author | huffman |
Wed, 03 Jun 2009 09:58:11 -0700 | |
changeset 31417 | c12b25b7f015 |
parent 31414 | 8514775606e0 |
child 31446 | 2d91b2416de8 |
permissions | -rw-r--r-- |
29993 | 1 |
(* Title: 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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0b6aff7451b2
Main is (Complex_Main) base entry point in library theories
haftmann
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imports Complex_Main FrechetDeriv |
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begin |
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subsection {* Real inner product spaces *} |
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class real_inner = real_vector + sgn_div_norm + dist_norm + topo_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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and inner_left_distrib: "inner (x + y) z = inner x z + inner y z" |
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and inner_scaleR_left: "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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using inner_left_distrib [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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using inner_left_distrib [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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by (simp add: diff_minus inner_left_distrib) |
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text {* Transfer distributivity rules to right argument. *} |
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lemma inner_right_distrib: "inner x (y + z) = inner x y + inner x z" |
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using inner_left_distrib [of y z x] by (simp only: inner_commute) |
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lemma inner_scaleR_right: "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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lemmas inner_distrib = inner_left_distrib inner_right_distrib |
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lemmas inner_diff = inner_diff_left inner_diff_right |
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lemmas inner_scaleR = inner_scaleR_left inner_scaleR_right |
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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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by (simp add: inner_diff inner_scaleR) |
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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 "0 \<le> norm x" |
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unfolding norm_eq_sqrt_inner by simp |
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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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by (simp add: inner_distrib inner_commute) |
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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 (simp add: add_nonneg_nonneg) |
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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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by (simp add: inner_scaleR power2_eq_square mult_assoc) |
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qed |
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end |
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interpretation/interpret: prefixes are mandatory by default;
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parents:
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interpretation 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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by (rule inner_left_distrib) |
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show "inner x (y + z) = inner x y + inner x z" |
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by (rule inner_right_distrib) |
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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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interpretation/interpret: prefixes are mandatory by default;
wenzelm
parents:
30663
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changeset
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interpretation inner_left: |
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bounded_linear "\<lambda>x::'a::real_inner. inner x y" |
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by (rule inner.bounded_linear_left) |
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461ee3e49ad3
interpretation/interpret: prefixes are mandatory by default;
wenzelm
parents:
30663
diff
changeset
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interpretation inner_right: |
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bounded_linear "\<lambda>y::'a::real_inner. inner x y" |
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by (rule inner.bounded_linear_right) |
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subsection {* Class instances *} |
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instantiation real :: real_inner |
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begin |
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definition inner_real_def [simp]: "inner = op *" |
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instance proof |
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fix x y z r :: real |
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show "inner x y = inner y x" |
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unfolding inner_real_def by (rule mult_commute) |
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show "inner (x + y) z = inner x z + inner y z" |
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unfolding inner_real_def by (rule left_distrib) |
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show "inner (scaleR r x) y = r * inner x y" |
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unfolding inner_real_def real_scaleR_def by (rule mult_assoc) |
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show "0 \<le> inner x x" |
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unfolding inner_real_def by simp |
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show "inner x x = 0 \<longleftrightarrow> x = 0" |
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unfolding inner_real_def by simp |
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show "norm x = sqrt (inner x x)" |
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unfolding inner_real_def by simp |
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qed |
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end |
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instantiation complex :: real_inner |
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begin |
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definition inner_complex_def: |
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"inner x y = Re x * Re y + Im x * Im y" |
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instance proof |
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fix x y z :: complex and r :: real |
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show "inner x y = inner y x" |
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unfolding inner_complex_def by (simp add: mult_commute) |
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show "inner (x + y) z = inner x z + inner y z" |
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unfolding inner_complex_def by (simp add: left_distrib) |
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show "inner (scaleR r x) y = r * inner x y" |
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unfolding inner_complex_def by (simp add: right_distrib) |
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show "0 \<le> inner x x" |
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unfolding inner_complex_def by (simp add: add_nonneg_nonneg) |
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show "inner x x = 0 \<longleftrightarrow> x = 0" |
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unfolding inner_complex_def |
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by (simp add: add_nonneg_eq_0_iff complex_Re_Im_cancel_iff) |
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show "norm x = sqrt (inner x x)" |
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unfolding inner_complex_def complex_norm_def |
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by (simp add: power2_eq_square) |
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qed |
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end |
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subsection {* Gradient derivative *} |
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definition |
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gderiv :: |
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"['a::real_inner \<Rightarrow> real, 'a, 'a] \<Rightarrow> bool" |
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("(GDERIV (_)/ (_)/ :> (_))" [1000, 1000, 60] 60) |
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where |
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"GDERIV f x :> D \<longleftrightarrow> FDERIV f x :> (\<lambda>h. inner h D)" |
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lemma deriv_fderiv: "DERIV f x :> D \<longleftrightarrow> FDERIV f x :> (\<lambda>h. h * D)" |
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by (simp only: deriv_def field_fderiv_def) |
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lemma gderiv_deriv [simp]: "GDERIV f x :> D \<longleftrightarrow> DERIV f x :> D" |
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by (simp only: gderiv_def deriv_fderiv inner_real_def) |
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lemma GDERIV_DERIV_compose: |
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"\<lbrakk>GDERIV f x :> df; DERIV g (f x) :> dg\<rbrakk> |
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\<Longrightarrow> GDERIV (\<lambda>x. g (f x)) x :> scaleR dg df" |
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unfolding gderiv_def deriv_fderiv |
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apply (drule (1) FDERIV_compose) |
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apply (simp add: inner_scaleR_right mult_ac) |
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done |
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lemma FDERIV_subst: "\<lbrakk>FDERIV f x :> df; df = d\<rbrakk> \<Longrightarrow> FDERIV f x :> d" |
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by simp |
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lemma GDERIV_subst: "\<lbrakk>GDERIV f x :> df; df = d\<rbrakk> \<Longrightarrow> GDERIV f x :> d" |
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by simp |
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lemma GDERIV_const: "GDERIV (\<lambda>x. k) x :> 0" |
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unfolding gderiv_def inner_right.zero by (rule FDERIV_const) |
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lemma GDERIV_add: |
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"\<lbrakk>GDERIV f x :> df; GDERIV g x :> dg\<rbrakk> |
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\<Longrightarrow> GDERIV (\<lambda>x. f x + g x) x :> df + dg" |
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unfolding gderiv_def inner_right.add by (rule FDERIV_add) |
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lemma GDERIV_minus: |
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"GDERIV f x :> df \<Longrightarrow> GDERIV (\<lambda>x. - f x) x :> - df" |
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unfolding gderiv_def inner_right.minus by (rule FDERIV_minus) |
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lemma GDERIV_diff: |
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"\<lbrakk>GDERIV f x :> df; GDERIV g x :> dg\<rbrakk> |
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\<Longrightarrow> GDERIV (\<lambda>x. f x - g x) x :> df - dg" |
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unfolding gderiv_def inner_right.diff by (rule FDERIV_diff) |
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lemma GDERIV_scaleR: |
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"\<lbrakk>DERIV f x :> df; GDERIV g x :> dg\<rbrakk> |
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\<Longrightarrow> GDERIV (\<lambda>x. scaleR (f x) (g x)) x |
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:> (scaleR (f x) dg + scaleR df (g x))" |
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unfolding gderiv_def deriv_fderiv inner_right.add inner_right.scaleR |
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apply (rule FDERIV_subst) |
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apply (erule (1) scaleR.FDERIV) |
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apply (simp add: mult_ac) |
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done |
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lemma GDERIV_mult: |
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"\<lbrakk>GDERIV f x :> df; GDERIV g x :> dg\<rbrakk> |
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\<Longrightarrow> GDERIV (\<lambda>x. f x * g x) x :> scaleR (f x) dg + scaleR (g x) df" |
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unfolding gderiv_def |
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apply (rule FDERIV_subst) |
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apply (erule (1) FDERIV_mult) |
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apply (simp add: inner_distrib inner_scaleR mult_ac) |
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done |
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lemma GDERIV_inverse: |
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"\<lbrakk>GDERIV f x :> df; f x \<noteq> 0\<rbrakk> |
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\<Longrightarrow> GDERIV (\<lambda>x. inverse (f x)) x :> - (inverse (f x))\<twosuperior> *\<^sub>R df" |
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apply (erule GDERIV_DERIV_compose) |
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apply (erule DERIV_inverse [folded numeral_2_eq_2]) |
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done |
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lemma GDERIV_norm: |
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assumes "x \<noteq> 0" shows "GDERIV (\<lambda>x. norm x) x :> sgn x" |
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proof - |
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have 1: "FDERIV (\<lambda>x. inner x x) x :> (\<lambda>h. inner x h + inner h x)" |
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by (intro inner.FDERIV FDERIV_ident) |
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have 2: "(\<lambda>h. inner x h + inner h x) = (\<lambda>h. inner h (scaleR 2 x))" |
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by (simp add: expand_fun_eq inner_scaleR inner_commute) |
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have "0 < inner x x" using `x \<noteq> 0` by simp |
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then have 3: "DERIV sqrt (inner x x) :> (inverse (sqrt (inner x x)) / 2)" |
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by (rule DERIV_real_sqrt) |
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have 4: "(inverse (sqrt (inner x x)) / 2) *\<^sub>R 2 *\<^sub>R x = sgn x" |
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by (simp add: sgn_div_norm norm_eq_sqrt_inner) |
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show ?thesis |
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unfolding norm_eq_sqrt_inner |
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apply (rule GDERIV_subst [OF _ 4]) |
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apply (rule GDERIV_DERIV_compose [where g=sqrt and df="scaleR 2 x"]) |
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apply (subst gderiv_def) |
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apply (rule FDERIV_subst [OF _ 2]) |
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apply (rule 1) |
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apply (rule 3) |
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done |
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qed |
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lemmas FDERIV_norm = GDERIV_norm [unfolded gderiv_def] |
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end |