src/HOL/Library/Product_Vector.thy
author huffman
Fri Jun 12 16:23:07 2009 -0700 (2009-06-12)
changeset 31590 776d6a4c1327
parent 31587 a7e187205fc0
child 34110 4c113c744b86
permissions -rw-r--r--
declare inner_add, inner_diff [algebra_simps]; declare inner_scaleR [simp]
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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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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" thus "open (S \<inter> T)"
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    unfolding open_prod_def
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    apply clarify
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    apply (drule (1) bspec)+
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    apply (clarify, rename_tac Sa Ta Sb Tb)
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    apply (rule_tac x="Sa \<inter> Ta" in exI)
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    apply (rule_tac x="Sb \<inter> Tb" in exI)
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    apply (simp add: open_Int)
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    apply fast
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    done
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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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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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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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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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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
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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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  proof
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    assume "open S" thus "\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S"
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    unfolding open_prod_def open_dist
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    apply safe
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    apply (drule (1) bspec)
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    apply clarify
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    apply (drule (1) bspec)+
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    apply (clarify, rename_tac r s)
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    apply (rule_tac x="min r s" in exI, simp)
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    apply (clarify, rename_tac c d)
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    apply (erule subsetD)
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    apply (simp add: dist_Pair_Pair)
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    apply (rule conjI)
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    apply (drule spec, erule mp)
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    apply (erule le_less_trans [OF real_sqrt_sum_squares_ge1])
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    apply (drule spec, erule mp)
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    apply (erule le_less_trans [OF real_sqrt_sum_squares_ge2])
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    done
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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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    unfolding open_prod_def open_dist
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    apply safe
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    apply (drule (1) bspec)
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    apply clarify
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    apply (subgoal_tac "\<exists>r>0. \<exists>s>0. e = sqrt (r\<twosuperior> + s\<twosuperior>)")
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    apply clarify
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    apply (rule_tac x="{y. dist y a < r}" in exI)
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    apply (rule_tac x="{y. dist y b < s}" in exI)
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    apply (rule conjI)
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    apply clarify
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    apply (rule_tac x="r - dist x a" in exI, rule conjI, simp)
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    apply clarify
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    apply (simp add: less_diff_eq)
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    apply (erule le_less_trans [OF dist_triangle])
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    apply (rule conjI)
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    apply clarify
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    apply (rule_tac x="s - dist x b" in exI, rule conjI, simp)
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    apply clarify
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    apply (simp add: less_diff_eq)
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    apply (erule le_less_trans [OF dist_triangle])
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    apply (rule conjI)
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    apply simp
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    apply (clarify, rename_tac c d)
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    apply (drule spec, erule mp)
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    apply (simp add: dist_Pair_Pair add_strict_mono power_strict_mono)
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    apply (rule_tac x="e / sqrt 2" in exI, simp add: divide_pos_pos)
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    apply (rule_tac x="e / sqrt 2" in exI, simp add: divide_pos_pos)
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    apply (simp add: power_divide)
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    done
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  qed
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qed
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end
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subsection {* Continuity of operations *}
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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 simp
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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 simp
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lemma tendsto_fst [tendsto_intros]:
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  assumes "(f ---> a) net"
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  shows "((\<lambda>x. fst (f x)) ---> fst a) net"
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proof (rule topological_tendstoI)
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  fix S assume "open S" "fst a \<in> S"
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  then have "open (fst -` S)" "a \<in> fst -` S"
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    unfolding open_prod_def
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    apply simp_all
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    apply clarify
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    apply (rule exI, erule conjI)
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    apply (rule exI, rule conjI [OF open_UNIV])
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    apply auto
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    done
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  with assms have "eventually (\<lambda>x. f x \<in> fst -` S) net"
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    by (rule topological_tendstoD)
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  then show "eventually (\<lambda>x. fst (f x) \<in> S) net"
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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) net"
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  shows "((\<lambda>x. snd (f x)) ---> snd a) net"
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proof (rule topological_tendstoI)
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  fix S assume "open S" "snd a \<in> S"
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  then have "open (snd -` S)" "a \<in> snd -` S"
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    unfolding open_prod_def
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    apply simp_all
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    apply clarify
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    apply (rule exI, rule conjI [OF open_UNIV])
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    apply (rule exI, erule conjI)
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    apply auto
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    done
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  with assms have "eventually (\<lambda>x. f x \<in> snd -` S) net"
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    by (rule topological_tendstoD)
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  then show "eventually (\<lambda>x. snd (f x) \<in> S) net"
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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) net" and "(g ---> b) net"
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  shows "((\<lambda>x. (f x, g x)) ---> (a, b)) net"
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proof (rule topological_tendstoI)
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  fix S assume "open S" "(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 auto
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  have "eventually (\<lambda>x. f x \<in> A) net"
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    using `(f ---> a) net` `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) net"
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    using `(g ---> b) net` `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) net"
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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 LIMSEQ_fst: "(X ----> a) \<Longrightarrow> (\<lambda>n. fst (X n)) ----> fst a"
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unfolding LIMSEQ_conv_tendsto by (rule tendsto_fst)
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lemma LIMSEQ_snd: "(X ----> a) \<Longrightarrow> (\<lambda>n. snd (X n)) ----> snd a"
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unfolding LIMSEQ_conv_tendsto by (rule tendsto_snd)
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lemma LIMSEQ_Pair:
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  assumes "X ----> a" and "Y ----> b"
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  shows "(\<lambda>n. (X n, Y n)) ----> (a, b)"
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using assms unfolding LIMSEQ_conv_tendsto
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by (rule tendsto_Pair)
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lemma LIM_fst: "f -- x --> a \<Longrightarrow> (\<lambda>x. fst (f x)) -- x --> fst a"
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unfolding LIM_conv_tendsto by (rule tendsto_fst)
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lemma LIM_snd: "f -- x --> a \<Longrightarrow> (\<lambda>x. snd (f x)) -- x --> snd a"
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unfolding LIM_conv_tendsto by (rule tendsto_snd)
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lemma LIM_Pair:
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  assumes "f -- x --> a" and "g -- x --> b"
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  shows "(\<lambda>x. (f x, g x)) -- x --> (a, b)"
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using assms unfolding LIM_conv_tendsto
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by (rule tendsto_Pair)
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lemma Cauchy_fst: "Cauchy X \<Longrightarrow> Cauchy (\<lambda>n. fst (X n))"
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unfolding Cauchy_def by (fast elim: le_less_trans [OF dist_fst_le])
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lemma Cauchy_snd: "Cauchy X \<Longrightarrow> Cauchy (\<lambda>n. snd (X n))"
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unfolding Cauchy_def by (fast elim: le_less_trans [OF dist_snd_le])
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lemma Cauchy_Pair:
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  assumes "Cauchy X" and "Cauchy Y"
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  shows "Cauchy (\<lambda>n. (X n, Y n))"
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proof (rule metric_CauchyI)
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  fix r :: real assume "0 < r"
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  then have "0 < r / sqrt 2" (is "0 < ?s")
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    by (simp add: divide_pos_pos)
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  obtain M where M: "\<forall>m\<ge>M. \<forall>n\<ge>M. dist (X m) (X n) < ?s"
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    using metric_CauchyD [OF `Cauchy X` `0 < ?s`] ..
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  obtain N where N: "\<forall>m\<ge>N. \<forall>n\<ge>N. dist (Y m) (Y n) < ?s"
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    using metric_CauchyD [OF `Cauchy Y` `0 < ?s`] ..
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  have "\<forall>m\<ge>max M N. \<forall>n\<ge>max M N. dist (X m, Y m) (X n, Y n) < r"
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    using M N by (simp add: real_sqrt_sum_squares_less dist_Pair_Pair)
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  then show "\<exists>n0. \<forall>m\<ge>n0. \<forall>n\<ge>n0. dist (X m, Y m) (X n, Y n) < r" ..
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qed
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lemma isCont_Pair [simp]:
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  "\<lbrakk>isCont f x; isCont g x\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. (f x, g x)) x"
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  unfolding isCont_def by (rule LIM_Pair)
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subsection {* Product is a complete metric space *}
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instance "*" :: (complete_space, complete_space) complete_space
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proof
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  fix X :: "nat \<Rightarrow> 'a \<times> 'b" assume "Cauchy X"
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  have 1: "(\<lambda>n. fst (X n)) ----> lim (\<lambda>n. fst (X n))"
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   310
    using Cauchy_fst [OF `Cauchy X`]
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   311
    by (simp add: Cauchy_convergent_iff convergent_LIMSEQ_iff)
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   312
  have 2: "(\<lambda>n. snd (X n)) ----> lim (\<lambda>n. snd (X n))"
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   313
    using Cauchy_snd [OF `Cauchy X`]
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   314
    by (simp add: Cauchy_convergent_iff convergent_LIMSEQ_iff)
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   315
  have "X ----> (lim (\<lambda>n. fst (X n)), lim (\<lambda>n. snd (X n)))"
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   316
    using LIMSEQ_Pair [OF 1 2] by simp
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   317
  then show "convergent X"
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   318
    by (rule convergentI)
huffman@31405
   319
qed
huffman@31405
   320
huffman@30019
   321
subsection {* Product is a normed vector space *}
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   322
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   323
instantiation
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   324
  "*" :: (real_normed_vector, real_normed_vector) real_normed_vector
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   325
begin
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   326
huffman@30019
   327
definition norm_prod_def:
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   328
  "norm x = sqrt ((norm (fst x))\<twosuperior> + (norm (snd x))\<twosuperior>)"
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   329
huffman@30019
   330
definition sgn_prod_def:
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   331
  "sgn (x::'a \<times> 'b) = scaleR (inverse (norm x)) x"
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   332
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   333
lemma norm_Pair: "norm (a, b) = sqrt ((norm a)\<twosuperior> + (norm b)\<twosuperior>)"
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   334
  unfolding norm_prod_def by simp
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   335
huffman@30019
   336
instance proof
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   337
  fix r :: real and x y :: "'a \<times> 'b"
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   338
  show "0 \<le> norm x"
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   339
    unfolding norm_prod_def by simp
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   340
  show "norm x = 0 \<longleftrightarrow> x = 0"
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   341
    unfolding norm_prod_def
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   342
    by (simp add: expand_prod_eq)
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   343
  show "norm (x + y) \<le> norm x + norm y"
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   344
    unfolding norm_prod_def
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   345
    apply (rule order_trans [OF _ real_sqrt_sum_squares_triangle_ineq])
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   346
    apply (simp add: add_mono power_mono norm_triangle_ineq)
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   347
    done
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   348
  show "norm (scaleR r x) = \<bar>r\<bar> * norm x"
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   349
    unfolding norm_prod_def
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   350
    apply (simp add: power_mult_distrib)
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   351
    apply (simp add: right_distrib [symmetric])
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   352
    apply (simp add: real_sqrt_mult_distrib)
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   353
    done
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   354
  show "sgn x = scaleR (inverse (norm x)) x"
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   355
    by (rule sgn_prod_def)
huffman@31290
   356
  show "dist x y = norm (x - y)"
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   357
    unfolding dist_prod_def norm_prod_def
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   358
    by (simp add: dist_norm)
huffman@30019
   359
qed
huffman@30019
   360
huffman@30019
   361
end
huffman@30019
   362
huffman@31405
   363
instance "*" :: (banach, banach) banach ..
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   364
huffman@30019
   365
subsection {* Product is an inner product space *}
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   366
huffman@30019
   367
instantiation "*" :: (real_inner, real_inner) real_inner
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   368
begin
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   369
huffman@30019
   370
definition inner_prod_def:
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   371
  "inner x y = inner (fst x) (fst y) + inner (snd x) (snd y)"
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   372
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   373
lemma inner_Pair [simp]: "inner (a, b) (c, d) = inner a c + inner b d"
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   374
  unfolding inner_prod_def by simp
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   375
huffman@30019
   376
instance proof
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   377
  fix r :: real
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   378
  fix x y z :: "'a::real_inner * 'b::real_inner"
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   379
  show "inner x y = inner y x"
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   380
    unfolding inner_prod_def
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   381
    by (simp add: inner_commute)
huffman@30019
   382
  show "inner (x + y) z = inner x z + inner y z"
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   383
    unfolding inner_prod_def
huffman@31590
   384
    by (simp add: inner_add_left)
huffman@30019
   385
  show "inner (scaleR r x) y = r * inner x y"
huffman@30019
   386
    unfolding inner_prod_def
huffman@31590
   387
    by (simp add: right_distrib)
huffman@30019
   388
  show "0 \<le> inner x x"
huffman@30019
   389
    unfolding inner_prod_def
huffman@30019
   390
    by (intro add_nonneg_nonneg inner_ge_zero)
huffman@30019
   391
  show "inner x x = 0 \<longleftrightarrow> x = 0"
huffman@30019
   392
    unfolding inner_prod_def expand_prod_eq
huffman@30019
   393
    by (simp add: add_nonneg_eq_0_iff)
huffman@30019
   394
  show "norm x = sqrt (inner x x)"
huffman@30019
   395
    unfolding norm_prod_def inner_prod_def
huffman@30019
   396
    by (simp add: power2_norm_eq_inner)
huffman@30019
   397
qed
huffman@30019
   398
huffman@30019
   399
end
huffman@30019
   400
huffman@31405
   401
subsection {* Pair operations are linear *}
huffman@30019
   402
wenzelm@30729
   403
interpretation fst: bounded_linear fst
huffman@30019
   404
  apply (unfold_locales)
huffman@30019
   405
  apply (rule fst_add)
huffman@30019
   406
  apply (rule fst_scaleR)
huffman@30019
   407
  apply (rule_tac x="1" in exI, simp add: norm_Pair)
huffman@30019
   408
  done
huffman@30019
   409
wenzelm@30729
   410
interpretation snd: bounded_linear snd
huffman@30019
   411
  apply (unfold_locales)
huffman@30019
   412
  apply (rule snd_add)
huffman@30019
   413
  apply (rule snd_scaleR)
huffman@30019
   414
  apply (rule_tac x="1" in exI, simp add: norm_Pair)
huffman@30019
   415
  done
huffman@30019
   416
huffman@30019
   417
text {* TODO: move to NthRoot *}
huffman@30019
   418
lemma sqrt_add_le_add_sqrt:
huffman@30019
   419
  assumes x: "0 \<le> x" and y: "0 \<le> y"
huffman@30019
   420
  shows "sqrt (x + y) \<le> sqrt x + sqrt y"
huffman@30019
   421
apply (rule power2_le_imp_le)
huffman@30019
   422
apply (simp add: real_sum_squared_expand add_nonneg_nonneg x y)
huffman@30019
   423
apply (simp add: mult_nonneg_nonneg x y)
huffman@30019
   424
apply (simp add: add_nonneg_nonneg x y)
huffman@30019
   425
done
huffman@30019
   426
huffman@30019
   427
lemma bounded_linear_Pair:
huffman@30019
   428
  assumes f: "bounded_linear f"
huffman@30019
   429
  assumes g: "bounded_linear g"
huffman@30019
   430
  shows "bounded_linear (\<lambda>x. (f x, g x))"
huffman@30019
   431
proof
huffman@30019
   432
  interpret f: bounded_linear f by fact
huffman@30019
   433
  interpret g: bounded_linear g by fact
huffman@30019
   434
  fix x y and r :: real
huffman@30019
   435
  show "(f (x + y), g (x + y)) = (f x, g x) + (f y, g y)"
huffman@30019
   436
    by (simp add: f.add g.add)
huffman@30019
   437
  show "(f (r *\<^sub>R x), g (r *\<^sub>R x)) = r *\<^sub>R (f x, g x)"
huffman@30019
   438
    by (simp add: f.scaleR g.scaleR)
huffman@30019
   439
  obtain Kf where "0 < Kf" and norm_f: "\<And>x. norm (f x) \<le> norm x * Kf"
huffman@30019
   440
    using f.pos_bounded by fast
huffman@30019
   441
  obtain Kg where "0 < Kg" and norm_g: "\<And>x. norm (g x) \<le> norm x * Kg"
huffman@30019
   442
    using g.pos_bounded by fast
huffman@30019
   443
  have "\<forall>x. norm (f x, g x) \<le> norm x * (Kf + Kg)"
huffman@30019
   444
    apply (rule allI)
huffman@30019
   445
    apply (simp add: norm_Pair)
huffman@30019
   446
    apply (rule order_trans [OF sqrt_add_le_add_sqrt], simp, simp)
huffman@30019
   447
    apply (simp add: right_distrib)
huffman@30019
   448
    apply (rule add_mono [OF norm_f norm_g])
huffman@30019
   449
    done
huffman@30019
   450
  then show "\<exists>K. \<forall>x. norm (f x, g x) \<le> norm x * K" ..
huffman@30019
   451
qed
huffman@30019
   452
huffman@30019
   453
subsection {* Frechet derivatives involving pairs *}
huffman@30019
   454
huffman@30019
   455
lemma FDERIV_Pair:
huffman@30019
   456
  assumes f: "FDERIV f x :> f'" and g: "FDERIV g x :> g'"
huffman@30019
   457
  shows "FDERIV (\<lambda>x. (f x, g x)) x :> (\<lambda>h. (f' h, g' h))"
huffman@30019
   458
apply (rule FDERIV_I)
huffman@30019
   459
apply (rule bounded_linear_Pair)
huffman@30019
   460
apply (rule FDERIV_bounded_linear [OF f])
huffman@30019
   461
apply (rule FDERIV_bounded_linear [OF g])
huffman@30019
   462
apply (simp add: norm_Pair)
huffman@30019
   463
apply (rule real_LIM_sandwich_zero)
huffman@30019
   464
apply (rule LIM_add_zero)
huffman@30019
   465
apply (rule FDERIV_D [OF f])
huffman@30019
   466
apply (rule FDERIV_D [OF g])
huffman@30019
   467
apply (rename_tac h)
huffman@30019
   468
apply (simp add: divide_nonneg_pos)
huffman@30019
   469
apply (rename_tac h)
huffman@30019
   470
apply (subst add_divide_distrib [symmetric])
huffman@30019
   471
apply (rule divide_right_mono [OF _ norm_ge_zero])
huffman@30019
   472
apply (rule order_trans [OF sqrt_add_le_add_sqrt])
huffman@30019
   473
apply simp
huffman@30019
   474
apply simp
huffman@30019
   475
apply simp
huffman@30019
   476
done
huffman@30019
   477
huffman@30019
   478
end