src/HOL/Multivariate_Analysis/Linear_Algebra.thy
author paulson
Mon Jul 20 23:12:50 2015 +0100 (2015-07-20)
changeset 60762 bf0c76ccee8d
parent 60420 884f54e01427
child 60800 7d04351c795a
permissions -rw-r--r--
new material for multivariate analysis, etc.
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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_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 (simp add: linear_iff)
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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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subsection \<open>Bilinear functions.\<close>
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definition "bilinear f \<longleftrightarrow> (\<forall>x. linear (\<lambda>y. f x y)) \<and> (\<forall>y. linear (\<lambda>x. f x y))"
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lemma bilinear_ladd: "bilinear h \<Longrightarrow> h (x + y) z = h x z + h y z"
huffman@53600
   309
  by (simp add: bilinear_def linear_iff)
wenzelm@49663
   310
wenzelm@53406
   311
lemma bilinear_radd: "bilinear h \<Longrightarrow> h x (y + z) = h x y + h x z"
huffman@53600
   312
  by (simp add: bilinear_def linear_iff)
huffman@44133
   313
wenzelm@53406
   314
lemma bilinear_lmul: "bilinear h \<Longrightarrow> h (c *\<^sub>R x) y = c *\<^sub>R h x y"
huffman@53600
   315
  by (simp add: bilinear_def linear_iff)
huffman@44133
   316
wenzelm@53406
   317
lemma bilinear_rmul: "bilinear h \<Longrightarrow> h x (c *\<^sub>R y) = c *\<^sub>R h x y"
huffman@53600
   318
  by (simp add: bilinear_def linear_iff)
huffman@44133
   319
wenzelm@53406
   320
lemma bilinear_lneg: "bilinear h \<Longrightarrow> h (- x) y = - h x y"
haftmann@54489
   321
  by (drule bilinear_lmul [of _ "- 1"]) simp
huffman@44133
   322
wenzelm@53406
   323
lemma bilinear_rneg: "bilinear h \<Longrightarrow> h x (- y) = - h x y"
haftmann@54489
   324
  by (drule bilinear_rmul [of _ _ "- 1"]) simp
huffman@44133
   325
wenzelm@53406
   326
lemma (in ab_group_add) eq_add_iff: "x = x + y \<longleftrightarrow> y = 0"
haftmann@59557
   327
  using add_left_imp_eq[of x y 0] by auto
huffman@44133
   328
wenzelm@53406
   329
lemma bilinear_lzero:
wenzelm@53406
   330
  assumes "bilinear h"
wenzelm@53406
   331
  shows "h 0 x = 0"
wenzelm@49663
   332
  using bilinear_ladd [OF assms, of 0 0 x] by (simp add: eq_add_iff field_simps)
wenzelm@49663
   333
wenzelm@53406
   334
lemma bilinear_rzero:
wenzelm@53406
   335
  assumes "bilinear h"
wenzelm@53406
   336
  shows "h x 0 = 0"
wenzelm@49663
   337
  using bilinear_radd [OF assms, of x 0 0 ] by (simp add: eq_add_iff field_simps)
huffman@44133
   338
wenzelm@53406
   339
lemma bilinear_lsub: "bilinear h \<Longrightarrow> h (x - y) z = h x z - h y z"
haftmann@54230
   340
  using bilinear_ladd [of h x "- y"] by (simp add: bilinear_lneg)
huffman@44133
   341
wenzelm@53406
   342
lemma bilinear_rsub: "bilinear h \<Longrightarrow> h z (x - y) = h z x - h z y"
haftmann@54230
   343
  using bilinear_radd [of h _ x "- y"] by (simp add: bilinear_rneg)
huffman@44133
   344
huffman@44133
   345
lemma bilinear_setsum:
wenzelm@49663
   346
  assumes bh: "bilinear h"
wenzelm@49663
   347
    and fS: "finite S"
wenzelm@49663
   348
    and fT: "finite T"
huffman@44133
   349
  shows "h (setsum f S) (setsum g T) = setsum (\<lambda>(i,j). h (f i) (g j)) (S \<times> T) "
wenzelm@49522
   350
proof -
huffman@44133
   351
  have "h (setsum f S) (setsum g T) = setsum (\<lambda>x. h (f x) (setsum g T)) S"
huffman@44133
   352
    apply (rule linear_setsum[unfolded o_def])
wenzelm@53406
   353
    using bh fS
wenzelm@53406
   354
    apply (auto simp add: bilinear_def)
wenzelm@49522
   355
    done
huffman@44133
   356
  also have "\<dots> = setsum (\<lambda>x. setsum (\<lambda>y. h (f x) (g y)) T) S"
haftmann@57418
   357
    apply (rule setsum.cong, simp)
huffman@44133
   358
    apply (rule linear_setsum[unfolded o_def])
wenzelm@49522
   359
    using bh fT
wenzelm@49522
   360
    apply (auto simp add: bilinear_def)
wenzelm@49522
   361
    done
wenzelm@53406
   362
  finally show ?thesis
haftmann@57418
   363
    unfolding setsum.cartesian_product .
huffman@44133
   364
qed
huffman@44133
   365
wenzelm@49522
   366
wenzelm@60420
   367
subsection \<open>Adjoints.\<close>
huffman@44133
   368
huffman@44133
   369
definition "adjoint f = (SOME f'. \<forall>x y. f x \<bullet> y = x \<bullet> f' y)"
huffman@44133
   370
huffman@44133
   371
lemma adjoint_unique:
huffman@44133
   372
  assumes "\<forall>x y. inner (f x) y = inner x (g y)"
huffman@44133
   373
  shows "adjoint f = g"
wenzelm@49522
   374
  unfolding adjoint_def
huffman@44133
   375
proof (rule some_equality)
wenzelm@53406
   376
  show "\<forall>x y. inner (f x) y = inner x (g y)"
wenzelm@53406
   377
    by (rule assms)
huffman@44133
   378
next
wenzelm@53406
   379
  fix h
wenzelm@53406
   380
  assume "\<forall>x y. inner (f x) y = inner x (h y)"
wenzelm@53406
   381
  then have "\<forall>x y. inner x (g y) = inner x (h y)"
wenzelm@53406
   382
    using assms by simp
wenzelm@53406
   383
  then have "\<forall>x y. inner x (g y - h y) = 0"
wenzelm@53406
   384
    by (simp add: inner_diff_right)
wenzelm@53406
   385
  then have "\<forall>y. inner (g y - h y) (g y - h y) = 0"
wenzelm@53406
   386
    by simp
wenzelm@53406
   387
  then have "\<forall>y. h y = g y"
wenzelm@53406
   388
    by simp
wenzelm@49652
   389
  then show "h = g" by (simp add: ext)
huffman@44133
   390
qed
huffman@44133
   391
wenzelm@60420
   392
text \<open>TODO: The following lemmas about adjoints should hold for any
hoelzl@50526
   393
Hilbert space (i.e. complete inner product space).
wenzelm@54703
   394
(see @{url "http://en.wikipedia.org/wiki/Hermitian_adjoint"})
wenzelm@60420
   395
\<close>
hoelzl@50526
   396
hoelzl@50526
   397
lemma adjoint_works:
wenzelm@56444
   398
  fixes f :: "'n::euclidean_space \<Rightarrow> 'm::euclidean_space"
hoelzl@50526
   399
  assumes lf: "linear f"
hoelzl@50526
   400
  shows "x \<bullet> adjoint f y = f x \<bullet> y"
hoelzl@50526
   401
proof -
hoelzl@50526
   402
  have "\<forall>y. \<exists>w. \<forall>x. f x \<bullet> y = x \<bullet> w"
hoelzl@50526
   403
  proof (intro allI exI)
hoelzl@50526
   404
    fix y :: "'m" and x
hoelzl@50526
   405
    let ?w = "(\<Sum>i\<in>Basis. (f i \<bullet> y) *\<^sub>R i) :: 'n"
hoelzl@50526
   406
    have "f x \<bullet> y = f (\<Sum>i\<in>Basis. (x \<bullet> i) *\<^sub>R i) \<bullet> y"
hoelzl@50526
   407
      by (simp add: euclidean_representation)
hoelzl@50526
   408
    also have "\<dots> = (\<Sum>i\<in>Basis. (x \<bullet> i) *\<^sub>R f i) \<bullet> y"
huffman@56196
   409
      unfolding linear_setsum[OF lf]
hoelzl@50526
   410
      by (simp add: linear_cmul[OF lf])
hoelzl@50526
   411
    finally show "f x \<bullet> y = x \<bullet> ?w"
haftmann@57512
   412
      by (simp add: inner_setsum_left inner_setsum_right mult.commute)
hoelzl@50526
   413
  qed
hoelzl@50526
   414
  then show ?thesis
hoelzl@50526
   415
    unfolding adjoint_def choice_iff
hoelzl@50526
   416
    by (intro someI2_ex[where Q="\<lambda>f'. x \<bullet> f' y = f x \<bullet> y"]) auto
hoelzl@50526
   417
qed
hoelzl@50526
   418
hoelzl@50526
   419
lemma adjoint_clauses:
wenzelm@56444
   420
  fixes f :: "'n::euclidean_space \<Rightarrow> 'm::euclidean_space"
hoelzl@50526
   421
  assumes lf: "linear f"
hoelzl@50526
   422
  shows "x \<bullet> adjoint f y = f x \<bullet> y"
hoelzl@50526
   423
    and "adjoint f y \<bullet> x = y \<bullet> f x"
hoelzl@50526
   424
  by (simp_all add: adjoint_works[OF lf] inner_commute)
hoelzl@50526
   425
hoelzl@50526
   426
lemma adjoint_linear:
wenzelm@56444
   427
  fixes f :: "'n::euclidean_space \<Rightarrow> 'm::euclidean_space"
hoelzl@50526
   428
  assumes lf: "linear f"
hoelzl@50526
   429
  shows "linear (adjoint f)"
huffman@53600
   430
  by (simp add: lf linear_iff euclidean_eq_iff[where 'a='n] euclidean_eq_iff[where 'a='m]
huffman@53939
   431
    adjoint_clauses[OF lf] inner_distrib)
hoelzl@50526
   432
hoelzl@50526
   433
lemma adjoint_adjoint:
wenzelm@56444
   434
  fixes f :: "'n::euclidean_space \<Rightarrow> 'm::euclidean_space"
hoelzl@50526
   435
  assumes lf: "linear f"
hoelzl@50526
   436
  shows "adjoint (adjoint f) = f"
hoelzl@50526
   437
  by (rule adjoint_unique, simp add: adjoint_clauses [OF lf])
hoelzl@50526
   438
wenzelm@53406
   439
wenzelm@60420
   440
subsection \<open>Interlude: Some properties of real sets\<close>
huffman@44133
   441
wenzelm@53406
   442
lemma seq_mono_lemma:
wenzelm@53406
   443
  assumes "\<forall>(n::nat) \<ge> m. (d n :: real) < e n"
wenzelm@53406
   444
    and "\<forall>n \<ge> m. e n \<le> e m"
huffman@44133
   445
  shows "\<forall>n \<ge> m. d n < e m"
wenzelm@53406
   446
  using assms
wenzelm@53406
   447
  apply auto
huffman@44133
   448
  apply (erule_tac x="n" in allE)
huffman@44133
   449
  apply (erule_tac x="n" in allE)
huffman@44133
   450
  apply auto
huffman@44133
   451
  done
huffman@44133
   452
wenzelm@53406
   453
lemma infinite_enumerate:
wenzelm@53406
   454
  assumes fS: "infinite S"
huffman@44133
   455
  shows "\<exists>r. subseq r \<and> (\<forall>n. r n \<in> S)"
wenzelm@49525
   456
  unfolding subseq_def
wenzelm@49525
   457
  using enumerate_in_set[OF fS] enumerate_mono[of _ _ S] fS by auto
huffman@44133
   458
wenzelm@56444
   459
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
   460
  apply auto
wenzelm@49522
   461
  apply (rule_tac x="d/2" in exI)
wenzelm@49522
   462
  apply auto
wenzelm@49522
   463
  done
huffman@44133
   464
paulson@60762
   465
lemma approachable_lt_le2:  --{*like the above, but pushes aside an extra formula*}
paulson@60762
   466
    "(\<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
   467
  apply auto
paulson@60762
   468
  apply (rule_tac x="d/2" in exI, auto)
paulson@60762
   469
  done
paulson@60762
   470
huffman@44133
   471
lemma triangle_lemma:
wenzelm@53406
   472
  fixes x y z :: real
wenzelm@53406
   473
  assumes x: "0 \<le> x"
wenzelm@53406
   474
    and y: "0 \<le> y"
wenzelm@53406
   475
    and z: "0 \<le> z"
wenzelm@53406
   476
    and xy: "x\<^sup>2 \<le> y\<^sup>2 + z\<^sup>2"
wenzelm@53406
   477
  shows "x \<le> y + z"
wenzelm@49522
   478
proof -
wenzelm@56444
   479
  have "y\<^sup>2 + z\<^sup>2 \<le> y\<^sup>2 + 2 * y * z + z\<^sup>2"
nipkow@56536
   480
    using z y by simp
wenzelm@53406
   481
  with xy have th: "x\<^sup>2 \<le> (y + z)\<^sup>2"
wenzelm@53406
   482
    by (simp add: power2_eq_square field_simps)
wenzelm@53406
   483
  from y z have yz: "y + z \<ge> 0"
wenzelm@53406
   484
    by arith
huffman@44133
   485
  from power2_le_imp_le[OF th yz] show ?thesis .
huffman@44133
   486
qed
huffman@44133
   487
wenzelm@49522
   488
wenzelm@60420
   489
subsection \<open>A generic notion of "hull" (convex, affine, conic hull and closure).\<close>
huffman@44133
   490
wenzelm@53406
   491
definition hull :: "('a set \<Rightarrow> bool) \<Rightarrow> 'a set \<Rightarrow> 'a set"  (infixl "hull" 75)
wenzelm@53406
   492
  where "S hull s = \<Inter>{t. S t \<and> s \<subseteq> t}"
huffman@44170
   493
huffman@44170
   494
lemma hull_same: "S s \<Longrightarrow> S hull s = s"
huffman@44133
   495
  unfolding hull_def by auto
huffman@44133
   496
wenzelm@53406
   497
lemma hull_in: "(\<And>T. Ball T S \<Longrightarrow> S (\<Inter>T)) \<Longrightarrow> S (S hull s)"
wenzelm@49522
   498
  unfolding hull_def Ball_def by auto
huffman@44170
   499
wenzelm@53406
   500
lemma hull_eq: "(\<And>T. Ball T S \<Longrightarrow> S (\<Inter>T)) \<Longrightarrow> (S hull s) = s \<longleftrightarrow> S s"
wenzelm@49522
   501
  using hull_same[of S s] hull_in[of S s] by metis
huffman@44133
   502
huffman@44133
   503
lemma hull_hull: "S hull (S hull s) = S hull s"
huffman@44133
   504
  unfolding hull_def by blast
huffman@44133
   505
huffman@44133
   506
lemma hull_subset[intro]: "s \<subseteq> (S hull s)"
huffman@44133
   507
  unfolding hull_def by blast
huffman@44133
   508
wenzelm@53406
   509
lemma hull_mono: "s \<subseteq> t \<Longrightarrow> (S hull s) \<subseteq> (S hull t)"
huffman@44133
   510
  unfolding hull_def by blast
huffman@44133
   511
wenzelm@53406
   512
lemma hull_antimono: "\<forall>x. S x \<longrightarrow> T x \<Longrightarrow> (T hull s) \<subseteq> (S hull s)"
huffman@44133
   513
  unfolding hull_def by blast
huffman@44133
   514
wenzelm@53406
   515
lemma hull_minimal: "s \<subseteq> t \<Longrightarrow> S t \<Longrightarrow> (S hull s) \<subseteq> t"
huffman@44133
   516
  unfolding hull_def by blast
huffman@44133
   517
wenzelm@53406
   518
lemma subset_hull: "S t \<Longrightarrow> S hull s \<subseteq> t \<longleftrightarrow> s \<subseteq> t"
huffman@44133
   519
  unfolding hull_def by blast
huffman@44133
   520
huffman@53596
   521
lemma hull_UNIV: "S hull UNIV = UNIV"
huffman@53596
   522
  unfolding hull_def by auto
huffman@53596
   523
wenzelm@53406
   524
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
   525
  unfolding hull_def by auto
huffman@44133
   526
huffman@44133
   527
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
   528
  using hull_minimal[of S "{x. P x}" Q]
huffman@44170
   529
  by (auto simp add: subset_eq)
huffman@44133
   530
wenzelm@49522
   531
lemma hull_inc: "x \<in> S \<Longrightarrow> x \<in> P hull S"
wenzelm@49522
   532
  by (metis hull_subset subset_eq)
huffman@44133
   533
huffman@44133
   534
lemma hull_union_subset: "(S hull s) \<union> (S hull t) \<subseteq> (S hull (s \<union> t))"
wenzelm@49522
   535
  unfolding Un_subset_iff by (metis hull_mono Un_upper1 Un_upper2)
wenzelm@49522
   536
wenzelm@49522
   537
lemma hull_union:
wenzelm@53406
   538
  assumes T: "\<And>T. Ball T S \<Longrightarrow> S (\<Inter>T)"
huffman@44133
   539
  shows "S hull (s \<union> t) = S hull (S hull s \<union> S hull t)"
wenzelm@49522
   540
  apply rule
wenzelm@49522
   541
  apply (rule hull_mono)
wenzelm@49522
   542
  unfolding Un_subset_iff
wenzelm@49522
   543
  apply (metis hull_subset Un_upper1 Un_upper2 subset_trans)
wenzelm@49522
   544
  apply (rule hull_minimal)
wenzelm@49522
   545
  apply (metis hull_union_subset)
wenzelm@49522
   546
  apply (metis hull_in T)
wenzelm@49522
   547
  done
huffman@44133
   548
wenzelm@56444
   549
lemma hull_redundant_eq: "a \<in> (S hull s) \<longleftrightarrow> S hull (insert a s) = S hull s"
huffman@44133
   550
  unfolding hull_def by blast
huffman@44133
   551
wenzelm@56444
   552
lemma hull_redundant: "a \<in> (S hull s) \<Longrightarrow> S hull (insert a s) = S hull s"
wenzelm@49522
   553
  by (metis hull_redundant_eq)
wenzelm@49522
   554
huffman@44133
   555
wenzelm@60420
   556
subsection \<open>Archimedean properties and useful consequences\<close>
huffman@44133
   557
wenzelm@56444
   558
lemma real_arch_simple: "\<exists>n::nat. x \<le> real n"
huffman@44666
   559
  unfolding real_of_nat_def by (rule ex_le_of_nat)
huffman@44133
   560
huffman@44133
   561
lemma real_arch_inv: "0 < e \<longleftrightarrow> (\<exists>n::nat. n \<noteq> 0 \<and> 0 < inverse (real n) \<and> inverse (real n) < e)"
hoelzl@56480
   562
  using reals_Archimedean[of e] less_trans[of 0 "1 / real n" e for n::nat]
hoelzl@56480
   563
  by (auto simp add: field_simps cong: conj_cong)
huffman@44133
   564
wenzelm@53406
   565
lemma real_pow_lbound: "0 \<le> x \<Longrightarrow> 1 + real n * x \<le> (1 + x) ^ n"
wenzelm@49522
   566
proof (induct n)
wenzelm@49522
   567
  case 0
wenzelm@49522
   568
  then show ?case by simp
huffman@44133
   569
next
huffman@44133
   570
  case (Suc n)
wenzelm@53406
   571
  then have h: "1 + real n * x \<le> (1 + x) ^ n"
wenzelm@53406
   572
    by simp
wenzelm@53406
   573
  from h have p: "1 \<le> (1 + x) ^ n"
wenzelm@53406
   574
    using Suc.prems by simp
wenzelm@53406
   575
  from h have "1 + real n * x + x \<le> (1 + x) ^ n + x"
wenzelm@53406
   576
    by simp
wenzelm@53406
   577
  also have "\<dots> \<le> (1 + x) ^ Suc n"
wenzelm@53406
   578
    apply (subst diff_le_0_iff_le[symmetric])
lp15@60162
   579
    using mult_left_mono[OF p Suc.prems]
huffman@44133
   580
    apply (simp add: field_simps)
wenzelm@49522
   581
    done
wenzelm@53406
   582
  finally show ?case
wenzelm@53406
   583
    by (simp add: real_of_nat_Suc field_simps)
huffman@44133
   584
qed
huffman@44133
   585
wenzelm@53406
   586
lemma real_arch_pow:
wenzelm@53406
   587
  fixes x :: real
wenzelm@53406
   588
  assumes x: "1 < x"
wenzelm@53406
   589
  shows "\<exists>n. y < x^n"
wenzelm@49522
   590
proof -
wenzelm@53406
   591
  from x have x0: "x - 1 > 0"
wenzelm@53406
   592
    by arith
huffman@44666
   593
  from reals_Archimedean3[OF x0, rule_format, of y]
wenzelm@53406
   594
  obtain n :: nat where n: "y < real n * (x - 1)" by metis
huffman@44133
   595
  from x0 have x00: "x- 1 \<ge> 0" by arith
huffman@44133
   596
  from real_pow_lbound[OF x00, of n] n
huffman@44133
   597
  have "y < x^n" by auto
huffman@44133
   598
  then show ?thesis by metis
huffman@44133
   599
qed
huffman@44133
   600
wenzelm@53406
   601
lemma real_arch_pow2:
wenzelm@53406
   602
  fixes x :: real
wenzelm@53406
   603
  shows "\<exists>n. x < 2^ n"
huffman@44133
   604
  using real_arch_pow[of 2 x] by simp
huffman@44133
   605
wenzelm@49522
   606
lemma real_arch_pow_inv:
wenzelm@53406
   607
  fixes x y :: real
wenzelm@53406
   608
  assumes y: "y > 0"
wenzelm@53406
   609
    and x1: "x < 1"
huffman@44133
   610
  shows "\<exists>n. x^n < y"
wenzelm@53406
   611
proof (cases "x > 0")
wenzelm@53406
   612
  case True
wenzelm@53406
   613
  with x1 have ix: "1 < 1/x" by (simp add: field_simps)
wenzelm@53406
   614
  from real_arch_pow[OF ix, of "1/y"]
wenzelm@53406
   615
  obtain n where n: "1/y < (1/x)^n" by blast
wenzelm@60420
   616
  then show ?thesis using y \<open>x > 0\<close>
hoelzl@56480
   617
    by (auto simp add: field_simps)
wenzelm@53406
   618
next
wenzelm@53406
   619
  case False
wenzelm@53406
   620
  with y x1 show ?thesis
wenzelm@53406
   621
    apply auto
wenzelm@53406
   622
    apply (rule exI[where x=1])
wenzelm@53406
   623
    apply auto
wenzelm@53406
   624
    done
huffman@44133
   625
qed
huffman@44133
   626
wenzelm@49522
   627
lemma forall_pos_mono:
wenzelm@53406
   628
  "(\<And>d e::real. d < e \<Longrightarrow> P d \<Longrightarrow> P e) \<Longrightarrow>
wenzelm@53406
   629
    (\<And>n::nat. n \<noteq> 0 \<Longrightarrow> P (inverse (real n))) \<Longrightarrow> (\<And>e. 0 < e \<Longrightarrow> P e)"
huffman@44133
   630
  by (metis real_arch_inv)
huffman@44133
   631
wenzelm@49522
   632
lemma forall_pos_mono_1:
wenzelm@53406
   633
  "(\<And>d e::real. d < e \<Longrightarrow> P d \<Longrightarrow> P e) \<Longrightarrow>
wenzelm@53716
   634
    (\<And>n. P (inverse (real (Suc n)))) \<Longrightarrow> 0 < e \<Longrightarrow> P e"
huffman@44133
   635
  apply (rule forall_pos_mono)
huffman@44133
   636
  apply auto
huffman@44133
   637
  apply (atomize)
huffman@44133
   638
  apply (erule_tac x="n - 1" in allE)
huffman@44133
   639
  apply auto
huffman@44133
   640
  done
huffman@44133
   641
wenzelm@49522
   642
lemma real_archimedian_rdiv_eq_0:
wenzelm@53406
   643
  assumes x0: "x \<ge> 0"
wenzelm@53406
   644
    and c: "c \<ge> 0"
wenzelm@56444
   645
    and xc: "\<forall>(m::nat) > 0. real m * x \<le> c"
huffman@44133
   646
  shows "x = 0"
wenzelm@53406
   647
proof (rule ccontr)
wenzelm@53406
   648
  assume "x \<noteq> 0"
wenzelm@53406
   649
  with x0 have xp: "x > 0" by arith
wenzelm@53406
   650
  from reals_Archimedean3[OF xp, rule_format, of c]
wenzelm@53406
   651
  obtain n :: nat where n: "c < real n * x"
wenzelm@53406
   652
    by blast
wenzelm@53406
   653
  with xc[rule_format, of n] have "n = 0"
wenzelm@53406
   654
    by arith
wenzelm@53406
   655
  with n c show False
wenzelm@53406
   656
    by simp
huffman@44133
   657
qed
huffman@44133
   658
wenzelm@49522
   659
wenzelm@60420
   660
subsection\<open>A bit of linear algebra.\<close>
huffman@44133
   661
wenzelm@49522
   662
definition (in real_vector) subspace :: "'a set \<Rightarrow> bool"
wenzelm@56444
   663
  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
   664
huffman@44133
   665
definition (in real_vector) "span S = (subspace hull S)"
wenzelm@53716
   666
definition (in real_vector) "dependent S \<longleftrightarrow> (\<exists>a \<in> S. a \<in> span (S - {a}))"
wenzelm@53406
   667
abbreviation (in real_vector) "independent s \<equiv> \<not> dependent s"
huffman@44133
   668
wenzelm@60420
   669
text \<open>Closure properties of subspaces.\<close>
huffman@44133
   670
wenzelm@53406
   671
lemma subspace_UNIV[simp]: "subspace UNIV"
wenzelm@53406
   672
  by (simp add: subspace_def)
wenzelm@53406
   673
wenzelm@53406
   674
lemma (in real_vector) subspace_0: "subspace S \<Longrightarrow> 0 \<in> S"
wenzelm@53406
   675
  by (metis subspace_def)
wenzelm@53406
   676
wenzelm@53406
   677
lemma (in real_vector) subspace_add: "subspace S \<Longrightarrow> x \<in> S \<Longrightarrow> y \<in> S \<Longrightarrow> x + y \<in> S"
huffman@44133
   678
  by (metis subspace_def)
huffman@44133
   679
huffman@44133
   680
lemma (in real_vector) subspace_mul: "subspace S \<Longrightarrow> x \<in> S \<Longrightarrow> c *\<^sub>R x \<in> S"
huffman@44133
   681
  by (metis subspace_def)
huffman@44133
   682
huffman@44133
   683
lemma subspace_neg: "subspace S \<Longrightarrow> x \<in> S \<Longrightarrow> - x \<in> S"
huffman@44133
   684
  by (metis scaleR_minus1_left subspace_mul)
huffman@44133
   685
huffman@44133
   686
lemma subspace_sub: "subspace S \<Longrightarrow> x \<in> S \<Longrightarrow> y \<in> S \<Longrightarrow> x - y \<in> S"
haftmann@54230
   687
  using subspace_add [of S x "- y"] by (simp add: subspace_neg)
huffman@44133
   688
huffman@44133
   689
lemma (in real_vector) subspace_setsum:
wenzelm@53406
   690
  assumes sA: "subspace A"
huffman@56196
   691
    and f: "\<forall>x\<in>B. f x \<in> A"
huffman@44133
   692
  shows "setsum f B \<in> A"
huffman@56196
   693
proof (cases "finite B")
huffman@56196
   694
  case True
huffman@56196
   695
  then show ?thesis
huffman@56196
   696
    using f by induct (simp_all add: subspace_0 [OF sA] subspace_add [OF sA])
huffman@56196
   697
qed (simp add: subspace_0 [OF sA])
huffman@44133
   698
huffman@44133
   699
lemma subspace_linear_image:
wenzelm@53406
   700
  assumes lf: "linear f"
wenzelm@53406
   701
    and sS: "subspace S"
wenzelm@53406
   702
  shows "subspace (f ` S)"
huffman@44133
   703
  using lf sS linear_0[OF lf]
huffman@53600
   704
  unfolding linear_iff subspace_def
huffman@44133
   705
  apply (auto simp add: image_iff)
wenzelm@53406
   706
  apply (rule_tac x="x + y" in bexI)
wenzelm@53406
   707
  apply auto
wenzelm@53406
   708
  apply (rule_tac x="c *\<^sub>R x" in bexI)
wenzelm@53406
   709
  apply auto
huffman@44133
   710
  done
huffman@44133
   711
huffman@44521
   712
lemma subspace_linear_vimage: "linear f \<Longrightarrow> subspace S \<Longrightarrow> subspace (f -` S)"
huffman@53600
   713
  by (auto simp add: subspace_def linear_iff linear_0[of f])
huffman@44521
   714
wenzelm@53406
   715
lemma subspace_linear_preimage: "linear f \<Longrightarrow> subspace S \<Longrightarrow> subspace {x. f x \<in> S}"
huffman@53600
   716
  by (auto simp add: subspace_def linear_iff linear_0[of f])
huffman@44133
   717
huffman@44133
   718
lemma subspace_trivial: "subspace {0}"
huffman@44133
   719
  by (simp add: subspace_def)
huffman@44133
   720
wenzelm@53406
   721
lemma (in real_vector) subspace_inter: "subspace A \<Longrightarrow> subspace B \<Longrightarrow> subspace (A \<inter> B)"
huffman@44133
   722
  by (simp add: subspace_def)
huffman@44133
   723
wenzelm@53406
   724
lemma subspace_Times: "subspace A \<Longrightarrow> subspace B \<Longrightarrow> subspace (A \<times> B)"
huffman@44521
   725
  unfolding subspace_def zero_prod_def by simp
huffman@44521
   726
wenzelm@60420
   727
text \<open>Properties of span.\<close>
huffman@44521
   728
wenzelm@53406
   729
lemma (in real_vector) span_mono: "A \<subseteq> B \<Longrightarrow> span A \<subseteq> span B"
huffman@44133
   730
  by (metis span_def hull_mono)
huffman@44133
   731
wenzelm@53406
   732
lemma (in real_vector) subspace_span: "subspace (span S)"
huffman@44133
   733
  unfolding span_def
huffman@44170
   734
  apply (rule hull_in)
huffman@44133
   735
  apply (simp only: subspace_def Inter_iff Int_iff subset_eq)
huffman@44133
   736
  apply auto
huffman@44133
   737
  done
huffman@44133
   738
huffman@44133
   739
lemma (in real_vector) span_clauses:
wenzelm@53406
   740
  "a \<in> S \<Longrightarrow> a \<in> span S"
huffman@44133
   741
  "0 \<in> span S"
wenzelm@53406
   742
  "x\<in> span S \<Longrightarrow> y \<in> span S \<Longrightarrow> x + y \<in> span S"
huffman@44133
   743
  "x \<in> span S \<Longrightarrow> c *\<^sub>R x \<in> span S"
wenzelm@53406
   744
  by (metis span_def hull_subset subset_eq) (metis subspace_span subspace_def)+
huffman@44133
   745
huffman@44521
   746
lemma span_unique:
wenzelm@49522
   747
  "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
   748
  unfolding span_def by (rule hull_unique)
huffman@44521
   749
huffman@44521
   750
lemma span_minimal: "S \<subseteq> T \<Longrightarrow> subspace T \<Longrightarrow> span S \<subseteq> T"
huffman@44521
   751
  unfolding span_def by (rule hull_minimal)
huffman@44521
   752
huffman@44521
   753
lemma (in real_vector) span_induct:
wenzelm@49522
   754
  assumes x: "x \<in> span S"
wenzelm@49522
   755
    and P: "subspace P"
wenzelm@53406
   756
    and SP: "\<And>x. x \<in> S \<Longrightarrow> x \<in> P"
huffman@44521
   757
  shows "x \<in> P"
wenzelm@49522
   758
proof -
wenzelm@53406
   759
  from SP have SP': "S \<subseteq> P"
wenzelm@53406
   760
    by (simp add: subset_eq)
huffman@44170
   761
  from x hull_minimal[where S=subspace, OF SP' P, unfolded span_def[symmetric]]
wenzelm@53406
   762
  show "x \<in> P"
wenzelm@53406
   763
    by (metis subset_eq)
huffman@44133
   764
qed
huffman@44133
   765
huffman@44133
   766
lemma span_empty[simp]: "span {} = {0}"
huffman@44133
   767
  apply (simp add: span_def)
huffman@44133
   768
  apply (rule hull_unique)
huffman@44170
   769
  apply (auto simp add: subspace_def)
huffman@44133
   770
  done
huffman@44133
   771
huffman@44133
   772
lemma (in real_vector) independent_empty[intro]: "independent {}"
huffman@44133
   773
  by (simp add: dependent_def)
huffman@44133
   774
wenzelm@49522
   775
lemma dependent_single[simp]: "dependent {x} \<longleftrightarrow> x = 0"
huffman@44133
   776
  unfolding dependent_def by auto
huffman@44133
   777
wenzelm@53406
   778
lemma (in real_vector) independent_mono: "independent A \<Longrightarrow> B \<subseteq> A \<Longrightarrow> independent B"
huffman@44133
   779
  apply (clarsimp simp add: dependent_def span_mono)
huffman@44133
   780
  apply (subgoal_tac "span (B - {a}) \<le> span (A - {a})")
huffman@44133
   781
  apply force
huffman@44133
   782
  apply (rule span_mono)
huffman@44133
   783
  apply auto
huffman@44133
   784
  done
huffman@44133
   785
huffman@44133
   786
lemma (in real_vector) span_subspace: "A \<subseteq> B \<Longrightarrow> B \<le> span A \<Longrightarrow>  subspace B \<Longrightarrow> span A = B"
huffman@44170
   787
  by (metis order_antisym span_def hull_minimal)
huffman@44133
   788
wenzelm@49711
   789
lemma (in real_vector) span_induct':
wenzelm@49711
   790
  assumes SP: "\<forall>x \<in> S. P x"
wenzelm@49711
   791
    and P: "subspace {x. P x}"
wenzelm@49711
   792
  shows "\<forall>x \<in> span S. P x"
huffman@44133
   793
  using span_induct SP P by blast
huffman@44133
   794
wenzelm@56444
   795
inductive_set (in real_vector) span_induct_alt_help for S :: "'a set"
wenzelm@53406
   796
where
huffman@44170
   797
  span_induct_alt_help_0: "0 \<in> span_induct_alt_help S"
wenzelm@49522
   798
| span_induct_alt_help_S:
wenzelm@53406
   799
    "x \<in> S \<Longrightarrow> z \<in> span_induct_alt_help S \<Longrightarrow>
wenzelm@53406
   800
      (c *\<^sub>R x + z) \<in> span_induct_alt_help S"
huffman@44133
   801
huffman@44133
   802
lemma span_induct_alt':
wenzelm@53406
   803
  assumes h0: "h 0"
wenzelm@53406
   804
    and hS: "\<And>c x y. x \<in> S \<Longrightarrow> h y \<Longrightarrow> h (c *\<^sub>R x + y)"
wenzelm@49522
   805
  shows "\<forall>x \<in> span S. h x"
wenzelm@49522
   806
proof -
wenzelm@53406
   807
  {
wenzelm@53406
   808
    fix x :: 'a
wenzelm@53406
   809
    assume x: "x \<in> span_induct_alt_help S"
huffman@44133
   810
    have "h x"
huffman@44133
   811
      apply (rule span_induct_alt_help.induct[OF x])
huffman@44133
   812
      apply (rule h0)
wenzelm@53406
   813
      apply (rule hS)
wenzelm@53406
   814
      apply assumption
wenzelm@53406
   815
      apply assumption
wenzelm@53406
   816
      done
wenzelm@53406
   817
  }
huffman@44133
   818
  note th0 = this
wenzelm@53406
   819
  {
wenzelm@53406
   820
    fix x
wenzelm@53406
   821
    assume x: "x \<in> span S"
huffman@44170
   822
    have "x \<in> span_induct_alt_help S"
wenzelm@49522
   823
    proof (rule span_induct[where x=x and S=S])
wenzelm@53406
   824
      show "x \<in> span S" by (rule x)
wenzelm@49522
   825
    next
wenzelm@53406
   826
      fix x
wenzelm@53406
   827
      assume xS: "x \<in> S"
wenzelm@53406
   828
      from span_induct_alt_help_S[OF xS span_induct_alt_help_0, of 1]
wenzelm@53406
   829
      show "x \<in> span_induct_alt_help S"
wenzelm@53406
   830
        by simp
wenzelm@49522
   831
    next
wenzelm@49522
   832
      have "0 \<in> span_induct_alt_help S" by (rule span_induct_alt_help_0)
wenzelm@49522
   833
      moreover
wenzelm@53406
   834
      {
wenzelm@53406
   835
        fix x y
wenzelm@49522
   836
        assume h: "x \<in> span_induct_alt_help S" "y \<in> span_induct_alt_help S"
wenzelm@49522
   837
        from h have "(x + y) \<in> span_induct_alt_help S"
wenzelm@49522
   838
          apply (induct rule: span_induct_alt_help.induct)
wenzelm@49522
   839
          apply simp
haftmann@57512
   840
          unfolding add.assoc
wenzelm@49522
   841
          apply (rule span_induct_alt_help_S)
wenzelm@49522
   842
          apply assumption
wenzelm@49522
   843
          apply simp
wenzelm@53406
   844
          done
wenzelm@53406
   845
      }
wenzelm@49522
   846
      moreover
wenzelm@53406
   847
      {
wenzelm@53406
   848
        fix c x
wenzelm@49522
   849
        assume xt: "x \<in> span_induct_alt_help S"
wenzelm@49522
   850
        then have "(c *\<^sub>R x) \<in> span_induct_alt_help S"
wenzelm@49522
   851
          apply (induct rule: span_induct_alt_help.induct)
wenzelm@49522
   852
          apply (simp add: span_induct_alt_help_0)
wenzelm@49522
   853
          apply (simp add: scaleR_right_distrib)
wenzelm@49522
   854
          apply (rule span_induct_alt_help_S)
wenzelm@49522
   855
          apply assumption
wenzelm@49522
   856
          apply simp
wenzelm@49522
   857
          done }
wenzelm@53406
   858
      ultimately show "subspace (span_induct_alt_help S)"
wenzelm@49522
   859
        unfolding subspace_def Ball_def by blast
wenzelm@53406
   860
    qed
wenzelm@53406
   861
  }
huffman@44133
   862
  with th0 show ?thesis by blast
huffman@44133
   863
qed
huffman@44133
   864
huffman@44133
   865
lemma span_induct_alt:
wenzelm@53406
   866
  assumes h0: "h 0"
wenzelm@53406
   867
    and hS: "\<And>c x y. x \<in> S \<Longrightarrow> h y \<Longrightarrow> h (c *\<^sub>R x + y)"
wenzelm@53406
   868
    and x: "x \<in> span S"
huffman@44133
   869
  shows "h x"
wenzelm@49522
   870
  using span_induct_alt'[of h S] h0 hS x by blast
huffman@44133
   871
wenzelm@60420
   872
text \<open>Individual closure properties.\<close>
huffman@44133
   873
huffman@44133
   874
lemma span_span: "span (span A) = span A"
huffman@44133
   875
  unfolding span_def hull_hull ..
huffman@44133
   876
wenzelm@53406
   877
lemma (in real_vector) span_superset: "x \<in> S \<Longrightarrow> x \<in> span S"
wenzelm@53406
   878
  by (metis span_clauses(1))
wenzelm@53406
   879
wenzelm@53406
   880
lemma (in real_vector) span_0: "0 \<in> span S"
wenzelm@53406
   881
  by (metis subspace_span subspace_0)
huffman@44133
   882
huffman@44133
   883
lemma span_inc: "S \<subseteq> span S"
huffman@44133
   884
  by (metis subset_eq span_superset)
huffman@44133
   885
wenzelm@53406
   886
lemma (in real_vector) dependent_0:
wenzelm@53406
   887
  assumes "0 \<in> A"
wenzelm@53406
   888
  shows "dependent A"
wenzelm@53406
   889
  unfolding dependent_def
wenzelm@53406
   890
  using assms span_0
lp15@60162
   891
  by auto
wenzelm@53406
   892
wenzelm@53406
   893
lemma (in real_vector) span_add: "x \<in> span S \<Longrightarrow> y \<in> span S \<Longrightarrow> x + y \<in> span S"
huffman@44133
   894
  by (metis subspace_add subspace_span)
huffman@44133
   895
wenzelm@53406
   896
lemma (in real_vector) span_mul: "x \<in> span S \<Longrightarrow> c *\<^sub>R x \<in> span S"
huffman@44133
   897
  by (metis subspace_span subspace_mul)
huffman@44133
   898
wenzelm@53406
   899
lemma span_neg: "x \<in> span S \<Longrightarrow> - x \<in> span S"
huffman@44133
   900
  by (metis subspace_neg subspace_span)
huffman@44133
   901
wenzelm@53406
   902
lemma span_sub: "x \<in> span S \<Longrightarrow> y \<in> span S \<Longrightarrow> x - y \<in> span S"
huffman@44133
   903
  by (metis subspace_span subspace_sub)
huffman@44133
   904
huffman@56196
   905
lemma (in real_vector) span_setsum: "\<forall>x\<in>A. f x \<in> span S \<Longrightarrow> setsum f A \<in> span S"
huffman@56196
   906
  by (rule subspace_setsum [OF subspace_span])
huffman@44133
   907
huffman@44133
   908
lemma span_add_eq: "x \<in> span S \<Longrightarrow> x + y \<in> span S \<longleftrightarrow> y \<in> span S"
lp15@55775
   909
  by (metis add_minus_cancel scaleR_minus1_left subspace_def subspace_span)
huffman@44133
   910
wenzelm@60420
   911
text \<open>Mapping under linear image.\<close>
huffman@44133
   912
huffman@44521
   913
lemma span_linear_image:
huffman@44521
   914
  assumes lf: "linear f"
wenzelm@56444
   915
  shows "span (f ` S) = f ` span S"
huffman@44521
   916
proof (rule span_unique)
huffman@44521
   917
  show "f ` S \<subseteq> f ` span S"
huffman@44521
   918
    by (intro image_mono span_inc)
huffman@44521
   919
  show "subspace (f ` span S)"
huffman@44521
   920
    using lf subspace_span by (rule subspace_linear_image)
huffman@44521
   921
next
wenzelm@53406
   922
  fix T
wenzelm@53406
   923
  assume "f ` S \<subseteq> T" and "subspace T"
wenzelm@49522
   924
  then show "f ` span S \<subseteq> T"
huffman@44521
   925
    unfolding image_subset_iff_subset_vimage
huffman@44521
   926
    by (intro span_minimal subspace_linear_vimage lf)
huffman@44521
   927
qed
huffman@44521
   928
huffman@44521
   929
lemma span_union: "span (A \<union> B) = (\<lambda>(a, b). a + b) ` (span A \<times> span B)"
huffman@44521
   930
proof (rule span_unique)
huffman@44521
   931
  show "A \<union> B \<subseteq> (\<lambda>(a, b). a + b) ` (span A \<times> span B)"
huffman@44521
   932
    by safe (force intro: span_clauses)+
huffman@44521
   933
next
huffman@44521
   934
  have "linear (\<lambda>(a, b). a + b)"
huffman@53600
   935
    by (simp add: linear_iff scaleR_add_right)
huffman@44521
   936
  moreover have "subspace (span A \<times> span B)"
huffman@44521
   937
    by (intro subspace_Times subspace_span)
huffman@44521
   938
  ultimately show "subspace ((\<lambda>(a, b). a + b) ` (span A \<times> span B))"
huffman@44521
   939
    by (rule subspace_linear_image)
huffman@44521
   940
next
wenzelm@49711
   941
  fix T
wenzelm@49711
   942
  assume "A \<union> B \<subseteq> T" and "subspace T"
wenzelm@49522
   943
  then show "(\<lambda>(a, b). a + b) ` (span A \<times> span B) \<subseteq> T"
huffman@44521
   944
    by (auto intro!: subspace_add elim: span_induct)
huffman@44133
   945
qed
huffman@44133
   946
wenzelm@60420
   947
text \<open>The key breakdown property.\<close>
huffman@44133
   948
huffman@44521
   949
lemma span_singleton: "span {x} = range (\<lambda>k. k *\<^sub>R x)"
huffman@44521
   950
proof (rule span_unique)
huffman@44521
   951
  show "{x} \<subseteq> range (\<lambda>k. k *\<^sub>R x)"
huffman@44521
   952
    by (fast intro: scaleR_one [symmetric])
huffman@44521
   953
  show "subspace (range (\<lambda>k. k *\<^sub>R x))"
huffman@44521
   954
    unfolding subspace_def
huffman@44521
   955
    by (auto intro: scaleR_add_left [symmetric])
wenzelm@53406
   956
next
wenzelm@53406
   957
  fix T
wenzelm@53406
   958
  assume "{x} \<subseteq> T" and "subspace T"
wenzelm@53406
   959
  then show "range (\<lambda>k. k *\<^sub>R x) \<subseteq> T"
huffman@44521
   960
    unfolding subspace_def by auto
huffman@44521
   961
qed
huffman@44521
   962
wenzelm@49522
   963
lemma span_insert: "span (insert a S) = {x. \<exists>k. (x - k *\<^sub>R a) \<in> span S}"
huffman@44521
   964
proof -
huffman@44521
   965
  have "span ({a} \<union> S) = {x. \<exists>k. (x - k *\<^sub>R a) \<in> span S}"
huffman@44521
   966
    unfolding span_union span_singleton
huffman@44521
   967
    apply safe
huffman@44521
   968
    apply (rule_tac x=k in exI, simp)
huffman@44521
   969
    apply (erule rev_image_eqI [OF SigmaI [OF rangeI]])
haftmann@54230
   970
    apply auto
huffman@44521
   971
    done
wenzelm@49522
   972
  then show ?thesis by simp
huffman@44521
   973
qed
huffman@44521
   974
huffman@44133
   975
lemma span_breakdown:
wenzelm@53406
   976
  assumes bS: "b \<in> S"
wenzelm@53406
   977
    and aS: "a \<in> span S"
huffman@44521
   978
  shows "\<exists>k. a - k *\<^sub>R b \<in> span (S - {b})"
huffman@44521
   979
  using assms span_insert [of b "S - {b}"]
huffman@44521
   980
  by (simp add: insert_absorb)
huffman@44133
   981
wenzelm@53406
   982
lemma span_breakdown_eq: "x \<in> span (insert a S) \<longleftrightarrow> (\<exists>k. x - k *\<^sub>R a \<in> span S)"
huffman@44521
   983
  by (simp add: span_insert)
huffman@44133
   984
wenzelm@60420
   985
text \<open>Hence some "reversal" results.\<close>
huffman@44133
   986
huffman@44133
   987
lemma in_span_insert:
wenzelm@49711
   988
  assumes a: "a \<in> span (insert b S)"
wenzelm@49711
   989
    and na: "a \<notin> span S"
huffman@44133
   990
  shows "b \<in> span (insert a S)"
wenzelm@49663
   991
proof -
huffman@55910
   992
  from a obtain k where k: "a - k *\<^sub>R b \<in> span S"
huffman@55910
   993
    unfolding span_insert by fast
wenzelm@53406
   994
  show ?thesis
wenzelm@53406
   995
  proof (cases "k = 0")
wenzelm@53406
   996
    case True
huffman@55910
   997
    with k have "a \<in> span S" by simp
huffman@55910
   998
    with na show ?thesis by simp
wenzelm@53406
   999
  next
wenzelm@53406
  1000
    case False
huffman@55910
  1001
    from k have "(- inverse k) *\<^sub>R (a - k *\<^sub>R b) \<in> span S"
huffman@44133
  1002
      by (rule span_mul)
huffman@55910
  1003
    then have "b - inverse k *\<^sub>R a \<in> span S"
wenzelm@60420
  1004
      using \<open>k \<noteq> 0\<close> by (simp add: scaleR_diff_right)
huffman@55910
  1005
    then show ?thesis
huffman@55910
  1006
      unfolding span_insert by fast
wenzelm@53406
  1007
  qed
huffman@44133
  1008
qed
huffman@44133
  1009
huffman@44133
  1010
lemma in_span_delete:
huffman@44133
  1011
  assumes a: "a \<in> span S"
wenzelm@53716
  1012
    and na: "a \<notin> span (S - {b})"
huffman@44133
  1013
  shows "b \<in> span (insert a (S - {b}))"
huffman@44133
  1014
  apply (rule in_span_insert)
huffman@44133
  1015
  apply (rule set_rev_mp)
huffman@44133
  1016
  apply (rule a)
huffman@44133
  1017
  apply (rule span_mono)
huffman@44133
  1018
  apply blast
huffman@44133
  1019
  apply (rule na)
huffman@44133
  1020
  done
huffman@44133
  1021
wenzelm@60420
  1022
text \<open>Transitivity property.\<close>
huffman@44133
  1023
huffman@44521
  1024
lemma span_redundant: "x \<in> span S \<Longrightarrow> span (insert x S) = span S"
huffman@44521
  1025
  unfolding span_def by (rule hull_redundant)
huffman@44521
  1026
huffman@44133
  1027
lemma span_trans:
wenzelm@53406
  1028
  assumes x: "x \<in> span S"
wenzelm@53406
  1029
    and y: "y \<in> span (insert x S)"
huffman@44133
  1030
  shows "y \<in> span S"
huffman@44521
  1031
  using assms by (simp only: span_redundant)
huffman@44133
  1032
huffman@44133
  1033
lemma span_insert_0[simp]: "span (insert 0 S) = span S"
huffman@44521
  1034
  by (simp only: span_redundant span_0)
huffman@44133
  1035
wenzelm@60420
  1036
text \<open>An explicit expansion is sometimes needed.\<close>
huffman@44133
  1037
huffman@44133
  1038
lemma span_explicit:
huffman@44133
  1039
  "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
  1040
  (is "_ = ?E" is "_ = {y. ?h y}" is "_ = {y. \<exists>S u. ?Q S u y}")
wenzelm@49663
  1041
proof -
wenzelm@53406
  1042
  {
wenzelm@53406
  1043
    fix x
huffman@55910
  1044
    assume "?h x"
huffman@55910
  1045
    then obtain S u where "finite S" and "S \<subseteq> P" and "setsum (\<lambda>v. u v *\<^sub>R v) S = x"
huffman@44133
  1046
      by blast
huffman@55910
  1047
    then have "x \<in> span P"
huffman@55910
  1048
      by (auto intro: span_setsum span_mul span_superset)
wenzelm@53406
  1049
  }
huffman@44133
  1050
  moreover
huffman@55910
  1051
  have "\<forall>x \<in> span P. ?h x"
wenzelm@49522
  1052
  proof (rule span_induct_alt')
huffman@55910
  1053
    show "?h 0"
huffman@55910
  1054
      by (rule exI[where x="{}"], simp)
huffman@44133
  1055
  next
huffman@44133
  1056
    fix c x y
wenzelm@53406
  1057
    assume x: "x \<in> P"
huffman@55910
  1058
    assume hy: "?h y"
huffman@44133
  1059
    from hy obtain S u where fS: "finite S" and SP: "S\<subseteq>P"
huffman@44133
  1060
      and u: "setsum (\<lambda>v. u v *\<^sub>R v) S = y" by blast
huffman@44133
  1061
    let ?S = "insert x S"
wenzelm@49522
  1062
    let ?u = "\<lambda>y. if y = x then (if x \<in> S then u y + c else c) else u y"
wenzelm@53406
  1063
    from fS SP x have th0: "finite (insert x S)" "insert x S \<subseteq> P"
wenzelm@53406
  1064
      by blast+
wenzelm@53406
  1065
    have "?Q ?S ?u (c*\<^sub>R x + y)"
wenzelm@53406
  1066
    proof cases
wenzelm@53406
  1067
      assume xS: "x \<in> S"
huffman@55910
  1068
      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
  1069
        using xS by (simp add: setsum.remove [OF fS xS] insert_absorb)
huffman@44133
  1070
      also have "\<dots> = (\<Sum>v\<in>S. u v *\<^sub>R v) + c *\<^sub>R x"
huffman@55910
  1071
        by (simp add: setsum.remove [OF fS xS] algebra_simps)
huffman@44133
  1072
      also have "\<dots> = c*\<^sub>R x + y"
haftmann@57512
  1073
        by (simp add: add.commute u)
huffman@44133
  1074
      finally have "setsum (\<lambda>v. ?u v *\<^sub>R v) ?S = c*\<^sub>R x + y" .
wenzelm@53406
  1075
      then show ?thesis using th0 by blast
wenzelm@53406
  1076
    next
wenzelm@53406
  1077
      assume xS: "x \<notin> S"
wenzelm@49522
  1078
      have th00: "(\<Sum>v\<in>S. (if v = x then c else u v) *\<^sub>R v) = y"
wenzelm@49522
  1079
        unfolding u[symmetric]
haftmann@57418
  1080
        apply (rule setsum.cong)
wenzelm@53406
  1081
        using xS
wenzelm@53406
  1082
        apply auto
wenzelm@49522
  1083
        done
wenzelm@53406
  1084
      show ?thesis using fS xS th0
haftmann@57512
  1085
        by (simp add: th00 add.commute cong del: if_weak_cong)
wenzelm@53406
  1086
    qed
huffman@55910
  1087
    then show "?h (c*\<^sub>R x + y)"
huffman@55910
  1088
      by fast
huffman@44133
  1089
  qed
huffman@44133
  1090
  ultimately show ?thesis by blast
huffman@44133
  1091
qed
huffman@44133
  1092
huffman@44133
  1093
lemma dependent_explicit:
wenzelm@49522
  1094
  "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
  1095
  (is "?lhs = ?rhs")
wenzelm@49522
  1096
proof -
wenzelm@53406
  1097
  {
wenzelm@53406
  1098
    assume dP: "dependent P"
huffman@44133
  1099
    then obtain a S u where aP: "a \<in> P" and fS: "finite S"
huffman@44133
  1100
      and SP: "S \<subseteq> P - {a}" and ua: "setsum (\<lambda>v. u v *\<^sub>R v) S = a"
huffman@44133
  1101
      unfolding dependent_def span_explicit by blast
huffman@44133
  1102
    let ?S = "insert a S"
huffman@44133
  1103
    let ?u = "\<lambda>y. if y = a then - 1 else u y"
huffman@44133
  1104
    let ?v = a
wenzelm@53406
  1105
    from aP SP have aS: "a \<notin> S"
wenzelm@53406
  1106
      by blast
wenzelm@53406
  1107
    from fS SP aP have th0: "finite ?S" "?S \<subseteq> P" "?v \<in> ?S" "?u ?v \<noteq> 0"
wenzelm@53406
  1108
      by auto
huffman@44133
  1109
    have s0: "setsum (\<lambda>v. ?u v *\<^sub>R v) ?S = 0"
huffman@44133
  1110
      using fS aS
huffman@55910
  1111
      apply simp
huffman@44133
  1112
      apply (subst (2) ua[symmetric])
haftmann@57418
  1113
      apply (rule setsum.cong)
wenzelm@49522
  1114
      apply auto
wenzelm@49522
  1115
      done
huffman@55910
  1116
    with th0 have ?rhs by fast
wenzelm@49522
  1117
  }
huffman@44133
  1118
  moreover
wenzelm@53406
  1119
  {
wenzelm@53406
  1120
    fix S u v
wenzelm@49522
  1121
    assume fS: "finite S"
wenzelm@53406
  1122
      and SP: "S \<subseteq> P"
wenzelm@53406
  1123
      and vS: "v \<in> S"
wenzelm@53406
  1124
      and uv: "u v \<noteq> 0"
wenzelm@49522
  1125
      and u: "setsum (\<lambda>v. u v *\<^sub>R v) S = 0"
huffman@44133
  1126
    let ?a = v
huffman@44133
  1127
    let ?S = "S - {v}"
huffman@44133
  1128
    let ?u = "\<lambda>i. (- u i) / u v"
wenzelm@53406
  1129
    have th0: "?a \<in> P" "finite ?S" "?S \<subseteq> P"
wenzelm@53406
  1130
      using fS SP vS by auto
wenzelm@53406
  1131
    have "setsum (\<lambda>v. ?u v *\<^sub>R v) ?S =
wenzelm@53406
  1132
      setsum (\<lambda>v. (- (inverse (u ?a))) *\<^sub>R (u v *\<^sub>R v)) S - ?u v *\<^sub>R v"
hoelzl@56480
  1133
      using fS vS uv by (simp add: setsum_diff1 field_simps)
wenzelm@53406
  1134
    also have "\<dots> = ?a"
hoelzl@56479
  1135
      unfolding scaleR_right.setsum [symmetric] u using uv by simp
wenzelm@53406
  1136
    finally have "setsum (\<lambda>v. ?u v *\<^sub>R v) ?S = ?a" .
huffman@44133
  1137
    with th0 have ?lhs
huffman@44133
  1138
      unfolding dependent_def span_explicit
huffman@44133
  1139
      apply -
huffman@44133
  1140
      apply (rule bexI[where x= "?a"])
huffman@44133
  1141
      apply (simp_all del: scaleR_minus_left)
huffman@44133
  1142
      apply (rule exI[where x= "?S"])
wenzelm@49522
  1143
      apply (auto simp del: scaleR_minus_left)
wenzelm@49522
  1144
      done
wenzelm@49522
  1145
  }
huffman@44133
  1146
  ultimately show ?thesis by blast
huffman@44133
  1147
qed
huffman@44133
  1148
huffman@44133
  1149
huffman@44133
  1150
lemma span_finite:
huffman@44133
  1151
  assumes fS: "finite S"
huffman@44133
  1152
  shows "span S = {y. \<exists>u. setsum (\<lambda>v. u v *\<^sub>R v) S = y}"
huffman@44133
  1153
  (is "_ = ?rhs")
wenzelm@49522
  1154
proof -
wenzelm@53406
  1155
  {
wenzelm@53406
  1156
    fix y
wenzelm@49711
  1157
    assume y: "y \<in> span S"
wenzelm@53406
  1158
    from y obtain S' u where fS': "finite S'"
wenzelm@53406
  1159
      and SS': "S' \<subseteq> S"
wenzelm@53406
  1160
      and u: "setsum (\<lambda>v. u v *\<^sub>R v) S' = y"
wenzelm@53406
  1161
      unfolding span_explicit by blast
huffman@44133
  1162
    let ?u = "\<lambda>x. if x \<in> S' then u x else 0"
huffman@44133
  1163
    have "setsum (\<lambda>v. ?u v *\<^sub>R v) S = setsum (\<lambda>v. u v *\<^sub>R v) S'"
haftmann@57418
  1164
      using SS' fS by (auto intro!: setsum.mono_neutral_cong_right)
wenzelm@49522
  1165
    then have "setsum (\<lambda>v. ?u v *\<^sub>R v) S = y" by (metis u)
wenzelm@53406
  1166
    then have "y \<in> ?rhs" by auto
wenzelm@53406
  1167
  }
huffman@44133
  1168
  moreover
wenzelm@53406
  1169
  {
wenzelm@53406
  1170
    fix y u
wenzelm@49522
  1171
    assume u: "setsum (\<lambda>v. u v *\<^sub>R v) S = y"
wenzelm@53406
  1172
    then have "y \<in> span S" using fS unfolding span_explicit by auto
wenzelm@53406
  1173
  }
huffman@44133
  1174
  ultimately show ?thesis by blast
huffman@44133
  1175
qed
huffman@44133
  1176
wenzelm@60420
  1177
text \<open>This is useful for building a basis step-by-step.\<close>
huffman@44133
  1178
huffman@44133
  1179
lemma independent_insert:
wenzelm@53406
  1180
  "independent (insert a S) \<longleftrightarrow>
wenzelm@53406
  1181
    (if a \<in> S then independent S else independent S \<and> a \<notin> span S)"
wenzelm@53406
  1182
  (is "?lhs \<longleftrightarrow> ?rhs")
wenzelm@53406
  1183
proof (cases "a \<in> S")
wenzelm@53406
  1184
  case True
wenzelm@53406
  1185
  then show ?thesis
wenzelm@53406
  1186
    using insert_absorb[OF True] by simp
wenzelm@53406
  1187
next
wenzelm@53406
  1188
  case False
wenzelm@53406
  1189
  show ?thesis
wenzelm@53406
  1190
  proof
wenzelm@53406
  1191
    assume i: ?lhs
wenzelm@53406
  1192
    then show ?rhs
wenzelm@53406
  1193
      using False
wenzelm@53406
  1194
      apply simp
wenzelm@53406
  1195
      apply (rule conjI)
wenzelm@53406
  1196
      apply (rule independent_mono)
wenzelm@53406
  1197
      apply assumption
wenzelm@53406
  1198
      apply blast
wenzelm@53406
  1199
      apply (simp add: dependent_def)
wenzelm@53406
  1200
      done
wenzelm@53406
  1201
  next
wenzelm@53406
  1202
    assume i: ?rhs
wenzelm@53406
  1203
    show ?lhs
wenzelm@53406
  1204
      using i False
wenzelm@53406
  1205
      apply (auto simp add: dependent_def)
lp15@55775
  1206
      by (metis in_span_insert insert_Diff insert_Diff_if insert_iff)
wenzelm@53406
  1207
  qed
huffman@44133
  1208
qed
huffman@44133
  1209
wenzelm@60420
  1210
text \<open>The degenerate case of the Exchange Lemma.\<close>
huffman@44133
  1211
huffman@44133
  1212
lemma spanning_subset_independent:
wenzelm@49711
  1213
  assumes BA: "B \<subseteq> A"
wenzelm@49711
  1214
    and iA: "independent A"
wenzelm@49522
  1215
    and AsB: "A \<subseteq> span B"
huffman@44133
  1216
  shows "A = B"
huffman@44133
  1217
proof
wenzelm@49663
  1218
  show "B \<subseteq> A" by (rule BA)
wenzelm@49663
  1219
huffman@44133
  1220
  from span_mono[OF BA] span_mono[OF AsB]
huffman@44133
  1221
  have sAB: "span A = span B" unfolding span_span by blast
huffman@44133
  1222
wenzelm@53406
  1223
  {
wenzelm@53406
  1224
    fix x
wenzelm@53406
  1225
    assume x: "x \<in> A"
huffman@44133
  1226
    from iA have th0: "x \<notin> span (A - {x})"
huffman@44133
  1227
      unfolding dependent_def using x by blast
wenzelm@53406
  1228
    from x have xsA: "x \<in> span A"
wenzelm@53406
  1229
      by (blast intro: span_superset)
huffman@44133
  1230
    have "A - {x} \<subseteq> A" by blast
wenzelm@53406
  1231
    then have th1: "span (A - {x}) \<subseteq> span A"
wenzelm@53406
  1232
      by (metis span_mono)
wenzelm@53406
  1233
    {
wenzelm@53406
  1234
      assume xB: "x \<notin> B"
wenzelm@53406
  1235
      from xB BA have "B \<subseteq> A - {x}"
wenzelm@53406
  1236
        by blast
wenzelm@53406
  1237
      then have "span B \<subseteq> span (A - {x})"
wenzelm@53406
  1238
        by (metis span_mono)
wenzelm@53406
  1239
      with th1 th0 sAB have "x \<notin> span A"
wenzelm@53406
  1240
        by blast
wenzelm@53406
  1241
      with x have False
wenzelm@53406
  1242
        by (metis span_superset)
wenzelm@53406
  1243
    }
wenzelm@53406
  1244
    then have "x \<in> B" by blast
wenzelm@53406
  1245
  }
huffman@44133
  1246
  then show "A \<subseteq> B" by blast
huffman@44133
  1247
qed
huffman@44133
  1248
wenzelm@60420
  1249
text \<open>The general case of the Exchange Lemma, the key to what follows.\<close>
huffman@44133
  1250
huffman@44133
  1251
lemma exchange_lemma:
wenzelm@49711
  1252
  assumes f:"finite t"
wenzelm@49711
  1253
    and i: "independent s"
wenzelm@49711
  1254
    and sp: "s \<subseteq> span t"
wenzelm@53406
  1255
  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
  1256
  using f i sp
wenzelm@49522
  1257
proof (induct "card (t - s)" arbitrary: s t rule: less_induct)
huffman@44133
  1258
  case less
wenzelm@60420
  1259
  note ft = \<open>finite t\<close> and s = \<open>independent s\<close> and sp = \<open>s \<subseteq> span t\<close>
wenzelm@53406
  1260
  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
  1261
  let ?ths = "\<exists>t'. ?P t'"
wenzelm@53406
  1262
  {
lp15@55775
  1263
    assume "s \<subseteq> t"
lp15@55775
  1264
    then have ?ths
lp15@55775
  1265
      by (metis ft Un_commute sp sup_ge1)
wenzelm@53406
  1266
  }
huffman@44133
  1267
  moreover
wenzelm@53406
  1268
  {
wenzelm@53406
  1269
    assume st: "t \<subseteq> s"
wenzelm@53406
  1270
    from spanning_subset_independent[OF st s sp] st ft span_mono[OF st]
wenzelm@53406
  1271
    have ?ths
lp15@55775
  1272
      by (metis Un_absorb sp)
wenzelm@53406
  1273
  }
huffman@44133
  1274
  moreover
wenzelm@53406
  1275
  {
wenzelm@53406
  1276
    assume st: "\<not> s \<subseteq> t" "\<not> t \<subseteq> s"
wenzelm@53406
  1277
    from st(2) obtain b where b: "b \<in> t" "b \<notin> s"
wenzelm@53406
  1278
      by blast
wenzelm@53406
  1279
    from b have "t - {b} - s \<subset> t - s"
wenzelm@53406
  1280
      by blast
wenzelm@53406
  1281
    then have cardlt: "card (t - {b} - s) < card (t - s)"
wenzelm@53406
  1282
      using ft by (auto intro: psubset_card_mono)
wenzelm@53406
  1283
    from b ft have ct0: "card t \<noteq> 0"
wenzelm@53406
  1284
      by auto
wenzelm@53406
  1285
    have ?ths
wenzelm@53406
  1286
    proof cases
wenzelm@53716
  1287
      assume stb: "s \<subseteq> span (t - {b})"
wenzelm@53716
  1288
      from ft have ftb: "finite (t - {b})"
wenzelm@53406
  1289
        by auto
huffman@44133
  1290
      from less(1)[OF cardlt ftb s stb]
wenzelm@53716
  1291
      obtain u where u: "card u = card (t - {b})" "s \<subseteq> u" "u \<subseteq> s \<union> (t - {b})" "s \<subseteq> span u"
wenzelm@49522
  1292
        and fu: "finite u" by blast
huffman@44133
  1293
      let ?w = "insert b u"
wenzelm@53406
  1294
      have th0: "s \<subseteq> insert b u"
wenzelm@53406
  1295
        using u by blast
wenzelm@53406
  1296
      from u(3) b have "u \<subseteq> s \<union> t"
wenzelm@53406
  1297
        by blast
wenzelm@53406
  1298
      then have th1: "insert b u \<subseteq> s \<union> t"
wenzelm@53406
  1299
        using u b by blast
wenzelm@53406
  1300
      have bu: "b \<notin> u"
wenzelm@53406
  1301
        using b u by blast
wenzelm@53406
  1302
      from u(1) ft b have "card u = (card t - 1)"
wenzelm@53406
  1303
        by auto
wenzelm@49522
  1304
      then have th2: "card (insert b u) = card t"
huffman@44133
  1305
        using card_insert_disjoint[OF fu bu] ct0 by auto
huffman@44133
  1306
      from u(4) have "s \<subseteq> span u" .
wenzelm@53406
  1307
      also have "\<dots> \<subseteq> span (insert b u)"
wenzelm@53406
  1308
        by (rule span_mono) blast
huffman@44133
  1309
      finally have th3: "s \<subseteq> span (insert b u)" .
wenzelm@53406
  1310
      from th0 th1 th2 th3 fu have th: "?P ?w"
wenzelm@53406
  1311
        by blast
wenzelm@53406
  1312
      from th show ?thesis by blast
wenzelm@53406
  1313
    next
wenzelm@53716
  1314
      assume stb: "\<not> s \<subseteq> span (t - {b})"
wenzelm@53406
  1315
      from stb obtain a where a: "a \<in> s" "a \<notin> span (t - {b})"
wenzelm@53406
  1316
        by blast
wenzelm@53406
  1317
      have ab: "a \<noteq> b"
wenzelm@53406
  1318
        using a b by blast
wenzelm@53406
  1319
      have at: "a \<notin> t"
wenzelm@53406
  1320
        using a ab span_superset[of a "t- {b}"] by auto
huffman@44133
  1321
      have mlt: "card ((insert a (t - {b})) - s) < card (t - s)"
huffman@44133
  1322
        using cardlt ft a b by auto
wenzelm@53406
  1323
      have ft': "finite (insert a (t - {b}))"
wenzelm@53406
  1324
        using ft by auto
wenzelm@53406
  1325
      {
wenzelm@53406
  1326
        fix x
wenzelm@53406
  1327
        assume xs: "x \<in> s"
wenzelm@53406
  1328
        have t: "t \<subseteq> insert b (insert a (t - {b}))"
wenzelm@53406
  1329
          using b by auto
wenzelm@53406
  1330
        from b(1) have "b \<in> span t"
wenzelm@53406
  1331
          by (simp add: span_superset)
wenzelm@53406
  1332
        have bs: "b \<in> span (insert a (t - {b}))"
wenzelm@53406
  1333
          apply (rule in_span_delete)
wenzelm@53406
  1334
          using a sp unfolding subset_eq
wenzelm@53406
  1335
          apply auto
wenzelm@53406
  1336
          done
wenzelm@53406
  1337
        from xs sp have "x \<in> span t"
wenzelm@53406
  1338
          by blast
wenzelm@53406
  1339
        with span_mono[OF t] have x: "x \<in> span (insert b (insert a (t - {b})))" ..
wenzelm@53406
  1340
        from span_trans[OF bs x] have "x \<in> span (insert a (t - {b}))" .
wenzelm@53406
  1341
      }
wenzelm@53406
  1342
      then have sp': "s \<subseteq> span (insert a (t - {b}))"
wenzelm@53406
  1343
        by blast
wenzelm@53406
  1344
      from less(1)[OF mlt ft' s sp'] obtain u where u:
wenzelm@53716
  1345
        "card u = card (insert a (t - {b}))"
wenzelm@53716
  1346
        "finite u" "s \<subseteq> u" "u \<subseteq> s \<union> insert a (t - {b})"
wenzelm@53406
  1347
        "s \<subseteq> span u" by blast
wenzelm@53406
  1348
      from u a b ft at ct0 have "?P u"
wenzelm@53406
  1349
        by auto
wenzelm@53406
  1350
      then show ?thesis by blast
wenzelm@53406
  1351
    qed
huffman@44133
  1352
  }
wenzelm@49522
  1353
  ultimately show ?ths by blast
huffman@44133
  1354
qed
huffman@44133
  1355
wenzelm@60420
  1356
text \<open>This implies corresponding size bounds.\<close>
huffman@44133
  1357
huffman@44133
  1358
lemma independent_span_bound:
wenzelm@53406
  1359
  assumes f: "finite t"
wenzelm@53406
  1360
    and i: "independent s"
wenzelm@53406
  1361
    and sp: "s \<subseteq> span t"
huffman@44133
  1362
  shows "finite s \<and> card s \<le> card t"
huffman@44133
  1363
  by (metis exchange_lemma[OF f i sp] finite_subset card_mono)
huffman@44133
  1364
huffman@44133
  1365
lemma finite_Atleast_Atmost_nat[simp]: "finite {f x |x. x\<in> (UNIV::'a::finite set)}"
wenzelm@49522
  1366
proof -
wenzelm@53406
  1367
  have eq: "{f x |x. x\<in> UNIV} = f ` UNIV"
wenzelm@53406
  1368
    by auto
huffman@44133
  1369
  show ?thesis unfolding eq
huffman@44133
  1370
    apply (rule finite_imageI)
huffman@44133
  1371
    apply (rule finite)
huffman@44133
  1372
    done
huffman@44133
  1373
qed
huffman@44133
  1374
wenzelm@53406
  1375
wenzelm@60420
  1376
subsection \<open>Euclidean Spaces as Typeclass\<close>
huffman@44133
  1377
hoelzl@50526
  1378
lemma independent_Basis: "independent Basis"
hoelzl@50526
  1379
  unfolding dependent_def
hoelzl@50526
  1380
  apply (subst span_finite)
hoelzl@50526
  1381
  apply simp
huffman@44133
  1382
  apply clarify
hoelzl@50526
  1383
  apply (drule_tac f="inner a" in arg_cong)
hoelzl@50526
  1384
  apply (simp add: inner_Basis inner_setsum_right eq_commute)
hoelzl@50526
  1385
  done
hoelzl@50526
  1386
huffman@53939
  1387
lemma span_Basis [simp]: "span Basis = UNIV"
huffman@53939
  1388
  unfolding span_finite [OF finite_Basis]
huffman@53939
  1389
  by (fast intro: euclidean_representation)
huffman@44133
  1390
hoelzl@50526
  1391
lemma in_span_Basis: "x \<in> span Basis"
hoelzl@50526
  1392
  unfolding span_Basis ..
hoelzl@50526
  1393
hoelzl@50526
  1394
lemma Basis_le_norm: "b \<in> Basis \<Longrightarrow> \<bar>x \<bullet> b\<bar> \<le> norm x"
hoelzl@50526
  1395
  by (rule order_trans [OF Cauchy_Schwarz_ineq2]) simp
hoelzl@50526
  1396
hoelzl@50526
  1397
lemma norm_bound_Basis_le: "b \<in> Basis \<Longrightarrow> norm x \<le> e \<Longrightarrow> \<bar>x \<bullet> b\<bar> \<le> e"
hoelzl@50526
  1398
  by (metis Basis_le_norm order_trans)
hoelzl@50526
  1399
hoelzl@50526
  1400
lemma norm_bound_Basis_lt: "b \<in> Basis \<Longrightarrow> norm x < e \<Longrightarrow> \<bar>x \<bullet> b\<bar> < e"
huffman@53595
  1401
  by (metis Basis_le_norm le_less_trans)
hoelzl@50526
  1402
hoelzl@50526
  1403
lemma norm_le_l1: "norm x \<le> (\<Sum>b\<in>Basis. \<bar>x \<bullet> b\<bar>)"
hoelzl@50526
  1404
  apply (subst euclidean_representation[of x, symmetric])
huffman@44176
  1405
  apply (rule order_trans[OF norm_setsum])
wenzelm@49522
  1406
  apply (auto intro!: setsum_mono)
wenzelm@49522
  1407
  done
huffman@44133
  1408
huffman@44133
  1409
lemma setsum_norm_allsubsets_bound:
wenzelm@56444
  1410
  fixes f :: "'a \<Rightarrow> 'n::euclidean_space"
wenzelm@53406
  1411
  assumes fP: "finite P"
wenzelm@53406
  1412
    and fPs: "\<And>Q. Q \<subseteq> P \<Longrightarrow> norm (setsum f Q) \<le> e"
hoelzl@50526
  1413
  shows "(\<Sum>x\<in>P. norm (f x)) \<le> 2 * real DIM('n) * e"
wenzelm@49522
  1414
proof -
hoelzl@50526
  1415
  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
  1416
    by (rule setsum_mono) (rule norm_le_l1)
hoelzl@50526
  1417
  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
  1418
    by (rule setsum.commute)
hoelzl@50526
  1419
  also have "\<dots> \<le> of_nat (card (Basis :: 'n set)) * (2 * e)"
wenzelm@49522
  1420
  proof (rule setsum_bounded)
wenzelm@53406
  1421
    fix i :: 'n
wenzelm@53406
  1422
    assume i: "i \<in> Basis"
wenzelm@53406
  1423
    have "norm (\<Sum>x\<in>P. \<bar>f x \<bullet> i\<bar>) \<le>
hoelzl@50526
  1424
      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
  1425
      by (simp add: abs_real_def setsum.If_cases[OF fP] setsum_negf norm_triangle_ineq4 inner_setsum_left
wenzelm@56444
  1426
        del: real_norm_def)
wenzelm@53406
  1427
    also have "\<dots> \<le> e + e"
wenzelm@53406
  1428
      unfolding real_norm_def
hoelzl@50526
  1429
      by (intro add_mono norm_bound_Basis_le i fPs) auto
hoelzl@50526
  1430
    finally show "(\<Sum>x\<in>P. \<bar>f x \<bullet> i\<bar>) \<le> 2*e" by simp
huffman@44133
  1431
  qed
hoelzl@50526
  1432
  also have "\<dots> = 2 * real DIM('n) * e"
hoelzl@50526
  1433
    by (simp add: real_of_nat_def)
huffman@44133
  1434
  finally show ?thesis .
huffman@44133
  1435
qed
huffman@44133
  1436
wenzelm@53406
  1437
wenzelm@60420
  1438
subsection \<open>Linearity and Bilinearity continued\<close>
huffman@44133
  1439
huffman@44133
  1440
lemma linear_bounded:
wenzelm@56444
  1441
  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
huffman@44133
  1442
  assumes lf: "linear f"
huffman@44133
  1443
  shows "\<exists>B. \<forall>x. norm (f x) \<le> B * norm x"
huffman@53939
  1444
proof
hoelzl@50526
  1445
  let ?B = "\<Sum>b\<in>Basis. norm (f b)"
huffman@53939
  1446
  show "\<forall>x. norm (f x) \<le> ?B * norm x"
huffman@53939
  1447
  proof
wenzelm@53406
  1448
    fix x :: 'a
hoelzl@50526
  1449
    let ?g = "\<lambda>b. (x \<bullet> b) *\<^sub>R f b"
hoelzl@50526
  1450
    have "norm (f x) = norm (f (\<Sum>b\<in>Basis. (x \<bullet> b) *\<^sub>R b))"
hoelzl@50526
  1451
      unfolding euclidean_representation ..
hoelzl@50526
  1452
    also have "\<dots> = norm (setsum ?g Basis)"
huffman@53939
  1453
      by (simp add: linear_setsum [OF lf] linear_cmul [OF lf])
hoelzl@50526
  1454
    finally have th0: "norm (f x) = norm (setsum ?g Basis)" .
huffman@53939
  1455
    have th: "\<forall>b\<in>Basis. norm (?g b) \<le> norm (f b) * norm x"
huffman@53939
  1456
    proof
wenzelm@53406
  1457
      fix i :: 'a
wenzelm@53406
  1458
      assume i: "i \<in> Basis"
hoelzl@50526
  1459
      from Basis_le_norm[OF i, of x]
huffman@53939
  1460
      show "norm (?g i) \<le> norm (f i) * norm x"
wenzelm@49663
  1461
        unfolding norm_scaleR
haftmann@57512
  1462
        apply (subst mult.commute)
wenzelm@49663
  1463
        apply (rule mult_mono)
wenzelm@49663
  1464
        apply (auto simp add: field_simps)
wenzelm@53406
  1465
        done
huffman@53939
  1466
    qed
hoelzl@50526
  1467
    from setsum_norm_le[of _ ?g, OF th]
huffman@53939
  1468
    show "norm (f x) \<le> ?B * norm x"
wenzelm@53406
  1469
      unfolding th0 setsum_left_distrib by metis
huffman@53939
  1470
  qed
huffman@44133
  1471
qed
huffman@44133
  1472
huffman@44133
  1473
lemma linear_conv_bounded_linear:
huffman@44133
  1474
  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
huffman@44133
  1475
  shows "linear f \<longleftrightarrow> bounded_linear f"
huffman@44133
  1476
proof
huffman@44133
  1477
  assume "linear f"
huffman@53939
  1478
  then interpret f: linear f .
huffman@44133
  1479
  show "bounded_linear f"
huffman@44133
  1480
  proof
huffman@44133
  1481
    have "\<exists>B. \<forall>x. norm (f x) \<le> B * norm x"
wenzelm@60420
  1482
      using \<open>linear f\<close> by (rule linear_bounded)
wenzelm@49522
  1483
    then show "\<exists>K. \<forall>x. norm (f x) \<le> norm x * K"
haftmann@57512
  1484
      by (simp add: mult.commute)
huffman@44133
  1485
  qed
huffman@44133
  1486
next
huffman@44133
  1487
  assume "bounded_linear f"
huffman@44133
  1488
  then interpret f: bounded_linear f .
huffman@53939
  1489
  show "linear f" ..
huffman@53939
  1490
qed
huffman@53939
  1491
huffman@53939
  1492
lemma linear_bounded_pos:
wenzelm@56444
  1493
  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
huffman@53939
  1494
  assumes lf: "linear f"
huffman@53939
  1495
  shows "\<exists>B > 0. \<forall>x. norm (f x) \<le> B * norm x"
huffman@53939
  1496
proof -
huffman@53939
  1497
  have "\<exists>B > 0. \<forall>x. norm (f x) \<le> norm x * B"
huffman@53939
  1498
    using lf unfolding linear_conv_bounded_linear
huffman@53939
  1499
    by (rule bounded_linear.pos_bounded)
huffman@53939
  1500
  then show ?thesis
haftmann@57512
  1501
    by (simp only: mult.commute)
huffman@44133
  1502
qed
huffman@44133
  1503
wenzelm@49522
  1504
lemma bounded_linearI':
wenzelm@56444
  1505
  fixes f ::"'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
wenzelm@53406
  1506
  assumes "\<And>x y. f (x + y) = f x + f y"
wenzelm@53406
  1507
    and "\<And>c x. f (c *\<^sub>R x) = c *\<^sub>R f x"
wenzelm@49522
  1508
  shows "bounded_linear f"
wenzelm@53406
  1509
  unfolding linear_conv_bounded_linear[symmetric]
wenzelm@49522
  1510
  by (rule linearI[OF assms])
huffman@44133
  1511
huffman@44133
  1512
lemma bilinear_bounded:
wenzelm@56444
  1513
  fixes h :: "'m::euclidean_space \<Rightarrow> 'n::euclidean_space \<Rightarrow> 'k::real_normed_vector"
huffman@44133
  1514
  assumes bh: "bilinear h"
huffman@44133
  1515
  shows "\<exists>B. \<forall>x y. norm (h x y) \<le> B * norm x * norm y"
hoelzl@50526
  1516
proof (clarify intro!: exI[of _ "\<Sum>i\<in>Basis. \<Sum>j\<in>Basis. norm (h i j)"])
wenzelm@53406
  1517
  fix x :: 'm
wenzelm@53406
  1518
  fix y :: 'n
wenzelm@53406
  1519
  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
  1520
    apply (subst euclidean_representation[where 'a='m])
wenzelm@53406
  1521
    apply (subst euclidean_representation[where 'a='n])
hoelzl@50526
  1522
    apply rule
hoelzl@50526
  1523
    done
wenzelm@53406
  1524
  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
  1525
    unfolding bilinear_setsum[OF bh finite_Basis finite_Basis] ..
hoelzl@50526
  1526
  finally have th: "norm (h x y) = \<dots>" .
hoelzl@50526
  1527
  show "norm (h x y) \<le> (\<Sum>i\<in>Basis. \<Sum>j\<in>Basis. norm (h i j)) * norm x * norm y"
haftmann@57418
  1528
    apply (auto simp add: setsum_left_distrib th setsum.cartesian_product)
wenzelm@53406
  1529
    apply (rule setsum_norm_le)
wenzelm@53406
  1530
    apply simp
wenzelm@53406
  1531
    apply (auto simp add: bilinear_rmul[OF bh] bilinear_lmul[OF bh]
wenzelm@53406
  1532
      field_simps simp del: scaleR_scaleR)
wenzelm@53406
  1533
    apply (rule mult_mono)
wenzelm@53406
  1534
    apply (auto simp add: zero_le_mult_iff Basis_le_norm)
wenzelm@53406
  1535
    apply (rule mult_mono)
wenzelm@53406
  1536
    apply (auto simp add: zero_le_mult_iff Basis_le_norm)
wenzelm@53406
  1537
    done
huffman@44133
  1538
qed
huffman@44133
  1539
huffman@44133
  1540
lemma bilinear_conv_bounded_bilinear:
huffman@44133
  1541
  fixes h :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space \<Rightarrow> 'c::real_normed_vector"
huffman@44133
  1542
  shows "bilinear h \<longleftrightarrow> bounded_bilinear h"
huffman@44133
  1543
proof
huffman@44133
  1544
  assume "bilinear h"
huffman@44133
  1545
  show "bounded_bilinear h"
huffman@44133
  1546
  proof
wenzelm@53406
  1547
    fix x y z
wenzelm@53406
  1548
    show "h (x + y) z = h x z + h y z"
wenzelm@60420
  1549
      using \<open>bilinear h\<close> unfolding bilinear_def linear_iff by simp
huffman@44133
  1550
  next
wenzelm@53406
  1551
    fix x y z
wenzelm@53406
  1552
    show "h x (y + z) = h x y + h x z"
wenzelm@60420
  1553
      using \<open>bilinear h\<close> unfolding bilinear_def linear_iff by simp
huffman@44133
  1554
  next
wenzelm@53406
  1555
    fix r x y
wenzelm@53406
  1556
    show "h (scaleR r x) y = scaleR r (h x y)"
wenzelm@60420
  1557
      using \<open>bilinear h\<close> unfolding bilinear_def linear_iff
huffman@44133
  1558
      by simp
huffman@44133
  1559
  next
wenzelm@53406
  1560
    fix r x y
wenzelm@53406
  1561
    show "h x (scaleR r y) = scaleR r (h x y)"
wenzelm@60420
  1562
      using \<open>bilinear h\<close> unfolding bilinear_def linear_iff
huffman@44133
  1563
      by simp
huffman@44133
  1564
  next
huffman@44133
  1565
    have "\<exists>B. \<forall>x y. norm (h x y) \<le> B * norm x * norm y"
wenzelm@60420
  1566
      using \<open>bilinear h\<close> by (rule bilinear_bounded)
wenzelm@49522
  1567
    then show "\<exists>K. \<forall>x y. norm (h x y) \<le> norm x * norm y * K"
haftmann@57514
  1568
      by (simp add: ac_simps)
huffman@44133
  1569
  qed
huffman@44133
  1570
next
huffman@44133
  1571
  assume "bounded_bilinear h"
huffman@44133
  1572
  then interpret h: bounded_bilinear h .
huffman@44133
  1573
  show "bilinear h"
huffman@44133
  1574
    unfolding bilinear_def linear_conv_bounded_linear
wenzelm@49522
  1575
    using h.bounded_linear_left h.bounded_linear_right by simp
huffman@44133
  1576
qed
huffman@44133
  1577
huffman@53939
  1578
lemma bilinear_bounded_pos:
wenzelm@56444
  1579
  fixes h :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space \<Rightarrow> 'c::real_normed_vector"
huffman@53939
  1580
  assumes bh: "bilinear h"
huffman@53939
  1581
  shows "\<exists>B > 0. \<forall>x y. norm (h x y) \<le> B * norm x * norm y"
huffman@53939
  1582
proof -
huffman@53939
  1583
  have "\<exists>B > 0. \<forall>x y. norm (h x y) \<le> norm x * norm y * B"
huffman@53939
  1584
    using bh [unfolded bilinear_conv_bounded_bilinear]
huffman@53939
  1585
    by (rule bounded_bilinear.pos_bounded)
huffman@53939
  1586
  then show ?thesis
haftmann@57514
  1587
    by (simp only: ac_simps)
huffman@53939
  1588
qed
huffman@53939
  1589
wenzelm@49522
  1590
wenzelm@60420
  1591
subsection \<open>We continue.\<close>
huffman@44133
  1592
huffman@44133
  1593
lemma independent_bound:
wenzelm@53716
  1594
  fixes S :: "'a::euclidean_space set"
wenzelm@53716
  1595
  shows "independent S \<Longrightarrow> finite S \<and> card S \<le> DIM('a)"
hoelzl@50526
  1596
  using independent_span_bound[OF finite_Basis, of S] by auto
huffman@44133
  1597
paulson@60303
  1598
corollary 
paulson@60303
  1599
  fixes S :: "'a::euclidean_space set"
paulson@60303
  1600
  assumes "independent S"
paulson@60303
  1601
  shows independent_imp_finite: "finite S" and independent_card_le:"card S \<le> DIM('a)"
paulson@60303
  1602
using assms independent_bound by auto
paulson@60303
  1603
  
wenzelm@49663
  1604
lemma dependent_biggerset:
wenzelm@56444
  1605
  fixes S :: "'a::euclidean_space set"
wenzelm@56444
  1606
  shows "(finite S \<Longrightarrow> card S > DIM('a)) \<Longrightarrow> dependent S"
huffman@44133
  1607
  by (metis independent_bound not_less)
huffman@44133
  1608
wenzelm@60420
  1609
text \<open>Hence we can create a maximal independent subset.\<close>
huffman@44133
  1610
huffman@44133
  1611
lemma maximal_independent_subset_extend:
wenzelm@53406
  1612
  fixes S :: "'a::euclidean_space set"
wenzelm@53406
  1613
  assumes sv: "S \<subseteq> V"
wenzelm@49663
  1614
    and iS: "independent S"
huffman@44133
  1615
  shows "\<exists>B. S \<subseteq> B \<and> B \<subseteq> V \<and> independent B \<and> V \<subseteq> span B"
huffman@44133
  1616
  using sv iS
wenzelm@49522
  1617
proof (induct "DIM('a) - card S" arbitrary: S rule: less_induct)
huffman@44133
  1618
  case less
wenzelm@60420
  1619
  note sv = \<open>S \<subseteq> V\<close> and i = \<open>independent S\<close>
huffman@44133
  1620
  let ?P = "\<lambda>B. S \<subseteq> B \<and> B \<subseteq> V \<and> independent B \<and> V \<subseteq> span B"
huffman@44133
  1621
  let ?ths = "\<exists>x. ?P x"
huffman@44133
  1622
  let ?d = "DIM('a)"
wenzelm@53406
  1623
  show ?ths
wenzelm@53406
  1624
  proof (cases "V \<subseteq> span S")
wenzelm@53406
  1625
    case True
wenzelm@53406
  1626
    then show ?thesis
wenzelm@53406
  1627
      using sv i by blast
wenzelm@53406
  1628
  next
wenzelm@53406
  1629
    case False
wenzelm@53406
  1630
    then obtain a where a: "a \<in> V" "a \<notin> span S"
wenzelm@53406
  1631
      by blast
wenzelm@53406
  1632
    from a have aS: "a \<notin> S"
wenzelm@53406
  1633
      by (auto simp add: span_superset)
wenzelm@53406
  1634
    have th0: "insert a S \<subseteq> V"
wenzelm@53406
  1635
      using a sv by blast
huffman@44133
  1636
    from independent_insert[of a S]  i a
wenzelm@53406
  1637
    have th1: "independent (insert a S)"
wenzelm@53406
  1638
      by auto
huffman@44133
  1639
    have mlt: "?d - card (insert a S) < ?d - card S"
wenzelm@49522
  1640
      using aS a independent_bound[OF th1] by auto
huffman@44133
  1641
huffman@44133
  1642
    from less(1)[OF mlt th0 th1]
huffman@44133
  1643
    obtain B where B: "insert a S \<subseteq> B" "B \<subseteq> V" "independent B" " V \<subseteq> span B"
huffman@44133
  1644
      by blast
huffman@44133
  1645
    from B have "?P B" by auto
wenzelm@53406
  1646
    then show ?thesis by blast
wenzelm@53406
  1647
  qed
huffman@44133
  1648
qed
huffman@44133
  1649
huffman@44133
  1650
lemma maximal_independent_subset:
huffman@44133
  1651
  "\<exists>(B:: ('a::euclidean_space) set). B\<subseteq> V \<and> independent B \<and> V \<subseteq> span B"
wenzelm@49522
  1652
  by (metis maximal_independent_subset_extend[of "{}:: ('a::euclidean_space) set"]
wenzelm@49522
  1653
    empty_subsetI independent_empty)
huffman@44133
  1654
huffman@44133
  1655
wenzelm@60420
  1656
text \<open>Notion of dimension.\<close>
huffman@44133
  1657
wenzelm@53406
  1658
definition "dim V = (SOME n. \<exists>B. B \<subseteq> V \<and> independent B \<and> V \<subseteq> span B \<and> card B = n)"
huffman@44133
  1659
wenzelm@49522
  1660
lemma basis_exists:
wenzelm@49522
  1661
  "\<exists>B. (B :: ('a::euclidean_space) set) \<subseteq> V \<and> independent B \<and> V \<subseteq> span B \<and> (card B = dim V)"
wenzelm@49522
  1662
  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
  1663
  using maximal_independent_subset[of V] independent_bound
wenzelm@49522
  1664
  by auto
huffman@44133
  1665
lp15@60307
  1666
corollary dim_le_card:
lp15@60307
  1667
  fixes s :: "'a::euclidean_space set"
lp15@60307
  1668
  shows "finite s \<Longrightarrow> dim s \<le> card s"
lp15@60307
  1669
by (metis basis_exists card_mono)
lp15@60307
  1670
wenzelm@60420
  1671
text \<open>Consequences of independence or spanning for cardinality.\<close>
huffman@44133
  1672
wenzelm@53406
  1673
lemma independent_card_le_dim:
wenzelm@53406
  1674
  fixes B :: "'a::euclidean_space set"
wenzelm@53406
  1675
  assumes "B \<subseteq> V"
wenzelm@53406
  1676
    and "independent B"
wenzelm@49522
  1677
  shows "card B \<le> dim V"
huffman@44133
  1678
proof -
wenzelm@60420
  1679
  from basis_exists[of V] \<open>B \<subseteq> V\<close>
wenzelm@53406
  1680
  obtain B' where "independent B'"
wenzelm@53406
  1681
    and "B \<subseteq> span B'"
wenzelm@53406
  1682
    and "card B' = dim V"
wenzelm@53406
  1683
    by blast
wenzelm@60420
  1684
  with independent_span_bound[OF _ \<open>independent B\<close> \<open>B \<subseteq> span B'\<close>] independent_bound[of B']
huffman@44133
  1685
  show ?thesis by auto
huffman@44133
  1686
qed
huffman@44133
  1687
wenzelm@49522
  1688
lemma span_card_ge_dim:
wenzelm@53406
  1689
  fixes B :: "'a::euclidean_space set"
wenzelm@53406
  1690
  shows "B \<subseteq> V \<Longrightarrow> V \<subseteq> span B \<Longrightarrow> finite B \<Longrightarrow> dim V \<le> card B"
huffman@44133
  1691
  by (metis basis_exists[of V] independent_span_bound subset_trans)
huffman@44133
  1692
huffman@44133
  1693
lemma basis_card_eq_dim:
wenzelm@53406
  1694
  fixes V :: "'a::euclidean_space set"
wenzelm@53406
  1695
  shows "B \<subseteq> V \<Longrightarrow> V \<subseteq> span B \<Longrightarrow> independent B \<Longrightarrow> finite B \<and> card B = dim V"
huffman@44133
  1696
  by (metis order_eq_iff independent_card_le_dim span_card_ge_dim independent_bound)
huffman@44133
  1697
wenzelm@53406
  1698
lemma dim_unique:
wenzelm@53406
  1699
  fixes B :: "'a::euclidean_space set"
wenzelm@53406
  1700
  shows "B \<subseteq> V \<Longrightarrow> V \<subseteq> span B \<Longrightarrow> independent B \<Longrightarrow> card B = n \<Longrightarrow> dim V = n"
huffman@44133
  1701
  by (metis basis_card_eq_dim)
huffman@44133
  1702
wenzelm@60420
  1703
text \<open>More lemmas about dimension.\<close>
huffman@44133
  1704
wenzelm@53406
  1705
lemma dim_UNIV: "dim (UNIV :: 'a::euclidean_space set) = DIM('a)"
hoelzl@50526
  1706
  using independent_Basis
hoelzl@50526
  1707
  by (intro dim_unique[of Basis]) auto
huffman@44133
  1708
huffman@44133
  1709
lemma dim_subset:
wenzelm@53406
  1710
  fixes S :: "'a::euclidean_space set"
wenzelm@53406
  1711
  shows "S \<subseteq> T \<Longrightarrow> dim S \<le> dim T"
huffman@44133
  1712
  using basis_exists[of T] basis_exists[of S]
huffman@44133
  1713
  by (metis independent_card_le_dim subset_trans)
huffman@44133
  1714
wenzelm@53406
  1715
lemma dim_subset_UNIV:
wenzelm@53406
  1716
  fixes S :: "'a::euclidean_space set"
wenzelm@53406
  1717
  shows "dim S \<le> DIM('a)"
huffman@44133
  1718
  by (metis dim_subset subset_UNIV dim_UNIV)
huffman@44133
  1719
wenzelm@60420
  1720
text \<open>Converses to those.\<close>
huffman@44133
  1721
huffman@44133
  1722
lemma card_ge_dim_independent:
wenzelm@53406
  1723
  fixes B :: "'a::euclidean_space set"
wenzelm@53406
  1724
  assumes BV: "B \<subseteq> V"
wenzelm@53406
  1725
    and iB: "independent B"
wenzelm@53406
  1726
    and dVB: "dim V \<le> card B"
huffman@44133
  1727
  shows "V \<subseteq> span B"
wenzelm@53406
  1728
proof
wenzelm@53406
  1729
  fix a
wenzelm@53406
  1730
  assume aV: "a \<in> V"
wenzelm@53406
  1731
  {
wenzelm@53406
  1732
    assume aB: "a \<notin> span B"
wenzelm@53406
  1733
    then have iaB: "independent (insert a B)"
wenzelm@53406
  1734
      using iB aV BV by (simp add: independent_insert)
wenzelm@53406
  1735
    from aV BV have th0: "insert a B \<subseteq> V"
wenzelm@53406
  1736
      by blast
wenzelm@53406
  1737
    from aB have "a \<notin>B"
wenzelm@53406
  1738
      by (auto simp add: span_superset)
wenzelm@53406
  1739
    with independent_card_le_dim[OF th0 iaB] dVB independent_bound[OF iB]
wenzelm@53406
  1740
    have False by auto
wenzelm@53406
  1741
  }
wenzelm@53406
  1742
  then show "a \<in> span B" by blast
huffman@44133
  1743
qed
huffman@44133
  1744
huffman@44133
  1745
lemma card_le_dim_spanning:
wenzelm@49663
  1746
  assumes BV: "(B:: ('a::euclidean_space) set) \<subseteq> V"
wenzelm@49663
  1747
    and VB: "V \<subseteq> span B"
wenzelm@49663
  1748
    and fB: "finite B"
wenzelm@49663
  1749
    and dVB: "dim V \<ge> card B"
huffman@44133
  1750
  shows "independent B"
wenzelm@49522
  1751
proof -
wenzelm@53406
  1752
  {
wenzelm@53406
  1753
    fix a
wenzelm@53716
  1754
    assume a: "a \<in> B" "a \<in> span (B - {a})"
wenzelm@53406
  1755
    from a fB have c0: "card B \<noteq> 0"
wenzelm@53406
  1756
      by auto
wenzelm@53716
  1757
    from a fB have cb: "card (B - {a}) = card B - 1"
wenzelm@53406
  1758
      by auto
wenzelm@53716
  1759
    from BV a have th0: "B - {a} \<subseteq> V"
wenzelm@53406
  1760
      by blast
wenzelm@53406
  1761
    {
wenzelm@53406
  1762
      fix x
wenzelm@53406
  1763
      assume x: "x \<in> V"
wenzelm@53716
  1764
      from a have eq: "insert a (B - {a}) = B"
wenzelm@53406
  1765
        by blast
wenzelm@53406
  1766
      from x VB have x': "x \<in> span B"
wenzelm@53406
  1767
        by blast
huffman@44133
  1768
      from span_trans[OF a(2), unfolded eq, OF x']
wenzelm@53716
  1769
      have "x \<in> span (B - {a})" .
wenzelm@53406
  1770
    }
wenzelm@53716
  1771
    then have th1: "V \<subseteq> span (B - {a})"
wenzelm@53406
  1772
      by blast
wenzelm@53716
  1773
    have th2: "finite (B - {a})"
wenzelm@53406
  1774
      using fB by auto
huffman@44133
  1775
    from span_card_ge_dim[OF th0 th1 th2]
wenzelm@53716
  1776
    have c: "dim V \<le> card (B - {a})" .
wenzelm@53406
  1777
    from c c0 dVB cb have False by simp
wenzelm@53406
  1778
  }
wenzelm@53406
  1779
  then show ?thesis
wenzelm@53406
  1780
    unfolding dependent_def by blast
huffman@44133
  1781
qed
huffman@44133
  1782
wenzelm@53406
  1783
lemma card_eq_dim:
wenzelm@53406
  1784
  fixes B :: "'a::euclidean_space set"
wenzelm@53406
  1785
  shows "B \<subseteq> V \<Longrightarrow> card B = dim V \<Longrightarrow> finite B \<Longrightarrow> independent B \<longleftrightarrow> V \<subseteq> span B"
wenzelm@49522
  1786
  by (metis order_eq_iff card_le_dim_spanning card_ge_dim_independent)
huffman@44133
  1787
wenzelm@60420
  1788
text \<open>More general size bound lemmas.\<close>
huffman@44133
  1789
huffman@44133
  1790
lemma independent_bound_general:
wenzelm@53406
  1791
  fixes S :: "'a::euclidean_space set"
wenzelm@53406
  1792
  shows "independent S \<Longrightarrow> finite S \<and> card S \<le> dim S"
huffman@44133
  1793
  by (metis independent_card_le_dim independent_bound subset_refl)
huffman@44133
  1794
wenzelm@49522
  1795
lemma dependent_biggerset_general:
wenzelm@53406
  1796
  fixes S :: "'a::euclidean_space set"
wenzelm@53406
  1797
  shows "(finite S \<Longrightarrow> card S > dim S) \<Longrightarrow> dependent S"
huffman@44133
  1798
  using independent_bound_general[of S] by (metis linorder_not_le)
huffman@44133
  1799
paulson@60303
  1800
lemma dim_span [simp]:
wenzelm@53406
  1801
  fixes S :: "'a::euclidean_space set"
wenzelm@53406
  1802
  shows "dim (span S) = dim S"
wenzelm@49522
  1803
proof -
huffman@44133
  1804
  have th0: "dim S \<le> dim (span S)"
huffman@44133
  1805
    by (auto simp add: subset_eq intro: dim_subset span_superset)
huffman@44133
  1806
  from basis_exists[of S]
wenzelm@53406
  1807
  obtain B where B: "B \<subseteq> S" "independent B" "S \<subseteq> span B" "card B = dim S"
wenzelm@53406
  1808
    by blast
wenzelm@53406
  1809
  from B have fB: "finite B" "card B = dim S"
wenzelm@53406
  1810
    using independent_bound by blast+
wenzelm@53406
  1811
  have bSS: "B \<subseteq> span S"
wenzelm@53406
  1812
    using B(1) by (metis subset_eq span_inc)
wenzelm@53406
  1813
  have sssB: "span S \<subseteq> span B"
wenzelm@53406
  1814
    using span_mono[OF B(3)] by (simp add: span_span)
huffman@44133
  1815
  from span_card_ge_dim[OF bSS sssB fB(1)] th0 show ?thesis
wenzelm@49522
  1816
    using fB(2) by arith
huffman@44133
  1817
qed
huffman@44133
  1818
wenzelm@53406
  1819
lemma subset_le_dim:
wenzelm@53406
  1820
  fixes S :: "'a::euclidean_space set"
wenzelm@53406
  1821
  shows "S \<subseteq> span T \<Longrightarrow> dim S \<le> dim T"
huffman@44133
  1822
  by (metis dim_span dim_subset)
huffman@44133
  1823
wenzelm@53406
  1824
lemma span_eq_dim:
wenzelm@56444
  1825
  fixes S :: "'a::euclidean_space set"
wenzelm@53406
  1826
  shows "span S = span T \<Longrightarrow> dim S = dim T"
huffman@44133
  1827
  by (metis dim_span)
huffman@44133
  1828
huffman@44133
  1829
lemma spans_image:
wenzelm@49663
  1830
  assumes lf: "linear f"
wenzelm@49663
  1831
    and VB: "V \<subseteq> span B"
huffman@44133
  1832
  shows "f ` V \<subseteq> span (f ` B)"
wenzelm@49522
  1833
  unfolding span_linear_image[OF lf] by (metis VB image_mono)
huffman@44133
  1834
huffman@44133
  1835
lemma dim_image_le:
huffman@44133
  1836
  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
wenzelm@49663
  1837
  assumes lf: "linear f"
wenzelm@49663
  1838
  shows "dim (f ` S) \<le> dim (S)"
wenzelm@49522
  1839
proof -
huffman@44133
  1840
  from basis_exists[of S] obtain B where
huffman@44133
  1841
    B: "B \<subseteq> S" "independent B" "S \<subseteq> span B" "card B = dim S" by blast
wenzelm@53406
  1842
  from B have fB: "finite B" "card B = dim S"
wenzelm@53406
  1843
    using independent_bound by blast+
huffman@44133
  1844
  have "dim (f ` S) \<le> card (f ` B)"
huffman@44133
  1845
    apply (rule span_card_ge_dim)
wenzelm@53406
  1846
    using lf B fB
wenzelm@53406
  1847
    apply (auto simp add: span_linear_image spans_image subset_image_iff)
wenzelm@49522
  1848
    done
wenzelm@53406
  1849
  also have "\<dots> \<le> dim S"
wenzelm@53406
  1850
    using card_image_le[OF fB(1)] fB by simp
huffman@44133
  1851
  finally show ?thesis .
huffman@44133
  1852
qed
huffman@44133
  1853
wenzelm@60420
  1854
text \<open>Relation between bases and injectivity/surjectivity of map.\<close>
huffman@44133
  1855
huffman@44133
  1856
lemma spanning_surjective_image:
huffman@44133
  1857
  assumes us: "UNIV \<subseteq> span S"
wenzelm@53406
  1858
    and lf: "linear f"
wenzelm@53406
  1859
    and sf: "surj f"
huffman@44133
  1860
  shows "UNIV \<subseteq> span (f ` S)"
wenzelm@49663
  1861
proof -
wenzelm@53406
  1862
  have "UNIV \<subseteq> f ` UNIV"
wenzelm@53406
  1863
    using sf by (auto simp add: surj_def)
wenzelm@53406
  1864
  also have " \<dots> \<subseteq> span (f ` S)"
wenzelm@53406
  1865
    using spans_image[OF lf us] .
wenzelm@53406
  1866
  finally show ?thesis .
huffman@44133
  1867
qed
huffman@44133
  1868
huffman@44133
  1869
lemma independent_injective_image:
wenzelm@49663
  1870
  assumes iS: "independent S"
wenzelm@49663
  1871
    and lf: "linear f"
wenzelm@49663
  1872
    and fi: "inj f"
huffman@44133
  1873
  shows "independent (f ` S)"
wenzelm@49663
  1874
proof -
wenzelm@53406
  1875
  {
wenzelm@53406
  1876
    fix a
wenzelm@49663
  1877
    assume a: "a \<in> S" "f a \<in> span (f ` S - {f a})"
wenzelm@53406
  1878
    have eq: "f ` S - {f a} = f ` (S - {a})"
wenzelm@53406
  1879
      using fi by (auto simp add: inj_on_def)
wenzelm@53716
  1880
    from a have "f a \<in> f ` span (S - {a})"
wenzelm@53406
  1881
      unfolding eq span_linear_image[OF lf, of "S - {a}"] by blast
wenzelm@53716
  1882
    then have "a \<in> span (S - {a})"
wenzelm@53406
  1883
      using fi by (auto simp add: inj_on_def)
wenzelm@53406
  1884
    with a(1) iS have False
wenzelm@53406
  1885
      by (simp add: dependent_def)
wenzelm@53406
  1886
  }
wenzelm@53406
  1887
  then show ?thesis
wenzelm@53406
  1888
    unfolding dependent_def by blast
huffman@44133
  1889
qed
huffman@44133
  1890
wenzelm@60420
  1891
text \<open>Picking an orthogonal replacement for a spanning set.\<close>
huffman@44133
  1892
wenzelm@53406
  1893
(* FIXME : Move to some general theory ?*)
huffman@44133
  1894
definition "pairwise R S \<longleftrightarrow> (\<forall>x \<in> S. \<forall>y\<in> S. x\<noteq>y \<longrightarrow> R x y)"
huffman@44133
  1895
wenzelm@53406
  1896
lemma vector_sub_project_orthogonal:
wenzelm@53406
  1897
  fixes b x :: "'a::euclidean_space"
wenzelm@53406
  1898
  shows "b \<bullet> (x - ((b \<bullet> x) / (b \<bullet> b)) *\<^sub>R b) = 0"
huffman@44133
  1899
  unfolding inner_simps by auto
huffman@44133
  1900
huffman@44528
  1901
lemma pairwise_orthogonal_insert:
huffman@44528
  1902
  assumes "pairwise orthogonal S"
wenzelm@49522
  1903
    and "\<And>y. y \<in> S \<Longrightarrow> orthogonal x y"
huffman@44528
  1904
  shows "pairwise orthogonal (insert x S)"
huffman@44528
  1905
  using assms unfolding pairwise_def
huffman@44528
  1906
  by (auto simp add: orthogonal_commute)
huffman@44528
  1907
huffman@44133
  1908
lemma basis_orthogonal:
wenzelm@53406
  1909
  fixes B :: "'a::real_inner set"
huffman@44133
  1910
  assumes fB: "finite B"
huffman@44133
  1911
  shows "\<exists>C. finite C \<and> card C \<le> card B \<and> span C = span B \<and> pairwise orthogonal C"
huffman@44133
  1912
  (is " \<exists>C. ?P B C")
wenzelm@49522
  1913
  using fB
wenzelm@49522
  1914
proof (induct rule: finite_induct)
wenzelm@49522
  1915
  case empty
wenzelm@53406
  1916
  then show ?case
wenzelm@53406
  1917
    apply (rule exI[where x="{}"])
wenzelm@53406
  1918
    apply (auto simp add: pairwise_def)
wenzelm@53406
  1919
    done
huffman@44133
  1920
next
wenzelm@49522
  1921
  case (insert a B)
wenzelm@60420
  1922
  note fB = \<open>finite B\<close> and aB = \<open>a \<notin> B\<close>
wenzelm@60420
  1923
  from \<open>\<exists>C. finite C \<and> card C \<le> card B \<and> span C = span B \<and> pairwise orthogonal C\<close>
huffman@44133
  1924
  obtain C where C: "finite C" "card C \<le> card B"
huffman@44133
  1925
    "span C = span B" "pairwise orthogonal C" by blast
huffman@44133
  1926
  let ?a = "a - setsum (\<lambda>x. (x \<bullet> a / (x \<bullet> x)) *\<^sub>R x) C"
huffman@44133
  1927
  let ?C = "insert ?a C"
wenzelm@53406
  1928
  from C(1) have fC: "finite ?C"
wenzelm@53406
  1929
    by simp
wenzelm@49522
  1930
  from fB aB C(1,2) have cC: "card ?C \<le> card (insert a B)"
wenzelm@49522
  1931
    by (simp add: card_insert_if)
wenzelm@53406
  1932
  {
wenzelm@53406
  1933
    fix x k
wenzelm@49522
  1934
    have th0: "\<And>(a::'a) b c. a - (b - c) = c + (a - b)"
wenzelm@49522
  1935
      by (simp add: field_simps)
huffman@44133
  1936
    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
  1937
      apply (simp only: scaleR_right_diff_distrib th0)
huffman@44133
  1938
      apply (rule span_add_eq)
huffman@44133
  1939
      apply (rule span_mul)
huffman@56196
  1940
      apply (rule span_setsum)
huffman@44133
  1941
      apply clarify
huffman@44133
  1942
      apply (rule span_mul)
wenzelm@49522
  1943
      apply (rule span_superset)
wenzelm@49522
  1944
      apply assumption
wenzelm@53406
  1945
      done
wenzelm@53406
  1946
  }
huffman@44133
  1947
  then have SC: "span ?C = span (insert a B)"
huffman@44133
  1948
    unfolding set_eq_iff span_breakdown_eq C(3)[symmetric] by auto
wenzelm@53406
  1949
  {
wenzelm@53406
  1950
    fix y
wenzelm@53406
  1951
    assume yC: "y \<in> C"
wenzelm@53406
  1952
    then have Cy: "C = insert y (C - {y})"
wenzelm@53406
  1953
      by blast
wenzelm@53406
  1954
    have fth: "finite (C - {y})"
wenzelm@53406
  1955
      using C by simp
huffman@44528
  1956
    have "orthogonal ?a y"
huffman@44528
  1957
      unfolding orthogonal_def
haftmann@54230
  1958
      unfolding inner_diff inner_setsum_left right_minus_eq
wenzelm@60420
  1959
      unfolding setsum.remove [OF \<open>finite C\<close> \<open>y \<in> C\<close>]
huffman@44528
  1960
      apply (clarsimp simp add: inner_commute[of y a])
haftmann@57418
  1961
      apply (rule setsum.neutral)
huffman@44528
  1962
      apply clarsimp
huffman@44528
  1963
      apply (rule C(4)[unfolded pairwise_def orthogonal_def, rule_format])
wenzelm@60420
  1964
      using \<open>y \<in> C\<close> by auto
wenzelm@53406
  1965
  }
wenzelm@60420
  1966
  with \<open>pairwise orthogonal C\<close> have CPO: "pairwise orthogonal ?C"
huffman@44528
  1967
    by (rule pairwise_orthogonal_insert)
wenzelm@53406
  1968
  from fC cC SC CPO have "?P (insert a B) ?C"
wenzelm@53406
  1969
    by blast
huffman@44133
  1970
  then show ?case by blast
huffman@44133
  1971
qed
huffman@44133
  1972
huffman@44133
  1973
lemma orthogonal_basis_exists:
huffman@44133
  1974
  fixes V :: "('a::euclidean_space) set"
huffman@44133
  1975
  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
  1976
proof -
wenzelm@49522
  1977
  from basis_exists[of V] obtain B where
wenzelm@53406
  1978
    B: "B \<subseteq> V" "independent B" "V \<subseteq> span B" "card B = dim V"
wenzelm@53406
  1979
    by blast
wenzelm@53406
  1980
  from B have fB: "finite B" "card B = dim V"
wenzelm@53406
  1981
    using independent_bound by auto
huffman@44133
  1982
  from basis_orthogonal[OF fB(1)] obtain C where
wenzelm@53406
  1983
    C: "finite C" "card C \<le> card B" "span C = span B" "pairwise orthogonal C"
wenzelm@53406
  1984
    by blast
wenzelm@53406
  1985
  from C B have CSV: "C \<subseteq> span V"
wenzelm@53406
  1986
    by (metis span_inc span_mono subset_trans)
wenzelm@53406
  1987
  from span_mono[OF B(3)] C have SVC: "span V \<subseteq> span C"
wenzelm@53406
  1988
    by (simp add: span_span)
huffman@44133
  1989
  from card_le_dim_spanning[OF CSV SVC C(1)] C(2,3) fB
wenzelm@53406
  1990
  have iC: "independent C"
huffman@44133
  1991
    by (simp add: dim_span)
wenzelm@53406
  1992
  from C fB have "card C \<le> dim V"
wenzelm@53406
  1993
    by simp
wenzelm@53406
  1994
  moreover have "dim V \<le> card C"
wenzelm@53406
  1995
    using span_card_ge_dim[OF CSV SVC C(1)]
wenzelm@53406
  1996
    by (simp add: dim_span)
wenzelm@53406
  1997
  ultimately have CdV: "card C = dim V"
wenzelm@53406
  1998
    using C(1) by simp
wenzelm@53406
  1999
  from C B CSV CdV iC show ?thesis
wenzelm@53406
  2000
    by auto
huffman@44133
  2001
qed
huffman@44133
  2002
huffman@44133
  2003
lemma span_eq: "span S = span T \<longleftrightarrow> S \<subseteq> span T \<and> T \<subseteq> span S"
huffman@44133
  2004
  using span_inc[unfolded subset_eq] using span_mono[of T "span S"] span_mono[of S "span T"]
wenzelm@49522
  2005
  by (auto simp add: span_span)
huffman@44133
  2006
wenzelm@60420
  2007
text \<open>Low-dimensional subset is in a hyperplane (weak orthogonal complement).\<close>
huffman@44133
  2008
wenzelm@49522
  2009
lemma span_not_univ_orthogonal:
wenzelm@53406
  2010
  fixes S :: "'a::euclidean_space set"
huffman@44133
  2011
  assumes sU: "span S \<noteq> UNIV"
wenzelm@56444
  2012
  shows "\<exists>a::'a. a \<noteq> 0 \<and> (\<forall>x \<in> span S. a \<bullet> x = 0)"
wenzelm@49522
  2013
proof -
wenzelm@53406
  2014
  from sU obtain a where a: "a \<notin> span S"
wenzelm@53406
  2015
    by blast
huffman@44133
  2016
  from orthogonal_basis_exists obtain B where
huffman@44133
  2017
    B: "independent B" "B \<subseteq> span S" "S \<subseteq> span B" "card B = dim S" "pairwise orthogonal B"
huffman@44133
  2018
    by blast
wenzelm@53406
  2019
  from B have fB: "finite B" "card B = dim S"
wenzelm@53406
  2020
    using independent_bound by auto
huffman@44133
  2021
  from span_mono[OF B(2)] span_mono[OF B(3)]
wenzelm@53406
  2022
  have sSB: "span S = span B"
wenzelm@53406
  2023
    by (simp add: span_span)
huffman@44133
  2024
  let ?a = "a - setsum (\<lambda>b. (a \<bullet> b / (b \<bullet> b)) *\<^sub>R b) B"
huffman@44133
  2025
  have "setsum (\<lambda>b. (a \<bullet> b / (b \<bullet> b)) *\<^sub>R b) B \<in> span S"
huffman@44133
  2026
    unfolding sSB
huffman@56196
  2027
    apply (rule span_setsum)
huffman@44133
  2028
    apply clarsimp
huffman@44133
  2029
    apply (rule span_mul)
wenzelm@49522
  2030
    apply (rule span_superset)
wenzelm@49522
  2031
    apply assumption
wenzelm@49522
  2032
    done
wenzelm@53406
  2033
  with a have a0:"?a  \<noteq> 0"
wenzelm@53406
  2034
    by auto
huffman@44133
  2035
  have "\<forall>x\<in>span B. ?a \<bullet> x = 0"
wenzelm@49522
  2036
  proof (rule span_induct')
wenzelm@49522
  2037
    show "subspace {x. ?a \<bullet> x = 0}"
wenzelm@49522
  2038
      by (auto simp add: subspace_def inner_add)
wenzelm@49522
  2039
  next
wenzelm@53406
  2040
    {
wenzelm@53406
  2041
      fix x
wenzelm@53406
  2042
      assume x: "x \<in> B"
wenzelm@53406
  2043
      from x have B': "B = insert x (B - {x})"
wenzelm@53406
  2044
        by blast
wenzelm@53406
  2045
      have fth: "finite (B - {x})"
wenzelm@53406
  2046
        using fB by simp
huffman@44133
  2047
      have "?a \<bullet> x = 0"
wenzelm@53406
  2048
        apply (subst B')
wenzelm@53406
  2049
        using fB fth
huffman@44133
  2050
        unfolding setsum_clauses(2)[OF fth]
huffman@44133
  2051
        apply simp unfolding inner_simps
huffman@44527
  2052
        apply (clarsimp simp add: inner_add inner_setsum_left)
haftmann@57418
  2053
        apply (rule setsum.neutral, rule ballI)
huffman@44133
  2054
        unfolding inner_commute
wenzelm@49711
  2055
        apply (auto simp add: x field_simps
wenzelm@49711
  2056
          intro: B(5)[unfolded pairwise_def orthogonal_def, rule_format])
wenzelm@53406
  2057
        done
wenzelm@53406
  2058
    }
wenzelm@53406
  2059
    then show "\<forall>x \<in> B. ?a \<bullet> x = 0"
wenzelm@53406
  2060
      by blast
huffman@44133
  2061
  qed
wenzelm@53406
  2062
  with a0 show ?thesis
wenzelm@53406
  2063
    unfolding sSB by (auto intro: exI[where x="?a"])
huffman@44133
  2064
qed
huffman@44133
  2065
huffman@44133
  2066
lemma span_not_univ_subset_hyperplane:
wenzelm@53406
  2067
  fixes S :: "'a::euclidean_space set"
wenzelm@53406
  2068
  assumes SU: "span S \<noteq> UNIV"
huffman@44133
  2069
  shows "\<exists> a. a \<noteq>0 \<and> span S \<subseteq> {x. a \<bullet> x = 0}"
huffman@44133
  2070
  using span_not_univ_orthogonal[OF SU] by auto
huffman@44133
  2071
wenzelm@49663
  2072
lemma lowdim_subset_hyperplane:
wenzelm@53406
  2073
  fixes S :: "'a::euclidean_space set"
huffman@44133
  2074
  assumes d: "dim S < DIM('a)"
wenzelm@56444
  2075
  shows "\<exists>a::'a. a \<noteq> 0 \<and> span S \<subseteq> {x. a \<bullet> x = 0}"
wenzelm@49522
  2076
proof -
wenzelm@53406
  2077
  {
wenzelm@53406
  2078
    assume "span S = UNIV"
wenzelm@53406
  2079
    then have "dim (span S) = dim (UNIV :: ('a) set)"
wenzelm@53406
  2080
      by simp
wenzelm@53406
  2081
    then have "dim S = DIM('a)"
wenzelm@53406
  2082
      by (simp add: dim_span dim_UNIV)
wenzelm@53406
  2083
    with d have False by arith
wenzelm@53406
  2084
  }
wenzelm@53406
  2085
  then have th: "span S \<noteq> UNIV"
wenzelm@53406
  2086
    by blast
huffman@44133
  2087
  from span_not_univ_subset_hyperplane[OF th] show ?thesis .
huffman@44133
  2088
qed
huffman@44133
  2089
wenzelm@60420
  2090
text \<open>We can extend a linear basis-basis injection to the whole set.\<close>
huffman@44133
  2091
huffman@44133
  2092
lemma linear_indep_image_lemma:
wenzelm@49663
  2093
  assumes lf: "linear f"
wenzelm@49663
  2094
    and fB: "finite B"
wenzelm@49522
  2095
    and ifB: "independent (f ` B)"
wenzelm@49663
  2096
    and fi: "inj_on f B"
wenzelm@49663
  2097
    and xsB: "x \<in> span B"
wenzelm@49522
  2098
    and fx: "f x = 0"
huffman@44133
  2099
  shows "x = 0"
huffman@44133
  2100
  using fB ifB fi xsB fx
wenzelm@49522
  2101
proof (induct arbitrary: x rule: finite_induct[OF fB])
wenzelm@49663
  2102
  case 1
wenzelm@49663
  2103
  then show ?case by auto
huffman@44133
  2104
next
huffman@44133
  2105
  case (2 a b x)
huffman@44133
  2106
  have fb: "finite b" using "2.prems" by simp
huffman@44133
  2107
  have th0: "f ` b \<subseteq> f ` (insert a b)"
wenzelm@53406
  2108
    apply (rule image_mono)
wenzelm@53406
  2109
    apply blast
wenzelm@53406
  2110
    done
huffman@44133
  2111
  from independent_mono[ OF "2.prems"(2) th0]
huffman@44133
  2112
  have ifb: "independent (f ` b)"  .
huffman@44133
  2113
  have fib: "inj_on f b"
huffman@44133
  2114
    apply (rule subset_inj_on [OF "2.prems"(3)])
wenzelm@49522
  2115
    apply blast
wenzelm@49522
  2116
    done
huffman@44133
  2117
  from span_breakdown[of a "insert a b", simplified, OF "2.prems"(4)]
wenzelm@53406
  2118
  obtain k where k: "x - k*\<^sub>R a \<in> span (b - {a})"
wenzelm@53406
  2119
    by blast
huffman@44133
  2120
  have "f (x - k*\<^sub>R a) \<in> span (f ` b)"
huffman@44133
  2121
    unfolding span_linear_image[OF lf]
huffman@44133
  2122
    apply (rule imageI)
wenzelm@53716
  2123
    using k span_mono[of "b - {a}" b]
wenzelm@53406
  2124
    apply blast
wenzelm@49522
  2125
    done
wenzelm@49522
  2126
  then have "f x - k*\<^sub>R f a \<in> span (f ` b)"
huffman@44133
  2127
    by (simp add: linear_sub[OF lf] linear_cmul[OF lf])
wenzelm@49522
  2128
  then have th: "-k *\<^sub>R f a \<in> span (f ` b)"
huffman@44133
  2129
    using "2.prems"(5) by simp
wenzelm@53406
  2130
  have xsb: "x \<in> span b"
wenzelm@53406
  2131
  proof (cases "k = 0")
wenzelm@53406
  2132
    case True
wenzelm@53716
  2133
    with k have "x \<in> span (b - {a})" by simp
wenzelm@53716
  2134
    then show ?thesis using span_mono[of "b - {a}" b]
wenzelm@53406
  2135
      by blast
wenzelm@53406
  2136
  next
wenzelm@53406
  2137
    case False
wenzelm@53406
  2138
    with span_mul[OF th, of "- 1/ k"]
huffman@44133
  2139
    have th1: "f a \<in> span (f ` b)"
hoelzl@56479
  2140
      by auto
huffman@44133
  2141
    from inj_on_image_set_diff[OF "2.prems"(3), of "insert a b " "{a}", symmetric]
huffman@44133
  2142
    have tha: "f ` insert a b - f ` {a} = f ` (insert a b - {a})" by blast
huffman@44133
  2143
    from "2.prems"(2) [unfolded dependent_def bex_simps(8), rule_format, of "f a"]
huffman@44133
  2144
    have "f a \<notin> span (f ` b)" using tha
huffman@44133
  2145
      using "2.hyps"(2)
huffman@44133
  2146
      "2.prems"(3) by auto
huffman@44133
  2147
    with th1 have False by blast
wenzelm@53406
  2148
    then show ?thesis by blast
wenzelm@53406
  2149
  qed
wenzelm@53406
  2150
  from "2.hyps"(3)[OF fb ifb fib xsb "2.prems"(5)] show "x = 0" .
huffman@44133
  2151
qed
huffman@44133
  2152
wenzelm@60420
  2153
text \<open>We can extend a linear mapping from basis.\<close>
huffman@44133
  2154
huffman@44133
  2155
lemma linear_independent_extend_lemma:
huffman@44133
  2156
  fixes f :: "'a::real_vector \<Rightarrow> 'b::real_vector"
wenzelm@53406
  2157
  assumes fi: "finite B"
wenzelm@53406
  2158
    and ib: "independent B"
wenzelm@53406
  2159
  shows "\<exists>g.
wenzelm@53406
  2160
    (\<forall>x\<in> span B. \<forall>y\<in> span B. g (x + y) = g x + g y) \<and>
wenzelm@53406
  2161
    (\<forall>x\<in> span B. \<forall>c. g (c*\<^sub>R x) = c *\<^sub>R g x) \<and>
wenzelm@53406
  2162
    (\<forall>x\<in> B. g x = f x)"
wenzelm@49663
  2163
  using ib fi
wenzelm@49522
  2164
proof (induct rule: finite_induct[OF fi])
wenzelm@49663
  2165
  case 1
wenzelm@49663
  2166
  then show ?case by auto
huffman@44133
  2167
next
huffman@44133
  2168
  case (2 a b)
huffman@44133
  2169
  from "2.prems" "2.hyps" have ibf: "independent b" "finite b"
huffman@44133
  2170
    by (simp_all add: independent_insert)
huffman@44133
  2171
  from "2.hyps"(3)[OF ibf] obtain g where
huffman@44133
  2172
    g: "\<forall>x\<in>span b. \<forall>y\<in>span b. g (x + y) = g x + g y"
huffman@44133
  2173
    "\<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
  2174
  let ?h = "\<lambda>z. SOME k. (z - k *\<^sub>R a) \<in> span b"
wenzelm@53406
  2175
  {
wenzelm@53406
  2176
    fix z
wenzelm@53406
  2177
    assume z: "z \<in> span (insert a b)"
huffman@44133
  2178
    have th0: "z - ?h z *\<^sub>R a \<in> span b"
huffman@44133
  2179
      apply (rule someI_ex)
huffman@44133
  2180
      unfolding span_breakdown_eq[symmetric]
wenzelm@53406
  2181
      apply (rule z)
wenzelm@53406
  2182
      done
wenzelm@53406
  2183
    {
wenzelm@53406
  2184
      fix k
wenzelm@53406
  2185
      assume k: "z - k *\<^sub>R a \<in> span b"
huffman@44133
  2186
      have eq: "z - ?h z *\<^sub>R a - (z - k*\<^sub>R a) = (k - ?h z) *\<^sub>R a"
huffman@44133
  2187
        by (simp add: field_simps scaleR_left_distrib [symmetric])
wenzelm@53406
  2188
      from span_sub[OF th0 k] have khz: "(k - ?h z) *\<^sub>R a \<in> span b"
wenzelm@53406
  2189
        by (simp add: eq)
wenzelm@53406
  2190
      {
wenzelm@53406
  2191
        assume "k \<noteq> ?h z"
wenzelm@53406
  2192
        then have k0: "k - ?h z \<noteq> 0" by simp
huffman@44133
  2193
        from k0 span_mul[OF khz, of "1 /(k - ?h z)"]
huffman@44133
  2194
        have "a \<in> span b" by simp
huffman@44133
  2195
        with "2.prems"(1) "2.hyps"(2) have False
wenzelm@53406
  2196
          by (auto simp add: dependent_def)
wenzelm@53406
  2197
      }
wenzelm@53406
  2198
      then have "k = ?h z" by blast
wenzelm@53406
  2199
    }
wenzelm@53406
  2200
    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
  2201
      by blast
wenzelm@53406
  2202
  }
huffman@44133
  2203
  note h = this
huffman@44133
  2204
  let ?g = "\<lambda>z. ?h z *\<^sub>R f a + g (z - ?h z *\<^sub>R a)"
wenzelm@53406
  2205
  {
wenzelm@53406
  2206
    fix x y
wenzelm@53406
  2207
    assume x: "x \<in> span (insert a b)"
wenzelm@53406
  2208
      and y: "y \<in> span (insert a b)"
huffman@44133
  2209
    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
  2210
      by (simp add: algebra_simps)
huffman@44133
  2211
    have addh: "?h (x + y) = ?h x + ?h y"
huffman@44133
  2212
      apply (rule conjunct2[OF h, rule_format, symmetric])
huffman@44133
  2213
      apply (rule span_add[OF x y])
huffman@44133
  2214
      unfolding tha
wenzelm@53406
  2215
      apply (metis span_add x y conjunct1[OF h, rule_format])
wenzelm@53406
  2216
      done
huffman@44133
  2217
    have "?g (x + y) = ?g x + ?g y"
huffman@44133
  2218
      unfolding addh tha
huffman@44133
  2219
      g(1)[rule_format,OF conjunct1[OF h, OF x] conjunct1[OF h, OF y]]
huffman@44133
  2220
      by (simp add: scaleR_left_distrib)}
huffman@44133
  2221
  moreover
wenzelm@53406
  2222
  {
wenzelm@53406
  2223
    fix x :: "'a"
wenzelm@53406
  2224
    fix c :: real
wenzelm@49522
  2225
    assume x: "x \<in> span (insert a b)"
huffman@44133
  2226
    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
  2227
      by (simp add: algebra_simps)
huffman@44133
  2228
    have hc: "?h (c *\<^sub>R x) = c * ?h x"
huffman@44133
  2229
      apply (rule conjunct2[OF h, rule_format, symmetric])
huffman@44133
  2230
      apply (metis span_mul x)
wenzelm@49522
  2231
      apply (metis tha span_mul x conjunct1[OF h])
wenzelm@49522
  2232
      done
huffman@44133
  2233
    have "?g (c *\<^sub>R x) = c*\<^sub>R ?g x"
huffman@44133
  2234
      unfolding hc tha g(2)[rule_format, OF conjunct1[OF h, OF x]]
wenzelm@53406
  2235
      by (simp add: algebra_simps)
wenzelm@53406
  2236
  }
huffman@44133
  2237
  moreover
wenzelm@53406
  2238
  {
wenzelm@53406
  2239
    fix x
wenzelm@53406
  2240
    assume x: "x \<in> insert a b"
wenzelm@53406
  2241
    {
wenzelm@53406
  2242
      assume xa: "x = a"
huffman@44133
  2243
      have ha1: "1 = ?h a"
huffman@44133
  2244
        apply (rule conjunct2[OF h, rule_format])
huffman@44133
  2245
        apply (metis span_superset insertI1)
huffman@44133
  2246
        using conjunct1[OF h, OF span_superset, OF insertI1]
wenzelm@49522
  2247
        apply (auto simp add: span_0)
wenzelm@49522
  2248
        done
huffman@44133
  2249
      from xa ha1[symmetric] have "?g x = f x"
huffman@44133
  2250
        apply simp
huffman@44133
  2251
        using g(2)[rule_format, OF span_0, of 0]
wenzelm@49522
  2252
        apply simp
wenzelm@53406
  2253
        done
wenzelm@53406
  2254
    }
huffman@44133
  2255
    moreover
wenzelm@53406
  2256
    {
wenzelm@53406
  2257
      assume xb: "x \<in> b"
huffman@44133
  2258
      have h0: "0 = ?h x"
huffman@44133
  2259
        apply (rule conjunct2[OF h, rule_format])
huffman@44133
  2260
        apply (metis  span_superset x)
huffman@44133
  2261
        apply simp
huffman@44133
  2262
        apply (metis span_superset xb)
huffman@44133
  2263
        done
huffman@44133
  2264
      have "?g x = f x"
wenzelm@53406
  2265
        by (simp add: h0[symmetric] g(3)[rule_format, OF xb])
wenzelm@53406
  2266
    }
wenzelm@53406
  2267
    ultimately have "?g x = f x"
wenzelm@53406
  2268
      using x by blast
wenzelm@53406
  2269
  }
wenzelm@49663
  2270
  ultimately show ?case
wenzelm@49663
  2271
    apply -
wenzelm@49663
  2272
    apply (rule exI[where x="?g"])
wenzelm@49663
  2273
    apply blast
wenzelm@49663
  2274
    done
huffman@44133
  2275
qed
huffman@44133
  2276
huffman@44133
  2277
lemma linear_independent_extend:
wenzelm@53406
  2278
  fixes B :: "'a::euclidean_space set"
wenzelm@53406
  2279
  assumes iB: "independent B"
huffman@44133
  2280
  shows "\<exists>g. linear g \<and> (\<forall>x\<in>B. g x = f x)"
wenzelm@49522
  2281
proof -
huffman@44133
  2282
  from maximal_independent_subset_extend[of B UNIV] iB
wenzelm@53406
  2283
  obtain C where C: "B \<subseteq> C" "independent C" "\<And>x. x \<in> span C"
wenzelm@53406
  2284
    by auto
huffman@44133
  2285
huffman@44133
  2286
  from C(2) independent_bound[of C] linear_independent_extend_lemma[of C f]
wenzelm@53406
  2287
  obtain g where g:
wenzelm@53406
  2288
    "(\<forall>x\<in> span C. \<forall>y\<in> span C. g (x + y) = g x + g y) \<and>
wenzelm@53406
  2289
     (\<forall>x\<in> span C. \<forall>c. g (c*\<^sub>R x) = c *\<^sub>R g x) \<and>
wenzelm@53406
  2290
     (\<forall>x\<in> C. g x = f x)" by blast
wenzelm@53406
  2291
  from g show ?thesis
huffman@53600
  2292
    unfolding linear_iff
wenzelm@53406
  2293
    using C
wenzelm@49663
  2294
    apply clarsimp
wenzelm@49663
  2295
    apply blast
wenzelm@49663
  2296
    done
huffman@44133
  2297
qed
huffman@44133
  2298
wenzelm@60420
  2299
text \<open>Can construct an isomorphism between spaces of same dimension.\<close>
huffman@44133
  2300
huffman@44133
  2301
lemma subspace_isomorphism:
wenzelm@53406
  2302
  fixes S :: "'a::euclidean_space set"
wenzelm@53406
  2303
    and T :: "'b::euclidean_space set"
wenzelm@53406
  2304
  assumes s: "subspace S"
wenzelm@53406
  2305
    and t: "subspace T"
wenzelm@49522
  2306
    and d: "dim S = dim T"
huffman@44133
  2307
  shows "\<exists>f. linear f \<and> f ` S = T \<and> inj_on f S"
wenzelm@49522
  2308
proof -
wenzelm@53406
  2309
  from basis_exists[of S] independent_bound
wenzelm@53406
  2310
  obtain B where B: "B \<subseteq> S" "independent B" "S \<subseteq> span B" "card B = dim S" and fB: "finite B"
wenzelm@53406
  2311
    by blast
wenzelm@53406
  2312
  from basis_exists[of T] independent_bound
wenzelm@53406
  2313
  obtain C where C: "C \<subseteq> T" "independent C" "T \<subseteq> span C" "card C = dim T" and fC: "finite C"
wenzelm@53406
  2314
    by blast
wenzelm@53406
  2315
  from B(4) C(4) card_le_inj[of B C] d
wenzelm@60420
  2316
  obtain f where f: "f ` B \<subseteq> C" "inj_on f B" using \<open>finite B\<close> \<open>finite C\<close>
wenzelm@53406
  2317
    by auto
wenzelm@53406
  2318
  from linear_independent_extend[OF B(2)]
wenzelm@53406
  2319
  obtain g where g: "linear g" "\<forall>x\<in> B. g x = f x"
wenzelm@53406
  2320
    by blast
wenzelm@53406
  2321
  from inj_on_iff_eq_card[OF fB, of f] f(2) have "card (f ` B) = card B"
huffman@44133
  2322
    by simp
wenzelm@53406
  2323
  with B(4) C(4) have ceq: "card (f ` B) = card C"
wenzelm@53406
  2324
    using d by simp
wenzelm@53406
  2325
  have "g ` B = f ` B"
wenzelm@53406
  2326
    using g(2) by (auto simp add: image_iff)
huffman@44133
  2327
  also have "\<dots> = C" using card_subset_eq[OF fC f(1) ceq] .
huffman@44133
  2328
  finally have gBC: "g ` B = C" .
wenzelm@53406
  2329
  have gi: "inj_on g B"
wenzelm@53406
  2330
    using f(2) g(2) by (auto simp add: inj_on_def)
huffman@44133
  2331
  note g0 = linear_indep_image_lemma[OF g(1) fB, unfolded gBC, OF C(2) gi]
wenzelm@53406
  2332
  {
wenzelm@53406
  2333
    fix x y
wenzelm@53406
  2334
    assume x: "x \<in> S" and y: "y \<in> S" and gxy: "g x = g y"
wenzelm@53406
  2335
    from B(3) x y have x': "x \<in> span B" and y': "y \<in> span B"
wenzelm@53406
  2336
      by blast+
wenzelm@53406
  2337
    from gxy have th0: "g (x - y) = 0"
wenzelm@53406
  2338
      by (simp add: linear_sub[OF g(1)])
wenzelm@53406
  2339
    have th1: "x - y \<in> span B"
wenzelm@53406
  2340
      using x' y' by (metis span_sub)
wenzelm@53406
  2341
    have "x = y"
wenzelm@53406
  2342
      using g0[OF th1 th0] by simp
wenzelm@53406
  2343
  }
huffman@44133
  2344
  then have giS: "inj_on g S"
huffman@44133
  2345
    unfolding inj_on_def by blast
wenzelm@53406
  2346
  from span_subspace[OF B(1,3) s] have "g ` S = span (g ` B)"
wenzelm@53406
  2347
    by (simp add: span_linear_image[OF g(1)])
huffman@44133
  2348
  also have "\<dots> = span C" unfolding gBC ..
huffman@44133
  2349
  also have "\<dots> = T" using span_subspace[OF C(1,3) t] .
huffman@44133
  2350
  finally have gS: "g ` S = T" .
wenzelm@53406
  2351
  from g(1) gS giS show ?thesis
wenzelm@53406
  2352
    by blast
huffman@44133
  2353
qed
huffman@44133
  2354
wenzelm@60420
  2355
text \<open>Linear functions are equal on a subspace if they are on a spanning set.\<close>
huffman@44133
  2356
huffman@44133
  2357
lemma subspace_kernel:
huffman@44133
  2358
  assumes lf: "linear f"
huffman@44133
  2359
  shows "subspace {x. f x = 0}"
wenzelm@49522
  2360
  apply (simp add: subspace_def)
wenzelm@49522
  2361
  apply (simp add: linear_add[OF lf] linear_cmul[OF lf] linear_0[OF lf])
wenzelm@49522
  2362
  done
huffman@44133
  2363
huffman@44133
  2364
lemma linear_eq_0_span:
huffman@44133
  2365
  assumes lf: "linear f" and f0: "\<forall>x\<in>B. f x = 0"
huffman@44133
  2366
  shows "\<forall>x \<in> span B. f x = 0"
huffman@44170
  2367
  using f0 subspace_kernel[OF lf]
huffman@44170
  2368
  by (rule span_induct')
huffman@44133
  2369
huffman@44133
  2370
lemma linear_eq_0:
wenzelm@49663
  2371
  assumes lf: "linear f"
wenzelm@49663
  2372
    and SB: "S \<subseteq> span B"
wenzelm@49663
  2373
    and f0: "\<forall>x\<in>B. f x = 0"
huffman@44133
  2374
  shows "\<forall>x \<in> S. f x = 0"
huffman@44133
  2375
  by (metis linear_eq_0_span[OF lf] subset_eq SB f0)
huffman@44133
  2376
huffman@44133
  2377
lemma linear_eq:
wenzelm@49663
  2378
  assumes lf: "linear f"
wenzelm@49663
  2379
    and lg: "linear g"
wenzelm@49663
  2380
    and S: "S \<subseteq> span B"
wenzelm@49522
  2381
    and fg: "\<forall> x\<in> B. f x = g x"
huffman@44133
  2382
  shows "\<forall>x\<in> S. f x = g x"
wenzelm@49663
  2383
proof -
huffman@44133
  2384
  let ?h = "\<lambda>x. f x - g x"
huffman@44133
  2385
  from fg have fg': "\<forall>x\<in> B. ?h x = 0" by simp
huffman@44133
  2386
  from linear_eq_0[OF linear_compose_sub[OF lf lg] S fg']
huffman@44133
  2387
  show ?thesis by simp
huffman@44133
  2388
qed
huffman@44133
  2389
huffman@44133
  2390
lemma linear_eq_stdbasis:
wenzelm@56444
  2391
  fixes f :: "'a::euclidean_space \<Rightarrow> _"
wenzelm@56444
  2392
  assumes lf: "linear f"
wenzelm@49663
  2393
    and lg: "linear g"
hoelzl@50526
  2394
    and fg: "\<forall>b\<in>Basis. f b = g b"
huffman@44133
  2395
  shows "f = g"
hoelzl@50526
  2396
  using linear_eq[OF lf lg, of _ Basis] fg by auto
huffman@44133
  2397
wenzelm@60420
  2398
text \<open>Similar results for bilinear functions.\<close>
huffman@44133
  2399
huffman@44133
  2400
lemma bilinear_eq:
huffman@44133
  2401
  assumes bf: "bilinear f"
wenzelm@49522
  2402
    and bg: "bilinear g"
wenzelm@53406
  2403
    and SB: "S \<subseteq> span B"
wenzelm@53406
  2404
    and TC: "T \<subseteq> span C"
wenzelm@49522
  2405
    and fg: "\<forall>x\<in> B. \<forall>y\<in> C. f x y = g x y"
huffman@44133
  2406
  shows "\<forall>x\<in>S. \<forall>y\<in>T. f x y = g x y "
wenzelm@49663
  2407
proof -
huffman@44170
  2408
  let ?P = "{x. \<forall>y\<in> span C. f x y = g x y}"
huffman@44133
  2409
  from bf bg have sp: "subspace ?P"
huffman@53600
  2410
    unfolding bilinear_def linear_iff subspace_def bf bg
wenzelm@49663
  2411
    by (auto simp add: span_0 bilinear_lzero[OF bf] bilinear_lzero[OF bg] span_add Ball_def
wenzelm@49663
  2412
      intro: bilinear_ladd[OF bf])
huffman@44133
  2413
huffman@44133
  2414
  have "\<forall>x \<in> span B. \<forall>y\<in> span C. f x y = g x y"
huffman@44170
  2415
    apply (rule span_induct' [OF _ sp])
huffman@44133
  2416
    apply (rule ballI)
huffman@44170
  2417
    apply (rule span_induct')
huffman@44170
  2418
    apply (simp add: fg)
huffman@44133
  2419
    apply (auto simp add: subspace_def)
huffman@53600
  2420
    using bf bg unfolding bilinear_def linear_iff
wenzelm@49522
  2421
    apply (auto simp add: span_0 bilinear_rzero[OF bf] bilinear_rzero[OF bg] span_add Ball_def
wenzelm@49663
  2422
      intro: bilinear_ladd[OF bf])
wenzelm@49522
  2423
    done
wenzelm@53406
  2424
  then show ?thesis
wenzelm@53406
  2425
    using SB TC by auto
huffman@44133
  2426
qed
huffman@44133
  2427
wenzelm@49522
  2428
lemma bilinear_eq_stdbasis:
wenzelm@53406
  2429
  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space \<Rightarrow> _"
huffman@44133
  2430
  assumes bf: "bilinear f"
wenzelm@49522
  2431
    and bg: "bilinear g"
hoelzl@50526
  2432
    and fg: "\<forall>i\<in>Basis. \<forall>j\<in>Basis. f i j = g i j"
huffman@44133
  2433
  shows "f = g"
hoelzl@50526
  2434
  using bilinear_eq[OF bf bg equalityD2[OF span_Basis] equalityD2[OF span_Basis] fg] by blast
huffman@44133
  2435
wenzelm@60420
  2436
text \<open>Detailed theorems about left and right invertibility in general case.\<close>
huffman@44133
  2437
wenzelm@49522
  2438
lemma linear_injective_left_inverse:
wenzelm@56444
  2439
  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
wenzelm@56444
  2440
  assumes lf: "linear f"
wenzelm@56444
  2441
    and fi: "inj f"
wenzelm@56444
  2442
  shows "\<exists>g. linear g \<and> g \<circ> f = id"
wenzelm@49522
  2443
proof -
hoelzl@50526
  2444
  from linear_independent_extend[OF independent_injective_image, OF independent_Basis, OF lf fi]
wenzelm@56444
  2445
  obtain h :: "'b \<Rightarrow> 'a" where h: "linear h" "\<forall>x \<in> f ` Basis. h x = inv f x"
wenzelm@53406
  2446
    by blast
hoelzl@50526
  2447
  from h(2) have th: "\<forall>i\<in>Basis. (h \<circ> f) i = id i"
huffman@44133
  2448
    using inv_o_cancel[OF fi, unfolded fun_eq_iff id_def o_def]
huffman@44133
  2449
    by auto
huffman@44133
  2450
  from linear_eq_stdbasis[OF linear_compose[OF lf h(1)] linear_id th]
wenzelm@56444
  2451
  have "h \<circ> f = id" .
wenzelm@53406
  2452
  then show ?thesis
wenzelm@53406
  2453
    using h(1) by blast
huffman@44133
  2454
qed
huffman@44133
  2455
wenzelm@49522
  2456
lemma linear_surjective_right_inverse:
wenzelm@53406
  2457
  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
wenzelm@53406
  2458
  assumes lf: "linear f"
wenzelm@53406
  2459
    and sf: "surj f"
wenzelm@56444
  2460
  shows "\<exists>g. linear g \<and> f \<circ> g = id"
wenzelm@49522
  2461
proof -
hoelzl@50526
  2462
  from linear_independent_extend[OF independent_Basis[where 'a='b],of "inv f"]
wenzelm@56444
  2463
  obtain h :: "'b \<Rightarrow> 'a" where h: "linear h" "\<forall>x\<in>Basis. h x = inv f x"