src/HOL/Multivariate_Analysis/Linear_Algebra.thy
author haftmann
Sat, 05 Jul 2014 11:01:53 +0200
changeset 57514 bdc2c6b40bf2
parent 57512 cc97b347b301
child 58877 262572d90bc6
permissions -rw-r--r--
prefer ac_simps collections over separate name bindings for add and mult
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(*  Title:      HOL/Multivariate_Analysis/Linear_Algebra.thy
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    Author:     Amine Chaieb, University of Cambridge
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*)
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header {* Elementary linear algebra on Euclidean spaces *}
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theory Linear_Algebra
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imports
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  Euclidean_Space
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  "~~/src/HOL/Library/Infinite_Set"
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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 isCont_ident, of x, unfolded isCont_def LIM_eq, rule_format, of e 2]
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  apply (auto simp add: power2_eq_square)
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  apply (rule_tac x="s" in exI)
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  apply auto
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  apply (erule_tac x=y in allE)
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  apply auto
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  done
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text{* Hence derive more interesting properties of the norm. *}
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lemma norm_eq_0_dot: "norm x = 0 \<longleftrightarrow> x \<bullet> x = (0::real)"
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  by simp (* TODO: delete *)
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lemma norm_triangle_sub:
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  fixes x y :: "'a::real_normed_vector"
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  shows "norm x \<le> norm y + norm (x - y)"
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  using norm_triangle_ineq[of "y" "x - y"] by (simp add: field_simps)
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lemma norm_le: "norm x \<le> norm y \<longleftrightarrow> x \<bullet> x \<le> y \<bullet> y"
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  by (simp add: norm_eq_sqrt_inner)
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lemma norm_lt: "norm x < norm y \<longleftrightarrow> x \<bullet> x < y \<bullet> y"
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  by (simp add: norm_eq_sqrt_inner)
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lemma norm_eq: "norm x = norm y \<longleftrightarrow> x \<bullet> x = y \<bullet> y"
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  apply (subst order_eq_iff)
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  apply (auto simp: norm_le)
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  done
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lemma norm_eq_1: "norm x = 1 \<longleftrightarrow> x \<bullet> x = 1"
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  by (simp add: norm_eq_sqrt_inner)
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text{* Squaring equations and inequalities involving norms.  *}
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lemma dot_square_norm: "x \<bullet> x = (norm x)\<^sup>2"
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  by (simp only: power2_norm_eq_inner) (* TODO: move? *)
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lemma norm_eq_square: "norm x = a \<longleftrightarrow> 0 \<le> a \<and> x \<bullet> x = a\<^sup>2"
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  by (auto simp add: norm_eq_sqrt_inner)
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lemma real_abs_le_square_iff: "\<bar>x\<bar> \<le> \<bar>y\<bar> \<longleftrightarrow> (x::real)\<^sup>2 \<le> y\<^sup>2"
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proof
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  assume "\<bar>x\<bar> \<le> \<bar>y\<bar>"
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  then have "\<bar>x\<bar>\<^sup>2 \<le> \<bar>y\<bar>\<^sup>2" by (rule power_mono, simp)
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  then show "x\<^sup>2 \<le> y\<^sup>2" by simp
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next
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  assume "x\<^sup>2 \<le> y\<^sup>2"
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  then have "sqrt (x\<^sup>2) \<le> sqrt (y\<^sup>2)" by (rule real_sqrt_le_mono)
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  then show "\<bar>x\<bar> \<le> \<bar>y\<bar>" by simp
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qed
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lemma norm_le_square: "norm x \<le> a \<longleftrightarrow> 0 \<le> a \<and> x \<bullet> x \<le> a\<^sup>2"
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  apply (simp add: dot_square_norm real_abs_le_square_iff[symmetric])
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  using norm_ge_zero[of x]
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  apply arith
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  done
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lemma norm_ge_square: "norm x \<ge> a \<longleftrightarrow> a \<le> 0 \<or> x \<bullet> x \<ge> a\<^sup>2"
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  apply (simp add: dot_square_norm real_abs_le_square_iff[symmetric])
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  using norm_ge_zero[of x]
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  apply arith
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  done
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lemma norm_lt_square: "norm x < a \<longleftrightarrow> 0 < a \<and> x \<bullet> x < a\<^sup>2"
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  by (metis not_le norm_ge_square)
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lemma norm_gt_square: "norm x > a \<longleftrightarrow> a < 0 \<or> x \<bullet> x > a\<^sup>2"
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  by (metis norm_le_square not_less)
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text{* Dot product in terms of the norm rather than conversely. *}
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lemmas inner_simps = inner_add_left inner_add_right inner_diff_right inner_diff_left
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  inner_scaleR_left inner_scaleR_right
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lemma dot_norm: "x \<bullet> y = ((norm (x + y))\<^sup>2 - (norm x)\<^sup>2 - (norm y)\<^sup>2) / 2"
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  unfolding power2_norm_eq_inner inner_simps inner_commute by auto
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lemma dot_norm_neg: "x \<bullet> y = (((norm x)\<^sup>2 + (norm y)\<^sup>2) - (norm (x - y))\<^sup>2) / 2"
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  unfolding power2_norm_eq_inner inner_simps inner_commute
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  by (auto simp add: algebra_simps)
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text{* Equality of vectors in terms of @{term "op \<bullet>"} products.    *}
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lemma vector_eq: "x = y \<longleftrightarrow> x \<bullet> x = x \<bullet> y \<and> y \<bullet> y = x \<bullet> x"
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  (is "?lhs \<longleftrightarrow> ?rhs")
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proof
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  assume ?lhs
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  then show ?rhs by simp
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next
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  assume ?rhs
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  then have "x \<bullet> x - x \<bullet> y = 0 \<and> x \<bullet> y - y \<bullet> y = 0"
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    by simp
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  then have "x \<bullet> (x - y) = 0 \<and> y \<bullet> (x - y) = 0"
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    by (simp add: inner_diff inner_commute)
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  then have "(x - y) \<bullet> (x - y) = 0"
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    by (simp add: field_simps inner_diff inner_commute)
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  then show "x = y" by simp
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qed
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lemma norm_triangle_half_r:
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  "norm (y - x1) < e / 2 \<Longrightarrow> norm (y - x2) < e / 2 \<Longrightarrow> norm (x1 - x2) < e"
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  using dist_triangle_half_r unfolding dist_norm[symmetric] by auto
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lemma norm_triangle_half_l:
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  assumes "norm (x - y) < e / 2"
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    and "norm (x' - y) < e / 2"
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  shows "norm (x - x') < e"
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  using dist_triangle_half_l[OF assms[unfolded dist_norm[symmetric]]]
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  unfolding dist_norm[symmetric] .
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lemma norm_triangle_le: "norm x + norm y \<le> e \<Longrightarrow> norm (x + y) \<le> e"
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  by (rule norm_triangle_ineq [THEN order_trans])
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lemma norm_triangle_lt: "norm x + norm y < e \<Longrightarrow> norm (x + y) < e"
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  by (rule norm_triangle_ineq [THEN le_less_trans])
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lemma setsum_clauses:
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  shows "setsum f {} = 0"
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    and "finite S \<Longrightarrow> setsum f (insert x S) = (if x \<in> S then setsum f S else f x + setsum f S)"
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  by (auto simp add: insert_absorb)
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691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
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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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   167
  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"
53406
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  then have "\<forall>z. (x - y) \<bullet> z = 0"
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   186
    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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49522
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355f3d076924 tuned proofs;
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subsection {* Orthogonality. *}
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691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
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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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691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
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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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   210
  unfolding orthogonal_def inner_add inner_diff by auto
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   211
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end
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691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
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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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49522
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355f3d076924 tuned proofs;
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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)"
53600
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  (is "linear f \<longleftrightarrow> ?rhs")
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   223
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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   228
  assume "?rhs"
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  then show "linear f" by unfold_locales simp_all
53600
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qed
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53406
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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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53406
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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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53406
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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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53406
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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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53406
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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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53406
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lemma linear_id: "linear id"
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  by (simp add: linear_iff id_def)
53406
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   249
d4374a69ddff tuned proofs;
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lemma linear_zero: "linear (\<lambda>x. 0)"
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  by (simp add: linear_iff)
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691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
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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)"
56196
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   256
proof (cases "finite S")
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   257
  case True
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   258
  then show ?thesis
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   259
    using lS by induct (simp_all add: linear_zero linear_compose_add)
56444
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   260
next
f944ae8c80a3 tuned proofs;
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   261
  case False
f944ae8c80a3 tuned proofs;
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   262
  then show ?thesis
f944ae8c80a3 tuned proofs;
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   263
    by (simp add: linear_zero)
f944ae8c80a3 tuned proofs;
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   264
qed
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   265
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
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   266
lemma linear_0: "linear f \<Longrightarrow> f 0 = 0"
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   267
  unfolding linear_iff
44133
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   268
  apply clarsimp
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huffman
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   269
  apply (erule allE[where x="0::'a"])
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huffman
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   270
  apply simp
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   271
  done
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   272
53406
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   273
lemma linear_cmul: "linear f \<Longrightarrow> f (c *\<^sub>R x) = c *\<^sub>R f x"
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   274
  by (simp add: linear_iff)
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53406
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   276
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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   278
53716
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   279
lemma linear_add: "linear f \<Longrightarrow> f (x + y) = f x + f y"
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   280
  by (metis linear_iff)
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   281
53716
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   282
lemma linear_sub: "linear f \<Longrightarrow> f (x - y) = f x - f y"
54230
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parents: 53939
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   283
  using linear_add [of f x "- y"] by (simp add: linear_neg)
44133
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   284
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
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   285
lemma linear_setsum:
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   286
  assumes f: "linear f"
53406
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   287
  shows "f (setsum g S) = setsum (f \<circ> g) S"
56196
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huffman
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diff changeset
   288
proof (cases "finite S")
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   289
  case True
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   290
  then show ?thesis
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   291
    by induct (simp_all add: linear_0 [OF f] linear_add [OF f])
56444
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   292
next
f944ae8c80a3 tuned proofs;
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   293
  case False
f944ae8c80a3 tuned proofs;
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   294
  then show ?thesis
f944ae8c80a3 tuned proofs;
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   295
    by (simp add: linear_0 [OF f])
f944ae8c80a3 tuned proofs;
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   296
qed
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   297
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
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   298
lemma linear_setsum_mul:
53406
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   299
  assumes lin: "linear f"
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   300
  shows "f (setsum (\<lambda>i. c i *\<^sub>R v i) S) = setsum (\<lambda>i. c i *\<^sub>R f (v i)) S"
56196
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diff changeset
   301
  using linear_setsum[OF lin, of "\<lambda>i. c i *\<^sub>R v i" , unfolded o_def] linear_cmul[OF lin]
49522
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diff changeset
   302
  by simp
44133
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diff changeset
   303
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   304
lemma linear_injective_0:
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   305
  assumes lin: "linear f"
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   306
  shows "inj f \<longleftrightarrow> (\<forall>x. f x = 0 \<longrightarrow> x = 0)"
49663
b84fafaea4bb tuned proofs;
wenzelm
parents: 49652
diff changeset
   307
proof -
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   308
  have "inj f \<longleftrightarrow> (\<forall> x y. f x = f y \<longrightarrow> x = y)"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   309
    by (simp add: inj_on_def)
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   310
  also have "\<dots> \<longleftrightarrow> (\<forall> x y. f x - f y = 0 \<longrightarrow> x - y = 0)"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   311
    by simp
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   312
  also have "\<dots> \<longleftrightarrow> (\<forall> x y. f (x - y) = 0 \<longrightarrow> x - y = 0)"
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   313
    by (simp add: linear_sub[OF lin])
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   314
  also have "\<dots> \<longleftrightarrow> (\<forall> x. f x = 0 \<longrightarrow> x = 0)"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   315
    by auto
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   316
  finally show ?thesis .
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   317
qed
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   318
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
   319
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
   320
subsection {* Bilinear functions. *}
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   321
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   322
definition "bilinear f \<longleftrightarrow> (\<forall>x. linear (\<lambda>y. f x y)) \<and> (\<forall>y. linear (\<lambda>x. f x y))"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   323
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   324
lemma bilinear_ladd: "bilinear h \<Longrightarrow> h (x + y) z = h x z + h y z"
53600
8fda7ad57466 make 'linear' into a sublocale of 'bounded_linear';
huffman
parents: 53596
diff changeset
   325
  by (simp add: bilinear_def linear_iff)
49663
b84fafaea4bb tuned proofs;
wenzelm
parents: 49652
diff changeset
   326
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   327
lemma bilinear_radd: "bilinear h \<Longrightarrow> h x (y + z) = h x y + h x z"
53600
8fda7ad57466 make 'linear' into a sublocale of 'bounded_linear';
huffman
parents: 53596
diff changeset
   328
  by (simp add: bilinear_def linear_iff)
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   329
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   330
lemma bilinear_lmul: "bilinear h \<Longrightarrow> h (c *\<^sub>R x) y = c *\<^sub>R h x y"
53600
8fda7ad57466 make 'linear' into a sublocale of 'bounded_linear';
huffman
parents: 53596
diff changeset
   331
  by (simp add: bilinear_def linear_iff)
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   332
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   333
lemma bilinear_rmul: "bilinear h \<Longrightarrow> h x (c *\<^sub>R y) = c *\<^sub>R h x y"
53600
8fda7ad57466 make 'linear' into a sublocale of 'bounded_linear';
huffman
parents: 53596
diff changeset
   334
  by (simp add: bilinear_def linear_iff)
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   335
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   336
lemma bilinear_lneg: "bilinear h \<Longrightarrow> h (- x) y = - h x y"
54489
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54413
diff changeset
   337
  by (drule bilinear_lmul [of _ "- 1"]) simp
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   338
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   339
lemma bilinear_rneg: "bilinear h \<Longrightarrow> h x (- y) = - h x y"
54489
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54413
diff changeset
   340
  by (drule bilinear_rmul [of _ _ "- 1"]) simp
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   341
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   342
lemma (in ab_group_add) eq_add_iff: "x = x + y \<longleftrightarrow> y = 0"
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   343
  using add_imp_eq[of x y 0] by auto
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   344
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   345
lemma bilinear_lzero:
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   346
  assumes "bilinear h"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   347
  shows "h 0 x = 0"
49663
b84fafaea4bb tuned proofs;
wenzelm
parents: 49652
diff changeset
   348
  using bilinear_ladd [OF assms, of 0 0 x] by (simp add: eq_add_iff field_simps)
b84fafaea4bb tuned proofs;
wenzelm
parents: 49652
diff changeset
   349
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   350
lemma bilinear_rzero:
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   351
  assumes "bilinear h"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   352
  shows "h x 0 = 0"
49663
b84fafaea4bb tuned proofs;
wenzelm
parents: 49652
diff changeset
   353
  using bilinear_radd [OF assms, of x 0 0 ] by (simp add: eq_add_iff field_simps)
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   354
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   355
lemma bilinear_lsub: "bilinear h \<Longrightarrow> h (x - y) z = h x z - h y z"
54230
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 53939
diff changeset
   356
  using bilinear_ladd [of h x "- y"] by (simp add: bilinear_lneg)
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   357
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   358
lemma bilinear_rsub: "bilinear h \<Longrightarrow> h z (x - y) = h z x - h z y"
54230
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 53939
diff changeset
   359
  using bilinear_radd [of h _ x "- y"] by (simp add: bilinear_rneg)
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   360
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   361
lemma bilinear_setsum:
49663
b84fafaea4bb tuned proofs;
wenzelm
parents: 49652
diff changeset
   362
  assumes bh: "bilinear h"
b84fafaea4bb tuned proofs;
wenzelm
parents: 49652
diff changeset
   363
    and fS: "finite S"
b84fafaea4bb tuned proofs;
wenzelm
parents: 49652
diff changeset
   364
    and fT: "finite T"
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   365
  shows "h (setsum f S) (setsum g T) = setsum (\<lambda>(i,j). h (f i) (g j)) (S \<times> T) "
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
   366
proof -
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   367
  have "h (setsum f S) (setsum g T) = setsum (\<lambda>x. h (f x) (setsum g T)) S"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   368
    apply (rule linear_setsum[unfolded o_def])
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   369
    using bh fS
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   370
    apply (auto simp add: bilinear_def)
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
   371
    done
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   372
  also have "\<dots> = setsum (\<lambda>x. setsum (\<lambda>y. h (f x) (g y)) T) S"
57418
6ab1c7cb0b8d fact consolidation
haftmann
parents: 56536
diff changeset
   373
    apply (rule setsum.cong, simp)
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   374
    apply (rule linear_setsum[unfolded o_def])
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
   375
    using bh fT
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
   376
    apply (auto simp add: bilinear_def)
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
   377
    done
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   378
  finally show ?thesis
57418
6ab1c7cb0b8d fact consolidation
haftmann
parents: 56536
diff changeset
   379
    unfolding setsum.cartesian_product .
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   380
qed
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   381
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
   382
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
   383
subsection {* Adjoints. *}
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   384
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   385
definition "adjoint f = (SOME f'. \<forall>x y. f x \<bullet> y = x \<bullet> f' y)"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   386
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   387
lemma adjoint_unique:
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   388
  assumes "\<forall>x y. inner (f x) y = inner x (g y)"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   389
  shows "adjoint f = g"
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
   390
  unfolding adjoint_def
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   391
proof (rule some_equality)
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   392
  show "\<forall>x y. inner (f x) y = inner x (g y)"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   393
    by (rule assms)
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   394
next
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   395
  fix h
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   396
  assume "\<forall>x y. inner (f x) y = inner x (h y)"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   397
  then have "\<forall>x y. inner x (g y) = inner x (h y)"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   398
    using assms by simp
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   399
  then have "\<forall>x y. inner x (g y - h y) = 0"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   400
    by (simp add: inner_diff_right)
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   401
  then have "\<forall>y. inner (g y - h y) (g y - h y) = 0"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   402
    by simp
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   403
  then have "\<forall>y. h y = g y"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   404
    by simp
49652
2b82d495b586 tuned proofs;
wenzelm
parents: 49525
diff changeset
   405
  then show "h = g" by (simp add: ext)
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   406
qed
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   407
50526
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 50105
diff changeset
   408
text {* TODO: The following lemmas about adjoints should hold for any
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 50105
diff changeset
   409
Hilbert space (i.e. complete inner product space).
54703
499f92dc6e45 more antiquotations;
wenzelm
parents: 54489
diff changeset
   410
(see @{url "http://en.wikipedia.org/wiki/Hermitian_adjoint"})
50526
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 50105
diff changeset
   411
*}
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 50105
diff changeset
   412
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 50105
diff changeset
   413
lemma adjoint_works:
56444
f944ae8c80a3 tuned proofs;
wenzelm
parents: 56409
diff changeset
   414
  fixes f :: "'n::euclidean_space \<Rightarrow> 'm::euclidean_space"
50526
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 50105
diff changeset
   415
  assumes lf: "linear f"
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 50105
diff changeset
   416
  shows "x \<bullet> adjoint f y = f x \<bullet> y"
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 50105
diff changeset
   417
proof -
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 50105
diff changeset
   418
  have "\<forall>y. \<exists>w. \<forall>x. f x \<bullet> y = x \<bullet> w"
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 50105
diff changeset
   419
  proof (intro allI exI)
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 50105
diff changeset
   420
    fix y :: "'m" and x
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 50105
diff changeset
   421
    let ?w = "(\<Sum>i\<in>Basis. (f i \<bullet> y) *\<^sub>R i) :: 'n"
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 50105
diff changeset
   422
    have "f x \<bullet> y = f (\<Sum>i\<in>Basis. (x \<bullet> i) *\<^sub>R i) \<bullet> y"
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 50105
diff changeset
   423
      by (simp add: euclidean_representation)
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 50105
diff changeset
   424
    also have "\<dots> = (\<Sum>i\<in>Basis. (x \<bullet> i) *\<^sub>R f i) \<bullet> y"
56196
32b7eafc5a52 remove unnecessary finiteness assumptions from lemmas about setsum
huffman
parents: 56166
diff changeset
   425
      unfolding linear_setsum[OF lf]
50526
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 50105
diff changeset
   426
      by (simp add: linear_cmul[OF lf])
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 50105
diff changeset
   427
    finally show "f x \<bullet> y = x \<bullet> ?w"
57512
cc97b347b301 reduced name variants for assoc and commute on plus and mult
haftmann
parents: 57418
diff changeset
   428
      by (simp add: inner_setsum_left inner_setsum_right mult.commute)
50526
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 50105
diff changeset
   429
  qed
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 50105
diff changeset
   430
  then show ?thesis
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 50105
diff changeset
   431
    unfolding adjoint_def choice_iff
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 50105
diff changeset
   432
    by (intro someI2_ex[where Q="\<lambda>f'. x \<bullet> f' y = f x \<bullet> y"]) auto
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 50105
diff changeset
   433
qed
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 50105
diff changeset
   434
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 50105
diff changeset
   435
lemma adjoint_clauses:
56444
f944ae8c80a3 tuned proofs;
wenzelm
parents: 56409
diff changeset
   436
  fixes f :: "'n::euclidean_space \<Rightarrow> 'm::euclidean_space"
50526
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 50105
diff changeset
   437
  assumes lf: "linear f"
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 50105
diff changeset
   438
  shows "x \<bullet> adjoint f y = f x \<bullet> y"
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 50105
diff changeset
   439
    and "adjoint f y \<bullet> x = y \<bullet> f x"
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 50105
diff changeset
   440
  by (simp_all add: adjoint_works[OF lf] inner_commute)
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 50105
diff changeset
   441
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 50105
diff changeset
   442
lemma adjoint_linear:
56444
f944ae8c80a3 tuned proofs;
wenzelm
parents: 56409
diff changeset
   443
  fixes f :: "'n::euclidean_space \<Rightarrow> 'm::euclidean_space"
50526
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 50105
diff changeset
   444
  assumes lf: "linear f"
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 50105
diff changeset
   445
  shows "linear (adjoint f)"
53600
8fda7ad57466 make 'linear' into a sublocale of 'bounded_linear';
huffman
parents: 53596
diff changeset
   446
  by (simp add: lf linear_iff euclidean_eq_iff[where 'a='n] euclidean_eq_iff[where 'a='m]
53939
eb25bddf6a22 tuned proofs
huffman
parents: 53938
diff changeset
   447
    adjoint_clauses[OF lf] inner_distrib)
50526
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 50105
diff changeset
   448
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 50105
diff changeset
   449
lemma adjoint_adjoint:
56444
f944ae8c80a3 tuned proofs;
wenzelm
parents: 56409
diff changeset
   450
  fixes f :: "'n::euclidean_space \<Rightarrow> 'm::euclidean_space"
50526
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 50105
diff changeset
   451
  assumes lf: "linear f"
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 50105
diff changeset
   452
  shows "adjoint (adjoint f) = f"
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 50105
diff changeset
   453
  by (rule adjoint_unique, simp add: adjoint_clauses [OF lf])
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 50105
diff changeset
   454
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   455
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
   456
subsection {* Interlude: Some properties of real sets *}
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   457
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   458
lemma seq_mono_lemma:
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   459
  assumes "\<forall>(n::nat) \<ge> m. (d n :: real) < e n"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   460
    and "\<forall>n \<ge> m. e n \<le> e m"
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   461
  shows "\<forall>n \<ge> m. d n < e m"
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   462
  using assms
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   463
  apply auto
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   464
  apply (erule_tac x="n" in allE)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   465
  apply (erule_tac x="n" in allE)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   466
  apply auto
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   467
  done
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   468
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   469
lemma infinite_enumerate:
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   470
  assumes fS: "infinite S"
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   471
  shows "\<exists>r. subseq r \<and> (\<forall>n. r n \<in> S)"
49525
e87b42a26991 misc tuning;
wenzelm
parents: 49522
diff changeset
   472
  unfolding subseq_def
e87b42a26991 misc tuning;
wenzelm
parents: 49522
diff changeset
   473
  using enumerate_in_set[OF fS] enumerate_mono[of _ _ S] fS by auto
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   474
56444
f944ae8c80a3 tuned proofs;
wenzelm
parents: 56409
diff changeset
   475
lemma approachable_lt_le: "(\<exists>(d::real) > 0. \<forall>x. f x < d \<longrightarrow> P x) \<longleftrightarrow> (\<exists>d>0. \<forall>x. f x \<le> d \<longrightarrow> P x)"
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
   476
  apply auto
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
   477
  apply (rule_tac x="d/2" in exI)
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
   478
  apply auto
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
   479
  done
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   480
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   481
lemma triangle_lemma:
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   482
  fixes x y z :: real
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   483
  assumes x: "0 \<le> x"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   484
    and y: "0 \<le> y"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   485
    and z: "0 \<le> z"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   486
    and xy: "x\<^sup>2 \<le> y\<^sup>2 + z\<^sup>2"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   487
  shows "x \<le> y + z"
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
   488
proof -
56444
f944ae8c80a3 tuned proofs;
wenzelm
parents: 56409
diff changeset
   489
  have "y\<^sup>2 + z\<^sup>2 \<le> y\<^sup>2 + 2 * y * z + z\<^sup>2"
56536
aefb4a8da31f made mult_nonneg_nonneg a simp rule
nipkow
parents: 56480
diff changeset
   490
    using z y by simp
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   491
  with xy have th: "x\<^sup>2 \<le> (y + z)\<^sup>2"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   492
    by (simp add: power2_eq_square field_simps)
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   493
  from y z have yz: "y + z \<ge> 0"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   494
    by arith
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   495
  from power2_le_imp_le[OF th yz] show ?thesis .
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   496
qed
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   497
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
   498
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   499
subsection {* A generic notion of "hull" (convex, affine, conic hull and closure). *}
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   500
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   501
definition hull :: "('a set \<Rightarrow> bool) \<Rightarrow> 'a set \<Rightarrow> 'a set"  (infixl "hull" 75)
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   502
  where "S hull s = \<Inter>{t. S t \<and> s \<subseteq> t}"
44170
510ac30f44c0 make Multivariate_Analysis work with separate set type
huffman
parents: 44166
diff changeset
   503
510ac30f44c0 make Multivariate_Analysis work with separate set type
huffman
parents: 44166
diff changeset
   504
lemma hull_same: "S s \<Longrightarrow> S hull s = s"
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   505
  unfolding hull_def by auto
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   506
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   507
lemma hull_in: "(\<And>T. Ball T S \<Longrightarrow> S (\<Inter>T)) \<Longrightarrow> S (S hull s)"
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
   508
  unfolding hull_def Ball_def by auto
44170
510ac30f44c0 make Multivariate_Analysis work with separate set type
huffman
parents: 44166
diff changeset
   509
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   510
lemma hull_eq: "(\<And>T. Ball T S \<Longrightarrow> S (\<Inter>T)) \<Longrightarrow> (S hull s) = s \<longleftrightarrow> S s"
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
   511
  using hull_same[of S s] hull_in[of S s] by metis
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   512
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   513
lemma hull_hull: "S hull (S hull s) = S hull s"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   514
  unfolding hull_def by blast
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   515
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   516
lemma hull_subset[intro]: "s \<subseteq> (S hull s)"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   517
  unfolding hull_def by blast
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   518
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   519
lemma hull_mono: "s \<subseteq> t \<Longrightarrow> (S hull s) \<subseteq> (S hull t)"
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   520
  unfolding hull_def by blast
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   521
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   522
lemma hull_antimono: "\<forall>x. S x \<longrightarrow> T x \<Longrightarrow> (T hull s) \<subseteq> (S hull s)"
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   523
  unfolding hull_def by blast
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   524
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   525
lemma hull_minimal: "s \<subseteq> t \<Longrightarrow> S t \<Longrightarrow> (S hull s) \<subseteq> t"
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   526
  unfolding hull_def by blast
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   527
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   528
lemma subset_hull: "S t \<Longrightarrow> S hull s \<subseteq> t \<longleftrightarrow> s \<subseteq> t"
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   529
  unfolding hull_def by blast
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   530
53596
d29d63460d84 new lemmas
huffman
parents: 53595
diff changeset
   531
lemma hull_UNIV: "S hull UNIV = UNIV"
d29d63460d84 new lemmas
huffman
parents: 53595
diff changeset
   532
  unfolding hull_def by auto
d29d63460d84 new lemmas
huffman
parents: 53595
diff changeset
   533
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   534
lemma hull_unique: "s \<subseteq> t \<Longrightarrow> S t \<Longrightarrow> (\<And>t'. s \<subseteq> t' \<Longrightarrow> S t' \<Longrightarrow> t \<subseteq> t') \<Longrightarrow> (S hull s = t)"
49652
2b82d495b586 tuned proofs;
wenzelm
parents: 49525
diff changeset
   535
  unfolding hull_def by auto
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   536
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   537
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"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   538
  using hull_minimal[of S "{x. P x}" Q]
44170
510ac30f44c0 make Multivariate_Analysis work with separate set type
huffman
parents: 44166
diff changeset
   539
  by (auto simp add: subset_eq)
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   540
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
   541
lemma hull_inc: "x \<in> S \<Longrightarrow> x \<in> P hull S"
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
   542
  by (metis hull_subset subset_eq)
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   543
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   544
lemma hull_union_subset: "(S hull s) \<union> (S hull t) \<subseteq> (S hull (s \<union> t))"
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
   545
  unfolding Un_subset_iff by (metis hull_mono Un_upper1 Un_upper2)
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
   546
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
   547
lemma hull_union:
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   548
  assumes T: "\<And>T. Ball T S \<Longrightarrow> S (\<Inter>T)"
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   549
  shows "S hull (s \<union> t) = S hull (S hull s \<union> S hull t)"
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
   550
  apply rule
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
   551
  apply (rule hull_mono)
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
   552
  unfolding Un_subset_iff
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
   553
  apply (metis hull_subset Un_upper1 Un_upper2 subset_trans)
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
   554
  apply (rule hull_minimal)
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
   555
  apply (metis hull_union_subset)
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
   556
  apply (metis hull_in T)
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
   557
  done
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   558
56444
f944ae8c80a3 tuned proofs;
wenzelm
parents: 56409
diff changeset
   559
lemma hull_redundant_eq: "a \<in> (S hull s) \<longleftrightarrow> S hull (insert a s) = S hull s"
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   560
  unfolding hull_def by blast
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   561
56444
f944ae8c80a3 tuned proofs;
wenzelm
parents: 56409
diff changeset
   562
lemma hull_redundant: "a \<in> (S hull s) \<Longrightarrow> S hull (insert a s) = S hull s"
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
   563
  by (metis hull_redundant_eq)
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
   564
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   565
44666
8670a39d4420 remove more duplicate lemmas
huffman
parents: 44646
diff changeset
   566
subsection {* Archimedean properties and useful consequences *}
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   567
56444
f944ae8c80a3 tuned proofs;
wenzelm
parents: 56409
diff changeset
   568
lemma real_arch_simple: "\<exists>n::nat. x \<le> real n"
44666
8670a39d4420 remove more duplicate lemmas
huffman
parents: 44646
diff changeset
   569
  unfolding real_of_nat_def by (rule ex_le_of_nat)
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   570
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   571
lemma real_arch_inv: "0 < e \<longleftrightarrow> (\<exists>n::nat. n \<noteq> 0 \<and> 0 < inverse (real n) \<and> inverse (real n) < e)"
56480
093ea91498e6 field_simps: better support for negation and division, and power
hoelzl
parents: 56479
diff changeset
   572
  using reals_Archimedean[of e] less_trans[of 0 "1 / real n" e for n::nat]
093ea91498e6 field_simps: better support for negation and division, and power
hoelzl
parents: 56479
diff changeset
   573
  by (auto simp add: field_simps cong: conj_cong)
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   574
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   575
lemma real_pow_lbound: "0 \<le> x \<Longrightarrow> 1 + real n * x \<le> (1 + x) ^ n"
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
   576
proof (induct n)
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
   577
  case 0
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
   578
  then show ?case by simp
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   579
next
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   580
  case (Suc n)
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   581
  then have h: "1 + real n * x \<le> (1 + x) ^ n"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   582
    by simp
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   583
  from h have p: "1 \<le> (1 + x) ^ n"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   584
    using Suc.prems by simp
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   585
  from h have "1 + real n * x + x \<le> (1 + x) ^ n + x"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   586
    by simp
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   587
  also have "\<dots> \<le> (1 + x) ^ Suc n"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   588
    apply (subst diff_le_0_iff_le[symmetric])
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   589
    apply (simp add: field_simps)
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   590
    using mult_left_mono[OF p Suc.prems]
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   591
    apply simp
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
   592
    done
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   593
  finally show ?case
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   594
    by (simp add: real_of_nat_Suc field_simps)
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   595
qed
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   596
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   597
lemma real_arch_pow:
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   598
  fixes x :: real
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   599
  assumes x: "1 < x"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   600
  shows "\<exists>n. y < x^n"
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
   601
proof -
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   602
  from x have x0: "x - 1 > 0"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   603
    by arith
44666
8670a39d4420 remove more duplicate lemmas
huffman
parents: 44646
diff changeset
   604
  from reals_Archimedean3[OF x0, rule_format, of y]
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   605
  obtain n :: nat where n: "y < real n * (x - 1)" by metis
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   606
  from x0 have x00: "x- 1 \<ge> 0" by arith
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   607
  from real_pow_lbound[OF x00, of n] n
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   608
  have "y < x^n" by auto
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   609
  then show ?thesis by metis
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   610
qed
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   611
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   612
lemma real_arch_pow2:
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   613
  fixes x :: real
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   614
  shows "\<exists>n. x < 2^ n"
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   615
  using real_arch_pow[of 2 x] by simp
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   616
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
   617
lemma real_arch_pow_inv:
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   618
  fixes x y :: real
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   619
  assumes y: "y > 0"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   620
    and x1: "x < 1"
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   621
  shows "\<exists>n. x^n < y"
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   622
proof (cases "x > 0")
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   623
  case True
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   624
  with x1 have ix: "1 < 1/x" by (simp add: field_simps)
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   625
  from real_arch_pow[OF ix, of "1/y"]
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   626
  obtain n where n: "1/y < (1/x)^n" by blast
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   627
  then show ?thesis using y `x > 0`
56480
093ea91498e6 field_simps: better support for negation and division, and power
hoelzl
parents: 56479
diff changeset
   628
    by (auto simp add: field_simps)
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   629
next
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   630
  case False
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   631
  with y x1 show ?thesis
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   632
    apply auto
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   633
    apply (rule exI[where x=1])
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   634
    apply auto
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   635
    done
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   636
qed
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   637
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
   638
lemma forall_pos_mono:
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   639
  "(\<And>d e::real. d < e \<Longrightarrow> P d \<Longrightarrow> P e) \<Longrightarrow>
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   640
    (\<And>n::nat. n \<noteq> 0 \<Longrightarrow> P (inverse (real n))) \<Longrightarrow> (\<And>e. 0 < e \<Longrightarrow> P e)"
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   641
  by (metis real_arch_inv)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   642
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
   643
lemma forall_pos_mono_1:
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   644
  "(\<And>d e::real. d < e \<Longrightarrow> P d \<Longrightarrow> P e) \<Longrightarrow>
53716
b42d9a71fc1a tuned proofs;
wenzelm
parents: 53600
diff changeset
   645
    (\<And>n. P (inverse (real (Suc n)))) \<Longrightarrow> 0 < e \<Longrightarrow> P e"
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   646
  apply (rule forall_pos_mono)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   647
  apply auto
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   648
  apply (atomize)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   649
  apply (erule_tac x="n - 1" in allE)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   650
  apply auto
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   651
  done
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   652
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
   653
lemma real_archimedian_rdiv_eq_0:
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   654
  assumes x0: "x \<ge> 0"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   655
    and c: "c \<ge> 0"
56444
f944ae8c80a3 tuned proofs;
wenzelm
parents: 56409
diff changeset
   656
    and xc: "\<forall>(m::nat) > 0. real m * x \<le> c"
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   657
  shows "x = 0"
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   658
proof (rule ccontr)
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   659
  assume "x \<noteq> 0"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   660
  with x0 have xp: "x > 0" by arith
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   661
  from reals_Archimedean3[OF xp, rule_format, of c]
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   662
  obtain n :: nat where n: "c < real n * x"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   663
    by blast
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   664
  with xc[rule_format, of n] have "n = 0"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   665
    by arith
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   666
  with n c show False
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   667
    by simp
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   668
qed
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   669
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
   670
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   671
subsection{* A bit of linear algebra. *}
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   672
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
   673
definition (in real_vector) subspace :: "'a set \<Rightarrow> bool"
56444
f944ae8c80a3 tuned proofs;
wenzelm
parents: 56409
diff changeset
   674
  where "subspace S \<longleftrightarrow> 0 \<in> S \<and> (\<forall>x \<in> S. \<forall>y \<in> S. x + y \<in> S) \<and> (\<forall>c. \<forall>x \<in> S. c *\<^sub>R x \<in> S)"
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   675
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   676
definition (in real_vector) "span S = (subspace hull S)"
53716
b42d9a71fc1a tuned proofs;
wenzelm
parents: 53600
diff changeset
   677
definition (in real_vector) "dependent S \<longleftrightarrow> (\<exists>a \<in> S. a \<in> span (S - {a}))"
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   678
abbreviation (in real_vector) "independent s \<equiv> \<not> dependent s"
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   679
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   680
text {* Closure properties of subspaces. *}
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   681
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   682
lemma subspace_UNIV[simp]: "subspace UNIV"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   683
  by (simp add: subspace_def)
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   684
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   685
lemma (in real_vector) subspace_0: "subspace S \<Longrightarrow> 0 \<in> S"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   686
  by (metis subspace_def)
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   687
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   688
lemma (in real_vector) subspace_add: "subspace S \<Longrightarrow> x \<in> S \<Longrightarrow> y \<in> S \<Longrightarrow> x + y \<in> S"
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   689
  by (metis subspace_def)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   690
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   691
lemma (in real_vector) subspace_mul: "subspace S \<Longrightarrow> x \<in> S \<Longrightarrow> c *\<^sub>R x \<in> S"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   692
  by (metis subspace_def)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   693
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   694
lemma subspace_neg: "subspace S \<Longrightarrow> x \<in> S \<Longrightarrow> - x \<in> S"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   695
  by (metis scaleR_minus1_left subspace_mul)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   696
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   697
lemma subspace_sub: "subspace S \<Longrightarrow> x \<in> S \<Longrightarrow> y \<in> S \<Longrightarrow> x - y \<in> S"
54230
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 53939
diff changeset
   698
  using subspace_add [of S x "- y"] by (simp add: subspace_neg)
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   699
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   700
lemma (in real_vector) subspace_setsum:
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   701
  assumes sA: "subspace A"
56196
32b7eafc5a52 remove unnecessary finiteness assumptions from lemmas about setsum
huffman
parents: 56166
diff changeset
   702
    and f: "\<forall>x\<in>B. f x \<in> A"
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   703
  shows "setsum f B \<in> A"
56196
32b7eafc5a52 remove unnecessary finiteness assumptions from lemmas about setsum
huffman
parents: 56166
diff changeset
   704
proof (cases "finite B")
32b7eafc5a52 remove unnecessary finiteness assumptions from lemmas about setsum
huffman
parents: 56166
diff changeset
   705
  case True
32b7eafc5a52 remove unnecessary finiteness assumptions from lemmas about setsum
huffman
parents: 56166
diff changeset
   706
  then show ?thesis
32b7eafc5a52 remove unnecessary finiteness assumptions from lemmas about setsum
huffman
parents: 56166
diff changeset
   707
    using f by induct (simp_all add: subspace_0 [OF sA] subspace_add [OF sA])
32b7eafc5a52 remove unnecessary finiteness assumptions from lemmas about setsum
huffman
parents: 56166
diff changeset
   708
qed (simp add: subspace_0 [OF sA])
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   709
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   710
lemma subspace_linear_image:
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   711
  assumes lf: "linear f"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   712
    and sS: "subspace S"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   713
  shows "subspace (f ` S)"
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   714
  using lf sS linear_0[OF lf]
53600
8fda7ad57466 make 'linear' into a sublocale of 'bounded_linear';
huffman
parents: 53596
diff changeset
   715
  unfolding linear_iff subspace_def
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   716
  apply (auto simp add: image_iff)
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   717
  apply (rule_tac x="x + y" in bexI)
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   718
  apply auto
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   719
  apply (rule_tac x="c *\<^sub>R x" in bexI)
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   720
  apply auto
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   721
  done
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   722
44521
083eedb37a37 simplify many proofs about subspace and span;
huffman
parents: 44517
diff changeset
   723
lemma subspace_linear_vimage: "linear f \<Longrightarrow> subspace S \<Longrightarrow> subspace (f -` S)"
53600
8fda7ad57466 make 'linear' into a sublocale of 'bounded_linear';
huffman
parents: 53596
diff changeset
   724
  by (auto simp add: subspace_def linear_iff linear_0[of f])
44521
083eedb37a37 simplify many proofs about subspace and span;
huffman
parents: 44517
diff changeset
   725
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   726
lemma subspace_linear_preimage: "linear f \<Longrightarrow> subspace S \<Longrightarrow> subspace {x. f x \<in> S}"
53600
8fda7ad57466 make 'linear' into a sublocale of 'bounded_linear';
huffman
parents: 53596
diff changeset
   727
  by (auto simp add: subspace_def linear_iff linear_0[of f])
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   728
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   729
lemma subspace_trivial: "subspace {0}"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   730
  by (simp add: subspace_def)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   731
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   732
lemma (in real_vector) subspace_inter: "subspace A \<Longrightarrow> subspace B \<Longrightarrow> subspace (A \<inter> B)"
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   733
  by (simp add: subspace_def)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   734
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   735
lemma subspace_Times: "subspace A \<Longrightarrow> subspace B \<Longrightarrow> subspace (A \<times> B)"
44521
083eedb37a37 simplify many proofs about subspace and span;
huffman
parents: 44517
diff changeset
   736
  unfolding subspace_def zero_prod_def by simp
083eedb37a37 simplify many proofs about subspace and span;
huffman
parents: 44517
diff changeset
   737
083eedb37a37 simplify many proofs about subspace and span;
huffman
parents: 44517
diff changeset
   738
text {* Properties of span. *}
083eedb37a37 simplify many proofs about subspace and span;
huffman
parents: 44517
diff changeset
   739
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   740
lemma (in real_vector) span_mono: "A \<subseteq> B \<Longrightarrow> span A \<subseteq> span B"
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   741
  by (metis span_def hull_mono)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   742
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   743
lemma (in real_vector) subspace_span: "subspace (span S)"
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   744
  unfolding span_def
44170
510ac30f44c0 make Multivariate_Analysis work with separate set type
huffman
parents: 44166
diff changeset
   745
  apply (rule hull_in)
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   746
  apply (simp only: subspace_def Inter_iff Int_iff subset_eq)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   747
  apply auto
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   748
  done
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   749
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   750
lemma (in real_vector) span_clauses:
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   751
  "a \<in> S \<Longrightarrow> a \<in> span S"
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   752
  "0 \<in> span S"
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   753
  "x\<in> span S \<Longrightarrow> y \<in> span S \<Longrightarrow> x + y \<in> span S"
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   754
  "x \<in> span S \<Longrightarrow> c *\<^sub>R x \<in> span S"
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   755
  by (metis span_def hull_subset subset_eq) (metis subspace_span subspace_def)+
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   756
44521
083eedb37a37 simplify many proofs about subspace and span;
huffman
parents: 44517
diff changeset
   757
lemma span_unique:
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
   758
  "S \<subseteq> T \<Longrightarrow> subspace T \<Longrightarrow> (\<And>T'. S \<subseteq> T' \<Longrightarrow> subspace T' \<Longrightarrow> T \<subseteq> T') \<Longrightarrow> span S = T"
44521
083eedb37a37 simplify many proofs about subspace and span;
huffman
parents: 44517
diff changeset
   759
  unfolding span_def by (rule hull_unique)
083eedb37a37 simplify many proofs about subspace and span;
huffman
parents: 44517
diff changeset
   760
083eedb37a37 simplify many proofs about subspace and span;
huffman
parents: 44517
diff changeset
   761
lemma span_minimal: "S \<subseteq> T \<Longrightarrow> subspace T \<Longrightarrow> span S \<subseteq> T"
083eedb37a37 simplify many proofs about subspace and span;
huffman
parents: 44517
diff changeset
   762
  unfolding span_def by (rule hull_minimal)
083eedb37a37 simplify many proofs about subspace and span;
huffman
parents: 44517
diff changeset
   763
083eedb37a37 simplify many proofs about subspace and span;
huffman
parents: 44517
diff changeset
   764
lemma (in real_vector) span_induct:
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
   765
  assumes x: "x \<in> span S"
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
   766
    and P: "subspace P"
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   767
    and SP: "\<And>x. x \<in> S \<Longrightarrow> x \<in> P"
44521
083eedb37a37 simplify many proofs about subspace and span;
huffman
parents: 44517
diff changeset
   768
  shows "x \<in> P"
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
   769
proof -
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   770
  from SP have SP': "S \<subseteq> P"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   771
    by (simp add: subset_eq)
44170
510ac30f44c0 make Multivariate_Analysis work with separate set type
huffman
parents: 44166
diff changeset
   772
  from x hull_minimal[where S=subspace, OF SP' P, unfolded span_def[symmetric]]
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   773
  show "x \<in> P"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   774
    by (metis subset_eq)
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   775
qed
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   776
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   777
lemma span_empty[simp]: "span {} = {0}"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   778
  apply (simp add: span_def)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   779
  apply (rule hull_unique)
44170
510ac30f44c0 make Multivariate_Analysis work with separate set type
huffman
parents: 44166
diff changeset
   780
  apply (auto simp add: subspace_def)
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   781
  done
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   782
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   783
lemma (in real_vector) independent_empty[intro]: "independent {}"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   784
  by (simp add: dependent_def)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   785
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
   786
lemma dependent_single[simp]: "dependent {x} \<longleftrightarrow> x = 0"
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   787
  unfolding dependent_def by auto
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   788
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   789
lemma (in real_vector) independent_mono: "independent A \<Longrightarrow> B \<subseteq> A \<Longrightarrow> independent B"
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   790
  apply (clarsimp simp add: dependent_def span_mono)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   791
  apply (subgoal_tac "span (B - {a}) \<le> span (A - {a})")
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   792
  apply force
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   793
  apply (rule span_mono)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   794
  apply auto
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   795
  done
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   796
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   797
lemma (in real_vector) span_subspace: "A \<subseteq> B \<Longrightarrow> B \<le> span A \<Longrightarrow>  subspace B \<Longrightarrow> span A = B"
44170
510ac30f44c0 make Multivariate_Analysis work with separate set type
huffman
parents: 44166
diff changeset
   798
  by (metis order_antisym span_def hull_minimal)
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   799
49711
e5aaae7eadc9 tuned proofs;
wenzelm
parents: 49663
diff changeset
   800
lemma (in real_vector) span_induct':
e5aaae7eadc9 tuned proofs;
wenzelm
parents: 49663
diff changeset
   801
  assumes SP: "\<forall>x \<in> S. P x"
e5aaae7eadc9 tuned proofs;
wenzelm
parents: 49663
diff changeset
   802
    and P: "subspace {x. P x}"
e5aaae7eadc9 tuned proofs;
wenzelm
parents: 49663
diff changeset
   803
  shows "\<forall>x \<in> span S. P x"
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   804
  using span_induct SP P by blast
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   805
56444
f944ae8c80a3 tuned proofs;
wenzelm
parents: 56409
diff changeset
   806
inductive_set (in real_vector) span_induct_alt_help for S :: "'a set"
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   807
where
44170
510ac30f44c0 make Multivariate_Analysis work with separate set type
huffman
parents: 44166
diff changeset
   808
  span_induct_alt_help_0: "0 \<in> span_induct_alt_help S"
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
   809
| span_induct_alt_help_S:
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   810
    "x \<in> S \<Longrightarrow> z \<in> span_induct_alt_help S \<Longrightarrow>
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   811
      (c *\<^sub>R x + z) \<in> span_induct_alt_help S"
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   812
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   813
lemma span_induct_alt':
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   814
  assumes h0: "h 0"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   815
    and hS: "\<And>c x y. x \<in> S \<Longrightarrow> h y \<Longrightarrow> h (c *\<^sub>R x + y)"
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
   816
  shows "\<forall>x \<in> span S. h x"
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
   817
proof -
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   818
  {
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   819
    fix x :: 'a
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   820
    assume x: "x \<in> span_induct_alt_help S"
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   821
    have "h x"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   822
      apply (rule span_induct_alt_help.induct[OF x])
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   823
      apply (rule h0)
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   824
      apply (rule hS)
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   825
      apply assumption
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   826
      apply assumption
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   827
      done
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   828
  }
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   829
  note th0 = this
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   830
  {
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   831
    fix x
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   832
    assume x: "x \<in> span S"
44170
510ac30f44c0 make Multivariate_Analysis work with separate set type
huffman
parents: 44166
diff changeset
   833
    have "x \<in> span_induct_alt_help S"
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
   834
    proof (rule span_induct[where x=x and S=S])
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   835
      show "x \<in> span S" by (rule x)
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
   836
    next
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   837
      fix x
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   838
      assume xS: "x \<in> S"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   839
      from span_induct_alt_help_S[OF xS span_induct_alt_help_0, of 1]
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   840
      show "x \<in> span_induct_alt_help S"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   841
        by simp
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
   842
    next
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
   843
      have "0 \<in> span_induct_alt_help S" by (rule span_induct_alt_help_0)
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
   844
      moreover
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   845
      {
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   846
        fix x y
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
   847
        assume h: "x \<in> span_induct_alt_help S" "y \<in> span_induct_alt_help S"
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
   848
        from h have "(x + y) \<in> span_induct_alt_help S"
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
   849
          apply (induct rule: span_induct_alt_help.induct)
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
   850
          apply simp
57512
cc97b347b301 reduced name variants for assoc and commute on plus and mult
haftmann
parents: 57418
diff changeset
   851
          unfolding add.assoc
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
   852
          apply (rule span_induct_alt_help_S)
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
   853
          apply assumption
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
   854
          apply simp
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   855
          done
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   856
      }
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
   857
      moreover
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   858
      {
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   859
        fix c x
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
   860
        assume xt: "x \<in> span_induct_alt_help S"
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
   861
        then have "(c *\<^sub>R x) \<in> span_induct_alt_help S"
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
   862
          apply (induct rule: span_induct_alt_help.induct)
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
   863
          apply (simp add: span_induct_alt_help_0)
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
   864
          apply (simp add: scaleR_right_distrib)
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
   865
          apply (rule span_induct_alt_help_S)
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
   866
          apply assumption
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
   867
          apply simp
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
   868
          done }
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   869
      ultimately show "subspace (span_induct_alt_help S)"
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
   870
        unfolding subspace_def Ball_def by blast
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   871
    qed
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   872
  }
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   873
  with th0 show ?thesis by blast
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   874
qed
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   875
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   876
lemma span_induct_alt:
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   877
  assumes h0: "h 0"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   878
    and hS: "\<And>c x y. x \<in> S \<Longrightarrow> h y \<Longrightarrow> h (c *\<^sub>R x + y)"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   879
    and x: "x \<in> span S"
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   880
  shows "h x"
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
   881
  using span_induct_alt'[of h S] h0 hS x by blast
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   882
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   883
text {* Individual closure properties. *}
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   884
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   885
lemma span_span: "span (span A) = span A"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   886
  unfolding span_def hull_hull ..
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   887
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   888
lemma (in real_vector) span_superset: "x \<in> S \<Longrightarrow> x \<in> span S"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   889
  by (metis span_clauses(1))
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   890
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   891
lemma (in real_vector) span_0: "0 \<in> span S"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   892
  by (metis subspace_span subspace_0)
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   893
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   894
lemma span_inc: "S \<subseteq> span S"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   895
  by (metis subset_eq span_superset)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   896
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   897
lemma (in real_vector) dependent_0:
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   898
  assumes "0 \<in> A"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   899
  shows "dependent A"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   900
  unfolding dependent_def
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   901
  apply (rule_tac x=0 in bexI)
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   902
  using assms span_0
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   903
  apply auto
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   904
  done
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   905
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   906
lemma (in real_vector) span_add: "x \<in> span S \<Longrightarrow> y \<in> span S \<Longrightarrow> x + y \<in> span S"
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   907
  by (metis subspace_add subspace_span)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   908
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   909
lemma (in real_vector) span_mul: "x \<in> span S \<Longrightarrow> c *\<^sub>R x \<in> span S"
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   910
  by (metis subspace_span subspace_mul)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   911
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   912
lemma span_neg: "x \<in> span S \<Longrightarrow> - x \<in> span S"
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   913
  by (metis subspace_neg subspace_span)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   914
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   915
lemma span_sub: "x \<in> span S \<Longrightarrow> y \<in> span S \<Longrightarrow> x - y \<in> span S"
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   916
  by (metis subspace_span subspace_sub)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   917
56196
32b7eafc5a52 remove unnecessary finiteness assumptions from lemmas about setsum
huffman
parents: 56166
diff changeset
   918
lemma (in real_vector) span_setsum: "\<forall>x\<in>A. f x \<in> span S \<Longrightarrow> setsum f A \<in> span S"
32b7eafc5a52 remove unnecessary finiteness assumptions from lemmas about setsum
huffman
parents: 56166
diff changeset
   919
  by (rule subspace_setsum [OF subspace_span])
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   920
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   921
lemma span_add_eq: "x \<in> span S \<Longrightarrow> x + y \<in> span S \<longleftrightarrow> y \<in> span S"
55775
1557a391a858 A bit of tidying up
paulson <lp15@cam.ac.uk>
parents: 55136
diff changeset
   922
  by (metis add_minus_cancel scaleR_minus1_left subspace_def subspace_span)
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   923
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   924
text {* Mapping under linear image. *}
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   925
44521
083eedb37a37 simplify many proofs about subspace and span;
huffman
parents: 44517
diff changeset
   926
lemma span_linear_image:
083eedb37a37 simplify many proofs about subspace and span;
huffman
parents: 44517
diff changeset
   927
  assumes lf: "linear f"
56444
f944ae8c80a3 tuned proofs;
wenzelm
parents: 56409
diff changeset
   928
  shows "span (f ` S) = f ` span S"
44521
083eedb37a37 simplify many proofs about subspace and span;
huffman
parents: 44517
diff changeset
   929
proof (rule span_unique)
083eedb37a37 simplify many proofs about subspace and span;
huffman
parents: 44517
diff changeset
   930
  show "f ` S \<subseteq> f ` span S"
083eedb37a37 simplify many proofs about subspace and span;
huffman
parents: 44517
diff changeset
   931
    by (intro image_mono span_inc)
083eedb37a37 simplify many proofs about subspace and span;
huffman
parents: 44517
diff changeset
   932
  show "subspace (f ` span S)"
083eedb37a37 simplify many proofs about subspace and span;
huffman
parents: 44517
diff changeset
   933
    using lf subspace_span by (rule subspace_linear_image)
083eedb37a37 simplify many proofs about subspace and span;
huffman
parents: 44517
diff changeset
   934
next
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   935
  fix T
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   936
  assume "f ` S \<subseteq> T" and "subspace T"
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
   937
  then show "f ` span S \<subseteq> T"
44521
083eedb37a37 simplify many proofs about subspace and span;
huffman
parents: 44517
diff changeset
   938
    unfolding image_subset_iff_subset_vimage
083eedb37a37 simplify many proofs about subspace and span;
huffman
parents: 44517
diff changeset
   939
    by (intro span_minimal subspace_linear_vimage lf)
083eedb37a37 simplify many proofs about subspace and span;
huffman
parents: 44517
diff changeset
   940
qed
083eedb37a37 simplify many proofs about subspace and span;
huffman
parents: 44517
diff changeset
   941
083eedb37a37 simplify many proofs about subspace and span;
huffman
parents: 44517
diff changeset
   942
lemma span_union: "span (A \<union> B) = (\<lambda>(a, b). a + b) ` (span A \<times> span B)"
083eedb37a37 simplify many proofs about subspace and span;
huffman
parents: 44517
diff changeset
   943
proof (rule span_unique)
083eedb37a37 simplify many proofs about subspace and span;
huffman
parents: 44517
diff changeset
   944
  show "A \<union> B \<subseteq> (\<lambda>(a, b). a + b) ` (span A \<times> span B)"
083eedb37a37 simplify many proofs about subspace and span;
huffman
parents: 44517
diff changeset
   945
    by safe (force intro: span_clauses)+
083eedb37a37 simplify many proofs about subspace and span;
huffman
parents: 44517
diff changeset
   946
next
083eedb37a37 simplify many proofs about subspace and span;
huffman
parents: 44517
diff changeset
   947
  have "linear (\<lambda>(a, b). a + b)"
53600
8fda7ad57466 make 'linear' into a sublocale of 'bounded_linear';
huffman
parents: 53596
diff changeset
   948
    by (simp add: linear_iff scaleR_add_right)
44521
083eedb37a37 simplify many proofs about subspace and span;
huffman
parents: 44517
diff changeset
   949
  moreover have "subspace (span A \<times> span B)"
083eedb37a37 simplify many proofs about subspace and span;
huffman
parents: 44517
diff changeset
   950
    by (intro subspace_Times subspace_span)
083eedb37a37 simplify many proofs about subspace and span;
huffman
parents: 44517
diff changeset
   951
  ultimately show "subspace ((\<lambda>(a, b). a + b) ` (span A \<times> span B))"
083eedb37a37 simplify many proofs about subspace and span;
huffman
parents: 44517
diff changeset
   952
    by (rule subspace_linear_image)
083eedb37a37 simplify many proofs about subspace and span;
huffman
parents: 44517
diff changeset
   953
next
49711
e5aaae7eadc9 tuned proofs;
wenzelm
parents: 49663
diff changeset
   954
  fix T
e5aaae7eadc9 tuned proofs;
wenzelm
parents: 49663
diff changeset
   955
  assume "A \<union> B \<subseteq> T" and "subspace T"
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
   956
  then show "(\<lambda>(a, b). a + b) ` (span A \<times> span B) \<subseteq> T"
44521
083eedb37a37 simplify many proofs about subspace and span;
huffman
parents: 44517
diff changeset
   957
    by (auto intro!: subspace_add elim: span_induct)
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   958
qed
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   959
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   960
text {* The key breakdown property. *}
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   961
44521
083eedb37a37 simplify many proofs about subspace and span;
huffman
parents: 44517
diff changeset
   962
lemma span_singleton: "span {x} = range (\<lambda>k. k *\<^sub>R x)"
083eedb37a37 simplify many proofs about subspace and span;
huffman
parents: 44517
diff changeset
   963
proof (rule span_unique)
083eedb37a37 simplify many proofs about subspace and span;
huffman
parents: 44517
diff changeset
   964
  show "{x} \<subseteq> range (\<lambda>k. k *\<^sub>R x)"
083eedb37a37 simplify many proofs about subspace and span;
huffman
parents: 44517
diff changeset
   965
    by (fast intro: scaleR_one [symmetric])
083eedb37a37 simplify many proofs about subspace and span;
huffman
parents: 44517
diff changeset
   966
  show "subspace (range (\<lambda>k. k *\<^sub>R x))"
083eedb37a37 simplify many proofs about subspace and span;
huffman
parents: 44517
diff changeset
   967
    unfolding subspace_def
083eedb37a37 simplify many proofs about subspace and span;
huffman
parents: 44517
diff changeset
   968
    by (auto intro: scaleR_add_left [symmetric])
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   969
next
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   970
  fix T
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   971
  assume "{x} \<subseteq> T" and "subspace T"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   972
  then show "range (\<lambda>k. k *\<^sub>R x) \<subseteq> T"
44521
083eedb37a37 simplify many proofs about subspace and span;
huffman
parents: 44517
diff changeset
   973
    unfolding subspace_def by auto
083eedb37a37 simplify many proofs about subspace and span;
huffman
parents: 44517
diff changeset
   974
qed
083eedb37a37 simplify many proofs about subspace and span;
huffman
parents: 44517
diff changeset
   975
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
   976
lemma span_insert: "span (insert a S) = {x. \<exists>k. (x - k *\<^sub>R a) \<in> span S}"
44521
083eedb37a37 simplify many proofs about subspace and span;
huffman
parents: 44517
diff changeset
   977
proof -
083eedb37a37 simplify many proofs about subspace and span;
huffman
parents: 44517
diff changeset
   978
  have "span ({a} \<union> S) = {x. \<exists>k. (x - k *\<^sub>R a) \<in> span S}"
083eedb37a37 simplify many proofs about subspace and span;
huffman
parents: 44517
diff changeset
   979
    unfolding span_union span_singleton
083eedb37a37 simplify many proofs about subspace and span;
huffman
parents: 44517
diff changeset
   980
    apply safe
083eedb37a37 simplify many proofs about subspace and span;
huffman
parents: 44517
diff changeset
   981
    apply (rule_tac x=k in exI, simp)
083eedb37a37 simplify many proofs about subspace and span;
huffman
parents: 44517
diff changeset
   982
    apply (erule rev_image_eqI [OF SigmaI [OF rangeI]])
54230
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 53939
diff changeset
   983
    apply auto
44521
083eedb37a37 simplify many proofs about subspace and span;
huffman
parents: 44517
diff changeset
   984
    done
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
   985
  then show ?thesis by simp
44521
083eedb37a37 simplify many proofs about subspace and span;
huffman
parents: 44517
diff changeset
   986
qed
083eedb37a37 simplify many proofs about subspace and span;
huffman
parents: 44517
diff changeset
   987
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   988
lemma span_breakdown:
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   989
  assumes bS: "b \<in> S"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   990
    and aS: "a \<in> span S"
44521
083eedb37a37 simplify many proofs about subspace and span;
huffman
parents: 44517
diff changeset
   991
  shows "\<exists>k. a - k *\<^sub>R b \<in> span (S - {b})"
083eedb37a37 simplify many proofs about subspace and span;
huffman
parents: 44517
diff changeset
   992
  using assms span_insert [of b "S - {b}"]
083eedb37a37 simplify many proofs about subspace and span;
huffman
parents: 44517
diff changeset
   993
  by (simp add: insert_absorb)
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   994
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
   995
lemma span_breakdown_eq: "x \<in> span (insert a S) \<longleftrightarrow> (\<exists>k. x - k *\<^sub>R a \<in> span S)"
44521
083eedb37a37 simplify many proofs about subspace and span;
huffman
parents: 44517
diff changeset
   996
  by (simp add: span_insert)
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   997
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   998
text {* Hence some "reversal" results. *}
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   999
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1000
lemma in_span_insert:
49711
e5aaae7eadc9 tuned proofs;
wenzelm
parents: 49663
diff changeset
  1001
  assumes a: "a \<in> span (insert b S)"
e5aaae7eadc9 tuned proofs;
wenzelm
parents: 49663
diff changeset
  1002
    and na: "a \<notin> span S"
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1003
  shows "b \<in> span (insert a S)"
49663
b84fafaea4bb tuned proofs;
wenzelm
parents: 49652
diff changeset
  1004
proof -
55910
0a756571c7a4 tuned proof
huffman
parents: 55775
diff changeset
  1005
  from a obtain k where k: "a - k *\<^sub>R b \<in> span S"
0a756571c7a4 tuned proof
huffman
parents: 55775
diff changeset
  1006
    unfolding span_insert by fast
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1007
  show ?thesis
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1008
  proof (cases "k = 0")
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1009
    case True
55910
0a756571c7a4 tuned proof
huffman
parents: 55775
diff changeset
  1010
    with k have "a \<in> span S" by simp
0a756571c7a4 tuned proof
huffman
parents: 55775
diff changeset
  1011
    with na show ?thesis by simp
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1012
  next
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1013
    case False
55910
0a756571c7a4 tuned proof
huffman
parents: 55775
diff changeset
  1014
    from k have "(- inverse k) *\<^sub>R (a - k *\<^sub>R b) \<in> span S"
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1015
      by (rule span_mul)
55910
0a756571c7a4 tuned proof
huffman
parents: 55775
diff changeset
  1016
    then have "b - inverse k *\<^sub>R a \<in> span S"
0a756571c7a4 tuned proof
huffman
parents: 55775
diff changeset
  1017
      using `k \<noteq> 0` by (simp add: scaleR_diff_right)
0a756571c7a4 tuned proof
huffman
parents: 55775
diff changeset
  1018
    then show ?thesis
0a756571c7a4 tuned proof
huffman
parents: 55775
diff changeset
  1019
      unfolding span_insert by fast
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1020
  qed
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1021
qed
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1022
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1023
lemma in_span_delete:
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1024
  assumes a: "a \<in> span S"
53716
b42d9a71fc1a tuned proofs;
wenzelm
parents: 53600
diff changeset
  1025
    and na: "a \<notin> span (S - {b})"
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1026
  shows "b \<in> span (insert a (S - {b}))"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1027
  apply (rule in_span_insert)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1028
  apply (rule set_rev_mp)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1029
  apply (rule a)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1030
  apply (rule span_mono)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1031
  apply blast
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1032
  apply (rule na)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1033
  done
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1034
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1035
text {* Transitivity property. *}
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1036
44521
083eedb37a37 simplify many proofs about subspace and span;
huffman
parents: 44517
diff changeset
  1037
lemma span_redundant: "x \<in> span S \<Longrightarrow> span (insert x S) = span S"
083eedb37a37 simplify many proofs about subspace and span;
huffman
parents: 44517
diff changeset
  1038
  unfolding span_def by (rule hull_redundant)
083eedb37a37 simplify many proofs about subspace and span;
huffman
parents: 44517
diff changeset
  1039
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1040
lemma span_trans:
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1041
  assumes x: "x \<in> span S"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1042
    and y: "y \<in> span (insert x S)"
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1043
  shows "y \<in> span S"
44521
083eedb37a37 simplify many proofs about subspace and span;
huffman
parents: 44517
diff changeset
  1044
  using assms by (simp only: span_redundant)
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1045
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1046
lemma span_insert_0[simp]: "span (insert 0 S) = span S"
44521
083eedb37a37 simplify many proofs about subspace and span;
huffman
parents: 44517
diff changeset
  1047
  by (simp only: span_redundant span_0)
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1048
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1049
text {* An explicit expansion is sometimes needed. *}
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1050
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1051
lemma span_explicit:
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1052
  "span P = {y. \<exists>S u. finite S \<and> S \<subseteq> P \<and> setsum (\<lambda>v. u v *\<^sub>R v) S = y}"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1053
  (is "_ = ?E" is "_ = {y. ?h y}" is "_ = {y. \<exists>S u. ?Q S u y}")
49663
b84fafaea4bb tuned proofs;
wenzelm
parents: 49652
diff changeset
  1054
proof -
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1055
  {
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1056
    fix x
55910
0a756571c7a4 tuned proof
huffman
parents: 55775
diff changeset
  1057
    assume "?h x"
0a756571c7a4 tuned proof
huffman
parents: 55775
diff changeset
  1058
    then obtain S u where "finite S" and "S \<subseteq> P" and "setsum (\<lambda>v. u v *\<^sub>R v) S = x"
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1059
      by blast
55910
0a756571c7a4 tuned proof
huffman
parents: 55775
diff changeset
  1060
    then have "x \<in> span P"
0a756571c7a4 tuned proof
huffman
parents: 55775
diff changeset
  1061
      by (auto intro: span_setsum span_mul span_superset)
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1062
  }
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1063
  moreover
55910
0a756571c7a4 tuned proof
huffman
parents: 55775
diff changeset
  1064
  have "\<forall>x \<in> span P. ?h x"
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  1065
  proof (rule span_induct_alt')
55910
0a756571c7a4 tuned proof
huffman
parents: 55775
diff changeset
  1066
    show "?h 0"
0a756571c7a4 tuned proof
huffman
parents: 55775
diff changeset
  1067
      by (rule exI[where x="{}"], simp)
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1068
  next
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1069
    fix c x y
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1070
    assume x: "x \<in> P"
55910
0a756571c7a4 tuned proof
huffman
parents: 55775
diff changeset
  1071
    assume hy: "?h y"
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1072
    from hy obtain S u where fS: "finite S" and SP: "S\<subseteq>P"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1073
      and u: "setsum (\<lambda>v. u v *\<^sub>R v) S = y" by blast
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1074
    let ?S = "insert x S"
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  1075
    let ?u = "\<lambda>y. if y = x then (if x \<in> S then u y + c else c) else u y"
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1076
    from fS SP x have th0: "finite (insert x S)" "insert x S \<subseteq> P"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1077
      by blast+
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1078
    have "?Q ?S ?u (c*\<^sub>R x + y)"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1079
    proof cases
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1080
      assume xS: "x \<in> S"
55910
0a756571c7a4 tuned proof
huffman
parents: 55775
diff changeset
  1081
      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"
0a756571c7a4 tuned proof
huffman
parents: 55775
diff changeset
  1082
        using xS by (simp add: setsum.remove [OF fS xS] insert_absorb)
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1083
      also have "\<dots> = (\<Sum>v\<in>S. u v *\<^sub>R v) + c *\<^sub>R x"
55910
0a756571c7a4 tuned proof
huffman
parents: 55775
diff changeset
  1084
        by (simp add: setsum.remove [OF fS xS] algebra_simps)
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1085
      also have "\<dots> = c*\<^sub>R x + y"
57512
cc97b347b301 reduced name variants for assoc and commute on plus and mult
haftmann
parents: 57418
diff changeset
  1086
        by (simp add: add.commute u)
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1087
      finally have "setsum (\<lambda>v. ?u v *\<^sub>R v) ?S = c*\<^sub>R x + y" .
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1088
      then show ?thesis using th0 by blast
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1089
    next
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1090
      assume xS: "x \<notin> S"
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  1091
      have th00: "(\<Sum>v\<in>S. (if v = x then c else u v) *\<^sub>R v) = y"
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  1092
        unfolding u[symmetric]
57418
6ab1c7cb0b8d fact consolidation
haftmann
parents: 56536
diff changeset
  1093
        apply (rule setsum.cong)
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1094
        using xS
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1095
        apply auto
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  1096
        done
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1097
      show ?thesis using fS xS th0
57512
cc97b347b301 reduced name variants for assoc and commute on plus and mult
haftmann
parents: 57418
diff changeset
  1098
        by (simp add: th00 add.commute cong del: if_weak_cong)
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1099
    qed
55910
0a756571c7a4 tuned proof
huffman
parents: 55775
diff changeset
  1100
    then show "?h (c*\<^sub>R x + y)"
0a756571c7a4 tuned proof
huffman
parents: 55775
diff changeset
  1101
      by fast
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1102
  qed
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1103
  ultimately show ?thesis by blast
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1104
qed
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1105
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1106
lemma dependent_explicit:
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  1107
  "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))"
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  1108
  (is "?lhs = ?rhs")
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  1109
proof -
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1110
  {
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1111
    assume dP: "dependent P"
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1112
    then obtain a S u where aP: "a \<in> P" and fS: "finite S"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1113
      and SP: "S \<subseteq> P - {a}" and ua: "setsum (\<lambda>v. u v *\<^sub>R v) S = a"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1114
      unfolding dependent_def span_explicit by blast
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1115
    let ?S = "insert a S"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1116
    let ?u = "\<lambda>y. if y = a then - 1 else u y"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1117
    let ?v = a
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1118
    from aP SP have aS: "a \<notin> S"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1119
      by blast
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1120
    from fS SP aP have th0: "finite ?S" "?S \<subseteq> P" "?v \<in> ?S" "?u ?v \<noteq> 0"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1121
      by auto
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1122
    have s0: "setsum (\<lambda>v. ?u v *\<^sub>R v) ?S = 0"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1123
      using fS aS
55910
0a756571c7a4 tuned proof
huffman
parents: 55775
diff changeset
  1124
      apply simp
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1125
      apply (subst (2) ua[symmetric])
57418
6ab1c7cb0b8d fact consolidation
haftmann
parents: 56536
diff changeset
  1126
      apply (rule setsum.cong)
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  1127
      apply auto
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  1128
      done
55910
0a756571c7a4 tuned proof
huffman
parents: 55775
diff changeset
  1129
    with th0 have ?rhs by fast
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  1130
  }
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1131
  moreover
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1132
  {
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1133
    fix S u v
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  1134
    assume fS: "finite S"
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1135
      and SP: "S \<subseteq> P"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1136
      and vS: "v \<in> S"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1137
      and uv: "u v \<noteq> 0"
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  1138
      and u: "setsum (\<lambda>v. u v *\<^sub>R v) S = 0"
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1139
    let ?a = v
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1140
    let ?S = "S - {v}"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1141
    let ?u = "\<lambda>i. (- u i) / u v"
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1142
    have th0: "?a \<in> P" "finite ?S" "?S \<subseteq> P"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1143
      using fS SP vS by auto
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1144
    have "setsum (\<lambda>v. ?u v *\<^sub>R v) ?S =
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1145
      setsum (\<lambda>v. (- (inverse (u ?a))) *\<^sub>R (u v *\<^sub>R v)) S - ?u v *\<^sub>R v"
56480
093ea91498e6 field_simps: better support for negation and division, and power
hoelzl
parents: 56479
diff changeset
  1146
      using fS vS uv by (simp add: setsum_diff1 field_simps)
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1147
    also have "\<dots> = ?a"
56479
91958d4b30f7 revert c1bbd3e22226, a14831ac3023, and 36489d77c484: divide_minus_left/right are again simp rules
hoelzl
parents: 56444
diff changeset
  1148
      unfolding scaleR_right.setsum [symmetric] u using uv by simp
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1149
    finally have "setsum (\<lambda>v. ?u v *\<^sub>R v) ?S = ?a" .
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1150
    with th0 have ?lhs
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1151
      unfolding dependent_def span_explicit
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1152
      apply -
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1153
      apply (rule bexI[where x= "?a"])
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1154
      apply (simp_all del: scaleR_minus_left)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1155
      apply (rule exI[where x= "?S"])
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  1156
      apply (auto simp del: scaleR_minus_left)
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  1157
      done
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  1158
  }
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1159
  ultimately show ?thesis by blast
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1160
qed
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1161
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1162
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1163
lemma span_finite:
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1164
  assumes fS: "finite S"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1165
  shows "span S = {y. \<exists>u. setsum (\<lambda>v. u v *\<^sub>R v) S = y}"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1166
  (is "_ = ?rhs")
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  1167
proof -
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1168
  {
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1169
    fix y
49711
e5aaae7eadc9 tuned proofs;
wenzelm
parents: 49663
diff changeset
  1170
    assume y: "y \<in> span S"
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1171
    from y obtain S' u where fS': "finite S'"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1172
      and SS': "S' \<subseteq> S"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1173
      and u: "setsum (\<lambda>v. u v *\<^sub>R v) S' = y"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1174
      unfolding span_explicit by blast
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1175
    let ?u = "\<lambda>x. if x \<in> S' then u x else 0"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1176
    have "setsum (\<lambda>v. ?u v *\<^sub>R v) S = setsum (\<lambda>v. u v *\<^sub>R v) S'"
57418
6ab1c7cb0b8d fact consolidation
haftmann
parents: 56536
diff changeset
  1177
      using SS' fS by (auto intro!: setsum.mono_neutral_cong_right)
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  1178
    then have "setsum (\<lambda>v. ?u v *\<^sub>R v) S = y" by (metis u)
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1179
    then have "y \<in> ?rhs" by auto
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1180
  }
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1181
  moreover
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1182
  {
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1183
    fix y u
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  1184
    assume u: "setsum (\<lambda>v. u v *\<^sub>R v) S = y"
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1185
    then have "y \<in> span S" using fS unfolding span_explicit by auto
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1186
  }
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1187
  ultimately show ?thesis by blast
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1188
qed
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1189
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1190
text {* This is useful for building a basis step-by-step. *}
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1191
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1192
lemma independent_insert:
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1193
  "independent (insert a S) \<longleftrightarrow>
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1194
    (if a \<in> S then independent S else independent S \<and> a \<notin> span S)"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1195
  (is "?lhs \<longleftrightarrow> ?rhs")
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1196
proof (cases "a \<in> S")
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1197
  case True
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1198
  then show ?thesis
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1199
    using insert_absorb[OF True] by simp
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1200
next
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1201
  case False
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1202
  show ?thesis
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1203
  proof
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1204
    assume i: ?lhs
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1205
    then show ?rhs
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1206
      using False
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1207
      apply simp
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1208
      apply (rule conjI)
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1209
      apply (rule independent_mono)
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1210
      apply assumption
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1211
      apply blast
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1212
      apply (simp add: dependent_def)
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1213
      done
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1214
  next
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1215
    assume i: ?rhs
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1216
    show ?lhs
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1217
      using i False
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1218
      apply (auto simp add: dependent_def)
55775
1557a391a858 A bit of tidying up
paulson <lp15@cam.ac.uk>
parents: 55136
diff changeset
  1219
      by (metis in_span_insert insert_Diff insert_Diff_if insert_iff)
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1220
  qed
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1221
qed
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1222
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1223
text {* The degenerate case of the Exchange Lemma. *}
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1224
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1225
lemma spanning_subset_independent:
49711
e5aaae7eadc9 tuned proofs;
wenzelm
parents: 49663
diff changeset
  1226
  assumes BA: "B \<subseteq> A"
e5aaae7eadc9 tuned proofs;
wenzelm
parents: 49663
diff changeset
  1227
    and iA: "independent A"
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  1228
    and AsB: "A \<subseteq> span B"
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1229
  shows "A = B"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1230
proof
49663
b84fafaea4bb tuned proofs;
wenzelm
parents: 49652
diff changeset
  1231
  show "B \<subseteq> A" by (rule BA)
b84fafaea4bb tuned proofs;
wenzelm
parents: 49652
diff changeset
  1232
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1233
  from span_mono[OF BA] span_mono[OF AsB]
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1234
  have sAB: "span A = span B" unfolding span_span by blast
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1235
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1236
  {
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1237
    fix x
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1238
    assume x: "x \<in> A"
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1239
    from iA have th0: "x \<notin> span (A - {x})"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1240
      unfolding dependent_def using x by blast
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1241
    from x have xsA: "x \<in> span A"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1242
      by (blast intro: span_superset)
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1243
    have "A - {x} \<subseteq> A" by blast
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1244
    then have th1: "span (A - {x}) \<subseteq> span A"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1245
      by (metis span_mono)
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1246
    {
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1247
      assume xB: "x \<notin> B"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1248
      from xB BA have "B \<subseteq> A - {x}"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1249
        by blast
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1250
      then have "span B \<subseteq> span (A - {x})"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1251
        by (metis span_mono)
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1252
      with th1 th0 sAB have "x \<notin> span A"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1253
        by blast
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1254
      with x have False
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1255
        by (metis span_superset)
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1256
    }
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1257
    then have "x \<in> B" by blast
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1258
  }
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1259
  then show "A \<subseteq> B" by blast
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1260
qed
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1261
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1262
text {* The general case of the Exchange Lemma, the key to what follows. *}
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1263
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1264
lemma exchange_lemma:
49711
e5aaae7eadc9 tuned proofs;
wenzelm
parents: 49663
diff changeset
  1265
  assumes f:"finite t"
e5aaae7eadc9 tuned proofs;
wenzelm
parents: 49663
diff changeset
  1266
    and i: "independent s"
e5aaae7eadc9 tuned proofs;
wenzelm
parents: 49663
diff changeset
  1267
    and sp: "s \<subseteq> span t"
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1268
  shows "\<exists>t'. card t' = card t \<and> finite t' \<and> s \<subseteq> t' \<and> t' \<subseteq> s \<union> t \<and> s \<subseteq> span t'"
49663
b84fafaea4bb tuned proofs;
wenzelm
parents: 49652
diff changeset
  1269
  using f i sp
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  1270
proof (induct "card (t - s)" arbitrary: s t rule: less_induct)
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1271
  case less
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1272
  note ft = `finite t` and s = `independent s` and sp = `s \<subseteq> span t`
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1273
  let ?P = "\<lambda>t'. card t' = card t \<and> finite t' \<and> s \<subseteq> t' \<and> t' \<subseteq> s \<union> t \<and> s \<subseteq> span t'"
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1274
  let ?ths = "\<exists>t'. ?P t'"
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1275
  {
55775
1557a391a858 A bit of tidying up
paulson <lp15@cam.ac.uk>
parents: 55136
diff changeset
  1276
    assume "s \<subseteq> t"
1557a391a858 A bit of tidying up
paulson <lp15@cam.ac.uk>
parents: 55136
diff changeset
  1277
    then have ?ths
1557a391a858 A bit of tidying up
paulson <lp15@cam.ac.uk>
parents: 55136
diff changeset
  1278
      by (metis ft Un_commute sp sup_ge1)
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1279
  }
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1280
  moreover
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1281
  {
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1282
    assume st: "t \<subseteq> s"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1283
    from spanning_subset_independent[OF st s sp] st ft span_mono[OF st]
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1284
    have ?ths
55775
1557a391a858 A bit of tidying up
paulson <lp15@cam.ac.uk>
parents: 55136
diff changeset
  1285
      by (metis Un_absorb sp)
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1286
  }
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1287
  moreover
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1288
  {
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1289
    assume st: "\<not> s \<subseteq> t" "\<not> t \<subseteq> s"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1290
    from st(2) obtain b where b: "b \<in> t" "b \<notin> s"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1291
      by blast
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1292
    from b have "t - {b} - s \<subset> t - s"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1293
      by blast
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1294
    then have cardlt: "card (t - {b} - s) < card (t - s)"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1295
      using ft by (auto intro: psubset_card_mono)
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1296
    from b ft have ct0: "card t \<noteq> 0"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1297
      by auto
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1298
    have ?ths
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1299
    proof cases
53716
b42d9a71fc1a tuned proofs;
wenzelm
parents: 53600
diff changeset
  1300
      assume stb: "s \<subseteq> span (t - {b})"
b42d9a71fc1a tuned proofs;
wenzelm
parents: 53600
diff changeset
  1301
      from ft have ftb: "finite (t - {b})"
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1302
        by auto
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1303
      from less(1)[OF cardlt ftb s stb]
53716
b42d9a71fc1a tuned proofs;
wenzelm
parents: 53600
diff changeset
  1304
      obtain u where u: "card u = card (t - {b})" "s \<subseteq> u" "u \<subseteq> s \<union> (t - {b})" "s \<subseteq> span u"
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  1305
        and fu: "finite u" by blast
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1306
      let ?w = "insert b u"
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1307
      have th0: "s \<subseteq> insert b u"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1308
        using u by blast
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1309
      from u(3) b have "u \<subseteq> s \<union> t"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1310
        by blast
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1311
      then have th1: "insert b u \<subseteq> s \<union> t"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1312
        using u b by blast
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1313
      have bu: "b \<notin> u"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1314
        using b u by blast
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1315
      from u(1) ft b have "card u = (card t - 1)"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1316
        by auto
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  1317
      then have th2: "card (insert b u) = card t"
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1318
        using card_insert_disjoint[OF fu bu] ct0 by auto
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1319
      from u(4) have "s \<subseteq> span u" .
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1320
      also have "\<dots> \<subseteq> span (insert b u)"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1321
        by (rule span_mono) blast
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1322
      finally have th3: "s \<subseteq> span (insert b u)" .
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1323
      from th0 th1 th2 th3 fu have th: "?P ?w"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1324
        by blast
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1325
      from th show ?thesis by blast
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1326
    next
53716
b42d9a71fc1a tuned proofs;
wenzelm
parents: 53600
diff changeset
  1327
      assume stb: "\<not> s \<subseteq> span (t - {b})"
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1328
      from stb obtain a where a: "a \<in> s" "a \<notin> span (t - {b})"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1329
        by blast
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1330
      have ab: "a \<noteq> b"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1331
        using a b by blast
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1332
      have at: "a \<notin> t"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1333
        using a ab span_superset[of a "t- {b}"] by auto
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1334
      have mlt: "card ((insert a (t - {b})) - s) < card (t - s)"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1335
        using cardlt ft a b by auto
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1336
      have ft': "finite (insert a (t - {b}))"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1337
        using ft by auto
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1338
      {
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1339
        fix x
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1340
        assume xs: "x \<in> s"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1341
        have t: "t \<subseteq> insert b (insert a (t - {b}))"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1342
          using b by auto
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1343
        from b(1) have "b \<in> span t"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1344
          by (simp add: span_superset)
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1345
        have bs: "b \<in> span (insert a (t - {b}))"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1346
          apply (rule in_span_delete)
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1347
          using a sp unfolding subset_eq
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1348
          apply auto
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1349
          done
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1350
        from xs sp have "x \<in> span t"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1351
          by blast
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1352
        with span_mono[OF t] have x: "x \<in> span (insert b (insert a (t - {b})))" ..
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1353
        from span_trans[OF bs x] have "x \<in> span (insert a (t - {b}))" .
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1354
      }
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1355
      then have sp': "s \<subseteq> span (insert a (t - {b}))"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1356
        by blast
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1357
      from less(1)[OF mlt ft' s sp'] obtain u where u:
53716
b42d9a71fc1a tuned proofs;
wenzelm
parents: 53600
diff changeset
  1358
        "card u = card (insert a (t - {b}))"
b42d9a71fc1a tuned proofs;
wenzelm
parents: 53600
diff changeset
  1359
        "finite u" "s \<subseteq> u" "u \<subseteq> s \<union> insert a (t - {b})"
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1360
        "s \<subseteq> span u" by blast
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1361
      from u a b ft at ct0 have "?P u"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1362
        by auto
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1363
      then show ?thesis by blast
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1364
    qed
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1365
  }
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  1366
  ultimately show ?ths by blast
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1367
qed
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1368
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1369
text {* This implies corresponding size bounds. *}
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1370
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1371
lemma independent_span_bound:
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1372
  assumes f: "finite t"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1373
    and i: "independent s"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1374
    and sp: "s \<subseteq> span t"
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1375
  shows "finite s \<and> card s \<le> card t"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1376
  by (metis exchange_lemma[OF f i sp] finite_subset card_mono)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1377
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1378
lemma finite_Atleast_Atmost_nat[simp]: "finite {f x |x. x\<in> (UNIV::'a::finite set)}"
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  1379
proof -
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1380
  have eq: "{f x |x. x\<in> UNIV} = f ` UNIV"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1381
    by auto
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1382
  show ?thesis unfolding eq
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1383
    apply (rule finite_imageI)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1384
    apply (rule finite)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1385
    done
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1386
qed
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1387
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1388
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1389
subsection {* Euclidean Spaces as Typeclass *}
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1390
50526
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 50105
diff changeset
  1391
lemma independent_Basis: "independent Basis"
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 50105
diff changeset
  1392
  unfolding dependent_def
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 50105
diff changeset
  1393
  apply (subst span_finite)
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 50105
diff changeset
  1394
  apply simp
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1395
  apply clarify
50526
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 50105
diff changeset
  1396
  apply (drule_tac f="inner a" in arg_cong)
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 50105
diff changeset
  1397
  apply (simp add: inner_Basis inner_setsum_right eq_commute)
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 50105
diff changeset
  1398
  done
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 50105
diff changeset
  1399
53939
eb25bddf6a22 tuned proofs
huffman
parents: 53938
diff changeset
  1400
lemma span_Basis [simp]: "span Basis = UNIV"
eb25bddf6a22 tuned proofs
huffman
parents: 53938
diff changeset
  1401
  unfolding span_finite [OF finite_Basis]
eb25bddf6a22 tuned proofs
huffman
parents: 53938
diff changeset
  1402
  by (fast intro: euclidean_representation)
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1403
50526
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 50105
diff changeset
  1404
lemma in_span_Basis: "x \<in> span Basis"
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 50105
diff changeset
  1405
  unfolding span_Basis ..
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 50105
diff changeset
  1406
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 50105
diff changeset
  1407
lemma Basis_le_norm: "b \<in> Basis \<Longrightarrow> \<bar>x \<bullet> b\<bar> \<le> norm x"
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 50105
diff changeset
  1408
  by (rule order_trans [OF Cauchy_Schwarz_ineq2]) simp
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 50105
diff changeset
  1409
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 50105
diff changeset
  1410
lemma norm_bound_Basis_le: "b \<in> Basis \<Longrightarrow> norm x \<le> e \<Longrightarrow> \<bar>x \<bullet> b\<bar> \<le> e"
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 50105
diff changeset
  1411
  by (metis Basis_le_norm order_trans)
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 50105
diff changeset
  1412
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 50105
diff changeset
  1413
lemma norm_bound_Basis_lt: "b \<in> Basis \<Longrightarrow> norm x < e \<Longrightarrow> \<bar>x \<bullet> b\<bar> < e"
53595
5078034ade16 prefer theorem name over 'long_thm_list(n)'
huffman
parents: 53406
diff changeset
  1414
  by (metis Basis_le_norm le_less_trans)
50526
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 50105
diff changeset
  1415
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 50105
diff changeset
  1416
lemma norm_le_l1: "norm x \<le> (\<Sum>b\<in>Basis. \<bar>x \<bullet> b\<bar>)"
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 50105
diff changeset
  1417
  apply (subst euclidean_representation[of x, symmetric])
44176
eda112e9cdee remove redundant lemma setsum_norm in favor of norm_setsum;
huffman
parents: 44170
diff changeset
  1418
  apply (rule order_trans[OF norm_setsum])
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  1419
  apply (auto intro!: setsum_mono)
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  1420
  done
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1421
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1422
lemma setsum_norm_allsubsets_bound:
56444
f944ae8c80a3 tuned proofs;
wenzelm
parents: 56409
diff changeset
  1423
  fixes f :: "'a \<Rightarrow> 'n::euclidean_space"
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1424
  assumes fP: "finite P"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1425
    and fPs: "\<And>Q. Q \<subseteq> P \<Longrightarrow> norm (setsum f Q) \<le> e"
50526
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 50105
diff changeset
  1426
  shows "(\<Sum>x\<in>P. norm (f x)) \<le> 2 * real DIM('n) * e"
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  1427
proof -
50526
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 50105
diff changeset
  1428
  have "(\<Sum>x\<in>P. norm (f x)) \<le> (\<Sum>x\<in>P. \<Sum>b\<in>Basis. \<bar>f x \<bullet> b\<bar>)"
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 50105
diff changeset
  1429
    by (rule setsum_mono) (rule norm_le_l1)
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 50105
diff changeset
  1430
  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>)"
57418
6ab1c7cb0b8d fact consolidation
haftmann
parents: 56536
diff changeset
  1431
    by (rule setsum.commute)
50526
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 50105
diff changeset
  1432
  also have "\<dots> \<le> of_nat (card (Basis :: 'n set)) * (2 * e)"
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  1433
  proof (rule setsum_bounded)
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1434
    fix i :: 'n
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1435
    assume i: "i \<in> Basis"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1436
    have "norm (\<Sum>x\<in>P. \<bar>f x \<bullet> i\<bar>) \<le>
50526
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 50105
diff changeset
  1437
      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)"
57418
6ab1c7cb0b8d fact consolidation
haftmann
parents: 56536
diff changeset
  1438
      by (simp add: abs_real_def setsum.If_cases[OF fP] setsum_negf norm_triangle_ineq4 inner_setsum_left
56444
f944ae8c80a3 tuned proofs;
wenzelm
parents: 56409
diff changeset
  1439
        del: real_norm_def)
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1440
    also have "\<dots> \<le> e + e"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1441
      unfolding real_norm_def
50526
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 50105
diff changeset
  1442
      by (intro add_mono norm_bound_Basis_le i fPs) auto
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 50105
diff changeset
  1443
    finally show "(\<Sum>x\<in>P. \<bar>f x \<bullet> i\<bar>) \<le> 2*e" by simp
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1444
  qed
50526
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 50105
diff changeset
  1445
  also have "\<dots> = 2 * real DIM('n) * e"
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 50105
diff changeset
  1446
    by (simp add: real_of_nat_def)
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1447
  finally show ?thesis .
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1448
qed
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1449
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1450
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1451
subsection {* Linearity and Bilinearity continued *}
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1452
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1453
lemma linear_bounded:
56444
f944ae8c80a3 tuned proofs;
wenzelm
parents: 56409
diff changeset
  1454
  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1455
  assumes lf: "linear f"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1456
  shows "\<exists>B. \<forall>x. norm (f x) \<le> B * norm x"
53939
eb25bddf6a22 tuned proofs
huffman
parents: 53938
diff changeset
  1457
proof
50526
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 50105
diff changeset
  1458
  let ?B = "\<Sum>b\<in>Basis. norm (f b)"
53939
eb25bddf6a22 tuned proofs
huffman
parents: 53938
diff changeset
  1459
  show "\<forall>x. norm (f x) \<le> ?B * norm x"
eb25bddf6a22 tuned proofs
huffman
parents: 53938
diff changeset
  1460
  proof
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1461
    fix x :: 'a
50526
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 50105
diff changeset
  1462
    let ?g = "\<lambda>b. (x \<bullet> b) *\<^sub>R f b"
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 50105
diff changeset
  1463
    have "norm (f x) = norm (f (\<Sum>b\<in>Basis. (x \<bullet> b) *\<^sub>R b))"
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 50105
diff changeset
  1464
      unfolding euclidean_representation ..
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 50105
diff changeset
  1465
    also have "\<dots> = norm (setsum ?g Basis)"
53939
eb25bddf6a22 tuned proofs
huffman
parents: 53938
diff changeset
  1466
      by (simp add: linear_setsum [OF lf] linear_cmul [OF lf])
50526
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 50105
diff changeset
  1467
    finally have th0: "norm (f x) = norm (setsum ?g Basis)" .
53939
eb25bddf6a22 tuned proofs
huffman
parents: 53938
diff changeset
  1468
    have th: "\<forall>b\<in>Basis. norm (?g b) \<le> norm (f b) * norm x"
eb25bddf6a22 tuned proofs
huffman
parents: 53938
diff changeset
  1469
    proof
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1470
      fix i :: 'a
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1471
      assume i: "i \<in> Basis"
50526
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 50105
diff changeset
  1472
      from Basis_le_norm[OF i, of x]
53939
eb25bddf6a22 tuned proofs
huffman
parents: 53938
diff changeset
  1473
      show "norm (?g i) \<le> norm (f i) * norm x"
49663
b84fafaea4bb tuned proofs;
wenzelm
parents: 49652
diff changeset
  1474
        unfolding norm_scaleR
57512
cc97b347b301 reduced name variants for assoc and commute on plus and mult
haftmann
parents: 57418
diff changeset
  1475
        apply (subst mult.commute)
49663
b84fafaea4bb tuned proofs;
wenzelm
parents: 49652
diff changeset
  1476
        apply (rule mult_mono)
b84fafaea4bb tuned proofs;
wenzelm
parents: 49652
diff changeset
  1477
        apply (auto simp add: field_simps)
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1478
        done
53939
eb25bddf6a22 tuned proofs
huffman
parents: 53938
diff changeset
  1479
    qed
50526
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 50105
diff changeset
  1480
    from setsum_norm_le[of _ ?g, OF th]
53939
eb25bddf6a22 tuned proofs
huffman
parents: 53938
diff changeset
  1481
    show "norm (f x) \<le> ?B * norm x"
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1482
      unfolding th0 setsum_left_distrib by metis
53939
eb25bddf6a22 tuned proofs
huffman
parents: 53938
diff changeset
  1483
  qed
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1484
qed
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1485
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1486
lemma linear_conv_bounded_linear:
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1487
  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1488
  shows "linear f \<longleftrightarrow> bounded_linear f"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1489
proof
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1490
  assume "linear f"
53939
eb25bddf6a22 tuned proofs
huffman
parents: 53938
diff changeset
  1491
  then interpret f: linear f .
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1492
  show "bounded_linear f"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1493
  proof
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1494
    have "\<exists>B. \<forall>x. norm (f x) \<le> B * norm x"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1495
      using `linear f` by (rule linear_bounded)
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  1496
    then show "\<exists>K. \<forall>x. norm (f x) \<le> norm x * K"
57512
cc97b347b301 reduced name variants for assoc and commute on plus and mult
haftmann
parents: 57418
diff changeset
  1497
      by (simp add: mult.commute)
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1498
  qed
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1499
next
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1500
  assume "bounded_linear f"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1501
  then interpret f: bounded_linear f .
53939
eb25bddf6a22 tuned proofs
huffman
parents: 53938
diff changeset
  1502
  show "linear f" ..
eb25bddf6a22 tuned proofs
huffman
parents: 53938
diff changeset
  1503
qed
eb25bddf6a22 tuned proofs
huffman
parents: 53938
diff changeset
  1504
eb25bddf6a22 tuned proofs
huffman
parents: 53938
diff changeset
  1505
lemma linear_bounded_pos:
56444
f944ae8c80a3 tuned proofs;
wenzelm
parents: 56409
diff changeset
  1506
  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
53939
eb25bddf6a22 tuned proofs
huffman
parents: 53938
diff changeset
  1507
  assumes lf: "linear f"
eb25bddf6a22 tuned proofs
huffman
parents: 53938
diff changeset
  1508
  shows "\<exists>B > 0. \<forall>x. norm (f x) \<le> B * norm x"
eb25bddf6a22 tuned proofs
huffman
parents: 53938
diff changeset
  1509
proof -
eb25bddf6a22 tuned proofs
huffman
parents: 53938
diff changeset
  1510
  have "\<exists>B > 0. \<forall>x. norm (f x) \<le> norm x * B"
eb25bddf6a22 tuned proofs
huffman
parents: 53938
diff changeset
  1511
    using lf unfolding linear_conv_bounded_linear
eb25bddf6a22 tuned proofs
huffman
parents: 53938
diff changeset
  1512
    by (rule bounded_linear.pos_bounded)
eb25bddf6a22 tuned proofs
huffman
parents: 53938
diff changeset
  1513
  then show ?thesis
57512
cc97b347b301 reduced name variants for assoc and commute on plus and mult
haftmann
parents: 57418
diff changeset
  1514
    by (simp only: mult.commute)
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1515
qed
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1516
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  1517
lemma bounded_linearI':
56444
f944ae8c80a3 tuned proofs;
wenzelm
parents: 56409
diff changeset
  1518
  fixes f ::"'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1519
  assumes "\<And>x y. f (x + y) = f x + f y"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1520
    and "\<And>c x. f (c *\<^sub>R x) = c *\<^sub>R f x"
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  1521
  shows "bounded_linear f"
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1522
  unfolding linear_conv_bounded_linear[symmetric]
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  1523
  by (rule linearI[OF assms])
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1524
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1525
lemma bilinear_bounded:
56444
f944ae8c80a3 tuned proofs;
wenzelm
parents: 56409
diff changeset
  1526
  fixes h :: "'m::euclidean_space \<Rightarrow> 'n::euclidean_space \<Rightarrow> 'k::real_normed_vector"
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1527
  assumes bh: "bilinear h"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1528
  shows "\<exists>B. \<forall>x y. norm (h x y) \<le> B * norm x * norm y"
50526
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 50105
diff changeset
  1529
proof (clarify intro!: exI[of _ "\<Sum>i\<in>Basis. \<Sum>j\<in>Basis. norm (h i j)"])
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1530
  fix x :: 'm
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1531
  fix y :: 'n
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1532
  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))"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1533
    apply (subst euclidean_representation[where 'a='m])
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1534
    apply (subst euclidean_representation[where 'a='n])
50526
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 50105
diff changeset
  1535
    apply rule
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 50105
diff changeset
  1536
    done
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1537
  also have "\<dots> = norm (setsum (\<lambda> (i,j). h ((x \<bullet> i) *\<^sub>R i) ((y \<bullet> j) *\<^sub>R j)) (Basis \<times> Basis))"
50526
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 50105
diff changeset
  1538
    unfolding bilinear_setsum[OF bh finite_Basis finite_Basis] ..
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 50105
diff changeset
  1539
  finally have th: "norm (h x y) = \<dots>" .
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 50105
diff changeset
  1540
  show "norm (h x y) \<le> (\<Sum>i\<in>Basis. \<Sum>j\<in>Basis. norm (h i j)) * norm x * norm y"
57418
6ab1c7cb0b8d fact consolidation
haftmann
parents: 56536
diff changeset
  1541
    apply (auto simp add: setsum_left_distrib th setsum.cartesian_product)
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1542
    apply (rule setsum_norm_le)
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1543
    apply simp
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1544
    apply (auto simp add: bilinear_rmul[OF bh] bilinear_lmul[OF bh]
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1545
      field_simps simp del: scaleR_scaleR)
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1546
    apply (rule mult_mono)
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1547
    apply (auto simp add: zero_le_mult_iff Basis_le_norm)
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1548
    apply (rule mult_mono)
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1549
    apply (auto simp add: zero_le_mult_iff Basis_le_norm)
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1550
    done
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1551
qed
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1552
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1553
lemma bilinear_conv_bounded_bilinear:
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1554
  fixes h :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space \<Rightarrow> 'c::real_normed_vector"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1555
  shows "bilinear h \<longleftrightarrow> bounded_bilinear h"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1556
proof
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1557
  assume "bilinear h"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1558
  show "bounded_bilinear h"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1559
  proof
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1560
    fix x y z
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1561
    show "h (x + y) z = h x z + h y z"
53600
8fda7ad57466 make 'linear' into a sublocale of 'bounded_linear';
huffman
parents: 53596
diff changeset
  1562
      using `bilinear h` unfolding bilinear_def linear_iff by simp
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1563
  next
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1564
    fix x y z
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1565
    show "h x (y + z) = h x y + h x z"
53600
8fda7ad57466 make 'linear' into a sublocale of 'bounded_linear';
huffman
parents: 53596
diff changeset
  1566
      using `bilinear h` unfolding bilinear_def linear_iff by simp
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1567
  next
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1568
    fix r x y
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1569
    show "h (scaleR r x) y = scaleR r (h x y)"
53600
8fda7ad57466 make 'linear' into a sublocale of 'bounded_linear';
huffman
parents: 53596
diff changeset
  1570
      using `bilinear h` unfolding bilinear_def linear_iff
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1571
      by simp
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1572
  next
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1573
    fix r x y
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1574
    show "h x (scaleR r y) = scaleR r (h x y)"
53600
8fda7ad57466 make 'linear' into a sublocale of 'bounded_linear';
huffman
parents: 53596
diff changeset
  1575
      using `bilinear h` unfolding bilinear_def linear_iff
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1576
      by simp
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1577
  next
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1578
    have "\<exists>B. \<forall>x y. norm (h x y) \<le> B * norm x * norm y"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1579
      using `bilinear h` by (rule bilinear_bounded)
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  1580
    then show "\<exists>K. \<forall>x y. norm (h x y) \<le> norm x * norm y * K"
57514
bdc2c6b40bf2 prefer ac_simps collections over separate name bindings for add and mult
haftmann
parents: 57512
diff changeset
  1581
      by (simp add: ac_simps)
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1582
  qed
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1583
next
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1584
  assume "bounded_bilinear h"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1585
  then interpret h: bounded_bilinear h .
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1586
  show "bilinear h"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1587
    unfolding bilinear_def linear_conv_bounded_linear
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  1588
    using h.bounded_linear_left h.bounded_linear_right by simp
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1589
qed
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1590
53939
eb25bddf6a22 tuned proofs
huffman
parents: 53938
diff changeset
  1591
lemma bilinear_bounded_pos:
56444
f944ae8c80a3 tuned proofs;
wenzelm
parents: 56409
diff changeset
  1592
  fixes h :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space \<Rightarrow> 'c::real_normed_vector"
53939
eb25bddf6a22 tuned proofs
huffman
parents: 53938
diff changeset
  1593
  assumes bh: "bilinear h"
eb25bddf6a22 tuned proofs
huffman
parents: 53938
diff changeset
  1594
  shows "\<exists>B > 0. \<forall>x y. norm (h x y) \<le> B * norm x * norm y"
eb25bddf6a22 tuned proofs
huffman
parents: 53938
diff changeset
  1595
proof -
eb25bddf6a22 tuned proofs
huffman
parents: 53938
diff changeset
  1596
  have "\<exists>B > 0. \<forall>x y. norm (h x y) \<le> norm x * norm y * B"
eb25bddf6a22 tuned proofs
huffman
parents: 53938
diff changeset
  1597
    using bh [unfolded bilinear_conv_bounded_bilinear]
eb25bddf6a22 tuned proofs
huffman
parents: 53938
diff changeset
  1598
    by (rule bounded_bilinear.pos_bounded)
eb25bddf6a22 tuned proofs
huffman
parents: 53938
diff changeset
  1599
  then show ?thesis
57514
bdc2c6b40bf2 prefer ac_simps collections over separate name bindings for add and mult
haftmann
parents: 57512
diff changeset
  1600
    by (simp only: ac_simps)
53939
eb25bddf6a22 tuned proofs
huffman
parents: 53938
diff changeset
  1601
qed
eb25bddf6a22 tuned proofs
huffman
parents: 53938
diff changeset
  1602
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  1603
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1604
subsection {* We continue. *}
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1605
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1606
lemma independent_bound:
53716
b42d9a71fc1a tuned proofs;
wenzelm
parents: 53600
diff changeset
  1607
  fixes S :: "'a::euclidean_space set"
b42d9a71fc1a tuned proofs;
wenzelm
parents: 53600
diff changeset
  1608
  shows "independent S \<Longrightarrow> finite S \<and> card S \<le> DIM('a)"
50526
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 50105
diff changeset
  1609
  using independent_span_bound[OF finite_Basis, of S] by auto
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1610
49663
b84fafaea4bb tuned proofs;
wenzelm
parents: 49652
diff changeset
  1611
lemma dependent_biggerset:
56444
f944ae8c80a3 tuned proofs;
wenzelm
parents: 56409
diff changeset
  1612
  fixes S :: "'a::euclidean_space set"
f944ae8c80a3 tuned proofs;
wenzelm
parents: 56409
diff changeset
  1613
  shows "(finite S \<Longrightarrow> card S > DIM('a)) \<Longrightarrow> dependent S"
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1614
  by (metis independent_bound not_less)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1615
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1616
text {* Hence we can create a maximal independent subset. *}
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1617
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1618
lemma maximal_independent_subset_extend:
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1619
  fixes S :: "'a::euclidean_space set"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1620
  assumes sv: "S \<subseteq> V"
49663
b84fafaea4bb tuned proofs;
wenzelm
parents: 49652
diff changeset
  1621
    and iS: "independent S"
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1622
  shows "\<exists>B. S \<subseteq> B \<and> B \<subseteq> V \<and> independent B \<and> V \<subseteq> span B"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1623
  using sv iS
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  1624
proof (induct "DIM('a) - card S" arbitrary: S rule: less_induct)
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1625
  case less
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1626
  note sv = `S \<subseteq> V` and i = `independent S`
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1627
  let ?P = "\<lambda>B. S \<subseteq> B \<and> B \<subseteq> V \<and> independent B \<and> V \<subseteq> span B"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1628
  let ?ths = "\<exists>x. ?P x"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1629
  let ?d = "DIM('a)"
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1630
  show ?ths
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1631
  proof (cases "V \<subseteq> span S")
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1632
    case True
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1633
    then show ?thesis
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1634
      using sv i by blast
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1635
  next
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1636
    case False
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1637
    then obtain a where a: "a \<in> V" "a \<notin> span S"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1638
      by blast
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1639
    from a have aS: "a \<notin> S"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1640
      by (auto simp add: span_superset)
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1641
    have th0: "insert a S \<subseteq> V"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1642
      using a sv by blast
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1643
    from independent_insert[of a S]  i a
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1644
    have th1: "independent (insert a S)"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1645
      by auto
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1646
    have mlt: "?d - card (insert a S) < ?d - card S"
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  1647
      using aS a independent_bound[OF th1] by auto
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1648
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1649
    from less(1)[OF mlt th0 th1]
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1650
    obtain B where B: "insert a S \<subseteq> B" "B \<subseteq> V" "independent B" " V \<subseteq> span B"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1651
      by blast
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1652
    from B have "?P B" by auto
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1653
    then show ?thesis by blast
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1654
  qed
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1655
qed
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1656
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1657
lemma maximal_independent_subset:
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1658
  "\<exists>(B:: ('a::euclidean_space) set). B\<subseteq> V \<and> independent B \<and> V \<subseteq> span B"
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  1659
  by (metis maximal_independent_subset_extend[of "{}:: ('a::euclidean_space) set"]
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  1660
    empty_subsetI independent_empty)
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1661
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1662
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1663
text {* Notion of dimension. *}
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1664
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1665
definition "dim V = (SOME n. \<exists>B. B \<subseteq> V \<and> independent B \<and> V \<subseteq> span B \<and> card B = n)"
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1666
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  1667
lemma basis_exists:
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  1668
  "\<exists>B. (B :: ('a::euclidean_space) set) \<subseteq> V \<and> independent B \<and> V \<subseteq> span B \<and> (card B = dim V)"
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  1669
  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)"]
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  1670
  using maximal_independent_subset[of V] independent_bound
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  1671
  by auto
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1672
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1673
text {* Consequences of independence or spanning for cardinality. *}
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1674
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1675
lemma independent_card_le_dim:
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1676
  fixes B :: "'a::euclidean_space set"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1677
  assumes "B \<subseteq> V"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1678
    and "independent B"
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  1679
  shows "card B \<le> dim V"
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1680
proof -
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1681
  from basis_exists[of V] `B \<subseteq> V`
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1682
  obtain B' where "independent B'"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1683
    and "B \<subseteq> span B'"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1684
    and "card B' = dim V"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1685
    by blast
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1686
  with independent_span_bound[OF _ `independent B` `B \<subseteq> span B'`] independent_bound[of B']
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1687
  show ?thesis by auto
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1688
qed
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1689
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  1690
lemma span_card_ge_dim:
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1691
  fixes B :: "'a::euclidean_space set"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1692
  shows "B \<subseteq> V \<Longrightarrow> V \<subseteq> span B \<Longrightarrow> finite B \<Longrightarrow> dim V \<le> card B"
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1693
  by (metis basis_exists[of V] independent_span_bound subset_trans)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1694
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1695
lemma basis_card_eq_dim:
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1696
  fixes V :: "'a::euclidean_space set"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1697
  shows "B \<subseteq> V \<Longrightarrow> V \<subseteq> span B \<Longrightarrow> independent B \<Longrightarrow> finite B \<and> card B = dim V"
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1698
  by (metis order_eq_iff independent_card_le_dim span_card_ge_dim independent_bound)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1699
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1700
lemma dim_unique:
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1701
  fixes B :: "'a::euclidean_space set"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1702
  shows "B \<subseteq> V \<Longrightarrow> V \<subseteq> span B \<Longrightarrow> independent B \<Longrightarrow> card B = n \<Longrightarrow> dim V = n"
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1703
  by (metis basis_card_eq_dim)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1704
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1705
text {* More lemmas about dimension. *}
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1706
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1707
lemma dim_UNIV: "dim (UNIV :: 'a::euclidean_space set) = DIM('a)"
50526
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 50105
diff changeset
  1708
  using independent_Basis
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 50105
diff changeset
  1709
  by (intro dim_unique[of Basis]) auto
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1710
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1711
lemma dim_subset:
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1712
  fixes S :: "'a::euclidean_space set"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1713
  shows "S \<subseteq> T \<Longrightarrow> dim S \<le> dim T"
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1714
  using basis_exists[of T] basis_exists[of S]
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1715
  by (metis independent_card_le_dim subset_trans)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1716
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1717
lemma dim_subset_UNIV:
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1718
  fixes S :: "'a::euclidean_space set"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1719
  shows "dim S \<le> DIM('a)"
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1720
  by (metis dim_subset subset_UNIV dim_UNIV)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1721
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1722
text {* Converses to those. *}
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1723
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1724
lemma card_ge_dim_independent:
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1725
  fixes B :: "'a::euclidean_space set"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1726
  assumes BV: "B \<subseteq> V"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1727
    and iB: "independent B"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1728
    and dVB: "dim V \<le> card B"
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1729
  shows "V \<subseteq> span B"
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1730
proof
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1731
  fix a
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1732
  assume aV: "a \<in> V"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1733
  {
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1734
    assume aB: "a \<notin> span B"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1735
    then have iaB: "independent (insert a B)"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1736
      using iB aV BV by (simp add: independent_insert)
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1737
    from aV BV have th0: "insert a B \<subseteq> V"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1738
      by blast
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1739
    from aB have "a \<notin>B"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1740
      by (auto simp add: span_superset)
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1741
    with independent_card_le_dim[OF th0 iaB] dVB independent_bound[OF iB]
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1742
    have False by auto
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1743
  }
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1744
  then show "a \<in> span B" by blast
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1745
qed
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1746
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1747
lemma card_le_dim_spanning:
49663
b84fafaea4bb tuned proofs;
wenzelm
parents: 49652
diff changeset
  1748
  assumes BV: "(B:: ('a::euclidean_space) set) \<subseteq> V"
b84fafaea4bb tuned proofs;
wenzelm
parents: 49652
diff changeset
  1749
    and VB: "V \<subseteq> span B"
b84fafaea4bb tuned proofs;
wenzelm
parents: 49652
diff changeset
  1750
    and fB: "finite B"
b84fafaea4bb tuned proofs;
wenzelm
parents: 49652
diff changeset
  1751
    and dVB: "dim V \<ge> card B"
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1752
  shows "independent B"
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  1753
proof -
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1754
  {
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1755
    fix a
53716
b42d9a71fc1a tuned proofs;
wenzelm
parents: 53600
diff changeset
  1756
    assume a: "a \<in> B" "a \<in> span (B - {a})"
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1757
    from a fB have c0: "card B \<noteq> 0"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1758
      by auto
53716
b42d9a71fc1a tuned proofs;
wenzelm
parents: 53600
diff changeset
  1759
    from a fB have cb: "card (B - {a}) = card B - 1"
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1760
      by auto
53716
b42d9a71fc1a tuned proofs;
wenzelm
parents: 53600
diff changeset
  1761
    from BV a have th0: "B - {a} \<subseteq> V"
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1762
      by blast
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1763
    {
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1764
      fix x
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1765
      assume x: "x \<in> V"
53716
b42d9a71fc1a tuned proofs;
wenzelm
parents: 53600
diff changeset
  1766
      from a have eq: "insert a (B - {a}) = B"
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1767
        by blast
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1768
      from x VB have x': "x \<in> span B"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1769
        by blast
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1770
      from span_trans[OF a(2), unfolded eq, OF x']
53716
b42d9a71fc1a tuned proofs;
wenzelm
parents: 53600
diff changeset
  1771
      have "x \<in> span (B - {a})" .
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1772
    }
53716
b42d9a71fc1a tuned proofs;
wenzelm
parents: 53600
diff changeset
  1773
    then have th1: "V \<subseteq> span (B - {a})"
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1774
      by blast
53716
b42d9a71fc1a tuned proofs;
wenzelm
parents: 53600
diff changeset
  1775
    have th2: "finite (B - {a})"
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1776
      using fB by auto
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1777
    from span_card_ge_dim[OF th0 th1 th2]
53716
b42d9a71fc1a tuned proofs;
wenzelm
parents: 53600
diff changeset
  1778
    have c: "dim V \<le> card (B - {a})" .
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1779
    from c c0 dVB cb have False by simp
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1780
  }
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1781
  then show ?thesis
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1782
    unfolding dependent_def by blast
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1783
qed
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1784
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1785
lemma card_eq_dim:
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1786
  fixes B :: "'a::euclidean_space set"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1787
  shows "B \<subseteq> V \<Longrightarrow> card B = dim V \<Longrightarrow> finite B \<Longrightarrow> independent B \<longleftrightarrow> V \<subseteq> span B"
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  1788
  by (metis order_eq_iff card_le_dim_spanning card_ge_dim_independent)
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1789
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1790
text {* More general size bound lemmas. *}
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1791
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1792
lemma independent_bound_general:
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1793
  fixes S :: "'a::euclidean_space set"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1794
  shows "independent S \<Longrightarrow> finite S \<and> card S \<le> dim S"
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1795
  by (metis independent_card_le_dim independent_bound subset_refl)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1796
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  1797
lemma dependent_biggerset_general:
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1798
  fixes S :: "'a::euclidean_space set"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1799
  shows "(finite S \<Longrightarrow> card S > dim S) \<Longrightarrow> dependent S"
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1800
  using independent_bound_general[of S] by (metis linorder_not_le)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1801
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1802
lemma dim_span:
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1803
  fixes S :: "'a::euclidean_space set"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1804
  shows "dim (span S) = dim S"
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  1805
proof -
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1806
  have th0: "dim S \<le> dim (span S)"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1807
    by (auto simp add: subset_eq intro: dim_subset span_superset)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1808
  from basis_exists[of S]
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1809
  obtain B where B: "B \<subseteq> S" "independent B" "S \<subseteq> span B" "card B = dim S"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1810
    by blast
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1811
  from B have fB: "finite B" "card B = dim S"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1812
    using independent_bound by blast+
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1813
  have bSS: "B \<subseteq> span S"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1814
    using B(1) by (metis subset_eq span_inc)
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1815
  have sssB: "span S \<subseteq> span B"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1816
    using span_mono[OF B(3)] by (simp add: span_span)
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1817
  from span_card_ge_dim[OF bSS sssB fB(1)] th0 show ?thesis
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  1818
    using fB(2) by arith
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1819
qed
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1820
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1821
lemma subset_le_dim:
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1822
  fixes S :: "'a::euclidean_space set"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1823
  shows "S \<subseteq> span T \<Longrightarrow> dim S \<le> dim T"
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1824
  by (metis dim_span dim_subset)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1825
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1826
lemma span_eq_dim:
56444
f944ae8c80a3 tuned proofs;
wenzelm
parents: 56409
diff changeset
  1827
  fixes S :: "'a::euclidean_space set"
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1828
  shows "span S = span T \<Longrightarrow> dim S = dim T"
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1829
  by (metis dim_span)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1830
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1831
lemma spans_image:
49663
b84fafaea4bb tuned proofs;
wenzelm
parents: 49652
diff changeset
  1832
  assumes lf: "linear f"
b84fafaea4bb tuned proofs;
wenzelm
parents: 49652
diff changeset
  1833
    and VB: "V \<subseteq> span B"
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1834
  shows "f ` V \<subseteq> span (f ` B)"
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  1835
  unfolding span_linear_image[OF lf] by (metis VB image_mono)
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1836
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1837
lemma dim_image_le:
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1838
  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
49663
b84fafaea4bb tuned proofs;
wenzelm
parents: 49652
diff changeset
  1839
  assumes lf: "linear f"
b84fafaea4bb tuned proofs;
wenzelm
parents: 49652
diff changeset
  1840
  shows "dim (f ` S) \<le> dim (S)"
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  1841
proof -
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1842
  from basis_exists[of S] obtain B where
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1843
    B: "B \<subseteq> S" "independent B" "S \<subseteq> span B" "card B = dim S" by blast
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1844
  from B have fB: "finite B" "card B = dim S"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1845
    using independent_bound by blast+
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1846
  have "dim (f ` S) \<le> card (f ` B)"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1847
    apply (rule span_card_ge_dim)
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1848
    using lf B fB
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1849
    apply (auto simp add: span_linear_image spans_image subset_image_iff)
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  1850
    done
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1851
  also have "\<dots> \<le> dim S"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1852
    using card_image_le[OF fB(1)] fB by simp
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1853
  finally show ?thesis .
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1854
qed
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1855
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1856
text {* Relation between bases and injectivity/surjectivity of map. *}
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1857
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1858
lemma spanning_surjective_image:
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1859
  assumes us: "UNIV \<subseteq> span S"
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1860
    and lf: "linear f"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1861
    and sf: "surj f"
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1862
  shows "UNIV \<subseteq> span (f ` S)"
49663
b84fafaea4bb tuned proofs;
wenzelm
parents: 49652
diff changeset
  1863
proof -
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1864
  have "UNIV \<subseteq> f ` UNIV"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1865
    using sf by (auto simp add: surj_def)
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1866
  also have " \<dots> \<subseteq> span (f ` S)"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1867
    using spans_image[OF lf us] .
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1868
  finally show ?thesis .
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1869
qed
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1870
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1871
lemma independent_injective_image:
49663
b84fafaea4bb tuned proofs;
wenzelm
parents: 49652
diff changeset
  1872
  assumes iS: "independent S"
b84fafaea4bb tuned proofs;
wenzelm
parents: 49652
diff changeset
  1873
    and lf: "linear f"
b84fafaea4bb tuned proofs;
wenzelm
parents: 49652
diff changeset
  1874
    and fi: "inj f"
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1875
  shows "independent (f ` S)"
49663
b84fafaea4bb tuned proofs;
wenzelm
parents: 49652
diff changeset
  1876
proof -
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1877
  {
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1878
    fix a
49663
b84fafaea4bb tuned proofs;
wenzelm
parents: 49652
diff changeset
  1879
    assume a: "a \<in> S" "f a \<in> span (f ` S - {f a})"
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1880
    have eq: "f ` S - {f a} = f ` (S - {a})"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1881
      using fi by (auto simp add: inj_on_def)
53716
b42d9a71fc1a tuned proofs;
wenzelm
parents: 53600
diff changeset
  1882
    from a have "f a \<in> f ` span (S - {a})"
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1883
      unfolding eq span_linear_image[OF lf, of "S - {a}"] by blast
53716
b42d9a71fc1a tuned proofs;
wenzelm
parents: 53600
diff changeset
  1884
    then have "a \<in> span (S - {a})"
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1885
      using fi by (auto simp add: inj_on_def)
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1886
    with a(1) iS have False
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1887
      by (simp add: dependent_def)
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1888
  }
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1889
  then show ?thesis
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1890
    unfolding dependent_def by blast
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1891
qed
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1892
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1893
text {* Picking an orthogonal replacement for a spanning set. *}
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1894
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1895
(* FIXME : Move to some general theory ?*)
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1896
definition "pairwise R S \<longleftrightarrow> (\<forall>x \<in> S. \<forall>y\<in> S. x\<noteq>y \<longrightarrow> R x y)"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1897
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1898
lemma vector_sub_project_orthogonal:
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1899
  fixes b x :: "'a::euclidean_space"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1900
  shows "b \<bullet> (x - ((b \<bullet> x) / (b \<bullet> b)) *\<^sub>R b) = 0"
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1901
  unfolding inner_simps by auto
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1902
44528
0b8e0dbb2bdd generalize and shorten proof of basis_orthogonal
huffman
parents: 44527
diff changeset
  1903
lemma pairwise_orthogonal_insert:
0b8e0dbb2bdd generalize and shorten proof of basis_orthogonal
huffman
parents: 44527
diff changeset
  1904
  assumes "pairwise orthogonal S"
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  1905
    and "\<And>y. y \<in> S \<Longrightarrow> orthogonal x y"
44528
0b8e0dbb2bdd generalize and shorten proof of basis_orthogonal
huffman
parents: 44527
diff changeset
  1906
  shows "pairwise orthogonal (insert x S)"
0b8e0dbb2bdd generalize and shorten proof of basis_orthogonal
huffman
parents: 44527
diff changeset
  1907
  using assms unfolding pairwise_def
0b8e0dbb2bdd generalize and shorten proof of basis_orthogonal
huffman
parents: 44527
diff changeset
  1908
  by (auto simp add: orthogonal_commute)
0b8e0dbb2bdd generalize and shorten proof of basis_orthogonal
huffman
parents: 44527
diff changeset
  1909
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1910
lemma basis_orthogonal:
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1911
  fixes B :: "'a::real_inner set"
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1912
  assumes fB: "finite B"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1913
  shows "\<exists>C. finite C \<and> card C \<le> card B \<and> span C = span B \<and> pairwise orthogonal C"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1914
  (is " \<exists>C. ?P B C")
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  1915
  using fB
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  1916
proof (induct rule: finite_induct)
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  1917
  case empty
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1918
  then show ?case
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1919
    apply (rule exI[where x="{}"])
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1920
    apply (auto simp add: pairwise_def)
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1921
    done
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1922
next
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  1923
  case (insert a B)
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1924
  note fB = `finite B` and aB = `a \<notin> B`
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1925
  from `\<exists>C. finite C \<and> card C \<le> card B \<and> span C = span B \<and> pairwise orthogonal C`
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1926
  obtain C where C: "finite C" "card C \<le> card B"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1927
    "span C = span B" "pairwise orthogonal C" by blast
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1928
  let ?a = "a - setsum (\<lambda>x. (x \<bullet> a / (x \<bullet> x)) *\<^sub>R x) C"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1929
  let ?C = "insert ?a C"
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1930
  from C(1) have fC: "finite ?C"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1931
    by simp
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  1932
  from fB aB C(1,2) have cC: "card ?C \<le> card (insert a B)"
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  1933
    by (simp add: card_insert_if)
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1934
  {
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1935
    fix x k
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  1936
    have th0: "\<And>(a::'a) b c. a - (b - c) = c + (a - b)"
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  1937
      by (simp add: field_simps)
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1938
    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"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1939
      apply (simp only: scaleR_right_diff_distrib th0)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1940
      apply (rule span_add_eq)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1941
      apply (rule span_mul)
56196
32b7eafc5a52 remove unnecessary finiteness assumptions from lemmas about setsum
huffman
parents: 56166
diff changeset
  1942
      apply (rule span_setsum)
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1943
      apply clarify
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1944
      apply (rule span_mul)
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  1945
      apply (rule span_superset)
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  1946
      apply assumption
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1947
      done
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1948
  }
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1949
  then have SC: "span ?C = span (insert a B)"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1950
    unfolding set_eq_iff span_breakdown_eq C(3)[symmetric] by auto
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1951
  {
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1952
    fix y
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1953
    assume yC: "y \<in> C"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1954
    then have Cy: "C = insert y (C - {y})"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1955
      by blast
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1956
    have fth: "finite (C - {y})"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1957
      using C by simp
44528
0b8e0dbb2bdd generalize and shorten proof of basis_orthogonal
huffman
parents: 44527
diff changeset
  1958
    have "orthogonal ?a y"
0b8e0dbb2bdd generalize and shorten proof of basis_orthogonal
huffman
parents: 44527
diff changeset
  1959
      unfolding orthogonal_def
54230
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 53939
diff changeset
  1960
      unfolding inner_diff inner_setsum_left right_minus_eq
57418
6ab1c7cb0b8d fact consolidation
haftmann
parents: 56536
diff changeset
  1961
      unfolding setsum.remove [OF `finite C` `y \<in> C`]
44528
0b8e0dbb2bdd generalize and shorten proof of basis_orthogonal
huffman
parents: 44527
diff changeset
  1962
      apply (clarsimp simp add: inner_commute[of y a])
57418
6ab1c7cb0b8d fact consolidation
haftmann
parents: 56536
diff changeset
  1963
      apply (rule setsum.neutral)
44528
0b8e0dbb2bdd generalize and shorten proof of basis_orthogonal
huffman
parents: 44527
diff changeset
  1964
      apply clarsimp
0b8e0dbb2bdd generalize and shorten proof of basis_orthogonal
huffman
parents: 44527
diff changeset
  1965
      apply (rule C(4)[unfolded pairwise_def orthogonal_def, rule_format])
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1966
      using `y \<in> C` by auto
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1967
  }
44528
0b8e0dbb2bdd generalize and shorten proof of basis_orthogonal
huffman
parents: 44527
diff changeset
  1968
  with `pairwise orthogonal C` have CPO: "pairwise orthogonal ?C"
0b8e0dbb2bdd generalize and shorten proof of basis_orthogonal
huffman
parents: 44527
diff changeset
  1969
    by (rule pairwise_orthogonal_insert)
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1970
  from fC cC SC CPO have "?P (insert a B) ?C"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1971
    by blast
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1972
  then show ?case by blast
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1973
qed
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1974
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1975
lemma orthogonal_basis_exists:
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1976
  fixes V :: "('a::euclidean_space) set"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1977
  shows "\<exists>B. independent B \<and> B \<subseteq> span V \<and> V \<subseteq> span B \<and> (card B = dim V) \<and> pairwise orthogonal B"
49663
b84fafaea4bb tuned proofs;
wenzelm
parents: 49652
diff changeset
  1978
proof -
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  1979
  from basis_exists[of V] obtain B where
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1980
    B: "B \<subseteq> V" "independent B" "V \<subseteq> span B" "card B = dim V"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1981
    by blast
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1982
  from B have fB: "finite B" "card B = dim V"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1983
    using independent_bound by auto
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1984
  from basis_orthogonal[OF fB(1)] obtain C where
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1985
    C: "finite C" "card C \<le> card B" "span C = span B" "pairwise orthogonal C"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1986
    by blast
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1987
  from C B have CSV: "C \<subseteq> span V"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1988
    by (metis span_inc span_mono subset_trans)
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1989
  from span_mono[OF B(3)] C have SVC: "span V \<subseteq> span C"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1990
    by (simp add: span_span)
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1991
  from card_le_dim_spanning[OF CSV SVC C(1)] C(2,3) fB
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1992
  have iC: "independent C"
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1993
    by (simp add: dim_span)
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1994
  from C fB have "card C \<le> dim V"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1995
    by simp
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1996
  moreover have "dim V \<le> card C"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1997
    using span_card_ge_dim[OF CSV SVC C(1)]
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1998
    by (simp add: dim_span)
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  1999
  ultimately have CdV: "card C = dim V"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2000
    using C(1) by simp
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2001
  from C B CSV CdV iC show ?thesis
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2002
    by auto
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2003
qed
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2004
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2005
lemma span_eq: "span S = span T \<longleftrightarrow> S \<subseteq> span T \<and> T \<subseteq> span S"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2006
  using span_inc[unfolded subset_eq] using span_mono[of T "span S"] span_mono[of S "span T"]
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  2007
  by (auto simp add: span_span)
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2008
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2009
text {* Low-dimensional subset is in a hyperplane (weak orthogonal complement). *}
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2010
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  2011
lemma span_not_univ_orthogonal:
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2012
  fixes S :: "'a::euclidean_space set"
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2013
  assumes sU: "span S \<noteq> UNIV"
56444
f944ae8c80a3 tuned proofs;
wenzelm
parents: 56409
diff changeset
  2014
  shows "\<exists>a::'a. a \<noteq> 0 \<and> (\<forall>x \<in> span S. a \<bullet> x = 0)"
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  2015
proof -
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2016
  from sU obtain a where a: "a \<notin> span S"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2017
    by blast
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2018
  from orthogonal_basis_exists obtain B where
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2019
    B: "independent B" "B \<subseteq> span S" "S \<subseteq> span B" "card B = dim S" "pairwise orthogonal B"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2020
    by blast
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2021
  from B have fB: "finite B" "card B = dim S"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2022
    using independent_bound by auto
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2023
  from span_mono[OF B(2)] span_mono[OF B(3)]
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2024
  have sSB: "span S = span B"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2025
    by (simp add: span_span)
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2026
  let ?a = "a - setsum (\<lambda>b. (a \<bullet> b / (b \<bullet> b)) *\<^sub>R b) B"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2027
  have "setsum (\<lambda>b. (a \<bullet> b / (b \<bullet> b)) *\<^sub>R b) B \<in> span S"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2028
    unfolding sSB
56196
32b7eafc5a52 remove unnecessary finiteness assumptions from lemmas about setsum
huffman
parents: 56166
diff changeset
  2029
    apply (rule span_setsum)
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2030
    apply clarsimp
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2031
    apply (rule span_mul)
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  2032
    apply (rule span_superset)
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  2033
    apply assumption
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  2034
    done
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2035
  with a have a0:"?a  \<noteq> 0"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2036
    by auto
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2037
  have "\<forall>x\<in>span B. ?a \<bullet> x = 0"
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  2038
  proof (rule span_induct')
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  2039
    show "subspace {x. ?a \<bullet> x = 0}"
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  2040
      by (auto simp add: subspace_def inner_add)
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  2041
  next
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2042
    {
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2043
      fix x
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2044
      assume x: "x \<in> B"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2045
      from x have B': "B = insert x (B - {x})"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2046
        by blast
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2047
      have fth: "finite (B - {x})"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2048
        using fB by simp
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2049
      have "?a \<bullet> x = 0"
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2050
        apply (subst B')
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2051
        using fB fth
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2052
        unfolding setsum_clauses(2)[OF fth]
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2053
        apply simp unfolding inner_simps
44527
bf8014b4f933 remove dot_lsum and dot_rsum in favor of inner_setsum_{left,right}
huffman
parents: 44521
diff changeset
  2054
        apply (clarsimp simp add: inner_add inner_setsum_left)
57418
6ab1c7cb0b8d fact consolidation
haftmann
parents: 56536
diff changeset
  2055
        apply (rule setsum.neutral, rule ballI)
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2056
        unfolding inner_commute
49711
e5aaae7eadc9 tuned proofs;
wenzelm
parents: 49663
diff changeset
  2057
        apply (auto simp add: x field_simps
e5aaae7eadc9 tuned proofs;
wenzelm
parents: 49663
diff changeset
  2058
          intro: B(5)[unfolded pairwise_def orthogonal_def, rule_format])
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2059
        done
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2060
    }
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2061
    then show "\<forall>x \<in> B. ?a \<bullet> x = 0"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2062
      by blast
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2063
  qed
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2064
  with a0 show ?thesis
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2065
    unfolding sSB by (auto intro: exI[where x="?a"])
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2066
qed
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2067
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2068
lemma span_not_univ_subset_hyperplane:
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2069
  fixes S :: "'a::euclidean_space set"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2070
  assumes SU: "span S \<noteq> UNIV"
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2071
  shows "\<exists> a. a \<noteq>0 \<and> span S \<subseteq> {x. a \<bullet> x = 0}"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2072
  using span_not_univ_orthogonal[OF SU] by auto
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2073
49663
b84fafaea4bb tuned proofs;
wenzelm
parents: 49652
diff changeset
  2074
lemma lowdim_subset_hyperplane:
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2075
  fixes S :: "'a::euclidean_space set"
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2076
  assumes d: "dim S < DIM('a)"
56444
f944ae8c80a3 tuned proofs;
wenzelm
parents: 56409
diff changeset
  2077
  shows "\<exists>a::'a. a \<noteq> 0 \<and> span S \<subseteq> {x. a \<bullet> x = 0}"
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  2078
proof -
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2079
  {
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2080
    assume "span S = UNIV"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2081
    then have "dim (span S) = dim (UNIV :: ('a) set)"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2082
      by simp
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2083
    then have "dim S = DIM('a)"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2084
      by (simp add: dim_span dim_UNIV)
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2085
    with d have False by arith
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2086
  }
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2087
  then have th: "span S \<noteq> UNIV"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2088
    by blast
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2089
  from span_not_univ_subset_hyperplane[OF th] show ?thesis .
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2090
qed
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2091
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2092
text {* We can extend a linear basis-basis injection to the whole set. *}
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2093
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2094
lemma linear_indep_image_lemma:
49663
b84fafaea4bb tuned proofs;
wenzelm
parents: 49652
diff changeset
  2095
  assumes lf: "linear f"
b84fafaea4bb tuned proofs;
wenzelm
parents: 49652
diff changeset
  2096
    and fB: "finite B"
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  2097
    and ifB: "independent (f ` B)"
49663
b84fafaea4bb tuned proofs;
wenzelm
parents: 49652
diff changeset
  2098
    and fi: "inj_on f B"
b84fafaea4bb tuned proofs;
wenzelm
parents: 49652
diff changeset
  2099
    and xsB: "x \<in> span B"
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  2100
    and fx: "f x = 0"
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2101
  shows "x = 0"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2102
  using fB ifB fi xsB fx
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  2103
proof (induct arbitrary: x rule: finite_induct[OF fB])
49663
b84fafaea4bb tuned proofs;
wenzelm
parents: 49652
diff changeset
  2104
  case 1
b84fafaea4bb tuned proofs;
wenzelm
parents: 49652
diff changeset
  2105
  then show ?case by auto
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2106
next
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2107
  case (2 a b x)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2108
  have fb: "finite b" using "2.prems" by simp
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2109
  have th0: "f ` b \<subseteq> f ` (insert a b)"
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2110
    apply (rule image_mono)
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2111
    apply blast
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2112
    done
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2113
  from independent_mono[ OF "2.prems"(2) th0]
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2114
  have ifb: "independent (f ` b)"  .
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2115
  have fib: "inj_on f b"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2116
    apply (rule subset_inj_on [OF "2.prems"(3)])
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  2117
    apply blast
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  2118
    done
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2119
  from span_breakdown[of a "insert a b", simplified, OF "2.prems"(4)]
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2120
  obtain k where k: "x - k*\<^sub>R a \<in> span (b - {a})"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2121
    by blast
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2122
  have "f (x - k*\<^sub>R a) \<in> span (f ` b)"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2123
    unfolding span_linear_image[OF lf]
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2124
    apply (rule imageI)
53716
b42d9a71fc1a tuned proofs;
wenzelm
parents: 53600
diff changeset
  2125
    using k span_mono[of "b - {a}" b]
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2126
    apply blast
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  2127
    done
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  2128
  then have "f x - k*\<^sub>R f a \<in> span (f ` b)"
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2129
    by (simp add: linear_sub[OF lf] linear_cmul[OF lf])
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  2130
  then have th: "-k *\<^sub>R f a \<in> span (f ` b)"
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2131
    using "2.prems"(5) by simp
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2132
  have xsb: "x \<in> span b"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2133
  proof (cases "k = 0")
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2134
    case True
53716
b42d9a71fc1a tuned proofs;
wenzelm
parents: 53600
diff changeset
  2135
    with k have "x \<in> span (b - {a})" by simp
b42d9a71fc1a tuned proofs;
wenzelm
parents: 53600
diff changeset
  2136
    then show ?thesis using span_mono[of "b - {a}" b]
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2137
      by blast
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2138
  next
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2139
    case False
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2140
    with span_mul[OF th, of "- 1/ k"]
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2141
    have th1: "f a \<in> span (f ` b)"
56479
91958d4b30f7 revert c1bbd3e22226, a14831ac3023, and 36489d77c484: divide_minus_left/right are again simp rules
hoelzl
parents: 56444
diff changeset
  2142
      by auto
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2143
    from inj_on_image_set_diff[OF "2.prems"(3), of "insert a b " "{a}", symmetric]
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2144
    have tha: "f ` insert a b - f ` {a} = f ` (insert a b - {a})" by blast
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2145
    from "2.prems"(2) [unfolded dependent_def bex_simps(8), rule_format, of "f a"]
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2146
    have "f a \<notin> span (f ` b)" using tha
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2147
      using "2.hyps"(2)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2148
      "2.prems"(3) by auto
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2149
    with th1 have False by blast
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2150
    then show ?thesis by blast
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2151
  qed
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2152
  from "2.hyps"(3)[OF fb ifb fib xsb "2.prems"(5)] show "x = 0" .
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2153
qed
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2154
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2155
text {* We can extend a linear mapping from basis. *}
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2156
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2157
lemma linear_independent_extend_lemma:
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2158
  fixes f :: "'a::real_vector \<Rightarrow> 'b::real_vector"
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2159
  assumes fi: "finite B"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2160
    and ib: "independent B"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2161
  shows "\<exists>g.
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2162
    (\<forall>x\<in> span B. \<forall>y\<in> span B. g (x + y) = g x + g y) \<and>
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2163
    (\<forall>x\<in> span B. \<forall>c. g (c*\<^sub>R x) = c *\<^sub>R g x) \<and>
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2164
    (\<forall>x\<in> B. g x = f x)"
49663
b84fafaea4bb tuned proofs;
wenzelm
parents: 49652
diff changeset
  2165
  using ib fi
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  2166
proof (induct rule: finite_induct[OF fi])
49663
b84fafaea4bb tuned proofs;
wenzelm
parents: 49652
diff changeset
  2167
  case 1
b84fafaea4bb tuned proofs;
wenzelm
parents: 49652
diff changeset
  2168
  then show ?case by auto
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2169
next
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2170
  case (2 a b)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2171
  from "2.prems" "2.hyps" have ibf: "independent b" "finite b"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2172
    by (simp_all add: independent_insert)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2173
  from "2.hyps"(3)[OF ibf] obtain g where
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2174
    g: "\<forall>x\<in>span b. \<forall>y\<in>span b. g (x + y) = g x + g y"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2175
    "\<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
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2176
  let ?h = "\<lambda>z. SOME k. (z - k *\<^sub>R a) \<in> span b"
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2177
  {
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2178
    fix z
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2179
    assume z: "z \<in> span (insert a b)"
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2180
    have th0: "z - ?h z *\<^sub>R a \<in> span b"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2181
      apply (rule someI_ex)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2182
      unfolding span_breakdown_eq[symmetric]
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2183
      apply (rule z)
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2184
      done
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2185
    {
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2186
      fix k
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2187
      assume k: "z - k *\<^sub>R a \<in> span b"
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2188
      have eq: "z - ?h z *\<^sub>R a - (z - k*\<^sub>R a) = (k - ?h z) *\<^sub>R a"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2189
        by (simp add: field_simps scaleR_left_distrib [symmetric])
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2190
      from span_sub[OF th0 k] have khz: "(k - ?h z) *\<^sub>R a \<in> span b"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2191
        by (simp add: eq)
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2192
      {
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2193
        assume "k \<noteq> ?h z"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2194
        then have k0: "k - ?h z \<noteq> 0" by simp
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2195
        from k0 span_mul[OF khz, of "1 /(k - ?h z)"]
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2196
        have "a \<in> span b" by simp
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2197
        with "2.prems"(1) "2.hyps"(2) have False
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2198
          by (auto simp add: dependent_def)
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2199
      }
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2200
      then have "k = ?h z" by blast
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2201
    }
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2202
    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)"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2203
      by blast
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2204
  }
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2205
  note h = this
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2206
  let ?g = "\<lambda>z. ?h z *\<^sub>R f a + g (z - ?h z *\<^sub>R a)"
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2207
  {
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2208
    fix x y
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2209
    assume x: "x \<in> span (insert a b)"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2210
      and y: "y \<in> span (insert a b)"
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2211
    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)"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2212
      by (simp add: algebra_simps)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2213
    have addh: "?h (x + y) = ?h x + ?h y"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2214
      apply (rule conjunct2[OF h, rule_format, symmetric])
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2215
      apply (rule span_add[OF x y])
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2216
      unfolding tha
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2217
      apply (metis span_add x y conjunct1[OF h, rule_format])
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2218
      done
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2219
    have "?g (x + y) = ?g x + ?g y"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2220
      unfolding addh tha
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2221
      g(1)[rule_format,OF conjunct1[OF h, OF x] conjunct1[OF h, OF y]]
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2222
      by (simp add: scaleR_left_distrib)}
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2223
  moreover
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2224
  {
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2225
    fix x :: "'a"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2226
    fix c :: real
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  2227
    assume x: "x \<in> span (insert a b)"
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2228
    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)"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2229
      by (simp add: algebra_simps)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2230
    have hc: "?h (c *\<^sub>R x) = c * ?h x"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2231
      apply (rule conjunct2[OF h, rule_format, symmetric])
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2232
      apply (metis span_mul x)
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  2233
      apply (metis tha span_mul x conjunct1[OF h])
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  2234
      done
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2235
    have "?g (c *\<^sub>R x) = c*\<^sub>R ?g x"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2236
      unfolding hc tha g(2)[rule_format, OF conjunct1[OF h, OF x]]
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2237
      by (simp add: algebra_simps)
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2238
  }
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2239
  moreover
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2240
  {
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2241
    fix x
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2242
    assume x: "x \<in> insert a b"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2243
    {
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2244
      assume xa: "x = a"
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2245
      have ha1: "1 = ?h a"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2246
        apply (rule conjunct2[OF h, rule_format])
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2247
        apply (metis span_superset insertI1)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2248
        using conjunct1[OF h, OF span_superset, OF insertI1]
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  2249
        apply (auto simp add: span_0)
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  2250
        done
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2251
      from xa ha1[symmetric] have "?g x = f x"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2252
        apply simp
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2253
        using g(2)[rule_format, OF span_0, of 0]
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  2254
        apply simp
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2255
        done
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2256
    }
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2257
    moreover
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2258
    {
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2259
      assume xb: "x \<in> b"
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2260
      have h0: "0 = ?h x"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2261
        apply (rule conjunct2[OF h, rule_format])
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2262
        apply (metis  span_superset x)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2263
        apply simp
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2264
        apply (metis span_superset xb)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2265
        done
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2266
      have "?g x = f x"
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2267
        by (simp add: h0[symmetric] g(3)[rule_format, OF xb])
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2268
    }
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2269
    ultimately have "?g x = f x"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2270
      using x by blast
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2271
  }
49663
b84fafaea4bb tuned proofs;
wenzelm
parents: 49652
diff changeset
  2272
  ultimately show ?case
b84fafaea4bb tuned proofs;
wenzelm
parents: 49652
diff changeset
  2273
    apply -
b84fafaea4bb tuned proofs;
wenzelm
parents: 49652
diff changeset
  2274
    apply (rule exI[where x="?g"])
b84fafaea4bb tuned proofs;
wenzelm
parents: 49652
diff changeset
  2275
    apply blast
b84fafaea4bb tuned proofs;
wenzelm
parents: 49652
diff changeset
  2276
    done
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2277
qed
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2278
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2279
lemma linear_independent_extend:
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2280
  fixes B :: "'a::euclidean_space set"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2281
  assumes iB: "independent B"
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2282
  shows "\<exists>g. linear g \<and> (\<forall>x\<in>B. g x = f x)"
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  2283
proof -
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2284
  from maximal_independent_subset_extend[of B UNIV] iB
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2285
  obtain C where C: "B \<subseteq> C" "independent C" "\<And>x. x \<in> span C"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2286
    by auto
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2287
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2288
  from C(2) independent_bound[of C] linear_independent_extend_lemma[of C f]
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2289
  obtain g where g:
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2290
    "(\<forall>x\<in> span C. \<forall>y\<in> span C. g (x + y) = g x + g y) \<and>
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2291
     (\<forall>x\<in> span C. \<forall>c. g (c*\<^sub>R x) = c *\<^sub>R g x) \<and>
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2292
     (\<forall>x\<in> C. g x = f x)" by blast
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2293
  from g show ?thesis
53600
8fda7ad57466 make 'linear' into a sublocale of 'bounded_linear';
huffman
parents: 53596
diff changeset
  2294
    unfolding linear_iff
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2295
    using C
49663
b84fafaea4bb tuned proofs;
wenzelm
parents: 49652
diff changeset
  2296
    apply clarsimp
b84fafaea4bb tuned proofs;
wenzelm
parents: 49652
diff changeset
  2297
    apply blast
b84fafaea4bb tuned proofs;
wenzelm
parents: 49652
diff changeset
  2298
    done
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2299
qed
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2300
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2301
text {* Can construct an isomorphism between spaces of same dimension. *}
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2302
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2303
lemma subspace_isomorphism:
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2304
  fixes S :: "'a::euclidean_space set"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2305
    and T :: "'b::euclidean_space set"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2306
  assumes s: "subspace S"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2307
    and t: "subspace T"
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  2308
    and d: "dim S = dim T"
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2309
  shows "\<exists>f. linear f \<and> f ` S = T \<and> inj_on f S"
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  2310
proof -
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2311
  from basis_exists[of S] independent_bound
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2312
  obtain B where B: "B \<subseteq> S" "independent B" "S \<subseteq> span B" "card B = dim S" and fB: "finite B"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2313
    by blast
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2314
  from basis_exists[of T] independent_bound
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2315
  obtain C where C: "C \<subseteq> T" "independent C" "T \<subseteq> span C" "card C = dim T" and fC: "finite C"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2316
    by blast
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2317
  from B(4) C(4) card_le_inj[of B C] d
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2318
  obtain f where f: "f ` B \<subseteq> C" "inj_on f B" using `finite B` `finite C`
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2319
    by auto
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2320
  from linear_independent_extend[OF B(2)]
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2321
  obtain g where g: "linear g" "\<forall>x\<in> B. g x = f x"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2322
    by blast
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2323
  from inj_on_iff_eq_card[OF fB, of f] f(2) have "card (f ` B) = card B"
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2324
    by simp
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2325
  with B(4) C(4) have ceq: "card (f ` B) = card C"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2326
    using d by simp
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2327
  have "g ` B = f ` B"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2328
    using g(2) by (auto simp add: image_iff)
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2329
  also have "\<dots> = C" using card_subset_eq[OF fC f(1) ceq] .
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2330
  finally have gBC: "g ` B = C" .
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2331
  have gi: "inj_on g B"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2332
    using f(2) g(2) by (auto simp add: inj_on_def)
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2333
  note g0 = linear_indep_image_lemma[OF g(1) fB, unfolded gBC, OF C(2) gi]
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2334
  {
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2335
    fix x y
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2336
    assume x: "x \<in> S" and y: "y \<in> S" and gxy: "g x = g y"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2337
    from B(3) x y have x': "x \<in> span B" and y': "y \<in> span B"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2338
      by blast+
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2339
    from gxy have th0: "g (x - y) = 0"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2340
      by (simp add: linear_sub[OF g(1)])
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2341
    have th1: "x - y \<in> span B"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2342
      using x' y' by (metis span_sub)
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2343
    have "x = y"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2344
      using g0[OF th1 th0] by simp
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2345
  }
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2346
  then have giS: "inj_on g S"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2347
    unfolding inj_on_def by blast
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2348
  from span_subspace[OF B(1,3) s] have "g ` S = span (g ` B)"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2349
    by (simp add: span_linear_image[OF g(1)])
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2350
  also have "\<dots> = span C" unfolding gBC ..
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2351
  also have "\<dots> = T" using span_subspace[OF C(1,3) t] .
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2352
  finally have gS: "g ` S = T" .
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2353
  from g(1) gS giS show ?thesis
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2354
    by blast
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2355
qed
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2356
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2357
text {* Linear functions are equal on a subspace if they are on a spanning set. *}
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2358
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2359
lemma subspace_kernel:
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2360
  assumes lf: "linear f"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2361
  shows "subspace {x. f x = 0}"
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  2362
  apply (simp add: subspace_def)
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  2363
  apply (simp add: linear_add[OF lf] linear_cmul[OF lf] linear_0[OF lf])
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  2364
  done
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2365
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2366
lemma linear_eq_0_span:
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2367
  assumes lf: "linear f" and f0: "\<forall>x\<in>B. f x = 0"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2368
  shows "\<forall>x \<in> span B. f x = 0"
44170
510ac30f44c0 make Multivariate_Analysis work with separate set type
huffman
parents: 44166
diff changeset
  2369
  using f0 subspace_kernel[OF lf]
510ac30f44c0 make Multivariate_Analysis work with separate set type
huffman
parents: 44166
diff changeset
  2370
  by (rule span_induct')
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2371
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2372
lemma linear_eq_0:
49663
b84fafaea4bb tuned proofs;
wenzelm
parents: 49652
diff changeset
  2373
  assumes lf: "linear f"
b84fafaea4bb tuned proofs;
wenzelm
parents: 49652
diff changeset
  2374
    and SB: "S \<subseteq> span B"
b84fafaea4bb tuned proofs;
wenzelm
parents: 49652
diff changeset
  2375
    and f0: "\<forall>x\<in>B. f x = 0"
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2376
  shows "\<forall>x \<in> S. f x = 0"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2377
  by (metis linear_eq_0_span[OF lf] subset_eq SB f0)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2378
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2379
lemma linear_eq:
49663
b84fafaea4bb tuned proofs;
wenzelm
parents: 49652
diff changeset
  2380
  assumes lf: "linear f"
b84fafaea4bb tuned proofs;
wenzelm
parents: 49652
diff changeset
  2381
    and lg: "linear g"
b84fafaea4bb tuned proofs;
wenzelm
parents: 49652
diff changeset
  2382
    and S: "S \<subseteq> span B"
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  2383
    and fg: "\<forall> x\<in> B. f x = g x"
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2384
  shows "\<forall>x\<in> S. f x = g x"
49663
b84fafaea4bb tuned proofs;
wenzelm
parents: 49652
diff changeset
  2385
proof -
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2386
  let ?h = "\<lambda>x. f x - g x"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2387
  from fg have fg': "\<forall>x\<in> B. ?h x = 0" by simp
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2388
  from linear_eq_0[OF linear_compose_sub[OF lf lg] S fg']
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2389
  show ?thesis by simp
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2390
qed
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2391
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2392
lemma linear_eq_stdbasis:
56444
f944ae8c80a3 tuned proofs;
wenzelm
parents: 56409
diff changeset
  2393
  fixes f :: "'a::euclidean_space \<Rightarrow> _"
f944ae8c80a3 tuned proofs;
wenzelm
parents: 56409
diff changeset
  2394
  assumes lf: "linear f"
49663
b84fafaea4bb tuned proofs;
wenzelm
parents: 49652
diff changeset
  2395
    and lg: "linear g"
50526
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 50105
diff changeset
  2396
    and fg: "\<forall>b\<in>Basis. f b = g b"
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2397
  shows "f = g"
50526
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 50105
diff changeset
  2398
  using linear_eq[OF lf lg, of _ Basis] fg by auto
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2399
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2400
text {* Similar results for bilinear functions. *}
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2401
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2402
lemma bilinear_eq:
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2403
  assumes bf: "bilinear f"
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  2404
    and bg: "bilinear g"
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2405
    and SB: "S \<subseteq> span B"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2406
    and TC: "T \<subseteq> span C"
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  2407
    and fg: "\<forall>x\<in> B. \<forall>y\<in> C. f x y = g x y"
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2408
  shows "\<forall>x\<in>S. \<forall>y\<in>T. f x y = g x y "
49663
b84fafaea4bb tuned proofs;
wenzelm
parents: 49652
diff changeset
  2409
proof -
44170
510ac30f44c0 make Multivariate_Analysis work with separate set type
huffman
parents: 44166
diff changeset
  2410
  let ?P = "{x. \<forall>y\<in> span C. f x y = g x y}"
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2411
  from bf bg have sp: "subspace ?P"
53600
8fda7ad57466 make 'linear' into a sublocale of 'bounded_linear';
huffman
parents: 53596
diff changeset
  2412
    unfolding bilinear_def linear_iff subspace_def bf bg
49663
b84fafaea4bb tuned proofs;
wenzelm
parents: 49652
diff changeset
  2413
    by (auto simp add: span_0 bilinear_lzero[OF bf] bilinear_lzero[OF bg] span_add Ball_def
b84fafaea4bb tuned proofs;
wenzelm
parents: 49652
diff changeset
  2414
      intro: bilinear_ladd[OF bf])
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2415
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2416
  have "\<forall>x \<in> span B. \<forall>y\<in> span C. f x y = g x y"
44170
510ac30f44c0 make Multivariate_Analysis work with separate set type
huffman
parents: 44166
diff changeset
  2417
    apply (rule span_induct' [OF _ sp])
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2418
    apply (rule ballI)
44170
510ac30f44c0 make Multivariate_Analysis work with separate set type
huffman
parents: 44166
diff changeset
  2419
    apply (rule span_induct')
510ac30f44c0 make Multivariate_Analysis work with separate set type
huffman
parents: 44166
diff changeset
  2420
    apply (simp add: fg)
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2421
    apply (auto simp add: subspace_def)
53600
8fda7ad57466 make 'linear' into a sublocale of 'bounded_linear';
huffman
parents: 53596
diff changeset
  2422
    using bf bg unfolding bilinear_def linear_iff
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  2423
    apply (auto simp add: span_0 bilinear_rzero[OF bf] bilinear_rzero[OF bg] span_add Ball_def
49663
b84fafaea4bb tuned proofs;
wenzelm
parents: 49652
diff changeset
  2424
      intro: bilinear_ladd[OF bf])
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  2425
    done
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2426
  then show ?thesis
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2427
    using SB TC by auto
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2428
qed
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2429
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  2430
lemma bilinear_eq_stdbasis:
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2431
  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space \<Rightarrow> _"
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2432
  assumes bf: "bilinear f"
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  2433
    and bg: "bilinear g"
50526
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 50105
diff changeset
  2434
    and fg: "\<forall>i\<in>Basis. \<forall>j\<in>Basis. f i j = g i j"
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2435
  shows "f = g"
50526
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 50105
diff changeset
  2436
  using bilinear_eq[OF bf bg equalityD2[OF span_Basis] equalityD2[OF span_Basis] fg] by blast
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2437
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2438
text {* Detailed theorems about left and right invertibility in general case. *}
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2439
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  2440
lemma linear_injective_left_inverse:
56444
f944ae8c80a3 tuned proofs;
wenzelm
parents: 56409
diff changeset
  2441
  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
f944ae8c80a3 tuned proofs;
wenzelm
parents: 56409
diff changeset
  2442
  assumes lf: "linear f"
f944ae8c80a3 tuned proofs;
wenzelm
parents: 56409
diff changeset
  2443
    and fi: "inj f"
f944ae8c80a3 tuned proofs;
wenzelm
parents: 56409
diff changeset
  2444
  shows "\<exists>g. linear g \<and> g \<circ> f = id"
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  2445
proof -
50526
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 50105
diff changeset
  2446
  from linear_independent_extend[OF independent_injective_image, OF independent_Basis, OF lf fi]
56444
f944ae8c80a3 tuned proofs;
wenzelm
parents: 56409
diff changeset
  2447
  obtain h :: "'b \<Rightarrow> 'a" where h: "linear h" "\<forall>x \<in> f ` Basis. h x = inv f x"
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2448
    by blast
50526
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 50105
diff changeset
  2449
  from h(2) have th: "\<forall>i\<in>Basis. (h \<circ> f) i = id i"
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2450
    using inv_o_cancel[OF fi, unfolded fun_eq_iff id_def o_def]
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2451
    by auto
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2452
  from linear_eq_stdbasis[OF linear_compose[OF lf h(1)] linear_id th]
56444
f944ae8c80a3 tuned proofs;
wenzelm
parents: 56409
diff changeset
  2453
  have "h \<circ> f = id" .
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2454
  then show ?thesis
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2455
    using h(1) by blast
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2456
qed
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2457
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  2458
lemma linear_surjective_right_inverse:
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2459
  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2460
  assumes lf: "linear f"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2461
    and sf: "surj f"
56444
f944ae8c80a3 tuned proofs;
wenzelm
parents: 56409
diff changeset
  2462
  shows "\<exists>g. linear g \<and> f \<circ> g = id"
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  2463
proof -
50526
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 50105
diff changeset
  2464
  from linear_independent_extend[OF independent_Basis[where 'a='b],of "inv f"]
56444
f944ae8c80a3 tuned proofs;
wenzelm
parents: 56409
diff changeset
  2465
  obtain h :: "'b \<Rightarrow> 'a" where h: "linear h" "\<forall>x\<in>Basis. h x = inv f x"
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2466
    by blast
56444
f944ae8c80a3 tuned proofs;
wenzelm
parents: 56409
diff changeset
  2467
  from h(2) have th: "\<forall>i\<in>Basis. (f \<circ> h) i = id i"
50526
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 50105
diff changeset
  2468
    using sf by (auto simp add: surj_iff_all)
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2469
  from linear_eq_stdbasis[OF linear_compose[OF h(1) lf] linear_id th]
56444
f944ae8c80a3 tuned proofs;
wenzelm
parents: 56409
diff changeset
  2470
  have "f \<circ> h = id" .
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2471
  then show ?thesis
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2472
    using h(1) by blast
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2473
qed
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2474
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2475
text {* An injective map @{typ "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"} is also surjective. *}
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2476
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  2477
lemma linear_injective_imp_surjective:
56444
f944ae8c80a3 tuned proofs;
wenzelm
parents: 56409
diff changeset
  2478
  fixes f :: "'a::euclidean_space \<Rightarrow> 'a::euclidean_space"
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2479
  assumes lf: "linear f"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2480
    and fi: "inj f"
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2481
  shows "surj f"
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  2482
proof -
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2483
  let ?U = "UNIV :: 'a set"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2484
  from basis_exists[of ?U] obtain B
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2485
    where B: "B \<subseteq> ?U" "independent B" "?U \<subseteq> span B" "card B = dim ?U"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2486
    by blast
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2487
  from B(4) have d: "dim ?U = card B"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2488
    by simp
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2489
  have th: "?U \<subseteq> span (f ` B)"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2490
    apply (rule card_ge_dim_independent)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2491
    apply blast
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2492
    apply (rule independent_injective_image[OF B(2) lf fi])
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2493
    apply (rule order_eq_refl)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2494
    apply (rule sym)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2495
    unfolding d
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2496
    apply (rule card_image)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2497
    apply (rule subset_inj_on[OF fi])
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  2498
    apply blast
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  2499
    done
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2500
  from th show ?thesis
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2501
    unfolding span_linear_image[OF lf] surj_def
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2502
    using B(3) by blast
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2503
qed
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2504
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2505
text {* And vice versa. *}
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2506
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2507
lemma surjective_iff_injective_gen:
49663
b84fafaea4bb tuned proofs;
wenzelm
parents: 49652
diff changeset
  2508
  assumes fS: "finite S"
b84fafaea4bb tuned proofs;
wenzelm
parents: 49652
diff changeset
  2509
    and fT: "finite T"
b84fafaea4bb tuned proofs;
wenzelm
parents: 49652
diff changeset
  2510
    and c: "card S = card T"
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  2511
    and ST: "f ` S \<subseteq> T"
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2512
  shows "(\<forall>y \<in> T. \<exists>x \<in> S. f x = y) \<longleftrightarrow> inj_on f S"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2513
  (is "?lhs \<longleftrightarrow> ?rhs")
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2514
proof
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2515
  assume h: "?lhs"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2516
  {
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2517
    fix x y
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2518
    assume x: "x \<in> S"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2519
    assume y: "y \<in> S"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2520
    assume f: "f x = f y"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2521
    from x fS have S0: "card S \<noteq> 0"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2522
      by auto
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2523
    have "x = y"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2524
    proof (rule ccontr)
53716
b42d9a71fc1a tuned proofs;
wenzelm
parents: 53600
diff changeset
  2525
      assume xy: "\<not> ?thesis"
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2526
      have th: "card S \<le> card (f ` (S - {y}))"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2527
        unfolding c
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2528
        apply (rule card_mono)
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2529
        apply (rule finite_imageI)
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2530
        using fS apply simp
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2531
        using h xy x y f unfolding subset_eq image_iff
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2532
        apply auto
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2533
        apply (case_tac "xa = f x")
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2534
        apply (rule bexI[where x=x])
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2535
        apply auto
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2536
        done
53716
b42d9a71fc1a tuned proofs;
wenzelm
parents: 53600
diff changeset
  2537
      also have " \<dots> \<le> card (S - {y})"
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2538
        apply (rule card_image_le)
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2539
        using fS by simp
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2540
      also have "\<dots> \<le> card S - 1" using y fS by simp
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2541
      finally show False using S0 by arith
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2542
    qed
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2543
  }
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2544
  then show ?rhs
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2545
    unfolding inj_on_def by blast
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2546
next
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2547
  assume h: ?rhs
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2548
  have "f ` S = T"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2549
    apply (rule card_subset_eq[OF fT ST])
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2550
    unfolding card_image[OF h]
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2551
    apply (rule c)
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2552
    done
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2553
  then show ?lhs by blast
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2554
qed
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2555
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  2556
lemma linear_surjective_imp_injective:
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2557
  fixes f :: "'a::euclidean_space \<Rightarrow> 'a::euclidean_space"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2558
  assumes lf: "linear f"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2559
    and sf: "surj f"
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2560
  shows "inj f"
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  2561
proof -
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2562
  let ?U = "UNIV :: 'a set"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2563
  from basis_exists[of ?U] obtain B
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2564
    where B: "B \<subseteq> ?U" "independent B" "?U \<subseteq> span B" and d: "card B = dim ?U"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2565
    by blast
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2566
  {
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2567
    fix x
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2568
    assume x: "x \<in> span B"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2569
    assume fx: "f x = 0"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2570
    from B(2) have fB: "finite B"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2571
      using independent_bound by auto
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2572
    have fBi: "independent (f ` B)"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2573
      apply (rule card_le_dim_spanning[of "f ` B" ?U])
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2574
      apply blast
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2575
      using sf B(3)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2576
      unfolding span_linear_image[OF lf] surj_def subset_eq image_iff
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2577
      apply blast
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2578
      using fB apply blast
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2579
      unfolding d[symmetric]
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2580
      apply (rule card_image_le)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2581
      apply (rule fB)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2582
      done
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2583
    have th0: "dim ?U \<le> card (f ` B)"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2584
      apply (rule span_card_ge_dim)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2585
      apply blast
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2586
      unfolding span_linear_image[OF lf]
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2587
      apply (rule subset_trans[where B = "f ` UNIV"])
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2588
      using sf unfolding surj_def
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2589
      apply blast
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2590
      apply (rule image_mono)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2591
      apply (rule B(3))
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2592
      apply (metis finite_imageI fB)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2593
      done
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2594
    moreover have "card (f ` B) \<le> card B"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2595
      by (rule card_image_le, rule fB)
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2596
    ultimately have th1: "card B = card (f ` B)"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2597
      unfolding d by arith
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2598
    have fiB: "inj_on f B"
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  2599
      unfolding surjective_iff_injective_gen[OF fB finite_imageI[OF fB] th1 subset_refl, symmetric]
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  2600
      by blast
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2601
    from linear_indep_image_lemma[OF lf fB fBi fiB x] fx
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2602
    have "x = 0" by blast
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2603
  }
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2604
  then show ?thesis
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2605
    unfolding linear_injective_0[OF lf]
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2606
    using B(3)
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2607
    by blast
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2608
qed
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2609
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2610
text {* Hence either is enough for isomorphism. *}
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2611
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2612
lemma left_right_inverse_eq:
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2613
  assumes fg: "f \<circ> g = id"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2614
    and gh: "g \<circ> h = id"
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2615
  shows "f = h"
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  2616
proof -
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2617
  have "f = f \<circ> (g \<circ> h)"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2618
    unfolding gh by simp
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2619
  also have "\<dots> = (f \<circ> g) \<circ> h"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2620
    by (simp add: o_assoc)
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2621
  finally show "f = h"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2622
    unfolding fg by simp
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2623
qed
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2624
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2625
lemma isomorphism_expand:
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2626
  "f \<circ> g = id \<and> g \<circ> f = id \<longleftrightarrow> (\<forall>x. f (g x) = x) \<and> (\<forall>x. g (f x) = x)"
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2627
  by (simp add: fun_eq_iff o_def id_def)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2628
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  2629
lemma linear_injective_isomorphism:
56444
f944ae8c80a3 tuned proofs;
wenzelm
parents: 56409
diff changeset
  2630
  fixes f :: "'a::euclidean_space \<Rightarrow> 'a::euclidean_space"
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2631
  assumes lf: "linear f"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2632
    and fi: "inj f"
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2633
  shows "\<exists>f'. linear f' \<and> (\<forall>x. f' (f x) = x) \<and> (\<forall>x. f (f' x) = x)"
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  2634
  unfolding isomorphism_expand[symmetric]
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  2635
  using linear_surjective_right_inverse[OF lf linear_injective_imp_surjective[OF lf fi]]
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  2636
    linear_injective_left_inverse[OF lf fi]
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  2637
  by (metis left_right_inverse_eq)
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2638
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2639
lemma linear_surjective_isomorphism:
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2640
  fixes f :: "'a::euclidean_space \<Rightarrow> 'a::euclidean_space"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2641
  assumes lf: "linear f"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2642
    and sf: "surj f"
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2643
  shows "\<exists>f'. linear f' \<and> (\<forall>x. f' (f x) = x) \<and> (\<forall>x. f (f' x) = x)"
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  2644
  unfolding isomorphism_expand[symmetric]
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  2645
  using linear_surjective_right_inverse[OF lf sf]
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  2646
    linear_injective_left_inverse[OF lf linear_surjective_imp_injective[OF lf sf]]
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  2647
  by (metis left_right_inverse_eq)
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2648
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2649
text {* Left and right inverses are the same for
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2650
  @{typ "'a::euclidean_space \<Rightarrow> 'a::euclidean_space"}. *}
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2651
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  2652
lemma linear_inverse_left:
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2653
  fixes f :: "'a::euclidean_space \<Rightarrow> 'a::euclidean_space"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2654
  assumes lf: "linear f"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2655
    and lf': "linear f'"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2656
  shows "f \<circ> f' = id \<longleftrightarrow> f' \<circ> f = id"
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  2657
proof -
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2658
  {
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2659
    fix f f':: "'a \<Rightarrow> 'a"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2660
    assume lf: "linear f" "linear f'"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2661
    assume f: "f \<circ> f' = id"
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2662
    from f have sf: "surj f"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2663
      apply (auto simp add: o_def id_def surj_def)
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  2664
      apply metis
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  2665
      done
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2666
    from linear_surjective_isomorphism[OF lf(1) sf] lf f
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2667
    have "f' \<circ> f = id"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2668
      unfolding fun_eq_iff o_def id_def by metis
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2669
  }
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2670
  then show ?thesis
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2671
    using lf lf' by metis
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2672
qed
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2673
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2674
text {* Moreover, a one-sided inverse is automatically linear. *}
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2675
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  2676
lemma left_inverse_linear:
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2677
  fixes f :: "'a::euclidean_space \<Rightarrow> 'a::euclidean_space"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2678
  assumes lf: "linear f"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2679
    and gf: "g \<circ> f = id"
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2680
  shows "linear g"
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  2681
proof -
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  2682
  from gf have fi: "inj f"
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  2683
    apply (auto simp add: inj_on_def o_def id_def fun_eq_iff)
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  2684
    apply metis
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  2685
    done
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2686
  from linear_injective_isomorphism[OF lf fi]
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2687
  obtain h :: "'a \<Rightarrow> 'a" where h: "linear h" "\<forall>x. h (f x) = x" "\<forall>x. f (h x) = x"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2688
    by blast
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  2689
  have "h = g"
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  2690
    apply (rule ext) using gf h(2,3)
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2691
    apply (simp add: o_def id_def fun_eq_iff)
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  2692
    apply metis
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  2693
    done
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2694
  with h(1) show ?thesis by blast
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2695
qed
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2696
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  2697
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2698
subsection {* Infinity norm *}
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2699
56444
f944ae8c80a3 tuned proofs;
wenzelm
parents: 56409
diff changeset
  2700
definition "infnorm (x::'a::euclidean_space) = Sup {\<bar>x \<bullet> b\<bar> |b. b \<in> Basis}"
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2701
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2702
lemma infnorm_set_image:
53716
b42d9a71fc1a tuned proofs;
wenzelm
parents: 53600
diff changeset
  2703
  fixes x :: "'a::euclidean_space"
56444
f944ae8c80a3 tuned proofs;
wenzelm
parents: 56409
diff changeset
  2704
  shows "{\<bar>x \<bullet> i\<bar> |i. i \<in> Basis} = (\<lambda>i. \<bar>x \<bullet> i\<bar>) ` Basis"
50526
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 50105
diff changeset
  2705
  by blast
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2706
53716
b42d9a71fc1a tuned proofs;
wenzelm
parents: 53600
diff changeset
  2707
lemma infnorm_Max:
b42d9a71fc1a tuned proofs;
wenzelm
parents: 53600
diff changeset
  2708
  fixes x :: "'a::euclidean_space"
56444
f944ae8c80a3 tuned proofs;
wenzelm
parents: 56409
diff changeset
  2709
  shows "infnorm x = Max ((\<lambda>i. \<bar>x \<bullet> i\<bar>) ` Basis)"
56166
9a241bc276cd normalising simp rules for compound operators
haftmann
parents: 55910
diff changeset
  2710
  by (simp add: infnorm_def infnorm_set_image cSup_eq_Max del: Sup_image_eq)
51475
ebf9d4fd00ba introduct the conditional_complete_lattice type class; generalize theorems about real Sup and Inf to it
hoelzl
parents: 50526
diff changeset
  2711
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2712
lemma infnorm_set_lemma:
53716
b42d9a71fc1a tuned proofs;
wenzelm
parents: 53600
diff changeset
  2713
  fixes x :: "'a::euclidean_space"
56444
f944ae8c80a3 tuned proofs;
wenzelm
parents: 56409
diff changeset
  2714
  shows "finite {\<bar>x \<bullet> i\<bar> |i. i \<in> Basis}"
f944ae8c80a3 tuned proofs;
wenzelm
parents: 56409
diff changeset
  2715
    and "{\<bar>x \<bullet> i\<bar> |i. i \<in> Basis} \<noteq> {}"
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2716
  unfolding infnorm_set_image
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2717
  by auto
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2718
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2719
lemma infnorm_pos_le:
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2720
  fixes x :: "'a::euclidean_space"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2721
  shows "0 \<le> infnorm x"
51475
ebf9d4fd00ba introduct the conditional_complete_lattice type class; generalize theorems about real Sup and Inf to it
hoelzl
parents: 50526
diff changeset
  2722
  by (simp add: infnorm_Max Max_ge_iff ex_in_conv)
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2723
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2724
lemma infnorm_triangle:
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2725
  fixes x :: "'a::euclidean_space"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2726
  shows "infnorm (x + y) \<le> infnorm x + infnorm y"
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  2727
proof -
51475
ebf9d4fd00ba introduct the conditional_complete_lattice type class; generalize theorems about real Sup and Inf to it
hoelzl
parents: 50526
diff changeset
  2728
  have *: "\<And>a b c d :: real. \<bar>a\<bar> \<le> c \<Longrightarrow> \<bar>b\<bar> \<le> d \<Longrightarrow> \<bar>a + b\<bar> \<le> c + d"
ebf9d4fd00ba introduct the conditional_complete_lattice type class; generalize theorems about real Sup and Inf to it
hoelzl
parents: 50526
diff changeset
  2729
    by simp
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2730
  show ?thesis
51475
ebf9d4fd00ba introduct the conditional_complete_lattice type class; generalize theorems about real Sup and Inf to it
hoelzl
parents: 50526
diff changeset
  2731
    by (auto simp: infnorm_Max inner_add_left intro!: *)
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2732
qed
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2733
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2734
lemma infnorm_eq_0:
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2735
  fixes x :: "'a::euclidean_space"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2736
  shows "infnorm x = 0 \<longleftrightarrow> x = 0"
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  2737
proof -
51475
ebf9d4fd00ba introduct the conditional_complete_lattice type class; generalize theorems about real Sup and Inf to it
hoelzl
parents: 50526
diff changeset
  2738
  have "infnorm x \<le> 0 \<longleftrightarrow> x = 0"
ebf9d4fd00ba introduct the conditional_complete_lattice type class; generalize theorems about real Sup and Inf to it
hoelzl
parents: 50526
diff changeset
  2739
    unfolding infnorm_Max by (simp add: euclidean_all_zero_iff)
ebf9d4fd00ba introduct the conditional_complete_lattice type class; generalize theorems about real Sup and Inf to it
hoelzl
parents: 50526
diff changeset
  2740
  then show ?thesis
ebf9d4fd00ba introduct the conditional_complete_lattice type class; generalize theorems about real Sup and Inf to it
hoelzl
parents: 50526
diff changeset
  2741
    using infnorm_pos_le[of x] by simp
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2742
qed
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2743
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2744
lemma infnorm_0: "infnorm 0 = 0"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2745
  by (simp add: infnorm_eq_0)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2746
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2747
lemma infnorm_neg: "infnorm (- x) = infnorm x"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2748
  unfolding infnorm_def
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2749
  apply (rule cong[of "Sup" "Sup"])
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  2750
  apply blast
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  2751
  apply auto
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  2752
  done
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2753
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2754
lemma infnorm_sub: "infnorm (x - y) = infnorm (y - x)"
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  2755
proof -
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2756
  have "y - x = - (x - y)" by simp
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2757
  then show ?thesis
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2758
    by (metis infnorm_neg)
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2759
qed
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2760
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2761
lemma real_abs_sub_infnorm: "\<bar>infnorm x - infnorm y\<bar> \<le> infnorm (x - y)"
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  2762
proof -
56444
f944ae8c80a3 tuned proofs;
wenzelm
parents: 56409
diff changeset
  2763
  have th: "\<And>(nx::real) n ny. nx \<le> n + ny \<Longrightarrow> ny \<le> n + nx \<Longrightarrow> \<bar>nx - ny\<bar> \<le> n"
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2764
    by arith
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2765
  from infnorm_triangle[of "x - y" " y"] infnorm_triangle[of "x - y" "-x"]
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2766
  have ths: "infnorm x \<le> infnorm (x - y) + infnorm y"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2767
    "infnorm y \<le> infnorm (x - y) + infnorm x"
44454
6f28f96a09bf avoid warnings
huffman
parents: 44451
diff changeset
  2768
    by (simp_all add: field_simps infnorm_neg)
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2769
  from th[OF ths] show ?thesis .
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2770
qed
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2771
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2772
lemma real_abs_infnorm: "\<bar>infnorm x\<bar> = infnorm x"
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2773
  using infnorm_pos_le[of x] by arith
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2774
50526
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 50105
diff changeset
  2775
lemma Basis_le_infnorm:
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2776
  fixes x :: "'a::euclidean_space"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2777
  shows "b \<in> Basis \<Longrightarrow> \<bar>x \<bullet> b\<bar> \<le> infnorm x"
51475
ebf9d4fd00ba introduct the conditional_complete_lattice type class; generalize theorems about real Sup and Inf to it
hoelzl
parents: 50526
diff changeset
  2778
  by (simp add: infnorm_Max)
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2779
56444
f944ae8c80a3 tuned proofs;
wenzelm
parents: 56409
diff changeset
  2780
lemma infnorm_mul: "infnorm (a *\<^sub>R x) = \<bar>a\<bar> * infnorm x"
51475
ebf9d4fd00ba introduct the conditional_complete_lattice type class; generalize theorems about real Sup and Inf to it
hoelzl
parents: 50526
diff changeset
  2781
  unfolding infnorm_Max
ebf9d4fd00ba introduct the conditional_complete_lattice type class; generalize theorems about real Sup and Inf to it
hoelzl
parents: 50526
diff changeset
  2782
proof (safe intro!: Max_eqI)
ebf9d4fd00ba introduct the conditional_complete_lattice type class; generalize theorems about real Sup and Inf to it
hoelzl
parents: 50526
diff changeset
  2783
  let ?B = "(\<lambda>i. \<bar>x \<bullet> i\<bar>) ` Basis"
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2784
  {
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2785
    fix b :: 'a
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2786
    assume "b \<in> Basis"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2787
    then show "\<bar>a *\<^sub>R x \<bullet> b\<bar> \<le> \<bar>a\<bar> * Max ?B"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2788
      by (simp add: abs_mult mult_left_mono)
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2789
  next
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2790
    from Max_in[of ?B] obtain b where "b \<in> Basis" "Max ?B = \<bar>x \<bullet> b\<bar>"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2791
      by (auto simp del: Max_in)
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2792
    then show "\<bar>a\<bar> * Max ((\<lambda>i. \<bar>x \<bullet> i\<bar>) ` Basis) \<in> (\<lambda>i. \<bar>a *\<^sub>R x \<bullet> i\<bar>) ` Basis"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2793
      by (intro image_eqI[where x=b]) (auto simp: abs_mult)
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2794
  }
51475
ebf9d4fd00ba introduct the conditional_complete_lattice type class; generalize theorems about real Sup and Inf to it
hoelzl
parents: 50526
diff changeset
  2795
qed simp
ebf9d4fd00ba introduct the conditional_complete_lattice type class; generalize theorems about real Sup and Inf to it
hoelzl
parents: 50526
diff changeset
  2796
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2797
lemma infnorm_mul_lemma: "infnorm (a *\<^sub>R x) \<le> \<bar>a\<bar> * infnorm x"
51475
ebf9d4fd00ba introduct the conditional_complete_lattice type class; generalize theorems about real Sup and Inf to it
hoelzl
parents: 50526
diff changeset
  2798
  unfolding infnorm_mul ..
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2799
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2800
lemma infnorm_pos_lt: "infnorm x > 0 \<longleftrightarrow> x \<noteq> 0"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2801
  using infnorm_pos_le[of x] infnorm_eq_0[of x] by arith
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2802
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2803
text {* Prove that it differs only up to a bound from Euclidean norm. *}
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2804
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2805
lemma infnorm_le_norm: "infnorm x \<le> norm x"
51475
ebf9d4fd00ba introduct the conditional_complete_lattice type class; generalize theorems about real Sup and Inf to it
hoelzl
parents: 50526
diff changeset
  2806
  by (simp add: Basis_le_norm infnorm_Max)
50526
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 50105
diff changeset
  2807
54776
db890d9fc5c2 ordered_euclidean_space compatible with more standard pointwise ordering on products; conditionally complete lattice with product order
immler
parents: 54703
diff changeset
  2808
lemma (in euclidean_space) euclidean_inner: "inner x y = (\<Sum>b\<in>Basis. (x \<bullet> b) * (y \<bullet> b))"
57418
6ab1c7cb0b8d fact consolidation
haftmann
parents: 56536
diff changeset
  2809
  by (subst (1 2) euclidean_representation [symmetric])
6ab1c7cb0b8d fact consolidation
haftmann
parents: 56536
diff changeset
  2810
    (simp add: inner_setsum_right inner_Basis ac_simps)
50526
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 50105
diff changeset
  2811
53716
b42d9a71fc1a tuned proofs;
wenzelm
parents: 53600
diff changeset
  2812
lemma norm_le_infnorm:
b42d9a71fc1a tuned proofs;
wenzelm
parents: 53600
diff changeset
  2813
  fixes x :: "'a::euclidean_space"
b42d9a71fc1a tuned proofs;
wenzelm
parents: 53600
diff changeset
  2814
  shows "norm x \<le> sqrt DIM('a) * infnorm x"
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  2815
proof -
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2816
  let ?d = "DIM('a)"
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2817
  have "real ?d \<ge> 0"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2818
    by simp
53077
a1b3784f8129 more symbols;
wenzelm
parents: 53015
diff changeset
  2819
  then have d2: "(sqrt (real ?d))\<^sup>2 = real ?d"
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2820
    by (auto intro: real_sqrt_pow2)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2821
  have th: "sqrt (real ?d) * infnorm x \<ge> 0"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2822
    by (simp add: zero_le_mult_iff infnorm_pos_le)
53077
a1b3784f8129 more symbols;
wenzelm
parents: 53015
diff changeset
  2823
  have th1: "x \<bullet> x \<le> (sqrt (real ?d) * infnorm x)\<^sup>2"
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2824
    unfolding power_mult_distrib d2
50526
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 50105
diff changeset
  2825
    unfolding real_of_nat_def
53716
b42d9a71fc1a tuned proofs;
wenzelm
parents: 53600
diff changeset
  2826
    apply (subst euclidean_inner)
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2827
    apply (subst power2_abs[symmetric])
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51478
diff changeset
  2828
    apply (rule order_trans[OF setsum_bounded[where K="\<bar>infnorm x\<bar>\<^sup>2"]])
49663
b84fafaea4bb tuned proofs;
wenzelm
parents: 49652
diff changeset
  2829
    apply (auto simp add: power2_eq_square[symmetric])
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2830
    apply (subst power2_abs[symmetric])
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2831
    apply (rule power_mono)
51475
ebf9d4fd00ba introduct the conditional_complete_lattice type class; generalize theorems about real Sup and Inf to it
hoelzl
parents: 50526
diff changeset
  2832
    apply (auto simp: infnorm_Max)
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  2833
    done
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2834
  from real_le_lsqrt[OF inner_ge_zero th th1]
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2835
  show ?thesis
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2836
    unfolding norm_eq_sqrt_inner id_def .
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2837
qed
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2838
44646
a6047ddd9377 add lemma tendsto_infnorm
huffman
parents: 44629
diff changeset
  2839
lemma tendsto_infnorm [tendsto_intros]:
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  2840
  assumes "(f ---> a) F"
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  2841
  shows "((\<lambda>x. infnorm (f x)) ---> infnorm a) F"
44646
a6047ddd9377 add lemma tendsto_infnorm
huffman
parents: 44629
diff changeset
  2842
proof (rule tendsto_compose [OF LIM_I assms])
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2843
  fix r :: real
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2844
  assume "r > 0"
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  2845
  then show "\<exists>s>0. \<forall>x. x \<noteq> a \<and> norm (x - a) < s \<longrightarrow> norm (infnorm x - infnorm a) < r"
44646
a6047ddd9377 add lemma tendsto_infnorm
huffman
parents: 44629
diff changeset
  2846
    by (metis real_norm_def le_less_trans real_abs_sub_infnorm infnorm_le_norm)
a6047ddd9377 add lemma tendsto_infnorm
huffman
parents: 44629
diff changeset
  2847
qed
a6047ddd9377 add lemma tendsto_infnorm
huffman
parents: 44629
diff changeset
  2848
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2849
text {* Equality in Cauchy-Schwarz and triangle inequalities. *}
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2850
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2851
lemma norm_cauchy_schwarz_eq: "x \<bullet> y = norm x * norm y \<longleftrightarrow> norm x *\<^sub>R y = norm y *\<^sub>R x"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2852
  (is "?lhs \<longleftrightarrow> ?rhs")
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  2853
proof -
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2854
  {
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2855
    assume h: "x = 0"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2856
    then have ?thesis by simp
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2857
  }
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2858
  moreover
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2859
  {
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2860
    assume h: "y = 0"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2861
    then have ?thesis by simp
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2862
  }
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2863
  moreover
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2864
  {
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2865
    assume x: "x \<noteq> 0" and y: "y \<noteq> 0"
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2866
    from inner_eq_zero_iff[of "norm y *\<^sub>R x - norm x *\<^sub>R y"]
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  2867
    have "?rhs \<longleftrightarrow>
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  2868
      (norm y * (norm y * norm x * norm x - norm x * (x \<bullet> y)) -
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  2869
        norm x * (norm y * (y \<bullet> x) - norm x * norm y * norm y) =  0)"
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2870
      using x y
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2871
      unfolding inner_simps
54230
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 53939
diff changeset
  2872
      unfolding power2_norm_eq_inner[symmetric] power2_eq_square right_minus_eq
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  2873
      apply (simp add: inner_commute)
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  2874
      apply (simp add: field_simps)
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  2875
      apply metis
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  2876
      done
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2877
    also have "\<dots> \<longleftrightarrow> (2 * norm x * norm y * (norm x * norm y - x \<bullet> y) = 0)" using x y
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2878
      by (simp add: field_simps inner_commute)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2879
    also have "\<dots> \<longleftrightarrow> ?lhs" using x y
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2880
      apply simp
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  2881
      apply metis
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  2882
      done
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2883
    finally have ?thesis by blast
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2884
  }
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2885
  ultimately show ?thesis by blast
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2886
qed
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2887
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2888
lemma norm_cauchy_schwarz_abs_eq:
56444
f944ae8c80a3 tuned proofs;
wenzelm
parents: 56409
diff changeset
  2889
  "\<bar>x \<bullet> y\<bar> = norm x * norm y \<longleftrightarrow>
53716
b42d9a71fc1a tuned proofs;
wenzelm
parents: 53600
diff changeset
  2890
    norm x *\<^sub>R y = norm y *\<^sub>R x \<or> norm x *\<^sub>R y = - norm y *\<^sub>R x"
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2891
  (is "?lhs \<longleftrightarrow> ?rhs")
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  2892
proof -
56444
f944ae8c80a3 tuned proofs;
wenzelm
parents: 56409
diff changeset
  2893
  have th: "\<And>(x::real) a. a \<ge> 0 \<Longrightarrow> \<bar>x\<bar> = a \<longleftrightarrow> x = a \<or> x = - a"
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2894
    by arith
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2895
  have "?rhs \<longleftrightarrow> norm x *\<^sub>R y = norm y *\<^sub>R x \<or> norm (- x) *\<^sub>R y = norm y *\<^sub>R (- x)"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2896
    by simp
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2897
  also have "\<dots> \<longleftrightarrow>(x \<bullet> y = norm x * norm y \<or> (- x) \<bullet> y = norm x * norm y)"
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2898
    unfolding norm_cauchy_schwarz_eq[symmetric]
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2899
    unfolding norm_minus_cancel norm_scaleR ..
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2900
  also have "\<dots> \<longleftrightarrow> ?lhs"
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2901
    unfolding th[OF mult_nonneg_nonneg, OF norm_ge_zero[of x] norm_ge_zero[of y]] inner_simps
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2902
    by auto
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2903
  finally show ?thesis ..
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2904
qed
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2905
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2906
lemma norm_triangle_eq:
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2907
  fixes x y :: "'a::real_inner"
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2908
  shows "norm (x + y) = norm x + norm y \<longleftrightarrow> norm x *\<^sub>R y = norm y *\<^sub>R x"
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  2909
proof -
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2910
  {
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2911
    assume x: "x = 0 \<or> y = 0"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2912
    then have ?thesis
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2913
      by (cases "x = 0") simp_all
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2914
  }
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2915
  moreover
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2916
  {
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2917
    assume x: "x \<noteq> 0" and y: "y \<noteq> 0"
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  2918
    then have "norm x \<noteq> 0" "norm y \<noteq> 0"
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2919
      by simp_all
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  2920
    then have n: "norm x > 0" "norm y > 0"
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  2921
      using norm_ge_zero[of x] norm_ge_zero[of y] by arith+
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2922
    have th: "\<And>(a::real) b c. a + b + c \<noteq> 0 \<Longrightarrow> a = b + c \<longleftrightarrow> a\<^sup>2 = (b + c)\<^sup>2"
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  2923
      by algebra
53077
a1b3784f8129 more symbols;
wenzelm
parents: 53015
diff changeset
  2924
    have "norm (x + y) = norm x + norm y \<longleftrightarrow> (norm (x + y))\<^sup>2 = (norm x + norm y)\<^sup>2"
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2925
      apply (rule th)
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2926
      using n norm_ge_zero[of "x + y"]
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  2927
      apply arith
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  2928
      done
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2929
    also have "\<dots> \<longleftrightarrow> norm x *\<^sub>R y = norm y *\<^sub>R x"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2930
      unfolding norm_cauchy_schwarz_eq[symmetric]
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2931
      unfolding power2_norm_eq_inner inner_simps
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2932
      by (simp add: power2_norm_eq_inner[symmetric] power2_eq_square inner_commute field_simps)
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2933
    finally have ?thesis .
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2934
  }
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2935
  ultimately show ?thesis by blast
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2936
qed
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2937
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  2938
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2939
subsection {* Collinearity *}
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2940
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  2941
definition collinear :: "'a::real_vector set \<Rightarrow> bool"
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  2942
  where "collinear S \<longleftrightarrow> (\<exists>u. \<forall>x \<in> S. \<forall> y \<in> S. \<exists>c. x - y = c *\<^sub>R u)"
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2943
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2944
lemma collinear_empty: "collinear {}"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2945
  by (simp add: collinear_def)
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2946
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2947
lemma collinear_sing: "collinear {x}"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2948
  by (simp add: collinear_def)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2949
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2950
lemma collinear_2: "collinear {x, y}"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2951
  apply (simp add: collinear_def)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2952
  apply (rule exI[where x="x - y"])
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2953
  apply auto
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2954
  apply (rule exI[where x=1], simp)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2955
  apply (rule exI[where x="- 1"], simp)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2956
  done
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2957
56444
f944ae8c80a3 tuned proofs;
wenzelm
parents: 56409
diff changeset
  2958
lemma collinear_lemma: "collinear {0, x, y} \<longleftrightarrow> x = 0 \<or> y = 0 \<or> (\<exists>c. y = c *\<^sub>R x)"
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2959
  (is "?lhs \<longleftrightarrow> ?rhs")
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  2960
proof -
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2961
  {
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2962
    assume "x = 0 \<or> y = 0"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2963
    then have ?thesis
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2964
      by (cases "x = 0") (simp_all add: collinear_2 insert_commute)
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2965
  }
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2966
  moreover
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2967
  {
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2968
    assume x: "x \<noteq> 0" and y: "y \<noteq> 0"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2969
    have ?thesis
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2970
    proof
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2971
      assume h: "?lhs"
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  2972
      then obtain u where u: "\<forall> x\<in> {0,x,y}. \<forall>y\<in> {0,x,y}. \<exists>c. x - y = c *\<^sub>R u"
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  2973
        unfolding collinear_def by blast
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2974
      from u[rule_format, of x 0] u[rule_format, of y 0]
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2975
      obtain cx and cy where
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2976
        cx: "x = cx *\<^sub>R u" and cy: "y = cy *\<^sub>R u"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2977
        by auto
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2978
      from cx x have cx0: "cx \<noteq> 0" by auto
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2979
      from cy y have cy0: "cy \<noteq> 0" by auto
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2980
      let ?d = "cy / cx"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2981
      from cx cy cx0 have "y = ?d *\<^sub>R x"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2982
        by simp
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2983
      then show ?rhs using x y by blast
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2984
    next
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2985
      assume h: "?rhs"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2986
      then obtain c where c: "y = c *\<^sub>R x"
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2987
        using x y by blast
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2988
      show ?lhs
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2989
        unfolding collinear_def c
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2990
        apply (rule exI[where x=x])
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2991
        apply auto
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2992
        apply (rule exI[where x="- 1"], simp)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2993
        apply (rule exI[where x= "-c"], simp)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2994
        apply (rule exI[where x=1], simp)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2995
        apply (rule exI[where x="1 - c"], simp add: scaleR_left_diff_distrib)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  2996
        apply (rule exI[where x="c - 1"], simp add: scaleR_left_diff_distrib)
53406
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2997
        done
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2998
    qed
d4374a69ddff tuned proofs;
wenzelm
parents: 53077
diff changeset
  2999
  }
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  3000
  ultimately show ?thesis by blast
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  3001
qed
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  3002
56444
f944ae8c80a3 tuned proofs;
wenzelm
parents: 56409
diff changeset
  3003
lemma norm_cauchy_schwarz_equal: "\<bar>x \<bullet> y\<bar> = norm x * norm y \<longleftrightarrow> collinear {0, x, y}"
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  3004
  unfolding norm_cauchy_schwarz_abs_eq
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  3005
  apply (cases "x=0", simp_all add: collinear_2)
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  3006
  apply (cases "y=0", simp_all add: collinear_2 insert_commute)
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  3007
  unfolding collinear_lemma
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  3008
  apply simp
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  3009
  apply (subgoal_tac "norm x \<noteq> 0")
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  3010
  apply (subgoal_tac "norm y \<noteq> 0")
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  3011
  apply (rule iffI)
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  3012
  apply (cases "norm x *\<^sub>R y = norm y *\<^sub>R x")
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  3013
  apply (rule exI[where x="(1/norm x) * norm y"])
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  3014
  apply (drule sym)
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  3015
  unfolding scaleR_scaleR[symmetric]
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  3016
  apply (simp add: field_simps)
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  3017
  apply (rule exI[where x="(1/norm x) * - norm y"])
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  3018
  apply clarify
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  3019
  apply (drule sym)
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  3020
  unfolding scaleR_scaleR[symmetric]
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  3021
  apply (simp add: field_simps)
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  3022
  apply (erule exE)
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  3023
  apply (erule ssubst)
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  3024
  unfolding scaleR_scaleR
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  3025
  unfolding norm_scaleR
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  3026
  apply (subgoal_tac "norm x * c = \<bar>c\<bar> * norm x \<or> norm x * c = - \<bar>c\<bar> * norm x")
55775
1557a391a858 A bit of tidying up
paulson <lp15@cam.ac.uk>
parents: 55136
diff changeset
  3027
  apply (auto simp add: field_simps)
49522
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  3028
  done
355f3d076924 tuned proofs;
wenzelm
parents: 44890
diff changeset
  3029
54776
db890d9fc5c2 ordered_euclidean_space compatible with more standard pointwise ordering on products; conditionally complete lattice with product order
immler
parents: 54703
diff changeset
  3030
end