(* Title: HOL/Library/Product_Vector.thy Author: Brian Huffman*)header {* Cartesian Products as Vector Spaces *}theory Product_Vectorimports Inner_Product Product_plusbeginsubsection {* Product is a real vector space *}instantiation prod :: (real_vector, real_vector) real_vectorbegindefinition scaleR_prod_def: "scaleR r A = (scaleR r (fst A), scaleR r (snd A))"lemma fst_scaleR [simp]: "fst (scaleR r A) = scaleR r (fst A)" unfolding scaleR_prod_def by simplemma snd_scaleR [simp]: "snd (scaleR r A) = scaleR r (snd A)" unfolding scaleR_prod_def by simplemma scaleR_Pair [simp]: "scaleR r (a, b) = (scaleR r a, scaleR r b)" unfolding scaleR_prod_def by simpinstance proof fix a b :: real and x y :: "'a \<times> 'b" show "scaleR a (x + y) = scaleR a x + scaleR a y" by (simp add: prod_eq_iff scaleR_right_distrib) show "scaleR (a + b) x = scaleR a x + scaleR b x" by (simp add: prod_eq_iff scaleR_left_distrib) show "scaleR a (scaleR b x) = scaleR (a * b) x" by (simp add: prod_eq_iff) show "scaleR 1 x = x" by (simp add: prod_eq_iff)qedendsubsection {* Product is a topological space *}instantiation prod :: (topological_space, topological_space) topological_spacebegindefinition open_prod_def: "open (S :: ('a \<times> 'b) set) \<longleftrightarrow> (\<forall>x\<in>S. \<exists>A B. open A \<and> open B \<and> x \<in> A \<times> B \<and> A \<times> B \<subseteq> S)"lemma open_prod_elim: assumes "open S" and "x \<in> S" obtains A B where "open A" and "open B" and "x \<in> A \<times> B" and "A \<times> B \<subseteq> S"using assms unfolding open_prod_def by fastlemma open_prod_intro: 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" shows "open S"using assms unfolding open_prod_def by fastinstance proof show "open (UNIV :: ('a \<times> 'b) set)" unfolding open_prod_def by autonext fix S T :: "('a \<times> 'b) set" assume "open S" "open T" show "open (S \<inter> T)" proof (rule open_prod_intro) fix x assume x: "x \<in> S \<inter> T" from x have "x \<in> S" by simp obtain Sa Sb where A: "open Sa" "open Sb" "x \<in> Sa \<times> Sb" "Sa \<times> Sb \<subseteq> S" using `open S` and `x \<in> S` by (rule open_prod_elim) from x have "x \<in> T" by simp obtain Ta Tb where B: "open Ta" "open Tb" "x \<in> Ta \<times> Tb" "Ta \<times> Tb \<subseteq> T" using `open T` and `x \<in> T` by (rule open_prod_elim) let ?A = "Sa \<inter> Ta" and ?B = "Sb \<inter> Tb" have "open ?A \<and> open ?B \<and> x \<in> ?A \<times> ?B \<and> ?A \<times> ?B \<subseteq> S \<inter> T" using A B by (auto simp add: open_Int) thus "\<exists>A B. open A \<and> open B \<and> x \<in> A \<times> B \<and> A \<times> B \<subseteq> S \<inter> T" by fast qednext fix K :: "('a \<times> 'b) set set" assume "\<forall>S\<in>K. open S" thus "open (\<Union>K)" unfolding open_prod_def by fastqedendlemma open_Times: "open S \<Longrightarrow> open T \<Longrightarrow> open (S \<times> T)"unfolding open_prod_def by autolemma fst_vimage_eq_Times: "fst -` S = S \<times> UNIV"by autolemma snd_vimage_eq_Times: "snd -` S = UNIV \<times> S"by autolemma open_vimage_fst: "open S \<Longrightarrow> open (fst -` S)"by (simp add: fst_vimage_eq_Times open_Times)lemma open_vimage_snd: "open S \<Longrightarrow> open (snd -` S)"by (simp add: snd_vimage_eq_Times open_Times)lemma closed_vimage_fst: "closed S \<Longrightarrow> closed (fst -` S)"unfolding closed_open vimage_Compl [symmetric]by (rule open_vimage_fst)lemma closed_vimage_snd: "closed S \<Longrightarrow> closed (snd -` S)"unfolding closed_open vimage_Compl [symmetric]by (rule open_vimage_snd)lemma closed_Times: "closed S \<Longrightarrow> closed T \<Longrightarrow> closed (S \<times> T)"proof - have "S \<times> T = (fst -` S) \<inter> (snd -` T)" by auto thus "closed S \<Longrightarrow> closed T \<Longrightarrow> closed (S \<times> T)" by (simp add: closed_vimage_fst closed_vimage_snd closed_Int)qedlemma openI: (* TODO: move *) assumes "\<And>x. x \<in> S \<Longrightarrow> \<exists>T. open T \<and> x \<in> T \<and> T \<subseteq> S" shows "open S"proof - have "open (\<Union>{T. open T \<and> T \<subseteq> S})" by auto moreover have "\<Union>{T. open T \<and> T \<subseteq> S} = S" by (auto dest!: assms) ultimately show "open S" by simpqedlemma subset_fst_imageI: "A \<times> B \<subseteq> S \<Longrightarrow> y \<in> B \<Longrightarrow> A \<subseteq> fst ` S" unfolding image_def subset_eq by forcelemma subset_snd_imageI: "A \<times> B \<subseteq> S \<Longrightarrow> x \<in> A \<Longrightarrow> B \<subseteq> snd ` S" unfolding image_def subset_eq by forcelemma open_image_fst: assumes "open S" shows "open (fst ` S)"proof (rule openI) fix x assume "x \<in> fst ` S" then obtain y where "(x, y) \<in> S" by auto then obtain A B where "open A" "open B" "x \<in> A" "y \<in> B" "A \<times> B \<subseteq> S" using `open S` unfolding open_prod_def by auto from `A \<times> B \<subseteq> S` `y \<in> B` have "A \<subseteq> fst ` S" by (rule subset_fst_imageI) with `open A` `x \<in> A` have "open A \<and> x \<in> A \<and> A \<subseteq> fst ` S" by simp then show "\<exists>T. open T \<and> x \<in> T \<and> T \<subseteq> fst ` S" by - (rule exI)qedlemma open_image_snd: assumes "open S" shows "open (snd ` S)"proof (rule openI) fix y assume "y \<in> snd ` S" then obtain x where "(x, y) \<in> S" by auto then obtain A B where "open A" "open B" "x \<in> A" "y \<in> B" "A \<times> B \<subseteq> S" using `open S` unfolding open_prod_def by auto from `A \<times> B \<subseteq> S` `x \<in> A` have "B \<subseteq> snd ` S" by (rule subset_snd_imageI) with `open B` `y \<in> B` have "open B \<and> y \<in> B \<and> B \<subseteq> snd ` S" by simp then show "\<exists>T. open T \<and> y \<in> T \<and> T \<subseteq> snd ` S" by - (rule exI)qedsubsubsection {* Continuity of operations *}lemma tendsto_fst [tendsto_intros]: assumes "(f ---> a) F" shows "((\<lambda>x. fst (f x)) ---> fst a) F"proof (rule topological_tendstoI) fix S assume "open S" and "fst a \<in> S" then have "open (fst -` S)" and "a \<in> fst -` S" by (simp_all add: open_vimage_fst) with assms have "eventually (\<lambda>x. f x \<in> fst -` S) F" by (rule topological_tendstoD) then show "eventually (\<lambda>x. fst (f x) \<in> S) F" by simpqedlemma tendsto_snd [tendsto_intros]: assumes "(f ---> a) F" shows "((\<lambda>x. snd (f x)) ---> snd a) F"proof (rule topological_tendstoI) fix S assume "open S" and "snd a \<in> S" then have "open (snd -` S)" and "a \<in> snd -` S" by (simp_all add: open_vimage_snd) with assms have "eventually (\<lambda>x. f x \<in> snd -` S) F" by (rule topological_tendstoD) then show "eventually (\<lambda>x. snd (f x) \<in> S) F" by simpqedlemma tendsto_Pair [tendsto_intros]: assumes "(f ---> a) F" and "(g ---> b) F" shows "((\<lambda>x. (f x, g x)) ---> (a, b)) F"proof (rule topological_tendstoI) fix S assume "open S" and "(a, b) \<in> S" then obtain A B where "open A" "open B" "a \<in> A" "b \<in> B" "A \<times> B \<subseteq> S" unfolding open_prod_def by fast have "eventually (\<lambda>x. f x \<in> A) F" using `(f ---> a) F` `open A` `a \<in> A` by (rule topological_tendstoD) moreover have "eventually (\<lambda>x. g x \<in> B) F" using `(g ---> b) F` `open B` `b \<in> B` by (rule topological_tendstoD) ultimately show "eventually (\<lambda>x. (f x, g x) \<in> S) F" by (rule eventually_elim2) (simp add: subsetD [OF `A \<times> B \<subseteq> S`])qedlemma isCont_fst [simp]: "isCont f a \<Longrightarrow> isCont (\<lambda>x. fst (f x)) a" unfolding isCont_def by (rule tendsto_fst)lemma isCont_snd [simp]: "isCont f a \<Longrightarrow> isCont (\<lambda>x. snd (f x)) a" unfolding isCont_def by (rule tendsto_snd)lemma isCont_Pair [simp]: "\<lbrakk>isCont f a; isCont g a\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. (f x, g x)) a" unfolding isCont_def by (rule tendsto_Pair)subsubsection {* Separation axioms *}lemma mem_Times_iff: "x \<in> A \<times> B \<longleftrightarrow> fst x \<in> A \<and> snd x \<in> B" by (induct x) simp (* TODO: move elsewhere *)instance prod :: (t0_space, t0_space) t0_spaceproof fix x y :: "'a \<times> 'b" assume "x \<noteq> y" hence "fst x \<noteq> fst y \<or> snd x \<noteq> snd y" by (simp add: prod_eq_iff) thus "\<exists>U. open U \<and> (x \<in> U) \<noteq> (y \<in> U)" apply (rule disjE) apply (drule t0_space, erule exE, rule_tac x="U \<times> UNIV" in exI) apply (simp add: open_Times mem_Times_iff) apply (drule t0_space, erule exE, rule_tac x="UNIV \<times> U" in exI) apply (simp add: open_Times mem_Times_iff) doneqedinstance prod :: (t1_space, t1_space) t1_spaceproof fix x y :: "'a \<times> 'b" assume "x \<noteq> y" hence "fst x \<noteq> fst y \<or> snd x \<noteq> snd y" by (simp add: prod_eq_iff) thus "\<exists>U. open U \<and> x \<in> U \<and> y \<notin> U" apply (rule disjE) apply (drule t1_space, erule exE, rule_tac x="U \<times> UNIV" in exI) apply (simp add: open_Times mem_Times_iff) apply (drule t1_space, erule exE, rule_tac x="UNIV \<times> U" in exI) apply (simp add: open_Times mem_Times_iff) doneqedinstance prod :: (t2_space, t2_space) t2_spaceproof fix x y :: "'a \<times> 'b" assume "x \<noteq> y" hence "fst x \<noteq> fst y \<or> snd x \<noteq> snd y" by (simp add: prod_eq_iff) thus "\<exists>U V. open U \<and> open V \<and> x \<in> U \<and> y \<in> V \<and> U \<inter> V = {}" apply (rule disjE) apply (drule hausdorff, clarify) apply (rule_tac x="U \<times> UNIV" in exI, rule_tac x="V \<times> UNIV" in exI) apply (simp add: open_Times mem_Times_iff disjoint_iff_not_equal) apply (drule hausdorff, clarify) apply (rule_tac x="UNIV \<times> U" in exI, rule_tac x="UNIV \<times> V" in exI) apply (simp add: open_Times mem_Times_iff disjoint_iff_not_equal) doneqedsubsection {* Product is a metric space *}instantiation prod :: (metric_space, metric_space) metric_spacebegindefinition dist_prod_def: "dist x y = sqrt ((dist (fst x) (fst y))\<twosuperior> + (dist (snd x) (snd y))\<twosuperior>)"lemma dist_Pair_Pair: "dist (a, b) (c, d) = sqrt ((dist a c)\<twosuperior> + (dist b d)\<twosuperior>)" unfolding dist_prod_def by simplemma dist_fst_le: "dist (fst x) (fst y) \<le> dist x y"unfolding dist_prod_def by (rule real_sqrt_sum_squares_ge1)lemma dist_snd_le: "dist (snd x) (snd y) \<le> dist x y"unfolding dist_prod_def by (rule real_sqrt_sum_squares_ge2)instance proof fix x y :: "'a \<times> 'b" show "dist x y = 0 \<longleftrightarrow> x = y" unfolding dist_prod_def prod_eq_iff by simpnext fix x y z :: "'a \<times> 'b" show "dist x y \<le> dist x z + dist y z" unfolding dist_prod_def by (intro order_trans [OF _ real_sqrt_sum_squares_triangle_ineq] real_sqrt_le_mono add_mono power_mono dist_triangle2 zero_le_dist)next fix S :: "('a \<times> 'b) set" show "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S)" proof assume "open S" show "\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S" proof fix x assume "x \<in> S" obtain A B where "open A" "open B" "x \<in> A \<times> B" "A \<times> B \<subseteq> S" using `open S` and `x \<in> S` by (rule open_prod_elim) obtain r where r: "0 < r" "\<forall>y. dist y (fst x) < r \<longrightarrow> y \<in> A" using `open A` and `x \<in> A \<times> B` unfolding open_dist by auto obtain s where s: "0 < s" "\<forall>y. dist y (snd x) < s \<longrightarrow> y \<in> B" using `open B` and `x \<in> A \<times> B` unfolding open_dist by auto let ?e = "min r s" have "0 < ?e \<and> (\<forall>y. dist y x < ?e \<longrightarrow> y \<in> S)" proof (intro allI impI conjI) show "0 < min r s" by (simp add: r(1) s(1)) next fix y assume "dist y x < min r s" hence "dist y x < r" and "dist y x < s" by simp_all hence "dist (fst y) (fst x) < r" and "dist (snd y) (snd x) < s" by (auto intro: le_less_trans dist_fst_le dist_snd_le) hence "fst y \<in> A" and "snd y \<in> B" by (simp_all add: r(2) s(2)) hence "y \<in> A \<times> B" by (induct y, simp) with `A \<times> B \<subseteq> S` show "y \<in> S" .. qed thus "\<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S" .. qed next assume *: "\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S" show "open S" proof (rule open_prod_intro) fix x assume "x \<in> S" then obtain e where "0 < e" and S: "\<forall>y. dist y x < e \<longrightarrow> y \<in> S" using * by fast def r \<equiv> "e / sqrt 2" and s \<equiv> "e / sqrt 2" from `0 < e` have "0 < r" and "0 < s" unfolding r_def s_def by (simp_all add: divide_pos_pos) from `0 < e` have "e = sqrt (r\<twosuperior> + s\<twosuperior>)" unfolding r_def s_def by (simp add: power_divide) def A \<equiv> "{y. dist (fst x) y < r}" and B \<equiv> "{y. dist (snd x) y < s}" have "open A" and "open B" unfolding A_def B_def by (simp_all add: open_ball) moreover have "x \<in> A \<times> B" unfolding A_def B_def mem_Times_iff using `0 < r` and `0 < s` by simp moreover have "A \<times> B \<subseteq> S" proof (clarify) fix a b assume "a \<in> A" and "b \<in> B" hence "dist a (fst x) < r" and "dist b (snd x) < s" unfolding A_def B_def by (simp_all add: dist_commute) hence "dist (a, b) x < e" unfolding dist_prod_def `e = sqrt (r\<twosuperior> + s\<twosuperior>)` by (simp add: add_strict_mono power_strict_mono) thus "(a, b) \<in> S" by (simp add: S) qed ultimately show "\<exists>A B. open A \<and> open B \<and> x \<in> A \<times> B \<and> A \<times> B \<subseteq> S" by fast qed qedqedendlemma Cauchy_fst: "Cauchy X \<Longrightarrow> Cauchy (\<lambda>n. fst (X n))"unfolding Cauchy_def by (fast elim: le_less_trans [OF dist_fst_le])lemma Cauchy_snd: "Cauchy X \<Longrightarrow> Cauchy (\<lambda>n. snd (X n))"unfolding Cauchy_def by (fast elim: le_less_trans [OF dist_snd_le])lemma Cauchy_Pair: assumes "Cauchy X" and "Cauchy Y" shows "Cauchy (\<lambda>n. (X n, Y n))"proof (rule metric_CauchyI) fix r :: real assume "0 < r" then have "0 < r / sqrt 2" (is "0 < ?s") by (simp add: divide_pos_pos) obtain M where M: "\<forall>m\<ge>M. \<forall>n\<ge>M. dist (X m) (X n) < ?s" using metric_CauchyD [OF `Cauchy X` `0 < ?s`] .. obtain N where N: "\<forall>m\<ge>N. \<forall>n\<ge>N. dist (Y m) (Y n) < ?s" using metric_CauchyD [OF `Cauchy Y` `0 < ?s`] .. have "\<forall>m\<ge>max M N. \<forall>n\<ge>max M N. dist (X m, Y m) (X n, Y n) < r" using M N by (simp add: real_sqrt_sum_squares_less dist_Pair_Pair) then show "\<exists>n0. \<forall>m\<ge>n0. \<forall>n\<ge>n0. dist (X m, Y m) (X n, Y n) < r" ..qedsubsection {* Product is a complete metric space *}instance prod :: (complete_space, complete_space) complete_spaceproof fix X :: "nat \<Rightarrow> 'a \<times> 'b" assume "Cauchy X" have 1: "(\<lambda>n. fst (X n)) ----> lim (\<lambda>n. fst (X n))" using Cauchy_fst [OF `Cauchy X`] by (simp add: Cauchy_convergent_iff convergent_LIMSEQ_iff) have 2: "(\<lambda>n. snd (X n)) ----> lim (\<lambda>n. snd (X n))" using Cauchy_snd [OF `Cauchy X`] by (simp add: Cauchy_convergent_iff convergent_LIMSEQ_iff) have "X ----> (lim (\<lambda>n. fst (X n)), lim (\<lambda>n. snd (X n)))" using tendsto_Pair [OF 1 2] by simp then show "convergent X" by (rule convergentI)qedsubsection {* Product is a normed vector space *}instantiation prod :: (real_normed_vector, real_normed_vector) real_normed_vectorbegindefinition norm_prod_def: "norm x = sqrt ((norm (fst x))\<twosuperior> + (norm (snd x))\<twosuperior>)"definition sgn_prod_def: "sgn (x::'a \<times> 'b) = scaleR (inverse (norm x)) x"lemma norm_Pair: "norm (a, b) = sqrt ((norm a)\<twosuperior> + (norm b)\<twosuperior>)" unfolding norm_prod_def by simpinstance proof fix r :: real and x y :: "'a \<times> 'b" show "0 \<le> norm x" unfolding norm_prod_def by simp show "norm x = 0 \<longleftrightarrow> x = 0" unfolding norm_prod_def by (simp add: prod_eq_iff) show "norm (x + y) \<le> norm x + norm y" unfolding norm_prod_def apply (rule order_trans [OF _ real_sqrt_sum_squares_triangle_ineq]) apply (simp add: add_mono power_mono norm_triangle_ineq) done show "norm (scaleR r x) = \<bar>r\<bar> * norm x" unfolding norm_prod_def apply (simp add: power_mult_distrib) apply (simp add: right_distrib [symmetric]) apply (simp add: real_sqrt_mult_distrib) done show "sgn x = scaleR (inverse (norm x)) x" by (rule sgn_prod_def) show "dist x y = norm (x - y)" unfolding dist_prod_def norm_prod_def by (simp add: dist_norm)qedendinstance prod :: (banach, banach) banach ..subsubsection {* Pair operations are linear *}lemma bounded_linear_fst: "bounded_linear fst" using fst_add fst_scaleR by (rule bounded_linear_intro [where K=1], simp add: norm_prod_def)lemma bounded_linear_snd: "bounded_linear snd" using snd_add snd_scaleR by (rule bounded_linear_intro [where K=1], simp add: norm_prod_def)text {* TODO: move to NthRoot *}lemma sqrt_add_le_add_sqrt: assumes x: "0 \<le> x" and y: "0 \<le> y" shows "sqrt (x + y) \<le> sqrt x + sqrt y"apply (rule power2_le_imp_le)apply (simp add: power2_sum x y)apply (simp add: mult_nonneg_nonneg x y)apply (simp add: x y)donelemma bounded_linear_Pair: assumes f: "bounded_linear f" assumes g: "bounded_linear g" shows "bounded_linear (\<lambda>x. (f x, g x))"proof interpret f: bounded_linear f by fact interpret g: bounded_linear g by fact fix x y and r :: real show "(f (x + y), g (x + y)) = (f x, g x) + (f y, g y)" by (simp add: f.add g.add) show "(f (r *\<^sub>R x), g (r *\<^sub>R x)) = r *\<^sub>R (f x, g x)" by (simp add: f.scaleR g.scaleR) obtain Kf where "0 < Kf" and norm_f: "\<And>x. norm (f x) \<le> norm x * Kf" using f.pos_bounded by fast obtain Kg where "0 < Kg" and norm_g: "\<And>x. norm (g x) \<le> norm x * Kg" using g.pos_bounded by fast have "\<forall>x. norm (f x, g x) \<le> norm x * (Kf + Kg)" apply (rule allI) apply (simp add: norm_Pair) apply (rule order_trans [OF sqrt_add_le_add_sqrt], simp, simp) apply (simp add: right_distrib) apply (rule add_mono [OF norm_f norm_g]) done then show "\<exists>K. \<forall>x. norm (f x, g x) \<le> norm x * K" ..qedsubsubsection {* Frechet derivatives involving pairs *}lemma FDERIV_Pair: assumes f: "FDERIV f x :> f'" and g: "FDERIV g x :> g'" shows "FDERIV (\<lambda>x. (f x, g x)) x :> (\<lambda>h. (f' h, g' h))"proof (rule FDERIV_I) show "bounded_linear (\<lambda>h. (f' h, g' h))" using f g by (intro bounded_linear_Pair FDERIV_bounded_linear) let ?Rf = "\<lambda>h. f (x + h) - f x - f' h" let ?Rg = "\<lambda>h. g (x + h) - g x - g' h" let ?R = "\<lambda>h. ((f (x + h), g (x + h)) - (f x, g x) - (f' h, g' h))" show "(\<lambda>h. norm (?R h) / norm h) -- 0 --> 0" proof (rule real_LIM_sandwich_zero) show "(\<lambda>h. norm (?Rf h) / norm h + norm (?Rg h) / norm h) -- 0 --> 0" using f g by (intro tendsto_add_zero FDERIV_D) fix h :: 'a assume "h \<noteq> 0" thus "0 \<le> norm (?R h) / norm h" by (simp add: divide_nonneg_pos) show "norm (?R h) / norm h \<le> norm (?Rf h) / norm h + norm (?Rg h) / norm h" unfolding add_divide_distrib [symmetric] by (simp add: norm_Pair divide_right_mono order_trans [OF sqrt_add_le_add_sqrt]) qedqedsubsection {* Product is an inner product space *}instantiation prod :: (real_inner, real_inner) real_innerbegindefinition inner_prod_def: "inner x y = inner (fst x) (fst y) + inner (snd x) (snd y)"lemma inner_Pair [simp]: "inner (a, b) (c, d) = inner a c + inner b d" unfolding inner_prod_def by simpinstance proof fix r :: real fix x y z :: "'a::real_inner \<times> 'b::real_inner" show "inner x y = inner y x" unfolding inner_prod_def by (simp add: inner_commute) show "inner (x + y) z = inner x z + inner y z" unfolding inner_prod_def by (simp add: inner_add_left) show "inner (scaleR r x) y = r * inner x y" unfolding inner_prod_def by (simp add: right_distrib) show "0 \<le> inner x x" unfolding inner_prod_def by (intro add_nonneg_nonneg inner_ge_zero) show "inner x x = 0 \<longleftrightarrow> x = 0" unfolding inner_prod_def prod_eq_iff by (simp add: add_nonneg_eq_0_iff) show "norm x = sqrt (inner x x)" unfolding norm_prod_def inner_prod_def by (simp add: power2_norm_eq_inner)qedendend