src/HOL/Hyperreal/SEQ.thy
author huffman
Sun Apr 08 18:35:19 2007 +0200 (2007-04-08)
changeset 22614 17644bc9cee4
parent 22608 092a3353241e
child 22615 d650e51b5970
permissions -rw-r--r--
rearranged sections
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(*  Title       : SEQ.thy
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    Author      : Jacques D. Fleuriot
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    Copyright   : 1998  University of Cambridge
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    Description : Convergence of sequences and series
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    Conversion to Isar and new proofs by Lawrence C Paulson, 2004
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    Additional contributions by Jeremy Avigad and Brian Huffman
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*)
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header {* Sequences and Series *}
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theory SEQ
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imports NatStar
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begin
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definition
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  Zseq :: "[nat \<Rightarrow> 'a::real_normed_vector] \<Rightarrow> bool" where
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    --{*Standard definition of sequence converging to zero*}
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  "Zseq X = (\<forall>r>0. \<exists>no. \<forall>n\<ge>no. norm (X n) < r)"
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definition
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  LIMSEQ :: "[nat => 'a::real_normed_vector, 'a] => bool"
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    ("((_)/ ----> (_))" [60, 60] 60) where
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    --{*Standard definition of convergence of sequence*}
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  "X ----> L = (\<forall>r. 0 < r --> (\<exists>no. \<forall>n. no \<le> n --> norm (X n - L) < r))"
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definition
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  NSLIMSEQ :: "[nat => 'a::real_normed_vector, 'a] => bool"
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    ("((_)/ ----NS> (_))" [60, 60] 60) where
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    --{*Nonstandard definition of convergence of sequence*}
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  "X ----NS> L = (\<forall>N \<in> HNatInfinite. ( *f* X) N \<approx> star_of L)"
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definition
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  lim :: "(nat => 'a::real_normed_vector) => 'a" where
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    --{*Standard definition of limit using choice operator*}
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  "lim X = (THE L. X ----> L)"
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definition
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  nslim :: "(nat => 'a::real_normed_vector) => 'a" where
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    --{*Nonstandard definition of limit using choice operator*}
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  "nslim X = (THE L. X ----NS> L)"
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definition
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  convergent :: "(nat => 'a::real_normed_vector) => bool" where
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    --{*Standard definition of convergence*}
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  "convergent X = (\<exists>L. X ----> L)"
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definition
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  NSconvergent :: "(nat => 'a::real_normed_vector) => bool" where
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    --{*Nonstandard definition of convergence*}
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  "NSconvergent X = (\<exists>L. X ----NS> L)"
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definition
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  Bseq :: "(nat => 'a::real_normed_vector) => bool" where
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    --{*Standard definition for bounded sequence*}
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  "Bseq X = (\<exists>K>0.\<forall>n. norm (X n) \<le> K)"
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definition
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  NSBseq :: "(nat => 'a::real_normed_vector) => bool" where
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    --{*Nonstandard definition for bounded sequence*}
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  "NSBseq X = (\<forall>N \<in> HNatInfinite. ( *f* X) N : HFinite)"
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definition
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  monoseq :: "(nat=>real)=>bool" where
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    --{*Definition for monotonicity*}
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  "monoseq X = ((\<forall>m. \<forall>n\<ge>m. X m \<le> X n) | (\<forall>m. \<forall>n\<ge>m. X n \<le> X m))"
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definition
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  subseq :: "(nat => nat) => bool" where
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    --{*Definition of subsequence*}
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  "subseq f = (\<forall>m. \<forall>n>m. (f m) < (f n))"
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definition
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  Cauchy :: "(nat => 'a::real_normed_vector) => bool" where
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    --{*Standard definition of the Cauchy condition*}
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  "Cauchy X = (\<forall>e>0. \<exists>M. \<forall>m \<ge> M. \<forall>n \<ge> M. norm (X m - X n) < e)"
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definition
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  NSCauchy :: "(nat => 'a::real_normed_vector) => bool" where
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    --{*Nonstandard definition*}
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  "NSCauchy X = (\<forall>M \<in> HNatInfinite. \<forall>N \<in> HNatInfinite. ( *f* X) M \<approx> ( *f* X) N)"
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subsection {* Bounded Sequences *}
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lemma BseqI: assumes K: "\<And>n. norm (X n) \<le> K" shows "Bseq X"
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unfolding Bseq_def
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proof (intro exI conjI allI)
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  show "0 < max K 1" by simp
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next
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  fix n::nat
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  have "norm (X n) \<le> K" by (rule K)
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  thus "norm (X n) \<le> max K 1" by simp
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qed
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lemma BseqD: "Bseq X \<Longrightarrow> \<exists>K>0. \<forall>n. norm (X n) \<le> K"
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unfolding Bseq_def by simp
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lemma BseqE: "\<lbrakk>Bseq X; \<And>K. \<lbrakk>0 < K; \<forall>n. norm (X n) \<le> K\<rbrakk> \<Longrightarrow> Q\<rbrakk> \<Longrightarrow> Q"
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unfolding Bseq_def by auto
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lemma BseqI2: assumes K: "\<forall>n\<ge>N. norm (X n) \<le> K" shows "Bseq X"
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proof (rule BseqI)
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  let ?A = "norm ` X ` {..N}"
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  have 1: "finite ?A" by simp
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  have 2: "?A \<noteq> {}" by auto
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  fix n::nat
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  show "norm (X n) \<le> max K (Max ?A)"
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  proof (cases rule: linorder_le_cases)
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    assume "n \<ge> N"
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    hence "norm (X n) \<le> K" using K by simp
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    thus "norm (X n) \<le> max K (Max ?A)" by simp
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  next
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    assume "n \<le> N"
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    hence "norm (X n) \<in> ?A" by simp
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    with 1 2 have "norm (X n) \<le> Max ?A" by (rule Max_ge)
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    thus "norm (X n) \<le> max K (Max ?A)" by simp
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  qed
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qed
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lemma Bseq_ignore_initial_segment: "Bseq X \<Longrightarrow> Bseq (\<lambda>n. X (n + k))"
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unfolding Bseq_def by auto
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lemma Bseq_offset: "Bseq (\<lambda>n. X (n + k)) \<Longrightarrow> Bseq X"
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apply (erule BseqE)
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apply (rule_tac N="k" and K="K" in BseqI2)
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apply clarify
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apply (drule_tac x="n - k" in spec, simp)
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done
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subsection {* Sequences That Converge to Zero *}
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lemma ZseqI:
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  "(\<And>r. 0 < r \<Longrightarrow> \<exists>no. \<forall>n\<ge>no. norm (X n) < r) \<Longrightarrow> Zseq X"
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unfolding Zseq_def by simp
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lemma ZseqD:
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  "\<lbrakk>Zseq X; 0 < r\<rbrakk> \<Longrightarrow> \<exists>no. \<forall>n\<ge>no. norm (X n) < r"
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unfolding Zseq_def by simp
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lemma Zseq_zero: "Zseq (\<lambda>n. 0)"
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unfolding Zseq_def by simp
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lemma Zseq_const_iff: "Zseq (\<lambda>n. k) = (k = 0)"
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unfolding Zseq_def by force
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lemma Zseq_norm_iff: "Zseq (\<lambda>n. norm (X n)) = Zseq (\<lambda>n. X n)"
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unfolding Zseq_def by simp
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lemma Zseq_imp_Zseq:
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  assumes X: "Zseq X"
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  assumes Y: "\<And>n. norm (Y n) \<le> norm (X n) * K"
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  shows "Zseq (\<lambda>n. Y n)"
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proof (cases)
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  assume K: "0 < K"
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  show ?thesis
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  proof (rule ZseqI)
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    fix r::real assume "0 < r"
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    hence "0 < r / K"
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      using K by (rule divide_pos_pos)
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    then obtain N where "\<forall>n\<ge>N. norm (X n) < r / K"
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      using ZseqD [OF X] by fast
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    hence "\<forall>n\<ge>N. norm (X n) * K < r"
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      by (simp add: pos_less_divide_eq K)
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    hence "\<forall>n\<ge>N. norm (Y n) < r"
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      by (simp add: order_le_less_trans [OF Y])
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    thus "\<exists>N. \<forall>n\<ge>N. norm (Y n) < r" ..
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  qed
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next
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  assume "\<not> 0 < K"
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  hence K: "K \<le> 0" by (simp only: linorder_not_less)
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  {
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    fix n::nat
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    have "norm (Y n) \<le> norm (X n) * K" by (rule Y)
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    also have "\<dots> \<le> norm (X n) * 0"
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      using K norm_ge_zero by (rule mult_left_mono)
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    finally have "norm (Y n) = 0" by simp
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  }
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  thus ?thesis by (simp add: Zseq_zero)
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qed
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lemma Zseq_le: "\<lbrakk>Zseq Y; \<forall>n. norm (X n) \<le> norm (Y n)\<rbrakk> \<Longrightarrow> Zseq X"
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by (erule_tac K="1" in Zseq_imp_Zseq, simp)
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lemma Zseq_add:
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  assumes X: "Zseq X"
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  assumes Y: "Zseq Y"
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  shows "Zseq (\<lambda>n. X n + Y n)"
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proof (rule ZseqI)
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  fix r::real assume "0 < r"
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  hence r: "0 < r / 2" by simp
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  obtain M where M: "\<forall>n\<ge>M. norm (X n) < r/2"
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    using ZseqD [OF X r] by fast
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  obtain N where N: "\<forall>n\<ge>N. norm (Y n) < r/2"
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    using ZseqD [OF Y r] by fast
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  show "\<exists>N. \<forall>n\<ge>N. norm (X n + Y n) < r"
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  proof (intro exI allI impI)
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    fix n assume n: "max M N \<le> n"
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    have "norm (X n + Y n) \<le> norm (X n) + norm (Y n)"
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      by (rule norm_triangle_ineq)
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    also have "\<dots> < r/2 + r/2"
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    proof (rule add_strict_mono)
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      from M n show "norm (X n) < r/2" by simp
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      from N n show "norm (Y n) < r/2" by simp
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    qed
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    finally show "norm (X n + Y n) < r" by simp
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  qed
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qed
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lemma Zseq_minus: "Zseq X \<Longrightarrow> Zseq (\<lambda>n. - X n)"
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unfolding Zseq_def by simp
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lemma Zseq_diff: "\<lbrakk>Zseq X; Zseq Y\<rbrakk> \<Longrightarrow> Zseq (\<lambda>n. X n - Y n)"
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by (simp only: diff_minus Zseq_add Zseq_minus)
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lemma (in bounded_linear) Zseq:
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  assumes X: "Zseq X"
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  shows "Zseq (\<lambda>n. f (X n))"
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proof -
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  obtain K where "\<And>x. norm (f x) \<le> norm x * K"
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    using bounded by fast
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  with X show ?thesis
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    by (rule Zseq_imp_Zseq)
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qed
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lemma (in bounded_bilinear) Zseq_prod:
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  assumes X: "Zseq X"
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  assumes Y: "Zseq Y"
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  shows "Zseq (\<lambda>n. X n ** Y n)"
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proof (rule ZseqI)
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  fix r::real assume r: "0 < r"
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  obtain K where K: "0 < K"
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    and norm_le: "\<And>x y. norm (x ** y) \<le> norm x * norm y * K"
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    using pos_bounded by fast
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  from K have K': "0 < inverse K"
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    by (rule positive_imp_inverse_positive)
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  obtain M where M: "\<forall>n\<ge>M. norm (X n) < r"
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    using ZseqD [OF X r] by fast
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  obtain N where N: "\<forall>n\<ge>N. norm (Y n) < inverse K"
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    using ZseqD [OF Y K'] by fast
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  show "\<exists>N. \<forall>n\<ge>N. norm (X n ** Y n) < r"
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  proof (intro exI allI impI)
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    fix n assume n: "max M N \<le> n"
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    have "norm (X n ** Y n) \<le> norm (X n) * norm (Y n) * K"
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      by (rule norm_le)
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    also have "norm (X n) * norm (Y n) * K < r * inverse K * K"
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    proof (intro mult_strict_right_mono mult_strict_mono' norm_ge_zero K)
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      from M n show Xn: "norm (X n) < r" by simp
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      from N n show Yn: "norm (Y n) < inverse K" by simp
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    qed
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    also from K have "r * inverse K * K = r" by simp
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    finally show "norm (X n ** Y n) < r" .
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  qed
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qed
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lemma (in bounded_bilinear) Zseq_prod_Bseq:
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  assumes X: "Zseq X"
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  assumes Y: "Bseq Y"
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  shows "Zseq (\<lambda>n. X n ** Y n)"
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proof -
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  obtain K where K: "0 \<le> K"
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    and norm_le: "\<And>x y. norm (x ** y) \<le> norm x * norm y * K"
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    using nonneg_bounded by fast
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  obtain B where B: "0 < B"
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    and norm_Y: "\<And>n. norm (Y n) \<le> B"
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    using Y [unfolded Bseq_def] by fast
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  from X show ?thesis
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  proof (rule Zseq_imp_Zseq)
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    fix n::nat
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    have "norm (X n ** Y n) \<le> norm (X n) * norm (Y n) * K"
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      by (rule norm_le)
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    also have "\<dots> \<le> norm (X n) * B * K"
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      by (intro mult_mono' order_refl norm_Y norm_ge_zero
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                mult_nonneg_nonneg K)
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    also have "\<dots> = norm (X n) * (B * K)"
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      by (rule mult_assoc)
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    finally show "norm (X n ** Y n) \<le> norm (X n) * (B * K)" .
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  qed
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qed
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lemma (in bounded_bilinear) Bseq_prod_Zseq:
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  assumes X: "Bseq X"
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  assumes Y: "Zseq Y"
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  shows "Zseq (\<lambda>n. X n ** Y n)"
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proof -
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  obtain K where K: "0 \<le> K"
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    and norm_le: "\<And>x y. norm (x ** y) \<le> norm x * norm y * K"
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    using nonneg_bounded by fast
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  obtain B where B: "0 < B"
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    and norm_X: "\<And>n. norm (X n) \<le> B"
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    using X [unfolded Bseq_def] by fast
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  from Y show ?thesis
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  proof (rule Zseq_imp_Zseq)
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    fix n::nat
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    have "norm (X n ** Y n) \<le> norm (X n) * norm (Y n) * K"
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      by (rule norm_le)
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    also have "\<dots> \<le> B * norm (Y n) * K"
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      by (intro mult_mono' order_refl norm_X norm_ge_zero
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                mult_nonneg_nonneg K)
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    also have "\<dots> = norm (Y n) * (B * K)"
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      by (simp only: mult_ac)
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    finally show "norm (X n ** Y n) \<le> norm (Y n) * (B * K)" .
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  qed
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qed
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lemma (in bounded_bilinear) Zseq_prod_left:
huffman@22608
   307
  "Zseq X \<Longrightarrow> Zseq (\<lambda>n. X n ** a)"
huffman@22608
   308
by (rule bounded_linear_left [THEN bounded_linear.Zseq])
huffman@22608
   309
huffman@22608
   310
lemma (in bounded_bilinear) Zseq_prod_right:
huffman@22608
   311
  "Zseq X \<Longrightarrow> Zseq (\<lambda>n. a ** X n)"
huffman@22608
   312
by (rule bounded_linear_right [THEN bounded_linear.Zseq])
huffman@22608
   313
huffman@22608
   314
lemmas Zseq_mult = bounded_bilinear_mult.Zseq_prod
huffman@22608
   315
lemmas Zseq_mult_right = bounded_bilinear_mult.Zseq_prod_right
huffman@22608
   316
lemmas Zseq_mult_left = bounded_bilinear_mult.Zseq_prod_left
huffman@22608
   317
huffman@22608
   318
huffman@20696
   319
subsection {* Limits of Sequences *}
huffman@20696
   320
huffman@20696
   321
subsubsection {* Purely standard proofs *}
paulson@15082
   322
paulson@15082
   323
lemma LIMSEQ_iff:
huffman@20563
   324
      "(X ----> L) = (\<forall>r>0. \<exists>no. \<forall>n \<ge> no. norm (X n - L) < r)"
huffman@22608
   325
by (rule LIMSEQ_def)
huffman@22608
   326
huffman@22608
   327
lemma LIMSEQ_Zseq_iff: "((\<lambda>n. X n) ----> L) = Zseq (\<lambda>n. X n - L)"
huffman@22608
   328
by (simp only: LIMSEQ_def Zseq_def)
paulson@15082
   329
huffman@20751
   330
lemma LIMSEQ_I:
huffman@20751
   331
  "(\<And>r. 0 < r \<Longrightarrow> \<exists>no. \<forall>n\<ge>no. norm (X n - L) < r) \<Longrightarrow> X ----> L"
huffman@20751
   332
by (simp add: LIMSEQ_def)
huffman@20751
   333
huffman@20751
   334
lemma LIMSEQ_D:
huffman@20751
   335
  "\<lbrakk>X ----> L; 0 < r\<rbrakk> \<Longrightarrow> \<exists>no. \<forall>n\<ge>no. norm (X n - L) < r"
huffman@20751
   336
by (simp add: LIMSEQ_def)
huffman@20751
   337
huffman@22608
   338
lemma LIMSEQ_const: "(\<lambda>n. k) ----> k"
huffman@20696
   339
by (simp add: LIMSEQ_def)
huffman@20696
   340
huffman@22608
   341
lemma LIMSEQ_const_iff: "(\<lambda>n. k) ----> l = (k = l)"
huffman@22608
   342
by (simp add: LIMSEQ_Zseq_iff Zseq_const_iff)
huffman@22608
   343
huffman@20696
   344
lemma LIMSEQ_norm: "X ----> a \<Longrightarrow> (\<lambda>n. norm (X n)) ----> norm a"
huffman@20696
   345
apply (simp add: LIMSEQ_def, safe)
huffman@20696
   346
apply (drule_tac x="r" in spec, safe)
huffman@20696
   347
apply (rule_tac x="no" in exI, safe)
huffman@20696
   348
apply (drule_tac x="n" in spec, safe)
huffman@20696
   349
apply (erule order_le_less_trans [OF norm_triangle_ineq3])
huffman@20696
   350
done
huffman@20696
   351
huffman@20696
   352
lemma LIMSEQ_ignore_initial_segment: "f ----> a 
huffman@20696
   353
  ==> (%n. f(n + k)) ----> a"
huffman@20696
   354
  apply (unfold LIMSEQ_def) 
huffman@20696
   355
  apply (clarify)
huffman@20696
   356
  apply (drule_tac x = r in spec)
huffman@20696
   357
  apply (clarify)
huffman@20696
   358
  apply (rule_tac x = "no + k" in exI)
huffman@20696
   359
  by auto
huffman@20696
   360
huffman@20696
   361
lemma LIMSEQ_offset: "(%x. f (x + k)) ----> a ==>
huffman@20696
   362
    f ----> a"
huffman@20696
   363
  apply (unfold LIMSEQ_def)
huffman@20696
   364
  apply clarsimp
huffman@20696
   365
  apply (drule_tac x = r in spec)
huffman@20696
   366
  apply clarsimp
huffman@20696
   367
  apply (rule_tac x = "no + k" in exI)
huffman@20696
   368
  apply clarsimp
huffman@20696
   369
  apply (drule_tac x = "n - k" in spec)
huffman@20696
   370
  apply (frule mp)
huffman@20696
   371
  apply arith
huffman@20696
   372
  apply simp
huffman@20696
   373
done
huffman@20696
   374
huffman@22608
   375
lemma add_diff_add:
huffman@22608
   376
  fixes a b c d :: "'a::ab_group_add"
huffman@22608
   377
  shows "(a + c) - (b + d) = (a - b) + (c - d)"
huffman@22608
   378
by simp
huffman@22608
   379
huffman@22608
   380
lemma minus_diff_minus:
huffman@22608
   381
  fixes a b :: "'a::ab_group_add"
huffman@22608
   382
  shows "(- a) - (- b) = - (a - b)"
huffman@22608
   383
by simp
huffman@22608
   384
huffman@22608
   385
lemma LIMSEQ_add: "\<lbrakk>X ----> a; Y ----> b\<rbrakk> \<Longrightarrow> (\<lambda>n. X n + Y n) ----> a + b"
huffman@22608
   386
by (simp only: LIMSEQ_Zseq_iff add_diff_add Zseq_add)
huffman@22608
   387
huffman@22608
   388
lemma LIMSEQ_minus: "X ----> a \<Longrightarrow> (\<lambda>n. - X n) ----> - a"
huffman@22608
   389
by (simp only: LIMSEQ_Zseq_iff minus_diff_minus Zseq_minus)
huffman@22608
   390
huffman@22608
   391
lemma LIMSEQ_minus_cancel: "(\<lambda>n. - X n) ----> - a \<Longrightarrow> X ----> a"
huffman@22608
   392
by (drule LIMSEQ_minus, simp)
huffman@22608
   393
huffman@22608
   394
lemma LIMSEQ_diff: "\<lbrakk>X ----> a; Y ----> b\<rbrakk> \<Longrightarrow> (\<lambda>n. X n - Y n) ----> a - b"
huffman@22608
   395
by (simp add: diff_minus LIMSEQ_add LIMSEQ_minus)
huffman@22608
   396
huffman@22608
   397
lemma LIMSEQ_unique: "\<lbrakk>X ----> a; X ----> b\<rbrakk> \<Longrightarrow> a = b"
huffman@22608
   398
by (drule (1) LIMSEQ_diff, simp add: LIMSEQ_const_iff)
huffman@22608
   399
huffman@22608
   400
lemma (in bounded_linear) LIMSEQ:
huffman@22608
   401
  "X ----> a \<Longrightarrow> (\<lambda>n. f (X n)) ----> f a"
huffman@22608
   402
by (simp only: LIMSEQ_Zseq_iff diff [symmetric] Zseq)
huffman@22608
   403
huffman@22608
   404
lemma (in bounded_bilinear) LIMSEQ:
huffman@22608
   405
  "\<lbrakk>X ----> a; Y ----> b\<rbrakk> \<Longrightarrow> (\<lambda>n. X n ** Y n) ----> a ** b"
huffman@22608
   406
by (simp only: LIMSEQ_Zseq_iff prod_diff_prod
huffman@22608
   407
               Zseq_add Zseq_prod Zseq_prod_left Zseq_prod_right)
huffman@22608
   408
huffman@22608
   409
lemma LIMSEQ_mult:
huffman@22608
   410
  fixes a b :: "'a::real_normed_algebra"
huffman@22608
   411
  shows "[| X ----> a; Y ----> b |] ==> (%n. X n * Y n) ----> a * b"
huffman@22608
   412
by (rule bounded_bilinear_mult.LIMSEQ)
huffman@22608
   413
huffman@22608
   414
lemma inverse_diff_inverse:
huffman@22608
   415
  "\<lbrakk>(a::'a::division_ring) \<noteq> 0; b \<noteq> 0\<rbrakk>
huffman@22608
   416
   \<Longrightarrow> inverse a - inverse b = - (inverse a * (a - b) * inverse b)"
huffman@22608
   417
by (simp add: right_diff_distrib left_diff_distrib mult_assoc)
huffman@22608
   418
huffman@22608
   419
lemma Bseq_inverse_lemma:
huffman@22608
   420
  fixes x :: "'a::real_normed_div_algebra"
huffman@22608
   421
  shows "\<lbrakk>r \<le> norm x; 0 < r\<rbrakk> \<Longrightarrow> norm (inverse x) \<le> inverse r"
huffman@22608
   422
apply (subst nonzero_norm_inverse, clarsimp)
huffman@22608
   423
apply (erule (1) le_imp_inverse_le)
huffman@22608
   424
done
huffman@22608
   425
huffman@22608
   426
lemma Bseq_inverse:
huffman@22608
   427
  fixes a :: "'a::real_normed_div_algebra"
huffman@22608
   428
  assumes X: "X ----> a"
huffman@22608
   429
  assumes a: "a \<noteq> 0"
huffman@22608
   430
  shows "Bseq (\<lambda>n. inverse (X n))"
huffman@22608
   431
proof -
huffman@22608
   432
  from a have "0 < norm a" by simp
huffman@22608
   433
  hence "\<exists>r>0. r < norm a" by (rule dense)
huffman@22608
   434
  then obtain r where r1: "0 < r" and r2: "r < norm a" by fast
huffman@22608
   435
  obtain N where N: "\<And>n. N \<le> n \<Longrightarrow> norm (X n - a) < r"
huffman@22608
   436
    using LIMSEQ_D [OF X r1] by fast
huffman@22608
   437
  show ?thesis
huffman@22608
   438
  proof (rule BseqI2 [rule_format])
huffman@22608
   439
    fix n assume n: "N \<le> n"
huffman@22608
   440
    hence 1: "norm (X n - a) < r" by (rule N)
huffman@22608
   441
    hence 2: "X n \<noteq> 0" using r2 by auto
huffman@22608
   442
    hence "norm (inverse (X n)) = inverse (norm (X n))"
huffman@22608
   443
      by (rule nonzero_norm_inverse)
huffman@22608
   444
    also have "\<dots> \<le> inverse (norm a - r)"
huffman@22608
   445
    proof (rule le_imp_inverse_le)
huffman@22608
   446
      show "0 < norm a - r" using r2 by simp
huffman@22608
   447
    next
huffman@22608
   448
      have "norm a - norm (X n) \<le> norm (a - X n)"
huffman@22608
   449
        by (rule norm_triangle_ineq2)
huffman@22608
   450
      also have "\<dots> = norm (X n - a)"
huffman@22608
   451
        by (rule norm_minus_commute)
huffman@22608
   452
      also have "\<dots> < r" using 1 .
huffman@22608
   453
      finally show "norm a - r \<le> norm (X n)" by simp
huffman@22608
   454
    qed
huffman@22608
   455
    finally show "norm (inverse (X n)) \<le> inverse (norm a - r)" .
huffman@22608
   456
  qed
huffman@22608
   457
qed
huffman@22608
   458
huffman@22608
   459
lemma LIMSEQ_inverse_lemma:
huffman@22608
   460
  fixes a :: "'a::real_normed_div_algebra"
huffman@22608
   461
  shows "\<lbrakk>X ----> a; a \<noteq> 0; \<forall>n. X n \<noteq> 0\<rbrakk>
huffman@22608
   462
         \<Longrightarrow> (\<lambda>n. inverse (X n)) ----> inverse a"
huffman@22608
   463
apply (subst LIMSEQ_Zseq_iff)
huffman@22608
   464
apply (simp add: inverse_diff_inverse nonzero_imp_inverse_nonzero)
huffman@22608
   465
apply (rule Zseq_minus)
huffman@22608
   466
apply (rule Zseq_mult_left)
huffman@22608
   467
apply (rule bounded_bilinear_mult.Bseq_prod_Zseq)
huffman@22608
   468
apply (erule (1) Bseq_inverse)
huffman@22608
   469
apply (simp add: LIMSEQ_Zseq_iff)
huffman@22608
   470
done
huffman@22608
   471
huffman@22608
   472
lemma LIMSEQ_inverse:
huffman@22608
   473
  fixes a :: "'a::real_normed_div_algebra"
huffman@22608
   474
  assumes X: "X ----> a"
huffman@22608
   475
  assumes a: "a \<noteq> 0"
huffman@22608
   476
  shows "(\<lambda>n. inverse (X n)) ----> inverse a"
huffman@22608
   477
proof -
huffman@22608
   478
  from a have "0 < norm a" by simp
huffman@22608
   479
  then obtain k where "\<forall>n\<ge>k. norm (X n - a) < norm a"
huffman@22608
   480
    using LIMSEQ_D [OF X] by fast
huffman@22608
   481
  hence "\<forall>n\<ge>k. X n \<noteq> 0" by auto
huffman@22608
   482
  hence k: "\<forall>n. X (n + k) \<noteq> 0" by simp
huffman@22608
   483
huffman@22608
   484
  from X have "(\<lambda>n. X (n + k)) ----> a"
huffman@22608
   485
    by (rule LIMSEQ_ignore_initial_segment)
huffman@22608
   486
  hence "(\<lambda>n. inverse (X (n + k))) ----> inverse a"
huffman@22608
   487
    using a k by (rule LIMSEQ_inverse_lemma)
huffman@22608
   488
  thus "(\<lambda>n. inverse (X n)) ----> inverse a"
huffman@22608
   489
    by (rule LIMSEQ_offset)
huffman@22608
   490
qed
huffman@22608
   491
huffman@22608
   492
lemma LIMSEQ_divide:
huffman@22608
   493
  fixes a b :: "'a::real_normed_field"
huffman@22608
   494
  shows "\<lbrakk>X ----> a; Y ----> b; b \<noteq> 0\<rbrakk> \<Longrightarrow> (\<lambda>n. X n / Y n) ----> a / b"
huffman@22608
   495
by (simp add: LIMSEQ_mult LIMSEQ_inverse divide_inverse)
huffman@22608
   496
huffman@22608
   497
lemma LIMSEQ_pow:
huffman@22608
   498
  fixes a :: "'a::{real_normed_algebra,recpower}"
huffman@22608
   499
  shows "X ----> a \<Longrightarrow> (\<lambda>n. (X n) ^ m) ----> a ^ m"
huffman@22608
   500
by (induct m) (simp_all add: power_Suc LIMSEQ_const LIMSEQ_mult)
huffman@22608
   501
huffman@22608
   502
lemma LIMSEQ_setsum:
huffman@22608
   503
  assumes n: "\<And>n. n \<in> S \<Longrightarrow> X n ----> L n"
huffman@22608
   504
  shows "(\<lambda>m. \<Sum>n\<in>S. X n m) ----> (\<Sum>n\<in>S. L n)"
huffman@22608
   505
proof (cases "finite S")
huffman@22608
   506
  case True
huffman@22608
   507
  thus ?thesis using n
huffman@22608
   508
  proof (induct)
huffman@22608
   509
    case empty
huffman@22608
   510
    show ?case
huffman@22608
   511
      by (simp add: LIMSEQ_const)
huffman@22608
   512
  next
huffman@22608
   513
    case insert
huffman@22608
   514
    thus ?case
huffman@22608
   515
      by (simp add: LIMSEQ_add)
huffman@22608
   516
  qed
huffman@22608
   517
next
huffman@22608
   518
  case False
huffman@22608
   519
  thus ?thesis
huffman@22608
   520
    by (simp add: LIMSEQ_const)
huffman@22608
   521
qed
huffman@22608
   522
huffman@22608
   523
lemma LIMSEQ_setprod:
huffman@22608
   524
  fixes L :: "'a \<Rightarrow> 'b::{real_normed_algebra,comm_ring_1}"
huffman@22608
   525
  assumes n: "\<And>n. n \<in> S \<Longrightarrow> X n ----> L n"
huffman@22608
   526
  shows "(\<lambda>m. \<Prod>n\<in>S. X n m) ----> (\<Prod>n\<in>S. L n)"
huffman@22608
   527
proof (cases "finite S")
huffman@22608
   528
  case True
huffman@22608
   529
  thus ?thesis using n
huffman@22608
   530
  proof (induct)
huffman@22608
   531
    case empty
huffman@22608
   532
    show ?case
huffman@22608
   533
      by (simp add: LIMSEQ_const)
huffman@22608
   534
  next
huffman@22608
   535
    case insert
huffman@22608
   536
    thus ?case
huffman@22608
   537
      by (simp add: LIMSEQ_mult)
huffman@22608
   538
  qed
huffman@22608
   539
next
huffman@22608
   540
  case False
huffman@22608
   541
  thus ?thesis
huffman@22608
   542
    by (simp add: setprod_def LIMSEQ_const)
huffman@22608
   543
qed
huffman@22608
   544
huffman@22614
   545
lemma LIMSEQ_add_const: "f ----> a ==> (%n.(f n + b)) ----> a + b"
huffman@22614
   546
by (simp add: LIMSEQ_add LIMSEQ_const)
huffman@22614
   547
huffman@22614
   548
(* FIXME: delete *)
huffman@22614
   549
lemma LIMSEQ_add_minus:
huffman@22614
   550
     "[| X ----> a; Y ----> b |] ==> (%n. X n + -Y n) ----> a + -b"
huffman@22614
   551
by (simp only: LIMSEQ_add LIMSEQ_minus)
huffman@22614
   552
huffman@22614
   553
lemma LIMSEQ_diff_const: "f ----> a ==> (%n.(f n  - b)) ----> a - b"
huffman@22614
   554
by (simp add: LIMSEQ_diff LIMSEQ_const)
huffman@22614
   555
huffman@22614
   556
lemma LIMSEQ_diff_approach_zero: 
huffman@22614
   557
  "g ----> L ==> (%x. f x - g x) ----> 0  ==>
huffman@22614
   558
     f ----> L"
huffman@22614
   559
  apply (drule LIMSEQ_add)
huffman@22614
   560
  apply assumption
huffman@22614
   561
  apply simp
huffman@22614
   562
done
huffman@22614
   563
huffman@22614
   564
lemma LIMSEQ_diff_approach_zero2: 
huffman@22614
   565
  "f ----> L ==> (%x. f x - g x) ----> 0  ==>
huffman@22614
   566
     g ----> L";
huffman@22614
   567
  apply (drule LIMSEQ_diff)
huffman@22614
   568
  apply assumption
huffman@22614
   569
  apply simp
huffman@22614
   570
done
huffman@22614
   571
huffman@22614
   572
text{*A sequence tends to zero iff its abs does*}
huffman@22614
   573
lemma LIMSEQ_norm_zero: "((\<lambda>n. norm (X n)) ----> 0) = (X ----> 0)"
huffman@22614
   574
by (simp add: LIMSEQ_def)
huffman@22614
   575
huffman@22614
   576
lemma LIMSEQ_rabs_zero: "((%n. \<bar>f n\<bar>) ----> 0) = (f ----> (0::real))"
huffman@22614
   577
by (simp add: LIMSEQ_def)
huffman@22614
   578
huffman@22614
   579
lemma LIMSEQ_imp_rabs: "f ----> (l::real) ==> (%n. \<bar>f n\<bar>) ----> \<bar>l\<bar>"
huffman@22614
   580
by (drule LIMSEQ_norm, simp)
huffman@22614
   581
huffman@22614
   582
text{*An unbounded sequence's inverse tends to 0*}
huffman@22614
   583
huffman@22614
   584
text{* standard proof seems easier *}
huffman@22614
   585
lemma LIMSEQ_inverse_zero:
huffman@22614
   586
      "\<forall>y::real. \<exists>N. \<forall>n \<ge> N. y < f(n) ==> (%n. inverse(f n)) ----> 0"
huffman@22614
   587
apply (simp add: LIMSEQ_def, safe)
huffman@22614
   588
apply (drule_tac x = "inverse r" in spec, safe)
huffman@22614
   589
apply (rule_tac x = N in exI, safe)
huffman@22614
   590
apply (drule spec, auto)
huffman@22614
   591
apply (frule positive_imp_inverse_positive)
huffman@22614
   592
apply (frule order_less_trans, assumption)
huffman@22614
   593
apply (frule_tac a = "f n" in positive_imp_inverse_positive)
huffman@22614
   594
apply (simp add: abs_if) 
huffman@22614
   595
apply (rule_tac t = r in inverse_inverse_eq [THEN subst])
huffman@22614
   596
apply (auto intro: inverse_less_iff_less [THEN iffD2]
huffman@22614
   597
            simp del: inverse_inverse_eq)
huffman@22614
   598
done
huffman@22614
   599
huffman@22614
   600
text{*The sequence @{term "1/n"} tends to 0 as @{term n} tends to infinity*}
huffman@22614
   601
huffman@22614
   602
lemma LIMSEQ_inverse_real_of_nat: "(%n. inverse(real(Suc n))) ----> 0"
huffman@22614
   603
apply (rule LIMSEQ_inverse_zero, safe)
huffman@22614
   604
apply (cut_tac x = y in reals_Archimedean2)
huffman@22614
   605
apply (safe, rule_tac x = n in exI)
huffman@22614
   606
apply (auto simp add: real_of_nat_Suc)
huffman@22614
   607
done
huffman@22614
   608
huffman@22614
   609
text{*The sequence @{term "r + 1/n"} tends to @{term r} as @{term n} tends to
huffman@22614
   610
infinity is now easily proved*}
huffman@22614
   611
huffman@22614
   612
lemma LIMSEQ_inverse_real_of_nat_add:
huffman@22614
   613
     "(%n. r + inverse(real(Suc n))) ----> r"
huffman@22614
   614
by (cut_tac LIMSEQ_add [OF LIMSEQ_const LIMSEQ_inverse_real_of_nat], auto)
huffman@22614
   615
huffman@22614
   616
lemma LIMSEQ_inverse_real_of_nat_add_minus:
huffman@22614
   617
     "(%n. r + -inverse(real(Suc n))) ----> r"
huffman@22614
   618
by (cut_tac LIMSEQ_add_minus [OF LIMSEQ_const LIMSEQ_inverse_real_of_nat], auto)
huffman@22614
   619
huffman@22614
   620
lemma LIMSEQ_inverse_real_of_nat_add_minus_mult:
huffman@22614
   621
     "(%n. r*( 1 + -inverse(real(Suc n)))) ----> r"
huffman@22614
   622
by (cut_tac b=1 in
huffman@22614
   623
        LIMSEQ_mult [OF LIMSEQ_const LIMSEQ_inverse_real_of_nat_add_minus], auto)
huffman@22614
   624
huffman@20696
   625
subsubsection {* Purely nonstandard proofs *}
huffman@20696
   626
paulson@15082
   627
lemma NSLIMSEQ_iff:
huffman@20552
   628
    "(X ----NS> L) = (\<forall>N \<in> HNatInfinite. ( *f* X) N \<approx> star_of L)"
paulson@15082
   629
by (simp add: NSLIMSEQ_def)
paulson@15082
   630
huffman@20751
   631
lemma NSLIMSEQ_I:
huffman@20751
   632
  "(\<And>N. N \<in> HNatInfinite \<Longrightarrow> starfun X N \<approx> star_of L) \<Longrightarrow> X ----NS> L"
huffman@20751
   633
by (simp add: NSLIMSEQ_def)
huffman@20751
   634
huffman@20751
   635
lemma NSLIMSEQ_D:
huffman@20751
   636
  "\<lbrakk>X ----NS> L; N \<in> HNatInfinite\<rbrakk> \<Longrightarrow> starfun X N \<approx> star_of L"
huffman@20751
   637
by (simp add: NSLIMSEQ_def)
huffman@20751
   638
huffman@20696
   639
lemma NSLIMSEQ_const: "(%n. k) ----NS> k"
huffman@20696
   640
by (simp add: NSLIMSEQ_def)
huffman@20696
   641
huffman@20696
   642
lemma NSLIMSEQ_add:
huffman@20696
   643
      "[| X ----NS> a; Y ----NS> b |] ==> (%n. X n + Y n) ----NS> a + b"
huffman@20696
   644
by (auto intro: approx_add simp add: NSLIMSEQ_def starfun_add [symmetric])
huffman@20696
   645
huffman@20696
   646
lemma NSLIMSEQ_add_const: "f ----NS> a ==> (%n.(f n + b)) ----NS> a + b"
huffman@20696
   647
by (simp only: NSLIMSEQ_add NSLIMSEQ_const)
huffman@20696
   648
huffman@20696
   649
lemma NSLIMSEQ_mult:
huffman@20696
   650
  fixes a b :: "'a::real_normed_algebra"
huffman@20696
   651
  shows "[| X ----NS> a; Y ----NS> b |] ==> (%n. X n * Y n) ----NS> a * b"
huffman@20696
   652
by (auto intro!: approx_mult_HFinite simp add: NSLIMSEQ_def)
huffman@20696
   653
huffman@20696
   654
lemma NSLIMSEQ_minus: "X ----NS> a ==> (%n. -(X n)) ----NS> -a"
huffman@20696
   655
by (auto simp add: NSLIMSEQ_def)
huffman@20696
   656
huffman@20696
   657
lemma NSLIMSEQ_minus_cancel: "(%n. -(X n)) ----NS> -a ==> X ----NS> a"
huffman@20696
   658
by (drule NSLIMSEQ_minus, simp)
huffman@20696
   659
huffman@20696
   660
(* FIXME: delete *)
huffman@20696
   661
lemma NSLIMSEQ_add_minus:
huffman@20696
   662
     "[| X ----NS> a; Y ----NS> b |] ==> (%n. X n + -Y n) ----NS> a + -b"
huffman@20696
   663
by (simp add: NSLIMSEQ_add NSLIMSEQ_minus)
huffman@20696
   664
huffman@20696
   665
lemma NSLIMSEQ_diff:
huffman@20696
   666
     "[| X ----NS> a; Y ----NS> b |] ==> (%n. X n - Y n) ----NS> a - b"
huffman@20696
   667
by (simp add: diff_minus NSLIMSEQ_add NSLIMSEQ_minus)
huffman@20696
   668
huffman@20696
   669
lemma NSLIMSEQ_diff_const: "f ----NS> a ==> (%n.(f n - b)) ----NS> a - b"
huffman@20696
   670
by (simp add: NSLIMSEQ_diff NSLIMSEQ_const)
huffman@20696
   671
huffman@20696
   672
lemma NSLIMSEQ_inverse:
huffman@20696
   673
  fixes a :: "'a::real_normed_div_algebra"
huffman@20696
   674
  shows "[| X ----NS> a;  a ~= 0 |] ==> (%n. inverse(X n)) ----NS> inverse(a)"
huffman@20696
   675
by (simp add: NSLIMSEQ_def star_of_approx_inverse)
huffman@20696
   676
huffman@20696
   677
lemma NSLIMSEQ_mult_inverse:
huffman@20696
   678
  fixes a b :: "'a::real_normed_field"
huffman@20696
   679
  shows
huffman@20696
   680
     "[| X ----NS> a;  Y ----NS> b;  b ~= 0 |] ==> (%n. X n / Y n) ----NS> a/b"
huffman@20696
   681
by (simp add: NSLIMSEQ_mult NSLIMSEQ_inverse divide_inverse)
huffman@20696
   682
huffman@20696
   683
lemma starfun_hnorm: "\<And>x. hnorm (( *f* f) x) = ( *f* (\<lambda>x. norm (f x))) x"
huffman@20696
   684
by transfer simp
huffman@20696
   685
huffman@20696
   686
lemma NSLIMSEQ_norm: "X ----NS> a \<Longrightarrow> (\<lambda>n. norm (X n)) ----NS> norm a"
huffman@20696
   687
by (simp add: NSLIMSEQ_def starfun_hnorm [symmetric] approx_hnorm)
huffman@20696
   688
huffman@20696
   689
text{*Uniqueness of limit*}
huffman@20696
   690
lemma NSLIMSEQ_unique: "[| X ----NS> a; X ----NS> b |] ==> a = b"
huffman@20696
   691
apply (simp add: NSLIMSEQ_def)
huffman@20696
   692
apply (drule HNatInfinite_whn [THEN [2] bspec])+
huffman@20696
   693
apply (auto dest: approx_trans3)
huffman@20696
   694
done
huffman@20696
   695
huffman@20696
   696
lemma NSLIMSEQ_pow [rule_format]:
huffman@20696
   697
  fixes a :: "'a::{real_normed_algebra,recpower}"
huffman@20696
   698
  shows "(X ----NS> a) --> ((%n. (X n) ^ m) ----NS> a ^ m)"
huffman@20696
   699
apply (induct "m")
huffman@20696
   700
apply (auto simp add: power_Suc intro: NSLIMSEQ_mult NSLIMSEQ_const)
huffman@20696
   701
done
huffman@20696
   702
huffman@22614
   703
text{*We can now try and derive a few properties of sequences,
huffman@22614
   704
     starting with the limit comparison property for sequences.*}
huffman@22614
   705
huffman@22614
   706
lemma NSLIMSEQ_le:
huffman@22614
   707
       "[| f ----NS> l; g ----NS> m;
huffman@22614
   708
           \<exists>N. \<forall>n \<ge> N. f(n) \<le> g(n)
huffman@22614
   709
        |] ==> l \<le> (m::real)"
huffman@22614
   710
apply (simp add: NSLIMSEQ_def, safe)
huffman@22614
   711
apply (drule starfun_le_mono)
huffman@22614
   712
apply (drule HNatInfinite_whn [THEN [2] bspec])+
huffman@22614
   713
apply (drule_tac x = whn in spec)
huffman@22614
   714
apply (drule bex_Infinitesimal_iff2 [THEN iffD2])+
huffman@22614
   715
apply clarify
huffman@22614
   716
apply (auto intro: hypreal_of_real_le_add_Infininitesimal_cancel2)
huffman@22614
   717
done
huffman@22614
   718
huffman@22614
   719
lemma NSLIMSEQ_le_const: "[| X ----NS> (r::real); \<forall>n. a \<le> X n |] ==> a \<le> r"
huffman@22614
   720
by (erule NSLIMSEQ_le [OF NSLIMSEQ_const], auto)
huffman@22614
   721
huffman@22614
   722
lemma NSLIMSEQ_le_const2: "[| X ----NS> (r::real); \<forall>n. X n \<le> a |] ==> r \<le> a"
huffman@22614
   723
by (erule NSLIMSEQ_le [OF _ NSLIMSEQ_const], auto)
huffman@22614
   724
huffman@22614
   725
text{*Shift a convergent series by 1:
huffman@22614
   726
  By the equivalence between Cauchiness and convergence and because
huffman@22614
   727
  the successor of an infinite hypernatural is also infinite.*}
huffman@22614
   728
huffman@22614
   729
lemma NSLIMSEQ_Suc: "f ----NS> l ==> (%n. f(Suc n)) ----NS> l"
huffman@22614
   730
apply (unfold NSLIMSEQ_def, safe)
huffman@22614
   731
apply (drule_tac x="N + 1" in bspec)
huffman@22614
   732
apply (erule HNatInfinite_add)
huffman@22614
   733
apply (simp add: starfun_shift_one)
huffman@22614
   734
done
huffman@22614
   735
huffman@22614
   736
lemma NSLIMSEQ_imp_Suc: "(%n. f(Suc n)) ----NS> l ==> f ----NS> l"
huffman@22614
   737
apply (unfold NSLIMSEQ_def, safe)
huffman@22614
   738
apply (drule_tac x="N - 1" in bspec) 
huffman@22614
   739
apply (erule Nats_1 [THEN [2] HNatInfinite_diff])
huffman@22614
   740
apply (simp add: starfun_shift_one one_le_HNatInfinite)
huffman@22614
   741
done
huffman@22614
   742
huffman@22614
   743
lemma NSLIMSEQ_Suc_iff: "((%n. f(Suc n)) ----NS> l) = (f ----NS> l)"
huffman@22614
   744
by (blast intro: NSLIMSEQ_imp_Suc NSLIMSEQ_Suc)
huffman@22614
   745
huffman@20696
   746
subsubsection {* Equivalence of @{term LIMSEQ} and @{term NSLIMSEQ} *}
paulson@15082
   747
paulson@15082
   748
lemma LIMSEQ_NSLIMSEQ:
huffman@20751
   749
  assumes X: "X ----> L" shows "X ----NS> L"
huffman@20751
   750
proof (rule NSLIMSEQ_I)
huffman@20751
   751
  fix N assume N: "N \<in> HNatInfinite"
huffman@20751
   752
  have "starfun X N - star_of L \<in> Infinitesimal"
huffman@20751
   753
  proof (rule InfinitesimalI2)
huffman@20751
   754
    fix r::real assume r: "0 < r"
huffman@20751
   755
    from LIMSEQ_D [OF X r]
huffman@20751
   756
    obtain no where "\<forall>n\<ge>no. norm (X n - L) < r" ..
huffman@20751
   757
    hence "\<forall>n\<ge>star_of no. hnorm (starfun X n - star_of L) < star_of r"
huffman@20751
   758
      by transfer
huffman@20751
   759
    thus "hnorm (starfun X N - star_of L) < star_of r"
huffman@20751
   760
      using N by (simp add: star_of_le_HNatInfinite)
huffman@20751
   761
  qed
huffman@20751
   762
  thus "starfun X N \<approx> star_of L"
huffman@20751
   763
    by (unfold approx_def)
huffman@20751
   764
qed
paulson@15082
   765
huffman@20751
   766
lemma NSLIMSEQ_LIMSEQ:
huffman@20751
   767
  assumes X: "X ----NS> L" shows "X ----> L"
huffman@20751
   768
proof (rule LIMSEQ_I)
huffman@20751
   769
  fix r::real assume r: "0 < r"
huffman@20751
   770
  have "\<exists>no. \<forall>n\<ge>no. hnorm (starfun X n - star_of L) < star_of r"
huffman@20751
   771
  proof (intro exI allI impI)
huffman@20751
   772
    fix n assume "whn \<le> n"
huffman@20751
   773
    with HNatInfinite_whn have "n \<in> HNatInfinite"
huffman@20751
   774
      by (rule HNatInfinite_upward_closed)
huffman@20751
   775
    with X have "starfun X n \<approx> star_of L"
huffman@20751
   776
      by (rule NSLIMSEQ_D)
huffman@20751
   777
    hence "starfun X n - star_of L \<in> Infinitesimal"
huffman@20751
   778
      by (unfold approx_def)
huffman@20751
   779
    thus "hnorm (starfun X n - star_of L) < star_of r"
huffman@20751
   780
      using r by (rule InfinitesimalD2)
huffman@20751
   781
  qed
huffman@20751
   782
  thus "\<exists>no. \<forall>n\<ge>no. norm (X n - L) < r"
huffman@20751
   783
    by transfer
huffman@20751
   784
qed
paulson@15082
   785
huffman@20751
   786
theorem LIMSEQ_NSLIMSEQ_iff: "(f ----> L) = (f ----NS> L)"
huffman@20751
   787
by (blast intro: LIMSEQ_NSLIMSEQ NSLIMSEQ_LIMSEQ)
paulson@15082
   788
huffman@20751
   789
(* Used once by Integration/Rats.thy in AFP *)
paulson@15082
   790
lemma NSLIMSEQ_finite_set:
paulson@15082
   791
     "!!(f::nat=>nat). \<forall>n. n \<le> f n ==> finite {n. f n \<le> u}"
huffman@20751
   792
by (rule_tac B="{..u}" in finite_subset, auto intro: order_trans)
paulson@15082
   793
huffman@20696
   794
subsubsection {* Derived theorems about @{term LIMSEQ} *}
paulson@15082
   795
huffman@22614
   796
lemma LIMSEQ_le:
huffman@22614
   797
     "[| f ----> l; g ----> m; \<exists>N. \<forall>n \<ge> N. f(n) \<le> g(n) |]
huffman@22614
   798
      ==> l \<le> (m::real)"
huffman@22614
   799
by (simp add: LIMSEQ_NSLIMSEQ_iff NSLIMSEQ_le)
avigad@16819
   800
huffman@22614
   801
lemma LIMSEQ_le_const: "[| X ----> (r::real); \<forall>n. a \<le> X n |] ==> a \<le> r"
huffman@22614
   802
by (erule LIMSEQ_le [OF LIMSEQ_const], auto)
paulson@15082
   803
huffman@22614
   804
lemma LIMSEQ_le_const2: "[| X ----> (r::real); \<forall>n. X n \<le> a |] ==> r \<le> a"
huffman@22614
   805
by (erule LIMSEQ_le [OF _ LIMSEQ_const], auto)
avigad@16819
   806
huffman@22614
   807
lemma LIMSEQ_Suc: "f ----> l ==> (%n. f(Suc n)) ----> l"
huffman@22614
   808
by (simp add: LIMSEQ_NSLIMSEQ_iff NSLIMSEQ_Suc)
huffman@22614
   809
huffman@22614
   810
lemma LIMSEQ_imp_Suc: "(%n. f(Suc n)) ----> l ==> f ----> l"
huffman@22614
   811
apply (simp add: LIMSEQ_NSLIMSEQ_iff)
huffman@22614
   812
apply (erule NSLIMSEQ_imp_Suc)
avigad@16819
   813
done
avigad@16819
   814
huffman@22614
   815
lemma LIMSEQ_Suc_iff: "((%n. f(Suc n)) ----> l) = (f ----> l)"
huffman@22614
   816
by (blast intro: LIMSEQ_imp_Suc LIMSEQ_Suc)
huffman@22614
   817
huffman@22614
   818
text{*We prove the NS version from the standard one, since the NS proof
huffman@22614
   819
   seems more complicated than the standard one above!*}
huffman@22614
   820
lemma NSLIMSEQ_norm_zero: "((\<lambda>n. norm (X n)) ----NS> 0) = (X ----NS> 0)"
huffman@22614
   821
by (simp add: LIMSEQ_NSLIMSEQ_iff [symmetric] LIMSEQ_norm_zero)
huffman@22614
   822
huffman@22614
   823
lemma NSLIMSEQ_rabs_zero: "((%n. \<bar>f n\<bar>) ----NS> 0) = (f ----NS> (0::real))"
huffman@22614
   824
by (simp add: LIMSEQ_NSLIMSEQ_iff [symmetric] LIMSEQ_rabs_zero)
huffman@22614
   825
huffman@22614
   826
text{*Generalization to other limits*}
huffman@22614
   827
lemma NSLIMSEQ_imp_rabs: "f ----NS> (l::real) ==> (%n. \<bar>f n\<bar>) ----NS> \<bar>l\<bar>"
huffman@22614
   828
apply (simp add: NSLIMSEQ_def)
huffman@22614
   829
apply (auto intro: approx_hrabs 
huffman@22614
   830
            simp add: starfun_abs)
avigad@16819
   831
done
avigad@16819
   832
huffman@22614
   833
lemma NSLIMSEQ_inverse_zero:
huffman@22614
   834
     "\<forall>y::real. \<exists>N. \<forall>n \<ge> N. y < f(n)
huffman@22614
   835
      ==> (%n. inverse(f n)) ----NS> 0"
huffman@22614
   836
by (simp add: LIMSEQ_NSLIMSEQ_iff [symmetric] LIMSEQ_inverse_zero)
huffman@22614
   837
huffman@22614
   838
lemma NSLIMSEQ_inverse_real_of_nat: "(%n. inverse(real(Suc n))) ----NS> 0"
huffman@22614
   839
by (simp add: LIMSEQ_NSLIMSEQ_iff [symmetric] LIMSEQ_inverse_real_of_nat)
huffman@22614
   840
huffman@22614
   841
lemma NSLIMSEQ_inverse_real_of_nat_add:
huffman@22614
   842
     "(%n. r + inverse(real(Suc n))) ----NS> r"
huffman@22614
   843
by (simp add: LIMSEQ_NSLIMSEQ_iff [symmetric] LIMSEQ_inverse_real_of_nat_add)
huffman@22614
   844
huffman@22614
   845
lemma NSLIMSEQ_inverse_real_of_nat_add_minus:
huffman@22614
   846
     "(%n. r + -inverse(real(Suc n))) ----NS> r"
huffman@22614
   847
by (simp add: LIMSEQ_NSLIMSEQ_iff [symmetric] LIMSEQ_inverse_real_of_nat_add_minus)
huffman@22614
   848
huffman@22614
   849
lemma NSLIMSEQ_inverse_real_of_nat_add_minus_mult:
huffman@22614
   850
     "(%n. r*( 1 + -inverse(real(Suc n)))) ----NS> r"
huffman@22614
   851
by (simp add: LIMSEQ_NSLIMSEQ_iff [symmetric] LIMSEQ_inverse_real_of_nat_add_minus_mult)
huffman@22614
   852
paulson@15082
   853
huffman@20696
   854
subsection {* Convergence *}
paulson@15082
   855
paulson@15082
   856
lemma limI: "X ----> L ==> lim X = L"
paulson@15082
   857
apply (simp add: lim_def)
paulson@15082
   858
apply (blast intro: LIMSEQ_unique)
paulson@15082
   859
done
paulson@15082
   860
paulson@15082
   861
lemma nslimI: "X ----NS> L ==> nslim X = L"
paulson@15082
   862
apply (simp add: nslim_def)
paulson@15082
   863
apply (blast intro: NSLIMSEQ_unique)
paulson@15082
   864
done
paulson@15082
   865
paulson@15082
   866
lemma lim_nslim_iff: "lim X = nslim X"
paulson@15082
   867
by (simp add: lim_def nslim_def LIMSEQ_NSLIMSEQ_iff)
paulson@15082
   868
paulson@15082
   869
lemma convergentD: "convergent X ==> \<exists>L. (X ----> L)"
paulson@15082
   870
by (simp add: convergent_def)
paulson@15082
   871
paulson@15082
   872
lemma convergentI: "(X ----> L) ==> convergent X"
paulson@15082
   873
by (auto simp add: convergent_def)
paulson@15082
   874
paulson@15082
   875
lemma NSconvergentD: "NSconvergent X ==> \<exists>L. (X ----NS> L)"
paulson@15082
   876
by (simp add: NSconvergent_def)
paulson@15082
   877
paulson@15082
   878
lemma NSconvergentI: "(X ----NS> L) ==> NSconvergent X"
paulson@15082
   879
by (auto simp add: NSconvergent_def)
paulson@15082
   880
paulson@15082
   881
lemma convergent_NSconvergent_iff: "convergent X = NSconvergent X"
paulson@15082
   882
by (simp add: convergent_def NSconvergent_def LIMSEQ_NSLIMSEQ_iff)
paulson@15082
   883
paulson@15082
   884
lemma NSconvergent_NSLIMSEQ_iff: "NSconvergent X = (X ----NS> nslim X)"
huffman@20682
   885
by (auto intro: theI NSLIMSEQ_unique simp add: NSconvergent_def nslim_def)
paulson@15082
   886
paulson@15082
   887
lemma convergent_LIMSEQ_iff: "convergent X = (X ----> lim X)"
huffman@20682
   888
by (auto intro: theI LIMSEQ_unique simp add: convergent_def lim_def)
paulson@15082
   889
huffman@20696
   890
lemma convergent_minus_iff: "(convergent X) = (convergent (%n. -(X n)))"
huffman@20696
   891
apply (simp add: convergent_def)
huffman@20696
   892
apply (auto dest: LIMSEQ_minus)
huffman@20696
   893
apply (drule LIMSEQ_minus, auto)
huffman@20696
   894
done
huffman@20696
   895
huffman@20696
   896
huffman@20696
   897
subsection {* Bounded Monotonic Sequences *}
huffman@20696
   898
paulson@15082
   899
text{*Subsequence (alternative definition, (e.g. Hoskins)*}
paulson@15082
   900
paulson@15082
   901
lemma subseq_Suc_iff: "subseq f = (\<forall>n. (f n) < (f (Suc n)))"
paulson@15082
   902
apply (simp add: subseq_def)
paulson@15082
   903
apply (auto dest!: less_imp_Suc_add)
paulson@15082
   904
apply (induct_tac k)
paulson@15082
   905
apply (auto intro: less_trans)
paulson@15082
   906
done
paulson@15082
   907
paulson@15082
   908
lemma monoseq_Suc:
paulson@15082
   909
   "monoseq X = ((\<forall>n. X n \<le> X (Suc n))
paulson@15082
   910
                 | (\<forall>n. X (Suc n) \<le> X n))"
paulson@15082
   911
apply (simp add: monoseq_def)
paulson@15082
   912
apply (auto dest!: le_imp_less_or_eq)
paulson@15082
   913
apply (auto intro!: lessI [THEN less_imp_le] dest!: less_imp_Suc_add)
paulson@15082
   914
apply (induct_tac "ka")
paulson@15082
   915
apply (auto intro: order_trans)
wenzelm@18585
   916
apply (erule contrapos_np)
paulson@15082
   917
apply (induct_tac "k")
paulson@15082
   918
apply (auto intro: order_trans)
paulson@15082
   919
done
paulson@15082
   920
nipkow@15360
   921
lemma monoI1: "\<forall>m. \<forall> n \<ge> m. X m \<le> X n ==> monoseq X"
paulson@15082
   922
by (simp add: monoseq_def)
paulson@15082
   923
nipkow@15360
   924
lemma monoI2: "\<forall>m. \<forall> n \<ge> m. X n \<le> X m ==> monoseq X"
paulson@15082
   925
by (simp add: monoseq_def)
paulson@15082
   926
paulson@15082
   927
lemma mono_SucI1: "\<forall>n. X n \<le> X (Suc n) ==> monoseq X"
paulson@15082
   928
by (simp add: monoseq_Suc)
paulson@15082
   929
paulson@15082
   930
lemma mono_SucI2: "\<forall>n. X (Suc n) \<le> X n ==> monoseq X"
paulson@15082
   931
by (simp add: monoseq_Suc)
paulson@15082
   932
huffman@20696
   933
text{*Bounded Sequence*}
paulson@15082
   934
huffman@20552
   935
lemma BseqD: "Bseq X ==> \<exists>K. 0 < K & (\<forall>n. norm (X n) \<le> K)"
paulson@15082
   936
by (simp add: Bseq_def)
paulson@15082
   937
huffman@20552
   938
lemma BseqI: "[| 0 < K; \<forall>n. norm (X n) \<le> K |] ==> Bseq X"
paulson@15082
   939
by (auto simp add: Bseq_def)
paulson@15082
   940
paulson@15082
   941
lemma lemma_NBseq_def:
huffman@20552
   942
     "(\<exists>K > 0. \<forall>n. norm (X n) \<le> K) =
huffman@20552
   943
      (\<exists>N. \<forall>n. norm (X n) \<le> real(Suc N))"
paulson@15082
   944
apply auto
paulson@15082
   945
 prefer 2 apply force
paulson@15082
   946
apply (cut_tac x = K in reals_Archimedean2, clarify)
paulson@15082
   947
apply (rule_tac x = n in exI, clarify)
paulson@15082
   948
apply (drule_tac x = na in spec)
paulson@15082
   949
apply (auto simp add: real_of_nat_Suc)
paulson@15082
   950
done
paulson@15082
   951
paulson@15082
   952
text{* alternative definition for Bseq *}
huffman@20552
   953
lemma Bseq_iff: "Bseq X = (\<exists>N. \<forall>n. norm (X n) \<le> real(Suc N))"
paulson@15082
   954
apply (simp add: Bseq_def)
paulson@15082
   955
apply (simp (no_asm) add: lemma_NBseq_def)
paulson@15082
   956
done
paulson@15082
   957
paulson@15082
   958
lemma lemma_NBseq_def2:
huffman@20552
   959
     "(\<exists>K > 0. \<forall>n. norm (X n) \<le> K) = (\<exists>N. \<forall>n. norm (X n) < real(Suc N))"
paulson@15082
   960
apply (subst lemma_NBseq_def, auto)
paulson@15082
   961
apply (rule_tac x = "Suc N" in exI)
paulson@15082
   962
apply (rule_tac [2] x = N in exI)
paulson@15082
   963
apply (auto simp add: real_of_nat_Suc)
paulson@15082
   964
 prefer 2 apply (blast intro: order_less_imp_le)
paulson@15082
   965
apply (drule_tac x = n in spec, simp)
paulson@15082
   966
done
paulson@15082
   967
paulson@15082
   968
(* yet another definition for Bseq *)
huffman@20552
   969
lemma Bseq_iff1a: "Bseq X = (\<exists>N. \<forall>n. norm (X n) < real(Suc N))"
paulson@15082
   970
by (simp add: Bseq_def lemma_NBseq_def2)
paulson@15082
   971
huffman@17318
   972
lemma NSBseqD: "[| NSBseq X;  N: HNatInfinite |] ==> ( *f* X) N : HFinite"
paulson@15082
   973
by (simp add: NSBseq_def)
paulson@15082
   974
huffman@21842
   975
lemma Standard_subset_HFinite: "Standard \<subseteq> HFinite"
huffman@21842
   976
unfolding Standard_def by auto
huffman@21842
   977
huffman@21842
   978
lemma NSBseqD2: "NSBseq X \<Longrightarrow> ( *f* X) N \<in> HFinite"
huffman@21842
   979
apply (cases "N \<in> HNatInfinite")
huffman@21842
   980
apply (erule (1) NSBseqD)
huffman@21842
   981
apply (rule subsetD [OF Standard_subset_HFinite])
huffman@21842
   982
apply (simp add: HNatInfinite_def Nats_eq_Standard)
huffman@21842
   983
done
huffman@21842
   984
huffman@17318
   985
lemma NSBseqI: "\<forall>N \<in> HNatInfinite. ( *f* X) N : HFinite ==> NSBseq X"
paulson@15082
   986
by (simp add: NSBseq_def)
paulson@15082
   987
paulson@15082
   988
text{*The standard definition implies the nonstandard definition*}
paulson@15082
   989
huffman@20552
   990
lemma lemma_Bseq: "\<forall>n. norm (X n) \<le> K ==> \<forall>n. norm(X((f::nat=>nat) n)) \<le> K"
paulson@15082
   991
by auto
paulson@15082
   992
paulson@15082
   993
lemma Bseq_NSBseq: "Bseq X ==> NSBseq X"
huffman@21139
   994
proof (unfold NSBseq_def, safe)
huffman@21139
   995
  assume X: "Bseq X"
huffman@21139
   996
  fix N assume N: "N \<in> HNatInfinite"
huffman@21139
   997
  from BseqD [OF X] obtain K where "\<forall>n. norm (X n) \<le> K" by fast
huffman@21139
   998
  hence "\<forall>N. hnorm (starfun X N) \<le> star_of K" by transfer
huffman@21139
   999
  hence "hnorm (starfun X N) \<le> star_of K" by simp
huffman@21139
  1000
  also have "star_of K < star_of (K + 1)" by simp
huffman@21139
  1001
  finally have "\<exists>x\<in>Reals. hnorm (starfun X N) < x" by (rule bexI, simp)
huffman@21139
  1002
  thus "starfun X N \<in> HFinite" by (simp add: HFinite_def)
huffman@21139
  1003
qed
paulson@15082
  1004
paulson@15082
  1005
text{*The nonstandard definition implies the standard definition*}
paulson@15082
  1006
huffman@21842
  1007
lemma SReal_less_omega: "r \<in> \<real> \<Longrightarrow> r < \<omega>"
huffman@21842
  1008
apply (insert HInfinite_omega)
huffman@21842
  1009
apply (simp add: HInfinite_def)
huffman@21842
  1010
apply (simp add: order_less_imp_le)
paulson@15082
  1011
done
paulson@15082
  1012
huffman@21842
  1013
lemma NSBseq_Bseq: "NSBseq X \<Longrightarrow> Bseq X"
huffman@21842
  1014
proof (rule ccontr)
huffman@21842
  1015
  let ?n = "\<lambda>K. LEAST n. K < norm (X n)"
huffman@21842
  1016
  assume "NSBseq X"
huffman@21842
  1017
  hence finite: "( *f* X) (( *f* ?n) \<omega>) \<in> HFinite"
huffman@21842
  1018
    by (rule NSBseqD2)
huffman@21842
  1019
  assume "\<not> Bseq X"
huffman@21842
  1020
  hence "\<forall>K>0. \<exists>n. K < norm (X n)"
huffman@21842
  1021
    by (simp add: Bseq_def linorder_not_le)
huffman@21842
  1022
  hence "\<forall>K>0. K < norm (X (?n K))"
huffman@21842
  1023
    by (auto intro: LeastI_ex)
huffman@21842
  1024
  hence "\<forall>K>0. K < hnorm (( *f* X) (( *f* ?n) K))"
huffman@21842
  1025
    by transfer
huffman@21842
  1026
  hence "\<omega> < hnorm (( *f* X) (( *f* ?n) \<omega>))"
huffman@21842
  1027
    by simp
huffman@21842
  1028
  hence "\<forall>r\<in>\<real>. r < hnorm (( *f* X) (( *f* ?n) \<omega>))"
huffman@21842
  1029
    by (simp add: order_less_trans [OF SReal_less_omega])
huffman@21842
  1030
  hence "( *f* X) (( *f* ?n) \<omega>) \<in> HInfinite"
huffman@21842
  1031
    by (simp add: HInfinite_def)
huffman@21842
  1032
  with finite show "False"
huffman@21842
  1033
    by (simp add: HFinite_HInfinite_iff)
huffman@21842
  1034
qed
paulson@15082
  1035
paulson@15082
  1036
text{* Equivalence of nonstandard and standard definitions
paulson@15082
  1037
  for a bounded sequence*}
paulson@15082
  1038
lemma Bseq_NSBseq_iff: "(Bseq X) = (NSBseq X)"
paulson@15082
  1039
by (blast intro!: NSBseq_Bseq Bseq_NSBseq)
paulson@15082
  1040
paulson@15082
  1041
text{*A convergent sequence is bounded: 
paulson@15082
  1042
 Boundedness as a necessary condition for convergence. 
paulson@15082
  1043
 The nonstandard version has no existential, as usual *}
paulson@15082
  1044
paulson@15082
  1045
lemma NSconvergent_NSBseq: "NSconvergent X ==> NSBseq X"
paulson@15082
  1046
apply (simp add: NSconvergent_def NSBseq_def NSLIMSEQ_def)
huffman@20552
  1047
apply (blast intro: HFinite_star_of approx_sym approx_HFinite)
paulson@15082
  1048
done
paulson@15082
  1049
paulson@15082
  1050
text{*Standard Version: easily now proved using equivalence of NS and
paulson@15082
  1051
 standard definitions *}
paulson@15082
  1052
lemma convergent_Bseq: "convergent X ==> Bseq X"
paulson@15082
  1053
by (simp add: NSconvergent_NSBseq convergent_NSconvergent_iff Bseq_NSBseq_iff)
paulson@15082
  1054
huffman@20696
  1055
subsubsection{*Upper Bounds and Lubs of Bounded Sequences*}
paulson@15082
  1056
paulson@15082
  1057
lemma Bseq_isUb:
paulson@15082
  1058
  "!!(X::nat=>real). Bseq X ==> \<exists>U. isUb (UNIV::real set) {x. \<exists>n. X n = x} U"
paulson@15082
  1059
by (auto intro: isUbI setleI simp add: Bseq_def abs_le_interval_iff)
paulson@15082
  1060
paulson@15082
  1061
paulson@15082
  1062
text{* Use completeness of reals (supremum property)
paulson@15082
  1063
   to show that any bounded sequence has a least upper bound*}
paulson@15082
  1064
paulson@15082
  1065
lemma Bseq_isLub:
paulson@15082
  1066
  "!!(X::nat=>real). Bseq X ==>
paulson@15082
  1067
   \<exists>U. isLub (UNIV::real set) {x. \<exists>n. X n = x} U"
paulson@15082
  1068
by (blast intro: reals_complete Bseq_isUb)
paulson@15082
  1069
huffman@20552
  1070
lemma NSBseq_isUb: "NSBseq X ==> \<exists>U::real. isUb UNIV {x. \<exists>n. X n = x} U"
paulson@15082
  1071
by (simp add: Bseq_NSBseq_iff [symmetric] Bseq_isUb)
paulson@15082
  1072
huffman@20552
  1073
lemma NSBseq_isLub: "NSBseq X ==> \<exists>U::real. isLub UNIV {x. \<exists>n. X n = x} U"
paulson@15082
  1074
by (simp add: Bseq_NSBseq_iff [symmetric] Bseq_isLub)
paulson@15082
  1075
paulson@15082
  1076
huffman@20696
  1077
subsubsection{*A Bounded and Monotonic Sequence Converges*}
paulson@15082
  1078
paulson@15082
  1079
lemma lemma_converg1:
nipkow@15360
  1080
     "!!(X::nat=>real). [| \<forall>m. \<forall> n \<ge> m. X m \<le> X n;
paulson@15082
  1081
                  isLub (UNIV::real set) {x. \<exists>n. X n = x} (X ma)
nipkow@15360
  1082
               |] ==> \<forall>n \<ge> ma. X n = X ma"
paulson@15082
  1083
apply safe
paulson@15082
  1084
apply (drule_tac y = "X n" in isLubD2)
paulson@15082
  1085
apply (blast dest: order_antisym)+
paulson@15082
  1086
done
paulson@15082
  1087
paulson@15082
  1088
text{* The best of both worlds: Easier to prove this result as a standard
paulson@15082
  1089
   theorem and then use equivalence to "transfer" it into the
paulson@15082
  1090
   equivalent nonstandard form if needed!*}
paulson@15082
  1091
paulson@15082
  1092
lemma Bmonoseq_LIMSEQ: "\<forall>n. m \<le> n --> X n = X m ==> \<exists>L. (X ----> L)"
paulson@15082
  1093
apply (simp add: LIMSEQ_def)
paulson@15082
  1094
apply (rule_tac x = "X m" in exI, safe)
paulson@15082
  1095
apply (rule_tac x = m in exI, safe)
paulson@15082
  1096
apply (drule spec, erule impE, auto)
paulson@15082
  1097
done
paulson@15082
  1098
paulson@15082
  1099
text{*Now, the same theorem in terms of NS limit *}
nipkow@15360
  1100
lemma Bmonoseq_NSLIMSEQ: "\<forall>n \<ge> m. X n = X m ==> \<exists>L. (X ----NS> L)"
paulson@15082
  1101
by (auto dest!: Bmonoseq_LIMSEQ simp add: LIMSEQ_NSLIMSEQ_iff)
paulson@15082
  1102
paulson@15082
  1103
lemma lemma_converg2:
paulson@15082
  1104
   "!!(X::nat=>real).
paulson@15082
  1105
    [| \<forall>m. X m ~= U;  isLub UNIV {x. \<exists>n. X n = x} U |] ==> \<forall>m. X m < U"
paulson@15082
  1106
apply safe
paulson@15082
  1107
apply (drule_tac y = "X m" in isLubD2)
paulson@15082
  1108
apply (auto dest!: order_le_imp_less_or_eq)
paulson@15082
  1109
done
paulson@15082
  1110
paulson@15082
  1111
lemma lemma_converg3: "!!(X ::nat=>real). \<forall>m. X m \<le> U ==> isUb UNIV {x. \<exists>n. X n = x} U"
paulson@15082
  1112
by (rule setleI [THEN isUbI], auto)
paulson@15082
  1113
paulson@15082
  1114
text{* FIXME: @{term "U - T < U"} is redundant *}
paulson@15082
  1115
lemma lemma_converg4: "!!(X::nat=> real).
paulson@15082
  1116
               [| \<forall>m. X m ~= U;
paulson@15082
  1117
                  isLub UNIV {x. \<exists>n. X n = x} U;
paulson@15082
  1118
                  0 < T;
paulson@15082
  1119
                  U + - T < U
paulson@15082
  1120
               |] ==> \<exists>m. U + -T < X m & X m < U"
paulson@15082
  1121
apply (drule lemma_converg2, assumption)
paulson@15082
  1122
apply (rule ccontr, simp)
paulson@15082
  1123
apply (simp add: linorder_not_less)
paulson@15082
  1124
apply (drule lemma_converg3)
paulson@15082
  1125
apply (drule isLub_le_isUb, assumption)
paulson@15082
  1126
apply (auto dest: order_less_le_trans)
paulson@15082
  1127
done
paulson@15082
  1128
paulson@15082
  1129
text{*A standard proof of the theorem for monotone increasing sequence*}
paulson@15082
  1130
paulson@15082
  1131
lemma Bseq_mono_convergent:
huffman@20552
  1132
     "[| Bseq X; \<forall>m. \<forall>n \<ge> m. X m \<le> X n |] ==> convergent (X::nat=>real)"
paulson@15082
  1133
apply (simp add: convergent_def)
paulson@15082
  1134
apply (frule Bseq_isLub, safe)
paulson@15082
  1135
apply (case_tac "\<exists>m. X m = U", auto)
paulson@15082
  1136
apply (blast dest: lemma_converg1 Bmonoseq_LIMSEQ)
paulson@15082
  1137
(* second case *)
paulson@15082
  1138
apply (rule_tac x = U in exI)
paulson@15082
  1139
apply (subst LIMSEQ_iff, safe)
paulson@15082
  1140
apply (frule lemma_converg2, assumption)
paulson@15082
  1141
apply (drule lemma_converg4, auto)
paulson@15082
  1142
apply (rule_tac x = m in exI, safe)
paulson@15082
  1143
apply (subgoal_tac "X m \<le> X n")
paulson@15082
  1144
 prefer 2 apply blast
paulson@15082
  1145
apply (drule_tac x=n and P="%m. X m < U" in spec, arith)
paulson@15082
  1146
done
paulson@15082
  1147
paulson@15082
  1148
text{*Nonstandard version of the theorem*}
paulson@15082
  1149
paulson@15082
  1150
lemma NSBseq_mono_NSconvergent:
huffman@20552
  1151
     "[| NSBseq X; \<forall>m. \<forall>n \<ge> m. X m \<le> X n |] ==> NSconvergent (X::nat=>real)"
paulson@15082
  1152
by (auto intro: Bseq_mono_convergent 
paulson@15082
  1153
         simp add: convergent_NSconvergent_iff [symmetric] 
paulson@15082
  1154
                   Bseq_NSBseq_iff [symmetric])
paulson@15082
  1155
paulson@15082
  1156
lemma Bseq_minus_iff: "Bseq (%n. -(X n)) = Bseq X"
paulson@15082
  1157
by (simp add: Bseq_def)
paulson@15082
  1158
paulson@15082
  1159
text{*Main monotonicity theorem*}
paulson@15082
  1160
lemma Bseq_monoseq_convergent: "[| Bseq X; monoseq X |] ==> convergent X"
paulson@15082
  1161
apply (simp add: monoseq_def, safe)
paulson@15082
  1162
apply (rule_tac [2] convergent_minus_iff [THEN ssubst])
paulson@15082
  1163
apply (drule_tac [2] Bseq_minus_iff [THEN ssubst])
paulson@15082
  1164
apply (auto intro!: Bseq_mono_convergent)
paulson@15082
  1165
done
paulson@15082
  1166
huffman@20696
  1167
subsubsection{*A Few More Equivalence Theorems for Boundedness*}
paulson@15082
  1168
paulson@15082
  1169
text{*alternative formulation for boundedness*}
huffman@20552
  1170
lemma Bseq_iff2: "Bseq X = (\<exists>k > 0. \<exists>x. \<forall>n. norm (X(n) + -x) \<le> k)"
paulson@15082
  1171
apply (unfold Bseq_def, safe)
huffman@20552
  1172
apply (rule_tac [2] x = "k + norm x" in exI)
nipkow@15360
  1173
apply (rule_tac x = K in exI, simp)
paulson@15221
  1174
apply (rule exI [where x = 0], auto)
huffman@20552
  1175
apply (erule order_less_le_trans, simp)
huffman@20552
  1176
apply (drule_tac x=n in spec, fold diff_def)
huffman@20552
  1177
apply (drule order_trans [OF norm_triangle_ineq2])
huffman@20552
  1178
apply simp
paulson@15082
  1179
done
paulson@15082
  1180
paulson@15082
  1181
text{*alternative formulation for boundedness*}
huffman@20552
  1182
lemma Bseq_iff3: "Bseq X = (\<exists>k > 0. \<exists>N. \<forall>n. norm(X(n) + -X(N)) \<le> k)"
paulson@15082
  1183
apply safe
paulson@15082
  1184
apply (simp add: Bseq_def, safe)
huffman@20552
  1185
apply (rule_tac x = "K + norm (X N)" in exI)
paulson@15082
  1186
apply auto
huffman@20552
  1187
apply (erule order_less_le_trans, simp)
paulson@15082
  1188
apply (rule_tac x = N in exI, safe)
huffman@20552
  1189
apply (drule_tac x = n in spec)
huffman@20552
  1190
apply (rule order_trans [OF norm_triangle_ineq], simp)
paulson@15082
  1191
apply (auto simp add: Bseq_iff2)
paulson@15082
  1192
done
paulson@15082
  1193
huffman@20552
  1194
lemma BseqI2: "(\<forall>n. k \<le> f n & f n \<le> (K::real)) ==> Bseq f"
paulson@15082
  1195
apply (simp add: Bseq_def)
paulson@15221
  1196
apply (rule_tac x = " (\<bar>k\<bar> + \<bar>K\<bar>) + 1" in exI, auto)
webertj@20217
  1197
apply (drule_tac x = n in spec, arith)
paulson@15082
  1198
done
paulson@15082
  1199
paulson@15082
  1200
huffman@20696
  1201
subsection {* Cauchy Sequences *}
paulson@15082
  1202
huffman@20751
  1203
lemma CauchyI:
huffman@20751
  1204
  "(\<And>e. 0 < e \<Longrightarrow> \<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. norm (X m - X n) < e) \<Longrightarrow> Cauchy X"
huffman@20751
  1205
by (simp add: Cauchy_def)
huffman@20751
  1206
huffman@20751
  1207
lemma CauchyD:
huffman@20751
  1208
  "\<lbrakk>Cauchy X; 0 < e\<rbrakk> \<Longrightarrow> \<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. norm (X m - X n) < e"
huffman@20751
  1209
by (simp add: Cauchy_def)
huffman@20751
  1210
huffman@20751
  1211
lemma NSCauchyI:
huffman@20751
  1212
  "(\<And>M N. \<lbrakk>M \<in> HNatInfinite; N \<in> HNatInfinite\<rbrakk> \<Longrightarrow> starfun X M \<approx> starfun X N)
huffman@20751
  1213
   \<Longrightarrow> NSCauchy X"
huffman@20751
  1214
by (simp add: NSCauchy_def)
huffman@20751
  1215
huffman@20751
  1216
lemma NSCauchyD:
huffman@20751
  1217
  "\<lbrakk>NSCauchy X; M \<in> HNatInfinite; N \<in> HNatInfinite\<rbrakk>
huffman@20751
  1218
   \<Longrightarrow> starfun X M \<approx> starfun X N"
huffman@20751
  1219
by (simp add: NSCauchy_def)
huffman@20751
  1220
huffman@20696
  1221
subsubsection{*Equivalence Between NS and Standard*}
huffman@20696
  1222
huffman@20751
  1223
lemma Cauchy_NSCauchy:
huffman@20751
  1224
  assumes X: "Cauchy X" shows "NSCauchy X"
huffman@20751
  1225
proof (rule NSCauchyI)
huffman@20751
  1226
  fix M assume M: "M \<in> HNatInfinite"
huffman@20751
  1227
  fix N assume N: "N \<in> HNatInfinite"
huffman@20751
  1228
  have "starfun X M - starfun X N \<in> Infinitesimal"
huffman@20751
  1229
  proof (rule InfinitesimalI2)
huffman@20751
  1230
    fix r :: real assume r: "0 < r"
huffman@20751
  1231
    from CauchyD [OF X r]
huffman@20751
  1232
    obtain k where "\<forall>m\<ge>k. \<forall>n\<ge>k. norm (X m - X n) < r" ..
huffman@20751
  1233
    hence "\<forall>m\<ge>star_of k. \<forall>n\<ge>star_of k.
huffman@20751
  1234
           hnorm (starfun X m - starfun X n) < star_of r"
huffman@20751
  1235
      by transfer
huffman@20751
  1236
    thus "hnorm (starfun X M - starfun X N) < star_of r"
huffman@20751
  1237
      using M N by (simp add: star_of_le_HNatInfinite)
huffman@20751
  1238
  qed
huffman@20751
  1239
  thus "starfun X M \<approx> starfun X N"
huffman@20751
  1240
    by (unfold approx_def)
huffman@20751
  1241
qed
paulson@15082
  1242
huffman@20751
  1243
lemma NSCauchy_Cauchy:
huffman@20751
  1244
  assumes X: "NSCauchy X" shows "Cauchy X"
huffman@20751
  1245
proof (rule CauchyI)
huffman@20751
  1246
  fix r::real assume r: "0 < r"
huffman@20751
  1247
  have "\<exists>k. \<forall>m\<ge>k. \<forall>n\<ge>k. hnorm (starfun X m - starfun X n) < star_of r"
huffman@20751
  1248
  proof (intro exI allI impI)
huffman@20751
  1249
    fix M assume "whn \<le> M"
huffman@20751
  1250
    with HNatInfinite_whn have M: "M \<in> HNatInfinite"
huffman@20751
  1251
      by (rule HNatInfinite_upward_closed)
huffman@20751
  1252
    fix N assume "whn \<le> N"
huffman@20751
  1253
    with HNatInfinite_whn have N: "N \<in> HNatInfinite"
huffman@20751
  1254
      by (rule HNatInfinite_upward_closed)
huffman@20751
  1255
    from X M N have "starfun X M \<approx> starfun X N"
huffman@20751
  1256
      by (rule NSCauchyD)
huffman@20751
  1257
    hence "starfun X M - starfun X N \<in> Infinitesimal"
huffman@20751
  1258
      by (unfold approx_def)
huffman@20751
  1259
    thus "hnorm (starfun X M - starfun X N) < star_of r"
huffman@20751
  1260
      using r by (rule InfinitesimalD2)
huffman@20751
  1261
  qed
huffman@20751
  1262
  thus "\<exists>k. \<forall>m\<ge>k. \<forall>n\<ge>k. norm (X m - X n) < r"
huffman@20751
  1263
    by transfer
huffman@20751
  1264
qed
paulson@15082
  1265
paulson@15082
  1266
theorem NSCauchy_Cauchy_iff: "NSCauchy X = Cauchy X"
paulson@15082
  1267
by (blast intro!: NSCauchy_Cauchy Cauchy_NSCauchy)
paulson@15082
  1268
huffman@20696
  1269
subsubsection {* Cauchy Sequences are Bounded *}
huffman@20696
  1270
paulson@15082
  1271
text{*A Cauchy sequence is bounded -- this is the standard
paulson@15082
  1272
  proof mechanization rather than the nonstandard proof*}
paulson@15082
  1273
huffman@20563
  1274
lemma lemmaCauchy: "\<forall>n \<ge> M. norm (X M - X n) < (1::real)
huffman@20552
  1275
          ==>  \<forall>n \<ge> M. norm (X n :: 'a::real_normed_vector) < 1 + norm (X M)"
huffman@20552
  1276
apply (clarify, drule spec, drule (1) mp)
huffman@20563
  1277
apply (simp only: norm_minus_commute)
huffman@20552
  1278
apply (drule order_le_less_trans [OF norm_triangle_ineq2])
huffman@20552
  1279
apply simp
huffman@20552
  1280
done
paulson@15082
  1281
paulson@15082
  1282
lemma Cauchy_Bseq: "Cauchy X ==> Bseq X"
huffman@20552
  1283
apply (simp add: Cauchy_def)
huffman@20552
  1284
apply (drule spec, drule mp, rule zero_less_one, safe)
huffman@20552
  1285
apply (drule_tac x="M" in spec, simp)
paulson@15082
  1286
apply (drule lemmaCauchy)
huffman@22608
  1287
apply (rule_tac k="M" in Bseq_offset)
huffman@20552
  1288
apply (simp add: Bseq_def)
huffman@20552
  1289
apply (rule_tac x="1 + norm (X M)" in exI)
huffman@20552
  1290
apply (rule conjI, rule order_less_le_trans [OF zero_less_one], simp)
huffman@20552
  1291
apply (simp add: order_less_imp_le)
paulson@15082
  1292
done
paulson@15082
  1293
paulson@15082
  1294
text{*A Cauchy sequence is bounded -- nonstandard version*}
paulson@15082
  1295
paulson@15082
  1296
lemma NSCauchy_NSBseq: "NSCauchy X ==> NSBseq X"
paulson@15082
  1297
by (simp add: Cauchy_Bseq Bseq_NSBseq_iff [symmetric] NSCauchy_Cauchy_iff)
paulson@15082
  1298
huffman@20696
  1299
subsubsection {* Cauchy Sequences are Convergent *}
paulson@15082
  1300
huffman@20830
  1301
axclass banach \<subseteq> real_normed_vector
huffman@20830
  1302
  Cauchy_convergent: "Cauchy X \<Longrightarrow> convergent X"
huffman@20830
  1303
paulson@15082
  1304
text{*Equivalence of Cauchy criterion and convergence:
paulson@15082
  1305
  We will prove this using our NS formulation which provides a
paulson@15082
  1306
  much easier proof than using the standard definition. We do not
paulson@15082
  1307
  need to use properties of subsequences such as boundedness,
paulson@15082
  1308
  monotonicity etc... Compare with Harrison's corresponding proof
paulson@15082
  1309
  in HOL which is much longer and more complicated. Of course, we do
paulson@15082
  1310
  not have problems which he encountered with guessing the right
paulson@15082
  1311
  instantiations for his 'espsilon-delta' proof(s) in this case
paulson@15082
  1312
  since the NS formulations do not involve existential quantifiers.*}
paulson@15082
  1313
huffman@20691
  1314
lemma NSconvergent_NSCauchy: "NSconvergent X \<Longrightarrow> NSCauchy X"
huffman@20691
  1315
apply (simp add: NSconvergent_def NSLIMSEQ_def NSCauchy_def, safe)
huffman@20691
  1316
apply (auto intro: approx_trans2)
huffman@20691
  1317
done
huffman@20691
  1318
huffman@20691
  1319
lemma convergent_Cauchy: "convergent X \<Longrightarrow> Cauchy X"
huffman@20691
  1320
apply (rule NSconvergent_NSCauchy [THEN NSCauchy_Cauchy])
huffman@20691
  1321
apply (simp add: convergent_NSconvergent_iff)
huffman@20691
  1322
done
huffman@20691
  1323
huffman@20830
  1324
lemma real_NSCauchy_NSconvergent:
huffman@20830
  1325
  fixes X :: "nat \<Rightarrow> real"
huffman@20830
  1326
  shows "NSCauchy X \<Longrightarrow> NSconvergent X"
huffman@20830
  1327
apply (simp add: NSconvergent_def NSLIMSEQ_def)
paulson@15082
  1328
apply (frule NSCauchy_NSBseq)
huffman@20830
  1329
apply (simp add: NSBseq_def NSCauchy_def)
paulson@15082
  1330
apply (drule HNatInfinite_whn [THEN [2] bspec])
paulson@15082
  1331
apply (drule HNatInfinite_whn [THEN [2] bspec])
paulson@15082
  1332
apply (auto dest!: st_part_Ex simp add: SReal_iff)
paulson@15082
  1333
apply (blast intro: approx_trans3)
paulson@15082
  1334
done
paulson@15082
  1335
paulson@15082
  1336
text{*Standard proof for free*}
huffman@20830
  1337
lemma real_Cauchy_convergent:
huffman@20830
  1338
  fixes X :: "nat \<Rightarrow> real"
huffman@20830
  1339
  shows "Cauchy X \<Longrightarrow> convergent X"
huffman@20830
  1340
apply (drule Cauchy_NSCauchy [THEN real_NSCauchy_NSconvergent])
huffman@20830
  1341
apply (erule convergent_NSconvergent_iff [THEN iffD2])
huffman@20830
  1342
done
huffman@20830
  1343
huffman@20830
  1344
instance real :: banach
huffman@20830
  1345
by intro_classes (rule real_Cauchy_convergent)
huffman@20830
  1346
huffman@20830
  1347
lemma NSCauchy_NSconvergent:
huffman@20830
  1348
  fixes X :: "nat \<Rightarrow> 'a::banach"
huffman@20830
  1349
  shows "NSCauchy X \<Longrightarrow> NSconvergent X"
huffman@20830
  1350
apply (drule NSCauchy_Cauchy [THEN Cauchy_convergent])
huffman@20830
  1351
apply (erule convergent_NSconvergent_iff [THEN iffD1])
huffman@20830
  1352
done
huffman@20830
  1353
huffman@20830
  1354
lemma NSCauchy_NSconvergent_iff:
huffman@20830
  1355
  fixes X :: "nat \<Rightarrow> 'a::banach"
huffman@20830
  1356
  shows "NSCauchy X = NSconvergent X"
huffman@20830
  1357
by (fast intro: NSCauchy_NSconvergent NSconvergent_NSCauchy)
huffman@20830
  1358
huffman@20830
  1359
lemma Cauchy_convergent_iff:
huffman@20830
  1360
  fixes X :: "nat \<Rightarrow> 'a::banach"
huffman@20830
  1361
  shows "Cauchy X = convergent X"
huffman@20830
  1362
by (fast intro: Cauchy_convergent convergent_Cauchy)
paulson@15082
  1363
paulson@15082
  1364
huffman@20696
  1365
subsection {* Power Sequences *}
paulson@15082
  1366
paulson@15082
  1367
text{*The sequence @{term "x^n"} tends to 0 if @{term "0\<le>x"} and @{term
paulson@15082
  1368
"x<1"}.  Proof will use (NS) Cauchy equivalence for convergence and
paulson@15082
  1369
  also fact that bounded and monotonic sequence converges.*}
paulson@15082
  1370
huffman@20552
  1371
lemma Bseq_realpow: "[| 0 \<le> (x::real); x \<le> 1 |] ==> Bseq (%n. x ^ n)"
paulson@15082
  1372
apply (simp add: Bseq_def)
paulson@15082
  1373
apply (rule_tac x = 1 in exI)
paulson@15082
  1374
apply (simp add: power_abs)
paulson@15082
  1375
apply (auto dest: power_mono intro: order_less_imp_le simp add: abs_if)
paulson@15082
  1376
done
paulson@15082
  1377
paulson@15082
  1378
lemma monoseq_realpow: "[| 0 \<le> x; x \<le> 1 |] ==> monoseq (%n. x ^ n)"
paulson@15082
  1379
apply (clarify intro!: mono_SucI2)
paulson@15082
  1380
apply (cut_tac n = n and N = "Suc n" and a = x in power_decreasing, auto)
paulson@15082
  1381
done
paulson@15082
  1382
huffman@20552
  1383
lemma convergent_realpow:
huffman@20552
  1384
  "[| 0 \<le> (x::real); x \<le> 1 |] ==> convergent (%n. x ^ n)"
paulson@15082
  1385
by (blast intro!: Bseq_monoseq_convergent Bseq_realpow monoseq_realpow)
paulson@15082
  1386
paulson@15082
  1387
text{* We now use NS criterion to bring proof of theorem through *}
paulson@15082
  1388
huffman@20552
  1389
lemma NSLIMSEQ_realpow_zero:
huffman@20552
  1390
  "[| 0 \<le> (x::real); x < 1 |] ==> (%n. x ^ n) ----NS> 0"
paulson@15082
  1391
apply (simp add: NSLIMSEQ_def)
paulson@15082
  1392
apply (auto dest!: convergent_realpow simp add: convergent_NSconvergent_iff)
paulson@15082
  1393
apply (frule NSconvergentD)
huffman@17318
  1394
apply (auto simp add: NSLIMSEQ_def NSCauchy_NSconvergent_iff [symmetric] NSCauchy_def starfun_pow)
paulson@15082
  1395
apply (frule HNatInfinite_add_one)
paulson@15082
  1396
apply (drule bspec, assumption)
paulson@15082
  1397
apply (drule bspec, assumption)
paulson@15082
  1398
apply (drule_tac x = "N + (1::hypnat) " in bspec, assumption)
paulson@15082
  1399
apply (simp add: hyperpow_add)
huffman@21810
  1400
apply (drule approx_mult_subst_star_of, assumption)
paulson@15082
  1401
apply (drule approx_trans3, assumption)
huffman@17318
  1402
apply (auto simp del: star_of_mult simp add: star_of_mult [symmetric])
paulson@15082
  1403
done
paulson@15082
  1404
paulson@15082
  1405
text{* standard version *}
huffman@20552
  1406
lemma LIMSEQ_realpow_zero:
huffman@20552
  1407
  "[| 0 \<le> (x::real); x < 1 |] ==> (%n. x ^ n) ----> 0"
paulson@15082
  1408
by (simp add: NSLIMSEQ_realpow_zero LIMSEQ_NSLIMSEQ_iff)
paulson@15082
  1409
huffman@20685
  1410
lemma LIMSEQ_power_zero:
huffman@20685
  1411
  fixes x :: "'a::{real_normed_div_algebra,recpower}"
huffman@20685
  1412
  shows "norm x < 1 \<Longrightarrow> (\<lambda>n. x ^ n) ----> 0"
huffman@20685
  1413
apply (drule LIMSEQ_realpow_zero [OF norm_ge_zero])
huffman@20685
  1414
apply (simp add: norm_power [symmetric] LIMSEQ_norm_zero)
huffman@20685
  1415
done
huffman@20685
  1416
huffman@20552
  1417
lemma LIMSEQ_divide_realpow_zero:
huffman@20552
  1418
  "1 < (x::real) ==> (%n. a / (x ^ n)) ----> 0"
paulson@15082
  1419
apply (cut_tac a = a and x1 = "inverse x" in
paulson@15082
  1420
        LIMSEQ_mult [OF LIMSEQ_const LIMSEQ_realpow_zero])
paulson@15082
  1421
apply (auto simp add: divide_inverse power_inverse)
paulson@15082
  1422
apply (simp add: inverse_eq_divide pos_divide_less_eq)
paulson@15082
  1423
done
paulson@15082
  1424
paulson@15102
  1425
text{*Limit of @{term "c^n"} for @{term"\<bar>c\<bar> < 1"}*}
paulson@15082
  1426
huffman@20552
  1427
lemma LIMSEQ_rabs_realpow_zero: "\<bar>c\<bar> < (1::real) ==> (%n. \<bar>c\<bar> ^ n) ----> 0"
huffman@20685
  1428
by (rule LIMSEQ_realpow_zero [OF abs_ge_zero])
paulson@15082
  1429
huffman@20552
  1430
lemma NSLIMSEQ_rabs_realpow_zero: "\<bar>c\<bar> < (1::real) ==> (%n. \<bar>c\<bar> ^ n) ----NS> 0"
paulson@15082
  1431
by (simp add: LIMSEQ_rabs_realpow_zero LIMSEQ_NSLIMSEQ_iff [symmetric])
paulson@15082
  1432
huffman@20552
  1433
lemma LIMSEQ_rabs_realpow_zero2: "\<bar>c\<bar> < (1::real) ==> (%n. c ^ n) ----> 0"
paulson@15082
  1434
apply (rule LIMSEQ_rabs_zero [THEN iffD1])
paulson@15082
  1435
apply (auto intro: LIMSEQ_rabs_realpow_zero simp add: power_abs)
paulson@15082
  1436
done
paulson@15082
  1437
huffman@20552
  1438
lemma NSLIMSEQ_rabs_realpow_zero2: "\<bar>c\<bar> < (1::real) ==> (%n. c ^ n) ----NS> 0"
paulson@15082
  1439
by (simp add: LIMSEQ_rabs_realpow_zero2 LIMSEQ_NSLIMSEQ_iff [symmetric])
paulson@15082
  1440
paulson@15082
  1441
(***---------------------------------------------------------------
paulson@15082
  1442
    Theorems proved by Harrison in HOL that we do not need
paulson@15082
  1443
    in order to prove equivalence between Cauchy criterion
paulson@15082
  1444
    and convergence:
paulson@15082
  1445
 -- Show that every sequence contains a monotonic subsequence
paulson@15082
  1446
Goal "\<exists>f. subseq f & monoseq (%n. s (f n))"
paulson@15082
  1447
 -- Show that a subsequence of a bounded sequence is bounded
paulson@15082
  1448
Goal "Bseq X ==> Bseq (%n. X (f n))";
paulson@15082
  1449
 -- Show we can take subsequential terms arbitrarily far
paulson@15082
  1450
    up a sequence
paulson@15082
  1451
Goal "subseq f ==> n \<le> f(n)";
paulson@15082
  1452
Goal "subseq f ==> \<exists>n. N1 \<le> n & N2 \<le> f(n)";
paulson@15082
  1453
 ---------------------------------------------------------------***)
paulson@15082
  1454
paulson@10751
  1455
end