
@ ( tl @ A
@ ( dropWhile @ A
@ ^ [Y6: A] : Y6 != X
@ Xs ) ) ) ) ) ) ).
% takeWhile_neq_rev
thf(fact_6842_infinite__cartesian__product,axiom,
! [B: $tType,A: $tType,A4: set @ A,B2: set @ B] :
( ~ ( finite_finite2 @ A @ A4 )
=> ( ~ ( finite_finite2 @ B @ B2 )
=> ~ ( finite_finite2 @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ A4
@ ^ [Uu3: A] : B2 ) ) ) ) ).
% infinite_cartesian_product
thf(fact_6843_partition__filter__conv,axiom,
! [A: $tType] :
( ( partition @ A )
= ( ^ [F3: A > $o,Xs3: list @ A] : ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( filter2 @ A @ F3 @ Xs3 ) @ ( filter2 @ A @ ( comp @ $o @ $o @ A @ (~) @ F3 ) @ Xs3 ) ) ) ) ).
% partition_filter_conv
thf(fact_6844_partition_Osimps_I1_J,axiom,
! [A: $tType,P: A > $o] :
( ( partition @ A @ P @ ( nil @ A ) )
= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( nil @ A ) ) ) ).
% partition.simps(1)
thf(fact_6845_partition__P,axiom,
! [A: $tType,P: A > $o,Xs: list @ A,Yes: list @ A,No4: list @ A] :
( ( ( partition @ A @ P @ Xs )
= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Yes @ No4 ) )
=> ( ! [X2: A] :
( ( member @ A @ X2 @ ( set2 @ A @ Yes ) )
=> ( P @ X2 ) )
& ! [X2: A] :
( ( member @ A @ X2 @ ( set2 @ A @ No4 ) )
=> ~ ( P @ X2 ) ) ) ) ).
% partition_P
thf(fact_6846_partition_Osimps_I2_J,axiom,
! [A: $tType,P: A > $o,X: A,Xs: list @ A] :
( ( partition @ A @ P @ ( cons @ A @ X @ Xs ) )
= ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) )
@ ^ [Yes2: list @ A,No3: list @ A] : ( if @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( P @ X ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X @ Yes2 ) @ No3 ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Yes2 @ ( cons @ A @ X @ No3 ) ) )
@ ( partition @ A @ P @ Xs ) ) ) ).
% partition.simps(2)
thf(fact_6847_pairs__le__eq__Sigma,axiom,
! [M2: nat] :
( ( collect @ ( product_prod @ nat @ nat )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [I3: nat,J2: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ I3 @ J2 ) @ M2 ) ) )
= ( product_Sigma @ nat @ nat @ ( set_ord_atMost @ nat @ M2 )
@ ^ [R4: nat] : ( set_ord_atMost @ nat @ ( minus_minus @ nat @ M2 @ R4 ) ) ) ) ).
% pairs_le_eq_Sigma
thf(fact_6848_partition__set,axiom,
! [A: $tType,P: A > $o,Xs: list @ A,Yes: list @ A,No4: list @ A] :
( ( ( partition @ A @ P @ Xs )
= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Yes @ No4 ) )
=> ( ( sup_sup @ ( set @ A ) @ ( set2 @ A @ Yes ) @ ( set2 @ A @ No4 ) )
= ( set2 @ A @ Xs ) ) ) ).
% partition_set
thf(fact_6849_relChain__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ( ( bNF_Ca3754400796208372196lChain @ A @ B )
= ( ^ [R4: set @ ( product_prod @ A @ A ),As4: A > B] :
! [I3: A,J2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ I3 @ J2 ) @ R4 )
=> ( ord_less_eq @ B @ ( As4 @ I3 ) @ ( As4 @ J2 ) ) ) ) ) ) ).
% relChain_def
thf(fact_6850_natLess__def,axiom,
( bNF_Ca8459412986667044542atLess
= ( collect @ ( product_prod @ nat @ nat ) @ ( product_case_prod @ nat @ nat @ $o @ ( ord_less @ nat ) ) ) ) ).
% natLess_def
thf(fact_6851_Gr__incl,axiom,
! [A: $tType,B: $tType,A4: set @ A,F2: A > B,B2: set @ B] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( bNF_Gr @ A @ B @ A4 @ F2 )
@ ( product_Sigma @ A @ B @ A4
@ ^ [Uu3: A] : B2 ) )
= ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F2 @ A4 ) @ B2 ) ) ).
% Gr_incl
thf(fact_6852_ntrancl__Suc,axiom,
! [A: $tType,N2: nat,R: set @ ( product_prod @ A @ A )] :
( ( transitive_ntrancl @ A @ ( suc @ N2 ) @ R )
= ( relcomp @ A @ A @ A @ ( transitive_ntrancl @ A @ N2 @ R ) @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ ( id2 @ A ) @ R ) ) ) ).
% ntrancl_Suc
thf(fact_6853_rtrancl__empty,axiom,
! [A: $tType] :
( ( transitive_rtrancl @ A @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
= ( id2 @ A ) ) ).
% rtrancl_empty
thf(fact_6854_relpow_Osimps_I1_J,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( zero_zero @ nat ) @ R )
= ( id2 @ A ) ) ).
% relpow.simps(1)
thf(fact_6855_Restr__natLeq,axiom,
! [N2: nat] :
( ( inf_inf @ ( set @ ( product_prod @ nat @ nat ) ) @ bNF_Ca8665028551170535155natLeq
@ ( product_Sigma @ nat @ nat
@ ( collect @ nat
@ ^ [X4: nat] : ( ord_less @ nat @ X4 @ N2 ) )
@ ^ [Uu3: nat] :
( collect @ nat
@ ^ [X4: nat] : ( ord_less @ nat @ X4 @ N2 ) ) ) )
= ( collect @ ( product_prod @ nat @ nat )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [X4: nat,Y6: nat] :
( ( ord_less @ nat @ X4 @ N2 )
& ( ord_less @ nat @ Y6 @ N2 )
& ( ord_less_eq @ nat @ X4 @ Y6 ) ) ) ) ) ).
% Restr_natLeq
thf(fact_6856_max__ext__eq,axiom,
! [A: $tType] :
( ( max_ext @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A )] :
( collect @ ( product_prod @ ( set @ A ) @ ( set @ A ) )
@ ( product_case_prod @ ( set @ A ) @ ( set @ A ) @ $o
@ ^ [X6: set @ A,Y8: set @ A] :
( ( finite_finite2 @ A @ X6 )
& ( finite_finite2 @ A @ Y8 )
& ( Y8
!= ( bot_bot @ ( set @ A ) ) )
& ! [X4: A] :
( ( member @ A @ X4 @ X6 )
=> ? [Y6: A] :
( ( member @ A @ Y6 @ Y8 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y6 ) @ R5 ) ) ) ) ) ) ) ) ).
% max_ext_eq
thf(fact_6857_bex__empty,axiom,
! [A: $tType,P: A > $o] :
~ ? [X2: A] :
( ( member @ A @ X2 @ ( bot_bot @ ( set @ A ) ) )
& ( P @ X2 ) ) ).
% bex_empty
thf(fact_6858_finite__Collect__bex,axiom,
! [B: $tType,A: $tType,A4: set @ A,Q: B > A > $o] :
( ( finite_finite2 @ A @ A4 )
=> ( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X4: B] :
? [Y6: A] :
( ( member @ A @ Y6 @ A4 )
& ( Q @ X4 @ Y6 ) ) ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ A4 )
=> ( finite_finite2 @ B
@ ( collect @ B
@ ^ [Y6: B] : ( Q @ Y6 @ X4 ) ) ) ) ) ) ) ).
% finite_Collect_bex
thf(fact_6859_bex__UNIV,axiom,
! [A: $tType,P: A > $o] :
( ( ? [X4: A] :
( ( member @ A @ X4 @ ( top_top @ ( set @ A ) ) )
& ( P @ X4 ) ) )
= ( ? [X6: A] : ( P @ X6 ) ) ) ).
% bex_UNIV
thf(fact_6860_Bex__def,axiom,
! [A: $tType] :
( ( bex @ A )
= ( ^ [A7: set @ A,P3: A > $o] :
? [X4: A] :
( ( member @ A @ X4 @ A7 )
& ( P3 @ X4 ) ) ) ) ).
% Bex_def
thf(fact_6861_image__def,axiom,
! [B: $tType,A: $tType] :
( ( image2 @ A @ B )
= ( ^ [F3: A > B,A7: set @ A] :
( collect @ B
@ ^ [Y6: B] :
? [X4: A] :
( ( member @ A @ X4 @ A7 )
& ( Y6
= ( F3 @ X4 ) ) ) ) ) ) ).
% image_def
thf(fact_6862_Bex__fold,axiom,
! [A: $tType,A4: set @ A,P: A > $o] :
( ( finite_finite2 @ A @ A4 )
=> ( ( ? [X4: A] :
( ( member @ A @ X4 @ A4 )
& ( P @ X4 ) ) )
= ( finite_fold @ A @ $o
@ ^ [K2: A,S7: $o] :
( S7
| ( P @ K2 ) )
@ $false
@ A4 ) ) ) ).
% Bex_fold
thf(fact_6863_natLeq__def,axiom,
( bNF_Ca8665028551170535155natLeq
= ( collect @ ( product_prod @ nat @ nat ) @ ( product_case_prod @ nat @ nat @ $o @ ( ord_less_eq @ nat ) ) ) ) ).
% natLeq_def
thf(fact_6864_nths__nths,axiom,
! [A: $tType,Xs: list @ A,A4: set @ nat,B2: set @ nat] :
( ( nths @ A @ ( nths @ A @ Xs @ A4 ) @ B2 )
= ( nths @ A @ Xs
@ ( collect @ nat
@ ^ [I3: nat] :
( ( member @ nat @ I3 @ A4 )
& ( member @ nat
@ ( finite_card @ nat
@ ( collect @ nat
@ ^ [I10: nat] :
( ( member @ nat @ I10 @ A4 )
& ( ord_less @ nat @ I10 @ I3 ) ) ) )
@ B2 ) ) ) ) ) ).
% nths_nths
thf(fact_6865_min__ext__def,axiom,
! [A: $tType] :
( ( min_ext @ A )
= ( ^ [R4: set @ ( product_prod @ A @ A )] :
( collect @ ( product_prod @ ( set @ A ) @ ( set @ A ) )
@ ^ [Uu3: product_prod @ ( set @ A ) @ ( set @ A )] :
? [X6: set @ A,Y8: set @ A] :
( ( Uu3
= ( product_Pair @ ( set @ A ) @ ( set @ A ) @ X6 @ Y8 ) )
& ( X6
!= ( bot_bot @ ( set @ A ) ) )
& ! [X4: A] :
( ( member @ A @ X4 @ Y8 )
=> ? [Y6: A] :
( ( member @ A @ Y6 @ X6 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y6 @ X4 ) @ R4 ) ) ) ) ) ) ) ).
% min_ext_def
thf(fact_6866_map__project__def,axiom,
! [B: $tType,A: $tType] :
( ( map_project @ A @ B )
= ( ^ [F3: A > ( option @ B ),A7: set @ A] :
( collect @ B
@ ^ [B4: B] :
? [X4: A] :
( ( member @ A @ X4 @ A7 )
& ( ( F3 @ X4 )
= ( some @ B @ B4 ) ) ) ) ) ) ).
% map_project_def
thf(fact_6867_Restr__natLeq2,axiom,
! [N2: nat] :
( ( inf_inf @ ( set @ ( product_prod @ nat @ nat ) ) @ bNF_Ca8665028551170535155natLeq
@ ( product_Sigma @ nat @ nat @ ( order_underS @ nat @ bNF_Ca8665028551170535155natLeq @ N2 )
@ ^ [Uu3: nat] : ( order_underS @ nat @ bNF_Ca8665028551170535155natLeq @ N2 ) ) )
= ( collect @ ( product_prod @ nat @ nat )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [X4: nat,Y6: nat] :
( ( ord_less @ nat @ X4 @ N2 )
& ( ord_less @ nat @ Y6 @ N2 )
& ( ord_less_eq @ nat @ X4 @ Y6 ) ) ) ) ) ).
% Restr_natLeq2
thf(fact_6868_natLeq__underS__less,axiom,
! [N2: nat] :
( ( order_underS @ nat @ bNF_Ca8665028551170535155natLeq @ N2 )
= ( collect @ nat
@ ^ [X4: nat] : ( ord_less @ nat @ X4 @ N2 ) ) ) ).
% natLeq_underS_less
thf(fact_6869_max__extp_Ocases,axiom,
! [A: $tType,R: A > A > $o,A1: set @ A,A22: set @ A] :
( ( max_extp @ A @ R @ A1 @ A22 )
=> ~ ( ( finite_finite2 @ A @ A1 )
=> ( ( finite_finite2 @ A @ A22 )
=> ( ( A22
!= ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) )
=> ~ ! [X2: A] :
( ( member @ A @ X2 @ A1 )
=> ? [Xa3: A] :
( ( member @ A @ Xa3 @ A22 )
& ( R @ X2 @ Xa3 ) ) ) ) ) ) ) ).
% max_extp.cases
thf(fact_6870_max__extp_Osimps,axiom,
! [A: $tType] :
( ( max_extp @ A )
= ( ^ [R5: A > A > $o,A12: set @ A,A23: set @ A] :
( ( finite_finite2 @ A @ A12 )
& ( finite_finite2 @ A @ A23 )
& ( A23
!= ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) )
& ! [X4: A] :
( ( member @ A @ X4 @ A12 )
=> ? [Y6: A] :
( ( member @ A @ Y6 @ A23 )
& ( R5 @ X4 @ Y6 ) ) ) ) ) ) ).
% max_extp.simps
thf(fact_6871_max__extp_Omax__extI,axiom,
! [A: $tType,X8: set @ A,Y7: set @ A,R: A > A > $o] :
( ( finite_finite2 @ A @ X8 )
=> ( ( finite_finite2 @ A @ Y7 )
=> ( ( Y7
!= ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ X8 )
=> ? [Xa: A] :
( ( member @ A @ Xa @ Y7 )
& ( R @ X5 @ Xa ) ) )
=> ( max_extp @ A @ R @ X8 @ Y7 ) ) ) ) ) ).
% max_extp.max_extI
thf(fact_6872_lists__empty,axiom,
! [A: $tType] :
( ( lists @ A @ ( bot_bot @ ( set @ A ) ) )
= ( insert2 @ ( list @ A ) @ ( nil @ A ) @ ( bot_bot @ ( set @ ( list @ A ) ) ) ) ) ).
% lists_empty
thf(fact_6873_card__quotient__disjoint,axiom,
! [A: $tType,A4: set @ A,R2: set @ ( product_prod @ A @ A )] :
( ( finite_finite2 @ A @ A4 )
=> ( ( inj_on @ A @ ( set @ ( set @ A ) )
@ ^ [X4: A] : ( equiv_quotient @ A @ ( insert2 @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) @ R2 )
@ A4 )
=> ( ( finite_card @ ( set @ A ) @ ( equiv_quotient @ A @ A4 @ R2 ) )
= ( finite_card @ A @ A4 ) ) ) ) ).
% card_quotient_disjoint
thf(fact_6874_Cons__in__lists__iff,axiom,
! [A: $tType,X: A,Xs: list @ A,A4: set @ A] :
( ( member @ ( list @ A ) @ ( cons @ A @ X @ Xs ) @ ( lists @ A @ A4 ) )
= ( ( member @ A @ X @ A4 )
& ( member @ ( list @ A ) @ Xs @ ( lists @ A @ A4 ) ) ) ) ).
% Cons_in_lists_iff
thf(fact_6875_in__listsI,axiom,
! [A: $tType,Xs: list @ A,A4: set @ A] :
( ! [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Xs ) )
=> ( member @ A @ X5 @ A4 ) )
=> ( member @ ( list @ A ) @ Xs @ ( lists @ A @ A4 ) ) ) ).
% in_listsI
thf(fact_6876_lists__Int__eq,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] :
( ( lists @ A @ ( inf_inf @ ( set @ A ) @ A4 @ B2 ) )
= ( inf_inf @ ( set @ ( list @ A ) ) @ ( lists @ A @ A4 ) @ ( lists @ A @ B2 ) ) ) ).
% lists_Int_eq
thf(fact_6877_append__in__lists__conv,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,A4: set @ A] :
( ( member @ ( list @ A ) @ ( append @ A @ Xs @ Ys ) @ ( lists @ A @ A4 ) )
= ( ( member @ ( list @ A ) @ Xs @ ( lists @ A @ A4 ) )
& ( member @ ( list @ A ) @ Ys @ ( lists @ A @ A4 ) ) ) ) ).
% append_in_lists_conv
thf(fact_6878_lists__UNIV,axiom,
! [A: $tType] :
( ( lists @ A @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ ( list @ A ) ) ) ) ).
% lists_UNIV
thf(fact_6879_lists__IntI,axiom,
! [A: $tType,L: list @ A,A4: set @ A,B2: set @ A] :
( ( member @ ( list @ A ) @ L @ ( lists @ A @ A4 ) )
=> ( ( member @ ( list @ A ) @ L @ ( lists @ A @ B2 ) )
=> ( member @ ( list @ A ) @ L @ ( lists @ A @ ( inf_inf @ ( set @ A ) @ A4 @ B2 ) ) ) ) ) ).
% lists_IntI
thf(fact_6880_listsE,axiom,
! [A: $tType,X: A,L: list @ A,A4: set @ A] :
( ( member @ ( list @ A ) @ ( cons @ A @ X @ L ) @ ( lists @ A @ A4 ) )
=> ~ ( ( member @ A @ X @ A4 )
=> ~ ( member @ ( list @ A ) @ L @ ( lists @ A @ A4 ) ) ) ) ).
% listsE
thf(fact_6881_lists_OCons,axiom,
! [A: $tType,A3: A,A4: set @ A,L: list @ A] :
( ( member @ A @ A3 @ A4 )
=> ( ( member @ ( list @ A ) @ L @ ( lists @ A @ A4 ) )
=> ( member @ ( list @ A ) @ ( cons @ A @ A3 @ L ) @ ( lists @ A @ A4 ) ) ) ) ).
% lists.Cons
thf(fact_6882_lists_ONil,axiom,
! [A: $tType,A4: set @ A] : ( member @ ( list @ A ) @ ( nil @ A ) @ ( lists @ A @ A4 ) ) ).
% lists.Nil
thf(fact_6883_lists_Osimps,axiom,
! [A: $tType,A3: list @ A,A4: set @ A] :
( ( member @ ( list @ A ) @ A3 @ ( lists @ A @ A4 ) )
= ( ( A3
= ( nil @ A ) )
| ? [A5: A,L2: list @ A] :
( ( A3
= ( cons @ A @ A5 @ L2 ) )
& ( member @ A @ A5 @ A4 )
& ( member @ ( list @ A ) @ L2 @ ( lists @ A @ A4 ) ) ) ) ) ).
% lists.simps
thf(fact_6884_lists_Ocases,axiom,
! [A: $tType,A3: list @ A,A4: set @ A] :
( ( member @ ( list @ A ) @ A3 @ ( lists @ A @ A4 ) )
=> ( ( A3
!= ( nil @ A ) )
=> ~ ! [A6: A,L4: list @ A] :
( ( A3
= ( cons @ A @ A6 @ L4 ) )
=> ( ( member @ A @ A6 @ A4 )
=> ~ ( member @ ( list @ A ) @ L4 @ ( lists @ A @ A4 ) ) ) ) ) ) ).
% lists.cases
thf(fact_6885_in__lists__conv__set,axiom,
! [A: $tType,Xs: list @ A,A4: set @ A] :
( ( member @ ( list @ A ) @ Xs @ ( lists @ A @ A4 ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
=> ( member @ A @ X4 @ A4 ) ) ) ) ).
% in_lists_conv_set
thf(fact_6886_in__listsD,axiom,
! [A: $tType,Xs: list @ A,A4: set @ A] :
( ( member @ ( list @ A ) @ Xs @ ( lists @ A @ A4 ) )
=> ! [X2: A] :
( ( member @ A @ X2 @ ( set2 @ A @ Xs ) )
=> ( member @ A @ X2 @ A4 ) ) ) ).
% in_listsD
thf(fact_6887_listrel__refl__on,axiom,
! [A: $tType,A4: set @ A,R2: set @ ( product_prod @ A @ A )] :
( ( refl_on @ A @ A4 @ R2 )
=> ( refl_on @ ( list @ A ) @ ( lists @ A @ A4 ) @ ( listrel @ A @ A @ R2 ) ) ) ).
% listrel_refl_on
thf(fact_6888_lists__eq__set,axiom,
! [A: $tType] :
( ( lists @ A )
= ( ^ [A7: set @ A] :
( collect @ ( list @ A )
@ ^ [Xs3: list @ A] : ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs3 ) @ A7 ) ) ) ) ).
% lists_eq_set
thf(fact_6889_lists__mono,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B2 )
=> ( ord_less_eq @ ( set @ ( list @ A ) ) @ ( lists @ A @ A4 ) @ ( lists @ A @ B2 ) ) ) ).
% lists_mono
thf(fact_6890_lists__image,axiom,
! [A: $tType,B: $tType,F2: B > A,A4: set @ B] :
( ( lists @ A @ ( image2 @ B @ A @ F2 @ A4 ) )
= ( image2 @ ( list @ B ) @ ( list @ A ) @ ( map @ B @ A @ F2 ) @ ( lists @ B @ A4 ) ) ) ).
% lists_image
thf(fact_6891_Collect__finite__eq__lists,axiom,
! [A: $tType] :
( ( collect @ ( set @ A ) @ ( finite_finite2 @ A ) )
= ( image2 @ ( list @ A ) @ ( set @ A ) @ ( set2 @ A ) @ ( lists @ A @ ( top_top @ ( set @ A ) ) ) ) ) ).
% Collect_finite_eq_lists
thf(fact_6892_Collect__finite__subset__eq__lists,axiom,
! [A: $tType,T4: set @ A] :
( ( collect @ ( set @ A )
@ ^ [A7: set @ A] :
( ( finite_finite2 @ A @ A7 )
& ( ord_less_eq @ ( set @ A ) @ A7 @ T4 ) ) )
= ( image2 @ ( list @ A ) @ ( set @ A ) @ ( set2 @ A ) @ ( lists @ A @ T4 ) ) ) ).
% Collect_finite_subset_eq_lists
thf(fact_6893_listrel__subset,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A4: set @ A] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R2
@ ( product_Sigma @ A @ A @ A4
@ ^ [Uu3: A] : A4 ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) @ ( listrel @ A @ A @ R2 )
@ ( product_Sigma @ ( list @ A ) @ ( list @ A ) @ ( lists @ A @ A4 )
@ ^ [Uu3: list @ A] : ( lists @ A @ A4 ) ) ) ) ).
% listrel_subset
thf(fact_6894_quotient__empty,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( equiv_quotient @ A @ ( bot_bot @ ( set @ A ) ) @ R2 )
= ( bot_bot @ ( set @ ( set @ A ) ) ) ) ).
% quotient_empty
thf(fact_6895_quotient__is__empty,axiom,
! [A: $tType,A4: set @ A,R2: set @ ( product_prod @ A @ A )] :
( ( ( equiv_quotient @ A @ A4 @ R2 )
= ( bot_bot @ ( set @ ( set @ A ) ) ) )
= ( A4
= ( bot_bot @ ( set @ A ) ) ) ) ).
% quotient_is_empty
thf(fact_6896_quotient__is__empty2,axiom,
! [A: $tType,A4: set @ A,R2: set @ ( product_prod @ A @ A )] :
( ( ( bot_bot @ ( set @ ( set @ A ) ) )
= ( equiv_quotient @ A @ A4 @ R2 ) )
= ( A4
= ( bot_bot @ ( set @ A ) ) ) ) ).
% quotient_is_empty2
thf(fact_6897_finite__equiv__class,axiom,
! [A: $tType,A4: set @ A,R2: set @ ( product_prod @ A @ A ),X8: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R2
@ ( product_Sigma @ A @ A @ A4
@ ^ [Uu3: A] : A4 ) )
=> ( ( member @ ( set @ A ) @ X8 @ ( equiv_quotient @ A @ A4 @ R2 ) )
=> ( finite_finite2 @ A @ X8 ) ) ) ) ).
% finite_equiv_class
thf(fact_6898_finite__quotient,axiom,
! [A: $tType,A4: set @ A,R2: set @ ( product_prod @ A @ A )] :
( ( finite_finite2 @ A @ A4 )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R2
@ ( product_Sigma @ A @ A @ A4
@ ^ [Uu3: A] : A4 ) )
=> ( finite_finite2 @ ( set @ A ) @ ( equiv_quotient @ A @ A4 @ R2 ) ) ) ) ).
% finite_quotient
thf(fact_6899_quotient__diff1,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A4: set @ A,A3: A] :
( ( inj_on @ A @ ( set @ ( set @ A ) )
@ ^ [A5: A] : ( equiv_quotient @ A @ ( insert2 @ A @ A5 @ ( bot_bot @ ( set @ A ) ) ) @ R2 )
@ A4 )
=> ( ( member @ A @ A3 @ A4 )
=> ( ( equiv_quotient @ A @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) @ R2 )
= ( minus_minus @ ( set @ ( set @ A ) ) @ ( equiv_quotient @ A @ A4 @ R2 ) @ ( equiv_quotient @ A @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) @ R2 ) ) ) ) ) ).
% quotient_diff1
thf(fact_6900_INF__set__fold,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F2: B > A,Xs: list @ B] :
( ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F2 @ ( set2 @ B @ Xs ) ) )
= ( fold @ B @ A @ ( comp @ A @ ( A > A ) @ B @ ( inf_inf @ A ) @ F2 ) @ Xs @ ( top_top @ A ) ) ) ) ).
% INF_set_fold
thf(fact_6901_SUP__set__fold,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F2: B > A,Xs: list @ B] :
( ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F2 @ ( set2 @ B @ Xs ) ) )
= ( fold @ B @ A @ ( comp @ A @ ( A > A ) @ B @ ( sup_sup @ A ) @ F2 ) @ Xs @ ( bot_bot @ A ) ) ) ) ).
% SUP_set_fold
thf(fact_6902_fold__append,axiom,
! [A: $tType,B: $tType,F2: B > A > A,Xs: list @ B,Ys: list @ B] :
( ( fold @ B @ A @ F2 @ ( append @ B @ Xs @ Ys ) )
= ( comp @ A @ A @ A @ ( fold @ B @ A @ F2 @ Ys ) @ ( fold @ B @ A @ F2 @ Xs ) ) ) ).
% fold_append
thf(fact_6903_fold__replicate,axiom,
! [A: $tType,B: $tType,F2: B > A > A,N2: nat,X: B] :
( ( fold @ B @ A @ F2 @ ( replicate @ B @ N2 @ X ) )
= ( compow @ ( A > A ) @ N2 @ ( F2 @ X ) ) ) ).
% fold_replicate
thf(fact_6904_fold__map,axiom,
! [B: $tType,A: $tType,C: $tType,G: B > A > A,F2: C > B,Xs: list @ C] :
( ( fold @ B @ A @ G @ ( map @ C @ B @ F2 @ Xs ) )
= ( fold @ C @ A @ ( comp @ B @ ( A > A ) @ C @ G @ F2 ) @ Xs ) ) ).
% fold_map
thf(fact_6905_fold__Cons,axiom,
! [B: $tType,A: $tType,F2: A > B > B,X: A,Xs: list @ A] :
( ( fold @ A @ B @ F2 @ ( cons @ A @ X @ Xs ) )
= ( comp @ B @ B @ B @ ( fold @ A @ B @ F2 @ Xs ) @ ( F2 @ X ) ) ) ).
% fold_Cons
thf(fact_6906_fold__commute,axiom,
! [A: $tType,C: $tType,B: $tType,Xs: list @ A,H2: B > C,G: A > B > B,F2: A > C > C] :
( ! [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Xs ) )
=> ( ( comp @ B @ C @ B @ H2 @ ( G @ X5 ) )
= ( comp @ C @ C @ B @ ( F2 @ X5 ) @ H2 ) ) )
=> ( ( comp @ B @ C @ B @ H2 @ ( fold @ A @ B @ G @ Xs ) )
= ( comp @ C @ C @ B @ ( fold @ A @ C @ F2 @ Xs ) @ H2 ) ) ) ).
% fold_commute
thf(fact_6907_fold__commute__apply,axiom,
! [A: $tType,C: $tType,B: $tType,Xs: list @ A,H2: B > C,G: A > B > B,F2: A > C > C,S: B] :
( ! [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Xs ) )
=> ( ( comp @ B @ C @ B @ H2 @ ( G @ X5 ) )
= ( comp @ C @ C @ B @ ( F2 @ X5 ) @ H2 ) ) )
=> ( ( H2 @ ( fold @ A @ B @ G @ Xs @ S ) )
= ( fold @ A @ C @ F2 @ Xs @ ( H2 @ S ) ) ) ) ).
% fold_commute_apply
thf(fact_6908_fold__simps_I2_J,axiom,
! [B: $tType,A: $tType,F2: B > A > A,X: B,Xs: list @ B,S: A] :
( ( fold @ B @ A @ F2 @ ( cons @ B @ X @ Xs ) @ S )
= ( fold @ B @ A @ F2 @ Xs @ ( F2 @ X @ S ) ) ) ).
% fold_simps(2)
thf(fact_6909_fold__Nil,axiom,
! [A: $tType,B: $tType,F2: A > B > B] :
( ( fold @ A @ B @ F2 @ ( nil @ A ) )
= ( id @ B ) ) ).
% fold_Nil
thf(fact_6910_fold__simps_I1_J,axiom,
! [B: $tType,A: $tType,F2: B > A > A,S: A] :
( ( fold @ B @ A @ F2 @ ( nil @ B ) @ S )
= S ) ).
% fold_simps(1)
thf(fact_6911_fold__id,axiom,
! [A: $tType,B: $tType,Xs: list @ A,F2: A > B > B] :
( ! [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Xs ) )
=> ( ( F2 @ X5 )
= ( id @ B ) ) )
=> ( ( fold @ A @ B @ F2 @ Xs )
= ( id @ B ) ) ) ).
% fold_id
thf(fact_6912_List_Ofold__cong,axiom,
! [B: $tType,A: $tType,A3: A,B3: A,Xs: list @ B,Ys: list @ B,F2: B > A > A,G: B > A > A] :
( ( A3 = B3 )
=> ( ( Xs = Ys )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ ( set2 @ B @ Xs ) )
=> ( ( F2 @ X5 )
= ( G @ X5 ) ) )
=> ( ( fold @ B @ A @ F2 @ Xs @ A3 )
= ( fold @ B @ A @ G @ Ys @ B3 ) ) ) ) ) ).
% List.fold_cong
thf(fact_6913_fold__invariant,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Q: A > $o,P: B > $o,S: B,F2: A > B > B] :
( ! [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Xs ) )
=> ( Q @ X5 ) )
=> ( ( P @ S )
=> ( ! [X5: A,S3: B] :
( ( Q @ X5 )
=> ( ( P @ S3 )
=> ( P @ ( F2 @ X5 @ S3 ) ) ) )
=> ( P @ ( fold @ A @ B @ F2 @ Xs @ S ) ) ) ) ) ).
% fold_invariant
thf(fact_6914_foldr__conv__fold,axiom,
! [A: $tType,B: $tType] :
( ( foldr @ B @ A )
= ( ^ [F3: B > A > A,Xs3: list @ B] : ( fold @ B @ A @ F3 @ ( rev @ B @ Xs3 ) ) ) ) ).
% foldr_conv_fold
thf(fact_6915_fold__filter,axiom,
! [A: $tType,B: $tType,F2: B > A > A,P: B > $o,Xs: list @ B] :
( ( fold @ B @ A @ F2 @ ( filter2 @ B @ P @ Xs ) )
= ( fold @ B @ A
@ ^ [X4: B] : ( if @ ( A > A ) @ ( P @ X4 ) @ ( F2 @ X4 ) @ ( id @ A ) )
@ Xs ) ) ).
% fold_filter
thf(fact_6916_union__set__fold,axiom,
! [A: $tType,Xs: list @ A,A4: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( set2 @ A @ Xs ) @ A4 )
= ( fold @ A @ ( set @ A ) @ ( insert2 @ A ) @ Xs @ A4 ) ) ).
% union_set_fold
thf(fact_6917_rev__conv__fold,axiom,
! [A: $tType] :
( ( rev @ A )
= ( ^ [Xs3: list @ A] : ( fold @ A @ ( list @ A ) @ ( cons @ A ) @ Xs3 @ ( nil @ A ) ) ) ) ).
% rev_conv_fold
thf(fact_6918_fold__Cons__rev,axiom,
! [A: $tType,Xs: list @ A] :
( ( fold @ A @ ( list @ A ) @ ( cons @ A ) @ Xs )
= ( append @ A @ ( rev @ A @ Xs ) ) ) ).
% fold_Cons_rev
thf(fact_6919_fold__rev,axiom,
! [B: $tType,A: $tType,Xs: list @ A,F2: A > B > B] :
( ! [X5: A,Y3: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Xs ) )
=> ( ( member @ A @ Y3 @ ( set2 @ A @ Xs ) )
=> ( ( comp @ B @ B @ B @ ( F2 @ Y3 ) @ ( F2 @ X5 ) )
= ( comp @ B @ B @ B @ ( F2 @ X5 ) @ ( F2 @ Y3 ) ) ) ) )
=> ( ( fold @ A @ B @ F2 @ ( rev @ A @ Xs ) )
= ( fold @ A @ B @ F2 @ Xs ) ) ) ).
% fold_rev
thf(fact_6920_fold__plus__sum__list__rev,axiom,
! [A: $tType] :
( ( monoid_add @ A )
=> ! [Xs: list @ A] :
( ( fold @ A @ A @ ( plus_plus @ A ) @ Xs )
= ( plus_plus @ A @ ( groups8242544230860333062m_list @ A @ ( rev @ A @ Xs ) ) ) ) ) ).
% fold_plus_sum_list_rev
thf(fact_6921_fold__remove1__split,axiom,
! [B: $tType,A: $tType,Xs: list @ A,F2: A > B > B,X: A] :
( ! [X5: A,Y3: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Xs ) )
=> ( ( member @ A @ Y3 @ ( set2 @ A @ Xs ) )
=> ( ( comp @ B @ B @ B @ ( F2 @ X5 ) @ ( F2 @ Y3 ) )
= ( comp @ B @ B @ B @ ( F2 @ Y3 ) @ ( F2 @ X5 ) ) ) ) )
=> ( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ( fold @ A @ B @ F2 @ Xs )
= ( comp @ B @ B @ B @ ( fold @ A @ B @ F2 @ ( remove1 @ A @ X @ Xs ) ) @ ( F2 @ X ) ) ) ) ) ).
% fold_remove1_split
thf(fact_6922_foldr__fold,axiom,
! [B: $tType,A: $tType,Xs: list @ A,F2: A > B > B] :
( ! [X5: A,Y3: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Xs ) )
=> ( ( member @ A @ Y3 @ ( set2 @ A @ Xs ) )
=> ( ( comp @ B @ B @ B @ ( F2 @ Y3 ) @ ( F2 @ X5 ) )
= ( comp @ B @ B @ B @ ( F2 @ X5 ) @ ( F2 @ Y3 ) ) ) ) )
=> ( ( foldr @ A @ B @ F2 @ Xs )
= ( fold @ A @ B @ F2 @ Xs ) ) ) ).
% foldr_fold
thf(fact_6923_minus__set__fold,axiom,
! [A: $tType,A4: set @ A,Xs: list @ A] :
( ( minus_minus @ ( set @ A ) @ A4 @ ( set2 @ A @ Xs ) )
= ( fold @ A @ ( set @ A ) @ ( remove @ A ) @ Xs @ A4 ) ) ).
% minus_set_fold
thf(fact_6924_fold__append__concat__rev,axiom,
! [A: $tType,Xss: list @ ( list @ A )] :
( ( fold @ ( list @ A ) @ ( list @ A ) @ ( append @ A ) @ Xss )
= ( append @ A @ ( concat @ A @ ( rev @ ( list @ A ) @ Xss ) ) ) ) ).
% fold_append_concat_rev
thf(fact_6925_Sup__set__fold,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [Xs: list @ A] :
( ( complete_Sup_Sup @ A @ ( set2 @ A @ Xs ) )
= ( fold @ A @ A @ ( sup_sup @ A ) @ Xs @ ( bot_bot @ A ) ) ) ) ).
% Sup_set_fold
thf(fact_6926_Inf__set__fold,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [Xs: list @ A] :
( ( complete_Inf_Inf @ A @ ( set2 @ A @ Xs ) )
= ( fold @ A @ A @ ( inf_inf @ A ) @ Xs @ ( top_top @ A ) ) ) ) ).
% Inf_set_fold
thf(fact_6927_Inf__fin_Oset__eq__fold,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X: A,Xs: list @ A] :
( ( lattic7752659483105999362nf_fin @ A @ ( set2 @ A @ ( cons @ A @ X @ Xs ) ) )
= ( fold @ A @ A @ ( inf_inf @ A ) @ Xs @ X ) ) ) ).
% Inf_fin.set_eq_fold
thf(fact_6928_Sup__fin_Oset__eq__fold,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X: A,Xs: list @ A] :
( ( lattic5882676163264333800up_fin @ A @ ( set2 @ A @ ( cons @ A @ X @ Xs ) ) )
= ( fold @ A @ A @ ( sup_sup @ A ) @ Xs @ X ) ) ) ).
% Sup_fin.set_eq_fold
thf(fact_6929_Max_Oset__eq__fold,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Xs: list @ A] :
( ( lattic643756798349783984er_Max @ A @ ( set2 @ A @ ( cons @ A @ X @ Xs ) ) )
= ( fold @ A @ A @ ( ord_max @ A ) @ Xs @ X ) ) ) ).
% Max.set_eq_fold
thf(fact_6930_Min_Oset__eq__fold,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Xs: list @ A] :
( ( lattic643756798350308766er_Min @ A @ ( set2 @ A @ ( cons @ A @ X @ Xs ) ) )
= ( fold @ A @ A @ ( ord_min @ A ) @ Xs @ X ) ) ) ).
% Min.set_eq_fold
thf(fact_6931_comp__fun__idem__on_Ofold__set__fold,axiom,
! [A: $tType,B: $tType,S2: set @ A,F2: A > B > B,Xs: list @ A,Y: B] :
( ( finite673082921795544331dem_on @ A @ B @ S2 @ F2 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ S2 )
=> ( ( finite_fold @ A @ B @ F2 @ Y @ ( set2 @ A @ Xs ) )
= ( fold @ A @ B @ F2 @ Xs @ Y ) ) ) ) ).
% comp_fun_idem_on.fold_set_fold
thf(fact_6932_comp__fun__commute__on_Ofold__set__fold__remdups,axiom,
! [A: $tType,B: $tType,S2: set @ A,F2: A > B > B,Xs: list @ A,Y: B] :
( ( finite4664212375090638736ute_on @ A @ B @ S2 @ F2 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ S2 )
=> ( ( finite_fold @ A @ B @ F2 @ Y @ ( set2 @ A @ Xs ) )
= ( fold @ A @ B @ F2 @ ( remdups @ A @ Xs ) @ Y ) ) ) ) ).
% comp_fun_commute_on.fold_set_fold_remdups
thf(fact_6933_finite__sequence__to__countable__set,axiom,
! [A: $tType,X8: set @ A] :
( ( countable_countable @ A @ X8 )
=> ~ ! [F6: nat > ( set @ A )] :
( ! [I5: nat] : ( ord_less_eq @ ( set @ A ) @ ( F6 @ I5 ) @ X8 )
=> ( ! [I5: nat] : ( ord_less_eq @ ( set @ A ) @ ( F6 @ I5 ) @ ( F6 @ ( suc @ I5 ) ) )
=> ( ! [I5: nat] : ( finite_finite2 @ A @ ( F6 @ I5 ) )
=> ( ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ F6 @ ( top_top @ ( set @ nat ) ) ) )
!= X8 ) ) ) ) ) ).
% finite_sequence_to_countable_set
thf(fact_6934_butlast__power,axiom,
! [A: $tType,N2: nat,Xs: list @ A] :
( ( compow @ ( ( list @ A ) > ( list @ A ) ) @ N2 @ ( butlast @ A ) @ Xs )
= ( take @ A @ ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ N2 ) @ Xs ) ) ).
% butlast_power
thf(fact_6935_countable__empty,axiom,
! [A: $tType] : ( countable_countable @ A @ ( bot_bot @ ( set @ A ) ) ) ).
% countable_empty
thf(fact_6936_butlast__rev,axiom,
! [A: $tType,Xs: list @ A] :
( ( butlast @ A @ ( rev @ A @ Xs ) )
= ( rev @ A @ ( tl @ A @ Xs ) ) ) ).
% butlast_rev
thf(fact_6937_countable__Diff__eq,axiom,
! [A: $tType,A4: set @ A,X: A] :
( ( countable_countable @ A @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( countable_countable @ A @ A4 ) ) ).
% countable_Diff_eq
thf(fact_6938_butlast__snoc,axiom,
! [A: $tType,Xs: list @ A,X: A] :
( ( butlast @ A @ ( append @ A @ Xs @ ( cons @ A @ X @ ( nil @ A ) ) ) )
= Xs ) ).
% butlast_snoc
thf(fact_6939_length__butlast,axiom,
! [A: $tType,Xs: list @ A] :
( ( size_size @ ( list @ A ) @ ( butlast @ A @ Xs ) )
= ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) ) ) ).
% length_butlast
thf(fact_6940_map__butlast,axiom,
! [A: $tType,B: $tType,F2: B > A,Xs: list @ B] :
( ( map @ B @ A @ F2 @ ( butlast @ B @ Xs ) )
= ( butlast @ A @ ( map @ B @ A @ F2 @ Xs ) ) ) ).
% map_butlast
thf(fact_6941_ccSup__upper2,axiom,
! [A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A4: set @ A,U: A,V2: A] :
( ( countable_countable @ A @ A4 )
=> ( ( member @ A @ U @ A4 )
=> ( ( ord_less_eq @ A @ V2 @ U )
=> ( ord_less_eq @ A @ V2 @ ( complete_Sup_Sup @ A @ A4 ) ) ) ) ) ) ).
% ccSup_upper2
thf(fact_6942_ccSup__le__iff,axiom,
! [A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A4: set @ A,B3: A] :
( ( countable_countable @ A @ A4 )
=> ( ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A4 ) @ B3 )
= ( ! [X4: A] :
( ( member @ A @ X4 @ A4 )
=> ( ord_less_eq @ A @ X4 @ B3 ) ) ) ) ) ) ).
% ccSup_le_iff
thf(fact_6943_ccSup__upper,axiom,
! [A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A4: set @ A,X: A] :
( ( countable_countable @ A @ A4 )
=> ( ( member @ A @ X @ A4 )
=> ( ord_less_eq @ A @ X @ ( complete_Sup_Sup @ A @ A4 ) ) ) ) ) ).
% ccSup_upper
thf(fact_6944_ccSup__least,axiom,
! [A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A4: set @ A,Z3: A] :
( ( countable_countable @ A @ A4 )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ A4 )
=> ( ord_less_eq @ A @ X5 @ Z3 ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A4 ) @ Z3 ) ) ) ) ).
% ccSup_least
thf(fact_6945_ccSup__mono,axiom,
! [A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [B2: set @ A,A4: set @ A] :
( ( countable_countable @ A @ B2 )
=> ( ( countable_countable @ A @ A4 )
=> ( ! [A6: A] :
( ( member @ A @ A6 @ A4 )
=> ? [X2: A] :
( ( member @ A @ X2 @ B2 )
& ( ord_less_eq @ A @ A6 @ X2 ) ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A4 ) @ ( complete_Sup_Sup @ A @ B2 ) ) ) ) ) ) ).
% ccSup_mono
thf(fact_6946_infinite__countable__subset_H,axiom,
! [A: $tType,X8: set @ A] :
( ~ ( finite_finite2 @ A @ X8 )
=> ? [C7: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ C7 @ X8 )
& ( countable_countable @ A @ C7 )
& ~ ( finite_finite2 @ A @ C7 ) ) ) ).
% infinite_countable_subset'
thf(fact_6947_countable__Collect__finite__subset,axiom,
! [A: $tType,T4: set @ A] :
( ( countable_countable @ A @ T4 )
=> ( countable_countable @ ( set @ A )
@ ( collect @ ( set @ A )
@ ^ [A7: set @ A] :
( ( finite_finite2 @ A @ A7 )
& ( ord_less_eq @ ( set @ A ) @ A7 @ T4 ) ) ) ) ) ).
% countable_Collect_finite_subset
thf(fact_6948_countable__subset,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B2 )
=> ( ( countable_countable @ A @ B2 )
=> ( countable_countable @ A @ A4 ) ) ) ).
% countable_subset
thf(fact_6949_ccInf__mono,axiom,
! [A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [B2: set @ A,A4: set @ A] :
( ( countable_countable @ A @ B2 )
=> ( ( countable_countable @ A @ A4 )
=> ( ! [B5: A] :
( ( member @ A @ B5 @ B2 )
=> ? [X2: A] :
( ( member @ A @ X2 @ A4 )
& ( ord_less_eq @ A @ X2 @ B5 ) ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A4 ) @ ( complete_Inf_Inf @ A @ B2 ) ) ) ) ) ) ).
% ccInf_mono
thf(fact_6950_ccInf__lower,axiom,
! [A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A4: set @ A,X: A] :
( ( countable_countable @ A @ A4 )
=> ( ( member @ A @ X @ A4 )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A4 ) @ X ) ) ) ) ).
% ccInf_lower
thf(fact_6951_ccInf__lower2,axiom,
! [A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A4: set @ A,U: A,V2: A] :
( ( countable_countable @ A @ A4 )
=> ( ( member @ A @ U @ A4 )
=> ( ( ord_less_eq @ A @ U @ V2 )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A4 ) @ V2 ) ) ) ) ) ).
% ccInf_lower2
thf(fact_6952_le__ccInf__iff,axiom,
! [A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A4: set @ A,B3: A] :
( ( countable_countable @ A @ A4 )
=> ( ( ord_less_eq @ A @ B3 @ ( complete_Inf_Inf @ A @ A4 ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ A4 )
=> ( ord_less_eq @ A @ B3 @ X4 ) ) ) ) ) ) ).
% le_ccInf_iff
thf(fact_6953_ccInf__greatest,axiom,
! [A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A4: set @ A,Z3: A] :
( ( countable_countable @ A @ A4 )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ A4 )
=> ( ord_less_eq @ A @ Z3 @ X5 ) )
=> ( ord_less_eq @ A @ Z3 @ ( complete_Inf_Inf @ A @ A4 ) ) ) ) ) ).
% ccInf_greatest
thf(fact_6954_in__set__butlast__appendI,axiom,
! [A: $tType,X: A,Xs: list @ A,Ys: list @ A] :
( ( ( member @ A @ X @ ( set2 @ A @ ( butlast @ A @ Xs ) ) )
| ( member @ A @ X @ ( set2 @ A @ ( butlast @ A @ Ys ) ) ) )
=> ( member @ A @ X @ ( set2 @ A @ ( butlast @ A @ ( append @ A @ Xs @ Ys ) ) ) ) ) ).
% in_set_butlast_appendI
thf(fact_6955_distinct__butlast,axiom,
! [A: $tType,Xs: list @ A] :
( ( distinct @ A @ Xs )
=> ( distinct @ A @ ( butlast @ A @ Xs ) ) ) ).
% distinct_butlast
thf(fact_6956_butlast__append,axiom,
! [A: $tType,Ys: list @ A,Xs: list @ A] :
( ( ( Ys
= ( nil @ A ) )
=> ( ( butlast @ A @ ( append @ A @ Xs @ Ys ) )
= ( butlast @ A @ Xs ) ) )
& ( ( Ys
!= ( nil @ A ) )
=> ( ( butlast @ A @ ( append @ A @ Xs @ Ys ) )
= ( append @ A @ Xs @ ( butlast @ A @ Ys ) ) ) ) ) ).
% butlast_append
thf(fact_6957_in__set__butlastD,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( member @ A @ X @ ( set2 @ A @ ( butlast @ A @ Xs ) ) )
=> ( member @ A @ X @ ( set2 @ A @ Xs ) ) ) ).
% in_set_butlastD
thf(fact_6958_butlast__tl,axiom,
! [A: $tType,Xs: list @ A] :
( ( butlast @ A @ ( tl @ A @ Xs ) )
= ( tl @ A @ ( butlast @ A @ Xs ) ) ) ).
% butlast_tl
thf(fact_6959_drop__butlast,axiom,
! [A: $tType,N2: nat,Xs: list @ A] :
( ( drop @ A @ N2 @ ( butlast @ A @ Xs ) )
= ( butlast @ A @ ( drop @ A @ N2 @ Xs ) ) ) ).
% drop_butlast
thf(fact_6960_butlast_Osimps_I1_J,axiom,
! [A: $tType] :
( ( butlast @ A @ ( nil @ A ) )
= ( nil @ A ) ) ).
% butlast.simps(1)
thf(fact_6961_ccInf__less__iff,axiom,
! [A: $tType] :
( ( ( counta3822494911875563373attice @ A )
& ( linorder @ A ) )
=> ! [S2: set @ A,A3: A] :
( ( countable_countable @ A @ S2 )
=> ( ( ord_less @ A @ ( complete_Inf_Inf @ A @ S2 ) @ A3 )
= ( ? [X4: A] :
( ( member @ A @ X4 @ S2 )
& ( ord_less @ A @ X4 @ A3 ) ) ) ) ) ) ).
% ccInf_less_iff
thf(fact_6962_countable__Collect__finite,axiom,
! [A: $tType] :
( ( countable @ A )
=> ( countable_countable @ ( set @ A ) @ ( collect @ ( set @ A ) @ ( finite_finite2 @ A ) ) ) ) ).
% countable_Collect_finite
thf(fact_6963_countable__finite,axiom,
! [A: $tType,S2: set @ A] :
( ( finite_finite2 @ A @ S2 )
=> ( countable_countable @ A @ S2 ) ) ).
% countable_finite
thf(fact_6964_uncountable__infinite,axiom,
! [A: $tType,A4: set @ A] :
( ~ ( countable_countable @ A @ A4 )
=> ~ ( finite_finite2 @ A @ A4 ) ) ).
% uncountable_infinite
thf(fact_6965_less__ccSup__iff,axiom,
! [A: $tType] :
( ( ( counta3822494911875563373attice @ A )
& ( linorder @ A ) )
=> ! [S2: set @ A,A3: A] :
( ( countable_countable @ A @ S2 )
=> ( ( ord_less @ A @ A3 @ ( complete_Sup_Sup @ A @ S2 ) )
= ( ? [X4: A] :
( ( member @ A @ X4 @ S2 )
& ( ord_less @ A @ A3 @ X4 ) ) ) ) ) ) ).
% less_ccSup_iff
thf(fact_6966_to__nat__on__surj,axiom,
! [A: $tType,A4: set @ A,N2: nat] :
( ( countable_countable @ A @ A4 )
=> ( ~ ( finite_finite2 @ A @ A4 )
=> ? [X5: A] :
( ( member @ A @ X5 @ A4 )
& ( ( countable_to_nat_on @ A @ A4 @ X5 )
= N2 ) ) ) ) ).
% to_nat_on_surj
thf(fact_6967_butlast_Osimps_I2_J,axiom,
! [A: $tType,Xs: list @ A,X: A] :
( ( ( Xs
= ( nil @ A ) )
=> ( ( butlast @ A @ ( cons @ A @ X @ Xs ) )
= ( nil @ A ) ) )
& ( ( Xs
!= ( nil @ A ) )
=> ( ( butlast @ A @ ( cons @ A @ X @ Xs ) )
= ( cons @ A @ X @ ( butlast @ A @ Xs ) ) ) ) ) ).
% butlast.simps(2)
thf(fact_6968_all__countable__subset__image,axiom,
! [A: $tType,B: $tType,F2: B > A,S2: set @ B,P: ( set @ A ) > $o] :
( ( ! [T10: set @ A] :
( ( ( countable_countable @ A @ T10 )
& ( ord_less_eq @ ( set @ A ) @ T10 @ ( image2 @ B @ A @ F2 @ S2 ) ) )
=> ( P @ T10 ) ) )
= ( ! [T10: set @ B] :
( ( ( countable_countable @ B @ T10 )
& ( ord_less_eq @ ( set @ B ) @ T10 @ S2 ) )
=> ( P @ ( image2 @ B @ A @ F2 @ T10 ) ) ) ) ) ).
% all_countable_subset_image
thf(fact_6969_ex__countable__subset__image,axiom,
! [A: $tType,B: $tType,F2: B > A,S2: set @ B,P: ( set @ A ) > $o] :
( ( ? [T10: set @ A] :
( ( countable_countable @ A @ T10 )
& ( ord_less_eq @ ( set @ A ) @ T10 @ ( image2 @ B @ A @ F2 @ S2 ) )
& ( P @ T10 ) ) )
= ( ? [T10: set @ B] :
( ( countable_countable @ B @ T10 )
& ( ord_less_eq @ ( set @ B ) @ T10 @ S2 )
& ( P @ ( image2 @ B @ A @ F2 @ T10 ) ) ) ) ) ).
% ex_countable_subset_image
thf(fact_6970_countable__subset__image,axiom,
! [A: $tType,B: $tType,B2: set @ A,F2: B > A,A4: set @ B] :
( ( ( countable_countable @ A @ B2 )
& ( ord_less_eq @ ( set @ A ) @ B2 @ ( image2 @ B @ A @ F2 @ A4 ) ) )
= ( ? [A17: set @ B] :
( ( countable_countable @ B @ A17 )
& ( ord_less_eq @ ( set @ B ) @ A17 @ A4 )
& ( B2
= ( image2 @ B @ A @ F2 @ A17 ) ) ) ) ) ).
% countable_subset_image
thf(fact_6971_countable__image__eq,axiom,
! [A: $tType,B: $tType,F2: B > A,S2: set @ B] :
( ( countable_countable @ A @ ( image2 @ B @ A @ F2 @ S2 ) )
= ( ? [T10: set @ B] :
( ( countable_countable @ B @ T10 )
& ( ord_less_eq @ ( set @ B ) @ T10 @ S2 )
& ( ( image2 @ B @ A @ F2 @ S2 )
= ( image2 @ B @ A @ F2 @ T10 ) ) ) ) ) ).
% countable_image_eq
thf(fact_6972_ccSup__subset__mono,axiom,
! [A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [B2: set @ A,A4: set @ A] :
( ( countable_countable @ A @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ A4 @ B2 )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A4 ) @ ( complete_Sup_Sup @ A @ B2 ) ) ) ) ) ).
% ccSup_subset_mono
thf(fact_6973_ccInf__superset__mono,axiom,
! [A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A4: set @ A,B2: set @ A] :
( ( countable_countable @ A @ A4 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ A4 )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A4 ) @ ( complete_Inf_Inf @ A @ B2 ) ) ) ) ) ).
% ccInf_superset_mono
thf(fact_6974_all__countable__subset__image__inj,axiom,
! [A: $tType,B: $tType,F2: B > A,S2: set @ B,P: ( set @ A ) > $o] :
( ( ! [T10: set @ A] :
( ( ( countable_countable @ A @ T10 )
& ( ord_less_eq @ ( set @ A ) @ T10 @ ( image2 @ B @ A @ F2 @ S2 ) ) )
=> ( P @ T10 ) ) )
= ( ! [T10: set @ B] :
( ( ( countable_countable @ B @ T10 )
& ( ord_less_eq @ ( set @ B ) @ T10 @ S2 )
& ( inj_on @ B @ A @ F2 @ T10 ) )
=> ( P @ ( image2 @ B @ A @ F2 @ T10 ) ) ) ) ) ).
% all_countable_subset_image_inj
thf(fact_6975_ex__countable__subset__image__inj,axiom,
! [A: $tType,B: $tType,F2: B > A,S2: set @ B,P: ( set @ A ) > $o] :
( ( ? [T10: set @ A] :
( ( countable_countable @ A @ T10 )
& ( ord_less_eq @ ( set @ A ) @ T10 @ ( image2 @ B @ A @ F2 @ S2 ) )
& ( P @ T10 ) ) )
= ( ? [T10: set @ B] :
( ( countable_countable @ B @ T10 )
& ( ord_less_eq @ ( set @ B ) @ T10 @ S2 )
& ( inj_on @ B @ A @ F2 @ T10 )
& ( P @ ( image2 @ B @ A @ F2 @ T10 ) ) ) ) ) ).
% ex_countable_subset_image_inj
thf(fact_6976_countable__image__eq__inj,axiom,
! [A: $tType,B: $tType,F2: B > A,S2: set @ B] :
( ( countable_countable @ A @ ( image2 @ B @ A @ F2 @ S2 ) )
= ( ? [T10: set @ B] :
( ( countable_countable @ B @ T10 )
& ( ord_less_eq @ ( set @ B ) @ T10 @ S2 )
& ( ( image2 @ B @ A @ F2 @ S2 )
= ( image2 @ B @ A @ F2 @ T10 ) )
& ( inj_on @ B @ A @ F2 @ T10 ) ) ) ) ).
% countable_image_eq_inj
thf(fact_6977_uncountable__def,axiom,
! [A: $tType,A4: set @ A] :
( ( ~ ( countable_countable @ A @ A4 ) )
= ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
& ~ ? [F3: nat > A] :
( ( image2 @ nat @ A @ F3 @ ( top_top @ ( set @ nat ) ) )
= A4 ) ) ) ).
% uncountable_def
thf(fact_6978_ccSUP__mono,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A4: set @ B,B2: set @ C,F2: B > A,G: C > A] :
( ( countable_countable @ B @ A4 )
=> ( ( countable_countable @ C @ B2 )
=> ( ! [N3: B] :
( ( member @ B @ N3 @ A4 )
=> ? [X2: C] :
( ( member @ C @ X2 @ B2 )
& ( ord_less_eq @ A @ ( F2 @ N3 ) @ ( G @ X2 ) ) ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F2 @ A4 ) ) @ ( complete_Sup_Sup @ A @ ( image2 @ C @ A @ G @ B2 ) ) ) ) ) ) ) ).
% ccSUP_mono
thf(fact_6979_ccSUP__least,axiom,
! [B: $tType,A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A4: set @ B,F2: B > A,U: A] :
( ( countable_countable @ B @ A4 )
=> ( ! [I4: B] :
( ( member @ B @ I4 @ A4 )
=> ( ord_less_eq @ A @ ( F2 @ I4 ) @ U ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F2 @ A4 ) ) @ U ) ) ) ) ).
% ccSUP_least
thf(fact_6980_ccSUP__upper,axiom,
! [A: $tType,B: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A4: set @ B,I: B,F2: B > A] :
( ( countable_countable @ B @ A4 )
=> ( ( member @ B @ I @ A4 )
=> ( ord_less_eq @ A @ ( F2 @ I ) @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F2 @ A4 ) ) ) ) ) ) ).
% ccSUP_upper
thf(fact_6981_ccSUP__le__iff,axiom,
! [A: $tType,B: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A4: set @ B,F2: B > A,U: A] :
( ( countable_countable @ B @ A4 )
=> ( ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F2 @ A4 ) ) @ U )
= ( ! [X4: B] :
( ( member @ B @ X4 @ A4 )
=> ( ord_less_eq @ A @ ( F2 @ X4 ) @ U ) ) ) ) ) ) ).
% ccSUP_le_iff
thf(fact_6982_ccSUP__upper2,axiom,
! [A: $tType,B: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A4: set @ B,I: B,U: A,F2: B > A] :
( ( countable_countable @ B @ A4 )
=> ( ( member @ B @ I @ A4 )
=> ( ( ord_less_eq @ A @ U @ ( F2 @ I ) )
=> ( ord_less_eq @ A @ U @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F2 @ A4 ) ) ) ) ) ) ) ).
% ccSUP_upper2
thf(fact_6983_countable__infiniteE_H,axiom,
! [A: $tType,A4: set @ A] :
( ( countable_countable @ A @ A4 )
=> ( ~ ( finite_finite2 @ A @ A4 )
=> ~ ! [G7: nat > A] :
~ ( bij_betw @ nat @ A @ G7 @ ( top_top @ ( set @ nat ) ) @ A4 ) ) ) ).
% countable_infiniteE'
thf(fact_6984_less__ccSUP__iff,axiom,
! [A: $tType,B: $tType] :
( ( ( counta3822494911875563373attice @ A )
& ( linorder @ A ) )
=> ! [A4: set @ B,A3: A,F2: B > A] :
( ( countable_countable @ B @ A4 )
=> ( ( ord_less @ A @ A3 @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F2 @ A4 ) ) )
= ( ? [X4: B] :
( ( member @ B @ X4 @ A4 )
& ( ord_less @ A @ A3 @ ( F2 @ X4 ) ) ) ) ) ) ) ).
% less_ccSUP_iff
thf(fact_6985_ccINF__mono,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A4: set @ B,B2: set @ C,F2: B > A,G: C > A] :
( ( countable_countable @ B @ A4 )
=> ( ( countable_countable @ C @ B2 )
=> ( ! [M3: C] :
( ( member @ C @ M3 @ B2 )
=> ? [X2: B] :
( ( member @ B @ X2 @ A4 )
& ( ord_less_eq @ A @ ( F2 @ X2 ) @ ( G @ M3 ) ) ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F2 @ A4 ) ) @ ( complete_Inf_Inf @ A @ ( image2 @ C @ A @ G @ B2 ) ) ) ) ) ) ) ).
% ccINF_mono
thf(fact_6986_ccINF__lower,axiom,
! [A: $tType,B: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A4: set @ B,I: B,F2: B > A] :
( ( countable_countable @ B @ A4 )
=> ( ( member @ B @ I @ A4 )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F2 @ A4 ) ) @ ( F2 @ I ) ) ) ) ) ).
% ccINF_lower
thf(fact_6987_ccINF__lower2,axiom,
! [B: $tType,A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A4: set @ B,I: B,F2: B > A,U: A] :
( ( countable_countable @ B @ A4 )
=> ( ( member @ B @ I @ A4 )
=> ( ( ord_less_eq @ A @ ( F2 @ I ) @ U )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F2 @ A4 ) ) @ U ) ) ) ) ) ).
% ccINF_lower2
thf(fact_6988_le__ccINF__iff,axiom,
! [A: $tType,B: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A4: set @ B,U: A,F2: B > A] :
( ( countable_countable @ B @ A4 )
=> ( ( ord_less_eq @ A @ U @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F2 @ A4 ) ) )
= ( ! [X4: B] :
( ( member @ B @ X4 @ A4 )
=> ( ord_less_eq @ A @ U @ ( F2 @ X4 ) ) ) ) ) ) ) ).
% le_ccINF_iff
thf(fact_6989_ccINF__greatest,axiom,
! [A: $tType,B: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A4: set @ B,U: A,F2: B > A] :
( ( countable_countable @ B @ A4 )
=> ( ! [I4: B] :
( ( member @ B @ I4 @ A4 )
=> ( ord_less_eq @ A @ U @ ( F2 @ I4 ) ) )
=> ( ord_less_eq @ A @ U @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F2 @ A4 ) ) ) ) ) ) ).
% ccINF_greatest
thf(fact_6990_ccINF__less__iff,axiom,
! [A: $tType,B: $tType] :
( ( ( counta3822494911875563373attice @ A )
& ( linorder @ A ) )
=> ! [A4: set @ B,F2: B > A,A3: A] :
( ( countable_countable @ B @ A4 )
=> ( ( ord_less @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F2 @ A4 ) ) @ A3 )
= ( ? [X4: B] :
( ( member @ B @ X4 @ A4 )
& ( ord_less @ A @ ( F2 @ X4 ) @ A3 ) ) ) ) ) ) ).
% ccINF_less_iff
thf(fact_6991_countableE__infinite,axiom,
! [A: $tType,S2: set @ A] :
( ( countable_countable @ A @ S2 )
=> ( ~ ( finite_finite2 @ A @ S2 )
=> ~ ! [E: A > nat] :
~ ( bij_betw @ A @ nat @ E @ S2 @ ( top_top @ ( set @ nat ) ) ) ) ) ).
% countableE_infinite
thf(fact_6992_sorted__butlast,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( butlast @ A @ Xs ) ) ) ) ) ).
% sorted_butlast
thf(fact_6993_nth__butlast,axiom,
! [A: $tType,N2: nat,Xs: list @ A] :
( ( ord_less @ nat @ N2 @ ( size_size @ ( list @ A ) @ ( butlast @ A @ Xs ) ) )
=> ( ( nth @ A @ ( butlast @ A @ Xs ) @ N2 )
= ( nth @ A @ Xs @ N2 ) ) ) ).
% nth_butlast
thf(fact_6994_take__butlast,axiom,
! [A: $tType,N2: nat,Xs: list @ A] :
( ( ord_less @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( take @ A @ N2 @ ( butlast @ A @ Xs ) )
= ( take @ A @ N2 @ Xs ) ) ) ).
% take_butlast
thf(fact_6995_ccSup__inter__less__eq,axiom,
! [A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A4: set @ A,B2: set @ A] :
( ( countable_countable @ A @ A4 )
=> ( ( countable_countable @ A @ B2 )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( inf_inf @ ( set @ A ) @ A4 @ B2 ) ) @ ( inf_inf @ A @ ( complete_Sup_Sup @ A @ A4 ) @ ( complete_Sup_Sup @ A @ B2 ) ) ) ) ) ) ).
% ccSup_inter_less_eq
thf(fact_6996_less__eq__ccInf__inter,axiom,
! [A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A4: set @ A,B2: set @ A] :
( ( countable_countable @ A @ A4 )
=> ( ( countable_countable @ A @ B2 )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ ( complete_Inf_Inf @ A @ A4 ) @ ( complete_Inf_Inf @ A @ B2 ) ) @ ( complete_Inf_Inf @ A @ ( inf_inf @ ( set @ A ) @ A4 @ B2 ) ) ) ) ) ) ).
% less_eq_ccInf_inter
thf(fact_6997_ccSUP__subset__mono,axiom,
! [A: $tType,B: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [B2: set @ B,A4: set @ B,F2: B > A,G: B > A] :
( ( countable_countable @ B @ B2 )
=> ( ( ord_less_eq @ ( set @ B ) @ A4 @ B2 )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ A4 )
=> ( ord_less_eq @ A @ ( F2 @ X5 ) @ ( G @ X5 ) ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F2 @ A4 ) ) @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ G @ B2 ) ) ) ) ) ) ) ).
% ccSUP_subset_mono
thf(fact_6998_ccINF__superset__mono,axiom,
! [A: $tType,B: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A4: set @ B,B2: set @ B,F2: B > A,G: B > A] :
( ( countable_countable @ B @ A4 )
=> ( ( ord_less_eq @ ( set @ B ) @ B2 @ A4 )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ B2 )
=> ( ord_less_eq @ A @ ( F2 @ X5 ) @ ( G @ X5 ) ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F2 @ A4 ) ) @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ G @ B2 ) ) ) ) ) ) ) ).
% ccINF_superset_mono
thf(fact_6999_mono__ccSup,axiom,
! [B: $tType,A: $tType] :
( ( ( counta4013691401010221786attice @ A )
& ( counta3822494911875563373attice @ B ) )
=> ! [F2: A > B,A4: set @ A] :
( ( order_mono @ A @ B @ F2 )
=> ( ( countable_countable @ A @ A4 )
=> ( ord_less_eq @ B @ ( complete_Sup_Sup @ B @ ( image2 @ A @ B @ F2 @ A4 ) ) @ ( F2 @ ( complete_Sup_Sup @ A @ A4 ) ) ) ) ) ) ).
% mono_ccSup
thf(fact_7000_mono__ccSUP,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( ( counta4013691401010221786attice @ A )
& ( counta3822494911875563373attice @ B ) )
=> ! [F2: A > B,I6: set @ C,A4: C > A] :
( ( order_mono @ A @ B @ F2 )
=> ( ( countable_countable @ C @ I6 )
=> ( ord_less_eq @ B
@ ( complete_Sup_Sup @ B
@ ( image2 @ C @ B
@ ^ [X4: C] : ( F2 @ ( A4 @ X4 ) )
@ I6 ) )
@ ( F2 @ ( complete_Sup_Sup @ A @ ( image2 @ C @ A @ A4 @ I6 ) ) ) ) ) ) ) ).
% mono_ccSUP
thf(fact_7001_mono__ccINF,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( ( counta4013691401010221786attice @ A )
& ( counta3822494911875563373attice @ B ) )
=> ! [F2: A > B,I6: set @ C,A4: C > A] :
( ( order_mono @ A @ B @ F2 )
=> ( ( countable_countable @ C @ I6 )
=> ( ord_less_eq @ B @ ( F2 @ ( complete_Inf_Inf @ A @ ( image2 @ C @ A @ A4 @ I6 ) ) )
@ ( complete_Inf_Inf @ B
@ ( image2 @ C @ B
@ ^ [X4: C] : ( F2 @ ( A4 @ X4 ) )
@ I6 ) ) ) ) ) ) ).
% mono_ccINF
thf(fact_7002_mono__ccInf,axiom,
! [B: $tType,A: $tType] :
( ( ( counta4013691401010221786attice @ A )
& ( counta3822494911875563373attice @ B ) )
=> ! [F2: A > B,A4: set @ A] :
( ( order_mono @ A @ B @ F2 )
=> ( ( countable_countable @ A @ A4 )
=> ( ord_less_eq @ B @ ( F2 @ ( complete_Inf_Inf @ A @ A4 ) ) @ ( complete_Inf_Inf @ B @ ( image2 @ A @ B @ F2 @ A4 ) ) ) ) ) ) ).
% mono_ccInf
thf(fact_7003_countable__as__injective__image,axiom,
! [A: $tType,A4: set @ A] :
( ( countable_countable @ A @ A4 )
=> ( ~ ( finite_finite2 @ A @ A4 )
=> ~ ! [F4: nat > A] :
( ( A4
= ( image2 @ nat @ A @ F4 @ ( top_top @ ( set @ nat ) ) ) )
=> ~ ( inj_on @ nat @ A @ F4 @ ( top_top @ ( set @ nat ) ) ) ) ) ) ).
% countable_as_injective_image
thf(fact_7004_image__to__nat__on,axiom,
! [A: $tType,A4: set @ A] :
( ( countable_countable @ A @ A4 )
=> ( ~ ( finite_finite2 @ A @ A4 )
=> ( ( image2 @ A @ nat @ ( countable_to_nat_on @ A @ A4 ) @ A4 )
= ( top_top @ ( set @ nat ) ) ) ) ) ).
% image_to_nat_on
thf(fact_7005_to__nat__on__infinite,axiom,
! [A: $tType,S2: set @ A] :
( ( countable_countable @ A @ S2 )
=> ( ~ ( finite_finite2 @ A @ S2 )
=> ( bij_betw @ A @ nat @ ( countable_to_nat_on @ A @ S2 ) @ S2 @ ( top_top @ ( set @ nat ) ) ) ) ) ).
% to_nat_on_infinite
thf(fact_7006_butlast__conv__take,axiom,
! [A: $tType] :
( ( butlast @ A )
= ( ^ [Xs3: list @ A] : ( take @ A @ ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs3 ) @ ( one_one @ nat ) ) @ Xs3 ) ) ) ).
% butlast_conv_take
thf(fact_7007_butlast__list__update,axiom,
! [A: $tType,K: nat,Xs: list @ A,X: A] :
( ( ( K
= ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) ) )
=> ( ( butlast @ A @ ( list_update @ A @ Xs @ K @ X ) )
= ( butlast @ A @ Xs ) ) )
& ( ( K
!= ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) ) )
=> ( ( butlast @ A @ ( list_update @ A @ Xs @ K @ X ) )
= ( list_update @ A @ ( butlast @ A @ Xs ) @ K @ X ) ) ) ) ).
% butlast_list_update
thf(fact_7008_countable__enum__cases,axiom,
! [A: $tType,S2: set @ A] :
( ( countable_countable @ A @ S2 )
=> ( ( ( finite_finite2 @ A @ S2 )
=> ! [F4: A > nat] :
~ ( bij_betw @ A @ nat @ F4 @ S2 @ ( set_ord_lessThan @ nat @ ( finite_card @ A @ S2 ) ) ) )
=> ~ ( ~ ( finite_finite2 @ A @ S2 )
=> ! [F4: A > nat] :
~ ( bij_betw @ A @ nat @ F4 @ S2 @ ( top_top @ ( set @ nat ) ) ) ) ) ) ).
% countable_enum_cases
thf(fact_7009_butlast__take,axiom,
! [A: $tType,N2: nat,Xs: list @ A] :
( ( ord_less_eq @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( butlast @ A @ ( take @ A @ N2 @ Xs ) )
= ( take @ A @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) @ Xs ) ) ) ).
% butlast_take
thf(fact_7010_sort__key__conv__fold,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F2: B > A,Xs: list @ B] :
( ( inj_on @ B @ A @ F2 @ ( set2 @ B @ Xs ) )
=> ( ( linorder_sort_key @ B @ A @ F2 @ Xs )
= ( fold @ B @ ( list @ B ) @ ( linorder_insort_key @ B @ A @ F2 ) @ Xs @ ( nil @ B ) ) ) ) ) ).
% sort_key_conv_fold
thf(fact_7011_range__from__nat__into,axiom,
! [A: $tType,A4: set @ A] :
( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( countable_countable @ A @ A4 )
=> ( ( image2 @ nat @ A @ ( counta4804993851260445106t_into @ A @ A4 ) @ ( top_top @ ( set @ nat ) ) )
= A4 ) ) ) ).
% range_from_nat_into
thf(fact_7012_sort__upt,axiom,
! [M2: nat,N2: nat] :
( ( linorder_sort_key @ nat @ nat
@ ^ [X4: nat] : X4
@ ( upt @ M2 @ N2 ) )
= ( upt @ M2 @ N2 ) ) ).
% sort_upt
thf(fact_7013_sort__upto,axiom,
! [I: int,J: int] :
( ( linorder_sort_key @ int @ int
@ ^ [X4: int] : X4
@ ( upto @ I @ J ) )
= ( upto @ I @ J ) ) ).
% sort_upto
thf(fact_7014_sort__key__simps_I1_J,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F2: B > A] :
( ( linorder_sort_key @ B @ A @ F2 @ ( nil @ B ) )
= ( nil @ B ) ) ) ).
% sort_key_simps(1)
thf(fact_7015_set__sort,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F2: B > A,Xs: list @ B] :
( ( set2 @ B @ ( linorder_sort_key @ B @ A @ F2 @ Xs ) )
= ( set2 @ B @ Xs ) ) ) ).
% set_sort
thf(fact_7016_length__sort,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F2: B > A,Xs: list @ B] :
( ( size_size @ ( list @ B ) @ ( linorder_sort_key @ B @ A @ F2 @ Xs ) )
= ( size_size @ ( list @ B ) @ Xs ) ) ) ).
% length_sort
thf(fact_7017_distinct__sort,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F2: B > A,Xs: list @ B] :
( ( distinct @ B @ ( linorder_sort_key @ B @ A @ F2 @ Xs ) )
= ( distinct @ B @ Xs ) ) ) ).
% distinct_sort
thf(fact_7018_from__nat__into__inject,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] :
( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( countable_countable @ A @ A4 )
=> ( ( B2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( countable_countable @ A @ B2 )
=> ( ( ( counta4804993851260445106t_into @ A @ A4 )
= ( counta4804993851260445106t_into @ A @ B2 ) )
= ( A4 = B2 ) ) ) ) ) ) ).
% from_nat_into_inject
thf(fact_7019_from__nat__into__inj__infinite,axiom,
! [A: $tType,A4: set @ A,M2: nat,N2: nat] :
( ( countable_countable @ A @ A4 )
=> ( ~ ( finite_finite2 @ A @ A4 )
=> ( ( ( counta4804993851260445106t_into @ A @ A4 @ M2 )
= ( counta4804993851260445106t_into @ A @ A4 @ N2 ) )
= ( M2 = N2 ) ) ) ) ).
% from_nat_into_inj_infinite
thf(fact_7020_sort__key__simps_I2_J,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F2: B > A,X: B,Xs: list @ B] :
( ( linorder_sort_key @ B @ A @ F2 @ ( cons @ B @ X @ Xs ) )
= ( linorder_insort_key @ B @ A @ F2 @ X @ ( linorder_sort_key @ B @ A @ F2 @ Xs ) ) ) ) ).
% sort_key_simps(2)
thf(fact_7021_to__nat__on__from__nat__into__infinite,axiom,
! [A: $tType,A4: set @ A,N2: nat] :
( ( countable_countable @ A @ A4 )
=> ( ~ ( finite_finite2 @ A @ A4 )
=> ( ( countable_to_nat_on @ A @ A4 @ ( counta4804993851260445106t_into @ A @ A4 @ N2 ) )
= N2 ) ) ) ).
% to_nat_on_from_nat_into_infinite
thf(fact_7022_sort__key__const,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ B )
=> ! [C2: B,Xs: list @ A] :
( ( linorder_sort_key @ A @ B
@ ^ [X4: A] : C2
@ Xs )
= Xs ) ) ).
% sort_key_const
thf(fact_7023_sort__key__stable,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ B )
=> ! [F2: A > B,K: B,Xs: list @ A] :
( ( filter2 @ A
@ ^ [Y6: A] :
( ( F2 @ Y6 )
= K )
@ ( linorder_sort_key @ A @ B @ F2 @ Xs ) )
= ( filter2 @ A
@ ^ [Y6: A] :
( ( F2 @ Y6 )
= K )
@ Xs ) ) ) ).
% sort_key_stable
thf(fact_7024_filter__sort,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [P: B > $o,F2: B > A,Xs: list @ B] :
( ( filter2 @ B @ P @ ( linorder_sort_key @ B @ A @ F2 @ Xs ) )
= ( linorder_sort_key @ B @ A @ F2 @ ( filter2 @ B @ P @ Xs ) ) ) ) ).
% filter_sort
thf(fact_7025_from__nat__into,axiom,
! [A: $tType,A4: set @ A,N2: nat] :
( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( member @ A @ ( counta4804993851260445106t_into @ A @ A4 @ N2 ) @ A4 ) ) ).
% from_nat_into
thf(fact_7026_inj__on__from__nat__into,axiom,
! [A: $tType] :
( inj_on @ ( set @ A ) @ ( nat > A ) @ ( counta4804993851260445106t_into @ A )
@ ( collect @ ( set @ A )
@ ^ [A7: set @ A] :
( ( A7
!= ( bot_bot @ ( set @ A ) ) )
& ( countable_countable @ A @ A7 ) ) ) ) ).
% inj_on_from_nat_into
thf(fact_7027_sorted__sort__id,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
=> ( ( linorder_sort_key @ A @ A
@ ^ [X4: A] : X4
@ Xs )
= Xs ) ) ) ).
% sorted_sort_id
thf(fact_7028_sorted__sort,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A] :
( sorted_wrt @ A @ ( ord_less_eq @ A )
@ ( linorder_sort_key @ A @ A
@ ^ [X4: A] : X4
@ Xs ) ) ) ).
% sorted_sort
thf(fact_7029_sorted__sort__key,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F2: B > A,Xs: list @ B] : ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F2 @ ( linorder_sort_key @ B @ A @ F2 @ Xs ) ) ) ) ).
% sorted_sort_key
thf(fact_7030_sorted__list__of__set__sort__remdups,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A] :
( ( linord4507533701916653071of_set @ A @ ( set2 @ A @ Xs ) )
= ( linorder_sort_key @ A @ A
@ ^ [X4: A] : X4
@ ( remdups @ A @ Xs ) ) ) ) ).
% sorted_list_of_set_sort_remdups
thf(fact_7031_range__from__nat__into__subset,axiom,
! [A: $tType,A4: set @ A] :
( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ord_less_eq @ ( set @ A ) @ ( image2 @ nat @ A @ ( counta4804993851260445106t_into @ A @ A4 ) @ ( top_top @ ( set @ nat ) ) ) @ A4 ) ) ).
% range_from_nat_into_subset
thf(fact_7032_subset__range__from__nat__into,axiom,
! [A: $tType,A4: set @ A] :
( ( countable_countable @ A @ A4 )
=> ( ord_less_eq @ ( set @ A ) @ A4 @ ( image2 @ nat @ A @ ( counta4804993851260445106t_into @ A @ A4 ) @ ( top_top @ ( set @ nat ) ) ) ) ) ).
% subset_range_from_nat_into
thf(fact_7033_sort__conv__fold,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A] :
( ( linorder_sort_key @ A @ A
@ ^ [X4: A] : X4
@ Xs )
= ( fold @ A @ ( list @ A )
@ ( linorder_insort_key @ A @ A
@ ^ [X4: A] : X4 )
@ Xs
@ ( nil @ A ) ) ) ) ).
% sort_conv_fold
thf(fact_7034_bij__betw__from__nat__into__finite,axiom,
! [A: $tType,S2: set @ A] :
( ( finite_finite2 @ A @ S2 )
=> ( bij_betw @ nat @ A @ ( counta4804993851260445106t_into @ A @ S2 ) @ ( set_ord_lessThan @ nat @ ( finite_card @ A @ S2 ) ) @ S2 ) ) ).
% bij_betw_from_nat_into_finite
thf(fact_7035_bij__betw__from__nat__into,axiom,
! [A: $tType,S2: set @ A] :
( ( countable_countable @ A @ S2 )
=> ( ~ ( finite_finite2 @ A @ S2 )
=> ( bij_betw @ nat @ A @ ( counta4804993851260445106t_into @ A @ S2 ) @ ( top_top @ ( set @ nat ) ) @ S2 ) ) ) ).
% bij_betw_from_nat_into
thf(fact_7036_sort__key__def,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ( ( linorder_sort_key @ B @ A )
= ( ^ [F3: B > A,Xs3: list @ B] : ( foldr @ B @ ( list @ B ) @ ( linorder_insort_key @ B @ A @ F3 ) @ Xs3 @ ( nil @ B ) ) ) ) ) ).
% sort_key_def
thf(fact_7037_Bleast__code,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,P: A > $o] :
( ( bleast @ A @ ( set2 @ A @ Xs ) @ P )
= ( case_list @ A @ A @ ( abort_Bleast @ A @ ( set2 @ A @ Xs ) @ P )
@ ^ [X4: A,Xs3: list @ A] : X4
@ ( filter2 @ A @ P
@ ( linorder_sort_key @ A @ A
@ ^ [X4: A] : X4
@ Xs ) ) ) ) ) ).
% Bleast_code
thf(fact_7038_inter__coset__fold,axiom,
! [A: $tType,A4: set @ A,Xs: list @ A] :
( ( inf_inf @ ( set @ A ) @ A4 @ ( coset @ A @ Xs ) )
= ( fold @ A @ ( set @ A ) @ ( remove @ A ) @ Xs @ A4 ) ) ).
% inter_coset_fold
thf(fact_7039_UNIV__coset,axiom,
! [A: $tType] :
( ( top_top @ ( set @ A ) )
= ( coset @ A @ ( nil @ A ) ) ) ).
% UNIV_coset
thf(fact_7040_subset__code_I2_J,axiom,
! [B: $tType,A4: set @ B,Ys: list @ B] :
( ( ord_less_eq @ ( set @ B ) @ A4 @ ( coset @ B @ Ys ) )
= ( ! [X4: B] :
( ( member @ B @ X4 @ ( set2 @ B @ Ys ) )
=> ~ ( member @ B @ X4 @ A4 ) ) ) ) ).
% subset_code(2)
thf(fact_7041_coset__def,axiom,
! [A: $tType] :
( ( coset @ A )
= ( ^ [Xs3: list @ A] : ( uminus_uminus @ ( set @ A ) @ ( set2 @ A @ Xs3 ) ) ) ) ).
% coset_def
thf(fact_7042_compl__coset,axiom,
! [A: $tType,Xs: list @ A] :
( ( uminus_uminus @ ( set @ A ) @ ( coset @ A @ Xs ) )
= ( set2 @ A @ Xs ) ) ).
% compl_coset
thf(fact_7043_insert__code_I2_J,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( insert2 @ A @ X @ ( coset @ A @ Xs ) )
= ( coset @ A @ ( removeAll @ A @ X @ Xs ) ) ) ).
% insert_code(2)
thf(fact_7044_union__coset__filter,axiom,
! [A: $tType,Xs: list @ A,A4: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( coset @ A @ Xs ) @ A4 )
= ( coset @ A
@ ( filter2 @ A
@ ^ [X4: A] :
~ ( member @ A @ X4 @ A4 )
@ Xs ) ) ) ).
% union_coset_filter
thf(fact_7045_subset__code_I3_J,axiom,
! [C: $tType] :
~ ( ord_less_eq @ ( set @ C ) @ ( coset @ C @ ( nil @ C ) ) @ ( set2 @ C @ ( nil @ C ) ) ) ).
% subset_code(3)
thf(fact_7046_minus__coset__filter,axiom,
! [A: $tType,A4: set @ A,Xs: list @ A] :
( ( minus_minus @ ( set @ A ) @ A4 @ ( coset @ A @ Xs ) )
= ( set2 @ A
@ ( filter2 @ A
@ ^ [X4: A] : ( member @ A @ X4 @ A4 )
@ Xs ) ) ) ).
% minus_coset_filter
thf(fact_7047_comp__fun__idem__on__def,axiom,
! [B: $tType,A: $tType] :
( ( finite673082921795544331dem_on @ A @ B )
= ( ^ [S6: set @ A,F3: A > B > B] :
( ( finite4664212375090638736ute_on @ A @ B @ S6 @ F3 )
& ( finite4980608107308702382axioms @ A @ B @ S6 @ F3 ) ) ) ) ).
% comp_fun_idem_on_def
thf(fact_7048_comp__fun__idem__on_Ointro,axiom,
! [B: $tType,A: $tType,S2: set @ A,F2: A > B > B] :
( ( finite4664212375090638736ute_on @ A @ B @ S2 @ F2 )
=> ( ( finite4980608107308702382axioms @ A @ B @ S2 @ F2 )
=> ( finite673082921795544331dem_on @ A @ B @ S2 @ F2 ) ) ) ).
% comp_fun_idem_on.intro
thf(fact_7049_comp__fun__idem__on__axioms_Ointro,axiom,
! [B: $tType,A: $tType,S2: set @ A,F2: A > B > B] :
( ! [X5: A] :
( ( member @ A @ X5 @ S2 )
=> ( ( comp @ B @ B @ B @ ( F2 @ X5 ) @ ( F2 @ X5 ) )
= ( F2 @ X5 ) ) )
=> ( finite4980608107308702382axioms @ A @ B @ S2 @ F2 ) ) ).
% comp_fun_idem_on_axioms.intro
thf(fact_7050_comp__fun__idem__on__axioms__def,axiom,
! [B: $tType,A: $tType] :
( ( finite4980608107308702382axioms @ A @ B )
= ( ^ [S6: set @ A,F3: A > B > B] :
! [X4: A] :
( ( member @ A @ X4 @ S6 )
=> ( ( comp @ B @ B @ B @ ( F3 @ X4 ) @ ( F3 @ X4 ) )
= ( F3 @ X4 ) ) ) ) ) ).
% comp_fun_idem_on_axioms_def
thf(fact_7051_comp__fun__idem__on_Oaxioms_I2_J,axiom,
! [B: $tType,A: $tType,S2: set @ A,F2: A > B > B] :
( ( finite673082921795544331dem_on @ A @ B @ S2 @ F2 )
=> ( finite4980608107308702382axioms @ A @ B @ S2 @ F2 ) ) ).
% comp_fun_idem_on.axioms(2)
thf(fact_7052_complete__uniform,axiom,
! [A: $tType] :
( ( topolo7287701948861334536_space @ A )
=> ( ( topolo2479028161051973599mplete @ A )
= ( ^ [S6: set @ A] :
! [F10: filter @ A] :
( ( ord_less_eq @ ( filter @ A ) @ F10 @ ( principal @ A @ S6 ) )
=> ( ( F10
!= ( bot_bot @ ( filter @ A ) ) )
=> ( ( topolo6773858410816713723filter @ A @ F10 )
=> ? [X4: A] :
( ( member @ A @ X4 @ S6 )
& ( ord_less_eq @ ( filter @ A ) @ F10 @ ( topolo7230453075368039082e_nhds @ A @ X4 ) ) ) ) ) ) ) ) ) ).
% complete_uniform
thf(fact_7053_shuffles_Opelims,axiom,
! [A: $tType,X: list @ A,Xa2: list @ A,Y: set @ ( list @ A )] :
( ( ( shuffles @ A @ X @ Xa2 )
= Y )
=> ( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( shuffles_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Xa2 ) )
=> ( ( ( X
= ( nil @ A ) )
=> ( ( Y
= ( insert2 @ ( list @ A ) @ Xa2 @ ( bot_bot @ ( set @ ( list @ A ) ) ) ) )
=> ~ ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( shuffles_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Xa2 ) ) ) )
=> ( ( ( Xa2
= ( nil @ A ) )
=> ( ( Y
= ( insert2 @ ( list @ A ) @ X @ ( bot_bot @ ( set @ ( list @ A ) ) ) ) )
=> ~ ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( shuffles_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ ( nil @ A ) ) ) ) )
=> ~ ! [X5: A,Xs2: list @ A] :
( ( X
= ( cons @ A @ X5 @ Xs2 ) )
=> ! [Y3: A,Ys4: list @ A] :
( ( Xa2
= ( cons @ A @ Y3 @ Ys4 ) )
=> ( ( Y
= ( sup_sup @ ( set @ ( list @ A ) ) @ ( image2 @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X5 ) @ ( shuffles @ A @ Xs2 @ ( cons @ A @ Y3 @ Ys4 ) ) ) @ ( image2 @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ Y3 ) @ ( shuffles @ A @ ( cons @ A @ X5 @ Xs2 ) @ Ys4 ) ) ) )
=> ~ ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( shuffles_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X5 @ Xs2 ) @ ( cons @ A @ Y3 @ Ys4 ) ) ) ) ) ) ) ) ) ) ).
% shuffles.pelims
thf(fact_7054_shuffles_Opinduct,axiom,
! [A: $tType,A0: list @ A,A1: list @ A,P: ( list @ A ) > ( list @ A ) > $o] :
( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( shuffles_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ A0 @ A1 ) )
=> ( ! [Ys4: list @ A] :
( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( shuffles_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Ys4 ) )
=> ( P @ ( nil @ A ) @ Ys4 ) )
=> ( ! [Xs2: list @ A] :
( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( shuffles_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ ( nil @ A ) ) )
=> ( P @ Xs2 @ ( nil @ A ) ) )
=> ( ! [X5: A,Xs2: list @ A,Y3: A,Ys4: list @ A] :
( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( shuffles_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X5 @ Xs2 ) @ ( cons @ A @ Y3 @ Ys4 ) ) )
=> ( ( P @ Xs2 @ ( cons @ A @ Y3 @ Ys4 ) )
=> ( ( P @ ( cons @ A @ X5 @ Xs2 ) @ Ys4 )
=> ( P @ ( cons @ A @ X5 @ Xs2 ) @ ( cons @ A @ Y3 @ Ys4 ) ) ) ) )
=> ( P @ A0 @ A1 ) ) ) ) ) ).
% shuffles.pinduct
thf(fact_7055_shuffles_Opsimps_I1_J,axiom,
! [A: $tType,Ys: list @ A] :
( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( shuffles_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Ys ) )
=> ( ( shuffles @ A @ ( nil @ A ) @ Ys )
= ( insert2 @ ( list @ A ) @ Ys @ ( bot_bot @ ( set @ ( list @ A ) ) ) ) ) ) ).
% shuffles.psimps(1)
thf(fact_7056_shuffles_Opsimps_I2_J,axiom,
! [A: $tType,Xs: list @ A] :
( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( shuffles_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ ( nil @ A ) ) )
=> ( ( shuffles @ A @ Xs @ ( nil @ A ) )
= ( insert2 @ ( list @ A ) @ Xs @ ( bot_bot @ ( set @ ( list @ A ) ) ) ) ) ) ).
% shuffles.psimps(2)
thf(fact_7057_shuffles_Opsimps_I3_J,axiom,
! [A: $tType,X: A,Xs: list @ A,Y: A,Ys: list @ A] :
( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( shuffles_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y @ Ys ) ) )
=> ( ( shuffles @ A @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y @ Ys ) )
= ( sup_sup @ ( set @ ( list @ A ) ) @ ( image2 @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X ) @ ( shuffles @ A @ Xs @ ( cons @ A @ Y @ Ys ) ) ) @ ( image2 @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ Y ) @ ( shuffles @ A @ ( cons @ A @ X @ Xs ) @ Ys ) ) ) ) ) ).
% shuffles.psimps(3)
thf(fact_7058_splice_Opinduct,axiom,
! [A: $tType,A0: list @ A,A1: list @ A,P: ( list @ A ) > ( list @ A ) > $o] :
( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( splice_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ A0 @ A1 ) )
=> ( ! [Ys4: list @ A] :
( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( splice_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Ys4 ) )
=> ( P @ ( nil @ A ) @ Ys4 ) )
=> ( ! [X5: A,Xs2: list @ A,Ys4: list @ A] :
( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( splice_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X5 @ Xs2 ) @ Ys4 ) )
=> ( ( P @ Ys4 @ Xs2 )
=> ( P @ ( cons @ A @ X5 @ Xs2 ) @ Ys4 ) ) )
=> ( P @ A0 @ A1 ) ) ) ) ).
% splice.pinduct
thf(fact_7059_Field__insert,axiom,
! [A: $tType,A3: A,B3: A,R2: set @ ( product_prod @ A @ A )] :
( ( field2 @ A @ ( insert2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B3 ) @ R2 ) )
= ( sup_sup @ ( set @ A ) @ ( insert2 @ A @ A3 @ ( insert2 @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( field2 @ A @ R2 ) ) ) ).
% Field_insert
thf(fact_7060_Field__empty,axiom,
! [A: $tType] :
( ( field2 @ A @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Field_empty
thf(fact_7061_finite__Field,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( finite_finite2 @ ( product_prod @ A @ A ) @ R2 )
=> ( finite_finite2 @ A @ ( field2 @ A @ R2 ) ) ) ).
% finite_Field
thf(fact_7062_mono__Field,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),S: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ S )
=> ( ord_less_eq @ ( set @ A ) @ ( field2 @ A @ R2 ) @ ( field2 @ A @ S ) ) ) ).
% mono_Field
thf(fact_7063_Field__Restr__subset,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A4: set @ A] :
( ord_less_eq @ ( set @ A )
@ ( field2 @ A
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R2
@ ( product_Sigma @ A @ A @ A4
@ ^ [Uu3: A] : A4 ) ) )
@ A4 ) ).
% Field_Restr_subset
thf(fact_7064_Field__natLeq__on,axiom,
! [N2: nat] :
( ( field2 @ nat
@ ( collect @ ( product_prod @ nat @ nat )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [X4: nat,Y6: nat] :
( ( ord_less @ nat @ X4 @ N2 )
& ( ord_less @ nat @ Y6 @ N2 )
& ( ord_less_eq @ nat @ X4 @ Y6 ) ) ) ) )
= ( collect @ nat
@ ^ [X4: nat] : ( ord_less @ nat @ X4 @ N2 ) ) ) ).
% Field_natLeq_on
thf(fact_7065_Refl__Field__Restr2,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A4: set @ A] :
( ( refl_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( ord_less_eq @ ( set @ A ) @ A4 @ ( field2 @ A @ R2 ) )
=> ( ( field2 @ A
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R2
@ ( product_Sigma @ A @ A @ A4
@ ^ [Uu3: A] : A4 ) ) )
= A4 ) ) ) ).
% Refl_Field_Restr2
thf(fact_7066_Total__subset__Id,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( total_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ ( id2 @ A ) )
=> ( ( R2
= ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
| ? [A6: A] :
( R2
= ( insert2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A6 @ A6 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ) ) ) ).
% Total_subset_Id
thf(fact_7067_underS__Field3,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A3: A] :
( ( ( field2 @ A @ R2 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ord_less @ ( set @ A ) @ ( order_underS @ A @ R2 @ A3 ) @ ( field2 @ A @ R2 ) ) ) ).
% underS_Field3
thf(fact_7068_underS__empty,axiom,
! [A: $tType,A3: A,R2: set @ ( product_prod @ A @ A )] :
( ~ ( member @ A @ A3 @ ( field2 @ A @ R2 ) )
=> ( ( order_underS @ A @ R2 @ A3 )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% underS_empty
thf(fact_7069_Order__Relation_OunderS__Field,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A3: A] : ( ord_less_eq @ ( set @ A ) @ ( order_underS @ A @ R2 @ A3 ) @ ( field2 @ A @ R2 ) ) ).
% Order_Relation.underS_Field
thf(fact_7070_underS__Field2,axiom,
! [A: $tType,A3: A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ A @ A3 @ ( field2 @ A @ R2 ) )
=> ( ord_less @ ( set @ A ) @ ( order_underS @ A @ R2 @ A3 ) @ ( field2 @ A @ R2 ) ) ) ).
% underS_Field2
thf(fact_7071_splice_Opelims,axiom,
! [A: $tType,X: list @ A,Xa2: list @ A,Y: list @ A] :
( ( ( splice @ A @ X @ Xa2 )
= Y )
=> ( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( splice_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Xa2 ) )
=> ( ( ( X
= ( nil @ A ) )
=> ( ( Y = Xa2 )
=> ~ ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( splice_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Xa2 ) ) ) )
=> ~ ! [X5: A,Xs2: list @ A] :
( ( X
= ( cons @ A @ X5 @ Xs2 ) )
=> ( ( Y
= ( cons @ A @ X5 @ ( splice @ A @ Xa2 @ Xs2 ) ) )
=> ~ ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( splice_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X5 @ Xs2 ) @ Xa2 ) ) ) ) ) ) ) ).
% splice.pelims
thf(fact_7072_Refl__under__underS,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A3: A] :
( ( refl_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( member @ A @ A3 @ ( field2 @ A @ R2 ) )
=> ( ( order_under @ A @ R2 @ A3 )
= ( sup_sup @ ( set @ A ) @ ( order_underS @ A @ R2 @ A3 ) @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% Refl_under_underS
thf(fact_7073_split__Nil__iff,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( ( splice @ A @ Xs @ Ys )
= ( nil @ A ) )
= ( ( Xs
= ( nil @ A ) )
& ( Ys
= ( nil @ A ) ) ) ) ).
% split_Nil_iff
thf(fact_7074_splice__Nil2,axiom,
! [A: $tType,Xs: list @ A] :
( ( splice @ A @ Xs @ ( nil @ A ) )
= Xs ) ).
% splice_Nil2
thf(fact_7075_splice__in__shuffles,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] : ( member @ ( list @ A ) @ ( splice @ A @ Xs @ Ys ) @ ( shuffles @ A @ Xs @ Ys ) ) ).
% splice_in_shuffles
thf(fact_7076_length__splice,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( size_size @ ( list @ A ) @ ( splice @ A @ Xs @ Ys ) )
= ( plus_plus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ A ) @ Ys ) ) ) ).
% length_splice
thf(fact_7077_splice__replicate,axiom,
! [A: $tType,M2: nat,X: A,N2: nat] :
( ( splice @ A @ ( replicate @ A @ M2 @ X ) @ ( replicate @ A @ N2 @ X ) )
= ( replicate @ A @ ( plus_plus @ nat @ M2 @ N2 ) @ X ) ) ).
% splice_replicate
thf(fact_7078_under__Field,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A3: A] : ( ord_less_eq @ ( set @ A ) @ ( order_under @ A @ R2 @ A3 ) @ ( field2 @ A @ R2 ) ) ).
% under_Field
thf(fact_7079_underS__subset__under,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A3: A] : ( ord_less_eq @ ( set @ A ) @ ( order_underS @ A @ R2 @ A3 ) @ ( order_under @ A @ R2 @ A3 ) ) ).
% underS_subset_under
thf(fact_7080_splice_Osimps_I2_J,axiom,
! [A: $tType,X: A,Xs: list @ A,Ys: list @ A] :
( ( splice @ A @ ( cons @ A @ X @ Xs ) @ Ys )
= ( cons @ A @ X @ ( splice @ A @ Ys @ Xs ) ) ) ).
% splice.simps(2)
thf(fact_7081_splice_Osimps_I1_J,axiom,
! [A: $tType,Ys: list @ A] :
( ( splice @ A @ ( nil @ A ) @ Ys )
= Ys ) ).
% splice.simps(1)
thf(fact_7082_splice_Oelims,axiom,
! [A: $tType,X: list @ A,Xa2: list @ A,Y: list @ A] :
( ( ( splice @ A @ X @ Xa2 )
= Y )
=> ( ( ( X
= ( nil @ A ) )
=> ( Y != Xa2 ) )
=> ~ ! [X5: A,Xs2: list @ A] :
( ( X
= ( cons @ A @ X5 @ Xs2 ) )
=> ( Y
!= ( cons @ A @ X5 @ ( splice @ A @ Xa2 @ Xs2 ) ) ) ) ) ) ).
% splice.elims
thf(fact_7083_splice_Opsimps_I2_J,axiom,
! [A: $tType,X: A,Xs: list @ A,Ys: list @ A] :
( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( splice_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X @ Xs ) @ Ys ) )
=> ( ( splice @ A @ ( cons @ A @ X @ Xs ) @ Ys )
= ( cons @ A @ X @ ( splice @ A @ Ys @ Xs ) ) ) ) ).
% splice.psimps(2)
thf(fact_7084_splice_Opsimps_I1_J,axiom,
! [A: $tType,Ys: list @ A] :
( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( splice_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Ys ) )
=> ( ( splice @ A @ ( nil @ A ) @ Ys )
= Ys ) ) ).
% splice.psimps(1)
thf(fact_7085_linear__order__on__singleton,axiom,
! [A: $tType,X: A] : ( order_679001287576687338der_on @ A @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) @ ( insert2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ X ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ).
% linear_order_on_singleton
thf(fact_7086_bsqr__def,axiom,
! [A: $tType] :
( ( bNF_Wellorder_bsqr @ A )
= ( ^ [R4: set @ ( product_prod @ A @ A )] :
( collect @ ( product_prod @ ( product_prod @ A @ A ) @ ( product_prod @ A @ A ) )
@ ( product_case_prod @ ( product_prod @ A @ A ) @ ( product_prod @ A @ A ) @ $o
@ ( product_case_prod @ A @ A @ ( ( product_prod @ A @ A ) > $o )
@ ^ [A12: A,A23: A] :
( product_case_prod @ A @ A @ $o
@ ^ [B14: A,B23: A] :
( ( ord_less_eq @ ( set @ A ) @ ( insert2 @ A @ A12 @ ( insert2 @ A @ A23 @ ( insert2 @ A @ B14 @ ( insert2 @ A @ B23 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) @ ( field2 @ A @ R4 ) )
& ( ( ( A12 = B14 )
& ( A23 = B23 ) )
| ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( bNF_We1388413361240627857o_max2 @ A @ R4 @ A12 @ A23 ) @ ( bNF_We1388413361240627857o_max2 @ A @ R4 @ B14 @ B23 ) ) @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R4 @ ( id2 @ A ) ) )
| ( ( ( bNF_We1388413361240627857o_max2 @ A @ R4 @ A12 @ A23 )
= ( bNF_We1388413361240627857o_max2 @ A @ R4 @ B14 @ B23 ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A12 @ B14 ) @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R4 @ ( id2 @ A ) ) ) )
| ( ( ( bNF_We1388413361240627857o_max2 @ A @ R4 @ A12 @ A23 )
= ( bNF_We1388413361240627857o_max2 @ A @ R4 @ B14 @ B23 ) )
& ( A12 = B14 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A23 @ B23 ) @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R4 @ ( id2 @ A ) ) ) ) ) ) ) ) ) ) ) ) ).
% bsqr_def
thf(fact_7087_lnear__order__on__empty,axiom,
! [A: $tType] : ( order_679001287576687338der_on @ A @ ( bot_bot @ ( set @ A ) ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ).
% lnear_order_on_empty
thf(fact_7088_underS__incl__iff,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A3: A,B3: A] :
( ( order_679001287576687338der_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( member @ A @ A3 @ ( field2 @ A @ R2 ) )
=> ( ( member @ A @ B3 @ ( field2 @ A @ R2 ) )
=> ( ( ord_less_eq @ ( set @ A ) @ ( order_underS @ A @ R2 @ A3 ) @ ( order_underS @ A @ R2 @ B3 ) )
= ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B3 ) @ R2 ) ) ) ) ) ).
% underS_incl_iff
thf(fact_7089_finite__Linear__order__induct,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),X: A,P: A > $o] :
( ( order_679001287576687338der_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( member @ A @ X @ ( field2 @ A @ R2 ) )
=> ( ( finite_finite2 @ ( product_prod @ A @ A ) @ R2 )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ ( field2 @ A @ R2 ) )
=> ( ! [Y4: A] :
( ( member @ A @ Y4 @ ( order_aboveS @ A @ R2 @ X5 ) )
=> ( P @ Y4 ) )
=> ( P @ X5 ) ) )
=> ( P @ X ) ) ) ) ) ).
% finite_Linear_order_induct
thf(fact_7090_Linear__order__wf__diff__Id,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( order_679001287576687338der_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( wf @ A @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ ( id2 @ A ) ) )
= ( ! [A7: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A7 @ ( field2 @ A @ R2 ) )
=> ( ( A7
!= ( bot_bot @ ( set @ A ) ) )
=> ? [X4: A] :
( ( member @ A @ X4 @ A7 )
& ! [Y6: A] :
( ( member @ A @ Y6 @ A7 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y6 ) @ R2 ) ) ) ) ) ) ) ) ).
% Linear_order_wf_diff_Id
thf(fact_7091_wf__empty,axiom,
! [A: $tType] : ( wf @ A @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ).
% wf_empty
thf(fact_7092_wf__listrel1__iff,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( wf @ ( list @ A ) @ ( listrel1 @ A @ R2 ) )
= ( wf @ A @ R2 ) ) ).
% wf_listrel1_iff
thf(fact_7093_wf__lex,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( wf @ A @ R2 )
=> ( wf @ ( list @ A ) @ ( lex @ A @ R2 ) ) ) ).
% wf_lex
thf(fact_7094_wf__lenlex,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( wf @ A @ R2 )
=> ( wf @ ( list @ A ) @ ( lenlex @ A @ R2 ) ) ) ).
% wf_lenlex
thf(fact_7095_wf__if__measure,axiom,
! [A: $tType,P: A > $o,F2: A > nat,G: A > A] :
( ! [X5: A] :
( ( P @ X5 )
=> ( ord_less @ nat @ ( F2 @ ( G @ X5 ) ) @ ( F2 @ X5 ) ) )
=> ( wf @ A
@ ( collect @ ( product_prod @ A @ A )
@ ( product_case_prod @ A @ A @ $o
@ ^ [Y6: A,X4: A] :
( ( P @ X4 )
& ( Y6
= ( G @ X4 ) ) ) ) ) ) ) ).
% wf_if_measure
thf(fact_7096_wf,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ( wf @ A @ ( collect @ ( product_prod @ A @ A ) @ ( product_case_prod @ A @ A @ $o @ ( ord_less @ A ) ) ) ) ) ).
% wf
thf(fact_7097_wf__less,axiom,
wf @ nat @ ( collect @ ( product_prod @ nat @ nat ) @ ( product_case_prod @ nat @ nat @ $o @ ( ord_less @ nat ) ) ) ).
% wf_less
thf(fact_7098_wf__int__ge__less__than2,axiom,
! [D2: int] : ( wf @ int @ ( int_ge_less_than2 @ D2 ) ) ).
% wf_int_ge_less_than2
thf(fact_7099_wf__int__ge__less__than,axiom,
! [D2: int] : ( wf @ int @ ( int_ge_less_than @ D2 ) ) ).
% wf_int_ge_less_than
thf(fact_7100_wf__lexn,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),N2: nat] :
( ( wf @ A @ R2 )
=> ( wf @ ( list @ A ) @ ( lexn @ A @ R2 @ N2 ) ) ) ).
% wf_lexn
thf(fact_7101_wf__no__infinite__down__chainE,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),F2: nat > A] :
( ( wf @ A @ R2 )
=> ~ ! [K3: nat] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( F2 @ ( suc @ K3 ) ) @ ( F2 @ K3 ) ) @ R2 ) ) ).
% wf_no_infinite_down_chainE
thf(fact_7102_wf__iff__no__infinite__down__chain,axiom,
! [A: $tType] :
( ( wf @ A )
= ( ^ [R4: set @ ( product_prod @ A @ A )] :
~ ? [F3: nat > A] :
! [I3: nat] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( F3 @ ( suc @ I3 ) ) @ ( F3 @ I3 ) ) @ R4 ) ) ) ).
% wf_iff_no_infinite_down_chain
thf(fact_7103_wfE__min_H,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),Q: set @ A] :
( ( wf @ A @ R )
=> ( ( Q
!= ( bot_bot @ ( set @ A ) ) )
=> ~ ! [Z: A] :
( ( member @ A @ Z @ Q )
=> ~ ! [Y4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Z ) @ R )
=> ~ ( member @ A @ Y4 @ Q ) ) ) ) ) ).
% wfE_min'
thf(fact_7104_wf__no__loop,axiom,
! [B: $tType,R: set @ ( product_prod @ B @ B )] :
( ( ( relcomp @ B @ B @ B @ R @ R )
= ( bot_bot @ ( set @ ( product_prod @ B @ B ) ) ) )
=> ( wf @ B @ R ) ) ).
% wf_no_loop
thf(fact_7105_wf__bounded__measure,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),Ub: A > nat,F2: A > nat] :
( ! [A6: A,B5: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B5 @ A6 ) @ R2 )
=> ( ( ord_less_eq @ nat @ ( Ub @ B5 ) @ ( Ub @ A6 ) )
& ( ord_less_eq @ nat @ ( F2 @ B5 ) @ ( Ub @ A6 ) )
& ( ord_less @ nat @ ( F2 @ A6 ) @ ( F2 @ B5 ) ) ) )
=> ( wf @ A @ R2 ) ) ).
% wf_bounded_measure
thf(fact_7106_wf__linord__ex__has__least,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ A ),P: B > $o,K: B,M2: B > A] :
( ( wf @ A @ R2 )
=> ( ! [X5: A,Y3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X5 @ Y3 ) @ ( transitive_trancl @ A @ R2 ) )
= ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ X5 ) @ ( transitive_rtrancl @ A @ R2 ) ) ) )
=> ( ( P @ K )
=> ? [X5: B] :
( ( P @ X5 )
& ! [Y4: B] :
( ( P @ Y4 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( M2 @ X5 ) @ ( M2 @ Y4 ) ) @ ( transitive_rtrancl @ A @ R2 ) ) ) ) ) ) ) ).
% wf_linord_ex_has_least
thf(fact_7107_wf__eq__minimal2,axiom,
! [A: $tType] :
( ( wf @ A )
= ( ^ [R4: set @ ( product_prod @ A @ A )] :
! [A7: set @ A] :
( ( ( ord_less_eq @ ( set @ A ) @ A7 @ ( field2 @ A @ R4 ) )
& ( A7
!= ( bot_bot @ ( set @ A ) ) ) )
=> ? [X4: A] :
( ( member @ A @ X4 @ A7 )
& ! [Y6: A] :
( ( member @ A @ Y6 @ A7 )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y6 @ X4 ) @ R4 ) ) ) ) ) ) ).
% wf_eq_minimal2
thf(fact_7108_wf__bounded__set,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ A ),Ub: A > ( set @ B ),F2: A > ( set @ B )] :
( ! [A6: A,B5: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B5 @ A6 ) @ R2 )
=> ( ( finite_finite2 @ B @ ( Ub @ A6 ) )
& ( ord_less_eq @ ( set @ B ) @ ( Ub @ B5 ) @ ( Ub @ A6 ) )
& ( ord_less_eq @ ( set @ B ) @ ( F2 @ B5 ) @ ( Ub @ A6 ) )
& ( ord_less @ ( set @ B ) @ ( F2 @ A6 ) @ ( F2 @ B5 ) ) ) )
=> ( wf @ A @ R2 ) ) ).
% wf_bounded_set
thf(fact_7109_finite__subset__wf,axiom,
! [A: $tType,A4: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( wf @ ( set @ A )
@ ( collect @ ( product_prod @ ( set @ A ) @ ( set @ A ) )
@ ( product_case_prod @ ( set @ A ) @ ( set @ A ) @ $o
@ ^ [X6: set @ A,Y8: set @ A] :
( ( ord_less @ ( set @ A ) @ X6 @ Y8 )
& ( ord_less_eq @ ( set @ A ) @ Y8 @ A4 ) ) ) ) ) ) ).
% finite_subset_wf
thf(fact_7110_partial__order__on__well__order__on,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A4: set @ A] :
( ( finite_finite2 @ ( product_prod @ A @ A ) @ R2 )
=> ( ( order_7125193373082350890der_on @ A @ A4 @ R2 )
=> ( wf @ A @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ ( id2 @ A ) ) ) ) ) ).
% partial_order_on_well_order_on
thf(fact_7111_comp__fun__commute__on_Ofold__graph__insertE__aux,axiom,
! [A: $tType,B: $tType,S2: set @ A,F2: A > B > B,A4: set @ A,Z3: B,Y: B,A3: A] :
( ( finite4664212375090638736ute_on @ A @ B @ S2 @ F2 )
=> ( ( ord_less_eq @ ( set @ A ) @ A4 @ S2 )
=> ( ( finite_fold_graph @ A @ B @ F2 @ Z3 @ A4 @ Y )
=> ( ( member @ A @ A3 @ A4 )
=> ? [Y9: B] :
( ( Y
= ( F2 @ A3 @ Y9 ) )
& ( finite_fold_graph @ A @ B @ F2 @ Z3 @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) @ Y9 ) ) ) ) ) ) ).
% comp_fun_commute_on.fold_graph_insertE_aux
thf(fact_7112_fold__graph_Ocases,axiom,
! [A: $tType,B: $tType,F2: A > B > B,Z3: B,A1: set @ A,A22: B] :
( ( finite_fold_graph @ A @ B @ F2 @ Z3 @ A1 @ A22 )
=> ( ( ( A1
= ( bot_bot @ ( set @ A ) ) )
=> ( A22 != Z3 ) )
=> ~ ! [X5: A,A8: set @ A] :
( ( A1
= ( insert2 @ A @ X5 @ A8 ) )
=> ! [Y3: B] :
( ( A22
= ( F2 @ X5 @ Y3 ) )
=> ( ~ ( member @ A @ X5 @ A8 )
=> ~ ( finite_fold_graph @ A @ B @ F2 @ Z3 @ A8 @ Y3 ) ) ) ) ) ) ).
% fold_graph.cases
thf(fact_7113_fold__graph_Osimps,axiom,
! [B: $tType,A: $tType] :
( ( finite_fold_graph @ A @ B )
= ( ^ [F3: A > B > B,Z6: B,A12: set @ A,A23: B] :
( ( ( A12
= ( bot_bot @ ( set @ A ) ) )
& ( A23 = Z6 ) )
| ? [X4: A,A7: set @ A,Y6: B] :
( ( A12
= ( insert2 @ A @ X4 @ A7 ) )
& ( A23
= ( F3 @ X4 @ Y6 ) )
& ~ ( member @ A @ X4 @ A7 )
& ( finite_fold_graph @ A @ B @ F3 @ Z6 @ A7 @ Y6 ) ) ) ) ) ).
% fold_graph.simps
thf(fact_7114_fold__graph_OinsertI,axiom,
! [A: $tType,B: $tType,X: A,A4: set @ A,F2: A > B > B,Z3: B,Y: B] :
( ~ ( member @ A @ X @ A4 )
=> ( ( finite_fold_graph @ A @ B @ F2 @ Z3 @ A4 @ Y )
=> ( finite_fold_graph @ A @ B @ F2 @ Z3 @ ( insert2 @ A @ X @ A4 ) @ ( F2 @ X @ Y ) ) ) ) ).
% fold_graph.insertI
thf(fact_7115_empty__fold__graphE,axiom,
! [A: $tType,B: $tType,F2: A > B > B,Z3: B,X: B] :
( ( finite_fold_graph @ A @ B @ F2 @ Z3 @ ( bot_bot @ ( set @ A ) ) @ X )
=> ( X = Z3 ) ) ).
% empty_fold_graphE
thf(fact_7116_fold__graph_OemptyI,axiom,
! [A: $tType,B: $tType,F2: A > B > B,Z3: B] : ( finite_fold_graph @ A @ B @ F2 @ Z3 @ ( bot_bot @ ( set @ A ) ) @ Z3 ) ).
% fold_graph.emptyI
thf(fact_7117_fold__graph__closed__eq,axiom,
! [B: $tType,A: $tType,A4: set @ A,B2: set @ B,F2: A > B > B,G: A > B > B,Z3: B] :
( ! [A6: A,B5: B] :
( ( member @ A @ A6 @ A4 )
=> ( ( member @ B @ B5 @ B2 )
=> ( ( F2 @ A6 @ B5 )
= ( G @ A6 @ B5 ) ) ) )
=> ( ! [A6: A,B5: B] :
( ( member @ A @ A6 @ A4 )
=> ( ( member @ B @ B5 @ B2 )
=> ( member @ B @ ( G @ A6 @ B5 ) @ B2 ) ) )
=> ( ( member @ B @ Z3 @ B2 )
=> ( ( finite_fold_graph @ A @ B @ F2 @ Z3 @ A4 )
= ( finite_fold_graph @ A @ B @ G @ Z3 @ A4 ) ) ) ) ) ).
% fold_graph_closed_eq
thf(fact_7118_fold__graph__closed__lemma,axiom,
! [A: $tType,B: $tType,G: A > B > B,Z3: B,A4: set @ A,X: B,B2: set @ B,F2: A > B > B] :
( ( finite_fold_graph @ A @ B @ G @ Z3 @ A4 @ X )
=> ( ! [A6: A,B5: B] :
( ( member @ A @ A6 @ A4 )
=> ( ( member @ B @ B5 @ B2 )
=> ( ( F2 @ A6 @ B5 )
= ( G @ A6 @ B5 ) ) ) )
=> ( ! [A6: A,B5: B] :
( ( member @ A @ A6 @ A4 )
=> ( ( member @ B @ B5 @ B2 )
=> ( member @ B @ ( G @ A6 @ B5 ) @ B2 ) ) )
=> ( ( member @ B @ Z3 @ B2 )
=> ( ( finite_fold_graph @ A @ B @ F2 @ Z3 @ A4 @ X )
& ( member @ B @ X @ B2 ) ) ) ) ) ) ).
% fold_graph_closed_lemma
thf(fact_7119_finite__imp__fold__graph,axiom,
! [A: $tType,B: $tType,A4: set @ A,F2: A > B > B,Z3: B] :
( ( finite_finite2 @ A @ A4 )
=> ? [X_12: B] : ( finite_fold_graph @ A @ B @ F2 @ Z3 @ A4 @ X_12 ) ) ).
% finite_imp_fold_graph
thf(fact_7120_comp__fun__commute__on_Ofold__graph__determ,axiom,
! [A: $tType,B: $tType,S2: set @ A,F2: A > B > B,A4: set @ A,Z3: B,X: B,Y: B] :
( ( finite4664212375090638736ute_on @ A @ B @ S2 @ F2 )
=> ( ( ord_less_eq @ ( set @ A ) @ A4 @ S2 )
=> ( ( finite_fold_graph @ A @ B @ F2 @ Z3 @ A4 @ X )
=> ( ( finite_fold_graph @ A @ B @ F2 @ Z3 @ A4 @ Y )
=> ( Y = X ) ) ) ) ) ).
% comp_fun_commute_on.fold_graph_determ
thf(fact_7121_comp__fun__commute__on_Ofold__graph__finite,axiom,
! [B: $tType,A: $tType,S2: set @ A,F2: A > B > B,Z3: B,A4: set @ A,Y: B] :
( ( finite4664212375090638736ute_on @ A @ B @ S2 @ F2 )
=> ( ( finite_fold_graph @ A @ B @ F2 @ Z3 @ A4 @ Y )
=> ( finite_finite2 @ A @ A4 ) ) ) ).
% comp_fun_commute_on.fold_graph_finite
thf(fact_7122_partial__order__on__empty,axiom,
! [A: $tType] : ( order_7125193373082350890der_on @ A @ ( bot_bot @ ( set @ A ) ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ).
% partial_order_on_empty
thf(fact_7123_fold__graph__image,axiom,
! [C: $tType,B: $tType,A: $tType,G: A > B,A4: set @ A,F2: B > C > C,Z3: C] :
( ( inj_on @ A @ B @ G @ A4 )
=> ( ( finite_fold_graph @ B @ C @ F2 @ Z3 @ ( image2 @ A @ B @ G @ A4 ) )
= ( finite_fold_graph @ A @ C @ ( comp @ B @ ( C > C ) @ A @ F2 @ G ) @ Z3 @ A4 ) ) ) ).
% fold_graph_image
thf(fact_7124_comp__fun__commute__on_Ofold__graph__insertE,axiom,
! [A: $tType,B: $tType,S2: set @ A,F2: A > B > B,X: A,A4: set @ A,Z3: B,V2: B] :
( ( finite4664212375090638736ute_on @ A @ B @ S2 @ F2 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( insert2 @ A @ X @ A4 ) @ S2 )
=> ( ( finite_fold_graph @ A @ B @ F2 @ Z3 @ ( insert2 @ A @ X @ A4 ) @ V2 )
=> ( ~ ( member @ A @ X @ A4 )
=> ~ ! [Y3: B] :
( ( V2
= ( F2 @ X @ Y3 ) )
=> ~ ( finite_fold_graph @ A @ B @ F2 @ Z3 @ A4 @ Y3 ) ) ) ) ) ) ).
% comp_fun_commute_on.fold_graph_insertE
thf(fact_7125_comp__fun__commute__on_Ofold__equality,axiom,
! [A: $tType,B: $tType,S2: set @ A,F2: A > B > B,A4: set @ A,Z3: B,Y: B] :
( ( finite4664212375090638736ute_on @ A @ B @ S2 @ F2 )
=> ( ( ord_less_eq @ ( set @ A ) @ A4 @ S2 )
=> ( ( finite_fold_graph @ A @ B @ F2 @ Z3 @ A4 @ Y )
=> ( ( finite_fold @ A @ B @ F2 @ Z3 @ A4 )
= Y ) ) ) ) ).
% comp_fun_commute_on.fold_equality
thf(fact_7126_comp__fun__commute__on_Ofold__graph__fold,axiom,
! [B: $tType,A: $tType,S2: set @ A,F2: A > B > B,A4: set @ A,Z3: B] :
( ( finite4664212375090638736ute_on @ A @ B @ S2 @ F2 )
=> ( ( ord_less_eq @ ( set @ A ) @ A4 @ S2 )
=> ( ( finite_finite2 @ A @ A4 )
=> ( finite_fold_graph @ A @ B @ F2 @ Z3 @ A4 @ ( finite_fold @ A @ B @ F2 @ Z3 @ A4 ) ) ) ) ) ).
% comp_fun_commute_on.fold_graph_fold
thf(fact_7127_finite__Partial__order__induct,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),X: A,P: A > $o] :
( ( order_7125193373082350890der_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( member @ A @ X @ ( field2 @ A @ R2 ) )
=> ( ( finite_finite2 @ ( product_prod @ A @ A ) @ R2 )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ ( field2 @ A @ R2 ) )
=> ( ! [Y4: A] :
( ( member @ A @ Y4 @ ( order_aboveS @ A @ R2 @ X5 ) )
=> ( P @ Y4 ) )
=> ( P @ X5 ) ) )
=> ( P @ X ) ) ) ) ) ).
% finite_Partial_order_induct
thf(fact_7128_increasing__Bseq__subseq__iff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A,G: nat > nat] :
( ! [X5: nat,Y3: nat] :
( ( ord_less_eq @ nat @ X5 @ Y3 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( F2 @ X5 ) ) @ ( real_V7770717601297561774m_norm @ A @ ( F2 @ Y3 ) ) ) )
=> ( ( order_strict_mono @ nat @ nat @ G )
=> ( ( bfun @ nat @ A
@ ^ [X4: nat] : ( F2 @ ( G @ X4 ) )
@ ( at_top @ nat ) )
= ( bfun @ nat @ A @ F2 @ ( at_top @ nat ) ) ) ) ) ) ).
% increasing_Bseq_subseq_iff
thf(fact_7129_power_Opower__eq__if,axiom,
! [A: $tType] :
( ( power2 @ A )
= ( ^ [One: A,Times: A > A > A,P6: A,M: nat] :
( if @ A
@ ( M
= ( zero_zero @ nat ) )
@ One
@ ( Times @ P6 @ ( power2 @ A @ One @ Times @ P6 @ ( minus_minus @ nat @ M @ ( one_one @ nat ) ) ) ) ) ) ) ).
% power.power_eq_if
thf(fact_7130_power_Opower_Opower__0,axiom,
! [A: $tType,One2: A,Times2: A > A > A,A3: A] :
( ( power2 @ A @ One2 @ Times2 @ A3 @ ( zero_zero @ nat ) )
= One2 ) ).
% power.power.power_0
thf(fact_7131_infinite__enumerate,axiom,
! [S2: set @ nat] :
( ~ ( finite_finite2 @ nat @ S2 )
=> ? [R3: nat > nat] :
( ( order_strict_mono @ nat @ nat @ R3 )
& ! [N7: nat] : ( member @ nat @ ( R3 @ N7 ) @ S2 ) ) ) ).
% infinite_enumerate
thf(fact_7132_strict__mono__add,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [K: A] :
( order_strict_mono @ A @ A
@ ^ [N: A] : ( plus_plus @ A @ N @ K ) ) ) ).
% strict_mono_add
thf(fact_7133_strict__mono__less,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( order @ B ) )
=> ! [F2: A > B,X: A,Y: A] :
( ( order_strict_mono @ A @ B @ F2 )
=> ( ( ord_less @ B @ ( F2 @ X ) @ ( F2 @ Y ) )
= ( ord_less @ A @ X @ Y ) ) ) ) ).
% strict_mono_less
thf(fact_7134_strict__mono__def,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ( ( order_strict_mono @ A @ B )
= ( ^ [F3: A > B] :
! [X4: A,Y6: A] :
( ( ord_less @ A @ X4 @ Y6 )
=> ( ord_less @ B @ ( F3 @ X4 ) @ ( F3 @ Y6 ) ) ) ) ) ) ).
% strict_mono_def
thf(fact_7135_strict__monoI,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [F2: A > B] :
( ! [X5: A,Y3: A] :
( ( ord_less @ A @ X5 @ Y3 )
=> ( ord_less @ B @ ( F2 @ X5 ) @ ( F2 @ Y3 ) ) )
=> ( order_strict_mono @ A @ B @ F2 ) ) ) ).
% strict_monoI
thf(fact_7136_strict__monoD,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [F2: A > B,X: A,Y: A] :
( ( order_strict_mono @ A @ B @ F2 )
=> ( ( ord_less @ A @ X @ Y )
=> ( ord_less @ B @ ( F2 @ X ) @ ( F2 @ Y ) ) ) ) ) ).
% strict_monoD
thf(fact_7137_strict__mono__eq,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( order @ B ) )
=> ! [F2: A > B,X: A,Y: A] :
( ( order_strict_mono @ A @ B @ F2 )
=> ( ( ( F2 @ X )
= ( F2 @ Y ) )
= ( X = Y ) ) ) ) ).
% strict_mono_eq
thf(fact_7138_power_Opower_Opower__Suc,axiom,
! [A: $tType,One2: A,Times2: A > A > A,A3: A,N2: nat] :
( ( power2 @ A @ One2 @ Times2 @ A3 @ ( suc @ N2 ) )
= ( Times2 @ A3 @ ( power2 @ A @ One2 @ Times2 @ A3 @ N2 ) ) ) ).
% power.power.power_Suc
thf(fact_7139_strict__mono__mono,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [F2: A > B] :
( ( order_strict_mono @ A @ B @ F2 )
=> ( order_mono @ A @ B @ F2 ) ) ) ).
% strict_mono_mono
thf(fact_7140_strict__mono__leD,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [R2: A > B,M2: A,N2: A] :
( ( order_strict_mono @ A @ B @ R2 )
=> ( ( ord_less_eq @ A @ M2 @ N2 )
=> ( ord_less_eq @ B @ ( R2 @ M2 ) @ ( R2 @ N2 ) ) ) ) ) ).
% strict_mono_leD
thf(fact_7141_strict__mono__less__eq,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( order @ B ) )
=> ! [F2: A > B,X: A,Y: A] :
( ( order_strict_mono @ A @ B @ F2 )
=> ( ( ord_less_eq @ B @ ( F2 @ X ) @ ( F2 @ Y ) )
= ( ord_less_eq @ A @ X @ Y ) ) ) ) ).
% strict_mono_less_eq
thf(fact_7142_strict__mono__imp__increasing,axiom,
! [F2: nat > nat,N2: nat] :
( ( order_strict_mono @ nat @ nat @ F2 )
=> ( ord_less_eq @ nat @ N2 @ ( F2 @ N2 ) ) ) ).
% strict_mono_imp_increasing
thf(fact_7143_strict__mono__Suc__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( order_strict_mono @ nat @ A )
= ( ^ [F3: nat > A] :
! [N: nat] : ( ord_less @ A @ ( F3 @ N ) @ ( F3 @ ( suc @ N ) ) ) ) ) ) ).
% strict_mono_Suc_iff
thf(fact_7144_summable__mono__reindex,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [G: nat > nat,F2: nat > A] :
( ( order_strict_mono @ nat @ nat @ G )
=> ( ! [N3: nat] :
( ~ ( member @ nat @ N3 @ ( image2 @ nat @ nat @ G @ ( top_top @ ( set @ nat ) ) ) )
=> ( ( F2 @ N3 )
= ( zero_zero @ A ) ) )
=> ( ( summable @ A
@ ^ [N: nat] : ( F2 @ ( G @ N ) ) )
= ( summable @ A @ F2 ) ) ) ) ) ).
% summable_mono_reindex
thf(fact_7145_sums__mono__reindex,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [G: nat > nat,F2: nat > A,C2: A] :
( ( order_strict_mono @ nat @ nat @ G )
=> ( ! [N3: nat] :
( ~ ( member @ nat @ N3 @ ( image2 @ nat @ nat @ G @ ( top_top @ ( set @ nat ) ) ) )
=> ( ( F2 @ N3 )
= ( zero_zero @ A ) ) )
=> ( ( sums @ A
@ ^ [N: nat] : ( F2 @ ( G @ N ) )
@ C2 )
= ( sums @ A @ F2 @ C2 ) ) ) ) ) ).
% sums_mono_reindex
thf(fact_7146_suminf__mono__reindex,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topological_t2_space @ A ) )
=> ! [G: nat > nat,F2: nat > A] :
( ( order_strict_mono @ nat @ nat @ G )
=> ( ! [N3: nat] :
( ~ ( member @ nat @ N3 @ ( image2 @ nat @ nat @ G @ ( top_top @ ( set @ nat ) ) ) )
=> ( ( F2 @ N3 )
= ( zero_zero @ A ) ) )
=> ( ( suminf @ A
@ ^ [N: nat] : ( F2 @ ( G @ N ) ) )
= ( suminf @ A @ F2 ) ) ) ) ) ).
% suminf_mono_reindex
thf(fact_7147_finite__def,axiom,
! [A: $tType] :
( ( finite_finite2 @ A )
= ( complete_lattice_lfp @ ( ( set @ A ) > $o )
@ ^ [P6: ( set @ A ) > $o,X4: set @ A] :
( ( X4
= ( bot_bot @ ( set @ A ) ) )
| ? [A7: set @ A,A5: A] :
( ( X4
= ( insert2 @ A @ A5 @ A7 ) )
& ( P6 @ A7 ) ) ) ) ) ).
% finite_def
thf(fact_7148_finite__refines__card__le,axiom,
! [A: $tType,A4: set @ A,R: set @ ( product_prod @ A @ A ),S2: set @ ( product_prod @ A @ A )] :
( ( finite_finite2 @ ( set @ A ) @ ( equiv_quotient @ A @ A4 @ R ) )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R @ S2 )
=> ( ( equiv_equiv @ A @ A4 @ R )
=> ( ( equiv_equiv @ A @ A4 @ S2 )
=> ( ord_less_eq @ nat @ ( finite_card @ ( set @ A ) @ ( equiv_quotient @ A @ A4 @ S2 ) ) @ ( finite_card @ ( set @ A ) @ ( equiv_quotient @ A @ A4 @ R ) ) ) ) ) ) ) ).
% finite_refines_card_le
thf(fact_7149_lfp__def,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ( ( complete_lattice_lfp @ A )
= ( ^ [F3: A > A] :
( complete_Inf_Inf @ A
@ ( collect @ A
@ ^ [U2: A] : ( ord_less_eq @ A @ ( F3 @ U2 ) @ U2 ) ) ) ) ) ) ).
% lfp_def
thf(fact_7150_equiv__listrel,axiom,
! [A: $tType,A4: set @ A,R2: set @ ( product_prod @ A @ A )] :
( ( equiv_equiv @ A @ A4 @ R2 )
=> ( equiv_equiv @ ( list @ A ) @ ( lists @ A @ A4 ) @ ( listrel @ A @ A @ R2 ) ) ) ).
% equiv_listrel
thf(fact_7151_lfp__eqI,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F5: A > A,X: A] :
( ( order_mono @ A @ A @ F5 )
=> ( ( ( F5 @ X )
= X )
=> ( ! [Z: A] :
( ( ( F5 @ Z )
= Z )
=> ( ord_less_eq @ A @ X @ Z ) )
=> ( ( complete_lattice_lfp @ A @ F5 )
= X ) ) ) ) ) ).
% lfp_eqI
thf(fact_7152_lfp__lfp,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F2: A > A > A] :
( ! [X5: A,Y3: A,W: A,Z: A] :
( ( ord_less_eq @ A @ X5 @ Y3 )
=> ( ( ord_less_eq @ A @ W @ Z )
=> ( ord_less_eq @ A @ ( F2 @ X5 @ W ) @ ( F2 @ Y3 @ Z ) ) ) )
=> ( ( complete_lattice_lfp @ A
@ ^ [X4: A] : ( complete_lattice_lfp @ A @ ( F2 @ X4 ) ) )
= ( complete_lattice_lfp @ A
@ ^ [X4: A] : ( F2 @ X4 @ X4 ) ) ) ) ) ).
% lfp_lfp
thf(fact_7153_lfp__greatest,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F2: A > A,A4: A] :
( ! [U4: A] :
( ( ord_less_eq @ A @ ( F2 @ U4 ) @ U4 )
=> ( ord_less_eq @ A @ A4 @ U4 ) )
=> ( ord_less_eq @ A @ A4 @ ( complete_lattice_lfp @ A @ F2 ) ) ) ) ).
% lfp_greatest
thf(fact_7154_lfp__lowerbound,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F2: A > A,A4: A] :
( ( ord_less_eq @ A @ ( F2 @ A4 ) @ A4 )
=> ( ord_less_eq @ A @ ( complete_lattice_lfp @ A @ F2 ) @ A4 ) ) ) ).
% lfp_lowerbound
thf(fact_7155_lfp__mono,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F2: A > A,G: A > A] :
( ! [Z10: A] : ( ord_less_eq @ A @ ( F2 @ Z10 ) @ ( G @ Z10 ) )
=> ( ord_less_eq @ A @ ( complete_lattice_lfp @ A @ F2 ) @ ( complete_lattice_lfp @ A @ G ) ) ) ) ).
% lfp_mono
thf(fact_7156_in__quotient__imp__non__empty,axiom,
! [A: $tType,A4: set @ A,R2: set @ ( product_prod @ A @ A ),X8: set @ A] :
( ( equiv_equiv @ A @ A4 @ R2 )
=> ( ( member @ ( set @ A ) @ X8 @ ( equiv_quotient @ A @ A4 @ R2 ) )
=> ( X8
!= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% in_quotient_imp_non_empty
thf(fact_7157_in__quotient__imp__subset,axiom,
! [A: $tType,A4: set @ A,R2: set @ ( product_prod @ A @ A ),X8: set @ A] :
( ( equiv_equiv @ A @ A4 @ R2 )
=> ( ( member @ ( set @ A ) @ X8 @ ( equiv_quotient @ A @ A4 @ R2 ) )
=> ( ord_less_eq @ ( set @ A ) @ X8 @ A4 ) ) ) ).
% in_quotient_imp_subset
thf(fact_7158_quotient__disj,axiom,
! [A: $tType,A4: set @ A,R2: set @ ( product_prod @ A @ A ),X8: set @ A,Y7: set @ A] :
( ( equiv_equiv @ A @ A4 @ R2 )
=> ( ( member @ ( set @ A ) @ X8 @ ( equiv_quotient @ A @ A4 @ R2 ) )
=> ( ( member @ ( set @ A ) @ Y7 @ ( equiv_quotient @ A @ A4 @ R2 ) )
=> ( ( X8 = Y7 )
| ( ( inf_inf @ ( set @ A ) @ X8 @ Y7 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% quotient_disj
thf(fact_7159_lfp__induct,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F2: A > A,P: A] :
( ( order_mono @ A @ A @ F2 )
=> ( ( ord_less_eq @ A @ ( F2 @ ( inf_inf @ A @ ( complete_lattice_lfp @ A @ F2 ) @ P ) ) @ P )
=> ( ord_less_eq @ A @ ( complete_lattice_lfp @ A @ F2 ) @ P ) ) ) ) ).
% lfp_induct
thf(fact_7160_def__lfp__induct,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A4: A,F2: A > A,P: A] :
( ( A4
= ( complete_lattice_lfp @ A @ F2 ) )
=> ( ( order_mono @ A @ A @ F2 )
=> ( ( ord_less_eq @ A @ ( F2 @ ( inf_inf @ A @ A4 @ P ) ) @ P )
=> ( ord_less_eq @ A @ A4 @ P ) ) ) ) ) ).
% def_lfp_induct
thf(fact_7161_lfp__ordinal__induct,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F2: A > A,P: A > $o] :
( ( order_mono @ A @ A @ F2 )
=> ( ! [S4: A] :
( ( P @ S4 )
=> ( ( ord_less_eq @ A @ S4 @ ( complete_lattice_lfp @ A @ F2 ) )
=> ( P @ ( F2 @ S4 ) ) ) )
=> ( ! [M8: set @ A] :
( ! [X2: A] :
( ( member @ A @ X2 @ M8 )
=> ( P @ X2 ) )
=> ( P @ ( complete_Sup_Sup @ A @ M8 ) ) )
=> ( P @ ( complete_lattice_lfp @ A @ F2 ) ) ) ) ) ) ).
% lfp_ordinal_induct
thf(fact_7162_finite__refines__finite,axiom,
! [A: $tType,A4: set @ A,R: set @ ( product_prod @ A @ A ),S2: set @ ( product_prod @ A @ A )] :
( ( finite_finite2 @ ( set @ A ) @ ( equiv_quotient @ A @ A4 @ R ) )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R @ S2 )
=> ( ( equiv_equiv @ A @ A4 @ R )
=> ( ( equiv_equiv @ A @ A4 @ S2 )
=> ( finite_finite2 @ ( set @ A ) @ ( equiv_quotient @ A @ A4 @ S2 ) ) ) ) ) ) ).
% finite_refines_finite
thf(fact_7163_lfp__funpow,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F2: A > A,N2: nat] :
( ( order_mono @ A @ A @ F2 )
=> ( ( complete_lattice_lfp @ A @ ( compow @ ( A > A ) @ ( suc @ N2 ) @ F2 ) )
= ( complete_lattice_lfp @ A @ F2 ) ) ) ) ).
% lfp_funpow
thf(fact_7164_eq__equiv__class__iff2,axiom,
! [A: $tType,A4: set @ A,R2: set @ ( product_prod @ A @ A ),X: A,Y: A] :
( ( equiv_equiv @ A @ A4 @ R2 )
=> ( ( member @ A @ X @ A4 )
=> ( ( member @ A @ Y @ A4 )
=> ( ( ( equiv_quotient @ A @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) @ R2 )
= ( equiv_quotient @ A @ ( insert2 @ A @ Y @ ( bot_bot @ ( set @ A ) ) ) @ R2 ) )
= ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R2 ) ) ) ) ) ).
% eq_equiv_class_iff2
thf(fact_7165_equiv__imp__dvd__card,axiom,
! [A: $tType,A4: set @ A,R2: set @ ( product_prod @ A @ A ),K: nat] :
( ( finite_finite2 @ A @ A4 )
=> ( ( equiv_equiv @ A @ A4 @ R2 )
=> ( ! [X9: set @ A] :
( ( member @ ( set @ A ) @ X9 @ ( equiv_quotient @ A @ A4 @ R2 ) )
=> ( dvd_dvd @ nat @ K @ ( finite_card @ A @ X9 ) ) )
=> ( dvd_dvd @ nat @ K @ ( finite_card @ A @ A4 ) ) ) ) ) ).
% equiv_imp_dvd_card
thf(fact_7166_lfp__Kleene__iter,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F2: A > A,K: nat] :
( ( order_mono @ A @ A @ F2 )
=> ( ( ( compow @ ( A > A ) @ ( suc @ K ) @ F2 @ ( bot_bot @ A ) )
= ( compow @ ( A > A ) @ K @ F2 @ ( bot_bot @ A ) ) )
=> ( ( complete_lattice_lfp @ A @ F2 )
= ( compow @ ( A > A ) @ K @ F2 @ ( bot_bot @ A ) ) ) ) ) ) ).
% lfp_Kleene_iter
thf(fact_7167_in__quotient__imp__in__rel,axiom,
! [A: $tType,A4: set @ A,R2: set @ ( product_prod @ A @ A ),X8: set @ A,X: A,Y: A] :
( ( equiv_equiv @ A @ A4 @ R2 )
=> ( ( member @ ( set @ A ) @ X8 @ ( equiv_quotient @ A @ A4 @ R2 ) )
=> ( ( ord_less_eq @ ( set @ A ) @ ( insert2 @ A @ X @ ( insert2 @ A @ Y @ ( bot_bot @ ( set @ A ) ) ) ) @ X8 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R2 ) ) ) ) ).
% in_quotient_imp_in_rel
thf(fact_7168_iteratesp__def,axiom,
! [A: $tType] :
( ( comple9053668089753744459l_ccpo @ A )
=> ( ( comple7512665784863727008ratesp @ A )
= ( ^ [F3: A > A] :
( complete_lattice_lfp @ ( A > $o )
@ ^ [P6: A > $o,X4: A] :
( ? [Y6: A] :
( ( X4
= ( F3 @ Y6 ) )
& ( P6 @ Y6 ) )
| ? [M9: set @ A] :
( ( X4
= ( complete_Sup_Sup @ A @ M9 ) )
& ( comple1602240252501008431_chain @ A @ ( ord_less_eq @ A ) @ M9 )
& ! [Y6: A] :
( ( member @ A @ Y6 @ M9 )
=> ( P6 @ Y6 ) ) ) ) ) ) ) ) ).
% iteratesp_def
thf(fact_7169_lfp__transfer__bounded,axiom,
! [A: $tType,B: $tType] :
( ( ( comple6319245703460814977attice @ B )
& ( comple6319245703460814977attice @ A ) )
=> ! [P: A > $o,F2: A > A,Alpha: A > B,G: B > B] :
( ( P @ ( bot_bot @ A ) )
=> ( ! [X5: A] :
( ( P @ X5 )
=> ( P @ ( F2 @ X5 ) ) )
=> ( ! [M8: nat > A] :
( ! [I5: nat] : ( P @ ( M8 @ I5 ) )
=> ( P @ ( complete_Sup_Sup @ A @ ( image2 @ nat @ A @ M8 @ ( top_top @ ( set @ nat ) ) ) ) ) )
=> ( ! [M8: nat > A] :
( ( order_mono @ nat @ A @ M8 )
=> ( ! [I5: nat] : ( P @ ( M8 @ I5 ) )
=> ( ( Alpha @ ( complete_Sup_Sup @ A @ ( image2 @ nat @ A @ M8 @ ( top_top @ ( set @ nat ) ) ) ) )
= ( complete_Sup_Sup @ B
@ ( image2 @ nat @ B
@ ^ [I3: nat] : ( Alpha @ ( M8 @ I3 ) )
@ ( top_top @ ( set @ nat ) ) ) ) ) ) )
=> ( ( order_sup_continuous @ A @ A @ F2 )
=> ( ( order_sup_continuous @ B @ B @ G )
=> ( ! [X5: A] :
( ( P @ X5 )
=> ( ( ord_less_eq @ A @ X5 @ ( complete_lattice_lfp @ A @ F2 ) )
=> ( ( Alpha @ ( F2 @ X5 ) )
= ( G @ ( Alpha @ X5 ) ) ) ) )
=> ( ! [X5: B] : ( ord_less_eq @ B @ ( Alpha @ ( bot_bot @ A ) ) @ ( G @ X5 ) )
=> ( ( Alpha @ ( complete_lattice_lfp @ A @ F2 ) )
= ( complete_lattice_lfp @ B @ G ) ) ) ) ) ) ) ) ) ) ) ).
% lfp_transfer_bounded
thf(fact_7170_lfp__transfer,axiom,
! [A: $tType,B: $tType] :
( ( ( comple6319245703460814977attice @ B )
& ( comple6319245703460814977attice @ A ) )
=> ! [Alpha: A > B,F2: A > A,G: B > B] :
( ( order_sup_continuous @ A @ B @ Alpha )
=> ( ( order_sup_continuous @ A @ A @ F2 )
=> ( ( order_sup_continuous @ B @ B @ G )
=> ( ! [X5: B] : ( ord_less_eq @ B @ ( Alpha @ ( bot_bot @ A ) ) @ ( G @ X5 ) )
=> ( ! [X5: A] :
( ( ord_less_eq @ A @ X5 @ ( complete_lattice_lfp @ A @ F2 ) )
=> ( ( Alpha @ ( F2 @ X5 ) )
= ( G @ ( Alpha @ X5 ) ) ) )
=> ( ( Alpha @ ( complete_lattice_lfp @ A @ F2 ) )
= ( complete_lattice_lfp @ B @ G ) ) ) ) ) ) ) ) ).
% lfp_transfer
thf(fact_7171_cclfp__transfer,axiom,
! [A: $tType,B: $tType] :
( ( ( counta3822494911875563373attice @ B )
& ( counta3822494911875563373attice @ A ) )
=> ! [Alpha: A > B,F2: A > A,G: B > B] :
( ( order_sup_continuous @ A @ B @ Alpha )
=> ( ( order_mono @ A @ A @ F2 )
=> ( ( ( Alpha @ ( bot_bot @ A ) )
= ( bot_bot @ B ) )
=> ( ! [X5: A] :
( ( Alpha @ ( F2 @ X5 ) )
= ( G @ ( Alpha @ X5 ) ) )
=> ( ( Alpha @ ( order_532582986084564980_cclfp @ A @ F2 ) )
= ( order_532582986084564980_cclfp @ B @ G ) ) ) ) ) ) ) ).
% cclfp_transfer
thf(fact_7172_iteratesp_OSup,axiom,
! [A: $tType] :
( ( comple9053668089753744459l_ccpo @ A )
=> ! [M7: set @ A,F2: A > A] :
( ( comple1602240252501008431_chain @ A @ ( ord_less_eq @ A ) @ M7 )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ M7 )
=> ( comple7512665784863727008ratesp @ A @ F2 @ X5 ) )
=> ( comple7512665784863727008ratesp @ A @ F2 @ ( complete_Sup_Sup @ A @ M7 ) ) ) ) ) ).
% iteratesp.Sup
thf(fact_7173_iteratesp_Ocases,axiom,
! [A: $tType] :
( ( comple9053668089753744459l_ccpo @ A )
=> ! [F2: A > A,A3: A] :
( ( comple7512665784863727008ratesp @ A @ F2 @ A3 )
=> ( ! [X5: A] :
( ( A3
= ( F2 @ X5 ) )
=> ~ ( comple7512665784863727008ratesp @ A @ F2 @ X5 ) )
=> ~ ! [M8: set @ A] :
( ( A3
= ( complete_Sup_Sup @ A @ M8 ) )
=> ( ( comple1602240252501008431_chain @ A @ ( ord_less_eq @ A ) @ M8 )
=> ~ ! [X2: A] :
( ( member @ A @ X2 @ M8 )
=> ( comple7512665784863727008ratesp @ A @ F2 @ X2 ) ) ) ) ) ) ) ).
% iteratesp.cases
thf(fact_7174_iteratesp_Osimps,axiom,
! [A: $tType] :
( ( comple9053668089753744459l_ccpo @ A )
=> ( ( comple7512665784863727008ratesp @ A )
= ( ^ [F3: A > A,A5: A] :
( ? [X4: A] :
( ( A5
= ( F3 @ X4 ) )
& ( comple7512665784863727008ratesp @ A @ F3 @ X4 ) )
| ? [M9: set @ A] :
( ( A5
= ( complete_Sup_Sup @ A @ M9 ) )
& ( comple1602240252501008431_chain @ A @ ( ord_less_eq @ A ) @ M9 )
& ! [X4: A] :
( ( member @ A @ X4 @ M9 )
=> ( comple7512665784863727008ratesp @ A @ F3 @ X4 ) ) ) ) ) ) ) ).
% iteratesp.simps
thf(fact_7175_sup__continuous__lfp,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F5: A > A] :
( ( order_sup_continuous @ A @ A @ F5 )
=> ( ( complete_lattice_lfp @ A @ F5 )
= ( complete_Sup_Sup @ A
@ ( image2 @ nat @ A
@ ^ [I3: nat] : ( compow @ ( A > A ) @ I3 @ F5 @ ( bot_bot @ A ) )
@ ( top_top @ ( set @ nat ) ) ) ) ) ) ) ).
% sup_continuous_lfp
thf(fact_7176_proj__iff,axiom,
! [A: $tType,A4: set @ A,R2: set @ ( product_prod @ A @ A ),X: A,Y: A] :
( ( equiv_equiv @ A @ A4 @ R2 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( insert2 @ A @ X @ ( insert2 @ A @ Y @ ( bot_bot @ ( set @ A ) ) ) ) @ A4 )
=> ( ( ( equiv_proj @ A @ A @ R2 @ X )
= ( equiv_proj @ A @ A @ R2 @ Y ) )
= ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R2 ) ) ) ) ).
% proj_iff
thf(fact_7177_ord__class_Olexordp__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( ord_lexordp @ A )
= ( complete_lattice_lfp @ ( ( list @ A ) > ( list @ A ) > $o )
@ ^ [P6: ( list @ A ) > ( list @ A ) > $o,X15: list @ A,X24: list @ A] :
( ? [Y6: A,Ys3: list @ A] :
( ( X15
= ( nil @ A ) )
& ( X24
= ( cons @ A @ Y6 @ Ys3 ) ) )
| ? [X4: A,Y6: A,Xs3: list @ A,Ys3: list @ A] :
( ( X15
= ( cons @ A @ X4 @ Xs3 ) )
& ( X24
= ( cons @ A @ Y6 @ Ys3 ) )
& ( ord_less @ A @ X4 @ Y6 ) )
| ? [X4: A,Y6: A,Xs3: list @ A,Ys3: list @ A] :
( ( X15
= ( cons @ A @ X4 @ Xs3 ) )
& ( X24
= ( cons @ A @ Y6 @ Ys3 ) )
& ~ ( ord_less @ A @ X4 @ Y6 )
& ~ ( ord_less @ A @ Y6 @ X4 )
& ( P6 @ Xs3 @ Ys3 ) ) ) ) ) ) ).
% ord_class.lexordp_def
thf(fact_7178_lexordp__simps_I1_J,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [Ys: list @ A] :
( ( ord_lexordp @ A @ ( nil @ A ) @ Ys )
= ( Ys
!= ( nil @ A ) ) ) ) ).
% lexordp_simps(1)
thf(fact_7179_lexordp__simps_I2_J,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [Xs: list @ A] :
~ ( ord_lexordp @ A @ Xs @ ( nil @ A ) ) ) ).
% lexordp_simps(2)
thf(fact_7180_lexordp__simps_I3_J,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X: A,Xs: list @ A,Y: A,Ys: list @ A] :
( ( ord_lexordp @ A @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y @ Ys ) )
= ( ( ord_less @ A @ X @ Y )
| ( ~ ( ord_less @ A @ Y @ X )
& ( ord_lexordp @ A @ Xs @ Ys ) ) ) ) ) ).
% lexordp_simps(3)
thf(fact_7181_lexordp_ONil,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [Y: A,Ys: list @ A] : ( ord_lexordp @ A @ ( nil @ A ) @ ( cons @ A @ Y @ Ys ) ) ) ).
% lexordp.Nil
thf(fact_7182_lexordp__irreflexive,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [Xs: list @ A] :
( ! [X5: A] :
~ ( ord_less @ A @ X5 @ X5 )
=> ~ ( ord_lexordp @ A @ Xs @ Xs ) ) ) ).
% lexordp_irreflexive
thf(fact_7183_lexordp__antisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [Xs: list @ A,Ys: list @ A] :
( ( ord_lexordp @ A @ Xs @ Ys )
=> ~ ( ord_lexordp @ A @ Ys @ Xs ) ) ) ).
% lexordp_antisym
thf(fact_7184_lexordp__trans,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,Ys: list @ A,Zs: list @ A] :
( ( ord_lexordp @ A @ Xs @ Ys )
=> ( ( ord_lexordp @ A @ Ys @ Zs )
=> ( ord_lexordp @ A @ Xs @ Zs ) ) ) ) ).
% lexordp_trans
thf(fact_7185_lexordp__linear,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,Ys: list @ A] :
( ( ord_lexordp @ A @ Xs @ Ys )
| ( Xs = Ys )
| ( ord_lexordp @ A @ Ys @ Xs ) ) ) ).
% lexordp_linear
thf(fact_7186_lexordp__irreflexive_H,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [Xs: list @ A] :
~ ( ord_lexordp @ A @ Xs @ Xs ) ) ).
% lexordp_irreflexive'
thf(fact_7187_lexordp_OCons,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X: A,Y: A,Xs: list @ A,Ys: list @ A] :
( ( ord_less @ A @ X @ Y )
=> ( ord_lexordp @ A @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y @ Ys ) ) ) ) ).
% lexordp.Cons
thf(fact_7188_lexordp_OCons__eq,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X: A,Y: A,Xs: list @ A,Ys: list @ A] :
( ~ ( ord_less @ A @ X @ Y )
=> ( ~ ( ord_less @ A @ Y @ X )
=> ( ( ord_lexordp @ A @ Xs @ Ys )
=> ( ord_lexordp @ A @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y @ Ys ) ) ) ) ) ) ).
% lexordp.Cons_eq
thf(fact_7189_lexordp__append__leftI,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [Us: list @ A,Vs: list @ A,Xs: list @ A] :
( ( ord_lexordp @ A @ Us @ Vs )
=> ( ord_lexordp @ A @ ( append @ A @ Xs @ Us ) @ ( append @ A @ Xs @ Vs ) ) ) ) ).
% lexordp_append_leftI
thf(fact_7190_lexordp__append__leftD,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [Xs: list @ A,Us: list @ A,Vs: list @ A] :
( ( ord_lexordp @ A @ ( append @ A @ Xs @ Us ) @ ( append @ A @ Xs @ Vs ) )
=> ( ! [A6: A] :
~ ( ord_less @ A @ A6 @ A6 )
=> ( ord_lexordp @ A @ Us @ Vs ) ) ) ) ).
% lexordp_append_leftD
thf(fact_7191_lexordp__append__rightI,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [Ys: list @ A,Xs: list @ A] :
( ( Ys
!= ( nil @ A ) )
=> ( ord_lexordp @ A @ Xs @ ( append @ A @ Xs @ Ys ) ) ) ) ).
% lexordp_append_rightI
thf(fact_7192_lexordp_Ocases,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A1: list @ A,A22: list @ A] :
( ( ord_lexordp @ A @ A1 @ A22 )
=> ( ( ( A1
= ( nil @ A ) )
=> ! [Y3: A,Ys4: list @ A] :
( A22
!= ( cons @ A @ Y3 @ Ys4 ) ) )
=> ( ! [X5: A] :
( ? [Xs2: list @ A] :
( A1
= ( cons @ A @ X5 @ Xs2 ) )
=> ! [Y3: A] :
( ? [Ys4: list @ A] :
( A22
= ( cons @ A @ Y3 @ Ys4 ) )
=> ~ ( ord_less @ A @ X5 @ Y3 ) ) )
=> ~ ! [X5: A,Y3: A,Xs2: list @ A] :
( ( A1
= ( cons @ A @ X5 @ Xs2 ) )
=> ! [Ys4: list @ A] :
( ( A22
= ( cons @ A @ Y3 @ Ys4 ) )
=> ( ~ ( ord_less @ A @ X5 @ Y3 )
=> ( ~ ( ord_less @ A @ Y3 @ X5 )
=> ~ ( ord_lexordp @ A @ Xs2 @ Ys4 ) ) ) ) ) ) ) ) ) ).
% lexordp.cases
thf(fact_7193_lexordp_Osimps,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( ord_lexordp @ A )
= ( ^ [A12: list @ A,A23: list @ A] :
( ? [Y6: A,Ys3: list @ A] :
( ( A12
= ( nil @ A ) )
& ( A23
= ( cons @ A @ Y6 @ Ys3 ) ) )
| ? [X4: A,Y6: A,Xs3: list @ A,Ys3: list @ A] :
( ( A12
= ( cons @ A @ X4 @ Xs3 ) )
& ( A23
= ( cons @ A @ Y6 @ Ys3 ) )
& ( ord_less @ A @ X4 @ Y6 ) )
| ? [X4: A,Y6: A,Xs3: list @ A,Ys3: list @ A] :
( ( A12
= ( cons @ A @ X4 @ Xs3 ) )
& ( A23
= ( cons @ A @ Y6 @ Ys3 ) )
& ~ ( ord_less @ A @ X4 @ Y6 )
& ~ ( ord_less @ A @ Y6 @ X4 )
& ( ord_lexordp @ A @ Xs3 @ Ys3 ) ) ) ) ) ) ).
% lexordp.simps
thf(fact_7194_lexordp__cases,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,Ys: list @ A] :
( ( ord_lexordp @ A @ Xs @ Ys )
=> ( ( ( Xs
= ( nil @ A ) )
=> ! [Y3: A,Ys5: list @ A] :
( Ys
!= ( cons @ A @ Y3 @ Ys5 ) ) )
=> ( ! [X5: A] :
( ? [Xs4: list @ A] :
( Xs
= ( cons @ A @ X5 @ Xs4 ) )
=> ! [Y3: A] :
( ? [Ys5: list @ A] :
( Ys
= ( cons @ A @ Y3 @ Ys5 ) )
=> ~ ( ord_less @ A @ X5 @ Y3 ) ) )
=> ~ ! [X5: A,Xs4: list @ A] :
( ( Xs
= ( cons @ A @ X5 @ Xs4 ) )
=> ! [Ys5: list @ A] :
( ( Ys
= ( cons @ A @ X5 @ Ys5 ) )
=> ~ ( ord_lexordp @ A @ Xs4 @ Ys5 ) ) ) ) ) ) ) ).
% lexordp_cases
thf(fact_7195_lexordp__induct,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,Ys: list @ A,P: ( list @ A ) > ( list @ A ) > $o] :
( ( ord_lexordp @ A @ Xs @ Ys )
=> ( ! [Y3: A,Ys4: list @ A] : ( P @ ( nil @ A ) @ ( cons @ A @ Y3 @ Ys4 ) )
=> ( ! [X5: A,Xs2: list @ A,Y3: A,Ys4: list @ A] :
( ( ord_less @ A @ X5 @ Y3 )
=> ( P @ ( cons @ A @ X5 @ Xs2 ) @ ( cons @ A @ Y3 @ Ys4 ) ) )
=> ( ! [X5: A,Xs2: list @ A,Ys4: list @ A] :
( ( ord_lexordp @ A @ Xs2 @ Ys4 )
=> ( ( P @ Xs2 @ Ys4 )
=> ( P @ ( cons @ A @ X5 @ Xs2 ) @ ( cons @ A @ X5 @ Ys4 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ) ).
% lexordp_induct
thf(fact_7196_lexordp__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_lexordp @ A )
= ( ^ [Xs3: list @ A,Ys3: list @ A] :
( ? [X4: A,Vs3: list @ A] :
( Ys3
= ( append @ A @ Xs3 @ ( cons @ A @ X4 @ Vs3 ) ) )
| ? [Us2: list @ A,A5: A,B4: A,Vs3: list @ A,Ws3: list @ A] :
( ( ord_less @ A @ A5 @ B4 )
& ( Xs3
= ( append @ A @ Us2 @ ( cons @ A @ A5 @ Vs3 ) ) )
& ( Ys3
= ( append @ A @ Us2 @ ( cons @ A @ B4 @ Ws3 ) ) ) ) ) ) ) ) ).
% lexordp_iff
thf(fact_7197_lexordp__append__left__rightI,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X: A,Y: A,Us: list @ A,Xs: list @ A,Ys: list @ A] :
( ( ord_less @ A @ X @ Y )
=> ( ord_lexordp @ A @ ( append @ A @ Us @ ( cons @ A @ X @ Xs ) ) @ ( append @ A @ Us @ ( cons @ A @ Y @ Ys ) ) ) ) ) ).
% lexordp_append_left_rightI
thf(fact_7198_lexordp__conv__lexord,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_lexordp @ A )
= ( ^ [Xs3: list @ A,Ys3: list @ A] : ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs3 @ Ys3 ) @ ( lexord @ A @ ( collect @ ( product_prod @ A @ A ) @ ( product_case_prod @ A @ A @ $o @ ( ord_less @ A ) ) ) ) ) ) ) ) ).
% lexordp_conv_lexord
thf(fact_7199_ord_Olexordp__def,axiom,
! [A: $tType] :
( ( lexordp2 @ A )
= ( ^ [Less2: A > A > $o] :
( complete_lattice_lfp @ ( ( list @ A ) > ( list @ A ) > $o )
@ ^ [P6: ( list @ A ) > ( list @ A ) > $o,X15: list @ A,X24: list @ A] :
( ? [Y6: A,Ys3: list @ A] :
( ( X15
= ( nil @ A ) )
& ( X24
= ( cons @ A @ Y6 @ Ys3 ) ) )
| ? [X4: A,Y6: A,Xs3: list @ A,Ys3: list @ A] :
( ( X15
= ( cons @ A @ X4 @ Xs3 ) )
& ( X24
= ( cons @ A @ Y6 @ Ys3 ) )
& ( Less2 @ X4 @ Y6 ) )
| ? [X4: A,Y6: A,Xs3: list @ A,Ys3: list @ A] :
( ( X15
= ( cons @ A @ X4 @ Xs3 ) )
& ( X24
= ( cons @ A @ Y6 @ Ys3 ) )
& ~ ( Less2 @ X4 @ Y6 )
& ~ ( Less2 @ Y6 @ X4 )
& ( P6 @ Xs3 @ Ys3 ) ) ) ) ) ) ).
% ord.lexordp_def
thf(fact_7200_finite__enumerate__initial__segment,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [S2: set @ A,N2: nat,S: A] :
( ( finite_finite2 @ A @ S2 )
=> ( ( ord_less @ nat @ N2 @ ( finite_card @ A @ ( inf_inf @ ( set @ A ) @ S2 @ ( set_ord_lessThan @ A @ S ) ) ) )
=> ( ( infini527867602293511546merate @ A @ ( inf_inf @ ( set @ A ) @ S2 @ ( set_ord_lessThan @ A @ S ) ) @ N2 )
= ( infini527867602293511546merate @ A @ S2 @ N2 ) ) ) ) ) ).
% finite_enumerate_initial_segment
thf(fact_7201_ord_Olexordp__simps_I3_J,axiom,
! [A: $tType,Less: A > A > $o,X: A,Xs: list @ A,Y: A,Ys: list @ A] :
( ( lexordp2 @ A @ Less @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y @ Ys ) )
= ( ( Less @ X @ Y )
| ( ~ ( Less @ Y @ X )
& ( lexordp2 @ A @ Less @ Xs @ Ys ) ) ) ) ).
% ord.lexordp_simps(3)
thf(fact_7202_ord_Olexordp__simps_I2_J,axiom,
! [A: $tType,Less: A > A > $o,Xs: list @ A] :
~ ( lexordp2 @ A @ Less @ Xs @ ( nil @ A ) ) ).
% ord.lexordp_simps(2)
thf(fact_7203_ord_Olexordp__simps_I1_J,axiom,
! [A: $tType,Less: A > A > $o,Ys: list @ A] :
( ( lexordp2 @ A @ Less @ ( nil @ A ) @ Ys )
= ( Ys
!= ( nil @ A ) ) ) ).
% ord.lexordp_simps(1)
thf(fact_7204_enumerate__mono__iff,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [S2: set @ A,M2: nat,N2: nat] :
( ~ ( finite_finite2 @ A @ S2 )
=> ( ( ord_less @ A @ ( infini527867602293511546merate @ A @ S2 @ M2 ) @ ( infini527867602293511546merate @ A @ S2 @ N2 ) )
= ( ord_less @ nat @ M2 @ N2 ) ) ) ) ).
% enumerate_mono_iff
thf(fact_7205_finite__enumerate__mono__iff,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [S2: set @ A,M2: nat,N2: nat] :
( ( finite_finite2 @ A @ S2 )
=> ( ( ord_less @ nat @ M2 @ ( finite_card @ A @ S2 ) )
=> ( ( ord_less @ nat @ N2 @ ( finite_card @ A @ S2 ) )
=> ( ( ord_less @ A @ ( infini527867602293511546merate @ A @ S2 @ M2 ) @ ( infini527867602293511546merate @ A @ S2 @ N2 ) )
= ( ord_less @ nat @ M2 @ N2 ) ) ) ) ) ) ).
% finite_enumerate_mono_iff
thf(fact_7206_ord_Olexordp__append__left__rightI,axiom,
! [A: $tType,Less: A > A > $o,X: A,Y: A,Us: list @ A,Xs: list @ A,Ys: list @ A] :
( ( Less @ X @ Y )
=> ( lexordp2 @ A @ Less @ ( append @ A @ Us @ ( cons @ A @ X @ Xs ) ) @ ( append @ A @ Us @ ( cons @ A @ Y @ Ys ) ) ) ) ).
% ord.lexordp_append_left_rightI
thf(fact_7207_ord_Olexordp__append__rightI,axiom,
! [A: $tType,Ys: list @ A,Less: A > A > $o,Xs: list @ A] :
( ( Ys
!= ( nil @ A ) )
=> ( lexordp2 @ A @ Less @ Xs @ ( append @ A @ Xs @ Ys ) ) ) ).
% ord.lexordp_append_rightI
thf(fact_7208_ord_Olexordp_Osimps,axiom,
! [A: $tType] :
( ( lexordp2 @ A )
= ( ^ [Less2: A > A > $o,A12: list @ A,A23: list @ A] :
( ? [Y6: A,Ys3: list @ A] :
( ( A12
= ( nil @ A ) )
& ( A23
= ( cons @ A @ Y6 @ Ys3 ) ) )
| ? [X4: A,Y6: A,Xs3: list @ A,Ys3: list @ A] :
( ( A12
= ( cons @ A @ X4 @ Xs3 ) )
& ( A23
= ( cons @ A @ Y6 @ Ys3 ) )
& ( Less2 @ X4 @ Y6 ) )
| ? [X4: A,Y6: A,Xs3: list @ A,Ys3: list @ A] :
( ( A12
= ( cons @ A @ X4 @ Xs3 ) )
& ( A23
= ( cons @ A @ Y6 @ Ys3 ) )
& ~ ( Less2 @ X4 @ Y6 )
& ~ ( Less2 @ Y6 @ X4 )
& ( lexordp2 @ A @ Less2 @ Xs3 @ Ys3 ) ) ) ) ) ).
% ord.lexordp.simps
thf(fact_7209_ord_Olexordp_Ocases,axiom,
! [A: $tType,Less: A > A > $o,A1: list @ A,A22: list @ A] :
( ( lexordp2 @ A @ Less @ A1 @ A22 )
=> ( ( ( A1
= ( nil @ A ) )
=> ! [Y3: A,Ys4: list @ A] :
( A22
!= ( cons @ A @ Y3 @ Ys4 ) ) )
=> ( ! [X5: A] :
( ? [Xs2: list @ A] :
( A1
= ( cons @ A @ X5 @ Xs2 ) )
=> ! [Y3: A] :
( ? [Ys4: list @ A] :
( A22
= ( cons @ A @ Y3 @ Ys4 ) )
=> ~ ( Less @ X5 @ Y3 ) ) )
=> ~ ! [X5: A,Y3: A,Xs2: list @ A] :
( ( A1
= ( cons @ A @ X5 @ Xs2 ) )
=> ! [Ys4: list @ A] :
( ( A22
= ( cons @ A @ Y3 @ Ys4 ) )
=> ( ~ ( Less @ X5 @ Y3 )
=> ( ~ ( Less @ Y3 @ X5 )
=> ~ ( lexordp2 @ A @ Less @ Xs2 @ Ys4 ) ) ) ) ) ) ) ) ).
% ord.lexordp.cases
thf(fact_7210_ord_Olexordp_ONil,axiom,
! [A: $tType,Less: A > A > $o,Y: A,Ys: list @ A] : ( lexordp2 @ A @ Less @ ( nil @ A ) @ ( cons @ A @ Y @ Ys ) ) ).
% ord.lexordp.Nil
thf(fact_7211_ord_Olexordp__append__leftD,axiom,
! [A: $tType,Less: A > A > $o,Xs: list @ A,Us: list @ A,Vs: list @ A] :
( ( lexordp2 @ A @ Less @ ( append @ A @ Xs @ Us ) @ ( append @ A @ Xs @ Vs ) )
=> ( ! [A6: A] :
~ ( Less @ A6 @ A6 )
=> ( lexordp2 @ A @ Less @ Us @ Vs ) ) ) ).
% ord.lexordp_append_leftD
thf(fact_7212_ord_Olexordp__append__leftI,axiom,
! [A: $tType,Less: A > A > $o,Us: list @ A,Vs: list @ A,Xs: list @ A] :
( ( lexordp2 @ A @ Less @ Us @ Vs )
=> ( lexordp2 @ A @ Less @ ( append @ A @ Xs @ Us ) @ ( append @ A @ Xs @ Vs ) ) ) ).
% ord.lexordp_append_leftI
thf(fact_7213_ord_Olexordp_OCons__eq,axiom,
! [A: $tType,Less: A > A > $o,X: A,Y: A,Xs: list @ A,Ys: list @ A] :
( ~ ( Less @ X @ Y )
=> ( ~ ( Less @ Y @ X )
=> ( ( lexordp2 @ A @ Less @ Xs @ Ys )
=> ( lexordp2 @ A @ Less @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y @ Ys ) ) ) ) ) ).
% ord.lexordp.Cons_eq
thf(fact_7214_ord_Olexordp_OCons,axiom,
! [A: $tType,Less: A > A > $o,X: A,Y: A,Xs: list @ A,Ys: list @ A] :
( ( Less @ X @ Y )
=> ( lexordp2 @ A @ Less @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y @ Ys ) ) ) ).
% ord.lexordp.Cons
thf(fact_7215_enumerate__in__set,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [S2: set @ A,N2: nat] :
( ~ ( finite_finite2 @ A @ S2 )
=> ( member @ A @ ( infini527867602293511546merate @ A @ S2 @ N2 ) @ S2 ) ) ) ).
% enumerate_in_set
thf(fact_7216_enumerate__Ex,axiom,
! [S2: set @ nat,S: nat] :
( ~ ( finite_finite2 @ nat @ S2 )
=> ( ( member @ nat @ S @ S2 )
=> ? [N3: nat] :
( ( infini527867602293511546merate @ nat @ S2 @ N3 )
= S ) ) ) ).
% enumerate_Ex
thf(fact_7217_ord_Olexordp__irreflexive,axiom,
! [A: $tType,Less: A > A > $o,Xs: list @ A] :
( ! [X5: A] :
~ ( Less @ X5 @ X5 )
=> ~ ( lexordp2 @ A @ Less @ Xs @ Xs ) ) ).
% ord.lexordp_irreflexive
thf(fact_7218_ord_Olexordp_Ocong,axiom,
! [A: $tType] :
( ( lexordp2 @ A )
= ( lexordp2 @ A ) ) ).
% ord.lexordp.cong
thf(fact_7219_le__enumerate,axiom,
! [S2: set @ nat,N2: nat] :
( ~ ( finite_finite2 @ nat @ S2 )
=> ( ord_less_eq @ nat @ N2 @ ( infini527867602293511546merate @ nat @ S2 @ N2 ) ) ) ).
% le_enumerate
thf(fact_7220_strict__mono__enumerate,axiom,
! [S2: set @ nat] :
( ~ ( finite_finite2 @ nat @ S2 )
=> ( order_strict_mono @ nat @ nat @ ( infini527867602293511546merate @ nat @ S2 ) ) ) ).
% strict_mono_enumerate
thf(fact_7221_enumerate__step,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [S2: set @ A,N2: nat] :
( ~ ( finite_finite2 @ A @ S2 )
=> ( ord_less @ A @ ( infini527867602293511546merate @ A @ S2 @ N2 ) @ ( infini527867602293511546merate @ A @ S2 @ ( suc @ N2 ) ) ) ) ) ).
% enumerate_step
thf(fact_7222_enumerate__mono,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [M2: nat,N2: nat,S2: set @ A] :
( ( ord_less @ nat @ M2 @ N2 )
=> ( ~ ( finite_finite2 @ A @ S2 )
=> ( ord_less @ A @ ( infini527867602293511546merate @ A @ S2 @ M2 ) @ ( infini527867602293511546merate @ A @ S2 @ N2 ) ) ) ) ) ).
% enumerate_mono
thf(fact_7223_finite__enumerate__in__set,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [S2: set @ A,N2: nat] :
( ( finite_finite2 @ A @ S2 )
=> ( ( ord_less @ nat @ N2 @ ( finite_card @ A @ S2 ) )
=> ( member @ A @ ( infini527867602293511546merate @ A @ S2 @ N2 ) @ S2 ) ) ) ) ).
% finite_enumerate_in_set
thf(fact_7224_finite__enumerate__Ex,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [S2: set @ A,S: A] :
( ( finite_finite2 @ A @ S2 )
=> ( ( member @ A @ S @ S2 )
=> ? [N3: nat] :
( ( ord_less @ nat @ N3 @ ( finite_card @ A @ S2 ) )
& ( ( infini527867602293511546merate @ A @ S2 @ N3 )
= S ) ) ) ) ) ).
% finite_enumerate_Ex
thf(fact_7225_finite__enum__ext,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [X8: set @ A,Y7: set @ A] :
( ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( finite_card @ A @ X8 ) )
=> ( ( infini527867602293511546merate @ A @ X8 @ I4 )
= ( infini527867602293511546merate @ A @ Y7 @ I4 ) ) )
=> ( ( finite_finite2 @ A @ X8 )
=> ( ( finite_finite2 @ A @ Y7 )
=> ( ( ( finite_card @ A @ X8 )
= ( finite_card @ A @ Y7 ) )
=> ( X8 = Y7 ) ) ) ) ) ) ).
% finite_enum_ext
thf(fact_7226_range__enumerate,axiom,
! [S2: set @ nat] :
( ~ ( finite_finite2 @ nat @ S2 )
=> ( ( image2 @ nat @ nat @ ( infini527867602293511546merate @ nat @ S2 ) @ ( top_top @ ( set @ nat ) ) )
= S2 ) ) ).
% range_enumerate
thf(fact_7227_inj__enumerate,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [S2: set @ A] :
( ~ ( finite_finite2 @ A @ S2 )
=> ( inj_on @ nat @ A @ ( infini527867602293511546merate @ A @ S2 ) @ ( top_top @ ( set @ nat ) ) ) ) ) ).
% inj_enumerate
thf(fact_7228_finite__enumerate__mono,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [M2: nat,N2: nat,S2: set @ A] :
( ( ord_less @ nat @ M2 @ N2 )
=> ( ( finite_finite2 @ A @ S2 )
=> ( ( ord_less @ nat @ N2 @ ( finite_card @ A @ S2 ) )
=> ( ord_less @ A @ ( infini527867602293511546merate @ A @ S2 @ M2 ) @ ( infini527867602293511546merate @ A @ S2 @ N2 ) ) ) ) ) ) ).
% finite_enumerate_mono
thf(fact_7229_finite__le__enumerate,axiom,
! [S2: set @ nat,N2: nat] :
( ( finite_finite2 @ nat @ S2 )
=> ( ( ord_less @ nat @ N2 @ ( finite_card @ nat @ S2 ) )
=> ( ord_less_eq @ nat @ N2 @ ( infini527867602293511546merate @ nat @ S2 @ N2 ) ) ) ) ).
% finite_le_enumerate
thf(fact_7230_bij__enumerate,axiom,
! [S2: set @ nat] :
( ~ ( finite_finite2 @ nat @ S2 )
=> ( bij_betw @ nat @ nat @ ( infini527867602293511546merate @ nat @ S2 ) @ ( top_top @ ( set @ nat ) ) @ S2 ) ) ).
% bij_enumerate
thf(fact_7231_finite__bij__enumerate,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [S2: set @ A] :
( ( finite_finite2 @ A @ S2 )
=> ( bij_betw @ nat @ A @ ( infini527867602293511546merate @ A @ S2 ) @ ( set_ord_lessThan @ nat @ ( finite_card @ A @ S2 ) ) @ S2 ) ) ) ).
% finite_bij_enumerate
thf(fact_7232_finite__enumerate__step,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [S2: set @ A,N2: nat] :
( ( finite_finite2 @ A @ S2 )
=> ( ( ord_less @ nat @ ( suc @ N2 ) @ ( finite_card @ A @ S2 ) )
=> ( ord_less @ A @ ( infini527867602293511546merate @ A @ S2 @ N2 ) @ ( infini527867602293511546merate @ A @ S2 @ ( suc @ N2 ) ) ) ) ) ) ).
% finite_enumerate_step
thf(fact_7233_enumerate__Suc_H,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [S2: set @ A,N2: nat] :
( ( infini527867602293511546merate @ A @ S2 @ ( suc @ N2 ) )
= ( infini527867602293511546merate @ A @ ( minus_minus @ ( set @ A ) @ S2 @ ( insert2 @ A @ ( infini527867602293511546merate @ A @ S2 @ ( zero_zero @ nat ) ) @ ( bot_bot @ ( set @ A ) ) ) ) @ N2 ) ) ) ).
% enumerate_Suc'
thf(fact_7234_finite__enum__subset,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [X8: set @ A,Y7: set @ A] :
( ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( finite_card @ A @ X8 ) )
=> ( ( infini527867602293511546merate @ A @ X8 @ I4 )
= ( infini527867602293511546merate @ A @ Y7 @ I4 ) ) )
=> ( ( finite_finite2 @ A @ X8 )
=> ( ( finite_finite2 @ A @ Y7 )
=> ( ( ord_less_eq @ nat @ ( finite_card @ A @ X8 ) @ ( finite_card @ A @ Y7 ) )
=> ( ord_less_eq @ ( set @ A ) @ X8 @ Y7 ) ) ) ) ) ) ).
% finite_enum_subset
thf(fact_7235_finite__enumerate__Suc_H_H,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [S2: set @ A,N2: nat] :
( ( finite_finite2 @ A @ S2 )
=> ( ( ord_less @ nat @ ( suc @ N2 ) @ ( finite_card @ A @ S2 ) )
=> ( ( infini527867602293511546merate @ A @ S2 @ ( suc @ N2 ) )
= ( ord_Least @ A
@ ^ [S7: A] :
( ( member @ A @ S7 @ S2 )
& ( ord_less @ A @ ( infini527867602293511546merate @ A @ S2 @ N2 ) @ S7 ) ) ) ) ) ) ) ).
% finite_enumerate_Suc''
thf(fact_7236_enumerate__Suc,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [S2: set @ A,N2: nat] :
( ( infini527867602293511546merate @ A @ S2 @ ( suc @ N2 ) )
= ( infini527867602293511546merate @ A
@ ( minus_minus @ ( set @ A ) @ S2
@ ( insert2 @ A
@ ( ord_Least @ A
@ ^ [N: A] : ( member @ A @ N @ S2 ) )
@ ( bot_bot @ ( set @ A ) ) ) )
@ N2 ) ) ) ).
% enumerate_Suc
thf(fact_7237_Least__eq__0,axiom,
! [P: nat > $o] :
( ( P @ ( zero_zero @ nat ) )
=> ( ( ord_Least @ nat @ P )
= ( zero_zero @ nat ) ) ) ).
% Least_eq_0
thf(fact_7238_Least__le,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [P: A > $o,K: A] :
( ( P @ K )
=> ( ord_less_eq @ A @ ( ord_Least @ A @ P ) @ K ) ) ) ).
% Least_le
thf(fact_7239_Least1I,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [P: A > $o] :
( ? [X2: A] :
( ( P @ X2 )
& ! [Y3: A] :
( ( P @ Y3 )
=> ( ord_less_eq @ A @ X2 @ Y3 ) )
& ! [Y3: A] :
( ( ( P @ Y3 )
& ! [Ya2: A] :
( ( P @ Ya2 )
=> ( ord_less_eq @ A @ Y3 @ Ya2 ) ) )
=> ( Y3 = X2 ) ) )
=> ( P @ ( ord_Least @ A @ P ) ) ) ) ).
% Least1I
thf(fact_7240_Least1__le,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [P: A > $o,Z3: A] :
( ? [X2: A] :
( ( P @ X2 )
& ! [Y3: A] :
( ( P @ Y3 )
=> ( ord_less_eq @ A @ X2 @ Y3 ) )
& ! [Y3: A] :
( ( ( P @ Y3 )
& ! [Ya2: A] :
( ( P @ Ya2 )
=> ( ord_less_eq @ A @ Y3 @ Ya2 ) ) )
=> ( Y3 = X2 ) ) )
=> ( ( P @ Z3 )
=> ( ord_less_eq @ A @ ( ord_Least @ A @ P ) @ Z3 ) ) ) ) ).
% Least1_le
thf(fact_7241_LeastI2__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [P: A > $o,X: A,Q: A > $o] :
( ( P @ X )
=> ( ! [Y3: A] :
( ( P @ Y3 )
=> ( ord_less_eq @ A @ X @ Y3 ) )
=> ( ! [X5: A] :
( ( P @ X5 )
=> ( ! [Y4: A] :
( ( P @ Y4 )
=> ( ord_less_eq @ A @ X5 @ Y4 ) )
=> ( Q @ X5 ) ) )
=> ( Q @ ( ord_Least @ A @ P ) ) ) ) ) ) ).
% LeastI2_order
thf(fact_7242_Least__equality,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [P: A > $o,X: A] :
( ( P @ X )
=> ( ! [Y3: A] :
( ( P @ Y3 )
=> ( ord_less_eq @ A @ X @ Y3 ) )
=> ( ( ord_Least @ A @ P )
= X ) ) ) ) ).
% Least_equality
thf(fact_7243_LeastI2__wellorder,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [P: A > $o,A3: A,Q: A > $o] :
( ( P @ A3 )
=> ( ! [A6: A] :
( ( P @ A6 )
=> ( ! [B10: A] :
( ( P @ B10 )
=> ( ord_less_eq @ A @ A6 @ B10 ) )
=> ( Q @ A6 ) ) )
=> ( Q @ ( ord_Least @ A @ P ) ) ) ) ) ).
% LeastI2_wellorder
thf(fact_7244_LeastI2__wellorder__ex,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [P: A > $o,Q: A > $o] :
( ? [X_1: A] : ( P @ X_1 )
=> ( ! [A6: A] :
( ( P @ A6 )
=> ( ! [B10: A] :
( ( P @ B10 )
=> ( ord_less_eq @ A @ A6 @ B10 ) )
=> ( Q @ A6 ) ) )
=> ( Q @ ( ord_Least @ A @ P ) ) ) ) ) ).
% LeastI2_wellorder_ex
thf(fact_7245_not__less__Least,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [K: A,P: A > $o] :
( ( ord_less @ A @ K @ ( ord_Least @ A @ P ) )
=> ~ ( P @ K ) ) ) ).
% not_less_Least
thf(fact_7246_LeastI2,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [P: A > $o,A3: A,Q: A > $o] :
( ( P @ A3 )
=> ( ! [X5: A] :
( ( P @ X5 )
=> ( Q @ X5 ) )
=> ( Q @ ( ord_Least @ A @ P ) ) ) ) ) ).
% LeastI2
thf(fact_7247_LeastI__ex,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [P: A > $o] :
( ? [X_1: A] : ( P @ X_1 )
=> ( P @ ( ord_Least @ A @ P ) ) ) ) ).
% LeastI_ex
thf(fact_7248_LeastI2__ex,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [P: A > $o,Q: A > $o] :
( ? [X_1: A] : ( P @ X_1 )
=> ( ! [X5: A] :
( ( P @ X5 )
=> ( Q @ X5 ) )
=> ( Q @ ( ord_Least @ A @ P ) ) ) ) ) ).
% LeastI2_ex
thf(fact_7249_LeastI,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [P: A > $o,K: A] :
( ( P @ K )
=> ( P @ ( ord_Least @ A @ P ) ) ) ) ).
% LeastI
thf(fact_7250_Least__Suc2,axiom,
! [P: nat > $o,N2: nat,Q: nat > $o,M2: nat] :
( ( P @ N2 )
=> ( ( Q @ M2 )
=> ( ~ ( P @ ( zero_zero @ nat ) )
=> ( ! [K3: nat] :
( ( P @ ( suc @ K3 ) )
= ( Q @ K3 ) )
=> ( ( ord_Least @ nat @ P )
= ( suc @ ( ord_Least @ nat @ Q ) ) ) ) ) ) ) ).
% Least_Suc2
thf(fact_7251_Least__Suc,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ N2 )
=> ( ~ ( P @ ( zero_zero @ nat ) )
=> ( ( ord_Least @ nat @ P )
= ( suc
@ ( ord_Least @ nat
@ ^ [M: nat] : ( P @ ( suc @ M ) ) ) ) ) ) ) ).
% Least_Suc
thf(fact_7252_Least__Min,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > $o] :
( ( finite_finite2 @ A @ ( collect @ A @ P ) )
=> ( ? [X_1: A] : ( P @ X_1 )
=> ( ( ord_Least @ A @ P )
= ( lattic643756798350308766er_Min @ A @ ( collect @ A @ P ) ) ) ) ) ) ).
% Least_Min
thf(fact_7253_Bleast__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( bleast @ A )
= ( ^ [S6: set @ A,P3: A > $o] :
( ord_Least @ A
@ ^ [X4: A] :
( ( member @ A @ X4 @ S6 )
& ( P3 @ X4 ) ) ) ) ) ) ).
% Bleast_def
thf(fact_7254_abort__Bleast__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( abort_Bleast @ A )
= ( ^ [S6: set @ A,P3: A > $o] :
( ord_Least @ A
@ ^ [X4: A] :
( ( member @ A @ X4 @ S6 )
& ( P3 @ X4 ) ) ) ) ) ) ).
% abort_Bleast_def
thf(fact_7255_enumerate__0,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [S2: set @ A] :
( ( infini527867602293511546merate @ A @ S2 @ ( zero_zero @ nat ) )
= ( ord_Least @ A
@ ^ [N: A] : ( member @ A @ N @ S2 ) ) ) ) ).
% enumerate_0
thf(fact_7256_Least__mono,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [F2: A > B,S2: set @ A] :
( ( order_mono @ A @ B @ F2 )
=> ( ? [X2: A] :
( ( member @ A @ X2 @ S2 )
& ! [Xa3: A] :
( ( member @ A @ Xa3 @ S2 )
=> ( ord_less_eq @ A @ X2 @ Xa3 ) ) )
=> ( ( ord_Least @ B
@ ^ [Y6: B] : ( member @ B @ Y6 @ ( image2 @ A @ B @ F2 @ S2 ) ) )
= ( F2
@ ( ord_Least @ A
@ ^ [X4: A] : ( member @ A @ X4 @ S2 ) ) ) ) ) ) ) ).
% Least_mono
thf(fact_7257_enumerate__Suc_H_H,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [S2: set @ A,N2: nat] :
( ~ ( finite_finite2 @ A @ S2 )
=> ( ( infini527867602293511546merate @ A @ S2 @ ( suc @ N2 ) )
= ( ord_Least @ A
@ ^ [S7: A] :
( ( member @ A @ S7 @ S2 )
& ( ord_less @ A @ ( infini527867602293511546merate @ A @ S2 @ N2 ) @ S7 ) ) ) ) ) ) ).
% enumerate_Suc''
thf(fact_7258_fold__atLeastAtMost__nat_Opelims,axiom,
! [A: $tType,X: nat > A > A,Xa2: nat,Xb: nat,Xc: A,Y: A] :
( ( ( set_fo6178422350223883121st_nat @ A @ X @ Xa2 @ Xb @ Xc )
= Y )
=> ( ( accp @ ( product_prod @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) ) @ ( set_fo1817059534552279752at_rel @ A ) @ ( product_Pair @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) @ X @ ( product_Pair @ nat @ ( product_prod @ nat @ A ) @ Xa2 @ ( product_Pair @ nat @ A @ Xb @ Xc ) ) ) )
=> ~ ( ( ( ( ord_less @ nat @ Xb @ Xa2 )
=> ( Y = Xc ) )
& ( ~ ( ord_less @ nat @ Xb @ Xa2 )
=> ( Y
= ( set_fo6178422350223883121st_nat @ A @ X @ ( plus_plus @ nat @ Xa2 @ ( one_one @ nat ) ) @ Xb @ ( X @ Xa2 @ Xc ) ) ) ) )
=> ~ ( accp @ ( product_prod @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) ) @ ( set_fo1817059534552279752at_rel @ A ) @ ( product_Pair @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) @ X @ ( product_Pair @ nat @ ( product_prod @ nat @ A ) @ Xa2 @ ( product_Pair @ nat @ A @ Xb @ Xc ) ) ) ) ) ) ) ).
% fold_atLeastAtMost_nat.pelims
thf(fact_7259_fold__atLeastAtMost__nat_Opsimps,axiom,
! [A: $tType,F2: nat > A > A,A3: nat,B3: nat,Acc3: A] :
( ( accp @ ( product_prod @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) ) @ ( set_fo1817059534552279752at_rel @ A ) @ ( product_Pair @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) @ F2 @ ( product_Pair @ nat @ ( product_prod @ nat @ A ) @ A3 @ ( product_Pair @ nat @ A @ B3 @ Acc3 ) ) ) )
=> ( ( ( ord_less @ nat @ B3 @ A3 )
=> ( ( set_fo6178422350223883121st_nat @ A @ F2 @ A3 @ B3 @ Acc3 )
= Acc3 ) )
& ( ~ ( ord_less @ nat @ B3 @ A3 )
=> ( ( set_fo6178422350223883121st_nat @ A @ F2 @ A3 @ B3 @ Acc3 )
= ( set_fo6178422350223883121st_nat @ A @ F2 @ ( plus_plus @ nat @ A3 @ ( one_one @ nat ) ) @ B3 @ ( F2 @ A3 @ Acc3 ) ) ) ) ) ) ).
% fold_atLeastAtMost_nat.psimps
thf(fact_7260_fold__atLeastAtMost__nat_Opinduct,axiom,
! [A: $tType,A0: nat > A > A,A1: nat,A22: nat,A32: A,P: ( nat > A > A ) > nat > nat > A > $o] :
( ( accp @ ( product_prod @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) ) @ ( set_fo1817059534552279752at_rel @ A ) @ ( product_Pair @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) @ A0 @ ( product_Pair @ nat @ ( product_prod @ nat @ A ) @ A1 @ ( product_Pair @ nat @ A @ A22 @ A32 ) ) ) )
=> ( ! [F4: nat > A > A,A6: nat,B5: nat,Acc: A] :
( ( accp @ ( product_prod @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) ) @ ( set_fo1817059534552279752at_rel @ A ) @ ( product_Pair @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) @ F4 @ ( product_Pair @ nat @ ( product_prod @ nat @ A ) @ A6 @ ( product_Pair @ nat @ A @ B5 @ Acc ) ) ) )
=> ( ( ~ ( ord_less @ nat @ B5 @ A6 )
=> ( P @ F4 @ ( plus_plus @ nat @ A6 @ ( one_one @ nat ) ) @ B5 @ ( F4 @ A6 @ Acc ) ) )
=> ( P @ F4 @ A6 @ B5 @ Acc ) ) )
=> ( P @ A0 @ A1 @ A22 @ A32 ) ) ) ).
% fold_atLeastAtMost_nat.pinduct
thf(fact_7261_stable__sort__key__def,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ( ( linord3483353639454293061rt_key @ B @ A )
= ( ^ [Sk: ( B > A ) > ( list @ B ) > ( list @ B )] :
! [F3: B > A,Xs3: list @ B,K2: A] :
( ( filter2 @ B
@ ^ [Y6: B] :
( ( F3 @ Y6 )
= K2 )
@ ( Sk @ F3 @ Xs3 ) )
= ( filter2 @ B
@ ^ [Y6: B] :
( ( F3 @ Y6 )
= K2 )
@ Xs3 ) ) ) ) ) ).
% stable_sort_key_def
thf(fact_7262_independent__explicit__finite__subsets,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [A4: set @ A] :
( ( ~ ( real_V358717886546972837endent @ A @ A4 ) )
= ( ! [S6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ S6 @ A4 )
=> ( ( finite_finite2 @ A @ S6 )
=> ! [U2: A > real] :
( ( ( groups7311177749621191930dd_sum @ A @ A
@ ^ [V5: A] : ( real_V8093663219630862766scaleR @ A @ ( U2 @ V5 ) @ V5 )
@ S6 )
= ( zero_zero @ A ) )
=> ! [X4: A] :
( ( member @ A @ X4 @ S6 )
=> ( ( U2 @ X4 )
= ( zero_zero @ real ) ) ) ) ) ) ) ) ) ).
% independent_explicit_finite_subsets
thf(fact_7263_independent__empty,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ~ ( real_V358717886546972837endent @ A @ ( bot_bot @ ( set @ A ) ) ) ) ).
% independent_empty
thf(fact_7264_dependent__single,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [X: A] :
( ( real_V358717886546972837endent @ A @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( X
= ( zero_zero @ A ) ) ) ) ).
% dependent_single
thf(fact_7265_independent__Union__directed,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [C5: set @ ( set @ A )] :
( ! [C3: set @ A,D5: set @ A] :
( ( member @ ( set @ A ) @ C3 @ C5 )
=> ( ( member @ ( set @ A ) @ D5 @ C5 )
=> ( ( ord_less_eq @ ( set @ A ) @ C3 @ D5 )
| ( ord_less_eq @ ( set @ A ) @ D5 @ C3 ) ) ) )
=> ( ! [C3: set @ A] :
( ( member @ ( set @ A ) @ C3 @ C5 )
=> ~ ( real_V358717886546972837endent @ A @ C3 ) )
=> ~ ( real_V358717886546972837endent @ A @ ( complete_Sup_Sup @ ( set @ A ) @ C5 ) ) ) ) ) ).
% independent_Union_directed
thf(fact_7266_independent__mono,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [A4: set @ A,B2: set @ A] :
( ~ ( real_V358717886546972837endent @ A @ A4 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ A4 )
=> ~ ( real_V358717886546972837endent @ A @ B2 ) ) ) ) ).
% independent_mono
thf(fact_7267_dependent__mono,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [B2: set @ A,A4: set @ A] :
( ( real_V358717886546972837endent @ A @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ A4 )
=> ( real_V358717886546972837endent @ A @ A4 ) ) ) ) ).
% dependent_mono
thf(fact_7268_dependent__zero,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [A4: set @ A] :
( ( member @ A @ ( zero_zero @ A ) @ A4 )
=> ( real_V358717886546972837endent @ A @ A4 ) ) ) ).
% dependent_zero
thf(fact_7269_unique__representation,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [Basis: set @ A,F2: A > real,G: A > real] :
( ~ ( real_V358717886546972837endent @ A @ Basis )
=> ( ! [V3: A] :
( ( ( F2 @ V3 )
!= ( zero_zero @ real ) )
=> ( member @ A @ V3 @ Basis ) )
=> ( ! [V3: A] :
( ( ( G @ V3 )
!= ( zero_zero @ real ) )
=> ( member @ A @ V3 @ Basis ) )
=> ( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [V5: A] :
( ( F2 @ V5 )
!= ( zero_zero @ real ) ) ) )
=> ( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [V5: A] :
( ( G @ V5 )
!= ( zero_zero @ real ) ) ) )
=> ( ( ( groups7311177749621191930dd_sum @ A @ A
@ ^ [V5: A] : ( real_V8093663219630862766scaleR @ A @ ( F2 @ V5 ) @ V5 )
@ ( collect @ A
@ ^ [V5: A] :
( ( F2 @ V5 )
!= ( zero_zero @ real ) ) ) )
= ( groups7311177749621191930dd_sum @ A @ A
@ ^ [V5: A] : ( real_V8093663219630862766scaleR @ A @ ( G @ V5 ) @ V5 )
@ ( collect @ A
@ ^ [V5: A] :
( ( G @ V5 )
!= ( zero_zero @ real ) ) ) ) )
=> ( F2 = G ) ) ) ) ) ) ) ) ).
% unique_representation
thf(fact_7270_dependent__finite,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [S2: set @ A] :
( ( finite_finite2 @ A @ S2 )
=> ( ( real_V358717886546972837endent @ A @ S2 )
= ( ? [U2: A > real] :
( ? [X4: A] :
( ( member @ A @ X4 @ S2 )
& ( ( U2 @ X4 )
!= ( zero_zero @ real ) ) )
& ( ( groups7311177749621191930dd_sum @ A @ A
@ ^ [V5: A] : ( real_V8093663219630862766scaleR @ A @ ( U2 @ V5 ) @ V5 )
@ S2 )
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% dependent_finite
thf(fact_7271_independent__if__scalars__zero,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [A4: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( ! [F4: A > real,X5: A] :
( ( ( groups7311177749621191930dd_sum @ A @ A
@ ^ [Y6: A] : ( real_V8093663219630862766scaleR @ A @ ( F4 @ Y6 ) @ Y6 )
@ A4 )
= ( zero_zero @ A ) )
=> ( ( member @ A @ X5 @ A4 )
=> ( ( F4 @ X5 )
= ( zero_zero @ real ) ) ) )
=> ~ ( real_V358717886546972837endent @ A @ A4 ) ) ) ) ).
% independent_if_scalars_zero
thf(fact_7272_independentD__unique,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [B2: set @ A,X8: A > real,Y7: A > real] :
( ~ ( real_V358717886546972837endent @ A @ B2 )
=> ( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X4: A] :
( ( X8 @ X4 )
!= ( zero_zero @ real ) ) ) )
=> ( ( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [X4: A] :
( ( X8 @ X4 )
!= ( zero_zero @ real ) ) )
@ B2 )
=> ( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X4: A] :
( ( Y7 @ X4 )
!= ( zero_zero @ real ) ) ) )
=> ( ( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [X4: A] :
( ( Y7 @ X4 )
!= ( zero_zero @ real ) ) )
@ B2 )
=> ( ( ( groups7311177749621191930dd_sum @ A @ A
@ ^ [X4: A] : ( real_V8093663219630862766scaleR @ A @ ( X8 @ X4 ) @ X4 )
@ ( collect @ A
@ ^ [X4: A] :
( ( X8 @ X4 )
!= ( zero_zero @ real ) ) ) )
= ( groups7311177749621191930dd_sum @ A @ A
@ ^ [X4: A] : ( real_V8093663219630862766scaleR @ A @ ( Y7 @ X4 ) @ X4 )
@ ( collect @ A
@ ^ [X4: A] :
( ( Y7 @ X4 )
!= ( zero_zero @ real ) ) ) ) )
=> ( X8 = Y7 ) ) ) ) ) ) ) ) ).
% independentD_unique
thf(fact_7273_stable__sort__key__sort__key,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ B )
=> ( linord3483353639454293061rt_key @ A @ B @ ( linorder_sort_key @ A @ B ) ) ) ).
% stable_sort_key_sort_key
thf(fact_7274_independentD,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [S: set @ A,T2: set @ A,U: A > real,V2: A] :
( ~ ( real_V358717886546972837endent @ A @ S )
=> ( ( finite_finite2 @ A @ T2 )
=> ( ( ord_less_eq @ ( set @ A ) @ T2 @ S )
=> ( ( ( groups7311177749621191930dd_sum @ A @ A
@ ^ [V5: A] : ( real_V8093663219630862766scaleR @ A @ ( U @ V5 ) @ V5 )
@ T2 )
= ( zero_zero @ A ) )
=> ( ( member @ A @ V2 @ T2 )
=> ( ( U @ V2 )
= ( zero_zero @ real ) ) ) ) ) ) ) ) ).
% independentD
thf(fact_7275_dependent__alt,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ( ( real_V358717886546972837endent @ A )
= ( ^ [B6: set @ A] :
? [X6: A > real] :
( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X4: A] :
( ( X6 @ X4 )
!= ( zero_zero @ real ) ) ) )
& ( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [X4: A] :
( ( X6 @ X4 )
!= ( zero_zero @ real ) ) )
@ B6 )
& ( ( groups7311177749621191930dd_sum @ A @ A
@ ^ [X4: A] : ( real_V8093663219630862766scaleR @ A @ ( X6 @ X4 ) @ X4 )
@ ( collect @ A
@ ^ [X4: A] :
( ( X6 @ X4 )
!= ( zero_zero @ real ) ) ) )
= ( zero_zero @ A ) )
& ? [X4: A] :
( ( X6 @ X4 )
!= ( zero_zero @ real ) ) ) ) ) ) ).
% dependent_alt
thf(fact_7276_independent__alt,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [B2: set @ A] :
( ( ~ ( real_V358717886546972837endent @ A @ B2 ) )
= ( ! [X6: A > real] :
( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X4: A] :
( ( X6 @ X4 )
!= ( zero_zero @ real ) ) ) )
=> ( ( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [X4: A] :
( ( X6 @ X4 )
!= ( zero_zero @ real ) ) )
@ B2 )
=> ( ( ( groups7311177749621191930dd_sum @ A @ A
@ ^ [X4: A] : ( real_V8093663219630862766scaleR @ A @ ( X6 @ X4 ) @ X4 )
@ ( collect @ A
@ ^ [X4: A] :
( ( X6 @ X4 )
!= ( zero_zero @ real ) ) ) )
= ( zero_zero @ A ) )
=> ! [X4: A] :
( ( X6 @ X4 )
= ( zero_zero @ real ) ) ) ) ) ) ) ) ).
% independent_alt
thf(fact_7277_independentD__alt,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [B2: set @ A,X8: A > real,X: A] :
( ~ ( real_V358717886546972837endent @ A @ B2 )
=> ( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X4: A] :
( ( X8 @ X4 )
!= ( zero_zero @ real ) ) ) )
=> ( ( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [X4: A] :
( ( X8 @ X4 )
!= ( zero_zero @ real ) ) )
@ B2 )
=> ( ( ( groups7311177749621191930dd_sum @ A @ A
@ ^ [X4: A] : ( real_V8093663219630862766scaleR @ A @ ( X8 @ X4 ) @ X4 )
@ ( collect @ A
@ ^ [X4: A] :
( ( X8 @ X4 )
!= ( zero_zero @ real ) ) ) )
= ( zero_zero @ A ) )
=> ( ( X8 @ X )
= ( zero_zero @ real ) ) ) ) ) ) ) ).
% independentD_alt
thf(fact_7278_dependent__explicit,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ( ( real_V358717886546972837endent @ A )
= ( ^ [S7: set @ A] :
? [T3: set @ A] :
( ( finite_finite2 @ A @ T3 )
& ( ord_less_eq @ ( set @ A ) @ T3 @ S7 )
& ? [U2: A > real] :
( ( ( groups7311177749621191930dd_sum @ A @ A
@ ^ [V5: A] : ( real_V8093663219630862766scaleR @ A @ ( U2 @ V5 ) @ V5 )
@ T3 )
= ( zero_zero @ A ) )
& ? [X4: A] :
( ( member @ A @ X4 @ T3 )
& ( ( U2 @ X4 )
!= ( zero_zero @ real ) ) ) ) ) ) ) ) ).
% dependent_explicit
thf(fact_7279_independent__explicit__module,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [S: set @ A] :
( ( ~ ( real_V358717886546972837endent @ A @ S ) )
= ( ! [T3: set @ A,U2: A > real,V5: A] :
( ( finite_finite2 @ A @ T3 )
=> ( ( ord_less_eq @ ( set @ A ) @ T3 @ S )
=> ( ( ( groups7311177749621191930dd_sum @ A @ A
@ ^ [W3: A] : ( real_V8093663219630862766scaleR @ A @ ( U2 @ W3 ) @ W3 )
@ T3 )
= ( zero_zero @ A ) )
=> ( ( member @ A @ V5 @ T3 )
=> ( ( U2 @ V5 )
= ( zero_zero @ real ) ) ) ) ) ) ) ) ) ).
% independent_explicit_module
thf(fact_7280_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_7281_possible__bit__def,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ( ( bit_se6407376104438227557le_bit @ A )
= ( ^ [Tyrep: itself @ A,N: nat] :
( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N )
!= ( zero_zero @ A ) ) ) ) ) ).
% possible_bit_def
thf(fact_7282_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_7283_possible__bit__less__imp,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [Tyrep2: itself @ A,I: nat,J: nat] :
( ( bit_se6407376104438227557le_bit @ A @ Tyrep2 @ I )
=> ( ( ord_less_eq @ nat @ J @ I )
=> ( bit_se6407376104438227557le_bit @ A @ Tyrep2 @ J ) ) ) ) ).
% possible_bit_less_imp
thf(fact_7284_dbl__inc__def,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( ( neg_numeral_dbl_inc @ A )
= ( ^ [X4: A] : ( plus_plus @ A @ ( plus_plus @ A @ X4 @ X4 ) @ ( one_one @ A ) ) ) ) ) ).
% dbl_inc_def
thf(fact_7285_drop__bit__exp__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M2: nat,N2: nat] :
( ( bit_se4197421643247451524op_bit @ A @ M2 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) )
= ( times_times @ A
@ ( zero_neq_one_of_bool @ A
@ ( ( ord_less_eq @ nat @ M2 @ N2 )
& ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N2 ) ) )
@ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ N2 @ M2 ) ) ) ) ) ).
% drop_bit_exp_eq
thf(fact_7286_bit__minus__2__iff,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N2: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ N2 )
= ( ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N2 )
& ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ) ).
% bit_minus_2_iff
thf(fact_7287_CHAR__eq__0,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ( ( semiri4206861660011772517g_char @ A @ ( type2 @ A ) )
= ( zero_zero @ nat ) ) ) ).
% CHAR_eq_0
thf(fact_7288_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_7289_bit__mask__iff,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M2: nat,N2: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( bit_se2239418461657761734s_mask @ A @ M2 ) @ N2 )
= ( ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N2 )
& ( ord_less @ nat @ N2 @ M2 ) ) ) ) ).
% bit_mask_iff
thf(fact_7290_CHAR__eqI,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [C2: nat] :
( ( ( semiring_1_of_nat @ A @ C2 )
= ( zero_zero @ A ) )
=> ( ! [X5: nat] :
( ( ( semiring_1_of_nat @ A @ X5 )
= ( zero_zero @ A ) )
=> ( dvd_dvd @ nat @ C2 @ X5 ) )
=> ( ( semiri4206861660011772517g_char @ A @ ( type2 @ A ) )
= C2 ) ) ) ) ).
% CHAR_eqI
thf(fact_7291_of__nat__eq__0__iff__char__dvd,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [N2: nat] :
( ( ( semiring_1_of_nat @ A @ N2 )
= ( zero_zero @ A ) )
= ( dvd_dvd @ nat @ ( semiri4206861660011772517g_char @ A @ ( type2 @ A ) ) @ N2 ) ) ) ).
% of_nat_eq_0_iff_char_dvd
thf(fact_7292_CHAR__eq0__iff,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ( ( ( semiri4206861660011772517g_char @ A @ ( type2 @ A ) )
= ( zero_zero @ nat ) )
= ( ! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( semiring_1_of_nat @ A @ N )
!= ( zero_zero @ A ) ) ) ) ) ) ).
% CHAR_eq0_iff
thf(fact_7293_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 ) )
=> ( ! [X5: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ X5 )
=> ( ( ord_less @ nat @ X5 @ C2 )
=> ( ( semiring_1_of_nat @ A @ X5 )
!= ( zero_zero @ A ) ) ) )
=> ( ( semiri4206861660011772517g_char @ A @ ( type2 @ A ) )
= C2 ) ) ) ) ) ).
% CHAR_eq_posI
thf(fact_7294_CHAR__pos__iff,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( semiri4206861660011772517g_char @ A @ ( type2 @ A ) ) )
= ( ? [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
& ( ( semiring_1_of_nat @ A @ N )
= ( zero_zero @ A ) ) ) ) ) ) ).
% CHAR_pos_iff
thf(fact_7295_bit__push__bit__iff,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M2: nat,A3: A,N2: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( bit_se4730199178511100633sh_bit @ A @ M2 @ A3 ) @ N2 )
= ( ( ord_less_eq @ nat @ M2 @ N2 )
& ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N2 )
& ( bit_se5641148757651400278ts_bit @ A @ A3 @ ( minus_minus @ nat @ N2 @ M2 ) ) ) ) ) ).
% bit_push_bit_iff
thf(fact_7296_fold__possible__bit,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [N2: nat] :
( ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 )
= ( zero_zero @ A ) )
= ( ~ ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N2 ) ) ) ) ).
% fold_possible_bit
thf(fact_7297_bit__minus__exp__iff,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [M2: nat,N2: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( uminus_uminus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M2 ) ) @ N2 )
= ( ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N2 )
& ( ord_less_eq @ nat @ M2 @ N2 ) ) ) ) ).
% bit_minus_exp_iff
thf(fact_7298_bit__mask__sub__iff,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [M2: nat,N2: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M2 ) @ ( one_one @ A ) ) @ N2 )
= ( ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N2 )
& ( ord_less @ nat @ N2 @ M2 ) ) ) ) ).
% bit_mask_sub_iff
thf(fact_7299_bit__double__iff,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A3: A,N2: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) @ N2 )
= ( ( bit_se5641148757651400278ts_bit @ A @ A3 @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) )
& ( N2
!= ( zero_zero @ nat ) )
& ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N2 ) ) ) ) ).
% bit_double_iff
thf(fact_7300_Image__fold,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S2: set @ A] :
( ( finite_finite2 @ ( product_prod @ A @ B ) @ R )
=> ( ( image @ A @ B @ R @ S2 )
= ( finite_fold @ ( product_prod @ A @ B ) @ ( set @ B )
@ ( product_case_prod @ A @ B @ ( ( set @ B ) > ( set @ B ) )
@ ^ [X4: A,Y6: B,A7: set @ B] : ( if @ ( set @ B ) @ ( member @ A @ X4 @ S2 ) @ ( insert2 @ B @ Y6 @ A7 ) @ A7 ) )
@ ( bot_bot @ ( set @ B ) )
@ R ) ) ) ).
% Image_fold
thf(fact_7301_map__rec,axiom,
! [A: $tType,B: $tType] :
( ( map @ B @ A )
= ( ^ [F3: B > A] :
( rec_list @ ( list @ A ) @ B @ ( nil @ A )
@ ^ [X4: B,Uu3: list @ B] : ( cons @ A @ ( F3 @ X4 ) ) ) ) ) ).
% map_rec
thf(fact_7302_Image__empty2,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ B @ A )] :
( ( image @ B @ A @ R @ ( bot_bot @ ( set @ B ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Image_empty2
thf(fact_7303_Image__empty1,axiom,
! [B: $tType,A: $tType,X8: set @ B] :
( ( image @ B @ A @ ( bot_bot @ ( set @ ( product_prod @ B @ A ) ) ) @ X8 )
= ( bot_bot @ ( set @ A ) ) ) ).
% Image_empty1
thf(fact_7304_Image__singleton__iff,axiom,
! [A: $tType,B: $tType,B3: A,R2: set @ ( product_prod @ B @ A ),A3: B] :
( ( member @ A @ B3 @ ( image @ B @ A @ R2 @ ( insert2 @ B @ A3 @ ( bot_bot @ ( set @ B ) ) ) ) )
= ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ A3 @ B3 ) @ R2 ) ) ).
% Image_singleton_iff
thf(fact_7305_listrel__Nil,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ B @ A )] :
( ( image @ ( list @ B ) @ ( list @ A ) @ ( listrel @ B @ A @ R2 ) @ ( insert2 @ ( list @ B ) @ ( nil @ B ) @ ( bot_bot @ ( set @ ( list @ B ) ) ) ) )
= ( insert2 @ ( list @ A ) @ ( nil @ A ) @ ( bot_bot @ ( set @ ( list @ A ) ) ) ) ) ).
% listrel_Nil
thf(fact_7306_wfI__pf,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ! [A8: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A8 @ ( image @ A @ A @ R @ A8 ) )
=> ( A8
= ( bot_bot @ ( set @ A ) ) ) )
=> ( wf @ A @ R ) ) ).
% wfI_pf
thf(fact_7307_wfE__pf,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),A4: set @ A] :
( ( wf @ A @ R )
=> ( ( ord_less_eq @ ( set @ A ) @ A4 @ ( image @ A @ A @ R @ A4 ) )
=> ( A4
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% wfE_pf
thf(fact_7308_countable__Image,axiom,
! [B: $tType,A: $tType,Y7: set @ A,X8: set @ ( product_prod @ A @ B )] :
( ! [Y3: A] :
( ( member @ A @ Y3 @ Y7 )
=> ( countable_countable @ B @ ( image @ A @ B @ X8 @ ( insert2 @ A @ Y3 @ ( bot_bot @ ( set @ A ) ) ) ) ) )
=> ( ( countable_countable @ A @ Y7 )
=> ( countable_countable @ B @ ( image @ A @ B @ X8 @ Y7 ) ) ) ) ).
% countable_Image
thf(fact_7309_Image__Int__subset,axiom,
! [A: $tType,B: $tType,R: set @ ( product_prod @ B @ A ),A4: set @ B,B2: set @ B] : ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ R @ ( inf_inf @ ( set @ B ) @ A4 @ B2 ) ) @ ( inf_inf @ ( set @ A ) @ ( image @ B @ A @ R @ A4 ) @ ( image @ B @ A @ R @ B2 ) ) ) ).
% Image_Int_subset
thf(fact_7310_list_Osimps_I7_J,axiom,
! [C: $tType,A: $tType,F1: C,F22: A > ( list @ A ) > C > C,X21: A,X222: list @ A] :
( ( rec_list @ C @ A @ F1 @ F22 @ ( cons @ A @ X21 @ X222 ) )
= ( F22 @ X21 @ X222 @ ( rec_list @ C @ A @ F1 @ F22 @ X222 ) ) ) ).
% list.simps(7)
thf(fact_7311_finite__Image,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),A4: set @ A] :
( ( finite_finite2 @ ( product_prod @ A @ B ) @ R )
=> ( finite_finite2 @ B @ ( image @ A @ B @ R @ A4 ) ) ) ).
% finite_Image
thf(fact_7312_list_Osimps_I6_J,axiom,
! [A: $tType,C: $tType,F1: C,F22: A > ( list @ A ) > C > C] :
( ( rec_list @ C @ A @ F1 @ F22 @ ( nil @ A ) )
= F1 ) ).
% list.simps(6)
thf(fact_7313_Image__mono,axiom,
! [B: $tType,A: $tType,R6: set @ ( product_prod @ A @ B ),R2: set @ ( product_prod @ A @ B ),A16: set @ A,A4: set @ A] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R6 @ R2 )
=> ( ( ord_less_eq @ ( set @ A ) @ A16 @ A4 )
=> ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ R6 @ A16 ) @ ( image @ A @ B @ R2 @ A4 ) ) ) ) ).
% Image_mono
thf(fact_7314_Image__closed__trancl,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),X8: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ R2 @ X8 ) @ X8 )
=> ( ( image @ A @ A @ ( transitive_rtrancl @ A @ R2 ) @ X8 )
= X8 ) ) ).
% Image_closed_trancl
thf(fact_7315_equiv__class__self,axiom,
! [A: $tType,A4: set @ A,R2: set @ ( product_prod @ A @ A ),A3: A] :
( ( equiv_equiv @ A @ A4 @ R2 )
=> ( ( member @ A @ A3 @ A4 )
=> ( member @ A @ A3 @ ( image @ A @ A @ R2 @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% equiv_class_self
thf(fact_7316_quotientE,axiom,
! [A: $tType,X8: set @ A,A4: set @ A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( set @ A ) @ X8 @ ( equiv_quotient @ A @ A4 @ R2 ) )
=> ~ ! [X5: A] :
( ( X8
= ( image @ A @ A @ R2 @ ( insert2 @ A @ X5 @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ~ ( member @ A @ X5 @ A4 ) ) ) ).
% quotientE
thf(fact_7317_quotientI,axiom,
! [A: $tType,X: A,A4: set @ A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ A @ X @ A4 )
=> ( member @ ( set @ A ) @ ( image @ A @ A @ R2 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) @ ( equiv_quotient @ A @ A4 @ R2 ) ) ) ).
% quotientI
thf(fact_7318_finite__rtrancl__Image,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),A4: set @ A] :
( ( finite_finite2 @ ( product_prod @ A @ A ) @ R )
=> ( ( finite_finite2 @ A @ A4 )
=> ( finite_finite2 @ A @ ( image @ A @ A @ ( transitive_rtrancl @ A @ R ) @ A4 ) ) ) ) ).
% finite_rtrancl_Image
thf(fact_7319_Image__singleton,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ B @ A ),A3: B] :
( ( image @ B @ A @ R2 @ ( insert2 @ B @ A3 @ ( bot_bot @ ( set @ B ) ) ) )
= ( collect @ A
@ ^ [B4: A] : ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ A3 @ B4 ) @ R2 ) ) ) ).
% Image_singleton
thf(fact_7320_proj__def,axiom,
! [A: $tType,B: $tType] :
( ( equiv_proj @ B @ A )
= ( ^ [R4: set @ ( product_prod @ B @ A ),X4: B] : ( image @ B @ A @ R4 @ ( insert2 @ B @ X4 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ).
% proj_def
thf(fact_7321_Image__INT__subset,axiom,
! [A: $tType,B: $tType,C: $tType,R2: set @ ( product_prod @ B @ A ),B2: C > ( set @ B ),A4: set @ C] :
( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ R2 @ ( complete_Inf_Inf @ ( set @ B ) @ ( image2 @ C @ ( set @ B ) @ B2 @ A4 ) ) )
@ ( complete_Inf_Inf @ ( set @ A )
@ ( image2 @ C @ ( set @ A )
@ ^ [X4: C] : ( image @ B @ A @ R2 @ ( B2 @ X4 ) )
@ A4 ) ) ) ).
% Image_INT_subset
thf(fact_7322_Image__subset,axiom,
! [A: $tType,B: $tType,R2: set @ ( product_prod @ A @ B ),A4: set @ A,B2: set @ B,C5: set @ A] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R2
@ ( product_Sigma @ A @ B @ A4
@ ^ [Uu3: A] : B2 ) )
=> ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ R2 @ C5 ) @ B2 ) ) ).
% Image_subset
thf(fact_7323_listrel__Cons,axiom,
! [A: $tType,B: $tType,R2: set @ ( product_prod @ B @ A ),X: B,Xs: list @ B] :
( ( image @ ( list @ B ) @ ( list @ A ) @ ( listrel @ B @ A @ R2 ) @ ( insert2 @ ( list @ B ) @ ( cons @ B @ X @ Xs ) @ ( bot_bot @ ( set @ ( list @ B ) ) ) ) )
= ( set_Cons @ A @ ( image @ B @ A @ R2 @ ( insert2 @ B @ X @ ( bot_bot @ ( set @ B ) ) ) ) @ ( image @ ( list @ B ) @ ( list @ A ) @ ( listrel @ B @ A @ R2 ) @ ( insert2 @ ( list @ B ) @ Xs @ ( bot_bot @ ( set @ ( list @ B ) ) ) ) ) ) ) ).
% listrel_Cons
thf(fact_7324_equiv__class__eq__iff,axiom,
! [A: $tType,A4: set @ A,R2: set @ ( product_prod @ A @ A ),X: A,Y: A] :
( ( equiv_equiv @ A @ A4 @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R2 )
= ( ( ( image @ A @ A @ R2 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( image @ A @ A @ R2 @ ( insert2 @ A @ Y @ ( bot_bot @ ( set @ A ) ) ) ) )
& ( member @ A @ X @ A4 )
& ( member @ A @ Y @ A4 ) ) ) ) ).
% equiv_class_eq_iff
thf(fact_7325_eq__equiv__class__iff,axiom,
! [A: $tType,A4: set @ A,R2: set @ ( product_prod @ A @ A ),X: A,Y: A] :
( ( equiv_equiv @ A @ A4 @ R2 )
=> ( ( member @ A @ X @ A4 )
=> ( ( member @ A @ Y @ A4 )
=> ( ( ( image @ A @ A @ R2 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( image @ A @ A @ R2 @ ( insert2 @ A @ Y @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R2 ) ) ) ) ) ).
% eq_equiv_class_iff
thf(fact_7326_equiv__class__eq,axiom,
! [A: $tType,A4: set @ A,R2: set @ ( product_prod @ A @ A ),A3: A,B3: A] :
( ( equiv_equiv @ A @ A4 @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B3 ) @ R2 )
=> ( ( image @ A @ A @ R2 @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( image @ A @ A @ R2 @ ( insert2 @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% equiv_class_eq
thf(fact_7327_eq__equiv__class,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A3: A,B3: A,A4: set @ A] :
( ( ( image @ A @ A @ R2 @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( image @ A @ A @ R2 @ ( insert2 @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( ( equiv_equiv @ A @ A4 @ R2 )
=> ( ( member @ A @ B3 @ A4 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B3 ) @ R2 ) ) ) ) ).
% eq_equiv_class
thf(fact_7328_refines__equiv__class__eq,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),S2: set @ ( product_prod @ A @ A ),A4: set @ A,A3: A] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R @ S2 )
=> ( ( equiv_equiv @ A @ A4 @ R )
=> ( ( equiv_equiv @ A @ A4 @ S2 )
=> ( ( image @ A @ A @ R @ ( image @ A @ A @ S2 @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( image @ A @ A @ S2 @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% refines_equiv_class_eq
thf(fact_7329_refines__equiv__class__eq2,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),S2: set @ ( product_prod @ A @ A ),A4: set @ A,A3: A] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R @ S2 )
=> ( ( equiv_equiv @ A @ A4 @ R )
=> ( ( equiv_equiv @ A @ A4 @ S2 )
=> ( ( image @ A @ A @ S2 @ ( image @ A @ A @ R @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( image @ A @ A @ S2 @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% refines_equiv_class_eq2
thf(fact_7330_Partial__order__eq__Image1__Image1__iff,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A3: A,B3: A] :
( ( order_7125193373082350890der_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( member @ A @ A3 @ ( field2 @ A @ R2 ) )
=> ( ( member @ A @ B3 @ ( field2 @ A @ R2 ) )
=> ( ( ( image @ A @ A @ R2 @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( image @ A @ A @ R2 @ ( insert2 @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( A3 = B3 ) ) ) ) ) ).
% Partial_order_eq_Image1_Image1_iff
thf(fact_7331_Image__eq__UN,axiom,
! [A: $tType,B: $tType] :
( ( image @ B @ A )
= ( ^ [R4: set @ ( product_prod @ B @ A ),B6: set @ B] :
( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [Y6: B] : ( image @ B @ A @ R4 @ ( insert2 @ B @ Y6 @ ( bot_bot @ ( set @ B ) ) ) )
@ B6 ) ) ) ) ).
% Image_eq_UN
thf(fact_7332_subset__equiv__class,axiom,
! [A: $tType,A4: set @ A,R2: set @ ( product_prod @ A @ A ),B3: A,A3: A] :
( ( equiv_equiv @ A @ A4 @ R2 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ R2 @ ( insert2 @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( image @ A @ A @ R2 @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( ( member @ A @ B3 @ A4 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B3 ) @ R2 ) ) ) ) ).
% subset_equiv_class
thf(fact_7333_equiv__class__subset,axiom,
! [A: $tType,A4: set @ A,R2: set @ ( product_prod @ A @ A ),A3: A,B3: A] :
( ( equiv_equiv @ A @ A4 @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B3 ) @ R2 )
=> ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ R2 @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( image @ A @ A @ R2 @ ( insert2 @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% equiv_class_subset
thf(fact_7334_equiv__class__nondisjoint,axiom,
! [A: $tType,A4: set @ A,R2: set @ ( product_prod @ A @ A ),X: A,A3: A,B3: A] :
( ( equiv_equiv @ A @ A4 @ R2 )
=> ( ( member @ A @ X @ ( inf_inf @ ( set @ A ) @ ( image @ A @ A @ R2 @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( image @ A @ A @ R2 @ ( insert2 @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B3 ) @ R2 ) ) ) ).
% equiv_class_nondisjoint
thf(fact_7335_singleton__quotient,axiom,
! [A: $tType,X: A,R2: set @ ( product_prod @ A @ A )] :
( ( equiv_quotient @ A @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) @ R2 )
= ( insert2 @ ( set @ A ) @ ( image @ A @ A @ R2 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) @ ( bot_bot @ ( set @ ( set @ A ) ) ) ) ) ).
% singleton_quotient
thf(fact_7336_list_Orec__o__map,axiom,
! [C: $tType,B: $tType,A: $tType,G: C,Ga: B > ( list @ B ) > C > C,F2: A > B] :
( ( comp @ ( list @ B ) @ C @ ( list @ A ) @ ( rec_list @ C @ B @ G @ Ga ) @ ( map @ A @ B @ F2 ) )
= ( rec_list @ C @ A @ G
@ ^ [X4: A,Xa4: list @ A] : ( Ga @ ( F2 @ X4 ) @ ( map @ A @ B @ F2 @ Xa4 ) ) ) ) ).
% list.rec_o_map
thf(fact_7337_quotient__def,axiom,
! [A: $tType] :
( ( equiv_quotient @ A )
= ( ^ [A7: set @ A,R4: set @ ( product_prod @ A @ A )] :
( complete_Sup_Sup @ ( set @ ( set @ A ) )
@ ( image2 @ A @ ( set @ ( set @ A ) )
@ ^ [X4: A] : ( insert2 @ ( set @ A ) @ ( image @ A @ A @ R4 @ ( insert2 @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( bot_bot @ ( set @ ( set @ A ) ) ) )
@ A7 ) ) ) ) ).
% quotient_def
thf(fact_7338_UN__equiv__class2,axiom,
! [A: $tType,C: $tType,B: $tType,A18: set @ A,R1: set @ ( product_prod @ A @ A ),A26: set @ B,R22: set @ ( product_prod @ B @ B ),F2: A > B > ( set @ C ),A1: A,A22: B] :
( ( equiv_equiv @ A @ A18 @ R1 )
=> ( ( equiv_equiv @ B @ A26 @ R22 )
=> ( ( equiv_congruent2 @ A @ B @ ( set @ C ) @ R1 @ R22 @ F2 )
=> ( ( member @ A @ A1 @ A18 )
=> ( ( member @ B @ A22 @ A26 )
=> ( ( complete_Sup_Sup @ ( set @ C )
@ ( image2 @ A @ ( set @ C )
@ ^ [X15: A] : ( complete_Sup_Sup @ ( set @ C ) @ ( image2 @ B @ ( set @ C ) @ ( F2 @ X15 ) @ ( image @ B @ B @ R22 @ ( insert2 @ B @ A22 @ ( bot_bot @ ( set @ B ) ) ) ) ) )
@ ( image @ A @ A @ R1 @ ( insert2 @ A @ A1 @ ( bot_bot @ ( set @ A ) ) ) ) ) )
= ( F2 @ A1 @ A22 ) ) ) ) ) ) ) ).
% UN_equiv_class2
thf(fact_7339_UN__equiv__class,axiom,
! [B: $tType,A: $tType,A4: set @ A,R2: set @ ( product_prod @ A @ A ),F2: A > ( set @ B ),A3: A] :
( ( equiv_equiv @ A @ A4 @ R2 )
=> ( ( equiv_congruent @ A @ ( set @ B ) @ R2 @ F2 )
=> ( ( member @ A @ A3 @ A4 )
=> ( ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ F2 @ ( image @ A @ A @ R2 @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) )
= ( F2 @ A3 ) ) ) ) ) ).
% UN_equiv_class
thf(fact_7340_congruent2__implies__congruent__UN,axiom,
! [B: $tType,C: $tType,A: $tType,A18: set @ A,R1: set @ ( product_prod @ A @ A ),A26: set @ B,R22: set @ ( product_prod @ B @ B ),F2: A > B > ( set @ C ),A3: B] :
( ( equiv_equiv @ A @ A18 @ R1 )
=> ( ( equiv_equiv @ B @ A26 @ R22 )
=> ( ( equiv_congruent2 @ A @ B @ ( set @ C ) @ R1 @ R22 @ F2 )
=> ( ( member @ B @ A3 @ A26 )
=> ( equiv_congruent @ A @ ( set @ C ) @ R1
@ ^ [X15: A] : ( complete_Sup_Sup @ ( set @ C ) @ ( image2 @ B @ ( set @ C ) @ ( F2 @ X15 ) @ ( image @ B @ B @ R22 @ ( insert2 @ B @ A3 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ) ) ).
% congruent2_implies_congruent_UN
thf(fact_7341_set__rec,axiom,
! [A: $tType] :
( ( set2 @ A )
= ( rec_list @ ( set @ A ) @ A @ ( bot_bot @ ( set @ A ) )
@ ^ [X4: A,Uu3: list @ A] : ( insert2 @ A @ X4 ) ) ) ).
% set_rec
thf(fact_7342_subset__Image1__Image1__iff,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A3: A,B3: A] :
( ( order_preorder_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( member @ A @ A3 @ ( field2 @ A @ R2 ) )
=> ( ( member @ A @ B3 @ ( field2 @ A @ R2 ) )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ R2 @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( image @ A @ A @ R2 @ ( insert2 @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B3 @ A3 ) @ R2 ) ) ) ) ) ).
% subset_Image1_Image1_iff
thf(fact_7343_subset__Image__Image__iff,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A4: set @ A,B2: set @ A] :
( ( order_preorder_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( ord_less_eq @ ( set @ A ) @ A4 @ ( field2 @ A @ R2 ) )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ ( field2 @ A @ R2 ) )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ R2 @ A4 ) @ ( image @ A @ A @ R2 @ B2 ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ A4 )
=> ? [Y6: A] :
( ( member @ A @ Y6 @ B2 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y6 @ X4 ) @ R2 ) ) ) ) ) ) ) ) ).
% subset_Image_Image_iff
thf(fact_7344_preorder__on__empty,axiom,
! [A: $tType] : ( order_preorder_on @ A @ ( bot_bot @ ( set @ A ) ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ).
% preorder_on_empty
thf(fact_7345_Refl__antisym__eq__Image1__Image1__iff,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A3: A,B3: A] :
( ( refl_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( antisym @ A @ R2 )
=> ( ( member @ A @ A3 @ ( field2 @ A @ R2 ) )
=> ( ( member @ A @ B3 @ ( field2 @ A @ R2 ) )
=> ( ( ( image @ A @ A @ R2 @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( image @ A @ A @ R2 @ ( insert2 @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( A3 = B3 ) ) ) ) ) ) ).
% Refl_antisym_eq_Image1_Image1_iff
thf(fact_7346_append__butlast__last__id,axiom,
! [A: $tType,Xs: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( append @ A @ ( butlast @ A @ Xs ) @ ( cons @ A @ ( last @ A @ Xs ) @ ( nil @ A ) ) )
= Xs ) ) ).
% append_butlast_last_id
thf(fact_7347_last__remdups__adj,axiom,
! [A: $tType,Xs: list @ A] :
( ( last @ A @ ( remdups_adj @ A @ Xs ) )
= ( last @ A @ Xs ) ) ).
% last_remdups_adj
thf(fact_7348_last__appendL,axiom,
! [A: $tType,Ys: list @ A,Xs: list @ A] :
( ( Ys
= ( nil @ A ) )
=> ( ( last @ A @ ( append @ A @ Xs @ Ys ) )
= ( last @ A @ Xs ) ) ) ).
% last_appendL
thf(fact_7349_last__appendR,axiom,
! [A: $tType,Ys: list @ A,Xs: list @ A] :
( ( Ys
!= ( nil @ A ) )
=> ( ( last @ A @ ( append @ A @ Xs @ Ys ) )
= ( last @ A @ Ys ) ) ) ).
% last_appendR
thf(fact_7350_last__replicate,axiom,
! [A: $tType,N2: nat,X: A] :
( ( N2
!= ( zero_zero @ nat ) )
=> ( ( last @ A @ ( replicate @ A @ N2 @ X ) )
= X ) ) ).
% last_replicate
thf(fact_7351_last__upt,axiom,
! [I: nat,J: nat] :
( ( ord_less @ nat @ I @ J )
=> ( ( last @ nat @ ( upt @ I @ J ) )
= ( minus_minus @ nat @ J @ ( one_one @ nat ) ) ) ) ).
% last_upt
thf(fact_7352_last__snoc,axiom,
! [A: $tType,Xs: list @ A,X: A] :
( ( last @ A @ ( append @ A @ Xs @ ( cons @ A @ X @ ( nil @ A ) ) ) )
= X ) ).
% last_snoc
thf(fact_7353_last__drop,axiom,
! [A: $tType,N2: nat,Xs: list @ A] :
( ( ord_less @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( last @ A @ ( drop @ A @ N2 @ Xs ) )
= ( last @ A @ Xs ) ) ) ).
% last_drop
thf(fact_7354_hd__rev,axiom,
! [A: $tType,Xs: list @ A] :
( ( hd @ A @ ( rev @ A @ Xs ) )
= ( last @ A @ Xs ) ) ).
% hd_rev
thf(fact_7355_last__rev,axiom,
! [A: $tType,Xs: list @ A] :
( ( last @ A @ ( rev @ A @ Xs ) )
= ( hd @ A @ Xs ) ) ).
% last_rev
thf(fact_7356_last__map,axiom,
! [B: $tType,A: $tType,Xs: list @ A,F2: A > B] :
( ( Xs
!= ( nil @ A ) )
=> ( ( last @ B @ ( map @ A @ B @ F2 @ Xs ) )
= ( F2 @ ( last @ A @ Xs ) ) ) ) ).
% last_map
thf(fact_7357_antisym__empty,axiom,
! [A: $tType] : ( antisym @ A @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ).
% antisym_empty
thf(fact_7358_last__zip,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B] :
( ( Xs
!= ( nil @ A ) )
=> ( ( Ys
!= ( nil @ B ) )
=> ( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( last @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys ) )
= ( product_Pair @ A @ B @ ( last @ A @ Xs ) @ ( last @ B @ Ys ) ) ) ) ) ) ).
% last_zip
thf(fact_7359_hd__Nil__eq__last,axiom,
! [A: $tType] :
( ( hd @ A @ ( nil @ A ) )
= ( last @ A @ ( nil @ A ) ) ) ).
% hd_Nil_eq_last
thf(fact_7360_last__tl,axiom,
! [A: $tType,Xs: list @ A] :
( ( ( Xs
= ( nil @ A ) )
| ( ( tl @ A @ Xs )
!= ( nil @ A ) ) )
=> ( ( last @ A @ ( tl @ A @ Xs ) )
= ( last @ A @ Xs ) ) ) ).
% last_tl
thf(fact_7361_last__in__set,axiom,
! [A: $tType,As: list @ A] :
( ( As
!= ( nil @ A ) )
=> ( member @ A @ ( last @ A @ As ) @ ( set2 @ A @ As ) ) ) ).
% last_in_set
thf(fact_7362_last_Osimps,axiom,
! [A: $tType,Xs: list @ A,X: A] :
( ( ( Xs
= ( nil @ A ) )
=> ( ( last @ A @ ( cons @ A @ X @ Xs ) )
= X ) )
& ( ( Xs
!= ( nil @ A ) )
=> ( ( last @ A @ ( cons @ A @ X @ Xs ) )
= ( last @ A @ Xs ) ) ) ) ).
% last.simps
thf(fact_7363_last__ConsL,axiom,
! [A: $tType,Xs: list @ A,X: A] :
( ( Xs
= ( nil @ A ) )
=> ( ( last @ A @ ( cons @ A @ X @ Xs ) )
= X ) ) ).
% last_ConsL
thf(fact_7364_last__ConsR,axiom,
! [A: $tType,Xs: list @ A,X: A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( last @ A @ ( cons @ A @ X @ Xs ) )
= ( last @ A @ Xs ) ) ) ).
% last_ConsR
thf(fact_7365_last__append,axiom,
! [A: $tType,Ys: list @ A,Xs: list @ A] :
( ( ( Ys
= ( nil @ A ) )
=> ( ( last @ A @ ( append @ A @ Xs @ Ys ) )
= ( last @ A @ Xs ) ) )
& ( ( Ys
!= ( nil @ A ) )
=> ( ( last @ A @ ( append @ A @ Xs @ Ys ) )
= ( last @ A @ Ys ) ) ) ) ).
% last_append
thf(fact_7366_longest__common__suffix,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
? [Ss: list @ A,Xs4: list @ A,Ys5: list @ A] :
( ( Xs
= ( append @ A @ Xs4 @ Ss ) )
& ( Ys
= ( append @ A @ Ys5 @ Ss ) )
& ( ( Xs4
= ( nil @ A ) )
| ( Ys5
= ( nil @ A ) )
| ( ( last @ A @ Xs4 )
!= ( last @ A @ Ys5 ) ) ) ) ).
% longest_common_suffix
thf(fact_7367_dropWhile__last,axiom,
! [A: $tType,X: A,Xs: list @ A,P: A > $o] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ~ ( P @ X )
=> ( ( last @ A @ ( dropWhile @ A @ P @ Xs ) )
= ( last @ A @ Xs ) ) ) ) ).
% dropWhile_last
thf(fact_7368_antisym__singleton,axiom,
! [A: $tType,X: product_prod @ A @ A] : ( antisym @ A @ ( insert2 @ ( product_prod @ A @ A ) @ X @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ).
% antisym_singleton
thf(fact_7369_takeWhile__not__last,axiom,
! [A: $tType,Xs: list @ A] :
( ( distinct @ A @ Xs )
=> ( ( takeWhile @ A
@ ^ [Y6: A] :
( Y6
!= ( last @ A @ Xs ) )
@ Xs )
= ( butlast @ A @ Xs ) ) ) ).
% takeWhile_not_last
thf(fact_7370_snoc__eq__iff__butlast,axiom,
! [A: $tType,Xs: list @ A,X: A,Ys: list @ A] :
( ( ( append @ A @ Xs @ ( cons @ A @ X @ ( nil @ A ) ) )
= Ys )
= ( ( Ys
!= ( nil @ A ) )
& ( ( butlast @ A @ Ys )
= Xs )
& ( ( last @ A @ Ys )
= X ) ) ) ).
% snoc_eq_iff_butlast
thf(fact_7371_remdups__adj__append_H,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( ( Xs
= ( nil @ A ) )
| ( Ys
= ( nil @ A ) )
| ( ( last @ A @ Xs )
!= ( hd @ A @ Ys ) ) )
=> ( ( remdups_adj @ A @ ( append @ A @ Xs @ Ys ) )
= ( append @ A @ ( remdups_adj @ A @ Xs ) @ ( remdups_adj @ A @ Ys ) ) ) ) ).
% remdups_adj_append'
thf(fact_7372_remdups__adj__append_H_H,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( remdups_adj @ A @ ( append @ A @ Xs @ Ys ) )
= ( append @ A @ ( remdups_adj @ A @ Xs )
@ ( remdups_adj @ A
@ ( dropWhile @ A
@ ^ [Y6: A] :
( Y6
= ( last @ A @ Xs ) )
@ Ys ) ) ) ) ) ).
% remdups_adj_append''
thf(fact_7373_last__conv__nth,axiom,
! [A: $tType,Xs: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( last @ A @ Xs )
= ( nth @ A @ Xs @ ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) ) ) ) ) ).
% last_conv_nth
thf(fact_7374_last__list__update,axiom,
! [A: $tType,Xs: list @ A,K: nat,X: A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( ( K
= ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) ) )
=> ( ( last @ A @ ( list_update @ A @ Xs @ K @ X ) )
= X ) )
& ( ( K
!= ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) ) )
=> ( ( last @ A @ ( list_update @ A @ Xs @ K @ X ) )
= ( last @ A @ Xs ) ) ) ) ) ).
% last_list_update
thf(fact_7375_numeral__xor__num,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [M2: num,N2: num] :
( ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ M2 ) @ ( numeral_numeral @ A @ N2 ) )
= ( case_option @ A @ num @ ( zero_zero @ A ) @ ( numeral_numeral @ A ) @ ( bit_un2480387367778600638or_num @ M2 @ N2 ) ) ) ) ).
% numeral_xor_num
thf(fact_7376_ran__map__add,axiom,
! [B: $tType,A: $tType,M1: A > ( option @ B ),M22: A > ( option @ B )] :
( ( ( inf_inf @ ( set @ A ) @ ( dom @ A @ B @ M1 ) @ ( dom @ A @ B @ M22 ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( ran @ A @ B @ ( map_add @ A @ B @ M1 @ M22 ) )
= ( sup_sup @ ( set @ B ) @ ( ran @ A @ B @ M1 ) @ ( ran @ A @ B @ M22 ) ) ) ) ).
% ran_map_add
thf(fact_7377_map__add__comm,axiom,
! [B: $tType,A: $tType,M1: A > ( option @ B ),M22: A > ( option @ B )] :
( ( ( inf_inf @ ( set @ A ) @ ( dom @ A @ B @ M1 ) @ ( dom @ A @ B @ M22 ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( map_add @ A @ B @ M1 @ M22 )
= ( map_add @ A @ B @ M22 @ M1 ) ) ) ).
% map_add_comm
thf(fact_7378_finite__range__map__of__map__add,axiom,
! [A: $tType,B: $tType,F2: B > ( option @ A ),L: list @ ( product_prod @ B @ A )] :
( ( finite_finite2 @ ( option @ A ) @ ( image2 @ B @ ( option @ A ) @ F2 @ ( top_top @ ( set @ B ) ) ) )
=> ( finite_finite2 @ ( option @ A ) @ ( image2 @ B @ ( option @ A ) @ ( map_add @ B @ A @ F2 @ ( map_of @ B @ A @ L ) ) @ ( top_top @ ( set @ B ) ) ) ) ) ).
% finite_range_map_of_map_add
thf(fact_7379_xor__num__eq__None__iff,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [M2: num,N2: num] :
( ( ( bit_un2480387367778600638or_num @ M2 @ N2 )
= ( none @ num ) )
= ( ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ M2 ) @ ( numeral_numeral @ A @ N2 ) )
= ( zero_zero @ A ) ) ) ) ).
% xor_num_eq_None_iff
thf(fact_7380_graph__map__add,axiom,
! [B: $tType,A: $tType,M1: A > ( option @ B ),M22: A > ( option @ B )] :
( ( ( inf_inf @ ( set @ A ) @ ( dom @ A @ B @ M1 ) @ ( dom @ A @ B @ M22 ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( graph @ A @ B @ ( map_add @ A @ B @ M1 @ M22 ) )
= ( sup_sup @ ( set @ ( product_prod @ A @ B ) ) @ ( graph @ A @ B @ M1 ) @ ( graph @ A @ B @ M22 ) ) ) ) ).
% graph_map_add
thf(fact_7381_List_Ounion__def,axiom,
! [A: $tType] :
( ( union @ A )
= ( fold @ A @ ( list @ A ) @ ( insert @ A ) ) ) ).
% List.union_def
thf(fact_7382_in__set__insert,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ( insert @ A @ X @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_7383_distinct__insert,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( distinct @ A @ ( insert @ A @ X @ Xs ) )
= ( distinct @ A @ Xs ) ) ).
% distinct_insert
thf(fact_7384_insert__Nil,axiom,
! [A: $tType,X: A] :
( ( insert @ A @ X @ ( nil @ A ) )
= ( cons @ A @ X @ ( nil @ A ) ) ) ).
% insert_Nil
thf(fact_7385_not__in__set__insert,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ~ ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ( insert @ A @ X @ Xs )
= ( cons @ A @ X @ Xs ) ) ) ).
% not_in_set_insert
thf(fact_7386_List_Oset__insert,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( set2 @ A @ ( insert @ A @ X @ Xs ) )
= ( insert2 @ A @ X @ ( set2 @ A @ Xs ) ) ) ).
% List.set_insert
thf(fact_7387_graph__empty,axiom,
! [B: $tType,A: $tType] :
( ( graph @ A @ B
@ ^ [X4: A] : ( none @ B ) )
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% graph_empty
thf(fact_7388_finite__graph__iff__finite__dom,axiom,
! [B: $tType,A: $tType,M2: A > ( option @ B )] :
( ( finite_finite2 @ ( product_prod @ A @ B ) @ ( graph @ A @ B @ M2 ) )
= ( finite_finite2 @ A @ ( dom @ A @ B @ M2 ) ) ) ).
% finite_graph_iff_finite_dom
thf(fact_7389_finite__graph__map__of,axiom,
! [B: $tType,A: $tType,Al: list @ ( product_prod @ A @ B )] : ( finite_finite2 @ ( product_prod @ A @ B ) @ ( graph @ A @ B @ ( map_of @ A @ B @ Al ) ) ) ).
% finite_graph_map_of
thf(fact_7390_insert__remdups,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( insert @ A @ X @ ( remdups @ A @ Xs ) )
= ( remdups @ A @ ( insert @ A @ X @ Xs ) ) ) ).
% insert_remdups
thf(fact_7391_List_Oinsert__def,axiom,
! [A: $tType] :
( ( insert @ A )
= ( ^ [X4: A,Xs3: list @ A] : ( if @ ( list @ A ) @ ( member @ A @ X4 @ ( set2 @ A @ Xs3 ) ) @ Xs3 @ ( cons @ A @ X4 @ Xs3 ) ) ) ) ).
% List.insert_def
thf(fact_7392_remove__code_I2_J,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( remove @ A @ X @ ( coset @ A @ Xs ) )
= ( coset @ A @ ( insert @ A @ X @ Xs ) ) ) ).
% remove_code(2)
thf(fact_7393_inv__image__partition,axiom,
! [A: $tType,Xs: list @ A,P: A > $o,Ys: list @ A] :
( ! [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Xs ) )
=> ( P @ X5 ) )
=> ( ! [Y3: A] :
( ( member @ A @ Y3 @ ( set2 @ A @ Ys ) )
=> ~ ( P @ Y3 ) )
=> ( ( vimage @ ( list @ A ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( partition @ A @ P ) @ ( insert2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( bot_bot @ ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) ) ) )
= ( shuffles @ A @ Xs @ Ys ) ) ) ) ).
% inv_image_partition
thf(fact_7394_numeral__and__num,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [M2: num,N2: num] :
( ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ M2 ) @ ( numeral_numeral @ A @ N2 ) )
= ( case_option @ A @ num @ ( zero_zero @ A ) @ ( numeral_numeral @ A ) @ ( bit_un7362597486090784418nd_num @ M2 @ N2 ) ) ) ) ).
% numeral_and_num
thf(fact_7395_vimage__eq,axiom,
! [A: $tType,B: $tType,A3: A,F2: A > B,B2: set @ B] :
( ( member @ A @ A3 @ ( vimage @ A @ B @ F2 @ B2 ) )
= ( member @ B @ ( F2 @ A3 ) @ B2 ) ) ).
% vimage_eq
thf(fact_7396_vimageI,axiom,
! [B: $tType,A: $tType,F2: B > A,A3: B,B3: A,B2: set @ A] :
( ( ( F2 @ A3 )
= B3 )
=> ( ( member @ A @ B3 @ B2 )
=> ( member @ B @ A3 @ ( vimage @ B @ A @ F2 @ B2 ) ) ) ) ).
% vimageI
thf(fact_7397_vimage__Collect__eq,axiom,
! [B: $tType,A: $tType,F2: A > B,P: B > $o] :
( ( vimage @ A @ B @ F2 @ ( collect @ B @ P ) )
= ( collect @ A
@ ^ [Y6: A] : ( P @ ( F2 @ Y6 ) ) ) ) ).
% vimage_Collect_eq
thf(fact_7398_vimage__ident,axiom,
! [A: $tType,Y7: set @ A] :
( ( vimage @ A @ A
@ ^ [X4: A] : X4
@ Y7 )
= Y7 ) ).
% vimage_ident
thf(fact_7399_vimage__UNIV,axiom,
! [B: $tType,A: $tType,F2: A > B] :
( ( vimage @ A @ B @ F2 @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) ) ).
% vimage_UNIV
thf(fact_7400_vimage__empty,axiom,
! [B: $tType,A: $tType,F2: A > B] :
( ( vimage @ A @ B @ F2 @ ( bot_bot @ ( set @ B ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% vimage_empty
thf(fact_7401_vimage__Int,axiom,
! [A: $tType,B: $tType,F2: A > B,A4: set @ B,B2: set @ B] :
( ( vimage @ A @ B @ F2 @ ( inf_inf @ ( set @ B ) @ A4 @ B2 ) )
= ( inf_inf @ ( set @ A ) @ ( vimage @ A @ B @ F2 @ A4 ) @ ( vimage @ A @ B @ F2 @ B2 ) ) ) ).
% vimage_Int
thf(fact_7402_vimage__Un,axiom,
! [A: $tType,B: $tType,F2: A > B,A4: set @ B,B2: set @ B] :
( ( vimage @ A @ B @ F2 @ ( sup_sup @ ( set @ B ) @ A4 @ B2 ) )
= ( sup_sup @ ( set @ A ) @ ( vimage @ A @ B @ F2 @ A4 ) @ ( vimage @ A @ B @ F2 @ B2 ) ) ) ).
% vimage_Un
thf(fact_7403_vimage__const,axiom,
! [B: $tType,A: $tType,C2: B,A4: set @ B] :
( ( ( member @ B @ C2 @ A4 )
=> ( ( vimage @ A @ B
@ ^ [X4: A] : C2
@ A4 )
= ( top_top @ ( set @ A ) ) ) )
& ( ~ ( member @ B @ C2 @ A4 )
=> ( ( vimage @ A @ B
@ ^ [X4: A] : C2
@ A4 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% vimage_const
thf(fact_7404_image__vimage__eq,axiom,
! [A: $tType,B: $tType,F2: B > A,A4: set @ A] :
( ( image2 @ B @ A @ F2 @ ( vimage @ B @ A @ F2 @ A4 ) )
= ( inf_inf @ ( set @ A ) @ A4 @ ( image2 @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) ) ) ) ).
% image_vimage_eq
thf(fact_7405_vimage__if,axiom,
! [B: $tType,A: $tType,C2: B,A4: set @ B,D2: B,B2: set @ A] :
( ( ( member @ B @ C2 @ A4 )
=> ( ( ( member @ B @ D2 @ A4 )
=> ( ( vimage @ A @ B
@ ^ [X4: A] : ( if @ B @ ( member @ A @ X4 @ B2 ) @ C2 @ D2 )
@ A4 )
= ( top_top @ ( set @ A ) ) ) )
& ( ~ ( member @ B @ D2 @ A4 )
=> ( ( vimage @ A @ B
@ ^ [X4: A] : ( if @ B @ ( member @ A @ X4 @ B2 ) @ C2 @ D2 )
@ A4 )
= B2 ) ) ) )
& ( ~ ( member @ B @ C2 @ A4 )
=> ( ( ( member @ B @ D2 @ A4 )
=> ( ( vimage @ A @ B
@ ^ [X4: A] : ( if @ B @ ( member @ A @ X4 @ B2 ) @ C2 @ D2 )
@ A4 )
= ( uminus_uminus @ ( set @ A ) @ B2 ) ) )
& ( ~ ( member @ B @ D2 @ A4 )
=> ( ( vimage @ A @ B
@ ^ [X4: A] : ( if @ B @ ( member @ A @ X4 @ B2 ) @ C2 @ D2 )
@ A4 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% vimage_if
thf(fact_7406_vimage__image__eq,axiom,
! [B: $tType,A: $tType,F2: A > B,A4: set @ A] :
( ( vimage @ A @ B @ F2 @ ( image2 @ A @ B @ F2 @ A4 ) )
= ( collect @ A
@ ^ [Y6: A] :
? [X4: A] :
( ( member @ A @ X4 @ A4 )
& ( ( F2 @ X4 )
= ( F2 @ Y6 ) ) ) ) ) ).
% vimage_image_eq
thf(fact_7407_vimage__subsetD,axiom,
! [A: $tType,B: $tType,F2: B > A,B2: set @ A,A4: set @ B] :
( ( ( image2 @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ B ) @ ( vimage @ B @ A @ F2 @ B2 ) @ A4 )
=> ( ord_less_eq @ ( set @ A ) @ B2 @ ( image2 @ B @ A @ F2 @ A4 ) ) ) ) ).
% vimage_subsetD
thf(fact_7408_finite__vimageD,axiom,
! [A: $tType,B: $tType,H2: A > B,F5: set @ B] :
( ( finite_finite2 @ A @ ( vimage @ A @ B @ H2 @ F5 ) )
=> ( ( ( image2 @ A @ B @ H2 @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ B ) ) )
=> ( finite_finite2 @ B @ F5 ) ) ) ).
% finite_vimageD
thf(fact_7409_surj__vimage__empty,axiom,
! [B: $tType,A: $tType,F2: B > A,A4: set @ A] :
( ( ( image2 @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
=> ( ( ( vimage @ B @ A @ F2 @ A4 )
= ( bot_bot @ ( set @ B ) ) )
= ( A4
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% surj_vimage_empty
thf(fact_7410_image__subset__iff__subset__vimage,axiom,
! [B: $tType,A: $tType,F2: B > A,A4: set @ B,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( image2 @ B @ A @ F2 @ A4 ) @ B2 )
= ( ord_less_eq @ ( set @ B ) @ A4 @ ( vimage @ B @ A @ F2 @ B2 ) ) ) ).
% image_subset_iff_subset_vimage
thf(fact_7411_image__vimage__subset,axiom,
! [B: $tType,A: $tType,F2: B > A,A4: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( image2 @ B @ A @ F2 @ ( vimage @ B @ A @ F2 @ A4 ) ) @ A4 ) ).
% image_vimage_subset
thf(fact_7412_vimage__Diff,axiom,
! [A: $tType,B: $tType,F2: A > B,A4: set @ B,B2: set @ B] :
( ( vimage @ A @ B @ F2 @ ( minus_minus @ ( set @ B ) @ A4 @ B2 ) )
= ( minus_minus @ ( set @ A ) @ ( vimage @ A @ B @ F2 @ A4 ) @ ( vimage @ A @ B @ F2 @ B2 ) ) ) ).
% vimage_Diff
thf(fact_7413_finite__vimageI,axiom,
! [B: $tType,A: $tType,F5: set @ A,H2: B > A] :
( ( finite_finite2 @ A @ F5 )
=> ( ( inj_on @ B @ A @ H2 @ ( top_top @ ( set @ B ) ) )
=> ( finite_finite2 @ B @ ( vimage @ B @ A @ H2 @ F5 ) ) ) ) ).
% finite_vimageI
thf(fact_7414_finite__vimage__IntI,axiom,
! [A: $tType,B: $tType,F5: set @ A,H2: B > A,A4: set @ B] :
( ( finite_finite2 @ A @ F5 )
=> ( ( inj_on @ B @ A @ H2 @ A4 )
=> ( finite_finite2 @ B @ ( inf_inf @ ( set @ B ) @ ( vimage @ B @ A @ H2 @ F5 ) @ A4 ) ) ) ) ).
% finite_vimage_IntI
thf(fact_7415_Pair__vimage__Sigma,axiom,
! [B: $tType,A: $tType,X: B,A4: set @ B,F2: B > ( set @ A )] :
( ( ( member @ B @ X @ A4 )
=> ( ( vimage @ A @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X ) @ ( product_Sigma @ B @ A @ A4 @ F2 ) )
= ( F2 @ X ) ) )
& ( ~ ( member @ B @ X @ A4 )
=> ( ( vimage @ A @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X ) @ ( product_Sigma @ B @ A @ A4 @ F2 ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Pair_vimage_Sigma
thf(fact_7416_vimage__inter__cong,axiom,
! [B: $tType,A: $tType,S2: set @ A,F2: A > B,G: A > B,Y: set @ B] :
( ! [W: A] :
( ( member @ A @ W @ S2 )
=> ( ( F2 @ W )
= ( G @ W ) ) )
=> ( ( inf_inf @ ( set @ A ) @ ( vimage @ A @ B @ F2 @ Y ) @ S2 )
= ( inf_inf @ ( set @ A ) @ ( vimage @ A @ B @ G @ Y ) @ S2 ) ) ) ).
% vimage_inter_cong
thf(fact_7417_continuous__imp__open__vimage,axiom,
! [A: $tType,B: $tType] :
( ( ( topolo4958980785337419405_space @ B )
& ( topolo4958980785337419405_space @ A ) )
=> ! [S: set @ A,F2: A > B,B2: set @ B] :
( ( topolo81223032696312382ous_on @ A @ B @ S @ F2 )
=> ( ( topolo1002775350975398744n_open @ A @ S )
=> ( ( topolo1002775350975398744n_open @ B @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( vimage @ A @ B @ F2 @ B2 ) @ S )
=> ( topolo1002775350975398744n_open @ A @ ( vimage @ A @ B @ F2 @ B2 ) ) ) ) ) ) ) ).
% continuous_imp_open_vimage
thf(fact_7418_finite__vimage__iff,axiom,
! [A: $tType,B: $tType,H2: A > B,F5: set @ B] :
( ( bij_betw @ A @ B @ H2 @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
=> ( ( finite_finite2 @ A @ ( vimage @ A @ B @ H2 @ F5 ) )
= ( finite_finite2 @ B @ F5 ) ) ) ).
% finite_vimage_iff
thf(fact_7419_vimage__def,axiom,
! [B: $tType,A: $tType] :
( ( vimage @ A @ B )
= ( ^ [F3: A > B,B6: set @ B] :
( collect @ A
@ ^ [X4: A] : ( member @ B @ ( F3 @ X4 ) @ B6 ) ) ) ) ).
% vimage_def
thf(fact_7420_vimage__Collect,axiom,
! [B: $tType,A: $tType,P: B > $o,F2: A > B,Q: A > $o] :
( ! [X5: A] :
( ( P @ ( F2 @ X5 ) )
= ( Q @ X5 ) )
=> ( ( vimage @ A @ B @ F2 @ ( collect @ B @ P ) )
= ( collect @ A @ Q ) ) ) ).
% vimage_Collect
thf(fact_7421_vimageI2,axiom,
! [B: $tType,A: $tType,F2: B > A,A3: B,A4: set @ A] :
( ( member @ A @ ( F2 @ A3 ) @ A4 )
=> ( member @ B @ A3 @ ( vimage @ B @ A @ F2 @ A4 ) ) ) ).
% vimageI2
thf(fact_7422_vimageE,axiom,
! [A: $tType,B: $tType,A3: A,F2: A > B,B2: set @ B] :
( ( member @ A @ A3 @ ( vimage @ A @ B @ F2 @ B2 ) )
=> ( member @ B @ ( F2 @ A3 ) @ B2 ) ) ).
% vimageE
thf(fact_7423_vimageD,axiom,
! [A: $tType,B: $tType,A3: A,F2: A > B,A4: set @ B] :
( ( member @ A @ A3 @ ( vimage @ A @ B @ F2 @ A4 ) )
=> ( member @ B @ ( F2 @ A3 ) @ A4 ) ) ).
% vimageD
thf(fact_7424_vimage__Compl,axiom,
! [A: $tType,B: $tType,F2: A > B,A4: set @ B] :
( ( vimage @ A @ B @ F2 @ ( uminus_uminus @ ( set @ B ) @ A4 ) )
= ( uminus_uminus @ ( set @ A ) @ ( vimage @ A @ B @ F2 @ A4 ) ) ) ).
% vimage_Compl
thf(fact_7425_vimage__mono,axiom,
! [B: $tType,A: $tType,A4: set @ A,B2: set @ A,F2: B > A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B2 )
=> ( ord_less_eq @ ( set @ B ) @ ( vimage @ B @ A @ F2 @ A4 ) @ ( vimage @ B @ A @ F2 @ B2 ) ) ) ).
% vimage_mono
thf(fact_7426_subset__vimage__iff,axiom,
! [B: $tType,A: $tType,A4: set @ A,F2: A > B,B2: set @ B] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ ( vimage @ A @ B @ F2 @ B2 ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ A4 )
=> ( member @ B @ ( F2 @ X4 ) @ B2 ) ) ) ) ).
% subset_vimage_iff
thf(fact_7427_vimage__singleton__eq,axiom,
! [A: $tType,B: $tType,A3: A,F2: A > B,B3: B] :
( ( member @ A @ A3 @ ( vimage @ A @ B @ F2 @ ( insert2 @ B @ B3 @ ( bot_bot @ ( set @ B ) ) ) ) )
= ( ( F2 @ A3 )
= B3 ) ) ).
% vimage_singleton_eq
thf(fact_7428_vimage__insert,axiom,
! [A: $tType,B: $tType,F2: A > B,A3: B,B2: set @ B] :
( ( vimage @ A @ B @ F2 @ ( insert2 @ B @ A3 @ B2 ) )
= ( sup_sup @ ( set @ A ) @ ( vimage @ A @ B @ F2 @ ( insert2 @ B @ A3 @ ( bot_bot @ ( set @ B ) ) ) ) @ ( vimage @ A @ B @ F2 @ B2 ) ) ) ).
% vimage_insert
thf(fact_7429_finite__vimageD_H,axiom,
! [A: $tType,B: $tType,F2: A > B,A4: set @ B] :
( ( finite_finite2 @ A @ ( vimage @ A @ B @ F2 @ A4 ) )
=> ( ( ord_less_eq @ ( set @ B ) @ A4 @ ( image2 @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) ) )
=> ( finite_finite2 @ B @ A4 ) ) ) ).
% finite_vimageD'
thf(fact_7430_inf__img__fin__domE,axiom,
! [B: $tType,A: $tType,F2: B > A,A4: set @ B] :
( ( finite_finite2 @ A @ ( image2 @ B @ A @ F2 @ A4 ) )
=> ( ~ ( finite_finite2 @ B @ A4 )
=> ~ ! [Y3: A] :
( ( member @ A @ Y3 @ ( image2 @ B @ A @ F2 @ A4 ) )
=> ( finite_finite2 @ B @ ( vimage @ B @ A @ F2 @ ( insert2 @ A @ Y3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% inf_img_fin_domE
thf(fact_7431_inf__img__fin__dom,axiom,
! [B: $tType,A: $tType,F2: B > A,A4: set @ B] :
( ( finite_finite2 @ A @ ( image2 @ B @ A @ F2 @ A4 ) )
=> ( ~ ( finite_finite2 @ B @ A4 )
=> ? [X5: A] :
( ( member @ A @ X5 @ ( image2 @ B @ A @ F2 @ A4 ) )
& ~ ( finite_finite2 @ B @ ( vimage @ B @ A @ F2 @ ( insert2 @ A @ X5 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% inf_img_fin_dom
thf(fact_7432_vimage__subsetI,axiom,
! [B: $tType,A: $tType,F2: A > B,B2: set @ B,A4: set @ A] :
( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ B ) @ B2 @ ( image2 @ A @ B @ F2 @ A4 ) )
=> ( ord_less_eq @ ( set @ A ) @ ( vimage @ A @ B @ F2 @ B2 ) @ A4 ) ) ) ).
% vimage_subsetI
thf(fact_7433_finite__finite__vimage__IntI,axiom,
! [A: $tType,B: $tType,F5: set @ A,H2: B > A,A4: set @ B] :
( ( finite_finite2 @ A @ F5 )
=> ( ! [Y3: A] :
( ( member @ A @ Y3 @ F5 )
=> ( finite_finite2 @ B @ ( inf_inf @ ( set @ B ) @ ( vimage @ B @ A @ H2 @ ( insert2 @ A @ Y3 @ ( bot_bot @ ( set @ A ) ) ) ) @ A4 ) ) )
=> ( finite_finite2 @ B @ ( inf_inf @ ( set @ B ) @ ( vimage @ B @ A @ H2 @ F5 ) @ A4 ) ) ) ) ).
% finite_finite_vimage_IntI
thf(fact_7434_countable__vimage,axiom,
! [B: $tType,A: $tType,B2: set @ A,F2: B > A] :
( ( ord_less_eq @ ( set @ A ) @ B2 @ ( image2 @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) ) )
=> ( ( countable_countable @ B @ ( vimage @ B @ A @ F2 @ B2 ) )
=> ( countable_countable @ A @ B2 ) ) ) ).
% countable_vimage
thf(fact_7435_vimage__subset__eq,axiom,
! [B: $tType,A: $tType,F2: A > B,B2: set @ B,A4: set @ A] :
( ( bij_betw @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
=> ( ( ord_less_eq @ ( set @ A ) @ ( vimage @ A @ B @ F2 @ B2 ) @ A4 )
= ( ord_less_eq @ ( set @ B ) @ B2 @ ( image2 @ A @ B @ F2 @ A4 ) ) ) ) ).
% vimage_subset_eq
thf(fact_7436_vimage__eq__UN,axiom,
! [B: $tType,A: $tType] :
( ( vimage @ A @ B )
= ( ^ [F3: A > B,B6: set @ B] :
( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [Y6: B] : ( vimage @ A @ B @ F3 @ ( insert2 @ B @ Y6 @ ( bot_bot @ ( set @ B ) ) ) )
@ B6 ) ) ) ) ).
% vimage_eq_UN
thf(fact_7437_inf__img__fin__domE_H,axiom,
! [A: $tType,B: $tType,F2: B > A,A4: set @ B] :
( ( finite_finite2 @ A @ ( image2 @ B @ A @ F2 @ A4 ) )
=> ( ~ ( finite_finite2 @ B @ A4 )
=> ~ ! [Y3: A] :
( ( member @ A @ Y3 @ ( image2 @ B @ A @ F2 @ A4 ) )
=> ( finite_finite2 @ B @ ( inf_inf @ ( set @ B ) @ ( vimage @ B @ A @ F2 @ ( insert2 @ A @ Y3 @ ( bot_bot @ ( set @ A ) ) ) ) @ A4 ) ) ) ) ) ).
% inf_img_fin_domE'
thf(fact_7438_inf__img__fin__dom_H,axiom,
! [A: $tType,B: $tType,F2: B > A,A4: set @ B] :
( ( finite_finite2 @ A @ ( image2 @ B @ A @ F2 @ A4 ) )
=> ( ~ ( finite_finite2 @ B @ A4 )
=> ? [X5: A] :
( ( member @ A @ X5 @ ( image2 @ B @ A @ F2 @ A4 ) )
& ~ ( finite_finite2 @ B @ ( inf_inf @ ( set @ B ) @ ( vimage @ B @ A @ F2 @ ( insert2 @ A @ X5 @ ( bot_bot @ ( set @ A ) ) ) ) @ A4 ) ) ) ) ) ).
% inf_img_fin_dom'
thf(fact_7439_card__vimage__inj,axiom,
! [A: $tType,B: $tType,F2: A > B,A4: set @ B] :
( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ B ) @ A4 @ ( image2 @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) ) )
=> ( ( finite_card @ A @ ( vimage @ A @ B @ F2 @ A4 ) )
= ( finite_card @ B @ A4 ) ) ) ) ).
% card_vimage_inj
thf(fact_7440_card__vimage__inj__on__le,axiom,
! [A: $tType,B: $tType,F2: A > B,D6: set @ A,A4: set @ B] :
( ( inj_on @ A @ B @ F2 @ D6 )
=> ( ( finite_finite2 @ B @ A4 )
=> ( ord_less_eq @ nat @ ( finite_card @ A @ ( inf_inf @ ( set @ A ) @ ( vimage @ A @ B @ F2 @ A4 ) @ D6 ) ) @ ( finite_card @ B @ A4 ) ) ) ) ).
% card_vimage_inj_on_le
thf(fact_7441_and__num__eq__None__iff,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [M2: num,N2: num] :
( ( ( bit_un7362597486090784418nd_num @ M2 @ N2 )
= ( none @ num ) )
= ( ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ M2 ) @ ( numeral_numeral @ A @ N2 ) )
= ( zero_zero @ A ) ) ) ) ).
% and_num_eq_None_iff
thf(fact_7442_image__split__eq__Sigma,axiom,
! [C: $tType,B: $tType,A: $tType,F2: C > A,G: C > B,A4: set @ C] :
( ( image2 @ C @ ( product_prod @ A @ B )
@ ^ [X4: C] : ( product_Pair @ A @ B @ ( F2 @ X4 ) @ ( G @ X4 ) )
@ A4 )
= ( product_Sigma @ A @ B @ ( image2 @ C @ A @ F2 @ A4 )
@ ^ [X4: A] : ( image2 @ C @ B @ G @ ( inf_inf @ ( set @ C ) @ ( vimage @ C @ A @ F2 @ ( insert2 @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) ) @ A4 ) ) ) ) ).
% image_split_eq_Sigma
thf(fact_7443_inj__on__vimage__singleton,axiom,
! [B: $tType,A: $tType,F2: A > B,A4: set @ A,A3: B] :
( ( inj_on @ A @ B @ F2 @ A4 )
=> ( ord_less_eq @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ ( vimage @ A @ B @ F2 @ ( insert2 @ B @ A3 @ ( bot_bot @ ( set @ B ) ) ) ) @ A4 )
@ ( insert2 @ A
@ ( the @ A
@ ^ [X4: A] :
( ( member @ A @ X4 @ A4 )
& ( ( F2 @ X4 )
= A3 ) ) )
@ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% inj_on_vimage_singleton
thf(fact_7444_inj__vimage__singleton,axiom,
! [B: $tType,A: $tType,F2: A > B,A3: B] :
( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
=> ( ord_less_eq @ ( set @ A ) @ ( vimage @ A @ B @ F2 @ ( insert2 @ B @ A3 @ ( bot_bot @ ( set @ B ) ) ) )
@ ( insert2 @ A
@ ( the @ A
@ ^ [X4: A] :
( ( F2 @ X4 )
= A3 ) )
@ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% inj_vimage_singleton
thf(fact_7445_vimage__Suc__insert__0,axiom,
! [A4: set @ nat] :
( ( vimage @ nat @ nat @ suc @ ( insert2 @ nat @ ( zero_zero @ nat ) @ A4 ) )
= ( vimage @ nat @ nat @ suc @ A4 ) ) ).
% vimage_Suc_insert_0
thf(fact_7446_finite__vimage__Suc__iff,axiom,
! [F5: set @ nat] :
( ( finite_finite2 @ nat @ ( vimage @ nat @ nat @ suc @ F5 ) )
= ( finite_finite2 @ nat @ F5 ) ) ).
% finite_vimage_Suc_iff
thf(fact_7447_vimage__Suc__insert__Suc,axiom,
! [N2: nat,A4: set @ nat] :
( ( vimage @ nat @ nat @ suc @ ( insert2 @ nat @ ( suc @ N2 ) @ A4 ) )
= ( insert2 @ nat @ N2 @ ( vimage @ nat @ nat @ suc @ A4 ) ) ) ).
% vimage_Suc_insert_Suc
thf(fact_7448_Least__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( ord_Least @ A )
= ( ^ [P3: A > $o] :
( the @ A
@ ^ [X4: A] :
( ( P3 @ X4 )
& ! [Y6: A] :
( ( P3 @ Y6 )
=> ( ord_less_eq @ A @ X4 @ Y6 ) ) ) ) ) ) ) ).
% Least_def
thf(fact_7449_Finite__Set_Ofold__def,axiom,
! [B: $tType,A: $tType] :
( ( finite_fold @ A @ B )
= ( ^ [F3: A > B > B,Z6: B,A7: set @ A] : ( if @ B @ ( finite_finite2 @ A @ A7 ) @ ( the @ B @ ( finite_fold_graph @ A @ B @ F3 @ Z6 @ A7 ) ) @ Z6 ) ) ) ).
% Finite_Set.fold_def
thf(fact_7450_the__elem__def,axiom,
! [A: $tType] :
( ( the_elem @ A )
= ( ^ [X6: set @ A] :
( the @ A
@ ^ [X4: A] :
( X6
= ( insert2 @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% the_elem_def
thf(fact_7451_set__decode__div__2,axiom,
! [X: nat] :
( ( nat_set_decode @ ( divide_divide @ nat @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( vimage @ nat @ nat @ suc @ ( nat_set_decode @ X ) ) ) ).
% set_decode_div_2
thf(fact_7452_set__encode__vimage__Suc,axiom,
! [A4: set @ nat] :
( ( nat_set_encode @ ( vimage @ nat @ nat @ suc @ A4 ) )
= ( divide_divide @ nat @ ( nat_set_encode @ A4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% set_encode_vimage_Suc
thf(fact_7453_old_Orec__nat__def,axiom,
! [T: $tType] :
( ( rec_nat @ T )
= ( ^ [F14: T,F23: nat > T > T,X4: nat] : ( the @ T @ ( rec_set_nat @ T @ F14 @ F23 @ X4 ) ) ) ) ).
% old.rec_nat_def
thf(fact_7454_Greatest__def,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( order_Greatest @ A )
= ( ^ [P3: A > $o] :
( the @ A
@ ^ [X4: A] :
( ( P3 @ X4 )
& ! [Y6: A] :
( ( P3 @ Y6 )
=> ( ord_less_eq @ A @ Y6 @ X4 ) ) ) ) ) ) ) ).
% Greatest_def
thf(fact_7455_GreatestI__nat,axiom,
! [P: nat > $o,K: nat,B3: nat] :
( ( P @ K )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq @ nat @ Y3 @ B3 ) )
=> ( P @ ( order_Greatest @ nat @ P ) ) ) ) ).
% GreatestI_nat
thf(fact_7456_Greatest__le__nat,axiom,
! [P: nat > $o,K: nat,B3: nat] :
( ( P @ K )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq @ nat @ Y3 @ B3 ) )
=> ( ord_less_eq @ nat @ K @ ( order_Greatest @ nat @ P ) ) ) ) ).
% Greatest_le_nat
thf(fact_7457_GreatestI__ex__nat,axiom,
! [P: nat > $o,B3: nat] :
( ? [X_1: nat] : ( P @ X_1 )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq @ nat @ Y3 @ B3 ) )
=> ( P @ ( order_Greatest @ nat @ P ) ) ) ) ).
% GreatestI_ex_nat
thf(fact_7458_GreatestI2__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [P: A > $o,X: A,Q: A > $o] :
( ( P @ X )
=> ( ! [Y3: A] :
( ( P @ Y3 )
=> ( ord_less_eq @ A @ Y3 @ X ) )
=> ( ! [X5: A] :
( ( P @ X5 )
=> ( ! [Y4: A] :
( ( P @ Y4 )
=> ( ord_less_eq @ A @ Y4 @ X5 ) )
=> ( Q @ X5 ) ) )
=> ( Q @ ( order_Greatest @ A @ P ) ) ) ) ) ) ).
% GreatestI2_order
thf(fact_7459_Greatest__equality,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [P: A > $o,X: A] :
( ( P @ X )
=> ( ! [Y3: A] :
( ( P @ Y3 )
=> ( ord_less_eq @ A @ Y3 @ X ) )
=> ( ( order_Greatest @ A @ P )
= X ) ) ) ) ).
% Greatest_equality
thf(fact_7460_flat__lub__def,axiom,
! [A: $tType] :
( ( partial_flat_lub @ A )
= ( ^ [B4: A,A7: set @ A] :
( if @ A @ ( ord_less_eq @ ( set @ A ) @ A7 @ ( insert2 @ A @ B4 @ ( bot_bot @ ( set @ A ) ) ) ) @ B4
@ ( the @ A
@ ^ [X4: A] : ( member @ A @ X4 @ ( minus_minus @ ( set @ A ) @ A7 @ ( insert2 @ A @ B4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ).
% flat_lub_def
thf(fact_7461_ord_OLeast__def,axiom,
! [A: $tType] :
( ( least @ A )
= ( ^ [Less_eq: A > A > $o,P3: A > $o] :
( the @ A
@ ^ [X4: A] :
( ( P3 @ X4 )
& ! [Y6: A] :
( ( P3 @ Y6 )
=> ( Less_eq @ X4 @ Y6 ) ) ) ) ) ) ).
% ord.Least_def
thf(fact_7462_ord_OLeast_Ocong,axiom,
! [A: $tType] :
( ( least @ A )
= ( least @ A ) ) ).
% ord.Least.cong
thf(fact_7463_Gcd__fin__0__iff,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A4: set @ A] :
( ( ( semiring_gcd_Gcd_fin @ A @ A4 )
= ( zero_zero @ A ) )
= ( ( ord_less_eq @ ( set @ A ) @ A4 @ ( insert2 @ A @ ( zero_zero @ A ) @ ( bot_bot @ ( set @ A ) ) ) )
& ( finite_finite2 @ A @ A4 ) ) ) ) ).
% Gcd_fin_0_iff
thf(fact_7464_connected__closed,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ( ( topolo1966860045006549960nected @ A )
= ( ^ [S7: set @ A] :
~ ? [A7: set @ A,B6: set @ A] :
( ( topolo7761053866217962861closed @ A @ A7 )
& ( topolo7761053866217962861closed @ A @ B6 )
& ( ord_less_eq @ ( set @ A ) @ S7 @ ( sup_sup @ ( set @ A ) @ A7 @ B6 ) )
& ( ( inf_inf @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A7 @ B6 ) @ S7 )
= ( bot_bot @ ( set @ A ) ) )
& ( ( inf_inf @ ( set @ A ) @ A7 @ S7 )
!= ( bot_bot @ ( set @ A ) ) )
& ( ( inf_inf @ ( set @ A ) @ B6 @ S7 )
!= ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% connected_closed
thf(fact_7465_Gcd__fin_Oempty,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ( ( semiring_gcd_Gcd_fin @ A @ ( bot_bot @ ( set @ A ) ) )
= ( zero_zero @ A ) ) ) ).
% Gcd_fin.empty
thf(fact_7466_Gcd__fin_Oinfinite,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A4: set @ A] :
( ~ ( finite_finite2 @ A @ A4 )
=> ( ( semiring_gcd_Gcd_fin @ A @ A4 )
= ( one_one @ A ) ) ) ) ).
% Gcd_fin.infinite
thf(fact_7467_Gcd__fin__eq__Gcd,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [A4: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( semiring_gcd_Gcd_fin @ A @ A4 )
= ( gcd_Gcd @ A @ A4 ) ) ) ) ).
% Gcd_fin_eq_Gcd
thf(fact_7468_connected__Times__eq,axiom,
! [A: $tType,B: $tType] :
( ( ( topolo4958980785337419405_space @ B )
& ( topolo4958980785337419405_space @ A ) )
=> ! [S2: set @ A,T4: set @ B] :
( ( topolo1966860045006549960nected @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ S2
@ ^ [Uu3: A] : T4 ) )
= ( ( S2
= ( bot_bot @ ( set @ A ) ) )
| ( T4
= ( bot_bot @ ( set @ B ) ) )
| ( ( topolo1966860045006549960nected @ A @ S2 )
& ( topolo1966860045006549960nected @ B @ T4 ) ) ) ) ) ).
% connected_Times_eq
thf(fact_7469_dvd__Gcd__fin__iff,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A4: set @ A,B3: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( dvd_dvd @ A @ B3 @ ( semiring_gcd_Gcd_fin @ A @ A4 ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ A4 )
=> ( dvd_dvd @ A @ B3 @ X4 ) ) ) ) ) ) ).
% dvd_Gcd_fin_iff
thf(fact_7470_Gcd__fin__greatest,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A4: set @ A,A3: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ! [B5: A] :
( ( member @ A @ B5 @ A4 )
=> ( dvd_dvd @ A @ A3 @ B5 ) )
=> ( dvd_dvd @ A @ A3 @ ( semiring_gcd_Gcd_fin @ A @ A4 ) ) ) ) ) ).
% Gcd_fin_greatest
thf(fact_7471_dvd__gcd__list__iff,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [B3: A,Xs: list @ A] :
( ( dvd_dvd @ A @ B3 @ ( semiring_gcd_Gcd_fin @ A @ ( set2 @ A @ Xs ) ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
=> ( dvd_dvd @ A @ B3 @ X4 ) ) ) ) ) ).
% dvd_gcd_list_iff
thf(fact_7472_gcd__list__greatest,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [Bs: list @ A,A3: A] :
( ! [B5: A] :
( ( member @ A @ B5 @ ( set2 @ A @ Bs ) )
=> ( dvd_dvd @ A @ A3 @ B5 ) )
=> ( dvd_dvd @ A @ A3 @ ( semiring_gcd_Gcd_fin @ A @ ( set2 @ A @ Bs ) ) ) ) ) ).
% gcd_list_greatest
thf(fact_7473_connected__contains__Ioo,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [A4: set @ A,A3: A,B3: A] :
( ( topolo1966860045006549960nected @ A @ A4 )
=> ( ( member @ A @ A3 @ A4 )
=> ( ( member @ A @ B3 @ A4 )
=> ( ord_less_eq @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ A3 @ B3 ) @ A4 ) ) ) ) ) ).
% connected_contains_Ioo
thf(fact_7474_connected__contains__Icc,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [A4: set @ A,A3: A,B3: A] :
( ( topolo1966860045006549960nected @ A @ A4 )
=> ( ( member @ A @ A3 @ A4 )
=> ( ( member @ A @ B3 @ A4 )
=> ( ord_less_eq @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A3 @ B3 ) @ A4 ) ) ) ) ) ).
% connected_contains_Icc
thf(fact_7475_connectedD__interval,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [U3: set @ A,X: A,Y: A,Z3: A] :
( ( topolo1966860045006549960nected @ A @ U3 )
=> ( ( member @ A @ X @ U3 )
=> ( ( member @ A @ Y @ U3 )
=> ( ( ord_less_eq @ A @ X @ Z3 )
=> ( ( ord_less_eq @ A @ Z3 @ Y )
=> ( member @ A @ Z3 @ U3 ) ) ) ) ) ) ) ).
% connectedD_interval
thf(fact_7476_connectedI__interval,axiom,
! [A: $tType] :
( ( topolo8458572112393995274pology @ A )
=> ! [U3: set @ A] :
( ! [X5: A,Y3: A,Z: A] :
( ( member @ A @ X5 @ U3 )
=> ( ( member @ A @ Y3 @ U3 )
=> ( ( ord_less_eq @ A @ X5 @ Z )
=> ( ( ord_less_eq @ A @ Z @ Y3 )
=> ( member @ A @ Z @ U3 ) ) ) ) )
=> ( topolo1966860045006549960nected @ A @ U3 ) ) ) ).
% connectedI_interval
thf(fact_7477_connected__iff__interval,axiom,
! [A: $tType] :
( ( topolo8458572112393995274pology @ A )
=> ( ( topolo1966860045006549960nected @ A )
= ( ^ [U6: set @ A] :
! [X4: A] :
( ( member @ A @ X4 @ U6 )
=> ! [Y6: A] :
( ( member @ A @ Y6 @ U6 )
=> ! [Z6: A] :
( ( ord_less_eq @ A @ X4 @ Z6 )
=> ( ( ord_less_eq @ A @ Z6 @ Y6 )
=> ( member @ A @ Z6 @ U6 ) ) ) ) ) ) ) ) ).
% connected_iff_interval
thf(fact_7478_connected__empty,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ( topolo1966860045006549960nected @ A @ ( bot_bot @ ( set @ A ) ) ) ) ).
% connected_empty
thf(fact_7479_connected__sing,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [X: A] : ( topolo1966860045006549960nected @ A @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% connected_sing
thf(fact_7480_connected__Un,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [S: set @ A,T2: set @ A] :
( ( topolo1966860045006549960nected @ A @ S )
=> ( ( topolo1966860045006549960nected @ A @ T2 )
=> ( ( ( inf_inf @ ( set @ A ) @ S @ T2 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( topolo1966860045006549960nected @ A @ ( sup_sup @ ( set @ A ) @ S @ T2 ) ) ) ) ) ) ).
% connected_Un
thf(fact_7481_connected__Union,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [S2: set @ ( set @ A )] :
( ! [S3: set @ A] :
( ( member @ ( set @ A ) @ S3 @ S2 )
=> ( topolo1966860045006549960nected @ A @ S3 ) )
=> ( ( ( complete_Inf_Inf @ ( set @ A ) @ S2 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( topolo1966860045006549960nected @ A @ ( complete_Sup_Sup @ ( set @ A ) @ S2 ) ) ) ) ) ).
% connected_Union
thf(fact_7482_not__in__connected__cases,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [S2: set @ A,X: A] :
( ( topolo1966860045006549960nected @ A @ S2 )
=> ( ~ ( member @ A @ X @ S2 )
=> ( ( S2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ( condit941137186595557371_above @ A @ S2 )
=> ~ ! [Y4: A] :
( ( member @ A @ Y4 @ S2 )
=> ( ord_less_eq @ A @ Y4 @ X ) ) )
=> ~ ( ( condit1013018076250108175_below @ A @ S2 )
=> ~ ! [Y4: A] :
( ( member @ A @ Y4 @ S2 )
=> ( ord_less_eq @ A @ X @ Y4 ) ) ) ) ) ) ) ) ).
% not_in_connected_cases
thf(fact_7483_connected__diff__open__from__closed,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [S: set @ A,T2: set @ A,U: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ S @ T2 )
=> ( ( ord_less_eq @ ( set @ A ) @ T2 @ U )
=> ( ( topolo1002775350975398744n_open @ A @ S )
=> ( ( topolo7761053866217962861closed @ A @ T2 )
=> ( ( topolo1966860045006549960nected @ A @ U )
=> ( ( topolo1966860045006549960nected @ A @ ( minus_minus @ ( set @ A ) @ T2 @ S ) )
=> ( topolo1966860045006549960nected @ A @ ( minus_minus @ ( set @ A ) @ U @ S ) ) ) ) ) ) ) ) ) ).
% connected_diff_open_from_closed
thf(fact_7484_connected__def,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ( ( topolo1966860045006549960nected @ A )
= ( ^ [S6: set @ A] :
~ ? [A7: set @ A,B6: set @ A] :
( ( topolo1002775350975398744n_open @ A @ A7 )
& ( topolo1002775350975398744n_open @ A @ B6 )
& ( ord_less_eq @ ( set @ A ) @ S6 @ ( sup_sup @ ( set @ A ) @ A7 @ B6 ) )
& ( ( inf_inf @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A7 @ B6 ) @ S6 )
= ( bot_bot @ ( set @ A ) ) )
& ( ( inf_inf @ ( set @ A ) @ A7 @ S6 )
!= ( bot_bot @ ( set @ A ) ) )
& ( ( inf_inf @ ( set @ A ) @ B6 @ S6 )
!= ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% connected_def
thf(fact_7485_connectedI,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [U3: set @ A] :
( ! [A8: set @ A] :
( ( topolo1002775350975398744n_open @ A @ A8 )
=> ! [B9: set @ A] :
( ( topolo1002775350975398744n_open @ A @ B9 )
=> ( ( ( inf_inf @ ( set @ A ) @ A8 @ U3 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ( inf_inf @ ( set @ A ) @ B9 @ U3 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ( inf_inf @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A8 @ B9 ) @ U3 )
= ( bot_bot @ ( set @ A ) ) )
=> ~ ( ord_less_eq @ ( set @ A ) @ U3 @ ( sup_sup @ ( set @ A ) @ A8 @ B9 ) ) ) ) ) ) )
=> ( topolo1966860045006549960nected @ A @ U3 ) ) ) ).
% connectedI
thf(fact_7486_connectedD,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [A4: set @ A,U3: set @ A,V: set @ A] :
( ( topolo1966860045006549960nected @ A @ A4 )
=> ( ( topolo1002775350975398744n_open @ A @ U3 )
=> ( ( topolo1002775350975398744n_open @ A @ V )
=> ( ( ( inf_inf @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ U3 @ V ) @ A4 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ A ) @ A4 @ ( sup_sup @ ( set @ A ) @ U3 @ V ) )
=> ( ( ( inf_inf @ ( set @ A ) @ U3 @ A4 )
= ( bot_bot @ ( set @ A ) ) )
| ( ( inf_inf @ ( set @ A ) @ V @ A4 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ).
% connectedD
thf(fact_7487_connected__closedD,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [S: set @ A,A4: set @ A,B2: set @ A] :
( ( topolo1966860045006549960nected @ A @ S )
=> ( ( ( inf_inf @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A4 @ B2 ) @ S )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ A ) @ S @ ( sup_sup @ ( set @ A ) @ A4 @ B2 ) )
=> ( ( topolo7761053866217962861closed @ A @ A4 )
=> ( ( topolo7761053866217962861closed @ A @ B2 )
=> ( ( ( inf_inf @ ( set @ A ) @ A4 @ S )
= ( bot_bot @ ( set @ A ) ) )
| ( ( inf_inf @ ( set @ A ) @ B2 @ S )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ).
% connected_closedD
thf(fact_7488_disjnt__equiv__class,axiom,
! [A: $tType,A4: set @ A,R2: set @ ( product_prod @ A @ A ),A3: A,B3: A] :
( ( equiv_equiv @ A @ A4 @ R2 )
=> ( ( disjnt @ A @ ( image @ A @ A @ R2 @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( image @ A @ A @ R2 @ ( insert2 @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B3 ) @ R2 ) ) ) ) ).
% disjnt_equiv_class
thf(fact_7489_map__tailrec__rev,axiom,
! [A: $tType,B: $tType] :
( ( map_tailrec_rev @ B @ A )
= ( ^ [F3: B > A,As5: list @ B] : ( append @ A @ ( rev @ A @ ( map @ B @ A @ F3 @ As5 ) ) ) ) ) ).
% map_tailrec_rev
thf(fact_7490_disjnt__self__iff__empty,axiom,
! [A: $tType,S2: set @ A] :
( ( disjnt @ A @ S2 @ S2 )
= ( S2
= ( bot_bot @ ( set @ A ) ) ) ) ).
% disjnt_self_iff_empty
thf(fact_7491_disjnt__insert2,axiom,
! [A: $tType,Y7: set @ A,A3: A,X8: set @ A] :
( ( disjnt @ A @ Y7 @ ( insert2 @ A @ A3 @ X8 ) )
= ( ~ ( member @ A @ A3 @ Y7 )
& ( disjnt @ A @ Y7 @ X8 ) ) ) ).
% disjnt_insert2
thf(fact_7492_disjnt__insert1,axiom,
! [A: $tType,A3: A,X8: set @ A,Y7: set @ A] :
( ( disjnt @ A @ ( insert2 @ A @ A3 @ X8 ) @ Y7 )
= ( ~ ( member @ A @ A3 @ Y7 )
& ( disjnt @ A @ X8 @ Y7 ) ) ) ).
% disjnt_insert1
thf(fact_7493_disjnt__Un1,axiom,
! [A: $tType,A4: set @ A,B2: set @ A,C5: set @ A] :
( ( disjnt @ A @ ( sup_sup @ ( set @ A ) @ A4 @ B2 ) @ C5 )
= ( ( disjnt @ A @ A4 @ C5 )
& ( disjnt @ A @ B2 @ C5 ) ) ) ).
% disjnt_Un1
thf(fact_7494_disjnt__Un2,axiom,
! [A: $tType,C5: set @ A,A4: set @ A,B2: set @ A] :
( ( disjnt @ A @ C5 @ ( sup_sup @ ( set @ A ) @ A4 @ B2 ) )
= ( ( disjnt @ A @ C5 @ A4 )
& ( disjnt @ A @ C5 @ B2 ) ) ) ).
% disjnt_Un2
thf(fact_7495_disjnt__Times1__iff,axiom,
! [A: $tType,B: $tType,C5: set @ A,A4: set @ B,B2: set @ B] :
( ( disjnt @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ C5
@ ^ [Uu3: A] : A4 )
@ ( product_Sigma @ A @ B @ C5
@ ^ [Uu3: A] : B2 ) )
= ( ( C5
= ( bot_bot @ ( set @ A ) ) )
| ( disjnt @ B @ A4 @ B2 ) ) ) ).
% disjnt_Times1_iff
thf(fact_7496_disjnt__Times2__iff,axiom,
! [B: $tType,A: $tType,A4: set @ A,C5: set @ B,B2: set @ A] :
( ( disjnt @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ A4
@ ^ [Uu3: A] : C5 )
@ ( product_Sigma @ A @ B @ B2
@ ^ [Uu3: A] : C5 ) )
= ( ( C5
= ( bot_bot @ ( set @ B ) ) )
| ( disjnt @ A @ A4 @ B2 ) ) ) ).
% disjnt_Times2_iff
thf(fact_7497_disjnt__insert,axiom,
! [A: $tType,X: A,N6: set @ A,M7: set @ A] :
( ~ ( member @ A @ X @ N6 )
=> ( ( disjnt @ A @ M7 @ N6 )
=> ( disjnt @ A @ ( insert2 @ A @ X @ M7 ) @ N6 ) ) ) ).
% disjnt_insert
thf(fact_7498_disjnt__empty1,axiom,
! [A: $tType,A4: set @ A] : ( disjnt @ A @ ( bot_bot @ ( set @ A ) ) @ A4 ) ).
% disjnt_empty1
thf(fact_7499_disjnt__empty2,axiom,
! [A: $tType,A4: set @ A] : ( disjnt @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) ).
% disjnt_empty2
thf(fact_7500_disjnt__subset2,axiom,
! [A: $tType,X8: set @ A,Y7: set @ A,Z5: set @ A] :
( ( disjnt @ A @ X8 @ Y7 )
=> ( ( ord_less_eq @ ( set @ A ) @ Z5 @ Y7 )
=> ( disjnt @ A @ X8 @ Z5 ) ) ) ).
% disjnt_subset2
thf(fact_7501_disjnt__subset1,axiom,
! [A: $tType,X8: set @ A,Y7: set @ A,Z5: set @ A] :
( ( disjnt @ A @ X8 @ Y7 )
=> ( ( ord_less_eq @ ( set @ A ) @ Z5 @ X8 )
=> ( disjnt @ A @ Z5 @ Y7 ) ) ) ).
% disjnt_subset1
thf(fact_7502_map__tailrec__rev_Osimps_I1_J,axiom,
! [A: $tType,B: $tType,F2: A > B,Bs: list @ B] :
( ( map_tailrec_rev @ A @ B @ F2 @ ( nil @ A ) @ Bs )
= Bs ) ).
% map_tailrec_rev.simps(1)
thf(fact_7503_disjnt__iff,axiom,
! [A: $tType] :
( ( disjnt @ A )
= ( ^ [A7: set @ A,B6: set @ A] :
! [X4: A] :
~ ( ( member @ A @ X4 @ A7 )
& ( member @ A @ X4 @ B6 ) ) ) ) ).
% disjnt_iff
thf(fact_7504_disjnt__sym,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] :
( ( disjnt @ A @ A4 @ B2 )
=> ( disjnt @ A @ B2 @ A4 ) ) ).
% disjnt_sym
thf(fact_7505_map__tailrec__rev_Osimps_I2_J,axiom,
! [A: $tType,B: $tType,F2: A > B,A3: A,As: list @ A,Bs: list @ B] :
( ( map_tailrec_rev @ A @ B @ F2 @ ( cons @ A @ A3 @ As ) @ Bs )
= ( map_tailrec_rev @ A @ B @ F2 @ As @ ( cons @ B @ ( F2 @ A3 ) @ Bs ) ) ) ).
% map_tailrec_rev.simps(2)
thf(fact_7506_disjnt__def,axiom,
! [A: $tType] :
( ( disjnt @ A )
= ( ^ [A7: set @ A,B6: set @ A] :
( ( inf_inf @ ( set @ A ) @ A7 @ B6 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% disjnt_def
thf(fact_7507_disjnt__Sigma__iff,axiom,
! [B: $tType,A: $tType,A4: set @ A,C5: A > ( set @ B ),B2: set @ A] :
( ( disjnt @ ( product_prod @ A @ B ) @ ( product_Sigma @ A @ B @ A4 @ C5 ) @ ( product_Sigma @ A @ B @ B2 @ C5 ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ ( inf_inf @ ( set @ A ) @ A4 @ B2 ) )
=> ( ( C5 @ X4 )
= ( bot_bot @ ( set @ B ) ) ) )
| ( disjnt @ A @ A4 @ B2 ) ) ) ).
% disjnt_Sigma_iff
thf(fact_7508_disjnt__ge__max,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Y7: set @ A,X8: set @ A] :
( ( finite_finite2 @ A @ Y7 )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ X8 )
=> ( ord_less @ A @ ( lattic643756798349783984er_Max @ A @ Y7 ) @ X5 ) )
=> ( disjnt @ A @ X8 @ Y7 ) ) ) ) ).
% disjnt_ge_max
thf(fact_7509_map__tailrec__rev_Oelims,axiom,
! [A: $tType,B: $tType,X: A > B,Xa2: list @ A,Xb: list @ B,Y: list @ B] :
( ( ( map_tailrec_rev @ A @ B @ X @ Xa2 @ Xb )
= Y )
=> ( ( ( Xa2
= ( nil @ A ) )
=> ( Y != Xb ) )
=> ~ ! [A6: A,As2: list @ A] :
( ( Xa2
= ( cons @ A @ A6 @ As2 ) )
=> ( Y
!= ( map_tailrec_rev @ A @ B @ X @ As2 @ ( cons @ B @ ( X @ A6 ) @ Xb ) ) ) ) ) ) ).
% map_tailrec_rev.elims
thf(fact_7510_card__Un__disjnt,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( finite_finite2 @ A @ B2 )
=> ( ( disjnt @ A @ A4 @ B2 )
=> ( ( finite_card @ A @ ( sup_sup @ ( set @ A ) @ A4 @ B2 ) )
= ( plus_plus @ nat @ ( finite_card @ A @ A4 ) @ ( finite_card @ A @ B2 ) ) ) ) ) ) ).
% card_Un_disjnt
thf(fact_7511_map__tailrec__def,axiom,
! [B: $tType,A: $tType] :
( ( map_tailrec @ A @ B )
= ( ^ [F3: A > B,As5: list @ A] : ( rev @ B @ ( map_tailrec_rev @ A @ B @ F3 @ As5 @ ( nil @ B ) ) ) ) ) ).
% map_tailrec_def
thf(fact_7512_sum__card__image,axiom,
! [B: $tType,A: $tType,A4: set @ A,F2: A > ( set @ B )] :
( ( finite_finite2 @ A @ A4 )
=> ( ( pairwise @ A
@ ^ [S7: A,T3: A] : ( disjnt @ B @ ( F2 @ S7 ) @ ( F2 @ T3 ) )
@ A4 )
=> ( ( groups7311177749621191930dd_sum @ ( set @ B ) @ nat @ ( finite_card @ B ) @ ( image2 @ A @ ( set @ B ) @ F2 @ A4 ) )
= ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [A5: A] : ( finite_card @ B @ ( F2 @ A5 ) )
@ A4 ) ) ) ) ).
% sum_card_image
thf(fact_7513_pairwise__imageI,axiom,
! [B: $tType,A: $tType,A4: set @ A,F2: A > B,P: B > B > $o] :
( ! [X5: A,Y3: A] :
( ( member @ A @ X5 @ A4 )
=> ( ( member @ A @ Y3 @ A4 )
=> ( ( X5 != Y3 )
=> ( ( ( F2 @ X5 )
!= ( F2 @ Y3 ) )
=> ( P @ ( F2 @ X5 ) @ ( F2 @ Y3 ) ) ) ) ) )
=> ( pairwise @ B @ P @ ( image2 @ A @ B @ F2 @ A4 ) ) ) ).
% pairwise_imageI
thf(fact_7514_pairwise__image,axiom,
! [A: $tType,B: $tType,R2: A > A > $o,F2: B > A,S: set @ B] :
( ( pairwise @ A @ R2 @ ( image2 @ B @ A @ F2 @ S ) )
= ( pairwise @ B
@ ^ [X4: B,Y6: B] :
( ( ( F2 @ X4 )
!= ( F2 @ Y6 ) )
=> ( R2 @ ( F2 @ X4 ) @ ( F2 @ Y6 ) ) )
@ S ) ) ).
% pairwise_image
thf(fact_7515_pairwise__trivial,axiom,
! [A: $tType,I6: set @ A] :
( pairwise @ A
@ ^ [I3: A,J2: A] : J2 != I3
@ I6 ) ).
% pairwise_trivial
thf(fact_7516_pairwise__def,axiom,
! [A: $tType] :
( ( pairwise @ A )
= ( ^ [R5: A > A > $o,S6: set @ A] :
! [X4: A] :
( ( member @ A @ X4 @ S6 )
=> ! [Y6: A] :
( ( member @ A @ Y6 @ S6 )
=> ( ( X4 != Y6 )
=> ( R5 @ X4 @ Y6 ) ) ) ) ) ) ).
% pairwise_def
thf(fact_7517_pairwiseI,axiom,
! [A: $tType,S2: set @ A,R: A > A > $o] :
( ! [X5: A,Y3: A] :
( ( member @ A @ X5 @ S2 )
=> ( ( member @ A @ Y3 @ S2 )
=> ( ( X5 != Y3 )
=> ( R @ X5 @ Y3 ) ) ) )
=> ( pairwise @ A @ R @ S2 ) ) ).
% pairwiseI
thf(fact_7518_pairwiseD,axiom,
! [A: $tType,R: A > A > $o,S2: set @ A,X: A,Y: A] :
( ( pairwise @ A @ R @ S2 )
=> ( ( member @ A @ X @ S2 )
=> ( ( member @ A @ Y @ S2 )
=> ( ( X != Y )
=> ( R @ X @ Y ) ) ) ) ) ).
% pairwiseD
thf(fact_7519_pairwise__subset,axiom,
! [A: $tType,P: A > A > $o,S2: set @ A,T4: set @ A] :
( ( pairwise @ A @ P @ S2 )
=> ( ( ord_less_eq @ ( set @ A ) @ T4 @ S2 )
=> ( pairwise @ A @ P @ T4 ) ) ) ).
% pairwise_subset
thf(fact_7520_pairwise__mono,axiom,
! [A: $tType,P: A > A > $o,A4: set @ A,Q: A > A > $o,B2: set @ A] :
( ( pairwise @ A @ P @ A4 )
=> ( ! [X5: A,Y3: A] :
( ( P @ X5 @ Y3 )
=> ( Q @ X5 @ Y3 ) )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ A4 )
=> ( pairwise @ A @ Q @ B2 ) ) ) ) ).
% pairwise_mono
thf(fact_7521_pairwise__empty,axiom,
! [A: $tType,P: A > A > $o] : ( pairwise @ A @ P @ ( bot_bot @ ( set @ A ) ) ) ).
% pairwise_empty
thf(fact_7522_pairwise__insert,axiom,
! [A: $tType,R2: A > A > $o,X: A,S: set @ A] :
( ( pairwise @ A @ R2 @ ( insert2 @ A @ X @ S ) )
= ( ! [Y6: A] :
( ( ( member @ A @ Y6 @ S )
& ( Y6 != X ) )
=> ( ( R2 @ X @ Y6 )
& ( R2 @ Y6 @ X ) ) )
& ( pairwise @ A @ R2 @ S ) ) ) ).
% pairwise_insert
thf(fact_7523_pairwise__singleton,axiom,
! [A: $tType,P: A > A > $o,A4: A] : ( pairwise @ A @ P @ ( insert2 @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% pairwise_singleton
thf(fact_7524_pairwise__alt,axiom,
! [A: $tType] :
( ( pairwise @ A )
= ( ^ [R5: A > A > $o,S6: set @ A] :
! [X4: A] :
( ( member @ A @ X4 @ S6 )
=> ! [Y6: A] :
( ( member @ A @ Y6 @ ( minus_minus @ ( set @ A ) @ S6 @ ( insert2 @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( R5 @ X4 @ Y6 ) ) ) ) ) ).
% pairwise_alt
thf(fact_7525_disjoint__image__subset,axiom,
! [A: $tType,A20: set @ ( set @ A ),F2: ( set @ A ) > ( set @ A )] :
( ( pairwise @ ( set @ A ) @ ( disjnt @ A ) @ A20 )
=> ( ! [X9: set @ A] :
( ( member @ ( set @ A ) @ X9 @ A20 )
=> ( ord_less_eq @ ( set @ A ) @ ( F2 @ X9 ) @ X9 ) )
=> ( pairwise @ ( set @ A ) @ ( disjnt @ A ) @ ( image2 @ ( set @ A ) @ ( set @ A ) @ F2 @ A20 ) ) ) ) ).
% disjoint_image_subset
thf(fact_7526_map__eq__map__tailrec,axiom,
! [B: $tType,A: $tType] :
( ( map @ A @ B )
= ( map_tailrec @ A @ B ) ) ).
% map_eq_map_tailrec
thf(fact_7527_card__Union__disjoint,axiom,
! [A: $tType,C5: set @ ( set @ A )] :
( ( pairwise @ ( set @ A ) @ ( disjnt @ A ) @ C5 )
=> ( ! [A8: set @ A] :
( ( member @ ( set @ A ) @ A8 @ C5 )
=> ( finite_finite2 @ A @ A8 ) )
=> ( ( finite_card @ A @ ( complete_Sup_Sup @ ( set @ A ) @ C5 ) )
= ( groups7311177749621191930dd_sum @ ( set @ A ) @ nat @ ( finite_card @ A ) @ C5 ) ) ) ) ).
% card_Union_disjoint
thf(fact_7528_infinite__infinite__partition,axiom,
! [A: $tType,A4: set @ A] :
( ~ ( finite_finite2 @ A @ A4 )
=> ~ ! [C7: nat > ( set @ A )] :
( ( pairwise @ nat
@ ^ [I3: nat,J2: nat] : ( disjnt @ A @ ( C7 @ I3 ) @ ( C7 @ J2 ) )
@ ( top_top @ ( set @ nat ) ) )
=> ( ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ C7 @ ( top_top @ ( set @ nat ) ) ) ) @ A4 )
=> ~ ! [I5: nat] :
~ ( finite_finite2 @ A @ ( C7 @ I5 ) ) ) ) ) ).
% infinite_infinite_partition
thf(fact_7529_insort__insert__key__def,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ( ( linord329482645794927042rt_key @ B @ A )
= ( ^ [F3: B > A,X4: B,Xs3: list @ B] : ( if @ ( list @ B ) @ ( member @ A @ ( F3 @ X4 ) @ ( image2 @ B @ A @ F3 @ ( set2 @ B @ Xs3 ) ) ) @ Xs3 @ ( linorder_insort_key @ B @ A @ F3 @ X4 @ Xs3 ) ) ) ) ) ).
% insort_insert_key_def
thf(fact_7530_insort__insert__insort__key,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F2: B > A,X: B,Xs: list @ B] :
( ~ ( member @ A @ ( F2 @ X ) @ ( image2 @ B @ A @ F2 @ ( set2 @ B @ Xs ) ) )
=> ( ( linord329482645794927042rt_key @ B @ A @ F2 @ X @ Xs )
= ( linorder_insort_key @ B @ A @ F2 @ X @ Xs ) ) ) ) ).
% insort_insert_insort_key
thf(fact_7531_insort__insert__triv,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Xs: list @ A] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ( linord329482645794927042rt_key @ A @ A
@ ^ [X4: A] : X4
@ X
@ Xs )
= Xs ) ) ) ).
% insort_insert_triv
thf(fact_7532_distinct__insort__insert,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ B,F2: B > A,X: B] :
( ( distinct @ B @ Xs )
=> ( distinct @ B @ ( linord329482645794927042rt_key @ B @ A @ F2 @ X @ Xs ) ) ) ) ).
% distinct_insort_insert
thf(fact_7533_insort__insert__key__triv,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F2: B > A,X: B,Xs: list @ B] :
( ( member @ A @ ( F2 @ X ) @ ( image2 @ B @ A @ F2 @ ( set2 @ B @ Xs ) ) )
=> ( ( linord329482645794927042rt_key @ B @ A @ F2 @ X @ Xs )
= Xs ) ) ) ).
% insort_insert_key_triv
thf(fact_7534_set__insort__insert,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Xs: list @ A] :
( ( set2 @ A
@ ( linord329482645794927042rt_key @ A @ A
@ ^ [X4: A] : X4
@ X
@ Xs ) )
= ( insert2 @ A @ X @ ( set2 @ A @ Xs ) ) ) ) ).
% set_insort_insert
thf(fact_7535_sorted__insort__insert,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,X: A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A )
@ ( linord329482645794927042rt_key @ A @ A
@ ^ [X4: A] : X4
@ X
@ Xs ) ) ) ) ).
% sorted_insort_insert
thf(fact_7536_insort__insert__insort,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Xs: list @ A] :
( ~ ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ( linord329482645794927042rt_key @ A @ A
@ ^ [X4: A] : X4
@ X
@ Xs )
= ( linorder_insort_key @ A @ A
@ ^ [X4: A] : X4
@ X
@ Xs ) ) ) ) ).
% insort_insert_insort
thf(fact_7537_sorted__insort__insert__key,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F2: B > A,Xs: list @ B,X: B] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F2 @ Xs ) )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F2 @ ( linord329482645794927042rt_key @ B @ A @ F2 @ X @ Xs ) ) ) ) ) ).
% sorted_insort_insert_key
thf(fact_7538_differentiable__cmult__right__iff,axiom,
! [A: $tType,B: $tType] :
( ( ( real_V822414075346904944vector @ B )
& ( real_V3459762299906320749_field @ A ) )
=> ! [Q3: B > A,C2: A,T2: B] :
( ( differentiable @ B @ A
@ ^ [T3: B] : ( times_times @ A @ ( Q3 @ T3 ) @ C2 )
@ ( topolo174197925503356063within @ B @ T2 @ ( top_top @ ( set @ B ) ) ) )
= ( ( C2
= ( zero_zero @ A ) )
| ( differentiable @ B @ A @ Q3 @ ( topolo174197925503356063within @ B @ T2 @ ( top_top @ ( set @ B ) ) ) ) ) ) ) ).
% differentiable_cmult_right_iff
thf(fact_7539_differentiable__cmult__left__iff,axiom,
! [A: $tType,B: $tType] :
( ( ( real_V822414075346904944vector @ B )
& ( real_V3459762299906320749_field @ A ) )
=> ! [C2: A,Q3: B > A,T2: B] :
( ( differentiable @ B @ A
@ ^ [T3: B] : ( times_times @ A @ C2 @ ( Q3 @ T3 ) )
@ ( topolo174197925503356063within @ B @ T2 @ ( top_top @ ( set @ B ) ) ) )
= ( ( C2
= ( zero_zero @ A ) )
| ( differentiable @ B @ A @ Q3 @ ( topolo174197925503356063within @ B @ T2 @ ( top_top @ ( set @ B ) ) ) ) ) ) ) ).
% differentiable_cmult_left_iff
thf(fact_7540_differentiable__add,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,F5: filter @ A,G: A > B] :
( ( differentiable @ A @ B @ F2 @ F5 )
=> ( ( differentiable @ A @ B @ G @ F5 )
=> ( differentiable @ A @ B
@ ^ [X4: A] : ( plus_plus @ B @ ( F2 @ X4 ) @ ( G @ X4 ) )
@ F5 ) ) ) ) ).
% differentiable_add
thf(fact_7541_differentiable__within__subset,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,X: A,S: set @ A,T2: set @ A] :
( ( differentiable @ A @ B @ F2 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( ( ord_less_eq @ ( set @ A ) @ T2 @ S )
=> ( differentiable @ A @ B @ F2 @ ( topolo174197925503356063within @ A @ X @ T2 ) ) ) ) ) ).
% differentiable_within_subset
thf(fact_7542_differentiable__sum,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( ( real_V822414075346904944vector @ B )
& ( real_V822414075346904944vector @ C ) )
=> ! [S: set @ A,F2: A > B > C,Net: filter @ B] :
( ( finite_finite2 @ A @ S )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ S )
=> ( differentiable @ B @ C @ ( F2 @ X5 ) @ Net ) )
=> ( differentiable @ B @ C
@ ^ [X4: B] :
( groups7311177749621191930dd_sum @ A @ C
@ ^ [A5: A] : ( F2 @ A5 @ X4 )
@ S )
@ Net ) ) ) ) ).
% differentiable_sum
thf(fact_7543_differentiable__divide,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V3459762299906320749_field @ B ) )
=> ! [F2: A > B,X: A,S: set @ A,G: A > B] :
( ( differentiable @ A @ B @ F2 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( ( differentiable @ A @ B @ G @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( ( ( G @ X )
!= ( zero_zero @ B ) )
=> ( differentiable @ A @ B
@ ^ [X4: A] : ( divide_divide @ B @ ( F2 @ X4 ) @ ( G @ X4 ) )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ) ).
% differentiable_divide
thf(fact_7544_differentiable__inverse,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V3459762299906320749_field @ B ) )
=> ! [F2: A > B,X: A,S: set @ A] :
( ( differentiable @ A @ B @ F2 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( ( ( F2 @ X )
!= ( zero_zero @ B ) )
=> ( differentiable @ A @ B
@ ^ [X4: A] : ( inverse_inverse @ B @ ( F2 @ X4 ) )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ).
% differentiable_inverse
thf(fact_7545_differentiable__power__int,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V3459762299906320749_field @ B ) )
=> ! [F2: A > B,X: A,S: set @ A,N2: int] :
( ( differentiable @ A @ B @ F2 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( ( ( F2 @ X )
!= ( zero_zero @ B ) )
=> ( differentiable @ A @ B
@ ^ [X4: A] : ( power_int @ B @ ( F2 @ X4 ) @ N2 )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ).
% differentiable_power_int
thf(fact_7546_arg__min__inj__eq,axiom,
! [B: $tType,A: $tType] :
( ( order @ B )
=> ! [F2: A > B,P: A > $o,A3: A] :
( ( inj_on @ A @ B @ F2 @ ( collect @ A @ P ) )
=> ( ( P @ A3 )
=> ( ! [Y3: A] :
( ( P @ Y3 )
=> ( ord_less_eq @ B @ ( F2 @ A3 ) @ ( F2 @ Y3 ) ) )
=> ( ( lattices_ord_arg_min @ A @ B @ F2 @ P )
= A3 ) ) ) ) ) ).
% arg_min_inj_eq
thf(fact_7547_concat__map__maps,axiom,
! [A: $tType,B: $tType,F2: B > ( list @ A ),Xs: list @ B] :
( ( concat @ A @ ( map @ B @ ( list @ A ) @ F2 @ Xs ) )
= ( maps @ B @ A @ F2 @ Xs ) ) ).
% concat_map_maps
thf(fact_7548_maps__simps_I2_J,axiom,
! [B: $tType,A: $tType,F2: B > ( list @ A )] :
( ( maps @ B @ A @ F2 @ ( nil @ B ) )
= ( nil @ A ) ) ).
% maps_simps(2)
thf(fact_7549_arg__minI,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [P: A > $o,X: A,F2: A > B,Q: A > $o] :
( ( P @ X )
=> ( ! [Y3: A] :
( ( P @ Y3 )
=> ~ ( ord_less @ B @ ( F2 @ Y3 ) @ ( F2 @ X ) ) )
=> ( ! [X5: A] :
( ( P @ X5 )
=> ( ! [Y4: A] :
( ( P @ Y4 )
=> ~ ( ord_less @ B @ ( F2 @ Y4 ) @ ( F2 @ X5 ) ) )
=> ( Q @ X5 ) ) )
=> ( Q @ ( lattices_ord_arg_min @ A @ B @ F2 @ P ) ) ) ) ) ) ).
% arg_minI
thf(fact_7550_arg__min__equality,axiom,
! [A: $tType,C: $tType] :
( ( order @ A )
=> ! [P: C > $o,K: C,F2: C > A] :
( ( P @ K )
=> ( ! [X5: C] :
( ( P @ X5 )
=> ( ord_less_eq @ A @ ( F2 @ K ) @ ( F2 @ X5 ) ) )
=> ( ( F2 @ ( lattices_ord_arg_min @ C @ A @ F2 @ P ) )
= ( F2 @ K ) ) ) ) ) ).
% arg_min_equality
thf(fact_7551_arg__min__nat__lemma,axiom,
! [A: $tType,P: A > $o,K: A,M2: A > nat] :
( ( P @ K )
=> ( ( P @ ( lattices_ord_arg_min @ A @ nat @ M2 @ P ) )
& ! [Y4: A] :
( ( P @ Y4 )
=> ( ord_less_eq @ nat @ ( M2 @ ( lattices_ord_arg_min @ A @ nat @ M2 @ P ) ) @ ( M2 @ Y4 ) ) ) ) ) ).
% arg_min_nat_lemma
thf(fact_7552_arg__min__nat__le,axiom,
! [A: $tType,P: A > $o,X: A,M2: A > nat] :
( ( P @ X )
=> ( ord_less_eq @ nat @ ( M2 @ ( lattices_ord_arg_min @ A @ nat @ M2 @ P ) ) @ ( M2 @ X ) ) ) ).
% arg_min_nat_le
thf(fact_7553_arg__min__on__def,axiom,
! [A: $tType,B: $tType] :
( ( ord @ A )
=> ( ( lattic7623131987881927897min_on @ B @ A )
= ( ^ [F3: B > A,S6: set @ B] :
( lattices_ord_arg_min @ B @ A @ F3
@ ^ [X4: B] : ( member @ B @ X4 @ S6 ) ) ) ) ) ).
% arg_min_on_def
thf(fact_7554_maps__simps_I1_J,axiom,
! [A: $tType,B: $tType,F2: B > ( list @ A ),X: B,Xs: list @ B] :
( ( maps @ B @ A @ F2 @ ( cons @ B @ X @ Xs ) )
= ( append @ A @ ( F2 @ X ) @ ( maps @ B @ A @ F2 @ Xs ) ) ) ).
% maps_simps(1)
thf(fact_7555_maps__def,axiom,
! [B: $tType,A: $tType] :
( ( maps @ A @ B )
= ( ^ [F3: A > ( list @ B ),Xs3: list @ A] : ( concat @ B @ ( map @ A @ ( list @ B ) @ F3 @ Xs3 ) ) ) ) ).
% maps_def
thf(fact_7556_arg__min__def,axiom,
! [A: $tType,B: $tType] :
( ( ord @ A )
=> ( ( lattices_ord_arg_min @ B @ A )
= ( ^ [F3: B > A,P3: B > $o] : ( fChoice @ B @ ( lattic501386751177426532rg_min @ B @ A @ F3 @ P3 ) ) ) ) ) ).
% arg_min_def
thf(fact_7557_in__measures_I2_J,axiom,
! [A: $tType,X: A,Y: A,F2: A > nat,Fs: list @ ( A > nat )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( measures @ A @ ( cons @ ( A > nat ) @ F2 @ Fs ) ) )
= ( ( ord_less @ nat @ ( F2 @ X ) @ ( F2 @ Y ) )
| ( ( ( F2 @ X )
= ( F2 @ Y ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( measures @ A @ Fs ) ) ) ) ) ).
% in_measures(2)
thf(fact_7558_in__measures_I1_J,axiom,
! [A: $tType,X: A,Y: A] :
~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( measures @ A @ ( nil @ ( A > nat ) ) ) ) ).
% in_measures(1)
thf(fact_7559_arg__min__natI,axiom,
! [A: $tType,P: A > $o,K: A,M2: A > nat] :
( ( P @ K )
=> ( P @ ( lattices_ord_arg_min @ A @ nat @ M2 @ P ) ) ) ).
% arg_min_natI
thf(fact_7560_is__arg__min__arg__min__nat,axiom,
! [A: $tType,P: A > $o,X: A,M2: A > nat] :
( ( P @ X )
=> ( lattic501386751177426532rg_min @ A @ nat @ M2 @ P @ ( lattices_ord_arg_min @ A @ nat @ M2 @ P ) ) ) ).
% is_arg_min_arg_min_nat
thf(fact_7561_wf__measures,axiom,
! [A: $tType,Fs: list @ ( A > nat )] : ( wf @ A @ ( measures @ A @ Fs ) ) ).
% wf_measures
thf(fact_7562_is__arg__min__def,axiom,
! [A: $tType,B: $tType] :
( ( ord @ A )
=> ( ( lattic501386751177426532rg_min @ B @ A )
= ( ^ [F3: B > A,P3: B > $o,X4: B] :
( ( P3 @ X4 )
& ~ ? [Y6: B] :
( ( P3 @ Y6 )
& ( ord_less @ A @ ( F3 @ Y6 ) @ ( F3 @ X4 ) ) ) ) ) ) ) ).
% is_arg_min_def
thf(fact_7563_is__arg__min__antimono,axiom,
! [B: $tType,A: $tType] :
( ( order @ B )
=> ! [F2: A > B,P: A > $o,X: A,Y: A] :
( ( lattic501386751177426532rg_min @ A @ B @ F2 @ P @ X )
=> ( ( ord_less_eq @ B @ ( F2 @ Y ) @ ( F2 @ X ) )
=> ( ( P @ Y )
=> ( lattic501386751177426532rg_min @ A @ B @ F2 @ P @ Y ) ) ) ) ) ).
% is_arg_min_antimono
thf(fact_7564_is__arg__min__linorder,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ B )
=> ( ( lattic501386751177426532rg_min @ A @ B )
= ( ^ [F3: A > B,P3: A > $o,X4: A] :
( ( P3 @ X4 )
& ! [Y6: A] :
( ( P3 @ Y6 )
=> ( ord_less_eq @ B @ ( F3 @ X4 ) @ ( F3 @ Y6 ) ) ) ) ) ) ) ).
% is_arg_min_linorder
thf(fact_7565_measures__less,axiom,
! [A: $tType,F2: A > nat,X: A,Y: A,Fs: list @ ( A > nat )] :
( ( ord_less @ nat @ ( F2 @ X ) @ ( F2 @ Y ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( measures @ A @ ( cons @ ( A > nat ) @ F2 @ Fs ) ) ) ) ).
% measures_less
thf(fact_7566_measures__lesseq,axiom,
! [A: $tType,F2: A > nat,X: A,Y: A,Fs: list @ ( A > nat )] :
( ( ord_less_eq @ nat @ ( F2 @ X ) @ ( F2 @ Y ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( measures @ A @ Fs ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( measures @ A @ ( cons @ ( A > nat ) @ F2 @ Fs ) ) ) ) ) ).
% measures_lesseq
thf(fact_7567_ex__is__arg__min__if__finite,axiom,
! [B: $tType,A: $tType] :
( ( order @ B )
=> ! [S2: set @ A,F2: A > B] :
( ( finite_finite2 @ A @ S2 )
=> ( ( S2
!= ( bot_bot @ ( set @ A ) ) )
=> ? [X_12: A] :
( lattic501386751177426532rg_min @ A @ B @ F2
@ ^ [X4: A] : ( member @ A @ X4 @ S2 )
@ X_12 ) ) ) ) ).
% ex_is_arg_min_if_finite
thf(fact_7568_chains__extend,axiom,
! [A: $tType,C2: set @ ( set @ A ),S2: set @ ( set @ A ),Z3: set @ A] :
( ( member @ ( set @ ( set @ A ) ) @ C2 @ ( chains @ A @ S2 ) )
=> ( ( member @ ( set @ A ) @ Z3 @ S2 )
=> ( ! [X5: set @ A] :
( ( member @ ( set @ A ) @ X5 @ C2 )
=> ( ord_less_eq @ ( set @ A ) @ X5 @ Z3 ) )
=> ( member @ ( set @ ( set @ A ) ) @ ( sup_sup @ ( set @ ( set @ A ) ) @ ( insert2 @ ( set @ A ) @ Z3 @ ( bot_bot @ ( set @ ( set @ A ) ) ) ) @ C2 ) @ ( chains @ A @ S2 ) ) ) ) ) ).
% chains_extend
thf(fact_7569_wo__rel_Ocases__Total3,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A3: A,B3: A,Phi: A > A > $o] :
( ( bNF_Wellorder_wo_rel @ A @ R2 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( insert2 @ A @ A3 @ ( insert2 @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( field2 @ A @ R2 ) )
=> ( ( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B3 ) @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ ( id2 @ A ) ) )
| ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B3 @ A3 ) @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ ( id2 @ A ) ) ) )
=> ( Phi @ A3 @ B3 ) )
=> ( ( ( A3 = B3 )
=> ( Phi @ A3 @ B3 ) )
=> ( Phi @ A3 @ B3 ) ) ) ) ) ).
% wo_rel.cases_Total3
thf(fact_7570_Zorn__Lemma2,axiom,
! [A: $tType,A4: set @ ( set @ A )] :
( ! [X5: set @ ( set @ A )] :
( ( member @ ( set @ ( set @ A ) ) @ X5 @ ( chains @ A @ A4 ) )
=> ? [Xa: set @ A] :
( ( member @ ( set @ A ) @ Xa @ A4 )
& ! [Xb2: set @ A] :
( ( member @ ( set @ A ) @ Xb2 @ X5 )
=> ( ord_less_eq @ ( set @ A ) @ Xb2 @ Xa ) ) ) )
=> ? [X5: set @ A] :
( ( member @ ( set @ A ) @ X5 @ A4 )
& ! [Xa: set @ A] :
( ( member @ ( set @ A ) @ Xa @ A4 )
=> ( ( ord_less_eq @ ( set @ A ) @ X5 @ Xa )
=> ( Xa = X5 ) ) ) ) ) ).
% Zorn_Lemma2
thf(fact_7571_chainsD,axiom,
! [A: $tType,C2: set @ ( set @ A ),S2: set @ ( set @ A ),X: set @ A,Y: set @ A] :
( ( member @ ( set @ ( set @ A ) ) @ C2 @ ( chains @ A @ S2 ) )
=> ( ( member @ ( set @ A ) @ X @ C2 )
=> ( ( member @ ( set @ A ) @ Y @ C2 )
=> ( ( ord_less_eq @ ( set @ A ) @ X @ Y )
| ( ord_less_eq @ ( set @ A ) @ Y @ X ) ) ) ) ) ).
% chainsD
thf(fact_7572_Zorn__Lemma,axiom,
! [A: $tType,A4: set @ ( set @ A )] :
( ! [X5: set @ ( set @ A )] :
( ( member @ ( set @ ( set @ A ) ) @ X5 @ ( chains @ A @ A4 ) )
=> ( member @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ X5 ) @ A4 ) )
=> ? [X5: set @ A] :
( ( member @ ( set @ A ) @ X5 @ A4 )
& ! [Xa: set @ A] :
( ( member @ ( set @ A ) @ Xa @ A4 )
=> ( ( ord_less_eq @ ( set @ A ) @ X5 @ Xa )
=> ( Xa = X5 ) ) ) ) ) ).
% Zorn_Lemma
thf(fact_7573_dependent__wellorder__choice,axiom,
! [B: $tType,A: $tType] :
( ( wellorder @ A )
=> ! [P: ( A > B ) > A > B > $o] :
( ! [R3: B,F4: A > B,G7: A > B,X5: A] :
( ! [Y4: A] :
( ( ord_less @ A @ Y4 @ X5 )
=> ( ( F4 @ Y4 )
= ( G7 @ Y4 ) ) )
=> ( ( P @ F4 @ X5 @ R3 )
= ( P @ G7 @ X5 @ R3 ) ) )
=> ( ! [X5: A,F4: A > B] :
( ! [Y4: A] :
( ( ord_less @ A @ Y4 @ X5 )
=> ( P @ F4 @ Y4 @ ( F4 @ Y4 ) ) )
=> ? [X_1: B] : ( P @ F4 @ X5 @ X_1 ) )
=> ? [F4: A > B] :
! [X2: A] : ( P @ F4 @ X2 @ ( F4 @ X2 ) ) ) ) ) ).
% dependent_wellorder_choice
thf(fact_7574_wo__rel_Omax2__among,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A3: A,B3: A] :
( ( bNF_Wellorder_wo_rel @ A @ R2 )
=> ( ( member @ A @ A3 @ ( field2 @ A @ R2 ) )
=> ( ( member @ A @ B3 @ ( field2 @ A @ R2 ) )
=> ( member @ A @ ( bNF_We1388413361240627857o_max2 @ A @ R2 @ A3 @ B3 ) @ ( insert2 @ A @ A3 @ ( insert2 @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% wo_rel.max2_among
thf(fact_7575_wo__rel_Ocases__Total,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A3: A,B3: A,Phi: A > A > $o] :
( ( bNF_Wellorder_wo_rel @ A @ R2 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( insert2 @ A @ A3 @ ( insert2 @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( field2 @ A @ R2 ) )
=> ( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B3 ) @ R2 )
=> ( Phi @ A3 @ B3 ) )
=> ( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B3 @ A3 ) @ R2 )
=> ( Phi @ A3 @ B3 ) )
=> ( Phi @ A3 @ B3 ) ) ) ) ) ).
% wo_rel.cases_Total
thf(fact_7576_natLeq__on__wo__rel,axiom,
! [N2: nat] :
( bNF_Wellorder_wo_rel @ nat
@ ( collect @ ( product_prod @ nat @ nat )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [X4: nat,Y6: nat] :
( ( ord_less @ nat @ X4 @ N2 )
& ( ord_less @ nat @ Y6 @ N2 )
& ( ord_less_eq @ nat @ X4 @ Y6 ) ) ) ) ) ).
% natLeq_on_wo_rel
thf(fact_7577_wo__rel_Omax2__greater__among,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A3: A,B3: A] :
( ( bNF_Wellorder_wo_rel @ A @ R2 )
=> ( ( member @ A @ A3 @ ( field2 @ A @ R2 ) )
=> ( ( member @ A @ B3 @ ( field2 @ A @ R2 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ ( bNF_We1388413361240627857o_max2 @ A @ R2 @ A3 @ B3 ) ) @ R2 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B3 @ ( bNF_We1388413361240627857o_max2 @ A @ R2 @ A3 @ B3 ) ) @ R2 )
& ( member @ A @ ( bNF_We1388413361240627857o_max2 @ A @ R2 @ A3 @ B3 ) @ ( insert2 @ A @ A3 @ ( insert2 @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ).
% wo_rel.max2_greater_among
thf(fact_7578_wo__rel_OWell__order__isMinim__exists,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),B2: set @ A] :
( ( bNF_Wellorder_wo_rel @ A @ R2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ ( field2 @ A @ R2 ) )
=> ( ( B2
!= ( bot_bot @ ( set @ A ) ) )
=> ? [X_12: A] : ( bNF_We4791949203932849705sMinim @ A @ R2 @ B2 @ X_12 ) ) ) ) ).
% wo_rel.Well_order_isMinim_exists
thf(fact_7579_wo__rel_Ominim__inField,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),B2: set @ A] :
( ( bNF_Wellorder_wo_rel @ A @ R2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ ( field2 @ A @ R2 ) )
=> ( ( B2
!= ( bot_bot @ ( set @ A ) ) )
=> ( member @ A @ ( bNF_We6954850376910717587_minim @ A @ R2 @ B2 ) @ ( field2 @ A @ R2 ) ) ) ) ) ).
% wo_rel.minim_inField
thf(fact_7580_wo__rel_Ominim__isMinim,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),B2: set @ A] :
( ( bNF_Wellorder_wo_rel @ A @ R2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ ( field2 @ A @ R2 ) )
=> ( ( B2
!= ( bot_bot @ ( set @ A ) ) )
=> ( bNF_We4791949203932849705sMinim @ A @ R2 @ B2 @ ( bNF_We6954850376910717587_minim @ A @ R2 @ B2 ) ) ) ) ) ).
% wo_rel.minim_isMinim
thf(fact_7581_wo__rel_Ominim__least,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),B2: set @ A,B3: A] :
( ( bNF_Wellorder_wo_rel @ A @ R2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ ( field2 @ A @ R2 ) )
=> ( ( member @ A @ B3 @ B2 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( bNF_We6954850376910717587_minim @ A @ R2 @ B2 ) @ B3 ) @ R2 ) ) ) ) ).
% wo_rel.minim_least
thf(fact_7582_wo__rel_Oequals__minim,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),B2: set @ A,A3: A] :
( ( bNF_Wellorder_wo_rel @ A @ R2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ ( field2 @ A @ R2 ) )
=> ( ( member @ A @ A3 @ B2 )
=> ( ! [B5: A] :
( ( member @ A @ B5 @ B2 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B5 ) @ R2 ) )
=> ( A3
= ( bNF_We6954850376910717587_minim @ A @ R2 @ B2 ) ) ) ) ) ) ).
% wo_rel.equals_minim
thf(fact_7583_wo__rel_Ominim__in,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),B2: set @ A] :
( ( bNF_Wellorder_wo_rel @ A @ R2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ ( field2 @ A @ R2 ) )
=> ( ( B2
!= ( bot_bot @ ( set @ A ) ) )
=> ( member @ A @ ( bNF_We6954850376910717587_minim @ A @ R2 @ B2 ) @ B2 ) ) ) ) ).
% wo_rel.minim_in
thf(fact_7584_Suc__0__mod__numeral,axiom,
! [K: num] :
( ( modulo_modulo @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( numeral_numeral @ nat @ K ) )
= ( product_snd @ nat @ nat @ ( unique8689654367752047608divmod @ nat @ one2 @ K ) ) ) ).
% Suc_0_mod_numeral
thf(fact_7585_Suc__0__div__numeral,axiom,
! [K: num] :
( ( divide_divide @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( numeral_numeral @ nat @ K ) )
= ( product_fst @ nat @ nat @ ( unique8689654367752047608divmod @ nat @ one2 @ K ) ) ) ).
% Suc_0_div_numeral
thf(fact_7586_map__snd__enumerate,axiom,
! [A: $tType,N2: nat,Xs: list @ A] :
( ( map @ ( product_prod @ nat @ A ) @ A @ ( product_snd @ nat @ A ) @ ( enumerate @ A @ N2 @ Xs ) )
= Xs ) ).
% map_snd_enumerate
thf(fact_7587_map__fst__zip,axiom,
! [B: $tType,A: $tType,Xs: list @ A,Ys: list @ B] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys ) )
= Xs ) ) ).
% map_fst_zip
thf(fact_7588_map__snd__zip,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( map @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys ) )
= Ys ) ) ).
% map_snd_zip
thf(fact_7589_fst__image__times,axiom,
! [B: $tType,A: $tType,B2: set @ B,A4: set @ A] :
( ( ( B2
= ( bot_bot @ ( set @ B ) ) )
=> ( ( image2 @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B )
@ ( product_Sigma @ A @ B @ A4
@ ^ [Uu3: A] : B2 ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( B2
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( image2 @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B )
@ ( product_Sigma @ A @ B @ A4
@ ^ [Uu3: A] : B2 ) )
= A4 ) ) ) ).
% fst_image_times
thf(fact_7590_snd__image__times,axiom,
! [B: $tType,A: $tType,A4: set @ B,B2: set @ A] :
( ( ( A4
= ( bot_bot @ ( set @ B ) ) )
=> ( ( image2 @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A )
@ ( product_Sigma @ B @ A @ A4
@ ^ [Uu3: B] : B2 ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( A4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( image2 @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A )
@ ( product_Sigma @ B @ A @ A4
@ ^ [Uu3: B] : B2 ) )
= B2 ) ) ) ).
% snd_image_times
thf(fact_7591_map__fst__enumerate,axiom,
! [A: $tType,N2: nat,Xs: list @ A] :
( ( map @ ( product_prod @ nat @ A ) @ nat @ ( product_fst @ nat @ A ) @ ( enumerate @ A @ N2 @ Xs ) )
= ( upt @ N2 @ ( plus_plus @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ).
% map_fst_enumerate
thf(fact_7592_nths__shift__lemma,axiom,
! [A: $tType,A4: set @ nat,Xs: list @ A,I: nat] :
( ( map @ ( product_prod @ A @ nat ) @ A @ ( product_fst @ A @ nat )
@ ( filter2 @ ( product_prod @ A @ nat )
@ ^ [P6: product_prod @ A @ nat] : ( member @ nat @ ( product_snd @ A @ nat @ P6 ) @ A4 )
@ ( zip @ A @ nat @ Xs @ ( upt @ I @ ( plus_plus @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ) )
= ( map @ ( product_prod @ A @ nat ) @ A @ ( product_fst @ A @ nat )
@ ( filter2 @ ( product_prod @ A @ nat )
@ ^ [P6: product_prod @ A @ nat] : ( member @ nat @ ( plus_plus @ nat @ ( product_snd @ A @ nat @ P6 ) @ I ) @ A4 )
@ ( zip @ A @ nat @ Xs @ ( upt @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ) ) ).
% nths_shift_lemma
thf(fact_7593_nths__def,axiom,
! [A: $tType] :
( ( nths @ A )
= ( ^ [Xs3: list @ A,A7: set @ nat] :
( map @ ( product_prod @ A @ nat ) @ A @ ( product_fst @ A @ nat )
@ ( filter2 @ ( product_prod @ A @ nat )
@ ^ [P6: product_prod @ A @ nat] : ( member @ nat @ ( product_snd @ A @ nat @ P6 ) @ A7 )
@ ( zip @ A @ nat @ Xs3 @ ( upt @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Xs3 ) ) ) ) ) ) ) ).
% nths_def
thf(fact_7594_nths__shift__lemma__Suc,axiom,
! [A: $tType,P: nat > $o,Xs: list @ A,Is: list @ nat] :
( ( map @ ( product_prod @ A @ nat ) @ A @ ( product_fst @ A @ nat )
@ ( filter2 @ ( product_prod @ A @ nat )
@ ^ [P6: product_prod @ A @ nat] : ( P @ ( suc @ ( product_snd @ A @ nat @ P6 ) ) )
@ ( zip @ A @ nat @ Xs @ Is ) ) )
= ( map @ ( product_prod @ A @ nat ) @ A @ ( product_fst @ A @ nat )
@ ( filter2 @ ( product_prod @ A @ nat )
@ ^ [P6: product_prod @ A @ nat] : ( P @ ( product_snd @ A @ nat @ P6 ) )
@ ( zip @ A @ nat @ Xs @ ( map @ nat @ nat @ suc @ Is ) ) ) ) ) ).
% nths_shift_lemma_Suc
thf(fact_7595_zip__map__fst__snd,axiom,
! [B: $tType,A: $tType,Zs: list @ ( product_prod @ A @ B )] :
( ( zip @ A @ B @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Zs ) @ ( map @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ Zs ) )
= Zs ) ).
% zip_map_fst_snd
thf(fact_7596_pair__list__eqI,axiom,
! [B: $tType,A: $tType,Xs: list @ ( product_prod @ A @ B ),Ys: list @ ( product_prod @ A @ B )] :
( ( ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Xs )
= ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Ys ) )
=> ( ( ( map @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ Xs )
= ( map @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ Ys ) )
=> ( Xs = Ys ) ) ) ).
% pair_list_eqI
thf(fact_7597_zip__eq__conv,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B,Zs: list @ ( product_prod @ A @ B )] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( ( zip @ A @ B @ Xs @ Ys )
= Zs )
= ( ( ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Zs )
= Xs )
& ( ( map @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ Zs )
= Ys ) ) ) ) ).
% zip_eq_conv
thf(fact_7598_in__set__zip,axiom,
! [B: $tType,A: $tType,P5: product_prod @ A @ B,Xs: list @ A,Ys: list @ B] :
( ( member @ ( product_prod @ A @ B ) @ P5 @ ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys ) ) )
= ( ? [N: nat] :
( ( ( nth @ A @ Xs @ N )
= ( product_fst @ A @ B @ P5 ) )
& ( ( nth @ B @ Ys @ N )
= ( product_snd @ A @ B @ P5 ) )
& ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
& ( ord_less @ nat @ N @ ( size_size @ ( list @ B ) @ Ys ) ) ) ) ) ).
% in_set_zip
thf(fact_7599_update__zip,axiom,
! [B: $tType,A: $tType,Xs: list @ A,Ys: list @ B,I: nat,Xy: product_prod @ A @ B] :
( ( list_update @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys ) @ I @ Xy )
= ( zip @ A @ B @ ( list_update @ A @ Xs @ I @ ( product_fst @ A @ B @ Xy ) ) @ ( list_update @ B @ Ys @ I @ ( product_snd @ A @ B @ Xy ) ) ) ) ).
% update_zip
thf(fact_7600_divides__aux__def,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ( ( unique5940410009612947441es_aux @ A )
= ( ^ [Qr: product_prod @ A @ A] :
( ( product_snd @ A @ A @ Qr )
= ( zero_zero @ A ) ) ) ) ) ).
% divides_aux_def
thf(fact_7601_zip__takeWhile__fst,axiom,
! [A: $tType,B: $tType,P: A > $o,Xs: list @ A,Ys: list @ B] :
( ( zip @ A @ B @ ( takeWhile @ A @ P @ Xs ) @ Ys )
= ( takeWhile @ ( product_prod @ A @ B ) @ ( comp @ A @ $o @ ( product_prod @ A @ B ) @ P @ ( product_fst @ A @ B ) ) @ ( zip @ A @ B @ Xs @ Ys ) ) ) ).
% zip_takeWhile_fst
thf(fact_7602_zip__takeWhile__snd,axiom,
! [A: $tType,B: $tType,Xs: list @ A,P: B > $o,Ys: list @ B] :
( ( zip @ A @ B @ Xs @ ( takeWhile @ B @ P @ Ys ) )
= ( takeWhile @ ( product_prod @ A @ B ) @ ( comp @ B @ $o @ ( product_prod @ A @ B ) @ P @ ( product_snd @ A @ B ) ) @ ( zip @ A @ B @ Xs @ Ys ) ) ) ).
% zip_takeWhile_snd
thf(fact_7603_map__of__eqI,axiom,
! [B: $tType,A: $tType,Xs: list @ ( product_prod @ A @ B ),Ys: list @ ( product_prod @ A @ B )] :
( ( ( set2 @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Xs ) )
= ( set2 @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Ys ) ) )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Xs ) ) )
=> ( ( map_of @ A @ B @ Xs @ X5 )
= ( map_of @ A @ B @ Ys @ X5 ) ) )
=> ( ( map_of @ A @ B @ Xs )
= ( map_of @ A @ B @ Ys ) ) ) ) ).
% map_of_eqI
thf(fact_7604_in__set__enumerate__eq,axiom,
! [A: $tType,P5: product_prod @ nat @ A,N2: nat,Xs: list @ A] :
( ( member @ ( product_prod @ nat @ A ) @ P5 @ ( set2 @ ( product_prod @ nat @ A ) @ ( enumerate @ A @ N2 @ Xs ) ) )
= ( ( ord_less_eq @ nat @ N2 @ ( product_fst @ nat @ A @ P5 ) )
& ( ord_less @ nat @ ( product_fst @ nat @ A @ P5 ) @ ( plus_plus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ N2 ) )
& ( ( nth @ A @ Xs @ ( minus_minus @ nat @ ( product_fst @ nat @ A @ P5 ) @ N2 ) )
= ( product_snd @ nat @ A @ P5 ) ) ) ) ).
% in_set_enumerate_eq
thf(fact_7605_fst__image__Sigma,axiom,
! [B: $tType,A: $tType,A4: set @ A,B2: A > ( set @ B )] :
( ( image2 @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ ( product_Sigma @ A @ B @ A4 @ B2 ) )
= ( collect @ A
@ ^ [X4: A] :
( ( member @ A @ X4 @ A4 )
& ( ( B2 @ X4 )
!= ( bot_bot @ ( set @ B ) ) ) ) ) ) ).
% fst_image_Sigma
thf(fact_7606_sorted__enumerate,axiom,
! [A: $tType,N2: nat,Xs: list @ A] : ( sorted_wrt @ nat @ ( ord_less_eq @ nat ) @ ( map @ ( product_prod @ nat @ A ) @ nat @ ( product_fst @ nat @ A ) @ ( enumerate @ A @ N2 @ Xs ) ) ) ).
% sorted_enumerate
thf(fact_7607_insert__relcomp__fold,axiom,
! [C: $tType,B: $tType,A: $tType,S2: set @ ( product_prod @ A @ B ),X: product_prod @ C @ A,R: set @ ( product_prod @ C @ A )] :
( ( finite_finite2 @ ( product_prod @ A @ B ) @ S2 )
=> ( ( relcomp @ C @ A @ B @ ( insert2 @ ( product_prod @ C @ A ) @ X @ R ) @ S2 )
= ( finite_fold @ ( product_prod @ A @ B ) @ ( set @ ( product_prod @ C @ B ) )
@ ( product_case_prod @ A @ B @ ( ( set @ ( product_prod @ C @ B ) ) > ( set @ ( product_prod @ C @ B ) ) )
@ ^ [W3: A,Z6: B,A17: set @ ( product_prod @ C @ B )] :
( if @ ( set @ ( product_prod @ C @ B ) )
@ ( ( product_snd @ C @ A @ X )
= W3 )
@ ( insert2 @ ( product_prod @ C @ B ) @ ( product_Pair @ C @ B @ ( product_fst @ C @ A @ X ) @ Z6 ) @ A17 )
@ A17 ) )
@ ( relcomp @ C @ A @ B @ R @ S2 )
@ S2 ) ) ) ).
% insert_relcomp_fold
thf(fact_7608_set__relcomp,axiom,
! [B: $tType,C: $tType,A: $tType,Xys: list @ ( product_prod @ A @ C ),Yzs: list @ ( product_prod @ C @ B )] :
( ( relcomp @ A @ C @ B @ ( set2 @ ( product_prod @ A @ C ) @ Xys ) @ ( set2 @ ( product_prod @ C @ B ) @ Yzs ) )
= ( set2 @ ( product_prod @ A @ B )
@ ( concat @ ( product_prod @ A @ B )
@ ( map @ ( product_prod @ A @ C ) @ ( list @ ( product_prod @ A @ B ) )
@ ^ [Xy2: product_prod @ A @ C] :
( concat @ ( product_prod @ A @ B )
@ ( map @ ( product_prod @ C @ B ) @ ( list @ ( product_prod @ A @ B ) )
@ ^ [Yz: product_prod @ C @ B] :
( if @ ( list @ ( product_prod @ A @ B ) )
@ ( ( product_snd @ A @ C @ Xy2 )
= ( product_fst @ C @ B @ Yz ) )
@ ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ ( product_fst @ A @ C @ Xy2 ) @ ( product_snd @ C @ B @ Yz ) ) @ ( nil @ ( product_prod @ A @ B ) ) )
@ ( nil @ ( product_prod @ A @ B ) ) )
@ Yzs ) )
@ Xys ) ) ) ) ).
% set_relcomp
thf(fact_7609_subset__snd__imageI,axiom,
! [B: $tType,A: $tType,A4: set @ A,B2: set @ B,S2: set @ ( product_prod @ A @ B ),X: A] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) )
@ ( product_Sigma @ A @ B @ A4
@ ^ [Uu3: A] : B2 )
@ S2 )
=> ( ( member @ A @ X @ A4 )
=> ( ord_less_eq @ ( set @ B ) @ B2 @ ( image2 @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ S2 ) ) ) ) ).
% subset_snd_imageI
thf(fact_7610_subset__fst__imageI,axiom,
! [B: $tType,A: $tType,A4: set @ A,B2: set @ B,S2: set @ ( product_prod @ A @ B ),Y: B] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) )
@ ( product_Sigma @ A @ B @ A4
@ ^ [Uu3: A] : B2 )
@ S2 )
=> ( ( member @ B @ Y @ B2 )
=> ( ord_less_eq @ ( set @ A ) @ A4 @ ( image2 @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ S2 ) ) ) ) ).
% subset_fst_imageI
thf(fact_7611_insert__relcomp__union__fold,axiom,
! [C: $tType,B: $tType,A: $tType,S2: set @ ( product_prod @ A @ B ),X: product_prod @ C @ A,X8: set @ ( product_prod @ C @ B )] :
( ( finite_finite2 @ ( product_prod @ A @ B ) @ S2 )
=> ( ( sup_sup @ ( set @ ( product_prod @ C @ B ) ) @ ( relcomp @ C @ A @ B @ ( insert2 @ ( product_prod @ C @ A ) @ X @ ( bot_bot @ ( set @ ( product_prod @ C @ A ) ) ) ) @ S2 ) @ X8 )
= ( finite_fold @ ( product_prod @ A @ B ) @ ( set @ ( product_prod @ C @ B ) )
@ ( product_case_prod @ A @ B @ ( ( set @ ( product_prod @ C @ B ) ) > ( set @ ( product_prod @ C @ B ) ) )
@ ^ [W3: A,Z6: B,A17: set @ ( product_prod @ C @ B )] :
( if @ ( set @ ( product_prod @ C @ B ) )
@ ( ( product_snd @ C @ A @ X )
= W3 )
@ ( insert2 @ ( product_prod @ C @ B ) @ ( product_Pair @ C @ B @ ( product_fst @ C @ A @ X ) @ Z6 ) @ A17 )
@ A17 ) )
@ X8
@ S2 ) ) ) ).
% insert_relcomp_union_fold
thf(fact_7612_map__snd__zip__take,axiom,
! [B: $tType,A: $tType,Xs: list @ B,Ys: list @ A] :
( ( map @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A ) @ ( zip @ B @ A @ Xs @ Ys ) )
= ( take @ A @ ( ord_min @ nat @ ( size_size @ ( list @ B ) @ Xs ) @ ( size_size @ ( list @ A ) @ Ys ) ) @ Ys ) ) ).
% map_snd_zip_take
thf(fact_7613_map__fst__zip__take,axiom,
! [B: $tType,A: $tType,Xs: list @ A,Ys: list @ B] :
( ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys ) )
= ( take @ A @ ( ord_min @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ B ) @ Ys ) ) @ Xs ) ) ).
% map_fst_zip_take
thf(fact_7614_size__prod__simp,axiom,
! [B: $tType,A: $tType] :
( ( basic_BNF_size_prod @ A @ B )
= ( ^ [F3: A > nat,G2: B > nat,P6: product_prod @ A @ B] : ( plus_plus @ nat @ ( plus_plus @ nat @ ( F3 @ ( product_fst @ A @ B @ P6 ) ) @ ( G2 @ ( product_snd @ A @ B @ P6 ) ) ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ).
% size_prod_simp
thf(fact_7615_list_Oin__rel,axiom,
! [B: $tType,A: $tType] :
( ( list_all2 @ A @ B )
= ( ^ [R5: A > B > $o,A5: list @ A,B4: list @ B] :
? [Z6: list @ ( product_prod @ A @ B )] :
( ( member @ ( list @ ( product_prod @ A @ B ) ) @ Z6
@ ( collect @ ( list @ ( product_prod @ A @ B ) )
@ ^ [X4: list @ ( product_prod @ A @ B )] : ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( set2 @ ( product_prod @ A @ B ) @ X4 ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R5 ) ) ) ) )
& ( ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Z6 )
= A5 )
& ( ( map @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ Z6 )
= B4 ) ) ) ) ).
% list.in_rel
thf(fact_7616_list__all2__Nil2,axiom,
! [B: $tType,A: $tType,P: A > B > $o,Xs: list @ A] :
( ( list_all2 @ A @ B @ P @ Xs @ ( nil @ B ) )
= ( Xs
= ( nil @ A ) ) ) ).
% list_all2_Nil2
thf(fact_7617_list__all2__Nil,axiom,
! [A: $tType,B: $tType,P: A > B > $o,Ys: list @ B] :
( ( list_all2 @ A @ B @ P @ ( nil @ A ) @ Ys )
= ( Ys
= ( nil @ B ) ) ) ).
% list_all2_Nil
thf(fact_7618_list__all2__rev,axiom,
! [A: $tType,B: $tType,P: A > B > $o,Xs: list @ A,Ys: list @ B] :
( ( list_all2 @ A @ B @ P @ ( rev @ A @ Xs ) @ ( rev @ B @ Ys ) )
= ( list_all2 @ A @ B @ P @ Xs @ Ys ) ) ).
% list_all2_rev
thf(fact_7619_partition__filter1,axiom,
! [A: $tType,P: A > $o,Xs: list @ A] :
( ( product_fst @ ( list @ A ) @ ( list @ A ) @ ( partition @ A @ P @ Xs ) )
= ( filter2 @ A @ P @ Xs ) ) ).
% partition_filter1
thf(fact_7620_list_Orel__distinct_I2_J,axiom,
! [A: $tType,B: $tType,R: A > B > $o,Y21: A,Y222: list @ A] :
~ ( list_all2 @ A @ B @ R @ ( cons @ A @ Y21 @ Y222 ) @ ( nil @ B ) ) ).
% list.rel_distinct(2)
thf(fact_7621_list_Orel__distinct_I1_J,axiom,
! [A: $tType,B: $tType,R: A > B > $o,Y21: B,Y222: list @ B] :
~ ( list_all2 @ A @ B @ R @ ( nil @ A ) @ ( cons @ B @ Y21 @ Y222 ) ) ).
% list.rel_distinct(1)
thf(fact_7622_list_Orel__cases,axiom,
! [A: $tType,B: $tType,R: A > B > $o,A3: list @ A,B3: list @ B] :
( ( list_all2 @ A @ B @ R @ A3 @ B3 )
=> ( ( ( A3
= ( nil @ A ) )
=> ( B3
!= ( nil @ B ) ) )
=> ~ ! [X1: A,X23: list @ A] :
( ( A3
= ( cons @ A @ X1 @ X23 ) )
=> ! [Y1: B,Y22: list @ B] :
( ( B3
= ( cons @ B @ Y1 @ Y22 ) )
=> ( ( R @ X1 @ Y1 )
=> ~ ( list_all2 @ A @ B @ R @ X23 @ Y22 ) ) ) ) ) ) ).
% list.rel_cases
thf(fact_7623_list_Orel__induct,axiom,
! [A: $tType,B: $tType,R: A > B > $o,X: list @ A,Y: list @ B,Q: ( list @ A ) > ( list @ B ) > $o] :
( ( list_all2 @ A @ B @ R @ X @ Y )
=> ( ( Q @ ( nil @ A ) @ ( nil @ B ) )
=> ( ! [A21: A,A222: list @ A,B21: B,B222: list @ B] :
( ( R @ A21 @ B21 )
=> ( ( Q @ A222 @ B222 )
=> ( Q @ ( cons @ A @ A21 @ A222 ) @ ( cons @ B @ B21 @ B222 ) ) ) )
=> ( Q @ X @ Y ) ) ) ) ).
% list.rel_induct
thf(fact_7624_list__all2__induct,axiom,
! [A: $tType,B: $tType,P: A > B > $o,Xs: list @ A,Ys: list @ B,R: ( list @ A ) > ( list @ B ) > $o] :
( ( list_all2 @ A @ B @ P @ Xs @ Ys )
=> ( ( R @ ( nil @ A ) @ ( nil @ B ) )
=> ( ! [X5: A,Xs2: list @ A,Y3: B,Ys4: list @ B] :
( ( P @ X5 @ Y3 )
=> ( ( list_all2 @ A @ B @ P @ Xs2 @ Ys4 )
=> ( ( R @ Xs2 @ Ys4 )
=> ( R @ ( cons @ A @ X5 @ Xs2 ) @ ( cons @ B @ Y3 @ Ys4 ) ) ) ) )
=> ( R @ Xs @ Ys ) ) ) ) ).
% list_all2_induct
thf(fact_7625_list_Octr__transfer_I1_J,axiom,
! [A: $tType,B: $tType,R: A > B > $o] : ( list_all2 @ A @ B @ R @ ( nil @ A ) @ ( nil @ B ) ) ).
% list.ctr_transfer(1)
thf(fact_7626_list_Orel__sel,axiom,
! [B: $tType,A: $tType] :
( ( list_all2 @ A @ B )
= ( ^ [R5: A > B > $o,A5: list @ A,B4: list @ B] :
( ( ( A5
= ( nil @ A ) )
= ( B4
= ( nil @ B ) ) )
& ( ( A5
!= ( nil @ A ) )
=> ( ( B4
!= ( nil @ B ) )
=> ( ( R5 @ ( hd @ A @ A5 ) @ ( hd @ B @ B4 ) )
& ( list_all2 @ A @ B @ R5 @ ( tl @ A @ A5 ) @ ( tl @ B @ B4 ) ) ) ) ) ) ) ) ).
% list.rel_sel
thf(fact_7627_list__all2__appendI,axiom,
! [A: $tType,B: $tType,P: A > B > $o,A3: list @ A,B3: list @ B,C2: list @ A,D2: list @ B] :
( ( list_all2 @ A @ B @ P @ A3 @ B3 )
=> ( ( list_all2 @ A @ B @ P @ C2 @ D2 )
=> ( list_all2 @ A @ B @ P @ ( append @ A @ A3 @ C2 ) @ ( append @ B @ B3 @ D2 ) ) ) ) ).
% list_all2_appendI
thf(fact_7628_list__all2__antisym,axiom,
! [A: $tType,P: A > A > $o,Q: A > A > $o,Xs: list @ A,Ys: list @ A] :
( ! [X5: A,Y3: A] :
( ( P @ X5 @ Y3 )
=> ( ( Q @ Y3 @ X5 )
=> ( X5 = Y3 ) ) )
=> ( ( list_all2 @ A @ A @ P @ Xs @ Ys )
=> ( ( list_all2 @ A @ A @ Q @ Ys @ Xs )
=> ( Xs = Ys ) ) ) ) ).
% list_all2_antisym
thf(fact_7629_list__all2__trans,axiom,
! [B: $tType,A: $tType,C: $tType,P1: A > B > $o,P22: B > C > $o,P32: A > C > $o,As: list @ A,Bs: list @ B,Cs: list @ C] :
( ! [A6: A,B5: B,C3: C] :
( ( P1 @ A6 @ B5 )
=> ( ( P22 @ B5 @ C3 )
=> ( P32 @ A6 @ C3 ) ) )
=> ( ( list_all2 @ A @ B @ P1 @ As @ Bs )
=> ( ( list_all2 @ B @ C @ P22 @ Bs @ Cs )
=> ( list_all2 @ A @ C @ P32 @ As @ Cs ) ) ) ) ).
% list_all2_trans
thf(fact_7630_list__all2__refl,axiom,
! [A: $tType,P: A > A > $o,Xs: list @ A] :
( ! [X5: A] : ( P @ X5 @ X5 )
=> ( list_all2 @ A @ A @ P @ Xs @ Xs ) ) ).
% list_all2_refl
thf(fact_7631_list__all2__mono,axiom,
! [A: $tType,B: $tType,P: A > B > $o,Xs: list @ A,Ys: list @ B,Q: A > B > $o] :
( ( list_all2 @ A @ B @ P @ Xs @ Ys )
=> ( ! [Xs2: A,Ys4: B] :
( ( P @ Xs2 @ Ys4 )
=> ( Q @ Xs2 @ Ys4 ) )
=> ( list_all2 @ A @ B @ Q @ Xs @ Ys ) ) ) ).
% list_all2_mono
thf(fact_7632_list__all2__eq,axiom,
! [A: $tType] :
( ( ^ [Y5: list @ A,Z2: list @ A] : Y5 = Z2 )
= ( list_all2 @ A @ A
@ ^ [Y5: A,Z2: A] : Y5 = Z2 ) ) ).
% list_all2_eq
thf(fact_7633_list_Orel__refl,axiom,
! [B: $tType,Ra: B > B > $o,X: list @ B] :
( ! [X5: B] : ( Ra @ X5 @ X5 )
=> ( list_all2 @ B @ B @ Ra @ X @ X ) ) ).
% list.rel_refl
thf(fact_7634_list_Orel__eq,axiom,
! [A: $tType] :
( ( list_all2 @ A @ A
@ ^ [Y5: A,Z2: A] : Y5 = Z2 )
= ( ^ [Y5: list @ A,Z2: list @ A] : Y5 = Z2 ) ) ).
% list.rel_eq
thf(fact_7635_list__all2__update__cong,axiom,
! [A: $tType,B: $tType,P: A > B > $o,Xs: list @ A,Ys: list @ B,X: A,Y: B,I: nat] :
( ( list_all2 @ A @ B @ P @ Xs @ Ys )
=> ( ( P @ X @ Y )
=> ( list_all2 @ A @ B @ P @ ( list_update @ A @ Xs @ I @ X ) @ ( list_update @ B @ Ys @ I @ Y ) ) ) ) ).
% list_all2_update_cong
thf(fact_7636_list__all2__dropI,axiom,
! [A: $tType,B: $tType,P: A > B > $o,Xs: list @ A,Ys: list @ B,N2: nat] :
( ( list_all2 @ A @ B @ P @ Xs @ Ys )
=> ( list_all2 @ A @ B @ P @ ( drop @ A @ N2 @ Xs ) @ ( drop @ B @ N2 @ Ys ) ) ) ).
% list_all2_dropI
thf(fact_7637_list__all2__takeI,axiom,
! [A: $tType,B: $tType,P: A > B > $o,Xs: list @ A,Ys: list @ B,N2: nat] :
( ( list_all2 @ A @ B @ P @ Xs @ Ys )
=> ( list_all2 @ A @ B @ P @ ( take @ A @ N2 @ Xs ) @ ( take @ B @ N2 @ Ys ) ) ) ).
% list_all2_takeI
thf(fact_7638_list__all2__append,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B,P: A > B > $o,Us: list @ A,Vs: list @ B] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( list_all2 @ A @ B @ P @ ( append @ A @ Xs @ Us ) @ ( append @ B @ Ys @ Vs ) )
= ( ( list_all2 @ A @ B @ P @ Xs @ Ys )
& ( list_all2 @ A @ B @ P @ Us @ Vs ) ) ) ) ).
% list_all2_append
thf(fact_7639_list__all2__append1,axiom,
! [A: $tType,B: $tType,P: A > B > $o,Xs: list @ A,Ys: list @ A,Zs: list @ B] :
( ( list_all2 @ A @ B @ P @ ( append @ A @ Xs @ Ys ) @ Zs )
= ( ? [Us2: list @ B,Vs3: list @ B] :
( ( Zs
= ( append @ B @ Us2 @ Vs3 ) )
& ( ( size_size @ ( list @ B ) @ Us2 )
= ( size_size @ ( list @ A ) @ Xs ) )
& ( ( size_size @ ( list @ B ) @ Vs3 )
= ( size_size @ ( list @ A ) @ Ys ) )
& ( list_all2 @ A @ B @ P @ Xs @ Us2 )
& ( list_all2 @ A @ B @ P @ Ys @ Vs3 ) ) ) ) ).
% list_all2_append1
thf(fact_7640_list__all2__append2,axiom,
! [B: $tType,A: $tType,P: A > B > $o,Xs: list @ A,Ys: list @ B,Zs: list @ B] :
( ( list_all2 @ A @ B @ P @ Xs @ ( append @ B @ Ys @ Zs ) )
= ( ? [Us2: list @ A,Vs3: list @ A] :
( ( Xs
= ( append @ A @ Us2 @ Vs3 ) )
& ( ( size_size @ ( list @ A ) @ Us2 )
= ( size_size @ ( list @ B ) @ Ys ) )
& ( ( size_size @ ( list @ A ) @ Vs3 )
= ( size_size @ ( list @ B ) @ Zs ) )
& ( list_all2 @ A @ B @ P @ Us2 @ Ys )
& ( list_all2 @ A @ B @ P @ Vs3 @ Zs ) ) ) ) ).
% list_all2_append2
thf(fact_7641_list__all2__lengthD,axiom,
! [A: $tType,B: $tType,P: A > B > $o,Xs: list @ A,Ys: list @ B] :
( ( list_all2 @ A @ B @ P @ Xs @ Ys )
=> ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys ) ) ) ).
% list_all2_lengthD
thf(fact_7642_list_Orel__cong,axiom,
! [A: $tType,B: $tType,X: list @ A,Ya: list @ A,Y: list @ B,Xa2: list @ B,R: A > B > $o,Ra: A > B > $o] :
( ( X = Ya )
=> ( ( Y = Xa2 )
=> ( ! [Z: A,Yb: B] :
( ( member @ A @ Z @ ( set2 @ A @ Ya ) )
=> ( ( member @ B @ Yb @ ( set2 @ B @ Xa2 ) )
=> ( ( R @ Z @ Yb )
= ( Ra @ Z @ Yb ) ) ) )
=> ( ( list_all2 @ A @ B @ R @ X @ Y )
= ( list_all2 @ A @ B @ Ra @ Ya @ Xa2 ) ) ) ) ) ).
% list.rel_cong
thf(fact_7643_list_Orel__mono__strong,axiom,
! [A: $tType,B: $tType,R: A > B > $o,X: list @ A,Y: list @ B,Ra: A > B > $o] :
( ( list_all2 @ A @ B @ R @ X @ Y )
=> ( ! [Z: A,Yb: B] :
( ( member @ A @ Z @ ( set2 @ A @ X ) )
=> ( ( member @ B @ Yb @ ( set2 @ B @ Y ) )
=> ( ( R @ Z @ Yb )
=> ( Ra @ Z @ Yb ) ) ) )
=> ( list_all2 @ A @ B @ Ra @ X @ Y ) ) ) ).
% list.rel_mono_strong
thf(fact_7644_list_Orel__refl__strong,axiom,
! [A: $tType,X: list @ A,Ra: A > A > $o] :
( ! [Z: A] :
( ( member @ A @ Z @ ( set2 @ A @ X ) )
=> ( Ra @ Z @ Z ) )
=> ( list_all2 @ A @ A @ Ra @ X @ X ) ) ).
% list.rel_refl_strong
thf(fact_7645_list__all2__same,axiom,
! [A: $tType,P: A > A > $o,Xs: list @ A] :
( ( list_all2 @ A @ A @ P @ Xs @ Xs )
= ( ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
=> ( P @ X4 @ X4 ) ) ) ) ).
% list_all2_same
thf(fact_7646_list_Orel__mono,axiom,
! [B: $tType,A: $tType,R: A > B > $o,Ra: A > B > $o] :
( ( ord_less_eq @ ( A > B > $o ) @ R @ Ra )
=> ( ord_less_eq @ ( ( list @ A ) > ( list @ B ) > $o ) @ ( list_all2 @ A @ B @ R ) @ ( list_all2 @ A @ B @ Ra ) ) ) ).
% list.rel_mono
thf(fact_7647_list__all2__map2,axiom,
! [A: $tType,B: $tType,C: $tType,P: A > B > $o,As: list @ A,F2: C > B,Bs: list @ C] :
( ( list_all2 @ A @ B @ P @ As @ ( map @ C @ B @ F2 @ Bs ) )
= ( list_all2 @ A @ C
@ ^ [X4: A,Y6: C] : ( P @ X4 @ ( F2 @ Y6 ) )
@ As
@ Bs ) ) ).
% list_all2_map2
thf(fact_7648_list__all2__map1,axiom,
! [C: $tType,A: $tType,B: $tType,P: A > B > $o,F2: C > A,As: list @ C,Bs: list @ B] :
( ( list_all2 @ A @ B @ P @ ( map @ C @ A @ F2 @ As ) @ Bs )
= ( list_all2 @ C @ B
@ ^ [X4: C] : ( P @ ( F2 @ X4 ) )
@ As
@ Bs ) ) ).
% list_all2_map1
thf(fact_7649_list_Orel__map_I1_J,axiom,
! [A: $tType,C: $tType,B: $tType,Sb: C > B > $o,I: A > C,X: list @ A,Y: list @ B] :
( ( list_all2 @ C @ B @ Sb @ ( map @ A @ C @ I @ X ) @ Y )
= ( list_all2 @ A @ B
@ ^ [X4: A] : ( Sb @ ( I @ X4 ) )
@ X
@ Y ) ) ).
% list.rel_map(1)
thf(fact_7650_list_Orel__map_I2_J,axiom,
! [A: $tType,C: $tType,B: $tType,Sa: A > C > $o,X: list @ A,G: B > C,Y: list @ B] :
( ( list_all2 @ A @ C @ Sa @ X @ ( map @ B @ C @ G @ Y ) )
= ( list_all2 @ A @ B
@ ^ [X4: A,Y6: B] : ( Sa @ X4 @ ( G @ Y6 ) )
@ X
@ Y ) ) ).
% list.rel_map(2)
thf(fact_7651_list__all2__rev1,axiom,
! [A: $tType,B: $tType,P: A > B > $o,Xs: list @ A,Ys: list @ B] :
( ( list_all2 @ A @ B @ P @ ( rev @ A @ Xs ) @ Ys )
= ( list_all2 @ A @ B @ P @ Xs @ ( rev @ B @ Ys ) ) ) ).
% list_all2_rev1
thf(fact_7652_list__all2__conv__all__nth,axiom,
! [B: $tType,A: $tType] :
( ( list_all2 @ A @ B )
= ( ^ [P3: A > B > $o,Xs3: list @ A,Ys3: list @ B] :
( ( ( size_size @ ( list @ A ) @ Xs3 )
= ( size_size @ ( list @ B ) @ Ys3 ) )
& ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs3 ) )
=> ( P3 @ ( nth @ A @ Xs3 @ I3 ) @ ( nth @ B @ Ys3 @ I3 ) ) ) ) ) ) ).
% list_all2_conv_all_nth
thf(fact_7653_list__all2__all__nthI,axiom,
! [A: $tType,B: $tType,A3: list @ A,B3: list @ B,P: A > B > $o] :
( ( ( size_size @ ( list @ A ) @ A3 )
= ( size_size @ ( list @ B ) @ B3 ) )
=> ( ! [N3: nat] :
( ( ord_less @ nat @ N3 @ ( size_size @ ( list @ A ) @ A3 ) )
=> ( P @ ( nth @ A @ A3 @ N3 ) @ ( nth @ B @ B3 @ N3 ) ) )
=> ( list_all2 @ A @ B @ P @ A3 @ B3 ) ) ) ).
% list_all2_all_nthI
thf(fact_7654_list__all2__nthD2,axiom,
! [A: $tType,B: $tType,P: A > B > $o,Xs: list @ A,Ys: list @ B,P5: nat] :
( ( list_all2 @ A @ B @ P @ Xs @ Ys )
=> ( ( ord_less @ nat @ P5 @ ( size_size @ ( list @ B ) @ Ys ) )
=> ( P @ ( nth @ A @ Xs @ P5 ) @ ( nth @ B @ Ys @ P5 ) ) ) ) ).
% list_all2_nthD2
thf(fact_7655_list__all2__nthD,axiom,
! [A: $tType,B: $tType,P: A > B > $o,Xs: list @ A,Ys: list @ B,P5: nat] :
( ( list_all2 @ A @ B @ P @ Xs @ Ys )
=> ( ( ord_less @ nat @ P5 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( P @ ( nth @ A @ Xs @ P5 ) @ ( nth @ B @ Ys @ P5 ) ) ) ) ).
% list_all2_nthD
thf(fact_7656_list_Orel__inject_I2_J,axiom,
! [A: $tType,B: $tType,R: A > B > $o,X21: A,X222: list @ A,Y21: B,Y222: list @ B] :
( ( list_all2 @ A @ B @ R @ ( cons @ A @ X21 @ X222 ) @ ( cons @ B @ Y21 @ Y222 ) )
= ( ( R @ X21 @ Y21 )
& ( list_all2 @ A @ B @ R @ X222 @ Y222 ) ) ) ).
% list.rel_inject(2)
thf(fact_7657_list_Orel__intros_I2_J,axiom,
! [A: $tType,B: $tType,R: A > B > $o,X21: A,Y21: B,X222: list @ A,Y222: list @ B] :
( ( R @ X21 @ Y21 )
=> ( ( list_all2 @ A @ B @ R @ X222 @ Y222 )
=> ( list_all2 @ A @ B @ R @ ( cons @ A @ X21 @ X222 ) @ ( cons @ B @ Y21 @ Y222 ) ) ) ) ).
% list.rel_intros(2)
thf(fact_7658_list__all2__Cons,axiom,
! [A: $tType,B: $tType,P: A > B > $o,X: A,Xs: list @ A,Y: B,Ys: list @ B] :
( ( list_all2 @ A @ B @ P @ ( cons @ A @ X @ Xs ) @ ( cons @ B @ Y @ Ys ) )
= ( ( P @ X @ Y )
& ( list_all2 @ A @ B @ P @ Xs @ Ys ) ) ) ).
% list_all2_Cons
thf(fact_7659_list__all2__Cons1,axiom,
! [A: $tType,B: $tType,P: A > B > $o,X: A,Xs: list @ A,Ys: list @ B] :
( ( list_all2 @ A @ B @ P @ ( cons @ A @ X @ Xs ) @ Ys )
= ( ? [Z6: B,Zs3: list @ B] :
( ( Ys
= ( cons @ B @ Z6 @ Zs3 ) )
& ( P @ X @ Z6 )
& ( list_all2 @ A @ B @ P @ Xs @ Zs3 ) ) ) ) ).
% list_all2_Cons1
thf(fact_7660_list__all2__Cons2,axiom,
! [B: $tType,A: $tType,P: A > B > $o,Xs: list @ A,Y: B,Ys: list @ B] :
( ( list_all2 @ A @ B @ P @ Xs @ ( cons @ B @ Y @ Ys ) )
= ( ? [Z6: A,Zs3: list @ A] :
( ( Xs
= ( cons @ A @ Z6 @ Zs3 ) )
& ( P @ Z6 @ Y )
& ( list_all2 @ A @ B @ P @ Zs3 @ Ys ) ) ) ) ).
% list_all2_Cons2
thf(fact_7661_product__lists__set,axiom,
! [A: $tType,Xss: list @ ( list @ A )] :
( ( set2 @ ( list @ A ) @ ( product_lists @ A @ Xss ) )
= ( collect @ ( list @ A )
@ ^ [Xs3: list @ A] :
( list_all2 @ A @ ( list @ A )
@ ^ [X4: A,Ys3: list @ A] : ( member @ A @ X4 @ ( set2 @ A @ Ys3 ) )
@ Xs3
@ Xss ) ) ) ).
% product_lists_set
thf(fact_7662_partition__filter2,axiom,
! [A: $tType,P: A > $o,Xs: list @ A] :
( ( product_snd @ ( list @ A ) @ ( list @ A ) @ ( partition @ A @ P @ Xs ) )
= ( filter2 @ A @ ( comp @ $o @ $o @ A @ (~) @ P ) @ Xs ) ) ).
% partition_filter2
thf(fact_7663_list__all2I,axiom,
! [A: $tType,B: $tType,A3: list @ A,B3: list @ B,P: A > B > $o] :
( ! [X5: product_prod @ A @ B] :
( ( member @ ( product_prod @ A @ B ) @ X5 @ ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ A3 @ B3 ) ) )
=> ( product_case_prod @ A @ B @ $o @ P @ X5 ) )
=> ( ( ( size_size @ ( list @ A ) @ A3 )
= ( size_size @ ( list @ B ) @ B3 ) )
=> ( list_all2 @ A @ B @ P @ A3 @ B3 ) ) ) ).
% list_all2I
thf(fact_7664_bezw__non__0,axiom,
! [Y: nat,X: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ Y )
=> ( ( bezw @ X @ Y )
= ( product_Pair @ int @ int @ ( product_snd @ int @ int @ ( bezw @ Y @ ( modulo_modulo @ nat @ X @ Y ) ) ) @ ( minus_minus @ int @ ( product_fst @ int @ int @ ( bezw @ Y @ ( modulo_modulo @ nat @ X @ Y ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ ( bezw @ Y @ ( modulo_modulo @ nat @ X @ Y ) ) ) @ ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ X @ Y ) ) ) ) ) ) ) ).
% bezw_non_0
thf(fact_7665_bezw_Oelims,axiom,
! [X: nat,Xa2: nat,Y: product_prod @ int @ int] :
( ( ( bezw @ X @ Xa2 )
= Y )
=> ( ( ( Xa2
= ( zero_zero @ nat ) )
=> ( Y
= ( product_Pair @ int @ int @ ( one_one @ int ) @ ( zero_zero @ int ) ) ) )
& ( ( Xa2
!= ( zero_zero @ nat ) )
=> ( Y
= ( product_Pair @ int @ int @ ( product_snd @ int @ int @ ( bezw @ Xa2 @ ( modulo_modulo @ nat @ X @ Xa2 ) ) ) @ ( minus_minus @ int @ ( product_fst @ int @ int @ ( bezw @ Xa2 @ ( modulo_modulo @ nat @ X @ Xa2 ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ ( bezw @ Xa2 @ ( modulo_modulo @ nat @ X @ Xa2 ) ) ) @ ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ X @ Xa2 ) ) ) ) ) ) ) ) ) ).
% bezw.elims
thf(fact_7666_bezw_Osimps,axiom,
( bezw
= ( ^ [X4: nat,Y6: nat] :
( if @ ( product_prod @ int @ int )
@ ( Y6
= ( zero_zero @ nat ) )
@ ( product_Pair @ int @ int @ ( one_one @ int ) @ ( zero_zero @ int ) )
@ ( product_Pair @ int @ int @ ( product_snd @ int @ int @ ( bezw @ Y6 @ ( modulo_modulo @ nat @ X4 @ Y6 ) ) ) @ ( minus_minus @ int @ ( product_fst @ int @ int @ ( bezw @ Y6 @ ( modulo_modulo @ nat @ X4 @ Y6 ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ ( bezw @ Y6 @ ( modulo_modulo @ nat @ X4 @ Y6 ) ) ) @ ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ X4 @ Y6 ) ) ) ) ) ) ) ) ).
% bezw.simps
thf(fact_7667_list__all2__iff,axiom,
! [B: $tType,A: $tType] :
( ( list_all2 @ A @ B )
= ( ^ [P3: A > B > $o,Xs3: list @ A,Ys3: list @ B] :
( ( ( size_size @ ( list @ A ) @ Xs3 )
= ( size_size @ ( list @ B ) @ Ys3 ) )
& ! [X4: product_prod @ A @ B] :
( ( member @ ( product_prod @ A @ B ) @ X4 @ ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs3 @ Ys3 ) ) )
=> ( product_case_prod @ A @ B @ $o @ P3 @ X4 ) ) ) ) ) ).
% list_all2_iff
thf(fact_7668_bezw_Opelims,axiom,
! [X: nat,Xa2: nat,Y: product_prod @ int @ int] :
( ( ( bezw @ X @ Xa2 )
= Y )
=> ( ( accp @ ( product_prod @ nat @ nat ) @ bezw_rel @ ( product_Pair @ nat @ nat @ X @ Xa2 ) )
=> ~ ( ( ( ( Xa2
= ( zero_zero @ nat ) )
=> ( Y
= ( product_Pair @ int @ int @ ( one_one @ int ) @ ( zero_zero @ int ) ) ) )
& ( ( Xa2
!= ( zero_zero @ nat ) )
=> ( Y
= ( product_Pair @ int @ int @ ( product_snd @ int @ int @ ( bezw @ Xa2 @ ( modulo_modulo @ nat @ X @ Xa2 ) ) ) @ ( minus_minus @ int @ ( product_fst @ int @ int @ ( bezw @ Xa2 @ ( modulo_modulo @ nat @ X @ Xa2 ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ ( bezw @ Xa2 @ ( modulo_modulo @ nat @ X @ Xa2 ) ) ) @ ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ X @ Xa2 ) ) ) ) ) ) ) )
=> ~ ( accp @ ( product_prod @ nat @ nat ) @ bezw_rel @ ( product_Pair @ nat @ nat @ X @ Xa2 ) ) ) ) ) ).
% bezw.pelims
thf(fact_7669_pair__lessI2,axiom,
! [A3: nat,B3: nat,S: nat,T2: nat] :
( ( ord_less_eq @ nat @ A3 @ B3 )
=> ( ( ord_less @ nat @ S @ T2 )
=> ( member @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ A3 @ S ) @ ( product_Pair @ nat @ nat @ B3 @ T2 ) ) @ fun_pair_less ) ) ) ).
% pair_lessI2
thf(fact_7670_pair__less__iff1,axiom,
! [X: nat,Y: nat,Z3: nat] :
( ( member @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ X @ Y ) @ ( product_Pair @ nat @ nat @ X @ Z3 ) ) @ fun_pair_less )
= ( ord_less @ nat @ Y @ Z3 ) ) ).
% pair_less_iff1
thf(fact_7671_pair__lessI1,axiom,
! [A3: nat,B3: nat,S: nat,T2: nat] :
( ( ord_less @ nat @ A3 @ B3 )
=> ( member @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ A3 @ S ) @ ( product_Pair @ nat @ nat @ B3 @ T2 ) ) @ fun_pair_less ) ) ).
% pair_lessI1
thf(fact_7672_prod__decode__aux_Opelims,axiom,
! [X: nat,Xa2: nat,Y: product_prod @ nat @ nat] :
( ( ( nat_prod_decode_aux @ X @ Xa2 )
= Y )
=> ( ( accp @ ( product_prod @ nat @ nat ) @ nat_pr5047031295181774490ux_rel @ ( product_Pair @ nat @ nat @ X @ Xa2 ) )
=> ~ ( ( ( ( ord_less_eq @ nat @ Xa2 @ X )
=> ( Y
= ( product_Pair @ nat @ nat @ Xa2 @ ( minus_minus @ nat @ X @ Xa2 ) ) ) )
& ( ~ ( ord_less_eq @ nat @ Xa2 @ X )
=> ( Y
= ( nat_prod_decode_aux @ ( suc @ X ) @ ( minus_minus @ nat @ Xa2 @ ( suc @ X ) ) ) ) ) )
=> ~ ( accp @ ( product_prod @ nat @ nat ) @ nat_pr5047031295181774490ux_rel @ ( product_Pair @ nat @ nat @ X @ Xa2 ) ) ) ) ) ).
% prod_decode_aux.pelims
thf(fact_7673_pair__leqI2,axiom,
! [A3: nat,B3: nat,S: nat,T2: nat] :
( ( ord_less_eq @ nat @ A3 @ B3 )
=> ( ( ord_less_eq @ nat @ S @ T2 )
=> ( member @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ A3 @ S ) @ ( product_Pair @ nat @ nat @ B3 @ T2 ) ) @ fun_pair_leq ) ) ) ).
% pair_leqI2
thf(fact_7674_pair__leqI1,axiom,
! [A3: nat,B3: nat,S: nat,T2: nat] :
( ( ord_less @ nat @ A3 @ B3 )
=> ( member @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ A3 @ S ) @ ( product_Pair @ nat @ nat @ B3 @ T2 ) ) @ fun_pair_leq ) ) ).
% pair_leqI1
thf(fact_7675_folding__idem__def_H,axiom,
! [B: $tType,A: $tType] :
( ( finite_folding_idem @ A @ B )
= ( finite1890593828518410140dem_on @ A @ B @ ( top_top @ ( set @ A ) ) ) ) ).
% folding_idem_def'
thf(fact_7676_admissible__chfin,axiom,
! [A: $tType] :
( ( comple9053668089753744459l_ccpo @ A )
=> ! [P: A > $o] :
( ! [S4: set @ A] :
( ( comple1602240252501008431_chain @ A @ ( ord_less_eq @ A ) @ S4 )
=> ( finite_finite2 @ A @ S4 ) )
=> ( comple1908693960933563346ssible @ A @ ( complete_Sup_Sup @ A ) @ ( ord_less_eq @ A ) @ P ) ) ) ).
% admissible_chfin
thf(fact_7677_folding__idem_Ocomp__fun__idem,axiom,
! [B: $tType,A: $tType,F2: A > B > B,X: A] :
( ( finite_folding_idem @ A @ B @ F2 )
=> ( ( comp @ B @ B @ B @ ( F2 @ X ) @ ( F2 @ X ) )
= ( F2 @ X ) ) ) ).
% folding_idem.comp_fun_idem
thf(fact_7678_ccpo_OadmissibleD,axiom,
! [A: $tType,Lub: ( set @ A ) > A,Ord: A > A > $o,P: A > $o,A4: set @ A] :
( ( comple1908693960933563346ssible @ A @ Lub @ Ord @ P )
=> ( ( comple1602240252501008431_chain @ A @ Ord @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ A4 )
=> ( P @ X5 ) )
=> ( P @ ( Lub @ A4 ) ) ) ) ) ) ).
% ccpo.admissibleD
thf(fact_7679_ccpo_OadmissibleI,axiom,
! [A: $tType,Ord: A > A > $o,P: A > $o,Lub: ( set @ A ) > A] :
( ! [A8: set @ A] :
( ( comple1602240252501008431_chain @ A @ Ord @ A8 )
=> ( ( A8
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X2: A] :
( ( member @ A @ X2 @ A8 )
=> ( P @ X2 ) )
=> ( P @ ( Lub @ A8 ) ) ) ) )
=> ( comple1908693960933563346ssible @ A @ Lub @ Ord @ P ) ) ).
% ccpo.admissibleI
thf(fact_7680_ccpo_Oadmissible__def,axiom,
! [A: $tType] :
( ( comple1908693960933563346ssible @ A )
= ( ^ [Lub2: ( set @ A ) > A,Ord2: A > A > $o,P3: A > $o] :
! [A7: set @ A] :
( ( comple1602240252501008431_chain @ A @ Ord2 @ A7 )
=> ( ( A7
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ A7 )
=> ( P3 @ X4 ) )
=> ( P3 @ ( Lub2 @ A7 ) ) ) ) ) ) ) ).
% ccpo.admissible_def
thf(fact_7681_admissible__disj,axiom,
! [A: $tType] :
( ( comple9053668089753744459l_ccpo @ A )
=> ! [P: A > $o,Q: A > $o] :
( ( comple1908693960933563346ssible @ A @ ( complete_Sup_Sup @ A ) @ ( ord_less_eq @ A ) @ P )
=> ( ( comple1908693960933563346ssible @ A @ ( complete_Sup_Sup @ A ) @ ( ord_less_eq @ A ) @ Q )
=> ( comple1908693960933563346ssible @ A @ ( complete_Sup_Sup @ A ) @ ( ord_less_eq @ A )
@ ^ [X4: A] :
( ( P @ X4 )
| ( Q @ X4 ) ) ) ) ) ) ).
% admissible_disj
thf(fact_7682_distinct__adj__append__iff,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( distinct_adj @ A @ ( append @ A @ Xs @ Ys ) )
= ( ( distinct_adj @ A @ Xs )
& ( distinct_adj @ A @ Ys )
& ( ( Xs
= ( nil @ A ) )
| ( Ys
= ( nil @ A ) )
| ( ( last @ A @ Xs )
!= ( hd @ A @ Ys ) ) ) ) ) ).
% distinct_adj_append_iff
thf(fact_7683_bot_Oordering__top__axioms,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ( ordering_top @ A
@ ^ [X4: A,Y6: A] : ( ord_less_eq @ A @ Y6 @ X4 )
@ ^ [X4: A,Y6: A] : ( ord_less @ A @ Y6 @ X4 )
@ ( bot_bot @ A ) ) ) ).
% bot.ordering_top_axioms
thf(fact_7684_distinct__adj__Cons__Cons,axiom,
! [B: $tType,X: B,Y: B,Xs: list @ B] :
( ( distinct_adj @ B @ ( cons @ B @ X @ ( cons @ B @ Y @ Xs ) ) )
= ( ( X != Y )
& ( distinct_adj @ B @ ( cons @ B @ Y @ Xs ) ) ) ) ).
% distinct_adj_Cons_Cons
thf(fact_7685_distinct__adj__rev,axiom,
! [A: $tType,Xs: list @ A] :
( ( distinct_adj @ A @ ( rev @ A @ Xs ) )
= ( distinct_adj @ A @ Xs ) ) ).
% distinct_adj_rev
thf(fact_7686_distinct__adj__singleton,axiom,
! [B: $tType,X: B] : ( distinct_adj @ B @ ( cons @ B @ X @ ( nil @ B ) ) ) ).
% distinct_adj_singleton
thf(fact_7687_distinct__adj__Nil,axiom,
! [A: $tType] : ( distinct_adj @ A @ ( nil @ A ) ) ).
% distinct_adj_Nil
thf(fact_7688_distinct__adj__appendD1,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( distinct_adj @ A @ ( append @ A @ Xs @ Ys ) )
=> ( distinct_adj @ A @ Xs ) ) ).
% distinct_adj_appendD1
thf(fact_7689_distinct__adj__appendD2,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( distinct_adj @ A @ ( append @ A @ Xs @ Ys ) )
=> ( distinct_adj @ A @ Ys ) ) ).
% distinct_adj_appendD2
thf(fact_7690_distinct__adj__altdef,axiom,
! [A: $tType] :
( ( distinct_adj @ A )
= ( ^ [Xs3: list @ A] :
( ( remdups_adj @ A @ Xs3 )
= Xs3 ) ) ) ).
% distinct_adj_altdef
thf(fact_7691_distinct__adj__remdups__adj,axiom,
! [A: $tType,Xs: list @ A] : ( distinct_adj @ A @ ( remdups_adj @ A @ Xs ) ) ).
% distinct_adj_remdups_adj
thf(fact_7692_ordering__top_Oextremum__uniqueI,axiom,
! [A: $tType,Less_eq2: A > A > $o,Less: A > A > $o,Top: A,A3: A] :
( ( ordering_top @ A @ Less_eq2 @ Less @ Top )
=> ( ( Less_eq2 @ Top @ A3 )
=> ( A3 = Top ) ) ) ).
% ordering_top.extremum_uniqueI
thf(fact_7693_ordering__top_Onot__eq__extremum,axiom,
! [A: $tType,Less_eq2: A > A > $o,Less: A > A > $o,Top: A,A3: A] :
( ( ordering_top @ A @ Less_eq2 @ Less @ Top )
=> ( ( A3 != Top )
= ( Less @ A3 @ Top ) ) ) ).
% ordering_top.not_eq_extremum
thf(fact_7694_ordering__top_Oextremum__unique,axiom,
! [A: $tType,Less_eq2: A > A > $o,Less: A > A > $o,Top: A,A3: A] :
( ( ordering_top @ A @ Less_eq2 @ Less @ Top )
=> ( ( Less_eq2 @ Top @ A3 )
= ( A3 = Top ) ) ) ).
% ordering_top.extremum_unique
thf(fact_7695_ordering__top_Oextremum__strict,axiom,
! [A: $tType,Less_eq2: A > A > $o,Less: A > A > $o,Top: A,A3: A] :
( ( ordering_top @ A @ Less_eq2 @ Less @ Top )
=> ~ ( Less @ Top @ A3 ) ) ).
% ordering_top.extremum_strict
thf(fact_7696_ordering__top_Oextremum,axiom,
! [A: $tType,Less_eq2: A > A > $o,Less: A > A > $o,Top: A,A3: A] :
( ( ordering_top @ A @ Less_eq2 @ Less @ Top )
=> ( Less_eq2 @ A3 @ Top ) ) ).
% ordering_top.extremum
thf(fact_7697_distinct__adj__ConsD,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( distinct_adj @ A @ ( cons @ A @ X @ Xs ) )
=> ( distinct_adj @ A @ Xs ) ) ).
% distinct_adj_ConsD
thf(fact_7698_distinct__adj__mapD,axiom,
! [A: $tType,B: $tType,F2: B > A,Xs: list @ B] :
( ( distinct_adj @ A @ ( map @ B @ A @ F2 @ Xs ) )
=> ( distinct_adj @ B @ Xs ) ) ).
% distinct_adj_mapD
thf(fact_7699_gcd__nat_Oordering__top__axioms,axiom,
( ordering_top @ nat @ ( dvd_dvd @ nat )
@ ^ [M: nat,N: nat] :
( ( dvd_dvd @ nat @ M @ N )
& ( M != N ) )
@ ( zero_zero @ nat ) ) ).
% gcd_nat.ordering_top_axioms
thf(fact_7700_distinct__adj__map__iff,axiom,
! [B: $tType,A: $tType,F2: A > B,Xs: list @ A] :
( ( inj_on @ A @ B @ F2 @ ( set2 @ A @ Xs ) )
=> ( ( distinct_adj @ B @ ( map @ A @ B @ F2 @ Xs ) )
= ( distinct_adj @ A @ Xs ) ) ) ).
% distinct_adj_map_iff
thf(fact_7701_distinct__adj__mapI,axiom,
! [B: $tType,A: $tType,Xs: list @ A,F2: A > B] :
( ( distinct_adj @ A @ Xs )
=> ( ( inj_on @ A @ B @ F2 @ ( set2 @ A @ Xs ) )
=> ( distinct_adj @ B @ ( map @ A @ B @ F2 @ Xs ) ) ) ) ).
% distinct_adj_mapI
thf(fact_7702_distinct__adj__Cons,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( distinct_adj @ A @ ( cons @ A @ X @ Xs ) )
= ( ( Xs
= ( nil @ A ) )
| ( ( X
!= ( hd @ A @ Xs ) )
& ( distinct_adj @ A @ Xs ) ) ) ) ).
% distinct_adj_Cons
thf(fact_7703_top_Oordering__top__axioms,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ( ordering_top @ A @ ( ord_less_eq @ A ) @ ( ord_less @ A ) @ ( top_top @ A ) ) ) ).
% top.ordering_top_axioms
thf(fact_7704_bot__nat__0_Oordering__top__axioms,axiom,
( ordering_top @ nat
@ ^ [X4: nat,Y6: nat] : ( ord_less_eq @ nat @ Y6 @ X4 )
@ ^ [X4: nat,Y6: nat] : ( ord_less @ nat @ Y6 @ X4 )
@ ( zero_zero @ nat ) ) ).
% bot_nat_0.ordering_top_axioms
thf(fact_7705_span__singleton,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [X: A] :
( ( real_Vector_span @ A @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( image2 @ real @ A
@ ^ [K2: real] : ( real_V8093663219630862766scaleR @ A @ K2 @ X )
@ ( top_top @ ( set @ real ) ) ) ) ) ).
% span_singleton
thf(fact_7706_span__explicit,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ( ( real_Vector_span @ A )
= ( ^ [B4: set @ A] :
( collect @ A
@ ^ [Uu3: A] :
? [T3: set @ A,R4: A > real] :
( ( Uu3
= ( groups7311177749621191930dd_sum @ A @ A
@ ^ [A5: A] : ( real_V8093663219630862766scaleR @ A @ ( R4 @ A5 ) @ A5 )
@ T3 ) )
& ( finite_finite2 @ A @ T3 )
& ( ord_less_eq @ ( set @ A ) @ T3 @ B4 ) ) ) ) ) ) ).
% span_explicit
thf(fact_7707_span__insert__0,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [S2: set @ A] :
( ( real_Vector_span @ A @ ( insert2 @ A @ ( zero_zero @ A ) @ S2 ) )
= ( real_Vector_span @ A @ S2 ) ) ) ).
% span_insert_0
thf(fact_7708_span__empty,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ( ( real_Vector_span @ A @ ( bot_bot @ ( set @ A ) ) )
= ( insert2 @ A @ ( zero_zero @ A ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% span_empty
thf(fact_7709_span__delete__0,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [S2: set @ A] :
( ( real_Vector_span @ A @ ( minus_minus @ ( set @ A ) @ S2 @ ( insert2 @ A @ ( zero_zero @ A ) @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( real_Vector_span @ A @ S2 ) ) ) ).
% span_delete_0
thf(fact_7710_in__span__delete,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [A3: A,S2: set @ A,B3: A] :
( ( member @ A @ A3 @ ( real_Vector_span @ A @ S2 ) )
=> ( ~ ( member @ A @ A3 @ ( real_Vector_span @ A @ ( minus_minus @ ( set @ A ) @ S2 @ ( insert2 @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) ) )
=> ( member @ A @ B3 @ ( real_Vector_span @ A @ ( insert2 @ A @ A3 @ ( minus_minus @ ( set @ A ) @ S2 @ ( insert2 @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ).
% in_span_delete
thf(fact_7711_span__induct__alt,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [X: A,S2: set @ A,H2: A > $o] :
( ( member @ A @ X @ ( real_Vector_span @ A @ S2 ) )
=> ( ( H2 @ ( zero_zero @ A ) )
=> ( ! [C3: real,X5: A,Y3: A] :
( ( member @ A @ X5 @ S2 )
=> ( ( H2 @ Y3 )
=> ( H2 @ ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ C3 @ X5 ) @ Y3 ) ) ) )
=> ( H2 @ X ) ) ) ) ) ).
% span_induct_alt
thf(fact_7712_span__0,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [S2: set @ A] : ( member @ A @ ( zero_zero @ A ) @ ( real_Vector_span @ A @ S2 ) ) ) ).
% span_0
thf(fact_7713_span__add__eq2,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [Y: A,S2: set @ A,X: A] :
( ( member @ A @ Y @ ( real_Vector_span @ A @ S2 ) )
=> ( ( member @ A @ ( plus_plus @ A @ X @ Y ) @ ( real_Vector_span @ A @ S2 ) )
= ( member @ A @ X @ ( real_Vector_span @ A @ S2 ) ) ) ) ) ).
% span_add_eq2
thf(fact_7714_span__add__eq,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [X: A,S2: set @ A,Y: A] :
( ( member @ A @ X @ ( real_Vector_span @ A @ S2 ) )
=> ( ( member @ A @ ( plus_plus @ A @ X @ Y ) @ ( real_Vector_span @ A @ S2 ) )
= ( member @ A @ Y @ ( real_Vector_span @ A @ S2 ) ) ) ) ) ).
% span_add_eq
thf(fact_7715_span__add,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [X: A,S2: set @ A,Y: A] :
( ( member @ A @ X @ ( real_Vector_span @ A @ S2 ) )
=> ( ( member @ A @ Y @ ( real_Vector_span @ A @ S2 ) )
=> ( member @ A @ ( plus_plus @ A @ X @ Y ) @ ( real_Vector_span @ A @ S2 ) ) ) ) ) ).
% span_add
thf(fact_7716_span__Un,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [S2: set @ A,T4: set @ A] :
( ( real_Vector_span @ A @ ( sup_sup @ ( set @ A ) @ S2 @ T4 ) )
= ( collect @ A
@ ^ [Uu3: A] :
? [X4: A,Y6: A] :
( ( Uu3
= ( plus_plus @ A @ X4 @ Y6 ) )
& ( member @ A @ X4 @ ( real_Vector_span @ A @ S2 ) )
& ( member @ A @ Y6 @ ( real_Vector_span @ A @ T4 ) ) ) ) ) ) ).
% span_Un
thf(fact_7717_maximal__independent__subset__extend,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [S2: set @ A,V: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ S2 @ V )
=> ( ~ ( real_V358717886546972837endent @ A @ S2 )
=> ~ ! [B9: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ S2 @ B9 )
=> ( ( ord_less_eq @ ( set @ A ) @ B9 @ V )
=> ( ~ ( real_V358717886546972837endent @ A @ B9 )
=> ~ ( ord_less_eq @ ( set @ A ) @ V @ ( real_Vector_span @ A @ B9 ) ) ) ) ) ) ) ) ).
% maximal_independent_subset_extend
thf(fact_7718_spanning__subset__independent,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [B2: set @ A,A4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ B2 @ A4 )
=> ( ~ ( real_V358717886546972837endent @ A @ A4 )
=> ( ( ord_less_eq @ ( set @ A ) @ A4 @ ( real_Vector_span @ A @ B2 ) )
=> ( A4 = B2 ) ) ) ) ) ).
% spanning_subset_independent
thf(fact_7719_maximal__independent__subset,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [V: set @ A] :
~ ! [B9: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ B9 @ V )
=> ( ~ ( real_V358717886546972837endent @ A @ B9 )
=> ~ ( ord_less_eq @ ( set @ A ) @ V @ ( real_Vector_span @ A @ B9 ) ) ) ) ) ).
% maximal_independent_subset
thf(fact_7720_span__eq,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [S2: set @ A,T4: set @ A] :
( ( ( real_Vector_span @ A @ S2 )
= ( real_Vector_span @ A @ T4 ) )
= ( ( ord_less_eq @ ( set @ A ) @ S2 @ ( real_Vector_span @ A @ T4 ) )
& ( ord_less_eq @ ( set @ A ) @ T4 @ ( real_Vector_span @ A @ S2 ) ) ) ) ) ).
% span_eq
thf(fact_7721_span__mono,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [A4: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B2 )
=> ( ord_less_eq @ ( set @ A ) @ ( real_Vector_span @ A @ A4 ) @ ( real_Vector_span @ A @ B2 ) ) ) ) ).
% span_mono
thf(fact_7722_span__superset,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [S2: set @ A] : ( ord_less_eq @ ( set @ A ) @ S2 @ ( real_Vector_span @ A @ S2 ) ) ) ).
% span_superset
thf(fact_7723_span__finite,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [S2: set @ A] :
( ( finite_finite2 @ A @ S2 )
=> ( ( real_Vector_span @ A @ S2 )
= ( image2 @ ( A > real ) @ A
@ ^ [U2: A > real] :
( groups7311177749621191930dd_sum @ A @ A
@ ^ [V5: A] : ( real_V8093663219630862766scaleR @ A @ ( U2 @ V5 ) @ V5 )
@ S2 )
@ ( top_top @ ( set @ ( A > real ) ) ) ) ) ) ) ).
% span_finite
thf(fact_7724_dependent__def,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ( ( real_V358717886546972837endent @ A )
= ( ^ [P3: set @ A] :
? [X4: A] :
( ( member @ A @ X4 @ P3 )
& ( member @ A @ X4 @ ( real_Vector_span @ A @ ( minus_minus @ ( set @ A ) @ P3 @ ( insert2 @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ).
% dependent_def
thf(fact_7725_span__image__scale,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [S2: set @ A,C2: A > real] :
( ( finite_finite2 @ A @ S2 )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ S2 )
=> ( ( C2 @ X5 )
!= ( zero_zero @ real ) ) )
=> ( ( real_Vector_span @ A
@ ( image2 @ A @ A
@ ^ [X4: A] : ( real_V8093663219630862766scaleR @ A @ ( C2 @ X4 ) @ X4 )
@ S2 ) )
= ( real_Vector_span @ A @ S2 ) ) ) ) ) ).
% span_image_scale
thf(fact_7726_span__breakdown,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [B3: A,S2: set @ A,A3: A] :
( ( member @ A @ B3 @ S2 )
=> ( ( member @ A @ A3 @ ( real_Vector_span @ A @ S2 ) )
=> ? [K3: real] : ( member @ A @ ( minus_minus @ A @ A3 @ ( real_V8093663219630862766scaleR @ A @ K3 @ B3 ) ) @ ( real_Vector_span @ A @ ( minus_minus @ ( set @ A ) @ S2 @ ( insert2 @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ).
% span_breakdown
thf(fact_7727_independent__span__bound,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [T4: set @ A,S2: set @ A] :
( ( finite_finite2 @ A @ T4 )
=> ( ~ ( real_V358717886546972837endent @ A @ S2 )
=> ( ( ord_less_eq @ ( set @ A ) @ S2 @ ( real_Vector_span @ A @ T4 ) )
=> ( ( finite_finite2 @ A @ S2 )
& ( ord_less_eq @ nat @ ( finite_card @ A @ S2 ) @ ( finite_card @ A @ T4 ) ) ) ) ) ) ) ).
% independent_span_bound
thf(fact_7728_exchange__lemma,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [T4: set @ A,S2: set @ A] :
( ( finite_finite2 @ A @ T4 )
=> ( ~ ( real_V358717886546972837endent @ A @ S2 )
=> ( ( ord_less_eq @ ( set @ A ) @ S2 @ ( real_Vector_span @ A @ T4 ) )
=> ? [T13: set @ A] :
( ( ( finite_card @ A @ T13 )
= ( finite_card @ A @ T4 ) )
& ( finite_finite2 @ A @ T13 )
& ( ord_less_eq @ ( set @ A ) @ S2 @ T13 )
& ( ord_less_eq @ ( set @ A ) @ T13 @ ( sup_sup @ ( set @ A ) @ S2 @ T4 ) )
& ( ord_less_eq @ ( set @ A ) @ S2 @ ( real_Vector_span @ A @ T13 ) ) ) ) ) ) ) ).
% exchange_lemma
thf(fact_7729_span__alt,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ( ( real_Vector_span @ A )
= ( ^ [B6: set @ A] :
( collect @ A
@ ^ [Uu3: A] :
? [F3: A > real] :
( ( Uu3
= ( groups7311177749621191930dd_sum @ A @ A
@ ^ [X4: A] : ( real_V8093663219630862766scaleR @ A @ ( F3 @ X4 ) @ X4 )
@ ( collect @ A
@ ^ [X4: A] :
( ( F3 @ X4 )
!= ( zero_zero @ real ) ) ) ) )
& ( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [X4: A] :
( ( F3 @ X4 )
!= ( zero_zero @ real ) ) )
@ B6 )
& ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X4: A] :
( ( F3 @ X4 )
!= ( zero_zero @ real ) ) ) ) ) ) ) ) ) ).
% span_alt
thf(fact_7730_span__explicit_H,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ( ( real_Vector_span @ A )
= ( ^ [B4: set @ A] :
( collect @ A
@ ^ [Uu3: A] :
? [F3: A > real] :
( ( Uu3
= ( groups7311177749621191930dd_sum @ A @ A
@ ^ [V5: A] : ( real_V8093663219630862766scaleR @ A @ ( F3 @ V5 ) @ V5 )
@ ( collect @ A
@ ^ [V5: A] :
( ( F3 @ V5 )
!= ( zero_zero @ real ) ) ) ) )
& ( finite_finite2 @ A
@ ( collect @ A
@ ^ [V5: A] :
( ( F3 @ V5 )
!= ( zero_zero @ real ) ) ) )
& ! [V5: A] :
( ( ( F3 @ V5 )
!= ( zero_zero @ real ) )
=> ( member @ A @ V5 @ B4 ) ) ) ) ) ) ) ).
% span_explicit'
thf(fact_7731_representation__def,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ( ( real_V7696804695334737415tation @ A )
= ( ^ [Basis2: set @ A,V5: A] :
( if @ ( A > real )
@ ( ~ ( real_V358717886546972837endent @ A @ Basis2 )
& ( member @ A @ V5 @ ( real_Vector_span @ A @ Basis2 ) ) )
@ ( fChoice @ ( A > real )
@ ^ [F3: A > real] :
( ! [W3: A] :
( ( ( F3 @ W3 )
!= ( zero_zero @ real ) )
=> ( member @ A @ W3 @ Basis2 ) )
& ( finite_finite2 @ A
@ ( collect @ A
@ ^ [W3: A] :
( ( F3 @ W3 )
!= ( zero_zero @ real ) ) ) )
& ( ( groups7311177749621191930dd_sum @ A @ A
@ ^ [W3: A] : ( real_V8093663219630862766scaleR @ A @ ( F3 @ W3 ) @ W3 )
@ ( collect @ A
@ ^ [W3: A] :
( ( F3 @ W3 )
!= ( zero_zero @ real ) ) ) )
= V5 ) ) )
@ ^ [B4: A] : ( zero_zero @ real ) ) ) ) ) ).
% representation_def
thf(fact_7732_extend__basis__def,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ( ( real_V4986007116245087402_basis @ A )
= ( ^ [B6: set @ A] :
( fChoice @ ( set @ A )
@ ^ [B15: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ B6 @ B15 )
& ~ ( real_V358717886546972837endent @ A @ B15 )
& ( ( real_Vector_span @ A @ B15 )
= ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ).
% extend_basis_def
thf(fact_7733_finite__representation,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [Basis: set @ A,V2: A] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [B4: A] :
( ( real_V7696804695334737415tation @ A @ Basis @ V2 @ B4 )
!= ( zero_zero @ real ) ) ) ) ) ).
% finite_representation
thf(fact_7734_representation__zero,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [Basis: set @ A] :
( ( real_V7696804695334737415tation @ A @ Basis @ ( zero_zero @ A ) )
= ( ^ [B4: A] : ( zero_zero @ real ) ) ) ) ).
% representation_zero
thf(fact_7735_representation__extend,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [Basis: set @ A,V2: A,Basis3: set @ A] :
( ~ ( real_V358717886546972837endent @ A @ Basis )
=> ( ( member @ A @ V2 @ ( real_Vector_span @ A @ Basis3 ) )
=> ( ( ord_less_eq @ ( set @ A ) @ Basis3 @ Basis )
=> ( ( real_V7696804695334737415tation @ A @ Basis @ V2 )
= ( real_V7696804695334737415tation @ A @ Basis3 @ V2 ) ) ) ) ) ) ).
% representation_extend
thf(fact_7736_extend__basis__superset,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [B2: set @ A] :
( ~ ( real_V358717886546972837endent @ A @ B2 )
=> ( ord_less_eq @ ( set @ A ) @ B2 @ ( real_V4986007116245087402_basis @ A @ B2 ) ) ) ) ).
% extend_basis_superset
thf(fact_7737_representation__add,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [Basis: set @ A,V2: A,U: A] :
( ~ ( real_V358717886546972837endent @ A @ Basis )
=> ( ( member @ A @ V2 @ ( real_Vector_span @ A @ Basis ) )
=> ( ( member @ A @ U @ ( real_Vector_span @ A @ Basis ) )
=> ( ( real_V7696804695334737415tation @ A @ Basis @ ( plus_plus @ A @ U @ V2 ) )
= ( ^ [B4: A] : ( plus_plus @ real @ ( real_V7696804695334737415tation @ A @ Basis @ U @ B4 ) @ ( real_V7696804695334737415tation @ A @ Basis @ V2 @ B4 ) ) ) ) ) ) ) ) ).
% representation_add
thf(fact_7738_sum__representation__eq,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [Basis: set @ A,V2: A,B2: set @ A] :
( ~ ( real_V358717886546972837endent @ A @ Basis )
=> ( ( member @ A @ V2 @ ( real_Vector_span @ A @ Basis ) )
=> ( ( finite_finite2 @ A @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ Basis @ B2 )
=> ( ( groups7311177749621191930dd_sum @ A @ A
@ ^ [B4: A] : ( real_V8093663219630862766scaleR @ A @ ( real_V7696804695334737415tation @ A @ Basis @ V2 @ B4 ) @ B4 )
@ B2 )
= V2 ) ) ) ) ) ) ).
% sum_representation_eq
thf(fact_7739_representation__eqI,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [Basis: set @ A,V2: A,F2: A > real] :
( ~ ( real_V358717886546972837endent @ A @ Basis )
=> ( ( member @ A @ V2 @ ( real_Vector_span @ A @ Basis ) )
=> ( ! [B5: A] :
( ( ( F2 @ B5 )
!= ( zero_zero @ real ) )
=> ( member @ A @ B5 @ Basis ) )
=> ( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [B4: A] :
( ( F2 @ B4 )
!= ( zero_zero @ real ) ) ) )
=> ( ( ( groups7311177749621191930dd_sum @ A @ A
@ ^ [B4: A] : ( real_V8093663219630862766scaleR @ A @ ( F2 @ B4 ) @ B4 )
@ ( collect @ A
@ ^ [B4: A] :
( ( F2 @ B4 )
!= ( zero_zero @ real ) ) ) )
= V2 )
=> ( ( real_V7696804695334737415tation @ A @ Basis @ V2 )
= F2 ) ) ) ) ) ) ) ).
% representation_eqI
thf(fact_7740_construct__def,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V4867850818363320053vector @ A )
& ( real_V4867850818363320053vector @ B ) )
=> ( ( real_V4425403222259421789struct @ A @ B )
= ( ^ [B6: set @ A,G2: A > B,V5: A] :
( groups7311177749621191930dd_sum @ A @ B
@ ^ [B4: A] : ( real_V8093663219630862766scaleR @ B @ ( real_V7696804695334737415tation @ A @ ( real_V4986007116245087402_basis @ A @ B6 ) @ V5 @ B4 ) @ ( if @ B @ ( member @ A @ B4 @ B6 ) @ ( G2 @ B4 ) @ ( zero_zero @ B ) ) )
@ ( collect @ A
@ ^ [B4: A] :
( ( real_V7696804695334737415tation @ A @ ( real_V4986007116245087402_basis @ A @ B6 ) @ V5 @ B4 )
!= ( zero_zero @ real ) ) ) ) ) ) ) ).
% construct_def
thf(fact_7741_dim__def,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ( ( real_Vector_dim @ A )
= ( ^ [V7: set @ A] :
( if @ nat
@ ? [B4: set @ A] :
( ~ ( real_V358717886546972837endent @ A @ B4 )
& ( ( real_Vector_span @ A @ B4 )
= ( real_Vector_span @ A @ V7 ) ) )
@ ( finite_card @ A
@ ( fChoice @ ( set @ A )
@ ^ [B4: set @ A] :
( ~ ( real_V358717886546972837endent @ A @ B4 )
& ( ( real_Vector_span @ A @ B4 )
= ( real_Vector_span @ A @ V7 ) ) ) ) )
@ ( zero_zero @ nat ) ) ) ) ) ).
% dim_def
thf(fact_7742_construct__add,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V4867850818363320053vector @ A )
& ( real_V4867850818363320053vector @ B ) )
=> ! [B2: set @ A,F2: A > B,G: A > B,V2: A] :
( ~ ( real_V358717886546972837endent @ A @ B2 )
=> ( ( real_V4425403222259421789struct @ A @ B @ B2
@ ^ [X4: A] : ( plus_plus @ B @ ( F2 @ X4 ) @ ( G @ X4 ) )
@ V2 )
= ( plus_plus @ B @ ( real_V4425403222259421789struct @ A @ B @ B2 @ F2 @ V2 ) @ ( real_V4425403222259421789struct @ A @ B @ B2 @ G @ V2 ) ) ) ) ) ).
% construct_add
thf(fact_7743_dim__le__card_H,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [S: set @ A] :
( ( finite_finite2 @ A @ S )
=> ( ord_less_eq @ nat @ ( real_Vector_dim @ A @ S ) @ ( finite_card @ A @ S ) ) ) ) ).
% dim_le_card'
thf(fact_7744_basis__card__eq__dim,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [B2: set @ A,V: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ B2 @ V )
=> ( ( ord_less_eq @ ( set @ A ) @ V @ ( real_Vector_span @ A @ B2 ) )
=> ( ~ ( real_V358717886546972837endent @ A @ B2 )
=> ( ( finite_card @ A @ B2 )
= ( real_Vector_dim @ A @ V ) ) ) ) ) ) ).
% basis_card_eq_dim
thf(fact_7745_basis__exists,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [V: set @ A] :
~ ! [B9: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ B9 @ V )
=> ( ~ ( real_V358717886546972837endent @ A @ B9 )
=> ( ( ord_less_eq @ ( set @ A ) @ V @ ( real_Vector_span @ A @ B9 ) )
=> ( ( finite_card @ A @ B9 )
!= ( real_Vector_dim @ A @ V ) ) ) ) ) ) ).
% basis_exists
thf(fact_7746_dim__unique,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [B2: set @ A,V: set @ A,N2: nat] :
( ( ord_less_eq @ ( set @ A ) @ B2 @ V )
=> ( ( ord_less_eq @ ( set @ A ) @ V @ ( real_Vector_span @ A @ B2 ) )
=> ( ~ ( real_V358717886546972837endent @ A @ B2 )
=> ( ( ( finite_card @ A @ B2 )
= N2 )
=> ( ( real_Vector_dim @ A @ V )
= N2 ) ) ) ) ) ) ).
% dim_unique
thf(fact_7747_span__card__ge__dim,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [B2: set @ A,V: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ B2 @ V )
=> ( ( ord_less_eq @ ( set @ A ) @ V @ ( real_Vector_span @ A @ B2 ) )
=> ( ( finite_finite2 @ A @ B2 )
=> ( ord_less_eq @ nat @ ( real_Vector_dim @ A @ V ) @ ( finite_card @ A @ B2 ) ) ) ) ) ) ).
% span_card_ge_dim
thf(fact_7748_dim__le__card,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [V: set @ A,W4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ V @ ( real_Vector_span @ A @ W4 ) )
=> ( ( finite_finite2 @ A @ W4 )
=> ( ord_less_eq @ nat @ ( real_Vector_dim @ A @ V ) @ ( finite_card @ A @ W4 ) ) ) ) ) ).
% dim_le_card
thf(fact_7749_construct__outside,axiom,
! [A: $tType,B: $tType] :
( ( ( real_V4867850818363320053vector @ B )
& ( real_V4867850818363320053vector @ A ) )
=> ! [B2: set @ A,V2: A,F2: A > B] :
( ~ ( real_V358717886546972837endent @ A @ B2 )
=> ( ( member @ A @ V2 @ ( real_Vector_span @ A @ ( minus_minus @ ( set @ A ) @ ( real_V4986007116245087402_basis @ A @ B2 ) @ B2 ) ) )
=> ( ( real_V4425403222259421789struct @ A @ B @ B2 @ F2 @ V2 )
= ( zero_zero @ B ) ) ) ) ) ).
% construct_outside
thf(fact_7750_linear__indep__image__lemma,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V4867850818363320053vector @ A )
& ( real_V4867850818363320053vector @ B ) )
=> ! [F2: A > B,B2: set @ A,X: A] :
( ( real_Vector_linear @ A @ B @ F2 )
=> ( ( finite_finite2 @ A @ B2 )
=> ( ~ ( real_V358717886546972837endent @ B @ ( image2 @ A @ B @ F2 @ B2 ) )
=> ( ( inj_on @ A @ B @ F2 @ B2 )
=> ( ( member @ A @ X @ ( real_Vector_span @ A @ B2 ) )
=> ( ( ( F2 @ X )
= ( zero_zero @ B ) )
=> ( X
= ( zero_zero @ A ) ) ) ) ) ) ) ) ) ).
% linear_indep_image_lemma
thf(fact_7751_euclidean__size__times__nonunit,axiom,
! [A: $tType] :
( ( euclid3725896446679973847miring @ A )
=> ! [A3: A,B3: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( B3
!= ( zero_zero @ A ) )
=> ( ~ ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ord_less @ nat @ ( euclid6346220572633701492n_size @ A @ B3 ) @ ( euclid6346220572633701492n_size @ A @ ( times_times @ A @ A3 @ B3 ) ) ) ) ) ) ) ).
% euclidean_size_times_nonunit
thf(fact_7752_euclidean__size__eq__0__iff,axiom,
! [A: $tType] :
( ( euclid3725896446679973847miring @ A )
=> ! [B3: A] :
( ( ( euclid6346220572633701492n_size @ A @ B3 )
= ( zero_zero @ nat ) )
= ( B3
= ( zero_zero @ A ) ) ) ) ).
% euclidean_size_eq_0_iff
thf(fact_7753_size__0,axiom,
! [A: $tType] :
( ( euclid3725896446679973847miring @ A )
=> ( ( euclid6346220572633701492n_size @ A @ ( zero_zero @ A ) )
= ( zero_zero @ nat ) ) ) ).
% size_0
thf(fact_7754_euclidean__size__greater__0__iff,axiom,
! [A: $tType] :
( ( euclid3725896446679973847miring @ A )
=> ! [B3: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( euclid6346220572633701492n_size @ A @ B3 ) )
= ( B3
!= ( zero_zero @ A ) ) ) ) ).
% euclidean_size_greater_0_iff
thf(fact_7755_linear__eq__0__on__span,axiom,
! [A: $tType,B: $tType] :
( ( ( real_V4867850818363320053vector @ B )
& ( real_V4867850818363320053vector @ A ) )
=> ! [F2: A > B,B3: set @ A,X: A] :
( ( real_Vector_linear @ A @ B @ F2 )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ B3 )
=> ( ( F2 @ X5 )
= ( zero_zero @ B ) ) )
=> ( ( member @ A @ X @ ( real_Vector_span @ A @ B3 ) )
=> ( ( F2 @ X )
= ( zero_zero @ B ) ) ) ) ) ) ).
% linear_eq_0_on_span
thf(fact_7756_linear__compose__add,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V4867850818363320053vector @ A )
& ( real_V4867850818363320053vector @ B ) )
=> ! [F2: A > B,G: A > B] :
( ( real_Vector_linear @ A @ B @ F2 )
=> ( ( real_Vector_linear @ A @ B @ G )
=> ( real_Vector_linear @ A @ B
@ ^ [X4: A] : ( plus_plus @ B @ ( F2 @ X4 ) @ ( G @ X4 ) ) ) ) ) ) ).
% linear_compose_add
thf(fact_7757_module__hom__zero,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V4867850818363320053vector @ A )
& ( real_V4867850818363320053vector @ B ) )
=> ( real_Vector_linear @ A @ B
@ ^ [X4: A] : ( zero_zero @ B ) ) ) ).
% module_hom_zero
thf(fact_7758_linear__add,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V4867850818363320053vector @ A )
& ( real_V4867850818363320053vector @ B ) )
=> ! [F2: A > B,B16: A,B24: A] :
( ( real_Vector_linear @ A @ B @ F2 )
=> ( ( F2 @ ( plus_plus @ A @ B16 @ B24 ) )
= ( plus_plus @ B @ ( F2 @ B16 ) @ ( F2 @ B24 ) ) ) ) ) ).
% linear_add
thf(fact_7759_linear__0,axiom,
! [A: $tType,B: $tType] :
( ( ( real_V4867850818363320053vector @ B )
& ( real_V4867850818363320053vector @ A ) )
=> ! [F2: A > B] :
( ( real_Vector_linear @ A @ B @ F2 )
=> ( ( F2 @ ( zero_zero @ A ) )
= ( zero_zero @ B ) ) ) ) ).
% linear_0
thf(fact_7760_Real__Vector__Spaces_Olinear__iff,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V4867850818363320053vector @ A )
& ( real_V4867850818363320053vector @ B ) )
=> ( ( real_Vector_linear @ A @ B )
= ( ^ [F3: A > B] :
( ! [X4: A,Y6: A] :
( ( F3 @ ( plus_plus @ A @ X4 @ Y6 ) )
= ( plus_plus @ B @ ( F3 @ X4 ) @ ( F3 @ Y6 ) ) )
& ! [C4: real,X4: A] :
( ( F3 @ ( real_V8093663219630862766scaleR @ A @ C4 @ X4 ) )
= ( real_V8093663219630862766scaleR @ B @ C4 @ ( F3 @ X4 ) ) ) ) ) ) ) ).
% Real_Vector_Spaces.linear_iff
thf(fact_7761_linearI,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V4867850818363320053vector @ A )
& ( real_V4867850818363320053vector @ B ) )
=> ! [F2: A > B] :
( ! [B17: A,B25: A] :
( ( F2 @ ( plus_plus @ A @ B17 @ B25 ) )
= ( plus_plus @ B @ ( F2 @ B17 ) @ ( F2 @ B25 ) ) )
=> ( ! [R3: real,B5: A] :
( ( F2 @ ( real_V8093663219630862766scaleR @ A @ R3 @ B5 ) )
= ( real_V8093663219630862766scaleR @ B @ R3 @ ( F2 @ B5 ) ) )
=> ( real_Vector_linear @ A @ B @ F2 ) ) ) ) ).
% linearI
thf(fact_7762_dvd__euclidean__size__eq__imp__dvd,axiom,
! [A: $tType] :
( ( euclid3725896446679973847miring @ A )
=> ! [A3: A,B3: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( ( euclid6346220572633701492n_size @ A @ A3 )
= ( euclid6346220572633701492n_size @ A @ B3 ) )
=> ( ( dvd_dvd @ A @ B3 @ A3 )
=> ( dvd_dvd @ A @ A3 @ B3 ) ) ) ) ) ).
% dvd_euclidean_size_eq_imp_dvd
thf(fact_7763_linear__injective__0,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V4867850818363320053vector @ A )
& ( real_V4867850818363320053vector @ B ) )
=> ! [F2: A > B] :
( ( real_Vector_linear @ A @ B @ F2 )
=> ( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
= ( ! [X4: A] :
( ( ( F2 @ X4 )
= ( zero_zero @ B ) )
=> ( X4
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% linear_injective_0
thf(fact_7764_linear__spans__image,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V4867850818363320053vector @ A )
& ( real_V4867850818363320053vector @ B ) )
=> ! [F2: A > B,V: set @ A,B2: set @ A] :
( ( real_Vector_linear @ A @ B @ F2 )
=> ( ( ord_less_eq @ ( set @ A ) @ V @ ( real_Vector_span @ A @ B2 ) )
=> ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F2 @ V ) @ ( real_Vector_span @ B @ ( image2 @ A @ B @ F2 @ B2 ) ) ) ) ) ) ).
% linear_spans_image
thf(fact_7765_unit__iff__euclidean__size,axiom,
! [A: $tType] :
( ( euclid3725896446679973847miring @ A )
=> ! [A3: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
= ( ( ( euclid6346220572633701492n_size @ A @ A3 )
= ( euclid6346220572633701492n_size @ A @ ( one_one @ A ) ) )
& ( A3
!= ( zero_zero @ A ) ) ) ) ) ).
% unit_iff_euclidean_size
thf(fact_7766_size__mult__mono_H,axiom,
! [A: $tType] :
( ( euclid3725896446679973847miring @ A )
=> ! [B3: A,A3: A] :
( ( B3
!= ( zero_zero @ A ) )
=> ( ord_less_eq @ nat @ ( euclid6346220572633701492n_size @ A @ A3 ) @ ( euclid6346220572633701492n_size @ A @ ( times_times @ A @ B3 @ A3 ) ) ) ) ) ).
% size_mult_mono'
thf(fact_7767_size__mult__mono,axiom,
! [A: $tType] :
( ( euclid3725896446679973847miring @ A )
=> ! [B3: A,A3: A] :
( ( B3
!= ( zero_zero @ A ) )
=> ( ord_less_eq @ nat @ ( euclid6346220572633701492n_size @ A @ A3 ) @ ( euclid6346220572633701492n_size @ A @ ( times_times @ A @ A3 @ B3 ) ) ) ) ) ).
% size_mult_mono
thf(fact_7768_dvd__proper__imp__size__less,axiom,
! [A: $tType] :
( ( euclid3725896446679973847miring @ A )
=> ! [A3: A,B3: A] :
( ( dvd_dvd @ A @ A3 @ B3 )
=> ( ~ ( dvd_dvd @ A @ B3 @ A3 )
=> ( ( B3
!= ( zero_zero @ A ) )
=> ( ord_less @ nat @ ( euclid6346220572633701492n_size @ A @ A3 ) @ ( euclid6346220572633701492n_size @ A @ B3 ) ) ) ) ) ) ).
% dvd_proper_imp_size_less
thf(fact_7769_dvd__imp__size__le,axiom,
! [A: $tType] :
( ( euclid3725896446679973847miring @ A )
=> ! [A3: A,B3: A] :
( ( dvd_dvd @ A @ A3 @ B3 )
=> ( ( B3
!= ( zero_zero @ A ) )
=> ( ord_less_eq @ nat @ ( euclid6346220572633701492n_size @ A @ A3 ) @ ( euclid6346220572633701492n_size @ A @ B3 ) ) ) ) ) ).
% dvd_imp_size_le
thf(fact_7770_mod__size__less,axiom,
! [A: $tType] :
( ( euclid3725896446679973847miring @ A )
=> ! [B3: A,A3: A] :
( ( B3
!= ( zero_zero @ A ) )
=> ( ord_less @ nat @ ( euclid6346220572633701492n_size @ A @ ( modulo_modulo @ A @ A3 @ B3 ) ) @ ( euclid6346220572633701492n_size @ A @ B3 ) ) ) ) ).
% mod_size_less
thf(fact_7771_linear__spanning__surjective__image,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V4867850818363320053vector @ A )
& ( real_V4867850818363320053vector @ B ) )
=> ! [F2: A > B,S2: set @ A] :
( ( real_Vector_linear @ A @ B @ F2 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( top_top @ ( set @ A ) ) @ ( real_Vector_span @ A @ S2 ) )
=> ( ( ( image2 @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ B ) ) )
=> ( ord_less_eq @ ( set @ B ) @ ( top_top @ ( set @ B ) ) @ ( real_Vector_span @ B @ ( image2 @ A @ B @ F2 @ S2 ) ) ) ) ) ) ) ).
% linear_spanning_surjective_image
thf(fact_7772_linear__surj__right__inverse,axiom,
! [A: $tType,B: $tType] :
( ( ( real_V4867850818363320053vector @ B )
& ( real_V4867850818363320053vector @ A ) )
=> ! [F2: A > B,T4: set @ B,S2: set @ A] :
( ( real_Vector_linear @ A @ B @ F2 )
=> ( ( ord_less_eq @ ( set @ B ) @ ( real_Vector_span @ B @ T4 ) @ ( image2 @ A @ B @ F2 @ ( real_Vector_span @ A @ S2 ) ) )
=> ? [G7: B > A] :
( ( ord_less_eq @ ( set @ A ) @ ( image2 @ B @ A @ G7 @ ( top_top @ ( set @ B ) ) ) @ ( real_Vector_span @ A @ S2 ) )
& ( real_Vector_linear @ B @ A @ G7 )
& ! [X2: B] :
( ( member @ B @ X2 @ ( real_Vector_span @ B @ T4 ) )
=> ( ( F2 @ ( G7 @ X2 ) )
= X2 ) ) ) ) ) ) ).
% linear_surj_right_inverse
thf(fact_7773_linear__inj__on__left__inverse,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V4867850818363320053vector @ A )
& ( real_V4867850818363320053vector @ B ) )
=> ! [F2: A > B,S2: set @ A] :
( ( real_Vector_linear @ A @ B @ F2 )
=> ( ( inj_on @ A @ B @ F2 @ ( real_Vector_span @ A @ S2 ) )
=> ? [G7: B > A] :
( ( ord_less_eq @ ( set @ A ) @ ( image2 @ B @ A @ G7 @ ( top_top @ ( set @ B ) ) ) @ ( real_Vector_span @ A @ S2 ) )
& ( real_Vector_linear @ B @ A @ G7 )
& ! [X2: A] :
( ( member @ A @ X2 @ ( real_Vector_span @ A @ S2 ) )
=> ( ( G7 @ ( F2 @ X2 ) )
= X2 ) ) ) ) ) ) ).
% linear_inj_on_left_inverse
thf(fact_7774_finite__basis__to__basis__subspace__isomorphism,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V4867850818363320053vector @ A )
& ( real_V4867850818363320053vector @ B ) )
=> ! [S2: set @ A,T4: set @ B,B2: set @ A,C5: set @ B] :
( ( real_Vector_subspace @ A @ S2 )
=> ( ( real_Vector_subspace @ B @ T4 )
=> ( ( ( real_Vector_dim @ A @ S2 )
= ( real_Vector_dim @ B @ T4 ) )
=> ( ( finite_finite2 @ A @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ S2 )
=> ( ~ ( real_V358717886546972837endent @ A @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ S2 @ ( real_Vector_span @ A @ B2 ) )
=> ( ( ( finite_card @ A @ B2 )
= ( real_Vector_dim @ A @ S2 ) )
=> ( ( finite_finite2 @ B @ C5 )
=> ( ( ord_less_eq @ ( set @ B ) @ C5 @ T4 )
=> ( ~ ( real_V358717886546972837endent @ B @ C5 )
=> ( ( ord_less_eq @ ( set @ B ) @ T4 @ ( real_Vector_span @ B @ C5 ) )
=> ( ( ( finite_card @ B @ C5 )
= ( real_Vector_dim @ B @ T4 ) )
=> ? [F4: A > B] :
( ( real_Vector_linear @ A @ B @ F4 )
& ( ( image2 @ A @ B @ F4 @ B2 )
= C5 )
& ( ( image2 @ A @ B @ F4 @ S2 )
= T4 )
& ( inj_on @ A @ B @ F4 @ S2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% finite_basis_to_basis_subspace_isomorphism
thf(fact_7775_divmod__cases,axiom,
! [A: $tType] :
( ( euclid3128863361964157862miring @ A )
=> ! [B3: A,A3: A] :
( ( ( B3
!= ( zero_zero @ A ) )
=> ( ( ( modulo_modulo @ A @ A3 @ B3 )
= ( zero_zero @ A ) )
=> ( A3
!= ( times_times @ A @ ( divide_divide @ A @ A3 @ B3 ) @ B3 ) ) ) )
=> ( ( ( B3
!= ( zero_zero @ A ) )
=> ! [Q2: A,R3: A] :
( ( ( euclid7384307370059645450egment @ A @ R3 )
= ( euclid7384307370059645450egment @ A @ B3 ) )
=> ( ( ord_less @ nat @ ( euclid6346220572633701492n_size @ A @ R3 ) @ ( euclid6346220572633701492n_size @ A @ B3 ) )
=> ( ( R3
!= ( zero_zero @ A ) )
=> ( ( ( divide_divide @ A @ A3 @ B3 )
= Q2 )
=> ( ( ( modulo_modulo @ A @ A3 @ B3 )
= R3 )
=> ( A3
!= ( plus_plus @ A @ ( times_times @ A @ Q2 @ B3 ) @ R3 ) ) ) ) ) ) ) )
=> ( B3
= ( zero_zero @ A ) ) ) ) ) ).
% divmod_cases
thf(fact_7776_linear__subspace__kernel,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V4867850818363320053vector @ A )
& ( real_V4867850818363320053vector @ B ) )
=> ! [F2: A > B] :
( ( real_Vector_linear @ A @ B @ F2 )
=> ( real_Vector_subspace @ A
@ ( collect @ A
@ ^ [X4: A] :
( ( F2 @ X4 )
= ( zero_zero @ B ) ) ) ) ) ) ).
% linear_subspace_kernel
thf(fact_7777_division__segment__mult,axiom,
! [A: $tType] :
( ( euclid3128863361964157862miring @ A )
=> ! [A3: A,B3: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( B3
!= ( zero_zero @ A ) )
=> ( ( euclid7384307370059645450egment @ A @ ( times_times @ A @ A3 @ B3 ) )
= ( times_times @ A @ ( euclid7384307370059645450egment @ A @ A3 ) @ ( euclid7384307370059645450egment @ A @ B3 ) ) ) ) ) ) ).
% division_segment_mult
thf(fact_7778_subspace__add,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [S2: set @ A,X: A,Y: A] :
( ( real_Vector_subspace @ A @ S2 )
=> ( ( member @ A @ X @ S2 )
=> ( ( member @ A @ Y @ S2 )
=> ( member @ A @ ( plus_plus @ A @ X @ Y ) @ S2 ) ) ) ) ) ).
% subspace_add
thf(fact_7779_subspace__0,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [S2: set @ A] :
( ( real_Vector_subspace @ A @ S2 )
=> ( member @ A @ ( zero_zero @ A ) @ S2 ) ) ) ).
% subspace_0
thf(fact_7780_division__segment__not__0,axiom,
! [A: $tType] :
( ( euclid3128863361964157862miring @ A )
=> ! [A3: A] :
( ( euclid7384307370059645450egment @ A @ A3 )
!= ( zero_zero @ A ) ) ) ).
% division_segment_not_0
thf(fact_7781_span__subspace,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [A4: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ ( real_Vector_span @ A @ A4 ) )
=> ( ( real_Vector_subspace @ A @ B2 )
=> ( ( real_Vector_span @ A @ A4 )
= B2 ) ) ) ) ) ).
% span_subspace
thf(fact_7782_span__minimal,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [S2: set @ A,T4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ S2 @ T4 )
=> ( ( real_Vector_subspace @ A @ T4 )
=> ( ord_less_eq @ ( set @ A ) @ ( real_Vector_span @ A @ S2 ) @ T4 ) ) ) ) ).
% span_minimal
thf(fact_7783_span__unique,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [S2: set @ A,T4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ S2 @ T4 )
=> ( ( real_Vector_subspace @ A @ T4 )
=> ( ! [T14: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ S2 @ T14 )
=> ( ( real_Vector_subspace @ A @ T14 )
=> ( ord_less_eq @ ( set @ A ) @ T4 @ T14 ) ) )
=> ( ( real_Vector_span @ A @ S2 )
= T4 ) ) ) ) ) ).
% span_unique
thf(fact_7784_subspace__sums,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [S2: set @ A,T4: set @ A] :
( ( real_Vector_subspace @ A @ S2 )
=> ( ( real_Vector_subspace @ A @ T4 )
=> ( real_Vector_subspace @ A
@ ( collect @ A
@ ^ [Uu3: A] :
? [X4: A,Y6: A] :
( ( Uu3
= ( plus_plus @ A @ X4 @ Y6 ) )
& ( member @ A @ X4 @ S2 )
& ( member @ A @ Y6 @ T4 ) ) ) ) ) ) ) ).
% subspace_sums
thf(fact_7785_subspace__single__0,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ( real_Vector_subspace @ A @ ( insert2 @ A @ ( zero_zero @ A ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% subspace_single_0
thf(fact_7786_subspace__def,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ( ( real_Vector_subspace @ A )
= ( ^ [S6: set @ A] :
( ( member @ A @ ( zero_zero @ A ) @ S6 )
& ! [X4: A] :
( ( member @ A @ X4 @ S6 )
=> ! [Y6: A] :
( ( member @ A @ Y6 @ S6 )
=> ( member @ A @ ( plus_plus @ A @ X4 @ Y6 ) @ S6 ) ) )
& ! [C4: real,X4: A] :
( ( member @ A @ X4 @ S6 )
=> ( member @ A @ ( real_V8093663219630862766scaleR @ A @ C4 @ X4 ) @ S6 ) ) ) ) ) ) ).
% subspace_def
thf(fact_7787_subspaceI,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [S2: set @ A] :
( ( member @ A @ ( zero_zero @ A ) @ S2 )
=> ( ! [X5: A,Y3: A] :
( ( member @ A @ X5 @ S2 )
=> ( ( member @ A @ Y3 @ S2 )
=> ( member @ A @ ( plus_plus @ A @ X5 @ Y3 ) @ S2 ) ) )
=> ( ! [C3: real,X5: A] :
( ( member @ A @ X5 @ S2 )
=> ( member @ A @ ( real_V8093663219630862766scaleR @ A @ C3 @ X5 ) @ S2 ) )
=> ( real_Vector_subspace @ A @ S2 ) ) ) ) ) ).
% subspaceI
thf(fact_7788_linear__injective__on__subspace__0,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V4867850818363320053vector @ A )
& ( real_V4867850818363320053vector @ B ) )
=> ! [F2: A > B,S: set @ A] :
( ( real_Vector_linear @ A @ B @ F2 )
=> ( ( real_Vector_subspace @ A @ S )
=> ( ( inj_on @ A @ B @ F2 @ S )
= ( ! [X4: A] :
( ( member @ A @ X4 @ S )
=> ( ( ( F2 @ X4 )
= ( zero_zero @ B ) )
=> ( X4
= ( zero_zero @ A ) ) ) ) ) ) ) ) ) ).
% linear_injective_on_subspace_0
thf(fact_7789_division__segment__mod,axiom,
! [A: $tType] :
( ( euclid3128863361964157862miring @ A )
=> ! [B3: A,A3: A] :
( ( B3
!= ( zero_zero @ A ) )
=> ( ~ ( dvd_dvd @ A @ B3 @ A3 )
=> ( ( euclid7384307370059645450egment @ A @ ( modulo_modulo @ A @ A3 @ B3 ) )
= ( euclid7384307370059645450egment @ A @ B3 ) ) ) ) ) ).
% division_segment_mod
thf(fact_7790_linear__exists__right__inverse__on,axiom,
! [A: $tType,B: $tType] :
( ( ( real_V4867850818363320053vector @ B )
& ( real_V4867850818363320053vector @ A ) )
=> ! [F2: A > B,V: set @ A] :
( ( real_Vector_linear @ A @ B @ F2 )
=> ( ( real_Vector_subspace @ A @ V )
=> ? [G7: B > A] :
( ( ord_less_eq @ ( set @ A ) @ ( image2 @ B @ A @ G7 @ ( top_top @ ( set @ B ) ) ) @ V )
& ( real_Vector_linear @ B @ A @ G7 )
& ! [X2: B] :
( ( member @ B @ X2 @ ( image2 @ A @ B @ F2 @ V ) )
=> ( ( F2 @ ( G7 @ X2 ) )
= X2 ) ) ) ) ) ) ).
% linear_exists_right_inverse_on
thf(fact_7791_unique__euclidean__semiring__class_Odiv__eq__0__iff,axiom,
! [A: $tType] :
( ( euclid3128863361964157862miring @ A )
=> ! [A3: A,B3: A] :
( ( ( euclid7384307370059645450egment @ A @ A3 )
= ( euclid7384307370059645450egment @ A @ B3 ) )
=> ( ( ( divide_divide @ A @ A3 @ B3 )
= ( zero_zero @ A ) )
= ( ( ord_less @ nat @ ( euclid6346220572633701492n_size @ A @ A3 ) @ ( euclid6346220572633701492n_size @ A @ B3 ) )
| ( B3
= ( zero_zero @ A ) ) ) ) ) ) ).
% unique_euclidean_semiring_class.div_eq_0_iff
thf(fact_7792_linear__exists__left__inverse__on,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V4867850818363320053vector @ A )
& ( real_V4867850818363320053vector @ B ) )
=> ! [F2: A > B,V: set @ A] :
( ( real_Vector_linear @ A @ B @ F2 )
=> ( ( real_Vector_subspace @ A @ V )
=> ( ( inj_on @ A @ B @ F2 @ V )
=> ? [G7: B > A] :
( ( ord_less_eq @ ( set @ A ) @ ( image2 @ B @ A @ G7 @ ( top_top @ ( set @ B ) ) ) @ V )
& ( real_Vector_linear @ B @ A @ G7 )
& ! [X2: A] :
( ( member @ A @ X2 @ V )
=> ( ( G7 @ ( F2 @ X2 ) )
= X2 ) ) ) ) ) ) ) ).
% linear_exists_left_inverse_on
thf(fact_7793_div__bounded,axiom,
! [A: $tType] :
( ( euclid3128863361964157862miring @ A )
=> ! [B3: A,R2: A,Q3: A] :
( ( B3
!= ( zero_zero @ A ) )
=> ( ( ( euclid7384307370059645450egment @ A @ R2 )
= ( euclid7384307370059645450egment @ A @ B3 ) )
=> ( ( ord_less @ nat @ ( euclid6346220572633701492n_size @ A @ R2 ) @ ( euclid6346220572633701492n_size @ A @ B3 ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ ( times_times @ A @ Q3 @ B3 ) @ R2 ) @ B3 )
= Q3 ) ) ) ) ) ).
% div_bounded
thf(fact_7794_div__eqI,axiom,
! [A: $tType] :
( ( euclid3128863361964157862miring @ A )
=> ! [B3: A,R2: A,Q3: A,A3: A] :
( ( B3
!= ( zero_zero @ A ) )
=> ( ( ( euclid7384307370059645450egment @ A @ R2 )
= ( euclid7384307370059645450egment @ A @ B3 ) )
=> ( ( ord_less @ nat @ ( euclid6346220572633701492n_size @ A @ R2 ) @ ( euclid6346220572633701492n_size @ A @ B3 ) )
=> ( ( ( plus_plus @ A @ ( times_times @ A @ Q3 @ B3 ) @ R2 )
= A3 )
=> ( ( divide_divide @ A @ A3 @ B3 )
= Q3 ) ) ) ) ) ) ).
% div_eqI
thf(fact_7795_mod__eqI,axiom,
! [A: $tType] :
( ( euclid3128863361964157862miring @ A )
=> ! [B3: A,R2: A,Q3: A,A3: A] :
( ( B3
!= ( zero_zero @ A ) )
=> ( ( ( euclid7384307370059645450egment @ A @ R2 )
= ( euclid7384307370059645450egment @ A @ B3 ) )
=> ( ( ord_less @ nat @ ( euclid6346220572633701492n_size @ A @ R2 ) @ ( euclid6346220572633701492n_size @ A @ B3 ) )
=> ( ( ( plus_plus @ A @ ( times_times @ A @ Q3 @ B3 ) @ R2 )
= A3 )
=> ( ( modulo_modulo @ A @ A3 @ B3 )
= R2 ) ) ) ) ) ) ).
% mod_eqI
thf(fact_7796_less__eq__enat__def,axiom,
( ( ord_less_eq @ extended_enat )
= ( ^ [M: extended_enat] :
( extended_case_enat @ $o
@ ^ [N1: nat] :
( extended_case_enat @ $o
@ ^ [M12: nat] : ( ord_less_eq @ nat @ M12 @ N1 )
@ $false
@ M )
@ $true ) ) ) ).
% less_eq_enat_def
thf(fact_7797_sorted__wrt__iff__nth__Suc__transp,axiom,
! [A: $tType,P: A > A > $o,Xs: list @ A] :
( ( transp @ A @ P )
=> ( ( sorted_wrt @ A @ P @ Xs )
= ( ! [I3: nat] :
( ( ord_less @ nat @ ( suc @ I3 ) @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( P @ ( nth @ A @ Xs @ I3 ) @ ( nth @ A @ Xs @ ( suc @ I3 ) ) ) ) ) ) ) ).
% sorted_wrt_iff_nth_Suc_transp
thf(fact_7798_list_Orel__transp,axiom,
! [A: $tType,R: A > A > $o] :
( ( transp @ A @ R )
=> ( transp @ ( list @ A ) @ ( list_all2 @ A @ A @ R ) ) ) ).
% list.rel_transp
thf(fact_7799_transp__ge,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( transp @ A
@ ^ [X4: A,Y6: A] : ( ord_less_eq @ A @ Y6 @ X4 ) ) ) ).
% transp_ge
thf(fact_7800_transp__le,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( transp @ A @ ( ord_less_eq @ A ) ) ) ).
% transp_le
thf(fact_7801_transp__gr,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( transp @ A
@ ^ [X4: A,Y6: A] : ( ord_less @ A @ Y6 @ X4 ) ) ) ).
% transp_gr
thf(fact_7802_transp__less,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( transp @ A @ ( ord_less @ A ) ) ) ).
% transp_less
thf(fact_7803_sorted__wrt2,axiom,
! [A: $tType,P: A > A > $o,X: A,Y: A,Zs: list @ A] :
( ( transp @ A @ P )
=> ( ( sorted_wrt @ A @ P @ ( cons @ A @ X @ ( cons @ A @ Y @ Zs ) ) )
= ( ( P @ X @ Y )
& ( sorted_wrt @ A @ P @ ( cons @ A @ Y @ Zs ) ) ) ) ) ).
% sorted_wrt2
thf(fact_7804_less__enat__def,axiom,
( ( ord_less @ extended_enat )
= ( ^ [M: extended_enat,N: extended_enat] :
( extended_case_enat @ $o
@ ^ [M12: nat] : ( extended_case_enat @ $o @ ( ord_less @ nat @ M12 ) @ $true @ N )
@ $false
@ M ) ) ) ).
% less_enat_def
thf(fact_7805_bounded__bilinear__def,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B )
& ( real_V822414075346904944vector @ C ) )
=> ( ( real_V2442710119149674383linear @ A @ B @ C )
= ( ^ [Prod: A > B > C] :
( ! [A5: A,A27: A,B4: B] :
( ( Prod @ ( plus_plus @ A @ A5 @ A27 ) @ B4 )
= ( plus_plus @ C @ ( Prod @ A5 @ B4 ) @ ( Prod @ A27 @ B4 ) ) )
& ! [A5: A,B4: B,B18: B] :
( ( Prod @ A5 @ ( plus_plus @ B @ B4 @ B18 ) )
= ( plus_plus @ C @ ( Prod @ A5 @ B4 ) @ ( Prod @ A5 @ B18 ) ) )
& ! [R4: real,A5: A,B4: B] :
( ( Prod @ ( real_V8093663219630862766scaleR @ A @ R4 @ A5 ) @ B4 )
= ( real_V8093663219630862766scaleR @ C @ R4 @ ( Prod @ A5 @ B4 ) ) )
& ! [A5: A,R4: real,B4: B] :
( ( Prod @ A5 @ ( real_V8093663219630862766scaleR @ B @ R4 @ B4 ) )
= ( real_V8093663219630862766scaleR @ C @ R4 @ ( Prod @ A5 @ B4 ) ) )
& ? [K6: real] :
! [A5: A,B4: B] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ C @ ( Prod @ A5 @ B4 ) ) @ ( times_times @ real @ ( times_times @ real @ ( real_V7770717601297561774m_norm @ A @ A5 ) @ ( real_V7770717601297561774m_norm @ B @ B4 ) ) @ K6 ) ) ) ) ) ) ).
% bounded_bilinear_def
thf(fact_7806_bounded__bilinear_Ointro,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B )
& ( real_V822414075346904944vector @ C ) )
=> ! [Prod2: A > B > C] :
( ! [A6: A,A28: A,B5: B] :
( ( Prod2 @ ( plus_plus @ A @ A6 @ A28 ) @ B5 )
= ( plus_plus @ C @ ( Prod2 @ A6 @ B5 ) @ ( Prod2 @ A28 @ B5 ) ) )
=> ( ! [A6: A,B5: B,B19: B] :
( ( Prod2 @ A6 @ ( plus_plus @ B @ B5 @ B19 ) )
= ( plus_plus @ C @ ( Prod2 @ A6 @ B5 ) @ ( Prod2 @ A6 @ B19 ) ) )
=> ( ! [R3: real,A6: A,B5: B] :
( ( Prod2 @ ( real_V8093663219630862766scaleR @ A @ R3 @ A6 ) @ B5 )
= ( real_V8093663219630862766scaleR @ C @ R3 @ ( Prod2 @ A6 @ B5 ) ) )
=> ( ! [A6: A,R3: real,B5: B] :
( ( Prod2 @ A6 @ ( real_V8093663219630862766scaleR @ B @ R3 @ B5 ) )
= ( real_V8093663219630862766scaleR @ C @ R3 @ ( Prod2 @ A6 @ B5 ) ) )
=> ( ? [K8: real] :
! [A6: A,B5: B] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ C @ ( Prod2 @ A6 @ B5 ) ) @ ( times_times @ real @ ( times_times @ real @ ( real_V7770717601297561774m_norm @ A @ A6 ) @ ( real_V7770717601297561774m_norm @ B @ B5 ) ) @ K8 ) )
=> ( real_V2442710119149674383linear @ A @ B @ C @ Prod2 ) ) ) ) ) ) ) ).
% bounded_bilinear.intro
thf(fact_7807_bounded__bilinear_Oadd__right,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( ( real_V822414075346904944vector @ B )
& ( real_V822414075346904944vector @ C )
& ( real_V822414075346904944vector @ A ) )
=> ! [Prod2: A > B > C,A3: A,B3: B,B8: B] :
( ( real_V2442710119149674383linear @ A @ B @ C @ Prod2 )
=> ( ( Prod2 @ A3 @ ( plus_plus @ B @ B3 @ B8 ) )
= ( plus_plus @ C @ ( Prod2 @ A3 @ B3 ) @ ( Prod2 @ A3 @ B8 ) ) ) ) ) ).
% bounded_bilinear.add_right
thf(fact_7808_bounded__bilinear_Oadd__left,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( ( real_V822414075346904944vector @ B )
& ( real_V822414075346904944vector @ C )
& ( real_V822414075346904944vector @ A ) )
=> ! [Prod2: A > B > C,A3: A,A9: A,B3: B] :
( ( real_V2442710119149674383linear @ A @ B @ C @ Prod2 )
=> ( ( Prod2 @ ( plus_plus @ A @ A3 @ A9 ) @ B3 )
= ( plus_plus @ C @ ( Prod2 @ A3 @ B3 ) @ ( Prod2 @ A9 @ B3 ) ) ) ) ) ).
% bounded_bilinear.add_left
thf(fact_7809_bounded__bilinear_Ozero__right,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [Prod2: A > B > C,A3: A] :
( ( real_V2442710119149674383linear @ A @ B @ C @ Prod2 )
=> ( ( Prod2 @ A3 @ ( zero_zero @ B ) )
= ( zero_zero @ C ) ) ) ) ).
% bounded_bilinear.zero_right
thf(fact_7810_bounded__bilinear_Ozero__left,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [Prod2: A > B > C,B3: B] :
( ( real_V2442710119149674383linear @ A @ B @ C @ Prod2 )
=> ( ( Prod2 @ ( zero_zero @ A ) @ B3 )
= ( zero_zero @ C ) ) ) ) ).
% bounded_bilinear.zero_left
thf(fact_7811_bounded__bilinear_Oprod__diff__prod,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( ( real_V822414075346904944vector @ B )
& ( real_V822414075346904944vector @ C )
& ( real_V822414075346904944vector @ A ) )
=> ! [Prod2: A > B > C,X: A,Y: B,A3: A,B3: B] :
( ( real_V2442710119149674383linear @ A @ B @ C @ Prod2 )
=> ( ( minus_minus @ C @ ( Prod2 @ X @ Y ) @ ( Prod2 @ A3 @ B3 ) )
= ( plus_plus @ C @ ( plus_plus @ C @ ( Prod2 @ ( minus_minus @ A @ X @ A3 ) @ ( minus_minus @ B @ Y @ B3 ) ) @ ( Prod2 @ ( minus_minus @ A @ X @ A3 ) @ B3 ) ) @ ( Prod2 @ A3 @ ( minus_minus @ B @ Y @ B3 ) ) ) ) ) ) ).
% bounded_bilinear.prod_diff_prod
thf(fact_7812_bounded__bilinear_Otendsto__zero,axiom,
! [C: $tType,B: $tType,A: $tType,D: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B )
& ( real_V822414075346904944vector @ C ) )
=> ! [Prod2: A > B > C,F2: D > A,F5: filter @ D,G: D > B] :
( ( real_V2442710119149674383linear @ A @ B @ C @ Prod2 )
=> ( ( filterlim @ D @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ F5 )
=> ( ( filterlim @ D @ B @ G @ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) ) @ F5 )
=> ( filterlim @ D @ C
@ ^ [X4: D] : ( Prod2 @ ( F2 @ X4 ) @ ( G @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ C @ ( zero_zero @ C ) )
@ F5 ) ) ) ) ) ).
% bounded_bilinear.tendsto_zero
thf(fact_7813_bounded__bilinear_Otendsto__left__zero,axiom,
! [C: $tType,B: $tType,A: $tType,D: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B )
& ( real_V822414075346904944vector @ C ) )
=> ! [Prod2: A > B > C,F2: D > A,F5: filter @ D,C2: B] :
( ( real_V2442710119149674383linear @ A @ B @ C @ Prod2 )
=> ( ( filterlim @ D @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ F5 )
=> ( filterlim @ D @ C
@ ^ [X4: D] : ( Prod2 @ ( F2 @ X4 ) @ C2 )
@ ( topolo7230453075368039082e_nhds @ C @ ( zero_zero @ C ) )
@ F5 ) ) ) ) ).
% bounded_bilinear.tendsto_left_zero
thf(fact_7814_bounded__bilinear_Otendsto__right__zero,axiom,
! [C: $tType,B: $tType,A: $tType,D: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B )
& ( real_V822414075346904944vector @ C ) )
=> ! [Prod2: A > B > C,F2: D > B,F5: filter @ D,C2: A] :
( ( real_V2442710119149674383linear @ A @ B @ C @ Prod2 )
=> ( ( filterlim @ D @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) ) @ F5 )
=> ( filterlim @ D @ C
@ ^ [X4: D] : ( Prod2 @ C2 @ ( F2 @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ C @ ( zero_zero @ C ) )
@ F5 ) ) ) ) ).
% bounded_bilinear.tendsto_right_zero
thf(fact_7815_bounded__bilinear_OFDERIV,axiom,
! [A: $tType,C: $tType,B: $tType,D: $tType] :
( ( ( real_V822414075346904944vector @ D )
& ( real_V822414075346904944vector @ B )
& ( real_V822414075346904944vector @ C )
& ( real_V822414075346904944vector @ A ) )
=> ! [Prod2: A > B > C,F2: D > A,F8: D > A,X: D,S: set @ D,G: D > B,G6: D > B] :
( ( real_V2442710119149674383linear @ A @ B @ C @ Prod2 )
=> ( ( has_derivative @ D @ A @ F2 @ F8 @ ( topolo174197925503356063within @ D @ X @ S ) )
=> ( ( has_derivative @ D @ B @ G @ G6 @ ( topolo174197925503356063within @ D @ X @ S ) )
=> ( has_derivative @ D @ C
@ ^ [X4: D] : ( Prod2 @ ( F2 @ X4 ) @ ( G @ X4 ) )
@ ^ [H: D] : ( plus_plus @ C @ ( Prod2 @ ( F2 @ X ) @ ( G6 @ H ) ) @ ( Prod2 @ ( F8 @ H ) @ ( G @ X ) ) )
@ ( topolo174197925503356063within @ D @ X @ S ) ) ) ) ) ) ).
% bounded_bilinear.FDERIV
thf(fact_7816_prod__decode__triangle__add,axiom,
! [K: nat,M2: nat] :
( ( nat_prod_decode @ ( plus_plus @ nat @ ( nat_triangle @ K ) @ M2 ) )
= ( nat_prod_decode_aux @ K @ M2 ) ) ).
% prod_decode_triangle_add
thf(fact_7817_Max_Osemilattice__order__set__axioms,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( lattic4895041142388067077er_set @ A @ ( ord_max @ A )
@ ^ [X4: A,Y6: A] : ( ord_less_eq @ A @ Y6 @ X4 )
@ ^ [X4: A,Y6: A] : ( ord_less @ A @ Y6 @ X4 ) ) ) ).
% Max.semilattice_order_set_axioms
thf(fact_7818_prod__decode__def,axiom,
( nat_prod_decode
= ( nat_prod_decode_aux @ ( zero_zero @ nat ) ) ) ).
% prod_decode_def
thf(fact_7819_Inf__fin_Osemilattice__order__set__axioms,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( lattic4895041142388067077er_set @ A @ ( inf_inf @ A ) @ ( ord_less_eq @ A ) @ ( ord_less @ A ) ) ) ).
% Inf_fin.semilattice_order_set_axioms
thf(fact_7820_Min_Osemilattice__order__set__axioms,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( lattic4895041142388067077er_set @ A @ ( ord_min @ A ) @ ( ord_less_eq @ A ) @ ( ord_less @ A ) ) ) ).
% Min.semilattice_order_set_axioms
thf(fact_7821_Sup__fin_Osemilattice__order__set__axioms,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( lattic4895041142388067077er_set @ A @ ( sup_sup @ A )
@ ^ [X4: A,Y6: A] : ( ord_less_eq @ A @ Y6 @ X4 )
@ ^ [X4: A,Y6: A] : ( ord_less @ A @ Y6 @ X4 ) ) ) ).
% Sup_fin.semilattice_order_set_axioms
thf(fact_7822_list__decode_Opinduct,axiom,
! [A0: nat,P: nat > $o] :
( ( accp @ nat @ nat_list_decode_rel @ A0 )
=> ( ( ( accp @ nat @ nat_list_decode_rel @ ( zero_zero @ nat ) )
=> ( P @ ( zero_zero @ nat ) ) )
=> ( ! [N3: nat] :
( ( accp @ nat @ nat_list_decode_rel @ ( suc @ N3 ) )
=> ( ! [X2: nat,Y4: nat] :
( ( ( product_Pair @ nat @ nat @ X2 @ Y4 )
= ( nat_prod_decode @ N3 ) )
=> ( P @ Y4 ) )
=> ( P @ ( suc @ N3 ) ) ) )
=> ( P @ A0 ) ) ) ) ).
% list_decode.pinduct
thf(fact_7823_list__decode_Oelims,axiom,
! [X: nat,Y: list @ nat] :
( ( ( nat_list_decode @ X )
= Y )
=> ( ( ( X
= ( zero_zero @ nat ) )
=> ( Y
!= ( nil @ nat ) ) )
=> ~ ! [N3: nat] :
( ( X
= ( suc @ N3 ) )
=> ( Y
!= ( product_case_prod @ nat @ nat @ ( list @ nat )
@ ^ [X4: nat,Y6: nat] : ( cons @ nat @ X4 @ ( nat_list_decode @ Y6 ) )
@ ( nat_prod_decode @ N3 ) ) ) ) ) ) ).
% list_decode.elims
thf(fact_7824_list__decode_Opsimps_I1_J,axiom,
( ( accp @ nat @ nat_list_decode_rel @ ( zero_zero @ nat ) )
=> ( ( nat_list_decode @ ( zero_zero @ nat ) )
= ( nil @ nat ) ) ) ).
% list_decode.psimps(1)
thf(fact_7825_list__decode_Osimps_I1_J,axiom,
( ( nat_list_decode @ ( zero_zero @ nat ) )
= ( nil @ nat ) ) ).
% list_decode.simps(1)
thf(fact_7826_list__decode_Opsimps_I2_J,axiom,
! [N2: nat] :
( ( accp @ nat @ nat_list_decode_rel @ ( suc @ N2 ) )
=> ( ( nat_list_decode @ ( suc @ N2 ) )
= ( product_case_prod @ nat @ nat @ ( list @ nat )
@ ^ [X4: nat,Y6: nat] : ( cons @ nat @ X4 @ ( nat_list_decode @ Y6 ) )
@ ( nat_prod_decode @ N2 ) ) ) ) ).
% list_decode.psimps(2)
thf(fact_7827_list__decode_Osimps_I2_J,axiom,
! [N2: nat] :
( ( nat_list_decode @ ( suc @ N2 ) )
= ( product_case_prod @ nat @ nat @ ( list @ nat )
@ ^ [X4: nat,Y6: nat] : ( cons @ nat @ X4 @ ( nat_list_decode @ Y6 ) )
@ ( nat_prod_decode @ N2 ) ) ) ).
% list_decode.simps(2)
thf(fact_7828_list__decode_Opelims,axiom,
! [X: nat,Y: list @ nat] :
( ( ( nat_list_decode @ X )
= Y )
=> ( ( accp @ nat @ nat_list_decode_rel @ X )
=> ( ( ( X
= ( zero_zero @ nat ) )
=> ( ( Y
= ( nil @ nat ) )
=> ~ ( accp @ nat @ nat_list_decode_rel @ ( zero_zero @ nat ) ) ) )
=> ~ ! [N3: nat] :
( ( X
= ( suc @ N3 ) )
=> ( ( Y
= ( product_case_prod @ nat @ nat @ ( list @ nat )
@ ^ [X4: nat,Y6: nat] : ( cons @ nat @ X4 @ ( nat_list_decode @ Y6 ) )
@ ( nat_prod_decode @ N3 ) ) )
=> ~ ( accp @ nat @ nat_list_decode_rel @ ( suc @ N3 ) ) ) ) ) ) ) ).
% list_decode.pelims
thf(fact_7829_arg__min__list_Opelims,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ B )
=> ! [X: A > B,Xa2: list @ A,Y: A] :
( ( ( arg_min_list @ A @ B @ X @ Xa2 )
= Y )
=> ( ( accp @ ( product_prod @ ( A > B ) @ ( list @ A ) ) @ ( arg_min_list_rel @ A @ B ) @ ( product_Pair @ ( A > B ) @ ( list @ A ) @ X @ Xa2 ) )
=> ( ! [X5: A] :
( ( Xa2
= ( cons @ A @ X5 @ ( nil @ A ) ) )
=> ( ( Y = X5 )
=> ~ ( accp @ ( product_prod @ ( A > B ) @ ( list @ A ) ) @ ( arg_min_list_rel @ A @ B ) @ ( product_Pair @ ( A > B ) @ ( list @ A ) @ X @ ( cons @ A @ X5 @ ( nil @ A ) ) ) ) ) )
=> ( ! [X5: A,Y3: A,Zs2: list @ A] :
( ( Xa2
= ( cons @ A @ X5 @ ( cons @ A @ Y3 @ Zs2 ) ) )
=> ( ( Y
= ( if @ A @ ( ord_less_eq @ B @ ( X @ X5 ) @ ( X @ ( arg_min_list @ A @ B @ X @ ( cons @ A @ Y3 @ Zs2 ) ) ) ) @ X5 @ ( arg_min_list @ A @ B @ X @ ( cons @ A @ Y3 @ Zs2 ) ) ) )
=> ~ ( accp @ ( product_prod @ ( A > B ) @ ( list @ A ) ) @ ( arg_min_list_rel @ A @ B ) @ ( product_Pair @ ( A > B ) @ ( list @ A ) @ X @ ( cons @ A @ X5 @ ( cons @ A @ Y3 @ Zs2 ) ) ) ) ) )
=> ~ ( ( Xa2
= ( nil @ A ) )
=> ( ( Y
= ( undefined @ A ) )
=> ~ ( accp @ ( product_prod @ ( A > B ) @ ( list @ A ) ) @ ( arg_min_list_rel @ A @ B ) @ ( product_Pair @ ( A > B ) @ ( list @ A ) @ X @ ( nil @ A ) ) ) ) ) ) ) ) ) ) ).
% arg_min_list.pelims
thf(fact_7830_char__of__take__bit__eq,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [N2: nat,M2: A] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) @ N2 )
=> ( ( unique5772411509450598832har_of @ A @ ( bit_se2584673776208193580ke_bit @ A @ N2 @ M2 ) )
= ( unique5772411509450598832har_of @ A @ M2 ) ) ) ) ).
% char_of_take_bit_eq
thf(fact_7831_UNIV__char__of__nat,axiom,
( ( top_top @ ( set @ char ) )
= ( image2 @ nat @ char @ ( unique5772411509450598832har_of @ nat ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ).
% UNIV_char_of_nat
thf(fact_7832_inj__on__char__of__nat,axiom,
inj_on @ nat @ char @ ( unique5772411509450598832har_of @ nat ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% inj_on_char_of_nat
thf(fact_7833_sorted__wrt_Opelims_I2_J,axiom,
! [A: $tType,X: A > A > $o,Xa2: list @ A] :
( ( sorted_wrt @ A @ X @ Xa2 )
=> ( ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( sorted_wrt_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X @ Xa2 ) )
=> ( ( ( Xa2
= ( nil @ A ) )
=> ~ ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( sorted_wrt_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X @ ( nil @ A ) ) ) )
=> ~ ! [X5: A,Ys4: list @ A] :
( ( Xa2
= ( cons @ A @ X5 @ Ys4 ) )
=> ( ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( sorted_wrt_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X @ ( cons @ A @ X5 @ Ys4 ) ) )
=> ~ ( ! [Xa: A] :
( ( member @ A @ Xa @ ( set2 @ A @ Ys4 ) )
=> ( X @ X5 @ Xa ) )
& ( sorted_wrt @ A @ X @ Ys4 ) ) ) ) ) ) ) ).
% sorted_wrt.pelims(2)
thf(fact_7834_sorted__wrt_Opelims_I1_J,axiom,
! [A: $tType,X: A > A > $o,Xa2: list @ A,Y: $o] :
( ( ( sorted_wrt @ A @ X @ Xa2 )
= Y )
=> ( ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( sorted_wrt_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X @ Xa2 ) )
=> ( ( ( Xa2
= ( nil @ A ) )
=> ( Y
=> ~ ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( sorted_wrt_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X @ ( nil @ A ) ) ) ) )
=> ~ ! [X5: A,Ys4: list @ A] :
( ( Xa2
= ( cons @ A @ X5 @ Ys4 ) )
=> ( ( Y
= ( ! [Y6: A] :
( ( member @ A @ Y6 @ ( set2 @ A @ Ys4 ) )
=> ( X @ X5 @ Y6 ) )
& ( sorted_wrt @ A @ X @ Ys4 ) ) )
=> ~ ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( sorted_wrt_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X @ ( cons @ A @ X5 @ Ys4 ) ) ) ) ) ) ) ) ).
% sorted_wrt.pelims(1)
thf(fact_7835_sorted__wrt_Opelims_I3_J,axiom,
! [A: $tType,X: A > A > $o,Xa2: list @ A] :
( ~ ( sorted_wrt @ A @ X @ Xa2 )
=> ( ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( sorted_wrt_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X @ Xa2 ) )
=> ~ ! [X5: A,Ys4: list @ A] :
( ( Xa2
= ( cons @ A @ X5 @ Ys4 ) )
=> ( ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( sorted_wrt_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X @ ( cons @ A @ X5 @ Ys4 ) ) )
=> ( ! [Xa3: A] :
( ( member @ A @ Xa3 @ ( set2 @ A @ Ys4 ) )
=> ( X @ X5 @ Xa3 ) )
& ( sorted_wrt @ A @ X @ Ys4 ) ) ) ) ) ) ).
% sorted_wrt.pelims(3)
thf(fact_7836_range__nat__of__char,axiom,
( ( image2 @ char @ nat @ ( comm_s6883823935334413003f_char @ nat ) @ ( top_top @ ( set @ char ) ) )
= ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ).
% range_nat_of_char
thf(fact_7837_one__int_Otransfer,axiom,
pcr_int @ ( product_Pair @ nat @ nat @ ( one_one @ nat ) @ ( zero_zero @ nat ) ) @ ( one_one @ int ) ).
% one_int.transfer
thf(fact_7838_zero__int_Otransfer,axiom,
pcr_int @ ( product_Pair @ nat @ nat @ ( zero_zero @ nat ) @ ( zero_zero @ nat ) ) @ ( zero_zero @ int ) ).
% zero_int.transfer
thf(fact_7839_nat__of__char__less__256,axiom,
! [C2: char] : ( ord_less @ nat @ ( comm_s6883823935334413003f_char @ nat @ C2 ) @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% nat_of_char_less_256
thf(fact_7840_tendsto__add__Pair,axiom,
! [A: $tType] :
( ( topolo6943815403480290642id_add @ A )
=> ! [A3: A,B3: A] :
( filterlim @ ( product_prod @ A @ A ) @ A
@ ^ [X4: product_prod @ A @ A] : ( plus_plus @ A @ ( product_fst @ A @ A @ X4 ) @ ( product_snd @ A @ A @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( plus_plus @ A @ A3 @ B3 ) )
@ ( prod_filter @ A @ A @ ( topolo7230453075368039082e_nhds @ A @ A3 ) @ ( topolo7230453075368039082e_nhds @ A @ B3 ) ) ) ) ).
% tendsto_add_Pair
thf(fact_7841_Rats__abs__nat__div__natE,axiom,
! [X: real] :
( ( member @ real @ X @ ( field_char_0_Rats @ real ) )
=> ~ ! [M3: nat,N3: nat] :
( ( N3
!= ( zero_zero @ nat ) )
=> ( ( ( abs_abs @ real @ X )
= ( divide_divide @ real @ ( semiring_1_of_nat @ real @ M3 ) @ ( semiring_1_of_nat @ real @ N3 ) ) )
=> ~ ( algebr8660921524188924756oprime @ nat @ M3 @ N3 ) ) ) ) ).
% Rats_abs_nat_div_natE
thf(fact_7842_coprime__self,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A] :
( ( algebr8660921524188924756oprime @ A @ A3 @ A3 )
= ( dvd_dvd @ A @ A3 @ ( one_one @ A ) ) ) ) ).
% coprime_self
thf(fact_7843_coprime__mod__left__iff,axiom,
! [A: $tType] :
( ( euclid3725896446679973847miring @ A )
=> ! [B3: A,A3: A] :
( ( B3
!= ( zero_zero @ A ) )
=> ( ( algebr8660921524188924756oprime @ A @ ( modulo_modulo @ A @ A3 @ B3 ) @ B3 )
= ( algebr8660921524188924756oprime @ A @ A3 @ B3 ) ) ) ) ).
% coprime_mod_left_iff
thf(fact_7844_coprime__mod__right__iff,axiom,
! [A: $tType] :
( ( euclid3725896446679973847miring @ A )
=> ! [A3: A,B3: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( algebr8660921524188924756oprime @ A @ A3 @ ( modulo_modulo @ A @ B3 @ A3 ) )
= ( algebr8660921524188924756oprime @ A @ A3 @ B3 ) ) ) ) ).
% coprime_mod_right_iff
thf(fact_7845_coprime__power__left__iff,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,N2: nat,B3: A] :
( ( algebr8660921524188924756oprime @ A @ ( power_power @ A @ A3 @ N2 ) @ B3 )
= ( ( algebr8660921524188924756oprime @ A @ A3 @ B3 )
| ( N2
= ( zero_zero @ nat ) ) ) ) ) ).
% coprime_power_left_iff
thf(fact_7846_coprime__power__right__iff,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B3: A,N2: nat] :
( ( algebr8660921524188924756oprime @ A @ A3 @ ( power_power @ A @ B3 @ N2 ) )
= ( ( algebr8660921524188924756oprime @ A @ A3 @ B3 )
| ( N2
= ( zero_zero @ nat ) ) ) ) ) ).
% coprime_power_right_iff
thf(fact_7847_coprime__0__left__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A] :
( ( algebr8660921524188924756oprime @ A @ ( zero_zero @ A ) @ A3 )
= ( dvd_dvd @ A @ A3 @ ( one_one @ A ) ) ) ) ).
% coprime_0_left_iff
thf(fact_7848_coprime__0__right__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A] :
( ( algebr8660921524188924756oprime @ A @ A3 @ ( zero_zero @ A ) )
= ( dvd_dvd @ A @ A3 @ ( one_one @ A ) ) ) ) ).
% coprime_0_right_iff
thf(fact_7849_coprime__mult__self__right__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,C2: A,B3: A] :
( ( algebr8660921524188924756oprime @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B3 @ C2 ) )
= ( ( dvd_dvd @ A @ C2 @ ( one_one @ A ) )
& ( algebr8660921524188924756oprime @ A @ A3 @ B3 ) ) ) ) ).
% coprime_mult_self_right_iff
thf(fact_7850_coprime__mult__self__left__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [C2: A,A3: A,B3: A] :
( ( algebr8660921524188924756oprime @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B3 ) )
= ( ( dvd_dvd @ A @ C2 @ ( one_one @ A ) )
& ( algebr8660921524188924756oprime @ A @ A3 @ B3 ) ) ) ) ).
% coprime_mult_self_left_iff
thf(fact_7851_prod__list__coprime__right,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [Xs: list @ A,A3: A] :
( ! [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Xs ) )
=> ( algebr8660921524188924756oprime @ A @ A3 @ X5 ) )
=> ( algebr8660921524188924756oprime @ A @ A3 @ ( groups5270119922927024881d_list @ A @ Xs ) ) ) ) ).
% prod_list_coprime_right
thf(fact_7852_prod__list__coprime__left,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [Xs: list @ A,A3: A] :
( ! [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Xs ) )
=> ( algebr8660921524188924756oprime @ A @ X5 @ A3 ) )
=> ( algebr8660921524188924756oprime @ A @ ( groups5270119922927024881d_list @ A @ Xs ) @ A3 ) ) ) ).
% prod_list_coprime_left
thf(fact_7853_prod__filter__eq__bot,axiom,
! [A: $tType,B: $tType,A4: filter @ A,B2: filter @ B] :
( ( ( prod_filter @ A @ B @ A4 @ B2 )
= ( bot_bot @ ( filter @ ( product_prod @ A @ B ) ) ) )
= ( ( A4
= ( bot_bot @ ( filter @ A ) ) )
| ( B2
= ( bot_bot @ ( filter @ B ) ) ) ) ) ).
% prod_filter_eq_bot
thf(fact_7854_is__unit__right__imp__coprime,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B3: A,A3: A] :
( ( dvd_dvd @ A @ B3 @ ( one_one @ A ) )
=> ( algebr8660921524188924756oprime @ A @ A3 @ B3 ) ) ) ).
% is_unit_right_imp_coprime
thf(fact_7855_is__unit__left__imp__coprime,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B3: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( algebr8660921524188924756oprime @ A @ A3 @ B3 ) ) ) ).
% is_unit_left_imp_coprime
thf(fact_7856_coprime__common__divisor,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B3: A,C2: A] :
( ( algebr8660921524188924756oprime @ A @ A3 @ B3 )
=> ( ( dvd_dvd @ A @ C2 @ A3 )
=> ( ( dvd_dvd @ A @ C2 @ B3 )
=> ( dvd_dvd @ A @ C2 @ ( one_one @ A ) ) ) ) ) ) ).
% coprime_common_divisor
thf(fact_7857_coprime__absorb__right,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [Y: A,X: A] :
( ( dvd_dvd @ A @ Y @ X )
=> ( ( algebr8660921524188924756oprime @ A @ X @ Y )
= ( dvd_dvd @ A @ Y @ ( one_one @ A ) ) ) ) ) ).
% coprime_absorb_right
thf(fact_7858_coprime__imp__coprime,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [C2: A,D2: A,A3: A,B3: A] :
( ( algebr8660921524188924756oprime @ A @ C2 @ D2 )
=> ( ! [E: A] :
( ~ ( dvd_dvd @ A @ E @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ E @ A3 )
=> ( ( dvd_dvd @ A @ E @ B3 )
=> ( dvd_dvd @ A @ E @ C2 ) ) ) )
=> ( ! [E: A] :
( ~ ( dvd_dvd @ A @ E @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ E @ A3 )
=> ( ( dvd_dvd @ A @ E @ B3 )
=> ( dvd_dvd @ A @ E @ D2 ) ) ) )
=> ( algebr8660921524188924756oprime @ A @ A3 @ B3 ) ) ) ) ) ).
% coprime_imp_coprime
thf(fact_7859_coprime__absorb__left,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [X: A,Y: A] :
( ( dvd_dvd @ A @ X @ Y )
=> ( ( algebr8660921524188924756oprime @ A @ X @ Y )
= ( dvd_dvd @ A @ X @ ( one_one @ A ) ) ) ) ) ).
% coprime_absorb_left
thf(fact_7860_not__coprimeI,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [C2: A,A3: A,B3: A] :
( ( dvd_dvd @ A @ C2 @ A3 )
=> ( ( dvd_dvd @ A @ C2 @ B3 )
=> ( ~ ( dvd_dvd @ A @ C2 @ ( one_one @ A ) )
=> ~ ( algebr8660921524188924756oprime @ A @ A3 @ B3 ) ) ) ) ) ).
% not_coprimeI
thf(fact_7861_not__coprimeE,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B3: A] :
( ~ ( algebr8660921524188924756oprime @ A @ A3 @ B3 )
=> ~ ! [C3: A] :
( ( dvd_dvd @ A @ C3 @ A3 )
=> ( ( dvd_dvd @ A @ C3 @ B3 )
=> ( dvd_dvd @ A @ C3 @ ( one_one @ A ) ) ) ) ) ) ).
% not_coprimeE
thf(fact_7862_coprime__def,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ( ( algebr8660921524188924756oprime @ A )
= ( ^ [A5: A,B4: A] :
! [C4: A] :
( ( dvd_dvd @ A @ C4 @ A5 )
=> ( ( dvd_dvd @ A @ C4 @ B4 )
=> ( dvd_dvd @ A @ C4 @ ( one_one @ A ) ) ) ) ) ) ) ).
% coprime_def
thf(fact_7863_coprimeI,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B3: A] :
( ! [C3: A] :
( ( dvd_dvd @ A @ C3 @ A3 )
=> ( ( dvd_dvd @ A @ C3 @ B3 )
=> ( dvd_dvd @ A @ C3 @ ( one_one @ A ) ) ) )
=> ( algebr8660921524188924756oprime @ A @ A3 @ B3 ) ) ) ).
% coprimeI
thf(fact_7864_coprime__1__right,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A] : ( algebr8660921524188924756oprime @ A @ A3 @ ( one_one @ A ) ) ) ).
% coprime_1_right
thf(fact_7865_coprime__1__left,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A] : ( algebr8660921524188924756oprime @ A @ ( one_one @ A ) @ A3 ) ) ).
% coprime_1_left
thf(fact_7866_coprime__divisors,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,C2: A,B3: A,D2: A] :
( ( dvd_dvd @ A @ A3 @ C2 )
=> ( ( dvd_dvd @ A @ B3 @ D2 )
=> ( ( algebr8660921524188924756oprime @ A @ C2 @ D2 )
=> ( algebr8660921524188924756oprime @ A @ A3 @ B3 ) ) ) ) ) ).
% coprime_divisors
thf(fact_7867_coprime__Suc__left__nat,axiom,
! [N2: nat] : ( algebr8660921524188924756oprime @ nat @ ( suc @ N2 ) @ N2 ) ).
% coprime_Suc_left_nat
thf(fact_7868_coprime__Suc__right__nat,axiom,
! [N2: nat] : ( algebr8660921524188924756oprime @ nat @ N2 @ ( suc @ N2 ) ) ).
% coprime_Suc_right_nat
thf(fact_7869_coprime__commute,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ( ( algebr8660921524188924756oprime @ A )
= ( ^ [B4: A,A5: A] : ( algebr8660921524188924756oprime @ A @ A5 @ B4 ) ) ) ) ).
% coprime_commute
thf(fact_7870_coprime__Suc__0__left,axiom,
! [N2: nat] : ( algebr8660921524188924756oprime @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N2 ) ).
% coprime_Suc_0_left
thf(fact_7871_coprime__Suc__0__right,axiom,
! [N2: nat] : ( algebr8660921524188924756oprime @ nat @ N2 @ ( suc @ ( zero_zero @ nat ) ) ) ).
% coprime_Suc_0_right
thf(fact_7872_coprime__add__one__left,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A] : ( algebr8660921524188924756oprime @ A @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) @ A3 ) ) ).
% coprime_add_one_left
thf(fact_7873_coprime__add__one__right,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A] : ( algebr8660921524188924756oprime @ A @ A3 @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) ) ) ).
% coprime_add_one_right
thf(fact_7874_prod__filter__mono__iff,axiom,
! [A: $tType,B: $tType,A4: filter @ A,B2: filter @ B,C5: filter @ A,D6: filter @ B] :
( ( A4
!= ( bot_bot @ ( filter @ A ) ) )
=> ( ( B2
!= ( bot_bot @ ( filter @ B ) ) )
=> ( ( ord_less_eq @ ( filter @ ( product_prod @ A @ B ) ) @ ( prod_filter @ A @ B @ A4 @ B2 ) @ ( prod_filter @ A @ B @ C5 @ D6 ) )
= ( ( ord_less_eq @ ( filter @ A ) @ A4 @ C5 )
& ( ord_less_eq @ ( filter @ B ) @ B2 @ D6 ) ) ) ) ) ).
% prod_filter_mono_iff
thf(fact_7875_eventually__prod__sequentially,axiom,
! [P: ( product_prod @ nat @ nat ) > $o] :
( ( eventually @ ( product_prod @ nat @ nat ) @ P @ ( prod_filter @ nat @ nat @ ( at_top @ nat ) @ ( at_top @ nat ) ) )
= ( ? [N5: nat] :
! [M: nat] :
( ( ord_less_eq @ nat @ N5 @ M )
=> ! [N: nat] :
( ( ord_less_eq @ nat @ N5 @ N )
=> ( P @ ( product_Pair @ nat @ nat @ N @ M ) ) ) ) ) ) ).
% eventually_prod_sequentially
thf(fact_7876_coprime__diff__one__left__nat,axiom,
! [N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( algebr8660921524188924756oprime @ nat @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) @ N2 ) ) ).
% coprime_diff_one_left_nat
thf(fact_7877_coprime__diff__one__right__nat,axiom,
! [N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( algebr8660921524188924756oprime @ nat @ N2 @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) ) ) ).
% coprime_diff_one_right_nat
thf(fact_7878_eventually__prod2,axiom,
! [A: $tType,B: $tType,A4: filter @ A,P: B > $o,B2: filter @ B] :
( ( A4
!= ( bot_bot @ ( filter @ A ) ) )
=> ( ( eventually @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [X4: A] : P )
@ ( prod_filter @ A @ B @ A4 @ B2 ) )
= ( eventually @ B @ P @ B2 ) ) ) ).
% eventually_prod2
thf(fact_7879_eventually__prod1,axiom,
! [A: $tType,B: $tType,B2: filter @ A,P: B > $o,A4: filter @ B] :
( ( B2
!= ( bot_bot @ ( filter @ A ) ) )
=> ( ( eventually @ ( product_prod @ B @ A )
@ ( product_case_prod @ B @ A @ $o
@ ^ [X4: B,Y6: A] : ( P @ X4 ) )
@ ( prod_filter @ B @ A @ A4 @ B2 ) )
= ( eventually @ B @ P @ A4 ) ) ) ).
% eventually_prod1
thf(fact_7880_prod__filter__INF,axiom,
! [C: $tType,D: $tType,B: $tType,A: $tType,I6: set @ A,J4: set @ B,A4: A > ( filter @ C ),B2: B > ( filter @ D )] :
( ( I6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( J4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( prod_filter @ C @ D @ ( complete_Inf_Inf @ ( filter @ C ) @ ( image2 @ A @ ( filter @ C ) @ A4 @ I6 ) ) @ ( complete_Inf_Inf @ ( filter @ D ) @ ( image2 @ B @ ( filter @ D ) @ B2 @ J4 ) ) )
= ( complete_Inf_Inf @ ( filter @ ( product_prod @ C @ D ) )
@ ( image2 @ A @ ( filter @ ( product_prod @ C @ D ) )
@ ^ [I3: A] :
( complete_Inf_Inf @ ( filter @ ( product_prod @ C @ D ) )
@ ( image2 @ B @ ( filter @ ( product_prod @ C @ D ) )
@ ^ [J2: B] : ( prod_filter @ C @ D @ ( A4 @ I3 ) @ ( B2 @ J2 ) )
@ J4 ) )
@ I6 ) ) ) ) ) ).
% prod_filter_INF
thf(fact_7881_prod__filter__INF1,axiom,
! [B: $tType,C: $tType,A: $tType,I6: set @ A,A4: A > ( filter @ B ),B2: filter @ C] :
( ( I6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( prod_filter @ B @ C @ ( complete_Inf_Inf @ ( filter @ B ) @ ( image2 @ A @ ( filter @ B ) @ A4 @ I6 ) ) @ B2 )
= ( complete_Inf_Inf @ ( filter @ ( product_prod @ B @ C ) )
@ ( image2 @ A @ ( filter @ ( product_prod @ B @ C ) )
@ ^ [I3: A] : ( prod_filter @ B @ C @ ( A4 @ I3 ) @ B2 )
@ I6 ) ) ) ) ).
% prod_filter_INF1
thf(fact_7882_prod__filter__INF2,axiom,
! [B: $tType,C: $tType,A: $tType,J4: set @ A,A4: filter @ B,B2: A > ( filter @ C )] :
( ( J4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( prod_filter @ B @ C @ A4 @ ( complete_Inf_Inf @ ( filter @ C ) @ ( image2 @ A @ ( filter @ C ) @ B2 @ J4 ) ) )
= ( complete_Inf_Inf @ ( filter @ ( product_prod @ B @ C ) )
@ ( image2 @ A @ ( filter @ ( product_prod @ B @ C ) )
@ ^ [I3: A] : ( prod_filter @ B @ C @ A4 @ ( B2 @ I3 ) )
@ J4 ) ) ) ) ).
% prod_filter_INF2
thf(fact_7883_prod__filter__principal__singleton2,axiom,
! [B: $tType,A: $tType,F5: filter @ A,X: B] :
( ( prod_filter @ A @ B @ F5 @ ( principal @ B @ ( insert2 @ B @ X @ ( bot_bot @ ( set @ B ) ) ) ) )
= ( filtermap @ A @ ( product_prod @ A @ B )
@ ^ [A5: A] : ( product_Pair @ A @ B @ A5 @ X )
@ F5 ) ) ).
% prod_filter_principal_singleton2
thf(fact_7884_prod__filter__principal__singleton,axiom,
! [A: $tType,B: $tType,X: A,F5: filter @ B] :
( ( prod_filter @ A @ B @ ( principal @ A @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) @ F5 )
= ( filtermap @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X ) @ F5 ) ) ).
% prod_filter_principal_singleton
thf(fact_7885_filtermap__bot,axiom,
! [B: $tType,A: $tType,F2: B > A] :
( ( filtermap @ B @ A @ F2 @ ( bot_bot @ ( filter @ B ) ) )
= ( bot_bot @ ( filter @ A ) ) ) ).
% filtermap_bot
thf(fact_7886_filtermap__bot__iff,axiom,
! [A: $tType,B: $tType,F2: B > A,F5: filter @ B] :
( ( ( filtermap @ B @ A @ F2 @ F5 )
= ( bot_bot @ ( filter @ A ) ) )
= ( F5
= ( bot_bot @ ( filter @ B ) ) ) ) ).
% filtermap_bot_iff
thf(fact_7887_filtermap__nhds__times,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [C2: A,A3: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( filtermap @ A @ A @ ( times_times @ A @ C2 ) @ ( topolo7230453075368039082e_nhds @ A @ A3 ) )
= ( topolo7230453075368039082e_nhds @ A @ ( times_times @ A @ C2 @ A3 ) ) ) ) ) ).
% filtermap_nhds_times
thf(fact_7888_at__to__0,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [A3: A] :
( ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) )
= ( filtermap @ A @ A
@ ^ [X4: A] : ( plus_plus @ A @ X4 @ A3 )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% at_to_0
thf(fact_7889_filtermap__times__pos__at__right,axiom,
! [A: $tType] :
( ( ( linordered_field @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [C2: A,P5: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( filtermap @ A @ A @ ( times_times @ A @ C2 ) @ ( topolo174197925503356063within @ A @ P5 @ ( set_ord_greaterThan @ A @ P5 ) ) )
= ( topolo174197925503356063within @ A @ ( times_times @ A @ C2 @ P5 ) @ ( set_ord_greaterThan @ A @ ( times_times @ A @ C2 @ P5 ) ) ) ) ) ) ).
% filtermap_times_pos_at_right
thf(fact_7890_at__to__infinity,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ( ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) )
= ( filtermap @ A @ A @ ( inverse_inverse @ A ) @ ( at_infinity @ A ) ) ) ) ).
% at_to_infinity
thf(fact_7891_totally__bounded__def,axiom,
! [A: $tType] :
( ( topolo7287701948861334536_space @ A )
=> ( ( topolo6688025880775521714ounded @ A )
= ( ^ [S6: set @ A] :
! [E6: ( product_prod @ A @ A ) > $o] :
( ( eventually @ ( product_prod @ A @ A ) @ E6 @ ( topolo7806501430040627800ormity @ A ) )
=> ? [X6: set @ A] :
( ( finite_finite2 @ A @ X6 )
& ! [X4: A] :
( ( member @ A @ X4 @ S6 )
=> ? [Y6: A] :
( ( member @ A @ Y6 @ X6 )
& ( E6 @ ( product_Pair @ A @ A @ Y6 @ X4 ) ) ) ) ) ) ) ) ) ).
% totally_bounded_def
thf(fact_7892_Cauchy__uniform__iff,axiom,
! [A: $tType] :
( ( topolo7287701948861334536_space @ A )
=> ( ( topolo3814608138187158403Cauchy @ A )
= ( ^ [X6: nat > A] :
! [P3: ( product_prod @ A @ A ) > $o] :
( ( eventually @ ( product_prod @ A @ A ) @ P3 @ ( topolo7806501430040627800ormity @ A ) )
=> ? [N5: nat] :
! [N: nat] :
( ( ord_less_eq @ nat @ N5 @ N )
=> ! [M: nat] :
( ( ord_less_eq @ nat @ N5 @ M )
=> ( P3 @ ( product_Pair @ A @ A @ ( X6 @ N ) @ ( X6 @ M ) ) ) ) ) ) ) ) ) ).
% Cauchy_uniform_iff
thf(fact_7893_uniformity__bot,axiom,
! [A: $tType] :
( ( topolo7287701948861334536_space @ A )
=> ( ( topolo7806501430040627800ormity @ A )
!= ( bot_bot @ ( filter @ ( product_prod @ A @ A ) ) ) ) ) ).
% uniformity_bot
thf(fact_7894_filtermap__sequentually__ne__bot,axiom,
! [A: $tType,F2: nat > A] :
( ( filtermap @ nat @ A @ F2 @ ( at_top @ nat ) )
!= ( bot_bot @ ( filter @ A ) ) ) ).
% filtermap_sequentually_ne_bot
thf(fact_7895_of__int_Otransfer,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ int @ A @ A @ pcr_int
@ ^ [Y5: A,Z2: A] : Y5 = Z2
@ ( product_case_prod @ nat @ nat @ A
@ ^ [I3: nat,J2: nat] : ( minus_minus @ A @ ( semiring_1_of_nat @ A @ I3 ) @ ( semiring_1_of_nat @ A @ J2 ) ) )
@ ( ring_1_of_int @ A ) ) ) ).
% of_int.transfer
thf(fact_7896_finite__subset__Union__chain,axiom,
! [A: $tType,A4: set @ A,B11: set @ ( set @ A ),A20: set @ ( set @ A )] :
( ( finite_finite2 @ A @ A4 )
=> ( ( ord_less_eq @ ( set @ A ) @ A4 @ ( complete_Sup_Sup @ ( set @ A ) @ B11 ) )
=> ( ( B11
!= ( bot_bot @ ( set @ ( set @ A ) ) ) )
=> ( ( pred_chain @ ( set @ A ) @ A20 @ ( ord_less @ ( set @ A ) ) @ B11 )
=> ~ ! [B9: set @ A] :
( ( member @ ( set @ A ) @ B9 @ B11 )
=> ~ ( ord_less_eq @ ( set @ A ) @ A4 @ B9 ) ) ) ) ) ) ).
% finite_subset_Union_chain
thf(fact_7897_foldr__transfer,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,A4: A > C > $o,B2: B > D > $o] : ( bNF_rel_fun @ ( A > B > B ) @ ( C > D > D ) @ ( ( list @ A ) > B > B ) @ ( ( list @ C ) > D > D ) @ ( bNF_rel_fun @ A @ C @ ( B > B ) @ ( D > D ) @ A4 @ ( bNF_rel_fun @ B @ D @ B @ D @ B2 @ B2 ) ) @ ( bNF_rel_fun @ ( list @ A ) @ ( list @ C ) @ ( B > B ) @ ( D > D ) @ ( list_all2 @ A @ C @ A4 ) @ ( bNF_rel_fun @ B @ D @ B @ D @ B2 @ B2 ) ) @ ( foldr @ A @ B ) @ ( foldr @ C @ D ) ) ).
% foldr_transfer
thf(fact_7898_List_Ofold__transfer,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,A4: A > C > $o,B2: B > D > $o] : ( bNF_rel_fun @ ( A > B > B ) @ ( C > D > D ) @ ( ( list @ A ) > B > B ) @ ( ( list @ C ) > D > D ) @ ( bNF_rel_fun @ A @ C @ ( B > B ) @ ( D > D ) @ A4 @ ( bNF_rel_fun @ B @ D @ B @ D @ B2 @ B2 ) ) @ ( bNF_rel_fun @ ( list @ A ) @ ( list @ C ) @ ( B > B ) @ ( D > D ) @ ( list_all2 @ A @ C @ A4 ) @ ( bNF_rel_fun @ B @ D @ B @ D @ B2 @ B2 ) ) @ ( fold @ A @ B ) @ ( fold @ C @ D ) ) ).
% List.fold_transfer
thf(fact_7899_horner__sum__transfer,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType] :
( ( ( comm_semiring_0 @ B )
& ( comm_semiring_0 @ A ) )
=> ! [A4: A > B > $o,B2: C > D > $o] :
( ( A4 @ ( zero_zero @ A ) @ ( zero_zero @ B ) )
=> ( ( bNF_rel_fun @ A @ B @ ( A > A ) @ ( B > B ) @ A4 @ ( bNF_rel_fun @ A @ B @ A @ B @ A4 @ A4 ) @ ( plus_plus @ A ) @ ( plus_plus @ B ) )
=> ( ( bNF_rel_fun @ A @ B @ ( A > A ) @ ( B > B ) @ A4 @ ( bNF_rel_fun @ A @ B @ A @ B @ A4 @ A4 ) @ ( times_times @ A ) @ ( times_times @ B ) )
=> ( bNF_rel_fun @ ( C > A ) @ ( D > B ) @ ( A > ( list @ C ) > A ) @ ( B > ( list @ D ) > B ) @ ( bNF_rel_fun @ C @ D @ A @ B @ B2 @ A4 ) @ ( bNF_rel_fun @ A @ B @ ( ( list @ C ) > A ) @ ( ( list @ D ) > B ) @ A4 @ ( bNF_rel_fun @ ( list @ C ) @ ( list @ D ) @ A @ B @ ( list_all2 @ C @ D @ B2 ) @ A4 ) ) @ ( groups4207007520872428315er_sum @ C @ A ) @ ( groups4207007520872428315er_sum @ D @ B ) ) ) ) ) ) ).
% horner_sum_transfer
thf(fact_7900_sum__list__transfer,axiom,
! [A: $tType,B: $tType] :
( ( ( monoid_add @ B )
& ( monoid_add @ A ) )
=> ! [A4: A > B > $o] :
( ( A4 @ ( zero_zero @ A ) @ ( zero_zero @ B ) )
=> ( ( bNF_rel_fun @ A @ B @ ( A > A ) @ ( B > B ) @ A4 @ ( bNF_rel_fun @ A @ B @ A @ B @ A4 @ A4 ) @ ( plus_plus @ A ) @ ( plus_plus @ B ) )
=> ( bNF_rel_fun @ ( list @ A ) @ ( list @ B ) @ A @ B @ ( list_all2 @ A @ B @ A4 ) @ A4 @ ( groups8242544230860333062m_list @ A ) @ ( groups8242544230860333062m_list @ B ) ) ) ) ) ).
% sum_list_transfer
thf(fact_7901_list_Omap__transfer,axiom,
! [A: $tType,B: $tType,F: $tType,E4: $tType,Rb: A > E4 > $o,Sd: B > F > $o] : ( bNF_rel_fun @ ( A > B ) @ ( E4 > F ) @ ( ( list @ A ) > ( list @ B ) ) @ ( ( list @ E4 ) > ( list @ F ) ) @ ( bNF_rel_fun @ A @ E4 @ B @ F @ Rb @ Sd ) @ ( bNF_rel_fun @ ( list @ A ) @ ( list @ E4 ) @ ( list @ B ) @ ( list @ F ) @ ( list_all2 @ A @ E4 @ Rb ) @ ( list_all2 @ B @ F @ Sd ) ) @ ( map @ A @ B ) @ ( map @ E4 @ F ) ) ).
% list.map_transfer
thf(fact_7902_list_Orec__transfer,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,S2: C > D > $o,R: A > B > $o] : ( bNF_rel_fun @ C @ D @ ( ( A > ( list @ A ) > C > C ) > ( list @ A ) > C ) @ ( ( B > ( list @ B ) > D > D ) > ( list @ B ) > D ) @ S2 @ ( bNF_rel_fun @ ( A > ( list @ A ) > C > C ) @ ( B > ( list @ B ) > D > D ) @ ( ( list @ A ) > C ) @ ( ( list @ B ) > D ) @ ( bNF_rel_fun @ A @ B @ ( ( list @ A ) > C > C ) @ ( ( list @ B ) > D > D ) @ R @ ( bNF_rel_fun @ ( list @ A ) @ ( list @ B ) @ ( C > C ) @ ( D > D ) @ ( list_all2 @ A @ B @ R ) @ ( bNF_rel_fun @ C @ D @ C @ D @ S2 @ S2 ) ) ) @ ( bNF_rel_fun @ ( list @ A ) @ ( list @ B ) @ C @ D @ ( list_all2 @ A @ B @ R ) @ S2 ) ) @ ( rec_list @ C @ A ) @ ( rec_list @ D @ B ) ) ).
% list.rec_transfer
thf(fact_7903_subset_Ochain__total,axiom,
! [A: $tType,A4: set @ ( set @ A ),C5: set @ ( set @ A ),X: set @ A,Y: set @ A] :
( ( pred_chain @ ( set @ A ) @ A4 @ ( ord_less @ ( set @ A ) ) @ C5 )
=> ( ( member @ ( set @ A ) @ X @ C5 )
=> ( ( member @ ( set @ A ) @ Y @ C5 )
=> ( ( sup_sup @ ( ( set @ A ) > ( set @ A ) > $o ) @ ( ord_less @ ( set @ A ) )
@ ^ [Y5: set @ A,Z2: set @ A] : Y5 = Z2
@ X
@ Y )
| ( sup_sup @ ( ( set @ A ) > ( set @ A ) > $o ) @ ( ord_less @ ( set @ A ) )
@ ^ [Y5: set @ A,Z2: set @ A] : Y5 = Z2
@ Y
@ X ) ) ) ) ) ).
% subset.chain_total
thf(fact_7904_chains__alt__def,axiom,
! [A: $tType] :
( ( chains @ A )
= ( ^ [A7: set @ ( set @ A )] : ( collect @ ( set @ ( set @ A ) ) @ ( pred_chain @ ( set @ A ) @ A7 @ ( ord_less @ ( set @ A ) ) ) ) ) ) ).
% chains_alt_def
thf(fact_7905_transfer__rule__numeral,axiom,
! [A: $tType,B: $tType] :
( ( ( monoid_add @ B )
& ( semiring_numeral @ B )
& ( monoid_add @ A )
& ( semiring_numeral @ A ) )
=> ! [R: A > B > $o] :
( ( R @ ( zero_zero @ A ) @ ( zero_zero @ B ) )
=> ( ( R @ ( one_one @ A ) @ ( one_one @ B ) )
=> ( ( bNF_rel_fun @ A @ B @ ( A > A ) @ ( B > B ) @ R @ ( bNF_rel_fun @ A @ B @ A @ B @ R @ R ) @ ( plus_plus @ A ) @ ( plus_plus @ B ) )
=> ( bNF_rel_fun @ num @ num @ A @ B
@ ^ [Y5: num,Z2: num] : Y5 = Z2
@ R
@ ( numeral_numeral @ A )
@ ( numeral_numeral @ B ) ) ) ) ) ) ).
% transfer_rule_numeral
thf(fact_7906_transfer__rule__of__int,axiom,
! [A: $tType,B: $tType] :
( ( ( ring_1 @ B )
& ( ring_1 @ A ) )
=> ! [R: A > B > $o] :
( ( R @ ( zero_zero @ A ) @ ( zero_zero @ B ) )
=> ( ( R @ ( one_one @ A ) @ ( one_one @ B ) )
=> ( ( bNF_rel_fun @ A @ B @ ( A > A ) @ ( B > B ) @ R @ ( bNF_rel_fun @ A @ B @ A @ B @ R @ R ) @ ( plus_plus @ A ) @ ( plus_plus @ B ) )
=> ( ( bNF_rel_fun @ A @ B @ A @ B @ R @ R @ ( uminus_uminus @ A ) @ ( uminus_uminus @ B ) )
=> ( bNF_rel_fun @ int @ int @ A @ B
@ ^ [Y5: int,Z2: int] : Y5 = Z2
@ R
@ ( ring_1_of_int @ A )
@ ( ring_1_of_int @ B ) ) ) ) ) ) ) ).
% transfer_rule_of_int
thf(fact_7907_minus__int_Otransfer,axiom,
( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ int @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) ) @ ( int > int ) @ pcr_int @ ( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ int @ ( product_prod @ nat @ nat ) @ int @ pcr_int @ pcr_int )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
@ ^ [X4: nat,Y6: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V5: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X4 @ V5 ) @ ( plus_plus @ nat @ Y6 @ U2 ) ) ) )
@ ( minus_minus @ int ) ) ).
% minus_int.transfer
thf(fact_7908_plus__int_Otransfer,axiom,
( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ int @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) ) @ ( int > int ) @ pcr_int @ ( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ int @ ( product_prod @ nat @ nat ) @ int @ pcr_int @ pcr_int )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
@ ^ [X4: nat,Y6: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V5: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X4 @ U2 ) @ ( plus_plus @ nat @ Y6 @ V5 ) ) ) )
@ ( plus_plus @ int ) ) ).
% plus_int.transfer
thf(fact_7909_less__int_Otransfer,axiom,
( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ int @ ( ( product_prod @ nat @ nat ) > $o ) @ ( int > $o ) @ pcr_int
@ ( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ int @ $o @ $o @ pcr_int
@ ^ [Y5: $o,Z2: $o] : Y5 = Z2 )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [X4: nat,Y6: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U2: nat,V5: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ X4 @ V5 ) @ ( plus_plus @ nat @ U2 @ Y6 ) ) ) )
@ ( ord_less @ int ) ) ).
% less_int.transfer
thf(fact_7910_less__eq__int_Otransfer,axiom,
( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ int @ ( ( product_prod @ nat @ nat ) > $o ) @ ( int > $o ) @ pcr_int
@ ( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ int @ $o @ $o @ pcr_int
@ ^ [Y5: $o,Z2: $o] : Y5 = Z2 )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [X4: nat,Y6: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U2: nat,V5: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ X4 @ V5 ) @ ( plus_plus @ nat @ U2 @ Y6 ) ) ) )
@ ( ord_less_eq @ int ) ) ).
% less_eq_int.transfer
thf(fact_7911_subset__chain__insert,axiom,
! [A: $tType,A20: set @ ( set @ A ),B2: set @ A,B11: set @ ( set @ A )] :
( ( pred_chain @ ( set @ A ) @ A20 @ ( ord_less @ ( set @ A ) ) @ ( insert2 @ ( set @ A ) @ B2 @ B11 ) )
= ( ( member @ ( set @ A ) @ B2 @ A20 )
& ! [X4: set @ A] :
( ( member @ ( set @ A ) @ X4 @ B11 )
=> ( ( ord_less_eq @ ( set @ A ) @ X4 @ B2 )
| ( ord_less_eq @ ( set @ A ) @ B2 @ X4 ) ) )
& ( pred_chain @ ( set @ A ) @ A20 @ ( ord_less @ ( set @ A ) ) @ B11 ) ) ) ).
% subset_chain_insert
thf(fact_7912_subset_Ochain__empty,axiom,
! [A: $tType,A4: set @ ( set @ A )] : ( pred_chain @ ( set @ A ) @ A4 @ ( ord_less @ ( set @ A ) ) @ ( bot_bot @ ( set @ ( set @ A ) ) ) ) ).
% subset.chain_empty
thf(fact_7913_pred__on_Ochain__empty,axiom,
! [A: $tType,A4: set @ A,P: A > A > $o] : ( pred_chain @ A @ A4 @ P @ ( bot_bot @ ( set @ A ) ) ) ).
% pred_on.chain_empty
thf(fact_7914_subset__Zorn,axiom,
! [A: $tType,A4: set @ ( set @ A )] :
( ! [C7: set @ ( set @ A )] :
( ( pred_chain @ ( set @ A ) @ A4 @ ( ord_less @ ( set @ A ) ) @ C7 )
=> ? [X2: set @ A] :
( ( member @ ( set @ A ) @ X2 @ A4 )
& ! [Xa3: set @ A] :
( ( member @ ( set @ A ) @ Xa3 @ C7 )
=> ( ord_less_eq @ ( set @ A ) @ Xa3 @ X2 ) ) ) )
=> ? [X5: set @ A] :
( ( member @ ( set @ A ) @ X5 @ A4 )
& ! [Xa: set @ A] :
( ( member @ ( set @ A ) @ Xa @ A4 )
=> ( ( ord_less_eq @ ( set @ A ) @ X5 @ Xa )
=> ( Xa = X5 ) ) ) ) ) ).
% subset_Zorn
thf(fact_7915_subset_Ochain__def,axiom,
! [A: $tType,A4: set @ ( set @ A ),C5: set @ ( set @ A )] :
( ( pred_chain @ ( set @ A ) @ A4 @ ( ord_less @ ( set @ A ) ) @ C5 )
= ( ( ord_less_eq @ ( set @ ( set @ A ) ) @ C5 @ A4 )
& ! [X4: set @ A] :
( ( member @ ( set @ A ) @ X4 @ C5 )
=> ! [Y6: set @ A] :
( ( member @ ( set @ A ) @ Y6 @ C5 )
=> ( ( sup_sup @ ( ( set @ A ) > ( set @ A ) > $o ) @ ( ord_less @ ( set @ A ) )
@ ^ [Y5: set @ A,Z2: set @ A] : Y5 = Z2
@ X4
@ Y6 )
| ( sup_sup @ ( ( set @ A ) > ( set @ A ) > $o ) @ ( ord_less @ ( set @ A ) )
@ ^ [Y5: set @ A,Z2: set @ A] : Y5 = Z2
@ Y6
@ X4 ) ) ) ) ) ) ).
% subset.chain_def
thf(fact_7916_subset_OchainI,axiom,
! [A: $tType,C5: set @ ( set @ A ),A4: set @ ( set @ A )] :
( ( ord_less_eq @ ( set @ ( set @ A ) ) @ C5 @ A4 )
=> ( ! [X5: set @ A,Y3: set @ A] :
( ( member @ ( set @ A ) @ X5 @ C5 )
=> ( ( member @ ( set @ A ) @ Y3 @ C5 )
=> ( ( sup_sup @ ( ( set @ A ) > ( set @ A ) > $o ) @ ( ord_less @ ( set @ A ) )
@ ^ [Y5: set @ A,Z2: set @ A] : Y5 = Z2
@ X5
@ Y3 )
| ( sup_sup @ ( ( set @ A ) > ( set @ A ) > $o ) @ ( ord_less @ ( set @ A ) )
@ ^ [Y5: set @ A,Z2: set @ A] : Y5 = Z2
@ Y3
@ X5 ) ) ) )
=> ( pred_chain @ ( set @ A ) @ A4 @ ( ord_less @ ( set @ A ) ) @ C5 ) ) ) ).
% subset.chainI
thf(fact_7917_pred__on_OchainI,axiom,
! [A: $tType,C5: set @ A,A4: set @ A,P: A > A > $o] :
( ( ord_less_eq @ ( set @ A ) @ C5 @ A4 )
=> ( ! [X5: A,Y3: A] :
( ( member @ A @ X5 @ C5 )
=> ( ( member @ A @ Y3 @ C5 )
=> ( ( sup_sup @ ( A > A > $o ) @ P
@ ^ [Y5: A,Z2: A] : Y5 = Z2
@ X5
@ Y3 )
| ( sup_sup @ ( A > A > $o ) @ P
@ ^ [Y5: A,Z2: A] : Y5 = Z2
@ Y3
@ X5 ) ) ) )
=> ( pred_chain @ A @ A4 @ P @ C5 ) ) ) ).
% pred_on.chainI
thf(fact_7918_pred__on_Ochain__def,axiom,
! [A: $tType] :
( ( pred_chain @ A )
= ( ^ [A7: set @ A,P3: A > A > $o,C6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ C6 @ A7 )
& ! [X4: A] :
( ( member @ A @ X4 @ C6 )
=> ! [Y6: A] :
( ( member @ A @ Y6 @ C6 )
=> ( ( sup_sup @ ( A > A > $o ) @ P3
@ ^ [Y5: A,Z2: A] : Y5 = Z2
@ X4
@ Y6 )
| ( sup_sup @ ( A > A > $o ) @ P3
@ ^ [Y5: A,Z2: A] : Y5 = Z2
@ Y6
@ X4 ) ) ) ) ) ) ) ).
% pred_on.chain_def
thf(fact_7919_subset__Zorn_H,axiom,
! [A: $tType,A4: set @ ( set @ A )] :
( ! [C7: set @ ( set @ A )] :
( ( pred_chain @ ( set @ A ) @ A4 @ ( ord_less @ ( set @ A ) ) @ C7 )
=> ( member @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ C7 ) @ A4 ) )
=> ? [X5: set @ A] :
( ( member @ ( set @ A ) @ X5 @ A4 )
& ! [Xa: set @ A] :
( ( member @ ( set @ A ) @ Xa @ A4 )
=> ( ( ord_less_eq @ ( set @ A ) @ X5 @ Xa )
=> ( Xa = X5 ) ) ) ) ) ).
% subset_Zorn'
thf(fact_7920_subset__chain__def,axiom,
! [A: $tType,A20: set @ ( set @ A ),C10: set @ ( set @ A )] :
( ( pred_chain @ ( set @ A ) @ A20 @ ( ord_less @ ( set @ A ) ) @ C10 )
= ( ( ord_less_eq @ ( set @ ( set @ A ) ) @ C10 @ A20 )
& ! [X4: set @ A] :
( ( member @ ( set @ A ) @ X4 @ C10 )
=> ! [Y6: set @ A] :
( ( member @ ( set @ A ) @ Y6 @ C10 )
=> ( ( ord_less_eq @ ( set @ A ) @ X4 @ Y6 )
| ( ord_less_eq @ ( set @ A ) @ Y6 @ X4 ) ) ) ) ) ) ).
% subset_chain_def
thf(fact_7921_times__int_Otransfer,axiom,
( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ int @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) ) @ ( int > int ) @ pcr_int @ ( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ int @ ( product_prod @ nat @ nat ) @ int @ pcr_int @ pcr_int )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
@ ^ [X4: nat,Y6: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V5: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ X4 @ U2 ) @ ( times_times @ nat @ Y6 @ V5 ) ) @ ( plus_plus @ nat @ ( times_times @ nat @ X4 @ V5 ) @ ( times_times @ nat @ Y6 @ U2 ) ) ) ) )
@ ( times_times @ int ) ) ).
% times_int.transfer
thf(fact_7922_pred__on_Ochain__extend,axiom,
! [A: $tType,A4: set @ A,P: A > A > $o,C5: set @ A,Z3: A] :
( ( pred_chain @ A @ A4 @ P @ C5 )
=> ( ( member @ A @ Z3 @ A4 )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ C5 )
=> ( sup_sup @ ( A > A > $o ) @ P
@ ^ [Y5: A,Z2: A] : Y5 = Z2
@ X5
@ Z3 ) )
=> ( pred_chain @ A @ A4 @ P @ ( sup_sup @ ( set @ A ) @ ( insert2 @ A @ Z3 @ ( bot_bot @ ( set @ A ) ) ) @ C5 ) ) ) ) ) ).
% pred_on.chain_extend
thf(fact_7923_subset__Zorn__nonempty,axiom,
! [A: $tType,A20: set @ ( set @ A )] :
( ( A20
!= ( bot_bot @ ( set @ ( set @ A ) ) ) )
=> ( ! [C11: set @ ( set @ A )] :
( ( C11
!= ( bot_bot @ ( set @ ( set @ A ) ) ) )
=> ( ( pred_chain @ ( set @ A ) @ A20 @ ( ord_less @ ( set @ A ) ) @ C11 )
=> ( member @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ C11 ) @ A20 ) ) )
=> ? [X5: set @ A] :
( ( member @ ( set @ A ) @ X5 @ A20 )
& ! [Xa: set @ A] :
( ( member @ ( set @ A ) @ Xa @ A20 )
=> ( ( ord_less_eq @ ( set @ A ) @ X5 @ Xa )
=> ( Xa = X5 ) ) ) ) ) ) ).
% subset_Zorn_nonempty
thf(fact_7924_Union__in__chain,axiom,
! [A: $tType,B11: set @ ( set @ A ),A20: set @ ( set @ A )] :
( ( finite_finite2 @ ( set @ A ) @ B11 )
=> ( ( B11
!= ( bot_bot @ ( set @ ( set @ A ) ) ) )
=> ( ( pred_chain @ ( set @ A ) @ A20 @ ( ord_less @ ( set @ A ) ) @ B11 )
=> ( member @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ B11 ) @ B11 ) ) ) ) ).
% Union_in_chain
thf(fact_7925_Inter__in__chain,axiom,
! [A: $tType,B11: set @ ( set @ A ),A20: set @ ( set @ A )] :
( ( finite_finite2 @ ( set @ A ) @ B11 )
=> ( ( B11
!= ( bot_bot @ ( set @ ( set @ A ) ) ) )
=> ( ( pred_chain @ ( set @ A ) @ A20 @ ( ord_less @ ( set @ A ) ) @ B11 )
=> ( member @ ( set @ A ) @ ( complete_Inf_Inf @ ( set @ A ) @ B11 ) @ B11 ) ) ) ) ).
% Inter_in_chain
thf(fact_7926_uminus__int_Otransfer,axiom,
( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ int @ ( product_prod @ nat @ nat ) @ int @ pcr_int @ pcr_int
@ ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [X4: nat,Y6: nat] : ( product_Pair @ nat @ nat @ Y6 @ X4 ) )
@ ( uminus_uminus @ int ) ) ).
% uminus_int.transfer
thf(fact_7927_nat_Otransfer,axiom,
( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ int @ nat @ nat @ pcr_int
@ ^ [Y5: nat,Z2: nat] : Y5 = Z2
@ ( product_case_prod @ nat @ nat @ nat @ ( minus_minus @ nat ) )
@ nat2 ) ).
% nat.transfer
thf(fact_7928_subset_Ochain__extend,axiom,
! [A: $tType,A4: set @ ( set @ A ),C5: set @ ( set @ A ),Z3: set @ A] :
( ( pred_chain @ ( set @ A ) @ A4 @ ( ord_less @ ( set @ A ) ) @ C5 )
=> ( ( member @ ( set @ A ) @ Z3 @ A4 )
=> ( ! [X5: set @ A] :
( ( member @ ( set @ A ) @ X5 @ C5 )
=> ( sup_sup @ ( ( set @ A ) > ( set @ A ) > $o ) @ ( ord_less @ ( set @ A ) )
@ ^ [Y5: set @ A,Z2: set @ A] : Y5 = Z2
@ X5
@ Z3 ) )
=> ( pred_chain @ ( set @ A ) @ A4 @ ( ord_less @ ( set @ A ) ) @ ( sup_sup @ ( set @ ( set @ A ) ) @ ( insert2 @ ( set @ A ) @ Z3 @ ( bot_bot @ ( set @ ( set @ A ) ) ) ) @ C5 ) ) ) ) ) ).
% subset.chain_extend
thf(fact_7929_transfer__rule__of__nat,axiom,
! [A: $tType,B: $tType] :
( ( ( semiring_1 @ B )
& ( semiring_1 @ A ) )
=> ! [R: A > B > $o] :
( ( R @ ( zero_zero @ A ) @ ( zero_zero @ B ) )
=> ( ( R @ ( one_one @ A ) @ ( one_one @ B ) )
=> ( ( bNF_rel_fun @ A @ B @ ( A > A ) @ ( B > B ) @ R @ ( bNF_rel_fun @ A @ B @ A @ B @ R @ R ) @ ( plus_plus @ A ) @ ( plus_plus @ B ) )
=> ( bNF_rel_fun @ nat @ nat @ A @ B
@ ^ [Y5: nat,Z2: nat] : Y5 = Z2
@ R
@ ( semiring_1_of_nat @ A )
@ ( semiring_1_of_nat @ B ) ) ) ) ) ) ).
% transfer_rule_of_nat
thf(fact_7930_chain__subset__alt__def,axiom,
! [A: $tType] :
( ( chain_subset @ A )
= ( pred_chain @ ( set @ A ) @ ( top_top @ ( set @ ( set @ A ) ) ) @ ( ord_less @ ( set @ A ) ) ) ) ).
% chain_subset_alt_def
thf(fact_7931_list_Odisc__transfer_I1_J,axiom,
! [A: $tType,B: $tType,R: A > B > $o] :
( bNF_rel_fun @ ( list @ A ) @ ( list @ B ) @ $o @ $o @ ( list_all2 @ A @ B @ R )
@ ^ [Y5: $o,Z2: $o] : Y5 = Z2
@ ^ [List3: list @ A] :
( List3
= ( nil @ A ) )
@ ^ [List3: list @ B] :
( List3
= ( nil @ B ) ) ) ).
% list.disc_transfer(1)
thf(fact_7932_list_Odisc__transfer_I2_J,axiom,
! [A: $tType,B: $tType,R: A > B > $o] :
( bNF_rel_fun @ ( list @ A ) @ ( list @ B ) @ $o @ $o @ ( list_all2 @ A @ B @ R )
@ ^ [Y5: $o,Z2: $o] : Y5 = Z2
@ ^ [List3: list @ A] :
( List3
!= ( nil @ A ) )
@ ^ [List3: list @ B] :
( List3
!= ( nil @ B ) ) ) ).
% list.disc_transfer(2)
thf(fact_7933_butlast__transfer,axiom,
! [A: $tType,B: $tType,A4: A > B > $o] : ( bNF_rel_fun @ ( list @ A ) @ ( list @ B ) @ ( list @ A ) @ ( list @ B ) @ ( list_all2 @ A @ B @ A4 ) @ ( list_all2 @ A @ B @ A4 ) @ ( butlast @ A ) @ ( butlast @ B ) ) ).
% butlast_transfer
thf(fact_7934_dropWhile__transfer,axiom,
! [A: $tType,B: $tType,A4: A > B > $o] :
( bNF_rel_fun @ ( A > $o ) @ ( B > $o ) @ ( ( list @ A ) > ( list @ A ) ) @ ( ( list @ B ) > ( list @ B ) )
@ ( bNF_rel_fun @ A @ B @ $o @ $o @ A4
@ ^ [Y5: $o,Z2: $o] : Y5 = Z2 )
@ ( bNF_rel_fun @ ( list @ A ) @ ( list @ B ) @ ( list @ A ) @ ( list @ B ) @ ( list_all2 @ A @ B @ A4 ) @ ( list_all2 @ A @ B @ A4 ) )
@ ( dropWhile @ A )
@ ( dropWhile @ B ) ) ).
% dropWhile_transfer
thf(fact_7935_takeWhile__transfer,axiom,
! [A: $tType,B: $tType,A4: A > B > $o] :
( bNF_rel_fun @ ( A > $o ) @ ( B > $o ) @ ( ( list @ A ) > ( list @ A ) ) @ ( ( list @ B ) > ( list @ B ) )
@ ( bNF_rel_fun @ A @ B @ $o @ $o @ A4
@ ^ [Y5: $o,Z2: $o] : Y5 = Z2 )
@ ( bNF_rel_fun @ ( list @ A ) @ ( list @ B ) @ ( list @ A ) @ ( list @ B ) @ ( list_all2 @ A @ B @ A4 ) @ ( list_all2 @ A @ B @ A4 ) )
@ ( takeWhile @ A )
@ ( takeWhile @ B ) ) ).
% takeWhile_transfer
thf(fact_7936_append__transfer,axiom,
! [A: $tType,B: $tType,A4: A > B > $o] : ( bNF_rel_fun @ ( list @ A ) @ ( list @ B ) @ ( ( list @ A ) > ( list @ A ) ) @ ( ( list @ B ) > ( list @ B ) ) @ ( list_all2 @ A @ B @ A4 ) @ ( bNF_rel_fun @ ( list @ A ) @ ( list @ B ) @ ( list @ A ) @ ( list @ B ) @ ( list_all2 @ A @ B @ A4 ) @ ( list_all2 @ A @ B @ A4 ) ) @ ( append @ A ) @ ( append @ B ) ) ).
% append_transfer
thf(fact_7937_product__lists__transfer,axiom,
! [A: $tType,B: $tType,A4: A > B > $o] : ( bNF_rel_fun @ ( list @ ( list @ A ) ) @ ( list @ ( list @ B ) ) @ ( list @ ( list @ A ) ) @ ( list @ ( list @ B ) ) @ ( list_all2 @ ( list @ A ) @ ( list @ B ) @ ( list_all2 @ A @ B @ A4 ) ) @ ( list_all2 @ ( list @ A ) @ ( list @ B ) @ ( list_all2 @ A @ B @ A4 ) ) @ ( product_lists @ A ) @ ( product_lists @ B ) ) ).
% product_lists_transfer
thf(fact_7938_rotate1__transfer,axiom,
! [A: $tType,B: $tType,A4: A > B > $o] : ( bNF_rel_fun @ ( list @ A ) @ ( list @ B ) @ ( list @ A ) @ ( list @ B ) @ ( list_all2 @ A @ B @ A4 ) @ ( list_all2 @ A @ B @ A4 ) @ ( rotate1 @ A ) @ ( rotate1 @ B ) ) ).
% rotate1_transfer
thf(fact_7939_list_Orel__transfer,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,Sa: A > C > $o,Sc: B > D > $o] :
( bNF_rel_fun @ ( A > B > $o ) @ ( C > D > $o ) @ ( ( list @ A ) > ( list @ B ) > $o ) @ ( ( list @ C ) > ( list @ D ) > $o )
@ ( bNF_rel_fun @ A @ C @ ( B > $o ) @ ( D > $o ) @ Sa
@ ( bNF_rel_fun @ B @ D @ $o @ $o @ Sc
@ ^ [Y5: $o,Z2: $o] : Y5 = Z2 ) )
@ ( bNF_rel_fun @ ( list @ A ) @ ( list @ C ) @ ( ( list @ B ) > $o ) @ ( ( list @ D ) > $o ) @ ( list_all2 @ A @ C @ Sa )
@ ( bNF_rel_fun @ ( list @ B ) @ ( list @ D ) @ $o @ $o @ ( list_all2 @ B @ D @ Sc )
@ ^ [Y5: $o,Z2: $o] : Y5 = Z2 ) )
@ ( list_all2 @ A @ B )
@ ( list_all2 @ C @ D ) ) ).
% list.rel_transfer
thf(fact_7940_subseqs__transfer,axiom,
! [A: $tType,B: $tType,A4: A > B > $o] : ( bNF_rel_fun @ ( list @ A ) @ ( list @ B ) @ ( list @ ( list @ A ) ) @ ( list @ ( list @ B ) ) @ ( list_all2 @ A @ B @ A4 ) @ ( list_all2 @ ( list @ A ) @ ( list @ B ) @ ( list_all2 @ A @ B @ A4 ) ) @ ( subseqs @ A ) @ ( subseqs @ B ) ) ).
% subseqs_transfer
thf(fact_7941_list__update__transfer,axiom,
! [A: $tType,B: $tType,A4: A > B > $o] :
( bNF_rel_fun @ ( list @ A ) @ ( list @ B ) @ ( nat > A > ( list @ A ) ) @ ( nat > B > ( list @ B ) ) @ ( list_all2 @ A @ B @ A4 )
@ ( bNF_rel_fun @ nat @ nat @ ( A > ( list @ A ) ) @ ( B > ( list @ B ) )
@ ^ [Y5: nat,Z2: nat] : Y5 = Z2
@ ( bNF_rel_fun @ A @ B @ ( list @ A ) @ ( list @ B ) @ A4 @ ( list_all2 @ A @ B @ A4 ) ) )
@ ( list_update @ A )
@ ( list_update @ B ) ) ).
% list_update_transfer
thf(fact_7942_replicate__transfer,axiom,
! [A: $tType,B: $tType,A4: A > B > $o] :
( bNF_rel_fun @ nat @ nat @ ( A > ( list @ A ) ) @ ( B > ( list @ B ) )
@ ^ [Y5: nat,Z2: nat] : Y5 = Z2
@ ( bNF_rel_fun @ A @ B @ ( list @ A ) @ ( list @ B ) @ A4 @ ( list_all2 @ A @ B @ A4 ) )
@ ( replicate @ A )
@ ( replicate @ B ) ) ).
% replicate_transfer
thf(fact_7943_rotate__transfer,axiom,
! [A: $tType,B: $tType,A4: A > B > $o] :
( bNF_rel_fun @ nat @ nat @ ( ( list @ A ) > ( list @ A ) ) @ ( ( list @ B ) > ( list @ B ) )
@ ^ [Y5: nat,Z2: nat] : Y5 = Z2
@ ( bNF_rel_fun @ ( list @ A ) @ ( list @ B ) @ ( list @ A ) @ ( list @ B ) @ ( list_all2 @ A @ B @ A4 ) @ ( list_all2 @ A @ B @ A4 ) )
@ ( rotate @ A )
@ ( rotate @ B ) ) ).
% rotate_transfer
thf(fact_7944_drop__transfer,axiom,
! [A: $tType,B: $tType,A4: A > B > $o] :
( bNF_rel_fun @ nat @ nat @ ( ( list @ A ) > ( list @ A ) ) @ ( ( list @ B ) > ( list @ B ) )
@ ^ [Y5: nat,Z2: nat] : Y5 = Z2
@ ( bNF_rel_fun @ ( list @ A ) @ ( list @ B ) @ ( list @ A ) @ ( list @ B ) @ ( list_all2 @ A @ B @ A4 ) @ ( list_all2 @ A @ B @ A4 ) )
@ ( drop @ A )
@ ( drop @ B ) ) ).
% drop_transfer
thf(fact_7945_take__transfer,axiom,
! [A: $tType,B: $tType,A4: A > B > $o] :
( bNF_rel_fun @ nat @ nat @ ( ( list @ A ) > ( list @ A ) ) @ ( ( list @ B ) > ( list @ B ) )
@ ^ [Y5: nat,Z2: nat] : Y5 = Z2
@ ( bNF_rel_fun @ ( list @ A ) @ ( list @ B ) @ ( list @ A ) @ ( list @ B ) @ ( list_all2 @ A @ B @ A4 ) @ ( list_all2 @ A @ B @ A4 ) )
@ ( take @ A )
@ ( take @ B ) ) ).
% take_transfer
thf(fact_7946_tl__transfer,axiom,
! [A: $tType,B: $tType,A4: A > B > $o] : ( bNF_rel_fun @ ( list @ A ) @ ( list @ B ) @ ( list @ A ) @ ( list @ B ) @ ( list_all2 @ A @ B @ A4 ) @ ( list_all2 @ A @ B @ A4 ) @ ( tl @ A ) @ ( tl @ B ) ) ).
% tl_transfer
thf(fact_7947_length__transfer,axiom,
! [A: $tType,B: $tType,A4: A > B > $o] :
( bNF_rel_fun @ ( list @ A ) @ ( list @ B ) @ nat @ nat @ ( list_all2 @ A @ B @ A4 )
@ ^ [Y5: nat,Z2: nat] : Y5 = Z2
@ ( size_size @ ( list @ A ) )
@ ( size_size @ ( list @ B ) ) ) ).
% length_transfer
thf(fact_7948_concat__transfer,axiom,
! [A: $tType,B: $tType,A4: A > B > $o] : ( bNF_rel_fun @ ( list @ ( list @ A ) ) @ ( list @ ( list @ B ) ) @ ( list @ A ) @ ( list @ B ) @ ( list_all2 @ ( list @ A ) @ ( list @ B ) @ ( list_all2 @ A @ B @ A4 ) ) @ ( list_all2 @ A @ B @ A4 ) @ ( concat @ A ) @ ( concat @ B ) ) ).
% concat_transfer
thf(fact_7949_List_Ofilter__transfer,axiom,
! [A: $tType,B: $tType,A4: A > B > $o] :
( bNF_rel_fun @ ( A > $o ) @ ( B > $o ) @ ( ( list @ A ) > ( list @ A ) ) @ ( ( list @ B ) > ( list @ B ) )
@ ( bNF_rel_fun @ A @ B @ $o @ $o @ A4
@ ^ [Y5: $o,Z2: $o] : Y5 = Z2 )
@ ( bNF_rel_fun @ ( list @ A ) @ ( list @ B ) @ ( list @ A ) @ ( list @ B ) @ ( list_all2 @ A @ B @ A4 ) @ ( list_all2 @ A @ B @ A4 ) )
@ ( filter2 @ A )
@ ( filter2 @ B ) ) ).
% List.filter_transfer
thf(fact_7950_rev__transfer,axiom,
! [A: $tType,B: $tType,A4: A > B > $o] : ( bNF_rel_fun @ ( list @ A ) @ ( list @ B ) @ ( list @ A ) @ ( list @ B ) @ ( list_all2 @ A @ B @ A4 ) @ ( list_all2 @ A @ B @ A4 ) @ ( rev @ A ) @ ( rev @ B ) ) ).
% rev_transfer
thf(fact_7951_list_Ocase__transfer,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,S2: C > D > $o,R: A > B > $o] : ( bNF_rel_fun @ C @ D @ ( ( A > ( list @ A ) > C ) > ( list @ A ) > C ) @ ( ( B > ( list @ B ) > D ) > ( list @ B ) > D ) @ S2 @ ( bNF_rel_fun @ ( A > ( list @ A ) > C ) @ ( B > ( list @ B ) > D ) @ ( ( list @ A ) > C ) @ ( ( list @ B ) > D ) @ ( bNF_rel_fun @ A @ B @ ( ( list @ A ) > C ) @ ( ( list @ B ) > D ) @ R @ ( bNF_rel_fun @ ( list @ A ) @ ( list @ B ) @ C @ D @ ( list_all2 @ A @ B @ R ) @ S2 ) ) @ ( bNF_rel_fun @ ( list @ A ) @ ( list @ B ) @ C @ D @ ( list_all2 @ A @ B @ R ) @ S2 ) ) @ ( case_list @ C @ A ) @ ( case_list @ D @ B ) ) ).
% list.case_transfer
thf(fact_7952_splice__transfer,axiom,
! [A: $tType,B: $tType,A4: A > B > $o] : ( bNF_rel_fun @ ( list @ A ) @ ( list @ B ) @ ( ( list @ A ) > ( list @ A ) ) @ ( ( list @ B ) > ( list @ B ) ) @ ( list_all2 @ A @ B @ A4 ) @ ( bNF_rel_fun @ ( list @ A ) @ ( list @ B ) @ ( list @ A ) @ ( list @ B ) @ ( list_all2 @ A @ B @ A4 ) @ ( list_all2 @ A @ B @ A4 ) ) @ ( splice @ A ) @ ( splice @ B ) ) ).
% splice_transfer
thf(fact_7953_list_Octr__transfer_I2_J,axiom,
! [A: $tType,B: $tType,R: A > B > $o] : ( bNF_rel_fun @ A @ B @ ( ( list @ A ) > ( list @ A ) ) @ ( ( list @ B ) > ( list @ B ) ) @ R @ ( bNF_rel_fun @ ( list @ A ) @ ( list @ B ) @ ( list @ A ) @ ( list @ B ) @ ( list_all2 @ A @ B @ R ) @ ( list_all2 @ A @ B @ R ) ) @ ( cons @ A ) @ ( cons @ B ) ) ).
% list.ctr_transfer(2)
thf(fact_7954_transfer__rule__of__bool,axiom,
! [A: $tType,B: $tType] :
( ( ( zero_neq_one @ B )
& ( zero_neq_one @ A ) )
=> ! [R: A > B > $o] :
( ( R @ ( zero_zero @ A ) @ ( zero_zero @ B ) )
=> ( ( R @ ( one_one @ A ) @ ( one_one @ B ) )
=> ( bNF_rel_fun @ $o @ $o @ A @ B
@ ^ [Y5: $o,Z2: $o] : Y5 = Z2
@ R
@ ( zero_neq_one_of_bool @ A )
@ ( zero_neq_one_of_bool @ B ) ) ) ) ) ).
% transfer_rule_of_bool
thf(fact_7955_chain__subset__def,axiom,
! [A: $tType] :
( ( chain_subset @ A )
= ( ^ [C6: set @ ( set @ A )] :
! [X4: set @ A] :
( ( member @ ( set @ A ) @ X4 @ C6 )
=> ! [Y6: set @ A] :
( ( member @ ( set @ A ) @ Y6 @ C6 )
=> ( ( ord_less_eq @ ( set @ A ) @ X4 @ Y6 )
| ( ord_less_eq @ ( set @ A ) @ Y6 @ X4 ) ) ) ) ) ) ).
% chain_subset_def
thf(fact_7956_int__transfer,axiom,
( bNF_rel_fun @ nat @ nat @ ( product_prod @ nat @ nat ) @ int
@ ^ [Y5: nat,Z2: nat] : Y5 = Z2
@ pcr_int
@ ^ [N: nat] : ( product_Pair @ nat @ nat @ N @ ( zero_zero @ nat ) )
@ ( semiring_1_of_nat @ int ) ) ).
% int_transfer
thf(fact_7957_foldl__transfer,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,B2: A > C > $o,A4: B > D > $o] : ( bNF_rel_fun @ ( A > B > A ) @ ( C > D > C ) @ ( A > ( list @ B ) > A ) @ ( C > ( list @ D ) > C ) @ ( bNF_rel_fun @ A @ C @ ( B > A ) @ ( D > C ) @ B2 @ ( bNF_rel_fun @ B @ D @ A @ C @ A4 @ B2 ) ) @ ( bNF_rel_fun @ A @ C @ ( ( list @ B ) > A ) @ ( ( list @ D ) > C ) @ B2 @ ( bNF_rel_fun @ ( list @ B ) @ ( list @ D ) @ A @ C @ ( list_all2 @ B @ D @ A4 ) @ B2 ) ) @ ( foldl @ A @ B ) @ ( foldl @ C @ D ) ) ).
% foldl_transfer
thf(fact_7958_pred__on_Onot__maxchain__Some,axiom,
! [A: $tType,A4: set @ A,P: A > A > $o,C5: set @ A] :
( ( pred_chain @ A @ A4 @ P @ C5 )
=> ( ~ ( pred_maxchain @ A @ A4 @ P @ C5 )
=> ( ( pred_chain @ A @ A4 @ P
@ ( fChoice @ ( set @ A )
@ ^ [D7: set @ A] :
( ( pred_chain @ A @ A4 @ P @ D7 )
& ( ord_less @ ( set @ A ) @ C5 @ D7 ) ) ) )
& ( ord_less @ ( set @ A ) @ C5
@ ( fChoice @ ( set @ A )
@ ^ [D7: set @ A] :
( ( pred_chain @ A @ A4 @ P @ D7 )
& ( ord_less @ ( set @ A ) @ C5 @ D7 ) ) ) ) ) ) ) ).
% pred_on.not_maxchain_Some
thf(fact_7959_foldl__append,axiom,
! [A: $tType,B: $tType,F2: A > B > A,A3: A,Xs: list @ B,Ys: list @ B] :
( ( foldl @ A @ B @ F2 @ A3 @ ( append @ B @ Xs @ Ys ) )
= ( foldl @ A @ B @ F2 @ ( foldl @ A @ B @ F2 @ A3 @ Xs ) @ Ys ) ) ).
% foldl_append
thf(fact_7960_subset_Omaxchain__imp__chain,axiom,
! [A: $tType,A4: set @ ( set @ A ),C5: set @ ( set @ A )] :
( ( pred_maxchain @ ( set @ A ) @ A4 @ ( ord_less @ ( set @ A ) ) @ C5 )
=> ( pred_chain @ ( set @ A ) @ A4 @ ( ord_less @ ( set @ A ) ) @ C5 ) ) ).
% subset.maxchain_imp_chain
thf(fact_7961_subset_Omaxchain__def,axiom,
! [A: $tType,A4: set @ ( set @ A ),C5: set @ ( set @ A )] :
( ( pred_maxchain @ ( set @ A ) @ A4 @ ( ord_less @ ( set @ A ) ) @ C5 )
= ( ( pred_chain @ ( set @ A ) @ A4 @ ( ord_less @ ( set @ A ) ) @ C5 )
& ~ ? [S6: set @ ( set @ A )] :
( ( pred_chain @ ( set @ A ) @ A4 @ ( ord_less @ ( set @ A ) ) @ S6 )
& ( ord_less @ ( set @ ( set @ A ) ) @ C5 @ S6 ) ) ) ) ).
% subset.maxchain_def
thf(fact_7962_foldl__Nil,axiom,
! [A: $tType,B: $tType,F2: B > A > B,A3: B] :
( ( foldl @ B @ A @ F2 @ A3 @ ( nil @ A ) )
= A3 ) ).
% foldl_Nil
thf(fact_7963_subset_OHausdorff,axiom,
! [A: $tType,A4: set @ ( set @ A )] :
? [X_12: set @ ( set @ A )] : ( pred_maxchain @ ( set @ A ) @ A4 @ ( ord_less @ ( set @ A ) ) @ X_12 ) ).
% subset.Hausdorff
thf(fact_7964_foldl__conv__fold,axiom,
! [B: $tType,A: $tType] :
( ( foldl @ A @ B )
= ( ^ [F3: A > B > A,S7: A,Xs3: list @ B] :
( fold @ B @ A
@ ^ [X4: B,T3: A] : ( F3 @ T3 @ X4 )
@ Xs3
@ S7 ) ) ) ).
% foldl_conv_fold
thf(fact_7965_foldl__Cons,axiom,
! [B: $tType,A: $tType,F2: B > A > B,A3: B,X: A,Xs: list @ A] :
( ( foldl @ B @ A @ F2 @ A3 @ ( cons @ A @ X @ Xs ) )
= ( foldl @ B @ A @ F2 @ ( F2 @ A3 @ X ) @ Xs ) ) ).
% foldl_Cons
thf(fact_7966_foldl__map,axiom,
! [A: $tType,B: $tType,C: $tType,G: A > B > A,A3: A,F2: C > B,Xs: list @ C] :
( ( foldl @ A @ B @ G @ A3 @ ( map @ C @ B @ F2 @ Xs ) )
= ( foldl @ A @ C
@ ^ [A5: A,X4: C] : ( G @ A5 @ ( F2 @ X4 ) )
@ A3
@ Xs ) ) ).
% foldl_map
thf(fact_7967_foldl__cong,axiom,
! [A: $tType,B: $tType,A3: A,B3: A,L: list @ B,K: list @ B,F2: A > B > A,G: A > B > A] :
( ( A3 = B3 )
=> ( ( L = K )
=> ( ! [A6: A,X5: B] :
( ( member @ B @ X5 @ ( set2 @ B @ L ) )
=> ( ( F2 @ A6 @ X5 )
= ( G @ A6 @ X5 ) ) )
=> ( ( foldl @ A @ B @ F2 @ A3 @ L )
= ( foldl @ A @ B @ G @ B3 @ K ) ) ) ) ) ).
% foldl_cong
thf(fact_7968_subset_Onot__maxchain__Some,axiom,
! [A: $tType,A4: set @ ( set @ A ),C5: set @ ( set @ A )] :
( ( pred_chain @ ( set @ A ) @ A4 @ ( ord_less @ ( set @ A ) ) @ C5 )
=> ( ~ ( pred_maxchain @ ( set @ A ) @ A4 @ ( ord_less @ ( set @ A ) ) @ C5 )
=> ( ( pred_chain @ ( set @ A ) @ A4 @ ( ord_less @ ( set @ A ) )
@ ( fChoice @ ( set @ ( set @ A ) )
@ ^ [D7: set @ ( set @ A )] :
( ( pred_chain @ ( set @ A ) @ A4 @ ( ord_less @ ( set @ A ) ) @ D7 )
& ( ord_less @ ( set @ ( set @ A ) ) @ C5 @ D7 ) ) ) )
& ( ord_less @ ( set @ ( set @ A ) ) @ C5
@ ( fChoice @ ( set @ ( set @ A ) )
@ ^ [D7: set @ ( set @ A )] :
( ( pred_chain @ ( set @ A ) @ A4 @ ( ord_less @ ( set @ A ) ) @ D7 )
& ( ord_less @ ( set @ ( set @ A ) ) @ C5 @ D7 ) ) ) ) ) ) ) ).
% subset.not_maxchain_Some
thf(fact_7969_foldl__conv__foldr,axiom,
! [B: $tType,A: $tType] :
( ( foldl @ A @ B )
= ( ^ [F3: A > B > A,A5: A,Xs3: list @ B] :
( foldr @ B @ A
@ ^ [X4: B,Y6: A] : ( F3 @ Y6 @ X4 )
@ ( rev @ B @ Xs3 )
@ A5 ) ) ) ).
% foldl_conv_foldr
thf(fact_7970_foldr__conv__foldl,axiom,
! [A: $tType,B: $tType] :
( ( foldr @ B @ A )
= ( ^ [F3: B > A > A,Xs3: list @ B,A5: A] :
( foldl @ A @ B
@ ^ [X4: A,Y6: B] : ( F3 @ Y6 @ X4 )
@ A5
@ ( rev @ B @ Xs3 ) ) ) ) ).
% foldr_conv_foldl
thf(fact_7971_pred__on_Omaxchain__def,axiom,
! [A: $tType] :
( ( pred_maxchain @ A )
= ( ^ [A7: set @ A,P3: A > A > $o,C6: set @ A] :
( ( pred_chain @ A @ A7 @ P3 @ C6 )
& ~ ? [S6: set @ A] :
( ( pred_chain @ A @ A7 @ P3 @ S6 )
& ( ord_less @ ( set @ A ) @ C6 @ S6 ) ) ) ) ) ).
% pred_on.maxchain_def
thf(fact_7972_subset__maxchain__max,axiom,
! [A: $tType,A4: set @ ( set @ A ),C5: set @ ( set @ A ),X8: set @ A] :
( ( pred_maxchain @ ( set @ A ) @ A4 @ ( ord_less @ ( set @ A ) ) @ C5 )
=> ( ( member @ ( set @ A ) @ X8 @ A4 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ C5 ) @ X8 )
=> ( ( complete_Sup_Sup @ ( set @ A ) @ C5 )
= X8 ) ) ) ) ).
% subset_maxchain_max
thf(fact_7973_times__int_Orsp,axiom,
( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) ) @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) ) @ intrel @ ( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ intrel @ intrel )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
@ ^ [X4: nat,Y6: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V5: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ X4 @ U2 ) @ ( times_times @ nat @ Y6 @ V5 ) ) @ ( plus_plus @ nat @ ( times_times @ nat @ X4 @ V5 ) @ ( times_times @ nat @ Y6 @ U2 ) ) ) ) )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
@ ^ [X4: nat,Y6: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V5: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ X4 @ U2 ) @ ( times_times @ nat @ Y6 @ V5 ) ) @ ( plus_plus @ nat @ ( times_times @ nat @ X4 @ V5 ) @ ( times_times @ nat @ Y6 @ U2 ) ) ) ) ) ) ).
% times_int.rsp
thf(fact_7974_pred__on_Osuc__def,axiom,
! [A: $tType] :
( ( pred_suc @ A )
= ( ^ [A7: set @ A,P3: A > A > $o,C6: set @ A] :
( if @ ( set @ A )
@ ( ~ ( pred_chain @ A @ A7 @ P3 @ C6 )
| ( pred_maxchain @ A @ A7 @ P3 @ C6 ) )
@ C6
@ ( fChoice @ ( set @ A )
@ ^ [D7: set @ A] :
( ( pred_chain @ A @ A7 @ P3 @ D7 )
& ( ord_less @ ( set @ A ) @ C6 @ D7 ) ) ) ) ) ) ).
% pred_on.suc_def
thf(fact_7975_subset_Onot__chain__suc,axiom,
! [A: $tType,A4: set @ ( set @ A ),X8: set @ ( set @ A )] :
( ~ ( pred_chain @ ( set @ A ) @ A4 @ ( ord_less @ ( set @ A ) ) @ X8 )
=> ( ( pred_suc @ ( set @ A ) @ A4 @ ( ord_less @ ( set @ A ) ) @ X8 )
= X8 ) ) ).
% subset.not_chain_suc
thf(fact_7976_subset_Omaxchain__suc,axiom,
! [A: $tType,A4: set @ ( set @ A ),X8: set @ ( set @ A )] :
( ( pred_maxchain @ ( set @ A ) @ A4 @ ( ord_less @ ( set @ A ) ) @ X8 )
=> ( ( pred_suc @ ( set @ A ) @ A4 @ ( ord_less @ ( set @ A ) ) @ X8 )
= X8 ) ) ).
% subset.maxchain_suc
thf(fact_7977_intrel__iff,axiom,
! [X: nat,Y: nat,U: nat,V2: nat] :
( ( intrel @ ( product_Pair @ nat @ nat @ X @ Y ) @ ( product_Pair @ nat @ nat @ U @ V2 ) )
= ( ( plus_plus @ nat @ X @ V2 )
= ( plus_plus @ nat @ U @ Y ) ) ) ).
% intrel_iff
thf(fact_7978_nat_Orsp,axiom,
( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ nat @ nat @ intrel
@ ^ [Y5: nat,Z2: nat] : Y5 = Z2
@ ( product_case_prod @ nat @ nat @ nat @ ( minus_minus @ nat ) )
@ ( product_case_prod @ nat @ nat @ nat @ ( minus_minus @ nat ) ) ) ).
% nat.rsp
thf(fact_7979_uminus__int_Orsp,axiom,
( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ intrel @ intrel
@ ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [X4: nat,Y6: nat] : ( product_Pair @ nat @ nat @ Y6 @ X4 ) )
@ ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [X4: nat,Y6: nat] : ( product_Pair @ nat @ nat @ Y6 @ X4 ) ) ) ).
% uminus_int.rsp
thf(fact_7980_subset_Osuc__not__equals,axiom,
! [A: $tType,A4: set @ ( set @ A ),C5: set @ ( set @ A )] :
( ( pred_chain @ ( set @ A ) @ A4 @ ( ord_less @ ( set @ A ) ) @ C5 )
=> ( ~ ( pred_maxchain @ ( set @ A ) @ A4 @ ( ord_less @ ( set @ A ) ) @ C5 )
=> ( ( pred_suc @ ( set @ A ) @ A4 @ ( ord_less @ ( set @ A ) ) @ C5 )
!= C5 ) ) ) ).
% subset.suc_not_equals
thf(fact_7981_subset_Osuc__def,axiom,
! [A: $tType,A4: set @ ( set @ A ),C5: set @ ( set @ A )] :
( ( ( ~ ( pred_chain @ ( set @ A ) @ A4 @ ( ord_less @ ( set @ A ) ) @ C5 )
| ( pred_maxchain @ ( set @ A ) @ A4 @ ( ord_less @ ( set @ A ) ) @ C5 ) )
=> ( ( pred_suc @ ( set @ A ) @ A4 @ ( ord_less @ ( set @ A ) ) @ C5 )
= C5 ) )
& ( ~ ( ~ ( pred_chain @ ( set @ A ) @ A4 @ ( ord_less @ ( set @ A ) ) @ C5 )
| ( pred_maxchain @ ( set @ A ) @ A4 @ ( ord_less @ ( set @ A ) ) @ C5 ) )
=> ( ( pred_suc @ ( set @ A ) @ A4 @ ( ord_less @ ( set @ A ) ) @ C5 )
= ( fChoice @ ( set @ ( set @ A ) )
@ ^ [D7: set @ ( set @ A )] :
( ( pred_chain @ ( set @ A ) @ A4 @ ( ord_less @ ( set @ A ) ) @ D7 )
& ( ord_less @ ( set @ ( set @ A ) ) @ C5 @ D7 ) ) ) ) ) ) ).
% subset.suc_def
thf(fact_7982_subset_Osubset__suc,axiom,
! [A: $tType,X8: set @ ( set @ A ),Y7: set @ ( set @ A ),A4: set @ ( set @ A )] :
( ( ord_less_eq @ ( set @ ( set @ A ) ) @ X8 @ Y7 )
=> ( ord_less_eq @ ( set @ ( set @ A ) ) @ X8 @ ( pred_suc @ ( set @ A ) @ A4 @ ( ord_less @ ( set @ A ) ) @ Y7 ) ) ) ).
% subset.subset_suc
thf(fact_7983_subset_Osuc__subset,axiom,
! [A: $tType,X8: set @ ( set @ A ),A4: set @ ( set @ A )] : ( ord_less_eq @ ( set @ ( set @ A ) ) @ X8 @ ( pred_suc @ ( set @ A ) @ A4 @ ( ord_less @ ( set @ A ) ) @ X8 ) ) ).
% subset.suc_subset
thf(fact_7984_subset_Osuc__in__carrier,axiom,
! [A: $tType,X8: set @ ( set @ A ),A4: set @ ( set @ A )] :
( ( ord_less_eq @ ( set @ ( set @ A ) ) @ X8 @ A4 )
=> ( ord_less_eq @ ( set @ ( set @ A ) ) @ ( pred_suc @ ( set @ A ) @ A4 @ ( ord_less @ ( set @ A ) ) @ X8 ) @ A4 ) ) ).
% subset.suc_in_carrier
thf(fact_7985_pred__on_Osubset__suc,axiom,
! [A: $tType,X8: set @ A,Y7: set @ A,A4: set @ A,P: A > A > $o] :
( ( ord_less_eq @ ( set @ A ) @ X8 @ Y7 )
=> ( ord_less_eq @ ( set @ A ) @ X8 @ ( pred_suc @ A @ A4 @ P @ Y7 ) ) ) ).
% pred_on.subset_suc
thf(fact_7986_pred__on_Osuc__subset,axiom,
! [A: $tType,X8: set @ A,A4: set @ A,P: A > A > $o] : ( ord_less_eq @ ( set @ A ) @ X8 @ ( pred_suc @ A @ A4 @ P @ X8 ) ) ).
% pred_on.suc_subset
thf(fact_7987_pred__on_Osuc__in__carrier,axiom,
! [A: $tType,X8: set @ A,A4: set @ A,P: A > A > $o] :
( ( ord_less_eq @ ( set @ A ) @ X8 @ A4 )
=> ( ord_less_eq @ ( set @ A ) @ ( pred_suc @ A @ A4 @ P @ X8 ) @ A4 ) ) ).
% pred_on.suc_in_carrier
thf(fact_7988_int_Oabs__eq__iff,axiom,
! [X: product_prod @ nat @ nat,Y: product_prod @ nat @ nat] :
( ( ( abs_Integ @ X )
= ( abs_Integ @ Y ) )
= ( intrel @ X @ Y ) ) ).
% int.abs_eq_iff
thf(fact_7989_zero__int_Orsp,axiom,
intrel @ ( product_Pair @ nat @ nat @ ( zero_zero @ nat ) @ ( zero_zero @ nat ) ) @ ( product_Pair @ nat @ nat @ ( zero_zero @ nat ) @ ( zero_zero @ nat ) ) ).
% zero_int.rsp
thf(fact_7990_pred__on_Ochain__sucD,axiom,
! [A: $tType,A4: set @ A,P: A > A > $o,X8: set @ A] :
( ( pred_chain @ A @ A4 @ P @ X8 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( pred_suc @ A @ A4 @ P @ X8 ) @ A4 )
& ( pred_chain @ A @ A4 @ P @ ( pred_suc @ A @ A4 @ P @ X8 ) ) ) ) ).
% pred_on.chain_sucD
thf(fact_7991_subset_Ochain__suc,axiom,
! [A: $tType,A4: set @ ( set @ A ),X8: set @ ( set @ A )] :
( ( pred_chain @ ( set @ A ) @ A4 @ ( ord_less @ ( set @ A ) ) @ X8 )
=> ( pred_chain @ ( set @ A ) @ A4 @ ( ord_less @ ( set @ A ) ) @ ( pred_suc @ ( set @ A ) @ A4 @ ( ord_less @ ( set @ A ) ) @ X8 ) ) ) ).
% subset.chain_suc
thf(fact_7992_one__int_Orsp,axiom,
intrel @ ( product_Pair @ nat @ nat @ ( one_one @ nat ) @ ( zero_zero @ nat ) ) @ ( product_Pair @ nat @ nat @ ( one_one @ nat ) @ ( zero_zero @ nat ) ) ).
% one_int.rsp
thf(fact_7993_subset_Ochain__sucD,axiom,
! [A: $tType,A4: set @ ( set @ A ),X8: set @ ( set @ A )] :
( ( pred_chain @ ( set @ A ) @ A4 @ ( ord_less @ ( set @ A ) ) @ X8 )
=> ( ( ord_less_eq @ ( set @ ( set @ A ) ) @ ( pred_suc @ ( set @ A ) @ A4 @ ( ord_less @ ( set @ A ) ) @ X8 ) @ A4 )
& ( pred_chain @ ( set @ A ) @ A4 @ ( ord_less @ ( set @ A ) ) @ ( pred_suc @ ( set @ A ) @ A4 @ ( ord_less @ ( set @ A ) ) @ X8 ) ) ) ) ).
% subset.chain_sucD
thf(fact_7994_of__int_Orsp,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ A @ A @ intrel
@ ^ [Y5: A,Z2: A] : Y5 = Z2
@ ( product_case_prod @ nat @ nat @ A
@ ^ [I3: nat,J2: nat] : ( minus_minus @ A @ ( semiring_1_of_nat @ A @ I3 ) @ ( semiring_1_of_nat @ A @ J2 ) ) )
@ ( product_case_prod @ nat @ nat @ A
@ ^ [I3: nat,J2: nat] : ( minus_minus @ A @ ( semiring_1_of_nat @ A @ I3 ) @ ( semiring_1_of_nat @ A @ J2 ) ) ) ) ) ).
% of_int.rsp
thf(fact_7995_intrel__def,axiom,
( intrel
= ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [X4: nat,Y6: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U2: nat,V5: nat] :
( ( plus_plus @ nat @ X4 @ V5 )
= ( plus_plus @ nat @ U2 @ Y6 ) ) ) ) ) ).
% intrel_def
thf(fact_7996_less__int_Orsp,axiom,
( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( ( product_prod @ nat @ nat ) > $o ) @ ( ( product_prod @ nat @ nat ) > $o ) @ intrel
@ ( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ $o @ $o @ intrel
@ ^ [Y5: $o,Z2: $o] : Y5 = Z2 )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [X4: nat,Y6: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U2: nat,V5: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ X4 @ V5 ) @ ( plus_plus @ nat @ U2 @ Y6 ) ) ) )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [X4: nat,Y6: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U2: nat,V5: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ X4 @ V5 ) @ ( plus_plus @ nat @ U2 @ Y6 ) ) ) ) ) ).
% less_int.rsp
thf(fact_7997_less__eq__int_Orsp,axiom,
( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( ( product_prod @ nat @ nat ) > $o ) @ ( ( product_prod @ nat @ nat ) > $o ) @ intrel
@ ( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ $o @ $o @ intrel
@ ^ [Y5: $o,Z2: $o] : Y5 = Z2 )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [X4: nat,Y6: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U2: nat,V5: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ X4 @ V5 ) @ ( plus_plus @ nat @ U2 @ Y6 ) ) ) )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [X4: nat,Y6: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U2: nat,V5: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ X4 @ V5 ) @ ( plus_plus @ nat @ U2 @ Y6 ) ) ) ) ) ).
% less_eq_int.rsp
thf(fact_7998_int_Orel__eq__transfer,axiom,
( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ int @ ( ( product_prod @ nat @ nat ) > $o ) @ ( int > $o ) @ pcr_int
@ ( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ int @ $o @ $o @ pcr_int
@ ^ [Y5: $o,Z2: $o] : Y5 = Z2 )
@ intrel
@ ^ [Y5: int,Z2: int] : Y5 = Z2 ) ).
% int.rel_eq_transfer
thf(fact_7999_minus__int_Orsp,axiom,
( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) ) @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) ) @ intrel @ ( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ intrel @ intrel )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
@ ^ [X4: nat,Y6: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V5: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X4 @ V5 ) @ ( plus_plus @ nat @ Y6 @ U2 ) ) ) )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
@ ^ [X4: nat,Y6: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V5: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X4 @ V5 ) @ ( plus_plus @ nat @ Y6 @ U2 ) ) ) ) ) ).
% minus_int.rsp
thf(fact_8000_plus__int_Orsp,axiom,
( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) ) @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) ) @ intrel @ ( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ intrel @ intrel )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
@ ^ [X4: nat,Y6: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V5: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X4 @ U2 ) @ ( plus_plus @ nat @ Y6 @ V5 ) ) ) )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
@ ^ [X4: nat,Y6: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V5: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X4 @ U2 ) @ ( plus_plus @ nat @ Y6 @ V5 ) ) ) ) ) ).
% plus_int.rsp
thf(fact_8001_null__transfer,axiom,
! [A: $tType,B: $tType,A4: A > B > $o] :
( bNF_rel_fun @ ( list @ A ) @ ( list @ B ) @ $o @ $o @ ( list_all2 @ A @ B @ A4 )
@ ^ [Y5: $o,Z2: $o] : Y5 = Z2
@ ( null @ A )
@ ( null @ B ) ) ).
% null_transfer
thf(fact_8002_list__ex__transfer,axiom,
! [A: $tType,B: $tType,A4: A > B > $o] :
( bNF_rel_fun @ ( A > $o ) @ ( B > $o ) @ ( ( list @ A ) > $o ) @ ( ( list @ B ) > $o )
@ ( bNF_rel_fun @ A @ B @ $o @ $o @ A4
@ ^ [Y5: $o,Z2: $o] : Y5 = Z2 )
@ ( bNF_rel_fun @ ( list @ A ) @ ( list @ B ) @ $o @ $o @ ( list_all2 @ A @ B @ A4 )
@ ^ [Y5: $o,Z2: $o] : Y5 = Z2 )
@ ( list_ex @ A )
@ ( list_ex @ B ) ) ).
% list_ex_transfer
thf(fact_8003_list__ex__simps_I1_J,axiom,
! [A: $tType,P: A > $o,X: A,Xs: list @ A] :
( ( list_ex @ A @ P @ ( cons @ A @ X @ Xs ) )
= ( ( P @ X )
| ( list_ex @ A @ P @ Xs ) ) ) ).
% list_ex_simps(1)
thf(fact_8004_list__ex__simps_I2_J,axiom,
! [A: $tType,P: A > $o] :
~ ( list_ex @ A @ P @ ( nil @ A ) ) ).
% list_ex_simps(2)
thf(fact_8005_list__ex__append,axiom,
! [A: $tType,P: A > $o,Xs: list @ A,Ys: list @ A] :
( ( list_ex @ A @ P @ ( append @ A @ Xs @ Ys ) )
= ( ( list_ex @ A @ P @ Xs )
| ( list_ex @ A @ P @ Ys ) ) ) ).
% list_ex_append
thf(fact_8006_list__ex__rev,axiom,
! [A: $tType,P: A > $o,Xs: list @ A] :
( ( list_ex @ A @ P @ ( rev @ A @ Xs ) )
= ( list_ex @ A @ P @ Xs ) ) ).
% list_ex_rev
thf(fact_8007_eq__Nil__null,axiom,
! [A: $tType,Xs: list @ A] :
( ( Xs
= ( nil @ A ) )
= ( null @ A @ Xs ) ) ).
% eq_Nil_null
thf(fact_8008_null__rec_I2_J,axiom,
! [B: $tType] : ( null @ B @ ( nil @ B ) ) ).
% null_rec(2)
thf(fact_8009_null__rec_I1_J,axiom,
! [A: $tType,X: A,Xs: list @ A] :
~ ( null @ A @ ( cons @ A @ X @ Xs ) ) ).
% null_rec(1)
thf(fact_8010_list__ex__cong,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,F2: A > $o,G: A > $o] :
( ( Xs = Ys )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Ys ) )
=> ( ( F2 @ X5 )
= ( G @ X5 ) ) )
=> ( ( list_ex @ A @ F2 @ Xs )
= ( list_ex @ A @ G @ Ys ) ) ) ) ).
% list_ex_cong
thf(fact_8011_Bex__set__list__ex,axiom,
! [A: $tType,Xs: list @ A,P: A > $o] :
( ( ? [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
& ( P @ X4 ) ) )
= ( list_ex @ A @ P @ Xs ) ) ).
% Bex_set_list_ex
thf(fact_8012_list__ex__length,axiom,
! [A: $tType] :
( ( list_ex @ A )
= ( ^ [P3: A > $o,Xs3: list @ A] :
? [N: nat] :
( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs3 ) )
& ( P3 @ ( nth @ A @ Xs3 @ N ) ) ) ) ) ).
% list_ex_length
thf(fact_8013_is__empty__set,axiom,
! [A: $tType,Xs: list @ A] :
( ( is_empty @ A @ ( set2 @ A @ Xs ) )
= ( null @ A @ Xs ) ) ).
% is_empty_set
thf(fact_8014_list__all__transfer,axiom,
! [A: $tType,B: $tType,A4: A > B > $o] :
( bNF_rel_fun @ ( A > $o ) @ ( B > $o ) @ ( ( list @ A ) > $o ) @ ( ( list @ B ) > $o )
@ ( bNF_rel_fun @ A @ B @ $o @ $o @ A4
@ ^ [Y5: $o,Z2: $o] : Y5 = Z2 )
@ ( bNF_rel_fun @ ( list @ A ) @ ( list @ B ) @ $o @ $o @ ( list_all2 @ A @ B @ A4 )
@ ^ [Y5: $o,Z2: $o] : Y5 = Z2 )
@ ( list_all @ A )
@ ( list_all @ B ) ) ).
% list_all_transfer
thf(fact_8015_list_Opred__inject_I2_J,axiom,
! [A: $tType,P: A > $o,A3: A,Aa2: list @ A] :
( ( list_all @ A @ P @ ( cons @ A @ A3 @ Aa2 ) )
= ( ( P @ A3 )
& ( list_all @ A @ P @ Aa2 ) ) ) ).
% list.pred_inject(2)
thf(fact_8016_list__all__simps_I1_J,axiom,
! [A: $tType,P: A > $o,X: A,Xs: list @ A] :
( ( list_all @ A @ P @ ( cons @ A @ X @ Xs ) )
= ( ( P @ X )
& ( list_all @ A @ P @ Xs ) ) ) ).
% list_all_simps(1)
thf(fact_8017_list__all__simps_I2_J,axiom,
! [A: $tType,P: A > $o] : ( list_all @ A @ P @ ( nil @ A ) ) ).
% list_all_simps(2)
thf(fact_8018_list__all__append,axiom,
! [A: $tType,P: A > $o,Xs: list @ A,Ys: list @ A] :
( ( list_all @ A @ P @ ( append @ A @ Xs @ Ys ) )
= ( ( list_all @ A @ P @ Xs )
& ( list_all @ A @ P @ Ys ) ) ) ).
% list_all_append
thf(fact_8019_list__all__rev,axiom,
! [A: $tType,P: A > $o,Xs: list @ A] :
( ( list_all @ A @ P @ ( rev @ A @ Xs ) )
= ( list_all @ A @ P @ Xs ) ) ).
% list_all_rev
thf(fact_8020_list_Opred__mono,axiom,
! [A: $tType,P: A > $o,Pa: A > $o] :
( ( ord_less_eq @ ( A > $o ) @ P @ Pa )
=> ( ord_less_eq @ ( ( list @ A ) > $o ) @ ( list_all @ A @ P ) @ ( list_all @ A @ Pa ) ) ) ).
% list.pred_mono
thf(fact_8021_list_Opred__map,axiom,
! [B: $tType,A: $tType,Q: B > $o,F2: A > B,X: list @ A] :
( ( list_all @ B @ Q @ ( map @ A @ B @ F2 @ X ) )
= ( list_all @ A @ ( comp @ B @ $o @ A @ Q @ F2 ) @ X ) ) ).
% list.pred_map
thf(fact_8022_list__all__cong,axiom,
! [A: $tType,X: list @ A,Ya: list @ A,P: A > $o,Pa: A > $o] :
( ( X = Ya )
=> ( ! [Z: A] :
( ( member @ A @ Z @ ( set2 @ A @ Ya ) )
=> ( ( P @ Z )
= ( Pa @ Z ) ) )
=> ( ( list_all @ A @ P @ X )
= ( list_all @ A @ Pa @ Ya ) ) ) ) ).
% list_all_cong
thf(fact_8023_list_Opred__mono__strong,axiom,
! [A: $tType,P: A > $o,X: list @ A,Pa: A > $o] :
( ( list_all @ A @ P @ X )
=> ( ! [Z: A] :
( ( member @ A @ Z @ ( set2 @ A @ X ) )
=> ( ( P @ Z )
=> ( Pa @ Z ) ) )
=> ( list_all @ A @ Pa @ X ) ) ) ).
% list.pred_mono_strong
thf(fact_8024_Ball__set__list__all,axiom,
! [A: $tType,Xs: list @ A,P: A > $o] :
( ( ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
=> ( P @ X4 ) ) )
= ( list_all @ A @ P @ Xs ) ) ).
% Ball_set_list_all
thf(fact_8025_list__all__iff,axiom,
! [A: $tType] :
( ( list_all @ A )
= ( ^ [P3: A > $o,X4: list @ A] :
! [Y6: A] :
( ( member @ A @ Y6 @ ( set2 @ A @ X4 ) )
=> ( P3 @ Y6 ) ) ) ) ).
% list_all_iff
thf(fact_8026_list_Opred__set,axiom,
! [A: $tType] :
( ( list_all @ A )
= ( ^ [P3: A > $o,X4: list @ A] :
! [Y6: A] :
( ( member @ A @ Y6 @ ( set2 @ A @ X4 ) )
=> ( P3 @ Y6 ) ) ) ) ).
% list.pred_set
thf(fact_8027_list_Omap__cong__pred,axiom,
! [B: $tType,A: $tType,X: list @ A,Ya: list @ A,F2: A > B,G: A > B] :
( ( X = Ya )
=> ( ( list_all @ A
@ ^ [Z6: A] :
( ( F2 @ Z6 )
= ( G @ Z6 ) )
@ Ya )
=> ( ( map @ A @ B @ F2 @ X )
= ( map @ A @ B @ G @ Ya ) ) ) ) ).
% list.map_cong_pred
thf(fact_8028_list_Opred__True,axiom,
! [A: $tType] :
( ( list_all @ A
@ ^ [Uu3: A] : $true )
= ( ^ [Uu3: list @ A] : $true ) ) ).
% list.pred_True
thf(fact_8029_list_Opred__inject_I1_J,axiom,
! [A: $tType,P: A > $o] : ( list_all @ A @ P @ ( nil @ A ) ) ).
% list.pred_inject(1)
thf(fact_8030_list__all__length,axiom,
! [A: $tType] :
( ( list_all @ A )
= ( ^ [P3: A > $o,Xs3: list @ A] :
! [N: nat] :
( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs3 ) )
=> ( P3 @ ( nth @ A @ Xs3 @ N ) ) ) ) ) ).
% list_all_length
thf(fact_8031_Set_Ois__empty__def,axiom,
! [A: $tType] :
( ( is_empty @ A )
= ( ^ [A7: set @ A] :
( A7
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Set.is_empty_def
thf(fact_8032_list_Opred__transfer,axiom,
! [A: $tType,B: $tType,R: A > B > $o] :
( bNF_rel_fun @ ( A > $o ) @ ( B > $o ) @ ( ( list @ A ) > $o ) @ ( ( list @ B ) > $o )
@ ( bNF_rel_fun @ A @ B @ $o @ $o @ R
@ ^ [Y5: $o,Z2: $o] : Y5 = Z2 )
@ ( bNF_rel_fun @ ( list @ A ) @ ( list @ B ) @ $o @ $o @ ( list_all2 @ A @ B @ R )
@ ^ [Y5: $o,Z2: $o] : Y5 = Z2 )
@ ( list_all @ A )
@ ( list_all @ B ) ) ).
% list.pred_transfer
thf(fact_8033_list__ex1__simps_I2_J,axiom,
! [A: $tType,P: A > $o,X: A,Xs: list @ A] :
( ( list_ex1 @ A @ P @ ( cons @ A @ X @ Xs ) )
= ( ( ( P @ X )
=> ( list_all @ A
@ ^ [Y6: A] :
( ~ ( P @ Y6 )
| ( X = Y6 ) )
@ Xs ) )
& ( ~ ( P @ X )
=> ( list_ex1 @ A @ P @ Xs ) ) ) ) ).
% list_ex1_simps(2)
thf(fact_8034_mono__transfer,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ D )
& ( order @ C )
& ( order @ A ) )
=> ! [A4: A > B > $o,B2: C > D > $o] :
( ( bi_total @ A @ B @ A4 )
=> ( ( bNF_rel_fun @ A @ B @ ( A > $o ) @ ( B > $o ) @ A4
@ ( bNF_rel_fun @ A @ B @ $o @ $o @ A4
@ ^ [Y5: $o,Z2: $o] : Y5 = Z2 )
@ ( ord_less_eq @ A )
@ ( ord_less_eq @ B ) )
=> ( ( bNF_rel_fun @ C @ D @ ( C > $o ) @ ( D > $o ) @ B2
@ ( bNF_rel_fun @ C @ D @ $o @ $o @ B2
@ ^ [Y5: $o,Z2: $o] : Y5 = Z2 )
@ ( ord_less_eq @ C )
@ ( ord_less_eq @ D ) )
=> ( bNF_rel_fun @ ( A > C ) @ ( B > D ) @ $o @ $o @ ( bNF_rel_fun @ A @ B @ C @ D @ A4 @ B2 )
@ ^ [Y5: $o,Z2: $o] : Y5 = Z2
@ ( order_mono @ A @ C )
@ ( order_mono @ B @ D ) ) ) ) ) ) ).
% mono_transfer
thf(fact_8035_list__ex1__simps_I1_J,axiom,
! [A: $tType,P: A > $o] :
~ ( list_ex1 @ A @ P @ ( nil @ A ) ) ).
% list_ex1_simps(1)
thf(fact_8036_list__ex1__iff,axiom,
! [A: $tType] :
( ( list_ex1 @ A )
= ( ^ [P3: A > $o,Xs3: list @ A] :
? [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs3 ) )
& ( P3 @ X4 )
& ! [Y6: A] :
( ( ( member @ A @ Y6 @ ( set2 @ A @ Xs3 ) )
& ( P3 @ Y6 ) )
=> ( Y6 = X4 ) ) ) ) ) ).
% list_ex1_iff
thf(fact_8037_int_Obi__total,axiom,
bi_total @ ( product_prod @ nat @ nat ) @ int @ pcr_int ).
% int.bi_total
thf(fact_8038_list_Obi__total__rel,axiom,
! [B: $tType,A: $tType,R: A > B > $o] :
( ( bi_total @ A @ B @ R )
=> ( bi_total @ ( list @ A ) @ ( list @ B ) @ ( list_all2 @ A @ B @ R ) ) ) ).
% list.bi_total_rel
thf(fact_8039_can__select__set__list__ex1,axiom,
! [A: $tType,P: A > $o,A4: list @ A] :
( ( can_select @ A @ P @ ( set2 @ A @ A4 ) )
= ( list_ex1 @ A @ P @ A4 ) ) ).
% can_select_set_list_ex1
thf(fact_8040_monotone__parametric,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,A4: A > B > $o,B2: C > D > $o] :
( ( bi_total @ A @ B @ A4 )
=> ( bNF_rel_fun @ ( A > A > $o ) @ ( B > B > $o ) @ ( ( C > C > $o ) > ( A > C ) > $o ) @ ( ( D > D > $o ) > ( B > D ) > $o )
@ ( bNF_rel_fun @ A @ B @ ( A > $o ) @ ( B > $o ) @ A4
@ ( bNF_rel_fun @ A @ B @ $o @ $o @ A4
@ ^ [Y5: $o,Z2: $o] : Y5 = Z2 ) )
@ ( bNF_rel_fun @ ( C > C > $o ) @ ( D > D > $o ) @ ( ( A > C ) > $o ) @ ( ( B > D ) > $o )
@ ( bNF_rel_fun @ C @ D @ ( C > $o ) @ ( D > $o ) @ B2
@ ( bNF_rel_fun @ C @ D @ $o @ $o @ B2
@ ^ [Y5: $o,Z2: $o] : Y5 = Z2 ) )
@ ( bNF_rel_fun @ ( A > C ) @ ( B > D ) @ $o @ $o @ ( bNF_rel_fun @ A @ B @ C @ D @ A4 @ B2 )
@ ^ [Y5: $o,Z2: $o] : Y5 = Z2 ) )
@ ( comple7038119648293358887notone @ A @ C )
@ ( comple7038119648293358887notone @ B @ D ) ) ) ).
% monotone_parametric
thf(fact_8041_can__select__def,axiom,
! [A: $tType] :
( ( can_select @ A )
= ( ^ [P3: A > $o,A7: set @ A] :
? [X4: A] :
( ( member @ A @ X4 @ A7 )
& ( P3 @ X4 )
& ! [Y6: A] :
( ( ( member @ A @ Y6 @ A7 )
& ( P3 @ Y6 ) )
=> ( Y6 = X4 ) ) ) ) ) ).
% can_select_def
thf(fact_8042_fun__ord__parametric,axiom,
! [C: $tType,D: $tType,A: $tType,B: $tType,F: $tType,E4: $tType,C5: A > B > $o,A4: C > E4 > $o,B2: D > F > $o] :
( ( bi_total @ A @ B @ C5 )
=> ( bNF_rel_fun @ ( C > D > $o ) @ ( E4 > F > $o ) @ ( ( A > C ) > ( A > D ) > $o ) @ ( ( B > E4 ) > ( B > F ) > $o )
@ ( bNF_rel_fun @ C @ E4 @ ( D > $o ) @ ( F > $o ) @ A4
@ ( bNF_rel_fun @ D @ F @ $o @ $o @ B2
@ ^ [Y5: $o,Z2: $o] : Y5 = Z2 ) )
@ ( bNF_rel_fun @ ( A > C ) @ ( B > E4 ) @ ( ( A > D ) > $o ) @ ( ( B > F ) > $o ) @ ( bNF_rel_fun @ A @ B @ C @ E4 @ C5 @ A4 )
@ ( bNF_rel_fun @ ( A > D ) @ ( B > F ) @ $o @ $o @ ( bNF_rel_fun @ A @ B @ D @ F @ C5 @ B2 )
@ ^ [Y5: $o,Z2: $o] : Y5 = Z2 ) )
@ ( partial_fun_ord @ C @ D @ A )
@ ( partial_fun_ord @ E4 @ F @ B ) ) ) ).
% fun_ord_parametric
thf(fact_8043_fixp__induct,axiom,
! [A: $tType] :
( ( comple9053668089753744459l_ccpo @ A )
=> ! [P: A > $o,F2: A > A] :
( ( comple1908693960933563346ssible @ A @ ( complete_Sup_Sup @ A ) @ ( ord_less_eq @ A ) @ P )
=> ( ( comple7038119648293358887notone @ A @ A @ ( ord_less_eq @ A ) @ ( ord_less_eq @ A ) @ F2 )
=> ( ( P @ ( complete_Sup_Sup @ A @ ( bot_bot @ ( set @ A ) ) ) )
=> ( ! [X5: A] :
( ( P @ X5 )
=> ( P @ ( F2 @ X5 ) ) )
=> ( P @ ( comple115746919287870866o_fixp @ A @ F2 ) ) ) ) ) ) ) ).
% fixp_induct
thf(fact_8044_fixp__lowerbound,axiom,
! [A: $tType] :
( ( comple9053668089753744459l_ccpo @ A )
=> ! [F2: A > A,Z3: A] :
( ( comple7038119648293358887notone @ A @ A @ ( ord_less_eq @ A ) @ ( ord_less_eq @ A ) @ F2 )
=> ( ( ord_less_eq @ A @ ( F2 @ Z3 ) @ Z3 )
=> ( ord_less_eq @ A @ ( comple115746919287870866o_fixp @ A @ F2 ) @ Z3 ) ) ) ) ).
% fixp_lowerbound
thf(fact_8045_fixp__unfold,axiom,
! [A: $tType] :
( ( comple9053668089753744459l_ccpo @ A )
=> ! [F2: A > A] :
( ( comple7038119648293358887notone @ A @ A @ ( ord_less_eq @ A ) @ ( ord_less_eq @ A ) @ F2 )
=> ( ( comple115746919287870866o_fixp @ A @ F2 )
= ( F2 @ ( comple115746919287870866o_fixp @ A @ F2 ) ) ) ) ) ).
% fixp_unfold
thf(fact_8046_irrefl__tranclI,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),X: A] :
( ( ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ ( converse @ A @ A @ R2 ) @ ( transitive_rtrancl @ A @ R2 ) )
= ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ X ) @ ( transitive_trancl @ A @ R2 ) ) ) ).
% irrefl_tranclI
thf(fact_8047_less__than__iff,axiom,
! [X: nat,Y: nat] :
( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ X @ Y ) @ less_than )
= ( ord_less @ nat @ X @ Y ) ) ).
% less_than_iff
thf(fact_8048_converse__empty,axiom,
! [B: $tType,A: $tType] :
( ( converse @ B @ A @ ( bot_bot @ ( set @ ( product_prod @ B @ A ) ) ) )
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% converse_empty
thf(fact_8049_finite__converse,axiom,
! [A: $tType,B: $tType,R2: set @ ( product_prod @ B @ A )] :
( ( finite_finite2 @ ( product_prod @ A @ B ) @ ( converse @ B @ A @ R2 ) )
= ( finite_finite2 @ ( product_prod @ B @ A ) @ R2 ) ) ).
% finite_converse
thf(fact_8050_Image__subset__eq,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ B @ A ),A4: set @ B,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ R2 @ A4 ) @ B2 )
= ( ord_less_eq @ ( set @ B ) @ A4 @ ( uminus_uminus @ ( set @ B ) @ ( image @ A @ B @ ( converse @ B @ A @ R2 ) @ ( uminus_uminus @ ( set @ A ) @ B2 ) ) ) ) ) ).
% Image_subset_eq
thf(fact_8051_listrel1__converse,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( listrel1 @ A @ ( converse @ A @ A @ R2 ) )
= ( converse @ ( list @ A ) @ ( list @ A ) @ ( listrel1 @ A @ R2 ) ) ) ).
% listrel1_converse
thf(fact_8052_in__listrel1__converse,axiom,
! [A: $tType,X: list @ A,Y: list @ A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Y ) @ ( listrel1 @ A @ ( converse @ A @ A @ R2 ) ) )
= ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Y ) @ ( converse @ ( list @ A ) @ ( list @ A ) @ ( listrel1 @ A @ R2 ) ) ) ) ).
% in_listrel1_converse
thf(fact_8053_Image__INT__eq,axiom,
! [A: $tType,B: $tType,C: $tType,R2: set @ ( product_prod @ B @ A ),A4: set @ C,B2: C > ( set @ B )] :
( ( single_valued @ A @ B @ ( converse @ B @ A @ R2 ) )
=> ( ( A4
!= ( bot_bot @ ( set @ C ) ) )
=> ( ( image @ B @ A @ R2 @ ( complete_Inf_Inf @ ( set @ B ) @ ( image2 @ C @ ( set @ B ) @ B2 @ A4 ) ) )
= ( complete_Inf_Inf @ ( set @ A )
@ ( image2 @ C @ ( set @ A )
@ ^ [X4: C] : ( image @ B @ A @ R2 @ ( B2 @ X4 ) )
@ A4 ) ) ) ) ) ).
% Image_INT_eq
thf(fact_8054_trans__wf__iff,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( trans @ A @ R2 )
=> ( ( wf @ A @ R2 )
= ( ! [A5: A] :
( wf @ A
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R2
@ ( product_Sigma @ A @ A @ ( image @ A @ A @ ( converse @ A @ A @ R2 ) @ ( insert2 @ A @ A5 @ ( bot_bot @ ( set @ A ) ) ) )
@ ^ [Uu3: A] : ( image @ A @ A @ ( converse @ A @ A @ R2 ) @ ( insert2 @ A @ A5 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ).
% trans_wf_iff
thf(fact_8055_lenlex__transI,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( trans @ A @ R2 )
=> ( trans @ ( list @ A ) @ ( lenlex @ A @ R2 ) ) ) ).
% lenlex_transI
thf(fact_8056_lexord__trans,axiom,
! [A: $tType,X: list @ A,Y: list @ A,R2: set @ ( product_prod @ A @ A ),Z3: list @ A] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Y ) @ ( lexord @ A @ R2 ) )
=> ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Y @ Z3 ) @ ( lexord @ A @ R2 ) )
=> ( ( trans @ A @ R2 )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Z3 ) @ ( lexord @ A @ R2 ) ) ) ) ) ).
% lexord_trans
thf(fact_8057_lenlex__trans,axiom,
! [A: $tType,X: list @ A,Y: list @ A,R2: set @ ( product_prod @ A @ A ),Z3: list @ A] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Y ) @ ( lenlex @ A @ R2 ) )
=> ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Y @ Z3 ) @ ( lenlex @ A @ R2 ) )
=> ( ( trans @ A @ R2 )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Z3 ) @ ( lenlex @ A @ R2 ) ) ) ) ) ).
% lenlex_trans
thf(fact_8058_lex__transI,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( trans @ A @ R2 )
=> ( trans @ ( list @ A ) @ ( lex @ A @ R2 ) ) ) ).
% lex_transI
thf(fact_8059_lexord__transI,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( trans @ A @ R2 )
=> ( trans @ ( list @ A ) @ ( lexord @ A @ R2 ) ) ) ).
% lexord_transI
thf(fact_8060_listrel__trans,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( trans @ A @ R2 )
=> ( trans @ ( list @ A ) @ ( listrel @ A @ A @ R2 ) ) ) ).
% listrel_trans
thf(fact_8061_lexn__transI,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),N2: nat] :
( ( trans @ A @ R2 )
=> ( trans @ ( list @ A ) @ ( lexn @ A @ R2 @ N2 ) ) ) ).
% lexn_transI
thf(fact_8062_trans__empty,axiom,
! [A: $tType] : ( trans @ A @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ).
% trans_empty
thf(fact_8063_single__valued__empty,axiom,
! [B: $tType,A: $tType] : ( single_valued @ A @ B @ ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% single_valued_empty
thf(fact_8064_trans__singleton,axiom,
! [A: $tType,A3: A] : ( trans @ A @ ( insert2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ A3 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ).
% trans_singleton
thf(fact_8065_under__incr,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A3: A,B3: A] :
( ( trans @ A @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B3 ) @ R2 )
=> ( ord_less_eq @ ( set @ A ) @ ( order_under @ A @ R2 @ A3 ) @ ( order_under @ A @ R2 @ B3 ) ) ) ) ).
% under_incr
thf(fact_8066_underS__incr,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A3: A,B3: A] :
( ( trans @ A @ R2 )
=> ( ( antisym @ A @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B3 ) @ R2 )
=> ( ord_less_eq @ ( set @ A ) @ ( order_underS @ A @ R2 @ A3 ) @ ( order_underS @ A @ R2 @ B3 ) ) ) ) ) ).
% underS_incr
thf(fact_8067_wf__converse,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( irrefl @ A @ R2 )
=> ( ( trans @ A @ R2 )
=> ( ( finite_finite2 @ ( product_prod @ A @ A ) @ R2 )
=> ( wf @ A @ ( converse @ A @ A @ R2 ) ) ) ) ) ).
% wf_converse
thf(fact_8068_wf__finite__segments,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( irrefl @ A @ R2 )
=> ( ( trans @ A @ R2 )
=> ( ! [X5: A] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [Y6: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y6 @ X5 ) @ R2 ) ) )
=> ( wf @ A @ R2 ) ) ) ) ).
% wf_finite_segments
thf(fact_8069_measures__def,axiom,
! [A: $tType] :
( ( measures @ A )
= ( ^ [Fs2: list @ ( A > nat )] :
( inv_image @ ( list @ nat ) @ A @ ( lex @ nat @ less_than )
@ ^ [A5: A] :
( map @ ( A > nat ) @ nat
@ ^ [F3: A > nat] : ( F3 @ A5 )
@ Fs2 ) ) ) ) ).
% measures_def
thf(fact_8070_elimnum,axiom,
! [Info: option @ ( product_prod @ nat @ nat ),Deg: nat,TreeList2: list @ vEBT_VEBT,Summary: vEBT_VEBT,N2: nat] :
( ( vEBT_invar_vebt @ ( vEBT_Node @ Info @ Deg @ TreeList2 @ Summary ) @ N2 )
=> ( ( vEBT_VEBT_elim_dead @ ( vEBT_Node @ Info @ Deg @ TreeList2 @ Summary ) @ ( extended_enat2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) )
= ( vEBT_Node @ Info @ Deg @ TreeList2 @ Summary ) ) ) ).
% elimnum
thf(fact_8071_idiff__enat__0__right,axiom,
! [N2: extended_enat] :
( ( minus_minus @ extended_enat @ N2 @ ( extended_enat2 @ ( zero_zero @ nat ) ) )
= N2 ) ).
% idiff_enat_0_right
thf(fact_8072_idiff__enat__0,axiom,
! [N2: extended_enat] :
( ( minus_minus @ extended_enat @ ( extended_enat2 @ ( zero_zero @ nat ) ) @ N2 )
= ( extended_enat2 @ ( zero_zero @ nat ) ) ) ).
% idiff_enat_0
thf(fact_8073_plus__enat__simps_I1_J,axiom,
! [M2: nat,N2: nat] :
( ( plus_plus @ extended_enat @ ( extended_enat2 @ M2 ) @ ( extended_enat2 @ N2 ) )
= ( extended_enat2 @ ( plus_plus @ nat @ M2 @ N2 ) ) ) ).
% plus_enat_simps(1)
thf(fact_8074_enat__ord__simps_I2_J,axiom,
! [M2: nat,N2: nat] :
( ( ord_less @ extended_enat @ ( extended_enat2 @ M2 ) @ ( extended_enat2 @ N2 ) )
= ( ord_less @ nat @ M2 @ N2 ) ) ).
% enat_ord_simps(2)
thf(fact_8075_enat__ord__simps_I1_J,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq @ extended_enat @ ( extended_enat2 @ M2 ) @ ( extended_enat2 @ N2 ) )
= ( ord_less_eq @ nat @ M2 @ N2 ) ) ).
% enat_ord_simps(1)
thf(fact_8076_numeral__less__enat__iff,axiom,
! [M2: num,N2: nat] :
( ( ord_less @ extended_enat @ ( numeral_numeral @ extended_enat @ M2 ) @ ( extended_enat2 @ N2 ) )
= ( ord_less @ nat @ ( numeral_numeral @ nat @ M2 ) @ N2 ) ) ).
% numeral_less_enat_iff
thf(fact_8077_numeral__le__enat__iff,axiom,
! [M2: num,N2: nat] :
( ( ord_less_eq @ extended_enat @ ( numeral_numeral @ extended_enat @ M2 ) @ ( extended_enat2 @ N2 ) )
= ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ M2 ) @ N2 ) ) ).
% numeral_le_enat_iff
thf(fact_8078_Suc__ile__eq,axiom,
! [M2: nat,N2: extended_enat] :
( ( ord_less_eq @ extended_enat @ ( extended_enat2 @ ( suc @ M2 ) ) @ N2 )
= ( ord_less @ extended_enat @ ( extended_enat2 @ M2 ) @ N2 ) ) ).
% Suc_ile_eq
thf(fact_8079_finite__enat__bounded,axiom,
! [A4: set @ extended_enat,N2: nat] :
( ! [Y3: extended_enat] :
( ( member @ extended_enat @ Y3 @ A4 )
=> ( ord_less_eq @ extended_enat @ Y3 @ ( extended_enat2 @ N2 ) ) )
=> ( finite_finite2 @ extended_enat @ A4 ) ) ).
% finite_enat_bounded
thf(fact_8080_iadd__le__enat__iff,axiom,
! [X: extended_enat,Y: extended_enat,N2: nat] :
( ( ord_less_eq @ extended_enat @ ( plus_plus @ extended_enat @ X @ Y ) @ ( extended_enat2 @ N2 ) )
= ( ? [Y10: nat,X10: nat] :
( ( X
= ( extended_enat2 @ X10 ) )
& ( Y
= ( extended_enat2 @ Y10 ) )
& ( ord_less_eq @ nat @ ( plus_plus @ nat @ X10 @ Y10 ) @ N2 ) ) ) ) ).
% iadd_le_enat_iff
thf(fact_8081_enat__0__iff_I2_J,axiom,
! [X: nat] :
( ( ( zero_zero @ extended_enat )
= ( extended_enat2 @ X ) )
= ( X
= ( zero_zero @ nat ) ) ) ).
% enat_0_iff(2)
thf(fact_8082_enat__0__iff_I1_J,axiom,
! [X: nat] :
( ( ( extended_enat2 @ X )
= ( zero_zero @ extended_enat ) )
= ( X
= ( zero_zero @ nat ) ) ) ).
% enat_0_iff(1)
thf(fact_8083_zero__enat__def,axiom,
( ( zero_zero @ extended_enat )
= ( extended_enat2 @ ( zero_zero @ nat ) ) ) ).
% zero_enat_def
thf(fact_8084_less__enatE,axiom,
! [N2: extended_enat,M2: nat] :
( ( ord_less @ extended_enat @ N2 @ ( extended_enat2 @ M2 ) )
=> ~ ! [K3: nat] :
( ( N2
= ( extended_enat2 @ K3 ) )
=> ~ ( ord_less @ nat @ K3 @ M2 ) ) ) ).
% less_enatE
thf(fact_8085_elimcomplete,axiom,
! [Info: option @ ( product_prod @ nat @ nat ),Deg: nat,TreeList2: list @ vEBT_VEBT,Summary: vEBT_VEBT,N2: nat] :
( ( vEBT_invar_vebt @ ( vEBT_Node @ Info @ Deg @ TreeList2 @ Summary ) @ N2 )
=> ( ( vEBT_VEBT_elim_dead @ ( vEBT_Node @ Info @ Deg @ TreeList2 @ Summary ) @ ( extend4730790105801354508finity @ extended_enat ) )
= ( vEBT_Node @ Info @ Deg @ TreeList2 @ Summary ) ) ) ).
% elimcomplete
thf(fact_8086_times__enat__simps_I4_J,axiom,
! [M2: nat] :
( ( ( M2
= ( zero_zero @ nat ) )
=> ( ( times_times @ extended_enat @ ( extended_enat2 @ M2 ) @ ( extend4730790105801354508finity @ extended_enat ) )
= ( zero_zero @ extended_enat ) ) )
& ( ( M2
!= ( zero_zero @ nat ) )
=> ( ( times_times @ extended_enat @ ( extended_enat2 @ M2 ) @ ( extend4730790105801354508finity @ extended_enat ) )
= ( extend4730790105801354508finity @ extended_enat ) ) ) ) ).
% times_enat_simps(4)
thf(fact_8087_times__enat__simps_I3_J,axiom,
! [N2: nat] :
( ( ( N2
= ( zero_zero @ nat ) )
=> ( ( times_times @ extended_enat @ ( extend4730790105801354508finity @ extended_enat ) @ ( extended_enat2 @ N2 ) )
= ( zero_zero @ extended_enat ) ) )
& ( ( N2
!= ( zero_zero @ nat ) )
=> ( ( times_times @ extended_enat @ ( extend4730790105801354508finity @ extended_enat ) @ ( extended_enat2 @ N2 ) )
= ( extend4730790105801354508finity @ extended_enat ) ) ) ) ).
% times_enat_simps(3)
thf(fact_8088_Inf__enat__def,axiom,
( ( complete_Inf_Inf @ extended_enat )
= ( ^ [A7: set @ extended_enat] :
( if @ extended_enat
@ ( A7
= ( bot_bot @ ( set @ extended_enat ) ) )
@ ( extend4730790105801354508finity @ extended_enat )
@ ( ord_Least @ extended_enat
@ ^ [X4: extended_enat] : ( member @ extended_enat @ X4 @ A7 ) ) ) ) ) ).
% Inf_enat_def
thf(fact_8089_bot__enat__def,axiom,
( ( bot_bot @ extended_enat )
= ( zero_zero @ extended_enat ) ) ).
% bot_enat_def
thf(fact_8090_Sup__enat__def,axiom,
( ( complete_Sup_Sup @ extended_enat )
= ( ^ [A7: set @ extended_enat] :
( if @ extended_enat
@ ( A7
= ( bot_bot @ ( set @ extended_enat ) ) )
@ ( zero_zero @ extended_enat )
@ ( if @ extended_enat @ ( finite_finite2 @ extended_enat @ A7 ) @ ( lattic643756798349783984er_Max @ extended_enat @ A7 ) @ ( extend4730790105801354508finity @ extended_enat ) ) ) ) ) ).
% Sup_enat_def
thf(fact_8091_times__enat__def,axiom,
( ( times_times @ extended_enat )
= ( ^ [M: extended_enat,N: extended_enat] :
( extended_case_enat @ extended_enat
@ ^ [O: nat] :
( extended_case_enat @ extended_enat
@ ^ [P6: nat] : ( extended_enat2 @ ( times_times @ nat @ O @ P6 ) )
@ ( if @ extended_enat
@ ( O
= ( zero_zero @ nat ) )
@ ( zero_zero @ extended_enat )
@ ( extend4730790105801354508finity @ extended_enat ) )
@ N )
@ ( if @ extended_enat
@ ( N
= ( zero_zero @ extended_enat ) )
@ ( zero_zero @ extended_enat )
@ ( extend4730790105801354508finity @ extended_enat ) )
@ M ) ) ) ).
% times_enat_def
thf(fact_8092_plus__enat__def,axiom,
( ( plus_plus @ extended_enat )
= ( ^ [M: extended_enat,N: extended_enat] :
( extended_case_enat @ extended_enat
@ ^ [O: nat] :
( extended_case_enat @ extended_enat
@ ^ [P6: nat] : ( extended_enat2 @ ( plus_plus @ nat @ O @ P6 ) )
@ ( extend4730790105801354508finity @ extended_enat )
@ N )
@ ( extend4730790105801354508finity @ extended_enat )
@ M ) ) ) ).
% plus_enat_def
thf(fact_8093_eSuc__Max,axiom,
! [A4: set @ extended_enat] :
( ( finite_finite2 @ extended_enat @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ extended_enat ) ) )
=> ( ( extended_eSuc @ ( lattic643756798349783984er_Max @ extended_enat @ A4 ) )
= ( lattic643756798349783984er_Max @ extended_enat @ ( image2 @ extended_enat @ extended_enat @ extended_eSuc @ A4 ) ) ) ) ) ).
% eSuc_Max
thf(fact_8094_eSuc__def,axiom,
( extended_eSuc
= ( extended_case_enat @ extended_enat
@ ^ [N: nat] : ( extended_enat2 @ ( suc @ N ) )
@ ( extend4730790105801354508finity @ extended_enat ) ) ) ).
% eSuc_def
thf(fact_8095_enat__eSuc__iff,axiom,
! [Y: nat,X: extended_enat] :
( ( ( extended_enat2 @ Y )
= ( extended_eSuc @ X ) )
= ( ? [N: nat] :
( ( Y
= ( suc @ N ) )
& ( ( extended_enat2 @ N )
= X ) ) ) ) ).
% enat_eSuc_iff
thf(fact_8096_eSuc__enat__iff,axiom,
! [X: extended_enat,Y: nat] :
( ( ( extended_eSuc @ X )
= ( extended_enat2 @ Y ) )
= ( ? [N: nat] :
( ( Y
= ( suc @ N ) )
& ( X
= ( extended_enat2 @ N ) ) ) ) ) ).
% eSuc_enat_iff
thf(fact_8097_eSuc__enat,axiom,
! [N2: nat] :
( ( extended_eSuc @ ( extended_enat2 @ N2 ) )
= ( extended_enat2 @ ( suc @ N2 ) ) ) ).
% eSuc_enat
thf(fact_8098_eSuc__Sup,axiom,
! [A4: set @ extended_enat] :
( ( A4
!= ( bot_bot @ ( set @ extended_enat ) ) )
=> ( ( extended_eSuc @ ( complete_Sup_Sup @ extended_enat @ A4 ) )
= ( complete_Sup_Sup @ extended_enat @ ( image2 @ extended_enat @ extended_enat @ extended_eSuc @ A4 ) ) ) ) ).
% eSuc_Sup
thf(fact_8099_wo__rel_Oofilter__def,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A4: set @ A] :
( ( bNF_Wellorder_wo_rel @ A @ R2 )
=> ( ( order_ofilter @ A @ R2 @ A4 )
= ( ( ord_less_eq @ ( set @ A ) @ A4 @ ( field2 @ A @ R2 ) )
& ! [X4: A] :
( ( member @ A @ X4 @ A4 )
=> ( ord_less_eq @ ( set @ A ) @ ( order_under @ A @ R2 @ X4 ) @ A4 ) ) ) ) ) ).
% wo_rel.ofilter_def
thf(fact_8100_has__vector__derivative__scaleR,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: real > real,F8: real,X: real,S: set @ real,G: real > A,G6: A] :
( ( has_field_derivative @ real @ F2 @ F8 @ ( topolo174197925503356063within @ real @ X @ S ) )
=> ( ( has_ve8173657378732805170vative @ A @ G @ G6 @ ( topolo174197925503356063within @ real @ X @ S ) )
=> ( has_ve8173657378732805170vative @ A
@ ^ [X4: real] : ( real_V8093663219630862766scaleR @ A @ ( F2 @ X4 ) @ ( G @ X4 ) )
@ ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ ( F2 @ X ) @ G6 ) @ ( real_V8093663219630862766scaleR @ A @ F8 @ ( G @ X ) ) )
@ ( topolo174197925503356063within @ real @ X @ S ) ) ) ) ) ).
% has_vector_derivative_scaleR
thf(fact_8101_has__vector__derivative__add__const,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [G: real > A,Z3: A,F8: A,Net: filter @ real] :
( ( has_ve8173657378732805170vative @ A
@ ^ [T3: real] : ( plus_plus @ A @ ( G @ T3 ) @ Z3 )
@ F8
@ Net )
= ( has_ve8173657378732805170vative @ A @ G @ F8 @ Net ) ) ) ).
% has_vector_derivative_add_const
thf(fact_8102_has__vector__derivative__add,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: real > A,F8: A,Net: filter @ real,G: real > A,G6: A] :
( ( has_ve8173657378732805170vative @ A @ F2 @ F8 @ Net )
=> ( ( has_ve8173657378732805170vative @ A @ G @ G6 @ Net )
=> ( has_ve8173657378732805170vative @ A
@ ^ [X4: real] : ( plus_plus @ A @ ( F2 @ X4 ) @ ( G @ X4 ) )
@ ( plus_plus @ A @ F8 @ G6 )
@ Net ) ) ) ) ).
% has_vector_derivative_add
thf(fact_8103_has__vector__derivative__const,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [C2: A,Net: filter @ real] :
( has_ve8173657378732805170vative @ A
@ ^ [X4: real] : C2
@ ( zero_zero @ A )
@ Net ) ) ).
% has_vector_derivative_const
thf(fact_8104_wo__rel_Oofilter__linord,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A4: set @ A,B2: set @ A] :
( ( bNF_Wellorder_wo_rel @ A @ R2 )
=> ( ( order_ofilter @ A @ R2 @ A4 )
=> ( ( order_ofilter @ A @ R2 @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ A4 @ B2 )
| ( ord_less_eq @ ( set @ A ) @ B2 @ A4 ) ) ) ) ) ).
% wo_rel.ofilter_linord
thf(fact_8105_ofilter__def,axiom,
! [A: $tType] :
( ( order_ofilter @ A )
= ( ^ [R4: set @ ( product_prod @ A @ A ),A7: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A7 @ ( field2 @ A @ R4 ) )
& ! [X4: A] :
( ( member @ A @ X4 @ A7 )
=> ( ord_less_eq @ ( set @ A ) @ ( order_under @ A @ R4 @ X4 ) @ A7 ) ) ) ) ) ).
% ofilter_def
thf(fact_8106_has__vector__derivative__mult,axiom,
! [A: $tType] :
( ( real_V4412858255891104859lgebra @ A )
=> ! [F2: real > A,F8: A,X: real,S: set @ real,G: real > A,G6: A] :
( ( has_ve8173657378732805170vative @ A @ F2 @ F8 @ ( topolo174197925503356063within @ real @ X @ S ) )
=> ( ( has_ve8173657378732805170vative @ A @ G @ G6 @ ( topolo174197925503356063within @ real @ X @ S ) )
=> ( has_ve8173657378732805170vative @ A
@ ^ [X4: real] : ( times_times @ A @ ( F2 @ X4 ) @ ( G @ X4 ) )
@ ( plus_plus @ A @ ( times_times @ A @ ( F2 @ X ) @ G6 ) @ ( times_times @ A @ F8 @ ( G @ X ) ) )
@ ( topolo174197925503356063within @ real @ X @ S ) ) ) ) ) ).
% has_vector_derivative_mult
thf(fact_8107_bounded__bilinear_Ohas__vector__derivative,axiom,
! [B: $tType,C: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ C )
& ( real_V822414075346904944vector @ B ) )
=> ! [Prod2: A > B > C,F2: real > A,F8: A,X: real,S: set @ real,G: real > B,G6: B] :
( ( real_V2442710119149674383linear @ A @ B @ C @ Prod2 )
=> ( ( has_ve8173657378732805170vative @ A @ F2 @ F8 @ ( topolo174197925503356063within @ real @ X @ S ) )
=> ( ( has_ve8173657378732805170vative @ B @ G @ G6 @ ( topolo174197925503356063within @ real @ X @ S ) )
=> ( has_ve8173657378732805170vative @ C
@ ^ [X4: real] : ( Prod2 @ ( F2 @ X4 ) @ ( G @ X4 ) )
@ ( plus_plus @ C @ ( Prod2 @ ( F2 @ X ) @ G6 ) @ ( Prod2 @ F8 @ ( G @ X ) ) )
@ ( topolo174197925503356063within @ real @ X @ S ) ) ) ) ) ) ).
% bounded_bilinear.has_vector_derivative
thf(fact_8108_ofilterIncl__def,axiom,
! [A: $tType] :
( ( bNF_We413866401316099525erIncl @ A )
= ( ^ [R4: set @ ( product_prod @ A @ A )] :
( collect @ ( product_prod @ ( set @ A ) @ ( set @ A ) )
@ ( product_case_prod @ ( set @ A ) @ ( set @ A ) @ $o
@ ^ [A7: set @ A,B6: set @ A] :
( ( order_ofilter @ A @ R4 @ A7 )
& ( A7
!= ( field2 @ A @ R4 ) )
& ( order_ofilter @ A @ R4 @ B6 )
& ( B6
!= ( field2 @ A @ R4 ) )
& ( ord_less @ ( set @ A ) @ A7 @ B6 ) ) ) ) ) ) ).
% ofilterIncl_def
thf(fact_8109_bsqr__ofilter,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),D6: set @ ( product_prod @ A @ A )] :
( ( order_well_order_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( order_ofilter @ ( product_prod @ A @ A ) @ ( bNF_Wellorder_bsqr @ A @ R2 ) @ D6 )
=> ( ( ord_less @ ( set @ ( product_prod @ A @ A ) ) @ D6
@ ( product_Sigma @ A @ A @ ( field2 @ A @ R2 )
@ ^ [Uu3: A] : ( field2 @ A @ R2 ) ) )
=> ( ~ ? [A6: A] :
( ( field2 @ A @ R2 )
= ( order_under @ A @ R2 @ A6 ) )
=> ? [A8: set @ A] :
( ( order_ofilter @ A @ R2 @ A8 )
& ( ord_less @ ( set @ A ) @ A8 @ ( field2 @ A @ R2 ) )
& ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ D6
@ ( product_Sigma @ A @ A @ A8
@ ^ [Uu3: A] : A8 ) ) ) ) ) ) ) ).
% bsqr_ofilter
thf(fact_8110_well__order__on__empty,axiom,
! [A: $tType] : ( order_well_order_on @ A @ ( bot_bot @ ( set @ A ) ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ).
% well_order_on_empty
thf(fact_8111_natLeq__on__well__order__on,axiom,
! [N2: nat] :
( order_well_order_on @ nat
@ ( collect @ nat
@ ^ [X4: nat] : ( ord_less @ nat @ X4 @ N2 ) )
@ ( collect @ ( product_prod @ nat @ nat )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [X4: nat,Y6: nat] :
( ( ord_less @ nat @ X4 @ N2 )
& ( ord_less @ nat @ Y6 @ N2 )
& ( ord_less_eq @ nat @ X4 @ Y6 ) ) ) ) ) ).
% natLeq_on_well_order_on
thf(fact_8112_linear__order__on__well__order__on,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A4: set @ A] :
( ( finite_finite2 @ ( product_prod @ A @ A ) @ R2 )
=> ( ( order_679001287576687338der_on @ A @ A4 @ R2 )
= ( order_well_order_on @ A @ A4 @ R2 ) ) ) ).
% linear_order_on_well_order_on
thf(fact_8113_well__order__on__Restr,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A4: set @ A] :
( ( order_well_order_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( ord_less_eq @ ( set @ A ) @ A4 @ ( field2 @ A @ R2 ) )
=> ( order_well_order_on @ A @ A4
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R2
@ ( product_Sigma @ A @ A @ A4
@ ^ [Uu3: A] : A4 ) ) ) ) ) ).
% well_order_on_Restr
thf(fact_8114_natLeq__on__Well__order,axiom,
! [N2: nat] :
( order_well_order_on @ nat
@ ( field2 @ nat
@ ( collect @ ( product_prod @ nat @ nat )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [X4: nat,Y6: nat] :
( ( ord_less @ nat @ X4 @ N2 )
& ( ord_less @ nat @ Y6 @ N2 )
& ( ord_less_eq @ nat @ X4 @ Y6 ) ) ) ) )
@ ( collect @ ( product_prod @ nat @ nat )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [X4: nat,Y6: nat] :
( ( ord_less @ nat @ X4 @ N2 )
& ( ord_less @ nat @ Y6 @ N2 )
& ( ord_less_eq @ nat @ X4 @ Y6 ) ) ) ) ) ).
% natLeq_on_Well_order
thf(fact_8115_Linear__order__Well__order__iff,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( order_679001287576687338der_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( order_well_order_on @ A @ ( field2 @ A @ R2 ) @ R2 )
= ( ! [A7: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A7 @ ( field2 @ A @ R2 ) )
=> ( ( A7
!= ( bot_bot @ ( set @ A ) ) )
=> ? [X4: A] :
( ( member @ A @ X4 @ A7 )
& ! [Y6: A] :
( ( member @ A @ Y6 @ A7 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y6 ) @ R2 ) ) ) ) ) ) ) ) ).
% Linear_order_Well_order_iff
thf(fact_8116_ofilter__Restr__subset,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A4: set @ A,B2: set @ A] :
( ( order_well_order_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( order_ofilter @ A @ R2 @ A4 )
=> ( ( ord_less_eq @ ( set @ A ) @ A4 @ B2 )
=> ( order_ofilter @ A
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R2
@ ( product_Sigma @ A @ A @ B2
@ ^ [Uu3: A] : B2 ) )
@ A4 ) ) ) ) ).
% ofilter_Restr_subset
thf(fact_8117_ofilter__subset__embedS,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A4: set @ A,B2: set @ A] :
( ( order_well_order_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( order_ofilter @ A @ R2 @ A4 )
=> ( ( order_ofilter @ A @ R2 @ B2 )
=> ( ( ord_less @ ( set @ A ) @ A4 @ B2 )
= ( bNF_Wellorder_embedS @ A @ A
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R2
@ ( product_Sigma @ A @ A @ A4
@ ^ [Uu3: A] : A4 ) )
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R2
@ ( product_Sigma @ A @ A @ B2
@ ^ [Uu3: A] : B2 ) )
@ ( id @ A ) ) ) ) ) ) ).
% ofilter_subset_embedS
thf(fact_8118_ofilter__subset__embed,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A4: set @ A,B2: set @ A] :
( ( order_well_order_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( order_ofilter @ A @ R2 @ A4 )
=> ( ( order_ofilter @ A @ R2 @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ A4 @ B2 )
= ( bNF_Wellorder_embed @ A @ A
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R2
@ ( product_Sigma @ A @ A @ A4
@ ^ [Uu3: A] : A4 ) )
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R2
@ ( product_Sigma @ A @ A @ B2
@ ^ [Uu3: A] : B2 ) )
@ ( id @ A ) ) ) ) ) ) ).
% ofilter_subset_embed
thf(fact_8119_ofilter__embed,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A4: set @ A] :
( ( order_well_order_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( order_ofilter @ A @ R2 @ A4 )
= ( ( ord_less_eq @ ( set @ A ) @ A4 @ ( field2 @ A @ R2 ) )
& ( bNF_Wellorder_embed @ A @ A
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R2
@ ( product_Sigma @ A @ A @ A4
@ ^ [Uu3: A] : A4 ) )
@ R2
@ ( id @ A ) ) ) ) ) ).
% ofilter_embed
thf(fact_8120_embedS__iff,axiom,
! [A: $tType,B: $tType,R2: set @ ( product_prod @ A @ A ),R6: set @ ( product_prod @ B @ B ),F2: A > B] :
( ( order_well_order_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( bNF_Wellorder_embed @ A @ B @ R2 @ R6 @ F2 )
=> ( ( bNF_Wellorder_embedS @ A @ B @ R2 @ R6 @ F2 )
= ( ord_less @ ( set @ B ) @ ( image2 @ A @ B @ F2 @ ( field2 @ A @ R2 ) ) @ ( field2 @ B @ R6 ) ) ) ) ) ).
% embedS_iff
thf(fact_8121_embedS__Field,axiom,
! [A: $tType,B: $tType,R2: set @ ( product_prod @ A @ A ),R6: set @ ( product_prod @ B @ B ),F2: A > B] :
( ( order_well_order_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( bNF_Wellorder_embedS @ A @ B @ R2 @ R6 @ F2 )
=> ( ord_less @ ( set @ B ) @ ( image2 @ A @ B @ F2 @ ( field2 @ A @ R2 ) ) @ ( field2 @ B @ R6 ) ) ) ) ).
% embedS_Field
thf(fact_8122_embed__Field,axiom,
! [A: $tType,B: $tType,R2: set @ ( product_prod @ A @ A ),R6: set @ ( product_prod @ B @ B ),F2: A > B] :
( ( bNF_Wellorder_embed @ A @ B @ R2 @ R6 @ F2 )
=> ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F2 @ ( field2 @ A @ R2 ) ) @ ( field2 @ B @ R6 ) ) ) ).
% embed_Field
thf(fact_8123_ofilter__subset__embedS__iso,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A4: set @ A,B2: set @ A] :
( ( order_well_order_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( order_ofilter @ A @ R2 @ A4 )
=> ( ( order_ofilter @ A @ R2 @ B2 )
=> ( ( ( ord_less @ ( set @ A ) @ A4 @ B2 )
= ( bNF_Wellorder_embedS @ A @ A
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R2
@ ( product_Sigma @ A @ A @ A4
@ ^ [Uu3: A] : A4 ) )
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R2
@ ( product_Sigma @ A @ A @ B2
@ ^ [Uu3: A] : B2 ) )
@ ( id @ A ) ) )
& ( ( A4 = B2 )
= ( bNF_Wellorder_iso @ A @ A
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R2
@ ( product_Sigma @ A @ A @ A4
@ ^ [Uu3: A] : A4 ) )
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R2
@ ( product_Sigma @ A @ A @ B2
@ ^ [Uu3: A] : B2 ) )
@ ( id @ A ) ) ) ) ) ) ) ).
% ofilter_subset_embedS_iso
thf(fact_8124_ofilter__subset__ordLess,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A4: set @ A,B2: set @ A] :
( ( order_well_order_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( order_ofilter @ A @ R2 @ A4 )
=> ( ( order_ofilter @ A @ R2 @ B2 )
=> ( ( ord_less @ ( set @ A ) @ A4 @ B2 )
= ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) )
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R2
@ ( product_Sigma @ A @ A @ A4
@ ^ [Uu3: A] : A4 ) )
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R2
@ ( product_Sigma @ A @ A @ B2
@ ^ [Uu3: A] : B2 ) ) )
@ ( bNF_We4044943003108391690rdLess @ A @ A ) ) ) ) ) ) ).
% ofilter_subset_ordLess
thf(fact_8125_finite__ordLess__infinite,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ A ),R6: set @ ( product_prod @ B @ B )] :
( ( order_well_order_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( order_well_order_on @ B @ ( field2 @ B @ R6 ) @ R6 )
=> ( ( finite_finite2 @ A @ ( field2 @ A @ R2 ) )
=> ( ~ ( finite_finite2 @ B @ ( field2 @ B @ R6 ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R2 @ R6 ) @ ( bNF_We4044943003108391690rdLess @ A @ B ) ) ) ) ) ) ).
% finite_ordLess_infinite
thf(fact_8126_underS__Restr__ordLess,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A3: A] :
( ( order_well_order_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( ( field2 @ A @ R2 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) )
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R2
@ ( product_Sigma @ A @ A @ ( order_underS @ A @ R2 @ A3 )
@ ^ [Uu3: A] : ( order_underS @ A @ R2 @ A3 ) ) )
@ R2 )
@ ( bNF_We4044943003108391690rdLess @ A @ A ) ) ) ) ).
% underS_Restr_ordLess
thf(fact_8127_ofilter__ordLess,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A4: set @ A] :
( ( order_well_order_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( order_ofilter @ A @ R2 @ A4 )
=> ( ( ord_less @ ( set @ A ) @ A4 @ ( field2 @ A @ R2 ) )
= ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) )
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R2
@ ( product_Sigma @ A @ A @ A4
@ ^ [Uu3: A] : A4 ) )
@ R2 )
@ ( bNF_We4044943003108391690rdLess @ A @ A ) ) ) ) ) ).
% ofilter_ordLess
thf(fact_8128_ofilter__subset__ordLeq,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A4: set @ A,B2: set @ A] :
( ( order_well_order_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( order_ofilter @ A @ R2 @ A4 )
=> ( ( order_ofilter @ A @ R2 @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ A4 @ B2 )
= ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) )
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R2
@ ( product_Sigma @ A @ A @ A4
@ ^ [Uu3: A] : A4 ) )
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R2
@ ( product_Sigma @ A @ A @ B2
@ ^ [Uu3: A] : B2 ) ) )
@ ( bNF_Wellorder_ordLeq @ A @ A ) ) ) ) ) ) ).
% ofilter_subset_ordLeq
thf(fact_8129_successively__transfer,axiom,
! [A: $tType,B: $tType,A4: A > B > $o] :
( bNF_rel_fun @ ( A > A > $o ) @ ( B > B > $o ) @ ( ( list @ A ) > $o ) @ ( ( list @ B ) > $o )
@ ( bNF_rel_fun @ A @ B @ ( A > $o ) @ ( B > $o ) @ A4
@ ( bNF_rel_fun @ A @ B @ $o @ $o @ A4
@ ^ [Y5: $o,Z2: $o] : Y5 = Z2 ) )
@ ( bNF_rel_fun @ ( list @ A ) @ ( list @ B ) @ $o @ $o @ ( list_all2 @ A @ B @ A4 )
@ ^ [Y5: $o,Z2: $o] : Y5 = Z2 )
@ ( successively @ A )
@ ( successively @ B ) ) ).
% successively_transfer
thf(fact_8130_successively__rev,axiom,
! [A: $tType,P: A > A > $o,Xs: list @ A] :
( ( successively @ A @ P @ ( rev @ A @ Xs ) )
= ( successively @ A
@ ^ [X4: A,Y6: A] : ( P @ Y6 @ X4 )
@ Xs ) ) ).
% successively_rev
thf(fact_8131_exists__minim__Well__order,axiom,
! [A: $tType,R: set @ ( set @ ( product_prod @ A @ A ) )] :
( ( R
!= ( bot_bot @ ( set @ ( set @ ( product_prod @ A @ A ) ) ) ) )
=> ( ! [X5: set @ ( product_prod @ A @ A )] :
( ( member @ ( set @ ( product_prod @ A @ A ) ) @ X5 @ R )
=> ( order_well_order_on @ A @ ( field2 @ A @ X5 ) @ X5 ) )
=> ? [X5: set @ ( product_prod @ A @ A )] :
( ( member @ ( set @ ( product_prod @ A @ A ) ) @ X5 @ R )
& ! [Xa: set @ ( product_prod @ A @ A )] :
( ( member @ ( set @ ( product_prod @ A @ A ) ) @ Xa @ R )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ X5 @ Xa ) @ ( bNF_Wellorder_ordLeq @ A @ A ) ) ) ) ) ) ).
% exists_minim_Well_order
thf(fact_8132_successively__map,axiom,
! [A: $tType,B: $tType,P: A > A > $o,F2: B > A,Xs: list @ B] :
( ( successively @ A @ P @ ( map @ B @ A @ F2 @ Xs ) )
= ( successively @ B
@ ^ [X4: B,Y6: B] : ( P @ ( F2 @ X4 ) @ ( F2 @ Y6 ) )
@ Xs ) ) ).
% successively_map
thf(fact_8133_successively__conv__sorted__wrt,axiom,
! [A: $tType,P: A > A > $o,Xs: list @ A] :
( ( transp @ A @ P )
=> ( ( successively @ A @ P @ Xs )
= ( sorted_wrt @ A @ P @ Xs ) ) ) ).
% successively_conv_sorted_wrt
thf(fact_8134_successively_Osimps_I2_J,axiom,
! [A: $tType,P: A > A > $o,X: A] : ( successively @ A @ P @ ( cons @ A @ X @ ( nil @ A ) ) ) ).
% successively.simps(2)
thf(fact_8135_successively_Oelims_I1_J,axiom,
! [A: $tType,X: A > A > $o,Xa2: list @ A,Y: $o] :
( ( ( successively @ A @ X @ Xa2 )
= Y )
=> ( ( ( Xa2
= ( nil @ A ) )
=> ~ Y )
=> ( ( ? [X5: A] :
( Xa2
= ( cons @ A @ X5 @ ( nil @ A ) ) )
=> ~ Y )
=> ~ ! [X5: A,Y3: A,Xs2: list @ A] :
( ( Xa2
= ( cons @ A @ X5 @ ( cons @ A @ Y3 @ Xs2 ) ) )
=> ( Y
= ( ~ ( ( X @ X5 @ Y3 )
& ( successively @ A @ X @ ( cons @ A @ Y3 @ Xs2 ) ) ) ) ) ) ) ) ) ).
% successively.elims(1)
thf(fact_8136_successively_Oelims_I2_J,axiom,
! [A: $tType,X: A > A > $o,Xa2: list @ A] :
( ( successively @ A @ X @ Xa2 )
=> ( ( Xa2
!= ( nil @ A ) )
=> ( ! [X5: A] :
( Xa2
!= ( cons @ A @ X5 @ ( nil @ A ) ) )
=> ~ ! [X5: A,Y3: A,Xs2: list @ A] :
( ( Xa2
= ( cons @ A @ X5 @ ( cons @ A @ Y3 @ Xs2 ) ) )
=> ~ ( ( X @ X5 @ Y3 )
& ( successively @ A @ X @ ( cons @ A @ Y3 @ Xs2 ) ) ) ) ) ) ) ).
% successively.elims(2)
thf(fact_8137_successively_Osimps_I1_J,axiom,
! [A: $tType,P: A > A > $o] : ( successively @ A @ P @ ( nil @ A ) ) ).
% successively.simps(1)
thf(fact_8138_successively__remdups__adjI,axiom,
! [A: $tType,P: A > A > $o,Xs: list @ A] :
( ( successively @ A @ P @ Xs )
=> ( successively @ A @ P @ ( remdups_adj @ A @ Xs ) ) ) ).
% successively_remdups_adjI
thf(fact_8139_successively__if__sorted__wrt,axiom,
! [A: $tType,P: A > A > $o,Xs: list @ A] :
( ( sorted_wrt @ A @ P @ Xs )
=> ( successively @ A @ P @ Xs ) ) ).
% successively_if_sorted_wrt
thf(fact_8140_successively__altdef,axiom,
! [A: $tType] :
( ( successively @ A )
= ( ^ [P3: A > A > $o] :
( rec_list @ $o @ A @ $true
@ ^ [X4: A,Xs3: list @ A,B4: $o] :
( case_list @ $o @ A @ $true
@ ^ [Y6: A,Xa4: list @ A] :
( ( P3 @ X4 @ Y6 )
& B4 )
@ Xs3 ) ) ) ) ).
% successively_altdef
thf(fact_8141_successively_Oelims_I3_J,axiom,
! [A: $tType,X: A > A > $o,Xa2: list @ A] :
( ~ ( successively @ A @ X @ Xa2 )
=> ~ ! [X5: A,Y3: A,Xs2: list @ A] :
( ( Xa2
= ( cons @ A @ X5 @ ( cons @ A @ Y3 @ Xs2 ) ) )
=> ( ( X @ X5 @ Y3 )
& ( successively @ A @ X @ ( cons @ A @ Y3 @ Xs2 ) ) ) ) ) ).
% successively.elims(3)
thf(fact_8142_successively_Osimps_I3_J,axiom,
! [A: $tType,P: A > A > $o,X: A,Y: A,Xs: list @ A] :
( ( successively @ A @ P @ ( cons @ A @ X @ ( cons @ A @ Y @ Xs ) ) )
= ( ( P @ X @ Y )
& ( successively @ A @ P @ ( cons @ A @ Y @ Xs ) ) ) ) ).
% successively.simps(3)
thf(fact_8143_successively__iff__sorted__wrt__strong,axiom,
! [A: $tType,Xs: list @ A,P: A > A > $o] :
( ! [X5: A,Y3: A,Z: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Xs ) )
=> ( ( member @ A @ Y3 @ ( set2 @ A @ Xs ) )
=> ( ( member @ A @ Z @ ( set2 @ A @ Xs ) )
=> ( ( P @ X5 @ Y3 )
=> ( ( P @ Y3 @ Z )
=> ( P @ X5 @ Z ) ) ) ) ) )
=> ( ( successively @ A @ P @ Xs )
= ( sorted_wrt @ A @ P @ Xs ) ) ) ).
% successively_iff_sorted_wrt_strong
thf(fact_8144_successively__cong,axiom,
! [A: $tType,Xs: list @ A,P: A > A > $o,Q: A > A > $o,Ys: list @ A] :
( ! [X5: A,Y3: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Xs ) )
=> ( ( member @ A @ Y3 @ ( set2 @ A @ Xs ) )
=> ( ( P @ X5 @ Y3 )
= ( Q @ X5 @ Y3 ) ) ) )
=> ( ( Xs = Ys )
=> ( ( successively @ A @ P @ Xs )
= ( successively @ A @ Q @ Ys ) ) ) ) ).
% successively_cong
thf(fact_8145_successively__mono,axiom,
! [A: $tType,P: A > A > $o,Xs: list @ A,Q: A > A > $o] :
( ( successively @ A @ P @ Xs )
=> ( ! [X5: A,Y3: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Xs ) )
=> ( ( member @ A @ Y3 @ ( set2 @ A @ Xs ) )
=> ( ( P @ X5 @ Y3 )
=> ( Q @ X5 @ Y3 ) ) ) )
=> ( successively @ A @ Q @ Xs ) ) ) ).
% successively_mono
thf(fact_8146_successively__remdups__adj__iff,axiom,
! [A: $tType,Xs: list @ A,P: A > A > $o] :
( ! [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Xs ) )
=> ( P @ X5 @ X5 ) )
=> ( ( successively @ A @ P @ ( remdups_adj @ A @ Xs ) )
= ( successively @ A @ P @ Xs ) ) ) ).
% successively_remdups_adj_iff
thf(fact_8147_distinct__adj__def,axiom,
! [A: $tType] :
( ( distinct_adj @ A )
= ( successively @ A
@ ^ [X4: A,Y6: A] : X4 != Y6 ) ) ).
% distinct_adj_def
thf(fact_8148_successively__Cons,axiom,
! [A: $tType,P: A > A > $o,X: A,Xs: list @ A] :
( ( successively @ A @ P @ ( cons @ A @ X @ Xs ) )
= ( ( Xs
= ( nil @ A ) )
| ( ( P @ X @ ( hd @ A @ Xs ) )
& ( successively @ A @ P @ Xs ) ) ) ) ).
% successively_Cons
thf(fact_8149_successively__append__iff,axiom,
! [A: $tType,P: A > A > $o,Xs: list @ A,Ys: list @ A] :
( ( successively @ A @ P @ ( append @ A @ Xs @ Ys ) )
= ( ( successively @ A @ P @ Xs )
& ( successively @ A @ P @ Ys )
& ( ( Xs
= ( nil @ A ) )
| ( Ys
= ( nil @ A ) )
| ( P @ ( last @ A @ Xs ) @ ( hd @ A @ Ys ) ) ) ) ) ).
% successively_append_iff
thf(fact_8150_successively_Opelims_I1_J,axiom,
! [A: $tType,X: A > A > $o,Xa2: list @ A,Y: $o] :
( ( ( successively @ A @ X @ Xa2 )
= Y )
=> ( ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( successively_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X @ Xa2 ) )
=> ( ( ( Xa2
= ( nil @ A ) )
=> ( Y
=> ~ ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( successively_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X @ ( nil @ A ) ) ) ) )
=> ( ! [X5: A] :
( ( Xa2
= ( cons @ A @ X5 @ ( nil @ A ) ) )
=> ( Y
=> ~ ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( successively_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X @ ( cons @ A @ X5 @ ( nil @ A ) ) ) ) ) )
=> ~ ! [X5: A,Y3: A,Xs2: list @ A] :
( ( Xa2
= ( cons @ A @ X5 @ ( cons @ A @ Y3 @ Xs2 ) ) )
=> ( ( Y
= ( ( X @ X5 @ Y3 )
& ( successively @ A @ X @ ( cons @ A @ Y3 @ Xs2 ) ) ) )
=> ~ ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( successively_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X @ ( cons @ A @ X5 @ ( cons @ A @ Y3 @ Xs2 ) ) ) ) ) ) ) ) ) ) ).
% successively.pelims(1)
thf(fact_8151_successively_Opelims_I2_J,axiom,
! [A: $tType,X: A > A > $o,Xa2: list @ A] :
( ( successively @ A @ X @ Xa2 )
=> ( ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( successively_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X @ Xa2 ) )
=> ( ( ( Xa2
= ( nil @ A ) )
=> ~ ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( successively_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X @ ( nil @ A ) ) ) )
=> ( ! [X5: A] :
( ( Xa2
= ( cons @ A @ X5 @ ( nil @ A ) ) )
=> ~ ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( successively_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X @ ( cons @ A @ X5 @ ( nil @ A ) ) ) ) )
=> ~ ! [X5: A,Y3: A,Xs2: list @ A] :
( ( Xa2
= ( cons @ A @ X5 @ ( cons @ A @ Y3 @ Xs2 ) ) )
=> ( ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( successively_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X @ ( cons @ A @ X5 @ ( cons @ A @ Y3 @ Xs2 ) ) ) )
=> ~ ( ( X @ X5 @ Y3 )
& ( successively @ A @ X @ ( cons @ A @ Y3 @ Xs2 ) ) ) ) ) ) ) ) ) ).
% successively.pelims(2)
thf(fact_8152_successively_Opelims_I3_J,axiom,
! [A: $tType,X: A > A > $o,Xa2: list @ A] :
( ~ ( successively @ A @ X @ Xa2 )
=> ( ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( successively_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X @ Xa2 ) )
=> ~ ! [X5: A,Y3: A,Xs2: list @ A] :
( ( Xa2
= ( cons @ A @ X5 @ ( cons @ A @ Y3 @ Xs2 ) ) )
=> ( ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( successively_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X @ ( cons @ A @ X5 @ ( cons @ A @ Y3 @ Xs2 ) ) ) )
=> ( ( X @ X5 @ Y3 )
& ( successively @ A @ X @ ( cons @ A @ Y3 @ Xs2 ) ) ) ) ) ) ) ).
% successively.pelims(3)
thf(fact_8153_ord__to__filter__compat,axiom,
! [A: $tType,R0: set @ ( product_prod @ A @ A )] :
( bNF_Wellorder_compat @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ A )
@ ( inf_inf @ ( set @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) ) @ ( bNF_We4044943003108391690rdLess @ A @ A )
@ ( product_Sigma @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( image @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( converse @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_We4044943003108391690rdLess @ A @ A ) ) @ ( insert2 @ ( set @ ( product_prod @ A @ A ) ) @ R0 @ ( bot_bot @ ( set @ ( set @ ( product_prod @ A @ A ) ) ) ) ) )
@ ^ [Uu3: set @ ( product_prod @ A @ A )] : ( image @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( converse @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_We4044943003108391690rdLess @ A @ A ) ) @ ( insert2 @ ( set @ ( product_prod @ A @ A ) ) @ R0 @ ( bot_bot @ ( set @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ) ) )
@ ( bNF_We413866401316099525erIncl @ A @ R0 )
@ ( bNF_We8469521843155493636filter @ A @ R0 ) ) ).
% ord_to_filter_compat
thf(fact_8154_card__of__UNION__ordLeq__infinite,axiom,
! [B: $tType,A: $tType,C: $tType,B2: set @ A,I6: set @ B,A4: B > ( set @ C )] :
( ~ ( finite_finite2 @ A @ B2 )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ I6 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ B2 ) ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ I6 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ C @ ( A4 @ X5 ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ B2 ) ) @ ( bNF_Wellorder_ordLeq @ C @ A ) ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ C @ ( complete_Sup_Sup @ ( set @ C ) @ ( image2 @ B @ ( set @ C ) @ A4 @ I6 ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ B2 ) ) @ ( bNF_Wellorder_ordLeq @ C @ A ) ) ) ) ) ).
% card_of_UNION_ordLeq_infinite
thf(fact_8155_card__of__Times2,axiom,
! [A: $tType,B: $tType,A4: set @ A,B2: set @ B] :
( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B2 )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ A4
@ ^ [Uu3: A] : B2 ) ) )
@ ( bNF_Wellorder_ordLeq @ B @ ( product_prod @ A @ B ) ) ) ) ).
% card_of_Times2
thf(fact_8156_card__of__Times1,axiom,
! [A: $tType,B: $tType,A4: set @ A,B2: set @ B] :
( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ A ) @ ( product_prod @ B @ A ) ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ A ) @ ( product_prod @ B @ A ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B2 )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ B @ A )
@ ( product_Sigma @ B @ A @ B2
@ ^ [Uu3: B] : A4 ) ) )
@ ( bNF_Wellorder_ordLeq @ B @ ( product_prod @ B @ A ) ) ) ) ).
% card_of_Times1
thf(fact_8157_card__of__empty,axiom,
! [B: $tType,A: $tType,A4: set @ B] : ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ ( bot_bot @ ( set @ A ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ A4 ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) ) ).
% card_of_empty
thf(fact_8158_card__of__empty3,axiom,
! [B: $tType,A: $tType,A4: set @ A] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ ( bot_bot @ ( set @ B ) ) ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( A4
= ( bot_bot @ ( set @ A ) ) ) ) ).
% card_of_empty3
thf(fact_8159_finite__iff__ordLess__natLeq,axiom,
! [A: $tType] :
( ( finite_finite2 @ A )
= ( ^ [A7: set @ A] : ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A7 ) @ bNF_Ca8665028551170535155natLeq ) @ ( bNF_We4044943003108391690rdLess @ A @ nat ) ) ) ) ).
% finite_iff_ordLess_natLeq
thf(fact_8160_card__of__mono1,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B2 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ B2 ) ) @ ( bNF_Wellorder_ordLeq @ A @ A ) ) ) ).
% card_of_mono1
thf(fact_8161_card__of__Sigma__ordLeq__infinite,axiom,
! [A: $tType,C: $tType,B: $tType,B2: set @ A,I6: set @ B,A4: B > ( set @ C )] :
( ~ ( finite_finite2 @ A @ B2 )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ I6 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ B2 ) ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ I6 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ C @ ( A4 @ X5 ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ B2 ) ) @ ( bNF_Wellorder_ordLeq @ C @ A ) ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ B @ C ) @ ( product_prod @ B @ C ) ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ B @ C ) @ ( product_prod @ B @ C ) ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ B @ C ) @ ( product_Sigma @ B @ C @ I6 @ A4 ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ B2 ) ) @ ( bNF_Wellorder_ordLeq @ ( product_prod @ B @ C ) @ A ) ) ) ) ) ).
% card_of_Sigma_ordLeq_infinite
thf(fact_8162_infinite__iff__natLeq__ordLeq,axiom,
! [A: $tType,A4: set @ A] :
( ( finite_finite2 @ A @ A4 )
!= ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ A @ A ) ) @ bNF_Ca8665028551170535155natLeq @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) ) @ ( bNF_Wellorder_ordLeq @ nat @ A ) ) ) ).
% infinite_iff_natLeq_ordLeq
thf(fact_8163_card__of__ordLeq__finite,axiom,
! [B: $tType,A: $tType,A4: set @ A,B2: set @ B] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B2 ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ( finite_finite2 @ B @ B2 )
=> ( finite_finite2 @ A @ A4 ) ) ) ).
% card_of_ordLeq_finite
thf(fact_8164_card__of__ordLeq__infinite,axiom,
! [A: $tType,B: $tType,A4: set @ A,B2: set @ B] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B2 ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ~ ( finite_finite2 @ A @ A4 )
=> ~ ( finite_finite2 @ B @ B2 ) ) ) ).
% card_of_ordLeq_infinite
thf(fact_8165_infinite__iff__card__of__nat,axiom,
! [A: $tType,A4: set @ A] :
( ( ~ ( finite_finite2 @ A @ A4 ) )
= ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ nat @ ( top_top @ ( set @ nat ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) ) @ ( bNF_Wellorder_ordLeq @ nat @ A ) ) ) ).
% infinite_iff_card_of_nat
thf(fact_8166_card__of__ordLeq2,axiom,
! [B: $tType,A: $tType,A4: set @ A,B2: set @ B] :
( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ? [G2: B > A] :
( ( image2 @ B @ A @ G2 @ B2 )
= A4 ) )
= ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B2 ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) ) ) ) ).
% card_of_ordLeq2
thf(fact_8167_surj__imp__ordLeq,axiom,
! [B: $tType,A: $tType,B2: set @ A,F2: B > A,A4: set @ B] :
( ( ord_less_eq @ ( set @ A ) @ B2 @ ( image2 @ B @ A @ F2 @ A4 ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ B2 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ A4 ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) ) ) ).
% surj_imp_ordLeq
thf(fact_8168_card__of__singl__ordLeq,axiom,
! [A: $tType,B: $tType,A4: set @ A,B3: B] :
( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ ( insert2 @ B @ B3 @ ( bot_bot @ ( set @ B ) ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) ) @ ( bNF_Wellorder_ordLeq @ B @ A ) ) ) ).
% card_of_singl_ordLeq
thf(fact_8169_card__of__ordLess2,axiom,
! [A: $tType,B: $tType,B2: set @ A,A4: set @ B] :
( ( B2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ~ ? [F3: B > A] :
( ( image2 @ B @ A @ F3 @ A4 )
= B2 ) )
= ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ A4 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ B2 ) ) @ ( bNF_We4044943003108391690rdLess @ B @ A ) ) ) ) ).
% card_of_ordLess2
thf(fact_8170_card__of__ordLeq,axiom,
! [B: $tType,A: $tType,A4: set @ A,B2: set @ B] :
( ( ? [F3: A > B] :
( ( inj_on @ A @ B @ F3 @ A4 )
& ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F3 @ A4 ) @ B2 ) ) )
= ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B2 ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) ) ) ).
% card_of_ordLeq
thf(fact_8171_card__of__ordLess,axiom,
! [A: $tType,B: $tType,A4: set @ A,B2: set @ B] :
( ( ~ ? [F3: A > B] :
( ( inj_on @ A @ B @ F3 @ A4 )
& ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F3 @ A4 ) @ B2 ) ) )
= ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B2 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) ) @ ( bNF_We4044943003108391690rdLess @ B @ A ) ) ) ).
% card_of_ordLess
thf(fact_8172_ordLeq3__finite__infinite,axiom,
! [B: $tType,A: $tType,A4: set @ A,B2: set @ B] :
( ( finite_finite2 @ A @ A4 )
=> ( ~ ( finite_finite2 @ B @ B2 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B2 ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) ) ) ) ).
% ordLeq3_finite_infinite
thf(fact_8173_card__of__Plus__Times__aux,axiom,
! [B: $tType,A: $tType,A1: A,A22: A,A4: set @ A,B2: set @ B] :
( ( ( A1 != A22 )
& ( ord_less_eq @ ( set @ A ) @ ( insert2 @ A @ A1 @ ( insert2 @ A @ A22 @ ( bot_bot @ ( set @ A ) ) ) ) @ A4 ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B2 ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ A @ B ) @ ( sum_Plus @ A @ B @ A4 @ B2 ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ A4
@ ^ [Uu3: A] : B2 ) ) )
@ ( bNF_Wellorder_ordLeq @ ( sum_sum @ A @ B ) @ ( product_prod @ A @ B ) ) ) ) ) ).
% card_of_Plus_Times_aux
thf(fact_8174_finite__Plus__iff,axiom,
! [A: $tType,B: $tType,A4: set @ A,B2: set @ B] :
( ( finite_finite2 @ ( sum_sum @ A @ B ) @ ( sum_Plus @ A @ B @ A4 @ B2 ) )
= ( ( finite_finite2 @ A @ A4 )
& ( finite_finite2 @ B @ B2 ) ) ) ).
% finite_Plus_iff
thf(fact_8175_card__Plus,axiom,
! [A: $tType,B: $tType,A4: set @ A,B2: set @ B] :
( ( finite_finite2 @ A @ A4 )
=> ( ( finite_finite2 @ B @ B2 )
=> ( ( finite_card @ ( sum_sum @ A @ B ) @ ( sum_Plus @ A @ B @ A4 @ B2 ) )
= ( plus_plus @ nat @ ( finite_card @ A @ A4 ) @ ( finite_card @ B @ B2 ) ) ) ) ) ).
% card_Plus
thf(fact_8176_finite__Plus,axiom,
! [A: $tType,B: $tType,A4: set @ A,B2: set @ B] :
( ( finite_finite2 @ A @ A4 )
=> ( ( finite_finite2 @ B @ B2 )
=> ( finite_finite2 @ ( sum_sum @ A @ B ) @ ( sum_Plus @ A @ B @ A4 @ B2 ) ) ) ) ).
% finite_Plus
thf(fact_8177_finite__PlusD_I1_J,axiom,
! [B: $tType,A: $tType,A4: set @ A,B2: set @ B] :
( ( finite_finite2 @ ( sum_sum @ A @ B ) @ ( sum_Plus @ A @ B @ A4 @ B2 ) )
=> ( finite_finite2 @ A @ A4 ) ) ).
% finite_PlusD(1)
thf(fact_8178_finite__PlusD_I2_J,axiom,
! [A: $tType,B: $tType,A4: set @ A,B2: set @ B] :
( ( finite_finite2 @ ( sum_sum @ A @ B ) @ ( sum_Plus @ A @ B @ A4 @ B2 ) )
=> ( finite_finite2 @ B @ B2 ) ) ).
% finite_PlusD(2)
thf(fact_8179_card__Plus__conv__if,axiom,
! [B: $tType,A: $tType,A4: set @ A,B2: set @ B] :
( ( ( ( finite_finite2 @ A @ A4 )
& ( finite_finite2 @ B @ B2 ) )
=> ( ( finite_card @ ( sum_sum @ A @ B ) @ ( sum_Plus @ A @ B @ A4 @ B2 ) )
= ( plus_plus @ nat @ ( finite_card @ A @ A4 ) @ ( finite_card @ B @ B2 ) ) ) )
& ( ~ ( ( finite_finite2 @ A @ A4 )
& ( finite_finite2 @ B @ B2 ) )
=> ( ( finite_card @ ( sum_sum @ A @ B ) @ ( sum_Plus @ A @ B @ A4 @ B2 ) )
= ( zero_zero @ nat ) ) ) ) ).
% card_Plus_conv_if
thf(fact_8180_card__of__Plus__ordLess__infinite,axiom,
! [A: $tType,C: $tType,B: $tType,C5: set @ A,A4: set @ B,B2: set @ C] :
( ~ ( finite_finite2 @ A @ C5 )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ A4 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ C5 ) ) @ ( bNF_We4044943003108391690rdLess @ B @ A ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ C @ B2 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ C5 ) ) @ ( bNF_We4044943003108391690rdLess @ C @ A ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ B @ C ) @ ( sum_sum @ B @ C ) ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ B @ C ) @ ( sum_sum @ B @ C ) ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ B @ C ) @ ( sum_Plus @ B @ C @ A4 @ B2 ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ C5 ) ) @ ( bNF_We4044943003108391690rdLess @ ( sum_sum @ B @ C ) @ A ) ) ) ) ) ).
% card_of_Plus_ordLess_infinite
thf(fact_8181_card__of__Plus__Times,axiom,
! [B: $tType,A: $tType,A1: A,A22: A,A4: set @ A,B16: B,B24: B,B2: set @ B] :
( ( ( A1 != A22 )
& ( ord_less_eq @ ( set @ A ) @ ( insert2 @ A @ A1 @ ( insert2 @ A @ A22 @ ( bot_bot @ ( set @ A ) ) ) ) @ A4 ) )
=> ( ( ( B16 != B24 )
& ( ord_less_eq @ ( set @ B ) @ ( insert2 @ B @ B16 @ ( insert2 @ B @ B24 @ ( bot_bot @ ( set @ B ) ) ) ) @ B2 ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ A @ B ) @ ( sum_Plus @ A @ B @ A4 @ B2 ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ A4
@ ^ [Uu3: A] : B2 ) ) )
@ ( bNF_Wellorder_ordLeq @ ( sum_sum @ A @ B ) @ ( product_prod @ A @ B ) ) ) ) ) ).
% card_of_Plus_Times
thf(fact_8182_Plus__eq__empty__conv,axiom,
! [A: $tType,B: $tType,A4: set @ A,B2: set @ B] :
( ( ( sum_Plus @ A @ B @ A4 @ B2 )
= ( bot_bot @ ( set @ ( sum_sum @ A @ B ) ) ) )
= ( ( A4
= ( bot_bot @ ( set @ A ) ) )
& ( B2
= ( bot_bot @ ( set @ B ) ) ) ) ) ).
% Plus_eq_empty_conv
thf(fact_8183_card__of__Sigma__ordLeq__infinite__Field,axiom,
! [A: $tType,C: $tType,B: $tType,R2: set @ ( product_prod @ A @ A ),I6: set @ B,A4: B > ( set @ C )] :
( ~ ( finite_finite2 @ A @ ( field2 @ A @ R2 ) )
=> ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ I6 ) @ R2 ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ I6 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ C @ ( A4 @ X5 ) ) @ R2 ) @ ( bNF_Wellorder_ordLeq @ C @ A ) ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ B @ C ) @ ( product_prod @ B @ C ) ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ B @ C ) @ ( product_prod @ B @ C ) ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ B @ C ) @ ( product_Sigma @ B @ C @ I6 @ A4 ) ) @ R2 ) @ ( bNF_Wellorder_ordLeq @ ( product_prod @ B @ C ) @ A ) ) ) ) ) ) ).
% card_of_Sigma_ordLeq_infinite_Field
thf(fact_8184_infinite__Card__order__limit,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A3: A] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ~ ( finite_finite2 @ A @ ( field2 @ A @ R2 ) )
=> ( ( member @ A @ A3 @ ( field2 @ A @ R2 ) )
=> ? [X5: A] :
( ( member @ A @ X5 @ ( field2 @ A @ R2 ) )
& ( A3 != X5 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ X5 ) @ R2 ) ) ) ) ) ).
% infinite_Card_order_limit
thf(fact_8185_Card__order__infinite__not__under,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ~ ( finite_finite2 @ A @ ( field2 @ A @ R2 ) )
=> ~ ? [A11: A] :
( ( field2 @ A @ R2 )
= ( order_under @ A @ R2 @ A11 ) ) ) ) ).
% Card_order_infinite_not_under
thf(fact_8186_exists__minim__Card__order,axiom,
! [A: $tType,R: set @ ( set @ ( product_prod @ A @ A ) )] :
( ( R
!= ( bot_bot @ ( set @ ( set @ ( product_prod @ A @ A ) ) ) ) )
=> ( ! [X5: set @ ( product_prod @ A @ A )] :
( ( member @ ( set @ ( product_prod @ A @ A ) ) @ X5 @ R )
=> ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ X5 ) @ X5 ) )
=> ? [X5: set @ ( product_prod @ A @ A )] :
( ( member @ ( set @ ( product_prod @ A @ A ) ) @ X5 @ R )
& ! [Xa: set @ ( product_prod @ A @ A )] :
( ( member @ ( set @ ( product_prod @ A @ A ) ) @ Xa @ R )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ X5 @ Xa ) @ ( bNF_Wellorder_ordLeq @ A @ A ) ) ) ) ) ) ).
% exists_minim_Card_order
% Type constructors (632)
thf(tcon_fun___Countable__Complete__Lattices_Ocountable__complete__distrib__lattice,axiom,
! [A10: $tType,A29: $tType] :
( ( comple592849572758109894attice @ A29 )
=> ( counta4013691401010221786attice @ ( A10 > A29 ) ) ) ).
thf(tcon_fun___Conditionally__Complete__Lattices_Oconditionally__complete__lattice,axiom,
! [A10: $tType,A29: $tType] :
( ( comple6319245703460814977attice @ A29 )
=> ( condit1219197933456340205attice @ ( A10 > A29 ) ) ) ).
thf(tcon_fun___Countable__Complete__Lattices_Ocountable__complete__lattice,axiom,
! [A10: $tType,A29: $tType] :
( ( counta3822494911875563373attice @ A29 )
=> ( counta3822494911875563373attice @ ( A10 > A29 ) ) ) ).
thf(tcon_fun___Complete__Lattices_Ocomplete__distrib__lattice,axiom,
! [A10: $tType,A29: $tType] :
( ( comple592849572758109894attice @ A29 )
=> ( comple592849572758109894attice @ ( A10 > A29 ) ) ) ).
thf(tcon_fun___Lattices_Obounded__semilattice__sup__bot,axiom,
! [A10: $tType,A29: $tType] :
( ( bounded_lattice @ A29 )
=> ( bounde4967611905675639751up_bot @ ( A10 > A29 ) ) ) ).
thf(tcon_fun___Lattices_Obounded__semilattice__inf__top,axiom,
! [A10: $tType,A29: $tType] :
( ( bounded_lattice @ A29 )
=> ( bounde4346867609351753570nf_top @ ( A10 > A29 ) ) ) ).
thf(tcon_fun___Complete__Lattices_Ocomplete__lattice,axiom,
! [A10: $tType,A29: $tType] :
( ( comple6319245703460814977attice @ A29 )
=> ( comple6319245703460814977attice @ ( A10 > A29 ) ) ) ).
thf(tcon_fun___Boolean__Algebras_Oboolean__algebra,axiom,
! [A10: $tType,A29: $tType] :
( ( boolea8198339166811842893lgebra @ A29 )
=> ( boolea8198339166811842893lgebra @ ( A10 > A29 ) ) ) ).
thf(tcon_fun___Lattices_Obounded__lattice__bot,axiom,
! [A10: $tType,A29: $tType] :
( ( bounded_lattice @ A29 )
=> ( bounded_lattice_bot @ ( A10 > A29 ) ) ) ).
thf(tcon_fun___Complete__Partial__Order_Occpo,axiom,
! [A10: $tType,A29: $tType] :
( ( comple6319245703460814977attice @ A29 )
=> ( comple9053668089753744459l_ccpo @ ( A10 > A29 ) ) ) ).
thf(tcon_fun___Lattices_Osemilattice__sup,axiom,
! [A10: $tType,A29: $tType] :
( ( semilattice_sup @ A29 )
=> ( semilattice_sup @ ( A10 > A29 ) ) ) ).
thf(tcon_fun___Lattices_Osemilattice__inf,axiom,
! [A10: $tType,A29: $tType] :
( ( semilattice_inf @ A29 )
=> ( semilattice_inf @ ( A10 > A29 ) ) ) ).
thf(tcon_fun___Lattices_Odistrib__lattice,axiom,
! [A10: $tType,A29: $tType] :
( ( distrib_lattice @ A29 )
=> ( distrib_lattice @ ( A10 > A29 ) ) ) ).
thf(tcon_fun___Lattices_Obounded__lattice,axiom,
! [A10: $tType,A29: $tType] :
( ( bounded_lattice @ A29 )
=> ( bounded_lattice @ ( A10 > A29 ) ) ) ).
thf(tcon_fun___Orderings_Oorder__top,axiom,
! [A10: $tType,A29: $tType] :
( ( order_top @ A29 )
=> ( order_top @ ( A10 > A29 ) ) ) ).
thf(tcon_fun___Orderings_Oorder__bot,axiom,
! [A10: $tType,A29: $tType] :
( ( order_bot @ A29 )
=> ( order_bot @ ( A10 > A29 ) ) ) ).
thf(tcon_fun___Countable_Ocountable,axiom,
! [A10: $tType,A29: $tType] :
( ( ( finite_finite @ A10 )
& ( countable @ A29 ) )
=> ( countable @ ( A10 > A29 ) ) ) ).
thf(tcon_fun___Orderings_Opreorder,axiom,
! [A10: $tType,A29: $tType] :
( ( preorder @ A29 )
=> ( preorder @ ( A10 > A29 ) ) ) ).
thf(tcon_fun___Finite__Set_Ofinite,axiom,
! [A10: $tType,A29: $tType] :
( ( ( finite_finite @ A10 )
& ( finite_finite @ A29 ) )
=> ( finite_finite @ ( A10 > A29 ) ) ) ).
thf(tcon_fun___Lattices_Olattice,axiom,
! [A10: $tType,A29: $tType] :
( ( lattice @ A29 )
=> ( lattice @ ( A10 > A29 ) ) ) ).
thf(tcon_fun___Orderings_Oorder,axiom,
! [A10: $tType,A29: $tType] :
( ( order @ A29 )
=> ( order @ ( A10 > A29 ) ) ) ).
thf(tcon_fun___Orderings_Otop,axiom,
! [A10: $tType,A29: $tType] :
( ( top @ A29 )
=> ( top @ ( A10 > A29 ) ) ) ).
thf(tcon_fun___Orderings_Oord,axiom,
! [A10: $tType,A29: $tType] :
( ( ord @ A29 )
=> ( ord @ ( A10 > A29 ) ) ) ).
thf(tcon_fun___Orderings_Obot,axiom,
! [A10: $tType,A29: $tType] :
( ( bot @ A29 )
=> ( bot @ ( A10 > A29 ) ) ) ).
thf(tcon_fun___Groups_Ouminus,axiom,
! [A10: $tType,A29: $tType] :
( ( uminus @ A29 )
=> ( uminus @ ( A10 > A29 ) ) ) ).
thf(tcon_Int_Oint___Conditionally__Complete__Lattices_Oconditionally__complete__linorder,axiom,
condit6923001295902523014norder @ int ).
thf(tcon_Int_Oint___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_1,axiom,
condit1219197933456340205attice @ int ).
thf(tcon_Int_Oint___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations,axiom,
bit_un5681908812861735899ations @ int ).
thf(tcon_Int_Oint___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct,axiom,
semiri1453513574482234551roduct @ int ).
thf(tcon_Int_Oint___Euclidean__Division_Ounique__euclidean__semiring__with__nat,axiom,
euclid5411537665997757685th_nat @ int ).
thf(tcon_Int_Oint___Groups_Oordered__ab__semigroup__monoid__add__imp__le,axiom,
ordere1937475149494474687imp_le @ int ).
thf(tcon_Int_Oint___Euclidean__Division_Ounique__euclidean__semiring,axiom,
euclid3128863361964157862miring @ int ).
thf(tcon_Int_Oint___Euclidean__Division_Oeuclidean__semiring__cancel,axiom,
euclid4440199948858584721cancel @ int ).
thf(tcon_Int_Oint___Divides_Ounique__euclidean__semiring__numeral,axiom,
unique1627219031080169319umeral @ int ).
thf(tcon_Int_Oint___Euclidean__Division_Oeuclidean__ring__cancel,axiom,
euclid8851590272496341667cancel @ int ).
thf(tcon_Int_Oint___Rings_Osemiring__no__zero__divisors__cancel,axiom,
semiri6575147826004484403cancel @ int ).
thf(tcon_Int_Oint___Groups_Ostrict__ordered__ab__semigroup__add,axiom,
strict9044650504122735259up_add @ int ).
thf(tcon_Int_Oint___Groups_Oordered__cancel__ab__semigroup__add,axiom,
ordere580206878836729694up_add @ int ).
thf(tcon_Int_Oint___Groups_Oordered__ab__semigroup__add__imp__le,axiom,
ordere2412721322843649153imp_le @ int ).
thf(tcon_Int_Oint___Bit__Operations_Osemiring__bit__operations,axiom,
bit_se359711467146920520ations @ int ).
thf(tcon_Int_Oint___Rings_Olinordered__comm__semiring__strict,axiom,
linord2810124833399127020strict @ int ).
thf(tcon_Int_Oint___Groups_Ostrict__ordered__comm__monoid__add,axiom,
strict7427464778891057005id_add @ int ).
thf(tcon_Int_Oint___Groups_Oordered__cancel__comm__monoid__add,axiom,
ordere8940638589300402666id_add @ int ).
thf(tcon_Int_Oint___Euclidean__Division_Oeuclidean__semiring,axiom,
euclid3725896446679973847miring @ int ).
thf(tcon_Int_Oint___Topological__Spaces_Otopological__space,axiom,
topolo4958980785337419405_space @ int ).
thf(tcon_Int_Oint___Topological__Spaces_Olinorder__topology,axiom,
topolo1944317154257567458pology @ int ).
thf(tcon_Int_Oint___Topological__Spaces_Odiscrete__topology,axiom,
topolo8865339358273720382pology @ int ).
thf(tcon_Int_Oint___Rings_Olinordered__semiring__1__strict,axiom,
linord715952674999750819strict @ int ).
thf(tcon_Int_Oint___Limits_Otopological__comm__monoid__add,axiom,
topolo5987344860129210374id_add @ int ).
thf(tcon_Int_Oint___Groups_Olinordered__ab__semigroup__add,axiom,
linord4140545234300271783up_add @ int ).
thf(tcon_Int_Oint___Bit__Operations_Oring__bit__operations,axiom,
bit_ri3973907225187159222ations @ int ).
thf(tcon_Int_Oint___Topological__Spaces_Oorder__topology,axiom,
topolo2564578578187576103pology @ int ).
thf(tcon_Int_Oint___Rings_Osemiring__1__no__zero__divisors,axiom,
semiri2026040879449505780visors @ int ).
thf(tcon_Int_Oint___Rings_Olinordered__nonzero__semiring,axiom,
linord181362715937106298miring @ int ).
thf(tcon_Int_Oint___Limits_Otopological__semigroup__mult,axiom,
topolo4211221413907600880p_mult @ int ).
thf(tcon_Int_Oint___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___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___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_2,axiom,
semilattice_sup @ int ).
thf(tcon_Int_Oint___Lattices_Osemilattice__inf_3,axiom,
semilattice_inf @ int ).
thf(tcon_Int_Oint___Lattices_Odistrib__lattice_4,axiom,
distrib_lattice @ 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___Rings_Oordered__semiring,axiom,
ordered_semiring @ int ).
thf(tcon_Int_Oint___Rings_Oordered__ring__abs,axiom,
ordered_ring_abs @ int ).
thf(tcon_Int_Oint___Parity_Osemiring__parity,axiom,
semiring_parity @ int ).
thf(tcon_Int_Oint___Groups_Ocomm__monoid__add,axiom,
comm_monoid_add @ int ).
thf(tcon_Int_Oint___Rings_Osemiring__modulo,axiom,
semiring_modulo @ int ).
thf(tcon_Int_Oint___Rings_Olinordered__ring,axiom,
linordered_ring @ int ).
thf(tcon_Int_Oint___Rings_Olinordered__idom,axiom,
linordered_idom @ int ).
thf(tcon_Int_Oint___Rings_Ocomm__semiring__1,axiom,
comm_semiring_1 @ int ).
thf(tcon_Int_Oint___Rings_Ocomm__semiring__0,axiom,
comm_semiring_0 @ int ).
thf(tcon_Int_Oint___Groups_Osemigroup__mult,axiom,
semigroup_mult @ int ).
thf(tcon_Int_Oint___Rings_Osemidom__modulo,axiom,
semidom_modulo @ int ).
thf(tcon_Int_Oint___Rings_Osemidom__divide,axiom,
semidom_divide @ int ).
thf(tcon_Int_Oint___Num_Osemiring__numeral,axiom,
semiring_numeral @ int ).
thf(tcon_Int_Oint___Groups_Osemigroup__add,axiom,
semigroup_add @ int ).
thf(tcon_Int_Oint___Rings_Ozero__less__one,axiom,
zero_less_one @ int ).
thf(tcon_Int_Oint___Rings_Ocomm__semiring,axiom,
comm_semiring @ int ).
thf(tcon_Int_Oint___Nat_Osemiring__char__0,axiom,
semiring_char_0 @ int ).
thf(tcon_Int_Oint___Groups_Oab__group__add,axiom,
ab_group_add @ int ).
thf(tcon_Int_Oint___Countable_Ocountable_5,axiom,
countable @ 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_6,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_7,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___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_8,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_9,axiom,
ord @ int ).
thf(tcon_Int_Oint___Groups_Ouminus_10,axiom,
uminus @ int ).
thf(tcon_Int_Oint___Rings_Oring__1,axiom,
ring_1 @ int ).
thf(tcon_Int_Oint___Rings_Oabs__if,axiom,
abs_if @ int ).
thf(tcon_Int_Oint___Power_Opower,axiom,
power @ int ).
thf(tcon_Int_Oint___Num_Onumeral,axiom,
numeral @ int ).
thf(tcon_Int_Oint___Groups_Ozero,axiom,
zero @ int ).
thf(tcon_Int_Oint___Groups_Oplus,axiom,
plus @ int ).
thf(tcon_Int_Oint___Rings_Oring,axiom,
ring @ int ).
thf(tcon_Int_Oint___Rings_Oidom,axiom,
idom @ int ).
thf(tcon_Int_Oint___Groups_Oone,axiom,
one @ int ).
thf(tcon_Int_Oint___Rings_Odvd,axiom,
dvd @ int ).
thf(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_11,axiom,
condit6923001295902523014norder @ nat ).
thf(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_12,axiom,
condit1219197933456340205attice @ nat ).
thf(tcon_Nat_Onat___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_13,axiom,
bit_un5681908812861735899ations @ nat ).
thf(tcon_Nat_Onat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_14,axiom,
semiri1453513574482234551roduct @ nat ).
thf(tcon_Nat_Onat___Euclidean__Division_Ounique__euclidean__semiring__with__nat_15,axiom,
euclid5411537665997757685th_nat @ nat ).
thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__monoid__add__imp__le_16,axiom,
ordere1937475149494474687imp_le @ nat ).
thf(tcon_Nat_Onat___Euclidean__Division_Ounique__euclidean__semiring_17,axiom,
euclid3128863361964157862miring @ 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__add_32,axiom,
topolo5987344860129210374id_add @ nat ).
thf(tcon_Nat_Onat___Groups_Olinordered__ab__semigroup__add_33,axiom,
linord4140545234300271783up_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___Groups_Oordered__comm__monoid__add_41,axiom,
ordere6911136660526730532id_add @ nat ).
thf(tcon_Nat_Onat___Groups_Ocancel__ab__semigroup__add_42,axiom,
cancel2418104881723323429up_add @ nat ).
thf(tcon_Nat_Onat___Limits_Otopological__monoid__add_43,axiom,
topolo6943815403480290642id_add @ nat ).
thf(tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add_44,axiom,
cancel1802427076303600483id_add @ nat ).
thf(tcon_Nat_Onat___Rings_Ocomm__semiring__1__cancel_45,axiom,
comm_s4317794764714335236cancel @ nat ).
thf(tcon_Nat_Onat___Bit__Operations_Osemiring__bits_46,axiom,
bit_semiring_bits @ nat ).
thf(tcon_Nat_Onat___Topological__Spaces_Ot2__space_47,axiom,
topological_t2_space @ nat ).
thf(tcon_Nat_Onat___Topological__Spaces_Ot1__space_48,axiom,
topological_t1_space @ nat ).
thf(tcon_Nat_Onat___Rings_Oordered__comm__semiring_49,axiom,
ordere2520102378445227354miring @ nat ).
thf(tcon_Nat_Onat___Groups_Ocancel__semigroup__add_50,axiom,
cancel_semigroup_add @ nat ).
thf(tcon_Nat_Onat___Rings_Olinordered__semiring_51,axiom,
linordered_semiring @ nat ).
thf(tcon_Nat_Onat___Rings_Oordered__semiring__0_52,axiom,
ordered_semiring_0 @ nat ).
thf(tcon_Nat_Onat___Rings_Olinordered__semidom_53,axiom,
linordered_semidom @ nat ).
thf(tcon_Nat_Onat___Lattices_Osemilattice__sup_54,axiom,
semilattice_sup @ nat ).
thf(tcon_Nat_Onat___Lattices_Osemilattice__inf_55,axiom,
semilattice_inf @ nat ).
thf(tcon_Nat_Onat___Lattices_Odistrib__lattice_56,axiom,
distrib_lattice @ nat ).
thf(tcon_Nat_Onat___Groups_Oab__semigroup__mult_57,axiom,
ab_semigroup_mult @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__1__cancel_58,axiom,
semiring_1_cancel @ nat ).
thf(tcon_Nat_Onat___Rings_Oalgebraic__semidom_59,axiom,
algebraic_semidom @ nat ).
thf(tcon_Nat_Onat___Groups_Ocomm__monoid__mult_60,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_61,axiom,
ab_semigroup_add @ nat ).
thf(tcon_Nat_Onat___Rings_Oordered__semiring_62,axiom,
ordered_semiring @ nat ).
thf(tcon_Nat_Onat___Parity_Osemiring__parity_63,axiom,
semiring_parity @ nat ).
thf(tcon_Nat_Onat___Groups_Ocomm__monoid__add_64,axiom,
comm_monoid_add @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__modulo_65,axiom,
semiring_modulo @ nat ).
thf(tcon_Nat_Onat___Rings_Ocomm__semiring__1_66,axiom,
comm_semiring_1 @ nat ).
thf(tcon_Nat_Onat___Rings_Ocomm__semiring__0_67,axiom,
comm_semiring_0 @ nat ).
thf(tcon_Nat_Onat___Groups_Osemigroup__mult_68,axiom,
semigroup_mult @ nat ).
thf(tcon_Nat_Onat___Rings_Osemidom__modulo_69,axiom,
semidom_modulo @ nat ).
thf(tcon_Nat_Onat___Rings_Osemidom__divide_70,axiom,
semidom_divide @ nat ).
thf(tcon_Nat_Onat___Num_Osemiring__numeral_71,axiom,
semiring_numeral @ nat ).
thf(tcon_Nat_Onat___Groups_Osemigroup__add_72,axiom,
semigroup_add @ nat ).
thf(tcon_Nat_Onat___Rings_Ozero__less__one_73,axiom,
zero_less_one @ nat ).
thf(tcon_Nat_Onat___Rings_Ocomm__semiring_74,axiom,
comm_semiring @ nat ).
thf(tcon_Nat_Onat___Orderings_Owellorder,axiom,
wellorder @ nat ).
thf(tcon_Nat_Onat___Orderings_Oorder__bot_75,axiom,
order_bot @ nat ).
thf(tcon_Nat_Onat___Nat_Osemiring__char__0_76,axiom,
semiring_char_0 @ nat ).
thf(tcon_Nat_Onat___Countable_Ocountable_77,axiom,
countable @ nat ).
thf(tcon_Nat_Onat___Rings_Ozero__neq__one_78,axiom,
zero_neq_one @ nat ).
thf(tcon_Nat_Onat___Orderings_Opreorder_79,axiom,
preorder @ nat ).
thf(tcon_Nat_Onat___Orderings_Olinorder_80,axiom,
linorder @ nat ).
thf(tcon_Nat_Onat___Groups_Omonoid__mult_81,axiom,
monoid_mult @ nat ).
thf(tcon_Nat_Onat___Groups_Omonoid__add_82,axiom,
monoid_add @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__1_83,axiom,
semiring_1 @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__0_84,axiom,
semiring_0 @ nat ).
thf(tcon_Nat_Onat___Orderings_Ono__top_85,axiom,
no_top @ nat ).
thf(tcon_Nat_Onat___Lattices_Olattice_86,axiom,
lattice @ nat ).
thf(tcon_Nat_Onat___GCD_Osemiring__gcd_87,axiom,
semiring_gcd @ nat ).
thf(tcon_Nat_Onat___GCD_Osemiring__Gcd_88,axiom,
semiring_Gcd @ nat ).
thf(tcon_Nat_Onat___Rings_Omult__zero_89,axiom,
mult_zero @ nat ).
thf(tcon_Nat_Onat___Orderings_Oorder_90,axiom,
order @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring_91,axiom,
semiring @ nat ).
thf(tcon_Nat_Onat___Rings_Osemidom_92,axiom,
semidom @ nat ).
thf(tcon_Nat_Onat___Orderings_Oord_93,axiom,
ord @ nat ).
thf(tcon_Nat_Onat___Orderings_Obot_94,axiom,
bot @ nat ).
thf(tcon_Nat_Onat___Power_Opower_95,axiom,
power @ nat ).
thf(tcon_Nat_Onat___Num_Onumeral_96,axiom,
numeral @ nat ).
thf(tcon_Nat_Onat___Groups_Ozero_97,axiom,
zero @ nat ).
thf(tcon_Nat_Onat___Groups_Oplus_98,axiom,
plus @ nat ).
thf(tcon_Nat_Onat___Groups_Oone_99,axiom,
one @ nat ).
thf(tcon_Nat_Onat___Rings_Odvd_100,axiom,
dvd @ nat ).
thf(tcon_Nat_Onat___Nat_Osize,axiom,
size @ nat ).
thf(tcon_Num_Onum___Orderings_Opreorder_101,axiom,
preorder @ num ).
thf(tcon_Num_Onum___Orderings_Olinorder_102,axiom,
linorder @ num ).
thf(tcon_Num_Onum___Orderings_Oorder_103,axiom,
order @ num ).
thf(tcon_Num_Onum___Orderings_Oord_104,axiom,
ord @ num ).
thf(tcon_Num_Onum___Groups_Oplus_105,axiom,
plus @ num ).
thf(tcon_Num_Onum___Nat_Osize_106,axiom,
size @ num ).
thf(tcon_Set_Oset___Countable__Complete__Lattices_Ocountable__complete__distrib__lattice_107,axiom,
! [A10: $tType] : ( counta4013691401010221786attice @ ( set @ A10 ) ) ).
thf(tcon_Set_Oset___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_108,axiom,
! [A10: $tType] : ( condit1219197933456340205attice @ ( set @ A10 ) ) ).
thf(tcon_Set_Oset___Countable__Complete__Lattices_Ocountable__complete__lattice_109,axiom,
! [A10: $tType] : ( counta3822494911875563373attice @ ( set @ A10 ) ) ).
thf(tcon_Set_Oset___Complete__Lattices_Ocomplete__distrib__lattice_110,axiom,
! [A10: $tType] : ( comple592849572758109894attice @ ( set @ A10 ) ) ).
thf(tcon_Set_Oset___Lattices_Obounded__semilattice__sup__bot_111,axiom,
! [A10: $tType] : ( bounde4967611905675639751up_bot @ ( set @ A10 ) ) ).
thf(tcon_Set_Oset___Lattices_Obounded__semilattice__inf__top_112,axiom,
! [A10: $tType] : ( bounde4346867609351753570nf_top @ ( set @ A10 ) ) ).
thf(tcon_Set_Oset___Complete__Lattices_Ocomplete__lattice_113,axiom,
! [A10: $tType] : ( comple6319245703460814977attice @ ( set @ A10 ) ) ).
thf(tcon_Set_Oset___Boolean__Algebras_Oboolean__algebra_114,axiom,
! [A10: $tType] : ( boolea8198339166811842893lgebra @ ( set @ A10 ) ) ).
thf(tcon_Set_Oset___Lattices_Obounded__lattice__bot_115,axiom,
! [A10: $tType] : ( bounded_lattice_bot @ ( set @ A10 ) ) ).
thf(tcon_Set_Oset___Complete__Partial__Order_Occpo_116,axiom,
! [A10: $tType] : ( comple9053668089753744459l_ccpo @ ( set @ A10 ) ) ).
thf(tcon_Set_Oset___Lattices_Osemilattice__sup_117,axiom,
! [A10: $tType] : ( semilattice_sup @ ( set @ A10 ) ) ).
thf(tcon_Set_Oset___Lattices_Osemilattice__inf_118,axiom,
! [A10: $tType] : ( semilattice_inf @ ( set @ A10 ) ) ).
thf(tcon_Set_Oset___Lattices_Odistrib__lattice_119,axiom,
! [A10: $tType] : ( distrib_lattice @ ( set @ A10 ) ) ).
thf(tcon_Set_Oset___Lattices_Obounded__lattice_120,axiom,
! [A10: $tType] : ( bounded_lattice @ ( set @ A10 ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder__top_121,axiom,
! [A10: $tType] : ( order_top @ ( set @ A10 ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder__bot_122,axiom,
! [A10: $tType] : ( order_bot @ ( set @ A10 ) ) ).
thf(tcon_Set_Oset___Countable_Ocountable_123,axiom,
! [A10: $tType] :
( ( finite_finite @ A10 )
=> ( countable @ ( set @ A10 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Opreorder_124,axiom,
! [A10: $tType] : ( preorder @ ( set @ A10 ) ) ).
thf(tcon_Set_Oset___Finite__Set_Ofinite_125,axiom,
! [A10: $tType] :
( ( finite_finite @ A10 )
=> ( finite_finite @ ( set @ A10 ) ) ) ).
thf(tcon_Set_Oset___Lattices_Olattice_126,axiom,
! [A10: $tType] : ( lattice @ ( set @ A10 ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder_127,axiom,
! [A10: $tType] : ( order @ ( set @ A10 ) ) ).
thf(tcon_Set_Oset___Orderings_Otop_128,axiom,
! [A10: $tType] : ( top @ ( set @ A10 ) ) ).
thf(tcon_Set_Oset___Orderings_Oord_129,axiom,
! [A10: $tType] : ( ord @ ( set @ A10 ) ) ).
thf(tcon_Set_Oset___Orderings_Obot_130,axiom,
! [A10: $tType] : ( bot @ ( set @ A10 ) ) ).
thf(tcon_Set_Oset___Groups_Ouminus_131,axiom,
! [A10: $tType] : ( uminus @ ( set @ A10 ) ) ).
thf(tcon_HOL_Obool___Countable__Complete__Lattices_Ocountable__complete__distrib__lattice_132,axiom,
counta4013691401010221786attice @ $o ).
thf(tcon_HOL_Obool___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_133,axiom,
condit1219197933456340205attice @ $o ).
thf(tcon_HOL_Obool___Countable__Complete__Lattices_Ocountable__complete__lattice_134,axiom,
counta3822494911875563373attice @ $o ).
thf(tcon_HOL_Obool___Complete__Lattices_Ocomplete__distrib__lattice_135,axiom,
comple592849572758109894attice @ $o ).
thf(tcon_HOL_Obool___Topological__Spaces_Otopological__space_136,axiom,
topolo4958980785337419405_space @ $o ).
thf(tcon_HOL_Obool___Topological__Spaces_Olinorder__topology_137,axiom,
topolo1944317154257567458pology @ $o ).
thf(tcon_HOL_Obool___Topological__Spaces_Odiscrete__topology_138,axiom,
topolo8865339358273720382pology @ $o ).
thf(tcon_HOL_Obool___Lattices_Obounded__semilattice__sup__bot_139,axiom,
bounde4967611905675639751up_bot @ $o ).
thf(tcon_HOL_Obool___Lattices_Obounded__semilattice__inf__top_140,axiom,
bounde4346867609351753570nf_top @ $o ).
thf(tcon_HOL_Obool___Complete__Lattices_Ocomplete__lattice_141,axiom,
comple6319245703460814977attice @ $o ).
thf(tcon_HOL_Obool___Topological__Spaces_Oorder__topology_142,axiom,
topolo2564578578187576103pology @ $o ).
thf(tcon_HOL_Obool___Boolean__Algebras_Oboolean__algebra_143,axiom,
boolea8198339166811842893lgebra @ $o ).
thf(tcon_HOL_Obool___Lattices_Obounded__lattice__bot_144,axiom,
bounded_lattice_bot @ $o ).
thf(tcon_HOL_Obool___Topological__Spaces_Ot2__space_145,axiom,
topological_t2_space @ $o ).
thf(tcon_HOL_Obool___Topological__Spaces_Ot1__space_146,axiom,
topological_t1_space @ $o ).
thf(tcon_HOL_Obool___Complete__Partial__Order_Occpo_147,axiom,
comple9053668089753744459l_ccpo @ $o ).
thf(tcon_HOL_Obool___Lattices_Osemilattice__sup_148,axiom,
semilattice_sup @ $o ).
thf(tcon_HOL_Obool___Lattices_Osemilattice__inf_149,axiom,
semilattice_inf @ $o ).
thf(tcon_HOL_Obool___Lattices_Odistrib__lattice_150,axiom,
distrib_lattice @ $o ).
thf(tcon_HOL_Obool___Lattices_Obounded__lattice_151,axiom,
bounded_lattice @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder__top_152,axiom,
order_top @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder__bot_153,axiom,
order_bot @ $o ).
thf(tcon_HOL_Obool___Countable_Ocountable_154,axiom,
countable @ $o ).
thf(tcon_HOL_Obool___Orderings_Opreorder_155,axiom,
preorder @ $o ).
thf(tcon_HOL_Obool___Orderings_Olinorder_156,axiom,
linorder @ $o ).
thf(tcon_HOL_Obool___Finite__Set_Ofinite_157,axiom,
finite_finite @ $o ).
thf(tcon_HOL_Obool___Lattices_Olattice_158,axiom,
lattice @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder_159,axiom,
order @ $o ).
thf(tcon_HOL_Obool___Orderings_Otop_160,axiom,
top @ $o ).
thf(tcon_HOL_Obool___Orderings_Oord_161,axiom,
ord @ $o ).
thf(tcon_HOL_Obool___Orderings_Obot_162,axiom,
bot @ $o ).
thf(tcon_HOL_Obool___Groups_Ouminus_163,axiom,
uminus @ $o ).
thf(tcon_List_Olist___Countable_Ocountable_164,axiom,
! [A10: $tType] :
( ( countable @ A10 )
=> ( countable @ ( list @ A10 ) ) ) ).
thf(tcon_List_Olist___Nat_Osize_165,axiom,
! [A10: $tType] : ( size @ ( list @ A10 ) ) ).
thf(tcon_Real_Oreal___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_166,axiom,
condit6923001295902523014norder @ real ).
thf(tcon_Real_Oreal___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_167,axiom,
condit1219197933456340205attice @ real ).
thf(tcon_Real_Oreal___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_168,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_169,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_170,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_171,axiom,
strict9044650504122735259up_add @ real ).
thf(tcon_Real_Oreal___Groups_Oordered__cancel__ab__semigroup__add_172,axiom,
ordere580206878836729694up_add @ real ).
thf(tcon_Real_Oreal___Groups_Oordered__ab__semigroup__add__imp__le_173,axiom,
ordere2412721322843649153imp_le @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__comm__semiring__strict_174,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_175,axiom,
strict7427464778891057005id_add @ real ).
thf(tcon_Real_Oreal___Groups_Oordered__cancel__comm__monoid__add_176,axiom,
ordere8940638589300402666id_add @ real ).
thf(tcon_Real_Oreal___Topological__Spaces_Otopological__space_177,axiom,
topolo4958980785337419405_space @ real ).
thf(tcon_Real_Oreal___Topological__Spaces_Olinorder__topology_178,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,axiom,
archim462609752435547400_field @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__semiring__1__strict_179,axiom,
linord715952674999750819strict @ real ).
thf(tcon_Real_Oreal___Orderings_Ounbounded__dense__linorder,axiom,
unboun7993243217541854897norder @ real ).
thf(tcon_Real_Oreal___Limits_Otopological__comm__monoid__add_180,axiom,
topolo5987344860129210374id_add @ real ).
thf(tcon_Real_Oreal___Groups_Olinordered__ab__semigroup__add_181,axiom,
linord4140545234300271783up_add @ real ).
thf(tcon_Real_Oreal___Topological__Spaces_Oorder__topology_182,axiom,
topolo2564578578187576103pology @ real ).
thf(tcon_Real_Oreal___Rings_Osemiring__1__no__zero__divisors_183,axiom,
semiri2026040879449505780visors @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__nonzero__semiring_184,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_185,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_186,axiom,
linord8928482502909563296strict @ real ).
thf(tcon_Real_Oreal___Rings_Osemiring__no__zero__divisors_187,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_188,axiom,
ordere6658533253407199908up_add @ real ).
thf(tcon_Real_Oreal___Groups_Oordered__ab__group__add__abs_189,axiom,
ordere166539214618696060dd_abs @ real ).
thf(tcon_Real_Oreal___Archimedean__Field_Ofloor__ceiling,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_190,axiom,
ordere6911136660526730532id_add @ real ).
thf(tcon_Real_Oreal___Groups_Olinordered__ab__group__add_191,axiom,
linord5086331880401160121up_add @ real ).
thf(tcon_Real_Oreal___Groups_Ocancel__ab__semigroup__add_192,axiom,
cancel2418104881723323429up_add @ real ).
thf(tcon_Real_Oreal___Rings_Oring__1__no__zero__divisors_193,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_194,axiom,
topolo6943815403480290642id_add @ real ).
thf(tcon_Real_Oreal___Groups_Ocancel__comm__monoid__add_195,axiom,
cancel1802427076303600483id_add @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__ring__strict_196,axiom,
linord4710134922213307826strict @ real ).
thf(tcon_Real_Oreal___Rings_Ocomm__semiring__1__cancel_197,axiom,
comm_s4317794764714335236cancel @ real ).
thf(tcon_Real_Oreal___Limits_Otopological__group__add,axiom,
topolo1633459387980952147up_add @ real ).
thf(tcon_Real_Oreal___Topological__Spaces_Ot2__space_198,axiom,
topological_t2_space @ real ).
thf(tcon_Real_Oreal___Topological__Spaces_Ot1__space_199,axiom,
topological_t1_space @ real ).
thf(tcon_Real_Oreal___Rings_Oordered__comm__semiring_200,axiom,
ordere2520102378445227354miring @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__semiring__1_201,axiom,
linord6961819062388156250ring_1 @ real ).
thf(tcon_Real_Oreal___Groups_Oordered__ab__group__add_202,axiom,
ordered_ab_group_add @ real ).
thf(tcon_Real_Oreal___Groups_Ocancel__semigroup__add_203,axiom,
cancel_semigroup_add @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__semiring_204,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_205,axiom,
ordered_semiring_0 @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__semidom_206,axiom,
linordered_semidom @ real ).
thf(tcon_Real_Oreal___Orderings_Odense__linorder,axiom,
dense_linorder @ real ).
thf(tcon_Real_Oreal___Lattices_Osemilattice__sup_207,axiom,
semilattice_sup @ real ).
thf(tcon_Real_Oreal___Lattices_Osemilattice__inf_208,axiom,
semilattice_inf @ real ).
thf(tcon_Real_Oreal___Lattices_Odistrib__lattice_209,axiom,
distrib_lattice @ real ).
thf(tcon_Real_Oreal___Groups_Oab__semigroup__mult_210,axiom,
ab_semigroup_mult @ real ).
thf(tcon_Real_Oreal___Rings_Osemiring__1__cancel_211,axiom,
semiring_1_cancel @ real ).
thf(tcon_Real_Oreal___Groups_Ocomm__monoid__mult_212,axiom,
comm_monoid_mult @ real ).
thf(tcon_Real_Oreal___Groups_Oab__semigroup__add_213,axiom,
ab_semigroup_add @ real ).
thf(tcon_Real_Oreal___Fields_Olinordered__field,axiom,
linordered_field @ real ).
thf(tcon_Real_Oreal___Rings_Oordered__semiring_214,axiom,
ordered_semiring @ real ).
thf(tcon_Real_Oreal___Rings_Oordered__ring__abs_215,axiom,
ordered_ring_abs @ real ).
thf(tcon_Real_Oreal___Groups_Ocomm__monoid__add_216,axiom,
comm_monoid_add @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__ring_217,axiom,
linordered_ring @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__idom_218,axiom,
linordered_idom @ real ).
thf(tcon_Real_Oreal___Rings_Ocomm__semiring__1_219,axiom,
comm_semiring_1 @ real ).
thf(tcon_Real_Oreal___Rings_Ocomm__semiring__0_220,axiom,
comm_semiring_0 @ real ).
thf(tcon_Real_Oreal___Orderings_Odense__order,axiom,
dense_order @ real ).
thf(tcon_Real_Oreal___Groups_Osemigroup__mult_221,axiom,
semigroup_mult @ real ).
thf(tcon_Real_Oreal___Rings_Osemidom__divide_222,axiom,
semidom_divide @ real ).
thf(tcon_Real_Oreal___Num_Osemiring__numeral_223,axiom,
semiring_numeral @ real ).
thf(tcon_Real_Oreal___Groups_Osemigroup__add_224,axiom,
semigroup_add @ real ).
thf(tcon_Real_Oreal___Fields_Ofield__abs__sgn,axiom,
field_abs_sgn @ real ).
thf(tcon_Real_Oreal___Fields_Odivision__ring,axiom,
division_ring @ real ).
thf(tcon_Real_Oreal___Rings_Ozero__less__one_225,axiom,
zero_less_one @ real ).
thf(tcon_Real_Oreal___Rings_Ocomm__semiring_226,axiom,
comm_semiring @ real ).
thf(tcon_Real_Oreal___Nat_Osemiring__char__0_227,axiom,
semiring_char_0 @ real ).
thf(tcon_Real_Oreal___Groups_Oab__group__add_228,axiom,
ab_group_add @ real ).
thf(tcon_Real_Oreal___Fields_Ofield__char__0,axiom,
field_char_0 @ real ).
thf(tcon_Real_Oreal___Rings_Ozero__neq__one_229,axiom,
zero_neq_one @ real ).
thf(tcon_Real_Oreal___Rings_Oordered__ring_230,axiom,
ordered_ring @ real ).
thf(tcon_Real_Oreal___Rings_Oidom__abs__sgn_231,axiom,
idom_abs_sgn @ real ).
thf(tcon_Real_Oreal___Orderings_Opreorder_232,axiom,
preorder @ real ).
thf(tcon_Real_Oreal___Orderings_Olinorder_233,axiom,
linorder @ real ).
thf(tcon_Real_Oreal___Groups_Omonoid__mult_234,axiom,
monoid_mult @ real ).
thf(tcon_Real_Oreal___Transcendental_Oln,axiom,
ln @ real ).
thf(tcon_Real_Oreal___Rings_Oidom__divide_235,axiom,
idom_divide @ real ).
thf(tcon_Real_Oreal___Rings_Ocomm__ring__1_236,axiom,
comm_ring_1 @ real ).
thf(tcon_Real_Oreal___Groups_Omonoid__add_237,axiom,
monoid_add @ real ).
thf(tcon_Real_Oreal___Rings_Osemiring__1_238,axiom,
semiring_1 @ real ).
thf(tcon_Real_Oreal___Rings_Osemiring__0_239,axiom,
semiring_0 @ real ).
thf(tcon_Real_Oreal___Orderings_Ono__top_240,axiom,
no_top @ real ).
thf(tcon_Real_Oreal___Orderings_Ono__bot_241,axiom,
no_bot @ real ).
thf(tcon_Real_Oreal___Lattices_Olattice_242,axiom,
lattice @ real ).
thf(tcon_Real_Oreal___Groups_Ogroup__add_243,axiom,
group_add @ real ).
thf(tcon_Real_Oreal___Rings_Omult__zero_244,axiom,
mult_zero @ real ).
thf(tcon_Real_Oreal___Rings_Ocomm__ring_245,axiom,
comm_ring @ real ).
thf(tcon_Real_Oreal___Orderings_Oorder_246,axiom,
order @ real ).
thf(tcon_Real_Oreal___Num_Oneg__numeral_247,axiom,
neg_numeral @ real ).
thf(tcon_Real_Oreal___Nat_Oring__char__0_248,axiom,
ring_char_0 @ real ).
thf(tcon_Real_Oreal___Rings_Osemiring_249,axiom,
semiring @ real ).
thf(tcon_Real_Oreal___Fields_Oinverse,axiom,
inverse @ real ).
thf(tcon_Real_Oreal___Rings_Osemidom_250,axiom,
semidom @ real ).
thf(tcon_Real_Oreal___Orderings_Oord_251,axiom,
ord @ real ).
thf(tcon_Real_Oreal___Groups_Ouminus_252,axiom,
uminus @ real ).
thf(tcon_Real_Oreal___Rings_Oring__1_253,axiom,
ring_1 @ real ).
thf(tcon_Real_Oreal___Rings_Oabs__if_254,axiom,
abs_if @ real ).
thf(tcon_Real_Oreal___Fields_Ofield,axiom,
field @ real ).
thf(tcon_Real_Oreal___Power_Opower_255,axiom,
power @ real ).
thf(tcon_Real_Oreal___Num_Onumeral_256,axiom,
numeral @ real ).
thf(tcon_Real_Oreal___Groups_Ozero_257,axiom,
zero @ real ).
thf(tcon_Real_Oreal___Groups_Oplus_258,axiom,
plus @ real ).
thf(tcon_Real_Oreal___Rings_Oring_259,axiom,
ring @ real ).
thf(tcon_Real_Oreal___Rings_Oidom_260,axiom,
idom @ real ).
thf(tcon_Real_Oreal___Groups_Oone_261,axiom,
one @ real ).
thf(tcon_Real_Oreal___Rings_Odvd_262,axiom,
dvd @ real ).
thf(tcon_String_Ochar___Countable_Ocountable_263,axiom,
countable @ char ).
thf(tcon_String_Ochar___Finite__Set_Ofinite_264,axiom,
finite_finite @ char ).
thf(tcon_String_Ochar___Nat_Osize_265,axiom,
size @ char ).
thf(tcon_Sum__Type_Osum___Countable_Ocountable_266,axiom,
! [A10: $tType,A29: $tType] :
( ( ( countable @ A10 )
& ( countable @ A29 ) )
=> ( countable @ ( sum_sum @ A10 @ A29 ) ) ) ).
thf(tcon_Sum__Type_Osum___Finite__Set_Ofinite_267,axiom,
! [A10: $tType,A29: $tType] :
( ( ( finite_finite @ A10 )
& ( finite_finite @ A29 ) )
=> ( finite_finite @ ( sum_sum @ A10 @ A29 ) ) ) ).
thf(tcon_Sum__Type_Osum___Nat_Osize_268,axiom,
! [A10: $tType,A29: $tType] : ( size @ ( sum_sum @ A10 @ A29 ) ) ).
thf(tcon_Filter_Ofilter___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_269,axiom,
! [A10: $tType] : ( condit1219197933456340205attice @ ( filter @ A10 ) ) ).
thf(tcon_Filter_Ofilter___Countable__Complete__Lattices_Ocountable__complete__lattice_270,axiom,
! [A10: $tType] : ( counta3822494911875563373attice @ ( filter @ A10 ) ) ).
thf(tcon_Filter_Ofilter___Lattices_Obounded__semilattice__sup__bot_271,axiom,
! [A10: $tType] : ( bounde4967611905675639751up_bot @ ( filter @ A10 ) ) ).
thf(tcon_Filter_Ofilter___Lattices_Obounded__semilattice__inf__top_272,axiom,
! [A10: $tType] : ( bounde4346867609351753570nf_top @ ( filter @ A10 ) ) ).
thf(tcon_Filter_Ofilter___Complete__Lattices_Ocomplete__lattice_273,axiom,
! [A10: $tType] : ( comple6319245703460814977attice @ ( filter @ A10 ) ) ).
thf(tcon_Filter_Ofilter___Lattices_Obounded__lattice__bot_274,axiom,
! [A10: $tType] : ( bounded_lattice_bot @ ( filter @ A10 ) ) ).
thf(tcon_Filter_Ofilter___Complete__Partial__Order_Occpo_275,axiom,
! [A10: $tType] : ( comple9053668089753744459l_ccpo @ ( filter @ A10 ) ) ).
thf(tcon_Filter_Ofilter___Lattices_Osemilattice__sup_276,axiom,
! [A10: $tType] : ( semilattice_sup @ ( filter @ A10 ) ) ).
thf(tcon_Filter_Ofilter___Lattices_Osemilattice__inf_277,axiom,
! [A10: $tType] : ( semilattice_inf @ ( filter @ A10 ) ) ).
thf(tcon_Filter_Ofilter___Lattices_Odistrib__lattice_278,axiom,
! [A10: $tType] : ( distrib_lattice @ ( filter @ A10 ) ) ).
thf(tcon_Filter_Ofilter___Lattices_Obounded__lattice_279,axiom,
! [A10: $tType] : ( bounded_lattice @ ( filter @ A10 ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Oorder__top_280,axiom,
! [A10: $tType] : ( order_top @ ( filter @ A10 ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Oorder__bot_281,axiom,
! [A10: $tType] : ( order_bot @ ( filter @ A10 ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Opreorder_282,axiom,
! [A10: $tType] : ( preorder @ ( filter @ A10 ) ) ).
thf(tcon_Filter_Ofilter___Lattices_Olattice_283,axiom,
! [A10: $tType] : ( lattice @ ( filter @ A10 ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Oorder_284,axiom,
! [A10: $tType] : ( order @ ( filter @ A10 ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Otop_285,axiom,
! [A10: $tType] : ( top @ ( filter @ A10 ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Oord_286,axiom,
! [A10: $tType] : ( ord @ ( filter @ A10 ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Obot_287,axiom,
! [A10: $tType] : ( bot @ ( filter @ A10 ) ) ).
thf(tcon_Option_Ooption___Countable_Ocountable_288,axiom,
! [A10: $tType] :
( ( countable @ A10 )
=> ( countable @ ( option @ A10 ) ) ) ).
thf(tcon_Option_Ooption___Finite__Set_Ofinite_289,axiom,
! [A10: $tType] :
( ( finite_finite @ A10 )
=> ( finite_finite @ ( option @ A10 ) ) ) ).
thf(tcon_Option_Ooption___Nat_Osize_290,axiom,
! [A10: $tType] : ( size @ ( option @ A10 ) ) ).
thf(tcon_Complex_Ocomplex___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_291,axiom,
semiri1453513574482234551roduct @ complex ).
thf(tcon_Complex_Ocomplex___Topological__Spaces_Ofirst__countable__topology_292,axiom,
topolo3112930676232923870pology @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__div__algebra_293,axiom,
real_V8999393235501362500lgebra @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__algebra__1_294,axiom,
real_V2822296259951069270ebra_1 @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Osemiring__no__zero__divisors__cancel_295,axiom,
semiri6575147826004484403cancel @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__algebra_296,axiom,
real_V4412858255891104859lgebra @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__vector_297,axiom,
real_V822414075346904944vector @ complex ).
thf(tcon_Complex_Ocomplex___Topological__Spaces_Otopological__space_298,axiom,
topolo4958980785337419405_space @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__field_299,axiom,
real_V3459762299906320749_field @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__div__algebra_300,axiom,
real_V5047593784448816457lgebra @ complex ).
thf(tcon_Complex_Ocomplex___Limits_Otopological__comm__monoid__add_301,axiom,
topolo5987344860129210374id_add @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Osemiring__1__no__zero__divisors_302,axiom,
semiri2026040879449505780visors @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__algebra__1_303,axiom,
real_V2191834092415804123ebra_1 @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Ocomplete__space_304,axiom,
real_V8037385150606011577_space @ complex ).
thf(tcon_Complex_Ocomplex___Limits_Otopological__semigroup__mult_305,axiom,
topolo4211221413907600880p_mult @ complex ).
thf(tcon_Complex_Ocomplex___Topological__Spaces_Ouniform__space_306,axiom,
topolo7287701948861334536_space @ complex ).
thf(tcon_Complex_Ocomplex___Topological__Spaces_Operfect__space_307,axiom,
topolo8386298272705272623_space @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Osemiring__no__zero__divisors_308,axiom,
semiri3467727345109120633visors @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Ometric__space_309,axiom,
real_V7819770556892013058_space @ complex ).
thf(tcon_Complex_Ocomplex___Limits_Otopological__ab__group__add_310,axiom,
topolo1287966508704411220up_add @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__vector_311,axiom,
real_V4867850818363320053vector @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Ocancel__ab__semigroup__add_312,axiom,
cancel2418104881723323429up_add @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Oring__1__no__zero__divisors_313,axiom,
ring_15535105094025558882visors @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__field_314,axiom,
real_V7773925162809079976_field @ complex ).
thf(tcon_Complex_Ocomplex___Limits_Otopological__monoid__add_315,axiom,
topolo6943815403480290642id_add @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Ocancel__comm__monoid__add_316,axiom,
cancel1802427076303600483id_add @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Ocomm__semiring__1__cancel_317,axiom,
comm_s4317794764714335236cancel @ complex ).
thf(tcon_Complex_Ocomplex___Limits_Otopological__group__add_318,axiom,
topolo1633459387980952147up_add @ complex ).
thf(tcon_Complex_Ocomplex___Topological__Spaces_Ot2__space_319,axiom,
topological_t2_space @ complex ).
thf(tcon_Complex_Ocomplex___Topological__Spaces_Ot1__space_320,axiom,
topological_t1_space @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Ocancel__semigroup__add_321,axiom,
cancel_semigroup_add @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Obanach_322,axiom,
real_Vector_banach @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Oab__semigroup__mult_323,axiom,
ab_semigroup_mult @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Osemiring__1__cancel_324,axiom,
semiring_1_cancel @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Ocomm__monoid__mult_325,axiom,
comm_monoid_mult @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Oab__semigroup__add_326,axiom,
ab_semigroup_add @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Ocomm__monoid__add_327,axiom,
comm_monoid_add @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Ocomm__semiring__1_328,axiom,
comm_semiring_1 @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Ocomm__semiring__0_329,axiom,
comm_semiring_0 @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Osemigroup__mult_330,axiom,
semigroup_mult @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Osemidom__divide_331,axiom,
semidom_divide @ complex ).
thf(tcon_Complex_Ocomplex___Num_Osemiring__numeral_332,axiom,
semiring_numeral @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Osemigroup__add_333,axiom,
semigroup_add @ complex ).
thf(tcon_Complex_Ocomplex___Fields_Ofield__abs__sgn_334,axiom,
field_abs_sgn @ complex ).
thf(tcon_Complex_Ocomplex___Fields_Odivision__ring_335,axiom,
division_ring @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Ocomm__semiring_336,axiom,
comm_semiring @ complex ).
thf(tcon_Complex_Ocomplex___Nat_Osemiring__char__0_337,axiom,
semiring_char_0 @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Oab__group__add_338,axiom,
ab_group_add @ complex ).
thf(tcon_Complex_Ocomplex___Fields_Ofield__char__0_339,axiom,
field_char_0 @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Ozero__neq__one_340,axiom,
zero_neq_one @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Oidom__abs__sgn_341,axiom,
idom_abs_sgn @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Omonoid__mult_342,axiom,
monoid_mult @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Oidom__divide_343,axiom,
idom_divide @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Ocomm__ring__1_344,axiom,
comm_ring_1 @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Omonoid__add_345,axiom,
monoid_add @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Osemiring__1_346,axiom,
semiring_1 @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Osemiring__0_347,axiom,
semiring_0 @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Ogroup__add_348,axiom,
group_add @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Omult__zero_349,axiom,
mult_zero @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Ocomm__ring_350,axiom,
comm_ring @ complex ).
thf(tcon_Complex_Ocomplex___Num_Oneg__numeral_351,axiom,
neg_numeral @ complex ).
thf(tcon_Complex_Ocomplex___Nat_Oring__char__0_352,axiom,
ring_char_0 @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Osemiring_353,axiom,
semiring @ complex ).
thf(tcon_Complex_Ocomplex___Fields_Oinverse_354,axiom,
inverse @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Osemidom_355,axiom,
semidom @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Ouminus_356,axiom,
uminus @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Oring__1_357,axiom,
ring_1 @ complex ).
thf(tcon_Complex_Ocomplex___Fields_Ofield_358,axiom,
field @ complex ).
thf(tcon_Complex_Ocomplex___Power_Opower_359,axiom,
power @ complex ).
thf(tcon_Complex_Ocomplex___Num_Onumeral_360,axiom,
numeral @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Ozero_361,axiom,
zero @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Oplus_362,axiom,
plus @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Oring_363,axiom,
ring @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Oidom_364,axiom,
idom @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Oone_365,axiom,
one @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Odvd_366,axiom,
dvd @ complex ).
thf(tcon_Extended__Nat_Oenat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_367,axiom,
condit6923001295902523014norder @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Countable__Complete__Lattices_Ocountable__complete__distrib__lattice_368,axiom,
counta4013691401010221786attice @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_369,axiom,
condit1219197933456340205attice @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Countable__Complete__Lattices_Ocountable__complete__lattice_370,axiom,
counta3822494911875563373attice @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Complete__Lattices_Ocomplete__distrib__lattice_371,axiom,
comple592849572758109894attice @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Ostrict__ordered__ab__semigroup__add_372,axiom,
strict9044650504122735259up_add @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Ostrict__ordered__comm__monoid__add_373,axiom,
strict7427464778891057005id_add @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Ocanonically__ordered__monoid__add_374,axiom,
canoni5634975068530333245id_add @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Lattices_Obounded__semilattice__sup__bot_375,axiom,
bounde4967611905675639751up_bot @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Lattices_Obounded__semilattice__inf__top_376,axiom,
bounde4346867609351753570nf_top @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Complete__Lattices_Ocomplete__linorder,axiom,
comple5582772986160207858norder @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Olinordered__ab__semigroup__add_377,axiom,
linord4140545234300271783up_add @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Complete__Lattices_Ocomplete__lattice_378,axiom,
comple6319245703460814977attice @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Olinordered__nonzero__semiring_379,axiom,
linord181362715937106298miring @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Osemiring__no__zero__divisors_380,axiom,
semiri3467727345109120633visors @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Oordered__ab__semigroup__add_381,axiom,
ordere6658533253407199908up_add @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Oordered__comm__monoid__add_382,axiom,
ordere6911136660526730532id_add @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Lattices_Obounded__lattice__bot_383,axiom,
bounded_lattice_bot @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Oordered__comm__semiring_384,axiom,
ordere2520102378445227354miring @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Complete__Partial__Order_Occpo_385,axiom,
comple9053668089753744459l_ccpo @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Lattices_Osemilattice__sup_386,axiom,
semilattice_sup @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Lattices_Osemilattice__inf_387,axiom,
semilattice_inf @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Lattices_Odistrib__lattice_388,axiom,
distrib_lattice @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Lattices_Obounded__lattice_389,axiom,
bounded_lattice @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Oab__semigroup__mult_390,axiom,
ab_semigroup_mult @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Ocomm__monoid__mult_391,axiom,
comm_monoid_mult @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Oab__semigroup__add_392,axiom,
ab_semigroup_add @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Oordered__semiring_393,axiom,
ordered_semiring @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Ocomm__monoid__add_394,axiom,
comm_monoid_add @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Ocomm__semiring__1_395,axiom,
comm_semiring_1 @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Ocomm__semiring__0_396,axiom,
comm_semiring_0 @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Osemigroup__mult_397,axiom,
semigroup_mult @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Num_Osemiring__numeral_398,axiom,
semiring_numeral @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Osemigroup__add_399,axiom,
semigroup_add @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Ozero__less__one_400,axiom,
zero_less_one @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Ocomm__semiring_401,axiom,
comm_semiring @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Orderings_Owellorder_402,axiom,
wellorder @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Orderings_Oorder__top_403,axiom,
order_top @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Orderings_Oorder__bot_404,axiom,
order_bot @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Nat_Osemiring__char__0_405,axiom,
semiring_char_0 @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Countable_Ocountable_406,axiom,
countable @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Ozero__neq__one_407,axiom,
zero_neq_one @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Orderings_Opreorder_408,axiom,
preorder @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Orderings_Olinorder_409,axiom,
linorder @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Omonoid__mult_410,axiom,
monoid_mult @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Omonoid__add_411,axiom,
monoid_add @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Osemiring__1_412,axiom,
semiring_1 @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Osemiring__0_413,axiom,
semiring_0 @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Lattices_Olattice_414,axiom,
lattice @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Omult__zero_415,axiom,
mult_zero @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Orderings_Oorder_416,axiom,
order @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Osemiring_417,axiom,
semiring @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Orderings_Otop_418,axiom,
top @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Orderings_Oord_419,axiom,
ord @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Orderings_Obot_420,axiom,
bot @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Power_Opower_421,axiom,
power @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Num_Onumeral_422,axiom,
numeral @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Ozero_423,axiom,
zero @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Oplus_424,axiom,
plus @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Oone_425,axiom,
one @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Odvd_426,axiom,
dvd @ extended_enat ).
thf(tcon_Product__Type_Oprod___Topological__Spaces_Otopological__space_427,axiom,
! [A10: $tType,A29: $tType] :
( ( ( topolo4958980785337419405_space @ A10 )
& ( topolo4958980785337419405_space @ A29 ) )
=> ( topolo4958980785337419405_space @ ( product_prod @ A10 @ A29 ) ) ) ).
thf(tcon_Product__Type_Oprod___Topological__Spaces_Ot2__space_428,axiom,
! [A10: $tType,A29: $tType] :
( ( ( topological_t2_space @ A10 )
& ( topological_t2_space @ A29 ) )
=> ( topological_t2_space @ ( product_prod @ A10 @ A29 ) ) ) ).
thf(tcon_Product__Type_Oprod___Topological__Spaces_Ot1__space_429,axiom,
! [A10: $tType,A29: $tType] :
( ( ( topological_t1_space @ A10 )
& ( topological_t1_space @ A29 ) )
=> ( topological_t1_space @ ( product_prod @ A10 @ A29 ) ) ) ).
thf(tcon_Product__Type_Oprod___Countable_Ocountable_430,axiom,
! [A10: $tType,A29: $tType] :
( ( ( countable @ A10 )
& ( countable @ A29 ) )
=> ( countable @ ( product_prod @ A10 @ A29 ) ) ) ).
thf(tcon_Product__Type_Oprod___Finite__Set_Ofinite_431,axiom,
! [A10: $tType,A29: $tType] :
( ( ( finite_finite @ A10 )
& ( finite_finite @ A29 ) )
=> ( finite_finite @ ( product_prod @ A10 @ A29 ) ) ) ).
thf(tcon_Product__Type_Oprod___Nat_Osize_432,axiom,
! [A10: $tType,A29: $tType] : ( size @ ( product_prod @ A10 @ A29 ) ) ).
thf(tcon_Product__Type_Ounit___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_433,axiom,
condit6923001295902523014norder @ product_unit ).
thf(tcon_Product__Type_Ounit___Countable__Complete__Lattices_Ocountable__complete__distrib__lattice_434,axiom,
counta4013691401010221786attice @ product_unit ).
thf(tcon_Product__Type_Ounit___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_435,axiom,
condit1219197933456340205attice @ product_unit ).
thf(tcon_Product__Type_Ounit___Countable__Complete__Lattices_Ocountable__complete__lattice_436,axiom,
counta3822494911875563373attice @ product_unit ).
thf(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__distrib__lattice_437,axiom,
comple592849572758109894attice @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Obounded__semilattice__sup__bot_438,axiom,
bounde4967611905675639751up_bot @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Obounded__semilattice__inf__top_439,axiom,
bounde4346867609351753570nf_top @ product_unit ).
thf(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__linorder_440,axiom,
comple5582772986160207858norder @ product_unit ).
thf(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__lattice_441,axiom,
comple6319245703460814977attice @ product_unit ).
thf(tcon_Product__Type_Ounit___Boolean__Algebras_Oboolean__algebra_442,axiom,
boolea8198339166811842893lgebra @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Obounded__lattice__bot_443,axiom,
bounded_lattice_bot @ product_unit ).
thf(tcon_Product__Type_Ounit___Complete__Partial__Order_Occpo_444,axiom,
comple9053668089753744459l_ccpo @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Osemilattice__sup_445,axiom,
semilattice_sup @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Osemilattice__inf_446,axiom,
semilattice_inf @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Odistrib__lattice_447,axiom,
distrib_lattice @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Obounded__lattice_448,axiom,
bounded_lattice @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Owellorder_449,axiom,
wellorder @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Oorder__top_450,axiom,
order_top @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Oorder__bot_451,axiom,
order_bot @ product_unit ).
thf(tcon_Product__Type_Ounit___Countable_Ocountable_452,axiom,
countable @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Opreorder_453,axiom,
preorder @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Olinorder_454,axiom,
linorder @ product_unit ).
thf(tcon_Product__Type_Ounit___Finite__Set_Ofinite_455,axiom,
finite_finite @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Olattice_456,axiom,
lattice @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Oorder_457,axiom,
order @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Otop_458,axiom,
top @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Oord_459,axiom,
ord @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Obot_460,axiom,
bot @ product_unit ).
thf(tcon_Product__Type_Ounit___Groups_Ouminus_461,axiom,
uminus @ product_unit ).
thf(tcon_VEBT__Definitions_OVEBT___Nat_Osize_462,axiom,
size @ vEBT_VEBT ).
% Helper facts (4)
thf(help_If_3_1_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_T,axiom,
! [A: $tType,X: A,Y: A] :
( ( if @ A @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_T,axiom,
! [A: $tType,X: A,Y: A] :
( ( if @ A @ $true @ X @ Y )
= X ) ).
thf(help_fChoice_1_1_T,axiom,
! [A: $tType,P: A > $o] :
( ( P @ ( fChoice @ A @ P ) )
= ( ? [X6: A] : ( P @ X6 ) ) ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( nth @ vEBT_VEBT @ ( list_update @ vEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ mi @ na ) @ ( vEBT_vebt_insert @ ( nth @ vEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ mi @ na ) ) @ ( vEBT_VEBT_low @ mi @ na ) ) ) @ i )
= ( vEBT_vebt_insert @ ( nth @ vEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ mi @ na ) ) @ ( vEBT_VEBT_low @ mi @ na ) ) ) ).
%------------------------------------------------------------------------------