author | immler |
Mon, 16 Dec 2013 17:08:22 +0100 | |
changeset 54776 | db890d9fc5c2 |
parent 54703 | 499f92dc6e45 |
child 54778 | 13f08c876899 |
permissions | -rw-r--r-- |
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(* Title: HOL/Multivariate_Analysis/Linear_Algebra.thy |
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Author: Amine Chaieb, University of Cambridge |
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*) |
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header {* Elementary linear algebra on Euclidean spaces *} |
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theory Linear_Algebra |
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imports |
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Euclidean_Space |
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"~~/src/HOL/Library/Infinite_Set" |
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ordered_euclidean_space compatible with more standard pointwise ordering on products; conditionally complete lattice with product order
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"~~/src/HOL/Library/Product_Order" |
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begin |
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lemma cond_application_beta: "(if b then f else g) x = (if b then f x else g x)" |
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by auto |
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notation inner (infix "\<bullet>" 70) |
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lemma square_bound_lemma: |
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fixes x :: real |
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shows "x < (1 + x) * (1 + x)" |
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proof - |
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have "(x + 1/2)\<^sup>2 + 3/4 > 0" |
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using zero_le_power2[of "x+1/2"] by arith |
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then show ?thesis |
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by (simp add: field_simps power2_eq_square) |
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qed |
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lemma square_continuous: |
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fixes e :: real |
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shows "e > 0 \<Longrightarrow> \<exists>d. 0 < d \<and> (\<forall>y. abs (y - x) < d \<longrightarrow> abs (y * y - x * x) < e)" |
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using isCont_power[OF isCont_ident, of x, unfolded isCont_def LIM_eq, rule_format, of e 2] |
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apply (auto simp add: power2_eq_square) |
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apply (rule_tac x="s" in exI) |
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apply auto |
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apply (erule_tac x=y in allE) |
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apply auto |
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done |
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text{* Hence derive more interesting properties of the norm. *} |
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lemma norm_eq_0_dot: "norm x = 0 \<longleftrightarrow> x \<bullet> x = (0::real)" |
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by simp (* TODO: delete *) |
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lemma norm_triangle_sub: |
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fixes x y :: "'a::real_normed_vector" |
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shows "norm x \<le> norm y + norm (x - y)" |
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using norm_triangle_ineq[of "y" "x - y"] by (simp add: field_simps) |
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lemma norm_le: "norm x \<le> norm y \<longleftrightarrow> x \<bullet> x \<le> y \<bullet> y" |
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by (simp add: norm_eq_sqrt_inner) |
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lemma norm_lt: "norm x < norm y \<longleftrightarrow> x \<bullet> x < y \<bullet> y" |
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by (simp add: norm_eq_sqrt_inner) |
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lemma norm_eq: "norm x = norm y \<longleftrightarrow> x \<bullet> x = y \<bullet> y" |
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apply (subst order_eq_iff) |
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apply (auto simp: norm_le) |
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done |
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lemma norm_eq_1: "norm x = 1 \<longleftrightarrow> x \<bullet> x = 1" |
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by (simp add: norm_eq_sqrt_inner) |
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text{* Squaring equations and inequalities involving norms. *} |
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lemma dot_square_norm: "x \<bullet> x = (norm x)\<^sup>2" |
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by (simp only: power2_norm_eq_inner) (* TODO: move? *) |
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lemma norm_eq_square: "norm x = a \<longleftrightarrow> 0 \<le> a \<and> x \<bullet> x = a\<^sup>2" |
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by (auto simp add: norm_eq_sqrt_inner) |
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lemma real_abs_le_square_iff: "\<bar>x\<bar> \<le> \<bar>y\<bar> \<longleftrightarrow> (x::real)\<^sup>2 \<le> y\<^sup>2" |
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proof |
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assume "\<bar>x\<bar> \<le> \<bar>y\<bar>" |
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then have "\<bar>x\<bar>\<^sup>2 \<le> \<bar>y\<bar>\<^sup>2" by (rule power_mono, simp) |
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then show "x\<^sup>2 \<le> y\<^sup>2" by simp |
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next |
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assume "x\<^sup>2 \<le> y\<^sup>2" |
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then have "sqrt (x\<^sup>2) \<le> sqrt (y\<^sup>2)" by (rule real_sqrt_le_mono) |
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then show "\<bar>x\<bar> \<le> \<bar>y\<bar>" by simp |
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qed |
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lemma norm_le_square: "norm x \<le> a \<longleftrightarrow> 0 \<le> a \<and> x \<bullet> x \<le> a\<^sup>2" |
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apply (simp add: dot_square_norm real_abs_le_square_iff[symmetric]) |
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using norm_ge_zero[of x] |
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apply arith |
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done |
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lemma norm_ge_square: "norm x \<ge> a \<longleftrightarrow> a \<le> 0 \<or> x \<bullet> x \<ge> a\<^sup>2" |
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apply (simp add: dot_square_norm real_abs_le_square_iff[symmetric]) |
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using norm_ge_zero[of x] |
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apply arith |
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done |
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lemma norm_lt_square: "norm x < a \<longleftrightarrow> 0 < a \<and> x \<bullet> x < a\<^sup>2" |
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by (metis not_le norm_ge_square) |
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lemma norm_gt_square: "norm x > a \<longleftrightarrow> a < 0 \<or> x \<bullet> x > a\<^sup>2" |
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by (metis norm_le_square not_less) |
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text{* Dot product in terms of the norm rather than conversely. *} |
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lemmas inner_simps = inner_add_left inner_add_right inner_diff_right inner_diff_left |
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inner_scaleR_left inner_scaleR_right |
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lemma dot_norm: "x \<bullet> y = ((norm (x + y))\<^sup>2 - (norm x)\<^sup>2 - (norm y)\<^sup>2) / 2" |
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unfolding power2_norm_eq_inner inner_simps inner_commute by auto |
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lemma dot_norm_neg: "x \<bullet> y = (((norm x)\<^sup>2 + (norm y)\<^sup>2) - (norm (x - y))\<^sup>2) / 2" |
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unfolding power2_norm_eq_inner inner_simps inner_commute |
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by (auto simp add: algebra_simps) |
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text{* Equality of vectors in terms of @{term "op \<bullet>"} products. *} |
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lemma vector_eq: "x = y \<longleftrightarrow> x \<bullet> x = x \<bullet> y \<and> y \<bullet> y = x \<bullet> x" |
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(is "?lhs \<longleftrightarrow> ?rhs") |
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proof |
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assume ?lhs |
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then show ?rhs by simp |
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next |
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assume ?rhs |
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then have "x \<bullet> x - x \<bullet> y = 0 \<and> x \<bullet> y - y \<bullet> y = 0" |
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by simp |
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then have "x \<bullet> (x - y) = 0 \<and> y \<bullet> (x - y) = 0" |
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by (simp add: inner_diff inner_commute) |
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then have "(x - y) \<bullet> (x - y) = 0" |
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by (simp add: field_simps inner_diff inner_commute) |
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then show "x = y" by simp |
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qed |
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lemma norm_triangle_half_r: |
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"norm (y - x1) < e / 2 \<Longrightarrow> norm (y - x2) < e / 2 \<Longrightarrow> norm (x1 - x2) < e" |
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using dist_triangle_half_r unfolding dist_norm[symmetric] by auto |
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lemma norm_triangle_half_l: |
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assumes "norm (x - y) < e / 2" |
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and "norm (x' - y) < e / 2" |
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shows "norm (x - x') < e" |
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using dist_triangle_half_l[OF assms[unfolded dist_norm[symmetric]]] |
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unfolding dist_norm[symmetric] . |
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lemma norm_triangle_le: "norm x + norm y \<le> e \<Longrightarrow> norm (x + y) \<le> e" |
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by (rule norm_triangle_ineq [THEN order_trans]) |
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lemma norm_triangle_lt: "norm x + norm y < e \<Longrightarrow> norm (x + y) < e" |
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by (rule norm_triangle_ineq [THEN le_less_trans]) |
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lemma setsum_clauses: |
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shows "setsum f {} = 0" |
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and "finite S \<Longrightarrow> setsum f (insert x S) = (if x \<in> S then setsum f S else f x + setsum f S)" |
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by (auto simp add: insert_absorb) |
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lemma setsum_norm_le: |
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fixes f :: "'a \<Rightarrow> 'b::real_normed_vector" |
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assumes fg: "\<forall>x \<in> S. norm (f x) \<le> g x" |
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shows "norm (setsum f S) \<le> setsum g S" |
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by (rule order_trans [OF norm_setsum setsum_mono]) (simp add: fg) |
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lemma setsum_norm_bound: |
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fixes f :: "'a \<Rightarrow> 'b::real_normed_vector" |
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assumes fS: "finite S" |
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and K: "\<forall>x \<in> S. norm (f x) \<le> K" |
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shows "norm (setsum f S) \<le> of_nat (card S) * K" |
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using setsum_norm_le[OF K] setsum_constant[symmetric] |
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by simp |
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lemma setsum_group: |
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assumes fS: "finite S" and fT: "finite T" and fST: "f ` S \<subseteq> T" |
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shows "setsum (\<lambda>y. setsum g {x. x \<in> S \<and> f x = y}) T = setsum g S" |
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apply (subst setsum_image_gen[OF fS, of g f]) |
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apply (rule setsum_mono_zero_right[OF fT fST]) |
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apply (auto intro: setsum_0') |
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done |
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lemma vector_eq_ldot: "(\<forall>x. x \<bullet> y = x \<bullet> z) \<longleftrightarrow> y = z" |
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proof |
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assume "\<forall>x. x \<bullet> y = x \<bullet> z" |
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then have "\<forall>x. x \<bullet> (y - z) = 0" |
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by (simp add: inner_diff) |
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then have "(y - z) \<bullet> (y - z) = 0" .. |
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then show "y = z" by simp |
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qed simp |
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lemma vector_eq_rdot: "(\<forall>z. x \<bullet> z = y \<bullet> z) \<longleftrightarrow> x = y" |
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proof |
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assume "\<forall>z. x \<bullet> z = y \<bullet> z" |
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then have "\<forall>z. (x - y) \<bullet> z = 0" |
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by (simp add: inner_diff) |
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then have "(x - y) \<bullet> (x - y) = 0" .. |
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then show "x = y" by simp |
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qed simp |
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subsection {* Orthogonality. *} |
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context real_inner |
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begin |
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definition "orthogonal x y \<longleftrightarrow> x \<bullet> y = 0" |
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lemma orthogonal_clauses: |
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"orthogonal a 0" |
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"orthogonal a x \<Longrightarrow> orthogonal a (c *\<^sub>R x)" |
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"orthogonal a x \<Longrightarrow> orthogonal a (- x)" |
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"orthogonal a x \<Longrightarrow> orthogonal a y \<Longrightarrow> orthogonal a (x + y)" |
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"orthogonal a x \<Longrightarrow> orthogonal a y \<Longrightarrow> orthogonal a (x - y)" |
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"orthogonal 0 a" |
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"orthogonal x a \<Longrightarrow> orthogonal (c *\<^sub>R x) a" |
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"orthogonal x a \<Longrightarrow> orthogonal (- x) a" |
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"orthogonal x a \<Longrightarrow> orthogonal y a \<Longrightarrow> orthogonal (x + y) a" |
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"orthogonal x a \<Longrightarrow> orthogonal y a \<Longrightarrow> orthogonal (x - y) a" |
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unfolding orthogonal_def inner_add inner_diff by auto |
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end |
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lemma orthogonal_commute: "orthogonal x y \<longleftrightarrow> orthogonal y x" |
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by (simp add: orthogonal_def inner_commute) |
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subsection {* Linear functions. *} |
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lemma linear_iff: |
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"linear f \<longleftrightarrow> (\<forall>x y. f (x + y) = f x + f y) \<and> (\<forall>c x. f (c *\<^sub>R x) = c *\<^sub>R f x)" |
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(is "linear f \<longleftrightarrow> ?rhs") |
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proof |
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assume "linear f" then interpret f: linear f . |
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show "?rhs" by (simp add: f.add f.scaleR) |
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next |
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assume "?rhs" then show "linear f" by unfold_locales simp_all |
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qed |
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lemma linear_compose_cmul: "linear f \<Longrightarrow> linear (\<lambda>x. c *\<^sub>R f x)" |
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by (simp add: linear_iff algebra_simps) |
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lemma linear_compose_neg: "linear f \<Longrightarrow> linear (\<lambda>x. - f x)" |
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by (simp add: linear_iff) |
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lemma linear_compose_add: "linear f \<Longrightarrow> linear g \<Longrightarrow> linear (\<lambda>x. f x + g x)" |
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by (simp add: linear_iff algebra_simps) |
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lemma linear_compose_sub: "linear f \<Longrightarrow> linear g \<Longrightarrow> linear (\<lambda>x. f x - g x)" |
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by (simp add: linear_iff algebra_simps) |
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lemma linear_compose: "linear f \<Longrightarrow> linear g \<Longrightarrow> linear (g \<circ> f)" |
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by (simp add: linear_iff) |
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lemma linear_id: "linear id" |
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by (simp add: linear_iff id_def) |
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lemma linear_zero: "linear (\<lambda>x. 0)" |
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by (simp add: linear_iff) |
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lemma linear_compose_setsum: |
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assumes fS: "finite S" |
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and lS: "\<forall>a \<in> S. linear (f a)" |
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shows "linear (\<lambda>x. setsum (\<lambda>a. f a x) S)" |
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using lS |
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apply (induct rule: finite_induct[OF fS]) |
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apply (auto simp add: linear_zero intro: linear_compose_add) |
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done |
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lemma linear_0: "linear f \<Longrightarrow> f 0 = 0" |
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unfolding linear_iff |
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apply clarsimp |
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apply (erule allE[where x="0::'a"]) |
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apply simp |
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267 |
done |
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lemma linear_cmul: "linear f \<Longrightarrow> f (c *\<^sub>R x) = c *\<^sub>R f x" |
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by (simp add: linear_iff) |
44133 | 271 |
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lemma linear_neg: "linear f \<Longrightarrow> f (- x) = - f x" |
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using linear_cmul [where c="-1"] by simp |
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lemma linear_add: "linear f \<Longrightarrow> f (x + y) = f x + f y" |
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by (metis linear_iff) |
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lemma linear_sub: "linear f \<Longrightarrow> f (x - y) = f x - f y" |
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using linear_add [of f x "- y"] by (simp add: linear_neg) |
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lemma linear_setsum: |
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assumes lin: "linear f" |
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and fin: "finite S" |
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shows "f (setsum g S) = setsum (f \<circ> g) S" |
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using fin |
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proof induct |
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case empty |
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then show ?case |
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by (simp add: linear_0[OF lin]) |
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next |
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case (insert x F) |
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have "f (setsum g (insert x F)) = f (g x + setsum g F)" |
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using insert.hyps by simp |
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also have "\<dots> = f (g x) + f (setsum g F)" |
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using linear_add[OF lin] by simp |
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also have "\<dots> = setsum (f \<circ> g) (insert x F)" |
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using insert.hyps by simp |
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finally show ?case . |
299 |
qed |
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lemma linear_setsum_mul: |
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53406 | 302 |
assumes lin: "linear f" |
303 |
and fin: "finite S" |
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shows "f (setsum (\<lambda>i. c i *\<^sub>R v i) S) = setsum (\<lambda>i. c i *\<^sub>R f (v i)) S" |
53406 | 305 |
using linear_setsum[OF lin fin, of "\<lambda>i. c i *\<^sub>R v i" , unfolded o_def] linear_cmul[OF lin] |
49522 | 306 |
by simp |
44133 | 307 |
|
308 |
lemma linear_injective_0: |
|
53406 | 309 |
assumes lin: "linear f" |
44133 | 310 |
shows "inj f \<longleftrightarrow> (\<forall>x. f x = 0 \<longrightarrow> x = 0)" |
49663 | 311 |
proof - |
53406 | 312 |
have "inj f \<longleftrightarrow> (\<forall> x y. f x = f y \<longrightarrow> x = y)" |
313 |
by (simp add: inj_on_def) |
|
314 |
also have "\<dots> \<longleftrightarrow> (\<forall> x y. f x - f y = 0 \<longrightarrow> x - y = 0)" |
|
315 |
by simp |
|
44133 | 316 |
also have "\<dots> \<longleftrightarrow> (\<forall> x y. f (x - y) = 0 \<longrightarrow> x - y = 0)" |
53406 | 317 |
by (simp add: linear_sub[OF lin]) |
318 |
also have "\<dots> \<longleftrightarrow> (\<forall> x. f x = 0 \<longrightarrow> x = 0)" |
|
319 |
by auto |
|
44133 | 320 |
finally show ?thesis . |
321 |
qed |
|
322 |
||
49522 | 323 |
|
324 |
subsection {* Bilinear functions. *} |
|
44133 | 325 |
|
53406 | 326 |
definition "bilinear f \<longleftrightarrow> (\<forall>x. linear (\<lambda>y. f x y)) \<and> (\<forall>y. linear (\<lambda>x. f x y))" |
327 |
||
328 |
lemma bilinear_ladd: "bilinear h \<Longrightarrow> h (x + y) z = h x z + h y z" |
|
53600
8fda7ad57466
make 'linear' into a sublocale of 'bounded_linear';
huffman
parents:
53596
diff
changeset
|
329 |
by (simp add: bilinear_def linear_iff) |
49663 | 330 |
|
53406 | 331 |
lemma bilinear_radd: "bilinear h \<Longrightarrow> h x (y + z) = h x y + h x z" |
53600
8fda7ad57466
make 'linear' into a sublocale of 'bounded_linear';
huffman
parents:
53596
diff
changeset
|
332 |
by (simp add: bilinear_def linear_iff) |
44133 | 333 |
|
53406 | 334 |
lemma bilinear_lmul: "bilinear h \<Longrightarrow> h (c *\<^sub>R x) y = c *\<^sub>R h x y" |
53600
8fda7ad57466
make 'linear' into a sublocale of 'bounded_linear';
huffman
parents:
53596
diff
changeset
|
335 |
by (simp add: bilinear_def linear_iff) |
44133 | 336 |
|
53406 | 337 |
lemma bilinear_rmul: "bilinear h \<Longrightarrow> h x (c *\<^sub>R y) = c *\<^sub>R h x y" |
53600
8fda7ad57466
make 'linear' into a sublocale of 'bounded_linear';
huffman
parents:
53596
diff
changeset
|
338 |
by (simp add: bilinear_def linear_iff) |
44133 | 339 |
|
53406 | 340 |
lemma bilinear_lneg: "bilinear h \<Longrightarrow> h (- x) y = - h x y" |
54489
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54413
diff
changeset
|
341 |
by (drule bilinear_lmul [of _ "- 1"]) simp |
44133 | 342 |
|
53406 | 343 |
lemma bilinear_rneg: "bilinear h \<Longrightarrow> h x (- y) = - h x y" |
54489
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54413
diff
changeset
|
344 |
by (drule bilinear_rmul [of _ _ "- 1"]) simp |
44133 | 345 |
|
53406 | 346 |
lemma (in ab_group_add) eq_add_iff: "x = x + y \<longleftrightarrow> y = 0" |
44133 | 347 |
using add_imp_eq[of x y 0] by auto |
348 |
||
53406 | 349 |
lemma bilinear_lzero: |
350 |
assumes "bilinear h" |
|
351 |
shows "h 0 x = 0" |
|
49663 | 352 |
using bilinear_ladd [OF assms, of 0 0 x] by (simp add: eq_add_iff field_simps) |
353 |
||
53406 | 354 |
lemma bilinear_rzero: |
355 |
assumes "bilinear h" |
|
356 |
shows "h x 0 = 0" |
|
49663 | 357 |
using bilinear_radd [OF assms, of x 0 0 ] by (simp add: eq_add_iff field_simps) |
44133 | 358 |
|
53406 | 359 |
lemma bilinear_lsub: "bilinear h \<Longrightarrow> h (x - y) z = h x z - h y z" |
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
53939
diff
changeset
|
360 |
using bilinear_ladd [of h x "- y"] by (simp add: bilinear_lneg) |
44133 | 361 |
|
53406 | 362 |
lemma bilinear_rsub: "bilinear h \<Longrightarrow> h z (x - y) = h z x - h z y" |
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
53939
diff
changeset
|
363 |
using bilinear_radd [of h _ x "- y"] by (simp add: bilinear_rneg) |
44133 | 364 |
|
365 |
lemma bilinear_setsum: |
|
49663 | 366 |
assumes bh: "bilinear h" |
367 |
and fS: "finite S" |
|
368 |
and fT: "finite T" |
|
44133 | 369 |
shows "h (setsum f S) (setsum g T) = setsum (\<lambda>(i,j). h (f i) (g j)) (S \<times> T) " |
49522 | 370 |
proof - |
44133 | 371 |
have "h (setsum f S) (setsum g T) = setsum (\<lambda>x. h (f x) (setsum g T)) S" |
372 |
apply (rule linear_setsum[unfolded o_def]) |
|
53406 | 373 |
using bh fS |
374 |
apply (auto simp add: bilinear_def) |
|
49522 | 375 |
done |
44133 | 376 |
also have "\<dots> = setsum (\<lambda>x. setsum (\<lambda>y. h (f x) (g y)) T) S" |
377 |
apply (rule setsum_cong, simp) |
|
378 |
apply (rule linear_setsum[unfolded o_def]) |
|
49522 | 379 |
using bh fT |
380 |
apply (auto simp add: bilinear_def) |
|
381 |
done |
|
53406 | 382 |
finally show ?thesis |
383 |
unfolding setsum_cartesian_product . |
|
44133 | 384 |
qed |
385 |
||
49522 | 386 |
|
387 |
subsection {* Adjoints. *} |
|
44133 | 388 |
|
389 |
definition "adjoint f = (SOME f'. \<forall>x y. f x \<bullet> y = x \<bullet> f' y)" |
|
390 |
||
391 |
lemma adjoint_unique: |
|
392 |
assumes "\<forall>x y. inner (f x) y = inner x (g y)" |
|
393 |
shows "adjoint f = g" |
|
49522 | 394 |
unfolding adjoint_def |
44133 | 395 |
proof (rule some_equality) |
53406 | 396 |
show "\<forall>x y. inner (f x) y = inner x (g y)" |
397 |
by (rule assms) |
|
44133 | 398 |
next |
53406 | 399 |
fix h |
400 |
assume "\<forall>x y. inner (f x) y = inner x (h y)" |
|
401 |
then have "\<forall>x y. inner x (g y) = inner x (h y)" |
|
402 |
using assms by simp |
|
403 |
then have "\<forall>x y. inner x (g y - h y) = 0" |
|
404 |
by (simp add: inner_diff_right) |
|
405 |
then have "\<forall>y. inner (g y - h y) (g y - h y) = 0" |
|
406 |
by simp |
|
407 |
then have "\<forall>y. h y = g y" |
|
408 |
by simp |
|
49652 | 409 |
then show "h = g" by (simp add: ext) |
44133 | 410 |
qed |
411 |
||
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
412 |
text {* TODO: The following lemmas about adjoints should hold for any |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
413 |
Hilbert space (i.e. complete inner product space). |
54703 | 414 |
(see @{url "http://en.wikipedia.org/wiki/Hermitian_adjoint"}) |
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
415 |
*} |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
416 |
|
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
417 |
lemma adjoint_works: |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
418 |
fixes f:: "'n::euclidean_space \<Rightarrow> 'm::euclidean_space" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
419 |
assumes lf: "linear f" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
420 |
shows "x \<bullet> adjoint f y = f x \<bullet> y" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
421 |
proof - |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
422 |
have "\<forall>y. \<exists>w. \<forall>x. f x \<bullet> y = x \<bullet> w" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
423 |
proof (intro allI exI) |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
424 |
fix y :: "'m" and x |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
425 |
let ?w = "(\<Sum>i\<in>Basis. (f i \<bullet> y) *\<^sub>R i) :: 'n" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
426 |
have "f x \<bullet> y = f (\<Sum>i\<in>Basis. (x \<bullet> i) *\<^sub>R i) \<bullet> y" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
427 |
by (simp add: euclidean_representation) |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
428 |
also have "\<dots> = (\<Sum>i\<in>Basis. (x \<bullet> i) *\<^sub>R f i) \<bullet> y" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
429 |
unfolding linear_setsum[OF lf finite_Basis] |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
430 |
by (simp add: linear_cmul[OF lf]) |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
431 |
finally show "f x \<bullet> y = x \<bullet> ?w" |
53406 | 432 |
by (simp add: inner_setsum_left inner_setsum_right mult_commute) |
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
433 |
qed |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
434 |
then show ?thesis |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
435 |
unfolding adjoint_def choice_iff |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
436 |
by (intro someI2_ex[where Q="\<lambda>f'. x \<bullet> f' y = f x \<bullet> y"]) auto |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
437 |
qed |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
438 |
|
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
439 |
lemma adjoint_clauses: |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
440 |
fixes f:: "'n::euclidean_space \<Rightarrow> 'm::euclidean_space" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
441 |
assumes lf: "linear f" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
442 |
shows "x \<bullet> adjoint f y = f x \<bullet> y" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
443 |
and "adjoint f y \<bullet> x = y \<bullet> f x" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
444 |
by (simp_all add: adjoint_works[OF lf] inner_commute) |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
445 |
|
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
446 |
lemma adjoint_linear: |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
447 |
fixes f:: "'n::euclidean_space \<Rightarrow> 'm::euclidean_space" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
448 |
assumes lf: "linear f" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
449 |
shows "linear (adjoint f)" |
53600
8fda7ad57466
make 'linear' into a sublocale of 'bounded_linear';
huffman
parents:
53596
diff
changeset
|
450 |
by (simp add: lf linear_iff euclidean_eq_iff[where 'a='n] euclidean_eq_iff[where 'a='m] |
53939 | 451 |
adjoint_clauses[OF lf] inner_distrib) |
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
452 |
|
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
453 |
lemma adjoint_adjoint: |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
454 |
fixes f:: "'n::euclidean_space \<Rightarrow> 'm::euclidean_space" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
455 |
assumes lf: "linear f" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
456 |
shows "adjoint (adjoint f) = f" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
457 |
by (rule adjoint_unique, simp add: adjoint_clauses [OF lf]) |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
458 |
|
53406 | 459 |
|
49522 | 460 |
subsection {* Interlude: Some properties of real sets *} |
44133 | 461 |
|
53406 | 462 |
lemma seq_mono_lemma: |
463 |
assumes "\<forall>(n::nat) \<ge> m. (d n :: real) < e n" |
|
464 |
and "\<forall>n \<ge> m. e n \<le> e m" |
|
44133 | 465 |
shows "\<forall>n \<ge> m. d n < e m" |
53406 | 466 |
using assms |
467 |
apply auto |
|
44133 | 468 |
apply (erule_tac x="n" in allE) |
469 |
apply (erule_tac x="n" in allE) |
|
470 |
apply auto |
|
471 |
done |
|
472 |
||
53406 | 473 |
lemma infinite_enumerate: |
474 |
assumes fS: "infinite S" |
|
44133 | 475 |
shows "\<exists>r. subseq r \<and> (\<forall>n. r n \<in> S)" |
49525 | 476 |
unfolding subseq_def |
477 |
using enumerate_in_set[OF fS] enumerate_mono[of _ _ S] fS by auto |
|
44133 | 478 |
|
479 |
lemma approachable_lt_le: "(\<exists>(d::real)>0. \<forall>x. f x < d \<longrightarrow> P x) \<longleftrightarrow> (\<exists>d>0. \<forall>x. f x \<le> d \<longrightarrow> P x)" |
|
49522 | 480 |
apply auto |
481 |
apply (rule_tac x="d/2" in exI) |
|
482 |
apply auto |
|
483 |
done |
|
44133 | 484 |
|
485 |
lemma triangle_lemma: |
|
53406 | 486 |
fixes x y z :: real |
487 |
assumes x: "0 \<le> x" |
|
488 |
and y: "0 \<le> y" |
|
489 |
and z: "0 \<le> z" |
|
490 |
and xy: "x\<^sup>2 \<le> y\<^sup>2 + z\<^sup>2" |
|
491 |
shows "x \<le> y + z" |
|
49522 | 492 |
proof - |
53406 | 493 |
have "y\<^sup>2 + z\<^sup>2 \<le> y\<^sup>2 + 2 *y * z + z\<^sup>2" |
494 |
using z y by (simp add: mult_nonneg_nonneg) |
|
495 |
with xy have th: "x\<^sup>2 \<le> (y + z)\<^sup>2" |
|
496 |
by (simp add: power2_eq_square field_simps) |
|
497 |
from y z have yz: "y + z \<ge> 0" |
|
498 |
by arith |
|
44133 | 499 |
from power2_le_imp_le[OF th yz] show ?thesis . |
500 |
qed |
|
501 |
||
49522 | 502 |
|
44133 | 503 |
subsection {* A generic notion of "hull" (convex, affine, conic hull and closure). *} |
504 |
||
53406 | 505 |
definition hull :: "('a set \<Rightarrow> bool) \<Rightarrow> 'a set \<Rightarrow> 'a set" (infixl "hull" 75) |
506 |
where "S hull s = \<Inter>{t. S t \<and> s \<subseteq> t}" |
|
44170
510ac30f44c0
make Multivariate_Analysis work with separate set type
huffman
parents:
44166
diff
changeset
|
507 |
|
510ac30f44c0
make Multivariate_Analysis work with separate set type
huffman
parents:
44166
diff
changeset
|
508 |
lemma hull_same: "S s \<Longrightarrow> S hull s = s" |
44133 | 509 |
unfolding hull_def by auto |
510 |
||
53406 | 511 |
lemma hull_in: "(\<And>T. Ball T S \<Longrightarrow> S (\<Inter>T)) \<Longrightarrow> S (S hull s)" |
49522 | 512 |
unfolding hull_def Ball_def by auto |
44170
510ac30f44c0
make Multivariate_Analysis work with separate set type
huffman
parents:
44166
diff
changeset
|
513 |
|
53406 | 514 |
lemma hull_eq: "(\<And>T. Ball T S \<Longrightarrow> S (\<Inter>T)) \<Longrightarrow> (S hull s) = s \<longleftrightarrow> S s" |
49522 | 515 |
using hull_same[of S s] hull_in[of S s] by metis |
44133 | 516 |
|
517 |
lemma hull_hull: "S hull (S hull s) = S hull s" |
|
518 |
unfolding hull_def by blast |
|
519 |
||
520 |
lemma hull_subset[intro]: "s \<subseteq> (S hull s)" |
|
521 |
unfolding hull_def by blast |
|
522 |
||
53406 | 523 |
lemma hull_mono: "s \<subseteq> t \<Longrightarrow> (S hull s) \<subseteq> (S hull t)" |
44133 | 524 |
unfolding hull_def by blast |
525 |
||
53406 | 526 |
lemma hull_antimono: "\<forall>x. S x \<longrightarrow> T x \<Longrightarrow> (T hull s) \<subseteq> (S hull s)" |
44133 | 527 |
unfolding hull_def by blast |
528 |
||
53406 | 529 |
lemma hull_minimal: "s \<subseteq> t \<Longrightarrow> S t \<Longrightarrow> (S hull s) \<subseteq> t" |
44133 | 530 |
unfolding hull_def by blast |
531 |
||
53406 | 532 |
lemma subset_hull: "S t \<Longrightarrow> S hull s \<subseteq> t \<longleftrightarrow> s \<subseteq> t" |
44133 | 533 |
unfolding hull_def by blast |
534 |
||
53596 | 535 |
lemma hull_UNIV: "S hull UNIV = UNIV" |
536 |
unfolding hull_def by auto |
|
537 |
||
53406 | 538 |
lemma hull_unique: "s \<subseteq> t \<Longrightarrow> S t \<Longrightarrow> (\<And>t'. s \<subseteq> t' \<Longrightarrow> S t' \<Longrightarrow> t \<subseteq> t') \<Longrightarrow> (S hull s = t)" |
49652 | 539 |
unfolding hull_def by auto |
44133 | 540 |
|
541 |
lemma hull_induct: "(\<And>x. x\<in> S \<Longrightarrow> P x) \<Longrightarrow> Q {x. P x} \<Longrightarrow> \<forall>x\<in> Q hull S. P x" |
|
542 |
using hull_minimal[of S "{x. P x}" Q] |
|
44170
510ac30f44c0
make Multivariate_Analysis work with separate set type
huffman
parents:
44166
diff
changeset
|
543 |
by (auto simp add: subset_eq) |
44133 | 544 |
|
49522 | 545 |
lemma hull_inc: "x \<in> S \<Longrightarrow> x \<in> P hull S" |
546 |
by (metis hull_subset subset_eq) |
|
44133 | 547 |
|
548 |
lemma hull_union_subset: "(S hull s) \<union> (S hull t) \<subseteq> (S hull (s \<union> t))" |
|
49522 | 549 |
unfolding Un_subset_iff by (metis hull_mono Un_upper1 Un_upper2) |
550 |
||
551 |
lemma hull_union: |
|
53406 | 552 |
assumes T: "\<And>T. Ball T S \<Longrightarrow> S (\<Inter>T)" |
44133 | 553 |
shows "S hull (s \<union> t) = S hull (S hull s \<union> S hull t)" |
49522 | 554 |
apply rule |
555 |
apply (rule hull_mono) |
|
556 |
unfolding Un_subset_iff |
|
557 |
apply (metis hull_subset Un_upper1 Un_upper2 subset_trans) |
|
558 |
apply (rule hull_minimal) |
|
559 |
apply (metis hull_union_subset) |
|
560 |
apply (metis hull_in T) |
|
561 |
done |
|
44133 | 562 |
|
563 |
lemma hull_redundant_eq: "a \<in> (S hull s) \<longleftrightarrow> (S hull (insert a s) = S hull s)" |
|
564 |
unfolding hull_def by blast |
|
565 |
||
53406 | 566 |
lemma hull_redundant: "a \<in> (S hull s) \<Longrightarrow> (S hull (insert a s) = S hull s)" |
49522 | 567 |
by (metis hull_redundant_eq) |
568 |
||
44133 | 569 |
|
44666 | 570 |
subsection {* Archimedean properties and useful consequences *} |
44133 | 571 |
|
53406 | 572 |
lemma real_arch_simple: "\<exists>n. x \<le> real (n::nat)" |
44666 | 573 |
unfolding real_of_nat_def by (rule ex_le_of_nat) |
44133 | 574 |
|
575 |
lemma real_arch_inv: "0 < e \<longleftrightarrow> (\<exists>n::nat. n \<noteq> 0 \<and> 0 < inverse (real n) \<and> inverse (real n) < e)" |
|
576 |
using reals_Archimedean |
|
577 |
apply (auto simp add: field_simps) |
|
578 |
apply (subgoal_tac "inverse (real n) > 0") |
|
579 |
apply arith |
|
580 |
apply simp |
|
581 |
done |
|
582 |
||
53406 | 583 |
lemma real_pow_lbound: "0 \<le> x \<Longrightarrow> 1 + real n * x \<le> (1 + x) ^ n" |
49522 | 584 |
proof (induct n) |
585 |
case 0 |
|
586 |
then show ?case by simp |
|
44133 | 587 |
next |
588 |
case (Suc n) |
|
53406 | 589 |
then have h: "1 + real n * x \<le> (1 + x) ^ n" |
590 |
by simp |
|
591 |
from h have p: "1 \<le> (1 + x) ^ n" |
|
592 |
using Suc.prems by simp |
|
593 |
from h have "1 + real n * x + x \<le> (1 + x) ^ n + x" |
|
594 |
by simp |
|
595 |
also have "\<dots> \<le> (1 + x) ^ Suc n" |
|
596 |
apply (subst diff_le_0_iff_le[symmetric]) |
|
44133 | 597 |
apply (simp add: field_simps) |
53406 | 598 |
using mult_left_mono[OF p Suc.prems] |
599 |
apply simp |
|
49522 | 600 |
done |
53406 | 601 |
finally show ?case |
602 |
by (simp add: real_of_nat_Suc field_simps) |
|
44133 | 603 |
qed |
604 |
||
53406 | 605 |
lemma real_arch_pow: |
606 |
fixes x :: real |
|
607 |
assumes x: "1 < x" |
|
608 |
shows "\<exists>n. y < x^n" |
|
49522 | 609 |
proof - |
53406 | 610 |
from x have x0: "x - 1 > 0" |
611 |
by arith |
|
44666 | 612 |
from reals_Archimedean3[OF x0, rule_format, of y] |
53406 | 613 |
obtain n :: nat where n: "y < real n * (x - 1)" by metis |
44133 | 614 |
from x0 have x00: "x- 1 \<ge> 0" by arith |
615 |
from real_pow_lbound[OF x00, of n] n |
|
616 |
have "y < x^n" by auto |
|
617 |
then show ?thesis by metis |
|
618 |
qed |
|
619 |
||
53406 | 620 |
lemma real_arch_pow2: |
621 |
fixes x :: real |
|
622 |
shows "\<exists>n. x < 2^ n" |
|
44133 | 623 |
using real_arch_pow[of 2 x] by simp |
624 |
||
49522 | 625 |
lemma real_arch_pow_inv: |
53406 | 626 |
fixes x y :: real |
627 |
assumes y: "y > 0" |
|
628 |
and x1: "x < 1" |
|
44133 | 629 |
shows "\<exists>n. x^n < y" |
53406 | 630 |
proof (cases "x > 0") |
631 |
case True |
|
632 |
with x1 have ix: "1 < 1/x" by (simp add: field_simps) |
|
633 |
from real_arch_pow[OF ix, of "1/y"] |
|
634 |
obtain n where n: "1/y < (1/x)^n" by blast |
|
635 |
then show ?thesis using y `x > 0` |
|
636 |
by (auto simp add: field_simps power_divide) |
|
637 |
next |
|
638 |
case False |
|
639 |
with y x1 show ?thesis |
|
640 |
apply auto |
|
641 |
apply (rule exI[where x=1]) |
|
642 |
apply auto |
|
643 |
done |
|
44133 | 644 |
qed |
645 |
||
49522 | 646 |
lemma forall_pos_mono: |
53406 | 647 |
"(\<And>d e::real. d < e \<Longrightarrow> P d \<Longrightarrow> P e) \<Longrightarrow> |
648 |
(\<And>n::nat. n \<noteq> 0 \<Longrightarrow> P (inverse (real n))) \<Longrightarrow> (\<And>e. 0 < e \<Longrightarrow> P e)" |
|
44133 | 649 |
by (metis real_arch_inv) |
650 |
||
49522 | 651 |
lemma forall_pos_mono_1: |
53406 | 652 |
"(\<And>d e::real. d < e \<Longrightarrow> P d \<Longrightarrow> P e) \<Longrightarrow> |
53716 | 653 |
(\<And>n. P (inverse (real (Suc n)))) \<Longrightarrow> 0 < e \<Longrightarrow> P e" |
44133 | 654 |
apply (rule forall_pos_mono) |
655 |
apply auto |
|
656 |
apply (atomize) |
|
657 |
apply (erule_tac x="n - 1" in allE) |
|
658 |
apply auto |
|
659 |
done |
|
660 |
||
49522 | 661 |
lemma real_archimedian_rdiv_eq_0: |
53406 | 662 |
assumes x0: "x \<ge> 0" |
663 |
and c: "c \<ge> 0" |
|
664 |
and xc: "\<forall>(m::nat)>0. real m * x \<le> c" |
|
44133 | 665 |
shows "x = 0" |
53406 | 666 |
proof (rule ccontr) |
667 |
assume "x \<noteq> 0" |
|
668 |
with x0 have xp: "x > 0" by arith |
|
669 |
from reals_Archimedean3[OF xp, rule_format, of c] |
|
670 |
obtain n :: nat where n: "c < real n * x" |
|
671 |
by blast |
|
672 |
with xc[rule_format, of n] have "n = 0" |
|
673 |
by arith |
|
674 |
with n c show False |
|
675 |
by simp |
|
44133 | 676 |
qed |
677 |
||
49522 | 678 |
|
44133 | 679 |
subsection{* A bit of linear algebra. *} |
680 |
||
49522 | 681 |
definition (in real_vector) subspace :: "'a set \<Rightarrow> bool" |
682 |
where "subspace S \<longleftrightarrow> 0 \<in> S \<and> (\<forall>x\<in> S. \<forall>y \<in>S. x + y \<in> S) \<and> (\<forall>c. \<forall>x \<in>S. c *\<^sub>R x \<in>S )" |
|
44133 | 683 |
|
684 |
definition (in real_vector) "span S = (subspace hull S)" |
|
53716 | 685 |
definition (in real_vector) "dependent S \<longleftrightarrow> (\<exists>a \<in> S. a \<in> span (S - {a}))" |
53406 | 686 |
abbreviation (in real_vector) "independent s \<equiv> \<not> dependent s" |
44133 | 687 |
|
688 |
text {* Closure properties of subspaces. *} |
|
689 |
||
53406 | 690 |
lemma subspace_UNIV[simp]: "subspace UNIV" |
691 |
by (simp add: subspace_def) |
|
692 |
||
693 |
lemma (in real_vector) subspace_0: "subspace S \<Longrightarrow> 0 \<in> S" |
|
694 |
by (metis subspace_def) |
|
695 |
||
696 |
lemma (in real_vector) subspace_add: "subspace S \<Longrightarrow> x \<in> S \<Longrightarrow> y \<in> S \<Longrightarrow> x + y \<in> S" |
|
44133 | 697 |
by (metis subspace_def) |
698 |
||
699 |
lemma (in real_vector) subspace_mul: "subspace S \<Longrightarrow> x \<in> S \<Longrightarrow> c *\<^sub>R x \<in> S" |
|
700 |
by (metis subspace_def) |
|
701 |
||
702 |
lemma subspace_neg: "subspace S \<Longrightarrow> x \<in> S \<Longrightarrow> - x \<in> S" |
|
703 |
by (metis scaleR_minus1_left subspace_mul) |
|
704 |
||
705 |
lemma subspace_sub: "subspace S \<Longrightarrow> x \<in> S \<Longrightarrow> y \<in> S \<Longrightarrow> x - y \<in> S" |
|
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
53939
diff
changeset
|
706 |
using subspace_add [of S x "- y"] by (simp add: subspace_neg) |
44133 | 707 |
|
708 |
lemma (in real_vector) subspace_setsum: |
|
53406 | 709 |
assumes sA: "subspace A" |
710 |
and fB: "finite B" |
|
49522 | 711 |
and f: "\<forall>x\<in> B. f x \<in> A" |
44133 | 712 |
shows "setsum f B \<in> A" |
713 |
using fB f sA |
|
49522 | 714 |
by (induct rule: finite_induct[OF fB]) |
715 |
(simp add: subspace_def sA, auto simp add: sA subspace_add) |
|
44133 | 716 |
|
717 |
lemma subspace_linear_image: |
|
53406 | 718 |
assumes lf: "linear f" |
719 |
and sS: "subspace S" |
|
720 |
shows "subspace (f ` S)" |
|
44133 | 721 |
using lf sS linear_0[OF lf] |
53600
8fda7ad57466
make 'linear' into a sublocale of 'bounded_linear';
huffman
parents:
53596
diff
changeset
|
722 |
unfolding linear_iff subspace_def |
44133 | 723 |
apply (auto simp add: image_iff) |
53406 | 724 |
apply (rule_tac x="x + y" in bexI) |
725 |
apply auto |
|
726 |
apply (rule_tac x="c *\<^sub>R x" in bexI) |
|
727 |
apply auto |
|
44133 | 728 |
done |
729 |
||
44521 | 730 |
lemma subspace_linear_vimage: "linear f \<Longrightarrow> subspace S \<Longrightarrow> subspace (f -` S)" |
53600
8fda7ad57466
make 'linear' into a sublocale of 'bounded_linear';
huffman
parents:
53596
diff
changeset
|
731 |
by (auto simp add: subspace_def linear_iff linear_0[of f]) |
44521 | 732 |
|
53406 | 733 |
lemma subspace_linear_preimage: "linear f \<Longrightarrow> subspace S \<Longrightarrow> subspace {x. f x \<in> S}" |
53600
8fda7ad57466
make 'linear' into a sublocale of 'bounded_linear';
huffman
parents:
53596
diff
changeset
|
734 |
by (auto simp add: subspace_def linear_iff linear_0[of f]) |
44133 | 735 |
|
736 |
lemma subspace_trivial: "subspace {0}" |
|
737 |
by (simp add: subspace_def) |
|
738 |
||
53406 | 739 |
lemma (in real_vector) subspace_inter: "subspace A \<Longrightarrow> subspace B \<Longrightarrow> subspace (A \<inter> B)" |
44133 | 740 |
by (simp add: subspace_def) |
741 |
||
53406 | 742 |
lemma subspace_Times: "subspace A \<Longrightarrow> subspace B \<Longrightarrow> subspace (A \<times> B)" |
44521 | 743 |
unfolding subspace_def zero_prod_def by simp |
744 |
||
745 |
text {* Properties of span. *} |
|
746 |
||
53406 | 747 |
lemma (in real_vector) span_mono: "A \<subseteq> B \<Longrightarrow> span A \<subseteq> span B" |
44133 | 748 |
by (metis span_def hull_mono) |
749 |
||
53406 | 750 |
lemma (in real_vector) subspace_span: "subspace (span S)" |
44133 | 751 |
unfolding span_def |
44170
510ac30f44c0
make Multivariate_Analysis work with separate set type
huffman
parents:
44166
diff
changeset
|
752 |
apply (rule hull_in) |
44133 | 753 |
apply (simp only: subspace_def Inter_iff Int_iff subset_eq) |
754 |
apply auto |
|
755 |
done |
|
756 |
||
757 |
lemma (in real_vector) span_clauses: |
|
53406 | 758 |
"a \<in> S \<Longrightarrow> a \<in> span S" |
44133 | 759 |
"0 \<in> span S" |
53406 | 760 |
"x\<in> span S \<Longrightarrow> y \<in> span S \<Longrightarrow> x + y \<in> span S" |
44133 | 761 |
"x \<in> span S \<Longrightarrow> c *\<^sub>R x \<in> span S" |
53406 | 762 |
by (metis span_def hull_subset subset_eq) (metis subspace_span subspace_def)+ |
44133 | 763 |
|
44521 | 764 |
lemma span_unique: |
49522 | 765 |
"S \<subseteq> T \<Longrightarrow> subspace T \<Longrightarrow> (\<And>T'. S \<subseteq> T' \<Longrightarrow> subspace T' \<Longrightarrow> T \<subseteq> T') \<Longrightarrow> span S = T" |
44521 | 766 |
unfolding span_def by (rule hull_unique) |
767 |
||
768 |
lemma span_minimal: "S \<subseteq> T \<Longrightarrow> subspace T \<Longrightarrow> span S \<subseteq> T" |
|
769 |
unfolding span_def by (rule hull_minimal) |
|
770 |
||
771 |
lemma (in real_vector) span_induct: |
|
49522 | 772 |
assumes x: "x \<in> span S" |
773 |
and P: "subspace P" |
|
53406 | 774 |
and SP: "\<And>x. x \<in> S \<Longrightarrow> x \<in> P" |
44521 | 775 |
shows "x \<in> P" |
49522 | 776 |
proof - |
53406 | 777 |
from SP have SP': "S \<subseteq> P" |
778 |
by (simp add: subset_eq) |
|
44170
510ac30f44c0
make Multivariate_Analysis work with separate set type
huffman
parents:
44166
diff
changeset
|
779 |
from x hull_minimal[where S=subspace, OF SP' P, unfolded span_def[symmetric]] |
53406 | 780 |
show "x \<in> P" |
781 |
by (metis subset_eq) |
|
44133 | 782 |
qed |
783 |
||
784 |
lemma span_empty[simp]: "span {} = {0}" |
|
785 |
apply (simp add: span_def) |
|
786 |
apply (rule hull_unique) |
|
44170
510ac30f44c0
make Multivariate_Analysis work with separate set type
huffman
parents:
44166
diff
changeset
|
787 |
apply (auto simp add: subspace_def) |
44133 | 788 |
done |
789 |
||
790 |
lemma (in real_vector) independent_empty[intro]: "independent {}" |
|
791 |
by (simp add: dependent_def) |
|
792 |
||
49522 | 793 |
lemma dependent_single[simp]: "dependent {x} \<longleftrightarrow> x = 0" |
44133 | 794 |
unfolding dependent_def by auto |
795 |
||
53406 | 796 |
lemma (in real_vector) independent_mono: "independent A \<Longrightarrow> B \<subseteq> A \<Longrightarrow> independent B" |
44133 | 797 |
apply (clarsimp simp add: dependent_def span_mono) |
798 |
apply (subgoal_tac "span (B - {a}) \<le> span (A - {a})") |
|
799 |
apply force |
|
800 |
apply (rule span_mono) |
|
801 |
apply auto |
|
802 |
done |
|
803 |
||
804 |
lemma (in real_vector) span_subspace: "A \<subseteq> B \<Longrightarrow> B \<le> span A \<Longrightarrow> subspace B \<Longrightarrow> span A = B" |
|
44170
510ac30f44c0
make Multivariate_Analysis work with separate set type
huffman
parents:
44166
diff
changeset
|
805 |
by (metis order_antisym span_def hull_minimal) |
44133 | 806 |
|
49711 | 807 |
lemma (in real_vector) span_induct': |
808 |
assumes SP: "\<forall>x \<in> S. P x" |
|
809 |
and P: "subspace {x. P x}" |
|
810 |
shows "\<forall>x \<in> span S. P x" |
|
44133 | 811 |
using span_induct SP P by blast |
812 |
||
44170
510ac30f44c0
make Multivariate_Analysis work with separate set type
huffman
parents:
44166
diff
changeset
|
813 |
inductive_set (in real_vector) span_induct_alt_help for S:: "'a set" |
53406 | 814 |
where |
44170
510ac30f44c0
make Multivariate_Analysis work with separate set type
huffman
parents:
44166
diff
changeset
|
815 |
span_induct_alt_help_0: "0 \<in> span_induct_alt_help S" |
49522 | 816 |
| span_induct_alt_help_S: |
53406 | 817 |
"x \<in> S \<Longrightarrow> z \<in> span_induct_alt_help S \<Longrightarrow> |
818 |
(c *\<^sub>R x + z) \<in> span_induct_alt_help S" |
|
44133 | 819 |
|
820 |
lemma span_induct_alt': |
|
53406 | 821 |
assumes h0: "h 0" |
822 |
and hS: "\<And>c x y. x \<in> S \<Longrightarrow> h y \<Longrightarrow> h (c *\<^sub>R x + y)" |
|
49522 | 823 |
shows "\<forall>x \<in> span S. h x" |
824 |
proof - |
|
53406 | 825 |
{ |
826 |
fix x :: 'a |
|
827 |
assume x: "x \<in> span_induct_alt_help S" |
|
44133 | 828 |
have "h x" |
829 |
apply (rule span_induct_alt_help.induct[OF x]) |
|
830 |
apply (rule h0) |
|
53406 | 831 |
apply (rule hS) |
832 |
apply assumption |
|
833 |
apply assumption |
|
834 |
done |
|
835 |
} |
|
44133 | 836 |
note th0 = this |
53406 | 837 |
{ |
838 |
fix x |
|
839 |
assume x: "x \<in> span S" |
|
44170
510ac30f44c0
make Multivariate_Analysis work with separate set type
huffman
parents:
44166
diff
changeset
|
840 |
have "x \<in> span_induct_alt_help S" |
49522 | 841 |
proof (rule span_induct[where x=x and S=S]) |
53406 | 842 |
show "x \<in> span S" by (rule x) |
49522 | 843 |
next |
53406 | 844 |
fix x |
845 |
assume xS: "x \<in> S" |
|
846 |
from span_induct_alt_help_S[OF xS span_induct_alt_help_0, of 1] |
|
847 |
show "x \<in> span_induct_alt_help S" |
|
848 |
by simp |
|
49522 | 849 |
next |
850 |
have "0 \<in> span_induct_alt_help S" by (rule span_induct_alt_help_0) |
|
851 |
moreover |
|
53406 | 852 |
{ |
853 |
fix x y |
|
49522 | 854 |
assume h: "x \<in> span_induct_alt_help S" "y \<in> span_induct_alt_help S" |
855 |
from h have "(x + y) \<in> span_induct_alt_help S" |
|
856 |
apply (induct rule: span_induct_alt_help.induct) |
|
857 |
apply simp |
|
858 |
unfolding add_assoc |
|
859 |
apply (rule span_induct_alt_help_S) |
|
860 |
apply assumption |
|
861 |
apply simp |
|
53406 | 862 |
done |
863 |
} |
|
49522 | 864 |
moreover |
53406 | 865 |
{ |
866 |
fix c x |
|
49522 | 867 |
assume xt: "x \<in> span_induct_alt_help S" |
868 |
then have "(c *\<^sub>R x) \<in> span_induct_alt_help S" |
|
869 |
apply (induct rule: span_induct_alt_help.induct) |
|
870 |
apply (simp add: span_induct_alt_help_0) |
|
871 |
apply (simp add: scaleR_right_distrib) |
|
872 |
apply (rule span_induct_alt_help_S) |
|
873 |
apply assumption |
|
874 |
apply simp |
|
875 |
done } |
|
53406 | 876 |
ultimately show "subspace (span_induct_alt_help S)" |
49522 | 877 |
unfolding subspace_def Ball_def by blast |
53406 | 878 |
qed |
879 |
} |
|
44133 | 880 |
with th0 show ?thesis by blast |
881 |
qed |
|
882 |
||
883 |
lemma span_induct_alt: |
|
53406 | 884 |
assumes h0: "h 0" |
885 |
and hS: "\<And>c x y. x \<in> S \<Longrightarrow> h y \<Longrightarrow> h (c *\<^sub>R x + y)" |
|
886 |
and x: "x \<in> span S" |
|
44133 | 887 |
shows "h x" |
49522 | 888 |
using span_induct_alt'[of h S] h0 hS x by blast |
44133 | 889 |
|
890 |
text {* Individual closure properties. *} |
|
891 |
||
892 |
lemma span_span: "span (span A) = span A" |
|
893 |
unfolding span_def hull_hull .. |
|
894 |
||
53406 | 895 |
lemma (in real_vector) span_superset: "x \<in> S \<Longrightarrow> x \<in> span S" |
896 |
by (metis span_clauses(1)) |
|
897 |
||
898 |
lemma (in real_vector) span_0: "0 \<in> span S" |
|
899 |
by (metis subspace_span subspace_0) |
|
44133 | 900 |
|
901 |
lemma span_inc: "S \<subseteq> span S" |
|
902 |
by (metis subset_eq span_superset) |
|
903 |
||
53406 | 904 |
lemma (in real_vector) dependent_0: |
905 |
assumes "0 \<in> A" |
|
906 |
shows "dependent A" |
|
907 |
unfolding dependent_def |
|
908 |
apply (rule_tac x=0 in bexI) |
|
909 |
using assms span_0 |
|
910 |
apply auto |
|
911 |
done |
|
912 |
||
913 |
lemma (in real_vector) span_add: "x \<in> span S \<Longrightarrow> y \<in> span S \<Longrightarrow> x + y \<in> span S" |
|
44133 | 914 |
by (metis subspace_add subspace_span) |
915 |
||
53406 | 916 |
lemma (in real_vector) span_mul: "x \<in> span S \<Longrightarrow> c *\<^sub>R x \<in> span S" |
44133 | 917 |
by (metis subspace_span subspace_mul) |
918 |
||
53406 | 919 |
lemma span_neg: "x \<in> span S \<Longrightarrow> - x \<in> span S" |
44133 | 920 |
by (metis subspace_neg subspace_span) |
921 |
||
53406 | 922 |
lemma span_sub: "x \<in> span S \<Longrightarrow> y \<in> span S \<Longrightarrow> x - y \<in> span S" |
44133 | 923 |
by (metis subspace_span subspace_sub) |
924 |
||
53406 | 925 |
lemma (in real_vector) span_setsum: "finite A \<Longrightarrow> \<forall>x \<in> A. f x \<in> span S \<Longrightarrow> setsum f A \<in> span S" |
44133 | 926 |
by (rule subspace_setsum, rule subspace_span) |
927 |
||
928 |
lemma span_add_eq: "x \<in> span S \<Longrightarrow> x + y \<in> span S \<longleftrightarrow> y \<in> span S" |
|
929 |
apply (auto simp only: span_add span_sub) |
|
53406 | 930 |
apply (subgoal_tac "(x + y) - x \<in> span S") |
931 |
apply simp |
|
49522 | 932 |
apply (simp only: span_add span_sub) |
933 |
done |
|
44133 | 934 |
|
935 |
text {* Mapping under linear image. *} |
|
936 |
||
44521 | 937 |
lemma image_subset_iff_subset_vimage: "f ` A \<subseteq> B \<longleftrightarrow> A \<subseteq> f -` B" |
938 |
by auto (* TODO: move *) |
|
939 |
||
940 |
lemma span_linear_image: |
|
941 |
assumes lf: "linear f" |
|
44133 | 942 |
shows "span (f ` S) = f ` (span S)" |
44521 | 943 |
proof (rule span_unique) |
944 |
show "f ` S \<subseteq> f ` span S" |
|
945 |
by (intro image_mono span_inc) |
|
946 |
show "subspace (f ` span S)" |
|
947 |
using lf subspace_span by (rule subspace_linear_image) |
|
948 |
next |
|
53406 | 949 |
fix T |
950 |
assume "f ` S \<subseteq> T" and "subspace T" |
|
49522 | 951 |
then show "f ` span S \<subseteq> T" |
44521 | 952 |
unfolding image_subset_iff_subset_vimage |
953 |
by (intro span_minimal subspace_linear_vimage lf) |
|
954 |
qed |
|
955 |
||
956 |
lemma span_union: "span (A \<union> B) = (\<lambda>(a, b). a + b) ` (span A \<times> span B)" |
|
957 |
proof (rule span_unique) |
|
958 |
show "A \<union> B \<subseteq> (\<lambda>(a, b). a + b) ` (span A \<times> span B)" |
|
959 |
by safe (force intro: span_clauses)+ |
|
960 |
next |
|
961 |
have "linear (\<lambda>(a, b). a + b)" |
|
53600
8fda7ad57466
make 'linear' into a sublocale of 'bounded_linear';
huffman
parents:
53596
diff
changeset
|
962 |
by (simp add: linear_iff scaleR_add_right) |
44521 | 963 |
moreover have "subspace (span A \<times> span B)" |
964 |
by (intro subspace_Times subspace_span) |
|
965 |
ultimately show "subspace ((\<lambda>(a, b). a + b) ` (span A \<times> span B))" |
|
966 |
by (rule subspace_linear_image) |
|
967 |
next |
|
49711 | 968 |
fix T |
969 |
assume "A \<union> B \<subseteq> T" and "subspace T" |
|
49522 | 970 |
then show "(\<lambda>(a, b). a + b) ` (span A \<times> span B) \<subseteq> T" |
44521 | 971 |
by (auto intro!: subspace_add elim: span_induct) |
44133 | 972 |
qed |
973 |
||
974 |
text {* The key breakdown property. *} |
|
975 |
||
44521 | 976 |
lemma span_singleton: "span {x} = range (\<lambda>k. k *\<^sub>R x)" |
977 |
proof (rule span_unique) |
|
978 |
show "{x} \<subseteq> range (\<lambda>k. k *\<^sub>R x)" |
|
979 |
by (fast intro: scaleR_one [symmetric]) |
|
980 |
show "subspace (range (\<lambda>k. k *\<^sub>R x))" |
|
981 |
unfolding subspace_def |
|
982 |
by (auto intro: scaleR_add_left [symmetric]) |
|
53406 | 983 |
next |
984 |
fix T |
|
985 |
assume "{x} \<subseteq> T" and "subspace T" |
|
986 |
then show "range (\<lambda>k. k *\<^sub>R x) \<subseteq> T" |
|
44521 | 987 |
unfolding subspace_def by auto |
988 |
qed |
|
989 |
||
49522 | 990 |
lemma span_insert: "span (insert a S) = {x. \<exists>k. (x - k *\<^sub>R a) \<in> span S}" |
44521 | 991 |
proof - |
992 |
have "span ({a} \<union> S) = {x. \<exists>k. (x - k *\<^sub>R a) \<in> span S}" |
|
993 |
unfolding span_union span_singleton |
|
994 |
apply safe |
|
995 |
apply (rule_tac x=k in exI, simp) |
|
996 |
apply (erule rev_image_eqI [OF SigmaI [OF rangeI]]) |
|
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
53939
diff
changeset
|
997 |
apply auto |
44521 | 998 |
done |
49522 | 999 |
then show ?thesis by simp |
44521 | 1000 |
qed |
1001 |
||
44133 | 1002 |
lemma span_breakdown: |
53406 | 1003 |
assumes bS: "b \<in> S" |
1004 |
and aS: "a \<in> span S" |
|
44521 | 1005 |
shows "\<exists>k. a - k *\<^sub>R b \<in> span (S - {b})" |
1006 |
using assms span_insert [of b "S - {b}"] |
|
1007 |
by (simp add: insert_absorb) |
|
44133 | 1008 |
|
53406 | 1009 |
lemma span_breakdown_eq: "x \<in> span (insert a S) \<longleftrightarrow> (\<exists>k. x - k *\<^sub>R a \<in> span S)" |
44521 | 1010 |
by (simp add: span_insert) |
44133 | 1011 |
|
1012 |
text {* Hence some "reversal" results. *} |
|
1013 |
||
1014 |
lemma in_span_insert: |
|
49711 | 1015 |
assumes a: "a \<in> span (insert b S)" |
1016 |
and na: "a \<notin> span S" |
|
44133 | 1017 |
shows "b \<in> span (insert a S)" |
49663 | 1018 |
proof - |
44133 | 1019 |
from span_breakdown[of b "insert b S" a, OF insertI1 a] |
1020 |
obtain k where k: "a - k*\<^sub>R b \<in> span (S - {b})" by auto |
|
53406 | 1021 |
show ?thesis |
1022 |
proof (cases "k = 0") |
|
1023 |
case True |
|
44133 | 1024 |
with k have "a \<in> span S" |
1025 |
apply (simp) |
|
1026 |
apply (rule set_rev_mp) |
|
1027 |
apply assumption |
|
1028 |
apply (rule span_mono) |
|
1029 |
apply blast |
|
1030 |
done |
|
53406 | 1031 |
with na show ?thesis by blast |
1032 |
next |
|
1033 |
case False |
|
44133 | 1034 |
have eq: "b = (1/k) *\<^sub>R a - ((1/k) *\<^sub>R a - b)" by simp |
53406 | 1035 |
from False have eq': "(1/k) *\<^sub>R (a - k*\<^sub>R b) = (1/k) *\<^sub>R a - b" |
44133 | 1036 |
by (simp add: algebra_simps) |
1037 |
from k have "(1/k) *\<^sub>R (a - k*\<^sub>R b) \<in> span (S - {b})" |
|
1038 |
by (rule span_mul) |
|
49652 | 1039 |
then have th: "(1/k) *\<^sub>R a - b \<in> span (S - {b})" |
44133 | 1040 |
unfolding eq' . |
53406 | 1041 |
from k show ?thesis |
44133 | 1042 |
apply (subst eq) |
1043 |
apply (rule span_sub) |
|
1044 |
apply (rule span_mul) |
|
1045 |
apply (rule span_superset) |
|
1046 |
apply blast |
|
1047 |
apply (rule set_rev_mp) |
|
1048 |
apply (rule th) |
|
1049 |
apply (rule span_mono) |
|
53406 | 1050 |
using na |
1051 |
apply blast |
|
1052 |
done |
|
1053 |
qed |
|
44133 | 1054 |
qed |
1055 |
||
1056 |
lemma in_span_delete: |
|
1057 |
assumes a: "a \<in> span S" |
|
53716 | 1058 |
and na: "a \<notin> span (S - {b})" |
44133 | 1059 |
shows "b \<in> span (insert a (S - {b}))" |
1060 |
apply (rule in_span_insert) |
|
1061 |
apply (rule set_rev_mp) |
|
1062 |
apply (rule a) |
|
1063 |
apply (rule span_mono) |
|
1064 |
apply blast |
|
1065 |
apply (rule na) |
|
1066 |
done |
|
1067 |
||
1068 |
text {* Transitivity property. *} |
|
1069 |
||
44521 | 1070 |
lemma span_redundant: "x \<in> span S \<Longrightarrow> span (insert x S) = span S" |
1071 |
unfolding span_def by (rule hull_redundant) |
|
1072 |
||
44133 | 1073 |
lemma span_trans: |
53406 | 1074 |
assumes x: "x \<in> span S" |
1075 |
and y: "y \<in> span (insert x S)" |
|
44133 | 1076 |
shows "y \<in> span S" |
44521 | 1077 |
using assms by (simp only: span_redundant) |
44133 | 1078 |
|
1079 |
lemma span_insert_0[simp]: "span (insert 0 S) = span S" |
|
44521 | 1080 |
by (simp only: span_redundant span_0) |
44133 | 1081 |
|
1082 |
text {* An explicit expansion is sometimes needed. *} |
|
1083 |
||
1084 |
lemma span_explicit: |
|
1085 |
"span P = {y. \<exists>S u. finite S \<and> S \<subseteq> P \<and> setsum (\<lambda>v. u v *\<^sub>R v) S = y}" |
|
1086 |
(is "_ = ?E" is "_ = {y. ?h y}" is "_ = {y. \<exists>S u. ?Q S u y}") |
|
49663 | 1087 |
proof - |
53406 | 1088 |
{ |
1089 |
fix x |
|
1090 |
assume x: "x \<in> ?E" |
|
44133 | 1091 |
then obtain S u where fS: "finite S" and SP: "S\<subseteq>P" and u: "setsum (\<lambda>v. u v *\<^sub>R v) S = x" |
1092 |
by blast |
|
1093 |
have "x \<in> span P" |
|
1094 |
unfolding u[symmetric] |
|
1095 |
apply (rule span_setsum[OF fS]) |
|
1096 |
using span_mono[OF SP] |
|
49522 | 1097 |
apply (auto intro: span_superset span_mul) |
53406 | 1098 |
done |
1099 |
} |
|
44133 | 1100 |
moreover |
1101 |
have "\<forall>x \<in> span P. x \<in> ?E" |
|
49522 | 1102 |
proof (rule span_induct_alt') |
44170
510ac30f44c0
make Multivariate_Analysis work with separate set type
huffman
parents:
44166
diff
changeset
|
1103 |
show "0 \<in> Collect ?h" |
510ac30f44c0
make Multivariate_Analysis work with separate set type
huffman
parents:
44166
diff
changeset
|
1104 |
unfolding mem_Collect_eq |
49522 | 1105 |
apply (rule exI[where x="{}"]) |
1106 |
apply simp |
|
1107 |
done |
|
44133 | 1108 |
next |
1109 |
fix c x y |
|
53406 | 1110 |
assume x: "x \<in> P" |
1111 |
assume hy: "y \<in> Collect ?h" |
|
44133 | 1112 |
from hy obtain S u where fS: "finite S" and SP: "S\<subseteq>P" |
1113 |
and u: "setsum (\<lambda>v. u v *\<^sub>R v) S = y" by blast |
|
1114 |
let ?S = "insert x S" |
|
49522 | 1115 |
let ?u = "\<lambda>y. if y = x then (if x \<in> S then u y + c else c) else u y" |
53406 | 1116 |
from fS SP x have th0: "finite (insert x S)" "insert x S \<subseteq> P" |
1117 |
by blast+ |
|
1118 |
have "?Q ?S ?u (c*\<^sub>R x + y)" |
|
1119 |
proof cases |
|
1120 |
assume xS: "x \<in> S" |
|
44133 | 1121 |
have S1: "S = (S - {x}) \<union> {x}" |
53716 | 1122 |
and Sss:"finite (S - {x})" "finite {x}" "(S - {x}) \<inter> {x} = {}" |
53406 | 1123 |
using xS fS by auto |
44133 | 1124 |
have "setsum (\<lambda>v. ?u v *\<^sub>R v) ?S =(\<Sum>v\<in>S - {x}. u v *\<^sub>R v) + (u x + c) *\<^sub>R x" |
1125 |
using xS |
|
1126 |
by (simp add: setsum_Un_disjoint[OF Sss, unfolded S1[symmetric]] |
|
1127 |
setsum_clauses(2)[OF fS] cong del: if_weak_cong) |
|
1128 |
also have "\<dots> = (\<Sum>v\<in>S. u v *\<^sub>R v) + c *\<^sub>R x" |
|
1129 |
apply (simp add: setsum_Un_disjoint[OF Sss, unfolded S1[symmetric]]) |
|
49522 | 1130 |
apply (simp add: algebra_simps) |
1131 |
done |
|
44133 | 1132 |
also have "\<dots> = c*\<^sub>R x + y" |
1133 |
by (simp add: add_commute u) |
|
1134 |
finally have "setsum (\<lambda>v. ?u v *\<^sub>R v) ?S = c*\<^sub>R x + y" . |
|
53406 | 1135 |
then show ?thesis using th0 by blast |
1136 |
next |
|
1137 |
assume xS: "x \<notin> S" |
|
49522 | 1138 |
have th00: "(\<Sum>v\<in>S. (if v = x then c else u v) *\<^sub>R v) = y" |
1139 |
unfolding u[symmetric] |
|
1140 |
apply (rule setsum_cong2) |
|
53406 | 1141 |
using xS |
1142 |
apply auto |
|
49522 | 1143 |
done |
53406 | 1144 |
show ?thesis using fS xS th0 |
1145 |
by (simp add: th00 setsum_clauses add_commute cong del: if_weak_cong) |
|
1146 |
qed |
|
44170
510ac30f44c0
make Multivariate_Analysis work with separate set type
huffman
parents:
44166
diff
changeset
|
1147 |
then show "(c*\<^sub>R x + y) \<in> Collect ?h" |
510ac30f44c0
make Multivariate_Analysis work with separate set type
huffman
parents:
44166
diff
changeset
|
1148 |
unfolding mem_Collect_eq |
44133 | 1149 |
apply - |
1150 |
apply (rule exI[where x="?S"]) |
|
49522 | 1151 |
apply (rule exI[where x="?u"]) |
1152 |
apply metis |
|
1153 |
done |
|
44133 | 1154 |
qed |
1155 |
ultimately show ?thesis by blast |
|
1156 |
qed |
|
1157 |
||
1158 |
lemma dependent_explicit: |
|
49522 | 1159 |
"dependent P \<longleftrightarrow> (\<exists>S u. finite S \<and> S \<subseteq> P \<and> (\<exists>v\<in>S. u v \<noteq> 0 \<and> setsum (\<lambda>v. u v *\<^sub>R v) S = 0))" |
1160 |
(is "?lhs = ?rhs") |
|
1161 |
proof - |
|
53406 | 1162 |
{ |
1163 |
assume dP: "dependent P" |
|
44133 | 1164 |
then obtain a S u where aP: "a \<in> P" and fS: "finite S" |
1165 |
and SP: "S \<subseteq> P - {a}" and ua: "setsum (\<lambda>v. u v *\<^sub>R v) S = a" |
|
1166 |
unfolding dependent_def span_explicit by blast |
|
1167 |
let ?S = "insert a S" |
|
1168 |
let ?u = "\<lambda>y. if y = a then - 1 else u y" |
|
1169 |
let ?v = a |
|
53406 | 1170 |
from aP SP have aS: "a \<notin> S" |
1171 |
by blast |
|
1172 |
from fS SP aP have th0: "finite ?S" "?S \<subseteq> P" "?v \<in> ?S" "?u ?v \<noteq> 0" |
|
1173 |
by auto |
|
44133 | 1174 |
have s0: "setsum (\<lambda>v. ?u v *\<^sub>R v) ?S = 0" |
1175 |
using fS aS |
|
1176 |
apply (simp add: setsum_clauses field_simps) |
|
1177 |
apply (subst (2) ua[symmetric]) |
|
1178 |
apply (rule setsum_cong2) |
|
49522 | 1179 |
apply auto |
1180 |
done |
|
44133 | 1181 |
with th0 have ?rhs |
1182 |
apply - |
|
1183 |
apply (rule exI[where x= "?S"]) |
|
1184 |
apply (rule exI[where x= "?u"]) |
|
49522 | 1185 |
apply auto |
1186 |
done |
|
1187 |
} |
|
44133 | 1188 |
moreover |
53406 | 1189 |
{ |
1190 |
fix S u v |
|
49522 | 1191 |
assume fS: "finite S" |
53406 | 1192 |
and SP: "S \<subseteq> P" |
1193 |
and vS: "v \<in> S" |
|
1194 |
and uv: "u v \<noteq> 0" |
|
49522 | 1195 |
and u: "setsum (\<lambda>v. u v *\<^sub>R v) S = 0" |
44133 | 1196 |
let ?a = v |
1197 |
let ?S = "S - {v}" |
|
1198 |
let ?u = "\<lambda>i. (- u i) / u v" |
|
53406 | 1199 |
have th0: "?a \<in> P" "finite ?S" "?S \<subseteq> P" |
1200 |
using fS SP vS by auto |
|
1201 |
have "setsum (\<lambda>v. ?u v *\<^sub>R v) ?S = |
|
1202 |
setsum (\<lambda>v. (- (inverse (u ?a))) *\<^sub>R (u v *\<^sub>R v)) S - ?u v *\<^sub>R v" |
|
49522 | 1203 |
using fS vS uv by (simp add: setsum_diff1 divide_inverse field_simps) |
53406 | 1204 |
also have "\<dots> = ?a" |
1205 |
unfolding scaleR_right.setsum [symmetric] u using uv by simp |
|
1206 |
finally have "setsum (\<lambda>v. ?u v *\<^sub>R v) ?S = ?a" . |
|
44133 | 1207 |
with th0 have ?lhs |
1208 |
unfolding dependent_def span_explicit |
|
1209 |
apply - |
|
1210 |
apply (rule bexI[where x= "?a"]) |
|
1211 |
apply (simp_all del: scaleR_minus_left) |
|
1212 |
apply (rule exI[where x= "?S"]) |
|
49522 | 1213 |
apply (auto simp del: scaleR_minus_left) |
1214 |
done |
|
1215 |
} |
|
44133 | 1216 |
ultimately show ?thesis by blast |
1217 |
qed |
|
1218 |
||
1219 |
||
1220 |
lemma span_finite: |
|
1221 |
assumes fS: "finite S" |
|
1222 |
shows "span S = {y. \<exists>u. setsum (\<lambda>v. u v *\<^sub>R v) S = y}" |
|
1223 |
(is "_ = ?rhs") |
|
49522 | 1224 |
proof - |
53406 | 1225 |
{ |
1226 |
fix y |
|
49711 | 1227 |
assume y: "y \<in> span S" |
53406 | 1228 |
from y obtain S' u where fS': "finite S'" |
1229 |
and SS': "S' \<subseteq> S" |
|
1230 |
and u: "setsum (\<lambda>v. u v *\<^sub>R v) S' = y" |
|
1231 |
unfolding span_explicit by blast |
|
44133 | 1232 |
let ?u = "\<lambda>x. if x \<in> S' then u x else 0" |
1233 |
have "setsum (\<lambda>v. ?u v *\<^sub>R v) S = setsum (\<lambda>v. u v *\<^sub>R v) S'" |
|
1234 |
using SS' fS by (auto intro!: setsum_mono_zero_cong_right) |
|
49522 | 1235 |
then have "setsum (\<lambda>v. ?u v *\<^sub>R v) S = y" by (metis u) |
53406 | 1236 |
then have "y \<in> ?rhs" by auto |
1237 |
} |
|
44133 | 1238 |
moreover |
53406 | 1239 |
{ |
1240 |
fix y u |
|
49522 | 1241 |
assume u: "setsum (\<lambda>v. u v *\<^sub>R v) S = y" |
53406 | 1242 |
then have "y \<in> span S" using fS unfolding span_explicit by auto |
1243 |
} |
|
44133 | 1244 |
ultimately show ?thesis by blast |
1245 |
qed |
|
1246 |
||
1247 |
text {* This is useful for building a basis step-by-step. *} |
|
1248 |
||
1249 |
lemma independent_insert: |
|
53406 | 1250 |
"independent (insert a S) \<longleftrightarrow> |
1251 |
(if a \<in> S then independent S else independent S \<and> a \<notin> span S)" |
|
1252 |
(is "?lhs \<longleftrightarrow> ?rhs") |
|
1253 |
proof (cases "a \<in> S") |
|
1254 |
case True |
|
1255 |
then show ?thesis |
|
1256 |
using insert_absorb[OF True] by simp |
|
1257 |
next |
|
1258 |
case False |
|
1259 |
show ?thesis |
|
1260 |
proof |
|
1261 |
assume i: ?lhs |
|
1262 |
then show ?rhs |
|
1263 |
using False |
|
1264 |
apply simp |
|
1265 |
apply (rule conjI) |
|
1266 |
apply (rule independent_mono) |
|
1267 |
apply assumption |
|
1268 |
apply blast |
|
1269 |
apply (simp add: dependent_def) |
|
1270 |
done |
|
1271 |
next |
|
1272 |
assume i: ?rhs |
|
1273 |
show ?lhs |
|
1274 |
using i False |
|
1275 |
apply simp |
|
1276 |
apply (auto simp add: dependent_def) |
|
1277 |
apply (case_tac "aa = a") |
|
1278 |
apply auto |
|
1279 |
apply (subgoal_tac "insert a S - {aa} = insert a (S - {aa})") |
|
1280 |
apply simp |
|
1281 |
apply (subgoal_tac "a \<in> span (insert aa (S - {aa}))") |
|
1282 |
apply (subgoal_tac "insert aa (S - {aa}) = S") |
|
1283 |
apply simp |
|
1284 |
apply blast |
|
1285 |
apply (rule in_span_insert) |
|
1286 |
apply assumption |
|
1287 |
apply blast |
|
1288 |
apply blast |
|
1289 |
done |
|
1290 |
qed |
|
44133 | 1291 |
qed |
1292 |
||
1293 |
text {* The degenerate case of the Exchange Lemma. *} |
|
1294 |
||
1295 |
lemma mem_delete: "x \<in> (A - {a}) \<longleftrightarrow> x \<noteq> a \<and> x \<in> A" |
|
1296 |
by blast |
|
1297 |
||
1298 |
lemma spanning_subset_independent: |
|
49711 | 1299 |
assumes BA: "B \<subseteq> A" |
1300 |
and iA: "independent A" |
|
49522 | 1301 |
and AsB: "A \<subseteq> span B" |
44133 | 1302 |
shows "A = B" |
1303 |
proof |
|
49663 | 1304 |
show "B \<subseteq> A" by (rule BA) |
1305 |
||
44133 | 1306 |
from span_mono[OF BA] span_mono[OF AsB] |
1307 |
have sAB: "span A = span B" unfolding span_span by blast |
|
1308 |
||
53406 | 1309 |
{ |
1310 |
fix x |
|
1311 |
assume x: "x \<in> A" |
|
44133 | 1312 |
from iA have th0: "x \<notin> span (A - {x})" |
1313 |
unfolding dependent_def using x by blast |
|
53406 | 1314 |
from x have xsA: "x \<in> span A" |
1315 |
by (blast intro: span_superset) |
|
44133 | 1316 |
have "A - {x} \<subseteq> A" by blast |
53406 | 1317 |
then have th1: "span (A - {x}) \<subseteq> span A" |
1318 |
by (metis span_mono) |
|
1319 |
{ |
|
1320 |
assume xB: "x \<notin> B" |
|
1321 |
from xB BA have "B \<subseteq> A - {x}" |
|
1322 |
by blast |
|
1323 |
then have "span B \<subseteq> span (A - {x})" |
|
1324 |
by (metis span_mono) |
|
1325 |
with th1 th0 sAB have "x \<notin> span A" |
|
1326 |
by blast |
|
1327 |
with x have False |
|
1328 |
by (metis span_superset) |
|
1329 |
} |
|
1330 |
then have "x \<in> B" by blast |
|
1331 |
} |
|
44133 | 1332 |
then show "A \<subseteq> B" by blast |
1333 |
qed |
|
1334 |
||
1335 |
text {* The general case of the Exchange Lemma, the key to what follows. *} |
|
1336 |
||
1337 |
lemma exchange_lemma: |
|
49711 | 1338 |
assumes f:"finite t" |
1339 |
and i: "independent s" |
|
1340 |
and sp: "s \<subseteq> span t" |
|
53406 | 1341 |
shows "\<exists>t'. card t' = card t \<and> finite t' \<and> s \<subseteq> t' \<and> t' \<subseteq> s \<union> t \<and> s \<subseteq> span t'" |
49663 | 1342 |
using f i sp |
49522 | 1343 |
proof (induct "card (t - s)" arbitrary: s t rule: less_induct) |
44133 | 1344 |
case less |
1345 |
note ft = `finite t` and s = `independent s` and sp = `s \<subseteq> span t` |
|
53406 | 1346 |
let ?P = "\<lambda>t'. card t' = card t \<and> finite t' \<and> s \<subseteq> t' \<and> t' \<subseteq> s \<union> t \<and> s \<subseteq> span t'" |
44133 | 1347 |
let ?ths = "\<exists>t'. ?P t'" |
53406 | 1348 |
{ |
1349 |
assume st: "s \<subseteq> t" |
|
1350 |
from st ft span_mono[OF st] |
|
1351 |
have ?ths |
|
1352 |
apply - |
|
1353 |
apply (rule exI[where x=t]) |
|
49522 | 1354 |
apply (auto intro: span_superset) |
53406 | 1355 |
done |
1356 |
} |
|
44133 | 1357 |
moreover |
53406 | 1358 |
{ |
1359 |
assume st: "t \<subseteq> s" |
|
1360 |
from spanning_subset_independent[OF st s sp] st ft span_mono[OF st] |
|
1361 |
have ?ths |
|
1362 |
apply - |
|
1363 |
apply (rule exI[where x=t]) |
|
1364 |
apply (auto intro: span_superset) |
|
1365 |
done |
|
1366 |
} |
|
44133 | 1367 |
moreover |
53406 | 1368 |
{ |
1369 |
assume st: "\<not> s \<subseteq> t" "\<not> t \<subseteq> s" |
|
1370 |
from st(2) obtain b where b: "b \<in> t" "b \<notin> s" |
|
1371 |
by blast |
|
1372 |
from b have "t - {b} - s \<subset> t - s" |
|
1373 |
by blast |
|
1374 |
then have cardlt: "card (t - {b} - s) < card (t - s)" |
|
1375 |
using ft by (auto intro: psubset_card_mono) |
|
1376 |
from b ft have ct0: "card t \<noteq> 0" |
|
1377 |
by auto |
|
1378 |
have ?ths |
|
1379 |
proof cases |
|
53716 | 1380 |
assume stb: "s \<subseteq> span (t - {b})" |
1381 |
from ft have ftb: "finite (t - {b})" |
|
53406 | 1382 |
by auto |
44133 | 1383 |
from less(1)[OF cardlt ftb s stb] |
53716 | 1384 |
obtain u where u: "card u = card (t - {b})" "s \<subseteq> u" "u \<subseteq> s \<union> (t - {b})" "s \<subseteq> span u" |
49522 | 1385 |
and fu: "finite u" by blast |
44133 | 1386 |
let ?w = "insert b u" |
53406 | 1387 |
have th0: "s \<subseteq> insert b u" |
1388 |
using u by blast |
|
1389 |
from u(3) b have "u \<subseteq> s \<union> t" |
|
1390 |
by blast |
|
1391 |
then have th1: "insert b u \<subseteq> s \<union> t" |
|
1392 |
using u b by blast |
|
1393 |
have bu: "b \<notin> u" |
|
1394 |
using b u by blast |
|
1395 |
from u(1) ft b have "card u = (card t - 1)" |
|
1396 |
by auto |
|
49522 | 1397 |
then have th2: "card (insert b u) = card t" |
44133 | 1398 |
using card_insert_disjoint[OF fu bu] ct0 by auto |
1399 |
from u(4) have "s \<subseteq> span u" . |
|
53406 | 1400 |
also have "\<dots> \<subseteq> span (insert b u)" |
1401 |
by (rule span_mono) blast |
|
44133 | 1402 |
finally have th3: "s \<subseteq> span (insert b u)" . |
53406 | 1403 |
from th0 th1 th2 th3 fu have th: "?P ?w" |
1404 |
by blast |
|
1405 |
from th show ?thesis by blast |
|
1406 |
next |
|
53716 | 1407 |
assume stb: "\<not> s \<subseteq> span (t - {b})" |
53406 | 1408 |
from stb obtain a where a: "a \<in> s" "a \<notin> span (t - {b})" |
1409 |
by blast |
|
1410 |
have ab: "a \<noteq> b" |
|
1411 |
using a b by blast |
|
1412 |
have at: "a \<notin> t" |
|
1413 |
using a ab span_superset[of a "t- {b}"] by auto |
|
44133 | 1414 |
have mlt: "card ((insert a (t - {b})) - s) < card (t - s)" |
1415 |
using cardlt ft a b by auto |
|
53406 | 1416 |
have ft': "finite (insert a (t - {b}))" |
1417 |
using ft by auto |
|
1418 |
{ |
|
1419 |
fix x |
|
1420 |
assume xs: "x \<in> s" |
|
1421 |
have t: "t \<subseteq> insert b (insert a (t - {b}))" |
|
1422 |
using b by auto |
|
1423 |
from b(1) have "b \<in> span t" |
|
1424 |
by (simp add: span_superset) |
|
1425 |
have bs: "b \<in> span (insert a (t - {b}))" |
|
1426 |
apply (rule in_span_delete) |
|
1427 |
using a sp unfolding subset_eq |
|
1428 |
apply auto |
|
1429 |
done |
|
1430 |
from xs sp have "x \<in> span t" |
|
1431 |
by blast |
|
1432 |
with span_mono[OF t] have x: "x \<in> span (insert b (insert a (t - {b})))" .. |
|
1433 |
from span_trans[OF bs x] have "x \<in> span (insert a (t - {b}))" . |
|
1434 |
} |
|
1435 |
then have sp': "s \<subseteq> span (insert a (t - {b}))" |
|
1436 |
by blast |
|
1437 |
from less(1)[OF mlt ft' s sp'] obtain u where u: |
|
53716 | 1438 |
"card u = card (insert a (t - {b}))" |
1439 |
"finite u" "s \<subseteq> u" "u \<subseteq> s \<union> insert a (t - {b})" |
|
53406 | 1440 |
"s \<subseteq> span u" by blast |
1441 |
from u a b ft at ct0 have "?P u" |
|
1442 |
by auto |
|
1443 |
then show ?thesis by blast |
|
1444 |
qed |
|
44133 | 1445 |
} |
49522 | 1446 |
ultimately show ?ths by blast |
44133 | 1447 |
qed |
1448 |
||
1449 |
text {* This implies corresponding size bounds. *} |
|
1450 |
||
1451 |
lemma independent_span_bound: |
|
53406 | 1452 |
assumes f: "finite t" |
1453 |
and i: "independent s" |
|
1454 |
and sp: "s \<subseteq> span t" |
|
44133 | 1455 |
shows "finite s \<and> card s \<le> card t" |
1456 |
by (metis exchange_lemma[OF f i sp] finite_subset card_mono) |
|
1457 |
||
1458 |
lemma finite_Atleast_Atmost_nat[simp]: "finite {f x |x. x\<in> (UNIV::'a::finite set)}" |
|
49522 | 1459 |
proof - |
53406 | 1460 |
have eq: "{f x |x. x\<in> UNIV} = f ` UNIV" |
1461 |
by auto |
|
44133 | 1462 |
show ?thesis unfolding eq |
1463 |
apply (rule finite_imageI) |
|
1464 |
apply (rule finite) |
|
1465 |
done |
|
1466 |
qed |
|
1467 |
||
53406 | 1468 |
|
1469 |
subsection {* Euclidean Spaces as Typeclass *} |
|
44133 | 1470 |
|
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
1471 |
lemma independent_Basis: "independent Basis" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
1472 |
unfolding dependent_def |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
1473 |
apply (subst span_finite) |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
1474 |
apply simp |
44133 | 1475 |
apply clarify |
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
1476 |
apply (drule_tac f="inner a" in arg_cong) |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
1477 |
apply (simp add: inner_Basis inner_setsum_right eq_commute) |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
1478 |
done |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
1479 |
|
53939 | 1480 |
lemma span_Basis [simp]: "span Basis = UNIV" |
1481 |
unfolding span_finite [OF finite_Basis] |
|
1482 |
by (fast intro: euclidean_representation) |
|
44133 | 1483 |
|
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
1484 |
lemma in_span_Basis: "x \<in> span Basis" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
1485 |
unfolding span_Basis .. |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
1486 |
|
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
1487 |
lemma Basis_le_norm: "b \<in> Basis \<Longrightarrow> \<bar>x \<bullet> b\<bar> \<le> norm x" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
1488 |
by (rule order_trans [OF Cauchy_Schwarz_ineq2]) simp |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
1489 |
|
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
1490 |
lemma norm_bound_Basis_le: "b \<in> Basis \<Longrightarrow> norm x \<le> e \<Longrightarrow> \<bar>x \<bullet> b\<bar> \<le> e" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
1491 |
by (metis Basis_le_norm order_trans) |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
1492 |
|
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
1493 |
lemma norm_bound_Basis_lt: "b \<in> Basis \<Longrightarrow> norm x < e \<Longrightarrow> \<bar>x \<bullet> b\<bar> < e" |
53595 | 1494 |
by (metis Basis_le_norm le_less_trans) |
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
1495 |
|
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
1496 |
lemma norm_le_l1: "norm x \<le> (\<Sum>b\<in>Basis. \<bar>x \<bullet> b\<bar>)" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
1497 |
apply (subst euclidean_representation[of x, symmetric]) |
44176
eda112e9cdee
remove redundant lemma setsum_norm in favor of norm_setsum;
huffman
parents:
44170
diff
changeset
|
1498 |
apply (rule order_trans[OF norm_setsum]) |
49522 | 1499 |
apply (auto intro!: setsum_mono) |
1500 |
done |
|
44133 | 1501 |
|
1502 |
lemma setsum_norm_allsubsets_bound: |
|
1503 |
fixes f:: "'a \<Rightarrow> 'n::euclidean_space" |
|
53406 | 1504 |
assumes fP: "finite P" |
1505 |
and fPs: "\<And>Q. Q \<subseteq> P \<Longrightarrow> norm (setsum f Q) \<le> e" |
|
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
1506 |
shows "(\<Sum>x\<in>P. norm (f x)) \<le> 2 * real DIM('n) * e" |
49522 | 1507 |
proof - |
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
1508 |
have "(\<Sum>x\<in>P. norm (f x)) \<le> (\<Sum>x\<in>P. \<Sum>b\<in>Basis. \<bar>f x \<bullet> b\<bar>)" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
1509 |
by (rule setsum_mono) (rule norm_le_l1) |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
1510 |
also have "(\<Sum>x\<in>P. \<Sum>b\<in>Basis. \<bar>f x \<bullet> b\<bar>) = (\<Sum>b\<in>Basis. \<Sum>x\<in>P. \<bar>f x \<bullet> b\<bar>)" |
44133 | 1511 |
by (rule setsum_commute) |
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
1512 |
also have "\<dots> \<le> of_nat (card (Basis :: 'n set)) * (2 * e)" |
49522 | 1513 |
proof (rule setsum_bounded) |
53406 | 1514 |
fix i :: 'n |
1515 |
assume i: "i \<in> Basis" |
|
1516 |
have "norm (\<Sum>x\<in>P. \<bar>f x \<bullet> i\<bar>) \<le> |
|
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
1517 |
norm ((\<Sum>x\<in>P \<inter> - {x. f x \<bullet> i < 0}. f x) \<bullet> i) + norm ((\<Sum>x\<in>P \<inter> {x. f x \<bullet> i < 0}. f x) \<bullet> i)" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
1518 |
by (simp add: abs_real_def setsum_cases[OF fP] setsum_negf uminus_add_conv_diff |
53406 | 1519 |
norm_triangle_ineq4 inner_setsum_left del: real_norm_def) |
1520 |
also have "\<dots> \<le> e + e" |
|
1521 |
unfolding real_norm_def |
|
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
1522 |
by (intro add_mono norm_bound_Basis_le i fPs) auto |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
1523 |
finally show "(\<Sum>x\<in>P. \<bar>f x \<bullet> i\<bar>) \<le> 2*e" by simp |
44133 | 1524 |
qed |
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
1525 |
also have "\<dots> = 2 * real DIM('n) * e" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
1526 |
by (simp add: real_of_nat_def) |
44133 | 1527 |
finally show ?thesis . |
1528 |
qed |
|
1529 |
||
53406 | 1530 |
|
44133 | 1531 |
subsection {* Linearity and Bilinearity continued *} |
1532 |
||
1533 |
lemma linear_bounded: |
|
1534 |
fixes f:: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector" |
|
1535 |
assumes lf: "linear f" |
|
1536 |
shows "\<exists>B. \<forall>x. norm (f x) \<le> B * norm x" |
|
53939 | 1537 |
proof |
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
1538 |
let ?B = "\<Sum>b\<in>Basis. norm (f b)" |
53939 | 1539 |
show "\<forall>x. norm (f x) \<le> ?B * norm x" |
1540 |
proof |
|
53406 | 1541 |
fix x :: 'a |
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
1542 |
let ?g = "\<lambda>b. (x \<bullet> b) *\<^sub>R f b" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
1543 |
have "norm (f x) = norm (f (\<Sum>b\<in>Basis. (x \<bullet> b) *\<^sub>R b))" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
1544 |
unfolding euclidean_representation .. |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
1545 |
also have "\<dots> = norm (setsum ?g Basis)" |
53939 | 1546 |
by (simp add: linear_setsum [OF lf] linear_cmul [OF lf]) |
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
1547 |
finally have th0: "norm (f x) = norm (setsum ?g Basis)" . |
53939 | 1548 |
have th: "\<forall>b\<in>Basis. norm (?g b) \<le> norm (f b) * norm x" |
1549 |
proof |
|
53406 | 1550 |
fix i :: 'a |
1551 |
assume i: "i \<in> Basis" |
|
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
1552 |
from Basis_le_norm[OF i, of x] |
53939 | 1553 |
show "norm (?g i) \<le> norm (f i) * norm x" |
49663 | 1554 |
unfolding norm_scaleR |
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
1555 |
apply (subst mult_commute) |
49663 | 1556 |
apply (rule mult_mono) |
1557 |
apply (auto simp add: field_simps) |
|
53406 | 1558 |
done |
53939 | 1559 |
qed |
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
1560 |
from setsum_norm_le[of _ ?g, OF th] |
53939 | 1561 |
show "norm (f x) \<le> ?B * norm x" |
53406 | 1562 |
unfolding th0 setsum_left_distrib by metis |
53939 | 1563 |
qed |
44133 | 1564 |
qed |
1565 |
||
1566 |
lemma linear_conv_bounded_linear: |
|
1567 |
fixes f :: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector" |
|
1568 |
shows "linear f \<longleftrightarrow> bounded_linear f" |
|
1569 |
proof |
|
1570 |
assume "linear f" |
|
53939 | 1571 |
then interpret f: linear f . |
44133 | 1572 |
show "bounded_linear f" |
1573 |
proof |
|
1574 |
have "\<exists>B. \<forall>x. norm (f x) \<le> B * norm x" |
|
1575 |
using `linear f` by (rule linear_bounded) |
|
49522 | 1576 |
then show "\<exists>K. \<forall>x. norm (f x) \<le> norm x * K" |
44133 | 1577 |
by (simp add: mult_commute) |
1578 |
qed |
|
1579 |
next |
|
1580 |
assume "bounded_linear f" |
|
1581 |
then interpret f: bounded_linear f . |
|
53939 | 1582 |
show "linear f" .. |
1583 |
qed |
|
1584 |
||
1585 |
lemma linear_bounded_pos: |
|
1586 |
fixes f:: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector" |
|
1587 |
assumes lf: "linear f" |
|
1588 |
shows "\<exists>B > 0. \<forall>x. norm (f x) \<le> B * norm x" |
|
1589 |
proof - |
|
1590 |
have "\<exists>B > 0. \<forall>x. norm (f x) \<le> norm x * B" |
|
1591 |
using lf unfolding linear_conv_bounded_linear |
|
1592 |
by (rule bounded_linear.pos_bounded) |
|
1593 |
then show ?thesis |
|
1594 |
by (simp only: mult_commute) |
|
44133 | 1595 |
qed |
1596 |
||
49522 | 1597 |
lemma bounded_linearI': |
1598 |
fixes f::"'a::euclidean_space \<Rightarrow> 'b::real_normed_vector" |
|
53406 | 1599 |
assumes "\<And>x y. f (x + y) = f x + f y" |
1600 |
and "\<And>c x. f (c *\<^sub>R x) = c *\<^sub>R f x" |
|
49522 | 1601 |
shows "bounded_linear f" |
53406 | 1602 |
unfolding linear_conv_bounded_linear[symmetric] |
49522 | 1603 |
by (rule linearI[OF assms]) |
44133 | 1604 |
|
1605 |
lemma bilinear_bounded: |
|
1606 |
fixes h:: "'m::euclidean_space \<Rightarrow> 'n::euclidean_space \<Rightarrow> 'k::real_normed_vector" |
|
1607 |
assumes bh: "bilinear h" |
|
1608 |
shows "\<exists>B. \<forall>x y. norm (h x y) \<le> B * norm x * norm y" |
|
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
1609 |
proof (clarify intro!: exI[of _ "\<Sum>i\<in>Basis. \<Sum>j\<in>Basis. norm (h i j)"]) |
53406 | 1610 |
fix x :: 'm |
1611 |
fix y :: 'n |
|
1612 |
have "norm (h x y) = norm (h (setsum (\<lambda>i. (x \<bullet> i) *\<^sub>R i) Basis) (setsum (\<lambda>i. (y \<bullet> i) *\<^sub>R i) Basis))" |
|
1613 |
apply (subst euclidean_representation[where 'a='m]) |
|
1614 |
apply (subst euclidean_representation[where 'a='n]) |
|
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
1615 |
apply rule |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
1616 |
done |
53406 | 1617 |
also have "\<dots> = norm (setsum (\<lambda> (i,j). h ((x \<bullet> i) *\<^sub>R i) ((y \<bullet> j) *\<^sub>R j)) (Basis \<times> Basis))" |
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
1618 |
unfolding bilinear_setsum[OF bh finite_Basis finite_Basis] .. |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
1619 |
finally have th: "norm (h x y) = \<dots>" . |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
1620 |
show "norm (h x y) \<le> (\<Sum>i\<in>Basis. \<Sum>j\<in>Basis. norm (h i j)) * norm x * norm y" |
53406 | 1621 |
apply (auto simp add: setsum_left_distrib th setsum_cartesian_product) |
1622 |
apply (rule setsum_norm_le) |
|
1623 |
apply simp |
|
1624 |
apply (auto simp add: bilinear_rmul[OF bh] bilinear_lmul[OF bh] |
|
1625 |
field_simps simp del: scaleR_scaleR) |
|
1626 |
apply (rule mult_mono) |
|
1627 |
apply (auto simp add: zero_le_mult_iff Basis_le_norm) |
|
1628 |
apply (rule mult_mono) |
|
1629 |
apply (auto simp add: zero_le_mult_iff Basis_le_norm) |
|
1630 |
done |
|
44133 | 1631 |
qed |
1632 |
||
1633 |
lemma bilinear_conv_bounded_bilinear: |
|
1634 |
fixes h :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space \<Rightarrow> 'c::real_normed_vector" |
|
1635 |
shows "bilinear h \<longleftrightarrow> bounded_bilinear h" |
|
1636 |
proof |
|
1637 |
assume "bilinear h" |
|
1638 |
show "bounded_bilinear h" |
|
1639 |
proof |
|
53406 | 1640 |
fix x y z |
1641 |
show "h (x + y) z = h x z + h y z" |
|
53600
8fda7ad57466
make 'linear' into a sublocale of 'bounded_linear';
huffman
parents:
53596
diff
changeset
|
1642 |
using `bilinear h` unfolding bilinear_def linear_iff by simp |
44133 | 1643 |
next |
53406 | 1644 |
fix x y z |
1645 |
show "h x (y + z) = h x y + h x z" |
|
53600
8fda7ad57466
make 'linear' into a sublocale of 'bounded_linear';
huffman
parents:
53596
diff
changeset
|
1646 |
using `bilinear h` unfolding bilinear_def linear_iff by simp |
44133 | 1647 |
next |
53406 | 1648 |
fix r x y |
1649 |
show "h (scaleR r x) y = scaleR r (h x y)" |
|
53600
8fda7ad57466
make 'linear' into a sublocale of 'bounded_linear';
huffman
parents:
53596
diff
changeset
|
1650 |
using `bilinear h` unfolding bilinear_def linear_iff |
44133 | 1651 |
by simp |
1652 |
next |
|
53406 | 1653 |
fix r x y |
1654 |
show "h x (scaleR r y) = scaleR r (h x y)" |
|
53600
8fda7ad57466
make 'linear' into a sublocale of 'bounded_linear';
huffman
parents:
53596
diff
changeset
|
1655 |
using `bilinear h` unfolding bilinear_def linear_iff |
44133 | 1656 |
by simp |
1657 |
next |
|
1658 |
have "\<exists>B. \<forall>x y. norm (h x y) \<le> B * norm x * norm y" |
|
1659 |
using `bilinear h` by (rule bilinear_bounded) |
|
49522 | 1660 |
then show "\<exists>K. \<forall>x y. norm (h x y) \<le> norm x * norm y * K" |
44133 | 1661 |
by (simp add: mult_ac) |
1662 |
qed |
|
1663 |
next |
|
1664 |
assume "bounded_bilinear h" |
|
1665 |
then interpret h: bounded_bilinear h . |
|
1666 |
show "bilinear h" |
|
1667 |
unfolding bilinear_def linear_conv_bounded_linear |
|
49522 | 1668 |
using h.bounded_linear_left h.bounded_linear_right by simp |
44133 | 1669 |
qed |
1670 |
||
53939 | 1671 |
lemma bilinear_bounded_pos: |
1672 |
fixes h:: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space \<Rightarrow> 'c::real_normed_vector" |
|
1673 |
assumes bh: "bilinear h" |
|
1674 |
shows "\<exists>B > 0. \<forall>x y. norm (h x y) \<le> B * norm x * norm y" |
|
1675 |
proof - |
|
1676 |
have "\<exists>B > 0. \<forall>x y. norm (h x y) \<le> norm x * norm y * B" |
|
1677 |
using bh [unfolded bilinear_conv_bounded_bilinear] |
|
1678 |
by (rule bounded_bilinear.pos_bounded) |
|
1679 |
then show ?thesis |
|
1680 |
by (simp only: mult_ac) |
|
1681 |
qed |
|
1682 |
||
49522 | 1683 |
|
44133 | 1684 |
subsection {* We continue. *} |
1685 |
||
1686 |
lemma independent_bound: |
|
53716 | 1687 |
fixes S :: "'a::euclidean_space set" |
1688 |
shows "independent S \<Longrightarrow> finite S \<and> card S \<le> DIM('a)" |
|
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
1689 |
using independent_span_bound[OF finite_Basis, of S] by auto |
44133 | 1690 |
|
49663 | 1691 |
lemma dependent_biggerset: |
53406 | 1692 |
"(finite (S::('a::euclidean_space) set) \<Longrightarrow> card S > DIM('a)) \<Longrightarrow> dependent S" |
44133 | 1693 |
by (metis independent_bound not_less) |
1694 |
||
1695 |
text {* Hence we can create a maximal independent subset. *} |
|
1696 |
||
1697 |
lemma maximal_independent_subset_extend: |
|
53406 | 1698 |
fixes S :: "'a::euclidean_space set" |
1699 |
assumes sv: "S \<subseteq> V" |
|
49663 | 1700 |
and iS: "independent S" |
44133 | 1701 |
shows "\<exists>B. S \<subseteq> B \<and> B \<subseteq> V \<and> independent B \<and> V \<subseteq> span B" |
1702 |
using sv iS |
|
49522 | 1703 |
proof (induct "DIM('a) - card S" arbitrary: S rule: less_induct) |
44133 | 1704 |
case less |
1705 |
note sv = `S \<subseteq> V` and i = `independent S` |
|
1706 |
let ?P = "\<lambda>B. S \<subseteq> B \<and> B \<subseteq> V \<and> independent B \<and> V \<subseteq> span B" |
|
1707 |
let ?ths = "\<exists>x. ?P x" |
|
1708 |
let ?d = "DIM('a)" |
|
53406 | 1709 |
show ?ths |
1710 |
proof (cases "V \<subseteq> span S") |
|
1711 |
case True |
|
1712 |
then show ?thesis |
|
1713 |
using sv i by blast |
|
1714 |
next |
|
1715 |
case False |
|
1716 |
then obtain a where a: "a \<in> V" "a \<notin> span S" |
|
1717 |
by blast |
|
1718 |
from a have aS: "a \<notin> S" |
|
1719 |
by (auto simp add: span_superset) |
|
1720 |
have th0: "insert a S \<subseteq> V" |
|
1721 |
using a sv by blast |
|
44133 | 1722 |
from independent_insert[of a S] i a |
53406 | 1723 |
have th1: "independent (insert a S)" |
1724 |
by auto |
|
44133 | 1725 |
have mlt: "?d - card (insert a S) < ?d - card S" |
49522 | 1726 |
using aS a independent_bound[OF th1] by auto |
44133 | 1727 |
|
1728 |
from less(1)[OF mlt th0 th1] |
|
1729 |
obtain B where B: "insert a S \<subseteq> B" "B \<subseteq> V" "independent B" " V \<subseteq> span B" |
|
1730 |
by blast |
|
1731 |
from B have "?P B" by auto |
|
53406 | 1732 |
then show ?thesis by blast |
1733 |
qed |
|
44133 | 1734 |
qed |
1735 |
||
1736 |
lemma maximal_independent_subset: |
|
1737 |
"\<exists>(B:: ('a::euclidean_space) set). B\<subseteq> V \<and> independent B \<and> V \<subseteq> span B" |
|
49522 | 1738 |
by (metis maximal_independent_subset_extend[of "{}:: ('a::euclidean_space) set"] |
1739 |
empty_subsetI independent_empty) |
|
44133 | 1740 |
|
1741 |
||
1742 |
text {* Notion of dimension. *} |
|
1743 |
||
53406 | 1744 |
definition "dim V = (SOME n. \<exists>B. B \<subseteq> V \<and> independent B \<and> V \<subseteq> span B \<and> card B = n)" |
44133 | 1745 |
|
49522 | 1746 |
lemma basis_exists: |
1747 |
"\<exists>B. (B :: ('a::euclidean_space) set) \<subseteq> V \<and> independent B \<and> V \<subseteq> span B \<and> (card B = dim V)" |
|
1748 |
unfolding dim_def some_eq_ex[of "\<lambda>n. \<exists>B. B \<subseteq> V \<and> independent B \<and> V \<subseteq> span B \<and> (card B = n)"] |
|
1749 |
using maximal_independent_subset[of V] independent_bound |
|
1750 |
by auto |
|
44133 | 1751 |
|
1752 |
text {* Consequences of independence or spanning for cardinality. *} |
|
1753 |
||
53406 | 1754 |
lemma independent_card_le_dim: |
1755 |
fixes B :: "'a::euclidean_space set" |
|
1756 |
assumes "B \<subseteq> V" |
|
1757 |
and "independent B" |
|
49522 | 1758 |
shows "card B \<le> dim V" |
44133 | 1759 |
proof - |
1760 |
from basis_exists[of V] `B \<subseteq> V` |
|
53406 | 1761 |
obtain B' where "independent B'" |
1762 |
and "B \<subseteq> span B'" |
|
1763 |
and "card B' = dim V" |
|
1764 |
by blast |
|
44133 | 1765 |
with independent_span_bound[OF _ `independent B` `B \<subseteq> span B'`] independent_bound[of B'] |
1766 |
show ?thesis by auto |
|
1767 |
qed |
|
1768 |
||
49522 | 1769 |
lemma span_card_ge_dim: |
53406 | 1770 |
fixes B :: "'a::euclidean_space set" |
1771 |
shows "B \<subseteq> V \<Longrightarrow> V \<subseteq> span B \<Longrightarrow> finite B \<Longrightarrow> dim V \<le> card B" |
|
44133 | 1772 |
by (metis basis_exists[of V] independent_span_bound subset_trans) |
1773 |
||
1774 |
lemma basis_card_eq_dim: |
|
53406 | 1775 |
fixes V :: "'a::euclidean_space set" |
1776 |
shows "B \<subseteq> V \<Longrightarrow> V \<subseteq> span B \<Longrightarrow> independent B \<Longrightarrow> finite B \<and> card B = dim V" |
|
44133 | 1777 |
by (metis order_eq_iff independent_card_le_dim span_card_ge_dim independent_bound) |
1778 |
||
53406 | 1779 |
lemma dim_unique: |
1780 |
fixes B :: "'a::euclidean_space set" |
|
1781 |
shows "B \<subseteq> V \<Longrightarrow> V \<subseteq> span B \<Longrightarrow> independent B \<Longrightarrow> card B = n \<Longrightarrow> dim V = n" |
|
44133 | 1782 |
by (metis basis_card_eq_dim) |
1783 |
||
1784 |
text {* More lemmas about dimension. *} |
|
1785 |
||
53406 | 1786 |
lemma dim_UNIV: "dim (UNIV :: 'a::euclidean_space set) = DIM('a)" |
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
1787 |
using independent_Basis |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
1788 |
by (intro dim_unique[of Basis]) auto |
44133 | 1789 |
|
1790 |
lemma dim_subset: |
|
53406 | 1791 |
fixes S :: "'a::euclidean_space set" |
1792 |
shows "S \<subseteq> T \<Longrightarrow> dim S \<le> dim T" |
|
44133 | 1793 |
using basis_exists[of T] basis_exists[of S] |
1794 |
by (metis independent_card_le_dim subset_trans) |
|
1795 |
||
53406 | 1796 |
lemma dim_subset_UNIV: |
1797 |
fixes S :: "'a::euclidean_space set" |
|
1798 |
shows "dim S \<le> DIM('a)" |
|
44133 | 1799 |
by (metis dim_subset subset_UNIV dim_UNIV) |
1800 |
||
1801 |
text {* Converses to those. *} |
|
1802 |
||
1803 |
lemma card_ge_dim_independent: |
|
53406 | 1804 |
fixes B :: "'a::euclidean_space set" |
1805 |
assumes BV: "B \<subseteq> V" |
|
1806 |
and iB: "independent B" |
|
1807 |
and dVB: "dim V \<le> card B" |
|
44133 | 1808 |
shows "V \<subseteq> span B" |
53406 | 1809 |
proof |
1810 |
fix a |
|
1811 |
assume aV: "a \<in> V" |
|
1812 |
{ |
|
1813 |
assume aB: "a \<notin> span B" |
|
1814 |
then have iaB: "independent (insert a B)" |
|
1815 |
using iB aV BV by (simp add: independent_insert) |
|
1816 |
from aV BV have th0: "insert a B \<subseteq> V" |
|
1817 |
by blast |
|
1818 |
from aB have "a \<notin>B" |
|
1819 |
by (auto simp add: span_superset) |
|
1820 |
with independent_card_le_dim[OF th0 iaB] dVB independent_bound[OF iB] |
|
1821 |
have False by auto |
|
1822 |
} |
|
1823 |
then show "a \<in> span B" by blast |
|
44133 | 1824 |
qed |
1825 |
||
1826 |
lemma card_le_dim_spanning: |
|
49663 | 1827 |
assumes BV: "(B:: ('a::euclidean_space) set) \<subseteq> V" |
1828 |
and VB: "V \<subseteq> span B" |
|
1829 |
and fB: "finite B" |
|
1830 |
and dVB: "dim V \<ge> card B" |
|
44133 | 1831 |
shows "independent B" |
49522 | 1832 |
proof - |
53406 | 1833 |
{ |
1834 |
fix a |
|
53716 | 1835 |
assume a: "a \<in> B" "a \<in> span (B - {a})" |
53406 | 1836 |
from a fB have c0: "card B \<noteq> 0" |
1837 |
by auto |
|
53716 | 1838 |
from a fB have cb: "card (B - {a}) = card B - 1" |
53406 | 1839 |
by auto |
53716 | 1840 |
from BV a have th0: "B - {a} \<subseteq> V" |
53406 | 1841 |
by blast |
1842 |
{ |
|
1843 |
fix x |
|
1844 |
assume x: "x \<in> V" |
|
53716 | 1845 |
from a have eq: "insert a (B - {a}) = B" |
53406 | 1846 |
by blast |
1847 |
from x VB have x': "x \<in> span B" |
|
1848 |
by blast |
|
44133 | 1849 |
from span_trans[OF a(2), unfolded eq, OF x'] |
53716 | 1850 |
have "x \<in> span (B - {a})" . |
53406 | 1851 |
} |
53716 | 1852 |
then have th1: "V \<subseteq> span (B - {a})" |
53406 | 1853 |
by blast |
53716 | 1854 |
have th2: "finite (B - {a})" |
53406 | 1855 |
using fB by auto |
44133 | 1856 |
from span_card_ge_dim[OF th0 th1 th2] |
53716 | 1857 |
have c: "dim V \<le> card (B - {a})" . |
53406 | 1858 |
from c c0 dVB cb have False by simp |
1859 |
} |
|
1860 |
then show ?thesis |
|
1861 |
unfolding dependent_def by blast |
|
44133 | 1862 |
qed |
1863 |
||
53406 | 1864 |
lemma card_eq_dim: |
1865 |
fixes B :: "'a::euclidean_space set" |
|
1866 |
shows "B \<subseteq> V \<Longrightarrow> card B = dim V \<Longrightarrow> finite B \<Longrightarrow> independent B \<longleftrightarrow> V \<subseteq> span B" |
|
49522 | 1867 |
by (metis order_eq_iff card_le_dim_spanning card_ge_dim_independent) |
44133 | 1868 |
|
1869 |
text {* More general size bound lemmas. *} |
|
1870 |
||
1871 |
lemma independent_bound_general: |
|
53406 | 1872 |
fixes S :: "'a::euclidean_space set" |
1873 |
shows "independent S \<Longrightarrow> finite S \<and> card S \<le> dim S" |
|
44133 | 1874 |
by (metis independent_card_le_dim independent_bound subset_refl) |
1875 |
||
49522 | 1876 |
lemma dependent_biggerset_general: |
53406 | 1877 |
fixes S :: "'a::euclidean_space set" |
1878 |
shows "(finite S \<Longrightarrow> card S > dim S) \<Longrightarrow> dependent S" |
|
44133 | 1879 |
using independent_bound_general[of S] by (metis linorder_not_le) |
1880 |
||
53406 | 1881 |
lemma dim_span: |
1882 |
fixes S :: "'a::euclidean_space set" |
|
1883 |
shows "dim (span S) = dim S" |
|
49522 | 1884 |
proof - |
44133 | 1885 |
have th0: "dim S \<le> dim (span S)" |
1886 |
by (auto simp add: subset_eq intro: dim_subset span_superset) |
|
1887 |
from basis_exists[of S] |
|
53406 | 1888 |
obtain B where B: "B \<subseteq> S" "independent B" "S \<subseteq> span B" "card B = dim S" |
1889 |
by blast |
|
1890 |
from B have fB: "finite B" "card B = dim S" |
|
1891 |
using independent_bound by blast+ |
|
1892 |
have bSS: "B \<subseteq> span S" |
|
1893 |
using B(1) by (metis subset_eq span_inc) |
|
1894 |
have sssB: "span S \<subseteq> span B" |
|
1895 |
using span_mono[OF B(3)] by (simp add: span_span) |
|
44133 | 1896 |
from span_card_ge_dim[OF bSS sssB fB(1)] th0 show ?thesis |
49522 | 1897 |
using fB(2) by arith |
44133 | 1898 |
qed |
1899 |
||
53406 | 1900 |
lemma subset_le_dim: |
1901 |
fixes S :: "'a::euclidean_space set" |
|
1902 |
shows "S \<subseteq> span T \<Longrightarrow> dim S \<le> dim T" |
|
44133 | 1903 |
by (metis dim_span dim_subset) |
1904 |
||
53406 | 1905 |
lemma span_eq_dim: |
1906 |
fixes S:: "'a::euclidean_space set" |
|
1907 |
shows "span S = span T \<Longrightarrow> dim S = dim T" |
|
44133 | 1908 |
by (metis dim_span) |
1909 |
||
1910 |
lemma spans_image: |
|
49663 | 1911 |
assumes lf: "linear f" |
1912 |
and VB: "V \<subseteq> span B" |
|
44133 | 1913 |
shows "f ` V \<subseteq> span (f ` B)" |
49522 | 1914 |
unfolding span_linear_image[OF lf] by (metis VB image_mono) |
44133 | 1915 |
|
1916 |
lemma dim_image_le: |
|
1917 |
fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space" |
|
49663 | 1918 |
assumes lf: "linear f" |
1919 |
shows "dim (f ` S) \<le> dim (S)" |
|
49522 | 1920 |
proof - |
44133 | 1921 |
from basis_exists[of S] obtain B where |
1922 |
B: "B \<subseteq> S" "independent B" "S \<subseteq> span B" "card B = dim S" by blast |
|
53406 | 1923 |
from B have fB: "finite B" "card B = dim S" |
1924 |
using independent_bound by blast+ |
|
44133 | 1925 |
have "dim (f ` S) \<le> card (f ` B)" |
1926 |
apply (rule span_card_ge_dim) |
|
53406 | 1927 |
using lf B fB |
1928 |
apply (auto simp add: span_linear_image spans_image subset_image_iff) |
|
49522 | 1929 |
done |
53406 | 1930 |
also have "\<dots> \<le> dim S" |
1931 |
using card_image_le[OF fB(1)] fB by simp |
|
44133 | 1932 |
finally show ?thesis . |
1933 |
qed |
|
1934 |
||
1935 |
text {* Relation between bases and injectivity/surjectivity of map. *} |
|
1936 |
||
1937 |
lemma spanning_surjective_image: |
|
1938 |
assumes us: "UNIV \<subseteq> span S" |
|
53406 | 1939 |
and lf: "linear f" |
1940 |
and sf: "surj f" |
|
44133 | 1941 |
shows "UNIV \<subseteq> span (f ` S)" |
49663 | 1942 |
proof - |
53406 | 1943 |
have "UNIV \<subseteq> f ` UNIV" |
1944 |
using sf by (auto simp add: surj_def) |
|
1945 |
also have " \<dots> \<subseteq> span (f ` S)" |
|
1946 |
using spans_image[OF lf us] . |
|
1947 |
finally show ?thesis . |
|
44133 | 1948 |
qed |
1949 |
||
1950 |
lemma independent_injective_image: |
|
49663 | 1951 |
assumes iS: "independent S" |
1952 |
and lf: "linear f" |
|
1953 |
and fi: "inj f" |
|
44133 | 1954 |
shows "independent (f ` S)" |
49663 | 1955 |
proof - |
53406 | 1956 |
{ |
1957 |
fix a |
|
49663 | 1958 |
assume a: "a \<in> S" "f a \<in> span (f ` S - {f a})" |
53406 | 1959 |
have eq: "f ` S - {f a} = f ` (S - {a})" |
1960 |
using fi by (auto simp add: inj_on_def) |
|
53716 | 1961 |
from a have "f a \<in> f ` span (S - {a})" |
53406 | 1962 |
unfolding eq span_linear_image[OF lf, of "S - {a}"] by blast |
53716 | 1963 |
then have "a \<in> span (S - {a})" |
53406 | 1964 |
using fi by (auto simp add: inj_on_def) |
1965 |
with a(1) iS have False |
|
1966 |
by (simp add: dependent_def) |
|
1967 |
} |
|
1968 |
then show ?thesis |
|
1969 |
unfolding dependent_def by blast |
|
44133 | 1970 |
qed |
1971 |
||
1972 |
text {* Picking an orthogonal replacement for a spanning set. *} |
|
1973 |
||
53406 | 1974 |
(* FIXME : Move to some general theory ?*) |
44133 | 1975 |
definition "pairwise R S \<longleftrightarrow> (\<forall>x \<in> S. \<forall>y\<in> S. x\<noteq>y \<longrightarrow> R x y)" |
1976 |
||
53406 | 1977 |
lemma vector_sub_project_orthogonal: |
1978 |
fixes b x :: "'a::euclidean_space" |
|
1979 |
shows "b \<bullet> (x - ((b \<bullet> x) / (b \<bullet> b)) *\<^sub>R b) = 0" |
|
44133 | 1980 |
unfolding inner_simps by auto |
1981 |
||
44528 | 1982 |
lemma pairwise_orthogonal_insert: |
1983 |
assumes "pairwise orthogonal S" |
|
49522 | 1984 |
and "\<And>y. y \<in> S \<Longrightarrow> orthogonal x y" |
44528 | 1985 |
shows "pairwise orthogonal (insert x S)" |
1986 |
using assms unfolding pairwise_def |
|
1987 |
by (auto simp add: orthogonal_commute) |
|
1988 |
||
44133 | 1989 |
lemma basis_orthogonal: |
53406 | 1990 |
fixes B :: "'a::real_inner set" |
44133 | 1991 |
assumes fB: "finite B" |
1992 |
shows "\<exists>C. finite C \<and> card C \<le> card B \<and> span C = span B \<and> pairwise orthogonal C" |
|
1993 |
(is " \<exists>C. ?P B C") |
|
49522 | 1994 |
using fB |
1995 |
proof (induct rule: finite_induct) |
|
1996 |
case empty |
|
53406 | 1997 |
then show ?case |
1998 |
apply (rule exI[where x="{}"]) |
|
1999 |
apply (auto simp add: pairwise_def) |
|
2000 |
done |
|
44133 | 2001 |
next |
49522 | 2002 |
case (insert a B) |
44133 | 2003 |
note fB = `finite B` and aB = `a \<notin> B` |
2004 |
from `\<exists>C. finite C \<and> card C \<le> card B \<and> span C = span B \<and> pairwise orthogonal C` |
|
2005 |
obtain C where C: "finite C" "card C \<le> card B" |
|
2006 |
"span C = span B" "pairwise orthogonal C" by blast |
|
2007 |
let ?a = "a - setsum (\<lambda>x. (x \<bullet> a / (x \<bullet> x)) *\<^sub>R x) C" |
|
2008 |
let ?C = "insert ?a C" |
|
53406 | 2009 |
from C(1) have fC: "finite ?C" |
2010 |
by simp |
|
49522 | 2011 |
from fB aB C(1,2) have cC: "card ?C \<le> card (insert a B)" |
2012 |
by (simp add: card_insert_if) |
|
53406 | 2013 |
{ |
2014 |
fix x k |
|
49522 | 2015 |
have th0: "\<And>(a::'a) b c. a - (b - c) = c + (a - b)" |
2016 |
by (simp add: field_simps) |
|
44133 | 2017 |
have "x - k *\<^sub>R (a - (\<Sum>x\<in>C. (x \<bullet> a / (x \<bullet> x)) *\<^sub>R x)) \<in> span C \<longleftrightarrow> x - k *\<^sub>R a \<in> span C" |
2018 |
apply (simp only: scaleR_right_diff_distrib th0) |
|
2019 |
apply (rule span_add_eq) |
|
2020 |
apply (rule span_mul) |
|
2021 |
apply (rule span_setsum[OF C(1)]) |
|
2022 |
apply clarify |
|
2023 |
apply (rule span_mul) |
|
49522 | 2024 |
apply (rule span_superset) |
2025 |
apply assumption |
|
53406 | 2026 |
done |
2027 |
} |
|
44133 | 2028 |
then have SC: "span ?C = span (insert a B)" |
2029 |
unfolding set_eq_iff span_breakdown_eq C(3)[symmetric] by auto |
|
53406 | 2030 |
{ |
2031 |
fix y |
|
2032 |
assume yC: "y \<in> C" |
|
2033 |
then have Cy: "C = insert y (C - {y})" |
|
2034 |
by blast |
|
2035 |
have fth: "finite (C - {y})" |
|
2036 |
using C by simp |
|
44528 | 2037 |
have "orthogonal ?a y" |
2038 |
unfolding orthogonal_def |
|
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
53939
diff
changeset
|
2039 |
unfolding inner_diff inner_setsum_left right_minus_eq |
44528 | 2040 |
unfolding setsum_diff1' [OF `finite C` `y \<in> C`] |
2041 |
apply (clarsimp simp add: inner_commute[of y a]) |
|
2042 |
apply (rule setsum_0') |
|
2043 |
apply clarsimp |
|
2044 |
apply (rule C(4)[unfolded pairwise_def orthogonal_def, rule_format]) |
|
53406 | 2045 |
using `y \<in> C` by auto |
2046 |
} |
|
44528 | 2047 |
with `pairwise orthogonal C` have CPO: "pairwise orthogonal ?C" |
2048 |
by (rule pairwise_orthogonal_insert) |
|
53406 | 2049 |
from fC cC SC CPO have "?P (insert a B) ?C" |
2050 |
by blast |
|
44133 | 2051 |
then show ?case by blast |
2052 |
qed |
|
2053 |
||
2054 |
lemma orthogonal_basis_exists: |
|
2055 |
fixes V :: "('a::euclidean_space) set" |
|
2056 |
shows "\<exists>B. independent B \<and> B \<subseteq> span V \<and> V \<subseteq> span B \<and> (card B = dim V) \<and> pairwise orthogonal B" |
|
49663 | 2057 |
proof - |
49522 | 2058 |
from basis_exists[of V] obtain B where |
53406 | 2059 |
B: "B \<subseteq> V" "independent B" "V \<subseteq> span B" "card B = dim V" |
2060 |
by blast |
|
2061 |
from B have fB: "finite B" "card B = dim V" |
|
2062 |
using independent_bound by auto |
|
44133 | 2063 |
from basis_orthogonal[OF fB(1)] obtain C where |
53406 | 2064 |
C: "finite C" "card C \<le> card B" "span C = span B" "pairwise orthogonal C" |
2065 |
by blast |
|
2066 |
from C B have CSV: "C \<subseteq> span V" |
|
2067 |
by (metis span_inc span_mono subset_trans) |
|
2068 |
from span_mono[OF B(3)] C have SVC: "span V \<subseteq> span C" |
|
2069 |
by (simp add: span_span) |
|
44133 | 2070 |
from card_le_dim_spanning[OF CSV SVC C(1)] C(2,3) fB |
53406 | 2071 |
have iC: "independent C" |
44133 | 2072 |
by (simp add: dim_span) |
53406 | 2073 |
from C fB have "card C \<le> dim V" |
2074 |
by simp |
|
2075 |
moreover have "dim V \<le> card C" |
|
2076 |
using span_card_ge_dim[OF CSV SVC C(1)] |
|
2077 |
by (simp add: dim_span) |
|
2078 |
ultimately have CdV: "card C = dim V" |
|
2079 |
using C(1) by simp |
|
2080 |
from C B CSV CdV iC show ?thesis |
|
2081 |
by auto |
|
44133 | 2082 |
qed |
2083 |
||
2084 |
lemma span_eq: "span S = span T \<longleftrightarrow> S \<subseteq> span T \<and> T \<subseteq> span S" |
|
2085 |
using span_inc[unfolded subset_eq] using span_mono[of T "span S"] span_mono[of S "span T"] |
|
49522 | 2086 |
by (auto simp add: span_span) |
44133 | 2087 |
|
2088 |
text {* Low-dimensional subset is in a hyperplane (weak orthogonal complement). *} |
|
2089 |
||
49522 | 2090 |
lemma span_not_univ_orthogonal: |
53406 | 2091 |
fixes S :: "'a::euclidean_space set" |
44133 | 2092 |
assumes sU: "span S \<noteq> UNIV" |
2093 |
shows "\<exists>(a::'a). a \<noteq>0 \<and> (\<forall>x \<in> span S. a \<bullet> x = 0)" |
|
49522 | 2094 |
proof - |
53406 | 2095 |
from sU obtain a where a: "a \<notin> span S" |
2096 |
by blast |
|
44133 | 2097 |
from orthogonal_basis_exists obtain B where |
2098 |
B: "independent B" "B \<subseteq> span S" "S \<subseteq> span B" "card B = dim S" "pairwise orthogonal B" |
|
2099 |
by blast |
|
53406 | 2100 |
from B have fB: "finite B" "card B = dim S" |
2101 |
using independent_bound by auto |
|
44133 | 2102 |
from span_mono[OF B(2)] span_mono[OF B(3)] |
53406 | 2103 |
have sSB: "span S = span B" |
2104 |
by (simp add: span_span) |
|
44133 | 2105 |
let ?a = "a - setsum (\<lambda>b. (a \<bullet> b / (b \<bullet> b)) *\<^sub>R b) B" |
2106 |
have "setsum (\<lambda>b. (a \<bullet> b / (b \<bullet> b)) *\<^sub>R b) B \<in> span S" |
|
2107 |
unfolding sSB |
|
2108 |
apply (rule span_setsum[OF fB(1)]) |
|
2109 |
apply clarsimp |
|
2110 |
apply (rule span_mul) |
|
49522 | 2111 |
apply (rule span_superset) |
2112 |
apply assumption |
|
2113 |
done |
|
53406 | 2114 |
with a have a0:"?a \<noteq> 0" |
2115 |
by auto |
|
44133 | 2116 |
have "\<forall>x\<in>span B. ?a \<bullet> x = 0" |
49522 | 2117 |
proof (rule span_induct') |
2118 |
show "subspace {x. ?a \<bullet> x = 0}" |
|
2119 |
by (auto simp add: subspace_def inner_add) |
|
2120 |
next |
|
53406 | 2121 |
{ |
2122 |
fix x |
|
2123 |
assume x: "x \<in> B" |
|
2124 |
from x have B': "B = insert x (B - {x})" |
|
2125 |
by blast |
|
2126 |
have fth: "finite (B - {x})" |
|
2127 |
using fB by simp |
|
44133 | 2128 |
have "?a \<bullet> x = 0" |
53406 | 2129 |
apply (subst B') |
2130 |
using fB fth |
|
44133 | 2131 |
unfolding setsum_clauses(2)[OF fth] |
2132 |
apply simp unfolding inner_simps |
|
44527
bf8014b4f933
remove dot_lsum and dot_rsum in favor of inner_setsum_{left,right}
huffman
parents:
44521
diff
changeset
|
2133 |
apply (clarsimp simp add: inner_add inner_setsum_left) |
44133 | 2134 |
apply (rule setsum_0', rule ballI) |
2135 |
unfolding inner_commute |
|
49711 | 2136 |
apply (auto simp add: x field_simps |
2137 |
intro: B(5)[unfolded pairwise_def orthogonal_def, rule_format]) |
|
53406 | 2138 |
done |
2139 |
} |
|
2140 |
then show "\<forall>x \<in> B. ?a \<bullet> x = 0" |
|
2141 |
by blast |
|
44133 | 2142 |
qed |
53406 | 2143 |
with a0 show ?thesis |
2144 |
unfolding sSB by (auto intro: exI[where x="?a"]) |
|
44133 | 2145 |
qed |
2146 |
||
2147 |
lemma span_not_univ_subset_hyperplane: |
|
53406 | 2148 |
fixes S :: "'a::euclidean_space set" |
2149 |
assumes SU: "span S \<noteq> UNIV" |
|
44133 | 2150 |
shows "\<exists> a. a \<noteq>0 \<and> span S \<subseteq> {x. a \<bullet> x = 0}" |
2151 |
using span_not_univ_orthogonal[OF SU] by auto |
|
2152 |
||
49663 | 2153 |
lemma lowdim_subset_hyperplane: |
53406 | 2154 |
fixes S :: "'a::euclidean_space set" |
44133 | 2155 |
assumes d: "dim S < DIM('a)" |
2156 |
shows "\<exists>(a::'a). a \<noteq> 0 \<and> span S \<subseteq> {x. a \<bullet> x = 0}" |
|
49522 | 2157 |
proof - |
53406 | 2158 |
{ |
2159 |
assume "span S = UNIV" |
|
2160 |
then have "dim (span S) = dim (UNIV :: ('a) set)" |
|
2161 |
by simp |
|
2162 |
then have "dim S = DIM('a)" |
|
2163 |
by (simp add: dim_span dim_UNIV) |
|
2164 |
with d have False by arith |
|
2165 |
} |
|
2166 |
then have th: "span S \<noteq> UNIV" |
|
2167 |
by blast |
|
44133 | 2168 |
from span_not_univ_subset_hyperplane[OF th] show ?thesis . |
2169 |
qed |
|
2170 |
||
2171 |
text {* We can extend a linear basis-basis injection to the whole set. *} |
|
2172 |
||
2173 |
lemma linear_indep_image_lemma: |
|
49663 | 2174 |
assumes lf: "linear f" |
2175 |
and fB: "finite B" |
|
49522 | 2176 |
and ifB: "independent (f ` B)" |
49663 | 2177 |
and fi: "inj_on f B" |
2178 |
and xsB: "x \<in> span B" |
|
49522 | 2179 |
and fx: "f x = 0" |
44133 | 2180 |
shows "x = 0" |
2181 |
using fB ifB fi xsB fx |
|
49522 | 2182 |
proof (induct arbitrary: x rule: finite_induct[OF fB]) |
49663 | 2183 |
case 1 |
2184 |
then show ?case by auto |
|
44133 | 2185 |
next |
2186 |
case (2 a b x) |
|
2187 |
have fb: "finite b" using "2.prems" by simp |
|
2188 |
have th0: "f ` b \<subseteq> f ` (insert a b)" |
|
53406 | 2189 |
apply (rule image_mono) |
2190 |
apply blast |
|
2191 |
done |
|
44133 | 2192 |
from independent_mono[ OF "2.prems"(2) th0] |
2193 |
have ifb: "independent (f ` b)" . |
|
2194 |
have fib: "inj_on f b" |
|
2195 |
apply (rule subset_inj_on [OF "2.prems"(3)]) |
|
49522 | 2196 |
apply blast |
2197 |
done |
|
44133 | 2198 |
from span_breakdown[of a "insert a b", simplified, OF "2.prems"(4)] |
53406 | 2199 |
obtain k where k: "x - k*\<^sub>R a \<in> span (b - {a})" |
2200 |
by blast |
|
44133 | 2201 |
have "f (x - k*\<^sub>R a) \<in> span (f ` b)" |
2202 |
unfolding span_linear_image[OF lf] |
|
2203 |
apply (rule imageI) |
|
53716 | 2204 |
using k span_mono[of "b - {a}" b] |
53406 | 2205 |
apply blast |
49522 | 2206 |
done |
2207 |
then have "f x - k*\<^sub>R f a \<in> span (f ` b)" |
|
44133 | 2208 |
by (simp add: linear_sub[OF lf] linear_cmul[OF lf]) |
49522 | 2209 |
then have th: "-k *\<^sub>R f a \<in> span (f ` b)" |
44133 | 2210 |
using "2.prems"(5) by simp |
53406 | 2211 |
have xsb: "x \<in> span b" |
2212 |
proof (cases "k = 0") |
|
2213 |
case True |
|
53716 | 2214 |
with k have "x \<in> span (b - {a})" by simp |
2215 |
then show ?thesis using span_mono[of "b - {a}" b] |
|
53406 | 2216 |
by blast |
2217 |
next |
|
2218 |
case False |
|
2219 |
with span_mul[OF th, of "- 1/ k"] |
|
44133 | 2220 |
have th1: "f a \<in> span (f ` b)" |
2221 |
by auto |
|
2222 |
from inj_on_image_set_diff[OF "2.prems"(3), of "insert a b " "{a}", symmetric] |
|
2223 |
have tha: "f ` insert a b - f ` {a} = f ` (insert a b - {a})" by blast |
|
2224 |
from "2.prems"(2) [unfolded dependent_def bex_simps(8), rule_format, of "f a"] |
|
2225 |
have "f a \<notin> span (f ` b)" using tha |
|
2226 |
using "2.hyps"(2) |
|
2227 |
"2.prems"(3) by auto |
|
2228 |
with th1 have False by blast |
|
53406 | 2229 |
then show ?thesis by blast |
2230 |
qed |
|
2231 |
from "2.hyps"(3)[OF fb ifb fib xsb "2.prems"(5)] show "x = 0" . |
|
44133 | 2232 |
qed |
2233 |
||
2234 |
text {* We can extend a linear mapping from basis. *} |
|
2235 |
||
2236 |
lemma linear_independent_extend_lemma: |
|
2237 |
fixes f :: "'a::real_vector \<Rightarrow> 'b::real_vector" |
|
53406 | 2238 |
assumes fi: "finite B" |
2239 |
and ib: "independent B" |
|
2240 |
shows "\<exists>g. |
|
2241 |
(\<forall>x\<in> span B. \<forall>y\<in> span B. g (x + y) = g x + g y) \<and> |
|
2242 |
(\<forall>x\<in> span B. \<forall>c. g (c*\<^sub>R x) = c *\<^sub>R g x) \<and> |
|
2243 |
(\<forall>x\<in> B. g x = f x)" |
|
49663 | 2244 |
using ib fi |
49522 | 2245 |
proof (induct rule: finite_induct[OF fi]) |
49663 | 2246 |
case 1 |
2247 |
then show ?case by auto |
|
44133 | 2248 |
next |
2249 |
case (2 a b) |
|
2250 |
from "2.prems" "2.hyps" have ibf: "independent b" "finite b" |
|
2251 |
by (simp_all add: independent_insert) |
|
2252 |
from "2.hyps"(3)[OF ibf] obtain g where |
|
2253 |
g: "\<forall>x\<in>span b. \<forall>y\<in>span b. g (x + y) = g x + g y" |
|
2254 |
"\<forall>x\<in>span b. \<forall>c. g (c *\<^sub>R x) = c *\<^sub>R g x" "\<forall>x\<in>b. g x = f x" by blast |
|
2255 |
let ?h = "\<lambda>z. SOME k. (z - k *\<^sub>R a) \<in> span b" |
|
53406 | 2256 |
{ |
2257 |
fix z |
|
2258 |
assume z: "z \<in> span (insert a b)" |
|
44133 | 2259 |
have th0: "z - ?h z *\<^sub>R a \<in> span b" |
2260 |
apply (rule someI_ex) |
|
2261 |
unfolding span_breakdown_eq[symmetric] |
|
53406 | 2262 |
apply (rule z) |
2263 |
done |
|
2264 |
{ |
|
2265 |
fix k |
|
2266 |
assume k: "z - k *\<^sub>R a \<in> span b" |
|
44133 | 2267 |
have eq: "z - ?h z *\<^sub>R a - (z - k*\<^sub>R a) = (k - ?h z) *\<^sub>R a" |
2268 |
by (simp add: field_simps scaleR_left_distrib [symmetric]) |
|
53406 | 2269 |
from span_sub[OF th0 k] have khz: "(k - ?h z) *\<^sub>R a \<in> span b" |
2270 |
by (simp add: eq) |
|
2271 |
{ |
|
2272 |
assume "k \<noteq> ?h z" |
|
2273 |
then have k0: "k - ?h z \<noteq> 0" by simp |
|
44133 | 2274 |
from k0 span_mul[OF khz, of "1 /(k - ?h z)"] |
2275 |
have "a \<in> span b" by simp |
|
2276 |
with "2.prems"(1) "2.hyps"(2) have False |
|
53406 | 2277 |
by (auto simp add: dependent_def) |
2278 |
} |
|
2279 |
then have "k = ?h z" by blast |
|
2280 |
} |
|
2281 |
with th0 have "z - ?h z *\<^sub>R a \<in> span b \<and> (\<forall>k. z - k *\<^sub>R a \<in> span b \<longrightarrow> k = ?h z)" |
|
2282 |
by blast |
|
2283 |
} |
|
44133 | 2284 |
note h = this |
2285 |
let ?g = "\<lambda>z. ?h z *\<^sub>R f a + g (z - ?h z *\<^sub>R a)" |
|
53406 | 2286 |
{ |
2287 |
fix x y |
|
2288 |
assume x: "x \<in> span (insert a b)" |
|
2289 |
and y: "y \<in> span (insert a b)" |
|
44133 | 2290 |
have tha: "\<And>(x::'a) y a k l. (x + y) - (k + l) *\<^sub>R a = (x - k *\<^sub>R a) + (y - l *\<^sub>R a)" |
2291 |
by (simp add: algebra_simps) |
|
2292 |
have addh: "?h (x + y) = ?h x + ?h y" |
|
2293 |
apply (rule conjunct2[OF h, rule_format, symmetric]) |
|
2294 |
apply (rule span_add[OF x y]) |
|
2295 |
unfolding tha |
|
53406 | 2296 |
apply (metis span_add x y conjunct1[OF h, rule_format]) |
2297 |
done |
|
44133 | 2298 |
have "?g (x + y) = ?g x + ?g y" |
2299 |
unfolding addh tha |
|
2300 |
g(1)[rule_format,OF conjunct1[OF h, OF x] conjunct1[OF h, OF y]] |
|
2301 |
by (simp add: scaleR_left_distrib)} |
|
2302 |
moreover |
|
53406 | 2303 |
{ |
2304 |
fix x :: "'a" |
|
2305 |
fix c :: real |
|
49522 | 2306 |
assume x: "x \<in> span (insert a b)" |
44133 | 2307 |
have tha: "\<And>(x::'a) c k a. c *\<^sub>R x - (c * k) *\<^sub>R a = c *\<^sub>R (x - k *\<^sub>R a)" |
2308 |
by (simp add: algebra_simps) |
|
2309 |
have hc: "?h (c *\<^sub>R x) = c * ?h x" |
|
2310 |
apply (rule conjunct2[OF h, rule_format, symmetric]) |
|
2311 |
apply (metis span_mul x) |
|
49522 | 2312 |
apply (metis tha span_mul x conjunct1[OF h]) |
2313 |
done |
|
44133 | 2314 |
have "?g (c *\<^sub>R x) = c*\<^sub>R ?g x" |
2315 |
unfolding hc tha g(2)[rule_format, OF conjunct1[OF h, OF x]] |
|
53406 | 2316 |
by (simp add: algebra_simps) |
2317 |
} |
|
44133 | 2318 |
moreover |
53406 | 2319 |
{ |
2320 |
fix x |
|
2321 |
assume x: "x \<in> insert a b" |
|
2322 |
{ |
|
2323 |
assume xa: "x = a" |
|
44133 | 2324 |
have ha1: "1 = ?h a" |
2325 |
apply (rule conjunct2[OF h, rule_format]) |
|
2326 |
apply (metis span_superset insertI1) |
|
2327 |
using conjunct1[OF h, OF span_superset, OF insertI1] |
|
49522 | 2328 |
apply (auto simp add: span_0) |
2329 |
done |
|
44133 | 2330 |
from xa ha1[symmetric] have "?g x = f x" |
2331 |
apply simp |
|
2332 |
using g(2)[rule_format, OF span_0, of 0] |
|
49522 | 2333 |
apply simp |
53406 | 2334 |
done |
2335 |
} |
|
44133 | 2336 |
moreover |
53406 | 2337 |
{ |
2338 |
assume xb: "x \<in> b" |
|
44133 | 2339 |
have h0: "0 = ?h x" |
2340 |
apply (rule conjunct2[OF h, rule_format]) |
|
2341 |
apply (metis span_superset x) |
|
2342 |
apply simp |
|
2343 |
apply (metis span_superset xb) |
|
2344 |
done |
|
2345 |
have "?g x = f x" |
|
53406 | 2346 |
by (simp add: h0[symmetric] g(3)[rule_format, OF xb]) |
2347 |
} |
|
2348 |
ultimately have "?g x = f x" |
|
2349 |
using x by blast |
|
2350 |
} |
|
49663 | 2351 |
ultimately show ?case |
2352 |
apply - |
|
2353 |
apply (rule exI[where x="?g"]) |
|
2354 |
apply blast |
|
2355 |
done |
|
44133 | 2356 |
qed |
2357 |
||
2358 |
lemma linear_independent_extend: |
|
53406 | 2359 |
fixes B :: "'a::euclidean_space set" |
2360 |
assumes iB: "independent B" |
|
44133 | 2361 |
shows "\<exists>g. linear g \<and> (\<forall>x\<in>B. g x = f x)" |
49522 | 2362 |
proof - |
44133 | 2363 |
from maximal_independent_subset_extend[of B UNIV] iB |
53406 | 2364 |
obtain C where C: "B \<subseteq> C" "independent C" "\<And>x. x \<in> span C" |
2365 |
by auto |
|
44133 | 2366 |
|
2367 |
from C(2) independent_bound[of C] linear_independent_extend_lemma[of C f] |
|
53406 | 2368 |
obtain g where g: |
2369 |
"(\<forall>x\<in> span C. \<forall>y\<in> span C. g (x + y) = g x + g y) \<and> |
|
2370 |
(\<forall>x\<in> span C. \<forall>c. g (c*\<^sub>R x) = c *\<^sub>R g x) \<and> |
|
2371 |
(\<forall>x\<in> C. g x = f x)" by blast |
|
2372 |
from g show ?thesis |
|
53600
8fda7ad57466
make 'linear' into a sublocale of 'bounded_linear';
huffman
parents:
53596
diff
changeset
|
2373 |
unfolding linear_iff |
53406 | 2374 |
using C |
49663 | 2375 |
apply clarsimp |
2376 |
apply blast |
|
2377 |
done |
|
44133 | 2378 |
qed |
2379 |
||
2380 |
text {* Can construct an isomorphism between spaces of same dimension. *} |
|
2381 |
||
2382 |
lemma subspace_isomorphism: |
|
53406 | 2383 |
fixes S :: "'a::euclidean_space set" |
2384 |
and T :: "'b::euclidean_space set" |
|
2385 |
assumes s: "subspace S" |
|
2386 |
and t: "subspace T" |
|
49522 | 2387 |
and d: "dim S = dim T" |
44133 | 2388 |
shows "\<exists>f. linear f \<and> f ` S = T \<and> inj_on f S" |
49522 | 2389 |
proof - |
53406 | 2390 |
from basis_exists[of S] independent_bound |
2391 |
obtain B where B: "B \<subseteq> S" "independent B" "S \<subseteq> span B" "card B = dim S" and fB: "finite B" |
|
2392 |
by blast |
|
2393 |
from basis_exists[of T] independent_bound |
|
2394 |
obtain C where C: "C \<subseteq> T" "independent C" "T \<subseteq> span C" "card C = dim T" and fC: "finite C" |
|
2395 |
by blast |
|
2396 |
from B(4) C(4) card_le_inj[of B C] d |
|
2397 |
obtain f where f: "f ` B \<subseteq> C" "inj_on f B" using `finite B` `finite C` |
|
2398 |
by auto |
|
2399 |
from linear_independent_extend[OF B(2)] |
|
2400 |
obtain g where g: "linear g" "\<forall>x\<in> B. g x = f x" |
|
2401 |
by blast |
|
2402 |
from inj_on_iff_eq_card[OF fB, of f] f(2) have "card (f ` B) = card B" |
|
44133 | 2403 |
by simp |
53406 | 2404 |
with B(4) C(4) have ceq: "card (f ` B) = card C" |
2405 |
using d by simp |
|
2406 |
have "g ` B = f ` B" |
|
2407 |
using g(2) by (auto simp add: image_iff) |
|
44133 | 2408 |
also have "\<dots> = C" using card_subset_eq[OF fC f(1) ceq] . |
2409 |
finally have gBC: "g ` B = C" . |
|
53406 | 2410 |
have gi: "inj_on g B" |
2411 |
using f(2) g(2) by (auto simp add: inj_on_def) |
|
44133 | 2412 |
note g0 = linear_indep_image_lemma[OF g(1) fB, unfolded gBC, OF C(2) gi] |
53406 | 2413 |
{ |
2414 |
fix x y |
|
2415 |
assume x: "x \<in> S" and y: "y \<in> S" and gxy: "g x = g y" |
|
2416 |
from B(3) x y have x': "x \<in> span B" and y': "y \<in> span B" |
|
2417 |
by blast+ |
|
2418 |
from gxy have th0: "g (x - y) = 0" |
|
2419 |
by (simp add: linear_sub[OF g(1)]) |
|
2420 |
have th1: "x - y \<in> span B" |
|
2421 |
using x' y' by (metis span_sub) |
|
2422 |
have "x = y" |
|
2423 |
using g0[OF th1 th0] by simp |
|
2424 |
} |
|
44133 | 2425 |
then have giS: "inj_on g S" |
2426 |
unfolding inj_on_def by blast |
|
53406 | 2427 |
from span_subspace[OF B(1,3) s] have "g ` S = span (g ` B)" |
2428 |
by (simp add: span_linear_image[OF g(1)]) |
|
44133 | 2429 |
also have "\<dots> = span C" unfolding gBC .. |
2430 |
also have "\<dots> = T" using span_subspace[OF C(1,3) t] . |
|
2431 |
finally have gS: "g ` S = T" . |
|
53406 | 2432 |
from g(1) gS giS show ?thesis |
2433 |
by blast |
|
44133 | 2434 |
qed |
2435 |
||
2436 |
text {* Linear functions are equal on a subspace if they are on a spanning set. *} |
|
2437 |
||
2438 |
lemma subspace_kernel: |
|
2439 |
assumes lf: "linear f" |
|
2440 |
shows "subspace {x. f x = 0}" |
|
49522 | 2441 |
apply (simp add: subspace_def) |
2442 |
apply (simp add: linear_add[OF lf] linear_cmul[OF lf] linear_0[OF lf]) |
|
2443 |
done |
|
44133 | 2444 |
|
2445 |
lemma linear_eq_0_span: |
|
2446 |
assumes lf: "linear f" and f0: "\<forall>x\<in>B. f x = 0" |
|
2447 |
shows "\<forall>x \<in> span B. f x = 0" |
|
44170
510ac30f44c0
make Multivariate_Analysis work with separate set type
huffman
parents:
44166
diff
changeset
|
2448 |
using f0 subspace_kernel[OF lf] |
510ac30f44c0
make Multivariate_Analysis work with separate set type
huffman
parents:
44166
diff
changeset
|
2449 |
by (rule span_induct') |
44133 | 2450 |
|
2451 |
lemma linear_eq_0: |
|
49663 | 2452 |
assumes lf: "linear f" |
2453 |
and SB: "S \<subseteq> span B" |
|
2454 |
and f0: "\<forall>x\<in>B. f x = 0" |
|
44133 | 2455 |
shows "\<forall>x \<in> S. f x = 0" |
2456 |
by (metis linear_eq_0_span[OF lf] subset_eq SB f0) |
|
2457 |
||
2458 |
lemma linear_eq: |
|
49663 | 2459 |
assumes lf: "linear f" |
2460 |
and lg: "linear g" |
|
2461 |
and S: "S \<subseteq> span B" |
|
49522 | 2462 |
and fg: "\<forall> x\<in> B. f x = g x" |
44133 | 2463 |
shows "\<forall>x\<in> S. f x = g x" |
49663 | 2464 |
proof - |
44133 | 2465 |
let ?h = "\<lambda>x. f x - g x" |
2466 |
from fg have fg': "\<forall>x\<in> B. ?h x = 0" by simp |
|
2467 |
from linear_eq_0[OF linear_compose_sub[OF lf lg] S fg'] |
|
2468 |
show ?thesis by simp |
|
2469 |
qed |
|
2470 |
||
2471 |
lemma linear_eq_stdbasis: |
|
49663 | 2472 |
assumes lf: "linear (f::'a::euclidean_space \<Rightarrow> _)" |
2473 |
and lg: "linear g" |
|
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
2474 |
and fg: "\<forall>b\<in>Basis. f b = g b" |
44133 | 2475 |
shows "f = g" |
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
2476 |
using linear_eq[OF lf lg, of _ Basis] fg by auto |
44133 | 2477 |
|
2478 |
text {* Similar results for bilinear functions. *} |
|
2479 |
||
2480 |
lemma bilinear_eq: |
|
2481 |
assumes bf: "bilinear f" |
|
49522 | 2482 |
and bg: "bilinear g" |
53406 | 2483 |
and SB: "S \<subseteq> span B" |
2484 |
and TC: "T \<subseteq> span C" |
|
49522 | 2485 |
and fg: "\<forall>x\<in> B. \<forall>y\<in> C. f x y = g x y" |
44133 | 2486 |
shows "\<forall>x\<in>S. \<forall>y\<in>T. f x y = g x y " |
49663 | 2487 |
proof - |
44170
510ac30f44c0
make Multivariate_Analysis work with separate set type
huffman
parents:
44166
diff
changeset
|
2488 |
let ?P = "{x. \<forall>y\<in> span C. f x y = g x y}" |
44133 | 2489 |
from bf bg have sp: "subspace ?P" |
53600
8fda7ad57466
make 'linear' into a sublocale of 'bounded_linear';
huffman
parents:
53596
diff
changeset
|
2490 |
unfolding bilinear_def linear_iff subspace_def bf bg |
49663 | 2491 |
by (auto simp add: span_0 bilinear_lzero[OF bf] bilinear_lzero[OF bg] span_add Ball_def |
2492 |
intro: bilinear_ladd[OF bf]) |
|
44133 | 2493 |
|
2494 |
have "\<forall>x \<in> span B. \<forall>y\<in> span C. f x y = g x y" |
|
44170
510ac30f44c0
make Multivariate_Analysis work with separate set type
huffman
parents:
44166
diff
changeset
|
2495 |
apply (rule span_induct' [OF _ sp]) |
44133 | 2496 |
apply (rule ballI) |
44170
510ac30f44c0
make Multivariate_Analysis work with separate set type
huffman
parents:
44166
diff
changeset
|
2497 |
apply (rule span_induct') |
510ac30f44c0
make Multivariate_Analysis work with separate set type
huffman
parents:
44166
diff
changeset
|
2498 |
apply (simp add: fg) |
44133 | 2499 |
apply (auto simp add: subspace_def) |
53600
8fda7ad57466
make 'linear' into a sublocale of 'bounded_linear';
huffman
parents:
53596
diff
changeset
|
2500 |
using bf bg unfolding bilinear_def linear_iff |
49522 | 2501 |
apply (auto simp add: span_0 bilinear_rzero[OF bf] bilinear_rzero[OF bg] span_add Ball_def |
49663 | 2502 |
intro: bilinear_ladd[OF bf]) |
49522 | 2503 |
done |
53406 | 2504 |
then show ?thesis |
2505 |
using SB TC by auto |
|
44133 | 2506 |
qed |
2507 |
||
49522 | 2508 |
lemma bilinear_eq_stdbasis: |
53406 | 2509 |
fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space \<Rightarrow> _" |
44133 | 2510 |
assumes bf: "bilinear f" |
49522 | 2511 |
and bg: "bilinear g" |
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
2512 |
and fg: "\<forall>i\<in>Basis. \<forall>j\<in>Basis. f i j = g i j" |
44133 | 2513 |
shows "f = g" |
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
2514 |
using bilinear_eq[OF bf bg equalityD2[OF span_Basis] equalityD2[OF span_Basis] fg] by blast |
44133 | 2515 |
|
2516 |
text {* Detailed theorems about left and right invertibility in general case. *} |
|
2517 |
||
49522 | 2518 |
lemma linear_injective_left_inverse: |
53406 | 2519 |
fixes f::"'a::euclidean_space \<Rightarrow> 'b::euclidean_space" |
44133 | 2520 |
assumes lf: "linear f" and fi: "inj f" |
2521 |
shows "\<exists>g. linear g \<and> g o f = id" |
|
49522 | 2522 |
proof - |
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
2523 |
from linear_independent_extend[OF independent_injective_image, OF independent_Basis, OF lf fi] |
53406 | 2524 |
obtain h:: "'b \<Rightarrow> 'a" where h: "linear h" "\<forall>x \<in> f ` Basis. h x = inv f x" |
2525 |
by blast |
|
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
2526 |
from h(2) have th: "\<forall>i\<in>Basis. (h \<circ> f) i = id i" |
44133 | 2527 |
using inv_o_cancel[OF fi, unfolded fun_eq_iff id_def o_def] |
2528 |
by auto |
|
2529 |
from linear_eq_stdbasis[OF linear_compose[OF lf h(1)] linear_id th] |
|
2530 |
have "h o f = id" . |
|
53406 | 2531 |
then show ?thesis |
2532 |
using h(1) by blast |
|
44133 | 2533 |
qed |
2534 |
||
49522 | 2535 |
lemma linear_surjective_right_inverse: |
53406 | 2536 |
fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space" |
2537 |
assumes lf: "linear f" |
|
2538 |
and sf: "surj f" |
|
44133 | 2539 |
shows "\<exists>g. linear g \<and> f o g = id" |
49522 | 2540 |
proof - |
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
2541 |
from linear_independent_extend[OF independent_Basis[where 'a='b],of "inv f"] |
53406 | 2542 |
obtain h:: "'b \<Rightarrow> 'a" where h: "linear h" "\<forall>x\<in>Basis. h x = inv f x" |
2543 |
by blast |
|
2544 |
from h(2) have th: "\<forall>i\<in>Basis. (f o h) i = id i" |
|
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
2545 |
using sf by (auto simp add: surj_iff_all) |
44133 | 2546 |
from linear_eq_stdbasis[OF linear_compose[OF h(1) lf] linear_id th] |
2547 |
have "f o h = id" . |
|
53406 | 2548 |
then show ?thesis |
2549 |
using h(1) by blast |
|
44133 | 2550 |
qed |
2551 |
||
2552 |
text {* An injective map @{typ "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"} is also surjective. *} |
|
2553 |
||
49522 | 2554 |
lemma linear_injective_imp_surjective: |
53406 | 2555 |
fixes f::"'a::euclidean_space \<Rightarrow> 'a::euclidean_space" |
2556 |
assumes lf: "linear f" |
|
2557 |
and fi: "inj f" |
|
44133 | 2558 |
shows "surj f" |
49522 | 2559 |
proof - |
44133 | 2560 |
let ?U = "UNIV :: 'a set" |
2561 |
from basis_exists[of ?U] obtain B |
|
2562 |
where B: "B \<subseteq> ?U" "independent B" "?U \<subseteq> span B" "card B = dim ?U" |
|
2563 |
by blast |
|
53406 | 2564 |
from B(4) have d: "dim ?U = card B" |
2565 |
by simp |
|
44133 | 2566 |
have th: "?U \<subseteq> span (f ` B)" |
2567 |
apply (rule card_ge_dim_independent) |
|
2568 |
apply blast |
|
2569 |
apply (rule independent_injective_image[OF B(2) lf fi]) |
|
2570 |
apply (rule order_eq_refl) |
|
2571 |
apply (rule sym) |
|
2572 |
unfolding d |
|
2573 |
apply (rule card_image) |
|
2574 |
apply (rule subset_inj_on[OF fi]) |
|
49522 | 2575 |
apply blast |
2576 |
done |
|
44133 | 2577 |
from th show ?thesis |
2578 |
unfolding span_linear_image[OF lf] surj_def |
|
2579 |
using B(3) by blast |
|
2580 |
qed |
|
2581 |
||
2582 |
text {* And vice versa. *} |
|
2583 |
||
2584 |
lemma surjective_iff_injective_gen: |
|
49663 | 2585 |
assumes fS: "finite S" |
2586 |
and fT: "finite T" |
|
2587 |
and c: "card S = card T" |
|
49522 | 2588 |
and ST: "f ` S \<subseteq> T" |
53406 | 2589 |
shows "(\<forall>y \<in> T. \<exists>x \<in> S. f x = y) \<longleftrightarrow> inj_on f S" |
2590 |
(is "?lhs \<longleftrightarrow> ?rhs") |
|
2591 |
proof |
|
2592 |
assume h: "?lhs" |
|
2593 |
{ |
|
2594 |
fix x y |
|
2595 |
assume x: "x \<in> S" |
|
2596 |
assume y: "y \<in> S" |
|
2597 |
assume f: "f x = f y" |
|
2598 |
from x fS have S0: "card S \<noteq> 0" |
|
2599 |
by auto |
|
2600 |
have "x = y" |
|
2601 |
proof (rule ccontr) |
|
53716 | 2602 |
assume xy: "\<not> ?thesis" |
53406 | 2603 |
have th: "card S \<le> card (f ` (S - {y}))" |
2604 |
unfolding c |
|
2605 |
apply (rule card_mono) |
|
2606 |
apply (rule finite_imageI) |
|
2607 |
using fS apply simp |
|
2608 |
using h xy x y f unfolding subset_eq image_iff |
|
2609 |
apply auto |
|
2610 |
apply (case_tac "xa = f x") |
|
2611 |
apply (rule bexI[where x=x]) |
|
2612 |
apply auto |
|
2613 |
done |
|
53716 | 2614 |
also have " \<dots> \<le> card (S - {y})" |
53406 | 2615 |
apply (rule card_image_le) |
2616 |
using fS by simp |
|
2617 |
also have "\<dots> \<le> card S - 1" using y fS by simp |
|
2618 |
finally show False using S0 by arith |
|
2619 |
qed |
|
2620 |
} |
|
2621 |
then show ?rhs |
|
2622 |
unfolding inj_on_def by blast |
|
2623 |
next |
|
2624 |
assume h: ?rhs |
|
2625 |
have "f ` S = T" |
|
2626 |
apply (rule card_subset_eq[OF fT ST]) |
|
2627 |
unfolding card_image[OF h] |
|
2628 |
apply (rule c) |
|
2629 |
done |
|
2630 |
then show ?lhs by blast |
|
44133 | 2631 |
qed |
2632 |
||
49522 | 2633 |
lemma linear_surjective_imp_injective: |
53406 | 2634 |
fixes f :: "'a::euclidean_space \<Rightarrow> 'a::euclidean_space" |
2635 |
assumes lf: "linear f" |
|
2636 |
and sf: "surj f" |
|
44133 | 2637 |
shows "inj f" |
49522 | 2638 |
proof - |
44133 | 2639 |
let ?U = "UNIV :: 'a set" |
2640 |
from basis_exists[of ?U] obtain B |
|
2641 |
where B: "B \<subseteq> ?U" "independent B" "?U \<subseteq> span B" and d: "card B = dim ?U" |
|
2642 |
by blast |
|
53406 | 2643 |
{ |
2644 |
fix x |
|
2645 |
assume x: "x \<in> span B" |
|
2646 |
assume fx: "f x = 0" |
|
2647 |
from B(2) have fB: "finite B" |
|
2648 |
using independent_bound by auto |
|
44133 | 2649 |
have fBi: "independent (f ` B)" |
2650 |
apply (rule card_le_dim_spanning[of "f ` B" ?U]) |
|
2651 |
apply blast |
|
2652 |
using sf B(3) |
|
2653 |
unfolding span_linear_image[OF lf] surj_def subset_eq image_iff |
|
2654 |
apply blast |
|
2655 |
using fB apply blast |
|
2656 |
unfolding d[symmetric] |
|
2657 |
apply (rule card_image_le) |
|
2658 |
apply (rule fB) |
|
2659 |
done |
|
2660 |
have th0: "dim ?U \<le> card (f ` B)" |
|
2661 |
apply (rule span_card_ge_dim) |
|
2662 |
apply blast |
|
2663 |
unfolding span_linear_image[OF lf] |
|
2664 |
apply (rule subset_trans[where B = "f ` UNIV"]) |
|
53406 | 2665 |
using sf unfolding surj_def |
2666 |
apply blast |
|
44133 | 2667 |
apply (rule image_mono) |
2668 |
apply (rule B(3)) |
|
2669 |
apply (metis finite_imageI fB) |
|
2670 |
done |
|
2671 |
moreover have "card (f ` B) \<le> card B" |
|
2672 |
by (rule card_image_le, rule fB) |
|
53406 | 2673 |
ultimately have th1: "card B = card (f ` B)" |
2674 |
unfolding d by arith |
|
44133 | 2675 |
have fiB: "inj_on f B" |
49522 | 2676 |
unfolding surjective_iff_injective_gen[OF fB finite_imageI[OF fB] th1 subset_refl, symmetric] |
2677 |
by blast |
|
44133 | 2678 |
from linear_indep_image_lemma[OF lf fB fBi fiB x] fx |
53406 | 2679 |
have "x = 0" by blast |
2680 |
} |
|
2681 |
then show ?thesis |
|
2682 |
unfolding linear_injective_0[OF lf] |
|
2683 |
using B(3) |
|
2684 |
by blast |
|
44133 | 2685 |
qed |
2686 |
||
2687 |
text {* Hence either is enough for isomorphism. *} |
|
2688 |
||
2689 |
lemma left_right_inverse_eq: |
|
53406 | 2690 |
assumes fg: "f \<circ> g = id" |
2691 |
and gh: "g \<circ> h = id" |
|
44133 | 2692 |
shows "f = h" |
49522 | 2693 |
proof - |
53406 | 2694 |
have "f = f \<circ> (g \<circ> h)" |
2695 |
unfolding gh by simp |
|
2696 |
also have "\<dots> = (f \<circ> g) \<circ> h" |
|
2697 |
by (simp add: o_assoc) |
|
2698 |
finally show "f = h" |
|
2699 |
unfolding fg by simp |
|
44133 | 2700 |
qed |
2701 |
||
2702 |
lemma isomorphism_expand: |
|
53406 | 2703 |
"f \<circ> g = id \<and> g \<circ> f = id \<longleftrightarrow> (\<forall>x. f (g x) = x) \<and> (\<forall>x. g (f x) = x)" |
44133 | 2704 |
by (simp add: fun_eq_iff o_def id_def) |
2705 |
||
49522 | 2706 |
lemma linear_injective_isomorphism: |
53406 | 2707 |
fixes f::"'a::euclidean_space \<Rightarrow> 'a::euclidean_space" |
2708 |
assumes lf: "linear f" |
|
2709 |
and fi: "inj f" |
|
44133 | 2710 |
shows "\<exists>f'. linear f' \<and> (\<forall>x. f' (f x) = x) \<and> (\<forall>x. f (f' x) = x)" |
49522 | 2711 |
unfolding isomorphism_expand[symmetric] |
2712 |
using linear_surjective_right_inverse[OF lf linear_injective_imp_surjective[OF lf fi]] |
|
2713 |
linear_injective_left_inverse[OF lf fi] |
|
2714 |
by (metis left_right_inverse_eq) |
|
44133 | 2715 |
|
53406 | 2716 |
lemma linear_surjective_isomorphism: |
2717 |
fixes f :: "'a::euclidean_space \<Rightarrow> 'a::euclidean_space" |
|
2718 |
assumes lf: "linear f" |
|
2719 |
and sf: "surj f" |
|
44133 | 2720 |
shows "\<exists>f'. linear f' \<and> (\<forall>x. f' (f x) = x) \<and> (\<forall>x. f (f' x) = x)" |
49522 | 2721 |
unfolding isomorphism_expand[symmetric] |
2722 |
using linear_surjective_right_inverse[OF lf sf] |
|
2723 |
linear_injective_left_inverse[OF lf linear_surjective_imp_injective[OF lf sf]] |
|
2724 |
by (metis left_right_inverse_eq) |
|
44133 | 2725 |
|
53406 | 2726 |
text {* Left and right inverses are the same for |
2727 |
@{typ "'a::euclidean_space \<Rightarrow> 'a::euclidean_space"}. *} |
|
44133 | 2728 |
|
49522 | 2729 |
lemma linear_inverse_left: |
53406 | 2730 |
fixes f :: "'a::euclidean_space \<Rightarrow> 'a::euclidean_space" |
2731 |
assumes lf: "linear f" |
|
2732 |
and lf': "linear f'" |
|
2733 |
shows "f \<circ> f' = id \<longleftrightarrow> f' \<circ> f = id" |
|
49522 | 2734 |
proof - |
53406 | 2735 |
{ |
2736 |
fix f f':: "'a \<Rightarrow> 'a" |
|
2737 |
assume lf: "linear f" "linear f'" |
|
2738 |
assume f: "f \<circ> f' = id" |
|
44133 | 2739 |
from f have sf: "surj f" |
2740 |
apply (auto simp add: o_def id_def surj_def) |
|
49522 | 2741 |
apply metis |
2742 |
done |
|
44133 | 2743 |
from linear_surjective_isomorphism[OF lf(1) sf] lf f |
53406 | 2744 |
have "f' \<circ> f = id" |
2745 |
unfolding fun_eq_iff o_def id_def by metis |
|
2746 |
} |
|
2747 |
then show ?thesis |
|
2748 |
using lf lf' by metis |
|
44133 | 2749 |
qed |
2750 |
||
2751 |
text {* Moreover, a one-sided inverse is automatically linear. *} |
|
2752 |
||
49522 | 2753 |
lemma left_inverse_linear: |
53406 | 2754 |
fixes f :: "'a::euclidean_space \<Rightarrow> 'a::euclidean_space" |
2755 |
assumes lf: "linear f" |
|
2756 |
and gf: "g \<circ> f = id" |
|
44133 | 2757 |
shows "linear g" |
49522 | 2758 |
proof - |
2759 |
from gf have fi: "inj f" |
|
2760 |
apply (auto simp add: inj_on_def o_def id_def fun_eq_iff) |
|
2761 |
apply metis |
|
2762 |
done |
|
44133 | 2763 |
from linear_injective_isomorphism[OF lf fi] |
53406 | 2764 |
obtain h :: "'a \<Rightarrow> 'a" where h: "linear h" "\<forall>x. h (f x) = x" "\<forall>x. f (h x) = x" |
2765 |
by blast |
|
49522 | 2766 |
have "h = g" |
2767 |
apply (rule ext) using gf h(2,3) |
|
44133 | 2768 |
apply (simp add: o_def id_def fun_eq_iff) |
49522 | 2769 |
apply metis |
2770 |
done |
|
44133 | 2771 |
with h(1) show ?thesis by blast |
2772 |
qed |
|
2773 |
||
49522 | 2774 |
|
44133 | 2775 |
subsection {* Infinity norm *} |
2776 |
||
53716 | 2777 |
definition "infnorm (x::'a::euclidean_space) = Sup {abs (x \<bullet> b) |b. b \<in> Basis}" |
44133 | 2778 |
|
2779 |
lemma infnorm_set_image: |
|
53716 | 2780 |
fixes x :: "'a::euclidean_space" |
2781 |
shows "{abs (x \<bullet> i) |i. i \<in> Basis} = (\<lambda>i. abs (x \<bullet> i)) ` Basis" |
|
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
2782 |
by blast |
44133 | 2783 |
|
53716 | 2784 |
lemma infnorm_Max: |
2785 |
fixes x :: "'a::euclidean_space" |
|
2786 |
shows "infnorm x = Max ((\<lambda>i. abs (x \<bullet> i)) ` Basis)" |
|
51475
ebf9d4fd00ba
introduct the conditional_complete_lattice type class; generalize theorems about real Sup and Inf to it
hoelzl
parents:
50526
diff
changeset
|
2787 |
by (simp add: infnorm_def infnorm_set_image cSup_eq_Max) |
ebf9d4fd00ba
introduct the conditional_complete_lattice type class; generalize theorems about real Sup and Inf to it
hoelzl
parents:
50526
diff
changeset
|
2788 |
|
44133 | 2789 |
lemma infnorm_set_lemma: |
53716 | 2790 |
fixes x :: "'a::euclidean_space" |
2791 |
shows "finite {abs (x \<bullet> i) |i. i \<in> Basis}" |
|
2792 |
and "{abs (x \<bullet> i) |i. i \<in> Basis} \<noteq> {}" |
|
44133 | 2793 |
unfolding infnorm_set_image |
2794 |
by auto |
|
2795 |
||
53406 | 2796 |
lemma infnorm_pos_le: |
2797 |
fixes x :: "'a::euclidean_space" |
|
2798 |
shows "0 \<le> infnorm x" |
|
51475
ebf9d4fd00ba
introduct the conditional_complete_lattice type class; generalize theorems about real Sup and Inf to it
hoelzl
parents:
50526
diff
changeset
|
2799 |
by (simp add: infnorm_Max Max_ge_iff ex_in_conv) |
44133 | 2800 |
|
53406 | 2801 |
lemma infnorm_triangle: |
2802 |
fixes x :: "'a::euclidean_space" |
|
2803 |
shows "infnorm (x + y) \<le> infnorm x + infnorm y" |
|
49522 | 2804 |
proof - |
51475
ebf9d4fd00ba
introduct the conditional_complete_lattice type class; generalize theorems about real Sup and Inf to it
hoelzl
parents:
50526
diff
changeset
|
2805 |
have *: "\<And>a b c d :: real. \<bar>a\<bar> \<le> c \<Longrightarrow> \<bar>b\<bar> \<le> d \<Longrightarrow> \<bar>a + b\<bar> \<le> c + d" |
ebf9d4fd00ba
introduct the conditional_complete_lattice type class; generalize theorems about real Sup and Inf to it
hoelzl
parents:
50526
diff
changeset
|
2806 |
by simp |
44133 | 2807 |
show ?thesis |
51475
ebf9d4fd00ba
introduct the conditional_complete_lattice type class; generalize theorems about real Sup and Inf to it
hoelzl
parents:
50526
diff
changeset
|
2808 |
by (auto simp: infnorm_Max inner_add_left intro!: *) |
44133 | 2809 |
qed |
2810 |
||
53406 | 2811 |
lemma infnorm_eq_0: |
2812 |
fixes x :: "'a::euclidean_space" |
|
2813 |
shows "infnorm x = 0 \<longleftrightarrow> x = 0" |
|
49522 | 2814 |
proof - |
51475
ebf9d4fd00ba
introduct the conditional_complete_lattice type class; generalize theorems about real Sup and Inf to it
hoelzl
parents:
50526
diff
changeset
|
2815 |
have "infnorm x \<le> 0 \<longleftrightarrow> x = 0" |
ebf9d4fd00ba
introduct the conditional_complete_lattice type class; generalize theorems about real Sup and Inf to it
hoelzl
parents:
50526
diff
changeset
|
2816 |
unfolding infnorm_Max by (simp add: euclidean_all_zero_iff) |
ebf9d4fd00ba
introduct the conditional_complete_lattice type class; generalize theorems about real Sup and Inf to it
hoelzl
parents:
50526
diff
changeset
|
2817 |
then show ?thesis |
ebf9d4fd00ba
introduct the conditional_complete_lattice type class; generalize theorems about real Sup and Inf to it
hoelzl
parents:
50526
diff
changeset
|
2818 |
using infnorm_pos_le[of x] by simp |
44133 | 2819 |
qed |
2820 |
||
2821 |
lemma infnorm_0: "infnorm 0 = 0" |
|
2822 |
by (simp add: infnorm_eq_0) |
|
2823 |
||
2824 |
lemma infnorm_neg: "infnorm (- x) = infnorm x" |
|
2825 |
unfolding infnorm_def |
|
2826 |
apply (rule cong[of "Sup" "Sup"]) |
|
49522 | 2827 |
apply blast |
2828 |
apply auto |
|
2829 |
done |
|
44133 | 2830 |
|
2831 |
lemma infnorm_sub: "infnorm (x - y) = infnorm (y - x)" |
|
49522 | 2832 |
proof - |
44133 | 2833 |
have "y - x = - (x - y)" by simp |
53406 | 2834 |
then show ?thesis |
2835 |
by (metis infnorm_neg) |
|
44133 | 2836 |
qed |
2837 |
||
53406 | 2838 |
lemma real_abs_sub_infnorm: "\<bar>infnorm x - infnorm y\<bar> \<le> infnorm (x - y)" |
49522 | 2839 |
proof - |
53406 | 2840 |
have th: "\<And>(nx::real) n ny. nx \<le> n + ny \<Longrightarrow> ny <= n + nx \<Longrightarrow> \<bar>nx - ny\<bar> \<le> n" |
44133 | 2841 |
by arith |
2842 |
from infnorm_triangle[of "x - y" " y"] infnorm_triangle[of "x - y" "-x"] |
|
2843 |
have ths: "infnorm x \<le> infnorm (x - y) + infnorm y" |
|
2844 |
"infnorm y \<le> infnorm (x - y) + infnorm x" |
|
44454 | 2845 |
by (simp_all add: field_simps infnorm_neg) |
53406 | 2846 |
from th[OF ths] show ?thesis . |
44133 | 2847 |
qed |
2848 |
||
53406 | 2849 |
lemma real_abs_infnorm: "\<bar>infnorm x\<bar> = infnorm x" |
44133 | 2850 |
using infnorm_pos_le[of x] by arith |
2851 |
||
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
2852 |
lemma Basis_le_infnorm: |
53406 | 2853 |
fixes x :: "'a::euclidean_space" |
2854 |
shows "b \<in> Basis \<Longrightarrow> \<bar>x \<bullet> b\<bar> \<le> infnorm x" |
|
51475
ebf9d4fd00ba
introduct the conditional_complete_lattice type class; generalize theorems about real Sup and Inf to it
hoelzl
parents:
50526
diff
changeset
|
2855 |
by (simp add: infnorm_Max) |
44133 | 2856 |
|
53716 | 2857 |
lemma infnorm_mul: "infnorm (a *\<^sub>R x) = abs a * infnorm x" |
51475
ebf9d4fd00ba
introduct the conditional_complete_lattice type class; generalize theorems about real Sup and Inf to it
hoelzl
parents:
50526
diff
changeset
|
2858 |
unfolding infnorm_Max |
ebf9d4fd00ba
introduct the conditional_complete_lattice type class; generalize theorems about real Sup and Inf to it
hoelzl
parents:
50526
diff
changeset
|
2859 |
proof (safe intro!: Max_eqI) |
ebf9d4fd00ba
introduct the conditional_complete_lattice type class; generalize theorems about real Sup and Inf to it
hoelzl
parents:
50526
diff
changeset
|
2860 |
let ?B = "(\<lambda>i. \<bar>x \<bullet> i\<bar>) ` Basis" |
53406 | 2861 |
{ |
2862 |
fix b :: 'a |
|
2863 |
assume "b \<in> Basis" |
|
2864 |
then show "\<bar>a *\<^sub>R x \<bullet> b\<bar> \<le> \<bar>a\<bar> * Max ?B" |
|
2865 |
by (simp add: abs_mult mult_left_mono) |
|
2866 |
next |
|
2867 |
from Max_in[of ?B] obtain b where "b \<in> Basis" "Max ?B = \<bar>x \<bullet> b\<bar>" |
|
2868 |
by (auto simp del: Max_in) |
|
2869 |
then show "\<bar>a\<bar> * Max ((\<lambda>i. \<bar>x \<bullet> i\<bar>) ` Basis) \<in> (\<lambda>i. \<bar>a *\<^sub>R x \<bullet> i\<bar>) ` Basis" |
|
2870 |
by (intro image_eqI[where x=b]) (auto simp: abs_mult) |
|
2871 |
} |
|
51475
ebf9d4fd00ba
introduct the conditional_complete_lattice type class; generalize theorems about real Sup and Inf to it
hoelzl
parents:
50526
diff
changeset
|
2872 |
qed simp |
ebf9d4fd00ba
introduct the conditional_complete_lattice type class; generalize theorems about real Sup and Inf to it
hoelzl
parents:
50526
diff
changeset
|
2873 |
|
53406 | 2874 |
lemma infnorm_mul_lemma: "infnorm (a *\<^sub>R x) \<le> \<bar>a\<bar> * infnorm x" |
51475
ebf9d4fd00ba
introduct the conditional_complete_lattice type class; generalize theorems about real Sup and Inf to it
hoelzl
parents:
50526
diff
changeset
|
2875 |
unfolding infnorm_mul .. |
44133 | 2876 |
|
2877 |
lemma infnorm_pos_lt: "infnorm x > 0 \<longleftrightarrow> x \<noteq> 0" |
|
2878 |
using infnorm_pos_le[of x] infnorm_eq_0[of x] by arith |
|
2879 |
||
2880 |
text {* Prove that it differs only up to a bound from Euclidean norm. *} |
|
2881 |
||
2882 |
lemma infnorm_le_norm: "infnorm x \<le> norm x" |
|
51475
ebf9d4fd00ba
introduct the conditional_complete_lattice type class; generalize theorems about real Sup and Inf to it
hoelzl
parents:
50526
diff
changeset
|
2883 |
by (simp add: Basis_le_norm infnorm_Max) |
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
2884 |
|
54776
db890d9fc5c2
ordered_euclidean_space compatible with more standard pointwise ordering on products; conditionally complete lattice with product order
immler
parents:
54703
diff
changeset
|
2885 |
lemma (in euclidean_space) euclidean_inner: "inner x y = (\<Sum>b\<in>Basis. (x \<bullet> b) * (y \<bullet> b))" |
db890d9fc5c2
ordered_euclidean_space compatible with more standard pointwise ordering on products; conditionally complete lattice with product order
immler
parents:
54703
diff
changeset
|
2886 |
by (subst (1 2) euclidean_representation[symmetric]) |
db890d9fc5c2
ordered_euclidean_space compatible with more standard pointwise ordering on products; conditionally complete lattice with product order
immler
parents:
54703
diff
changeset
|
2887 |
(simp add: inner_setsum_left inner_setsum_right setsum_cases inner_Basis ac_simps if_distrib) |
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
2888 |
|
53716 | 2889 |
lemma norm_le_infnorm: |
2890 |
fixes x :: "'a::euclidean_space" |
|
2891 |
shows "norm x \<le> sqrt DIM('a) * infnorm x" |
|
49522 | 2892 |
proof - |
44133 | 2893 |
let ?d = "DIM('a)" |
53406 | 2894 |
have "real ?d \<ge> 0" |
2895 |
by simp |
|
53077 | 2896 |
then have d2: "(sqrt (real ?d))\<^sup>2 = real ?d" |
44133 | 2897 |
by (auto intro: real_sqrt_pow2) |
2898 |
have th: "sqrt (real ?d) * infnorm x \<ge> 0" |
|
2899 |
by (simp add: zero_le_mult_iff infnorm_pos_le) |
|
53077 | 2900 |
have th1: "x \<bullet> x \<le> (sqrt (real ?d) * infnorm x)\<^sup>2" |
44133 | 2901 |
unfolding power_mult_distrib d2 |
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
2902 |
unfolding real_of_nat_def |
53716 | 2903 |
apply (subst euclidean_inner) |
44133 | 2904 |
apply (subst power2_abs[symmetric]) |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
51478
diff
changeset
|
2905 |
apply (rule order_trans[OF setsum_bounded[where K="\<bar>infnorm x\<bar>\<^sup>2"]]) |
49663 | 2906 |
apply (auto simp add: power2_eq_square[symmetric]) |
44133 | 2907 |
apply (subst power2_abs[symmetric]) |
2908 |
apply (rule power_mono) |
|
51475
ebf9d4fd00ba
introduct the conditional_complete_lattice type class; generalize theorems about real Sup and Inf to it
hoelzl
parents:
50526
diff
changeset
|
2909 |
apply (auto simp: infnorm_Max) |
49522 | 2910 |
done |
44133 | 2911 |
from real_le_lsqrt[OF inner_ge_zero th th1] |
53406 | 2912 |
show ?thesis |
2913 |
unfolding norm_eq_sqrt_inner id_def . |
|
44133 | 2914 |
qed |
2915 |
||
44646 | 2916 |
lemma tendsto_infnorm [tendsto_intros]: |
49522 | 2917 |
assumes "(f ---> a) F" |
2918 |
shows "((\<lambda>x. infnorm (f x)) ---> infnorm a) F" |
|
44646 | 2919 |
proof (rule tendsto_compose [OF LIM_I assms]) |
53406 | 2920 |
fix r :: real |
2921 |
assume "r > 0" |
|
49522 | 2922 |
then show "\<exists>s>0. \<forall>x. x \<noteq> a \<and> norm (x - a) < s \<longrightarrow> norm (infnorm x - infnorm a) < r" |
44646 | 2923 |
by (metis real_norm_def le_less_trans real_abs_sub_infnorm infnorm_le_norm) |
2924 |
qed |
|
2925 |
||
44133 | 2926 |
text {* Equality in Cauchy-Schwarz and triangle inequalities. *} |
2927 |
||
53406 | 2928 |
lemma norm_cauchy_schwarz_eq: "x \<bullet> y = norm x * norm y \<longleftrightarrow> norm x *\<^sub>R y = norm y *\<^sub>R x" |
2929 |
(is "?lhs \<longleftrightarrow> ?rhs") |
|
49522 | 2930 |
proof - |
53406 | 2931 |
{ |
2932 |
assume h: "x = 0" |
|
2933 |
then have ?thesis by simp |
|
2934 |
} |
|
44133 | 2935 |
moreover |
53406 | 2936 |
{ |
2937 |
assume h: "y = 0" |
|
2938 |
then have ?thesis by simp |
|
2939 |
} |
|
44133 | 2940 |
moreover |
53406 | 2941 |
{ |
2942 |
assume x: "x \<noteq> 0" and y: "y \<noteq> 0" |
|
44133 | 2943 |
from inner_eq_zero_iff[of "norm y *\<^sub>R x - norm x *\<^sub>R y"] |
49522 | 2944 |
have "?rhs \<longleftrightarrow> |
2945 |
(norm y * (norm y * norm x * norm x - norm x * (x \<bullet> y)) - |
|
2946 |
norm x * (norm y * (y \<bullet> x) - norm x * norm y * norm y) = 0)" |
|
44133 | 2947 |
using x y |
2948 |
unfolding inner_simps |
|
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
53939
diff
changeset
|
2949 |
unfolding power2_norm_eq_inner[symmetric] power2_eq_square right_minus_eq |
49522 | 2950 |
apply (simp add: inner_commute) |
2951 |
apply (simp add: field_simps) |
|
2952 |
apply metis |
|
2953 |
done |
|
44133 | 2954 |
also have "\<dots> \<longleftrightarrow> (2 * norm x * norm y * (norm x * norm y - x \<bullet> y) = 0)" using x y |
2955 |
by (simp add: field_simps inner_commute) |
|
2956 |
also have "\<dots> \<longleftrightarrow> ?lhs" using x y |
|
2957 |
apply simp |
|
49522 | 2958 |
apply metis |
2959 |
done |
|
53406 | 2960 |
finally have ?thesis by blast |
2961 |
} |
|
44133 | 2962 |
ultimately show ?thesis by blast |
2963 |
qed |
|
2964 |
||
2965 |
lemma norm_cauchy_schwarz_abs_eq: |
|
53716 | 2966 |
"abs (x \<bullet> y) = norm x * norm y \<longleftrightarrow> |
2967 |
norm x *\<^sub>R y = norm y *\<^sub>R x \<or> norm x *\<^sub>R y = - norm y *\<^sub>R x" |
|
53406 | 2968 |
(is "?lhs \<longleftrightarrow> ?rhs") |
49522 | 2969 |
proof - |
53406 | 2970 |
have th: "\<And>(x::real) a. a \<ge> 0 \<Longrightarrow> abs x = a \<longleftrightarrow> x = a \<or> x = - a" |
2971 |
by arith |
|
44133 | 2972 |
have "?rhs \<longleftrightarrow> norm x *\<^sub>R y = norm y *\<^sub>R x \<or> norm (- x) *\<^sub>R y = norm y *\<^sub>R (- x)" |
2973 |
by simp |
|
53406 | 2974 |
also have "\<dots> \<longleftrightarrow>(x \<bullet> y = norm x * norm y \<or> (- x) \<bullet> y = norm x * norm y)" |
44133 | 2975 |
unfolding norm_cauchy_schwarz_eq[symmetric] |
2976 |
unfolding norm_minus_cancel norm_scaleR .. |
|
2977 |
also have "\<dots> \<longleftrightarrow> ?lhs" |
|
53406 | 2978 |
unfolding th[OF mult_nonneg_nonneg, OF norm_ge_zero[of x] norm_ge_zero[of y]] inner_simps |
2979 |
by auto |
|
44133 | 2980 |
finally show ?thesis .. |
2981 |
qed |
|
2982 |
||
2983 |
lemma norm_triangle_eq: |
|
2984 |
fixes x y :: "'a::real_inner" |
|
53406 | 2985 |
shows "norm (x + y) = norm x + norm y \<longleftrightarrow> norm x *\<^sub>R y = norm y *\<^sub>R x" |
49522 | 2986 |
proof - |
53406 | 2987 |
{ |
2988 |
assume x: "x = 0 \<or> y = 0" |
|
2989 |
then have ?thesis |
|
2990 |
by (cases "x = 0") simp_all |
|
2991 |
} |
|
44133 | 2992 |
moreover |
53406 | 2993 |
{ |
2994 |
assume x: "x \<noteq> 0" and y: "y \<noteq> 0" |
|
49522 | 2995 |
then have "norm x \<noteq> 0" "norm y \<noteq> 0" |
44133 | 2996 |
by simp_all |
49522 | 2997 |
then have n: "norm x > 0" "norm y > 0" |
2998 |
using norm_ge_zero[of x] norm_ge_zero[of y] by arith+ |
|
53406 | 2999 |
have th: "\<And>(a::real) b c. a + b + c \<noteq> 0 \<Longrightarrow> a = b + c \<longleftrightarrow> a\<^sup>2 = (b + c)\<^sup>2" |
49522 | 3000 |
by algebra |
53077 | 3001 |
have "norm (x + y) = norm x + norm y \<longleftrightarrow> (norm (x + y))\<^sup>2 = (norm x + norm y)\<^sup>2" |
53406 | 3002 |
apply (rule th) |
3003 |
using n norm_ge_zero[of "x + y"] |
|
49522 | 3004 |
apply arith |
3005 |
done |
|
44133 | 3006 |
also have "\<dots> \<longleftrightarrow> norm x *\<^sub>R y = norm y *\<^sub>R x" |
3007 |
unfolding norm_cauchy_schwarz_eq[symmetric] |
|
3008 |
unfolding power2_norm_eq_inner inner_simps |
|
3009 |
by (simp add: power2_norm_eq_inner[symmetric] power2_eq_square inner_commute field_simps) |
|
53406 | 3010 |
finally have ?thesis . |
3011 |
} |
|
44133 | 3012 |
ultimately show ?thesis by blast |
3013 |
qed |
|
3014 |
||
49522 | 3015 |
|
44133 | 3016 |
subsection {* Collinearity *} |
3017 |
||
49522 | 3018 |
definition collinear :: "'a::real_vector set \<Rightarrow> bool" |
3019 |
where "collinear S \<longleftrightarrow> (\<exists>u. \<forall>x \<in> S. \<forall> y \<in> S. \<exists>c. x - y = c *\<^sub>R u)" |
|
44133 | 3020 |
|
53406 | 3021 |
lemma collinear_empty: "collinear {}" |
3022 |
by (simp add: collinear_def) |
|
44133 | 3023 |
|
3024 |
lemma collinear_sing: "collinear {x}" |
|
3025 |
by (simp add: collinear_def) |
|
3026 |
||
3027 |
lemma collinear_2: "collinear {x, y}" |
|
3028 |
apply (simp add: collinear_def) |
|
3029 |
apply (rule exI[where x="x - y"]) |
|
3030 |
apply auto |
|
3031 |
apply (rule exI[where x=1], simp) |
|
3032 |
apply (rule exI[where x="- 1"], simp) |
|
3033 |
done |
|
3034 |
||
53406 | 3035 |
lemma collinear_lemma: "collinear {0,x,y} \<longleftrightarrow> x = 0 \<or> y = 0 \<or> (\<exists>c. y = c *\<^sub>R x)" |
3036 |
(is "?lhs \<longleftrightarrow> ?rhs") |
|
49522 | 3037 |
proof - |
53406 | 3038 |
{ |
3039 |
assume "x = 0 \<or> y = 0" |
|
3040 |
then have ?thesis |
|
3041 |
by (cases "x = 0") (simp_all add: collinear_2 insert_commute) |
|
3042 |
} |
|
44133 | 3043 |
moreover |
53406 | 3044 |
{ |
3045 |
assume x: "x \<noteq> 0" and y: "y \<noteq> 0" |
|
3046 |
have ?thesis |
|
3047 |
proof |
|
3048 |
assume h: "?lhs" |
|
49522 | 3049 |
then obtain u where u: "\<forall> x\<in> {0,x,y}. \<forall>y\<in> {0,x,y}. \<exists>c. x - y = c *\<^sub>R u" |
3050 |
unfolding collinear_def by blast |
|
44133 | 3051 |
from u[rule_format, of x 0] u[rule_format, of y 0] |
3052 |
obtain cx and cy where |
|
3053 |
cx: "x = cx *\<^sub>R u" and cy: "y = cy *\<^sub>R u" |
|
3054 |
by auto |
|
3055 |
from cx x have cx0: "cx \<noteq> 0" by auto |
|
3056 |
from cy y have cy0: "cy \<noteq> 0" by auto |
|
3057 |
let ?d = "cy / cx" |
|
3058 |
from cx cy cx0 have "y = ?d *\<^sub>R x" |
|
3059 |
by simp |
|
53406 | 3060 |
then show ?rhs using x y by blast |
3061 |
next |
|
3062 |
assume h: "?rhs" |
|
3063 |
then obtain c where c: "y = c *\<^sub>R x" |
|
3064 |
using x y by blast |
|
3065 |
show ?lhs |
|
3066 |
unfolding collinear_def c |
|
44133 | 3067 |
apply (rule exI[where x=x]) |
3068 |
apply auto |
|
3069 |
apply (rule exI[where x="- 1"], simp) |
|
3070 |
apply (rule exI[where x= "-c"], simp) |
|
3071 |
apply (rule exI[where x=1], simp) |
|
3072 |
apply (rule exI[where x="1 - c"], simp add: scaleR_left_diff_distrib) |
|
3073 |
apply (rule exI[where x="c - 1"], simp add: scaleR_left_diff_distrib) |
|
53406 | 3074 |
done |
3075 |
qed |
|
3076 |
} |
|
44133 | 3077 |
ultimately show ?thesis by blast |
3078 |
qed |
|
3079 |
||
53406 | 3080 |
lemma norm_cauchy_schwarz_equal: "abs (x \<bullet> y) = norm x * norm y \<longleftrightarrow> collinear {0, x, y}" |
49522 | 3081 |
unfolding norm_cauchy_schwarz_abs_eq |
3082 |
apply (cases "x=0", simp_all add: collinear_2) |
|
3083 |
apply (cases "y=0", simp_all add: collinear_2 insert_commute) |
|
3084 |
unfolding collinear_lemma |
|
3085 |
apply simp |
|
3086 |
apply (subgoal_tac "norm x \<noteq> 0") |
|
3087 |
apply (subgoal_tac "norm y \<noteq> 0") |
|
3088 |
apply (rule iffI) |
|
3089 |
apply (cases "norm x *\<^sub>R y = norm y *\<^sub>R x") |
|
3090 |
apply (rule exI[where x="(1/norm x) * norm y"]) |
|
3091 |
apply (drule sym) |
|
3092 |
unfolding scaleR_scaleR[symmetric] |
|
3093 |
apply (simp add: field_simps) |
|
3094 |
apply (rule exI[where x="(1/norm x) * - norm y"]) |
|
3095 |
apply clarify |
|
3096 |
apply (drule sym) |
|
3097 |
unfolding scaleR_scaleR[symmetric] |
|
3098 |
apply (simp add: field_simps) |
|
3099 |
apply (erule exE) |
|
3100 |
apply (erule ssubst) |
|
3101 |
unfolding scaleR_scaleR |
|
3102 |
unfolding norm_scaleR |
|
3103 |
apply (subgoal_tac "norm x * c = \<bar>c\<bar> * norm x \<or> norm x * c = - \<bar>c\<bar> * norm x") |
|
53406 | 3104 |
apply (case_tac "c \<le> 0", simp add: field_simps) |
49522 | 3105 |
apply (simp add: field_simps) |
53406 | 3106 |
apply (case_tac "c \<le> 0", simp add: field_simps) |
49522 | 3107 |
apply (simp add: field_simps) |
3108 |
apply simp |
|
3109 |
apply simp |
|
3110 |
done |
|
3111 |
||
44133 | 3112 |
|
44529
d4d9ea33703c
arrange everything related to ordered_euclidean_space class together
huffman
parents:
44528
diff
changeset
|
3113 |
subsection {* An ordering on euclidean spaces that will allow us to talk about intervals *} |
d4d9ea33703c
arrange everything related to ordered_euclidean_space class together
huffman
parents:
44528
diff
changeset
|
3114 |
|
54776
db890d9fc5c2
ordered_euclidean_space compatible with more standard pointwise ordering on products; conditionally complete lattice with product order
immler
parents:
54703
diff
changeset
|
3115 |
class ordered_euclidean_space = ord + inf + sup + abs + Inf + Sup + euclidean_space + |
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
3116 |
assumes eucl_le: "x \<le> y \<longleftrightarrow> (\<forall>i\<in>Basis. x \<bullet> i \<le> y \<bullet> i)" |
54776
db890d9fc5c2
ordered_euclidean_space compatible with more standard pointwise ordering on products; conditionally complete lattice with product order
immler
parents:
54703
diff
changeset
|
3117 |
assumes eucl_less_le_not_le: "x < y \<longleftrightarrow> x \<le> y \<and> \<not> y \<le> x" |
db890d9fc5c2
ordered_euclidean_space compatible with more standard pointwise ordering on products; conditionally complete lattice with product order
immler
parents:
54703
diff
changeset
|
3118 |
assumes eucl_inf: "inf x y = (\<Sum>i\<in>Basis. inf (x \<bullet> i) (y \<bullet> i) *\<^sub>R i)" |
db890d9fc5c2
ordered_euclidean_space compatible with more standard pointwise ordering on products; conditionally complete lattice with product order
immler
parents:
54703
diff
changeset
|
3119 |
assumes eucl_sup: "sup x y = (\<Sum>i\<in>Basis. sup (x \<bullet> i) (y \<bullet> i) *\<^sub>R i)" |
db890d9fc5c2
ordered_euclidean_space compatible with more standard pointwise ordering on products; conditionally complete lattice with product order
immler
parents:
54703
diff
changeset
|
3120 |
assumes eucl_Inf: "Inf X = (\<Sum>i\<in>Basis. (INF x:X. x \<bullet> i) *\<^sub>R i)" |
db890d9fc5c2
ordered_euclidean_space compatible with more standard pointwise ordering on products; conditionally complete lattice with product order
immler
parents:
54703
diff
changeset
|
3121 |
assumes eucl_Sup: "Sup X = (\<Sum>i\<in>Basis. (SUP x:X. x \<bullet> i) *\<^sub>R i)" |
db890d9fc5c2
ordered_euclidean_space compatible with more standard pointwise ordering on products; conditionally complete lattice with product order
immler
parents:
54703
diff
changeset
|
3122 |
assumes eucl_abs: "abs x = (\<Sum>i\<in>Basis. abs (x \<bullet> i) *\<^sub>R i)" |
db890d9fc5c2
ordered_euclidean_space compatible with more standard pointwise ordering on products; conditionally complete lattice with product order
immler
parents:
54703
diff
changeset
|
3123 |
begin |
db890d9fc5c2
ordered_euclidean_space compatible with more standard pointwise ordering on products; conditionally complete lattice with product order
immler
parents:
54703
diff
changeset
|
3124 |
|
db890d9fc5c2
ordered_euclidean_space compatible with more standard pointwise ordering on products; conditionally complete lattice with product order
immler
parents:
54703
diff
changeset
|
3125 |
subclass order |
db890d9fc5c2
ordered_euclidean_space compatible with more standard pointwise ordering on products; conditionally complete lattice with product order
immler
parents:
54703
diff
changeset
|
3126 |
by default |
db890d9fc5c2
ordered_euclidean_space compatible with more standard pointwise ordering on products; conditionally complete lattice with product order
immler
parents:
54703
diff
changeset
|
3127 |
(auto simp: eucl_le eucl_less_le_not_le intro!: euclidean_eqI antisym intro: order.trans) |
db890d9fc5c2
ordered_euclidean_space compatible with more standard pointwise ordering on products; conditionally complete lattice with product order
immler
parents:
54703
diff
changeset
|
3128 |
|
db890d9fc5c2
ordered_euclidean_space compatible with more standard pointwise ordering on products; conditionally complete lattice with product order
immler
parents:
54703
diff
changeset
|
3129 |
subclass ordered_ab_group_add_abs |
db890d9fc5c2
ordered_euclidean_space compatible with more standard pointwise ordering on products; conditionally complete lattice with product order
immler
parents:
54703
diff
changeset
|
3130 |
by default (auto simp: eucl_le inner_add_left eucl_abs abs_leI) |
db890d9fc5c2
ordered_euclidean_space compatible with more standard pointwise ordering on products; conditionally complete lattice with product order
immler
parents:
54703
diff
changeset
|
3131 |
|
db890d9fc5c2
ordered_euclidean_space compatible with more standard pointwise ordering on products; conditionally complete lattice with product order
immler
parents:
54703
diff
changeset
|
3132 |
subclass lattice |
db890d9fc5c2
ordered_euclidean_space compatible with more standard pointwise ordering on products; conditionally complete lattice with product order
immler
parents:
54703
diff
changeset
|
3133 |
by default (auto simp: eucl_inf eucl_sup eucl_le) |
db890d9fc5c2
ordered_euclidean_space compatible with more standard pointwise ordering on products; conditionally complete lattice with product order
immler
parents:
54703
diff
changeset
|
3134 |
|
db890d9fc5c2
ordered_euclidean_space compatible with more standard pointwise ordering on products; conditionally complete lattice with product order
immler
parents:
54703
diff
changeset
|
3135 |
subclass distrib_lattice |
db890d9fc5c2
ordered_euclidean_space compatible with more standard pointwise ordering on products; conditionally complete lattice with product order
immler
parents:
54703
diff
changeset
|
3136 |
by default (auto simp: eucl_inf eucl_sup sup_inf_distrib1 intro!: euclidean_eqI) |
db890d9fc5c2
ordered_euclidean_space compatible with more standard pointwise ordering on products; conditionally complete lattice with product order
immler
parents:
54703
diff
changeset
|
3137 |
|
db890d9fc5c2
ordered_euclidean_space compatible with more standard pointwise ordering on products; conditionally complete lattice with product order
immler
parents:
54703
diff
changeset
|
3138 |
subclass conditionally_complete_lattice |
db890d9fc5c2
ordered_euclidean_space compatible with more standard pointwise ordering on products; conditionally complete lattice with product order
immler
parents:
54703
diff
changeset
|
3139 |
proof |
db890d9fc5c2
ordered_euclidean_space compatible with more standard pointwise ordering on products; conditionally complete lattice with product order
immler
parents:
54703
diff
changeset
|
3140 |
fix z::'a and X::"'a set" |
db890d9fc5c2
ordered_euclidean_space compatible with more standard pointwise ordering on products; conditionally complete lattice with product order
immler
parents:
54703
diff
changeset
|
3141 |
assume "X \<noteq> {}" |
db890d9fc5c2
ordered_euclidean_space compatible with more standard pointwise ordering on products; conditionally complete lattice with product order
immler
parents:
54703
diff
changeset
|
3142 |
hence "\<And>i. (\<lambda>x. x \<bullet> i) ` X \<noteq> {}" by simp |
db890d9fc5c2
ordered_euclidean_space compatible with more standard pointwise ordering on products; conditionally complete lattice with product order
immler
parents:
54703
diff
changeset
|
3143 |
thus "(\<And>x. x \<in> X \<Longrightarrow> z \<le> x) \<Longrightarrow> z \<le> Inf X" "(\<And>x. x \<in> X \<Longrightarrow> x \<le> z) \<Longrightarrow> Sup X \<le> z" |
db890d9fc5c2
ordered_euclidean_space compatible with more standard pointwise ordering on products; conditionally complete lattice with product order
immler
parents:
54703
diff
changeset
|
3144 |
by (auto simp: eucl_Inf eucl_Sup eucl_le Inf_class.INF_def Sup_class.SUP_def |
db890d9fc5c2
ordered_euclidean_space compatible with more standard pointwise ordering on products; conditionally complete lattice with product order
immler
parents:
54703
diff
changeset
|
3145 |
intro!: cInf_greatest cSup_least) |
db890d9fc5c2
ordered_euclidean_space compatible with more standard pointwise ordering on products; conditionally complete lattice with product order
immler
parents:
54703
diff
changeset
|
3146 |
qed (force intro!: cInf_lower cSup_upper |
db890d9fc5c2
ordered_euclidean_space compatible with more standard pointwise ordering on products; conditionally complete lattice with product order
immler
parents:
54703
diff
changeset
|
3147 |
simp: bdd_below_def bdd_above_def preorder_class.bdd_below_def preorder_class.bdd_above_def |
db890d9fc5c2
ordered_euclidean_space compatible with more standard pointwise ordering on products; conditionally complete lattice with product order
immler
parents:
54703
diff
changeset
|
3148 |
eucl_Inf eucl_Sup eucl_le Inf_class.INF_def Sup_class.SUP_def)+ |
db890d9fc5c2
ordered_euclidean_space compatible with more standard pointwise ordering on products; conditionally complete lattice with product order
immler
parents:
54703
diff
changeset
|
3149 |
|
db890d9fc5c2
ordered_euclidean_space compatible with more standard pointwise ordering on products; conditionally complete lattice with product order
immler
parents:
54703
diff
changeset
|
3150 |
lemma inner_Basis_inf_left: "i \<in> Basis \<Longrightarrow> inf x y \<bullet> i = inf (x \<bullet> i) (y \<bullet> i)" |
db890d9fc5c2
ordered_euclidean_space compatible with more standard pointwise ordering on products; conditionally complete lattice with product order
immler
parents:
54703
diff
changeset
|
3151 |
and inner_Basis_sup_left: "i \<in> Basis \<Longrightarrow> sup x y \<bullet> i = sup (x \<bullet> i) (y \<bullet> i)" |
db890d9fc5c2
ordered_euclidean_space compatible with more standard pointwise ordering on products; conditionally complete lattice with product order
immler
parents:
54703
diff
changeset
|
3152 |
by (simp_all add: eucl_inf eucl_sup inner_setsum_left inner_Basis if_distrib setsum_delta |
db890d9fc5c2
ordered_euclidean_space compatible with more standard pointwise ordering on products; conditionally complete lattice with product order
immler
parents:
54703
diff
changeset
|
3153 |
cong: if_cong) |
db890d9fc5c2
ordered_euclidean_space compatible with more standard pointwise ordering on products; conditionally complete lattice with product order
immler
parents:
54703
diff
changeset
|
3154 |
|
db890d9fc5c2
ordered_euclidean_space compatible with more standard pointwise ordering on products; conditionally complete lattice with product order
immler
parents:
54703
diff
changeset
|
3155 |
lemma inner_Basis_INF_left: "i \<in> Basis \<Longrightarrow> (INF x:X. f x) \<bullet> i = (INF x:X. f x \<bullet> i)" |
db890d9fc5c2
ordered_euclidean_space compatible with more standard pointwise ordering on products; conditionally complete lattice with product order
immler
parents:
54703
diff
changeset
|
3156 |
and inner_Basis_SUP_left: "i \<in> Basis \<Longrightarrow> (SUP x:X. f x) \<bullet> i = (SUP x:X. f x \<bullet> i)" |
db890d9fc5c2
ordered_euclidean_space compatible with more standard pointwise ordering on products; conditionally complete lattice with product order
immler
parents:
54703
diff
changeset
|
3157 |
by (simp_all add: INF_def SUP_def eucl_Sup eucl_Inf) |
db890d9fc5c2
ordered_euclidean_space compatible with more standard pointwise ordering on products; conditionally complete lattice with product order
immler
parents:
54703
diff
changeset
|
3158 |
|
db890d9fc5c2
ordered_euclidean_space compatible with more standard pointwise ordering on products; conditionally complete lattice with product order
immler
parents:
54703
diff
changeset
|
3159 |
lemma abs_inner: "i \<in> Basis \<Longrightarrow> abs x \<bullet> i = abs (x \<bullet> i)" |
db890d9fc5c2
ordered_euclidean_space compatible with more standard pointwise ordering on products; conditionally complete lattice with product order
immler
parents:
54703
diff
changeset
|
3160 |
by (auto simp: eucl_abs) |
db890d9fc5c2
ordered_euclidean_space compatible with more standard pointwise ordering on products; conditionally complete lattice with product order
immler
parents:
54703
diff
changeset
|
3161 |
|
db890d9fc5c2
ordered_euclidean_space compatible with more standard pointwise ordering on products; conditionally complete lattice with product order
immler
parents:
54703
diff
changeset
|
3162 |
lemma |
db890d9fc5c2
ordered_euclidean_space compatible with more standard pointwise ordering on products; conditionally complete lattice with product order
immler
parents:
54703
diff
changeset
|
3163 |
abs_scaleR: "\<bar>a *\<^sub>R b\<bar> = \<bar>a\<bar> *\<^sub>R \<bar>b\<bar>" |
db890d9fc5c2
ordered_euclidean_space compatible with more standard pointwise ordering on products; conditionally complete lattice with product order
immler
parents:
54703
diff
changeset
|
3164 |
by (auto simp: eucl_abs abs_mult intro!: euclidean_eqI) |
db890d9fc5c2
ordered_euclidean_space compatible with more standard pointwise ordering on products; conditionally complete lattice with product order
immler
parents:
54703
diff
changeset
|
3165 |
|
db890d9fc5c2
ordered_euclidean_space compatible with more standard pointwise ordering on products; conditionally complete lattice with product order
immler
parents:
54703
diff
changeset
|
3166 |
lemma interval_inner_leI: |
db890d9fc5c2
ordered_euclidean_space compatible with more standard pointwise ordering on products; conditionally complete lattice with product order
immler
parents:
54703
diff
changeset
|
3167 |
assumes "x \<in> {a .. b}" "0 \<le> i" |
db890d9fc5c2
ordered_euclidean_space compatible with more standard pointwise ordering on products; conditionally complete lattice with product order
immler
parents:
54703
diff
changeset
|
3168 |
shows "a\<bullet>i \<le> x\<bullet>i" "x\<bullet>i \<le> b\<bullet>i" |
db890d9fc5c2
ordered_euclidean_space compatible with more standard pointwise ordering on products; conditionally complete lattice with product order
immler
parents:
54703
diff
changeset
|
3169 |
using assms |
db890d9fc5c2
ordered_euclidean_space compatible with more standard pointwise ordering on products; conditionally complete lattice with product order
immler
parents:
54703
diff
changeset
|
3170 |
unfolding euclidean_inner[of a i] euclidean_inner[of x i] euclidean_inner[of b i] |
db890d9fc5c2
ordered_euclidean_space compatible with more standard pointwise ordering on products; conditionally complete lattice with product order
immler
parents:
54703
diff
changeset
|
3171 |
by (auto intro!: setsum_mono mult_right_mono simp: eucl_le) |
db890d9fc5c2
ordered_euclidean_space compatible with more standard pointwise ordering on products; conditionally complete lattice with product order
immler
parents:
54703
diff
changeset
|
3172 |
|
db890d9fc5c2
ordered_euclidean_space compatible with more standard pointwise ordering on products; conditionally complete lattice with product order
immler
parents:
54703
diff
changeset
|
3173 |
lemma inner_nonneg_nonneg: |
db890d9fc5c2
ordered_euclidean_space compatible with more standard pointwise ordering on products; conditionally complete lattice with product order
immler
parents:
54703
diff
changeset
|
3174 |
shows "0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> 0 \<le> a \<bullet> b" |
db890d9fc5c2
ordered_euclidean_space compatible with more standard pointwise ordering on products; conditionally complete lattice with product order
immler
parents:
54703
diff
changeset
|
3175 |
using interval_inner_leI[of a 0 a b] |
db890d9fc5c2
ordered_euclidean_space compatible with more standard pointwise ordering on products; conditionally complete lattice with product order
immler
parents:
54703
diff
changeset
|
3176 |
by auto |
db890d9fc5c2
ordered_euclidean_space compatible with more standard pointwise ordering on products; conditionally complete lattice with product order
immler
parents:
54703
diff
changeset
|
3177 |
|
db890d9fc5c2
ordered_euclidean_space compatible with more standard pointwise ordering on products; conditionally complete lattice with product order
immler
parents:
54703
diff
changeset
|
3178 |
lemma inner_Basis_mono: |
db890d9fc5c2
ordered_euclidean_space compatible with more standard pointwise ordering on products; conditionally complete lattice with product order
immler
parents:
54703
diff
changeset
|
3179 |
shows "a \<le> b \<Longrightarrow> c \<in> Basis \<Longrightarrow> a \<bullet> c \<le> b \<bullet> c" |
db890d9fc5c2
ordered_euclidean_space compatible with more standard pointwise ordering on products; conditionally complete lattice with product order
immler
parents:
54703
diff
changeset
|
3180 |
by (simp add: eucl_le) |
db890d9fc5c2
ordered_euclidean_space compatible with more standard pointwise ordering on products; conditionally complete lattice with product order
immler
parents:
54703
diff
changeset
|
3181 |
|
db890d9fc5c2
ordered_euclidean_space compatible with more standard pointwise ordering on products; conditionally complete lattice with product order
immler
parents:
54703
diff
changeset
|
3182 |
lemma Basis_nonneg[intro, simp]: "i \<in> Basis \<Longrightarrow> 0 \<le> i" |
db890d9fc5c2
ordered_euclidean_space compatible with more standard pointwise ordering on products; conditionally complete lattice with product order
immler
parents:
54703
diff
changeset
|
3183 |
by (auto simp: eucl_le inner_Basis) |
db890d9fc5c2
ordered_euclidean_space compatible with more standard pointwise ordering on products; conditionally complete lattice with product order
immler
parents:
54703
diff
changeset
|
3184 |
|
db890d9fc5c2
ordered_euclidean_space compatible with more standard pointwise ordering on products; conditionally complete lattice with product order
immler
parents:
54703
diff
changeset
|
3185 |
end |
db890d9fc5c2
ordered_euclidean_space compatible with more standard pointwise ordering on products; conditionally complete lattice with product order
immler
parents:
54703
diff
changeset
|
3186 |
|
db890d9fc5c2
ordered_euclidean_space compatible with more standard pointwise ordering on products; conditionally complete lattice with product order
immler
parents:
54703
diff
changeset
|
3187 |
lemma (in order) atLeastatMost_empty'[simp]: |
db890d9fc5c2
ordered_euclidean_space compatible with more standard pointwise ordering on products; conditionally complete lattice with product order
immler
parents:
54703
diff
changeset
|
3188 |
"(~ a <= b) \<Longrightarrow> {a..b} = {}" |
db890d9fc5c2
ordered_euclidean_space compatible with more standard pointwise ordering on products; conditionally complete lattice with product order
immler
parents:
54703
diff
changeset
|
3189 |
by (auto) |
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
3190 |
|
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
3191 |
instance real :: ordered_euclidean_space |
54776
db890d9fc5c2
ordered_euclidean_space compatible with more standard pointwise ordering on products; conditionally complete lattice with product order
immler
parents:
54703
diff
changeset
|
3192 |
by default (auto simp: INF_def SUP_def) |
db890d9fc5c2
ordered_euclidean_space compatible with more standard pointwise ordering on products; conditionally complete lattice with product order
immler
parents:
54703
diff
changeset
|
3193 |
|
db890d9fc5c2
ordered_euclidean_space compatible with more standard pointwise ordering on products; conditionally complete lattice with product order
immler
parents:
54703
diff
changeset
|
3194 |
lemma in_Basis_prod_iff: |
db890d9fc5c2
ordered_euclidean_space compatible with more standard pointwise ordering on products; conditionally complete lattice with product order
immler
parents:
54703
diff
changeset
|
3195 |
fixes i::"'a::euclidean_space*'b::euclidean_space" |
db890d9fc5c2
ordered_euclidean_space compatible with more standard pointwise ordering on products; conditionally complete lattice with product order
immler
parents:
54703
diff
changeset
|
3196 |
shows "i \<in> Basis \<longleftrightarrow> fst i = 0 \<and> snd i \<in> Basis \<or> snd i = 0 \<and> fst i \<in> Basis" |
db890d9fc5c2
ordered_euclidean_space compatible with more standard pointwise ordering on products; conditionally complete lattice with product order
immler
parents:
54703
diff
changeset
|
3197 |
by (cases i) (auto simp: Basis_prod_def) |
db890d9fc5c2
ordered_euclidean_space compatible with more standard pointwise ordering on products; conditionally complete lattice with product order
immler
parents:
54703
diff
changeset
|
3198 |
|
db890d9fc5c2
ordered_euclidean_space compatible with more standard pointwise ordering on products; conditionally complete lattice with product order
immler
parents:
54703
diff
changeset
|
3199 |
instantiation prod::(abs, abs) abs |
44133 | 3200 |
begin |
3201 |
||
54776
db890d9fc5c2
ordered_euclidean_space compatible with more standard pointwise ordering on products; conditionally complete lattice with product order
immler
parents:
54703
diff
changeset
|
3202 |
definition "abs x = (abs (fst x), abs (snd x))" |
db890d9fc5c2
ordered_euclidean_space compatible with more standard pointwise ordering on products; conditionally complete lattice with product order
immler
parents:
54703
diff
changeset
|
3203 |
|
db890d9fc5c2
ordered_euclidean_space compatible with more standard pointwise ordering on products; conditionally complete lattice with product order
immler
parents:
54703
diff
changeset
|
3204 |
instance proof qed |
db890d9fc5c2
ordered_euclidean_space compatible with more standard pointwise ordering on products; conditionally complete lattice with product order
immler
parents:
54703
diff
changeset
|
3205 |
end |
db890d9fc5c2
ordered_euclidean_space compatible with more standard pointwise ordering on products; conditionally complete lattice with product order
immler
parents:
54703
diff
changeset
|
3206 |
|
db890d9fc5c2
ordered_euclidean_space compatible with more standard pointwise ordering on products; conditionally complete lattice with product order
immler
parents:
54703
diff
changeset
|
3207 |
instance prod :: (ordered_euclidean_space, ordered_euclidean_space) ordered_euclidean_space |
db890d9fc5c2
ordered_euclidean_space compatible with more standard pointwise ordering on products; conditionally complete lattice with product order
immler
parents:
54703
diff
changeset
|
3208 |
by default |
db890d9fc5c2
ordered_euclidean_space compatible with more standard pointwise ordering on products; conditionally complete lattice with product order
immler
parents:
54703
diff
changeset
|
3209 |
(auto intro!: add_mono simp add: euclidean_representation_setsum' Ball_def inner_prod_def |
db890d9fc5c2
ordered_euclidean_space compatible with more standard pointwise ordering on products; conditionally complete lattice with product order
immler
parents:
54703
diff
changeset
|
3210 |
in_Basis_prod_iff inner_Basis_inf_left inner_Basis_sup_left inner_Basis_INF_left Inf_prod_def |
db890d9fc5c2
ordered_euclidean_space compatible with more standard pointwise ordering on products; conditionally complete lattice with product order
immler
parents:
54703
diff
changeset
|
3211 |
inner_Basis_SUP_left Sup_prod_def less_prod_def less_eq_prod_def eucl_le[where 'a='a] |
db890d9fc5c2
ordered_euclidean_space compatible with more standard pointwise ordering on products; conditionally complete lattice with product order
immler
parents:
54703
diff
changeset
|
3212 |
eucl_le[where 'a='b] abs_prod_def abs_inner) |
49522 | 3213 |
|
44133 | 3214 |
end |
3215 |