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products/Sources/formale Sprachen/Isabelle/HOL/Nonstandard_Analysis/ (Beweissystem Isabelle Version 2025-1^©) Datei vom 16.11.2025 mit Größe 23 kB

Quelle HSEQ.thy Sprache: Isabelle

(*  Title:      HOL/Nonstandard_Analysis/HSEQ.thy
    Author:     Jacques D. Fleuriot
    Copyright:  1998  University of Cambridge

Convergence of sequences and series.

Conversion to Isar and new proofs by Lawrence C Paulson, 2004
Additional contributions by Jeremy Avigad and Brian Huffman.
*)

section \<open>Sequences and Convergence (Nonstandard)\<close>

   "X \ L"
proofrule)
  abbrevs:real
begin

definition NSLIMSEQ :: "(nat \ 'a::real_normed_vector) \ 'a \ bool"
    (\<open>(\<open>notation=\<open>mixfix NSLIMSEQ\<close>\<close>(_)/ \<longlonglongrightarrow>\<^sub>N\<^sub>S (_))\<close> [60, 60] 60) whereassume:0<rjava.lang.StringIndexOutOfBoundsException: Index 19 out of bounds for length 19
    \<comment> \<open>Nonstandard definition of convergence of sequence\<close>
" \\<^sub>N\<^sub>S L \ (\N \ HNatInfinite. ( *f* X) N \ star_of L)"

       ( HNatInfinite_upward_closed)
  wherewith havestarfunn\<approx> star_of L"
  \<comment> \<open>Nonstandard definition of limit using choice operator\<close>

definitionby ( only approx_def
  whereNSconvergentX \<longleftrightarrow> (\<exists>L. X \<longlonglongrightarrow>\<^sub>N\<^sub>S L)"

definitionNSBseq :: "nat \ 'a::real_normed_vector) \ bool"
  where "NSBseq X \ (\N \ HNatInfinite. ( *f* X) N \ HFinite)"
  \<comment> \<open>Nonstandard definition for bounded sequence\<close>

definitionNSCauchy ::"(nat 'a::real_normed_vector) \ bool"
  where "NSCauchy X \ (\M \ HNatInfinite. \N \ HNatInfinite. ( *f* X) M \ ( *f* X) N)"
<comment

subsection \<open>Limits of Sequences\<close>

lemmaby( intro  NSLIMSEQ_LIMSEQjava.lang.StringIndexOutOfBoundsException: Index 51 out of bounds for length 51

textjava.lang.StringIndexOutOfBoundsException: Index 77 out of bounds for length 77
  by (simp add  seems complicated than standard one!\<close>

lemma : "(\n. k) \\<^sub>N\<^sub>S k"
  by (simp (simp add:LIMSEQ_NSLIMSEQ_iffsymmetric)

lemma NSLIMSEQ_add: "X \\<^sub>N\<^sub>S a \ Y \\<^sub>N\<^sub>S b \ (\n. X n + Y n) \\<^sub>N\<^sub>S a + b"
  by auto: approx_add  add: NSLIMSEQ_def

lemma NSLIMSEQ_add_const:  by( add: LIMSEQ_NSLIMSEQ_iffsymmetrictendsto_rabs_zero_iff)
text

lemma NSLIMSEQ_multlemmaNSLIMSEQ_imp_rabsapprox_hrabssimp add)
  for a b :
  mmaNSLIMSEQ_inverse_zero "<>y::real.\

lemma NSLIMSEQ_minusby( add:  [symmetricLIMSEQ_inverse_zero
  by (lemma NSLIMSEQ_inverse_real_of_nat

lemma   (simp add:LIMSEQ_NSLIMSEQ_iffsymmetricLIMSEQ_inverse_real_of_nat: of_nat_Suc)
  by

lemma: "(\n. r + inverse (real (Suc n))) \\<^sub>N\<^sub>S r"
using [ofXa" Y simpadd fun_Compl_def)

lemma NSLIMSEQ_diff_const: "f \\<^sub>N\<^sub>S a \ (\n. f n - b) \\<^sub>N\<^sub>S a - b"
  by (simp add: NSLIMSEQ_diff NSLIMSEQ_const)

lemma NSLIMSEQ_inverse: "X \\<^sub>N\<^sub>S a \ a \ 0 \ (\n. inverse (X n)) \\<^sub>N\<^sub>S inverse a"
  for a :
  by (lemma NSLIMSEQ_inverse_real_of_nat_add_minus:"(n. r + - inverse (real (Suc n))) \\<^sub>N\<^sub>S r"

lemmaNSLIMSEQ_mult_inverse X <>\<^sub>N\<^sub>S a \<Longrightarrow> Y \<longlonglongrightarrow>\<^sub>N\<^sub>S b \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> (\<lambda>n. X n / Y n) \<longlonglongrightarrow>\<^sub>N\<^sub>S a / b"
  for:"'a::real_normed_field"

lemma NSLIMSEQ_diff_const: "f \<longlonglongrightarrow>\<^sub>N\<^sub>S a \<Longrightarrow> (\<lambda>n. f n - b) \<longlonglongrightarrow>\<^sub>N\<^sub>S a - b"
  by (simp add: NSLIMSEQ_diff NSLIMSEQ_const)

lemma NSLIMSEQ_inverse: "X \<longlonglongrightarrow>\<^sub>N\<^sub>S a \<Longrightarrow> a \<noteq> 0 \<Longrightarrow> (\<lambda>n. inverse (X n)) \<longlonglongrightarrow>\<^sub>N\<^sub>S inverse a"
  for a :: "'a::real_normed_div_algebra"
  by (simp add: NSLIMSEQ_def star_of_approx_inverse)

lemma NSLIMSEQ_mult_inverse: "X \<longlonglongrightarrow>\<^sub>N\<^sub>S a \<Longrightarrow> Y \<longlonglongrightarrow>\<^sub>N\<^sub>S b \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> (\<lambda>n. X n / Y n) \<longlonglongrightarrow>\<^sub>N\<^sub>S a / b"
  for a b :: "'a::real_normed_field"
  by (simp add: NSLIMSEQ_mult NSLIMSEQ_inverse divide_inverse)

lemma starfun_hnorm: "\<And>x. hnorm (( *f* f) x) = ( *f* (\<lambda>x. norm (f x))) x"
  by transfer simp

lemma NSLIMSEQ_norm: "X \<longlonglongrightarrow>\<^sub>N\<^sub>S a \<Longrightarrow> (\<lambda>n. norm (X n)) \<longlonglongrightarrow>\<^sub>N\<^sub>S norm a"
  by (simp add: NSLIMSEQ_def starfun_hnorm [symmetric] approx_hnorm)

text \<open>Uniqueness of limit.\<close>
lemma NSLIMSEQ_unique: "X \<longlonglongrightarrow>\<^sub>N\<^sub>S a \<Longrightarrow> X \<longlonglongrightarrow>\<^sub>N\<^sub>S b \<Longrightarrow> a = b"
  unfolding NSLIMSEQ_def
  using HNatInfinite_whn approx_trans3 star_of_approx_iff by blast

lemma NSLIMSEQ_pow [rule_format]: "(X \<longlonglongrightarrow>\<^sub>N\<^sub>S a) \<longrightarrow> ((\<lambda>n. (X n) ^ m) \<longlonglongrightarrow>\<^sub>N\<^sub>S a ^ m)"
  for a :: "'a::{real_normed_algebra,power}"
  by (induct m) (auto intro: NSLIMSEQ_mult NSLIMSEQ_const)

text \<open>We can now try and derive a few properties of sequences,
  starting with the limit comparison property for sequences.\<close>

lemma NSLIMSEQ_le: "f \<longlonglongrightarrow>\<^sub>N\<^sub>S l \<Longrightarrow> g \<longlonglongrightarrow>\<^sub>N\<^sub>S m \<Longrightarrow> \<exists>N. \<forall>n \<ge> N. f n \<le> g n \<Longrightarrow> l \<le> m"
  for l m :: real
  unfolding NSLIMSEQ_def
  by (metis HNatInfinite_whn bex_Infinitesimal_iff2 hypnat_of_nat_le_whn hypreal_of_real_le_add_Infininitesimal_cancel2 starfun_le_mono)

lemma NSLIMSEQ_le_const: "X \<longlonglongrightarrow>\<^sub>N\<^sub>S r \<Longrightarrow> \<forall>n. a \<le> X n \<Longrightarrow> a \<le> r"
  for a r :: real
  by (erule NSLIMSEQ_le [OF NSLIMSEQ_const]) auto

lemma NSLIMSEQ_le_const2: "X \<longlonglongrightarrow>\<^sub>N\<^sub>S r \<Longrightarrow> \<forall>n. X n \<le> a \<Longrightarrow> r \<le> a"
  for a r :: real
  by (erule NSLIMSEQ_le [OF _ NSLIMSEQ_const]) auto

text \<open>Shift a convergent series by 1:
  By the equivalence between Cauchiness and convergence and because
  the successor of an infinite hypernatural is also infinite.\<close>

lemma NSLIMSEQ_Suc_iff: "((\<lambda>n. f (Suc n)) \<longlonglongrightarrow>\<^sub>N\<^sub>S l) \<longleftrightarrow> (f \<longlonglongrightarrow>\<^sub>N\<^sub>S l)"
proof
  assume *: "f \<longlonglongrightarrow>\<^sub>N\<^sub>S l"
  show "(\<lambda>n. f(Suc n)) \<longlonglongrightarrow>\<^sub>N\<^sub>S l"
  proof (rule NSLIMSEQ_I)
    fix N
    assume "N \<in> HNatInfinite"
    then have "(*f* f) (N + 1) \<approx> star_of l"
      by (simp add: HNatInfinite_add NSLIMSEQ_D *)
by add)
       NSLIMSEQ_const \>nk java.lang.StringIndexOutOfBoundsException: Index 83 out of bounds for length 83
   auto simp:)
next

rightarrow
  proof (rule onlyNSLIMSEQ_const
java.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 9

    then have "(*f* (\n. f (Suc n))) (N - 1) \ star_of l"
     using* simp: NSLIMSEQ_D
    then   by( simp:NSLIMSEQ_def

  qed
qed

subsubsection \<open>Equivalence of \<^term>\<open>LIMSEQ\<close> and \<^term>\<open>NSLIMSEQ\<close>\<close>

:
  assumes X:
   "X \\<^sub>N\<^sub>S L"
proofrule)
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
assume: in
  have "starfun X N - java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  fora: ':real_normed_div_algebrajava.lang.StringIndexOutOfBoundsException: Index 40 out of bounds for length 40
    fixr: java.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 17
  0<r
    from
then java.lang.StringIndexOutOfBoundsException: Index 86 out of bounds for length 86
      by transfer
    then " (starfun XN- L) < star_of rjava.lang.StringIndexOutOfBoundsException: Index 59 out of bounds for length 59
N simp)
  qed
  then
     simp: approx_def)
qed

lemma NSLIMSEQ_LIMSEQ:
  lemmaNSLIMSEQ_unique: X
  shows " NSLIMSEQ_def
proofrule)
  fix
lemmaNSLIMSEQ_pow]: "X\\<^sub>N\<^sub>S a) \ ((\n. (X n) ^ m) \\<^sub>N\<^sub>S a ^ m)"
  have "\no. \n\no. hnorm (starfun X n - star_of L) < star_of r"
  prooffor :: 'a:{power"
    fix n
    assume "whn \ n"
    with HNatInfinite_whn have "n \ HNatInfinite"
HNatInfinite_upward_closed
    with X java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
byrule)
    then have "starfun X n - star_of L
       simp: approx_def
    then show   for :real
       r by( InfinitesimalD2
: "X \\<^sub>N\<^sub>S r \ \n. a \ X n \ a \ r"
then"<>no. \n\no. norm (X n - L) < r"

qed

lemma: "X\
     erule [ _ ]) auto

\<open>Derived theorems about \<^term>\<open>NSLIMSEQ\<close>\<close> successor an hypernatural also.\<close>

text
   more thanthe standard one above!\<close>
lemma NSLIMSEQ_norm_zero: "(\n. norm (X n)) \\<^sub>N\<^sub>S 0 \ X \\<^sub>N\<^sub>S 0"
  by (simp: LIMSEQ_NSLIMSEQ_iffsymmetrictendsto_norm_zero_iff)

lemma   "(\n. f(Suc n)) \\<^sub>N\<^sub>S l"
  by (simp symmetric)

text \<open>Generalization to other limits.\<close>
lemmaNSLIMSEQ_imp_rabsf\<longlonglongrightarrow>\<^sub>N\<^sub>S l \<Longrightarrow> (\<lambda>n. \<bar>f n\<bar>) \<longlonglongrightarrow>\<^sub>N\<^sub>S \<bar>l\<bar>"
  bysimp:HNatInfinite_addNSLIMSEQ_D
   simp:N) (auto:  simp: starfun_abs

lemma      bysimp: starfun_shift_one
by( addLIMSEQ_NSLIMSEQ_iff [ymmetric)

lemma NSLIMSEQ_inverse_real_of_nat "f \<^sub>N\<^sub>S l"
   ( NSLIMSEQ_Ijava.lang.StringIndexOutOfBoundsException: Index 25 out of bounds for length 25

lemma NSLIMSEQ_inverse_real_of_nat_add: "(\n. r + inverse (real (Suc n))) \\<^sub>N\<^sub>S r"
simpLIMSEQ_NSLIMSEQ_iff [symmetricLIMSEQ_inverse_real_of_nat_add: of_nat_Suc

qed


lemma
  "(\n. r * (1 + - inverse (real (Suc n)))) \\<^sub>N\<^sub>S r"
  using LIMSEQ_inverse_real_of_nat_add_minus_mult LIMSEQ_NSLIMSEQ
  by ( add LIMSEQ_NSLIMSEQ_iffsymmetric

subsection \<open>Convergence\<close>

lemma nslimI: "X \\<^sub>N\<^sub>S L \ nslim X = L"
by( add) (blast: )

lemma lim_nslim_iff  fixjava.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
  by (simpreal

lemma NSconvergentD: "NSconvergent X \ \L. X \\<^sub>N\<^sub>S L"
  by (simp add: NSconvergent_def    fromLIMSEQ_DOF r  no "\n\no. norm (X n - L) < r" ..

lemmatransfer
  by (auto simp add: NSconvergent_def)

lemma convergent_NSconvergent_iff:     thenshow"norm(starfunX N star_ofL star_of r"
  by (simp add: convergent_def NSconvergent_def LIMSEQ_NSLIMSEQ_iff)

  qed
  by (auto intro: theI NSLIMSEQ_unique simp add: NSconvergent_def nslim_def)

subsection \<open>Bounded Monotonic Sequences\<close>

lemmaby( only:approx_def
  lemma:

Standard_subset_HFiniteStandard
  by  shows java.lang.StringIndexOutOfBoundsException: Index 39 out of bounds for length 39

:" X \ ( *f* X) N \ HFinite"
  using HNatInfinite_def  proofintro allI)

lemma     HNatInfinite_whn "\ HNatInfinite"
      by ( HNatInfinite_upward_closed

text \<open>The standard definition implies the nonstandard definition.\<close>
lemma Bseq_NSBseq: "Bseq X \ NSBseq X"
        by rule)
proofsafe
  assume X: "Bseq X"
  fix N
  assume N: "N \ HNatInfinite"
  from BseqD [OF X] obtain K where "\n. norm (X n) \ K"
by
then \>  (  )\le  "
    by transfer
then" (starfun XN \ star_of K"
    bythen "existsno.\n\no. norm (X n - L) < r"
  alsojava.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
simp


   showXjava.lang.NullPointerException
text
qed

text \<open>The nonstandard definition implies the standard definition.\<close> more  than  standard above\<close>
lemma SReal_less_omega: "r \ \ \ r < \"
  using   (imp:  [symmetric)
lemma: "(\n. \f n\) \\<^sub>N\<^sub>S 0 \ f \\<^sub>N\<^sub>S (0::real)"

lemma NSBseq_Bseq: "NSBseq X \ Bseq X"
  by (simp: LIMSEQ_NSLIMSEQ_iff] tendsto_rabs_zero_iff
text \<open>Generalization to other limits.\<close>
   " X"
  then l : real
    by( NSBseqD2
  assume "lemmaNSLIMSEQ_inverse_zero:"<>y:. \<exists>N. \<forall>n \<ge> N. y < f n \<Longrightarrow> (\<lambda>n. inverse (f n)) \<longlonglongrightarrow>\<^sub>N\<^sub>S 0" :  [symmetric
  then    (impadd: LIMSEQ_NSLIMSEQ_iffsymmetric )

  then NSLIMSEQ_inverse_real_of_nat_add(<lambda +( Suc longlonglongrightarrow
    by auto:LeastI_ex
lemmaNSLIMSEQ_inverse_real_of_nat_add_minus(
    by simp:  [symmetric
  then NSLIMSEQ_inverse_real_of_nat_add_minus_multjava.lang.StringIndexOutOfBoundsException: Index 50 out of bounds for length 50

  then have
    bysimp:order_less_trans SReal_less_omega
  then
    by( add)
  with finite   simp:nslim_def intro: NSLIMSEQ_unique)
    bylemma lim_nslim_iff " X = X"
qed

\<open>Equivalence of nonstandard and standard definitions for a bounded sequence.\<close> : )
emma: BseqNSBseq Xjava.lang.StringIndexOutOfBoundsException: Index 42 out of bounds for length 42
(blastjava.lang.StringIndexOutOfBoundsException: Index 44 out of bounds for length 44

: "NSconvergentX X \\<^sub>N\<^sub>S nslim X"
  Boundedness as a necessary condition for convergence.
  The nonstandard version has no existential, as usual.\<close>  byauto:  NSLIMSEQ_unique addNSconvergent_def nslim_def)
NSconvergent_NSBseq"SconvergentX\Longrightarrow "
lemma: " X \ N \ HNatInfinite \ ( *f* X) N \ HFinite"
    blast: approx_sym)

textjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
definitions

lemma convergent_Bseq: "convergent X \ Bseq X"
  forlemma NSBseqI "<>N\in . ( ** X) N \ HFinite \ NSBseq X"
  by simp: )

subsubsection \<open>Upper Bounds and Lubs of Bounded Sequences\<close>

X\<Longrightarrow> \<exists>U::real. isUb UNIV {x. \<exists>n. X n = x} U"
( add [symmetric)

NSBseq_isLub NSBseqX  \<Longrightarrow> \<exists>U::real. isLub UNIV {x. \<exists>n. X n = x} U"
byadd]java.lang.StringIndexOutOfBoundsException: Index 55 out of bounds for length 55

subsubsection \<open>A Bounded and Monotonic Sequence Converges\<close>

text     by (rule bexI) simp
   theorem and then equivalence transfer into
   equivalent nonstandard formby( addHFinite_def)

lemma Bmonoseq_NSLIMSEQ: "\\<^sub>F k in sequentially. X k = X m \ X \\<^sub>N\<^sub>S X m"
  java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
  bysimp:eventually_monoeventually_nhds_x_imp_x )

lemma NSBseq_mono_NSconvergentlemma: "r \ \ r < \"
   X : "simp)
  by (auto
      : convergent_NSconvergent_iff[] Bseq_NSBseq_iff])

subsection n \>. .   (X n)"

lemma NSCauchyI:
  "(\M N. M \ HNatInfinite \ N \ HNatInfinite \ starfun X M \ starfun X N) \ NSCauchy X"
(:java.lang.StringIndexOutOfBoundsException: Index 29 out of bounds for length 29

lemma NSCauchyD:
  "NSCauchy X \ M \ HNatInfinite \ N \ HNatInfinite \ starfun X M \ starfun X N"
  by (

subsubsection \<open>Equivalence Between NS and Standard\<close>

lemma  then have"K>.
  assumesCauchy
  showsthen "
proof (rule NSCauchyI)
   M
  assume M: " then \omega < hnorm (( **X ( f n
  fix    bysimp
assumeN " HNatInfinite"
  have "starfun X M - starfun X N \ Infinitesimal"
  proof (rule InfinitesimalI2)
    fix:
    assume r: "0 < r"
from OF]obtainwhere
    then have "\m\star_of k. \n\star_of k. hnorm (starfun X m - starfun X n) < star_of r"
      byjava.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 17
    then show "hnorm (starfun X M - starfun X N) < star_of r"
MNbysimp:)
  qed
  then show "starfun X M \ starfun X N"
    by (simp: approx_def
qed

lemma <>Equivalence nonstandard  definitionsforabounded.
  lemma: "Bseq X = NSBseq "
  shows "Cauchy X"
proof (rule CauchyI)
  fix r :: real
r 0<r
  have
  proofintro allI)
       asa necessary for.
     "\ M"
    with lemmaNSconvergent_NSBseq " X \ NSBseq X"
by( HNatInfinite_upward_closed
    fix N
assumewhn
    withHNatInfinite_whn :"
  definitions\<close>
    from X M N have "starfun X M \ starfun X N"
      by (rule NSCauchyD
     have" X M starfun N\<> Infinitesimal"
( only)
    then show "hnorm ( (simpadd:NSconvergent_NSBseq convergent_NSconvergent_iff Bseq_NSBseq_iff)
      using  by ( InfinitesimalD2
  qed
  then show "\k. \m\k. \n\k. norm (X m - X n) < r"
    by transfer
qed

theorem NSCauchy_Cauchy_iff: "NSCauchy X = Cauchy X"
  by lemmaNSBseq_isUbNSBseqjava.lang.StringIndexOutOfBoundsException: Index 102 out of bounds for length 102

   ( add Bseq_NSBseq_iffsymmetric)

text

lemma NSCauchy_NSBseq" X \ NSBseq X"
  bytext\<open>The best of both worlds: Easier to prove this result as a standard

subsubsection     nonstandard if!\<close>

text LIMSEQ_NSLIMSEQ_iff]
   simp:  eventually_nhds_x_imp_x)
  much easier
  need use ofsubsequences such boundedness
monotonicity.. Compare 's proof
  in  intro
  not have      : convergent_NSconvergent_iffsymmetric [symmetric
  subsection
  java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

lemma
   ( add NSLIMSEQ_def) ( introapprox_trans2

lemma" X\ M \ HNatInfinite \ N \ HNatInfinite \ starfun X M \ starfun X N"
  fixes X :: "nat \ real"
  assumes "NSCauchy X" shows "NSconvergent X"

subsubs \<open>Equivalence Between NS and Standard\<close>
  haveassumes" X"
    by( add NSCauchy_NSBseq)

  M M\<in> HNatInfinite"
  ssume "\
by dest: simp: SReal_iff:approx_trans3
qed

lemma NSCauchy_NSconvergent: "NSCauchy X \ NSconvergent X"
  for X     r ::java.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 17
    NSCauchy_Cauchy by

lemma NSCauchy_NSconvergent_iff:     have
for : " \ 'a::banach"
     show   XN < r"

subsection \<open>Power Sequences\<close>

using  simpstar_of_le_HNatInfinite
also that and  converges

\<open>We now use NS criterion to bring proof of theorem through.\<close>
:
   X: " X"
  assumes
proof -
   ( **^ )
    if  fix : real
  proof
    have  have"<> forall>m\k. \n\k. hnorm (starfun X m - starfun X n) < star_of r"
      by (metis HNatInfinite_add N NSCauchy_NSconvergent_iff NSCauchy_def starfun_pow x)
  L  L:"x pow N \ hypreal_of_real L"
      using NSconvergentD [OF x]     fixM
    ultimatelyM Mjava.lang.NullPointerException
       simp: hyperpow_add
    then have    fixjava.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 9
      using L approx_trans3XM  " X M approx> starfun X N"
show
      bythen starfun-
java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
       r by( InfinitesimalD2
    by  then "\k. \m\k. \n\k. norm (X m - X n) < r"
qed

java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  for c :: real
   (imp: LIMSEQ_abs_realpow_zeroLIMSEQ_NSLIMSEQ_iffsymmetric])

lemmajava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  for:real
  by (simp add: LIMSEQ_abs_realpow_zero2 java.lang.StringIndexOutOfBoundsException: Index 50 out of bounds for length 0

end

quality99%

;prove    by (force dest!: st_part_Exqed
  much  for X :: "nat \ 'a::banach"
  needlemma NSCauchy_NSconvergent_iff: "NSCauchy X = java.lang.StringIndexOutOfBoundsException: Index 49 out of bounds for length 41
  monotonicitylemma NSLIMSEQ_realpow_zero  fixes x   assumes "0 \ x" "x < 1" shows "(\n. x ^ n) \\<^sub>N\<^sub>S 0"
  in HOL    if N: "N \ HNatInfinite" and x: "NSconvergent ((^) x)" for N
  not       by (metis HNatInfinite_add N NSCauchy_NSconvergent_iff NSCauchy_def    moreover obtain L where L: "hypreal_of_real using NSconvergentD [OF x] N by (auto simp add: NSLIMSEQ_def starfun_pow ultimately have "hypreal_of_real x pow N \<approx> hypreal_of_real L * hypreal_of_real x"
  instantiations for his 'espsilon-delta' proof(s) in  qed
  since the NS formulationsqed

lemma NSconvergent_NSCauchy: "NSconvergent X \ NSCauchy X"
  by (simp add: java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

lemma real_NSCauchy_NSconvergent:
  fixes X :: "nat \ real"
  assumes "NSCauchy X" shows "NSconvergent X"
  unfolding NSconvergent_def NSLIMSEQ_def
proof -
  have "( *f* X) whn \ HFinite"
    by (simp add: NSBseqD2 NSCauchy_NSBseq assms)
  moreover have "\N\HNatInfinite. ( *f* X) whn \ ( *f* X) N"
    using HNatInfinite_whn NSCauchy_def assms by blast
  ultimately show "\L. \N\HNatInfinite. ( *f* X) N \ hypreal_of_real L"
    by (force dest!: st_part_Ex simp add: SReal_iff intro: approx_trans3)
qed

lemma NSCauchy_NSconvergent: "NSCauchy X \ NSconvergent X"
  for X :: "nat \ 'a::banach"
  using Cauchy_convergent NSCauchy_Cauchy convergent_NSconvergent_iff by auto

lemma NSCauchy_NSconvergent_iff: "NSCauchy X = NSconvergent X"
  for X :: "nat \ 'a::banach"
  by (fast intro: NSCauchy_NSconvergent NSconvergent_NSCauchy)

subsection \<open>Power Sequences\<close>

text \<open>The sequence \<^term>\<open>x^n\<close> tends to 0 if \<^term>\<open>0\<le>x\<close> and \<^term>\<open>x<1\<close>.  Proof will use (NS) Cauchy equivalence for convergence and
  also fact that bounded and monotonic sequence converges.\<close>

text \<open>We now use NS criterion to bring proof of theorem through.\<close>
lemma NSLIMSEQ_realpow_zero:
  fixes x :: real
  assumes "0 \ x" "x < 1" shows "(\n. x ^ n) \\<^sub>N\<^sub>S 0"
proof -
  have "( *f* (^) x) N \ 0"
    if N: "N \ HNatInfinite" and x: "NSconvergent ((^) x)" for N
  proof -
    have "hypreal_of_real x pow N \ hypreal_of_real x pow (N + 1)"
      by (metis HNatInfinite_add N NSCauchy_NSconvergent_iff NSCauchy_def starfun_pow x)
    moreover obtain L where L: "hypreal_of_real x pow N \ hypreal_of_real L"
      using NSconvergentD [OF x] N by (auto simp add: NSLIMSEQ_def starfun_pow)
    ultimately have "hypreal_of_real x pow N \ hypreal_of_real L * hypreal_of_real x"
      by (simp add: approx_mult_subst_star_of hyperpow_add)
    then have "hypreal_of_real L \ hypreal_of_real L * hypreal_of_real x"
      using L approx_trans3 by blast
    then show ?thesis
      by (metis L \<open>x < 1\<close> hyperpow_def less_irrefl mult.right_neutral mult_left_cancel star_of_approx_iff star_of_mult star_of_simps(9) starfun2_star_of)
  qed
  with assms show ?thesis
    by (force dest!: convergent_realpow simp add: NSLIMSEQ_def convergent_NSconvergent_iff)
qed

lemma NSLIMSEQ_abs_realpow_zero: "\c\ < 1 \ (\n. \c\ ^ n) \\<^sub>N\<^sub>S 0"
  for c :: real
  by (simp add: LIMSEQ_abs_realpow_zero LIMSEQ_NSLIMSEQ_iff [symmetric])

lemma NSLIMSEQ_abs_realpow_zero2: "\c\ < 1 \ (\n. c ^ n) \\<^sub>N\<^sub>S 0"
  for c :: real
  by (simp add: LIMSEQ_abs_realpow_zero2 LIMSEQ_NSLIMSEQ_iff [symmetric])

end

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