author | wenzelm |
Fri, 12 Jul 2013 16:19:05 +0200 | |
changeset 52622 | e0ff1625e96d |
parent 51642 | 400ec5ae7f8f |
child 53015 | a1119cf551e8 |
permissions | -rw-r--r-- |
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(* Title: HOL/Library/Inner_Product.thy |
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Author: Brian Huffman |
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*) |
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header {* Inner Product Spaces and the Gradient Derivative *} |
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theory Inner_Product |
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imports "~~/src/HOL/Complex_Main" |
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begin |
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subsection {* Real inner product spaces *} |
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text {* |
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Temporarily relax type constraints for @{term "open"}, |
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@{term dist}, and @{term norm}. |
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*} |
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setup {* Sign.add_const_constraint |
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(@{const_name "open"}, SOME @{typ "'a::open set \<Rightarrow> bool"}) *} |
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setup {* Sign.add_const_constraint |
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(@{const_name dist}, SOME @{typ "'a::dist \<Rightarrow> 'a \<Rightarrow> real"}) *} |
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setup {* Sign.add_const_constraint |
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(@{const_name norm}, SOME @{typ "'a::norm \<Rightarrow> real"}) *} |
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class real_inner = real_vector + sgn_div_norm + dist_norm + open_dist + |
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fixes inner :: "'a \<Rightarrow> 'a \<Rightarrow> real" |
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assumes inner_commute: "inner x y = inner y x" |
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declare inner_add, inner_diff [algebra_simps]; declare inner_scaleR [simp]
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and inner_add_left: "inner (x + y) z = inner x z + inner y z" |
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declare inner_add, inner_diff [algebra_simps]; declare inner_scaleR [simp]
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and inner_scaleR_left [simp]: "inner (scaleR r x) y = r * (inner x y)" |
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and inner_ge_zero [simp]: "0 \<le> inner x x" |
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and inner_eq_zero_iff [simp]: "inner x x = 0 \<longleftrightarrow> x = 0" |
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and norm_eq_sqrt_inner: "norm x = sqrt (inner x x)" |
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begin |
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lemma inner_zero_left [simp]: "inner 0 x = 0" |
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declare inner_add, inner_diff [algebra_simps]; declare inner_scaleR [simp]
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using inner_add_left [of 0 0 x] by simp |
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lemma inner_minus_left [simp]: "inner (- x) y = - inner x y" |
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declare inner_add, inner_diff [algebra_simps]; declare inner_scaleR [simp]
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using inner_add_left [of x "- x" y] by simp |
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lemma inner_diff_left: "inner (x - y) z = inner x z - inner y z" |
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declare inner_add, inner_diff [algebra_simps]; declare inner_scaleR [simp]
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by (simp add: diff_minus inner_add_left) |
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lemma inner_setsum_left: "inner (\<Sum>x\<in>A. f x) y = (\<Sum>x\<in>A. inner (f x) y)" |
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by (cases "finite A", induct set: finite, simp_all add: inner_add_left) |
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text {* Transfer distributivity rules to right argument. *} |
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declare inner_add, inner_diff [algebra_simps]; declare inner_scaleR [simp]
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lemma inner_add_right: "inner x (y + z) = inner x y + inner x z" |
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declare inner_add, inner_diff [algebra_simps]; declare inner_scaleR [simp]
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using inner_add_left [of y z x] by (simp only: inner_commute) |
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declare inner_add, inner_diff [algebra_simps]; declare inner_scaleR [simp]
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lemma inner_scaleR_right [simp]: "inner x (scaleR r y) = r * (inner x y)" |
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using inner_scaleR_left [of r y x] by (simp only: inner_commute) |
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lemma inner_zero_right [simp]: "inner x 0 = 0" |
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using inner_zero_left [of x] by (simp only: inner_commute) |
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lemma inner_minus_right [simp]: "inner x (- y) = - inner x y" |
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using inner_minus_left [of y x] by (simp only: inner_commute) |
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lemma inner_diff_right: "inner x (y - z) = inner x y - inner x z" |
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using inner_diff_left [of y z x] by (simp only: inner_commute) |
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lemma inner_setsum_right: "inner x (\<Sum>y\<in>A. f y) = (\<Sum>y\<in>A. inner x (f y))" |
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using inner_setsum_left [of f A x] by (simp only: inner_commute) |
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declare inner_add, inner_diff [algebra_simps]; declare inner_scaleR [simp]
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lemmas inner_add [algebra_simps] = inner_add_left inner_add_right |
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declare inner_add, inner_diff [algebra_simps]; declare inner_scaleR [simp]
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lemmas inner_diff [algebra_simps] = inner_diff_left inner_diff_right |
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declare inner_add, inner_diff [algebra_simps]; declare inner_scaleR [simp]
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lemmas inner_scaleR = inner_scaleR_left inner_scaleR_right |
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declare inner_add, inner_diff [algebra_simps]; declare inner_scaleR [simp]
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declare inner_add, inner_diff [algebra_simps]; declare inner_scaleR [simp]
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text {* Legacy theorem names *} |
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declare inner_add, inner_diff [algebra_simps]; declare inner_scaleR [simp]
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lemmas inner_left_distrib = inner_add_left |
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declare inner_add, inner_diff [algebra_simps]; declare inner_scaleR [simp]
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lemmas inner_right_distrib = inner_add_right |
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lemmas inner_distrib = inner_left_distrib inner_right_distrib |
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lemma inner_gt_zero_iff [simp]: "0 < inner x x \<longleftrightarrow> x \<noteq> 0" |
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by (simp add: order_less_le) |
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lemma power2_norm_eq_inner: "(norm x)\<twosuperior> = inner x x" |
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by (simp add: norm_eq_sqrt_inner) |
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lemma Cauchy_Schwarz_ineq: |
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"(inner x y)\<twosuperior> \<le> inner x x * inner y y" |
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proof (cases) |
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assume "y = 0" |
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thus ?thesis by simp |
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next |
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assume y: "y \<noteq> 0" |
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let ?r = "inner x y / inner y y" |
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have "0 \<le> inner (x - scaleR ?r y) (x - scaleR ?r y)" |
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by (rule inner_ge_zero) |
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also have "\<dots> = inner x x - inner y x * ?r" |
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declare inner_add, inner_diff [algebra_simps]; declare inner_scaleR [simp]
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by (simp add: inner_diff) |
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also have "\<dots> = inner x x - (inner x y)\<twosuperior> / inner y y" |
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by (simp add: power2_eq_square inner_commute) |
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finally have "0 \<le> inner x x - (inner x y)\<twosuperior> / inner y y" . |
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hence "(inner x y)\<twosuperior> / inner y y \<le> inner x x" |
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by (simp add: le_diff_eq) |
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thus "(inner x y)\<twosuperior> \<le> inner x x * inner y y" |
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by (simp add: pos_divide_le_eq y) |
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qed |
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lemma Cauchy_Schwarz_ineq2: |
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"\<bar>inner x y\<bar> \<le> norm x * norm y" |
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proof (rule power2_le_imp_le) |
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have "(inner x y)\<twosuperior> \<le> inner x x * inner y y" |
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using Cauchy_Schwarz_ineq . |
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thus "\<bar>inner x y\<bar>\<twosuperior> \<le> (norm x * norm y)\<twosuperior>" |
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by (simp add: power_mult_distrib power2_norm_eq_inner) |
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show "0 \<le> norm x * norm y" |
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unfolding norm_eq_sqrt_inner |
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by (intro mult_nonneg_nonneg real_sqrt_ge_zero inner_ge_zero) |
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qed |
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subclass real_normed_vector |
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proof |
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fix a :: real and x y :: 'a |
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show "norm x = 0 \<longleftrightarrow> x = 0" |
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unfolding norm_eq_sqrt_inner by simp |
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show "norm (x + y) \<le> norm x + norm y" |
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proof (rule power2_le_imp_le) |
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have "inner x y \<le> norm x * norm y" |
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by (rule order_trans [OF abs_ge_self Cauchy_Schwarz_ineq2]) |
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thus "(norm (x + y))\<twosuperior> \<le> (norm x + norm y)\<twosuperior>" |
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unfolding power2_sum power2_norm_eq_inner |
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declare inner_add, inner_diff [algebra_simps]; declare inner_scaleR [simp]
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by (simp add: inner_add inner_commute) |
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show "0 \<le> norm x + norm y" |
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unfolding norm_eq_sqrt_inner by simp |
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qed |
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have "sqrt (a\<twosuperior> * inner x x) = \<bar>a\<bar> * sqrt (inner x x)" |
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by (simp add: real_sqrt_mult_distrib) |
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then show "norm (a *\<^sub>R x) = \<bar>a\<bar> * norm x" |
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unfolding norm_eq_sqrt_inner |
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declare inner_add, inner_diff [algebra_simps]; declare inner_scaleR [simp]
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by (simp add: power2_eq_square mult_assoc) |
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qed |
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end |
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text {* |
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Re-enable constraints for @{term "open"}, |
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@{term dist}, and @{term norm}. |
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*} |
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setup {* Sign.add_const_constraint |
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(@{const_name "open"}, SOME @{typ "'a::topological_space set \<Rightarrow> bool"}) *} |
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setup {* Sign.add_const_constraint |
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(@{const_name dist}, SOME @{typ "'a::metric_space \<Rightarrow> 'a \<Rightarrow> real"}) *} |
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setup {* Sign.add_const_constraint |
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(@{const_name norm}, SOME @{typ "'a::real_normed_vector \<Rightarrow> real"}) *} |
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lemma bounded_bilinear_inner: |
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"bounded_bilinear (inner::'a::real_inner \<Rightarrow> 'a \<Rightarrow> real)" |
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proof |
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fix x y z :: 'a and r :: real |
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show "inner (x + y) z = inner x z + inner y z" |
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declare inner_add, inner_diff [algebra_simps]; declare inner_scaleR [simp]
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by (rule inner_add_left) |
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show "inner x (y + z) = inner x y + inner x z" |
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declare inner_add, inner_diff [algebra_simps]; declare inner_scaleR [simp]
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by (rule inner_add_right) |
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show "inner (scaleR r x) y = scaleR r (inner x y)" |
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unfolding real_scaleR_def by (rule inner_scaleR_left) |
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show "inner x (scaleR r y) = scaleR r (inner x y)" |
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unfolding real_scaleR_def by (rule inner_scaleR_right) |
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show "\<exists>K. \<forall>x y::'a. norm (inner x y) \<le> norm x * norm y * K" |
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proof |
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show "\<forall>x y::'a. norm (inner x y) \<le> norm x * norm y * 1" |
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by (simp add: Cauchy_Schwarz_ineq2) |
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qed |
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qed |
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lemmas tendsto_inner [tendsto_intros] = |
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bounded_bilinear.tendsto [OF bounded_bilinear_inner] |
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lemmas isCont_inner [simp] = |
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bounded_bilinear.isCont [OF bounded_bilinear_inner] |
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lemmas FDERIV_inner [FDERIV_intros] = |
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bounded_bilinear.FDERIV [OF bounded_bilinear_inner] |
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lemmas bounded_linear_inner_left = |
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bounded_bilinear.bounded_linear_left [OF bounded_bilinear_inner] |
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lemmas bounded_linear_inner_right = |
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bounded_bilinear.bounded_linear_right [OF bounded_bilinear_inner] |
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lemmas FDERIV_inner_right [FDERIV_intros] = |
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bounded_linear.FDERIV [OF bounded_linear_inner_right] |
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|
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lemmas FDERIV_inner_left [FDERIV_intros] = |
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bounded_linear.FDERIV [OF bounded_linear_inner_left] |
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|
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lemma differentiable_inner [simp]: |
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"f differentiable x in s \<Longrightarrow> g differentiable x in s \<Longrightarrow> (\<lambda>x. inner (f x) (g x)) differentiable x in s" |
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unfolding isDiff_def by (blast intro: FDERIV_inner) |
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subsection {* Class instances *} |
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instantiation real :: real_inner |
|
202 |
begin |
|
203 |
||
204 |
definition inner_real_def [simp]: "inner = op *" |
|
205 |
||
206 |
instance proof |
|
207 |
fix x y z r :: real |
|
208 |
show "inner x y = inner y x" |
|
209 |
unfolding inner_real_def by (rule mult_commute) |
|
210 |
show "inner (x + y) z = inner x z + inner y z" |
|
49962
a8cc904a6820
Renamed {left,right}_distrib to distrib_{right,left}.
webertj
parents:
44902
diff
changeset
|
211 |
unfolding inner_real_def by (rule distrib_right) |
29993 | 212 |
show "inner (scaleR r x) y = r * inner x y" |
213 |
unfolding inner_real_def real_scaleR_def by (rule mult_assoc) |
|
214 |
show "0 \<le> inner x x" |
|
215 |
unfolding inner_real_def by simp |
|
216 |
show "inner x x = 0 \<longleftrightarrow> x = 0" |
|
217 |
unfolding inner_real_def by simp |
|
218 |
show "norm x = sqrt (inner x x)" |
|
219 |
unfolding inner_real_def by simp |
|
220 |
qed |
|
221 |
||
222 |
end |
|
223 |
||
224 |
instantiation complex :: real_inner |
|
225 |
begin |
|
226 |
||
227 |
definition inner_complex_def: |
|
228 |
"inner x y = Re x * Re y + Im x * Im y" |
|
229 |
||
230 |
instance proof |
|
231 |
fix x y z :: complex and r :: real |
|
232 |
show "inner x y = inner y x" |
|
233 |
unfolding inner_complex_def by (simp add: mult_commute) |
|
234 |
show "inner (x + y) z = inner x z + inner y z" |
|
49962
a8cc904a6820
Renamed {left,right}_distrib to distrib_{right,left}.
webertj
parents:
44902
diff
changeset
|
235 |
unfolding inner_complex_def by (simp add: distrib_right) |
29993 | 236 |
show "inner (scaleR r x) y = r * inner x y" |
49962
a8cc904a6820
Renamed {left,right}_distrib to distrib_{right,left}.
webertj
parents:
44902
diff
changeset
|
237 |
unfolding inner_complex_def by (simp add: distrib_left) |
29993 | 238 |
show "0 \<le> inner x x" |
44126 | 239 |
unfolding inner_complex_def by simp |
29993 | 240 |
show "inner x x = 0 \<longleftrightarrow> x = 0" |
241 |
unfolding inner_complex_def |
|
242 |
by (simp add: add_nonneg_eq_0_iff complex_Re_Im_cancel_iff) |
|
243 |
show "norm x = sqrt (inner x x)" |
|
244 |
unfolding inner_complex_def complex_norm_def |
|
245 |
by (simp add: power2_eq_square) |
|
246 |
qed |
|
247 |
||
248 |
end |
|
249 |
||
44902
9ba11d41cd1f
move lemmas about complex number 'i' to Complex.thy and Library/Inner_Product.thy
huffman
parents:
44282
diff
changeset
|
250 |
lemma complex_inner_1 [simp]: "inner 1 x = Re x" |
9ba11d41cd1f
move lemmas about complex number 'i' to Complex.thy and Library/Inner_Product.thy
huffman
parents:
44282
diff
changeset
|
251 |
unfolding inner_complex_def by simp |
9ba11d41cd1f
move lemmas about complex number 'i' to Complex.thy and Library/Inner_Product.thy
huffman
parents:
44282
diff
changeset
|
252 |
|
9ba11d41cd1f
move lemmas about complex number 'i' to Complex.thy and Library/Inner_Product.thy
huffman
parents:
44282
diff
changeset
|
253 |
lemma complex_inner_1_right [simp]: "inner x 1 = Re x" |
9ba11d41cd1f
move lemmas about complex number 'i' to Complex.thy and Library/Inner_Product.thy
huffman
parents:
44282
diff
changeset
|
254 |
unfolding inner_complex_def by simp |
9ba11d41cd1f
move lemmas about complex number 'i' to Complex.thy and Library/Inner_Product.thy
huffman
parents:
44282
diff
changeset
|
255 |
|
9ba11d41cd1f
move lemmas about complex number 'i' to Complex.thy and Library/Inner_Product.thy
huffman
parents:
44282
diff
changeset
|
256 |
lemma complex_inner_ii_left [simp]: "inner ii x = Im x" |
9ba11d41cd1f
move lemmas about complex number 'i' to Complex.thy and Library/Inner_Product.thy
huffman
parents:
44282
diff
changeset
|
257 |
unfolding inner_complex_def by simp |
9ba11d41cd1f
move lemmas about complex number 'i' to Complex.thy and Library/Inner_Product.thy
huffman
parents:
44282
diff
changeset
|
258 |
|
9ba11d41cd1f
move lemmas about complex number 'i' to Complex.thy and Library/Inner_Product.thy
huffman
parents:
44282
diff
changeset
|
259 |
lemma complex_inner_ii_right [simp]: "inner x ii = Im x" |
9ba11d41cd1f
move lemmas about complex number 'i' to Complex.thy and Library/Inner_Product.thy
huffman
parents:
44282
diff
changeset
|
260 |
unfolding inner_complex_def by simp |
9ba11d41cd1f
move lemmas about complex number 'i' to Complex.thy and Library/Inner_Product.thy
huffman
parents:
44282
diff
changeset
|
261 |
|
29993 | 262 |
|
263 |
subsection {* Gradient derivative *} |
|
264 |
||
265 |
definition |
|
266 |
gderiv :: |
|
267 |
"['a::real_inner \<Rightarrow> real, 'a, 'a] \<Rightarrow> bool" |
|
268 |
("(GDERIV (_)/ (_)/ :> (_))" [1000, 1000, 60] 60) |
|
269 |
where |
|
270 |
"GDERIV f x :> D \<longleftrightarrow> FDERIV f x :> (\<lambda>h. inner h D)" |
|
271 |
||
272 |
lemma gderiv_deriv [simp]: "GDERIV f x :> D \<longleftrightarrow> DERIV f x :> D" |
|
273 |
by (simp only: gderiv_def deriv_fderiv inner_real_def) |
|
274 |
||
275 |
lemma GDERIV_DERIV_compose: |
|
276 |
"\<lbrakk>GDERIV f x :> df; DERIV g (f x) :> dg\<rbrakk> |
|
277 |
\<Longrightarrow> GDERIV (\<lambda>x. g (f x)) x :> scaleR dg df" |
|
278 |
unfolding gderiv_def deriv_fderiv |
|
279 |
apply (drule (1) FDERIV_compose) |
|
31590
776d6a4c1327
declare inner_add, inner_diff [algebra_simps]; declare inner_scaleR [simp]
huffman
parents:
31492
diff
changeset
|
280 |
apply (simp add: mult_ac) |
29993 | 281 |
done |
282 |
||
283 |
lemma FDERIV_subst: "\<lbrakk>FDERIV f x :> df; df = d\<rbrakk> \<Longrightarrow> FDERIV f x :> d" |
|
284 |
by simp |
|
285 |
||
286 |
lemma GDERIV_subst: "\<lbrakk>GDERIV f x :> df; df = d\<rbrakk> \<Longrightarrow> GDERIV f x :> d" |
|
287 |
by simp |
|
288 |
||
289 |
lemma GDERIV_const: "GDERIV (\<lambda>x. k) x :> 0" |
|
44282
f0de18b62d63
remove bounded_(bi)linear locale interpretations, to avoid duplicating so many lemmas
huffman
parents:
44233
diff
changeset
|
290 |
unfolding gderiv_def inner_zero_right by (rule FDERIV_const) |
29993 | 291 |
|
292 |
lemma GDERIV_add: |
|
293 |
"\<lbrakk>GDERIV f x :> df; GDERIV g x :> dg\<rbrakk> |
|
294 |
\<Longrightarrow> GDERIV (\<lambda>x. f x + g x) x :> df + dg" |
|
44282
f0de18b62d63
remove bounded_(bi)linear locale interpretations, to avoid duplicating so many lemmas
huffman
parents:
44233
diff
changeset
|
295 |
unfolding gderiv_def inner_add_right by (rule FDERIV_add) |
29993 | 296 |
|
297 |
lemma GDERIV_minus: |
|
298 |
"GDERIV f x :> df \<Longrightarrow> GDERIV (\<lambda>x. - f x) x :> - df" |
|
44282
f0de18b62d63
remove bounded_(bi)linear locale interpretations, to avoid duplicating so many lemmas
huffman
parents:
44233
diff
changeset
|
299 |
unfolding gderiv_def inner_minus_right by (rule FDERIV_minus) |
29993 | 300 |
|
301 |
lemma GDERIV_diff: |
|
302 |
"\<lbrakk>GDERIV f x :> df; GDERIV g x :> dg\<rbrakk> |
|
303 |
\<Longrightarrow> GDERIV (\<lambda>x. f x - g x) x :> df - dg" |
|
44282
f0de18b62d63
remove bounded_(bi)linear locale interpretations, to avoid duplicating so many lemmas
huffman
parents:
44233
diff
changeset
|
304 |
unfolding gderiv_def inner_diff_right by (rule FDERIV_diff) |
29993 | 305 |
|
306 |
lemma GDERIV_scaleR: |
|
307 |
"\<lbrakk>DERIV f x :> df; GDERIV g x :> dg\<rbrakk> |
|
308 |
\<Longrightarrow> GDERIV (\<lambda>x. scaleR (f x) (g x)) x |
|
309 |
:> (scaleR (f x) dg + scaleR df (g x))" |
|
44282
f0de18b62d63
remove bounded_(bi)linear locale interpretations, to avoid duplicating so many lemmas
huffman
parents:
44233
diff
changeset
|
310 |
unfolding gderiv_def deriv_fderiv inner_add_right inner_scaleR_right |
29993 | 311 |
apply (rule FDERIV_subst) |
44282
f0de18b62d63
remove bounded_(bi)linear locale interpretations, to avoid duplicating so many lemmas
huffman
parents:
44233
diff
changeset
|
312 |
apply (erule (1) FDERIV_scaleR) |
29993 | 313 |
apply (simp add: mult_ac) |
314 |
done |
|
315 |
||
316 |
lemma GDERIV_mult: |
|
317 |
"\<lbrakk>GDERIV f x :> df; GDERIV g x :> dg\<rbrakk> |
|
318 |
\<Longrightarrow> GDERIV (\<lambda>x. f x * g x) x :> scaleR (f x) dg + scaleR (g x) df" |
|
319 |
unfolding gderiv_def |
|
320 |
apply (rule FDERIV_subst) |
|
321 |
apply (erule (1) FDERIV_mult) |
|
31590
776d6a4c1327
declare inner_add, inner_diff [algebra_simps]; declare inner_scaleR [simp]
huffman
parents:
31492
diff
changeset
|
322 |
apply (simp add: inner_add mult_ac) |
29993 | 323 |
done |
324 |
||
325 |
lemma GDERIV_inverse: |
|
326 |
"\<lbrakk>GDERIV f x :> df; f x \<noteq> 0\<rbrakk> |
|
327 |
\<Longrightarrow> GDERIV (\<lambda>x. inverse (f x)) x :> - (inverse (f x))\<twosuperior> *\<^sub>R df" |
|
328 |
apply (erule GDERIV_DERIV_compose) |
|
329 |
apply (erule DERIV_inverse [folded numeral_2_eq_2]) |
|
330 |
done |
|
331 |
||
332 |
lemma GDERIV_norm: |
|
333 |
assumes "x \<noteq> 0" shows "GDERIV (\<lambda>x. norm x) x :> sgn x" |
|
334 |
proof - |
|
335 |
have 1: "FDERIV (\<lambda>x. inner x x) x :> (\<lambda>h. inner x h + inner h x)" |
|
44282
f0de18b62d63
remove bounded_(bi)linear locale interpretations, to avoid duplicating so many lemmas
huffman
parents:
44233
diff
changeset
|
336 |
by (intro FDERIV_inner FDERIV_ident) |
29993 | 337 |
have 2: "(\<lambda>h. inner x h + inner h x) = (\<lambda>h. inner h (scaleR 2 x))" |
39302
d7728f65b353
renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents:
39198
diff
changeset
|
338 |
by (simp add: fun_eq_iff inner_commute) |
29993 | 339 |
have "0 < inner x x" using `x \<noteq> 0` by simp |
340 |
then have 3: "DERIV sqrt (inner x x) :> (inverse (sqrt (inner x x)) / 2)" |
|
341 |
by (rule DERIV_real_sqrt) |
|
342 |
have 4: "(inverse (sqrt (inner x x)) / 2) *\<^sub>R 2 *\<^sub>R x = sgn x" |
|
343 |
by (simp add: sgn_div_norm norm_eq_sqrt_inner) |
|
344 |
show ?thesis |
|
345 |
unfolding norm_eq_sqrt_inner |
|
346 |
apply (rule GDERIV_subst [OF _ 4]) |
|
347 |
apply (rule GDERIV_DERIV_compose [where g=sqrt and df="scaleR 2 x"]) |
|
348 |
apply (subst gderiv_def) |
|
349 |
apply (rule FDERIV_subst [OF _ 2]) |
|
350 |
apply (rule 1) |
|
351 |
apply (rule 3) |
|
352 |
done |
|
353 |
qed |
|
354 |
||
355 |
lemmas FDERIV_norm = GDERIV_norm [unfolded gderiv_def] |
|
356 |
||
357 |
end |