huffman@30019: (*  Title:      HOL/Library/Product_Vector.thy
huffman@30019:     Author:     Brian Huffman
huffman@30019: *)
huffman@30019: 
huffman@30019: header {* Cartesian Products as Vector Spaces *}
huffman@30019: 
huffman@30019: theory Product_Vector
huffman@30019: imports Inner_Product Product_plus
huffman@30019: begin
huffman@30019: 
huffman@30019: subsection {* Product is a real vector space *}
huffman@30019: 
haftmann@37678: instantiation prod :: (real_vector, real_vector) real_vector
huffman@30019: begin
huffman@30019: 
huffman@30019: definition scaleR_prod_def:
huffman@30019:   "scaleR r A = (scaleR r (fst A), scaleR r (snd A))"
huffman@30019: 
huffman@30019: lemma fst_scaleR [simp]: "fst (scaleR r A) = scaleR r (fst A)"
huffman@30019:   unfolding scaleR_prod_def by simp
huffman@30019: 
huffman@30019: lemma snd_scaleR [simp]: "snd (scaleR r A) = scaleR r (snd A)"
huffman@30019:   unfolding scaleR_prod_def by simp
huffman@30019: 
huffman@30019: lemma scaleR_Pair [simp]: "scaleR r (a, b) = (scaleR r a, scaleR r b)"
huffman@30019:   unfolding scaleR_prod_def by simp
huffman@30019: 
huffman@30019: instance proof
huffman@30019:   fix a b :: real and x y :: "'a \<times> 'b"
huffman@30019:   show "scaleR a (x + y) = scaleR a x + scaleR a y"
huffman@44066:     by (simp add: prod_eq_iff scaleR_right_distrib)
huffman@30019:   show "scaleR (a + b) x = scaleR a x + scaleR b x"
huffman@44066:     by (simp add: prod_eq_iff scaleR_left_distrib)
huffman@30019:   show "scaleR a (scaleR b x) = scaleR (a * b) x"
huffman@44066:     by (simp add: prod_eq_iff)
huffman@30019:   show "scaleR 1 x = x"
huffman@44066:     by (simp add: prod_eq_iff)
huffman@30019: qed
huffman@30019: 
huffman@30019: end
huffman@30019: 
huffman@31415: subsection {* Product is a topological space *}
huffman@31415: 
haftmann@37678: instantiation prod :: (topological_space, topological_space) topological_space
huffman@31415: begin
huffman@31415: 
huffman@31492: definition open_prod_def:
huffman@31492:   "open (S :: ('a \<times> 'b) set) \<longleftrightarrow>
huffman@31492:     (\<forall>x\<in>S. \<exists>A B. open A \<and> open B \<and> x \<in> A \<times> B \<and> A \<times> B \<subseteq> S)"
huffman@31415: 
huffman@36332: lemma open_prod_elim:
huffman@36332:   assumes "open S" and "x \<in> S"
huffman@36332:   obtains A B where "open A" and "open B" and "x \<in> A \<times> B" and "A \<times> B \<subseteq> S"
huffman@36332: using assms unfolding open_prod_def by fast
huffman@36332: 
huffman@36332: lemma open_prod_intro:
huffman@36332:   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"
huffman@36332:   shows "open S"
huffman@36332: using assms unfolding open_prod_def by fast
huffman@36332: 
huffman@31415: instance proof
huffman@31492:   show "open (UNIV :: ('a \<times> 'b) set)"
huffman@31492:     unfolding open_prod_def by auto
huffman@31415: next
huffman@31415:   fix S T :: "('a \<times> 'b) set"
huffman@36332:   assume "open S" "open T"
huffman@36332:   show "open (S \<inter> T)"
huffman@36332:   proof (rule open_prod_intro)
huffman@36332:     fix x assume x: "x \<in> S \<inter> T"
huffman@36332:     from x have "x \<in> S" by simp
huffman@36332:     obtain Sa Sb where A: "open Sa" "open Sb" "x \<in> Sa \<times> Sb" "Sa \<times> Sb \<subseteq> S"
huffman@36332:       using `open S` and `x \<in> S` by (rule open_prod_elim)
huffman@36332:     from x have "x \<in> T" by simp
huffman@36332:     obtain Ta Tb where B: "open Ta" "open Tb" "x \<in> Ta \<times> Tb" "Ta \<times> Tb \<subseteq> T"
huffman@36332:       using `open T` and `x \<in> T` by (rule open_prod_elim)
huffman@36332:     let ?A = "Sa \<inter> Ta" and ?B = "Sb \<inter> Tb"
huffman@36332:     have "open ?A \<and> open ?B \<and> x \<in> ?A \<times> ?B \<and> ?A \<times> ?B \<subseteq> S \<inter> T"
huffman@36332:       using A B by (auto simp add: open_Int)
huffman@36332:     thus "\<exists>A B. open A \<and> open B \<and> x \<in> A \<times> B \<and> A \<times> B \<subseteq> S \<inter> T"
huffman@36332:       by fast
huffman@36332:   qed
huffman@31415: next
huffman@31492:   fix K :: "('a \<times> 'b) set set"
huffman@31492:   assume "\<forall>S\<in>K. open S" thus "open (\<Union>K)"
huffman@31492:     unfolding open_prod_def by fast
huffman@31415: qed
huffman@31415: 
huffman@31415: end
huffman@31415: 
huffman@31562: lemma open_Times: "open S \<Longrightarrow> open T \<Longrightarrow> open (S \<times> T)"
huffman@31562: unfolding open_prod_def by auto
huffman@31562: 
huffman@31562: lemma fst_vimage_eq_Times: "fst -` S = S \<times> UNIV"
huffman@31562: by auto
huffman@31562: 
huffman@31562: lemma snd_vimage_eq_Times: "snd -` S = UNIV \<times> S"
huffman@31562: by auto
huffman@31562: 
huffman@31562: lemma open_vimage_fst: "open S \<Longrightarrow> open (fst -` S)"
huffman@31562: by (simp add: fst_vimage_eq_Times open_Times)
huffman@31562: 
huffman@31562: lemma open_vimage_snd: "open S \<Longrightarrow> open (snd -` S)"
huffman@31562: by (simp add: snd_vimage_eq_Times open_Times)
huffman@31562: 
huffman@31568: lemma closed_vimage_fst: "closed S \<Longrightarrow> closed (fst -` S)"
huffman@31568: unfolding closed_open vimage_Compl [symmetric]
huffman@31568: by (rule open_vimage_fst)
huffman@31568: 
huffman@31568: lemma closed_vimage_snd: "closed S \<Longrightarrow> closed (snd -` S)"
huffman@31568: unfolding closed_open vimage_Compl [symmetric]
huffman@31568: by (rule open_vimage_snd)
huffman@31568: 
huffman@31568: lemma closed_Times: "closed S \<Longrightarrow> closed T \<Longrightarrow> closed (S \<times> T)"
huffman@31568: proof -
huffman@31568:   have "S \<times> T = (fst -` S) \<inter> (snd -` T)" by auto
huffman@31568:   thus "closed S \<Longrightarrow> closed T \<Longrightarrow> closed (S \<times> T)"
huffman@31568:     by (simp add: closed_vimage_fst closed_vimage_snd closed_Int)
huffman@31568: qed
huffman@31568: 
huffman@34110: lemma openI: (* TODO: move *)
huffman@34110:   assumes "\<And>x. x \<in> S \<Longrightarrow> \<exists>T. open T \<and> x \<in> T \<and> T \<subseteq> S"
huffman@34110:   shows "open S"
huffman@34110: proof -
huffman@34110:   have "open (\<Union>{T. open T \<and> T \<subseteq> S})" by auto
huffman@34110:   moreover have "\<Union>{T. open T \<and> T \<subseteq> S} = S" by (auto dest!: assms)
huffman@34110:   ultimately show "open S" by simp
huffman@34110: qed
huffman@34110: 
huffman@34110: lemma subset_fst_imageI: "A \<times> B \<subseteq> S \<Longrightarrow> y \<in> B \<Longrightarrow> A \<subseteq> fst ` S"
huffman@34110:   unfolding image_def subset_eq by force
huffman@34110: 
huffman@34110: lemma subset_snd_imageI: "A \<times> B \<subseteq> S \<Longrightarrow> x \<in> A \<Longrightarrow> B \<subseteq> snd ` S"
huffman@34110:   unfolding image_def subset_eq by force
huffman@34110: 
huffman@34110: lemma open_image_fst: assumes "open S" shows "open (fst ` S)"
huffman@34110: proof (rule openI)
huffman@34110:   fix x assume "x \<in> fst ` S"
huffman@34110:   then obtain y where "(x, y) \<in> S" by auto
huffman@34110:   then obtain A B where "open A" "open B" "x \<in> A" "y \<in> B" "A \<times> B \<subseteq> S"
huffman@34110:     using `open S` unfolding open_prod_def by auto
huffman@34110:   from `A \<times> B \<subseteq> S` `y \<in> B` have "A \<subseteq> fst ` S" by (rule subset_fst_imageI)
huffman@34110:   with `open A` `x \<in> A` have "open A \<and> x \<in> A \<and> A \<subseteq> fst ` S" by simp
huffman@34110:   then show "\<exists>T. open T \<and> x \<in> T \<and> T \<subseteq> fst ` S" by - (rule exI)
huffman@34110: qed
huffman@34110: 
huffman@34110: lemma open_image_snd: assumes "open S" shows "open (snd ` S)"
huffman@34110: proof (rule openI)
huffman@34110:   fix y assume "y \<in> snd ` S"
huffman@34110:   then obtain x where "(x, y) \<in> S" by auto
huffman@34110:   then obtain A B where "open A" "open B" "x \<in> A" "y \<in> B" "A \<times> B \<subseteq> S"
huffman@34110:     using `open S` unfolding open_prod_def by auto
huffman@34110:   from `A \<times> B \<subseteq> S` `x \<in> A` have "B \<subseteq> snd ` S" by (rule subset_snd_imageI)
huffman@34110:   with `open B` `y \<in> B` have "open B \<and> y \<in> B \<and> B \<subseteq> snd ` S" by simp
huffman@34110:   then show "\<exists>T. open T \<and> y \<in> T \<and> T \<subseteq> snd ` S" by - (rule exI)
huffman@34110: qed
huffman@31568: 
huffman@44575: subsubsection {* Continuity of operations *}
huffman@44575: 
huffman@44575: lemma tendsto_fst [tendsto_intros]:
huffman@44575:   assumes "(f ---> a) F"
huffman@44575:   shows "((\<lambda>x. fst (f x)) ---> fst a) F"
huffman@44575: proof (rule topological_tendstoI)
huffman@44575:   fix S assume "open S" and "fst a \<in> S"
huffman@44575:   then have "open (fst -` S)" and "a \<in> fst -` S"
huffman@44575:     by (simp_all add: open_vimage_fst)
huffman@44575:   with assms have "eventually (\<lambda>x. f x \<in> fst -` S) F"
huffman@44575:     by (rule topological_tendstoD)
huffman@44575:   then show "eventually (\<lambda>x. fst (f x) \<in> S) F"
huffman@44575:     by simp
huffman@44575: qed
huffman@44575: 
huffman@44575: lemma tendsto_snd [tendsto_intros]:
huffman@44575:   assumes "(f ---> a) F"
huffman@44575:   shows "((\<lambda>x. snd (f x)) ---> snd a) F"
huffman@44575: proof (rule topological_tendstoI)
huffman@44575:   fix S assume "open S" and "snd a \<in> S"
huffman@44575:   then have "open (snd -` S)" and "a \<in> snd -` S"
huffman@44575:     by (simp_all add: open_vimage_snd)
huffman@44575:   with assms have "eventually (\<lambda>x. f x \<in> snd -` S) F"
huffman@44575:     by (rule topological_tendstoD)
huffman@44575:   then show "eventually (\<lambda>x. snd (f x) \<in> S) F"
huffman@44575:     by simp
huffman@44575: qed
huffman@44575: 
huffman@44575: lemma tendsto_Pair [tendsto_intros]:
huffman@44575:   assumes "(f ---> a) F" and "(g ---> b) F"
huffman@44575:   shows "((\<lambda>x. (f x, g x)) ---> (a, b)) F"
huffman@44575: proof (rule topological_tendstoI)
huffman@44575:   fix S assume "open S" and "(a, b) \<in> S"
huffman@44575:   then obtain A B where "open A" "open B" "a \<in> A" "b \<in> B" "A \<times> B \<subseteq> S"
huffman@44575:     unfolding open_prod_def by fast
huffman@44575:   have "eventually (\<lambda>x. f x \<in> A) F"
huffman@44575:     using `(f ---> a) F` `open A` `a \<in> A`
huffman@44575:     by (rule topological_tendstoD)
huffman@44575:   moreover
huffman@44575:   have "eventually (\<lambda>x. g x \<in> B) F"
huffman@44575:     using `(g ---> b) F` `open B` `b \<in> B`
huffman@44575:     by (rule topological_tendstoD)
huffman@44575:   ultimately
huffman@44575:   show "eventually (\<lambda>x. (f x, g x) \<in> S) F"
huffman@44575:     by (rule eventually_elim2)
huffman@44575:        (simp add: subsetD [OF `A \<times> B \<subseteq> S`])
huffman@44575: qed
huffman@44575: 
huffman@44575: lemma isCont_fst [simp]: "isCont f a \<Longrightarrow> isCont (\<lambda>x. fst (f x)) a"
huffman@44575:   unfolding isCont_def by (rule tendsto_fst)
huffman@44575: 
huffman@44575: lemma isCont_snd [simp]: "isCont f a \<Longrightarrow> isCont (\<lambda>x. snd (f x)) a"
huffman@44575:   unfolding isCont_def by (rule tendsto_snd)
huffman@44575: 
huffman@44575: lemma isCont_Pair [simp]:
huffman@44575:   "\<lbrakk>isCont f a; isCont g a\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. (f x, g x)) a"
huffman@44575:   unfolding isCont_def by (rule tendsto_Pair)
huffman@44575: 
huffman@44575: subsubsection {* Separation axioms *}
huffman@44214: 
huffman@44214: lemma mem_Times_iff: "x \<in> A \<times> B \<longleftrightarrow> fst x \<in> A \<and> snd x \<in> B"
huffman@44214:   by (induct x) simp (* TODO: move elsewhere *)
huffman@44214: 
huffman@44214: instance prod :: (t0_space, t0_space) t0_space
huffman@44214: proof
huffman@44214:   fix x y :: "'a \<times> 'b" assume "x \<noteq> y"
huffman@44214:   hence "fst x \<noteq> fst y \<or> snd x \<noteq> snd y"
huffman@44214:     by (simp add: prod_eq_iff)
huffman@44214:   thus "\<exists>U. open U \<and> (x \<in> U) \<noteq> (y \<in> U)"
huffman@44214:     apply (rule disjE)
huffman@44214:     apply (drule t0_space, erule exE, rule_tac x="U \<times> UNIV" in exI)
huffman@44214:     apply (simp add: open_Times mem_Times_iff)
huffman@44214:     apply (drule t0_space, erule exE, rule_tac x="UNIV \<times> U" in exI)
huffman@44214:     apply (simp add: open_Times mem_Times_iff)
huffman@44214:     done
huffman@44214: qed
huffman@44214: 
huffman@44214: instance prod :: (t1_space, t1_space) t1_space
huffman@44214: proof
huffman@44214:   fix x y :: "'a \<times> 'b" assume "x \<noteq> y"
huffman@44214:   hence "fst x \<noteq> fst y \<or> snd x \<noteq> snd y"
huffman@44214:     by (simp add: prod_eq_iff)
huffman@44214:   thus "\<exists>U. open U \<and> x \<in> U \<and> y \<notin> U"
huffman@44214:     apply (rule disjE)
huffman@44214:     apply (drule t1_space, erule exE, rule_tac x="U \<times> UNIV" in exI)
huffman@44214:     apply (simp add: open_Times mem_Times_iff)
huffman@44214:     apply (drule t1_space, erule exE, rule_tac x="UNIV \<times> U" in exI)
huffman@44214:     apply (simp add: open_Times mem_Times_iff)
huffman@44214:     done
huffman@44214: qed
huffman@44214: 
huffman@44214: instance prod :: (t2_space, t2_space) t2_space
huffman@44214: proof
huffman@44214:   fix x y :: "'a \<times> 'b" assume "x \<noteq> y"
huffman@44214:   hence "fst x \<noteq> fst y \<or> snd x \<noteq> snd y"
huffman@44214:     by (simp add: prod_eq_iff)
huffman@44214:   thus "\<exists>U V. open U \<and> open V \<and> x \<in> U \<and> y \<in> V \<and> U \<inter> V = {}"
huffman@44214:     apply (rule disjE)
huffman@44214:     apply (drule hausdorff, clarify)
huffman@44214:     apply (rule_tac x="U \<times> UNIV" in exI, rule_tac x="V \<times> UNIV" in exI)
huffman@44214:     apply (simp add: open_Times mem_Times_iff disjoint_iff_not_equal)
huffman@44214:     apply (drule hausdorff, clarify)
huffman@44214:     apply (rule_tac x="UNIV \<times> U" in exI, rule_tac x="UNIV \<times> V" in exI)
huffman@44214:     apply (simp add: open_Times mem_Times_iff disjoint_iff_not_equal)
huffman@44214:     done
huffman@44214: qed
huffman@44214: 
huffman@31339: subsection {* Product is a metric space *}
huffman@31339: 
haftmann@37678: instantiation prod :: (metric_space, metric_space) metric_space
huffman@31339: begin
huffman@31339: 
huffman@31339: definition dist_prod_def:
huffman@44214:   "dist x y = sqrt ((dist (fst x) (fst y))\<twosuperior> + (dist (snd x) (snd y))\<twosuperior>)"
huffman@31339: 
huffman@31339: lemma dist_Pair_Pair: "dist (a, b) (c, d) = sqrt ((dist a c)\<twosuperior> + (dist b d)\<twosuperior>)"
huffman@31339:   unfolding dist_prod_def by simp
huffman@31339: 
huffman@36332: lemma dist_fst_le: "dist (fst x) (fst y) \<le> dist x y"
huffman@36332: unfolding dist_prod_def by (rule real_sqrt_sum_squares_ge1)
huffman@36332: 
huffman@36332: lemma dist_snd_le: "dist (snd x) (snd y) \<le> dist x y"
huffman@36332: unfolding dist_prod_def by (rule real_sqrt_sum_squares_ge2)
huffman@36332: 
huffman@31339: instance proof
huffman@31339:   fix x y :: "'a \<times> 'b"
huffman@31339:   show "dist x y = 0 \<longleftrightarrow> x = y"
huffman@44066:     unfolding dist_prod_def prod_eq_iff by simp
huffman@31339: next
huffman@31339:   fix x y z :: "'a \<times> 'b"
huffman@31339:   show "dist x y \<le> dist x z + dist y z"
huffman@31339:     unfolding dist_prod_def
huffman@31563:     by (intro order_trans [OF _ real_sqrt_sum_squares_triangle_ineq]
huffman@31563:         real_sqrt_le_mono add_mono power_mono dist_triangle2 zero_le_dist)
huffman@31415: next
huffman@31492:   fix S :: "('a \<times> 'b) set"
huffman@31492:   show "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S)"
huffman@31563:   proof
huffman@36332:     assume "open S" show "\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S"
huffman@36332:     proof
huffman@36332:       fix x assume "x \<in> S"
huffman@36332:       obtain A B where "open A" "open B" "x \<in> A \<times> B" "A \<times> B \<subseteq> S"
huffman@36332:         using `open S` and `x \<in> S` by (rule open_prod_elim)
huffman@36332:       obtain r where r: "0 < r" "\<forall>y. dist y (fst x) < r \<longrightarrow> y \<in> A"
huffman@36332:         using `open A` and `x \<in> A \<times> B` unfolding open_dist by auto
huffman@36332:       obtain s where s: "0 < s" "\<forall>y. dist y (snd x) < s \<longrightarrow> y \<in> B"
huffman@36332:         using `open B` and `x \<in> A \<times> B` unfolding open_dist by auto
huffman@36332:       let ?e = "min r s"
huffman@36332:       have "0 < ?e \<and> (\<forall>y. dist y x < ?e \<longrightarrow> y \<in> S)"
huffman@36332:       proof (intro allI impI conjI)
huffman@36332:         show "0 < min r s" by (simp add: r(1) s(1))
huffman@36332:       next
huffman@36332:         fix y assume "dist y x < min r s"
huffman@36332:         hence "dist y x < r" and "dist y x < s"
huffman@36332:           by simp_all
huffman@36332:         hence "dist (fst y) (fst x) < r" and "dist (snd y) (snd x) < s"
huffman@36332:           by (auto intro: le_less_trans dist_fst_le dist_snd_le)
huffman@36332:         hence "fst y \<in> A" and "snd y \<in> B"
huffman@36332:           by (simp_all add: r(2) s(2))
huffman@36332:         hence "y \<in> A \<times> B" by (induct y, simp)
huffman@36332:         with `A \<times> B \<subseteq> S` show "y \<in> S" ..
huffman@36332:       qed
huffman@36332:       thus "\<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S" ..
huffman@36332:     qed
huffman@31563:   next
huffman@44575:     assume *: "\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S" show "open S"
huffman@44575:     proof (rule open_prod_intro)
huffman@44575:       fix x assume "x \<in> S"
huffman@44575:       then obtain e where "0 < e" and S: "\<forall>y. dist y x < e \<longrightarrow> y \<in> S"
huffman@44575:         using * by fast
huffman@44575:       def r \<equiv> "e / sqrt 2" and s \<equiv> "e / sqrt 2"
huffman@44575:       from `0 < e` have "0 < r" and "0 < s"
huffman@44575:         unfolding r_def s_def by (simp_all add: divide_pos_pos)
huffman@44575:       from `0 < e` have "e = sqrt (r\<twosuperior> + s\<twosuperior>)"
huffman@44575:         unfolding r_def s_def by (simp add: power_divide)
huffman@44575:       def A \<equiv> "{y. dist (fst x) y < r}" and B \<equiv> "{y. dist (snd x) y < s}"
huffman@44575:       have "open A" and "open B"
huffman@44575:         unfolding A_def B_def by (simp_all add: open_ball)
huffman@44575:       moreover have "x \<in> A \<times> B"
huffman@44575:         unfolding A_def B_def mem_Times_iff
huffman@44575:         using `0 < r` and `0 < s` by simp
huffman@44575:       moreover have "A \<times> B \<subseteq> S"
huffman@44575:       proof (clarify)
huffman@44575:         fix a b assume "a \<in> A" and "b \<in> B"
huffman@44575:         hence "dist a (fst x) < r" and "dist b (snd x) < s"
huffman@44575:           unfolding A_def B_def by (simp_all add: dist_commute)
huffman@44575:         hence "dist (a, b) x < e"
huffman@44575:           unfolding dist_prod_def `e = sqrt (r\<twosuperior> + s\<twosuperior>)`
huffman@44575:           by (simp add: add_strict_mono power_strict_mono)
huffman@44575:         thus "(a, b) \<in> S"
huffman@44575:           by (simp add: S)
huffman@44575:       qed
huffman@44575:       ultimately show "\<exists>A B. open A \<and> open B \<and> x \<in> A \<times> B \<and> A \<times> B \<subseteq> S" by fast
huffman@44575:     qed
huffman@31563:   qed
huffman@31339: qed
huffman@31339: 
huffman@31339: end
huffman@31339: 
huffman@31405: lemma Cauchy_fst: "Cauchy X \<Longrightarrow> Cauchy (\<lambda>n. fst (X n))"
huffman@31405: unfolding Cauchy_def by (fast elim: le_less_trans [OF dist_fst_le])
huffman@31405: 
huffman@31405: lemma Cauchy_snd: "Cauchy X \<Longrightarrow> Cauchy (\<lambda>n. snd (X n))"
huffman@31405: unfolding Cauchy_def by (fast elim: le_less_trans [OF dist_snd_le])
huffman@31405: 
huffman@31405: lemma Cauchy_Pair:
huffman@31405:   assumes "Cauchy X" and "Cauchy Y"
huffman@31405:   shows "Cauchy (\<lambda>n. (X n, Y n))"
huffman@31405: proof (rule metric_CauchyI)
huffman@31405:   fix r :: real assume "0 < r"
huffman@31405:   then have "0 < r / sqrt 2" (is "0 < ?s")
huffman@31405:     by (simp add: divide_pos_pos)
huffman@31405:   obtain M where M: "\<forall>m\<ge>M. \<forall>n\<ge>M. dist (X m) (X n) < ?s"
huffman@31405:     using metric_CauchyD [OF `Cauchy X` `0 < ?s`] ..
huffman@31405:   obtain N where N: "\<forall>m\<ge>N. \<forall>n\<ge>N. dist (Y m) (Y n) < ?s"
huffman@31405:     using metric_CauchyD [OF `Cauchy Y` `0 < ?s`] ..
huffman@31405:   have "\<forall>m\<ge>max M N. \<forall>n\<ge>max M N. dist (X m, Y m) (X n, Y n) < r"
huffman@31405:     using M N by (simp add: real_sqrt_sum_squares_less dist_Pair_Pair)
huffman@31405:   then show "\<exists>n0. \<forall>m\<ge>n0. \<forall>n\<ge>n0. dist (X m, Y m) (X n, Y n) < r" ..
huffman@31405: qed
huffman@31405: 
huffman@31405: subsection {* Product is a complete metric space *}
huffman@31405: 
haftmann@37678: instance prod :: (complete_space, complete_space) complete_space
huffman@31405: proof
huffman@31405:   fix X :: "nat \<Rightarrow> 'a \<times> 'b" assume "Cauchy X"
huffman@31405:   have 1: "(\<lambda>n. fst (X n)) ----> lim (\<lambda>n. fst (X n))"
huffman@31405:     using Cauchy_fst [OF `Cauchy X`]
huffman@31405:     by (simp add: Cauchy_convergent_iff convergent_LIMSEQ_iff)
huffman@31405:   have 2: "(\<lambda>n. snd (X n)) ----> lim (\<lambda>n. snd (X n))"
huffman@31405:     using Cauchy_snd [OF `Cauchy X`]
huffman@31405:     by (simp add: Cauchy_convergent_iff convergent_LIMSEQ_iff)
huffman@31405:   have "X ----> (lim (\<lambda>n. fst (X n)), lim (\<lambda>n. snd (X n)))"
huffman@36660:     using tendsto_Pair [OF 1 2] by simp
huffman@31405:   then show "convergent X"
huffman@31405:     by (rule convergentI)
huffman@31405: qed
huffman@31405: 
huffman@30019: subsection {* Product is a normed vector space *}
huffman@30019: 
haftmann@37678: instantiation prod :: (real_normed_vector, real_normed_vector) real_normed_vector
huffman@30019: begin
huffman@30019: 
huffman@30019: definition norm_prod_def:
huffman@30019:   "norm x = sqrt ((norm (fst x))\<twosuperior> + (norm (snd x))\<twosuperior>)"
huffman@30019: 
huffman@30019: definition sgn_prod_def:
huffman@30019:   "sgn (x::'a \<times> 'b) = scaleR (inverse (norm x)) x"
huffman@30019: 
huffman@30019: lemma norm_Pair: "norm (a, b) = sqrt ((norm a)\<twosuperior> + (norm b)\<twosuperior>)"
huffman@30019:   unfolding norm_prod_def by simp
huffman@30019: 
huffman@30019: instance proof
huffman@30019:   fix r :: real and x y :: "'a \<times> 'b"
huffman@30019:   show "norm x = 0 \<longleftrightarrow> x = 0"
huffman@30019:     unfolding norm_prod_def
huffman@44066:     by (simp add: prod_eq_iff)
huffman@30019:   show "norm (x + y) \<le> norm x + norm y"
huffman@30019:     unfolding norm_prod_def
huffman@30019:     apply (rule order_trans [OF _ real_sqrt_sum_squares_triangle_ineq])
huffman@30019:     apply (simp add: add_mono power_mono norm_triangle_ineq)
huffman@30019:     done
huffman@30019:   show "norm (scaleR r x) = \<bar>r\<bar> * norm x"
huffman@30019:     unfolding norm_prod_def
huffman@31587:     apply (simp add: power_mult_distrib)
webertj@49962:     apply (simp add: distrib_left [symmetric])
huffman@30019:     apply (simp add: real_sqrt_mult_distrib)
huffman@30019:     done
huffman@30019:   show "sgn x = scaleR (inverse (norm x)) x"
huffman@30019:     by (rule sgn_prod_def)
huffman@31290:   show "dist x y = norm (x - y)"
huffman@31339:     unfolding dist_prod_def norm_prod_def
huffman@31339:     by (simp add: dist_norm)
huffman@30019: qed
huffman@30019: 
huffman@30019: end
huffman@30019: 
haftmann@37678: instance prod :: (banach, banach) banach ..
huffman@31405: 
huffman@44575: subsubsection {* Pair operations are linear *}
huffman@30019: 
huffman@44282: lemma bounded_linear_fst: "bounded_linear fst"
huffman@44127:   using fst_add fst_scaleR
huffman@44127:   by (rule bounded_linear_intro [where K=1], simp add: norm_prod_def)
huffman@30019: 
huffman@44282: lemma bounded_linear_snd: "bounded_linear snd"
huffman@44127:   using snd_add snd_scaleR
huffman@44127:   by (rule bounded_linear_intro [where K=1], simp add: norm_prod_def)
huffman@30019: 
huffman@30019: text {* TODO: move to NthRoot *}
huffman@30019: lemma sqrt_add_le_add_sqrt:
huffman@30019:   assumes x: "0 \<le> x" and y: "0 \<le> y"
huffman@30019:   shows "sqrt (x + y) \<le> sqrt x + sqrt y"
huffman@30019: apply (rule power2_le_imp_le)
huffman@44749: apply (simp add: power2_sum x y)
huffman@30019: apply (simp add: mult_nonneg_nonneg x y)
huffman@44126: apply (simp add: x y)
huffman@30019: done
huffman@30019: 
huffman@30019: lemma bounded_linear_Pair:
huffman@30019:   assumes f: "bounded_linear f"
huffman@30019:   assumes g: "bounded_linear g"
huffman@30019:   shows "bounded_linear (\<lambda>x. (f x, g x))"
huffman@30019: proof
huffman@30019:   interpret f: bounded_linear f by fact
huffman@30019:   interpret g: bounded_linear g by fact
huffman@30019:   fix x y and r :: real
huffman@30019:   show "(f (x + y), g (x + y)) = (f x, g x) + (f y, g y)"
huffman@30019:     by (simp add: f.add g.add)
huffman@30019:   show "(f (r *\<^sub>R x), g (r *\<^sub>R x)) = r *\<^sub>R (f x, g x)"
huffman@30019:     by (simp add: f.scaleR g.scaleR)
huffman@30019:   obtain Kf where "0 < Kf" and norm_f: "\<And>x. norm (f x) \<le> norm x * Kf"
huffman@30019:     using f.pos_bounded by fast
huffman@30019:   obtain Kg where "0 < Kg" and norm_g: "\<And>x. norm (g x) \<le> norm x * Kg"
huffman@30019:     using g.pos_bounded by fast
huffman@30019:   have "\<forall>x. norm (f x, g x) \<le> norm x * (Kf + Kg)"
huffman@30019:     apply (rule allI)
huffman@30019:     apply (simp add: norm_Pair)
huffman@30019:     apply (rule order_trans [OF sqrt_add_le_add_sqrt], simp, simp)
webertj@49962:     apply (simp add: distrib_left)
huffman@30019:     apply (rule add_mono [OF norm_f norm_g])
huffman@30019:     done
huffman@30019:   then show "\<exists>K. \<forall>x. norm (f x, g x) \<le> norm x * K" ..
huffman@30019: qed
huffman@30019: 
huffman@44575: subsubsection {* Frechet derivatives involving pairs *}
huffman@30019: 
huffman@30019: lemma FDERIV_Pair:
huffman@30019:   assumes f: "FDERIV f x :> f'" and g: "FDERIV g x :> g'"
huffman@30019:   shows "FDERIV (\<lambda>x. (f x, g x)) x :> (\<lambda>h. (f' h, g' h))"
huffman@44575: proof (rule FDERIV_I)
huffman@44575:   show "bounded_linear (\<lambda>h. (f' h, g' h))"
huffman@44575:     using f g by (intro bounded_linear_Pair FDERIV_bounded_linear)
huffman@44575:   let ?Rf = "\<lambda>h. f (x + h) - f x - f' h"
huffman@44575:   let ?Rg = "\<lambda>h. g (x + h) - g x - g' h"
huffman@44575:   let ?R = "\<lambda>h. ((f (x + h), g (x + h)) - (f x, g x) - (f' h, g' h))"
huffman@44575:   show "(\<lambda>h. norm (?R h) / norm h) -- 0 --> 0"
huffman@44575:   proof (rule real_LIM_sandwich_zero)
huffman@44575:     show "(\<lambda>h. norm (?Rf h) / norm h + norm (?Rg h) / norm h) -- 0 --> 0"
huffman@44575:       using f g by (intro tendsto_add_zero FDERIV_D)
huffman@44575:     fix h :: 'a assume "h \<noteq> 0"
huffman@44575:     thus "0 \<le> norm (?R h) / norm h"
huffman@44575:       by (simp add: divide_nonneg_pos)
huffman@44575:     show "norm (?R h) / norm h \<le> norm (?Rf h) / norm h + norm (?Rg h) / norm h"
huffman@44575:       unfolding add_divide_distrib [symmetric]
huffman@44575:       by (simp add: norm_Pair divide_right_mono
huffman@44575:         order_trans [OF sqrt_add_le_add_sqrt])
huffman@44575:   qed
huffman@44575: qed
huffman@44575: 
huffman@44575: subsection {* Product is an inner product space *}
huffman@44575: 
huffman@44575: instantiation prod :: (real_inner, real_inner) real_inner
huffman@44575: begin
huffman@44575: 
huffman@44575: definition inner_prod_def:
huffman@44575:   "inner x y = inner (fst x) (fst y) + inner (snd x) (snd y)"
huffman@44575: 
huffman@44575: lemma inner_Pair [simp]: "inner (a, b) (c, d) = inner a c + inner b d"
huffman@44575:   unfolding inner_prod_def by simp
huffman@44575: 
huffman@44575: instance proof
huffman@44575:   fix r :: real
huffman@44575:   fix x y z :: "'a::real_inner \<times> 'b::real_inner"
huffman@44575:   show "inner x y = inner y x"
huffman@44575:     unfolding inner_prod_def
huffman@44575:     by (simp add: inner_commute)
huffman@44575:   show "inner (x + y) z = inner x z + inner y z"
huffman@44575:     unfolding inner_prod_def
huffman@44575:     by (simp add: inner_add_left)
huffman@44575:   show "inner (scaleR r x) y = r * inner x y"
huffman@44575:     unfolding inner_prod_def
webertj@49962:     by (simp add: distrib_left)
huffman@44575:   show "0 \<le> inner x x"
huffman@44575:     unfolding inner_prod_def
huffman@44575:     by (intro add_nonneg_nonneg inner_ge_zero)
huffman@44575:   show "inner x x = 0 \<longleftrightarrow> x = 0"
huffman@44575:     unfolding inner_prod_def prod_eq_iff
huffman@44575:     by (simp add: add_nonneg_eq_0_iff)
huffman@44575:   show "norm x = sqrt (inner x x)"
huffman@44575:     unfolding norm_prod_def inner_prod_def
huffman@44575:     by (simp add: power2_norm_eq_inner)
huffman@44575: qed
huffman@30019: 
huffman@30019: end
huffman@44575: 
huffman@44575: end