src/HOL/Multivariate_Analysis/Linear_Algebra.thy
author paulson <lp15@cam.ac.uk>
Thu May 28 14:33:35 2015 +0100 (2015-05-28)
changeset 60307 75e1aa7a450e
parent 60303 00c06f1315d0
child 60420 884f54e01427
permissions -rw-r--r--
Convex hulls: theorems about interior, etc. And a few simple lemmas.
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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 {* Elementary linear algebra on Euclidean spaces *}
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theory Linear_Algebra
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imports
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  Euclidean_Space
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  "~~/src/HOL/Library/Infinite_Set"
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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{* Hence derive more interesting properties of the norm. *}
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lemma norm_eq_0_dot: "norm x = 0 \<longleftrightarrow> x \<bullet> x = (0::real)"
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  by simp (* TODO: delete *)
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lemma norm_triangle_sub:
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  fixes x y :: "'a::real_normed_vector"
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  shows "norm x \<le> norm y + norm (x - y)"
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  using norm_triangle_ineq[of "y" "x - y"] by (simp add: field_simps)
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lemma norm_le: "norm x \<le> norm y \<longleftrightarrow> x \<bullet> x \<le> y \<bullet> y"
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  by (simp add: norm_eq_sqrt_inner)
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lemma norm_lt: "norm x < norm y \<longleftrightarrow> x \<bullet> x < y \<bullet> y"
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  by (simp add: norm_eq_sqrt_inner)
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lemma norm_eq: "norm x = norm y \<longleftrightarrow> x \<bullet> x = y \<bullet> y"
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  apply (subst order_eq_iff)
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  apply (auto simp: norm_le)
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  done
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lemma norm_eq_1: "norm x = 1 \<longleftrightarrow> x \<bullet> x = 1"
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  by (simp add: norm_eq_sqrt_inner)
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text{* Squaring equations and inequalities involving norms.  *}
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lemma dot_square_norm: "x \<bullet> x = (norm x)\<^sup>2"
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  by (simp only: power2_norm_eq_inner) (* TODO: move? *)
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lemma norm_eq_square: "norm x = a \<longleftrightarrow> 0 \<le> a \<and> x \<bullet> x = a\<^sup>2"
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  by (auto simp add: norm_eq_sqrt_inner)
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lemma 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{* Dot product in terms of the norm rather than conversely. *}
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lemmas inner_simps = inner_add_left inner_add_right inner_diff_right inner_diff_left
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  inner_scaleR_left inner_scaleR_right
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lemma dot_norm: "x \<bullet> y = ((norm (x + y))\<^sup>2 - (norm x)\<^sup>2 - (norm y)\<^sup>2) / 2"
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  unfolding power2_norm_eq_inner inner_simps inner_commute by auto
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lemma dot_norm_neg: "x \<bullet> y = (((norm x)\<^sup>2 + (norm y)\<^sup>2) - (norm (x - y))\<^sup>2) / 2"
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  unfolding power2_norm_eq_inner inner_simps inner_commute
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  by (auto simp add: algebra_simps)
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text{* Equality of vectors in terms of @{term "op \<bullet>"} products.    *}
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lemma vector_eq: "x = y \<longleftrightarrow> x \<bullet> x = x \<bullet> y \<and> y \<bullet> y = x \<bullet> x"
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  (is "?lhs \<longleftrightarrow> ?rhs")
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proof
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  assume ?lhs
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  then show ?rhs by simp
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next
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  assume ?rhs
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  then have "x \<bullet> x - x \<bullet> y = 0 \<and> x \<bullet> y - y \<bullet> y = 0"
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    by simp
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  then have "x \<bullet> (x - y) = 0 \<and> y \<bullet> (x - y) = 0"
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    by (simp add: inner_diff inner_commute)
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  then have "(x - y) \<bullet> (x - y) = 0"
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    by (simp add: field_simps inner_diff inner_commute)
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  then show "x = y" by simp
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qed
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lemma norm_triangle_half_r:
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  "norm (y - x1) < e / 2 \<Longrightarrow> norm (y - x2) < e / 2 \<Longrightarrow> norm (x1 - x2) < e"
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  using dist_triangle_half_r unfolding dist_norm[symmetric] by auto
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lemma norm_triangle_half_l:
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  assumes "norm (x - y) < e / 2"
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    and "norm (x' - y) < e / 2"
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  shows "norm (x - x') < e"
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  using dist_triangle_half_l[OF assms[unfolded dist_norm[symmetric]]]
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  unfolding dist_norm[symmetric] .
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lemma norm_triangle_le: "norm x + norm y \<le> e \<Longrightarrow> norm (x + y) \<le> e"
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  by (rule norm_triangle_ineq [THEN order_trans])
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lemma norm_triangle_lt: "norm x + norm y < e \<Longrightarrow> norm (x + y) < e"
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  by (rule norm_triangle_ineq [THEN le_less_trans])
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lemma setsum_clauses:
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  shows "setsum f {} = 0"
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    and "finite S \<Longrightarrow> setsum f (insert x S) = (if x \<in> S then setsum f S else f x + setsum f S)"
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  by (auto simp add: insert_absorb)
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lemma setsum_norm_le:
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  fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
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  assumes fg: "\<forall>x \<in> S. norm (f x) \<le> g x"
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  shows "norm (setsum f S) \<le> setsum g S"
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  by (rule order_trans [OF norm_setsum setsum_mono]) (simp add: fg)
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lemma setsum_norm_bound:
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  fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
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  assumes 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 {* Orthogonality. *}
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context real_inner
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begin
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definition "orthogonal x y \<longleftrightarrow> x \<bullet> y = 0"
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lemma orthogonal_clauses:
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  "orthogonal a 0"
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  "orthogonal a x \<Longrightarrow> orthogonal a (c *\<^sub>R x)"
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  "orthogonal a x \<Longrightarrow> orthogonal a (- x)"
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  "orthogonal a x \<Longrightarrow> orthogonal a y \<Longrightarrow> orthogonal a (x + y)"
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  "orthogonal a x \<Longrightarrow> orthogonal a y \<Longrightarrow> orthogonal a (x - y)"
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  "orthogonal 0 a"
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  "orthogonal x a \<Longrightarrow> orthogonal (c *\<^sub>R x) a"
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  "orthogonal x a \<Longrightarrow> orthogonal (- x) a"
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  "orthogonal x a \<Longrightarrow> orthogonal y a \<Longrightarrow> orthogonal (x + y) a"
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  "orthogonal x a \<Longrightarrow> orthogonal y a \<Longrightarrow> orthogonal (x - y) a"
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  unfolding orthogonal_def inner_add inner_diff by auto
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end
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lemma orthogonal_commute: "orthogonal x y \<longleftrightarrow> orthogonal y x"
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  by (simp add: orthogonal_def inner_commute)
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subsection {* Linear functions. *}
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lemma linear_iff:
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  "linear f \<longleftrightarrow> (\<forall>x y. f (x + y) = f x + f y) \<and> (\<forall>c x. f (c *\<^sub>R x) = c *\<^sub>R f x)"
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  (is "linear f \<longleftrightarrow> ?rhs")
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proof
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  assume "linear f"
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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 {* Bilinear functions. *}
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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@49522
   367
subsection {* Adjoints. *}
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
hoelzl@50526
   392
text {* 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"})
hoelzl@50526
   395
*}
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@49522
   440
subsection {* Interlude: Some properties of real sets *}
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
huffman@44133
   465
lemma triangle_lemma:
wenzelm@53406
   466
  fixes x y z :: real
wenzelm@53406
   467
  assumes x: "0 \<le> x"
wenzelm@53406
   468
    and y: "0 \<le> y"
wenzelm@53406
   469
    and z: "0 \<le> z"
wenzelm@53406
   470
    and xy: "x\<^sup>2 \<le> y\<^sup>2 + z\<^sup>2"
wenzelm@53406
   471
  shows "x \<le> y + z"
wenzelm@49522
   472
proof -
wenzelm@56444
   473
  have "y\<^sup>2 + z\<^sup>2 \<le> y\<^sup>2 + 2 * y * z + z\<^sup>2"
nipkow@56536
   474
    using z y by simp
wenzelm@53406
   475
  with xy have th: "x\<^sup>2 \<le> (y + z)\<^sup>2"
wenzelm@53406
   476
    by (simp add: power2_eq_square field_simps)
wenzelm@53406
   477
  from y z have yz: "y + z \<ge> 0"
wenzelm@53406
   478
    by arith
huffman@44133
   479
  from power2_le_imp_le[OF th yz] show ?thesis .
huffman@44133
   480
qed
huffman@44133
   481
wenzelm@49522
   482
huffman@44133
   483
subsection {* A generic notion of "hull" (convex, affine, conic hull and closure). *}
huffman@44133
   484
wenzelm@53406
   485
definition hull :: "('a set \<Rightarrow> bool) \<Rightarrow> 'a set \<Rightarrow> 'a set"  (infixl "hull" 75)
wenzelm@53406
   486
  where "S hull s = \<Inter>{t. S t \<and> s \<subseteq> t}"
huffman@44170
   487
huffman@44170
   488
lemma hull_same: "S s \<Longrightarrow> S hull s = s"
huffman@44133
   489
  unfolding hull_def by auto
huffman@44133
   490
wenzelm@53406
   491
lemma hull_in: "(\<And>T. Ball T S \<Longrightarrow> S (\<Inter>T)) \<Longrightarrow> S (S hull s)"
wenzelm@49522
   492
  unfolding hull_def Ball_def by auto
huffman@44170
   493
wenzelm@53406
   494
lemma hull_eq: "(\<And>T. Ball T S \<Longrightarrow> S (\<Inter>T)) \<Longrightarrow> (S hull s) = s \<longleftrightarrow> S s"
wenzelm@49522
   495
  using hull_same[of S s] hull_in[of S s] by metis
huffman@44133
   496
huffman@44133
   497
lemma hull_hull: "S hull (S hull s) = S hull s"
huffman@44133
   498
  unfolding hull_def by blast
huffman@44133
   499
huffman@44133
   500
lemma hull_subset[intro]: "s \<subseteq> (S hull s)"
huffman@44133
   501
  unfolding hull_def by blast
huffman@44133
   502
wenzelm@53406
   503
lemma hull_mono: "s \<subseteq> t \<Longrightarrow> (S hull s) \<subseteq> (S hull t)"
huffman@44133
   504
  unfolding hull_def by blast
huffman@44133
   505
wenzelm@53406
   506
lemma hull_antimono: "\<forall>x. S x \<longrightarrow> T x \<Longrightarrow> (T hull s) \<subseteq> (S hull s)"
huffman@44133
   507
  unfolding hull_def by blast
huffman@44133
   508
wenzelm@53406
   509
lemma hull_minimal: "s \<subseteq> t \<Longrightarrow> S t \<Longrightarrow> (S hull s) \<subseteq> t"
huffman@44133
   510
  unfolding hull_def by blast
huffman@44133
   511
wenzelm@53406
   512
lemma subset_hull: "S t \<Longrightarrow> S hull s \<subseteq> t \<longleftrightarrow> s \<subseteq> t"
huffman@44133
   513
  unfolding hull_def by blast
huffman@44133
   514
huffman@53596
   515
lemma hull_UNIV: "S hull UNIV = UNIV"
huffman@53596
   516
  unfolding hull_def by auto
huffman@53596
   517
wenzelm@53406
   518
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
   519
  unfolding hull_def by auto
huffman@44133
   520
huffman@44133
   521
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
   522
  using hull_minimal[of S "{x. P x}" Q]
huffman@44170
   523
  by (auto simp add: subset_eq)
huffman@44133
   524
wenzelm@49522
   525
lemma hull_inc: "x \<in> S \<Longrightarrow> x \<in> P hull S"
wenzelm@49522
   526
  by (metis hull_subset subset_eq)
huffman@44133
   527
huffman@44133
   528
lemma hull_union_subset: "(S hull s) \<union> (S hull t) \<subseteq> (S hull (s \<union> t))"
wenzelm@49522
   529
  unfolding Un_subset_iff by (metis hull_mono Un_upper1 Un_upper2)
wenzelm@49522
   530
wenzelm@49522
   531
lemma hull_union:
wenzelm@53406
   532
  assumes T: "\<And>T. Ball T S \<Longrightarrow> S (\<Inter>T)"
huffman@44133
   533
  shows "S hull (s \<union> t) = S hull (S hull s \<union> S hull t)"
wenzelm@49522
   534
  apply rule
wenzelm@49522
   535
  apply (rule hull_mono)
wenzelm@49522
   536
  unfolding Un_subset_iff
wenzelm@49522
   537
  apply (metis hull_subset Un_upper1 Un_upper2 subset_trans)
wenzelm@49522
   538
  apply (rule hull_minimal)
wenzelm@49522
   539
  apply (metis hull_union_subset)
wenzelm@49522
   540
  apply (metis hull_in T)
wenzelm@49522
   541
  done
huffman@44133
   542
wenzelm@56444
   543
lemma hull_redundant_eq: "a \<in> (S hull s) \<longleftrightarrow> S hull (insert a s) = S hull s"
huffman@44133
   544
  unfolding hull_def by blast
huffman@44133
   545
wenzelm@56444
   546
lemma hull_redundant: "a \<in> (S hull s) \<Longrightarrow> S hull (insert a s) = S hull s"
wenzelm@49522
   547
  by (metis hull_redundant_eq)
wenzelm@49522
   548
huffman@44133
   549
huffman@44666
   550
subsection {* Archimedean properties and useful consequences *}
huffman@44133
   551
wenzelm@56444
   552
lemma real_arch_simple: "\<exists>n::nat. x \<le> real n"
huffman@44666
   553
  unfolding real_of_nat_def by (rule ex_le_of_nat)
huffman@44133
   554
huffman@44133
   555
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
   556
  using reals_Archimedean[of e] less_trans[of 0 "1 / real n" e for n::nat]
hoelzl@56480
   557
  by (auto simp add: field_simps cong: conj_cong)
huffman@44133
   558
wenzelm@53406
   559
lemma real_pow_lbound: "0 \<le> x \<Longrightarrow> 1 + real n * x \<le> (1 + x) ^ n"
wenzelm@49522
   560
proof (induct n)
wenzelm@49522
   561
  case 0
wenzelm@49522
   562
  then show ?case by simp
huffman@44133
   563
next
huffman@44133
   564
  case (Suc n)
wenzelm@53406
   565
  then have h: "1 + real n * x \<le> (1 + x) ^ n"
wenzelm@53406
   566
    by simp
wenzelm@53406
   567
  from h have p: "1 \<le> (1 + x) ^ n"
wenzelm@53406
   568
    using Suc.prems by simp
wenzelm@53406
   569
  from h have "1 + real n * x + x \<le> (1 + x) ^ n + x"
wenzelm@53406
   570
    by simp
wenzelm@53406
   571
  also have "\<dots> \<le> (1 + x) ^ Suc n"
wenzelm@53406
   572
    apply (subst diff_le_0_iff_le[symmetric])
lp15@60162
   573
    using mult_left_mono[OF p Suc.prems]
huffman@44133
   574
    apply (simp add: field_simps)
wenzelm@49522
   575
    done
wenzelm@53406
   576
  finally show ?case
wenzelm@53406
   577
    by (simp add: real_of_nat_Suc field_simps)
huffman@44133
   578
qed
huffman@44133
   579
wenzelm@53406
   580
lemma real_arch_pow:
wenzelm@53406
   581
  fixes x :: real
wenzelm@53406
   582
  assumes x: "1 < x"
wenzelm@53406
   583
  shows "\<exists>n. y < x^n"
wenzelm@49522
   584
proof -
wenzelm@53406
   585
  from x have x0: "x - 1 > 0"
wenzelm@53406
   586
    by arith
huffman@44666
   587
  from reals_Archimedean3[OF x0, rule_format, of y]
wenzelm@53406
   588
  obtain n :: nat where n: "y < real n * (x - 1)" by metis
huffman@44133
   589
  from x0 have x00: "x- 1 \<ge> 0" by arith
huffman@44133
   590
  from real_pow_lbound[OF x00, of n] n
huffman@44133
   591
  have "y < x^n" by auto
huffman@44133
   592
  then show ?thesis by metis
huffman@44133
   593
qed
huffman@44133
   594
wenzelm@53406
   595
lemma real_arch_pow2:
wenzelm@53406
   596
  fixes x :: real
wenzelm@53406
   597
  shows "\<exists>n. x < 2^ n"
huffman@44133
   598
  using real_arch_pow[of 2 x] by simp
huffman@44133
   599
wenzelm@49522
   600
lemma real_arch_pow_inv:
wenzelm@53406
   601
  fixes x y :: real
wenzelm@53406
   602
  assumes y: "y > 0"
wenzelm@53406
   603
    and x1: "x < 1"
huffman@44133
   604
  shows "\<exists>n. x^n < y"
wenzelm@53406
   605
proof (cases "x > 0")
wenzelm@53406
   606
  case True
wenzelm@53406
   607
  with x1 have ix: "1 < 1/x" by (simp add: field_simps)
wenzelm@53406
   608
  from real_arch_pow[OF ix, of "1/y"]
wenzelm@53406
   609
  obtain n where n: "1/y < (1/x)^n" by blast
wenzelm@53406
   610
  then show ?thesis using y `x > 0`
hoelzl@56480
   611
    by (auto simp add: field_simps)
wenzelm@53406
   612
next
wenzelm@53406
   613
  case False
wenzelm@53406
   614
  with y x1 show ?thesis
wenzelm@53406
   615
    apply auto
wenzelm@53406
   616
    apply (rule exI[where x=1])
wenzelm@53406
   617
    apply auto
wenzelm@53406
   618
    done
huffman@44133
   619
qed
huffman@44133
   620
wenzelm@49522
   621
lemma forall_pos_mono:
wenzelm@53406
   622
  "(\<And>d e::real. d < e \<Longrightarrow> P d \<Longrightarrow> P e) \<Longrightarrow>
wenzelm@53406
   623
    (\<And>n::nat. n \<noteq> 0 \<Longrightarrow> P (inverse (real n))) \<Longrightarrow> (\<And>e. 0 < e \<Longrightarrow> P e)"
huffman@44133
   624
  by (metis real_arch_inv)
huffman@44133
   625
wenzelm@49522
   626
lemma forall_pos_mono_1:
wenzelm@53406
   627
  "(\<And>d e::real. d < e \<Longrightarrow> P d \<Longrightarrow> P e) \<Longrightarrow>
wenzelm@53716
   628
    (\<And>n. P (inverse (real (Suc n)))) \<Longrightarrow> 0 < e \<Longrightarrow> P e"
huffman@44133
   629
  apply (rule forall_pos_mono)
huffman@44133
   630
  apply auto
huffman@44133
   631
  apply (atomize)
huffman@44133
   632
  apply (erule_tac x="n - 1" in allE)
huffman@44133
   633
  apply auto
huffman@44133
   634
  done
huffman@44133
   635
wenzelm@49522
   636
lemma real_archimedian_rdiv_eq_0:
wenzelm@53406
   637
  assumes x0: "x \<ge> 0"
wenzelm@53406
   638
    and c: "c \<ge> 0"
wenzelm@56444
   639
    and xc: "\<forall>(m::nat) > 0. real m * x \<le> c"
huffman@44133
   640
  shows "x = 0"
wenzelm@53406
   641
proof (rule ccontr)
wenzelm@53406
   642
  assume "x \<noteq> 0"
wenzelm@53406
   643
  with x0 have xp: "x > 0" by arith
wenzelm@53406
   644
  from reals_Archimedean3[OF xp, rule_format, of c]
wenzelm@53406
   645
  obtain n :: nat where n: "c < real n * x"
wenzelm@53406
   646
    by blast
wenzelm@53406
   647
  with xc[rule_format, of n] have "n = 0"
wenzelm@53406
   648
    by arith
wenzelm@53406
   649
  with n c show False
wenzelm@53406
   650
    by simp
huffman@44133
   651
qed
huffman@44133
   652
wenzelm@49522
   653
huffman@44133
   654
subsection{* A bit of linear algebra. *}
huffman@44133
   655
wenzelm@49522
   656
definition (in real_vector) subspace :: "'a set \<Rightarrow> bool"
wenzelm@56444
   657
  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
   658
huffman@44133
   659
definition (in real_vector) "span S = (subspace hull S)"
wenzelm@53716
   660
definition (in real_vector) "dependent S \<longleftrightarrow> (\<exists>a \<in> S. a \<in> span (S - {a}))"
wenzelm@53406
   661
abbreviation (in real_vector) "independent s \<equiv> \<not> dependent s"
huffman@44133
   662
huffman@44133
   663
text {* Closure properties of subspaces. *}
huffman@44133
   664
wenzelm@53406
   665
lemma subspace_UNIV[simp]: "subspace UNIV"
wenzelm@53406
   666
  by (simp add: subspace_def)
wenzelm@53406
   667
wenzelm@53406
   668
lemma (in real_vector) subspace_0: "subspace S \<Longrightarrow> 0 \<in> S"
wenzelm@53406
   669
  by (metis subspace_def)
wenzelm@53406
   670
wenzelm@53406
   671
lemma (in real_vector) subspace_add: "subspace S \<Longrightarrow> x \<in> S \<Longrightarrow> y \<in> S \<Longrightarrow> x + y \<in> S"
huffman@44133
   672
  by (metis subspace_def)
huffman@44133
   673
huffman@44133
   674
lemma (in real_vector) subspace_mul: "subspace S \<Longrightarrow> x \<in> S \<Longrightarrow> c *\<^sub>R x \<in> S"
huffman@44133
   675
  by (metis subspace_def)
huffman@44133
   676
huffman@44133
   677
lemma subspace_neg: "subspace S \<Longrightarrow> x \<in> S \<Longrightarrow> - x \<in> S"
huffman@44133
   678
  by (metis scaleR_minus1_left subspace_mul)
huffman@44133
   679
huffman@44133
   680
lemma subspace_sub: "subspace S \<Longrightarrow> x \<in> S \<Longrightarrow> y \<in> S \<Longrightarrow> x - y \<in> S"
haftmann@54230
   681
  using subspace_add [of S x "- y"] by (simp add: subspace_neg)
huffman@44133
   682
huffman@44133
   683
lemma (in real_vector) subspace_setsum:
wenzelm@53406
   684
  assumes sA: "subspace A"
huffman@56196
   685
    and f: "\<forall>x\<in>B. f x \<in> A"
huffman@44133
   686
  shows "setsum f B \<in> A"
huffman@56196
   687
proof (cases "finite B")
huffman@56196
   688
  case True
huffman@56196
   689
  then show ?thesis
huffman@56196
   690
    using f by induct (simp_all add: subspace_0 [OF sA] subspace_add [OF sA])
huffman@56196
   691
qed (simp add: subspace_0 [OF sA])
huffman@44133
   692
huffman@44133
   693
lemma subspace_linear_image:
wenzelm@53406
   694
  assumes lf: "linear f"
wenzelm@53406
   695
    and sS: "subspace S"
wenzelm@53406
   696
  shows "subspace (f ` S)"
huffman@44133
   697
  using lf sS linear_0[OF lf]
huffman@53600
   698
  unfolding linear_iff subspace_def
huffman@44133
   699
  apply (auto simp add: image_iff)
wenzelm@53406
   700
  apply (rule_tac x="x + y" in bexI)
wenzelm@53406
   701
  apply auto
wenzelm@53406
   702
  apply (rule_tac x="c *\<^sub>R x" in bexI)
wenzelm@53406
   703
  apply auto
huffman@44133
   704
  done
huffman@44133
   705
huffman@44521
   706
lemma subspace_linear_vimage: "linear f \<Longrightarrow> subspace S \<Longrightarrow> subspace (f -` S)"
huffman@53600
   707
  by (auto simp add: subspace_def linear_iff linear_0[of f])
huffman@44521
   708
wenzelm@53406
   709
lemma subspace_linear_preimage: "linear f \<Longrightarrow> subspace S \<Longrightarrow> subspace {x. f x \<in> S}"
huffman@53600
   710
  by (auto simp add: subspace_def linear_iff linear_0[of f])
huffman@44133
   711
huffman@44133
   712
lemma subspace_trivial: "subspace {0}"
huffman@44133
   713
  by (simp add: subspace_def)
huffman@44133
   714
wenzelm@53406
   715
lemma (in real_vector) subspace_inter: "subspace A \<Longrightarrow> subspace B \<Longrightarrow> subspace (A \<inter> B)"
huffman@44133
   716
  by (simp add: subspace_def)
huffman@44133
   717
wenzelm@53406
   718
lemma subspace_Times: "subspace A \<Longrightarrow> subspace B \<Longrightarrow> subspace (A \<times> B)"
huffman@44521
   719
  unfolding subspace_def zero_prod_def by simp
huffman@44521
   720
huffman@44521
   721
text {* Properties of span. *}
huffman@44521
   722
wenzelm@53406
   723
lemma (in real_vector) span_mono: "A \<subseteq> B \<Longrightarrow> span A \<subseteq> span B"
huffman@44133
   724
  by (metis span_def hull_mono)
huffman@44133
   725
wenzelm@53406
   726
lemma (in real_vector) subspace_span: "subspace (span S)"
huffman@44133
   727
  unfolding span_def
huffman@44170
   728
  apply (rule hull_in)
huffman@44133
   729
  apply (simp only: subspace_def Inter_iff Int_iff subset_eq)
huffman@44133
   730
  apply auto
huffman@44133
   731
  done
huffman@44133
   732
huffman@44133
   733
lemma (in real_vector) span_clauses:
wenzelm@53406
   734
  "a \<in> S \<Longrightarrow> a \<in> span S"
huffman@44133
   735
  "0 \<in> span S"
wenzelm@53406
   736
  "x\<in> span S \<Longrightarrow> y \<in> span S \<Longrightarrow> x + y \<in> span S"
huffman@44133
   737
  "x \<in> span S \<Longrightarrow> c *\<^sub>R x \<in> span S"
wenzelm@53406
   738
  by (metis span_def hull_subset subset_eq) (metis subspace_span subspace_def)+
huffman@44133
   739
huffman@44521
   740
lemma span_unique:
wenzelm@49522
   741
  "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
   742
  unfolding span_def by (rule hull_unique)
huffman@44521
   743
huffman@44521
   744
lemma span_minimal: "S \<subseteq> T \<Longrightarrow> subspace T \<Longrightarrow> span S \<subseteq> T"
huffman@44521
   745
  unfolding span_def by (rule hull_minimal)
huffman@44521
   746
huffman@44521
   747
lemma (in real_vector) span_induct:
wenzelm@49522
   748
  assumes x: "x \<in> span S"
wenzelm@49522
   749
    and P: "subspace P"
wenzelm@53406
   750
    and SP: "\<And>x. x \<in> S \<Longrightarrow> x \<in> P"
huffman@44521
   751
  shows "x \<in> P"
wenzelm@49522
   752
proof -
wenzelm@53406
   753
  from SP have SP': "S \<subseteq> P"
wenzelm@53406
   754
    by (simp add: subset_eq)
huffman@44170
   755
  from x hull_minimal[where S=subspace, OF SP' P, unfolded span_def[symmetric]]
wenzelm@53406
   756
  show "x \<in> P"
wenzelm@53406
   757
    by (metis subset_eq)
huffman@44133
   758
qed
huffman@44133
   759
huffman@44133
   760
lemma span_empty[simp]: "span {} = {0}"
huffman@44133
   761
  apply (simp add: span_def)
huffman@44133
   762
  apply (rule hull_unique)
huffman@44170
   763
  apply (auto simp add: subspace_def)
huffman@44133
   764
  done
huffman@44133
   765
huffman@44133
   766
lemma (in real_vector) independent_empty[intro]: "independent {}"
huffman@44133
   767
  by (simp add: dependent_def)
huffman@44133
   768
wenzelm@49522
   769
lemma dependent_single[simp]: "dependent {x} \<longleftrightarrow> x = 0"
huffman@44133
   770
  unfolding dependent_def by auto
huffman@44133
   771
wenzelm@53406
   772
lemma (in real_vector) independent_mono: "independent A \<Longrightarrow> B \<subseteq> A \<Longrightarrow> independent B"
huffman@44133
   773
  apply (clarsimp simp add: dependent_def span_mono)
huffman@44133
   774
  apply (subgoal_tac "span (B - {a}) \<le> span (A - {a})")
huffman@44133
   775
  apply force
huffman@44133
   776
  apply (rule span_mono)
huffman@44133
   777
  apply auto
huffman@44133
   778
  done
huffman@44133
   779
huffman@44133
   780
lemma (in real_vector) span_subspace: "A \<subseteq> B \<Longrightarrow> B \<le> span A \<Longrightarrow>  subspace B \<Longrightarrow> span A = B"
huffman@44170
   781
  by (metis order_antisym span_def hull_minimal)
huffman@44133
   782
wenzelm@49711
   783
lemma (in real_vector) span_induct':
wenzelm@49711
   784
  assumes SP: "\<forall>x \<in> S. P x"
wenzelm@49711
   785
    and P: "subspace {x. P x}"
wenzelm@49711
   786
  shows "\<forall>x \<in> span S. P x"
huffman@44133
   787
  using span_induct SP P by blast
huffman@44133
   788
wenzelm@56444
   789
inductive_set (in real_vector) span_induct_alt_help for S :: "'a set"
wenzelm@53406
   790
where
huffman@44170
   791
  span_induct_alt_help_0: "0 \<in> span_induct_alt_help S"
wenzelm@49522
   792
| span_induct_alt_help_S:
wenzelm@53406
   793
    "x \<in> S \<Longrightarrow> z \<in> span_induct_alt_help S \<Longrightarrow>
wenzelm@53406
   794
      (c *\<^sub>R x + z) \<in> span_induct_alt_help S"
huffman@44133
   795
huffman@44133
   796
lemma span_induct_alt':
wenzelm@53406
   797
  assumes h0: "h 0"
wenzelm@53406
   798
    and hS: "\<And>c x y. x \<in> S \<Longrightarrow> h y \<Longrightarrow> h (c *\<^sub>R x + y)"
wenzelm@49522
   799
  shows "\<forall>x \<in> span S. h x"
wenzelm@49522
   800
proof -
wenzelm@53406
   801
  {
wenzelm@53406
   802
    fix x :: 'a
wenzelm@53406
   803
    assume x: "x \<in> span_induct_alt_help S"
huffman@44133
   804
    have "h x"
huffman@44133
   805
      apply (rule span_induct_alt_help.induct[OF x])
huffman@44133
   806
      apply (rule h0)
wenzelm@53406
   807
      apply (rule hS)
wenzelm@53406
   808
      apply assumption
wenzelm@53406
   809
      apply assumption
wenzelm@53406
   810
      done
wenzelm@53406
   811
  }
huffman@44133
   812
  note th0 = this
wenzelm@53406
   813
  {
wenzelm@53406
   814
    fix x
wenzelm@53406
   815
    assume x: "x \<in> span S"
huffman@44170
   816
    have "x \<in> span_induct_alt_help S"
wenzelm@49522
   817
    proof (rule span_induct[where x=x and S=S])
wenzelm@53406
   818
      show "x \<in> span S" by (rule x)
wenzelm@49522
   819
    next
wenzelm@53406
   820
      fix x
wenzelm@53406
   821
      assume xS: "x \<in> S"
wenzelm@53406
   822
      from span_induct_alt_help_S[OF xS span_induct_alt_help_0, of 1]
wenzelm@53406
   823
      show "x \<in> span_induct_alt_help S"
wenzelm@53406
   824
        by simp
wenzelm@49522
   825
    next
wenzelm@49522
   826
      have "0 \<in> span_induct_alt_help S" by (rule span_induct_alt_help_0)
wenzelm@49522
   827
      moreover
wenzelm@53406
   828
      {
wenzelm@53406
   829
        fix x y
wenzelm@49522
   830
        assume h: "x \<in> span_induct_alt_help S" "y \<in> span_induct_alt_help S"
wenzelm@49522
   831
        from h have "(x + y) \<in> span_induct_alt_help S"
wenzelm@49522
   832
          apply (induct rule: span_induct_alt_help.induct)
wenzelm@49522
   833
          apply simp
haftmann@57512
   834
          unfolding add.assoc
wenzelm@49522
   835
          apply (rule span_induct_alt_help_S)
wenzelm@49522
   836
          apply assumption
wenzelm@49522
   837
          apply simp
wenzelm@53406
   838
          done
wenzelm@53406
   839
      }
wenzelm@49522
   840
      moreover
wenzelm@53406
   841
      {
wenzelm@53406
   842
        fix c x
wenzelm@49522
   843
        assume xt: "x \<in> span_induct_alt_help S"
wenzelm@49522
   844
        then have "(c *\<^sub>R x) \<in> span_induct_alt_help S"
wenzelm@49522
   845
          apply (induct rule: span_induct_alt_help.induct)
wenzelm@49522
   846
          apply (simp add: span_induct_alt_help_0)
wenzelm@49522
   847
          apply (simp add: scaleR_right_distrib)
wenzelm@49522
   848
          apply (rule span_induct_alt_help_S)
wenzelm@49522
   849
          apply assumption
wenzelm@49522
   850
          apply simp
wenzelm@49522
   851
          done }
wenzelm@53406
   852
      ultimately show "subspace (span_induct_alt_help S)"
wenzelm@49522
   853
        unfolding subspace_def Ball_def by blast
wenzelm@53406
   854
    qed
wenzelm@53406
   855
  }
huffman@44133
   856
  with th0 show ?thesis by blast
huffman@44133
   857
qed
huffman@44133
   858
huffman@44133
   859
lemma span_induct_alt:
wenzelm@53406
   860
  assumes h0: "h 0"
wenzelm@53406
   861
    and hS: "\<And>c x y. x \<in> S \<Longrightarrow> h y \<Longrightarrow> h (c *\<^sub>R x + y)"
wenzelm@53406
   862
    and x: "x \<in> span S"
huffman@44133
   863
  shows "h x"
wenzelm@49522
   864
  using span_induct_alt'[of h S] h0 hS x by blast
huffman@44133
   865
huffman@44133
   866
text {* Individual closure properties. *}
huffman@44133
   867
huffman@44133
   868
lemma span_span: "span (span A) = span A"
huffman@44133
   869
  unfolding span_def hull_hull ..
huffman@44133
   870
wenzelm@53406
   871
lemma (in real_vector) span_superset: "x \<in> S \<Longrightarrow> x \<in> span S"
wenzelm@53406
   872
  by (metis span_clauses(1))
wenzelm@53406
   873
wenzelm@53406
   874
lemma (in real_vector) span_0: "0 \<in> span S"
wenzelm@53406
   875
  by (metis subspace_span subspace_0)
huffman@44133
   876
huffman@44133
   877
lemma span_inc: "S \<subseteq> span S"
huffman@44133
   878
  by (metis subset_eq span_superset)
huffman@44133
   879
wenzelm@53406
   880
lemma (in real_vector) dependent_0:
wenzelm@53406
   881
  assumes "0 \<in> A"
wenzelm@53406
   882
  shows "dependent A"
wenzelm@53406
   883
  unfolding dependent_def
wenzelm@53406
   884
  using assms span_0
lp15@60162
   885
  by auto
wenzelm@53406
   886
wenzelm@53406
   887
lemma (in real_vector) span_add: "x \<in> span S \<Longrightarrow> y \<in> span S \<Longrightarrow> x + y \<in> span S"
huffman@44133
   888
  by (metis subspace_add subspace_span)
huffman@44133
   889
wenzelm@53406
   890
lemma (in real_vector) span_mul: "x \<in> span S \<Longrightarrow> c *\<^sub>R x \<in> span S"
huffman@44133
   891
  by (metis subspace_span subspace_mul)
huffman@44133
   892
wenzelm@53406
   893
lemma span_neg: "x \<in> span S \<Longrightarrow> - x \<in> span S"
huffman@44133
   894
  by (metis subspace_neg subspace_span)
huffman@44133
   895
wenzelm@53406
   896
lemma span_sub: "x \<in> span S \<Longrightarrow> y \<in> span S \<Longrightarrow> x - y \<in> span S"
huffman@44133
   897
  by (metis subspace_span subspace_sub)
huffman@44133
   898
huffman@56196
   899
lemma (in real_vector) span_setsum: "\<forall>x\<in>A. f x \<in> span S \<Longrightarrow> setsum f A \<in> span S"
huffman@56196
   900
  by (rule subspace_setsum [OF subspace_span])
huffman@44133
   901
huffman@44133
   902
lemma span_add_eq: "x \<in> span S \<Longrightarrow> x + y \<in> span S \<longleftrightarrow> y \<in> span S"
lp15@55775
   903
  by (metis add_minus_cancel scaleR_minus1_left subspace_def subspace_span)
huffman@44133
   904
huffman@44133
   905
text {* Mapping under linear image. *}
huffman@44133
   906
huffman@44521
   907
lemma span_linear_image:
huffman@44521
   908
  assumes lf: "linear f"
wenzelm@56444
   909
  shows "span (f ` S) = f ` span S"
huffman@44521
   910
proof (rule span_unique)
huffman@44521
   911
  show "f ` S \<subseteq> f ` span S"
huffman@44521
   912
    by (intro image_mono span_inc)
huffman@44521
   913
  show "subspace (f ` span S)"
huffman@44521
   914
    using lf subspace_span by (rule subspace_linear_image)
huffman@44521
   915
next
wenzelm@53406
   916
  fix T
wenzelm@53406
   917
  assume "f ` S \<subseteq> T" and "subspace T"
wenzelm@49522
   918
  then show "f ` span S \<subseteq> T"
huffman@44521
   919
    unfolding image_subset_iff_subset_vimage
huffman@44521
   920
    by (intro span_minimal subspace_linear_vimage lf)
huffman@44521
   921
qed
huffman@44521
   922
huffman@44521
   923
lemma span_union: "span (A \<union> B) = (\<lambda>(a, b). a + b) ` (span A \<times> span B)"
huffman@44521
   924
proof (rule span_unique)
huffman@44521
   925
  show "A \<union> B \<subseteq> (\<lambda>(a, b). a + b) ` (span A \<times> span B)"
huffman@44521
   926
    by safe (force intro: span_clauses)+
huffman@44521
   927
next
huffman@44521
   928
  have "linear (\<lambda>(a, b). a + b)"
huffman@53600
   929
    by (simp add: linear_iff scaleR_add_right)
huffman@44521
   930
  moreover have "subspace (span A \<times> span B)"
huffman@44521
   931
    by (intro subspace_Times subspace_span)
huffman@44521
   932
  ultimately show "subspace ((\<lambda>(a, b). a + b) ` (span A \<times> span B))"
huffman@44521
   933
    by (rule subspace_linear_image)
huffman@44521
   934
next
wenzelm@49711
   935
  fix T
wenzelm@49711
   936
  assume "A \<union> B \<subseteq> T" and "subspace T"
wenzelm@49522
   937
  then show "(\<lambda>(a, b). a + b) ` (span A \<times> span B) \<subseteq> T"
huffman@44521
   938
    by (auto intro!: subspace_add elim: span_induct)
huffman@44133
   939
qed
huffman@44133
   940
huffman@44133
   941
text {* The key breakdown property. *}
huffman@44133
   942
huffman@44521
   943
lemma span_singleton: "span {x} = range (\<lambda>k. k *\<^sub>R x)"
huffman@44521
   944
proof (rule span_unique)
huffman@44521
   945
  show "{x} \<subseteq> range (\<lambda>k. k *\<^sub>R x)"
huffman@44521
   946
    by (fast intro: scaleR_one [symmetric])
huffman@44521
   947
  show "subspace (range (\<lambda>k. k *\<^sub>R x))"
huffman@44521
   948
    unfolding subspace_def
huffman@44521
   949
    by (auto intro: scaleR_add_left [symmetric])
wenzelm@53406
   950
next
wenzelm@53406
   951
  fix T
wenzelm@53406
   952
  assume "{x} \<subseteq> T" and "subspace T"
wenzelm@53406
   953
  then show "range (\<lambda>k. k *\<^sub>R x) \<subseteq> T"
huffman@44521
   954
    unfolding subspace_def by auto
huffman@44521
   955
qed
huffman@44521
   956
wenzelm@49522
   957
lemma span_insert: "span (insert a S) = {x. \<exists>k. (x - k *\<^sub>R a) \<in> span S}"
huffman@44521
   958
proof -
huffman@44521
   959
  have "span ({a} \<union> S) = {x. \<exists>k. (x - k *\<^sub>R a) \<in> span S}"
huffman@44521
   960
    unfolding span_union span_singleton
huffman@44521
   961
    apply safe
huffman@44521
   962
    apply (rule_tac x=k in exI, simp)
huffman@44521
   963
    apply (erule rev_image_eqI [OF SigmaI [OF rangeI]])
haftmann@54230
   964
    apply auto
huffman@44521
   965
    done
wenzelm@49522
   966
  then show ?thesis by simp
huffman@44521
   967
qed
huffman@44521
   968
huffman@44133
   969
lemma span_breakdown:
wenzelm@53406
   970
  assumes bS: "b \<in> S"
wenzelm@53406
   971
    and aS: "a \<in> span S"
huffman@44521
   972
  shows "\<exists>k. a - k *\<^sub>R b \<in> span (S - {b})"
huffman@44521
   973
  using assms span_insert [of b "S - {b}"]
huffman@44521
   974
  by (simp add: insert_absorb)
huffman@44133
   975
wenzelm@53406
   976
lemma span_breakdown_eq: "x \<in> span (insert a S) \<longleftrightarrow> (\<exists>k. x - k *\<^sub>R a \<in> span S)"
huffman@44521
   977
  by (simp add: span_insert)
huffman@44133
   978
huffman@44133
   979
text {* Hence some "reversal" results. *}
huffman@44133
   980
huffman@44133
   981
lemma in_span_insert:
wenzelm@49711
   982
  assumes a: "a \<in> span (insert b S)"
wenzelm@49711
   983
    and na: "a \<notin> span S"
huffman@44133
   984
  shows "b \<in> span (insert a S)"
wenzelm@49663
   985
proof -
huffman@55910
   986
  from a obtain k where k: "a - k *\<^sub>R b \<in> span S"
huffman@55910
   987
    unfolding span_insert by fast
wenzelm@53406
   988
  show ?thesis
wenzelm@53406
   989
  proof (cases "k = 0")
wenzelm@53406
   990
    case True
huffman@55910
   991
    with k have "a \<in> span S" by simp
huffman@55910
   992
    with na show ?thesis by simp
wenzelm@53406
   993
  next
wenzelm@53406
   994
    case False
huffman@55910
   995
    from k have "(- inverse k) *\<^sub>R (a - k *\<^sub>R b) \<in> span S"
huffman@44133
   996
      by (rule span_mul)
huffman@55910
   997
    then have "b - inverse k *\<^sub>R a \<in> span S"
huffman@55910
   998
      using `k \<noteq> 0` by (simp add: scaleR_diff_right)
huffman@55910
   999
    then show ?thesis
huffman@55910
  1000
      unfolding span_insert by fast
wenzelm@53406
  1001
  qed
huffman@44133
  1002
qed
huffman@44133
  1003
huffman@44133
  1004
lemma in_span_delete:
huffman@44133
  1005
  assumes a: "a \<in> span S"
wenzelm@53716
  1006
    and na: "a \<notin> span (S - {b})"
huffman@44133
  1007
  shows "b \<in> span (insert a (S - {b}))"
huffman@44133
  1008
  apply (rule in_span_insert)
huffman@44133
  1009
  apply (rule set_rev_mp)
huffman@44133
  1010
  apply (rule a)
huffman@44133
  1011
  apply (rule span_mono)
huffman@44133
  1012
  apply blast
huffman@44133
  1013
  apply (rule na)
huffman@44133
  1014
  done
huffman@44133
  1015
huffman@44133
  1016
text {* Transitivity property. *}
huffman@44133
  1017
huffman@44521
  1018
lemma span_redundant: "x \<in> span S \<Longrightarrow> span (insert x S) = span S"
huffman@44521
  1019
  unfolding span_def by (rule hull_redundant)
huffman@44521
  1020
huffman@44133
  1021
lemma span_trans:
wenzelm@53406
  1022
  assumes x: "x \<in> span S"
wenzelm@53406
  1023
    and y: "y \<in> span (insert x S)"
huffman@44133
  1024
  shows "y \<in> span S"
huffman@44521
  1025
  using assms by (simp only: span_redundant)
huffman@44133
  1026
huffman@44133
  1027
lemma span_insert_0[simp]: "span (insert 0 S) = span S"
huffman@44521
  1028
  by (simp only: span_redundant span_0)
huffman@44133
  1029
huffman@44133
  1030
text {* An explicit expansion is sometimes needed. *}
huffman@44133
  1031
huffman@44133
  1032
lemma span_explicit:
huffman@44133
  1033
  "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
  1034
  (is "_ = ?E" is "_ = {y. ?h y}" is "_ = {y. \<exists>S u. ?Q S u y}")
wenzelm@49663
  1035
proof -
wenzelm@53406
  1036
  {
wenzelm@53406
  1037
    fix x
huffman@55910
  1038
    assume "?h x"
huffman@55910
  1039
    then obtain S u where "finite S" and "S \<subseteq> P" and "setsum (\<lambda>v. u v *\<^sub>R v) S = x"
huffman@44133
  1040
      by blast
huffman@55910
  1041
    then have "x \<in> span P"
huffman@55910
  1042
      by (auto intro: span_setsum span_mul span_superset)
wenzelm@53406
  1043
  }
huffman@44133
  1044
  moreover
huffman@55910
  1045
  have "\<forall>x \<in> span P. ?h x"
wenzelm@49522
  1046
  proof (rule span_induct_alt')
huffman@55910
  1047
    show "?h 0"
huffman@55910
  1048
      by (rule exI[where x="{}"], simp)
huffman@44133
  1049
  next
huffman@44133
  1050
    fix c x y
wenzelm@53406
  1051
    assume x: "x \<in> P"
huffman@55910
  1052
    assume hy: "?h y"
huffman@44133
  1053
    from hy obtain S u where fS: "finite S" and SP: "S\<subseteq>P"
huffman@44133
  1054
      and u: "setsum (\<lambda>v. u v *\<^sub>R v) S = y" by blast
huffman@44133
  1055
    let ?S = "insert x S"
wenzelm@49522
  1056
    let ?u = "\<lambda>y. if y = x then (if x \<in> S then u y + c else c) else u y"
wenzelm@53406
  1057
    from fS SP x have th0: "finite (insert x S)" "insert x S \<subseteq> P"
wenzelm@53406
  1058
      by blast+
wenzelm@53406
  1059
    have "?Q ?S ?u (c*\<^sub>R x + y)"
wenzelm@53406
  1060
    proof cases
wenzelm@53406
  1061
      assume xS: "x \<in> S"
huffman@55910
  1062
      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
  1063
        using xS by (simp add: setsum.remove [OF fS xS] insert_absorb)
huffman@44133
  1064
      also have "\<dots> = (\<Sum>v\<in>S. u v *\<^sub>R v) + c *\<^sub>R x"
huffman@55910
  1065
        by (simp add: setsum.remove [OF fS xS] algebra_simps)
huffman@44133
  1066
      also have "\<dots> = c*\<^sub>R x + y"
haftmann@57512
  1067
        by (simp add: add.commute u)
huffman@44133
  1068
      finally have "setsum (\<lambda>v. ?u v *\<^sub>R v) ?S = c*\<^sub>R x + y" .
wenzelm@53406
  1069
      then show ?thesis using th0 by blast
wenzelm@53406
  1070
    next
wenzelm@53406
  1071
      assume xS: "x \<notin> S"
wenzelm@49522
  1072
      have th00: "(\<Sum>v\<in>S. (if v = x then c else u v) *\<^sub>R v) = y"
wenzelm@49522
  1073
        unfolding u[symmetric]
haftmann@57418
  1074
        apply (rule setsum.cong)
wenzelm@53406
  1075
        using xS
wenzelm@53406
  1076
        apply auto
wenzelm@49522
  1077
        done
wenzelm@53406
  1078
      show ?thesis using fS xS th0
haftmann@57512
  1079
        by (simp add: th00 add.commute cong del: if_weak_cong)
wenzelm@53406
  1080
    qed
huffman@55910
  1081
    then show "?h (c*\<^sub>R x + y)"
huffman@55910
  1082
      by fast
huffman@44133
  1083
  qed
huffman@44133
  1084
  ultimately show ?thesis by blast
huffman@44133
  1085
qed
huffman@44133
  1086
huffman@44133
  1087
lemma dependent_explicit:
wenzelm@49522
  1088
  "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
  1089
  (is "?lhs = ?rhs")
wenzelm@49522
  1090
proof -
wenzelm@53406
  1091
  {
wenzelm@53406
  1092
    assume dP: "dependent P"
huffman@44133
  1093
    then obtain a S u where aP: "a \<in> P" and fS: "finite S"
huffman@44133
  1094
      and SP: "S \<subseteq> P - {a}" and ua: "setsum (\<lambda>v. u v *\<^sub>R v) S = a"
huffman@44133
  1095
      unfolding dependent_def span_explicit by blast
huffman@44133
  1096
    let ?S = "insert a S"
huffman@44133
  1097
    let ?u = "\<lambda>y. if y = a then - 1 else u y"
huffman@44133
  1098
    let ?v = a
wenzelm@53406
  1099
    from aP SP have aS: "a \<notin> S"
wenzelm@53406
  1100
      by blast
wenzelm@53406
  1101
    from fS SP aP have th0: "finite ?S" "?S \<subseteq> P" "?v \<in> ?S" "?u ?v \<noteq> 0"
wenzelm@53406
  1102
      by auto
huffman@44133
  1103
    have s0: "setsum (\<lambda>v. ?u v *\<^sub>R v) ?S = 0"
huffman@44133
  1104
      using fS aS
huffman@55910
  1105
      apply simp
huffman@44133
  1106
      apply (subst (2) ua[symmetric])
haftmann@57418
  1107
      apply (rule setsum.cong)
wenzelm@49522
  1108
      apply auto
wenzelm@49522
  1109
      done
huffman@55910
  1110
    with th0 have ?rhs by fast
wenzelm@49522
  1111
  }
huffman@44133
  1112
  moreover
wenzelm@53406
  1113
  {
wenzelm@53406
  1114
    fix S u v
wenzelm@49522
  1115
    assume fS: "finite S"
wenzelm@53406
  1116
      and SP: "S \<subseteq> P"
wenzelm@53406
  1117
      and vS: "v \<in> S"
wenzelm@53406
  1118
      and uv: "u v \<noteq> 0"
wenzelm@49522
  1119
      and u: "setsum (\<lambda>v. u v *\<^sub>R v) S = 0"
huffman@44133
  1120
    let ?a = v
huffman@44133
  1121
    let ?S = "S - {v}"
huffman@44133
  1122
    let ?u = "\<lambda>i. (- u i) / u v"
wenzelm@53406
  1123
    have th0: "?a \<in> P" "finite ?S" "?S \<subseteq> P"
wenzelm@53406
  1124
      using fS SP vS by auto
wenzelm@53406
  1125
    have "setsum (\<lambda>v. ?u v *\<^sub>R v) ?S =
wenzelm@53406
  1126
      setsum (\<lambda>v. (- (inverse (u ?a))) *\<^sub>R (u v *\<^sub>R v)) S - ?u v *\<^sub>R v"
hoelzl@56480
  1127
      using fS vS uv by (simp add: setsum_diff1 field_simps)
wenzelm@53406
  1128
    also have "\<dots> = ?a"
hoelzl@56479
  1129
      unfolding scaleR_right.setsum [symmetric] u using uv by simp
wenzelm@53406
  1130
    finally have "setsum (\<lambda>v. ?u v *\<^sub>R v) ?S = ?a" .
huffman@44133
  1131
    with th0 have ?lhs
huffman@44133
  1132
      unfolding dependent_def span_explicit
huffman@44133
  1133
      apply -
huffman@44133
  1134
      apply (rule bexI[where x= "?a"])
huffman@44133
  1135
      apply (simp_all del: scaleR_minus_left)
huffman@44133
  1136
      apply (rule exI[where x= "?S"])
wenzelm@49522
  1137
      apply (auto simp del: scaleR_minus_left)
wenzelm@49522
  1138
      done
wenzelm@49522
  1139
  }
huffman@44133
  1140
  ultimately show ?thesis by blast
huffman@44133
  1141
qed
huffman@44133
  1142
huffman@44133
  1143
huffman@44133
  1144
lemma span_finite:
huffman@44133
  1145
  assumes fS: "finite S"
huffman@44133
  1146
  shows "span S = {y. \<exists>u. setsum (\<lambda>v. u v *\<^sub>R v) S = y}"
huffman@44133
  1147
  (is "_ = ?rhs")
wenzelm@49522
  1148
proof -
wenzelm@53406
  1149
  {
wenzelm@53406
  1150
    fix y
wenzelm@49711
  1151
    assume y: "y \<in> span S"
wenzelm@53406
  1152
    from y obtain S' u where fS': "finite S'"
wenzelm@53406
  1153
      and SS': "S' \<subseteq> S"
wenzelm@53406
  1154
      and u: "setsum (\<lambda>v. u v *\<^sub>R v) S' = y"
wenzelm@53406
  1155
      unfolding span_explicit by blast
huffman@44133
  1156
    let ?u = "\<lambda>x. if x \<in> S' then u x else 0"
huffman@44133
  1157
    have "setsum (\<lambda>v. ?u v *\<^sub>R v) S = setsum (\<lambda>v. u v *\<^sub>R v) S'"
haftmann@57418
  1158
      using SS' fS by (auto intro!: setsum.mono_neutral_cong_right)
wenzelm@49522
  1159
    then have "setsum (\<lambda>v. ?u v *\<^sub>R v) S = y" by (metis u)
wenzelm@53406
  1160
    then have "y \<in> ?rhs" by auto
wenzelm@53406
  1161
  }
huffman@44133
  1162
  moreover
wenzelm@53406
  1163
  {
wenzelm@53406
  1164
    fix y u
wenzelm@49522
  1165
    assume u: "setsum (\<lambda>v. u v *\<^sub>R v) S = y"
wenzelm@53406
  1166
    then have "y \<in> span S" using fS unfolding span_explicit by auto
wenzelm@53406
  1167
  }
huffman@44133
  1168
  ultimately show ?thesis by blast
huffman@44133
  1169
qed
huffman@44133
  1170
huffman@44133
  1171
text {* This is useful for building a basis step-by-step. *}
huffman@44133
  1172
huffman@44133
  1173
lemma independent_insert:
wenzelm@53406
  1174
  "independent (insert a S) \<longleftrightarrow>
wenzelm@53406
  1175
    (if a \<in> S then independent S else independent S \<and> a \<notin> span S)"
wenzelm@53406
  1176
  (is "?lhs \<longleftrightarrow> ?rhs")
wenzelm@53406
  1177
proof (cases "a \<in> S")
wenzelm@53406
  1178
  case True
wenzelm@53406
  1179
  then show ?thesis
wenzelm@53406
  1180
    using insert_absorb[OF True] by simp
wenzelm@53406
  1181
next
wenzelm@53406
  1182
  case False
wenzelm@53406
  1183
  show ?thesis
wenzelm@53406
  1184
  proof
wenzelm@53406
  1185
    assume i: ?lhs
wenzelm@53406
  1186
    then show ?rhs
wenzelm@53406
  1187
      using False
wenzelm@53406
  1188
      apply simp
wenzelm@53406
  1189
      apply (rule conjI)
wenzelm@53406
  1190
      apply (rule independent_mono)
wenzelm@53406
  1191
      apply assumption
wenzelm@53406
  1192
      apply blast
wenzelm@53406
  1193
      apply (simp add: dependent_def)
wenzelm@53406
  1194
      done
wenzelm@53406
  1195
  next
wenzelm@53406
  1196
    assume i: ?rhs
wenzelm@53406
  1197
    show ?lhs
wenzelm@53406
  1198
      using i False
wenzelm@53406
  1199
      apply (auto simp add: dependent_def)
lp15@55775
  1200
      by (metis in_span_insert insert_Diff insert_Diff_if insert_iff)
wenzelm@53406
  1201
  qed
huffman@44133
  1202
qed
huffman@44133
  1203
huffman@44133
  1204
text {* The degenerate case of the Exchange Lemma. *}
huffman@44133
  1205
huffman@44133
  1206
lemma spanning_subset_independent:
wenzelm@49711
  1207
  assumes BA: "B \<subseteq> A"
wenzelm@49711
  1208
    and iA: "independent A"
wenzelm@49522
  1209
    and AsB: "A \<subseteq> span B"
huffman@44133
  1210
  shows "A = B"
huffman@44133
  1211
proof
wenzelm@49663
  1212
  show "B \<subseteq> A" by (rule BA)
wenzelm@49663
  1213
huffman@44133
  1214
  from span_mono[OF BA] span_mono[OF AsB]
huffman@44133
  1215
  have sAB: "span A = span B" unfolding span_span by blast
huffman@44133
  1216
wenzelm@53406
  1217
  {
wenzelm@53406
  1218
    fix x
wenzelm@53406
  1219
    assume x: "x \<in> A"
huffman@44133
  1220
    from iA have th0: "x \<notin> span (A - {x})"
huffman@44133
  1221
      unfolding dependent_def using x by blast
wenzelm@53406
  1222
    from x have xsA: "x \<in> span A"
wenzelm@53406
  1223
      by (blast intro: span_superset)
huffman@44133
  1224
    have "A - {x} \<subseteq> A" by blast
wenzelm@53406
  1225
    then have th1: "span (A - {x}) \<subseteq> span A"
wenzelm@53406
  1226
      by (metis span_mono)
wenzelm@53406
  1227
    {
wenzelm@53406
  1228
      assume xB: "x \<notin> B"
wenzelm@53406
  1229
      from xB BA have "B \<subseteq> A - {x}"
wenzelm@53406
  1230
        by blast
wenzelm@53406
  1231
      then have "span B \<subseteq> span (A - {x})"
wenzelm@53406
  1232
        by (metis span_mono)
wenzelm@53406
  1233
      with th1 th0 sAB have "x \<notin> span A"
wenzelm@53406
  1234
        by blast
wenzelm@53406
  1235
      with x have False
wenzelm@53406
  1236
        by (metis span_superset)
wenzelm@53406
  1237
    }
wenzelm@53406
  1238
    then have "x \<in> B" by blast
wenzelm@53406
  1239
  }
huffman@44133
  1240
  then show "A \<subseteq> B" by blast
huffman@44133
  1241
qed
huffman@44133
  1242
huffman@44133
  1243
text {* The general case of the Exchange Lemma, the key to what follows. *}
huffman@44133
  1244
huffman@44133
  1245
lemma exchange_lemma:
wenzelm@49711
  1246
  assumes f:"finite t"
wenzelm@49711
  1247
    and i: "independent s"
wenzelm@49711
  1248
    and sp: "s \<subseteq> span t"
wenzelm@53406
  1249
  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
  1250
  using f i sp
wenzelm@49522
  1251
proof (induct "card (t - s)" arbitrary: s t rule: less_induct)
huffman@44133
  1252
  case less
huffman@44133
  1253
  note ft = `finite t` and s = `independent s` and sp = `s \<subseteq> span t`
wenzelm@53406
  1254
  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
  1255
  let ?ths = "\<exists>t'. ?P t'"
wenzelm@53406
  1256
  {
lp15@55775
  1257
    assume "s \<subseteq> t"
lp15@55775
  1258
    then have ?ths
lp15@55775
  1259
      by (metis ft Un_commute sp sup_ge1)
wenzelm@53406
  1260
  }
huffman@44133
  1261
  moreover
wenzelm@53406
  1262
  {
wenzelm@53406
  1263
    assume st: "t \<subseteq> s"
wenzelm@53406
  1264
    from spanning_subset_independent[OF st s sp] st ft span_mono[OF st]
wenzelm@53406
  1265
    have ?ths
lp15@55775
  1266
      by (metis Un_absorb sp)
wenzelm@53406
  1267
  }
huffman@44133
  1268
  moreover
wenzelm@53406
  1269
  {
wenzelm@53406
  1270
    assume st: "\<not> s \<subseteq> t" "\<not> t \<subseteq> s"
wenzelm@53406
  1271
    from st(2) obtain b where b: "b \<in> t" "b \<notin> s"
wenzelm@53406
  1272
      by blast
wenzelm@53406
  1273
    from b have "t - {b} - s \<subset> t - s"
wenzelm@53406
  1274
      by blast
wenzelm@53406
  1275
    then have cardlt: "card (t - {b} - s) < card (t - s)"
wenzelm@53406
  1276
      using ft by (auto intro: psubset_card_mono)
wenzelm@53406
  1277
    from b ft have ct0: "card t \<noteq> 0"
wenzelm@53406
  1278
      by auto
wenzelm@53406
  1279
    have ?ths
wenzelm@53406
  1280
    proof cases
wenzelm@53716
  1281
      assume stb: "s \<subseteq> span (t - {b})"
wenzelm@53716
  1282
      from ft have ftb: "finite (t - {b})"
wenzelm@53406
  1283
        by auto
huffman@44133
  1284
      from less(1)[OF cardlt ftb s stb]
wenzelm@53716
  1285
      obtain u where u: "card u = card (t - {b})" "s \<subseteq> u" "u \<subseteq> s \<union> (t - {b})" "s \<subseteq> span u"
wenzelm@49522
  1286
        and fu: "finite u" by blast
huffman@44133
  1287
      let ?w = "insert b u"
wenzelm@53406
  1288
      have th0: "s \<subseteq> insert b u"
wenzelm@53406
  1289
        using u by blast
wenzelm@53406
  1290
      from u(3) b have "u \<subseteq> s \<union> t"
wenzelm@53406
  1291
        by blast
wenzelm@53406
  1292
      then have th1: "insert b u \<subseteq> s \<union> t"
wenzelm@53406
  1293
        using u b by blast
wenzelm@53406
  1294
      have bu: "b \<notin> u"
wenzelm@53406
  1295
        using b u by blast
wenzelm@53406
  1296
      from u(1) ft b have "card u = (card t - 1)"
wenzelm@53406
  1297
        by auto
wenzelm@49522
  1298
      then have th2: "card (insert b u) = card t"
huffman@44133
  1299
        using card_insert_disjoint[OF fu bu] ct0 by auto
huffman@44133
  1300
      from u(4) have "s \<subseteq> span u" .
wenzelm@53406
  1301
      also have "\<dots> \<subseteq> span (insert b u)"
wenzelm@53406
  1302
        by (rule span_mono) blast
huffman@44133
  1303
      finally have th3: "s \<subseteq> span (insert b u)" .
wenzelm@53406
  1304
      from th0 th1 th2 th3 fu have th: "?P ?w"
wenzelm@53406
  1305
        by blast
wenzelm@53406
  1306
      from th show ?thesis by blast
wenzelm@53406
  1307
    next
wenzelm@53716
  1308
      assume stb: "\<not> s \<subseteq> span (t - {b})"
wenzelm@53406
  1309
      from stb obtain a where a: "a \<in> s" "a \<notin> span (t - {b})"
wenzelm@53406
  1310
        by blast
wenzelm@53406
  1311
      have ab: "a \<noteq> b"
wenzelm@53406
  1312
        using a b by blast
wenzelm@53406
  1313
      have at: "a \<notin> t"
wenzelm@53406
  1314
        using a ab span_superset[of a "t- {b}"] by auto
huffman@44133
  1315
      have mlt: "card ((insert a (t - {b})) - s) < card (t - s)"
huffman@44133
  1316
        using cardlt ft a b by auto
wenzelm@53406
  1317
      have ft': "finite (insert a (t - {b}))"
wenzelm@53406
  1318
        using ft by auto
wenzelm@53406
  1319
      {
wenzelm@53406
  1320
        fix x
wenzelm@53406
  1321
        assume xs: "x \<in> s"
wenzelm@53406
  1322
        have t: "t \<subseteq> insert b (insert a (t - {b}))"
wenzelm@53406
  1323
          using b by auto
wenzelm@53406
  1324
        from b(1) have "b \<in> span t"
wenzelm@53406
  1325
          by (simp add: span_superset)
wenzelm@53406
  1326
        have bs: "b \<in> span (insert a (t - {b}))"
wenzelm@53406
  1327
          apply (rule in_span_delete)
wenzelm@53406
  1328
          using a sp unfolding subset_eq
wenzelm@53406
  1329
          apply auto
wenzelm@53406
  1330
          done
wenzelm@53406
  1331
        from xs sp have "x \<in> span t"
wenzelm@53406
  1332
          by blast
wenzelm@53406
  1333
        with span_mono[OF t] have x: "x \<in> span (insert b (insert a (t - {b})))" ..
wenzelm@53406
  1334
        from span_trans[OF bs x] have "x \<in> span (insert a (t - {b}))" .
wenzelm@53406
  1335
      }
wenzelm@53406
  1336
      then have sp': "s \<subseteq> span (insert a (t - {b}))"
wenzelm@53406
  1337
        by blast
wenzelm@53406
  1338
      from less(1)[OF mlt ft' s sp'] obtain u where u:
wenzelm@53716
  1339
        "card u = card (insert a (t - {b}))"
wenzelm@53716
  1340
        "finite u" "s \<subseteq> u" "u \<subseteq> s \<union> insert a (t - {b})"
wenzelm@53406
  1341
        "s \<subseteq> span u" by blast
wenzelm@53406
  1342
      from u a b ft at ct0 have "?P u"
wenzelm@53406
  1343
        by auto
wenzelm@53406
  1344
      then show ?thesis by blast
wenzelm@53406
  1345
    qed
huffman@44133
  1346
  }
wenzelm@49522
  1347
  ultimately show ?ths by blast
huffman@44133
  1348
qed
huffman@44133
  1349
huffman@44133
  1350
text {* This implies corresponding size bounds. *}
huffman@44133
  1351
huffman@44133
  1352
lemma independent_span_bound:
wenzelm@53406
  1353
  assumes f: "finite t"
wenzelm@53406
  1354
    and i: "independent s"
wenzelm@53406
  1355
    and sp: "s \<subseteq> span t"
huffman@44133
  1356
  shows "finite s \<and> card s \<le> card t"
huffman@44133
  1357
  by (metis exchange_lemma[OF f i sp] finite_subset card_mono)
huffman@44133
  1358
huffman@44133
  1359
lemma finite_Atleast_Atmost_nat[simp]: "finite {f x |x. x\<in> (UNIV::'a::finite set)}"
wenzelm@49522
  1360
proof -
wenzelm@53406
  1361
  have eq: "{f x |x. x\<in> UNIV} = f ` UNIV"
wenzelm@53406
  1362
    by auto
huffman@44133
  1363
  show ?thesis unfolding eq
huffman@44133
  1364
    apply (rule finite_imageI)
huffman@44133
  1365
    apply (rule finite)
huffman@44133
  1366
    done
huffman@44133
  1367
qed
huffman@44133
  1368
wenzelm@53406
  1369
wenzelm@53406
  1370
subsection {* Euclidean Spaces as Typeclass *}
huffman@44133
  1371
hoelzl@50526
  1372
lemma independent_Basis: "independent Basis"
hoelzl@50526
  1373
  unfolding dependent_def
hoelzl@50526
  1374
  apply (subst span_finite)
hoelzl@50526
  1375
  apply simp
huffman@44133
  1376
  apply clarify
hoelzl@50526
  1377
  apply (drule_tac f="inner a" in arg_cong)
hoelzl@50526
  1378
  apply (simp add: inner_Basis inner_setsum_right eq_commute)
hoelzl@50526
  1379
  done
hoelzl@50526
  1380
huffman@53939
  1381
lemma span_Basis [simp]: "span Basis = UNIV"
huffman@53939
  1382
  unfolding span_finite [OF finite_Basis]
huffman@53939
  1383
  by (fast intro: euclidean_representation)
huffman@44133
  1384
hoelzl@50526
  1385
lemma in_span_Basis: "x \<in> span Basis"
hoelzl@50526
  1386
  unfolding span_Basis ..
hoelzl@50526
  1387
hoelzl@50526
  1388
lemma Basis_le_norm: "b \<in> Basis \<Longrightarrow> \<bar>x \<bullet> b\<bar> \<le> norm x"
hoelzl@50526
  1389
  by (rule order_trans [OF Cauchy_Schwarz_ineq2]) simp
hoelzl@50526
  1390
hoelzl@50526
  1391
lemma norm_bound_Basis_le: "b \<in> Basis \<Longrightarrow> norm x \<le> e \<Longrightarrow> \<bar>x \<bullet> b\<bar> \<le> e"
hoelzl@50526
  1392
  by (metis Basis_le_norm order_trans)
hoelzl@50526
  1393
hoelzl@50526
  1394
lemma norm_bound_Basis_lt: "b \<in> Basis \<Longrightarrow> norm x < e \<Longrightarrow> \<bar>x \<bullet> b\<bar> < e"
huffman@53595
  1395
  by (metis Basis_le_norm le_less_trans)
hoelzl@50526
  1396
hoelzl@50526
  1397
lemma norm_le_l1: "norm x \<le> (\<Sum>b\<in>Basis. \<bar>x \<bullet> b\<bar>)"
hoelzl@50526
  1398
  apply (subst euclidean_representation[of x, symmetric])
huffman@44176
  1399
  apply (rule order_trans[OF norm_setsum])
wenzelm@49522
  1400
  apply (auto intro!: setsum_mono)
wenzelm@49522
  1401
  done
huffman@44133
  1402
huffman@44133
  1403
lemma setsum_norm_allsubsets_bound:
wenzelm@56444
  1404
  fixes f :: "'a \<Rightarrow> 'n::euclidean_space"
wenzelm@53406
  1405
  assumes fP: "finite P"
wenzelm@53406
  1406
    and fPs: "\<And>Q. Q \<subseteq> P \<Longrightarrow> norm (setsum f Q) \<le> e"
hoelzl@50526
  1407
  shows "(\<Sum>x\<in>P. norm (f x)) \<le> 2 * real DIM('n) * e"
wenzelm@49522
  1408
proof -
hoelzl@50526
  1409
  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
  1410
    by (rule setsum_mono) (rule norm_le_l1)
hoelzl@50526
  1411
  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
  1412
    by (rule setsum.commute)
hoelzl@50526
  1413
  also have "\<dots> \<le> of_nat (card (Basis :: 'n set)) * (2 * e)"
wenzelm@49522
  1414
  proof (rule setsum_bounded)
wenzelm@53406
  1415
    fix i :: 'n
wenzelm@53406
  1416
    assume i: "i \<in> Basis"
wenzelm@53406
  1417
    have "norm (\<Sum>x\<in>P. \<bar>f x \<bullet> i\<bar>) \<le>
hoelzl@50526
  1418
      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
  1419
      by (simp add: abs_real_def setsum.If_cases[OF fP] setsum_negf norm_triangle_ineq4 inner_setsum_left
wenzelm@56444
  1420
        del: real_norm_def)
wenzelm@53406
  1421
    also have "\<dots> \<le> e + e"
wenzelm@53406
  1422
      unfolding real_norm_def
hoelzl@50526
  1423
      by (intro add_mono norm_bound_Basis_le i fPs) auto
hoelzl@50526
  1424
    finally show "(\<Sum>x\<in>P. \<bar>f x \<bullet> i\<bar>) \<le> 2*e" by simp
huffman@44133
  1425
  qed
hoelzl@50526
  1426
  also have "\<dots> = 2 * real DIM('n) * e"
hoelzl@50526
  1427
    by (simp add: real_of_nat_def)
huffman@44133
  1428
  finally show ?thesis .
huffman@44133
  1429
qed
huffman@44133
  1430
wenzelm@53406
  1431
huffman@44133
  1432
subsection {* Linearity and Bilinearity continued *}
huffman@44133
  1433
huffman@44133
  1434
lemma linear_bounded:
wenzelm@56444
  1435
  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
huffman@44133
  1436
  assumes lf: "linear f"
huffman@44133
  1437
  shows "\<exists>B. \<forall>x. norm (f x) \<le> B * norm x"
huffman@53939
  1438
proof
hoelzl@50526
  1439
  let ?B = "\<Sum>b\<in>Basis. norm (f b)"
huffman@53939
  1440
  show "\<forall>x. norm (f x) \<le> ?B * norm x"
huffman@53939
  1441
  proof
wenzelm@53406
  1442
    fix x :: 'a
hoelzl@50526
  1443
    let ?g = "\<lambda>b. (x \<bullet> b) *\<^sub>R f b"
hoelzl@50526
  1444
    have "norm (f x) = norm (f (\<Sum>b\<in>Basis. (x \<bullet> b) *\<^sub>R b))"
hoelzl@50526
  1445
      unfolding euclidean_representation ..
hoelzl@50526
  1446
    also have "\<dots> = norm (setsum ?g Basis)"
huffman@53939
  1447
      by (simp add: linear_setsum [OF lf] linear_cmul [OF lf])
hoelzl@50526
  1448
    finally have th0: "norm (f x) = norm (setsum ?g Basis)" .
huffman@53939
  1449
    have th: "\<forall>b\<in>Basis. norm (?g b) \<le> norm (f b) * norm x"
huffman@53939
  1450
    proof
wenzelm@53406
  1451
      fix i :: 'a
wenzelm@53406
  1452
      assume i: "i \<in> Basis"
hoelzl@50526
  1453
      from Basis_le_norm[OF i, of x]
huffman@53939
  1454
      show "norm (?g i) \<le> norm (f i) * norm x"
wenzelm@49663
  1455
        unfolding norm_scaleR
haftmann@57512
  1456
        apply (subst mult.commute)
wenzelm@49663
  1457
        apply (rule mult_mono)
wenzelm@49663
  1458
        apply (auto simp add: field_simps)
wenzelm@53406
  1459
        done
huffman@53939
  1460
    qed
hoelzl@50526
  1461
    from setsum_norm_le[of _ ?g, OF th]
huffman@53939
  1462
    show "norm (f x) \<le> ?B * norm x"
wenzelm@53406
  1463
      unfolding th0 setsum_left_distrib by metis
huffman@53939
  1464
  qed
huffman@44133
  1465
qed
huffman@44133
  1466
huffman@44133
  1467
lemma linear_conv_bounded_linear:
huffman@44133
  1468
  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
huffman@44133
  1469
  shows "linear f \<longleftrightarrow> bounded_linear f"
huffman@44133
  1470
proof
huffman@44133
  1471
  assume "linear f"
huffman@53939
  1472
  then interpret f: linear f .
huffman@44133
  1473
  show "bounded_linear f"
huffman@44133
  1474
  proof
huffman@44133
  1475
    have "\<exists>B. \<forall>x. norm (f x) \<le> B * norm x"
huffman@44133
  1476
      using `linear f` by (rule linear_bounded)
wenzelm@49522
  1477
    then show "\<exists>K. \<forall>x. norm (f x) \<le> norm x * K"
haftmann@57512
  1478
      by (simp add: mult.commute)
huffman@44133
  1479
  qed
huffman@44133
  1480
next
huffman@44133
  1481
  assume "bounded_linear f"
huffman@44133
  1482
  then interpret f: bounded_linear f .
huffman@53939
  1483
  show "linear f" ..
huffman@53939
  1484
qed
huffman@53939
  1485
huffman@53939
  1486
lemma linear_bounded_pos:
wenzelm@56444
  1487
  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
huffman@53939
  1488
  assumes lf: "linear f"
huffman@53939
  1489
  shows "\<exists>B > 0. \<forall>x. norm (f x) \<le> B * norm x"
huffman@53939
  1490
proof -
huffman@53939
  1491
  have "\<exists>B > 0. \<forall>x. norm (f x) \<le> norm x * B"
huffman@53939
  1492
    using lf unfolding linear_conv_bounded_linear
huffman@53939
  1493
    by (rule bounded_linear.pos_bounded)
huffman@53939
  1494
  then show ?thesis
haftmann@57512
  1495
    by (simp only: mult.commute)
huffman@44133
  1496
qed
huffman@44133
  1497
wenzelm@49522
  1498
lemma bounded_linearI':
wenzelm@56444
  1499
  fixes f ::"'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
wenzelm@53406
  1500
  assumes "\<And>x y. f (x + y) = f x + f y"
wenzelm@53406
  1501
    and "\<And>c x. f (c *\<^sub>R x) = c *\<^sub>R f x"
wenzelm@49522
  1502
  shows "bounded_linear f"
wenzelm@53406
  1503
  unfolding linear_conv_bounded_linear[symmetric]
wenzelm@49522
  1504
  by (rule linearI[OF assms])
huffman@44133
  1505
huffman@44133
  1506
lemma bilinear_bounded:
wenzelm@56444
  1507
  fixes h :: "'m::euclidean_space \<Rightarrow> 'n::euclidean_space \<Rightarrow> 'k::real_normed_vector"
huffman@44133
  1508
  assumes bh: "bilinear h"
huffman@44133
  1509
  shows "\<exists>B. \<forall>x y. norm (h x y) \<le> B * norm x * norm y"
hoelzl@50526
  1510
proof (clarify intro!: exI[of _ "\<Sum>i\<in>Basis. \<Sum>j\<in>Basis. norm (h i j)"])
wenzelm@53406
  1511
  fix x :: 'm
wenzelm@53406
  1512
  fix y :: 'n
wenzelm@53406
  1513
  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
  1514
    apply (subst euclidean_representation[where 'a='m])
wenzelm@53406
  1515
    apply (subst euclidean_representation[where 'a='n])
hoelzl@50526
  1516
    apply rule
hoelzl@50526
  1517
    done
wenzelm@53406
  1518
  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
  1519
    unfolding bilinear_setsum[OF bh finite_Basis finite_Basis] ..
hoelzl@50526
  1520
  finally have th: "norm (h x y) = \<dots>" .
hoelzl@50526
  1521
  show "norm (h x y) \<le> (\<Sum>i\<in>Basis. \<Sum>j\<in>Basis. norm (h i j)) * norm x * norm y"
haftmann@57418
  1522
    apply (auto simp add: setsum_left_distrib th setsum.cartesian_product)
wenzelm@53406
  1523
    apply (rule setsum_norm_le)
wenzelm@53406
  1524
    apply simp
wenzelm@53406
  1525
    apply (auto simp add: bilinear_rmul[OF bh] bilinear_lmul[OF bh]
wenzelm@53406
  1526
      field_simps simp del: scaleR_scaleR)
wenzelm@53406
  1527
    apply (rule mult_mono)
wenzelm@53406
  1528
    apply (auto simp add: zero_le_mult_iff Basis_le_norm)
wenzelm@53406
  1529
    apply (rule mult_mono)
wenzelm@53406
  1530
    apply (auto simp add: zero_le_mult_iff Basis_le_norm)
wenzelm@53406
  1531
    done
huffman@44133
  1532
qed
huffman@44133
  1533
huffman@44133
  1534
lemma bilinear_conv_bounded_bilinear:
huffman@44133
  1535
  fixes h :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space \<Rightarrow> 'c::real_normed_vector"
huffman@44133
  1536
  shows "bilinear h \<longleftrightarrow> bounded_bilinear h"
huffman@44133
  1537
proof
huffman@44133
  1538
  assume "bilinear h"
huffman@44133
  1539
  show "bounded_bilinear h"
huffman@44133
  1540
  proof
wenzelm@53406
  1541
    fix x y z
wenzelm@53406
  1542
    show "h (x + y) z = h x z + h y z"
huffman@53600
  1543
      using `bilinear h` unfolding bilinear_def linear_iff by simp
huffman@44133
  1544
  next
wenzelm@53406
  1545
    fix x y z
wenzelm@53406
  1546
    show "h x (y + z) = h x y + h x z"
huffman@53600
  1547
      using `bilinear h` unfolding bilinear_def linear_iff by simp
huffman@44133
  1548
  next
wenzelm@53406
  1549
    fix r x y
wenzelm@53406
  1550
    show "h (scaleR r x) y = scaleR r (h x y)"
huffman@53600
  1551
      using `bilinear h` unfolding bilinear_def linear_iff
huffman@44133
  1552
      by simp
huffman@44133
  1553
  next
wenzelm@53406
  1554
    fix r x y
wenzelm@53406
  1555
    show "h x (scaleR r y) = scaleR r (h x y)"
huffman@53600
  1556
      using `bilinear h` unfolding bilinear_def linear_iff
huffman@44133
  1557
      by simp
huffman@44133
  1558
  next
huffman@44133
  1559
    have "\<exists>B. \<forall>x y. norm (h x y) \<le> B * norm x * norm y"
huffman@44133
  1560
      using `bilinear h` by (rule bilinear_bounded)
wenzelm@49522
  1561
    then show "\<exists>K. \<forall>x y. norm (h x y) \<le> norm x * norm y * K"
haftmann@57514
  1562
      by (simp add: ac_simps)
huffman@44133
  1563
  qed
huffman@44133
  1564
next
huffman@44133
  1565
  assume "bounded_bilinear h"
huffman@44133
  1566
  then interpret h: bounded_bilinear h .
huffman@44133
  1567
  show "bilinear h"
huffman@44133
  1568
    unfolding bilinear_def linear_conv_bounded_linear
wenzelm@49522
  1569
    using h.bounded_linear_left h.bounded_linear_right by simp
huffman@44133
  1570
qed
huffman@44133
  1571
huffman@53939
  1572
lemma bilinear_bounded_pos:
wenzelm@56444
  1573
  fixes h :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space \<Rightarrow> 'c::real_normed_vector"
huffman@53939
  1574
  assumes bh: "bilinear h"
huffman@53939
  1575
  shows "\<exists>B > 0. \<forall>x y. norm (h x y) \<le> B * norm x * norm y"
huffman@53939
  1576
proof -
huffman@53939
  1577
  have "\<exists>B > 0. \<forall>x y. norm (h x y) \<le> norm x * norm y * B"
huffman@53939
  1578
    using bh [unfolded bilinear_conv_bounded_bilinear]
huffman@53939
  1579
    by (rule bounded_bilinear.pos_bounded)
huffman@53939
  1580
  then show ?thesis
haftmann@57514
  1581
    by (simp only: ac_simps)
huffman@53939
  1582
qed
huffman@53939
  1583
wenzelm@49522
  1584
huffman@44133
  1585
subsection {* We continue. *}
huffman@44133
  1586
huffman@44133
  1587
lemma independent_bound:
wenzelm@53716
  1588
  fixes S :: "'a::euclidean_space set"
wenzelm@53716
  1589
  shows "independent S \<Longrightarrow> finite S \<and> card S \<le> DIM('a)"
hoelzl@50526
  1590
  using independent_span_bound[OF finite_Basis, of S] by auto
huffman@44133
  1591
paulson@60303
  1592
corollary 
paulson@60303
  1593
  fixes S :: "'a::euclidean_space set"
paulson@60303
  1594
  assumes "independent S"
paulson@60303
  1595
  shows independent_imp_finite: "finite S" and independent_card_le:"card S \<le> DIM('a)"
paulson@60303
  1596
using assms independent_bound by auto
paulson@60303
  1597
  
wenzelm@49663
  1598
lemma dependent_biggerset:
wenzelm@56444
  1599
  fixes S :: "'a::euclidean_space set"
wenzelm@56444
  1600
  shows "(finite S \<Longrightarrow> card S > DIM('a)) \<Longrightarrow> dependent S"
huffman@44133
  1601
  by (metis independent_bound not_less)
huffman@44133
  1602
huffman@44133
  1603
text {* Hence we can create a maximal independent subset. *}
huffman@44133
  1604
huffman@44133
  1605
lemma maximal_independent_subset_extend:
wenzelm@53406
  1606
  fixes S :: "'a::euclidean_space set"
wenzelm@53406
  1607
  assumes sv: "S \<subseteq> V"
wenzelm@49663
  1608
    and iS: "independent S"
huffman@44133
  1609
  shows "\<exists>B. S \<subseteq> B \<and> B \<subseteq> V \<and> independent B \<and> V \<subseteq> span B"
huffman@44133
  1610
  using sv iS
wenzelm@49522
  1611
proof (induct "DIM('a) - card S" arbitrary: S rule: less_induct)
huffman@44133
  1612
  case less
huffman@44133
  1613
  note sv = `S \<subseteq> V` and i = `independent S`
huffman@44133
  1614
  let ?P = "\<lambda>B. S \<subseteq> B \<and> B \<subseteq> V \<and> independent B \<and> V \<subseteq> span B"
huffman@44133
  1615
  let ?ths = "\<exists>x. ?P x"
huffman@44133
  1616
  let ?d = "DIM('a)"
wenzelm@53406
  1617
  show ?ths
wenzelm@53406
  1618
  proof (cases "V \<subseteq> span S")
wenzelm@53406
  1619
    case True
wenzelm@53406
  1620
    then show ?thesis
wenzelm@53406
  1621
      using sv i by blast
wenzelm@53406
  1622
  next
wenzelm@53406
  1623
    case False
wenzelm@53406
  1624
    then obtain a where a: "a \<in> V" "a \<notin> span S"
wenzelm@53406
  1625
      by blast
wenzelm@53406
  1626
    from a have aS: "a \<notin> S"
wenzelm@53406
  1627
      by (auto simp add: span_superset)
wenzelm@53406
  1628
    have th0: "insert a S \<subseteq> V"
wenzelm@53406
  1629
      using a sv by blast
huffman@44133
  1630
    from independent_insert[of a S]  i a
wenzelm@53406
  1631
    have th1: "independent (insert a S)"
wenzelm@53406
  1632
      by auto
huffman@44133
  1633
    have mlt: "?d - card (insert a S) < ?d - card S"
wenzelm@49522
  1634
      using aS a independent_bound[OF th1] by auto
huffman@44133
  1635
huffman@44133
  1636
    from less(1)[OF mlt th0 th1]
huffman@44133
  1637
    obtain B where B: "insert a S \<subseteq> B" "B \<subseteq> V" "independent B" " V \<subseteq> span B"
huffman@44133
  1638
      by blast
huffman@44133
  1639
    from B have "?P B" by auto
wenzelm@53406
  1640
    then show ?thesis by blast
wenzelm@53406
  1641
  qed
huffman@44133
  1642
qed
huffman@44133
  1643
huffman@44133
  1644
lemma maximal_independent_subset:
huffman@44133
  1645
  "\<exists>(B:: ('a::euclidean_space) set). B\<subseteq> V \<and> independent B \<and> V \<subseteq> span B"
wenzelm@49522
  1646
  by (metis maximal_independent_subset_extend[of "{}:: ('a::euclidean_space) set"]
wenzelm@49522
  1647
    empty_subsetI independent_empty)
huffman@44133
  1648
huffman@44133
  1649
huffman@44133
  1650
text {* Notion of dimension. *}
huffman@44133
  1651
wenzelm@53406
  1652
definition "dim V = (SOME n. \<exists>B. B \<subseteq> V \<and> independent B \<and> V \<subseteq> span B \<and> card B = n)"
huffman@44133
  1653
wenzelm@49522
  1654
lemma basis_exists:
wenzelm@49522
  1655
  "\<exists>B. (B :: ('a::euclidean_space) set) \<subseteq> V \<and> independent B \<and> V \<subseteq> span B \<and> (card B = dim V)"
wenzelm@49522
  1656
  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
  1657
  using maximal_independent_subset[of V] independent_bound
wenzelm@49522
  1658
  by auto
huffman@44133
  1659
lp15@60307
  1660
corollary dim_le_card:
lp15@60307
  1661
  fixes s :: "'a::euclidean_space set"
lp15@60307
  1662
  shows "finite s \<Longrightarrow> dim s \<le> card s"
lp15@60307
  1663
by (metis basis_exists card_mono)
lp15@60307
  1664
huffman@44133
  1665
text {* Consequences of independence or spanning for cardinality. *}
huffman@44133
  1666
wenzelm@53406
  1667
lemma independent_card_le_dim:
wenzelm@53406
  1668
  fixes B :: "'a::euclidean_space set"
wenzelm@53406
  1669
  assumes "B \<subseteq> V"
wenzelm@53406
  1670
    and "independent B"
wenzelm@49522
  1671
  shows "card B \<le> dim V"
huffman@44133
  1672
proof -
huffman@44133
  1673
  from basis_exists[of V] `B \<subseteq> V`
wenzelm@53406
  1674
  obtain B' where "independent B'"
wenzelm@53406
  1675
    and "B \<subseteq> span B'"
wenzelm@53406
  1676
    and "card B' = dim V"
wenzelm@53406
  1677
    by blast
huffman@44133
  1678
  with independent_span_bound[OF _ `independent B` `B \<subseteq> span B'`] independent_bound[of B']
huffman@44133
  1679
  show ?thesis by auto
huffman@44133
  1680
qed
huffman@44133
  1681
wenzelm@49522
  1682
lemma span_card_ge_dim:
wenzelm@53406
  1683
  fixes B :: "'a::euclidean_space set"
wenzelm@53406
  1684
  shows "B \<subseteq> V \<Longrightarrow> V \<subseteq> span B \<Longrightarrow> finite B \<Longrightarrow> dim V \<le> card B"
huffman@44133
  1685
  by (metis basis_exists[of V] independent_span_bound subset_trans)
huffman@44133
  1686
huffman@44133
  1687
lemma basis_card_eq_dim:
wenzelm@53406
  1688
  fixes V :: "'a::euclidean_space set"
wenzelm@53406
  1689
  shows "B \<subseteq> V \<Longrightarrow> V \<subseteq> span B \<Longrightarrow> independent B \<Longrightarrow> finite B \<and> card B = dim V"
huffman@44133
  1690
  by (metis order_eq_iff independent_card_le_dim span_card_ge_dim independent_bound)
huffman@44133
  1691
wenzelm@53406
  1692
lemma dim_unique:
wenzelm@53406
  1693
  fixes B :: "'a::euclidean_space set"
wenzelm@53406
  1694
  shows "B \<subseteq> V \<Longrightarrow> V \<subseteq> span B \<Longrightarrow> independent B \<Longrightarrow> card B = n \<Longrightarrow> dim V = n"
huffman@44133
  1695
  by (metis basis_card_eq_dim)
huffman@44133
  1696
huffman@44133
  1697
text {* More lemmas about dimension. *}
huffman@44133
  1698
wenzelm@53406
  1699
lemma dim_UNIV: "dim (UNIV :: 'a::euclidean_space set) = DIM('a)"
hoelzl@50526
  1700
  using independent_Basis
hoelzl@50526
  1701
  by (intro dim_unique[of Basis]) auto
huffman@44133
  1702
huffman@44133
  1703
lemma dim_subset:
wenzelm@53406
  1704
  fixes S :: "'a::euclidean_space set"
wenzelm@53406
  1705
  shows "S \<subseteq> T \<Longrightarrow> dim S \<le> dim T"
huffman@44133
  1706
  using basis_exists[of T] basis_exists[of S]
huffman@44133
  1707
  by (metis independent_card_le_dim subset_trans)
huffman@44133
  1708
wenzelm@53406
  1709
lemma dim_subset_UNIV:
wenzelm@53406
  1710
  fixes S :: "'a::euclidean_space set"
wenzelm@53406
  1711
  shows "dim S \<le> DIM('a)"
huffman@44133
  1712
  by (metis dim_subset subset_UNIV dim_UNIV)
huffman@44133
  1713
huffman@44133
  1714
text {* Converses to those. *}
huffman@44133
  1715
huffman@44133
  1716
lemma card_ge_dim_independent:
wenzelm@53406
  1717
  fixes B :: "'a::euclidean_space set"
wenzelm@53406
  1718
  assumes BV: "B \<subseteq> V"
wenzelm@53406
  1719
    and iB: "independent B"
wenzelm@53406
  1720
    and dVB: "dim V \<le> card B"
huffman@44133
  1721
  shows "V \<subseteq> span B"
wenzelm@53406
  1722
proof
wenzelm@53406
  1723
  fix a
wenzelm@53406
  1724
  assume aV: "a \<in> V"
wenzelm@53406
  1725
  {
wenzelm@53406
  1726
    assume aB: "a \<notin> span B"
wenzelm@53406
  1727
    then have iaB: "independent (insert a B)"
wenzelm@53406
  1728
      using iB aV BV by (simp add: independent_insert)
wenzelm@53406
  1729
    from aV BV have th0: "insert a B \<subseteq> V"
wenzelm@53406
  1730
      by blast
wenzelm@53406
  1731
    from aB have "a \<notin>B"
wenzelm@53406
  1732
      by (auto simp add: span_superset)
wenzelm@53406
  1733
    with independent_card_le_dim[OF th0 iaB] dVB independent_bound[OF iB]
wenzelm@53406
  1734
    have False by auto
wenzelm@53406
  1735
  }
wenzelm@53406
  1736
  then show "a \<in> span B" by blast
huffman@44133
  1737
qed
huffman@44133
  1738
huffman@44133
  1739
lemma card_le_dim_spanning:
wenzelm@49663
  1740
  assumes BV: "(B:: ('a::euclidean_space) set) \<subseteq> V"
wenzelm@49663
  1741
    and VB: "V \<subseteq> span B"
wenzelm@49663
  1742
    and fB: "finite B"
wenzelm@49663
  1743
    and dVB: "dim V \<ge> card B"
huffman@44133
  1744
  shows "independent B"
wenzelm@49522
  1745
proof -
wenzelm@53406
  1746
  {
wenzelm@53406
  1747
    fix a
wenzelm@53716
  1748
    assume a: "a \<in> B" "a \<in> span (B - {a})"
wenzelm@53406
  1749
    from a fB have c0: "card B \<noteq> 0"
wenzelm@53406
  1750
      by auto
wenzelm@53716
  1751
    from a fB have cb: "card (B - {a}) = card B - 1"
wenzelm@53406
  1752
      by auto
wenzelm@53716
  1753
    from BV a have th0: "B - {a} \<subseteq> V"
wenzelm@53406
  1754
      by blast
wenzelm@53406
  1755
    {
wenzelm@53406
  1756
      fix x
wenzelm@53406
  1757
      assume x: "x \<in> V"
wenzelm@53716
  1758
      from a have eq: "insert a (B - {a}) = B"
wenzelm@53406
  1759
        by blast
wenzelm@53406
  1760
      from x VB have x': "x \<in> span B"
wenzelm@53406
  1761
        by blast
huffman@44133
  1762
      from span_trans[OF a(2), unfolded eq, OF x']
wenzelm@53716
  1763
      have "x \<in> span (B - {a})" .
wenzelm@53406
  1764
    }
wenzelm@53716
  1765
    then have th1: "V \<subseteq> span (B - {a})"
wenzelm@53406
  1766
      by blast
wenzelm@53716
  1767
    have th2: "finite (B - {a})"
wenzelm@53406
  1768
      using fB by auto
huffman@44133
  1769
    from span_card_ge_dim[OF th0 th1 th2]
wenzelm@53716
  1770
    have c: "dim V \<le> card (B - {a})" .
wenzelm@53406
  1771
    from c c0 dVB cb have False by simp
wenzelm@53406
  1772
  }
wenzelm@53406
  1773
  then show ?thesis
wenzelm@53406
  1774
    unfolding dependent_def by blast
huffman@44133
  1775
qed
huffman@44133
  1776
wenzelm@53406
  1777
lemma card_eq_dim:
wenzelm@53406
  1778
  fixes B :: "'a::euclidean_space set"
wenzelm@53406
  1779
  shows "B \<subseteq> V \<Longrightarrow> card B = dim V \<Longrightarrow> finite B \<Longrightarrow> independent B \<longleftrightarrow> V \<subseteq> span B"
wenzelm@49522
  1780
  by (metis order_eq_iff card_le_dim_spanning card_ge_dim_independent)
huffman@44133
  1781
huffman@44133
  1782
text {* More general size bound lemmas. *}
huffman@44133
  1783
huffman@44133
  1784
lemma independent_bound_general:
wenzelm@53406
  1785
  fixes S :: "'a::euclidean_space set"
wenzelm@53406
  1786
  shows "independent S \<Longrightarrow> finite S \<and> card S \<le> dim S"
huffman@44133
  1787
  by (metis independent_card_le_dim independent_bound subset_refl)
huffman@44133
  1788
wenzelm@49522
  1789
lemma dependent_biggerset_general:
wenzelm@53406
  1790
  fixes S :: "'a::euclidean_space set"
wenzelm@53406
  1791
  shows "(finite S \<Longrightarrow> card S > dim S) \<Longrightarrow> dependent S"
huffman@44133
  1792
  using independent_bound_general[of S] by (metis linorder_not_le)
huffman@44133
  1793
paulson@60303
  1794
lemma dim_span [simp]:
wenzelm@53406
  1795
  fixes S :: "'a::euclidean_space set"
wenzelm@53406
  1796
  shows "dim (span S) = dim S"
wenzelm@49522
  1797
proof -
huffman@44133
  1798
  have th0: "dim S \<le> dim (span S)"
huffman@44133
  1799
    by (auto simp add: subset_eq intro: dim_subset span_superset)
huffman@44133
  1800
  from basis_exists[of S]
wenzelm@53406
  1801
  obtain B where B: "B \<subseteq> S" "independent B" "S \<subseteq> span B" "card B = dim S"
wenzelm@53406
  1802
    by blast
wenzelm@53406
  1803
  from B have fB: "finite B" "card B = dim S"
wenzelm@53406
  1804
    using independent_bound by blast+
wenzelm@53406
  1805
  have bSS: "B \<subseteq> span S"
wenzelm@53406
  1806
    using B(1) by (metis subset_eq span_inc)
wenzelm@53406
  1807
  have sssB: "span S \<subseteq> span B"
wenzelm@53406
  1808
    using span_mono[OF B(3)] by (simp add: span_span)
huffman@44133
  1809
  from span_card_ge_dim[OF bSS sssB fB(1)] th0 show ?thesis
wenzelm@49522
  1810
    using fB(2) by arith
huffman@44133
  1811
qed
huffman@44133
  1812
wenzelm@53406
  1813
lemma subset_le_dim:
wenzelm@53406
  1814
  fixes S :: "'a::euclidean_space set"
wenzelm@53406
  1815
  shows "S \<subseteq> span T \<Longrightarrow> dim S \<le> dim T"
huffman@44133
  1816
  by (metis dim_span dim_subset)
huffman@44133
  1817
wenzelm@53406
  1818
lemma span_eq_dim:
wenzelm@56444
  1819
  fixes S :: "'a::euclidean_space set"
wenzelm@53406
  1820
  shows "span S = span T \<Longrightarrow> dim S = dim T"
huffman@44133
  1821
  by (metis dim_span)
huffman@44133
  1822
huffman@44133
  1823
lemma spans_image:
wenzelm@49663
  1824
  assumes lf: "linear f"
wenzelm@49663
  1825
    and VB: "V \<subseteq> span B"
huffman@44133
  1826
  shows "f ` V \<subseteq> span (f ` B)"
wenzelm@49522
  1827
  unfolding span_linear_image[OF lf] by (metis VB image_mono)
huffman@44133
  1828
huffman@44133
  1829
lemma dim_image_le:
huffman@44133
  1830
  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
wenzelm@49663
  1831
  assumes lf: "linear f"
wenzelm@49663
  1832
  shows "dim (f ` S) \<le> dim (S)"
wenzelm@49522
  1833
proof -
huffman@44133
  1834
  from basis_exists[of S] obtain B where
huffman@44133
  1835
    B: "B \<subseteq> S" "independent B" "S \<subseteq> span B" "card B = dim S" by blast
wenzelm@53406
  1836
  from B have fB: "finite B" "card B = dim S"
wenzelm@53406
  1837
    using independent_bound by blast+
huffman@44133
  1838
  have "dim (f ` S) \<le> card (f ` B)"
huffman@44133
  1839
    apply (rule span_card_ge_dim)
wenzelm@53406
  1840
    using lf B fB
wenzelm@53406
  1841
    apply (auto simp add: span_linear_image spans_image subset_image_iff)
wenzelm@49522
  1842
    done
wenzelm@53406
  1843
  also have "\<dots> \<le> dim S"
wenzelm@53406
  1844
    using card_image_le[OF fB(1)] fB by simp
huffman@44133
  1845
  finally show ?thesis .
huffman@44133
  1846
qed
huffman@44133
  1847
huffman@44133
  1848
text {* Relation between bases and injectivity/surjectivity of map. *}
huffman@44133
  1849
huffman@44133
  1850
lemma spanning_surjective_image:
huffman@44133
  1851
  assumes us: "UNIV \<subseteq> span S"
wenzelm@53406
  1852
    and lf: "linear f"
wenzelm@53406
  1853
    and sf: "surj f"
huffman@44133
  1854
  shows "UNIV \<subseteq> span (f ` S)"
wenzelm@49663
  1855
proof -
wenzelm@53406
  1856
  have "UNIV \<subseteq> f ` UNIV"
wenzelm@53406
  1857
    using sf by (auto simp add: surj_def)
wenzelm@53406
  1858
  also have " \<dots> \<subseteq> span (f ` S)"
wenzelm@53406
  1859
    using spans_image[OF lf us] .
wenzelm@53406
  1860
  finally show ?thesis .
huffman@44133
  1861
qed
huffman@44133
  1862
huffman@44133
  1863
lemma independent_injective_image:
wenzelm@49663
  1864
  assumes iS: "independent S"
wenzelm@49663
  1865
    and lf: "linear f"
wenzelm@49663
  1866
    and fi: "inj f"
huffman@44133
  1867
  shows "independent (f ` S)"
wenzelm@49663
  1868
proof -
wenzelm@53406
  1869
  {
wenzelm@53406
  1870
    fix a
wenzelm@49663
  1871
    assume a: "a \<in> S" "f a \<in> span (f ` S - {f a})"
wenzelm@53406
  1872
    have eq: "f ` S - {f a} = f ` (S - {a})"
wenzelm@53406
  1873
      using fi by (auto simp add: inj_on_def)
wenzelm@53716
  1874
    from a have "f a \<in> f ` span (S - {a})"
wenzelm@53406
  1875
      unfolding eq span_linear_image[OF lf, of "S - {a}"] by blast
wenzelm@53716
  1876
    then have "a \<in> span (S - {a})"
wenzelm@53406
  1877
      using fi by (auto simp add: inj_on_def)
wenzelm@53406
  1878
    with a(1) iS have False
wenzelm@53406
  1879
      by (simp add: dependent_def)
wenzelm@53406
  1880
  }
wenzelm@53406
  1881
  then show ?thesis
wenzelm@53406
  1882
    unfolding dependent_def by blast
huffman@44133
  1883
qed
huffman@44133
  1884
huffman@44133
  1885
text {* Picking an orthogonal replacement for a spanning set. *}
huffman@44133
  1886
wenzelm@53406
  1887
(* FIXME : Move to some general theory ?*)
huffman@44133
  1888
definition "pairwise R S \<longleftrightarrow> (\<forall>x \<in> S. \<forall>y\<in> S. x\<noteq>y \<longrightarrow> R x y)"
huffman@44133
  1889
wenzelm@53406
  1890
lemma vector_sub_project_orthogonal:
wenzelm@53406
  1891
  fixes b x :: "'a::euclidean_space"
wenzelm@53406
  1892
  shows "b \<bullet> (x - ((b \<bullet> x) / (b \<bullet> b)) *\<^sub>R b) = 0"
huffman@44133
  1893
  unfolding inner_simps by auto
huffman@44133
  1894
huffman@44528
  1895
lemma pairwise_orthogonal_insert:
huffman@44528
  1896
  assumes "pairwise orthogonal S"
wenzelm@49522
  1897
    and "\<And>y. y \<in> S \<Longrightarrow> orthogonal x y"
huffman@44528
  1898
  shows "pairwise orthogonal (insert x S)"
huffman@44528
  1899
  using assms unfolding pairwise_def
huffman@44528
  1900
  by (auto simp add: orthogonal_commute)
huffman@44528
  1901
huffman@44133
  1902
lemma basis_orthogonal:
wenzelm@53406
  1903
  fixes B :: "'a::real_inner set"
huffman@44133
  1904
  assumes fB: "finite B"
huffman@44133
  1905
  shows "\<exists>C. finite C \<and> card C \<le> card B \<and> span C = span B \<and> pairwise orthogonal C"
huffman@44133
  1906
  (is " \<exists>C. ?P B C")
wenzelm@49522
  1907
  using fB
wenzelm@49522
  1908
proof (induct rule: finite_induct)
wenzelm@49522
  1909
  case empty
wenzelm@53406
  1910
  then show ?case
wenzelm@53406
  1911
    apply (rule exI[where x="{}"])
wenzelm@53406
  1912
    apply (auto simp add: pairwise_def)
wenzelm@53406
  1913
    done
huffman@44133
  1914
next
wenzelm@49522
  1915
  case (insert a B)
huffman@44133
  1916
  note fB = `finite B` and aB = `a \<notin> B`
huffman@44133
  1917
  from `\<exists>C. finite C \<and> card C \<le> card B \<and> span C = span B \<and> pairwise orthogonal C`
huffman@44133
  1918
  obtain C where C: "finite C" "card C \<le> card B"
huffman@44133
  1919
    "span C = span B" "pairwise orthogonal C" by blast
huffman@44133
  1920
  let ?a = "a - setsum (\<lambda>x. (x \<bullet> a / (x \<bullet> x)) *\<^sub>R x) C"
huffman@44133
  1921
  let ?C = "insert ?a C"
wenzelm@53406
  1922
  from C(1) have fC: "finite ?C"
wenzelm@53406
  1923
    by simp
wenzelm@49522
  1924
  from fB aB C(1,2) have cC: "card ?C \<le> card (insert a B)"
wenzelm@49522
  1925
    by (simp add: card_insert_if)
wenzelm@53406
  1926
  {
wenzelm@53406
  1927
    fix x k
wenzelm@49522
  1928
    have th0: "\<And>(a::'a) b c. a - (b - c) = c + (a - b)"
wenzelm@49522
  1929
      by (simp add: field_simps)
huffman@44133
  1930
    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
  1931
      apply (simp only: scaleR_right_diff_distrib th0)
huffman@44133
  1932
      apply (rule span_add_eq)
huffman@44133
  1933
      apply (rule span_mul)
huffman@56196
  1934
      apply (rule span_setsum)
huffman@44133
  1935
      apply clarify
huffman@44133
  1936
      apply (rule span_mul)
wenzelm@49522
  1937
      apply (rule span_superset)
wenzelm@49522
  1938
      apply assumption
wenzelm@53406
  1939
      done
wenzelm@53406
  1940
  }
huffman@44133
  1941
  then have SC: "span ?C = span (insert a B)"
huffman@44133
  1942
    unfolding set_eq_iff span_breakdown_eq C(3)[symmetric] by auto
wenzelm@53406
  1943
  {
wenzelm@53406
  1944
    fix y
wenzelm@53406
  1945
    assume yC: "y \<in> C"
wenzelm@53406
  1946
    then have Cy: "C = insert y (C - {y})"
wenzelm@53406
  1947
      by blast
wenzelm@53406
  1948
    have fth: "finite (C - {y})"
wenzelm@53406
  1949
      using C by simp
huffman@44528
  1950
    have "orthogonal ?a y"
huffman@44528
  1951
      unfolding orthogonal_def
haftmann@54230
  1952
      unfolding inner_diff inner_setsum_left right_minus_eq
haftmann@57418
  1953
      unfolding setsum.remove [OF `finite C` `y \<in> C`]
huffman@44528
  1954
      apply (clarsimp simp add: inner_commute[of y a])
haftmann@57418
  1955
      apply (rule setsum.neutral)
huffman@44528
  1956
      apply clarsimp
huffman@44528
  1957
      apply (rule C(4)[unfolded pairwise_def orthogonal_def, rule_format])
wenzelm@53406
  1958
      using `y \<in> C` by auto
wenzelm@53406
  1959
  }
huffman@44528
  1960
  with `pairwise orthogonal C` have CPO: "pairwise orthogonal ?C"
huffman@44528
  1961
    by (rule pairwise_orthogonal_insert)
wenzelm@53406
  1962
  from fC cC SC CPO have "?P (insert a B) ?C"
wenzelm@53406
  1963
    by blast
huffman@44133
  1964
  then show ?case by blast
huffman@44133
  1965
qed
huffman@44133
  1966
huffman@44133
  1967
lemma orthogonal_basis_exists:
huffman@44133
  1968
  fixes V :: "('a::euclidean_space) set"
huffman@44133
  1969
  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
  1970
proof -
wenzelm@49522
  1971
  from basis_exists[of V] obtain B where
wenzelm@53406
  1972
    B: "B \<subseteq> V" "independent B" "V \<subseteq> span B" "card B = dim V"
wenzelm@53406
  1973
    by blast
wenzelm@53406
  1974
  from B have fB: "finite B" "card B = dim V"
wenzelm@53406
  1975
    using independent_bound by auto
huffman@44133
  1976
  from basis_orthogonal[OF fB(1)] obtain C where
wenzelm@53406
  1977
    C: "finite C" "card C \<le> card B" "span C = span B" "pairwise orthogonal C"
wenzelm@53406
  1978
    by blast
wenzelm@53406
  1979
  from C B have CSV: "C \<subseteq> span V"
wenzelm@53406
  1980
    by (metis span_inc span_mono subset_trans)
wenzelm@53406
  1981
  from span_mono[OF B(3)] C have SVC: "span V \<subseteq> span C"
wenzelm@53406
  1982
    by (simp add: span_span)
huffman@44133
  1983
  from card_le_dim_spanning[OF CSV SVC C(1)] C(2,3) fB
wenzelm@53406
  1984
  have iC: "independent C"
huffman@44133
  1985
    by (simp add: dim_span)
wenzelm@53406
  1986
  from C fB have "card C \<le> dim V"
wenzelm@53406
  1987
    by simp
wenzelm@53406
  1988
  moreover have "dim V \<le> card C"
wenzelm@53406
  1989
    using span_card_ge_dim[OF CSV SVC C(1)]
wenzelm@53406
  1990
    by (simp add: dim_span)
wenzelm@53406
  1991
  ultimately have CdV: "card C = dim V"
wenzelm@53406
  1992
    using C(1) by simp
wenzelm@53406
  1993
  from C B CSV CdV iC show ?thesis
wenzelm@53406
  1994
    by auto
huffman@44133
  1995
qed
huffman@44133
  1996
huffman@44133
  1997
lemma span_eq: "span S = span T \<longleftrightarrow> S \<subseteq> span T \<and> T \<subseteq> span S"
huffman@44133
  1998
  using span_inc[unfolded subset_eq] using span_mono[of T "span S"] span_mono[of S "span T"]
wenzelm@49522
  1999
  by (auto simp add: span_span)
huffman@44133
  2000
huffman@44133
  2001
text {* Low-dimensional subset is in a hyperplane (weak orthogonal complement). *}
huffman@44133
  2002
wenzelm@49522
  2003
lemma span_not_univ_orthogonal:
wenzelm@53406
  2004
  fixes S :: "'a::euclidean_space set"
huffman@44133
  2005
  assumes sU: "span S \<noteq> UNIV"
wenzelm@56444
  2006
  shows "\<exists>a::'a. a \<noteq> 0 \<and> (\<forall>x \<in> span S. a \<bullet> x = 0)"
wenzelm@49522
  2007
proof -
wenzelm@53406
  2008
  from sU obtain a where a: "a \<notin> span S"
wenzelm@53406
  2009
    by blast
huffman@44133
  2010
  from orthogonal_basis_exists obtain B where
huffman@44133
  2011
    B: "independent B" "B \<subseteq> span S" "S \<subseteq> span B" "card B = dim S" "pairwise orthogonal B"
huffman@44133
  2012
    by blast
wenzelm@53406
  2013
  from B have fB: "finite B" "card B = dim S"
wenzelm@53406
  2014
    using independent_bound by auto
huffman@44133
  2015
  from span_mono[OF B(2)] span_mono[OF B(3)]
wenzelm@53406
  2016
  have sSB: "span S = span B"
wenzelm@53406
  2017
    by (simp add: span_span)
huffman@44133
  2018
  let ?a = "a - setsum (\<lambda>b. (a \<bullet> b / (b \<bullet> b)) *\<^sub>R b) B"
huffman@44133
  2019
  have "setsum (\<lambda>b. (a \<bullet> b / (b \<bullet> b)) *\<^sub>R b) B \<in> span S"
huffman@44133
  2020
    unfolding sSB
huffman@56196
  2021
    apply (rule span_setsum)
huffman@44133
  2022
    apply clarsimp
huffman@44133
  2023
    apply (rule span_mul)
wenzelm@49522
  2024
    apply (rule span_superset)
wenzelm@49522
  2025
    apply assumption
wenzelm@49522
  2026
    done
wenzelm@53406
  2027
  with a have a0:"?a  \<noteq> 0"
wenzelm@53406
  2028
    by auto
huffman@44133
  2029
  have "\<forall>x\<in>span B. ?a \<bullet> x = 0"
wenzelm@49522
  2030
  proof (rule span_induct')
wenzelm@49522
  2031
    show "subspace {x. ?a \<bullet> x = 0}"
wenzelm@49522
  2032
      by (auto simp add: subspace_def inner_add)
wenzelm@49522
  2033
  next
wenzelm@53406
  2034
    {
wenzelm@53406
  2035
      fix x
wenzelm@53406
  2036
      assume x: "x \<in> B"
wenzelm@53406
  2037
      from x have B': "B = insert x (B - {x})"
wenzelm@53406
  2038
        by blast
wenzelm@53406
  2039
      have fth: "finite (B - {x})"
wenzelm@53406
  2040
        using fB by simp
huffman@44133
  2041
      have "?a \<bullet> x = 0"
wenzelm@53406
  2042
        apply (subst B')
wenzelm@53406
  2043
        using fB fth
huffman@44133
  2044
        unfolding setsum_clauses(2)[OF fth]
huffman@44133
  2045
        apply simp unfolding inner_simps
huffman@44527
  2046
        apply (clarsimp simp add: inner_add inner_setsum_left)
haftmann@57418
  2047
        apply (rule setsum.neutral, rule ballI)
huffman@44133
  2048
        unfolding inner_commute
wenzelm@49711
  2049
        apply (auto simp add: x field_simps
wenzelm@49711
  2050
          intro: B(5)[unfolded pairwise_def orthogonal_def, rule_format])
wenzelm@53406
  2051
        done
wenzelm@53406
  2052
    }
wenzelm@53406
  2053
    then show "\<forall>x \<in> B. ?a \<bullet> x = 0"
wenzelm@53406
  2054
      by blast
huffman@44133
  2055
  qed
wenzelm@53406
  2056
  with a0 show ?thesis
wenzelm@53406
  2057
    unfolding sSB by (auto intro: exI[where x="?a"])
huffman@44133
  2058
qed
huffman@44133
  2059
huffman@44133
  2060
lemma span_not_univ_subset_hyperplane:
wenzelm@53406
  2061
  fixes S :: "'a::euclidean_space set"
wenzelm@53406
  2062
  assumes SU: "span S \<noteq> UNIV"
huffman@44133
  2063
  shows "\<exists> a. a \<noteq>0 \<and> span S \<subseteq> {x. a \<bullet> x = 0}"
huffman@44133
  2064
  using span_not_univ_orthogonal[OF SU] by auto
huffman@44133
  2065
wenzelm@49663
  2066
lemma lowdim_subset_hyperplane:
wenzelm@53406
  2067
  fixes S :: "'a::euclidean_space set"
huffman@44133
  2068
  assumes d: "dim S < DIM('a)"
wenzelm@56444
  2069
  shows "\<exists>a::'a. a \<noteq> 0 \<and> span S \<subseteq> {x. a \<bullet> x = 0}"
wenzelm@49522
  2070
proof -
wenzelm@53406
  2071
  {
wenzelm@53406
  2072
    assume "span S = UNIV"
wenzelm@53406
  2073
    then have "dim (span S) = dim (UNIV :: ('a) set)"
wenzelm@53406
  2074
      by simp
wenzelm@53406
  2075
    then have "dim S = DIM('a)"
wenzelm@53406
  2076
      by (simp add: dim_span dim_UNIV)
wenzelm@53406
  2077
    with d have False by arith
wenzelm@53406
  2078
  }
wenzelm@53406
  2079
  then have th: "span S \<noteq> UNIV"
wenzelm@53406
  2080
    by blast
huffman@44133
  2081
  from span_not_univ_subset_hyperplane[OF th] show ?thesis .
huffman@44133
  2082
qed
huffman@44133
  2083
huffman@44133
  2084
text {* We can extend a linear basis-basis injection to the whole set. *}
huffman@44133
  2085
huffman@44133
  2086
lemma linear_indep_image_lemma:
wenzelm@49663
  2087
  assumes lf: "linear f"
wenzelm@49663
  2088
    and fB: "finite B"
wenzelm@49522
  2089
    and ifB: "independent (f ` B)"
wenzelm@49663
  2090
    and fi: "inj_on f B"
wenzelm@49663
  2091
    and xsB: "x \<in> span B"
wenzelm@49522
  2092
    and fx: "f x = 0"
huffman@44133
  2093
  shows "x = 0"
huffman@44133
  2094
  using fB ifB fi xsB fx
wenzelm@49522
  2095
proof (induct arbitrary: x rule: finite_induct[OF fB])
wenzelm@49663
  2096
  case 1
wenzelm@49663
  2097
  then show ?case by auto
huffman@44133
  2098
next
huffman@44133
  2099
  case (2 a b x)
huffman@44133
  2100
  have fb: "finite b" using "2.prems" by simp
huffman@44133
  2101
  have th0: "f ` b \<subseteq> f ` (insert a b)"
wenzelm@53406
  2102
    apply (rule image_mono)
wenzelm@53406
  2103
    apply blast
wenzelm@53406
  2104
    done
huffman@44133
  2105
  from independent_mono[ OF "2.prems"(2) th0]
huffman@44133
  2106
  have ifb: "independent (f ` b)"  .
huffman@44133
  2107
  have fib: "inj_on f b"
huffman@44133
  2108
    apply (rule subset_inj_on [OF "2.prems"(3)])
wenzelm@49522
  2109
    apply blast
wenzelm@49522
  2110
    done
huffman@44133
  2111
  from span_breakdown[of a "insert a b", simplified, OF "2.prems"(4)]
wenzelm@53406
  2112
  obtain k where k: "x - k*\<^sub>R a \<in> span (b - {a})"
wenzelm@53406
  2113
    by blast
huffman@44133
  2114
  have "f (x - k*\<^sub>R a) \<in> span (f ` b)"
huffman@44133
  2115
    unfolding span_linear_image[OF lf]
huffman@44133
  2116
    apply (rule imageI)
wenzelm@53716
  2117
    using k span_mono[of "b - {a}" b]
wenzelm@53406
  2118
    apply blast
wenzelm@49522
  2119
    done
wenzelm@49522
  2120
  then have "f x - k*\<^sub>R f a \<in> span (f ` b)"
huffman@44133
  2121
    by (simp add: linear_sub[OF lf] linear_cmul[OF lf])
wenzelm@49522
  2122
  then have th: "-k *\<^sub>R f a \<in> span (f ` b)"
huffman@44133
  2123
    using "2.prems"(5) by simp
wenzelm@53406
  2124
  have xsb: "x \<in> span b"
wenzelm@53406
  2125
  proof (cases "k = 0")
wenzelm@53406
  2126
    case True
wenzelm@53716
  2127
    with k have "x \<in> span (b - {a})" by simp
wenzelm@53716
  2128
    then show ?thesis using span_mono[of "b - {a}" b]
wenzelm@53406
  2129
      by blast
wenzelm@53406
  2130
  next
wenzelm@53406
  2131
    case False
wenzelm@53406
  2132
    with span_mul[OF th, of "- 1/ k"]
huffman@44133
  2133
    have th1: "f a \<in> span (f ` b)"
hoelzl@56479
  2134
      by auto
huffman@44133
  2135
    from inj_on_image_set_diff[OF "2.prems"(3), of "insert a b " "{a}", symmetric]
huffman@44133
  2136
    have tha: "f ` insert a b - f ` {a} = f ` (insert a b - {a})" by blast
huffman@44133
  2137
    from "2.prems"(2) [unfolded dependent_def bex_simps(8), rule_format, of "f a"]
huffman@44133
  2138
    have "f a \<notin> span (f ` b)" using tha
huffman@44133
  2139
      using "2.hyps"(2)
huffman@44133
  2140
      "2.prems"(3) by auto
huffman@44133
  2141
    with th1 have False by blast
wenzelm@53406
  2142
    then show ?thesis by blast
wenzelm@53406
  2143
  qed
wenzelm@53406
  2144
  from "2.hyps"(3)[OF fb ifb fib xsb "2.prems"(5)] show "x = 0" .
huffman@44133
  2145
qed
huffman@44133
  2146
huffman@44133
  2147
text {* We can extend a linear mapping from basis. *}
huffman@44133
  2148
huffman@44133
  2149
lemma linear_independent_extend_lemma:
huffman@44133
  2150
  fixes f :: "'a::real_vector \<Rightarrow> 'b::real_vector"
wenzelm@53406
  2151
  assumes fi: "finite B"
wenzelm@53406
  2152
    and ib: "independent B"
wenzelm@53406
  2153
  shows "\<exists>g.
wenzelm@53406
  2154
    (\<forall>x\<in> span B. \<forall>y\<in> span B. g (x + y) = g x + g y) \<and>
wenzelm@53406
  2155
    (\<forall>x\<in> span B. \<forall>c. g (c*\<^sub>R x) = c *\<^sub>R g x) \<and>
wenzelm@53406
  2156
    (\<forall>x\<in> B. g x = f x)"
wenzelm@49663
  2157
  using ib fi
wenzelm@49522
  2158
proof (induct rule: finite_induct[OF fi])
wenzelm@49663
  2159
  case 1
wenzelm@49663
  2160
  then show ?case by auto
huffman@44133
  2161
next
huffman@44133
  2162
  case (2 a b)
huffman@44133
  2163
  from "2.prems" "2.hyps" have ibf: "independent b" "finite b"
huffman@44133
  2164
    by (simp_all add: independent_insert)
huffman@44133
  2165
  from "2.hyps"(3)[OF ibf] obtain g where
huffman@44133
  2166
    g: "\<forall>x\<in>span b. \<forall>y\<in>span b. g (x + y) = g x + g y"
huffman@44133
  2167
    "\<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
  2168
  let ?h = "\<lambda>z. SOME k. (z - k *\<^sub>R a) \<in> span b"
wenzelm@53406
  2169
  {
wenzelm@53406
  2170
    fix z
wenzelm@53406
  2171
    assume z: "z \<in> span (insert a b)"
huffman@44133
  2172
    have th0: "z - ?h z *\<^sub>R a \<in> span b"
huffman@44133
  2173
      apply (rule someI_ex)
huffman@44133
  2174
      unfolding span_breakdown_eq[symmetric]
wenzelm@53406
  2175
      apply (rule z)
wenzelm@53406
  2176
      done
wenzelm@53406
  2177
    {
wenzelm@53406
  2178
      fix k
wenzelm@53406
  2179
      assume k: "z - k *\<^sub>R a \<in> span b"
huffman@44133
  2180
      have eq: "z - ?h z *\<^sub>R a - (z - k*\<^sub>R a) = (k - ?h z) *\<^sub>R a"
huffman@44133
  2181
        by (simp add: field_simps scaleR_left_distrib [symmetric])
wenzelm@53406
  2182
      from span_sub[OF th0 k] have khz: "(k - ?h z) *\<^sub>R a \<in> span b"
wenzelm@53406
  2183
        by (simp add: eq)
wenzelm@53406
  2184
      {
wenzelm@53406
  2185
        assume "k \<noteq> ?h z"
wenzelm@53406
  2186
        then have k0: "k - ?h z \<noteq> 0" by simp
huffman@44133
  2187
        from k0 span_mul[OF khz, of "1 /(k - ?h z)"]
huffman@44133
  2188
        have "a \<in> span b" by simp
huffman@44133
  2189
        with "2.prems"(1) "2.hyps"(2) have False
wenzelm@53406
  2190
          by (auto simp add: dependent_def)
wenzelm@53406
  2191
      }
wenzelm@53406
  2192
      then have "k = ?h z" by blast
wenzelm@53406
  2193
    }
wenzelm@53406
  2194
    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
  2195
      by blast
wenzelm@53406
  2196
  }
huffman@44133
  2197
  note h = this
huffman@44133
  2198
  let ?g = "\<lambda>z. ?h z *\<^sub>R f a + g (z - ?h z *\<^sub>R a)"
wenzelm@53406
  2199
  {
wenzelm@53406
  2200
    fix x y
wenzelm@53406
  2201
    assume x: "x \<in> span (insert a b)"
wenzelm@53406
  2202
      and y: "y \<in> span (insert a b)"
huffman@44133
  2203
    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
  2204
      by (simp add: algebra_simps)
huffman@44133
  2205
    have addh: "?h (x + y) = ?h x + ?h y"
huffman@44133
  2206
      apply (rule conjunct2[OF h, rule_format, symmetric])
huffman@44133
  2207
      apply (rule span_add[OF x y])
huffman@44133
  2208
      unfolding tha
wenzelm@53406
  2209
      apply (metis span_add x y conjunct1[OF h, rule_format])
wenzelm@53406
  2210
      done
huffman@44133
  2211
    have "?g (x + y) = ?g x + ?g y"
huffman@44133
  2212
      unfolding addh tha
huffman@44133
  2213
      g(1)[rule_format,OF conjunct1[OF h, OF x] conjunct1[OF h, OF y]]
huffman@44133
  2214
      by (simp add: scaleR_left_distrib)}
huffman@44133
  2215
  moreover
wenzelm@53406
  2216
  {
wenzelm@53406
  2217
    fix x :: "'a"
wenzelm@53406
  2218
    fix c :: real
wenzelm@49522
  2219
    assume x: "x \<in> span (insert a b)"
huffman@44133
  2220
    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
  2221
      by (simp add: algebra_simps)
huffman@44133
  2222
    have hc: "?h (c *\<^sub>R x) = c * ?h x"
huffman@44133
  2223
      apply (rule conjunct2[OF h, rule_format, symmetric])
huffman@44133
  2224
      apply (metis span_mul x)
wenzelm@49522
  2225
      apply (metis tha span_mul x conjunct1[OF h])
wenzelm@49522
  2226
      done
huffman@44133
  2227
    have "?g (c *\<^sub>R x) = c*\<^sub>R ?g x"
huffman@44133
  2228
      unfolding hc tha g(2)[rule_format, OF conjunct1[OF h, OF x]]
wenzelm@53406
  2229
      by (simp add: algebra_simps)
wenzelm@53406
  2230
  }
huffman@44133
  2231
  moreover
wenzelm@53406
  2232
  {
wenzelm@53406
  2233
    fix x
wenzelm@53406
  2234
    assume x: "x \<in> insert a b"
wenzelm@53406
  2235
    {
wenzelm@53406
  2236
      assume xa: "x = a"
huffman@44133
  2237
      have ha1: "1 = ?h a"
huffman@44133
  2238
        apply (rule conjunct2[OF h, rule_format])
huffman@44133
  2239
        apply (metis span_superset insertI1)
huffman@44133
  2240
        using conjunct1[OF h, OF span_superset, OF insertI1]
wenzelm@49522
  2241
        apply (auto simp add: span_0)
wenzelm@49522
  2242
        done
huffman@44133
  2243
      from xa ha1[symmetric] have "?g x = f x"
huffman@44133
  2244
        apply simp
huffman@44133
  2245
        using g(2)[rule_format, OF span_0, of 0]
wenzelm@49522
  2246
        apply simp
wenzelm@53406
  2247
        done
wenzelm@53406
  2248
    }
huffman@44133
  2249
    moreover
wenzelm@53406
  2250
    {
wenzelm@53406
  2251
      assume xb: "x \<in> b"
huffman@44133
  2252
      have h0: "0 = ?h x"
huffman@44133
  2253
        apply (rule conjunct2[OF h, rule_format])
huffman@44133
  2254
        apply (metis  span_superset x)
huffman@44133
  2255
        apply simp
huffman@44133
  2256
        apply (metis span_superset xb)
huffman@44133
  2257
        done
huffman@44133
  2258
      have "?g x = f x"
wenzelm@53406
  2259
        by (simp add: h0[symmetric] g(3)[rule_format, OF xb])
wenzelm@53406
  2260
    }
wenzelm@53406
  2261
    ultimately have "?g x = f x"
wenzelm@53406
  2262
      using x by blast
wenzelm@53406
  2263
  }
wenzelm@49663
  2264
  ultimately show ?case
wenzelm@49663
  2265
    apply -
wenzelm@49663
  2266
    apply (rule exI[where x="?g"])
wenzelm@49663
  2267
    apply blast
wenzelm@49663
  2268
    done
huffman@44133
  2269
qed
huffman@44133
  2270
huffman@44133
  2271
lemma linear_independent_extend:
wenzelm@53406
  2272
  fixes B :: "'a::euclidean_space set"
wenzelm@53406
  2273
  assumes iB: "independent B"
huffman@44133
  2274
  shows "\<exists>g. linear g \<and> (\<forall>x\<in>B. g x = f x)"
wenzelm@49522
  2275
proof -
huffman@44133
  2276
  from maximal_independent_subset_extend[of B UNIV] iB
wenzelm@53406
  2277
  obtain C where C: "B \<subseteq> C" "independent C" "\<And>x. x \<in> span C"
wenzelm@53406
  2278
    by auto
huffman@44133
  2279
huffman@44133
  2280
  from C(2) independent_bound[of C] linear_independent_extend_lemma[of C f]
wenzelm@53406
  2281
  obtain g where g:
wenzelm@53406
  2282
    "(\<forall>x\<in> span C. \<forall>y\<in> span C. g (x + y) = g x + g y) \<and>
wenzelm@53406
  2283
     (\<forall>x\<in> span C. \<forall>c. g (c*\<^sub>R x) = c *\<^sub>R g x) \<and>
wenzelm@53406
  2284
     (\<forall>x\<in> C. g x = f x)" by blast
wenzelm@53406
  2285
  from g show ?thesis
huffman@53600
  2286
    unfolding linear_iff
wenzelm@53406
  2287
    using C
wenzelm@49663
  2288
    apply clarsimp
wenzelm@49663
  2289
    apply blast
wenzelm@49663
  2290
    done
huffman@44133
  2291
qed
huffman@44133
  2292
huffman@44133
  2293
text {* Can construct an isomorphism between spaces of same dimension. *}
huffman@44133
  2294
huffman@44133
  2295
lemma subspace_isomorphism:
wenzelm@53406
  2296
  fixes S :: "'a::euclidean_space set"
wenzelm@53406
  2297
    and T :: "'b::euclidean_space set"
wenzelm@53406
  2298
  assumes s: "subspace S"
wenzelm@53406
  2299
    and t: "subspace T"
wenzelm@49522
  2300
    and d: "dim S = dim T"
huffman@44133
  2301
  shows "\<exists>f. linear f \<and> f ` S = T \<and> inj_on f S"
wenzelm@49522
  2302
proof -
wenzelm@53406
  2303
  from basis_exists[of S] independent_bound
wenzelm@53406
  2304
  obtain B where B: "B \<subseteq> S" "independent B" "S \<subseteq> span B" "card B = dim S" and fB: "finite B"
wenzelm@53406
  2305
    by blast
wenzelm@53406
  2306
  from basis_exists[of T] independent_bound
wenzelm@53406
  2307
  obtain C where C: "C \<subseteq> T" "independent C" "T \<subseteq> span C" "card C = dim T" and fC: "finite C"
wenzelm@53406
  2308
    by blast
wenzelm@53406
  2309
  from B(4) C(4) card_le_inj[of B C] d
wenzelm@53406
  2310
  obtain f where f: "f ` B \<subseteq> C" "inj_on f B" using `finite B` `finite C`
wenzelm@53406
  2311
    by auto
wenzelm@53406
  2312
  from linear_independent_extend[OF B(2)]
wenzelm@53406
  2313
  obtain g where g: "linear g" "\<forall>x\<in> B. g x = f x"
wenzelm@53406
  2314
    by blast
wenzelm@53406
  2315
  from inj_on_iff_eq_card[OF fB, of f] f(2) have "card (f ` B) = card B"
huffman@44133
  2316
    by simp
wenzelm@53406
  2317
  with B(4) C(4) have ceq: "card (f ` B) = card C"
wenzelm@53406
  2318
    using d by simp
wenzelm@53406
  2319
  have "g ` B = f ` B"
wenzelm@53406
  2320
    using g(2) by (auto simp add: image_iff)
huffman@44133
  2321
  also have "\<dots> = C" using card_subset_eq[OF fC f(1) ceq] .
huffman@44133
  2322
  finally have gBC: "g ` B = C" .
wenzelm@53406
  2323
  have gi: "inj_on g B"
wenzelm@53406
  2324
    using f(2) g(2) by (auto simp add: inj_on_def)
huffman@44133
  2325
  note g0 = linear_indep_image_lemma[OF g(1) fB, unfolded gBC, OF C(2) gi]
wenzelm@53406
  2326
  {
wenzelm@53406
  2327
    fix x y
wenzelm@53406
  2328
    assume x: "x \<in> S" and y: "y \<in> S" and gxy: "g x = g y"
wenzelm@53406
  2329
    from B(3) x y have x': "x \<in> span B" and y': "y \<in> span B"
wenzelm@53406
  2330
      by blast+
wenzelm@53406
  2331
    from gxy have th0: "g (x - y) = 0"
wenzelm@53406
  2332
      by (simp add: linear_sub[OF g(1)])
wenzelm@53406
  2333
    have th1: "x - y \<in> span B"
wenzelm@53406
  2334
      using x' y' by (metis span_sub)
wenzelm@53406
  2335
    have "x = y"
wenzelm@53406
  2336
      using g0[OF th1 th0] by simp
wenzelm@53406
  2337
  }
huffman@44133
  2338
  then have giS: "inj_on g S"
huffman@44133
  2339
    unfolding inj_on_def by blast
wenzelm@53406
  2340
  from span_subspace[OF B(1,3) s] have "g ` S = span (g ` B)"
wenzelm@53406
  2341
    by (simp add: span_linear_image[OF g(1)])
huffman@44133
  2342
  also have "\<dots> = span C" unfolding gBC ..
huffman@44133
  2343
  also have "\<dots> = T" using span_subspace[OF C(1,3) t] .
huffman@44133
  2344
  finally have gS: "g ` S = T" .
wenzelm@53406
  2345
  from g(1) gS giS show ?thesis
wenzelm@53406
  2346
    by blast
huffman@44133
  2347
qed
huffman@44133
  2348
huffman@44133
  2349
text {* Linear functions are equal on a subspace if they are on a spanning set. *}
huffman@44133
  2350
huffman@44133
  2351
lemma subspace_kernel:
huffman@44133
  2352
  assumes lf: "linear f"
huffman@44133
  2353
  shows "subspace {x. f x = 0}"
wenzelm@49522
  2354
  apply (simp add: subspace_def)
wenzelm@49522
  2355
  apply (simp add: linear_add[OF lf] linear_cmul[OF lf] linear_0[OF lf])
wenzelm@49522
  2356
  done
huffman@44133
  2357
huffman@44133
  2358
lemma linear_eq_0_span:
huffman@44133
  2359
  assumes lf: "linear f" and f0: "\<forall>x\<in>B. f x = 0"
huffman@44133
  2360
  shows "\<forall>x \<in> span B. f x = 0"
huffman@44170
  2361
  using f0 subspace_kernel[OF lf]
huffman@44170
  2362
  by (rule span_induct')
huffman@44133
  2363
huffman@44133
  2364
lemma linear_eq_0:
wenzelm@49663
  2365
  assumes lf: "linear f"
wenzelm@49663
  2366
    and SB: "S \<subseteq> span B"
wenzelm@49663
  2367
    and f0: "\<forall>x\<in>B. f x = 0"
huffman@44133
  2368
  shows "\<forall>x \<in> S. f x = 0"
huffman@44133
  2369
  by (metis linear_eq_0_span[OF lf] subset_eq SB f0)
huffman@44133
  2370
huffman@44133
  2371
lemma linear_eq:
wenzelm@49663
  2372
  assumes lf: "linear f"
wenzelm@49663
  2373
    and lg: "linear g"
wenzelm@49663
  2374
    and S: "S \<subseteq> span B"
wenzelm@49522
  2375
    and fg: "\<forall> x\<in> B. f x = g x"
huffman@44133
  2376
  shows "\<forall>x\<in> S. f x = g x"
wenzelm@49663
  2377
proof -
huffman@44133
  2378
  let ?h = "\<lambda>x. f x - g x"
huffman@44133
  2379
  from fg have fg': "\<forall>x\<in> B. ?h x = 0" by simp
huffman@44133
  2380
  from linear_eq_0[OF linear_compose_sub[OF lf lg] S fg']
huffman@44133
  2381
  show ?thesis by simp
huffman@44133
  2382
qed
huffman@44133
  2383
huffman@44133
  2384
lemma linear_eq_stdbasis:
wenzelm@56444
  2385
  fixes f :: "'a::euclidean_space \<Rightarrow> _"
wenzelm@56444
  2386
  assumes lf: "linear f"
wenzelm@49663
  2387
    and lg: "linear g"
hoelzl@50526
  2388
    and fg: "\<forall>b\<in>Basis. f b = g b"
huffman@44133
  2389
  shows "f = g"
hoelzl@50526
  2390
  using linear_eq[OF lf lg, of _ Basis] fg by auto
huffman@44133
  2391
huffman@44133
  2392
text {* Similar results for bilinear functions. *}
huffman@44133
  2393
huffman@44133
  2394
lemma bilinear_eq:
huffman@44133
  2395
  assumes bf: "bilinear f"
wenzelm@49522
  2396
    and bg: "bilinear g"
wenzelm@53406
  2397
    and SB: "S \<subseteq> span B"
wenzelm@53406
  2398
    and TC: "T \<subseteq> span C"
wenzelm@49522
  2399
    and fg: "\<forall>x\<in> B. \<forall>y\<in> C. f x y = g x y"
huffman@44133
  2400
  shows "\<forall>x\<in>S. \<forall>y\<in>T. f x y = g x y "
wenzelm@49663
  2401
proof -
huffman@44170
  2402
  let ?P = "{x. \<forall>y\<in> span C. f x y = g x y}"
huffman@44133
  2403
  from bf bg have sp: "subspace ?P"
huffman@53600
  2404
    unfolding bilinear_def linear_iff subspace_def bf bg
wenzelm@49663
  2405
    by (auto simp add: span_0 bilinear_lzero[OF bf] bilinear_lzero[OF bg] span_add Ball_def
wenzelm@49663
  2406
      intro: bilinear_ladd[OF bf])
huffman@44133
  2407
huffman@44133
  2408
  have "\<forall>x \<in> span B. \<forall>y\<in> span C. f x y = g x y"
huffman@44170
  2409
    apply (rule span_induct' [OF _ sp])
huffman@44133
  2410
    apply (rule ballI)
huffman@44170
  2411
    apply (rule span_induct')
huffman@44170
  2412
    apply (simp add: fg)
huffman@44133
  2413
    apply (auto simp add: subspace_def)
huffman@53600
  2414
    using bf bg unfolding bilinear_def linear_iff
wenzelm@49522
  2415
    apply (auto simp add: span_0 bilinear_rzero[OF bf] bilinear_rzero[OF bg] span_add Ball_def
wenzelm@49663
  2416
      intro: bilinear_ladd[OF bf])
wenzelm@49522
  2417
    done
wenzelm@53406
  2418
  then show ?thesis
wenzelm@53406
  2419
    using SB TC by auto
huffman@44133
  2420
qed
huffman@44133
  2421
wenzelm@49522
  2422
lemma bilinear_eq_stdbasis:
wenzelm@53406
  2423
  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space \<Rightarrow> _"
huffman@44133
  2424
  assumes bf: "bilinear f"
wenzelm@49522
  2425
    and bg: "bilinear g"
hoelzl@50526
  2426
    and fg: "\<forall>i\<in>Basis. \<forall>j\<in>Basis. f i j = g i j"
huffman@44133
  2427
  shows "f = g"
hoelzl@50526
  2428
  using bilinear_eq[OF bf bg equalityD2[OF span_Basis] equalityD2[OF span_Basis] fg] by blast
huffman@44133
  2429
huffman@44133
  2430
text {* Detailed theorems about left and right invertibility in general case. *}
huffman@44133
  2431
wenzelm@49522
  2432
lemma linear_injective_left_inverse:
wenzelm@56444
  2433
  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
wenzelm@56444
  2434
  assumes lf: "linear f"
wenzelm@56444
  2435
    and fi: "inj f"
wenzelm@56444
  2436
  shows "\<exists>g. linear g \<and> g \<circ> f = id"
wenzelm@49522
  2437
proof -
hoelzl@50526
  2438
  from linear_independent_extend[OF independent_injective_image, OF independent_Basis, OF lf fi]
wenzelm@56444
  2439
  obtain h :: "'b \<Rightarrow> 'a" where h: "linear h" "\<forall>x \<in> f ` Basis. h x = inv f x"
wenzelm@53406
  2440
    by blast
hoelzl@50526
  2441
  from h(2) have th: "\<forall>i\<in>Basis. (h \<circ> f) i = id i"
huffman@44133
  2442
    using inv_o_cancel[OF fi, unfolded fun_eq_iff id_def o_def]
huffman@44133
  2443
    by auto
huffman@44133
  2444
  from linear_eq_stdbasis[OF linear_compose[OF lf h(1)] linear_id th]
wenzelm@56444
  2445
  have "h \<circ> f = id" .
wenzelm@53406
  2446
  then show ?thesis
wenzelm@53406
  2447
    using h(1) by blast
huffman@44133
  2448
qed
huffman@44133
  2449
wenzelm@49522
  2450
lemma linear_surjective_right_inverse:
wenzelm@53406
  2451
  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
wenzelm@53406
  2452
  assumes lf: "linear f"
wenzelm@53406
  2453
    and sf: "surj f"
wenzelm@56444
  2454
  shows "\<exists>g. linear g \<and> f \<circ> g = id"
wenzelm@49522
  2455
proof -
hoelzl@50526
  2456
  from linear_independent_extend[OF independent_Basis[where 'a='b],of "inv f"]
wenzelm@56444
  2457
  obtain h :: "'b \<Rightarrow> 'a" where h: "linear h" "\<forall>x\<in>Basis. h x = inv f x"
wenzelm@53406
  2458
    by blast
wenzelm@56444
  2459
  from h(2) have th: "\<forall>i\<in>Basis. (f \<circ> h) i = id i"
hoelzl@50526
  2460
    using sf by (auto simp add: surj_iff_all)
huffman@44133
  2461
  from linear_eq_stdbasis[OF linear_compose[OF h(1) lf] linear_id th]
wenzelm@56444
  2462
  have "f \<circ> h = id" .
wenzelm@53406
  2463
  then show ?thesis
wenzelm@53406
  2464
    using h(1) by blast