src/HOL/SEQ.thy
author huffman
Sun, 31 May 2009 21:59:33 -0700
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permissions -rw-r--r--
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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 Convergence *}
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theory SEQ
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imports Limits
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begin
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definition
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  sequentially :: "nat filter" where
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  "sequentially = Abs_filter (\<lambda>P. \<exists>N. \<forall>n\<ge>N. P n)"
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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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  [code del]: "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 \<Rightarrow> 'a::metric_space, 'a] \<Rightarrow> bool"
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    ("((_)/ ----> (_))" [60, 60] 60) where
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    --{*Standard definition of convergence of sequence*}
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  [code del]: "X ----> L = (\<forall>r>0. \<exists>no. \<forall>n\<ge>no. dist (X n) L < r)"
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definition
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  lim :: "(nat \<Rightarrow> 'a::real_normed_vector) \<Rightarrow> '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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  convergent :: "(nat \<Rightarrow> 'a::metric_space) \<Rightarrow> 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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  Bseq :: "(nat => 'a::real_normed_vector) => bool" where
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    --{*Standard definition for bounded sequence*}
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  [code del]: "Bseq X = (\<exists>K>0.\<forall>n. norm (X n) \<le> K)"
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definition
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  monoseq :: "(nat=>real)=>bool" where
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    --{*Definition of monotonicity. 
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        The use of disjunction here complicates proofs considerably. 
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        One alternative is to add a Boolean argument to indicate the direction. 
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        Another is to develop the notions of increasing and decreasing first.*}
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  [code del]: "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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  incseq :: "(nat=>real)=>bool" where
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    --{*Increasing sequence*}
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  [code del]: "incseq X = (\<forall>m. \<forall>n\<ge>m. X m \<le> X n)"
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definition
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  decseq :: "(nat=>real)=>bool" where
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    --{*Increasing sequence*}
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  [code del]: "decseq X = (\<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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  [code del]:   "subseq f = (\<forall>m. \<forall>n>m. (f m) < (f n))"
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definition
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  Cauchy :: "(nat \<Rightarrow> 'a::metric_space) \<Rightarrow> bool" where
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    --{*Standard definition of the Cauchy condition*}
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  [code del]: "Cauchy X = (\<forall>e>0. \<exists>M. \<forall>m \<ge> M. \<forall>n \<ge> M. dist (X m) (X n) < e)"
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subsection {* Sequentially *}
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lemma eventually_sequentially:
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  "eventually P sequentially \<longleftrightarrow> (\<exists>N. \<forall>n\<ge>N. P n)"
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unfolding sequentially_def
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apply (rule eventually_Abs_filter)
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apply simp
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apply (clarify, rule_tac x=N in exI, simp)
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apply (clarify, rename_tac M N)
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apply (rule_tac x="max M N" in exI, simp)
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done
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lemma Zseq_conv_Zfun: "Zseq X \<longleftrightarrow> Zfun X sequentially"
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unfolding Zseq_def Zfun_def eventually_sequentially ..
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lemma LIMSEQ_conv_tendsto: "(X ----> L) \<longleftrightarrow> tendsto X L sequentially"
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unfolding LIMSEQ_def tendsto_def eventually_sequentially ..
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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 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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  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 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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   154
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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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using assms unfolding Zseq_conv_Zfun by (rule Zfun_imp_Zfun)
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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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  "Zseq X \<Longrightarrow> Zseq Y \<Longrightarrow> Zseq (\<lambda>n. X n + Y n)"
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unfolding Zseq_conv_Zfun by (rule Zfun_add)
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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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  "Zseq X \<Longrightarrow> Zseq (\<lambda>n. f (X n))"
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unfolding Zseq_conv_Zfun by (rule Zfun)
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lemma (in bounded_bilinear) Zseq:
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  "Zseq X \<Longrightarrow> Zseq Y \<Longrightarrow> Zseq (\<lambda>n. X n ** Y n)"
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unfolding Zseq_conv_Zfun by (rule Zfun)
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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_left:
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  "Zseq X \<Longrightarrow> Zseq (\<lambda>n. X n ** a)"
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by (rule bounded_linear_left [THEN bounded_linear.Zseq])
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lemma (in bounded_bilinear) Zseq_right:
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  "Zseq X \<Longrightarrow> Zseq (\<lambda>n. a ** X n)"
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by (rule bounded_linear_right [THEN bounded_linear.Zseq])
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lemmas Zseq_mult = mult.Zseq
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lemmas Zseq_mult_right = mult.Zseq_right
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lemmas Zseq_mult_left = mult.Zseq_left
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subsection {* Limits of Sequences *}
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lemma LIMSEQ_iff:
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  fixes L :: "'a::real_normed_vector"
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  shows "(X ----> L) = (\<forall>r>0. \<exists>no. \<forall>n \<ge> no. norm (X n - L) < r)"
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unfolding LIMSEQ_def dist_norm ..
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lemma LIMSEQ_Zseq_iff: "((\<lambda>n. X n) ----> L) = Zseq (\<lambda>n. X n - L)"
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by (simp only: LIMSEQ_iff Zseq_def)
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lemma metric_LIMSEQ_I:
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  "(\<And>r. 0 < r \<Longrightarrow> \<exists>no. \<forall>n\<ge>no. dist (X n) L < r) \<Longrightarrow> X ----> L"
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by (simp add: LIMSEQ_def)
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lemma metric_LIMSEQ_D:
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  "\<lbrakk>X ----> L; 0 < r\<rbrakk> \<Longrightarrow> \<exists>no. \<forall>n\<ge>no. dist (X n) L < r"
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by (simp add: LIMSEQ_def)
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lemma LIMSEQ_I:
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  fixes L :: "'a::real_normed_vector"
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  shows "(\<And>r. 0 < r \<Longrightarrow> \<exists>no. \<forall>n\<ge>no. norm (X n - L) < r) \<Longrightarrow> X ----> L"
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by (simp add: LIMSEQ_iff)
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lemma LIMSEQ_D:
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  fixes L :: "'a::real_normed_vector"
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  shows "\<lbrakk>X ----> L; 0 < r\<rbrakk> \<Longrightarrow> \<exists>no. \<forall>n\<ge>no. norm (X n - L) < r"
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by (simp add: LIMSEQ_iff)
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   272
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lemma LIMSEQ_const: "(\<lambda>n. k) ----> k"
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by (simp add: LIMSEQ_def)
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lemma LIMSEQ_const_iff: "(\<lambda>n. k) ----> l \<longleftrightarrow> k = l"
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apply (safe intro!: LIMSEQ_const)
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apply (rule ccontr)
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apply (drule_tac r="dist k l" in metric_LIMSEQ_D)
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apply (simp add: zero_less_dist_iff)
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apply auto
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done
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lemma LIMSEQ_norm:
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  fixes a :: "'a::real_normed_vector"
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   286
  shows "X ----> a \<Longrightarrow> (\<lambda>n. norm (X n)) ----> norm a"
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   287
unfolding LIMSEQ_conv_tendsto by (rule tendsto_norm)
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   288
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lemma LIMSEQ_ignore_initial_segment:
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  "f ----> a \<Longrightarrow> (\<lambda>n. f (n + k)) ----> a"
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apply (rule metric_LIMSEQ_I)
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   292
apply (drule (1) metric_LIMSEQ_D)
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   293
apply (erule exE, rename_tac N)
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   294
apply (rule_tac x=N in exI)
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   295
apply simp
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done
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   297
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lemma LIMSEQ_offset:
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  "(\<lambda>n. f (n + k)) ----> a \<Longrightarrow> f ----> a"
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apply (rule metric_LIMSEQ_I)
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   301
apply (drule (1) metric_LIMSEQ_D)
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diff changeset
   302
apply (erule exE, rename_tac N)
d650e51b5970 new standard proofs of some LIMSEQ lemmas
huffman
parents: 22614
diff changeset
   303
apply (rule_tac x="N + k" in exI)
d650e51b5970 new standard proofs of some LIMSEQ lemmas
huffman
parents: 22614
diff changeset
   304
apply clarify
d650e51b5970 new standard proofs of some LIMSEQ lemmas
huffman
parents: 22614
diff changeset
   305
apply (drule_tac x="n - k" in spec)
d650e51b5970 new standard proofs of some LIMSEQ lemmas
huffman
parents: 22614
diff changeset
   306
apply (simp add: le_diff_conv2)
20696
3b887ad7d196 reorganized subsection headings
huffman
parents: 20695
diff changeset
   307
done
3b887ad7d196 reorganized subsection headings
huffman
parents: 20695
diff changeset
   308
22615
d650e51b5970 new standard proofs of some LIMSEQ lemmas
huffman
parents: 22614
diff changeset
   309
lemma LIMSEQ_Suc: "f ----> l \<Longrightarrow> (\<lambda>n. f (Suc n)) ----> l"
30082
43c5b7bfc791 make more proofs work whether or not One_nat_def is a simp rule
huffman
parents: 29803
diff changeset
   310
by (drule_tac k="Suc 0" in LIMSEQ_ignore_initial_segment, simp)
22615
d650e51b5970 new standard proofs of some LIMSEQ lemmas
huffman
parents: 22614
diff changeset
   311
d650e51b5970 new standard proofs of some LIMSEQ lemmas
huffman
parents: 22614
diff changeset
   312
lemma LIMSEQ_imp_Suc: "(\<lambda>n. f (Suc n)) ----> l \<Longrightarrow> f ----> l"
30082
43c5b7bfc791 make more proofs work whether or not One_nat_def is a simp rule
huffman
parents: 29803
diff changeset
   313
by (rule_tac k="Suc 0" in LIMSEQ_offset, simp)
22615
d650e51b5970 new standard proofs of some LIMSEQ lemmas
huffman
parents: 22614
diff changeset
   314
d650e51b5970 new standard proofs of some LIMSEQ lemmas
huffman
parents: 22614
diff changeset
   315
lemma LIMSEQ_Suc_iff: "(\<lambda>n. f (Suc n)) ----> l = f ----> l"
d650e51b5970 new standard proofs of some LIMSEQ lemmas
huffman
parents: 22614
diff changeset
   316
by (blast intro: LIMSEQ_imp_Suc LIMSEQ_Suc)
d650e51b5970 new standard proofs of some LIMSEQ lemmas
huffman
parents: 22614
diff changeset
   317
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
   318
lemma LIMSEQ_linear: "\<lbrakk> X ----> x ; l > 0 \<rbrakk> \<Longrightarrow> (\<lambda> n. X (n * l)) ----> x"
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
   319
  unfolding LIMSEQ_def
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
   320
  by (metis div_le_dividend div_mult_self1_is_m le_trans nat_mult_commute)
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
   321
31336
e17f13cd1280 generalize constants in SEQ.thy to class metric_space
huffman
parents: 31017
diff changeset
   322
lemma LIMSEQ_add:
e17f13cd1280 generalize constants in SEQ.thy to class metric_space
huffman
parents: 31017
diff changeset
   323
  fixes a b :: "'a::real_normed_vector"
e17f13cd1280 generalize constants in SEQ.thy to class metric_space
huffman
parents: 31017
diff changeset
   324
  shows "\<lbrakk>X ----> a; Y ----> b\<rbrakk> \<Longrightarrow> (\<lambda>n. X n + Y n) ----> a + b"
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents: 31336
diff changeset
   325
unfolding LIMSEQ_conv_tendsto by (rule tendsto_add)
22608
092a3353241e add new standard proofs for limits of sequences
huffman
parents: 21842
diff changeset
   326
31336
e17f13cd1280 generalize constants in SEQ.thy to class metric_space
huffman
parents: 31017
diff changeset
   327
lemma LIMSEQ_minus:
e17f13cd1280 generalize constants in SEQ.thy to class metric_space
huffman
parents: 31017
diff changeset
   328
  fixes a :: "'a::real_normed_vector"
e17f13cd1280 generalize constants in SEQ.thy to class metric_space
huffman
parents: 31017
diff changeset
   329
  shows "X ----> a \<Longrightarrow> (\<lambda>n. - X n) ----> - a"
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents: 31336
diff changeset
   330
unfolding LIMSEQ_conv_tendsto by (rule tendsto_minus)
22608
092a3353241e add new standard proofs for limits of sequences
huffman
parents: 21842
diff changeset
   331
31336
e17f13cd1280 generalize constants in SEQ.thy to class metric_space
huffman
parents: 31017
diff changeset
   332
lemma LIMSEQ_minus_cancel:
e17f13cd1280 generalize constants in SEQ.thy to class metric_space
huffman
parents: 31017
diff changeset
   333
  fixes a :: "'a::real_normed_vector"
e17f13cd1280 generalize constants in SEQ.thy to class metric_space
huffman
parents: 31017
diff changeset
   334
  shows "(\<lambda>n. - X n) ----> - a \<Longrightarrow> X ----> a"
22608
092a3353241e add new standard proofs for limits of sequences
huffman
parents: 21842
diff changeset
   335
by (drule LIMSEQ_minus, simp)
092a3353241e add new standard proofs for limits of sequences
huffman
parents: 21842
diff changeset
   336
31336
e17f13cd1280 generalize constants in SEQ.thy to class metric_space
huffman
parents: 31017
diff changeset
   337
lemma LIMSEQ_diff:
e17f13cd1280 generalize constants in SEQ.thy to class metric_space
huffman
parents: 31017
diff changeset
   338
  fixes a b :: "'a::real_normed_vector"
e17f13cd1280 generalize constants in SEQ.thy to class metric_space
huffman
parents: 31017
diff changeset
   339
  shows "\<lbrakk>X ----> a; Y ----> b\<rbrakk> \<Longrightarrow> (\<lambda>n. X n - Y n) ----> a - b"
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents: 31336
diff changeset
   340
unfolding LIMSEQ_conv_tendsto by (rule tendsto_diff)
22608
092a3353241e add new standard proofs for limits of sequences
huffman
parents: 21842
diff changeset
   341
092a3353241e add new standard proofs for limits of sequences
huffman
parents: 21842
diff changeset
   342
lemma LIMSEQ_unique: "\<lbrakk>X ----> a; X ----> b\<rbrakk> \<Longrightarrow> a = b"
31336
e17f13cd1280 generalize constants in SEQ.thy to class metric_space
huffman
parents: 31017
diff changeset
   343
apply (rule ccontr)
e17f13cd1280 generalize constants in SEQ.thy to class metric_space
huffman
parents: 31017
diff changeset
   344
apply (drule_tac r="dist a b / 2" in metric_LIMSEQ_D, simp add: zero_less_dist_iff)
e17f13cd1280 generalize constants in SEQ.thy to class metric_space
huffman
parents: 31017
diff changeset
   345
apply (drule_tac r="dist a b / 2" in metric_LIMSEQ_D, simp add: zero_less_dist_iff)
e17f13cd1280 generalize constants in SEQ.thy to class metric_space
huffman
parents: 31017
diff changeset
   346
apply (clarify, rename_tac M N)
e17f13cd1280 generalize constants in SEQ.thy to class metric_space
huffman
parents: 31017
diff changeset
   347
apply (subgoal_tac "dist a b < dist a b / 2 + dist a b / 2", simp)
e17f13cd1280 generalize constants in SEQ.thy to class metric_space
huffman
parents: 31017
diff changeset
   348
apply (subgoal_tac "dist a b \<le> dist (X (max M N)) a + dist (X (max M N)) b")
e17f13cd1280 generalize constants in SEQ.thy to class metric_space
huffman
parents: 31017
diff changeset
   349
apply (erule le_less_trans, rule add_strict_mono, simp, simp)
e17f13cd1280 generalize constants in SEQ.thy to class metric_space
huffman
parents: 31017
diff changeset
   350
apply (subst dist_commute, rule dist_triangle)
e17f13cd1280 generalize constants in SEQ.thy to class metric_space
huffman
parents: 31017
diff changeset
   351
done
22608
092a3353241e add new standard proofs for limits of sequences
huffman
parents: 21842
diff changeset
   352
092a3353241e add new standard proofs for limits of sequences
huffman
parents: 21842
diff changeset
   353
lemma (in bounded_linear) LIMSEQ:
092a3353241e add new standard proofs for limits of sequences
huffman
parents: 21842
diff changeset
   354
  "X ----> a \<Longrightarrow> (\<lambda>n. f (X n)) ----> f a"
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents: 31336
diff changeset
   355
unfolding LIMSEQ_conv_tendsto by (rule tendsto)
22608
092a3353241e add new standard proofs for limits of sequences
huffman
parents: 21842
diff changeset
   356
092a3353241e add new standard proofs for limits of sequences
huffman
parents: 21842
diff changeset
   357
lemma (in bounded_bilinear) LIMSEQ:
092a3353241e add new standard proofs for limits of sequences
huffman
parents: 21842
diff changeset
   358
  "\<lbrakk>X ----> a; Y ----> b\<rbrakk> \<Longrightarrow> (\<lambda>n. X n ** Y n) ----> a ** b"
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents: 31336
diff changeset
   359
unfolding LIMSEQ_conv_tendsto by (rule tendsto)
22608
092a3353241e add new standard proofs for limits of sequences
huffman
parents: 21842
diff changeset
   360
092a3353241e add new standard proofs for limits of sequences
huffman
parents: 21842
diff changeset
   361
lemma LIMSEQ_mult:
092a3353241e add new standard proofs for limits of sequences
huffman
parents: 21842
diff changeset
   362
  fixes a b :: "'a::real_normed_algebra"
092a3353241e add new standard proofs for limits of sequences
huffman
parents: 21842
diff changeset
   363
  shows "[| X ----> a; Y ----> b |] ==> (%n. X n * Y n) ----> a * b"
23127
56ee8105c002 simplify names of locale interpretations
huffman
parents: 22998
diff changeset
   364
by (rule mult.LIMSEQ)
22608
092a3353241e add new standard proofs for limits of sequences
huffman
parents: 21842
diff changeset
   365
092a3353241e add new standard proofs for limits of sequences
huffman
parents: 21842
diff changeset
   366
lemma inverse_diff_inverse:
092a3353241e add new standard proofs for limits of sequences
huffman
parents: 21842
diff changeset
   367
  "\<lbrakk>(a::'a::division_ring) \<noteq> 0; b \<noteq> 0\<rbrakk>
092a3353241e add new standard proofs for limits of sequences
huffman
parents: 21842
diff changeset
   368
   \<Longrightarrow> inverse a - inverse b = - (inverse a * (a - b) * inverse b)"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29197
diff changeset
   369
by (simp add: algebra_simps)
22608
092a3353241e add new standard proofs for limits of sequences
huffman
parents: 21842
diff changeset
   370
092a3353241e add new standard proofs for limits of sequences
huffman
parents: 21842
diff changeset
   371
lemma Bseq_inverse_lemma:
092a3353241e add new standard proofs for limits of sequences
huffman
parents: 21842
diff changeset
   372
  fixes x :: "'a::real_normed_div_algebra"
092a3353241e add new standard proofs for limits of sequences
huffman
parents: 21842
diff changeset
   373
  shows "\<lbrakk>r \<le> norm x; 0 < r\<rbrakk> \<Longrightarrow> norm (inverse x) \<le> inverse r"
092a3353241e add new standard proofs for limits of sequences
huffman
parents: 21842
diff changeset
   374
apply (subst nonzero_norm_inverse, clarsimp)
092a3353241e add new standard proofs for limits of sequences
huffman
parents: 21842
diff changeset
   375
apply (erule (1) le_imp_inverse_le)
092a3353241e add new standard proofs for limits of sequences
huffman
parents: 21842
diff changeset
   376
done
092a3353241e add new standard proofs for limits of sequences
huffman
parents: 21842
diff changeset
   377
092a3353241e add new standard proofs for limits of sequences
huffman
parents: 21842
diff changeset
   378
lemma Bseq_inverse:
092a3353241e add new standard proofs for limits of sequences
huffman
parents: 21842
diff changeset
   379
  fixes a :: "'a::real_normed_div_algebra"
092a3353241e add new standard proofs for limits of sequences
huffman
parents: 21842
diff changeset
   380
  assumes X: "X ----> a"
092a3353241e add new standard proofs for limits of sequences
huffman
parents: 21842
diff changeset
   381
  assumes a: "a \<noteq> 0"
092a3353241e add new standard proofs for limits of sequences
huffman
parents: 21842
diff changeset
   382
  shows "Bseq (\<lambda>n. inverse (X n))"
092a3353241e add new standard proofs for limits of sequences
huffman
parents: 21842
diff changeset
   383
proof -
092a3353241e add new standard proofs for limits of sequences
huffman
parents: 21842
diff changeset
   384
  from a have "0 < norm a" by simp
092a3353241e add new standard proofs for limits of sequences
huffman
parents: 21842
diff changeset
   385
  hence "\<exists>r>0. r < norm a" by (rule dense)
092a3353241e add new standard proofs for limits of sequences
huffman
parents: 21842
diff changeset
   386
  then obtain r where r1: "0 < r" and r2: "r < norm a" by fast
092a3353241e add new standard proofs for limits of sequences
huffman
parents: 21842
diff changeset
   387
  obtain N where N: "\<And>n. N \<le> n \<Longrightarrow> norm (X n - a) < r"
092a3353241e add new standard proofs for limits of sequences
huffman
parents: 21842
diff changeset
   388
    using LIMSEQ_D [OF X r1] by fast
092a3353241e add new standard proofs for limits of sequences
huffman
parents: 21842
diff changeset
   389
  show ?thesis
26312
e9a65675e5e8 avoid rebinding of existing facts;
wenzelm
parents: 23482
diff changeset
   390
  proof (rule BseqI2' [rule_format])
22608
092a3353241e add new standard proofs for limits of sequences
huffman
parents: 21842
diff changeset
   391
    fix n assume n: "N \<le> n"
092a3353241e add new standard proofs for limits of sequences
huffman
parents: 21842
diff changeset
   392
    hence 1: "norm (X n - a) < r" by (rule N)
092a3353241e add new standard proofs for limits of sequences
huffman
parents: 21842
diff changeset
   393
    hence 2: "X n \<noteq> 0" using r2 by auto
092a3353241e add new standard proofs for limits of sequences
huffman
parents: 21842
diff changeset
   394
    hence "norm (inverse (X n)) = inverse (norm (X n))"
092a3353241e add new standard proofs for limits of sequences
huffman
parents: 21842
diff changeset
   395
      by (rule nonzero_norm_inverse)
092a3353241e add new standard proofs for limits of sequences
huffman
parents: 21842
diff changeset
   396
    also have "\<dots> \<le> inverse (norm a - r)"
092a3353241e add new standard proofs for limits of sequences
huffman
parents: 21842
diff changeset
   397
    proof (rule le_imp_inverse_le)
092a3353241e add new standard proofs for limits of sequences
huffman
parents: 21842
diff changeset
   398
      show "0 < norm a - r" using r2 by simp
092a3353241e add new standard proofs for limits of sequences
huffman
parents: 21842
diff changeset
   399
    next
092a3353241e add new standard proofs for limits of sequences
huffman
parents: 21842
diff changeset
   400
      have "norm a - norm (X n) \<le> norm (a - X n)"
092a3353241e add new standard proofs for limits of sequences
huffman
parents: 21842
diff changeset
   401
        by (rule norm_triangle_ineq2)
092a3353241e add new standard proofs for limits of sequences
huffman
parents: 21842
diff changeset
   402
      also have "\<dots> = norm (X n - a)"
092a3353241e add new standard proofs for limits of sequences
huffman
parents: 21842
diff changeset
   403
        by (rule norm_minus_commute)
092a3353241e add new standard proofs for limits of sequences
huffman
parents: 21842
diff changeset
   404
      also have "\<dots> < r" using 1 .
092a3353241e add new standard proofs for limits of sequences
huffman
parents: 21842
diff changeset
   405
      finally show "norm a - r \<le> norm (X n)" by simp
092a3353241e add new standard proofs for limits of sequences
huffman
parents: 21842
diff changeset
   406
    qed
092a3353241e add new standard proofs for limits of sequences
huffman
parents: 21842
diff changeset
   407
    finally show "norm (inverse (X n)) \<le> inverse (norm a - r)" .
092a3353241e add new standard proofs for limits of sequences
huffman
parents: 21842
diff changeset
   408
  qed
092a3353241e add new standard proofs for limits of sequences
huffman
parents: 21842
diff changeset
   409
qed
092a3353241e add new standard proofs for limits of sequences
huffman
parents: 21842
diff changeset
   410
092a3353241e add new standard proofs for limits of sequences
huffman
parents: 21842
diff changeset
   411
lemma LIMSEQ_inverse_lemma:
092a3353241e add new standard proofs for limits of sequences
huffman
parents: 21842
diff changeset
   412
  fixes a :: "'a::real_normed_div_algebra"
092a3353241e add new standard proofs for limits of sequences
huffman
parents: 21842
diff changeset
   413
  shows "\<lbrakk>X ----> a; a \<noteq> 0; \<forall>n. X n \<noteq> 0\<rbrakk>
092a3353241e add new standard proofs for limits of sequences
huffman
parents: 21842
diff changeset
   414
         \<Longrightarrow> (\<lambda>n. inverse (X n)) ----> inverse a"
092a3353241e add new standard proofs for limits of sequences
huffman
parents: 21842
diff changeset
   415
apply (subst LIMSEQ_Zseq_iff)
092a3353241e add new standard proofs for limits of sequences
huffman
parents: 21842
diff changeset
   416
apply (simp add: inverse_diff_inverse nonzero_imp_inverse_nonzero)
092a3353241e add new standard proofs for limits of sequences
huffman
parents: 21842
diff changeset
   417
apply (rule Zseq_minus)
092a3353241e add new standard proofs for limits of sequences
huffman
parents: 21842
diff changeset
   418
apply (rule Zseq_mult_left)
23127
56ee8105c002 simplify names of locale interpretations
huffman
parents: 22998
diff changeset
   419
apply (rule mult.Bseq_prod_Zseq)
22608
092a3353241e add new standard proofs for limits of sequences
huffman
parents: 21842
diff changeset
   420
apply (erule (1) Bseq_inverse)
092a3353241e add new standard proofs for limits of sequences
huffman
parents: 21842
diff changeset
   421
apply (simp add: LIMSEQ_Zseq_iff)
092a3353241e add new standard proofs for limits of sequences
huffman
parents: 21842
diff changeset
   422
done
092a3353241e add new standard proofs for limits of sequences
huffman
parents: 21842
diff changeset
   423
092a3353241e add new standard proofs for limits of sequences
huffman
parents: 21842
diff changeset
   424
lemma LIMSEQ_inverse:
092a3353241e add new standard proofs for limits of sequences
huffman
parents: 21842
diff changeset
   425
  fixes a :: "'a::real_normed_div_algebra"
092a3353241e add new standard proofs for limits of sequences
huffman
parents: 21842
diff changeset
   426
  assumes X: "X ----> a"
092a3353241e add new standard proofs for limits of sequences
huffman
parents: 21842
diff changeset
   427
  assumes a: "a \<noteq> 0"
092a3353241e add new standard proofs for limits of sequences
huffman
parents: 21842
diff changeset
   428
  shows "(\<lambda>n. inverse (X n)) ----> inverse a"
092a3353241e add new standard proofs for limits of sequences
huffman
parents: 21842
diff changeset
   429
proof -
092a3353241e add new standard proofs for limits of sequences
huffman
parents: 21842
diff changeset
   430
  from a have "0 < norm a" by simp
092a3353241e add new standard proofs for limits of sequences
huffman
parents: 21842
diff changeset
   431
  then obtain k where "\<forall>n\<ge>k. norm (X n - a) < norm a"
092a3353241e add new standard proofs for limits of sequences
huffman
parents: 21842
diff changeset
   432
    using LIMSEQ_D [OF X] by fast
092a3353241e add new standard proofs for limits of sequences
huffman
parents: 21842
diff changeset
   433
  hence "\<forall>n\<ge>k. X n \<noteq> 0" by auto
092a3353241e add new standard proofs for limits of sequences
huffman
parents: 21842
diff changeset
   434
  hence k: "\<forall>n. X (n + k) \<noteq> 0" by simp
092a3353241e add new standard proofs for limits of sequences
huffman
parents: 21842
diff changeset
   435
092a3353241e add new standard proofs for limits of sequences
huffman
parents: 21842
diff changeset
   436
  from X have "(\<lambda>n. X (n + k)) ----> a"
092a3353241e add new standard proofs for limits of sequences
huffman
parents: 21842
diff changeset
   437
    by (rule LIMSEQ_ignore_initial_segment)
092a3353241e add new standard proofs for limits of sequences
huffman
parents: 21842
diff changeset
   438
  hence "(\<lambda>n. inverse (X (n + k))) ----> inverse a"
092a3353241e add new standard proofs for limits of sequences
huffman
parents: 21842
diff changeset
   439
    using a k by (rule LIMSEQ_inverse_lemma)
092a3353241e add new standard proofs for limits of sequences
huffman
parents: 21842
diff changeset
   440
  thus "(\<lambda>n. inverse (X n)) ----> inverse a"
092a3353241e add new standard proofs for limits of sequences
huffman
parents: 21842
diff changeset
   441
    by (rule LIMSEQ_offset)
092a3353241e add new standard proofs for limits of sequences
huffman
parents: 21842
diff changeset
   442
qed
092a3353241e add new standard proofs for limits of sequences
huffman
parents: 21842
diff changeset
   443
092a3353241e add new standard proofs for limits of sequences
huffman
parents: 21842
diff changeset
   444
lemma LIMSEQ_divide:
092a3353241e add new standard proofs for limits of sequences
huffman
parents: 21842
diff changeset
   445
  fixes a b :: "'a::real_normed_field"
092a3353241e add new standard proofs for limits of sequences
huffman
parents: 21842
diff changeset
   446
  shows "\<lbrakk>X ----> a; Y ----> b; b \<noteq> 0\<rbrakk> \<Longrightarrow> (\<lambda>n. X n / Y n) ----> a / b"
092a3353241e add new standard proofs for limits of sequences
huffman
parents: 21842
diff changeset
   447
by (simp add: LIMSEQ_mult LIMSEQ_inverse divide_inverse)
092a3353241e add new standard proofs for limits of sequences
huffman
parents: 21842
diff changeset
   448
092a3353241e add new standard proofs for limits of sequences
huffman
parents: 21842
diff changeset
   449
lemma LIMSEQ_pow:
31017
2c227493ea56 stripped class recpower further
haftmann
parents: 30730
diff changeset
   450
  fixes a :: "'a::{power, real_normed_algebra}"
22608
092a3353241e add new standard proofs for limits of sequences
huffman
parents: 21842
diff changeset
   451
  shows "X ----> a \<Longrightarrow> (\<lambda>n. (X n) ^ m) ----> a ^ m"
30273
ecd6f0ca62ea declare power_Suc [simp]; remove redundant type-specific versions of power_Suc
huffman
parents: 30242
diff changeset
   452
by (induct m) (simp_all add: LIMSEQ_const LIMSEQ_mult)
22608
092a3353241e add new standard proofs for limits of sequences
huffman
parents: 21842
diff changeset
   453
092a3353241e add new standard proofs for limits of sequences
huffman
parents: 21842
diff changeset
   454
lemma LIMSEQ_setsum:
31336
e17f13cd1280 generalize constants in SEQ.thy to class metric_space
huffman
parents: 31017
diff changeset
   455
  fixes L :: "'a \<Rightarrow> 'b::real_normed_vector"
22608
092a3353241e add new standard proofs for limits of sequences
huffman
parents: 21842
diff changeset
   456
  assumes n: "\<And>n. n \<in> S \<Longrightarrow> X n ----> L n"
092a3353241e add new standard proofs for limits of sequences
huffman
parents: 21842
diff changeset
   457
  shows "(\<lambda>m. \<Sum>n\<in>S. X n m) ----> (\<Sum>n\<in>S. L n)"
092a3353241e add new standard proofs for limits of sequences
huffman
parents: 21842
diff changeset
   458
proof (cases "finite S")
092a3353241e add new standard proofs for limits of sequences
huffman
parents: 21842
diff changeset
   459
  case True
092a3353241e add new standard proofs for limits of sequences
huffman
parents: 21842
diff changeset
   460
  thus ?thesis using n
092a3353241e add new standard proofs for limits of sequences
huffman
parents: 21842
diff changeset
   461
  proof (induct)
092a3353241e add new standard proofs for limits of sequences
huffman
parents: 21842
diff changeset
   462
    case empty
092a3353241e add new standard proofs for limits of sequences
huffman
parents: 21842
diff changeset
   463
    show ?case
092a3353241e add new standard proofs for limits of sequences
huffman
parents: 21842
diff changeset
   464
      by (simp add: LIMSEQ_const)
092a3353241e add new standard proofs for limits of sequences
huffman
parents: 21842
diff changeset
   465
  next
092a3353241e add new standard proofs for limits of sequences
huffman
parents: 21842
diff changeset
   466
    case insert
092a3353241e add new standard proofs for limits of sequences
huffman
parents: 21842
diff changeset
   467
    thus ?case
092a3353241e add new standard proofs for limits of sequences
huffman
parents: 21842
diff changeset
   468
      by (simp add: LIMSEQ_add)
092a3353241e add new standard proofs for limits of sequences
huffman
parents: 21842
diff changeset
   469
  qed
092a3353241e add new standard proofs for limits of sequences
huffman
parents: 21842
diff changeset
   470
next
092a3353241e add new standard proofs for limits of sequences
huffman
parents: 21842
diff changeset
   471
  case False
092a3353241e add new standard proofs for limits of sequences
huffman
parents: 21842
diff changeset
   472
  thus ?thesis
092a3353241e add new standard proofs for limits of sequences
huffman
parents: 21842
diff changeset
   473
    by (simp add: LIMSEQ_const)
092a3353241e add new standard proofs for limits of sequences
huffman
parents: 21842
diff changeset
   474
qed
092a3353241e add new standard proofs for limits of sequences
huffman
parents: 21842
diff changeset
   475
092a3353241e add new standard proofs for limits of sequences
huffman
parents: 21842
diff changeset
   476
lemma LIMSEQ_setprod:
092a3353241e add new standard proofs for limits of sequences
huffman
parents: 21842
diff changeset
   477
  fixes L :: "'a \<Rightarrow> 'b::{real_normed_algebra,comm_ring_1}"
092a3353241e add new standard proofs for limits of sequences
huffman
parents: 21842
diff changeset
   478
  assumes n: "\<And>n. n \<in> S \<Longrightarrow> X n ----> L n"
092a3353241e add new standard proofs for limits of sequences
huffman
parents: 21842
diff changeset
   479
  shows "(\<lambda>m. \<Prod>n\<in>S. X n m) ----> (\<Prod>n\<in>S. L n)"
092a3353241e add new standard proofs for limits of sequences
huffman
parents: 21842
diff changeset
   480
proof (cases "finite S")
092a3353241e add new standard proofs for limits of sequences
huffman
parents: 21842
diff changeset
   481
  case True
092a3353241e add new standard proofs for limits of sequences
huffman
parents: 21842
diff changeset
   482
  thus ?thesis using n
092a3353241e add new standard proofs for limits of sequences
huffman
parents: 21842
diff changeset
   483
  proof (induct)
092a3353241e add new standard proofs for limits of sequences
huffman
parents: 21842
diff changeset
   484
    case empty
092a3353241e add new standard proofs for limits of sequences
huffman
parents: 21842
diff changeset
   485
    show ?case
092a3353241e add new standard proofs for limits of sequences
huffman
parents: 21842
diff changeset
   486
      by (simp add: LIMSEQ_const)
092a3353241e add new standard proofs for limits of sequences
huffman
parents: 21842
diff changeset
   487
  next
092a3353241e add new standard proofs for limits of sequences
huffman
parents: 21842
diff changeset
   488
    case insert
092a3353241e add new standard proofs for limits of sequences
huffman
parents: 21842
diff changeset
   489
    thus ?case
092a3353241e add new standard proofs for limits of sequences
huffman
parents: 21842
diff changeset
   490
      by (simp add: LIMSEQ_mult)
092a3353241e add new standard proofs for limits of sequences
huffman
parents: 21842
diff changeset
   491
  qed
092a3353241e add new standard proofs for limits of sequences
huffman
parents: 21842
diff changeset
   492
next
092a3353241e add new standard proofs for limits of sequences
huffman
parents: 21842
diff changeset
   493
  case False
092a3353241e add new standard proofs for limits of sequences
huffman
parents: 21842
diff changeset
   494
  thus ?thesis
092a3353241e add new standard proofs for limits of sequences
huffman
parents: 21842
diff changeset
   495
    by (simp add: setprod_def LIMSEQ_const)
092a3353241e add new standard proofs for limits of sequences
huffman
parents: 21842
diff changeset
   496
qed
092a3353241e add new standard proofs for limits of sequences
huffman
parents: 21842
diff changeset
   497
31336
e17f13cd1280 generalize constants in SEQ.thy to class metric_space
huffman
parents: 31017
diff changeset
   498
lemma LIMSEQ_add_const:
e17f13cd1280 generalize constants in SEQ.thy to class metric_space
huffman
parents: 31017
diff changeset
   499
  fixes a :: "'a::real_normed_vector"
e17f13cd1280 generalize constants in SEQ.thy to class metric_space
huffman
parents: 31017
diff changeset
   500
  shows "f ----> a ==> (%n.(f n + b)) ----> a + b"
22614
17644bc9cee4 rearranged sections
huffman
parents: 22608
diff changeset
   501
by (simp add: LIMSEQ_add LIMSEQ_const)
17644bc9cee4 rearranged sections
huffman
parents: 22608
diff changeset
   502
17644bc9cee4 rearranged sections
huffman
parents: 22608
diff changeset
   503
(* FIXME: delete *)
17644bc9cee4 rearranged sections
huffman
parents: 22608
diff changeset
   504
lemma LIMSEQ_add_minus:
31336
e17f13cd1280 generalize constants in SEQ.thy to class metric_space
huffman
parents: 31017
diff changeset
   505
  fixes a b :: "'a::real_normed_vector"
e17f13cd1280 generalize constants in SEQ.thy to class metric_space
huffman
parents: 31017
diff changeset
   506
  shows "[| X ----> a; Y ----> b |] ==> (%n. X n + -Y n) ----> a + -b"
22614
17644bc9cee4 rearranged sections
huffman
parents: 22608
diff changeset
   507
by (simp only: LIMSEQ_add LIMSEQ_minus)
17644bc9cee4 rearranged sections
huffman
parents: 22608
diff changeset
   508
31336
e17f13cd1280 generalize constants in SEQ.thy to class metric_space
huffman
parents: 31017
diff changeset
   509
lemma LIMSEQ_diff_const:
e17f13cd1280 generalize constants in SEQ.thy to class metric_space
huffman
parents: 31017
diff changeset
   510
  fixes a b :: "'a::real_normed_vector"
e17f13cd1280 generalize constants in SEQ.thy to class metric_space
huffman
parents: 31017
diff changeset
   511
  shows "f ----> a ==> (%n.(f n  - b)) ----> a - b"
22614
17644bc9cee4 rearranged sections
huffman
parents: 22608
diff changeset
   512
by (simp add: LIMSEQ_diff LIMSEQ_const)
17644bc9cee4 rearranged sections
huffman
parents: 22608
diff changeset
   513
31336
e17f13cd1280 generalize constants in SEQ.thy to class metric_space
huffman
parents: 31017
diff changeset
   514
lemma LIMSEQ_diff_approach_zero:
e17f13cd1280 generalize constants in SEQ.thy to class metric_space
huffman
parents: 31017
diff changeset
   515
  fixes L :: "'a::real_normed_vector"
e17f13cd1280 generalize constants in SEQ.thy to class metric_space
huffman
parents: 31017
diff changeset
   516
  shows "g ----> L ==> (%x. f x - g x) ----> 0 ==> f ----> L"
e17f13cd1280 generalize constants in SEQ.thy to class metric_space
huffman
parents: 31017
diff changeset
   517
by (drule (1) LIMSEQ_add, simp)
22614
17644bc9cee4 rearranged sections
huffman
parents: 22608
diff changeset
   518
31336
e17f13cd1280 generalize constants in SEQ.thy to class metric_space
huffman
parents: 31017
diff changeset
   519
lemma LIMSEQ_diff_approach_zero2:
e17f13cd1280 generalize constants in SEQ.thy to class metric_space
huffman
parents: 31017
diff changeset
   520
  fixes L :: "'a::real_normed_vector"
e17f13cd1280 generalize constants in SEQ.thy to class metric_space
huffman
parents: 31017
diff changeset
   521
  shows "f ----> L ==> (%x. f x - g x) ----> 0 ==> g ----> L";
e17f13cd1280 generalize constants in SEQ.thy to class metric_space
huffman
parents: 31017
diff changeset
   522
by (drule (1) LIMSEQ_diff, simp)
22614
17644bc9cee4 rearranged sections
huffman
parents: 22608
diff changeset
   523
17644bc9cee4 rearranged sections
huffman
parents: 22608
diff changeset
   524
text{*A sequence tends to zero iff its abs does*}
31336
e17f13cd1280 generalize constants in SEQ.thy to class metric_space
huffman
parents: 31017
diff changeset
   525
lemma LIMSEQ_norm_zero:
e17f13cd1280 generalize constants in SEQ.thy to class metric_space
huffman
parents: 31017
diff changeset
   526
  fixes X :: "nat \<Rightarrow> 'a::real_normed_vector"
e17f13cd1280 generalize constants in SEQ.thy to class metric_space
huffman
parents: 31017
diff changeset
   527
  shows "((\<lambda>n. norm (X n)) ----> 0) \<longleftrightarrow> (X ----> 0)"
e17f13cd1280 generalize constants in SEQ.thy to class metric_space
huffman
parents: 31017
diff changeset
   528
by (simp add: LIMSEQ_iff)
22614
17644bc9cee4 rearranged sections
huffman
parents: 22608
diff changeset
   529
17644bc9cee4 rearranged sections
huffman
parents: 22608
diff changeset
   530
lemma LIMSEQ_rabs_zero: "((%n. \<bar>f n\<bar>) ----> 0) = (f ----> (0::real))"
31336
e17f13cd1280 generalize constants in SEQ.thy to class metric_space
huffman
parents: 31017
diff changeset
   531
by (simp add: LIMSEQ_iff)
22614
17644bc9cee4 rearranged sections
huffman
parents: 22608
diff changeset
   532
17644bc9cee4 rearranged sections
huffman
parents: 22608
diff changeset
   533
lemma LIMSEQ_imp_rabs: "f ----> (l::real) ==> (%n. \<bar>f n\<bar>) ----> \<bar>l\<bar>"
17644bc9cee4 rearranged sections
huffman
parents: 22608
diff changeset
   534
by (drule LIMSEQ_norm, simp)
17644bc9cee4 rearranged sections
huffman
parents: 22608
diff changeset
   535
17644bc9cee4 rearranged sections
huffman
parents: 22608
diff changeset
   536
text{*An unbounded sequence's inverse tends to 0*}
17644bc9cee4 rearranged sections
huffman
parents: 22608
diff changeset
   537
17644bc9cee4 rearranged sections
huffman
parents: 22608
diff changeset
   538
lemma LIMSEQ_inverse_zero:
22974
08b0fa905ea0 tuned proofs
huffman
parents: 22631
diff changeset
   539
  "\<forall>r::real. \<exists>N. \<forall>n\<ge>N. r < X n \<Longrightarrow> (\<lambda>n. inverse (X n)) ----> 0"
08b0fa905ea0 tuned proofs
huffman
parents: 22631
diff changeset
   540
apply (rule LIMSEQ_I)
08b0fa905ea0 tuned proofs
huffman
parents: 22631
diff changeset
   541
apply (drule_tac x="inverse r" in spec, safe)
08b0fa905ea0 tuned proofs
huffman
parents: 22631
diff changeset
   542
apply (rule_tac x="N" in exI, safe)
08b0fa905ea0 tuned proofs
huffman
parents: 22631
diff changeset
   543
apply (drule_tac x="n" in spec, safe)
22614
17644bc9cee4 rearranged sections
huffman
parents: 22608
diff changeset
   544
apply (frule positive_imp_inverse_positive)
22974
08b0fa905ea0 tuned proofs
huffman
parents: 22631
diff changeset
   545
apply (frule (1) less_imp_inverse_less)
08b0fa905ea0 tuned proofs
huffman
parents: 22631
diff changeset
   546
apply (subgoal_tac "0 < X n", simp)
08b0fa905ea0 tuned proofs
huffman
parents: 22631
diff changeset
   547
apply (erule (1) order_less_trans)
22614
17644bc9cee4 rearranged sections
huffman
parents: 22608
diff changeset
   548
done
17644bc9cee4 rearranged sections
huffman
parents: 22608
diff changeset
   549
17644bc9cee4 rearranged sections
huffman
parents: 22608
diff changeset
   550
text{*The sequence @{term "1/n"} tends to 0 as @{term n} tends to infinity*}
17644bc9cee4 rearranged sections
huffman
parents: 22608
diff changeset
   551
17644bc9cee4 rearranged sections
huffman
parents: 22608
diff changeset
   552
lemma LIMSEQ_inverse_real_of_nat: "(%n. inverse(real(Suc n))) ----> 0"
17644bc9cee4 rearranged sections
huffman
parents: 22608
diff changeset
   553
apply (rule LIMSEQ_inverse_zero, safe)
22974
08b0fa905ea0 tuned proofs
huffman
parents: 22631
diff changeset
   554
apply (cut_tac x = r in reals_Archimedean2)
22614
17644bc9cee4 rearranged sections
huffman
parents: 22608
diff changeset
   555
apply (safe, rule_tac x = n in exI)
17644bc9cee4 rearranged sections
huffman
parents: 22608
diff changeset
   556
apply (auto simp add: real_of_nat_Suc)
17644bc9cee4 rearranged sections
huffman
parents: 22608
diff changeset
   557
done
17644bc9cee4 rearranged sections
huffman
parents: 22608
diff changeset
   558
17644bc9cee4 rearranged sections
huffman
parents: 22608
diff changeset
   559
text{*The sequence @{term "r + 1/n"} tends to @{term r} as @{term n} tends to
17644bc9cee4 rearranged sections
huffman
parents: 22608
diff changeset
   560
infinity is now easily proved*}
17644bc9cee4 rearranged sections
huffman
parents: 22608
diff changeset
   561
17644bc9cee4 rearranged sections
huffman
parents: 22608
diff changeset
   562
lemma LIMSEQ_inverse_real_of_nat_add:
17644bc9cee4 rearranged sections
huffman
parents: 22608
diff changeset
   563
     "(%n. r + inverse(real(Suc n))) ----> r"
17644bc9cee4 rearranged sections
huffman
parents: 22608
diff changeset
   564
by (cut_tac LIMSEQ_add [OF LIMSEQ_const LIMSEQ_inverse_real_of_nat], auto)
17644bc9cee4 rearranged sections
huffman
parents: 22608
diff changeset
   565
17644bc9cee4 rearranged sections
huffman
parents: 22608
diff changeset
   566
lemma LIMSEQ_inverse_real_of_nat_add_minus:
17644bc9cee4 rearranged sections
huffman
parents: 22608
diff changeset
   567
     "(%n. r + -inverse(real(Suc n))) ----> r"
17644bc9cee4 rearranged sections
huffman
parents: 22608
diff changeset
   568
by (cut_tac LIMSEQ_add_minus [OF LIMSEQ_const LIMSEQ_inverse_real_of_nat], auto)
17644bc9cee4 rearranged sections
huffman
parents: 22608
diff changeset
   569
17644bc9cee4 rearranged sections
huffman
parents: 22608
diff changeset
   570
lemma LIMSEQ_inverse_real_of_nat_add_minus_mult:
17644bc9cee4 rearranged sections
huffman
parents: 22608
diff changeset
   571
     "(%n. r*( 1 + -inverse(real(Suc n)))) ----> r"
17644bc9cee4 rearranged sections
huffman
parents: 22608
diff changeset
   572
by (cut_tac b=1 in
17644bc9cee4 rearranged sections
huffman
parents: 22608
diff changeset
   573
        LIMSEQ_mult [OF LIMSEQ_const LIMSEQ_inverse_real_of_nat_add_minus], auto)
17644bc9cee4 rearranged sections
huffman
parents: 22608
diff changeset
   574
22615
d650e51b5970 new standard proofs of some LIMSEQ lemmas
huffman
parents: 22614
diff changeset
   575
lemma LIMSEQ_le_const:
d650e51b5970 new standard proofs of some LIMSEQ lemmas
huffman
parents: 22614
diff changeset
   576
  "\<lbrakk>X ----> (x::real); \<exists>N. \<forall>n\<ge>N. a \<le> X n\<rbrakk> \<Longrightarrow> a \<le> x"
d650e51b5970 new standard proofs of some LIMSEQ lemmas
huffman
parents: 22614
diff changeset
   577
apply (rule ccontr, simp only: linorder_not_le)
d650e51b5970 new standard proofs of some LIMSEQ lemmas
huffman
parents: 22614
diff changeset
   578
apply (drule_tac r="a - x" in LIMSEQ_D, simp)
d650e51b5970 new standard proofs of some LIMSEQ lemmas
huffman
parents: 22614
diff changeset
   579
apply clarsimp
d650e51b5970 new standard proofs of some LIMSEQ lemmas
huffman
parents: 22614
diff changeset
   580
apply (drule_tac x="max N no" in spec, drule mp, rule le_maxI1)
d650e51b5970 new standard proofs of some LIMSEQ lemmas
huffman
parents: 22614
diff changeset
   581
apply (drule_tac x="max N no" in spec, drule mp, rule le_maxI2)
d650e51b5970 new standard proofs of some LIMSEQ lemmas
huffman
parents: 22614
diff changeset
   582
apply simp
d650e51b5970 new standard proofs of some LIMSEQ lemmas
huffman
parents: 22614
diff changeset
   583
done
d650e51b5970 new standard proofs of some LIMSEQ lemmas
huffman
parents: 22614
diff changeset
   584
d650e51b5970 new standard proofs of some LIMSEQ lemmas
huffman
parents: 22614
diff changeset
   585
lemma LIMSEQ_le_const2:
d650e51b5970 new standard proofs of some LIMSEQ lemmas
huffman
parents: 22614
diff changeset
   586
  "\<lbrakk>X ----> (x::real); \<exists>N. \<forall>n\<ge>N. X n \<le> a\<rbrakk> \<Longrightarrow> x \<le> a"
d650e51b5970 new standard proofs of some LIMSEQ lemmas
huffman
parents: 22614
diff changeset
   587
apply (subgoal_tac "- a \<le> - x", simp)
d650e51b5970 new standard proofs of some LIMSEQ lemmas
huffman
parents: 22614
diff changeset
   588
apply (rule LIMSEQ_le_const)
d650e51b5970 new standard proofs of some LIMSEQ lemmas
huffman
parents: 22614
diff changeset
   589
apply (erule LIMSEQ_minus)
d650e51b5970 new standard proofs of some LIMSEQ lemmas
huffman
parents: 22614
diff changeset
   590
apply simp
d650e51b5970 new standard proofs of some LIMSEQ lemmas
huffman
parents: 22614
diff changeset
   591
done
d650e51b5970 new standard proofs of some LIMSEQ lemmas
huffman
parents: 22614
diff changeset
   592
d650e51b5970 new standard proofs of some LIMSEQ lemmas
huffman
parents: 22614
diff changeset
   593
lemma LIMSEQ_le:
d650e51b5970 new standard proofs of some LIMSEQ lemmas
huffman
parents: 22614
diff changeset
   594
  "\<lbrakk>X ----> x; Y ----> y; \<exists>N. \<forall>n\<ge>N. X n \<le> Y n\<rbrakk> \<Longrightarrow> x \<le> (y::real)"
d650e51b5970 new standard proofs of some LIMSEQ lemmas
huffman
parents: 22614
diff changeset
   595
apply (subgoal_tac "0 \<le> y - x", simp)
d650e51b5970 new standard proofs of some LIMSEQ lemmas
huffman
parents: 22614
diff changeset
   596
apply (rule LIMSEQ_le_const)
d650e51b5970 new standard proofs of some LIMSEQ lemmas
huffman
parents: 22614
diff changeset
   597
apply (erule (1) LIMSEQ_diff)
d650e51b5970 new standard proofs of some LIMSEQ lemmas
huffman
parents: 22614
diff changeset
   598
apply (simp add: le_diff_eq)
d650e51b5970 new standard proofs of some LIMSEQ lemmas
huffman
parents: 22614
diff changeset
   599
done
d650e51b5970 new standard proofs of some LIMSEQ lemmas
huffman
parents: 22614
diff changeset
   600
15082
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
   601
20696
3b887ad7d196 reorganized subsection headings
huffman
parents: 20695
diff changeset
   602
subsection {* Convergence *}
15082
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
   603
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
   604
lemma limI: "X ----> L ==> lim X = L"
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
   605
apply (simp add: lim_def)
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
   606
apply (blast intro: LIMSEQ_unique)
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
   607
done
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
   608
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
   609
lemma convergentD: "convergent X ==> \<exists>L. (X ----> L)"
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
   610
by (simp add: convergent_def)
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
   611
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
   612
lemma convergentI: "(X ----> L) ==> convergent X"
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
   613
by (auto simp add: convergent_def)
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
   614
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
   615
lemma convergent_LIMSEQ_iff: "convergent X = (X ----> lim X)"
20682
cecff1f51431 define constants with THE instead of SOME
huffman
parents: 20653
diff changeset
   616
by (auto intro: theI LIMSEQ_unique simp add: convergent_def lim_def)
15082
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
   617
31336
e17f13cd1280 generalize constants in SEQ.thy to class metric_space
huffman
parents: 31017
diff changeset
   618
lemma convergent_minus_iff:
e17f13cd1280 generalize constants in SEQ.thy to class metric_space
huffman
parents: 31017
diff changeset
   619
  fixes X :: "nat \<Rightarrow> 'a::real_normed_vector"
e17f13cd1280 generalize constants in SEQ.thy to class metric_space
huffman
parents: 31017
diff changeset
   620
  shows "convergent X \<longleftrightarrow> convergent (\<lambda>n. - X n)"
20696
3b887ad7d196 reorganized subsection headings
huffman
parents: 20695
diff changeset
   621
apply (simp add: convergent_def)
3b887ad7d196 reorganized subsection headings
huffman
parents: 20695
diff changeset
   622
apply (auto dest: LIMSEQ_minus)
3b887ad7d196 reorganized subsection headings
huffman
parents: 20695
diff changeset
   623
apply (drule LIMSEQ_minus, auto)
3b887ad7d196 reorganized subsection headings
huffman
parents: 20695
diff changeset
   624
done
3b887ad7d196 reorganized subsection headings
huffman
parents: 20695
diff changeset
   625
30196
6ffaa79c352c Moved a few theorems about monotonic sequences from Fundamental_Theorem_Algebra to SEQ.thy
chaieb
parents: 30082
diff changeset
   626
text{* Given a binary function @{text "f:: nat \<Rightarrow> 'a \<Rightarrow> 'a"}, its values are uniquely determined by a function g *}
20696
3b887ad7d196 reorganized subsection headings
huffman
parents: 20695
diff changeset
   627
30196
6ffaa79c352c Moved a few theorems about monotonic sequences from Fundamental_Theorem_Algebra to SEQ.thy
chaieb
parents: 30082
diff changeset
   628
lemma nat_function_unique: "EX! g. g 0 = e \<and> (\<forall>n. g (Suc n) = f n (g n))"
6ffaa79c352c Moved a few theorems about monotonic sequences from Fundamental_Theorem_Algebra to SEQ.thy
chaieb
parents: 30082
diff changeset
   629
  unfolding Ex1_def
6ffaa79c352c Moved a few theorems about monotonic sequences from Fundamental_Theorem_Algebra to SEQ.thy
chaieb
parents: 30082
diff changeset
   630
  apply (rule_tac x="nat_rec e f" in exI)
6ffaa79c352c Moved a few theorems about monotonic sequences from Fundamental_Theorem_Algebra to SEQ.thy
chaieb
parents: 30082
diff changeset
   631
  apply (rule conjI)+
6ffaa79c352c Moved a few theorems about monotonic sequences from Fundamental_Theorem_Algebra to SEQ.thy
chaieb
parents: 30082
diff changeset
   632
apply (rule def_nat_rec_0, simp)
6ffaa79c352c Moved a few theorems about monotonic sequences from Fundamental_Theorem_Algebra to SEQ.thy
chaieb
parents: 30082
diff changeset
   633
apply (rule allI, rule def_nat_rec_Suc, simp)
6ffaa79c352c Moved a few theorems about monotonic sequences from Fundamental_Theorem_Algebra to SEQ.thy
chaieb
parents: 30082
diff changeset
   634
apply (rule allI, rule impI, rule ext)
6ffaa79c352c Moved a few theorems about monotonic sequences from Fundamental_Theorem_Algebra to SEQ.thy
chaieb
parents: 30082
diff changeset
   635
apply (erule conjE)
6ffaa79c352c Moved a few theorems about monotonic sequences from Fundamental_Theorem_Algebra to SEQ.thy
chaieb
parents: 30082
diff changeset
   636
apply (induct_tac x)
6ffaa79c352c Moved a few theorems about monotonic sequences from Fundamental_Theorem_Algebra to SEQ.thy
chaieb
parents: 30082
diff changeset
   637
apply (simp add: nat_rec_0)
6ffaa79c352c Moved a few theorems about monotonic sequences from Fundamental_Theorem_Algebra to SEQ.thy
chaieb
parents: 30082
diff changeset
   638
apply (erule_tac x="n" in allE)
6ffaa79c352c Moved a few theorems about monotonic sequences from Fundamental_Theorem_Algebra to SEQ.thy
chaieb
parents: 30082
diff changeset
   639
apply (simp)
6ffaa79c352c Moved a few theorems about monotonic sequences from Fundamental_Theorem_Algebra to SEQ.thy
chaieb
parents: 30082
diff changeset
   640
done
20696
3b887ad7d196 reorganized subsection headings
huffman
parents: 20695
diff changeset
   641
15082
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
   642
text{*Subsequence (alternative definition, (e.g. Hoskins)*}
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
   643
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
   644
lemma subseq_Suc_iff: "subseq f = (\<forall>n. (f n) < (f (Suc n)))"
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
   645
apply (simp add: subseq_def)
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
   646
apply (auto dest!: less_imp_Suc_add)
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
   647
apply (induct_tac k)
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
   648
apply (auto intro: less_trans)
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
   649
done
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
   650
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
   651
lemma monoseq_Suc:
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
   652
   "monoseq X = ((\<forall>n. X n \<le> X (Suc n))
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
   653
                 | (\<forall>n. X (Suc n) \<le> X n))"
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
   654
apply (simp add: monoseq_def)
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
   655
apply (auto dest!: le_imp_less_or_eq)
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
   656
apply (auto intro!: lessI [THEN less_imp_le] dest!: less_imp_Suc_add)
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
   657
apply (induct_tac "ka")
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
   658
apply (auto intro: order_trans)
18585
5d379fe2eb74 replaced swap by contrapos_np;
wenzelm
parents: 17439
diff changeset
   659
apply (erule contrapos_np)
15082
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
   660
apply (induct_tac "k")
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
   661
apply (auto intro: order_trans)
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
   662
done
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
   663
15360
300e09825d8b Added "ALL x > y" and relatives.
nipkow
parents: 15312
diff changeset
   664
lemma monoI1: "\<forall>m. \<forall> n \<ge> m. X m \<le> X n ==> monoseq X"
15082
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
   665
by (simp add: monoseq_def)
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
   666
15360
300e09825d8b Added "ALL x > y" and relatives.
nipkow
parents: 15312
diff changeset
   667
lemma monoI2: "\<forall>m. \<forall> n \<ge> m. X n \<le> X m ==> monoseq X"
15082
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
   668
by (simp add: monoseq_def)
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
   669
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
   670
lemma mono_SucI1: "\<forall>n. X n \<le> X (Suc n) ==> monoseq X"
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
   671
by (simp add: monoseq_Suc)
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
   672
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
   673
lemma mono_SucI2: "\<forall>n. X (Suc n) \<le> X n ==> monoseq X"
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
   674
by (simp add: monoseq_Suc)
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
   675
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
   676
lemma monoseq_minus: assumes "monoseq a"
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
   677
  shows "monoseq (\<lambda> n. - a n)"
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
   678
proof (cases "\<forall> m. \<forall> n \<ge> m. a m \<le> a n")
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
   679
  case True
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
   680
  hence "\<forall> m. \<forall> n \<ge> m. - a n \<le> - a m" by auto
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
   681
  thus ?thesis by (rule monoI2)
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
   682
next
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
   683
  case False
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
   684
  hence "\<forall> m. \<forall> n \<ge> m. - a m \<le> - a n" using `monoseq a`[unfolded monoseq_def] by auto
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
   685
  thus ?thesis by (rule monoI1)
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
   686
qed
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
   687
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
   688
lemma monoseq_le: assumes "monoseq a" and "a ----> x"
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
   689
  shows "((\<forall> n. a n \<le> x) \<and> (\<forall>m. \<forall>n\<ge>m. a m \<le> a n)) \<or> 
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
   690
         ((\<forall> n. x \<le> a n) \<and> (\<forall>m. \<forall>n\<ge>m. a n \<le> a m))"
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
   691
proof -
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
   692
  { fix x n fix a :: "nat \<Rightarrow> real"
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
   693
    assume "a ----> x" and "\<forall> m. \<forall> n \<ge> m. a m \<le> a n"
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
   694
    hence monotone: "\<And> m n. m \<le> n \<Longrightarrow> a m \<le> a n" by auto
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
   695
    have "a n \<le> x"
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
   696
    proof (rule ccontr)
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
   697
      assume "\<not> a n \<le> x" hence "x < a n" by auto
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
   698
      hence "0 < a n - x" by auto
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
   699
      from `a ----> x`[THEN LIMSEQ_D, OF this]
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
   700
      obtain no where "\<And>n'. no \<le> n' \<Longrightarrow> norm (a n' - x) < a n - x" by blast
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
   701
      hence "norm (a (max no n) - x) < a n - x" by auto
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
   702
      moreover
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
   703
      { fix n' have "n \<le> n' \<Longrightarrow> x < a n'" using monotone[where m=n and n=n'] and `x < a n` by auto }
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
   704
      hence "x < a (max no n)" by auto
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
   705
      ultimately
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
   706
      have "a (max no n) < a n" by auto
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
   707
      with monotone[where m=n and n="max no n"]
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
   708
      show False by auto
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
   709
    qed
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
   710
  } note top_down = this
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
   711
  { fix x n m fix a :: "nat \<Rightarrow> real"
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
   712
    assume "a ----> x" and "monoseq a" and "a m < x"
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
   713
    have "a n \<le> x \<and> (\<forall> m. \<forall> n \<ge> m. a m \<le> a n)"
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
   714
    proof (cases "\<forall> m. \<forall> n \<ge> m. a m \<le> a n")
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
   715
      case True with top_down and `a ----> x` show ?thesis by auto
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
   716
    next
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
   717
      case False with `monoseq a`[unfolded monoseq_def] have "\<forall> m. \<forall> n \<ge> m. - a m \<le> - a n" by auto
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
   718
      hence "- a m \<le> - x" using top_down[OF LIMSEQ_minus[OF `a ----> x`]] by blast
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
   719
      hence False using `a m < x` by auto
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
   720
      thus ?thesis ..
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
   721
    qed
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
   722
  } note when_decided = this
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
   723
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
   724
  show ?thesis
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
   725
  proof (cases "\<exists> m. a m \<noteq> x")
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
   726
    case True then obtain m where "a m \<noteq> x" by auto
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
   727
    show ?thesis
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
   728
    proof (cases "a m < x")
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
   729
      case True with when_decided[OF `a ----> x` `monoseq a`, where m2=m]
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
   730
      show ?thesis by blast
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
   731
    next
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
   732
      case False hence "- a m < - x" using `a m \<noteq> x` by auto
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
   733
      with when_decided[OF LIMSEQ_minus[OF `a ----> x`] monoseq_minus[OF `monoseq a`], where m2=m]
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
   734
      show ?thesis by auto
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
   735
    qed
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
   736
  qed auto
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
   737
qed
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
   738
30196
6ffaa79c352c Moved a few theorems about monotonic sequences from Fundamental_Theorem_Algebra to SEQ.thy
chaieb
parents: 30082
diff changeset
   739
text{* for any sequence, there is a mootonic subsequence *}
6ffaa79c352c Moved a few theorems about monotonic sequences from Fundamental_Theorem_Algebra to SEQ.thy
chaieb
parents: 30082
diff changeset
   740
lemma seq_monosub: "\<exists>f. subseq f \<and> monoseq (\<lambda> n. (s (f n)))"
6ffaa79c352c Moved a few theorems about monotonic sequences from Fundamental_Theorem_Algebra to SEQ.thy
chaieb
parents: 30082
diff changeset
   741
proof-
6ffaa79c352c Moved a few theorems about monotonic sequences from Fundamental_Theorem_Algebra to SEQ.thy
chaieb
parents: 30082
diff changeset
   742
  {assume H: "\<forall>n. \<exists>p >n. \<forall> m\<ge>p. s m \<le> s p"
6ffaa79c352c Moved a few theorems about monotonic sequences from Fundamental_Theorem_Algebra to SEQ.thy
chaieb
parents: 30082
diff changeset
   743
    let ?P = "\<lambda> p n. p > n \<and> (\<forall>m \<ge> p. s m \<le> s p)"
6ffaa79c352c Moved a few theorems about monotonic sequences from Fundamental_Theorem_Algebra to SEQ.thy
chaieb
parents: 30082
diff changeset
   744
    from nat_function_unique[of "SOME p. ?P p 0" "\<lambda>p n. SOME p. ?P p n"]
6ffaa79c352c Moved a few theorems about monotonic sequences from Fundamental_Theorem_Algebra to SEQ.thy
chaieb
parents: 30082
diff changeset
   745
    obtain f where f: "f 0 = (SOME p. ?P p 0)" "\<forall>n. f (Suc n) = (SOME p. ?P p (f n))" by blast
6ffaa79c352c Moved a few theorems about monotonic sequences from Fundamental_Theorem_Algebra to SEQ.thy
chaieb
parents: 30082
diff changeset
   746
    have "?P (f 0) 0"  unfolding f(1) some_eq_ex[of "\<lambda>p. ?P p 0"]
6ffaa79c352c Moved a few theorems about monotonic sequences from Fundamental_Theorem_Algebra to SEQ.thy
chaieb
parents: 30082
diff changeset
   747
      using H apply - 
6ffaa79c352c Moved a few theorems about monotonic sequences from Fundamental_Theorem_Algebra to SEQ.thy
chaieb
parents: 30082
diff changeset
   748
      apply (erule allE[where x=0], erule exE, rule_tac x="p" in exI) 
6ffaa79c352c Moved a few theorems about monotonic sequences from Fundamental_Theorem_Algebra to SEQ.thy
chaieb
parents: 30082
diff changeset
   749
      unfolding order_le_less by blast 
6ffaa79c352c Moved a few theorems about monotonic sequences from Fundamental_Theorem_Algebra to SEQ.thy
chaieb
parents: 30082
diff changeset
   750
    hence f0: "f 0 > 0" "\<forall>m \<ge> f 0. s m \<le> s (f 0)" by blast+
6ffaa79c352c Moved a few theorems about monotonic sequences from Fundamental_Theorem_Algebra to SEQ.thy
chaieb
parents: 30082
diff changeset
   751
    {fix n
6ffaa79c352c Moved a few theorems about monotonic sequences from Fundamental_Theorem_Algebra to SEQ.thy
chaieb
parents: 30082
diff changeset
   752
      have "?P (f (Suc n)) (f n)" 
6ffaa79c352c Moved a few theorems about monotonic sequences from Fundamental_Theorem_Algebra to SEQ.thy
chaieb
parents: 30082
diff changeset
   753
	unfolding f(2)[rule_format, of n] some_eq_ex[of "\<lambda>p. ?P p (f n)"]
6ffaa79c352c Moved a few theorems about monotonic sequences from Fundamental_Theorem_Algebra to SEQ.thy
chaieb
parents: 30082
diff changeset
   754
	using H apply - 
6ffaa79c352c Moved a few theorems about monotonic sequences from Fundamental_Theorem_Algebra to SEQ.thy
chaieb
parents: 30082
diff changeset
   755
      apply (erule allE[where x="f n"], erule exE, rule_tac x="p" in exI) 
6ffaa79c352c Moved a few theorems about monotonic sequences from Fundamental_Theorem_Algebra to SEQ.thy
chaieb
parents: 30082
diff changeset
   756
      unfolding order_le_less by blast 
6ffaa79c352c Moved a few theorems about monotonic sequences from Fundamental_Theorem_Algebra to SEQ.thy
chaieb
parents: 30082
diff changeset
   757
    hence "f (Suc n) > f n" "\<forall>m \<ge> f (Suc n). s m \<le> s (f (Suc n))" by blast+}
6ffaa79c352c Moved a few theorems about monotonic sequences from Fundamental_Theorem_Algebra to SEQ.thy
chaieb
parents: 30082
diff changeset
   758
  note fSuc = this
6ffaa79c352c Moved a few theorems about monotonic sequences from Fundamental_Theorem_Algebra to SEQ.thy
chaieb
parents: 30082
diff changeset
   759
    {fix p q assume pq: "p \<ge> f q"
6ffaa79c352c Moved a few theorems about monotonic sequences from Fundamental_Theorem_Algebra to SEQ.thy
chaieb
parents: 30082
diff changeset
   760
      have "s p \<le> s(f(q))"  using f0(2)[rule_format, of p] pq fSuc
6ffaa79c352c Moved a few theorems about monotonic sequences from Fundamental_Theorem_Algebra to SEQ.thy
chaieb
parents: 30082
diff changeset
   761
	by (cases q, simp_all) }
6ffaa79c352c Moved a few theorems about monotonic sequences from Fundamental_Theorem_Algebra to SEQ.thy
chaieb
parents: 30082
diff changeset
   762
    note pqth = this
6ffaa79c352c Moved a few theorems about monotonic sequences from Fundamental_Theorem_Algebra to SEQ.thy
chaieb
parents: 30082
diff changeset
   763
    {fix q
6ffaa79c352c Moved a few theorems about monotonic sequences from Fundamental_Theorem_Algebra to SEQ.thy
chaieb
parents: 30082
diff changeset
   764
      have "f (Suc q) > f q" apply (induct q) 
6ffaa79c352c Moved a few theorems about monotonic sequences from Fundamental_Theorem_Algebra to SEQ.thy
chaieb
parents: 30082
diff changeset
   765
	using f0(1) fSuc(1)[of 0] apply simp by (rule fSuc(1))}
6ffaa79c352c Moved a few theorems about monotonic sequences from Fundamental_Theorem_Algebra to SEQ.thy
chaieb
parents: 30082
diff changeset
   766
    note fss = this
6ffaa79c352c Moved a few theorems about monotonic sequences from Fundamental_Theorem_Algebra to SEQ.thy
chaieb
parents: 30082
diff changeset
   767
    from fss have th1: "subseq f" unfolding subseq_Suc_iff ..
6ffaa79c352c Moved a few theorems about monotonic sequences from Fundamental_Theorem_Algebra to SEQ.thy
chaieb
parents: 30082
diff changeset
   768
    {fix a b 
6ffaa79c352c Moved a few theorems about monotonic sequences from Fundamental_Theorem_Algebra to SEQ.thy
chaieb
parents: 30082
diff changeset
   769
      have "f a \<le> f (a + b)"
6ffaa79c352c Moved a few theorems about monotonic sequences from Fundamental_Theorem_Algebra to SEQ.thy
chaieb
parents: 30082
diff changeset
   770
      proof(induct b)
6ffaa79c352c Moved a few theorems about monotonic sequences from Fundamental_Theorem_Algebra to SEQ.thy
chaieb
parents: 30082
diff changeset
   771
	case 0 thus ?case by simp
6ffaa79c352c Moved a few theorems about monotonic sequences from Fundamental_Theorem_Algebra to SEQ.thy
chaieb
parents: 30082
diff changeset
   772
      next
6ffaa79c352c Moved a few theorems about monotonic sequences from Fundamental_Theorem_Algebra to SEQ.thy
chaieb
parents: 30082
diff changeset
   773
	case (Suc b)
6ffaa79c352c Moved a few theorems about monotonic sequences from Fundamental_Theorem_Algebra to SEQ.thy
chaieb
parents: 30082
diff changeset
   774
	from fSuc(1)[of "a + b"] Suc.hyps show ?case by simp
6ffaa79c352c Moved a few theorems about monotonic sequences from Fundamental_Theorem_Algebra to SEQ.thy
chaieb
parents: 30082
diff changeset
   775
      qed}
6ffaa79c352c Moved a few theorems about monotonic sequences from Fundamental_Theorem_Algebra to SEQ.thy
chaieb
parents: 30082
diff changeset
   776
    note fmon0 = this
6ffaa79c352c Moved a few theorems about monotonic sequences from Fundamental_Theorem_Algebra to SEQ.thy
chaieb
parents: 30082
diff changeset
   777
    have "monoseq (\<lambda>n. s (f n))" 
6ffaa79c352c Moved a few theorems about monotonic sequences from Fundamental_Theorem_Algebra to SEQ.thy
chaieb
parents: 30082
diff changeset
   778
    proof-
6ffaa79c352c Moved a few theorems about monotonic sequences from Fundamental_Theorem_Algebra to SEQ.thy
chaieb
parents: 30082
diff changeset
   779
      {fix n
6ffaa79c352c Moved a few theorems about monotonic sequences from Fundamental_Theorem_Algebra to SEQ.thy
chaieb
parents: 30082
diff changeset
   780
	have "s (f n) \<ge> s (f (Suc n))" 
6ffaa79c352c Moved a few theorems about monotonic sequences from Fundamental_Theorem_Algebra to SEQ.thy
chaieb
parents: 30082
diff changeset
   781
	proof(cases n)
6ffaa79c352c Moved a few theorems about monotonic sequences from Fundamental_Theorem_Algebra to SEQ.thy
chaieb
parents: 30082
diff changeset
   782
	  case 0
6ffaa79c352c Moved a few theorems about monotonic sequences from Fundamental_Theorem_Algebra to SEQ.thy
chaieb
parents: 30082
diff changeset
   783
	  assume n0: "n = 0"
6ffaa79c352c Moved a few theorems about monotonic sequences from Fundamental_Theorem_Algebra to SEQ.thy
chaieb
parents: 30082
diff changeset
   784
	  from fSuc(1)[of 0] have th0: "f 0 \<le> f (Suc 0)" by simp
6ffaa79c352c Moved a few theorems about monotonic sequences from Fundamental_Theorem_Algebra to SEQ.thy
chaieb
parents: 30082
diff changeset
   785
	  from f0(2)[rule_format, OF th0] show ?thesis  using n0 by simp
6ffaa79c352c Moved a few theorems about monotonic sequences from Fundamental_Theorem_Algebra to SEQ.thy
chaieb
parents: 30082
diff changeset
   786
	next
6ffaa79c352c Moved a few theorems about monotonic sequences from Fundamental_Theorem_Algebra to SEQ.thy
chaieb
parents: 30082
diff changeset
   787
	  case (Suc m)
6ffaa79c352c Moved a few theorems about monotonic sequences from Fundamental_Theorem_Algebra to SEQ.thy
chaieb
parents: 30082
diff changeset
   788
	  assume m: "n = Suc m"
6ffaa79c352c Moved a few theorems about monotonic sequences from Fundamental_Theorem_Algebra to SEQ.thy
chaieb
parents: 30082
diff changeset
   789
	  from fSuc(1)[of n] m have th0: "f (Suc m) \<le> f (Suc (Suc m))" by simp
6ffaa79c352c Moved a few theorems about monotonic sequences from Fundamental_Theorem_Algebra to SEQ.thy
chaieb
parents: 30082
diff changeset
   790
	  from m fSuc(2)[rule_format, OF th0] show ?thesis by simp 
6ffaa79c352c Moved a few theorems about monotonic sequences from Fundamental_Theorem_Algebra to SEQ.thy
chaieb
parents: 30082
diff changeset
   791
	qed}
6ffaa79c352c Moved a few theorems about monotonic sequences from Fundamental_Theorem_Algebra to SEQ.thy
chaieb
parents: 30082
diff changeset
   792
      thus "monoseq (\<lambda>n. s (f n))" unfolding monoseq_Suc by blast 
6ffaa79c352c Moved a few theorems about monotonic sequences from Fundamental_Theorem_Algebra to SEQ.thy
chaieb
parents: 30082
diff changeset
   793
    qed
6ffaa79c352c Moved a few theorems about monotonic sequences from Fundamental_Theorem_Algebra to SEQ.thy
chaieb
parents: 30082
diff changeset
   794
    with th1 have ?thesis by blast}
6ffaa79c352c Moved a few theorems about monotonic sequences from Fundamental_Theorem_Algebra to SEQ.thy
chaieb
parents: 30082
diff changeset
   795
  moreover
6ffaa79c352c Moved a few theorems about monotonic sequences from Fundamental_Theorem_Algebra to SEQ.thy
chaieb
parents: 30082
diff changeset
   796
  {fix N assume N: "\<forall>p >N. \<exists> m\<ge>p. s m > s p"
6ffaa79c352c Moved a few theorems about monotonic sequences from Fundamental_Theorem_Algebra to SEQ.thy
chaieb
parents: 30082
diff changeset
   797
    {fix p assume p: "p \<ge> Suc N" 
6ffaa79c352c Moved a few theorems about monotonic sequences from Fundamental_Theorem_Algebra to SEQ.thy
chaieb
parents: 30082
diff changeset
   798
      hence pN: "p > N" by arith with N obtain m where m: "m \<ge> p" "s m > s p" by blast
6ffaa79c352c Moved a few theorems about monotonic sequences from Fundamental_Theorem_Algebra to SEQ.thy
chaieb
parents: 30082
diff changeset
   799
      have "m \<noteq> p" using m(2) by auto 
6ffaa79c352c Moved a few theorems about monotonic sequences from Fundamental_Theorem_Algebra to SEQ.thy
chaieb
parents: 30082
diff changeset
   800
      with m have "\<exists>m>p. s p < s m" by - (rule exI[where x=m], auto)}
6ffaa79c352c Moved a few theorems about monotonic sequences from Fundamental_Theorem_Algebra to SEQ.thy
chaieb
parents: 30082
diff changeset
   801
    note th0 = this
6ffaa79c352c Moved a few theorems about monotonic sequences from Fundamental_Theorem_Algebra to SEQ.thy
chaieb
parents: 30082
diff changeset
   802
    let ?P = "\<lambda>m x. m > x \<and> s x < s m"
6ffaa79c352c Moved a few theorems about monotonic sequences from Fundamental_Theorem_Algebra to SEQ.thy
chaieb
parents: 30082
diff changeset
   803
    from nat_function_unique[of "SOME x. ?P x (Suc N)" "\<lambda>m x. SOME y. ?P y x"]
6ffaa79c352c Moved a few theorems about monotonic sequences from Fundamental_Theorem_Algebra to SEQ.thy
chaieb
parents: 30082
diff changeset
   804
    obtain f where f: "f 0 = (SOME x. ?P x (Suc N))" 
6ffaa79c352c Moved a few theorems about monotonic sequences from Fundamental_Theorem_Algebra to SEQ.thy
chaieb
parents: 30082
diff changeset
   805
      "\<forall>n. f (Suc n) = (SOME m. ?P m (f n))" by blast
6ffaa79c352c Moved a few theorems about monotonic sequences from Fundamental_Theorem_Algebra to SEQ.thy
chaieb
parents: 30082
diff changeset
   806
    have "?P (f 0) (Suc N)"  unfolding f(1) some_eq_ex[of "\<lambda>p. ?P p (Suc N)"]
6ffaa79c352c Moved a few theorems about monotonic sequences from Fundamental_Theorem_Algebra to SEQ.thy
chaieb
parents: 30082
diff changeset
   807
      using N apply - 
6ffaa79c352c Moved a few theorems about monotonic sequences from Fundamental_Theorem_Algebra to SEQ.thy
chaieb
parents: 30082
diff changeset
   808
      apply (erule allE[where x="Suc N"], clarsimp)
6ffaa79c352c Moved a few theorems about monotonic sequences from Fundamental_Theorem_Algebra to SEQ.thy
chaieb
parents: 30082
diff changeset
   809
      apply (rule_tac x="m" in exI)
6ffaa79c352c Moved a few theorems about monotonic sequences from Fundamental_Theorem_Algebra to SEQ.thy
chaieb
parents: 30082
diff changeset
   810
      apply auto
6ffaa79c352c Moved a few theorems about monotonic sequences from Fundamental_Theorem_Algebra to SEQ.thy
chaieb
parents: 30082
diff changeset
   811
      apply (subgoal_tac "Suc N \<noteq> m")
6ffaa79c352c Moved a few theorems about monotonic sequences from Fundamental_Theorem_Algebra to SEQ.thy
chaieb
parents: 30082
diff changeset
   812
      apply simp
6ffaa79c352c Moved a few theorems about monotonic sequences from Fundamental_Theorem_Algebra to SEQ.thy
chaieb
parents: 30082
diff changeset
   813
      apply (rule ccontr, simp)
6ffaa79c352c Moved a few theorems about monotonic sequences from Fundamental_Theorem_Algebra to SEQ.thy
chaieb
parents: 30082
diff changeset
   814
      done
6ffaa79c352c Moved a few theorems about monotonic sequences from Fundamental_Theorem_Algebra to SEQ.thy
chaieb
parents: 30082
diff changeset
   815
    hence f0: "f 0 > Suc N" "s (Suc N) < s (f 0)" by blast+
6ffaa79c352c Moved a few theorems about monotonic sequences from Fundamental_Theorem_Algebra to SEQ.thy
chaieb
parents: 30082
diff changeset
   816
    {fix n
6ffaa79c352c Moved a few theorems about monotonic sequences from Fundamental_Theorem_Algebra to SEQ.thy
chaieb
parents: 30082
diff changeset
   817
      have "f n > N \<and> ?P (f (Suc n)) (f n)"
6ffaa79c352c Moved a few theorems about monotonic sequences from Fundamental_Theorem_Algebra to SEQ.thy
chaieb
parents: 30082
diff changeset
   818
	unfolding f(2)[rule_format, of n] some_eq_ex[of "\<lambda>p. ?P p (f n)"]
6ffaa79c352c Moved a few theorems about monotonic sequences from Fundamental_Theorem_Algebra to SEQ.thy
chaieb
parents: 30082
diff changeset
   819
      proof (induct n)
6ffaa79c352c Moved a few theorems about monotonic sequences from Fundamental_Theorem_Algebra to SEQ.thy
chaieb
parents: 30082
diff changeset
   820
	case 0 thus ?case
6ffaa79c352c Moved a few theorems about monotonic sequences from Fundamental_Theorem_Algebra to SEQ.thy
chaieb
parents: 30082
diff changeset
   821
	  using f0 N apply auto 
6ffaa79c352c Moved a few theorems about monotonic sequences from Fundamental_Theorem_Algebra to SEQ.thy
chaieb
parents: 30082
diff changeset
   822
	  apply (erule allE[where x="f 0"], clarsimp) 
6ffaa79c352c Moved a few theorems about monotonic sequences from Fundamental_Theorem_Algebra to SEQ.thy
chaieb
parents: 30082
diff changeset
   823
	  apply (rule_tac x="m" in exI, simp)
6ffaa79c352c Moved a few theorems about monotonic sequences from Fundamental_Theorem_Algebra to SEQ.thy
chaieb
parents: 30082
diff changeset
   824
	  by (subgoal_tac "f 0 \<noteq> m", auto)
6ffaa79c352c Moved a few theorems about monotonic sequences from Fundamental_Theorem_Algebra to SEQ.thy
chaieb
parents: 30082
diff changeset
   825
      next
6ffaa79c352c Moved a few theorems about monotonic sequences from Fundamental_Theorem_Algebra to SEQ.thy
chaieb
parents: 30082
diff changeset
   826
	case (Suc n)
6ffaa79c352c Moved a few theorems about monotonic sequences from Fundamental_Theorem_Algebra to SEQ.thy
chaieb
parents: 30082
diff changeset
   827
	from Suc.hyps have Nfn: "N < f n" by blast
6ffaa79c352c Moved a few theorems about monotonic sequences from Fundamental_Theorem_Algebra to SEQ.thy
chaieb
parents: 30082
diff changeset
   828
	from Suc.hyps obtain m where m: "m > f n" "s (f n) < s m" by blast
6ffaa79c352c Moved a few theorems about monotonic sequences from Fundamental_Theorem_Algebra to SEQ.thy
chaieb
parents: 30082
diff changeset
   829
	with Nfn have mN: "m > N" by arith
6ffaa79c352c Moved a few theorems about monotonic sequences from Fundamental_Theorem_Algebra to SEQ.thy
chaieb
parents: 30082
diff changeset
   830
	note key = Suc.hyps[unfolded some_eq_ex[of "\<lambda>p. ?P p (f n)", symmetric] f(2)[rule_format, of n, symmetric]]
6ffaa79c352c Moved a few theorems about monotonic sequences from Fundamental_Theorem_Algebra to SEQ.thy
chaieb
parents: 30082
diff changeset
   831
	
6ffaa79c352c Moved a few theorems about monotonic sequences from Fundamental_Theorem_Algebra to SEQ.thy
chaieb
parents: 30082
diff changeset
   832
	from key have th0: "f (Suc n) > N" by simp
6ffaa79c352c Moved a few theorems about monotonic sequences from Fundamental_Theorem_Algebra to SEQ.thy
chaieb
parents: 30082
diff changeset
   833
	from N[rule_format, OF th0]
6ffaa79c352c Moved a few theorems about monotonic sequences from Fundamental_Theorem_Algebra to SEQ.thy
chaieb
parents: 30082
diff changeset
   834
	obtain m' where m': "m' \<ge> f (Suc n)" "s (f (Suc n)) < s m'" by blast
6ffaa79c352c Moved a few theorems about monotonic sequences from Fundamental_Theorem_Algebra to SEQ.thy
chaieb
parents: 30082
diff changeset
   835
	have "m' \<noteq> f (Suc (n))" apply (rule ccontr) using m'(2) by auto
6ffaa79c352c Moved a few theorems about monotonic sequences from Fundamental_Theorem_Algebra to SEQ.thy
chaieb
parents: 30082
diff changeset
   836
	hence "m' > f (Suc n)" using m'(1) by simp
6ffaa79c352c Moved a few theorems about monotonic sequences from Fundamental_Theorem_Algebra to SEQ.thy
chaieb
parents: 30082
diff changeset
   837
	with key m'(2) show ?case by auto
6ffaa79c352c Moved a few theorems about monotonic sequences from Fundamental_Theorem_Algebra to SEQ.thy
chaieb
parents: 30082
diff changeset
   838
      qed}
6ffaa79c352c Moved a few theorems about monotonic sequences from Fundamental_Theorem_Algebra to SEQ.thy
chaieb
parents: 30082
diff changeset
   839
    note fSuc = this
6ffaa79c352c Moved a few theorems about monotonic sequences from Fundamental_Theorem_Algebra to SEQ.thy
chaieb
parents: 30082
diff changeset
   840
    {fix n
6ffaa79c352c Moved a few theorems about monotonic sequences from Fundamental_Theorem_Algebra to SEQ.thy
chaieb
parents: 30082
diff changeset
   841
      have "f n \<ge> Suc N \<and> f(Suc n) > f n \<and> s(f n) < s(f(Suc n))" using fSuc[of n] by auto 
6ffaa79c352c Moved a few theorems about monotonic sequences from Fundamental_Theorem_Algebra to SEQ.thy
chaieb
parents: 30082
diff changeset
   842
      hence "f n \<ge> Suc N" "f(Suc n) > f n" "s(f n) < s(f(Suc n))" by blast+}
6ffaa79c352c Moved a few theorems about monotonic sequences from Fundamental_Theorem_Algebra to SEQ.thy
chaieb
parents: 30082
diff changeset
   843
    note thf = this
6ffaa79c352c Moved a few theorems about monotonic sequences from Fundamental_Theorem_Algebra to SEQ.thy
chaieb
parents: 30082
diff changeset
   844
    have sqf: "subseq f" unfolding subseq_Suc_iff using thf by simp
6ffaa79c352c Moved a few theorems about monotonic sequences from Fundamental_Theorem_Algebra to SEQ.thy
chaieb
parents: 30082
diff changeset
   845
    have "monoseq (\<lambda>n. s (f n))"  unfolding monoseq_Suc using thf
6ffaa79c352c Moved a few theorems about monotonic sequences from Fundamental_Theorem_Algebra to SEQ.thy
chaieb
parents: 30082
diff changeset
   846
      apply -
6ffaa79c352c Moved a few theorems about monotonic sequences from Fundamental_Theorem_Algebra to SEQ.thy
chaieb
parents: 30082
diff changeset
   847
      apply (rule disjI1)
6ffaa79c352c Moved a few theorems about monotonic sequences from Fundamental_Theorem_Algebra to SEQ.thy
chaieb
parents: 30082
diff changeset
   848
      apply auto
6ffaa79c352c Moved a few theorems about monotonic sequences from Fundamental_Theorem_Algebra to SEQ.thy
chaieb
parents: 30082
diff changeset
   849
      apply (rule order_less_imp_le)
6ffaa79c352c Moved a few theorems about monotonic sequences from Fundamental_Theorem_Algebra to SEQ.thy
chaieb
parents: 30082
diff changeset
   850
      apply blast
6ffaa79c352c Moved a few theorems about monotonic sequences from Fundamental_Theorem_Algebra to SEQ.thy
chaieb
parents: 30082
diff changeset
   851
      done
6ffaa79c352c Moved a few theorems about monotonic sequences from Fundamental_Theorem_Algebra to SEQ.thy
chaieb
parents: 30082
diff changeset
   852
    then have ?thesis  using sqf by blast}
6ffaa79c352c Moved a few theorems about monotonic sequences from Fundamental_Theorem_Algebra to SEQ.thy
chaieb
parents: 30082
diff changeset
   853
  ultimately show ?thesis unfolding linorder_not_less[symmetric] by blast
6ffaa79c352c Moved a few theorems about monotonic sequences from Fundamental_Theorem_Algebra to SEQ.thy
chaieb
parents: 30082
diff changeset
   854
qed
6ffaa79c352c Moved a few theorems about monotonic sequences from Fundamental_Theorem_Algebra to SEQ.thy
chaieb
parents: 30082
diff changeset
   855
6ffaa79c352c Moved a few theorems about monotonic sequences from Fundamental_Theorem_Algebra to SEQ.thy
chaieb
parents: 30082
diff changeset
   856
lemma seq_suble: assumes sf: "subseq f" shows "n \<le> f n"
6ffaa79c352c Moved a few theorems about monotonic sequences from Fundamental_Theorem_Algebra to SEQ.thy
chaieb
parents: 30082
diff changeset
   857
proof(induct n)
6ffaa79c352c Moved a few theorems about monotonic sequences from Fundamental_Theorem_Algebra to SEQ.thy
chaieb
parents: 30082
diff changeset
   858
  case 0 thus ?case by simp
6ffaa79c352c Moved a few theorems about monotonic sequences from Fundamental_Theorem_Algebra to SEQ.thy
chaieb
parents: 30082
diff changeset
   859
next
6ffaa79c352c Moved a few theorems about monotonic sequences from Fundamental_Theorem_Algebra to SEQ.thy
chaieb
parents: 30082
diff changeset
   860
  case (Suc n)
6ffaa79c352c Moved a few theorems about monotonic sequences from Fundamental_Theorem_Algebra to SEQ.thy
chaieb
parents: 30082
diff changeset
   861
  from sf[unfolded subseq_Suc_iff, rule_format, of n] Suc.hyps
6ffaa79c352c Moved a few theorems about monotonic sequences from Fundamental_Theorem_Algebra to SEQ.thy
chaieb
parents: 30082
diff changeset
   862
  have "n < f (Suc n)" by arith 
6ffaa79c352c Moved a few theorems about monotonic sequences from Fundamental_Theorem_Algebra to SEQ.thy
chaieb
parents: 30082
diff changeset
   863
  thus ?case by arith
6ffaa79c352c Moved a few theorems about monotonic sequences from Fundamental_Theorem_Algebra to SEQ.thy
chaieb
parents: 30082
diff changeset
   864
qed
6ffaa79c352c Moved a few theorems about monotonic sequences from Fundamental_Theorem_Algebra to SEQ.thy
chaieb
parents: 30082
diff changeset
   865
30730
4d3565f2cb0e New theorems mostly concerning infinite series.
paulson
parents: 30273
diff changeset
   866
lemma LIMSEQ_subseq_LIMSEQ:
4d3565f2cb0e New theorems mostly concerning infinite series.
paulson
parents: 30273
diff changeset
   867
  "\<lbrakk> X ----> L; subseq f \<rbrakk> \<Longrightarrow> (X o f) ----> L"
4d3565f2cb0e New theorems mostly concerning infinite series.
paulson
parents: 30273
diff changeset
   868
apply (auto simp add: LIMSEQ_def) 
4d3565f2cb0e New theorems mostly concerning infinite series.
paulson
parents: 30273
diff changeset
   869
apply (drule_tac x=r in spec, clarify)  
4d3565f2cb0e New theorems mostly concerning infinite series.
paulson
parents: 30273
diff changeset
   870
apply (rule_tac x=no in exI, clarify) 
4d3565f2cb0e New theorems mostly concerning infinite series.
paulson
parents: 30273
diff changeset
   871
apply (blast intro: seq_suble le_trans dest!: spec) 
4d3565f2cb0e New theorems mostly concerning infinite series.
paulson
parents: 30273
diff changeset
   872
done
4d3565f2cb0e New theorems mostly concerning infinite series.
paulson
parents: 30273
diff changeset
   873
30196
6ffaa79c352c Moved a few theorems about monotonic sequences from Fundamental_Theorem_Algebra to SEQ.thy
chaieb
parents: 30082
diff changeset
   874
subsection {* Bounded Monotonic Sequences *}
6ffaa79c352c Moved a few theorems about monotonic sequences from Fundamental_Theorem_Algebra to SEQ.thy
chaieb
parents: 30082
diff changeset
   875
6ffaa79c352c Moved a few theorems about monotonic sequences from Fundamental_Theorem_Algebra to SEQ.thy
chaieb
parents: 30082
diff changeset
   876
20696
3b887ad7d196 reorganized subsection headings
huffman
parents: 20695
diff changeset
   877
text{*Bounded Sequence*}
15082
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
   878
20552
2c31dd358c21 generalized types of many constants to work over arbitrary vector spaces;
huffman
parents: 20408
diff changeset
   879
lemma BseqD: "Bseq X ==> \<exists>K. 0 < K & (\<forall>n. norm (X n) \<le> K)"
15082
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
   880
by (simp add: Bseq_def)
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
   881
20552
2c31dd358c21 generalized types of many constants to work over arbitrary vector spaces;
huffman
parents: 20408
diff changeset
   882
lemma BseqI: "[| 0 < K; \<forall>n. norm (X n) \<le> K |] ==> Bseq X"
15082
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
   883
by (auto simp add: Bseq_def)
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
   884
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
   885
lemma lemma_NBseq_def:
20552
2c31dd358c21 generalized types of many constants to work over arbitrary vector spaces;
huffman
parents: 20408
diff changeset
   886
     "(\<exists>K > 0. \<forall>n. norm (X n) \<le> K) =
2c31dd358c21 generalized types of many constants to work over arbitrary vector spaces;
huffman
parents: 20408
diff changeset
   887
      (\<exists>N. \<forall>n. norm (X n) \<le> real(Suc N))"
15082
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
   888
apply auto
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
   889
 prefer 2 apply force
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
   890
apply (cut_tac x = K in reals_Archimedean2, clarify)
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
   891
apply (rule_tac x = n in exI, clarify)
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
   892
apply (drule_tac x = na in spec)
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
   893
apply (auto simp add: real_of_nat_Suc)
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
   894
done
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
   895
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
   896
text{* alternative definition for Bseq *}
20552
2c31dd358c21 generalized types of many constants to work over arbitrary vector spaces;
huffman
parents: 20408
diff changeset
   897
lemma Bseq_iff: "Bseq X = (\<exists>N. \<forall>n. norm (X n) \<le> real(Suc N))"
15082
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
   898
apply (simp add: Bseq_def)
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
   899
apply (simp (no_asm) add: lemma_NBseq_def)
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
   900
done
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
   901
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
   902
lemma lemma_NBseq_def2:
20552
2c31dd358c21 generalized types of many constants to work over arbitrary vector spaces;
huffman
parents: 20408
diff changeset
   903
     "(\<exists>K > 0. \<forall>n. norm (X n) \<le> K) = (\<exists>N. \<forall>n. norm (X n) < real(Suc N))"
15082
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
   904
apply (subst lemma_NBseq_def, auto)
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
   905
apply (rule_tac x = "Suc N" in exI)
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
   906
apply (rule_tac [2] x = N in exI)
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
   907
apply (auto simp add: real_of_nat_Suc)
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
   908
 prefer 2 apply (blast intro: order_less_imp_le)
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
   909
apply (drule_tac x = n in spec, simp)
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
   910
done
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
   911
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
   912
(* yet another definition for Bseq *)
20552
2c31dd358c21 generalized types of many constants to work over arbitrary vector spaces;
huffman
parents: 20408
diff changeset
   913
lemma Bseq_iff1a: "Bseq X = (\<exists>N. \<forall>n. norm (X n) < real(Suc N))"
15082
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
   914
by (simp add: Bseq_def lemma_NBseq_def2)
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
   915
20696
3b887ad7d196 reorganized subsection headings
huffman
parents: 20695
diff changeset
   916
subsubsection{*Upper Bounds and Lubs of Bounded Sequences*}
15082
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
   917
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
   918
lemma Bseq_isUb:
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
   919
  "!!(X::nat=>real). Bseq X ==> \<exists>U. isUb (UNIV::real set) {x. \<exists>n. X n = x} U"
22998
97e1f9c2cc46 avoid using redundant lemmas from RealDef.thy
huffman
parents: 22974
diff changeset
   920
by (auto intro: isUbI setleI simp add: Bseq_def abs_le_iff)
15082
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
   921
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
   922
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
   923
text{* Use completeness of reals (supremum property)
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
   924
   to show that any bounded sequence has a least upper bound*}
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
   925
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
   926
lemma Bseq_isLub:
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
   927
  "!!(X::nat=>real). Bseq X ==>
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
   928
   \<exists>U. isLub (UNIV::real set) {x. \<exists>n. X n = x} U"
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
   929
by (blast intro: reals_complete Bseq_isUb)
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
   930
20696
3b887ad7d196 reorganized subsection headings
huffman
parents: 20695
diff changeset
   931
subsubsection{*A Bounded and Monotonic Sequence Converges*}
15082
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
   932
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
   933
lemma lemma_converg1:
15360
300e09825d8b Added "ALL x > y" and relatives.
nipkow
parents: 15312
diff changeset
   934
     "!!(X::nat=>real). [| \<forall>m. \<forall> n \<ge> m. X m \<le> X n;
15082
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
   935
                  isLub (UNIV::real set) {x. \<exists>n. X n = x} (X ma)
15360
300e09825d8b Added "ALL x > y" and relatives.
nipkow
parents: 15312
diff changeset
   936
               |] ==> \<forall>n \<ge> ma. X n = X ma"
15082
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
   937
apply safe
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
   938
apply (drule_tac y = "X n" in isLubD2)
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
   939
apply (blast dest: order_antisym)+
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
   940
done
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
   941
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
   942
text{* The best of both worlds: Easier to prove this result as a standard
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
   943
   theorem and then use equivalence to "transfer" it into the
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
   944
   equivalent nonstandard form if needed!*}
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
   945
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
   946
lemma Bmonoseq_LIMSEQ: "\<forall>n. m \<le> n --> X n = X m ==> \<exists>L. (X ----> L)"
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
   947
apply (simp add: LIMSEQ_def)
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
   948
apply (rule_tac x = "X m" in exI, safe)
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
   949
apply (rule_tac x = m in exI, safe)
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
   950
apply (drule spec, erule impE, auto)
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
   951
done
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
   952
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
   953
lemma lemma_converg2:
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
   954
   "!!(X::nat=>real).
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
   955
    [| \<forall>m. X m ~= U;  isLub UNIV {x. \<exists>n. X n = x} U |] ==> \<forall>m. X m < U"
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
   956
apply safe
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
   957
apply (drule_tac y = "X m" in isLubD2)
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
   958
apply (auto dest!: order_le_imp_less_or_eq)
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
   959
done
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
   960
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
   961
lemma lemma_converg3: "!!(X ::nat=>real). \<forall>m. X m \<le> U ==> isUb UNIV {x. \<exists>n. X n = x} U"
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
   962
by (rule setleI [THEN isUbI], auto)
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
   963
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
   964
text{* FIXME: @{term "U - T < U"} is redundant *}
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
   965
lemma lemma_converg4: "!!(X::nat=> real).
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
   966
               [| \<forall>m. X m ~= U;
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
   967
                  isLub UNIV {x. \<exists>n. X n = x} U;
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
   968
                  0 < T;
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
   969
                  U + - T < U
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
   970
               |] ==> \<exists>m. U + -T < X m & X m < U"
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
   971
apply (drule lemma_converg2, assumption)
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
   972
apply (rule ccontr, simp)
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
   973
apply (simp add: linorder_not_less)
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
   974
apply (drule lemma_converg3)
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
   975
apply (drule isLub_le_isUb, assumption)
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
   976
apply (auto dest: order_less_le_trans)
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
   977
done
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
   978
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
   979
text{*A standard proof of the theorem for monotone increasing sequence*}
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
   980
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
   981
lemma Bseq_mono_convergent:
20552
2c31dd358c21 generalized types of many constants to work over arbitrary vector spaces;
huffman
parents: 20408
diff changeset
   982
     "[| Bseq X; \<forall>m. \<forall>n \<ge> m. X m \<le> X n |] ==> convergent (X::nat=>real)"
15082
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
   983
apply (simp add: convergent_def)
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
   984
apply (frule Bseq_isLub, safe)
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
   985
apply (case_tac "\<exists>m. X m = U", auto)
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
   986
apply (blast dest: lemma_converg1 Bmonoseq_LIMSEQ)
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
   987
(* second case *)
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
   988
apply (rule_tac x = U in exI)
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
   989
apply (subst LIMSEQ_iff, safe)
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
   990
apply (frule lemma_converg2, assumption)
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
   991
apply (drule lemma_converg4, auto)
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
   992
apply (rule_tac x = m in exI, safe)
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
   993
apply (subgoal_tac "X m \<le> X n")
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
   994
 prefer 2 apply blast
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
   995
apply (drule_tac x=n and P="%m. X m < U" in spec, arith)
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
   996
done
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
   997
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
   998
lemma Bseq_minus_iff: "Bseq (%n. -(X n)) = Bseq X"
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
   999
by (simp add: Bseq_def)
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
  1000
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
  1001
text{*Main monotonicity theorem*}
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
  1002
lemma Bseq_monoseq_convergent: "[| Bseq X; monoseq X |] ==> convergent X"
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
  1003
apply (simp add: monoseq_def, safe)
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
  1004
apply (rule_tac [2] convergent_minus_iff [THEN ssubst])
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
  1005
apply (drule_tac [2] Bseq_minus_iff [THEN ssubst])
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
  1006
apply (auto intro!: Bseq_mono_convergent)
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
  1007
done
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
  1008
30730
4d3565f2cb0e New theorems mostly concerning infinite series.
paulson
parents: 30273
diff changeset
  1009
subsubsection{*Increasing and Decreasing Series*}
4d3565f2cb0e New theorems mostly concerning infinite series.
paulson
parents: 30273
diff changeset
  1010
4d3565f2cb0e New theorems mostly concerning infinite series.
paulson
parents: 30273
diff changeset
  1011
lemma incseq_imp_monoseq:  "incseq X \<Longrightarrow> monoseq X"
4d3565f2cb0e New theorems mostly concerning infinite series.
paulson
parents: 30273
diff changeset
  1012
  by (simp add: incseq_def monoseq_def) 
4d3565f2cb0e New theorems mostly concerning infinite series.
paulson
parents: 30273
diff changeset
  1013
4d3565f2cb0e New theorems mostly concerning infinite series.
paulson
parents: 30273
diff changeset
  1014
lemma incseq_le: assumes inc: "incseq X" and lim: "X ----> L" shows "X n \<le> L"
4d3565f2cb0e New theorems mostly concerning infinite series.
paulson
parents: 30273
diff changeset
  1015
  using monoseq_le [OF incseq_imp_monoseq [OF inc] lim]
4d3565f2cb0e New theorems mostly concerning infinite series.
paulson
parents: 30273
diff changeset
  1016
proof
4d3565f2cb0e New theorems mostly concerning infinite series.
paulson
parents: 30273
diff changeset
  1017
  assume "(\<forall>n. X n \<le> L) \<and> (\<forall>m n. m \<le> n \<longrightarrow> X m \<le> X n)"
4d3565f2cb0e New theorems mostly concerning infinite series.
paulson
parents: 30273
diff changeset
  1018
  thus ?thesis by simp
4d3565f2cb0e New theorems mostly concerning infinite series.
paulson
parents: 30273
diff changeset
  1019
next
4d3565f2cb0e New theorems mostly concerning infinite series.
paulson
parents: 30273
diff changeset
  1020
  assume "(\<forall>n. L \<le> X n) \<and> (\<forall>m n. m \<le> n \<longrightarrow> X n \<le> X m)"
4d3565f2cb0e New theorems mostly concerning infinite series.
paulson
parents: 30273
diff changeset
  1021
  hence const: "(!!m n. m \<le> n \<Longrightarrow> X n = X m)" using inc
4d3565f2cb0e New theorems mostly concerning infinite series.
paulson
parents: 30273
diff changeset
  1022
    by (auto simp add: incseq_def intro: order_antisym)
4d3565f2cb0e New theorems mostly concerning infinite series.
paulson
parents: 30273
diff changeset
  1023
  have X: "!!n. X n = X 0"
4d3565f2cb0e New theorems mostly concerning infinite series.
paulson
parents: 30273
diff changeset
  1024
    by (blast intro: const [of 0]) 
4d3565f2cb0e New theorems mostly concerning infinite series.
paulson
parents: 30273
diff changeset
  1025
  have "X = (\<lambda>n. X 0)"
4d3565f2cb0e New theorems mostly concerning infinite series.
paulson
parents: 30273
diff changeset
  1026
    by (blast intro: ext X)
4d3565f2cb0e New theorems mostly concerning infinite series.
paulson
parents: 30273
diff changeset
  1027
  hence "L = X 0" using LIMSEQ_const [of "X 0"]
4d3565f2cb0e New theorems mostly concerning infinite series.
paulson
parents: 30273
diff changeset
  1028
    by (auto intro: LIMSEQ_unique lim) 
4d3565f2cb0e New theorems mostly concerning infinite series.
paulson
parents: 30273
diff changeset
  1029
  thus ?thesis
4d3565f2cb0e New theorems mostly concerning infinite series.
paulson
parents: 30273
diff changeset
  1030
    by (blast intro: eq_refl X)
4d3565f2cb0e New theorems mostly concerning infinite series.
paulson
parents: 30273
diff changeset
  1031
qed
4d3565f2cb0e New theorems mostly concerning infinite series.
paulson
parents: 30273
diff changeset
  1032
4d3565f2cb0e New theorems mostly concerning infinite series.
paulson
parents: 30273
diff changeset
  1033
lemma decseq_imp_monoseq:  "decseq X \<Longrightarrow> monoseq X"
4d3565f2cb0e New theorems mostly concerning infinite series.
paulson
parents: 30273
diff changeset
  1034
  by (simp add: decseq_def monoseq_def)
4d3565f2cb0e New theorems mostly concerning infinite series.
paulson
parents: 30273
diff changeset
  1035
4d3565f2cb0e New theorems mostly concerning infinite series.
paulson
parents: 30273
diff changeset
  1036
lemma decseq_eq_incseq: "decseq X = incseq (\<lambda>n. - X n)" 
4d3565f2cb0e New theorems mostly concerning infinite series.
paulson
parents: 30273
diff changeset
  1037
  by (simp add: decseq_def incseq_def)
4d3565f2cb0e New theorems mostly concerning infinite series.
paulson
parents: 30273
diff changeset
  1038
4d3565f2cb0e New theorems mostly concerning infinite series.
paulson
parents: 30273
diff changeset
  1039
4d3565f2cb0e New theorems mostly concerning infinite series.
paulson
parents: 30273
diff changeset
  1040
lemma decseq_le: assumes dec: "decseq X" and lim: "X ----> L" shows "L \<le> X n"
4d3565f2cb0e New theorems mostly concerning infinite series.
paulson
parents: 30273
diff changeset
  1041
proof -
4d3565f2cb0e New theorems mostly concerning infinite series.
paulson
parents: 30273
diff changeset
  1042
  have inc: "incseq (\<lambda>n. - X n)" using dec
4d3565f2cb0e New theorems mostly concerning infinite series.
paulson
parents: 30273
diff changeset
  1043
    by (simp add: decseq_eq_incseq)
4d3565f2cb0e New theorems mostly concerning infinite series.
paulson
parents: 30273
diff changeset
  1044
  have "- X n \<le> - L" 
4d3565f2cb0e New theorems mostly concerning infinite series.
paulson
parents: 30273
diff changeset
  1045
    by (blast intro: incseq_le [OF inc] LIMSEQ_minus lim) 
4d3565f2cb0e New theorems mostly concerning infinite series.
paulson
parents: 30273
diff changeset
  1046
  thus ?thesis
4d3565f2cb0e New theorems mostly concerning infinite series.
paulson
parents: 30273
diff changeset
  1047
    by simp
4d3565f2cb0e New theorems mostly concerning infinite series.
paulson
parents: 30273
diff changeset
  1048
qed
4d3565f2cb0e New theorems mostly concerning infinite series.
paulson
parents: 30273
diff changeset
  1049
20696
3b887ad7d196 reorganized subsection headings
huffman
parents: 20695
diff changeset
  1050
subsubsection{*A Few More Equivalence Theorems for Boundedness*}
15082
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
  1051
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
  1052
text{*alternative formulation for boundedness*}
20552
2c31dd358c21 generalized types of many constants to work over arbitrary vector spaces;
huffman
parents: 20408
diff changeset
  1053
lemma Bseq_iff2: "Bseq X = (\<exists>k > 0. \<exists>x. \<forall>n. norm (X(n) + -x) \<le> k)"
15082
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
  1054
apply (unfold Bseq_def, safe)
20552
2c31dd358c21 generalized types of many constants to work over arbitrary vector spaces;
huffman
parents: 20408
diff changeset
  1055
apply (rule_tac [2] x = "k + norm x" in exI)
15360
300e09825d8b Added "ALL x > y" and relatives.
nipkow
parents: 15312
diff changeset
  1056
apply (rule_tac x = K in exI, simp)
15221
8412cfdf3287 tweaking of arithmetic proofs
paulson
parents: 15140
diff changeset
  1057
apply (rule exI [where x = 0], auto)
20552
2c31dd358c21 generalized types of many constants to work over arbitrary vector spaces;
huffman
parents: 20408
diff changeset
  1058
apply (erule order_less_le_trans, simp)
2c31dd358c21 generalized types of many constants to work over arbitrary vector spaces;
huffman
parents: 20408
diff changeset
  1059
apply (drule_tac x=n in spec, fold diff_def)
2c31dd358c21 generalized types of many constants to work over arbitrary vector spaces;
huffman
parents: 20408
diff changeset
  1060
apply (drule order_trans [OF norm_triangle_ineq2])
2c31dd358c21 generalized types of many constants to work over arbitrary vector spaces;
huffman
parents: 20408
diff changeset
  1061
apply simp
15082
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
  1062
done
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
  1063
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
  1064
text{*alternative formulation for boundedness*}
20552
2c31dd358c21 generalized types of many constants to work over arbitrary vector spaces;
huffman
parents: 20408
diff changeset
  1065
lemma Bseq_iff3: "Bseq X = (\<exists>k > 0. \<exists>N. \<forall>n. norm(X(n) + -X(N)) \<le> k)"
15082
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
  1066
apply safe
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
  1067
apply (simp add: Bseq_def, safe)
20552
2c31dd358c21 generalized types of many constants to work over arbitrary vector spaces;
huffman
parents: 20408
diff changeset
  1068
apply (rule_tac x = "K + norm (X N)" in exI)
15082
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
  1069
apply auto
20552
2c31dd358c21 generalized types of many constants to work over arbitrary vector spaces;
huffman
parents: 20408
diff changeset
  1070
apply (erule order_less_le_trans, simp)
15082
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
  1071
apply (rule_tac x = N in exI, safe)
20552
2c31dd358c21 generalized types of many constants to work over arbitrary vector spaces;
huffman
parents: 20408
diff changeset
  1072
apply (drule_tac x = n in spec)
2c31dd358c21 generalized types of many constants to work over arbitrary vector spaces;
huffman
parents: 20408
diff changeset
  1073
apply (rule order_trans [OF norm_triangle_ineq], simp)
15082
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
  1074
apply (auto simp add: Bseq_iff2)
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
  1075
done
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
  1076
20552
2c31dd358c21 generalized types of many constants to work over arbitrary vector spaces;
huffman
parents: 20408
diff changeset
  1077
lemma BseqI2: "(\<forall>n. k \<le> f n & f n \<le> (K::real)) ==> Bseq f"
15082
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
  1078
apply (simp add: Bseq_def)
15221
8412cfdf3287 tweaking of arithmetic proofs
paulson
parents: 15140
diff changeset
  1079
apply (rule_tac x = " (\<bar>k\<bar> + \<bar>K\<bar>) + 1" in exI, auto)
20217
25b068a99d2b linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents: 19765
diff changeset
  1080
apply (drule_tac x = n in spec, arith)
15082
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
  1081
done
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
  1082
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
  1083
20696
3b887ad7d196 reorganized subsection headings
huffman
parents: 20695
diff changeset
  1084
subsection {* Cauchy Sequences *}
15082
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
  1085
31336
e17f13cd1280 generalize constants in SEQ.thy to class metric_space
huffman
parents: 31017
diff changeset
  1086
lemma metric_CauchyI:
e17f13cd1280 generalize constants in SEQ.thy to class metric_space
huffman
parents: 31017
diff changeset
  1087
  "(\<And>e. 0 < e \<Longrightarrow> \<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. dist (X m) (X n) < e) \<Longrightarrow> Cauchy X"
e17f13cd1280 generalize constants in SEQ.thy to class metric_space
huffman
parents: 31017
diff changeset
  1088
by (simp add: Cauchy_def)
e17f13cd1280 generalize constants in SEQ.thy to class metric_space
huffman
parents: 31017
diff changeset
  1089
e17f13cd1280 generalize constants in SEQ.thy to class metric_space
huffman
parents: 31017
diff changeset
  1090
lemma metric_CauchyD:
e17f13cd1280 generalize constants in SEQ.thy to class metric_space
huffman
parents: 31017
diff changeset
  1091
  "\<lbrakk>Cauchy X; 0 < e\<rbrakk> \<Longrightarrow> \<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. dist (X m) (X n) < e"
20751
93271c59d211 add intro/dest rules for (NS)LIMSEQ and (NS)Cauchy; rewrite equivalence proofs using transfer
huffman
parents: 20740
diff changeset
  1092
by (simp add: Cauchy_def)
93271c59d211 add intro/dest rules for (NS)LIMSEQ and (NS)Cauchy; rewrite equivalence proofs using transfer
huffman
parents: 20740
diff changeset
  1093
31336
e17f13cd1280 generalize constants in SEQ.thy to class metric_space
huffman
parents: 31017
diff changeset
  1094
lemma Cauchy_iff:
e17f13cd1280 generalize constants in SEQ.thy to class metric_space
huffman
parents: 31017
diff changeset
  1095
  fixes X :: "nat \<Rightarrow> 'a::real_normed_vector"
e17f13cd1280 generalize constants in SEQ.thy to class metric_space
huffman
parents: 31017
diff changeset
  1096
  shows "Cauchy X \<longleftrightarrow> (\<forall>e>0. \<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. norm (X m - X n) < e)"
e17f13cd1280 generalize constants in SEQ.thy to class metric_space
huffman
parents: 31017
diff changeset
  1097
unfolding Cauchy_def dist_norm ..
e17f13cd1280 generalize constants in SEQ.thy to class metric_space
huffman
parents: 31017
diff changeset
  1098
e17f13cd1280 generalize constants in SEQ.thy to class metric_space
huffman
parents: 31017
diff changeset
  1099
lemma CauchyI:
e17f13cd1280 generalize constants in SEQ.thy to class metric_space
huffman
parents: 31017
diff changeset
  1100
  fixes X :: "nat \<Rightarrow> 'a::real_normed_vector"
e17f13cd1280 generalize constants in SEQ.thy to class metric_space
huffman
parents: 31017
diff changeset
  1101
  shows "(\<And>e. 0 < e \<Longrightarrow> \<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. norm (X m - X n) < e) \<Longrightarrow> Cauchy X"
e17f13cd1280 generalize constants in SEQ.thy to class metric_space
huffman
parents: 31017
diff changeset
  1102
by (simp add: Cauchy_iff)
e17f13cd1280 generalize constants in SEQ.thy to class metric_space
huffman
parents: 31017
diff changeset
  1103
20751
93271c59d211 add intro/dest rules for (NS)LIMSEQ and (NS)Cauchy; rewrite equivalence proofs using transfer
huffman
parents: 20740
diff changeset
  1104
lemma CauchyD:
31336
e17f13cd1280 generalize constants in SEQ.thy to class metric_space
huffman
parents: 31017
diff changeset
  1105
  fixes X :: "nat \<Rightarrow> 'a::real_normed_vector"
e17f13cd1280 generalize constants in SEQ.thy to class metric_space
huffman
parents: 31017
diff changeset
  1106
  shows "\<lbrakk>Cauchy X; 0 < e\<rbrakk> \<Longrightarrow> \<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. norm (X m - X n) < e"
e17f13cd1280 generalize constants in SEQ.thy to class metric_space
huffman
parents: 31017
diff changeset
  1107
by (simp add: Cauchy_iff)
20751
93271c59d211 add intro/dest rules for (NS)LIMSEQ and (NS)Cauchy; rewrite equivalence proofs using transfer
huffman
parents: 20740
diff changeset
  1108
30730
4d3565f2cb0e New theorems mostly concerning infinite series.
paulson
parents: 30273
diff changeset
  1109
lemma Cauchy_subseq_Cauchy:
4d3565f2cb0e New theorems mostly concerning infinite series.
paulson
parents: 30273
diff changeset
  1110
  "\<lbrakk> Cauchy X; subseq f \<rbrakk> \<Longrightarrow> Cauchy (X o f)"
31336
e17f13cd1280 generalize constants in SEQ.thy to class metric_space
huffman
parents: 31017
diff changeset
  1111
apply (auto simp add: Cauchy_def)
e17f13cd1280 generalize constants in SEQ.thy to class metric_space
huffman
parents: 31017
diff changeset
  1112
apply (drule_tac x=e in spec, clarify)
e17f13cd1280 generalize constants in SEQ.thy to class metric_space
huffman
parents: 31017
diff changeset
  1113
apply (rule_tac x=M in exI, clarify)
e17f13cd1280 generalize constants in SEQ.thy to class metric_space
huffman
parents: 31017
diff changeset
  1114
apply (blast intro: le_trans [OF _ seq_suble] dest!: spec)
30730
4d3565f2cb0e New theorems mostly concerning infinite series.
paulson
parents: 30273
diff changeset
  1115
done
4d3565f2cb0e New theorems mostly concerning infinite series.
paulson
parents: 30273
diff changeset
  1116
20696
3b887ad7d196 reorganized subsection headings
huffman
parents: 20695
diff changeset
  1117
subsubsection {* Cauchy Sequences are Bounded *}
3b887ad7d196 reorganized subsection headings
huffman
parents: 20695
diff changeset
  1118
15082
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
  1119
text{*A Cauchy sequence is bounded -- this is the standard
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
  1120
  proof mechanization rather than the nonstandard proof*}
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
  1121
20563
44eda2314aab replace (x + - y) with (x - y)
huffman
parents: 20552
diff changeset
  1122
lemma lemmaCauchy: "\<forall>n \<ge> M. norm (X M - X n) < (1::real)
20552
2c31dd358c21 generalized types of many constants to work over arbitrary vector spaces;
huffman
parents: 20408
diff changeset
  1123
          ==>  \<forall>n \<ge> M. norm (X n :: 'a::real_normed_vector) < 1 + norm (X M)"
2c31dd358c21 generalized types of many constants to work over arbitrary vector spaces;
huffman
parents: 20408
diff changeset
  1124
apply (clarify, drule spec, drule (1) mp)
20563
44eda2314aab replace (x + - y) with (x - y)
huffman
parents: 20552
diff changeset
  1125
apply (simp only: norm_minus_commute)
20552
2c31dd358c21 generalized types of many constants to work over arbitrary vector spaces;
huffman
parents: 20408
diff changeset
  1126
apply (drule order_le_less_trans [OF norm_triangle_ineq2])
2c31dd358c21 generalized types of many constants to work over arbitrary vector spaces;
huffman
parents: 20408
diff changeset
  1127
apply simp
2c31dd358c21 generalized types of many constants to work over arbitrary vector spaces;
huffman
parents: 20408
diff changeset
  1128
done
15082
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
  1129
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
  1130
lemma Cauchy_Bseq: "Cauchy X ==> Bseq X"
31336
e17f13cd1280 generalize constants in SEQ.thy to class metric_space
huffman
parents: 31017
diff changeset
  1131
apply (simp add: Cauchy_iff)
20552
2c31dd358c21 generalized types of many constants to work over arbitrary vector spaces;
huffman
parents: 20408
diff changeset
  1132
apply (drule spec, drule mp, rule zero_less_one, safe)
2c31dd358c21 generalized types of many constants to work over arbitrary vector spaces;
huffman
parents: 20408
diff changeset
  1133
apply (drule_tac x="M" in spec, simp)
15082
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
  1134
apply (drule lemmaCauchy)
22608
092a3353241e add new standard proofs for limits of sequences
huffman
parents: 21842
diff changeset
  1135
apply (rule_tac k="M" in Bseq_offset)
20552
2c31dd358c21 generalized types of many constants to work over arbitrary vector spaces;
huffman
parents: 20408
diff changeset
  1136
apply (simp add: Bseq_def)
2c31dd358c21 generalized types of many constants to work over arbitrary vector spaces;
huffman
parents: 20408
diff changeset
  1137
apply (rule_tac x="1 + norm (X M)" in exI)
2c31dd358c21 generalized types of many constants to work over arbitrary vector spaces;
huffman
parents: 20408
diff changeset
  1138
apply (rule conjI, rule order_less_le_trans [OF zero_less_one], simp)
2c31dd358c21 generalized types of many constants to work over arbitrary vector spaces;
huffman
parents: 20408
diff changeset
  1139
apply (simp add: order_less_imp_le)
15082
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
  1140
done
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
  1141
20696
3b887ad7d196 reorganized subsection headings
huffman
parents: 20695
diff changeset
  1142
subsubsection {* Cauchy Sequences are Convergent *}
15082
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
  1143
20830
65ba80cae6df add axclass banach for complete normed vector spaces
huffman
parents: 20829
diff changeset
  1144
axclass banach \<subseteq> real_normed_vector
65ba80cae6df add axclass banach for complete normed vector spaces
huffman
parents: 20829
diff changeset
  1145
  Cauchy_convergent: "Cauchy X \<Longrightarrow> convergent X"
65ba80cae6df add axclass banach for complete normed vector spaces
huffman
parents: 20829
diff changeset
  1146
22629
73771f454861 new standard proof of convergent = Cauchy
huffman
parents: 22628
diff changeset
  1147
theorem LIMSEQ_imp_Cauchy:
73771f454861 new standard proof of convergent = Cauchy
huffman
parents: 22628
diff changeset
  1148
  assumes X: "X ----> a" shows "Cauchy X"
31336
e17f13cd1280 generalize constants in SEQ.thy to class metric_space
huffman
parents: 31017
diff changeset
  1149
proof (rule metric_CauchyI)
22629
73771f454861 new standard proof of convergent = Cauchy
huffman
parents: 22628
diff changeset
  1150
  fix e::real assume "0 < e"
73771f454861 new standard proof of convergent = Cauchy
huffman
parents: 22628
diff changeset
  1151
  hence "0 < e/2" by simp
31336
e17f13cd1280 generalize constants in SEQ.thy to class metric_space
huffman
parents: 31017
diff changeset
  1152
  with X have "\<exists>N. \<forall>n\<ge>N. dist (X n) a < e/2" by (rule metric_LIMSEQ_D)
e17f13cd1280 generalize constants in SEQ.thy to class metric_space
huffman
parents: 31017
diff changeset
  1153
  then obtain N where N: "\<forall>n\<ge>N. dist (X n) a < e/2" ..
e17f13cd1280 generalize constants in SEQ.thy to class metric_space
huffman
parents: 31017
diff changeset
  1154
  show "\<exists>N. \<forall>m\<ge>N. \<forall>n\<ge>N. dist (X m) (X n) < e"
22629
73771f454861 new standard proof of convergent = Cauchy
huffman
parents: 22628
diff changeset
  1155
  proof (intro exI allI impI)
73771f454861 new standard proof of convergent = Cauchy
huffman
parents: 22628
diff changeset
  1156
    fix m assume "N \<le> m"
31336
e17f13cd1280 generalize constants in SEQ.thy to class metric_space
huffman
parents: 31017
diff changeset
  1157
    hence m: "dist (X m) a < e/2" using N by fast
22629
73771f454861 new standard proof of convergent = Cauchy
huffman
parents: 22628
diff changeset
  1158
    fix n assume "N \<le> n"
31336
e17f13cd1280 generalize constants in SEQ.thy to class metric_space
huffman
parents: 31017
diff changeset
  1159
    hence n: "dist (X n) a < e/2" using N by fast
e17f13cd1280 generalize constants in SEQ.thy to class metric_space
huffman
parents: 31017
diff changeset
  1160
    have "dist (X m) (X n) \<le> dist (X m) a + dist (X n) a"
e17f13cd1280 generalize constants in SEQ.thy to class metric_space
huffman
parents: 31017
diff changeset
  1161
      by (rule dist_triangle2)
e17f13cd1280 generalize constants in SEQ.thy to class metric_space
huffman
parents: 31017
diff changeset
  1162
    also from m n have "\<dots> < e" by simp
e17f13cd1280 generalize constants in SEQ.thy to class metric_space
huffman
parents: 31017
diff changeset
  1163
    finally show "dist (X m) (X n) < e" .
22629
73771f454861 new standard proof of convergent = Cauchy
huffman
parents: 22628
diff changeset
  1164
  qed
73771f454861 new standard proof of convergent = Cauchy
huffman
parents: 22628
diff changeset
  1165
qed
73771f454861 new standard proof of convergent = Cauchy
huffman
parents: 22628
diff changeset
  1166
20691
53cbea20e4d9 add lemma convergent_Cauchy
huffman
parents: 20685
diff changeset
  1167
lemma convergent_Cauchy: "convergent X \<Longrightarrow> Cauchy X"
22629
73771f454861 new standard proof of convergent = Cauchy
huffman
parents: 22628
diff changeset
  1168
unfolding convergent_def
73771f454861 new standard proof of convergent = Cauchy
huffman
parents: 22628
diff changeset
  1169
by (erule exE, erule LIMSEQ_imp_Cauchy)
20691
53cbea20e4d9 add lemma convergent_Cauchy
huffman
parents: 20685
diff changeset
  1170
22629
73771f454861 new standard proof of convergent = Cauchy
huffman
parents: 22628
diff changeset
  1171
text {*
73771f454861 new standard proof of convergent = Cauchy
huffman
parents: 22628
diff changeset
  1172
Proof that Cauchy sequences converge based on the one from
73771f454861 new standard proof of convergent = Cauchy
huffman
parents: 22628
diff changeset
  1173
http://pirate.shu.edu/~wachsmut/ira/numseq/proofs/cauconv.html
73771f454861 new standard proof of convergent = Cauchy
huffman
parents: 22628
diff changeset
  1174
*}
73771f454861 new standard proof of convergent = Cauchy
huffman
parents: 22628
diff changeset
  1175
73771f454861 new standard proof of convergent = Cauchy
huffman
parents: 22628
diff changeset
  1176
text {*
73771f454861 new standard proof of convergent = Cauchy
huffman
parents: 22628
diff changeset
  1177
  If sequence @{term "X"} is Cauchy, then its limit is the lub of
73771f454861 new standard proof of convergent = Cauchy
huffman
parents: 22628
diff changeset
  1178
  @{term "{r::real. \<exists>N. \<forall>n\<ge>N. r < X n}"}
73771f454861 new standard proof of convergent = Cauchy
huffman
parents: 22628
diff changeset
  1179
*}
73771f454861 new standard proof of convergent = Cauchy
huffman
parents: 22628
diff changeset
  1180
73771f454861 new standard proof of convergent = Cauchy
huffman
parents: 22628
diff changeset
  1181
lemma isUb_UNIV_I: "(\<And>y. y \<in> S \<Longrightarrow> y \<le> u) \<Longrightarrow> isUb UNIV S u"
73771f454861 new standard proof of convergent = Cauchy
huffman
parents: 22628
diff changeset
  1182
by (simp add: isUbI setleI)
73771f454861 new standard proof of convergent = Cauchy
huffman
parents: 22628
diff changeset
  1183
73771f454861 new standard proof of convergent = Cauchy
huffman
parents: 22628
diff changeset
  1184
lemma real_abs_diff_less_iff:
73771f454861 new standard proof of convergent = Cauchy
huffman
parents: 22628
diff changeset
  1185
  "(\<bar>x - a\<bar> < (r::real)) = (a - r < x \<and> x < a + r)"
73771f454861 new standard proof of convergent = Cauchy
huffman
parents: 22628
diff changeset
  1186
by auto
73771f454861 new standard proof of convergent = Cauchy
huffman
parents: 22628
diff changeset
  1187
27681
8cedebf55539 dropped locale (open)
haftmann
parents: 27543
diff changeset
  1188
locale real_Cauchy =
22629
73771f454861 new standard proof of convergent = Cauchy
huffman
parents: 22628
diff changeset
  1189
  fixes X :: "nat \<Rightarrow> real"
73771f454861 new standard proof of convergent = Cauchy
huffman
parents: 22628
diff changeset
  1190
  assumes X: "Cauchy X"
73771f454861 new standard proof of convergent = Cauchy
huffman
parents: 22628
diff changeset
  1191
  fixes S :: "real set"
73771f454861 new standard proof of convergent = Cauchy
huffman
parents: 22628
diff changeset
  1192
  defines S_def: "S \<equiv> {x::real. \<exists>N. \<forall>n\<ge>N. x < X n}"
73771f454861 new standard proof of convergent = Cauchy
huffman
parents: 22628
diff changeset
  1193
27681
8cedebf55539 dropped locale (open)
haftmann
parents: 27543
diff changeset
  1194
lemma real_CauchyI:
8cedebf55539 dropped locale (open)
haftmann
parents: 27543
diff changeset
  1195
  assumes "Cauchy X"
8cedebf55539 dropped locale (open)
haftmann
parents: 27543
diff changeset
  1196
  shows "real_Cauchy X"
28823
dcbef866c9e2 tuned unfold_locales invocation
haftmann
parents: 28562
diff changeset
  1197
  proof qed (fact assms)
27681
8cedebf55539 dropped locale (open)
haftmann
parents: 27543
diff changeset
  1198
22629
73771f454861 new standard proof of convergent = Cauchy
huffman
parents: 22628
diff changeset
  1199
lemma (in real_Cauchy) mem_S: "\<forall>n\<ge>N. x < X n \<Longrightarrow> x \<in> S"
73771f454861 new standard proof of convergent = Cauchy
huffman
parents: 22628
diff changeset
  1200
by (unfold S_def, auto)
73771f454861 new standard proof of convergent = Cauchy
huffman
parents: 22628
diff changeset
  1201
73771f454861 new standard proof of convergent = Cauchy
huffman
parents: 22628
diff changeset
  1202
lemma (in real_Cauchy) bound_isUb:
73771f454861 new standard proof of convergent = Cauchy
huffman
parents: 22628
diff changeset
  1203
  assumes N: "\<forall>n\<ge>N. X n < x"
73771f454861 new standard proof of convergent = Cauchy
huffman
parents: 22628
diff changeset
  1204
  shows "isUb UNIV S x"
73771f454861 new standard proof of convergent = Cauchy
huffman
parents: 22628
diff changeset
  1205
proof (rule isUb_UNIV_I)
73771f454861 new standard proof of convergent = Cauchy
huffman
parents: 22628
diff changeset
  1206
  fix y::real assume "y \<in> S"
73771f454861 new standard proof of convergent = Cauchy
huffman
parents: 22628
diff changeset
  1207
  hence "\<exists>M. \<forall>n\<ge>M. y < X n"
73771f454861 new standard proof of convergent = Cauchy
huffman
parents: 22628
diff changeset
  1208
    by (simp add: S_def)
73771f454861 new standard proof of convergent = Cauchy
huffman
parents: 22628
diff changeset
  1209
  then obtain M where "\<forall>n\<ge>M. y < X n" ..
73771f454861 new standard proof of convergent = Cauchy
huffman
parents: 22628
diff changeset
  1210
  hence "y < X (max M N)" by simp
73771f454861 new standard proof of convergent = Cauchy
huffman
parents: 22628
diff changeset
  1211
  also have "\<dots> < x" using N by simp
73771f454861 new standard proof of convergent = Cauchy
huffman
parents: 22628
diff changeset
  1212
  finally show "y \<le> x"
73771f454861 new standard proof of convergent = Cauchy
huffman
parents: 22628
diff changeset
  1213
    by (rule order_less_imp_le)
73771f454861 new standard proof of convergent = Cauchy
huffman
parents: 22628
diff changeset
  1214
qed
73771f454861 new standard proof of convergent = Cauchy
huffman
parents: 22628
diff changeset
  1215
73771f454861 new standard proof of convergent = Cauchy
huffman
parents: 22628
diff changeset
  1216
lemma (in real_Cauchy) isLub_ex: "\<exists>u. isLub UNIV S u"
73771f454861 new standard proof of convergent = Cauchy
huffman
parents: 22628
diff changeset
  1217
proof (rule reals_complete)
73771f454861 new standard proof of convergent = Cauchy
huffman
parents: 22628
diff changeset
  1218
  obtain N where "\<forall>m\<ge>N. \<forall>n\<ge>N. norm (X m - X n) < 1"
73771f454861 new standard proof of convergent = Cauchy
huffman
parents: 22628
diff changeset
  1219
    using CauchyD [OF X zero_less_one] by fast
73771f454861 new standard proof of convergent = Cauchy
huffman
parents: 22628
diff changeset
  1220
  hence N: "\<forall>n\<ge>N. norm (X n - X N) < 1" by simp
73771f454861 new standard proof of convergent = Cauchy
huffman
parents: 22628
diff changeset
  1221
  show "\<exists>x. x \<in> S"
73771f454861 new standard proof of convergent = Cauchy
huffman
parents: 22628
diff changeset
  1222
  proof
73771f454861 new standard proof of convergent = Cauchy
huffman
parents: 22628
diff changeset
  1223
    from N have "\<forall>n\<ge>N. X N - 1 < X n"
73771f454861 new standard proof of convergent = Cauchy
huffman
parents: 22628
diff changeset
  1224
      by (simp add: real_abs_diff_less_iff)
73771f454861 new standard proof of convergent = Cauchy
huffman
parents: 22628
diff changeset
  1225
    thus "X N - 1 \<in> S" by (rule mem_S)
73771f454861 new standard proof of convergent = Cauchy
huffman
parents: 22628
diff changeset
  1226
  qed
73771f454861 new standard proof of convergent = Cauchy
huffman
parents: 22628
diff changeset
  1227
  show "\<exists>u. isUb UNIV S u"
73771f454861 new standard proof of convergent = Cauchy
huffman
parents: 22628
diff changeset
  1228
  proof
73771f454861 new standard proof of convergent = Cauchy
huffman
parents: 22628
diff changeset
  1229
    from N have "\<forall>n\<ge>N. X n < X N + 1"
73771f454861 new standard proof of convergent = Cauchy
huffman
parents: 22628
diff changeset
  1230
      by (simp add: real_abs_diff_less_iff)
73771f454861 new standard proof of convergent = Cauchy
huffman
parents: 22628
diff changeset
  1231
    thus "isUb UNIV S (X N + 1)"
73771f454861 new standard proof of convergent = Cauchy
huffman
parents: 22628
diff changeset
  1232
      by (rule bound_isUb)
73771f454861 new standard proof of convergent = Cauchy
huffman
parents: 22628
diff changeset
  1233
  qed
73771f454861 new standard proof of convergent = Cauchy
huffman
parents: 22628
diff changeset
  1234
qed
73771f454861 new standard proof of convergent = Cauchy
huffman
parents: 22628
diff changeset
  1235
73771f454861 new standard proof of convergent = Cauchy
huffman
parents: 22628
diff changeset
  1236
lemma (in real_Cauchy) isLub_imp_LIMSEQ:
73771f454861 new standard proof of convergent = Cauchy
huffman
parents: 22628
diff changeset
  1237
  assumes x: "isLub UNIV S x"
73771f454861 new standard proof of convergent = Cauchy
huffman
parents: 22628
diff changeset
  1238
  shows "X ----> x"
73771f454861 new standard proof of convergent = Cauchy
huffman
parents: 22628
diff changeset
  1239
proof (rule LIMSEQ_I)
73771f454861 new standard proof of convergent = Cauchy
huffman
parents: 22628
diff changeset
  1240
  fix r::real assume "0 < r"
73771f454861 new standard proof of convergent = Cauchy
huffman
parents: 22628
diff changeset
  1241
  hence r: "0 < r/2" by simp
73771f454861 new standard proof of convergent = Cauchy
huffman
parents: 22628
diff changeset
  1242
  obtain N where "\<forall>n\<ge>N. \<forall>m\<ge>N. norm (X n - X m) < r/2"
73771f454861 new standard proof of convergent = Cauchy
huffman
parents: 22628
diff changeset
  1243
    using CauchyD [OF X r] by fast
73771f454861 new standard proof of convergent = Cauchy
huffman
parents: 22628
diff changeset
  1244
  hence "\<forall>n\<ge>N. norm (X n - X N) < r/2" by simp
73771f454861 new standard proof of convergent = Cauchy
huffman
parents: 22628
diff changeset
  1245
  hence N: "\<forall>n\<ge>N. X N - r/2 < X n \<and> X n < X N + r/2"
73771f454861 new standard proof of convergent = Cauchy
huffman
parents: 22628
diff changeset
  1246
    by (simp only: real_norm_def real_abs_diff_less_iff)
73771f454861 new standard proof of convergent = Cauchy
huffman
parents: 22628
diff changeset
  1247
73771f454861 new standard proof of convergent = Cauchy
huffman
parents: 22628
diff changeset
  1248
  from N have "\<forall>n\<ge>N. X N - r/2 < X n" by fast
73771f454861 new standard proof of convergent = Cauchy
huffman
parents: 22628
diff changeset
  1249
  hence "X N - r/2 \<in> S" by (rule mem_S)
23482
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1250
  hence 1: "X N - r/2 \<le> x" using x isLub_isUb isUbD by fast
22629
73771f454861 new standard proof of convergent = Cauchy
huffman
parents: 22628
diff changeset
  1251
73771f454861 new standard proof of convergent = Cauchy
huffman
parents: 22628
diff changeset
  1252
  from N have "\<forall>n\<ge>N. X n < X N + r/2" by fast
73771f454861 new standard proof of convergent = Cauchy
huffman
parents: 22628
diff changeset
  1253
  hence "isUb UNIV S (X N + r/2)" by (rule bound_isUb)
23482
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1254
  hence 2: "x \<le> X N + r/2" using x isLub_le_isUb by fast
22629
73771f454861 new standard proof of convergent = Cauchy
huffman
parents: 22628
diff changeset
  1255
73771f454861 new standard proof of convergent = Cauchy
huffman
parents: 22628
diff changeset
  1256
  show "\<exists>N. \<forall>n\<ge>N. norm (X n - x) < r"
73771f454861 new standard proof of convergent = Cauchy
huffman
parents: 22628
diff changeset
  1257
  proof (intro exI allI impI)
73771f454861 new standard proof of convergent = Cauchy
huffman
parents: 22628
diff changeset
  1258
    fix n assume n: "N \<le> n"
23482
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1259
    from N n have "X n < X N + r/2" and "X N - r/2 < X n" by simp+
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1260
    thus "norm (X n - x) < r" using 1 2
22629
73771f454861 new standard proof of convergent = Cauchy
huffman
parents: 22628
diff changeset
  1261
      by (simp add: real_abs_diff_less_iff)
73771f454861 new standard proof of convergent = Cauchy
huffman
parents: 22628
diff changeset
  1262
  qed
73771f454861 new standard proof of convergent = Cauchy
huffman
parents: 22628
diff changeset
  1263
qed
73771f454861 new standard proof of convergent = Cauchy
huffman
parents: 22628
diff changeset
  1264
73771f454861 new standard proof of convergent = Cauchy
huffman
parents: 22628
diff changeset
  1265
lemma (in real_Cauchy) LIMSEQ_ex: "\<exists>x. X ----> x"
73771f454861 new standard proof of convergent = Cauchy
huffman
parents: 22628
diff changeset
  1266
proof -
73771f454861 new standard proof of convergent = Cauchy
huffman
parents: 22628
diff changeset
  1267
  obtain x where "isLub UNIV S x"
73771f454861 new standard proof of convergent = Cauchy
huffman
parents: 22628
diff changeset
  1268
    using isLub_ex by fast
73771f454861 new standard proof of convergent = Cauchy
huffman
parents: 22628
diff changeset
  1269
  hence "X ----> x"
73771f454861 new standard proof of convergent = Cauchy
huffman
parents: 22628
diff changeset
  1270
    by (rule isLub_imp_LIMSEQ)
73771f454861 new standard proof of convergent = Cauchy
huffman
parents: 22628
diff changeset
  1271
  thus ?thesis ..
73771f454861 new standard proof of convergent = Cauchy
huffman
parents: 22628
diff changeset
  1272
qed
73771f454861 new standard proof of convergent = Cauchy
huffman
parents: 22628
diff changeset
  1273
20830
65ba80cae6df add axclass banach for complete normed vector spaces
huffman
parents: 20829
diff changeset
  1274
lemma real_Cauchy_convergent:
65ba80cae6df add axclass banach for complete normed vector spaces
huffman
parents: 20829
diff changeset
  1275
  fixes X :: "nat \<Rightarrow> real"
65ba80cae6df add axclass banach for complete normed vector spaces
huffman
parents: 20829
diff changeset
  1276
  shows "Cauchy X \<Longrightarrow> convergent X"
27681
8cedebf55539 dropped locale (open)
haftmann
parents: 27543
diff changeset
  1277
unfolding convergent_def
8cedebf55539 dropped locale (open)
haftmann
parents: 27543
diff changeset
  1278
by (rule real_Cauchy.LIMSEQ_ex)
8cedebf55539 dropped locale (open)
haftmann
parents: 27543
diff changeset
  1279
 (rule real_CauchyI)
20830
65ba80cae6df add axclass banach for complete normed vector spaces
huffman
parents: 20829
diff changeset
  1280
65ba80cae6df add axclass banach for complete normed vector spaces
huffman
parents: 20829
diff changeset
  1281
instance real :: banach
65ba80cae6df add axclass banach for complete normed vector spaces
huffman
parents: 20829
diff changeset
  1282
by intro_classes (rule real_Cauchy_convergent)
65ba80cae6df add axclass banach for complete normed vector spaces
huffman
parents: 20829
diff changeset
  1283
65ba80cae6df add axclass banach for complete normed vector spaces
huffman
parents: 20829
diff changeset
  1284
lemma Cauchy_convergent_iff:
65ba80cae6df add axclass banach for complete normed vector spaces
huffman
parents: 20829
diff changeset
  1285
  fixes X :: "nat \<Rightarrow> 'a::banach"
65ba80cae6df add axclass banach for complete normed vector spaces
huffman
parents: 20829
diff changeset
  1286
  shows "Cauchy X = convergent X"
65ba80cae6df add axclass banach for complete normed vector spaces
huffman
parents: 20829
diff changeset
  1287
by (fast intro: Cauchy_convergent convergent_Cauchy)
15082
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
  1288
30730
4d3565f2cb0e New theorems mostly concerning infinite series.
paulson
parents: 30273
diff changeset
  1289
lemma convergent_subseq_convergent:
4d3565f2cb0e New theorems mostly concerning infinite series.
paulson
parents: 30273
diff changeset
  1290
  fixes X :: "nat \<Rightarrow> 'a::banach"
4d3565f2cb0e New theorems mostly concerning infinite series.
paulson
parents: 30273
diff changeset
  1291
  shows "\<lbrakk> convergent X; subseq f \<rbrakk> \<Longrightarrow> convergent (X o f)"
31336
e17f13cd1280 generalize constants in SEQ.thy to class metric_space
huffman
parents: 31017
diff changeset
  1292
  by (simp add: Cauchy_subseq_Cauchy Cauchy_convergent_iff [symmetric])
30730
4d3565f2cb0e New theorems mostly concerning infinite series.
paulson
parents: 30273
diff changeset
  1293
15082
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
  1294
20696
3b887ad7d196 reorganized subsection headings
huffman
parents: 20695
diff changeset
  1295
subsection {* Power Sequences *}
15082
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
  1296
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
  1297
text{*The sequence @{term "x^n"} tends to 0 if @{term "0\<le>x"} and @{term
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff changeset
  1298
"x<1"}.  Proof will use (NS) Cauchy equivalence for convergence and
6c3276a2735b conversion of SEQ.ML to Isar script
paulson
parents: 13810
diff