src/HOL/Multivariate_Analysis/Linear_Algebra.thy
author paulson <lp15@cam.ac.uk>
Mon Apr 18 14:30:32 2016 +0100 (2016-04-18)
changeset 63007 aa894a49f77d
parent 62948 7700f467892b
child 63050 ca4cce24c75d
permissions -rw-r--r--
new theorems about convex hulls, etc.; also, renamed some theorems
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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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section \<open>Elementary linear algebra on Euclidean spaces\<close>
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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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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. \<bar>y - x\<bar> < d \<longrightarrow> \<bar>y * y - x * x\<bar> < e)"
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  using isCont_power[OF continuous_ident, of x, unfolded isCont_def LIM_eq, rule_format, of e 2]
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  by (force simp add: power2_eq_square)
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text\<open>Hence derive more interesting properties of the norm.\<close>
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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\<open>Squaring equations and inequalities involving norms.\<close>
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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 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 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 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\<open>Dot product in terms of the norm rather than conversely.\<close>
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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\<open>Equality of vectors in terms of @{term "op \<bullet>"} products.\<close>
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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 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_neutral_right[OF fT fST])
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  apply (auto intro: setsum.neutral)
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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 \<open>Orthogonality.\<close>
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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 \<open>Linear functions.\<close>
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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"
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  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"
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  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_scaleR: "linear f \<Longrightarrow> linear (\<lambda>x. f x *\<^sub>R c)"
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  by (simp add: linear_iff scaleR_add_left)
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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 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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proof (cases "finite S")
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  case True
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  then show ?thesis
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    using lS by induct (simp_all add: linear_zero linear_compose_add)
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next
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  case False
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  then show ?thesis
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    by (simp add: linear_zero)
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qed
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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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  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 (rule linear.scaleR)
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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 f: "linear f"
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  shows "f (setsum g S) = setsum (f \<circ> g) S"
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proof (cases "finite S")
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  case True
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  then show ?thesis
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    by induct (simp_all add: linear_0 [OF f] linear_add [OF f])
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next
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  case False
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  then show ?thesis
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    by (simp add: linear_0 [OF f])
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qed
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lemma linear_setsum_mul:
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  assumes lin: "linear f"
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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"
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  using linear_setsum[OF lin, of "\<lambda>i. c i *\<^sub>R v i" , unfolded o_def] linear_cmul[OF lin]
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  by simp
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lemma linear_injective_0:
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  assumes lin: "linear f"
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  shows "inj f \<longleftrightarrow> (\<forall>x. f x = 0 \<longrightarrow> x = 0)"
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proof -
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  have "inj f \<longleftrightarrow> (\<forall> x y. f x = f y \<longrightarrow> x = y)"
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    by (simp add: inj_on_def)
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  also have "\<dots> \<longleftrightarrow> (\<forall> x y. f x - f y = 0 \<longrightarrow> x - y = 0)"
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    by simp
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  also have "\<dots> \<longleftrightarrow> (\<forall> x y. f (x - y) = 0 \<longrightarrow> x - y = 0)"
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    by (simp add: linear_sub[OF lin])
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  also have "\<dots> \<longleftrightarrow> (\<forall> x. f x = 0 \<longrightarrow> x = 0)"
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    by auto
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  finally show ?thesis .
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qed
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lemma linear_scaleR  [simp]: "linear (\<lambda>x. scaleR c x)"
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  by (simp add: linear_iff scaleR_add_right)
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lp15@61520
   309
lemma linear_scaleR_left [simp]: "linear (\<lambda>r. scaleR r x)"
lp15@61520
   310
  by (simp add: linear_iff scaleR_add_left)
lp15@61520
   311
lp15@61520
   312
lemma injective_scaleR: "c \<noteq> 0 \<Longrightarrow> inj (\<lambda>x::'a::real_vector. scaleR c x)"
lp15@61520
   313
  by (simp add: inj_on_def)
lp15@61520
   314
lp15@61520
   315
lemma linear_add_cmul:
lp15@61520
   316
  assumes "linear f"
lp15@61520
   317
  shows "f (a *\<^sub>R x + b *\<^sub>R y) = a *\<^sub>R f x +  b *\<^sub>R f y"
lp15@61520
   318
  using linear_add[of f] linear_cmul[of f] assms by simp
lp15@61520
   319
immler@61915
   320
lemma linear_componentwise:
immler@61915
   321
  fixes f:: "'a::euclidean_space \<Rightarrow> 'b::real_inner"
immler@61915
   322
  assumes lf: "linear f"
immler@61915
   323
  shows "(f x) \<bullet> j = (\<Sum>i\<in>Basis. (x\<bullet>i) * (f i\<bullet>j))" (is "?lhs = ?rhs")
immler@61915
   324
proof -
immler@61915
   325
  have "?rhs = (\<Sum>i\<in>Basis. (x\<bullet>i) *\<^sub>R (f i))\<bullet>j"
immler@61915
   326
    by (simp add: inner_setsum_left)
immler@61915
   327
  then show ?thesis
immler@61915
   328
    unfolding linear_setsum_mul[OF lf, symmetric]
immler@61915
   329
    unfolding euclidean_representation ..
immler@61915
   330
qed
immler@61915
   331
wenzelm@49522
   332
wenzelm@60420
   333
subsection \<open>Bilinear functions.\<close>
huffman@44133
   334
wenzelm@53406
   335
definition "bilinear f \<longleftrightarrow> (\<forall>x. linear (\<lambda>y. f x y)) \<and> (\<forall>y. linear (\<lambda>x. f x y))"
wenzelm@53406
   336
wenzelm@53406
   337
lemma bilinear_ladd: "bilinear h \<Longrightarrow> h (x + y) z = h x z + h y z"
huffman@53600
   338
  by (simp add: bilinear_def linear_iff)
wenzelm@49663
   339
wenzelm@53406
   340
lemma bilinear_radd: "bilinear h \<Longrightarrow> h x (y + z) = h x y + h x z"
huffman@53600
   341
  by (simp add: bilinear_def linear_iff)
huffman@44133
   342
wenzelm@53406
   343
lemma bilinear_lmul: "bilinear h \<Longrightarrow> h (c *\<^sub>R x) y = c *\<^sub>R h x y"
huffman@53600
   344
  by (simp add: bilinear_def linear_iff)
huffman@44133
   345
wenzelm@53406
   346
lemma bilinear_rmul: "bilinear h \<Longrightarrow> h x (c *\<^sub>R y) = c *\<^sub>R h x y"
huffman@53600
   347
  by (simp add: bilinear_def linear_iff)
huffman@44133
   348
wenzelm@53406
   349
lemma bilinear_lneg: "bilinear h \<Longrightarrow> h (- x) y = - h x y"
haftmann@54489
   350
  by (drule bilinear_lmul [of _ "- 1"]) simp
huffman@44133
   351
wenzelm@53406
   352
lemma bilinear_rneg: "bilinear h \<Longrightarrow> h x (- y) = - h x y"
haftmann@54489
   353
  by (drule bilinear_rmul [of _ _ "- 1"]) simp
huffman@44133
   354
wenzelm@53406
   355
lemma (in ab_group_add) eq_add_iff: "x = x + y \<longleftrightarrow> y = 0"
haftmann@59557
   356
  using add_left_imp_eq[of x y 0] by auto
huffman@44133
   357
wenzelm@53406
   358
lemma bilinear_lzero:
wenzelm@53406
   359
  assumes "bilinear h"
wenzelm@53406
   360
  shows "h 0 x = 0"
wenzelm@49663
   361
  using bilinear_ladd [OF assms, of 0 0 x] by (simp add: eq_add_iff field_simps)
wenzelm@49663
   362
wenzelm@53406
   363
lemma bilinear_rzero:
wenzelm@53406
   364
  assumes "bilinear h"
wenzelm@53406
   365
  shows "h x 0 = 0"
wenzelm@49663
   366
  using bilinear_radd [OF assms, of x 0 0 ] by (simp add: eq_add_iff field_simps)
huffman@44133
   367
wenzelm@53406
   368
lemma bilinear_lsub: "bilinear h \<Longrightarrow> h (x - y) z = h x z - h y z"
haftmann@54230
   369
  using bilinear_ladd [of h x "- y"] by (simp add: bilinear_lneg)
huffman@44133
   370
wenzelm@53406
   371
lemma bilinear_rsub: "bilinear h \<Longrightarrow> h z (x - y) = h z x - h z y"
haftmann@54230
   372
  using bilinear_radd [of h _ x "- y"] by (simp add: bilinear_rneg)
huffman@44133
   373
huffman@44133
   374
lemma bilinear_setsum:
wenzelm@49663
   375
  assumes bh: "bilinear h"
wenzelm@49663
   376
    and fS: "finite S"
wenzelm@49663
   377
    and fT: "finite T"
huffman@44133
   378
  shows "h (setsum f S) (setsum g T) = setsum (\<lambda>(i,j). h (f i) (g j)) (S \<times> T) "
wenzelm@49522
   379
proof -
huffman@44133
   380
  have "h (setsum f S) (setsum g T) = setsum (\<lambda>x. h (f x) (setsum g T)) S"
huffman@44133
   381
    apply (rule linear_setsum[unfolded o_def])
wenzelm@53406
   382
    using bh fS
wenzelm@53406
   383
    apply (auto simp add: bilinear_def)
wenzelm@49522
   384
    done
huffman@44133
   385
  also have "\<dots> = setsum (\<lambda>x. setsum (\<lambda>y. h (f x) (g y)) T) S"
haftmann@57418
   386
    apply (rule setsum.cong, simp)
huffman@44133
   387
    apply (rule linear_setsum[unfolded o_def])
wenzelm@49522
   388
    using bh fT
wenzelm@49522
   389
    apply (auto simp add: bilinear_def)
wenzelm@49522
   390
    done
wenzelm@53406
   391
  finally show ?thesis
haftmann@57418
   392
    unfolding setsum.cartesian_product .
huffman@44133
   393
qed
huffman@44133
   394
wenzelm@49522
   395
wenzelm@60420
   396
subsection \<open>Adjoints.\<close>
huffman@44133
   397
huffman@44133
   398
definition "adjoint f = (SOME f'. \<forall>x y. f x \<bullet> y = x \<bullet> f' y)"
huffman@44133
   399
huffman@44133
   400
lemma adjoint_unique:
huffman@44133
   401
  assumes "\<forall>x y. inner (f x) y = inner x (g y)"
huffman@44133
   402
  shows "adjoint f = g"
wenzelm@49522
   403
  unfolding adjoint_def
huffman@44133
   404
proof (rule some_equality)
wenzelm@53406
   405
  show "\<forall>x y. inner (f x) y = inner x (g y)"
wenzelm@53406
   406
    by (rule assms)
huffman@44133
   407
next
wenzelm@53406
   408
  fix h
wenzelm@53406
   409
  assume "\<forall>x y. inner (f x) y = inner x (h y)"
wenzelm@53406
   410
  then have "\<forall>x y. inner x (g y) = inner x (h y)"
wenzelm@53406
   411
    using assms by simp
wenzelm@53406
   412
  then have "\<forall>x y. inner x (g y - h y) = 0"
wenzelm@53406
   413
    by (simp add: inner_diff_right)
wenzelm@53406
   414
  then have "\<forall>y. inner (g y - h y) (g y - h y) = 0"
wenzelm@53406
   415
    by simp
wenzelm@53406
   416
  then have "\<forall>y. h y = g y"
wenzelm@53406
   417
    by simp
wenzelm@49652
   418
  then show "h = g" by (simp add: ext)
huffman@44133
   419
qed
huffman@44133
   420
wenzelm@60420
   421
text \<open>TODO: The following lemmas about adjoints should hold for any
hoelzl@50526
   422
Hilbert space (i.e. complete inner product space).
wenzelm@54703
   423
(see @{url "http://en.wikipedia.org/wiki/Hermitian_adjoint"})
wenzelm@60420
   424
\<close>
hoelzl@50526
   425
hoelzl@50526
   426
lemma adjoint_works:
wenzelm@56444
   427
  fixes f :: "'n::euclidean_space \<Rightarrow> 'm::euclidean_space"
hoelzl@50526
   428
  assumes lf: "linear f"
hoelzl@50526
   429
  shows "x \<bullet> adjoint f y = f x \<bullet> y"
hoelzl@50526
   430
proof -
hoelzl@50526
   431
  have "\<forall>y. \<exists>w. \<forall>x. f x \<bullet> y = x \<bullet> w"
hoelzl@50526
   432
  proof (intro allI exI)
hoelzl@50526
   433
    fix y :: "'m" and x
hoelzl@50526
   434
    let ?w = "(\<Sum>i\<in>Basis. (f i \<bullet> y) *\<^sub>R i) :: 'n"
hoelzl@50526
   435
    have "f x \<bullet> y = f (\<Sum>i\<in>Basis. (x \<bullet> i) *\<^sub>R i) \<bullet> y"
hoelzl@50526
   436
      by (simp add: euclidean_representation)
hoelzl@50526
   437
    also have "\<dots> = (\<Sum>i\<in>Basis. (x \<bullet> i) *\<^sub>R f i) \<bullet> y"
huffman@56196
   438
      unfolding linear_setsum[OF lf]
hoelzl@50526
   439
      by (simp add: linear_cmul[OF lf])
hoelzl@50526
   440
    finally show "f x \<bullet> y = x \<bullet> ?w"
haftmann@57512
   441
      by (simp add: inner_setsum_left inner_setsum_right mult.commute)
hoelzl@50526
   442
  qed
hoelzl@50526
   443
  then show ?thesis
hoelzl@50526
   444
    unfolding adjoint_def choice_iff
hoelzl@50526
   445
    by (intro someI2_ex[where Q="\<lambda>f'. x \<bullet> f' y = f x \<bullet> y"]) auto
hoelzl@50526
   446
qed
hoelzl@50526
   447
hoelzl@50526
   448
lemma adjoint_clauses:
wenzelm@56444
   449
  fixes f :: "'n::euclidean_space \<Rightarrow> 'm::euclidean_space"
hoelzl@50526
   450
  assumes lf: "linear f"
hoelzl@50526
   451
  shows "x \<bullet> adjoint f y = f x \<bullet> y"
hoelzl@50526
   452
    and "adjoint f y \<bullet> x = y \<bullet> f x"
hoelzl@50526
   453
  by (simp_all add: adjoint_works[OF lf] inner_commute)
hoelzl@50526
   454
hoelzl@50526
   455
lemma adjoint_linear:
wenzelm@56444
   456
  fixes f :: "'n::euclidean_space \<Rightarrow> 'm::euclidean_space"
hoelzl@50526
   457
  assumes lf: "linear f"
hoelzl@50526
   458
  shows "linear (adjoint f)"
huffman@53600
   459
  by (simp add: lf linear_iff euclidean_eq_iff[where 'a='n] euclidean_eq_iff[where 'a='m]
huffman@53939
   460
    adjoint_clauses[OF lf] inner_distrib)
hoelzl@50526
   461
hoelzl@50526
   462
lemma adjoint_adjoint:
wenzelm@56444
   463
  fixes f :: "'n::euclidean_space \<Rightarrow> 'm::euclidean_space"
hoelzl@50526
   464
  assumes lf: "linear f"
hoelzl@50526
   465
  shows "adjoint (adjoint f) = f"
hoelzl@50526
   466
  by (rule adjoint_unique, simp add: adjoint_clauses [OF lf])
hoelzl@50526
   467
wenzelm@53406
   468
wenzelm@60420
   469
subsection \<open>Interlude: Some properties of real sets\<close>
huffman@44133
   470
wenzelm@53406
   471
lemma seq_mono_lemma:
wenzelm@53406
   472
  assumes "\<forall>(n::nat) \<ge> m. (d n :: real) < e n"
wenzelm@53406
   473
    and "\<forall>n \<ge> m. e n \<le> e m"
huffman@44133
   474
  shows "\<forall>n \<ge> m. d n < e m"
wenzelm@53406
   475
  using assms
wenzelm@53406
   476
  apply auto
huffman@44133
   477
  apply (erule_tac x="n" in allE)
huffman@44133
   478
  apply (erule_tac x="n" in allE)
huffman@44133
   479
  apply auto
huffman@44133
   480
  done
huffman@44133
   481
wenzelm@53406
   482
lemma infinite_enumerate:
wenzelm@53406
   483
  assumes fS: "infinite S"
huffman@44133
   484
  shows "\<exists>r. subseq r \<and> (\<forall>n. r n \<in> S)"
wenzelm@49525
   485
  unfolding subseq_def
wenzelm@49525
   486
  using enumerate_in_set[OF fS] enumerate_mono[of _ _ S] fS by auto
huffman@44133
   487
wenzelm@56444
   488
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)"
wenzelm@49522
   489
  apply auto
wenzelm@49522
   490
  apply (rule_tac x="d/2" in exI)
wenzelm@49522
   491
  apply auto
wenzelm@49522
   492
  done
huffman@44133
   493
wenzelm@61808
   494
lemma approachable_lt_le2:  \<comment>\<open>like the above, but pushes aside an extra formula\<close>
paulson@60762
   495
    "(\<exists>(d::real) > 0. \<forall>x. Q x \<longrightarrow> f x < d \<longrightarrow> P x) \<longleftrightarrow> (\<exists>d>0. \<forall>x. f x \<le> d \<longrightarrow> Q x \<longrightarrow> P x)"
paulson@60762
   496
  apply auto
paulson@60762
   497
  apply (rule_tac x="d/2" in exI, auto)
paulson@60762
   498
  done
paulson@60762
   499
huffman@44133
   500
lemma triangle_lemma:
wenzelm@53406
   501
  fixes x y z :: real
wenzelm@53406
   502
  assumes x: "0 \<le> x"
wenzelm@53406
   503
    and y: "0 \<le> y"
wenzelm@53406
   504
    and z: "0 \<le> z"
wenzelm@53406
   505
    and xy: "x\<^sup>2 \<le> y\<^sup>2 + z\<^sup>2"
wenzelm@53406
   506
  shows "x \<le> y + z"
wenzelm@49522
   507
proof -
wenzelm@56444
   508
  have "y\<^sup>2 + z\<^sup>2 \<le> y\<^sup>2 + 2 * y * z + z\<^sup>2"
nipkow@56536
   509
    using z y by simp
wenzelm@53406
   510
  with xy have th: "x\<^sup>2 \<le> (y + z)\<^sup>2"
wenzelm@53406
   511
    by (simp add: power2_eq_square field_simps)
wenzelm@53406
   512
  from y z have yz: "y + z \<ge> 0"
wenzelm@53406
   513
    by arith
huffman@44133
   514
  from power2_le_imp_le[OF th yz] show ?thesis .
huffman@44133
   515
qed
huffman@44133
   516
wenzelm@49522
   517
wenzelm@60420
   518
subsection \<open>A generic notion of "hull" (convex, affine, conic hull and closure).\<close>
huffman@44133
   519
wenzelm@53406
   520
definition hull :: "('a set \<Rightarrow> bool) \<Rightarrow> 'a set \<Rightarrow> 'a set"  (infixl "hull" 75)
wenzelm@53406
   521
  where "S hull s = \<Inter>{t. S t \<and> s \<subseteq> t}"
huffman@44170
   522
huffman@44170
   523
lemma hull_same: "S s \<Longrightarrow> S hull s = s"
huffman@44133
   524
  unfolding hull_def by auto
huffman@44133
   525
wenzelm@53406
   526
lemma hull_in: "(\<And>T. Ball T S \<Longrightarrow> S (\<Inter>T)) \<Longrightarrow> S (S hull s)"
wenzelm@49522
   527
  unfolding hull_def Ball_def by auto
huffman@44170
   528
wenzelm@53406
   529
lemma hull_eq: "(\<And>T. Ball T S \<Longrightarrow> S (\<Inter>T)) \<Longrightarrow> (S hull s) = s \<longleftrightarrow> S s"
wenzelm@49522
   530
  using hull_same[of S s] hull_in[of S s] by metis
huffman@44133
   531
lp15@63007
   532
lemma hull_hull [simp]: "S hull (S hull s) = S hull s"
huffman@44133
   533
  unfolding hull_def by blast
huffman@44133
   534
huffman@44133
   535
lemma hull_subset[intro]: "s \<subseteq> (S hull s)"
huffman@44133
   536
  unfolding hull_def by blast
huffman@44133
   537
wenzelm@53406
   538
lemma hull_mono: "s \<subseteq> t \<Longrightarrow> (S hull s) \<subseteq> (S hull t)"
huffman@44133
   539
  unfolding hull_def by blast
huffman@44133
   540
wenzelm@53406
   541
lemma hull_antimono: "\<forall>x. S x \<longrightarrow> T x \<Longrightarrow> (T hull s) \<subseteq> (S hull s)"
huffman@44133
   542
  unfolding hull_def by blast
huffman@44133
   543
wenzelm@53406
   544
lemma hull_minimal: "s \<subseteq> t \<Longrightarrow> S t \<Longrightarrow> (S hull s) \<subseteq> t"
huffman@44133
   545
  unfolding hull_def by blast
huffman@44133
   546
wenzelm@53406
   547
lemma subset_hull: "S t \<Longrightarrow> S hull s \<subseteq> t \<longleftrightarrow> s \<subseteq> t"
huffman@44133
   548
  unfolding hull_def by blast
huffman@44133
   549
lp15@63007
   550
lemma hull_UNIV [simp]: "S hull UNIV = UNIV"
huffman@53596
   551
  unfolding hull_def by auto
huffman@53596
   552
wenzelm@53406
   553
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)"
wenzelm@49652
   554
  unfolding hull_def by auto
huffman@44133
   555
huffman@44133
   556
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"
huffman@44133
   557
  using hull_minimal[of S "{x. P x}" Q]
huffman@44170
   558
  by (auto simp add: subset_eq)
huffman@44133
   559
wenzelm@49522
   560
lemma hull_inc: "x \<in> S \<Longrightarrow> x \<in> P hull S"
wenzelm@49522
   561
  by (metis hull_subset subset_eq)
huffman@44133
   562
huffman@44133
   563
lemma hull_union_subset: "(S hull s) \<union> (S hull t) \<subseteq> (S hull (s \<union> t))"
wenzelm@49522
   564
  unfolding Un_subset_iff by (metis hull_mono Un_upper1 Un_upper2)
wenzelm@49522
   565
wenzelm@49522
   566
lemma hull_union:
wenzelm@53406
   567
  assumes T: "\<And>T. Ball T S \<Longrightarrow> S (\<Inter>T)"
huffman@44133
   568
  shows "S hull (s \<union> t) = S hull (S hull s \<union> S hull t)"
wenzelm@49522
   569
  apply rule
wenzelm@49522
   570
  apply (rule hull_mono)
wenzelm@49522
   571
  unfolding Un_subset_iff
wenzelm@49522
   572
  apply (metis hull_subset Un_upper1 Un_upper2 subset_trans)
wenzelm@49522
   573
  apply (rule hull_minimal)
wenzelm@49522
   574
  apply (metis hull_union_subset)
wenzelm@49522
   575
  apply (metis hull_in T)
wenzelm@49522
   576
  done
huffman@44133
   577
wenzelm@56444
   578
lemma hull_redundant_eq: "a \<in> (S hull s) \<longleftrightarrow> S hull (insert a s) = S hull s"
huffman@44133
   579
  unfolding hull_def by blast
huffman@44133
   580
wenzelm@56444
   581
lemma hull_redundant: "a \<in> (S hull s) \<Longrightarrow> S hull (insert a s) = S hull s"
wenzelm@49522
   582
  by (metis hull_redundant_eq)
wenzelm@49522
   583
huffman@44133
   584
wenzelm@60420
   585
subsection \<open>Archimedean properties and useful consequences\<close>
huffman@44133
   586
wenzelm@61222
   587
text\<open>Bernoulli's inequality\<close>
lp15@62623
   588
proposition Bernoulli_inequality:
lp15@60974
   589
  fixes x :: real
lp15@60974
   590
  assumes "-1 \<le> x"
lp15@60974
   591
    shows "1 + n * x \<le> (1 + x) ^ n"
wenzelm@49522
   592
proof (induct n)
wenzelm@49522
   593
  case 0
wenzelm@49522
   594
  then show ?case by simp
huffman@44133
   595
next
huffman@44133
   596
  case (Suc n)
lp15@60974
   597
  have "1 + Suc n * x \<le> 1 + (Suc n)*x + n * x^2"
lp15@60974
   598
    by (simp add: algebra_simps)
lp15@60974
   599
  also have "... = (1 + x) * (1 + n*x)"
lp15@61609
   600
    by (auto simp: power2_eq_square algebra_simps  of_nat_Suc)
lp15@60974
   601
  also have "... \<le> (1 + x) ^ Suc n"
lp15@60974
   602
    using Suc.hyps assms mult_left_mono by fastforce
lp15@60974
   603
  finally show ?case .
lp15@60974
   604
qed
lp15@60974
   605
lp15@62623
   606
corollary Bernoulli_inequality_even:
lp15@60974
   607
  fixes x :: real
lp15@60974
   608
  assumes "even n"
lp15@60974
   609
    shows "1 + n * x \<le> (1 + x) ^ n"
lp15@60974
   610
proof (cases "-1 \<le> x \<or> n=0")
lp15@60974
   611
  case True
lp15@60974
   612
  then show ?thesis
lp15@60974
   613
    by (auto simp: Bernoulli_inequality)
lp15@60974
   614
next
lp15@60974
   615
  case False
lp15@60974
   616
  then have "real n \<ge> 1"
wenzelm@53406
   617
    by simp
lp15@60974
   618
  with False have "n * x \<le> -1"
lp15@60974
   619
    by (metis linear minus_zero mult.commute mult.left_neutral mult_left_mono_neg neg_le_iff_le order_trans zero_le_one)
lp15@60974
   620
  then have "1 + n * x \<le> 0"
lp15@60974
   621
    by auto
lp15@60974
   622
  also have "... \<le> (1 + x) ^ n"
lp15@60974
   623
    using assms
lp15@60974
   624
    using zero_le_even_power by blast
lp15@60974
   625
  finally show ?thesis .
huffman@44133
   626
qed
huffman@44133
   627
lp15@62623
   628
corollary real_arch_pow:
wenzelm@53406
   629
  fixes x :: real
wenzelm@53406
   630
  assumes x: "1 < x"
wenzelm@53406
   631
  shows "\<exists>n. y < x^n"
wenzelm@49522
   632
proof -
wenzelm@53406
   633
  from x have x0: "x - 1 > 0"
wenzelm@53406
   634
    by arith
huffman@44666
   635
  from reals_Archimedean3[OF x0, rule_format, of y]
wenzelm@53406
   636
  obtain n :: nat where n: "y < real n * (x - 1)" by metis
lp15@60974
   637
  from x0 have x00: "x- 1 \<ge> -1" by arith
lp15@60974
   638
  from Bernoulli_inequality[OF x00, of n] n
huffman@44133
   639
  have "y < x^n" by auto
huffman@44133
   640
  then show ?thesis by metis
huffman@44133
   641
qed
huffman@44133
   642
lp15@62623
   643
corollary real_arch_pow_inv:
wenzelm@53406
   644
  fixes x y :: real
wenzelm@53406
   645
  assumes y: "y > 0"
wenzelm@53406
   646
    and x1: "x < 1"
huffman@44133
   647
  shows "\<exists>n. x^n < y"
wenzelm@53406
   648
proof (cases "x > 0")
wenzelm@53406
   649
  case True
wenzelm@53406
   650
  with x1 have ix: "1 < 1/x" by (simp add: field_simps)
wenzelm@53406
   651
  from real_arch_pow[OF ix, of "1/y"]
wenzelm@53406
   652
  obtain n where n: "1/y < (1/x)^n" by blast
wenzelm@60420
   653
  then show ?thesis using y \<open>x > 0\<close>
hoelzl@56480
   654
    by (auto simp add: field_simps)
wenzelm@53406
   655
next
wenzelm@53406
   656
  case False
wenzelm@53406
   657
  with y x1 show ?thesis
wenzelm@53406
   658
    apply auto
wenzelm@53406
   659
    apply (rule exI[where x=1])
wenzelm@53406
   660
    apply auto
wenzelm@53406
   661
    done
huffman@44133
   662
qed
huffman@44133
   663
wenzelm@49522
   664
lemma forall_pos_mono:
wenzelm@53406
   665
  "(\<And>d e::real. d < e \<Longrightarrow> P d \<Longrightarrow> P e) \<Longrightarrow>
wenzelm@53406
   666
    (\<And>n::nat. n \<noteq> 0 \<Longrightarrow> P (inverse (real n))) \<Longrightarrow> (\<And>e. 0 < e \<Longrightarrow> P e)"
lp15@62623
   667
  by (metis real_arch_inverse)
huffman@44133
   668
wenzelm@49522
   669
lemma forall_pos_mono_1:
wenzelm@53406
   670
  "(\<And>d e::real. d < e \<Longrightarrow> P d \<Longrightarrow> P e) \<Longrightarrow>
wenzelm@53716
   671
    (\<And>n. P (inverse (real (Suc n)))) \<Longrightarrow> 0 < e \<Longrightarrow> P e"
huffman@44133
   672
  apply (rule forall_pos_mono)
huffman@44133
   673
  apply auto
lp15@61609
   674
  apply (metis Suc_pred of_nat_Suc)
huffman@44133
   675
  done
huffman@44133
   676
wenzelm@49522
   677
wenzelm@60420
   678
subsection\<open>A bit of linear algebra.\<close>
huffman@44133
   679
wenzelm@49522
   680
definition (in real_vector) subspace :: "'a set \<Rightarrow> bool"
wenzelm@56444
   681
  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)"
huffman@44133
   682
huffman@44133
   683
definition (in real_vector) "span S = (subspace hull S)"
wenzelm@53716
   684
definition (in real_vector) "dependent S \<longleftrightarrow> (\<exists>a \<in> S. a \<in> span (S - {a}))"
wenzelm@53406
   685
abbreviation (in real_vector) "independent s \<equiv> \<not> dependent s"
huffman@44133
   686
wenzelm@60420
   687
text \<open>Closure properties of subspaces.\<close>
huffman@44133
   688
wenzelm@53406
   689
lemma subspace_UNIV[simp]: "subspace UNIV"
wenzelm@53406
   690
  by (simp add: subspace_def)
wenzelm@53406
   691
wenzelm@53406
   692
lemma (in real_vector) subspace_0: "subspace S \<Longrightarrow> 0 \<in> S"
wenzelm@53406
   693
  by (metis subspace_def)
wenzelm@53406
   694
wenzelm@53406
   695
lemma (in real_vector) subspace_add: "subspace S \<Longrightarrow> x \<in> S \<Longrightarrow> y \<in> S \<Longrightarrow> x + y \<in> S"
huffman@44133
   696
  by (metis subspace_def)
huffman@44133
   697
huffman@44133
   698
lemma (in real_vector) subspace_mul: "subspace S \<Longrightarrow> x \<in> S \<Longrightarrow> c *\<^sub>R x \<in> S"
huffman@44133
   699
  by (metis subspace_def)
huffman@44133
   700
huffman@44133
   701
lemma subspace_neg: "subspace S \<Longrightarrow> x \<in> S \<Longrightarrow> - x \<in> S"
huffman@44133
   702
  by (metis scaleR_minus1_left subspace_mul)
huffman@44133
   703
huffman@44133
   704
lemma subspace_sub: "subspace S \<Longrightarrow> x \<in> S \<Longrightarrow> y \<in> S \<Longrightarrow> x - y \<in> S"
haftmann@54230
   705
  using subspace_add [of S x "- y"] by (simp add: subspace_neg)
huffman@44133
   706
huffman@44133
   707
lemma (in real_vector) subspace_setsum:
wenzelm@53406
   708
  assumes sA: "subspace A"
huffman@56196
   709
    and f: "\<forall>x\<in>B. f x \<in> A"
huffman@44133
   710
  shows "setsum f B \<in> A"
huffman@56196
   711
proof (cases "finite B")
huffman@56196
   712
  case True
huffman@56196
   713
  then show ?thesis
huffman@56196
   714
    using f by induct (simp_all add: subspace_0 [OF sA] subspace_add [OF sA])
huffman@56196
   715
qed (simp add: subspace_0 [OF sA])
huffman@44133
   716
huffman@44133
   717
lemma subspace_linear_image:
wenzelm@53406
   718
  assumes lf: "linear f"
wenzelm@53406
   719
    and sS: "subspace S"
wenzelm@53406
   720
  shows "subspace (f ` S)"
huffman@44133
   721
  using lf sS linear_0[OF lf]
huffman@53600
   722
  unfolding linear_iff subspace_def
huffman@44133
   723
  apply (auto simp add: image_iff)
wenzelm@53406
   724
  apply (rule_tac x="x + y" in bexI)
wenzelm@53406
   725
  apply auto
wenzelm@53406
   726
  apply (rule_tac x="c *\<^sub>R x" in bexI)
wenzelm@53406
   727
  apply auto
huffman@44133
   728
  done
huffman@44133
   729
huffman@44521
   730
lemma subspace_linear_vimage: "linear f \<Longrightarrow> subspace S \<Longrightarrow> subspace (f -` S)"
huffman@53600
   731
  by (auto simp add: subspace_def linear_iff linear_0[of f])
huffman@44521
   732
wenzelm@53406
   733
lemma subspace_linear_preimage: "linear f \<Longrightarrow> subspace S \<Longrightarrow> subspace {x. f x \<in> S}"
huffman@53600
   734
  by (auto simp add: subspace_def linear_iff linear_0[of f])
huffman@44133
   735
huffman@44133
   736
lemma subspace_trivial: "subspace {0}"
huffman@44133
   737
  by (simp add: subspace_def)
huffman@44133
   738
wenzelm@53406
   739
lemma (in real_vector) subspace_inter: "subspace A \<Longrightarrow> subspace B \<Longrightarrow> subspace (A \<inter> B)"
huffman@44133
   740
  by (simp add: subspace_def)
huffman@44133
   741
wenzelm@53406
   742
lemma subspace_Times: "subspace A \<Longrightarrow> subspace B \<Longrightarrow> subspace (A \<times> B)"
huffman@44521
   743
  unfolding subspace_def zero_prod_def by simp
huffman@44521
   744
wenzelm@60420
   745
text \<open>Properties of span.\<close>
huffman@44521
   746
wenzelm@53406
   747
lemma (in real_vector) span_mono: "A \<subseteq> B \<Longrightarrow> span A \<subseteq> span B"
huffman@44133
   748
  by (metis span_def hull_mono)
huffman@44133
   749
wenzelm@53406
   750
lemma (in real_vector) subspace_span: "subspace (span S)"
huffman@44133
   751
  unfolding span_def
huffman@44170
   752
  apply (rule hull_in)
huffman@44133
   753
  apply (simp only: subspace_def Inter_iff Int_iff subset_eq)
huffman@44133
   754
  apply auto
huffman@44133
   755
  done
huffman@44133
   756
huffman@44133
   757
lemma (in real_vector) span_clauses:
wenzelm@53406
   758
  "a \<in> S \<Longrightarrow> a \<in> span S"
huffman@44133
   759
  "0 \<in> span S"
wenzelm@53406
   760
  "x\<in> span S \<Longrightarrow> y \<in> span S \<Longrightarrow> x + y \<in> span S"
huffman@44133
   761
  "x \<in> span S \<Longrightarrow> c *\<^sub>R x \<in> span S"
wenzelm@53406
   762
  by (metis span_def hull_subset subset_eq) (metis subspace_span subspace_def)+
huffman@44133
   763
huffman@44521
   764
lemma span_unique:
wenzelm@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"
huffman@44521
   766
  unfolding span_def by (rule hull_unique)
huffman@44521
   767
huffman@44521
   768
lemma span_minimal: "S \<subseteq> T \<Longrightarrow> subspace T \<Longrightarrow> span S \<subseteq> T"
huffman@44521
   769
  unfolding span_def by (rule hull_minimal)
huffman@44521
   770
huffman@44521
   771
lemma (in real_vector) span_induct:
wenzelm@49522
   772
  assumes x: "x \<in> span S"
wenzelm@49522
   773
    and P: "subspace P"
wenzelm@53406
   774
    and SP: "\<And>x. x \<in> S \<Longrightarrow> x \<in> P"
huffman@44521
   775
  shows "x \<in> P"
wenzelm@49522
   776
proof -
wenzelm@53406
   777
  from SP have SP': "S \<subseteq> P"
wenzelm@53406
   778
    by (simp add: subset_eq)
huffman@44170
   779
  from x hull_minimal[where S=subspace, OF SP' P, unfolded span_def[symmetric]]
wenzelm@53406
   780
  show "x \<in> P"
wenzelm@53406
   781
    by (metis subset_eq)
huffman@44133
   782
qed
huffman@44133
   783
huffman@44133
   784
lemma span_empty[simp]: "span {} = {0}"
huffman@44133
   785
  apply (simp add: span_def)
huffman@44133
   786
  apply (rule hull_unique)
huffman@44170
   787
  apply (auto simp add: subspace_def)
huffman@44133
   788
  done
huffman@44133
   789
lp15@62948
   790
lemma (in real_vector) independent_empty [iff]: "independent {}"
huffman@44133
   791
  by (simp add: dependent_def)
huffman@44133
   792
wenzelm@49522
   793
lemma dependent_single[simp]: "dependent {x} \<longleftrightarrow> x = 0"
huffman@44133
   794
  unfolding dependent_def by auto
huffman@44133
   795
wenzelm@53406
   796
lemma (in real_vector) independent_mono: "independent A \<Longrightarrow> B \<subseteq> A \<Longrightarrow> independent B"
huffman@44133
   797
  apply (clarsimp simp add: dependent_def span_mono)
huffman@44133
   798
  apply (subgoal_tac "span (B - {a}) \<le> span (A - {a})")
huffman@44133
   799
  apply force
huffman@44133
   800
  apply (rule span_mono)
huffman@44133
   801
  apply auto
huffman@44133
   802
  done
huffman@44133
   803
huffman@44133
   804
lemma (in real_vector) span_subspace: "A \<subseteq> B \<Longrightarrow> B \<le> span A \<Longrightarrow>  subspace B \<Longrightarrow> span A = B"
huffman@44170
   805
  by (metis order_antisym span_def hull_minimal)
huffman@44133
   806
wenzelm@49711
   807
lemma (in real_vector) span_induct':
wenzelm@49711
   808
  assumes SP: "\<forall>x \<in> S. P x"
wenzelm@49711
   809
    and P: "subspace {x. P x}"
wenzelm@49711
   810
  shows "\<forall>x \<in> span S. P x"
huffman@44133
   811
  using span_induct SP P by blast
huffman@44133
   812
wenzelm@56444
   813
inductive_set (in real_vector) span_induct_alt_help for S :: "'a set"
wenzelm@53406
   814
where
huffman@44170
   815
  span_induct_alt_help_0: "0 \<in> span_induct_alt_help S"
wenzelm@49522
   816
| span_induct_alt_help_S:
wenzelm@53406
   817
    "x \<in> S \<Longrightarrow> z \<in> span_induct_alt_help S \<Longrightarrow>
wenzelm@53406
   818
      (c *\<^sub>R x + z) \<in> span_induct_alt_help S"
huffman@44133
   819
huffman@44133
   820
lemma span_induct_alt':
wenzelm@53406
   821
  assumes h0: "h 0"
wenzelm@53406
   822
    and hS: "\<And>c x y. x \<in> S \<Longrightarrow> h y \<Longrightarrow> h (c *\<^sub>R x + y)"
wenzelm@49522
   823
  shows "\<forall>x \<in> span S. h x"
wenzelm@49522
   824
proof -
wenzelm@53406
   825
  {
wenzelm@53406
   826
    fix x :: 'a
wenzelm@53406
   827
    assume x: "x \<in> span_induct_alt_help S"
huffman@44133
   828
    have "h x"
huffman@44133
   829
      apply (rule span_induct_alt_help.induct[OF x])
huffman@44133
   830
      apply (rule h0)
wenzelm@53406
   831
      apply (rule hS)
wenzelm@53406
   832
      apply assumption
wenzelm@53406
   833
      apply assumption
wenzelm@53406
   834
      done
wenzelm@53406
   835
  }
huffman@44133
   836
  note th0 = this
wenzelm@53406
   837
  {
wenzelm@53406
   838
    fix x
wenzelm@53406
   839
    assume x: "x \<in> span S"
huffman@44170
   840
    have "x \<in> span_induct_alt_help S"
wenzelm@49522
   841
    proof (rule span_induct[where x=x and S=S])
wenzelm@53406
   842
      show "x \<in> span S" by (rule x)
wenzelm@49522
   843
    next
wenzelm@53406
   844
      fix x
wenzelm@53406
   845
      assume xS: "x \<in> S"
wenzelm@53406
   846
      from span_induct_alt_help_S[OF xS span_induct_alt_help_0, of 1]
wenzelm@53406
   847
      show "x \<in> span_induct_alt_help S"
wenzelm@53406
   848
        by simp
wenzelm@49522
   849
    next
wenzelm@49522
   850
      have "0 \<in> span_induct_alt_help S" by (rule span_induct_alt_help_0)
wenzelm@49522
   851
      moreover
wenzelm@53406
   852
      {
wenzelm@53406
   853
        fix x y
wenzelm@49522
   854
        assume h: "x \<in> span_induct_alt_help S" "y \<in> span_induct_alt_help S"
wenzelm@49522
   855
        from h have "(x + y) \<in> span_induct_alt_help S"
wenzelm@49522
   856
          apply (induct rule: span_induct_alt_help.induct)
wenzelm@49522
   857
          apply simp
haftmann@57512
   858
          unfolding add.assoc
wenzelm@49522
   859
          apply (rule span_induct_alt_help_S)
wenzelm@49522
   860
          apply assumption
wenzelm@49522
   861
          apply simp
wenzelm@53406
   862
          done
wenzelm@53406
   863
      }
wenzelm@49522
   864
      moreover
wenzelm@53406
   865
      {
wenzelm@53406
   866
        fix c x
wenzelm@49522
   867
        assume xt: "x \<in> span_induct_alt_help S"
wenzelm@49522
   868
        then have "(c *\<^sub>R x) \<in> span_induct_alt_help S"
wenzelm@49522
   869
          apply (induct rule: span_induct_alt_help.induct)
wenzelm@49522
   870
          apply (simp add: span_induct_alt_help_0)
wenzelm@49522
   871
          apply (simp add: scaleR_right_distrib)
wenzelm@49522
   872
          apply (rule span_induct_alt_help_S)
wenzelm@49522
   873
          apply assumption
wenzelm@49522
   874
          apply simp
wenzelm@49522
   875
          done }
wenzelm@53406
   876
      ultimately show "subspace (span_induct_alt_help S)"
wenzelm@49522
   877
        unfolding subspace_def Ball_def by blast
wenzelm@53406
   878
    qed
wenzelm@53406
   879
  }
huffman@44133
   880
  with th0 show ?thesis by blast
huffman@44133
   881
qed
huffman@44133
   882
huffman@44133
   883
lemma span_induct_alt:
wenzelm@53406
   884
  assumes h0: "h 0"
wenzelm@53406
   885
    and hS: "\<And>c x y. x \<in> S \<Longrightarrow> h y \<Longrightarrow> h (c *\<^sub>R x + y)"
wenzelm@53406
   886
    and x: "x \<in> span S"
huffman@44133
   887
  shows "h x"
wenzelm@49522
   888
  using span_induct_alt'[of h S] h0 hS x by blast
huffman@44133
   889
wenzelm@60420
   890
text \<open>Individual closure properties.\<close>
huffman@44133
   891
huffman@44133
   892
lemma span_span: "span (span A) = span A"
huffman@44133
   893
  unfolding span_def hull_hull ..
huffman@44133
   894
wenzelm@53406
   895
lemma (in real_vector) span_superset: "x \<in> S \<Longrightarrow> x \<in> span S"
wenzelm@53406
   896
  by (metis span_clauses(1))
wenzelm@53406
   897
wenzelm@53406
   898
lemma (in real_vector) span_0: "0 \<in> span S"
wenzelm@53406
   899
  by (metis subspace_span subspace_0)
huffman@44133
   900
huffman@44133
   901
lemma span_inc: "S \<subseteq> span S"
huffman@44133
   902
  by (metis subset_eq span_superset)
huffman@44133
   903
wenzelm@53406
   904
lemma (in real_vector) dependent_0:
wenzelm@53406
   905
  assumes "0 \<in> A"
wenzelm@53406
   906
  shows "dependent A"
wenzelm@53406
   907
  unfolding dependent_def
wenzelm@53406
   908
  using assms span_0
lp15@60162
   909
  by auto
wenzelm@53406
   910
wenzelm@53406
   911
lemma (in real_vector) span_add: "x \<in> span S \<Longrightarrow> y \<in> span S \<Longrightarrow> x + y \<in> span S"
huffman@44133
   912
  by (metis subspace_add subspace_span)
huffman@44133
   913
wenzelm@53406
   914
lemma (in real_vector) span_mul: "x \<in> span S \<Longrightarrow> c *\<^sub>R x \<in> span S"
huffman@44133
   915
  by (metis subspace_span subspace_mul)
huffman@44133
   916
wenzelm@53406
   917
lemma span_neg: "x \<in> span S \<Longrightarrow> - x \<in> span S"
huffman@44133
   918
  by (metis subspace_neg subspace_span)
huffman@44133
   919
wenzelm@53406
   920
lemma span_sub: "x \<in> span S \<Longrightarrow> y \<in> span S \<Longrightarrow> x - y \<in> span S"
huffman@44133
   921
  by (metis subspace_span subspace_sub)
huffman@44133
   922
huffman@56196
   923
lemma (in real_vector) span_setsum: "\<forall>x\<in>A. f x \<in> span S \<Longrightarrow> setsum f A \<in> span S"
huffman@56196
   924
  by (rule subspace_setsum [OF subspace_span])
huffman@44133
   925
huffman@44133
   926
lemma span_add_eq: "x \<in> span S \<Longrightarrow> x + y \<in> span S \<longleftrightarrow> y \<in> span S"
lp15@55775
   927
  by (metis add_minus_cancel scaleR_minus1_left subspace_def subspace_span)
huffman@44133
   928
wenzelm@60420
   929
text \<open>Mapping under linear image.\<close>
huffman@44133
   930
huffman@44521
   931
lemma span_linear_image:
huffman@44521
   932
  assumes lf: "linear f"
wenzelm@56444
   933
  shows "span (f ` S) = f ` span S"
huffman@44521
   934
proof (rule span_unique)
huffman@44521
   935
  show "f ` S \<subseteq> f ` span S"
huffman@44521
   936
    by (intro image_mono span_inc)
huffman@44521
   937
  show "subspace (f ` span S)"
huffman@44521
   938
    using lf subspace_span by (rule subspace_linear_image)
huffman@44521
   939
next
wenzelm@53406
   940
  fix T
wenzelm@53406
   941
  assume "f ` S \<subseteq> T" and "subspace T"
wenzelm@49522
   942
  then show "f ` span S \<subseteq> T"
huffman@44521
   943
    unfolding image_subset_iff_subset_vimage
huffman@44521
   944
    by (intro span_minimal subspace_linear_vimage lf)
huffman@44521
   945
qed
huffman@44521
   946
huffman@44521
   947
lemma span_union: "span (A \<union> B) = (\<lambda>(a, b). a + b) ` (span A \<times> span B)"
huffman@44521
   948
proof (rule span_unique)
huffman@44521
   949
  show "A \<union> B \<subseteq> (\<lambda>(a, b). a + b) ` (span A \<times> span B)"
huffman@44521
   950
    by safe (force intro: span_clauses)+
huffman@44521
   951
next
huffman@44521
   952
  have "linear (\<lambda>(a, b). a + b)"
huffman@53600
   953
    by (simp add: linear_iff scaleR_add_right)
huffman@44521
   954
  moreover have "subspace (span A \<times> span B)"
huffman@44521
   955
    by (intro subspace_Times subspace_span)
huffman@44521
   956
  ultimately show "subspace ((\<lambda>(a, b). a + b) ` (span A \<times> span B))"
huffman@44521
   957
    by (rule subspace_linear_image)
huffman@44521
   958
next
wenzelm@49711
   959
  fix T
wenzelm@49711
   960
  assume "A \<union> B \<subseteq> T" and "subspace T"
wenzelm@49522
   961
  then show "(\<lambda>(a, b). a + b) ` (span A \<times> span B) \<subseteq> T"
huffman@44521
   962
    by (auto intro!: subspace_add elim: span_induct)
huffman@44133
   963
qed
huffman@44133
   964
wenzelm@60420
   965
text \<open>The key breakdown property.\<close>
huffman@44133
   966
huffman@44521
   967
lemma span_singleton: "span {x} = range (\<lambda>k. k *\<^sub>R x)"
huffman@44521
   968
proof (rule span_unique)
huffman@44521
   969
  show "{x} \<subseteq> range (\<lambda>k. k *\<^sub>R x)"
huffman@44521
   970
    by (fast intro: scaleR_one [symmetric])
huffman@44521
   971
  show "subspace (range (\<lambda>k. k *\<^sub>R x))"
huffman@44521
   972
    unfolding subspace_def
huffman@44521
   973
    by (auto intro: scaleR_add_left [symmetric])
wenzelm@53406
   974
next
wenzelm@53406
   975
  fix T
wenzelm@53406
   976
  assume "{x} \<subseteq> T" and "subspace T"
wenzelm@53406
   977
  then show "range (\<lambda>k. k *\<^sub>R x) \<subseteq> T"
huffman@44521
   978
    unfolding subspace_def by auto
huffman@44521
   979
qed
huffman@44521
   980
wenzelm@49522
   981
lemma span_insert: "span (insert a S) = {x. \<exists>k. (x - k *\<^sub>R a) \<in> span S}"
huffman@44521
   982
proof -
huffman@44521
   983
  have "span ({a} \<union> S) = {x. \<exists>k. (x - k *\<^sub>R a) \<in> span S}"
huffman@44521
   984
    unfolding span_union span_singleton
huffman@44521
   985
    apply safe
huffman@44521
   986
    apply (rule_tac x=k in exI, simp)
huffman@44521
   987
    apply (erule rev_image_eqI [OF SigmaI [OF rangeI]])
haftmann@54230
   988
    apply auto
huffman@44521
   989
    done
wenzelm@49522
   990
  then show ?thesis by simp
huffman@44521
   991
qed
huffman@44521
   992
huffman@44133
   993
lemma span_breakdown:
wenzelm@53406
   994
  assumes bS: "b \<in> S"
wenzelm@53406
   995
    and aS: "a \<in> span S"
huffman@44521
   996
  shows "\<exists>k. a - k *\<^sub>R b \<in> span (S - {b})"
huffman@44521
   997
  using assms span_insert [of b "S - {b}"]
huffman@44521
   998
  by (simp add: insert_absorb)
huffman@44133
   999
wenzelm@53406
  1000
lemma span_breakdown_eq: "x \<in> span (insert a S) \<longleftrightarrow> (\<exists>k. x - k *\<^sub>R a \<in> span S)"
huffman@44521
  1001
  by (simp add: span_insert)
huffman@44133
  1002
wenzelm@60420
  1003
text \<open>Hence some "reversal" results.\<close>
huffman@44133
  1004
huffman@44133
  1005
lemma in_span_insert:
wenzelm@49711
  1006
  assumes a: "a \<in> span (insert b S)"
wenzelm@49711
  1007
    and na: "a \<notin> span S"
huffman@44133
  1008
  shows "b \<in> span (insert a S)"
wenzelm@49663
  1009
proof -
huffman@55910
  1010
  from a obtain k where k: "a - k *\<^sub>R b \<in> span S"
huffman@55910
  1011
    unfolding span_insert by fast
wenzelm@53406
  1012
  show ?thesis
wenzelm@53406
  1013
  proof (cases "k = 0")
wenzelm@53406
  1014
    case True
huffman@55910
  1015
    with k have "a \<in> span S" by simp
huffman@55910
  1016
    with na show ?thesis by simp
wenzelm@53406
  1017
  next
wenzelm@53406
  1018
    case False
huffman@55910
  1019
    from k have "(- inverse k) *\<^sub>R (a - k *\<^sub>R b) \<in> span S"
huffman@44133
  1020
      by (rule span_mul)
huffman@55910
  1021
    then have "b - inverse k *\<^sub>R a \<in> span S"
wenzelm@60420
  1022
      using \<open>k \<noteq> 0\<close> by (simp add: scaleR_diff_right)
huffman@55910
  1023
    then show ?thesis
huffman@55910
  1024
      unfolding span_insert by fast
wenzelm@53406
  1025
  qed
huffman@44133
  1026
qed
huffman@44133
  1027
huffman@44133
  1028
lemma in_span_delete:
huffman@44133
  1029
  assumes a: "a \<in> span S"
wenzelm@53716
  1030
    and na: "a \<notin> span (S - {b})"
huffman@44133
  1031
  shows "b \<in> span (insert a (S - {b}))"
huffman@44133
  1032
  apply (rule in_span_insert)
huffman@44133
  1033
  apply (rule set_rev_mp)
huffman@44133
  1034
  apply (rule a)
huffman@44133
  1035
  apply (rule span_mono)
huffman@44133
  1036
  apply blast
huffman@44133
  1037
  apply (rule na)
huffman@44133
  1038
  done
huffman@44133
  1039
wenzelm@60420
  1040
text \<open>Transitivity property.\<close>
huffman@44133
  1041
huffman@44521
  1042
lemma span_redundant: "x \<in> span S \<Longrightarrow> span (insert x S) = span S"
huffman@44521
  1043
  unfolding span_def by (rule hull_redundant)
huffman@44521
  1044
huffman@44133
  1045
lemma span_trans:
wenzelm@53406
  1046
  assumes x: "x \<in> span S"
wenzelm@53406
  1047
    and y: "y \<in> span (insert x S)"
huffman@44133
  1048
  shows "y \<in> span S"
huffman@44521
  1049
  using assms by (simp only: span_redundant)
huffman@44133
  1050
huffman@44133
  1051
lemma span_insert_0[simp]: "span (insert 0 S) = span S"
huffman@44521
  1052
  by (simp only: span_redundant span_0)
huffman@44133
  1053
wenzelm@60420
  1054
text \<open>An explicit expansion is sometimes needed.\<close>
huffman@44133
  1055
huffman@44133
  1056
lemma span_explicit:
huffman@44133
  1057
  "span P = {y. \<exists>S u. finite S \<and> S \<subseteq> P \<and> setsum (\<lambda>v. u v *\<^sub>R v) S = y}"
huffman@44133
  1058
  (is "_ = ?E" is "_ = {y. ?h y}" is "_ = {y. \<exists>S u. ?Q S u y}")
wenzelm@49663
  1059
proof -
wenzelm@53406
  1060
  {
wenzelm@53406
  1061
    fix x
huffman@55910
  1062
    assume "?h x"
huffman@55910
  1063
    then obtain S u where "finite S" and "S \<subseteq> P" and "setsum (\<lambda>v. u v *\<^sub>R v) S = x"
huffman@44133
  1064
      by blast
huffman@55910
  1065
    then have "x \<in> span P"
huffman@55910
  1066
      by (auto intro: span_setsum span_mul span_superset)
wenzelm@53406
  1067
  }
huffman@44133
  1068
  moreover
huffman@55910
  1069
  have "\<forall>x \<in> span P. ?h x"
wenzelm@49522
  1070
  proof (rule span_induct_alt')
huffman@55910
  1071
    show "?h 0"
huffman@55910
  1072
      by (rule exI[where x="{}"], simp)
huffman@44133
  1073
  next
huffman@44133
  1074
    fix c x y
wenzelm@53406
  1075
    assume x: "x \<in> P"
huffman@55910
  1076
    assume hy: "?h y"
huffman@44133
  1077
    from hy obtain S u where fS: "finite S" and SP: "S\<subseteq>P"
huffman@44133
  1078
      and u: "setsum (\<lambda>v. u v *\<^sub>R v) S = y" by blast
huffman@44133
  1079
    let ?S = "insert x S"
wenzelm@49522
  1080
    let ?u = "\<lambda>y. if y = x then (if x \<in> S then u y + c else c) else u y"
wenzelm@53406
  1081
    from fS SP x have th0: "finite (insert x S)" "insert x S \<subseteq> P"
wenzelm@53406
  1082
      by blast+
wenzelm@53406
  1083
    have "?Q ?S ?u (c*\<^sub>R x + y)"
wenzelm@53406
  1084
    proof cases
wenzelm@53406
  1085
      assume xS: "x \<in> S"
huffman@55910
  1086
      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"
huffman@55910
  1087
        using xS by (simp add: setsum.remove [OF fS xS] insert_absorb)
huffman@44133
  1088
      also have "\<dots> = (\<Sum>v\<in>S. u v *\<^sub>R v) + c *\<^sub>R x"
huffman@55910
  1089
        by (simp add: setsum.remove [OF fS xS] algebra_simps)
huffman@44133
  1090
      also have "\<dots> = c*\<^sub>R x + y"
haftmann@57512
  1091
        by (simp add: add.commute u)
huffman@44133
  1092
      finally have "setsum (\<lambda>v. ?u v *\<^sub>R v) ?S = c*\<^sub>R x + y" .
wenzelm@53406
  1093
      then show ?thesis using th0 by blast
wenzelm@53406
  1094
    next
wenzelm@53406
  1095
      assume xS: "x \<notin> S"
wenzelm@49522
  1096
      have th00: "(\<Sum>v\<in>S. (if v = x then c else u v) *\<^sub>R v) = y"
wenzelm@49522
  1097
        unfolding u[symmetric]
haftmann@57418
  1098
        apply (rule setsum.cong)
wenzelm@53406
  1099
        using xS
wenzelm@53406
  1100
        apply auto
wenzelm@49522
  1101
        done
wenzelm@53406
  1102
      show ?thesis using fS xS th0
haftmann@57512
  1103
        by (simp add: th00 add.commute cong del: if_weak_cong)
wenzelm@53406
  1104
    qed
huffman@55910
  1105
    then show "?h (c*\<^sub>R x + y)"
huffman@55910
  1106
      by fast
huffman@44133
  1107
  qed
huffman@44133
  1108
  ultimately show ?thesis by blast
huffman@44133
  1109
qed
huffman@44133
  1110
huffman@44133
  1111
lemma dependent_explicit:
wenzelm@49522
  1112
  "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))"
wenzelm@49522
  1113
  (is "?lhs = ?rhs")
wenzelm@49522
  1114
proof -
wenzelm@53406
  1115
  {
wenzelm@53406
  1116
    assume dP: "dependent P"
huffman@44133
  1117
    then obtain a S u where aP: "a \<in> P" and fS: "finite S"
huffman@44133
  1118
      and SP: "S \<subseteq> P - {a}" and ua: "setsum (\<lambda>v. u v *\<^sub>R v) S = a"
huffman@44133
  1119
      unfolding dependent_def span_explicit by blast
huffman@44133
  1120
    let ?S = "insert a S"
huffman@44133
  1121
    let ?u = "\<lambda>y. if y = a then - 1 else u y"
huffman@44133
  1122
    let ?v = a
wenzelm@53406
  1123
    from aP SP have aS: "a \<notin> S"
wenzelm@53406
  1124
      by blast
wenzelm@53406
  1125
    from fS SP aP have th0: "finite ?S" "?S \<subseteq> P" "?v \<in> ?S" "?u ?v \<noteq> 0"
wenzelm@53406
  1126
      by auto
huffman@44133
  1127
    have s0: "setsum (\<lambda>v. ?u v *\<^sub>R v) ?S = 0"
huffman@44133
  1128
      using fS aS
huffman@55910
  1129
      apply simp
huffman@44133
  1130
      apply (subst (2) ua[symmetric])
haftmann@57418
  1131
      apply (rule setsum.cong)
wenzelm@49522
  1132
      apply auto
wenzelm@49522
  1133
      done
huffman@55910
  1134
    with th0 have ?rhs by fast
wenzelm@49522
  1135
  }
huffman@44133
  1136
  moreover
wenzelm@53406
  1137
  {
wenzelm@53406
  1138
    fix S u v
wenzelm@49522
  1139
    assume fS: "finite S"
wenzelm@53406
  1140
      and SP: "S \<subseteq> P"
wenzelm@53406
  1141
      and vS: "v \<in> S"
wenzelm@53406
  1142
      and uv: "u v \<noteq> 0"
wenzelm@49522
  1143
      and u: "setsum (\<lambda>v. u v *\<^sub>R v) S = 0"
huffman@44133
  1144
    let ?a = v
huffman@44133
  1145
    let ?S = "S - {v}"
huffman@44133
  1146
    let ?u = "\<lambda>i. (- u i) / u v"
wenzelm@53406
  1147
    have th0: "?a \<in> P" "finite ?S" "?S \<subseteq> P"
wenzelm@53406
  1148
      using fS SP vS by auto
wenzelm@53406
  1149
    have "setsum (\<lambda>v. ?u v *\<^sub>R v) ?S =
wenzelm@53406
  1150
      setsum (\<lambda>v. (- (inverse (u ?a))) *\<^sub>R (u v *\<^sub>R v)) S - ?u v *\<^sub>R v"
hoelzl@56480
  1151
      using fS vS uv by (simp add: setsum_diff1 field_simps)
wenzelm@53406
  1152
    also have "\<dots> = ?a"
hoelzl@56479
  1153
      unfolding scaleR_right.setsum [symmetric] u using uv by simp
wenzelm@53406
  1154
    finally have "setsum (\<lambda>v. ?u v *\<^sub>R v) ?S = ?a" .
huffman@44133
  1155
    with th0 have ?lhs
huffman@44133
  1156
      unfolding dependent_def span_explicit
huffman@44133
  1157
      apply -
huffman@44133
  1158
      apply (rule bexI[where x= "?a"])
huffman@44133
  1159
      apply (simp_all del: scaleR_minus_left)
huffman@44133
  1160
      apply (rule exI[where x= "?S"])
wenzelm@49522
  1161
      apply (auto simp del: scaleR_minus_left)
wenzelm@49522
  1162
      done
wenzelm@49522
  1163
  }
huffman@44133
  1164
  ultimately show ?thesis by blast
huffman@44133
  1165
qed
huffman@44133
  1166
huffman@44133
  1167
huffman@44133
  1168
lemma span_finite:
huffman@44133
  1169
  assumes fS: "finite S"
huffman@44133
  1170
  shows "span S = {y. \<exists>u. setsum (\<lambda>v. u v *\<^sub>R v) S = y}"
huffman@44133
  1171
  (is "_ = ?rhs")
wenzelm@49522
  1172
proof -
wenzelm@53406
  1173
  {
wenzelm@53406
  1174
    fix y
wenzelm@49711
  1175
    assume y: "y \<in> span S"
wenzelm@53406
  1176
    from y obtain S' u where fS': "finite S'"
wenzelm@53406
  1177
      and SS': "S' \<subseteq> S"
wenzelm@53406
  1178
      and u: "setsum (\<lambda>v. u v *\<^sub>R v) S' = y"
wenzelm@53406
  1179
      unfolding span_explicit by blast
huffman@44133
  1180
    let ?u = "\<lambda>x. if x \<in> S' then u x else 0"
huffman@44133
  1181
    have "setsum (\<lambda>v. ?u v *\<^sub>R v) S = setsum (\<lambda>v. u v *\<^sub>R v) S'"
haftmann@57418
  1182
      using SS' fS by (auto intro!: setsum.mono_neutral_cong_right)
wenzelm@49522
  1183
    then have "setsum (\<lambda>v. ?u v *\<^sub>R v) S = y" by (metis u)
wenzelm@53406
  1184
    then have "y \<in> ?rhs" by auto
wenzelm@53406
  1185
  }
huffman@44133
  1186
  moreover
wenzelm@53406
  1187
  {
wenzelm@53406
  1188
    fix y u
wenzelm@49522
  1189
    assume u: "setsum (\<lambda>v. u v *\<^sub>R v) S = y"
wenzelm@53406
  1190
    then have "y \<in> span S" using fS unfolding span_explicit by auto
wenzelm@53406
  1191
  }
huffman@44133
  1192
  ultimately show ?thesis by blast
huffman@44133
  1193
qed
huffman@44133
  1194
wenzelm@60420
  1195
text \<open>This is useful for building a basis step-by-step.\<close>
huffman@44133
  1196
huffman@44133
  1197
lemma independent_insert:
wenzelm@53406
  1198
  "independent (insert a S) \<longleftrightarrow>
wenzelm@53406
  1199
    (if a \<in> S then independent S else independent S \<and> a \<notin> span S)"
wenzelm@53406
  1200
  (is "?lhs \<longleftrightarrow> ?rhs")
wenzelm@53406
  1201
proof (cases "a \<in> S")
wenzelm@53406
  1202
  case True
wenzelm@53406
  1203
  then show ?thesis
wenzelm@53406
  1204
    using insert_absorb[OF True] by simp
wenzelm@53406
  1205
next
wenzelm@53406
  1206
  case False
wenzelm@53406
  1207
  show ?thesis
wenzelm@53406
  1208
  proof
wenzelm@53406
  1209
    assume i: ?lhs
wenzelm@53406
  1210
    then show ?rhs
wenzelm@53406
  1211
      using False
wenzelm@53406
  1212
      apply simp
wenzelm@53406
  1213
      apply (rule conjI)
wenzelm@53406
  1214
      apply (rule independent_mono)
wenzelm@53406
  1215
      apply assumption
wenzelm@53406
  1216
      apply blast
wenzelm@53406
  1217
      apply (simp add: dependent_def)
wenzelm@53406
  1218
      done
wenzelm@53406
  1219
  next
wenzelm@53406
  1220
    assume i: ?rhs
wenzelm@53406
  1221
    show ?lhs
wenzelm@53406
  1222
      using i False
wenzelm@53406
  1223
      apply (auto simp add: dependent_def)
lp15@60810
  1224
      by (metis in_span_insert insert_Diff_if insert_Diff_single insert_absorb)
wenzelm@53406
  1225
  qed
huffman@44133
  1226
qed
huffman@44133
  1227
wenzelm@60420
  1228
text \<open>The degenerate case of the Exchange Lemma.\<close>
huffman@44133
  1229
huffman@44133
  1230
lemma spanning_subset_independent:
wenzelm@49711
  1231
  assumes BA: "B \<subseteq> A"
wenzelm@49711
  1232
    and iA: "independent A"
wenzelm@49522
  1233
    and AsB: "A \<subseteq> span B"
huffman@44133
  1234
  shows "A = B"
huffman@44133
  1235
proof
wenzelm@49663
  1236
  show "B \<subseteq> A" by (rule BA)
wenzelm@49663
  1237
huffman@44133
  1238
  from span_mono[OF BA] span_mono[OF AsB]
huffman@44133
  1239
  have sAB: "span A = span B" unfolding span_span by blast
huffman@44133
  1240
wenzelm@53406
  1241
  {
wenzelm@53406
  1242
    fix x
wenzelm@53406
  1243
    assume x: "x \<in> A"
huffman@44133
  1244
    from iA have th0: "x \<notin> span (A - {x})"
huffman@44133
  1245
      unfolding dependent_def using x by blast
wenzelm@53406
  1246
    from x have xsA: "x \<in> span A"
wenzelm@53406
  1247
      by (blast intro: span_superset)
huffman@44133
  1248
    have "A - {x} \<subseteq> A" by blast
wenzelm@53406
  1249
    then have th1: "span (A - {x}) \<subseteq> span A"
wenzelm@53406
  1250
      by (metis span_mono)
wenzelm@53406
  1251
    {
wenzelm@53406
  1252
      assume xB: "x \<notin> B"
wenzelm@53406
  1253
      from xB BA have "B \<subseteq> A - {x}"
wenzelm@53406
  1254
        by blast
wenzelm@53406
  1255
      then have "span B \<subseteq> span (A - {x})"
wenzelm@53406
  1256
        by (metis span_mono)
wenzelm@53406
  1257
      with th1 th0 sAB have "x \<notin> span A"
wenzelm@53406
  1258
        by blast
wenzelm@53406
  1259
      with x have False
wenzelm@53406
  1260
        by (metis span_superset)
wenzelm@53406
  1261
    }
wenzelm@53406
  1262
    then have "x \<in> B" by blast
wenzelm@53406
  1263
  }
huffman@44133
  1264
  then show "A \<subseteq> B" by blast
huffman@44133
  1265
qed
huffman@44133
  1266
wenzelm@60420
  1267
text \<open>The general case of the Exchange Lemma, the key to what follows.\<close>
huffman@44133
  1268
huffman@44133
  1269
lemma exchange_lemma:
wenzelm@49711
  1270
  assumes f:"finite t"
wenzelm@49711
  1271
    and i: "independent s"
wenzelm@49711
  1272
    and sp: "s \<subseteq> span t"
wenzelm@53406
  1273
  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'"
wenzelm@49663
  1274
  using f i sp
wenzelm@49522
  1275
proof (induct "card (t - s)" arbitrary: s t rule: less_induct)
huffman@44133
  1276
  case less
wenzelm@60420
  1277
  note ft = \<open>finite t\<close> and s = \<open>independent s\<close> and sp = \<open>s \<subseteq> span t\<close>
wenzelm@53406
  1278
  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'"
huffman@44133
  1279
  let ?ths = "\<exists>t'. ?P t'"
wenzelm@53406
  1280
  {
lp15@55775
  1281
    assume "s \<subseteq> t"
lp15@55775
  1282
    then have ?ths
lp15@55775
  1283
      by (metis ft Un_commute sp sup_ge1)
wenzelm@53406
  1284
  }
huffman@44133
  1285
  moreover
wenzelm@53406
  1286
  {
wenzelm@53406
  1287
    assume st: "t \<subseteq> s"
wenzelm@53406
  1288
    from spanning_subset_independent[OF st s sp] st ft span_mono[OF st]
wenzelm@53406
  1289
    have ?ths
lp15@55775
  1290
      by (metis Un_absorb sp)
wenzelm@53406
  1291
  }
huffman@44133
  1292
  moreover
wenzelm@53406
  1293
  {
wenzelm@53406
  1294
    assume st: "\<not> s \<subseteq> t" "\<not> t \<subseteq> s"
wenzelm@53406
  1295
    from st(2) obtain b where b: "b \<in> t" "b \<notin> s"
wenzelm@53406
  1296
      by blast
wenzelm@53406
  1297
    from b have "t - {b} - s \<subset> t - s"
wenzelm@53406
  1298
      by blast
wenzelm@53406
  1299
    then have cardlt: "card (t - {b} - s) < card (t - s)"
wenzelm@53406
  1300
      using ft by (auto intro: psubset_card_mono)
wenzelm@53406
  1301
    from b ft have ct0: "card t \<noteq> 0"
wenzelm@53406
  1302
      by auto
wenzelm@53406
  1303
    have ?ths
wenzelm@53406
  1304
    proof cases
wenzelm@53716
  1305
      assume stb: "s \<subseteq> span (t - {b})"
wenzelm@53716
  1306
      from ft have ftb: "finite (t - {b})"
wenzelm@53406
  1307
        by auto
huffman@44133
  1308
      from less(1)[OF cardlt ftb s stb]
wenzelm@53716
  1309
      obtain u where u: "card u = card (t - {b})" "s \<subseteq> u" "u \<subseteq> s \<union> (t - {b})" "s \<subseteq> span u"
wenzelm@49522
  1310
        and fu: "finite u" by blast
huffman@44133
  1311
      let ?w = "insert b u"
wenzelm@53406
  1312
      have th0: "s \<subseteq> insert b u"
wenzelm@53406
  1313
        using u by blast
wenzelm@53406
  1314
      from u(3) b have "u \<subseteq> s \<union> t"
wenzelm@53406
  1315
        by blast
wenzelm@53406
  1316
      then have th1: "insert b u \<subseteq> s \<union> t"
wenzelm@53406
  1317
        using u b by blast
wenzelm@53406
  1318
      have bu: "b \<notin> u"
wenzelm@53406
  1319
        using b u by blast
wenzelm@53406
  1320
      from u(1) ft b have "card u = (card t - 1)"
wenzelm@53406
  1321
        by auto
wenzelm@49522
  1322
      then have th2: "card (insert b u) = card t"
huffman@44133
  1323
        using card_insert_disjoint[OF fu bu] ct0 by auto
huffman@44133
  1324
      from u(4) have "s \<subseteq> span u" .
wenzelm@53406
  1325
      also have "\<dots> \<subseteq> span (insert b u)"
wenzelm@53406
  1326
        by (rule span_mono) blast
huffman@44133
  1327
      finally have th3: "s \<subseteq> span (insert b u)" .
wenzelm@53406
  1328
      from th0 th1 th2 th3 fu have th: "?P ?w"
wenzelm@53406
  1329
        by blast
wenzelm@53406
  1330
      from th show ?thesis by blast
wenzelm@53406
  1331
    next
wenzelm@53716
  1332
      assume stb: "\<not> s \<subseteq> span (t - {b})"
wenzelm@53406
  1333
      from stb obtain a where a: "a \<in> s" "a \<notin> span (t - {b})"
wenzelm@53406
  1334
        by blast
wenzelm@53406
  1335
      have ab: "a \<noteq> b"
wenzelm@53406
  1336
        using a b by blast
wenzelm@53406
  1337
      have at: "a \<notin> t"
wenzelm@53406
  1338
        using a ab span_superset[of a "t- {b}"] by auto
huffman@44133
  1339
      have mlt: "card ((insert a (t - {b})) - s) < card (t - s)"
huffman@44133
  1340
        using cardlt ft a b by auto
wenzelm@53406
  1341
      have ft': "finite (insert a (t - {b}))"
wenzelm@53406
  1342
        using ft by auto
wenzelm@53406
  1343
      {
wenzelm@53406
  1344
        fix x
wenzelm@53406
  1345
        assume xs: "x \<in> s"
wenzelm@53406
  1346
        have t: "t \<subseteq> insert b (insert a (t - {b}))"
wenzelm@53406
  1347
          using b by auto
wenzelm@53406
  1348
        from b(1) have "b \<in> span t"
wenzelm@53406
  1349
          by (simp add: span_superset)
wenzelm@53406
  1350
        have bs: "b \<in> span (insert a (t - {b}))"
wenzelm@53406
  1351
          apply (rule in_span_delete)
wenzelm@53406
  1352
          using a sp unfolding subset_eq
wenzelm@53406
  1353
          apply auto
wenzelm@53406
  1354
          done
wenzelm@53406
  1355
        from xs sp have "x \<in> span t"
wenzelm@53406
  1356
          by blast
wenzelm@53406
  1357
        with span_mono[OF t] have x: "x \<in> span (insert b (insert a (t - {b})))" ..
wenzelm@53406
  1358
        from span_trans[OF bs x] have "x \<in> span (insert a (t - {b}))" .
wenzelm@53406
  1359
      }
wenzelm@53406
  1360
      then have sp': "s \<subseteq> span (insert a (t - {b}))"
wenzelm@53406
  1361
        by blast
wenzelm@53406
  1362
      from less(1)[OF mlt ft' s sp'] obtain u where u:
wenzelm@53716
  1363
        "card u = card (insert a (t - {b}))"
wenzelm@53716
  1364
        "finite u" "s \<subseteq> u" "u \<subseteq> s \<union> insert a (t - {b})"
wenzelm@53406
  1365
        "s \<subseteq> span u" by blast
wenzelm@53406
  1366
      from u a b ft at ct0 have "?P u"
wenzelm@53406
  1367
        by auto
wenzelm@53406
  1368
      then show ?thesis by blast
wenzelm@53406
  1369
    qed
huffman@44133
  1370
  }
wenzelm@49522
  1371
  ultimately show ?ths by blast
huffman@44133
  1372
qed
huffman@44133
  1373
wenzelm@60420
  1374
text \<open>This implies corresponding size bounds.\<close>
huffman@44133
  1375
huffman@44133
  1376
lemma independent_span_bound:
wenzelm@53406
  1377
  assumes f: "finite t"
wenzelm@53406
  1378
    and i: "independent s"
wenzelm@53406
  1379
    and sp: "s \<subseteq> span t"
huffman@44133
  1380
  shows "finite s \<and> card s \<le> card t"
huffman@44133
  1381
  by (metis exchange_lemma[OF f i sp] finite_subset card_mono)
huffman@44133
  1382
huffman@44133
  1383
lemma finite_Atleast_Atmost_nat[simp]: "finite {f x |x. x\<in> (UNIV::'a::finite set)}"
wenzelm@49522
  1384
proof -
wenzelm@53406
  1385
  have eq: "{f x |x. x\<in> UNIV} = f ` UNIV"
wenzelm@53406
  1386
    by auto
huffman@44133
  1387
  show ?thesis unfolding eq
huffman@44133
  1388
    apply (rule finite_imageI)
huffman@44133
  1389
    apply (rule finite)
huffman@44133
  1390
    done
huffman@44133
  1391
qed
huffman@44133
  1392
wenzelm@53406
  1393
wenzelm@60420
  1394
subsection \<open>Euclidean Spaces as Typeclass\<close>
huffman@44133
  1395
hoelzl@50526
  1396
lemma independent_Basis: "independent Basis"
hoelzl@50526
  1397
  unfolding dependent_def
hoelzl@50526
  1398
  apply (subst span_finite)
hoelzl@50526
  1399
  apply simp
huffman@44133
  1400
  apply clarify
hoelzl@50526
  1401
  apply (drule_tac f="inner a" in arg_cong)
hoelzl@50526
  1402
  apply (simp add: inner_Basis inner_setsum_right eq_commute)
hoelzl@50526
  1403
  done
hoelzl@50526
  1404
huffman@53939
  1405
lemma span_Basis [simp]: "span Basis = UNIV"
huffman@53939
  1406
  unfolding span_finite [OF finite_Basis]
huffman@53939
  1407
  by (fast intro: euclidean_representation)
huffman@44133
  1408
hoelzl@50526
  1409
lemma in_span_Basis: "x \<in> span Basis"
hoelzl@50526
  1410
  unfolding span_Basis ..
hoelzl@50526
  1411
hoelzl@50526
  1412
lemma Basis_le_norm: "b \<in> Basis \<Longrightarrow> \<bar>x \<bullet> b\<bar> \<le> norm x"
hoelzl@50526
  1413
  by (rule order_trans [OF Cauchy_Schwarz_ineq2]) simp
hoelzl@50526
  1414
hoelzl@50526
  1415
lemma norm_bound_Basis_le: "b \<in> Basis \<Longrightarrow> norm x \<le> e \<Longrightarrow> \<bar>x \<bullet> b\<bar> \<le> e"
hoelzl@50526
  1416
  by (metis Basis_le_norm order_trans)
hoelzl@50526
  1417
hoelzl@50526
  1418
lemma norm_bound_Basis_lt: "b \<in> Basis \<Longrightarrow> norm x < e \<Longrightarrow> \<bar>x \<bullet> b\<bar> < e"
huffman@53595
  1419
  by (metis Basis_le_norm le_less_trans)
hoelzl@50526
  1420
hoelzl@50526
  1421
lemma norm_le_l1: "norm x \<le> (\<Sum>b\<in>Basis. \<bar>x \<bullet> b\<bar>)"
hoelzl@50526
  1422
  apply (subst euclidean_representation[of x, symmetric])
huffman@44176
  1423
  apply (rule order_trans[OF norm_setsum])
wenzelm@49522
  1424
  apply (auto intro!: setsum_mono)
wenzelm@49522
  1425
  done
huffman@44133
  1426
huffman@44133
  1427
lemma setsum_norm_allsubsets_bound:
wenzelm@56444
  1428
  fixes f :: "'a \<Rightarrow> 'n::euclidean_space"
wenzelm@53406
  1429
  assumes fP: "finite P"
wenzelm@53406
  1430
    and fPs: "\<And>Q. Q \<subseteq> P \<Longrightarrow> norm (setsum f Q) \<le> e"
hoelzl@50526
  1431
  shows "(\<Sum>x\<in>P. norm (f x)) \<le> 2 * real DIM('n) * e"
wenzelm@49522
  1432
proof -
hoelzl@50526
  1433
  have "(\<Sum>x\<in>P. norm (f x)) \<le> (\<Sum>x\<in>P. \<Sum>b\<in>Basis. \<bar>f x \<bullet> b\<bar>)"
hoelzl@50526
  1434
    by (rule setsum_mono) (rule norm_le_l1)
hoelzl@50526
  1435
  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>)"
haftmann@57418
  1436
    by (rule setsum.commute)
hoelzl@50526
  1437
  also have "\<dots> \<le> of_nat (card (Basis :: 'n set)) * (2 * e)"
lp15@60974
  1438
  proof (rule setsum_bounded_above)
wenzelm@53406
  1439
    fix i :: 'n
wenzelm@53406
  1440
    assume i: "i \<in> Basis"
wenzelm@53406
  1441
    have "norm (\<Sum>x\<in>P. \<bar>f x \<bullet> i\<bar>) \<le>
hoelzl@50526
  1442
      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)"
haftmann@57418
  1443
      by (simp add: abs_real_def setsum.If_cases[OF fP] setsum_negf norm_triangle_ineq4 inner_setsum_left
wenzelm@56444
  1444
        del: real_norm_def)
wenzelm@53406
  1445
    also have "\<dots> \<le> e + e"
wenzelm@53406
  1446
      unfolding real_norm_def
hoelzl@50526
  1447
      by (intro add_mono norm_bound_Basis_le i fPs) auto
hoelzl@50526
  1448
    finally show "(\<Sum>x\<in>P. \<bar>f x \<bullet> i\<bar>) \<le> 2*e" by simp
huffman@44133
  1449
  qed
lp15@61609
  1450
  also have "\<dots> = 2 * real DIM('n) * e" by simp
huffman@44133
  1451
  finally show ?thesis .
huffman@44133
  1452
qed
huffman@44133
  1453
wenzelm@53406
  1454
wenzelm@60420
  1455
subsection \<open>Linearity and Bilinearity continued\<close>
huffman@44133
  1456
huffman@44133
  1457
lemma linear_bounded:
wenzelm@56444
  1458
  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
huffman@44133
  1459
  assumes lf: "linear f"
huffman@44133
  1460
  shows "\<exists>B. \<forall>x. norm (f x) \<le> B * norm x"
huffman@53939
  1461
proof
hoelzl@50526
  1462
  let ?B = "\<Sum>b\<in>Basis. norm (f b)"
huffman@53939
  1463
  show "\<forall>x. norm (f x) \<le> ?B * norm x"
huffman@53939
  1464
  proof
wenzelm@53406
  1465
    fix x :: 'a
hoelzl@50526
  1466
    let ?g = "\<lambda>b. (x \<bullet> b) *\<^sub>R f b"
hoelzl@50526
  1467
    have "norm (f x) = norm (f (\<Sum>b\<in>Basis. (x \<bullet> b) *\<^sub>R b))"
hoelzl@50526
  1468
      unfolding euclidean_representation ..
hoelzl@50526
  1469
    also have "\<dots> = norm (setsum ?g Basis)"
huffman@53939
  1470
      by (simp add: linear_setsum [OF lf] linear_cmul [OF lf])
hoelzl@50526
  1471
    finally have th0: "norm (f x) = norm (setsum ?g Basis)" .
huffman@53939
  1472
    have th: "\<forall>b\<in>Basis. norm (?g b) \<le> norm (f b) * norm x"
huffman@53939
  1473
    proof
wenzelm@53406
  1474
      fix i :: 'a
wenzelm@53406
  1475
      assume i: "i \<in> Basis"
hoelzl@50526
  1476
      from Basis_le_norm[OF i, of x]
huffman@53939
  1477
      show "norm (?g i) \<le> norm (f i) * norm x"
wenzelm@49663
  1478
        unfolding norm_scaleR
haftmann@57512
  1479
        apply (subst mult.commute)
wenzelm@49663
  1480
        apply (rule mult_mono)
wenzelm@49663
  1481
        apply (auto simp add: field_simps)
wenzelm@53406
  1482
        done
huffman@53939
  1483
    qed
hoelzl@50526
  1484
    from setsum_norm_le[of _ ?g, OF th]
huffman@53939
  1485
    show "norm (f x) \<le> ?B * norm x"
wenzelm@53406
  1486
      unfolding th0 setsum_left_distrib by metis
huffman@53939
  1487
  qed
huffman@44133
  1488
qed
huffman@44133
  1489
huffman@44133
  1490
lemma linear_conv_bounded_linear:
huffman@44133
  1491
  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
huffman@44133
  1492
  shows "linear f \<longleftrightarrow> bounded_linear f"
huffman@44133
  1493
proof
huffman@44133
  1494
  assume "linear f"
huffman@53939
  1495
  then interpret f: linear f .
huffman@44133
  1496
  show "bounded_linear f"
huffman@44133
  1497
  proof
huffman@44133
  1498
    have "\<exists>B. \<forall>x. norm (f x) \<le> B * norm x"
wenzelm@60420
  1499
      using \<open>linear f\<close> by (rule linear_bounded)
wenzelm@49522
  1500
    then show "\<exists>K. \<forall>x. norm (f x) \<le> norm x * K"
haftmann@57512
  1501
      by (simp add: mult.commute)
huffman@44133
  1502
  qed
huffman@44133
  1503
next
huffman@44133
  1504
  assume "bounded_linear f"
huffman@44133
  1505
  then interpret f: bounded_linear f .
huffman@53939
  1506
  show "linear f" ..
huffman@53939
  1507
qed
huffman@53939
  1508
paulson@61518
  1509
lemmas linear_linear = linear_conv_bounded_linear[symmetric]
paulson@61518
  1510
huffman@53939
  1511
lemma linear_bounded_pos:
wenzelm@56444
  1512
  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
huffman@53939
  1513
  assumes lf: "linear f"
huffman@53939
  1514
  shows "\<exists>B > 0. \<forall>x. norm (f x) \<le> B * norm x"
huffman@53939
  1515
proof -
huffman@53939
  1516
  have "\<exists>B > 0. \<forall>x. norm (f x) \<le> norm x * B"
huffman@53939
  1517
    using lf unfolding linear_conv_bounded_linear
huffman@53939
  1518
    by (rule bounded_linear.pos_bounded)
huffman@53939
  1519
  then show ?thesis
haftmann@57512
  1520
    by (simp only: mult.commute)
huffman@44133
  1521
qed
huffman@44133
  1522
wenzelm@49522
  1523
lemma bounded_linearI':
wenzelm@56444
  1524
  fixes f ::"'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
wenzelm@53406
  1525
  assumes "\<And>x y. f (x + y) = f x + f y"
wenzelm@53406
  1526
    and "\<And>c x. f (c *\<^sub>R x) = c *\<^sub>R f x"
wenzelm@49522
  1527
  shows "bounded_linear f"
wenzelm@53406
  1528
  unfolding linear_conv_bounded_linear[symmetric]
wenzelm@49522
  1529
  by (rule linearI[OF assms])
huffman@44133
  1530
huffman@44133
  1531
lemma bilinear_bounded:
wenzelm@56444
  1532
  fixes h :: "'m::euclidean_space \<Rightarrow> 'n::euclidean_space \<Rightarrow> 'k::real_normed_vector"
huffman@44133
  1533
  assumes bh: "bilinear h"
huffman@44133
  1534
  shows "\<exists>B. \<forall>x y. norm (h x y) \<le> B * norm x * norm y"
hoelzl@50526
  1535
proof (clarify intro!: exI[of _ "\<Sum>i\<in>Basis. \<Sum>j\<in>Basis. norm (h i j)"])
wenzelm@53406
  1536
  fix x :: 'm
wenzelm@53406
  1537
  fix y :: 'n
wenzelm@53406
  1538
  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))"
wenzelm@53406
  1539
    apply (subst euclidean_representation[where 'a='m])
wenzelm@53406
  1540
    apply (subst euclidean_representation[where 'a='n])
hoelzl@50526
  1541
    apply rule
hoelzl@50526
  1542
    done
wenzelm@53406
  1543
  also have "\<dots> = norm (setsum (\<lambda> (i,j). h ((x \<bullet> i) *\<^sub>R i) ((y \<bullet> j) *\<^sub>R j)) (Basis \<times> Basis))"
hoelzl@50526
  1544
    unfolding bilinear_setsum[OF bh finite_Basis finite_Basis] ..
hoelzl@50526
  1545
  finally have th: "norm (h x y) = \<dots>" .
hoelzl@50526
  1546
  show "norm (h x y) \<le> (\<Sum>i\<in>Basis. \<Sum>j\<in>Basis. norm (h i j)) * norm x * norm y"
haftmann@57418
  1547
    apply (auto simp add: setsum_left_distrib th setsum.cartesian_product)
wenzelm@53406
  1548
    apply (rule setsum_norm_le)
wenzelm@53406
  1549
    apply simp
wenzelm@53406
  1550
    apply (auto simp add: bilinear_rmul[OF bh] bilinear_lmul[OF bh]
wenzelm@53406
  1551
      field_simps simp del: scaleR_scaleR)
wenzelm@53406
  1552
    apply (rule mult_mono)
wenzelm@53406
  1553
    apply (auto simp add: zero_le_mult_iff Basis_le_norm)
wenzelm@53406
  1554
    apply (rule mult_mono)
wenzelm@53406
  1555
    apply (auto simp add: zero_le_mult_iff Basis_le_norm)
wenzelm@53406
  1556
    done
huffman@44133
  1557
qed
huffman@44133
  1558
huffman@44133
  1559
lemma bilinear_conv_bounded_bilinear:
huffman@44133
  1560
  fixes h :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space \<Rightarrow> 'c::real_normed_vector"
huffman@44133
  1561
  shows "bilinear h \<longleftrightarrow> bounded_bilinear h"
huffman@44133
  1562
proof
huffman@44133
  1563
  assume "bilinear h"
huffman@44133
  1564
  show "bounded_bilinear h"
huffman@44133
  1565
  proof
wenzelm@53406
  1566
    fix x y z
wenzelm@53406
  1567
    show "h (x + y) z = h x z + h y z"
wenzelm@60420
  1568
      using \<open>bilinear h\<close> unfolding bilinear_def linear_iff by simp
huffman@44133
  1569
  next
wenzelm@53406
  1570
    fix x y z
wenzelm@53406
  1571
    show "h x (y + z) = h x y + h x z"
wenzelm@60420
  1572
      using \<open>bilinear h\<close> unfolding bilinear_def linear_iff by simp
huffman@44133
  1573
  next
wenzelm@53406
  1574
    fix r x y
wenzelm@53406
  1575
    show "h (scaleR r x) y = scaleR r (h x y)"
wenzelm@60420
  1576
      using \<open>bilinear h\<close> unfolding bilinear_def linear_iff
huffman@44133
  1577
      by simp
huffman@44133
  1578
  next
wenzelm@53406
  1579
    fix r x y
wenzelm@53406
  1580
    show "h x (scaleR r y) = scaleR r (h x y)"
wenzelm@60420
  1581
      using \<open>bilinear h\<close> unfolding bilinear_def linear_iff
huffman@44133
  1582
      by simp
huffman@44133
  1583
  next
huffman@44133
  1584
    have "\<exists>B. \<forall>x y. norm (h x y) \<le> B * norm x * norm y"
wenzelm@60420
  1585
      using \<open>bilinear h\<close> by (rule bilinear_bounded)
wenzelm@49522
  1586
    then show "\<exists>K. \<forall>x y. norm (h x y) \<le> norm x * norm y * K"
haftmann@57514
  1587
      by (simp add: ac_simps)
huffman@44133
  1588
  qed
huffman@44133
  1589
next
huffman@44133
  1590
  assume "bounded_bilinear h"
huffman@44133
  1591
  then interpret h: bounded_bilinear h .
huffman@44133
  1592
  show "bilinear h"
huffman@44133
  1593
    unfolding bilinear_def linear_conv_bounded_linear
wenzelm@49522
  1594
    using h.bounded_linear_left h.bounded_linear_right by simp
huffman@44133
  1595
qed
huffman@44133
  1596
huffman@53939
  1597
lemma bilinear_bounded_pos:
wenzelm@56444
  1598
  fixes h :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space \<Rightarrow> 'c::real_normed_vector"
huffman@53939
  1599
  assumes bh: "bilinear h"
huffman@53939
  1600
  shows "\<exists>B > 0. \<forall>x y. norm (h x y) \<le> B * norm x * norm y"
huffman@53939
  1601
proof -
huffman@53939
  1602
  have "\<exists>B > 0. \<forall>x y. norm (h x y) \<le> norm x * norm y * B"
huffman@53939
  1603
    using bh [unfolded bilinear_conv_bounded_bilinear]
huffman@53939
  1604
    by (rule bounded_bilinear.pos_bounded)
huffman@53939
  1605
  then show ?thesis
haftmann@57514
  1606
    by (simp only: ac_simps)
huffman@53939
  1607
qed
huffman@53939
  1608
wenzelm@49522
  1609
wenzelm@60420
  1610
subsection \<open>We continue.\<close>
huffman@44133
  1611
huffman@44133
  1612
lemma independent_bound:
wenzelm@53716
  1613
  fixes S :: "'a::euclidean_space set"
wenzelm@53716
  1614
  shows "independent S \<Longrightarrow> finite S \<and> card S \<le> DIM('a)"
hoelzl@50526
  1615
  using independent_span_bound[OF finite_Basis, of S] by auto
huffman@44133
  1616
lp15@61609
  1617
corollary
paulson@60303
  1618
  fixes S :: "'a::euclidean_space set"
paulson@60303
  1619
  assumes "independent S"
paulson@60303
  1620
  shows independent_imp_finite: "finite S" and independent_card_le:"card S \<le> DIM('a)"
paulson@60303
  1621
using assms independent_bound by auto
lp15@61609
  1622
wenzelm@49663
  1623
lemma dependent_biggerset:
wenzelm@56444
  1624
  fixes S :: "'a::euclidean_space set"
wenzelm@56444
  1625
  shows "(finite S \<Longrightarrow> card S > DIM('a)) \<Longrightarrow> dependent S"
huffman@44133
  1626
  by (metis independent_bound not_less)
huffman@44133
  1627
wenzelm@60420
  1628
text \<open>Hence we can create a maximal independent subset.\<close>
huffman@44133
  1629
huffman@44133
  1630
lemma maximal_independent_subset_extend:
wenzelm@53406
  1631
  fixes S :: "'a::euclidean_space set"
wenzelm@53406
  1632
  assumes sv: "S \<subseteq> V"
wenzelm@49663
  1633
    and iS: "independent S"
huffman@44133
  1634
  shows "\<exists>B. S \<subseteq> B \<and> B \<subseteq> V \<and> independent B \<and> V \<subseteq> span B"
huffman@44133
  1635
  using sv iS
wenzelm@49522
  1636
proof (induct "DIM('a) - card S" arbitrary: S rule: less_induct)
huffman@44133
  1637
  case less
wenzelm@60420
  1638
  note sv = \<open>S \<subseteq> V\<close> and i = \<open>independent S\<close>
huffman@44133
  1639
  let ?P = "\<lambda>B. S \<subseteq> B \<and> B \<subseteq> V \<and> independent B \<and> V \<subseteq> span B"
huffman@44133
  1640
  let ?ths = "\<exists>x. ?P x"
huffman@44133
  1641
  let ?d = "DIM('a)"
wenzelm@53406
  1642
  show ?ths
wenzelm@53406
  1643
  proof (cases "V \<subseteq> span S")
wenzelm@53406
  1644
    case True
wenzelm@53406
  1645
    then show ?thesis
wenzelm@53406
  1646
      using sv i by blast
wenzelm@53406
  1647
  next
wenzelm@53406
  1648
    case False
wenzelm@53406
  1649
    then obtain a where a: "a \<in> V" "a \<notin> span S"
wenzelm@53406
  1650
      by blast
wenzelm@53406
  1651
    from a have aS: "a \<notin> S"
wenzelm@53406
  1652
      by (auto simp add: span_superset)
wenzelm@53406
  1653
    have th0: "insert a S \<subseteq> V"
wenzelm@53406
  1654
      using a sv by blast
huffman@44133
  1655
    from independent_insert[of a S]  i a
wenzelm@53406
  1656
    have th1: "independent (insert a S)"
wenzelm@53406
  1657
      by auto
huffman@44133
  1658
    have mlt: "?d - card (insert a S) < ?d - card S"
wenzelm@49522
  1659
      using aS a independent_bound[OF th1] by auto
huffman@44133
  1660
huffman@44133
  1661
    from less(1)[OF mlt th0 th1]
huffman@44133
  1662
    obtain B where B: "insert a S \<subseteq> B" "B \<subseteq> V" "independent B" " V \<subseteq> span B"
huffman@44133
  1663
      by blast
huffman@44133
  1664
    from B have "?P B" by auto
wenzelm@53406
  1665
    then show ?thesis by blast
wenzelm@53406
  1666
  qed
huffman@44133
  1667
qed
huffman@44133
  1668
huffman@44133
  1669
lemma maximal_independent_subset:
huffman@44133
  1670
  "\<exists>(B:: ('a::euclidean_space) set). B\<subseteq> V \<and> independent B \<and> V \<subseteq> span B"
wenzelm@49522
  1671
  by (metis maximal_independent_subset_extend[of "{}:: ('a::euclidean_space) set"]
wenzelm@49522
  1672
    empty_subsetI independent_empty)
huffman@44133
  1673
huffman@44133
  1674
wenzelm@60420
  1675
text \<open>Notion of dimension.\<close>
huffman@44133
  1676
wenzelm@53406
  1677
definition "dim V = (SOME n. \<exists>B. B \<subseteq> V \<and> independent B \<and> V \<subseteq> span B \<and> card B = n)"
huffman@44133
  1678
wenzelm@49522
  1679
lemma basis_exists:
wenzelm@49522
  1680
  "\<exists>B. (B :: ('a::euclidean_space) set) \<subseteq> V \<and> independent B \<and> V \<subseteq> span B \<and> (card B = dim V)"
wenzelm@49522
  1681
  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)"]
wenzelm@49522
  1682
  using maximal_independent_subset[of V] independent_bound
wenzelm@49522
  1683
  by auto
huffman@44133
  1684
lp15@60307
  1685
corollary dim_le_card:
lp15@60307
  1686
  fixes s :: "'a::euclidean_space set"
lp15@60307
  1687
  shows "finite s \<Longrightarrow> dim s \<le> card s"
lp15@60307
  1688
by (metis basis_exists card_mono)
lp15@60307
  1689
wenzelm@60420
  1690
text \<open>Consequences of independence or spanning for cardinality.\<close>
huffman@44133
  1691
wenzelm@53406
  1692
lemma independent_card_le_dim:
wenzelm@53406
  1693
  fixes B :: "'a::euclidean_space set"
wenzelm@53406
  1694
  assumes "B \<subseteq> V"
wenzelm@53406
  1695
    and "independent B"
wenzelm@49522
  1696
  shows "card B \<le> dim V"
huffman@44133
  1697
proof -
wenzelm@60420
  1698
  from basis_exists[of V] \<open>B \<subseteq> V\<close>
wenzelm@53406
  1699
  obtain B' where "independent B'"
wenzelm@53406
  1700
    and "B \<subseteq> span B'"
wenzelm@53406
  1701
    and "card B' = dim V"
wenzelm@53406
  1702
    by blast
wenzelm@60420
  1703
  with independent_span_bound[OF _ \<open>independent B\<close> \<open>B \<subseteq> span B'\<close>] independent_bound[of B']
huffman@44133
  1704
  show ?thesis by auto
huffman@44133
  1705
qed
huffman@44133
  1706
wenzelm@49522
  1707
lemma span_card_ge_dim:
wenzelm@53406
  1708
  fixes B :: "'a::euclidean_space set"
wenzelm@53406
  1709
  shows "B \<subseteq> V \<Longrightarrow> V \<subseteq> span B \<Longrightarrow> finite B \<Longrightarrow> dim V \<le> card B"
huffman@44133
  1710
  by (metis basis_exists[of V] independent_span_bound subset_trans)
huffman@44133
  1711
huffman@44133
  1712
lemma basis_card_eq_dim:
wenzelm@53406
  1713
  fixes V :: "'a::euclidean_space set"
wenzelm@53406
  1714
  shows "B \<subseteq> V \<Longrightarrow> V \<subseteq> span B \<Longrightarrow> independent B \<Longrightarrow> finite B \<and> card B = dim V"
huffman@44133
  1715
  by (metis order_eq_iff independent_card_le_dim span_card_ge_dim independent_bound)
huffman@44133
  1716
wenzelm@53406
  1717
lemma dim_unique:
wenzelm@53406
  1718
  fixes B :: "'a::euclidean_space set"
wenzelm@53406
  1719
  shows "B \<subseteq> V \<Longrightarrow> V \<subseteq> span B \<Longrightarrow> independent B \<Longrightarrow> card B = n \<Longrightarrow> dim V = n"
huffman@44133
  1720
  by (metis basis_card_eq_dim)
huffman@44133
  1721
wenzelm@60420
  1722
text \<open>More lemmas about dimension.\<close>
huffman@44133
  1723
wenzelm@53406
  1724
lemma dim_UNIV: "dim (UNIV :: 'a::euclidean_space set) = DIM('a)"
hoelzl@50526
  1725
  using independent_Basis
hoelzl@50526
  1726
  by (intro dim_unique[of Basis]) auto
huffman@44133
  1727
huffman@44133
  1728
lemma dim_subset:
wenzelm@53406
  1729
  fixes S :: "'a::euclidean_space set"
wenzelm@53406
  1730
  shows "S \<subseteq> T \<Longrightarrow> dim S \<le> dim T"
huffman@44133
  1731
  using basis_exists[of T] basis_exists[of S]
huffman@44133
  1732
  by (metis independent_card_le_dim subset_trans)
huffman@44133
  1733
wenzelm@53406
  1734
lemma dim_subset_UNIV:
wenzelm@53406
  1735
  fixes S :: "'a::euclidean_space set"
wenzelm@53406
  1736
  shows "dim S \<le> DIM('a)"
huffman@44133
  1737
  by (metis dim_subset subset_UNIV dim_UNIV)
huffman@44133
  1738
wenzelm@60420
  1739
text \<open>Converses to those.\<close>
huffman@44133
  1740
huffman@44133
  1741
lemma card_ge_dim_independent:
wenzelm@53406
  1742
  fixes B :: "'a::euclidean_space set"
wenzelm@53406
  1743
  assumes BV: "B \<subseteq> V"
wenzelm@53406
  1744
    and iB: "independent B"
wenzelm@53406
  1745
    and dVB: "dim V \<le> card B"
huffman@44133
  1746
  shows "V \<subseteq> span B"
wenzelm@53406
  1747
proof
wenzelm@53406
  1748
  fix a
wenzelm@53406
  1749
  assume aV: "a \<in> V"
wenzelm@53406
  1750
  {
wenzelm@53406
  1751
    assume aB: "a \<notin> span B"
wenzelm@53406
  1752
    then have iaB: "independent (insert a B)"
wenzelm@53406
  1753
      using iB aV BV by (simp add: independent_insert)
wenzelm@53406
  1754
    from aV BV have th0: "insert a B \<subseteq> V"
wenzelm@53406
  1755
      by blast
wenzelm@53406
  1756
    from aB have "a \<notin>B"
wenzelm@53406
  1757
      by (auto simp add: span_superset)
wenzelm@53406
  1758
    with independent_card_le_dim[OF th0 iaB] dVB independent_bound[OF iB]
wenzelm@53406
  1759
    have False by auto
wenzelm@53406
  1760
  }
wenzelm@53406
  1761
  then show "a \<in> span B" by blast
huffman@44133
  1762
qed
huffman@44133
  1763
huffman@44133
  1764
lemma card_le_dim_spanning:
wenzelm@49663
  1765
  assumes BV: "(B:: ('a::euclidean_space) set) \<subseteq> V"
wenzelm@49663
  1766
    and VB: "V \<subseteq> span B"
wenzelm@49663
  1767
    and fB: "finite B"
wenzelm@49663
  1768
    and dVB: "dim V \<ge> card B"
huffman@44133
  1769
  shows "independent B"
wenzelm@49522
  1770
proof -
wenzelm@53406
  1771
  {
wenzelm@53406
  1772
    fix a
wenzelm@53716
  1773
    assume a: "a \<in> B" "a \<in> span (B - {a})"
wenzelm@53406
  1774
    from a fB have c0: "card B \<noteq> 0"
wenzelm@53406
  1775
      by auto
wenzelm@53716
  1776
    from a fB have cb: "card (B - {a}) = card B - 1"
wenzelm@53406
  1777
      by auto
wenzelm@53716
  1778
    from BV a have th0: "B - {a} \<subseteq> V"
wenzelm@53406
  1779
      by blast
wenzelm@53406
  1780
    {
wenzelm@53406
  1781
      fix x
wenzelm@53406
  1782
      assume x: "x \<in> V"
wenzelm@53716
  1783
      from a have eq: "insert a (B - {a}) = B"
wenzelm@53406
  1784
        by blast
wenzelm@53406
  1785
      from x VB have x': "x \<in> span B"
wenzelm@53406
  1786
        by blast
huffman@44133
  1787
      from span_trans[OF a(2), unfolded eq, OF x']
wenzelm@53716
  1788
      have "x \<in> span (B - {a})" .
wenzelm@53406
  1789
    }
wenzelm@53716
  1790
    then have th1: "V \<subseteq> span (B - {a})"
wenzelm@53406
  1791
      by blast
wenzelm@53716
  1792
    have th2: "finite (B - {a})"
wenzelm@53406
  1793
      using fB by auto
huffman@44133
  1794
    from span_card_ge_dim[OF th0 th1 th2]
wenzelm@53716
  1795
    have c: "dim V \<le> card (B - {a})" .
wenzelm@53406
  1796
    from c c0 dVB cb have False by simp
wenzelm@53406
  1797
  }
wenzelm@53406
  1798
  then show ?thesis
wenzelm@53406
  1799
    unfolding dependent_def by blast
huffman@44133
  1800
qed
huffman@44133
  1801
wenzelm@53406
  1802
lemma card_eq_dim:
wenzelm@53406
  1803
  fixes B :: "'a::euclidean_space set"
wenzelm@53406
  1804
  shows "B \<subseteq> V \<Longrightarrow> card B = dim V \<Longrightarrow> finite B \<Longrightarrow> independent B \<longleftrightarrow> V \<subseteq> span B"
wenzelm@49522
  1805
  by (metis order_eq_iff card_le_dim_spanning card_ge_dim_independent)
huffman@44133
  1806
wenzelm@60420
  1807
text \<open>More general size bound lemmas.\<close>
huffman@44133
  1808
huffman@44133
  1809
lemma independent_bound_general:
wenzelm@53406
  1810
  fixes S :: "'a::euclidean_space set"
wenzelm@53406
  1811
  shows "independent S \<Longrightarrow> finite S \<and> card S \<le> dim S"
huffman@44133
  1812
  by (metis independent_card_le_dim independent_bound subset_refl)
huffman@44133
  1813
wenzelm@49522
  1814
lemma dependent_biggerset_general:
wenzelm@53406
  1815
  fixes S :: "'a::euclidean_space set"
wenzelm@53406
  1816
  shows "(finite S \<Longrightarrow> card S > dim S) \<Longrightarrow> dependent S"
huffman@44133
  1817
  using independent_bound_general[of S] by (metis linorder_not_le)
huffman@44133
  1818
paulson@60303
  1819
lemma dim_span [simp]:
wenzelm@53406
  1820
  fixes S :: "'a::euclidean_space set"
wenzelm@53406
  1821
  shows "dim (span S) = dim S"
wenzelm@49522
  1822
proof -
huffman@44133
  1823
  have th0: "dim S \<le> dim (span S)"
huffman@44133
  1824
    by (auto simp add: subset_eq intro: dim_subset span_superset)
huffman@44133
  1825
  from basis_exists[of S]
wenzelm@53406
  1826
  obtain B where B: "B \<subseteq> S" "independent B" "S \<subseteq> span B" "card B = dim S"
wenzelm@53406
  1827
    by blast
wenzelm@53406
  1828
  from B have fB: "finite B" "card B = dim S"
wenzelm@53406
  1829
    using independent_bound by blast+
wenzelm@53406
  1830
  have bSS: "B \<subseteq> span S"
wenzelm@53406
  1831
    using B(1) by (metis subset_eq span_inc)
wenzelm@53406
  1832
  have sssB: "span S \<subseteq> span B"
wenzelm@53406
  1833
    using span_mono[OF B(3)] by (simp add: span_span)
huffman@44133
  1834
  from span_card_ge_dim[OF bSS sssB fB(1)] th0 show ?thesis
wenzelm@49522
  1835
    using fB(2) by arith
huffman@44133
  1836
qed
huffman@44133
  1837
wenzelm@53406
  1838
lemma subset_le_dim:
wenzelm@53406
  1839
  fixes S :: "'a::euclidean_space set"
wenzelm@53406
  1840
  shows "S \<subseteq> span T \<Longrightarrow> dim S \<le> dim T"
huffman@44133
  1841
  by (metis dim_span dim_subset)
huffman@44133
  1842
wenzelm@53406
  1843
lemma span_eq_dim:
wenzelm@56444
  1844
  fixes S :: "'a::euclidean_space set"
wenzelm@53406
  1845
  shows "span S = span T \<Longrightarrow> dim S = dim T"
huffman@44133
  1846
  by (metis dim_span)
huffman@44133
  1847
huffman@44133
  1848
lemma spans_image:
wenzelm@49663
  1849
  assumes lf: "linear f"
wenzelm@49663
  1850
    and VB: "V \<subseteq> span B"
huffman@44133
  1851
  shows "f ` V \<subseteq> span (f ` B)"
wenzelm@49522
  1852
  unfolding span_linear_image[OF lf] by (metis VB image_mono)
huffman@44133
  1853
huffman@44133
  1854
lemma dim_image_le:
huffman@44133
  1855
  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
wenzelm@49663
  1856
  assumes lf: "linear f"
wenzelm@49663
  1857
  shows "dim (f ` S) \<le> dim (S)"
wenzelm@49522
  1858
proof -
huffman@44133
  1859
  from basis_exists[of S] obtain B where
huffman@44133
  1860
    B: "B \<subseteq> S" "independent B" "S \<subseteq> span B" "card B = dim S" by blast
wenzelm@53406
  1861
  from B have fB: "finite B" "card B = dim S"
wenzelm@53406
  1862
    using independent_bound by blast+
huffman@44133
  1863
  have "dim (f ` S) \<le> card (f ` B)"
huffman@44133
  1864
    apply (rule span_card_ge_dim)
wenzelm@53406
  1865
    using lf B fB
wenzelm@53406
  1866
    apply (auto simp add: span_linear_image spans_image subset_image_iff)
wenzelm@49522
  1867
    done
wenzelm@53406
  1868
  also have "\<dots> \<le> dim S"
wenzelm@53406
  1869
    using card_image_le[OF fB(1)] fB by simp
huffman@44133
  1870
  finally show ?thesis .
huffman@44133
  1871
qed
huffman@44133
  1872
wenzelm@60420
  1873
text \<open>Relation between bases and injectivity/surjectivity of map.\<close>
huffman@44133
  1874
huffman@44133
  1875
lemma spanning_surjective_image:
huffman@44133
  1876
  assumes us: "UNIV \<subseteq> span S"
wenzelm@53406
  1877
    and lf: "linear f"
wenzelm@53406
  1878
    and sf: "surj f"
huffman@44133
  1879
  shows "UNIV \<subseteq> span (f ` S)"
wenzelm@49663
  1880
proof -
wenzelm@53406
  1881
  have "UNIV \<subseteq> f ` UNIV"
wenzelm@53406
  1882
    using sf by (auto simp add: surj_def)
wenzelm@53406
  1883
  also have " \<dots> \<subseteq> span (f ` S)"
wenzelm@53406
  1884
    using spans_image[OF lf us] .
wenzelm@53406
  1885
  finally show ?thesis .
huffman@44133
  1886
qed
huffman@44133
  1887
huffman@44133
  1888
lemma independent_injective_image:
wenzelm@49663
  1889
  assumes iS: "independent S"
wenzelm@49663
  1890
    and lf: "linear f"
wenzelm@49663
  1891
    and fi: "inj f"
huffman@44133
  1892
  shows "independent (f ` S)"
wenzelm@49663
  1893
proof -
wenzelm@53406
  1894
  {
wenzelm@53406
  1895
    fix a
wenzelm@49663
  1896
    assume a: "a \<in> S" "f a \<in> span (f ` S - {f a})"
wenzelm@53406
  1897
    have eq: "f ` S - {f a} = f ` (S - {a})"
wenzelm@53406
  1898
      using fi by (auto simp add: inj_on_def)
wenzelm@53716
  1899
    from a have "f a \<in> f ` span (S - {a})"
wenzelm@53406
  1900
      unfolding eq span_linear_image[OF lf, of "S - {a}"] by blast
wenzelm@53716
  1901
    then have "a \<in> span (S - {a})"
wenzelm@53406
  1902
      using fi by (auto simp add: inj_on_def)
wenzelm@53406
  1903
    with a(1) iS have False
wenzelm@53406
  1904
      by (simp add: dependent_def)
wenzelm@53406
  1905
  }
wenzelm@53406
  1906
  then show ?thesis
wenzelm@53406
  1907
    unfolding dependent_def by blast
huffman@44133
  1908
qed
huffman@44133
  1909
wenzelm@60420
  1910
text \<open>Picking an orthogonal replacement for a spanning set.\<close>
huffman@44133
  1911
wenzelm@53406
  1912
lemma vector_sub_project_orthogonal:
wenzelm@53406
  1913
  fixes b x :: "'a::euclidean_space"
wenzelm@53406
  1914
  shows "b \<bullet> (x - ((b \<bullet> x) / (b \<bullet> b)) *\<^sub>R b) = 0"
huffman@44133
  1915
  unfolding inner_simps by auto
huffman@44133
  1916
huffman@44528
  1917
lemma pairwise_orthogonal_insert:
huffman@44528
  1918
  assumes "pairwise orthogonal S"
wenzelm@49522
  1919
    and "\<And>y. y \<in> S \<Longrightarrow> orthogonal x y"
huffman@44528
  1920
  shows "pairwise orthogonal (insert x S)"
huffman@44528
  1921
  using assms unfolding pairwise_def
huffman@44528
  1922
  by (auto simp add: orthogonal_commute)
huffman@44528
  1923
huffman@44133
  1924
lemma basis_orthogonal:
wenzelm@53406
  1925
  fixes B :: "'a::real_inner set"
huffman@44133
  1926
  assumes fB: "finite B"
huffman@44133
  1927
  shows "\<exists>C. finite C \<and> card C \<le> card B \<and> span C = span B \<and> pairwise orthogonal C"
huffman@44133
  1928
  (is " \<exists>C. ?P B C")
wenzelm@49522
  1929
  using fB
wenzelm@49522
  1930
proof (induct rule: finite_induct)
wenzelm@49522
  1931
  case empty
wenzelm@53406
  1932
  then show ?case
wenzelm@53406
  1933
    apply (rule exI[where x="{}"])
wenzelm@53406
  1934
    apply (auto simp add: pairwise_def)
wenzelm@53406
  1935
    done
huffman@44133
  1936
next
wenzelm@49522
  1937
  case (insert a B)
wenzelm@60420
  1938
  note fB = \<open>finite B\<close> and aB = \<open>a \<notin> B\<close>
wenzelm@60420
  1939
  from \<open>\<exists>C. finite C \<and> card C \<le> card B \<and> span C = span B \<and> pairwise orthogonal C\<close>
huffman@44133
  1940
  obtain C where C: "finite C" "card C \<le> card B"
huffman@44133
  1941
    "span C = span B" "pairwise orthogonal C" by blast
huffman@44133
  1942
  let ?a = "a - setsum (\<lambda>x. (x \<bullet> a / (x \<bullet> x)) *\<^sub>R x) C"
huffman@44133
  1943
  let ?C = "insert ?a C"
wenzelm@53406
  1944
  from C(1) have fC: "finite ?C"
wenzelm@53406
  1945
    by simp
wenzelm@49522
  1946
  from fB aB C(1,2) have cC: "card ?C \<le> card (insert a B)"
wenzelm@49522
  1947
    by (simp add: card_insert_if)
wenzelm@53406
  1948
  {
wenzelm@53406
  1949
    fix x k
wenzelm@49522
  1950
    have th0: "\<And>(a::'a) b c. a - (b - c) = c + (a - b)"
wenzelm@49522
  1951
      by (simp add: field_simps)
huffman@44133
  1952
    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"
huffman@44133
  1953
      apply (simp only: scaleR_right_diff_distrib th0)
huffman@44133
  1954
      apply (rule span_add_eq)
huffman@44133
  1955
      apply (rule span_mul)
huffman@56196
  1956
      apply (rule span_setsum)
huffman@44133
  1957
      apply clarify
huffman@44133
  1958
      apply (rule span_mul)
wenzelm@49522
  1959
      apply (rule span_superset)
wenzelm@49522
  1960
      apply assumption
wenzelm@53406
  1961
      done
wenzelm@53406
  1962
  }
huffman@44133
  1963
  then have SC: "span ?C = span (insert a B)"
huffman@44133
  1964
    unfolding set_eq_iff span_breakdown_eq C(3)[symmetric] by auto
wenzelm@53406
  1965
  {
wenzelm@53406
  1966
    fix y
wenzelm@53406
  1967
    assume yC: "y \<in> C"
wenzelm@53406
  1968
    then have Cy: "C = insert y (C - {y})"
wenzelm@53406
  1969
      by blast
wenzelm@53406
  1970
    have fth: "finite (C - {y})"
wenzelm@53406
  1971
      using C by simp
huffman@44528
  1972
    have "orthogonal ?a y"
huffman@44528
  1973
      unfolding orthogonal_def
haftmann@54230
  1974
      unfolding inner_diff inner_setsum_left right_minus_eq
wenzelm@60420
  1975
      unfolding setsum.remove [OF \<open>finite C\<close> \<open>y \<in> C\<close>]
huffman@44528
  1976
      apply (clarsimp simp add: inner_commute[of y a])
haftmann@57418
  1977
      apply (rule setsum.neutral)
huffman@44528
  1978
      apply clarsimp
huffman@44528
  1979
      apply (rule C(4)[unfolded pairwise_def orthogonal_def, rule_format])
wenzelm@60420
  1980
      using \<open>y \<in> C\<close> by auto
wenzelm@53406
  1981
  }
wenzelm@60420
  1982
  with \<open>pairwise orthogonal C\<close> have CPO: "pairwise orthogonal ?C"
huffman@44528
  1983
    by (rule pairwise_orthogonal_insert)
wenzelm@53406
  1984
  from fC cC SC CPO have "?P (insert a B) ?C"
wenzelm@53406
  1985
    by blast
huffman@44133
  1986
  then show ?case by blast
huffman@44133
  1987
qed
huffman@44133
  1988
huffman@44133
  1989
lemma orthogonal_basis_exists:
huffman@44133
  1990
  fixes V :: "('a::euclidean_space) set"
huffman@44133
  1991
  shows "\<exists>B. independent B \<and> B \<subseteq> span V \<and> V \<subseteq> span B \<and> (card B = dim V) \<and> pairwise orthogonal B"
wenzelm@49663
  1992
proof -
wenzelm@49522
  1993
  from basis_exists[of V] obtain B where
wenzelm@53406
  1994
    B: "B \<subseteq> V" "independent B" "V \<subseteq> span B" "card B = dim V"
wenzelm@53406
  1995
    by blast
wenzelm@53406
  1996
  from B have fB: "finite B" "card B = dim V"
wenzelm@53406
  1997
    using independent_bound by auto
huffman@44133
  1998
  from basis_orthogonal[OF fB(1)] obtain C where
wenzelm@53406
  1999
    C: "finite C" "card C \<le> card B" "span C = span B" "pairwise orthogonal C"
wenzelm@53406
  2000
    by blast
wenzelm@53406
  2001
  from C B have CSV: "C \<subseteq> span V"
wenzelm@53406
  2002
    by (metis span_inc span_mono subset_trans)
wenzelm@53406
  2003
  from span_mono[OF B(3)] C have SVC: "span V \<subseteq> span C"
wenzelm@53406
  2004
    by (simp add: span_span)
huffman@44133
  2005
  from card_le_dim_spanning[OF CSV SVC C(1)] C(2,3) fB
wenzelm@53406
  2006
  have iC: "independent C"
huffman@44133
  2007
    by (simp add: dim_span)
wenzelm@53406
  2008
  from C fB have "card C \<le> dim V"
wenzelm@53406
  2009
    by simp
wenzelm@53406
  2010
  moreover have "dim V \<le> card C"
wenzelm@53406
  2011
    using span_card_ge_dim[OF CSV SVC C(1)]
wenzelm@53406
  2012
    by (simp add: dim_span)
wenzelm@53406
  2013
  ultimately have CdV: "card C = dim V"
wenzelm@53406
  2014
    using C(1) by simp
wenzelm@53406
  2015
  from C B CSV CdV iC show ?thesis
wenzelm@53406
  2016
    by auto
huffman@44133
  2017
qed
huffman@44133
  2018
huffman@44133
  2019
lemma span_eq: "span S = span T \<longleftrightarrow> S \<subseteq> span T \<and> T \<subseteq> span S"
huffman@44133
  2020
  using span_inc[unfolded subset_eq] using span_mono[of T "span S"] span_mono[of S "span T"]
wenzelm@49522
  2021
  by (auto simp add: span_span)
huffman@44133
  2022
wenzelm@60420
  2023
text \<open>Low-dimensional subset is in a hyperplane (weak orthogonal complement).\<close>
huffman@44133
  2024
wenzelm@49522
  2025
lemma span_not_univ_orthogonal:
wenzelm@53406
  2026
  fixes S :: "'a::euclidean_space set"
huffman@44133
  2027
  assumes sU: "span S \<noteq> UNIV"
wenzelm@56444
  2028
  shows "\<exists>a::'a. a \<noteq> 0 \<and> (\<forall>x \<in> span S. a \<bullet> x = 0)"
wenzelm@49522
  2029
proof -
wenzelm@53406
  2030
  from sU obtain a where a: "a \<notin> span S"
wenzelm@53406
  2031
    by blast
huffman@44133
  2032
  from orthogonal_basis_exists obtain B where
huffman@44133
  2033
    B: "independent B" "B \<subseteq> span S" "S \<subseteq> span B" "card B = dim S" "pairwise orthogonal B"
huffman@44133
  2034
    by blast
wenzelm@53406
  2035
  from B have fB: "finite B" "card B = dim S"
wenzelm@53406
  2036
    using independent_bound by auto
huffman@44133
  2037
  from span_mono[OF B(2)] span_mono[OF B(3)]
wenzelm@53406
  2038
  have sSB: "span S = span B"
wenzelm@53406
  2039
    by (simp add: span_span)
huffman@44133
  2040
  let ?a = "a - setsum (\<lambda>b. (a \<bullet> b / (b \<bullet> b)) *\<^sub>R b) B"
huffman@44133
  2041
  have "setsum (\<lambda>b. (a \<bullet> b / (b \<bullet> b)) *\<^sub>R b) B \<in> span S"
huffman@44133
  2042
    unfolding sSB
huffman@56196
  2043
    apply (rule span_setsum)
huffman@44133
  2044
    apply clarsimp
huffman@44133
  2045
    apply (rule span_mul)
wenzelm@49522
  2046
    apply (rule span_superset)
wenzelm@49522
  2047
    apply assumption
wenzelm@49522
  2048
    done
wenzelm@53406
  2049
  with a have a0:"?a  \<noteq> 0"
wenzelm@53406
  2050
    by auto
huffman@44133
  2051
  have "\<forall>x\<in>span B. ?a \<bullet> x = 0"
wenzelm@49522
  2052
  proof (rule span_induct')
wenzelm@49522
  2053
    show "subspace {x. ?a \<bullet> x = 0}"
wenzelm@49522
  2054
      by (auto simp add: subspace_def inner_add)
wenzelm@49522
  2055
  next
wenzelm@53406
  2056
    {
wenzelm@53406
  2057
      fix x
wenzelm@53406
  2058
      assume x: "x \<in> B"
wenzelm@53406
  2059
      from x have B': "B = insert x (B - {x})"
wenzelm@53406
  2060
        by blast
wenzelm@53406
  2061
      have fth: "finite (B - {x})"
wenzelm@53406
  2062
        using fB by simp
huffman@44133
  2063
      have "?a \<bullet> x = 0"
wenzelm@53406
  2064
        apply (subst B')
wenzelm@53406
  2065
        using fB fth
huffman@44133
  2066
        unfolding setsum_clauses(2)[OF fth]
huffman@44133
  2067
        apply simp unfolding inner_simps
huffman@44527
  2068
        apply (clarsimp simp add: inner_add inner_setsum_left)
haftmann@57418
  2069
        apply (rule setsum.neutral, rule ballI)
huffman@44133
  2070
        unfolding inner_commute
wenzelm@49711
  2071
        apply (auto simp add: x field_simps
wenzelm@49711
  2072
          intro: B(5)[unfolded pairwise_def orthogonal_def, rule_format])
wenzelm@53406
  2073
        done
wenzelm@53406
  2074
    }
wenzelm@53406
  2075
    then show "\<forall>x \<in> B. ?a \<bullet> x = 0"
wenzelm@53406
  2076
      by blast
huffman@44133
  2077
  qed
wenzelm@53406
  2078
  with a0 show ?thesis
wenzelm@53406
  2079
    unfolding sSB by (auto intro: exI[where x="?a"])
huffman@44133
  2080
qed
huffman@44133
  2081
huffman@44133
  2082
lemma span_not_univ_subset_hyperplane:
wenzelm@53406
  2083
  fixes S :: "'a::euclidean_space set"
wenzelm@53406
  2084
  assumes SU: "span S \<noteq> UNIV"
huffman@44133
  2085
  shows "\<exists> a. a \<noteq>0 \<and> span S \<subseteq> {x. a \<bullet> x = 0}"
huffman@44133
  2086
  using span_not_univ_orthogonal[OF SU] by auto
huffman@44133
  2087
wenzelm@49663
  2088
lemma lowdim_subset_hyperplane:
wenzelm@53406
  2089
  fixes S :: "'a::euclidean_space set"
huffman@44133
  2090
  assumes d: "dim S < DIM('a)"
wenzelm@56444
  2091
  shows "\<exists>a::'a. a \<noteq> 0 \<and> span S \<subseteq> {x. a \<bullet> x = 0}"
wenzelm@49522
  2092
proof -
wenzelm@53406
  2093
  {
wenzelm@53406
  2094
    assume "span S = UNIV"
wenzelm@53406
  2095
    then have "dim (span S) = dim (UNIV :: ('a) set)"
wenzelm@53406
  2096
      by simp
wenzelm@53406
  2097
    then have "dim S = DIM('a)"
wenzelm@53406
  2098
      by (simp add: dim_span dim_UNIV)
wenzelm@53406
  2099
    with d have False by arith
wenzelm@53406
  2100
  }
wenzelm@53406
  2101
  then have th: "span S \<noteq> UNIV"
wenzelm@53406
  2102
    by blast
huffman@44133
  2103
  from span_not_univ_subset_hyperplane[OF th] show ?thesis .
huffman@44133
  2104
qed
huffman@44133
  2105
wenzelm@60420
  2106
text \<open>We can extend a linear basis-basis injection to the whole set.\<close>
huffman@44133
  2107
huffman@44133
  2108
lemma linear_indep_image_lemma:
wenzelm@49663
  2109
  assumes lf: "linear f"
wenzelm@49663
  2110
    and fB: "finite B"
wenzelm@49522
  2111
    and ifB: "independent (f ` B)"
wenzelm@49663
  2112
    and fi: "inj_on f B"
wenzelm@49663
  2113
    and xsB: "x \<in> span B"
wenzelm@49522
  2114
    and fx: "f x = 0"
huffman@44133
  2115
  shows "x = 0"
huffman@44133
  2116
  using fB ifB fi xsB fx
wenzelm@49522
  2117
proof (induct arbitrary: x rule: finite_induct[OF fB])
wenzelm@49663
  2118
  case 1
wenzelm@49663
  2119
  then show ?case by auto
huffman@44133
  2120
next
huffman@44133
  2121
  case (2 a b x)
huffman@44133
  2122
  have fb: "finite b" using "2.prems" by simp
huffman@44133
  2123
  have th0: "f ` b \<subseteq> f ` (insert a b)"
wenzelm@53406
  2124
    apply (rule image_mono)
wenzelm@53406
  2125
    apply blast
wenzelm@53406
  2126
    done
huffman@44133
  2127
  from independent_mono[ OF "2.prems"(2) th0]
huffman@44133
  2128
  have ifb: "independent (f ` b)"  .
huffman@44133
  2129
  have fib: "inj_on f b"
huffman@44133
  2130
    apply (rule subset_inj_on [OF "2.prems"(3)])
wenzelm@49522
  2131
    apply blast
wenzelm@49522
  2132
    done
huffman@44133
  2133
  from span_breakdown[of a "insert a b", simplified, OF "2.prems"(4)]
wenzelm@53406
  2134
  obtain k where k: "x - k*\<^sub>R a \<in> span (b - {a})"
wenzelm@53406
  2135
    by blast
huffman@44133
  2136
  have "f (x - k*\<^sub>R a) \<in> span (f ` b)"
huffman@44133
  2137
    unfolding span_linear_image[OF lf]
huffman@44133
  2138
    apply (rule imageI)
wenzelm@53716
  2139
    using k span_mono[of "b - {a}" b]
wenzelm@53406
  2140
    apply blast
wenzelm@49522
  2141
    done
wenzelm@49522
  2142
  then have "f x - k*\<^sub>R f a \<in> span (f ` b)"
huffman@44133
  2143
    by (simp add: linear_sub[OF lf] linear_cmul[OF lf])
wenzelm@49522
  2144
  then have th: "-k *\<^sub>R f a \<in> span (f ` b)"
huffman@44133
  2145
    using "2.prems"(5) by simp
wenzelm@53406
  2146
  have xsb: "x \<in> span b"
wenzelm@53406
  2147
  proof (cases "k = 0")
wenzelm@53406
  2148
    case True
wenzelm@53716
  2149
    with k have "x \<in> span (b - {a})" by simp
wenzelm@53716
  2150
    then show ?thesis using span_mono[of "b - {a}" b]
wenzelm@53406
  2151
      by blast
wenzelm@53406
  2152
  next
wenzelm@53406
  2153
    case False
wenzelm@53406
  2154
    with span_mul[OF th, of "- 1/ k"]
huffman@44133
  2155
    have th1: "f a \<in> span (f ` b)"
hoelzl@56479
  2156
      by auto
huffman@44133
  2157
    from inj_on_image_set_diff[OF "2.prems"(3), of "insert a b " "{a}", symmetric]
huffman@44133
  2158
    have tha: "f ` insert a b - f ` {a} = f ` (insert a b - {a})" by blast
huffman@44133
  2159
    from "2.prems"(2) [unfolded dependent_def bex_simps(8), rule_format, of "f a"]
huffman@44133
  2160
    have "f a \<notin> span (f ` b)" using tha
huffman@44133
  2161
      using "2.hyps"(2)
huffman@44133
  2162
      "2.prems"(3) by auto
huffman@44133
  2163
    with th1 have False by blast
wenzelm@53406
  2164
    then show ?thesis by blast
wenzelm@53406
  2165
  qed
wenzelm@53406
  2166
  from "2.hyps"(3)[OF fb ifb fib xsb "2.prems"(5)] show "x = 0" .
huffman@44133
  2167
qed
huffman@44133
  2168
wenzelm@60420
  2169
text \<open>We can extend a linear mapping from basis.\<close>
huffman@44133
  2170
huffman@44133
  2171
lemma linear_independent_extend_lemma:
huffman@44133
  2172
  fixes f :: "'a::real_vector \<Rightarrow> 'b::real_vector"
wenzelm@53406
  2173
  assumes fi: "finite B"
wenzelm@53406
  2174
    and ib: "independent B"
wenzelm@53406
  2175
  shows "\<exists>g.
wenzelm@53406
  2176
    (\<forall>x\<in> span B. \<forall>y\<in> span B. g (x + y) = g x + g y) \<and>
wenzelm@53406
  2177
    (\<forall>x\<in> span B. \<forall>c. g (c*\<^sub>R x) = c *\<^sub>R g x) \<and>
wenzelm@53406
  2178
    (\<forall>x\<in> B. g x = f x)"
wenzelm@49663
  2179
  using ib fi
wenzelm@49522
  2180
proof (induct rule: finite_induct[OF fi])
wenzelm@49663
  2181
  case 1
wenzelm@49663
  2182
  then show ?case by auto
huffman@44133
  2183
next
huffman@44133
  2184
  case (2 a b)
huffman@44133
  2185
  from "2.prems" "2.hyps" have ibf: "independent b" "finite b"
huffman@44133
  2186
    by (simp_all add: independent_insert)
huffman@44133
  2187
  from "2.hyps"(3)[OF ibf] obtain g where
huffman@44133
  2188
    g: "\<forall>x\<in>span b. \<forall>y\<in>span b. g (x + y) = g x + g y"
huffman@44133
  2189
    "\<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
huffman@44133
  2190
  let ?h = "\<lambda>z. SOME k. (z - k *\<^sub>R a) \<in> span b"
wenzelm@53406
  2191
  {
wenzelm@53406
  2192
    fix z
wenzelm@53406
  2193
    assume z: "z \<in> span (insert a b)"
huffman@44133
  2194
    have th0: "z - ?h z *\<^sub>R a \<in> span b"
huffman@44133
  2195
      apply (rule someI_ex)
huffman@44133
  2196
      unfolding span_breakdown_eq[symmetric]
wenzelm@53406
  2197
      apply (rule z)
wenzelm@53406
  2198
      done
wenzelm@53406
  2199
    {
wenzelm@53406
  2200
      fix k
wenzelm@53406
  2201
      assume k: "z - k *\<^sub>R a \<in> span b"
huffman@44133
  2202
      have eq: "z - ?h z *\<^sub>R a - (z - k*\<^sub>R a) = (k - ?h z) *\<^sub>R a"
huffman@44133
  2203
        by (simp add: field_simps scaleR_left_distrib [symmetric])
wenzelm@53406
  2204
      from span_sub[OF th0 k] have khz: "(k - ?h z) *\<^sub>R a \<in> span b"
wenzelm@53406
  2205
        by (simp add: eq)
wenzelm@53406
  2206
      {
wenzelm@53406
  2207
        assume "k \<noteq> ?h z"
wenzelm@53406
  2208
        then have k0: "k - ?h z \<noteq> 0" by simp
huffman@44133
  2209
        from k0 span_mul[OF khz, of "1 /(k - ?h z)"]
huffman@44133
  2210
        have "a \<in> span b" by simp
huffman@44133
  2211
        with "2.prems"(1) "2.hyps"(2) have False
wenzelm@53406
  2212
          by (auto simp add: dependent_def)
wenzelm@53406
  2213
      }
wenzelm@53406
  2214
      then have "k = ?h z" by blast
wenzelm@53406
  2215
    }
wenzelm@53406
  2216
    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)"
wenzelm@53406
  2217
      by blast
wenzelm@53406
  2218
  }
huffman@44133
  2219
  note h = this
huffman@44133
  2220
  let ?g = "\<lambda>z. ?h z *\<^sub>R f a + g (z - ?h z *\<^sub>R a)"
wenzelm@53406
  2221
  {
wenzelm@53406
  2222
    fix x y
wenzelm@53406
  2223
    assume x: "x \<in> span (insert a b)"
wenzelm@53406
  2224
      and y: "y \<in> span (insert a b)"
huffman@44133
  2225
    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)"
huffman@44133
  2226
      by (simp add: algebra_simps)
huffman@44133
  2227
    have addh: "?h (x + y) = ?h x + ?h y"
huffman@44133
  2228
      apply (rule conjunct2[OF h, rule_format, symmetric])
huffman@44133
  2229
      apply (rule span_add[OF x y])
huffman@44133
  2230
      unfolding tha
wenzelm@53406
  2231
      apply (metis span_add x y conjunct1[OF h, rule_format])
wenzelm@53406
  2232
      done
huffman@44133
  2233
    have "?g (x + y) = ?g x + ?g y"
huffman@44133
  2234
      unfolding addh tha
huffman@44133
  2235
      g(1)[rule_format,OF conjunct1[OF h, OF x] conjunct1[OF h, OF y]]
huffman@44133
  2236
      by (simp add: scaleR_left_distrib)}
huffman@44133
  2237
  moreover
wenzelm@53406
  2238
  {
wenzelm@53406
  2239
    fix x :: "'a"
wenzelm@53406
  2240
    fix c :: real
wenzelm@49522
  2241
    assume x: "x \<in> span (insert a b)"
huffman@44133
  2242
    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)"
huffman@44133
  2243
      by (simp add: algebra_simps)
huffman@44133
  2244
    have hc: "?h (c *\<^sub>R x) = c * ?h x"
huffman@44133
  2245
      apply (rule conjunct2[OF h, rule_format, symmetric])
huffman@44133
  2246
      apply (metis span_mul x)
wenzelm@49522
  2247
      apply (metis tha span_mul x conjunct1[OF h])
wenzelm@49522
  2248
      done
huffman@44133
  2249
    have "?g (c *\<^sub>R x) = c*\<^sub>R ?g x"
huffman@44133
  2250
      unfolding hc tha g(2)[rule_format, OF conjunct1[OF h, OF x]]
wenzelm@53406
  2251
      by (simp add: algebra_simps)
wenzelm@53406
  2252
  }
huffman@44133
  2253
  moreover
wenzelm@53406
  2254
  {
wenzelm@53406
  2255
    fix x
wenzelm@53406
  2256
    assume x: "x \<in> insert a b"
wenzelm@53406
  2257
    {
wenzelm@53406
  2258
      assume xa: "x = a"
huffman@44133
  2259
      have ha1: "1 = ?h a"
huffman@44133
  2260
        apply (rule conjunct2[OF h, rule_format])
huffman@44133
  2261
        apply (metis span_superset insertI1)
huffman@44133
  2262
        using conjunct1[OF h, OF span_superset, OF insertI1]
wenzelm@49522
  2263
        apply (auto simp add: span_0)
wenzelm@49522
  2264
        done
huffman@44133
  2265
      from xa ha1[symmetric] have "?g x = f x"
huffman@44133
  2266
        apply simp
huffman@44133
  2267
        using g(2)[rule_format, OF span_0, of 0]
wenzelm@49522
  2268
        apply simp
wenzelm@53406
  2269
        done
wenzelm@53406
  2270
    }
huffman@44133
  2271
    moreover
wenzelm@53406
  2272
    {
wenzelm@53406
  2273
      assume xb: "x \<in> b"
huffman@44133
  2274
      have h0: "0 = ?h x"
huffman@44133
  2275
        apply (rule conjunct2[OF h, rule_format])
huffman@44133
  2276
        apply (metis  span_superset x)
huffman@44133
  2277
        apply simp
huffman@44133
  2278
        apply (metis span_superset xb)
huffman@44133
  2279
        done
huffman@44133
  2280
      have "?g x = f x"
wenzelm@53406
  2281
        by (simp add: h0[symmetric] g(3)[rule_format, OF xb])
wenzelm@53406
  2282
    }
wenzelm@53406
  2283
    ultimately have "?g x = f x"
wenzelm@53406
  2284
      using x by blast
wenzelm@53406
  2285
  }
wenzelm@49663
  2286
  ultimately show ?case
wenzelm@49663
  2287
    apply -
wenzelm@49663
  2288
    apply (rule exI[where x="?g"])
wenzelm@49663
  2289
    apply blast
wenzelm@49663
  2290
    done
huffman@44133
  2291
qed
huffman@44133
  2292
huffman@44133
  2293
lemma linear_independent_extend:
wenzelm@53406
  2294
  fixes B :: "'a::euclidean_space set"
wenzelm@53406
  2295
  assumes iB: "independent B"
huffman@44133
  2296
  shows "\<exists>g. linear g \<and> (\<forall>x\<in>B. g x = f x)"
wenzelm@49522
  2297
proof -
huffman@44133
  2298
  from maximal_independent_subset_extend[of B UNIV] iB
wenzelm@53406
  2299
  obtain C where C: "B \<subseteq> C" "independent C" "\<And>x. x \<in> span C"
wenzelm@53406
  2300
    by auto
huffman@44133
  2301
huffman@44133
  2302
  from C(2) independent_bound[of C] linear_independent_extend_lemma[of C f]
wenzelm@53406
  2303
  obtain g where g:
wenzelm@53406
  2304
    "(\<forall>x\<in> span C. \<forall>y\<in> span C. g (x + y) = g x + g y) \<and>
wenzelm@53406
  2305
     (\<forall>x\<in> span C. \<forall>c. g (c*\<^sub>R x) = c *\<^sub>R g x) \<and>
wenzelm@53406
  2306
     (\<forall>x\<in> C. g x = f x)" by blast
wenzelm@53406
  2307
  from g show ?thesis
huffman@53600
  2308
    unfolding linear_iff
wenzelm@53406
  2309
    using C
wenzelm@49663
  2310
    apply clarsimp
wenzelm@49663
  2311
    apply blast
wenzelm@49663
  2312
    done
huffman@44133
  2313
qed
huffman@44133
  2314
wenzelm@60420
  2315
text \<open>Can construct an isomorphism between spaces of same dimension.\<close>
huffman@44133
  2316
huffman@44133
  2317
lemma subspace_isomorphism:
wenzelm@53406
  2318
  fixes S :: "'a::euclidean_space set"
wenzelm@53406
  2319
    and T :: "'b::euclidean_space set"
wenzelm@53406
  2320
  assumes s: "subspace S"
wenzelm@53406
  2321
    and t: "subspace T"
wenzelm@49522
  2322
    and d: "dim S = dim T"
huffman@44133
  2323
  shows "\<exists>f. linear f \<and> f ` S = T \<and> inj_on f S"
wenzelm@49522
  2324
proof -
wenzelm@53406
  2325
  from basis_exists[of S] independent_bound
wenzelm@53406
  2326
  obtain B where B: "B \<subseteq> S" "independent B" "S \<subseteq> span B" "card B = dim S" and fB: "finite B"
wenzelm@53406
  2327
    by blast
wenzelm@53406
  2328
  from basis_exists[of T] independent_bound
wenzelm@53406
  2329
  obtain C where C: "C \<subseteq> T" "independent C" "T \<subseteq> span C" "card C = dim T" and fC: "finite C"
wenzelm@53406
  2330
    by blast
wenzelm@53406
  2331
  from B(4) C(4) card_le_inj[of B C] d
wenzelm@60420
  2332
  obtain f where f: "f ` B \<subseteq> C" "inj_on f B" using \<open>finite B\<close> \<open>finite C\<close>
wenzelm@53406
  2333
    by auto
wenzelm@53406
  2334
  from linear_independent_extend[OF B(2)]
wenzelm@53406
  2335
  obtain g where g: "linear g" "\<forall>x\<in> B. g x = f x"
wenzelm@53406
  2336
    by blast
wenzelm@53406
  2337
  from inj_on_iff_eq_card[OF fB, of f] f(2) have "card (f ` B) = card B"
huffman@44133
  2338
    by simp
wenzelm@53406
  2339
  with B(4) C(4) have ceq: "card (f ` B) = card C"
wenzelm@53406
  2340
    using d by simp
wenzelm@53406
  2341
  have "g ` B = f ` B"
wenzelm@53406
  2342
    using g(2) by (auto simp add: image_iff)
huffman@44133
  2343
  also have "\<dots> = C" using card_subset_eq[OF fC f(1) ceq] .
huffman@44133
  2344
  finally have gBC: "g ` B = C" .
wenzelm@53406
  2345
  have gi: "inj_on g B"
wenzelm@53406
  2346
    using f(2) g(2) by (auto simp add: inj_on_def)
huffman@44133
  2347
  note g0 = linear_indep_image_lemma[OF g(1) fB, unfolded gBC, OF C(2) gi]
wenzelm@53406
  2348
  {
wenzelm@53406
  2349
    fix x y
wenzelm@53406
  2350
    assume x: "x \<in> S" and y: "y \<in> S" and gxy: "g x = g y"
wenzelm@53406
  2351
    from B(3) x y have x': "x \<in> span B" and y': "y \<in> span B"
wenzelm@53406
  2352
      by blast+
wenzelm@53406
  2353
    from gxy have th0: "g (x - y) = 0"
wenzelm@53406
  2354
      by (simp add: linear_sub[OF g(1)])
wenzelm@53406
  2355
    have th1: "x - y \<in> span B"
wenzelm@53406
  2356
      using x' y' by (metis span_sub)
wenzelm@53406
  2357
    have "x = y"
wenzelm@53406
  2358
      using g0[OF th1 th0] by simp
wenzelm@53406
  2359
  }
huffman@44133
  2360
  then have giS: "inj_on g S"
huffman@44133
  2361
    unfolding inj_on_def by blast
wenzelm@53406
  2362
  from span_subspace[OF B(1,3) s] have "g ` S = span (g ` B)"
wenzelm@53406
  2363
    by (simp add: span_linear_image[OF g(1)])
huffman@44133
  2364
  also have "\<dots> = span C" unfolding gBC ..
huffman@44133
  2365
  also have "\<dots> = T" using span_subspace[OF C(1,3) t] .
huffman@44133
  2366
  finally have gS: "g ` S = T" .
wenzelm@53406
  2367
  from g(1) gS giS show ?thesis
wenzelm@53406
  2368
    by blast
huffman@44133
  2369
qed
huffman@44133
  2370
wenzelm@60420
  2371
text \<open>Linear functions are equal on a subspace if they are on a spanning set.\<close>
huffman@44133
  2372
huffman@44133
  2373
lemma subspace_kernel:
huffman@44133
  2374
  assumes lf: "linear f"
huffman@44133
  2375
  shows "subspace {x. f x = 0}"
wenzelm@49522
  2376
  apply (simp add: subspace_def)
wenzelm@49522
  2377
  apply (simp add: linear_add[OF lf] linear_cmul[OF lf] linear_0[OF lf])
wenzelm@49522
  2378
  done
huffman@44133
  2379
huffman@44133
  2380
lemma linear_eq_0_span:
huffman@44133
  2381
  assumes lf: "linear f" and f0: "\<forall>x\<in>B. f x = 0"
huffman@44133
  2382
  shows "\<forall>x \<in> span B. f x = 0"
huffman@44170
  2383
  using f0 subspace_kernel[OF lf]
huffman@44170
  2384
  by (rule span_induct')
huffman@44133
  2385
huffman@44133
  2386
lemma linear_eq_0:
wenzelm@49663
  2387
  assumes lf: "linear f"
wenzelm@49663
  2388
    and SB: "S \<subseteq> span B"
wenzelm@49663
  2389
    and f0: "\<forall>x\<in>B. f x = 0"
huffman@44133
  2390
  shows "\<forall>x \<in> S. f x = 0"
huffman@44133
  2391
  by (metis linear_eq_0_span[OF lf] subset_eq SB f0)
huffman@44133
  2392
huffman@44133
  2393
lemma linear_eq:
wenzelm@49663
  2394
  assumes lf: "linear f"
wenzelm@49663
  2395
    and lg: "linear g"
wenzelm@49663
  2396
    and S: "S \<subseteq> span B"
wenzelm@49522
  2397
    and fg: "\<forall> x\<in> B. f x = g x"
huffman@44133
  2398
  shows "\<forall>x\<in> S. f x = g x"
wenzelm@49663
  2399
proof -
huffman@44133
  2400
  let ?h = "\<lambda>x. f x - g x"
huffman@44133
  2401
  from fg have fg': "\<forall>x\<in> B. ?h x = 0" by simp
huffman@44133
  2402
  from linear_eq_0[OF linear_compose_sub[OF lf lg] S fg']
huffman@44133
  2403
  show ?thesis by simp
huffman@44133
  2404
qed
huffman@44133
  2405
huffman@44133
  2406
lemma linear_eq_stdbasis:
wenzelm@56444
  2407
  fixes f :: "'a::euclidean_space \<Rightarrow> _"
wenzelm@56444
  2408
  assumes lf: "linear f"
wenzelm@49663
  2409
    and lg: "linear g"
hoelzl@50526
  2410
    and fg: "\<forall>b\<in>Basis. f b = g b"
huffman@44133
  2411
  shows "f = g"
hoelzl@50526
  2412
  using linear_eq[OF lf lg, of _ Basis] fg by auto
huffman@44133
  2413
wenzelm@60420
  2414
text \<open>Similar results for bilinear functions.\<close>
huffman@44133
  2415
huffman@44133
  2416
lemma bilinear_eq:
huffman@44133
  2417
  assumes bf: "bilinear f"
wenzelm@49522
  2418
    and bg: "bilinear g"
wenzelm@53406
  2419
    and SB: "S \<subseteq> span B"
wenzelm@53406
  2420
    and TC: "T \<subseteq> span C"
wenzelm@49522
  2421
    and fg: "\<forall>x\<in> B. \<forall>y\<in> C. f x y = g x y"
huffman@44133
  2422
  shows "\<forall>x\<in>S. \<forall>y\<in>T. f x y = g x y "
wenzelm@49663
  2423
proof -
huffman@44170
  2424
  let ?P = "{x. \<forall>y\<in> span C. f x y = g x y}"
huffman@44133
  2425
  from bf bg have sp: "subspace ?P"
huffman@53600
  2426
    unfolding bilinear_def linear_iff subspace_def bf bg
wenzelm@49663
  2427
    by (auto simp add: span_0 bilinear_lzero[OF bf] bilinear_lzero[OF bg] span_add Ball_def
wenzelm@49663
  2428
      intro: bilinear_ladd[OF bf])
huffman@44133
  2429
huffman@44133
  2430
  have "\<forall>x \<in> span B. \<forall>y\<in> span C. f x y = g x y"
huffman@44170
  2431
    apply (rule span_induct' [OF _ sp])
huffman@44133
  2432
    apply (rule ballI)
huffman@44170
  2433
    apply (rule span_induct')
huffman@44170
  2434
    apply (simp add: fg)
huffman@44133
  2435
    apply (auto simp add: subspace_def)
huffman@53600
  2436
    using bf bg unfolding bilinear_def linear_iff
wenzelm@49522
  2437
    apply (auto simp add: span_0 bilinear_rzero[OF bf] bilinear_rzero[OF bg] span_add Ball_def
wenzelm@49663
  2438
      intro: bilinear_ladd[OF bf])
wenzelm@49522
  2439
    done
wenzelm@53406
  2440
  then show ?thesis
wenzelm@53406
  2441
    using SB TC by auto
huffman@44133
  2442
qed
huffman@44133
  2443
wenzelm@49522
  2444
lemma bilinear_eq_stdbasis:
wenzelm@53406
  2445
  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space \<Rightarrow> _"
huffman@44133
  2446
  assumes bf: "bilinear f"
wenzelm@49522
  2447
    and bg: "bilinear g"
hoelzl@50526
  2448
    and fg: "\<forall>i\<in>Basis. \<forall>j\<in>Basis. f i j = g i j"
huffman@44133
  2449
  shows "f = g"
hoelzl@50526
  2450
  using bilinear_eq[OF bf bg equalityD2[OF span_Basis] equalityD2[OF span_Basis] fg] by blast
huffman@44133
  2451
wenzelm@60420
  2452
text \<open>Detailed theorems about left and right invertibility in general case.\<close>
huffman@44133
  2453
wenzelm@49522
  2454
lemma linear_injective_left_inverse:
wenzelm@56444
  2455
  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"