src/HOL/Library/Product_Vector.thy
author wenzelm
Wed Sep 12 13:42:28 2012 +0200 (2012-09-12)
changeset 49322 fbb320d02420
parent 44749 5b1e1432c320
child 49962 a8cc904a6820
permissions -rw-r--r--
tuned headers;
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(*  Title:      HOL/Library/Product_Vector.thy
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    Author:     Brian Huffman
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*)
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header {* Cartesian Products as Vector Spaces *}
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theory Product_Vector
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imports Inner_Product Product_plus
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begin
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subsection {* Product is a real vector space *}
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instantiation prod :: (real_vector, real_vector) real_vector
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begin
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definition scaleR_prod_def:
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  "scaleR r A = (scaleR r (fst A), scaleR r (snd A))"
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lemma fst_scaleR [simp]: "fst (scaleR r A) = scaleR r (fst A)"
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  unfolding scaleR_prod_def by simp
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lemma snd_scaleR [simp]: "snd (scaleR r A) = scaleR r (snd A)"
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  unfolding scaleR_prod_def by simp
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lemma scaleR_Pair [simp]: "scaleR r (a, b) = (scaleR r a, scaleR r b)"
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  unfolding scaleR_prod_def by simp
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instance proof
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  fix a b :: real and x y :: "'a \<times> 'b"
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  show "scaleR a (x + y) = scaleR a x + scaleR a y"
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    by (simp add: prod_eq_iff scaleR_right_distrib)
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  show "scaleR (a + b) x = scaleR a x + scaleR b x"
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    by (simp add: prod_eq_iff scaleR_left_distrib)
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  show "scaleR a (scaleR b x) = scaleR (a * b) x"
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    by (simp add: prod_eq_iff)
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  show "scaleR 1 x = x"
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    by (simp add: prod_eq_iff)
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qed
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end
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subsection {* Product is a topological space *}
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instantiation prod :: (topological_space, topological_space) topological_space
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begin
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definition open_prod_def:
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  "open (S :: ('a \<times> 'b) set) \<longleftrightarrow>
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    (\<forall>x\<in>S. \<exists>A B. open A \<and> open B \<and> x \<in> A \<times> B \<and> A \<times> B \<subseteq> S)"
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lemma open_prod_elim:
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  assumes "open S" and "x \<in> S"
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  obtains A B where "open A" and "open B" and "x \<in> A \<times> B" and "A \<times> B \<subseteq> S"
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using assms unfolding open_prod_def by fast
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lemma open_prod_intro:
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  assumes "\<And>x. x \<in> S \<Longrightarrow> \<exists>A B. open A \<and> open B \<and> x \<in> A \<times> B \<and> A \<times> B \<subseteq> S"
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  shows "open S"
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using assms unfolding open_prod_def by fast
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instance proof
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  show "open (UNIV :: ('a \<times> 'b) set)"
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    unfolding open_prod_def by auto
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next
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  fix S T :: "('a \<times> 'b) set"
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  assume "open S" "open T"
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  show "open (S \<inter> T)"
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  proof (rule open_prod_intro)
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    fix x assume x: "x \<in> S \<inter> T"
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    from x have "x \<in> S" by simp
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    obtain Sa Sb where A: "open Sa" "open Sb" "x \<in> Sa \<times> Sb" "Sa \<times> Sb \<subseteq> S"
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      using `open S` and `x \<in> S` by (rule open_prod_elim)
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    from x have "x \<in> T" by simp
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    obtain Ta Tb where B: "open Ta" "open Tb" "x \<in> Ta \<times> Tb" "Ta \<times> Tb \<subseteq> T"
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      using `open T` and `x \<in> T` by (rule open_prod_elim)
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    let ?A = "Sa \<inter> Ta" and ?B = "Sb \<inter> Tb"
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    have "open ?A \<and> open ?B \<and> x \<in> ?A \<times> ?B \<and> ?A \<times> ?B \<subseteq> S \<inter> T"
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      using A B by (auto simp add: open_Int)
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    thus "\<exists>A B. open A \<and> open B \<and> x \<in> A \<times> B \<and> A \<times> B \<subseteq> S \<inter> T"
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      by fast
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  qed
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next
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  fix K :: "('a \<times> 'b) set set"
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  assume "\<forall>S\<in>K. open S" thus "open (\<Union>K)"
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    unfolding open_prod_def by fast
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qed
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end
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lemma open_Times: "open S \<Longrightarrow> open T \<Longrightarrow> open (S \<times> T)"
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unfolding open_prod_def by auto
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lemma fst_vimage_eq_Times: "fst -` S = S \<times> UNIV"
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by auto
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lemma snd_vimage_eq_Times: "snd -` S = UNIV \<times> S"
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by auto
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lemma open_vimage_fst: "open S \<Longrightarrow> open (fst -` S)"
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by (simp add: fst_vimage_eq_Times open_Times)
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lemma open_vimage_snd: "open S \<Longrightarrow> open (snd -` S)"
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by (simp add: snd_vimage_eq_Times open_Times)
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lemma closed_vimage_fst: "closed S \<Longrightarrow> closed (fst -` S)"
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unfolding closed_open vimage_Compl [symmetric]
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by (rule open_vimage_fst)
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lemma closed_vimage_snd: "closed S \<Longrightarrow> closed (snd -` S)"
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unfolding closed_open vimage_Compl [symmetric]
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by (rule open_vimage_snd)
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lemma closed_Times: "closed S \<Longrightarrow> closed T \<Longrightarrow> closed (S \<times> T)"
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proof -
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  have "S \<times> T = (fst -` S) \<inter> (snd -` T)" by auto
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  thus "closed S \<Longrightarrow> closed T \<Longrightarrow> closed (S \<times> T)"
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    by (simp add: closed_vimage_fst closed_vimage_snd closed_Int)
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qed
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lemma openI: (* TODO: move *)
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  assumes "\<And>x. x \<in> S \<Longrightarrow> \<exists>T. open T \<and> x \<in> T \<and> T \<subseteq> S"
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  shows "open S"
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proof -
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  have "open (\<Union>{T. open T \<and> T \<subseteq> S})" by auto
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  moreover have "\<Union>{T. open T \<and> T \<subseteq> S} = S" by (auto dest!: assms)
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  ultimately show "open S" by simp
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qed
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lemma subset_fst_imageI: "A \<times> B \<subseteq> S \<Longrightarrow> y \<in> B \<Longrightarrow> A \<subseteq> fst ` S"
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  unfolding image_def subset_eq by force
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lemma subset_snd_imageI: "A \<times> B \<subseteq> S \<Longrightarrow> x \<in> A \<Longrightarrow> B \<subseteq> snd ` S"
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  unfolding image_def subset_eq by force
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lemma open_image_fst: assumes "open S" shows "open (fst ` S)"
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proof (rule openI)
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  fix x assume "x \<in> fst ` S"
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  then obtain y where "(x, y) \<in> S" by auto
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  then obtain A B where "open A" "open B" "x \<in> A" "y \<in> B" "A \<times> B \<subseteq> S"
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    using `open S` unfolding open_prod_def by auto
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  from `A \<times> B \<subseteq> S` `y \<in> B` have "A \<subseteq> fst ` S" by (rule subset_fst_imageI)
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  with `open A` `x \<in> A` have "open A \<and> x \<in> A \<and> A \<subseteq> fst ` S" by simp
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  then show "\<exists>T. open T \<and> x \<in> T \<and> T \<subseteq> fst ` S" by - (rule exI)
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qed
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lemma open_image_snd: assumes "open S" shows "open (snd ` S)"
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proof (rule openI)
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  fix y assume "y \<in> snd ` S"
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  then obtain x where "(x, y) \<in> S" by auto
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  then obtain A B where "open A" "open B" "x \<in> A" "y \<in> B" "A \<times> B \<subseteq> S"
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    using `open S` unfolding open_prod_def by auto
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  from `A \<times> B \<subseteq> S` `x \<in> A` have "B \<subseteq> snd ` S" by (rule subset_snd_imageI)
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  with `open B` `y \<in> B` have "open B \<and> y \<in> B \<and> B \<subseteq> snd ` S" by simp
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  then show "\<exists>T. open T \<and> y \<in> T \<and> T \<subseteq> snd ` S" by - (rule exI)
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qed
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subsubsection {* Continuity of operations *}
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lemma tendsto_fst [tendsto_intros]:
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  assumes "(f ---> a) F"
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  shows "((\<lambda>x. fst (f x)) ---> fst a) F"
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proof (rule topological_tendstoI)
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  fix S assume "open S" and "fst a \<in> S"
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  then have "open (fst -` S)" and "a \<in> fst -` S"
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    by (simp_all add: open_vimage_fst)
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  with assms have "eventually (\<lambda>x. f x \<in> fst -` S) F"
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    by (rule topological_tendstoD)
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  then show "eventually (\<lambda>x. fst (f x) \<in> S) F"
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    by simp
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qed
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lemma tendsto_snd [tendsto_intros]:
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  assumes "(f ---> a) F"
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  shows "((\<lambda>x. snd (f x)) ---> snd a) F"
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proof (rule topological_tendstoI)
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  fix S assume "open S" and "snd a \<in> S"
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  then have "open (snd -` S)" and "a \<in> snd -` S"
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    by (simp_all add: open_vimage_snd)
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  with assms have "eventually (\<lambda>x. f x \<in> snd -` S) F"
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    by (rule topological_tendstoD)
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  then show "eventually (\<lambda>x. snd (f x) \<in> S) F"
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    by simp
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qed
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lemma tendsto_Pair [tendsto_intros]:
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  assumes "(f ---> a) F" and "(g ---> b) F"
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  shows "((\<lambda>x. (f x, g x)) ---> (a, b)) F"
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proof (rule topological_tendstoI)
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  fix S assume "open S" and "(a, b) \<in> S"
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  then obtain A B where "open A" "open B" "a \<in> A" "b \<in> B" "A \<times> B \<subseteq> S"
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    unfolding open_prod_def by fast
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  have "eventually (\<lambda>x. f x \<in> A) F"
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    using `(f ---> a) F` `open A` `a \<in> A`
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    by (rule topological_tendstoD)
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  moreover
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  have "eventually (\<lambda>x. g x \<in> B) F"
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    using `(g ---> b) F` `open B` `b \<in> B`
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    by (rule topological_tendstoD)
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  ultimately
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  show "eventually (\<lambda>x. (f x, g x) \<in> S) F"
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    by (rule eventually_elim2)
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       (simp add: subsetD [OF `A \<times> B \<subseteq> S`])
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qed
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lemma isCont_fst [simp]: "isCont f a \<Longrightarrow> isCont (\<lambda>x. fst (f x)) a"
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  unfolding isCont_def by (rule tendsto_fst)
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lemma isCont_snd [simp]: "isCont f a \<Longrightarrow> isCont (\<lambda>x. snd (f x)) a"
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  unfolding isCont_def by (rule tendsto_snd)
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lemma isCont_Pair [simp]:
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  "\<lbrakk>isCont f a; isCont g a\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. (f x, g x)) a"
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  unfolding isCont_def by (rule tendsto_Pair)
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subsubsection {* Separation axioms *}
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lemma mem_Times_iff: "x \<in> A \<times> B \<longleftrightarrow> fst x \<in> A \<and> snd x \<in> B"
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  by (induct x) simp (* TODO: move elsewhere *)
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instance prod :: (t0_space, t0_space) t0_space
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proof
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  fix x y :: "'a \<times> 'b" assume "x \<noteq> y"
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  hence "fst x \<noteq> fst y \<or> snd x \<noteq> snd y"
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    by (simp add: prod_eq_iff)
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  thus "\<exists>U. open U \<and> (x \<in> U) \<noteq> (y \<in> U)"
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    apply (rule disjE)
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    apply (drule t0_space, erule exE, rule_tac x="U \<times> UNIV" in exI)
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    apply (simp add: open_Times mem_Times_iff)
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    apply (drule t0_space, erule exE, rule_tac x="UNIV \<times> U" in exI)
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    apply (simp add: open_Times mem_Times_iff)
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    done
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qed
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instance prod :: (t1_space, t1_space) t1_space
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proof
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  fix x y :: "'a \<times> 'b" assume "x \<noteq> y"
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  hence "fst x \<noteq> fst y \<or> snd x \<noteq> snd y"
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    by (simp add: prod_eq_iff)
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  thus "\<exists>U. open U \<and> x \<in> U \<and> y \<notin> U"
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    apply (rule disjE)
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    apply (drule t1_space, erule exE, rule_tac x="U \<times> UNIV" in exI)
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    apply (simp add: open_Times mem_Times_iff)
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    apply (drule t1_space, erule exE, rule_tac x="UNIV \<times> U" in exI)
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    apply (simp add: open_Times mem_Times_iff)
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    done
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qed
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instance prod :: (t2_space, t2_space) t2_space
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proof
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  fix x y :: "'a \<times> 'b" assume "x \<noteq> y"
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  hence "fst x \<noteq> fst y \<or> snd x \<noteq> snd y"
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    by (simp add: prod_eq_iff)
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  thus "\<exists>U V. open U \<and> open V \<and> x \<in> U \<and> y \<in> V \<and> U \<inter> V = {}"
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    apply (rule disjE)
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    apply (drule hausdorff, clarify)
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    apply (rule_tac x="U \<times> UNIV" in exI, rule_tac x="V \<times> UNIV" in exI)
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    apply (simp add: open_Times mem_Times_iff disjoint_iff_not_equal)
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    apply (drule hausdorff, clarify)
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    apply (rule_tac x="UNIV \<times> U" in exI, rule_tac x="UNIV \<times> V" in exI)
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    apply (simp add: open_Times mem_Times_iff disjoint_iff_not_equal)
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    done
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qed
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subsection {* Product is a metric space *}
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instantiation prod :: (metric_space, metric_space) metric_space
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begin
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definition dist_prod_def:
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  "dist x y = sqrt ((dist (fst x) (fst y))\<twosuperior> + (dist (snd x) (snd y))\<twosuperior>)"
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lemma dist_Pair_Pair: "dist (a, b) (c, d) = sqrt ((dist a c)\<twosuperior> + (dist b d)\<twosuperior>)"
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  unfolding dist_prod_def by simp
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lemma dist_fst_le: "dist (fst x) (fst y) \<le> dist x y"
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unfolding dist_prod_def by (rule real_sqrt_sum_squares_ge1)
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lemma dist_snd_le: "dist (snd x) (snd y) \<le> dist x y"
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unfolding dist_prod_def by (rule real_sqrt_sum_squares_ge2)
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instance proof
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  fix x y :: "'a \<times> 'b"
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  show "dist x y = 0 \<longleftrightarrow> x = y"
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    unfolding dist_prod_def prod_eq_iff by simp
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next
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  fix x y z :: "'a \<times> 'b"
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  show "dist x y \<le> dist x z + dist y z"
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    unfolding dist_prod_def
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    by (intro order_trans [OF _ real_sqrt_sum_squares_triangle_ineq]
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   290
        real_sqrt_le_mono add_mono power_mono dist_triangle2 zero_le_dist)
huffman@31415
   291
next
huffman@31492
   292
  fix S :: "('a \<times> 'b) set"
huffman@31492
   293
  show "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S)"
huffman@31563
   294
  proof
huffman@36332
   295
    assume "open S" show "\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S"
huffman@36332
   296
    proof
huffman@36332
   297
      fix x assume "x \<in> S"
huffman@36332
   298
      obtain A B where "open A" "open B" "x \<in> A \<times> B" "A \<times> B \<subseteq> S"
huffman@36332
   299
        using `open S` and `x \<in> S` by (rule open_prod_elim)
huffman@36332
   300
      obtain r where r: "0 < r" "\<forall>y. dist y (fst x) < r \<longrightarrow> y \<in> A"
huffman@36332
   301
        using `open A` and `x \<in> A \<times> B` unfolding open_dist by auto
huffman@36332
   302
      obtain s where s: "0 < s" "\<forall>y. dist y (snd x) < s \<longrightarrow> y \<in> B"
huffman@36332
   303
        using `open B` and `x \<in> A \<times> B` unfolding open_dist by auto
huffman@36332
   304
      let ?e = "min r s"
huffman@36332
   305
      have "0 < ?e \<and> (\<forall>y. dist y x < ?e \<longrightarrow> y \<in> S)"
huffman@36332
   306
      proof (intro allI impI conjI)
huffman@36332
   307
        show "0 < min r s" by (simp add: r(1) s(1))
huffman@36332
   308
      next
huffman@36332
   309
        fix y assume "dist y x < min r s"
huffman@36332
   310
        hence "dist y x < r" and "dist y x < s"
huffman@36332
   311
          by simp_all
huffman@36332
   312
        hence "dist (fst y) (fst x) < r" and "dist (snd y) (snd x) < s"
huffman@36332
   313
          by (auto intro: le_less_trans dist_fst_le dist_snd_le)
huffman@36332
   314
        hence "fst y \<in> A" and "snd y \<in> B"
huffman@36332
   315
          by (simp_all add: r(2) s(2))
huffman@36332
   316
        hence "y \<in> A \<times> B" by (induct y, simp)
huffman@36332
   317
        with `A \<times> B \<subseteq> S` show "y \<in> S" ..
huffman@36332
   318
      qed
huffman@36332
   319
      thus "\<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S" ..
huffman@36332
   320
    qed
huffman@31563
   321
  next
huffman@44575
   322
    assume *: "\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S" show "open S"
huffman@44575
   323
    proof (rule open_prod_intro)
huffman@44575
   324
      fix x assume "x \<in> S"
huffman@44575
   325
      then obtain e where "0 < e" and S: "\<forall>y. dist y x < e \<longrightarrow> y \<in> S"
huffman@44575
   326
        using * by fast
huffman@44575
   327
      def r \<equiv> "e / sqrt 2" and s \<equiv> "e / sqrt 2"
huffman@44575
   328
      from `0 < e` have "0 < r" and "0 < s"
huffman@44575
   329
        unfolding r_def s_def by (simp_all add: divide_pos_pos)
huffman@44575
   330
      from `0 < e` have "e = sqrt (r\<twosuperior> + s\<twosuperior>)"
huffman@44575
   331
        unfolding r_def s_def by (simp add: power_divide)
huffman@44575
   332
      def A \<equiv> "{y. dist (fst x) y < r}" and B \<equiv> "{y. dist (snd x) y < s}"
huffman@44575
   333
      have "open A" and "open B"
huffman@44575
   334
        unfolding A_def B_def by (simp_all add: open_ball)
huffman@44575
   335
      moreover have "x \<in> A \<times> B"
huffman@44575
   336
        unfolding A_def B_def mem_Times_iff
huffman@44575
   337
        using `0 < r` and `0 < s` by simp
huffman@44575
   338
      moreover have "A \<times> B \<subseteq> S"
huffman@44575
   339
      proof (clarify)
huffman@44575
   340
        fix a b assume "a \<in> A" and "b \<in> B"
huffman@44575
   341
        hence "dist a (fst x) < r" and "dist b (snd x) < s"
huffman@44575
   342
          unfolding A_def B_def by (simp_all add: dist_commute)
huffman@44575
   343
        hence "dist (a, b) x < e"
huffman@44575
   344
          unfolding dist_prod_def `e = sqrt (r\<twosuperior> + s\<twosuperior>)`
huffman@44575
   345
          by (simp add: add_strict_mono power_strict_mono)
huffman@44575
   346
        thus "(a, b) \<in> S"
huffman@44575
   347
          by (simp add: S)
huffman@44575
   348
      qed
huffman@44575
   349
      ultimately show "\<exists>A B. open A \<and> open B \<and> x \<in> A \<times> B \<and> A \<times> B \<subseteq> S" by fast
huffman@44575
   350
    qed
huffman@31563
   351
  qed
huffman@31339
   352
qed
huffman@31339
   353
huffman@31339
   354
end
huffman@31339
   355
huffman@31405
   356
lemma Cauchy_fst: "Cauchy X \<Longrightarrow> Cauchy (\<lambda>n. fst (X n))"
huffman@31405
   357
unfolding Cauchy_def by (fast elim: le_less_trans [OF dist_fst_le])
huffman@31405
   358
huffman@31405
   359
lemma Cauchy_snd: "Cauchy X \<Longrightarrow> Cauchy (\<lambda>n. snd (X n))"
huffman@31405
   360
unfolding Cauchy_def by (fast elim: le_less_trans [OF dist_snd_le])
huffman@31405
   361
huffman@31405
   362
lemma Cauchy_Pair:
huffman@31405
   363
  assumes "Cauchy X" and "Cauchy Y"
huffman@31405
   364
  shows "Cauchy (\<lambda>n. (X n, Y n))"
huffman@31405
   365
proof (rule metric_CauchyI)
huffman@31405
   366
  fix r :: real assume "0 < r"
huffman@31405
   367
  then have "0 < r / sqrt 2" (is "0 < ?s")
huffman@31405
   368
    by (simp add: divide_pos_pos)
huffman@31405
   369
  obtain M where M: "\<forall>m\<ge>M. \<forall>n\<ge>M. dist (X m) (X n) < ?s"
huffman@31405
   370
    using metric_CauchyD [OF `Cauchy X` `0 < ?s`] ..
huffman@31405
   371
  obtain N where N: "\<forall>m\<ge>N. \<forall>n\<ge>N. dist (Y m) (Y n) < ?s"
huffman@31405
   372
    using metric_CauchyD [OF `Cauchy Y` `0 < ?s`] ..
huffman@31405
   373
  have "\<forall>m\<ge>max M N. \<forall>n\<ge>max M N. dist (X m, Y m) (X n, Y n) < r"
huffman@31405
   374
    using M N by (simp add: real_sqrt_sum_squares_less dist_Pair_Pair)
huffman@31405
   375
  then show "\<exists>n0. \<forall>m\<ge>n0. \<forall>n\<ge>n0. dist (X m, Y m) (X n, Y n) < r" ..
huffman@31405
   376
qed
huffman@31405
   377
huffman@31405
   378
subsection {* Product is a complete metric space *}
huffman@31405
   379
haftmann@37678
   380
instance prod :: (complete_space, complete_space) complete_space
huffman@31405
   381
proof
huffman@31405
   382
  fix X :: "nat \<Rightarrow> 'a \<times> 'b" assume "Cauchy X"
huffman@31405
   383
  have 1: "(\<lambda>n. fst (X n)) ----> lim (\<lambda>n. fst (X n))"
huffman@31405
   384
    using Cauchy_fst [OF `Cauchy X`]
huffman@31405
   385
    by (simp add: Cauchy_convergent_iff convergent_LIMSEQ_iff)
huffman@31405
   386
  have 2: "(\<lambda>n. snd (X n)) ----> lim (\<lambda>n. snd (X n))"
huffman@31405
   387
    using Cauchy_snd [OF `Cauchy X`]
huffman@31405
   388
    by (simp add: Cauchy_convergent_iff convergent_LIMSEQ_iff)
huffman@31405
   389
  have "X ----> (lim (\<lambda>n. fst (X n)), lim (\<lambda>n. snd (X n)))"
huffman@36660
   390
    using tendsto_Pair [OF 1 2] by simp
huffman@31405
   391
  then show "convergent X"
huffman@31405
   392
    by (rule convergentI)
huffman@31405
   393
qed
huffman@31405
   394
huffman@30019
   395
subsection {* Product is a normed vector space *}
huffman@30019
   396
haftmann@37678
   397
instantiation prod :: (real_normed_vector, real_normed_vector) real_normed_vector
huffman@30019
   398
begin
huffman@30019
   399
huffman@30019
   400
definition norm_prod_def:
huffman@30019
   401
  "norm x = sqrt ((norm (fst x))\<twosuperior> + (norm (snd x))\<twosuperior>)"
huffman@30019
   402
huffman@30019
   403
definition sgn_prod_def:
huffman@30019
   404
  "sgn (x::'a \<times> 'b) = scaleR (inverse (norm x)) x"
huffman@30019
   405
huffman@30019
   406
lemma norm_Pair: "norm (a, b) = sqrt ((norm a)\<twosuperior> + (norm b)\<twosuperior>)"
huffman@30019
   407
  unfolding norm_prod_def by simp
huffman@30019
   408
huffman@30019
   409
instance proof
huffman@30019
   410
  fix r :: real and x y :: "'a \<times> 'b"
huffman@30019
   411
  show "0 \<le> norm x"
huffman@30019
   412
    unfolding norm_prod_def by simp
huffman@30019
   413
  show "norm x = 0 \<longleftrightarrow> x = 0"
huffman@30019
   414
    unfolding norm_prod_def
huffman@44066
   415
    by (simp add: prod_eq_iff)
huffman@30019
   416
  show "norm (x + y) \<le> norm x + norm y"
huffman@30019
   417
    unfolding norm_prod_def
huffman@30019
   418
    apply (rule order_trans [OF _ real_sqrt_sum_squares_triangle_ineq])
huffman@30019
   419
    apply (simp add: add_mono power_mono norm_triangle_ineq)
huffman@30019
   420
    done
huffman@30019
   421
  show "norm (scaleR r x) = \<bar>r\<bar> * norm x"
huffman@30019
   422
    unfolding norm_prod_def
huffman@31587
   423
    apply (simp add: power_mult_distrib)
huffman@30019
   424
    apply (simp add: right_distrib [symmetric])
huffman@30019
   425
    apply (simp add: real_sqrt_mult_distrib)
huffman@30019
   426
    done
huffman@30019
   427
  show "sgn x = scaleR (inverse (norm x)) x"
huffman@30019
   428
    by (rule sgn_prod_def)
huffman@31290
   429
  show "dist x y = norm (x - y)"
huffman@31339
   430
    unfolding dist_prod_def norm_prod_def
huffman@31339
   431
    by (simp add: dist_norm)
huffman@30019
   432
qed
huffman@30019
   433
huffman@30019
   434
end
huffman@30019
   435
haftmann@37678
   436
instance prod :: (banach, banach) banach ..
huffman@31405
   437
huffman@44575
   438
subsubsection {* Pair operations are linear *}
huffman@30019
   439
huffman@44282
   440
lemma bounded_linear_fst: "bounded_linear fst"
huffman@44127
   441
  using fst_add fst_scaleR
huffman@44127
   442
  by (rule bounded_linear_intro [where K=1], simp add: norm_prod_def)
huffman@30019
   443
huffman@44282
   444
lemma bounded_linear_snd: "bounded_linear snd"
huffman@44127
   445
  using snd_add snd_scaleR
huffman@44127
   446
  by (rule bounded_linear_intro [where K=1], simp add: norm_prod_def)
huffman@30019
   447
huffman@30019
   448
text {* TODO: move to NthRoot *}
huffman@30019
   449
lemma sqrt_add_le_add_sqrt:
huffman@30019
   450
  assumes x: "0 \<le> x" and y: "0 \<le> y"
huffman@30019
   451
  shows "sqrt (x + y) \<le> sqrt x + sqrt y"
huffman@30019
   452
apply (rule power2_le_imp_le)
huffman@44749
   453
apply (simp add: power2_sum x y)
huffman@30019
   454
apply (simp add: mult_nonneg_nonneg x y)
huffman@44126
   455
apply (simp add: x y)
huffman@30019
   456
done
huffman@30019
   457
huffman@30019
   458
lemma bounded_linear_Pair:
huffman@30019
   459
  assumes f: "bounded_linear f"
huffman@30019
   460
  assumes g: "bounded_linear g"
huffman@30019
   461
  shows "bounded_linear (\<lambda>x. (f x, g x))"
huffman@30019
   462
proof
huffman@30019
   463
  interpret f: bounded_linear f by fact
huffman@30019
   464
  interpret g: bounded_linear g by fact
huffman@30019
   465
  fix x y and r :: real
huffman@30019
   466
  show "(f (x + y), g (x + y)) = (f x, g x) + (f y, g y)"
huffman@30019
   467
    by (simp add: f.add g.add)
huffman@30019
   468
  show "(f (r *\<^sub>R x), g (r *\<^sub>R x)) = r *\<^sub>R (f x, g x)"
huffman@30019
   469
    by (simp add: f.scaleR g.scaleR)
huffman@30019
   470
  obtain Kf where "0 < Kf" and norm_f: "\<And>x. norm (f x) \<le> norm x * Kf"
huffman@30019
   471
    using f.pos_bounded by fast
huffman@30019
   472
  obtain Kg where "0 < Kg" and norm_g: "\<And>x. norm (g x) \<le> norm x * Kg"
huffman@30019
   473
    using g.pos_bounded by fast
huffman@30019
   474
  have "\<forall>x. norm (f x, g x) \<le> norm x * (Kf + Kg)"
huffman@30019
   475
    apply (rule allI)
huffman@30019
   476
    apply (simp add: norm_Pair)
huffman@30019
   477
    apply (rule order_trans [OF sqrt_add_le_add_sqrt], simp, simp)
huffman@30019
   478
    apply (simp add: right_distrib)
huffman@30019
   479
    apply (rule add_mono [OF norm_f norm_g])
huffman@30019
   480
    done
huffman@30019
   481
  then show "\<exists>K. \<forall>x. norm (f x, g x) \<le> norm x * K" ..
huffman@30019
   482
qed
huffman@30019
   483
huffman@44575
   484
subsubsection {* Frechet derivatives involving pairs *}
huffman@30019
   485
huffman@30019
   486
lemma FDERIV_Pair:
huffman@30019
   487
  assumes f: "FDERIV f x :> f'" and g: "FDERIV g x :> g'"
huffman@30019
   488
  shows "FDERIV (\<lambda>x. (f x, g x)) x :> (\<lambda>h. (f' h, g' h))"
huffman@44575
   489
proof (rule FDERIV_I)
huffman@44575
   490
  show "bounded_linear (\<lambda>h. (f' h, g' h))"
huffman@44575
   491
    using f g by (intro bounded_linear_Pair FDERIV_bounded_linear)
huffman@44575
   492
  let ?Rf = "\<lambda>h. f (x + h) - f x - f' h"
huffman@44575
   493
  let ?Rg = "\<lambda>h. g (x + h) - g x - g' h"
huffman@44575
   494
  let ?R = "\<lambda>h. ((f (x + h), g (x + h)) - (f x, g x) - (f' h, g' h))"
huffman@44575
   495
  show "(\<lambda>h. norm (?R h) / norm h) -- 0 --> 0"
huffman@44575
   496
  proof (rule real_LIM_sandwich_zero)
huffman@44575
   497
    show "(\<lambda>h. norm (?Rf h) / norm h + norm (?Rg h) / norm h) -- 0 --> 0"
huffman@44575
   498
      using f g by (intro tendsto_add_zero FDERIV_D)
huffman@44575
   499
    fix h :: 'a assume "h \<noteq> 0"
huffman@44575
   500
    thus "0 \<le> norm (?R h) / norm h"
huffman@44575
   501
      by (simp add: divide_nonneg_pos)
huffman@44575
   502
    show "norm (?R h) / norm h \<le> norm (?Rf h) / norm h + norm (?Rg h) / norm h"
huffman@44575
   503
      unfolding add_divide_distrib [symmetric]
huffman@44575
   504
      by (simp add: norm_Pair divide_right_mono
huffman@44575
   505
        order_trans [OF sqrt_add_le_add_sqrt])
huffman@44575
   506
  qed
huffman@44575
   507
qed
huffman@44575
   508
huffman@44575
   509
subsection {* Product is an inner product space *}
huffman@44575
   510
huffman@44575
   511
instantiation prod :: (real_inner, real_inner) real_inner
huffman@44575
   512
begin
huffman@44575
   513
huffman@44575
   514
definition inner_prod_def:
huffman@44575
   515
  "inner x y = inner (fst x) (fst y) + inner (snd x) (snd y)"
huffman@44575
   516
huffman@44575
   517
lemma inner_Pair [simp]: "inner (a, b) (c, d) = inner a c + inner b d"
huffman@44575
   518
  unfolding inner_prod_def by simp
huffman@44575
   519
huffman@44575
   520
instance proof
huffman@44575
   521
  fix r :: real
huffman@44575
   522
  fix x y z :: "'a::real_inner \<times> 'b::real_inner"
huffman@44575
   523
  show "inner x y = inner y x"
huffman@44575
   524
    unfolding inner_prod_def
huffman@44575
   525
    by (simp add: inner_commute)
huffman@44575
   526
  show "inner (x + y) z = inner x z + inner y z"
huffman@44575
   527
    unfolding inner_prod_def
huffman@44575
   528
    by (simp add: inner_add_left)
huffman@44575
   529
  show "inner (scaleR r x) y = r * inner x y"
huffman@44575
   530
    unfolding inner_prod_def
huffman@44575
   531
    by (simp add: right_distrib)
huffman@44575
   532
  show "0 \<le> inner x x"
huffman@44575
   533
    unfolding inner_prod_def
huffman@44575
   534
    by (intro add_nonneg_nonneg inner_ge_zero)
huffman@44575
   535
  show "inner x x = 0 \<longleftrightarrow> x = 0"
huffman@44575
   536
    unfolding inner_prod_def prod_eq_iff
huffman@44575
   537
    by (simp add: add_nonneg_eq_0_iff)
huffman@44575
   538
  show "norm x = sqrt (inner x x)"
huffman@44575
   539
    unfolding norm_prod_def inner_prod_def
huffman@44575
   540
    by (simp add: power2_norm_eq_inner)
huffman@44575
   541
qed
huffman@30019
   542
huffman@30019
   543
end
huffman@44575
   544
huffman@44575
   545
end