src/HOL/Library/Product_Vector.thy
author huffman
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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 "*" :: (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: expand_prod_eq 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: expand_prod_eq scaleR_left_distrib)
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  show "scaleR a (scaleR b x) = scaleR (a * b) x"
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    by (simp add: expand_prod_eq)
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  show "scaleR 1 x = x"
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    by (simp add: expand_prod_eq)
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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
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  "*" :: (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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   154
  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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subsection {* Product is a metric space *}
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instantiation
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  "*" :: (metric_space, metric_space) metric_space
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begin
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b4660351e8e7 instance * :: (metric_space, metric_space) metric_space; generalize lemmas to class metric_space
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definition dist_prod_def:
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  "dist (x::'a \<times> 'b) y = sqrt ((dist (fst x) (fst y))\<twosuperior> + (dist (snd x) (snd y))\<twosuperior>)"
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b4660351e8e7 instance * :: (metric_space, metric_space) metric_space; generalize lemmas to class metric_space
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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 expand_prod_eq 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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        real_sqrt_le_mono add_mono power_mono dist_triangle2 zero_le_dist)
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next
80686a815b59 instance * :: topological_space
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  (* FIXME: long proof! *)
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  (* Maybe it would be easier to define topological spaces *)
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  (* in terms of neighborhoods instead of open sets? *)
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  fix S :: "('a \<times> 'b) set"
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  show "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S)"
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   192
  proof
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    assume "open S" show "\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S"
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    proof
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      fix x assume "x \<in> S"
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      obtain A B where "open A" "open B" "x \<in> A \<times> B" "A \<times> B \<subseteq> S"
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   197
        using `open S` and `x \<in> S` by (rule open_prod_elim)
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      obtain r where r: "0 < r" "\<forall>y. dist y (fst x) < r \<longrightarrow> y \<in> A"
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   199
        using `open A` and `x \<in> A \<times> B` unfolding open_dist by auto
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      obtain s where s: "0 < s" "\<forall>y. dist y (snd x) < s \<longrightarrow> y \<in> B"
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        using `open B` and `x \<in> A \<times> B` unfolding open_dist by auto
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      let ?e = "min r s"
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      have "0 < ?e \<and> (\<forall>y. dist y x < ?e \<longrightarrow> y \<in> S)"
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      proof (intro allI impI conjI)
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        show "0 < min r s" by (simp add: r(1) s(1))
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      next
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   207
        fix y assume "dist y x < min r s"
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        hence "dist y x < r" and "dist y x < s"
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          by simp_all
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        hence "dist (fst y) (fst x) < r" and "dist (snd y) (snd x) < s"
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   211
          by (auto intro: le_less_trans dist_fst_le dist_snd_le)
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   212
        hence "fst y \<in> A" and "snd y \<in> B"
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   213
          by (simp_all add: r(2) s(2))
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        hence "y \<in> A \<times> B" by (induct y, simp)
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        with `A \<times> B \<subseteq> S` show "y \<in> S" ..
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   216
      qed
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      thus "\<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S" ..
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   218
    qed
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  next
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    assume "\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S" thus "open S"
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   221
    unfolding open_prod_def open_dist
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   222
    apply safe
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   223
    apply (drule (1) bspec)
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   224
    apply clarify
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   225
    apply (subgoal_tac "\<exists>r>0. \<exists>s>0. e = sqrt (r\<twosuperior> + s\<twosuperior>)")
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   226
    apply clarify
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    apply (rule_tac x="{y. dist y a < r}" in exI)
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   228
    apply (rule_tac x="{y. dist y b < s}" in exI)
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    apply (rule conjI)
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   230
    apply clarify
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   231
    apply (rule_tac x="r - dist x a" in exI, rule conjI, simp)
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   232
    apply clarify
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   233
    apply (simp add: less_diff_eq)
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   234
    apply (erule le_less_trans [OF dist_triangle])
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   235
    apply (rule conjI)
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   236
    apply clarify
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   237
    apply (rule_tac x="s - dist x b" in exI, rule conjI, simp)
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   238
    apply clarify
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   239
    apply (simp add: less_diff_eq)
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   240
    apply (erule le_less_trans [OF dist_triangle])
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   241
    apply (rule conjI)
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   242
    apply simp
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   243
    apply (clarify, rename_tac c d)
80686a815b59 instance * :: topological_space
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   244
    apply (drule spec, erule mp)
80686a815b59 instance * :: topological_space
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   245
    apply (simp add: dist_Pair_Pair add_strict_mono power_strict_mono)
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   246
    apply (rule_tac x="e / sqrt 2" in exI, simp add: divide_pos_pos)
80686a815b59 instance * :: topological_space
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   247
    apply (rule_tac x="e / sqrt 2" in exI, simp add: divide_pos_pos)
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   248
    apply (simp add: power_divide)
80686a815b59 instance * :: topological_space
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   249
    done
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   250
  qed
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   251
qed
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   252
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   253
end
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   254
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   255
subsection {* Continuity of operations *}
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   256
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   257
lemma tendsto_fst [tendsto_intros]:
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  assumes "(f ---> a) net"
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   259
  shows "((\<lambda>x. fst (f x)) ---> fst a) net"
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   260
proof (rule topological_tendstoI)
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   261
  fix S assume "open S" "fst a \<in> S"
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   262
  then have "open (fst -` S)" "a \<in> fst -` S"
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   263
    unfolding open_prod_def
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   264
    apply simp_all
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   265
    apply clarify
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   266
    apply (rule exI, erule conjI)
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   267
    apply (rule exI, rule conjI [OF open_UNIV])
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   268
    apply auto
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   269
    done
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   270
  with assms have "eventually (\<lambda>x. f x \<in> fst -` S) net"
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   271
    by (rule topological_tendstoD)
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  then show "eventually (\<lambda>x. fst (f x) \<in> S) net"
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   273
    by simp
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   274
qed
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   275
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   276
lemma tendsto_snd [tendsto_intros]:
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  assumes "(f ---> a) net"
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  shows "((\<lambda>x. snd (f x)) ---> snd a) net"
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   279
proof (rule topological_tendstoI)
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   280
  fix S assume "open S" "snd a \<in> S"
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   281
  then have "open (snd -` S)" "a \<in> snd -` S"
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   282
    unfolding open_prod_def
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   283
    apply simp_all
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   284
    apply clarify
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   285
    apply (rule exI, rule conjI [OF open_UNIV])
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   286
    apply (rule exI, erule conjI)
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   287
    apply auto
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   288
    done
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   289
  with assms have "eventually (\<lambda>x. f x \<in> snd -` S) net"
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   290
    by (rule topological_tendstoD)
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   291
  then show "eventually (\<lambda>x. snd (f x) \<in> S) net"
f7310185481d generalize tendsto lemmas for products
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   292
    by simp
31405
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   293
qed
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   294
31565
da5a5589418e theorem attribute [tendsto_intros]
huffman
parents: 31563
diff changeset
   295
lemma tendsto_Pair [tendsto_intros]:
31491
f7310185481d generalize tendsto lemmas for products
huffman
parents: 31417
diff changeset
   296
  assumes "(f ---> a) net" and "(g ---> b) net"
f7310185481d generalize tendsto lemmas for products
huffman
parents: 31417
diff changeset
   297
  shows "((\<lambda>x. (f x, g x)) ---> (a, b)) net"
f7310185481d generalize tendsto lemmas for products
huffman
parents: 31417
diff changeset
   298
proof (rule topological_tendstoI)
31492
5400beeddb55 replace 'topo' with 'open'; add extra type constraint for 'open'
huffman
parents: 31491
diff changeset
   299
  fix S assume "open S" "(a, b) \<in> S"
5400beeddb55 replace 'topo' with 'open'; add extra type constraint for 'open'
huffman
parents: 31491
diff changeset
   300
  then obtain A B where "open A" "open B" "a \<in> A" "b \<in> B" "A \<times> B \<subseteq> S"
5400beeddb55 replace 'topo' with 'open'; add extra type constraint for 'open'
huffman
parents: 31491
diff changeset
   301
    unfolding open_prod_def by auto
31491
f7310185481d generalize tendsto lemmas for products
huffman
parents: 31417
diff changeset
   302
  have "eventually (\<lambda>x. f x \<in> A) net"
31492
5400beeddb55 replace 'topo' with 'open'; add extra type constraint for 'open'
huffman
parents: 31491
diff changeset
   303
    using `(f ---> a) net` `open A` `a \<in> A`
31491
f7310185481d generalize tendsto lemmas for products
huffman
parents: 31417
diff changeset
   304
    by (rule topological_tendstoD)
31405
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   305
  moreover
31491
f7310185481d generalize tendsto lemmas for products
huffman
parents: 31417
diff changeset
   306
  have "eventually (\<lambda>x. g x \<in> B) net"
31492
5400beeddb55 replace 'topo' with 'open'; add extra type constraint for 'open'
huffman
parents: 31491
diff changeset
   307
    using `(g ---> b) net` `open B` `b \<in> B`
31491
f7310185481d generalize tendsto lemmas for products
huffman
parents: 31417
diff changeset
   308
    by (rule topological_tendstoD)
31405
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   309
  ultimately
31491
f7310185481d generalize tendsto lemmas for products
huffman
parents: 31417
diff changeset
   310
  show "eventually (\<lambda>x. (f x, g x) \<in> S) net"
31405
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   311
    by (rule eventually_elim2)
31491
f7310185481d generalize tendsto lemmas for products
huffman
parents: 31417
diff changeset
   312
       (simp add: subsetD [OF `A \<times> B \<subseteq> S`])
31405
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   313
qed
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   314
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   315
lemma Cauchy_fst: "Cauchy X \<Longrightarrow> Cauchy (\<lambda>n. fst (X n))"
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   316
unfolding Cauchy_def by (fast elim: le_less_trans [OF dist_fst_le])
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   317
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   318
lemma Cauchy_snd: "Cauchy X \<Longrightarrow> Cauchy (\<lambda>n. snd (X n))"
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   319
unfolding Cauchy_def by (fast elim: le_less_trans [OF dist_snd_le])
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   320
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   321
lemma Cauchy_Pair:
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   322
  assumes "Cauchy X" and "Cauchy Y"
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   323
  shows "Cauchy (\<lambda>n. (X n, Y n))"
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   324
proof (rule metric_CauchyI)
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   325
  fix r :: real assume "0 < r"
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   326
  then have "0 < r / sqrt 2" (is "0 < ?s")
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   327
    by (simp add: divide_pos_pos)
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   328
  obtain M where M: "\<forall>m\<ge>M. \<forall>n\<ge>M. dist (X m) (X n) < ?s"
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   329
    using metric_CauchyD [OF `Cauchy X` `0 < ?s`] ..
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   330
  obtain N where N: "\<forall>m\<ge>N. \<forall>n\<ge>N. dist (Y m) (Y n) < ?s"
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   331
    using metric_CauchyD [OF `Cauchy Y` `0 < ?s`] ..
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   332
  have "\<forall>m\<ge>max M N. \<forall>n\<ge>max M N. dist (X m, Y m) (X n, Y n) < r"
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   333
    using M N by (simp add: real_sqrt_sum_squares_less dist_Pair_Pair)
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   334
  then show "\<exists>n0. \<forall>m\<ge>n0. \<forall>n\<ge>n0. dist (X m, Y m) (X n, Y n) < r" ..
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   335
qed
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   336
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   337
lemma isCont_Pair [simp]:
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   338
  "\<lbrakk>isCont f x; isCont g x\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. (f x, g x)) x"
36661
0a5b7b818d65 make (f -- a --> x) an abbreviation for (f ---> x) (at a)
huffman
parents: 36660
diff changeset
   339
  unfolding isCont_def by (rule tendsto_Pair)
31405
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   340
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   341
subsection {* Product is a complete metric space *}
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   342
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   343
instance "*" :: (complete_space, complete_space) complete_space
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   344
proof
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   345
  fix X :: "nat \<Rightarrow> 'a \<times> 'b" assume "Cauchy X"
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   346
  have 1: "(\<lambda>n. fst (X n)) ----> lim (\<lambda>n. fst (X n))"
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   347
    using Cauchy_fst [OF `Cauchy X`]
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   348
    by (simp add: Cauchy_convergent_iff convergent_LIMSEQ_iff)
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   349
  have 2: "(\<lambda>n. snd (X n)) ----> lim (\<lambda>n. snd (X n))"
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   350
    using Cauchy_snd [OF `Cauchy X`]
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   351
    by (simp add: Cauchy_convergent_iff convergent_LIMSEQ_iff)
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   352
  have "X ----> (lim (\<lambda>n. fst (X n)), lim (\<lambda>n. snd (X n)))"
36660
1cc4ab4b7ff7 make (X ----> L) an abbreviation for (X ---> L) sequentially
huffman
parents: 36332
diff changeset
   353
    using tendsto_Pair [OF 1 2] by simp
31405
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   354
  then show "convergent X"
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   355
    by (rule convergentI)
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   356
qed
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   357
30019
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   358
subsection {* Product is a normed vector space *}
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   359
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   360
instantiation
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   361
  "*" :: (real_normed_vector, real_normed_vector) real_normed_vector
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   362
begin
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   363
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   364
definition norm_prod_def:
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   365
  "norm x = sqrt ((norm (fst x))\<twosuperior> + (norm (snd x))\<twosuperior>)"
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   366
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   367
definition sgn_prod_def:
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   368
  "sgn (x::'a \<times> 'b) = scaleR (inverse (norm x)) x"
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   369
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   370
lemma norm_Pair: "norm (a, b) = sqrt ((norm a)\<twosuperior> + (norm b)\<twosuperior>)"
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   371
  unfolding norm_prod_def by simp
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   372
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   373
instance proof
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   374
  fix r :: real and x y :: "'a \<times> 'b"
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   375
  show "0 \<le> norm x"
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   376
    unfolding norm_prod_def by simp
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   377
  show "norm x = 0 \<longleftrightarrow> x = 0"
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   378
    unfolding norm_prod_def
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   379
    by (simp add: expand_prod_eq)
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   380
  show "norm (x + y) \<le> norm x + norm y"
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   381
    unfolding norm_prod_def
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   382
    apply (rule order_trans [OF _ real_sqrt_sum_squares_triangle_ineq])
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   383
    apply (simp add: add_mono power_mono norm_triangle_ineq)
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   384
    done
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   385
  show "norm (scaleR r x) = \<bar>r\<bar> * norm x"
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   386
    unfolding norm_prod_def
31587
a7e187205fc0 remove simp add: norm_scaleR
huffman
parents: 31568
diff changeset
   387
    apply (simp add: power_mult_distrib)
30019
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   388
    apply (simp add: right_distrib [symmetric])
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   389
    apply (simp add: real_sqrt_mult_distrib)
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   390
    done
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   391
  show "sgn x = scaleR (inverse (norm x)) x"
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   392
    by (rule sgn_prod_def)
31290
f41c023d90bc define dist for products
huffman
parents: 30729
diff changeset
   393
  show "dist x y = norm (x - y)"
31339
b4660351e8e7 instance * :: (metric_space, metric_space) metric_space; generalize lemmas to class metric_space
huffman
parents: 31290
diff changeset
   394
    unfolding dist_prod_def norm_prod_def
b4660351e8e7 instance * :: (metric_space, metric_space) metric_space; generalize lemmas to class metric_space
huffman
parents: 31290
diff changeset
   395
    by (simp add: dist_norm)
30019
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   396
qed
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   397
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   398
end
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   399
31405
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   400
instance "*" :: (banach, banach) banach ..
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   401
30019
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   402
subsection {* Product is an inner product space *}
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   403
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   404
instantiation "*" :: (real_inner, real_inner) real_inner
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   405
begin
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   406
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   407
definition inner_prod_def:
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   408
  "inner x y = inner (fst x) (fst y) + inner (snd x) (snd y)"
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   409
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   410
lemma inner_Pair [simp]: "inner (a, b) (c, d) = inner a c + inner b d"
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   411
  unfolding inner_prod_def by simp
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   412
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   413
instance proof
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   414
  fix r :: real
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   415
  fix x y z :: "'a::real_inner * 'b::real_inner"
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   416
  show "inner x y = inner y x"
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   417
    unfolding inner_prod_def
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   418
    by (simp add: inner_commute)
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   419
  show "inner (x + y) z = inner x z + inner y z"
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   420
    unfolding inner_prod_def
31590
776d6a4c1327 declare inner_add, inner_diff [algebra_simps]; declare inner_scaleR [simp]
huffman
parents: 31587
diff changeset
   421
    by (simp add: inner_add_left)
30019
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   422
  show "inner (scaleR r x) y = r * inner x y"
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   423
    unfolding inner_prod_def
31590
776d6a4c1327 declare inner_add, inner_diff [algebra_simps]; declare inner_scaleR [simp]
huffman
parents: 31587
diff changeset
   424
    by (simp add: right_distrib)
30019
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   425
  show "0 \<le> inner x x"
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   426
    unfolding inner_prod_def
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   427
    by (intro add_nonneg_nonneg inner_ge_zero)
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   428
  show "inner x x = 0 \<longleftrightarrow> x = 0"
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   429
    unfolding inner_prod_def expand_prod_eq
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   430
    by (simp add: add_nonneg_eq_0_iff)
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   431
  show "norm x = sqrt (inner x x)"
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   432
    unfolding norm_prod_def inner_prod_def
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   433
    by (simp add: power2_norm_eq_inner)
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   434
qed
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   435
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   436
end
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   437
31405
1f72869f1a2e instance * :: complete_space; generalize continuity lemmas for fst, snd, Pair
huffman
parents: 31388
diff changeset
   438
subsection {* Pair operations are linear *}
30019
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   439
30729
461ee3e49ad3 interpretation/interpret: prefixes are mandatory by default;
wenzelm
parents: 30019
diff changeset
   440
interpretation fst: bounded_linear fst
30019
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   441
  apply (unfold_locales)
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   442
  apply (rule fst_add)
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   443
  apply (rule fst_scaleR)
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   444
  apply (rule_tac x="1" in exI, simp add: norm_Pair)
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   445
  done
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   446
30729
461ee3e49ad3 interpretation/interpret: prefixes are mandatory by default;
wenzelm
parents: 30019
diff changeset
   447
interpretation snd: bounded_linear snd
30019
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   448
  apply (unfold_locales)
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   449
  apply (rule snd_add)
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   450
  apply (rule snd_scaleR)
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   451
  apply (rule_tac x="1" in exI, simp add: norm_Pair)
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   452
  done
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   453
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   454
text {* TODO: move to NthRoot *}
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   455
lemma sqrt_add_le_add_sqrt:
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   456
  assumes x: "0 \<le> x" and y: "0 \<le> y"
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   457
  shows "sqrt (x + y) \<le> sqrt x + sqrt y"
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   458
apply (rule power2_le_imp_le)
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   459
apply (simp add: real_sum_squared_expand add_nonneg_nonneg x y)
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   460
apply (simp add: mult_nonneg_nonneg x y)
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   461
apply (simp add: add_nonneg_nonneg x y)
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   462
done
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   463
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   464
lemma bounded_linear_Pair:
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   465
  assumes f: "bounded_linear f"
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   466
  assumes g: "bounded_linear g"
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   467
  shows "bounded_linear (\<lambda>x. (f x, g x))"
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   468
proof
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   469
  interpret f: bounded_linear f by fact
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   470
  interpret g: bounded_linear g by fact
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   471
  fix x y and r :: real
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   472
  show "(f (x + y), g (x + y)) = (f x, g x) + (f y, g y)"
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   473
    by (simp add: f.add g.add)
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   474
  show "(f (r *\<^sub>R x), g (r *\<^sub>R x)) = r *\<^sub>R (f x, g x)"
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   475
    by (simp add: f.scaleR g.scaleR)
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   476
  obtain Kf where "0 < Kf" and norm_f: "\<And>x. norm (f x) \<le> norm x * Kf"
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   477
    using f.pos_bounded by fast
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   478
  obtain Kg where "0 < Kg" and norm_g: "\<And>x. norm (g x) \<le> norm x * Kg"
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   479
    using g.pos_bounded by fast
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   480
  have "\<forall>x. norm (f x, g x) \<le> norm x * (Kf + Kg)"
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   481
    apply (rule allI)
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   482
    apply (simp add: norm_Pair)
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   483
    apply (rule order_trans [OF sqrt_add_le_add_sqrt], simp, simp)
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   484
    apply (simp add: right_distrib)
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   485
    apply (rule add_mono [OF norm_f norm_g])
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   486
    done
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   487
  then show "\<exists>K. \<forall>x. norm (f x, g x) \<le> norm x * K" ..
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   488
qed
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   489
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   490
subsection {* Frechet derivatives involving pairs *}
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   491
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   492
lemma FDERIV_Pair:
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   493
  assumes f: "FDERIV f x :> f'" and g: "FDERIV g x :> g'"
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   494
  shows "FDERIV (\<lambda>x. (f x, g x)) x :> (\<lambda>h. (f' h, g' h))"
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   495
apply (rule FDERIV_I)
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   496
apply (rule bounded_linear_Pair)
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   497
apply (rule FDERIV_bounded_linear [OF f])
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   498
apply (rule FDERIV_bounded_linear [OF g])
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   499
apply (simp add: norm_Pair)
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   500
apply (rule real_LIM_sandwich_zero)
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   501
apply (rule LIM_add_zero)
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   502
apply (rule FDERIV_D [OF f])
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   503
apply (rule FDERIV_D [OF g])
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   504
apply (rename_tac h)
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   505
apply (simp add: divide_nonneg_pos)
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   506
apply (rename_tac h)
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   507
apply (subst add_divide_distrib [symmetric])
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   508
apply (rule divide_right_mono [OF _ norm_ge_zero])
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   509
apply (rule order_trans [OF sqrt_add_le_add_sqrt])
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   510
apply simp
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   511
apply simp
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   512
apply simp
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   513
done
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   514
a2f19e0a28b2 add theory of products as real vector spaces to Library
huffman
parents:
diff changeset
   515
end