src/HOL/SEQ.thy
author huffman
Tue Nov 30 08:58:47 2010 -0800 (2010-11-30)
changeset 40811 ab0a8cc7976a
parent 37887 2ae085b07f2f
child 41367 1b65137d598c
permissions -rw-r--r--
simplify proof of LIMSEQ_unique
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(*  Title:      HOL/SEQ.thy
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    Author:     Jacques D. Fleuriot, University of Cambridge
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    Author:     Lawrence C Paulson
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    Author:     Jeremy Avigad
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    Author:     Brian Huffman
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Convergence of sequences and series.
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*)
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header {* Sequences and Convergence *}
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theory SEQ
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imports Limits RComplete
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begin
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abbreviation
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  LIMSEQ :: "[nat \<Rightarrow> 'a::topological_space, 'a] \<Rightarrow> bool"
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    ("((_)/ ----> (_))" [60, 60] 60) where
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  "X ----> L \<equiv> (X ---> L) sequentially"
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definition
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  lim :: "(nat \<Rightarrow> 'a::metric_space) \<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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  "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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  "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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  "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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  "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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  "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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  "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 {* 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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lemma Bseq_conv_Bfun: "Bseq X \<longleftrightarrow> Bfun X sequentially"
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unfolding Bfun_def eventually_sequentially
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apply (rule iffI)
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apply (simp add: Bseq_def)
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apply (auto intro: BseqI2')
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done
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subsection {* Limits of Sequences *}
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lemma [trans]: "X=Y ==> Y ----> z ==> X ----> z"
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  by simp
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lemma LIMSEQ_def: "X ----> L = (\<forall>r>0. \<exists>no. \<forall>n\<ge>no. dist (X n) L < r)"
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unfolding tendsto_iff eventually_sequentially ..
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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_iff_nz: "X ----> L = (\<forall>r>0. \<exists>no>0. \<forall>n\<ge>no. dist (X n) L < r)"
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  unfolding LIMSEQ_def by (metis Suc_leD zero_less_Suc)
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lemma LIMSEQ_Zfun_iff: "((\<lambda>n. X n) ----> L) = Zfun (\<lambda>n. X n - L) sequentially"
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by (rule tendsto_Zfun_iff)
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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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lemma LIMSEQ_const: "(\<lambda>n. k) ----> k"
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by (rule tendsto_const)
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lemma LIMSEQ_const_iff:
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  fixes k l :: "'a::metric_space"
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  shows "(\<lambda>n. k) ----> l \<longleftrightarrow> k = l"
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by (rule tendsto_const_iff, rule sequentially_bot)
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lemma LIMSEQ_norm:
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  fixes a :: "'a::real_normed_vector"
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  shows "X ----> a \<Longrightarrow> (\<lambda>n. norm (X n)) ----> norm a"
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by (rule tendsto_norm)
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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 topological_tendstoI)
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apply (drule (2) topological_tendstoD)
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apply (simp only: eventually_sequentially)
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apply (erule exE, rename_tac N)
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apply (rule_tac x=N in exI)
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apply simp
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done
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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 topological_tendstoI)
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apply (drule (2) topological_tendstoD)
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apply (simp only: eventually_sequentially)
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apply (erule exE, rename_tac N)
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apply (rule_tac x="N + k" in exI)
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apply clarify
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apply (drule_tac x="n - k" in spec)
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apply (simp add: le_diff_conv2)
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done
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lemma LIMSEQ_Suc: "f ----> l \<Longrightarrow> (\<lambda>n. f (Suc n)) ----> l"
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by (drule_tac k="Suc 0" in LIMSEQ_ignore_initial_segment, simp)
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lemma LIMSEQ_imp_Suc: "(\<lambda>n. f (Suc n)) ----> l \<Longrightarrow> f ----> l"
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by (rule_tac k="Suc 0" in LIMSEQ_offset, simp)
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lemma LIMSEQ_Suc_iff: "(\<lambda>n. f (Suc n)) ----> l = f ----> l"
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by (blast intro: LIMSEQ_imp_Suc LIMSEQ_Suc)
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lemma LIMSEQ_linear: "\<lbrakk> X ----> x ; l > 0 \<rbrakk> \<Longrightarrow> (\<lambda> n. X (n * l)) ----> x"
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  unfolding tendsto_def eventually_sequentially
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  by (metis div_le_dividend div_mult_self1_is_m le_trans nat_mult_commute)
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lemma LIMSEQ_add:
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  fixes a b :: "'a::real_normed_vector"
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  shows "\<lbrakk>X ----> a; Y ----> b\<rbrakk> \<Longrightarrow> (\<lambda>n. X n + Y n) ----> a + b"
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by (rule tendsto_add)
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lemma LIMSEQ_minus:
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  fixes a :: "'a::real_normed_vector"
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  shows "X ----> a \<Longrightarrow> (\<lambda>n. - X n) ----> - a"
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by (rule tendsto_minus)
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lemma LIMSEQ_minus_cancel:
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  fixes a :: "'a::real_normed_vector"
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  shows "(\<lambda>n. - X n) ----> - a \<Longrightarrow> X ----> a"
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by (rule tendsto_minus_cancel)
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lemma LIMSEQ_diff:
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  fixes a b :: "'a::real_normed_vector"
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  shows "\<lbrakk>X ----> a; Y ----> b\<rbrakk> \<Longrightarrow> (\<lambda>n. X n - Y n) ----> a - b"
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by (rule tendsto_diff)
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lemma LIMSEQ_unique:
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  fixes a b :: "'a::metric_space"
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  shows "\<lbrakk>X ----> a; X ----> b\<rbrakk> \<Longrightarrow> a = b"
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by (drule (1) tendsto_dist, simp add: LIMSEQ_const_iff)
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lemma (in bounded_linear) LIMSEQ:
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  "X ----> a \<Longrightarrow> (\<lambda>n. f (X n)) ----> f a"
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by (rule tendsto)
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lemma (in bounded_bilinear) LIMSEQ:
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  "\<lbrakk>X ----> a; Y ----> b\<rbrakk> \<Longrightarrow> (\<lambda>n. X n ** Y n) ----> a ** b"
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by (rule tendsto)
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lemma LIMSEQ_mult:
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  fixes a b :: "'a::real_normed_algebra"
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  shows "[| X ----> a; Y ----> b |] ==> (%n. X n * Y n) ----> a * b"
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by (rule mult.tendsto)
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lemma increasing_LIMSEQ:
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  fixes f :: "nat \<Rightarrow> real"
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  assumes inc: "!!n. f n \<le> f (Suc n)"
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      and bdd: "!!n. f n \<le> l"
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      and en: "!!e. 0 < e \<Longrightarrow> \<exists>n. l \<le> f n + e"
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  shows "f ----> l"
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proof (auto simp add: LIMSEQ_def)
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  fix e :: real
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  assume e: "0 < e"
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  then obtain N where "l \<le> f N + e/2"
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    by (metis half_gt_zero e en that)
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  hence N: "l < f N + e" using e
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    by simp
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  { fix k
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    have [simp]: "!!n. \<bar>f n - l\<bar> = l - f n"
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      by (simp add: bdd) 
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    have "\<bar>f (N+k) - l\<bar> < e"
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    proof (induct k)
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      case 0 show ?case using N
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        by simp   
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    next
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      case (Suc k) thus ?case using N inc [of "N+k"]
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        by simp
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    qed 
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  } note 1 = this
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  { fix n
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    have "N \<le> n \<Longrightarrow> \<bar>f n - l\<bar> < e" using 1 [of "n-N"]
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      by simp 
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  } note [intro] = this
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  show " \<exists>no. \<forall>n\<ge>no. dist (f n) l < e"
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    by (auto simp add: dist_real_def) 
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  qed
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lemma Bseq_inverse_lemma:
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  fixes x :: "'a::real_normed_div_algebra"
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  shows "\<lbrakk>r \<le> norm x; 0 < r\<rbrakk> \<Longrightarrow> norm (inverse x) \<le> inverse r"
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apply (subst nonzero_norm_inverse, clarsimp)
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apply (erule (1) le_imp_inverse_le)
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done
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lemma Bseq_inverse:
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  fixes a :: "'a::real_normed_div_algebra"
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  shows "\<lbrakk>X ----> a; a \<noteq> 0\<rbrakk> \<Longrightarrow> Bseq (\<lambda>n. inverse (X n))"
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unfolding Bseq_conv_Bfun by (rule Bfun_inverse)
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lemma LIMSEQ_inverse:
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  fixes a :: "'a::real_normed_div_algebra"
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  shows "\<lbrakk>X ----> a; a \<noteq> 0\<rbrakk> \<Longrightarrow> (\<lambda>n. inverse (X n)) ----> inverse a"
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by (rule tendsto_inverse)
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lemma LIMSEQ_divide:
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  fixes a b :: "'a::real_normed_field"
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  shows "\<lbrakk>X ----> a; Y ----> b; b \<noteq> 0\<rbrakk> \<Longrightarrow> (\<lambda>n. X n / Y n) ----> a / b"
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by (rule tendsto_divide)
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lemma LIMSEQ_pow:
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  fixes a :: "'a::{power, real_normed_algebra}"
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  shows "X ----> a \<Longrightarrow> (\<lambda>n. (X n) ^ m) ----> a ^ m"
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by (induct m) (simp_all add: LIMSEQ_const LIMSEQ_mult)
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lemma LIMSEQ_setsum:
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  fixes L :: "'a \<Rightarrow> 'b::real_normed_vector"
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  assumes n: "\<And>n. n \<in> S \<Longrightarrow> X n ----> L n"
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  shows "(\<lambda>m. \<Sum>n\<in>S. X n m) ----> (\<Sum>n\<in>S. L n)"
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using assms by (rule tendsto_setsum)
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lemma LIMSEQ_setprod:
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  fixes L :: "'a \<Rightarrow> 'b::{real_normed_algebra,comm_ring_1}"
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   307
  assumes n: "\<And>n. n \<in> S \<Longrightarrow> X n ----> L n"
huffman@22608
   308
  shows "(\<lambda>m. \<Prod>n\<in>S. X n m) ----> (\<Prod>n\<in>S. L n)"
huffman@22608
   309
proof (cases "finite S")
huffman@22608
   310
  case True
huffman@22608
   311
  thus ?thesis using n
huffman@22608
   312
  proof (induct)
huffman@22608
   313
    case empty
huffman@22608
   314
    show ?case
huffman@22608
   315
      by (simp add: LIMSEQ_const)
huffman@22608
   316
  next
huffman@22608
   317
    case insert
huffman@22608
   318
    thus ?case
huffman@22608
   319
      by (simp add: LIMSEQ_mult)
huffman@22608
   320
  qed
huffman@22608
   321
next
huffman@22608
   322
  case False
huffman@22608
   323
  thus ?thesis
huffman@22608
   324
    by (simp add: setprod_def LIMSEQ_const)
huffman@22608
   325
qed
huffman@22608
   326
huffman@36660
   327
lemma LIMSEQ_add_const: (* FIXME: delete *)
huffman@31336
   328
  fixes a :: "'a::real_normed_vector"
huffman@31336
   329
  shows "f ----> a ==> (%n.(f n + b)) ----> a + b"
huffman@36660
   330
by (intro tendsto_intros)
huffman@22614
   331
huffman@22614
   332
(* FIXME: delete *)
huffman@22614
   333
lemma LIMSEQ_add_minus:
huffman@31336
   334
  fixes a b :: "'a::real_normed_vector"
huffman@31336
   335
  shows "[| X ----> a; Y ----> b |] ==> (%n. X n + -Y n) ----> a + -b"
huffman@36660
   336
by (intro tendsto_intros)
huffman@22614
   337
huffman@36660
   338
lemma LIMSEQ_diff_const: (* FIXME: delete *)
huffman@31336
   339
  fixes a b :: "'a::real_normed_vector"
huffman@31336
   340
  shows "f ----> a ==> (%n.(f n  - b)) ----> a - b"
huffman@36660
   341
by (intro tendsto_intros)
huffman@22614
   342
huffman@31336
   343
lemma LIMSEQ_diff_approach_zero:
huffman@31336
   344
  fixes L :: "'a::real_normed_vector"
huffman@31336
   345
  shows "g ----> L ==> (%x. f x - g x) ----> 0 ==> f ----> L"
huffman@31336
   346
by (drule (1) LIMSEQ_add, simp)
huffman@22614
   347
huffman@31336
   348
lemma LIMSEQ_diff_approach_zero2:
huffman@31336
   349
  fixes L :: "'a::real_normed_vector"
hoelzl@35292
   350
  shows "f ----> L ==> (%x. f x - g x) ----> 0 ==> g ----> L"
huffman@31336
   351
by (drule (1) LIMSEQ_diff, simp)
huffman@22614
   352
huffman@22614
   353
text{*A sequence tends to zero iff its abs does*}
huffman@31336
   354
lemma LIMSEQ_norm_zero:
huffman@31336
   355
  fixes X :: "nat \<Rightarrow> 'a::real_normed_vector"
huffman@31336
   356
  shows "((\<lambda>n. norm (X n)) ----> 0) \<longleftrightarrow> (X ----> 0)"
huffman@31336
   357
by (simp add: LIMSEQ_iff)
huffman@22614
   358
huffman@22614
   359
lemma LIMSEQ_rabs_zero: "((%n. \<bar>f n\<bar>) ----> 0) = (f ----> (0::real))"
huffman@31336
   360
by (simp add: LIMSEQ_iff)
huffman@22614
   361
huffman@22614
   362
lemma LIMSEQ_imp_rabs: "f ----> (l::real) ==> (%n. \<bar>f n\<bar>) ----> \<bar>l\<bar>"
huffman@22614
   363
by (drule LIMSEQ_norm, simp)
huffman@22614
   364
huffman@22614
   365
text{*An unbounded sequence's inverse tends to 0*}
huffman@22614
   366
huffman@22614
   367
lemma LIMSEQ_inverse_zero:
huffman@22974
   368
  "\<forall>r::real. \<exists>N. \<forall>n\<ge>N. r < X n \<Longrightarrow> (\<lambda>n. inverse (X n)) ----> 0"
huffman@22974
   369
apply (rule LIMSEQ_I)
huffman@22974
   370
apply (drule_tac x="inverse r" in spec, safe)
huffman@22974
   371
apply (rule_tac x="N" in exI, safe)
huffman@22974
   372
apply (drule_tac x="n" in spec, safe)
huffman@22614
   373
apply (frule positive_imp_inverse_positive)
huffman@22974
   374
apply (frule (1) less_imp_inverse_less)
huffman@22974
   375
apply (subgoal_tac "0 < X n", simp)
huffman@22974
   376
apply (erule (1) order_less_trans)
huffman@22614
   377
done
huffman@22614
   378
huffman@22614
   379
text{*The sequence @{term "1/n"} tends to 0 as @{term n} tends to infinity*}
huffman@22614
   380
huffman@22614
   381
lemma LIMSEQ_inverse_real_of_nat: "(%n. inverse(real(Suc n))) ----> 0"
huffman@22614
   382
apply (rule LIMSEQ_inverse_zero, safe)
huffman@22974
   383
apply (cut_tac x = r in reals_Archimedean2)
huffman@22614
   384
apply (safe, rule_tac x = n in exI)
huffman@22614
   385
apply (auto simp add: real_of_nat_Suc)
huffman@22614
   386
done
huffman@22614
   387
huffman@22614
   388
text{*The sequence @{term "r + 1/n"} tends to @{term r} as @{term n} tends to
huffman@22614
   389
infinity is now easily proved*}
huffman@22614
   390
huffman@22614
   391
lemma LIMSEQ_inverse_real_of_nat_add:
huffman@22614
   392
     "(%n. r + inverse(real(Suc n))) ----> r"
huffman@22614
   393
by (cut_tac LIMSEQ_add [OF LIMSEQ_const LIMSEQ_inverse_real_of_nat], auto)
huffman@22614
   394
huffman@22614
   395
lemma LIMSEQ_inverse_real_of_nat_add_minus:
huffman@22614
   396
     "(%n. r + -inverse(real(Suc n))) ----> r"
huffman@22614
   397
by (cut_tac LIMSEQ_add_minus [OF LIMSEQ_const LIMSEQ_inverse_real_of_nat], auto)
huffman@22614
   398
huffman@22614
   399
lemma LIMSEQ_inverse_real_of_nat_add_minus_mult:
huffman@22614
   400
     "(%n. r*( 1 + -inverse(real(Suc n)))) ----> r"
huffman@22614
   401
by (cut_tac b=1 in
huffman@22614
   402
        LIMSEQ_mult [OF LIMSEQ_const LIMSEQ_inverse_real_of_nat_add_minus], auto)
huffman@22614
   403
huffman@22615
   404
lemma LIMSEQ_le_const:
huffman@22615
   405
  "\<lbrakk>X ----> (x::real); \<exists>N. \<forall>n\<ge>N. a \<le> X n\<rbrakk> \<Longrightarrow> a \<le> x"
huffman@22615
   406
apply (rule ccontr, simp only: linorder_not_le)
huffman@22615
   407
apply (drule_tac r="a - x" in LIMSEQ_D, simp)
huffman@22615
   408
apply clarsimp
huffman@22615
   409
apply (drule_tac x="max N no" in spec, drule mp, rule le_maxI1)
huffman@22615
   410
apply (drule_tac x="max N no" in spec, drule mp, rule le_maxI2)
huffman@22615
   411
apply simp
huffman@22615
   412
done
huffman@22615
   413
huffman@22615
   414
lemma LIMSEQ_le_const2:
huffman@22615
   415
  "\<lbrakk>X ----> (x::real); \<exists>N. \<forall>n\<ge>N. X n \<le> a\<rbrakk> \<Longrightarrow> x \<le> a"
huffman@22615
   416
apply (subgoal_tac "- a \<le> - x", simp)
huffman@22615
   417
apply (rule LIMSEQ_le_const)
huffman@22615
   418
apply (erule LIMSEQ_minus)
huffman@22615
   419
apply simp
huffman@22615
   420
done
huffman@22615
   421
huffman@22615
   422
lemma LIMSEQ_le:
huffman@22615
   423
  "\<lbrakk>X ----> x; Y ----> y; \<exists>N. \<forall>n\<ge>N. X n \<le> Y n\<rbrakk> \<Longrightarrow> x \<le> (y::real)"
huffman@22615
   424
apply (subgoal_tac "0 \<le> y - x", simp)
huffman@22615
   425
apply (rule LIMSEQ_le_const)
huffman@22615
   426
apply (erule (1) LIMSEQ_diff)
huffman@22615
   427
apply (simp add: le_diff_eq)
huffman@22615
   428
done
huffman@22615
   429
paulson@15082
   430
huffman@20696
   431
subsection {* Convergence *}
paulson@15082
   432
paulson@15082
   433
lemma limI: "X ----> L ==> lim X = L"
paulson@15082
   434
apply (simp add: lim_def)
paulson@15082
   435
apply (blast intro: LIMSEQ_unique)
paulson@15082
   436
done
paulson@15082
   437
paulson@15082
   438
lemma convergentD: "convergent X ==> \<exists>L. (X ----> L)"
paulson@15082
   439
by (simp add: convergent_def)
paulson@15082
   440
paulson@15082
   441
lemma convergentI: "(X ----> L) ==> convergent X"
paulson@15082
   442
by (auto simp add: convergent_def)
paulson@15082
   443
paulson@15082
   444
lemma convergent_LIMSEQ_iff: "convergent X = (X ----> lim X)"
huffman@20682
   445
by (auto intro: theI LIMSEQ_unique simp add: convergent_def lim_def)
paulson@15082
   446
huffman@36625
   447
lemma convergent_const: "convergent (\<lambda>n. c)"
huffman@36625
   448
by (rule convergentI, rule LIMSEQ_const)
huffman@36625
   449
huffman@36625
   450
lemma convergent_add:
huffman@36625
   451
  fixes X Y :: "nat \<Rightarrow> 'a::real_normed_vector"
huffman@36625
   452
  assumes "convergent (\<lambda>n. X n)"
huffman@36625
   453
  assumes "convergent (\<lambda>n. Y n)"
huffman@36625
   454
  shows "convergent (\<lambda>n. X n + Y n)"
huffman@36625
   455
using assms unfolding convergent_def by (fast intro: LIMSEQ_add)
huffman@36625
   456
huffman@36625
   457
lemma convergent_setsum:
huffman@36625
   458
  fixes X :: "'a \<Rightarrow> nat \<Rightarrow> 'b::real_normed_vector"
hoelzl@36647
   459
  assumes "\<And>i. i \<in> A \<Longrightarrow> convergent (\<lambda>n. X i n)"
huffman@36625
   460
  shows "convergent (\<lambda>n. \<Sum>i\<in>A. X i n)"
hoelzl@36647
   461
proof (cases "finite A")
wenzelm@36650
   462
  case True from this and assms show ?thesis
hoelzl@36647
   463
    by (induct A set: finite) (simp_all add: convergent_const convergent_add)
hoelzl@36647
   464
qed (simp add: convergent_const)
huffman@36625
   465
huffman@36625
   466
lemma (in bounded_linear) convergent:
huffman@36625
   467
  assumes "convergent (\<lambda>n. X n)"
huffman@36625
   468
  shows "convergent (\<lambda>n. f (X n))"
huffman@36625
   469
using assms unfolding convergent_def by (fast intro: LIMSEQ)
huffman@36625
   470
huffman@36625
   471
lemma (in bounded_bilinear) convergent:
huffman@36625
   472
  assumes "convergent (\<lambda>n. X n)" and "convergent (\<lambda>n. Y n)"
huffman@36625
   473
  shows "convergent (\<lambda>n. X n ** Y n)"
huffman@36625
   474
using assms unfolding convergent_def by (fast intro: LIMSEQ)
huffman@36625
   475
huffman@31336
   476
lemma convergent_minus_iff:
huffman@31336
   477
  fixes X :: "nat \<Rightarrow> 'a::real_normed_vector"
huffman@31336
   478
  shows "convergent X \<longleftrightarrow> convergent (\<lambda>n. - X n)"
huffman@20696
   479
apply (simp add: convergent_def)
huffman@20696
   480
apply (auto dest: LIMSEQ_minus)
huffman@20696
   481
apply (drule LIMSEQ_minus, auto)
huffman@20696
   482
done
huffman@20696
   483
paulson@32707
   484
lemma lim_le:
paulson@32707
   485
  fixes x :: real
paulson@32707
   486
  assumes f: "convergent f" and fn_le: "!!n. f n \<le> x"
paulson@32707
   487
  shows "lim f \<le> x"
paulson@32707
   488
proof (rule classical)
paulson@32707
   489
  assume "\<not> lim f \<le> x"
paulson@32707
   490
  hence 0: "0 < lim f - x" by arith
paulson@32707
   491
  have 1: "f----> lim f"
paulson@32707
   492
    by (metis convergent_LIMSEQ_iff f) 
paulson@32707
   493
  thus ?thesis
paulson@32707
   494
    proof (simp add: LIMSEQ_iff)
paulson@32707
   495
      assume "\<forall>r>0. \<exists>no. \<forall>n\<ge>no. \<bar>f n - lim f\<bar> < r"
paulson@32707
   496
      hence "\<exists>no. \<forall>n\<ge>no. \<bar>f n - lim f\<bar> < lim f - x"
wenzelm@32960
   497
        by (metis 0)
paulson@32707
   498
      from this obtain no where "\<forall>n\<ge>no. \<bar>f n - lim f\<bar> < lim f - x"
wenzelm@32960
   499
        by blast
paulson@32707
   500
      thus "lim f \<le> x"
haftmann@37887
   501
        by (metis 1 LIMSEQ_le_const2 fn_le)
paulson@32707
   502
    qed
paulson@32707
   503
qed
paulson@32707
   504
chaieb@30196
   505
text{* Given a binary function @{text "f:: nat \<Rightarrow> 'a \<Rightarrow> 'a"}, its values are uniquely determined by a function g *}
huffman@20696
   506
chaieb@30196
   507
lemma nat_function_unique: "EX! g. g 0 = e \<and> (\<forall>n. g (Suc n) = f n (g n))"
chaieb@30196
   508
  unfolding Ex1_def
chaieb@30196
   509
  apply (rule_tac x="nat_rec e f" in exI)
chaieb@30196
   510
  apply (rule conjI)+
chaieb@30196
   511
apply (rule def_nat_rec_0, simp)
chaieb@30196
   512
apply (rule allI, rule def_nat_rec_Suc, simp)
chaieb@30196
   513
apply (rule allI, rule impI, rule ext)
chaieb@30196
   514
apply (erule conjE)
chaieb@30196
   515
apply (induct_tac x)
huffman@35216
   516
apply simp
chaieb@30196
   517
apply (erule_tac x="n" in allE)
chaieb@30196
   518
apply (simp)
chaieb@30196
   519
done
huffman@20696
   520
paulson@15082
   521
text{*Subsequence (alternative definition, (e.g. Hoskins)*}
paulson@15082
   522
paulson@15082
   523
lemma subseq_Suc_iff: "subseq f = (\<forall>n. (f n) < (f (Suc n)))"
paulson@15082
   524
apply (simp add: subseq_def)
paulson@15082
   525
apply (auto dest!: less_imp_Suc_add)
paulson@15082
   526
apply (induct_tac k)
paulson@15082
   527
apply (auto intro: less_trans)
paulson@15082
   528
done
paulson@15082
   529
paulson@15082
   530
lemma monoseq_Suc:
paulson@15082
   531
   "monoseq X = ((\<forall>n. X n \<le> X (Suc n))
paulson@15082
   532
                 | (\<forall>n. X (Suc n) \<le> X n))"
paulson@15082
   533
apply (simp add: monoseq_def)
paulson@15082
   534
apply (auto dest!: le_imp_less_or_eq)
paulson@15082
   535
apply (auto intro!: lessI [THEN less_imp_le] dest!: less_imp_Suc_add)
paulson@15082
   536
apply (induct_tac "ka")
paulson@15082
   537
apply (auto intro: order_trans)
wenzelm@18585
   538
apply (erule contrapos_np)
paulson@15082
   539
apply (induct_tac "k")
paulson@15082
   540
apply (auto intro: order_trans)
paulson@15082
   541
done
paulson@15082
   542
nipkow@15360
   543
lemma monoI1: "\<forall>m. \<forall> n \<ge> m. X m \<le> X n ==> monoseq X"
paulson@15082
   544
by (simp add: monoseq_def)
paulson@15082
   545
nipkow@15360
   546
lemma monoI2: "\<forall>m. \<forall> n \<ge> m. X n \<le> X m ==> monoseq X"
paulson@15082
   547
by (simp add: monoseq_def)
paulson@15082
   548
paulson@15082
   549
lemma mono_SucI1: "\<forall>n. X n \<le> X (Suc n) ==> monoseq X"
paulson@15082
   550
by (simp add: monoseq_Suc)
paulson@15082
   551
paulson@15082
   552
lemma mono_SucI2: "\<forall>n. X (Suc n) \<le> X n ==> monoseq X"
paulson@15082
   553
by (simp add: monoseq_Suc)
paulson@15082
   554
hoelzl@29803
   555
lemma monoseq_minus: assumes "monoseq a"
hoelzl@29803
   556
  shows "monoseq (\<lambda> n. - a n)"
hoelzl@29803
   557
proof (cases "\<forall> m. \<forall> n \<ge> m. a m \<le> a n")
hoelzl@29803
   558
  case True
hoelzl@29803
   559
  hence "\<forall> m. \<forall> n \<ge> m. - a n \<le> - a m" by auto
hoelzl@29803
   560
  thus ?thesis by (rule monoI2)
hoelzl@29803
   561
next
hoelzl@29803
   562
  case False
hoelzl@29803
   563
  hence "\<forall> m. \<forall> n \<ge> m. - a m \<le> - a n" using `monoseq a`[unfolded monoseq_def] by auto
hoelzl@29803
   564
  thus ?thesis by (rule monoI1)
hoelzl@29803
   565
qed
hoelzl@29803
   566
hoelzl@29803
   567
lemma monoseq_le: assumes "monoseq a" and "a ----> x"
hoelzl@29803
   568
  shows "((\<forall> n. a n \<le> x) \<and> (\<forall>m. \<forall>n\<ge>m. a m \<le> a n)) \<or> 
hoelzl@29803
   569
         ((\<forall> n. x \<le> a n) \<and> (\<forall>m. \<forall>n\<ge>m. a n \<le> a m))"
hoelzl@29803
   570
proof -
hoelzl@29803
   571
  { fix x n fix a :: "nat \<Rightarrow> real"
hoelzl@29803
   572
    assume "a ----> x" and "\<forall> m. \<forall> n \<ge> m. a m \<le> a n"
hoelzl@29803
   573
    hence monotone: "\<And> m n. m \<le> n \<Longrightarrow> a m \<le> a n" by auto
hoelzl@29803
   574
    have "a n \<le> x"
hoelzl@29803
   575
    proof (rule ccontr)
hoelzl@29803
   576
      assume "\<not> a n \<le> x" hence "x < a n" by auto
hoelzl@29803
   577
      hence "0 < a n - x" by auto
hoelzl@29803
   578
      from `a ----> x`[THEN LIMSEQ_D, OF this]
hoelzl@29803
   579
      obtain no where "\<And>n'. no \<le> n' \<Longrightarrow> norm (a n' - x) < a n - x" by blast
hoelzl@29803
   580
      hence "norm (a (max no n) - x) < a n - x" by auto
hoelzl@29803
   581
      moreover
hoelzl@29803
   582
      { fix n' have "n \<le> n' \<Longrightarrow> x < a n'" using monotone[where m=n and n=n'] and `x < a n` by auto }
hoelzl@29803
   583
      hence "x < a (max no n)" by auto
hoelzl@29803
   584
      ultimately
hoelzl@29803
   585
      have "a (max no n) < a n" by auto
hoelzl@29803
   586
      with monotone[where m=n and n="max no n"]
nipkow@32436
   587
      show False by (auto simp:max_def split:split_if_asm)
hoelzl@29803
   588
    qed
hoelzl@29803
   589
  } note top_down = this
hoelzl@29803
   590
  { fix x n m fix a :: "nat \<Rightarrow> real"
hoelzl@29803
   591
    assume "a ----> x" and "monoseq a" and "a m < x"
hoelzl@29803
   592
    have "a n \<le> x \<and> (\<forall> m. \<forall> n \<ge> m. a m \<le> a n)"
hoelzl@29803
   593
    proof (cases "\<forall> m. \<forall> n \<ge> m. a m \<le> a n")
hoelzl@29803
   594
      case True with top_down and `a ----> x` show ?thesis by auto
hoelzl@29803
   595
    next
hoelzl@29803
   596
      case False with `monoseq a`[unfolded monoseq_def] have "\<forall> m. \<forall> n \<ge> m. - a m \<le> - a n" by auto
hoelzl@29803
   597
      hence "- a m \<le> - x" using top_down[OF LIMSEQ_minus[OF `a ----> x`]] by blast
hoelzl@29803
   598
      hence False using `a m < x` by auto
hoelzl@29803
   599
      thus ?thesis ..
hoelzl@29803
   600
    qed
hoelzl@29803
   601
  } note when_decided = this
hoelzl@29803
   602
hoelzl@29803
   603
  show ?thesis
hoelzl@29803
   604
  proof (cases "\<exists> m. a m \<noteq> x")
hoelzl@29803
   605
    case True then obtain m where "a m \<noteq> x" by auto
hoelzl@29803
   606
    show ?thesis
hoelzl@29803
   607
    proof (cases "a m < x")
hoelzl@29803
   608
      case True with when_decided[OF `a ----> x` `monoseq a`, where m2=m]
hoelzl@29803
   609
      show ?thesis by blast
hoelzl@29803
   610
    next
hoelzl@29803
   611
      case False hence "- a m < - x" using `a m \<noteq> x` by auto
hoelzl@29803
   612
      with when_decided[OF LIMSEQ_minus[OF `a ----> x`] monoseq_minus[OF `monoseq a`], where m2=m]
hoelzl@29803
   613
      show ?thesis by auto
hoelzl@29803
   614
    qed
hoelzl@29803
   615
  qed auto
hoelzl@29803
   616
qed
hoelzl@29803
   617
chaieb@30196
   618
text{* for any sequence, there is a mootonic subsequence *}
chaieb@30196
   619
lemma seq_monosub: "\<exists>f. subseq f \<and> monoseq (\<lambda> n. (s (f n)))"
chaieb@30196
   620
proof-
chaieb@30196
   621
  {assume H: "\<forall>n. \<exists>p >n. \<forall> m\<ge>p. s m \<le> s p"
chaieb@30196
   622
    let ?P = "\<lambda> p n. p > n \<and> (\<forall>m \<ge> p. s m \<le> s p)"
chaieb@30196
   623
    from nat_function_unique[of "SOME p. ?P p 0" "\<lambda>p n. SOME p. ?P p n"]
chaieb@30196
   624
    obtain f where f: "f 0 = (SOME p. ?P p 0)" "\<forall>n. f (Suc n) = (SOME p. ?P p (f n))" by blast
chaieb@30196
   625
    have "?P (f 0) 0"  unfolding f(1) some_eq_ex[of "\<lambda>p. ?P p 0"]
chaieb@30196
   626
      using H apply - 
chaieb@30196
   627
      apply (erule allE[where x=0], erule exE, rule_tac x="p" in exI) 
chaieb@30196
   628
      unfolding order_le_less by blast 
chaieb@30196
   629
    hence f0: "f 0 > 0" "\<forall>m \<ge> f 0. s m \<le> s (f 0)" by blast+
chaieb@30196
   630
    {fix n
chaieb@30196
   631
      have "?P (f (Suc n)) (f n)" 
wenzelm@32960
   632
        unfolding f(2)[rule_format, of n] some_eq_ex[of "\<lambda>p. ?P p (f n)"]
wenzelm@32960
   633
        using H apply - 
chaieb@30196
   634
      apply (erule allE[where x="f n"], erule exE, rule_tac x="p" in exI) 
chaieb@30196
   635
      unfolding order_le_less by blast 
chaieb@30196
   636
    hence "f (Suc n) > f n" "\<forall>m \<ge> f (Suc n). s m \<le> s (f (Suc n))" by blast+}
chaieb@30196
   637
  note fSuc = this
chaieb@30196
   638
    {fix p q assume pq: "p \<ge> f q"
chaieb@30196
   639
      have "s p \<le> s(f(q))"  using f0(2)[rule_format, of p] pq fSuc
wenzelm@32960
   640
        by (cases q, simp_all) }
chaieb@30196
   641
    note pqth = this
chaieb@30196
   642
    {fix q
chaieb@30196
   643
      have "f (Suc q) > f q" apply (induct q) 
wenzelm@32960
   644
        using f0(1) fSuc(1)[of 0] apply simp by (rule fSuc(1))}
chaieb@30196
   645
    note fss = this
chaieb@30196
   646
    from fss have th1: "subseq f" unfolding subseq_Suc_iff ..
chaieb@30196
   647
    {fix a b 
chaieb@30196
   648
      have "f a \<le> f (a + b)"
chaieb@30196
   649
      proof(induct b)
wenzelm@32960
   650
        case 0 thus ?case by simp
chaieb@30196
   651
      next
wenzelm@32960
   652
        case (Suc b)
wenzelm@32960
   653
        from fSuc(1)[of "a + b"] Suc.hyps show ?case by simp
chaieb@30196
   654
      qed}
chaieb@30196
   655
    note fmon0 = this
chaieb@30196
   656
    have "monoseq (\<lambda>n. s (f n))" 
chaieb@30196
   657
    proof-
chaieb@30196
   658
      {fix n
wenzelm@32960
   659
        have "s (f n) \<ge> s (f (Suc n))" 
wenzelm@32960
   660
        proof(cases n)
wenzelm@32960
   661
          case 0
wenzelm@32960
   662
          assume n0: "n = 0"
wenzelm@32960
   663
          from fSuc(1)[of 0] have th0: "f 0 \<le> f (Suc 0)" by simp
wenzelm@32960
   664
          from f0(2)[rule_format, OF th0] show ?thesis  using n0 by simp
wenzelm@32960
   665
        next
wenzelm@32960
   666
          case (Suc m)
wenzelm@32960
   667
          assume m: "n = Suc m"
wenzelm@32960
   668
          from fSuc(1)[of n] m have th0: "f (Suc m) \<le> f (Suc (Suc m))" by simp
wenzelm@32960
   669
          from m fSuc(2)[rule_format, OF th0] show ?thesis by simp 
wenzelm@32960
   670
        qed}
chaieb@30196
   671
      thus "monoseq (\<lambda>n. s (f n))" unfolding monoseq_Suc by blast 
chaieb@30196
   672
    qed
chaieb@30196
   673
    with th1 have ?thesis by blast}
chaieb@30196
   674
  moreover
chaieb@30196
   675
  {fix N assume N: "\<forall>p >N. \<exists> m\<ge>p. s m > s p"
chaieb@30196
   676
    {fix p assume p: "p \<ge> Suc N" 
chaieb@30196
   677
      hence pN: "p > N" by arith with N obtain m where m: "m \<ge> p" "s m > s p" by blast
chaieb@30196
   678
      have "m \<noteq> p" using m(2) by auto 
chaieb@30196
   679
      with m have "\<exists>m>p. s p < s m" by - (rule exI[where x=m], auto)}
chaieb@30196
   680
    note th0 = this
chaieb@30196
   681
    let ?P = "\<lambda>m x. m > x \<and> s x < s m"
chaieb@30196
   682
    from nat_function_unique[of "SOME x. ?P x (Suc N)" "\<lambda>m x. SOME y. ?P y x"]
chaieb@30196
   683
    obtain f where f: "f 0 = (SOME x. ?P x (Suc N))" 
chaieb@30196
   684
      "\<forall>n. f (Suc n) = (SOME m. ?P m (f n))" by blast
chaieb@30196
   685
    have "?P (f 0) (Suc N)"  unfolding f(1) some_eq_ex[of "\<lambda>p. ?P p (Suc N)"]
chaieb@30196
   686
      using N apply - 
chaieb@30196
   687
      apply (erule allE[where x="Suc N"], clarsimp)
chaieb@30196
   688
      apply (rule_tac x="m" in exI)
chaieb@30196
   689
      apply auto
chaieb@30196
   690
      apply (subgoal_tac "Suc N \<noteq> m")
chaieb@30196
   691
      apply simp
chaieb@30196
   692
      apply (rule ccontr, simp)
chaieb@30196
   693
      done
chaieb@30196
   694
    hence f0: "f 0 > Suc N" "s (Suc N) < s (f 0)" by blast+
chaieb@30196
   695
    {fix n
chaieb@30196
   696
      have "f n > N \<and> ?P (f (Suc n)) (f n)"
wenzelm@32960
   697
        unfolding f(2)[rule_format, of n] some_eq_ex[of "\<lambda>p. ?P p (f n)"]
chaieb@30196
   698
      proof (induct n)
wenzelm@32960
   699
        case 0 thus ?case
wenzelm@32960
   700
          using f0 N apply auto 
wenzelm@32960
   701
          apply (erule allE[where x="f 0"], clarsimp) 
wenzelm@32960
   702
          apply (rule_tac x="m" in exI, simp)
wenzelm@32960
   703
          by (subgoal_tac "f 0 \<noteq> m", auto)
chaieb@30196
   704
      next
wenzelm@32960
   705
        case (Suc n)
wenzelm@32960
   706
        from Suc.hyps have Nfn: "N < f n" by blast
wenzelm@32960
   707
        from Suc.hyps obtain m where m: "m > f n" "s (f n) < s m" by blast
wenzelm@32960
   708
        with Nfn have mN: "m > N" by arith
wenzelm@32960
   709
        note key = Suc.hyps[unfolded some_eq_ex[of "\<lambda>p. ?P p (f n)", symmetric] f(2)[rule_format, of n, symmetric]]
wenzelm@32960
   710
        
wenzelm@32960
   711
        from key have th0: "f (Suc n) > N" by simp
wenzelm@32960
   712
        from N[rule_format, OF th0]
wenzelm@32960
   713
        obtain m' where m': "m' \<ge> f (Suc n)" "s (f (Suc n)) < s m'" by blast
wenzelm@32960
   714
        have "m' \<noteq> f (Suc (n))" apply (rule ccontr) using m'(2) by auto
wenzelm@32960
   715
        hence "m' > f (Suc n)" using m'(1) by simp
wenzelm@32960
   716
        with key m'(2) show ?case by auto
chaieb@30196
   717
      qed}
chaieb@30196
   718
    note fSuc = this
chaieb@30196
   719
    {fix n
chaieb@30196
   720
      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 
chaieb@30196
   721
      hence "f n \<ge> Suc N" "f(Suc n) > f n" "s(f n) < s(f(Suc n))" by blast+}
chaieb@30196
   722
    note thf = this
chaieb@30196
   723
    have sqf: "subseq f" unfolding subseq_Suc_iff using thf by simp
chaieb@30196
   724
    have "monoseq (\<lambda>n. s (f n))"  unfolding monoseq_Suc using thf
chaieb@30196
   725
      apply -
chaieb@30196
   726
      apply (rule disjI1)
chaieb@30196
   727
      apply auto
chaieb@30196
   728
      apply (rule order_less_imp_le)
chaieb@30196
   729
      apply blast
chaieb@30196
   730
      done
chaieb@30196
   731
    then have ?thesis  using sqf by blast}
chaieb@30196
   732
  ultimately show ?thesis unfolding linorder_not_less[symmetric] by blast
chaieb@30196
   733
qed
chaieb@30196
   734
chaieb@30196
   735
lemma seq_suble: assumes sf: "subseq f" shows "n \<le> f n"
chaieb@30196
   736
proof(induct n)
chaieb@30196
   737
  case 0 thus ?case by simp
chaieb@30196
   738
next
chaieb@30196
   739
  case (Suc n)
chaieb@30196
   740
  from sf[unfolded subseq_Suc_iff, rule_format, of n] Suc.hyps
chaieb@30196
   741
  have "n < f (Suc n)" by arith 
chaieb@30196
   742
  thus ?case by arith
chaieb@30196
   743
qed
chaieb@30196
   744
paulson@30730
   745
lemma LIMSEQ_subseq_LIMSEQ:
paulson@30730
   746
  "\<lbrakk> X ----> L; subseq f \<rbrakk> \<Longrightarrow> (X o f) ----> L"
huffman@36662
   747
apply (rule topological_tendstoI)
huffman@36662
   748
apply (drule (2) topological_tendstoD)
huffman@36662
   749
apply (simp only: eventually_sequentially)
huffman@36662
   750
apply (clarify, rule_tac x=N in exI, clarsimp)
paulson@30730
   751
apply (blast intro: seq_suble le_trans dest!: spec) 
paulson@30730
   752
done
paulson@30730
   753
chaieb@30196
   754
subsection {* Bounded Monotonic Sequences *}
chaieb@30196
   755
chaieb@30196
   756
huffman@20696
   757
text{*Bounded Sequence*}
paulson@15082
   758
huffman@20552
   759
lemma BseqD: "Bseq X ==> \<exists>K. 0 < K & (\<forall>n. norm (X n) \<le> K)"
paulson@15082
   760
by (simp add: Bseq_def)
paulson@15082
   761
huffman@20552
   762
lemma BseqI: "[| 0 < K; \<forall>n. norm (X n) \<le> K |] ==> Bseq X"
paulson@15082
   763
by (auto simp add: Bseq_def)
paulson@15082
   764
paulson@15082
   765
lemma lemma_NBseq_def:
huffman@20552
   766
     "(\<exists>K > 0. \<forall>n. norm (X n) \<le> K) =
huffman@20552
   767
      (\<exists>N. \<forall>n. norm (X n) \<le> real(Suc N))"
haftmann@32064
   768
proof auto
haftmann@32064
   769
  fix K :: real
haftmann@32064
   770
  from reals_Archimedean2 obtain n :: nat where "K < real n" ..
haftmann@32064
   771
  then have "K \<le> real (Suc n)" by auto
haftmann@32064
   772
  assume "\<forall>m. norm (X m) \<le> K"
haftmann@32064
   773
  have "\<forall>m. norm (X m) \<le> real (Suc n)"
haftmann@32064
   774
  proof
haftmann@32064
   775
    fix m :: 'a
haftmann@32064
   776
    from `\<forall>m. norm (X m) \<le> K` have "norm (X m) \<le> K" ..
haftmann@32064
   777
    with `K \<le> real (Suc n)` show "norm (X m) \<le> real (Suc n)" by auto
haftmann@32064
   778
  qed
haftmann@32064
   779
  then show "\<exists>N. \<forall>n. norm (X n) \<le> real (Suc N)" ..
haftmann@32064
   780
next
haftmann@32064
   781
  fix N :: nat
haftmann@32064
   782
  have "real (Suc N) > 0" by (simp add: real_of_nat_Suc)
haftmann@32064
   783
  moreover assume "\<forall>n. norm (X n) \<le> real (Suc N)"
haftmann@32064
   784
  ultimately show "\<exists>K>0. \<forall>n. norm (X n) \<le> K" by blast
haftmann@32064
   785
qed
haftmann@32064
   786
paulson@15082
   787
paulson@15082
   788
text{* alternative definition for Bseq *}
huffman@20552
   789
lemma Bseq_iff: "Bseq X = (\<exists>N. \<forall>n. norm (X n) \<le> real(Suc N))"
paulson@15082
   790
apply (simp add: Bseq_def)
paulson@15082
   791
apply (simp (no_asm) add: lemma_NBseq_def)
paulson@15082
   792
done
paulson@15082
   793
paulson@15082
   794
lemma lemma_NBseq_def2:
huffman@20552
   795
     "(\<exists>K > 0. \<forall>n. norm (X n) \<le> K) = (\<exists>N. \<forall>n. norm (X n) < real(Suc N))"
paulson@15082
   796
apply (subst lemma_NBseq_def, auto)
paulson@15082
   797
apply (rule_tac x = "Suc N" in exI)
paulson@15082
   798
apply (rule_tac [2] x = N in exI)
paulson@15082
   799
apply (auto simp add: real_of_nat_Suc)
paulson@15082
   800
 prefer 2 apply (blast intro: order_less_imp_le)
paulson@15082
   801
apply (drule_tac x = n in spec, simp)
paulson@15082
   802
done
paulson@15082
   803
paulson@15082
   804
(* yet another definition for Bseq *)
huffman@20552
   805
lemma Bseq_iff1a: "Bseq X = (\<exists>N. \<forall>n. norm (X n) < real(Suc N))"
paulson@15082
   806
by (simp add: Bseq_def lemma_NBseq_def2)
paulson@15082
   807
huffman@20696
   808
subsubsection{*Upper Bounds and Lubs of Bounded Sequences*}
paulson@15082
   809
paulson@15082
   810
lemma Bseq_isUb:
paulson@15082
   811
  "!!(X::nat=>real). Bseq X ==> \<exists>U. isUb (UNIV::real set) {x. \<exists>n. X n = x} U"
huffman@22998
   812
by (auto intro: isUbI setleI simp add: Bseq_def abs_le_iff)
paulson@15082
   813
paulson@15082
   814
paulson@15082
   815
text{* Use completeness of reals (supremum property)
paulson@15082
   816
   to show that any bounded sequence has a least upper bound*}
paulson@15082
   817
paulson@15082
   818
lemma Bseq_isLub:
paulson@15082
   819
  "!!(X::nat=>real). Bseq X ==>
paulson@15082
   820
   \<exists>U. isLub (UNIV::real set) {x. \<exists>n. X n = x} U"
paulson@15082
   821
by (blast intro: reals_complete Bseq_isUb)
paulson@15082
   822
huffman@20696
   823
subsubsection{*A Bounded and Monotonic Sequence Converges*}
paulson@15082
   824
paulson@15082
   825
lemma lemma_converg1:
nipkow@15360
   826
     "!!(X::nat=>real). [| \<forall>m. \<forall> n \<ge> m. X m \<le> X n;
paulson@15082
   827
                  isLub (UNIV::real set) {x. \<exists>n. X n = x} (X ma)
nipkow@15360
   828
               |] ==> \<forall>n \<ge> ma. X n = X ma"
paulson@15082
   829
apply safe
paulson@15082
   830
apply (drule_tac y = "X n" in isLubD2)
paulson@15082
   831
apply (blast dest: order_antisym)+
paulson@15082
   832
done
paulson@15082
   833
paulson@15082
   834
lemma Bmonoseq_LIMSEQ: "\<forall>n. m \<le> n --> X n = X m ==> \<exists>L. (X ----> L)"
huffman@36662
   835
unfolding tendsto_def eventually_sequentially
paulson@15082
   836
apply (rule_tac x = "X m" in exI, safe)
paulson@15082
   837
apply (rule_tac x = m in exI, safe)
paulson@15082
   838
apply (drule spec, erule impE, auto)
paulson@15082
   839
done
paulson@15082
   840
paulson@15082
   841
lemma lemma_converg2:
paulson@15082
   842
   "!!(X::nat=>real).
paulson@15082
   843
    [| \<forall>m. X m ~= U;  isLub UNIV {x. \<exists>n. X n = x} U |] ==> \<forall>m. X m < U"
paulson@15082
   844
apply safe
paulson@15082
   845
apply (drule_tac y = "X m" in isLubD2)
paulson@15082
   846
apply (auto dest!: order_le_imp_less_or_eq)
paulson@15082
   847
done
paulson@15082
   848
paulson@15082
   849
lemma lemma_converg3: "!!(X ::nat=>real). \<forall>m. X m \<le> U ==> isUb UNIV {x. \<exists>n. X n = x} U"
paulson@15082
   850
by (rule setleI [THEN isUbI], auto)
paulson@15082
   851
paulson@15082
   852
text{* FIXME: @{term "U - T < U"} is redundant *}
paulson@15082
   853
lemma lemma_converg4: "!!(X::nat=> real).
paulson@15082
   854
               [| \<forall>m. X m ~= U;
paulson@15082
   855
                  isLub UNIV {x. \<exists>n. X n = x} U;
paulson@15082
   856
                  0 < T;
paulson@15082
   857
                  U + - T < U
paulson@15082
   858
               |] ==> \<exists>m. U + -T < X m & X m < U"
paulson@15082
   859
apply (drule lemma_converg2, assumption)
paulson@15082
   860
apply (rule ccontr, simp)
paulson@15082
   861
apply (simp add: linorder_not_less)
paulson@15082
   862
apply (drule lemma_converg3)
paulson@15082
   863
apply (drule isLub_le_isUb, assumption)
paulson@15082
   864
apply (auto dest: order_less_le_trans)
paulson@15082
   865
done
paulson@15082
   866
paulson@15082
   867
text{*A standard proof of the theorem for monotone increasing sequence*}
paulson@15082
   868
paulson@15082
   869
lemma Bseq_mono_convergent:
huffman@20552
   870
     "[| Bseq X; \<forall>m. \<forall>n \<ge> m. X m \<le> X n |] ==> convergent (X::nat=>real)"
paulson@15082
   871
apply (simp add: convergent_def)
paulson@15082
   872
apply (frule Bseq_isLub, safe)
paulson@15082
   873
apply (case_tac "\<exists>m. X m = U", auto)
paulson@15082
   874
apply (blast dest: lemma_converg1 Bmonoseq_LIMSEQ)
paulson@15082
   875
(* second case *)
paulson@15082
   876
apply (rule_tac x = U in exI)
paulson@15082
   877
apply (subst LIMSEQ_iff, safe)
paulson@15082
   878
apply (frule lemma_converg2, assumption)
paulson@15082
   879
apply (drule lemma_converg4, auto)
paulson@15082
   880
apply (rule_tac x = m in exI, safe)
paulson@15082
   881
apply (subgoal_tac "X m \<le> X n")
paulson@15082
   882
 prefer 2 apply blast
paulson@15082
   883
apply (drule_tac x=n and P="%m. X m < U" in spec, arith)
paulson@15082
   884
done
paulson@15082
   885
paulson@15082
   886
lemma Bseq_minus_iff: "Bseq (%n. -(X n)) = Bseq X"
paulson@15082
   887
by (simp add: Bseq_def)
paulson@15082
   888
paulson@15082
   889
text{*Main monotonicity theorem*}
paulson@15082
   890
lemma Bseq_monoseq_convergent: "[| Bseq X; monoseq X |] ==> convergent X"
paulson@15082
   891
apply (simp add: monoseq_def, safe)
paulson@15082
   892
apply (rule_tac [2] convergent_minus_iff [THEN ssubst])
paulson@15082
   893
apply (drule_tac [2] Bseq_minus_iff [THEN ssubst])
paulson@15082
   894
apply (auto intro!: Bseq_mono_convergent)
paulson@15082
   895
done
paulson@15082
   896
paulson@30730
   897
subsubsection{*Increasing and Decreasing Series*}
paulson@30730
   898
paulson@30730
   899
lemma incseq_imp_monoseq:  "incseq X \<Longrightarrow> monoseq X"
paulson@30730
   900
  by (simp add: incseq_def monoseq_def) 
paulson@30730
   901
paulson@30730
   902
lemma incseq_le: assumes inc: "incseq X" and lim: "X ----> L" shows "X n \<le> L"
paulson@30730
   903
  using monoseq_le [OF incseq_imp_monoseq [OF inc] lim]
paulson@30730
   904
proof
paulson@30730
   905
  assume "(\<forall>n. X n \<le> L) \<and> (\<forall>m n. m \<le> n \<longrightarrow> X m \<le> X n)"
paulson@30730
   906
  thus ?thesis by simp
paulson@30730
   907
next
paulson@30730
   908
  assume "(\<forall>n. L \<le> X n) \<and> (\<forall>m n. m \<le> n \<longrightarrow> X n \<le> X m)"
paulson@30730
   909
  hence const: "(!!m n. m \<le> n \<Longrightarrow> X n = X m)" using inc
paulson@30730
   910
    by (auto simp add: incseq_def intro: order_antisym)
paulson@30730
   911
  have X: "!!n. X n = X 0"
paulson@30730
   912
    by (blast intro: const [of 0]) 
paulson@30730
   913
  have "X = (\<lambda>n. X 0)"
paulson@30730
   914
    by (blast intro: ext X)
paulson@30730
   915
  hence "L = X 0" using LIMSEQ_const [of "X 0"]
paulson@30730
   916
    by (auto intro: LIMSEQ_unique lim) 
paulson@30730
   917
  thus ?thesis
paulson@30730
   918
    by (blast intro: eq_refl X)
paulson@30730
   919
qed
paulson@30730
   920
hoelzl@35748
   921
lemma incseq_SucI:
hoelzl@35748
   922
  assumes "\<And>n. X n \<le> X (Suc n)"
hoelzl@35748
   923
  shows "incseq X" unfolding incseq_def
hoelzl@35748
   924
proof safe
hoelzl@35748
   925
  fix m n :: nat
hoelzl@35748
   926
  { fix d m :: nat
hoelzl@35748
   927
    have "X m \<le> X (m + d)"
hoelzl@35748
   928
    proof (induct d)
hoelzl@35748
   929
      case (Suc d)
hoelzl@35748
   930
      also have "X (m + d) \<le> X (m + Suc d)"
hoelzl@35748
   931
        using assms by simp
hoelzl@35748
   932
      finally show ?case .
hoelzl@35748
   933
    qed simp }
hoelzl@35748
   934
  note this[of m "n - m"]
hoelzl@35748
   935
  moreover assume "m \<le> n"
hoelzl@35748
   936
  ultimately show "X m \<le> X n" by simp
hoelzl@35748
   937
qed
hoelzl@35748
   938
paulson@30730
   939
lemma decseq_imp_monoseq:  "decseq X \<Longrightarrow> monoseq X"
paulson@30730
   940
  by (simp add: decseq_def monoseq_def)
paulson@30730
   941
paulson@30730
   942
lemma decseq_eq_incseq: "decseq X = incseq (\<lambda>n. - X n)" 
paulson@30730
   943
  by (simp add: decseq_def incseq_def)
paulson@30730
   944
paulson@30730
   945
paulson@30730
   946
lemma decseq_le: assumes dec: "decseq X" and lim: "X ----> L" shows "L \<le> X n"
paulson@30730
   947
proof -
paulson@30730
   948
  have inc: "incseq (\<lambda>n. - X n)" using dec
paulson@30730
   949
    by (simp add: decseq_eq_incseq)
paulson@30730
   950
  have "- X n \<le> - L" 
paulson@30730
   951
    by (blast intro: incseq_le [OF inc] LIMSEQ_minus lim) 
paulson@30730
   952
  thus ?thesis
paulson@30730
   953
    by simp
paulson@30730
   954
qed
paulson@30730
   955
huffman@20696
   956
subsubsection{*A Few More Equivalence Theorems for Boundedness*}
paulson@15082
   957
paulson@15082
   958
text{*alternative formulation for boundedness*}
huffman@20552
   959
lemma Bseq_iff2: "Bseq X = (\<exists>k > 0. \<exists>x. \<forall>n. norm (X(n) + -x) \<le> k)"
paulson@15082
   960
apply (unfold Bseq_def, safe)
huffman@20552
   961
apply (rule_tac [2] x = "k + norm x" in exI)
nipkow@15360
   962
apply (rule_tac x = K in exI, simp)
paulson@15221
   963
apply (rule exI [where x = 0], auto)
huffman@20552
   964
apply (erule order_less_le_trans, simp)
haftmann@37887
   965
apply (drule_tac x=n in spec, fold diff_minus)
huffman@20552
   966
apply (drule order_trans [OF norm_triangle_ineq2])
huffman@20552
   967
apply simp
paulson@15082
   968
done
paulson@15082
   969
paulson@15082
   970
text{*alternative formulation for boundedness*}
huffman@20552
   971
lemma Bseq_iff3: "Bseq X = (\<exists>k > 0. \<exists>N. \<forall>n. norm(X(n) + -X(N)) \<le> k)"
paulson@15082
   972
apply safe
paulson@15082
   973
apply (simp add: Bseq_def, safe)
huffman@20552
   974
apply (rule_tac x = "K + norm (X N)" in exI)
paulson@15082
   975
apply auto
huffman@20552
   976
apply (erule order_less_le_trans, simp)
paulson@15082
   977
apply (rule_tac x = N in exI, safe)
huffman@20552
   978
apply (drule_tac x = n in spec)
huffman@20552
   979
apply (rule order_trans [OF norm_triangle_ineq], simp)
paulson@15082
   980
apply (auto simp add: Bseq_iff2)
paulson@15082
   981
done
paulson@15082
   982
huffman@20552
   983
lemma BseqI2: "(\<forall>n. k \<le> f n & f n \<le> (K::real)) ==> Bseq f"
paulson@15082
   984
apply (simp add: Bseq_def)
paulson@15221
   985
apply (rule_tac x = " (\<bar>k\<bar> + \<bar>K\<bar>) + 1" in exI, auto)
webertj@20217
   986
apply (drule_tac x = n in spec, arith)
paulson@15082
   987
done
paulson@15082
   988
paulson@15082
   989
huffman@20696
   990
subsection {* Cauchy Sequences *}
paulson@15082
   991
huffman@31336
   992
lemma metric_CauchyI:
huffman@31336
   993
  "(\<And>e. 0 < e \<Longrightarrow> \<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. dist (X m) (X n) < e) \<Longrightarrow> Cauchy X"
huffman@31336
   994
by (simp add: Cauchy_def)
huffman@31336
   995
huffman@31336
   996
lemma metric_CauchyD:
huffman@31336
   997
  "\<lbrakk>Cauchy X; 0 < e\<rbrakk> \<Longrightarrow> \<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. dist (X m) (X n) < e"
huffman@20751
   998
by (simp add: Cauchy_def)
huffman@20751
   999
huffman@31336
  1000
lemma Cauchy_iff:
huffman@31336
  1001
  fixes X :: "nat \<Rightarrow> 'a::real_normed_vector"
huffman@31336
  1002
  shows "Cauchy X \<longleftrightarrow> (\<forall>e>0. \<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. norm (X m - X n) < e)"
huffman@31336
  1003
unfolding Cauchy_def dist_norm ..
huffman@31336
  1004
hoelzl@35292
  1005
lemma Cauchy_iff2:
hoelzl@35292
  1006
     "Cauchy X =
hoelzl@35292
  1007
      (\<forall>j. (\<exists>M. \<forall>m \<ge> M. \<forall>n \<ge> M. \<bar>X m - X n\<bar> < inverse(real (Suc j))))"
hoelzl@35292
  1008
apply (simp add: Cauchy_iff, auto)
hoelzl@35292
  1009
apply (drule reals_Archimedean, safe)
hoelzl@35292
  1010
apply (drule_tac x = n in spec, auto)
hoelzl@35292
  1011
apply (rule_tac x = M in exI, auto)
hoelzl@35292
  1012
apply (drule_tac x = m in spec, simp)
hoelzl@35292
  1013
apply (drule_tac x = na in spec, auto)
hoelzl@35292
  1014
done
hoelzl@35292
  1015
huffman@31336
  1016
lemma CauchyI:
huffman@31336
  1017
  fixes X :: "nat \<Rightarrow> 'a::real_normed_vector"
huffman@31336
  1018
  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"
huffman@31336
  1019
by (simp add: Cauchy_iff)
huffman@31336
  1020
huffman@20751
  1021
lemma CauchyD:
huffman@31336
  1022
  fixes X :: "nat \<Rightarrow> 'a::real_normed_vector"
huffman@31336
  1023
  shows "\<lbrakk>Cauchy X; 0 < e\<rbrakk> \<Longrightarrow> \<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. norm (X m - X n) < e"
huffman@31336
  1024
by (simp add: Cauchy_iff)
huffman@20751
  1025
paulson@30730
  1026
lemma Cauchy_subseq_Cauchy:
paulson@30730
  1027
  "\<lbrakk> Cauchy X; subseq f \<rbrakk> \<Longrightarrow> Cauchy (X o f)"
huffman@31336
  1028
apply (auto simp add: Cauchy_def)
huffman@31336
  1029
apply (drule_tac x=e in spec, clarify)
huffman@31336
  1030
apply (rule_tac x=M in exI, clarify)
huffman@31336
  1031
apply (blast intro: le_trans [OF _ seq_suble] dest!: spec)
paulson@30730
  1032
done
paulson@30730
  1033
huffman@20696
  1034
subsubsection {* Cauchy Sequences are Bounded *}
huffman@20696
  1035
paulson@15082
  1036
text{*A Cauchy sequence is bounded -- this is the standard
paulson@15082
  1037
  proof mechanization rather than the nonstandard proof*}
paulson@15082
  1038
huffman@20563
  1039
lemma lemmaCauchy: "\<forall>n \<ge> M. norm (X M - X n) < (1::real)
huffman@20552
  1040
          ==>  \<forall>n \<ge> M. norm (X n :: 'a::real_normed_vector) < 1 + norm (X M)"
huffman@20552
  1041
apply (clarify, drule spec, drule (1) mp)
huffman@20563
  1042
apply (simp only: norm_minus_commute)
huffman@20552
  1043
apply (drule order_le_less_trans [OF norm_triangle_ineq2])
huffman@20552
  1044
apply simp
huffman@20552
  1045
done
paulson@15082
  1046
paulson@15082
  1047
lemma Cauchy_Bseq: "Cauchy X ==> Bseq X"
huffman@31336
  1048
apply (simp add: Cauchy_iff)
huffman@20552
  1049
apply (drule spec, drule mp, rule zero_less_one, safe)
huffman@20552
  1050
apply (drule_tac x="M" in spec, simp)
paulson@15082
  1051
apply (drule lemmaCauchy)
huffman@22608
  1052
apply (rule_tac k="M" in Bseq_offset)
huffman@20552
  1053
apply (simp add: Bseq_def)
huffman@20552
  1054
apply (rule_tac x="1 + norm (X M)" in exI)
huffman@20552
  1055
apply (rule conjI, rule order_less_le_trans [OF zero_less_one], simp)
huffman@20552
  1056
apply (simp add: order_less_imp_le)
paulson@15082
  1057
done
paulson@15082
  1058
huffman@20696
  1059
subsubsection {* Cauchy Sequences are Convergent *}
paulson@15082
  1060
haftmann@33042
  1061
class complete_space =
haftmann@33042
  1062
  assumes Cauchy_convergent: "Cauchy X \<Longrightarrow> convergent X"
huffman@20830
  1063
haftmann@33042
  1064
class banach = real_normed_vector + complete_space
huffman@31403
  1065
huffman@22629
  1066
theorem LIMSEQ_imp_Cauchy:
huffman@22629
  1067
  assumes X: "X ----> a" shows "Cauchy X"
huffman@31336
  1068
proof (rule metric_CauchyI)
huffman@22629
  1069
  fix e::real assume "0 < e"
huffman@22629
  1070
  hence "0 < e/2" by simp
huffman@31336
  1071
  with X have "\<exists>N. \<forall>n\<ge>N. dist (X n) a < e/2" by (rule metric_LIMSEQ_D)
huffman@31336
  1072
  then obtain N where N: "\<forall>n\<ge>N. dist (X n) a < e/2" ..
huffman@31336
  1073
  show "\<exists>N. \<forall>m\<ge>N. \<forall>n\<ge>N. dist (X m) (X n) < e"
huffman@22629
  1074
  proof (intro exI allI impI)
huffman@22629
  1075
    fix m assume "N \<le> m"
huffman@31336
  1076
    hence m: "dist (X m) a < e/2" using N by fast
huffman@22629
  1077
    fix n assume "N \<le> n"
huffman@31336
  1078
    hence n: "dist (X n) a < e/2" using N by fast
huffman@31336
  1079
    have "dist (X m) (X n) \<le> dist (X m) a + dist (X n) a"
huffman@31336
  1080
      by (rule dist_triangle2)
huffman@31336
  1081
    also from m n have "\<dots> < e" by simp
huffman@31336
  1082
    finally show "dist (X m) (X n) < e" .
huffman@22629
  1083
  qed
huffman@22629
  1084
qed
huffman@22629
  1085
huffman@20691
  1086
lemma convergent_Cauchy: "convergent X \<Longrightarrow> Cauchy X"
huffman@22629
  1087
unfolding convergent_def
huffman@22629
  1088
by (erule exE, erule LIMSEQ_imp_Cauchy)
huffman@20691
  1089
huffman@31403
  1090
lemma Cauchy_convergent_iff:
huffman@31403
  1091
  fixes X :: "nat \<Rightarrow> 'a::complete_space"
huffman@31403
  1092
  shows "Cauchy X = convergent X"
huffman@31403
  1093
by (fast intro: Cauchy_convergent convergent_Cauchy)
huffman@31403
  1094
huffman@31403
  1095
lemma convergent_subseq_convergent:
huffman@31403
  1096
  fixes X :: "nat \<Rightarrow> 'a::complete_space"
huffman@31403
  1097
  shows "\<lbrakk> convergent X; subseq f \<rbrakk> \<Longrightarrow> convergent (X o f)"
huffman@31403
  1098
  by (simp add: Cauchy_subseq_Cauchy Cauchy_convergent_iff [symmetric])
huffman@31403
  1099
huffman@22629
  1100
text {*
huffman@22629
  1101
Proof that Cauchy sequences converge based on the one from
huffman@22629
  1102
http://pirate.shu.edu/~wachsmut/ira/numseq/proofs/cauconv.html
huffman@22629
  1103
*}
huffman@22629
  1104
huffman@22629
  1105
text {*
huffman@22629
  1106
  If sequence @{term "X"} is Cauchy, then its limit is the lub of
huffman@22629
  1107
  @{term "{r::real. \<exists>N. \<forall>n\<ge>N. r < X n}"}
huffman@22629
  1108
*}
huffman@22629
  1109
huffman@22629
  1110
lemma isUb_UNIV_I: "(\<And>y. y \<in> S \<Longrightarrow> y \<le> u) \<Longrightarrow> isUb UNIV S u"
huffman@22629
  1111
by (simp add: isUbI setleI)
huffman@22629
  1112
haftmann@27681
  1113
locale real_Cauchy =
huffman@22629
  1114
  fixes X :: "nat \<Rightarrow> real"
huffman@22629
  1115
  assumes X: "Cauchy X"
huffman@22629
  1116
  fixes S :: "real set"
huffman@22629
  1117
  defines S_def: "S \<equiv> {x::real. \<exists>N. \<forall>n\<ge>N. x < X n}"
huffman@22629
  1118
haftmann@27681
  1119
lemma real_CauchyI:
haftmann@27681
  1120
  assumes "Cauchy X"
haftmann@27681
  1121
  shows "real_Cauchy X"
haftmann@28823
  1122
  proof qed (fact assms)
haftmann@27681
  1123
huffman@22629
  1124
lemma (in real_Cauchy) mem_S: "\<forall>n\<ge>N. x < X n \<Longrightarrow> x \<in> S"
huffman@22629
  1125
by (unfold S_def, auto)
huffman@22629
  1126
huffman@22629
  1127
lemma (in real_Cauchy) bound_isUb:
huffman@22629
  1128
  assumes N: "\<forall>n\<ge>N. X n < x"
huffman@22629
  1129
  shows "isUb UNIV S x"
huffman@22629
  1130
proof (rule isUb_UNIV_I)
huffman@22629
  1131
  fix y::real assume "y \<in> S"
huffman@22629
  1132
  hence "\<exists>M. \<forall>n\<ge>M. y < X n"
huffman@22629
  1133
    by (simp add: S_def)
huffman@22629
  1134
  then obtain M where "\<forall>n\<ge>M. y < X n" ..
huffman@22629
  1135
  hence "y < X (max M N)" by simp
huffman@22629
  1136
  also have "\<dots> < x" using N by simp
huffman@22629
  1137
  finally show "y \<le> x"
huffman@22629
  1138
    by (rule order_less_imp_le)
huffman@22629
  1139
qed
huffman@22629
  1140
huffman@22629
  1141
lemma (in real_Cauchy) isLub_ex: "\<exists>u. isLub UNIV S u"
huffman@22629
  1142
proof (rule reals_complete)
huffman@22629
  1143
  obtain N where "\<forall>m\<ge>N. \<forall>n\<ge>N. norm (X m - X n) < 1"
haftmann@32064
  1144
    using CauchyD [OF X zero_less_one] by auto
huffman@22629
  1145
  hence N: "\<forall>n\<ge>N. norm (X n - X N) < 1" by simp
huffman@22629
  1146
  show "\<exists>x. x \<in> S"
huffman@22629
  1147
  proof
huffman@22629
  1148
    from N have "\<forall>n\<ge>N. X N - 1 < X n"
paulson@32707
  1149
      by (simp add: abs_diff_less_iff)
huffman@22629
  1150
    thus "X N - 1 \<in> S" by (rule mem_S)
huffman@22629
  1151
  qed
huffman@22629
  1152
  show "\<exists>u. isUb UNIV S u"
huffman@22629
  1153
  proof
huffman@22629
  1154
    from N have "\<forall>n\<ge>N. X n < X N + 1"
paulson@32707
  1155
      by (simp add: abs_diff_less_iff)
huffman@22629
  1156
    thus "isUb UNIV S (X N + 1)"
huffman@22629
  1157
      by (rule bound_isUb)
huffman@22629
  1158
  qed
huffman@22629
  1159
qed
huffman@22629
  1160
huffman@22629
  1161
lemma (in real_Cauchy) isLub_imp_LIMSEQ:
huffman@22629
  1162
  assumes x: "isLub UNIV S x"
huffman@22629
  1163
  shows "X ----> x"
huffman@22629
  1164
proof (rule LIMSEQ_I)
huffman@22629
  1165
  fix r::real assume "0 < r"
huffman@22629
  1166
  hence r: "0 < r/2" by simp
huffman@22629
  1167
  obtain N where "\<forall>n\<ge>N. \<forall>m\<ge>N. norm (X n - X m) < r/2"
haftmann@32064
  1168
    using CauchyD [OF X r] by auto
huffman@22629
  1169
  hence "\<forall>n\<ge>N. norm (X n - X N) < r/2" by simp
huffman@22629
  1170
  hence N: "\<forall>n\<ge>N. X N - r/2 < X n \<and> X n < X N + r/2"
paulson@32707
  1171
    by (simp only: real_norm_def abs_diff_less_iff)
huffman@22629
  1172
huffman@22629
  1173
  from N have "\<forall>n\<ge>N. X N - r/2 < X n" by fast
huffman@22629
  1174
  hence "X N - r/2 \<in> S" by (rule mem_S)
nipkow@23482
  1175
  hence 1: "X N - r/2 \<le> x" using x isLub_isUb isUbD by fast
huffman@22629
  1176
huffman@22629
  1177
  from N have "\<forall>n\<ge>N. X n < X N + r/2" by fast
huffman@22629
  1178
  hence "isUb UNIV S (X N + r/2)" by (rule bound_isUb)
nipkow@23482
  1179
  hence 2: "x \<le> X N + r/2" using x isLub_le_isUb by fast
huffman@22629
  1180
huffman@22629
  1181
  show "\<exists>N. \<forall>n\<ge>N. norm (X n - x) < r"
huffman@22629
  1182
  proof (intro exI allI impI)
huffman@22629
  1183
    fix n assume n: "N \<le> n"
nipkow@23482
  1184
    from N n have "X n < X N + r/2" and "X N - r/2 < X n" by simp+
nipkow@23482
  1185
    thus "norm (X n - x) < r" using 1 2
paulson@32707
  1186
      by (simp add: abs_diff_less_iff)
huffman@22629
  1187
  qed
huffman@22629
  1188
qed
huffman@22629
  1189
huffman@22629
  1190
lemma (in real_Cauchy) LIMSEQ_ex: "\<exists>x. X ----> x"
huffman@22629
  1191
proof -
huffman@22629
  1192
  obtain x where "isLub UNIV S x"
huffman@22629
  1193
    using isLub_ex by fast
huffman@22629
  1194
  hence "X ----> x"
huffman@22629
  1195
    by (rule isLub_imp_LIMSEQ)
huffman@22629
  1196
  thus ?thesis ..
huffman@22629
  1197
qed
huffman@22629
  1198
huffman@20830
  1199
lemma real_Cauchy_convergent:
huffman@20830
  1200
  fixes X :: "nat \<Rightarrow> real"
huffman@20830
  1201
  shows "Cauchy X \<Longrightarrow> convergent X"
haftmann@27681
  1202
unfolding convergent_def
haftmann@27681
  1203
by (rule real_Cauchy.LIMSEQ_ex)
haftmann@27681
  1204
 (rule real_CauchyI)
huffman@20830
  1205
huffman@20830
  1206
instance real :: banach
huffman@20830
  1207
by intro_classes (rule real_Cauchy_convergent)
huffman@20830
  1208
paulson@15082
  1209
huffman@20696
  1210
subsection {* Power Sequences *}
paulson@15082
  1211
paulson@15082
  1212
text{*The sequence @{term "x^n"} tends to 0 if @{term "0\<le>x"} and @{term
paulson@15082
  1213
"x<1"}.  Proof will use (NS) Cauchy equivalence for convergence and
paulson@15082
  1214
  also fact that bounded and monotonic sequence converges.*}
paulson@15082
  1215
huffman@20552
  1216
lemma Bseq_realpow: "[| 0 \<le> (x::real); x \<le> 1 |] ==> Bseq (%n. x ^ n)"
paulson@15082
  1217
apply (simp add: Bseq_def)
paulson@15082
  1218
apply (rule_tac x = 1 in exI)
paulson@15082
  1219
apply (simp add: power_abs)
huffman@22974
  1220
apply (auto dest: power_mono)
paulson@15082
  1221
done
paulson@15082
  1222
paulson@15082
  1223
lemma monoseq_realpow: "[| 0 \<le> x; x \<le> 1 |] ==> monoseq (%n. x ^ n)"
paulson@15082
  1224
apply (clarify intro!: mono_SucI2)
paulson@15082
  1225
apply (cut_tac n = n and N = "Suc n" and a = x in power_decreasing, auto)
paulson@15082
  1226
done
paulson@15082
  1227
huffman@20552
  1228
lemma convergent_realpow:
huffman@20552
  1229
  "[| 0 \<le> (x::real); x \<le> 1 |] ==> convergent (%n. x ^ n)"
paulson@15082
  1230
by (blast intro!: Bseq_monoseq_convergent Bseq_realpow monoseq_realpow)
paulson@15082
  1231
huffman@22628
  1232
lemma LIMSEQ_inverse_realpow_zero_lemma:
huffman@22628
  1233
  fixes x :: real
huffman@22628
  1234
  assumes x: "0 \<le> x"
huffman@22628
  1235
  shows "real n * x + 1 \<le> (x + 1) ^ n"
huffman@22628
  1236
apply (induct n)
huffman@22628
  1237
apply simp
huffman@22628
  1238
apply simp
huffman@22628
  1239
apply (rule order_trans)
huffman@22628
  1240
prefer 2
huffman@22628
  1241
apply (erule mult_left_mono)
huffman@22628
  1242
apply (rule add_increasing [OF x], simp)
huffman@22628
  1243
apply (simp add: real_of_nat_Suc)
nipkow@23477
  1244
apply (simp add: ring_distribs)
huffman@22628
  1245
apply (simp add: mult_nonneg_nonneg x)
huffman@22628
  1246
done
huffman@22628
  1247
huffman@22628
  1248
lemma LIMSEQ_inverse_realpow_zero:
huffman@22628
  1249
  "1 < (x::real) \<Longrightarrow> (\<lambda>n. inverse (x ^ n)) ----> 0"
huffman@22628
  1250
proof (rule LIMSEQ_inverse_zero [rule_format])
huffman@22628
  1251
  fix y :: real
huffman@22628
  1252
  assume x: "1 < x"
huffman@22628
  1253
  hence "0 < x - 1" by simp
huffman@22628
  1254
  hence "\<forall>y. \<exists>N::nat. y < real N * (x - 1)"
huffman@22628
  1255
    by (rule reals_Archimedean3)
huffman@22628
  1256
  hence "\<exists>N::nat. y < real N * (x - 1)" ..
huffman@22628
  1257
  then obtain N::nat where "y < real N * (x - 1)" ..
huffman@22628
  1258
  also have "\<dots> \<le> real N * (x - 1) + 1" by simp
huffman@22628
  1259
  also have "\<dots> \<le> (x - 1 + 1) ^ N"
huffman@22628
  1260
    by (rule LIMSEQ_inverse_realpow_zero_lemma, cut_tac x, simp)
huffman@22628
  1261
  also have "\<dots> = x ^ N" by simp
huffman@22628
  1262
  finally have "y < x ^ N" .
huffman@22628
  1263
  hence "\<forall>n\<ge>N. y < x ^ n"
huffman@22628
  1264
    apply clarify
huffman@22628
  1265
    apply (erule order_less_le_trans)
huffman@22628
  1266
    apply (erule power_increasing)
huffman@22628
  1267
    apply (rule order_less_imp_le [OF x])
huffman@22628
  1268
    done
huffman@22628
  1269
  thus "\<exists>N. \<forall>n\<ge>N. y < x ^ n" ..
huffman@22628
  1270
qed
huffman@22628
  1271
huffman@20552
  1272
lemma LIMSEQ_realpow_zero:
huffman@22628
  1273
  "\<lbrakk>0 \<le> (x::real); x < 1\<rbrakk> \<Longrightarrow> (\<lambda>n. x ^ n) ----> 0"
huffman@22628
  1274
proof (cases)
huffman@22628
  1275
  assume "x = 0"
huffman@22628
  1276
  hence "(\<lambda>n. x ^ Suc n) ----> 0" by (simp add: LIMSEQ_const)
huffman@22628
  1277
  thus ?thesis by (rule LIMSEQ_imp_Suc)
huffman@22628
  1278
next
huffman@22628
  1279
  assume "0 \<le> x" and "x \<noteq> 0"
huffman@22628
  1280
  hence x0: "0 < x" by simp
huffman@22628
  1281
  assume x1: "x < 1"
huffman@22628
  1282
  from x0 x1 have "1 < inverse x"
huffman@36776
  1283
    by (rule one_less_inverse)
huffman@22628
  1284
  hence "(\<lambda>n. inverse (inverse x ^ n)) ----> 0"
huffman@22628
  1285
    by (rule LIMSEQ_inverse_realpow_zero)
huffman@22628
  1286
  thus ?thesis by (simp add: power_inverse)
huffman@22628
  1287
qed
paulson@15082
  1288
huffman@20685
  1289
lemma LIMSEQ_power_zero:
haftmann@31017
  1290
  fixes x :: "'a::{real_normed_algebra_1}"
huffman@20685
  1291
  shows "norm x < 1 \<Longrightarrow> (\<lambda>n. x ^ n) ----> 0"
huffman@20685
  1292
apply (drule LIMSEQ_realpow_zero [OF norm_ge_zero])
huffman@36657
  1293
apply (simp only: LIMSEQ_Zfun_iff, erule Zfun_le)
huffman@22974
  1294
apply (simp add: power_abs norm_power_ineq)
huffman@20685
  1295
done
huffman@20685
  1296
huffman@20552
  1297
lemma LIMSEQ_divide_realpow_zero:
huffman@20552
  1298
  "1 < (x::real) ==> (%n. a / (x ^ n)) ----> 0"
paulson@15082
  1299
apply (cut_tac a = a and x1 = "inverse x" in
paulson@15082
  1300
        LIMSEQ_mult [OF LIMSEQ_const LIMSEQ_realpow_zero])
paulson@15082
  1301
apply (auto simp add: divide_inverse power_inverse)
paulson@15082
  1302
apply (simp add: inverse_eq_divide pos_divide_less_eq)
paulson@15082
  1303
done
paulson@15082
  1304
paulson@15102
  1305
text{*Limit of @{term "c^n"} for @{term"\<bar>c\<bar> < 1"}*}
paulson@15082
  1306
huffman@20552
  1307
lemma LIMSEQ_rabs_realpow_zero: "\<bar>c\<bar> < (1::real) ==> (%n. \<bar>c\<bar> ^ n) ----> 0"
huffman@20685
  1308
by (rule LIMSEQ_realpow_zero [OF abs_ge_zero])
paulson@15082
  1309
huffman@20552
  1310
lemma LIMSEQ_rabs_realpow_zero2: "\<bar>c\<bar> < (1::real) ==> (%n. c ^ n) ----> 0"
paulson@15082
  1311
apply (rule LIMSEQ_rabs_zero [THEN iffD1])
paulson@15082
  1312
apply (auto intro: LIMSEQ_rabs_realpow_zero simp add: power_abs)
paulson@15082
  1313
done
paulson@15082
  1314
paulson@10751
  1315
end